cubo a mathematical journal vol.16, no¯ 02, (49–52). june 2014 k-theory for the group c∗-algebras of a residually finite discrete group with kazhdan property t takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, senbaru 1, nishihara, okinawa 903-0213, japan. sudo@math.u-ryukyu.ac.jp abstract we compute the k-theory groups for the full and reduced group c∗-algebras of a residually finite, finitely generated discrete group with kazhdan property t. resumen calculamos los grupos de la k-teoŕıa para grupo de c∗-algebras de reducido y completo de un grupo discreto generado finitamente y residualmente finito con la propiedad t de kazhdan. keywords and phrases: group c*-algebra, k-theory, discrete group, projection. 2010 ams mathematics subject classification: 46l05, 46l80, 19k14. 50 takahiro sudo cubo 16, 2 (2014) 1 introduction in this paper, first of all, we compute the k-theory groups for the full group c∗-algebra of a residually finite, finitely generated discrete group with kazhdan property t, such as sln(z) (n ≥ 3) the n × n special linear groups over the integers. the highly non-trivial and interesting problem to compute the k-theory groups has been considered by the author [6], but it was found to be not completed, and to be corrected as [7] (perhaps partly). this time we obtain a sort of solution for this problem to settle the issue, without using the results of [5] used in [6], but more precisely k0 only, with a mysterious part left. we next compute the k-theory groups for the reduced group c∗-algebra of a residually finite, finitely generated discrete group with kazhdan property t, by using the six-term exact sequence of k-groups and the results obtained in the full case. 2 the main result proposition 1. let γ be a residually finite, finitely generated discrete group with kazhdan property t and c∗(γ) its full group c∗-algebra. then the k0-group k0(c ∗(γ)) has a direct summand isomorphic to the group generated by the infinite direct sum of copies of z and one copy of z corresponding to the unit. proof. since γ is residully finite, then there is a (separable) family of finite dimensional irreducible representations πλ of γ such that the intersection of their kernels is trivial (see [4, p. 480]). denote also by πλ the corresponding finite dimensional irreducible representations of c ∗(γ). then c∗(γ) has a ∗-homomorphism (which can not be injective in general, see [2]) into the direct product c∗-algebra πλmnλ(c) of the nλ × nλ matrix algebras mnλ(c) over c, where nλ = dim πλ, by the direct product representation πλπλ of c ∗(γ). the representation implies the k-theory homomorphism: k∗(c ∗(γ)) (πλπλ)∗ −−−−−−→ k∗(πλmnλ(c)) for ∗ = 0, 1, and k∗(πλmnλ(c)) ∼= πλk∗(mnλ(c)) with k0(mnλ(c)) ∼= z and k1(mnλ(c)) ∼= 0. note that since γ is discrete, c∗(γ) has the unit and that the map (πλπλ)∗ is unital. on the other hand, since γ has kazhdan propery t, then c∗(γ) has mnλ(c) as a direct summand (see [9]). hence k∗(c ∗(γ)) has k∗(mnλ(c)) as a direct summand. since k∗(mnλ(c)) is mapped injectively under the induced map (πλπλ)∗, it follows that both k∗(c ∗(γ)) and the image of k∗(c ∗(γ)) contain the infinite direct sum ⊕λk∗(mnλ(c)). furthermore, all or nothing principle tells us that the image of k0(c ∗(γ)) does not contain other classes corresponding to other non-trivial (infinite) projections in πλmnλ(c) except projections in the group generated by the direct sum ⊕λz and z of the unit class, because if it does contain, the principle implies that the image must be equal to πλk0(mnλ(c)), so that c ∗(γ) has πλmnλ(c) as a quotient, but the direct product is non-separable, while c∗(γ) is separable, a contradiction. indeed, we can not find the difference among those extra infinite projections in πλmnλ(c). hence the proof is completed. cubo 16, 2 (2014) k-theory for the group c∗-algebras of a residually finite discrete . . . 51 remark. unfortunately, we do not know about the mysterious kernel ker(πλπλ)∗ of the k-theory homomorphism in the k0 and k1-groups k∗(c ∗(γ)) which may not be trivial in general, so that we could not determine the k0 and k1-group. corollary 1. for n ≥ 3, the abelian group k0(c ∗(sln(z))) has a direct summand isomorphic to the group generated by an infinite direct sum of copies of z and one copy of z. proof. note that sln(z) for n ≥ 3 are residually finite, finitely generated groups with kazhdan property t. indeed, it is known that every finitely generated subgroup of sln(c) is residually finite (see [1] and also [4]) and that sln(z) have kazhdan property t (see [3, p. 34]). theorem 1. let γ be a non-amenable, residually finite, finitely generated discrete group with kazhdan property t and c∗r(γ) its reduced group c ∗-algebra. then k0(c ∗ r(γ)) ∼= z ⊕ q∗[ker(πλπλ)∗] and k1(c ∗ r(γ)) is a quotient of k1(c ∗(γ)), where this quotient and q∗ are induced from the canonical quotient map q : c∗(g) → c∗r(g). proof. denote by iγ the kernel of q. then we have the following six-term diagram: k0(iγ ) i∗ −−−−→ k0(c ∗(γ)) q∗ −−−−→ k0(c ∗ r(γ)) ! ⏐ ⏐ ⏐ ⏐ # k1(c ∗ r(γ)) q∗ ←−−−− k1(c ∗(γ)) i∗ ←−−−− k1(iγ ) where i∗ is induced by the inclusion i : iγ → c ∗(γ). note that the infinite direct sum of z in k0(c ∗(γ)) is mapped to zero by q∗ since γ is non-amenable, so that c ∗ r(γ) has no finite dimensional representation (a fact of the representation theory for γ), and the other copy of z in k0(c ∗(γ)) is mapped injectively. it follows that k0(iγ ) is isomorphic to the direct sum ⊕z. since i∗ on k0 is injective, the index map from k1(c ∗ r(γ)) is zero, so that q∗ on k1(c ∗(γ)) is surjective. since the class of the unit in k0(c ∗ r(γ)) is mapped to zero by the exactness of the diagram, it follows that k0(c ∗ r(γ)) ∼= z ⊕ q∗[ker(πλπλ)∗] and i∗ on k1(iγ ) is injective. corollary 2. for n ≥ 3, we have k0(c ∗ r(sln(z))) ∼= z ⊕ q∗[ker(πλπλ)∗], and k1(c ∗ r(sln(z))) is a quotient of k1(c ∗(sln(z))). remark. note that q∗[ker(πλπλ)∗] is not trivial. because if it is zero, then k0(c ∗ r(sln(z))) ∼= z, which implies that c∗r(sln(z)) does not contain non-trivial projections. but sln(z) has torsion since it contains sl2(z) ∼= z4 ∗z2 z6 as a subgroup, so that c ∗ r(sln(z)) has non-trivial projections, a contradiction. 52 takahiro sudo cubo 16, 2 (2014) the kadison-kaplansky conjecture is that if γ is a torsion free, discrete group, then c∗r(γ) has no non-trivial projections. see [8] about the conjecture. received: march 2013. revised: september 2013. references [1] r. alperin, an elementary account of selberg’s lemma, l’ensignement math. 33 (1987), 269-273. [2] m. b. bekka and n. louvet, some properties of c∗-algebras associated to discrete linear groups, c∗-algebras, springer (2000), 1-22. [3] p. de la harpe and a. vallette, la propriété (t) de kazhdan pour les groupes localement compacts, astérisque 175 (1989), soc. math. france. [4] e. kirchberg, on non-semisplit extensions, tensor products and exactness of group c∗algebras, invent. math. 112 (1993), 449-489. [5] c. soulé, the cohomology of sl3(z), topology, 17 (1978), 1-22. [6] t. sudo, k-theory for amalgams and multi-ones of c∗-algebras, ryukyu math. j. 21 (2008), 57-139. [7] t. sudo, erratum: k-theory for amalgams and multi-ones of c∗-algebras, ryukyu math. j. 21 (2008), 57-139, ryukyu math. j. 22 (2009), 115-117. [8] a. valette, the conjecture of idempotents: a survey of the c∗-algebraic approach, bull. soc. math. belgique 41 (1989), 485–521. [9] p. s. wang, on isolated points in the dual spaces of locally compact groups, math. ann. 218 (1975), 19-34. () cubo a mathematical journal vol.17, no¯ 01, (01–09). march 2015 instability to vector lienard equation with multiple delays cemil tunc department of mathematics, faculty of science yüzüncü yıl university 65080, van, turkey cemtunc@yahoo.com abstract by making use of a special lyapunov-krasovskii functional and applying krasovskii’s properties, we prove instability of zero solution of a modified vector lienard equation with multiple constant delays that includes van der pol, rayleigh and lienard equations, widely encountered in applications. resumen usando un funcional especial de lyapunov-krasovskii y aplicando propiedades de krasovskii, probamos la inestabilidad de la solución nula de una ecuación de lienard vectorial modificada con retardos constantes múltiples que incluyen a las ecuaciones de van der pol, rayleigh y liénard ampliamente encontradas en las aplicaciones. keywords and phrases: lienard, lyapunov-krasovskii functional, instability, delay. 2010 ams mathematics subject classification: 34k12, 34k20. 2 cemil tunc cubo 17, 1 (2015) 1 introduction in this paper, we consider the following modified vector lienard equation with multiple constant delays, τi > 0: x ′′ (t) + f(x(t), x ′ (t)) + g(x(t)) + n∑ i=1 hi(x(t − τi)) = 0, (1.1) where t ∈ r+, r+ = [0, ∞), x ∈ rn , τi > 0 are fixed constant delays, t−τi ≥ 0; f : rn×rn → rn is a continuous function; g : rn → rn and hi : rn → rn are continuously differentiable functions; f(x, 0) = 0, g(0) = 0, hi(0) = 0. we assume that the existence and uniqueness of the solutions hold for equation (1.1), (see [3]). making y = x ′ in equation (1.1), we obtain x ′ = y, y ′ = −f(x, y) − g(x) − n∑ k=1 hi(x) + t∫ t−τi jhi(x(s))y(s)ds. (1.2) let jg(x) and jhi(x) denote the linear operators from the vectors g and hi to the matrices jg(x) = ( ∂gi ∂xj ) , jh1(x) = ( ∂h1i ∂xj ) , ..., jhn(x) = ( ∂hni ∂xj ) , (i, j = 1, 2, ..., n), where (x1, ..., xn), (g1, ..., gn) and (h1i, ..., hni) are the components of x, g and hi, respectively. besides, it is also assumed as basic throughout this paper that the jacobian matrices jg(x) and jhi(x) exist, are symmetric and continuous. this research has been motivated by the paper of hale [4] and the recent papers of tunc [6 − 8] dealing with stability and instability of zero solution for certain scalar and vector differential equations of second order. first, in 1965, hale [4] studied instability of zero solution of the scalar lienard and rayleigh equations with a constant delay,r(> 0), respectively: x ′′ (t) + f(x ′ (t)) + g(x(t − r)) = 0 and x ′′ (t) − ε ( 1 − x ′2 (t) 3 ) x ′ (t) + g(x(t − r)) = 0, cubo 17, 1 (2015) instability to vector lienard equation with multiple delays 3 where ε is a positive constant. by defining lyapunov-krasovskii functionals, the author gave sufficient conditions to guarantee the instability of zero solution to these equations. later, tunc ([6], [7]) discussed the instability of the zero solution for the following modified scalar and vector lienard equation with multiple constant delays and constant delay, respectively: x ′′ (t) + f1(x(t), x ′ (t))x ′ (t) + f2(x(t))x ′ (t) + g0(x(t)) + n∑ i=1 gi(x(t − τi)) = 0 and x ′′ (t) + f(x(t), x ′ (t))x ′ (t) + h(x(t − τ)) = 0, where τi(> 0) and τ(> 0) are fixed constant delays. throughout this paper, the symbol < x, y > corresponding to any pair x and y in rn stands for the usual scalar product n∑ i=1 xiyi, that is, < x, y >= n∑ i=1 xiyi; thus < x, x >= ||x|| 2, and λi(a) are the eigenvalues of the real symmetric n × n matrix a. the following preliminary result is need in the proof. lemma(bellman [1]). let a be a real symmetric n × n matrix. then for any x ∈ rn, a ′ 〈x, x〉 ≥ 〈ax, x〉 ≥ a 〈x, x〉 and a ′ 2 〈x, x〉 ≥ 〈ax, x〉 ≥ a2 〈x, x〉 , where a ′ and a are, respectively, the least and greatest eigenvalues of the matrix a. it may also be useful to give basic information for general autonomous delay differential system with finite delay (see burton [2]). let r ≥ 0 be given, and let c = c([−r, 0], rn) with ||φ(s)|| = max −r≤s≤0 |φ(s)| , φ ∈ c. for h > 0 define ch ⊂ c by ch = {φ ∈ c : ||φ|| < h} . 4 cemil tunc cubo 17, 1 (2015) if x : [−r, t) → rn is continuous, 0 < t ≤ ∞, then, for each t in [0, t), xt in c is defined by xt(s) = x(t + s), −r ≤ s ≤ 0, t ≥ 0. let g be an open subset of c and consider the general autonomous delay differential system with finite delay ẋ = f(xt), f(0) = 0, xt = x(t + θ), −r ≤ θ ≤ 0, t ≥ 0, where f : g → rn is continuous and maps closed and bounded sets into bounded sets. it follows from these conditions on f that each initial value problem ẋ = f(xt), x0 = φ ∈ g has a unique solution defined on some interval [0, t), 0 < t ≤ ∞. this solution will be denoted by x(φ)(.) so that x0(φ) = φ. definition. the zero solution, x = 0, of ẋ = f(xt) is stable if for each ε > 0 there exists δ = δ(ε) > 0 such that ||φ|| < δ implies that ||x(φ)(t)|| < ε for all t ≥ 0. the zero solution is said to be unstable if it is not stable. consider the equations of perturbed motion dxi dt = xi(x1, ..., xn, t), (i = 1, 2, ..., n), where the functions xi(x1, ..., xn, t) are defined and continuous in the region ||x|| < h, −∞ < t < ∞, (h=constant or h = ∞) theorem a. let h1 < h. suppose that there exists a function v(x, t) which is periodic in the time or does not dependent explicitly on the time, such that (a) ν is defined in the region ||x|| < h, −∞ < t < ∞, (h=constant or h = ∞), (b) ν admits an infinitely small upper bound in the region ||x|| < h, −∞ < t < ∞, (c) dv dt ≥ 0 in the region ||x|| < h, −∞ < t < ∞,along a trajectory of dxi dt = xi(x1, ..., xn, t), (d) the set of the points m at which the derivative dv dt is 0 contains no non-trivial half trajectory x(x0, t0, t), (t0 ≤ t < ∞). suppose further that in every neighborhood of the point x = 0, there is a point x0 such that for arbitrary t0 ≥ 0 we have v(x0, t0) > 0. then the null solution x = 0 is unstable, and the trajectories x(x0, t0, t) for which v(x0, t0) > 0 leave the region ||x|| < h1 as the time t increases (see krasovskii [5, theorem 15.1]). cubo 17, 1 (2015) instability to vector lienard equation with multiple delays 5 2 main result the main result of this paper is the following. let p(x) = g(x) + n∑ i=1 hi(x). theorem.assume that there exist positive constants a, b, di such that for all x, y ∈ r we have (i)−yt f(x, y) ≥ a ||y|| 2 , (a = n∑ i=1 ai), (ii)xt jp(x)x ≥ b ||x|| 2 , (iii)jp(x) = j t p(x), (iv) √ λi(j t hi (x)jhi(x)) ≤ di , (i = 1, 2, ..., n), (v)x 6= 0 ⇒ p(x) 6= 0. if τ < a n∑ i=1 di , then the zero solution of equation (1.1) is unstable. proof.introducing a lyapunov-krasovskii functional v = v(xt, yt) by the formula v = n∑ i=1 ∫1 0 〈hi(σx), x〉dσ + ∫1 0 〈g(σx), x〉σ + 1 2 〈y, y〉 − n∑ i=1 µi ∫0 −τi ∫t t+s ||y(θ)|| 2 dθds, where s is a real variable such that the integrals ∫0 −τi ∫t t+s ||y(θ)|| 2 dθds are non-negative, and µi are certain positive constants to be determined later in the proof. we observe the existence of the following estimates: v(0, 0) = 0, ∂ ∂σ hi(σx) = jhi(σx)x 6 cemil tunc cubo 17, 1 (2015) ⇒ hi(x) = ∫1 0 jhi(σx)xdσ, ∂ ∂σ g(σx) = jg(σx)x ⇒ g(x) = ∫1 0 jg(σx)xdσ. then, ∫1 0 〈hi(σx), x〉dσ = ∫1 0 ∫1 0 〈σ1jhi(σ1σ2x)x, x〉 dσ2dσ1 and ∫1 0 〈g(σx), x〉dσ = ∫1 0 ∫1 0 〈σ1jg(σ1σ2x)x, x〉 dσ2dσ1. by noting (ii), we have n∑ i=1 ∫1 0 〈hi(σx), x〉dσ + ∫1 0 〈g(σx), x〉dσ = n∑ i=1 ∫1 0 ∫1 0 〈σ1jhi(σ1σ2x)x, x〉 dσ2dσ1 + ∫1 0 ∫1 0 〈σ1jg(σ1σ2x)x, x〉 dσ2dσ1 ≥ 1 2 b ||x|| 2 . hence, v ≥ 1 2 b ||x|| 2 + 1 2 ||y|| 2 − n∑ i=1 µi ∫0 −τi ∫t t+s ||y(θ)|| 2 dθds. let ξ̄ ∈ rn and ξ̄ = (ξ11, ..., ξ1n). cubo 17, 1 (2015) instability to vector lienard equation with multiple delays 7 then, the last estimate becomes v(ξ̄, 0) ≥ 1 2 b ∣ ∣ ∣ ∣ξ̄ ∣ ∣ ∣ ∣ 2 > 0 for all arbitrary ξ̄ 6= 0, ξ̄ ∈ rn. so, the first property of krasovskii [5] holds. let us compute the time derivative of v along the solution (x(t), y(t)) of system (1.2), v̇ = −〈f(x, y), y〉 + 〈 n∑ i=1 ∫t t−τi jhi(x(s))y(s)ds, y 〉 − 〈 n∑ i=1 (µiτi)y, y 〉 + n∑ i=1 µi ∫t t−τi ||y(θ)|| 2 dθ. using the assumptions of the theorem and elementary inequalities, we obtain −〈f(x, y), y〉 ≥ n∑ i=1 ai ||y|| 2 , 〈 n∑ i=1 ∫t t−τi jhi(x(s))y(s)ds, y 〉 ≥ − ||y|| ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫t t−τi jhi(x(s))y(s) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ds ≥ − di ||y|| ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫t t−τi y(s) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ds ≥ − di ||y|| ∫t t−τi | ||y(s)|| ds ≥ − 1 2 di ∫t t−τi { ||y(t)|| 2 + ||y(s)|| 2 } ds = − 1 2 diτi ||y|| 2 − 1 2 di ∫t t−τi ||y(s)|| 2 . therefore, v̇ ≥ n∑ i=1 ai ||y|| 2 − ( n∑ i=1 µiτi) ||y|| 2 − 1 2 ( n∑ i=1 diτi) ||y|| 2 + n∑ i=1 (µi − 1 2 di) ∫t t−τi ||y(s)|| 2 ds. let µi = 1 2 di and τ=max{τ1, τ2, ..., τn}. then, v̇ ≥ ( n∑ i=1 ai − n∑ i=1 diτ ) ||y|| 2 . 8 cemil tunc cubo 17, 1 (2015) if τ < an∑ i=1 di ,then v̇ ≥ α||y||2 > 0, where α is some positive constant. thus, the second property of krasovskii [5] holds. finally, it follows that v̇ = 0 ⇔ y = 0. in view of y = 0 and system (1.2), it follows that v̇ = 0 ⇔ g(x) + n∑ i=1 hi(x) = 0 and y = 0. by noting the assumptions of the theorem, x 6= 0 ⇒ p(x) 6= 0, we can conclude that g(x) + n∑ i=1 hi(x) = 0 ⇔ x = 0. this result shows that the only invariant set of system (1.2) for which v̇ = 0 is the solution x = y = 0. therefore, the third property of krasovskii [5] holds. this completes the proof of the theorem. 3 conclusion a functional vector lienard equation with multiple retardations has been considered. the instability of zero solution of that equation has been discussed by using the lyapunov-krasovskii functional approach. the obtained result extends and improve some well known results in the literature. received: april 2014. accepted: november 2014. references [1] r. bellman, introduction to matrix analysis. reprint of the second (1970) edition. with a foreword by gene golub. classics in applied mathematics, 19. society for industrial and applied mathematics (siam), philadelphia, pa, 1997. [2] t. a. burton, stability and periodic solutions of ordinary and functional differential equations, academic press, orlando, 1985. [3] l. e. elsgolts and s. b. norkin, introduction to the theory and application of differential equations with deviating arguments. translated from the russian by john l. casti. mathematics in science and engineering, vol. 105. academic press [a subsidiary of harcourt brace jovanovich, publishers], new york-london, 1973. [4] j. hale, sufficient conditions for stability and instability of autonomous functional-differential equations. j. differential equations 1 (1965), 452-482. [5] n. n. krasovskii, stability of motion. applications of lyapunov’s second method to differential systems and equations with delay, stanford, calif.: stanford university press 1963. cubo 17, 1 (2015) instability to vector lienard equation with multiple delays 9 [6] c. tunc, on the instability of solutions to a linard type equation with multiple deviating arguments. afr. mat. 25 (2014), no. 4, 1013-1021. [7] c. tunc, instability of solutions of vector linard equation with constant delay. bull. math. soc. sci. math. roumanie, (2012), (accepted). [8] c. tunc, stability to vector lienard equation with constant deviating argument. nonlinear dynam. 73(3), (2013), 1245-1251. introduction main result conclusion () cubo a mathematical journal vol.17, no¯ 02, (89–95). june 2015 submatrices of four dimensional summability matrices fatih nuray department of mathematics, afyonkocatepe university, afyonkarahisar, turkey, deparment of mathematics and statistics, university of north florida, jacksonville, fl, usa university of north florida, jacksonville, fl, usa fnuray@aku.edu.tr richard f. patterson department of mathematics and statistics, university of north florida jacksonville, florida, 32224 rpatters@unf.edu abstract in this paper, we show that a matrix that maps ℓ′′ into ℓ′′ can be obtained from any rh-regular matrix by the deletion of rows. also a four dimensional conservative matrix can be obtained by the deletion of rows from a matrix that preserves boundedness. we will use these techniques to derive a sufficient condition for a four dimensional matrix to sum an unbounded sequence. resumen en este trabajo probamos que una matriz que lleva ℓ′′ en ℓ′′ se puede obtener a partir de cualquier matriz rh-regular eliminando filas. también una matriz cuatro dimensional conservative se puede obteber eliminando filas en una matriz que preserva acotación. usamos estas técnicas para encontrar una condición suficiente para que una matriz cuatro dimensional sume una sucesión no acotada. keywords and phrases: submatrix, four dimensional summability matrix, double sequences, pringsheim limit, rh-regular matrix 2010 ams mathematics subject classification: 40c05, 40d05. 90 fatih nuray & richard f. patterson cubo 17, 2 (2015) 1 introduction the most well-known notion of convergence for double sequences is the convergence in the sense of pringsheim. recall that a double sequence x = {xk,l} of complex (or real) numbers is called convergent to a scalar ℓ in pringsheim’s sense (denoted by p-lim x = ℓ) if for every ǫ > 0 there exists an n ∈ n such that |xk,l − ℓ| < ǫ whenever k, l > n. such an x is described more briefly as ”p-convergent”. it is easy to verify that x = {xk,l} convergences in pringsheim’s sense if and only if for every ǫ > 0 there exists an integer n = n(ǫ) such that |xi,j − xk,l| < ǫ whenever min{i, j, k, l} ≥ n. a double sequence x = {xk,l} is bounded if there exists a positive number m such that |xk,l| ≤ m for all k and l, that is, if supk,l |xk,l| < ∞. a double sequence x = {xk,l} is said to convergence regularly if it converges in pringsheim’s sense and, in addition, the following finite limits exist: lim k→∞ xk,l = ℓl (l = 1, 2, ...), lim l→∞ xk,l = lk (k = 1, 2, ...). note that the main drawback of the pringsheim’s convergence is that a convergent sequence fails in general to be bounded. the notion of regular convergence lacks this disadvantage. a double sequence x is divergent in the pringsheim sense(p-divergent) provided that x does not convergence in the pringsheim sense. let a = (am,n,k,l) denote a four dimensional summability method that maps the complex double sequence x into the double sequence ax where the mn-th term to ax is as follows: (ax)m,n = ∞∑ k=0 ∞∑ l=0 am,n,k,lxk,l. in [12] robison presented the following notion of regularity for four-dimensional matrix transformation and a silverman-toeplitz type characterization of such notion. definition 1.1. the four-dimensional matrix a is said to be rh-regular if it maps every bounded p-convergent sequence into a p-convergent sequence with the same p-limit. the assumption of bounded was added because a double sequence which is p-convergent is not necessarily bounded. the four-dimensional matrix a is said to be rh-conservative if it maps every bounded pconvergent sequence into a p-convergent sequence. in [12], robison presented the following notion of a conservative four-dimensional matrix transformation and a silvermantoeplitz type characterization of such a notion. theorem 1.2. ([5],[12]) the four-dimensional matrix a is rh-conservative if and only if rh − c1: p-limm,n am,n,k,l = ckl for each k and l; cubo 17, 2 (2015) submatrices of four dimensional summability matrices 91 rh − c2: p-limm,n ∑ ∞,∞ k,l=0,0 am,n,k,l = α; rh − c3: p-limm,n ∑ ∞ k=0 |am,n,k,l| = 0 for each l; rh − c4: p-limm,n ∑ ∞ l=0 |am,n,k,l| = 0 for each k; rh − c5: ∑ ∞,∞ k,l=0,0 |am,n,k,l| is p-convergent; rh − c6: there exist finite positive integers ∆ and γ such that ∑ k,l>γ |am,n,k,l| < ∆. along these same lines, robison and hamilton presented a silverman-toeplitz type multidimensional characterization of regularity in [5] and [12]. theorem 1.3. (hamilton [5], robison [12]) the four-dimensional matrix a is rh-regular if and only if rh1: p-limm,n am,n,k,l = 0 for each k and l; rh2: p-limm,n ∑ ∞,∞ k,l=0,0 am,n,k,l = 1; rh3: p-limm,n ∑ ∞ k=0 |am,n,k,l| = 0 for each l; rh4: p-limm,n ∑ ∞ l=0 |am,n,k,l| = 0 for each k; rh5: ∑ ∞,∞ k,l=0,0 |am,n,k,l| is p-convergent; rh6: there exist finite positive integers ∆ and γ such that ∑ k,l>γ |am,n,k,l| < ∆. the set of all absolutely convergent double sequences will be denoted ℓ′′, that is ℓ′′ = {x = {xk,l} : ∞∑ m=0 ∞∑ n=0 |xk,l| < ∞}. in [9], patterson proved that the matrix a = (am,n,k,l) determines an ℓ ′′ − ℓ′′ method if and only if sup k,l ∞∑ m=0 ∞∑ n=0 |am,n,k,l| < ∞. (1) in this paper, firstly we will obtain a general correspondance between rh-regular matrices and ℓ′′ − ℓ′′ matrices by proving that every rh-regular matrix gives rise to an ℓ′′ − ℓ′′ matrix by the removing of appropriate rows. secondly we prove that a matrix from double bounded sequences to double bounded sequences contains a row-submatrix is conservative. lastly, in order to prove a criterion for the summability of an unbounded double sequence the row-selection technique will be replaced by a column-selection technique. 2 main results theorem 2.1. if a = (amnkl) is a four dimensional summability matrix in which each row and each column converge to zero and sup m,n |amnkl| = α < ∞ 92 fatih nuray & richard f. patterson cubo 17, 2 (2015) for each k and l then a contains a row-submatrix that is an ℓ′′ − ℓ′′ matrix. proof. first chose positive integer v0 and w0 satisfying |av0,w0,0,0| ≤ 1; then, using the assumption that p − lim k,l av0,w0,0,0 = 0, choose k0 and l0 so that k > k0 and l > l0implies |av0,w0,k,l| ≤ 1. having selected vi, wj, ki and lj for i vp−1 and wq > wq−1 so that k ≤ kp−1, l ≤ lq−1 implies |avp,wq,k,l| ≤ 2 −(p+q) then choose kp > kp−1 and lq > lq−1 so that k > kp, l > lq implies |avp,wq,k,l| ≤ 2 −(p+q). now define the submatrix b by bmnkl = avp,wq,k,l. the above construction guarantees that each column sequence of b is dominated, except for at most one term, by the sequence (2−(p+q)); that is, if kp−1 < k ≤ kp, lq−1 < l ≤ lq, i 6= p and j 6= q, then |bijkl| = |avi,wj,k,l| ≤ 2 −(i+j). since |avp,wq,k,l| ≤ α, it is clear that for each k, l, ∞∑ p=0 ∞∑ q=0 |bpqkl| ≤ 4 + α. hence, by (1.1), b is an ℓ′′ − ℓ′′ matrix. corollary 2.2. every four dimensional rh-regular matrix contains a row-submatrix that is an ℓ′′ − ℓ′′ matrix. theorem 2.3. if a maps double bounded sequences into itself, then a contains a conservative row-submatrix b. proof. since a maps double bounded sequences into itself, we have sup m,n ∑ k ∑ l |amnkl| < ∞. therefore the sequence of row sums ( ∑ k ∑ l amnkl) is bounded, so we can choose a convergent subsequence. this yields a row-submatrix a′ of a that satisfies properties (ii) and (iii). it remains to choose a row-submatrix of a′ whose columns are convergent sequences. but this is a special case of the familiar diagonal process that is used in the proof of the multidimensional analog of helly selection principle (see [13],[2])for we have a family of functions (the rows of a′) that are cubo 17, 2 (2015) submatrices of four dimensional summability matrices 93 uniformly bounded by supm,n ∑ k ∑ l |amnkl| on their countable domain    (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), ... (1, 0), (1, 1), (1, 2, ) (1, 3), (1, 4), ... (2, 0, ) (2, 1), (2, 2), (2, 3), (2, 4), ... ... ... ... ... ... ...    therefore we can select a sequence of these functions that converges at each k and l. this sequence of rows of a′ are then the rows of b. 3 summability of unbounded sequences in [8] patterson and savaş proved multidimensional generalization of agnew’s theorem. theorem 3.1. if the four dimensional matrix a = (amnkl) such that ∑ k ∑ l |amnkl| < ∞ and p − lim m,n→∞ sup k,l |amnkl| = 0, (2) then there exists at least one p-divergent double sequence of zeros and ones that is a summable . by modifying the proof of theorem 2.1 from row selection to column selection, we can prove a theorem in which we relax the regularity of a, weaken property (3.1), and construct an unbounded sequence that is summed by a. theorem 3.2. if a is a four dimensional summability matrix whose column sequences tend to zero and lim inf k,l {max m,n |amnkl}| = 0, (3) then a sums an unbounded sequence. proof. using (3.2), we choose increasing sequences of column indices (kp) and (lq) such that for each p and q, max m,n |amnkplq| < 2 −(p+q). (4) then choose increasing row indices (vp) and (wq) so that if k ≤ kp, l ≤ lq, m > vp and n > wq, then |amnkl| < 2 −(p+q). now define double sequence x by xkl = { (p + 1)(q + 1), if k = kp, l = lq for some p, q 0, otherwise. 94 fatih nuray & richard f. patterson cubo 17, 2 (2015) then m > vp and n > wq implies |(ax)m,n| = | ∞∑ i=0 ∞∑ j=0 amnkiljxkilj| ≤ p∑ i=0 q∑ j=0 (i + 1)(j + 1)2−(p+q) + ∞∑ i=p+1 ∞∑ j=q+1 (i + 1)(j + 1)2−(i+j) = ((p + 1)(p + 2)(q + 1)(q + 2)2−(p+q)−2 + rpq, where limp,q rpq = 0. hence, limm,n(ax)m,n = 0. we note that if the row sequences of a tend to zero then (3.1) implies lim k,l→∞ {max m,n |amnkl}| = 0, which stronger than (3.2). therefore theorem 3.2 does have a weaker hypothesis than theorem 3.1. received: january 2014. accepted: january 2015. references [1] r. p. agnew inclusion relations among methods of summability compounded form given matrix methods, ark. mat. 2, (1952), 361-374. [2] d. djurcič, l. d. r. kočinac, and m. r. žižović, double sequences and selections, abstract and applied analysis, volume 2012, article id 497594, 6 pages . [3] j. a. fridy, absolute summability matrices that are stronger than the identity mapping, proc. amer. math. soc. 47, (195), 112-118. [4] j. a. fridy, submatrices of summability matrices, internat. j. math. and math. sci., 1, (1978,) 519-524. [5] h. j. hamilton, transformations of multiple sequences, duke math. jour., 2 (1936), 29 60. [6] g. h. hardy, divergent series, oxford, (1949). [7] r. f. patterson and e. savaş, matrix summability of statistically p-convergence sequences, filomat 25, (4), (2011), 55-62. [8] r. f. patterson, a theorem on entire four dimensional summability methods, appl. math. comput., 219, (2013), 7777-7782. cubo 17, 2 (2015) submatrices of four dimensional summability matrices 95 [9] r. f. patterson, four dimensional matrix characterization of absolute summability, soochow journal of math., 30 (1), (2004), 21-26. [10] r. f. patterson, analogues of some fundamental theorems of summability theory, internat. j. math. & math. sci. 23 (1), (2000), 1-9. [11] a. pringsheim, zur theorie der zweifach unendlichen zahlenfolgen, mathematische annalen, 53, (1900), 28932. [12] g. m. robison, divergent double sequences and series, amer. math. soc. trans. 28, (1926), 50-73. [13] v. c. vyacheslav and m. caterina a pointwise selection principle for metric semigroup valued functions, j. math. anal. appl. 341, (2008), 613-625. introduction main results summability of unbounded sequences cubo a mathematical journal vol.16, no¯ 01, (73–80). march 2014 on certain functional equation in semiprime rings and standard operator algebras nejc širovnik 1 department of mathematics and computer science, faculty of natural sciences and mathematics, university of maribor, koroš ka cesta 160, 2000 maribor, slovenia nejc.sirovnik@uni-mb.si abstract the main purpose of this paper is to prove the following result, which is related to a classical result of chernoff. let x be a real or complex banach space, let l(x) be the algebra of all bounded linear operators on x and let a(x) ⊆ l(x) be a standard operator algebra. suppose there exists a linear mapping d : a(x) → l(x) satisfying the relation 2d(an) = d(an−1)a+an−1d(a)+d(a)an−1+ad(an−1) for all a ∈ a(x), where n ≥ 2 is some fixed integer. in this case d is of the form d(a) = [a, b] for all a ∈ a(x) and some fixed b ∈ l(x), which means that d is a linear derivation. in particular, d is continuous. resumen el propósito principal de este art́ıculo es probar el siguiente resultado, el cual se relaciona a un resultado clásico de chernoff. sea x un espacio de banach real o complejo, sea l(x) el álgebra de todos los operadores lineales acotados en x y sea a(x) ⊆ l(x) una álgebra de operadores estándar. supongamos que existe una aplicación lineal d : a(x) → l(x) satisfaciendo la relación 2d(an) = d(an−1)a + an−1d(a) + d(a)an−1 + ad(an−1) para todo a ∈ a(x), donde n ≥ 2 es algún entero fijo. en este caso d es de la forma d(a) = [a, b] para todo a ∈ a(x) y algún b ∈ l(x) fijo, lo que significa que d es una derivación lineal. en particular, d es continua. keywords and phrases: prime ring, semiprime ring, banach space, standard operator algebra, derivation, jordan derivation. 2010 ams mathematics subject classification: 16n60, 46b99, 39b42 1this research has been supported by the research council of slovenia. 74 nejc širovnik cubo 16, 1 (2014) this research has been motivated by the work of vukman [19]. throughout, r will represent an associative ring with center z(r). as usual we write [x, y] for xy − yx. given an integer n ≥ 2, a ring r is said to be n-torsion free if for x ∈ r, nx = 0 implies x = 0. recall that a ring r is prime if for a, b ∈ r, arb = (0) implies that either a = 0 or b = 0, and is semiprime in case ara = (0) implies a = 0. let a be an algebra over the real or complex field and let b be a subalgebra of a. a linear mapping d : b → a is called a linear derivation in case d(xy) = d(x)y + xd(y) holds for all pairs x, y ∈ b. in case we have a ring r, an additive mapping d : r → r is called a derivation if d(xy) = d(x)y + xd(y) holds for all pairs x, y ∈ r and is called a jordan derivation in case d(x2) = d(x)x + xd(x) is fulfilled for all x ∈ r. a derivation d is inner in case there exists such a ∈ r that d(x) = [x, a] holds for all x ∈ r. every derivation is a jordan derivation. the converse is in general not true. a classical result of herstein [9] asserts that any jordan derivation on a 2-torsion free prime ring is a derivation. a brief proof of herstein theorem can be found in [2]. cusack [7] generalized herstein theorem to 2-torsion free semiprime rings (see [3] for an alternative proof). herstein theorem has been fairly generalized by beidar, brešar, chebotar and martindale [1]. for results concerning derivations in rings and algebras we refer to [5, 11, 16, 17, 18, 19], where further references can be found. let x be a real or complex banach space and let l(x) and f(x) denote the algebra of all bounded linear operators on x and the ideal of all finite rank operators in l(x), respectively. an algebra a(x) ⊆ l(x) is said to be standard in case f(x) ⊂ a(x). let us point out that any standard operator algebra is prime. motivated by the work of brešar [4], vukman [19] has recently conjectured that in case we have an additive mapping d : r → r, where r is a 2-torsion free semiprime ring satisfying the relation 2d(xyx) = d(xy)x + xyd(x) + d(x)yx + xd(yx) (1) for all pairs x, y ∈ r, then d is a derivation. note that in case a ring has the identity element, the proof of vukman’s conjecture is immediate. namely, in this case the substitution y = e in the relation (1), where e stands for the identity element, gives that d is a jordan derivation and then it follows from cusack’s generalization of herstein theorem that d is a derivation. the substitution y = xn−2 in the relation (1) gives 2d(xn) = d(xn−1)x + xn−1d(x) + d(x)xn−1 + xd(xn−1), which leads to the following conjecture. conjecture 0.1. let r be a semiprime ring with suitable torsion restrictions and let d : r → r be an additive mapping. suppose that 2d(xn) = d(xn−1)x + xn−1d(x) + d(x)xn−1 + xd(xn−1) holds for all x ∈ r and some fixed integer n ≥ 2. in this case d is a derivation. cubo 16, 1 (2014) on certain functional equation in semiprime rings . . . 75 it is our aim in this paper to prove the conjecture above in case a ring has the identity element. theorem 0.2. let n ≥ 2 be some fixed integer, let r be a n!-torsion free semiprime ring with the identity element and let d : r → r be an additive mapping satisfying the relation 2d(xn) = d(xn−1)x + xn−1d(x) + d(x)xn−1 + xd(xn−1) for all x ∈ r. in this case d is a derivation. proof. we have the relation 2d(xn) = d(xn−1)x + xn−1d(x) + d(x)xn−1 + xd(xn−1) (2) and let us denote the identity element of r by e. putting e for x in the above relation, we obtain d(e) = 0. (3) let y be any element of the center z(r). putting x + y in the above relation, we obtain 2 n∑ i=0 ( n i ) d(xn−iyi) = ( n−1∑ i=0 ( n−1 i ) d(xn−1−iyi) ) (x + y) + ( n−1∑ i=0 ( n−1 i ) xn−1−iyi ) d(x + y) + d(x + y) ( n−1∑ i=0 ( n−1 i ) xn−1−iyi ) + (x + y) ( n−1∑ i=0 ( n−1 i ) d(xn−1−iyi) ) . using (2) in the above relation and rearranging it in sense of collecting together terms involving equal number of factors of y, we obtain n−1∑ i=1 fi(x, y) = 0, where fi(x, y) stands for the expression of terms involving i factors of y. replacing x by x + 2y, x + 3y, . . . , x + (n − 1)y in turn in the relation (2) and expressing the resulting system of n − 1 homogeneous equations of variables fi(x, y), i = 1, 2, . . . , n − 1, we see that the coefficient matrix of the system is a vandermonde matrix        1 1 . . . 1 2 22 . . . 2n−1 ... ... ... ... n − 1 (n − 1)2 . . . (n − 1)n−1        . 76 nejc širovnik cubo 16, 1 (2014) since the determinant of this matrix is different from zero, it follows that the system has only a trivial solution. in particular, fn−2(x, e) = 2 ( n n−2 ) d(x2) − ( n−1 n−2 ) d(x)x − ( n−1 n−3 ) d(x2) − ( n−1 n−2 ) xd(x) − ( n−1 n−3 ) x2a − ( n−1 n−2 ) d(x)x − ( n−1 n−3 ) ax2 − ( n−1 n−2 ) xd(x) − ( n−1 n−3 ) d(x2), where a denotes t(e). after some calculation and considering the relation (3), we obtain (n(n − 1) − (n − 1)(n − 2))d(x2) = 2(n − 1)(d(x)x + xd(x)). since r is 2(n − 1)-torsion free, the above relation reduces to d(x2) = d(x)x + xd(x) for all x ∈ r. in other words, d is a jordan derivation and cusack’s generalization of herstein theorem now implies that d is a derivation, which completes the proof. in the proof of theorem 0.2 we used methods similar to those used by vukman and kosi-ulbl in [10]. we proceed with the following result in the spirit of conjecture 0.1. theorem 0.3. let x be a real or complex banach space and let a(x) be a standard operator algebra on x. suppose there exists a linear mapping d : a(x) → l(x) satisfying the relation 2d(an) = d(an−1)a + an−1d(a) + d(a)an−1 + ad(an−1) for all a ∈ a(x) and some fixed integer n ≥ 2. in this case d is of the form d(a) = [a, b] for all a ∈ a(x) and some fixed b ∈ l(x), which means that d is a linear derivation. in case n = 3 the above relation reduces to theorem 4 in [19]. let us point out that in theorem 0.3 we obtain as a result the continuity of d under purely algebraic assumptions concerning d, which means that theorem 0.3 might be of some interest from the automatic continuity point of view. for results concerning automatic continuity we refer the reader to [8] and [13]. in the proof of theorem 0.3 we use herstein theorem, the result below and methods that are similar to those used by kosi-ulbl and vukman in [12]. theorem 0.4. let x be a real or complex banach space, let a(x) be a standard operator algebra on x and let d : a(x) → l(x) be a linear derivation. in this case d is of the form d(a) = [a, b] for all a ∈ a(x) and some fixed b ∈ l(x). theorem 0.4 has been proved by chernoff [6] (see also [14, 15]). proof of the theorem 0.3. we have the relation 2d(an) = d(an−1)a + an−1d(a) + d(a)an−1 + ad(an−1) (4) cubo 16, 1 (2014) on certain functional equation in semiprime rings . . . 77 for all a ∈ a(x). let us first restrict our attention on f(x). let a be from f(x) and let p ∈ f(x) be a projection with ap = pa = a. putting p for a in the relation (4), we obtain d(p) = d(p)p + pd(p). (5) putting a + p for a in the relation (4), we obtain, similary as in the proof of theorem 0.2, the relation 2 n∑ i=0 ( n i ) d(an−ipi) = ( n−1∑ i=0 ( n−1 i ) d(an−1−ipi) ) (a + p) + ( n−1∑ i=0 ( n−1 i ) an−1−ipi ) d(a + p) + d(a + p) ( n−1∑ i=0 ( n−1 i ) an−1−ipi ) + (a + p) ( n−1∑ i=0 ( n−1 i ) d(an−1−ipi) ) . using (4) and (5) in the above relation and rearranging it in sense of collecting together terms involving equal number of factors of p, we obtain n−1∑ i=1 fi(a, p) = 0, where fi(a, p) stands for the expression of terms involving i factors of p. replacing a by a + 2p, a + 3p, . . . , a + (n − 1)p in turn in the relation (4) and expressing the resulting system of n − 1 homogeneous equations of variables fi(a, p), i = 1, 2, . . . , n − 1, we see that the coefficient matrix of the system is a vandermonde matrix        1 1 . . . 1 2 22 . . . 2n−1 ... ... ... ... n − 1 (n − 1)2 . . . (n − 1)n−1        . since the determinant of this matrix is different from zero, it follows that the system has only a trivial solution. in particular, fn−1(a, p) = 2 ( n n−1 ) d(a) − ( n−1 n−1 ) d(p)a − ( n−1 n−2 ) d(a)p − ( n−1 n−1 ) pd(a) − ( n−1 n−2 ) ad(p) − ( n−1 n−1 ) d(a)p − ( n−1 n−2 ) d(p)a − ( n−1 n−1 ) ad(p) − ( n−1 n−2 ) pd(a). the above relation reduces to 2d(a) = d(a)p + ad(p) + d(p)a + pd(a) (6) 78 nejc širovnik cubo 16, 1 (2014) and putting a2 for a in the above relation, we obtain 2d(a2) = d(a2)p + a2d(p) + d(p)a2 + pd(a2). (7) as the previously mentioned system of n − 1 homogeneous equations has only a trivial solution, we also obtain fn−2(a, p) = 2 ( n n−2 ) d(a2) − ( n−1 n−2 ) d(a)a − ( n−1 n−3 ) d(a2)p − ( n−1 n−2 ) ad(a) − ( n−1 n−3 ) a2d(p) − ( n−1 n−2 ) d(a)a − ( n−1 n−3 ) d(p)a2 − ( n−1 n−2 ) ad(a) − ( n−1 n−3 ) pd(a2). the above relation now reduces to n(n − 1)d(a2) = 2(n − 1)(d(a)a + ad(a)) + + ( n−1 n−3 ) (d(a2)p + a2d(p) + d(p)a2 + pd(a2)). applying the relation (7) in the above relation, we obtain n(n − 1)d(a2) = 2(n − 1)(d(a)a + ad(a)) + (n − 1)(n − 2)d(a2), which reduces to d(a2) = d(a)a + ad(a). (8) from the relation (6) one can conclude that d maps f(x) into itself. we therefore have a linear mapping d, which maps f(x) into itself and satisfies the relation (8) for all a ∈ f(x). in other words, d is a jordan derivation on f(x) and since f(x) is prime, it follows, according to herstein theorem, that d is a derivation on f(x). applying theorem 0.4 one can conclude that d is of the form d(a) = [a, b] (9) for all a ∈ f(x) and some fixed b ∈ l(x). it remains to prove that (9) holds for all a ∈ a(x) as well. for this purpose we introduce d1 : a(x) → l(x) by d1(a) = [a, b] and consider the mapping d0 = d−d1. the mapping d0 is obviously linear, satisfies the relation (4) and vanishes on f(x). it is our aim to prove that d0 vanishes on a(x) as well. let a ∈ a(x), let p be a one-dimensional projection and let us introduce s ∈ a(x) by s = a + pap − (ap + pa). we have sp = ps = 0. obviously, d0(s) = d0(a). by the relation (4) we now have d0(s n−1)s + sn−1d0(s) + d0(s)s n−1 + sd0(s n−1) = 2d0(s n) = 2d0(s n + p) = 2d0((s + p) n) = d0((s + p) n−1)(s + p) + (s + p)n−1d0(s + p) + d0(s + p)(s + p) n−1 + (s + p)d0((s + p) n−1) = d0(s n−1 )s + d0(s n−1 )p + sn−1d0(s) + pd0(s) + d0(s)s n−1 + d0(s)p + sd0(s n−1 ) + pd0(s n−1 ). cubo 16, 1 (2014) on certain functional equation in semiprime rings . . . 79 from the above relation it follows that d0(s n−1)p + pd0(s) + d0(s)p + pd0(s n−1) = 0. since d0(s) = d0(a), we can rewrite the above relation as d0(a n−1 )p + pd0(a) + d0(a)p + pd0(a n−1 ) = 0. (10) putting 2a for a in the above relation, we obtain 2n−1d0(a n−1)p + 2pd0(a) + 2d0(a)p + 2 n−1pd0(a n−1) = 0. (11) in case n = 2, the relation (10) implies that pd0(a) + d0(a)p = 0. (12) in case n > 2, the relations (10) and (11) give the above relation (12). multiplying the above relation from both sides by p, we obtain pd0(a)p = 0. right multiplication by p in the relation (12) gives pd0(a)p + d0(a)p = 0, which is reduced by the above relation to d0(a)p = 0. since p is an arbitrary one-dimensional projection, it follows from the above relation that d0(a) = 0 for all a ∈ a(x), which completes the proof of the theorem. received: january 2013. accepted: february 2014. references [1] k. i. beidar, m. brešar, m. a. chebotar, w. s. martindale 3rd: on herstein’s lie map conjectures ii, j. algebra 238 (2001), 239-264. [2] m. brešar, j. vukman: jordan derivations on prime rings, bull. austral. math. soc. vol. 37 (1988), 321-322. [3] m. brešar: jordan derivations on semiprime rings, proc. amer. math. soc. 104 (1988), 1003-1006. [4] m. brešar: jordan mappings of semiprime rings, j. algebra 127 (1989), 218-228. [5] m. brešar, j. vukman: jordan (θ,φ)-derivations, glasnik mat. 16 (1991), 13-17. 80 nejc širovnik cubo 16, 1 (2014) [6] p. r. chernoff: representations, automorphisms and derivations of some operator algebras, j. funct. anal. 2 (1973), 275-289. [7] j. cusack: jordan derivations on rings, proc. amer. math. soc. 53 (1975), 321-324. [8] h. g. dales: automatic continuity, bull. london math. soc. 10 (1978), 129-183. [9] i. n. herstein: jordan derivations of prime rings, proc. amer. math. soc. 8 (1957), 11041119. [10] i. kosi-ulbl, j. vukman: a note on derivations in semiprime rings, int. j. math. & math. sci., 20 (2005), 3347-3350. [11] i. kosi-ulbl, j. vukman: on derivations in rings with involution, internat. math. j. vol. 6 (2005), 81-91. [12] i. kosi-ulbl, j. vukman: an identity related to derivations of standard operator algebras and semisimple h∗-algebras, cubo a mathematical journal, 12 (2010), 95-103. [13] a. m. sinclair: automatic continuity of linear operators, london math. soc. lecture note ser. 21, cambridge university press, cambridge, london, new york and melbourne (1976). [14] p. šemrl: ring derivations on standard operator algebras, j. funct. anal. vol. 112 (1993), 318-324. [15] j. vukman: on automorphisms and derivations of operator algebras, glasnik mat. vol. 19 (1984), 135-138. [16] j. vukman: on derivations of algebras with involution, acta math. hungar. 112 (3) (2006), 181-186. [17] j. vukman: on derivations of standard operator algebras and semisimple h∗-algebras, studia sci. math. hungar. 44 (2007), 57-63. [18] j. vukman: identities related to derivations and centralizers on standard operator algebras, taiwan. j. math. vol. 11 (2007), 255-265. [19] j. vukman: some remarks on derivations in semiprime rings and standard operator algebras, glasnik. mat. vol. 46 (2011), 43-48. cubo a mathematical journal vol.14, no¯ 02, (175–182). june 2012 on a condition for the nonexistence of w-solutions of nonlinear high-order equations with l1-data alexander a. kovalevsky institute of applied mathematics and mechanics, rosa luxemburg st. 74, 83114 donetsk, ukraine email: alexkvl@iamm.ac.donetsk.ua and francesco nicolosi department of mathematics and informatics, university of catania, 95125 catania, italy email: fnicolosi@dmi.unict.it abstract in a bounded open set of rn we consider the dirichlet problem for nonlinear 2m-order equations in divergence form with l1-right-hand sides. it is supposed that 2 6 m < n, and the coefficients of the equations admit the growth of rate p−1 > 0 with respect to the derivatives of order m of unknown function. we establish that under the condition p 6 2 − m/n for some l1-data the corresponding dirichlet problem does not have w-solutions. resumen en un conjunto abierto y acotado de rn consideramos el problema de dirichlet para ecuaciones no lineales de orden 2m en la forma divergente con lados l1-right-hand. se supone que 2 6 m < n, y los coeficientes de las ecuaciones admiten el radio de crecimiento p−1 > 0 con respecto a las derivadas de orden m de la función desconocida. establecemos que bajo la condición p 6 2 − m/n para algn l1data el problema de dirichlet correspondiente no tiene w-soluciones. keywords and phrases: nonlinear high-order equations in divergence form, l1-data, dirichlet problem, w-solution, nonexistence of w-solutions. 2010 ams mathematics subject classification: 35g30, 35j40, 35j60. 176 alexander a. kovalevsky and francesco nicolosi cubo 14, 2 (2012) 1 introduction it is known that nonlinear elliptic second-order equations in divergence form whose principal coefficients grow with respect to the gradient of unknown function u as |∇u|p−1 for some l1-right-hand sides may not have weak solutions if the exponent p is sufficiently close to 1. the fact of the nonexistence of weak solutions was observed in [1] by giving the following example: if ω is an open set of rn with n > 2 and 1 < p 6 2 − 1/n, then there exists a function f ∈ l1(ω) such that the problem u ∈ w1,1 loc (ω), −∆pu + u = f in d ′ (ω) does not have a solution. the given observation was one of motivations for the development of the theory of entropy solutions for nonlinear elliptic second-order equations with l1-data [1]. according to the results of [1], if 1 < p < n, under natural growth, coercivity and strict monotonicity conditions for coefficients of the equations under consideration an entropy solution exists and is unique for every l1-right-hand side. moreover, if p > 2 − 1/n, the entropy solution is a weak solution. analogous results on the existence of entropy and weak solutions for nonlinear elliptic highorder equations with coefficients satisfying a strengthened coercivity condition and l1-right-hand sides were obtained in [3, 4]. conditions of the existence of weak solutions for some classes of degenerate nonlinear elliptic high-order equations with strengthened coercivity and l1-data were given in [5, 6]. as far as nonlinear elliptic high-order equations with l1-right-hand sides and coefficients satisfying the natural coercivity condition are concerned, the question on their solvability on the whole is still open. it seems, for these equations the approaches which work in the cases of secondorder equations with l1-data and high-order equations with strengthened coercivity and l1-data are not suitable. on the other hand, the use of the known principle of uniform boundedness (see [2, chapter 2]) one can consider as a general functional tool for the study of conditions for the nonexistence of weak solutions of nonlinear arbitrary even order equations in divergence form with l1-data. using this principle, in the present article we give such a condition for high-order equations. 2 main results let m,n ∈ n be numbers such that 2 6 m < n. let ω be a bounded open set of rn. we shall use the following notation: λ is the set of all n-dimensional multi-indicies α such that |α| = m, rnm is the space of all functions ξ : λ → r; if u ∈ l 1 loc (ω) and the function u has the weak derivatives dαu, α ∈ λ, then ∇mu : ω → r n m is the mapping such that for every x ∈ ω cubo 14, 2 (2012) on a condition for the nonexistence of w-solutions ... 177 and for every α ∈ λ, (∇mu(x))α = d αu(x). let p > 1, c > 0, g ∈ l1/(p−1)(ω), g > 0 in ω, and let for every α ∈ λ, aα : ω × r n m → r be a carathéodory function. we shall assume that for almost every x ∈ ω and for every ξ ∈ rnm, ∑ α∈λ |aα(x,ξ)| 6 c ∑ α∈λ |ξα| p−1 + g(x). (2.1) for every f ∈ l1(ω) by (pf) we denote the following problem: ∑ α∈λ (−1)|α|dαaα(x,∇mu) = f in ω, dαu = 0, |α| 6 m − 1, on ∂ω. definition 2.1. let f ∈ l1(ω). a w-solution of problem (pf) is a function u ∈ ◦ wm,1(ω) such that (i) for every α ∈ λ, aα(x,∇mu) ∈ l 1(ω); (ii) for every ϕ ∈ c∞0 (ω), ∫ ω { ∑ α∈λ aα(x,∇mu)d αϕ } dx = ∫ ω fϕdx. theorem 2.1. suppose that p 6 2 − m n . (2.2) then there exists f ∈ l1(ω) such that problem (pf) does not have w-solutions. proof. let us assume that for every f ∈ l1(ω) there exists a w-solution of problem (pf). this implies that if f ∈ l1(ω), then there exists a function uf ∈ ◦ wm,1(ω) such that for every α ∈ λ, aα(x,∇muf) ∈ l 1(ω) and ∀ϕ ∈ c∞0 (ω), ∫ ω { ∑ α∈λ aα(x,∇muf)d αϕ } dx = ∫ ω fϕdx. (2.3) observe that due to the inequality p > 1 and (2.2) we have 0 < 2−p < 1. we set p1 = 1/(2−p). clearly, p1 > 1. using (2.1) and the inclusion g ∈ l1/(p−1)(ω), we establish that if f ∈ l1(ω), then for every α ∈ λ, aα(x,∇muf) ∈ l p1/(p1−1)(ω). for every f ∈ l1(ω) we define the functional hf : ◦ wm,p1(ω) → r by 〈hf,ϕ〉 = ∫ ω { ∑ α∈λ aα(x,∇muf)d αϕ } dx, ϕ ∈ ◦ wm,p1(ω). 178 alexander a. kovalevsky and francesco nicolosi cubo 14, 2 (2012) it is easy to see that ∀f ∈ l1(ω), hf ∈ ( ◦ wm,p1(ω))∗. (2.4) moreover, taking into account (2.3), for every f ∈ l1(ω) and for every ϕ ∈ c∞0 (ω) we get 〈hf,ϕ〉 = ∫ ω fϕdx. (2.5) from (2.4) and (2.5) it follows that for every f1,f2 ∈ l 1(ω), hf1+f2 = hf1 + hf2, (2.6) for every f ∈ l1(ω) and for every λ ∈ r, hλf = λhf. (2.7) next, let ϕ ∈ ◦ wm,p1(ω). we fix a sequence {ϕk} ⊂ c ∞ 0 (ω) such that ‖ϕk − ϕ‖wm,p1 (ω) → 0. (2.8) for every k ∈ n we define the functional fk : l 1(ω) → r by 〈fk,f〉 = |〈hf,ϕk〉|, f ∈ l 1 (ω). using (2.5), we establish the following fact: if k ∈ n and f1,f2 ∈ l 1(ω), then |〈fk,f1〉 − 〈fk,f2〉| 6 ( max ω |ϕk| ) ‖f1 − f2‖l1(ω). this implies that for every k ∈ n the functional fk is continuous on l 1(ω). moreover, with the use of (2.6) and (2.7) we obtain the next properties: (i) for every k ∈ n and for every f1,f2 ∈ l 1(ω), 〈fk,f1 + f2〉 6 〈fk,f1〉 + 〈fk,f2〉; (ii) for every k ∈ n, for every f ∈ l1(ω) and for every λ ∈ r, 〈fk,λf〉 = |λ|〈fk,f〉. finally, taking into account (2.4) and (2.8), we establish that for every f ∈ l1(ω) the sequence of the numbers 〈fk,f〉 is bounded. this along with the nonnegativity and continuity of the functionals fk, properties (i) and (ii) and the principle of uniform boundedness [2, chapter 2] allows us to conclude that there exists m > 0 such that for every k ∈ n and for every f ∈ l1(ω), 〈fk,f〉 6 m‖f‖l1(ω). from the result obtained, using the definition of the functionals fk and (2.4) and (2.8), we deduce that ∀f ∈ l1(ω), |〈hf,ϕ〉| 6 m‖f‖l1(ω). (2.9) cubo 14, 2 (2012) on a condition for the nonexistence of w-solutions ... 179 now let f : l1(ω) → r be the functional such that for every f ∈ l1(ω), 〈f,f〉 = 〈hf,ϕ〉. (2.10) owing to (2.6) and (2.7), the functional f is linear, and by virtue of (2.9) and (2.10), for every f ∈ l1(ω), |〈f,f〉| 6 m‖f‖l1(ω). therefore, f ∈ (l 1(ω))∗. then there exists a function ψ ∈ l∞(ω) such that for every f ∈ l1(ω), 〈f,f〉 = ∫ ω ψfdx. this and (2.10) imply that ∀f ∈ l1(ω), 〈hf,ϕ〉 = ∫ ω ψfdx. (2.11) let us show that ϕ = ψ a. e. in ω. in fact, let f ∈ l1/(p−1)(ω). then, by (2.5) and (2.8), 〈hf,ϕk〉 → ∫ ω fϕdx. (2.12) on the other hand, from (2.4), (2.8) and (2.11) it follows that 〈hf,ϕk〉 → ∫ ω fψdx. this and (2.12) imply that ∫ ω f(ϕ − ψ)dx = 0. hence, taking into account the arbitrariness of the function f in l1/(p−1)(ω), we obtain that ϕ = ψ a. e. in ω. therefore, ϕ ∈ l∞(ω). thus, we conclude that ◦ wm,p1(ω) ⊂ l∞(ω). (2.13) however, since, by (2.2), we have mp1 6 n, inclusion (2.13) is not true. for instance, if p < 2 − m/n, y ∈ ω, v : ω → r is a function such that for every x ∈ ω\{y}, v(x) = ln |x − y|, b is a closed ball in rn with the center y such that b ⊂ ω, ψ1 ∈ c ∞ 0 (ω) and ψ1 = 1 in b, then vψ1 ∈ ◦ wm,p1(ω)\l∞(ω). the contradiction obtained allows us to conclude that there exists a function f ∈ l1(ω) such that problem (pf) does not have w-solutions. this completes the proof of the theorem. now we give an analogous result for equations with lower-order terms. let c0 > 0, 0 < σ < n/(n − m), σ1 = n σ(n−m) , g0 ∈ l σ1(ω), g0 > 0 in ω, and let a : ω × r → r be a carathéodory function such that for almost every x ∈ ω and for every s ∈ r, |a(x,s)| 6 c0|s| σ + g0(x). (2.14) 180 alexander a. kovalevsky and francesco nicolosi cubo 14, 2 (2012) for every f ∈ l1(ω) by (pf) we denote the following problem: ∑ α∈λ (−1)|α|dαaα(x,∇mu) + a(x,u) = f in ω, dαu = 0, |α| 6 m − 1, on ∂ω. definition 2.2. let f ∈ l1(ω). a w-solution of problem (pf) is a function u ∈ ◦ wm,1(ω) such that (i) for every α ∈ λ, aα(x,∇mu) ∈ l 1(ω); (ii) a(x,u) ∈ l1(ω); (iii) for every ϕ ∈ c∞0 (ω), ∫ ω { ∑ α∈λ aα(x,∇mu)d αϕ + a(x,u)ϕ } dx = ∫ ω fϕdx. theorem 2.2. suppose that condition (2.2) is satisfied. then there exists f ∈ l1(ω) such that problem (pf) does not have w-solutions. proof. let us assume that for every f ∈ l1(ω) there exists a w-solution of problem (pf). this implies that if f ∈ l1(ω), then there exists a function uf ∈ ◦ wm,1(ω) such that for every α ∈ λ, aα(x,∇muf) ∈ l 1(ω), a(x,uf) ∈ l 1(ω) and ∀ϕ ∈ c∞0 (ω), ∫ ω { ∑ α∈λ aα(x,∇muf)d αϕ + a(x,uf)ϕ } dx = ∫ ω fϕdx. (2.15) we set p1 = 1/(2 − p). as in the proof of the previous theorem, we have: if f ∈ l 1(ω), then for every α ∈ λ, aα(x,∇muf) ∈ l p1/(p1−1)(ω). moreover, taking into account that, by sobolev embedding theorem, ◦ wm,1(ω) ⊂ ln/(n−m)(ω) and using the inclusion g0 ∈ l σ1(ω) and (2.14), we obtain that for every f ∈ l1(ω), a(x,uf) ∈ l σ1(ω). next, we define v = ◦ wm,p1(ω) ∩ lσ1/(σ1−1)(ω). the set v is a banach space with the norm ‖u‖v = ‖u‖wm,p1 (ω) + ‖u‖lσ1/(σ1−1)(ω). for every f ∈ l1(ω) we define the functional gf : v → r by 〈gf,ϕ〉 = ∫ ω { ∑ α∈λ aα(x,∇muf)d αϕ + a(x,uf)ϕ } dx, ϕ ∈ v. cubo 14, 2 (2012) on a condition for the nonexistence of w-solutions ... 181 clearly, ∀f ∈ l1(ω), gf ∈ v ∗. (2.16) furthermore, taking into account (2.15), for every f ∈ l1(ω) and for every ϕ ∈ c∞0 (ω) we get 〈gf,ϕ〉 = ∫ ω fϕdx. (2.17) we denote by v1 the closure of c ∞ 0 (ω) in v. using (2.16) and (2.17) and arguing by analogy with the proof of theorem 2.1, we establish that v1 ⊂ l ∞ (ω). (2.18) however, since, by condition (2.2), we have mp1 6 n, inclusion (2.18) is not true. for instance, if p < 2−m/n and v and ψ1 are the functions described at the end of the proof of theorem 2.1, then vψ1 ∈ v1\l ∞(ω). the contradiction obtained allows us to conclude that there exists f ∈ l1(ω) such that problem (pf) does not have w-solutions. this completes the proof of the theorem. 3 remarks simple examples of the functions aα and a satisfying inequalities (2.1) and (2.14) are as follows: (i) aα(x,ξ) = |ξα| p−1 or aα(x,ξ) = |ξα| p−1signξα if α ∈ λ; (ii) a(x,s) = as or a(x,s) = a|s|σ or a(x,s) = a|s|σsigns, where a ∈ r and 0 < σ < n/(n − m). finally, observe that if 1 0 and σ ∈ (0,n/(n − m)), then for every f ∈ lnp/(n−mp)(ω) problem (pf) has a wsolution u ∈ ◦ wm,p(ω). this fact simply follows from the known results of the theory of monotone operators (see for instance [7]). however, if 1 < p 6 2− m n , according to theorem 2.2, there exists f ∈ l1(ω) such that problem (pf) does not have w-solutions. received: march 2011. revised: december 2011. references [1] ph. bénilan, l. boccardo, t. gallouët, r. gariepy, m. pierre, and j.l. vazquez, an l1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, ann. scuola norm. sup. pisa cl. sci. (4) 22 (1995) 241–273. 182 alexander a. kovalevsky and francesco nicolosi cubo 14, 2 (2012) [2] n. dunford and j.t. schwartz, linear operators. part i. general theory, john wiley & sons, inc., new york, 1988. [3] a.a. kovalevskii, entropy solutions of the dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in l1, izv. math. 65 (2001) 231–283. [4] a. kovalevsky, entropy solutions of dirichlet problem for a class of nonlinear elliptic high-order equations with l1-data, nelinejnye granichnye zadachi 12 (2002) 119–127. [5] a. kovalevsky and f. nicolosi, solvability of dirichlet problem for a class of degenerate nonlinear high-order equations with l1-data, nonlinear anal. 47 (2001) 435–446. [6] a. kovalevsky and f. nicolosi, existence of solutions of some degenerate nonlinear elliptic fourth-order equations with l1-data, appl. anal. 81 (2002) 905–914. [7] j.-l. lions, quelques méthodes de résolution des problèmes aux limites non linéaires, dunod, gauthier-villars, paris, 1969. cubo a mathematical journal vol.15, no¯ 01, (159–169). march 2013 generalized ulam hyers stability of derivations of a aq functional equation m. arunkumar government arts college, department of mathematics, tiruvannamalai 606 603, tamilnadu, india. annarun2002@yahoo.co.in abstract in this paper, the author established the generalized ulam hyers stability of derivations of additive and quadratic (aq)functional equation f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y). resumen en este art́ıculo el autor establece la estabilidad generalizada ulam-hyers de derivaciones de la ecuación (aq)-funcional cuadrática y aditiva f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y). keywords and phrases: additive functional equations, quadratic functional equation, mixed type functional equation, additive derivations, quadratic derivations, ulam hyers stability 2010 ams mathematics subject classification: 39b52, 32b72, 32b82 160 m. arunkumar cubo 15, 1 (2013) 1 introduction the study of stability problems for functional equations is related to a question of ulam [33] concerning the stability of group homomorphisms and affirmatively answered for banach spaces by hyers [13]. it was further generalized and excellent results obtained by number of authors [2, 10, 24, 30, 32]. over the last six or seven decades, the above problem was tackled by numerous authors and its solutions via various forms of functional equations like additive, quadratic, cubic, quartic, mixed type functional equations which involves only these types of functional equations were discussed. we refer the interested readers for more information on such problems to the monographs [1, 9, 14, 19, 22, 31]. c. park [25] applied gavruta’s result to banach modules over a c∗−algebra. many authors have studied the structure of c∗− algebras for different types of functional equations in various settings one can refer [6, 8, 26, 28]. it seems that approximate derivations was first investigated by k.w. jun and d.w. park [16]. recently, the stability of derivations have been investigated in [7, 11, 12, 20, 27, 29] and references therein. the stability of cubic derivations was first time introduced and investigated by m.e. gordji et. al.,[12]. with the help of [12], the stability of quadratic derivations was discussed by m. arunkumar et. al., [3]. very recently m. arunkumar and j.m. rassias [5], established the generalized ulam hyers stability of an additive and quadratic (aq)-mixed type functional equation f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y) (1) in banach spaces. the solution and stability of several types of mixed type additive and quadratic type functional equations were discussed in [4, 17, 18, 21, 23] in this paper, the author first time established the generalized ulam hyers stability of mixed derivations of a additive quadratic (aq)functional equation (1). hereafter through out this paper, let us consider x and y to be a normed algebra and a banach algebra, respectively. 2 stability results: additive derivations in this section, the authors investigate the generalized ulam-hyers stability of additive derivations of the aq-functional equation (1). definition 2.1. a c−linear mapping a : x → x is called additive derivation on x if a satisfies a(xy) = a(x)y + xa(y) (1) for all x, y ∈ x. cubo 15, 1 (2013) generalized ulam hyers stability of derivations of a aq ... 161 theorem 2.1. let j = ±1. let fa : x → y be a odd mapping for which there exist a function α, β : x2 → [0, ∞) with the condition ∞∑ n=0 α ( 2njx, 2njy ) 2nj converges in r and lim n→∞ α ( 2njx, 2njy ) 2nj = 0 (2) ∞∑ n=0 β ( 2njx, 2njy ) 22nj converges in r and lim n→∞ α ( 2njx, 2njy ) 22nj = 0 (3) such that the functional inequalities ‖fa(x + y) + fa(x − y) − 2fa(x) − fa(y) − fa(−y)‖ ≤ α (x, y) (4) and ‖fa(xy) − fa(x)y − xfa(y)‖ ≤ β (x, y) (5) for all x, y ∈ x. then there exists a unique additive derivation mapping a : x → y satisfying the functional equation (1) and ‖fa(x) − a(x)‖ ≤ 1 2 ∞∑ k= 1−j 2 α(2kjx, 2kjx) 2kj (6) for all x ∈ x. the mapping a(x) is defined by a(x) = lim n→∞ fa(2 njx) 2nj (7) for all x ∈ x. proof. assume j = 1. replacing y by x in (4) and using oddness of f, we get ∥ ∥ ∥ ∥ fa(x) − fa(2x) 2 ∥ ∥ ∥ ∥ ≤ α (x, x) 2 (8) for all x ∈ x. now replacing x by 2x and dividing by 2 in (8), we get ∥ ∥ ∥ ∥ fa(2x) 2 − fa(2 2x) 22 ∥ ∥ ∥ ∥ ≤ α (2x, 2x) 22 (9) for all x ∈ x. from (8) and (9), we obtain ∥ ∥ ∥ ∥ fa(x) − fa(2 2x) 22 ∥ ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∥ fa(x) − fa(2x) 2 ∥ ∥ ∥ ∥ + ∥ ∥ ∥ ∥ fa(2x) 2 − fa(2 2x) 22 ∥ ∥ ∥ ∥ ≤ 1 2 [ α(x, x) + α(2x, 2x) 2 ] (10) 162 m. arunkumar cubo 15, 1 (2013) for all x ∈ x. in general for any positive integer n , we get ∥ ∥ ∥ ∥ fa(x) − fa(2 nx) 2n ∥ ∥ ∥ ∥ ≤ 1 2 n−1∑ k=0 α(2kx, 2kx) 2k (11) ≤ 1 2 ∞∑ k=0 α(2kx, 2kx) 2k for all x ∈ x. in order to prove the convergence of the sequence { fa(2 nx) 2n } , replace x by 2mx and dividing by 2m in (11), for any m, n > 0 , we deduce ∥ ∥ ∥ ∥ fa(2 mx) 2m − fa(2 n+mx) 2(n+m) ∥ ∥ ∥ ∥ = 1 2m ∥ ∥ ∥ ∥ fa(2 mx) − fa(2 n · 2mx) 2n ∥ ∥ ∥ ∥ ≤ 1 2 n−1∑ k=0 α(2k+mx, 2k+mx) 2k+m ≤ 1 2 ∞∑ k=0 α(2k+mx, 2k+mx) 2k+m → 0 as m → ∞ for all x ∈ x. hence the sequence { fa(2 nx) 2n } is cauchy sequence. since y is complete, there exists a mapping a : x → y such that a(x) = lim n→∞ fa(2 nx) 2n ∀ x ∈ x. letting n → ∞ in (11) we see that (6) holds for all x ∈ x. to prove that a satisfies (1), replacing (x, y) by (2nx, 2ny) and dividing by 2n in (4), we obtain 1 2n ∥ ∥ ∥ fa(2 nx + 2ny) + fa(2 nx − 2ny) − 2fa(2 nx) + fa(2 ny) + fa(−2 ny) ∥ ∥ ∥ ≤ 1 2n α(2nx, 2ny) for all x, y ∈ x. letting n → ∞ in the above inequality and using the definition of a(x), we see that a(x + y) + a(x − y) = 2a(x) + a(y) + a(−y). hence a satisfies (1) for all x, y ∈ x. it follows from (5) that ‖a(xy) − a(x)y − xa(y)‖ = 1 22n ‖fa(2 n(xy)) − fa(2 nx)(2ny) − (2nx)fa(2 ny)‖ ≤ 1 2n β (2nx, 2ny) → 0 as n → ∞ cubo 15, 1 (2013) generalized ulam hyers stability of derivations of a aq ... 163 for all x, y ∈ x. to prove that a is unique, let b(x) be another mapping satisfying (1) and (6), then ‖a(x) − b(x)‖ = 1 2n ‖a(2nx) − b(2nx)‖ ≤ 1 2n {‖a(2nx) − fa(2 nx)‖ + ‖fa(2 nx) − b(2nx)‖} ≤ ∞∑ k=0 α(2k+nx, 2k+nx) 2(k+n) → 0 as n → ∞ for all x ∈ x. hence a is unique. thus the mapping a : x → y is a unique additive derivation mapping satisfying (6). for j = −1, we can prove a similar stability result. this completes the proof of the theorem. the following corollary is an immediate consequence of theorem 2.1 concerning the stability of (1). corollary 2.2. let fa : x → y be a odd mapping and there exists real numbers λ and s such that ‖fa(x + y) + fa(x − y) − 2fa(x) − fa(y) − fa(−y)‖ ≤    λ, λ {||x||s + ||y||s} , s < 1 or s > 1; λ ||x||s||y||s, s < 1 2 or s > 1 2 ; λ { ||x||s||y||s + { ||x||2s + ||y||2s }} , s < 1 2 or s > 1 2 ; (12) ‖fa(xy) − fa(x)y − xfa(y)‖ ≤    λ, λ {||x||s + ||y||s} , s < 1 or s > 1; λ ||x||s||y||s, s < 1 2 or s > 1 2 ; λ { ||x||s||y||s + { ||x||2s + ||y||2s }} , s < 1 2 or s > 1 2 ; (13) for all x, y ∈ x. then there exists a unique additive derivation function a : x → y such that ‖fa(x) − a(x)‖ ≤    λ, 2λ||x||s |2 − 2s| , λ||x||2s |2 − 22s| 3λ||x||2s |2 − 22s| (14) for all x ∈ x. 164 m. arunkumar cubo 15, 1 (2013) 3 stability results: quadratic derivations in this section, the author establish the generalized ulam-hyers stability of quadratic derivations of the aq-functional equation (1). definition 3.1. quadratic derivation. a c−linear mapping q : x → x is called quadratic derivation on x if q satisfies q(xy) = q(x)y2 + x2q(y) (1) for all x, y ∈ x. theorem 3.1. let j = ±1. let fq : x → y be a even mapping for which there exist a function α, β : x2 → [0, ∞) with the condition ∞∑ n=0 α ( 2njx, 2njy ) 22nj converges in r and lim n→∞ α ( 2njx, 2njy ) 22nj = 0 (2) ∞∑ n=0 β ( 2njx, 2njy ) 24nj converges in r and lim n→∞ β ( 2njx, 2njy ) 24nj = 0 (3) such that the functional inequalities ‖fq(x + y) + fq(x − y) − 2fq(x) − fq(y) − fq(−y)‖ ≤ α (x, y) (4) and ∥ ∥fq(xy) − x 2fq(y) − fq(x)y 2 ∥ ∥ ≤ β (x, y) (5) for all x, y ∈ x. then there exists a unique quadratic derivation mapping q : x → y satisfying the functional equation (1) and ‖fq(x) − q(x)‖ ≤ 1 4 ∞∑ k= 1−j 2 α(2kjx, 2kjx) 22kj (6) for all x ∈ x. the mapping q(x) is defined by q(x) = lim n→∞ fq(2 njx) 22nj (7) for all x ∈ x. proof. it follows from (5) that ∥ ∥q(xy) − x2q(y) − q(x)y2 ∥ ∥ = 1 24n ∥ ∥fq(2 n(xy)) − (2nx)2fq(2 ny) − fq(2 nx)(2ny)2 ∥ ∥ ≤ 1 24n β (2nx, 2ny) → 0 as n → ∞ cubo 15, 1 (2013) generalized ulam hyers stability of derivations of a aq ... 165 for all x, y ∈ x. the rest of the proof is similar tracing to that of theorem 2.1. thus the mapping q : x → y is a unique quadratic derivation mapping satisfying (6). the following corollary is an immediate consequence of theorem 3.1 concerning the stability of (1). corollary 3.2. let fq : x → y be a even mapping and there exists real numbers λ and s such that ‖fq(x + y) + fq(x − y) − 2fq(x) − fq(y) − fq(−y)‖ ≤    λ, λ {||x||s + ||y||s} , s < 2 or s > 2; λ ||x||s||y||s, s < 1 or s > 1; λ { ||x||s||y||s + { ||x||2s + ||y||2s }} , s < 1 or s > 1; (8) ∥ ∥fq(xy) − x 2fq(y) − fq(x)y 2 ∥ ∥ ≤    λ, λ {||x||s + ||y||s} , s < 2 or s > 2; λ ||x||s||y||s, s < 1 or s > 1; λ { ||x||s||y||s + { ||x||2s + ||y||2s }} , s < 1 or s > 1; (9) for all x, y ∈ x. then there exists a unique quadratic deviation function q : x → y such that ‖fq(x) − q(x)‖ ≤    λ 3 , 2λ||x||s |4 − 2s| , λ||x||2s |4 − 22s| 3λ||x||2s |4 − 22s| (10) for all x ∈ x. 4 stability results: mixed derivations in this section, the author present the generalized ulam-hyers stability of mixed derivations of the aq-functional equation (1). theorem 4.1. let j = ±1. let f : x → y be a odd mapping for which there exist a function α, β : x2 → [0, ∞) with the conditions (2), (3), (2) and (3) such that the functional inequalities ‖f(x + y) + f(x − y) − 2f(x) − f(y) − f(−y)‖ ≤ α (x, y) (1) 166 m. arunkumar cubo 15, 1 (2013) (5) and (5) for all x, y ∈ x. then there exists a unique additive derivation mapping a : x → y and a unique quadratic derivation mapping q : x → y satisfying the functional equation (1) and ‖f(x) − a(x) − q(x)‖ ≤ 1 2   1 2 ∞∑ k= 1−j 2 ( α(2kjx, 2kjx) 2kj + α(−2kjx, −2kjx) 2kj ) + 1 4 ∞∑ k= 1−j 2 ( α(2kjx, 2kjx) 22kj + α(−2kjx, −2kjx) 22kj )   (2) for all x ∈ x. the mapping a(x) and q(x) are defined in (6) and (6) respectively for all x ∈ x. proof. let fo(x) = fa(x) − fa(−x) 2 for all x ∈ x. then fo(0) = 0 and fo(−x) = −fo(x) for all x ∈ x. hence ‖fo(x + y) + fo(x − y) − 2fo(x) − fo(y) − fo(−y)‖ ≤ α(x, y) 2 + α(−x, −y) 2 (3) by theorem 2.1, we have ‖fo(x) − a(x)‖ ≤ 1 4 ∞∑ k= 1−j 2 ( α(2kjx, 2kjx) 2kj + α(−2kjx, −2kjx) 2kj ) (4) for all x ∈ x. also, let fe(x) = fq(x) + fq(−x) 2 for all x ∈ x. then fe(0) = 0 and fe(−x) = fe(x) for all x ∈ x. hence ‖fe(x + y) + fe(x − y) − 2fe(x) − fe(y) − fe(−y)‖ ≤ α(x, y) 2 + α(−x, −y) 2 (5) by theorem 3.1, we have ‖fe(x) − q(x)‖ ≤ 1 8 ∞∑ k= 1−j 2 ( α(2kjx, 2kjx) 22kj + α(−2kjx, −2kjx) 22kj ) (6) for all x ∈ x. define f(x) = fe(x) + fo(x) (7) for all x ∈ x. from (4),(6) and (7), we arrive ‖f(x) − a(x) − q(x)‖ = ‖fe(x) + fo(x) − a(x) − q(x)‖ ≤ ‖fo(x) − a(x)‖ + ‖fe(x) − q(x)‖ ≤ 1 4 ∞∑ k= 1−j 2 ( α(2kjx, 2kjx) 2kj + α(−2kjx, −2kjx) 2kj ) + 1 8 ∞∑ k= 1−j 2 ( α(2kjx, 2kjx) 22kj + α(−2kjx, −2kjx) 22kj ) for all x ∈ x. hence the theorem is proved. cubo 15, 1 (2013) generalized ulam hyers stability of derivations of a aq ... 167 using corollaries 2.2 and 3.2 we have the following corollary concerning the stability of (1). corollary 4.1. let f : x → y be a mapping and there exits real numbers λ and s such that ‖f(x + y) + f(x − y) − 2f(x) − f(y) − f(−y)‖ ≤    λ, λ {||x||s + ||y||s} , s < 1 or s > 1; λ ||x||s||y||s, s < 1 2 or s > 1 2 ; λ { ||x||s||y||s + { ||x||2s + ||y||2s }} , s < 1 2 or s > 1 2 ; (8) and (13), (9) for all x, y ∈ x. then there exists a unique additive deviation function a : x → y and a unique quadratic deviation function q : x → y such that ‖f(x) − a(x) − q(x)‖ ≤    λ ( 1 + 1 3 ) , 2λ ( 1 |2 − 2s| + 1 |4 − 2s| ) ||x||s, λ ( 1 |2 − 22s| + 1 |4 − 22s| ) ||x||2s, 3λ ( 1 |2 − 22s| + 1 |4 − 22s| ) ||x||2s (9) for all x ∈ x. received: december 2012. revised: february 2013. references [1] j. aczel and j. dhombres, functional equations in several variables, cambridge univ, press, 1989. [2] t. aoki, on the stability of the linear transformation in banach spaces, j. math. soc. japan, 2 (1950), 64-66. [3] m. arunkumar, s. jayanthi, s. hema latha, stability of quadratic derivations of arunquadratic functional equation, international journal mathematical sciences and engineering applications, vol. 5 no. v, sept. 2011, 433-443. [4] m. arunkumar, s. karthikeyan, solution and stability of n−dimensional mixed type additive and quadratic functional equation, far east journal of applied mathematics, volume 54, number 1, 2011, 47-64. [5] m. arunkumar, john m. rassias, on the generalized ulam-hyers stability of an aqmixed type functional equation with counter examples, far east journal of applied mathematics volume 71, no. 2, (2012), 279-305. 168 m. arunkumar cubo 15, 1 (2013) [6] c. baak, d. boo, th.m. rassias, generalized additive mapping in banach modules and isomorphism between c∗−algebras, j. math. anal. appl. 314, (2006), 150-161. [7] r. badora, on approximate derivations, math. inequal. appl. 9 (2006), no. 1, 167-173. [8] l. brown and g. pedersen, c∗−algebras of real rank zero, j. funct. analysis. 99, (1991), 138-149. [9] s. czerwik, functional equations and inequalities in several variables, world scientific, river edge, nj, 2002. [10] p. gavruta, a generalization of the hyers-ulam-rassias stability of approximately additive mappings , j. math. anal. appl., 184 (1994), 431-436. [11] m.e.gordji, j.m. rassias, n. ghobadipour, generalized hyers-ulam stability of the generalized (n, k)−derivations, abs. appl. anal. volume 2009, article id 437931, [12] m.e.gordji, s. kaboli gharetapeh, m. b. savadkouhi, m. aghaei, t. karimi, on cubic derivations, int. journal of math. analysis, vol. 4, 2010, no. 51, 2501 2514 [13] d.h. hyers, on the stability of the linear functional equation, proc.nat. acad.sci.,u.s.a.,27 (1941) 222-224. 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[27] c. park, linear derivations on banach algebras, nonlinear funct. anal. appl. 9 (2004), no. 3, 359-368. [28] c. park, fixed points and hyers-ulam-rassias stability of cauchy-jensen functional equation in banach algebras, fixed point theory and applications, 2007, art id 50175. [29] c. park and j. hou, homomorphism and derivations in c∗−algebras, abstract appl. anal. 2007, art. id 80630. [30] th.m. rassias, on the stability of the linear mapping in banach spaces, proc.amer.math. soc., 72 (1978), 297-300. [31] th.m. rassias, functional equations, inequalities and applications, kluwer acedamic publishers, dordrecht, bostan london, 2003. [32] k. ravi, m. arunkumar and j.m. rassias, on the ulam stability for the orthogonally general euler-lagrange type functional equation, international journal of mathematical sciences, autumn 2008 vol.3, no. 08, 36-47. [33] s.m. ulam, problems in modern mathematics, science editions, wiley, newyork, 1964. cubo a mathematical journal vol.16, no¯ 01, (37–48). march 2014 existence of ψ-bounded solutions for linear matrix difference equations on z+ g.suresh kumar, ch.vasavi, t.s.rao koneru lakshmaiah university, department of mathematics, vaddeswaram, guntur dt., a.p., india. drgsk006@kluniversity.in and m.s.n.murty acharya nagarjuna university, department of mathematics, nagarjuna nagar-522510, guntur dt., a.p., india. drmsn2002@gmail.com abstract this paper deals with obtaining necessary and sufficient conditions for the existence of at least one ψ-bounded solution for the linear matrix difference equation x(n + 1) = a(n)x(n)b(n) + f(n), where f(n) is a ψ-summable matrix valued function on z+. finally, we prove a result relating to the asymptotic behavior of the ψ-bounded solutions of this equation on z+. resumen este art́ıculo se enfoca en obtener condiciones necesarias y suficientes para la existencia de la menos una solución ψ-acotada para la ecuación lineal en diferencias matricial x(n + 1) = a(n)x(n)b(n) + f(n), donde f(n) es una función ψ-sumable con valores matriciales en z+. finalmente, probamos un resultado relacionado al comportamiento asintótico de las soluciones ψ-acotadas de esta ecuación en z+. keywords and phrases: difference equations; fundamental matrix; ψ-bounded; ψ-summable, kronecker product. 2010 ams mathematics subject classification: 39a10, 39b42. 38 g.suresh kumar, ch.vasavi, t.s.rao & m.s.n.murty cubo 16, 1 (2014) 1 introduction difference equations serve as a natural description of observed evolution phenomena. the theory of difference equations is of immense use in the construction of dicrete mathematical models, which can explain better when compared to continuous models. one of the important fetures of difference equations is, they appear in the study of discretization methods for differential equations. difference equations play an important role in many scientific fields such as numerical analysis, finite element techniques, control theory, discrete mathematical structures and several problems of mathematical modelling [1, 2, 14]. due to the importance and rapid growth of research in this area, we confine our attention to the linear matrix difference equation x(n + 1) = a(n)x(n)b(n) + f(n), (1) where a(n), b(n),and f(n) are m × m matrix-valued functions on z+ = {1, 2, . . .}. the basic problem under consideration is the determination of necessary and sufficient conditions for the existence of a solution with some specified boundedness condition. a classical result of this type, for system of differential equations is given by coppel [6, theorem 2, chapter v]. the problem of ψ-boundedness of the solutions for systems of ordinary differential equations has been studied in many papers, [3, 4, 5, 7, 13, 16]. recently [10, 18, 19], extended the concept of ψ-boundedness of the solutions to lyapunov matrix differential equations. recently , han and hong [15], diamandescu [8, 11] extended the concept of ψ-bounded solutions of system of differential equations to difference equations. the existence and uniqueness of solutions of matrix difference equation (1) was studied by murty, anand and lakshmi prasannam [17]. the aim of present paper is to give a necessary and sufficient condition for the existence of ψ-bounded solution of the linear matrix difference equation (1) via ψ-summable sequence. the introduction of the matrix function ψ permits to obtain a mixed asymptotic behavior of the components of the solutions. here, ψ is a matrix-valued function. this paper include the results of han and hong [15] as a particular case when b = i, x and f are column vectors. 2 preliminaries in this section we present some basic definitions, notations and results which are useful for later discussion. let rm be the euclidean m-space. for u = (u1, u2, u3, . . . , um) t ∈ rm, let ‖u‖ = max{|u1|, |u2|, |u3|, . . . , |um|} be the norm of u. let r m×m be the linear space of all m × m real valued matrices. for a m × m real matrix a = [aij], we define the norm |a| = sup‖u‖≤1 ‖au‖. it is well-known that |a| = max 1≤i≤m { m∑ j=1 |aij|}. cubo 16, 1 (2014) existence of ψ-bounded solutions for linear matrix difference. . . 39 let ψk : z + → r − {0} (r − {0} is the set of all nonzero real numbers), k = 1, 2, . . . m, and let ψ = diag[ψ1, ψ2, . . . , ψm]. then the matrix ψ(n) is an invertible square matrix of order m, for all n ∈ z+. definition 2.1. [12] let a ∈ rp×q and b ∈ rr×s then the kronecker product of a and b written a ⊗ b is defined to be the partitioned matrix a ⊗ b =       a11b a12b . . . a1qb a21b a22b . . . a2qb . . . . . . ap1b ap2b . . . apqb       is an pr × qs matrix and is in rpr×qs. definition 2.2. [12] let a = [aij] ∈ r p×q, then the vectorization operator vec : rp×q → rpq, defined and denote by â = veca =          a.1 a.2 . . a.q          , where a.j =          a1j a2j . . apj          (1 ≤ j ≤ q) . lemma 2.3. the vectorization operator vec : rm×m → rm 2 , is a linear and one-to-one operator. in addition, vec and vec−1 are continuous operators. proof. the fact that the vectorization operator is linear and one-to-one is immediate. now, fora = [aij] ∈ r m×m, we have ‖vec(a)‖ = max 1≤i,j≤m {|aij|} ≤ max 1≤i≤m    m∑ j=1 |aij|    = |a| . thus, the vectorization operator is continuous and ‖vec‖ ≤ 1. in addition, for a = im (the identity m × m matrix) we have ‖vec(im)‖ = 1 = |im| and then, ‖vec‖ = 1. obviously, the inverse of the vectorization operator, vec−1 : rm 2 → rm×m, is defined by vec−1(u) =            u1 um+1 . . . um2−m+1 u2 um+2 . . . um2−m+2 . . . . . . . . . . . . . . . . . . um u2m . . . um2            . 40 g.suresh kumar, ch.vasavi, t.s.rao & m.s.n.murty cubo 16, 1 (2014) where u = (u1, u2, u3, ....., um2) t ∈ rm 2 . we have ∣ ∣vec−1(u) ∣ ∣ = max 1≤i≤m    m−1∑ j=0 |umj+i|    ≤ m. max 1≤i≤m {|ui|} = m. ‖u‖. thus, vec−1 is a continuous operator. also, if we take u = veca in the above inequality, then the following inequality holds |a| ≤ m‖veca‖, for every a ∈ rm×m. regarding properties and rules for kronecker product of matrices we refer to [12]. now by applying the vec operator to the linear nonhomogeneous matrix difference equation (1) and using kronecker product properties, we have x̂(n + 1) = g(n)x̂(n) + f̂(n), (2) where g(n) = bt (n) ⊗ a(n) is a m2 × m2 matrix and f̂(n) = vecf(n) is a column matrix of order m2. the equation (2) is called the kronecker product difference equation associated with (1). it is clear that, if x(n) is a solution of (1) if and only if x̂(n) = vecx(n) is a solution of (2). the corresponding homogeneous difference equation of (2) is x̂(n + 1) = g(n)x̂(n). (3) definition 2.4. [15] a sequence φ : z+ → rm is said to be ψbounded on z+ if ψ(n)φ(n) is bounded on z+ (i.e., there exists l > 0 such that ‖ψ(n)φ(n)‖ ≤ l, for all n ∈ z+). extend this definition to matrix functions. definition 2.5. a matrix sequence f : z+ → rm×m is said to be ψ-bounded on z+ if ψf is bounded on z+ (i.e., there exists l > 0 such that |ψ(n)f(n)| ≤ l, for all n ∈ z+). definition 2.6. [15] a sequence φ : z+ → rm is said to be ψ-summable on z+ if φ(n) ∈ l1 and ψ(n)φ(n) ∈ l1. ( i.e., lim p→∞ p∑ n=1 ‖ψ(n)φ(n)‖ < ∞ ) . extend this definition to matrix functions. definition 2.7. a matrix sequence f : z+ → rm×m is called ψ-summable on z+ if ∞∑ n=1 ψ(n)f(n) is convergent ( i.e., lim p→∞ p∑ n=1 |ψ(n)f(n)| < ∞ ) . now we shall assume that a(n) and b(n) are bounded m × m matrices on z+ and f(n) is a ψ-summable matrix function on z+. cubo 16, 1 (2014) existence of ψ-bounded solutions for linear matrix difference. . . 41 by a solution of (1), we mean a matrix function w(n) satisfying the equation (1) for all most all n ∈ z+. the following lemmas play a vital role in the proof of main result. lemma 2.8. the matrix function f : z+ → rm×m is ψ-summable on z+ if and only if the vector function vecf(n) is im ⊗ ψ-summable on z +. proof. from the proof of lemma 2.3, it follows that 1 m |a| ≤ ‖veca‖ rm 2 ≤ |a| , for every a ∈ rm×m. put a = ψ(n)f(n) in the above inequality, we have 1 m |ψ(n)f(n)| ≤ ‖(im ⊗ ψ(n)).vecf(n)‖ rm 2 ≤ |ψ(n)f(n)| , (4) n ∈ z+, for all matrix functions f(n). suppose that f(n) is ψ-summable on z+. from (4), we get ‖(im ⊗ ψ(n)).vecf(n)‖rm2 ≤ |ψ(n)f(n)| , which implies ∞∑ n=1 ‖(im ⊗ ψ(n)).vecf(n)‖rm2 ≤ ∞∑ n=1 |ψ(n)f(n)| . from comparison test, definitions 2.6 and 2.7, f̂(n) is im ⊗ ψ-summable on z +. conversely suppose that f̂(n) is im ⊗ ψ-summable on z +. again from (4), we get |ψ(n)f(n)| ≤ m ‖(im ⊗ ψ(n)).vecf(n)‖rm2 , which implies ∞∑ n=1 |ψ(n)f(n)| ≤ m ∞∑ n=1 ‖(im ⊗ ψ(n)).vecf(n)‖ rm 2 . from comparison test, definitions 2.6 and 2.7, f(n) is ψ-summable on z+. now the proof is complete. lemma 2.9. the matrix function f(n) is ψ bounded on z+ if and only if the vector function vecf(n) is im ⊗ ψ bounded on z +. proof. the proof easily follows from the inequality (4). 42 g.suresh kumar, ch.vasavi, t.s.rao & m.s.n.murty cubo 16, 1 (2014) lemma 2.10. if a(n), b(n) are invertible matrix functions and f(n) is a matrix function on z+. let y(n) and z(n) be the fundamental matrices for the matrix difference equations x(n + 1) = a(n)x(n), n ∈ z+ (5) and x(n + 1) = bt (n)x(n), n ∈ z+ (6) respectively. then the matrix z(n) ⊗ y(n) is a fundamental matrix of (3). proof. consider z(n + 1) ⊗ y(n + 1) = bt (n)z(n) ⊗ a(n)y(n) = (bt (n) ⊗ a(n))(z(n) ⊗ y(n)) = g(n)(z(n) ⊗ y(n)), for all n ∈ z+. on the other hand, the matrix z(n) ⊗ y(n) is an invertible matrix for all n ∈ z+ (because z(n) and y(n) are invertible matrices for all n ∈ z+). let x1 denote the subspace of r m×m consisting of all matrices which are values of ψ-bounded solution of x(n + 1) = a(n)x(n)b(n) on z+ at n = 1 and let x2 an arbitrary fixed subspace of r m×m, supplementary to x1. let p1, p2 denote the corresponding projections of r m×m onto x1, x2 respectively. then x1 denote the subspace of r m 2 consisting of all vectors which are values of im ⊗ ψbounded solution of (3) on z+ at n = 1 and x2 a fixed subspace of r m 2 , supplementary to x1. let q1, q2 denote the corresponding projections of r m 2 onto x1, x2 respectively. theorem 2.11. let y(n) and z(n) be the fundamental matrices for the systems (5) and (6). if x̂(n) = n−1∑ k=1 (z(n) ⊗ y(n))q1(z −1 (k + 1) ⊗ y−1(k + 1))f̂(k) − ∞∑ k=1 (z(n) ⊗ y(n))q2(z −1(k + 1) ⊗ y−1(k + 1))f̂(k) (7) is convergent, then it is a solution of (2) on z+. proof. it is easily seen that x̂(n) is the solution of (2) on z+. the following theorems are useful in the proofs of our main results. cubo 16, 1 (2014) existence of ψ-bounded solutions for linear matrix difference. . . 43 theorem 2.12. [15] let {a(n)} be bounded. then x(n + 1) = a(n)x(n) + f(n) (8) has at least one ψ-bounded solution on z+ for every ψ-summable sequence {f(n)} on z+ if and only if there is a positive constant k such that |ψ(n)y(n)p1y −1 (k + 1)ψ−1(k)| ≤ k, 1 ≤ k + 1 ≤ n |ψ(n)y(n)p2y −1(k + 1)ψ−1(k)| ≤ k, 1 ≤ n < k + 1. (9) theorem 2.13. [15] suppose that: (1) the fundamental matrix y(n) of x(n + 1) = a(n)x(n) satisfies conditions (a) lim n→∞ |ψ(n)y(n)p1| = 0, (b) condition (9) holds, where k is a positive constant, p1 and p2 are suplementary projections. (2) the sequence f : z+ → rm is ψ-summable on z+. then, every ψ-bounded solution x(n) of (8) satisfies lim n→∞ ‖ψ(n)x(n)‖ = 0. 3 main result the main theorems of this paper are proved in this section. theorem 3.1. let a(n) and b(n) be bounded matrices on z+, then (1) has at least one ψ-bounded solution on z+ for every ψ-summable matrix function f : z+ → rm×m on z+ if and only if there exists a positive constant k such that |(z(n) ⊗ ψ(n)y(n))q1(z −1(k + 1) ⊗ y−1(k + 1)ψ−1(k))| ≤ k, 1 ≤ k + 1 ≤ n, |(z(n) ⊗ ψ(n)y(n))q2(z −1(k + 1) ⊗ y−1(k + 1)ψ−1(k))| ≤ k, 1 ≤ n < k + 1. (10) proof. suppose that the equation (1) has at least one ψ-bounded solution on z+ for every ψsummable matrix function f : z+ → rm×m. let f̂ : z+ → rm 2 be im ⊗ ψ-summable function on z +. from lemma 2.8, it follows that the matrix function f(n) = vec−1f̂(n) is ψ summable matrix function on z+. from the hypothesis, the system (1) has at least one ψ bounded solution x(n) on z+. from lemma 2.9, it follows that the vector valued function x̂(n) = vecx(n) is a im ⊗ ψ-bounded solution of (2) on z +. thus, equation (2) has at least one im⊗ψ-bounded solution on z + for every im⊗ψ-summable function f̂ on z+. 44 g.suresh kumar, ch.vasavi, t.s.rao & m.s.n.murty cubo 16, 1 (2014) from theorem 2.12, there is a positive constant k such that the fundamental matrix t(n) = z(n) ⊗ y(n) of the system (3) satisfies the condition |(im ⊗ ψ(n))t(n)q1t −1(k + 1)(im ⊗ ψ −1(k))| ≤ k, 1 ≤ k + 1 ≤ n, |(im ⊗ ψ(n))t(n)q2t −1(k + 1)(im ⊗ ψ −1(k))| ≤ k, 1 ≤ n < k + 1. putting t(n) = z(n) ⊗ y(n) and using kronecker product properties, (10) holds. conversely suppose that (10) holds for some k > 0. let f : z+ → rm×m be a ψ-summable matrix function on z+. from lemma 2.8, it follows that the vector valued function f̂(n) = vecf(n) is a im ⊗ ψ-summable function on z +. since a(n), b(n) are bounded, then g(n) = bt (n) ⊗ a(n) is also bounded. now from theorem 2.12, the difference equation (2) has at least one im ⊗ ψ bounded solution on z +. let x(n) be this solution. from lemma 2.9, it follows that the matrix function x(n) = vec−1x(n) is a ψ-bounded solution of the equation (1) on z+ (because f(n) = vec−1f̂(n)). thus, the matrix difference equation (1) has at least one ψ-bounded solution on z+ for every ψ-summable matrix function f on z+. theorem 3.2. suppose that: (1) the fundamental matrices y(n) and z(n) of (5) and (6) satisfies: (a) lim n→∞ |(z(n) ⊗ ψ(n)y(n))q1| = 0; (b) condition (10) holds, for some k > 0. (2) the matrix function f : z+ → rm×m is ψ-summable on z+. then, every ψ-bounded solution x of (1) is such that lim n→∞ |ψ(n)x(n)| = 0. proof. let x(n) be a ψ-bounded solution of (1). from lemma 2.9, it follows that the function x̂(n) = vecx(n) is a im ⊗ ψbounded solution on z + of the difference equation (2). also from lemma 2.8, the function f̂(n) is im ⊗ ψ-summable on z +. from the theorem 2.13, it follows that lim n→∞ ∥ ∥(im ⊗ ψ(n)) x̂(n) ∥ ∥ = 0. now, from the inequality (4) we have |ψ(n)x(n)| ≤ m ∥ ∥(im ⊗ ψ(n)) x̂(n) ∥ ∥ , n ∈ z+ and, then lim n→∞ |ψ(n)x(n)| = 0. cubo 16, 1 (2014) existence of ψ-bounded solutions for linear matrix difference. . . 45 the following example illustrates the above theorems. example 3.1. consider the matrix difference equation (1) with a(n) = [ n+1 n 0 0 2 ] , b(n) = [ ( n+2 n+3 ) 1 4 0 0 n+3 n+1 ] and f(n) = [ n 3n 0 0 n 2 2 n−1 3n ] . then, y(n) = [ n 0 0 2n−1 ] and z(n) = [ ( 3 n+2 ) 1 4 0 0 (n+1)(n+2) 6 ] are the fundamental matrices for (5) and (6) respectively. consider ψ(n) = [ 1 n 0 0 21−n ] , for all n ∈ z+. there exist projections q1 = [ i2 o2 o2 o2 ] and q2 = [ o2 o2 o2 i2 ] such that conditions in (10) are satisfied with k = 2. in addition, the hypothesis (1a) and (2) of theorem 3.2 are satisfied. because |(z(n) ⊗ ψ(n)y(n))q1| = ( 3 n + 2 ) 1 4 , ψ(n)f(n) = [ n 2 3n 0 0 n 2 3n ] , and ∞∑ n=1 |ψ(n)f(n)| = ∞∑ n=1 n2 3n < ∞ (ratio test), the matrix function f is ψ-summable on z+. from theorems 3.1 and 3.2, the difference equation has at least one ψ-bounded solution and every ψ-bounded solution x of (1) is such that lim n→∞ |ψ(n)x(n)| = 0. remark 3.1. theorem 3.2 is no longer true if we require that the matrix function f be ψ-bounded on z+, instead of the condition (2) in the above theorem. this is shown in the following example. 46 g.suresh kumar, ch.vasavi, t.s.rao & m.s.n.murty cubo 16, 1 (2014) example 3.2. consider the matrix difference equation (1) with a(n) = [ 1 2 0 0 2 ] , b(n) = [ 1 0 0 3 ] and f(n) = [ 1 0 0 3n ] . then, y(n) = [ 21−n 0 0 2n−1 ] and z(n) = [ 1 0 0 3n−1 ] are the fundamental matrices for (5) and (6) respectively. consider ψ(n) = [ 1 0 0 31−n ] , for all n ∈ z+. there exist projections q1 = [ i2 o2 o2 o2 ] and q2 = [ o2 o2 o2 i2 ] , such that conditions in hypothesis (1) are satisfied with k = 2. also |ψ(n)f(n)| = 3, for n ∈ z+. therefore, f is ψ-bounded on z+. clearly, the function f is not ψ-summable on z+. the solutions of the equation (1) are x(n) = [ 21−n(c1 − 2) + 2 ( 3 2 )n−1c2 2n−1c3 6 n−1(c4 + 1) − 3 n−1 ] , where c1, c2, c3 and c4 are arbitrary constants. it is easily seen that, there is no solution x(n) of (1) for lim n→∞ |ψ(n)x(n)| = 0. received: october 2012. accepted: may 2013. references [1] agarwal, r.p. : difference equations and inequalities, second edition, marcel dekker, new york, 2000. [2] agarwal, r.p. and wong, p.j.y. : advanced topics in difference equations, kluwer, dordrecht, 1997. [3] akinyele, o. : on partial stability and boundedness of degree k, atti. accad. naz. lincei rend. cl. sci. fis. mat. natur., (8), 65(1978), 259-264. cubo 16, 1 (2014) existence of ψ-bounded solutions for linear matrix difference. . . 47 [4] avramescu, c. : asupra comportării asimptotice a soluţiilor unor ecuaţii funcţionale, analele universităţii din timiş oara, seria ştiinţe matematice-fizice, vol. vi (1968) 41-55. [5] constantin, a. : asymptotic properties of solutions of differential equations, analele universităţii din timişoara, seria ştiin ţe matematice, vol. xxx, fasc. 2-3 (1992) 183-225. [6] coppel, w.a. : stability and asymptotic behavior of differential equations, heath, boston,1963. [7] diamandescu, a. : existence of ψ bounded solutions for a system of differential equation, electronic journal of differential equations, 63 (2004), 1-6. 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[17] murty, k.n., anand, p.v.s. and lakshmi prasannam, v. : first order difference systems existence and uniqueness, proceedings of the american mathematical society, vol.125, no.12 (1997), 3533-3539. 48 g.suresh kumar, ch.vasavi, t.s.rao & m.s.n.murty cubo 16, 1 (2014) [18] murty, m.s.n. and suresh kumar, g. : on ψ-boundedness and ψ-stability of matrix lyapunov systems, journal of applied mathematics and computing, vol.26, (2008) 67-84. [19] murty, m.s.n. and suresh kumar, g. : on ψ-bounded solutions for non-homogeneous matrix lyapunov systems on r, electronic journal of qualitative theory of differential equations, vol.2009(2009), no.62, 1-12. () cubo a mathematical journal vol.16, no¯ 03, (55–65). october 2014 s-asymptotically ω-periodic solution for a nonlinear differential equation with piecewise constant argument in a banach space william dimbour & jean-claude mado laboratoire c.e.r.e.g.m.i.a. université des antilles et de la guyane, campus fouillole 97159 pointe-à-pitre guadeloupe (fwi) william.dimbour@univ-ag.fr, jean-claude.mado@univ-ag.fr abstract in this paper, we give some sufficient conditions for the existence and uniqueness of s-asymptotically ω-periodic (mild) solutions for a differential equation with piecewise constant argument, when ω is an integer. an example is also given in order to illustrate the result. resumen en este art́ıculo entregamos algunas condiciones suficientes para la existencia y unicidad de las soluciones mild ω-periódicas s-asintóticas para una ecuación diferencial semilineal con argumento constante por tramos en un espacio de banach cuando ω es un entero. luego, entregamos ejemplos para ilustrar nuestros resultados. keywords and phrases: s-asymptotically ω-periodic function, differential equations with piecewise constant argument, semigroup 2010 ams mathematics subject classification: 34k05; 34a12; 34a40. 56 william dimbour & jean-claude mado cubo 16, 3 (2014) 1 introduction let (x, ||.||) be a banach space. this work is concerned with the existence of s-asymptotically ω-periodic solutions to the differential equations with piecewise constant argument of the form (2) { x′(t) = ax(t) + a0x([t]) + g(t,x(t)) x(0) = c0 where a is the infitesimal generator of an exponentially stable c0-semigroup acting on x, [·] is the largest interger function and g : r+ × x → x is an appropriate function that will be defined later. they are a some papers dealing wiht s-asymptotically ω-periodic functions. qualitative properties of such functions are discussed for instance in [1] and [5]. in [5], a new composition theorem for such functions is also presented. in [7], lizama and n’guérékata created a chart establishing a general relationship between s-asymptotically ω-periodic functions and various subspaces of bc(r,x). [1], [3], [4],[5], [6],[7] study the existence of s-asymptotically ω-periodic solutions of diffenretial equations in finite as well infinite dimensional spaces. they are also some papers dealing with the existence of almost automorphic solutions for differential equation with piecewise constant argument. in [8], nguyen van minh and tran tat dat give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations wtih piecewise constant argument of the form x′(t) = ax(t) + f(t),t ∈ r, where a is a bounded linear operator in x and f in an x-valued almost automorphic function. in [2], dimbour generalize the work of nguyen van minh and tran tat dat, giving also sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations wtih piecewise constant argument of the form x′(t) = a(t)x(t) + f(t),t ∈ r, where a(t) is an almost automorphy operator and f in an x-valued almost automorphic function. following this work, we study in this paper s-asymptotically ω-periodic solutions of (2). we first study the linear system associated to (2). then using the banach’s theorem, we showed the existence of s-asymptotically ω-periodic solutions for the following equation (1) { x′(t) = ax(t) + a0x([t]) + f(t) x(0) = c0 the rest of the paper is organise as follows. in section 2, we recall some results on s-asymptotically ω-periodic functions. in section 3, first of all considering the c0 semigroup theory ([9]), we define a mild solution of (1). we give some sufficient conditions for the existence and uniqueness of s-asymptotically ω-periodic solutions of (1) and (2). these results are obtained by mean of the banach fixed point principle, when ω is an integer. in the section 4, we give an example. cubo 16, 3 (2014) s-asymptotically ω-periodic solution for a nonlinear differential . . . 57 2 preliminaries let x be a banach space. bc(r+,x) denotes the space of the continuous bounded functions from r + into x; endowed with the norm ||f||∞ := supt≥0 ||f(t)||, it is a banach space. c0(r +,x) denotes the space of the continuous functions from r into x such that lim t→∞ f(t) = 0; it is a banach subspace of bc(r+,x). when we fix a positive number ω, pω(x) denotes the space of all continuous ω-periodic functions from r+ into x; it is a banach subspace of bc(r+,x) under the sup norm. definition 1. let f ∈ bc(r+,x) and ω > 0. we say that f is asymptotically ω-periodic if f = g + h where g ∈ pω(x) and h ∈ c0(r +,x). we denote by ap(x) the set of all asymptotically ω-periodic functions from r+ to x. it is a banach space under the sup norm. definition 2. a function f ∈ bc(r+,x) is called s-asymptotically ω-periodic if there exists ω such that lim t→∞ (f(t + ω) − f(t)) = 0. in this case we say that ω is an asymptotic period of f and that f is s-asymptotically ω-periodic. we will denote by sapω(x), the set of all s-asymptotically ω-periodic functions from r + to x. then we have apω(x) ⊂ sapω(x). the inclusion is strict. indeed consider the function f : r+ → c0 where c0 = {x = (xn)n∈n : lim n→∞ xn = 0} equipped with the norm ||x|| = supn∈n |x(n)|, and (f(t) = 2nt 2 t2+n2 )n∈n. then f ∈ sapω(x) but f /∈ apω(x)(see [5] example 3.1). the following result is due to henriquez-pierri-tàboas; proposition 3.5 in [5]. theorem 1. endowed with the norm || · ||∞, sapω(x) is a banach space. corollary 1. (see [1], corollary 3.10 p.5) let x and y be two banach spaces, and let a ∈ l(x,y). then when f ∈ sapω(x), we have af := [t → af(t)] ∈ sapω(y). for the sequel we consider asymptotically ω-periodic functions with parameters. definition 3. (see [5]) a continuous function g : [0,∞[×x → x is said to be uniformly sasymptotically ω-periodic on bounded sets if for every bounded set k ⊂ x, the set {f(t,x) : t ≥ 0,x ∈ k} is bounded and limt→∞(f(t,x) − f(t + ω,x)) = 0 uniformly on x ∈ k. definition 4. (see [5]) a continuous function g : [0,∞[×x → x is said to be asymptotically uniformly continuous on bounded sets if for every ǫ > 0 and every bounded set k ⊂ x, there exist lǫ,k > 0 and δǫ,k > 0 such that ||f(t,x) − f(t,y)|| < ǫ for all t ≥ lǫ,k and all x,y ∈ k with ||x − y|| < δǫ,k. 58 william dimbour & jean-claude mado cubo 16, 3 (2014) theorem 2. (see [5]) let g : [0,∞[×x → x be a function which uniformly s-asymptotically ω-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. let u : [0,∞[→ x be s-asymptotically ω-periodic function. then the nemytskii function φ(.) := f(.,u(.)) is s-asymptotically ω-periodic function. 3 main result 3.1 the linear case definition 5. a solution of eq.(1) on r+ is a function x(t) that satisfies the conditions: 1-x(t) is continuous on r+. 2-the derivative x′(t) exists at each point t ∈ r+, with possible exception of the points [t] ∈ r+ where one-sided derivatives exists. 3-eq.(1) is satisfied on each interval [n,n + 1[ with n ∈ n. let t(t) be the c0 semigroup generated by a and x a solution of (1). we assume that f ∈ l1(r+,x). then the function g defined by g(s) = t(t − s)x(s) is differentiable for s < t. dg(s) ds = −at(t − s)x(s) + t(t − s)x′(s) = −at(t−s)x(s)+t(t−s)ax(s)+ t(t−s)a0x([s])+t(t−s)f(s) (3) = t(t − s)a0x([s]) + t(t − s)f(s) since f ∈ l1(r,x), t(t − s)f(s) is integrable on [0,t] with t ∈ r+. the function x([s]) is a step function. therefore x([s]) is integrable on [0,t] with t ∈ r+. integrating (3) on [0,t], we obtain that x(t) − t(t)x(0) = ∫t 0 t(t − s)a0x([s])ds + ∫t 0 t(t − s)f(s)ds. therefore, we define definition 6. let t(t) be the c0 semigroup generated by a and f ∈ l 1(r+,x). the function x ∈ c(r+,x) given by x(t) = t(t)c0 + ∫t 0 t(t − s)a0x([s])ds + ∫t 0 t(t − s)f(s)ds is the mild solution of the equation (1). now we make the following hypothesis. cubo 16, 3 (2014) s-asymptotically ω-periodic solution for a nonlinear differential . . . 59 (h.1) the operator a is the infinitesimal generator of an exponentially stable semigroup (t(t))t≥0 such that there exist constants m > 0 and δ > 0 with ||t(t)||b(x) ≤ me −δt, ∀t ≥ 0. lemma 1. we assume that the hypothesis (h.1) is satisfied. then the function l defined by l(t) = t(t)x(0) belongs to sapω(x). proof. ||l(t + ω) − l(t)|| = ||t(t + ω)x(0) − t(t)x(0)|| ≤ ||t(t + ω)x(0)|| + ||t(t)x(0)|| ≤ me−δ(t+ω) + me−δt since δ > 0, we deduce that lim t→∞ ||l(t + ω) − l(t)|| = 0. then l ∈ sapω(x).✷ lemma 2. we assume that the hypothesis (h.1) is satisfied. we assume also that a0 is a linear bounded operator and ω ∈ n. we define the nonlinear operator ∧1 by: for each φ ∈ sapω(x) (∧1φ)(t) = ∫t 0 t(t − s)a0φ([s])ds. then the operator ∧1 maps sapω(x) into itself. proof. we put v(t) = ∫t 0 t(t − s)a0φ([s])ds. for t ≥ 0, we have v(t + ω) − v(t) = ∫t+ω 0 t(t + ω − s)a0φ([s])ds − ∫t 0 t(t − s)a0φ([s])ds = ∫ω 0 t(t + ω − s)a0φ([s])ds + ∫t+ω ω t(t + ω − s)a0φ([s])ds − ∫t 0 t(t − s)a0φ(s)ds. then we have ||v(t + ω) − v(t)|| ≤ ||i1(t)|| + ||i2(t)|| where i1(t) = ∫ω 0 t(t + ω − s)a0φ([s])ds 60 william dimbour & jean-claude mado cubo 16, 3 (2014) and i2(t) = ∫t+ω ω t(t + ω − s)a0φ([s])ds − ∫t 0 t(t − s)a0φ([s])ds. observing that i1(t) = t(t) ∫ω 0 t(ω − s)a0φ([s])ds and using the fact that ( t(t) ) t≥0 is exponentially stable, we deduce that ||i1(t)|| ≤ me −δt||v(ω)||. therefore lim t→∞ i1(t) = 0. now, show that lim t→∞ ||φ([t + ω]) − φ([t])|| = 0. we have that lim t→∞ ||φ(t + ω) − φ(t)|| = 0. therefore: ∀ǫ > 0, ∃ t0ǫ ∈ r +,∀ t > t0ǫ ⇒ ||φ(t + ω) − φ(t)|| < ǫ. we put tǫ = [t 0 ǫ] + 1. let ǫ > 0. for t > tǫ, we observe that [t] ≥ tǫ because tǫ is an integer. we deduce so that ∀ǫ > 0, ∃tǫ ∈ r +,∀ t > tǫ ⇒ ||φ([t] + ω) − φ([t])|| < ǫ. since ω is an integer, we observe that ∀ǫ > 0, ∃tǫ ∈ r +,∀ t > tǫ ⇒ ||φ([t + ω]) − φ([t])|| < ǫ. let ǫ > 0, we can find tǫ sufficiently large such that ||φ([t + ω]) − φ([t])|| < δ m ||a0 || ǫ, for t > tǫ. let’s write i2(t) = ∫t 0 t(t − s))a0(φ([s + ω]) − φ([s]))ds. then we obtain ||i2(t)|| ≤ || ∫tǫ 0 t(t − s))a0(φ([s + ω]) − x([s]))ds|| + || ∫t tǫ t(t − s))a0(x([s + ω]) − x([s]))ds||. observing that || ∫tǫ 0 t(t − s)a0(φ([s + ω]) − φ([s]))ds|| ≤ ∫tǫ 0 ||t(t − s)|| ||a0|| ||φ([s + ω]) − φ([s])|| ≤ ∫tǫ 0 me−δ(t−s) ||a0||2||φ||∞ds cubo 16, 3 (2014) s-asymptotically ω-periodic solution for a nonlinear differential . . . 61 ≤ m ||a0||2||φ||∞ δ (e−δ(t−tǫ) − e−δt) we deduce that lim t→∞ ∫tǫ 0 t(t − s)(φ([s + ω]) − φ([s]))ds = 0. we have also that || ∫t tǫ t(t − s)a0(φ([s + ω]) − φ([s]))ds|| ≤ ∫t tǫ me−δ(t−s) ||a0|| δ m ||a0|| ǫds ≤ ǫ ∫t tǫ δe−δ(t−s)ds ≤ ǫ(1 − e−δ(t−tǫ)) ≤ ǫ therefore lim t→∞ ∫t tǫ t(t − s)a0(φ([s + ω]) − φ([s]))ds = 0. we deduce so that lim t→∞ i2(t) = 0, this proves that ∧1 ∈ sapω(x).✷ theorem 3. we assume that the hypothesis (h.1) is satisfied. let ω ∈ n. we assume also that f is a s asymptotically ω-periodic function. then the equation (1) has a unique s asymptotically ω-periodic solution if θ := m δ ||a0|| < 1. proof. define the nonlinear operator γ : sapω(x) 7→ sapω(x) (γu)(t) := l(t) + (∧1u)(t) + ∧2(t) for every u ∈ sapω(x), where (∧1u)(t) = ∫t 0 t(t − s)a0φ([s])ds and ∧2(t) = ∫t 0 t(t − s)f(s)ds. we satisfy that the nonlinear operator γ is well defined. the lemma 1 show that l(t) is is a s asymptotically ω-periodic. the lemma 2 show that the operator ∧1 maps sapω(x) into itself. then the nonlinear operator γ maps sapω(x) into itself. since ||l(t)|| ≤ me−δt, ∀t ≥ 0, we observe that l(t) ∈ c0(r +,x). for every φ,ψ ∈ sapω(x) ||γ(φ)(t) − γ(ψ)(t)|| 62 william dimbour & jean-claude mado cubo 16, 3 (2014) = ||t(t)c0 + ∫t 0 t(t − s)a0φ([s])ds + ∫t 0 t(t − s)f(s)ds −t(t)c0 − ∫t 0 t(t − s)a0ψ([s])ds − ∫t 0 t(t − s)f(s)ds|| ≤ || ∫t 0 t(t−s)a0 ( φ([s])−ψ([s]) ) ds|| ≤ ∫t 0 ||t(t−s)|| ||a0|| ||φ([s])−ψ([s])||ds ≤ ∫t 0 ||t(t−s)|| ||a0|| ||φ−ψ||∞ ds ≤ ∫t 0 me−δ(t−s)ds||a0|| ||φ − ψ||∞ ≤ m δ ||a0|| ||φ − ψ||∞. therefore, if θ < 1, then the equation (1) has a unique s asymptotically ω-periodic solution. 3.2 the nonlinear case definition 7. a solution of eq.(2) on r+ is a function x(t) that satisfies the conditions: 1-x(t) is continuous on r+. 2-the derivative x′(t) exists at each point t ∈ r+, with possible exception of the points [t] ∈ r+ where one-sided derivatives exists. 3-eq.(2) is satisfied on each interval [n,n + 1[ with n ∈ n. we now make the following assumption. (h.2) the function g : r+×x → x,(t,u) → g(t,u) is uniformly s-asymptotically ω-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. there exist constant kg ≥ 0 such that ||g(t,u) − g(t,v)|| ≤ kg||u − v|| for all t ∈ r+, and ∀u,v ∈ x. definition 8. let t(t) be the c0 semigroup generated by a. the function x ∈ c(r +,x) given by x(t) = t(t)c0 + ∫t 0 t(t − s)a0x([s])ds + ∫t 0 t(t − s)g(s,x(s))ds is the mild solution of the equation (2). cubo 16, 3 (2014) s-asymptotically ω-periodic solution for a nonlinear differential . . . 63 theorem 4. we assume that the hypothesis (h.1) and (h.2) are satisfied. let ω ∈ n. then the equation (2) has a unique s asymptotically ω-periodic solution if θ := m δ ( ||a0|| + kg ) < 1. proof. define the nonlinear operator γ : sapω(x) 7→ sapω(x) (γu)(t) := l(t) + (∧1u)(t) + ∧2(t) for every u ∈ sapω(x), where (∧1u)(t) = ∫t 0 t(t − s)a0φ([s])ds and ∧2(t) = ∫t 0 t(t − s)g(s,x(s))ds. since the hypothesis (h.2) is satisfied, the nonlinear operator γ is well defined. for every φ,ψ ∈ sapω(x) ||γ(φ)(t) − γ(ψ)(t)|| = ||t(t)c0+ ∫t 0 t(t−s)a0φ([s])ds+ ∫t 0 t(t−s)g(s,φ(s))ds −t(t)c0 − ∫t 0 t(t−s)a0ψ([s])ds− ∫t 0 t(t−s)g(s,ψ(s))ds|| ≤ || ∫t 0 t(t−s)a0 ( φ([s])−ψ([s]) ) ds||+|| ∫t 0 t(t−s) ( g(s,φ(s))−g(s,ψ(s)) ) ds ≤ ∫t 0 ||t(t−s)|| ||a0|| ||φ([s])−ψ([s])||ds+ ∫t 0 ||t(t−s)|| ||g(s,φ(s))−g(s,ψ(s))||ds ≤ ∫t 0 ||t(t−s)|| ||a0|| ||φ−ψ||∞ ds+ ∫t 0 ||t(t−s)||kg ||φ−ψ||∞ ds ≤ ∫t 0 me−δ(t−s)ds||a0|| ||φ−ψ||∞+ ∫t 0 mkge −δ(t−s)ds ||φ−ψ||∞ ≤ m δ ||a0|| ||φ−ψ||∞ + m δ kg ||φ−ψ||∞ ≤ m δ ( ||a0||+kg ) ||φ−ψ||∞ therefore, if θ < 1, then the equation (2) has a unique s asymptotically ω-periodic solution. 64 william dimbour & jean-claude mado cubo 16, 3 (2014) 4 application as an application, we consider (4)    ∂u ∂t (t,x) = ∂ 2 u ∂x2 (t,x) + αu([t],x) + g(t,u(t,x)) t ∈ r+,x ∈ [0,π], α ∈ r u(t,0) = u(t,π) = 0 t ∈ r+ u(0) = c0 ∈ x we assume that (x, || · ||) = (l2(0,π), || · ||2) and define d(a) = {u ∈ l2[0,π], u(0) = u(π) = 0} au(·) = △u = u′′(·), ∀u(·) ∈ d(a). a is the infinetesimal generator of a semigroup t(t) on l2[0,π] with ||t(t)|| ≤ e−t for t ≥ 0. put u(t) = u(t, ·) that is u(t)x = u(t,x), (t,x) ∈ r+ × (0,π). considering a0 : l 2[0,π] 7→ l2[0,π], y → αy, we observe that a0 is a linear bounded operator such that ||a0|| = |α| and a0u([t]) = αu([t], ·). theorem 5. we assume that ω ∈ n. then the system (4) has a unique mild solution s asymptotically ω-periodic if |α| < 1. proof. we have m = 1, δ = 1, ||a0|| = |α|. then we apply the theorem (4) for the system (4). received: april 2014. accepted: may 2014. references [1] j.blot, p.cieutat, g.n’guérékata s-asymptotically ω-periodic functions and applications to evolution equations afr. diaspora. j.math. 12 (2011), 113-121. [2] w. dimbour, almost automorphic solutions for a differential equations with piecewise constant argument in a banach space, nonlinear analysis, vol.74 (2011) 2351-2357. [3] w. dimbour, g. n’guérékata, s-asymptotically ω-periodic solutions to some classes of partial evolution equation, applied mathematics and computation, vol.218 (2012), 7622-7628. [4] w. dimbour, g. mophou, g. n’guérékata, s-asymptotically ω-periodic solutions for partial differential equations with finite delay, electronic journal of differential equation, vol.2011 (2011), 1-12. cubo 16, 3 (2014) s-asymptotically ω-periodic solution for a nonlinear differential . . . 65 [5] h.r.henŕıquez, m.pierre, p.táboas on s-asymptotically ω-periodic function on banach spaces and applications, j.math.anal.appl 343(2008), 1119-1130. [6] h.r.henŕıquez, m.pierre, p.táboas existence of s-asymptotically ω-periodic solutions for abstract neutral equations, bull.austr.math.soc 78(2008), 365-382. [7] c.lizama, g.n’guérékata bounded mild solutions for semilinear integrodifferential equations in banach space, integr.eq.oper. theory 68(2010) 207-227. [8] n.van minh, t. tat dat, on the almost automorphy of bounded solutions of differential equations with piecewise constant argument, journal of mathematical analysis and appliction 326(2007), 165-178. [9] k.yosida, functional analysis, springer-verlag (1968). introduction preliminaries main result the linear case the nonlinear case application cubo a mathematical journal vol.15, no¯ 02, (105–110). june 2013 an iterative method for finite family of hemi contractions in hilbert space balwant singh thakur school of studies in mathematics, pt.ravishankar shukla university, raipur,492010, india. balwantst@gmail.com abstract we consider the problem of finding a common fixed point of n hemicontractions defined on a compact convex subset of a hilbert space, an algorithm for solving this problem will be studied. we will prove strong convergence theorem for this algorithm. resumen consideramos el problema de búsqueda de un punto fijo común de n hemicontracciones definida sobre un subconjunto convexo compacto de un espacio de hilbert. se estudiará un algoritmo para resolver este problema. probaremos el teorema de convergencia fuerte para este algoritmo. keywords and phrases: hemicontraction, mann iteration, implicit iteration, common fixed point. 2010 ams mathematics subject classification: 47h09,47h10. 106 balwant singh thakur cubo 15, 2 (2013) 1 introduction let h be a hilbert space and let k be a nonempty subset of h. a map t : k → k is called nonexpansive if ‖tx − ty‖ ≤ ‖x − y‖ , ∀x, y ∈ k . an important generalization of nonexpansive mapping is pseudocontractive mapping. a mapping t : k → k is said to be pseudocontractive if, ∀x, y ∈ k , ‖tx − ty‖ 2 ≤ ‖x − y‖ 2 + ‖(i − t)x − (i − t)y‖ 2 holds. t is said to be strongly pseudocontractive if, there exists k ∈ (0, 1) such that, ‖tx − ty‖ 2 ≤ ‖x − y‖ 2 + k ‖(i − t)x − (i − t)y‖ 2 , ∀x, y ∈ k. for importance of fixed points of pseudocontractive mappings one may refer [1]. iterative methods for approximating fixed points of nonexpansive mappings have been extensively studied (see e.g. [2, 3, 4, 7]), but, iterative methods for approximating pseudocontractive mappings are far less developed than those of nonexpansive mappings. however, on the otherhand pseudocontractions have more powerful applications than nonexpansive mappings in solving nonlinear inverse problems. in recent years many authors have studied iterative approximation of fixed point of strongly pseudocontractive mappings. most of them used mann’s iteration process [6]. but in the case of pseudocontractive mapping, it is well known that mann’s iteration fails to converge to fixed point of lipschitz pseudocontractive mappings in a compact convex subset of a hilbert space. in 1974, ishikawa [5] introduced an iteration process which converges to a fixed point of lipschitz pseudocontractive mapping in a compact convex subset of a hilbert space. qihou [8], extended result of ishikawa to slightly more general class of lipschitz hemicontractive mappings. a mapping t : k → k is said to be hemicontractive if f(t) 6= ∅ and ‖tx − p‖ 2 ≤ ‖x − p‖ 2 + ‖x − tx‖ 2 , ∀x ∈ k, p ∈ f(t) where f(t) := {x ∈ k : tx = x} is the fixed point set of t. it is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings. more recently, rafiq [9], proposed mann type implicit iteration process to approximate fixed points of hemicontractive mapping defined in a compact convex subset of a hilbert space. for arbitrary chosen x0 ∈ k the iteration process is given by xn = αnxn−1 + (1 − αn)txn cubo 15, 2 (2013) an iterative method for finite family of hemi contractions . . . 107 where {αn} is a real sequence in [0, 1] satisfying some appropriate conditions. the purpose of this paper to study the problem of finding a point x such that x ∈ n ⋂ i=1 fix(ti) where n ≥ 1 is a positive integer and {ti} n i=1 are n hemicontractive mappings defined on a compact convex subset k of a hilbert space h. we study the strong convergence of the algorithm which generates a sequence {xn} in the following way: xn = αnxn−1 + (1 − αn) n∑ i=1 λ (n) i tixn , (1) where the sequence of weights {λ (n) i }ni=1 satisfies appropriate assumptions. 2 preliminaries following well known identity holds in a hilbert space h : ‖(1 − λ)x + λy‖ 2 = (1 − λ) ‖x‖ 2 + λ ‖y‖ 2 − λ(1 − λ) ‖x − y‖ 2 for all x, y ∈ h and λ ∈ [0, 1]. we shall use the following lemma to prove our main result: lemma 2.1. [10] suppose {ρn} and {σn} are two sequences of nonnegative numbers such that for some real number n0 ≥ 1, ρn+1 ≤ ρn + σn ∀n ≥ n0 . (a) if ∑ σn < ∞ then, lim ρn exists. (b) if ∑ ρn < ∞ and {ρn} has a subsequence converging to zero, then lim ρn = 0. given an integer n ≥ 1, for each 1 ≤ i ≤ n, assume that ti : k → k is a hemicontractive mapping. then the family {ti} n i=1 is said to satisfy condition b if n ⋂ i=1 fix(ti) 6= ∅ , and fix ( n∑ i=1 λiti ) = n ⋂ i=1 fix(ti) where {λi} is a positive sequence such that ∑n i=1 λi = 1. 108 balwant singh thakur cubo 15, 2 (2013) proposition 2.2. let n ≥ 1 be a given integer. for each 1 ≤ i ≤ n, assume that ti : k → k is a hemicontractive mapping and the family {ti} n i=1 satisfies the condition b. then ∑n i=1 λiti is a hemicontractive mapping. proof. let us consider the case of n = 2. set v = (1 − λ)t1 + λt2, where λ ∈ (0, 1), where t1, t2 are hemicontractions. we have to prove that ‖vx − p‖ 2 ≤ ‖x − p‖ 2 + ‖(i − v)x‖ 2 ∀x ∈ k , p ∈ fix(v) . we have ‖(i − v)x‖ 2 = ‖(i − ((1 − λ)t1 + λt2))x‖ 2 = (1 − λ) ‖(i − t1)x‖ 2 + λ ‖(i − t2)x‖ 2 − λ(1 − λ) ‖(t1 − t2)x‖ 2 so, ‖vx − p‖ 2 = ‖(1 − λ)(t1x − p) + λ(t2x − p)‖ 2 = (1 − λ) ‖t1x − p‖ 2 + λ ‖t2x − p‖ 2 − λ(1 − λ) ‖t1x − t2x‖ 2 ≤ (1 − λ) [ ‖x − p‖ 2 + ‖(i − t1)x‖ 2 ] + λ [ ‖x − p‖ 2 + ‖(i − t2)x‖ 2 ] − λ(1 − λ) ‖t1x − t2x‖ 2 = ‖x − p‖ 2 + ‖(i − v)x‖ 2 . hence v is a hemicontraction. the general case can be proved by induction. 3 main result theorem 3.1. let k be a compact convex subset of a hilbert space h. let n ≥ 1 be an integer. for each n ≥ 1, assume that {λ (n) i }ni=1 is a finite sequence of positive numbers such that ∑n i=1 λ (n) i = 1 and infn≥1 λ (n) i > 0 for all 1 ≤ i ≤ n. for each 1 ≤ i ≤ n, let ti : k → k is a hemicontractive mapping and the family {ti} n i=1 satisfies the condition b. for arbitrary chosen x0 ∈ k, let {xn} be a sequence generated by the algorithm (1), where the sequence {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1). then {xn} converges strongly to a common fixed point of the family {ti} n i=1. proof. write, for each n ≥ 1, sn = n∑ i=1 λ (n) i ti . by the proposition 2.2, each sn is hemicontractive on k, and the algorithm (1) can be rewritten as, xn = αnxn−1 + (1 − αn)snxn . (2) cubo 15, 2 (2013) an iterative method for finite family of hemi contractions . . . 109 for p ∈ f := ⋂n i=1 fix(ti), we have ‖xn − p‖ 2 = ‖αn(xn−1 − p) + (1 − αn)(snxn − p)‖ 2 = αn ‖xn−1 − p‖ 2 + (1 − αn) ‖snxn − p‖ 2 − αn(1 − αn) ‖xn−1 − snxn‖ 2 ≤ αn ‖xn−1 − p‖ 2 + (1 − αn) [ ‖xn − p‖ 2 + ‖xn − snxn‖ 2 ] − αn(1 − αn) ‖xn−1 − snxn‖ 2 (3) also, ‖xn − snxn‖ 2 = ‖αnxn−1 + (1 − αn)snxn − snxn‖ 2 = α2n ‖xn−1 − snxn‖ 2 . (4) using (3) and (4), we have ‖xn − p‖ 2 ≤ ‖xn−1 − p‖ 2 − (1 − αn) 2 ‖xn−1 − snxn‖ 2 . (5) from the condition {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1), we conclude that ‖xn − p‖ 2 ≤ ‖xn−1 − p‖ 2 − δ2 ‖xn−1 − snxn‖ 2 (6) holds for all p ∈ f. now, δ2 ‖xn−1 − snxn‖ 2 ≤ ‖xn−1 − p‖ 2 − ‖xn − p‖ 2 and hence, δ2 ∞∑ j=1 ‖xj−1 − sjxj‖ 2 ≤ ∞∑ j=1 ( ‖xj−1 − p‖ 2 − ‖xj − p‖ 2 ) = ‖x0 − p‖ 2 implies, δ2 ∞∑ j=1 ‖xj−1 − sjxj‖ 2 < ∞ . (7) so, lim n→∞ ‖xn−1 − snxn‖ = 0 . (8) from (4), we have lim n→∞ ‖xn − snxn‖ = 0 . (9) without loss of generality, we may assume that λ (nl) i → λi ( as l → ∞), 1 ≤ i ≤ n . it is easily seen that each λi > 0 and ∑n i=1 λi = 1. we also have snlx → sx ( as l → ∞), for all x ∈ k , 110 balwant singh thakur cubo 15, 2 (2013) where s = n∑ i=1 λiti . since k is compact, there is a subsequence {xnj} of {xn} which converges to a fixed point of s, say z. using (6), we have ‖xn − z‖ 2 ≤ ‖xn−1 − z‖ 2 − δ2 ‖xn−1 − sxn‖ 2 in view of lemma 2.1 and (7), we conclude that ‖xn − z‖ → 0 as n → ∞ i.e. xn → z as n → ∞. this completes the proof. acknowledgements the author is supported by a project (fn-41-1390/2012) of university grants commission of india. received: december 2011. accepted: september 2012. references [1] browder,f.e. nonlinear operators and nonlinear equations of evolution in banach spaces, proc. sympos. pure math., xviii (2) (1976). [2] browder,f.e. and petryshyn,w.v., construction of fixed points of nonlinear mappings in hilbert spaces, j. math. anal. appl., 20 (1967), 197-228. [3] chidume,c.e., li,j. and udomene,a., convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, proc. amer. math. soc., 133 (2) (2004), 473-480. [4] halpern,b., fixed points of nonexpansive maps’, bull. amer. math. soc., 73 (1967), 957-961. [5] ishikawa,s. fixed point by a new iteration method, proc. amer. math. soc., 4 (1974), 147-150. [6] mann,w.r. mean value methods in iteration, proc. amer. math. soc., 4 (1953), 506-510. [7] opial,z., weak convergence of the sequence of successive approximation for nonexpansive mappings, bull. amer. math. soc., 76 (1967), 591-597. [8] qihou,l. the convergence theorems of the sequences of ishikawa iterates for hemicontractive mappings, j. math. anal. appl., 148 (1990), 55-62. [9] rafiq,a. on mann iteration in hilbert spaces, nonlinear analysis, 66 (2007), 2230-2236. [10] tan,k.k. and xu,h.k., approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl., 178 (1993), 301-308. cubo a mathematical journal vol.15, no¯ 02, (71–77). june 2013 on the impossibility of the convolution of distributions ghislain r. franssens belgian institute for space aeronomy, ringlaan 3, b-1180 brussels, belgium, ghislain.franssens@aeronomy.be abstract certain incompatibilities are proved related to the prolongation of an associative derivation convolution algebra, defined for a subset of distributions, to a larger subset of distributions containing a derivation and the one distribution. this result is a twin of schwartz’ impossibility theorem, stating certain incompatibilities related to the prolongation of the multiplication product from the set of continuous functions to a larger subset of distributions containing a derivation and the delta distribution. the presented result shows that the non-associativity of a recently constructed derivation convolution algebra of associated homogeneous distributions with support in r cannot be avoided. resumen se prueban algunas incompatibilidades relacionadas con la prolongación de un álgebra de convolución de derivación asociativa, definida para un subconjunto de distribuciones a un subconjunto mayor de distribuciones que contienen una derivación y una distribución. este resultado es un gemelo del teorema de imposibilidad de schwartz declarando algunas incompatibilidades relacionadas a la prolongación del producto de multiplicación de un conjunto de funciones continuas a un subconjunto mayor de distribuciones conteniendo una derivación y una distribución delta. el resultado presente muestra que la no asociatividad de un álgebra de convolución de derivación construida recientemente de distribuciones homogéneas asociadas con soporte en r no puede evitarse. keywords and phrases: generalized function, distribution, convolution algebra, impossibility theorem. 2010 ams mathematics subject classification: 46f10, 46f30. 72 ghislain r. franssens cubo 15, 2 (2013) 1 introduction in a series of preceding papers, [3]–[9], the author embarked on an in-depth study of the set h′ (r) of associated homogeneous distributions (ahds) based on (i.e., with support in) the real line r, [11]. the elements of h′ (r) are the distributional analogues of power-log functions with domain in r and contain the majority of the distributions one encounters in (one-dimensional) physics applications (including the δ and η , 1 π x−1 distributions). for an introduction to ahds, an overview of their properties and possible applications of this work, the reader is referred to [3]. the main result of the above study was the construction of a convolution algebra and an isomorphic multiplication algebra of ahds on r. the multiplication algebra provides a non-trivial example of how a distributional product can be defined, for an important subset of distributions containing a derivation and the delta distribution, and how this is influenced by l. schwartz’ “impossibility theorem” [14]. both constructed algebras are non-commutative and non-associative, but in a minimal and interesting way, see [7], [8]. schwartz’ theorem, stating certain incompatibilities in a distributional derivation multiplication algebra, is well-known. one might be inclined to think that the existence of such incompatibilities is unique to the multiplication product of generalized functions. the aim of this note is to show that this is not the case. we state a similar impossibility theorem related to the prolongation of an associative derivation convolution algebra, defined for a subset of distributions d′ r ⊂ d′, to a larger (not necessarily proper) subset of distributions containing a derivation and the one distribution, [13], [10], [15], [11]. let us first recall schwartz’ impossibility theorem for the multiplication product. let ck denote the set of continuous (k = 0) or k-times continuously differentiable (k > 0) functions from r → r. theorem 1.1. denote by ( c0, +, .; r ) the algebra over r, consisting of the set c0 together with pointwise addition + and pointwise multiplication ., and 0 its +-identity element. (i) let (f′, +; r) be any linear space over r such that f′ ⊃ c0. (ii) let . : f′ ×f′ → f′ be an associative multiplication product, which coincides with the one defined on c0 and with common .-identity element 1. (iii) let d : f′ → f′ be a derivation, with respect to ., which coincides with the derivation defined on c1 and x ∈ c1\ {0} : dx = 1. then, ∄δ ∈ f′\ {0} : x.δ = 0. although this theorem is usually referred to as an impossibility theorem, it does not say that a multiplication product defined on d′ is not possible. it says that a multiplication product defined on c0, with properties as stated in condition (ii), and a derivation defined on c1, with properties as stated in condition (iii), cannot be faithfully prolonged to a superset f′ of c0 which contains the delta distribution δ. notice that f′ is allowed to be a (proper) subset of d′. cubo 15, 2 (2013) on the impossibility of the convolution of distributions 73 for schwartz’ linear space of distributions (d′, +; r), the multiplication defined for continuous functions is not prolongated to all distributions, so condition (ii) does not hold, and this allows the existence in d′ of a derivation d and a distribution δ 6= 0 : x.δ = 0 (e.g., δ = d2 ( 1 2 |x| ) ). in the construction of commutative, associative generalized function algebras, such as by egorov, [2], rosinger, [12] or colombeau, [1], schwartz’ theorem is evaded in a more subtle way. for instance, colombeau’s construction has led to an algebra of generalized functions g, which does not contain the algebra of the continuous functions as a subalgebra in the usual algebraically exact way. first, the set c0 is embedded in g as a subset c̃0 ⊂ g such that c0 is “associated” to c̃0 in some weak sense, which involves non-standard analysis. then, colombeau’s product ⊙ defined on g, when restricted to c̃0, agrees with the usual function product . defined on c0 only in his weak sense and the last clause in theorem 1.1 is circumvented since (i) does not hold. more explicitly, ∀f ∈ c0 associated to a f̃ ∈ g and ∀g ∈ c0 associated to a g̃ ∈ g holds that f.g ∈ c0 is associated to f̃ ⊙ g̃ ∈ g. we will proof hereafter a twin of schwartz’ impossibility theorem, stating certain incompatibilities in a distributional convolution algebra. we use the notation and definitions introduced in [3]. 2 the convolution of distributions definition 2.1. denote by h′ the set of associated homogeneous distributions (ahds) based on r and fz m a typical element, having degree of homogeneity z and order of association m. let d′ z − stand for the set of all finite sums over c of elements of h′ z − , { f−k−1 m ∈ h′, ∀k, m ∈ n } ⊂ h′, the set of ahds based on r with negative integer degrees. theorem 2.2. denote by ( d′ z − , +, ∗; c ) the algebra over c, consisting of the set d′ z − together with distributional addition + and distributional convolution ∗, and 0 its +-identity element. (i) let (f′, +; c) be any linear space over c such that f′ ⊃ d′ z − . (ii) let ∗ : f′ × f′ → f′ be an associative convolution product, which coincides with the one defined on d′ z − and with common identity element δ. (iii) let x : f′ → f′ be a derivation, with respect to ∗, which coincides with the derivation defined on d′ z − and −δ(1) ∈ d′ z − \ {0} : x ( −δ(1) ) = δ. then, ∄1 ∈ f′\ {0} : δ(1) ∗ 1 = 0. proof. it follows from the results obtained in [5] that ( d′ z − , +, ∗; c ) is an associative convolution algebra over c. a. consider the distribution f satisfying x ( δ(1) ∗ f ) + 2f = −δ(1). (1) 74 ghislain r. franssens cubo 15, 2 (2013) by [3, eq. (67)], f is readily seen to be an associated homogeneous distribution based on r having degree of homogeneity −2 and order of association 1, hence f ∈ d′ z − ⊂ f′. on the one hand, applying the derivation x to (1) and using the given property x ( −δ(1) ) = δ, we get x2 ( δ(1) ∗ f ) = −2 (xf) + δ. (2) on the other hand, since x is given to be a derivation with respect to the commutative convolution product ∗, we have by leibniz’ rule and due to the property x ( −δ(1) ) = δ, that x2 ( δ(1) ∗ f ) = ( x2δ(1) ) ∗ f + 2 ( xδ(1) ) ∗ (xf) + δ(1) ∗ ( x2f ) , = ( x2δ(1) ) ∗ f − 2 (xf) + δ(1) ∗ ( x2f ) . (3) since it is given that x is a derivation with respect to the commutative convolution product ∗ and that δ is the ∗-identity, it holds further by leibniz’ rule that xδ = x (δ ∗ δ) = 2 (xδ) ∗ δ = 2 (xδ) , so xδ = 0. this in turn implies that x2δ(1) = xδ = 0. then, (3) simplifies to x2 ( δ(1) ∗ f ) = −2 (xf) + δ(1) ∗ ( x2f ) . (4) combining (2) with (4) gives δ(1) ∗ ( x2f ) = δ, (5) which shows that x2f is a convolutional inverse of δ(1). consequently, x2f has degree 0, so x2f /∈ d′ z − . as it is given that x is an automorphism of f′, we must necessarily have that x2f ∈ f′. b. since it is given that δ is the ∗-identity, that ∗ in (f′, +, ∗; c) is associative, that δ(1)∗1 = 0, and using eq. (5), we obtain 1 = 1 ∗ δ = 1 ∗ ( δ(1) ∗ ( x2f )) = ( δ(1) ∗ 1 ) ∗ ( x2f ) = 0 ∗ ( x2f ) = 0. hence, ∄1 ∈ f′\ {0} : δ(1) ∗ 1 = 0. theorem 2.2 does not say that a convolution product of distributions is not possible. it says that a convolution product defined on d′ z − , with properties as stated in condition (ii), and a derivation defined on d′ z − , with properties as stated in condition (iii), cannot be faithfully prolonged to a superset f′ of d′ z − which contains the one distribution 1. notice that f′ is again allowed to be a subset of d′. for schwartz’ set of distributions d′, the convolution defined on d′ z − is not prolonged to d′, so condition (ii) does not hold, and this allows the existence in d′ of a derivation x and a distribution 1 6= 0 : δ(1) ∗ 1 = 0 (e.g., 1 = x2 ( −πη(1) ) ). cubo 15, 2 (2013) on the impossibility of the convolution of distributions 75 let f′ now stand for the set of all finite sums over c of elements of h′. we constructed earlier in [5]–[7] the convolution product in the algebra (f′, +, ∗; c). clearly, f′ ⊃ d′ z − . theorem 2.2 (in particular, part b of the proof) shows that the non-associativity of critical convolution products, obtained in [8] for the set h′, cannot be avoided. theorem 2.2 is for convolution algebras what schwartz’ theorem is for multiplication algebras. however, since the identity element δ of the convolution product ∗ is a distribution, there are no convolution algebras with ∗-identity for function sets. a corollary of schwartz’ theorem states that the set f′ in his theorem contains elements g that satisfy dg = 0 and for which g is not proportional to 1. we have the following analogue for the convolution product. corollary 2.3. the set f′ in theorem 2.2 contains elements g that satisfy xg = 0 and for which g is not proportional to δ. proof. consider the distribution h satisfying x ( δ(1) ∗ h ) + 2h = 0. (6) by [3, eq. (69)], f is readily seen to be an associated homogeneous distribution based on r having degree of homogeneity −2, hence f ∈ d′ z − ⊂ f′. on the one hand, applying the derivation x to (6) gives x2 ( δ(1) ∗ h ) = −2 (xh) . on the other hand, since it is given that x is a derivation with respect to the convolution product ∗, we have by leibniz’ rule and by using the property x ( −δ(1) ) = δ, given in theorem 2.2, that x2 ( δ(1) ∗ h ) = −2 (xh) + δ(1) ∗ ( x2h ) . combining both results gives δ(1) ∗ ( x2h ) = 0. theorem 2.2 states that in f′ necessarily must hold that x2h = 0. however, any homogeneous distribution h of degree −2 is of the form, [4, eq. (14)], h = aδ(1) + bη(1). we can choose a = 0 and b 6= 0, use xη(1) = η 6= δ, which is a particular result of [3, eq. (181)], and so obtain a distribution h for which g , xh is not of the form cδ. hence, xg can be zero for a distribution g that is not proportional to δ. in the non-associative convolution algebra, developed in [5]–[7], we have due to [3, eq. (181)] that x2η(1) = −1/π 6= 0. 76 ghislain r. franssens cubo 15, 2 (2013) 3 summary 3.1 multiplication on c0, a commutative and associative multiplication . can always be defined. hence, the operation x , x. (derivation with respect to ∗) is always possible on c0. the derivation x can be prolongated from c0 to any superset f′, since multiplication of a distribution by a polynomial is always defined. however, the multiplication . can not be prolongated from c0 to f′ with all its properties preserved. further, the derivation d , δ(1)∗ is not defined everywhere on c0 (regarded as a function space). it is however defined everywhere in the subset of regular distributions generated by c0, d′ c0 . then, the derivation d can be prolongated from d′ c0 to f′, since convolution of any distribution with a compact support distribution is always defined. 3.2 convolution on d′ z − , a commutative and associative convolution ∗ can always be defined. hence, the operation d , δ(1)∗ (derivation with respect to .) is always possible on d′ z − . the derivation d can be prolongated from d′ z − to any superset f′, since convolution of any distribution with a compact support distribution is always defined. however, the convolution ∗ can not be prolongated from d′ z − to f′ with all its properties preserved. further, the derivation x , x. is defined everywhere on d′ z − , since multiplication of any distribution by a smooth function is always defined. received: february 2012. accepted: september 2012. references [1] j.f. colombeau, new generalized functions and multiplication of distributions, math. studies 84, north holland, 1984. 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[8] g.r. franssens, multiplication product formula for associated homogeneous distributions on r, math. methods appl. sci., 34(12), pp. 1460–1471, 2011. [9] g.r. franssens, substructures in algebras of associated homogeneous distributions on r, bull. belg. math. society simon stevin, 18, pp. 1–17, 2011. [10] g. friedlander, m. joshi, introduction to the theory of distributions, 2nd ed., cambridge univ. press, cambridge, 1998. [11] i.m. gel’fand, g.e. shilov, generalized functions, vol. i, academic press, 1964. [12] e.e. rosinger, nonlinear partial differential equations. sequential and weak solutions, north holland, 1980. [13] l. schwartz, théorie des distributions, vols. i&ii, hermann, 1957. [14] l. schwartz, sur l’impossibilité de la multiplication des distributions, c. r. acad. sci. paris 239, pp. 847–848, 1954. [15] a.h. zemanian, distribution theory and transform analysis, dover, 1965. cubo a mathematical journal vol.15, no¯ 02, (43–52). june 2013 isometric weighted composition operators on weighted banach spaces of holomorphic functions defined on the unit ball of a complex banach space elke wolf university of paderborn, mathematical institute, d-33095 paderborn, germany. lichte@math.uni-paderborn.de abstract let x and y be complex banach spaces and bx resp. by the closed unit ball. analytic maps φ : by → bx and ψ : bx → c induce the weighted composition operator: cφ,ψ : h(by) → h(bx), f 7→ ψ(f ◦ φ), where h(by) resp. h(bx) denotes the collection of all analytic functions f : bx(resp.by) → c. we study when such operators acting between weighted spaces of analytic functions are isometric. resumen sea x y y espacios de banach complejos, bx y by las bolas unitarias cerradas correspondientes. las aplicaciones anaĺıticas φ : by → bx y ψ : bx → c inducen el operador de composición con pesos: cφ,ψ : h(by) → h(bx), f 7→ ψ(f ◦ φ), donde h(by) y h(bx) denotan la colección de todas las funciones anaĺıticas f : bx(resp.by) → c. estudiamos cuándo dichos operadores que actúan entre los espacios con peso de funciones anaĺıticas son isométricas. keywords and phrases: weighted composition operators, weighted spaces of holomorphic functions on the unit ball of a complex banach space. 2010 ams mathematics subject classification: 47b38, 47b33. 44 elke wolf cubo 15, 2 (2013) 1 introduction let d denote the open unit disk in the complex plane and h(d) the collection of all analytic functions on d. then, an analytic self-map φ of d induces through composition a linear composition operator cφ : h(d) → h(d), f 7→ f ◦ φ. since such operators appear naturally in a variety of problems and since they link in the classical setting of the hardy space h2 (see [10] and [24]) operator theoretical questions with classical results in complex analysis their study has a long and rich history. now, let ψ ∈ h(d). the next step is to combine the composition operator cφ with a multiplication operator mψ : h(d) → h(d), f 7→ ψf to obtain the so-called weighted composition operator cφ,ψ := mψcφ : h(d) → h(d), f 7→ ψ(f ◦ φ). for a bounded and continuous function (weight) v : d → (0,∞) we consider h∞v := {f ∈ h(d); ‖f‖v := sup z∈d v(z)|f(z)| < ∞}. endowed with norm ‖.‖v, these spaces are banach spaces and in the sequel we refer to them as weighted banach spaces of holomorphic functions. such spaces arise in functional analysis, partial differential equations and convolution equations as well as in distribution theory. they have been studied intensively in several articles, see e.g. [1], [2], [3], [4], [18], [19]. in [6] bonet, domański, lindström and taskinen characterized boundedness and compactness of operators cφ : h ∞ v → h ∞ w , f 7→ f ◦ φ in terms of the inducing symbol φ as well as the involved weights v and w. the same properties of the weighted composition operator cφ,ψ : h ∞ v → h ∞ w were analyzed independently by contreras and hernández-dı́az as well as montes-rodŕıguez. in [8] we investigated under which conditions the weighted composition operator cφ,ψ acting on h ∞ v is isometric. the work of bonet, domański, lindström and taskinen motivated garcia, maestre and sevilla-peris to study boundedness and compactness of composition operators in the following setting. let x be a complex banach space, bx its open unit ball and h(bx) the collection of all holomorphic functions f : bx → c. moreover, we consider continuous and bounded functions v : bx → (0,∞). such a map is called a weight. a weight v induces the space hv(bx) := { f ∈ h(bx); ‖f‖v = sup x∈bx v(x)|f(x)| < ∞ } which, endowed with the weighted sup-norm ‖.‖v is a banach space as in the onedimensional case. now, an analytic map φ : by → bx induces an operator cφ : h(by) → h(bx), f 7→ f ◦ φ. cubo 15, 2 (2013) isometric weighted composition operators on weighted banach ... 45 garcia, maestre and sevilla-peris, characterized when an operator cφ : hv(by) → hw(bx), f 7→ f ◦ φ is bounded and compact, i.e. they gave sufficient and necessary conditions in terms of the inducing map φ as well as of the involved weights v and w for a composition operator to be bounded resp. compact. in this article we are interested in weighted composition operators cφ,ψ : hv(by) → hw(by), f 7→ ψ(f ◦ φ). motivated by [14] we will investigate when such an operator is bounded. a full characterization when such an operator is bounded follows easily with a similar proof as given in [14]. the more interesting question (motivated by [8]) is the following: when is a bounded operator cφ,ψ acting on hv(bx) an isometry. 2 basics on weights and weighted spaces this section is devoted to collect some basic facts on weights and weighted spaces in the setting of a complex banach space x and its open unit ball bx. these can be found in [13] and [14]. we say that a set a ⊂ bx is bx-bounded if there exists 0 < r < 1 such that a ⊂ rbx. we write hb(bx) = {f ∈ h(bx); f bounded on the bx-bounded sets } . we consider hv(bx) = { f ∈ h(bx); ‖f‖v := sup x∈bx v(x)|f(x)| < ∞ } . with the norm ‖.‖v, the space hv(bx) is a banach space. a weight v is radial if v(λx) = v(x) for every λ ∈ c with |λ| = 1 and every x ∈ bx. a weight v satisfies condition i if infx∈rbx v(x) > 0 for every 0 < r < 1. if v satisfies condition i, then hv(bx) ⊂ hb(bx). if x is finite-dimensional, then all weights on bx enjoy condition i. in the sequel we will assume that each weight v satisfies the condition i. given any weight v we consider ṽ(z) = 1 sup{|f(z)|; ‖f‖v ≤ 1} . by [14] proposition 1.1 the following hold: (1) 0 < v ≤ ṽ and ṽ is bounded and continuous, i.e. ṽ is a weight. (2) ṽ is radial and decreasing whenever v is so. (3) ‖f‖v ≤ 1 ⇐⇒ ‖f‖ṽ ≤ 1. 46 elke wolf cubo 15, 2 (2013) (4) for every x ∈ bx there is fx ∈ h ∞ v with ‖f‖v ≤ 1 such that ṽ(x) = |fx(x)|. we say that a weight v is norm-radial if v(x) = v(y) for every x,y with ‖x‖ = ‖y‖. we need some extra condition on the weight -which in a sense is an analogon to the lusky condition (l1) which appeared during his studies on the isomorphism classes of h∞v , see [18]. let v be a normradial weight that is continuously differentiable w.r.t. x. then we say that v satisfies condition (b) if and only if (b) sup x∈bx (1 − ‖x‖)|v′(x)| v(x) < ∞. finally, to study isometries we need some geometric tools. the generalized pseudohyperbolic distance of two points z,p ∈ bx is given by d(z,p) := sup {ρ(h(z),h(p)); h : bx → d holomorphic } 3 boundedness as we said before the following proof is very similar to the proof of proposition 2.3 in [14]. nevertheless we give it here for the sake of completeness. proposition 3.1. let v,w be two weights and φ : bx → by be holomorphic. moreover, let ψ ∈ h(bx). then the following are equivalent: (a) cφ,ψ : hv(by) → hw(bx) is well-defined and bounded. (b) supx∈bx w(x)|ψ(x)| ṽ(φ(x)) < ∞. proof. let us first suppose that the operator is bounded. we assume to the contrary that (b) does not hold. then we can find a sequence (xn)n ⊂ bx such that w(xn)|ψ(xn)| ṽ(φ(xn)) ≥ n for every n ∈ n. now, for each n ∈ n we can select fn ∈ hv(bx) with ‖fn‖v ≤ 1 such that |fn(φ(xn))| = 1 ṽ(φ(xn)) . since cφ,ψ : hv(by) → hw(bx) is bounded, there is c > 0 such that c ≥ ‖cφ,ψfn‖w ≥ w(xn)|ψ(xn)| ṽ(φ(xn)) ≥ n for every n ∈ n, which is a contradiction. conversely, let f ∈ hv(by). then we obtain for every x ∈ bx w(x)|ψ(x)||f(φ(x))| = |ψ(x)|w(x) ṽ(φ(x)) ṽ(φ(x)) ≤ m‖f‖ṽ = m‖f‖v. thus, the claim follows. cubo 15, 2 (2013) isometric weighted composition operators on weighted banach ... 47 4 isometries we obtain the following lemma which was shown for the setting of the spaces h∞v in [7]. however, in this setting there occur several different phenomena. lemma 4.1. let v be a weight on bx such that v is norm-radial and satisfies condition (b). moreover, let f ∈ h∞v . then there is a finite constant m > 0 independent of f ∈ h ∞ v such that |v(a)f(a) − v(b)f(b)| ≤ m‖f‖vd(a,b) for every a,b ∈ bx. proof. we fix a,b ∈ bx with a 6= b. now, there are n1,n2 ∈ n such that ‖a‖x < 1 − 1 n1 and ‖b‖x < 1 − 1 n2 . then we can find ε > 0 such that h : d → bx, h(t) = (t − ε)b + (1 − (t − ε))a. moroever h(ε) = a and h(1 − ε) = b. now, by cauchy’s formula we obtain |(f ◦ h)′(ε)| = 1 2π ∣ ∣ ∣ ∣ ∣ ∫ |ξ−ε|=(1−|ε|)r (f ◦ h)(ξ) |ξ − ε| dξ ∣ ∣ ∣ ∣ ∣ ≤ 1 2πr 1 (1 − |ε|)2 ‖f‖v ∫ |ξ−ε|=(1−|ε|)r |dξ| v(h(ξ)) . now, since v(a) < m and ‖h(ξ)‖x ≤ r0 < 1 for every ξ with |ξ − ε| = (1 − |ε|)r. hence there is c > 0 such that v(a) v(h(ξ)) = v(h(ε)) v(h(ξ)) ≤ c for every ξ with |ξ − ε| = (1 − |ε|)r. thus, |(f ◦ h)′(ε)| ≤ c 2πr2 1 (1 − |ε|)2 ‖f‖v v(h(ε)) 2π(1 − |ε|)r = c‖f‖v r(1 − ε)v(h(ε)) . next, we consider k(q) := v(q)f(q) for every q ∈ bx. then the total differential of k ◦ h is given by d(k ◦ h) = ∂(k ◦ h) ∂t dt + ∂(k ◦ h) ∂t dt. 48 elke wolf cubo 15, 2 (2013) now, for every t ∈ d we obtain ∂(k ◦ h) ∂t = (v ◦ h)′(t)f(h(t)) + v(h(t))(v ◦ h)′(t) and ∂(k ◦ h) ∂t = 0 this yields |d(k ◦ h)(t)| ≤ [|(v ◦ h)′(t)|f(h(t))| + |v(h(t))||(f ◦ h)′(t)|] |dt| ≤ [ | (v ◦ h)′(t) v(h(t)) ‖f‖v + c‖f‖v r(1 − |t|)v(h(t)) ] |dt| by condition (b) we can find c1 > 0 such that |v′(h(t))| v(h(t)) ‖b − a‖ = |(v ◦ h)′(t)| (v ◦ h)(t)| ≤ c1 1 − |h(t)| . therefore |d(k ◦ h)(t)| ≤ ( c1 + c r ) ‖f‖v 1 − |t| |dt|. if d(h(p),h(q)) ≤ r, then ρ(p,q) ≤ r and by using 1 − ρ(p,q)2 = (1 − |q|2)(1 − |p|2) |1 − pq|2 we have that |q − p| 1 − |p| ∼ ρ(p,q). here the constants only depend on r. by integration on both sides we can find constants c2,c3 > 0 with |k(h(q)) − k(h(p))| ≤ c2‖f‖v 1 1 − |p| |q − p| ≤ c3‖f‖vρ(p,q) ≤ c3‖f‖vd(h(p),h(q)) for all p,q with d(h(p),h(q)) ≤ r. if d(h(p),h(q)) > r then |v(p)f(p) − v(q)f(q)| ≤ 2‖f‖v ≤ 2 r ‖f‖vd(p,q) and the claim follows. the main ideas of the proof of the following theorem are taken from [8] but there also occur new phenomena. theorem 4.2. let φ be an analytic self-map of bx and ψ ∈ h(bx). moreover, assume that v is a norm-radial weight satisfying condition (b) such that v is continuously differentiable. cubo 15, 2 (2013) isometric weighted composition operators on weighted banach ... 49 (a) if supx∈bx |ψ(x)|v(x) ṽ(φ(x)) ≤ 1 and (m) for every a ∈ bx there is (xn)n ⊂ bx such that d(φ(xn),a) → 0 and |ψ(xn)|v(xn) ṽ(φ(xn)) → 1 then cφ,ψ : hv(bx) → hv(bx) is an isometry. (b) let v be a norm-radial weight with v = ṽ such that for each h : bx → d holomorphic w(x) := v(x) 1−|h(x)|2)p for every x ∈ bx is a weight for some 0 < p < ∞ and w = w̃. if cφ,ψ : hv(bx) → hv(bx) is an isometry, then condition (m) holds and supx∈bx |ψ(x)|v(x) ṽ(φ(x)) ≤ 1. proof. we first show (a). for every f ∈ hv(bx) we have that ‖cφ,ψ‖v = sup z∈bx |ψ(x)|v(x) v(φ(x)) v(φ(x))|f(φ(x))| ≤ ‖f‖v. now, let f ∈ hv(bx). then ‖f‖v = limm→∞ v(am)|f(am)| for some sequence (am)m. let m ∈ n be fixed. hence, by condition (m), there is (xmn )n ⊂ bx such that d(φ(x m n ),am) → 0 and |ψ(x m n )|v(x m n ) v(φ(xmn )) → 1 when n → ∞. by the previous lemma, for all m and n |v(am)f(am) − v(φ(x m n ))f(φ(x m n ))| ≤ m‖f‖vd(am,φ(x m n )). hence ‖cφ,ψ‖v = sup x∈bx |ψ(x)|v(x) v(φ(x)) (|f(am)|v(am) − m‖f‖vd(φ(x m n ),am)) = v(am)|f(am)|. since this is true for all m, we have ‖cφ,ψf‖v ≥ ‖f‖v. next, we show (b). we choose p > 0 and fix h : bx → d holomorphic such that w(x) = v(x) (1−|h(x)|2)p is a weight on bx with w = w̃. by assumption ‖cφ,ψf‖v = ‖f‖v for all f ∈ hv(bx). thus, ‖cφ,ψ‖ = sup x∈bx |ψ(x)|v(x) ṽ(φ(x)) ≤ 1. next, fix a ∈ bx and h : bx → d. then there exists ga ∈ hw(bx) with ‖ga‖w ≤ 1 such that ga(a) = w̃(a). put fa(z) = ga(z) ( (1 − |h(a)|2) (1 − h(z)h(a))2 )p . now, ‖fa‖v = 1 since |fa(a)|v(a) = 1. this means, that we can pick a sequence (xn)n ⊂ bx so that |ψ(xn)|fa(φ(xn))|v(xn) → 1 when n → ∞. hence 1 ≥ |ψ(xn)|v(xn) ṽ(φ(xn)) ≥ |ψ(xn)|v(xn) ṽ(φ(xn)) |fa(φ(xn))|ṽ(φ(xn)) = |ψ(xn)|v(xn)|fa(φ(xn))|. finally, lim n→∞ |ψ(xn)|v(xn) ṽ(φ(xn)) = 1. 50 elke wolf cubo 15, 2 (2013) further, 1 ≥ (1 − |σh(a)(h(φ(zn))| 2)p = (1 − |h(a)|2)p(1 − |h(φ(xn))| 2)p |1 − h(φ(xn))h(a)| 2p = |fa(φ(xn))|v(φ(xn))(1 − |h(φ(xn))| 2)p ga(h(φ(xn)))v(φ(xn)) ≥ |fa(φ(xn))|v(φ(xn)). since, |fa(φ(xn))|ṽ(φ(xn)) → 1 when n → ∞, we conclude, as v = ṽ, that limn→∞(1 − σh(a)(h(φ(xn))| 2)p = 1 and ρ(h(φ(xn)),h(a)) → 0 when n → ∞. since h : bx → d holomorphic was arbitrary the claim follows. example 4.3. let x be an arbitrary complex banach space, h : bx → d be holomorphic and select v(x) = (1− |h(x)|2)p. for fixed b ∈ bx we put φ(x) := σh(b)(h(x)) and ψ(x) := (σh(b)) ′(h(x)) for every x ∈ bx. then easy calculations show that the corresponding weighted composition operator is an isometry. received: march 2012. accepted: september 2012. references [1] j.m. anderson, j. duncan, duals of banach spaces of entire functions, glasgow math. j. 32 (1990), no. 2, 215-220. [2] k.d. bierstedt, w.h. summers, biduals of weighted banach spaces of analytic functions, j. austral. math. soc. (series a) 54 (1993), 70-79. [3] k.d. bierstedt, j. bonet, j. taskinen, associated weights and spaces of holomorphic functions, studia math. 127 (1998), 137-168. [4] k.d. bierstedt, r. meise, w.h. summers, a projective description of weighted inductive limits, trans. amer. math. soc. 272 (1982), no. 1, 107-160. [5] j. bonet, p. domanski, m. lindström, essential norm and weak compactness of composition operators on weighted banach spaces of analytic functions, canad. math. bull. 42, no. 2, (1999), 139-148. [6] j. bonet, p. domański, m. lindström, j. taskinen, composition operators between weighted banach spaces of analytic functions, j. austral. math. soc. (serie a) 64 (1998), 101-118. 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[25] a.l. shields, d.l. williams, bounded projections, duality, and multipliers in spaces of harmonic functions, j. reine angew. math. 299/300 (1978), 256-279. 52 elke wolf cubo 15, 2 (2013) [26] a.l. shields, d.l. williams, bounded projections and the growth of harmonic conjugates in the disc, michigan math. j. 29 (1982), 3-25. [27] p. wojtaszczyk, banach spaces for analysts, cambridge, 1991. () cubo a mathematical journal vol.17, no¯ 01, (85–97). march 2015 measure of noncompactness on lp(rn) and applications a. aghajani school of mathematics, iran university of science and technology, narmak, tehran 16844-13114, iran. aghajani@iust.ac.ir d. o , regan school of mathematics, statistics and applied mathematics, national university of ireland, galway, ireland. nonlinear analysis and applied mathematics (naam), department of mathematics, king abdulaziz university, jeddah, saudi arabia. donal.oregan@nuigalway.ie a. shole haghighi school of mathematics, iran university of science and technology, narmak, tehran 16844-13114, iran. a.sholehaghighi@kiau.ac.ir abstract in this paper we define a new measure of noncompactness on lp(rn) (1 ≤ p < ∞) and study its properties. as an application we study the existence of solutions for a class of nonlinear functional integral equations using darbo’s fixed point theorem associated with this new measure of noncompactness. resumen en este art́ıculo definimos una nueva medida de no-compacidad sobre lp(rn) (1 ≤ p < ∞) y estudiamos sus propiedades. como aplicación, estudiamos la existencia de soluciones para una clase de ecuaciones integrales funcionales no lineales usando el teorema de punto fijo de darbo asociado a esta nueva medida de no-compacidad. keywords and phrases: measure of noncompactness, darbo’s fixed point theorem, fixed point. 2010 ams mathematics subject classification: 47h08, 47h10. 86 a. aghajani, d. o , regan & a. shole haghighi cubo 17, 1 (2015) 1 introduction measures of noncompactness and darbo’s fixed point theorem play major roles in fixed point theory and their applications. measures of noncompactness were introduced by kuratowski [19]. in 1955, darbo presented a fixed point theorem [12], using this notion. this result was used to establish the existence and behavior of solutions in c[a,b], bc(r+) and bc(r+ × r+) to many classes of integral equations; see [1, 2, 3, 4, 6, 9, 10, 16, 17] and the references cited therein. when one seeks solutions in unbounded domains there are particular difficulties. the aim of this paper is to construct a regular measure of noncompactness on the space lp(rn) (1 ≤ p < ∞) and investigate the existence of solutions of a particular nonlinear functional integral equation. let r+ = [0, + ∞) and (e,‖.‖) be a banach space. the symbols x and convx stand for the closure and closed convex hull of a subset x of e, respectively. now me denotes the family of all nonempty and bounded subsets of e and ne denotes the family of all nonempty and relatively compact subsets. definition 1.1. a mapping µ : me −→ r+ is said to be a measure of noncompactness in e if it satisfies the following conditions: 1◦ the family kerµ = {x ∈ me : µ(x) = 0} is nonempty and kerµ ⊆ ne. 2◦ x ⊂ y =⇒ µ(x) ≤ µ(y). 3◦ µ(x) = µ(x). 4◦ µ(convx) = µ(x). 5◦ µ(λx + (1 − λ)y) ≤ λµ(x) + (1 − λ)µ(y) for λ ∈ [0,1]. 6◦ if {xn} is a sequence of closed sets from me such that xn+1 ⊂ xn for n = 1,2, · · · and if lim n→∞ µ(xn) = 0 then x∞ = ∩ ∞ n=1xn 6= ∅. we say that a measure of noncompactness is regular [7] if it additionally satisfies the following conditions: 7◦ µ(x ∪ y) = max{µ(x),µ(y)}. 8◦ µ(x + y) ≤ µ(x) + µ(y). 9◦ µ(λx) = |λ|µ(x) for λ ∈ r. 10◦ kerµ = ne. the kuratowski and hausdorff measures of noncompactness have all the above properties (see [5, 7]). the following darbo’s fixed point theorem will be needed in section 3. cubo 17, 1 (2015) measure of noncompactness on lp(rn) and applications 87 theorem 1.2. [12] let ω be a nonempty, bounded, closed and convex subset of a banach space e and let f : ω −→ ω be a continuous mapping such that there exists a constant k ∈ [0,1) with the property µ(fx) ≤ kµ(x) (1) for any nonempty subset x of ω. then f has a fixed point in the set ω. integral equations of urysohn type in the space of lebesgue integrable functions on bounded and unbounded intervals and the concept of weak measure of noncompactness on l1(r+) was studied in [8, 13, 14]. in section 2, we define a new measure of noncompactness on lp(rn) and study its properties. in section 3, using the obtained results in section 2, we investigate the problem of existence of solutions for a class of nonlinear integral equations. 2 main results let lp(u) (u ⊆ rn) denote the space of lebesgue integrable functions on u with the standard norm ‖x‖lp(u) = ( ∫ u |x(t)|pdt ) 1 p . before introducing the new measures of noncompactness on lp(rn), we need to characterize the compact subsets of lp(rn). theorem 2.1. [11, 18] let f be a bounded set in lp(rn) with 1 ≤ p < ∞. the closure of f in lp(rn) is compact if and only if lim h−→0 ‖τhf − f‖lp(rn) = 0 uniformly in f ∈ f, (2) where τhf(x) = f(x+h) for all x,h ∈ r n. also for ǫ > 0 there is a bounded and measurable subset ω ⊂ rn such that ‖f‖lp(rn\ω) < ǫ for all f ∈ f. (3) now, we are ready to define a new measure of noncompactness on lp(rn). theorem 2.2. suppose 1 ≤ p < ∞ and x is a bounded subset of lp(rn). for x ∈ x and ǫ > 0 let ωt (x,ǫ) = sup{‖τhx − x‖lp(bt ) : ‖h‖rn < ǫ}, ωt (x,ǫ) = sup{ωt(x,ǫ) : x ∈ x}, ωt (x) = lim ǫ→0 ωt (x,ǫ), ω(x) = lim t→∞ ωt (x), d(x) = lim t→∞ sup{‖x‖lp(rn\bt ) : x ∈ x}, 88 a. aghajani, d. o , regan & a. shole haghighi cubo 17, 1 (2015) where bt = {a ∈ r n : ‖a‖rn ≤ t}. then ω0 : mlp(rn) −→ r given by ω0(x) = ω(x) + d(x) (4) defines a measure of noncompactness on lp(rn). proof. first we show that 1◦ holds. take x ∈ mlp(rn) such that ω0(x) = 0. let η > 0 be arbitrary. since ω0(x) = 0, then limt→∞ limǫ→0 ω t (x,ǫ) = 0 and thus, there exist δ > 0 and t > 0 such that ωt (x,δ) < η implies that ‖τhx − x‖lp(bt ) < η for all x ∈ x and h ∈ r n such that ‖h‖rn < δ. since η > 0 was arbitrary, we get lim h→0 ‖τhx − x‖lp(rn) = lim h→0 lim t→∞ ‖τhx − x‖lp(bt ) = 0 uniformly in x ∈ x. again, keeping in mind that ω0(x) = 0 we have lim t→∞ sup{‖x‖lp(rn\bt ) : x ∈ x} = 0 and so for ε > 0 there exists t > 0 such that ‖x‖lp(rn\bt ) < ǫ for all x ∈ x. thus, from theorem 2.1 we infer that the closure of x in lp(rn) is compact and kerω0 ⊆ ne. the proof of 2◦ is clear. now, suppose that x ∈ mlp(rn) and (xn) ⊂ x such that xn → x ∈ x in lp(rn) . from the definition of ωt (x,ǫ) we have ‖τhxn − xn‖lp(bt ) ≤ ω t (x,ǫ) for any n ∈ n, t > 0 and ‖h‖rn < ǫ. letting n → ∞ we get ‖τhx − x‖lp(bt ) ≤ ω t (x,ǫ) for any ‖h‖rn < ǫ and t > 0 , hence lim t→∞ lim ǫ→0 ωt (x,ǫ) ≤ lim t→∞ lim ǫ→0 ωt (x,ǫ), implies that ω(x) ≤ ω(x). (5) similarly, we can show that d(x) ≤ d(x) so from (5) and 2◦ we get ω0(x) = ω0(x), so ω0 satisfies condition 3◦ of definition 1.1. the proof of conditions 4◦ and 5◦ can be carried out similarly by using the inequality ‖λx + (1 − λ)y‖lp(bt ) ≤ λ‖x‖lp(bt ) + (1 − λ)‖y‖lp(bt ). to prove 6◦, suppose that {xn} is a sequence of closed and nonempty sets from me such that xn+1 ⊂ xn for n = 1,2, · · · , and lim n→∞ ω0(xn) = 0. now for any n ∈ n take an xn ∈ xn and set f = {xn}. we claim f is a compact set in l p(rn). to prove the claim, we need to check conditions (2) and (3) of theorem 2.1. let ε > 0 be fixed. since lim n→∞ ω0(xn) = 0 there exists k ∈ n such that ω0(xk) < ε. hence, we can find δ1 > 0 and t1 > 0 such that ωt1(xk,δ1) < ε, cubo 17, 1 (2015) measure of noncompactness on lp(rn) and applications 89 and sup{‖x‖lp(rn\bt1 ) : x ∈ xk} < ε. thus, for all n ≥ k and ‖h‖rn < δ1 we get ‖τhxn − xn‖lp(rn) ≤ ‖τhxn − xn‖lp(bt1) + ‖τhxn − xn‖lp(rn\bt1) ≤ ‖τhxn − xn‖lp(bt1) + 2‖xn‖lp(rn\bt1 ) < 3ε and ‖xn‖lp(rn\bt1 ) < ε. (6) the set {x1,x2, . . . ,xk−1} is compact, hence there exists δ2 > 0 such that ‖τhxn − xn‖lp(rn) < ε (7) for all n = 1,2, . . . ,k and ‖h‖rn < δ2, and there exists t2 > 0 such that ‖xn‖lp(rn\t2) < ε (8) for all n = 1,2, . . . ,k. therefore by (6) and (7) we obtain ‖τhxn − xn‖lp(rn) < 3ε for all n ∈ n and ‖h‖ < min{δ1,δ2}, and from (6), (8) we get ‖xn‖lp(rn\bt ) < ε (9) for all n ∈ n, where t = max{t1,t2}. thus all the hypotheses of theorem 2.1 are satisfied and so the claim is proved. hence there exist a subsequence {xnj} and x0 ∈ l p(rn) such that xnj → x0, and since xn ∈ xn, xn+1 ⊂ xn and xn is closed for all n ∈ n we get x0 ∈ ∞ ⋂ n=1 xn = x∞, and this finishes the proof of the theorem. ✷ now, we study the regularity of ω0. theorem 2.3. the measure of noncompactness ω0 defined in theorem 2.1 is regular. proof. suppose that x,y ∈ mlp(rn). since for all ε > 0, λ > 0 and t > 0 we have ωt (x ∪ y,ε) ≤ max{ωt (x,ε),ωt (y,ε)} , ωt (x + y,ε) ≤ ωt (x,ε) + ωt (y,ε), ωt (λx,ε) ≤ λωt (x,ε) 90 a. aghajani, d. o , regan & a. shole haghighi cubo 17, 1 (2015) and sup x∈x∪y ‖x‖lp(rn\bt ) ≤ max{sup x∈x ‖x‖lp(rn\bt ),sup x∈y ‖x‖lp(rn\bt )}, sup x∈x+y ‖x‖lp(rn\bt ) ≤ sup x∈x ‖x‖lp(rn\bt ) + sup x∈y ‖x‖lp(rn\bt ), sup x∈λx ‖x‖lp(rn\bt ) ≤ λ sup x∈x ‖x‖lp(rn\bt ), then the hypotheses 7◦, 8◦ and 9◦ hold. to show that 10◦ holds, suppose that x ∈ nlp(rn). thus, the closure of x in lp(rn) is compact and hence from theorem 2.1, for any ǫ > 0 there exists t > 0 such that ‖x‖lp(rn\bt ) < ǫ for all x ∈ x and also limh−→0 ‖τhx − x‖lp(rn) = 0 uniformly in x ∈ x. from the first conclusion, there exists δ > 0 such that ‖τhx − x‖lp(rn) < ǫ for any ‖h‖rn < δ. then for all x ∈ x we have ωt (x,δ) = sup{‖τhx − x‖lp(bt ) : ‖h‖rn < δ} ≤ ǫ. therefore, ωt (x,δ) = sup{‖ω(x,δ) : x ∈ x} ≤ ǫ, which proves lim t→∞ lim δ→0 ω(x,δ) = 0 (10) and lim t→∞ sup{‖x‖lp(rn\bt ) : x ∈ x} = 0. (11) now from (10) and (11) condition 10◦ holds. ✷ theorem 2.4. let q = {x ∈ lp(rn) : ‖x‖lp(rn) ≤ 1}. then ω0(q) = 3 proof. indeed, we have ‖τhx − x‖lp(rn) ≤ ‖τhx‖lp(rn) + ‖x‖lp(rn) ≤ 2 and ‖x‖lp(rn\bt ) ≤ ‖x‖lp(rn) ≤ 1 for all x ∈ q, h ∈ rn and t > 0. also for any ǫ > 0, t > 0 and x ∈ q we have ωt (x,ǫ) = sup{‖τhx − x‖lp(bt ) : ‖h‖ < ǫ} ≤ 2. therefore we obtain ω0(q) ≤ 3. now we prove that ω0(q) ≥ 3. for any k ∈ n there exists ek ⊂ r n such that m(ek) = 1 2k (m is the lebesgue measure on rn), diam(ek) ≤ 1 k , ek ∩bk = ∅ and ek ⊂ b2k. define fk : r n −→ r by fk(x) = { (2k) 1 p x ∈ ek 0 otherwise. (12) it is easy to verify that ‖fk‖lp(rn) = 1, ‖τ1 k fk − fk‖lp(b2k) = 2 and ‖fk‖lp(rn\bk) = 1 for all k ∈ n. thus, we get ω0(q) ≥ 3. this completes the proof. ✷ cubo 17, 1 (2015) measure of noncompactness on lp(rn) and applications 91 3 application in this section we show the applicability of our results. definition 3.1. we say that a function f : rn ×rm −→ r satisfies the carathéodory conditions if the function f(.,u) is measurable for any u ∈ rm and the function f(x,.) is continuous for almost all x ∈ rn. theorem 3.2. assume that the following conditions are satisfied: (i) f : rn × r −→ r satisfies the carathéodory conditions, and there exists a constant k ∈ [0,1) and a ∈ lp(rn) such that |f(x,u) − f(y,v)| ≤ |a(x) − a(y)| + k|u − v|, (13) for any u,v ∈ r and almost all x,y ∈ rn. (ii) f(.,0) ∈ lp(rn). (iii) k : rn × rn −→ r satisfies the carathéodory conditions and there exist g1,g2 ∈ lp(rn) and g ∈ lq(rn) ( 1 p + 1 q = 1) such that |k(x,y)| ≤ g(y)g1(x) for all x,y ∈ r n and |k(x1,y) − k(x2,y)| ≤ g(y)|g2(x1) − g2(x2)|. (14) (iv) the operator q acts continuously from the space lp(rn) into itself and there exists a nondecreasing function ψ : r+ −→ r+ such that ‖qu‖lp(rn) ≤ ψ(‖u‖lp(rn)) (15) for any u ∈ lp(rn). (v) there exists a positive solution r0 to the inequality kr + ψ(r)‖k‖1 + ‖f(.,0)‖lp(rn) ≤ r (16) where (ku)(t) = ∫ rn k(x,y)u(y)dy and ‖k‖1 = sup{‖ku‖lp(rn) : ‖u‖lp(rn) ≤ 1}. then the functional integral equation u(x) = f(x,u(x)) + ∫ rn k(x,y)(qu)(y)dy (17) has at least one solution in the space lp(rn). 92 a. aghajani, d. o , regan & a. shole haghighi cubo 17, 1 (2015) remark 3.3. the linear fredholm integral operator k : lp(rn) → lp(rn) is a continuous operator and ‖k‖1 < ∞. proof. first of all we define the operator f : lp(rn) → lp(rn) by f(u)(x) = f(x,u(x)) + ∫ rn k(x,y)(qu)(y)dy. (18) now fu is measurable for any u ∈ lp(rn). now we prove that fu ∈ lp(rn) for any u ∈ lp(rn). using conditions (i)-(iv), we have the following inequality |f(u)(x)| ≤ |f(x,u) − f(x,0)| + |f(x,0)| + | ∫ rn k(x,y)(qu)(y)ds| a.e. x ∈ rn. thus ‖fu‖lp(rn) ≤ k‖u‖lp(rn) + ‖f(.,0)‖lp(rn) + ‖k‖1ψ(‖u‖lp(rn)). (19) hence f(u) ∈ lp(rn) and f is well-defined and also from (19) we have f(br0) ⊆ br0, where r0 is the constant appearing in assumption (v). also, f is continuous in lp(rn), because f(t, .), k and q are continuous for a.e. x ∈ rn. now we show that for any nonempty set x ⊂ br0 we have ω0(f(x)) ≤ kω0(x). to do so, we fix arbitrary t > 0 and ε > 0. let us choose u ∈ x and for x,h ∈ bt with ‖h‖rn ≤ ǫ, we have |(fu)(x) − (fu)(x + h)| ≤ ∣ ∣ ∣ f(x,u(x)) + ∫ rn k(x,y)(qu)(y)dy − f(x + h,u(x + h)) + ∫ rn k(x + h,y)(qu)(y)dy ∣ ∣ ∣ ≤ |f(x,u(x)) − f(x + h,u(x))| + |f(x + h,u(x)) − f(x + h,u(x + h))| + | ∫ rn k(x,y)(qu)(y)dy − ∫ rn k(x + h,y)(qu)(y)dy| ≤ |a(x) − a(x + h)| + k|u(x) − u(x + h)| + ∫ rn |k(x,y) − k(x + h,y)||qu(y)|dy. cubo 17, 1 (2015) measure of noncompactness on lp(rn) and applications 93 therefore ( ∫ bt |(fu)(x + h) − (fu)(x)|pdt ) 1 p ≤ ( ∫ bt |a(x) − a(x + h)|pdt ) 1 p + k ( ∫ bt |u(x) − u(x + h)|pdt ) 1 p + ( ∫ bt ∣ ∣ ∫ rn |k(x,y) − k(x + h,y)||qu(y)|dy ∣ ∣ p dx ) 1 p ≤ ( ∫ bt |a(x) − a(x + h)|pdx ) 1 p + k ( ∫ bt |u(x) − u(x + h)|pdx ) 1 p + ( ∫ bt ( ∫ rn |k(x,y) − k(x + h,y)|qdy ) p q dx ) 1 p ‖qu‖lp(rn) ≤ ( ∫ bt |a(x) − a(x + h)|pdx ) 1 p + k ( ∫ bt |u(x) − u(x + h)|pdx ) 1 p + ( ∫ bt ( ∫ rn |g2(x) − g2(x + h)| q |g(y)|qdy ) p q dx ) 1 p ‖qu‖lp(rn) ≤ ‖τha − a‖lp(bt ) + k‖τhu − u‖lp(bt ) + ( ∫ bt |g2(x) − g2(x + h)| pdx ) 1 p ‖g‖lq(rn)‖qu‖lp(rn) ≤ ωt (a,ǫ) + kωt (u,ǫ) + ‖qu‖lp(rn)‖g‖lq(rn)ω t (g2,ǫ). thus we obtain ωt (fx,ǫ) ≤ ωt (a,ǫ) + kωt (x,ǫ) + ψ(r0)‖g‖lq(rn)ω t (g2,ǫ). also we have ωt (a,ǫ),ωt (g2,ǫ) → 0 as ǫ → 0. then we obtain ω(fx) ≤ kω(x). (20) next, let us fix an arbitrary number t > 0. then, taking into account our hypotheses, for an arbitrary function u ∈ x we have ( ∫ rn\bt |(fu)(x)|pdx ) 1 p ≤ ( ∫ rn\bt ∣ ∣ ∣ f(x,u(x)) + ∫ rn k(x,y)qu(y)dy ∣ ∣ ∣ p dt ) 1 p ≤ ( ∫ rn\bt |f(x,u(x)) − f(x,0)|pdx ) 1 p + ( ∫ (rn\bt |f(t,0)|pdx ) 1 p + ( ∫ rn\bt ∣ ∣ ∣ ∫ rn k(x,y)qu(y)dy ∣ ∣ ∣ p dx ) 1 p ≤ k ( ∫ rn\bt |u(x)|pdx ) 1 p + ( ∫ rn\bt |f(x,0)|pdx ) 1 p + ( ∫ rn\bt ( ∫ ∞ 0 |k(x,y)|qdy ) p q dx ) 1 p ‖qu‖lp(rn) ≤ k‖u‖lp(rn\bt ) + ‖f(.,0)‖lp(rn\bt ) + ‖g1‖lp(rn\bt )‖g‖lq(rn)ψ(‖u‖lp(rn)). also we have ‖f(.,0)‖lp(rn\bt ),‖g1‖lp(rn\bt ) −→ 0 94 a. aghajani, d. o , regan & a. shole haghighi cubo 17, 1 (2015) as t → ∞ and hence we deduce that d(fx) ≤ kd(x). (21) consequently from (20) and (21) we infer ω0(fx) ≤ kω0(x). (22) from (22) and theorem 1.2 we obtain that the operator f has a fixed-point u in br0 and thus the functional integral equation (17) has at least one solution in lp(rn). ✷ in the example below we will use the following well known result. theorem 3.4. [15] let ω ⊆ rn be a measure spaces and suppose k : ω × ω −→ r is an ω × ωmeasurable function for which there is constant c > 0 such that ∫ ω |k(x,y)|dx ≤ c for a.e. y ∈ ω and ∫ ω |k(x,y)|dy ≤ c for a.e. x ∈ ω. if k : lp(ω) −→ lp(ω) is defined by (kf)(x) = ∫ ω k(x,y)f(y)dy, (23) then k is a bounded and continuous operator and ‖k‖1 ≤ c. example 3.5. consider the integral equation u(x) = cosu(x) ‖x‖ + 2 + ∫ r3 e−(|x2|+|y2|+|y3|+1) (|x1| + 3) 2(|y1| + 2) 2(1 + |x3| 2) e−|u(y)|u(y)dy, (24) where x = (x1,x2,x3) ∈ r 3 and ‖x‖ is the euclidean norm. we study the solvability of the integral equation (24) on the space lp(rn) for p > 3. let f(x,u) = cosu ‖x‖ + 2 and note it satisfies hypothesis (i) with a(x) = 1 ‖x‖ + 2 and k = 1 2 . indeed, we have |f(x,u) − f(y,v)| = | cosu ‖x‖ + 2 − cosv ‖y‖ + 2 | ≤ | 1 ‖x‖ + 2 − 1 ‖y‖ + 2 || cosu| + 1 ‖y‖ + 2 | cosu − cosv| ≤ | 1 ‖x‖ + 2 − 1 ‖y‖ + 2 | + 1 2 |u − v| = |a(x) − a(y)| + k|u − v|. cubo 17, 1 (2015) measure of noncompactness on lp(rn) and applications 95 also, it is easily seen that f(.,0) satisfies assumption (ii) and ‖f(.,0)‖ p lp(r3) = ∫ r3 | 1 ‖x‖ + 2 |pdx = ∫2π 0 ∫π 0 ∫ ∞ 0 r2 sinϕ (r + 2)p drdϕdθ ≤ 4π ∫ ∞ 0 1 (r + 2)p−2 dr = 4π (p − 3)2p−3 for all p > 3. thus , we have ‖f(.,0)‖lp(r3) ≤ ( 4π p − 3 ) 1 p . moreover, taking k(x,y) = e−(|x2|+|y2|+|y3|+1) (|x1| + 3)2(|y1| + 2)2(1 + |x3|2) , g1(x) = g2(x) = e−|x2| (|x1| + 3) 2(1 + |x3| 2) and g(x) = e−(|x2|+|x3|) (|x1| + 2) 2 , we see that g1,g2,g ∈ l p(r3) for all 1 ≤ p < ∞ and k satisfies hypothesis (iii). also, we have ∫ r3 |k(x,y)|dx = ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ −∞ e−(|x2|+|y2|+|y3|+1) (|x1| + 3) 2(|y1| + 2) 2(1 + x2 3 ) dx1dx2dx3 ≤ π 3e , ∫ r3 |k(x,y)|dy = ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ −∞ e−(|x2|+|y2|+|y3|+1) (|x1| + 3) 2(|y1| + 2) 2(1 + |x3| 2) dy1dy2dy3 ≤ 4 9e and thus from theorem 3.2, ||k||1 ≤ π 3e . furthermore, q(u)(x) = e−|u(x)|u(x) satisfies hypothesis (iv) with ψ(t) = t. finally, the inequality from assumption (v), has the form kr + ψ(r)‖k‖1 + ‖f(.,0)‖lp(r3) = 1 2 r + π 3e r + ( 4π p − 3 ) 1 p = ( 1 2 + π 3e )r + ( 4π p − 3 ) 1 p ≤ r thus, for the number r0 we can take r0 = ( 4π p − 3 ) 1 p × 6e 3e − 2π . consequently, all the assumptions of theorem 3.2 are satisfied and thus equation (24) has at least one solution in the space lp(r3) if p > 3. received: december 2014. accepted: january 2015. references [1] r. agarwal, m. meehan, d. o’regan, fixed point theory and applications, cambridge university press 2004. 96 a. aghajani, d. o , regan & a. shole haghighi cubo 17, 1 (2015) [2] a. aghajani, j. banás, y. jalilian, existence of solution for a class nonlinear volterra sigular integral, comput. math. appl. 62 (2011), 1215-1227. 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[17] b.c. dhage, s.s. bellale, local asymptotic stability for nonlinear quadratic functional integral equations, electron. j. qual. theory differ. equ. 10 (2008), 1-13. cubo 17, 1 (2015) measure of noncompactness on lp(rn) and applications 97 [18] h. ha-olsen, h. holden, the kolmogorov-riesz compactness theorem , expo. math. 28 (2010), 385–394 [19] k. kuratowski, sur les espaces complets. fund. math. 15 (1930), 301-309. introduction main results application cubo a mathematical journal vol.14, no¯ 01, (49–54). march 2012 units in abelian group algebras over direct products of indecomposable rings peter danchev 13, general kutuzov str. 4003 plovdiv, bulgaria, email: pvdanchev@yahoo.com abstract let r be a commutative unitary ring of prime characteristic p which is a direct product of indecomposable subrings and let g be a multiplicative abelian group such that g0/gp is finite. we characterize the isomorphism class of the unit group u(rg) of the group algebra rg. this strengthens recent results due to mollov-nachev (commun. algebra, 2006) and danchev (studia babes bolyai mat., 2011). resumen sea r un anillo conmutativo y unitario de caracteŕıstica prima p, que es producto directo de subanillos indescomponibles y sea g un grupo multiplicativo y abeliano tal que g0/gp es finito. caracterizamos las clases de isomorfismo del grupo unitario u(rg) del álgebra del grupo rg. estos fuertes y recientes resultados se deben a mollov-nachev (commun. algebra, 2006) and danchev (studia babes bolyai mat., 2011). keywords and phrases: groups, rings, group rings, indecomposable rings, units, direct decompositions, isomorphisms. 2010 ams mathematics subject classification: 16s34, 16u60, 20k21. 50 peter danchev cubo 14, 1 (2012) 1 introduction throughout the current paper, suppose r is a commutative unitary (i.e., with identity) ring of prime characteristic p and suppose g is a multiplicative abelian group as is the custom when discussing group rings. for such r and g, we denote by rg the group ring of g over r with unit group u(rg), normalized subgroup v(rg) of units (with augmentation 1) and its idempotent subgroup id(rg). note that the decomposition u(rg) = v(rg)×u(r) is valid, where u(r) is the unit group of r. as usual, g0 is the maximal torsion subgroup of g with p-torsion component gp, and s(rg) = vp(rg) is the p-torsion component of v(rg). besides, for any natural number n, ζn denotes the primitive nth root of unity and r[ζn] is the free r-module, generated algebraically as a ring by ζn, with dimension [r[ζn] : r]. as it is well-known, a ring is said to be indecomposable if it cannot be decomposed into a direct sum of two or more non-trivial subrings (ideals), that is, this ring possesses only the trivial idempotents 0, 1. the algebraic structures of v(rg) and u(rg) have been very intensively explored in the past twenty years (see, e.g., [k]). in this aspect, some isomorphism description results were obtained in [da] and [mn], respectively. the purpose of this work is to improve considerably one of the central achievements in the second citation by giving a more direct and conceptual proof (some of parts of the proof of the corresponding result in [mn] are unnecessary intricated). likewise, we generalize the main result in [dg] to a ring which is an arbitrary direct product of indecomposable rings. notice that our method suggested below gives a new perspective for establishing some other results of this form, because it leads the general case to the p-mixed one. ii. main results as noted above, mollov and nachev obtained in ([mn], theorem 5.8) the following statement. theorem (2006). let r be a commutative indecomposable ring with identity of prime characteristic p and let g be a splitting abelian group. suppose that g0/gp is a finite group of exponent n and n ∈ u(r). then u(rg) ∼= ∐ d/n ∐ λ(d) u(r[ζd]) × ∐ b g/g0 × ∐ d/n ∐ λ(d) s(r[ζd](gp × g/g0)) where λ(d) = (g0/gp)(d) [r[ζd]:r] , with (g0/gp)(d) the number of elements of g0/gp of order d, and b = ∑ d/n λ(d). note that since char(r) = p is a prime integer, it is self-evident that exp(g0/gp) inverts in r, so that the condition n ∈ u(r) is always fulfilled and hence it is a superfluously stated in the theorem. cubo 14, 1 (2012) units in abelian group algebras over direct products . . . 51 in [dg] we dropped the limitation that g is a splitting group. specifically, we list the following: theorem (2011). suppose r is an indecomposable ring of char(r) = p and g is a group for which g0/gp is finite. then the following isomorphism is true: (*) u(rg) ∼= ∐ d/exp(g0/gp) ∐ a(d) [u(r[ζd]) × [(g/ ∐ q 6=p gq)vp(r[ζd](g/ ∐ q6=p gq))]] where a(d) = |{g∈g0/gp:order(g)=d}| [r[ζd]:r] . in particular: (1) if g is p-splitting, then u(rg) ∼= ∐ d/exp(g0/gp) ∐ a(d) [u(r[ζd]) × vp(r[ζd](g/ ∐ q 6=p gq))] × ∐ ∑ d/exp(g0/gp) a(d) g/g0. (2) if gp is a direct sum of cyclic groups, then u(rg) ∼= ∐ d/exp(g0/gp) ∐ a(d) [u(r[ζd]) × (vp(r[ζd](g/ ∐ q6=p gq))/(g/ ∐ q 6=p gq)p)]× × ∐ ∑ d/exp(g0/gp) a(d) g/ ∐ q 6=p gq. moreover, the quotient vp(r[ζd](g/ ∐ q 6=p gq))/(g/ ∐ q 6=p gq)p) is a direct sum of cyclic groups by [d] and can be characterized via the ulm-kaplansky invariants calculated in [df]. before stating and proving our chief attainment, we need two more preliminaries. proposition 1. let r = ∏ i∈i ri be a direct product of subrings ri where i is an index set, and f is a finite abelian group. then the following isomorphism holds: rf ∼= ∏ i∈i rif. 52 peter danchev cubo 14, 1 (2012) proof. it is straightforward and we leave it to the reader. 4 lemma 2. suppose g0/gp is bounded. then the following decomposition is true: g = m × b where m ∼= g/ ∐ q 6=p gq is p-mixed and b ∼= ∐ q 6=p gq ∼= g0/gp is bounded. proof. since ∐ q 6=p gq is bounded and is pure in g0 as its direct factor, whence pure in g, it follows that ∐ q 6=p gq is a direct factor of g as well. denoting b = ∐ q6=p gq, one may write g = b × m where m ∼= g/b. it is obvious that m is p-mixed, i.e., m0 = mp. 4 so, we come to our main achievement. theorem 3. let r be a ring of prime characteristic p which is a direct product of indecomposable rings ri for some index set i, and let g be an abelian group such that g0/gp is finite. then the following isomorphism formula is fulfilled: (*) u(rg) ∼= [ ∐ i∈i u(ri(g0/gp))] × [id(lm)vp(lm)] for some commutative unitary ring l of prime characteristic p which is a direct product of indecomposable rings, and where, for all indices i ∈ i, u(ri(g0/gp)) ∼= ∐ d/exp(g0/gp) ∐ ai(d) u(ri[ζd]) with ai(d) = |{g∈g0/gp:order(g)=d}| [ri[ζd]:ri] . in particular, the maximal divisible subgroup du(rg) of u(rg) is completely described up to isomorphism. proof. according to lemma 2 one may write g = f × m where f ∼= ∐ q 6=p gq is finite and m is p-mixed. thus rg = (rf)m = lm where we put rf = l. therefore, u(rg) = u(lm) = u(rf) × v(lm). concerning v(lm) we may write v(lm) = id(lm)vp(lm) (see, e.g., [dd] or [de]). on the other hand, owing to proposition 1, l = rf = ( ∏ i∈i ri)f ∼= ∏ i∈i rif where each ri is cubo 14, 1 (2012) units in abelian group algebras over direct products . . . 53 an indecomposable ring of characteristic p. furthermore, since f is finite of exponent that inverts in r, and hence it inverts in each ri, appealing to theorem 4.4 and remark 4.5 of [mn], every rif is a finite direct sum of indecomposable subrings. consequently, l is a commutative unitary ring of prime characteristic p which can be interpreted as a ring that is a direct product of indecomposable subrings. moreover, u(rf) ∼= ∐ i∈i u(rif), where u(rif) has an explicit description for any index i. thus formula (*) is deduced. finally, observe that du(rg) = du(rf)×dv(lm) ∼= ∐ i∈i du(rif)×dv(lm). since u(rif), and hence du(rif), is already characterized above, and dv(lm) is classified in [dd] and [de], we infer that the same can be said of du(rg). 4 remark. the proof of theorem 2.7 from [mmn] contains a gap and so it is uncomplete. in fact, the authors claimed that they will assume that the splitting group is p-mixed. the reason is that the k-algebras isomorphism kg ∼= kh yields that k(g/ ∐ q 6=p gq) ∼= k(h/ ∐ q6=p hq) whenever k is a field of char(k) = p. but they need to show that g being splitting ensures that so is g/ ∐ q 6=p gq. however, this was already done in [db]. we close the work with the following problem. conjecture. suppose r is an indecomposable ring and g is a finite group of exponent which inverts in r. then rg ∼= rh for some group h if, and only if, h is finite with the same exponent as that of g and rgp ∼= rhp for each prime number p. notice that the sufficiency is trivial, because g and h being both bounded implies that g = ∐ p gp and h = ∐ p hp, whence rg ∼= ⊗rrgp and rh ∼= ⊗rrhp. thus rgp ∼= rhp forces that rg ∼= rh, as desired. received: october 2010. revised: march 2011. references [d] p. v. danchev, commutative group algebras of σ-summable abelian groups, proc. amer. math. soc. (9) 125 (1997), 2559-2564. [da] p. v. danchev, normed units in abelian group rings, glasgow math. j. (3) 43 (2001), 365-373. [db] p. v. danchev, notes on the isomorphism and splitting problems for commutative modular group algebras, cubo math. j. (1) 9 (2007), 39-45. [dc] p. v. danchev, warfield invariants in commutative group rings, j. algebra appl. (6) 8 (2009), 829836. 54 peter danchev cubo 14, 1 (2012) [dd] p. v. danchev, maximal divisible subgroups in p-mixed modular abelian group rings, commun. algebra (6) 39 (2011), 2210-2215. [de] p. v. danchev, maximal divisible subgroups in modular group rings of p-mixed abelian groups, bull. braz. math. soc. (1) 41 (2010), 63-72. [df] p. v. danchev, ulm-kaplansky invariants in commutative modular group rings, j. algebra number theory academia (2) 1 (2011), 127-134. [dg] p. v. danchev, units in abelian group algebras over indecomposable rings, studia “babes bolyai” mat. (4) 56 (2011), 3-6. [k] g. karpilovsky, units of commutative group algebras, expo. math. 8 (1990), 247-287. [m] w. l. may, group algebras over finitely generated rings, j. algebra 39 (1976), 483-511. [mmn] w. l. may, t. zh. mollov, n. a. nachev, isomorphism of modular group algebras of p-mixed abelian groups, commun. algebra 38 (2010), 1988-1999. [mn] t. zh. mollov and n. a. nachev, unit groups of commutative group rings, commun. algebra 34 (2006), 3835-3857. cubo a mathematical journal vol.16, no¯ 01, (01–07). march 2014 stationary boltzmann equation and the nonlinear alternative of leray-schauder type rafael galeano andrades, pedro ortega palencia and john fredys cantillo palacio institute of applied mathematics, universidad de cartagena, cartagena, colombia. rgaleanoa@unicartagena.edu.co, portegap@unicartagena.edu.co, jcantillop@unicartagena.edu.co abstract by applying a nonlinear alternative of leray-schauder type, a fixed point of an operator is found, which, in turn, comes to be a solution of stationary boltzmann equation with boundary conditions of maxwellian type. resumen aplicando una alternativa no lineal del tipo leray-schauder, se encontró un punto fijo de un operador, el cual corresponde a la solución de una ecuación de boltzmann estacionaria con condiciones de frontera del tipo maxwelliano. keywords and phrases: nonlinear alternative of leray-schauder type, fixed point, solution of stationary boltzmann equation. 2010 ams mathematics subject classification: 35q20. 2 r. galeano, p. ortega & j. cantillo cubo 16, 1 (2014) 1 introduction let us consider the banach space e = { u ∈ l1(ω × b3r(0)) : vi ∂u ∂xi ∈ l1(ω × b3r(0)) } with norm ‖u‖ e := ‖u‖ l1(ω×b 3r (0)) + ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) and we expect to find u(x, v) ≥ 0, such that { v.∇xu = q(u, u), u ∈ be(0, r) u(x, v) = m(v) = e−|v| 2 (maxwellian), u ∈ ∂b e (0, r), (r > 0) (1.1) here q(u, u)(v) = ∫ b 3r (0) ∫ |p|=1 [p · (v − z)]p[u(x, z′)u(x, v′) − u(x, z)u(x, v)]dpdz, is the collision operator, ω bounded and regular and the speeds related by the following relations: { v′ = v − [p · (v − z)]p z′ = z + [p · (v − z)]p (1.2) (z, v) and ( z′, v′) son the pre-collision and post-collision speeds, respectively. it can be noted that if z, v ∈ b r (0) in rn, then z′, v′ ∈ b 3r (0). here, the following problem will be proved. theorem 1.1. let us suppose that q(u, u) ∈ b e (u, r/2); vi ∂u ∂xi ∈ b e (0, r/2n) y vi ∂un ∂xi ∈ b e ( vi ∂u ∂xi , r∗∗ ) , r∗∗ ≥ 0, for n = 1, 2, . . . , moreover 0 < ∫ b 3r (0) ∫ |p|=1 ‖v − z‖dpdz < ∞ and 0 < ∫ b 3r (0) ∫ |p|=1 ‖v − z‖dpdv < ∞, then there exits a solution for u ∈ b e (0, r) of problem (1.1). in these stationary problems, the flows of quantities as entropy control and compactness properties are under control, but they do not imply, per se, the desired results. anyway, energy control and similar properties are available from momentum flows and mass control that can be imposed on the problem to replace entropy-bounding non-availability. there are controls based on involution of entropy dissipation.. using such tools, in the last years arkeryd l. and noury a., [2] -[3] -[4][5], have made a development focused on the results of solutions existence in the l1 context for nonlinear equations boltzmann type and also for those presenting maxwellian equilibrium. the case of perturbation on the global maxwellian equilibrium has been typically studied since the 60’s. methods of general type as hilbert spaces and contraction mapping have been used, being the pioneers [6] -[7] -[9] -[10]; in [11] exposed, generally discussed the problem of boundary value for the stationary equation, in [12] and [14] is proved the theorem for the stationary equation povzner with certain spatial boundary conditions of type maxwellian and in [13] there are applications to dynamics of fluids, we present the main result five lemmas. cubo 16, 1 (2014) stationary boltzmann equation and the nonlinear alternative . . . 3 2 development the problem (1.1), v.∇xu = q(u, u) is equivalent to u + v.∇xu = u + q(u, u), this implies that u = u + v.∇xu − q(u, u) which suggests the following operator: j(u) := u + v.∇xu − q(u, u) defined on e = { u ∈ l1(ω × b3r(0)) : vi ∂u ∂xi ∈ l1(ω × b3r(0)) } , e is a banach space with the norm ‖u‖ e := ‖u‖ l1(ω×b 3r (0)) + ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) finding fixed points of j, coincides with finding solutions of (1.1). so we will work to find fixed points, via alternative leray-schauder type, in effect: let c = b e (0, r), this is a convex and closed set in e and u := b e (0, r), open ball centered in 0 and radius r. lemma 2.1. let us suppose that q(u, u) ∈ b e (u, r/2) and vi ∂u ∂xi ∈ b e (0, r/2n), u = 1, 2, . . . , then j sends c in c. proof. let u ∈ c = b e (0, r), as j(u) := u + v.∇xu − q(u, u), then |j(u)| ≤ |u − q(u, u)| + |v.∇xu| = |u − q(u, u)| + n∑ i=1 ∣ ∣ ∣ ∣ vi ∂u ∂xi ∣ ∣ ∣ ∣ , i.e, ‖j(u)‖ l1(ω×b 3r (0)) ≤ ‖u − q(u, u)‖ l1(ω×b 3r (0)) + n∑ i=1 ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) (2.1) now: ∂j(u) ∂xi = ∂u ∂xi + ∂ ∂xi [ n∑ i=1 vi ∂u ∂xi ] + ∂q(u, u) ∂xi , then ∣ ∣ ∣ ∣ ∂j(u) ∂xi ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∂ ∂xi ( u − q(u, u) ) ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∂ ∂xi n∑ i=1 vi ∂u ∂xi ∣ ∣ ∣ ∣ ∣ , i.e., ∥ ∥ ∥ ∥ ∂j(u) ∂xi ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) ≤ ∥ ∥ ∥ ∥ ∂ ∂xi ( u − q(u, u) ) ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) + n∑ i=1 ∥ ∥ ∥ ∥ ∂ ∂xi ( vi ∂u ∂xi ) ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) . (2.2) from (2.1) and (2.2) we conclude that: ‖j(u)‖ e ≤ ‖u − q(u, u)‖ e + n∑ i=1 ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ e ≤ r 2 + r 2 = r, that is to say, j(u) ∈ c = b e (0, r). 4 r. galeano, p. ortega & j. cantillo cubo 16, 1 (2014) lemma 2.2. if un ∈ be(u, r), such that |un(x, v)| ≤ r, |u(x, v)| ≤ r, for every n, x ∈ ω, v ∈ b3r(0), moreover 0 < ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdz < ∞ y 0 < ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdv < ∞, then q(un, un) ∈ be(q(u, u), r ∗r) with r∗ = 4r max { ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdz, ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdv } . (r∗ ≥ 0) proof. by definition of q(u, u), leads to ∣ ∣ ∣ q(un, un)(v)−q(u, u)(v) ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ b 3r (0) ∫ |p|=1 [p · (v − z)]p[un(x, z ′)un(x, v ′) − un(x, z)un(x, v)]dpdz − ∫ b 3r (0) ∫ |p|=1 [p · (v − z)]p[u(x, z′)u(x, v′) − u(x, z)u(x, v)]dpdz ∣ ∣ ∣ ∣ , that is to say: ∣ ∣ ∣ q(un, un)(v)−q(u, u)(v) ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ b 3r (0) ∫ |p|=1 [p · (v − z)]p [ un(x, z ′)un(x, v ′) − u(x, z′)u(x, v′) − un(x, z)un(x, v) + u(x, z)u(x, v) ] dpdz ∣ ∣ ∣ ∣ ≤ ∫ b 3r (0) ∫ |p|=1 |p · (v − z)| ∣ ∣ ∣ un(x, z ′)un(x, v ′) − un(x, z ′)u(x, v′) + un(x, z ′)u(x, v′) − u(x, z′)u(x, v′) + u(x, z)u(x, v) − u(x, z)un(x, v) + u(x, z)un(x, v) − un(x, z)un(x, v) ∣ ∣ ∣ dpdz then ∣ ∣ ∣ q(un, un)(v)−q(u, u)(v) ∣ ∣ ∣ ≤ ∫ b 3r (0) ∫ |p|=1 |p · (v − z)| [ ∣ ∣un(x, z ′) ∣ ∣ ∣ ∣un(x, v ′) − u(x, v′) ∣ ∣ + ∣ ∣u(x, v′) ∣ ∣ ∣ ∣un(x, z ′ ) − u(x, z′) ∣ ∣ + ∣ ∣u(x, z) ∣ ∣ ∣ ∣un(x, v) − u(x, v) ∣ ∣ + ∣ ∣un(x, v) ∣ ∣ ∣ ∣un(x, z) − u(x, z) ∣ ∣ ] dpdz cubo 16, 1 (2014) stationary boltzmann equation and the nonlinear alternative . . . 5 so, ∥ ∥ ∥ q(un, un) − q(u, u) ∥ ∥ ∥ l1(ω×b 3r (0)) ≤ ∫ ω ∫ b 3r (0) ∣ ∣ ∣ q(un, un)(v) − q(u, u)(v) ∣ ∣ ∣ dxdv ≤ ∫ ω ∫ b 3r (0) ∫ b 3r (0) ∫ |p|=1 |p · (v − z)| |un(x, z ′)| |un(x, v ′) − u(x, v′)|dpdzdvdx + ∫ ω ∫ b 3r (0) ∫ b 3r (0) ∫ |p|=1 ‖v − z‖ |u(x, v′)| |un(x, z ′) − u(x, z′)|dpdzdvdx + ∫ ω ∫ b 3r (0) ∫ b 3r (0) ∫ |p|=1 ‖v − z‖ |u(x, z)| |un(x, v) − u(x, v)|dpdzdvdx + ∫ ω ∫ b 3r (0) ∫ b 3r (0) ∫ |p|=1 ‖v − z‖ |un(x, z) − u(x, z)|dpdzdvdx ≤ ∫ b 3r (0) ∫ |p|=1 r‖v − z‖dpdz ∫ ω ∫ b 3r (0) |un(x, v ′) − u(x, v′)|dvdx + ∫ b 3r (0) ∫ |p|=1 r‖v − z‖dpdv ∫ ω ∫ b 3r (0) |un(x, z ′) − u(x, z′)|dzdx + ∫ b 3r (0) ∫ |p|=1 r‖v − z‖dpdz ∫ ω ∫ b 3r (0) |un(x, v) − u(x, v)|dvdx + ∫ b 3r (0) ∫ |p|=1 r‖v − z‖dpdv ∫ ω ∫ b 3r (0) |un(x, z) − u(x, z)|dzdx making the change of variables v′ → v y z′ → z, whose jacobians are 1, then: ∥ ∥ ∥ q(un, un) − q(u, u) ∥ ∥ ∥ l1(ω×b 3r (0)) ≤ 2r [ ∫ b 3r (0) ∫ |p|=1 ‖v − z‖dpdz + ∫ b 3r (0) ∫ |p|=1 ‖v − z‖dpdv ] ‖un − u‖ l1(ω×b 3r (0)) ≤ 2r [ ∫ b 3r (0) ∫ |p|=1 ‖v − z‖dpdz + ∫ b 3r (0) ∫ |p|=1 ‖v − z‖dpdv ] ‖un − u‖e ≤ 4r max { ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdz, ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdv } ‖un − u‖e. hence q(un, un) ∈ be(q(u, u), r ∗r). lemma 2.3. if un ∈ be(0, r) ∩ be(u, r), u ∈ be(0, r), and vi ∂un ∂xi ∈ b e ( vi ∂u ∂xi , r∗∗ ) , with r∗∗ = r∗r + r + nr∗∗ ≥ 0 such that 0 < ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdz < ∞ and 0 < ∫ b 3r ∫ |p|=1 ‖(v − z)‖dpdv < ∞, then j(un) ∈ be(j(u), r ∗∗) 6 r. galeano, p. ortega & j. cantillo cubo 16, 1 (2014) proof. |j(un) − j(u)| = |un + v.∇xun − q(un, un) − u − v.∇xu + q(u, u)| ≤ |un − u| + |q(un, un) − q(u, u)| + |v.∇xun − v.∇xu|, luego: ∥ ∥ ∥ j(un) − j(u) ∥ ∥ ∥ l1(ω×b 3r (0)) ≤ ∥ ∥ ∥ un − u ∥ ∥ ∥ l1(ω×b 3r (0)) + ∥ ∥ ∥ q(un, un) − q(u, u) ∥ ∥ ∥ l1(ω×b 3r (0)) + ∥ ∥ ∥ ∥ ∥ n∑ i=1 vi ∂ ∂xi (un − u) ∥ ∥ ∥ ∥ ∥ l1(ω×b 3r (0)) (2.3) now calculating ∂j(u) ∂xi , we obtain that: ‖j(un) − j(u)‖e ≤ ‖un − u‖e + ‖q(un, un) − q(u, u)‖e + n∑ i=1 ∥ ∥ ∥ ∥ vi ∂ ∂xi (un − u) ∥ ∥ ∥ ∥ e ≤ r∗r + r + nr∗∗ = r∗∗ therefore j(un) ∈ be(j(u), r ∗∗). lemma 2.4. the operator j : u −→ c is compact. dunford-pettis criterion will be applied, see [8], in fact: i) ∫ ω ∣ ∣ ∣ j(u) ∣ ∣ ∣ du ≤ ∫ ω 2r du = 2r m(ω) ≤ 2rδ, defining ε = 2rδ, then the existence of δ, such that if m(ω) ≤ δ, then ∫ ω ∣ ∣ ∣ j(u) ∣ ∣ ∣ du ≤ ε. ii) given ε∗ > 0, exists a closed, f ⊂ ω such that if m(f) < ∞, then ∫ ω−f ∣ ∣ ∣ j(u) ∣ ∣ ∣ du ≤ 2r m(ω − f) ≤ 2rε∗, if we defined 2rε∗ ≤ ε, then ∫ ω−f ∣ ∣ ∣ j(u) ∣ ∣ ∣ du ≤ ε. lemma 2.5. for every u ∈ ∂u you have u = j(u). in fact, if u ∈ ∂u, then u = e−|v| 2 , and: j(u) = u + v.∇xu − q(u, u) = u = e −|v| 2 then for every u ∈ ∂u y λ ∈ (0, 1) must be u 6= λj(u). therefore the nonlinear alternative leray-schauder type, see [1], page 48, we conclude that there is a fixed point of the operator j a solution resulting from (1.1). cubo 16, 1 (2014) stationary boltzmann equation and the nonlinear alternative . . . 7 received: october 2012. accepted: september 2013. references [1] agarwal r., meeman m. and oregan d., fixed point theory and applications, cambridge university press, (2004). 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[14] ukay s., yang t. and zhao h., stationary solutions to the exterior problems for the boltzmann equation existence, discrete and continuous dynamical systems., vol. 23, no.152, january and february (2009). cubo a mathemati al journal vol.15, n o 03, (89�103). o tober 2013 approximate solution of fra tional integro-di�erential equation by taylor expansion and legendre wavelets methods m.h.saleh, s.m.amer, m.a.mohamed and n.s.abdelrhman mathemati s department, fa ulty of s ien e, suez anal university, ismailia. nsae191088�yahoo. om abstract this paper, deals with the approximate solution of fra tional integro-di�erential equations of the type dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ i = [0,1] by taylor expansion and legendre wavelet methods.in addition, illustrative example are presented to demonstrate the e� ien y and a ura y of this methods. resumen este artí ulo onsidera la solu ión aproximada de e ua iones integro-diferen iales fra ionales del tipo dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ i = [0,1] por expansiones de taylor y métodos de ondeletas de legendre. además, un ejemplo ilustrativo se presenta para mostrar la e� ien ia y pre isión de este método. keywords and phrases: fra tional integro-di�erential equation, caputo fra tional derivative, taylor expansion method, legendre wavelets method. 2010 ams mathemati s subje t classi� ation: 45b05 , 45bxx , 65r10. 90 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) 1. introdu tion we study the approximate solution of an integro-di�erential equation with fra tional derivative of the type dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ i = [0,1], y(0) = α , (1.1) where 0 < q < 1,α ∈ re and the fun tions f,p,k are assumed to be su� iently smooth on their domains i and s (s = {(t,s) : 0 ≤ s ≤ t ≤ 1}.this kinds of equations arise in many modeling problems in mathemati al physi s su h as heat ondu tion in materials with memory .the existen e and uniqueness of solution of fra tional di�erential equation have been investigated in [5,7℄. re ently some attentions have been paid to the numeri al solution of equation (1.1) . rawashdeh [4℄ applied the ollo ation method to �nd a spline approximation, in [11℄ used the de omposition method to �nd an analyti solution. 2. basi de�nitions de�nition2.1. the riemann-liouville fra tional integral of order q ≥ 0 of a fun tion f ∈ cα,α ≥ −1 is de�ned by : jqf(x) = 1 γ(q) ∫x 0 (x − s)q−1f(s)ds where the real fun tion f(x) ∈ cα,α ∈ re,x > 0 is said to be in spa e if there exist a real number p > α su h that f(x) = xpf1(x) where f1(x) ∈ c[0,∞). de�nition2.2. let f ∈ ck−1,k ∈ n. then the caputo fra tional derivative of f is de�ned by : dq ∗ f(x) =    jk−qf(k)(x) ifk − 1 < q < k, f(k)(x) ifk = q . to obtain a numeri al s heme for the approximation of caputo derivative , we an use a representation that has been introdu ed by elliots [2℄; dq ∗ f(x) = 1 γ(−q) ∫x 0 f(s) − f(0) (x − s)1+q ds, (1.2) , where the integral in equation (1.2) is a hadamard �nite-part integral cubo 15, 3 (2013) approximate solution of fra tional integro-di�erential equation . . . 91 de�nition2.3. dq, denotes the fra tional di�erential operator of order q, de�ned by [5℄ as: dqy(x) =    1 γ(n−q) d n dtn ∫t 0 y(s) (x−s)q−n+1 ds, 0 ≤ n − 1 < q < n, d n y(x) dxn , q = n de�nition2.4. the following fun tions ψk,n(t) = |a0| k 2 ψ(ak0t − nb0) , form a family of dis rete wavelets , where a0 > 1,b0 > 0 and n,k are positive integers and ψ is given fun tion alled mother wavelet . moreover, the fun tions ψn,m(t) =    √ m + 1 2 2 k 2 pm(2 kt − n̂), n̂−1 2k ≤ t < n̂+1 2k , = 0 otherwise (1.3) are alled legendre wavelets polynomials where n̂ = 2n−1, n = 1, .......,2k −1, k ∈ n, t ∈ [0,1] and m is the order of the legendre polynomials pm . some basi properties of the aputo and fra tional operator an be found in [5℄. 3. taylor expansion method we onsider the following fra tional integro-di�erential equation dqy(t) = f(t) + p(t)y(t) + λ ∫t 0 k(t,s)y(s)ds, (3.1) subje ted to the initial onditions y(k)(0) = ck,k = 0,1, ....n−1, n−1 < q ≤ n, n ∈ n. to �nd the solution of eq.(3.1) , we integrate both sides of eq.(3.1) with respe t to s for n times by using de�nitions (2.2) , (2.3) . in−qy(t) = inf(t) + in(p(t)y(t)) + λin( ∫t 0 k(t,s)y(s)ds), (3.2) further ∫t 0 (t − s)n−q−1 γ(n − q) y(s)ds = ∫t 0 (t − s)n−1 (n − 1)! f(s)ds + ∫t 0 (t − s)n−1 (n − 1)! p(s)y(s)ds + λ (n − 1)! ∫t 0 y(s) ∫t s k(x,s)(x − s)n−1dsdx + qn(t). (3.3). 92 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) next , we assume that the desired solution y(s) is m + 1 times ontinuously di�erentiable on the interval i. consequently , for y ∈ cm+1,y(s) an be represented in terms of the mth order taylor expansion as y(s) = y(t) + y′(t)(s − t) + ... + y(m)(t) (s − t)m m! + y(m+1)(ξ) (s − t)m+1 (m + 1)! , (3.4) where ξ is between s and t.the lagrange reminder y(m+1)(ξ) (s−t) m+1 (m+1)! is small for a large enough m provided that y(m+1)(s) is uniformly bounded fun tion for any m on the interval i.consequently we will negle t the reminder and the trun ated taylor expansions y(x) as y(s) ≈ m∑ j=0 y(j)(t) (x − t)j j! . (3.5) we noti e that the lagrange reminder vanishes for a polynomial of degree equal to or less than m, this is implying that the above mth order taylor expansion is exa t . substituting the approximate expression (3.5) for y(t) into eq(3.2) , we get m∑ j=0 ∫t 0 (t − s)n−q−1 γ(n − q) yj(t) (s − t)j j! ds = ∫t 0 (t − s)n−1 (n − 1)! f(s)ds + m∑ j=0 ∫t 0 (t − s)n−1 (n − 1)! p(s)yj(t) (s − t)j j! ds + m∑ j=0 λ (n − 1)! ∫t 0 yj(t) (s − t)j j! ∫t 0 k(x,s)(x − s)n−1dsdx + qn(t), (3.6) or k00(t)y(t) + k01(t)y ′ (t) + .... + k0m(t)y (m) (t) = fn(t), (3.7) where k0j(t) = (−1)jtn+j−q (n + j − q)γ(n − q)j! − λ (n − 1)!j! ∫t 0 (s − t)j ∫t 0 k(x,s)(x − s)n−1dsdx − (−1)j (n − 1)!j! ∫t 0 p(s)(t − s)n+j−1ds, j = 0,1, ...,m. (3.8) f(n)(t) = 1 (n − 1)! ∫t 0 (t − s)n−1f(s)ds + qn(t). (3.9) thus eq.(3.3) be omes an mth order , linear , ordinary di�erential equation with variable oef� ients for y(t) and its derivatives up to m . we will determine y(t), ...,ym(t) by solving linear equations instead of solving analyti ally ordinary di�erential equation . by integrating both sides of eq.(3.3) with respe t to s and hanging the order of the integrations we shall obtain m independent linear equation for y(s), ...,ym(s). ∫t 0 (t − s)n−q γ(n + 1 − q) y(s)ds = ∫t 0 (t − s)n n! y(s)ds + ∫t 0 (t − s)n n! f(s) + qn(s)ds cubo 15, 3 (2013) approximate solution of fra tional integro-di�erential equation . . . 93 + λ n! ∫t 0 y(s) ∫t 0 k(x,s)(x − s)ndsdx (3.10) where we have repla ed x with t . applying the taylor expansion again and substituting (3.5) for y(s) into eq.(3.10) gives k10(t)y(t) + k11(t)y ′(t) + .... + k1m(t)y (m)(t) = fn+1(t), (3.11) k1j(t) = (−1)jtn+j+1−q (n + j + 1 − q)γ(n + 1 − q)j! − λ n!j! ∫t 0 (s − t)j ∫t 0 k(x,s)(x − s)ndsdx − (−1)j n!j! ∫t 0 p(s)(t − s)n+jds, j = 0,1, ...,m. (3.12) f(n+1)(t) = ∫t 0 (t − s)n n! f(s) + qn(t)ds. (3.13) now we have another linear equation for y(j)(t), j = 0, ...,m with y(0)(t) = y(t). by repeating the above integration pro ess for i (i ∈ n+,1 < i ≤ m) times, we get ki0(t)y(t) + ki1(t)y ′(t) + .... + kim(t)y (m)(t) = fn+i(t), , i ≤ m (3.14) where kij(t) = (−1)jtn+j+i−q (n + j + i − q)γ(n + i − q)j! − (−1)j (n + i − 1)!j! ∫t 0 p(s)(t − s)n+j+i−1ds − λ (n + i − 1)!j! ∫t 0 (s − t)j ∫t 0 k(x,s)(x − s)n+j−1dsdx, j = 0,1, ...,m. (3.15) f(r)(t) = ∫t 0 fr−1(s)ds, r > n + 1, r ∈ n +. (3.16) consequently, eqs.(3.7) , (3.11) and (3.14) form a system of m+1 unknown fun tions y(s), ....y(m)(s). this system an be written as kmm(t)ym(t) = fm(t), (3.17) 94 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) where kmm(t) is an (m + 1) × (m + 1) matrix fun tion in t , ym(t) and fm(t) are two ve tor de�ned as kmm =                             k00(t) k01(t) . . . k0m(t) k10(t) k11(t) . . . k1m(t) . . . . . . . . . . . . km0(t) km1(t) . . . kmm(t)                             , (3.18) ym(t) =                             y(t) y′(t) . . . y(m)                             , fm(t) =                             f(n)(t) f(n+1)(t) . . . f(n+m)(t)                             . (3.19) using ramer's rule , we obtain the m th-order approximate solution as y(t) = detmmm(t) detkmm(t) . (3.20) cubo 15, 3 (2013) approximate solution of fra tional integro-di�erential equation . . . 95 where mmm =                             fn(t) k01(t) . . . k0m(t) f(n+1)(t) k11(t) . . . k1m(t) . . . . . . . . . . . . fn+m(t) km1(t) . . . kmm(t)                             . (3.21) 4. legendre wavelets method we onsider the following fra tional integro-di�erential equation dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ i = [0,1] y(0) = α (4.1) the exa t solution of eq.(4.1) an be expanded as a legendre wavelets series as y(t) = ∞∑ n=1 ∞∑ m=0 cnmψn,m(t), where ψn,m(t) is given by eq.(1.3).we approximate the solution y(t) by the trun ated series yk,m(t) = 2 k−1 ∑ n=1 m−1∑ m=0 cnmψn,m(t), (4.2) then a total number of 2k−1m onditions exist for determination of 2k−1m oe� ients c10,c11, .....,c1m−1,c20, ....,c2m−1, ....,c2k−10, ....,c2k−1m−1. by the initial ondition we obtain , yk,m(0) = 2 k−1 ∑ n=1 m−1∑ m=0 cnmψn,m(0) = α. (4.3) 96 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) we must obtain 2k−1m − 1 extra onditions to re over the unknown oe� ients cnm. these onditions an be obtained by substituting eq.(4.2) in eq.(4.1). 2 k−1 ∑ n=1 m−1∑ m=0 cnmd q ∗ ψn,m(t) = f(t)+ 2 k−1 ∑ n=1 m−1∑ m=0 cnm p(t) ψn,m(t)+ 2 k−1 ∑ n=1 m−1∑ m=0 cnm ∫t 0 k(t,s) ψn,m(s)ds. (4.4) now we assume eq.(4.4) is exa t at 2k−1m − 1 points xi as : 2 k−1 ∑ n=1 m−1∑ m=0 cnmd q ∗ ψn,m(xi) = f(xi) + 2 k−1 ∑ n=1 m−1∑ m=0 cnm p(xi) ψn,m(xi) + 2 k−1 ∑ n=1 m−1∑ m=0 cnm ∫xi 0 k(xi,s) ψn,m(s)ds. (4.5) the best hoi e of the xi points are the zeros of the shifted hebyshev polynomials of degree 2k−1m − 1 in the interval [0,1] that is xi = si + 1 2 , si = ( (2i − 1)π 2k−1m − 1 ), i = 1, ...,2k−1m − 1. approximating d q ∗ ψn,m using diethhelm method [6℄ on the representation that has been given by eq.(1.2) , we get dq ∗ ψn,m(xi) = 1 γ(−q) ∫xi 0 ψn,m(s) − ψn,m(0) (xi − s) 1+q ds = x −q i γ(−q) ∫1 0 ψn,m(xi − xiw) − ψn,m(0) w1+q ds ≃ l∑ r=0 αr(ψn,m(xi − xir l ) − ψn,m(0)) where l ∈ n and the weights αr is given by q(1 − q)l−q γ(−q) x −q i αr =    −1, if r = 0, 2r1−q − (r − 1)1−q − (r + 1)1+q, if r = 1,2, ..., l − 1, (q − 1)r−q − (r − 1)1−q + r1−q, if r = l cubo 15, 3 (2013) approximate solution of fra tional integro-di�erential equation . . . 97 then eq.(4.5) be omes 2 k−1 ∑ n=1 m−1∑ m=0 l∑ r=0 αr(ψn,m(xi − xir l ) − ψn,m(0))cnm = f(xi) + 2 k−1 ∑ n=1 m−1∑ m=0 cnm p(xi) ψn,m(xi) + 2 k−1 ∑ n=1 m−1∑ m=0 cnm ∫xi 0 k(xi,s) ψn,m(s)ds. (4.6) combine eq.(4.3) and eq.(4.6) to obtain 2k−1m linear equations from whi h we an ompute the unknowns oe� ients , cnm 5. numeri al examples to show the e� ien y and the a ura y of the proposed methods , we onsider here some fra tional integro-di�erential equations . now we shall solve some examples by taylor expansion and legendre wavelet methods and ompare the results in tables . all results are obtained by using maple 15. example 1. consider the following fra tional integro-di�erential equation d0.75 ∗ y(t) = 2t1.25 γ(2.25) − t4 − t5 4 + t2y(t) + ∫t 0 tsy(s) ds , (5.1) with the initial ondition y(0) = 0 and the exa t solution y(t) = t2, with k=1 and m=6 . table 1. the results of example 1. 98 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) t exa t taylor method abs.e legendre method abs.e 0.1 0.01 0.009999996 4.086580027 × 10 −9 0.041204470 0.312044705 × 10 −2 0.2 0.04 0.039999686 3.137931423 × 10 −7 0.157896393 0.1178963927 0.3 0.09 0.089996776 3.224235647 × 10−6 0.270260094 0.1802600943 0.4 0.16 0.159983397 1.660282668 × 10 −5 0.3523832431 0.1923832431 0.5 0.25 0.249941528 5.847208088 × 10−5 0.429628144 0.179628144 0.6 0.36 0.359839516 1.604836601 × 10 −4 0.576003047 0.216003047 0.7 0.49 0.489634434 3.655662632 × 10 −4 0.911533447 0.421533447 0.8 0.64 0.639288648 7.113517648 × 10 −4 1.599633386 0.959633386 0.9 0.81 0.808822770 1.177230376 × 10 −3 2.844476755 2.034476755 figure 1: results of example 1 cubo 15, 3 (2013) approximate solution of fra tional integro-di�erential equation . . . 99 example 2. consider the following fra tional integro-di�erential equation d0.25 ∗ y(t) = 6t2.75 γ(3.75) − t4 − t2et 5 y(t) + ∫t 0 etsy(s) ds , (5.2) with the initial ondition y(0) = 0 and the exa t solution y(t) = t3, with k=2 and m=2 . table 2. the results of example 2. t exa t taylor method abs.e legendre method abs.e 0.1 0.001 0.001000055 5.5082 × 10−8 0.7846772718 × 10−2 6.846772718 × 10−3 0.2 0.008 0.008001757 1.756874 × 10 −6 0.015693545 7.69354543 × 10 −3 0.3 0.027 0.027011518 1.151813 × 10 −5 0.02354031813 3.45968187 × 10 −3 0.4 0.064 0.064035232 3.523224 × 10 −5 0.03138709084 3.261290916 × 10 −2 0.5 0.125 0.125051836 5.18360 × 10 −5 -1.668332670 1.793332670 0.6 0.216 0.215957399 4.26011 × 10−5 -0.882599797 1.098599797 0.7 0.343 0.342472414 5.275860 × 10 −4 -0.096866924 0.439866924 0.8 0.512 0.510019153 1.980847 × 10−3 0.688865948 0.176865948 0.9 0.729 0.723579567 5.4204329 × 10 −3 1.474598821 0.745598821 100 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) figure 2: results of example 2. example 3. consider the following fra tional integro-di�erential equation d0.5 ∗ y(t) = 2t1.5 γ(2.5) + t0.5 γ(1.5) +t(2−3 ost−tsint+t2 ost)−( ost−sint)y(t)+ ∫t 0 etsy(s) ds , (5.3) with the initial ondition y(0) = 0 and the exa t solution y(t) = t2 + t with k=2 and m=2 . similarly as in examples 1,2 applying the taylor expansion method and legendre wavelet method of the a omparison between the exa t solution and the approximate solution and the absolute error (abs.e) are given in table 3 and figures 3. cubo 15, 3 (2013) approximate solution of fra tional integro-di�erential equation . . . 101 table 3. the results of example 3. t exa t taylor method abs.e legendre method abs.e 0.1 0.11 0.109999426 5.740 × 10 −7 0.1373123038 2.73123038 × 10 −2 0.2 0.24 0.239986839 1.31607 × 10−5 0.2746246075 3.46246075 × 10−2 0.3 0.39 0.389914938 8.50616 × 10 −5 0.4119369112 2.19369112 × 10 −2 0.4 0.56 0.559673632 3.263682 × 10 −4 0.5492492149 1.07507851 × 10 −2 0.5 0.75 0.749057525 9.424746 × 10 −4 0.649795587 0.100204413 0.6 0.96 0.957730079 2.2699214 × 10 −3 0.995295259 3.5295259 × 10 −2 0.7 1.19 1.185190201 4.809799 × 10−3 1.340794930 0.150794930 0.8 1.44 1.430747819 9.252181 × 10 −3 1.686294602 0.246294602 0.9 1.71 1.693507524 1.6492476 × 10−2 2.031794273 0.321794273 figure 3: results of example 3. with taylor expansion and legendre wavelet method. 102 m.h.saleh, s.m.amer, m.a.mohamed & n.s.abdelrhman cubo 15, 3 (2013) 6. con lusion in this paper , we have applied the legendre wavelet and taylor expansion methods for solving the fra tional integro-di�erential equation . the integro-di�erential equations onverted to a system of linear equations by two methods . by omparing the errors in two methods we �nd that taylor expansion method gives results better than legendre wavelet method . re eived: mar h 2013. a epted: september 2013. referen es [1℄ a. arikoglu and i. ozkol ; solution of fra tional integro-di�erential equation by using fra tional di�erential transform method , chaos solitons and fra tals 40 (2009) , 521 529. 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() cubo a mathematical journal vol.13, no¯ 03, (185–196). october 2011 linear convergence analysis for general proximal point algorithms involving (h, η)−monotonicity frameworks ram u. verma texas a&m university at kingsville, department of mathematics, kingsville, texas 78363, usa. email: verma99@msn.com abstract general framework for the generalized proximal point algorithm, based on the notion of (h, η)− monotonicity, is developed. the linear convergence analysis for the generalized proximal point algorithm to the context of solving a class of nonlinear variational inclusions is examined, the obtained results generalize and unify a wide range of problems to the context of achieving the linear convergence for proximal point algorithms. resumen se desarrolla un marco general para el algoritmo de punto proximal generalizado, basado en la noción de (h, η)− monotonia. se examina el análisis de convergencia lineal para el algoritmo de punto proximal generalizado en el contexto de la resolución de una clase de inclusiones no lineales variacional. los resultados obtenidos generalizan y unifican una amplia gama de problemas en el contexto de lograr la convergencia lineal de los algoritmos punto proximal. keywords. general cocoerciveness, variational inclusions, maximal monotone mapping, (h, η)− monotone mapping, generalized proximal point algorithm, generalized resolvent operator. 186 ram u. verma cubo 13, 3 (2011) mathematics subject classification: 49j40, 47h10, 65b05. 1. introduction based on recent advances on linear convergence for proximal point algorithms, we are concerned to develop a general framework for the generalized proximal point algorithm [7] based on the notion of (h, η)− monotonicity introduced and studied by fang and huang [9], and then achieve a linear convergence to the context of solving a general variational inclusion problem. as a result, we establish a significant generalization on linear convergence analysis based on proximal point algorithms/relaxed proximal point algorithms. the generalized version of relaxed proximal point algorithm generalizes the proximal point algorithm of rockafellar [25, 26], that in turn generalizes the algorithm of martinet [18] for convex programming. it appears that a general class of problems of variational character, including minimization or maximization of functions, variational inequality problems, and minimax problems, can be unified into this form. let x be a real hilbert space with the norm ‖.‖ and the inner product 〈., .〉. we consider the inclusion problem: find a solution to 0 ∈ m(x), (1.1) where m : x → 2x is a set-valued mapping on x. in this communication, we first present the generalized version of proximal point algorithm based on the notion of (h, η)− monotonicity, and then apply it to approximate a solution to a general class of nonlinear inclusion problems involving (h, η)− monotone mappings in a hilbert space setting. second, we explore the linear convergence analysis for the generalized proximal point algorithms for solving a class of nonlinear inclusions. also, several results on the generalized cocoercive and generalized resolvent mappings are demonstrated . the results, thus obtained here, are significantly general in nature. for more details, we refer the reader [1-38]. 2. (h, η)− monotonicity and generalized cocoerciveness this section deals with some results based on basic properties of (h, η)− monotonicity, and other results involving (h, η)− monotonicity and the generalized cocoerciveness . let x denote a real hilbert space with the norm ‖.‖ and inner product < ., . > . let m : x → 2x be a multivalued mapping on x. we shall denote both the map m and its graph by m, that is, the set {(x, y) : y ∈ m(x)}. this is equivalent to stating that a mapping is any subset m of x × x, and m(x) = {y : (x, y) ∈ m}. if m is single-valued, we shall still use m(x) to represent the unique y such that (x, y) ∈ m rather than the singleton set {y}. this interpretation shall much depend on the context. the domain of a map m is defined (as its projection onto the first argument) by d(m) = {x ∈ x : ∃ y ∈ x : (x, y) ∈ m} = {x ∈ x : m(x) 6= ∅}. cubo 13, 3 (2011) linear convergence analysis for general . . . 187 d(t)=x, shall denote the full domain of m, and the range of m is defined by r(m) = {y ∈ x : ∃ x ∈ x : (x, y) ∈ m}. the inverse m−1 of m is {(y, x) : (x, y) ∈ m}. for a real number ρ and a mapping m, let ρm = (x, ρy) : (x, y) ∈ m}. if l and m are any mappings, we define l + m = {(x, y + z) : (x, y) ∈ l, (x, z) ∈ m}. definition 2.1. let m : x → 2x be a multivalued mapping on x. the map m is said to be: (i) (r)− strongly monotone if there exists a positive constant r such that 〈u∗ − v∗, u − v〉 ≥ r‖u − v‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (ii) (1)− strongly monotone if 〈u∗ − v∗, u − v〉 ≥ ‖u − v‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (iii) (m)−relaxed monotone if there exists a positive constant m such that 〈u∗ − v∗, u − v〉 ≥ (−m)‖u − v‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (iv) (c)− cocoercive if there is a positive constant c such that 〈u∗ − v∗, u − v〉 ≥ c‖u∗ − v∗‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). definition 2.2. a mapping m : x → 2x is said to be maximal (m)− relaxed monotone if (i) m is (m)−relaxed monotone, (ii) for (u, u∗) ∈ x × x, and 〈u∗ − v∗, u − v〉 ≥ (−m)‖u − v‖2 ∀ (v, v∗) ∈ graph(m), we have u∗ ∈ m(u). definition 2.3. let m : x → 2x be a mapping on x. the map m is said to be: (i) nonexpansive if ‖u∗ − v∗‖ ≤ ‖u − v‖ ∀ (u, u∗), (v, v∗) ∈ graph(m). 188 ram u. verma cubo 13, 3 (2011) (ii) firmly nonexpansive if ‖u∗ − v∗‖2 ≤ 〈u∗ − v∗, u − v〉 ∀ (u, u∗), (v, v∗) ∈ graph(m). (iii) (c)−firmly nonexpansive if there exists a constant c > 0 such that ‖u∗ − v∗‖2 ≤ ‖u − v‖2 − c‖u − v − (u∗ − v∗)‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (iv) (c)−firmly nonexpansive if there exists a constant c > 0 such that ‖u∗ − v∗‖2 ≤ c〈u∗ − v∗, u − v〉 ∀ (u, u∗), (v, v∗) ∈ graph(m). proposition 2.1. let h : x → x be an (r, η)−strongly monotone mapping and let m : x → 2x be an (h, η)−monotone mapping. then the operator (h + ρm)−1 is single-valued. definition 2.4. let h : x → x be an (r, η)−strongly monotone mapping and let m : x → 2x be an (h, η)− monotone mapping. then the generalized resolvent operator jmρ,h : x → x is defined by jmρ,h (u) = (h + ρm) −1(u). definition 2.5. let h, t : x → x be two mappings. then map t is said to be: (i) monotone with respect to h if 〈t (x) − t (y), h(x) − h(y)〉 ≥ 0 ∀ (x, y) ∈ x. (ii) (r) − strongly monotone with respect to h if there exists a positive constant r such that 〈t (x) − t (y), h(x) − h(y)〉 ≥ (r)‖x − y‖2 ∀ (x, y) ∈ x. (iii) (γ, α)-relaxed cocoercive with respect to h if there exist positive constants γ and α such that 〈t (x) − t (y), h(x) − h(y)〉 ≥ −γ‖t (x) − t (y)‖2 + α‖x − y‖2 ∀(x, y) ∈ x. definition 2.6. a map η : x × x → x is said to be: (i) (η)− monotone if 〈x − y, η(x, y)〉 ≥ 0 ∀ (x, y) ∈ x. cubo 13, 3 (2011) linear convergence analysis for general . . . 189 (ii) (t)-strongly monotone if there exists a positive constant t such that 〈x − y, η(x, y)〉 ≥ t‖x − y‖2 ∀ (x, y) ∈ x. (iii) (τ)-lipschitz continuous if there exists a positive constant τ such that ‖η(x, y)‖ ≤ τ‖x − y‖. definition 2.7. let m : x → 2x be a multivalued mapping on x, and let η : x×x → x be another mapping. the map m is said to be: (i) (η)− monotone if 〈u∗ − v∗, η(u, v)〉 ≥ 0 ∀ (u, u∗), (v, v∗) ∈ graph(m). (ii) (r, η)− strongly monotone if there exists a positive constant r such that 〈u∗ − v∗, η(u, v)〉 ≥ r‖u − v‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (iii) (1, η)− strongly monotone if 〈u∗ − v∗, η(u, v)〉 ≥ ‖u − v‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (iv) (m, η)−relaxed monotone if there exists a positive constant m such that 〈u∗ − v∗, η(u, v)〉 ≥ (−m)‖u − v‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). (v) (c, η)− cocoercive if there is a positive constant c such that 〈u∗ − v∗, η(u, v)〉 ≥ c‖u∗ − v∗‖2 ∀ (u, u∗), (v, v∗) ∈ graph(m). definition 2.8. [9] let h : x → x be (r)− strongly monotone. the map m : x → 2x is said to be h− monotone if (i) m is monotone, (ii) r(h + ρm) = x for ρ > 0. 190 ram u. verma cubo 13, 3 (2011) definition 2.9.let h : x → x be (r, η)− strongly monotone. the map m : x → 2x is said to be to (h, η)− monotone if (i) m is (η)− monotone, (ii) r(h + ρm) = x for ρ > 0. definition 2.10. a map m : x → 2x is said to be (η)monotone if (i) m is (η)− monotone, (ii) r(i + ρm) = x for ρ > 0. 3. generalized proximal point algorithms this section deals with the generalized proximal point algorithm and its application to approximation solvability of the inclusion problem (1) based on the (h, η)-monotonicity. furthermore, some results connecting the (h, η)− monotonicity and corresponding generalized resolvent operator are established, that generalize the results on the generalized cocoerciveness and h− monotonicity [9], while the auxiliary results on (h, η)− monotonicity and general maximal monotonicity are obtained. lemma 3.1. [9] let x be a real hilbert space, let h : x → x be (r)−strongly monotone, and let m : x → 2x be h− monotone. then the generalized resolvent operator associated with m and defined by jmρ,h (u) = (h + ρm) −1(u) ∀ u ∈ x, is (1/r)− lipschitz continuous. theorem 3.1. let x be a real hilbert space, let h : x → x be (r, η)−strongly monotone, and let m : x → 2x be (h, η)monotone. then the following statements are equivalent: (i) an element u ∈ x is a solution to (1). (ii) for an u ∈ x, we have u = jmρ,h (h (u)), where jmρ,h (u) = (h + ρm) −1(u). cubo 13, 3 (2011) linear convergence analysis for general . . . 191 next, we introduce a generalization to the relaxed proximal point algorithm. algorithm 3.1. let h : x → x be a single-valued mapping, let m : x → 2x be a set-valued (h, η)− monotone mapping on x with 0 ∈ range(m), and let the sequence {xk} be generated by the iterative procedure h(xk+1) = (1 − αk)h(x k) + αky k ∀k ≥ 0, (3.1) and yk satisfies ‖yk − h(jmρk,h(h(x k)))‖ ≤ δk‖y k − h(xk)‖, where jm ρk,h = (h + ρkm) −1, ∑∞ k=0 δk < ∞, δk → 0 and, {δk}, {αk}, {ρk} ⊆ [0, ∞) are scalar sequences. theorem 3.2 let x be a real hilbert space, let h : x → x be (r, η)−strongly monotone, and let m : x → 2x be (h, η)−monotone. let η : x × x → x be (τ)−lipschitz continuous. for an arbitrarily chosen initial point x0, suppose that the sequence {xk} is generated by algorithm 3.1. suppose that hojm ρk,h is (λ, η)−cocoercive for λ > 1, that is, for all u, v ∈ x, 〈h(jmρk,h(h(u))) − h(j m ρk,h (h(v))), η(h(u), h(v))〉 ≥ λ‖h(jmρk,h(h(u))) − h(j m ρk,h (h(v)))‖2. (3.2) then the sequence {xk} converges linearly to a solution of (1.1) with convergence rate √ 1 − 2α[1 − (1 − α)τ λ − ατ2 2λ2 − α 2 ] < 1, for λ > 1, τ < λ, ∑∞ k=0 δk < ∞, δk → 0, αk ≤ 1 and, {δk}, {αk}, {ρk} ⊆ (0, ∞) are scalar sequences. proof. suppose that x∗ is a zero of m. from theorem 3.1, it follows that any solution to (1) is a fixed point of jm ρk,h oh. for all k ≥ 0, we express h(zk+1) = (1 − αk)h(x k) + αkh(j m ρk,h (h(xk))). 192 ram u. verma cubo 13, 3 (2011) next, we find the estimate using the appropriate implications of (3.2) that ‖h(zk+1) − h(x∗)‖2 = ‖(1 − αk)h(x k) + αkh(j m ρk,h (h(xk))) − [(1 − αk)h(x ∗) + αkh(j m ρk,h (h(x∗)))‖2 = (1 − αk) 2‖h(xk) − h(x∗)‖2 + 2αk(1 − αk)〈h(x k) − h(x∗), h(jmρk,h(h(x k))) − h(jmρk,h(h(x ∗)))〉 + α2k‖h(j m ρk,h (h(xk))) − h(jmρk,h(h(x ∗)))‖2 ≤ (1 − αk) 2‖h(xk) − h(x∗)‖2 + 2αk(1 − αk) τ λ ‖h(xk) − h(x∗)‖2 + α2k‖h(j m ρk,h (h(xk))) − h(jmρk,h(h(x ∗)))‖2 ≤ (1 − αk) 2‖h(xk) − h(x∗)‖2 + 2αk(1 − αk) τ λ ‖h(xk) − h(x∗)‖2 + α2k‖h(j m ρk,h (h(xk))) − h(jmρk,h(h(x ∗)))‖2 ≤ (1 − αk) 2‖h(xk) − h(x∗)‖2 + 2αk(1 − αk) τ λ ‖h(xk) − h(x∗)‖2 + α2 k τ2 λ2 ‖h(xk) − h(x∗)‖2 = [1 − 2αk[1 − (1 − αk) τ λ − αkτ 2 2λ2 − αk 2 ]]‖h(xk) − h(x∗)‖2, where τ < λ. it follows from the above inequality that ‖h(zk+1) − h(x∗)‖ ≤ θk‖h(x k) − h(x∗)‖, (3.3) where θk = √ 1 − 2αk[1 − (1 − αk) τ λ − αkτ 2 2λ2 − αk 2 ]. since h(xk+1) = (1 − αk)h(x k) + αky k, it implies h(xk+1) − h(xk) = αk(y k − h(xk)). on the other hand, we have ‖h(xk+1) − h(zk+1)‖ = αk‖y k − h(jmρk,h(h(x k)))‖ ≤ αkδk‖y k − h(xk)‖. cubo 13, 3 (2011) linear convergence analysis for general . . . 193 finally, we estimate ‖h(xk+1) − h(x∗)‖ ≤ ‖h(zk+1) − h(x∗)‖ + ‖h(xk+1) − h(zk+1)‖ ≤ ‖h(zk+1) − h(x∗)‖ + αkδk‖y k − h(xk)‖ ≤ ‖h(zk+1) − h(x∗)‖ + δk‖h(x k+1) − h(xk)‖ ≤ ‖h(zk+1) − h(x∗)‖ + δk[‖h(x k+1) − h(x∗)‖ + ‖h(xk) − h(x∗)‖] ≤ θk‖h(x k) − h(x∗)‖ + δk‖h(x k+1) − h(x∗)‖ + δk‖h(x k) − h(x∗)‖. this implies that ‖h(xk+1) − h(x∗)‖ ≤ θk + δk 1 − δk ‖h(xk) − h(x∗)‖, (3.4) where ĺım sup θk + δk 1 − δk = ĺım sup θk = √ 1 − 2α[1 − (1 − α)τ λ − ατ2 2λ2 − α 2 ] < 1. it follows that ‖h(xk) − h(x∗)‖ → 0 as k → ∞. since h is (r, η)−strongly monotone, we have ‖h(xk) − h(x∗)‖ ≥ r τ ‖xk − x∗‖. therefore, we conclude that r τ ‖xk − x∗‖ ≤ ‖h(xk) − h(x∗)‖ → 0. this completes the proof. received: september 2010. revised: november 2010. references [1] r. p. agarwal, and r. u. verma, inexact a−proximal point algorithm and applications to nonlinear variational inclusion problems, journal of optimization theory and applications 144 (3) (2010), 431–444. 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[38] e. zeidler, nonlinear functional analysis and its applications ii/b, springer-verlag, new york, new york, 1990. introduction (h,)monotonicity and generalized cocoerciveness generalized proximal point algorithms () cubo a mathematical journal vol.13, no¯ 02, (151–161). june 2011 some new characterizations for pgl(2, q) b. khosravi 12, m. khatami2 and z. akhlaghi2 1 school of mathematics, institute for research in fundamental sciences (ipm), p.o. box: 19395-5746, tehran, iran. email: khosravibbb@yahoo.com and 2 dept. of pure math., faculty of math. and computer sci., amirkabir university of technology (tehran polytechnic), 424, hafez ave., tehran 15914, iran. abstract many authors introduced some characterizations for finite groups. in this paper as the main result we prove that the finite group pgl(2, q) is uniquely determined by its noncommuting graph. also we prove that pgl(2, q) is characterizable by its noncyclic graph. throughout the proof of these results we prove that pgl(2, q) is uniquely determined by its order components and using this fact we give positive answer to a conjecture of thompson and another conjecture of shi and bi for the group pgl(2, q). resumen muchos autores introdujeron algunas caracterizaciones de los grupos finitos. en este trabajo como principal resultado se demuestra que grupo finito pgl(2, q) es determinado nicamente por su gráfica no conmutativa. también se demuestra que pgl(2, q) 1the first author was supported in part by a grant from ipm (no. 89200113). 152 b. khosravi, m. khatami & z. akhlaghi cubo 13, 2 (2011) es caracterizable por su gráfico no ćıclico. a lo largo de la prueba de estos resultados se demuestra que pgl(2, q) es determinado únicamente por los componentes de su orden y con ello damos respuesta positiva a una conjetura de thompson y otra conjetura de shi bi y para el grupo pgl(2, q). keywords and phrases: noncommuting graph, prime graph, noncyclic graph, order components. mathematics subject classification: 20d05, 20d60. 1 introduction if n is an integer, then we denote by π(n) the set of all prime divisors of n. if g is a finite group, then π(|g|) is denoted by π(g). we construct the prime graph of g which is denoted by γ (g) as follows: the vertex set is π(g) and two distinct primes p and q are joined with an edge if and only if g contains an element of order pq. let t(g) be the number of connected components of γ (g) and let π1, π2, ..., πt(g) be the connected components of γ (g). if 2 ∈ π(g), then we assume that 2 ∈ π1. now we can express |g| as a product of coprime natural numbers mi, such that 1 ≤ i ≤ t(g) and π(mi) = πi. these integers are called order components of g. the set of order components of g is denoted by oc(g). one of the other graphs which associated with a non-abelian group g is the noncommuting graph that is denoted by ∇(g) and is constructed as follows: the vertex set of ∇(g) is g \ z(g) with two vertices x and y are joined by an edge whenever the commutator of x and y is not identity. in [1] the authors put forward the following conjecture: conjecture a. let s be a finite non-abelian simple group and g be a finite group such that ∇(g) ∼= ∇(s). then g ∼= s. the validity of this conjecture has been proved for all simple groups with non-connected prime graphs. also it is proved that some finite simple groups with connected prime graphs, say a10, u4(7), l4(8), l4(4) and l4(9), can be uniquely determined by their noncommuting graghs (see [19, 20, 21, 22]). in this paper as the main result we prove that the almost simple group pgl(2, q), where q = pn for a prime number p and a natural number n, is characterizable by its noncommuting graph. as a consequence of our results we prove the validity of a conjecture of thompson and another conjecture of shi and bi for the group pgl(2, q). let g be a noncyclic group and cyc(g) = {x ∈ g|〈x, y〉 is cyclic for all y ∈ g}. in [2], the authors introduced the cyclic graph of g, which is denoted by γ1(g) as follows: take g \ cyc(g) as the vertex set and join two vertices if they do not generate a cyclic subgroup. in this graph the degree of each vertex x is equal to |g| \ |cycg(x)|, where cycg(x) = {y ∈ g|〈x, y〉 is cyclic}. it is cubo 13, 2 (2011) some new characterizations for pgl(2, q) 153 proved that some finite simple groups, sn, d2k , d2n ,where n is odd, are characterizable by the noncyclic graph. we show that pgl(2, q) is uniquely determined by its noncyclic graph. in this paper, all groups are finite and by simple groups we mean non-abelian simple groups. all further unexplained notations are standard and refer to [6], for example. 2. preliminary results in this section we bring some preliminary lemmas which are necessary in the proof of the main theorem. remark 2.1. let n be a normal subgroup of g and p, q be incident vertices of γ (g/n). then p, q are incident in γ (g). in fact if xn is of order pq, then there exists a power of x which is of order pq. definition 2.2. ([8]) a finite group g is called a 2-frobenius group if it has a normal series 1 � h � k � g, where k and g/h are frobenius groups with kernels h and k/h, respectively. lemma 2.3. let g be a frobenius group of even order and let h, k be frobenius complement and frobenius kernel of g, respectively. then t(g) = 2, and the prime graph components of g are π(h), π(k) and g has one of the following structures: (a) 2 ∈ π(k) and all sylow subgroups of h are cyclic; (b) 2 ∈ π(h), k is an abelian group, h is a solvable group, the sylow subgroups of odd order of h are cyclic groups and the 2-sylow subgroups of h are cyclic or generalized quaternion groups; (c) 2 ∈ π(h), k is an abelian group and there exists h0 ≤ h such that |h : h0| ≤ 2, h0 = z × sl(2, 5), π(z) ∩ {2, 3, 5} = ∅ and the sylow subgroups of z are cyclic. also the next lemma follows from [8] and the properties of frobenius groups [9]: lemma 2.4. let g be a 2-frobenius group, i.e., g has a normal series 1 � h � k � g, such that k and g/h are frobenius groups with kernels h and k/h, respectively. then (a) t(g) = 2, π1 = π(g/k) ∪ π(h) and π2 = π(k/h); (b) g/k and k/h are cyclic, |g/k| | (|k/h| − 1) and g/k ≤ aut(k/h); (c) h is nilpotent and g is a solvable group. lemma 2.5. ([4, lemma 8]) let g be a finite group with t(g) ≥ 2 and let n be a normal subgroup of g. if n is a πi-group for some prime graph component of g and m1, m2, . . . , mr are some order components of g but not πi-numbers, then m1m2 · · · mr is a divisor of |n| − 1. lemma 2.6. ([3, lemma 1.4]) suppose g and m are two finite groups satisfying t(m) ≥ 2, n(g) = n(m), where n(g)={n | g has a conjugacy class of size n }, and z(g) = 1. then 154 b. khosravi, m. khatami & z. akhlaghi cubo 13, 2 (2011) |g| = |m|. lemma 2.7. ([3, lemma 1.5]) let g1 and g2 be finite groups satisfying |g1| = |g2| and n(g1) = n(g2). then t(g1) = t(g2) and oc(g1) = oc(g2). lemma 2.8. ([11]) let g be a finite group and m be a finite group with t(m) = 2 satisfying oc(g) = oc(m). let oc(m) = {m1, m2}. then one of the following holds: (a) g is a frobenius or 2-frobenius group; (b) g has a normal series 1 � h � k � g such that g/k is a π1-group, h is a nilpotent π1-group, and k/h is a non-abelian simple group. moreover oc(k/h) = {m′1, m ′ 2, . . . , m ′ s, m2}, where m′1m ′ 2 . . . m ′ s|m1. also g/k ≤ out(k/h). lemma 2.9. ([1]) let g be a finite non-abelian group. if h is a group such that ∇(g) ∼= ∇(h), then h is a finite non-abelian group such that |z(h)| divides gcd(|g| − |z(g)|, |g| − |cg(x)|, |cg(x)| − |z(g)| : x ∈ g \ z(g)). lemma 2.10. ([18]) let g be a non-abelian group such that ∇(g) ∼= ∇(psl(2, 2n)), where n is a natural number. then g ∼= psl(2, 2n). lemma 2.11.([7, remark 1]) the equation pm − qn = 1, where p and q are primes and m, n > 1 has only one solution, namely 32 − 23 = 1. lemma 2.12. ([2]) let g be a finite noncyclic group. if h is a group such that γ1(g) ∼= γ1(h), then h is a finite noncyclic group such that |cyc(h)| divides gcd(|g| − |cyc(g)|, |g| − |cycg(x)|, |cycg(x)| − |cyc(g)| : x ∈ g \ cyc(g)). lemma 2.13. ([2]) let g and h be two finite noncyclic groups such that γ1(g) ∼= γ1(h). if |g| = |h|, then πe(g) = πe(h). 3. main results we note that if q = 2n, then pgl(2, q) = psl(2, q) and we know that psl(2, q) is characterizable by its noncommuting graph (see [18]). therefore throughout this section we suppose m is the almost simple group pgl(2, q), where q = pn for an odd prime number p and a natural number n. theorem 3.1. let g be a group such that ∇(g) ∼= ∇(m). then |g| = |m|. proof. first note that g is a finite non-abelian group. since ∇(g) ∼= ∇(m), we have |g|−|z(g)| = |m| − |z(m)|. then it is enough to prove that |z(g)| = |z(m)|. cubo 13, 2 (2011) some new characterizations for pgl(2, q) 155 by lemma 2.9, |z(g)| divides |m| − |z(m)|. since |z(m)| = 1, we have |z(g)| divides q(q2 − 1) − 1. let p be a sylow p-subgroup of m. we know that z(p) 6= 1. so there exists 1 6= x ∈ z(p). we claim that cm(x) = p. it is obvious that p ≤ cm(x), since x ∈ z(p). on the contrary we suppose that y ∈ cm(x) \ p. so we can conclude that o(xy) = o(x)o(y). without lose of generality we suppose |y| = r, where r 6= p is a prime number. then m has an element of order rp. but p is an isolated vertex in γ (m), a contradiction. therefore our claim is proved. by lemma 2.9 we have |z(g)| divides |cm(x)| − |z(m)|. then |z(g)| divides q − 1. we know that z(g) divides q(q2 − 1) − 1, which implies that |z(g)| = 1 and so |g| = |m|. 2 theorem 3.2. let g be a group such that ∇(g) ∼= ∇(m), where m = pgl(2, q). then oc(g) = oc(m). proof. since ∇(g) ∼= ∇(m), the set of vertex degrees of two graphs are the same. therefore {|g| − |cg(x)| | x ∈ g} = {|m| − |cm(y)| | y ∈ m}. on the other hand theorem 3.1 implies that |g| = |m|, and so n(g) = n(m). now using lemma 2.7 we have oc(g) = oc(m). 2 theorem 3.3. let g be a finite group and oc(g) = oc(m). if q = pn 6= 3 then g is neither a frobenius group nor a 2-frobenius group. if q = 3 and g is a 2-frobenius group, then g ∼= s4. proof. if g is a frobenius group, then by lemma 2.3, oc(g) = {|h|, |k|} where k and h are frobenius kernel and frobenius complement of g, respectively. therefore oc(g) = {q, q2 − 1} and since |h| | (|k| − 1) it follows that |h| < |k| and so |h| = q and |k| = q2 − 1. also q | (q2 − 2) implies that q = 2, which is a contradiction, since q is odd. therefore g is not a frobenius group. let g be a 2-frobenius group. hence g = abc, where a and ab are normal subgroups of g; ab and bc are frobenius groups with kernels a, b and complements b, c, respectively. by lemma 2.4, we have |b| = q and |a||c| = q2 − 1. also |b| | (|a| − 1) and so |a| = qt + 1, for some t > 0. on the other hand, |a| | (q2 − 1), which implies that q2 − 1 = k(qt + 1), for some k > 0. therefore q | (k + 1) and so q − 1 ≤ k. if t > 1, then q2 − 1 = k(qt + 1) ≥ (q − 1)(qt + 1) > (q − 1)(q + 1), which is a contradiction. hence t = 1 and |a| = q + 1 and |c| = q − 1. if there exists an odd prime r such that r | (q + 1), then let r be a sylow r-subgroup of a. since a is a nilpotent group, it follows that r is a normal subgroup of g. now lemma 2.5, implies that q | (|r| − 1) and |r| | (q + 1)/2, which is a contradiction. therefore q + 1 = 2α, for some α > 0. similarly z(a) 6= 1 is a characteristic subgroup of a and hence a is abelian. let x = {x ∈ a|o(x) = 2} ∪ {1}. then x is a non-identity characteristic subgroup of a. therefore a is an elementary abelian 2-subgroup of g and |a| = 2α = q + 1. by lemma 2.11, if q = pn such that n > 1, then the equation 2α − q = 1 does not have any solution. 156 b. khosravi, m. khatami & z. akhlaghi cubo 13, 2 (2011) now let n = 1. suppose f = gf(2α) and so a is the additive group of f. also |b| = q = p = 2α − 1 and so b is the multiplicative group of f. now c acts by conjugation on a and similarly c acts by conjugation on b and this action is faithful. therefore c keeps the structure of the field f and so c is isomorphic to a subgroup of the automorphism group of f. hence |c| = 2α − 2 ≤ |aut(f)| = α. therefore α ≤ 2. if α = 2, then g = s4, the symmetric group on 4 letters. 2 lemma 3.4. let g be a finite group and m = pgl(2, q), where q > 3 or q = 3 and m is not a 2-frobenius group. if oc(g) = oc(m), then g has a normal series 1 � h � k � g such that h and g/k are π1-groups and k/h is a simple group. moreover the odd order component of m is equal to an odd order component of k/h. in particular, t(k/h) ≥ 2. also |g/h| divides |aut(k/h)|, and in fact g/h ≤ aut(k/h). proof. the first part of the lemma follows from lemma 2.8 and theorem 3.3, since the prime graph of g has two components. if k/h has an element of order pq, where p and q are primes, then by remark 2.1, k has an element of order pq. therefore g has an element of order pq. so by the definition of prime graph component, the odd order component of g is equal to an odd order component of k/h. also k/h � g/h and cg/h(k/h) = 1, which implies that g/h = ng/h(k/h) cg/h(k/h) ∼= t , t ≤ aut(k/h). 2 theorem 3.5. let g be a finite group such that oc(g) = oc(m), where m = pgl(2, q). then g ∼= pgl(2, q). proof. if q = 3 and g is a 2-frobenius group, then theorem 3.3 implies that g = s4 ∼= pgl(2, 3), as desired. otherwise lemma 3.4 implies that g has a normal series 1 � h � k � g such that h and g/k are π1-groups and k/h is a simple subgroup and t(k/h) ≥ 2. now using the classification of finite simple groups and the results in tables 1-3 in [10], we consider the following cases. case 1. let k/h ∼= am, where m = p ′, p′ + 1 or p′ + 2 and p′ ≥ 5 is a prime number and m and m − 2 are not primes at the same time. then q = p′, and consequently n = 1 and q = p = p′. on the other hand, |am| | |g| = p(p2 − 1). if m > p, then |am| > (p + 1)p(p − 1), which is a contradiction. therefore m = p and |ap| | |g| = p(p 2 − 1), and so |ap| = p!/2 ≤ p(p2 − 1). hence (p − 2)!/2 ≤ p + 1. but p ≥ 7, since p − 2 is not a prime. so (p − 2)(p − 3) < (p − 2)!/2 ≤ p + 1, which is a contradiction. this completes the proof. case 2. let k/h ∼= ap ′ , where p ′ and p′ − 2 are primes. cubo 13, 2 (2011) some new characterizations for pgl(2, q) 157 if p = p′, for p′ ≥ 7, then we can get a contradiction similarly to the previous case. so p = 5 and k/h ∼= a5 ∼= psl(2, 5). since k/h ≤ g/h ≤ aut(k/h), we have psl(2, 5) ≤ g/h ≤ pgl(2, 5). hence g/h is isomorphic to psl(2, 5) or pgl(2, 5). if g/h ∼= psl(2, 5), then |h| = 2. but h � g, which implies that h ⊆ z(g) and we get a contradiction. so g/h ∼= pgl(2, 5), which implies that h = 1 and g ∼= pgl(2, 5). let p = p′ − 2. since p′ | |ap ′ |, we have p ′ | |g| = p(p2 − 1). but we know that p = p′ − 2 is the greatest prime divisor of |g|, which is a contradiction. case 3. let k/h be a sporadic simple group. using the tables in [10] we see that the odd order components of sporadic simple groups are prime. let s be a sporadic simple group and k/h ∼= s. since q is equal to the greatest odd order component of k/h, we have q = mi, such that mi = max{m2, m3, ..., mt(s)}. so q is a prime number. if s = j4, then q = p = 43. since 11 2 | |k/h|, we have 112 | (p2 − 1) = 432 − 1, which is a contradiction. if s = co2, then q = p = 23. since 7 | |k/h|, we have 7 | (23 2 − 1), which is a contradiction. the proof of other cases are similar and we omit them for convenience. if k/h is isomorphic to 2a3(2), 2f4(2) ′, a2(4), 2a5(2), e7(2), e7(3) or 2e6(2), then similarly we get a contradiction. in the sequel of the proof we consider simple groups of lie type. since the proofs of these cases are similar we state only a few of them. in all of the following cases p′ is an odd prime number and q′ is a prime power. case 4. let k/h ∼= ap ′−1(q ′), where (p′, q′) 6= (3, 2), (3, 4). by hypothesis we have q = (q′p ′ − 1)/((q′ − 1)(p′, q′ − 1)). hence q < q′p ′ − 1 < q′p ′ . then q2 − 1 < q′2p ′ . on the other hand, we know q′p ′ (p ′ −1)/2 | (q2 − 1) and therefore q′p ′ (p ′ −1)/2 < q′2p ′ . so p′(p′ − 1)/2 < 2p′ and hence p′ < 5. so p′ = 3 and q = (q′2 + q′ + 1)/(3, q′ − 1), which implies that q < 2q′2. therefore q2 − 1 < 4q′4 − 1. on the other hand, q′3(q′2 − 1)(q′ − 1) | (q2 − 1) and consequently q′3(q′2 − 1)(q′ − 1) < 4q′4 − 1. so q′ = 2, 3 or 4. since (p′, q′) 6= (3, 2), (3, 4), we have q′ = 3 and q = 13. then 33(32 − 1)(3 − 1) | (132 − 1), which is a contradiction. case 5. let k/h ∼= 2ap ′ (q ′), where (q′ + 1) | (p′ + 1) and (p′, q′) 6= (3, 3), (5, 2). in this case we have q = (q′p ′ + 1)/(q′ + 1). therefore q < q′p ′ + 1 < 2q′p ′ ≤ q′p ′+1 and hence q2 − 1 < q′2(p ′ +1). on the other hand, we have q′p ′ (p ′ +1)/2 | (q2 − 1). so we conclude that q′p ′ (p ′ +1)/2 < q′2(p ′ +1). hence p′(p′ + 1)/2 < 2(p′ + 1), which implies that p′ = 3. then (q′ + 1) | 4 and hence q′ = 3. so (p′, q′) = (3, 3), which is impossible. 158 b. khosravi, m. khatami & z. akhlaghi cubo 13, 2 (2011) case 6. let k/h ∼= bn(q ′), where n = 2m ≥ 4 and q′ is odd. therefore q = (q′n + 1)/2. so q < 2q′n < q′n+1. therefore q2 − 1 < q′2(n+1). on the other hand, we have q′n 2 | (q2 − 1) and consequently q′n 2 < q′2(n+1). so n2 < 2(n + 1), which implies that n = 2, and this is a contradiction. case 7. let k/h ∼= cn(q ′), where n = 2m ≥ 2. then q = (q′n + 1)/(2, q′ − 1). therefore q ≤ q′n + 1 < 2q′n ≤ q′n+1, which implies that q2 − 1 < q′2(n+1). on the other hand, we have q′n 2 | (q2 − 1), which implies that q′n 2 < q′2(n+1). so we have n2 < 2(n + 1) and hence n = 2. therefore q < 2q′2 and so q′4(q′2 − 1) < q2 − 1 < 4q′4, which is impossible. case 8. let k/h ∼= 2dp ′ (3), where p ′ = 2n + 1 ≥ 5. so we have q = (3p ′ + 1)/4 or q = (3p ′ −1 + 1)/2. if q = (3p ′ + 1)/4, then q < 3p ′ +1. on the other hand, we have 3p ′ (p ′ −1) | (q2 − 1), which implies that 3p ′ (p ′ −1) ≤ q2 − 1 < 32(p ′+1). therefore p′(p′ − 1) < 2(p′ + 1), and hence p′ ≤ 3, which is impossible. if q = (3p ′ −1 + 1)/2, then q < 3p ′ . on the other hand, 3p ′ (p ′ −1) | (q2 − 1), which implies that 3p ′ (p ′ −1) < 32p ′ , and so p′(p′ − 1) < 2p′, which is impossible. case 9. let k/h ∼= 2b2(q ′), where q′ = 22n+1 > 2. in this case we have q = q′ ± √ 2q′ + 1 or q = q′ − 1. if q = q′ ± √ 2q′ + 1, then q2 − 1 = q′2 + 4q′ ± 2 √ 2q′(q′ + 1). on the other hand, we have q′2 | (q2 − 1) and so q′ | (q′2 + 4q′ ± 2 √ 2q′(q′ + 1)), which implies that q′ ≤ 2 √ 2q′. hence q′2 ≤ 8q′. therefore q′ = 8 and so q = 5 or 13, which is a contradiction by q′2 | (q2 − 1). if q = q′ − 1, then q′2|(q′2 − 2q′), which is a contradiction. case 10. let k/h ∼= 2f4(q ′), where q′ = 22n+1 > 2. in this case we have q = q′2 ± √ 2q′3 + q′ ± √ 2q′ + 1. therefore q < 4q′2 < q′3 and so q2 − 1 < q′6. on the other hand, q′12 | (q2 − 1), which is a contradiction. case 11. let k/h ∼= a1(q ′), where 4|q′. by hypothesis we have q = q′ − 1 or q = q′ + 1. if q = q′ − 1, then q2 − 1 = q′2 − 2q′. but we know q′(q′ + 1) | (q2 − 1), which is a contradiction. if q = q′ + 1, then q2 − 1 = q′2 + 2q′. since q′(q′ − 1) | (q2 − 1), we conclude that (q′ − 1) | 3. so q′ = 4 and hence k/h ∼= a1(4) ∼= a5. by the proof of case 2 we have k/h ∼= pgl(2, 5). case 12. if k/h ∼= a1(q ′), where 4|(q′ − 1), then q = (q′ + 1)/2 or q = q′. if q = (q′ + 1)/2, then q2 − 1 = (q′2 − 3 + 2q′)/4. on the other hand, q′(q′ − 1) | (q2 − 1) cubo 13, 2 (2011) some new characterizations for pgl(2, q) 159 and hence q′(q′ − 1) ≤ (q′2 − 3 + 2q′)/4. so q′2 − 2q′ + 1 ≤ 0, which is a contradiction. if q = q′, then k/h ∼= a1(q) = psl(2, q). since k/h ≤ g/h and |g| = 2|psl(2, q)|, we conclude that |h| = 1 or 2. let |h| = 2. since h � g we have h ⊆ z(g), which is a contradiction. so h = 1. by lemma 2.8, g/k ≤ out(k/h) and |g/k| = 2. if g/k contains a field automorphism, then 2p ∈ πe(g), which is a contradiction. if g/k contains a diagonal-field automorphism, then g is the non-split extension of psl(2, q) by z2 and we know that the prime graph of g is the prime graph of psl(2, q) (see [12]), which is a contradiction. so a diagonal automorphism generates g/k and consequently g ∼= pgl(2, q). if k/h ∼= a1(q ′), where 4|(q′ + 1), then similarly we conclude that g ∼= pgl(2, q). 2 theorem 3.6. let g be a group such that ∇(g) ∼= ∇(m), where m = pgl(2, q) and q is a prime power. then g ∼= m. proof. if q = 2n, where n is an integer, then pgl(2, q) ∼= psl(2, q) and so lemma 2.10 implies that g ∼= m. if q is odd, then obviously the theorem follows from theorems 3.2 and 3.5. 2 remark 3.7. it is a well known conjecture of j. g. thompson that if g is a finite group with z(g) = 1 and m is a non-abelian simple group satisfying n(g) = n(m), then g ∼= m. we can give a positive answer to this conjecture for the group pgl(2, q) by our characterization of this group. corollary 3.8. let g be a finite group with z(g) = 1 and m = pgl(2, q), where q is a prime power. if n(g) = n(m), then g ∼= m. proof. by lemmas 2.6 and 2.7, if g and m are two finite groups satisfying the conditions of corollary 3.8, then oc(g) = oc(m). so using theorem 3.5 we get the result. 2 remark 3.9. w. shi and j. bi in [16] put forward the following conjecture: conjecture. let g be a group and m be a finite simple group. then g ∼= m if and only if (i) |g| = |m|, and, (ii) πe(g) = πe(m), where πe(g) denotes the set of orders of elements in g. this conjecture is valid for sporadic simple groups [13], alternating groups [17], and some simple groups of lie type [14, 15, 16]. as a consequence of theorem 3.5, we prove the validity of this conjecture for the almost simple group pgl(2, q), where q is a prime power. corollary 3.10. let g be a finite group and m = pgl(2, q), where q is a prime power. if 160 b. khosravi, m. khatami & z. akhlaghi cubo 13, 2 (2011) |g| = |m| and πe(g) = πe(m), then g ∼= m. proof. by assumption we have oc(g) = oc(m). thus the corollary follows from theorem 3.5. 2 proposition 3.11. let g be a group such that γ1(g) ∼= γ1(m), where m = pgl(2, q) and q is a prime power. then g ∼= m. proof. first we show that |g| = |m|. by lemma 2.12 we have |cyc(g)| divides |m| − |cyc(m)|. since cyc(m) ≤ z(m) = 1, it follows that |cyc(g)| divides |m|−1. on the other hand, by lemma 2.12, |cyc(g)| divides |cycm(x)| − |cyc(m)|, where x ∈ m \ cyc(m). let x be a p-element of m. we claim that 〈x〉 = cycm(x). we know that 〈x〉 ⊆ cycm(x) and so it is enough to prove that cycm(x) ⊆ 〈x〉. on the contrary let y ∈ cycm(x) \ 〈x〉 and hence 〈y, x〉 is cyclic. if y is a p-element, then we know that 〈y, x〉 has only one subgroup of order p and so 〈x〉 = 〈y〉, which is a contradiction. therefore y is not a p-element. so we have an element of order po(y), which is a contradiction by the structure of γ (m). so p = |〈x〉| = |cycm(x)|. therefore |cyc(g)| divides p − 1 and p − 1 divides |m|. we know that |cyc(g)| divides |m| − 1 and so |cyc(g)| = 1 and |g| = |m|. now using lemma 2.13 we conclude that πe(g) = πe(m) and by corollary 3.10 the proof is complete. 2 remark 3.12. we note that in the main theorem of [5] it is proved that pgl(2, q) is uniquely determined by πe(g). received: february 2009. revised: august 2010. references [1] a. abdollahi, s. akbari and h. r. maimani, non-commuting graph of 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(in chinese) introduction cubo a mathematical journal vol.18, no¯ 01, (59–68). december 2016 submanifolds of a (k, µ)-contact manifold m.s. siddesha, c.s. bagewadi department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india. mssiddesha@gmail.com, prof bagewadi@yahoo.co.in abstract the object of the present paper is to study submanifolds of (k, µ)-contact manifolds. we find the necessary and sufficient conditions for a submanifolds of (k, µ)-contact manifolds to be invariant and anti-invariant. also, we examine the integrability of the distributions involved in the definition of cr-submanifolds of (k, µ)-contact manifolds. resumen el objeto del presente art́ıculo es estudiar subvariedades de variedades (k, µ)-contacto. encontramos las condiciones necesarias y suficientes para que subvariedades de variedades (k, µ)-contacto sean invariantes y anti-invariantes. también examinamos la integrabilidad de las distribuciones involucradas en la definición de subvariedades cr de variedades (k, µ)-contacto. keywords and phrases: (k, µ)-contact manifold; invariant submanifold; anti-invariant submanifold. 2010 ams mathematics subject classification: 53c15, 53c40. 60 m.s. siddesha, c.s. bagewadi cubo 18, 1 (2016) 1 introduction in 1995 blair, koufogiorgos and papantoniou [4] introduced the notion of contact metric manifolds with characteristic vector field ξ belonging to the (k, µ)-nullity distribution and such type of manifolds are called (k, µ)-contact manifolds. to study the geometry of an unknown manifold, it is sometime convenient and yet interesting to first imbed it into a rather known manifold and then study its geometry side by side that of the ambient manifold. this approach gave birth to the introduction of submanifold theory. the study of complex submanifolds of a kähler manifold from differential geometric points of view was initiated by calabi [7] in the early 1950’s and it was continued by several geometers like blair, ogiue [5], chen, verheyen [8], yano, kon [19], yano, ishihara [18], kon [11] and many others. from then onwards submanifolds of a contact manifold have been major area of research and these submanifolds are divided into several types, mainly invariant, anti-invariant and semi-invariant. the study of submanifolds of different contact manifolds is carried out from 1970 onwards by several authors, for example [9]-[12], while the study of submanifolds of (k, µ)-contact manifold have been done by montano et al [13], avjit sarkar et al [1], tripathi et al [16], siddesha and bagewadi [14] and others. in [13], the authors have shown that invariant submanifolds of (k, µ)-contact manifold carries a (k, µ) structure and prove the totally geodesicity of invariant submanifolds when the second fundamental form is parallel. later authors of [2] and [15] continued the work of above authors and they proved the totally geodesicity of recurrent, generalized recurrent of second fundamental form and semiparallel, pseudoparallel, ricci-generalized pseudoparallel submanifolds. motivated by these studies of the above authors [2, 13, 16], in the present paper we find the necessary and sufficient conditions for the submanifolds to be invariant and anti-invariant. also we study cr-submanifolds of (k, µ)-contact manifold and examine the integrability of the horizontal and vertical distributions involved in the definition of cr-submanifolds of (k, µ)-contact manifold. the paper is organized as follows: in section 2, we give a brief account of (k, µ)-contact manifolds and necessary details about submanifolds. in section 3, we show the existence of an invariant and anti-invariant submanifold, while, the section 4 deals with non-existence of an anti-invariant submanifold. lastly in section 5, we consider cr-submanifolds of (k, µ)-contact manifold with distributions d and d⊥, we find the conditions under which d⊥ is integrable or totally geodesic. 2 preliminaries a contact manifold is a c∞-(2n + 1) manifold m̃2n+1 equipped with a global 1-form η such that η ∧ (dη)n ̸= 0 everywhere on m̃2n+1. given a contact form η it is well known that there exists a unique vector field ξ, called the characteristic vector field of η, such that η(ξ) = 1 and dη(x, ξ) = 0 for every vector field x on m̃2n+1. a riemannian metric g is said to be associated cubo 18, 1 (2016) submanifolds of a (k, µ)-contact manifold 61 metric if there exists a tensor field φ of type (1,1) such that φ2x = −x + η(x)ξ, η(ξ) = 1, η ◦ φ = 0, φξ = 0, (2.1) g(φx, φy) = g(x, y) − η(x)η(y), g(x, ξ) = η(x), (2.2) for all vector fields x, y on m̃. then the structure (φ, ξ, η, g) on m̃ is called a contact metric structure and the manifold equipped with such a structure is called a contact metric manifold [3]. we now define a (1, 1) tensor field h by h = 1 2 lξφ, where l denotes the lie differentiation, then h is symmetric and satisfies hφ = −φh. further, a q-dimensional distribution on a manifold m is defined as a mapping d on m which assigns to each point p ∈ m, a q-dimensional subspace dp of tpm. the (k, µ)-nullity distribution of a contact metric manifold m̃(φ, ξ, η, g) is a distribution n(k, µ) : p → np(k, µ) = {z ∈ tpm : r̃(x, y)z = k[g̃(y, z)x − g̃(x, z)y] + µ[g̃(y, z)hx − g̃(x, z)hy]}, for all x, y ∈ tm̃. hence if the characteristic vector field ξ belongs to the (k, µ) nullity distribution, then we have r̃(x, y)ξ = k[η(y)x − η(x)y] + µ[η(y)hx − η(x)hy]. (2.3) the contact metric manifold satisfying the relation (2.3) is called (k, µ) contact metric manifold [4]. it consists of both k-nullity distribution for µ = 0 and sasakian for k = 1. a (k, µ)-contact metric manifold m̃(φ, ξ, η, g) satisfies (∇̃xφ)y = g(x + hx, y)ξ − η(y)(x + hx), (2.4) for all x, y ∈ tm̃, where ∇̃ denotes the riemannian connection with respect to g. from (2.4), we have ∇̃xξ = −φx − φhx, (2.5) for all x, y ∈ tm̃. again, if we put ω(x, y) = g(x, φy), then ω is a skew-symmetric (0, 2) tensor field [4]. thus we have from (2.5) ω(x + hx, y) = (∇̃xη)(y). (2.6) also from (2.4), it follows that (∇̃zω)(x, y) = g(x, (∇̃zφ)y) = −g((∇̃zφ)x, y), (2.7) (∇̃zω)(x, y) = g(z + hz, y)η(x) − η(y)g(x, z + hz), (2.8) for any x, y ∈ tm̃ let m be a riemannian submanifold of a (k, µ)-contact manifold m̃. then the gauss and wein62 m.s. siddesha, c.s. bagewadi cubo 18, 1 (2016) garten formulae are given by ∇̃xy = ∇xy + σ(x, y), (2.9) ∇̃xn = −anx + ∇ ⊥ xn, (2.10) for all x, y ∈ tm and each n ∈ t⊥m, where ∇ is the levi-civita connection on m, ∇⊥ is the normal connection on the normal bundle t⊥m, σ is the second fundamental form of m and a is the shape operator with respect to the normal connection n. then the shape operator a and the second fundamental form σ are related by g(σ(x, y), n) = g(anx, y), (2.11) for all x, y ∈ tm and n ∈ t⊥m. we denote by the same symbols g both metrics on m̃ and m. definition 1. a submanifold m is said to be (i) totally geodesic in m̃ if σ = 0 or equivalently an = 0 (2.12) for each n ∈ t⊥m. (ii) minimal in m̃ if the curvature vector h satisfies h = tr(σ) dimm = 0 (2.13) and (iii) totally umbilical if σ(x, y) = g(x, y)h. (2.14) put φx = tx+nx for any tangent vector field x, where tx (resp. nx) denotes the tangential (resp. normal) component of φx. similarly φv = tv + nv for any normal vector field v with tv tangent and nv normal to m. then from straightforward calculation and using (2.4), (2.9) and (2.10), we obtain lemma 2.1. let m be a submanifold of a (k, µ)-contact manifold (m̃, φ, ξ, η, g), then (∇xt)y − tσ(x, y) − anyx = g(x + hx, y)ξ − η(y)(x + hx), (2.15) (∇xn)y + σ(x, ty) − nσ(x, y) = 0, (2.16) for any vector fields x, y ∈ tm. 3 submanifolds of a (k, µ)-contact manifold in this section, we define invariant and anti-invariant submanifolds of (k, µ)-contact manifold and prove the existence. cubo 18, 1 (2016) submanifolds of a (k, µ)-contact manifold 63 a submanifold m of a (k, µ)-contact manifold m̃ is said to be invariant (resp. anti-invariant) submanifold of m̃ if for each x ∈ m, φ(txm) ⊂ txm (resp. φ(txm) ⊂ t ⊥ x m), here txm and t⊥x m are the tangent and normal bundles. we first prove the following lemma: lemma 3.1. for a submanifold m of a (k, µ)-contact manifold m̃, we have −φx − φhx = ∇xξ + σ(x, ξ), ξ ∈ tm, (3.1) −φx − φhx = −aξx + ∇ ⊥ xξ, ξ ∈ t ⊥m (3.2) η(anx) = 0, ξ ∈ t ⊥m (3.3) η(anx) = −g(φx + φhx, n), ξ ∈ tm (3.4) for each x ∈ tm and n ∈ t⊥m. proof. from (2.5) and (2.9), we get (3.1). also from (2.5) and (2.10), we obtain (3.2). again, in view of (2.2), (3.3) is obvious. now for ξ ∈ tm, and in view of (2.2), (2.5), (2.10) we get η(anx) = g(ξ, anx) = −g(ξ, ∇̃xn) = g(∇̃xξ, n) = −g(φx + φhx, n). this completes the proof of our lemma. theorem 3.1. let m be a submanifold of a (k, µ)-contact manifold m̃ such that the structure vector field ξ is tangent to m. then m is invariant if and only if σ(x, ξ) = 0, and m is antiinvariant if and only if ∇xξ = 0. since it is trivial from lemma 3.2., we omit to prove our theorem. theorem 3.2. if m is a totally umbilical submanifold of a (k, µ)-contact manifold m̃ such that the structure vector field ξ is tangent to m, then (i) m is necessarily minimal and consequently totally geodesic and (ii) m is an invariant submanifold of m̃ and ∇xξ ̸= 0. proof. let m be a totally umbilical. using (2.1), (2.2) and (3.1) in (2.14), we get 0 = σ(ξ, ξ) = g(ξ, ξ)h = h. hence in view of (2.13) and (2.14), we obtain (i). the second part follow from theorem 3.1. and the above (i). theorem 3.3. a submanifold m of a (k, µ)-contact manifold m̃ with structure vector field ξ normal to m is anti-invariant in m̃ if and only if aξx = 0. consequently, if m is totally geodesic, then it is anti-invariant. 64 m.s. siddesha, c.s. bagewadi cubo 18, 1 (2016) proof. since ξ is normal to m, by virtue of (2.10) and (3.2) yields g(−φx − φhx, y) = g(aξx, y) = g(σ(x, y), ξ), x, y ∈ tm, which provides the proof of our theorem. 4 non-existence of an anti-invariant distribution this section is devoted to study of anti-invariant distribution. a distribution d on a manifold m is said to be invariant under φ if φdx ⊂ dx for each x ∈ m and orthogonal complementary distribution d⊥ on m is said to be anti-invariant under φ if φd⊥x ⊂ d ⊥ x for each x ∈ m. now we define semi-invariant submanifold as follows: a submanifold m of a (k, µ)-contact manifold m̃ is said to be semi-invariant submanifold [6], if the following conditions are satisfied (i) tm = d⊕d⊥⊕{ξ}, where d, d⊥ are orthogonal distributions on m and {ξ} is the 1-dimensional distribution spanned by ξ, (ii) the distribution d is invariant by φ, (iii) the distribution d⊥ is anti-invariant under φ. the distribution d(resp. d⊥) is called the horizontal (resp. vertical) distribution. if both the distribution d and d⊥ are non-zero then the semi-invariant submanifold is called a proper semiinvariant submanifold. to prove the main result of this section first we prove the following lemmas: lemma 4.1. for a submanifold m of a (k, µ)-contact manifold m̃, we have (∇̃zω)(x, y) = g(aφyx, z) − ω(x, ∇zy) − ω(x, σ(z, y)) (4.1) for y ∈ d⊥, x, z ∈ tm. (∇̃zω)(x, y) = g(aφxy + aφyx, z) (4.2) for all x, y ∈ d⊥, z ∈ tm. proof. let y ∈ d⊥, z ∈ tm. then, by virtue of (2.11) and the fact φy ∈ t⊥m, we get (∇̃zφ)y = −aφyz + ∇ ⊥ zφy − φ(∇̃zy). (4.3) using this equation in (2.7), we can easily derive (4.1). next, in the special case of x ∈ d⊥, since φx ∈ t⊥m, (4.1) in view of (2.5) and (2.11) yields (4.2). lemma 4.2. let m be a submanifold of a (k, µ)-contact manifold m̃ and d⊥ ⊥ {ξ}. then we get (∇̃zω)(x, x) = 0, (4.4) cubo 18, 1 (2016) submanifolds of a (k, µ)-contact manifold 65 for x ∈ d⊥ and z ∈ tm,and consequently aφxx = 0, (4.5) for x ∈ d⊥. proof. since d⊥ ⊥ {ξ}, we have η(x) = 0 for any x ∈ d⊥ and hence in view of (2.8), we get (4.4). again (4.5) follows from (4.2) and (4.4). theorem 4.1. there does not exist any anti-invariant distribution d⊥ on a submanifold m of a (k, µ)-contact manifold m̃ if ξ is tangent to m and d⊥ ⊥ {ξ}. proof. since d⊥ ⊥ {ξ}, we get η(x) = 0 for any x ∈ d⊥. thus, from (2.2), (3.4) and (4.5), we have 0 = η(aφxx) = g(aφxx, ξ) = −g(φx, φx + φhx) = −g(x, x + hx), for any x ∈ d⊥. this implies g(x, x) = 0. hence, x must be zero vector. thus, if x is any arbitrary vector in d⊥ then we have x = 0. therefore, d⊥ = 0. this proves the theorem. hence by virtue of theorem 3.1, we have the following: corolary 1. a (k, µ)-contact manifold does not admit any proper semi-invariant submanifold. 5 cr-submanifolds of (k, µ)-contact manifold in this section, we shall see the integrability conditions of the involved distributions d and d⊥ in the definition of cr-submanifold m of a (k, µ)-contact manifold. a submanifold m is said to be cr-submanifold in m̃ if there exist two orthogonal complementary distributions d and d⊥ of tm such that ξ ∈ tm and (1) d is invariant by φ, i.e. φ(dp) ⊂ dp, ∀p ∈ m, (2) d⊥ is anti-invariant by φ, i.e. φ(d⊥p ) ⊂ t ⊥ p m, ∀p ∈ m. proposition 1. let m be a cr-submanifold of a (k, µ)-contact manifold m̃. then, d, d⊥ and d ⊕ d⊥ are ξ-parallel. proof. for any x ∈ d and y ∈ d⊥ g(∇ξx, ξ) = ξg(x, ξ) − g(x, ∇ξξ) = 0, g(∇ξx, y) = ξg(x, y) − g(x, ∇ξy) = g(t 2x, ∇ξy) = −g(tx, t∇ξy) = g(tx, ∇ξty) = 0, 66 m.s. siddesha, c.s. bagewadi cubo 18, 1 (2016) so ∇ξx ∈ d, that is d is ξ-parallel. similarly, we can proceed for d⊥. finally, if d and d⊥ are ξ-parallel, d ⊕ d⊥ also is. lemma 5.1. let m be a submanifold of a (k, µ)-contact manifold. then, 2g(x, ty) = η([x, y]) for all x, y orthogonal to ξ. proof. for a (k, µ)-contact manifold it holds that dη = φ. so 2g(x, ty) = 2φ(x, y) = 2dη(x, y) = η[x, y]. lemma 5.2. let m be a cr-submanifold of a (k, µ)-contact manifold. then, d⊥ is integrable if and only if dφ(x, y, z) = 0, for any x tangent to m, y, z ∈ d⊥. proof. consider x tangent to m and y, z ∈ d⊥. then, 3dφ(x, y, z) = x(φ(y, z)) + y(φ(z, x)) + z(φ(x, y)) −φ([x, y], z) − φ([z, x], y) − φ([y, z], x) = −g([y, z], φx) = g(φ[y, z], x), so dφ(x, y, z) = 0 if and only if [y, z] ∈ kert = d⊥⊕ < ξ >. this is equivalent to [y, z] + η[y, z]ξ ∈ d⊥, but, using lemma 5.5., η([y, z]) = 2g(x, ty) = 0. now we can state the following theorem: theorem 5.1. let m be a cr-submanifold of a (k, µ)-contact manifold. then, d⊥ is always integrable. proof. if m is a contact metric manifold, dφ = d2η = 0 and so, the result follows from lemma 5.6. lemma 5.3. let m be a cr-submanifold of a (k, µ)-contact manifold. then, d⊥⊕ < ξ > is integrable if and only if dφ(x, y, z) = 0, for any x tangent to m, y, z ∈ d⊥⊕ < ξ >. proof. given x ∈ tm, y, z ∈ d ⊕ d⊥, we have ty = tz = 0 and 3dφ(x, y, z) = −g([y, z], φx) = g(φ[y, z], x). so [y, z] is normal if and only if dφ(x, y, z) = 0, for all x tangent to m. again, from this lemma, we deduce: theorem 5.2. let m be a cr-submanifold of a (k, µ)-contact manifold. then, d⊥⊕ < ξ > is always integrable. cubo 18, 1 (2016) submanifolds of a (k, µ)-contact manifold 67 finally, we characterize the integrability of d⊕ < ξ > theorem 5.3. let m be a cr-submanifold of a (k, µ)-contact manifold. then, d⊕ < ξ > is integrable if and only if σ(x, ty) − σ(y, tx) = 0 for all x, y ∈ d⊕ < ξ >. proof. given x, y ∈ d⊕ < ξ >, [x, y] belongs to d⊕ < ξ > if and only if n[x, y] = 0. using (2.16), n[x, y] = n∇xy − n∇yx = ∇xny + σ(x, ty) − nσ(x, y) − ∇ynx − σ(y, tx) + nσ(x, y) = σ(x, ty) − σ(y, tx), from which the proof follows. theorem 5.4. let m be a cr-submanifold of a (k, µ)-contact manifold. then, m is locally the product m1 × m2, where m1 is a leaf of d⊕ < ξ > and m2 is a leaf of d ⊥ if and only if σ(x, ty) ∈ tm̃, for all x tangent to m, y ∈ d⊥. proof. we shall prove that both d⊕ < ξ > and d⊥ are involutive and their leaves are totally geodesic immersed in m, so m is locally the product of these leaves. for y ∈ d⊕ < ξ >, z ∈ d⊥, by virtue of (2.4) and (2.9), g(∇xy, z) = g(∇̃xy, z) = g(φ∇̃xy, φz) = g(∇̃xφy + g(x + hx, y)ξ − η(y)(x + hx), nz) = g(∇̃xty, nz) = g(σ(x, ty), nz). (5.1) so if x ∈ d⊕ < ξ >, ∇xy ∈ d⊕ < ξ > if and only if σ(x, ty) ∈ d̃. then d⊕ < ξ > is involutive and its leaf is totally geodesic immersed in m. similarly, from (5.1), if x ∈ d⊥, as g(∇xz, y) = −g(z, ∇xy) = −g(σ(x, ty), nz), we have that ∇xy ∈ d⊕ < ξ > if and only if σ(x, ty) ∈ d̃. in this case, we obtain that d⊕ < ξ > is also involutive and its leaf is totally geodesic immersed in m. acknowledgement: the authors are thankful to the referee for his/her valuable suggestions in improvement of the paper. references [1] avijit sarkar, md. showkat ali and dipankar biswas, on submanifolds of (k, µ)-contact metric manifolds, dhaka univ. j. sci. 61(1) (2013), 93-96. 68 m.s. siddesha, c.s. bagewadi cubo 18, 1 (2016) [2] avik de, a note on invariant submanifolds of (k, µ)-contact manifolds, ukranian mathematical j. 62(11) (2011), 1803-1809. [3] d.e. blair, contact manifolds in riemannian geometry, lecture notes in math., 509, springer-verlag, berlin (1976). 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[19] k. yano and m. kon, cr-submanifolds of kaehlerian and sasakian manifolds, birkhauser, boston-basel, (1983). cubo a mathematical journal vol.15, no¯ 03, (19–30). october 2013 composition operators in hyperbolic general besov-type spaces a. el-sayed ahmed 1,2 1sohag university, faculty of science, department of mathematics, 82524 sohag, egypt. 2taif university, faculty of science, mathematics, department, box 888 el-hawiyah, el-taif 5700, saudi arabia. ahsayed80@hotmail.com m. a. bakhit department of mathematics, faculty of science, assiut branch, al-azhar university, assiut 32861, egypt. mabakhit2007@hotmail.com abstract in this paper we introduce natural metrics in the hyperbolic α-bloch and hyperbolic general besov-type classes f∗(p, q, s). these classes are shown to be complete metric spaces with respect to the corresponding metrics. moreover, compact composition operators cφ acting from the hyperbolic α-bloch class to the class f ∗(p, q, s) are characterized by conditions depending on an analytic self-map φ : d → d. resumen en este art́ıculo introducimos una métrica natural en las clases hiperbólicas α-bloch y tipo besov generales. estas clases se muestra que son espacios métricos completos respecto de las métricas correspondientes. además se caracterizan los operadores de composición compactos cφ que actúan desde las clases hiperbólicas α-bloch en la clase f∗(p, q, s) por condiciones que dependen de la autoaplicación anaĺıtica φ : d → d. keywords and phrases: hyperbolic classes, composition operators, lipschitz continuous, αbloch space, f∗(p, q, s) class. 2010 ams mathematics subject classification: 47b38, 30d50, 30d45, 46e15. 20 a. el-sayed ahmed & m. a. bakhit cubo 15, 3 (2013) 1 introduction let d := {z ∈ c : |z| < 1} be the open unit disc of the complex plane c, ∂d it’s boundary. let h(d) denote the space of all analytic functions in d and let b(d) be the subset of h(d) consisting of those f ∈ h(d) for which |f(z)| < 1 for all z ∈ d. also, da(z) be the normalized area measure on d so that a(d) ≡ 1. let the green’s function of d be defined as g(z, a) = log 1 |ϕa(z)| , where ϕa(z) = a−z 1−āz , for z, a ∈ d is the möbius transformation related to the point a ∈ d. if (x, d) is a metric space, we denote the open and closed balls with center x and radius r > 0 by b(x, r) := {y ∈ x : d(y, x) < r} and b̄(x, r) := {y ∈ x : d(x, y) ≤ r}, respectively. hyperbolic function classes are usually defined by using either the hyperbolic derivative f∗(z) = |f ′ (z)| 1−|f(z)|2 of f ∈ b(d), or the hyperbolic distance ρ(f(z), 0) := 1 2 log ( 1+|f(z)| 1−|f(z)| ) between f(z) and zero. a function f ∈ b(d) is said to belong to the hyperbolic α-bloch class b∗α if ‖f‖b∗α = sup z∈d f∗(z)(1 − |z|2)α < ∞, the little hyperbolic bloch-type class b∗α,0 consists of all f ∈ b ∗ α such that lim |z|→1 f∗(z)(1 − |z|2)α = 0. the usual α-bloch spaces bα and bα,0 are defined as the sets of those f ∈ h(d) for which ‖f‖bα = sup z∈d |f′(z)|(1 − |z|2)α < ∞, and lim |z|→1 |f′(z)|(1 − |z|2)α = 0, respectively. it is obvious that b∗α is not a linear space since the sum of two functions in b(d) does not necessarily belong to b(d). we now turn to consider hyperbolic f(p, q, s) type classes, which will be called f∗(p, q, s). for 0 < p, s < ∞, −2 < q < ∞, the hyperbolic class f∗(p, q, s) consists of those functions f ∈ b(d) for which (see [7]) ‖f‖ p f∗(p,q,s) = sup a∈d ∫ d (f∗(z))p(1 − |z|2)qgs(z, a)da(z) < ∞. moreover, we say that f ∈ f∗(p, q, s) belongs to the class f∗0(p, q, s) if lim |a|→1 ∫ d (f∗(z))p(1 − |z|2)qgs(z, a)da(z) = 0. cubo 15, 3 (2013) composition operators in hyperbolic general besov-type spaces 21 the usual general besov-type spaces f(p, q, s) (defined using the conventional derivative f′ instead of f∗) and their norms are denoted by the same symbols but with f′. yamashita was probably the first one considered systematically hyperbolic function classes. he introduced and studied hyperbolic hardy, bmoa and dirichlet classes in [14, 15, 16] and others. more recently, smith studied inner functions in the hyperbolic little bloch-class [11], and the hyperbolic counterparts of the qp spaces were studied by li in [7] and li et. al. in [8]. further, hyperbolic qp classes and composition operators studied by pérez-gonzález et. al. in [10]. very recently the first author in [1], gave some characterizations of hyperbolic q(p, α) classes and the hyperbolic (p, α)-bloch classes by composition operators. in this paper we will study the hyperbolic α-bloch classes b∗α and the general hyperbolic f∗(p, q, s) type classes. we will also give some results to characterize lipschitz continuous and compact composition operators mapping from the hyperbolic α-bloch class b∗α to f ∗(p, q, s) class by conditions depending on the symbol φ only. note that the general hyperbolic f∗(p, q, s) type classes include the class of so-called q∗p classes and the class of (hyperbolic) besov class b∗p. thus, the results are generalizations of the recent results of pérez-gonzález, rättyä and taskinen [10]. for any holomorphic self-mapping φ of d. the symbol φ induces a linear composition operator cφ(f) = f ◦ φ from h(d) or b(d) into itself. the study of composition operator cφ acting on spaces of analytic functions has engaged many analysts for many years (see e.g. [2, 3, 4, 5, 8, 9, 17] and others). recall that a linear operator t : x → y is said to be bounded if there exists a constant c > 0 such that ‖t(f)‖y ≤ c‖f‖x for all maps f ∈ x. by elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the lipschitz-continuity. moreover, t : x → y is said to be compact if it takes bounded sets in x to sets in y which have compact closure. for banach spaces x and y contained in b(d) or h(d), t : x → y is compact if and only if for each bounded sequence {xn} ∈ x, the sequence {txn} ∈ y contains a subsequence converging to a function f ∈ y. definition 1.1. a composition operator cφ : b ∗ α → f ∗(p, q, s) is said to be bounded, if there is a positive constant c such that ‖cφf‖f∗(p,q,s) ≤ c‖f‖b∗α for all f ∈ b ∗ α. definition 1.2. a composition operator cφ : b ∗ α → f ∗(p, q, s) is said to be compact, if it maps any ball in b∗α onto a precompact set in f ∗(p, q, s). the following lemma follows by standard arguments similar to those outline in lemma 3.8 of [12]. hence we omit the proof. lemma 1.3. assume φ is a holomorphic mapping from d into itself. let 0 < p, s < ∞, −1 < 22 a. el-sayed ahmed & m. a. bakhit cubo 15, 3 (2013) q < ∞ and 0 < α < ∞. then cφ : b∗α → f ∗(p, q, s) is compact if and only if for any bounded sequence {fn}n∈n ∈ b ∗ α which converges to zero uniformly on compact subsets of d as n → ∞, we have lim n→∞ ‖cφfn‖f∗(p,q,s) = 0. the following lemma can be found in [6], theorem 2.1.1. lemma 1.4. let 0 < α < ∞, then there exist two holomorphic maps f, g : d → c such that for some constant c, ( f′(z) + g′(z) ) (1 − |z|2)α ≥ c > 0, for each z ∈ d. 2 hyperbolic classes and natural metrics in this section we introduce natural metrics on the hyperbolic α-bloch classes b∗α and the classes f∗(p, q, s). let 0 < p, s < ∞, −2 < q < ∞ and 0 < α < 1. first, we can find a natural metric in b∗α (see [10]) by defining d(f, g; b∗α) := db∗α(f, g) + ‖f − g‖bα + |f(0) − g(0)|, (1) where db∗α(f, g) := sup z∈d ∣ ∣ ∣ ∣ f′(z) 1 − |f(z)|2 − g′(z) 1 − |g(z)|2 ∣ ∣ ∣ ∣ (1 − |z|2)α, for f, g ∈ b∗α. the presence of the conventional α-bloch-norm here perhaps unexpected. it is motivated by example (see [10], example in section 7 ). it shows the phenomenon that, though trivially db∗ α (f, 0) ≥ ‖f‖bα for all f ∈ b ∗ α, the same does no more hold for the differences of two functions: there does not even exist a constant c > 0 such that sup z∈d ∣ ∣ ∣ ∣ f′(z) 1 − |f(z)|2 − g′(z) 1 − |g(z)|2 ∣ ∣ ∣ ∣ (1 − |z|2)α ≥ c‖f − g‖bα would hold for all f, g ∈ b∗α, 0 < α < 1. for f, g ∈ f∗(p, q, s), define their distance by d(f, g; f∗(p, q, s)) := df∗(f, g) + ‖f − g‖f(p,q,s) + |f(0) − g(0)|, where df∗ (f, g) := ( sup z∈d ∫ d ∣ ∣ ∣ ∣ f′(z) 1 − |f(z)|2 − g′(z) 1 − |g(z)|2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) ) 1 p . the following characterization of complete metric space d(., .; b∗µ) can be proved as in proposition 2.1 of [10]. cubo 15, 3 (2013) composition operators in hyperbolic general besov-type spaces 23 proposition 2.1. the class b∗α equipped with the metric d(., .; b ∗ α) is a complete metric space. moreover, b∗α,0 is a closed (and therefore complete) subspace of b ∗ α. now we prove the following proposition proposition 2.2. the class f∗(p, q, s) equipped with the metric d(., .; f∗(p, q, s)) is a complete metric space. moreover, f∗0(p, q, s) is a closed (and therefore complete) subspace of f ∗(p, q, s). proof. for f, g, h ∈ f∗(p, q, s), then clearly • d(f, g; f∗(p, q, s)) ≥ 0, • d(f, f; f∗(p, q, s)) = 0, • d(f, g; f∗(p, q, s)) = 0 implies f = g. • d(f, g; f∗(p, q, s)) = d(g, f; f∗(p, q, s)), • d(f, h; f∗(p, q, s)) ≤ d(f, g; f∗(p, q, s)) + d(g, h; f∗(p, q, s)). hence, d is metric on f∗(p, q, s). for the completeness proof, let (fn) ∞ n=0 be a cauchy sequence in the metric space f ∗(p, q, s), that is, for any ε > 0 there is an n = n(ε) ∈ n such that d(fn, fm) < ε, for all n, m > n. since fn ∈ b(d) such that fn converges to f uniformly on compact subsets of d. let m > n and 0 < r < 1. then fatou’s lemma yields ∫ d(0,r) ∣ ∣ ∣ ∣ f′(z) 1 − |f(z)|2 − f′m(z) 1 − |fm(z)| 2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) = ∫ d(0,r) lim n→∞ ∣ ∣ ∣ ∣ f′n(z) 1 − |fn(z)| 2 − f′m(z) 1 − |fm(z)| 2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) ≤ lim n→∞ ∫ d ∣ ∣ ∣ ∣ f′n(z) 1 − |fn(z)| 2 − f′m(z) 1 − |fm(z)| 2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) ≤ εp. (2) by letting r → 1−, it follows from inequalities (2) and (a + b)p ≤ 2p(ap + bp) that ∫ d (f∗(z))p(1 − |z|2)qgs(z, a)da(z) ≤ 2pεp + 2p ∫ d (f∗m(z)) p(1 − |z|2)qgs(z, a)da(z). (3) this yields ‖f‖ p f∗(p,q,s) ≤ 2pεp + 2p‖fm‖ p f∗(p,q,s) , and thus f ∈ f∗(p, q, s). we also find that fn → f with respect to the metric of f∗(p, q, s). the second part of the assertion follows by (3). 24 a. el-sayed ahmed & m. a. bakhit cubo 15, 3 (2013) 3 compactness of cφ in hyperbolic classes for 0 < p, s < ∞, −2 < q < ∞ and 0 < α < ∞. we define the following notations: φφ(p, q, s, a) = ∫ d |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) and ωφ,r(p, q, s, a) = ∫ |φ|≥r |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z). theorem 3.1. assume φ is a holomorphic mapping from d into itself. let 0 ≤ p < ∞, 0 ≤ s ≤ 1, −1 < q < ∞ and 0 < α ≤ 1. then the following are equivalent: (i) cφ : b ∗ α → f ∗(p, q, s) is bounded; (ii) cφ : b ∗ α → f ∗(p, q, s) is lipschitz continuous; (iii) sup a∈d φφ(p, q, s, a) < ∞. proof. first, assume that (i) holds, then there exists a constant c such that ‖cφf‖f∗(p,q,s) ≤ c‖f‖b∗α, for all f ∈ b ∗ α. for given f ∈ b∗α, the function ft(z) = f(tz), where 0 < t < 1, belongs to b ∗ α with the property ‖ft‖b∗α ≤ ‖f‖b∗α. let f, g be the functions from lemma 1.4, such that 1 (1 − |z|2)α ≤ f∗(z) + g∗(z), for all z ∈ d, so that |φ′(z)| (1 − |φ(z)|)α ≤ (f ◦ φ)∗(z) + (g ◦ φ)∗(z). thus, the inequalities ∫ d |tφ′(z)|p (1 − |tφ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) ≤ 2p ∫ d [ ( (f ◦ tφ)∗(z) )p + ( (g ◦ tφ)∗(z) )p ] (1 − |z|2)qgs(z, a)da(z) ≤ 2p‖cφ‖ p ( ‖f‖ p b∗α + ‖g‖ p b∗α ) . this estimate together with the fatou’s lemma implies (iii). cubo 15, 3 (2013) composition operators in hyperbolic general besov-type spaces 25 conversely, assuming that (iii) holds and that f ∈ b∗α, we see that sup a∈d ∫ d ( (f ◦ φ)∗(z) )p (1 − |z|2)qgs(z, a)da(z)) = sup a∈d ∫ d ( f∗(φ(z)) )p |φ′(z)|p(1 − |z|2)qgs(z, a)da(z) ≤ ‖f‖ p b∗α sup a∈d ∫ d |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z). hence, it follows that (i) holds. (ii)⇐⇒(iii). assume first that cφ : b∗α → f ∗(p, q, s) is lipschitz continuous, that is, there exists a positive constant c such that d(f ◦ φ, g ◦ φ; f∗(p, q, s)) ≤ cd(f, g; b∗α), for all f, g ∈ b ∗ α. taking g = 0, this implies ‖f ◦ φ‖f∗(p,q,s) ≤ c ( ‖f‖b∗α + ‖f‖bα + |f(0)| ) , for all f ∈ b∗α. (4) the assertion (iii) for α = 1 follows by choosing f(z) = z in (4). if 0 < α < 1, then |f(z)| = ∣ ∣ ∣ ∣ ∫z 0 f′(s)ds + f(0) ∣ ∣ ∣ ∣ ≤ ‖f‖bα ∫ |z| 0 dx (1 − x2)α + |f(0)| ≤ ‖f‖bα (1 − α) + |f(0)|, and |f(z)| ≤ tanh−1(|z|)‖f‖b1 + |f(0)|, where tanh −1(.) stands for inverse hyperbolic tangent function. then, for 0 < α < 1, we deduce that ∣ ∣f(φ(0)) − g(φ(0)) ∣ ∣ ≤ ‖f − g‖bα (1 − α) + |f(0) − g(0)|. (5) moreover, lemma 1.4 implies the existence of f, g ∈ b∗α such that ( f′(z) + g′(z) ) (1 − |z|2)α ≥ c > 0, for all z ∈ d. (6) combining (4) and (6) we obtain ‖f‖b∗α + ‖g‖b∗α + ‖f‖bα + ‖g‖bα + |f(0)| + |g(0)| ≥ c ∫ d |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) ≥ c φφ(α, p, q, s, a), for which the assertion (iii) follows. 26 a. el-sayed ahmed & m. a. bakhit cubo 15, 3 (2013) assume now that (iii) is satisfied, we have from (5) that d(f ◦ φ, g ◦ φ; f∗(p, q, s)) = df∗ (f ◦ φ, g ◦ φ) + ‖f ◦ φ − g ◦ φ‖f(p,q,s) + ∣ ∣f(φ(0)) − g(φ(0)) ∣ ∣ ≤ db∗α(f, g) ( sup a∈d ∫ d |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) ) 1 p +‖f − g‖bα ( sup a∈d ∫ d |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) ) 1 p + ‖f − g‖bα (1 − α) + |f(0) − g(0)| ≤ c′d(f, g; b∗α). thus cφ : b ∗ α → f ∗(p, q, s) is lipschitz continuous and the proof is completed. remark 3.2. theorem 3.1 shows that cφ : b ∗ α → f ∗(p, q, s) is bounded if and only if it is lipschitz-continuous, that is, if there exists a positive constant c such that d(f ◦ φ, g ◦ φ; f∗(p, q, s)) ≤ cd(f, g; b∗α), for all f, g ∈ b ∗ α. by elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the lipschitz-continuity. so, our result for composition operators in hyperbolic spaces is the correct and natural generalization of the linear operator theory. the following observation is sometimes useful. proposition 3.3. assume φ is a holomorphic mapping from d into itself. let 0 < p, s < ∞, −1 < q < ∞ and 0 < α < ∞. if cφ : b∗α → f ∗(p, q, s) is compact, it maps closed balls onto compact sets. proof. if b ⊂ b∗α is a closed ball and g ∈ f ∗(p, q, s) belongs to the closure of cφ(b), we can find a sequence (fn) ∞ n=1 ⊂ b such that fn ◦ φ converges to g ∈ f ∗(p, q, s) as n → ∞. but (fn)∞n=1 is a normal family, hence it has a subsequence (fnj) ∞ j=1 converging uniformly on the compact subsets of d to an analytic function f. as in earlier arguments of proposition 2.1 in [10], we get a positive estimate which shows that f must belong to the closed ball b. on the other hand, also the sequence (fnj ◦ φ) ∞ j=1 converges uniformly on compact subsets to an analytic function, which is g ∈ f∗(p, q, s). we get g = f◦φ, i.e. g belongs to cφ(b). thus, this set is closed and also compact. compactness of composition operators can be characterized in full analogy with the linear case. cubo 15, 3 (2013) composition operators in hyperbolic general besov-type spaces 27 theorem 3.4. assume φ is a holomorphic mapping from d into itself. let 0 < p < ∞, −1 < q < ∞, 0 ≤ s ≤ 1 and 0 < α ≤ 1. then the following are equivalent: (i) cφ : b ∗ α → f ∗(p, q, s) is compact; (ii) lim r→1− sup a∈d ωφ,r(p, q, s, a) = 0. proof. we first assume that (ii) holds. let b := b̄(g, δ) ⊂ b∗α, where g ∈ b ∗ α and δ > 0, be a closed ball, and let (fn) ∞ n=1 ⊂ b be any sequence. we show that its image has a convergent subsequence in f∗(p, q, s), which proves the compactness of cφ by definition. again, (fn) ∞ n=1 ⊂ b(d) implies that, there is a subsequence (fnj) ∞ j=1 which converges uniformly on the compact subsets of d to an analytic function f. by the cauchy formula for the derivative of an analytic function, also the sequence (f′nj) ∞ j=1 converges uniformly on compact subsets of d to f′. it follows that also the sequences (fnj ◦ φ) ∞ j=1 and (f ′ nj ◦ φ)∞j=1 converge uniformly on compact subsets of d to f◦φ and f′ ◦φ, respectively. moreover, f ∈ b ⊂ b∗α since for any fixed r, 0 < r < 1, the uniform convergence yield d(f, g; b∗α) ≤ δ (see [10] pp.130). let ε > 0. since (ii) is satisfied, we may fix r, 0 < r < 1, such that sup a∈d ∫ |φ(z)|≥r |φ(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) ≤ ε. by the uniform convergence, we may fix n1 ∈ n such that |fnj ◦ φ(0) − f ◦ φ(0)| ≤ ε, for all j ≥ n1. (7) the condition (ii) is known to imply the compactness of cφ : bα → f(p, q, s), hence, possibly to passing once more to a subsequence and adjusting the notations, we may assume that ‖fnj ◦ φ − f ◦ φ‖f(p,q,s) ≤ ε, for all j ≥ n2, for some n2 ∈ n. (8) now let i1(a, r) = sup a∈d ∫ |φ(z)|≥r [ (fnj ◦ φ) ∗(z) − (g ◦ φ)∗(z) ]p (1 − |z|2)qgs(z, a)da(z), and i2(a, r) = sup a∈d ∫ |φ(z)|≤r [ (fnj ◦ φ) ∗(z) − (g ◦ φ)∗(z) ]p (1 − |z|2)qgs(z, a)da(z). 28 a. el-sayed ahmed & m. a. bakhit cubo 15, 3 (2013) since (fnj) ∞ j=1 ⊂ b and f ∈ b, it follows from (1) that i1(a, r) = sup a∈d ∫ |φ(z)|≥r [ (fnj ◦ φ) ∗(z) − (g ◦ φ)∗(z) ]p (1 − |z|2)qgs(z, a)da(z) ≤ sup a∈d ∫ |φ(z)|≥r ∣ ∣ ∣ ∣ (fnj ◦ φ) ′(z) 1 − |(fnj ◦ φ)(z)| 2 − (g ◦ φ)′(z) 1 − |(g ◦ φ)(z)|2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) = sup a∈d ∫ |φ(z)|≥r m(fnj, g, φ; α, p)(1 − |z| 2)qgs(z, a)da(z) ≤ db∗α(fnj , f) sup a∈d ∫ |φ(z)|≥r |φ(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z), where m(fnj, g, φ; α, p) = ∣ ∣ ∣ ∣ ( f′nj(φ(z)) 1 − |fnj(φ(z))| 2 − g′(φ(z)) 1 − |g((φ(z)|2 ) (1 − |φ(z)|2)α ∣ ∣ ∣ ∣ p∣ ∣ ∣ ∣ φ′(z) (1 − |φ(z)|2)α ∣ ∣ ∣ ∣ p . hence, i1(a, r) ≤ 2δ ε. (9) on the other hand, by the uniform convergence on compact subsets of d, we can find an n3 ∈ n such that for all j ≥ n3, ∣ ∣ ∣ ∣ f′nj(φ(z)) 1 − |fnj(φ(z))| 2 − f′(φ(z)) 1 − |f(φ(z))|2 ∣ ∣ ∣ ∣ ≤ ε for all z with |φ(z)| ≤ r. hence, for such j, i2(a, r) = sup a∈d ∫ |φ(z)|≤r [ (fnj ◦ φ) ∗(z) − (g ◦ φ)∗(z) ]p (1 − |z|2)qgs(z, a)da(z) ≤ sup a∈d ∫ |φ(z)|≤r ∣ ∣ ∣ ∣ (fnj ◦ φ) ′(z) 1 − |(fnj ◦ φ)(z)| 2 − (g ◦ φ)′(z) 1 − |(g ◦ φ)(z)|2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) ≤ ε ( sup a∈d ∫ |φ(z)|≤r |φ(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z) ) 1 p ≤ cε, hence, i2(a, r) ≤ c ε. (10) where c is the bounded obtained from (iii) of theorem 3.1. combining (7), (8), (9) and (10) we deduce that fnj → f in f ∗(p, q, s). as for the converse direction, let fn(z) := 1 2 nα−1zn for all n ∈ n, n ≥ 2. then the sequence (fn) ∞ n=1 belongs to the ball b̄(0, 3) ⊂ b ∗ α(see [10] pp.131). we are assuming that cφ maps the closed ball b̄(0, 3) ⊂ b ∗ α into a compact subset of f ∗(p, q, s), hence, there exists an unbounded increasing subsequence (fnj ) ∞ j=1 such that the image subsequence (cφfnj) ∞ j=1 converges with respect to the norm. since, both (fn) ∞ n=1 and (cφfnj) ∞ j=1 converge to cubo 15, 3 (2013) composition operators in hyperbolic general besov-type spaces 29 the zero function uniformly on compact subsets of d, the limit of the latter sequence must be 0. hence, ‖nα−1j φ nj ‖f∗(p,q,s) → 0, as j → ∞. (11) now let rj = 1 − 1 nj . for all numbers a, rj ≤ a < 1, we have the estimate (see [10]) nαj a nj−1 1 − anj ≥ 1 e(1 − a)α (12) using (12) we obtain ‖nα−1j φ nj‖ p f∗(p,q,s) ≥ sup a∈d ∫ |φ|≥rj ∣ ∣ ∣ ∣ nj α(φ(z))nj−1φ′(z) 1 − |φnj(z)|2 ∣ ∣ ∣ ∣ p (1 − |z|2)qgs(z, a)da(z) ≥ 1 (2e)p sup a∈d ∫ |φ|≥rj |φ′(z)|p (1 − |φ(z)|2)αp (1 − |z|2)qgs(z, a)da(z). hence, the condition (ii) follows. received: february 2012. accepted: november 2012. references [1] a. el-sayed ahmed, natural metrics and composition operators in generalized hyperbolic function spaces, journal of inequalities and applications, 185(2012), 1-12. [2] a. el-sayed ahmed and m. a. bakhit, composition operators on some holomorphic banach function spaces, mathematica scandinavica, 104(2)(2009), 275-295. [3] a. el-sayed ahmed and m. a. bakhit, composition operators acting between some weighted möbius invariant spaces, ann. funct. anal. afa 2(2)(2011), 138-152. [4] c. cowen and b. d. maccluer, composition operators on spaces of analytic functions, studies in advanced mathematics. boca raton, fl: crc press. xii, 1995. [5] m. kotilainen, studies on composition operators and function spaces, report series. department of mathematics, university of joensuu 11. (dissertation) 2007. [6] p. lappan and j. xiao, q # α -bounded composition maps on normal classes, note di matematica, 20(1) (2000/2001), 65-72. [7] x. li, on hyperbolic q classes, dissertation, university of joensuu, joensuu, 2005, ann. acad. sci. fenn. math. diss. 145 (2005), 65 pp. [8] x. li, f. pérez-gonzález, and j. rättyä, composition operators in hyperbolic q-classes, ann. acad. sci. fenn. math. 31 (2006), 391-404. 30 a. el-sayed ahmed & m. a. bakhit cubo 15, 3 (2013) [9] s. makhmutov and m. tjani, composition operators on some möbius invariant banach spaces, bull. austral. math. soc. 62 (2000), 1-19. [10] f. pérez-gonzález, j. rättyä and j. taskinen, lipschitz continuous and compact composition operators in hyperbolic classes, mediterr. j. math. 8 (2011), 123-135. [11] w. smith, inner functions in the hyperbolic little bloch class, michigan math. j. 45(1) (1998), 103-114. [12] m. tjani, compact composition operators on besov spaces, trans. amer. math. soc. 355 (2003), 4683-4698. [13] j. xiao, holomorphic q classes, lecture notes in mathematics, 1767, springer-verlag, berlin, 2001. [14] s. yamashita, hyperbolic hardy classes and hyperbolically dirichlet-finite functions, hokkaido math. j., special issue 10 (1981), 709-722. [15] s. yamashita, functions with hp hyperbolic derivative, math. scand. 53 (2)(1983), 238-244. [16] s. yamashita, holomorphic functions of hyperbolic bounded mean oscillation, boll. un. math. ital. 5 b(6), (3)(1986), 983-1000. [17] j. zhou, composition operators from bα to qk type spaces, j. funct. spaces appl. 6 (1)(2008), 89-105. cubo a mathemati al journal vol.15, n o 03, (51�58). o tober 2013 generalization of new continuous fun tions in topologi al spa es p. g. patil department of mathemati s, sksvm agadi college of engg. and te hn., laxmeshwar-582 116, karnataka state, india. pgpatil01�gmail. om t. d. rayanagoudar department of mathemati s, govt.first grade college, annigeri-582 116, karnataka state, india. rgoudar1980�gmail. om s. s. ben halli department of mathemati s, karnatak university, dharwad-580 003 karnataka state, india. ben halliss�gmail. om abstract in this paper, ωαlosed sets and ωα-open sets are used to de�ne and investigate the new lasses of fun tions namely somewhat ωαontinuous fun tions and totally ωαontinuous fun tions. resumen en este artí ulo onjuntos errados-ωα y abiertos-ωα se usan para de�nir e investigar las lases de nuevas fun iones ontinuas ωα y totalmente ontinuas ωα. keywords and phrases: ωαlosed, ωα-open, ωαontinuous, somewhat ωα ontinuous and totally ωα ontinuous fun tions. 2010 ams mathemati s subje t classi� ation: 54c08, 54c10. 52 p.g.patil, t.d. rayanagoudar & s.s.ben halli cubo 15, 3 (2013) 1 introdu tion re ent progress in study of hara treization and generalization of ontinuity has been done by means of several generalized losed sets. as a generalization of losed sets ωαlosed sets were introdu ed and studied by ben halli.et.al[1℄. the on epts of feebly ontinuous fun tions and feebly open fun tions were introdu ed by zdenek frolik[2℄. gentry and hoyle[3℄ introdu ed and studied the on epts of somewhat ontinuous fun tions and somewhat open fun tions. re ently, santhileela and balasubramanian[8℄ introdu ed and studied the on epts of somewhat semi ontinuous fun tions and somewhat semi open fun tions. in this paper, we will ontinue the study of related fun tions with ωαlosed and ωα-open sets. we introdu e and hara terize the on ept of somewhat ωαontnuous and totally ωαontinuous fun tions. 2 preliminaries throughout this paper (x, τ), (y, σ) and (z, η)(or simply x,y and z) represent topologi al spa es on whi h no separation axioms are assumed unless otherwise mentioned.for a subset a of (x, τ), cl(a),int(a), αcl(a) and ac denote the losure of a, inerior of a, the αlosure of a and the ompliment of a in x respe ively. we re all the following de�nitions, whi h are usefull in the sequel.before entering into our work we re all the following de�nitions from various authors. de�nition 2.1. a subset a of a topologi al spa e (x, τ) is alled semi-open [5℄ (resp. α-open[6℄) if a ⊆ cl(int(a)) (resp a ⊆ int(cl(int(a))).the ompliment of semi-open (resp.α-open) is alled semilosed(resp.αlosed). de�nition 2.2. a subset a of a topologi al spa e (x, τ) is alled ωαlosed [1℄ if αcl(a) ⊆ u whenever a ⊂ u and u is ω-open in x. the ompliment of ωαlosed set is ωα-open. the family of all ωαlosed sets of x is denoted by τ∗ ωα . in [7℄, we showed that τ∗ ωα forms a topology on x. de�nition 2.3. a fun tion f : (x, τ) → (y, σ) is is said to be ωαontinuous [7℄ if the inverse image of every open set in y is ωα-open in x. de�nition 2.4. a fun tion f : (x, τ) → (y, σ) is is said to be perfe tly ωαontinuous [7℄ if the inverse image of every ωα open set in y is lopen in x. de�nition 2.5. a fun tion f : (x, τ) → (y, σ) is is said to be somewhatontinuous [3℄(resp.somewhat semiontinuous[8℄) if for u ∈ σ and f−1(u) 6= φ there exists an open (resp.semi open) set v in x su h that v 6= φ and v ⊆ f−1(u). cubo 15, 3 (2013) generalization of new continuous fun tions in topologi al spa es 53 remark 2.6. every somewhat ontinuous fun tion is somewhat semi ontinuous but onverse need not true in general[8℄. de�nition 2.7. a fun tion f : (x, τ) → (y, σ) is said to be somewhat-open [3℄(resp.somewhat semi-open[8℄) fun tion provided that for u ∈ τ and u 6= φ, there exists an open (resp.semi open) set v in y su h that v 6= φ and v ⊆ f−1(u). remark 2.8. every somewhat open fun tion is somewhat semi open fun tion but the onverse need not be true in general[8℄. 3 somewhat ωα continuous fun tions in this se tion, we introdu e a new lass of fun tions alled somewhat ωαontinuous fun tions using ωαlosed sets and obtain some of their hara terizations. de�nition 3.1. a fun tion f : (x, τ) → (y, σ) is said to be somewhat ωα ontinuous if for every open set u in y and f−1(u) 6= φ, there exists ωα-open set v in x su h that v 6= φ and v ⊆ f−1(u). example 3.2. let x = y = {p, q}, τ = {x, φ, } and σ = {x, φ, {p}}. the identity fun tion f : (x, τ) → (y, σ) is somewhat ωαontinuous fun tion. theorem 3.3. every somewhat ontinuous fun tion is somewhat ωα ontinuous but onverse need not true in general. example 3.4. in example 3.2, f is somewhat ωαontinuous but not somewhat ontinuous. remark 3.5. the on ept of somewhat ωαontinuous and somewhat semiontinuous fun tions are independet as seen from the following examples. example 3.6. in example 3.2,f is somewhat ωαontinuous but not somewhat-semi ontinuous. example 3.7. let x = y = {a, b, c}, τ = {x, φ, {a, b}} and σ = {x, φ, {a}}. then the identity map f : (x, τ) → (y, σ) is somewhat-semi ontinuous but not somewhat ωαontinuous. theorem 3.8. if f : (x, τ) → (y, σ) is somewhat ωαontinuous and g : (y, σ) → (z, η) is ontinuous fun tion,then their omposition gof is somewhat ωαontinuous fun tion. proof. let u be an open set in z.suppose that f−1(u) 6= φ. sin e u is open and g is ontinuous, g−1(u) ∈ η. suppose that f−1(g−1(u)) 6= φ. by hypothesis, there exists a ωα-open set v in y su h that v 6= φ and v ⊆ f−1(g−1(u)) = (gof)−1(v). therefore gof is somewhat ωαontinuous fun tion. remark 3.9. in the above theorem 3.8, if f is ontinuous and g is somewhat ωαontinuous then their omposition gof need not be somewhat ωαontinuous fun tion as seen from the following example. 54 p.g.patil, t.d. rayanagoudar & s.s.ben halli cubo 15, 3 (2013) example 3.10. let x = y = z = {p, q}, τ = {x, φ, {p}} , σ = {y, φ, {p}} and η = {z, φ, {q}} de�ne the fun tions f : (x, τ) → (y, σ) by f(p) = f(q) = q and g : (y, σ) → (z, η) by g(p) = q and g(q) = p.then learly f is ontinuous fun tion and g is somewhat ωαontinuous fun tion but their omoposition gof : (x, τ) → (z, η) is not somewhat ωαontinuous fun tion. de�nition 3.11. a subset m of a topologi al spa e x is said to be ωα-dense in x if there is no proper ωαlosed set f in x su h that m ⊂ f ⊂ x. theorem 3.12. the following statements are equivalent for a fun tion f : (x, τ) → (y, σ): (1) f is somewhat ωαontinuous fun tion (2) if f is a losed subset of y su h that f−1(f) 6= x,then there is a proper ωαlosed subset d of x su h that f−1(f) ⊂ d. (3) if m is a ωα-dense subset of x, then f(m) is a dense subset of y. proof. (1) ⇒ (2): let f be a losed subset of y su h that f−1(f) 6= x.then f−1(y−f) = x−f−1(f) 6= φ. then from (1) there exists ωα-open set v in x su h that v 6= φ and v ⊂ f−1(y − f) = x − f−1(f).this implies f−1(f) ⊂ x − v and x − v = d is a ωαlosed set inx. (2) ⇒ (3): let m be any ωα-dense set in x. suppose f(m) is not a dense subset of y, then there exists a proper losed set f in y su h that f(m) ⊂ f ⊂ y. this implies f−1(f) 6= x. then from (2) there exists a proper ωαlosed set d su h that m ⊂ f−1(f) ⊂ d ⊂ x. this ontradi ts the fa t that m is a ωα-dense set in x. (3) ⇒ (2): suppose (2) is not true.then there exists a losed setf in y su h that f−1(f) 6= x.but there is no proper ωαlosed set d in x su h that f−1(f) ⊆ d. this means that f−1(f) is ωα-dense in x. but from hypothesis f(f−1(f)) = f must be dense in y, whi h is ontradi tion to the hoi e of f. (2) ⇒ (1):let u be an open set in y and f−1(u) 6= φ. then f−1(y − u) = x − f−1(u) = φ. then by hypothesis, there exists a proper ωαlosed set d su h that f−1(y − u) ⊂ d. this implies that x − d ⊂ f−1(u) and x − d is ωα-open and x − d 6= φ. theorem 3.13. let f : (x, τ) → (y, σ) be a fun tion and x = a ∪ b, a and b are open subsets of x su h that (f/a) and (f/b) are somewhat ωαontinuous fun tions then f is somewhat ωαontinuous fun tion. proof. let u be an open set in y su h thatf−1(u) 6= φ. then (f/a)−1(u) 6= φ or (f/b)−1(u) 6= φ or both (f/a)−1(u) 6= φ and (f/b)−1(u) 6= φ . ase(i): suppose (f/a)−1(u) 6= φ. sin e f/a is somewhat ωαontinuous , then there exists ωα open set v in a su h that v 6= φ and v ⊂ (f/a)−1(u) ⊂ f−1(u). sin e v is ωα-open in a and a is open in x, v is ωα-open x . hen e f is somewhat ωαontinuous fun tion. ase(ii): suppose (f/b)−1(u) 6= φ. sin e f/b is somewhat ωαontinuous , then there exists ωα open set v in b su h that v 6= φ and v ⊂ (f/b)−1(u) ⊂ f−1(u). sin e v is ωα-open in b and b cubo 15, 3 (2013) generalization of new continuous fun tions in topologi al spa es 55 is open in x, v is ωα-open x . hen e f is somewhat ωαontinuous fun tion. ase(iii): suppose (f/a)−1(u) 6= φ and (f/b)−1(u) 6= φ. follows from ase(i) and ase(ii). theorem 3.14. if a be any set in x and f : (x, τ) → (y, σ) be somewhat ωαontinuous su h that f(a) is dense in y. then any extension f of f is somewhat ωαontinuous. proof. let u be an open set iny su h thatf−1(u) 6= φ. sin e f(a) ⊂ y is dense in y and u ∩ f(a) 6= φ. it follows that f−1(u) ∩ a 6= φ. that is f−1(u) ∩ a 6= φ.hen e by hypothesis there exists a ωα-open set v in a su h that v 6= φ and v ⊂ f−1(u) ⊂ f−1(u).this implies f is somewhat ωαontinuous. de�nition 3.15. a topologi al spa e x is said to be ωα-separable if there exists a ountable subset b of x whi h is ωα-dense in x. theorem 3.16. let f : (x, τ) → (y, σ) is somewhat ωαontinuous fun tion.if x is ωα-separable then y is separable. proof. let b be ountable subset of x whi h is ωα-dense in x. then from theorem 3.12,f(b) is dense in y. sin e b is ountable f(b) is also ountable whi h is dense in y. this implies that y is separable. 4 somewhat ωα-open fun tions in this se tion, we introdu e the on ept of somewhat ωα-open fun tions and study some of their hara terizations. de�nition 4.1. a fun tion f : (x, τ) → (y, σ) is somewhat ωα-open provided that for open set u in x and u 6= φ there exists a ωα -open set v in y su h that v 6= φ and v ⊆ f(u). example 4.2. let x = y = {a, b, c} and τ = {x, φ, {a} , {b, c}} and σ = {x, φ, {a}}. de�ne a fun tion f : (x, τ) → (y, σ) by f(a) = c, f(b) = a and f(c) = b. then learly f is somewhat ωα-open. theorem 4.3. every somewhat open fun tion is somewhat ωα-open fun tion but onverse need not be true in general. example 4.4. in example 4.2, f is somewhat ωα-open fun tion but not somewhat -open fun tion. remark 4.5. somewhat ωα-open and somewhat semi-open fun tions are independent of ea h other as seen from the following examples. example 4.6. in example 4.2, f is somewhat ωα-open fun tion but not somewhat semi-open fun tion. 56 p.g.patil, t.d. rayanagoudar & s.s.ben halli cubo 15, 3 (2013) example 4.7. let x = y = {a, b, c}, τ = {x, φ, {b} , {a, c}} and σ = {y, φ, {a} , {b} , {a, b}}. then the identity fun tion f : (x, τ) → (y, σ) is somewhat semi-open but not somewhat ωα-open fun tion. theorem 4.8. if f : (x, τ) → (y, σ) is open fun tion and g : (y, σ) → (z.η) is somewhat ωα-open fun tion,then their omposition gof is somewhat ωα-open fun tion. we have the following hara terization. theorem 4.9. the following statements are eqivalent for bije tive fun tion f : (x, τ) → (y, σ) (1) f is somewhat ωα-open fun tion (2) if f is losed subset of x su h that f(f) 6= y, then there exists a ωαlosed subset d of y su h that d 6= y and f(f) ⊂ d. proof. (1) ⇒ (2):let f be a losed subset of x su h that f(f) 6= y. from (1), there exists a ωα -open set v 6= φ in y su h that v ⊂ f(x − f). put d = y − v. clearly d is a ωαlosed in y and we laim that d 6= y. if d = y, then v = φ whi h is a ontradi tion. sin e v ⊂ f(x − f), d = y − v ⊂ y − [f(x − f)] = f(f). (2) ⇒ (1):let u be any non-empty open set in x. put f = x − u. then f is a losed subset of x and f(x − u) = f(f) = y − f(u) whi h implies f(f) 6= φ. therefore by (2) there is a ωαlosed subset d of y su h that d 6= y and f(f) ⊂ d. put v = x − d, learly v is ωα-open set and v 6= φ.further, v = x − d ⊂ y − f(f) = y − [y − f(u)] = f(u). theorem 4.10. if f : (x, τ) → (y, σ) is somewhat ωα-open fun tion and a be any open subset of x. thenf/a : (a, τ/a) → (y, σ) is also somewhat ωα-open fun tion. theorem 4.11. if f : (x, τ) → (y, σ)be a fun tion su h that f/a and f/b are somewhat ωα-open, then fis somewhat ωα-open fun tion, where x = a ∪ b, a and b are open subsets of x. 5 totally ωα continuous fun tions in this se tion, we introdu e a new lass of fun tions alled totally ωα ontinuous fun tions and study some of their properties. de�nition 5.1. a fun tion f : (x, τ) → (y, σ) is said to be totally ωα ontinuous, if the inverse image of every open subset of y is an ωαlopen subset of x. example 5.2. let x = y = {a, b, c}, τ = {x, φ, {a}} and σ = {y, φ, {a} , {b, c}}. de�ne a fun tion f : (x, τ) → (y, σ) byf(a) = b, f(b) = a and f(c) = c.then f is totally ωα ontinuous fun tion theorem 5.3. every perfe tly ωα ontinuous map is totally ωα ontinuous but onverse need not be true in general. cubo 15, 3 (2013) generalization of new continuous fun tions in topologi al spa es 57 proof. let f : (x, τ) → (y, σ) be a perfe tly ωα ontinuous. let u be an open set in y. then u is ωα-open in y. sin e f is a perfe tly ωα ontinuous, f−1(u) is lopen in x, implies that f−1(u) is ωαlopen in x. example 5.4. in example 5.2, f is totally ωα ontinuous but not perfe tly ωα ontinuous. theorem 5.5. every totally ωα ontinuous fun tion is ωα ontinuous but onverse need not be true in general. example 5.6. let x = y = {a, b, c}, τ = {x, φ, {a}} and σ = {y, φ, {a} , {a, c}}.then the identity fun tion f : (x, τ) → (y, σ) is ωα ontinuous fun tion but not totally ωα ontinuous fun tion. remark 5.7. it is lear that the totally ωα ontinuous fun tion is stronger than ωα ontinuous and weaker than perfe tly ωα ontinuous. theorem 5.8. if f : (x, τ) → (y, σ) is totally ωα ontinuous fun tion from an ωαonne ted spa e x in to y, then y is an indis rete spa e. proof. suppose that y is not indis rete spa e. let a be a proper non-empty open subset of y. then f−1(a) is a non-empty proper ωα lopen subset of x whi h is ontradi tion to the fa t that x is ωαonne ted. de�nition 5.9. a topologi al spa e x is said to be ωα2-spa e [7℄, if for every pair of distin t points x and y in x, there exists ωα-open sets m and n su h that x ∈ n , y ∈ m and m∩n = φ. theorem 5.10. let f : (x, τ) → (y, σ) be totally ωα ontinuous inje tion map. if y is t0, then x is ωα2-spa e. proof. let x and y be any pair of distin t points of x. then f(x) 6= f(y). then there exists an open set u ontaining f(x) but notf(y). sin e y is t0. then x /∈ f −1(u) and y /∈ f−1(u). sin e f is totally ωα ontinuous,f−1(u) is an ωαlopen subset of x. also x ∈ f−1(u) and y ∈ (f−1(u))c. hen e x is ωα2-spa e. theorem 5.11. a topologi al spa e x is ωα onne ted if and only if every totally ωα ontinuous fun tion from a spa e x in to any t0-spa e y is a onstant fun tion. theorem 5.12. let f : (x, τ) → (y, σ) is totally ωα ontinuous and y be a t1-spa e. if a is an ωαonne ted subset of x, then f(a) is a single point. theorem 5.13. a fun tion f : (x, τ) → (y, σ) is totally ωα ontinuous at a point x ∈ x if for ea h open subset v in y ontaining f(x), there exists a ωαlopen subset u in x ontaining x su h that f(u) ⊂ v. 58 p.g.patil, t.d. rayanagoudar & s.s.ben halli cubo 15, 3 (2013) proof. let v be an open subset of y and let x ∈ f−1(v). sin e f(x) ∈ v, there exists a ωαlopen set ux in x ontaining x su h that ux ∈ f −1(v). we obtain f−1(v) = ux∈f−1(v). sin e arbitrary union of ωα-open sets is ωα-open, f−1(v) is ωαlopen in x. de�nition 5.14. let x be a topologi al spa e. then the set of all points y in x su h that x and y annot be separated by a ωα-separation of x is said to be the quasi ωαomponent of x. theorem 5.15. let f : (x, τ) → (y, σ) is totally ωα ontinuous map from a topologi al spa e x in to a t1-spa e y, then f is onstant on ea h quasi ωαomponent of x. proof. let x and y be two points of x that lie in the some quasi ωαomponent of x. assume that f(x) = α 6= β = f(y). sin e y is t1, α is losed in y and so α c is an open subset in y. sin e f is totally ωα ontinuous,f−1(α) and f−1(αc) are disjoint ωαlopen subsets of x. further x ∈ f−1(α) and y ∈ f−1(α)c, whi h is a ontradi tion in view of the fa t that y must belong to every ωαlopen set ontaining x. re eived: april 2013. a epted: september 2013. referen es [1℄ s.s.ben halli, p.g.patil and t.d.rayanagoudar, ωα-closed sets in topologi al spa es,the global jl.of appl.math.and math.s ien es, v.2,1-2,(2009),53-63. 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() cubo a mathematical journal vol.13, no¯ 02, (59–72). june 2011 on λ strong homogeneity existence for cofinality logic saharon shelah 1 einstein institute of mathematics, the hebrew university of jerusalem, jerusalem, 91904, israel. email: shelah@math.huji.ac.il url: http://shelah.logic.at abstract let c ⊂6= reg be a non-empty class (of regular cardinals). then the logic l(q cf c) has additional nice properties: it has the homogeneous model existence property. resumen sea c ⊂6= reg una clase no vaćıa (de cardinales regulares). entonces la lógica l(q cf c) tiene propiedades adicionales: esta tiene la propiedad de modelo existencia homogénea. keywords and phrases: model theory, soft model theory, cofinality quantifier. mathematics subject classification: 03c95, 03c80. 1the author thanks alice leonhardt for the beautiful typing. the author would like to thank the israel science foundation for partial support of this research (grants no. 451/99 and 710/07). publication 750. 60 saharon shelah cubo 13, 2 (2011) 1. introduction we deal with logics gotten by strengthening of first order logic by generalized quantifiers, in particular compact ones. we continue [sh:199] (and [sh:43]) a natural quantifier is the cofinality quantifier, qcf≤λ (or q cf c), introduced in [sh:43] as the first example of compact logic (stronger than first order logic, of course). recall that the “uncountably many x’s”quantifier qcard≥ℵ1 , is ℵ0-compact but not compact. also note that l(q cf ≤λ) is a very nice logic, e.g. with a nice axiomatization (in particular finitely many schemes) like the one of l(qcard≥ℵ1 ) of keisler. by [sh:199], e.g. for λ = 2ℵ0 , its beth closure is compact, giving the first compact logic with the beth property (i.e. implicit definition implies explicit definition). earlier there were indications that having the beth property is rare for such logic, see e.g. in makowsky [mak85]. a weaker version of the beth property is the weak beth property dealing with implicit definition which always works; h. friedman claim that historically this was the question. mekler-shelah [mksh:166] prove that at least consistently, l(qcard≥ℵ1 ) satisfies the weak beth property. väänänen in the mid nineties motivated by the result of mekler-shelah [mksh:166] asked whether we can find a parallel proof for l(qcf≤λ) in zfc. a natural property for a logic l is definition 1. a logic l has the (strong) homogeneous model existence property when every theory t ⊆ l(τ), (so has a model) has a strongly (l, ℵ0)-homogeneous model m, so τm = τ and m is a model of t and m satisfies: if ā, b̄ ∈ ω>m realize the same l(τ)-type in m then there is an automorphism of m mapping ā to b̄. this property was introduced in [sh:199] being natural and also as it helps to investigate the weak beth property. in §1 we prove that l(qcfc) has the strongly ℵ0-saturated model existence property. the situation concerning the weak beth property is not clear. question 2. 1) does the logic l(qcfc) have the weak beth property? 2) does the logic l(qcf≤λ1,q cf ≤λ2 ) has the homogeneous model existence property? the first version of this work was done in 1996. notation 3. 1) τ denotes a vocabulary, l a logic, l(τ) the language for the logic l and the vocabulary τ. 2) let l be first order logic, l(q∗) be first order logic when we add the quantifier q∗. 3) for a model m and ultrafilter d on a cardinal λ, let mλ/d be the ultrapower and jm,d = j λ m,d be the canonical embedding of m into mλ/d; of course, we can replace λ by any set. 4) let lst (theorem/argument) stand for löwenheim-skolem-tarski (on existence of elementary submodels). cubo 13, 2 (2011) on λ strong homogeneity existence for cofinality logic 61 concerning 1, more generally definition 4. 1) m is strongly (l,θ)-saturated (in l = l we may write just θ) when (a) it is θ-saturated (i.e. every set of l(τm)-formulas with < θ parameters from m and < θ free variables which is finitely satisfiable in m is realized in m (b) if ζ < θ and ā, b̄ ∈ ζm realize the same l(τm)-type in m, then some automorphism of m maps ā to b̄. 2) m is a strongly sequence (l,θ)-homogeneous when clause (b) above holds. 3) m is sequence (∆,κ)-homogeneous when: ∆ ⊆ l(τm) and if ζ < κ,ā ∈ ζm,b̄ ∈ ζm and tp∆(ā,∅,m) = tp∆(b̄,∅,m) then for every c ∈ m for some d ∈ m we have tp∆(b̄^〈d〉,∅,m) = tp∆(ā^〈c〉,m). 3a) σ1(τ) is the set of formulas of the form ϕ(x̄) = (∃ȳ)ϑ(x,ȳ) where ϑ(x̄, ȳ) is quantifier free first order formula in the vocabulary τ. 4) we may omit “sequence”. definition 5. 1) the logic l has “the strong κ-homogeneous existence property”when every theory t ⊆ l(τ1) has a strongly (l,κ)-homogeneous model. 2) similarly “the strong κ-saturated existence property”, etc. 2. on strongly saturated models we prove that any theory in l(qcfc) has strongly (l(q cf c),θ)-saturated model when c\θ /∈ {∅, reg\θ} of course. definition 6. let ι ∈ {1,2} and c be a class of regular cardinals such that c 6= ∅, reg. 1) the quantifier q cf(ι) c is defined as follows: syntactically: it bounds two variables, i.e. we can form (q cf(ι) c x,y)ϕ, with its set of free variables being defined as fvar(ϕ)\{x,y}. semantically: m |= (q cf(ι) c x,y)ϕ(x,y,ā) iff (a) + (b) holds where (a) relevancy demand: the case ι = 1: the formula ϕ(−, −; ā)m define in m a linear order with no last element called ≤ϕm,ā on the non-empty set dom(≤ ϕ m,ā) = {b ∈ m : m |= (∃y)(ϕ(b,y; ā)} the case ι = 2: similarly but ≤ϕm,ā is a quasi linear order on its domain 62 saharon shelah cubo 13, 2 (2011) (b) the actual demand: ≤ϕm,ā has cofinality cf(≤ ϕ m,ā), (necessarily an infinite regular cardinal) which belongs to c. convention 7. 1) writing qcfc we mean that this holds for q cf(ι) c for ι = 1 and for ι = 2. 2) let ι-order mean order when ι = 1 and quasi order when ι = 2; but when we are using q cf(ι) c then order means ι-order. definition 8. 1) as {ψ ∈ l(qcfc) : ψ has a model} does not depend on c (and is compact, see [sh:43]) we may use the formal quantifier qcf, so the syntex is determined but not the semantics, i.e. the satisfaction relation |=. we shall write m |=c ψ or m |=c t for the interpretation of qcf as qcfc, but also can say “t ⊆ l(qcf)(τ) has model/is consistent”. 2) if c is clear from the context, then qcfℓ stands for q cf c if ℓ = 1 and q cf reg\c if ℓ = 0. convention 9. 1) t∗ is a complete (consistent ≡ has models) theory in l(qcf) which is closed under definitions i.e. every formula ϕ = ϕ(x̄) is equivalent to a predicate pϕ(x̄) so pϕ ∈ τ(t ∗), i.e. t∗ ⊢ (∀z̄)[ϕ(z̄) ≡ pϕ(z̄)]. 2) let t = t∗∩ (first order logic), i.e. t = t∗ ∩ l(τt ∗ ), it is a complete first order theory. 3) c ⊆ reg, we let c1 = c and c0 = reg\c, both non-empty. theorem 10. assume χ = cf(χ),µ = µ<θ ≥ 2|t | + χ + κ,θ ≤ λ,cf(θ) ≤ min{χ,κ},χ 6= κ = cf(κ) and µℓ = { χ ℓ = 0 κ ℓ = 1 then there is a τ(t)-model m such that (a) m |= t,‖m‖ = µ and m is θ-saturated (b) if ϕ(z̄) = (qcfℓ )ψ(x,y; z̄) then: m |= pϕ(x̄)[ā] iff ϕ(y,z; ā) define in m a linear order with no last element and cofinality µℓ (c) m is strongly2 θ-saturated model of t∗. remark 11. 1) we can now change χ,κ,µ and ‖m‖ by lst. almost till the end instead µ ≥ 2|t | + χ + κ just µ ≥ |t | + χ + κ suffice. the proof is broken to a series of definitions and claims. the “≥ 2|t |” is necessary for ℵ0-saturativity. 3) we can assume v satisfies gch high enough and then use lst. so µ+ = 2µ below is not a real burden. 2as t∗ has elimination of quantifiers, doing it for l(qcf c ) or for l is the same cubo 13, 2 (2011) on λ strong homogeneity existence for cofinality logic 63 definition 12. 0) modt is the class of models of t. 1) (a) k = {(m,n) : m ≺ n are from modt } (b) kα = {m̄ : m̄ = 〈mi : i < α〉 satisfies mi ∈ modt and i < j ⇒ mi ≺ mj} (so k = k2) (c) kαµ = {m̄ ∈ kα : ‖mi‖ ≤ µ for i < α}, but then we (naturally) assume α < µ + (d) let τα = τt ∪ {pβ : β < α} ∪ {rϕ(x,y,z̄),β : ϕ(x,y, z̄) ∈ l(τt ),β < α}, each pβ a unary predicate and each rϕ(x,y,z̄),β is an (ℓg(z̄)+1)-place predicate and no incidental identification (so pα /∈ τ, etc.) (e) for m̄ ∈ kα let m(m̄) be the τα-model m with • universe ∪{mβ : β < α} • m↾τt = ∪{mβ : β < α} • pmβ = mβ • rm ϕ(x,y,z̄),β = {〈c〉^ā : ϕ(x,y,ā) a linear order, ā ∈ ℓg(z̄)(pmβ ) such that m |= p(qcf 0 x,y)ϕ(x,y,z̄)[ā] and c ∈ dom(≤ϕm,ā) and [b ∈ dom(≤ ϕ m,ā) ∩ p m β ⇒ b ≤ ϕ m,ā c]} (f) let m0(m̄) be the τ-model ∪{mβ : β < α} so m0(m̄) = m(m̄)↾τ. 2) assume (mℓ,nℓ) ∈ k for ℓ = 1,2 let (m1,n1) ≤ (m2,n2) mean that clauses (a),(b),(c) below hold and let (m1,n1) ≤k (m 2,n2) mean that in addition clause (d) below holds, where: (a) m1 ≺ m2 (b) m2 ∩ n1 = m1 (c) n1 ≺ n2 (d) if m1 |= p(qcf 0 x,y)ϕ(x,y,z̄)[ā],c ∈ n 1,c ∈ dom(≤ϕn1,ā) and in n 1 the element c is ≤ϕ n1,ā above all d ∈ dom(≤ϕ m1,ā ), then in n2 the element c is ≤ϕ n2,ā -above all d ∈ dom(≤ϕ m2,ā ). 3) for m̄1,m̄2 ∈ kα let m̄ 1 ≤ m̄2 means γ < β < α ⇒ (m1γ,m 1 β) ≤ (m 2 γ,m 2 β); similarly m̄1 ≤kα m̄ 2 means m̄1,m̄2 ∈ kα and γ < β < α ⇒ (m1δ,m 1 β) ≤k (m 2 γ,m 2 β). 4) for m̄ ∈ kα,d an ultrafilter on λ we define n̄ = m̄ λ/d,jm,d = j λ m̄,d naturally: nβ = m λ β/d for β < α and jm̄,d = ∪{jmβ,d : β < α}, recalling 3. 64 saharon shelah cubo 13, 2 (2011) fact 13. 0) for m̄1,m̄2 ∈ kα we have (a) m̄1 ≤kα m̄ 2 iff m(m1) ⊆ m(m̄2) (b) (m(m̄ℓ)↾pmβ )↾τt = m ℓ β (c) m̄1 ≤kα m̄ 1 implies m̄1 ≤ m̄2. 1) (kα,≤) and (kα,≤kα ) are partial orders. 2a) if m̄1 ≤kα m̄ 2 in kα and 0 < γ < β ≤ α then ( ⋃ ε<γ m1ε, ⋃ ε<β m1ε) ≤ ( ⋃ ε<γ m2ε, ⋃ ε<β m2ε) moreover 〈 ⋃ i<1+ε m1i : 1 + ε ≤ α〉 ≤kξ 〈 ⋃ i<1+ε m2i : 1 + ε ≤ α〉 where ξ is α if α < ω and is α + 1 if α ≥ ω. 2b) if 〈m̄i : i < δ〉 is a ≤kα -increasing sequence (of members of kα) and we define m̄ δ = 〈mδε : ε < α〉 by mδε = ∪{m i ε : i < δ} then i < δ ⇒ m̄ i ≤kα m̄ δ and the sequence 〈m̄i : i ≤ δ〉 is continuous in δ. 3) in part (2b), if in addition i < δ ⇒ m̄i ≤kα n̄ so n̄ ∈ kα then m̄ δ ≤kα n̄. 4) in part (2b), if δ < µ+ and i < δ ⇒ m̄i ∈ kαµ then m̄ δ ∈ kαµ. 5) if m̄ ≤kα n̄ and yε ⊆ nε for ε < α and σ{‖mε‖ + |yε| : ε < α} + |τ| + |α| ≤ λ then there is n̄′ ∈ kαλ such that m̄ ≤kα n̄ ′ ≤kα n̄ and ε < α ⇒ yε ⊆ n ′ ε. 6) assume m̄i ∈ k α(i) µ for i < δ < µ +,〈α(i) : i < δ〉 is a non-decreasing sequence of ordinals and i < j < δ ⇒ m̄i ≤kα(i),m̄ j↾α(i) and we define α(δ) = ∪{α(i) : i < δ},m̄δ = 〈mδβ : β < α(δ)〉 where mδβ = ∪{m i β : β < δ satisfies β < α(i)} then m̄ δ ∈ k α(δ) µ and i < δ ⇒ m̄i ≤kα(i) m̄ δ↾α(i). 7) if m̄ℓ ≤kα n̄ for ℓ = 1,2 and [a ∈ m(m̄ 1) ⇒ a ∈ m(m̄2)] then m̄1 ≤kα m̄ 2. 8) parts (2)-(7) holds also when we replace ≤kα by ≤. demostración. check. �13 fact 14. 1) if (m0,m1) ∈ k 2 µ and (m0,m ′ 1) ∈ k 2 µ then there are m2,f such that (a) m′1 ≺ m2 ∈ kµ (b) f is an elementary embedding of m1 into m2 (c) f↾m0 = idm0 (d) (m0,m ′ 1) ≤k2 (f(m1),m2). cubo 13, 2 (2011) on λ strong homogeneity existence for cofinality logic 65 2) if m̄ ∈ kα, x̄ = 〈xε : ε < ζ〉 and γ is a set of first order formulas from l(τ + α ) in the variables x̄ with parameters from the model m(m̄) finitely satisfiable in m(m) such that ε < ζ ⇒ ∨ β<α pβ(xε) ∈ γ, then there is n̄ ∈ kα such that m̄ ≤kα n̄ and γ is realized in m(n̄). 3) if γ is a type over m0(m̄) of cardinality 3 < cf(α) then it is included in some γ ′ as in part (2). 4) if m̄ ∈ kαµ,d an ultrafilter on θ and m ′ β = (mβ) θ/d for β < α then (a) m̄′ = 〈m′β : β < α〉 ∈ kα (b) jθ m̄,d := ∪{jθmβ,d : β < α} is a ≤kα -embedding of m̄ into m̄ ′, i.e. (b)′ 〈jθmβ,d(mβ) : β < α〉 = m̄ ′ ≤kα 〈m ′ β : β < α〉, so (c) for many y ∈ [∪{m′β : β < α}] µ we have jθ m̄,d (m̄) ≤kα 〈m ′ β↾y : β < α〉 ∈ k α µ; see 13(5), 17(3). demostración. 1) see [sh:199, §4]; just let d be a regular ultrafilter on λ ≥ ‖m1‖ + |τ|, let g an elementary embedding of m1 into (m0) λ/d extending j = jλm0,d, necessarily exists. lastly, let m2 ≺ (m ′ 1) λ/d include jλm1,d(m ′ 1) ∪ g(m1) be of cardinality µ. identifying m ′ 1 with jλ m ′ 1 ,d (m′1) ≺ (m ′ 1) λ/d we are done. 2) similarly. 3) trivial. 4) should be clear. �14 definition 15. kecα is the class of m̄ ∈ kα such that: if m̄ ≤kα n̄ ∈ kα, then m(m̄) ≤σ1 m(n̄), i.e. (∗) below and kec,αλ = k ec α ∩ kλ where (∗) if a1, . . . ,an ∈ m(m̄),b1, . . . ,bk ∈ m(n̄),ϕ ∈ l(τ + α ) is quantifier free and m(n̄) |= ϕ[a1, . . . ,an,b1, . . . ,bk] then for some b ′ 1, . . . ,b ′ k ∈ ⋃ β<α mβ we have m(m̄) |= ϕ[a1, . . . ,an,b ′ 1, . . . ,b ′ k]. claim 16. 1) kec,αµ is dense in k α µ when µ ≥ |τt | + |α| of course. 2) kec,αµ is closed under union of increasing chains of length < µ +. 3) in definition 15, if |α| + |τt | ≤ µ and m̄ ∈ k α µ then without loss of generality n̄ ∈ k α µ. demostración. 1) given m̄0 ∈ k α µ we try to choose m̄ε ∈ k α µ by induction on ε < µ + such that 〈m̄ζ : ζ ≤ ε〉 is ≤kα -increasing continuous and ε = ζ + 1 ⇒ m(m̄ζ) �σ1 m(mε). for ε = 0 the sequence is given, for ε limit use 13(2), for ε = ζ + 1 if we cannot choose then by 13(5) we get 3also if cf(α) = 1, i.e. α is a successor ordinal 66 saharon shelah cubo 13, 2 (2011) m̄ζ ∈ k ec,α µ is as required. but if we succeed to choose 〈m̄ε : ε < µ +〉 we get contradiction by fodor lemma. 2) think on the definitions. 3) by lst. �16 claim 17. 1) if m̄,n̄ ∈ kαµ and m̄ ≤σ1 n̄ and n̄ ∈ k ec α then m̄ ∈ k ec α . 2) if n̄ ∈ kec,αµ ,y ⊆ m0(n̄) and λ = |τt | + |α| + |y| then there is m̄ ∈ k ec,α λ such that m̄ ≤kα n̄ and y ⊆ m0(m̄). 3) assume m̄ℓ ∈ kαµ and m̄ 0 ≤kα m̄ 1 and m̄0 ≤ m̄2. if m̄0 ∈ kec,αµ ,m̄ 0 ≤kα m̄ 2 or m(m̄0) ≤σ1 m(m̄2), then we can find (n̄,f2) such that: m̄1 ≤kα n̄ ∈ k α µ, moreover n̄ ∈ k ec,α µ and f2 is a ≤kα -embedding of m̄ 2 into n̄ over m̄0. demostración. 1) by part (3). 2) by part (1) and the lst argument. 3) by the definition of m̄0 ∈ kec,αµ in both cases we can assume m̄ 0 ≤σ1 m̄ 2. let ā = 〈aε : ε < ζ〉 list the elements of m(m̄2) and let γ = tpqf(ā,∅,m(m̄ 2)) = {ϕ(xε0, . . . ,xεn−1, b̄) : ϕ ∈ l(τ + α ) is quantifier free, b̄ ⊆ m(m̄0) and m(m̄2) |= ϕ[aε0, . . . ,aεn−1, b̄]}; note that pβ(xε)) t(ε,β) ∈ γ when β < α,ε < ζ and t(ε,β) is the truth value of aε ∈ m 2 β. now let d be a regular ultrafilter on λ = ‖m(m̄2)‖ and use 14(2),(3). this is fine to get (f2,n̄) with n̄ ∈ kα and by 13(5) without loss of generality n̄ ∈ k α µ and by 16(1) without loss of generality n̄ ∈ kec,αµ . �17 claim 18. 1) (kec,αµ ,≤kαµ ) has the jep. 2) suppose m̄1,m̄2 ∈ kαµ,β ≤ α,f is an elementary embedding of ⋃ γ<β m1γ into ⋃ γ<β m2γ such that 〈f(mγ) : γ < β〉 ≤kµ 〈m 2 γ : γ < β〉, equivalently f is an embedding of m(m̄ 1 ↾ β) into m(m̄2↾β) (so if β = 0 then f = ∅ and there is no demand). then we can find m̄3,f+ such that: (a) m̄2 ≤kµ m̄ 3 ∈ kαµ (b) f ⊆ f+ (c) f+ is an elementary embedding of ⋃ γ<α m1γ into ⋃ γ<α m3γ (d) 〈f+(m1γ)) : γ < α〉 ≤kα 〈m 3 γ : γ < α〉. demostración. 1) a special case of part (2) recalling 16(1). cubo 13, 2 (2011) on λ strong homogeneity existence for cofinality logic 67 2) by induction on α. α = 0: nothing to do. β = α: nothing to do. α = 1: so β = 0 which is trivial or β = α, a case done above. α successor: by the induction hypothesis and transitive nature of conclusion replacing m̄2 without loss of generality β = α − 1, then use 14(1). α limit: by α − β successive uses of induction hypothesis using 13(2b). �18 conclusion 19. (kecα ,≤kα ), or formally k = (kk,≤k) defined by kk := {m(m̄) : m̄ ∈ k ec α },m(m̄ 1) ≤k m(m̄2) ⇔ m(m1) ⊆ m(m̄2), is an a.e.c. with amalgamation, the jep and lst(k) ≤ |τt |+|α|+ℵ0. demostración. by the above, on a.e.c. see [sh:h, ch.i], i.e. [sh:88r] and history there. �19 fact 20. assume λ = λ<λ > |τt | + ℵ0 + |α|. then there is m̄ such that (a) m̄ ∈ kecα is universal for (k ec α ,≤kα ) in cardinality λ (b) m(m̄) is model homogeneous for (kecα ,≤kα ) of cardinality λ (c) m(m̄) is sequence (σ1(τ + α ),λ)-homogeneous, see 4(3). demostración. clause (a) + (b) are straight by 17 + 18(1), or use 19 and see [sh:h, ch.i,§2] = [sh:88r, §2]. now clause (c) follows: just think. �20 fact 21. assume m̄ ∈ kαµ,β + 1 < α,ℓ ∈ {0,1} and mβ |= p(qcf ℓ x,y)ϕ(x,y,z̄)[ā] then there are n̄,c such that m̄ ≤kα n̄ ∈ k α µ and: (∗)1 if ℓ = 1 then c ∈ dom(≤ ϕ nβ,ā ) and c is ≤ϕnγ,ā-above d ∈ dom(≤ ϕ mγ,ā ) for any γ ∈ [β,α) (∗)2 if ℓ = 0 then c ∈ dom(≤ ϕ nβ+1,ā ) and is ≤ϕnβ+1,ā-above any d ∈ dom(≤ϕnβ,ā). demostración. first assume ℓ = 1, without loss of generality β = 0 as we can let n̄↾β = m̄↾β. by 13(2a) wlog m̄ is increasing continuous; we prove by induction on α so easily without loss of generality α = 2. now this is obvious by [sh:43], [sh:199]; in details by [sh:43] there is a µ+saturated model m∗ of t such that m1 ≺ m∗ and m∗ |=c∗ t ∗ whenever, e.g. µ++ ∈ c∗ ∧µ + /∈ c∗. let {ϕi(x,y,ā ∗ i ) : i < µ} list {ϕ(x,y,ā ′) : ϕ ∈ l(τt ),m0 |= p(qcfx,y 0 )ϕ(x,y;z̄) [ā′]}, and for each i < µ let 〈ci,ε : ε < µ +〉 be ≤ϕi m∗,ā ∗ i -increasing and cofinal. for ε < µ+ let fε be an elementary embedding of m1 into m∗ over m0 such that: 68 saharon shelah cubo 13, 2 (2011) (∗) if c ∈ dom(≤ϕi m∗,ā ∗ i ) is a ≤ϕi m∗,ā ∗ i –upper bound of dom(≤ϕi m0,ā ∗ i ), then ci,ε ≤ ϕi m∗,ā ∗ i c. let c∗ ∈ m∗ be a ≤ ϕ m∗,ā -upper bound of dom(≤ϕm0,ā). choose n0 ≺ m∗ of cardinality µ be such that m0 ∪ {c∗} ⊆ n0 and choose ε < µ + large enough such that: (∗) if i < µ and d ∈ n0 is a ≤ ϕi m∗,āi -upper bound of dom(≤ϕi mi,āi ) then d ≤ϕi m∗,āi ci,ε. let n1 ≺ m∗ be of cardinality µ be such that n0 ∪ fε(m1) ⊆ n1. renaming, fε is the identity and (n0,n1) is as required. second, assume ℓ = 0 is even easier (again without loss of generality first, α = β + 2 and second β = 0,α = 2 and use n0 = m0,n1 satisfies m1 ≺ n1 and ‖n1‖ = µ and n1 realizes the relevant upper). �21 conclusion 22. in 20 the model m∗ = m(m̄∗) = ⋃ β<α m∗β satisfies (a) if m∗ |= p(qcf 1 x,y)ϕ[ā] then the order ≤ ϕ m∗,ā has cofinality λ (b) if α is a limit ordinal and m∗ |= p(qcf 0 x,y)ϕ[ā] then the linear order ≤ ϕ m∗,ā has cofinality cf(α) (c) m∗ is cf(α)-saturated (d) if λ ∈ c and cf(α) ∈ reg\c then m∗ is a model of t∗. claim 23. assume m̄ ∈ kecα . if ζ ≤ µ and ā, b̄ ∈ ζ(m∗0) realize the same type (equivalently q.f. type) in m0 then they realize the same σ1-type in m(m̄). demostración. we choose (nβ,fβ,gβ,hβ) by induction on β < α such that: (a) nβ is a model of t (b) nβ is ≺-increasing continuous with β (c) fβ,gβ are ≤k1+β -embedding of m̄↾(1 + β) into 〈nγ : γ < 1 + β〉 ∈ k1+β (d) f0(ā) = g0(b̄) (e) if γ < β then fγ ⊆ fβ,gγ ⊆ gβ. cubo 13, 2 (2011) on λ strong homogeneity existence for cofinality logic 69 for β = 0 this speaks just on modt . for β successor use 14. for β limit as in the successor case, recalling we translated it to the successor case (by 13(2a)). having carried the induction f = ∪{fβ : β < α} and g = ∪{gβ : β < α} are ≤kα -embedding of m̄ into n̄ = 〈nβ : β < α〉. by 16(1) there is n̄ ′ ∈ kecα which is ≤kα -above n̄. now as m̄ ∈ k ec α , the σ1-type of ā in m(m̄) is equal to the σ1-type of f(ā) in m(n̄ ′), and the σ1-type of b̄ in m(m̄) is equal to the σ1-type of f(ā) in m(n̄ ′). but f(ā) = f0(ā) = g0(b̄) = g(b̄), so we have gotten the promised equality of σ1-types. �23 observation 24. 1) if m̄ ∈ kecα and β < α then m̄ ′ : m̄↾[β,α) = 〈mβ+γ : γ < α − β〉 belongs to kecα−β. 2) if m̄ ∈ kα,β < α and m̄↾[β,α) ≤kα,β n̄ ′ then for some n̄ ∈ kα we have m̄ ≤kα n̄ and n̄↾[β,α) = n̄′. demostración. 1) if not, then there is n̄′ ∈ kα−β such that m̄ ′ ≤kα−β n̄ ′ but m(m̄′) �σ1 m(n̄ ′). define n̄ = 〈nγ : γ < α〉 by: nγ is mγ if γ < β and is n ′ γ−β if γ ∈ [β,α). easily m̄ ≤kα n̄ ∈ kα but m(m̄) �σ1 m(n̄), contradiction to the assumption m̄ ∈ k ec α . 2) the proof is included in the proof of part (1). �24 claim 25. in 20 for each β < α we have (a) 〈m∗β+γ : γ < α − β〉 is homogeneous universal for k α−β µ (b) if α = α−β, i.e. β+α = α then there is an isomorphism from m̄∗ onto 〈m∗β+γ : γ < α−β〉, in fact, we can determine f(ā) = b̄ if ā ∈ ζ(m∗0), b̄ ∈ ζ(m∗β) and tp(ā,∅,m ∗ β) = tp(b̄,∅,m ∗ β). demostración. chase arrows as usual recalling 24. �25 demostración. proof of theorem 10: without loss of generality there is σ = σθ ≥ µ such that 2σ = σ+ (why? let σ = σθ > µ be regular, work in vlevy(σ + ,2 σ ) and use absoluteness argument, or choose set a of ordinals such that p(µ) ∈ l[a] hence t,t∗ ∈ l[a] and regular θ large enough such that l[a] |= ‘‘2σ = σ+", work in l[a] a little more; and for the desired conclusion (there is a model of cardinality µ such that ...) it makes no difference). let α = κ and let m̄∗ ∈ kec,αλ be as in 20 for λ := σ + and let m∗ = ∪{m ∗ β : β < α}. now (∗)1 m∗ is a model of t ∗ by the {µ+}-interpretation. 70 saharon shelah cubo 13, 2 (2011) [why? by 22.] (∗)2 m∗ is θ-saturated. [why? clearly m∗β is θ-saturated for each β < θ. as θ is regular and 〈m ∗ β : β < θ〉 is increasing with union m∗, also m∗ is θ-saturated.] (∗)3 m∗ is strongly ℵ0-saturated and even strongly θ-saturated, see definition 4(1). [why? let ζ < θ and ā, b̄ ∈ ζ(m∗) realize the same q.f.-type (equivalently the first order type) in m∗. as ζ < θ for some β < θ we have ā, b̄ ∈ ζ(mβ). now by 25 we know that 〈m ∗ β+γ : γ < θ〉 ∼= 〈m∗γ : γ < θ〉, and by 23 the sequences ā, b̄ realize the same σ1-type in m(〈m ∗ β+γ : γ < θ〉) hence by clause (c) of 20 there is an automorphism π of it mapping ā to b̄. so π is also an automorphism of m∗ mapping ā to b̄ as required.] lastly, we have to go back to models of cardinality µ = µ<θ ≥ λ+κ+2|t |, this is done by the lst argument recalling 22. more fully, first let 〈m̄ε : ε < λ〉 be ≤kσχ-increasing continuous sequence with union m̄ ∗. for ζ < θ and ā, b̄ ∈ ζ(m∗) let fā,b̄ be an automorphism of m∗ mapping ā to b̄. now the set of δ < λ satisfying ⊛δ below is a club of λ hence if cf(δ) = χ then m = ∪{m ε β : β < λ} is as required except of being of cardinality µ, where ⊛δ (a) if ε < δ,ζ < θ and ā, b̄ ∈ ζ(∪{mεβ : β < α}) realize the same σ1-type in m̄ζ then ∪{mδβ : β < α} is closed under fā,b̄ and under f −1 ā,b̄ (b) the witnesses for the cofinality work, i.e. •1 if β < α,ā ∈ ω>(mδβ),m δ β |= p(qcf 0 y,z)ϕ(y,z,x̄)[ā] then for some ε < δ we have ā ⊆ m ε β and for every γ ∈ (β,α) there is c = cϕ,ā,γ ∈ m ε γ+1 which is a ≤ ϕ mε γ+1 ,ā-upper bound of dom(≤ϕmεγ,ā ), hence this holds for any ε′ ∈ [ε,λ) •2 if β < α,ā ∈ ω>(m γ β ) and mδβ |= p(qcf 1 y,z)ϕ(y,z,x̄)[ā] then for arbitrarily large ε < δ we have ā ⊆ mεβ and there is c = cϕ,ā ∈ m ε+1 β which is a (≤ϕ mε+1γ ,ā )-upper bound of dom(≤ϕmεγ,ā ) for every γ ∈ [β,α). by a similar use of the lst argument we get a model of t∗ of cardinal µ. �10 remark 26. if you do not like the use of (set theoretic absoluteness) you may do the following. use 27 below, which is legitimate as cubo 13, 2 (2011) on λ strong homogeneity existence for cofinality logic 71 (a) the class (kecα ,≤kα ) is an a.e.c. with lst number ≤ |t | + ℵ0 and amalgamation, so 27(1) apply (b) using σ1-types, it falls under [sh:3] more exactly [sh:54], so 27(3) apply (c) we can define k ec(ε) α by induction on ε ≤ ω ε = 0: kα ε = 1: kecα ε = n + 1: k ec(n+1) α = {m̄ ∈ k ec(n) α : if m̄ ⊆ n ∈ k ec(n) α then m(m) ≤σn+1 m(n̄)} ε = ω: k ec(ε) α = ∩{k ec(n) α : n < ω}. on k ec(ω) α apply 27(2). remark 27. 1) assume k = (kk,≤k) is a a.e.c. satisfying amalgamation and the jep with λ > lst(k) and µ = µ<λ. for any m ∈ kµ there is a strongly model λ-homogeneous n ∈ kµ which ≤k-extend m, which means: if m ∈ kk has cardinality < λ and f1,f2 are ≤k-embedding of m into n then for some automorphism g of n we have f2 = g ◦ f1. 2) let d be a good finite diagram as in [sh:3] and let kd be as below in part (3) for ∆ = l(τ). if λ = λ<θ ≥ |d| and m ∈ kd has cardinality λ then there is n ∈ kd of cardinality λ which ≺-extend m and is strongly (d,θ)-homogenous, i.e. (a) if ζ < θ,ā, b̄ ∈ ζn realizes the same type then some automorphism f of n maps ā to b̄ (b) d = {tp(ā,∅,n) : ā ∈ ω>n}. 3) assume ∆ ⊆ l(τ), not necessarily closed under negation, d is a set of ∆-types, kd is the class of τ-models such that ā ∈ ω>m ⇒ tp∆(ā,∅,m) ∈ d and m ≤d n iff m ⊆ n are from kd and ā ∈ ω>m ⇒ tp∆(ā,∅,m) = tp∆(ā,∅,n). assume further d is good, i.e. for every m ∈ kd and λ there is a sequence (d,λ)-homogeneous model n ∈ kd which ≤d-extends m. then for every λ = λ<θ > |t | + ℵ0 and m ∈ kd of cardinality λ there is a strongly sequence (∆,λ)-homogeneous. conclusion 28. 1) the logic l(qcfc) has the strong ℵ0-saturated model existence property (hence the strong ℵ0-homogeneous model existence property). 2) if κ = cf(κ) ≤ min(c) and κ ≤ min(reg\c) then in part (1) we can replace ℵ0 by κ. demostración. choose χ ∈ c,κ ∈ reg\c and apply 10. �28 received: april 2009. revised: december 2009. 72 saharon shelah cubo 13, 2 (2011) referencias [mak85] johann a. makowsky, compactnes, embeddings and definability, model-theoretic logics (j. barwise and s. feferman, eds.), springer-verlag, 1985, pp. 645–716. [sh:h] saharon shelah, classification theory for abstract elementary classes, studies in logic: mathematical logic and foundations, vol. 18, college publications, 2009. [sh:3] finite diagrams stable in power, annals of mathematical logic 2 (1970), 69–118. [sh:43] generalized quantifiers and compact logic, transactions of the american mathematical society 204 (1975), 342–364. [sh:54] the lazy model-theoretician’s guide to stability, logique et analyse 18 (1975), 241– 308. [sh:88r] abstract elementary classes near ℵ1, chapter i studies in logic, college publ. 18 (2009). 0705.4137. 0705.4137. [mksh:166] alan h. mekler and saharon shelah, stationary logic and its friends. i, notre dame journal of formal logic 26 (1985), 129–138, proceedings of the 1980/1 jerusalem model theory year. [sh:199] saharon shelah, remarks in abstract model theory, annals of pure and applied logic 29 (1985), 255–288. introduction on strongly saturated models cubo a mathematical journal vol.14, no¯ 03, (115–127). october 2012 a unique common coupled fixed point theorem for four maps under ψ φ contractive condition in partial metric spaces k.p.r.rao department of mathematics, nagarjuna nagar522 510, acharya nagarjuna univertsity guntur district,andhra pradesh,india kprrao2004@yahoo.com g.n.v.kishore department of mathematics, swarnandhra institute of engineering and technology, west godavari district, andhra pradesh, india kishore.apr2@gmail.com nguyen van luong department of natural sciences, hong duc university, thanh hoa , viet nam luonghdu@gmail.com abstract in this paper, we obtain a unique coupled common fixed point theorem for four maps in partial metric spaces. resumen en este art́ıculo obtenemos un teorema del punto fijo clásico acoplado único para cuatro aplicaciones en espacios métricos parciales. keywords and phrases: partial metric, weakly compatible maps, complete space. 2010 ams mathematics subject classification: 54h25, 47h10. 116 k.p.r.rao , g.n.v.kishore and nguyen van luong cubo 14, 3 (2012) 1 introduction and preliminaries the notion of partial metric space was introduced by s.g.matthews [13] as a part of the study of denotational semantics of data flow networks. in fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation ([6-10, 14-16], etc). s.g.matthews [13], sandra oltra and oscar valero[11] and salvador romaguera [12] and i.altun, ferhan sola, hakan simsek [1], t. abdeljawad, e. karapinar, k. tas [3], e. karapinar, i.m. erhan [5] proved fixed point theorems in partial metric spaces for a single map and a pair of maps. regarding the concept of coupled fixed points introduced by bhaskar and lakshmikantham [17], in [4], aydi proved some coupled fixed point theorems for the mappings satisfying contractive conditions in partial metric spaces. in this paper, we obtain a unique common coupled fixed point theorem for four self mappings satisfying a ψ − φ contractive condition in partial metric spaces. our result is inspired by the results of luong and thuan [18]. first we recall some definitions and lemmas of partial metric spaces. definition 1.1. [13]. a partial metric on a nonempty set x is a function p : x × x → r+ such that for all x,y,z ∈ x: (p1) x = y ⇔ p(x,x) = p(x,y) = p(y,y), (p2) p(x,x) ≤ p(x,y),p(y,y) ≤ p(x,y), (p3) p(x,y) = p(y,x), (p4) p(x,y) ≤ p(x,z) + p(z,y) − p(z,z). (x,p) is called a partial metric space. it is clear that p(x,y) = 0 implies x = y from (p1) and (p2). but if x = y, p(x,y) may not be zero. a basic example of a partial metric space is the pair (r+,p), where p(x,y) = max{x,y} for all x,y ∈ r+. each partial metric p on x generates τ0 topology τp on x which has a base the family of open p balls {bp(x,ε)|x ∈ x,ε > 0} for all x ∈ x and ε > 0, where bp(x,ε) = {y ∈ x/p(x,y) < p(x,x)+ε} for all x ∈ x and ε > 0. if p is a partial metric on x, then the function ps : x×x → r+ given by ps(x,y) = 2p(x,y) − p(x,x) − p(y,y) is a metric on x. definition 1.2. [13]. let (x, p) be a partial metric space. (i) a sequence {xn} in (x,p) is said to converge to a point x ∈ x if and only if p(x,x) = lim n→∞ p(x,xn). (ii) a sequence {xn} in (x,p) is said to be cauchy sequence if lim n,m→∞ p(xn,xm) exists and is finite . (iii) (x,p) is said to be complete if every cauchy sequence {xn} in x converges, w.r.to τp, to a point x ∈ x such that p(x,x) = lim n,m→∞ p(xn,xm). cubo 14, 3 (2012) a unique common coupled fixed point theorem ... 117 lemma 1.1. [13]. let (x,p) be a partial metric space. (a) {xn} is a cauchy sequence in (x,p) if and only if it is a cauchy sequence in the metric space (x,ps). (b) (x,p) is complete if and only if the metric space (x,ps) is complete. furthermore, lim n→∞ ps(xn,x) = 0 if and only if p(x,x) = lim n→∞ p(xn,x) = lim n,m→∞ p(xn,xm). remark 1.2. let (x,p) be a partial metric space. if {xn} converges to x in (x,p), then lim n→∞ p(xn,y) ≤ p(x,y), ∀ y ∈ x. proof. since {xn} converges to x we have p(x,x) = lim n→∞ p(xn,x). now p(xn,y) ≤ p(xn,x) + p(x,y) − p(x,x). letting n → ∞, lim n→∞ p(xn,y) ≤ lim n→∞ p(xn,x) + p(x,y) − p(x,x). thus lim n→∞ p(xn,y) ≤ p(x,y). definition 1.3. [17]. an element (x,y) ∈ x × x is called a coupled fixed point of mapping f : x × x → x if x = f(x,y) and y = f(y,x). definition 1.4. [2]. an element (x,y) ∈ x × x is called (g1) a coupled coincident point of mappings f : x × x → x and f : x → x if fx = f(x,y) and fy = f(y,x). (g2) a common coupled fixed point of mappings f : x × x → x and f : x → x if x = fx = f(x,y) and y = fy = f(y,x). definition 1.5. [2]. the mappings f : x × x → x and f : x → x are called w compatible if f(f(x,y)) = f(fx,fy) and f(f(y,x)) = f(fy,fx) whenever fx = f(x,y) and fy = f(y,x). using concept of coupled fixed points, luong and thuan in [18] proved some coupled fixed point theorems for a mapping f : x × x → x satisfying the following contractive condition in the partially ordered metric spaces (x,d,≤) ψ(d(f(x,y),f(u,v))) ≤ 1 2 ψ(d(x,u) + d(y,v)) − φ ( d(x,u) + d(y,v) 2 ) for all x,y,u,v ∈ x with x ≥ u and y ≤ v, with φ ∈ φ and ψ ∈ ψ, where ψ denotes the set of all functions ψ : [0,∞) → [0,∞) satisfying (ψ1) ψ is continuous and non-decreasing, (ψ2) ψ(t) = 0 if and only if t = 0, (ψ3) ψ(t + s) ≤ ψ(t) + ψ(s), for all t,s ∈ [0,∞), while φ denotes the set of all functions φ : [0,∞) → [0,∞) satisfying (φ1) limt→rφ(t) > 0 for all r > 0. 118 k.p.r.rao , g.n.v.kishore and nguyen van luong cubo 14, 3 (2012) (φ2) limt→0+φ(t) = 0. from (φ1), it is clear that φ(t) > 0 for all t > 0. now we prove our main result. 2 main result theorem 1. let (x,p) be a partial metric space and let f,g : x → x and f,g : x × x → x be such that (i) for all x,y,u,v ∈ x, ψ(p(f(x,y),g(u,v))) ≤ 1 2 ψ(p(fx,gu) + p(fy,gv)) − φ(p(fx,gu) + p(fy,gv)) , where ψ ∈ ψ and φ ∈ φ, (ii) f(x × x) ⊆ g(x),g(x × x) ⊆ f(x), (iii) either f(x) or g(x) is a complete subspace of x and (iv) the pairs (f,f) and (g,g) are w compatible. then f,g,f and g have a unique common coupled fixed point in x × x. moreover, the common coupled fixed point of f,g,f and g have the form (u,u). proof. let x0,y0 be arbitrary points in x. from(ii), there exist sequences {xn}, {yn}, {zn} and {wn} in x such that f(x2n,y2n) = gx2n+1 = z2n, f(y2n,x2n) = gy2n+1 = w2n, g(x2n+1,y2n+1) = fx2n+2 = z2n+1 and g(y2n+1,x2n+1) = fy2n+2 = w2n+1, n = 0,1,2, ....... we have ψ(p(z2n+1,z2n)) = ψ(p(f(x2n,y2n),g(x2n+1,y2n+1)) ≤ 1 2 ψ(p(z2n,z2n−1) + p(w2n,w2n−1)) −φ(p(z2n,z2n−1) + p(w2n,w2n−1)) (2.1) similarly, ψ(p(w2n+1,w2n)) ≤ 1 2 ψ(p(z2n,z2n−1) + p(w2n,w2n−1)) −φ(p(z2n,z2n−1) + p(w2n,w2n−1)) (2.2) cubo 14, 3 (2012) a unique common coupled fixed point theorem ... 119 from (2.1), (2.2) and (ψ3), we have ψ(p(z2n+1,z2n) + p(w2n+1,w2n)) ≤ ψ(p(z2n+1,z2n)) + ψ(p(w2n+1,w2n)) ≤ ψ(p(z2n,z2n−1) + p(w2n,w2n−1)) −2φ(p(z2n,z2n−1) + p(w2n,w2n−1)) (2.3) ≤ ψ(p(z2n,z2n−1) + p(w2n,w2n−1)) . since ψ is non decreasing, we have p(z2n+1,z2n) + p(w2n+1,w2n) ≤ p(z2n,z2n−1) + p(w2n,w2n−1). similarly, we can show that p(z2n,z2n−1) + p(w2n,w2n−1) ≤ p(z2n−1,z2n−2) + p(w2n−1,w2n−2). thus p(zn+1,zn) + p(wn+1,wn) ≤ p(zn,zn−1) + p(wn,wn−1). put δn = p(zn+1,zn) + p(wn+1,wn).then we have δn ≤ δn−1,n = 1,2,3, ... thus {δn} is a non increasing sequence of nonnegitive real numbers and must converge to a real number, say, δ ≥ 0. suppose δ > 0. letting n → ∞ in (2.3) and using the properties of ψ and φ, we get ψ(δ) ≤ ψ(δ) − 2 lim δ2n→δ φ(δ2n) < ψ(δ) which is a contradiction. hence δ = 0. thus lim n→∞ [p(zn+1,zn) + p(wn+1,wn)] = 0 (2.4) hence from (p2), lim n→∞ [p(zn,zn) + p(wn,wn)] = 0 (2.5) from (2.4) and (2.5) we have that lim n→∞ ps(zn+1,zn) = 0 (2.6) and lim n→∞ ps(wn+1,wn) = 0 (2.7) now we prove that {z2n} and {w2n} are cauchy sequences. on contrary, suppose that {z2n} or {w2n} is not cauchy.this implies that p s(z2m,z2n) 6→ 0 or 120 k.p.r.rao , g.n.v.kishore and nguyen van luong cubo 14, 3 (2012) ps(w2m,w2n) 6→ 0 as n,m → ∞. consequently max{ps(z2m,z2n),p s (w2m,w2n)} 6→ 0 as n,m → ∞. then there exist an ǫ > 0 and monotone increasing sequences of natural numbers {2mk} and {2nk} such that nk > mk, max{ps(z2mk,z2nk),p s(w2mk,w2nk)} ≥ ǫ (2.8) and max{ps(z2mk,z2nk−2),p s(w2m,w2nk−2)} < ǫ. (2.9) from (2.8) and (2.9), we have ǫ ≤ max{ps(z2mk,z2nk),p s (w2mk,w2nk)} ≤ max{ps(z2mk,z2nk−2),p s (w2mk,w2nk−2)} + max{ps(z2nk−2,z2nk−1),p s(w2nk−2,w2nk−1)} + max{ps(z2nk−1,z2nk),p s(w2nk−1,w2nk)} < ǫ + max{ps(z2nk−2,z2nk−1),p s(w2nk−2,w2nk−1)} + max{ps(z2nk−1,z2nk),p s(w2nk−1,w2nk)}. letting k → ∞ and using (2.6) and (2.7) we have lim k→∞ max{ps(z2mk,z2nk),p s(w2mk,w2nk)} = ǫ. (2.10) also, ǫ ≤ max{ps(z2mk,z2nk),p s (w2mk,w2nk)} ≤ max{ps(z2mk,z2mk−1),p s(w2mk,w2mk−1)} + max{ps(z2mk−1,z2nk),p s(w2mk−1,w2nk)} (2.11) ≤ max{ps(z2mk,z2mk−1),p s(w2mk,w2mk−1)} + max{ps(z2mk−1,z2mk),p s(w2mk−1,w2mk)} + max{ps(z2mk,z2nk),p s(w2mk,w2nk)} = 2max{ps(z2mk,z2mk−1),p s (w2mk,w2mk−1)} + max{ps(z2mk,z2nk),p s (w2mk,w2nk)}. letting k → ∞ and using (2.6), (2.7), (2.10) and (2.11), we have lim k→∞ max{ps(z2mk−1,z2nk),p s(w2mk−1,w2nk)} = ǫ. (2.12) on other hand we have max {ps(z2mk,z2nk),p s(w2mk,w2nk)} ≤ max {p s(z2mk,z2nk+1),p s(w2mk,w2nk+1)} + max {ps(z2nk+1,z2nk),p s(w2nk+1,w2nk)} cubo 14, 3 (2012) a unique common coupled fixed point theorem ... 121 letting k → ∞ and using (2.5),(2.6) and (2.7), we have ǫ ≤ lim k→∞ max{ps(z2mk,z2nk+1),p s(w2mk,w2nk+1)} + 0 ≤ lim k→∞ max { 2p(z2mk,z2nk+1) − p(z2mk,z2mk) − p(z2nk+1,z2nk+1), 2p(w2mk,w2nk+1) − p(w2mk,w2mk) − p(w2nk+1,w2nk+1) } ≤ 2 lim k→∞ max{p(z2mk,z2nk+1),p(w2mk,w2nk+1)} thus, ǫ 2 ≤ lim k→∞ max{p(z2mk,z2nk+1),p(w2mk,w2nk+1)} by the properties of ψ ψ ( ǫ 2 ) ≤ ψ ( lim k→∞ max{p(z2mk,z2nk+1),p(w2mk,w2nk+1)} ) = lim k→∞ max{ψ(p(z2mk,z2nk+1)),ψ(p(w2mk,w2nk+1))} (2.13) now ψ(p(z2mk,z2nk+1)) = ψ(p(f(x2mk,y2mk),g(x2nk+1,y2nk+1))) ≤ 1 2 ψ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) −φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) ≤ 1 2 [ψ(p(z2mk−1,z2nk)) + ψ(p(w2mk−1,w2nk))] −φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) ≤ max{ψ(p(z2mk−1,z2nk)),ψ(p(w2mk−1,w2nk))} −φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) = ψ(max{p(z2mk−1,z2nk),p(w2mk−1,w2nk)}) −φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) similarly ψ(p(w2mk,w2nk+1)) ≤ ψ(max{p(z2mk−1,z2nk),p(w2mk−1,w2nk)}) −φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) . 122 k.p.r.rao , g.n.v.kishore and nguyen van luong cubo 14, 3 (2012) hence from (2.13),(2.5) and (2.12), we have ψ ( ǫ 2 ) ≤ lim k→∞ { ψ(max{p(z2mk−1,z2nk),p(w2mk−1,w2nk)}) −φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) } ≤ lim k→∞ ψ       max    1 2 ( ps(z2mk−1,z2nk) + p(z2mk−1,z2mk−1) +p(z2nk,z2nk) ) , 1 2 ( ps(w2mk−1,w2nk) + p(w2mk−1,w2mk−1) +p(w2nk,w2nk) )          − lim k→∞ φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) = ψ ( ǫ 2 ) − lim k→∞ φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) = ψ ( ǫ 2 ) − lim k→∞ φ     1 2     ps(z2mk−1,z2nk) + p(z2mk−1,z2mk−1) +p(z2nk,z2nk) + p s(w2mk−1,w2nk) +p(w2mk−1,w2mk−1) + p(w2nk,w2nk)         = ψ ( ǫ 2 ) − lim t→ ǫ 2 φ(t) , where ǫ 2 = lim k→∞ 1 2     ps(z2mk−1,z2nk) + p(z2mk−1,z2mk−1) +p(z2nk,z2nk) + p s(w2mk−1,w2nk) +p(w2mk−1,w2mk−1) + p(w2nk,w2nk)     < ψ ( ǫ 2 ) , which is a contradiction. hence {z2n} and {w2n} are cauchy sequences in the metric space (x,p s). letting n,m → ∞ in |ps(z2n+1,z2m+1) − p s(z2n,z2m)| ≤ p s(z2n+1,z2n) + p s(z2m+1,z2m). we get lim n→∞ ps(z2n+1,z2m+1) = 0. letting n,m → ∞ in |ps(w2n+1,w2m+1) − p s(w2n,w2m)| ≤ p s(w2n+1,w2n) + p s(w2m+1,w2m) we get lim n→∞ ps(w2n+1,w2m+1) = 0. thus {z2n+1} and {w2n+1} are cauchy sequences in the metric space (x,p s). hence {zn} and {wn} are cauchy sequences in the metric space (x,p s). hence we have that lim n→∞ ps(zn,zm) = 0 = lim n→∞ ps(wn,wm). now from definition of ps and from (2.5) we have lim n→∞ p(zn,zm) = 0 (2.14) cubo 14, 3 (2012) a unique common coupled fixed point theorem ... 123 and lim n→∞ p(wn,wm) = 0. (2.15) suppose f(x) is complete. since {z2n+1} ⊆ f(x) and {w2n+1} ⊆ f(x) are cauchy sequences in the complete metric space (f(x),ps), it follows that the sequences {z2n+1} and {w2n+1} are convergent in (f(x),ps).thus lim n→∞ ps(z2n+1,u) = 0 and lim n→∞ ps(w2n+1,v) = 0 for some u and v in f(x). since u,v ∈ f(x). there exist s,t ∈ x such that u = fs and v = ft. since {zn} and {wn} are cauchy sequences in x and {z2n+1} → u and {w2n+1} → v, it follows that {z2n} → u and {w2n} → v. from lemma 1.1, we have p(u,u) = lim n→∞ p(z2n,u) = lim n→∞ p(z2n+1,u) = lim n, m→∞ p(zn,zm) (2.16) and p(v,v) = lim n→∞ p(w2n,v) = lim n→∞ p(w2n+1,v) = lim n, m→∞ p(wn,wm) (2.17) from (2.16), (2.17), (2.14) and (2.15) we have p(u,u) = 0 = p(v,v). (2.18) now, p(f(s,t),u) ≤ p(f(s,t),z2n+1) + p(z2n+1,u) − p(z2n+1,z2n+1) ≤ p(f(s,t),g(x2n+1,y2n+1)) + p(z2n+1,u). therefore, ψ(p(f(s,t),u)) ≤ ψ(p(f(s,t),g(x2n+1,y2n+1)) + p(z2n+1,u)) ≤ ψ(p(f(s,t),g(x2n+1,y2n+1))) + ψ(p(z2n+1,u)), from (ψ3) ≤ 1 2 ψ(p(u,z2n) + p(v,w2n)) − φ(p(u,z2n) + p(v,w2n)) + ψ(p(z2n+1,u)). letting n → ∞ and using (2.16), (2.17), (2.18) and (φ2), (ψ1) we get ψ(p(f(s,t),u)) ≤ 0. hence f(s,t) = u = fs (by (ψ2)). similarly, we have f(t,s) = v = ft. since the pair (f,f) is w compatible, we have fu = f(u,v) and fv = f(v,u). suppose that fu 6= u or fv 6= v. ps(fu,z2n) = 2p(fu,z2n) − p(fu,fu) − p(z2n,z2n). 124 k.p.r.rao , g.n.v.kishore and nguyen van luong cubo 14, 3 (2012) letting n → ∞, we get ps(fu,u) = 2 lim n→∞ p(fu,z2n) − p(fu,fu) − 0, from (2.5) or 2p(fu,u) − p(fu,fu) − p(u,u) = 2 lim n→∞ p(fu,z2n) − p(fu,fu) or p(fu,u) = lim n→∞ p(fu,z2n), from (2.18). similarly, we have p(fv,v) = lim n→∞ p(fv,w2n).thus lim n→∞ [p(fv,z2n) + p(fv,w2n)] = p(fu,u) + p(fv,v) > 0 (2.19) we have p(fu,u) ≤ p(fu,z2n+1) + p(z2n+1,u) − p(z2n+1,z2n+1) ≤ p(f(u,v),g(x2n+1,y2n+1)) + p(z2n+1,u). thus, ψ(p(fu,u)) ≤ ψ(p(f(u,v),g(x2n+1,y2n+1)) + ψ(p(z2n+1,u)), from (ψ3) ≤ 1 2 ψ(p(fu,z2n) + p(fv,w2n)) −φ(p(fu,z2n) + p(fv,w2n)) + ψ(p(z2n+1,u)). similarly, we have ψ(p(fv,v)) ≤ 1 2 ψ(p(fu,z2n) + p(fv,w2n)) −φ(p(fu,z2n) + p(fv,w2n)) + ψ(p(w2n+1,v)). hence ψ(p(fu,u) + p(fv,v)) ≤ ψ(p(fu,u)) + ψ(p(fv,v)), from (ψ3) ≤ ψ(p(fu,z2n) + p(fv,w2n)) −2φ(p(fu,z2n) + p(fv,w2n)) +ψ(p(z2n+1,u)) + ψ(p(w2n+1,v)). letting n → ∞ and using (2.19),(φ1),(2.16),(2.17) and (ψ1), we get ψ(p(fu,u) + p(fv,v)) < ψ(p(fu,u) + p(fv,v)). it is a contradiction. hence fu = u and fv = v. thus f(u,v) = fu = u and f(v,u) = fv = v. (2.20) cubo 14, 3 (2012) a unique common coupled fixed point theorem ... 125 since f(x × x) ⊆ g(x), there exist a,b ∈ x such that u = f(u,v) = ga and v = f(v,u) = gb. ψ(p(u,g(a,b))) = ψ(p(f(u,v),g(a,b))) ≤ 1 2 ψ(p(u,u) + p(v,v)) − φ(p(u,u) + p(v,v)) = 1 2 ψ(0) − φ(0), ( from (2.18)) ≤ 0, (since ψ(0) = 0 and φ(0) ≥ 0). hence ψ(p(u,g(a,b))) = 0, which implies that g(a,b) = u = ga. similarly, we have g(b,a) = v = gb. since the pair (g,g) is w compatible, we have gu = g(u,v) and gv = g(v,u). suppose gu 6= u or gv 6= v. we have ψ(p(u,gu)) = ψ(p(f(u,v),g(u,v))) ≤ 1 2 ψ(p(u,gu) + p(v,gv)) − φ(p(u,gu) + p(v,gv)) and ψ(p(v,gv)) = ψ(p(f(v,u),g(v,u))) ≤ 1 2 ψ(p(u,gu) + p(v,gv)) − φ(p(u,gu) + p(v,gv)) . hence ψ(p(u,gu) + p(v,gv)) ≤ ψ(p(u,gu)) + ψ(p(v,gv)) ≤ ψ(p(u,gu) + p(v,gv)) − 2φ(p(u,gu) + p(v,gv)) < ψ(p(u,gu) + p(v,gv)) (since φ(t) > 0 ∀ t > 0). hence gu = u and gv = v.thus, u = gu = g(u,v) and v = gv = g(v,u) (2.21) from (2.20) and (2.21), it follows that (u,v) is a common coupled fixed point of f,g,f and g. let (u∗,v∗) be another common coupled fixed point of f,g,f and g. we have ψ(p(u,u∗) + p(v,v∗)) ≤ ψ(p(u,u∗)) + ψ(p(v,v∗)) ≤ ψ(p(f(u,v),g(u∗,v∗))) + ψ(p(f(v,u),g(v∗,u∗))) ≤ 1 2 ψ(p(u,u∗) + p(v,v∗)) − φ(p(u,u∗) + p(v,v∗)) + 1 2 ψ(p(u,u∗) + p(v,v∗)) − φ(p(u,u∗) + p(v,v∗)) = ψ(p(u,u∗) + p(v,v∗)) − 2φ(p(u,u∗) + p(v,v∗)) < ψ(p(u,u∗) + p(v,v∗)), 126 k.p.r.rao , g.n.v.kishore and nguyen van luong cubo 14, 3 (2012) which is a contradiction. hence (u,v) is the unique common coupled fixed point of f,g,f and g. now we will show that u = v. suppose u 6= v. ψ(p(u,v)) = ψ(p(f(u,v),g(u,v))) ≤ 1 2 ψ(p(u,v) + p(v,u)) − φ(p(u,v) + p(v,u)) ≤ ψ(p(u,v)) − φ(p(u,v)) < ψ(p(u,v)). hence u = v. thus u = fu = f(u,u) = g(u,u) = gu, that is, the common coupled fixed point of f,g,f and g has the form (u,u). received: june 2011. revised: february 2012. references (1) ishak altun, ferhan sola and hakan simsek, generalized contractions on partial metric spaces, topology and its applications, 157, (2010), 2778 2785. 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() cubo a mathematical journal vol.13, no¯ 03, (1–15). october 2011 on the semilocal convergence of newton–type methods, when the derivative is not continuously invertible ioannis k. argyros department of mathematics sciences, cameron university, lawton, ok 73505, usa. email: iargyros@cameron.edu and säıd hilout laboratoire de mathématiques et applications, poitiers university, bd. pierre et marie curie, téléport 2, b.p. 30179 86962 futuroscope chasseneuil cedex, france email: said.hilout@math.univ--poitiers.fr abstract we provide a semilocal convergence analysis for newton–type methods to approximate a locally unique solution of a nonlinear equation in a banach space setting. the fréchet– derivative of the operator involved is not necessarily continuous invertible. this way we extend the applicability of newton–type methods [1]–[12]. we also provide weaker sufficient convergence conditions, and finer error bound on the distances involved (under the same computational cost) than [1]–[12], in some intersting cases. numerical examples are also provided in this study. 2 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) resumen ofrecemos un análisis de convergencia semilocal de los métodos de newton type para aproximar una solución local única de una ecuación no lineal en un entorno de un espacio de banach. l derivada de fréchet del operador en cuestión no es necesariamente invertible continua. de esta manera ampliamos la aplicabilidad de los métodos del tipo newton [1]–[12]. también proporcionamos condiciones suficientes mas dbiles de convergencia, y una cota de error más fina de las distancias involucradas que [1]–[12] (en el mismo coste computacional), en algunos casos interesantes. también presentamos ejemplos numéricos. keywords: newton–type methods, banach space, small divisors, non–invertible operators, semilocal convergence, newton–kantorovich–type hypothesis. mathematics subject classification: 65h10, 65g99, 65j15, 47h17, 49m15. 1 introduction in this study we are concerned with the problem of approximating a locally unique solution x⋆ of equation f(x) = 0, (1.1) where, f is a fréchet–differentiable operator defined on a convex subset d of a banach space x with values in a banach space y. a large number of problems in applied mathematics and also in engineering are solved by finding the solutions of certain equations. for example, dynamic systems are mathematically modeled by difference or differential equations, and their solutions usually represent the states of the systems. for the sake of simplicity, assume that a time–invariant system is driven by the equation ẋ = t (x), for some suitable operator t , where x is the state. then the equilibrium states are determined by solving equation (1.1). similar equations are used in the case of discrete systems. the unknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). except in special cases, the most commonly used solution methods are iterative–when starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. iteration methods are also applied for solving optimization problems. in such cases, the iteration sequences converge to an optimal solution of the problem at hand. since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework. cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 3 the most popular method for generating a sequence {xn} (n ≥ 0), approximating x⋆ is undoubtedly newton’s method: xn+1 = xn − f ′(xn) −1 f(xn) (n ≥ 0), (x0 ∈ d). (1.2) there is an extensive literature on local as well as semilocal convergence results for newton’s method [1]–[12]. however, there many problems for which newton’s method is not applicable in its original form. a case of interest occurs when the derivative is not continuously invertible, as for instance, dealing with problems involving small divisors, or other important examples [4], [6]–[10]. several newton–type methods have addressed this problem [1]–[12]. moret in [10] unified a large class of such newton–type methods, where, at each step, the inverse of the derivative, is replaced by a linear operator, which is obtained recursively from the previous one. two iterative schemes were provided in [10]: scheme 1. let the following be given: a banach space z, an operator valued mapping b : d −→ l(x , y), x0 ∈ d, s−1 ∈ l(z, y), r−1 ∈ l(z, x ). for n ≥ 0, let nn ∈ l(z, z), and set: sn = sn−1 nn + b(xn) rn−1, rn = rn−1 + rn−1 nn, xn+1 = xn + rn ∆n, ∆n being a possible approximate solution of sn ∆n = −f(xn). that is ∆n satisfies an equation of type sn ∆n = −(f(xn) + rn), for a suitable null sequence {rn} ⊂ y. moret [10] provided a semilocal convergence analysis for newton’s method under general conditions on the mapping b, the starting guesses x0, s−1, r−1, the operators nn, and the sequence {rn}. similar results were given for scheme 2. 4 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) there are two type of problems common to both schemes: the convergence may be too slow, and the schemes may not even be applicable. for example in both schemes a condition of the form: ‖ iy − t r−1 ‖< 1 (1.3) is required, where, t = f′(x0) (scheme 1) (1.4) or t = b(x0) (scheme 2). (1.5) there are simple numerical examples to show that condition (1.3) is violated. let scalar function f given by: f(x) = 1 128 x 2 − 1 64 cos x, (1.6) and choose x0 = 0. then f′(x0) = 0, which shows that (1.3) is violated for t = b(x0) = f′(x0). hence, schemes 1 and 2 cannot be used to approximate x⋆. in particular, the classical newton–moser method [4], [6]–[10], obtained from scheme 1, by setting z = y, b = f′, s−1 = iy , nn = iy − f′(xn) rn−1, and rn = 0 (n ≥ 0) cannot be used. to address these problems, we consider the newton–type methods of the form (ntm): xn+1 = xn − (f ′(xn) − a)−1 f(xn) (1.7) or more generally (nlm): xn+1 = xn − (c(xn) − a)−1 f(xn), (1.8) where, c(x) ∈ l(x , y) is an approximation of f′(x), (x ∈ d), and a a given linear operator. in the case of function f given in (1.6) (ntm) can be used to approximate x⋆, if a is an invertible operator. methods (1.7), and (1.8) can be combined into one, even more general (gntm): xn+1 = xn − a(xn)−1 f(xn), (1.9) where, a(x) ∈ l(x , y), (x ∈ d). cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 5 note that if: a(x) = f′(x) − a (1.10) or a(x) = c(x) − a, (1.11) we obtain (1.7), and (1.8), respectively. sufficient conditions for semilocal convergence of (gntm), and estimates on the distances ‖ xn+1 − xn ‖, ‖ xn − x⋆ ‖ have been given by several authors [1]–[6], [11], [12]. however, in the special case of (1.10) (or (1.11)), we can do better by a direct approach. that is, we can provide (under the same computational cost) weaker sufficient convergence conditions, and finer error estimates on ‖ xn+1 − xn ‖, ‖ xn − x⋆ ‖ (n ≥ 0). 2 semilocal convergence analysis of (gntm) let l0 > 0, l > 0, η > 0, a > 0, c > 1, be given constants. set: b = c 2 (l η + 2 a), r = c − 1 c l0 . we need the following result on majorizing sequences for (gntm): lemma 2.1. assume: c ∈ (1, 1 a ), (2.1) and η ≤ (c − 1) (1 − c a) c ( l0 + c − 1 2 l ) . (2.2) then, scalar sequence {tn} (n ≥ 0) given by t0 = 0, t1 = η, tn+2 = tn+1 + l (tn+1 − tn) + 2 a 2 (1 − l0 tn+1) (tn+1 − tn) (2.3) is non–decreasing, bounded by t ⋆⋆ = c − 1 c l0 , (2.4) 6 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) and converges to its unique least upper bound t⋆, satisfying: 0 ≤ t⋆ ≤ η 1 − b ≤ t⋆⋆. (2.5) moreover the following estimates hold for all n ≥ 0: tn+2 − tn+1 ≤ b (tn+1 − tn) ≤ bn+1 η. (2.6) proof. we shall show using induction on the integer k: 0 ≤ tk+2 − tk+1 = l (tk+1 − tk) + 2 a 2 (1 − l0 tk+1) (tk+1 − tk) ≤ b (tk+1 − tk), (2.7) and tk+1 < r0. (2.8) estimates (2.7), and (2.8) hold for k = 0, by the initial conditions. assume (2.7), and (2.8) hold for all m ≤ k. then, we have: 0 ≤ tk+2 − tk+1 ≤ b (tk+1 − tk) ≤ b (b (tk − tk−1)) = b2 (tk − tk−1) ≤ · · · ≤ bk+1 η and tk+1 ≤ tk + bk η ≤ tk−1 + bk−1 η + bk η ≤ t1 + b1 η + · · · + bk η = 1 − bk+1 1 − b η < η 1 − b ≤ r by (2.1), and (2.2), which complete the induction for (2.7), and (2.8). finally, sequence {tn} is non–decreasing, and bounded above by t ⋆⋆, and as such it converges to its unique least upper bound t⋆. that completes the proof of lemma 2.1. ♦ we shall show the following semilocal convergence theorem for (gntm) in the special case, when a is given by (1.10). theorem 2.1. let f : d ⊆ x −→ y be a fréchet–differentiable operator, and let a(x) ∈ l(x , y) be given by (1.10). assume that there exist an open convex subset d of x , x0 ∈ d, a bounded cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 7 inverse a(x0)−1 of a(x0), and constants l0 > 0, l > 0, a ≥ 0, and η > 0, such that for all x, y ∈ d: ‖ a(x0)−1 f(x0) ‖≤ η, (2.9) ‖ a(x0)−1 [f′(x) − f′(y)] ‖≤ l ‖ x − y ‖, (2.10) ‖ a(x0)−1 [f′(x) − a(x)] ‖≤ a, (2.11) ‖ a(x0)−1 [a(x) − a(x0)] ‖≤ l0 ‖ x − x0 ‖, (2.12) u(x0, t ⋆) = {x ∈ x , ‖ x − x0 ‖≤ t⋆} ⊆ d, (2.13) and the hypotheses of lemma 2.1 hold. then, sequence {xn} (n ≥ 0) generated by (gntm) is well defined, remains in u(x0, t⋆) for all n ≥ 0, and converges to a solution x⋆ of equation f(x) = 0 in u(x0, t⋆). moreover, the following estimates hold for all n ≥ 0: ‖ xn+1 − xn ‖≤ tn+1 − tn, (2.14) and ‖ xn − x⋆ ‖≤ t⋆ − tn, (2.15) where, sequence {tn} (n ≥ 0), and t⋆ are given in lemma 2.1. furthermore, the solution x⋆ of equation (1.1) is unique in u(x0, t ⋆) provided that: ( l 2 + a + l0 ) t ⋆ + a < 1. (2.16) proof. we shall show using induction on m ≥ 0: ‖ xm+1 − xm ‖≤ tm+1 − tm, (2.17) and u(xm+1, t ⋆ − tm+1) ⊆ u(xm, t⋆ − tm). (2.18) for every z ∈ u(x1, t⋆ − t1), ‖ z − x0 ‖ ≤ ‖ z − x1 ‖ + ‖ x1 − x0 ‖ ≤ t⋆ − t1 + t1 = t⋆ − t0, 8 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) implies z ∈ u(x0, t⋆ − t0). we also have ‖ x1 − x0 ‖=‖ a(x0)−1 f(x0) ‖≤ η = t1 − t0. that is (2.17), and (2.18) hold for m = 0. given they hold for n ≤ m, then: ‖ xm+1 − x0 ‖ ≤ m+1∑ i=1 ‖ xi − xi−1 ‖ ≤ m+1∑ i=1 (ti − ti−1) = tm+1 − t0 = tm+1, and ‖ xm + θ (xm+1 − xm) − x0 ‖ ≤ tm + θ (tm+1 − tm) ≤ t⋆, for all θ ∈ (0, 1). using (2.8), (2.12), the induction hypotheses, we get: ‖ a(x0)−1 [a(xm+1) − a(x0)] ‖ ≤ l0 ‖ xm+1 − x0 ‖ ≤ l0 (tm+1 − t0) ≤ l0 tm+1 < 1. (2.19) it follows from (2.19), and the banach lemma on invertible operators [4], that a(xm+1)−1 exists, and ‖ a(xm+1)−1 a(x0) ‖≤ (1 − l tm+1)−1. (2.20) using (1.9), we obtain the approximation: xm+2 − xm+1 = −a(xm+1)−1 f(xm+1) = −a(xm+1)−1 a(x0) a(x0)−1 ( ∫ 1 0 [f′(xm+1 + θ (xm − xm+1)) − f ′(xm)] (xm+1 − xm) dθ+ (f′(xm) − a(xm)) (xm+1 − xm) ) (2.21) using (2.10)–(2.12), (2.17) (2.20), and (2.21), and the induction hypotheses, we obtain in turn: cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 9 ‖ xm+2 − xm+1 ‖ ≤ (1 − l0 tm+1)−1 ( l 2 ‖ xm+1 − xm ‖2 +a ‖ xm+1 − xm ‖ ) ≤ (1 − l0 tm+1)−1 ( l 2 (tm+1 − tm) + a ) (tm+1 − tm) = tm+2 − tm+1, (2.22) which shows (2.17) for all m ≥ 0. thus, for every z ∈ u(xm+2, t⋆ − tm+2), we have: ‖ z − xm+1 ‖ ≤ ‖ z − xm+2 ‖ + ‖ xm+2 − xm+1 ‖ ≤ t⋆ − tm+2 + tm+2 − tm+1 = t⋆ − tm+1, which shows (2.18) for all m ≥ 0. lemma 2.1 implies that sequence {tn} is cauchy. moreover, it follows from (2.17) and (2.18) that {xn} (n ≥ 0) is also a cauchy sequence in a banach space x , and as such it converges to some x⋆ ∈ u(x0, t⋆) (since u(x0, t⋆) is a closed set). by letting m −→ ∞ in (2.22), we obtain f(x⋆) = 0. furthermore estimate (2.15) is obtained from (2.14) by using standard majorization techniques [1], [4]. finally to show that x⋆ is the unique solution of equation (1.1) in u(x0, t ⋆), as in (2.21) and (2.22), we get in turn for y⋆ ∈ u(x0, t⋆), with f(y⋆) = 0, the estimation: ‖ y⋆ − xm+1 ‖ ≤ ‖ a(xm)−1 a(x0) ‖ ( ∫ 1 0 ‖ a(x0)−1 (f′(xm + θ (y⋆ − xm)) − f′(xm)) ‖ dθ + ‖ a(x0)−1 [f′(xm) − a(xm)] ‖ ) ‖ y⋆ − xm ‖ ≤ (1 − l0 tm+1)−1 ( l 2 ‖ y⋆ − xm ‖2 +a ‖ y⋆ − xm ‖ ) ≤ (1 − l0 tm+1)−1 ( l 2 (t⋆ − tm) + a ) ‖ y⋆ − xm ‖ ≤ (1 − l0 t⋆)−1 ( l 2 (t⋆ − t0) + a ) ‖ x⋆ − xm ‖ < ‖ y⋆ − xm ‖ (2.23) by (2.20). it follows by (2.23) that lim m−→ ∞ xm = y ⋆. but we showed lim m−→ ∞ xm = x ⋆. hence, we deduce x⋆ = y⋆. 10 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) that completes the proof of theorem 2.1. ♦ note that t⋆ can be replaced by t⋆⋆ given by (2.4) in the uniqueness hypothesis provided that u(x0, t ⋆⋆) ⊆ d, or in all hypotheses of the theorem. 3 applications remark 3.1. according to [4], [6], [12], the sufficient convergence condition becomes η ≤ (1 − a)2 2 l , (3.1) and the majorizing iteration {vn} is given by: v0 = 0, v1 = η, vn+2 = vn+1 + l 2 v 2 n+1 − (1 − a) vn+1 + η 1 − l0 vn+1 (n ≥ 0). (3.2) in view of (2.2), and (3.1), our condition is weaker, if: (1 − a)2 2 l < (c − 1) (1 − c a) c (l0 + c − 1 2 l) . (3.3) let l0 = p l, for p ∈ [0, 1]. then (3.3) holds, if: (1 − a)2 2 < (c − 1) (1 − c a) c (p + c − 1 2 ) . (3.4) if p is close enough to zero, and e.g. c = 2, we have (3.4) holds provided: (1 − a)2 < 2 (1 − 2 a) or a ∈ (0, √ 2 − 1). as example, set p = .1, c = 2, a = .3, then condition (3.4) becomes: .245 < .333 (3.5) hence, for η ∈ (.245, .333), our results can apply, where as the ones in [4], [6], [12] cannot guarantee convergence, since (3.1) is violated. cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 11 concerning the error estimates, if: (l − 2 l0) η + 2 (a + 1) 1 − l0 η (1 − η) + l η < 1 + 2 a, (3.6) then, we have v0 = t0, v1 = t1, v2 = t2, t3 − t2 < v3 − v2. an inductive argument shows: tn < vn (n ≥ 3), tn+1 − tn < vn+1 − vn (n ≥ 2), t ⋆ − tn < v ⋆ − vn (n ≥ 2), and t ⋆ < v ⋆ . estimates (3.6) holds, for example, let l0 = .00005, l = .0001, η = .8, a = .0001 to obtain .400496017 < 1.0002. note also that (2.2), and (3.1) also hold, since: η ≤ 4999, and η ≤ 4999.00005, respectively. hence, the claims made in the introduction of this study are now justified. in practice, on will test (2.2), (3.1), (3.6), and then use the combination of the best results. let return back to the numerical examples in the introduction of this study. example 3.1. let d = [−4, 4], and choose c = 2, and a = .2. then, we have: l0 = l = 5 32 , η = 5 64 , (3.7) and (2.2) becomes: .15625 < 1.28. hence, the conclusions of theorem 2.1 apply to solve equation f(x) = 0. 12 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) example 3.2. define the scalar function f by f(x) = c0 x + c1 + c2 sin e c3 x, x0 = 0, where ci, i = 1, 2, 3 are given parameters. then it can easily be seen that for c3 large and c2 sufficiently small, l l0 can be arbitrarily large. that is (2.2) may be satisfied but not (3.1). we provide two examples, where l0 < l. example 3.3. let x = y = c[0, 1] be the space of real–valued continuous functions defined on the interval [0, 1] with norm ‖ x ‖= max 0≤s≤1 |x(s)|. let θ ∈ [0, 1] be a given parameter. consider the ”cubic” integral equation: u 3(s) + λ u(s) ∫ 1 0 q(s, t) u(t) dt + y(s) − θ = 0. (3.8) here the kernel q(s, t) is a continuous function of two variables defined on [0, 1] × [0, 1]; the parameter λ is a real number called the ”albedo” for scattering; y(s) is a given continuous function defined on [0, 1] and x(s) is the unknown function sought in c[0, 1]. equations of the form (3.8) arise in the kinetic theory of gasses [4], [5]. for simplicity, we choose u0(s) = 0, y(s) = 1, and q(s, t) = s s + t , for all s ∈ [0, 1], and t ∈ [0, 1], with s + t 6= 0. if we let d = u(u0, 1 − θ), and define the operator f on d by: f(x(s)) = 1 3000 x 3(s) + λ x(s) ∫ 1 0 q(s, t) x(t) dt + y(s) − θ, (3.9) for all s ∈ [0, 1], then every zero of f satisfies equation (3.8). we have the estimates: max 0≤s≤1 | ∫ s s + t dt| = ln 2. therefore, if we set a−1 =‖ a(u0)−1 ‖, then it follows from hypotheses of theorem 2.1: f ′(u0(s)) = 0, η = a−1 (|λ| ln 2 + 1 − θ), l = 2 a−1 (|λ| ln 2 + 2 − θ 1000 ) and l0 = a −1 (2 |λ| ln 2 + 3 − θ 1000 ). it follows from theorem 2.1 that if condition (2.2) holds, then problem (3.8) has a unique solution near u0. this assumption can be weaker than the one given before the newton–kantorovich hypothesis (3.1), since l0 < l for all θ ∈ [0, 1]. for λ = .001, a = .9, θ = .9, c = 1.1, cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 13 condition (2.2) becomes: .111188124 < .223204896. hence, the conclusions of theorem 2.1 apply to solve equation (3.9). example 3.4. consider the following nonlinear boundary value problem [4] { u′′ = −u3 − γ u2 u(0) = 0, u(1) = 1. it is well known that this problem can be formulated as the integral equation u(s) = s + ∫ 1 0 q(s, t) (u3(t) + γ u2(t)) dt (3.10) where, q is the green function: q(s, t) = { t (1 − s), t ≤ s s (1 − t), s < t. we observe that max 0≤s≤1 ∫ 1 0 |q(s, t)| = 1 8 . let x = y = c[0, 1], with norm ‖ x ‖= max 0≤s≤1 |x(s)|. then problem (3.10) is in the form (1.1), where, f : d −→ y is defined as [f(x)] (s) = x(s) − s − ∫ 1 0 q(s, t) (x3(t) + γ x2(t)) dt. it is easy to verify that the fréchet derivative of f is defined in the form [f′(x)v] (s) = v(s) − ∫ 1 0 q(s, t) (3 x2(t) + 2 γ x(t)) v(t) dt. if we set u0(s) = s, and d = u(u0, r), then since ‖ u0 ‖= 1, it is easy to verify that u(u0, r) ⊂ u(0, r + 1). it follows that 2 γ < 5, then ‖ i − f′(u0) ‖ ≤ 3 ‖ u0 ‖2 +2 γ ‖ u0 ‖ 8 = 3 + 2 γ 8 , ‖ f′(u0)−1 ‖ ≤ 1 1 − 3 + 2 γ 8 = 8 5 − 2 γ , ‖ f(u0) ‖ ≤ ‖ u0 ‖3 +γ ‖ u0 ‖2 8 = 1 + γ 8 , 14 ioannis k. argyros and säıd hilout cubo 13, 3 (2011) ‖ f(u0)−1 f(u0) ‖ ≤ 1 + γ 5 − 2 γ . on the other hand, for x, y ∈ d, we have [(f′(x) − f′(y))v] (s) = − ∫ 1 0 q(s, t) (3 x2(t) − 3 y2(t) + 2 γ (x(t) − y(t))) v(t) dt. consequently, ‖ f′(x) − f′(y) ‖ ≤ ‖ x − y ‖ (2 γ + 3 (‖ x ‖ + ‖ y ‖)) 8 ≤ ‖ x − y ‖ (2 γ + 6 r + 6 ‖ u0 ‖) 8 = γ + 6 r + 3 4 ‖ x − y ‖, ‖ f′(x) − f′(u0) ‖ ≤ ‖ x − u0 ‖ (2 γ + 3 (‖ x ‖ + ‖ u0 ‖)) 8 ≤ ‖ x − u0 ‖ (2 γ + 3 r + 6 ‖ u0 ‖) 8 = 2 γ + 3 r + 6 8 ‖ x − u0 ‖ . therefore, conditions of theorem 2.1 hold with η = 1 + γ 5 − 2 γ , l = γ + 6 r + 3 4 , l0 = 2 γ + 3 r + 6 8 . note also that l0 < l. conclusion we provided a semilocal convergence analysis for (gntm) method in order to approximate a locally unique solution of an equation in a banach space, when the derivative of the operator involved is not continuously invertible. we provided an analysis with the following advantages over the work in [1]–[12] weaker sufficient convergence conditions, and larger convergence domain. note that these advantages are obtained under the same computational cost as in [1]–[12]. numerical examples further validating our results are also provided in this study. received: september 2009. revised: october 2009. cubo 13, 3 (2011) on the semilocal convergence of newton–type · · · 15 references [1] argyros, i.k., the theory and application of abstract polynomial equations, st.lucie/crc/lewis publ. mathematics series, 1998, boca raton, florida, u.s.a. [2] argyros, i.k., on the newton–kantorovich hypothesis for solving equations, j. comput. appl. math., 169 (2004), 315–332. [3] argyros, i.k., a unifying local–semilocal convergence analysis and applications for two–point newton–like methods in banach space, j. math. anal. appl., 298 (2004), 374–397. [4] argyros, i.k., computational theory of iterative methods, series: studies in computational mathematics, 15, editors: c.k. chui and l. wuytack, elsevier publ. co., new york, usa, 2007. [5] chandrasekhar, s., radiative transfer, dover publ., new york, 1960. [6] nashed, m.z., chen, x., convergence of newton–like methods for singular operator equations using outer inverses, numer. math., 66 (1993), 235–257. [7] craven, b.d., nashed, m z., generalized implicit function theorems when the derivative has no bounded inverse, nonlinear anal., 6 (1982), 375–387. [8] hald, o.h., on a newton–moser type method, numer. math., 23 (1975), 411–426. [9] jerome, j.w., an adaptive newton algorithm based on numerical inversion: regularization as postconditioner., numer. math., 47 (1985), 123–138. [10] moret, i., on a general iterative scheme for newton–type methods, numer. funct. anal. optim., 9 (1987-1988), 1115–1137. [11] potra, f.a., sharp error bounds for a class of newton–like methods, libertas mathematica, 5 (1985), 71–84. [12] yamamoto, t., a convergence theorem for newton–like methods in banach spaces, numer. math., 51 (1987), 545–557. introduction semilocal convergence analysis of (gntm) applications cubo a mathematical journal vol.17, no¯ 01, (15–26). december 2016 positive asymptotically almost periodic solutions of an impulsive hematopoiesis model peng chen 1, hui-sheng ding 1, gaston m. n’guérékata 2 1 college of mathematics and information science, jiangxi normal university, nanchang, jiangxi 330022, people’s republic of china. 2 department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore, m.d. 21251, usa. 9972740999@qq.com, dinghs@mail.ustc.edu.cn, gaston.n’guerekata@morgan.edu abstract in this paper, we introduce the notion of impulsive asymptotically almost periodic functions and prove some basic properties of such functions. then, we discuss the existence and exponential stability of positive asymptotically almost periodic solution for an impulsive hematopoiesis model. an example is given to illustrate our results. resumen en este art́ıculo, introducimos la noción de funciones impulsivas asintóticamente casi periódicas y probamos algunas propiedades básicas para dichas funciones. luego, discutimos la existencia y estabilidad exponencial de soluciones positivas asintóticamente casi periódicas para un modelo impulsivo de hematopoyesis. un ejemplo es dado para ilustrar nuestros resultados. keywords and phrases: almost periodic, asymptotically almost periodic, impulsive, hematopoiesis. 2010 ams mathematics subject classification: 34k14. 16 peng chen, hui-sheng ding, gaston m. n’guérékata cubo 18, 1 (2016) 1 introduction and preliminaries in [8], mackey and glass proposed the following nonlinear delay differential equation h′(t) = −αh(t) + β 1 + hn(t − τ) (1.1) as an appropriate model of hematopoiesis that describes the process of production of all types of blood cells generated by a remarkable self-regulated system that is responsive to the demands put upon it. in medical terms, h(t) denotes the density of mature cells in blood circulation at time t and τ is the time delay between the production of immature cells in the bone marrow and their maturation for release in circulating bloodstream. it is assumed that the cells are lost from the circulation at a rate α, and the flux of the cells into the circulation from the stem cell compartment depends on the density of mature cells at the previous time t − τ. in this paper, we consider the existence and stability of asymptotically almost periodic solutions for the following impulsive hematopoiesis model: ⎧ ⎪⎨ ⎪⎩ x′(t) = −a(t)x(t) + b(t) 1+xn(t−τ) , t ̸= tk, ∆x|t=tk = ckx(tk) + ik(x(tk)), k ∈ z, (1.2) where ∆x|t=tk = x(tk + 0) − x(tk − 0), τ ≥ 0 is a constant and the coefficients satisfy some conditions, which will be listed in section 3. the direct impetus of this paper comes from two sources. the first source is some recent works on the almost periodic solutions for hematopoiesis models without impulse effect (see, e.g., [4, 7] and references therein); the second source is some recent works on periodic solutions and almost periodic solutions for impulsive hematopoiesis models (see, e.g., [10, 1, 12] and references therein). stimulated by these works, we aim to make further study on this topic. as one will see, there are two differences of our work from some earlier works on almost periodic solutions to equation (1.2) (cf. [1, 12]). the first difference is that we do not assume that inf t∈r a(t) > 0 and −1 ≤ ck ≤ 0 for all k ∈ z. in fact, we weaken these assumptions to m(a) := lim t→+∞ 1 t ∫t 0 a(t)dt > 0 and −1 ≤ ck for all k ∈ z. the second difference is that we investigate the existence and stability of asymptotically almost periodic solution to equation (1.2). to the best of our knowledge, it seems that until now there is no results concerning asymptotically almost periodic solution to equation (1.2). recall that in 1940s, m. fréchet [5] introduced the notion of asymptotically almost periodicity, which turns out to be one of the most interesting and important generalizations of almost periodicity. throughout this paper, we denote by r the set of real numbers, by r+ the set of nonnegative real numbers, by z the set of integers, and by n the set of positive integers. now, let us recall some basic notations about classical almost periodic type functions (for more details, we refer the reader to [2, 3]). cubo 18, 1 (2016) positive asymptotically almost periodic solutions of an impulsive . . . 17 definition 1.1. a set p ⊂ z (or r) is called relatively dense in z (or r) if there exists a number l ∈ n (or r+) such that ∀a ∈ z (or r), [a, a + l] ! p ̸= ∅. definition 1.2. a continuous function f : r → r is called almost periodic if for every ε > 0, p(ε, f) = {τ ∈ r : sup t∈r |f(t + τ) − f(t)| < ε} is relatively dense in r. we denote the set of all such functions by ap(r). definition 1.3. a sequence f : z → r is called almost periodic if for every ε > 0, p(ε, f) = {τ ∈ z : sup n∈z |f(n + τ) − f(n)| < ε} is relatively dense in z. we denote the set of all such sequences by ap(z). definition 1.4. a sequence f : z → r is called asymptotically almost periodic if f = g + h, where g ∈ ap(z) and h ∈ c0(z), where c0(z) is the set of all functions h : z → r with lim n→∞ h(n) = 0. we denote the set of all such sequences by aap(z). definition 1.5. a set of sequences fλ : z → r, λ belongs to some index set λ, is called equi-almost periodic if for every ε > 0, ! λ∈λ p(ε, fλ) is relatively dense in z. next, let us recall some basic notations and properties about impulsive almost periodic type functions (for more details, we refer the reader to [9, 11, 6]). let t be the set of all sequences {tk}k∈z ⊂ r satisfying tk < tk+1 for all k ∈ z, inf k∈z |tk+1−tk| > 0, and lim k→+∞ tk = +∞, lim k→−∞ tk = −∞. it is easy to see that for every t = {tk}k∈z ∈ t and a ∈ r, there holds t + a ∈ t . in addition, we denote by pt c(r) the set of all functions f : r → r such that f is continuous on r \ t, and for every tk ∈ t, f(tk − 0) = f(tk) and f(tk + 0) exists. definition 1.6. let t = {tk}k∈z ∈ t . a function f ∈ pt c(r) is said to be almost periodic if (i) the set of sequences {tj}j∈z is equi-almost periodic, where tj = {t j k : t j k = tk+j − tk, k ∈ z} for every j ∈ z; (ii) for every ε > 0, there exists δ > 0 such that if t′ and t′′ belong to the same interval of continuity of f and |t′ − t′′| < δ, then |f(t′) − f(t′′)| < ε; (iii) for every ε > 0, there exists a relatively dense set p(ε, f) in r such that |f(t + r) − f(t)| < ε for every r ∈ p(ε, f) and every t ∈ r with |t − tk| > ε for all k ∈ z. we denote the set of all such functions by pt ap(r). 18 peng chen, hui-sheng ding, gaston m. n’guérékata cubo 18, 1 (2016) lemma 1.7. let t = {tk}k∈z ∈ t , and f, g ∈ pt ap(r). then, the following assertions hold true: (i) f + g ∈ pt ap(r) and f · g ∈ pt ap(r). (ii) f/g ∈ pt ap(r) provided that inf t∈r |g(t)| > 0. (iii) k → f(tk) belongs to ap(z). (iv) pt ap(r) is a banach space under the supremum norm. (v) for every a ∈ r, f(· − a) ∈ pt+aap(r). proof. (i)-(iii) have been proved in [9]. moreover, by definition of pt ap(r), it is not difficult to prove (iv) and (v). here, we omit the details. remark 1.8. it is not difficult to show that a continuous function f ∈ pt ap(r) implies that f ∈ ap(r). we denote the set of all functions f ∈ pt c(r) with lim t→∞ f(t) = 0 by pt c0(r). next, let us introduce the notion of impulsive asymptotically almost periodic functions. definition 1.9. let t = {tk}k∈z ∈ t . a function f ∈ pt c(r) is said to be be asymptotically almost periodic if f = g + h, where g ∈ pt ap(r) and h ∈ pt c0(r). we denote the set of all such functions by pt aap(r). lemma 1.10. let t = {tk}k∈z ∈ t . the following assertions hold true: (i) let f = g + h ∈ pt aap(r), where g ∈ pt ap(r) and h ∈ pt c0(r). then {g(t) : t ∈ r} ⊂ {f(t) : t ∈ r}. (ii) the decomposition of a function f ∈ pt aap(r) is unique. (iii) pt aap(r) is a banach space under the supremum norm. (iv) let f1, f2 ∈ pt aap(r). then f1 + f2 ∈ pt aap(r), f1 · f2 ∈ pt aap(r) and f1(· − a) ∈ pt+aaap(r) for every a ∈ r. (v) let f1, f2 ∈ pt aap(r). then f1/f2 ∈ pt aap(r) provided that inf t∈r |f2(t)| > 0. (vi) let f ∈ pt aap(r). then k → f(tk) belongs to aap(z). proof. (i) since g is left continuous on r, it suffices to prove that {g(t) : t ∈ r, t /∈ t} ⊂ {f(t) : t ∈ r}. cubo 18, 1 (2016) positive asymptotically almost periodic solutions of an impulsive . . . 19 we prove it by contradiction. assume that there exists t′ /∈ t such that ε0 := inf t∈r |g(t′) − f(t)| > 0. let δ = min { inf k∈z |t′−tk| 2 , ε0 2 } . then, δ > 0. it follows from g ∈ pt ap(r) that there exists l > 0 such that for every n ∈ n, there is rn ∈ [n − t′, n − t′ + l] such that |g(t′ + rn) − g(t ′)| < δ ≤ ε0 2 . combing this with h ∈ pt c0(r), we have ε0 ≤ |g(t′) − f(t′ + rn)| ≤ |g(t′) − g(t′ + rn)| + |h(t′ + rn)| ≤ ε0 2 + |h(t′ + rn)| → ε0 2 , which is a contradiction. this completes the proof. (ii) it suffices to show that 0 has unique decomposition. in fact, letting 0 = g+h ∈ pt aap(r), where g ∈ pt ap(r) and h ∈ pt c0(r), it follows from (i) that g = 0 and thus h = 0. (iii) note that pt c(r), pt c0(r) and pt ap(r) are all banach spaces under the supremum norm. let {fn} ⊂ pt aap(r) be a cauchy sequence, and fn = gn + hn, where gn ∈ pt ap(r) and hn ∈ pt c0(r). then, it follows from (i) that {gn} and {hn} are both cauchy sequences. the remaining proof follows easily. (iv) the proof follows from (i) and (v) of lemma 1.7 and the boundedness of every function in pt ap(r) (or pt c0(r)). (v) it suffices to prove that 1/f ∈ pt aap(r) if f ∈ pt aap(r) with inf t∈r |f(t)| > 0. let f = g + h ∈ pt aap(r), where g ∈ pt ap(r) and h ∈ pt c0(r). moreover, inf t∈r |f(t)| > 0. by (i), there holds inf t∈r |g(t)| > 0. the remaining proof follows from (ii) of lemma 1.7 and 1 f = 1 g − h fg . (vi) the proof follows from (iii) of lemma 1.7 and lim k→∞ tk = ∞. 2 linear inhomogenous equation throughout the rest of this paper, if there is no special statement, we assume that t = {tk}k∈z ∈ t and the set of sequences {tj}j∈z is equi-almost periodic, where tj = {t j k : t j k = tk+j − tk, k ∈ z} for every j ∈ z. by [9, lemma 22], the following limit exists: lim t−s→+∞ i(s, t) t − s , 20 peng chen, hui-sheng ding, gaston m. n’guérékata cubo 18, 1 (2016) where i(s, t) is the number of the terms of t ∩ [s, t]. we denote p = lim t−s→+∞ i(s, t) t − s . now, let us first consider the following linear inhomogenous equation: ⎧ ⎪⎨ ⎪⎩ x′(t) = −a(t)x(t) + f(t), t ̸= tk, ∆x|t=tk = ckx(tk) + ik, k ∈ z, (2.1) where a ∈ ap(r), ck is an almost periodic sequence, and ik is an asymptotically almost periodic sequence. moreover, let t ′ ∈ t with t ⊂ t ′ and f ∈ pt ′aap(r). denote m(a) = lim t→+∞ 1 t ∫t 0 a(t)dt, β = sup k∈z |1 + ck|, and x(t, s) = ⎧ ⎪⎪⎨ ⎪⎪⎩ exp(− ∫t s a(u)du), tk−1 < s ≤ t ≤ tk, k∏ i=m (1 + ci) · exp(− ∫t s a(u)du), tm−1 < s ≤ tm ≤ tk < t ≤ tk+1. definition 2.1. let t = {tk}k∈z ∈ t , t ′ ∈ t with t ⊂ t ′ and f ∈ pt ′c(r). we call that x is a global solution of equation (2.1) if ⎧ ⎪⎨ ⎪⎩ x′(t) = −a(t)x(t) + f(t), t /∈ t ′, ∆x|t=tk = ckx(tk) + ik, k ∈ z. (2.2) we have the following results about equation (2.1): theorem 2.2. let m(a) > p · lnβ. then, equation (2.1) has a unique global solution x in pt aap(r). moreover, we have x(t) = ∫t −∞ x(t, s)f(s)ds + ∑ tk 0 such that |x(t, s)| ≤ me−ω(t−s) for all t, s ∈ r with t ≥ s. cubo 18, 1 (2016) positive asymptotically almost periodic solutions of an impulsive . . . 21 it suffices to prove the above inequality for t − s being sufficiently large. let α ∈ (0, m(a)) and q > p be such that ω := α − q lnβ > 0. it follows that α(t − s) ≤ ∫t s a(u)du, i(s, t) ≤ q(t − s), for all t, s ∈ r with t − s being sufficiently large. then, we have |x(t, s)| = ⎧ ⎪⎪⎨ ⎪⎪⎩ " " " e− ∫ t s a(u)du " " " ≤ e−α(t−s), tk−1 < s ≤ t ≤ tk " " " " k∏ i=m (1 + ci) · e− ∫ t s a(u)du " " " " ≤ βq(t−s) · e−α(t−s) = e−ω(t−s), tm−1 < s ≤ tm ≤ tk < t ≤ tk+1. step 2. x is a global solution of equation (2.1). by step 1 and direct calculations, one can obtain x′(t) = −a(t)x(t) + f(t), t /∈ t ′. moreover, it is not difficult to verify that ∆x|t=tk = ckx(tk) + ik for k ∈ z. in addition, for every t ∈ t ′\t, there holds x′+(t) = −a(t)x(t) + f(t + 0), x ′ −(t) = −a(t)x(t) + f(t). step 3. x ∈ pt aap(r). let f = g + h, where g ∈ pt ′ap(r) and h ∈ pt ′c0(r). moreover, let ik = jk + lk, where k → jk belongs to ap(z) and k → lk belongs to c0(z). then, we have x(t) = ∫t −∞ x(t, s)f(s)ds + ∑ tk 1. we first list some assumptions. (h1) a ∈ ap(r), and ck is an almost periodic sequence with ck ≥ −1 for all k ∈ z. (h2) b ∈ pt aap(r) is nonnegative, and for every x ∈ r+, k → ik(x) is a nonnegative asymptotically almost periodic sequence. moreover, there exists a constant l > 0 such that for all x, y ∈ r+ and k ∈ z, there holds |ik(x) − ik(y)| ≤ l|x − y|. (h3) there exist m,ω > 0 such that |x(t, s)| ≤ me−ω(t−s) for all t, s ∈ r with t ≥ s. (h4) m∥b∥ ω · n 2−1 4n n # n+1 n−1 + ml 1−e−ωθ < 1, where θ = inf k∈z |tk+1 − tk|. remark 3.1. it follows from step 1 of the proof for theorem 2.2 that (h3) holds if m(a) > p·lnβ. before presenting our results, we need to clarify that our definition of solution for equation (1.2) has a slight difference with the classical definition of solution for equation (1.2). definition 3.2. we call that a function x ∈ pt c(r) is a global solution of equation (1.2) if ⎧ ⎪⎨ ⎪⎩ x′(t) = −a(t)x(t) + b(t) 1+xn(t−τ) , t ̸= tk, t ̸= tk + τ, ∆x|t=tk = ckx(tk) + ik(x(tk)), k ∈ z. moreover, we need to recall a gronwall inequality to discuss the stability. lemma 3.3. [9, lemma 2] let u ∈ pt c(r) be nonnegative, s ∈ r and for all t ≥ s, u(t) ≤ c + ∫t s γu(x)dx + ∑ s≤tk 0 are all constants. then, there exists a constant c′ > 0 such that for all t ≥ s, u(t) ≤ c′(1 + β)i(s,t)eγ(t−s). theorem 3.4. assume that (h1)-(h4) hold. then, equation (1.2) has a unique nonnegative asymptotically almost periodic solution. moreover, the asymptotically almost periodic solution of equation (1.2) is exponentially stability provided that p ln(1 + lm) + n2 − 1 4n n $ n + 1 n − 1 meωτ∥b∥ < ω. (3.1) proof. let ϕ ∈ pt aap(r) with inf t∈r ϕ(t) ≥ 0. consider ⎧ ⎪⎨ ⎪⎩ x′(t) = −a(t)x(t) + b(t) 1+ϕn(t−τ) , t ̸= tk, ∆x|t=tk = ckx(tk) + ik(ϕ(tk)), k ∈ z. (3.2) by lemma 1.10, we have b(·) 1 + ϕn(· − τ) ∈ pt∪(t+τ)aap(r). again by lemma 1.10, we get k → ϕ(tk) belongs to aap(z). then, by (h2), it is not difficult to show that k → ik(ϕ(tk)) belongs to aap(z). now, by theorem 2.2, we know that for every ϕ ∈ pt aap(r) with inf t∈r ϕ(t) ≥ 0, equation (3.2) has a unique global solution xϕ ∈ pt aap(r), which satisfies xϕ(t) = ∫t −∞ x(t, s) b(s) 1 + ϕn(s − τ) ds + ∑ tk 0, x(0) = µx′(0), x′(+∞) = x′′(+∞) = 0 where µ ≥ 0. by using the upper and lower solution approach and the fixed point theory, the existence of positive solutions is proved under a monotonic condition on f. the nonlinearity f may be singular at x = 0. an example of application is included to illustrate the main existence result. resumen en este art́ıculo investigamos la existencia de una solución positiva de una clase de problema singular de valores de frontera de tercer-orden asociado con el operador φlaplaciano y colocado sobre la semirecta real positiva: { (φ(−x′′))′(t) + f(t, x(t)) = 0, t > 0, x(0) = µx′(0), x′(+∞) = x′′(+∞) = 0 donde µ ≥ 0. usando la técnica de sub y súper soluciones y la teoŕıa del punto fijo, se prueba la existencia de soluciones positivas bajo una condición de monotonicidad sobre f. la nolinealidad f puede ser singular en x = 0. se incluye un ejemplo de aplicación para ilustrar el resultado principal de existencia. keywords and phrases: third order, half-line, φ−laplacian, singular problem, positive solution, fixed point, upper and lower solution. 2010 ams mathematics subject classification: 34b15, 34b18, 34b40, 47h10.. 106 smäıl djebali & ouiza saifi cubo 16, 1 (2014) 1 introduction this paper is concerned with the existence of positive solutions to the following third-order boundary value problem posed on the half-line and associated with a φ−laplacian operator:    (φ(−x′′))′(t) + f(t, x(t)) = 0, t > 0, x(0) = µx′(0), x′(+∞) = x′′(+∞) = 0, (1) where µ ≥ 0 and f = f(t, x) : r+ × (0, +∞) −→ r+ is a continuous function which may have space singularity at x = 0 and r+ = [0, +∞). the map φ : r −→ r is a continuous, increasing homeomorphism such that φ(0) = 0 (for instance the p−laplacian ϕp(s) = |s| p−1s, p > 1). boundary value problems (bvps for short) on the half-line appear in many applied problems relating to various phenomena in physics, biology, and combustion theory (see, e.g., [1] and references therein). in the last couple of years, the mathematical investigation of such problems, especially second-order boundary value problems have attracted several authors (see, e.g., [4], [5], [6], [7], [8], [9] and the references therein). however, only some of them were interested in higher-order differential equations on [0, +∞) (see [9], [11], [12]). the aim of this work to study a third-order differential equation with a φ-laplacian derivative operator and posed on the positive half-line. our approach is based on the upper and lower solution method adapted to this class of problems combined with the schauder fixed point theorem. this papers essentially consists of three sections. section 2 is devoted to some preliminaries facts and basic notions needed in this paper. a fixed point formulation is also provided in this section. in section 3, we present our existence result of positive solutions when the nonlinearity f is monotonic with respect to x but may be singular at x = 0. the case f is not singular at x = 0 is also considered with less hypotheses. our existence theorem is illustrated by means of an example of application. a function x is said to be a solution of problem (1) if x ∈ x = {x | x ∈ c2((0, ∞), r)} and φ(−x′′) ∈ c1((0, ∞), r)} (2) and satisfies (1). in addition, x said to be a positive solution if x(t) > 0 for t ∈ (0, +∞). 2 auxiliary lemmas a mapping defined on a banach space is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets. let cl([0, ∞), r) = {x ∈ c([0, ∞), r) | lim t→+∞ x(t) exists}. for x ∈ cl([0, ∞), r), define ‖x‖l = sup t∈r+ |x(t)|· then (cl, ‖x‖l) is a banach space. cubo 16, 1 (2014) upper and lower solutions for φ−laplacian . . . 107 lemma 2.1. ([3], p. 62) let m ⊆ cl(r +, r). then m is relatively compact in cl(r +, r) if the following three conditions hold: (a) m is uniformly bounded in cl(r +, r). (b) the functions belonging to m are almost equicontinuous on r+, i.e., equicontinuous on every compact interval of r+. (c) the functions from m are equiconvergent, that is, for every ε > 0, there exists t(ε) > 0 such that |x(t) − x(+∞)| < ε for any t ≥ t(ε) and x ∈ m. note that the space e = {x ∈ c([0, ∞), r) | lim t→+∞ x(t) 1 + t exists}. (3) is also a banach space with the norm ‖x‖ = sup t∈r+ |x(t)| 1+t . from lemma 2.1, we easily deduce lemma 2.2. let m ⊆ e. then m is relatively compact in e if the following conditions hold: (a) m is bounded in e, (b) the functions belonging to {u| u(t) = x(t) 1+t , x ∈ m} are locally equicontinuous on [0, +∞), (c) the functions belonging to {u| u(t) = x(t) 1+t , x ∈ m} are equiconvergent at +∞. definition 2.3. let α, β ∈ x. then α is called a lower solution of (1) if α satisfies    (φ(−α′′(t)))′ + f(t, α(t)) ≥ 0, t > 0 α(0) ≤ µα′(0), lim t→+∞ α′(t) ≤ 0, lim t→+∞ α′′(t) ≥ 0. β is called an upper solution of (1) if the above inequalities are reversed. let g(t, s) = { s + µ, 0 ≤ s ≤ t < +∞ t + µ, 0 ≤ t ≤ s < +∞, be the green function of the linear problem −x′′ = x(0) − µx′(0) = x′(+∞) = 0. the following lemmas are straightforward; the proofs are omitted. lemma 2.4. assume that δ ∈ c(r+, r+) satisfies ∫+∞ 0 δ(s)ds < +∞ and let x(t) = ∫+∞ 0 g(t, s)δ(s)ds. then    x′′(t) + δ(t) = 0, t > 0, x(0) = µx′(0), lim t→+∞ x′(t) = 0. (4) lemma 2.5. assume that δ ∈ c(r+, r+) ∩ l1(r, +∞) for all r > 0 and ∫+∞ 0 φ−1 (∫+∞ s δ(τ)dτ ) ds < +∞. 108 smäıl djebali & ouiza saifi cubo 16, 1 (2014) if x(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s δ(τ)dτ ) ds, then x ∈ x and    (φ(−x′′(t)))′ + δ(t) = 0, t > 0, x(0) = µx′(0), x′(+∞) = x′′(+∞) = 0. (5) consider the positive cone s = {x ∈ c1[0, +∞) | x(t) ≥ 0, concave on [0, +∞), lim t→+∞ x′(t) = 0}. (6) in addition to the null function, s contains, e.g., ln(1 + t), so s is a nonempty subset; moreover s has the following properties: lemma 2.6. let x ∈ s \ {0}. then there exists a positive constant λx such that (a) for all θ > 1, x(t) ≥ λx 1 θ , ∀ t ∈ [1/θ, θ], (b) if ρ(t) = { t, t ∈ [0, 1] 1, t ≥ 1 (7) then x(t) ≥ λxρ(t), ∀ t ∈ [0, +∞). proof. (a) notice that every x ∈ s is nondecreasing and thus by l’hopital’s rule lim t→+∞ x(t) 1+t = 0; as a consequence, the function x(t) 1+t achieves its maximum at some point tm ∈ [0, +∞); let λx = x(tm) 1+tm = ‖x‖ > 0. by concavity of x, we have for t ∈ [1/θ, θ] x(t) ≥ min t∈[ 1 θ ,θ] x(t) = x( 1 θ ) = x(θ−1+θtm θ+θtm 1 θ−1+θtm + 1 θ+θtm tm) ≥ θ−1+θtm θ+θtm x( 1 θ−1+θtm ) + 1 θ+θtm x(tm) ≥ 1 θ+θtm x(tm) = 1 θ x(tm) 1+tm = λx 1 θ , whence the first part of the lemma. (b) fix t0 ∈ [0, +∞) and distinguish between four cases. (i) if t0 = 0, then x(0) ≥ 0 = λxρ(0). (ii) if t0 ∈ (0, 1), then 1 t0 ∈ (1, +∞). from part (a), x(s) ≥ λxt0, ∀ s ∈ [t0, 1t0 ]. in particular for s = t0, x(t0) ≥ λxt0 = λxρ(t0). cubo 16, 1 (2014) upper and lower solutions for φ−laplacian . . . 109 (iii) if t0 = 1, let {tn}n be a real sequence such that 0 < tn < 1 and tn → 1, as n → +∞. by (ii), we know that x(tn) ≥ λxtn, ∀ n ≥ 1. then x(1) = lim n→+∞ x(tn) ≥ λx lim n→+∞ tn = λx = λxρ(1). (iv) finally, let t0 ∈ (1, +∞), since x is nondecreasing, then x(t0) ≥ x(1) ≥ λx = λxρ(t0), ending the proof of the lemma. 3 main existence results first we list some assumptions: (h1) f ∈ c(r + × (0, +∞), r+) and f(t, x) is a nonincreasing relatively to the second argument. (h2) for every λ > 0, ∫+∞ 0 f(τ, λρ(τ))dτ < +∞, ∫+∞ 0 φ−1 (∫+∞ s f(τ, λρ(τ))dτ ) ds < +∞. (h3) there exists a function a ∈ s \ {0} such that for t ≥ 0    b(t) := ∫+∞ 0 g(t, s)φ−1 (∫+∞ s f(τ, a(τ))dτ ) ds ≥ a(t), ∫+∞ 0 g(t, s)φ−1 (∫+∞ s f(τ, b(τ))dτ ) ds ≥ a(t). for x ∈ s \ {0}, define a fixed point operator t by tx(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s f(τ, x(τ))dτ ) ds. we have lemma 3.1. assume (h1)-(h2) holds. then the operator t maps s \ {0} into x ∩ s. in addition    (φ(−(tx)′′))′(t) + f(t, x(t)) = 0, t > 0, (tx)(0) = µ(tx)′(0), (tx)′(+∞) = (tx)′′(+∞) = 0. (8) proof. (a) for λ > 0, let fλ(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s f(τ, λρ(τ))dτ ) ds, then lim t→+∞ fλ(t) 1 + t = 0. 110 smäıl djebali & ouiza saifi cubo 16, 1 (2014) indeed, using the convergence of the second integral in (h2), we get lim t→+∞ f′λ(t) = lim t→+∞ ∫+∞ t φ−1 (∫+∞ s f(τ, λρ(τ))dτ ) ds = 0, then fλ is nondecreasing and lim t→+∞ fλ(t) 1 + t =    0, if lim t→+∞ fλ(t) < ∞, lim t→+∞ f′λ(t) = 0, if lim t→+∞ fλ(t) = ∞. (b) given x ∈ s \ {0}, by lemma 2.6, there exists λx > 0 such that x(t) ≥ λxρ(t), t ∈ r +. by (h1), (h2), and part (a), we have tx(t) 1+t = ∫ +∞ 0 g(t,s)φ −1( ∫ +∞ s f(τ,x(τ))dτ)ds 1+t ≤ ∫ +∞ 0 g(t,s)φ −1( ∫ +∞ s f(τ,λxρ(τ))dτ)ds 1+t = fλx (t) 1+t . hence lim t→+∞ tx(t) 1+t = 0. then tx ∈ e and even tx ∈ x ∩ s. indeed tx(t) ≥ 0, (tx)′(t) = ∫+∞ t φ−1 (∫+∞ s f(τ, x(τ))dτ ) ds =⇒ lim t→+∞ (tx)′(t) = 0, (tx)′′(t) = −φ−1 (∫+∞ t f(τ, x(τ))dτ ) ≤ 0, and thus (8) is satisfied. now we state and prove our main existence result: theorem 3.2. assume that assumptions (h1) − (h3) hold. then the boundary value problem (1) has at least one positive solution x ∈ x which satisfies x(t) ≥ λ0ρ(t) for some λ0 > 0. proof. the proof is be split into three steps. step 1. we first determine appropriate upper and lower solution for the bvp (1). since a ∈ s\{0} and b(t) = ta(t), then by (h3) and lemma 3.1, we have b, tb ∈ s \ {0}. moreover t being nonincreasing relatively to x, we have a ≤ b ⇒ a ≤ tb ≤ ta = b. (9) therefore, for all t > 0    (φ(−(tb)′′))′(t) + f(t, tb(t)) ≥ (φ(−(tb)′′))′(t) + f(t, b(t)) = 0 (tb)(0) = µ(tb)′(0), (tb)′(+∞) = 0, (tb)′′(+∞) = 0 (10) cubo 16, 1 (2014) upper and lower solutions for φ−laplacian . . . 111 and    (φ(−(ta)′′))′(t) + f(t, ta(t)) ≤ (φ(−(ta)′′))′(t) + f(t, a(t)) = 0, (ta)(0) = µ(ta)′(0), (ta)′(+∞) = 0, (ta)′′(+∞) = 0. (11) the functions α(t) = tb(t) and β(t) = ta(t) are lower and upper solution of the bvp (1), respectively with α ≤ β. step 2. we claim that the following regular modified boundary value problem    (φ(−x′′))′(t) + f∗(t, x(t)) = 0, t > 0, x(0) = µx′(0), x′(+∞) = x′′(+∞) = 0 (12) has a positive solution, where f∗(t, x) =    f(t, α), x < α(t), f(t, x), α(t) ≤ x ≤ β(t), f(t, β), x > β(t), (13) to see this, consider the operator a : e → e defined by ax(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s f∗(τ, x(τ))dτ ) ds. it is clear that a fixed point of the operator a is a solution of the boundary value problem (12). since α ∈ s \ {0}, by lemma 2.6 (b), there exists a positive constant λα such that α(t) ≥ λαρ(t), ∀ t ∈ r +. moreover f(t, x) being nonincreasing in x, we have f∗(t, x) ≤ f(t, α(t)) ≤ f(t, λαρ(t)) (14) for all positive t. (a) a(e) ⊆ e. for x ∈ e and t ∈ r+, we have, using (14) ax(t) 1+t = ∫ +∞ 0 g(t,s)φ −1 ( ∫ +∞ s f ∗ (τ,x(τ))dτ)ds 1+t ≤ ∫ +∞ 0 g(t,s)φ −1 ( ∫ +∞ s f(τ,λαρ(τ))dτ)ds 1+t = fλα (t) 1+t , hence lim t→+∞ ax(t) 1+t = 0 and a(e) ⊆ e. (b) a is continuous. let some sequence {xn}n≥1 ⊆ e be such that lim n→+∞ xn = x0 ∈ e. then 112 smäıl djebali & ouiza saifi cubo 16, 1 (2014) we have ‖axn − ax0‖ = sup t∈r+ |axn(t)−ax0(t)| 1+t = sup t∈r+ ∫+∞ 0 g(t,s) 1+t |φ−1( ∫+∞ s f∗(τ, xn(τ))dτ) − φ −1( ∫+∞ s f∗(τ, x0(τ))dτ)|ds ≤ max(1, µ) ∫+∞ 0 φ−1( ∫+∞ s f∗(τ, xn(τ))dτ) − φ −1( ∫+∞ s f∗(τ, x0(τ))dτ)|ds. since ∣ ∣ ∣ φ−1 (∫+∞ s f∗(τ, xn(τ))dτ ) − φ−1 (∫+∞ s f∗(τ, x0(τ))dτ ) ∣ ∣ ∣ ≤ 2φ−1 (∫+∞ 0 f(τ, λαρ(τ) ) dτ, then the continuity of f∗, φ−1, (h2) and the lebesgue dominated convergence theorem, we deduce ‖axn − ax0‖ −→ 0, as n −→ +∞ (c) a(e) is relatively compact. indeed (i) a(e) is uniformly bounded. for x ∈ e, we have ‖ax‖ = sup t∈r+ |ax(t)| 1+t ≤ sup t∈r+ ∫+∞ 0 g(t,s) 1+t φ−1 (∫+∞ s f∗(τ, x(τ))dτ) ) ds ≤ max(1, µ) ∫+∞ 0 φ−1 (∫+∞ s f∗(τ, x(τ))dτ) ) ds ≤ max(1, µ) ∫+∞ 0 φ−1 (∫+∞ s f(τ, λαρ(τ))dτ) ) ds < +∞. (ii) { a(e) 1+t } is almost equicontinuous. for a given t > 0, x ∈ e, and t, t′ ∈ [0, t] (t > t′), we have ∣ ∣ ∣ ax(t) 1+t − ax(t ′ ) 1+t′ ∣ ∣ ∣ ≤ ∫+∞ 0 ∣ ∣ ∣ g(t,s) 1+t − g(t ′ ,s) 1+t′ ∣ ∣ ∣ φ−1 (∫+∞ s f∗(τ, x(τ))dτ) ) ds ≤ ∫t 0 ∣ ∣ ∣ g(t,s) 1+t − g(t ′ ,s) 1+t′ ∣ ∣ ∣ φ−1 (∫+∞ s f(τ, λαρ(τ))dτ) ) ds + ∣ ∣ ∣ t+µ 1+t − t ′ +µ 1+t′ ∣ ∣ ∣ ∫+∞ t φ−1 (∫+∞ s f(τ, λαρ(τ))dτ) ) ds, cubo 16, 1 (2014) upper and lower solutions for φ−laplacian . . . 113 then by (h2), for any ε > 0 and t > 0, there exists δ > 0 such that ∣ ∣ ∣ ax(t) 1+t − ax(t ′ ) 1+t′ ∣ ∣ ∣ < ε for all t, t′ ∈ [0, t] with |t − t′| < δ. hence { a(e) 1+t } are almost equicontiuous. (iii) { a(e) 1+t } is equiconvergent at +∞. since lim t→+∞ ax(t) 1+t = 0, then by (h2) we have lim t→+∞ sup x∈e | ax(t) 1+t − lim t→+∞ ax(t) 1+t | = lim t→+∞ sup x∈e ∫ +∞ 0 g(t,s)φ −1( ∫ +∞ s f ∗ (τ,x(τ))dτ))ds 1+t ≤ lim t→+∞ ∫ +∞ 0 g(t,s)φ −1( ∫ +∞ s f(τ,λαρ(τ))dτ))ds 1+t = lim t→+∞ fλα (t) 1+t = 0. lemma 2.2 guarantees that a(e) is relatively compact. finally by the schauder fixed point theorem (see, e.g., [2]), the operator a has at least one fixed point x ∈ e, which is further in x by lemma 3.1, solution of the bvp (12). step 3. next we will prove that the boundary value problem (1) has at least one positive solution. for this, we only need to check that α(t) ≤ x(t) ≤ β(t), ∀t ∈ r+. since x is a solution of the bvp (12) x(0) = µx′(0), lim t→+∞ x′(t) = lim t→+∞ x′′(t) = 0 (15) in addition, f(t, x) is nonincreasing in x f(t, β(t)) ≤ f∗(t, x) ≤ f(t, α(t)), ∀ t ∈ r+. (16) it follows from (9) and (h3) that f(t, b(t)) ≤ f∗(t, x) ≤ f(t, a(t)), ∀ t ∈ r+. (17) since a ∈ s \ {0}, by lemma 3.1 (φ(−β′′(t)))′ = (φ(−ta)′′(t)))′ = −f(t, a(t)), ∀ t ∈ r+. these, together with lemma 3.1 (9), (15)-(17) yield    (φ(−β′′(t)))′ − (φ(−x′′(t)))′ = −f(t, a(t)) + f∗(t, x(t)) ≤ 0, t ∈ r+ (β − x)(0) = µ(β − x)′(0), (β − x)′(+∞) = 0, (β − x)′′(+∞) = 0 (18) this implies that the function z defined by z(t) = (φ(−β′′(t))) − (φ(−x′′(t))) is a nonincreasing function in r+. moreover z(+∞) = 0 implies z(t) ≥ 0, ∀ t ≥ 0 and then (β − x)′′(t) ≤ 0, ∀ t ∈ r+ which means that (β − x)′ is nonincreasing in r+. now (β − x)′(+∞) = 0 then (β − x)′(t) ≥ 0, ∀t ∈ r+ and so β − x is nondecreasing on r+. finally the boundary condition (β − x)(0) = µ(β − x)′(0) ≥ 0 implies that x(t) ≤ β(t), for 114 smäıl djebali & ouiza saifi cubo 16, 1 (2014) all t ∈ r+. in a similar way, we can prove that x(t) ≥ α(t), for all t ∈ r+. therefore, x is a solution of the bvp (1). in addition, there existence of a positive constant λ0 = λα such that x(t) ≥ α(t) ≥ λ0ρ(t), ∀ t ∈ r +. the proof of theorem 3.2 is completed. however, when f(t, x) is nonsingular at x = 0, i.e. f : r+ × r+ −→ r+ is a continuous function, then for all x ≥ 0, f(t, x) ≤ f(t, 0). in this case, we have theorem 3.3. assume that assumption (h1) holds and (h2) ′ 0 < ∫+∞ 0 f(τ, 0)dτ < +∞ and ∫+∞ 0 φ−1 (∫+∞ s f(τ, 0)dτ ) ds < +∞. then the bvp (1) has at least one positive solution x ∈ x such that x(t) ≥ λ0ρ(t) for some λ0 > 0. the proof is similar to that of theorem 3.2. we only check that t(s) ⊂ s ∩ x and if we take a(t) = 0, ∀ t ≥ 0, then condition (h3) holds. finally the condition (h2) ′ implies that β = ta = b, α = tb belong to s \ {0}. example 3.4. consider the singular boundary value problem    (φ(−x′′(t)))′ + e−tm(t)g(x(t)) = 0, x(0) = µx′(0), lim t→+∞ x′(t) = lim t→+∞ x′′(t) = 0, (19) where 0 ≤ µ ≤ 8 3 , φ(x) = x 1 3 , f(t, x) = e−tm(t)g(x), g(x) = { 1 x , x ∈ (0, 1] 1, x ≥ 1, and m(t) = { t3, t ∈ [0, 1] 1 t2 , t ≥ 1, then, we have (h1) f ∈ c((0, +∞) × r+, r+) and f(t, x) is a nonincreasing with respect to x for every positive t. (h2) for all λ > 0, ∫+∞ 0 f(τ, λρ(τ))dτ ≤ max{1, 1 λ } < +∞, and ∫+∞ 0 φ−1 (∫+∞ s f(τ, λρ(τ))dτ ) ds < +∞. cubo 16, 1 (2014) upper and lower solutions for φ−laplacian . . . 115 (h3) let a0(t) = 1, then a0 ∈ s and if we put a(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)g(a0(τ))dτ ) ds, then a(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)dτ ) ds. moreover for all t ∈ r+ a(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)dτ ) ≤ ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τdτ ) ≤ ∫+∞ 0 (s + 8/3)φ−1(e−s)ds ≤ 1 = a0(t). hence b(t) = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)g(a(τ))dτ ) ds ≥ ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)g(a0(τ))dτ ) ds = a(t). finally, since g ≥ 1, we have ∫+∞ 0 g(t, s)φ−1 (∫+∞ s f(τ, b(τ))dτ ) ds = ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)g(b(τ))dτ ) ds ≥ ∫+∞ 0 g(t, s)φ−1 (∫+∞ s e−τm(τ)dτ ) ds = a(t). then all conditions of theorem 3.2 are fulfilled which guarantees that the bvp (19) has at least one positive solution. received: may 2013. accepted: february 2014. references [1] r.p. agarwal and d. o’regan, infinite interval problems for differential, difference and integral equations, kluwer academic publisher, dordrecht, 2001. [2] r.p. agarwal, m. meehan, and d. o’regan, fixed point theory and applications, cambridge tracts in mathematics, vol. 141, cambridge university press, 2001. [3] c. corduneanu, integral equations and stability of feedback systems, academic press, new york, 1973. 116 smäıl djebali & ouiza saifi cubo 16, 1 (2014) [4] s. djebali and k. mebarki, multiple positive solutions for singular bvps on the positive halfline, comput. math. appl. 55(12) (2008) 2940–2952. [5] s. djebali and k. mebarki, multiple unbounded positive solutions for three-point bvps with sign-changing nonlinearities on the positive half-line, acta appl. math. 109(2) (2010) 361– 388 [6] s. djebali and o. saifi, positive solutions for singular φ−laplacian bvps on the positive half-line, e.j.q.t.d.e. 56 (2009) 1–24. 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() cubo a mathematical journal vol.13, no¯ 02, (119–126). june 2011 degree theory for the sum of vmo maps and maximal monotone maps yuqing chen faculty of applied mathematics, guangdong university of technology, guangdong 510006, p. r. china, email: ychen64@163.com donal o’regan department of mathematics, national university of ireland, galway, ireland, email: donal.oregan@nuigalway.ie and ravi p. agarwal department of mathematical science, florida institute of technology melbourne, fl, 32901, usa, email: agarwal@fit.edu abstract let ω ⊂ rn be an open bounded domain, f : ω → rn a vmo map, and t : d(t ) ⊆ rn → rn a maximal monotone map with d(t ) ∩ ω 6= ∅. we construct a degree for the sum of f + t , which can be viewed as a generalization of the degree both for vmo maps and maximal monotone maps. 120 yuqing chen, donal o’regan & ravi p. agarwal cubo 13, 2 (2011) resumen sea ω ⊂ rn un dominio abierto, f : ω → rn un mapa vmo, y t : d(t ) ⊆ rn → rn un mapa monotono maximal con d(t ) ∩ ω 6= ∅. construimos un grado por la suma de f + t , que se puede ver como una generalización de la medida, tanto para los mapas de vmo y para los mapas monotono maximal. keywords and phrases: degree theory, maximal monotone map. mathematics subject classification: 47h11, 47h05 1. introduction degree theory for continuous maps in finite dimensional spaces has a long history and has been extensively studied. in the early 80’s of the last century a degree for some classes of noncontinuous maps was established (see [8,1,17,18] and the references therein). in 1995 and 1996, h. brezis and l. nirenberg [12], [13] invented a degree theory for vmo maps; see [2-6,9-11,19,21,22]. generally, vmo functions need not be continuous. another important class of non-continuous maps is the class of maximal monotone maps, and there is no relation between the vmo maps and the maximal monotone maps. in this paper, we consider the sum of a vmo map and a maximal monotone map, and we will define a degree theory for such a map. first we recall some definitions. let ω be an open bounded domain in rn. the class of bounded mean oscillation functions (see [20]) are defined as bmo(ω) = {f : ω → rn is locally integrable, and |f|bmo < ∞}, where |f|bmo = supb⊂ω 1 m(b) ∫ b |f(x)−f|dx, f = 1 m(b) ∫ b f(x)dx (here m(·) represents the lebesgue measure), and the class of vanishing mean oscillation functions (see [23]) are defined as vmo(ω) = {f : ω → rn is locally integrable, and limm(b)→ 0 1 m(b) ∫ b |f(x) − f|dx = 0}, where b ⊂ rn is an open ball with its closure contained in ω. it is well known that if f ∈ vmo, then fǫ(x) = 1 m(bǫ(x)) ∫ bǫ(x) f(y)dy is continuous in ǫ and x where it is defined. let t : d(t ) ⊂ rn → rn be a function. if (h − g, x − y) ≥ 0 for all x, y ∈ d(t ) and h ∈ tx, g ∈ ty, then t is said to be monotone. if t is monotone and t has no monotone extension in rn, then t is said to be maximal monotone. it is well known that t is maximal monotone iff t is monotone and t + ǫi is surjective for all ǫ > 0. if t is maximal monotone, we use tǫ = (t −1 + ǫi)−1 to represent the yosida approximation, and rǫ = i − ǫtǫ, the resolvent with respect to tǫ. for maximal monotone maps we refer the reader to [7]. let f : ω → rn be a vmo map, t : d(t ) ⊆ rn → rn a maximal monotone map, p ∈ rn, and d(t ) ∩ ω 6= ∅. under appropriate assumptions, see (2.1) below, we define the degree deg(f + t, ω ∩ d(t ), p). if t = 0, this degree coincides with the degree for vmo cubo 13, 2 (2011) degree theory for the sum of vmo maps and maximal monotone maps 121 maps in [13], and if f = 0, then it coincides with the degree for maximal monotone maps (see [14-16]). 2. results in this section, ω ⊂ rn is an bounded open domain, f ∈ vmo(ω), t : d(t ) ⊆ rn → rn is a maximal monotone map, p ∈ rn, and ω ∩ d(t ) 6= ∅. suppose there exists an open neighborhood u of ∂ω in ω and a constant β > 0 such that 1 m(bǫ(y)) ∫ bǫ(y) |f(x) + g − p|dx ≥ β (2.1) for all 0 < ǫ < 1 2 d(y, ∂ω), g ∈ tz, z ∈ d(t ) ∩ bǫ(y), where bǫ(y) is an open ball centered at y with radius ǫ such that bǫ(y) ⊂ u, and d(y, ∂ω) is the distance between y and ∂ω. we remark that if t = 0, then (2.1) was first used in [13]. if f = 0, then (2.1) is equivalent to |g − p| ≥ β for all z ∈ d(t ) ∩ u and g ∈ tz, and in this case proposition 2.1 below shows that the assumption p /∈ t (∂ω ∩ d(t )) will guarantee (2.1) holds. proposition 2.1. if p /∈ t (∂ω ∩ d(t )), then there exists d0 > 0, α0 > 0 such that d(p, tx) ≥ d0 for all x ∈ ω ∩ d(t ) with d(x, ∂ω) < α0. proof. suppose the conclusion is not true. there exist xn ∈ ω ∩ d(t ), gn ∈ txn such that d(xn, ∂ω) → 0, and gn − p → 0. without loss of generality, we may assume that xn → x0 ∈ ∂ω. since (gn − g, xn − x) ≥ 0 for all x ∈ d(t ), g ∈ tx, we have (p − g, x0 − x) ≥ 0, for all x ∈ d(t ), g ∈ tx. therefore x0 ∈ ∂ω ∩ d(t ), p ∈ tx0, which is a contradiction. as in [13], we define ωǫ = {x ∈ ω : d(x, ∂ω) > 2ǫ} for each ǫ > 0. by definition of vmo functions, there exists ǫ0 > 0 such that 1 m(bǫ(x)) ∫ bǫ(x) |f(y) − f|dy < β 2 (2.2) for all ǫ < ǫ0, x ∈ ω and ǫ < d(x,∂ω) 2 . we may also take ǫ0 such that {x ∈ ω : d(x, ∂ω) ≤ 3ǫ0} ⊂ u, where u is the same as in (2.1). now for 0 < ǫ < ǫ0, and x ∈ ∂ωǫ ∩ d(t ), g ∈ tx, by (2.1) and (2.2), we obtain |fǫ(x) + g − p| ≥ β 2 , (2.3) where fǫ(x) = 1 m(bǫ(x)) ∫ bǫ(x)) f(y)dy. 122 yuqing chen, donal o’regan & ravi p. agarwal cubo 13, 2 (2011) lemma 2.2. suppose |fǫ(x) + g − p| ≥ β 2 , for x ∈ ∂ωǫ ∩ d(t ), g ∈ tx. then there exists λ0(ǫ) > 0 such that p 6= fǫ(x) + tλ(x), for all x ∈ ∂ωǫ, λ ∈ (0, λ0(ǫ)). proof. if this is not true, there exist λn → 0 +, xn ∈ ∂ωǫ with xn → x0 ∈ ∂ωǫ, such that fǫxn + tλn xn = p, n ∈ {1, 2, · · · }. since fǫxn → fǫx0, rλn xn = xn − λntλn xn → x0, the maximal monotonicity of t implies that x0 ∈ d(t ), and p − fǫx0 ∈ tx0, which is a contradiction. now, assume that (2.1) holds. in view of (2.3) and lemma 2.2, we define the degree deg(f + t, ω ∩ d(t ), p) by deg(f + t, ω ∩ d(t ), p) = limǫ→ 0+ limλ→ 0+ deg(fǫ + tλ, ωǫ, p). (2.4) we claim this definition is reasonable. first, for each ǫ < ǫ0, and λ1, λ2 ∈ (0, λ0(ǫ)), since ttλ1+(1−t)λ2 x is continuous in (t, x) (see corollary 2.8 in [15]) we know that {fǫ+ttλ1+(1−t)λ2 }t∈[0,1] is a homotopy, so deg(fǫ + tλ1 , ωǫ, p) = deg(fǫ + tλ2 , ωǫ, p). now, for any ǫ ∈ (0, ǫ0), by the continuity of ft(x) in (t, x) and (2.3), there exists δ > 0 such that |ft(x) + g − p| > β 4 , for |t − ǫ| ≤ δ and x ∈ ∂ωǫ and g ∈ tx. the same proof as in lemma 2.2 guarantees that there exists λ1 > 0 such that p 6= ft(x) + tλ(x), for all x ∈ ∂ωǫ, |t − ǫ| ≤ δ, λ ∈ (0, λ1), so deg(ft + tλ, ωǫ, p) is well defined for λ ∈ (0, λ1), and |t − ǫ| ≤ δ. by homotopy invariance, we have deg(ft + tλ, ωǫ, p) = deg(fǫ + tλ, ωǫ, p), so the degree in (2.4) is well defined. for a measurable function f : ω → rn, we recall that the essential range of f is defined as the smallest closed subset essr(f) such that f(x) ∈ essr(f) a. e. x ∈ ω (see [12]). proposition 2.3. if deg(f + t, ω ∩ d(t ), p) 6= 0, then p ∈ essr(f) + t (ω ∩ d(t )). proof. suppose the conclusion is not true. then exists r > 0 such that b(p, r) ∩ essr(f) + t (ω ∩ d(t )) = ∅. set σ = rn \ (b(p, r) − t (ω ∩ d(t ))). clearly, essr(f) ⊂ σ. also f(x) ∈ essr(f), cubo 13, 2 (2011) degree theory for the sum of vmo maps and maximal monotone maps 123 a. e. x ∈ ω, and f ∈ vmo(ω), so we deduce that limǫ→ 0+ d(fǫ(x), σ) = 0 uniformly. therefore, there exists ǫ1 ∈ (0, ǫ0) such that |fǫ(x) − p + g| ≥ r 2 , for all x ∈ ω, z ∈ d(t ) ∩ ω, g ∈ tz, ǫ ∈ (0, ǫ1). thus deg(fǫ + tλ, ω, p) = 0 for all λ ∈ (0, λ0(ǫ)), and ǫ ∈ (0, ǫ1). consequently, it follows from the definition that deg(f + t, ω ∩ d(t ), p) = 0, which is a contradiction. proposition 2.4. let {ht(·)}t∈[0,1] be a family of functions in vmo(ω), and ht(·) depends continuously on the parameter t in the topology of bmo ∩ l1loc(ω). assume that there exists an open neighborhood u of ∂ω in ω and a constant β > 0 such that 1 m(bǫ(y)) ∫ bǫ(y) |ht(x) + g − p|dx ≥ β (2.5) for all 0 < ǫ < 1 2 d(y, ∂ω), g ∈ tz, z ∈ d(t ) ∩ bǫ(y), t ∈ [0, 1], where bǫ(y) is an open ball centered at y with radius ǫ such that bǫ(y) ⊂ u. then deg(ht + t, ω ∩ d(t ), p) does not depend on t ∈ [0, 1]. proof. since ht(·) depends continuously on the parameter t in the topology of bmo∩l 1 loc(ω), we have limm(b)→ 0 1 m(b) ∫ b |ht(x) − ht| = 0, (2.6) uniformly in t. from (2.5), (2.6), and using the same proof as in (2.3), we know that there exists ǫ0 > 0, such that |ht,ǫ(x) + g − p| ≥ β 2 , (2.7) for all x ∈ ∂ωǫ ∩ d(t ), g ∈ tx, t ∈ [0, 1], ǫ ∈ (0, ǫ0). by using the same proof as in lemma 2.2, we know that there exists λ(ǫ) > 0, such that p 6= ht,ǫ(x) + tλx, for all x ∈ ∂ωǫ, t ∈ [0, 1], λ ∈ (0, λ(ǫ)). thus deg(ht,ǫ + tλ, ωǫ, p) does not depend on t for each ǫ ∈ (0, ǫ0), λ ∈ (0, λ(ǫ)). thus deg(ht + t, ω ∩ d(t ), p) does not depend on t ∈ [0, 1]. corollary 2.5. let f1, f2 ∈ vmo(ω) satisfying (2.1). suppose there exists 0 < β0 < β such that 1 m(b) ∫ b |f1(x) − f2(x)|dx < β0, for all b ⊂ u. then deg(f1 + t, ω ∩ d(t ), p) = deg(f2 + t, ω ∩ d(t ), p). proof. set ht(x) = tf1(x) + (1 − t)f2(x) for t ∈ [0, 1], x ∈ ω. then it is easy to see that ht depends continuous on t in the topology of bmo ∩ l1loc(ω). also we have 1 m(bǫ(y)) ∫ bǫ(y) |ht(x) + g − p|dx ≥ β − β0 124 yuqing chen, donal o’regan & ravi p. agarwal cubo 13, 2 (2011) for all 0 < ǫ < 1 2 d(y, ∂ω), g ∈ tz, z ∈ d(t )∩bǫ(y), t ∈ [0, 1], where bǫ(y) is an open ball centered at y with radius ǫ such that bǫ(y) ⊂ u. therefore the conclusion follows from proposition 2.4. proposition 2.6. let ti : d ⊆ r n, i = 1, 2, be two maximal monotone maps. if tt1 + (1 − t)t2 is maximal monotone for each t ∈ [0, 1], and there exist an open neighborhood u of ∂ω in ω and a constant β > 0 such that 1 m(bǫ(y)) ∫ bǫ(y) |f(x) + gt − p|dx ≥ β (2.8) for all 0 < ǫ < 1 2 d(y, ∂ω), gt ∈ [tt1 + (1 − t)t2]z, z ∈ d ∩bǫ(y)), t ∈ [0, 1], where bǫ(y) is an open ball centered at y with radius ǫ such that bǫ(y) ⊂ u. then deg(f + [tt1 + (1 − t)t2], ω ∩ d, p) does not depend on t ∈ [0, 1]. proof. by (2.8), using the same proof as in (2.3), we know that there exists ǫ0 > 0, such that |fǫ(x) + gt − p| ≥ β 2 , (2.9) for all x ∈ ∂ωǫ ∩ d, gt ∈ tt1x + 1 − t)t2x, t ∈ [0, 1], ǫ ∈ (0, ǫ0). from (2.9), and using the same proof as in lemma 2.2, we know that there exists λ(ǫ) > 0, such that p 6= fǫ(x) + t t λx, for all x ∈ ∂ωǫ, t ∈ [0, 1], λ ∈ (0, λ(ǫ)), where t t λ is the yosida approximation of tt1 + (1 − t)t2. from lemma 2.7 in [15], we know deg(fǫ + t t λ, ωǫ, p) does not depend on t ∈ [0, 1], λ ∈ (0, λ(ǫ)). therefore, deg(f + [tt1 + (1 − t)t2], ω ∩ d, p) does not depend on t ∈ [0, 1]. acknowledgement: the first author was supported by a nsfc grant, grant no. 10871052. received: april 2009. revised: may 2010. referencias [1] a. boutet de monvel-berthier, v. georgescu and r. purice, a boundary value problem related to the ginzburg-landau model, comm. math. phys 1991 pag. 1-23. 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[23] d. sarason, functions of vanishing mean oscillation trans. amer. math. soc. 1975 pag. 391-405 introduction results () cubo a mathematical journal vol.17, no¯ 03, (01–14). october 2015 right general fractional monotone approximation george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a., ganastss@memphis.edu abstract here is introduced a right general fractional derivative caputo style with respect to a base absolutely continuous strictly increasing function g. we give various examples of such right fractional derivatives for different g. let f be p-times continuously differentiable function on [a, b], and let l be a linear right general fractional differential operator such that l (f) is non-negative over a critical closed subinterval j of [a, b]. we can find a sequence of polynomials qn of degree less-equal n such that l (qn) is non-negative over j, furthermore f is approximated uniformly by qn over [a, b] . the degree of this constrained approximation is given by an inequality using the first modulus of continuity of f(p). we finish we applications of the main right fractional monotone approximation theorem for different g. resumen aqúı introducimos una derivada fraccional derecha general al estilo de caputo con respecto a una base de funciones absolutamente continuas estrictamente crecientes g. damos varios ejemplos de dichas derivadas fraccionales derechas para diferentes g. sea f una función p-veces continuamente diferenciable en [a, b], y sea l un operador diferencial fraccional derecho general tal que l(f) es no-negativo en un subintervalo cerrado cŕıtico j de [a, b]. podemos encontrar una sucesión de polinomios l (qn) de grado menor o igual a n tal que l (qn) es no-negativo en j, más aún f es aproximada uniformemente por qn en [a, b] . el grado de esta aproximación restringida es dada por una desigualdad usando el primer módulo de continuidad de f(p). concluimos con aplicaciones del teorema principal de aproximación monótona fraccional derecha para diferentes g. keywords and phrases: right fractional monotone approximation, general right fractional derivative, linear general right fractional differential operator, modulus of continuity. 2010 ams mathematics subject classification: 26a33, 41a10, 41a17, 41a25, 41a29. 2 george a. anastassiou cubo 17, 3 (2015) 1 introduction and preparation the topic of monotone approximation started in [11] has become a major trend in approximation theory. a typical problem in this subject is: given a positive integer k, approximate a given function whose kth derivative is ≥ 0 by polynomials having this property. in [4] the authors replaced the kth derivative with a linear ordinary differential operator of order k. furthermore in [1], the author generalized the result of [4] for linear right fractional differential operators. to describe the motivating result here we need definition 1. ([5]) let α > 0 and ⌈α⌉ = m, (⌈·⌉ ceiling of the number). consider f ∈ cm ([−1, 1]). we define the right caputo fractional derivative of f of order α as follows: ( dα1−f ) (x) = (−1) m γ (m − α) ∫1 x (t − x) m−α−1 f(m) (t) dt, (1) for any x ∈ [−1, 1], where γ is the gamma function γ (ν) = ∫ ∞ 0 e−ttν−1dt, ν > 0. we set d01−f (x) = f (x) , (2) dm1−f (x) = (−1) m f(m) (x) , ∀ x ∈ [−1, 1] . (3) in [1] we proved theorem 1.1. let h, k, p be integers, h is even, 0 ≤ h ≤ k ≤ p and let f be a real function, f(p) continuous in [−1, 1] with modulus of continuity ω1 ( f(p), δ ) , δ > 0, there. let αj (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume for x ∈ [−1, 0] that αh (x) is either ≥ some number α > 0 or ≤ some number β < 0. let the real numbers α0 = 0 < α1 < 1 < α2 < 2 < ... < αp < p. here d αj 1− f stands for the right caputo fractional derivative of f of order αj anchored at 1. consider the linear right fractional differential operator l := k∑ j=h αj (x) [ d αj 1− ] (4) and suppose, throughout [−1, 0] , l (f) ≥ 0. (5) then, for any n ∈ n, there exists a real polynomial qn (x) of degree ≤ n such that l (qn) ≥ 0 throughout [−1, 0] , (6) and max −1≤x≤1 |f (x) − qn (x)| ≤ cn k−pω1 ( f(p), 1 n ) , (7) where c is independent of n or f. cubo 17, 3 (2015) right general fractional monotone approximation 3 notice above that the monotonicity property is only true on [−1, 0], see (5), (6). however the approximation property (7) it is true over the whole interval [−1, 1] . in this article we extend theorem 1.1 to much more general linear right fractional differential operators. we use here the following right generalised fractional integral. definition 2. (see also [8, p. 99]) the right generalised fractional integral of a function f with respect to given function g is defined as follows: let a, b ∈ r, a < b, α > 0. here g ∈ ac ([a, b]) (absolutely continuous functions) and is strictly increasing, f ∈ l ∞ ([a, b]). we set ( iαb−;gf ) (x) = 1 γ (α) ∫b x (g (t) − g (x)) α−1 g′ (t) f (t) dt, x ≤ b, (8) clearly ( iαb−;gf ) (b) = 0. when g is the identity function id, we get that iαb−;id = i α b−, the ordinary right riemannliouville fractional integral, where ( iαb−f ) (x) = 1 γ (α) ∫b x (t − x) α−1 f (t) dt, x ≤ b, (9) ( iαb−f ) (b) = 0. when g (x) = ln x on [a, b], 0 < a 0. the right hadamard fractional integral of order α is given by ( jαb−f ) (x) = 1 γ (α) ∫b x ( ln y x )α−1 f (y) y dy, x ≤ b, (10) where f ∈ l ∞ ([a, b]) . we mention definition 4. the right fractional exponential integral is defined as follows: let a, b ∈ r, a < b, α > 0, f ∈ l ∞ ([a, b]). we set ( iαb−;exf ) (x) = 1 γ (α) ∫b x ( et − ex )α−1 etf (t) dt, x ≤ b. (11) definition 5. let a, b ∈ r, a < b, α > 0, f ∈ l ∞ ([a, b]), a > 1. we introduce the right fractional integral ( iαb−;axf ) (x) = ln a γ (α) ∫b x ( at − ax )α−1 atf (t) dt, x ≤ b. (12) 4 george a. anastassiou cubo 17, 3 (2015) we also give definition 6. let α, σ > 0, 0 ≤ a 0 and ⌈α⌉ = m, (⌈·⌉ ceiling of the number). consider f ∈ acm ([a, b]) (space of functions f with f(m−1) ∈ ac ([a, b])). we define the right general fractional derivative of f of order α as follows ( dαb−;gf ) (x) = (−1) m γ (m − α) ∫b x (g (t) − g (x)) m−α−1 g′ (t) f(m) (t) dt, (14) for any x ∈ [a, b], where γ is the gamma function. we set dmb−;gf (x) = (−1) m f(m) (x) , (15) d0b−;gf (x) = f (x) , ∀ x ∈ [a, b] . (16) when g = id, then dαb−f = d α b−;idf is the right caputo fractional derivative. so we have the specific general right fractional derivatives. definition 8. dαb−;ln xf (x) = (−1) m γ (m − α) ∫b x ( ln y x )m−α−1 f(m) (y) y dy, 0 < a ≤ x ≤ b, (17) dαb−;exf (x) = (−1) m γ (m − α) ∫b x ( et − ex )m−α−1 etf(m) (t) dt, a ≤ x ≤ b, (18) and dαb−;axf (x) = (−1) m ln a γ (m − α) ∫b x ( at − ax )m−α−1 atf(m) (t) dt, a ≤ x ≤ b, (19) ( dαb−;xσf ) (x) = (−1) m γ (m − α) ∫b x (tσ − xσ) m−α−1 σtσ−1f(m) (t) dt, 0 ≤ a ≤ x ≤ b. (20) we mention theorem 1.2. (trigub, [12], [13]) let g ∈ cp ([−1, 1]), p ∈ n. then there exists real polynomial qn (x) of degree ≤ n, x ∈ [−1, 1], such that max −1≤x≤1 ∣ ∣ ∣ g(j) (x) − q(j)n (x) ∣ ∣ ∣ ≤ rpn j−pω1 ( g(p), 1 n ) , (21) j = 0, 1, ..., p, where rp is independent of n or g. cubo 17, 3 (2015) right general fractional monotone approximation 5 in [2], based on theorem 1.2 we proved the following useful here result theorem 1.3. let f ∈ cp ([a, b]), p ∈ n. then there exist real polynomials q∗n (x) of degree ≤ n ∈ n, x ∈ [a, b], such that max a≤x≤b ∣ ∣ ∣ f(j) (x) − q∗(j)n (x) ∣ ∣ ∣ ≤ rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) , (22) j = 0, 1, ..., p, where rp is independent of n or g. remark 1.4. here g ∈ ac ([a, b]) (absolutely continuous functions), g is increasing over [a, b], α > 0. let g (a) = c, g (b) = d. we want to calculate i = ∫b a (g (t) − g (a)) α−1 g′ (t) dt. (23) consider the function f (y) = (y − g (a)) α−1 = (y − c) α−1 , ∀ y ∈ [c, d] . (24) we have that f (y) ≥ 0, it may be +∞ when y = c and 0 < α < 1, but f is measurable on [c, d]. by [9], royden, p. 107, exercise 13 d, we get that (f ◦ g) (t) g′ (t) = (g (t) − g (a)) α−1 g′ (t) (25) is measurable on [a, b], and i = ∫d c (y − c) α−1 dy = (d − c) α α (26) (notice that (y − c) α−1 is riemann integrable). that is i = (g (b) − g (a)) α α . (27) similarly it holds ∫b x (g (t) − g (x)) α−1 g′ (t) dt = (g (b) − g (x)) α α , ∀ x ∈ [a, b] . (28) finally we will use theorem 1.5. let α > 0, n ∋ m = ⌈α⌉, and f ∈ cm ([a, b]). then ( dαb−;gf ) (x) is continuous in x ∈ [a, b], −∞ < a < b < ∞. proof. by [3], apostol, p. 78, we get that g−1 exists and it is strictly increasing on [g (a) , g (b)]. since g is continuous on [a, b], it implies that g−1 is continuous on [g (a) , g (b)]. hence f(m) ◦ g−1 is a continuous function on [g (a) , g (b)] . 6 george a. anastassiou cubo 17, 3 (2015) if α = m ∈ n, then the claim is trivial. we treat the case of 0 < α < m. it holds that ( dαb−;gf ) (x) = (−1) m γ (m − α) ∫b x (g (t) − g (x)) m−α−1 g′ (t) f(m) (t) dt = (−1) m γ (m − α) ∫b x (g (t) − g (x)) m−α−1 g′ (t) ( f(m) ◦ g−1 ) (g (t)) dt = (29) (−1) m γ (m − α) ∫g(b) g(x) (z − g (x)) m−α−1 ( f(m) ◦ g−1 ) (z) dz. an explanation follows. the function g (z) = (z − g (x)) m−α−1 ( f(m) ◦ g−1 ) (z) is integrable on [g (x) , g (b)], and by assumption g is absolutely continuous : [a, b] → [g (a) , g (b)]. since g is monotone (strictly increasing here) the function (g (t) − g (x)) m−α−1 g′ (t) ( f(m) ◦ g−1 ) (g (t)) is integrable on [x, b] (see [7]). furthermore it holds (see also [7]), (−1) m γ (m − α) ∫g(b) g(x) (z − g (x)) m−α−1 ( f(m) ◦ g−1 ) (z) dz = (−1) m γ (m − α) ∫b x (g (t) − g (x)) m−α−1 g′ (t) ( f(m) ◦ g−1 ) (g (t)) dt (30) = ( dαb−;gf ) (x) , ∀ x ∈ [a, b] . and we can write ( dαb−;gf ) (x) = (−1) m γ (m − α) ∫g(b) g(x) (z − g (x)) m−α−1 ( f(m) ◦ g−1 ) (z) dz, ( dαb−;gf ) (y) = (−1) m γ (m − α) ∫g(b) g(y) (z − g (y)) m−α−1 ( f(m) ◦ g−1 ) (z) dz. (31) here a ≤ y ≤ x ≤ b, and g (a) ≤ g (y) ≤ g (x) ≤ g (b), and 0 ≤ g (b) − g (x) ≤ g (b) − g (y) . let λ = z − g (x), then z = g (x) + λ. thus ( dαb−;gf ) (x) = (−1) m γ (m − α) ∫g(b)−g(x) 0 λm−α−1 ( f(m) ◦ g−1 ) (g (x) + λ) dλ. (32) clearly, see that g (x) ≤ z ≤ g (b), and 0 ≤ λ ≤ g (b) − g (x) . cubo 17, 3 (2015) right general fractional monotone approximation 7 similarly ( dαb−;gf ) (y) = (−1) m γ (m − α) ∫g(b)−g(y) 0 λm−α−1 ( f(m) ◦ g−1 ) (g (y) + λ) dλ. (33) hence it holds ( dαb−;gf ) (y) − ( dαb−;gf ) (x) = (−1) m γ (m − α) · [∫g(b)−g(x) 0 λm−α−1 (( f(m) ◦ g−1 ) (g (y) + λ) − ( f(m) ◦ g−1 ) (g (x) + λ) ) dλ+ ∫g(b)−g(y) g(b)−g(x) λm−α−1 ( f(m) ◦ g−1 ) (g (y) + λ) dλ ] . (34) thus we obtain ∣ ∣ ( dαb−;gf ) (y) − ( dαb−;gf ) (x) ∣ ∣ ≤ 1 γ (m − α) · [ (g (b) − g (x)) m−α m − α ω1 ( f(m) ◦ g−1, |g (y) − g (x)| ) + (35) ∥ ∥f(m) ◦ g−1 ∥ ∥ ∞,[g(a),g(b)] m − α ( (g (b) − g (y)) m−α − (g (b) − g (x)) m−α ) ] =: (ξ) . as y → x, then g (y) → g (x) (since g ∈ ac ([a, b])). so that (ξ) → 0. as a result ( dαb−;gf ) (y) → ( dαb−;gf ) (x) , (36) proving that ( dαb−;gf ) (x) is continuous in x ∈ [a, b] . 2 main result we present theorem 2.1. here we assume that g (b) − g (a) > 1. let h, k, p be integers, h is even, 0 ≤ h ≤ k ≤ p and let f ∈ cp ([a, b]), a < b, with modulus of continuity ω1 ( f(p), δ ) , 0 < δ ≤ b − a. let αj (x), j = h, h + 1, ..., k be real functions, defined and bounded on [a, b] and assume for x ∈ [ a, g−1 (g (b) − 1) ] that αh (x) is either ≥ some number α ∗ > 0, or ≤ some number β∗ < 0. let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < ... < αp ≤ p. consider the linear right general fractional differential operator l = k∑ j=h αj (x) [ d αj b−;g ] , (37) and suppose, throughout [ a, g−1 (g (b) − 1) ] , l (f) ≥ 0. (38) 8 george a. anastassiou cubo 17, 3 (2015) then, for any n ∈ n, there exists a real polynomial qn (x) of degree ≤ n such that l (qn) ≥ 0 throughout [ a, g−1 (g (b) − 1) ] , (39) and max x∈[a,b] |f (x) − qn (x)| ≤ cn k−pω1 ( f(p), b − a 2n ) , (40) where c is independent of n or f. proof. of theorem 2.1. here h, k, p ∈ z+, 0 ≤ h ≤ k ≤ p. let αj > 0, j = 1, ..., p, such that 0 < α1 ≤ 1 < α2 ≤ 2 < α3 ≤ 3... < ... < αp ≤ p. that is ⌈αj⌉ = j, j = 1, ..., p. let q∗n (x) be as in theorem 1.3. we have that ( d αj b−;gf ) (x) = (−1) j γ (j − αj) ∫b x (g (t) − g (x)) j−αj−1 g′ (t) f(j) (t) dt, (41) and ( d αj b−;g q∗n ) (x) = (−1) j γ (j − αj) ∫b x (g (t) − g (x)) j−αj−1 g′ (t) q∗n (j) (t) dt, (42) j = 1, ..., p. also it holds ( d j b−;g f ) (x) = (−1) j f(j) (x) , ( d j b−;g q∗n ) (x) = (−1) j q∗(j)n (x) , j = 1, ..., p. (43) by [10], we get that there exists g′ a.e., and g′ is measurable and non-negative. we notice that ∣ ∣ ∣ ( d αj b−;g f ) (x) − d αj b−;g q∗n (x) ∣ ∣ ∣ = 1 γ (j − αj) ∣ ∣ ∣ ∣ ∣ ∫b x (g (x) − g (t)) j−αj−1 g′ (t) ( f(j) (t) − q∗(j)n (t) ) dt ∣ ∣ ∣ ∣ ∣ ≤ 1 γ (j − αj) ∫b x (g (x) − g (t)) j−αj−1 g′ (t) ∣ ∣ ∣ f(j) (t) − q∗(j)n (t) ∣ ∣ ∣ dt (22) ≤ 1 γ (j − αj) (∫b x (g (x) − g (t)) j−αj−1 g′ (t) dt ) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) (28) = (g (b) − g (x)) j−αj γ (j − αj + 1) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) ≤ (g (b) − g (a)) j−αj γ (j − αj + 1) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) . (44) cubo 17, 3 (2015) right general fractional monotone approximation 9 hence ∀ x ∈ [a, b], it holds ∣ ∣ ∣ ( d αj b−;g f ) (x) − d αj b−;g q∗n (x) ∣ ∣ ∣ ≤ (g (b) − g (a)) j−αj γ (j − αj + 1) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) , (45) and max x∈[a,b] ∣ ∣ ∣ d αj b−;g f (x) − d αj b−;g q∗n (x) ∣ ∣ ∣ ≤ (g (b) − g (a)) j−αj γ (j − αj + 1) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) , (46) j = 0, 1, ..., p. above we set d0b−;gf (x) = f (x), d 0 b−;gq ∗ n (x) = q ∗ n (x), ∀ x ∈ [a, b], and α0 = 0, i.e. ⌈α0⌉ = 0. put sj = sup a≤x≤b ∣ ∣α−1h (x) αj (x) ∣ ∣ , j = h, ..., k, (47) and ηn = rpω1 ( f(p), b − a 2n )   k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2n )p−j   . (48) i. suppose, throughout [ a, g−1 (g (b) − 1) ] , αh (x) ≥ α ∗ > 0. let qn (x), x ∈ [a, b], be a real polynomial of degree ≤ n, according to theorem 1.3 and (46), so that max x∈[a,b] ∣ ∣ ∣ d αj b−;g ( f (x) + ηn (h!) −1 xh ) − ( d αj b−;g qn ) (x) ∣ ∣ ∣ ≤ (49) (g (b) − g (a)) j−αj γ (j − αj + 1) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) , j = 0, 1, ..., p. in particular (j = 0) holds max x∈[a,b] ∣ ∣ ∣ ( f (x) + ηn (h!) −1 xh ) − qn (x) ∣ ∣ ∣ ≤ rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) , (50) and max x∈[a,b] |f (x) − qn (x)| ≤ ηn (h!) −1 (max (|a| , |b|)) h + rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) = ηn (h!) −1 max ( |a| h , |b| h ) + rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) = (51) rpω1 ( f(p), b − a 2n )   k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2n )p−j  (h!) −1 max ( |a| h , |b| h ) 10 george a. anastassiou cubo 17, 3 (2015) +rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) ≤ rpω1 ( f(p), b − a 2n ) nk−p·     k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2 )p−j  (h!) −1 max ( |a| h , |b| h ) + ( b − a 2 )p   . (52) we have found that max x∈[a,b] |f (x) − qn (x)| ≤ rp [( b − a 2 )p + (h!) −1 max ( |a| h , |b| h ) ·   k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2 )p−j    nk−pω1 ( f(p), b − a 2n ) , (53) proving (40). notice for j = h + 1, ..., k, that ( d αj b−;g xh ) = (−1) j γ (j − αj) ∫b x (g (t) − g (x)) j−αj−1 g′ (t) ( th )(j) dt = 0. (54) here l = k∑ j=h αj (x) [ d αj b−;g ] , and suppose, throughout [ a, g−1 (g (b) − 1) ] , lf ≥ 0. so over a ≤ x ≤ g−1 (g (b) − 1), we get α−1h (x) l (qn (x)) (54) = α−1h (x) l (f (x)) + ηn h! ( d αh b−;g ( xh ) ) + k∑ j=h α−1h (x) αj (x) [ d αj b−;g qn (x) − d αj b−;g f (x) − ηn h! d αj b−;g xh ] (49) ≥ (55) ηn h! ( d αh b−;g ( xh ) ) −   k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2n )p−j  rpω1 ( f(p), b − a 2n ) (56) (48) = ηn h! ( d αh b−;g ( xh ) ) − ηn = ηn ( d αh b−;g ( xh ) h! − 1 ) = (57) ηn ( 1 γ (h − αh) h! ∫b x (g (t) − g (x)) h−αh−1 g′ (t) ( th )(h) dt − 1 ) = ηn ( h! h!γ (h − αh) ∫b x (g (t) − g (x)) h−αh−1 g′ (t) dt − 1 ) (28) = cubo 17, 3 (2015) right general fractional monotone approximation 11 ηn ( (g (b) − g (x)) h−αh γ (h − αh + 1) − 1 ) = (58) ηn ( (g (b) − g (x)) h−αh − γ (h − αh + 1) γ (h − αh + 1) ) ≥ ηn ( 1 − γ (h − αh + 1) γ (h − αh + 1) ) ≥ 0. (59) clearly here g (b) − g (x) ≥ 1. hence l (qn (x)) ≥ 0, for x ∈ [ a, g−1 (g (b) − 1) ] . (60) a further explanation follows: we know γ (1) = 1, γ (2) = 1, and γ is convex and positive on (0, ∞). here 0 ≤ h − αh < 1 and 1 ≤ h − αh + 1 < 2. thus γ (h − αh + 1) ≤ 1 and 1 − γ (h − αh + 1) ≥ 0. (61) ii. suppose, throughout [ a, g−1 (g (b) − 1) ] , αh (x) ≤ β ∗ < 0. let qn (x), x ∈ [a, b] be a real polynomial of degree ≤ n, according to theorem 1.3 and (46), so that max x∈[a,b] ∣ ∣ ∣ d αj b−;g ( f (x) − ηn (h!) −1 xh ) − ( d αj b−;g qn ) (x) ∣ ∣ ∣ ≤ (62) (g (b) − g (a)) j−αj γ (j − αj + 1) rp ( b − a 2n )p−j ω1 ( f(p), b − a 2n ) , j = 0, 1, ..., p. in particular (j = 0) holds max x∈[a,b] ∣ ∣ ∣ ( f (x) − ηn (h!) −1 xh ) − qn (x) ∣ ∣ ∣ ≤ rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) , (63) and max x∈[a,b] |f (x) − qn (x)| ≤ ηn (h!) −1 (max (|a| , |b|)) h + rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) = ηn (h!) −1 max ( |a| h , |b| h ) + rp ( b − a 2n )p ω1 ( f(p), b − a 2n ) , (64) etc. we find again that max x∈[a,b] |f (x) − qn (x)| ≤ rp [( b − a 2 )p + (h!) −1 max ( |a| h , |b| h ) · 12 george a. anastassiou cubo 17, 3 (2015)   k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2 )p−j    nk−pω1 ( f(p), b − a 2n ) , (65) reproving (40). here again l = k∑ j=h αj (x) [ d αj b−;g ] , and suppose, throughout [ a, g−1 (g (b) − 1) ] , lf ≥ 0. so over a ≤ x ≤ g−1 (g (b) − 1), we get α−1h (x) l (qn (x)) (54) = α−1h (x) l (f (x)) − ηn h! ( d αh b−;g ( xh ) ) + k∑ j=h α−1h (x) αj (x) [ d αj b−;g qn (x) − d αj b−;g f (x) + ηn h! d αj b−;g xh ] (62) ≤ (66) − ηn h! ( d αh b−;g ( xh ) ) +   k∑ j=h sj (g (b) − g (a)) j−αj γ (j − αj + 1) ( b − a 2n )p−j  rpω1 ( f(p), b − a 2n ) (67) (48) = − ηn h! ( d αh b−;g ( xh ) ) + ηn = ηn ( 1 − d αh b−;g ( xh ) h! ) = (68) ηn ( 1 − 1 γ (h − αh) h! ∫b x (g (t) − g (x)) h−αh−1 g′ (t) ( th )(h) dt ) = ηn ( 1 − h! h!γ (h − αh) ∫b x (g (t) − g (x)) h−αh−1 g′ (t) dt ) (28) = ηn ( 1 − (g (b) − g (x)) h−αh γ (h − αh + 1) ) = (69) ηn ( γ (h − αh + 1) − (g (b) − g (x)) h−αh γ (h − αh + 1) ) (61) ≤ ηn ( 1 − (g (b) − g (x)) h−αh γ (h − αh + 1) ) ≤ 0. (70) hence again l (qn (x)) ≥ 0, ∀ x ∈ [ a, g−1 (g (b) − 1) ] . the case of αh = h is trivially concluded from the above. the proof of the theorem is now over. we make cubo 17, 3 (2015) right general fractional monotone approximation 13 remark 2.2. by theorem 1.5 we have that d αj b−;g f are continuous functions, j = 0, 1, ..., p. suppose that αh (x) , ..., αk (x) are continuous functions on [a, b], and l (f) ≥ 0 on [ a, g−1 (g (b) − 1) ] is replaced by l (f) > 0 on [ a, g−1 (g (b) − 1) ] . disregard the assumption made in the main theorem on αh (x). for n ∈ n, let qn (x) be the q ∗ n (x) of theorem 1.3, and f as in theorem 1.3 (same as in theorem 2.1). then qn (x) converges to f (x) at the jackson rate 1 np+1 ([6], p. 18, theorem viii) and at the same time, since l (qn) converges uniformly to l (f) on [a, b], l (qn) > 0 on [ a, g−1 (g (b) − 1) ] for all n sufficiently large. 3 applications (to theorem 2.1) 1) when g (x) = ln x on [a, b], 0 < a ae, αh (x) restriction true on [ a, b e ] , and lf = k∑ j=h αj (x) [ d αj b−;ln x f ] ≥ 0, (72) throughout [ a, b e ] . then l (qn) ≥ 0 on [ a, b e ] . 2) when g (x) = ex on [a, b], a ln (1 + ea), αh (x) restriction true on [ a, ln ( eb − 1 )] , and lf = k∑ j=h αj (x) [ d αj b−;ex f ] ≥ 0, (73) throughout [ a, ln ( eb − 1 )] . then l (qn) ≥ 0 on [ a, ln ( eb − 1 )] . 3) when, a > 1, g (x) = ax on [a, b], a loga (1 + a a), αh (x) restriction true on [ a, loga ( ab − 1 )] , and lf = k∑ j=h αj (x) [ d αj b−;ax f ] ≥ 0, (74) throughout [ a, loga ( ab − 1 )] . then l (qn) ≥ 0 on [ a, loga ( ab − 1 )] . 4) when σ > 0, g (x) = xσ, 0 ≤ a (1 + aσ) 1 σ , αh (x) restriction true on [ a, (bσ − 1) 1 σ ] , and lf = k∑ j=h αj (x) [ d αj b−;xσ f ] ≥ 0 (75) 14 george a. anastassiou cubo 17, 3 (2015) throughout [ a, (bσ − 1) 1 σ ] . then l (qn) ≥ 0 on [ a, (bσ − 1) 1 σ ] . received: april 2015. accepted: july 2015. references [1] g.a. anastassiou, right fractional monotone approximation, j. applied functional analysis, vol. 10, no.’s 1-2 (2015), 117-124. 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[13] r.m. trigub, approximation of functions by polynomials with integer coeficients, izv. akad. nauk sssr ser. mat. 26 (1962), 261-280 [russian]. introduction and preparation main result applications (to theorem 2.1) () cubo a mathematical journal vol.16, no¯ 03, (97–117). october 2014 computing the inverse laplace transform for rational functions vanishing at infinity takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, senbaru 1, nishihara, okinawa 903-0213, japan. sudo@math.u-ryukyu.ac.jp abstract we compute explicitly the inverse laplace transform for rational functions vanishing at infinity in the general case. we also compute explicitly convolution product for continuous elementary functions involved in the general case. we then consider algebraic structure about the laplace transform via convolution product. resumen calculamos expĺıcitamente la transformada de laplace inversa para funciones racionales que se anulan en infinito en el caso general. además calculamos expĺıcitamente el producto de convolución para funciones elementales continuas que participan en el caso general. luego, consideramos estructuras algebraicas de la transformada de laplace por medio del producto de convolución. keywords and phrases: laplace transform, rational function, convolution. 2010 ams mathematics subject classification: 44a10, 44a35, 26c15, 26a06, 26a09. 98 takahiro sudo cubo 16, 3 (2014) 1 introduction the laplace transform l(f) of a real-valued, function f on the interval [0, ∞) is defined by l(f)(s) = ∫ ∞ 0 e−stf(t)dt for s ∈ c in the domain of convergence (cf. [2] or [3]). the laplace transform l for continuous functions f on [0, ∞) is injective. this fact is known as lerch’s theorem as a fundamental theorem in the laplace transform theory (or deduced from switching the laplace transform as to be the fourier transform), so that the inverse laplace transform is well defined as the inverse image of l(f): l−1(l(f))(t) = f(t). in this paper we consider real-valued, elementary functions f that are defined on the real line r and are continuous on r, as well as the injectivity of the laplace transform for these continuous functions is preserved. the reason for this assumption of r just comes from that we do make most of the statements simplified and also that this additional symmetry induces several symmetric results and it makes it to be possible to consider the usual algebraic structure about continuous elementary functions, and is perhaps more natural than cutting down functions to be zero on the negative part of r. but this assumption is not the same as the usual convention that f is assumed to be zero on the interval (−∞, 0). indeed, the inverse laplace trasform defined as f(t) = 1 2πi ∫α+i∞ α−i∞ estl(f)(s)ds known as bromwich integral requires that convention, where the complex line integral is taken for some real α and is computed by residue theorem as sums of residue functions. this says that even real-valued functions of one variable as well as the complex case are determined by singularities of their images under the laplace transform. however, we do not use the complex integral in what follows. in this paper, by elementary calculation we compute explicitly the inverse laplace transform for rational functions vanishing at infinity in the general case and determine the inverse image. we also compute explicitly convolution product for continuous elementary functions involved in the general case. we then consider algebraic structure about the laplace transform from our view point, which may not be written in the literature. as a reference, there is another inductive computation known as a recursive formula for multiples of convolution in general (see [1]), but without using it we compute more explictly multiples of convolution for certain concrete continuous elementary functions. there are 4 sections after this introduction as follows: 2. inverse laplace transform for rational functions in a special case; 3. inverse laplace transform for rational functions in another special case; 4. inverse laplace transform for rational functions; 5. algebraic structure. cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 99 our elemetary but explicit computation results obtained in the several general cases of those sections and our consideration and determination on the algebraic structure about the laplace transform via convolution product would be new as well as useful and helpful as a reference. notation. we denote by ex the exponential function to the base e for x ∈ r and by sin x and cos x the trigonometric functions for x ∈ r. we denote by ( n k ) the combination of k items from n items mutually different. 2 inverse laplace transform for rational functions in a special case as a well known fact, we have lemma 2.1. let s ∈ c with the real part re(s) > 0 and λ ∈ r a constant with λ 6= 0. then l−1 ( 1 (s2 + λ2)2 ) = 1 2λ3 sin λt − t 2λ2 cos λt (t ∈ r). proof. by using a fact that the laplace transform of the convolution product of two functions f(t) and g(t) is the pointwise multiplication of their laplace transforms: l(f ∗ g)(s) = l(f)(s) · l(g)(s) with f ∗ g(t) = ∫t 0 f(t − τ)g(τ)dτ, we compute l−1 ( 1 (s2 + λ2)2 ) = l−1 ( 1 s2 + λ2 · 1 s2 + λ2 ) = 1 λ2 sin λt ∗ sin λt = 1 λ2 ∫t 0 sin λ(t − τ) sin λτdτ = 1 2λ2 ∫t 0 {cos λ(t − 2τ) − cos λt}dτ = 1 2λ2 [ 1 −2λ sin λ(t − 2τ) − τ cosλt ]t τ=0 = 1 2λ3 sin λt − t 2λ2 cos λt. note that l(sin λt) = λ s2+λ2 well known. as the second step in induction, we have lemma 2.2. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then l−1 ( 1 (s2 + λ2)3 ) = ( 3 − λ2t2 8λ5 ) sin λt − 3t 8λ4 cos λt. 100 takahiro sudo cubo 16, 3 (2014) proof. we compute l−1 ( 1 (s2 + λ2)3 ) = l−1 ( 1 s2 + λ2 · 1 (s2 + λ2)2 ) = 1 λ sin λt ∗ l−1 ( 1 (s2 + λ2)2 ) (t). inserting the result of lemma 2.1 above for l−1( 1 (s2+λ2)2 ) and computing the convolution product by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. as the third step in induction, we have lemma 2.3. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then l−1 ( 1 (s2 + λ2)4 ) = ( 5 − 2λ2t2 16λ7 ) sin λt + ( 3−1λ2t3 − 5t 16λ6 ) cos λt. proof. we compute l−1 ( 1 (s2 + λ2)4 ) = l−1 ( 1 s2 + λ2 · 1 (s2 + λ2)3 ) = 1 λ sin λt ∗ l−1 ( 1 (s2 + λ2)3 ) (t). inserting the result of lemma 2.2 above for l−1( 1 (s2+λ2)3 ) and computing the convolution product by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. theorem 2.4. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then, for an integer n ≥ 1, l−1 ( 1 (s2 + λ2)2n ) = e2n−2(t) sin λt + o2n−1(t) cosλt where e2n−2(t) is an even polynomial of t with degree 2n − 2 and with real coefficients involving λ, and o2n−1(t) is an odd polynomial of t with degree 2n − 1 and with real coefficients involving λ. similarly, for an integer n ≥ 2, l−1 ( 1 (s2 + λ2)2n−1 ) = e2n−2(t) sin λt + o2n−3(t) cos λt. proof. by induction, suppose that the formula for 2n in the statement holds. we then compute l−1 ( 1 (s2 + λ2)2n+1 ) = l−1 ( 1 s2 + λ2 · 1 (s2 + λ2)2n ) = 1 λ sin λt ∗ l−1 ( 1 (s2 + λ2)2n ) (t) cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 101 (and inserting the formula assumed for l−1( 1 (s2+λ2)2n ) we have:) = 1 λ sin λt ∗ {e2n−2(t) sin λt + o2n−1(t) cos λt} = 1 λ ∫t 0 sin λ(t − τ){e2n−2(τ) sin λτ + o2n−1(τ) cos λτ}dτ (and by addition theorem of trigonometric functions, we have:) = 1 2λ ∫t 0 e2n−2(τ){cos λ(t − 2τ) − cos λt}dτ + 1 2λ ∫t 0 o2n−1(τ){sin λt + sin λ(t − 2τ)}dτ = 1 2λ [∫t 0 e2n−2(τ) cos λ(t − 2τ)dτ − ∫t 0 e2n−2(τ)dτ · cos λt ] + 1 2λ [∫t 0 o2n−1(τ)dτ · sin λt + ∫t 0 o2n−1(τ) sin λ(t − 2τ)dτ ] . by using integration by parts repeatedly, we compute the first integral term among four terms as: ∫t 0 e2n−2(τ) cos λ(t − 2τ)dτ = [ e2n−2(τ) sin λ(t − 2τ) −2λ ]t τ=0 + 1 2λ ∫t 0 e′2n−2(τ) sin λ(t − 2τ)dτ = { e2n−2(t) − e2n−2(0) 2λ } sin λt + 1 2λ ∫t 0 e′2n−2(τ) sin λ(t − 2τ)dτ and note that the first coefficient {·} is an even polynomial of t of degree 2n − 2, and the integral in the second term is computed as: ∫t 0 e′2n−2(τ) sin λ(t − 2τ)dτ = [ e′2n−2(τ) − cos λ(t − 2τ) −2λ ]t τ=0 − 1 2λ ∫t 0 e′′2n−2(τ) cos λ(t − 2τ)dτ = { e′2n−2(t) − e ′ 2n−2(0) 2λ } cos λt − 1 2λ ∫t 0 e′′2n−2(τ) cos λ(t − 2τ)dτ with the differential e′2n−2(0) = 0 and the coefficient {·} an odd polynomial of t of degree 2n − 1, and moreover, the last integral ∫t 0 e′′2n−2(τ) cos λ(t − 2τ)dτ is computed inductively and finitely by integration by parts to obtain the similar coefficients of sin λt and cos λt summed as even and odd polynomials of t with degrees less than 2n − 2 and 2n − 1, respectively. the same consideration as for the first integral is applied for the fourth integral: ∫t 0 o2n−1(τ) sin λ(t− 2τ)dτ to be computed as the sum of cos λt and sin λt with coefficients odd and even polynomials of t of degree 2n − 1 and 2n, respectively. as for ∫t 0 e2n−2(τ)dτ · cos λt and ∫t 0 o2n−1(τ)dτ · sin λt, the second and third integrals among four terms are computed to be odd and even polynomials of t with degrees 2n − 1 and 2n − 2, respectively. 102 takahiro sudo cubo 16, 3 (2014) summing up the computations above, we obtain l−1 ( 1 (s2 + λ2)2(n+1)−1 ) = e2(n+1)−2(t) sin λt + o2(n+1)−3(t) cos λt for some even e2n(t) and odd o2n−1(t). this is the case where n is replaced with n + 1 in the second formula in the statement. similarly, the case of 2n + 2 is deduced from the case of 2n + 1 that now we have proved above. remark. perhaps, the real coefficients involving λ of even and odd polynomials in general 2n or 2n − 1 could be determined explicitly as given in lemmas 2.1 to 2.3. corollary 2.5. for any integer n ≥ 1, l−1 ( 1 (s2 + λ2)n ) (t) is an odd function as for t ∈ r. 3 inverse laplace transform for rational functions in another special case as a well known fact, we have lemma 3.1. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then l−1 ( s (s2 + λ2)2 ) = 1 2λ t sin λt (t ∈ r). proof. we compute as in the previous section, l−1 ( s (s2 + λ2)2 ) = l−1 ( 1 s2 + λ2 · s s2 + λ2 ) = 1 λ sin λt ∗ cos λt = 1 λ ∫t 0 sin λ(t − τ) cos λτdτ = 1 2λ ∫t 0 {sin λ(t − 2τ) + sin λt}dτ = 1 2λ [ 1 2λ cos λ(t − 2τ) + τ sin λt ]t τ=0 = 1 2λ t sin λt. note that l(cos λt) = s s2+λ2 well known. as the second step in induction, we have lemma 3.2. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then l−1 ( s (s2 + λ2)3 ) = ( 1 + λ2t2 4λ3 ) sin λt − t 2λ cos λt. cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 103 proof. we compute l−1 ( s (s2 + λ2)3 ) = l−1 ( 1 s2 + λ2 · s (s2 + λ2)2 ) = 1 λ sin λt ∗ l−1 ( s (s2 + λ2)2 ) (t). inserting the result of lemma 3.1 above for l−1( s (s2+λ2)2 ) and computing the convolution product by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. one can use the following decomposition and lemma 2.1 in the previous section: l−1 ( s s2 + λ2 · 1 (s2 + λ2)2 ) = cos λt ∗ l−1 ( 1 (s2 + λ2)2 ) (t). as the third step in induction, we have lemma 3.3. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then l−1 ( s (s2 + λ2)4 ) = ( 2 + λ − λ3t2 16λ5 ) sin λt + ( −3λt − 3−12λ3t3 16λ4 ) cos λt. proof. we compute l−1 ( s (s2 + λ2)4 ) = l−1 ( 1 s2 + λ2 · s (s2 + λ2)3 ) = 1 λ sin λt ∗ l−1 ( s (s2 + λ2)3 ) (t). inserting the result of lemma 3.2 above for l−1( s (s2+λ2)3 ) and computing the convolution product by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. theorem 3.4. let s ∈ c with re(s) > 0 and λ ∈ r a constant with λ 6= 0. then, for an integer n ≥ 2, l−1 ( s (s2 + λ2)2n ) = e2n−2(t) sin λt + o2n−1(t) cos λt where e2n−2(t) is an even polynomial of t with degree 2n − 2 and with real coefficients involving λ, and o2n−1(t) is an odd polynomial of t with degree 2n − 1 and with real coefficients involving λ. similarly, for an integer n ≥ 2, l−1 ( s (s2 + λ2)2n−1 ) = e2n−2(t) sin λt + o2n−3(t) cos λt. 104 takahiro sudo cubo 16, 3 (2014) proof. by induction, suppose that two formula in the statement hold for 2n. we then compute l−1 ( s (s2 + λ2)2n+1 ) = l−1 ( 1 s2 + λ2 · s (s2 + λ2)2n ) = 1 λ sin λt ∗ l−1 ( s (s2 + λ2)2n ) (t) and inserting the formula assumed for l−1( s (s2+λ2)2n ) we have = 1 λ sin λt ∗ {e2n−2(t) sin λt + o2n−1(t) cos λt} = 1 λ ∫t 0 sin λ(t − τ){e2n−2(τ) sin λτ + o2n−1(τ) cos λτ}dτ. note that this integral is exactly the same as the integral in the case of l−1 ( 1 (s2+λ2)2n+1 ) in the proof of theorem 2.4. thus, we omit the rest of the proof. similarly, the case of 2n + 2 is deduced from the case of 2n + 1 that now we have proved. corollary 3.5. for any integer n ≥ 2, l−1 ( s (s2 + λ2)n ) (t) is an odd function as for t ∈ r. remark. note that the polynomials obtained in theorem 3.4 are not the same as those in theorem 2.4, but we use the same symbols for both of the polynomials. anyhow, combining both of theorem 2.4 and theorem 3.4 we get corollary 3.6. both l−1( 1 (s2+λ2)2n ) for n ≥ 1 and l−1( s (s2+λ2)2n ) for n ≥ 2 are written as the same form: e2n−2(t) sin λt + o2n−1(t) cos λt; and both l−1( 1 (s2+λ2)2n−1 ) for n ≥ 1 and l−1( s (s2+λ2)2n−1 ) for n ≥ 2 are written as the same form: e2n−2(t) sin λt + o2n−3(t) cos λt. 4 inverse laplace transform for rational functions it is well known that a rational function f of s ∈ c such that f(s) = p(s) q(s) with p(s), q(s) polynomials of s ∈ c with real coefficients and with deg p(s) < deg q(s) is decomposed into partial fractions as: f(s) = m0∑ j=1 c0j sj + l1∑ k=1 mk∑ j=1 ckj (s − ak) j + l2∑ k=1 nk∑ j=1 dkjs ((s − bk) 2 + c2k) j + l2∑ k=1 nk∑ j=1 ekj ((s − bk) 2 + c2k) j , where q(s) = q0s m0π l1 k=1 (s − ak) mkπ l2 k=1 ((s − bk) 2 + c2k) nk cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 105 the factorization of q(s) in real r with ak real roots of multiplicity mk and with bk ±ick imaginary roots of multiplicity nk, for some q0 6= 0 in r, m0 ≥ 0, mk ≥ 1, nk ≥ 1, and ak 6= 0, bk 6= 0 or 0, ck 6= 0 in r, and l1, l2 ∈ n, and c0j, ckj ∈ r but c0m0 6= 0 or ckmk 6= 0 corresponding to the highest terms if nonzero; and dkj, ekj ∈ r but either dknk 6= 0 or eknk 6= 0 in r corresponding to the highest terms if nonzero. therefore, using the basic facts of laplace transform for polynomials and translation and our results in the previous sections we obtain theorem 4.1. let f(s) be a rational function of s ∈ c vanishing at infinity with the factorization as above. assume that the real part of s satisfies the following inequality: re(s) > max{0, ak, bj | 1 ≤ k ≤ l1, 1 ≤ j ≤ l2} if m0 ≥ 1, l1 ≥ 1, and l2 ≥ 1, and otherwise, some elements of the set may be dropped. then, in general, l−1(f(s))(t) = m0∑ j=1 c0jt j−1 (j − 1)! + l1∑ k=1 mk∑ j=1 ckje akttj−1 (j − 1)! + l2∑ k=1 dk1 ck ebkt sin ckt + l2∑ k=1 dk2 2ck ebktt sin ckt + l2∑ k=1 ⌈ nk 2 ⌉ ∑ n=2 dk(2n−1)e bkt [ ek(2n−2)(t) sin ckt + ok(2n−3)(t) cos ckt ] + l2∑ k=1 ⌊ nk 2 ⌋ ∑ n=2 dk(2n)e bkt [ ek(2n−2)(t) sin ckt + ok(2n−1)(t) cos ckt ] + l2∑ k=1 ek1e bkt cos ckt + l2∑ k=1 ⌈ nk 2 ⌉ ∑ n=2 ek(2n−1)e bkt [ e∼k(2n−2)(t) sin ckt + o ∼ k(2n−3)(t) cos ckt ] + l2∑ k=1 ⌊ nk 2 ⌋ ∑ n=1 ek(2n)e bkt [ e∼k(2n−2)(t) sin ckt + o ∼ k(2n−1)(t) cos ckt ] , where ⌈x⌉ means the minimum integer y such that y ≥ x, and ⌊x⌋ means the maximum integer y such that y ≤ x, and ekj(t), okj(t), e ∼ kj(t), and o ∼ kj(t) are polynomials of t with degree j and with real coefficients involving ck. remark. the restriction on s ∈ c comes from the existence of the laplace transform l(g(t))(s) = f(s) for some g(t). namely, s ∈ c should belong to a domain of convergence of l(g(t))(s). note that the maximum in the statement is just that of real parts of poles of the rational function f(s) given. 106 takahiro sudo cubo 16, 3 (2014) remark. note that l(δ(t)) = 1 for δ(t) the dirac function, viewed as a distribution, i.e. a functional, so that l−1(1) = δ(t), where the constant unit function 1 on c is also viewed as a distribution. 5 algebraic structure in this section we consider algebraic structure about the laplace transform via convolution product. we denote by r0(c) the set of rational functions on c with real coefficients, vanishing at infinity. under pointwise addition and multiplication, r0(c) becomes a non-unital algebra over r. indeed, lemma 5.1. the set r0(c) is an algebra over r under point-wise operations. proof. let f, g ∈ r0(c) such that f(s) = p1(s) q1(s) and g(s) = p2(s) q2(s) for some polynomials pj(s), qj(s) with deg pj < deg qj (j = 1, 2). then f(s) + g(s) = p1(s)q2(s) + p2(s)q1(s) q1(s)q2(s) ∈ r0(c) and f(s)g(s) = p1(s)p2(s) q1(s)q2(s) ∈ r0(c) and other axioms can be also easily checked. we denote by a(r) the algebra over r generated by the sets of elementary continuous functions on r: {tn | n ∈ n = {0, 1, 2, · · · }} of monomials and {eµt | µ ∈ r} and {sin λt, cosλt | λ ∈ r} under point-wise addition and point-wise multiplication. under (extended) convolution product defined as: (f ∗ g)(t) = ∫t 0 f(t − τ)g(τ)dτ, (t ∈ r) for f, g ∈ a(r), which is a commutative and associative operation as well known as the usual case on [0, ∞), a(r) becomes an algebra over r. indeed, check it in details as follows: proposition 5.2. the real algebra a(r) under point-wise operations is viewed as an algebra over r under convolution product, as given in the following: (1) (tn ∗ tm)(t) = [ n∑ k=0 ( n k ) (−1)k k + m + 1 ] tn+m+1 for n, m ∈ n; and for µ, λ ∈ r with µ 6= λ, (2) (eµt ∗ eλt)(t) = { e λt −e µt λ−µ if µ 6= λ, teµt if µ = λ; cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 107 and for n ≥ 0 in n and µ ∈ r with µ 6= 0, (3) (tn ∗ eµt)(t) = n∑ k=0 ( n k ) tn−k(−k) ∫t 0 τkeµτdτ with the integral ∫t 0 τkeµτdτ equal to eµt k+1∑ l=1 (−1)l−1k! µl(k + 1 − l)! tk+1−l + ( (−1)k+1k! µk+1 ) ; and for λ, µ ∈ r, (4) (sin λt ∗ cos µt)(t) = { λ λ2−µ2 cos µt − λ λ2−µ2 cos λt if λ 6= ±µ, 1 2 t sin λt if λ = ±µ, and (cos λt ∗ cos µt)(t) = { µ λ2−µ2 sin µt − λ λ2−µ2 sin λt if λ 6= ±µ, 1 2 t cosλt + 1 2λ sin λt if λ = ±µ, and (sin λt ∗ sin µt)(t) = { λ λ2−µ2 sin µt − µ λ2−µ2 sin λt if λ 6= ±µ, −1 2 t cos λt + 1 2λ sin λt if λ = ±µ; and moreover, for µ, λ ∈ r with µ 6= 0, (5) (eµt ∗ sin λt)(t) = 1 λ2 + µ2 {λe−µt − µ sin λt − λ cos λt}. and (eµt ∗ cos λt)(t) = 1 λ2 + µ2 {µe−µt − µ cos λt + λ sin λt}; and furthermore, (6) (tn ∗ sin λt)(t) = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk sin λτdτ with ik = ∫t 0 τk sin λτdτ given by, for m ∈ n with m ≥ 0, if k = 2m, i2m = m+1∑ l=1 (−1)lk! λ2l−1(k − 2l + 2)! tk−2l+2 cos λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 sin λt + (−1)mk! λ2m+1 and if k = 2m + 1, i2m+1 = m+1∑ l=1 (−1)lk! λ2l−1(k − 2l + 2)! tk−2l+2 cos λt + m+1∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 sin λt 108 takahiro sudo cubo 16, 3 (2014) so that (tn ∗ sin λt)(t) = ⌊ n 2 ⌋ ∑ m=0 ( n 2m ) { m+1∑ l=1 (−1)2m+lk! λ2l−1(k − 2l + 2)! tn−2l+2 cos λt + m∑ l=1 (−1)(2m+l−1)k! λ2l(k − 2l + 1)! tn−2l+1 sin λt + (−1)3mk! λ2m+1 tn−2m} + ⌈ n 2 ⌉−1 ∑ m=0 ( n 2m + 1 ) { m+1∑ l=1 (−1)2m+1+lk! λ2l−1(k − 2l + 2)! tn−2l+2 cos λt + m+1∑ l=1 (−1)(2m+l)k! λ2l(k − 2l + 1)! tn−2l+1 sin λt} and also, (tn ∗ cos λt)(t) = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk cos λτdτ with jk ≡ ∫t 0 τk cos λτdτ given by, for m ∈ n with m ≥ 0, if k = 2m, j2m = m+1∑ l=1 (−1)(l−1)k! λ2l−1(k − 2l + 2)! tk−2l+2 sin λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 cos λt and if k = 2m + 1, j2m+1 = m+1∑ l=1 (−1)(l−1)k! λ2l−1(k − 2l + 2)! tk−2l+2 sin λt + m+1∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 cos λt + (−1)m+1k! λ2m+2 so that (tn ∗ cos λt)(t) = ⌊ n 2 ⌋ ∑ m=0 ( n 2m ) { m+1∑ l=1 (−1)(2m+l)k! λ2l−1(k − 2l + 2)! tn−2l+2 sin λt + m∑ l=1 (−1)(2m+l)k! λ2l(k − 2l + 1)! tn−2l+1 cos λt} + ⌈ n 2 ⌉−1 ∑ m=0 ( n 2m + 1 ) { m+1∑ l=1 (−1)(2m+l)k! λ2l−1(k − 2l + 2)! tn−2l+2 sin λt + m+1∑ l=1 (−1)(2m+l)k! λ2l(k − 2l + 1)! tn−2l+1 cos λt + (−1)3m+2k! λ2m+2 tn−2m−1}. cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 109 finally, the convolution of general monomials is given by, as an example, (7) (tneµ1t sin λ1t) ∗ (t meµ2t cos λ2t)(t) = eµ1t n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk+me(µ2−µ1)τ sin λ1(t − τ) cos λ2τdτ with sin λ1(t − τ) cos λ2τ = 1 2 {sin(λ1t − (λ1 − λ2)τ) + sin(λ1t − (λ1 + λ2)τ)}, and then the following integral is computed inductively as ik+m,sin ≡ ∫t 0 τk+me(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 tk+me(µ2−µ1)t sin(∓λ2t) + λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 tk+me(µ2−µ1)t cos(∓λ2t) − (k + m)(µ2 − µ1) (µ2 − µ1) 2 + (λ1 ± λ2)2 ∫t 0 τk+m−1e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ − (k + m)(λ1 ± λ2) (µ2 − µ1) 2 + (λ1 ± λ2)2 ∫t 0 τk+m−1e(µ2−µ1)τ cos(λ1t − (λ1 ± λ2)τ)dτ, where the last two integrals are denoted by ik+m−1,sin and ik+m−1,cos respectively, and the integrals can be inductively reduced to the cases of ij,sin and ij,cos for 1 ≤ j ≤ k + m − 2 and finally to the case of i0,sin and i0,cos that are given by i0,sin ≡ ∫t 0 e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t sin(∓λ2t) − sin(λ1t)} + λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t cos(∓λ2t) − cos(λ1t)} and i0,cos ≡ ∫t 0 e(µ2−µ1)τ cos(λ1t − (λ1 ± λ2)τ)dτ = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t cos(∓λ2t) − cos(λ1t)} − λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t sin(∓λ2t) − sin(λ1t)} other cases of convolution products of general monomials with sin and cos changed are also computed similarly, but omitted. 110 takahiro sudo cubo 16, 3 (2014) proof. for (1), check first that for n, m ≥ 0 in n, (tn ∗ tm)(t) = ∫t 0 (t − τ)nτmdτ = ∫t 0 n∑ k=0 ( n k ) tn−k(−1)kτk+mdτ = n∑ k=0 ( n k ) tn−k(−1)k tk+m+1 k + m + 1 = [ n∑ k=0 ( n k ) (−1)k k + m + 1 ] tn+m+1 ∈ a(r). also, for (2), for µ, λ ∈ r with µ 6= λ, (eµt ∗ eλt)(t) = ∫t 0 eµ(t−τ)eλτdτ = eµt ∫t 0 e(λ−µ)τdτ = eµt [ e(λ−µ)τ λ − µ ]t τ=0 = eλt − eµt λ − µ ∈ a(r). if µ = λ, then (eµt ∗ eλt)(t) = ∫t 0 eµtdτ = teµt ∈ a(r). moreover, for (3), for n ≥ 0 in n and µ ∈ r with µ 6= 0, (tn ∗ eµt)(t) = ∫t 0 (t − τ)neµτdτ = ∫t 0 n∑ k=0 ( n k ) tn−k(−k)τkeµτdτ = n∑ k=0 ( n k ) tn−k(−k) ∫t 0 τkeµτdτ. we then compute the following integral by integration by parts: ik ≡ ∫t 0 τkeµτdτ = 1 µ tkeµt − k µ ik−1 = 1 µ tkeµt − k µ2 tk−1eµt + k(k − 1) µ2 ik−2 = 1 µ tkeµt − k µ2 tk−1eµt + · · · + (−1)k−1k(k − 1) · · · 2 µk−1 ( t µ eµt − 1 µ i0 ) = eµt k+1∑ l=1 (−1)l−1k! µl(k + 1 − l)! tk+1−l + ( (−1)k+1k! µk+1 ) with i0 = ∫t 0 eµτdτ = 1 µ (eµt − 1). cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 111 next, for (4), for λ, µ ∈ r with λ 6= ±µ, (sin λt ∗ cos µt)(t) = ∫t 0 sin λ(t − τ) cos µτdτ = 1 2 ∫t 0 {sin(λt − (λ − µ)τ) + sin(λt − (λ + µ)τ)} dτ = 1 2 [ cos(λt − (λ − µ)τ) λ − µ + cos(λt − (λ + µ)τ) λ + µ ]t τ=0 = λ λ2 − µ2 cos µt − λ λ2 − µ2 cos λt ∈ a(r). if λ = ±µ, then (sin λt ∗ cos µt)(t) = 1 2 t sin λt ∈ a(r). moreover, for λ, µ ∈ r with λ 6= ±µ, (cos λt ∗ cos µt)(t) = ∫t 0 cos λ(t − τ) cos µτdτ = 1 2 ∫t 0 {cos(λt − (λ − µ)τ) + cos(λt − (λ + µ)τ)} dτ = 1 2 [ sin(λt − (λ − µ)τ) −(λ − µ) + sin(λt − (λ + µ)τ) −(λ + µ) ]t τ=0 = µ λ2 − µ2 sin µt − λ λ2 − µ2 sin λt ∈ a(r). if λ = ±µ, then (cos λt ∗ cos µt)(t) = 1 2 t cosλt + 1 2λ sin λt ∈ a(r). furthermore, for λ, µ ∈ r with λ 6= ±µ, (sin λt ∗ sin µt)(t) = ∫t 0 sin λ(t − τ) sin µτdτ = −1 2 ∫t 0 {cos(λt − (λ − µ)τ) − cos(λt − (λ + µ)τ)} dτ = −1 2 [ sin(λt − (λ − µ)τ) −(λ − µ) − sin(λt − (λ + µ)τ) −(λ + µ) ]t τ=0 = λ λ2 − µ2 sin µt − µ λ2 − µ2 sin λt ∈ a(r). if λ = ±µ, then (sin λt ∗ sin µt)(t) = −1 2 t cos λt + 1 2λ sin λt ∈ a(r). next, for (5), for µ, λ ∈ r with µ 6= 0, (eµt ∗ sin λt)(t) = eµt ∫t 0 e−µτ sin λτdτ 112 takahiro sudo cubo 16, 3 (2014) with the integral is ≡ ∫t 0 e−µτ sin λτdτ computed as is = [ e−µτ −µ sin λτ ]t τ=0 + λ µ ∫t 0 e−µτ cos λτdτ = e−µt −µ sin λt + λ µ2 (1 − eµt cos λt) − λ2 µ2 is so that is = 1 λ2 + µ2 {λ − e−µt(µ sin λt + λ cos λt)}. similarly, (eµt ∗ cos λt)(t) = eµt ∫t 0 e−µτ cos λτdτ with the integral ic ≡ ∫t 0 e−µτ cos λτdτ computed as ic = [ e−µτ −µ cos λτ ]t τ=0 − λ µ ∫t 0 e−µτ sin λτdτ = 1 µ (1 − e−µt cos λt) + λ µ2 e−µt sin λt − λ2 µ2 ic so that ic = 1 λ2 + µ2 {µ − e−µt(µ cos λt − λ sin λt)}. on the other hand, for (6), for n ≥ 0 in n and λ ∈ r with λ 6= 0, (tn ∗ sin λt)(t) = ∫t 0 (t − τ)n sin λτdτ = ∫t 0 n∑ k=0 ( n k ) tn−k(−1)kτk sin λτdτ = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk sin λτdτ. we then compute the following integral by integration by parts: ik ≡ ∫t 0 τk sin λτdτ = −1 λ tk cos λt + k λ ∫t 0 τk−1 cos λτdτ = −1 λ tk cos λt + k λ2 tk−1 sin λt − k(k − 1) λ2 ik−2. inductively, if k = 2m with m ∈ n and m ≥ 0, then i2m = m∑ l=1 (−1)lk! λ2l−1(k − 2l + 2)! tk−2l+2 cos λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 sin λt + (−1)mk! λ2m i0 cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 113 with i0 = ∫t 0 sin λτdτ = 1 λ − 1 λ cos λt, and hence, i2m = m+1∑ l=1 (−1)lk! λ2l−1(k − 2l + 2)! tk−2l+2 cos λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 sin λt + (−1)mk! λ2m+1 . if k = 2m + 1 with m ∈ n and m ≥ 0, then i2m+1 = m∑ l=1 (−1)lk! λ2l−1(k − 2l + 2)! tk−2l+2 cos λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 sin λt + (−1)mk! λ2m i1 with i1 = ∫t 0 τ sin λτdτ = −1 λ t cosλt + 1 λ2 sin λt, and hence, i2m+1 = m+1∑ l=1 (−1)lk! λ2l−1(k − 2l + 2)! tk−2l+2 cos λt + m+1∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 sin λt. similarly, for n ≥ 0 in n and λ ∈ r with λ 6= 0, (tn ∗ cos λt)(t) = ∫t 0 (t − τ)n cos λτdτ = ∫t 0 n∑ k=0 ( n k ) tn−k(−1)kτk cos λτdτ = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk cos λτdτ. we then compute the following integral by integration by parts: jk ≡ ∫t 0 τk cos λτdτ = 1 λ tk sin λt + −k λ ∫t 0 τk−1 sin λτdτ = 1 λ tk sin λt + k λ2 tk−1 cos λt − k(k − 1) λ2 jk−2. inductively, if k = 2m with m ∈ n and m ≥ 0, then j2m = m∑ l=1 (−1)(l−1)k! λ2l−1(k − 2l + 2)! tk−2l+2 sin λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 cos λt + (−1)mk! λ2m j0 114 takahiro sudo cubo 16, 3 (2014) with j0 = ∫t 0 cos λτdτ = 1 λ sin λt, and hence, j2m = m+1∑ l=1 (−1)(l−1)k! λ2l−1(k − 2l + 2)! tk−2l+2 sin λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 cos λt. if k = 2m + 1 with m ∈ n and m ≥ 0, then j2m+1 = m∑ l=1 (−1)(l−1)k! λ2l−1(k − 2l + 2)! tk−2l+2 sin λt + m∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 cos λt + (−1)mk! λ2m j1 with j1 = ∫t 0 τ cos λτdτ = 1 λ t sin λt + 1 λ2 (cos λt − 1), and hence, j2m+1 = m+1∑ l=1 (−1)(l−1)k! λ2l−1(k − 2l + 2)! tk−2l+2 sin λt + m+1∑ l=1 (−1)(l−1)k! λ2l(k − 2l + 1)! tk−2l+1 cos λt + (−1)m+1k! λ2m+2 . finally, for (7), the convolution of general monomials is given by, as an example, (tneµ1t sin λ1t) ∗ (t meµ2t cos λ2t)(t) = ∫t 0 (t − τ)neµ1(t−τ) sin λ1(t − τ)τ meµ2τ cos λ2τdτ = eµ1t n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk+me(µ2−µ1)τ sin λ1(t − τ) cos λ2τdτ with sin λ1(t − τ) cosλ2τ = 1 2 {sin(λ1t − (λ1 − λ2)τ) + sin(λ1t − (λ1 + λ2)τ)}, and then the following integral is computed inductively as ik+m,sin ≡ ∫t 0 τk+me(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ = [ τk+m ∫ e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ ]t τ=0 − (k + m) ∫t 0 τk+m−1 (∫ e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ ) dτ cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 115 where the indefinite integral is computed by using integration by parts twice as: ∫ e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ) + λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 e(µ2−µ1)τ cos(λ1t − (λ1 ± λ2)τ) and hence, we obtain ik+m,sin = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 tk+me(µ2−µ1)t sin(∓λ2t) + λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 tk+me(µ2−µ1)t cos(∓λ2t) − (k + m)(µ2 − µ1) (µ2 − µ1) 2 + (λ1 ± λ2)2 ∫t 0 τk+m−1e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ − (k + m)(λ1 ± λ2) (µ2 − µ1) 2 + (λ1 ± λ2)2 ∫t 0 τk+m−1e(µ2−µ1)τ cos(λ1t − (λ1 ± λ2)τ)dτ, where the last two integrals are denoted by ik+m−1,sin and ik+m−1,cos respectively, and the integrals can be inductively reduced to the case of i0,sin and i0,cos that are given by i0,sin ≡ ∫t 0 e(µ2−µ1)τ sin(λ1t − (λ1 ± λ2)τ)dτ = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t sin(∓λ2t) − sin(λ1t)} + λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t cos(∓λ2t) − cos(λ1t)} and i0,cos ≡ ∫t 0 e(µ2−µ1)τ cos(λ1t − (λ1 ± λ2)τ)dτ = µ2 − µ1 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t cos(∓λ2t) − cos(λ1t)} − λ1 ± λ2 (µ2 − µ1) 2 + (λ1 ± λ2)2 {e(µ2−µ1)t sin(∓λ2t) − sin(λ1t)}. other cases of convolution products of general monomials with sin and cos changed are also computed similarly. theorem 5.3. the laplace transform l is an algebra homomorphism from a(r) with convolution product to r0(c) with point-wise multiplication. also, the inverse laplace transform l−1 is an algebra homomorphism from r0(c) to a(r). then l−1 ◦ l = ida(r) and l ◦ l −1 = idr0(c) the identity maps on a(r) and r0(c). 116 takahiro sudo cubo 16, 3 (2014) remark. note that we identify a rational function g in r0(c) with the image of f ∈ a(r) under l with domain of convergence such that l(f) = g. indeed, the infimum of the real part of the domain of convergence for l(f) = g is defined to be the maximum of the real parts of poles of g, so that the domain of convergence for l(f) is determined by g uniquely. proof. it is well known that l(f ∗ g)(s) = l(f)(s) · l(g)(s) for f, g ∈ a(r). it follows by the convolution products checked explicitly in proposition 5.2, in particular, that any element of a(r), which is a linear combination of multiples of elementary continuous functions, is a continuous function on r, so that, as also a well known fact, the laplace transform is injective on a(r) but restricted to [0, ∞), and hence the inverse laplace transform l−1 is also injective on l(a(r)). note that real coefficients in such linear combinations are determined uniquely by the injectivity, so that we may extend the definition domains from [0, ∞) to r preserving the injectivity. it is clear that l(a(r)) is contained in r0(c) by using basic formulae in laplace transform and by proposition 5.2. for instance, check that l(eµttn sin λt)(s) = l(tn sin λt)(s − µ) = (−1)nl((−t)n sin λt)(s − µ) = (−1)n dn dsn ( λ (s − µ)2 + λ2 ) ∈ r0(c). the last belonging is proved by induction. indeed, if p(s) q(s) ∈ r0(c), then d ds ( p(s) q(s) ) = p ′ (s)q(s)−p(s)q ′ (s) q(s)2 ∈ r0(c). it is also checked explicitly in theorem 4.1 that any element of r0(c) is mapped to an element of a(r) under l−1. corollary 5.4. it follows that the algebra a(r) with convolution product is isomorphic to r0(c), as an algebra. it also follows that the algebra a(r) with point-wise multiplication is isomorphic to r0(c), as a real vector space. remark. the laplace transform l (as well as the inverse l−1) is linear but dose not preserve point-wise multiplication. for instance, l(t2) = 2 s3 6= l(t) · l(t) = 1 s2 · 1 s2 = 1 s4 . received: october 2013. accepted: august 2014. references [1] hiroshi fukawa, laplace transformation and ordinary differential equations (in japanese), sho-ko-dou (1995). cubo 16, 3 (2014) computing the inverse laplace transform for rational functions . . . 117 [2] m. s. j., mathematics dictionary, (sugaku jiten, in japanese), math. soc. japan, 4th edition, iwanami (2007). [3] a. vretblad, fourier analysis and its applications, gtm 223, springer (2003). introduction inverse laplace transform for rational functions in a special case inverse laplace transform for rational functions in another special case inverse laplace transform for rational functions algebraic structure cubo a mathematical journal vol.16, no¯ 02, (121–134). june 2014 some coupled coincidence point theorems in partially ordered uniform spaces aris aghanians, kamal fallahi, kourosh nourouzi faculty of mathematics, k. n. toosi university of technology, p.o. box 16315-1618, tehran, iran. nourouzi@kntu.ac.ir donal o’regan school of mathematics, statistics and applied mathematics, national university of ireland, galway, university road, galway, ireland. abstract in this paper we investigate the existence of coupled coincidence points for some contractions in partially ordered separated uniform spaces under the mixed g-monotone property. we generalize a known result in partially ordered metric spaces to uniform spaces and give new types of contractions and results in partially ordered uniform spaces. resumen en este art́ıculo investigamos la existencia de puntos de coincidencia acoplados de algunas contracciones en espacios uniformes separados ordenados parcialmente bajo la propiedad g-monótona de mezcla. generalizamos un resultado conocido en espacios métricos ordenados parcialmente a espacios uniformes y entregamos tipos nuevos de contracciones y resultados para espacios uniformes ordenados parcialmente. keywords and phrases: separated uniform space; mixed g-monotone property; coupled coincidence point. 2010 ams mathematics subject classification: 54h25, 54e15. 122 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) 1 introduction and preliminaries in [3], gnana bhaskar and lakshmikantham investigated coupled fixed points for mappings having the mixed monotone property in metric spaces endowed with a partial order and they applied their coupled fixed point results to periodic boundary value problems. lakshmikantham and ćirić [4] generalized the results in [3] by considering coupled coincidence points and mappings having the mixed g-monotone property. using compatible mappings in partially ordered metric spaces, choudhury and kundu [2] extended the coupled fixed point results in [3]. in this paper, we aim to give a new generalization of a fixed point result in [3] to partially ordered uniform spaces. also, some new results on coupled coincidence points are presented. we first start by recalling some notions in uniform spaces. an in-depth discussion of uniformity can be found in [6]. a sequence {xn} in a uniform space (x, u) (briefly, x) is said to be convergent to a point x ∈ x, denoted by xn → x, if for each entourage u ∈ u, there exists an n > 0 such that (xn, x) ∈ u for all n ≥ n and cauchy if for each entourage u ∈ u, there exists an n > 0 such that (xm, xn) ∈ u for all m, n ≥ n. the uniform space x is called sequentially complete if each cauchy sequence in x is convergent to some point of x. a uniformity u on a set x is separating if the intersection of all entourages in u is equal to the diagonal {(x, x) : x ∈ x}. in this case, x is is called a separated uniform space. for any pseudometric ρ on x and any r > 0, we set v(ρ, r) = { (x, y) ∈ x × x : ρ(x, y) < r } . let f be a family of (uniformly continuous) pseudometrics on x that generates the uniformity u (see [1], theorem 2.1). denote by v, the family of all sets of the form n ! i=1 v(ρi, ri), where, n ≥ 1 and ρi ∈ f, ri > 0 for each i. then v is a base for the uniformity u, and the elements of v are called the basic entourages of x. if v = n ! i=1 v(ρi, ri) ∈ v, then αv = n ! i=1 v(ρi, αri) ∈ v, for each positive number α. recall that for any two subsets u and v of x × x, we denote by u ◦ v the set of all pairs (x, z) ∈ x × x for which (x, y) ∈ v and (y, z) ∈ u for some y ∈ x. we shall need the following lemma. for more details, the reader is referred to [1]. cubo 16, 2 (2014) some coupled coincidence point theorems in partially . . . 123 lemma 1.1. [1] let x be a uniform space. i) if v is a basic entourage of x and 0 < α ≤ β, then αv ⊆ βv. ii) if ρ is a pseudometric on x and α, β > 0, then (x, y) ∈ αv(ρ, r1) ◦ βv(ρ, r2) implies ρ(x, y) < αr1 + βr2. iii) for each x, y ∈ x and each basic entourage v of x, there exists a positive number λ such that (x, y) ∈ λv. iv) each basic entourage v of x is of the form v(ρ, 1) for some pseudometric ρ (the minkowski’s pseudometric of v) on x. definition 1. [4] let (x, ≼) be a partially ordered set and let f : x × x → x and g : x → x be two mappings. i) the mapping f is said to have the mixed g-monotone property if f is g-nondecreasing and g-nonincreasing in its first and second arguments, respectively, that is, g(x1) ≼ g(x2) =⇒ f(x1, y) ≼ f(x2, y) (x1, x2 ∈ x), and g(y1) ≼ g(y2) =⇒ f(x, y2) ≼ f(x, y1) (y1, y2 ∈ x), for all x, y ∈ x. ii) an element (x, y) ∈ x × x is called a coupled coincidence point for f and g if f(x, y) = g(x) and f(y, x) = g(y). iii) the mappings f and g are called commutative if f " g(x), g(y) # = g " f(x, y) # (x, y ∈ x). setting g = ix (the identity mapping of x) in definition 1, we get the concepts of the mixed monotone property and coupled fixed point defined in [3]. 2 main results throughout this section, we suppose that the nonempty set x is equipped with a separating uniformity u and a partial order ≼ unless otherwise stated. also, we consider a partial order ⊑ on x × x defined by (x1, y1) ⊑ (x2, y2) ⇐⇒ x1 ≼ x2 and y2 ≼ y1. 124 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) by two comparable elements (x, y) and (u, v) of x×x, we mean either (x, y) ⊑ (u, v) or (u, v) ⊑ (x, y). furthermore, we assume that f is a family of (uniformly continuous) pseudometrics on x that generates the uniformity u. we denote by v, the family of all sets of the form $n i=1 v(ρi, ri) in which for each i, ρi ∈ f, ri > 0 and n ≥ 1. we have the following lemma: lemma 2.1. the minkowski’s pseudometric ρ of a basic entourage v is jointly continuous, i.e., xn → x and yn → y imply ρ(xn, yn) → ρ(x, y). proof. let ε > 0 be given. then there exists an n > 0 such that (xn, x) ∈ ε 2 v and (yn, y) ∈ ε 2 v (n ≥ n). on the other hand, for each n ≥ 1, ρ(x, y) ≤ ρ(x, xn) + ρ(xn, yn) + ρ(yn, y). (2.1) substituting x and y with xn and yn in (2.1), respectively, and combining the obtained inequalities yield % %ρ(xn, yn) − ρ(x, y) % % ≤ ρ(xn, x) + ρ(yn, y). hence, for n ≥ n, % %ρ(xn, yn) − ρ(x, y) % % < ε 2 + ε 2 = ε. thus, ρ(xn, yn) → ρ(x, y). to present our results, we need the following concept: definition 2. a mapping g : x → x is called sequentially continuous on x if for each x ∈ x and each sequence {xn} in x converging to x, we have g(xn) → g(x). similarly, a mapping f : x×x → x is called sequentially continuous on x if xn → x and yn → y imply f(xn, yn) → f(x, y). definition 3. a partially ordered uniform space x is called upper (lower) regular if for each nondecreasing (nonincreasing) sequence {xn} in x converging to x, one has xn ≼ x (x ≼ xn) for all n ≥ 1. hereafter, by a pair (f, g) we mean mappings f : x × x → x and g : x → x such that f has the mixed g-monotone property, the range of g contains the range of f and f(x × x) or g(x) is a sequentially complete uniform subspace of x unless otherwise stated. we present some examples of such pairs. example 1. consider x = [0, +∞) with the uniformity induced from the usual metric and define a partial order ≼ by x ≼ y ⇐⇒ & x = y or x, y ∈ [0, 1] with x ≤ y ' . cubo 16, 2 (2014) some coupled coincidence point theorems in partially . . . 125 define f : x × x → x and g : x → x by f(x, y) = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ x2 − y2 3 y ≼ x 0 otherwise and g(x) = x2 for all x, y ∈ x. then it is seen that the range of g contains the range of f since g is surjective on x, and because g(x1) ≼ g(x2) implies x1 ≼ x2, it follows that f has the mixed g-monotone property. example 2. let x = {1, 2, 3} and u be the discrete uniformity on x, that is, u = p(x × x) and note that each uniform subspace of x is sequentially complete. consider the partial order ≼= { (1, 1), (2, 2), (3, 3), (1, 2) } on x and define f and g by f = { " (1, 1), 1 # , " (1, 2), 3 # , " (1, 3), 1 # , " (2, 1), 2 # , " (2, 2), 3 # , " (2, 3), 1 # , " (3, 1), 2 # , " (3, 2), 3 # , " (3, 3), 2 # } , and g = {(1, 1), (2, 3), (3, 2)}. observe that f(x × x) ⊆ g(x); furthermore, g(x1) ≼ g(x2) implies either x1 = x2 or x1 = 1 and x2 = 3, and since f(1, y) ≼ f(3, y) and f(x, 3) ≼ f(x, 1) for all x, y ∈ x, it follows that f has the mixed g-monotone property. here, (1, 1), (1, 3), (3, 1) and (3, 3) are the coupled coincidence points for f and g. example 3. let x be a sequentially complete real topological vector space and c a pointed cone in x, that is, c ∩ (−c) = {0}. it is well-known that the topology of a topological vector space can be derived by a unique uniformity, i.e., every topological vector space is “uniformizable” in a unique way (for the details, see [5]). consider x with this uniformity and partial order ≼ on x induced by c as x ≼ y ⇐⇒ y − x ∈ c. define mappings f : x × x → x and g : x → x by f(x, y) = x − y, and g(x) = { x x ∈ c 2x x /∈ c for all x, y ∈ x. then using the properties of a cone, it is easy to check that g is surjective on x. to see that f has the mixed g-monotone property, note that g(x1) ≼ g(x2) implies x1 ≼ x2. therefore, if g(x1) ≼ g(x2), then f(x1, y) = x1 − y ≼ x2 − y = f(x2, y) (y ∈ x). similarly, from g(y1) ≼ g(y2) we get f(x, y2) ≼ f(x, y1) for all x ∈ x. in this example, the coupled coincidence points for f and g are (0, 0) and all the pairs (x, −x) with x, −x /∈ c. 126 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) theorem 2.1. suppose that the pair (f, g) satisfies the following conditions: i) there exist α, β > 0 with α + β < 1 such that " f(x, y), f(u, v) # ∈ αv1 ◦ βv2 (2.2) if v1, v2 ∈ v, (g(x), g(u)) ∈ v1, (g(y), g(v)) ∈ v2, and the pairs (g(x), g(y)) and (g(u), g(v)) are comparable, where x, y, u, v ∈ x; ii) there exist x0, y0 ∈ x such that g(x0) ≼ f(x0, y0) and f(y0, x0) ≼ g(y0). then f and g have a coupled coincidence point if one of the following statements holds: (∗) f and g are commutative and sequentially continuous on x; (∗∗) g(x) is upper and lower regular. proof. since f(x × x) ⊆ g(x), there exist x1, y1 ∈ x such that g(x1) = f(x0, y0) and g(y1) = f(y0, x0). we can also choose x2, y2 ∈ x such that g(x2) = f(x1, y1) and g(y2) = f(y1, x1). continuing this process, we get sequences {xn} and {yn} in x such that g(xn+1) = f(xn, yn) and g(yn+1) = f(yn, xn) (n ≥ 0). by induction, we now see that {g(xn)} and {g(yn)} are nondecreasing and nonincreasing sequences in g(x), respectively. in fact, g(x0) ≼ f(x0, y0) = g(x1) and g(y1) ≼ g(y0). if g(xn−1) ≼ g(xn) and g(yn) ≼ g(yn−1) for n ≥ 1, since f has the mixed g-monotone property, then g(xn) = f(xn−1, yn−1) ≼ f(xn, yn−1) ≼ f(xn, yn) = g(xn+1). similarly, g(yn+1) ≼ g(yn). now, let v ∈ v and suppose that ρ is the minkowski’s pseudometric of v. for given comparable elements (g(x), g(y)) and (g(u), g(v)) of x × x, where x, y, u, v ∈ x, write r1 = ρ(g(x), g(u)) and r2 = ρ(g(y), g(v)) and take ε > 0. then " g(x), g(u) # ∈ (r1 + ε)v and " g(y), g(v) # ∈ (r2 + ε)v, and, therefore, by (2.2), we have " f(x, y), f(u, v) # ∈ α(r1 + ε)v ◦ β(r2 + ε)v. from lemma 1.1 we have ρ " f(x, y), f(u, v) # < α(r1 + ε) + β(r2 + ε) = αr1 + βr2 + (α + β)ε. since ε > 0 was arbitrary, it follows that ρ " f(x, y), f(u, v) # ≤ αρ " g(x), g(u) # + βρ " g(y), g(v) # . (2.3) cubo 16, 2 (2014) some coupled coincidence point theorems in partially . . . 127 next, by lemma 1.1, let λ > 0 be such that " g(x1), g(x0) # , " g(y1), g(y0) # ∈ λv. because (g(xn), g(yn)) and (g(xn−1), g(yn−1)) are comparable, by (2.3), we have ρ " g(xn+1), g(xn) # = ρ " f(xn, yn), f(xn−1, yn−1) # ≤ αρ " g(xn), g(xn−1) # + βρ " g(yn), g(yn−1) # , (2.4) and similarly, ρ " g(yn+1), g(yn) # = ρ " f(yn, xn), f(yn−1, xn−1) # ≤ αρ " g(yn), g(yn−1) # + βρ " g(xn), g(xn−1) # . (2.5) therefore, setting ρn = ρ " g(xn+1), g(xn) # + ρ " g(yn+1), g(yn) # n = 0, 1, . . . , from (2.4) and (2.5) we obtain ρn = ρ " g(xn+1), g(xn) # + ρ " g(yn+1), g(yn) # ≤ (α + β) & ρ " g(xn), g(xn−1) # + ρ " g(yn), g(yn−1) # ' = δρn−1, where δ = α + β < 1. thus, by induction, the inequality ρn ≤ δ nρ0 holds for all n ≥ 0. hence, for sufficiently large m and n with m > n, we have ρ " g(xm), g(xn) # + ρ " g(ym), g(yn) # ≤ m∑ k=n+1 ( ρ " g(xk), g(xk−1) # + ρ " g(yk), g(yk−1) # ) = ρm−1 + · · · + ρn ≤ (δm−1 + · · · + δn)ρ0 < δn 1 − δ 2λ < 1, that is, " g(xm), g(xn) # , " g(ym), g(yn) # ∈ v. consequently, {g(xn)} and {g(yn)} are cauchy sequences in g(x), and so there exist x, y ∈ x such that g(xn) → g(x) and g(yn) → g(y). to see the existence of a coupled coincidence point for f and g, suppose first that (∗) holds. since x is separated, g2(xn+1) → g 2(x), with g2(xn+1) = g " f(xn, yn) # = f " g(xn), g(yn) # → f " g(x), g(y) # , 128 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) implies that g2(x) = f(g(x), g(y)). similarly, g2(y) = f(g(y), g(x)), that is, (g(x), g(y)) is a coupled coincidence point for f and g. on the other hand, if (∗∗) holds, then g(xn) ≼ g(x) and g(y) ≼ g(yn), for all n ≥ 0. thus, (g(x), g(y)) is comparable to each (g(xn), g(yn)). if v ∈ v and ρ is the minkowski’s pseudometric of v, then by (2.3) and lemma 2.1, for sufficiently large n we have ρ " f(x, y), g(x) # ≤ ρ " f(x, y), g(xn+1) # + ρ " g(xn+1), g(x) # = ρ " f(x, y), f(xn, yn) # + ρ " g(xn+1), g(x) # ≤ αρ " g(x), g(xn) # + βρ " g(y), g(yn) # + ρ " g(xn+1), g(x) # < 1, that is, (f(x, y), g(x)) ∈ v. since v is arbitrary and x is separated, we get f(x, y) = g(x). similarly, f(y, x) = g(y) and so, in this case, (x, y) is a coupled coincidence point for f and g. example 4. let x be a nonzero real vector space and c be a pointed cone in x. consider two arbitrary complete norms ∥ · ∥1 and ∥ · ∥2 on x and define ρ1 " (x1, x2), (y1, y2) # = ∥x1 − y1∥1, and ρ2 " (x1, x2), (y1, y2) # = ∥x2 − y2∥2 for all (x1, x2), (y1, y2) ∈ x 2 = x×x. it is easy to verify that the uniformity u on x2 generated by the two pseudometrics ρ1 and ρ2 is separating and sequentially complete. define a partial order ≼ on x2 by (x1, x2) ≼ (y1, y2) ⇐⇒ y1 − x1, x2 − y2 ∈ c " (x1, x2), (y1, y2) ∈ x 2 # . since the family f = {ρ1, ρ2}, which generates the uniformity u has finitely many elements, it follows that two mappings f : x2 × x2 → x2 and g : x2 → x2 defined by f " (x1, x2), (y1, y2) # = &1 3 (x1 − y1), 1 4 (x2 − y2) ' , and g " (x1, x2) # = (3x1, 2x2) for all (x1, x2), (y1, y2) ∈ x 2 satisfy (2.2) since they satisfy the contractive condition ρi & f " (x1, x2), (y1, y2) # , f " (u1, u2), (v1, v2) # ' ≤ 1 4 ρi & g " (x1, x2) # , g " (u1, u2) # ' + 1 4 ρi & g " (y1, y2) # , g " (v1, v2) # ' for all (x1, x2), (y1, y2), (u1, u2), (v1, v2) ∈ x 2 such that the pairs (g((x1, x2)), g((y1, y2))) and (g((u1, u2)), g((v1, v2))) are comparable, and i = 1, 2. moreover, f and g commute and are cubo 16, 2 (2014) some coupled coincidence point theorems in partially . . . 129 sequentially continuous on x2, the mapping f has the mixed g-monotone property and f(x2×x2) ⊆ g(x2) = x2. therefore, setting x0 = (−2x ∗, x∗) and y0 = (x ∗, −2x∗) where x∗ ∈ c, we see that the hypotheses of theorem 2.1 are fulfilled and hence f and g have a coupled coincidence point, namely (0, 0). setting g = ix in theorem 2.1, the following generalization of the gnana bhaskar and lakshmikantham’s result [3] to partially ordered uniform spaces is obtained. corolary 1. suppose that x is sequentially complete and a mapping f : x × x → x satisfies the following conditions: i) f has the mixed monotone property; ii) there exist α, β > 0 with α + β < 1 such that " f(x, y), f(u, v) # ∈ αv1 ◦ βv2 if v1, v2 ∈ v, (x, u) ∈ v1, (y, v) ∈ v2, and the pairs (x, y) and (u, v) are comparable, where x, y, u, v ∈ x; iii) there exist x0, y0 ∈ x such that x0 ≼ f(x0, y0) and f(y0, x0) ≼ y0. then f has a coupled fixed point if one of the following statements holds: a) f is sequentially continuous on x; b) x is upper and lower regular. remark 1. in addition to the hypotheses of theorem 2.1, suppose that g(x0) ≼ g(y0). suppose further that x and y are as in the proof of theorem 2.1. then g(x) = g(y). to see this, we first show that g(xn) ≼ g(yn) for all n ≥ 0. if g(xn) ≼ g(yn) for n ≥ 1, since f has the mixed g-monotone property, it follows that g(xn+1) = f(xn, yn) ≼ f(yn, yn) ≼ f(yn, xn) = g(yn+1). thus, by induction, g(xn) ≼ g(yn) for all n ≥ 0. now, let v ∈ v and ρ be the minkowski’s pseudometric of v. since (g(xn), g(yn)) and (g(yn), g(xn)) are comparable, by (2.3), we have ρ " g(x), g(y) # ≤ ρ " g(x), g(xn+1) # + ρ " g(xn+1), g(yn+1) # + ρ " g(yn+1), g(y) # = ρ " g(x), g(xn+1) # + ρ " f(xn, yn), f(yn, xn) # + ρ " g(yn+1), g(y) # ≤ ρ " g(x), g(xn+1) # + αρ " g(xn), g(yn) # + βρ " g(yn), g(xn) # + ρ " g(yn+1), g(y) # 130 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) = ρ " g(x), g(xn+1) # + δρ " g(xn), g(yn) # + ρ " g(yn+1), g(y) # ≤ ρ " g(x), g(xn+1) # + δρ " g(xn), g(x) # + δρ " g(x), g(y) # + δρ " g(y), g(yn) # + ρ " g(yn+1), g(y) # , where δ = α + β < 1. hence, the joint continuity of the minkowski’s pseudometrics yields ρ " g(x), g(y) # ≤ 1 1 − δ ρ " g(x), g(xn+1) # + δ 1 − δ ρ " g(xn), g(x) # + δ 1 − δ ρ " g(y), g(yn) # + 1 1 − δ ρ " g(yn+1), g(y) # < 1, for sufficiently large n, that is, (g(x), g(y)) ∈ v. since v is arbitrary and x is separated, we get g(x) = g(y). in particular, if g is injective, then f(x, x) = g(x). we next present two coupled coincidence point theorems for two different types of contractions in partially ordered uniform spaces. theorem 2.2. suppose that a pair (f, g) satisfies the following conditions: i) there exist positive numbers α and β with α+β < 1 such that for all v1, v2 ∈ v, if (f(x, y), g(x)) ∈ v1, (f(u, v), g(u)) ∈ v2, and (g(x), g(y)) and (g(u), g(v)) are comparable, then " f(x, y), f(u, v) # ∈ αv1 ◦ βv2, (2.6) where x, y, u, v ∈ x; ii) there exist x0, y0 ∈ x such that g(x0) ≼ f(x0, y0) and f(y0, x0) ≼ g(y0). then f and g have a coupled coincidence point if (∗) or (∗∗) holds. proof. consider the sequences {xn} and {yn} with initial points x0 and y0 constructed in the proof of theorem 2.1. let v ∈ v and suppose that ρ is the minkowski’s pseudometric of v. for given comparable elements (g(x), g(y)) and (g(u), g(v)) of x × x, where x, y, u, v ∈ x, write r1 = ρ(f(x, y), g(x)) and r2 = ρ(f(u, v), g(u)) and take ε > 0. then " f(x, y), g(x) # ∈ (r1 + ε)v and " f(u, v), g(u) # ∈ (r2 + ε)v. therefore, by (2.6), " f(x, y), f(u, v) # ∈ α(r1 + ε)v ◦ β(r2 + ε)v. by lemma 1.1, we have ρ " f(x, y), f(u, v) # < α(r1 + ε) + β(r2 + ε) = αr1 + βr2 + (α + β)ε. since ε > 0 was arbitrary, it follows that ρ " f(x, y), f(u, v) # ≤ αρ " f(x, y), g(x) # + βρ " f(u, v), g(u) # . (2.7) cubo 16, 2 (2014) some coupled coincidence point theorems in partially . . . 131 next, by lemma 1.1, choose a λ > 0 such that (g(x1), g(x0)) ∈ λv. because (g(xn), g(yn)) and (g(xn−1), g(yn−1)) are comparable, by (2.7), we have ρ " g(xn+1), g(xn) # = ρ " f(xn, yn), f(xn−1, yn−1) # ≤ αρ " f(xn, yn), g(xn) # + βρ " f(xn−1, yn−1), g(xn−1) # = αρ " g(xn+1), g(xn) # + βρ " g(xn), g(xn−1) # . thus, for each n ≥ 1, the inequality ρ " g(xn+1), g(xn) # ≤ δρ " g(xn), g(xn−1) # holds, where δ = β 1−α . clearly, 0 < δ < 1 and, by induction, we have ρ " g(xn+1), g(xn) # ≤ δnρ " g(x1), g(x0) # (n ≥ 0). hence, for sufficiently large m and n with m > n we have ρ " g(xm), g(xn) # ≤ ρ " g(xm), g(xm−1) # + · · · + ρ " g(xn+1), g(xn) # ≤ δm−1ρ " g(x1), g(x0) # + · · · + δnρ " g(x1), g(x0) # < δn 1 − δ λ < 1, that is, (g(xm), g(xn)) ∈ v. therefore, {g(xn)} is a cauchy sequence in g(x). similarly, {g(yn)} is cauchy, and so there exist x, y ∈ x such that g(xn) → g(x) and g(yn) → g(y). now, if (∗) holds, then an argument similar to that in the proof of theorem 2.1 establishes that (g(x), g(y)) is a coupled coincidence point if f and g. if (∗∗) holds, then g(xn) ≼ g(x) and g(y) ≼ g(yn), for all n ≥ 1. thus, (g(x), g(y)) is comparable to each (g(xn), g(yn)). now, suppose v ∈ v and ρ is the minkowski’s pseudometric of v. then, by (2.7), for each n ≥ 0 we have ρ " f(x, y), g(x) # ≤ ρ " f(x, y), g(xn+1) # + ρ " g(xn+1), g(x) # = ρ " f(x, y), f(xn, yn) # + ρ " g(xn+1), g(x) # ≤ αρ " f(x, y), g(x) # + βρ " f(xn, yn), g(xn) # + ρ " g(xn+1), g(x) # = αρ " f(x, y), g(x) # + βρ " g(xn+1), g(xn) # + ρ " g(xn+1), g(x) # . since the minkowoski’s pseudometrics are jointly continuous, hence for sufficiently large n we obtain ρ " f(x, y), g(x) # ≤ β 1 − α ρ " g(xn+1), g(xn) # + 1 1 − α ρ " g(xn+1), g(x) # < 1, that is, (f(x, y), g(x)) ∈ v. since v is arbitrary and x is separated, we get f(x, y) = g(x). similarly, f(y, x) = g(y) and so, (x, y) is a coupled coincidence point for f and g. 132 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) we easily get the following consequence of theorem 2.2 in partially ordered metric spaces: corolary 2. let (x, ≼) be a partially ordered set and d be a metric on x. suppose that the pair (f, g) satisfies the following conditions: i) there exist α, β > 0 with α + β < 1 such that " f(x, y), f(u, v) # ∈ αv(d, r1) ◦ βv(d, r2) if r1, r2 > 0, d(f(x, y), g(x)) < r1, d(f(u, v), g(u)) < r2, and the pairs (g(x), g(y)) and (g(u), g(v)) are comparable, where x, y, u, v ∈ x; ii) there exist x0, y0 ∈ x such that g(x0) ≼ f(x0, y0) and f(y0, x0) ≼ g(y0). then f and g have a coupled coincidence point if (∗) or (∗∗) holds. theorem 2.3. suppose that a pair (f, g) satisfies the following conditions: i) there exist positive numbers α and β with α+β < 1 such that for all v1, v2 ∈ v, if (f(x, y), g(u)) ∈ v1, (f(u, v), g(x)) ∈ v2, and (g(x), g(y)) and (g(u), g(v)) are comparable, then " f(x, y), f(u, v) # ∈ αv1 ◦ βv2, (2.8) where x, y, u, v ∈ x; ii) there exist x0, y0 ∈ x such that g(x0) ≼ f(x0, y0) and f(y0, x0) ≼ g(y0). then f and g have a coupled coincidence point if (∗) or (∗∗) holds. proof. again, we construct the sequences {xn} and {yn} with initial points x0 and y0 as in the proof of theorem 2.1. since α+β < 1, without loss of generality, we assume that α < 1 2 . let v ∈ v and suppose that ρ is the minkowski’s pseudometric of v. for given comparable elements (g(x), g(y)) and (g(u), g(v)) of x × x, where x, y, u, v ∈ x, write r1 = ρ(f(x, y), g(u)) and r2 = ρ(f(u, v), g(x)) and take ε > 0. then " f(x, y), g(u) # ∈ (r1 + ε)v and " f(u, v), g(x) # ∈ (r2 + ε)v. therefore, by (2.8), " f(x, y), f(u, v) # ∈ α(r1 + ε)v ◦ β(r2 + ε)v. by lemma 1.1, we have ρ " f(x, y), f(u, v) # < α(r1 + ε) + β(r2 + ε) = αr1 + βr2 + (α + β)ε. since ε > 0 was arbitrary, it follows that ρ " f(x, y), f(u, v) # ≤ αρ " f(x, y), g(u) # + βρ " f(u, v), g(x) # . (2.9) cubo 16, 2 (2014) some coupled coincidence point theorems in partially . . . 133 now, by lemma 1.1, let λ > 0 be such that (g(x1), g(x0)) ∈ λv. because (g(xn), g(yn)) and (g(xn−1), g(yn−1)) are comparable, by (2.9), we have ρ " g(xn+1), g(xn) # = ρ " f(xn, yn), f(xn−1, yn−1) # ≤ αρ " f(xn, yn), g(xn−1) # + βρ " f(xn−1, yn−1), g(xn) # ≤ αρ " f(xn, yn), g(xn) # + αρ " g(xn), g(xn−1) # = αρ " g(xn+1), g(xn) # + αρ " g(xn), g(xn−1) # . thus, for each n ≥ 1, the inequality ρ " g(xn+1), g(xn) # ≤ δρ " g(xn), g(xn−1) # holds, where δ = α 1−α . since α < 1 2 , hence 0 < δ < 1 and, by induction, we have ρ " g(xn+1), g(xn) # ≤ δnρ " g(x1), g(x0) # (n ≥ 0). therefore, for sufficiently large m and n with m > n we have ρ " g(xm), g(xn) # ≤ ρ " g(xm), g(xm−1) # + · · · + ρ " g(xn+1), g(xn) # ≤ δm−1ρ " g(x1), g(x0) # + · · · + δnρ " g(x1), g(x0) # < δn 1 − δ λ < 1, that is, (g(xm), g(xn)) ∈ v. consequently, {g(xn)} is a cauchy sequence in g(x). similarly, {g(yn)} is cauchy, and so there exist x, y ∈ x such that g(xn) → g(x) and g(yn) → g(y). if (∗) holds, then an argument similar to that in the proof of theorem 2.1 establishes that (g(x), g(y)) is a coupled coincidence point if f and g. if (∗∗) holds, then g(xn) ≼ g(x) and g(y) ≼ g(yn), for all n ≥ 0. thus, (g(x), g(y)) is comparable to each (g(xn), g(yn)). if v ∈ v and ρ is the minkowski’s pseudometric of v, then by (2.9), for each n ≥ 1 we have ρ " f(x, y), g(x) # ≤ ρ " f(x, y), g(xn+1) # + ρ " g(xn+1), g(x) # = ρ " f(x, y), f(xn, yn) # + ρ " g(xn+1), g(x) # ≤ αρ " f(x, y), g(xn) # + βρ " f(xn, yn), g(x) # + ρ " g(xn+1), g(x) # ≤ αρ " f(x, y), g(x) # + αρ " g(x), g(xn) # + (β + 1)ρ " g(xn+1), g(x) # . since the minkowoski’s pseudometrics are jointly continuous, hence for sufficiently large n we get ρ " f(x, y), g(x) # ≤ α 1 − α ρ " g(x), g(xn) # + β + 1 1 − α ρ " g(xn+1), g(x) # < 1, that is, (f(x, y), g(x)) ∈ v. since v is arbitrary and x is separated, we have f(x, y) = g(x). similarly, f(y, x) = g(y) and so, (x, y) is a coupled coincidence point for f and g. 134 aris aghanians, kamal fallahi, kourosh nourouzi & donal o’regan cubo 16, 2 (2014) corolary 3. let (x, ≼) be a partially ordered set and d be a metric on x. suppose that the pair (f, g) satisfies the following conditions: i) there exist α, β > 0 with α + β < 1 such that " f(x, y), f(u, v) # ∈ αv(d, r1) ◦ βv(d, r2) if r1, r2 > 0, d(f(x, y), g(u)) < r1, d(f(u, v), g(x)) < r2, and the pairs (g(x), g(y)) and (g(u), g(v)) are comparable, where x, y, u, v ∈ x; ii) there exist x0, y0 ∈ x such that g(x0) ≼ f(x0, y0) and f(y0, x0) ≼ g(y0). then f and g have a coupled coincidence point if (∗) or (∗∗) holds. received: december 2013. revised: april 2014. references [1] s. p. acharya, some results on fixed points in uniform spaces, yokohama math. j. 22 (1974) 105-116. [2] b. s. choudhury, a. kundu, a coupled coincidence point result in partially ordered metric spaces for compatible mappings, nonlinear anal. 73 (2010) 2524-2531. [3] t. gnana bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal. 65 (2006) 1379-1393. [4] v. lakshmikantham, l. ćirić, coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, nonlinear anal. 70 (2009) 4341-4349. [5] h. h. schaefer, topological vector spaces, springer-verlag, new york-berlin, 1971. [6] s. willard, general topology, addison-wesley publishing co. reading, mass.-london-don mills, ont., 1970. cubo a mathematical journal vol.14, no¯ 03, (71–83). october 2012 fractional voronovskaya type asymptotic expansions for quasi-interpolation neural network operators george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a. email: ganastss@memphis.edu abstract here we study further the quasi-interpolation of sigmoidal and hyperbolic tangent types neural network operators of one hidden layer. based on fractional calculus theory we derive fractional voronovskaya type asymptotic expansions for the error of approximation of these operators to the unit operator. resumen estudiamos la cuasi-interpolación de los operadores de redes neuronales de tipo tangencial hiperbólico y sigmoidal de una capa oculta. basados en la teoŕıa del cálculo fraccional, obtenemos expansiones asintóticas del tipo voronovskaya para el error en la aproximación de estos operadores hacia el operador unitario. keywords and phrases: neural network fractional approximation, voronovskaya asymptotic expansion, fractional derivative. 2010 ams mathematics subject classification: 26a33, 41a25, 41a36, 41a60. 72 george a. anastassiou cubo 14, 3 (2012) 1 background we need definition 1. let ν > 0, n = ⌈ν⌉ (⌈·⌉ is the ceiling of the number), f ∈ acn ([a, b]) (space of functions f with f(n−1) ∈ ac ([a, b]), absolutely continuous functions). we call left caputo fractional derivative (see [13], pp. 49-52) the function dν∗af (x) = 1 γ (n − ν) ∫x a (x − t) n−ν−1 f(n) (t) dt, (1) ∀ x ∈ [a, b], where γ is the gamma function γ (ν) = ∫∞ 0 e−ttν−1dt, ν > 0. notice dν∗af ∈ l1 ([a, b]) and dν∗af exists a.e.on [a, b]. we set d0∗af (x) = f (x), ∀ x ∈ [a, b] . definition 2. (see also [3], [14], [15]). let f ∈ acm ([a, b]), m = ⌈α⌉, α > 0. the right caputo fractional derivative of order α > 0 is given by dαb−f (x) = (−1) m γ (m − α) ∫b x (ζ − x) m−α−1 f(m) (ζ) dζ, (2) ∀ x ∈ [a, b]. we set d0b−f (x) = f (x) . notice d α b−f ∈ l1 ([a, b]) and d α b−f exists a.e.on [a, b] . convention 3. we assume that dα∗x0f (x) = 0, for x < x0, (3) and dαx0−f (x) = 0, for x > x0, (4) for all x, x0 ∈ [a, b] . we mention proposition 4. (by [5]) let f ∈ cn ([a, b]), n = ⌈ν⌉, ν > 0. then dν∗af (x) is continuous in x ∈ [a, b] . also we have proposition 5. (by [5]) let f ∈ cm ([a, b]), m = ⌈α⌉, α > 0. then dαb−f (x) is continuous in x ∈ [a, b] . theorem 6. ([5]) let f ∈ cm ([a, b]), m = ⌈α⌉, α > 0, x, x0 ∈ [a, b]. then d α ∗x0 f (x), dαx0−f (x) are jointly continuous functions in (x, x0) from [a, b] 2 into r. we mention the left caputo fractional taylor formula with integral remainder. theorem 7. ([13], p. 54) let f ∈ acm ([a, b]), [a, b] ⊂ r, m = ⌈α⌉, α > 0. then f (x) = m−1∑ k=0 f(k) (x0) k! (x − x0) k + 1 γ (α) ∫x x0 (x − j) α−1 dα∗x0f (j) dj, (5) ∀ x ≥ x0; x, x0 ∈ [a, b] . cubo 14, 3 (2012) fractional voronovskaya type asymptotic ... 73 also we mention the right caputo fractional taylor formula. theorem 8. ([3]) let f ∈ acm ([a, b]), [a, b] ⊂ r, m = ⌈α⌉, α > 0. then f (x) = m−1∑ j=0 f(k) (x0) k! (x − x0) k + 1 γ (α) ∫x0 x (j − x) α−1 dαx0−f (j) dj, (6) ∀ x ≤ x0; x, x0 ∈ [a, b] . for more on fractional calculus related to this work see [2], [4] and [7]. we consider here the sigmoidal function of logarithmic type s (x) = 1 1 + e−x , x ∈ r. it has the properties lim x→+∞ s (x) = 1 and lim x→−∞ s (x) = 0. this function plays the role of an activation function in the hidden layer of neural networks. as in [12], we consider φ (x) := 1 2 (s (x + 1) − s (x − 1)) , x ∈ r. (7) we notice the following properties: i) φ (x) > 0, ∀ x ∈ r, ii) ∑∞ k=−∞ φ (x − k) = 1, ∀ x ∈ r, iii) ∑∞ k=−∞ φ (nx − k) = 1, ∀ x ∈ r; n ∈ n, iv) ∫∞ −∞ φ (x) dx = 1, v) φ is a density function, vi) φ is even: φ (−x) = φ (x), x ≥ 0. we see that ([12]) φ (x) = ( e2 − 1 2e ) e−x (1 + e−x−1) (1 + e−x+1) = 8 (1.1) ( e2 − 1 2e2 ) 1 (1 + ex−1) (1 + e−x−1) . vii) by [12] φ is decreasing on r+, and increasing on r−. 74 george a. anastassiou cubo 14, 3 (2012) viii) by [11] for n ∈ n, 0 < β < 1, we get ∞∑    k = −∞ : |nx − k| > n1−β φ (nx − k) < ( e2 − 1 2 ) e−n (1−β) = 3.1992e−n (1−β) . (9) denote by ⌊·⌋ the integral part of a number. consider x ∈ [a, b] ⊂ r and n ∈ n such that ⌈na⌉ ≤ ⌊nb⌋. ix) by [11] it holds 1 ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) < 1 φ (1) = 5.250312578, ∀ x ∈ [a, b] . (10) x) by [11] it holds lim n→∞ ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) 6= 1, for at least some x ∈ [a, b]. let f ∈ c ([a, b]) and n ∈ n such that ⌈na⌉ ≤ ⌊nb⌋. we study further (see also [11]) the quasi-interpolation positive linear neural network operator gn (f, x) := ∑⌊nb⌋ k=⌈na⌉ f ( k n ) φ (nx − k) ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) , x ∈ [a, b] . (11) for large enough n we always obtain ⌈na⌉ ≤ ⌊nb⌋. also a ≤ k n ≤ b, iff ⌈na⌉ ≤ k ≤ ⌊nb⌋. we also consider here the hyperbolic tangent function tanh x, x ∈ r : tanh x := ex − e−x ex + e−x = e2x − 1 e2x + 1 . it has the properties tanh 0 = 0, −1 < tanh x < 1, ∀ x ∈ r, and tanh (−x) = − tanh x. furthermore tanh x → 1 as x → ∞, and tanh x → −1, as x → −∞, and it is strictly increasing on r. furthermore it holds d dx tanh x = 1 cosh2 x > 0. this function plays also the role of an activation function in the hidden layer of neural networks. we further consider ψ (x) := 1 4 (tanh (x + 1) − tanh (x − 1)) > 0, ∀ x ∈ r. (12) we easily see thatψ (−x) = ψ (x), that is ψ is even on r. obviously ψ is differentiable, thus continuous. here we follow [8] proposition 9. ψ (x) for x ≥ 0 is strictly decreasing. cubo 14, 3 (2012) fractional voronovskaya type asymptotic ... 75 obviously ψ (x) is strictly increasing for x ≤ 0. also it holds lim x→−∞ ψ (x) = 0 = lim x→∞ ψ (x) . infact ψ has the bell shape with horizontal asymptote the x-axis. so the maximum of ψ is at zero, ψ (0) = 0.3809297. theorem 10. we have that ∑∞ i=−∞ ψ (x − i) = 1, ∀ x ∈ r. thus ∞∑ i=−∞ ψ (nx − i) = 1, ∀ n ∈ n, ∀ x ∈ r. furthermore we get: since ψ is even it holds ∑∞ i=−∞ ψ (i − x) = 1, ∀x ∈ r. hence ∑∞ i=−∞ ψ (i + x) = 1, ∀ x ∈ r, and ∑∞ i=−∞ ψ (x + i) = 1, ∀ x ∈ r. theorem 11. it holds ∫∞ −∞ ψ (x) dx = 1. so ψ (x) is a density function on r. theorem 12. let 0 < β < 1 and n ∈ n. it holds ∞∑    k = −∞ : |nx − k| ≥ n1−β ψ (nx − k) ≤ e4 · e−2n (1−β) . (13) theorem 13. let x ∈ [a, b] ⊂ r and n ∈ n so that ⌈na⌉ ≤ ⌊nb⌋. it holds 1 ∑⌊nb⌋ k=⌈na⌉ ψ (nx − k) < 4.1488766 = 1 ψ (1) . (14) also by [8], we obtain lim n→∞ ⌊nb⌋∑ k=⌈na⌉ ψ (nx − k) 6= 1, (15) for at least some x ∈ [a, b]. definition 14. let f ∈ c ([a, b]) and n ∈ n such that ⌈na⌉ ≤ ⌊nb⌋. we further study, as in [8], the quasi-interpolation positive linear neural network operator fn (f, x) := ∑⌊nb⌋ k=⌈na⌉ f ( k n ) ψ (nx − k) ∑⌊nb⌋ k=⌈na⌉ ψ (nx − k) , x ∈ [a, b] . (16) we find here fractional voronovskaya type asymptotic expansions for gn (f, x) and fn (f, x), x ∈ [a, b]. for related work on neural networks also see [1], [6], [9] and [10]. for neural networks in general see [16], [17] and [18]. 76 george a. anastassiou cubo 14, 3 (2012) 2 main results we present our first main result theorem 15. let α > 0, n ∈ n, n = ⌈α⌉, f ∈ acn ([a, b]), 0 < β < 1, x ∈ [a, b], n ∈ n large enough. assume that ∥ ∥dαx−f ∥ ∥ ∞,[a,x] , ‖dα∗xf‖∞,[x,b] ≤ m, m > 0. then gn (f, x) − f (x) = n−1∑ j=1 f(j) (x) j! gn ( (· − x) j ) (x) + o ( 1 nβ(α−ε) ) , (17) where 0 < ε ≤ α. if n = 1, the sum in (17) collapses. the last (17) implies that nβ(α−ε)  gn (f, x) − f (x) − n−1∑ j=1 f(j) (x) j! gn ( (· − x) j ) (x)   → 0, (18) as n → ∞, 0 < ε ≤ α. when n = 1, or f(j) (x) = 0, j = 1, ..., n − 1, then we derive that nβ(α−ε) [gn (f, x) − f (x)] → 0 as n → ∞, 0 < ε ≤ α. of great interest is the case of α = 1 2 . proof. from [13], p. 54; (5), we get by the left caputo fractional taylor formula that f ( k n ) = n−1∑ j=0 f(j) (x) j! ( k n − x )j + 1 γ (α) ∫ k n x ( k n − j )α−1 dα∗xf (j) dj, (19) for all x ≤ k n ≤ b. also from [3]; (6), using the right caputo fractional taylor formula we get f ( k n ) = n−1∑ j=0 f(j) (x) j! ( k n − x )j + 1 γ (α) ∫x k n ( j − k n )α−1 dαx−f (j) dj, (20) for all a ≤ k n ≤ x. we call v (x) := ⌊nb⌋∑ k=⌈na⌉ φ (nx − k) . (21) hence we have f ( k n ) φ (nx − k) v (x) = n−1∑ j=0 f(j) (x) j! φ (nx − k) v (x) ( k n − x )j + (22) cubo 14, 3 (2012) fractional voronovskaya type asymptotic ... 77 φ (nx − k) v (x) γ (α) ∫ k n x ( k n − j )α−1 dα∗xf (j) dj, all x ≤ k n ≤ b, iff ⌈nx⌉ ≤ k ≤ ⌊nb⌋, and f ( k n ) φ (nx − k) v (x) = n−1∑ j=0 f(j) (x) j! φ (nx − k) v (x) ( k n − x )j + (23) φ (nx − k) v (x) γ (α) ∫x k n ( j − k n )α−1 dαx−f (j) dj, for all a ≤ k n ≤ x, iff ⌈na⌉ ≤ k ≤ ⌊nx⌋. we have that ⌈nx⌉ ≤ ⌊nx⌋ + 1. therefore it holds ⌊nb⌋∑ k=⌊nx⌋+1 f ( k n ) φ (nx − k) v (x) = n−1∑ j=0 f(j) (x) j! ⌊nb⌋∑ k=⌊nx⌋+1 φ (nx − k) ( k n − x )j v (x) + (24) 1 γ (α)   ∑⌊nb⌋ k=⌊nx⌋+1 φ (nx − k) v (x) ∫ k n x ( k n − j )α−1 dα∗xf (j) dj   , and ⌊nx⌋∑ k=⌈na⌉ f ( k n ) φ (nx − k) v (x) = n−1∑ j=0 f(j) (x) j! ⌊nx⌋∑ k=⌈na⌉ φ (nx − k) v (x) ( k n − x )j + (25) 1 γ (α)   ⌊nx⌋∑ k=⌈na⌉ φ (nx − k) v (x) ∫x k n ( j − k n )α−1 dαx−f (j) dj   . adding the last two equalities (24) and (25) we obtain gn (f, x) = ⌊nb⌋∑ k=⌈na⌉ f ( k n ) φ (nx − k) v (x) = (26) n−1∑ j=0 f(j) (x) j! ⌊nb⌋∑ k=⌈na⌉ φ (nx − k) v (x) ( k n − x )j + 1 γ (α) v (x)    ⌊nx⌋∑ k=⌈na⌉ φ (nx − k) ∫x k n ( j − k n )α−1 dαx−f (j) dj+ ⌊nb⌋∑ k=⌊nx⌋+1 φ (nx − k) ∫ k n x ( k n − j )α−1 (dα∗xf (j)) dj    . 78 george a. anastassiou cubo 14, 3 (2012) so we have derived t (x) := gn (f, x) − f (x) − n−1∑ j=1 f(j) (x) j! gn ( (· − x) j ) (x) = θ∗n (x) , (27) where θ∗n (x) := 1 γ (α) v (x)    ⌊nx⌋∑ k=⌈na⌉ φ (nx − k) ∫x k n ( j − k n )α−1 dαx−f (j) dj + ⌊nb⌋∑ k=⌊nx⌋+1 φ (nx − k) ∫ k n x ( k n − j )α−1 dα∗xf (j) dj    . (28) we set θ∗1n (x) := 1 γ (α)   ∑⌊nx⌋ k=⌈na⌉ φ (nx − k) v (x) ∫x k n ( j − k n )α−1 dαx−f (j) dj   , (29) and θ∗2n := 1 γ (α)   ∑⌊nb⌋ k=⌊nx⌋+1 φ (nx − k) v (x) ∫ k n x ( k n − j )α−1 dα∗xf (j) dj   , (30) i.e. θ∗n (x) = θ ∗ 1n (x) + θ ∗ 2n (x) . (31) we assume b − a > 1 nβ , 0 < β < 1, which is always the case for large enough n ∈ n, that is when n > ⌈ (b − a) − 1 β ⌉ . it is always true that either ∣ ∣ k n − x ∣ ∣ ≤ 1 nβ or ∣ ∣ k n − x ∣ ∣ > 1 nβ . for k = ⌈na⌉ , ..., ⌊nx⌋, we consider γ1k := ∣ ∣ ∣ ∣ ∣ ∫x k n ( j − k n )α−1 dαx−f (j) dj ∣ ∣ ∣ ∣ ∣ ≤ (32) ∫x k n ( j − k n )α−1 ∣ ∣dαx−f (j) ∣ ∣ dj ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] ( x − κ n )α α ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] (x − a) α α . (33) that is γ1k ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] (x − a) α α , (34) for k = ⌈na⌉ , ..., ⌊nx⌋ . also we have in case of ∣ ∣ k n − x ∣ ∣ ≤ 1 nβ that γ1k ≤ ∫x k n ( j − k n )α−1 ∣ ∣dαx−f (j) ∣ ∣ dj (35) cubo 14, 3 (2012) fractional voronovskaya type asymptotic ... 79 ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] ( x − κ n )α α ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] 1 nαβα . so that, when ( x − k n ) ≤ 1 nβ , we get γ1k ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] 1 αnaβ . (36) therefore |θ∗1n (x)| ≤ 1 γ (α)   ∑⌊nx⌋ k=⌈na⌉ φ (nx − k) v (x) γ1k   = 1 γ (α) ·    ∑⌊nx⌋    k = ⌈na⌉ : ∣ ∣ k n − x ∣ ∣ ≤ 1 nβ φ (nx − k) v (x) γ1k + ∑⌊nx⌋    k = ⌈na⌉ : ∣ ∣ k n − x ∣ ∣ > 1 nβ φ (nx − k) v (x) γ1k    ≤ 1 γ (α)             ∑⌊nx⌋    k = ⌈na⌉ : ∣ ∣ k n − x ∣ ∣ ≤ 1 nβ φ (nx − k) v (x)          ∥ ∥dαx−f ∥ ∥ ∞,[a,x] 1 αnαβ + 1 v (x)           ⌊nx⌋∑    k = ⌈na⌉ : ∣ ∣ k n − x ∣ ∣ > 1 nβ φ (nx − k)           ∥ ∥dαx−f ∥ ∥ ∞,[a,x] (x − a) α α    (by (9), (10)) ≤ (37) ∥ ∥dαx−f ∥ ∥ ∞,[a,x] γ (α + 1) { 1 nαβ + (5.250312578) (3.1992) e−n (1−β) (x − a) α } . therefore we proved |θ∗1n (x)| ≤ ∥ ∥dαx−f ∥ ∥ ∞,[a,x] γ (α + 1) { 1 nαβ + (16.7968) e−n (1−β) (x − a) α } . (38) but for large enough n ∈ n we get |θ∗1n (x)| ≤ 2 ∥ ∥dαx−f ∥ ∥ ∞,[a,x] γ (α + 1) nαβ . (39) 80 george a. anastassiou cubo 14, 3 (2012) similarly we have γ2k := ∣ ∣ ∣ ∣ ∣ ∫ k n x ( k n − j )α−1 dα∗xf (j) dj ∣ ∣ ∣ ∣ ∣ ≤ ∫ k n x ( k n − j )α−1 |dα∗xf (j)| dj ≤ ‖dα∗xf‖∞,[x,b] ( k n − x )α α ≤ ‖dα∗xf‖∞,[x,b] (b − x) α α . (40) that is γ2k ≤ ‖d α ∗xf‖∞,[x,b] (b − x) α α , (41) for k = ⌊nx⌋ + 1, ..., ⌊nb⌋ . also we have in case of ∣ ∣ k n − x ∣ ∣ ≤ 1 nβ that γ2k ≤ ‖dα∗xf‖∞,[x,b] αnαβ . (42) consequently it holds |θ∗2n (x)| ≤ 1 γ (α)   ∑⌊nb⌋ k=⌊nx⌋+1 φ (nx − k) v (x) γ2k   = 1 γ (α)             ∑⌊nb⌋    k = ⌊nx⌋ + 1 : ∣ ∣ k n − x ∣ ∣ ≤ 1 nβ φ (nx − k) v (x)          ‖dα∗xf‖∞,[x,b] αnαβ + 1 v (x)           ⌊nb⌋∑    k = ⌊nx⌋ + 1 : ∣ ∣ k n − x ∣ ∣ > 1 nβ φ (nx − k)           ‖dα∗xf‖∞,[x,b] (b − x) α α    ≤ ‖dα∗xf‖∞,[x,b] γ (α + 1) { 1 nαβ + (16.7968) e−n (1−β) (b − x) α } . (43) that is |θ∗2n (x)| ≤ ‖dα∗xf‖∞,[x,b] γ (α + 1) { 1 nαβ + (16.7968) e−n (1−β) (b − x) α } . (44) but for large enough n ∈ n we get |θ∗2n (x)| ≤ 2 ‖dα∗xf‖∞,[x,b] γ (α + 1) nαβ . (45) cubo 14, 3 (2012) fractional voronovskaya type asymptotic ... 81 since ∥ ∥dαx−f ∥ ∥ ∞,[a,x] , ‖dα∗xf‖∞,[x,b] ≤ m, m > 0, we derive |θ∗n (x)| ≤ |θ ∗ 1n (x)| + |θ ∗ 2n (x)| (by (39), (45)) ≤ 4m γ (α + 1) nαβ . (46) that is for large enough n ∈ n we get |t (x)| = |θ∗n (x)| ≤ ( 4m γ (α + 1) ) ( 1 nαβ ) , (47) resulting to |t (x)| = o ( 1 nαβ ) , (48) and |t (x)| = o (1) . (49) and, letting 0 < ε ≤ α, we derive |t (x)| ( 1 nβ(α−ε) ) ≤ ( 4m γ (α + 1) ) ( 1 nβε ) → 0, (50) as n → ∞. i.e. |t (x)| = o ( 1 nβ(α−ε) ) , (51) proving the claim. we present our second main result theorem 16. let α > 0, n ∈ n, n = ⌈α⌉, f ∈ acn ([a, b]), 0 < β < 1, x ∈ [a, b], n ∈ n large enough. assume that ∥ ∥dαx−f ∥ ∥ ∞,[a,x] , ‖dα∗xf‖∞,[x,b] ≤ m, m > 0. then fn (f, x) − f (x) = n−1∑ j=1 f(j) (x) j! fn ( (· − x) j ) (x) + o ( 1 nβ(α−ε) ) , (52) where 0 < ε ≤ α. if n = 1, the sum in (52) collapses. the last (52) implies that nβ(α−ε)  fn (f, x) − f (x) − n−1∑ j=1 f(j) (x) j! fn ( (· − x) j ) (x)   → 0, (53) as n → ∞, 0 < ε ≤ α. when n = 1, or f(j) (x) = 0, j = 1, ..., n − 1, then we derive that nβ(α−ε) [fn (f, x) − f (x)] → 0 as n → ∞, 0 < ε ≤ α. of great interest is the case of α = 1 2 . 82 george a. anastassiou cubo 14, 3 (2012) proof. similar to theorem 15, using (13) and (14). received: december 2011. revised: may 2012. references [1] g.a. anastassiou, rate of convergence of some neural network operators to the unit-univariate case, j. math. anal. appli. 212 (1997), 237-262. 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[16] s. haykin, neural networks: a comprehensive foundation (2 ed.), prentice hall, new york, 1998. [17] w. mcculloch and w. pitts, a logical calculus of the ideas immanent in nervous activity, bulletin of mathematical biophysics, 7 (1943), 115-133. [18] t.m. mitchell, machine learning, wcb-mcgraw-hill, new york, 1997. cubo a mathematical journal vol.14, no¯ 01, (09–19). march 2012 integral composition operators between weighted bergman spaces and weighted bloch type spaces elke wolf university of paderborn, mathematical institute, d-33095 paderborn, germany, email: lichte@math.uni-paderborn.de abstract we characterize boundedness and compactness of integral composition operators acting between weighted bergman spaces av,p and weighted bloch type spaces bw. resumen caracterizamos la acotación y compacidad de operadores integrales compuestos actuando entre espacios de bergman con peso av,p y espacios bw de tipo bloch con peso. keywords and phrases: weighted bergman spaces, integral composition operator, weighted bloch type spaces 2010 ams mathematics subject classification: 47b33, 47b38. 10 elke wolf cubo 14, 1 (2012) 1 introduction let h(d) denote the set of all analytic functions on the open unit disk d of the complex plane. a map g ∈ h(d) induces the volterra type or riemann-stieltjes operator jg : h(d) → h(d), f 7→ ∫z 0 f(ξ)g′(ξ) dξ, z ∈ d. this operator appears naturally in the study of pointwise multiplication operators since with the companion integral operator ig : h(d) → h(d), f 7→ ∫z 0 f′(ξ)g(ξ) dξ, z ∈ d, we have that jgf + igf = mgf − f(0)g(0), where mg denotes the pointwise multiplication operator given by mg : h(d) → h(d), (mgf)(z) = g(z)f(z), z ∈ d. see e.g. [1], [2], [3], [17] or [21]. moreover, let v and w be strictly positive bounded and continuous functions (weights) on d. then the weighted bergman space av,p is defined as follows av,p = {f ∈ h(d); ‖f‖v,p := (∫ d |f(z)|pv(z) da(z) ) 1 p < ∞}, 1 ≤ p < ∞, where da(z) is the area measure on d normalized so that area of d is 1. furthermore, we consider the weighted bloch type spaces bw of functions f ∈ h(d) satisfying ‖f‖bw := supz∈d w(z)|f′(z)| <∞. provided we identify functions that differ by a constant, ‖.‖bw becomes a norm and bw a banach space. let φ be an analytic self-map of d. in [13] li characterized boundedness and compactness of volterra composition operators (jg,φf)(z) = ∫z 0 (f ◦ φ)(ξ)(g ◦ φ)′(ξ) dξ, z ∈ d, and the integral composition operators (ig,φf)(z) = ∫z 0 (f ◦ φ)′(ξ)(g ◦ φ)(ξ) dξ, z ∈ d, acting between weighted bergman spaces and weighted bloch type spaces, both generated by standard weights. in [19] we generalized his results related to the volterra composition operators jg,φ cubo 14, 1 (2012) integral composition operators between . . . 11 to a more general setting. in this article our aim is to characterize boundedness and compactness of the integral composition operators ig,φ acting between weighted bergman spaces and weighted bloch type spaces generated by a quite general class of weights. 2 the setting this section is devoted to the description of the setting in which we are interested. let ν be a holomorphic function on d, non-vanishing, strictly positive on [0, 1[ and satisfying limr→1 ν(r) = 0. then we define the weight v as follows v(z) := ν(|z|2) for every z ∈ d. (2.1) next, we give some illustrating examples of weights of this type: (i) consider ν(z) = (1 − z)α, α > 0. then the corresponding weight is the so-called standard weight v(z) = (1 − |z|2)α. (ii) select ν(z) = e − 1 (1−z)α , α > 0. then we obtain the weight v(z) = e − 1 (1−|z|2)α . (iii) choose ν(z) = sin(1 − z) and the corresponding weight is given by v(z) = sin(1 − |z|2). (iv) let ν(z) = (1 − log(1 − z))β for some β < 0. then we get v(z) = (1 − log(1 − |z|2))β. for a fixed point a ∈ d we introduce a function va(z) := ν(az) for every z ∈ d. since ν is holomorphic on d, so is the function va. we say that a weight v is radial if v(z) = v(|z|) for every z ∈ d. moreover, radial weights are typical if additionally lim|z|→1 v(z) = 0 holds. thus, we introduced a class of typical weights. in [15] lusky studied weights satisfying the following condition (l1) which was renamed after the author: (l1) inf n∈n v(1 − 2−n−1) v(1 − 2−n) > 0. among others examples of weights satisfying condition (l1) are the standard weights (see example (i)) and the logarithmic weights (example (iv)). throughout this work condition (l1) will play a great role, and we will need the following condition (a) which is equivalent to (l1): (a) there are 0 < r < 1 and 1 < c < ∞ with v(z) v(p) ≤ c for all p, z ∈ d with ρ(p, z) ≤ r. the equivalence of the conditions (l1) and (a) was shown in [10]. see also [14]. 12 elke wolf cubo 14, 1 (2012) 3 basic facts we need some geometric data of the open unit disk. fix a ∈ d and consider the authomorphism ϕa(z) := z−a 1−az , z ∈ d, which interchanges 0 and a. moreover, we use the fact that ϕ′a(z) = |a|2 − 1 (1 − az)2 , z ∈ d. now, the pseudohyperbolic metric is given by ρ(z, a) = |ϕa(z)|, z, a ∈ d. one of the most important properties of the pseudohyperbolic metric is that it is möbius invariant, that is, ρ(σ(z), σ(a)) = ρ(z, a) for every automorphism σ of d, z, a ∈ d. the pseudohyperbolic metric is a true metric. in fact, it even satisfies a stronger version of the triangle inequality, more precisely, for every z, a, b ∈ d we have that ρ(z, a) ≤ ρ(z, b) + ρ(b, a) 1 + ρ(z, b)ρ(b, a) . 4 results before we are able to treat boundedness and compactness of operators ig,φ we need a number of auxiliary lemmas. the first lemma is taken from [18]. lemma 1. let v be a weight as defined in (2.1) such that supa∈d supz∈d v(z)|va(ϕa(z))| v(ϕa(z)) ≤ c < ∞. then |f(z)| ≤ c 1 p v(0) 1 p (1 − |z|2) 2 p v(z) 1 p ‖f‖v,p for all z ∈ d, f ∈ av,p. calculations show that the examples (i) -(iv) which were listed up above satisfy the assumptions of the previous lemma. the next lemma was shown in [20]. lemma 2. let v be a radial weight as defined in (2.1) such that v additionally satisfies condition (l1). then for every f ∈ av,p there is cv > 0 such that |f(z) − f(w)| ≤ cv‖f‖v,p max { 1 (1 − |z|2) 2 p v(z) 1 p , 1 (1 − |w|2) 2 p v(w) 1 p } ρ(z, w) for every z, w ∈ d. cubo 14, 1 (2012) integral composition operators between . . . 13 lemma 3. let v be a radial weight as defined in (2.1) such that v additionally satisfies condition (l1) and supa∈d supz∈d v(z)|va(ϕa(z))| v(ϕa(z)) ≤ c < ∞. then |f′(z)| ≤ c 1 p v(0) 1 p (1 − |z|2) 2 p +1 v(z) 1 p ‖f‖v,p for every z ∈ d and every f ∈ av,p. proof. lemma 2 yields that for every f ∈ av,p and every h, z ∈ d with z + h ∈ d, we have |f(z + h) − f(z)| ≤ cv‖f‖v,p max { 1 (1 − |z + h|2) 2 p v(z + h) 1 p , 1 (1 − |z|2) 2 p v(z) 1 p } |h| |1 − z(z + h)| . hence∣∣∣∣f(z + h) − f(z)h ∣∣∣∣ ≤ cv‖f‖v,p max { 1 (1 − |z + h|2) 2 p v(z + h) 1 p , 1 (1 − |z|2) 2 p v(z) 1 p } 1 |1 − z(z + h)| and finally |f′(z)| = ∣∣∣∣ lim h→0 f(z + h) − f(z) h ∣∣∣∣ ≤ lim h→0 cv‖f‖v,p max { 1 (1 − |z + h|2) 2 p v(z + h) 1 p , 1 (1 − |z|2) 2 p v(z) 1 p } 1 |1 − z(z + h)| = cv‖f‖v,p 1 (1 − |z|2) 2 p +1 v(z) 1 p for every z ∈ d, as desired. lemma 4. let v be a radial weight as in lemma 3. then there exist 0 < r < 1 and a constant m > 0 such that for f ∈ av,p |f′(z) − f′(w)| ≤ 4mc 1 p v(0) 1 p ‖f‖v,p r(1 − |z|2) 2 p +1 v(z) 1 p ρ(z, w) for every z, w ∈ d with ρ(z, w) ≤ r 2 . proof. by hypotesis, v has condition (l1), and, moreover, we know that (l1) is equivalent to condition (a). since the weight u(z) = 1 − |z|2 also satisfies condition (l1), we can find 0 < r < 1 and constants m1 < ∞ and m2 < ∞ such that v(z) v(w) ≤ m1 and 1 − |z|2 1 − |w|2 ≤ m2 for every z, w ∈ d with ρ(z, w) ≤ r. let w ∈ d be fixed. since ϕw(ϕw(z)) = z and ϕw(0) = w, 14 elke wolf cubo 14, 1 (2012) we get that |f′(z) − f′(w)| = |f′(ϕw(ϕw(z)) − f ′(ϕw(ϕw(w))|. for |z| = ρ(ϕw(z), w) ≤ r we obtain by using lemma 3 |f′(ϕw(z))| ≤ c 1 p ‖f‖v,p v(0) 1 p (1 − |ϕw(z)|2) 2 p +1 v(ϕw(z)) 1 p = c 1 p ‖f‖v,p v(0) 1 p (1 − |w|2) 2 p +1 v(w) 1 p (1 − |w|2) 2 p +1 v(w) 1 p (1 − |ϕw(z)|2) 2 p +1 v(ϕw(z)) 1 p ≤ c 1 p m 1 p 1 m 2 p +1 2 v(0) 1 p ‖f‖v,p (1 − |w|2) 2 p +1 v(w) 1 p . let us now consider gw(z) := f ′(ϕw(z)) for every z ∈ d. thus, for ρ(z, w) = |ϕw(z)| ≤ r2 we can find θ ∈ d with |θ| ≤ |ϕw(z)| ≤ r2 such that |f′(z) − f′(w)| = |gw(ϕw(z)) − gw(0)| ≤ |ϕw(z)| ∣∣∣∣∣ ∫1 0 [ ∂ ∂t gw ] (tϕw(z)) dt ∣∣∣∣∣ ≤ |ϕw(z)| ∣∣∣∣ ∂∂zgw(θ) ∣∣∣∣ = |ϕw(z)| 1 2π ∣∣∣∣∣ ∫ |ξ|=r gw(ξ) (ξ − θ)2 dθ ∣∣∣∣∣ finally, |f′(z) − f′(w)| ≤ c 1 p m 1 p 1 m 2 p +1 2 v(0) 1 p |ϕw(z)|r‖f‖v,p (r − |ϕw(z)|)2(1 − |w|2) 2 p +1 v(w) 1 p ≤ 4c 1 p m 1 p 1 m 2 p +1 2 v(0) 1 p ρ(z, w)‖f‖v,p r(1 − |w|2) 2 p +1 v(w) 1 p . we select m := m 1 p 1 m 2 p +1 2 and obtain the claim. lemma 5. let v be a weight as in lemma 3. then, there is cv > 0 such that for every f ∈ av,p |f′(z) − f′(w)| ≤ cv‖f‖v,p max { 1 (1 − |z|2) 2 p +1 v(z) 1 p , 1 (1 − |w|2) 2 p +1 v(w) 1 p } ρ(z, w) for every z, w ∈ d. proof. by lemma 4 we can find 0 < s < 1 and a constant m < ∞ such that |f′(z) − f′(w)| ≤ 4mc 1 p v(0) 1 p ‖f‖v,p s(1 − |z|2) 2 p +1 v(z) 1 p ρ(z, w) cubo 14, 1 (2012) integral composition operators between . . . 15 for every z, w ∈ d with ρ(z, w) ≤ s 2 . next, if ρ(z, w) > s 2 , then |f′(z) − f′(w)| ≤ 2 c 1 p v(0) 1 p ‖f‖v,p max { 1 (1 − |z|2) 2 p +1 v(z) 1 p , 1 (1 − |w|2) 2 p +1 v(w) 1 p } ≤ 4 s c 1 p v(0) 1 p ‖f‖v,p max { 1 (1 − |z|2) 2 p +1 v(z) 1 p , 1 (1 − |w|2) 2 p +1 v(w) 1 p } ρ(z, w). hence, with cv := max { 4mc 1 p v(0) 1 p s , 4c 1 p sv(0) 1 p } , we conclude |f′(z) − f′(w)| ≤ cv max { 1 (1 − |z|2) 2 p +1 v(z) 1 p , 1 (1 − |w|2) 2 p +1 v(w) 1 p } ρ(z, w) for every z, w ∈ d and the claim follows. inductively, we can show the following lemmas: lemma 6. let v be a weight as in lemma 3. then there is cv > 0 such that for every f ∈ av,p |f(n)(z)| ≤ cv (1 − |z|2) 2 p +n v(z) 1 p ‖f‖v,p for every z ∈ d and every n ∈ n0. lemma 7. let v be a weight as in lemma 3. then there exists cv > 0 such that for every f ∈ av,p |f(n)(z) − f(n)(w)| ≤ cv‖f‖v,p max { 1 (1 − |z|2) 2 p +n v(z) 1 p , 1 (1 − |w|2) 2 p +n v(w) 1 p } ρ(z, w) for every z, w ∈ d and every n ∈ n0. now, we turn our attention to the operators ig,φ and start with characterizing when they are bounded. theorem 8. let w be a weight and v be a weight as in lemma 3 with m := supa∈d supz∈d v(z) |ν(az)| <∞. if sup z∈d w(z)|φ′(z)g(φ(z))| (1 − |φ(z)|2) 2 p +1 v(φ(z)) 1 p < ∞, (4.1) then the operator ig,φ : av,p → bw is bounded. if we assume additionally that sup z∈d |ν′(|φ(z)|2)|w(z)|φ′(z)g(φ(z))| v(φ(z)) 1 p +1(1 − |φ(z)|2) 2 p < ∞, (4.2) then the converse is also true. 16 elke wolf cubo 14, 1 (2012) proof. we start with assuming that the operator ig,φ is bounded and that the condition (4.2) is satisfied. fix a point a ∈ d and set fa(z) := ϕ′a(z) 2 p ν(az) 1 p for every z ∈ d. then ‖f‖pv,p = ∫ d |ϕ′a(z)| 2 |ν(az)| v(z) da(z) ≤ sup z∈d v(z) |ν(az)| ∫ d |ϕ′a(z)| 2 da(z) ≤ sup z∈d v(z) |ν(az)| ≤ m, and the constant m is independent of the choice of the point a. for the derivative we have f′a(z) = 2 p ϕ′a(z) 2 p −1 ϕ′′a(z) ν(az) 1 p − 1 p aν′(az)ϕ′a(z) 2 p ν(az) 1 p +1 for every z ∈ d. hence we can find a constant c∗ > 0 such that∣∣∣∣∣ w(a)|φ ′(a)||g(φ(a))| (1 − |φ(a)|2) 2 p +1 v(φ(a)) 1 p − |ν′(|φ(a)|2)|w(a)|φ′(a)g(φ(a))| v(φ(a)) 1 p +1(1 − |φ(a)|2) 2 p ∣∣∣∣∣ ≤ ∣∣∣f′φ(a)(φ(a))|w(a)|g(φ(a))||φ′(a)|∣∣∣ ≤ |(ig,φfφ(a))′(a)|w(a) ≤ c∗‖jg,φ‖‖fφ(a)‖v,p. finally, since (4.2) is fulfilled and the operator ig,φ is bounded, the claim follows. conversely, an application of lemma 3 yields for f ∈ av,p sup z∈d |(ig,φf) ′(z)|w(z) = sup z∈d |f′(φ(z))||g(φ(z))||φ′(z)|w(z) ≤ sup z∈d c 1 p ‖f‖v,pw(z)|g(φ(z))||φ′(z)| v(0) 1 p (1 − φ(z)|2) 2 p +1 v(φ(z)) 1 p . hence the claim follows. next, we study, when such operators are compact. to do this we need a lemma which can easily be derived from [9] proposition 3.11. lemma 9. let v and w be weights. then the operator ig,φ : av,p → bw is compact if and only if it is bounded and for every bounded sequence (fn)n in av,p which converges to zero uniformly on the compact subsets of d, ig,φfn tends to zero in bw if n → ∞. theorem 10. let w be a weight and v be a weight as in theorem 8. moreover, we assume that ig,φ : av,p → bw is bounded. if lim r→1 sup|φ(z)|>r w(z)|φ′(z)g(φ(z))| (1 − |φ(z)|2) 2 p +1 v(φ(z)) 1 p = 0, (4.3) cubo 14, 1 (2012) integral composition operators between . . . 17 then the operator ig,φ : av,p → bw is compact. if we assume additionally lim r→1 sup|φ(z)|>r |ν′(|φ(z)|2)|w(z)|φ′(z)g(φ(z))| v(φ(z)) 1 p +1(1 − |φ(z)|2) 2 p = 0, (4.4) then the converse is also true. proof. assume that the operator ig,φ : av,p → bw is compact and that (4.4) is satisfied. to show (4.3) let (zn)n be a sequence with |φ(zn)| → 1 and put fk(z) := ϕ′ φ(zk) (z) 2 p ν(φ(zk)z) 1 p for every z ∈ d and every k ∈ n. analogously to the proof of theorem 8 we can show that (fn)n is a bounded sequence which tends to zero uniformly on the compact subsets of d. since ig,φ is compact, by lemma 9 ‖ig,φfn‖bw → 0 if n → ∞. thus, ‖ig,φfn‖bw ≥ ∣∣∣∣∣ w(zn)|φ ′(zn)||g(φ(zn)| (1 − |φ(zn)|2) 2 p +1 v(φ(zn)) 1 p − |ν′(|φ(zn)| 2)|w(zn)|φ ′(zn)g(φ(zn))| v(φ(zn)) 1 p +1(1 − |φ(zn)|2) 2 p ∣∣∣∣∣ , and, since (4.4) holds, condition (4.3) follows. conversely, suppose that (4.3) is satisfied. let (fn)n be a bounded sequence in av,p such that ‖fn‖v,p ≤ m1 < ∞ for every n ∈ n and such that (fn)n converges uniformly to zero on the compact subsets of d if n → ∞. for a fixed ε > 0 we can find 0 < r0 < 1 such that if |φ(z)| > r0, then w(z)|g(φ(z))||φ′(z)| (1 − |φ(z)|2) 2 p +1 v(φ(z)) 1 p < εv(0) 1 p 2c 1 p m1 . moreover, we can find m2 > 0 such that sup |φ(z)|≤r0 w(z)|g(φ(z))||φ′(z)| ≤ m2. there is n0 ∈ n such that sup |φ(z)|≤r0 |f′n(φ(z))| ≤ ε 2m2 for every n ≥ n0. 18 elke wolf cubo 14, 1 (2012) we obtain applying lemma 3 sup z∈d |(ig,φfn) ′(z)|w(z) = sup z∈d w(z)|f′n(φ(z))||g(φ(z))||φ ′(z)| ≤ sup |φ(z)|≤r0 w(z)|f′n(φ(z))||g(φ(z))||φ ′(z)| + sup |φ(z)|>r0 w(z)|f′n(φ(z))||g(φ(z))||φ ′(z)| ≤ sup |φ(z)|≤r0 |f′n(φ(z))| sup |φ(z)|≤r0 w(z)|g(φ(z))||φ′(z)| + sup |φ(z)|>r0 c 1 p ‖fn‖v,pw(z)|g(φ(z))||φ′(z)| v(0) 1 p (1 − |φ(z)|2) 2 p +1 v(φ(z)) 1 p ≤ ε, and the claim follows. received: march 2011. revised: april 2011. references [1] a. aleman, j.a. cima, an integral operator on hp and hardy’s inequality, j. anal. math. 85 (2001), 157-176 [2] a. aleman, a. g. siskakis, an integral operator on hp, complex variables theory appl. 28 (1995), no. 2, 149158. [3] a. aleman, a. g. siskakis, integration operators on bergman spaces, indiana university math. j. 46 (1997), no. 2, 337-356. [4] j. bonet, p. domański, m. lindström, essential norm and weak compactness of composition operators on weighted banach spaces of analytic functions, canad. math. bull. 42, no. 2, (1999), 139-148. [5] j. bonet, p. domański, m. lindström, j. taskinen, composition operators between weighted banach spaces of analytic functions, j. austral. math. soc. (serie a) 64 (1998), 101-118. [6] j. bonet, m. friz, e. jordá, composition operators between weighted inductive limits of spaces of holomorphic functions, publ. math. 67 (2005), no. 3-4, 333-348. [7] j. bonet, m. lindström, e. wolf, differences of composition operators between weighted banach spaces of holomorphic functions, to appear in j. austr. math. soc. [8] m.d. contreras, a.g. hernández-dı́az, weighted composition operators in weighted banach spaces of analytic functions, j. austral. math. soc. (serie a) 69 (2000), 41-60. cubo 14, 1 (2012) integral composition operators between . . . 19 [9] c. cowen, b. maccluer, composition operators on spaces of analytic functions, crc press, boca raton, 1995. [10] p. domański, m. lindström, sets of interpolation and sampling for weighted banach spaces of holomorphic functions, ann. pol. math (2002) 79, 233-264. [11] p. duren, a. schuster, bergman spaces, mathematical surveys and monographs 100, american mathematical society, providence, ri, 2004. [12] h. hedenmalm, b. korenblum, k. zhu, theory of bergman spaces, graduate texts in mathematics 199, springer-verlag, new york, 2000. [13] s. li, volterra composition operators between weighted bergman spaces and bloch type spaces, j. korean math. soc. 45(2008), no. 1, 229-248. [14] m. lindström, e. wolf, essential norm of the difference of weighted composition operators, monatsh. math (2008) 153, 133-143. [15] w. lusky, on the structure of hv0(d) and hv0(d), math. nachr. 159 (1992), 279-289. [16] j.h. shapiro, composition operators and classical function theory, springer, 1993. [17] a. g. sisakis, r. zhao, a volterra type oprator on spaces of analytic functions, functions spaces (edwardville il, 1998), 299-311, contemp. math. 232, amer. math. soc. providence, ri, 1999. [18] e. wolf, weighted composition operators between weighted bergman spaces, racsam rev. r. acad. cien. serie a mat. 103, no. 1, 11-15. [19] e. wolf, volterra composition operators between weighted bergman spaces and weighted bloch type spaces, collect math. 61 (2010), no. 1, 57-63. [20] e. wolf, differences of weighted compostiion operators between bergman spaces and weighted banach spaces of holomorphic functions, glasgow math. j. 52 (2010), 325-332. [21] j. xiao, riemann-stieltjes operators on weighted bloch and bergman spaces of the unit ball, j. london, math. soc. (2), 70 (2004), no. 1, 199-214. cubo a mathematical journal vol.15, no¯ 01, (119–130). march 2013 existence of entire solutions for quasilinear elliptic systems under keller-osserman condition yuan zhang and zuodong yang 1 institute of mathematics, school of mathematical sciences, nanjing normal university, jiangsu nanjing 210023, china. zdyang jin@263.net abstract in this paper, we study the existence of entire solutions for the following elliptic system △mu = p(x)f(v),△lv = q(x)g(u), x ∈ rn, where 1 < m,l < ∞, f,g are continuous and non-decreasing on [0,∞), satisfy f(t) > 0,g(t) > 0 for all t > 0 and the keller-osserman condition. we establish conditions on p and q that are necessary for the existence of positive solutions, bounded and unbounded, of the given equation. resumen en este art́ıculo estudiamos la existencia de soluciones enteras para el siguiente sistema eĺıptico △mu = p(x)f(v),△lv = q(x)g(u), x ∈ rn, donde 1 < m,l < ∞, f,g son continuas y no-decrecientes en [0,∞), satisfaciendo f(t) > 0,g(t) > 0 para todo t > 0 y la condición de keller-osserman. establecemos condiciones sobre p y qy que son necesarias para la existencia de soluciones positivas, acotadas y no acotadas de la ecuación dada. keywords and phrases: quasi-linear elliptic system; sub/super-solution; large solution; existence. 2010 ams mathematics subject classification: 35j50; 35j57; 35j62; 35j92. 1project supported by the national natural science foundation of china(no.11171092); the natural science foundation of the jiangsu higher education institutions of china(no.08kjb110005) 120 yuan zhang and zuodong yang cubo 15, 1 (2013) 1 introduction in this paper, we investigate the following quasilinear elliptic system { △mu = p(x)f(v), x ∈ rn, △lv = q(x)g(u), x ∈ rn. (1.1) where 1 < m,l < ∞, n ≥ max{m,l} + 1,△m· = div(|∇ · |m−2∇·). denote d = min{m,l}, and see that d > 1. by an entire large solution (u,v), we mean a pair of functions u,v ∈ c1(rn) that satisfies (1.1) at every point of rn and lim |x|→∞ u(x) = lim |x|→∞ v(x) = ∞. (1.2) first, we introduce the assumptions below: (h1) p,q : rn → [0,∞) and f,g : [0,∞) → [0,∞) are continuous and nontrival functions; (h2) f and g are nondecreasing on [0,∞) and f(t) > 0,g(t) > 0 for all t > 0; (h3) h(∞) = limr→∞ h(r) = ∞, where h(r) = ∫r c dt d √ f(t) + g(t) , r ≥ c > 0; f(t) = ∫t 0 f(s)ds, g(t) = ∫t 0 g(s)ds. and c is a positive constant. notice that h′(r) = 1 d √ f(r)+g(r) > 0,∀ r > c, so h has the inverse function h−1 on [0,∞). denote φ1(r) := max |x|=r p(x), φ2(r) := min |x|=r p(x), ψ1(r) := max |x|=r q(x), ψ2(r) := min |x|=r q(x). since 1980s, many important results have been obtained for quasilinear elliptic systems. we will introduce some results in the following. existence and non-existence of solutions of the quasilinear elliptic system { div(|∇u|m−2∇u) + f(u,v) = 0, x ∈ rn div(|∇v|l−2∇v) + g(u,v) = 0, x ∈ rn (1.3) has gained much attention recently. see, for example, [3,4,10,15,19,21,22]. when p = q = 2, system (1.3) becomes { △u + f(u,v) = 0, x ∈ rn △v + g(u,v) = 0, x ∈ rn for which the existence and the non-existence of positive solutions and positive boundary blow-up solutions have been investigated extensively. we list here, for example, [1,2,5,6,12-14,16] and refer to the references therein. cubo 15, 1 (2013) existence of entire solutions for quasilinear elliptic ... 121 when p = q = 2,f = −a(|x|)vα,g = −b(|x|)uβ, system (1.3) becomes { △u = a(|x|)vα, x ∈ rn △v = b(|x|)uβ, x ∈ rn (1.4) for which existence results for positive boundary blow-up solutions can be found in a recent paper by lair and wood [12]. lair and wood established that all positive entire radial solutions of (1.4) are boundary blow-up provided that ∫ ∞ 0 ta(t)dt = ∞, ∫ ∞ 0 tb(t)dt = ∞. if, on the other hand ∫ ∞ 0 ta(t)dt < ∞, ∫ ∞ 0 tb(t) < ∞, then all positive entire radial solutions of (1.4) are bounded. f. cirstea and v.d. radulescu [1], extended the above results to a larger class of systems { △u = a(|x|)g(v), x ∈ rn △v = b(|x|)f(u), x ∈ rn in recent years, zhijun zhang et al.[23] studied the following semilinear elliptic systems { △u = p(x)f(v), x ∈ rn,(n ≥ 3), △v = q(x)g(u), x ∈ rn. (1.5) they obtained the existence and nonexistence of solutions for (1.5) by considering a set of hypotheses on p,q,f and g. z.d. yang [19], extended the above results to a class of systems { div(|∇u|m−2∇u) = a(|x|)g(v), x ∈ rn, div(|∇v|l−2∇v) = b(|x|)f(u), x ∈ rn. motivated by the results of the papers [19-23]. in this paper, we consider the quasilinear elliptic system (1.1). we modify the method developed by zhang et al.[23] and extend partial results of [23] to a quasilinear elliptic system (1.1). 2 main results in order to establish our main result, we introduce the following hypotheses : (h4) rn−1(φ1(r) + ψ1(r)) is nondecreasing for large r; 122 yuan zhang and zuodong yang cubo 15, 1 (2013) (h5) there exists a positive constant ε such that ∫ ∞ 0 t 1+ε m−1 (φ1(t) + ψ1(t)) 1 m−1 dt < ∞, and ∫ ∞ 0 t 1+ε l−1 (φ1(t) + ψ1(t)) 1 l−1 dt < ∞. our main results are as the following: theorem 1. under the hypotheses (h1)-(h5), equation (1.1) has a positive entire bounded solution (u,v). from the above theorem, we get the following corollary corollary 1. suppose that p and q are spherically symmetric(i.e. p(x) = p(|x|,q(x) = q(|x|)). under hypotheses (h1)-(h3), (1.1) has one positive solution (u,v). suppose further that p(∞) = q(∞) = ∞, where p(∞) := lim r→∞ p(r),p(r) := ∫r 0 (t1−n ∫t 0 sn−1p(s)ds) 1 m−1 dt,r ≥ 0; q(∞) := lim r→∞ q(r),q(r) := ∫r 0 (t1−n ∫t 0 sn−1q(s)ds) 1 l−1 dt,r ≥ 0 then every positive radial entire solution (u,v) of (1.1) is large and satisfies u(r) ≥ u(0) + f(v(0))p(r),v(r) ≥ v(0) + g(u(0))q(r). ∀r ≥ 0. corollary 2. under the assumption (h1)-(h4), if (1.1) has a non-negative radial entire large solution, then at least one of the following two equations hold: ∫ ∞ 0 r 1+ε m−1 (p(r) + q(r)) 1 m−1 dr = ∞, ∀ε > 0. ∫ ∞ 0 r 1+ε l−1 (p(r) + q(r)) 1 l−1 dr = ∞, ∀ε > 0. remark 1. by (h1) and (h3), we have ∫ ∞ a ds d √ f(s) = ∫ ∞ a ds d √ g(s) = ∞. remark 2. when 2 ≤ d < ∞, ∫ ∞ 0 r 1 d−1 (p(x) + q(x)) 1 d−1 dr = ∞ implies ∫ ∞ 0 (t1−n ∫t 0 sn−1(p(x) + q(x))(s)ds) 1 d−1 dt = ∞ . cubo 15, 1 (2013) existence of entire solutions for quasilinear elliptic ... 123 remark 3. if ∫ ∞ a ds d √ f(s) < ∞,a > 0, then ∫ ∞ a dt (f(t)) 1 d−1 < ∞. in other words, if ∫ ∞ a dt (f(t)) 1 d−1 = ∞, then ∫ ∞ a ds d √ f(s) = ∞. proof. we only need to prove (f(t)) 1 d−1 s > δd (1.6) for ∀ δ > 0. then f(s) ≡ ∫s 0 f(t)dt ≤ sf(s) ≤ f d d−1 (s) δd , and (f(s))− 1 d ≥ δ (f(s)) 1 d−1 . we suppose that (1.6) is not true, then ∃ an increasing sequence {sj}, limj→∞sj = ∞ such that (f(sj)) 1 d−1 sj < 1 j , which equals to f(sj) ≤ ( sj j )d−1, then (f(sj)) − 1 d ≥ (sj j )− d−1 d . since f is nondecreasing, we get f(s) ≤ f(sj) for all s ∈ [0,sj], so f(s) ≤ sf(s) ≤ sf(sj) for all s ∈ [0,sj], and ∫sj s1 (f(s))− 1 d ds ≥ ∫sj s1 (sf(sj)) − 1 d ds ≥ ( sj j )− d−1 d ∫sj s1 s− 1 d ds = j1− 1 d (1 − ( s1 sj )1− 1 d ) → ∞ this is a contradiction. in order to prove the theorem 1, we give the following lemma. lemma 1. for any nonnegative a and b, we have (a + b)α ≤ aα + bα, α ∈ (0,1] (a + b)β ≤ 2β−1(aβ + bβ), β ∈ [1,∞) proof of theorem 1. first, we have to find a pair of super-solution, (ū, v̄) and sub-solution,(u,v), which satisfy u ≤ ū and v ≤ v̄. consider the following system of integral equation: u(r) = β + ∫r 0 (t1−n ∫t 0 sn−1φ1(s)f(v(s))ds) 1 m−1 dt,r ≥ 0 v(r) = β + ∫r 0 (t1−n ∫t 0 sn−1ψ1(s)g(u(s))ds) 1 l−1 dt,r ≥ 0 (1) where β ≥ c > 0, c is in (h3). let {vk}k≥0 and {uk}k≥1 be the sequence of positive continuous functions defined on [0,∞) by v0 = β, uk(r) = β + ∫r 0 (t1−n ∫t 0 sn−1φ1(s)f(vk−1(s))ds) 1 m−1 dt,r ≥ 0 vk(r) = β + ∫r 0 (t1−n ∫t 0 sn−1ψ1(s)g(uk−1(s))ds) 1 l−1 dt,r ≥ 0 (2) 124 yuan zhang and zuodong yang cubo 15, 1 (2013) then, v0 ≤ v1, uk(r) ≥ β, and vk(r) ≥ β for all r ≥ 0 , k ∈ n. using the non-decreasing property of f and g, we get u1(r) ≤ u2(r) for all r ≥ 0, then v1(r) ≤ v2(r) for all r ≥ 0. continuing this line, we obtain that the sequence {uk} and {vk} are increasing with respect to k for r ∈ [0,∞). besides, u′k(r) = (r 1−n ∫r 0 sn−1φ1(s)f(vk−1(s))ds) 1 m−1 ≥ 0, v′k(r) = (r 1−n ∫r 0 sn−1ψ1(s)g(uk−1(s))ds) 1 l−1 ≥ 0, for each r > 0, and (rn−1|u′k| m−2u′k) ′ = rn−1φ1(r)f(vk−1(r)) ≤ r n−1φ1(r)f(vk(r)) (3) let θ(r) = max 0≤t≤r (φ1(t) + ψ1(t)), using this and the fact that u′k ≥ 0, we note that (3) yields ((u′k(r)) m−1)′ ≤ θ(r)f(vk(r)), multiply this by u′k and integrate to get (u′k(r)) m ≤ m m − 1 θ(r) ∫vk(r)+uk(r) 2β f(s)ds in the same way, (v′k(r)) l ≤ l l − 1 θ(r) ∫vk(r)+uk(r) 2β g(s)ds then from the inequality (u′k + v ′ k) d ≤ 2d−1((u′k)d + (v′k)d), where d = min{m,l}, and the above two inequalities, we get (u′k + v ′ k) d ≤ 2d−1((u′k) d + (v′k) d) ≤ 2d−1((u′k) m + (v′k) l + 1) ≤ 2d−1( d d − 1 θ(r) ∫vk(r)+uk(r) 2β (f(s) + g(s))ds + 1) ≤ 2d−1( d d − 1 θ(r)(f(u + v) + g(u + v)) + 1) (4) which yields u′k + v ′ k ≤ 2 d−1 d ( d d − 1 θ(r)(f(uk + vk) + g(uk + vk)) + 1) 1 d ≤ d √ 2d−1d d − 1 θ(r)(f(uk + vk) + g(uk + vk)) 1 d + 2 d−1 d (5) cubo 15, 1 (2013) existence of entire solutions for quasilinear elliptic ... 125 integrating the above inequality, we get ∫r 0 u′k(t) + v ′ k(t) (f(uk(t) + vk(t)) + g(uk(t) + vk(t))) 1 d dt = ∫vk(r)+uk(r) 2β dτ d √ f(τ) + g(τ) ≤ ∫r 0 ( d √ 2d−1dθ(t) d − 1 + c)dt where c = 2 d−1 d f(2β)+g(2β) . we can easily get h(uk(r) + vk(r)) ≤ h(2β) + ∫r 0 ( d √ 2d−1dθ(t) d − 1 + c)dt as we know that h−1 is increasing on [0,∞), so uk(r) + vk(r) ≤ h−1(h(2β) + ∫r 0 d √ 2d−1dθ(t) d − 1 + c)dt, ∀r ≥ 0 following by the definition of uk(r) and vk(r) and (h3), we get that the sequence {uk} and {vk} are bounded and equi-continuous on [0,c0] for arbitrary c0 > 0. by arzela-ascoli theorem, {uk} and {vk} have subsequence converging uniformly to u and v on [0,c0]. by the arbitrariness of c0 > 0,we see that (u,v) is a positive entire solution of ∆mu = φ1(r)f(v) ≥ p(x)f(v), x ∈ rn (6) ∆lv = ψ1(r)g(u) ≥ q(x)g(v), x ∈ rn (7) then, we take conclusion that (u,v) is a positive entire sub-solution of (1.1). in order to prove (u,v) is bounded, choosing r > 0, so that rd(n−1)(φ1(r) + ψ1(r)) is nondecreasing on [r,∞) and u(r) > 0,v(r) > 0. this is possible because of (h4). since (u,v) satisfies (rn−1(u′)m−1)′ = rn−1φ1(r)f(v(r)), (8) (rn−1(v′)l−1)′ = rn−1ψ1(r)g(u(r)). (9) u′(r) ≥ 0 and v′(r) ≥ 0 for r ≥ 0, and (h2) hold, multiplying (8) and (9) by u′ and v′, respectively, and integrating from r to r. take (8) as an example, ∫r r (sn−1(u′)m−1)′u′(s)ds = ∫r r sn−1φ1(s)f(v(s))u ′(s)ds, which implies that m − 1 m rn−1(u′(r))m− m − 1 m rn−1(u′(r))m+ n − 1 m ∫r r sn−2(u′(s))mds = ∫r r sn−1φ1(s)f(v(s))u ′(s)ds 126 yuan zhang and zuodong yang cubo 15, 1 (2013) it follows that rn−1(u′(r))m ≤ rn−1(u′(r))m + m m − 1 ∫r r sn−1φ1(s)f(v(s))u ′(s)ds using the monotonicity of tn−1(φ1(t) + ψ1(t)) for t ≥ 0, we get rn−1(u′(r))m ≤ c̄ + m m − 1 rn−1(φ1(r) + ψ1(r))(f(u(r) + v(r)) + g(u(r) + v(r))) for r > r, where c̄ = rn−1(u′(r))m + rn−1(v′(r))l. which yields u′(r) ≤ m √ c̄r 1−n m + m √ m m − 1 (φ1(r) + ψ1(r))(f(u + v) + g(u + v)) 1 m so u′(r) + v′(r) ≤ c1(r 1−n m + r 1−n l ) + ( m √ m m − 1 (φ1(r) + ψ1(r)) + l √ l l − 1 (φ1(r) + ψ1(r)))(2(f(u + v) + g(u + v)) 1 d + 1) and d dr ∫u(r)+v(r) u(r)+v(r) dτ d √ f(τ) + g(τ) ≤ c1(r 1−n m + r 1−n l )(f(u + v) + g(u + v))− 1 d + h(r)(2 + (f(u + v) + g(u + v))− 1 d ) (10) where h(r) = m √ m m−1 (φ1(r) + ψ1(r)) + l √ l l−1 (φ1(r) + ψ1(r)). we notice the fact that f(u(r) + v(r)) + g(u(r) + v(r)) ≥ f(u(r) + v(r)) + g(u(r) + v(r)) = c2 for all r ≥ r, and m √ m m − 1 (φ1(r) + ψ1(r)) ≤ m m − 1 m √ r1+ε(φ1(r) + ψ1(r))r −1−ε using young’s inequality, we get m √ m m − 1 (φ1(r) + ψ1(r)) ≤ 1 m − 1 r−1−ε + r 1+ε m−1 (φ1(r) + ψ1(r)) 1 m−1 for ε > 0. in the same way, l √ l l − 1 (φ1(r) + ψ1(r)) ≤ 1 l − 1 r−1−ε + r 1+ε l−1 (φ1(r) + ψ1(r))) 1 l−1 for ε > 0. cubo 15, 1 (2013) existence of entire solutions for quasilinear elliptic ... 127 then integrate (10) from r to r, r ≥ r, h(u(r) + v(r)) − h(u(r) + v(r)) ≤ c3 + c4(( 1 m − 1 + 1 l − 1 ) r−ε ε + ∫r r t 1+ε m−1 (φ1 + ψ1) 1 m−1 dt + ∫r r t 1+ε l−1 (φ1 + ψ1) 1 l−1 dt) where c3 = d √ c2c1( mr 1+m−n m n−m−1 + lr 1+l−n l n−l−1 ), c4 = 2 + (f(2r) + g(2r)) − 1 d . from (h5), we know ∫r r t 1+ε m−1 (φ1 + ψ1) 1 m−1 dt < ∞, and ∫r r t 1+ε l−1 (φ1 + ψ1) 1 l−1 dt < ∞, so h(u(r) + v)(r) < ∞ letting r → ∞, since h satisfies (h3), we find that (u,v) is bounded . by now, we have find a pair of bounded sub-solution to (1.1). we still have to find (ū, v̄), which is a bounded super-solution of (1.1), and u(r) ≤ ū(r), v(r) ≤ v̄(r) for all r ≥ 0. actually, since (u,v) is nondecreasing and bounded, we have lim r→∞ u(r) = m1 > 0, lim r→∞ v(r) = m2 > 0. let ū(0) = v̄(0) = max{m1,m2}, ū ′(0) = v̄′(0) = 0, then, the following system ∆mū(x) = φ2(r)f(v̄(r)) r > 0 ∆lv̄(x) = ψ2(r)g(ū(r)) r > 0 has a bounded solution (ū, v̄) by the same argument, and it is a supersolution for (1.1). from the above process, we get conclusion that u(r) ≤ m1 ≤ ū(r), v(r) ≤ m2 ≤ v̄(r). ∀r ≥ 0. the standard super-sub solution principle [18,20] implies that (1.1) has a bounded solution (u,v) satisfying u(x) ≤ u(x) ≤ ū and v(x) ≤ v(x) ≤ v̄ on rn, which is the desired solution. this completes the proof. 3 conclusion the boundary value quasilinear differential equation systems (1.1) are mathematical models occurring in the studies of the m-laplace equation, generalized reaction-diffusion theory, nonnewtonian fluid theory, and the turbulent flow of a gas in porous medium. when m 6= 2, the 128 yuan zhang and zuodong yang cubo 15, 1 (2013) problem becomes more complicated since certain nice properties in herent to the case m = 2 seem to be lost or at least difficult to verify. the main differences between m = 2 and m 6= 2 can be founded in [8,9]. when m = 2, it is well known that all the positive solutions in c2(br) of the problem { △u + f(u) = 0 in br u(x) = 0 on ∂br are radially symmetric solutions for very general f(see [7]). unfortunately, this result does not apply to the case m 6= 2. kichenassary and smoller showed that there exist many positive nonradial solutions of the above problem for some f(see [11]). the major stumbling block in the case of m 6= 2 is that certain nice features inherent to the case m = 2 seem to be lost or at least difficult to verify. in this paper, we first give some necessary preliminary knowledge. secondly, we further study the existence of positive solutions to problem (1.1) which the right hand side functions are more general based on the method of sub-supersolution. received: october 2012. revised: march 2013. references. 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[23] zhijun zhang ,yongxiu shi and yanxing xue, existence of entire solutions for semilinear elliptic systems under keller-osserman condition, electronic journal of differential equation 39(2011), 1-9. () cubo a mathematical journal vol.17, no¯ 03, (71–90). october 2015 computing the laplace transform and the convolution for more functions adjoined takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, senbaru 1, nishihara, okinawa 903-0213, japan. sudo@math.u-ryukyu.ac.jp abstract we compute explicitly the laplace transform and the convolution for more functions on the real than the continuous functions that obtained as the inverse laplace transforms of rational functions on the complex plane vanishing at infinity. we also consider the algebraic structure with more functions adjoined. resumen calculamos expĺıcitamente la transformada de laplace y la convolución para más funciones en los reales que las funciones continuas obtenidas como transformadas de laplace inversas de funciones racionales en el plano complejo que se anulan en el infinito. también consideramos la estructura algebraica con más funciones adjuntadas. keywords and phrases: laplace tranform, gaussian function, dirac delta function, convolution. 2010 ams mathematics subject classification: 44a10, 44a35, 26c15, 26a06, 26a09. 72 takahiro sudo cubo 17, 3 (2015) 1 introduction this paper is a continuation from the paper [3] of the author, in which we compute explicitly the inverse laplace transform for rational functions on the complex plane vanishing at infinity and consider the algebraic structure for the corresponding algebra of continuous elementary functions on the real r for which the laplace transform on the infinite interval [0, ∞) is defined. in this paper, we compute explicitly the laplace transform for more functions on the real than the continuous functions that obtained as the inverse laplace transforms of rational functions on the complex plane vanishing at infinity. for this, we compute the convolution product with more functions adjoined, such as gaussian functions, the dirac delta functions, and the shifted delta functions. we also consider the algebraic structure with more functions adjoined. this paper after this introduction is organized as follows: 2 adjoining gaussian functions; 3 algebraic structure with gaussian functions; 4 adjoining the delta function; 5 algebraic structure with the delta function; 6 adjoining the shifted delta functions; 7 algebraic structure with the shifted delta functions; 8 more with the weak derivative. our elementary explicit computation results obtained and algebraic consideration results obtained should be useful as a reference. refer to [4] or [2] (or [1]) for some basics about the laplace transform. notation. we denote by a(r) the real algebra with convolution, generated by the three sets of elementary continuous functions on the real line r with t real variable: {tn | n ∈ n = {0, 1, 2, · · · }} of monomials and {eµt | µ ∈ r} of exponential functions based to e and {sin λt, cosλt | λ ∈ r} of trigonometric functions and their point-wise multiples such as tneµt sin λt. the (extended in our sense) convolution product for a(r) (but the same as the usual one by restriction to [0, ∞)) is defined as the (riemann) integral: f ∗ g = (f ∗ g)(t) = ∫t 0 f(t − τ)g(τ)dτ, (t ∈ r) for f, g ∈ a(r), which makes sense and is a commutative and associative operation as well known as the usual case on the interval [0, ∞). denote by c the complex plane. for f = f(t) a suitable function on r (as in a(r)), the (extended in our sense) laplace transform of f (but the same as the usual one of f restricted to [0, ∞)) is defined by the (riemann) integral: l(f(t)) = l(f(t))(s) = ∫ ∞ 0 e−stf(t)dt for s ∈ c (formally or in the domain of convergence). cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 73 2 adjoining gaussian functions quite well known, but first of all we consider gaussian functions as follows: lemma 2.1. for t ∈ r, s ∈ c and λ ∈ r with λ > 0, we have l(e−λt 2 ) = √ π 2 √ λ e s2 4λ , proof. check that e−ste−λt 2 = e−λ(t+ s 2λ ) 2 + s 2 4λ and then l(e−λt 2 ) = e s2 4λ ∫ ∞ 0 e−λ(t+ s 2λ ) 2 dt. using change of variables and cauchy’s theorem, we obtain that the integral is computed to be equal to ∫ s 2 √ λ +∞ s 2 √ λ +0 e−u 2 du √ λ with u = √ λ(t + s 2λ )) = 1 √ λ ∫ ∞ 0 e−u 2 du = √ π 2 √ λ . lemma 2.2. for t ∈ r, s ∈ c and λ, µ ∈ r positive, we have l(e−λt 2 ∗ e−µt 2 ) = π 4 √ λµ e λ+µ 4λµ s 2 . we need to check the convolution explicitly as follows. proposition 1. for t ∈ r and positive λ, ν ∈ r, (e−λt 2 ∗ e−µt 2 )(t) = 1 √ λ + µ e −λ(1− λ λ+µ )t 2 ∫ µt √ λ+µ −λt √ λ+µ e−u 2 du = e −λ(1− λ λ+µ )t 2 ∞∑ n=0 λ2n+1 + µ2n+1 (λ + µ)n+1 (−1)n n! 1 2n + 1 t2n+1 proof. we compute (e−λt 2 ∗ e−µt 2 )(t) = ∫t 0 e−λ(t−τ) 2 e−µτ 2 dτ = e−λt 2 ∫t 0 e −(λ+µ)(τ− λt λ+µ ) 2 + λ 2 t 2 λ+µ dτ = e −λ(1− λ λ+µ )t 2 ∫t 0 e −(λ+µ)(τ− λt λ+µ ) 2 dτ 74 takahiro sudo cubo 17, 3 (2015) with 1 − λ λ+µ > 0, and the integral is computed by change of variables as: g(t) ≡ ∫t 0 e −(λ+µ)(τ− λt λ+µ ) 2 dτ = 1 √ λ + µ ∫ µt √ λ+µ −λt √ λ+µ e−u 2 du with u = √ λ + µ(τ − λt λ + µ ). using taylor’s expansion of ey we compute the integral above as: ∫ µt √ λ+µ −λt √ λ+µ e−u 2 du = ∫ µt √ λ+µ −λt √ λ+µ ∞∑ n=0 (−1)n n! u2ndu = ∞∑ n=0 (−1)n n! ∫ µt √ λ+µ −λt √ λ+µ u2ndu = ∞∑ n=0 (−1)n n! µ2n+1 + λ2n+1 (λ + µ)n+ 1 2 1 2n + 1 t2n+1. corollary 1. the convolutions e−λt 2 ∗e−µt 2 for λ, µ positive reals are point-wise limits of elements of a(r). proof. note that any gaussian functions are point-wise limits of elements of a(r) by taylor series. proposition 2. for λ ∈ r positive, n ∈ n, and t ∈ r, (1) (tn ∗ e−λt 2 ) = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τke−λτ 2 dτ with ik ≡ ∫t 0 τke−λτ 2 dτ given by i2k+1 = −e −λt 2 k∑ l=1 (2k)!! (2λ)l(2(k − l + 1))!! t2(k−l+1) + (2k)!! (2λ)k+1 (1 − e−λt 2 ), and i2k = −e −λt 2 k∑ l=1 (2k − 1)!! (2λ)l(2(k − l + 1) − 1)!! t2(k−l+1)−1 + (2k − 1)!! (2λ)k ∞∑ m=0 (−λ)m m!(2m + 1) t2m+1. also, for µ ∈ r non-zero and λ ∈ r positive, (2) (eµt ∗ e−λt 2 ) = e µ 2 4λ eµt ∞∑ n=0 (−1)nλn+ 1 2 n!(2n + 1) {(t + µ 2λ ) 2n+1 − ( µ 2λ ) 2n+1 }. cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 75 also, for µ ∈ r non-zero and λ ∈ r positive, (3) (sin µt ∗ e−λt 2 ) = sin µt ∫t 0 cos µτe−λτ 2 dτ − cos µt ∫t 0 sin µτe−λτ 2 dτ with ∫t 0 cos µτe−λτ 2 dτ = ∞∑ n=0 (−λ)n n! ∫t 0 τ2n cos µτdτ, ∫t 0 sin µτe−λτ 2 dτ = ∞∑ n=0 (−λ)n n! ∫t 0 τ2n sin µτdτ and each integral term is computed inductively as: ic,2n ≡ ∫t 0 τ2n cos µτdτ = 1 µ t2n sin µt + 2n µ2 t2n−1 cos µτ − (2n)(2n − 1) µ2 ic,2n−2, is,2n ≡ ∫t 0 τ2n sin µτdτ = − 1 µ t2n cos µt + 2n µ2 t2n−1 sin µτ − (2n)(2n − 1) µ2 is,2n−2, and similarly, (4) (cos µt ∗ e−λt 2 ) = cos µt ∫t 0 cos µτe−λτ 2 dτ − sin µt ∫t 0 sin µτe−λτ 2 dτ with the integrals the same as above. proof. for (1) we compute (tn ∗ e−λt 2 )(t) = ∫t 0 (t − τ)ne−λτ 2 dτ = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τke−λτ 2 dτ and the integral is computed by integration by parts as: ik ≡ ∫t 0 τke−λτ 2 dτ = ∫t 0 τk−1 { 1 −2λ (e−λτ 2 )′ } dτ = − 1 2λ tk−1e−λt 2 + k − 1 2λ ik−2 76 takahiro sudo cubo 17, 3 (2015) so that, inductively, i2k+1 = − 1 2λ t2ke−λt 2 − 2k (2λ)2 t2k−2e−λt 2 − 2k(2k − 2) (2λ)3 t2k−4e−λt 2 − · · · − 2k(2k − 2) · · · 4 (2λ)k t2e−λt 2 + 2k(2k − 2) · · · 2 (2λ)k i1 with i1 ≡ ∫t 0 τe−λτ 2 dτ = 1 2λ (1 − e−λt 2 ), and i2k = − 1 2λ t2k−1e−λt 2 − 2k − 1 (2λ)2 t2k−3e−λt 2 − (2k − 1)(2k − 3) (2λ)3 t2k−5e−λt 2 − · · · − (2k − 1)(2k − 3) · · · 3 (2λ)k te−λt 2 + (2k − 1)(2k − 3) · · · 1 (2λ)k i0 with i0 ≡ ∫t 0 e−λτ 2 dτ = ∫t 0 ∞∑ m=0 (−λ)m m! τ2mdτ = ∞∑ m=0 (−λ)m m! ∫t 0 τ2mdτ = ∞∑ m=0 (−λ)m m!(2m + 1) t2m+1. for (2) we compute eµt ∗ e−λt 2 = ∫t 0 eµ(t−τ)e−λτ 2 dτ = e µ 2 4λ eµt ∫t 0 e−λ(τ+ µ 2λ ) 2 dτ. by change of variables as u = √ λ(τ + µ 2λ ), the integral above is computed as ∫t 0 e−λ(τ+ µ 2λ ) 2 dτ = ∫√λ(t+ µ 2λ ) √ λ( µ 2λ ) e−u 2 du = ∫√λ(t+ µ 2λ ) √ λ( µ 2λ ) ∞∑ n=0 1 n! (−u2)ndu = ∞∑ n=0 (−1)n n! ∫√λ(t+ µ 2λ ) √ λ( µ 2λ ) u2ndu = ∞∑ n=0 (−1)nλn+ 1 2 n!(2n + 1) {(t + µ 2λ )2n+1 − ( µ 2λ )2n+1}. for (3) we next compute (sin µt ∗ e−λt 2 )(t) = ∫t 0 sin µ(t − τ)e−λt 2 dτ = sin µt ∫t 0 cos µτe−λτ 2 dτ − cos µt ∫t 0 sin µτe−λτ 2 dτ, cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 77 and the first integral is computed as: ∫t 0 cos µτe−λτ 2 dτ = ∫t 0 cos µτ ∞∑ n=0 1 n! (−λτ2)ndτ = ∞∑ n=0 (−λ)n n! ∫t 0 τ2n cos µτdτ and each integral term is computed inductively as: ic,2n ≡ ∫t 0 τ2n cos µτdτ = 1 µ t2n sin µt + 2n µ2 t2n−1 cos µτ − (2n)(2n − 1) µ2 ic,2n−2 by using integration by parts. similarly, we obtain ∫t 0 sin µτe−λτ 2 dτ = ∞∑ n=0 (−λ)n n! ∫t 0 τ2n sin µτdτ and is,2n ≡ ∫t 0 τ2n sin µτdτ = − 1 µ t2n cos µt + 2n µ2 t2n−1 sin µτ − (2n)(2n − 1) µ2 is,2n−2. remark. we omit the case of convolutions with general monomials such as multiples tneµt sin µt, but which are certainly computable by the similar computation technique as above, with some more integration. refer to [3]. 3 algebraic structure with gaussian functions we denote by aw(r) the set of all functions on r that are written as point-wise limits in r of elements of a(r). lemma 3.1. aw(r) is an algebra over r with point-wise multiplication. also, aw(r) is an algebra over r with convolution. proof. suppose that f, g ∈ aw(r) with f = lim fn, g = lim gn (n → ∞) point-wise limits of fn, gn ∈ aw(r). since the limits at any t ∈ r are finite, then (f · g)(t) = lim fn(t) · lim gn(t) = lim(fn · gn)(t) ∈ r, and hence, f · g ∈ aw(r). 78 takahiro sudo cubo 17, 3 (2015) note also that we define the convolution of f, g ∈ aw(r) as f ∗ g = lim(fn ∗ gn) ∈ aw(r), where it is shown in general ([3]) that a(r) is closed under convolution. this is well defined, because if f = lim hn and g = lim kn another limits with hn, kn ∈ a(r), and since the limits at any t ∈ r are finite, then (fn ∗ gn)(t) − (hn ∗ kn)(t) = ((fn − f) ∗ gn)(t) + (f ∗ (gn − g))(t) + ((f − hn) ∗ g)(t) + (hn ∗ (g − kn))(t), and the right hand side goes to zero as n → ∞ by applying the dominated convergence theorem to each convolution, and hence, | lim(fn ∗ gn) − lim(hn ∗ kn)| ≤ | lim(fn ∗ gn) − fn ∗ gn| + |fn ∗ gn − hn ∗ kn| + |hn ∗ kn − lim(hn ∗ kn)|, which go to zero as n → ∞. we denote by g(r) the algebra generated by a(r) and the set {e−λt 2 | λ ∈ r, λ > 0}, with either point-wise multiplication or convolution as a product. corollary 2. the algebra g(r) is contained in aw(r) and is a subalgebra of aw(r) with either point-wise multiplication or convolution. we denote by r0(c) the algebra of rational functions on c vanishing at infinity with point-wise multiplication and by e(c) the algebra generated by both r0(c) and the set {e λs 2 | λ ∈ r, λ > 0} of inverse gaussian functions on c with s complex variable. corollary 3. the algebra g(r) with convolution is isomorphic to the algebra e(c) by the laplace transform and by the inverse laplace transform. proof. note that l(f ∗ g) = l(f) · l(g) and l is injective for continuous functions on r. 4 adjoining the delta function we consider the dirac delta function. define the delta function as δ(t) = lim λ→∞ gλ(t) ≡ lim λ→∞ 2 √ λ π e−λt 2 = { ∞ t = 0 0 t 6= 0 with λ ∈ r positive. note that the delta function has the meaning in integration only in the weak sense, i.e. as the weak limit of gλ(t) by taking the inner product or the convolution with test cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 79 functions, and the delta function is also viewed as a distribution, i.e., a functional (or a scalarvalued linear map) on the linear space of test functions. also, one can take other functions of different types to define the delta function. note that ∫ ∞ −∞ gλ(t)dt = 2 but we need to have the next: lemma 4.1. the functions gλ are in g(r), and for s ∈ c, l(gλ)(s) = e s 2 4λ , which goes to 1 as λ → ∞. proof. use lemma 2.1. lemma 4.2. for s ∈ c, we have l(gλ ∗ gµ)(s) = e (λ+µ)s2 4λµ , which goes to 1 as λ → ∞ and µ → ∞. proof. use lemma 4.1 and l(gλ ∗ gµ) = l(gλ) · l(gµ). proposition 3. for positive µ, λ ∈ r, (1) lim λ→∞ (e−µt 2 ∗ gλ)(t) =    e−µt 2 if t > 0, 0 if t = 0, −e−µt 2 if t < 0; and for n ∈ n and λ > 0, (2) lim λ→∞ (tn ∗ gλ)(t) =    tn if t > 0, 0 if t = 0, −tn if t < 0; and for µ ∈ r and λ > 0, (3) lim λ→∞ (eµt ∗ gλ)(t) =    eµt if t > 0, 0 if t = 0, −eµt if t < 0; and for µ ∈ r and λ > 0, (4) lim λ→∞ (sin µt ∗ gλ)(t) =    sin µt if t > 0, 0 if t = 0, − sin µt if t < 0; 80 takahiro sudo cubo 17, 3 (2015) and (5) lim λ→∞ (cos µt ∗ gλ)(t) =    cos µt if t > 0, 0 if t = 0, − cos µt if t < 0. proof. the first equation (1) follows from the first equation in proposition 1 by taking the limit with respect to λ. for (2) we next use proposition 2. it follows that lim λ→∞ (tn ∗ gλ)(t) = lim λ→∞ tn ∫t 0 gλ(τ)dτ, namely, other terms in the convolution go to zero as λ → ∞, and then ∫t 0 gλ(τ)dτ = ∫√λt 0 2√ π e−u 2 du (u = √ λτ), which goes to    1 if t > 0, 0 if t = 0, −1 if t < 0, as λ → ∞. for (3), it also follows as in parts of the proof of proposition 2 that (eµt ∗ gλ)(t) = e µ 2 4λ eµt ∫t 0 2 √ λ π e−λ(τ+ µ 2λ ) 2 dτ = e µ 2 4λ eµt ∫√λ(t+ µ 2λ ) µ 2 √ λ 2 √ π e−u 2 du (u = √ λ(τ + µ 2λ )), which goes to    eµt if t > 0, 0 if t = 0, −eµt if t < 0, as λ → ∞. for (4) we next compute by allowing complex coefficients for this case and using the euler formula as: (sin µt ∗ gλ)(t) = ( eµti − e−µti 2i ∗ gλ)(t) = 1 2i ∫t 0 eµ(t−τ)igλ(τ)dτ − 1 2i ∫t 0 e−µ(t−τ)igλ(τ)dτ = eµti 2i ∫t 0 e−µτigλ(τ)dτ − e−µti 2i ∫t 0 eµτigλ(τ)dτ cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 81 and the first integral is converted to ∫t 0 2 √ λ π e−λ(τ+ µi 2λ ) 2 − µ 2 4λ dτ = e− µ 2 4λ ∫√λ(t+ µi 2λ ) iµ 2 √ λ 2 √ π e−u 2 du (u = √ λ(τ + µi 2λ )) which goes to    1 if t > 0, 0 if t = 0, −1 if t < 0 as λ → ∞ via cauchy integral theorem. similarly, the second integral above is converted to ∫t 0 2 √ λ π e−λ(τ− µi 2λ ) 2 − µ 2 4λ dτ = e− µ 2 4λ ∫√λ(t− µi 2λ ) iµ 2 √ λ 2 √ π e−u 2 du (u = √ λ(τ − µi 2λ )) which goes to    1 if t > 0, 0 if t = 0, −1 if t < 0 as λ → ∞. therefore, we obtain one of the last two equations in the statement. the other case (5) is obtained similarly, by cos µt = e µti +e −µti 2 . 5 algebraic structure with the delta function we denote by gδ(r) the algebra generated by g(r) and the delta function δ(t) with convolution, where we assume that f ∗ δ = f on [0, ∞) for any f ∈ g(r) and δ ∗ δ = δ on [0, ∞), which follows from our definition of the delta function in the weak sense, so that gδ(r) restricted to [0, ∞) is isomorphic to the unitization of g(r) by r, restricted to [0, ∞), as a real algebra. we define the laplace transform for δ as l(δ(t))(s) = 1(s) the unit function on c. note that gδ(r) is also viewed as the generated by g(r) and the point-wise limit function of gaussian functions on r. we denote by e1(c) the algebra generated by e(c) and the unit function 1 on c with pointwise multiplication. 82 takahiro sudo cubo 17, 3 (2015) corollary 4. the algebra gδ(r) with convolution, restricted to [0, ∞), is isomorphic to the algebra e1(c) by the laplace transform and by the inverse laplace transform. remark. our proposition 3 says that the convolution f ∗ δ on r for f ∈ gδ(r) is defined as (f ∗ δ)(t) =    f(t) if t > 0, 0 if t = 0, −f(t) if t < 0. 6 adjoining the shifted delta functions it is also defined as that for t ∈ r and s ∈ c, l(δ(t − µ)) = e−µs for µ > 0. we extend this definition for µ ∈ r. indeed, check that lemma 6.1. we have lim λ→∞ l(gλ(t − µ))(s) = e −µs for any µ ∈ r. proof. we compute l(gλ(t − µ))(s) = ∫ ∞ 0 e−st2 √ λ π e−λ(t−µ) 2 dt = e−µse s 2 4λ ∫ ∞ 0 e−λ(t+ s 2λ −µ) 2 2 √ λ π dt and changing variables as u = √ λ(t + s 2λ − u) and using cauchy integral theorem and taking the limit as λ → ∞ we obtain the result in the statement. lemma 6.2. we have lim λ→∞ l(gλ(t − µ) ∗ gλ(t − ρ))(s) = e−(µ+ρ)s for any µ, ρ ∈ r. it follows by defining the left hand side to be equal to l(δ(t − µ) ∗ δ(t − ρ))(s) = e−(µ+ρ)s. proof. note that l(gλ(t − µ) ∗ gλ(t − ρ))(s) = l(gλ(t − µ))(s) · l(gλ(t − ρ))(s) and the both factors are computed in the proof of lemma 6.1. cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 83 proposition 4. for positive µ, λ ∈ r and ρ ∈ r, if ρ > 0, then (1)+ lim λ→∞ (e−µt 2 ∗ gλ(t − ρ))(t) =    2e−µ(t−ρ) 2 t > ρ, 1 t = ρ, 0 t < ρ, and if ρ < 0, then (1)− lim λ→∞ (e−µt 2 ∗ gλ(t − ρ))(t) =    0 t > ρ, −1 t = ρ, −2e−µ(t−ρ) 2 t < ρ; and for n ∈ n and ρ ∈ r, if ρ > 0, then (2)+ lim λ→∞ (tn ∗ gλ(t − ρ))(t) =    2tn t > ρ, ρn t = ρ, 0 t < ρ, and if ρ < 0, then (2)− lim λ→∞ (tn ∗ gλ(t − ρ))(t) =    0 t > ρ, −ρn t = ρ, −2tn t < ρ; and for µ ∈ r, if ρ > 0, then (3)+ lim λ→∞ (eµt ∗ gλ(t − ρ))(t) =    2eµ(t−ρ) if t > ρ, 1 if t = ρ, 0 if t < ρ, and if ρ < 0, then (3)− lim λ→∞ (eµt ∗ gλ(t − ρ))(t) =    0 if t > ρ, −1 if t = ρ, −2eµ(t−ρ) if t < ρ; and for µ ∈ r, if ρ > 0, then (4)+ lim λ→∞ (sin µt ∗ gλ(t − ρ)(t) =    2 sin µ(t − ρ) if t > ρ, 0 if t = ρ, 0 if t < ρ, and if ρ < 0, then (4)− lim λ→∞ (sin µt ∗ gλ(t − ρ)(t) =    0 if t > ρ, 0 if t = ρ, −2 sin µ(t − ρ) if t < ρ; 84 takahiro sudo cubo 17, 3 (2015) and if ρ > 0, then (5)+ lim λ→∞ (cos µt ∗ gλ(t − ρ))(t) =    2 cos µ(t − ρ) if t > ρ, 1 if t = ρ, 0 if t < ρ, and if ρ > 0, then (5)− lim λ→∞ (cos µt ∗ gλ(t − ρ))(t) =    0 if t > ρ, −1 if t = ρ, −2 cos µ(t − ρ) if t < ρ. proof. for (1)± we compute (e−µt 2 ∗ gλ(t − ρ))(t) = ∫t 0 2 √ λ π e−λ(t−τ−ρ) 2 e−µτ 2 dτ = e −λµ λ+µ (t−ρ) 2 ∫t 0 2 √ λ π e −(λ+µ)(τ− λ(t−ρ) λ+µ ) 2 dτ = e −λµ λ+µ (t−ρ) 2 ∫√λ+µ(t− λ(t−ρ) λ+µ ) − λ(t−ρ) √ λ+µ 2 √ λ π(λ + µ) e−u 2 du where u = √ λ + µ(τ− λ(t−ρ) λ+µ ) and if τ = t, then u = µt+λρ√ λ+µ , which goes to ∞ as λ → ∞ for ρ > 0. if t > ρ, then − λ(t−ρ)√ λ+µ goes to −∞ and if t < ρ, then it goes to ∞. therefore, if ρ > 0, then lim λ→∞ (e−µt 2 ∗ gλ(t − ρ))(t) =    2e−µ(t−ρ) 2 t > ρ, 1 t = ρ, 0 t < ρ, and if ρ < 0, then lim λ→∞ (e−µt 2 ∗ gλ(t − ρ))(t) =    0 t > ρ, −1 t = ρ, −2e−µ(t−ρ) 2 t < ρ, for (2)± we next compute (tn ∗ gλ(t − ρ))(t) = n∑ k=0 ( n k ) tn−k(−1)k ∫t 0 τk2 √ λ π e−λ(τ−ρ) 2 dρ and the following integrals are converted as: ∫t 0 τke−λ(τ−ρ) 2 dτ = ∫√λ(t−ρ) − √ λρ (ρ + u √ λ )ke−u 2 du √ λ cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 85 where u = √ λ(τ − ρ). we then use proposition 2 and it follows that if ρ > 0, then lim λ→∞ (tn ∗ gλ(t − ρ))(t) =    2tn t > ρ, ρn t = ρ, 0 t < ρ, and if ρ < 0, then lim λ→∞ (tn ∗ gλ(t − ρ))(t) =    0 t > ρ, −ρn t = ρ, −2tn t < ρ. for (3)± we next compute (eµt ∗ gλ(t − ρ))(t) = ∫t 0 eµ(t−τ)2 √ λ π e−λ(τ−ρ) 2 dτ = e µ 2−4λρµ 4λ eµt ∫t 0 2 √ λ π e−λ(τ+ µ−2λρ 2λ ) 2 dτ = e µ2−4λρµ 4λ eµt ∫√λ(t−ρ+ µ 2λ ) µ−2λρ 2 √ λ 2 √ π e−u 2 du (u = √ λ(τ + µ − 2λρ 2λ )), which goes to, if ρ > 0, then    2eµ(t−ρ) if t > ρ, 1 if t = ρ, 0 if t < ρ as λ → ∞, and if ρ < 0, then    0 if t > ρ, −1 if t = ρ, −2eµ(t−ρ) if t < ρ. for (4)± we finally compute by allowing complex coefficients for this case and using the euler formula as: (sin µt ∗ gλ(t − ρ))(t) = ( eµti − e−µti 2i ∗ gλ(t − ρ))(t) = 1 2i ∫t 0 eµ(t−τ)igλ(τ − ρ)dτ − 1 2i ∫t 0 e−µ(t−τ)igλ(τ − ρ)dτ = eµti 2i ∫t 0 e−µτigλ(τ − ρ)dτ − e−µti 2i ∫t 0 eµτigλ(τ − ρ)dτ 86 takahiro sudo cubo 17, 3 (2015) and the first integral is converted to e−ρµi− µ 2 4λ ∫t 0 2 √ λ π e−λ(τ−ρ+ µi 2λ ) 2 dτ = e−ρµi− µ 2 4λ ∫√λ(t−ρ+ µi 2λ ) √ λ(−ρ+ µi 2λ ) 2 √ π e−u 2 du (u = √ λ(τ − ρ + µi 2λ )) which goes to, if ρ > 0, then,    2e−ρµi if t > ρ, e−ρµi if t = ρ, 0 if t < ρ as λ → ∞ via cauchy integral theorem, and if ρ < 0, then,    0 if t > ρ, −e−ρµi if t = ρ, −2e−ρµi if t < ρ. similarly, the second integral above is converted to have the similar limits as above, where e−ρµi is replaced with eρµi since u is replaced with u = √ λ(τ − ρ − µi 2λ ). therefore, we obtain one of the last two equations in the statement. the other case (5)± is obtained similarly, by cos µt = e µti +e −µti 2 . remark. the heaviside unit function u(t) on r is defined by u(t) = 1 if t > 0 and u(t) = 0 if t < 0 as well as u(0) = 1 2 . for ρ ∈ r positive, the shifted heaviside unit function u(t − ρ) is defined similarly. it is known that l(u(t − ρ))(s) = e−ρs s . we may extend the definition of u(t − ρ) for ρ < 0. but we always have l(u(t − ρ))(s) = 1 s , ρ ≤ 0. note also that l−1( 1 s · e−ρs) = ∫t 0 1 · δ(τ − ρ)dτ = { 1 t > ρ, 0 t < ρ, where note that this shifted delta function comes from the usual gaussian functions, not equal to ours multiplied by 2. cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 87 7 algebraic structure with the shifted delta functions we denote by gsδ(r) the algebra generated by g(r) and the shifted delta functions δ(t − µ) for µ ∈ r, with convolution. for convenience, we define gsδ(r+) to be the algebra generated by g δ(r) and the positive shifted delta functions δ(t − µ) for µ > 0, restricted to [0, ∞). we define the following convolution as (f ∗ δ(t − µ))(t) = u(t − µ)f(t − µ) for f ∈ gsδ(r+) and µ > 0, where u(t − µ) is the shifted heaviside unit function on [0, ∞). note that this definition is not perfectly compatible with proposition 4 but either changing gλ(t − µ) with it multiplied by 1 2 on [0, ∞) or cutting off functions to be zero on (−∞, 0) make sense almost everywhere. in fact, such a scalar multiplication is allowed to define the convolution with the shifted delta functions in the weak sense. we denote by es1(c) the algebra generated by e1(c) and the set {e−ρs | ρ > 0}, with pointwise mulitiplication. we define the laplace transform l(f ∗ g) to be l(f) · l(g) for f, g ∈ gsδ(r+). note that it is known as the first shift law that l(u(t − µ)f(t − µ)) = e−µsl(f)(s) and l(δ(t − µ)) = e−µs. corollary 5. the algebra gsδ(r+) with convolution is isomorphic to e s1(c) with pointwise multiplication, as a real algebra, via the laplace transform and the inverse laplace transform. proof. the laplace transform l is injective on gδ(r). the injectiveness of l extends to gsδ(r) by definition. also, gsδ(r+) is mapped onto e s1(c) by l as an algebra homomorphism, by definition. note that, by definition, gsδ(r+) contains discontinuous functions such as the shifted heaviside unit functions. lemma 7.1. for f, g ∈ gsδ(r+) and for µ, s > 0, we have (u(t − µ)f(t − µ) ∗ u(t − s)g(t − s))(t) = { 0 if t < µ or t < s, ∫t−u s f(t − τ − µ)g(τ − s)dτ if t > µ and t > s. note that in the second non-trivial case, the integral is just the sub-integral of (f(t−µ)∗g(t− s))(t) = ∫t 0 f(t − τ − µ)g(τ − s)dτ restricted to the sub-interval [s, t − µ] of [0, t], so that we may call the convolution in the statement the sub-convolution. therefore, 88 takahiro sudo cubo 17, 3 (2015) corollary 6. the convolution in gsδ(r+) generates such sub-convolutions, which generate the algebra. remark. however, the convolution product in gsδ(r+) is somewhat difficult to see what it is, but the corresponding point-wise multiplication in es1(c) is relatively easy to see what it is. as well known in applications to solving ordinary differential equations with initial values, a problem related to gsδ(r+) is relatively easily solved in e s1(c) via the laplace transform, and then the last task is to compute the corresponding convolution via the inverse laplace transform. 8 more with the weak derivative recall that lemma 8.1. we have l(δ′(t)) = s in the weak sense, i.e., as a functional or a distribution, so that we may define l−1(s) = δ′(t) for t ∈ r and s ∈ c. proof. we compute l(δ′(t)) = ∫ ∞ 0 e−stδ′(t)dt = ∫ ∞ 0 (−1) ∂ ∂t e−st · δ(t)dt by definition = sl(δ(t)) = s. note that the algebra g(r) is closed under the usual derivative and that g(r) is contained in c∞(r) the algebra of all infinitely many times differentiable (or smooth) functions on r. lemma 8.2. for f ∈ g(r), we have (f ∗ δ′)(t) = f′(t) in the weak sense. proof. we compute (f ∗ δ′)(t) = ∫t 0 f(t − τ)δ′(τ)dτ = ∫t 0 (−1) ∂ ∂τ f(t − τ) · δ(τ)dτ = (f′ ∗ δ)(t) = f′(t). cubo 17, 3 (2015) computing the laplace transform and the convolution for more . . . 89 we define the n-th derivative δ(n) to be the n-times convolution with δ. for f ∈ g(r), we define f ∗ δ(n) to be f(n) the n-th derivative of f. we also define l(δ(n)) to be sn for s ∈ c. we denote by da(r) the algebra generated by a(r), the delta function δ, and the set {δ(n) | n ∈ z, n ≥ 1}, with convolution. we also denote by r(c) the algebra of all rational functions on c with point-wise multiplication. proposition 5. there is an algebra isomorphism by the laplace transform from da(r) to r(c). we denote by dg(r) the algebra defined by replacing a(r) with g(r) in the definition of da(r). we also denote by re(c) the algebra generated by r(c) and e(c). proposition 6. there is an algebra isomorphism by the laplace transform from dg(r) to re(c). lemma 8.3. for f ∈ gsδ(r+) and µ > 0, we have (f ∗ δ′(t − µ))(t) = u(t − µ)f′(t) almost everywhere in the weak sense. proof. we compute (f ∗ δ′(t − µ))(t) = ∫t 0 f(t − τ)δ′(τ − µ)dτ = ∫t 0 (−1) ∂ ∂τ f(t − τ) · δ(τ − µ)dτ = (f′ ∗ δ(t − µ))(t) = u(t − µ)f′(t) for f ∈ gsδ(r+), where note that f′ is defined almost everywhere (except some finite points of r+). we denote by dgsδ(r+) the algebra generated by g sδ(r+) and the weak derivatives δ (n)(t−µ) for n positive integers and µ > 0 reals. we also denote by res1(c) the algebra generated by r(c) and es1(c). proposition 7. there is an algebra isomorphism by the laplace transform from dgsδ(r+) to res1(c). received: may 2013. accepted: may 2013. 90 takahiro sudo cubo 17, 3 (2015) references [1] hiroshi fukawa, laplace transformation and ordinary differential equations (in japanese), sho-ko-dou (1995). [2] m. s. j., mathematics dictionary, (sugaku jiten, in japanese), math. soc. japan, 4th edition, iwanami (2007). [3] t. sudo, computing the inverse laplace transform for rational functions vanishing at infinity, cubo a math. j., 16, no 3 (2014), 97-117. [4] a. vretblad, fourier analysis and its applications, gtm 223, springer (2003). introduction adjoining gaussian functions algebraic structure with gaussian functions adjoining the delta function algebraic structure with the delta function adjoining the shifted delta functions algebraic structure with the shifted delta functions more with the weak derivative cubo a mathematical journal vol.19, no¯ 01, (17–38). march 2017 hilfer and hadamard functional random fractional differential inclusions säıd abbas1, mouffak benchohra2, jamal-eddine lazreg2 and gaston m. n’guérékata3 1 laboratory of mathematics, university of säıda, p.o. box 138, säıda 20000, algeria 2 laboratory of mathematics, djillali liabes university of sidi bel-abbes, p.o. box 89, sidi bel-abbès 22000, algeria. 3 department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore m.d. 21252, usa. benchohra@univ-sba.dz, lagregjamal@yahoo.fr, gaston.n’guerekata@morgan.edu abstract this paper deals with some existence and ulam stability results for some functional differential inclusions of hilfer and hilfer-hadamard type with convex and non-convex right hand side. we employ some multi-valued random fixed point theorems for the existence of random solutions. next we prove that our problems are generalized ulamhyers-rassias stable. resumen este art́ıculo estudia algunos resultados de existencia y estabilidad de ulam para algunas inclusiones funcionales diferenciales de tipos hilfer y hilfer-hadamard con lado derecho convexo y no-convexo. empleamos algunos teoremas aleatorios de punto fijo multi-valuados para la existencia de soluciones aleatorias. a continuación demostramos que nuestros problemas son ulam-hyers-rassias estables generalizados. keywords and phrases: functional random differential inclusion; left-sided riemann-liouville integral of fractional order; left-sided hadamard integral of fractional order; hilfer fractional derivative; convex; non-convex; hilfer-hadamard fractional derivative; existence; ulam stability; random solution; fixed point. 2010 ams mathematics subject classification: 26a33. 34a60. 18 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) 1 introduction fractional calculus is relative to the traditional integer order calculus put forward, which is the order of calculus from integer orders extended to any order of the mathematical promotion. from the theoretical point of view, the fractional differential calculus signal processing order extended to any number from an integer, the ways and means of information processing were extended. fractional order differential equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering, and other applied sciences [19, 35]. for some fundamental results in the theory of fractional calculus and fractional differential equations we refer the reader to the monographs of abbas et al. [7, 8], kilbas et al. [26] and zhou [41, 42], the papers by abbas et al. [1, 4, 5, 9, 10], benchohra et al. [11], and the references therein. the nature of a dynamic system in engineering or natural sciences depends on the accuracy of the information we have concerning the parameters that describe that system. if the knowledge about a dynamic system is precise then a deterministic dynamical system arises. unfortunately in most cases the available data for the description and evaluation of parameters of a dynamic system are inaccurate, imprecise or confusing. in other words, evaluation of parameters of a dynamical system is not without uncertainties. when our knowledge about the parameters of a dynamic system are of statistical nature, that is, the information is probabilistic, the common approach in mathematical modeling of such systems is the use of random differential equations or stochastic differential equations. random differential equations, as natural extensions of deterministic ones, arise in many applications and have been investigated by many mathematicians. we refer the reader to the monographs [12, 27, 37]. the stability of functional equations was originally raised by ulam [38]). next by hyers [21]. thereafter, this type of stability is called the ulam-hyers stability. in 1978, rassias [32] provided a remarkable generalization of the ulam-hyers stability of mappings by considering variables. the concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. considerable attention has been given to the study of the ulam-hyers and ulam-hyers-rassias stability of all kinds of functional equations; one can see the monographs of [8, 22], and the papers of abbas et al. [1, 2, 3, 4, 6, 9, 10], petru et al. [29], and rus [33, 34] discussed the ulam-hyers stability for operatorial equations and inclusions. more details from historical point of view, and recent developments of such stabilities are reported in [23, 33]. recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with hilfer fractional derivative; see [15, 16, 19, 20, 24, 36, 39]. motivated by the above papers, in this article we discuss the existence and the ulam stability of solutions for the following problem of random hilfer fractional differential cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 19 inclusions of the form ⎧ ⎪⎨ ⎪⎩ (d α,β 0 u)(t, w) ∈ f(t, u(t, w), w); t ∈ i := [0, t], (i 1−γ 0 u)(t, w)|t=0 = φ(w), w ∈ ω, (1.1) where α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, t > 0, (ω, a) is a measurable space, φ : ω → r is a measurable function, f : i × r → p(r) is a given multivalued map, p(r) is the family of all nonempty subsets of r, i 1−γ 0 is the left-sided riemann-liouville integral of order 1 − γ, and d α,β 0 is the hilfer fractional derivative of order α and type β. next, we consider the following problem of random hilfer-hadamard fractional differential inclusions of the form ⎧ ⎪⎨ ⎪⎩ (hd α,β 1 u)(t, w) ∈ g(t, u(t, w), w); t ∈ [1, t], (hi 1−γ 1 u)(1, w) = φ0(w), w ∈ ω, (1.2) where α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, t > 1, φ0 : ω → r is a measurable function, g : [1, t] × r → p(r) is a given multivalued map, hi1−γ 1 is the left-sided hadamard integral of order 1 − γ, and hd α,β 1 is the hilfer-hadamard fractional derivative of order α and type β. 2 preliminaries let c be the banach space of all continuous functions v from i into r with the supremum (uniform) norm ∥v∥∞ := sup t∈i |v(t)|. as usual, ac(i) denotes the space of absolutely continuous functions from i into r. we denote by ac1(i) the space defined by ac1(i) := {w : i → r : d dt w(t) ∈ ac(i)}. by l1(i), we denote the space of lebesgue-integrable functions v : i → r with the norm ∥v∥1 = ∫ t 0 |v(t)|dt. let l∞(i) be the banach space of measurable functions u : i → r which are essentially bounded, equipped with the norm ∥u∥l∞ = inf{c > 0 : |u(t)| ≤ c, a.e. t ∈ i}. by cγ(i) and c 1 γ (i), we denote the weighted spaces of continuous functions defined by cγ(i) = {w : (0, t] → r : t 1−γw(t) ∈ c}, 20 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) with the norm ∥w∥cγ := sup t∈i |t1−γw(t)|, and c1 γ (i) = {w ∈ c : dw dt ∈ cγ}, with the norm ∥w∥c1 γ := ∥w∥∞ + ∥w ′∥cγ. throughout this paper, we denote ∥w∥cγ by ∥w∥c. for each u ∈ cγ and w ∈ ω, define the set of selections of f by sf◦u(w) = {v : ω → l 1(i) : v(t, w) ∈ f(t, u(t, w), w); t ∈ i}. let e be a banach space, and denote pcl(e) = {a ∈ p(e) : a closed}, pcp,c(e) = {a ∈ p(e) : a compact and convex}. consider hd : p(e) × p(e) −→ [0, ∞) ∪ {∞} given by hd(a, b) = max { sup a∈a d(a, b), sup b∈b d(a, b) } , where d(a, b) = inf a∈a d(a, b), d(a, b) = inf b∈b d(a, b). then (pbd,cl(e), hd) is a hausdorff metric space. definition 1. a multifunction f : ω → e is called ameasurable if, for any open subset b of e, the set f−1(b) = {w ∈ ω : f(w) ∩ b ̸= ∅} ∈ a. note that if f(w) ∈ pcl(e) for all w ∈ ω, then f is measurable if and only if f−1(d) ∈ a for all d ∈ pcl(e). a measurable operator u : ω → e is called a measurable selector for a measurable multifunction f : ω → e, if u(w) ∈ f(w). let m ∈ pcl(e), then a mapping f : ω × m → e is called a random operator if, for each u ∈ m, the mapping f(., u) : ω → e is measurable. an operator u : ω → e is said to be a random fixed point of f if u is measurable and u(w) ∈ f(w, u(w)) for all w ∈ ω. definition 2. a multifunction f : ω × e → p(e) is called carathéodory if f(·, u) is measurable for all u ∈ e and f(w, ·) is continuous for all w ∈ ω. definition 3. a multivalued map f : i × e × ω → pcp(e) is said to be random carathéodory if (i) (t, w) *−→ f(t, u, w) is jointly measurable for each u ∈ e; and (ii) u *−→ f(t, u, w) is hausdorff continuous for almost each t ∈ i, w ∈ ω. definition 4. [17] let e be a separable banach space. if f : i× e → pcp(e) is carathéodory, then the multivalued mapping (t, u(t)) → f(t, u(t)), is jointly measurable for any measurable e-valued function u on i. cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 21 definition 5. a multivalued random operator n : ω × e → pcl(e) is called multivalued random contraction if there is a measurable function k : ω → [0, ∞) such that hd(n(w)u, n(w)v) ≤ k(w)∥u − v∥e, for all u, v ∈ e and w ∈ ω, where k(w) ∈ [0, 1) on ω. now, we give some results and properties of fractional calculus. definition 6. [7, 26] the riemann-liouville integral of order r > 0 of a function w ∈ l1(i) is defined by (ir0w)(t) = 1 γ(r) ∫t 0 (t − s)r−1w(s)ds; for a.e. t ∈ i, where γ(·) is the (euler’s) gamma function defined by γ(ξ) = ∫ ∞ 0 tξ−1e−tdt; ξ > 0. notice that for all r, r1, r2 > 0 and each w ∈ c, we have i r 0 w ∈ c, and (ir1 0 ir2 0 w)(t) = (ir1+r2 0 w)(t); for a.e. t ∈ i. definition 7. [7, 26] the riemann-liouville fractional derivative of order r ∈ (0, 1] of a function w ∈ l1(i) is defined by (dr 0 w)(t) = ! d dt i1−r 0 w " (t) = 1 γ(1 − r) d dt ∫t 0 (t − s)−rw(s)ds; for a.e. t ∈ i. let r ∈ (0, 1], γ ∈ [0, 1) and w ∈ c1−γ(i). then the following expression leads to the left inverse operator as follows. (dr 0 ir 0 w)(t) = w(t); for all t ∈ (0, t]. moreover, if i1−r 0 w ∈ c1 1−γ (i), then (ir0d r 0w)(t) = w(t) − (i1−r 0 w)(0+) γ(r) tr−1; for all t ∈ (0, t]. definition 8. [7, 26] the caputo fractional derivative of order r ∈ (0, 1] of a function w ∈ ac(i) is defined by (cdr0w)(t) = ! i1−r 0 d dt w " (t) = 1 γ(1 − r) ∫t 0 (t − s)−r d ds w(s)ds; for a.e. t ∈ i. 22 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) in [19], r. hilfer studied applications of a generalized fractional operator having the riemannliouville and the caputo derivatives as specific cases (see also [20, 24, 36]. definition 9. (hilfer derivative). let α ∈ (0, 1), β ∈ [0, 1], w ∈ l1(i), i (1−α)(1−β) 0 ∈ ac1(i). the hilfer fractional derivative of order α and type β of w is defined as (d α,β 0 w)(t) = ! i β(1−α) 0 d dt i (1−α)(1−β) 0 w " (t); for a.e. t ∈ i. (2.1) properties. let α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, and w ∈ l1(i). 1. the operator (d α,β 0 w)(t) can be written as (d α,β 0 w)(t) = ! i β(1−α) 0 d dt i 1−γ 0 w " (t) = # i β(1−α) 0 d γ 0 w $ (t); for a.e. t ∈ i. moreover, the parameter γ satisfies γ ∈ (0, 1], γ ≥ α, γ > β, 1 − γ < 1 − β(1 − α). 2. the generalization (2.1) for β = 0, coincides with the riemann-liouville derivative and for β = 1 with the caputo derivative. dα,00 = d α 0 , and d α,1 0 = cdα0 . 3. if d β(1−α) 0 w exists and in l1(i), then (d α,β 0 iα0 w)(t) = (i β(1−α) 0 d β(1−α) 0 w)(t); for a.e. t ∈ i. furthermore, if w ∈ cγ(i) and i 1−β(1−α) 0 w ∈ c1 γ (i), then (d α,β 0 iα0 w)(t) = w(t); for a.e. t ∈ i. 4. if d γ 0 w exists and in l1(i), then (iα0 d α,β 0 w)(t) = (i γ 0 d γ 0 w)(t) = w(t) − i 1−γ 0 (0+) γ(γ) tγ−1; for a.e. t ∈ i. corolary 1. let h ∈ cγ(i). then the cauchy problem ⎧ ⎪⎨ ⎪⎩ (d α,β 0 u)(t) = h(t); t ∈ i, (i 1−γ 0 u)(t)|t=0 = φ, has a unique solution u ∈ l1(i) given by u(t) = φ γ(γ) tγ−1 + (iα0 h)(t). cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 23 from the above corollary, we conclude the following lemma. lemma 2.1. let f : i× r×ω → p(r) be such that sf◦u(w) ⊂ cγ for any u ∈ cγ. then problem (1.1) is equivalent to the problem of the solutions of the integral equation u(t, w) = φ(w) γ(γ) tγ−1 + (iα0 v)(t, w), where v ∈ sf◦u(w). now, we consider the ulam stability for the problem (1.1). let ϵ > 0 and φ : i × ω → [0, ∞) be a continuous function. we consider the following inequalities hd((d α,β 0 u)(t, w), f(t, u(t, w), w)) ≤ ϵ; t ∈ i, w ∈ ω. (2.2) hd((d α,β 0 u)(t, w), f(t, u(t, w), w)) ≤ φ(t, w); t ∈ i, w ∈ ω. (2.3) hd((d α,β 0 u)(t, w), f(t, u(t, w), w)) ≤ ϵφ(t, w); t ∈ i, w ∈ ω. (2.4) definition 10. [7, 33] the problem (1.1) is ulam-hyers stable if there exists a real number cf > 0 such that for each ϵ > 0 and for each random solution u : ω → cγ of the inequality (2.2) there exists a random solution v : ω → cγ of (1.1) with |u(t, w) − v(t, w)| ≤ ϵcf; t ∈ i, w ∈ ω. definition 11. [7, 33] the problem (1.1) is generalized ulam-hyers stable if there exists cf : c([0, ∞), [0, ∞)) with cf(0) = 0 such that for each ϵ > 0 and for each random solution u : ω → cγ of the inequality (2.2) there exists a random solution v : ω → cγ of (1.1) with |u(t, w) − v(t, w)| ≤ cf(ϵ); t ∈ i, w ∈ ω. definition 12. [7, 33] the problem (1.1) is ulam-hyers-rassias stable with respect to φ if there exists a real number cf,φ > 0 such that for each ϵ > 0 and for each random solution u : ω → cγ of the inequality (2.4) there exists a random solution v : ω → cγ of (1.1) with |u(t, w) − v(t, w)| ≤ ϵcf,φφ(t, w); t ∈ i, w ∈ ω. definition 13. [7, 33] the problem (1.1) is generalized ulam-hyers-rassias stable with respect to φ if there exists a real number cf,φ > 0 such that for each random solution u : ω → cγ of the inequality (2.3), there exists a random solution v : ω → cγ of (1.1) with |u(t, w) − v(t, w)| ≤ cf,φφ(t, w); t ∈ i, w ∈ ω. remark 1. it is clear that (i) definition 10 ⇒ definition 11, 24 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) (ii) definition 12 ⇒ definition 13, (iii) definition 12 for φ(., .) = 1 ⇒ definition 10. one can have similar remarks for the inequalities (2.2) and (2.4). in the sequel, we employ the following random multi-valued fixed point theorems: theorem 2.1. [14]] let (ω, a) be a complete σ-finite measure space, x be a separable banach space, m(ω, x) be the space of all measurable x-valued functions defined on ω, and let n : ω×x → pcp,cv(x) be a continuous and condensing multi-valued random operator. if the set {u ∈ m(ω, x) : λu ∈ n(w)u} is bounded for each w ∈ ω and all λ > 1, then n(w) has a random fixed point. theorem 2.2. [28] let (ω, a) be a complete σ-finite measure space, e a separable banach space, and let n : ω× e → pcl(e) be a random multi-valued contraction. then n(w) has a random fixed point. we recall an integral inequality which based on an iteration argument. lemma 2.2. [40] suppose β > 0, a(t) is a nonnegative function locally integrable on 0 ≤ t < t (some t ≤ +∞) and g(t) is a nonnegative, nondecreasing continuous function defined on 0 ≤ t < t, g(t) ≤ m (constant), and suppose u(t) is nonnegative and locally integrable on 0 ≤ t < t with u(t) ≤ a(t) + g(t) ∫t 0 (t − s)β−1u(s)ds on this interval. then u(t) ≤ a(t) + ∫ t 0 % ∞∑ n=1 (g(t)γ(β))n γ(nβ) (t − s)nβ−1a(s) & ds, 0 ≤ t < t. from the above lemma, we concluded with the following lemma. lemma 2.3. suppose β > 0, a(t, w) is a nonnegative function locally integrable on [0, t) × ω (some t ≤ +∞) and g(t, w is a nonnegative, nondecreasing continuous function with respect to t defined on [0, t) × ω, g(t, w) ≤ m (constant), and suppose u(t, w) is nonnegative and locally integrable with respect to t on [0, t) × ω with u(t, w) ≤ a(t, w) + g(t, w) ∫t 0 (t − s)β−1u(s, w)ds on [0, t) × ω. then u(t, w) ≤ a(t, w) + ∫t 0 % ∞∑ n=1 (g(t, w)γ(β))n γ(nβ) (t − s)nβ−1a(s, w) & ds, (t, w) ∈ [0, t) × ω. cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 25 3 hilfer random fractional differential inclusions in this section, we are concerned with the existence and the ulam-hyers-rassias stability for problem (1.1). let us start by defining what we mean by a random solution of the problem (1.1). definition 14. by a random solution of the problem (1.1) we mean a measurable function u : ω → cγ that satisfies the condition (i 1−γ 0 u)(0+, w) = φ(w), and the equation (d α,β 0 u)(t, w) = v(t, w) on i × ω, where v ∈ sf◦u(w). 3.1 the convex case we present now some existence and ulam stabilities results for the problem (1.1) with convex valued right hand side. the following hypotheses will be used in the sequel. (h1) the multifunction f : i × r × ω → pcp,cv(r) is random carathéodory on i × r × ω, (h2) there exists a measurable and bounded function l : ω → l ∞(i, [0, ∞)) satisfying for each w ∈ ω, hd(f(t, u, w), f(t, u, w)) ≤ t 1−γl(t, w)|u − u|; for every t ∈ i and u, u ∈ r. and d(0, f(t, 0, w)) ≤ t1−γl(t, w); for t ∈ i, (h3) there exists λφ > 0 such that for each t ∈ i, and w ∈ ω, we have ∫ t 0 % ∞∑ n=1 (l∗)n γ(nα) (t − s)nα−1φ(s, w) & ds ≤ λφφ(t, w). remark 2. for each u : ω → c, the set sf,u(w) is nonempty since by (h1), f has a measurable selection (see [13], theorem iii.6). remark 3. the hypothesis (h2) implies that, for every t ∈ i, u ∈ r and w ∈ ω, we get hd(f(t, u, w), f(t, 0, w)) ≤ l(t, w)|u|, and hd(0, f(t, u, w)) ≤ hd(0, f(t, 0, w)) + hd(f(t, u, w), f(t, 0, w)) ≤ l(t, w)(1 + |u|). set l∗ = sup w∈ω ∥l(w)∥l∞ and φ ∗ = sup w∈ω |φ(w)|. 26 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) theorem 3.1. assume that the hypotheses (h1) and (h2) hold. then, the problem (1.1) has a random solution defined on i × ω. proof. define a multivalued operator n : ω × cγ → p(cγ) by: (n(w)u)(t) = { h : ω → cγ : h(t, w) = φ(w) γ(γ) tγ−1 + (iα0 v)(t, w); t ∈ i, v ∈ sf◦u(w) } . (3.1) the map φ is measurable for all w ∈ ω. again, as the indefinite integral is continuous on i, for each v ∈ sf◦u(w), then n(w) defines a multivalued mapping n : ω × cγ → p(cγ). thus u is a random solution for the problem (1.1) if and only if u ∈ n(w)u. we shall show that the multivalued operator n satisfies all conditions of theorem 2.1. the proof will be given in several steps. step 1. n(w) is a multi-valued random operator on c. since f(t, u, w) is random carathéodory, the map w → f(ty, u, w) is measurable in view of definition 4. similarly, the product (t − s)α−1v(s, w) of a continuous function and a measurable multifunction is again measurable for each v ∈ sf◦u(w). further, the integral is a limit of a finite sum of measurable functions, therefore, the map w *→ φ(w) γ(γ) tγ−1 + ∫ t 0 (t − s)α−1 γ(α) v(t, w)ds, is measurable. as a result, n(w) is a multi-valued random operator on cγ. step 2. n(w)u ∈ pcv(cγ) for each u ∈ cγ. indeed, if h1, h2 belong to n(w)u, then there exist v1, v2 ∈ sf◦u(w) such that for each t ∈ i and w ∈ ω, we have hi(t, w) = φ(w) γ(γ) tγ−1 + (iα0 vi)(t, w); i = 1, 2. let 0 ≤ d ≤ 1. then, for each t ∈ i and w ∈ ω, we get (dh1 + (1 − d)h2)(t, w) = φ(w) γ(γ) tγ−1 + (iα0 [dv1 + (1 − d)v2])(t, w). since sf◦u(w) is convex (because f has convex values), we get dh1 + (1 − d)h2 ∈ n(u). step 3. n(w) is continuous and n(w)u ∈ pcp(cγ) for each u ∈ cγ. the proof of this step will be given in several claims. claim 1: n(w) is continuous. let {un} be a sequence such that un → u in cγ. then from (h2), for each t ∈ i and w ∈ ω, we cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 27 have hd(f(t, un(t, w), w), f(t, u(t, w), w)) ≤ t1−γl(t, w)|un(t, w) − u(t, w)| ≤ l∗∥un − u∥c → 0 as n → ∞. thus, we obtain hd(f(t, un(t, w), w), f(t, u(t, w), w)) → 0 as n → ∞. claim 2: n(w) maps bounded sets into bounded sets in cγ. let bη∗ = {u ∈ cγ : ∥u∥c ≤ η ∗} be bounded set in cγ, and u ∈ bη∗ . then for each h ∈ n(w)u, there exists v ∈ sf◦u(w) such that h(t, w) = φ(w) γ(γ) tγ−1 + (iα 0 v)(t, w). by (h2), for each t ∈ i and w ∈ ω, we obtain |t1−γh(t, w)| ≤ |φ(w)| γ(γ) + t1−γ ∫t 0 (t − s)α−1 γ(α) |v(s, w)|ds ≤ |φ(w)| γ(γ) + t1−γ ∫t 0 (t − s)α−1 γ(α) |s1−γl(s, w)(1 + v(s, w))|ds ≤ φ∗ γ(γ) + l∗t1−γ ∫t 0 (t − s)α−1 γ(α) (t1−γ + ∥v(s, w)∥c)ds ≤ φ∗ γ(γ) + l∗t1+α−γ γ(1 + α) (t1−γ + η∗) := ℓ. claim 3: n(w) maps bounded sets into equicontinuous sets in cγ. let t1), t2 ∈ i, t1 < t2, and let bη∗ be a bounded set of cγ as in claim 2, and let u ∈ bη∗ and h ∈ n(w)u. then, there exists v ∈ sf◦u(w) such that for each w ∈ ω, we get |t 1−γ 2 h(t2, w) − t 1−γ 1 h(t1, w)| ≤ l∗t1−γ+α γ(1 + α) (t2 − t1) α + l∗(t1−γ + η∗) γ(α) ∫t1 0 |t 1−γ 2 (t2 − s) α−1 − t 1−γ 1 (t1 − s) α−1ds. as t1 → t2, the right-hand side of the above inequality tends to zero. as a consequence of claims 1 to 3, together with the arzela-ascoli theorem, we can conclude that n(w) is continuous and completely continuous multi-valued random operator. step 4: the set e := {u ∈ cγ : λu ∈ n(w)u} is bounded for some λ > 1. let u ∈ cγ be arbitrary and let w ∈ ω be fixed such that λu ∈ n(w)u for all λ > 1. then, there exists v ∈ sf◦u(w) such that for each t ∈ i, we have u(t, w) = φ(w) λγ(γ) tγ−1 + λ−1(iα 0 v)(t, w). 28 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) this implies by (h2) that, for each t ∈ i, we get |t1−γu(t, w)| ≤ φ∗ γ(γ) + ∫ t 0 (t − s)α−1 γ(α) l(s, w)(t1−γ + |s1−γv(s, w)|)ds ≤ φ∗ γ(γ) + l∗t1−γ+α γ(1 + α) + l∗ ∫ t 0 (t − s)α−1 γ(α) |s1−γv(s, w)|ds. from lemma 2.3, for each (t, w) ∈ [0, t) × ω, we have |t1−γu(t, w)| ≤ ' φ∗ γ(γ) + l∗t1−γ+α γ(1 + α) (% 1 + ∫ t 0 % ∞∑ n=1 (l∗)n γ(nα) (t − s)nα−1 & ds & ≤ ' φ∗ γ(γ) + l∗t1−γ+α γ(1 + α) (% 1 + ∞∑ n=1 tnα γ(1 + nα) & := m. thus, for all t ∈ i and w ∈ ω, we obtain ∥u∥∞ ≤ m. as a consequence of steps 1 to 4, together with the theorem 2.1, n has a random fixed point u which is a random solution to problem (1.1). now, we are concerned with the generalized ulam-hyers-rassias stability of our problem (1.1). theorem 3.2. assume that the hypotheses (h1)−(h3) hold. then the problem (1.1) is generalized ulam-hyers-rassias stable. proof. let u be a random solution of the inequality (2.3), and let us assume that v is a random solution of problem (1.1). thus, we have v(t, w) = φ(w) γ(γ) tγ−1 + ∫ t 0 (t − s)α−1 fv(s, w) γ(α) ds, where fv ∈ sf◦v(w). from the inequality (2.3) for each t ∈ i, and w ∈ ω, we have ) ) ) )u(t, w) − φ(w) γ(γ) tγ−1 − ∫ t 0 (t − s)α−1 f(s, w) γ(α) ds ) ) ) ) ≤ (i α 0 φ)(t, w), where f ∈ sf◦u(w). from hypotheses (h2) and (h3), for each t ∈ i, and w ∈ ω, we get |u(t, w) − v(t, w)| ≤ ) ) ) )u(t, w) − φ(w) γ(γ) tγ−1 − ∫ t 0 (t − s)α−1 f(s, w) γ(α) ds ) ) ) ) + ∫ t 0 (t − s)α−1 |f(s, w) − fv(s, w)| γ(α) ds ≤ (iα 0 φ)(t, w) + l∗t1−γ γ(α) ∫ t 0 (t − s)α−1|u(s, w) − v(s, w)|ds. cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 29 from lemma 2.3, we obtain ∥u(t, w) − v(t, w)∥ ≤ λφ l∗ % φ(t, w) + ∫ t 0 % ∞∑ n=1 (l∗)n γ(nα) (t − s)nα−1φ(s, w) & ds & ≤ λφ l∗ [1 + λφ]φ(t, w) := cf,φφ(t, w). finally, the problem (1.1) is generalized ulam-hyers-rassias stable. 3.2 the non-convex case we present now some existence and ulam stabilities results for the problem (1.1) with non-convex valued right hand side. the following hypotheses will be used in the sequel. (h01) the multifunction f : i × r × ω → pcp(r) is random carathéodory on i × r × ω, (h02) there exists a measurable and bounded function l : ω → l ∞(i, [0, ∞)) satisfying for each w ∈ ω, hd(f(t, u, w), f(t, u, w)) ≤ t 1−γl(t, w)|u − u|; for every t ∈ i and u, u ∈ r. set l∗ = sup w∈ω ∥l(w)∥l∞ . now, we shall prove the following theorem concerning the existence of random solutions of problem (1.1). theorem 3.3. assume that the hypotheses (h01) and (h02) hold. if l∗t1+α−γ γ(1 + α) < 1, (3.2) then the problem (1.1) has at least one random solution defined on i × ω. proof. letn : ω × cγ → p(cγ) be the multivalued operator defined in (3.1). we know that n(w) is a multi-valued random operator on cγ. we shall show that the multivalued operator n satisfies all conditions of theorem 2.2. the proof will be given in two steps. 30 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) step 1. n(w)u ∈ pcl(cγ) for each u ∈ cγ. let {un}n≥0 ∈ n(w)u such that un −→ ũ in cγ. then, ũ ∈ cγ and there exists fn(·, ·, ·) ∈ sf◦u(w) be such that, for each t ∈ i and w ∈ ω, we have un(t, w) = φ(w) γ(γ) tγ−1 + (iα 0 fn)(t, w). using the fact that f has compact values and from (h01), we may pass to a subsequence if necessary to get that fn(·, ·, ·) converges to f in l 1(i), and hence f ∈ sf◦u(w). then, for each t ∈ i and w ∈ ω, we get un(t, w) −→ ũ(t, w) = φ(w) γ(γ) tγ−1 + (iα 0 f)(t, w). so, ũ ∈ n(w)u. step 2. there exists 0 ≤ λ < 1 such that, for each w ∈ ω, hd(n(w)u, n(w)u) ≤ λ∥u − u∥c for each u, u ∈ cγ. let u, u ∈ cγ and h ∈ n(w)u. then, there exists f(t, w) ∈ f(t, u(t, w), w) such that for each t ∈ i and w ∈ ω, we have h(t, w) = φ(w) γ(γ) tγ−1 + (iα 0 f)(t, w). from (h02) it follows that hd(f(t, u(t, w), w), f(t, u(t, w), w)) ≤ t 1−γl(t, w)|u(t, w) − u(t, w)|. hence, there exists v ∈ sf◦u such that |f(t, w) − v(t, w)| ≤ t1−γl(t, w)|u(t, w) − u(t, w)|. consider u : i × ω → p(r) given by u(t, w) = {v(t, w) ∈ r : |f(t, w) − v(t, w)| ≤ t1−γl(t, w)|u(t, w) − u(t, w)|}. since the multivalued operator u(t, w) = u(t, w) ∩ f(t, u(t, w), w) is measurable (see proposition iii.4 in [13]), there exists a function f(t, w) which is a measurable selection for u. so, f(t, w) ∈ f(t, u(t, w), w), and for each t ∈ i and w ∈ ω, we get |f(t, w) − f(t, w)| ≤ t1−γl(t, w)|u(t, w) − u(t, w)|. let us define for each t ∈ i and w ∈ ω, h(t, w) = φ(w) γ(γ) tγ−1 + (iα 0 f)(t, w). cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 31 then for each t ∈ i and w ∈ ω, we obtain |t1−γh(t, w) − t1−γh(t, w)| ≤ t1−γiα0 |f(t, w) − f(t, w)| ≤ t1−γ γ(α) ∫t 0 (t − s)α−1l(s, w)|s1−γu(s, w) − s1−γu(s, w)|ds ≤ l∗t1−γ∥u − u∥c γ(α) ∫t 0 (t − s)α−1ds. hence ∥h − h∥c ≤ l∗t1+α−γ γ(1 + α) ∥u − u∥c. by an analogous relation, obtained by interchanging the roles of u and u, it follows that hd(n(w)u, n(w)u) ≤ l∗t1+α−γ γ(1 + α) ∥u − u∥c. so by (3.2), n is random contraction and thus, by theorem 2.2, n has a random fixed point u which is a random solution to problem (1.1). now, we can show the following generalized ulam-hyers-rassias stability result. theorem 3.4. assume that the hypotheses (h01), (h02), (h3) and the condition (3.2) hold, then the problem (1.1) is generalized ulam-hyers-rassias stable. 4 hilfer-hadamard fractional random differential inclusions now, we are concerned with the existence and the ulam-hyers-rassias stability for problem (1.2). set c := c([1, t]). denote the weighted space of continuous functions defined by cγ,ln([1, t]) = {w(t) : (ln t) 1−γw(t) ∈ c}, with the norm ∥w∥cγ,ln := sup t∈[1,t] |(ln t)1−rw(t)|. let us recall some definitions and properties of hadamard fractional integration and differentiation. we refer to [18, 26] for a more detailed analysis. definition 15. [18, 26] (hadamard fractional integral). the hadamard fractional integral of order q > 0 for a function g ∈ l1([1, t]), is defined as (hi q 1 g)(x) = 1 γ(q) ∫x 1 # ln x s $q−1 g(s) s ds, provided the integral exists. 32 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) example 1. let 0 < q < 1. then hi q 1 ln t = 1 γ(2 + q) (ln t)1+q, for a.e. t ∈ [0, e]. set δ = x d dx , q > 0, n = [q] + 1, and acnδ := {u : [1, t] → e : δ n−1[u(x)] ∈ ac(i)}. analogous to the riemann-liouville fractional calculus, the hadamard fractional derivative is defined in terms of the hadamard fractional integral in the following way: definition 16. [18, 26] (hadamard fractional derivative). the hadamard fractional derivative of order q > 0 applied to the function w ∈ acn δ is defined as (hd q 1 w)(x) = δn(hi n−q 1 w)(x). in particular, if q ∈ (0, 1], then (hd q 1 w)(x) = δ(hi 1−q 1 w)(x). example 2. let 0 < q < 1. then hd q 1 ln t = 1 γ(2 − q) (ln t)1−q, for a.e. t ∈ [0, e]. it has been proved (see e.g. kilbas [[25], theorem 4.8]) that in the space l1(i, e), the hadamard fractional derivative is the left-inverse operator to the hadamard fractional integral, i.e. (hd q 1 )(hi q 1 w)(x) = w(x). from theorem 2.3 of [26], we have (hi q 1 )(hd q 1 w)(x) = w(x) − (hi 1−q 1 w)(1) γ(q) (ln x)q−1. analogous to the hadamard fractional calculus, the caputo-hadamard fractional derivative is defined in the following way: definition 17. (caputo-hadamard fractional derivative). the caputo-hadamard fractional derivative of order q > 0 applied to the function w ∈ acn δ is defined as (hcd q 1 w)(x) = (hi n−q 1 δnw)(x). cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 33 in particular, if q ∈ (0, 1], then (hcd q 1 w)(x) = (hi 1−q 1 δw)(x). from the hadamard fractional integral, the hilfer-hadamard fractional derivative (introduced for the first time in [30]) is defined in the following way: definition 18. (hilfer-hadamard fractional derivative). let α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, w ∈ l1(i), and hi (1−α)(1−β) 1 w ∈ ac1(i). the hilfer-hadamard fractional derivative of order α and type β applied to the function w is defined as (hd α,β 1 w)(t) = # hi β(1−α) 1 (hd γ 1 w) $ (t) = # hi β(1−α) 1 δ(hi 1−γ 1 w) $ (t); for a.e. t ∈ [1, t]. (4.1) this new fractional derivative (4.1) may be viewed as interpolating the hadamard fractional derivative and the caputo-hadamard fractional derivative. indeed for β = 0 this derivative reduces to the hadamard fractional derivative and when β = 1, we recover the caputo-hadamard fractional derivative. hdα,0 1 = hdα 1 , and hdα,1 1 = hcdα 1 . from theorem 21 in [31], we concluded the following lemma lemma 4.1. let g : [1, t]×r×ω → p(r) be such that sg◦u(w) ∈ cγ,ln([1, t]) for any u(·, w) ∈ cγ,ln([1, t]). then problem (1.2) is equivalent to the following volterra integral equation u(t, w) = φ0(w) γ(γ) (ln t)γ−1 + (hiα 1 g(·, w))(t); w ∈ ω, where g ∈ sg◦u(w). definition 19. by a random solution of the problem (1.2) we mean a measurable function u ∈ cγ,ln that satisfies the condition ( hi 1−γ 1 u)(1+, w) = φ0(w), and the equation ( hd α,β 1 u)(t, w) = g(t, w) on [1, t] × ω, where g ∈ sg◦u(w). now we give (without proof) existence and ulam slability results for problem (1.2). the following hypotheses will be used in the sequel. (h′ 1 ) the multifunction g : [1, t] × r × ω → pcp,cv(r) is random carathéodory, (h′ 2 ) there exists a measurable and bounded function l : ω → l∞([1, t], [0, ∞)) satisfying for each w ∈ ω, hd(g(t, u, w), g(t, u, w)) ≤ t 1−γl(t, w)|u − u|; for every t ∈ [1, t] and u, u ∈ r. and d(0, g(t, 0, w)) ≤ (ln t)1−γl(t, w); for t ∈ [1, t], 34 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) (h′ 3 ) there exists λφ > 0 such that for each t ∈ [1, t], and w ∈ ω, we have ∫t 1 % ∞∑ n=1 (l∗)n γ(nα) ! ln t s "nα−1 φ(s, w) & ds s ≤ λφφ(t, w). theorem 4.1. assume that the hypotheses (h′ 1 ) and (h′ 2 ) hold. then, the problem (1.1) has a random solution defined on [1, t] × ω. moreover; if the hypothesis (h′ 3 ) holds, then the problem (1.1) is generalized ulam-hyers-rassias stable. finally, we give (without proof) existence and ulam stability results for problem (1.2) with nonconvex valued right hand side. the following hypotheses will be used in the sequel. (h′ 01 ) the multifunction g : [1, t] × r × ω → pcp(r) is random carathéeodory on [1, t] × r × ω, (h′ 02 ) there exists a measurable and bounded function p : ω → l∞([1, t], [0, ∞)) satisfying for each w ∈ ω, hd(g(t, u, w), g(t, u, w)) ≤ (ln t) 1−γp(t, w)|u − u|; for every t ∈ [1, t] and u, u ∈ r. set l∗ = sup w∈ω ∥l(w)∥l∞ . theorem 4.2. assume that the hypotheses (h′ 01 ) and (h′ 02 ) hold. if l∗(ln t)1+α−γ γ(1 + α) < 1, (4.2) then the problem (1.2) has at least one random solution defined on [1, t] × ω. moreover, if the hypothesis (h′ 3 ) holds, then the problem (1.2) is generalized ulam-hyers-rassias stable. 5 examples let ω = (−∞, 0) be equipped with the usual σ-algebra consisting of lebesgue measurable subsets of (−∞, 0). example 1. consider hilfer fractional differential inclusion of the form { (d 1 2 , 1 2 0 u)(t, w) ∈ f(t, u(t, w), w); t ∈ [0, 1], (i 1 4 0 u)(0, w) = 1, w ∈ ω, (5.1) where f(t, u(t, w), w) = {v : ω → c([0, 1], r) : |f1(t, u(t, w), w)| ≤ |v(w)| ≤ |f2(t, u(t, w), w)|}; cubo 19, 1 (2017) hilfer and hadamard functional random fractional differential . . . 35 t ∈ [0, 1], w ∈ ω, with f1, f2 : [0, 1] × r × ω → r, such that f1(t, u(t, w), w) = t2u (1 + w2 + |u|)e10+t , and f2(t, u(t, w), w) = t2u (1 + w2)e10+t . set α = β = 1 2 , then γ = 3 4 . we assume that f is closed and convex valued. a simple computation shows that conditions of theorem 3.1 are satisfied. hence, the problem (5.1) has at least one random solution defined on [0, 1]. also, the hypothesis (h3) is satisfied with φ(t, w) = e3 1 + w2 , and λφ = ∞∑ n=1 e−10n γ(1 + nα) . φ(t, w) = e3 1 + w2 , and λφ = 1 γ(1 + α) . indeed, for each t ∈ [0, 1], and w ∈ ω, we get (iα 0 φ)(t, w) ≤ e3 (1 + w2) ∞∑ n=1 e−10n γ(1 + nα) . = λφφ(t, w). consequently, theorem 3.2 implies that the problem (5.1) is generalized ulam-hyers-rassias stable. example 2. consider hilfer fractional differential inclusion of the form { (d 1 2 , 1 2 0 u)(t, w) ∈ f(t, u(t, w), w); t ∈ [0, 1], (i 1 4 0 u)(0, w) = 1, w ∈ ω, (5.2) where f(t, u(t, w), w) = t2 (1 + w2 + |u|)e10+t [u − 1, u]; t ∈ [0, 1], w ∈ ω. set α = β = 1 2 , then γ = 3 4 . we assume that f is closed valued. a simple computation shows that conditions of theorem 3.3 are satisfied. hence, the problem (5.2) has at least one random solution defined on [0, 1]. also, theorem 3.4 implies that the problem (5.2) is generalized ulamhyers-rassias stable. references [1] s. abbas, w. a. albarakati, m. benchohra and j. henderson, existence and ulam stabilities for hadamard fractional integral equations with random effects, electron. j. differential equations 2016 (2016), no. 25, pp 1-12. 36 s. abbas, m. benchohra j.-e. lazreg, g. m. n’guérékata cubo 19, 1 (2017) [2] s. abbas, w. albarakati, m. benchohra and g. m. n’guérékata, existence and ulam stabilities for hadamard fractional integral equations in fréchet spaces, j. frac. calc. appl. 7 (2) (2016), 1-12. 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() cubo a mathematical journal vol.13, no¯ 03, (57–67). october 2011 some uniqueness results on meromorphic functions sharing three sets ii abhijit banerjee 1 department of mathematics, west bengal state university, barasat, 24 parganas (north), west bengal, kolkata-700126, india. email: abanerjee kal@yahoo.co.in, abanerjee kal@rediffmail.com abstract with the help of the notion of weighted sharing we investigate the uniqueness of meromorphic functions concerning three set sharing and significantly improve two results of zhang [16] and as a corollary of the main result we improve a result of the present author [2] as well. resumen con la ayuda del concepto de peso repartido, investigamos la unicidad de funciones meromorfas sobre un conjunto compartido y mejoramos significativamente dos resultados de zhang [16] y como corolario del resultado principal que mejoramos también el resultado de la autora [2]. mathematics subject classification: 30d35. keywords. meromorphic functions, uniqueness, weighted sharing, shared set. 1the author dedicates the paper to the memory of his respected teacher late prof. b. k. lahiri who first germinated the inquisition for research work in the author’s mind. 58 abhijit banerjee cubo 13, 3 (2011) 1 introduction, definitions and results in this paper by meromorphic functions we will always mean meromorphic functions in the complex plane. let f and g be two non-constant meromorphic functions and let a be a finite complex number. we shall use the standard notations of value distribution theory : t (r, f), m(r, f), n(r, ∞; f), n(r, ∞; f), . . . (see [5]). for any constant a, we define θ(a; f) = 1 − lim sup r−→ ∞ n(r, a; f) t (r, f) . we say that f and g share a cm, provided that f − a and g − a have the same zeros with the same multiplicities. similarly, we say that f and g share a im, provided that f − a and g − a have the same zeros ignoring multiplicities. in addition we say that f and g share ∞ cm, if 1/f and 1/g share 0 cm, and we say that f and g share ∞ im, if 1/f and 1/g share 0 im. let s be a set of distinct elements of c ∪ {∞} and ef(s) = ⋃ a∈s{z : f(z) − a = 0}, where each zero is counted according to its multiplicity. if we do not count the multiplicity the set ef(s) = ⋃ a∈s{z : f(z) − a = 0} is denoted by ef(s). if ef(s) = eg(s) we say that f and g share the set s cm. on the other hand if ef(s) = eg(s), we say that f and g share the set s im. in [4] gross posed the following question: can one find two finite sets sj (j = 1, 2) such that any two non-constant entire functions f and g satisfying ef(sj) = eg(sj) for j = 1, 2 must be identical ? in the last couple of years or so several attempts have been made in many papers to answer the above question under weaker hypothesis (see [1], [2], [3], [9], [10], [13], [15], [16])). a recent increment to uniqueness theory has been to considering weighted sharing instead of sharing im/cm which implies a gradual change from sharing im to sharing cm. this notion of weighted sharing has been introduced by i. lahiri around 2001 in [7, 8] and since then this notion played a vital role on the uniqueness of meromorphic or entire functions sharing sets concerning the question of gross. below we are giving the definition. definition 1.1. [7, 8] let k be a nonnegative integer or infinity. for a ∈ c ∪ {∞} we denote by ek(a; f) the set of all a-points of f, where an a-point of multiplicity m is counted m times if m ≤ k and k + 1 times if m > k. if ek(a; f) = ek(a; g), we say that f, g share the value a with weight k. we write f, g share (a, k) to mean that f, g share the value a with weight k. clearly if f, g share (a, k) then f, g share (a, p) for any integer p, 0 ≤ p < k. also we note that f, g share a value a im or cm if and only if f, g share (a, 0) or (a, ∞) respectively. definition 1.2. [7] let s be a set of distinct elements of c ∪ {∞} and k be a nonnegative integer or ∞. let ef(s, k) = ⋃ a∈s ek(a; f). cubo 13, 3 (2011) some uniqueness results on meromorphic functions . . . 59 clearly ef(s) = ef(s, ∞) and ef(s) = ef(s, 0). improving the result of lahiri-banerjee [10] and yi-lin [15] the present author have recently proved the following result. theorem a. [1] let s1 = {z : z n + azn−1 + b = 0}, s2 = {0} and s3 = {∞}, where a, b are nonzero constants such that zn + azn−1 + b = 0 has no repeated root and n (≥ 4) is an integer. if for two non-constant meromorphic functions f and g ef(s1, 4) = eg(s1, 4), ef(s2, 0) = eg(s2, 0) and ef(s3, ∞) = eg(s3, ∞) and θ(∞; f) + θ(∞; g) > 0 then f ≡ g. in [2] the present author further improved theorem a as follows. theorem b. [2] let si, i = 1, 2, 3 be defined as in theorem a. if for two non-constant meromorphic functions f and g ef(s1, 4) = eg(s1, 4), ef(s2, 0) = eg(s2, 0) and ef(s3, 6) = eg(s3, 6) and θ(∞; f) + θ(∞; g) > 0 then f ≡ g. now it is quite natural to ask the following question. i) what happens in theorem b if no conditions over the ramification indexes of f and g are imposed ? in the direction of the above question some investigations have already been carried out by zhang [16] in the following theorems. theorem c. let s1 = {z : z n(z + a) − b = 0}, s2 = {0} and s3 = {∞}, where a, b are nonzero constants such that zn(z + a) − b = 0 has no repeated root and n (≥ 3) is an integer. if for two nonconstant meromorphic functions f and g ef(s1, ∞) = eg(s1, ∞), ef(s2, 0) = eg(s2, 0) and ef(s3, ∞) = eg(s3, ∞) then f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non-constant entire function. theorem d. let si, i = 1, 2, 3 be defined as in theorem c and n (≥ 4) is an integer. if for two non-constant meromorphic functions f and g ef(s1, ∞) = eg(s1, ∞), ef(s2, 0) = eg(s2, 0) and ef(s3, 0) = eg(s3, 0) then f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non-constant entire function. the following example shows that in theorems a-c a 6= 0 is necessary. example 1.1. let f(z) = ez and g(z) = e−z and s1 = {z : z 4 − 1 = 0}, s2 = {0}, s3 = {∞}. since f − ωl = g − ω4−l, where ω = cos 2π 4 + isin 2π 4 , 0 ≤ l ≤ 3, clearly ef(si, ∞) = eg(si, ∞) for i = 1, 2, 3 but f and g do not satisfy the conclusions of theorems a-b. regarding theorems a-c following example establishes the fact that the set s1 can not be replaced by any arbitrary set containing three distinct elements. however it still remains open for investigations whether the degree of the equation defining s1 in theorem a-c can be reduced to three or less. 60 abhijit banerjee cubo 13, 3 (2011) example 1.2. let f(z) = √ ab e √ abz and g(z) = √ ab e− √ abz and s1 = {a, b, √ ab}, s2 = {0}, s3 = {∞}, where a and b are nonzero complex numbers. clearly ef(si, ∞) = eg(si, ∞) for i = 1, 2, 3 but f and g do not satisfy the conclusions of theorems a-c. in the paper we also concentrate our attention to the above problem as investigated by zhang [16] and provide a better solution in this direction. we now state the following two theorems which are the main results of the paper. theorem 1.1. let s1 = {z : z n(z + a) − b = 0}, s2 = {0} and s3 = {∞}, where a, b are nonzero constants such that zn(z + a) − b = 0 has no repeated root and n (≥ 3) is an integer. if for two non-constant meromorphic functions f and g ef(s1, 3) = eg(s1, 3), ef(s2, 0) = eg(s2, 0) and ef(s3, 6) = eg(s3, 6) then f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non-constant entire function. corollary 1.1. let s1, s2 and s3 be defined as in theorem 1.1 and n(≥ 3) be an integer. if for two non-constant meromorphic functions f and g ef(s1, 3) = eg(s1, 3), ef(s2, 0) = eg(s2, 0) and ef(s3, 6) = eg(s3, 6) and θ(∞; f) + θ(∞; g) > 0 then f ≡ g theorem 1.2. let s1, s2 and s3 be defined as in theorem 1.1 and n(≥ 4) be an integer. if for two non-constant meromorphic functions f and g ef(s1, 3) = eg(s1, 3), ef(s2, 0) = eg(s2, 0) and ef(s3, 0) = eg(s3, 0) then the conclusion of theorem 1.1 holds . remark 1. theorem 1.1, corollary 1.1 and theorem 1.2 are respectively the improvements of theorems c, b and d respectively. we now explain some notations which are used in the paper. definition 1.3. [6] after a ∈ c∪{∞}, we denote by n(r, a; f |= 1) the counting function of simple a points of f. for a positive integer m we denote by n(r, a; f |≤ m) (n(r, a; f |≥ m)) the counting function of those a points of f whose multiplicities are not greater(less) than m where each a point is counted according to its multiplicity. n(r, a; f |≤ m) (n(r, a; f |≥ m)) are defined similarly, where in counting the a-points of f we ignore the multiplicities. also n(r, a; f |< m), n(r, a; f |> m), n(r, a; f |< m) and n(r, a; f |> m) are defined analogously. definition 1.4. [2] we denote by n(r, a; f |= k) the reduced counting function of those a-points of f whose multiplicities is exactly k, where k ≥ 2 is an integer. definition 1.5. [2] let f and g be two non-constant meromorphic functions such that f and g share (a, k) where a ∈ c ∪ {∞}. let z0 be a a-point of f with multiplicity p, a a-point of g with multiplicity q. we denote by nl(r, a; f) the counting function of those a-points of f and g where p > q, by n (k+1 e (r, a; f) the counting function of those a-points of f and g where p = q ≥ k+1; each point in these counting functions is counted only once. in the same way we can define nl(r, a; g) and n (k+1 e (r, a; g). cubo 13, 3 (2011) some uniqueness results on meromorphic functions . . . 61 definition 1.6. [8] we denote by n2(r, a; f) = n(r, a; f) + n(r, a; f |≥ 2) definition 1.7. [7, 8] let f, g share a value a im. we denote by n∗(r, a; f, g) the reduced counting function of those a-points of f whose multiplicities differ from the multiplicities of the corresponding a-points of g. clearly n∗(r, a; f, g) ≡ n∗(r, a; g, f) and n∗(r, a; f, g) = nl(r, a; f) + nl(r, a; g). definition 1.8. [11] let a, b ∈ c ∪ {∞}. we denote by n(r, a; f | g = b) the counting function of those a-points of f, counted according to multiplicity, which are b-points of g. definition 1.9. [11] let a, b1, b2, . . . , bq ∈ c ∪{∞}. we denote by n(r, a; f | g 6= b1, b2, . . . , bq) the counting function of those a-points of f, counted according to multiplicity, which are not the bi-points of g for i = 1, 2, . . . , q. 2 lemmas in this section we present some lemmas which will be needed in the sequel. let f and g be two non-constant meromorphic functions defined as follows. f = fn(f + a) b , g = gn(g + a) b . (2.1) henceforth we shall denote by h, φ and v the following three functions h = ( f ′′ f ′ − 2f ′ f − 1 ) − ( g ′′ g ′ − 2g ′ g − 1 ) , φ = f ′ f − 1 − g ′ g − 1 and v = ( f ′ f − 1 − f′ f ) − ( g ′ g − 1 − g′ g ) = f′ f(f − 1) − g′ g(g − 1) . lemma 2.1. let f, g share (1, 1) and h 6≡ 0. then n(r, 1; f |= 1) = n(r, 1; g |= 1) ≤ n(r, h) + s(r, f) + s(r, g). proof. the lemma can be proved in the line of proof of lemma 1 [8]. lemma 2.2. let s1, s2 and s3 be defined as in theorem 1.1 and f, g be given by (2.1). if for two non-constant meromorphic functions f and g ef(s1, 0) = eg(s1, 0), ef(s2, 0) = eg(s2, 0), ef(s3, 0) = eg(s3, 0) and h 6≡ 0 then n(r, h) ≤ n∗(r, 0, f, g) + n(r, 0; f + a |≥ 2) + n(r, 0; g + a |≥ 2) + n∗(r, 1; f, g) +n∗(r, ∞; f, g) + n0(r, 0; f ′ ) + n0(r, 0; g ′ ), 62 abhijit banerjee cubo 13, 3 (2011) where n0(r, 0; f ′ ) is the reduced counting function of those zeros of f ′ which are not the zeros of f(f − 1) and n0(r, 0; g ′ ) is similarly defined. proof. the lemma can be proved in the line of proof of lemma 2.2 [2]. lemma 2.3. [12] let f be a nonconstant meromorphic function and let r(f) = n∑ k=0 akf k m∑ j=0 bjf j be an irreducible rational function in f with constant coefficients {ak} and {bj} where an 6= 0 and bm 6= 0. then t (r, r(f)) = dt (r, f) + s(r, f), where d = max{n, m}. lemma 2.4. let f and g be given by (2.1), n ≥ 3 an integer and f 6≡ g. if f, g share (1, m), f, g share (0, p), (∞, k), where 0 ≤ p < ∞ then [np + n − 1] n(r, 0; f |≥ p + 1) ≤ n∗(r, 1; f, g) + n∗(r, ∞; f, g) + s(r, f) + s(r, g). proof. suppose 0 is an e.v.p. (picard exceptional value) of f and g then the lemma follows immediately. next suppose 0 is not an e.v.p. of f and g. if φ ≡ 0, then by integration we obtain f − 1 ≡ c(g − 1). it is clear that if z0 is a zero of f then it is a zero of g. so it follows that f(z0) = g(z0) = 0. so c = 1 which contradicts f 6≡ g. so φ 6≡ 0. since f, g share (0, p) it follows that a common zero of f and g of order r ≤ p is a zero of φ of order exactly nr − 1 where as a common zero of f and g of order r > p is a zero of φ of order at least np + n − 1. let z0 is a zero of f with multiplicity q and a zero of g with multiplicity t. from (2.1) we know that z0 is a zero of f with multiplicity nq and a zero of g with multiplicity nt. so from the definition of φ it is clear that [np + n − 1]n(r, 0; f |≥ p + 1) = [np + n − 1]n(r, 0; g |≥ p + 1) = [np + n − 1]n (r, 0; f |≥ n(p + 1)) ≤ n(r, 0; φ) ≤ n(r, ∞; φ) + s(r, f) + s(r, g) ≤ n∗(r, ∞; f, g) + n∗(r, 1; f, g) + s(r, f) + s(r, g). the lemma follows from above. cubo 13, 3 (2011) some uniqueness results on meromorphic functions . . . 63 lemma 2.5. let f, g be given by (2.1), f, g share (1, m), 0 ≤ m < ∞ and ω1, ω2, . . . , ωn are the distinct roots of the equation zn + azn−1 + b = 0 and n ≥ 3. then n∗(r, 1; f, g) ≤ 1 m [ n(r, 0; f) + n(r, ∞; f) − n⊗(r, 0; f ′ ) ] + s(r, f), where n⊗(r, 0; f ′ ) = n(r, 0; f ′ | f 6= 0, ω1, ω2, . . . , ωn) proof. we omit the proof since it can be proved in the line of proof of lemma 2.15 [2]. lemma 2.6. let f and g be given by (2.1), n ≥ 3 an integer and f 6≡ g. if f, g share (1, m), f, g share (0, 0), (∞, k) then n(r, 0; f) ≤ m mn − m − 1 n(r, ∞; f |≥ k + 1) + 1 mn − m − 1 n(r, ∞; f) + s(r, f) +s(r, g). proof. since using lemma 2.5 in lemma 2.4 we get for p = 0 that (n − 1)n(r, 0; f) ≤ n(r, ∞; f |≥ k + 1) + 1 m [n(r, 0; f) + n(r, ∞; f)] +s(r, f) + s(r, g), the lemma follows. lemma 2.7. let f, g be given by (2.1), n ≥ 3 an integer and f 6≡ g. if f, g share (0, 0), (∞, k), where 0 ≤ k < ∞, and f, g share (1, m) then the poles of f and g are the zeros of v and (i) nn(r, ∞; f |= 1) + (2n + 1)n(r, ∞; f |= 2) + . . . + [(n + 1)k − 1]n(r, ∞; f |= k) +[(n + 1)k + n]n(r, ∞; f |≥ k + 1) ≤ 1 n − 1 n(r, ∞; f |≥ k + 1) + n(r, 0; f + a) +n(r, 0; g + a) + n n − 1 n∗(r, 1; f, g) + s(r, f) + s(r, g). (ii) n(r, ∞; f |≥ k + 1) ≤ n − 1 (n − 1)[(n + 1)k + n] − 1 [n(r, 0; f + a) + n(r, 0; g + a)] + n (n − 1)[(n + 1)k + n] − 1 n∗(r, 1; f, g) + s(r, f) + s(r, g). proof. suppose ∞ is an e.v.p. of f and g then the lemma follows immediately. next suppose ∞ is not an e.v.p. of f and g. if v ≡ 0, then by integration we obtain 1 − 1 f ≡ a ( 1 − 1 g ) . if z0 is a pole of f then it is a pole of g. hence from the definition of f and g we have 1 f(z0) = 0 and 1 g(z0) = 0. so a = 1 which contradicts f 6≡ g. so v 6≡ 0. since f, g share (∞, k), we note that f and g have no pole of multiplicity q where (n + 1)k < q < (n + 1)(k + 1) and so it 64 abhijit banerjee cubo 13, 3 (2011) follows that f, g share (∞, (n + 1)k + n). so using lemma 2.3 and lemma 2.4 for p = 0 we get from the definition of v nn(r, ∞; f |= 1) + (2n + 1)n(r, ∞; f |= 2) + . . . + [(n + 1)k − 1]n(r, ∞; f |= k) (2.2) +[(n + 1)k + n]n(r, ∞; f |≥ k + 1) ≤ n(r, 0; v) ≤ n(r, ∞; v) + s(r, f) + s(r, g) ≤ n∗(r, 0; f, g) + n(r, 0; f + a) + n(r, 0; g + a) + n∗(r, 1; f, g) + s(r, f) + s(r, g) ≤ 1 n − 1 n(r, ∞; f |≥ k + 1) + n(r, 0; f + a) + n(r, 0; g + a) + n n − 1 n∗(r, 1; f, g) +s(r, f) + s(r, g), from which (i) follows. again from (2.2) we note that (n − 1)[(n + 1)k + n] − 1 n − 1 n(r, ∞; f |≥ k + 1) ≤ n(r, 0; f + a) + n(r, 0; g + a) + n n − 1 n∗(r, 1; f, g) + s(r, f) + s(r, g), from which (ii) follows. lemma 2.8. ([2], lemma 2.9) let f, g be given by (2.1) and they share (1, m). if f, g share (0, p), (∞, k) where 2 ≤ m < ∞ and h 6≡ 0. then t (r, f) ≤ n(r, 0; f) + n(r, 0; g) + n∗(r, 0; f, g) + n2(r, 0; f + a) + n2(r, 0; g + a) +n(r, ∞; f) + n(r, ∞; g) + n∗(r, ∞; f, g) − m(r, 1; g) − n(r, 1; f |= 3) − . . . − (m − 2)n(r, 1; f |= m) − (m − 2) nl(r, 1; f) − (m − 1)nl(r, 1; g) −(m − 1)n (m+1 e (r, 1; f) + s(r, f) + s(r, g) lemma 2.9. ([14], lemma 6) if h ≡ 0, then f, g share (1, ∞). if further f, g share (∞, 0) then f, g share (∞, ∞). 3 proofs of the theorems proof of theorem 1.1. let f, g be given by (2.1). then f and g share (1, 3), (∞, 7n + 6). we consider the following cases. case 1. let h 6≡ 0. then f 6≡ g. noting that f, g share (0, 0) and (∞, 6) implies n∗(r, 0; f, g) ≤ n(r, 0; f) = n(r, 0; g) and n∗(r, ∞; f, g) ≤ n(r, ∞; f |≥ 7) = n(r, ∞; g |≥ 7), cubo 13, 3 (2011) some uniqueness results on meromorphic functions . . . 65 using lemmas 2.3 and 2.6 for m = 3 in lemma 2.8 we obtain (n + 1){t (r, f) + t (r, g)} (3.1) ≤ 6n(r, 0; f) + 2t (r, f) + 2t (r, g) + 4n(r, ∞; f) + 2n(r, ∞; f |≥ 7) −3n∗(r, 1; f, g) + s(r, f) + s(r, g) ≤ 2t (r, f) + 2t (r, g) + ( 6n + 10 3n − 4 ) n(r, ∞; f |≥ 7) + ( 12n − 10 3n − 4 ) n(r, ∞; f) − 3n∗(r, 1; f, g) + s(r, f) + s(r, g) so using lemma 2.7 (i) for k = 6 in (3.1) we get (n − 1){t (r, f) + t (r, g)} (3.2) ≤ ( 6n + 10 (3n − 4)(7n + 6) )[ t (r, f) + t (r, g) + 1 n − 1 { n(r, ∞; f |≥ 7) + nn∗(r, 1; f, g) } ] + ( 12n − 10 n(3n − 4) )[ t (r, f) + t (r, g) + 1 n − 1 { n(r, ∞; f |≥ 7) + nn∗(r, 1; f, g) } ] −3n∗(r, 1; f, g) + s(r, f) + s(r, g) ≤ [ 6n + 10 (3n − 4)(7n + 6) + 12n − 10 n(3n − 4) ] {t (r, f) + t (r, g)} + 1 n − 1 [ n(6n + 10) (3n − 4)(7n + 6) + 12n − 10 3n − 4 ] n∗(r, 1; f, g) + 1 (n − 1) [ 6n + 10 (3n − 4)(7n + 6) + 12n − 10 n(3n − 4) ] n(r, ∞; f |≥ 7) −3n∗(r, 1; f, g) + s(r, f) + s(r, g). now using lemma 2.7 (ii) for k = 6 in (3.2) we get [ n − 1 − 6n + 10 (3n − 4)(7n + 6) − 12n − 10 n(3n − 4) ] {t (r, f) + t (r, g)} ≤ [ n(6n + 10) (n − 1)(3n − 4)(7n + 6) + 12n − 10 (n − 1)(3n − 4) ] n∗(r, 1; f, g) + [ 6n + 10 (3n − 4)(7n + 6) + 12n − 10 n(3n − 4) ][ 1 7n2 − n − 7 {t (r, f) + t (r, g)} + n (n − 1)(7n2 − n − 7) n∗(r, 1; f, g) ] − 3n∗(r, 1; f, g) + s(r, f) + s(r, g), from which we get a contradiction for n ≥ 3 . case 2. let h ≡ 0. now from lemma 2.9 we have f and g share (1, ∞) and (∞, ∞). this implies ef(s1, ∞) = eg(s1, ∞), ef(s2, 0) = eg(s2, 0) and ef(s3, ∞) = eg(s3, ∞). now the theorem follows from theorem c. proof of corollary 1.1. let f, g be given by (2.1). then f and g share (1, 3), (∞, 7n + 6). by theorem 1.1 we get either f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a nonconstant entire function. if f 6≡ g then using lemma 2.3 clearly θ(∞; f) = θ(∞; g) = 1 − 66 abhijit banerjee cubo 13, 3 (2011) lim sup r−→ ∞ n∑ k=1 n(r,uk;e γ ) nt (r,eγ) = 0, where uk = exp ( 2kπi n+1 ) for k = 1, 2, . . . , n and hence we deduce a contradiction. this proves the corollary. proof of theorem 1.2. let f, g be given by (2.1). then f and g share (1, 3), (∞, n). we consider the following cases. case 1. let h 6≡ 0. then f 6≡ g. noting that f, g share (0, 0) and (∞, 0) implies n∗(r, 0; f, g) ≤ n(r, 0; f) = n(r, 0; g) and n∗(r, ∞; f, g) ≤ n(r, ∞; f |≥ 7) = n(r, ∞; g |≥ 7), using lemmas 2.3 and 2.6 for m = 3 and k = 0 in lemma 2.8 we obtain (n + 1){t (r, f) + t (r, g)} (3.3) ≤ 6n(r, 0; f) + 2t (r, f) + 2t (r, g) + 6n(r, ∞; f) − 3n∗(r, 1; f, g) +s(r, f) + s(r, g) ≤ 2t (r, f) + 2t (r, g) + ( 18n 3n − 4 ) n(r, ∞; f) − 3n∗(r, 1; f, g) + s(r, f) + s(r, g) so using lemma 2.7 (ii) for k = 0 and lemma 2.5 in (3.3) we get (n − 1){t (r, f) + t (r, g)} (3.4) ≤ ( 18n(n − 1) (3n − 4)(n2 − n − 1) ) [t (r, f) + t (r, g)] + ( 18n2 (3n − 4)(n2 − n − 1) ) n∗(r, 1; f, g) −3n∗(r, 1; f, g) + s(r, f) + s(r, g) ≤ ( 18n(n − 1) (3n − 4)(n2 − n − 1) ) [t (r, f) + t (r, g)] + ( 18n2 6(3n − 4)(n2 − n − 1) − 1 2 ) { n(r, 0; f) + n(r, ∞; f) + n(r, 0; g) + n(r, ∞; g) } + s(r, f) + s(r, g) ≤ ( 18n(n − 1) (3n − 4)(n2 − n − 1) + 18n2 3(3n − 4)(n2 − n − 1) − 1 ) [t (r, f) + t (r, g)] +s(r, f) + s(r, g). clearly (3.4) implies a contradiction for n ≥ 4 . case 2. let h ≡ 0. now from lemma 2.9 we have f and g share (1, ∞) and (∞, ∞). this implies ef(s1, ∞) = eg(s1, ∞), ef(s2, 0) = eg(s2, 0) and ef(s3, ∞) = eg(s3, ∞). now the theorem follows from theorem c. received: january 2010. revised: august 2010. references [1] a.banerjee, on a question of gross, j. math. anal. appl. 327(2) (2007) 1273-1283. [2] a.banerjee, some uniqueness results on meromorphic functions sharing three sets, ann. polon. math., 92(3)(2007), 261-274. cubo 13, 3 (2011) some uniqueness results on meromorphic functions . . . 67 [3] m.fang and w. xu, a note on a problem of gross, chinese j. contemporary math., 18(4)(1997), 395-402. [4] f.gross, factorization of meromorphic functions and some open problems, proc. conf. univ. kentucky, leixngton, ky(1976); lecture notes in math., 599(1977), 51-69, springer(berlin). [5] w.k.hayman, meromorphic functions, the clarendon press, oxford (1964). [6] i.lahiri, value distribution of certain differential polynomials. int. j. math. math. sci., 28(2)(2001), 83-91. [7] i.lahiri, weighted sharing and uniqueness of meromorphic functions, nagoya math. j., 161(2001), 193-206. [8] i.lahiri, weighted value sharing and uniqueness of meromorphic functions, complex var. theory appl., 46(2001), 241-253. [9] i.lahiri, on a question of hong xun yi, arch. math. (brno), 38(2002), 119-128. [10] i.lahiri and a.banerjee, uniqueness of meromorphic functions with deficient poles, kyungpook math. j., 44(2004), 575-584. [11] i.lahiri and a.banerjee, weighted sharing of two sets, kyungpook math. j., 46(1) (2006), 79-87. [12] a.z.mohon’ko, on the nevanlinna characteristics of some meromorphic functions,. theory of functions. functional analysis and their applications, 14 (1971), 83-87. [13] h.qiu and m. fang, a unicity theorem for meromorphic functions, bull. malaysian math. sci. soc., 25(2002) 31-38. [14] h.x.yi, meromorphic functions that share one or two values ii, kodai math. j., 22 (1999), 264-272. [15] h.x.yi and w.c.lin, uniqueness theorems concerning a question of gross, proc. japan acad., 80, ser.a(2004), 136-140. [16] q. zhang, meromorphic functions that share three sets, northeast math. j., 23(2)(2007), 103-114. introduction, definitions and results lemmas proofs of the theorems cubo a mathematical journal vol.15, no¯ 01, (77–96). march 2013 existence and stability of almost periodic solutions to impulsive stochastic differential equations junwei liu and chuanyi zhang harbin institute of technology, department of mathematics, harbin institute of technology, harbin 150001, p.r. china. junweiliuhit@gmail.com abstract this paper introduces the concept of square-mean piecewise almost periodic for impulsive stochastic processes. the existence of square-mean piecewise almost periodic solutions for linear and nonlinear impulsive stochastic differential equations is established by using the theory of the semigroups of the operators and schauder fixed point theorem. the stability of the square-mean piecewise almost periodic solutions for nonlinear impulsive stochastic differential equations is investigated. resumen este art́ıculo introduce el concepto de periodicidad cuadrática media por tramos casi periódica para procesos estocásticos impulsivos. la existencia de soluciones de media cuadrática casi periódicas para ecuaciones diferenciales estocásticas impulsivas lineales y no lineales se establece usando la teoŕıa de semigrupos de los operadores y el teorema de punto fijo de schauder. se estudia la estabilidad de las soluciones de media cuadrática por tramos casi periódica para ecuaciones diferenciales estocásticas impulsivas no lineales. keywords and phrases: square-mean piecewise almost periodic; impulsive stochastic differential equation; the semigroups of the operators; schauder fixed point theorem; stability 2010 ams mathematics subject classification: 35b15; 35r12; 60h15; 37c75 78 junwei liu and chuanyi zhang cubo 15, 1 (2013) 1 introduction in recent years, stochastic differential systems have been extensively studied since stochastic modeling plays an important role in physics, engineering, finance, social science and so on. qualitative properties such as existence, uniqueness and stability for stochastic differential systems have attracted more and more researchers’ attention. the existence of periodic, almost periodic(automorphic), asymptotically almost periodic, pseudo almost periodic(automorphic) solutions for stochastic differential equations was obtained. we refer the reader to [14, 6, 7, 17, 16, 10, 8, 1, 11] and references therein. on the other hand, impulsive phenomenon arises from many different real processes and phenomena which appeared in physics, chemical technology, population dynamics, biotechnology, medicine and economics. there has been a significant development in the theory of impulsive differential equations. for example, the existence of almost periodic (mild) solutions of abstract impulsive differential equations have been considered in [23, 24, 25, 4, 18, 19]. in [26], the authors combined the two directions and derived firstly some sufficient conditions for the existence and uniqueness of almost periodic solutions for a class of impulsive stochastic differential equations with delay. however, these above results quoted concern the case where the activation functions satisfy lipschitz conditions. there are few authors have considered the problem of almost periodic solutions of impulsive stochastic differential equations without lipschitz activation functions. on the basis of this, this article is devoted to the discussion of this problem. moreover, the stability analysis on impulsive stochastic differential equations has been an important research topic (see [20, 22, 27]). while, because the mild solutions don’t have stochastic differentials, ito’s formula fails to deal with the stability of mild solution to stochastic differential equations (see [20, 9, 15]). in [9], the authors gave some properties of the stochastic convolution which ensure the exponential stability of mild solutions. motivated by the above discussion, we investigate the existence and stability of almost periodic solutions for impulsive stochastic differential equations. the paper is organized as follows, in section 2 we recall some definitions, the related notations and some useful lemmas. in sections 3 and 4, we present some criteria ensuring the existence of almost periodic solutions to some linear and nonlinear impulsive stochastic differential equations, respectively. in section 5, we discuss the stability of almost periodic solutions to some impulsive stochastic differential equations. 2 preliminaries throughout this paper, r denotes the set of real numbers, r+ denotes the set of nonnegative real numbers, z denotes the set of integers, z+ denotes the set of nonnegative integers. (h, || · ||) is assumed to be a real and separable hilbert space. let (ω,f,p) be a complete probability space and l2(p,h) be a space of the h-valued random variables x such that e||x||2 = ∫ ω ||x||2dp < ∞. cubo 15, 1 (2013) existence and stability of almost periodic solutions to impulsive ... 79 l2(p,h) is a hilbert space equipped with the norm ||x||2 = ( ∫ ω ||x||2dp)1/2. definition 2.1. a stochastic process x : r+ → l2(p,h) is said to be stochastically bounded if there exists m > 0 such that e||x(t)||2 ≤ m for all t ∈ r+. definition 2.2. a stochastic process x : r+ → l2(p,h) is said to be stochastically continuous in s ∈ r+, if limt→s e||x(t) − x(s)|| 2 = 0. let t be the set consisting of all real sequences {ti}i∈z+ such that γ = infi∈z+(ti+1 − ti) > 0, t0 = 0 and limi→∞ ti = ∞. x(t + i ) and x(t − i ) represent the right and left limits of x(t) at ti, i ∈ z +, respectively. for {ti}i∈z+ ∈ t, let pc(r +,l2(p,h)) be the space consisting of all stochastically bounded functions φ : r+ → l2(p,h) such that φ(·) is stochastically continuous at t for any t 6∈ {ti}i∈z+ and φ(ti) = φ(t − i ) for all i ∈ z +; let pc(r+ ×l2(p,h),l2(p,h)) be the space formed by all stochastic processes φ : r+ × l2(p,h) → l2(p,h) such that for any x ∈ l2(p,h), φ(·,x) is stochastically continuous at t for any t 6∈ {ti}i∈z+ and φ(ti,x) = φ(t − i ,x) for all i ∈ z + and for any t ∈ r+, φ(t, ·) is stochastically continuous at x ∈ l2(p,h). definition 2.3. for {ti}i∈z+ ∈ t, the function φ ∈ pc(r +,l2(p,h)) is said to be square-mean piecewise almost periodic if the following conditions are fulfilled: (1) {t j i = ti+j − ti}, j ∈ z +, is equipotentially almost periodic, that is, for any ǫ > 0, there exists a relatively dense set qǫ of r such that for each τ ∈ qǫ there is an integer q ∈ z such that |ti+q − ti − τ| < ǫ for all i ∈ z +. (2) for any ǫ > 0, there exists a positive number δ = δ(ǫ) such that if the points t′ and t′′ belong to a same interval of continuity of φ and |t′ − t′′| < δ, then e||φ(t′) − φ(t′′)||2 < ǫ. (3) for every ǫ > 0, there exists a relatively dense set ω(ǫ) in r such that if τ ∈ ω(ǫ), then e||φ(t + τ) − φ(t)||2 < ǫ for all t ∈ r+ satisfying the condition |t − ti| > ǫ, i ∈ z +. the number τ is called ǫ-translation number of φ. we denote by apt (r +,l2(p,h)) the collection of all the square-mean piecewise almost periodic processes, it thus is a banach space with the norm ||x||∞ = supt∈r+ ||x(t)||2 = supt∈r+(e||x(t)|| 2) 1 2 for x ∈ apt (r +,l2(p,h)). lemma 2.4. let f ∈ apt (r +,l2(p,h)), then, r(f), the range of f is a relatively compact set of l2(p,h). refer to [18] for the detailed proof of lemma 2.4. definition 2.5. for {ti}i∈z+ ∈ t, the function f(t,x) ∈ pc(r + × l2(p,h),l2(p,h)) is said to be square-mean piecewise almost periodic in t ∈ r+ and uniform on compact subset of l2(p,h) if for every ǫ > 0 and every compact subset k ⊆ l2(p,h), there exists a relatively dense subset ω of r such that e||f(t + τ,x) − f(t,x)||2 < ǫ, 80 junwei liu and chuanyi zhang cubo 15, 1 (2013) for all x ∈ k,τ ∈ ω,t ∈ r+ satisfying |t − ti| > ǫ. the collection of all such processes is denoted by apt (r + × l2(p,h),l2(p,h)). lemma 2.6. suppose that f(t,x) ∈ apt (r +×l2(p,h),l2(p,h)) and f(t, ·) is uniformly continuous on each compact subset k ⊆ l2(p,h) uniformly for t ∈ r. that is, for all ǫ > 0, there exists δ > 0 such that x,y ∈ k and e||x − y||2 < δ implies that e||f(t,x) − f(t,y)||2 < ǫ for all t ∈ r+. then f(·,x(·)) ∈ apt (r +,l2(p,h)) for any x ∈ apt (r +,l2(p,h)). proof. since x ∈ apt (r +,l2(p,h)), by lemma 2.4, r(x) is a relatively compact subset of l2(p,h). because f(t, ·) is uniformly continuous on each compact subset k ⊆ l2(p,h) uniformly for t ∈ r. then for any ǫ > 0, there exists number δ : 0 < δ ≤ ǫ 4 , such that e||f(t,x1) − f(t,x2)|| 2 < ǫ 4 , (1) where x1,x2 ∈ r(x) and e||x1 − x2|| 2 < δ, t ∈ r. by square-mean piecewise almost periodic of f and x, there exists a relatively set ω of r such that the following conditions hold: e||f(t + τ,x0) − f(t,x0)|| 2 < ǫ 4 , (2) e||x(t + τ) − x(t)||2 < ǫ 4 , (3) for every x0 ∈ r(x) and t ∈ r +, |t − ti| > ǫ, i ∈ z +, τ ∈ ω. note that (a + b)2 ≤ 2(a2 + b2) and e||f(t + τ,x(t + τ)) − f(t,x(t))||2 ≤2e||f(t + τ,x(t + τ)) − f(t + τ,x(t))||2 + 2e||f(t + τ,x(t)) − f(t,x(t))||2. combing (1), (2) and (3), it follows that e||f(t + τ,x(t + τ)) − f(t,x(t))||2 < ǫ, t ∈ r+, |t − ti| > ǫ, i ∈ z +, τ ∈ ω. the proof is complete. we obtain the following corollary as an immediate consequence of lemma 2.6. corollary 2.7. let f(t,x) ∈ apt (r +×l2(p,h),l2(p,h)) and f is lipschitz, i.e., there is a number l > 0 such that e||f(t,x) − f(t,y)||2 ≤ le||x − y||2, for all t ∈ r+ and x,y ∈ l2(p,h), if for any x ∈ apt (r +,l2(p,h)), then f(·,x(·)) ∈ apt (r +,l2(p,h)). definition 2.8. a sequence x : z+ → l2(p,h) is called a square-mean almost periodic sequence if the ǫ-translation set of x t(x;ǫ) = {τ ∈ z : e||x(n + τ) − x(t)||2 < ǫ, for all n ∈ z+} is a relatively dense set in z for all ǫ > 0. the collection of all square-mean almost periodic sequences x : z+ → l2(p,h) will be denoted by apt (z +,l2(p,h)). cubo 15, 1 (2013) existence and stability of almost periodic solutions to impulsive ... 81 remark 2.9. if x(n) ∈ apt (z +,l2(p,h)), then {x(n) : n ∈ z+} is stochastically bounded, that is, supn∈z+ e||x(n)|| 2 < ∞. in order to obtain our main results, we introduce the following lemmas. let h : r+ → r be a continuous function such that h(t) ≥ 1 for all t ∈ r+ and h(t) → ∞ as t → ∞. we consider the space (pc)0h(r +,l2(p,h)) = { u ∈ pc(r+,l2(p,h)) : lim t→∞ e||u(t)||2 h(t) = 0 } . endowed with the norm ||u||h = supt∈r+ e||u(t)|| 2 h(t) , it is a banach space. lemma 2.10. a set b ⊆ (pc)0h(r +,l2(p,h)) is a relatively compact set if and only if (1) limt→∞ e||x(t)|| 2 h(t) = 0 uniformly for x ∈ b. (2) b(t) = {x(t) : x ∈ b} is relatively compact in l2(p,h) for every t ∈ r+. (3) the set b is equicontinuous on each interval (ti,ti+1)(i ∈ z +). lemma 2.11. assume that f ∈ apt (r +,l2(p,h)), the sequence {xi : i ∈ z +} is almost periodic in l2(p,h) and {t j i}, j ∈ z +, is equipotentially almost periodic. then for each ǫ > 0 there are relatively dense sets ωǫ,f,xi of r and qǫ,f,xi of z such that the following conditions hold: (i) e||f(t + τ) − f(t)||2 < ǫ for all t ∈ r+, |t − ti| > ǫ, τ ∈ ωǫ,f,xi and i ∈ z +. (ii) e||xi+q − xi|| 2 < ǫ for all q ∈ qǫ,f,xi and i ∈ z +. (iii) for every τ ∈ ωǫ,f,xi, there exists at least one number q ∈ qǫ,f,xi such that |t q i − τ| < ǫ, i ∈ z +. lemma 2.10 and lemma 2.11 are stochastic generalized versions of lemma 4.1 in [12] and lemma 35 in [23], respectively, and one may refer to [23, 18, 19, 26, 2, 13, 12] for more details. here we omit the proofs. lemma 2.12. ([9]) for any r ≥ 1 and for arbitrary l2(p,h)-valued process φ(·) such that sup s∈[0,t] e ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫s 0 φ(u)dw(u) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2r ≤ cr ( ∫t 0 (e||φ(s)||2r) 1 r ds )r , t ≥ 0, where cr = (r(2r − 1)) r. 3 almost periodic solutions for linear impulsive stochastic differential equations to begin, consider the following linear impulsive stochastic differential equation: { dx(t) = [ax(t) + f(t)]dt + g(t)dw(t), t ≥ 0,t 6= ti, i ∈ z +, △x(ti) = x(t + i ) − x(t − i ) = βi, i ∈ z +, (4) 82 junwei liu and chuanyi zhang cubo 15, 1 (2013) where a is an infinitesimal generator which generates a c0-semigroup {t(t) : t ≥ 0} such that for all t ≥ 0, ||t(t)|| ≤ me−δt with m,δ > 0 and {t(t) : t > 0} is compact. furthermore, f,g : r → l2(p,h) are two stochastic processes, βi is a square-mean almost periodic sequence and w(t) is a two-sided standard one-dimensional brownian motion, which is defined on the filtered probability space (ω,f,p,fσ) with ft = σ{w(u) − w(v) : u,v ≤ t}. definition 3.1. an ft-progressive process x(t) is called a mild solution of system (4) if it satisfies the following stochastic integral equation x(t) = t(t)x0 + ∫t 0 t(t − s)f(s)ds + ∫t 0 t(t − s)g(s)dw(s) + ∑ 0 0, there exist relatively dense sets ωǫ,f,g,xi of r and qǫ,f,g,xi of z such that the following relations hold: (1) e||f(t + τ) − f(t)||2 < ǫ,t ∈ r+, |t − ti| > ǫ,i ∈ z +,τ ∈ ωǫ,f,g,xi. (2) e||g(t + τ) − g(t)||2 < ǫ,t ∈ r+, |t − ti| > ǫ,i ∈ z +,τ ∈ ωǫ,f,g,xi. (3) e||xi+q − xi|| 2 < ǫ,i ∈ z+,q ∈ qǫ,f,g,xi. (4) for each τ ∈ ωǫ,f,g,xi, ∃ q ∈ qǫ,f,g,xi, s.t. |ti+q − ti − τ| < ǫ, i ∈ z +. we write x(t) of (5) as x(t) = t(t)x0 + x1(t) + x2(t) + x3(t) where x1(t) = ∫t 0 t(t − s)f(s)ds, x2(t) = ∫t 0 t(t − s)g(s)dw(s), x3(t) = ∑ 0 ǫ,i ∈ z +, one obtains e||x1(t + τ) − x1(t)|| 2 =e ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫t 0 t(t − s)[f(s + τ) − f(s)]ds ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 ≤e [ ∫t 0 me−δ(t−s)||f(s + τ) − f(s)||ds ]2 ≤e [ ∫t 0 m2e−δ(t−s)ds ∫t 0 e−δ(t−s)||f(s + τ) − f(s)||2ds ] ≤ m2 δ ∫t 0 e−δ(t−s)e||f(s + τ) − f(s)||2ds ≤ m2 δ ∫t 0 e−δ(t−s)ǫds ≤ m2 δ2 ǫ. (ii) x2 ∈ apt (r +,l2(p,h)). let w̃(s) = w(s + τ) −w(τ) for each s ∈ r+. note that w̃ is also 84 junwei liu and chuanyi zhang cubo 15, 1 (2013) a brownian motion and has the same distribution as w. by lemma 2.12 and (2), we have e||x2(t + τ) − x2(t)|| 2 =e ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫t 0 t(t − s)g(s + τ)dw(s + τ) − ∫t −∞ t(t − s)g(s)dw(s) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 =e ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫t 0 t(t − s)[g(s + τ) − g(s)]dw̃(s) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 ≤ ∫t 0 e||t(t − s)[g(s + τ) − g(s)]||2ds ≤ ∫t 0 m2e−2δ(t−s)e||g(s + τ) − g(s)||2ds ≤ ∫t 0 m2e−2δ(t−s)ǫds = m2 2δ ǫ. (iii) x3 ∈ apt (r +,l2(p,h)). define r(t) = t(t − ti)βi, ti < t ≤ ti+1, i ∈ z +. for ti < t ≤ ti+1, |t − ti| > ǫ, |t − ti+1| > ǫ,i ∈ z +, by (4), we can get t + τ > ti + ǫ + τ > ti+q, and ti+q+1 > ti+1 + τ − ǫ > t + τ, that is, ti+q+1 > t + τ > ti+q. since (a + b) 2 ≤ 2(a2 + b2), one has e||r(t + τ) − r(t)||2 =e||t(t + τ − ti+q)βi+q − t(t − ti)βi|| 2 =e||[t(t + τ − ti+q) − t(t − ti)]βi+q + t(t − ti)[βi+q − βi]|| 2 ≤2e||[t(t + τ − ti+q) − t(t − ti)]βi+q|| 2 + 2e||t(t − ti)[βi+q − βi]|| 2 ≤2||t(t + τ − ti+q) − t(t − ti)|| 2e||βi+q|| 2 + 2||t(t − ti)||e||βi+q − βi|| 2 ≤2||t(t + τ − ti+q) − t(t − ti)|| 2e||βi+q|| 2 + 2m2ǫ, since {t(t) : t ≥ 0} is a c0-semigroup (see [21, 3]), for the above ǫ, there exists 0 < µ < ǫ < 1 such that 0 < s < µ implies ||t(t − ti + s) − t(t − ti)|| < ǫ. note that m0 = supi∈z e||βi|| 2 < ∞, so e||r(t + τ) − r(t)||2 ≤ 2m0ǫ 2 + 2m2ǫ. next we will prove that r is uniformly continuous on each interval (ti,ti+1)(i ∈ z +). let t,h ∈ r+ such that ti < t,t + h < ti+1, then e||r(t + h) − r(t)||2 ≤ ||t(t + h − ti) − t(t − ti)|| 2e||βi|| 2. cubo 15, 1 (2013) existence and stability of almost periodic solutions to impulsive ... 85 since {t(t) : t ≥ 0} is a c0-semigroup and m0 = supi∈z+ e||βi|| 2 < ∞, we conclude that e||r(t + h) − r(t)||2 → 0 as h → 0 independent of t and i. finally, by cauchy-schwarz inequality and (3), e ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∑ 0 0} is compact, by theorem 2.1 in [5], t(·)x0 ∈ apt (r +,l2(p,h)). by combing (i), (ii) and (iii), it follows that (5) is a square-mean piecewise almost periodic process, so system (4) has a square-mean piecewise almost periodic solution. the proof is complete. 4 almost periodic solutions for nonlinear impulsive stochastic differential equations consider the following nonlinear impulsive stochastic differential equation { dx(t) = [ax(t) + f(t,x(t))]dt + g(t,x(t))dw(t), t ≥ 0,t 6= ti, i ∈ z +, △x(ti) = x(t + i ) − x(t − i ) = ii(x(ti)), i ∈ z +, (6) where f,g : r+ × l2(p,h) → l2(p,h), ii : l 2(p,h) → l2(p,h), i ∈ z+ and w(t) is a two-sided standard one dimensional brownian motion defined on the filtered probability space (ω,f,p,fσ) with ft = σ{w(u) − w(v) : u,v ≤ t}. definition 4.1. an ft-progressive process x(t) is called a mild solution of system (6) if it satisfies 86 junwei liu and chuanyi zhang cubo 15, 1 (2013) the corresponding stochastic integral equation x(t) = t(t)x0 + ∫t 0 t(t−s)f(s,x(s))ds+ ∫t 0 t(t−s)g(s,x(s))dw(s) + ∑ 0 0. moreover, t(t) is compact for t > 0. (a2) f,g ∈ apt (r + × l2(p,h),l2(p,h)), for each compact set k ⊆ l2(p,h), g(t, ·),f(t, ·) are uniformly continuous in each compact set k ⊆ l2(p,h) uniformly for t ∈ r+. ii(x) is almost periodic in i ∈ z+ uniformly in x ∈ k and is a uniformly continuous function defined on the set k ⊆ l2(p,h) for all i ∈ z+. (a3) fl = sup{t∈r+, e||x||2≤l} e||f(t,x)|| 2 < ∞, gl = sup{t∈r+, e||x||2≤l} e||g(t,x)|| 2 < ∞, il = sup{i∈z+, e||x||2≤l} e||ii(x(ti))|| 2 < ∞, where l is an arbitrary positive number. moreover, there exist a number l0 > 0 such that 4m 2l0 + 4m 2 δ2 fl0 + 2m 2 δ gl0 + 4m 2 (1−e−δγ)2 il0 ≤ l0. theorem 4.2. assume that the conditions (a1)-(a3) are satisfied, then the impulsive stochastic differential equation (6) admits at least one square-mean piecewise almost periodic solution. proof. let b = {x ∈ apt (r +,l2(p,h)) : e||x||2 ≤ l0}. obviously, b is a closed set of apt (r +,l2(p,h)). define γ on (pc)0h(r +,l2(p,h)), γx(t) = t(t)x0 + ∫t 0 t(t − s)f(s,x(s))ds + ∫t 0 t(t − s)g(s,x(s))dw(s) + ∑ 0 0, and τ ′ is are discontinuity points of first type of the function u(t). then the following estimate holds for the function u(t), u(t) ≤ c ∏ t0<τi 0) such that ak = ā + kr and bk = ā1 + kr1, then in (1.3) the equality holds. for some generalizations of gruss’ inequality for isotonic linear functionals defined on certain spaces of mappings see chapter x of the book [39] where further references are given . for related results, see [1]-[21], [25]-[31] and [34]-[44]. cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 55 2 some facts on banach algebras in order to extend the above results for banach algebras, we need some preliminary facts as follows: let b be an algebra. an algebra norm on b is a map ∥·∥ : b→[0, ∞) such that (b, ∥·∥) is a normed space, and, further: ∥ab∥ ≤ ∥a∥ ∥b∥ for any a, b ∈ b. the normed algebra (b, ∥·∥) is a banach algebra if ∥·∥ is a complete norm. we assume that the banach algebra is unital, this means that b has an identity 1 and that ∥1∥ = 1. let b be a unital algebra. an element a ∈ b is invertible if there exists an element b ∈ b with ab = ba = 1. the element b is unique; it is called the inverse of a and written a−1 or 1 a . the set of invertible elements of b is denoted by invb. if a, b ∈invb then ab ∈invb and (ab) −1 = b−1a−1. for a unital banach algebra we also have: (i) if a ∈ b and limn→∞ ∥a n∥ 1/n < 1, then 1 − a ∈invb; (ii) {a ∈ b: ∥1 − a∥ < 1} ⊂invb; (iii) invb is an open subset of b; (iv) the map invb ∋ a &−→ a−1 ∈invb is continuous. for simplicity, we denote λ1, where λ ∈ c and 1 is the identity of b, by λ. the resolvent set of a ∈ b is defined by ρ(a) := {λ ∈ c : λ − a ∈ invb} ; the spectrum of a is σ(a) , the complement of ρ(a) in c, and the resolvent function of a is ra : ρ(a) →invb, ra (λ) := (λ − a) −1 . for each λ,γ ∈ ρ(a) we have the identity ra (γ) − ra (λ) = (λ − γ) ra (λ) ra (γ) . we also have that σ(a) ⊂ {λ ∈ c : |λ| ≤ ∥a∥} . the spectral radius of a is defined as ν(a) = sup {|λ| : λ ∈ σ(a)} . if a, b are commuting elements in b, i.e. ab = ba, then ν(ab) ≤ ν(a)ν(b) and ν(a + b) ≤ ν(a) + ν(b) . let f be an analytic functions on the open disk d (0, r) given by the power series f (λ) := ∑ ∞ j=0 αjλ j (|λ| < r) . if ν(a) < r, then the series ∑ ∞ j=0 αja j converges in the banach algebra b because ∑ ∞ j=0 |αj| ∥ ∥aj ∥ ∥ < ∞, and we can define f (a) to be its sum. clearly f (a) is well defined and 56 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) there are many examples of important functions on a banach algebra b that can be constructed in this way. for instance, the exponential map on b denoted exp and defined as exp a := ∞∑ j=0 1 j! aj for each a ∈ b. if b is not commutative, then many of the familiar properties of the exponential function from the scalar case do not hold. the following key formula is valid, however with the additional hypothesis of commutativity for a and b from b exp (a + b) = exp (a) exp (b) . in a general banach algebra b it is difficult to determine the elements in the range of the exponential map exp (b) , i.e. the element which have a ”logarithm”. however, it is easy to see that if a is an element in b such that ∥1 − a∥ < 1, then a is in exp (b) . that follows from the fact that if we set b = − ∞∑ n=1 1 n (1 − a) n , then the series converges absolutely and, as in the scalar case, substituting this series into the series expansion for exp (b) yields exp (b) = a. it is known that if x and y are commuting, i.e. xy = yx, then the exponential function satisfies the property exp (x) exp (y) = exp (y) exp (x) = exp (x + y) . also, if x is invertible and a, b ∈ r with a < b then ∫ b a exp (tx) dt = x−1 [exp (bx) − exp (ax)] . moreover, if x and y are commuting and y − x is invertible, then ∫ 1 0 exp ((1 − s) x + sy) ds = ∫ 1 0 exp (s (y − x)) exp (x) ds = (∫ 1 0 exp (s (y − x)) ds ) exp (x) = (y − x) −1 [exp (y − x) − i] exp (x) = (y − x) −1 [exp (y) − exp (x)] . inequalities for functions of operators in hilbert spaces may be found in the papers [12], [11] and in the recent monographs [22], [23], [32] and the references therein. the following inequality of grüss-lupaş type in banach algebras holds: cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 57 theorem 2.1. let b be a banach algebra over k (=r,c) , ai, bi ∈ b and αi ∈ k (i = 1, ..., n) . then we have the inequality: ∥ ∥ ∥ ∥ ∥ n∑ i=1 αi n∑ i=1 αiaibi − n∑ i=1 αiai n∑ i=1 αibi ∥ ∥ ∥ ∥ ∥ (2.1) ≤ max 1≤j≤n−1 ∥aj+1 − aj∥ max 1≤j≤n−1 ∥bj+1 − bj∥ × ⎡ ⎣ n∑ i=1 |αi| n∑ i=1 i2 |αi| − ( n∑ i=1 i |αi| )2 ⎤ ⎦ the inequality (2.1) is sharp in the sense that the multiplicative constant c = 1 in the right membership can not be replaced by a smaller one. let αn be nonzero complex numbers and let r := 1 lim sup |αn| 1 n . clearly 0 ≤ r ≤ ∞, but we consider only the case 0 < r ≤ ∞. denote by: d(0, r) = { {λ ∈ c : |λ| < r}, if r < ∞ c, if r = ∞, consider the functions: λ &→ f(λ) : d(0, r) → c, f(λ) := ∞∑ n=0 αnλ n and λ &→ fa(λ) : d(0, r) → c, fa(λ) := ∞∑ n=0 |αn|λ n. let b be a unital banach algebra and 1 its unity. denote by b(0, r) = { {x ∈ b : ∥x∥ < r}, if r < ∞ b, if r = ∞. we associate to f the map: x &→ f̃(x) : b(0, r) → b, f̃(x) := ∞∑ n=0 αnx n. 58 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) obviously, f̃ is correctly defined because the series ∑ ∞ n=0 αnx n is absolutely convergent, since ∑ ∞ n=0 ∥αnx n∥ ≤ ∑ ∞ n=0 |αn| ∥x∥ n . making use of theorem 2.1 we have the following inequality for power series: theorem 2.2. let f(λ) = ∑ ∞ n=0 αnλ n be a power series that is convergent on the open disk d(0, r), with r > 0. if x, y ∈ b with xy = yx and ∥x∥ , ∥y∥ ≤ 1, then we have for λ ∈ c with |λ| < r the inequality: ∥ ∥ ∥f̃ (λ · 1) f̃ (λxy) − f̃ (λx) f̃ (λy) ∥ ∥ ∥ (2.2) ≤ ∥x − 1∥ ∥y − 1∥ { fa (|λ|) [ |λ| f′a (|λ|) + |λ| 2 f′′a (|λ|) ] − [|λ| f′a (|λ|)] 2 } . motivated by the above results we establish in this paper other similar inequalities for the norm of the chebyshev difference f̃ (λ · 1) f̃ (λxy) − f̃ (λx) f̃ (λy) by the use of some discrete inequalities of grüss’ type. first we establish some identities of interest in banach algebras, 3 identities in banach algebras consider the chebyshev functional defined for p = (p1, ..., pn) ∈ k n, a = (a1, ..., an) ∈ b n and b = (b1, ..., bn) ∈ b n, where b is a banach algebra over the real or complex number field k: tn (p; a, b) := pn n∑ i=1 piaibi − n∑ i=1 piai n∑ i=1 pibi, (3.1) where pn := ∑n i=1 pi. the following particular identities for unweighted means hold as well, where tn (a, x) is defined as follows: tn (a, b) := 1 n n∑ i=1 αibi − 1 n n∑ i=1 ai 1 n n∑ i=1 bi. (3.2) if α,β are scalars and x, y are vectors in a banach algebra b, we denote by det ( α β x y ) := αy − βx ∈ b. the first result is embodied in the following cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 59 theorem 3.1. let p = (p1, ..., pn) ∈ k n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ b n. if we define pi := i∑ k=1 pk, p̄i := pn − pi, i ∈ {1, ..., n − 1} , ai (p) := i∑ k=1 pkak, āi (p) := an (p) − ai (p) , i ∈ {1, ..., n − 1} , then we have the identity tn (p; a, b) = n−1∑ i=1 det ( pi pn ai (p) an (p) ) ∆bi (3.3) = pn n−1∑ i=1 pi ( an (p) pn − ai (p) pi ) ∆bi (if pi ̸= 0, i ∈ {1, ..., n}) = n−1∑ i=1 pip̄i ( āi (p) p̄i − ai (p) pi ) ∆bi ( if pi, p̄i ̸= 0, i ∈ {1, ..., n − 1} ) ; where ∆bi := bi+1 − bi (i ∈ {1, ..., n − 1}) is the forward difference. proof. we use the following well known summation by parts formula q−1∑ l=p dl∆vl = dlvl| q p − q−1∑ l=p ∆dlvl+1, (3.4) where dl and vl are vectors in a linear space, l = p, ..., q (q > p; p, q are natural numbers) . if we choose in (3.4), p = 1, q = n, di = pian (p) − pnai (p) and vi = bi (i ∈ {1, ..., n − 1}) , 60 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) then we get n−1∑ i=1 (pian (p) − pnai (p))∆bi = [pian (p) − pnai (p)] bi| n 1 − n−1∑ i=1 ∆(pian (p) − pnai (p)) bi+1 = [pnan (p) − pnan (p)] bn − [p1an (p) − pna1 (p)] b1 − n−1∑ i=1 [pi+1an (p) − pnai+1 (p) − pian (p) + pnai (p)] bi+1 = pnp1a1b1 − p1an (p) b1 − n−1∑ i=1 (pi+1an (p) − pnpi+1ai+1) bi+1 = pnp1a1b1 − p1an (p) b1 − an (p) n−1∑ i=1 pi+1bi+1 + pn n−1∑ i=1 pi+1ai+1bi+1 = pn n∑ i=1 piaibi − n∑ i=1 piai n∑ i=1 pibi = tn (p; a, b) , which produce the first identity in (3.3) . the second and the third identities are obvious and we omit the details. before we prove the second result, we need the following lemma providing an identity that is interesting in itself as well. lemma 3.1. let p = (p1, ..., pn) ∈ k n and a = (a1, ..., an) ∈ b n. then we have the equality det ( pi pn ai (p) an (p) ) = n−1∑ j=1 pmin{i,j}p̄max{i,j}∆aj, (3.5) for each i ∈ {1, ..., n − 1} . proof. define, for i ∈ {1, ..., n − 1} , k (i) := n−1∑ j=1 pmin{i,j}p̄max{i,j}∆aj. cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 61 we have k (i) = i∑ j=1 pmin{i,j}p̄max{i,j}∆aj + n−1∑ j=i+1 pmin{i,j}p̄max{i,j}∆aj (3.6) = i∑ j=1 pjp̄i∆aj + n−1∑ j=i+1 pip̄j∆aj = p̄i i∑ j=1 pj∆aj + pi n−1∑ j=i+1 p̄j∆aj. using the summation by parts formula, we have i∑ j=1 pj∆aj = pjaj| i+1 1 − i∑ j=1 (pj+1 − pj) aj+1 (3.7) = pi+1ai+1 − p1a1 − i∑ j=1 pj+1aj+1 = pi+1ai+1 − i+1∑ j=1 pjaj and n−1∑ j=i+1 p̄j∆aj = p̄jaj ∣ ∣n i+1 − n−1∑ j=i+1 ( p̄j+1 − p̄j ) aj+1 (3.8) = p̄nan − p̄i+1ai+1 − n−1∑ j=i+1 (pn − pj+1 − pn + pj) aj+1 = −p̄i+1ai+1 + n−1∑ j=i+1 pj+1aj+1. using (3.7) and (3.8) we have k (i) = p̄i ⎛ ⎝pi+1ai+1 − i+1∑ j=1 pjaj ⎞ ⎠ + pi ⎛ ⎝ n−1∑ j=i+1 pj+1aj+1 − p̄i+1ai+1 ⎞ ⎠ = p̄ipi+1ai+1 − pip̄i+1ai+1 − p̄i i+1∑ j=1 pjaj + pi n−1∑ j=i+1 pj+1aj+1 = [(pn − pi) pi+1 − pi (pn − pi+1)] ai+1 + pi n−1∑ j=i+1 pj+1aj+1 − p̄i i+1∑ j=1 pjaj 62 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) = pnpi+1ai+1 + pi n−1∑ j=i+1 pj+1aj+1 − p̄i i+1∑ j=1 pjaj = ( pi + p̄i ) pi+1ai+1 + pi n−1∑ j=i+1 pj+1aj+1 − p̄i i+1∑ j=1 pjaj = pi n−1∑ j=i+1 pjaj − p̄i i∑ j=1 pjaj = piāi (p) − p̄iai (p) = det ( pi pn ai (p) an (p) ) ; and the identity is proved. we are able now to state and prove the second identity for the čebyšev functional theorem 3.2. with the assumptions of theorem 3.1, we have the equality tn (p; a, b) = n−1∑ i=1 n−1∑ j=1 pmin{i,j}p̄max{i,j}∆aj∆bi. (3.9) the proof is obvious by theorem 3.1 and lemma 3.1. 4 some new inequalities the following result holds theorem 4.1. let p = (p1, ..., pn) ∈ k n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ b n. then we have the inequalities ∥tn (p; a, b)∥ (4.1) ≤ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i≤n−1 ∥ ∥ ∥ ∥ ∥ det ( pi pn ai (p) an (p) )∥ ∥ ∥ ∥ ∥ ∑ n−1 i=1 ∥∆bi∥ ; ( ∑ n−1 i=1 ∥ ∥ ∥ ∥ ∥ det ( pi pn ai (p) an (p) )∥ ∥ ∥ ∥ ∥ q)1/q (∑ n−1 i=1 ∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑n−1 i=1 ∥ ∥ ∥ ∥ ∥ det ( pi pn ai (p) an (p) )∥ ∥ ∥ ∥ ∥ max1≤i≤n−1 ∥∆bi∥ . cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 63 all the inequalities in (4.1) are sharp in the sense that the constants 1 cannot be replaced by smaller constants. proof. using the first identity in (3.3), we have ∥tn (p; a, b)∥ ≤ n−1∑ i=1 ∥ ∥ ∥ ∥ ∥ det ( pi pn ai (p) an (p) )∥ ∥ ∥ ∥ ∥ ∥∆bi∥ . using hölder’s inequality, we deduce the desired result (4.1) . let prove, for instance, that the constant 1 in the second inequality is best possible. assume, for c > 0, we have that ∥tn (p; a, b)∥ ≤ c ( n−1∑ i=1 ∥ ∥ ∥ ∥ ∥ det ( pi pn ai (p) an (p) )∥ ∥ ∥ ∥ ∥ q)1/q ( n−1∑ i=1 ∥∆bi∥ p )1/p (4.2) for p > 1, 1 p + 1 q = 1, n ≥ 2. if we choose n = 2, then we get t2 (p; a, b) = p1p2 (a2 − a1) (b2 − b1) . also, for n = 2, ( n−1∑ i=1 ∥ ∥ ∥ ∥ ∥ det ( pi pn ai (p) an (p) )∥ ∥ ∥ ∥ ∥ q)1/q = |p1p2| ∥a2 − a1∥ and ⎛ ⎝ n−1∑ j=1 ∥∆bj∥ p ⎞ ⎠ 1/p = ∥b2 − b1∥ . then by (4.2), holding for n = 2, p1, p2 ̸= 0, a1 ̸= a2, b2 ̸= b1, we deduce c ≥ 1, proving that 1 is the best possible constant in that inequality. the following corollary for the uniform distribution of the probability p holds. 64 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) corolary 1. with the assumptions of theorem 4.1 for a and b, we have the inequalities ∥tn (a, b)∥ (4.3) ≤ 1 n2 × ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i≤n−1 ∥ ∥ ∥ ∥ ∥ det ( i n ∑ i k=1 ak ∑ n k=1 ak )∥ ∥ ∥ ∥ ∥ ∑ n−1 i=1 ∥∆bi∥ ; ( ∑ n−1 i=1 ∥ ∥ ∥ ∥ ∥ det ( i n ∑i k=1 ak ∑n k=1 ak )∥ ∥ ∥ ∥ ∥ q)1/q × (∑ n−1 i=1 ∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑n−1 i=1 ∥ ∥ ∥ ∥ ∥ det ( i n ∑i k=1 ak ∑n k=1 ak )∥ ∥ ∥ ∥ ∥ max1≤i≤n−1 ∥∆bi∥ . the following result may be stated as well. theorem 4.2. with the assumptions of theorem 4.1 and if pi ̸= 0 (i = 1, ..., n) , then we have the inequalities ∥tn (p; a, b)∥ (4.4) ≤ |pn| × ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i≤n−1 ∥ ∥ ∥ an(p) pn − ai(p) pi ∥ ∥ ∥ ∑ n−1 i=1 |pi| ∥∆bi∥ ; (∑ n−1 i=1 |pi| ∥ ∥ ∥ an(p) pn − ai(p) pi ∥ ∥ ∥ q)1/q (∑ n−1 i=1 |pi| ∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑ n−1 i=1 |pi| ∥ ∥ ∥ an(p) pn − ai(p) pi ∥ ∥ ∥max1≤i≤n−1 ∥∆bi∥ . all the inequalities in (4.4) are sharp in the sense that the constant 1 cannot be replaced by a smaller constant. proof. follows by the second identity in (3.3) and taking into account that ∥tn (p; a, b)∥ ≤ |pn| n−1∑ i=1 ∥ ∥ ∥ ∥ an (p) pn − ai (p) pi ∥ ∥ ∥ ∥ |pi| ∥∆bi∥ . using hölder’s weighted inequality, we easily deduce (4.4) . the sharpness of the constant may be shown in a similar manner. we omit the details. cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 65 the following corollary containing the unweighted inequalities holds. corolary 2. with the above assumptions for a and b one, has ∥tn (a, b)∥ (4.5) ≤ 1 n × ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i≤n−1 ∥ ∥ ∥ 1 n ∑ n k=1 ak − 1 i ∑ i k=1 ak ∥ ∥ ∥ ∑ n−1 i=1 i ∥∆bi∥ ; (∑ n−1 i=1 i ∥ ∥ ∥ 1 n ∑ n k=1 ak − 1 i ∑ i k=1 ak ∥ ∥ ∥ q)1/q (∑ n−1 i=1 i ∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑n−1 i=1 i ∥ ∥ ∥ 1 n ∑n k=1 ak − 1 i ∑i k=1 ak ∥ ∥ ∥max1≤i≤n−1 ∥∆bi∥ . the inequalities in (4.5) are sharp in the sense mentioned above. another type of inequalities may be stated if one uses the third identity in (3.3). theorem 4.3. with the assumptions in theorem 4.1 and if pi, pi ̸= 0, i ∈ {1, ..., n − 1} , then we have the inequalities ∥tn (p; a, b)∥ (4.6) ≤ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i≤n−1 ∥ ∥ ∥ ai(p) pi − ai(p) pi ∥ ∥ ∥ ∑ n−1 i=1 |pi| ∣ ∣pi ∣ ∣∥∆bi∥ ; (∑n−1 i=1 |pi| ∣ ∣pi ∣ ∣ ∥ ∥ ∥ ai(p) pi − ai(p) pi ∥ ∥ ∥ q)1/q (∑n−1 i=1 |pi| ∣ ∣pi ∣ ∣∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑ n−1 i=1 |pi| ∣ ∣pi ∣ ∣ ∥ ∥ ∥ ai(p) pi − ai(p) pi ∥ ∥ ∥max1≤i≤n−1 ∥∆bi∥ . 66 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) in particular, if pi = 1 n , i ∈ {1, ..., n} , then we have ∥tn (a, b)∥ (4.7) ≤ 1 n2 ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i≤n−1 ∥ ∥ ∥ 1 n−i ∑ n k=i+1 ak − 1 i ∑ i k=1 ak ∥ ∥ ∥ × ∑n−1 i=1 i (n − i) ∥∆bi∥ ; (∑n−1 i=1 i (n − i) ∥ ∥ ∥ 1 n−i ∑n k=i+1 ak − 1 i ∑i k=1 ak ∥ ∥ ∥ q)1/q × (∑n−1 i=1 i (n − i) ∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑n−1 i=1 i (n − i) ∥ ∥ ∥ 1 n−i ∑n k=i+1 ak − 1 i ∑i k=1 ak ∥ ∥ ∥ × max1≤i≤n−1 ∥∆bi∥ . the inequalities in (4.6)and (4.7) are sharp in the above mentioned sense. a different approach may be considered if one uses the representation in terms of double sums for the chebyshev functional provided by the theorem 3.2. the following result holds. theorem 4.4. with the assumptions in theorem 4.1, we have the inequalities ∥tn (p; a, b)∥ (4.8) ≤ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ max1≤i,j≤n−1 {∣ ∣pmin{i,j} ∣ ∣ ∣ ∣p̄max{i,j} ∣ ∣ } ∑n−1 i=1 ∥∆ai∥ ∑n−1 i=1 ∥∆bi∥ ; (∑n−1 i=1 ∑n−1 j=1 ∣ ∣pmin{i,j} ∣ ∣q ∣ ∣p̄max{i,j} ∣ ∣q )1/q × (∑n−1 i=1 ∥∆ai∥ p )1/p (∑n−1 i=1 ∥∆bi∥ p )1/p for p > 1, 1 p + 1 q = 1; ∑n−1 i=1 ∑n−1 j=1 ∣ ∣pmin{i,j} ∣ ∣ ∣ ∣p̄max{i,j} ∣ ∣ × max1≤i≤n−1 ∥∆ai∥ max1≤i≤n−1 ∥∆bi∥ . the inequalities are sharp in the sense mentioned above. the proof follows by the identity (3.9) on using hölder’s inequality for double sums and we omit the details. the above theorem has some important consequences as follows: cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 67 corolary 3. let p = (p1, ..., pn) ∈ k n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ b n. then we have the inequality ∥tn (p; a, b)∥ ≤ 1 4 ( n∑ k=1 |pk| )2 n−1∑ i=1 ∥∆ai∥ n−1∑ i=1 ∥∆bi∥ . (4.9) the constant 1 4 is best possible in (4.9). proof. we observe that ∣ ∣pmin{i,j} ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ min{i,j}∑ k=1 pk ∣ ∣ ∣ ∣ ∣ ∣ ≤ min{i,j}∑ k=1 |pk| and ∣ ∣p̄max{i,j} ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ n∑ k=max{i,j} pk ∣ ∣ ∣ ∣ ∣ ∣ ≤ n∑ k=max{i,j} |pk| for any 1 ≤ i, j ≤ n − 1. this implies that ∣ ∣pmin{i,j} ∣ ∣ ∣ ∣p̄max{i,j} ∣ ∣ ≤ min{i,j}∑ k=1 |pk| n∑ k=max{i,j} |pk| ≤ 1 4 ⎛ ⎝ min{i,j}∑ k=1 |pk| + n∑ k=max{i,j} |pk| ⎞ ⎠ 2 ≤ 1 4 ( n∑ k=1 |pk| )2 for any 1 ≤ i, j ≤ n − 1. therefore max 1≤i,j≤n−1 {∣ ∣pmin{i,j} ∣ ∣ ∣ ∣p̄max{i,j} ∣ ∣ } ≤ 1 4 ( n∑ k=1 |pk| )2 and by the first inequality in (4.8) we get (4.9). to prove the sharpness of the constant 1 4 , assume that (4.9) holds with a constant c > 0, i.e. ∥tn (p; a, b)∥ ≤ c ( n∑ k=1 |pk| )2 n−1∑ i=1 ∥∆ai∥ n−1∑ i=1 ∥∆bi∥ . (4.10) if we take in (4.10) n = 2, then we have ∥p1p2 (a2 − a1) (b2 − b1)∥ ≤ c (|p1| + |p2|) 2 ∥a2 − a1∥ ∥b2 − b1∥ . if we take p1 = p2 = 1 2 , a2 − a1 = α · 1, b2 − b1 = β · 1 with α,β ̸= 0 then we get c ≥ 1 4 , and the proof is completed. 68 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) we have: corolary 4. let p = (p1, ..., pn) ∈ k n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ b n. then we have the inequality ∥tn (p; a, b)∥ ≤ 1 4p (n + 1) 2p ⎡ ⎣ n∑ i=1 |pi| q n∑ i=1 i2 |pi| q − ( n∑ i=1 i |pi| q )2 ⎤ ⎦ 1/q (4.11) × ( n−1∑ i=1 ∥∆ai∥ p )1/p ( n−1∑ i=1 ∥∆bi∥ p )1/p , for p > 1, 1 p + 1 q = 1. proof. we observe that by hölder’s inequality we have ∣ ∣pmin{i,j} ∣ ∣q = ∣ ∣ ∣ ∣ ∣ ∣ min{i,j}∑ k=1 pk ∣ ∣ ∣ ∣ ∣ ∣ q ≤ (min {i, j}) q−1 min{i,j}∑ k=1 |pk| q and ∣ ∣p̄max{i,j} ∣ ∣q = ∣ ∣ ∣ ∣ ∣ ∣ n∑ k=max{i,j} pk ∣ ∣ ∣ ∣ ∣ ∣ q ≤ (n − max {i, j} + 1) q−1 n∑ k=max{i,j} |pk| q for any 1 ≤ i, j ≤ n − 1. then ∣ ∣pmin{i,j} ∣ ∣q ∣ ∣p̄max{i,j} ∣ ∣q (4.12) ≤ (n − max {i, j} + 1) q−1 (min {i, j}) q−1 min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q = [(n − max {i, j} + 1) (min {i, j})] q−1 min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q . observe that (n − max {i, j} + 1) (min {i, j}) ≤ 1 4 (n − max {i, j} + 1 + min {i, j}) 2 . since for any 1 ≤ i, j ≤ n − 1 we have max {i, j} − min {i, j} = |i − j| then (n − max {i, j} + 1) (min {i, j}) ≤ 1 4 (n + 1 − |i − j|) 2 ≤ 1 4 (n + 1) 2 cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 69 and by (4.12) we get ∣ ∣pmin{i,j} ∣ ∣q ∣ ∣p̄max{i,j} ∣ ∣q ≤ [ 1 4 (n + 1) 2 ]q−1 min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q or any 1 ≤ i, j ≤ n − 1. making use of the second inequality in (4.8) we have ∥tn (p; a, b)∥ (4.13) ≤ ⎛ ⎝ n−1∑ i=1 n−1∑ j=1 ∣ ∣pmin{i,j} ∣ ∣q ∣ ∣p̄max{i,j} ∣ ∣q ⎞ ⎠ 1/q × ( n−1∑ i=1 ∥∆ai∥ p )1/p ( n−1∑ i=1 ∥∆bi∥ p )1/p ≤ ⎛ ⎝ [ 1 4 (n + 1) 2 ]q−1 n−1∑ i=1 n−1∑ j=1 ⎛ ⎝ min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q ⎞ ⎠ ⎞ ⎠ 1/q × ( n−1∑ i=1 ∥∆ai∥ p )1/p ( n−1∑ i=1 ∥∆bi∥ p )1/p = [ 1 4 (n + 1) 2 ]q−1 q ⎛ ⎝ n−1∑ i=1 n−1∑ j=1 ⎛ ⎝ min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q ⎞ ⎠ ⎞ ⎠ 1/q × ( n−1∑ i=1 ∥∆ai∥ p )1/p ( n−1∑ i=1 ∥∆bi∥ p )1/p = 1 4p (n + 1) 2p ⎛ ⎝ n−1∑ i=1 n−1∑ j=1 ⎛ ⎝ min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q ⎞ ⎠ ⎞ ⎠ 1/q × ( n−1∑ i=1 ∥∆ai∥ p )1/p ( n−1∑ i=1 ∥∆bi∥ p )1/p . if we use the identity for real numbers tn (q; a, b) = n−1∑ i=1 n−1∑ j=1 qmin{i,j}q̄max{i,j}∆aj∆bi. and the choices aj = bj = j and qk = |pk| q , then we get n−1∑ i=1 n−1∑ j=1 ⎛ ⎝ min{i,j}∑ k=1 |pk| q n∑ k=max{i,j} |pk| q ⎞ ⎠ = n∑ i=1 |pi| q n∑ i=1 i2 |pi| q − ( n∑ i=1 i |pi| q )2 . 70 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) replacing this in (4.13) produces the desired result (4.11). the following corollary also holds. it was obtained earlier in the paper with a different proof. corolary 5. let p = (p1, ..., pn) ∈ k n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ b n. then we have the inequality ∥tn (p; a, b)∥ ≤ ⎡ ⎣ n∑ i=1 |pi| n∑ i=1 i2 |pi| − ( n∑ i=1 i |pi| )2 ⎤ ⎦ (4.14) × max 1≤i≤n−1 ∥∆ai∥ max 1≤i≤n−1 ∥∆bi∥ . the inequality is sharp. proof. from the third inequality in (4.8) we have ∥tn (p; a, b)∥ ≤ max 1≤i≤n−1 ∥∆ai∥ max 1≤i≤n−1 ∥∆bi∥ n−1∑ i=1 n−1∑ j=1 ∣ ∣pmin{i,j} ∣ ∣ ∣ ∣p̄max{i,j} ∣ ∣ = max 1≤i≤n−1 ∥∆ai∥ max 1≤i≤n−1 ∥∆bi∥ n−1∑ i=1 n−1∑ j=1 ∣ ∣ ∣ ∣ ∣ ∣ min{i,j}∑ k=1 pk ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n∑ k=max{i,j} pk ∣ ∣ ∣ ∣ ∣ ∣ ≤ max 1≤i≤n−1 ∥∆ai∥ max 1≤i≤n−1 ∥∆bi∥ n−1∑ i=1 n−1∑ j=1 ⎛ ⎝ min{i,j}∑ k=1 |pk| n∑ k=max{i,j} |pk| ⎞ ⎠ = ⎡ ⎣ n∑ i=1 |pi| n∑ i=1 i2 |pi| − ( n∑ i=1 i |pi| )2 ⎤ ⎦ max 1≤i≤n−1 ∥∆ai∥ max 1≤i≤n−1 ∥∆bi∥ and the desired inequality (4.14) is proved. the sharpness of the inequality follows as above and the details are omitted. 5 inequalities for power series we have: theorem 5.1. let f(λ) = ∑ ∞ n=0 αnλ n be a power series that is convergent on the open disk d(0, r), with r > 0. if x, y ∈ b with xy = yx and ∥x∥ , ∥y∥ < 1, then we have for λ ∈ c with |λ| < r the inequality: ∥ ∥ ∥f̃ (λ · 1) f̃ (λxy) − f̃ (λx) f̃ (λy) ∥ ∥ ∥ ≤ 1 4 · ∥x − 1∥ ∥y − 1∥ (1 − ∥x∥) (1 − ∥y∥) f2a (|λ|) . (5.1) cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 71 proof. utilising the inequality (4.9) we have for all n ≥ 1 ∥ ∥ ∥ ∥ ∥ n∑ i=0 αiλ i n∑ i=0 αiλ ixiyi − n∑ i=0 αiλ ixi n∑ i=0 αiλ iyi ∥ ∥ ∥ ∥ ∥ (5.2) ≤ 1 4 ( n∑ k=0 |αk| |λ| k )2 n−1∑ i=0 ∥ ∥xi+1 − xi ∥ ∥ n−1∑ i=0 ∥ ∥yi+1 − yi ∥ ∥ . observe that n−1∑ i=0 ∥ ∥xi+1 − xi ∥ ∥ = n−1∑ i=0 ∥ ∥xi (x − 1) ∥ ∥ ≤ ∥x − 1∥ n−1∑ i=0 ∥ ∥xi ∥ ∥ ≤ ∥x − 1∥ n−1∑ i=0 ∥x∥ i = ∥x − 1∥ 1 − ∥x∥ n 1 − ∥x∥ and, similarly, n−1∑ i=0 ∥ ∥yi+1 − yi ∥ ∥ ≤ ∥y − 1∥ 1 − ∥y∥ n 1 − ∥y∥ . utilizing (5.2) and the fact that xy = yx, we have ∥ ∥ ∥ ∥ ∥ n∑ i=0 αiλ i n∑ i=0 αiλ i (xy) i − n∑ i=0 αiλ ixi n∑ i=0 αiλ iyi ∥ ∥ ∥ ∥ ∥ (5.3) ≤ 1 4 ∥x − 1∥ ∥y − 1∥ ( n∑ k=0 |αk| |λ| k )2 1 − ∥x∥ n 1 − ∥x∥ 1 − ∥y∥ n 1 − ∥y∥ for any n ≥ 1. since all the series whose partial sums are involved in (5.3) are convergent, then by letting n → ∞ in (5.3) we deduce the desired inequality (5.1). corolary 6. let f(λ) = ∑ ∞ n=0 αnλ n be a power series that is convergent on the open disk d(0, r), with r > 0. if x ∈ b and ∥x∥ < 1, then we have for λ ∈ c with |λ| < r the inequality: ∥ ∥ ∥ ∥f̃ (λ · 1) f̃ ( λx2 ) − [ f̃ (λx) ]2 ∥ ∥ ∥ ∥ ≤ 1 4 · ∥x − 1∥ 2 (1 − ∥x∥) 2 f2 a (|λ|) . (5.4) 72 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) as some natural examples that are useful for applications, we can point out that, if f (λ) = ∞∑ n=1 (−1) n n λn = ln 1 1 + λ , λ ∈ d (0, 1) ; (5.5) g (λ) = ∞∑ n=0 (−1) n (2n) ! λ2n = cosλ, λ ∈ c; h (λ) = ∞∑ n=0 (−1) n (2n + 1) ! λ2n+1 = sinλ, λ ∈ c; l (λ) = ∞∑ n=0 (−1) n λn = 1 1 + λ , λ ∈ d (0, 1) ; then the corresponding functions constructed by the use of the absolute values of the coefficients are fa (λ) = ∞∑ n=1 1 n λn = ln 1 1 − λ , λ ∈ d (0, 1) ; (5.6) ga (λ) = ∞∑ n=0 1 (2n) ! λ2n = coshλ, λ ∈ c; ha (λ) = ∞∑ n=0 1 (2n + 1) ! λ2n+1 = sinhλ, λ ∈ c; la (λ) = ∞∑ n=0 λn = 1 1 − λ , λ ∈ d (0, 1) . other important examples of functions as power series representations with nonnegative coefficients are: exp (λ) = ∞∑ n=0 1 n! λn λ ∈ c, (5.7) 1 2 ln ( 1 + λ 1 − λ ) = ∞∑ n=1 1 2n − 1 λ2n−1, λ ∈ d (0, 1) ; sin −1 (λ) = ∞∑ n=0 γ ( n + 1 2 ) √ π(2n + 1) n! λ2n+1, λ ∈ d (0, 1) ; tanh −1 (λ) = ∞∑ n=1 1 2n − 1 λ2n−1, λ ∈ d (0, 1) 2f1 (α,β,γ,λ) = ∞∑ n=0 γ (n + α)γ (n + β)γ (γ) n!γ (α)γ (β)γ (n + γ) λn,α,β,γ > 0, λ ∈ d (0, 1) ; where γ is gamma function. cubo 19, 1 (2017) inequalities for čeby šev functional in banach algebras 73 example 1. a) if x, y ∈ b with xy = yx and ∥x∥ , ∥y∥ < 1, then we have for λ ∈ c the inequality: ∥exp [λ(1 + xy)] − exp [λ(x + y)]∥ ≤ 1 4 · ∥x − 1∥ ∥y − 1∥ (1 − ∥x∥) (1 − ∥y∥) exp (2 |λ|) . (5.8) in particular, we have ∥ ∥exp [ λ ( 1 + x2 )] − exp [2λx] ∥ ∥ ≤ 1 4 · ∥x − 1∥ 2 (1 − ∥x∥) 2 exp (2 |λ|) (5.9) and ∥ ∥exp [ λ ( 1 − x2 )] − 1 ∥ ∥ ≤ 1 4 · ∥x − 1∥ ∥x + 1∥ (1 − ∥x∥) 2 exp (2 |λ|) (5.10) for any x ∈ b with ∥x∥ < 1 and λ ∈ c. b) we have the inequality ∥ ∥ ∥(1 − λ) −1 (1 − λxy) −1 − (1 − λx) −1 (1 − λy) −1 ∥ ∥ ∥ (5.11) ≤ 1 4 · ∥x − 1∥ ∥y − 1∥ (1 − ∥x∥) (1 − ∥y∥) (1 − |λ|) 2 for any x, y ∈ b with xy = yx, ∥x∥ , ∥y∥ < 1 and λ ∈ c with |λ| < 1. in particular, we have ∥ ∥ ∥(1 − λ) −1 ( 1 − λx2 ) −1 − (1 − λx) −2 ∥ ∥ ∥ ≤ 1 4 · ∥x − 1∥ 2 (1 − ∥x∥) 2 (1 − |λ|) 2 (5.12) and ∥ ∥ ∥(1 − λ) −1 ( 1 + λx2 ) −1 − (1 − λx) −1 (1 + λx) −1 ∥ ∥ ∥ (5.13) ≤ 1 4 · ∥x − 1∥ ∥x + 1∥ (1 − ∥x∥) 2 (1 − |λ|) 2 for any x ∈ b with ∥x∥ < 1 and λ ∈ c with |λ| < 1. c) we have the inequality ∥ ∥ ∥ln (1 − λ) −1 ln (1 − λxy) −1 − ln (1 − λx) −1 ln (1 − λy) −1 ∥ ∥ ∥ (5.14) ≤ 1 4 · ∥x − 1∥ ∥y − 1∥ (1 − ∥x∥) (1 − ∥y∥) [ln (1 − |λ|)] 2 for any x, y ∈ b with xy = yx, ∥x∥ , ∥y∥ < 1 and λ ∈ c with |λ| < 1. in particular, we have ∥ ∥ ∥(1 − λ) −1 ( 1 − λx2 ) −1 − (1 − λx) −2 ∥ ∥ ∥ ≤ 1 4 · ∥x − 1∥ 2 (1 − ∥x∥) 2 [ln (1 − |λ|)] 2 (5.15) 74 s. s. dragomir, m. v. boldea, m. megan cubo 19, 1 (2017) and ∥ ∥ ∥(1 − λ) −1 ( 1 + λx2 ) −1 − (1 − λx) −1 (1 + λx) −1 ∥ ∥ ∥ (5.16) ≤ 1 4 · ∥x − 1∥ ∥x + 1∥ (1 − ∥x∥) 2 [ln (1 − |λ|)] 2 for any x ∈ b with ∥x∥ < 1 and λ ∈ c with |λ| < 1. acknowledgement. the authors would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper. references [1] g. a. anastassiou, grüss type inequalities for the stieltjes integral. nonlinear funct. anal. appl. 12 (2007), no. 4, 583–593. 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[43] j. pečarić, j. mićić and y. seo, inequalities between operator means based on the mondpečarić method. houston j. math. 30 (2004), no. 1, 191–207 [44] c.-j. zhao and w.-s. cheung, on multivariate grüss inequalities. j. inequal. appl. 2008, art. id 249438, 8 pp. () cubo a mathematical journal vol.16, no¯ 03, (21–35). october 2014 higher order multivariate fuzzy approximation by basic neural network operators george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract here are studied in terms of multivariate fuzzy high approximation to the multivariate unit basic sequences of multivariate fuzzy neural network operators. these operators are multivariate fuzzy analogs of earlier studied multivariate real ones. the produced results generalize earlier real ones into the fuzzy setting. here the high order multivariate fuzzy pointwise convergence with rates to the multivariate fuzzy unit operator is established through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the nth order (n ≥ 1) h-fuzzy partial derivatives, of the engaged multivariate fuzzy number valued function. resumen utilizando aproximaciones multivariadas difusas superiores, estudiamos la aplicación a secuencias básicas unitarias multivariadas de operadores de redes neuronales disfusas multivariadas. estos operadores son análogos difusos multivariados de los reales multivariados estudiados anteriormente. los resultados obtenidos generalizan los resultados reales anteriores en el marco difuso. la convergencia puntual difusa multivariada de orden superior con velocidades para los operadores unitarios difusos multivariados se establece a través de desigualdades difusas multivariadas que involucran los módulos de continuidad difusos multivariados de las derivadas parciales h-difusas de n-ésimo orden (n ≥ 1) de las funciones con valores numéricos difusos multivariados. keywords and phrases: multivariate fuzzy real analysis, multivariate fuzzy neural network operators, high order multivariate fuzzy approximation, multivariate fuzzy modulus of continuity and multivariate jackson type inequalities. 2010 ams mathematics subject classification: 26a15, 26e50, 41a17, 41a25, 41a99, 47s40. 22 george a. anastassiou cubo 16, 3 (2014) 1 fuzzy real analysis background we need the following background definition 1. (see [14]) let µ : r → [0, 1] with the following properties (i) is normal, i.e., ∃ x0 ∈ r; µ (x0) = 1. (ii) µ (λx + (1 − λ) y) ≥ min{µ (x) , µ (y)}, ∀ x, y ∈ r, ∀ λ ∈ [0, 1] (µ is called a convex fuzzy subset). (iii) µ is upper semicontinuous on r, i.e. ∀ x0 ∈ r and ∀ ε > 0, ∃ neighborhood v (x0) : µ (x) ≤ µ (x0) + ε, ∀ x ∈ v (x0) . (iv) the set sup p (µ) is compact in r, (where sup p (µ) := {x ∈ r : µ (x) > 0}). we call µ a fuzzy real number. denote the set of all µ with rf. e.g. χ{x0} ∈ rf, for any x0 ∈ r, where χ{x0} is the characteristic function at x0. for 0 < r ≤ 1 and µ ∈ rf define [µ] r := {x ∈ r : µ (x) ≥ r} (1) and [µ] 0 := {x ∈ r : µ (x) ≥ 0}. then it is well known that for each r ∈ [0, 1], [µ] r is a closed and bounded interval on r ([11]). for u, v ∈ rf and λ ∈ r, we define uniquely the sum u ⊕ v and the product λ ⊙ u by [u ⊕ v] r = [u] r + [v] r , [λ ⊙ u] r = λ [u] r , ∀ r ∈ [0, 1] , where [u] r + [v] r means the usual addition of two intervals (as subsets of r) and λ [u] r means the usual product between a scalar and a subset of r (see, e.g. [14]). notice 1 ⊙ u = u and it holds u ⊕ v = v ⊕ u, λ ⊙ u = u ⊙ λ. if 0 ≤ r1 ≤ r2 ≤ 1 then [u] r2 ⊆ [u] r1 . actually [u] r = [ u (r) − , u (r) + ] , where u (r) − ≤ u (r) + , u (r) − , u (r) + ∈ r, ∀ r ∈ [0, 1]. for λ > 0 one has λu (r) ± = (λ ⊙ u) (r) ± , respectively. define d : rf × rf → rf by d (u, v) := sup r∈[0,1] max {∣∣∣u(r)− − v (r) − ∣∣∣ , ∣∣∣u(r)+ − v (r) + ∣∣∣ } , (2) cubo 16, 3 (2014) higher order multivariate fuzzy approximation by basic neural . . . 23 where [v] r = [ v (r) − , v (r) + ] ; u, v ∈ rf. we have that d is a metric on rf. then (rf, d) is a complete metric space, see [14], [15]. let f, g : rm → rf. we define the distance d∗ (f, g) = sup x∈rm d (f (x) , g (x)) . here σ∗ stands for fuzzy summation and 0̃ := χ{0} ∈ rf is the neutral element with respect to ⊕, i.e., u ⊕ 0̃ = 0̃ ⊕ u = u, ∀ u ∈ rf. we need remark 2. ([5]). here r ∈ [0, 1], x (r) i , y (r) i ∈ r, i = 1, ..., m ∈ n. suppose that sup r∈[0,1] max ( x (r) i , y (r) i ) ∈ r, for i = 1, ..., m. then one sees easily that sup r∈[0,1] max ( m∑ i=1 x (r) i , m∑ i=1 y (r) i ) ≤ m∑ i=1 sup r∈[0,1] max ( x (r) i , y (r) i ) . (3) definition 3. let f ∈ c (rm), m ∈ n, which is bounded or uniformly continuous, we define (h > 0) ω1 (f, h) := sup all xi,x′i∈r,|xi−x ′ i |≤h, for i=1,...,m |f (x1, ..., xm) − f (x ′ 1, ..., x ′ m)| . (4) definition 4. let f : rm → rf , we define the fuzzy modulus of continuity of f by ω (f) 1 (f, δ) = sup x,y∈r,|xi−yi|≤δ, for i=1,...,m d (f (x) , f (y)) , δ > 0, (5) where x = (x1, ..., xm), y = (y1, ..., ym) . for f : rm → rf, we use [f] r = [ f (r) − , f (r) + ] , (6) where f (r) ± : r m → r, ∀ r ∈ [0, 1] . we need 24 george a. anastassiou cubo 16, 3 (2014) proposition 5. let f : rm → rf. assume that ω f 1 (f, δ), ω1 ( f (r) − , δ ) , ω1 ( f (r) + , δ ) are finite for any δ > 0, r ∈ [0, 1] . then ω (f) 1 (f, δ) = sup r∈[0,1] max { ω1 ( f (r) − , δ ) , ω1 ( f (r) + , δ )} . (7) proof. by proposition 1 of [8]. we define by cuf (r m) the space of fuzzy uniformly continuous functions from rm → rf, also cf (r m) is the space of fuzzy continuous functions on rm, and cb (r m, rf) is the fuzzy continuous and bounded functions. we mention proposition 6. ([7]) let f ∈ cuf (r m). then ω (f) 1 (f, δ) < ∞, for any δ > 0. proposition 7. ([7]) it holds lim δ→0 ω (f) 1 (f, δ) = ω (f) 1 (f, 0) = 0, (8) iff f ∈ cuf (r m) . proposition 8. ([7]) let f ∈ cf (r m). then f (r) ± are equicontinuous with respect to r ∈ [0, 1] over rm, respectively in ±. note: it is clear by propositions 5, 7, that if f ∈ cuf (r m), then f (r) ± ∈ cu (r m) (uniformly continuous on rm). we need definition 9. let x, y ∈ rf. if there exists z ∈ rf : x = y ⊕ z, then we call z the h-difference on x and y, denoted x − y. definition 10. ([14]) let t := [x0, x0 + β] ⊂ r, with β > 0. a function f : t → rf is h-difference at x ∈ t if there exists an f′ (x) ∈ rf such that the limits (with respect to d) lim h→0+ f (x + h) − f (x) h , lim h→0+ f (x) − f (x − h) h (9) exist and are equal to f′ (x) . we call f′ the h-derivative or fuzzy derivative of f at x. above is assumed that the h-differences f (x + h) − f (x), f (x) − f (x − h) exists in rf in a neighborhood of x. definition 11. we denote by cnf (r m), n ∈ n, the space of all n-times fuzzy continuously differentiable functions from rm into rf. cubo 16, 3 (2014) higher order multivariate fuzzy approximation by basic neural . . . 25 here fuzzy partial derivatives are defined via definition 10 in the obvious way as in the ordinary real case. we mention theorem 12. ([12]) let f : [a, b] ⊆ r → rf be h-fuzzy differentiable. let t ∈ [a, b], 0 ≤ r ≤ 1. clearly [f (t)] r = [ f (t) (r) − , f (t) (r) + ] ⊆ r. then (f (t)) (r) ± are differentiable and [f′ (t)] r = [( f (t) (r) − )′ , ( f (t) (r) + )′] . i.e. (f′) (r) ± = ( f (r) ± )′ , ∀ r ∈ [0, 1] . (10) remark 13. (se also [6]) let f ∈ cn (r, rf), n ≥ 1. then by theorem 12 we obtain f (r) ± ∈ cn (r) and [ f(i) (t) ]r = [( f (t) (r) − )(i) , ( f (t) (r) + )(i)] , for i = 0, 1, 2, ..., n, and in particular we have ( f(i) )(r) ± = ( f (r) ± )(i) , (11) for any r ∈ [0, 1] . let f ∈ cnf (r m), denote fα̃ := ∂ α̃f ∂xα̃ , where α̃ := (α̃1, ..., α̃m), α̃i ∈ z +, i = 1, ..., m and 0 < |α̃| := m∑ i=1 α̃i ≤ n, n > 1. then by theorem 12 we get that ( f (r) ± ) α̃ = (fα̃) (r) ± , ∀ r ∈ [0, 1] , (12) and any α̃ : |α̃| ≤ n. here f (r) ± ∈ c n (rm) . for the definition of general fuzzy integral we follow [13] next. definition 14. let (ω, σ, µ) be a complete σ-finite measure space. we call f : ω → rf measurable iff ∀ closed b ⊆ r the function f−1 (b) : ω → [0, 1] defined by f−1 (b) (w) := sup x∈b f (w) (x) , all w ∈ ω is measurable, see [13]. 26 george a. anastassiou cubo 16, 3 (2014) theorem 15. ([13]) for f : ω → rf, f (w) = {(f (r) − (w) , f (r) + (w))|0 ≤ r ≤ 1}, the following are equivalent (1) f is measurable, (2) ∀ r ∈ [0, 1], f (r) − , f (r) + are measurable. following [13], given that for each r ∈ [0, 1], f (r) − , f (r) + are integrable we have that the parametrized representation {(∫ a f (r) − dµ, ∫ a f (r) + dµ ) |0 ≤ r ≤ 1 } is a fuzzy real number for each a ∈ σ. the last fact leads to definition 16. ([13]) a measurable function f : ω → rf, f (w) = {(f (r) − (w) , f (r) + (w))|0 ≤ r ≤ 1} is integrable if for each r ∈ [0, 1], f (r) ± are integrable, or equivalently, if f (0) ± are integrable. in this case, the fuzzy integral of f over a ∈ σ is defined by ∫ a fdµ := {(∫ a f (r) − dµ, ∫ a f (r) + dµ ) |0 ≤ r ≤ 1 } . (13) by [13] f is integrable iff w → ‖f (w)‖f is real-valeud integrable. here ‖u‖f := d ( u, 0̃ ) , ∀ u ∈ rf. we need also theorem 17. ([13]) let f, g : ω → rf be integrable. then (1) let a, b ∈ r, then af + bg is integrable and for each a ∈ σ, ∫ a (af + bg) dµ = a ∫ a fdµ + b ∫ a gdµ; (2) d (f, g) is a real-valued integrable function and for each a ∈ σ, d (∫ a fdµ, ∫ a gdµ ) ≤ ∫ a d (f, g) dµ. (14) in particular, ∥∥∥∥ ∫ a fdµ ∥∥∥∥ f ≤ ∫ a ‖f‖f dµ. cubo 16, 3 (2014) higher order multivariate fuzzy approximation by basic neural . . . 27 above µ could be the lebesgue measure, with all the basic properties valid here too. basically here we have [∫ a fdµ ]r := [∫ a f (r) − dµ, ∫ a f (r) + dµ ] , (15) i.e. (∫ a fdµ )(r) ± = ∫ a f (r) ± dµ, (16) ∀ r ∈ [0, 1], respectively. we use notation 18. we denote ( 2∑ i=1 d ( ∂ ∂xi , 0̃ ))2 f ( −→ x ) := (17) d ( ∂2f (x1, x2) ∂x2 1 , 0̃ ) + d ( ∂2f (x1, x2) ∂x2 2 , 0̃ ) + 2d ( ∂2f (x1, x2) ∂x1∂x2 , 0̃ ) . in general we denote (j = 1, ..., n) ( m∑ i=1 d ( ∂ ∂xi , 0̃ ))j f ( −→ x ) := (18) ∑ (j1,...,jm)∈z m + : ∑ m i=1 ji=j j! j1!j2!...jm! d ( ∂jf (x1, ..., xm) ∂x j1 1 ∂x j2 2 ...∂x jm m , 0̃ ) . 2 convergence with rates of real multivariate neural network operators here we follow [9]. we need the following (see [10]) definitions. definition 19. a function b : r → r is said to be bell-shaped if b belongs to l1 and its integral is nonzero, if it is nondecreasing on (−∞, a) and nonincreasing on [a, +∞), where a belongs to r. in particular b (x) is a nonnegative number and at a, b takes a global maximum; it is the center of the bell-shaped function. a bell-shaped function is said to be centered if its center is zero. definition 20. (see [10]) a function b : rd → r (d ≥ 1) is said to be a d-dimensional bell-shaped function if it is integrable and its integral is not zero, and for all i = 1, ..., d, t → b (x1, ..., t, ..., xd) is a centered bell-shaped function, where −→x := (x1, ..., xd) ∈ r d arbitrary. 28 george a. anastassiou cubo 16, 3 (2014) example 21. (from [10]) let b be a centered bell-shaped function over r, then (x1, ..., xd) → b (x1) ...b (xd) is a d-dimensional bell-shaped function. assumption 22. here b ( −→ x ) is of compact support b := ∏d i=1 [−ti, ti], ti > 0 and it may have jump discontinuities there. let f : rd → r be a continuous and bounded function or a uniformly continuous function. here we mention the study ([9]) of poitwise convergence with rates over rd, to the unit operator i, of the ”normalized bell” real multivariate neural network operators mn (f) ( −→ x ) := (19) ∑n2 k1=−n 2 ... ∑n2 kd=−n 2 f ( k1 n , ...kd n ) b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) ∑n2 k1=−n 2 ... ∑n2 kd=−n 2 b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) , where 0 < α < 1 and −→x := (x1, ..., xd) ∈ r d, n ∈ n. clearly, mn is a positive linear operator. the terms in the ratio of multiple sums (19) can be nonzero iff simultaneously ∣∣∣∣n 1−α ( xi − ki n )∣∣∣∣ ≤ ti, all i = 1, ..., d, i.e., ∣∣xi − kin ∣∣ ≤ ti n1−α , all i = 1, ..., d, iff nxi − tin α ≤ ki ≤ nxi + tin α, all i = 1, .., d. (20) to have the order − n2 ≤ nxi − tin α ≤ ki ≤ nxi + tin α ≤ n2, (21) we need n ≥ ti + |xi|, all i = 1, .., d. so (21) is true when we take n ≥ max i∈{1,...,d} (ti + |xi|) . (22) when −→x ∈ b in order to have (21) it is enough to assume that n ≥ 2t∗, where t∗ := max{t1, ..., td} > 0. consider ĩi := [nxi − tin α, nxi + tin α ] , i = 1, ..., d, n ∈ n. the length of ĩi is 2tin α. by proposition 1 of [1], we get that the cardinality of ki ∈ z that belong to ĩi := card (ki) ≥ max (2tin α − 1, 0), any i ∈ {1, ..., d}. in order to have card (ki) ≥ 1, we need 2tin α − 1 ≥ 1 iff n ≥ t − 1 α i , any i ∈ {1, ..., d}. therefore, a sufficient condition in order to obtain the order (21) along with the interval ĩi to contain at least one integer for all i = 1, ..., d is that n ≥ max i∈{1,...,d} { ti + |xi| , t − 1 α i } . (23) cubo 16, 3 (2014) higher order multivariate fuzzy approximation by basic neural . . . 29 clearly as n → +∞ we get that card (ki) → +∞, all i = 1, ..., d. also notice that card (ki) equals to the cardinality of integers in [⌈nxi − tin α⌉ , [nxi + tin α]] for all i = 1, ..., d. here, [·] denotes the integral part of the number, while ⌈·⌉ denotes its ceiling. from now on, in this article we will assume (23). furthermore it holds (mn (f)) ( −→ x ) := ∑[nx1+t1nα] k1=⌈nx1−t1nα⌉ ... ∑[nxd+tdnα] kd=⌈nxd−tdnα⌉ f ( k1 n , ...kd n ) v ( −→ x ) (24) ·b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) all −→x := (x1, ..., xd) ∈ r d, where v ( −→ x ) := [nx1+t1n α ]∑ k1=⌈nx1−t1nα⌉ ... [nxd+tdn α ]∑ kd=⌈nxd−tdnα⌉ b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) . (25) from [9], we need and mention theorem 23. let −→x ∈ rd; then ∣∣∣(mn (f)) ( −→ x ) − f ( −→ x )∣∣∣ ≤ ω1 ( f, t∗ n1−α ) . (26) inequality (26) is attained by constant functions. inequalities (26) gives mn (f) ( −→ x ) → f ( −→ x ) , pointwise with rates, as n → +∞, where −→ x ∈ rd, d ≥ 1, provided that f is uniformly continuous on rd. in the last case it is clear that mn → i, uniformly. from [9], we also need and mention theorem 24. let −→x ∈ rd, f ∈ cn ( r d ) , n ∈ n, such that all of its partial derivatives fα̃ of order n, α̃ : |α̃| = n, are uniformly continuous or continuous are bounded. then ∣∣∣(mn (f)) ( −→ x ) − f ( −→ x )∣∣∣ ≤ (27)    n∑ j=1 (t∗) j j!nj(1−α)   ( d∑ i=1 ∣∣∣∣ ∂ ∂xi ∣∣∣∣ )j f ( −→ x )      + (t∗) n dn n!nn(1−α) · max α̃:|α̃|=n ω1 ( fα̃, t∗ n1−α ) . inequality (27) is attained by constant functions. also, (27) gives us with rates the pointwise convergences of mn (f) → f over r d, as n → +∞. 30 george a. anastassiou cubo 16, 3 (2014) 3 main results convergence with rates of fuzzy multivariate neural networks here b is as in definition 20. assumption 25. we suppose that b ( −→ x ) is of compact support b := ∏d i=1 [−ti, ti], ti > 0, and it may have jump discontinuities there. we consider f : rd → rf to be fuzzy continuous and fuzzy bounded function or fuzzy uniformly continuous function. in this section we study the d-metric pointwise convergence with rates over rd, to the fuzzy unit operator if, of the fuzzy multivariate neural network operators (0 < α < 1, −→ x := (x1, ..., xd) ∈ r d, n ∈ n) mfn (f) ( −→ x ) := (28) ∑n2∗ k1=−n 2 ... ∑n2∗ kd=−n 2 f ( k1 n , ...kd n ) ⊙ b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) ∑n2 k1=−n 2 ... ∑n2 kd=−n 2 b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) = [nx1+t1n α ]∗∑ k1=⌈nx1−t1nα⌉ ... [nxd+tdn α ]∗∑ kd=⌈nxd−tdnα⌉ f ( k1 n , ... kd n ) (29) ⊙ b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) v ( −→ x ) , where v ( −→ x ) as in (25) and under the assumption (23). we notice for r ∈ [0, 1] that [ mfn (f) ( −→ x )]r = [nx1+t1n α ]∑ k1=⌈nx1−t1nα⌉ ... [nxd+tdn α ]∑ kd=⌈nxd−tdnα⌉ [ f ( k1 n , ... kd n )]r · b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) v ( −→ x ) (30) = [nx1+t1n α ]∑ k1=⌈nx1−t1nα⌉ ... [nxd+tdn α ]∑ kd=⌈nxd−tdnα⌉ [ f (r) − ( k1 n , ... kd n ) , f (r) + ( k1 n , ... kd n )] · b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) v ( −→ x ) =   [nx1+t1n α ]∑ k1=⌈nx1−t1nα⌉ ... [nxd+tdn α ]∑ kd=⌈nxd−tdnα⌉ f (r) − ( k1 n , ... kd n ) cubo 16, 3 (2014) higher order multivariate fuzzy approximation by basic neural . . . 31 · b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) v ( −→ x ) , [nx1+t1n α ]∑ k1=⌈nx1−t1nα⌉ ... [nxd+tdn α ]∑ kd=⌈nxd−tdnα⌉ f (r) + ( k1 n , ... kd n ) · b ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) v ( −→ x )   (31) = [( mn ( f (r) − ))( −→ x ) , ( mn ( f (r) + ))( −→ x )] . we have proved that ( mfn (f) )(r) ± = mn ( f (r) ± ) , ∀ r ∈ [0, 1] , (32) respectively. we present theorem 26. let −→x ∈ rd; then d (( mfn (f) )(−→ x ) , f ( −→ x )) ≤ ω (f) 1 ( f, t∗ n1−α ) . (33) notice that (33) gives mfn d → if pointwise and uniformly, as n → ∞, when f ∈ c u f ( r d ) . proof. we observe that d (( mfn (f) )(−→ x ) , f ( −→ x )) = sup r∈[0,1] max{ ∣∣∣ ( mfn (f) )(r) − ( −→ x ) − f (r) − ( −→ x )∣∣∣ , ∣∣∣ ( mfn (f) )(r) + ( −→ x ) − f (r) + ( −→ x )∣∣∣} (32) = sup r∈[0,1] max{ ∣∣∣ ( mn ( f (r) − ))( −→ x ) − f (r) − ( −→ x )∣∣∣ , ∣∣∣ ( mn ( f (r) + ))( −→ x ) − f (r) + ( −→ x )∣∣∣} (26) ≤ sup r∈[0,1] max { ω1 ( f (r) − , t∗ n1−α ) , ω1 ( f (r) + , t∗ n1−α )} (7) = ω (f) 1 ( f, t∗ n1−α ) , proving the claim. we continue with theorem 27. let −→x ∈ rd, f ∈ cnf ( r d ) , n ∈ n,such that all of its fuzzy partial derivatives fα̃ of order n, α̃ : |α̃| = n, are fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded. then d (( mfn (f) )(−→ x ) , f ( −→ x )) ≤ (34) 32 george a. anastassiou cubo 16, 3 (2014)    n∑ j=1 (t∗) j j!nj(1−α)   ( d∑ i=1 d ( ∂ ∂xi , 0̃ ))j f ( −→ x )      + (t∗) n dn n!nn(1−α) max α̃:|α̃|=n ω (f) 1 ( fα̃, t∗ n1−α ) . as n → ∞, we get d (( mfn (f) )(−→ x ) , f ( −→ x )) → 0 pointwise with rates. proof. as before we have d (( mfn (f) )(−→ x ) , f ( −→ x )) (32) = sup r∈[0,1] max{ ∣∣∣ ( mn ( f (r) − ))( −→ x ) − f (r) − ( −→ x )∣∣∣ , ∣∣∣ ( mn ( f (r) + ))( −→ x ) − f (r) + ( −→ x )∣∣∣} (27) ≤ sup r∈[0,1] max       n∑ j=1 (t∗) j j!nj(1−α)   ( d∑ i=1 ∣∣∣∣ ∂ ∂xi ∣∣∣∣ )j f (r) − ( −→ x )      + (t∗) n dn n!nn(1−α) max α̃:|α̃|=n ω1 (( f (r) − ) α̃ , t∗ n1−α ) ,    n∑ j=1 (t∗) j j!nj(1−α)   ( d∑ i=1 ∣∣∣∣ ∂ ∂xi ∣∣∣∣ )j f (r) + ( −→ x )      + (t∗) n dn n!nn(1−α) max α̃:|α̃|=n ω1 (( f (r) + ) α̃ , t∗ n1−α )} (3) ≤ n∑ j=1 (t∗) j j!nj(1−α) · (36) sup r∈[0,1] max      ( d∑ i=1 ∣∣∣∣ ∂ ∂xi ∣∣∣∣ )j f (r) − ( −→ x )   ,   ( d∑ i=1 ∣∣∣∣ ∂ ∂xi ∣∣∣∣ )j f (r) + ( −→ x )      + (t∗) n dn n!nn(1−α) max α̃:|α̃|=n sup r∈[0,1] max { ω1 (( f (r) − ) α̃ , t∗ n1−α ) , ω1 (( f (r) + ) α̃ , t∗ n1−α )} (by (3), (7), (12), (18)) ≤    n∑ j=1 (t∗) j j!nj(1−α)   ( d∑ i=1 d ( ∂ ∂xi , 0̃ ))j f ( −→ x )      + (37) (t∗) n dn n!nn(1−α) max α̃:|α̃|=n ω (f) 1 ( fα̃, t∗ n1−α ) , proving the claim. cubo 16, 3 (2014) higher order multivariate fuzzy approximation by basic neural . . . 33 4 main results the fuzzy multivariate ”normalized squashing type operators” and their fuzzy convergence to the fuzzy unit with rates we give the following definition definition 28. let the nonnegative function s : rd → r, d ≥ 1, s has compact support b := d∏ i=1 [−ti, ti], ti > 0 and is nondecreasing there for each coordinate. s can be continuous only on either d∏ i=1 (−∞, ti] or b and can have jump discontinuities. we call s the multivariate ”squashing function” (see also [10]). let f : rd → rf be either fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded function. for −→x ∈ rd, we define the fuzzy multivariate ” normalized squashing type operator”, lfn (f) ( −→ x ) := (38) ∑n2∗ k1=−n 2 ... ∑n2∗ kd=−n 2 f ( k1 n , ...kd n ) ⊙ s ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) w ( −→ x ) , where 0 < α < 1 and n ∈ n: n ≥ max i∈{1,...,d} { ti + |xi| , t − 1 α i } , (39) and w ( −→ x ) := n 2∗∑ k1=−n 2 ... n 2∗∑ kd=−n 2 s ( n1−α ( x1 − k1 n ) , ..., n1−α ( xd − kd n )) . (40) it is clear that ( lfn (f) )(−→ x ) := [ n−→x + −→ t n α ] ∗ ∑ −→ k = ⌈ n−→x − −→ t nα ⌉ f (−→ k n ) ⊙ s ( n1−α ( −→ x − −→ k n )) φ ( −→ x ) , (41) where φ ( −→ x ) := [ n−→x + −→ t n α ] ∑ −→ k = ⌈ n−→x − −→ t nα ⌉ s  n1−α  −→x − −→ k n     . (42) here, we study the d-metric pointwise convergence with rates of ( lfn (f) )(−→ x ) → f ( −→ x ) , as n → +∞, −→x ∈ rd. this is given first by the next result. 34 george a. anastassiou cubo 16, 3 (2014) theorem 29. under the above terms and asumptions, we find that d (( lfn (f) )(−→ x ) , f ( −→ x )) ≤ ω (f) 1 ( f, t∗ n1−α ) . (43) notice that (43) gives lfn d → if pointwise and uniformly, as n → ∞, when f ∈ c u f ( r d ) . proof. similar to (33). we also give theorem 30. let −→x ∈ rd, f ∈ cnf ( r d ) , n ∈ n, such that all of its fuzzy partial derivatives fα̃ of order n, α̃ : |α̃| = n, are fuzzy uniformly continuous or fuzzy continuous and fuzzy bounded. then d (( lfn (f) )(−→ x ) , f ( −→ x )) ≤ (44)    n∑ j=1 (t∗) j j!nj(1−α)   ( d∑ i=1 d ( ∂ ∂xi , 0̃ ))j f ( −→ x )      + (t∗) n dn n!nn(1−α) max α̃:|α̃|=n ω (f) 1 ( fα̃, t∗ n1−α ) . inequality (44) gives us with rates the poitwise convergence of d( ( lfn (f) )(−→ x ) , f ( −→ x ) ) → 0 over r d, as n → ∞. proof. similar to (34). received: november 2012. accepted: may 2014. references [1] g.a. anastassiou, rate of convergence of some neural network operators to the unit-univariate case, journal of mathematical analysis and application, vol. 212 (1997), 237-262. 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[15] c. wu, m. ma, on embedding problem of fuzzy numer spaces: part 1, fuzzy sets and systems, 44 (1991), 33-38. fuzzy real analysis background convergence with rates of real multivariate neural network operators main results convergence with rates of fuzzy multivariate neural networks main results the fuzzy multivariate ''normalized squashing type operators'' and their fuzzy convergence to the fuzzy unit with rates cubo a mathematical journal vol.14, no¯ 01, (93–110). march 2012 new aspects on elementary functions in the context of quaternionic analysis s. georgiev department of differential equations, university of sofia, sofia, bulgaria, email: sgg2000bg@yahoo.com j. morais technical university of mining, freiberg, germany, email: joao.pedro.morais@ua.pt and w. spröß technical university of mining, freiberg, germany, email: sproessig@math.tu-freiberg.de abstract the main objective of this article is to give a survey on elementary functions in the context of quaternionic analysis. we define some of their more common properties, which as in the real and complex cases, will be familiar to the reader. this leads to the consideration of quaternion-valued functions depending on a quaternion variable, that is, functions whose input and output are quaternions. 94 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) resumen el objetivo principal de este art́ıculo es dar una visión general sobre funciones elementales en el contexto de análisis cuaterniónico. definimos algunas de sus propiedades más comunes, que como en los casos reales y complejos, serán familiares para el lector. esto lleva a la consideración de las funciones de valor-cuaterniónico dependiendo de una variable cuaterniónica, esto es, funciones las cuales de entrada y salida son cuaterniones. keywords and phrases: quaternionic analysis, elementary functions. 2010 ams mathematics subject classification: 30g35, 30a10. 1 introduction as is well known, quaternionic analysis generalizes the theory of holomorphic functions of one complex variable and also provides the foundations to refine the theory of harmonic functions in higher dimensions. methods of quaternionic analysis in combination with other classical and modern analytical methods (such as harmonic analysis, variational methods, and finite difference methods) have been playing an increasingly active part in the treatment of problems, mainly in mathematical physics, which involve the treatment of elementary functions. basic results were independently discovered and rediscovered by many people, among others: scheffers (1893), dixon (1904), lanczos (1919), moisil-teodorescu (1931), melijhzon (1948), iftimie (1965), hestenes (1968), delanghe (1970), and sudbery (1979), brackx, delanghe and sommen (1982), gürlebeck and w. sprößig (1989). meanwhile quaternionic analysis has became a well established branch in mathematics and greatly successful in many different directions. navigation, computer vision, robotics, signal and image processing, or efficient description of classical mechanics and electrical engineering are examples of fields where quaternions are used nowadays. a survey within the scope of quaternionic analysis and its applications is given in [10], and references therein. the organization of this paper is as follows. section 2 begins with a review of some definitions and basic properties of quaternionic analysis. we proceed in section 3 to study the quaternion exponential and logarithmic functions. although quaternion multiplication is not commutative, many formal properties of the complex exponential and logarithmic functions can be generalized within this framework. in the remaining sections the quaternion trigonometric, hyperbolic, and their inverse functions are covered. a brief discussion on the notions of multiple-valued functions and branches is also presented. there is no attempt to cover everything related to elementary quaternion functions and review the historical development of quaternions. general information is contained in the books [6, 7]. cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 95 2 basic properties and definitions for quaternions for all what follows we will work in h, the skew field of real quaternions discovered by hamilton in 1843. this means we can express each element p ∈ h uniquely in the form p = p0+p1i+p2j+p3k, where pi (i = 0,1,2,3) are ordinary numbers and the imaginary units i, j, and k stand for the elements of the basis of h, subject to the multiplication rules i2 = j2 = k2 = −1, ij = k = −ji, jk = i = −kj, ki = j = −ik. in this way the quaternionic algebra arises as a natural extension of the complex field c. for, we identify c with the set of degenerate quaternions with zero coefficients of j and k. we shall always assume the quaternion p = 0 + 0i + 0j + 0k := 0h to be the neutral element of addition, and p = 1+0i +0j +0k := 1h to be the multiplicative identity quaternion in the sequel. they are such that any quaternion added or multiplied by them remains unchanged. further, we denote by (i) sc(p) = p0 the scalar part, and vec(p) = p = p1i + p2j + p3k the vector part of p, (ii) p = p0 − p the conjugate of p, (iii) |p| = √ pp = √ pp = √ p20 + p 2 1 + p 2 2 + p 2 3 the norm of p, (iv) p−1 = p |p|2 , p 6= 0h the inverse of p, (v) sgn(p) = p |p| the quaternion sign. with the intention to obtain some insight for a forthcoming study of elementary quaternion functions, we now consider the classical polar form of a real quaternion with a modulus and an argument. theorem 2.1. every real quaternion p with p 6= (0,0,0) satisfies the trigonometric representation p = |p| ( cos θ + sgn(p) sin θ ) . since p20 + |p| 2 = |p|2 we find the relations: cos θ = p0 |p| , and sin θ = |p| |p| . the angle θ := arg(p) (with positive counterclockwise orientation) is called the quaternion argument of the quaternion p and is only determined up to an integer multiple of 2π. for p = 0h, arg(p) cannot be defined in any way that is meaningful. in practice, when p 6= 0h we use tan θ = |p| p0 to find θ, where the quadrant in which p0 and |p| lie must always be specified or be clearly understood. although the symbol arg(p) actually represents a set of values, the argument θ of a quaternion that lies in the interval [0,π] is called the principal argument of p, and is represented 96 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) by the symbol arg(p). we proceed with some fundamental definitions and notations which will be needed throughout the text. definition 2.1. the sequence {pn} ∞ n=1 of real quaternions pn is called convergent to the real quaternion p if for every � > 0, there exists a natural number n such that for every n > n we have |pn − p| < �. we will use the traditional notation: lim n−→∞ pn = p. definition 2.2. the sequence {pn} ∞ n=1 of real quaternions is called convergent to infinity if for every positive constant m, there exists a natural number n such that for every n > n we have |pn| ≥ m. for this one uses the notation: lim n−→∞ pn = ∞. definition 2.3. the sequence {pn = p0,n + p1,ni + p2,nj + p3,nk} ∞ n=1, pm,n ∈ r (m = 0,1,2,3) is called convergent to −∞ and we write lim n−→∞ pn = −∞ if for every constant a ∈ r, there exists n = n(a) such that for every n > n we have pm,n < a, m = 0,1,2,3. let us go on to the consideration of quaternion-valued functions. a quaternion function of a quaternion variable p, or, briefly, an h-valued function is a function whose input and output are quaternions. it is a mapping f : h −→ h such that f(p) = [f(p)]0 + [f(p)]1i + [f(p)]2j + [f(p)]3k, where the coordinate-functions [f]i (i = 0,1,2,3) are real-valued functions defined in h. as in the case with complex functions we have the standard operations on quaternion functions. in particular, given two functions f(p) and g(p), we define addition (subtraction), f(p) ± g(p), conjugation f(p) = f(p), multiplication f(p)g(p), and quotient f(p) g(p) provided with g(p) 6= 0h. we shall always write f(p) g(p) to mean f(p)g−1(p) in the sequel. note that this is generally different from g−1(p)f(p) since quaternion multiplication is not commutative. implicit in the previous arithmetic operations are simple but very important facts: [f]0 = 1 2 ( f + f ) and f = vec(f) = 1 2 ( f − f ) . on occasion the modulus of a quaternion function f is defined by |f| = √ ff = √ ff = √∑3 i=0[f] 2 i , and coincides with its corresponding euclidean norm as a vector in r4. 3 exponents and logarithms revisited in this section we revisit the quaternionic analogues of the complex exponential and logarithmic functions. it turns out that exponential and logarithmic quaternion functions can be defined because quaternions have a division algebra. the principal value of the quaternion logarithm will be defined to be a single-valued function whose argument lies in the interval [0,π]. this principal value quaternion function will be shown to be an inverse of the quaternion exponential function defined on a suitably restricted domain of the quaternion space. we therefore start our discussion with the following definition. cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 97 definition 3.1. the function ep defined by ep := ∑∞ k=0 pk k! is called the quaternion natural exponential function. in the case that p is a complex number, the definition of ep is naturally extended to comply with the usual exponential function of complex numbers. analogous to the complex case one may derive a closed-form representation for the quaternion exponential function. theorem 3.1. for the quaternion natural exponential function the following representation holds: ep = ep0 ( cos |p| + sgn(p) sin |p| ) . corollary 3.1. let {pn} ∞ n=1 be a sequence of elements of h such that lim n−→∞ pn = p. the usual limit representation works: lim n−→∞ ( 1 + pn n )n = ep. we now record some useful properties of the quaternion exponential function. corollary 3.2. (see [6, 13, 7]) the quaternion natural exponential function satisfies the following properties: (1) ep 6= 0h, for all p ∈ h, (2) e−pep = 1h, e sgn(p)π = −1h, (3) (ep)n = enp for n = 0,±1,±2, . . . (de moivre’s formula), (4) ep1ep2 6= ep1+p2 in general, unless p1 and p2 commute, (5) |ep| = esc(p), (6) ep = ep. in what follows we introduce the quaternion natural logarithm function ln(p), which is motivated as defining the ”inverse” of the quaternion natural exponential function ep. more precisely, we define ln(p) to be any quaternion number such that eln(p) = p and ln(ep) = p. this is too much to hope for. we shall discuss this matter in further detail in the remainder of the section. definition 3.2. the multiple-valued quaternion natural logarithm function ln(p) is defined by ln(p) = loge |p| + sgn(p) arg(p). (3.1) 98 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) here loge |p| is the usual real natural logarithm of the positive number |p|. by switching to polar form in (3.1), we obtain the following alternative description of the quaternion logarithm: ln(p) =   loge |p| + sgn(p) ( arccos p0 |p| + 2πn ) , |p| 6= 0 loge |p0| , |p| = 0 =   loge |p| + sgn(p) ( arctan |p| p0 + 2πn ) , p0 > 0 loge |p| + sgn(p) ( π 2 + 2πn ) , p0 = 0 where n = 0,±1,±2, . . . . theorem 3.2. let {pn} ∞ n=1 be a sequence of elements of h and n ∈ n. the following statements are valid: (1) if lim n−→∞ pn = ∞ then limn−→∞ epn = ∞, (2) if lim n−→∞ pn = ∞ then limn−→∞ ln(pn) = ∞ and limn−→∞ ln(pn)pn = 0, (3) if lim n−→∞ pn = 0 then limn−→∞ ln(pn) = −∞, (4) if lim n−→∞ pn = p then limn−→∞ ln(pn) = ln(p). proof. these statements follow from definitions 3.1 and 3.2. if we wish to define single-valued branches of ln(p), it would be more satisfactory to restrict arg(p) to its principal value arg(p). this yields to the following definition. definition 3.3. the principal value of the quaternion logarithm is denoted by the symbol ln(p), and is defined as ln(p) = loge |p| + sgn(p) arg(p). (3.2) at this stage we return to the principle mentioned at the beginning of the section. since ln(p) is one of the values of the quaternion logarithm ln(p), it follows from (3.2) that: eln(p) = p and ln(ep) = p for all nonzero quaternion p defined on the so-called fundamental region p0 > 0 and |p| < π. this suggests that the quaternion function ln(p) plays the role of an inverse function of the exponential quaternion function ep. to justify this claim, observe that for the quaternion p = πk, which is clearly not in the fundamental region, we have: eln(πk) = eloge |π|k = πk, but ln(eπk) = ln(−1) = 0. corollary 3.3. the principal value of the quaternion logarithm function retains the following properties: cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 99 (1) eln(p) = p and ln(ep) = p, for all nonzero quaternion p defined on the fundamental region p0 > 0 and |p| < π, (2) ln(1) = 0, ln(i) = π 2 i, ln(j) = π 2 j, and ln(k) = π 2 k, (3) ln(p1p2) 6= ln(p1) + ln(p2) in general, unless p1 and p2 commute, (4) ln(pn) = nln(p), for n = 0,±1,±2, . . . (de moivre’s formula). proof. the proofs follow immediately from definition 3.3. in this connection it is of interest to note: theorem 3.3. let p be a real quaternion such that |p| ≥ 1. the following inequalities are valid: (1) |ln(p)| ≤ |p| − 1 + π, (2) |ln(p)| ≤ 2|p| 3−9|p|2+18|p|−11 6 + π, (3) |ln(p)| ≤ 2n−1∑ k=1 (|p| − 1)k k + π, n ∈ n. proof. for the proof of statement 1. a first straightforward computation shows that |ln(p)| ≤ | loge(1 + |p| − 1)| + |sgn(p)arg(p)| ≤ |p| − 1 + π. the last step follows from the standard inequality ln(1 + x) ≤ x for every x ≥ 0. in a similar manner, to prove statement 2. we shall use the inequality ln(1+x) ≤ x− x 2 2 + x 3 3 for every x ≥ 0. equally clear is the following |ln(p)| ≤ (|p| − 1) − (|p| − 1)2 2 + (|p| − 1)3 3 + π = 2|p|3 − 9|p|2 + 18|p| − 11 6 + π. lastly, the proof of statement 3. is a consequence of the classical inequality ln(1+x) ≤ 2n−1∑ k=1 (−1)k+1 xk k for every x ≥ 0. to supplement our investigations, we shall now introduce a general quaternion power function. definition 3.4. if q is a real quaternion and p 6= 0h, then the quaternion power function pq is defined to be: pq = eln(p)q. 100 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) one fact that should be stressed here is that depending on q the quaternion power function will have either one, finitely many or infinitely many values: if q = n is an integer then pn assumes only one value. if q = a b is a rational number, where a and b are common factors, then p a b = |p| a besgn(p)arg(p) a b may have a finite number of values. if q is a nonzero real quaternion, then pq may always have an infinite number of values. corollary 3.4. quaternion powers satisfy the following properties: (1) (pq)n = pnq for n = 0,±1,±2, . . . , (2) pq1pq2 6= pq1+q2 in general, unless ln(p)q1 and ln(p)q2 commute. proof. the proof follows immediately from definition 3.4. we shall conclude our considerations with some of the more common properties of the abovementioned elementary quaternion functions, which as in the real and complex cases, will be familiar to the reader. theorem 3.4. let p be a real quaternion, and n ∈ n. then lim n−→∞ n ( n √ p − 1 ) = loge |p|. proof. using previous definitions a first straightforward computation shows that n √ p = e 1 n loge |p| [ cos ∣∣∣1 n sgn(p)arg(p) ∣∣∣ + sgn(1 n sgn(p)arg(p) ) sin ∣∣∣1 n sgn(p)arg(p) ∣∣∣] = e 1 n loge |p| [ cos ∣∣∣arg(p) n ∣∣∣ + sgn(p) sin∣∣∣arg(p) n ∣∣∣] . this yields lim n−→∞ n ( n √ p − 1 ) = lim n−→∞ e 1 n loge |p| [ cos ∣∣∣arg(p)n ∣∣∣ + sgn(p) sin∣∣∣arg(p)n ∣∣∣] − 1 1 n = loge |p|. theorem 3.5. let p be a real quaternion, and n ∈ n. then lim n−→∞ (1 + n√p 2 )n = √ |p|. cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 101 proof. obviously, one has 1 + n √ p 2 = 1 + n ( n √ p − 1 ) 2n . hence in accordance with the previous result, it follows lim n−→∞ (1 + n√p 2 )n = lim n−→∞ ( 1 + n( n √ p − 1) 2n )n = e loge |p| 2 = √ |p|. having treated these special cases, we may now pass to a somewhat more general case in which a finite sequence of quaternions takes a part in the definitions of the above-mentioned quaternion elementary functions. theorem 3.6. let k ∈ n, and {pν}kν=1 be a sequence of elements of h. then lim n−→∞ ( 1 k k∑ ν=1 n √ pν )n = k √√√√ k∏ ν=1 |pν|. proof. we have clearly 1 k k∑ ν=1 n √ pν = 1 + 1 kn [ n k∑ ν=1 ( n √ pν − 1 )] , and, as the above discussion shows, it follows lim n−→∞ 1 k [ n k∑ ν=1 ( n √ pν − 1 )] = 1 k k∑ ν=1 loge |pν| = loge k √√√√ k∏ ν=1 |pν|. with these calculations at hand, we set lim n−→∞ ( 1 k k∑ ν=1 n √ pν )n = eloge k √∏ k ν=1 |pν| = k √√√√ k∏ ν=1 |pν|. as concerns the definition of quaternion power function, we can formulate the next theorems. theorem 3.7. let p1 and p2 be two real quaternions, and n ∈ n. then lim n−→∞ ( n2 √ p1 + n 3√ p2 )n 2n = 1. 102 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) proof. from definition 3.4, a direct computation shows that n2 √ p1 = e 1 n2 loge |p1| [ cos ∣∣∣arg(p1) n2 ∣∣∣ + sgn(p1) sin∣∣∣arg(p1) n2 ∣∣∣] , n3 √ p2 = e 1 n3 loge |p2| [ cos ∣∣∣arg(p2) n3 ∣∣∣ + sgn(p2) sin∣∣∣arg(p2) n3 ∣∣∣] , and therefore, it follows lim n−→∞ n ( n2 √ p1 − 1 ) = 0, and lim n−→∞ n ( n3 √ p2 − 1 ) = 0. accordingly, this leads to lim n−→∞ ( n2 √ p1 + n 3√ p2 )n 2n = lim n−→∞  1 + n ( n2 √ p1 − 1 ) + n ( n3 √ p2 − 1 ) 2n   n = 1. theorem 3.8. let {pn} ∞ n=1 and {qn} ∞ n=1 be two sequences of elements of h (n ∈ n) such that lim n−→∞ pn = p, and limn−→∞ qn = q. then lim n−→∞ 1 2n ( n √ pn + n √ qn )n = √ |p||q|. proof. the proof follows immediately from theorem 3.6. theorem 3.9. let {pkn} ∞ n=1 be a sequence of elements of h (n,k ∈ n) such that lim n−→∞ pkn = pk. then lim n−→∞ ( 1 k k∑ ν=1 n √ pνn ) = k √√√√ k∏ ν=1 |pν|. proof. the proof is a consequence of theorem 3.6. 4 trigonometry revisited in this section we define quaternion trigonometric functions. analogously to the quaternion functions ep and ln(p), these functions will agree with their counterparts for complex input. in addition, we will notice that the quaternion trigonometric functions satisfy various of the same identities as the real and complex trigonometric functions. with the help of the exponential function, quaternionic analogues of the trigonometric functions can be introduced. cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 103 definition 4.1. the functions sin(p) and cos(p) defined respectively by sin(p) =   − 1 2 sgn(p) ( epsgn(p) − e−psgn(p) ) , |p| 6= 0 sin(p0) , |p| = 0 cos(p) =   1 2 ( epsgn(p) + e−psgn(p) ) , |p| 6= 0 cos(p0) , |p| = 0 are called the quaternion sine and cosine functions. corollary 4.1. (see [6, 7]) the quaternion sine and cosine functions admit the following representations: sin(p) = sin(p0) cos(p) + cos(p0) sin(p), cos(p) = cos(p0) cos(p) − sin(p0) sin(p). the usual identities obey, such as sin(p) = − sin(−p) and cos(p) = cos(−p). a straightforward computation shows that sin2(p)+cos2(p) = 1h. however, one has to pay attention to the fact that the quaternion trigonometric functions do in general not satisfy the sum and difference formulae: take for example p = i+j, and sin(i+j) = (i+j)√ 2 sinh( √ 2) 6= (i+j) sinh(1) cosh(1) = sin(i) cos(j)+ cos(i) sin(j). we may now have a look to the zeros of the quaternion sine and cosine functions. since sgn(p) = −sgn(p), it can be easily shown that | sin(p)|2 = sin2(p0) + sinh 2 |p| (4.1) and | cos(p)|2 = cos2(p0) + cosh 2 |p| − 1. (4.2) since sin2(p0) (resp. cos 2(p0) ) and sinh 2 |p| are both nonnegative real numbers, it is evident that the structural equations are satisfied if and only if sin(p0) = 0 (resp. cos(p0) = 0) and sinh |p| = 0. as is well known, sin(p0) = 0 when p0 = nπ, n = 0,±1,±2, . . . (resp. cos(p0) = 0 when p0 = (2n+1)π 2 , n = 0,±1,±2, . . . ), and sinh |p| = 0 only when |p| = 0. therefore the quaternion trigonometric functions have only the zeros known for the real functions. now we turn our attention to inequalities involving the quaternion trigonometric functions. theorem 4.1. let p be a real quaternion such that |p| ≤ ln(1 + √ 2). then | sin(p)| ≤ √ p20 + 1. 104 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) proof. notice that sin2(p0) ≤ p20, and sinh 2(|p|) ≤ 1 for every |p| ∈ [0, ln(1 + √ 2)]. based on the representation (4.1), it follows | sin(p)|2 ≤ p20 + 1. theorem 4.2. let p be a real quaternion such that |p| ≥ ln(1+ √ 2), and p0 ≤ − √ 6 or p0 ∈ [0, √ 6]. then | sin(p)| ≥ √ 1+ ( p0 − p30 6 )2 . proof. note, in passing, that sin2(p0) ≥ ( p0 − p30 6 )2 when p0 ≤ − √ 6 or p0 ∈ [0, √ 6], and sinh2(|p|) ≥ 1 for |p| ≤ ln(1 + √ 2). in summary, we set | sin(p)|2 ≥ 1 + ( p0 − p30 6 )2 . theorem 4.3. let p be a real quaternion such that p0 ∈ [0, √ 2]. then | cos(p)| ≥ 1 − p20 2 . proof. let p0 ∈ [0, √ 2], it is clear that cos2(p0) ≥ ( 1 − p20 2 )2 . based on the relation (4.2), we conclude that | cos(p)|2 ≥ ( 1 − p20 2 )2 + 1 − 1 = ( 1 − p20 2 )2 . in closing this section, we note that the usual definitions of the other trigonometric functions are taken: definition 4.2. for p ∈ h \ {(n + 1/2)π : n = 0,±1,±2, . . . }, the functions tan(p) and sec(p) defined respectively by tan(p) = sin(p) cos(p) , and sec(p) = 1 cos(p) , are called the quaternion tangent and secant functions. definition 4.3. for p ∈ h \ {nπ : n = 0,±1,±2, . . . }, the functions cot(p) and csc(p) defined respectively by cot(p) = cos(p) sin(p) , and csc(p) = 1 sin(p) , are called the quaternion cotangent and cosecant functions. cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 105 theorem 4.4. let p be a real quaternion such that |p| ≤ ln(1 + √ 2), and p0 ∈ [0, √ 2). then | tan(p)| ≤ 2 √ 1 + p20 2 − p20 . proof. the proof follows from theorems 4.1 and 4.3. theorem 4.5. let p be a real quaternion such that |p| ≤ ln(1 + √ 2), and p0 ∈ [0, √ 2]. then | cot(p)| ≥ 2 − p20 2 √ p20 + 1 . proof. it follows from theorems 4.1 and 4.3. 5 hyperbolic revisited the quaternion hyperbolic sine and cosine functions may be defined using the quaternion exponential function as follows: definition 5.1. the functions sinh(p) and cosh(p) defined respectively by sinh(p) = ep − e−p 2 , cosh(p) = ep + e−p 2 , are called the quaternion sine and cosine hyperbolic functions. similarly to their counterparts for real and complex input the quaternion hyperbolic sine and cosine are respectively, odd and even functions. a straightforward computation shows that cosh2(p) − sinh2(p) = 1h. however, we must keep clearly in mind that the quaternion hyperbolic functions do in general not satisfy the sum and difference formulae, unless p1 and p2 commute. it can be settled by a simple example which we sketch briefly. take for example p = i − j, then sinh(i − j) = (i−j)√ 2 sin( √ 2) 6= (i − j) cos(1) sin(1) = sinh(i) cosh(j) − cosh(i) sinh(j). remark 5.1. the zeros of the function sinh(p) are the solutions for p of the equation ep = e−p, which can be rewritten as e2p = 1h. since e 2p = 1h only when 2p is an integer multiple of 2πsgn(p), we conclude that the zeros of sinh(p) are the numbers nπsgn(p), n = 0,±1,±2, . . . . similarly, the zeros of cosh(p) are the solutions for p of the equation ep = −e−p, which can be rewritten as e2p = −1h, or as e 2p−sgn(p)π = 1h (since e sgn(p)π = −1h). it follows that the zeros of cosh(p) are the numbers ( n + 1 2 ) πsgn(p) with n = 0,±1,±2, . . . . we proceed to study inequalities involving the quaternion sine and cosine hyperbolic functions. 106 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) theorem 5.1. let p be a real quaternion. then | sinh(p)| ≤ 2 cosh(p0), and | cosh(p)| ≤ 2 cosh(p0). proof. from theorem 3.1, we set |ep| ≤ 2ep0. in particular, it holds |e−p| ≤ 2e−p0. now, relying on defintion 5.1 it follows | sinh(p)| ≤ 2 cosh(p0). in a similar manner, we obtain the second inequality of our theorem. analogously to quaternion trigonometric functions, we next define the quaternion tangent, cotangent, secant, and cosecant hyperbolic functions using the quaternion hyperbolic sine and cosine. definition 5.2. for p ∈ h \ {(n + 1 2 )πsgn(p) : n = 0,±1,±2, . . . }, the functions tanh(p) and sech(p) defined respectively by tanh(p) = sinh(p) cosh(p) , and sech(p) = 1 cosh(p) , are called the quaternion tangent and secant hyperbolic functions. definition 5.3. for p ∈ h \ {nπsgn(p) : n = 0,±1,±2, . . . }, the functions coth(p) and csch(p) defined respectively by coth(p) = cosh(p) sinh(p) , and csch(p) = 1 sinh(p) , are called the quaternion cotangent and cosecant functions. 6 inverse hyperbolic and trigonometric functions revisited the main focus of this section is to study the inverses of the quaternion trigonometric and hyperbolic functions, and their properties. since the quaternion trigonometric and hyperbolic functions are defined in terms of the quaternion exponential function ep, their inverses are necessarily multiple-valued and may be computed via the quaternion natural logarithm function ln(p). we summarize this discussion in the following definition. definition 6.1. the multiple-valued functions sinh−1(p) and cosh−1(p) defined respectively by sinh−1(p) = ln ( p + √ p2 + 1 ) , cosh−1(p) = ln ( p + √ p2 − 1 ) are called the quaternion inverse hyperbolic sine and cosine. these functions have two sources of multivaluedness; one due to the quaternion natural logarithm function ln(p), the other due to the involved quaternion power functions. it is evident that cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 107 the quaternion inverse hyperbolic sine and cosine can be made single-valued by specifying a single value of the quaternion logarithm and a single value of the functions (p2 + 1)1/2 and (p2 − 1)1/2, respectively. we see at the same time that a branch of a quaternion inverse hyperbolic function may be obtained by choosing a branch of the quaternion logarithm and a branch of a quaternion power function. in spite of its local multiplevaluedness, the ln(p) function has an infinite number of branches, hence so do the quaternion inverse hyperbolic sine and cosine. in addition, the quaternion inverse hyperbolic functions have branch point-solutions to the equations p2 ±1 = 0h, because the functions (p2 + 1)1/2 and (p2 − 1)1/2 have no solutions of (p2 + 1)1/2 = 0h and (p2 − 1)1/2 = 0h, respectively. theorem 6.1. let p be a real quaternion such that |p|4−|p|2+2p20 ≥ 0. the following inequalities are valid: (1) | sinh−1(p)| ≤ √ 1 + 2(|p|4 − |p|2 + 2p20) + π, (2) | cosh−1(p)| ≤ √ 1 + 2(|p|4 − |p|2 + 2p20) + π, (3) | sinh−1(p)| ≤ 1 + 2(|p|4 − |p|2 + 2p20) − ( √ 1+2(|p|4−|p|2+2p2 0 ))2 2 + ( √ 1+2(|p|4−|p|2+2p2 0 ))3 3 + π, (4) | cosh−1(p)| ≤ 1 + 2(|p|4 − |p|2 + 2p20) − ( √ 1+2(|p|4−|p|2+2p2 0 ))2 2 + ( √ 1+2(|p|4−|p|2+2p2 0 ))3 3 + π, (5) | sinh−1(p)| ≤ 2n−1∑ k=1 (−1)k+1 ( √ 1 + 2(|p|4 − |p|2 + 2p20)) k k + π, n ∈ n, (6) | cosh−1(p)| ≤ 2n−1∑ k=1 (−1)k+1 ( √ 1 + 2(|p|4 − |p|2 + 2p20)) k k + π, n ∈ n. proof. let p := p0 + p1i + p2j + p3k be a real quaternion. by definition 3.4 we find the following representation for p + √ p2 + 1: p + √ (p20 + 1 − |p| 2)2 + 4p20|p| 2 ( cos ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣ + sgn(p) sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣). (6.1) 108 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) with a denoting the term √ (p20 + 1 − |p| 2)2 + 4p20|p| 2, we have, on account of (6.1): |p + √ p2 + 1|2 = ( p0 + a cos ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 + ( p1 + a p1√ p21 + p 2 2 + p 2 3 sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 + ( p2 + a p2√ p21 + p 2 2 + p 2 3 sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 + ( p3 + a p3√ p21 + p 2 2 + p 2 3 sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 now we make use of the standard inequality: (a + b)2 ≤ 2(a2 + b2). hence we obtain |p + √ p2 + 1|2 ≤ 2 [ p20 + ( a cos ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 + p21 + ( a p1√ p21 + p 2 2 + p 2 3 sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 + p22 + ( a p2√ p21 + p 2 2 + p 2 3 sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2 + p23 + ( a p3√ p21 + p 2 2 + p 2 3 sin ∣∣∣1 2 arccos p20 + 1 − |p| 2 (p20 + 1 − |p| 2)2 + 4p20|p| 2 ∣∣∣)2   = 2 ( |p|4 − |p|2 + 2p20 + 1 ) , that is, |p + √ p2 + 1|2 ≤ 1 + 1 + 2 ( |p|4 − |p|2 + 2p20 ) . furthermore, using the inequality √ a + b ≤ √ a + √ b for a ≥ 0 and b ≥ 0, the following further inequality is now immediate: loge |p + √ p2 + 1| ≤ loge ( 1 + √ 1 + 2(|p|4 − |p|2 + 2p20) ) . we may now use the inequality ln(1 + x) ≤ x for x ≥ 0 to obtain: loge |p + √ p2 + 1| ≤ √ 1 + 2 ( |p|4 − |p|2 + 2p20 ) , cubo 14, 1 (2012) new aspects on elementary functions in the context . . . 109 from which we find, |ln(p + √ p2 + 1)| ≤ √ 1 + 2 ( |p|4 − |p|2 + 2p20 ) + π. we have thus obtain the following further inequality: | sinh−1(p)| ≤ √ 1 + 2 ( |p|4 − |p|2 + 2p20 ) + π. in a similar way we obtain the inequality for cosh−1(p). the proofs of the remaining statements follow from the aforementioned inequalities used in the proof of theorem 3.3. the quaternion inverse hyperbolic tangent can be now introduced. definition 6.2. the multiple-valued function tanh−1(p) defined respectively by tanh−1(p) = ln(1 + p) − ln(1 − p) 2 , is called the quaternion inverse hyperbolic tangent. all this being established, we can now introduce the inverse trigonometric sine, cosine and tangent quaternion functions. definition 6.3. the multiple-valued functions sin−1(p) and cos−1(p) defined respectively by sin−1(p) = sgn(p) sinh−1 ( p sgn(p) ) , cos−1(p) = sgn(p) cosh−1(p) are called the quaternion inverse sine and cosine. definition 6.4. the multiple-valued function tan−1(p) defined respectively by tan−1(p) = sgn(p) tanh−1 ( p sgn(p) ) is called the quaternion inverse tangent. 6.1 acknowledgements the first author’s research is supported by the ministry of science and education of bulgaria, grant dd-vu 02/90. the second author acknowledges financial support from foundation for science and technology (fct) via the post-doctoral grant sfrh/bpd/66342/2009. received: may 2011. revised: june 2011. 110 s. georgiev, j. morais and w. spröß cubo 14, 1 (2012) references [1] r. fueter. analytische funktionen einer quaternionenvariablen. comment. math. helv., 4, 9–20 (1932). [2] r. fueter: die funktionentheorie der differentialgleichungen ∆u = 0 und ∆∆u = 0 mit vier reellen variablen. comm. math. helv. 7: 307–330 (1935). [3] r. fueter. über die analytische darstellung der regulären funktionen einer quaternionenvariablen. comment. math. helv., 8, 371–378 (1935). [4] r. fueter. functions of a hyper complex variable. lecture notes written and supplemented by e. bareiss, math. inst. univ. zürich, fall semester, 1949. [5] k. gürlebeck and w. sprössig. quaternionic analysis and elliptic boundary value problems. akademie verlag, berlin, 1989. [6] k. gürlebeck and w. sprössig. quaternionic calculus for physicists and engineers. john wiley and sons, chichester, 1997. [7] k. gürlebeck, k. habetha and w. sprößig. holomorphic functions in the plane and ndimensional space, birkhäuser verlag, basel boston berlin, 2008. [8] v. kravchenko. applied quaternionic analysis. research and exposition in mathematics. lemgo: heldermann verlag, vol. 28, 2003. [9] h. leutwiler, quaternionic analysis in r3 versus its hyperbolic modification, brackx, f., chisholm, j.s.r. and soucek, v. (ed.). nato science series ii. mathematics, physics and chemistry, vol. 25, kluwer academic publishers, dordrecht, boston, london, 2001, pp. 193– 211. [10] h. malonek. quaternions in applied sciences: a historical perspective of a mathematical concept. in proceedings of the international conference on the applications of computer science and mathematics in architecture and civil engineering. bauhaus-universität weimar (2003). [11] m. shapiro and n. l. vasilevski. quaternionic ψ-hyperholomorphic functions, singular operators and boundary value problems i. complex variables, theory appl., 1995. [12] m. shapiro and n. l. vasilevski. quaternionic ψ-hyperholomorphic functions, singular operators and boundary value problems ii. complex variables, theory appl., 1995. [13] w. sprössig. on operators and elementary functions in clifford analysis. zeitschrift fr analysis und ihre anwendungen, vol. 18, no. 2, pp. 349–360 (1999). [14] a. sudbery. quaternionic analysis. math. proc. cambridge phil. soc. 85: pp. 199–225 (1979). introduction basic properties and definitions for quaternions exponents and logarithms revisited trigonometry revisited hyperbolic revisited inverse hyperbolic and trigonometric functions revisited acknowledgements cubo a mathematical journal vol.15, no¯ 01, (13–32). march 2013 solow models on time scales1 martin bohner and julius heim missouri university of science and technology, department of mathematics and statistics, department of economics, rolla, mo 65409-0020, usa bohner@mst.edu julius.heim@mst.edu ailian liu shandong university of finance and economics, school of mathematics and quantitative economics, jinan 250014, p. r. china ailianliu2002@163.com abstract we introduce a general solow model on time scales and derive a nonlinear first-order dynamic equation that describes such a model. we first assume that there is neither technological development nor a change in the population. we present the cobb– douglas production function on time scales and use it to give the solution for the equation that describes the model. next, we provide several applications of the generalized solow model. finally, we generalize our work by allowing technological development and population growth. the presented results not only unify the continuous and the discrete solow models but also extend them to other cases “in between”, e.g., a quantum calculus version of the solow model. finally it is also noted that our results even generalize the classical continuous and discrete solow models since we allow the savings rate, the depreciation factor of goods, the growth rate of the population, and the technological growth rates to be functions of time rather than taking constant values as in the classical solow models. keywords and phrases: time scales, solow model, dynamic equation, cobb–douglas production function, economics. 2010 ams mathematics subject classification: 91b64, 34c10, 39a10, 39a11, 39a12, 39a13. 1this paper is dedicated to professor gaston m. n’guérékata on the occasion of his 60th birthday 14 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) resumen introducimos un modelo general de solow en escalas de tiempo y derivamos una ecuación dinámica no lineal de primer orden que describe el mencionado modelo. primero asumimos que no existe ni desarrollo tecnológico ni un cambio en la población. presentamos la función de producción de cobb-douglas en escalas de tiempo y la utilizamos para entregar la solución de la ecuación que describe el modelo. luego, mostramos varias aplicaciones del modelo generalizado de solow. finalmente, generalizamos nuestro trabajo permitiendo desarrollo tecnológico y crecimiento de la población. los resultados presentados no sólo unifican los modelos de solow continuos y discretos, sino que además se extienden a otros casos “entre medio”, es decir, una versión del cálculo cuántico del modelo de solow. finalmente, también se menciona que nuestro resultado también generaliza los modelos clásicos continuos y discretos, ya que permitimos tasas de ahorro, el factor de depreciación de bienes, la razón de crecimiento de la población y las razones de crecimiento tecnológico por ser funciones del tiempo más que asumiendo valores constantes como es el caso de los modelos de solow clásico. 1 the classical solow model modern growth theory is mainly based on the works of solow [12] and swan [13]. in the solow model, it is assumed that the national income y depends on consumption c and investment i, i.e., y(t) = c(t) + i(t). moreover, it is assumed that the national income is a function of the capital stock k and the product of the technological progress a and the population n, i.e., y(t) = f (k(t), a(t)n(t)) , (1) where the production function f satisfies the following conditions: 1. f(λk, λl) = λf(k, l) for all λ, k, l ∈ r+ (constant returns to scale); 2. f(k, 0) = f(0, l) = 0 for all k, l ∈ r+; 3. ∂f ∂k > 0, ∂f ∂l > 0, ∂ 2 f ∂k2 < 0, ∂ 2 f ∂l2 < 0; 4. lim k→0+ ∂f ∂k = lim l→0+ ∂f ∂l = +∞, lim k→+∞ ∂f ∂k = lim l→+∞ ∂f ∂l = 0. furthermore, the change of the capital stock in a particular period does not only depend on the new investment but also on the depreciation of goods. in other words, we take for granted that k′(t) = i(t) − δk(t) (2) cubo 15, 1 (2013) solow models on time scales 15 with given initial capital stock k(0), where δ is the depreciation rate of the goods. as usual, we presume that s(t) = i(t) = sy(t), (3) where s is the savings rate, and thus the savings s is the portion of the national income which is not consumed. plugging (1) and (3) into (2), we obtain the nonlinear first-order differential equation k′(t) = sf (k(t), a(t)n(t)) − δk(t). (4) in this paper, we assume that the technological knowledge of the society grows exponentially with rate r, that is, a′(t) = ra(t), i.e., a(t) = erta(0), where a(0) is the initial technological standing of the nation. similarly, we suppose that the population of the nation grows with growth rate n and an initial population of n(0), hence n′(t) = nn(t), i.e., n(t) = entn(0). both n and r can be positive or negative depending on which nation we are talking about. of course, it seems quite unrealistic that r is negative. solow also took the capital stock per efficiency of labor k into account, which he defined as the stationary variable k(t) := k(t) a(t)n(t) . we use constant returns in order to define the intensive version of the production function to be y(t) := y(t) a(t)n(t) = f (k(t), a(t)n(t)) a(t)n(t) = f(k(t), 1) =: f(k(t)). this allows us now to rewrite (4) as k′(t) a(t)n(t) = sf(k(t)) − δk(t). (5) a simple calculation shows that k′(t) k(t) = k′(t) k(t) − r − n. therefore (5) turns into k′(t) = sf(k(t)) − (δ + n + r)k(t). (6) table 1 summarizes all the variables with their meanings that appear in the solow model. for a more detailed discussion of stability and qualitative analysis of the solow model, the reader might consult [9]. the discrete analogue of solow’s model is discussed in [11], featuring some results similar to those in the continuous solow model. so far this dynamic process was regarded either as solely continuous or solely discrete. in this paper, we generalize these two theories in such a way that the continuous and the discrete versions of the solow model are only special cases of 16 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) table 1: explanation of variables variable explanation y national income i induced investment c consumption s savings s savings rate k capital stock k0 initial capital stock δ depreciation factor of goods a level of technological knowledge a0 initial technological progress level r technological growth rate n level of population n0 initial population n growth rate of population k capital stock per efficiency unit of labor our generalized solow model. we do this using the time scales theory, an area of mathematics which was originally introduced by stefan hilger in his phd thesis [10]. the two books [7, 8] by bohner and peterson offer an introduction with applications to time scales calculus along with some advanced topics. applications of time scales calculus can be found in many areas, also in economics. tisdell and zaidi [14] in particular already generalized some economic topics, and also bohner et al. discussed multiplier-accelerator models and utility functions on time scales in [5, 6]. the set up of this paper is as follows. in section 2, we give a brief introduction to the time scales theory. in section 3, we present the solow model on time scales and derive the nonlinear first-order dynamic equation that describes this model. in section 4, we define the generalized cobb–douglas production function on time scales and provide examples for various time scales. furthermore, we state a theorem that gives the solution of the nonlinear first-order dynamic equation, and we also provide examples for several time scales. we finally state a result that addresses asymptotic stability of the solution. section 5 is used to show some important properties of the production function, called the inada conditions. finally, in section 6, we point out how our model can be extended, assuming the presence of technological and population growth (or decay), i.e., by assuming n 6= 0 as well as r 6= 0. we present the cobb–douglas production function for cubo 15, 1 (2013) solow models on time scales 17 this more general case and also derive the equilibrium solution for this extended model. it is also noted that our results even generalize the classical continuous and discrete solow models since we allow the savings rate, the depreciation factor of goods, the growth rate of the population, and the technological growth rates to be functions of time rather than considering constant values as in the classical solow models. 2 time scales preliminaries in this section, we introduce some elements of time scales calculus. for a more rigorous time scales introduction, we refer the reader to [7, 8]. let t be a time scale, i.e., a nonempty closed subset of r. for t ∈ t, the forward jump operator σ : t → t is defined by σ(t) := inf {s ∈ t : s > t} , while the backward jump operator ρ : t → t is defined by ρ(t) := sup {s ∈ t : s < t} . in this definition, we set inf ∅ = sup t (i.e., σ(t) = t if t has maximum t) and sup ∅ = inf t (i.e., ρ(t) = t if t has minimum t). if σ(t) > t, σ(t) = t, ρ(t) < t, and ρ(t) = t, then t is called right-scattered, right-dense, left-scattered and left-dense, respectively. the graininess function µ : t → [0, ∞) is defined by µ(t) := σ(t) − t. we also need the set tκ which is defined in the following way: if t has a left-scattered maximum m, then tκ = t − {m}. else, tκ = t. now let f : t → r be a function. if t ∈ tκ, then f∆(t) is defined as the number (provided that it exists) such that for every ε > 0, there exists a neighborhood u of t (i.e., u = (t − δ, t + δ) ∩ t for some δ > 0) such that ∣∣[f(σ(t)) − f(s)] − f∆(t)[σ(t) − s] ∣∣ ≤ ε|σ(t) − s| for all s ∈ u. we call this number f∆(t) the delta derivative of f at t. moreover, f is called rd-continuous provided it is continuous at right-dense points in t and its left sided limits exist (finite) at left-dense points in t. the function fσ : t → r is defined by fσ = f ◦ σ. in our calculations, we use the so-called “simple useful formula” fσ = f + µf∆. we denote the set of rd-continuous functions by crd = crd(t) = crd(t, r). next, f is said to be regressive given that 1 + µ(t)f(t) 6= 0 for all t ∈ t 18 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) holds. the set of all regressive and rd-continuous functions is denoted by r = r(t) = r(t, r). we also define the set r+ of all positively regressive elements by r+ = r+(t, r) = {f ∈ r : 1 + µ(t)p(t) > 0 for all t ∈ t} . let now p, q ∈ r. we define the “circle plus” addition ⊕ on r by (p ⊕ q)(t) := p(t) + q(t) + µ(t)p(t)q(t) for all t ∈ t and the “circle minus” subtraction ⊖ on r by (p ⊖ q)(t) := p(t) − q(t) 1 + µ(t)q(t) for all t ∈ t. we put r(α) := { r if α ∈ n, r+ if α ∈ r \ n. for α ∈ r and p ∈ r(α), we define (α ⊙ p)(t) := αp(t) ∫1 0 (1 + µ(t)p(t)h)α−1dh. (7) the time scales exponential function ep(·, t0) is defined for p ∈ r and t0 ∈ t as the unique solution of the initial value problem y∆ = p(t)y, y(t0) = 1 on t. we have ep(·, t0)eq(·, t0) = ep⊕q(·, t0) and ep(·, t0) eq(·, t0) = ep⊖q(·, t0). if α ∈ r and p ∈ r(α), then eα⊙p = e α p (see [8, theorem 2.44]). let α ∈ r \ {1}. we say that x∆ = [ q ⊖ ( 1 α − 1 ⊙ (gxα−1) )] x (8) (see [8, section 2.6]) is a bernoulli equation on time scales. 3 solow model on time scales assume that f and f are production functions as defined in section 1. we now introduce the generalized solow model on an arbitrary time scale:    y(t) = f(k(t), a(t)n(t)), k∆(t) = i(t) − δ(t)k(t), i(t) = s(t)y(t), a∆(t) = r(t)a(t), n∆(t) = n(t)n(t), (9) cubo 15, 1 (2013) solow models on time scales 19 where we require δ(t) > 0 and s(t) > 0 for all t ∈ t (10) and n, r ∈ r. (11) the economical meanings of δ, s, r, and n are the same as described in table 1. if (k, y, a, n, i) solves (9), then k∆(t) = s(t)y(t) − δ(t)k(t) = s(t)f(k(t), a(t)n(t)) − δ(t)k(t). (12) define k(t) := k(t) a(t)n(t) and y(t) := y(t) a(t)n(t) , (13) which are regarded as the capital stock per efficiency unit of labor and the production per efficiency unit of labor, respectively. by (12) and (13), we have k∆(t) a(t)n(t) = s(t)f(k(t)) − δ(t)k(t). (14) theorem 3.1. assume (9), (10), and (11). if k is defined as in (13), then k∆(t) = s(t) (1 + µ(t)r(t))(1 + µ(t)n(t)) f(k(t)) − ( δ(t) + n(t) (1 + µ(t)r(t))(1 + µ(t)n(t)) + r(t) 1 + µ(t)r(t) ) k(t). (15) proof. the time scales quotient rule [7, theorem 1.20 (v)] provides k∆ = ( k an )∆ = k∆an − k ( an∆ + a∆nσ ) anaσnσ = k∆ aσnσ − kn∆ naσnσ − ka∆ anaσ (9) = k∆ an(1 + µr)(1 + µn) − n (1 + µr)(1 + µn) k − r 1 + µr k (14) = s (1 + µr)(1 + µn) f ◦ k − ( δ + n (1 + µr)(1 + µn) + r 1 + µr ) k, i.e., (15) holds. example 3.2. if t = r, then σ(t) = t and µ(t) = 0 for all t ∈ t. thus (15) can be rewritten as k′(t) = s(t)f(k(t)) − (δ(t) + n(t) + r(t))k(t), which reduces to (6) provided δ, s, n, and r are constants. 20 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) theorem 3.3. assume (10) and (11). then (15) holds if and only if k∆(t) = s(t)f(k(t)) − δ(t)k(t) − (n ⊕ r)(t)k(σ(t)). (16) proof. suppose k solves (16). then we use the “simple useful formula” to obtain k∆ = s(f ◦ k) − δk − (n ⊕ r)kσ = s(f ◦ k) − δk − (n + r + µnr) ( k + µk∆ ) . hence (1 + µn)(1 + µr)k∆ = s(f ◦ k) − (δ + n + r + µnr) k. dividing by (1 + µn)(1 + µr) yields (15). if k solves (15), then (16) follows by reversing the above steps. theorem 3.4. assume (10) and (11). if (16) holds, then (1 + µ(t)n(t))(1 + µ(t)r(t))kσ(t) = µ(t)s(t)f(k(t)) + (1 − µ(t)δ(t))k(t). (17) if (17) holds and µ(t) 6= 0, then (16) holds. proof. suppose k solves (16). then we multiply (16) by µ(t) and use the simple useful formula to obtain k(σ(t)) − k(t) = µ(t)k∆(t) = µ(t)s(t)f(k(t)) − µ(t)δ(t)k(t) − µ(t)(n ⊕ r)(t)k(σ(t)). hence (1 + µ(t)(n ⊕ r)(t)) k(σ(t)) = µ(t)s(t)f(k(t)) + (1 − µ(t)δ(t))k(t), which results in (17). if (17) holds at t ∈ t such that µ(t) 6= 0, then the above steps can be reversed. example 3.5. if t = z, then σ(t) = t + 1 and µ(t) = 1 for all t ∈ t. thus (17) can be rewritten as (1 + n(t))(1 + r(t))k(t + 1) = s(t)f(k(t)) + (1 − δ(t))k(t). this equation can be found in [11]. example 3.6. if t = hz with h > 0, then σ(t) = t + h and µ(t) = h for all t ∈ t. thus (17) can be rewritten as (1 + hn(t))(1 + hr(t))k(t + h) = hs(t)f(k(t)) + (1 − hδ(t))k(t). example 3.7. if t = qn0 with q > 1, then σ(t) = qt and µ(t) = (q − 1)t for all t ∈ t. thus (17) can be rewritten as (1 + (q − 1)tn(t))(1 + (q − 1)tr(t))k(qt) = (q − 1)ts(t)f(k(t)) + (1 − (q − 1)tδ(t))k(t) . cubo 15, 1 (2013) solow models on time scales 21 4 analysis of the basic solow model in this section, we assume that (10) holds and that there is no technological development and no population change, i.e., n = r = 0. then (16) simplifies to k∆(t) = s(t)f(k(t)) − δ(t)k(t). (18) let 0 < α < 1, w(t) = ( 1 α − 1 ⊙ δg s ) (t), and g(t) = (1 − α)s(t). (19) if f̃(x) := δ(t) + ( w ⊖ ( 1 α−1 ⊙ (gxα−1) )) (t) s(t) is independent of t ∈ t, (20) then we define the generalized cobb–douglas production function on time scales by f(x) = xf̃(x). (21) theorem 4.1. let t ∈ t. if µ(t) = 0, then δ(t) + ( w ⊖ ( 1 α−1 ⊙ (gxα−1) )) (t) s(t) = xα−1. (22) proof. assume µ(t) = 0. then at t, we have δ(t) + ( w ⊖ ( 1 α−1 ⊙ (gxα−1) )) (t) s(t) = δ(t) + w(t) − g(t)x α−1 α−1 s(t) = δ(t) + δ(t)g(t) (α−1)s(t) − g(t)x α−1 α−1 s(t) = δ(t) − δ(t) + s(t)xα−1 s(t) = xα−1, which shows (22). example 4.2. if t = r, then f̃(x) = xα−1, and thus (20) holds. hence the cobb–douglas production function is defined and equals f(x) (21) = xf̃(x) = xxα−1 = xα. theorem 4.3. let t ∈ t. if µ(t) > 0, then δ(t) + ( w ⊖ ( 1 α − 1 ⊙ ( gxα−1 ))) (t) = 1 µ(t) { δ(t)µ(t) − 1 + ( 1 + (1 − α)µ(t)s(t)xα−1 1 + (1 − α)µ(t)δ(t) ) 1 1−α } . (23) 22 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) proof. assume µ(t) > 0. then at t, we have 1 α − 1 ⊙ (gxα−1) (7) = 1 α − 1 gxα−1 ∫1 0 ( 1 + µgxα−1h ) 1 α−1 −1 dh = ( 1 + µgxα−1 ) 1 α−1 − 1 µ , w = 1 α − 1 ⊙ δg s = 1 α − 1 ⊙ (δ(1 − α)) (7) = 1 α − 1 δ(1 − α) ∫1 0 (1 + µδ(1 − α)h) 1 α−1 −1 dh = (1 + µδ(1 − α)) 1 α−1 − 1 µ , and hence w ⊖ ( 1 α − 1 ⊙ (gxα−1) ) = w − (1+µgxα−1) 1 α−1 −1 µ 1 + µ (1+µgxα−1) 1 α−1 −1 µ = w − (1+µgxα−1) 1 α−1 −1 µ (1 + µgxα−1) 1 α−1 = (1+µδ(1−α)) 1 α−1 −1 µ − (1+µgxα−1) 1 α−1 −1 µ (1 + µgxα−1) 1 α−1 = 1 µ { −1 + ( 1 + µδ(1 − α) 1 + µ(1 − α)sxα−1 ) 1 α−1 } , which shows (23). theorem 4.4. let µ(t) > 0 for all t ∈ t. assume (10) and suppose s̃ := s(t)µ(t) and δ̃ := δ(t)µ(t) are independent of t ∈ t. (24) then (20) holds and the cobb–douglas production function is defined and equals f(x) = x s̃    δ̃ − 1 + ( 1 + (1 − α)s̃xα−1 1 + (1 − α)δ̃ ) 1 1−α    . cubo 15, 1 (2013) solow models on time scales 23 proof. using proposition 4.3, we see that δ(t) + ( w ⊖ ( 1 α−1 ⊙ (gxα−1) )) (t) s(t) (23) = 1 µ(t)s(t) { δ(t)µ(t) − 1 + ( 1 + (1 − α)µ(t)s(t)xα−1 1 + (1 − α)µ(t)δ(t) ) 1 1−α } = 1 s̃    δ̃ − 1 + ( 1 + (1 − α)s̃xα−1 1 + (1 − α)δ̃ ) 1 1−α    is independent of t and therefore equals f̃(x). by (21), f(x) = xf̃(x). example 4.5. if t = hz with h > 0 and δ, s are constants and satisfy (10), then proposition 4.4 gives us that (20) is satisfied and that the cobb–douglas production function is defined and equals (note that (24) is satisfied in this case with s̃ = sh and δ̃ = δh) f(x) = x hs { δh − 1 + ( 1 + (1 − α)hsxα−1 1 + (1 − α)hδ ) 1 1−α } . (25) example 4.6. if t = z and δ, s are constants and satisfy (10), then example 4.5 gives us that (20) is satisfied and that the cobb–douglas production function, i.e., the discrete version of the classical cobb–douglas production function, is defined and equals f(x) = x s { δ − 1 + ( 1 + (1 − α)sxα−1 1 + (1 − α)δ ) 1 1−α } . example 4.7. if t = qn0 with q > 1 and δ, s satisfy (10) and (24), i.e., s̃ := (q − 1)ts(t) and δ̃ := (q − 1)tδ(t) are independent of t ∈ t, then proposition 4.4 gives us that (20) is satisfied and that the cobb–douglas production function is defined and equals f(x) = x s̃    δ̃ − 1 + ( 1 + (1 − α)s̃xα−1 1 + (1 − α)δ̃ ) 1 1−α    . using (21), we rewrite equation (18) in the form (8), i.e., as a bernoulli equation on time scales. theorem 4.8. assume (10) and (20) and let f be defined by (21). then (18) holds if and only if k∆(t) = { w ⊖ ( 1 α − 1 ⊙ (gkα−1) )} (t)k(t). (26) 24 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) example 4.9. if t = r, then w = −δ, and equation (26) is k′(t) = ( skα−1(t) − δ ) k(t). hence equation (26) is indeed a generalized form of the continuous solow model with the cobb– douglas production function. with the generalized cobb–douglas function, we can find the solution of the solow model (18) on time scales. theorem 4.10. assume (10) and κ := s(t) δ(t) is independent of t ∈ t (27) and define p ∈ r by p(t) := (1 − α)δ(t) for all t ∈ t. (28) then the solution of (26) with initial condition k(t0) = k0 > 0, where t0 ∈ t, is given by k(t) = { κ + k1−α0 − κ ep(t, t0) } 1 1−α for all t ∈ t, (29) provided the quantity in curly braces in (29) is always positive. proof. suppose k solves (26) such that k(t0) = k0. define x̃ := k α−1. by [8, theorem 2.37], we have x̃∆ x̃ = (α − 1) ⊙ k∆ k = (α − 1) ⊙ { w ⊖ [ 1 α − 1 ⊙ ( gkα−1 )]} = [(α − 1) ⊙ w] ⊖ ( gkα−1 ) = (δ(1 − α)) ⊖ ( gkα−1 ) , so x̃∆ = (p ⊖ (gx̃)) x̃, which shows that x̃ solves the logistic equation on time scales (see [1] and [8, section 2.4]). define y := 1/x̃. then y∆ = ( 1 x̃ )∆ = −x̃∆ x̃x̃σ = − (p ⊖ (gx̃)) yσ = gx̃ − p 1 + µgx̃ yσ and hence (1 + µgx̃) y∆ = gx̃yσ − pyσ, i.e., using the “simple useful formula”, y∆ + (yσ − y) gx̃ = gx̃yσ − pyσ, cubo 15, 1 (2013) solow models on time scales 25 i.e., y∆ = −pyσ + g. (30) using g = (1 − α)s = (1 − α)δκ = κp and the variation of constants formula [7, theorem 2.74], the solution of (30) is given by y(t) = y0e⊖p(t, t0) + ∫t t0 g(τ)e⊖p(t, τ)∆τ = y0e⊖p(t, t0) + ∫t t0 κp(τ)ep(τ, t)∆τ = y0e⊖p(t, t0) + κ ∫t t0 p(τ)ep(τ, t)∆τ = y0e⊖p(t, t0) + κep(τ, t)| t t0 = y0e⊖p(t, t0) + κ(1 − e⊖p(t, t0)). from the substitutions we performed, we have that y0 = k 1−α 0 as well as k(t) = 1 y(t) 1 α−1 , which shows (29). conversely, k given by (29) is easily seen to be a solution of (26). using theorem 4.10, we obtain the asymptotic stability of the unique equilibrium point of (26). theorem 4.11. assume (10) and (27). if ∫∞ t0 δ(t)∆t = ∞, (31) then any solution k of (26) in the form (29) satisfies lim t→∞ k(t) = κ 1 1−α =: κ, and κ is the unique equilibrium point of (26). proof. we have p(t) > 0 for all t ∈ t, where p is defined in theorem 4.10. hence p ∈ r+. thus, by [4, remark 2], we have ep(t, t0) ≥ 1 + ∫t t0 p(τ)∆τ = 1 + (1 − α) ∫t t0 δ(τ)∆τ for all t ≥ t0. therefore, using (31), lim t→∞ ep(t, t0) = ∞, and thus lim t→∞ k(t) = { κ + k1−α0 − κ limt→∞ ep(t, t0) } 1 1−α = κ 1 1−α = κ. 26 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) now we show that κ is the unique nontrivial equilibrium point of (26): a point κ is a nontrivial equilibrium point of (26) if and only if w ⊖ ( 1 α − 1 ⊙ ( gκα−1 )) = 0, which holds if and only if (use the definition of w, (27), and the properties of ⊖) 1 α − 1 ⊙ g κ = 1 α − 1 ⊙ ( gκα−1 ) , which is true if and only if (use the properties of ⊙ and the definition of g) 1 κ = κα−1, i.e., κ = κ. this completes the proof. example 4.12. let t = r and assume (10) and (27). then (29) reads k(t) = { κ + k1−α0 − κ e (1−α) ∫ t t0 δ(τ)∆τ } 1 1−α . if, in addition, t0 = 0 and δ is constant (this implies that s is constant as well), then k(t) = { κ + k1−α0 − κ e(1−α)δt } 1 1−α . example 4.13. let t = hz with h > 0 and assume (10) and (27). then (29) reads k(t) =    κ + k1−α0 − κ t/h−1∏ i=t0/h (1 + (1 − α)hδ(ih))    1 1−α . if, in addition, t0 = 0 and δ is constant (this implies that s is constant as well), then k(t) = { κ + k1−α0 − κ (1 + (1 − α)hδ) t/h } 1 1−α . example 4.14. let t = qn0 with q > 1 and assume (10) and (27). then (29) reads k(t) =    κ + k1−α0 − κ logq t−1∏ i=logq t0 (1 + (q − 1)qi(1 − α)δ(qi))    1 1−α . cubo 15, 1 (2013) solow models on time scales 27 if, in addition, t0 = 1 and δ̃ := (q − 1)tδ(t) is constant (this implies that (q − 1)ts(t) is constant as well), then k(t) =    κ + k1−α0 − κ( 1 + (1 − α)δ̃ )logq t    1 1−α . 5 properties of the production function in this section, we show that our cobb–douglas production function f given in (21) satisfies the time scales inada conditions (see [2, 3])    f(x) > 0, f̃′(x) < 0, f′′(x) < 0 for all x > 0, f̃(x) > ζ(t) ≥ 0, f′(x) > ζ(t) ≥ 0 for all x > 0 and all t ∈ t, lim x→0+ f̃(x) = lim x→0+ f′(x) = ∞, lim x→∞ f̃(x) = lim x→∞ f′(x) = ζ(t) ≥ 0 for all t ∈ t, (32) where ζ : t → r is defined in the following lemma. lemma 5.1. assume (10) and define ζ(t) := 1 s(t) { δ(t) + ( 1 α − 1 ⊙ ((1 − α)δ) ) (t) } . then ζ(t) ≥ 0. proof. if µ(t) = 0, then ζ(t) = 0. if µ(t) > 0, then, as in the proof of proposition 4.3, we have ( 1 α − 1 ⊙ ((1 − α)δ) ) (t) = (1 + (1 − α)µ(t)δ(t)) 1 α−1 − 1 µ(t) . now using the well-known bernoulli inequality, we obtain (1 + (1 − α)µ(t)δ(t)) 1 α−1 ≥ 1 + 1 α − 1 (1 − α)µ(t)δ(t) = 1 − µ(t)δ(t). this proves the claim. theorem 5.2. assume (10) and (20) and define the cobb–douglas production function by (21). if there exists t ∈ t such that µ(t) = 0, then the cobb–douglas production function satisfies the inada conditions (32). proof. by theorem 4.1 and (20), f(x) = xα and f̃(x) = xα−1. (33) clearly, f given by (33) satisfies the inada conditions (32). 28 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) theorem 5.3. assume (10) and (20) and define the cobb–douglas production function by (21). if there exists t ∈ t such that µ(t) > 0, then the cobb–douglas production function satisfies the inada conditions (32). proof. by theorem 4.3 and (20), we have f̃(x) = 1 µ(t)s(t) { µ(t)δ(t) − 1 + ( 1 + (1 − α)µ(t)s(t)xα−1 1 + (1 − α)µ(t)δ(t) ) 1 1−α } . (34) in order to check that the inada conditions (32) are satisfied, we use the following notation:    δ̃ := µ(t)δ(t), κ := s(t) δ(t) , α̃ := (1 − α)δ̃, ζ := 1 δ̃κ { δ̃ − 1 + (1 + α̃) 1 α−1 } , z(x) := 1 + α̃κxα−1 1 + α̃ . (35) using (35), we rewrite (34) as f̃(x) = 1 δ̃κ { δ̃ − 1 + (z(x)) 1 1−α } . (36) we obviously have z(x) > 1 1 + α̃ for all x > 0. (37) using (37) in (36), we find f̃(x) > 1 δ̃κ { δ̃ − 1 + (1 + α̃) 1 α−1 } = ζ for all x > 0. by lemma 5.1, we also have ζ ≥ 0, and hence f̃(x) > 0 for all x > 0 so that f(x) = xf̃(x) > 0 for all x > 0. next, note that z′(x) = α̃(α − 1)κ 1 + α̃ xα−2. (38) using (38) in (36), we find f̃′(x) = 1 δ̃κ(1 − α) (z(x)) 1 1−α −1 z′(x) = 1 δ̃κ(1 − α) (z(x)) α 1−α α̃(α − 1)κxα−2 1 + α̃ = − α̃xα−2 (1 + α̃)δ̃ (z(x)) α 1−α < 0 for all x > 0. cubo 15, 1 (2013) solow models on time scales 29 using this, (36), (35), (37), and the product rule for f(x) = xf̃(x), we get f′(x) = f̃(x) + xf̃′(x) = 1 δ̃κ { δ̃ − 1 + (z(x)) 1 1−α } − α̃xα−1κ (1 + α̃)δ̃κ (z(x)) α 1−α = 1 δ̃κ { δ̃ − 1 + (z(x)) 1 1−α } − ( z(x) − 1 1 + α̃ ) 1 δ̃κ (z(x)) α 1−α = 1 δ̃κ { δ̃ − 1 + 1 1 + α̃ (z(x)) α 1−α } > 1 δ̃κ { δ̃ − 1 + 1 1 + α̃ (1 + α̃) α α−1 } = 1 δ̃κ { δ̃ − 1 + (1 + α̃) 1 α−1 } = ζ, i.e., f′(x) = 1 δ̃κ { δ̃ − 1 + 1 1 + α̃ (z(x)) α 1−α } > ζ for all x > 0. (39) also, since lim x→0+ z(x) = ∞ and lim x→∞ z(x) = 1 1 + α̃ , we find from (36) and (39) that lim x→0+ f̃(x) = lim x→0+ f′(x) = ∞ and lim x→∞ f̃(x) = lim x→∞ f′(x) = ζ ≥ 0. finally, using (39) and (38), we obtain f′′(x) = 1 δ̃κ(1 + α̃) α 1 − α (z(x)) α 1−α −1z′(x) = 1 δ̃κ(1 + α̃) α 1 − α α̃(α − 1)κ 1 + α̃ xα−2(z(x)) 2α−1 1−α = − αα̃ δ̃(1 + α̃)2 xα−2(z(x)) 2α−1 1−α < 0 for all x > 0. this shows that all conditions in (32) are satisfied. 6 general solow model on time scales let us now assume (10) and (11) so that we allow now that the technology develops exponentially and/or the population increases exponentially. all results from section 4 and section 5 still hold true when we replace s and δ in (18) by s 1 + µ(n ⊕ r) and δ + (n ⊕ r) 1 + µ(n ⊕ r) , 30 martin bohner, julius heim and ailian liu cubo 15, 1 (2013) respectively, as (18) in this case results in (15). this means that w and g from (19) are replaced by w(t) = ( 1 α − 1 ⊙ (δ + (n ⊕ r))g s ) (t) and g(t) = (1 − α)s(t) 1 + µ(t)(n ⊕ r)(t) , respectively. next, f̃(x) in (20) is replaced by δ(t) + (n ⊕ r)(t) + (1 + µ(t)(n ⊕ r)(t)) ( w ⊖ ( 1 α−1 ⊙ ( gxα−1 ))) s(t) . moreover, κ and p in (27) and (28) are replaced by s(t) δ(t) + (n ⊕ r)(t) and (1 − α) δ(t) + (n ⊕ r)(t) 1 + µ(t)(n ⊕ r)(t) , respectively. finally, condition (31) turns into ∫∞ t0 δ(t) + (n ⊕ r)(t) 1 + µ(t)(n ⊕ r)(t) ∆t = ∞. received: january 2013. revised: february 2013. references [1] e. akin-bohner and m. bohner. miscellaneous dynamic equations. methods appl. anal., 10(1):11–30, 2003. 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[8] m. bohner and a. peterson, editors. advances in dynamic equations on time scales. birkhäuser boston inc., boston, ma, 2003. cubo 15, 1 (2013) solow models on time scales 31 [9] g. gandolfo. economic dynamics. north-holland publishing co., 1997. [10] s. hilger. ein maßkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten. phd thesis, universität würzburg, 1988. [11] l. ljungqvist and t. j. sargent. recursive macroeconomic theory. mit press, 2000. [12] r. m. solow. a contribution to the theory of economic growth. the quarterly journal of economics, 70(1):65–94, february 1956. [13] swan. economic growth and capital accumulation. economic record, 32:334–361, 1956. [14] c. c. tisdell and a. zaidi. basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling. nonlinear anal., 68(11):3504–3524, 2008. cubo a mathematical journal vol.14, no¯ 03, (41–57). october 2012 spectral results for operators commuting with translations on banach spaces of sequences on zk and z+ violeta petkova université metz ufr mim, laboratoire de mathmatiques et applications de metz, umr 7122, ile du saulcy 57045 metz cedex 1, france email: petkova@univ-metz.fr abstract we study the spectrum of multipliers (bounded operators commuting with the shift operator s) on a banach space e of sequences on z. given a multiplier m, we prove that m̃(σ(s)) ⊂ σ(m) where m̃ is the symbol of m. we obtain a similar result for the spectrum of an operator commuting with the shift on a banach space of sequences on z+. we generalize the results for multipliers on banach spaces of sequences on zk. resumen estudiamos el espectro de los multiplicadores (operadores acotados que conmutan con el operador shift s) en un espacio de banach e de sucesiones en z. dado un multiplicador m, probamos que m̃(σ(s)) ⊂ σ(m) donde m̃ es el śımbolo de m. obtenemos un resultados similar para el espectro de un operador que conmuta con el shift en un espacio de banach de sucesiones en z+. generalizamos los resultados sobre multiplicadores en espacios de banach de sucesiones en zk. keywords and phrases: multiplier, toeplitz operator, shift operator, space of sequences, spectrum of multiplier, joint spectrum of translations. 2010 ams mathematics subject classification: primary 47b37, 47b35; secondary 47a10. 42 violeta petkova cubo 14, 3 (2012) 1 introduction let e ⊂ cz be a banach space of complex sequences (x(n))n∈z. denote by s : cz −→ cz, the shift operator defined by sx = (x(n − 1))n∈z, for x = (x(n))n∈z ∈ cz, so that s−1x = (x(n + 1))n∈z. let f(z) be the set of sequences on z, which have a finite number of non-zero elements and assume that f(z) ⊂ e. we will call a multiplier on e every bounded operator m on e such that msa = sma, for every a ∈ f(z). denote by µ(e) the space of multipliers on e. for z ∈ t = {z ∈ c : |z| = 1}, consider the map e ∋ x −→ ψz(x) given by ψz(x) = (x(n)zn)n∈z. notice that if we assume that ψz(e) ⊂ e for all z ∈ t and if for all n ∈ z, the map pn : e ∋ x −→ x(n) ∈ c is continuous, then from the closed graph theorem it follows that the map ψz is bounded on e. in this paper, we deal with banach spaces of sequences on z satisfying only the following three very natural hypothesis: (h1) the set f(z) is dense in e. (h2) for every n ∈ z, pn is continuous from e into c. (h3) we have ψz(e) ⊂ e, ∀z ∈ t and supz∈t ‖ψz‖ < +∞. we give some examples of spaces satisfying our hypothesis. example 1. let ω be a weight (a sequence of positive real numbers) on z. set lpω(z) = { (x(n))n∈z ∈ cz; ∑ n∈z |x(n)|pω(n)p < +∞ } , 1 ≤ p < +∞ and ‖x‖ω,p = (∑ n∈z |x(n)|pω(n)p ) 1 p . it is easy to see that the banach space l p ω(z) satisfies our hypothesis. moreover, the operator s (resp. s−1) is bounded on l p ω(z) if and only if sup n∈z ω(n + 1) ω(n) < +∞ ( resp. sup n∈z ω(n − 1) ω(n) < +∞ ) . example 2. let k be a convex, non-decreasing, continuous function on r+ such that k(0) = 0 and k(x) > 0, for x > 0. for example, k may be xp, for 1 ≤ p < +∞ or xp+sin(log(− log(x))), for p > 1 + √ 2. let ω be a weight on z. set lk,ω(z) = { (x(n))n∈z ∈ cz; ∑ n∈z k ( |x(n)| t ) ω(n) < +∞, for some t > 0 } and ‖x‖ = inf { t > 0, ∑ n∈z k ( |x(n)| t ) ω(n) ≤ 1 } . the space lk,ω(z), called a weighted orlicz space (see [2], [3]), is a banach space satisfying our hypothesis. example 3. let (q(n))n∈z be a real sequence such that q(n) ≥ 1, for all n ∈ z. for a = (a(n))n∈z ∈ cz, set ‖a‖{q} = inf { t > 0, ∑ n∈z ∣∣∣a(n)t ∣∣∣ q(n) ≤ 1 } . consider the space l{q} = {a ∈ cz; ‖a‖{q} < +∞}, which is a banach space (see [1]) satisfying our hypothesis. notice that if limn→+∞ |q(n + 1) − q(n)| 6= 0 and if supn∈z q(n) < +∞, then either s or s−1 is cubo 14, 3 (2012) spectral results for operators commuting with translations ... 43 not bounded (see [4]). it is easy to see that if s(e) ⊂ e, then by the closed graph theorem the restriction s|e of s to e is bounded from e into e. from now on we will say that s (resp. s−1) is bounded when s(e) ⊂ e (resp. s−1(e) ⊂ e). if s(e) ⊂ e, we will call σ(s) the spectrum of the operator s with domain e. if s is not bounded, denote by σ(s) (resp. ρ(s)) the spectrum (resp. the spectral radius) of s, where s is the smallest extension of s|f(z) as a closed operator. recall that the domain d(s) of s is given by d(s) = {x ∈ e, ∃(xn)n∈n ⊂ f(z) s.t. xn −→ x andsxn −→ y ∈ e} and for x ∈ d(s) we set sx = y. we will denote by ◦ a (resp. δ(a)) the interior (resp. the boundary) of the set a. denote by ek the sequence such that ek(n) = 0 (resp. 1), if n 6= k (resp. n = k). for a multiplier m, we set m̂ = m(e0) and it is easy to see that ma = m̂ ∗ a, ∀a ∈ f(z). (1) given a ∈ cz, define ã(z) = ∑ n∈z a(n)zn and notice that if a ∈ l2(z), then ã ∈ l2(t). denote by m̃ the function z −→ m̃(z) = ∑ n∈z m̂(n)zn. usually, m̃ is called the symbol of m. it is easy to see that on the space of formal laurent series we have the equality m̃a(z) = m̃(z)ã(z), ∀z ∈ c, ∀a ∈ f(z). (2) however, it is difficult to determine for which z ∈ c the series m̃(z) converges. for r > 0, let cr be the circle of center 0 and radius r. recall the following result established in [8] (theorem 1). theorem 1. 1) if s is not bounded, but s−1 is bounded, then ρ(s) = +∞ and if s is bounded, but s−1 is not bounded, then ρ(s−1) = +∞. 2) we have σ(s) = { z ∈ c, 1 ρ(s−1) ≤ |z| ≤ ρ(s) } . 3) let m ∈ µ(e). for r > 0 such that cr ⊂ σ(s), we have m̃ ∈ l∞(cr) and |m̃(z)| ≤ ‖m‖, a.e. on cr. 4) if ρ(s) > 1 ρ(s−1) , then m̃ is holomorphic on ◦ σ(s). if s and s−1 are bounded, denote by ie the interval [ 1 ρ(s−1) ,ρ(s) ] . if s (resp. s−1) is not bounded, denote by ie the interval [ 1 ρ(s−1) ,+∞ [ (resp. ]0,ρ(s)]). the purpose of this paper is to use the symbol of an operator m ∈ µ(e) in order to characterize its spectrum. we deal with three different setups. first we study the multipliers on e, next we examine toeplitz operators on a banach space of sequences on z+ and finally we deal with multipliers on a banach space of sequences on zk. for φ ∈ f(z), we denote by mφ the operator 44 violeta petkova cubo 14, 3 (2012) of convolution by φ on e. let s be the closure with respect to the operator norm topology of the algebra generated by the operators mφ, for φ ∈ f(z). our first result is theorem 2. 1) if m ∈ µ(e), we have m̃(σ(s)) ⊂ σ(m). 2) if m ∈ s , then σ(m) \ {0} ⊂ m̃(σ(s)) ⊂ σ(m). notice that here for a set a, we denote by a the closure of a. if m ∈ µ(e), then m̃(σ(s)) denotes the essential range of m̃ on σ(s). notice that m̃ is holomorphic on ◦ σ(s) and essentially bounded on the boundary of σ(s). in general we have not spectral calculus for the operators in µ(e) and it seems difficult to characterize the spectrum of m ∈ µ(e) without using its symbol. we also study a similar spectral problem for toeplitz operators. let e ⊂ cz+ be a banach space and let f(z+) (resp. f(z−)) be the space of the sequences on z+ (resp. z−) which have a finite number of non-zero elements. by convention, we will say that x ∈ f(z) is a sequence of f(z+) (resp. f(z−)) if x(n) = 0, for n < 0 (resp. n > 0). we will assume that e satisfies the following hypothesis: (h1) the set f(z+) is dense in e. (h2) for every n ∈ z+, the application pn : x −→ x(n) is continuous from e into c. (h3) for x = (x(n))n∈z+ ∈ e, we have γz(x) = (znx(n))n∈z+ ∈ e, for every z ∈ t and supz∈t ‖γz‖ < +∞. definition 1. we define on cz + the operators s1 and s−1 as follows. foru ∈ cz + , (s1(u))(n) = 0, ifn = 0and (s1(u))(n) = u(n − 1), ifn ≥ 1 (s−1(u))(n) = u(n + 1), forn ≥ 0. for simplicity, we note s instead of s1. it is easy to see that if s(e) ⊂ e, then by the closed graph theorem the restriction s|e of s to e is bounded from e into e. we will say that s (resp. s−1) is bounded when s(e) ⊂ e (resp s−1(e) ⊂ e). next, if s|e (resp. s−1|e) is bounded, σ(s) (resp. σ(s−1)) denotes the spectrum of s|e (resp. s−1|e). if s (resp. s−1) is not bounded, σ(s) (resp. σ(s−1)) denotes the spectrum of the smallest closed extension of s|f(z+) (resp. s−1|f(z+)). for u ∈ l2(z−) ⊕ e introduce (p+(u))(n) = u(n), ∀n ≥ 0 and (p+(u))(n) = 0, ∀n < 0. if s1 and s−1 are the shift and the backward shift on l 2(z−)⊕e, then s = p+s1 and s−1 = p+s−1. example 4. let w be a positive sequence on z+. set lpw(z +) = { (x(n))n∈z+ ∈ cz + ; ∑ n∈z+ |x(n)|pw(n)p < +∞ } , 1 ≤ p < +∞ cubo 14, 3 (2012) spectral results for operators commuting with translations ... 45 and ‖x‖w,p = (∑ n∈z+ |x(n)|pw(n)p ) 1 p . it is easy to see that the banach space l p w(z +) satisfies our hypothesis. the operator s (resp. s−1) is bounded on l p w(z +), if and only if, w satisfies sup n∈z+ w(n + 1) w(n) < +∞ ( resp. sup n∈z+ w(n) w(n + 1) < +∞ ) . definition 2. a bounded operator t on e is called a toeplitz operator, if we have: (s−1ts)u = tu, ∀u ∈ f(z+). denote by te the space of toeplitz operators on e. it is easy to see that if t commutes either with s or with s−1, then t is a toeplitz operator. indeed, if ts−1 = s−1t, then t = s−1ts. notice that if t ∈ te, we have tu = p+s−ntsnu, for all u ∈ f(z+) and all n > 1. here sn (resp. s−n) denotes (s1)n (resp. (s−1)n) where s1 (resp. s−1) is the shift (resp. the backward shift) on l 2(z−) ⊕ e. remark that we have s−1s = i, however ss−1 6= i and this is the main difficulty in the analysis of toeplitz operators. given a toeplitz operator t, set t̂(n) = (te0)(n) and t̂(−n) = (ten)(0), for n ≥ 0 and define t̂ = (t̂(n))n∈z. it is easy to see that we have tu = p+(t̂ ∗ u), ∀u ∈ f(z+). (3) set t̃(z) = ∑ n∈z t̂(n)zn, for z ∈ c. notice that the series t̃(z) could diverge. if s and s−1 are bounded, we will denote by ie the interval [ 1 ρ(s−1) ,ρ(s) ] . if s (resp. s−1) is not bounded, then ie denotes [ 1 ρ(s−1) ,+∞ [ ( resp. ] 0,ρ(s) ]) . if s and s−1 are bounded, denote by ue the set { z ∈ c, 1 ρ(s−1) ≤ |z| ≤ ρ(s) } . if s (resp. s−1) is not bounded then ue denotes { z ∈ c, 1 ρ(s−1) ≤ |z| } ( resp. { z ∈ c, |z| ≤ ρ(s) }) . we have the following result (see theorem 2 in [8]) theorem 3. let t be a toeplitz operator on e. 1) for r ∈ [ 1 ρ(s−1) ,ρ(s) ] , if ρ(s) < +∞ or for r ∈ [ 1 ρ(s−1) ,+∞ [ , if ρ(s) = +∞ we have t̃ ∈ l∞(cr) and |t̃(z)| ≤ ‖t‖, a.e. on cr. 2) if ◦ ue is not empty, t̃ ∈ h∞( ◦ ue), where h∞( ◦ ue) is the space of holomorphic and essentially bounded functions on ◦ ue. denote by µ(e) the set of bounded operators on e commuting with either s or s−1. as mentioned above µ(e) ⊂ te. it is clear that the operators (sn)n≥0 and ((s−1)n)n≥0 are included in µ(e). in this paper we prove the following 46 violeta petkova cubo 14, 3 (2012) theorem 4. if s and s−1 are bounded operators, we have σ(s) = {z ∈ c : |z| ≤ ρ(s)}. (4) σ(s−1) = {z ∈ c : |z| ≤ ρ(s−1)}. (5) for the right r and left l weighted shifts on l2(n) the results (4), (5) are classical (see for instance, [11]). moreover, it is well known that the spectrum of r and l have a circular symmetry ([12]). the proofs of these results for r and l use the structure of l2(z+) and the analysis of the point spectrum is given by a direct calculus. in the general situation we deal with such an approach is not possible and our results on the symbols of toeplitz operators play a crucial role. first we establish in proposition 1 the relation { 1 ρ(s−1) ≤ |z| ≤ ρ(s)} ⊂ σ(s) and next we obtain (4). it seems that theorem 4 is the first result concerning the description of σ(s) and σ(s−1) in the general setup when (h1)(h3) hold. for operators commuting either with s or s−1 we have the following theorem 5. suppose that s and s−1 are bounded. let t be a bounded operator on e commuting with s. then we have t̃( ◦ σ(s)) ⊂ σ(t). (6) if t is a bounded operator on e commuting with s−1, we have t̃( ◦ σ(s−1)) ⊂ σ(t). (7) for φ ∈ cz, define tφf = p +(φ ∗ f), ∀f ∈ e. if φ ∈ cz is such that tφ is bounded on e (it is the case if for example φ ∈ f(z+)), then tφ ∈ µ(e). the author has established similar results for multipliers and winer-hopf operators in weighted spaces l2ω(r) and l 2 w(r +) (see [6], [9]). the spaces considered in this paper are much more general then weighted l2ω(z) and l 2 ω(z +) spaces. here we consider not only hilbert spaces, but also banach spaces which may have a complicated structure (see example 2 and example 3). moreover, we study multipliers on spaces where the shift is not a bounded operator. in these general cases our spectral results are based heavily on the symbolic representation and this was the main motivation for proving the existence of symbols for the operators of the classes we consider. for ψ ∈ cc(r+), denote by tψ the operator defined on lpω(r+) by (tψf)(x) = p+(ψ ∗ f)(x), a.e. set β0 = limt→+∞ ln ‖st‖ 1 t . a recent result of the author (see [10]) shows that if tψ commutes with (st)t≥0 then σ(tψ) = ψ̂(v), cubo 14, 3 (2012) spectral results for operators commuting with translations ... 47 where v = {z ∈ c, imz ≤ β0}. it is natural to conjecture that σ(tφ) = φ̂(ue) for tφ with φ ∈ f(z) commuting with s. in section 4, we study the so called joint spectrum for translation operators on a banach space of sequences on zk and we generalize the results of section 2. in theorem 7 we prove that the spectrum of a multiplier (bounded operator commuting with the translations) on a very general banach space e of sequences on zk is related to the image under its symbol of the joint spectrum of the translations s1, ...,sk (see section 4 for the definitions). this joint spectrum denoted by σa(s1, ...,sk) (see section 4) is very important in our analysis. notice that σa(s1, ...,sk) ⊂ σ(s1) × ...×σ(sk) but in general the inclusion is strict. the fact that the symbol of a multiplier is holomorphic on the interior of σa(s1, ...,sk) plays a crucial role. to our best knowledge it seems that theorem 7 is the first result in the literature concerning the spectrum of operators commuting with translations on a banach space of sequences on zk. 2 spectrum of a multiplier first, consider the case of the multipliers on a banach space e satisfying (h1)-(h3) and suppose that s or s−1 is bounded on e. define {z ∈ c, |z| ∈ ie}.. to prove theorem 2, we will need the following lemma established in [8] (lemma 4). lemma 1. for φ ∈ f(z), we have |m̃φ(z)| ≤ ‖mφ‖, ∀z ∈ ue. definition 3. for a ∈ cz, and r ∈ r, define the sequence (a)r so that (a)r(n) = a(n)r n, ∀n ∈ z. lemma 2. let r ∈ ie and f ∈ e be such that (f)r ∈ l2(z). if m ∈ µ(e), we have (mf)r = (m̂r ∗ (f)r), (m̃f)r(z) = m̃(rz)(̃f)r(z), ∀z ∈ t and (̃mf)r ∈ l2(t). lemma 2 is a generalization of (1). proof. the proof uses the arguments exposed in [8] with some modifications. for the completeness we give here the details. let m ∈ µ(e). let (mk)k∈n be a sequence such that limk→+∞ ‖mkx − mx‖ = 0, ∀x ∈ e, ‖mk‖ ≤ ‖m‖ and mk = mφk, where φk ∈ f(z), ∀k ∈ n. the existence of this sequence is established in [8] (lemma 3). let r ∈ ie. we have |(̃φk)r(z)| ≤ ‖mφk‖ ≤ ‖m‖, ∀z ∈ t, ∀k ∈ n. we can extract from ( (̃φk)r ) k∈n a subsequence which converges 48 violeta petkova cubo 14, 3 (2012) with respect to the weak topology σ(l∞(t),l1(t)) to a function νr ∈ l∞(t). for simplicity, this subsequence will be denoted also by ( (̃φk)r ) k∈n . we obtain lim k→+∞ ∫ t ( (̃φk)r(z)g(z) − νr(z)g(z) ) dz = 0, ∀g ∈ l1(t) and ‖νr‖∞ ≤ ‖m‖. fix f ∈ e such that (f)r ∈ l2(z). it is clear that lim k→+∞ ∫ t ( (̃φk)r(z)(̃f)r(z)g(z) − νr(z)(̃f)r(z)g(z) ) dz = 0, ∀g ∈ l2(t). we observe that the sequence ( (̃φk)r(̃f)r ) k∈n converges with respect to the weak topology of l2(t) to νr(̃f)r. set ν̂r(n) = 1 2π ∫π −π νr(e it)e−itndt, for n ∈ z and let ν̂r = (ν̂r(n))n∈z be the sequence of the fourier coefficients of νr. the fourier transform from l 2(z) to l2(t) defined by f : l2(z) ∋ (f(n))n∈z −→ f̃|t ∈ l2(t) is unitary, so the sequence ( (mφkf)r ) k∈n = ( (φk)r ∗ (f)r ) k∈n converges to ν̂r ∗ (f)r with respect to the weak topology of l2(z). taking into account that e satisfies (h2), for n ∈ z we obtain lim k→+∞ |((mφkf)r − (mf)r)(n)| ≤ lim k→+∞ c‖mφkf − mf‖ = 0. thus we deduce that (mf)r(n) = (ν̂r ∗ (f)r)(n), ∀n ∈ z, ∀f ∈ e, such that (f)r ∈ l2(z). this implies (m̂)r ∗ (f)r = ν̂r ∗ (f)r,∀f ∈ f(z) and then we get (m̂)r = ν̂r. we conclude that (mf)r = (m̂)r ∗ (f)r, ∀f ∈ e such that (f)r ∈ l2(z) and then we have (m̃f)r(z) = m̃(rz)(̃f)r(z), ∀z ∈ t. since (̃f)r ∈ l2(t) and m̃ ∈ l∞(ue), it is clear that (̃mf)r ∈ l2(t). ✷ proof of theorem 2. let m ∈ µ(e). suppose that α /∈ σ(m). then we have (m − αi)−1 ∈ µ(e). for z ∈ σ(s), we obtain ( ˜(m − αi)−1f ) (z) = ( ∑ n∈z ̂(m − αi)−1(n)zn )( ∑ n∈z f(n)zn ) , for all f ∈ e, such that for all r ∈ ie, (f)r ∈ l2(z). if g ∈ f(z), following lemma 2, we may replace f by (m − αi)g. we get g̃(z) = ( ∑ n∈z ̂(m − αi)−1(n)zn )( ∑ n∈z ((m − αi)g)(n)zn ) cubo 14, 3 (2012) spectral results for operators commuting with translations ... 49 = ˜(m − αi)−1(z)(m̃g(z) − αg̃(z)) = ˜(m − αi)−1(z)(m̃(z) − α)g̃(z), ∀g ∈ f(z), for all z ∈ σ(s). this implies that for fixed r ∈ ie, ˜(m − αi)−1(rη)(m̃(rη) − α) = 1, ∀η ∈ t. since, ˜(m − αi)−1 is holomorphic on ◦ σ(s) and essentially bounded on δ(σ(s)) (see theorem 1), we obtain that m̃(z) 6= α, for every z ∈ ◦ σ(s) and for almost every z ∈ δ(σ(s)). we conclude that m̃(σ(s)) ⊂ σ(m), which proves the first part of the theorem. for te second one, let m ∈ s. then there exists a sequence (mφn) with φn ∈ f(z) such that limn→+∞ ‖mφn − m‖ = 0. notice that from lemma 1 it follows that |m̃φn(z)| ≤ ‖mφn‖ ≤ ‖m‖, ∀z ∈ ue. taking into account that |m̃φn(z)−m̃φk(z)| ≤ ‖mφn −mφk‖, ∀z ∈ ue, and the fact that (mφn) converges with respect to the norm operator theory, we conclude that (m̃φn) converges uniformly on ue to a function µm. we observe that (m̃φn)r converges to µm(r.) with respect to the weak topology σ(l∞(t),l1(t)). so we can identify m̃(rz) and µm(rz) for z ∈ t. consequently, m̃ is continuous on δ(ue). let λ ∈ σ(m) \ {0}. then there exists a character γ on s such that λ = γ(m). for k ∈ n∗, denote by sk the operator (s1)k. we have γ(mφn) = γ( ∑ k∈z φ̂n(k)sk) = ∑ k∈z φ̂n(k)γ(s) k and we get γ(m) = limn→+∞ γ(mφn) = limn→+∞ m̃φn(γ(s)) = m̃(γ(s)). we conclude that σ(m) \ {0} ⊂ m̃(σ(s)). ✷ now suppose that m /∈ s. if α ∈ σ(m) \ {0}, then α = γ(m), where γ is a character on the commutative banach algebra µ(e). following [8] (lemma 3), there exists a sequence (mφn), with φn ∈ f(z) such that limn→+∞ ‖mφna−ma‖ = 0, ∀a ∈ e and we have limn→+∞ m̃φn(z) = m̃(z), for all z ∈ ◦ σ(s) and for almost every z ∈ δ(σ(s)). if we suppose that γ(s) ∈ ◦ σ(s) we have lim n→+∞ γ(mφn) = lim n→+∞ m̃φn(γ(s)) = m̃(γ(s)), but in the general case we do not have lim n→+∞ γ(mφn) = γ(m), because (mφn) converges to m with respect to the strong operator theory and may be not for the norm operator topology. 50 violeta petkova cubo 14, 3 (2012) 3 spectrum of an operator commuting either with s or s−1 on e in this section we consider a banach space e satisfying the conditions (h1) − (h3). suppose that s and s−1 are bounded on e. notice that it is easy to see that, for φ ∈ f(z), if tφ commutes with s (resp. s−1) then φ ∈ f(z+) (resp. f(z−)). lemma 3. for t ∈ te, r ∈ ie and for a ∈ e such that (a)r ∈ l2(z+) we have (ta)r = p +((t̂)r ∗ (a)r) (8) and then (ta)r ∈ l2(z+). lemma 3 is a generalization of the property (3). proof. let t be a bounded operator in te and let (φk)k∈n ⊂ f(z) be such that lim k→+∞ ‖tφka − ta‖ = 0, ∀a ∈ e and ‖tφk‖ ≤ ‖t‖, ∀k ∈ n. the existence of the sequence (tφk) is established in [8] (lemma 5). fix r ∈ ie. we have (see lemma 6 in [8]), |(̃φk)r(z)| ≤ ‖tφk‖ ≤ ‖t‖, ∀z ∈ t, ∀k ∈ n. we can extract from ( (̃φk)r ) k∈n a subsequence which converges with respect to the weak topology σ(l∞(t),l1(t)) to a function νr ∈ l∞(t). for simplicity, this subsequence will be denoted also by( (̃φk)r ) k∈n . let a ∈ e be such that (a)r ∈ l2(z+). we conclude that, ( (̃φk)r(̃a)r ) k∈n converges with respect to the weak topology of l2(t) to νr(̃a)r. denote by ν̂r = (ν̂r(n))n∈z the sequence of the fourier coefficients of νr. since the fourier transform from l 2(z) to l2(t) is an isometry, the sequence (φk)r ∗ (a)r converges to ν̂r ∗ (a)r with respect to the weak topology of l2(z). on the other hand, ( tφka ) k∈n converges to ta with respect to the topology of e. consequently, since e satisfies (h2) we have lim k→+∞ |((tφka)r − (ta)r)(n)| ≤ lim k→+∞ c‖tφka − ta‖ = 0, ∀n ∈ n. we conclude that (ta)r = p +(ν̂r ∗ (a)r), ∀a ∈ e such that (a)r ∈ l2(z+). (9) since (ta)r = p + ((t̂ ∗ a)r), ∀a ∈ f(z+), it follows that t̂(n)rn = ν̂r(n), ∀n ∈ z. then (9) implies obviously (8). combining (8) with the fact that t̂ ∈ l∞(z), it is clear that if (a)r ∈ l2(z+), then (ta)r ∈ l2(z+). ✷ cubo 14, 3 (2012) spectral results for operators commuting with translations ... 51 for the proof of theorem 4 we need the following proposition 1. let t be a bounded operator in µ(e). then we have t̃( ◦ ue) ⊂ σ(t). proof. let t ∈ µ(e) and suppose that λ /∈ σ(t). first we will show that (t − λi)−1 ∈ te. if ts = st, then (t − λi)s = s(t − λi) and we obtain (t − λi)−1s = s(t − λi)−1. as we have mentioned above this implies that (t −λi)−1 is a toeplitz operator. in the same way we treat the case when ts−1 = s−1t. set h(n) = ̂(t − λi)−1(n) and fix r ∈ ie. for all g ∈ e such that (g)r ∈ l2(z+), applying lemma 3 with (t −λi) ∈ te and a = g we get (t −λi)g ∈ l2(z+). then allpying a second time lemma 3 with (t − λi)−1 ∈ te and a = (t − λi)g, we get (g)r = p + ( (h)r ∗ ((t − λi)g)r ) , ∀g ∈ e such that (g)r ∈ l2(z+). since (h)r and ((t − λi)g)r are in l 2(z+) (see lemma 3), we have ‖(̃g)r‖l2(t) = ‖(g)r‖l2(z+) = ‖p+((h)r ∗ ((t − λi)g)r)‖l2(z+) ≤ ‖p+‖‖(h)r ∗ ((t − λi)g)r)‖l2(z+) = ‖p+‖‖(̃h)r(t̃ − λ)(̃g)r‖l2(t) ≤ ‖p+‖‖(̃h)r‖l∞(t)‖(t̃ − λ)(̃g)r‖l2(t) ≤ c‖(t̃ − λ)(̃g)r‖l2(t), ∀g ∈ e such that (g)r ∈ l2(z+). (10) first suppose that 1 ∈ ◦ ie. then for r = 1, we get ‖g̃‖l2(t) ≤ c‖(t̃ − λ)g̃‖l2(t), ∀g ∈ e ∩ l2(z+). assume that λ = t̃(z0) for z0 ∈ t ⊂ ◦ ue. according to theorem 3, t̃ is continuous on t and it is easy to choose f ∈ l2(t) so that 2c‖(t̃ − λ)f‖l2(t) < ‖f‖l2(t). (11) in fact, if |t̃(z) − λ| ≤ δ for |z − z0| < η(δ), we take f such that f(z) = 0 for z s.t. |z − z0| ≥ η(δ) and ‖f‖l2(t) = 1. for δ > 0 such that 2cδ < 1 we get the inequality (11). let g ∈ l2(z) be such that f = g̃ and let β = c‖t̃ − λ‖∞. fix ǫ > 0 so that ‖g̃‖l2(t) > (2β + 2)ǫ. next let gǫ ∈ f(z) be such that ‖gǫ − g‖l2(z) ≤ ǫ. then we have c‖(t̃ − λ)g̃ǫ‖l2(t) ≤ c‖(t̃ − λ)(g̃ǫ − g̃)‖l2(t) + c‖(t̃ − λ)g̃‖l2(t) ≤ βǫ + 1 2 ‖g̃‖l2(t) < βǫ + 1 2 ‖g̃ − g̃ǫ‖l2(t) + 1 2 ‖g̃ǫ‖l2(t) < (β + 1 2 )ǫ + 1 2 ‖g̃ǫ‖l2(t). 52 violeta petkova cubo 14, 3 (2012) on the other hand, ‖g̃ǫ‖l2(t) ≥ ‖g̃‖l2(t) −ǫ ≥ 2β+ǫ, hence (β+ 12)ǫ ≤ 1 2 ‖g̃ǫ‖l2(t). this implies c‖(t̃ − λ)g̃ǫ‖l2(t) < ‖g̃ǫ‖l2(t). notice that for f ∈ l2(z) and n ∈ z+, we have s̃nf(z) = znf̃(z), ∀z ∈ t. set h = sngǫ, where n ∈ z+ is chosen so that sngǫ ∈ f(z+). we have c‖(t̃ − λ)h̃‖l2(t) = c‖(t̃ − λ)s̃ngǫ‖l2(t) = c‖(t̃ − λ)g̃ǫ‖l2(t) < ‖g̃ǫ‖l2(t) = ‖h̃‖l2(t). taking into account (10), we obtain a contradiction and then t̃(z) ∈ σ(t) for z ∈ t. now let r ∈ ◦ ie and r 6= 1. repeating the above argument, we choose g ∈ f(z+) so that c‖(t̃ − λ)g̃‖l2(t) < ‖g̃‖l2(t). let h be the sequence defined by h(n) = g(n)r−n, ∀n ∈ z+. then g = (h)r and h ∈ f(z+). we have c‖(t̃ − λ)(̃h)r‖l2(t) < ‖(̃h)r‖l2(t). by using (10) once more, we obtain a contradiction and then t̃(cr) ⊂ σ(t), where cr is the circle of center 0 and radius r and this completes the proof of the theorem. ✷ proof of theorem 4. the symbol of s is z −→ z and according to proposition 1, we have ue ⊂ σ(s). it remains to show that {z ∈ c, |z| < 1ρ(s−1)} ⊂ σ(s). we apply the argument of [9]. for 0 < |z| < 1 ρ(s−1) we write s−1 − 1 z i = − 1 z ( s−1(s − z)i ) (12) if z /∈ σ(s), then there exists g 6= 0 such that (s − z)g = e0. this implies (s−1 − 1z)g = 0 and we obtain a contradiction with the fact that 1 |z| > ρ(s−1). this completes the proof of (4). now we pass to the analysis of σ(s−1). as above assume that s−1 is bounded. following [9], we show first that for the approximative spectrum π(s) of s we have π(s) ⊂ {z ∈ c, 1 ρ(s−1) ≤ |z| ≤ ρ(s)}. in fact, for z 6= 0, if there exists a sequence fn, ‖fn‖ = 1 such that (s − z)fn → 0 as n → ∞, then from (12) we deduce that (s−1 − 1 z )fn → 0 and this yields 1 |z| ≤ ρ(s−1). on the other hand, if 0 ∈ π(s), there exists a sequence fn, ‖fn‖ = 1 such that sfn → 0 and this yields a contradiction with the equality fn = s−1sfn. for the proof of (5) we use for z 6= 0 the adjoint operators s∗, s∗−1 and the equality z ( 1 z i − s∗ ) = s∗ ( (s−1) ∗ − zi ) . cubo 14, 3 (2012) spectral results for operators commuting with translations ... 53 the symbol of s−1 is z −→ 1z and an application of proposition 1 yields { 1 ρ(s) ≤ |z| ≤ ρ(s−1)} ⊂ σ(s−1). next assume that 0 < |z| < 1 ρ(s) . we are going to repeat the argument of the proof of theorem 3 in [9] and for completeness we give the proof. first, 0 ∈ σr(s), σr(s) being the residual spectrum of s. in fact, if this is not true, 0 will be in π(s) and this is a contradiction. secondly, we deduce that 0 will be an eigenvalue of the adjoint operator s∗. let s∗g = 0 with g 6= 0. if (s−1)∗ − zi is surjective, than there exists f 6= 0 such that ((s−1)∗ − z)f = g and we get (1 z − s∗)f = 0, hence 1 |z| ≤ ρ(s∗) = ρ(s) which is impossible. thus z ∈ σ(s∗−1) and, passing to the adjoint, we complete the proof. ✷. for the proof of theorem 5 we need the following lemma 4. let φ ∈ f(z+) (resp. f(z−)). then for z ∈ σ(s) (resp. z ∈ σ(s−1)), we have |(̃φ)(z)| ≤ ‖tφ‖ ≤ ‖t‖. proof. suppose that |z| = ρ(s). then z is in π(s) and there exists a sequence (fn)n∈n ⊂ e such that ‖fn‖ = 1 and limn→+∞ ‖sfn − zfn‖ = 0. then for φ ∈ f(z+), we have for some n > 0, ‖φ ∗ fn − φ̃(z)fn‖ ≤ n∑ k=0 ( sup |k|≤n |φ(k)|)‖skfn − zkfn‖ and we obtain lim n→+∞ ‖φ ∗ fn − φ̃(z)fn‖ = 0. since |φ̃(z)| = ‖φ̃(z)fn‖ ≤ ‖φ̃(z)fn − φ ∗ fn‖ + ‖tφfn‖, it follows that |φ̃(z)| ≤ ‖tφ‖. by using the maximum principle for the analytic function φ̃ we complete the proof for z ∈ σ(s). for σ(s−1) we apply the same argument. ✷. lemma 5. let t be a bounded operator on e commuting with s. let r be such that there exists z ∈ σ(s) with r = |z|. then for a ∈ e such that (a)r ∈ l2(z+) we have (ta)r = p +((t̂)r ∗ (a)r) (13) and then (ta)r ∈ l2(z+). proof. for the proof we apply lemma 4 and the same arguments as those in the proof of lemma 3. ✷ by using lemmas 4-5 and repeating the arguments of the proof of proposition 1, we obtain theorem 5. we leave the details to the reader. 54 violeta petkova cubo 14, 3 (2012) 4 spectral results for multipliers on banach space of functions on zk let f(zk) be the space of sequences of zk with a finite number of not vanishing terms. let e be a banach space of sequences on zk satisfying the following conditions: (h1) f(z k) is dense in e. (h2) for every n ∈ zk, the application e ∋ x −→ x(n) ∈ ck is continuous. (h3) for every z ∈ tk, we have ψz(e) ⊂ e and supz∈tk ‖ψz‖ < +∞, where (ψz(x))(n1, ...,nk) = x(n1, ...,nk)z n1 1 ...z nk k , ∀n ∈ z k, x ∈ e. denote by µ(e) the space of bounded operators on e commuting with the translations. denote by si the operator of translation by ei, where ei(n) = 1, if ni = 1 and nj = 0, for j 6= i and else ei(n) = 0. suppose that the operator si is bounded on e for all i ∈ z. for m ∈ µ(e) define m̃(z) = ∑ n∈zk m̂(n1, ...,nk)z n1 1 ...z nk k , for z = (z1, ...,zk) ∈ ck, where m̂(n1, ...,nk) = m(e0)(n1, ...,nk). for a ∈ e, set ã(z) = ∑ n∈zk a(n)z n1 1 ...z nk k , ∀z ∈ ck. notice that for m ∈ µ(e) and a ∈ f(zk), we have ma = m̂ ∗ a, ∀a ∈ f(zk) and formally we get m̃a(z) = m̃(z)ã(z), a ∈ e, z ∈ ck. if φ ∈ f(zk) denote by mφ the operator given by mφf = φ ∗ f, ∀f ∈ e. define the set zke = { z ∈ ck, ∣∣∣ ∑ n∈zk φ(n)z n1 1 ..z nk k ∣∣∣ ≤ ‖mφ‖, ∀φ ∈ f(zk) } . denote by σa(b1, ...,bp) the joint spectrum of the elements b1,...,bp in a commutative banach algebra a. recall that σa(b1, ...,bp) is the set of (λ1, ...,λp) ∈ cp such that for all l ∈ a, the operator (b1 −λ1i)l+...+(bp −λpi)l is not invertible (see [13]). we have also the representation σa(b1, ...,bp) \ {0} = {(γ(b1), ...,γ(bp)) : γ isacharacterona}. it is clear that σa(b1, ...,bp) ⊂ σ(b1) × ... × σ(bp), but in general these two sets are not equal and the inclusion could be strict. cubo 14, 3 (2012) spectral results for operators commuting with translations ... 55 definition 4. denote by a the closure of the subalgebra generated by the operators mφ, φ ∈ f(zk), with respect to the operator norm topology. proposition 2. we have σa(s1, ....,sk) \ {0} = zke \ {0}. proof. let z ∈ ck be such that z = (γ(s1), ...,γ(sk)), where γ is a character on the algebra a. then, we have ∑ n∈zk φ(n)z n1 1 ...z nk k = ∑ n∈zk φ(n)γ(s1)n1...γ(sk)nk = γ(mφ),∀φ ∈ f(zk) and it is clear that |γ(mφ)| ≤ ‖mφ‖, ∀φ ∈ f(zk). it follows that σa(s1, ....,sk) ⊂ zke. on the other hand, if z ∈ zke \ {0}, we define γz : mφ −→ ∑ n∈zk φ(n)z n1 1 ...z nk k . the application γz is a character on a and this implies that z = (γz(s1), ...,γz(sk)) is in the joint spectrum of s1, ...,sk in a. so we have zke \ {0} = σa(s1, ....,sk) \ {0}. ✷ define ie = {r ∈ rk, r1t × ... × rkt ∈ ◦ zke }. for a ∈ e and r ∈ ck, denote by (a)r the sequence (a)r(n1, ...,nk) = a(n1, ...,nk)r n1 1 ...r nk k , ∀(n1, ...,nk) ∈ zk. the following theorem was established in [7] (theorem 4 and collorary 1). theorem 6. let e be a banach space of sequences on zk satisfying (h1), (h2) and (h3) and such that si is bounded on e for all i ∈ z. suppose that ◦ zke 6= ∅. then, for m ∈ µ(e), there exists θm ∈ h∞( ◦ zke) such that for f ∈ f(zk) we have m̃f(z) = θm(z)f̃(z), ∀z ∈ ◦ zke. following lemma 2 in [7] there exists a sequence (mm)m∈n ⊂ µ(e) such that: limm→+∞ ‖mma − ma‖ = 0, ∀a ∈ e, mm = mφm, where φm ∈ f(zk) and ‖mm‖ ≤ c‖m‖. notice that using the sequence (mm) and the same arguments as in the proof of lemma 2, we obtain that in fact θm = m̃ and, moreover, we get the following lemma 6. for m ∈ µ(e) and for f ∈ e such that (f)r ∈ l2(zk), for all r ∈ ie we have m̃f(z) = m̃(z)f̃(z), ∀z ∈ ◦ zke. now we obtain the following spectral result. theorem 7. 1) for m ∈ µ(e), we have m̃( ◦ zk e ) ⊂ σ(m). 2) for m ∈ a, we have σ(m) \ {0} ⊂ m̃(zke) ⊂ σ(m). 56 violeta petkova cubo 14, 3 (2012) proof. let m ∈ µ(e). suppose that α /∈ σ(m). then we have k = (m − αi)−1 ∈ µ(e) and k̃f(z) = k̃(z)f̃(z), ∀z ∈ ◦ z k e, ∀f ∈ e s.t. (f)r ∈ l2(zk), ∀r ∈ ie. (14) ( ˜(m − αi)−1f ) (z) = ( ∑ n∈z ̂(m − αi)−1(n)zn )( ∑ n∈z f(n)zn ) , ∀f ∈ e,s.t. ∀r ∈ ie, (f)r ∈ l2(zk). if g ∈ f(zk), following lemma 6, we may replace f by (m − αi)g in (14). we get g̃(z) = ( ∑ n∈z ̂(m − αi)−1(n)zn )( ∑ n∈z ((m − αi)g)(n)zn ) = k̃(z)(m̃g(z) − αg̃(z)) = k̃(z)(m̃(z) − α)g̃(z), for all z ∈ ◦ zke. this implies that for fixed r ∈ ie, k̃(rη)(m̃(rη) − α) = 1, ∀η ∈ tk. since, k̃ is continuous on ◦ zke, we obtain that m̃(z) 6= α, for every z ∈ ◦ zke. we conclude that m̃( ◦ zke) ⊂ σ(m), which proves part 1). now suppose that m = mφ, with φ ∈ f(zk). let λ ∈ σ(mφ) \ {0}. then there exists γ a character on µ(e) such that λ = γ(mφ) = ∑ n∈zk φ(n)γ(sn1,...,nk) = φ̃(γ(s1), ...,γ(sk)) ∈ φ̃(zke). the end of the proof of 2) is now very similar to the proof of 2) in theorem 2 and is left to the reader. ✷ received: december 2011. revised: january 2012. references [1] d.e. edumnds and a. nekvinda, averaging operators on l{pn} and lp(x), math. inequal. appl., 5, no. 2 (2002), p.235-246. [2] f. fernanda, weighted shift operators and analytic function theory, topics in operator theory (c. pearcy, ed.), math. surveys, no. 13, amer. math. soc., providence, ri, 1974, p.49-128. [3] j. lindenstrauss, l. tzafriri, on orlicz sequence spaces, israel. j. math. 10 (1971), p.379-390. [4] a. nekvinda, equivalence of l{pn} norms and shift operators, math. inequal. appl. 5, no. 4 (2002), p.711-723. [5] v. petkova, wiener-hopf operators on l2ω(r +), arch. math.(basel), 84 (2005), p.311-324. [6] v. petkova, spectral theorem for multipliers on l2ω(r), arch. math. (basel), 93 (2009), p.357368. cubo 14, 3 (2012) spectral results for operators commuting with translations ... 57 [7] v. petkova, multipliers on banach spaces of functions on a locally compact abelian group, j. london math. soc. 75 (2007), p.369-390. [8] v. petkova, multipliers and toeplitz operators on banach spaces of sequences, journal of operator theory, 63 (2010), p.283-300. [9] v. petkova, spectra of the translations and wiener-hopf operators on l2ω(r +), proc. amer. math. soc. to appear. [10] v. petkova, multipliers and wiener-hopf operators on weighted lp spaces, central european journal of mathematics, to appear., (arxiv:1112.4985v1). [11] w. c. ridge, approximative point spectrum of a weighted shift, trans. ams, 147 (1970), p.349-356. [12] w. c. ridge, spectrum of a composition operator, proc. ams, 37 (1973), p.121-127. [13] w. zelasko, banach algebra, elvesier publishing company, amsterdam (1973). cubo a mathematical journal vol.16, no¯ 02, (53–69). june 2014 existence of blow-up solutions for quasilinear elliptic equation with nonlinear gradient term. 1 fang li institute of mathematics, school of mathematical sciences, nanjing normal university, jiangsu nanjing 210023, china. lifang101216@126.com zuodong yang school of teacher education, nanjing normal university, jiangsu nanjing 210097, china zdyang jin@263.net. abstract in this paper, we consider the quasilinear elliptic equation in a smooth bounded domain. by using the method of lower and upper solutions, we study the existence, asymptotic behavior near the boundary and uniqueness of the positive blow-up solutions for quasilinear elliptic equation with nonlinear gradient term. resumen en este art́ıculo consideramos la ecuación eĺıptica cuasilineal en un dominio acotado suave. usando el método de sub y súper soluciones, estudiamos la existencia, comportamiento asintótico cerca de la frontera y la unicidad de soluciones explosivas para ecuaciones eĺıpticas cuasilineales con término del gradiente nolineal. keywords and phrases: quasilinear elliptic equation; blow-up solutions; asymptotic behavior of solutions; lower and upper solutions. 2010 ams mathematics subject classification: 35j65, 35j50. 1project supported by the national natural science foundation of china(no.11171092); the natural science foundation of the jiangsu higher education institutions of china(no.08kjb110005) 54 fang li & zuodong yang cubo 16, 2 (2014) 1 introduction and main results we shall establish the results on the existence, asymptotic behavior near the boundary and uniqueness near the boundary for the following quasilinear elliptic equation { △mu = b(x)u p(1 + |∇u|q), x ∈ ω, u = ∞, x ∈ ∂ω, (1.1) where ω is a c2 bounded domain with smooth boundary ∂ω in rn, △mu := div(|∇u|m−2∇u), m ≥ 2, p, q > 0, b(x) ∈ cµ(ω̄) for some 0 < µ < 1. problems like (1.1) are usually known in the literature as a boundary blow-up problems and its solutions are named ”blow-up solutions” or ”explosive solutions” or ”large solutions” of eq. (1.1). precisely, by a solution of (1.1) we mean a solution of (1.1) satisfying u(x) → ∞ as d(x,∂ω) → 0. semilinear elliptic problems involving a gradient term with boundary blow-up interested many authors. namely bandle and giarrusso[1] developed existence and asymptotic behavior results for large solutions of ∆u+|∇u(x)|a = g(u) in a bounded domain. in the case g(u) = p(x)uγ, a > 0, and γ > max(1, a). ghergu et al.[2] considered more general equation ∆u + q(x)|∇u(x)|a = p(x)g(u), where 0 ≤ a ≤ 2, p and q are hölder continuous functions on (0, ∞). more results about some extensions to this problems, we can see in[24]-[25]. recently, goncalves et al. [11] showed the existence of nonnegative solutions of the boundary blow-up problem { △u = ψ(x, u, ∇u), x ∈ ω, u = ∞, x ∈ ∂ω (1.2) under the condition a(x)g(t) ≤ ψ(x, t,ξ) ≤ h(t)(1 + λ|ξ|2), where λ > 0 is a constant, a, g and h are continuous functions, a(x) > 0 in ω, g and h are non-decreasing and satisfying g(0) = 0, g(t) > 0 for t > 0, h(0) ≥ 1, and g satisfies the so called keller-osserman condition, namely ∫ ∞ 1 1 ! g(t) dt < ∞, g(t) = ∫t 0 g(s)ds. the study of the following equation: { △mu = g(x)f(u), in ω u(x) → ∞, as x → ∂ω (1.3) cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 55 also has been many results, see [3],[4],[15]-[19] and the references therein. gladiali and porru [15] studied boundary asymptotic of solutions of this equation under some condition on f and when g(x) ≡ 1. related problems on asymptotic behavior and uniqueness were also studied in [16]. ahmed mohammed in [17] established boundary asymptotic estimate for solution of this equation under appropriate conditions on g and the nonlinearity f. they still allowed g to be unbounded on ω or to vanish on ∂ω. diaz and letelier [18] proved the existence and uniqueness of large solutions to the problem (1.3) both for f(u) = uγ,γ > m−1(super-linear case) and ∂ω being of the class c2. lu, yang and e.h.twizell [4] proved the existence of large solutions to the problem (1.1) both for f(u) = uγ,γ > m − 1,ω = rn or ω being a bounded domain (super-linear case) and γ ≤ m − 1,ω = rn(sub-linear case) respectively. z.yang et.al. [19] also established an explosive sub-supersolution method for the existence of solutions to (1.3). for the other results of large solutions to quasilinear elliptic problems (1.1) with nonlinear gradient terms, see [5]-[8] and the references therein. motivated by the results of the above cited papers, we shall attempt to treat such equation (1.1), the results of the semilinear equations are extended to the quasilinear ones. we can find the related results for m = 2 in [10]. to study (1.1),we first consider the existence of nonnegative solutions of the generary boundary blow-up problem { △mu = ψ(x, u, ∇u), x ∈ ω, u = ∞, x ∈ ∂ω. (1.4) our main results are summarized in the following and to our best knowledge, they are not covered by any of the ones referred to above. theorem 1.1. let ψ ∈ cµ(ω × r × rn), 0 < µ < 1, and ū, u ∈ w1,∞(ω) be the ordered weak upper and lower solutions of (1.4) and be bounded on any closed subdomain of ω. assume that there exist constants k > m − 1, c1 > 0, and two functions h1 ∈ c µ(ω) and g ∈ l∞loc([0, +∞)) for some 0 < µ < 1, such that |ψ(x, t,ξ)| ≤ h1(x) + g(t) + c1|ξ| k−1a.e.x ∈ ω, ∀ξ ∈ rn, t ∈ [u, ū]. (1.5) then there is a c1,β(ω)-solution u of (1.4) for some 0 < β < 1 such that u ≤ u ≤ ū in ω.the ordered weak upper and lower solutions will be defined by definition 2.1. theorem 1.2. suppose that m ≥ 2, p, q > 0 and p + q > m − 1. if there exist two constants γ ≥ 0 and β1 > 0 such that γ + m − q ≥ 0 and b(x) ≥ β1d γ(x). then the problem (1.1) has at least one nonnegative c1-solution. theorem 1.3. suppose that b(x) > 0 in ω, m ≥ 2, p, q > 0 and p+q > m−1. 56 fang li & zuodong yang cubo 16, 2 (2014) if there are two constants β > 0 and γ ≥ 0 satisfying γ + m − q > 0, such that lim d(x)→0 b(x) d(x)γ = β. then the problem (1.1) possesses a nonnegative solution and any nonnegative solution u(x) satisfies lim d(x)→0 u(x) d−α(x) = ( αm−1−q(α + 1)(m − 1) β ) 1 p+q+1−m , (1.6) where α = γ+m−q p+q−m+1 . furthermore, if p ≥ m − 1, then the nonnegative solution of (1.1) is unique. this work is organized as follows: in section 2, we give a comparison principle and prove theorem 1.1. in section 3, we first find out the blow-up rate in the radially symmetric case and then prove theorems 1.2 and 1.3. 2 proofs of theorem 1.1 firstly, we consider the second order quasilinear operator q of the form: q(u,ϕ) = ∫ ω (a(x, u, ∇u).∇ϕ − b(x, u, ∇u)ϕ)dx where x = (x1, ..., xn) is contained in the domain ω of r n,the functions a(x, z, p) and b(x, z, p) are assumed to be defined for all values of (x, z, p) in the set ω×r × rn,ϕ ∈ c∞0 (ω). from [23], we get the following comparison principle which plays an important role in the proofs of theorems 1.2 and 1.3. lemma 2.1.(comparison principle) let u, v ∈ c1(ω) satisfy qu ≥ 0 in ω, qv ≤ 0 in ω and u ≤ v on ∂ω, where the functions a, b are continuously differentiable with respect to the z, p variables in ω × r × rn, the operator q is elliptic in ω, and the function b is non-increasing in z for fixed (x, p) ∈ ω × rn. the, if either (i) the vector function a is independent of z; or (ii) the function b is independent of p. it follows that u ≤ v in ω. now, we consider the general equation △mu − ψ(x, u, ∇u) = 0 x ∈ ω. (2.1) cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 57 definition 2.1. let 1 < k ≤ +∞, functions ū, u ∈ w1,k(ω) are called the weak upper and lower solutions of (2.1), respectively, if ψ(·, ū(·), ∇ū(·)) ∈ lk ′ (ω),ψ(·, u(·), ∇u(·)) ∈ lk ′ (ω) with k′ = { k k−1 , k < ∞, 1, k = ∞, and ∫ ω |∇ū|m−2∇ū∇vdx ≥ − ∫ ω ψ(x, ū, ∇ū)vdx, ∀v ∈ w1,k0 (ω), v ≥ 0 a.e.in ω ∫ ω |∇u|m−2∇u∇vdx ≤ − ∫ ω ψ(x, u, ∇u)vdx, ∀v ∈ w1,k0 (ω), v ≥ 0 a.e.in ω. if u ≤ ū, we call that they are the ordered weak upper and lower solutions of (2.1). firstly, we consider the existence of weak solution to the problem { △mu − ψ(x, u, ∇u) = 0, x ∈ ω, u = φ(x), x ∈ ∂ω, (2.2) where ψ(·, u(·), ∇u(·)) ∈ lk ′ (ω),φ ∈ w1,k(ω). assume that ū ∈ w1,k(ω) is a weak upper solution (u ∈ w1,k(ω) is a weak lower solution) of (2.1). here by ū ≥ φ(u ≤ φ) on ∂ω, we mean (φ − u)+ := max{φ − u, 0} ∈ w1,k0 (ω) ((u − φ) + ∈ w1,k0 (ω)). if ū ≥ φ(u ≤ φ) on ∂ω, we call that ū(u) is a weak upper solution (lower solution) of (2.2). if u ≤ ū a.e. in ω, we call that they are ordered. lemma 2.2.([9,theorem 4.9]). let ū, u ∈ w1,k(ω) be the ordered weak upper and lower solutions of (2.2), respectively, and u ≤ ū a.e. in ω. assume that there exists a positive constant c1 and a function h1 ∈ l k′(ω) with k′ = k/(k − 1), such that |ψ(x, t,ξ)| ≤ h1(x) + c1|ξ| k−1, a.e. x ∈ ω, ∀ξ ∈ rn, t ∈ [u, ū]. (2.3) then there is a weak solution u ∈ w1,k(ω) of the problem (2.2) such that u ≤ u ≤ ū a.e. in ω. lemma 2.3. let ψ ∈ cµ(ω × r × rn), and ū, u ∈ w1,∞(ω) be the ordered weak upper and lower solutions of (2.2), φ ∈ c1+µ(ω), 0 < µ < 1, and u ≤ φ ≤ ū a.e. in ω. assume that there exists constants k > 1, c1 > 0, and a function h1 ∈ c µ(ω) " l∞(ω), such that (2.3) holds. then for some 0 < β < 1, there is a c1,β-solution u of (2.2) such that u ≤ u ≤ ū in ω. 58 fang li & zuodong yang cubo 16, 2 (2014) the proof of the lemma 2.3 is similar to [10], so we omit it here. definition 2.2. a domain ω is called satisfying the uniform outside spherical condition: if there exists a constant r > 0 such that there exists a sphere b whose radius is r in rn for any z ∈ ∂ω, such that b ∩ ω = {z}. noticing that any c2 bounded domain satisfies the uniform outside spherical condition. lemma 2.4.( theorem 4.2 in [20]) assume that ω is a bounded domain in rn satisfying the uniform outside spherical condition, then there exists a series of c∞ domains {ωn} ∞ 1 , such that ωn ⊂ ωn+1 ⊂ ω, # ∞ n=1 ωn = ω. proof of theorem 1.1. since ω is a c2 bounded domain, from lemma 2.4 we know that there exists a series of c∞ domains {ωn} ∞ 1 , such that ωn ⊂ ωn+1 ⊂ ω, # ∞ n=1 ωn = ω. now we consider the problem { △mu = ψ(x, u, ∇u), x ∈ ωn, u = ū, x ∈ ∂ωn. (2.4) since ū ∈ w1,∞(ωn) and u ∈ w 1,∞(ωn), by (1.5) we see that there is a constant c2 = c2(n) > 0 such that |ψ(x, t,ξ)| ≤ h1(x) + c2 + c1|ξ| k−1, a.e. x ∈ ωn, ∀ξ ∈ rn, t ∈ [u, ū]. it is obvious that ū|ω and u|ω are the ordered upper and lower solutions of (2.4), and h1 ∈ c(ωn). by lemma 2.3, there exists a solution un ∈ c1,β(ωn) of (2.4) for some 0 < β < 1 such that u ≤ un ≤ ū in ωn. now, we want to apply elliptic interior estimates together with a diagonal process to conclude: {un : n ≥ 1} has a subsequence {uni : ni ↑ ∞} such that {uni} converges to a function u in ω(pointwise) and this convergence is in c1 on every compact set in ω. (therefore, u ∈ c1 and div(|∇u|m−2∇u) = ψ(x, u, ∇u) with u(x) ≤ u(x) ≤ u(x), and this concludes the proof.) step 1. on ω2, {un : n ≥ 2} is uniformly bounded by u(x) and u(x). since both u(x) and u(x) are bounded functions on ω2, there exists m > 0 such that ∥u(x)∥l∞(ωn) ≤ m, for all n ≥ 2. from (2.4), un satisfies ∫ ω2 |∇un| m ≤ ∫ ω2 ψun. (2.5) therefore, ∫ ω2 |∇un| m ≤ m(measω2) 1/q′c1∥∇un∥m, (2.6) here 1/q′ +1/m = 1, and c1 is the sobolev embedding constant. so, ∥un∥1,m ≤ c2. when 1 < m < n, the embedding of w1,m0 (ω2) in l nm/(n−m)(ω2) implies that cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 59 uk ∈ l nm/(n−m)(ω2). applying theorem 7.1 in [21, page 286-287], we obtain the estimate sup{|un|; x ∈ ω2} ≤ c3, (2.7) here c3 = c3(∥ψ∥0). if m ≥ n, we get (2.7) from the sobolev embedding theorem. using theorem 1.1 in ([21], page 251), we see that un belongs to c α(ω2) for some 0 < α < 1, and ∥un∥cα ≤ c4, (2.8) here c4 is determined by c3. by proposition 3.7 in [22, page 806], we also know that un belongs to c 1,α(ω2) and ∥un∥c1,α ≤ c5. (2.9) here c5 is determined by c4. from the arguments above we see that there exists c > 0 such that ∥un∥c1+α(ω1) ≤ c, for all n ≥ 2. (2.10) since the embedding c1+α(ω1) → c1(ω1) is compact, there exists a sequence denoted by {un1j}j=1,2... (where n1j ↑ ∞), which converges in c 1(ω1). let u1(x) = limj→∞ un1j(x), for x ∈ ω1, then u1 is a solution of (2.1) with u(x) ≤ u1 ≤ u(x). step 2. repeat step 1 up to the existence of the sequence {un1j}j=1,2... to get a subsequence {un2i}i=1,2... converging in c 1(ω2) to a limit u2. then, likewise, u2 is a solution of (2.4) and u2|ω1 = u1. repeat step 1 again on ω3, ..., etc. in this way, we obtain a sequence {unnj}j=1,2... which converges in c 1(ωn) and is a subsequence of {unn−1)j}j=1,2.... let un = limj→∞ unnj, then, un is a solution of (2.4) in ωn and un|ωn−1 = un−1. step 3. by a diagonal process, {unll}l=1,2... is a subsequence of {unlj}j=1,2... for every l. thus, on ωn for each n we have lim l→∞ unll = un. so, if we define u(x) = limn→∞ un(x), then u(x) satisfies div(|∇u|m−2∇u) = ψ(x, u, ∇u), and u ≤ u(x) ≤ u (since u ≤ un(x) ≤ u) for every n. this completes the proof of theorem 1.1. 60 fang li & zuodong yang cubo 16, 2 (2014) 3 proofs of theorems 1.2 and 1.3 to get the existence of large solutions, we first find the blow-up lower and super solutions. furthermore, if the blow-up lower solution and upper solution have the same blow-up rate near the boundary, we could get the asymptotic behavior of large solutions near the boundary. the idea of this section mainly comes from [10],[12][14]. 3.1 blow-up rate in order to get the asymptotic behavior of large solutions in the general domain, we first study the radically symmetric case: { (φm(v ′))′ + n−1 r φp(v ′) = a(r)(r − r)γvp(1 + |v′|q), r ∈ (0, r), v′(0) = 0, lim r→r v(r) = ∞, (3.1) where φm(u) = |u| m−2u, m ≥ 2. to ascertain the blow-up rate of the solution of (3.1) at r > 0, we first find out the blow-up rate of the following one-dimensional problem { (φm(u ′))′ = a(r)(r − r)γup(1 + |u′|q), r ∈ (0, r), u′(0) = 0, lim r→r u(r) = ∞. (3.2) set u(r) = (r − r)−αψ(r), r ∈ [0, r] for some positive constant α which will be determined later, ψ(r) ∈ c2(0, r), then the problem (3.2) becomes (|u′|m−2u′)′ = (m − 1)(α(r − r)−α−1ψ(r) + (r − r)−αψ′(r))m−2 sgn(α(r − r)−α−1ψ(r) + (r − r)−αψ′(r)) [α(α + 1)(r − r)−α−2ψ(r) + 2α(r − r)−α−1ψ′(r) + (r − r)−αψ′′(r)] = (m − 1)(r − r)−α−2−(α+1)(m−2)(αψ(r) + (r − r)ψ′(r))m−2 sgn(α(r − r)−α−1ψ(r) + (r − r)−αψ′(r)) [α(α + 1)ψ(r) + 2α(r − r)ψ′(r) + (r − r)2ψ′′(r)] = a(r)(r − r)γ−αpψp(r) + a(r)(r − r)γ−αp−(α+1)q ψp(r)|αψ(r) + (r − r)ψ′(r)|q (3.3) with the boundary condition ψ(0) = 0, and ψ(r) ∈ (0, ∞). therefore, the constant α provides us with the exact blow-up rate of u at r. multiplying (3.3) by (r − r)α+2+(α+1)(m−2) we have cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 61 (m − 1)(αψ(r) + (r − r)ψ′(r))m−2sgn(α(r − r)−α−1ψ(r) +(r − r)−αψ′(r))[α(α + 1)ψ(r) + 2α(r − r)ψ′(r) + (r − r)2ψ′′(r)] = a(r)(r − r)γ−αp+α+2+(α+1)(m−2)ψp(r) +a(r)(r − r)γ−αp−(α+1)q+α+2+(α+1)(m−2)ψp(r)|αψ(r) + (r − r)ψ′(r)|q. assuming lim r→r (r − r)2ψ′′(r) = lim r→r (r − r)ψ′(r) = 0, we obtain α = γ + m − q p + q + 1 − m , ψ(r) = ( αm−1−q(α + 1)(m − 1) a(r) ) 1 p+q+1−m . (3.4) theorem 3.1. assume that r > 0, a ∈ c([0, r]; (0, ∞)), m ≥ 2, γ ≥ 0, p, q > 0, and γ+m−q > 0, p +q > m−1. let α and ψ(r) be defined by (3.4). then for each ϵ > 0, the problem (3.1) has at least one nonnegative c1-solution vϵ satisfying 1 − ϵ ≤ lim r→r inf vϵ(r) ψ(r)(r − r)−α ≤ lim r→r sup vϵ(r) ψ(r)(r − r)−α ≤ 1 + ϵ. (3.5) therefore, for each x0 ∈ r n, m ≥ 2, the function uϵ(x) := vϵ(r) with r := |x − x0| provides us with a radially symmetric nonnegative solution of the problem { △mu = a(r)d γ(x)up(1 + |∇u|q), x ∈ br(x0), u = ∞, x ∈ ∂br(x0), satisfying 1 − ϵ ≤ lim d(x)→0 inf uϵ(x) ψ(r)d−α(x) ≤ lim d(x)→0 sup uϵ(x) ψ(r)d−α(x) ≤ 1 + ϵ, where d(x) := dist(x,∂br(x0)) = r − |x − x0| = r − r. proof. firstly, we show that, for each ϵ > 0 sufficiently small, there exists a constant aϵ > 0, for each a > aϵ , v̄ϵ(r) := a + b+( r r )2(r − r)−α (3.6) provides us with a positive upper solution of (3.1), where α is defined in(3.4), b+ = (1 + ϵ)( αm−1−q(α + 1)(m − 1) a(r) ) 1 p+q+1−m . (3.7) 62 fang li & zuodong yang cubo 16, 2 (2014) indeed, v̄ϵ ′(0) = 0 and lim r→r v̄ϵ(r) = ∞. thus, v̄ϵ is an upper solution of (3.1) if and only if ( b+ r2 )m−1[2(m − 1)(r − r)2 + 4(m − 1)αr(r − r) + (m − 1)α(α + 1)r2 +2(n − 1)(r − r)2 + α(n − 1)r(r − r)]|2r(r − r)2 + αr2|m−2 ≤ a(r)(r − r)γ−αp+α+2+(α+1)(m−2)(a(r − r)α + b+( r r )2)p((r − r)(α+1)p +| 2b+ r2 r(r − r) + αb+( r r )2|q). (3.8) note that γ − αp + α + 2 + (α + 1)(m − 2) = 0, at r = r, (3.8) becomes (m − 1)αm−1(α + 1)bm−1+ ≤ b p+q + α qa(r) which is valid if and only if b+ ≥ ( αm−1−q(α + 1)(m − 1) a(r) ) 1 p+q+1−m . therefore, according to the choice of b+, inequality (3.8) is satisfied in a left neighborhood of r = r, say (r − δ, r] for some δ = δ(ϵ) > 0. finally, by choosing a sufficiently large, it is clear that the inequality is satisfied in the whole interval [0, r] since p > 0 and a is away from zero. this concludes the proof of the claim above. now we will construct a suitable lower solution for problem (3.1). we claim that for each sufficiently small ϵ > 0, there exists c < 0 such that vϵ(r) := max{0, c + b−(r/r) 2(r − r)−α} (3.9) provides us with a nonnegative lower solution of (3.1), here b− = (1 − ϵ)( αm−1−q(α + 1)(m − 1) a(r) ) 1 p+q+1−m . (3.10) indeed, vϵ is a lower solution of (3.1) where c + b−( r r )2(r − r)−α ≥ 0, (3.11) which implies cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 63 ( b− r2 )m−1[2(m − 1)(r − r)2 + 4(m − 1)αr(r − r) + (m − 1)α(α + 1)r2 +2(n − 1)(r − r)2 + α(n − 1)r(r − r)]|2r(r − r)2 + αr2|m−2 ≥ a(r)(r − r)γ−αp+α+2+(α+1)(m−2)(c(r − r)α + b−( r r )2)p(r − r)(α+1)p +| 2b− r2 r(r − r) + αb−( r r )2|q). (3.12) now, for each c < 0, we can find the constant z = z(c) ∈ (0, r), such that c + b−( r r )2(r − r)−α < 0 if r ∈ [0, z(c)), and c + b−( r r )2(r − r)−α > 0 if r ∈ (z(c), r). moreover, z(c) is decreasing and lim c→−∞ z(c) = r, lim c→0 z(c) = 0. at r = r, (3.12) becomes into (m − 1)αm−1(α + 1)bm−1− ≥ b p+q − α qa(r), which is valid if and only if b− ≤ ( αm−1−q(α + 1)(m − 1) a(r) ) 1 p+q+1−m . therefore, by making the choice (3.10), inequality (3.11) is satisfied in a left neighborhood of r = r, say (r −δ, r] for some δ = δ(ϵ) > 0. moreover, thanks to (3.12), there exists c < 0, such that z(c) = r − δ(ϵ). for this choice of c, vϵ provides us with a weak lower solution of (3.1). since v̄ϵ(r), v(r) ∈ w 1,∞(0, r) are the ordered weak lower and upper solutions of (3.1) and are bounded on any closed subdomain of [0, r), it is easy to see (1.5) holds owing to p, q > 0. so the existence of a c1-solution u of (3.1) is followed by theorem 1.1, and vϵ(r) ≤ u ≤ v̄ϵ(r) in ω. finally, since lim r→r v̄ϵ(r) b+(r − r)−α = lim r→r vϵ(r) b−(r − r)−α = 1, (3.13) where b+ and b− are the constants defined through (3.7)and (3.10), one can easily deduce the remaining assertions of theorem 3.1. the proof is completed. 64 fang li & zuodong yang cubo 16, 2 (2014) 3.2 proofs of theorems 1.2 and 1.3 proof of theorem 1.2. for n ≥ 1, we consider the following problem { △mu = b(x)u p(1 + |∇u|q), x ∈ ω, u = n, x ∈ ∂ω. (3.14) it is obvious that the function ψ(x, t,ξ) = b(x)tp(1 + |ξ|q) satisfies the condition (2.3) since b(x) ∈ cµ(ω̄). the constant functions u(x) = 0 and u(x) = n are the ordered lower and upper solutions of (3.14). by lemma 2.3, we see that the problem (3.14) has at least one nonnegative solution un(x) ∈ c 1(ω) and it satisfies 0 ≤ un(x) ≤ n < n + 1. by our assumption, b(x) ≥ β1d γ(x) > 0 in ω.so,u = un and ū = n + 1 are the order lower and upper solutions of the following problem { △mu = b(x)u p(1 + |∇u|q), x ∈ ω, u = n + 1, x ∈ ∂ω. therefore un ≤ un+1. now we fix a point x0 ∈ ω and consider a small ball b centered at x0 and contained properly in ω. by theorem 3.1, there exists a nonnegative c 1solution v(x) to the problem { △mu = β1(dist(x,∂b)) γup(1 + |∇u|q), x ∈ br(x0), u = ∞, x ∈ ∂br(x0) since △mun − b(x)u p n(1 + |∇un| q) ≤ △mun − β1d γ(x)upn(1 + |∇un| q) ≤ △mun − β1(dist(x,∂b)) γ(x)upn(1 + |∇un| q), x ∈ b, and un(x) ≤ v(x) = ∞ on ∂b, by the comparison principle (lemma 2.1), we have un(x0) ≤ v(x0) for all n. since un increases in ω as n increases, un(x) is uniformly bounded on any compact subset of ω. standard elliptic regularity arguments show that lim n→∞ un(x) = u ∗(x) exists and u∗(x) satisfies the differential equation of (1.1). to prove u∗(x) is a nonnegative solution of (1.1), it remains to verify u∗(x)|∂ω = ∞. if this is not true, then there exist a nonnegative constant m, a sequence {xj} ⊂ ω and x0 ∈ ∂ω, such that xj → x0 and un(xj) ≤ m. for any fixed k, note that un(xj) → u∗(xj) as n → ∞, it follows that there exists nj > 0, such that un(xj) ≤ 1 + m for all n ≥ nj. note that un is increasing in n, we have un(xj) ≤ 1 + m for every n > 0. now fix a n > 1 + m and let j → ∞ in the above cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 65 inequality, it follows un(x0) ≤ 1 + m < n since xj → x0, which is a contradiction with un(x0) = n. the theorem is proved. proof of theorem 1.3. in view of lim d(x)→0 b(x) d(x)γ = β > 0 and b(x) > 0 in ω, it is easy to see that b(x) ≥ β1d γ(x), x ∈ ω for some β1 > 0. thus, the nonnegative c 1-solution of problem (1.1) exists by theorem 1.2. now we prove the limit (1.6). since lim d(x)→0 b(x) d(x)γ = β > 0, for any small ϵ > 0, there exists δ = δ(ϵ) > 0, such that for all x ∈ ω with d(x) < 2δ, (β − ϵ)dγ(x) ≤ b(x) ≤ (β + ϵ)dγ(x). now we define ωδ = {x ∈ ω : d(x) < δ} with ∂ωδ = {x ∈ ω : d(x) = δ} and { u+(x) = b+(ϵ)(d(x) − σ) −α, x ∈ d+σ = ω2δ/ω̄σ, u−(x) = b−(ϵ)(d(x) + σ) −α, x ∈ d−σ = ω2δ−σ, (3.15) where 0 < σ < δ, and b+(ϵ) = (1 + ϵ)( αm−1−q(α + 1)(m − 1) β − ϵ ) 1 p+q+1−m , b−(ϵ) = (1 − ϵ)( αm−1−q(α + 1)(m − 1) β + ϵ ) 1 p+q+1−m . it is easy to prove that by diminishing δ > 0 if necessary, d(x) is a c2-function on the domain ω̄2δ and { △mu + − b(x)(u+)p(1 + |∇u+|q) ≤ 0, x ∈ d+σ, △mu − − b(x)(u−)p(1 + |∇u−|q) ≥ 0, x ∈ d−σ. (3.16) let u be any nonnegative solution of (1.1) and m1(δ) = max d(x)≥2δ u(x), m2(δ) = b−(2δ) −α. we see that { u(x) ≤ u+(x) + m1(δ), x ∈ ∂d + σ, u−(x) ≤ u(x) + m2(δ), x ∈ ∂d − σ. (3.17) 66 fang li & zuodong yang cubo 16, 2 (2014) on the other hand, by p > 0, we have ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ △m(u + + m1(δ)) − b(x)(u + + m1(δ)) p(1 + |∇(u+ + m1(δ))| q) < 0, x ∈ d+σ, △m(u + m2(δ)) − b(x)(u + m2(δ)) p(1 + |∇(u + m2(δ))| q) < 0, x ∈ d−σ. (3.18) note (3.15) − (3.18), it follows by lemma 2.1 that { u(x) ≤ u+(x) + m1(δ), x ∈ d + σ, u−(x) ≤ u(x) + m2(δ), x ∈ d − σ. (3.19) for any x ∈ ω2δ, there is a σ ∗ > 0, such that x ∈ d+δ " d−δ for all 0 < σ ≤ σ ∗. letting σ → 0 , (3.19) yields b−(ϵ)d −α(x) ≤ u + m2(δ) ≤ b+(ϵ)d −α(x) + m1(δ) + m2(δ), which implies b−(ϵ) ≤ lim d(x)→0 inf uϵ(x) d−α(x) ≤ lim d(x)→0 sup uϵ(x) d−α(x) ≤ b+(ϵ). (3.20) taking ϵ → 0,(3.20) yields lim d(x)→0 u(x) d−α(x) = ( αm−1−q(α + 1)(m − 1) β ) 1 p+q+1−m , which is accordance with (1.6). the final step is to prove the uniqueness. let u1 and u2 be two nonnegative solutions of(1.1), then by(1.6), we have lim d(x)→0 u1(x) u2(x) = 1. indeed, for θ > 0 arbitrary, set ωi = (1 + θ)ui, for i = 1, 2. it follows that lim d(x)→0 (u1 − ω2)(x) = lim d(x)→0 (u2 − ω1)(x) = −∞. when p ≥ m − 1, since q > 0, we have that △mωi − b(x)ω p i (1 + |∇ωi| q) < (1 + θ)m−1[△mui − b(x)u p i (1 + |∇ui| q)], x ∈ ω. cubo 16, 2 (2014) existence of blow-up solutions for quasilinear elliptic equation . . . 67 therefore, by lemma 2.1, we infer that u1 ≤ (1 + θ)u2, u2 ≤ (1 + θ)u1, x ∈ ω. 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[25] s.alarcon, j.garcia-melian, a. quaas, keller-osserman type conditions for some elliptic problems with gradient terms, j.diff.eqns. 252(2012), 886-914. cubo a mathematical journal vol.15, no¯ 02, (111–119). june 2013 existence and uniqueness solution of a class of quasilinear parabolic boundary control problems m. h. farag1, t. a. talaat2 and e. m. kamal3 minia university, department of mathematics, faculty of science, minia, egypt. farag5358@yahoo.com1 talaat.2008@yahoo.com2 esamkamal55@yahoo.com3 abstract this paper presents an optimal control of processes described by a quasilinear parabolic systems with controls in the coefficients of equation, in the boundary condition and in the right side of this equation. theorems concarning the existence and uniqueness for the solution of the cosidering problem are invistigated. resumen este art́ıculo presenta un control óptimo de procesos descritos por un sistema parabólico cuasilineal con control en los coeficientes de la ecuación, en la condición de frontera y en el lado derecho de esta ecuación. se investigan los teoremas relacionados con la existencia y unicidad para la solución del problema considerado keywords and phrases: optimal control, quasilinear parabolic equation, existence and uniquness theorems. 2010 ams mathematics subject classification: 49j20, 49k20, 49m29, 49m30 112 m. h. farag, t. a. talaat and e. m. kamal cubo 15, 2 (2013) 1 introduction optimal control problems for partial differential equations are currently of much interest. a larage amount of the theoretical concept which governed by quasilinear parabolic equations [1-5] has been investigated in the field of optimal control problems. these problems have dealt with the processes of hydroand gasdynamics, heatphysics, filtration, the physics of plasma and others [6-8]. the study and determination of the optimal regimes of heat conduction processes at a long interval of the change of temperture gives rise to optimal control problems with respect to a quasilinear equation of parabolic type. in this work, we consider a constrained optimal control problem with respect to a quasilinear parabolic equation with controls in the coefficients of the equation. the existence and uniqueness of the optimal control problem is proved. 2 statement of the problem let d is a bounded domain of the n-dimensional euclidean space en; γ be the boundary of d, assumed to be sufficiently smooth; ν is the exterior unit normal of γ; t > 0 be a fixed time ; ω = d × (0, t] ; s = γ × (0, t]. now we consider a class of optimal control problems governed by the following quasilinear parabolic system. l(v)y(x, t) = f(x, t, v2), (x, t) ∈ ω, y(x, 0) = φ(x), x ∈ d, ∑n i=1 λi(y, v0) ∂y ∂xi cos(ν, xi)|s = g(ζ, t), (x, t) ∈ s (1) where φ ∈ l2(d), g(ζ, t) ∈ l2(s) are given functions and the differential operator l takes the following form: l(v)z(x, t) = ∂z ∂t − n∑ i=1 ∂ ∂xi [λi(z, v0) ∂z ∂xi ] + n∑ i=1 bi(z, v1) ∂z ∂xi (2) y(x, t), v = (v0, v1, v2) are the state and the controls rspectively for the system (1). furthermore, we consider the functional of the form jβ(v) = ∫ s [y(ζ, t) − f0(ζ, t)] 2dζdt + β 2∑ m=0 ‖vm − ωm‖ 2 l2 , (3) which is to minimized under condition (1) and additional restricitions ν0 ≤ λi(y, v0) ≤ µ0, ν1 ≤ bi(y, v1) ≤ µ1, r1 ≤ y(x, t) ≤ r2, i = 1, n (4) cubo 15, 2 (2013) existence and uniqueness solution of a class of quasilinear . . . 113 over the class v = {v = (v0, v1, v2) : vm = (v0m, v1m, · · · , vim, · · · ) ∈ l2, ‖vm‖l2 ≤ rm, m = 0, 2} and f0(ζ, t) ∈ l2(s) is a given function and β ≥ 0, νj, µj, j = 1, 2, r1, r2,rm > 0 are positive numbers, ωm = (ω0m, ω1m, · · · , ωim, · · · ) ∈ l2, m = 0, 2 are given numbers. throughout this paper, we adopt the following assumptions. assumption 2.1: v is closed and bonded subset of l2. assumption 2.2: the functions bi(y, v1), λi(y, v0), i = 1, n are continuous on (y, v) ∈ [r1, r2] × l2 have continuous derivatives in y at ∀(y, v) ∈ [r1, r2] × l2 and ∂bi ∂y , ∂λi ∂y , i = 1, nare bounded. assumption 2.3: the function f(x, t, v2) is given function continuous in v2 on l2 for almost all (x, t) ∈ ω, bounded and measurable in x, t on ω ∀v2 ∈ l2. assumption 2.4:the functions bi(y, v1), λi(y, v0), i = 1, n, f(x, t, v2) satisfy a lipschitz condition for v1, v0, v2 ,then |bi(y(x, t), v1 + δv1) − bi(y(x, t), v1)| ≤ s0(x, t)‖δv1‖l2, i = 1, n |λi(y(x, t), v0 + δv0) − λi(y(x, t), v0)| ≤ s1(x, t)‖δv0‖l2, i = 1, n |f(x, t, v2 + δv2) − f(x, t, v2)| ≤ s2(x, t)‖δv2‖l2 for almost all (x, t) ∈ ω, ∀y ∈ [r1, r2], ∀vm, vm + δvm ∈ l2 such that ‖vm‖l2, ‖vm + δvm‖l2 ≤ rm where sm(x, t) ∈ l∞, m = 0, 2. assumption 2.5: the first derivatives of the functions bi(y, v0), λi(y, v0), i = 1, n and f(x, t, v2) with respect to v are continuous functions in [r1, r2] × l2 and for any vm ∈ l2 such that ‖vm‖l2 ≤ rm, m = 0, 2. definition 2.1: the problem of finding the function y = y(x, t) ∈ v0,12 (ω) from condition (1)-(2) at given v ∈ v is called the reduced problem. definition 2.2: a function y = y(x, t) ∈ v1,02 (ω) is said to be a solution of the problem 114 m. h. farag, t. a. talaat and e. m. kamal cubo 15, 2 (2013) (1)-(2), if for all η = η(x, t) ∈ w1,12 (ω) the equation ∫ ω [−y ∂η ∂t + ∑n i=1 λi(y, v0) ∂y ∂xi ∂η ∂xi − ∑n i=1 bi(y, v1)( ∂y ∂xi )η(x, t) −f(x, t, v2)η(x, t)]dxdt = ∫ d φ(x)η(x, 0)dx + ∫ s g(ζ, t)η(ζ, t)dζdt, (5) is valid and η(x, t) = 0. it is proved in [8] that, under the foregoing assumptions, a reduced problem (1)-(2) has a unique solution and | ∂y ∂xi | ≤ c1, i = 1, n almost at all (x, t) ∈ ω, ∀v ∈ v, where c1 is a certain constant. 3 the existence theorem optimal control problems of the coefficients of differential equations do not always have solution [9]. examples in [10] and elswhere of problems of the type (1)-(4) having no solution for β = 0. a problem of minimization of a functional is said to be unstable, when a minimizing sequare does not converge to an element minimizing the functional [6]. to begin with, we need theorem 3.1 under the above assumptions for every solution of the reduced problem (1)-(2) the following estimate is valid: ‖δy‖v1,0 2 (ω) ≤ c2[‖ √ √ √ √ n∑ i=1 (∆λi ∂y ∂xi )2‖l2(ω) + ‖∆f − n∑ i=1 ∆bi ∂y ∂xi ‖l2(ω)], (6) where δy(x, t) = y(x, t; v + δv) − y(x, t; v), δy(x, t) ∈ w1,12 (ω), ∆λi = λi(u, v0 + δv0) − λi(u, v0) ,∆bi = bi(u, v1 + δv1) − bi(u, v1), ∆f = f(x, t, v2 + δv2) − f(x, t, v2) and c2 ≥ 0 is a constant not dependent on δv = (δv0, δv1, δv2), δvm ∈ l2, m = 0, 2. ptoof set δy(x, t) = y(x, t, v + δv) − y(x, t; v), y = y(x, t; v), y = y(x, t; v + δv). from (5) it follows that ∫ ω [−δy ∂η ∂t + ∑n i=1 λi ∂δy ∂xi ∂η ∂xi + ∑n i=1 ∂λi(y+θ1i,v0+δv0) ∂y ∂y ∂xi ∂η ∂xi δy + ∑n i=1 ∆λi ∂y ∂xi ∂η ∂xi + ∑n i=1 bi ∂δy ∂xi η + ∑n i=1 ∆bi( ∂y ∂xi )η − ∑n i=1 ∂bi(y+θ2i,v1+δv1) ∂y ∂y ∂xi δyη − ∆fη]dxdt = 0 (7) cubo 15, 2 (2013) existence and uniqueness solution of a class of quasilinear . . . 115 for all η = η(x, t) ∈ w1,12 (ω) and η(x, t) = 0. here θ1i, θ2i ∈ (0, 1), i = 1, n is some number, λi ≡ λi(y + δy, v0 + δv0) ,∆λi ≡ λi(y, v0 + δv0) − λi(y, v0), bi ≡ bi(y + δy, v1 + δv1) ,∆bi ≡ bi(y, v1 + δv1) − λi(y, v1), i = 1, n, i = 1, n, ∆f ≡ f(x, t, v2 + δv2) − f(x, t, v2). let ηh(x, t) = 1 h ∫t t−h η(x, τ)dτ, 0 < h < τ where η = δy(x, t) at (x, t) ∈ ωt1, zero at t > t1(t1 ≤ t − h) and ωt1 = d × (0, t1]. in identity (5) put η(x, t) instead of ηh(x, t), and following the method in [11,p. 166-168] we obtain 1 2 ∫ d (δy)2dx + ∫ ωt1 [ ∑n i=1 λi( ∂δy ∂xi )2 + ∑n i=1 ∂λi(y+θ1i,v0+δv0) ∂y ∂y ∂xi ∂δy ∂xi δy]dxdt + ∫ ωt1 ∑n i=1 ∆λi ∂y ∂xi ∂δ ∂xi dxdt + ∑n i=1 ∂bi(y+θ2i,v1+δv1) ∂y ∂y ∂xi (δy)2dxdt + ∫ ωt1 ∑n i=1 bi ∂δy ∂xi δy + ∫ ωt1 ∑n i=1 ∆bi( ∂y ∂xi )δydxdt − ∫ ωt1 ∆fδydxdt = 0 (8) hence,from the above assumptions and applying cauchy bunyakoviskii inequality, we obtain 1 2 ∫ d (δy(x, t1) 2dx + ν0 ∫ ωt1 ∑n i=1 | ∂δy ∂xi |2dxdt ≤ (c3 + c4)( ∫ ωt1 ∑n i=1 | ∂δy ∂xi |2dxdt) 1 2 ( ∫ ωt1 (δy(x, t))2dxdt) 1 2 +{ ∫ ωt1 ∑n i=1 |∆λi ∂y ∂xi |2dxdt} 1 2 ( ∫ ωt1 ∑n i=1 | ∂δy ∂xi |2dxdt) 1 2 + c5 ∫ ωt1 (δy(x, t))2dxdt − ∫t1 0 { ∫ d |∆f − ∑n i=1 ∆bi( ∂y ∂xi )| 1 2 dx( ∫ d (δy)2dx) 1 2 }dt, (9) where c3, c4, c5 are positive constants not depending on δv. applying cauchy’s inequality with ε and combine similar terms, then multiply both sides by two, we obtain ‖δy(x, t1)‖ 2 l2(d) + ν0 2 ‖ ∑n i=1 ∂δy ∂xi ‖2 l2(ωt1 ) ≤ c6‖δy(x, t)‖ 2 l2(ωt1 ) +2{ ∫ ωt1 ∑n i=1 |∆λi ∂y ∂xi |2dxdt} 1 2 ‖ ∑n i=1 ∂δy ∂xi ‖2 l2(ωt1 ) +2 max0≤τ≤t1 ‖δy(x, τ‖l2(d) ∫t1 0 { ∫ d |∆f − ∑n i=1 ∆bi( ∂y ∂xi )|2dx} 1 2 dt (10) now we replace y(t1) = max 0≤τ≤t1 ‖δy(x, τ‖l2(d), ‖δy(x, t)‖ 2 l2(ωt1 ) = t1(y(t1)) 2. 116 m. h. farag, t. a. talaat and e. m. kamal cubo 15, 2 (2013) this gives us the inequality ‖δy(x, t1)‖ 2 l2(d) + ν0 2 ‖ ∑n i=1 ∂δy ∂xi ‖2 l2(ωt1 ) ≤ c6t1(y(t1)) 2 +2{ ∫ ωt1 ∑n i=1 |∆λi ∂y ∂xi |2dxdt} 1 2 ‖ ∑n i=1 ∂δy ∂xi ‖2 l2(ωt1 ) +2y(t1) ∫t1 0 { ∫ d |∆f − ∑n i=1 ∆bi( ∂y ∂xi )|2dx} 1 2 dt ≡ j(t1). (11) from this follows the two inequalities (y(t1)) 2 ≤ j(t1) (12) and ‖ n∑ i=1 ∂δy ∂xi ‖2l2(ωt1 ) ≤ 2 ν0 j(t1) (13) we take the square root of both sides of (12) and (13), add together the resulting inequalities and then majorize the right-hand side in the same way in [12] (pp. 117-118) and this proves the estimate (6). this completes the proof of the theorm. corollary 3.1 under the above assumptions, the right part of estimate (6) converges to zero at ∑2 m=0 ‖δvm‖l2 → 0, therefore ‖δy‖v1,0 2 (ω) → 0 at 2∑ m=0 ‖δvm‖l2 → 0. (14) hence from the theorem on trace [13] we get ‖δy‖l2(ω) → 0, ‖δy‖l2(s) → 0 at 2∑ m=0 ‖δvm‖l2 → 0. (15) now we consider the functional j0(v) = ∫ s [y(ζ, t) − f0(ζ, t)] 2dζdt. theorem 3.2 the functional j0(v) is continuous on v. proof let δv = (δv0, δv1, δv2), δvm ∈ l2, m = 0, 2 be an increment of control on an element v ∈ v such that v + δv ∈ v. for the increment of j0(v) we have ∆j0(v) = j0(v + δv) − j0(v) = 2 ∫ s [y(ζ, t) − f0(ζ, t)]δy(ζ, t)dζdt + ∫ s [δy(ζ, t)] 2 dζdt (16) cubo 15, 2 (2013) existence and uniqueness solution of a class of quasilinear . . . 117 applying the cauchy-bunyakovskii inequality, we obtain |∆j0(v)| ≤ 2‖y(ζ, t) − f0(ζ, t)‖l2(s)‖δy(ζ, t)‖l2(s) + ‖δy(ζ, t)‖ 2 l2(s) (17) an application of the corollary 3.1 completes the proof. theorem 3.3 for any β ≥ 0 the problem (1)-(4) has a least one solution. proof the set of v is closed and bounded in l2. since j0(v) is continuous on v by theorem 3.2, so is jβ(v) = j0(v) + β 2∑ m=0 ‖vm − wm‖ 2 l2 . (18) then from the weierstrass theorem [14] it follows that the problem (1)-(4) has a least one solution. this completes the proof of the theorm. 4 the uniqueness theorem according to the above discussions, we ca easily obtain a theorem concerning solution uniqueness for the considering optimal control problem (1)-(4). theorem 4.1 there exists a dense set k of l2 such that for any ωm ∈ k, m = 0, 2 the problem (1)-(4) for β > 0 has a unique solution. proof the functional j0(v) is bounded below, and the foreging establishes that it is continues on v. furthermore, l2 is uniformaly convex [12]. it thus follows from a theorm in [16] that the space l2 contains an everywhere-dense subset k such that the problem (1)-(4) has a unque solution when ωm ∈ k, m = 0, 2 and β > 0. this completes the proof of the theorm. 5 conclusion we have investigated a constrained optimal control problems governed by quasilinear parabolic equations with controls in the coefficients of the equation. the existence and uniqueness of the optimal control problem is proved. 118 m. h. farag, t. a. talaat and e. m. kamal cubo 15, 2 (2013) 6 acknowledgment the authers gratefully acknowledgment the referee, who made useful suggestions and remarks which helped to improve the paper. received: september 2011. accepted: september 2012. references [1] li chun-fa, xue yang and en-min feng,optimal control problem governed by semilinear parabolic equation and its algorithm,acta mathematicae sinica,186,2008,29-40. 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[16] m. goebel,on existence of optimal control,math. nuchr.,93,1979,67–73. cubo a mathematical journal vol.15, no¯ 02, (01–19). june 2013 uniform convergence with rates of general singular operators george a. anastassiou and razvan a. mezei the university of memphis department of mathematical sciences, memphis, tn 38152, u.s.a. ganastss@memphis.edu, rmezei@memphis.edu abstract in this article we study the approximation properties of general singular integral operators over the real line. we establish their convergence to the unit operator with rates. the estimates are mostly sharp and they are pointwise or uniform. the established inequalities involve the higher order modulus of smoothness. we apply this theory to the trigonometric singular operators. resumen en este art́ıculo estudiamos propiedades de aproximación de operadores integrales singulares generales sobre la recta real. establecemos su convergencia al operador unidad con las tasas correspondientes. las estimaciones son mayormente ajustadas y son tanto puntuales como uniformes. las desigualdades encontradas involucran el módulo de suavidad de alto orden. aplicamos esta teoŕıa a los operadores singulares trigonométricos. keywords and phrases: best constant, general singular integral, trigonometric singular integral, modulus of smoothness, sharp inequality. 2010 ams mathematics subject classification: 26a15, 41a17, 41a35. 2 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) 1 introduction the rate of convergence of singular integrals has been studied earlier in [5], [12], [13], [15], [3], [6], [7], [8], [9] and these articles motivate this work. here we consider some very general singular integral operators over r and we study the degree of approximation to the unit operator with rates over smooth functions. we establish related inequalities involving the higher modulus of smoothness with respect to ‖ · ‖∞. the estimates are pointwise and uniform. most of the times these are optimal in the sense that the inequalities are attained by basic functions. we give particular applications of these operators to the trigonometric singular integral operators over r. the discussed operators are not in general positive. other motivation comes from [1], [2]. 2 main results in the next we study the following smooth general singular integral operators θr,ξ(f, x) defined as follows. let ξ > 0 and let µξ be borel probability measures on r. for r ∈ n and n ∈ z+ we put αj =    (−1)r−j ( r j ) j−n, j = 1, . . . , r, 1− r∑ j=1 (−1)r−j ( r j ) j−n, j = 0, (1) that is r∑ j=0 αj = 1. let f : r → r be borel measurable, we define for x ∈ r, the integral θr,ξ(f; x) := ∫∞ −∞   r∑ j=0 αjf(x + jt)  dµξ(t). (2) we suppose that θr,ξ(f; x) ∈ r for all x ∈ r. we will use also that θr,ξ(f; x) = r∑ j=0 αj (∫∞ −∞ f(x + jt)dµξ(t) ) . (3) we notice that θr,ξ(c, x) = c, c constant, and θr,ξ(f; x) − f(x) = r∑ j=0 αj (∫∞ −∞ f(x + jt) − f(x) ) dµξ(t). (4) let f ∈ cn (r) , n ∈ z+ with the rth modulus of smoothness finite, i.e. ωr(f (n), h) := sup |t|≤h ‖∆rtf (n)(x)‖∞,x < ∞, h > 0, (5) cubo 15, 2 (2013) uniform convergence with rates of general singular operators 3 where ∆rtf (n)(x) := r∑ j=0 (−1)r−j ( r j ) f(n)(x + jt), (6) see [10], p. 44. we need to introduce δk := r∑ j=1 αjj k, k = 1, . . . , n ∈ n, (7) and the even function gn(t) := ∫ |t| 0 (|t| − w)n−1 (n − 1)! ωr(f (n), w)dw, n ∈ n (8) with g0(t) := ωr(f, |t|), t ∈ r. (9) denote by ⌊·⌋ the integral part. we present our first result theorem 1. the integrals ck,ξ := ∫∞ −∞ tkdµξ(t), k = 1, . . . , n, are assumed to be finite. then ∣ ∣ ∣ ∣ ∣ θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ ∣ ∣ ∣ ∣ ∣ ≤ ∫∞ −∞ gn(t)dµξ(t). (10) proof. by taylor’s formula we obtain f(x + jt) = n−1∑ k=0 f(k)(x) k! (jt)k + ∫jt 0 (jt − z)n−1 (n − 1)! f(n)(x + z)dz = n−1∑ k=0 f(k)(x) k! (jt)k + jn ∫t 0 (t − w)n−1 (n − 1)! f(n)(x + jw)dw. (11) multiplying both sides of (11) by αj and summing up we get r∑ j=0 αj(f(x + jt) − f(x)) = n∑ k=1 f(k)(x) k! δkt k + rn(0, t), (12) where rn(0, t) := ∫t 0 (t − w)n−1 (n − 1)! τ(w)dw, (13) with τ(w) := r∑ j=0 αjj nf(n)(x + jw) − δnf (n)(x). 4 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) notice also that − r∑ j=1 (−1)r−j ( r j ) = (−1)r ( r 0 ) . (14) according to [3], p. 306, [1], we get τ(w) = ∆rwf (n)(x). (15) therefore |τ(w)| ≤ ωr(f (n), |w|), (16) all w ∈ r independently of x. we do have after integration, see also (4), that θr,ξ(f; x) − f(x) = ∫∞ −∞   r∑ j=0 αj(f(x + jt) − f(x))  dµξ(t) = ∫∞ −∞ ( n∑ k=1 f(k)(x) k! δkt k + rn(0, t) ) dµξ(t) = n∑ k=1 f(k)(x) k! δk (∫∞ −∞ tkdµξ(t) ) + r∗n, (17) where r∗n := ∫∞ −∞ rn(0, t)dµξ(t). (18) here by (8) and (13) we get |rn(0, t)| ≤ ∫ |t| 0 (|t| − w)n−1 (n − 1)! |τ (sign(t)w) |dw ≤ gn(t), (19) see [5]. hence by (18) we find |r∗n| ≤ ∫∞ −∞ gn(t)dµξ(t). (20) we also have θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ = r ∗ n. (21) inequality (10) is now clear via (21) and (20). � corollary 2. assume ωr(f, ξ) < ∞, ξ > 0. then it holds for n = 0 that |θr,ξ(f; x) − f(x)| ≤ ∫∞ −∞ ωr (f, |t|) dµξ (t) . (22) cubo 15, 2 (2013) uniform convergence with rates of general singular operators 5 proof. we observe that θr,ξ(f; x) − f(x) = ∫∞ −∞   r∑ j=1 αj(f(x + jt) − f(x))  dµξ (t) = ∫∞ −∞   r∑ j=1 (−1)r−j ( r j ) (f(x + jt) − f(x))  dµξ (t) = ∫∞ −∞   r∑ j=1 (−1)r−j ( r j ) f(x + jt) −   r∑ j=1 (−1)r−j ( r j )  f(x)  dµξ (t) (14) = ∫∞ −∞   r∑ j=1 (−1)r−j ( r j ) f(x + jt) + (−1)r ( r 0 ) f(x) ) dµξ (t) = ∫∞ −∞   r∑ j=0 (−1)r−j ( r j ) f(x + jt)  dµξ (t) (6) = ∫∞ −∞ ((∆rtf)(x)) dµξ (t) . i.e. we have proved θr,ξ(f; x) − f(x) = ∫∞ −∞ (∆rtf(x)) dµξ (t) . (23) hence by (23) we derive |θr,ξ(f; x) − f(x)| ≤ ∫∞ −∞ |∆rtf(x)|dµξ (t) ≤ ∫∞ −∞ ωr(f, |t|)dµξ (t) . that is proving (22). � inequality (10) is sharp. theorem 3. inequality ( 10) at x = 0 is attained by f(x) = xr+n, r, n ∈ n with r + n even. proof. as in [3], p. 307, [1], [16], p. 54 and (5), (6) we obtain ωr(f (n), t) = (r + n)(r + n − 1) · . . . · (r + 1)r!tr, 6 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) t > 0. and gn(t) = r!t r+n, t ∈ r. also we have f(k)(0) = 0, k = 0, 1, . . . , n. thus the right hand side of (10) equals r! ∫∞ −∞ tr+ndµξ (t) . (24) the left hand side of (10) equals |θr,ξ(f; 0)| = ∣ ∣ ∣ ∣ ∣ ∣ ∫∞ −∞   r∑ j=0 αjf(jt)  dµξ (t) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ ∫∞ −∞   r∑ j=1 αj(jt) r+n  dµξ (t) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ ∫∞ −∞   r∑ j=1 (−1)r−j ( r j ) j−n(jt)r+n  dµξ (t) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣   r∑ j=0 (−1)r−j ( r j ) jr   (∫∞ −∞ tr+ndµξ (t) ) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ (∆r1x r)(0) ∫∞ −∞ tr+ndµξ (t) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ r! ∫∞ −∞ tr+ndµξ (t) ∣ ∣ ∣ ∣ = r! ∫∞ −∞ tr+ndµξ (t) . i.e. we have established |θr,ξ(f; 0)| = r! ∫∞ −∞ tr+ndµξ (t) . (25) thus by (24) and (25) we have established the claim of the theorem. � corollary 4. inequality ( 22) is sharp, that is attained at x = 0 by f(x) = xr, r even. proof. notice that ∆rtx r = r!tr and ωr(f (n), t) = r!tr, t > 0. thus r.h.s.(22) = r! ∫∞ −∞ trdµξ (t) . cubo 15, 2 (2013) uniform convergence with rates of general singular operators 7 also f(0) = 0. therefore l.h.s.(22) = |θr,ξ(f; 0)| = ∣ ∣ ∣ ∣ ∣ ∣ ∫∞ −∞   r∑ j=1 αjj rtr  dµξ (t) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ ∫∞ −∞   r∑ j=0 (−1)r−j ( r j ) jr  trdµξ (t) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ (∆r1x r)(0) ∫∞ −∞ trdµξ (t) ∣ ∣ ∣ ∣ = r! ∫∞ −∞ trdµξ (t) , proving the claim. � remark 5. on inequalities (10) and (22). we have the uniform estimates ∥ ∥ ∥ ∥ ∥ θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ ∫∞ −∞ gn(t)dµξ(t), n ∈ n, (26) and ‖θr,ξ(f; x) − f(x)‖∞,x ≤ ∫∞ −∞ ωr(f, |t|)dµξ(t), n = 0. (27) remark 6. the following regards the convergence of operators θr,ξ. from (8) we have gn(t) ≤ |t|n n! ωr(f (n), |t|). (28) therefore by (26) we get ∥ ∥ ∥ ∥ ∥ θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ 1 n! ∫∞ −∞ |t|nωr(f (n), |t|)dµξ(t). (29) next using ωr(f, λt) ≤ (λ + 1) r ωr(f, t), λ, t > 0, we get 1 n! ∫∞ −∞ |t|nωr(f (n), |t|)dµξ(t) = 1 n! ∫∞ −∞ |t|nωr ( f(n), ξ · |t| ξ ) dµξ(t) ≤ ωr(f (n), ξ) n! ∫∞ −∞ |t|n ( 1 + |t| ξ )r dµξ(t). so we have proved that k1 : = ∥ ∥ ∥ ∥ ∥ θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ ωr(f (n), ξ) n! ∫∞ −∞ |t|n ( 1 + |t| ξ )r dµξ(t). (30) 8 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) similarly we get ∫∞ −∞ ωr(f, |t|)dµξ(t) = ∫∞ −∞ ωr ( f, ξ · |t| ξ ) dµξ(t) ≤ ωr(f, ξ) ∫∞ −∞ ( 1 + |t| ξ )r dµξ(t). so that we have k2 : = ‖θr,ξ(f; x) − f(x)‖∞ ≤ ωr(f, ξ) ∫∞ −∞ ( 1 + |t| ξ )r dµξ(t). (31) in case that ∫∞ −∞ |t|n ( 1 + |t| ξ )r dµξ(t) ≤ λ1, ∀ξ > 0, we get that k1 → 0, as ξ → 0. in case that ∫∞ −∞ ( 1 + |t| ξ )r dµξ(t) ≤ λ2, ∀ξ > 0, we get that k2 → 0, that is θr,ξ → i (unit operator) uniformly, as ξ → 0. note 7. the operators θr,ξ are not in general positive and they are of convolution type. let r = 2, n = 3. then α0 = 23 8 , α1 = −2, α2 = 1 8 . consider f(t) = t2 ≥ 0 and x = 0. then θ2,ξ(t 2; 0) = −1.5 ∫∞ −∞ t2dµξ(t) < 0, where we assumed that ∫∞ −∞ t2dµξ(t) < ∞. remark 8. from [5] we get that gn(t) ≤ ωr(f (n), ξ) ξr { n−1∑ k=0 (−1)k k!(n − k − 1)!(k + r + 1) · [ (ξ + |t|) n+r − ξr+k+1 (ξ + |t|) n−k−1 ]} , (32) for ξ > 0, ∀t ∈ r. so by (26) we obtain ∥ ∥ ∥ ∥ ∥ θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ ωr(f (n), ξ) ξr { n−1∑ k=0 (−1)k k!(n − k − 1)!(k + r + 1) · [∫∞ −∞ (ξ + |t|) n+r dµξ(t) − ξ r+k+1 ∫∞ −∞ (ξ + |t|) n−k−1 dµξ(t) ]} . (33) cubo 15, 2 (2013) uniform convergence with rates of general singular operators 9 so from remarks 5, 6 we derive theorem 9. let f ∈ cn(r), n ∈ z+. set ck,ξ := ∫∞ −∞ tkdµξ(t), k = 1, . . . , n. assume also ωr(f (n), h) < ∞, ∀h > 0. it is also supposed that ∫∞ −∞ |t|n ( 1 + |t| ξ )r dµξ(t) < ∞. then ∥ ∥ ∥ ∥ ∥ θr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δkck,ξ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ ωr(f (n), ξ) n! ∫∞ −∞ |t|n ( 1 + |t| ξ )r dµξ(t). (34) when n = 0 the sum in l.h.s.(34) collapses. 3 applications to general trigonometric singular operators we make remark 10. we need the following preliminary result. let p and m be integers with 1 ≤ p ≤ m. we define the integral i(m; p) := ∫∞ −∞ (sin x)2m x2p dx = 2 ∫∞ 0 (sin x)2m x2p dx. (35) this is an (absolutely) convergent integral. according to [11], page 210, item 1033, we obtain i(m; p) = π (−1)p(2m)! 4m−p(2p − 1)! m∑ k=1 (−1)k k2p−1 (m − k)!(m + k)! . (36) in particular, for p = m the above formula becomes ∫∞ 0 (sin x)2m x2m dx = π(−1)mm m∑ k=1 (−1)k k2m−1 (m − k)!(m + k)! . (37) in this section we apply the general theory of this article to the trigonometric smooth general singular integral operators tr,ξ(f, x) defined as follows. let ξ > 0. 10 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) let f : r → r be borel measurable, we define for x ∈ r and β ∈ n, the integral tr,ξ(f; x) := 1 w ∫∞ −∞   r∑ j=0 αjf(x + jt)   ( sin (t/ξ) t )2β dt, (38) where w = ∫∞ −∞ ( sin (t/ξ) t )2β dt = 2ξ1−2β ∫∞ 0 ( sin t t )2β dt (37) = 2ξ1−2βπ(−1)ββ β∑ k=1 (−1)k k2β−1 (β − k)!(β + k)! . (39) we suppose that tr,ξ(f; x) ∈ r for all x ∈ r. we present our first result of this section theorem 11. let 1 ≤ n ≤ 2β − 2 and k = 1, . . . , n. the integrals ck,ξ : = 1 w ∫∞ −∞ tk ( sin (t/ξ) t )2β dt =    0, for k odd ξ k (−1) k 2 (2β−1)! 2k(2β−k−1)! ∑β j=1 (−1) j j 2β−k−1 (β−j)!(β+j)! ∑β j=1 (−1)j j2β−1 (β−j)!(β+j)! , for k even , are finite. moreover, it holds ∣ ∣ ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − ⌊n/2⌋∑ k=1 f(2k)(x) (2k)! δ2kc2k,ξ ∣ ∣ ∣ ∣ ∣ ∣ ≤ 1 w ∫∞ −∞ gn(t) ( sin (t/ξ) t )2β dt. (40) when n = 1 the sum in the l.h.s.(40) colapses. proof. we used theorem 1 and relations (36) and (39). � corollary 12. assume ωr(f, ξ) < ∞, ξ > 0. then it holds for n = 0 and β > 1 that |tr,ξ(f; x) − f(x)| ≤ 2 w ∫∞ 0 ωr (f, t) ( sin (t/ξ) t )2β dt. (41) proof. we are applying corollary 2 here. � note 13. the operators tr,ξ are not in general positive and they are of convolution type. cubo 15, 2 (2013) uniform convergence with rates of general singular operators 11 let r = 2, n = 3, and β ≥ 2. then α0 = 23 8 , α1 = −2, α2 = 1 8 . consider f(t) = t2 ≥ 0 and x = 0. then t2,ξ(t 2; 0) = −1.5 1 w ∫∞ −∞ t2 ( sin (t/ξ) t )2β dt < 0, since 1 w ∫∞ −∞ t2 ( sin (t/ξ) t )2β dt = ξ2(−1)(2β − 1) (β − 1) 2 ∑β j=1(−1) j j 2β−3 (β−j)!(β+j)! ∑β j=1(−1) j j 2β−1 (β−j)!(β+j)! < ∞, by theorem 11. � theorem 14. let f ∈ cn(r), n ∈ z+, and β ≥ 1+ ⌊ n+r+1 2 ⌋ . assume also ωr(f (n), h) < ∞, ∀h > 0. then ∥ ∥ ∥ ∥ ∥ ∥ tr,ξ(f; x) − f(x) − ⌊n/2⌋∑ k=1 f(2k)(x) (2k) ! δ2kc2k,ξ ∥ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ ξn n! ωr(f (n), ξ) π(−1)ββ [∑β k=1(−1) k k 2β−1 (β−k)!(β+k)! ] · [∫∞ 0 tn (1 + t) r ( sin t t )2β dt ] . (42) when n = 0, 1 the sum in l.h.s.(42) collapses. proof. for i = 0, ..., r we have that ∫∞ 0 tn+i ( sin t t )2β dt = π(−1)β− n+i 2 (2β)! 2n+i+1 [2β − n − i − 1] ! β∑ k=1 (−1)k k2β−n−i−1 (β − k)!(β + k)! < ∞ in the case n + i even (because β ≥ 1 + ⌊ n+r+1 2 ⌋ ≥ 1 + n+i 2 ). 12 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) furthermore, when n + i is odd we get ∫∞ 0 tn+i ( sin t t )2β dt = ∫1 0 tn+i ( sin t t )2β dt + ∫∞ 1 tn+i ( sin t t )2β dt ≤ ∫1 0 tn+i−1 ( sin t t )2β dt + ∫∞ 1 tn+i+1 ( sin t t )2β dt ≤ ∫∞ 0 tn+i−1 ( sin t t )2β dt + ∫∞ 0 tn+i+1 ( sin t t )2β dt = π(−1)β− n+i−1 2 (2β)! 2n+i [2β − n − i] ! β∑ k=1 (−1)k k2β−n−i (β − k)!(β + k)! + π(−1)β− n+i+1 2 (2β)! 2n+i+2 [2β − n − i − 2] ! β∑ k=1 (−1)k k2β−n−i−2 (β − k)!(β + k)! < ∞, by β ≥ 1 + ⌊ n+r+1 2 ⌋ ≥ 1 + n+i+1 2 . therefore it holds ∫∞ 0 r∑ i=0 ( r i ) tn+i ( sin t t )2β dt = r∑ i=0 ( r i )∫∞ 0 tn+i ( sin t t )2β dt < ∞. hence 1 w ∫∞ −∞ |t|n ( 1 + |t| ξ )r ( sin (t/ξ) t )2β dt = 2 w ∫∞ 0 tn ( 1 + t ξ )r ( sin (t/ξ) t )2β dt = 2ξn−2β+1 w ∫∞ 0 tn (1 + t) r ( sin t t )2β dt (39) = ξn πβ [ (−1)β ∑β k=1(−1) k k 2β−1 (β−k)!(β+k)! ] ∫∞ 0 tn (1 + t) r ( sin t t )2β dt = ξn πβ [ (−1)β ∑β k=1(−1) k k 2β−1 (β−k)!(β+k)! ] ∫∞ 0 r∑ i=0 ( r i ) tn+i ( sin t t )2β dt < ∞, which implies that 1 w ∫∞ −∞ |t|n ( 1 + |t| ξ )r ( sin (t/ξ) t )2β dt < ∞. applying theorem 9 here we obtain the inequality (42). � cubo 15, 2 (2013) uniform convergence with rates of general singular operators 13 4 applications to particular trigonometric singular operators in this section we work on the approximation results given in the previous section, for some particular values of n and β. case β = 1. we have the following value corresponding to formula (39) w = πξ−1. none of the previous results hold in this case. case β = 2. we have the following value corresponding to formula (39) w = 2π 3 ξ−3. theorem 15. it holds, for n = 1 |tr,ξ(f; x) − f(x)| ≤ 3 2π ξ3 ∫∞ −∞ g1(t) ( sin (t/ξ) t )4 dt, (43) and, for n = 2 ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 3 2π ξ3 ∫∞ −∞ g2(t) ( sin (t/ξ) t )4 dt. (44) proof. by theorem 11, with β = 2, n = 1, 2. � corollary 16. assume ωr(f, ξ) < ∞, ξ > 0. then it holds for n = 0 that |tr,ξ(f; x) − f(x)| ≤ 3 π ξ3 ∫∞ 0 ωr (f, t) ( sin (t/ξ) t )4 dt. (45) proof. by corollary 12. � theorem 17. let f ∈ c1(r). assume ω1(f ′, h) < ∞, ∀h > 0. then ‖t1,ξ(f) − f‖∞ ≤ 3ξ π [ ln 2 + π 4 ] ω1(f ′, ξ). (46) 14 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) proof. we are applying theorem 14 here for n = r = 1, etc. � case β = 3. we have the following value corresponding to formula (39) w = 11π 20 ξ−5. we have theorem 18. it holds |tr,ξ(f; x) − f(x)| ≤ 20 11π ξ5 ∫∞ −∞ g1(t) ( sin (t/ξ) t )6 dt. (47) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 20 11π ξ5 ∫∞ −∞ g2(t) ( sin (t/ξ) t )6 dt. (48) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 20 11π ξ5 ∫∞ −∞ g3(t) ( sin (t/ξ) t )6 dt. (49) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ − f(4)(x) 24 δ4c4,ξ ∣ ∣ ∣ ∣ ≤ 20 11π ξ5 ∫∞ −∞ g4(t) ( sin (t/ξ) t )6 dt. (50) proof. by theorem 11, for β = 3, and n = 1, 2, 3, 4. � corollary 19. assume ωr(f, ξ) < ∞, ξ > 0. then it holds for n = 0 that |tr,ξ(f; x) − f(x)| ≤ 40 11π ξ5 ∫∞ 0 ωr (f, t) ( sin (t/ξ) t )6 dt. (51) proof. we are applying corollary 12 here. � theorem 20. let f ∈ cn(r), n ∈ z+, and n+r = 1, 2, 3, 4. assume also ωr(f (n), h) < ∞, ∀h > 0. then ∥ ∥ ∥ ∥ ∥ ∥ tr,ξ(f; x) − f(x) − ⌊n/2⌋∑ k=1 f(2k)(x) (2k) ! δ2kc2k,ξ ∥ ∥ ∥ ∥ ∥ ∥ ∞,x (52) ≤ 40 11π [∫∞ 0 tn (1 + t) r ( sin t t )6 dt ] ωr(f (n), ξ)ξn n! . when n = 0, 1 the sum in l.h.s.(52) collapses. cubo 15, 2 (2013) uniform convergence with rates of general singular operators 15 proof. we are applying theorem 14 here. � case β = 4. we have the following value corresponding to formula (39) w = 151π 315 ξ−7. theorem 21. it holds |tr,ξ(f; x) − f(x)| ≤ 315 151π ξ7 ∫∞ −∞ g1(t) ( sin (t/ξ) t )8 dt. (53) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 315 151π ξ7 ∫∞ −∞ g2(t) ( sin (t/ξ) t )8 dt. (54) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 315 151π ξ7 ∫∞ −∞ g3(t) ( sin (t/ξ) t )8 dt. (55) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ − f(4)(x) 24 δ4c4,ξ − f(6)(x) 720 δ6c6,ξ ∣ ∣ ∣ ∣ (56) ≤ 315 151π ξ7 ∫∞ −∞ g6(t) ( sin (t/ξ) t )8 dt. proof. we used theorem 11, with β = 4, and n = 1, 2, 3, 6. � corollary 22. assume ωr(f, ξ) < ∞, ξ > 0. then for n = 0 it holds |tr,ξ(f; x) − f(x)| ≤ 630 151π ξ7 ∫∞ 0 ωr (f, t) ( sin (t/ξ) t )8 dt. (57) proof. we are applying corollary 12 here. � theorem 23. let f ∈ cn(r), n ∈ z+, and n + r = 1, 2, 3, 4, 5, 6. assume also ωr(f (n), h) < ∞, ∀h > 0. then ∥ ∥ ∥ ∥ ∥ ∥ tr,ξ(f; x) − f(x) − ⌊n/2⌋∑ k=1 f(2k)(x) (2k) ! δ2kc2k,ξ ∥ ∥ ∥ ∥ ∥ ∥ ∞,x (58) ≤ 630 151π [∫∞ 0 tn (1 + t) r ( sin t t )8 dt ] ωr(f (n), ξ)ξn n! . 16 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) when n = 0, 1 the sum in l.h.s.(58) collapses. proof. we are applying theorem 14 here. � case β = 6. we have the following value corresponding to formula (39) w = ∫∞ −∞ ( sin (t/ξ) t )12 dt = 655177π 1663200 ξ−11. theorem 24. it holds |tr,ξ(f; x) − f(x)| ≤ 1663200 655177π ξ11 ∫∞ −∞ g1(t) ( sin (t/ξ) t )12 dt. (59) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 1663200 655177π ξ11 ∫∞ −∞ g2(t) ( sin (t/ξ) t )12 dt. (60) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 1663200 655177π ξ11 ∫∞ −∞ g3(t) ( sin (t/ξ) t )12 dt. (61) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ − f(4)(x) 24 δ4c4,ξ − f(6)(x) 720 δ6c6,ξ ∣ ∣ ∣ ∣ (62) ≤ 1663200 655177π ξ11 ∫∞ −∞ g6(t) ( sin (t/ξ) t )12 dt. ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ − f(4)(x) 24 δ4c4,ξ − f(6)(x) 720 δ6c6,ξ − f(8)(x) 8! δ8c8,ξ − f(10)(x) 10! δ10c10,ξ ∣ ∣ ∣ ∣ (63) ≤ 1663200 655177π ξ11 ∫∞ −∞ g10(t) ( sin (t/ξ) t )12 dt. (64) proof. we used theorem 11, with β = 6, and n = 1, 2, 3, 6, 10. � corollary 25. assume ωr(f, ξ) < ∞, ξ > 0. then for n = 0 it holds |tr,ξ(f; x) − f(x)| ≤ 3326400 655177π ξ11 ∫∞ 0 ωr (f, t) ( sin (t/ξ) t )12 dt. (65) proof. we are applying corollary 12 here. � cubo 15, 2 (2013) uniform convergence with rates of general singular operators 17 theorem 26. let f ∈ cn(r), n ∈ z+, and n + r ∈ {1, 2, 3, . . . , 10} . assume also ωr(f (n), h) < ∞, ∀h > 0. then ∥ ∥ ∥ ∥ ∥ ∥ tr,ξ(f; x) − f(x) − ⌊n/2⌋∑ k=1 f(2k)(x) (2k) ! δ2kc2k,ξ ∥ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ 3326400 655177π (66) · [∫∞ 0 tn (1 + t) r ( sin t t )12 dt ] ωr(f (n), ξ)ξn n! . when n = 0, 1 the sum in l.h.s.(46) collapses. proof. we are applying theorem 14 here. � case β = 10. we have the following value corresponding to formula (39) w = ∫∞ −∞ ( sin (t/ξ) t )20 dt = 37307713155613π 121645100408832 ξ−19. theorem 27. it holds |tr,ξ(f; x) − f(x)| ≤ 121645100408832 37307713155613π ξ19 (67) · ∫∞ −∞ g1(t) ( sin (t/ξ) t )20 dt. ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 121645100408832 37307713155613π ξ19 (68) · ∫∞ −∞ g2(t) ( sin (t/ξ) t )20 dt. ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ ∣ ∣ ∣ ∣ ≤ 121645100408832 37307713155613π ξ19 (69) · ∫∞ −∞ g3(t) ( sin (t/ξ) t )20 dt. ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ − f(4)(x) 24 δ4c4,ξ − f(6)(x) 720 δ6c6,ξ ∣ ∣ ∣ ∣ (70) ≤ 121645100408832 37307713155613π ξ19 ∫∞ −∞ g6(t) ( sin (t/ξ) t )20 dt. 18 george a. anastassiou and razvan a. mezei cubo 15, 2 (2013) ∣ ∣ ∣ ∣ tr,ξ(f; x) − f(x) − f′′(x) 2 δ2c2,ξ − f(4)(x) 24 δ4c4,ξ − f(6)(x) 720 δ6c6,ξ − f(8)(x) 8! δ8c8,ξ − f(10)(x) 10! δ10c10,ξ ∣ ∣ ∣ ∣ ≤ 121645100408832 37307713155613π ξ19 ∫∞ −∞ g10(t) ( sin (t/ξ) t )20 dt. (71) proof. we used theorem 11, with β = 10, and n = 1, 2, 3, 6, 10. � corollary 28. assume ωr(f, ξ) < ∞, ξ > 0. then for n = 0 it holds |tr,ξ(f; x) − f(x)| ≤ 243290200817664 37307713155613π ξ19 ∫∞ 0 ωr (f, t) ( sin (t/ξ) t )20 dt. (72) proof. we are applying corollary 12 here. � theorem 29. let f ∈ cn(r), n ∈ z+, and n + r = {1, 2, . . . , 18} . assume also ωr(f (n), h) < ∞, ∀h > 0. then ∥ ∥ ∥ ∥ ∥ ∥ tr,ξ(f; x) − f(x) − ⌊n/2⌋∑ k=1 f(2k)(x) (2k) ! δ2kc2k,ξ ∥ ∥ ∥ ∥ ∥ ∥ ∞,x ≤ 243290200817664 37307713155613π (73) · [∫∞ 0 tn (1 + t) r ( sin t t )20 dt ] ωr(f (n), ξ)ξn n! . when n = 0, 1 the sum in l.h.s.(73) collapses. proof. we are applying theorem 14 here. � acknowledgement. the authors would like to thank professor v. papanicolaou of national technical university of athens, greece, for having fruitful discussions during the preparation of this article. received: february 2011. accepted: april 2011. references [1] g.a. anastassiou, rate of convergence of non-positive linear convolution type operators. a sharp inequality, j. math. anal. and appl., 142 (1989), 441–451. 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() cubo a mathematical journal vol.17, no¯ 02, (31–48). june 2015 refinements of the generalized trapezoid inequality in terms of the cumulative variation and applications s.s. dragomir 1,2 1mathematics, college of engineering & science victoria university, po box 14428, melbourne city, mc 8001, australia. 2school of computational & applied mathematics, university of the witwatersrand, private bag 3, johannesburg 2050, south africa, sever.dragomir@vu.edu.au abstract refinements of the generalized trapezoid inequality for functions of bounded variation in terms of the cumulative variation function are given. applications for selfadjoint operators on complex hilbert spaces are also provided. resumen damos refinamientos de la desigualdad trapezoidal generalizada para funciones de variación acotada en términos de la función de variación acumulativa. también probamos aplicaciones a operadores autoadjuntos en espacios de hilbert complejos. keywords and phrases: generalized trapezoid inequality, functions of bounded variation, cumulative variation, selfadjoint operators 2010 ams mathematics subject classification: 26d15, 47a63. 32 s.s. dragomir cubo 17, 2 (2015) 1 introduction the following generalized trapezoidal inequality was obtained in 1999 by the author [11, proposition 1] ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − (x − a) f (a) − (b − x) f (b) ∣ ∣ ∣ ∣ ∣ ≤ [ 1 2 (b − a) + ∣ ∣ ∣ ∣ x − a + b 2 ∣ ∣ ∣ ∣ ] b ∨ a (f) , (1.1) where x ∈ [a, b] . the constant 1 2 cannot be replaced by a smaller quantity. see also [9] for a different proof and other details. the best inequality one can derive from (1.1) is the trapezoid inequality ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − f (a) + f (b) 2 (b − a) ∣ ∣ ∣ ∣ ∣ ≤ 1 2 (b − a) b ∨ a (f) . (1.2) here the constant 1 2 is also best possible. for related results, see [1]-[4], [6]-[8], [10], [14], [15], [17], [18], [20], [22]-[27] and [29]-[32]. the main aim of the present paper is to provide some refinements of the inequalities (1.1) and (1.2) in terms of the cumulative variation function. applications for selfadjoint operators on complex hilbert spaces are also given. 2 refinements of the generalized trapezoid inequality for a function of bounded variation v : [a, b] → c we define the cumulative variation function (cvf) v : [a, b] → [0, ∞) by v (t) := t ∨ a (v) the total variation of v on the interval [a, t] with t ∈ [a, b] . it is know that the cvf is monotonic nondecreasing on [a, b] and is continuous in a point c ∈ [a, b] if and only if the generating function v is continuing in that point. if v is lipschitzian with the constant l > 0, i.e. |v (t) − v (s)| ≤ l |t − s| for any t, s ∈ [a, b] then v is also lipschitzian with the same constant. the following lemma is of interest in itself as well, see also [16]. for the sake of completeness, we give here a simple proof. cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 33 lemma 2.1. let f, u : [a, b] → c. if f is continuous on [a, b] and u is of bounded variation on [a, b] , then ∣ ∣ ∣ ∣ ∣ ∫b a f (t) du (t) ∣ ∣ ∣ ∣ ∣ ≤ ∫b a |f (t)| d ( t ∨ a (u) ) ≤ max t∈[a,b] |f (t)| b ∨ a (u) . (2.1) proof. since the riemann-stieltjes integral ∫b a f (t) du (t) exists, then for any division in : a = t0 < t1 < · · · < tn−1 < tn = b with the norm v (in) := max i∈{0,...,n−1} (ti+1 − ti) → 0 and for any intermediate points ξi ∈ [ti, ti+1], i ∈ {0, . . . , n − 1} we have: ∣ ∣ ∣ ∣ ∣ ∫b a f (t) du (t) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ lim v(in)→0 n−1∑ i=0 f (ξi) [u (ti+1) − u (ti)] ∣ ∣ ∣ ∣ ∣ ≤ lim v(in)→0 n−1∑ i=0 |f (ξi)| |u (ti+1) − u (ti)| . (2.2) however, |u (ti+1) − u (ti)| ≤ ti+1 ∨ ti (u) = ti+1 ∨ a (u) − ti ∨ a (u) , for any i ∈ {0, . . . , n − 1} , and by (2.2) we have ∣ ∣ ∣ ∣ ∣ ∫b a f (t) du (t) ∣ ∣ ∣ ∣ ∣ ≤ lim v(in)→0 n−1∑ i=0 |f (ξi)| [ ti+1 ∨ a (u) − ti ∨ a (u) ] = ∫b a |f (t)| d ( t ∨ a (u) ) , and the last riemann-stieltjes integral exists since |f| is continuous and ∨ · a is monotonic nondecreasing on [a, b] . the last part follows from : ∣ ∣ ∣ ∣ ∣ ∫b a g (t) dv (t) ∣ ∣ ∣ ∣ ∣ ≤ max t∈[a,b] |g (t)| [v (b) − v (a)] , holding for any function g continuous on [a, b] and v monotonic nondecreasing on [a, b] . the details are omitted. the following result may be stated. theorem 2.2. let f : [a, b] → c be a function of bounded variation on [a, b] . then 34 s.s. dragomir cubo 17, 2 (2015) ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − [f (a) (x − a) + f (b) (b − x)] ∣ ∣ ∣ ∣ ∣ ≤ ∫x a ( t ∨ a (f) ) dt + ∫b x ( b ∨ t (f) ) dt ≤ (x − a) x ∨ a (f) + (b − x) b ∨ x (f) ≤    [ 1 2 (b − a) + ∣ ∣x − a+b 2 ∣ ∣ ] ∨b a (f) , [ 1 2 ∨b a (f) + 1 2 ∣ ∣ ∣ ∨x a (f) − ∨b x (f) ∣ ∣ ∣ ] (b − a) , (2.3) for any x ∈ [a, b]. proof. we use the identity obtained in [9] f (a) (x − a) + f (b) (b − x) − ∫b a f (t) dt = ∫b a (t − x) df (t) , (2.4) which holds for any riemann integrable function f : [a, b] → r. this can be easily proved integrating by parts in the second integral. now, if f is of bounded variation on [a, b] , then on applying the first inequality in (2.1) we deduce that: ∣ ∣ ∣ ∣ ∣ ∫b a (t − x) df (t) ∣ ∣ ∣ ∣ ∣ ≤ ∫b a |t − x| d ( t ∨ a (f) ) = ∫x a (x − t) d ( t ∨ a (f) ) + ∫b x (t − x) d ( t ∨ a (f) ) (2.5) for any x ∈ [a, b] . integrating by parts in the riemann-stieltjes integral we have ∫x a (x − t) d ( t ∨ a (f) ) = (x − t) ( t ∨ a (f) ) ∣ ∣ ∣ ∣ ∣ x a + ∫x a ( t ∨ a (f) ) dt = ∫x a ( t ∨ a (f) ) dt (2.6) cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 35 and ∫b x (t − x) d ( t ∨ a (f) ) = (t − x) ( t ∨ a (f) ) ∣ ∣ ∣ ∣ ∣ b x − ∫b x ( t ∨ a (f) ) dt = (b − x) ( b ∨ a (f) ) − ∫b x ( t ∨ a (f) ) dt = ∫b x ( b ∨ a (f) − t ∨ a (f) ) dt = ∫b x ( b ∨ t (f) ) dt (2.7) for any x ∈ [a, b] . making use of (2.5)-(2.7) we get the first inequality in (2.3). since ∨ · a is monotonic nondecreasing on [a, b] while ∨b · is nonincreasing in the same interval, we have ∫x a ( t ∨ a (f) ) dt ≤ (x − a) x ∨ a (f) and ∫b x ( b ∨ t (f) ) dt ≤ (b − x) b ∨ x (f) , for any x ∈ [a, b], which gives the second inequality in (2.3). using the properties of the maximum, we have (x − a) x ∨ a (f) + (b − x) b ∨ x (f) ≤    max {x − a, b − x} ∨b a (f) max { ∨x a (f) , ∨b x (f) } (b − a) =    [ 1 2 (b − a) + ∣ ∣x − a+b 2 ∣ ∣ ] ∨b a (f) [ 1 2 ∨b a (f) + 1 2 ∣ ∣ ∣ ∨x a (f) − ∨b x (f) ∣ ∣ ∣ ] (b − a) for any x ∈ [a, b], and the proof is complete. an important particular case is where x = a+b 2 , giving: corollary 2.3. let f : [a, b] → c be a function of bounded variation on [a, b] . then ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − f (a) + f (b) 2 (b − a) ∣ ∣ ∣ ∣ ∣ ≤ ∫ a+b 2 a ( t ∨ a (f) ) dt + ∫b a+b 2 ( b ∨ t (f) ) dt ≤ 1 2 (b − a) b ∨ a (f) . (2.8) the first inequality in (2.8) is sharp. the constant 1 2 in the second inequality is best possible. 36 s.s. dragomir cubo 17, 2 (2015) proof. we must prove only the sharpness of the first inequality in (2.8) and the best constant. consider the function f : [a, b] → r given by f (t) :=    1, t = a 0, t ∈ (a, b) 1, t = b. we observe that f is of bounded variation and the cvf is given by t ∨ a (f) =    0, t = a 1, t ∈ (a, b) 2, t = b. if we replace this function in (2.8) and perform the calculation, we get the same quantity b − a in all three terms. corollary 2.4. let f : [a, b] → c be a function of bounded variation on [a, b] . if p ∈ (a, b) is a median point in variation, namely ∨p a (f) = ∨b p (f) , then we have the inequality ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − [f (a) (p − a) + f (b) (b − p)] ∣ ∣ ∣ ∣ ∣ ≤ ∫p a ( t ∨ a (f) ) dt + ∫b p ( b ∨ t (f) ) dt ≤ 1 2 (b − a) b ∨ a (f) . (2.9) the first inequality in (2.3) is useful when some properties for the cvf are available, like for instance below: corollary 2.5. let f : [a, b] → c be a function of bounded variation on [a, b] . if there exists the constants la, lb > 0 and α, β > −1 so that t ∨ a (f) ≤ la (t − a) α for any t ∈ (a, b] (2.10) and b ∨ t (f) ≤ lb (b − t) β for any t ∈ [a, b), (2.11) then ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − [f (a) (x − a) + f (b) (b − x)] ∣ ∣ ∣ ∣ ∣ ≤ 1 α + 1 la (x − a) α+1 + 1 β + 1 lb (b − t) β+1 (2.12) for any x ∈ [a, b]. cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 37 the inequality (2.12) follows by integrating the inequalities (2.10) and (2.12) via (2.3). corollary 2.6. let f : [a, b] → c be a function of bounded variation on [a, b] . if there exists the constant l > 0 so that t ∨ a (f) ≤ l (t − a) and b ∨ t (f) ≤ l (b − t) for any t ∈ [a, b] (2.13) then ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − [f (a) (x − a) + f (b) (b − x)] ∣ ∣ ∣ ∣ ∣ ≤ [ 1 4 + ( x − a+b 2 b − a )] l (b − a) 2 (2.14) for any x ∈ [a, b]. the constant 1 4 is best possible. in particular, we have ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − f (a) + f (b) 2 (b − a) ∣ ∣ ∣ ∣ ∣ ≤ 1 4 l (b − a) 2 . (2.15) the constant 1 4 is best possible. proof. first, we notice that if h : [a, b] → c is of bounded variation, then |h| : [a, b] → [0, ∞) is of bounded variation and b ∨ a (|h|) ≤ b ∨ a (h) . (2.16) indeed, by the continuity property of the modulus, we have that n−1∑ j=0 ||h (tj+1)| − |h (tj)|| ≤ n−1∑ j=0 |h (tj+1) − h (tj)| for any division a = t0 < t1 < ... < tn−1 < tn = b, which, by taking the supremum over all divisions of [a, b] , produces the desired inequality (2.16). if we consider the function f0 : [a, b] → r, f0 (s) := ∣ ∣s − a+b 2 ∣ ∣ then, by denoting with e the identity function on [a, b] , i.e. e (t) = t, t ∈ [a, b] , we have t ∨ a ( ∣ ∣ ∣ ∣ e− a + b 2 ∣ ∣ ∣ ∣ ) ≤ t ∨ a ( e− a + b 2 ) = t ∨ a (e) = t − a for any t ∈ [a, b] . similarly, b ∨ t ( ∣ ∣ ∣ ∣ e− a + b 2 ∣ ∣ ∣ ∣ ) ≤ b − t 38 s.s. dragomir cubo 17, 2 (2015) for any t ∈ [a, b] , showing that f0 satisfies the condition (2.13) with l = 1. since f0 (a) + f0 (b) 2 = b − a, ∫b a f0 (t) dt = 1 4 (b − a) 2 , then we get in both sides of (2.15) the same quantity 1 4 (b − a) 2 . this proves the sharpness of the constant 1 4 in (2.15) and therefore in (2.14). remark 2.7. the inequalities (2.14) and (2.15) are known in the case of lipschitzian functions with the constant l > 0. we obtained them here under weaker conditions for the function f. this show that the refinement in terms of the cvf for the trapezoid inequality (2.3) is also useful to extend known results to larger classes of functions. the following similar result also holds: theorem 2.8. let f : [a, b] → c be a function of bounded variation on [a, b] . then ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − [f (a) (x − a) + f (b) (b − x)] ∣ ∣ ∣ ∣ ∣ ≤ ∫x a ( t ∨ a (|f − f (a)|) ) dt + ∫b x ( b ∨ t (|f (b) − f|) ) dt ≤ (x − a) x ∨ a (|f − f (a)|) + (b − x) b ∨ x (|f (b) − f|) ≤ (x − a) x ∨ a (f) + (b − x) b ∨ x (f) ≤    [ 1 2 (b − a) + ∣ ∣x − a+b 2 ∣ ∣ ] ∨b a (f) , [ 1 2 ∨b a (f) + 1 2 ∣ ∣ ∣ ∨x a (f) − ∨b x (f) ∣ ∣ ∣ ] (b − a) , (2.17) for any x ∈ [a, b]. proof. we observe that ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − [f (a) (x − a) + f (b) (b − x)] ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∫x a (f (a) − f (t)) dt + ∫b x (f (b) − f (t)) dt ∣ ∣ ∣ ∣ ∣ ≤ ∫x a |f (a) − f (t)| dt + ∫b x |f (b) − f (t)| dt (2.18) for any x ∈ [a, b]. cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 39 for a given x ∈ (a, b) , define the function g : [a, b] → [0, ∞) given by g (t) :=    |f (t) − f (a)| , t ∈ [a, x] |f (b) − f (t)| , t ∈ (x, b]. we observe that g is of bounded variation on the intervals [a, x] and [x, b] , g (a) = g (b) = 0. moreover, ∫b a g (t) dt = ∫x a |f (a) − f (t)| dt + ∫b x |f (b) − f (t)| dt, for any x ∈ (a, b) . since g is nonnegative, observe also that ∫b a g (t) dt = ∣ ∣ ∣ ∣ ∣ ∫b a g (t) dt − (b − x) g (b) − (x − a) g (a) ∣ ∣ ∣ ∣ ∣ for any x ∈ (a, b) . now, if we apply theorem 2.2 for the function g (we should notice that the theorem 2.2 also holds if we assume the involved function is of bounded variation on the portions [a, x] and [x, b]), then we get ∣ ∣ ∣ ∣ ∣ ∫b a g (t) dt − (b − x) g (b) − (x − a) g (a) ∣ ∣ ∣ ∣ ∣ ≤ ∫x a ( t ∨ a (g) ) dt + ∫b x ( b ∨ t (g) ) dt ≤ (x − a) x ∨ a (g) + (b − x) b ∨ x (g) . (2.19) since ∫x a ( t ∨ a (g) ) dt + ∫b x ( b ∨ t (g) ) dt = ∫x a ( t ∨ a (|f − f (a)|) ) dt + ∫b x ( b ∨ t (|f (b) − f|) ) dt and (x − a) x ∨ a (g) + (b − x) b ∨ x (g) = (x − a) x ∨ a (|f − f (a)|) + (b − x) b ∨ x (|f (b) − f|) 40 s.s. dragomir cubo 17, 2 (2015) then we get from (2.18) and (2.19) the first two inequalities in (2.17). for the last part, we observe that x ∨ a (|f − f (a)|) ≤ x ∨ a (f − f (a)) = x ∨ a (f) and b ∨ x (|f (b) − f|) ≤ b ∨ x (f (b) − f) = b ∨ x (f) . for any x ∈ (a, b) . the proof is complete. corollary 2.9. let f : [a, b] → c be a function of bounded variation on [a, b] . then ∣ ∣ ∣ ∣ ∣ ∫b a f (t) dt − f (a) + f (b) 2 (b − a) ∣ ∣ ∣ ∣ ∣ ≤ ∫ a+b 2 a ( t ∨ a (|f − f (a)|) ) dt + ∫b a+b 2 ( b ∨ t (|f (b) − f|) ) dt ≤ 1 2 (b − a)   a+b 2 ∨ a (|f − f (a)|) + b ∨ a+b 2 (|f (b) − f|)   ≤ 1 2 (b − a) b ∨ a (f) . (2.20) the inequalities in (2.20) are sharp. proof. we must prove only the sharpness of the inequalities. consider the function of bounded variation f : [a, b] → r given by f (t) :=    1, t = a 0, t ∈ (a, b) 1, t = b. observe that |f (t) − f (a)| = |f (b) − f (t)| =    0, t = a 1, t ∈ (a, b) 0, t = b. then t ∨ a (|f − f (a)|) = 1, t ∈ [ a, a + b 2 ] cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 41 and b ∨ t (|f (b) − f|) = 1, t ∈ [ a + b 2 , b ] . replacing these values in (2.20) and performing the calculations we obtain the same quantity b−a in all terms. this proves the sharpness of all inequalities in (2.20). 3 applications for selfadjoint operators we denote by b (h) the banach algebra of all bounded linear operators on a complex hilbert space (h; 〈·, ·〉) . let a ∈ b (h) be selfadjoint and let ϕλ be defined for all λ ∈ r as follows ϕλ (s) :=    1, for − ∞ < s ≤ λ, 0, for λ < s < +∞. then for every λ ∈ r the operator eλ := ϕλ (a) (3.1) is a projection which reduces a. the properties of these projections are collected in the following fundamental result concerning the spectral representation of bounded selfadjoint operators in hilbert spaces, see for instance [21, p. 256]: theorem 3.1 (spectral representation theorem). let a be a bounded selfadjoint operator on the hilbert space h and let m = min {λ |λ ∈ sp (a)} =: min sp (a) and m = max {λ |λ ∈ sp (a)} =: max sp (a) . then there exists a family of projections {eλ}λ∈r, called the spectral family of a, with the following properties a) eλ ≤ eλ′ for λ ≤ λ ′; b) em−0 = 0, em = i and eλ+0 = eλ for all λ ∈ r; c) we have the representation a = ∫m m−0 λdeλ. more generally, for every continuous complex-valued function ϕ defined on r and for every ε > 0 there exists a δ > 0 such that ∥ ∥ ∥ ∥ ∥ ϕ (a) − n∑ k=1 ϕ (λ′k) [eλk − eλk−1] ∥ ∥ ∥ ∥ ∥ ≤ ε 42 s.s. dragomir cubo 17, 2 (2015) whenever    λ0 < m = λ1 < ... < λn−1 < λn = m, λk − λk−1 ≤ δ for 1 ≤ k ≤ n, λ′k ∈ [λk−1, λk] for 1 ≤ k ≤ n this means that ϕ (a) = ∫m m−0 ϕ (λ) deλ, (3.2) where the integral is of riemann-stieltjes type. corollary 3.2. with the assumptions of theorem 3.1 for a, eλ and ϕ we have the representations ϕ (a) x = ∫m m−0 ϕ (λ) deλx for all x ∈ h and 〈ϕ (a) x, y〉 = ∫m m−0 ϕ (λ) d 〈eλx, y〉 for all x, y ∈ h. (3.3) in particular, 〈ϕ (a) x, x〉 = ∫m m−0 ϕ (λ) d 〈eλx, x〉 for all x ∈ h. moreover, we have the equality ‖ϕ (a) x‖ 2 = ∫m m−0 |ϕ (λ)| 2 d ‖eλx‖ 2 for all x ∈ h. we need the following result that provides an upper bound for the total variation of the function r ∋ λ 7→ 〈eλx, y〉 ∈ c on an interval [α, β] , see [16]. for the sake of completeness, we give here a short proof. lemma 3.3. let {eλ}λ∈r be the spectral family of the bounded selfadjoint operator a. then for any x, y ∈ h and α < β we have the inequality [ β ∨ α (〈 e(·)x, y 〉) ]2 ≤ 〈(eβ − eα) x, x〉〈(eβ − eα) y, y〉 , (3.4) where β ∨ α (〈 e(·)x, y 〉) denotes the total variation of the function 〈 e(·)x, y 〉 on [α, β] . proof. if p is a positive selfadjoint operator on h, i.e., 〈px, x〉 ≥ 0 for any x ∈ h, then the following inequality is a generalization of the schwarz inequality in h |〈px, y〉| 2 ≤ 〈px, x〉〈py, y〉 , (3.5) cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 43 for any x, y ∈ h. now, if d : α = t0 < t1 < ... < tn−1 < tn = β is an arbitrary partition of the interval [α, β] , then we have by schwarz’s inequality for positive operators (3.5) that β ∨ α (〈 e(·)x, y 〉) = sup d { n−1∑ i=0 |〈(eti+1 − eti) x, y〉| } ≤ sup d { n−1∑ i=0 [ 〈(eti+1 − eti) x, x〉 1/2 〈(eti+1 − eti) y, y〉 1/2 ] } := b. (3.6) by the cauchy-bunyakovsky-schwarz inequality for sequences of real numbers we also have that n−1∑ i=0 [ 〈(eti+1 − eti) x, x〉 1/2 〈(eti+1 − eti) y, y〉 1/2 ] ≤ [ n−1∑ i=0 〈(eti+1 − eti) x, x〉 ]1/2 [ n−1∑ i=0 〈(eti+1 − eti) y, y〉 ]1/2 = [〈(eβ − eα) x, x〉] 1/2 [〈(eβ − eα) y, y〉] 1/2 (3.7) for any x, y ∈ h. taking the supremum over d in (3.7) we get b ≤ [〈(eβ − eα) x, x〉] 1/2 [〈(eβ − eα) y, y〉] 1/2 for any x, y ∈ h which together with (3.6) produce the desired result (3.4). remark 3.4. for α = m − ε with ε > 0 and β = m we get from (3.4) the inequality m ∨ m−ε (〈 e(·)x, y 〉) ≤ 〈(i − em−ε) x, x〉 1/2 〈(i − em−ε) y, y〉 1/2 (3.8) for any x, y ∈ h. this implies, for any x, y ∈ h, that m ∨ m−0 (〈 e(·)x, y 〉) ≤ ‖x‖ ‖y‖ , (3.9) where m ∨ m−0 (〈 e(·)x, y 〉) denotes the limit limε→0+ [ m ∨ m−ε (〈 e(·)x, y 〉) ] . the following result holds: 44 s.s. dragomir cubo 17, 2 (2015) theorem 3.5. let a be a bounded selfadjoint operator on the hilbert space h and let m = min {λ |λ ∈ sp (a)} =: min sp (a) and m = max {λ |λ ∈ sp (a)} =: max sp (a) . if {eλ}λ∈r is the spectral family of the bounded selfadjoint operator a, then for any v ∈ [m, m] we have |〈(a − vi) x, y〉| ≤ ∫v m−0 〈etx, x〉 1/2 〈ety, y〉 1/2 dt + ∫m v 〈(i − et) x, x〉 1/2 〈(i − et) y, y〉 1/2 dt ≤ (v − m) 〈evx, x〉 1/2 〈evy, y〉 1/2 + (m − v) 〈(i − ev) x, x〉 1/2 〈(i − ev) y, y〉 1/2 ≤ [ 1 2 (m − m) + ∣ ∣ ∣ ∣ v − a + b 2 ∣ ∣ ∣ ∣ ] × [ 〈evx, x〉 1/2 〈evy, y〉 1/2 + 〈(i − ev) x, x〉 1/2 〈(i − ev) y, y〉 1/2 ] ≤ [ 1 2 (m − m) + ∣ ∣ ∣ ∣ v − a + b 2 ∣ ∣ ∣ ∣ ] ‖x‖ ‖y‖ (3.10) for any x, y ∈ h. in particular, we have ∣ ∣ ∣ ∣ 〈( a − m + m 2 i ) x, y 〉 ∣ ∣ ∣ ∣ ≤ ∫ m+m 2 m−0 〈etx, x〉 1/2 〈ety, y〉 1/2 dt + ∫m m+m 2 〈(i − et) x, x〉 1/2 〈(i − et) y, y〉 1/2 dt ≤ 1 2 (m − m) [ 〈 e m+m 2 x, x 〉1/2 〈 e m+m 2 y, y 〉1/2 + 〈( i − e m+m 2 ) x, x 〉1/2 〈( i − e m+m 2 ) y, y 〉1/2 ] ≤ 1 2 (m − m) ‖x‖ ‖y‖ (3.11) for any x, y ∈ h. proof. utilising the representation in (3.3) we have |〈(a − vi) x, y〉| = ∣ ∣ ∣ ∣ ∣ ∫m m−0 (t − v) d 〈etx, y〉 ∣ ∣ ∣ ∣ ∣ for any v ∈ [m, m] and for any x, y ∈ h. cubo 17, 2 (2015) refinements of the generalized trapezoid . . . 45 for ε > 0, by utilizing an argument similar to the one in theorem 2.2 we have ∣ ∣ ∣ ∣ ∣ ∫m m−ε (t − v) d 〈etx, y〉 ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ ∫m m−ε |t − v| d ( t ∨ a 〈etx, y〉 ) ∣ ∣ ∣ ∣ ∣ ≤ ∫v m−ε ( t ∨ a 〈etx, y〉 ) dt + ∫m v ( m ∨ t 〈etx, y〉 ) dt for any v ∈ [m, m] and for any x, y ∈ h. from lemma 3.3 we have ∫v m−ε ( t ∨ a 〈etx, y〉 ) dt ≤ ∫v m−ε 〈(et − em−ε) x, x〉 1/2 〈(et − em−ε) y, y〉 1/2 dt and ∫m v ( m ∨ t 〈etx, y〉 ) dt ≤ ∫m v 〈(em − et) x, x〉 1/2 〈(em − et) y, y〉 1/2 dt = ∫m v 〈(i − et) x, x〉 1/2 〈(i − et) y, y〉 1/2 dt for any v ∈ [m, m] and for any x, y ∈ h. therefore, ∣ ∣ ∣ ∣ ∣ ∫m m−ε (t − v) d 〈etx, y〉 ∣ ∣ ∣ ∣ ∣ ≤ ∫v m−ε 〈(et − em−ε) x, x〉 1/2 〈(et − em−ε) y, y〉 1/2 dt + ∫m v 〈(i − et) x, x〉 1/2 〈(i − et) y, y〉 1/2 dt (3.12) for any ε > 0, for any v ∈ [m, m] and for any x, y ∈ h. 46 s.s. dragomir cubo 17, 2 (2015) taking the limit over ε → 0+ and since em−0 = 0, we get ∣ ∣ ∣ ∣ ∣ ∫m m−0 (t − v) d 〈etx, y〉 ∣ ∣ ∣ ∣ ∣ ≤ ∫v m−0 〈etx, x〉 1/2 〈ety, y〉 1/2 dt + ∫m v 〈(i − et) x, x〉 1/2 〈(i − et) y, y〉 1/2 dt (3.13) for any v ∈ [m, m] and for any x, y ∈ h, which proves the first inequality in (3.13). the rest is easy to see and we omit the details. received: march 2015. accepted: may 2015. references [1] n. s. barnett and s.s. dragomir, a perturbed trapezoid inequality in terms of the third derivative and applications. inequality theory and applications. vol. 5, 1–11, nova sci. publ., new york, 2007. 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[32] q. wu and s. yang, a note to ujević’s generalization of ostrowski’s inequality. appl. math. lett. 18 (2005), no. 6, 657–665. introduction refinements of the generalized trapezoid inequality applications for selfadjoint operators cubo a mathematical journal vol.16, no¯ 01, (49–61). march 2014 on a result of q. han, s. mori and k. tohge concerning uniquesness of meromorphic functions. indrajit lahiri department of mathematics, university of kalyani, kalyani, kalyani, west bengal 741235, india. ilahiri@hotmail.com nintu mandal department of mathematics, a. b. n. seal college, cooch behar, west bengal 736101, india. nintu311209@gmail.com abstract in the paper we prove a result on the uniqueness of meromorphic functions that is related to a result of q. han, s. mori and k. tohge and is originated from a result of h.ueda and two subsequent results of g. brosch. resumen en este art́ıculo probamos un resultado de unicidad de funciones meromórficas que se relaciona a un resultado de q. han, s. mori y k. tohge, y se origina de un resultado de h. ueda y dos resultados derivados de g. brosch. keywords and phrases: meromorphic function, uniqueness, weighted sharing. 2010 ams mathematics subject classification: 30d35. 50 indrajit lahiri & nintu mandal cubo 16, 1 (2014) 1 introduction, definitions and results let f and g be two non-constant meromorphic functions defined in the open complex plane c. for a ∈ c ∪ {∞} we say that f and g share the value a cm ( counting multiplicities ) if f, g have the same a-points with the same multiplicities. if we do not take the multiplicities into account then f, g are said to share the value a im ( ignoring multiplicities ). for the standard notations and definitions of the value distribution theory we refer to [5] and [15] . however we require following notations. definition 1. let k be a positive integer or infinity. for a ∈ c ∪ {∞} we denote by ek)(a; f) and ek)(a; f) the collection of those a-points of f whose multiplicities does not exceed k, with counting multiplicities and with ignoring multiplicities respectively. definition 2. let k be a positive integer and a ∈ c ∪ {∞}. then by n(r, a; f| ≤ k) we denote the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not greater than k. by n(r, a; f| ≤ k) we denote the corresponding reduced counting funcion. in an analogous manner we define n(r, a; f| ≥ k) and n(r, a; f| ≥ k). also by n(r, a; f| = k) and n(r, a; f| = k) we denote respectively the counting function and reduced counting function of those a-points of f whose multiplicities are exactly k. in 1980 h.ueda[14]{see also p. 327 [15]}prove the following result. theorem a. [14] let f and g be nonconstant entire functions sharing 0, 1 cm, and a(6= 0, 1, ∞) be a complex number. if e ∞)(a; f) ⊂ e∞)(a; g), then f is a bilinear transformation of g. improving theorem a in 1989 g.brosch[2] proved the following result. theorem b. [2] let f and g be two nonconstant meromorphic functions sharing 0, 1, ∞ cm, and a(6= 0, 1, ∞) be a complex number. if e ∞)(a; f) ⊂ e∞)(a; g), then f is a bilinear transformation of g. following example shows that in theorem b the condition e ∞)(a; f) ⊂ e∞)(a; g) cannot be replaced by e ∞)(a; f) ⊂ e∞)(b; g) for b 6= a, 0, 1, ∞. example 1. let f = e2z + ez + 1, g = e−2z + e−z + 1, a = 3 4 and b = 3. then f, g share 0, 1, ∞ cm and f − a = 1 4 (2ez + 1)2, g − b = e−2z(1 + 2ez)(1 − ez). so ē ∞)(a; f) ⊂ ē∞)(b; g) but f is not a bilinear transformation of g. considering the possibility a 6= b, g.brosch[2] proved the following theorem. theorem c. [2] let f and g be two nonconstant meromorphic functions sharing 0, 1, ∞ cm, and a, b be two complex numbers such that a, b 6∈ {0, 1, ∞} . if e ∞)(a; f) = e∞)(b; g), then f is a bilinear transformation of g. cubo 16, 1 (2014) on a result of q. han, s. mori and k. tohge concerning . . . 51 in 2001 the idea of weighted sharing of values was introduced {cf.[6], [7]} which provides a scaling between im sharing and cm sharing of values. we now explain this notion in the following definition. definition 3. [11] let k be a nonnegative integer or infinity. for a ∈ c ∪ {∞} we denote by ek(a, f) the set of all a-points of f, where an a-point with multiplicity m is counted m times if m ≤ k and k + 1 times if m > k. if ek(a, f) = ek(a, g), we say that f, g share the value a with weight k. the definition means that z0 is a zero of f − a with multiplicity m(≤ k) if and only if z0 is a zero of g with multiplicity m(≤ k) and z0 is a zero of f − a with multiplicity m(> k) if and only if z0 is a zero of g with multiplicity n(> k), where m is not necessarily equal to n. we write f, g share (a, k) to mean that f, g share the value a with weight k. clearly if f, g share (a, k) then f, g share (a, p) for all integers p, 0 ≤ p < k. also we note that f, g share a value a im or cm if and only if f, g share (a, 0) or (a, ∞) respectively. in 2004 using the idea of weighted value sharing t.c. alzahari and h.x.yi [1] improved theorem c in the following manner . theorem d. [1] let f, g be two nonconstant meromorphic functions sharing (a1, 1), (a2, ∞), (a3, ∞), where {a1, a2, a3} = {0, 1, ∞}, and let a, b be two finite complex numbers such that a, b 6∈ {0, 1} . if ē ∞)(a; f) = ē∞)(b; g), then f is a bilinear transformation of g. moreover f and g satisfy exactly one of the following relations: (i) f ≡ g; (ii) fg ≡ 1; (iii) bf ≡ ag; (iv) f + g ≡ 1; (v) f ≡ ag; (vi) f ≡ (1 − a)g + a; (vii) (1 − b)f ≡ (1 − a)g + (a − b); (viii) (1 − a + g)f ≡ ag; (ix) f{(b − a)g + (a − 1)b} ≡ a(b − 1)g; 52 indrajit lahiri & nintu mandal cubo 16, 1 (2014) (x) f(g − 1) ≡ g; the cases (ii) and (v) may occur if ab = 1, cases (iv) and (viii) may occur if a + b = 1, cases (vi) and (x) may occur if ab = a + b. improving theorem d recently i.lahiri and p.sahoo [12] proved the following theorem. theorem e. [12] let f, g be two distinct nonconstant meromorphic functions sharing (a1, 1), (a2, m), (a3, k), where {a1, a2, a3} = {0, 1, ∞} and (m − 1)(mk − 1) > (1 + m) 2.if for two values a, b 6∈ {0, 1, ∞} the functions f − a and g − b share (0, 0) then f, g share (0, ∞), (1, ∞), (∞, ∞) and f − a, g − b share (0, ∞). also there exists a non-constant entire function λ such that f and g are one of the following forms: (i) f = aeλ and g = be−λ, where ab = 1; (ii) f = 1 + aeλ and g = 1 + (1 − 1 b )e−λ, where ab = a + b; (iii) f = a a+eλ and g = e λ 1−b+eλ , where a + b = 1: (iv) f = e λ −a eλ−1 and g = be λ −1 eλ−1 , where ab = 1; (v) f = be λ −a beλ−b and g = be λ −a aeλ−a , where a 6= b; (vi) f = a 1−eλ and g = be λ eλ−1 , where ab = a + b; (vii) f = b−a (b−1)(1−eλ) and g = (b−a)e λ (a−1)(1−eλ) , where a 6= b; (viii) f = a + eλ and g = b(1 + 1−b eλ ), where a + b = 1; (ix) f = eλ − a(b−1) a−b and g = b(a−1) a−b {1 − a(b−1) (b−a)eλ }, where a 6= b; q.han, s.mori and k.tohge [4] further improved theorem c, theorem d, theorem e and proved the following. theorem f. [4] let f and g be two distinct nonconstant meromorphic functions sharing (a1, k1), (a2, k2) and (a3, k3) for three distinct values a1, a2, a3 ∈ c∪{∞}, where k1k2k3 > k1 +k2 +k3 +2. furthermore if ek)(a4; f) = ek)(a5; g) for values a4, a5 in c∪{∞}\{a1, a2, a3} and for some positive integer k(≥ 2), then f is a bilinear transformation of g. cubo 16, 1 (2014) on a result of q. han, s. mori and k. tohge concerning . . . 53 example 1 with a = b = 3 4 shows that the conclusion of theorem f does not hold for k = 1. this suggests that some further investigation is necessary for the case k = 1. in the paper we take up this problem and prove the following result. theorem 1.1. let f, g be two distinct nonconstant meromorphic functions sharing (a1, k1), (a2, k2), (a3, k3) where a1, a2, a3 ∈ c ∪ {∞} are distinct and k1k2k3 > k1 + k2 + k3 + 2. further let e1)(a; f) ⊂ e∞)(b; g) for two complex numbers a, b 6∈ {a1, a2, a3} and e1)(0; f ′) ⊂ e ∞)(0; g ′). then f is a bilinear transformation of g. if, in particular, {a1, a2, a3} = {0, 1, ∞},then there exists a non-constant entire function λ such that f and g assume exactly one of the following forms: (i) f = aeλ and g = be−λ where ab = 1; (ii) f = 1 + aeλ and g = 1 + (1 − 1 b )e−λ where ab = a + b; (iii) f = a a+eλ and g = e λ 1−b+eλ where a + b = 1: (iv) f = e λ −a eλ−1 and g = e λ −a aeλ−a where e ∞)(a; f) = φ; (v) f = be λ −a beλ−b and g = be λ −a aeλ−a where a 6= b; (vi) f = a 1−eλ and g = ae λ (1−a)(1−eλ) where e ∞)(a; f) = φ; (vii) f = b−a (b−1)(1−eλ) and g = (b−a)e λ (a−1)(1−eλ) where a 6= b; (viii) f = a + eλ and g = (1 − a)(1 + a eλ ) where e(a; f) = φ; (ix) f = eλ − a(b−1) a−b and g = b(a−1) a−b {1 − a(b−1) (b−a)eλ } where a 6= b; considering example 1 we see that the condition e1)(0; f ′) ⊂ e ∞)(0, g ′) is essential for theorem 1.1. 2 lemmas in the section we present some necessary lemmas. lemma 2.1. [3] let f and g share (0, 0), (1, 0), (∞, 0).then t(r, f) ≤ 3t(r, g)+s(r, f) and t(r, g) ≤ 3t(r, f) + s(r, f). from this we conclude that s(r, f) = s(r, g). henceforth we denote either of them by s(r). 54 indrajit lahiri & nintu mandal cubo 16, 1 (2014) lemma 2.2. [16] let f and g share (0, k1), (1, k2), (∞, k3) and f 6≡ g, where k1k2k3 > k1 + k2 + k3 + 2.then n(r, 0; f |≥ 2) + n(r, 1; f |≥ 2) + n(r, ∞; f |≥ 2) = s(r). following can be proved in the line of theorem 3.2 of [11]. lemma 2.3. let f and g be two distinct nonconstant meromorphic functions sharing (0, k1), (1, k2), (∞, k3), where k1k2k3 > k1 + k2 + k3 + 2. if n0(r) + n1(r) ≥ λt(r, f) + s(r) for some λ > 1 2 , then f is a bilinear transformation of g and n0(r) + n1(r) = t(r, f) + s(r) = t(r, g) + s(r), where n0(r)(n1(r)) denotes the counting function of those simple(multiple) zeros of f − g which are not the zeros of f(f − 1) and 1 f . lemma 2.4. [13] let f and g be two distinct noncostant meromorphic functions sharing (0, 0), (1, 0), (∞, 0). further suppose that f is a bilinear transformation of g and e1)(a; f) ⊂ e∞)(b; g),where a, b 6∈ {0, 1, ∞}. then there exists a nonconstant entire function λ such that f and g assume exactly one of the forms given in theorem1.1. following can be proved in the line of lemma 2.4 [13]. lemma 2.5. let f and g share (0, k1), (1, k2), (∞, k3) and f 6≡ g, where k1k2k3 > k1 +k2 +k3 +2. if f is not a bilinear transformation of g, then for a complex number a 6∈ {0, 1, ∞} each of the following holds: (i) n(r, a; f |≥ 3) + n(r, a; g |≥ 3) = s(r); (ii) t(r, f) = n(r, a; f ≤ 2) + s(r); (iii) t(r, g) = n(r, a; g ≤ 2) + s(r). in the line of lemma 5 [9] we can prove the following. lemma 2.6. let f, g share (0, k1), (1, k2), (∞, k3) and f 6≡ g, where k1k2k3 > k1 + k2 + k3 + 2. if α = f−1 g−1 and β = g f , then n(r, a; α) = s(r) and n(r, a; β) = s(r) for a = 0, ∞. following is an analogue of lemma 2.6 [13]. lemma 2.7. let f and g be two distinct meromorphic functions sharing (0, k1), (1, k2), (∞, k3), where k1k2k3 > k1 + k2 + k3 + 2.then t(r, α (p) α ) + t(r, β (p) β ) = s(r), where p is a positive integer and α, β are defined as in lemma 2.6. cubo 16, 1 (2014) on a result of q. han, s. mori and k. tohge concerning . . . 55 using the techniques of [8] and [10] we can prove the following. lemma 2.8. let f, g share (0, k1), (1, k2), (∞, k3) and f 6≡ g, where k1k2k3 > k1 + k2 + k3 + 2. if f is not a bilinear transformation of g, then each of the following holds : (i) t(r, f) + t(r, g) = n(r, 0; f |≤ 1) + n(r, 1; f |≤ 1) + n(r, ∞; f |≤ 1) + n0(r) + s(r), (ii) t(r, f) = n(r, 0; g′ |≤ 1) + n0(r) + s(r), (iii) t(r, g) = n(r, 0; f′ |≤ 1) + n0(r) + s(r), (iv) n1(r) = s(r), (v) n0(r, 0; g ′ |≥ 2) = s(r), (vi) n0(r, 0; f ′ |≥ 2) = s(r), (vii) n(r, 0; g′ |≥ 2) = s(r), (viii) n(r, 0; f′ |≥ 2) = s(r), (ix) n(r, 0; f − g |≥ 2) = s(r), (x) n(r, 0; f − g | f = ∞) = s(r), where n0(r, 0; g ′ |≥ 2)(n0(r, 0; f ′ |≥ 2)) is the counting function of those multiple zeros of g′(f′) which are not the zeros of f(f − 1) and n(r, 0; f − g | f = ∞) is the counting function of those zeros of f − g which are poles of f. 3 proof of theorem 1.1 proof. if necessary considering a bilinear transformation we may choose {a1, a2, a3} = {0, 1, ∞}. we now consider the following cases case 1. let a = b. if possible, we suppose that f is not a blinear transformation of g. we put φ = f′(f − a) f(f − 1) − g′(g − a) g(g − 1) . let φ 6≡ 0. since φ = aβ ′ β + (1 − a)α ′ α , by lemma 2.7 we get t(r, φ) = s(r). since e1)(a; f) ⊂ e ∞)(a; g) and e1)(0; f ′) ⊂ e ∞)(0; g ′), it follows that n(r, a; f |≤ 2) ≤ 2n(r, 0; φ) = s(r), which contradicts (ii) of lemma 2.5. therefore φ ≡ 0 and so f′(f − a) f(f − 1) = g′(g − a) g(g − 1) (3.1) 56 indrajit lahiri & nintu mandal cubo 16, 1 (2014) if z0 is a double zero of g − a, then from (3.1) we see that z0 is a common zero of f ′ and g′.hence z0 is a zero of α ′ α = f ′ f−1 − g ′ g−1 . so by (i) of lemma 2.5 and lemma 2.7 we get n(r, a; g |≥ 2) = 2n(r, 0; α′ α ) + s(r) = s(r). (3.2) again if z1 is a zero of g ′ which is not a zero of g(g−1)(g−a), then from (3.1) and the hypotheses of the theorem it follows that z1 is a zero of f ′ and so of α ′ α . hence from lemma 2.2, lemma 2.7 and (3.2) we get n(r, 0; g′ |≤ 1) ≤ n(r, a; g |≥ 2) + n(r, 0; f |≥ 2) + n(r, 1; f |≥ 2) + n(r, 0; α′ α ) = s(r). (3.3) now from (ii) and (iv) of lemma 2.8 and (3.3) we obtain n0(r) + n1(r) = t(r, f) + s(r), which is impossible by lemma 2.3. therefore f is a bilinear transformation of g and so by lemma 2.4 f and g take one of the forms (i)-(iv),(vi) and (viii). case 2. let a 6= b. if f is a bilinear transformation of g, then by lemma 2.4 f and g assume one of the forms (i) − (ix). so we suppose that f is not a bilinear transformation of g. following two subcases come up for consideration. subcase (i) let n(r, a; f |≥ 2) 6= s(r). we put ψ = f ′ (f−b) f(f−1) − g ′ (g−b) g(g−1) . since a double zero of f − a is a zero of f′ and so a zero of g′, if ψ 6≡ 0, then we get by lemma 2.5(i) and lemma 2.7, n(r, a; f |≥ 2) ≤ 2n(r, 0; ψ) + s(r) = s(r) which is a contradiction. hence ψ ≡ 0 and so f′(f − b) f(f − 1) = g′(g − b) g(g − 1) . this shows that f − a has no simple zero because e1)(a; f) ⊆ e∞)(b; g). since α ′ α = f ′ f−1 − g ′ g−1 . and e1)(0; f ′) ⊆ e ∞)(0; g ′), it follows that a double zero of f − a is a zero of α ′ α . so by lemma 2.7 we get n(r, a; f |= 2) ≤ 2n(r, 0; α ′ α ) = s(r), which contradicts (ii) of lemma 2.5. subcase (ii) let n(r, a; f |≥ 2) = s(r). since f is not a bilinear transformation of g, we see that α, β and αβ are non-constant. also we note that f = 1 − α 1 − αβ and g = (1 − α)β 1 − αβ . cubo 16, 1 (2014) on a result of q. han, s. mori and k. tohge concerning . . . 57 we put f = (f−a)(1−αβ) = aαβ−α+1−a and w = f′ f . also we note that f = (f−a) g − f f(g − 1) . since by lemma 2.6 n(r, ∞; f) = s(r) and w has only simple poles (if there is any), we get t(r, w) = m(r, w) + n(r, w) = n(r, 0; f) + s(r). (3.4) now by lemma 2.2 and (ix), (x) of lemma 2.8 we obtain n(r, 0; f |≥ 2) ≤ n(r, a; f |≥ 2) + n(r, 0; f − g |≥ 2) + n(r, ∞; f |≥ 2) +n(r, 0; f − g | f = ∞) = s(r). (3.5) hence from (3.4) and (3.5) we get t(r, w) = n(r, 0; f |≤ 1) + s(r) = n(r, a; f |≤ 1) + n0(r) + n2(r) + s(r), (3.6) where n2(r) is the counting function of those simple poles of f which are non-zero regular points of f − g. from the definitions of α and β we get { g − α′β (αβ)′ }( α′ α + β′ β ) ≡ f′(g − f) f(f − 1) . (3.7) from (3.7) we see that a simple pole of f which is a non-zero regular point of f − g is a regular point of { g − α′β (αβ)′ }( α′ α + β′ β ) . hence it is either a pole of α′β (αβ)′ or a zero of α′ α + β′ β . therefore by lemma 2.7 and the first fundamental theorem we get n2(r) ≤ t ( r, α′ α + β′ β ) + t ( r, α′β (αβ)′ ) ≤ t ( r, α′ α + β′ β ) + t ( r, 1 1 + αβ′ α′β ) ≤ 2t ( r, α′ α ) + 2t ( r, β′ β ) + o(1) = s(r). so from (3.6) we get t(r, w) = n(r, a; f |≤ 1) + n0(r) + s(r). (3.8) by (ii) of lemma 2.5 we get from (3.8) t(r, w) = t(r, f) + n0(r) + s(r). (3.9) 58 indrajit lahiri & nintu mandal cubo 16, 1 (2014) let τ1 = a − 1 b − 1 (ξ − bδ), τ2 = 1 2 · a − 1 b − 1 {ξ′ + ξ2 − b(δ′ + δ2)} and τ3 = 1 6 · a − 1 b − 1 {ξ′′ + 3ξξ′ + ξ3 − b(δ′′ + 3δδ′ + δ3)}, where ξ = α′ α and δ = α′ α + β′ β . by lemma 2.7 we see that t(r, ξ) = s(r) and t(r, δ) = s(r). if τ1 ≡ 0, from (3.7) we get (g − b)δ ≡ f′(g − f) f(f − 1) . (3.10) since e1)(a; f) ⊂ e(b; g), it follows from (3.10) that a simple zero of f − a, which is neither a zero nor a pole of δ, is a zero of g − b and so is a zero of f′. hence n(r, a; f |≤ 1) = s(r), which contradicts (ii) of lemma 2.5. therefore τ1 6≡ 0. let z0 be a simple zero of f − a and τ1(z0) 6= 0. then g(z0) = b and so α(z0) = a − 1 b − 1 and β(z0) = b a . expanding f around z0 in taylor’s series we get −f(z) = τ1(z0)(z − z0) + τ2(z0)(z − z0) 2 + τ3(z0)(z − z0) 3 + o((z − z0) 4). hence in some neighbourhood of z0 we obtain w(z) = 1 z − z0 + b(z0) 2 + c(z0)(z − z0) + o((z − z0) 2), where b = 2τ2 τ1 and c = 2τ3 τ1 − ( τ2 τ1 )2 . we put h = w′ + w2 − bw − a, (3.11) where a = 3c − b2 4 − b′. clearly t(r, a) + t(r, b) + t(r, c) = s(r) and since w = f′ f and f = (f − a) g − f f(g − 1) , we get by lemma 2.1 and (3.9) that s(r, w) = s(r). let h 6≡ 0. then it is easy to see that z0 is a zero of h. so n(r, a; f |≤ 1) ≤ n(r, 0; h) + s(r) ≤ t(r, h) + s(r) = n(r, h) + s(r). (3.12) cubo 16, 1 (2014) on a result of q. han, s. mori and k. tohge concerning . . . 59 from (ii) of lemma 2.5 and (3.12) we get t(r, f) ≤ n(r, h) + s(r). (3.13) let z1 be a pole of f. then z1 is a simple pole of w. so if z1 is not a pole of a and b, then z1 is at most a double pole of h. hence by lemma 2.6 we get n(r, ∞; h | f = ∞) ≤ 2n(r, ∞; f) + s(r) = s(r), (3.14) where n(r, ∞; h | f = ∞) denotes the counting function of those poles of h which are also poles of f. let z2 be a multiple zero of f. then z2 is a simple pole of w. so if z2 is not a pole of a and b, then z2 is a pole of h of multiplicity at most two. hence by (3.5) we get n(r, ∞; h | f = 0, ≥ 2) ≤ 2n(r, 0; f |≥ 2) + s(r) = s(r), (3.15) where n(r, ∞; h | f = 0, ≥ 2) denotes the counting function of those poles of h which are multiple zeros of f. let z3 be a simple zero of f which is not a pole of a and b. then in some neighbourhood of z3 we get f(z) = (z−z3)h(z), where h is analytic at z3 and h(z3) 6= 0. hence in some neightbourhood of z3 we obtain h(z) = ( 2h′ h − b ) 1 z − z3 + h1, where h1 = ( h′ h ) ′ + ( h′ h )2 − bh′ h − a. this shows that z3 is at most a simple pole of h. since a simple zero of f − a is a zero of h and n(r, 0; f | f = t) ≤ n(r, 0; f − g |≥ 2) for t = 0, 1 and f = (f − a) g − f f(g − 1) , we get from (3.14) and (3.15) in view of (ix) of lemma 2.8 n(r, h) = n(r, ∞; h | f = ∞) + n(r, ∞; h | f = 0) + s(r) ≤ n(r, 0; f |≤ 1) − n(r, a; f |≤ 1) + s(r) = n0(r) + n2(r) + s(r) = n0(r) + s(r), (3.16) where n(r, 0; f | f = t) denotes the counting function of those zeros of f which are zeros of f − t and n(r, ∞; h | f = 0) denotes the counting function of those poles of h which are zeros of f from (3.13) and (3.16) we obtain t(r, f) ≤ n0(r) + s(r), which by (iv) of lemma 2.8 and 60 indrajit lahiri & nintu mandal cubo 16, 1 (2014) lemma 2.3 implies a contradiction. therefore h ≡ 0 and so w′ + w2 − bw − a ≡ 0 i.e., w′ w ≡ a w − w + b i.e., f′′ ≡ af + bf′. since f′ = a(αβ)′ − α′ and f′′ = a(αβ)′′ − α′′, we get from above kαβ + lα ≡ a(f − a)(1 − αβ), (3.17) where k = a{ (αβ)′′ αβ − b (αβ)′ αβ } and l = b α′ α − α′′ α . by lemma 2.7 we see that t(r, k) = s(r) and t(r, l) = s(r). since αβ = g(f − 1) f(g − 1) and α = f − 1 g − 1 , we get from (3.17) kg + lf ≡ a(f − a)(g − f) (f − 1) (3.18) let z0 be a simple zero of f−a which is not a pole of a. since e1)(a; f) ⊂ e∞)(b; g), it follows from 3.18 that z0 is a zero of bk + al. hence n(r, a; f |≤ 1) ≤ n(r, 0; bk + al) + n(r, ∞; a) ≡ s(r), which contradicts (ii) of lemma 2.5.this proves the theorem. received: april 2012. accepted: september 2012. references [1] t. c. alzahary and h.x.yi, weighted sharing three values and uniqueness of meromorphic functions, j. math.anal. appl., vol. 295 (2004), pp.247-257. 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() cubo a mathematical journal vol.13, no¯ 03, (117–139). october 2011 pseudo-almost automorphic solutions to some second-order differential equations toka diagana department of mathematics, howard university, 2441 6th street n.w., washington, d.c. 20059 usa. email: tdiagana@howard.edu and ahmed mohamed department of mathematics, howard university, 2441 6th street n.w., washington, d.c. 20059 usa. email: ahmohamed@howard.edu abstract in this paper we study and obtain the existence of pseudo-almost automorphic solutions to some classes of second-order abstract differential equations on a hilbert space. to illustrate our abstract results, we discuss the existence of pseudo almost automorphic solutions to the n-dimensional sine-gordon boundary value problem. resumen en este trabajo se estudia y obtiene la existencia de soluciones casi-seudo automorfas a algunas clases de ecuaciones diferenciales abstractas de segundo orden en un espacio de hilbert. para ilustrar nuestros resultados abstractos, se discute la existencia de 118 toka diagana and ahmed mohamed cubo 13, 3 (2011) soluciones casi-seudo automorfas en el problema de contorno n-dimensional de sinegordon . keywords. exponential stability, sectorial operator, hyperbolic semigroup, almost automorphic; pseudo-almost automorphic; autonomous second-order differential equation; sine-gordon equation. amenability, banach modules, module amenability, weak module amenability, semigroup algebra, inverse semigroup. mathematics subject classification: 43a60; 34b05; 34c27; 42a75; 47d06; 35l90. 1 introduction in leiva [36], the existence of (exponentially stable) bounded solutions and almost periodic solutions to the second-order systems of differential equations given by u′′(t) + cu′(t) + dau + kh(u) = p(t), u ∈ rn, t ∈ r, (1.1) where a is an n×n-matrix whose eigenvalues are positive, c,d,k are positive constants, h : rn 7→ r n is a locally lipschitz function, p : r 7→ rn is a bounded continuous function, were established. in this paper, using techniques developped in [36], we obtain some reasonable sufficient conditions, which do guarantee the existence of pseudo-almost automorphic solutions to u′′(t) + au′(t) + bau = f(t,u), t ∈ r, (1.2) where a : d(a) ⊂ h 7→ h is a self-adjoint linear operator whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < ... < λn → ∞ with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the corresponding eigenspace, a,b > are constants, and the function f : r × h 7→ h is pseudo-almost automorphic function satisfying some additional conditions. for that, the main idea consists of rewriting eq. (1.2) as a first-order differential equation on x := h × h involving the 2×2-operator matrix b. indeed, if u is differential, setting z := ( u u′ ) , eq. (1.2) can be rewritten in the hilbert space x in the following form z′(t) = bz(t) + f(t,z(t)), t ∈ r, (1.3) where b is the 2×2-operator matrix defined by b =       0 ih −ba −aih       (1.4) cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 119 whose domain d(b) is given by d(b) = d(a) × h. moreover, the semilinear term f appearing in eq. (1.3) is defined on r × xα for some α ∈ (0,1) by f(t,u,v) = ( 0 f(t,u) ) , where xα is an intermediate space (see assumption(h.2)). under some reasonable assumptions, it will be shown that the linear operator matrix b is sectorial and that its associated semigroup is exponentially stable. the concept of pseudo almost automorphy is a powerful generalization of both the notion of almost automorphy due to bochner (see [46]) and that of pseudo almost periodicity due to zhang (see [21]), which has recently been introduced in the literature by liang et al. [39, 52, 53]. such a concept has recently generated several developments and extensions, see, e.g., [18], [20], [29], [30], and [40]. the existence of almost periodic solutions to second-order differential equations constitutes one of the most important topics in qualitative theory of differential equations due essentially to their applications such thermoelastic plate equations [12, 37] or telegraph equation [43] or sinegordon equations [36]. some contributions on the maximal regularity, bounded, almost periodic, asymptotically almost periodic solutions to abstract second-order differential and partial differential equations have recently been made, among them are [9], [10], [18], [20], [29], [30], [39], [40], [52], [53], [54], [55], and [56]. however, to the best of our knowledge, the existence of pseudo-almost automorphic solutions to second-order differential equations of the form eq. (1.2) is an untreated original question, which in fact is the main motivation of the present paper. the paper is organized as follows: section 2 is devoted to preliminaries facts needed in the sequel. in particular, facts related to sectorial operators and hyperbolic semigroups are discussed. in addition, basic definitions and classical results on the concept of pseudo-almost automorphy are also given. in sections 3 and 4, we prove the main result. in section 5, we provide the reader with a few examples to illustrate our main result. 2 preliminaries in the sequel, a : d(a) ⊂ h 7→ h stands for a self-adjoint (possibly unbounded) linear operator on the hilbert space h whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < ... < λn → ∞ with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the corresponding eigenspace. let {ekj } be a (complete) orthonormal sequence of eigenvectors associated with the 120 toka diagana and ahmed mohamed cubo 13, 3 (2011) eigenvalues {λj}j≥1. clearly, for each u ∈ d(a) := { u ∈ h : ∞∑ j=1 λ2j ∥ ∥ ∥ eju ∥ ∥ ∥ 2 < ∞ } , au = ∞∑ j=1 λj γj∑ k=1 〈u,ekj 〉e k j = ∞∑ j=1 λjeju where eju = γj∑ k=1 〈u,ekj 〉e k j . note that {ej}j≥1 is a sequence of orthogonal projections on h. moreover, each u ∈ h can written as follows: u = ∞∑ j=1 eju. it should also be mentioned that the operator −a is the infinitesimal generator of an analytic semigroup {s(t)}t≥0, which is explicitly expressed in terms of those orthogonal projections ej by, for all u ∈ h, s(t)u = ∞∑ j=1 e−λjteju. in addition, the fractional powers ar (r ≥ 0) of a exist and are given by d(ar) = { u ∈ h : ∞∑ j=1 λ2rj ∥ ∥ ∥ eju ∥ ∥ ∥ 2 < ∞ } and aru = ∞∑ j=1 λ2rj eju, ∀u ∈ d(a r). let (x, ∥ ∥ ∥ · ∥ ∥ ∥ ) be a banach space. if l is a linear operator on the banach space x, then, d(l), ρ(l), σ(l), n(l), and r(l), stand respectively for the domain, resolvent, spectrum, null-space or kernel, and range of the operator l. moreover, one sets r(λ,l) := (λi − l)−1 for all 〈∈ ρ(a). furthermore, we set q = i−p for a projection p. if y, z are banach spaces, then the space b(y, z) denotes the collection of all bounded linear operators from y into z equipped with its natural topology. this is simply denoted by b(y) when y = z. 3 sectorial linear operators definition 3.1. a linear operator l : d(l) ⊂ x 7→ x (not necessarily densely defined) is said to be sectorial if the following hold: there exist constants ω ∈ r, θ ∈ ( π 2 ,π ) , and m > 0 such that cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 121 ρ(l) ⊃ sθ,ω, sθ,ω := { λ ∈ c : λ 6= ω, ∣ ∣ ∣ arg(λ − ω) ∣ ∣ ∣ < θ } , (3.1) and ‖r(λ,l)‖ ≤ m ∣ ∣ ∣ λ − ω ∣ ∣ ∣ , λ ∈ sθ,ω. (3.2) the class of sectorial operators is very rich and contains most of classical operators encountered in the literature. two examples of sectorial operators are given below. example 3.1. let p ≥ 1 and let x = lp(0,1) be the lebesgue space equipped with its norm ∥ ∥ ∥ · ∥ ∥ ∥ p defined by ‖ϕ‖p = (∫ 1 0 ∣ ∣ ∣ ϕ(x) ∣ ∣ ∣ p dx )1/p . define the linear operator a on lp(0,1) by d(a) = { u ∈ w2,p(0,1) : u′(0) = u′(1) = 0 } , a(ϕ) = ϕ′′, ∀ϕ ∈ d(a). it can be checked that the operator a is sectorial on lp(0,1). example 3.2. let p ≥ 1 and let ω ⊂ rd be open bounded subset with c2 boundary ∂ω. let x := lp(ω) be the lebesgue space equipped with the norm, ‖ · ‖p defined by, ‖ϕ‖p = (∫ ω ∣ ∣ ∣ ϕ(x) ∣ ∣ ∣ p dx )1/p . define the operator a as follows: d(a) = w2,p(ω) ∩ w 1,p 0 (ω), a(ϕ) = ∆ϕ, ∀ϕ ∈ d(a), where ∆ = d∑ k=1 ∂2 ∂x2 k is the laplace operator. it can be checked that the operator a is sectorial on lp(ω). it is well-known that [41] if a is sectorial, then it generates an analytic semigroup (t(t))t≥0, which maps (0, ∞) into b(x) and such that there exist m0,m1 > 0 with ∥ ∥ ∥ t(t) ∥ ∥ ∥ ≤ m0e ωt, t > 0, (3.3) ∥ ∥ ∥ t(a − ω)t(t) ∥ ∥ ∥ ≤ m1e ωt, t > 0. (3.4) in this paper, we suppose that the semigroup (t(t))t≥0 is hyperbolic, that is, there exist a projection p and constants m,δ > 0 such that t(t) commutes with p, n(p) is invariant with respect to t(t), t(t) : r(q) 7→ r(q) is invertible, and the following hold ∥ ∥ ∥ t(t)px ∥ ∥ ∥ ≤ me−δt ∥ ∥ ∥ x ∥ ∥ ∥ for t ≥ 0, (3.5) 122 toka diagana and ahmed mohamed cubo 13, 3 (2011) ∥ ∥ ∥ t(t)qx ∥ ∥ ∥ ≤ meδt ∥ ∥ ∥ x ∥ ∥ ∥ for t ≤ 0, (3.6) where q := i − p and, for t ≤ 0, t(t) := (t(−t))−1. recall that the analytic semigroup (t(t))t≥0 associated with a is hyperbolic if and only if σ(a) ∩ ir = ∅, see details in [28, prop. 1.15, pp.305]. definition 3.2. let α ∈ (0,1). a banach space (xα, ∥ ∥ ∥ · ∥ ∥ ∥ α ) is said to be an intermediate space between d(a) and x, or a space of class jα, if d(a) ⊂ xα ⊂ x and there is a constant c > 0 such that ∥ ∥ ∥ x ∥ ∥ ∥ α ≤ c ∥ ∥ ∥ x ∥ ∥ ∥ 1−α∥ ∥ ∥ x ∥ ∥ ∥ α a , x ∈ d(a), (3.7) where ∥ ∥ ∥ · ∥ ∥ ∥ a is the graph norm of a. concrete examples of xα include d((−a α)) for α ∈ (0,1), the domains of the fractional powers of a, the real interpolation spaces da(α, ∞), α ∈ (0,1), defined as the space of all x ∈ x such [ x ] α = sup 0 0 such that ∥ ∥ ∥ t(t)qx ∥ ∥ ∥ α ≤ c(α)eδt ∥ ∥ ∥ x ∥ ∥ ∥ for t ≤ 0. (3.8) in addition to the above, the following holds ∥ ∥ ∥ t(t)px ∥ ∥ ∥ α ≤ ∥ ∥ ∥ t(1) ∥ ∥ ∥ b(x,xα) ∥ ∥ ∥ t(t − 1)px ∥ ∥ ∥ , t ≥ 1, and hence from eq. (3.5), one obtains ∥ ∥ ∥ t(t)px ∥ ∥ ∥ α ≤ m′e−δt ∥ ∥ ∥ x ∥ ∥ ∥ , t ≥ 1, cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 123 where m′ depends on α. for t ∈ (0,1], by eq. (3.4) and eq. (3.7), ∥ ∥ ∥ t(t)px ∥ ∥ ∥ α ≤ m′′t−α ∥ ∥ ∥ x ∥ ∥ ∥ . hence, there exist constants m(α) > 0 and γ > 0 such that ∥ ∥ ∥ t(t)px ∥ ∥ ∥ α ≤ m(α)t−αe−γt ∥ ∥ ∥ x ∥ ∥ ∥ for t > 0. (3.9) 3.1 pseudo-almost automorphic functions let bc(r, x) (respectively, bc(r × y, x)) denote the collection of all x-valued bounded continuous functions (respectively, the class of jointly bounded continuous functions f : r × y 7→ x). the space bc(r, x) equipped with the sup norm defined by ∥ ∥ ∥ u ∥ ∥ ∥ ∞ = sup t∈r ∥ ∥ ∥ u(t) ∥ ∥ ∥ , is a banach space. furthermore, c(r,y) (respectively, c(r×y, x)) denotes the class of continuous functions from r into y (respectively, the class of jointly continuous functions f : r × y 7→ x). definition 3.3. a function f ∈ c(r, x) is said to be almost automorphic if for every sequence of real numbers (s′n)n∈n, there exists a subsequence (sn)n∈n such that g(t) := lim n→ ∞ f(t + sn) is well defined for each t ∈ r, and lim n→ ∞ g(t − sn) = f(t) for each t ∈ r. if the convergence above is uniform in t ∈ r, then f is almost periodic in the classical bochner’s sense. denote by aa(x) the collection of all almost automorphic functions r 7→ x. note that aa(x) equipped with the sup-norm turns out to be a banach space. among other things, almost automorphic functions satisfy the following properties. theorem 3.1. [46] if f,f1,f2 ∈ aa(x), then (i) f1 + f2 ∈ aa(x), (ii) λf ∈ aa(x) for any scalar λ, (iii) fα ∈ aa(x) where fα : r → x is defined by fα(·) = f(· + α), (iv) the range rf := { f(t) : t ∈ r } is relatively compact in x, thus f is bounded in norm, (v) if fn → f uniformly on r where each fn ∈ aa(x), then f ∈ aa(x) too. 124 toka diagana and ahmed mohamed cubo 13, 3 (2011) in addition to the above-mentioned properties, we have the the following property due to bugajewski and diagana [15]: (vi) if g ∈ l1(r), then f ∗ g ∈ aa(r), where f ∗ g is the convolution of f with g on r. let (y, ∥ ∥ ∥ · ∥ ∥ y ) be another banach space. definition 3.4. a jointly continuous function f : r × y 7→ x is said to be almost automorphic in t ∈ r if t 7→ f(t,x) is almost automorphic for all x ∈ k (k ⊂ y being any bounded subset). equivalently, for every sequence of real numbers (s′n)n∈n, there exists a subsequence (sn)n∈n such that g(t,x) := lim n→ ∞ f(t + sn,x) is well defined in t ∈ r and for each x ∈ k, and lim n→ ∞ g(t − sn,x) = f(t,x) for all t ∈ r and x ∈ k. the collection of such functions will be denoted by aa(y, x). for more on almost automorphic functions and related issues, we refer the reader to the excellent book by n’guérékata [46]. define pap0(r, x) := { f ∈ bc(r, x) : lim t → ∞ 1 2t ∫ t −t ∥ ∥ ∥ f(s) ∥ ∥ ∥ ds = 0 } . similarly, pap0(y, x) will denote the collection of all bounded continuous functions f : r × y 7→ x such that lim t → ∞ 1 2t ∫ t −t ∥ ∥ ∥ f(s,x) ∥ ∥ ∥ ds = 0 uniformly in x ∈ k, where k ⊂ y is any bounded subset. definition 3.5. (liang et al. [39] and xiao et al. [52]) a function f ∈ bc(r, x) is called pseudo almost automorphic if it can be expressed as f = g + φ, where g ∈ aa(x) and φ ∈ pap0(x). the collection of such functions will be denoted by paa(x). the functions g and φ appearing in definition 3.5 are respectively called the almost automorphic and the ergodic perturbation components of f. definition 3.6. a bounded continuous function f : r × y 7→ x belongs to aa(y, x) whenever it can be expressed as f = g + φ, where g ∈ aa(y, x) and φ ∈ pap0(y, x). the collection of such functions will be denoted by paa(y, x). we now collect a few useful properties of pseudo almost automorphic functions. cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 125 proposition 3.1. if g ∈ l1(r), f ∈ paa(r), then f ∗ g ∈ paa(r), where f ∗ g is the convolution of f with g on r. the proof of proposition 3.1 is based upon [15] and [16]. a substantial result is the next theorem, which is due to xiao et al. [52]. theorem 3.2. [52] the space paa(x) equipped with the sup norm ∥ ∥ ∥ · ∥ ∥ ∥ ∞ is a banach space. the next composition result, that is theorem 3.3, is a consequence of [40, theorem 2.4] and is crucial for the proof of the main result of the paper. theorem 3.3. suppose f : r × y 7→ x belongs to paa(y, x); f = g + h, with x 7→ g(t,x) being uniformly continuous on each bounded subset k of y uniformly in t ∈ r, that is, for each ε > 0 there exists δ > 0 such that x,y ∈ k and ∥ ∥ ∥ x − y ∥ ∥ ∥ < δ yields ∥ ∥ ∥ g(t,x) − g(t,y) ∥ ∥ ∥ < ε for all t ∈ r. furthermore, we suppose that there exists l > 0 such that ∥ ∥ ∥ f(t,x) − f(t,y) ∥ ∥ ∥ ≤ l ∥ ∥ ∥ x − y ∥ ∥ ∥ y for all x,y ∈ y and t ∈ r. then the function defined by h(t) = f(t,ϕ(t)) belongs to paa(x) provided ϕ ∈ paa(y). we also have: theorem 3.4. [52] if f : r × y 7→ x belongs to paa(y, x) and if x 7→ f(t,x) is uniformly continuous on each bounded subset k of y uniformly in t ∈ r, then the function defined by h(t) = f(t,ϕ(t)) belongs to paa(x) provided ϕ ∈ paa(y). 4 main results consider the differential equation u′(t) = lu(t) + f(t,u(t)), t ∈ r, (4.1) where l : d(l) ⊂ x 7→ x is sectorial and f : r × xα 7→ x is jointly continuous. fix once and for all α,β such that 0 ≤ α < β < 1. to study the existence and uniqueness of pseudo-almost automorphic solutions to eq. (4.1) we make the following additional assumptions (h.1) the operator l is sectorial on x and generates a hyperbolic (analytic) semigroup (t(t))t≥0. (h.2) let 0 < α < 1. then xα = d((−a α)), or xα = da(α,p),1 ≤ p ≤ +∞, or xα = da(α), or xα = [x,d(a)]α. 126 toka diagana and ahmed mohamed cubo 13, 3 (2011) (h.3) the function f : r × x 7→ x is given such that u 7→ f(t,u) is unformly continuous on each bounded subset b of x uniformly in t ∈ r. furthermore, f is lipschitz in the following sense: there exists l > 0 for which ∥ ∥ ∥ f(t,u) − f(t,v) ∥ ∥ ∥ β ≤ l ∥ ∥ ∥ u − v ∥ ∥ ∥ α for all u,v ∈ x and t ∈ r. set s1u(t) := s11u(t) − s12u(t) and s2u = s22u − s23u, where s11u(t) := ∫ t −∞ t(t − s)pf1(s,u(s))ds, s12u(t) := ∫ ∞ t t(t − s)qf1(s,u(s))ds for all t ∈ r. definition 4.1. under assumption (h.1), a function u : r 7→ xα is said to be a mild solution to eq. (4.1) provided that u(t) = t(t − s)u(s) + ∫ t s t(t − r)f(r,u(r))dr (4.2) for each ∀t ≥ s, t,s ∈ r. consider the differential equation u′(t) = lu(t) + g(t), t ∈ r, (4.3) where g : r 7→ x is continuous. theorem 4.1. under assumptions (h.1)-(h.2), if g ∈ b(r, x), then we have: (i) eq.(4.3) has a unique bounded mild solution u : r 7→ xα, which can be explicitly given by u(t) = ∫ t −∞ t(t − s)pg(s)ds − ∫ ∞ t t(t − s)qg(s)ds. (4.4) (ii) if g ∈ paa(xα), then u ∈ paa(xα). proof. (i) since g is bounded, we can easily show that u given above is well-defined. moreover, u satisfies u(t) = t(t − s)u(s) + ∫ t s t(t − r)g(r)dr for each ∀t ≥ s, t,s ∈ r. cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 127 the continuity and uniqueness of u is also clear. for the boundedness in xα, using (3.8) and (3.9), we obtain ∥ ∥ ∥ u(t) ∥ ∥ ∥ α ≤ c ∥ ∥ ∥ u(t) ∥ ∥ ∥ β ≤ c ∫ t −∞ ∥ ∥ ∥ t(t − s)pg(s) ∥ ∥ ∥ β ds + c ∫ +∞ t ∥ ∥ ∥ t(t − s)qg(s) ∥ ∥ ∥ β ds ≤ cm(β) ∫ t −∞ e− δ 2 (t−s)(t − s)−β ∥ ∥ ∥ g(s) ∥ ∥ ∥ ds + cc(β) ∫ +∞ t e−δ(s−t) ∥ ∥ ∥ g(s) ∥ ∥ ∥ ds ≤ cm(β) ∥ ∥ ∥ g ∥ ∥ ∥ ∞ ∫ +∞ 0 e−σ ( 2σ δ )−β 2dσ δ + cc(β) ∥ ∥ ∥ g ∥ ∥ ∥ ∞ ∫ +∞ 0 e−δσdσ ≤ cm(β)δαγ(1 − β) ∥ ∥ ∥ g ∥ ∥ ∥ ∞ + cc(β)δ−1 ∥ ∥ ∥ g ∥ ∥ ∥ ∞ , and hence ∥ ∥ ∥ x(t) ∥ ∥ ∥ α ≤ c ∥ ∥ ∥ x(t) ∥ ∥ ∥ β ≤ c′c [ m(β)δβγ(1 − β) + c(β)δ−1 ]∥ ∥ ∥ g ∥ ∥ ∥ ∞ ,α . (4.5) it remains to prove (ii). for that, we first consider the first integral in the expression of eq. (4.4) and denote it su. now write g = φ + ζ where φ ∈ aa(xα) and ζ ∈ pap0(xα). clearly, su can be rewritten as (su)(t) = ∫ t −∞ t(t − s)pφ(s)ds + ∫ t −∞ tt − s)pζ(s)ds. set φ(t) = ∫ t −∞ t(t − s)pφ(s)ds, and ψ(t) = ∫ t −∞ t(t − s)pζ(s)ds for each t ∈ r. the next step consists of showing that φ ∈ aa(xα) and ψ ∈ pap0(xα). indeed, since φ ∈ aa(xα), for every sequence of real numbers (τ ′ n)n∈n there exists a subsequence (τn)n∈n such that ψ(t) := lim n→ ∞ φ(t + τn) is well defined for each t ∈ r, and lim n→ ∞ ψ(t − τn) = φ(t) for each t ∈ r. set φ1(t) = ∫ t −∞ t(t − s)pψ(s)ds for all t ∈ r. now 128 toka diagana and ahmed mohamed cubo 13, 3 (2011) φ(t + τn) − φ1(t) = ∫ t+τn −∞ t(t + τn − s)pφ(s)ds − ∫ t −∞ t(t − s)pψ(s)ds = ∫ t −∞ t(t − s)pφ(s + τn)ds − ∫ t −∞ t(t − s)pψds = ∫ t −∞ t(t − s)p ( φ(s + τn) − ψ(s) ) ds using lebesgue dominated convergence theorem, one can easily see that ∥ ∥ ∥ ∫ t −∞ t(t − s)p ( φ(s + τn) − ψ(s) ) ds ∥ ∥ ∥ α → 0 as n → ∞, t ∈ r. thus φ1(t) = lim n→ ∞ φ(t + τn), t ∈ r. similarly, one can easily see that φ(t) = lim n→ ∞ φ1(t − τn), t ∈ r. therefore, φ ∈ aa(xα). let us now show that ψ ∈ pap0(xα). first, note that s 7→ ψ(s) is a bounded continuous function. it remains to show that lim t → ∞ 1 2t ∫ t −t ∥ ∥ ∥ ψ(t) ∥ ∥ ∥ α dt = 0. again using eq. (3.9) it follows that lim t → ∞ 1 2t ∫ t −t ∥ ∥ ∥ ψ(t) ∥ ∥ ∥ α dt ≤ lim t → ∞ m(α) 2t ∫ t −t ∫ +∞ 0 s−αe− δ 2 s ∥ ∥ ∥ ζ(t − s) ∥ ∥ ∥ α dsdt ≤ lim t → ∞ m(α) ∫ +∞ 0 s−αe− δ 2 s 1 2t ∫ t −t ∥ ∥ ∥ ζ(t − s) ∥ ∥ ∥ α dtds. let γs(t) = 1 2t ∫ t −t ∥ ∥ ∥ ζ(t − s) ∥ ∥ ∥ α dt. since pap0(xα) is translation invariant it follows that t 7→ ζ(t − s) belongs to pap0(xα) for each s ∈ r, and hence lim t 7→ ∞ 1 2t ∫ t −t ∥ ∥ ∥ ζ(t − s) ∥ ∥ ∥ α dt = 0 for each s ∈ r. cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 129 one completes the proof by using the lebesgue dominated convergence theorem and the fact γs(t) 7→ 0 as t → ∞ for each s ∈ r. the proof for the second integral in the expression of eq. (4.4) is similar to that of su(·) and hence moitted. (in that case, one makes use of eq. (3.8) rather than eq. (3.9).) using the composition of pseudo-almost automorphic functions and theorem 4.1, it is easy to see that the following technical lemmas hold. lemma 4.1. under assumptions (h.1)-(h.2)-(h.3), then the integral operator s1 defined above maps paa(xα) into itself. lemma 4.2. under assumptions (h.1)-(h.2)-(h.3), the integral operator s1 defined above is a contraction whenever l is small enough. proof. let v,w ∈ paa(xα). now, ∥ ∥ ∥ s11v(t) − s11w(t) ∥ ∥ ∥ α ≤ ∫ t −∞ m(α)(t − s)−αe− δ 2 (t−s) ∥ ∥ ∥ f1(s,v(s)) − f1(s,w(s)) ∥ ∥ ∥ ds ≤ c ∫ t −∞ m(α)(t − s)−αe− δ 2 (t−s) ∥ ∥ ∥ f1(s,v(s)) − f1(s,w(s)) ∥ ∥ ∥ β ds ≤ lcm(α) ∫ t −∞ (t − s)−αe− δ 2 (t−s) ∥ ∥ ∥ v(s) − w(s) ∥ ∥ ∥ ds ≤ lc′cm(α) ∫ t −∞ (t − s)−αe− δ 2 (t−s) ∥ ∥ ∥ v(s) − w(s) ∥ ∥ ∥ α ds. similarly, ∥ ∥ ∥ s12v(t) − s12w(t) ∥ ∥ ∥ α ≤ ∫ ∞ t c(β)e−δ(t−s) ∥ ∥ ∥ f1(s,v(s)) − f1(s,w(s)) ∥ ∥ ∥ ds ≤ cc(β) ∫ ∞ t e−δ(t−s) ∥ ∥ ∥ f1(s,v(s)) − f1(s,w(s)) ∥ ∥ ∥ β ds ≤ lcc(β) ∫ ∞ t e−δ(t−s) ∥ ∥ ∥ v(s) − w(s) ∥ ∥ ∥ ds ≤ lcc′c(β) ∫ ∞ t e−δ(t−s) ∥ ∥ ∥ v(s) − w(s) ∥ ∥ ∥ α ds. consequently, ∥ ∥ ∥ s1v − s1w ∥ ∥ ∥ ∞ ,α ≤ lcc′ ( m(α)γ(1 − α)(2δ−1)1−α + c(β)δ−1 )∥ ∥ ∥ v − w ∥ ∥ ∥ ∞ ,α and hence s1 is a contraction whenever l is small enough. 130 toka diagana and ahmed mohamed cubo 13, 3 (2011) theorem 4.2. suppose assumptions (h.1)-(h.2)-(h.3) and that l is small enough, then the autonmous differential equation eq. (4.1) has a unique pseudo almost automorphic solution u satisfying u = s1u. proof. this is an immediate consequence of lemma 4.1, lemma 4, lemma 4.2, and the banach fixed point theorem. 5 pseudo almost automorphic solutions to some secondorder differential equations we have previously seen that each u ∈ h can be written in terms of the sequence of orthogonal projections en as follows: u = ∞∑ n=1 γn∑ k=1 〈u,ekn〉e k n = ∞∑ n=1 enu. moreover, for each u ∈ d(a), au = ∞∑ j=1 λj γj∑ k=1 〈u,ekj 〉e k j = ∞∑ j=1 λjeju. therefore, for all z := ( u v ) ∈ d = d(b) = d(a) × h, we obtain the following bz =     0 ih −ba −aih         u v     =     v −bau − av     =         ∞∑ n=1 env −b ∞∑ n=1 λnenu − a ∞∑ n=1 env         = ∞∑ n=1     0 1 −bλn −a         en 0 0 en         u v     = ∞∑ n=1 anpnz, cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 131 where pn :=     en 0 0 en     , n ≥ 1, and an :=     0 1 −bλn −a     , n ≥ 1. (5.1) now, the characteristic equation for an is given by λ2 + aλ + λnb = 0 (5.2) whose eigenvalues are given by λn1 := −a + √ a2 − 4λnb 2 and λn2 := −a − √ a2 − 4λnb 2 . since a > 0 it follows that all roots of eq. (5.2) are nonzero. moreover, the real part of each of its roots is: −a/2 < 0. therefore, there exists ω ∈ ( π 2 ,π ) such that ρ(l) ⊃ sω, where sω := { z ∈ c \ {0} : ∣ ∣ ∣ arg z ∣ ∣ ∣ < ω } . on the other hand, one can show without difficulty that an = k −1 n jnkn, where jn,kn and k−1n are respectively given by jn =     λn1 0 0 λn2     , kn =     1 1 λn1 (t) λ n 2     , and k−1n = 1 λn 1 − λn 2     −λn2 1 λn1 −1     . for λ ∈ sω and z ∈ x, one has r(λ,b)z = ∞∑ n=1 (λ − an) −1pnz = ∞∑ n=1 kn(λ − jnpn) −1k−1n pnz. 132 toka diagana and ahmed mohamed cubo 13, 3 (2011) hence, ∥ ∥ ∥ r(λ,b)z ∥ ∥ ∥ 2 ≤ ∞∑ n=1 ∥ ∥ ∥ knpn(λ − jnpn) −1k−1n pn ∥ ∥ ∥ 2 b(x) ∥ ∥ ∥ pnz ∥ ∥ ∥ 2 ≤ ∞∑ n=1 ∥ ∥ ∥ knpn ∥ ∥ ∥ 2 b(x) ∥ ∥ ∥ (λ − jnpn) −1 ∥ ∥ ∥ 2 b(x) ∥ ∥ ∥ k−1n pn ∥ ∥ ∥ 2 b(x) ∥ ∥ ∥ pnz ∥ ∥ ∥ 2 . moreover, for z := ( z1 z2 ) ∈ x, we obtain ∥ ∥ ∥ knpnz ∥ ∥ ∥ 2 = ∥ ∥ ∥ enz1 + enz2 ∥ ∥ ∥ 2 + ∥ ∥ ∥ λn1 enz1 + λ n 2 enz2 ∥ ∥ ∥ 2 ≤ 3 ( 1 + ∣ ∣ ∣ λn1 ∣ ∣ ∣ 2)∥ ∥ ∥ z ∥ ∥ ∥ 2 . thus, there exists c1 > 0 such that ∥ ∥ ∥ knpnz ∥ ∥ ∥ ≤ c1 ∣ ∣ ∣ λn1 ∣ ∣ ∣ ∥ ∥ ∥ z ∥ ∥ ∥ for all n ≥ 1. similarly, for z := ( z1 z2 ) ∈ x, one can show that there is c2 > 0 such that ∥ ∥ ∥ k−1n pnz ∥ ∥ ∥ ≤ c2 ∣ ∣ ∣ λn 1 ∣ ∣ ∣ ∥ ∥ ∥ z ∥ ∥ ∥ for all n ≥ 1. now, for z ∈ x, we have ∥ ∥ ∥ (λ − jnpn) −1z ∥ ∥ ∥ 2 = ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥       1 λ−λn 1 0 0 1 λ−λn 2             z1 z2       ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ 2 ≤ 1 |λ − λn 1 |2 ∥ ∥ ∥ z1 ∥ ∥ ∥ 2 + 1 |λ − λn 2 |2 ∥ ∥ ∥ z2 ∥ ∥ ∥ 2 . let λ0 > 0. define the function η(λ) := 1 + ∣ ∣ ∣ λ ∣ ∣ ∣ ∣ ∣ ∣ λ − λn 2 ∣ ∣ ∣ . it is clear that η is continuous and bounded on the closed set σ := { λ ∈ c : ∣ ∣ ∣ λ ∣ ∣ ∣ ≤ λ0, ∣ ∣ ∣ arg λ ∣ ∣ ∣ ≤ ω } . cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 133 on the other hand, it is clear that η is bounded for ∣ ∣ ∣ λ ∣ ∣ ∣ > λ0. thus η is bounded on sω. if we take n = sup    1 + ∣ ∣ ∣ λ ∣ ∣ ∣ ∣ ∣ ∣ λ − λ j n ∣ ∣ ∣ : λ ∈ sω,n ≥ 1 ; j = 1,2,    . therefore, ∥ ∥ ∥ (λ − jnpn) −1z ∥ ∥ ∥ ≤ n 1 + ∣ ∣ ∣ λ ∣ ∣ ∣ ‖z‖, λ ∈ sω. consequently, ∥ ∥ ∥ r(λ,b) ∥ ∥ ∥ ≤ k 1 + ∣ ∣ ∣ λ ∣ ∣ ∣ for all λ ∈ sω and t ∈ r. in view of the above, b is sectorial. let (eτb)τ≥0 be the nalytic semigroup associated with it. let us show that (eτb)τ≥0 is exponentially stable. now eτbz = ∞∑ n=0 k−1n pne τjnpnknpnz, z ∈ x. on the other hand, we have ∥ ∥ ∥ eτbz ∥ ∥ ∥ = ∞∑ n=0 ∥ ∥ ∥ k−1n pn ∥ ∥ ∥ b(x) ∥ ∥ ∥ eτjnpn ∥ ∥ ∥ b(x) ∥ ∥ ∥ knpn ∥ ∥ ∥ b(x) ∥ ∥ ∥ pnz ∥ ∥ ∥ , with for each z = ( z1 z2 ) ∥ ∥ ∥ eτjnpnz ∥ ∥ ∥ 2 = ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥       eλ n 1 τen 0 0 eλ n 2 τen             z1 z2       ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ 2 ≤ ∥ ∥ ∥ eλ n 1 τenz1 ∥ ∥ ∥ 2 + ∥ ∥ ∥ eλ n 2 τenz2 ∥ ∥ ∥ 2 ≤ e−aτ ∥ ∥ ∥ z ∥ ∥ ∥ 2 . therefore, ∥ ∥ ∥ eτb ∥ ∥ ∥ ≤ ce−aτ, τ ≥ 0. (5.3) it is now clear that if l is small enough, then the second-order differential equation eq. (1.3) has at most one solution ( u v ) ∈ xα = hα × h, which in addition is pseudo almost automorphic. therefore, eq. (1.2) has a unique solution u ∈ hα, which in addition is pseudo almost automorphic. 134 toka diagana and ahmed mohamed cubo 13, 3 (2011) 6 examples 6.1 1-dimensional sine-gordon equation let l > 0 and and let j = (0,l). let h = l2(j) be equipped with its natural topology. our main objective here is to study the existence of pseudo almost automorphic solutions to a somewhat general one-dimensional sine-gordon equation with dirichlet boundary conditions, which had been studied in the literature especially by leiva [36] in the following form ∂2u ∂t2 + c ∂u ∂t − d ∂2u ∂x2 + k sin u = p(t,x), t ∈ r, x ∈ j (6.1) u(t,0) = u(t,l) = 0, t ∈ r (6.2) where c,d,k are positive constants, p : r × j 7→ r is continuous and bounded. precisely, we are interested in the following system of second-order partial differential equations ∂2u ∂t2 + a ∂u ∂t − b ∂2u ∂x2 = q(t,x,u), t ∈ r, x ∈ j (6.3) u(t,0) = u(t,l) = 0, t ∈ r (6.4) where a,b > 0 and q : r × j × l2(j) 7→ l2(j) is pseudo-almost automorphic. let us take av = −v′′ for all u ∈ d(a) = h10(j) ∩ h 2(j) and suppose that q : r × j × l2(j) 7→ h β 0 (j) is pseudo-almost automorphic. moreover, q is lipschitz in the following sense: there exists l′′ > 0 for which ∥ ∥ ∥ q(t,x,u) − q(t,x,v) ∥ ∥ ∥ h β 0 (j) ≤ l′′ ∥ ∥ ∥ u − v ∥ ∥ ∥ 2 for all u,v ∈ l2(j), x ∈ j and t ∈ r. consequently, the system eq. (6.3) eq. (6.4) has at most one solution u ∈ paa(h10(j)) when l′′ is small enough. 6.2 n-dimensional sine-gordon equation let ω ⊂ rn (n ≥ 1) be a open bounded subset with c2 boundary γ = ∂ω and let h = l2(ω) equipped with its natural topology ‖ · ‖l2(ω). here, we are interested in the n-dimensional sinegordon studied in the previous example, that is, the system of second-order partial differential equations given by cubo 13, 3 (2011) pseudo-almost automorphic solutions . . . 135 ∂2u ∂t2 + a ∂u ∂t − b∆u = r(t,x,u), t ∈ r, x ∈ ω (6.5) u(t,x) = 0, t ∈ r, x ∈ ∂ω (6.6) where a,b > 0, and r : r × ω × l2(ω) 7→ l2(ω) is jointly continuous. define the linear operator a as follows: au = −∆u for all u ∈ d(a) = h10(ω) ∩ h 2(ω). for each µ ∈ (0,1), we take hµ = d((−∆) µ) = h µ 0 (ω) ∩ h2µ(ω) equipped with its µ-norm ‖ · ‖µ. suppose that r : r × ω × l2(ω) 7→ h β 0 (ω) is pseudo-almost automorphic. moreover, r is lipschitz in the following sense: there exists l′′′ > 0 for which ∥ ∥ ∥ r(t,x,u) − r(t,x,v) ∥ ∥ ∥ β ≤ l′′′ ∥ ∥ ∥ u − v ∥ ∥ ∥ 2 for all u,v ∈ l2(ω), x ∈ ω and t ∈ r. therefore, the system eq. 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[59] s. zaidman, topics in abstract differential equations, pitman research notes in mathematics ser. ii john wiley and sons, new york, 1994–1995. introduction preliminaries sectorial linear operators pseudo-almost automorphic functions main results pseudo almost automorphic solutions to some second-order differential equations examples 1-dimensional sine-gordon equation n-dimensional sine-gordon equation cubo a mathematical journal vol.14, no¯ 02, (183–196). june 2012 k-theory for the c∗-algebras of continuous functions on certain homogeneous spaces in semi-simple lie groups takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, senbaru 1, nishihara, okinawa 903-0213, japan. email: sudo@math.u-ryukyu.ac.jp abstract we study k-theory for the c∗-algebras of all continuous functions on certain homogeneous spaces in the semi-simple connected lie groups sln(r) by the discrete subgroups sln(z), mainly. as a byproduct, we also consider a certain nilpotent case similarly. resumen estudiamos la k-teoŕıa para las c∗-álgebras de todas las funciones continuas sobre ciertos espacios homogéneos, principalmente en los grupos de lie conexos semisimples sln(r) y subgrupos discretos sln(z). como subproducto consideramos un caso nilpotente en forma análoga. keywords and phrases: c*-algebra, k-theory, homogeneous space, semi-simple lie group, discrete subgroup. 2000 ams mathematics subject classification: primary 46l80, 22d25, 22e15. 184 takahiro sudo cubo 14, 2 (2012) 1 introduction this work is started with an attempt to find a candidate for the k-theory groups for the full or reduced group c∗-algebras of the discrete groups sln(z). our idea comes from the fact that k-theory for the group c∗-algebra of the discrete groups zn of integers is the same as that for the c∗-algebra of all continuous functions on the tori tn viewed as the quotient rn/zn, via the fourier transform, and that this picture should have some similar meanings in more general or noncommutative setting, at least in k-theory level. refer to [5] for some basics of k-theory and c∗-algebras. after a quick review in section 2 about the abelian case of commutative connected lie groups, we consider in section 3 homogeneous spaces in sl2(r) a semi-simple connected lie group and compute the k-theory groups of the c∗-algebras of all continuous functions on those spaces. moreover, we consider the case of sln(r) (n ≥ 3) in section 4. the results obtained would be useful for further research in this direction. furthermore, as a byproduct, we consider a certain nilpotent case of discrete heisenberg groups. 2 abelian case for convenience, recall that we have the following short exact sequence of abelian (or commutative lie) groups: 0 → zn → rn → tn → 0. consider their group c∗-algebras c∗(zn), c∗(rn), and c∗(tn). by fourier transform, they are isomorphic respectively to c(tn), c0(r n), and c0(z n) the c∗-algebras of all continuous functions on tn, on rn and zn vanishing at infinity. their k-theory groups are well known as follows ([5]): kj(c ∗ (z n )) ∼= kj(c(t n )) ∼= z 2 n−1 , (j = 0, 1); k0(c ∗(r2n)) ∼= k0(c0(r 2n)) ∼= k0(c) ∼= z, k1(c ∗(r2n)) ∼= k1(c) ∼= 0, k0(c ∗(r2n−1)) ∼= k0(c0(r 2n−1)) ∼= k1(c) ∼= 0, k1(c ∗(r2n−1)) ∼= k0(c) ∼= z, k0(c ∗(tn)) ∼= k0(c0(z n)) ∼= ⊕ z n z, k1(c ∗(tn)) ∼= k1(c0(z n)) ∼= 0, where ⊕k means the k-times direct sum. observe that k-theory of the group c∗-algebra of the discrete group zn is the same as that of the c∗-algebra of all continuous functions on the quotient t n = rn/zn. 3 homogeneous spaces in sl2(r) consider the following inclusion and its homogeneous space denoted as: 0 → sl2(z) → sl2(r), sl2(r)/sl2(z) ≡ h2. cubo 14, 2 (2012) k-theory for the c∗-algebras of continuous functions ... 185 let sl2(r) = kan be the iwasawa decomposition. more precisely, we have the following homeomorphism: sl2(r) ≈ kan = so(2)a2n2, where so(2) = {( cos θ − sin θ sin θ cos θ ) | θ ∈ r } ∼= s 1 = {eiθ | θ ∈ r}, a2 = {( a1 0 0 a2 ) | a1a2 = 1, a1 > 0, a2 > 0 } , n2 = {( 1 b 0 1 ) | b ∈ r } . it follows that sl2(z) ≈ kzaznz = so(2)za2,zn2,z, where so(2)z ∼= s 1 z = {eiθ ∈ z2 | θ ∈ r} = {(±1, 0), (0, ±1)}, a2,z = {( 1 0 0 1 )} , n2,z = {( 1 b 0 1 ) | b ∈ z } . it follows from considering quotient spaces that the homogeneous space h2 is homeomorphic to the following product space: h2 ≈ (⊔ 4 r)+ × r × t, where ⊔kr means the disjoint union of k copies of r, and x+ means the one-point compactification of x, and so(2)/so(2)z ≈ (⊔ 4 r)+, and a2 ≈ r, and n2/n2,z ≈ t. let c0(h2) be the c ∗-algebra of all continuous functions on h2 vanishing at infinity. we compute its k-theory groups as follows. first of all, we have kj(c0(h2)) ∼= kj(c0((⊔ 4 r)+ × r × t)) ∼= kj+1(c((⊔ 4 r)+ × t)), by the bott periodicity, where j + 1 (mod 2). consider the following short exact sequence of c∗-algebras: 0 −−−−→ c0((⊔ 4 r) × t) i −−−−→ c((⊔4r)+ × t) q −−−−→ c(t) −−−−→ 0. note that this extension of c∗-algebras splits, clearly. we then have the following six-term exact sequence of k-groups: k0(c0((⊔ 4 r) × t)) i∗ −−−−→ k0(c((⊔ 4 r)+ × t)) q∗ −−−−→ k0(c(t)) x     y k1(c(t)) q∗ ←−−−− k1(c((⊔ 4 r)+ × t)) i∗ ←−−−− k1(c0((⊔ 4 r) × t)), 186 takahiro sudo cubo 14, 2 (2012) with kj(c0((⊔ 4 r) × t)) ∼= ⊕ 4kj(c0(r × t)) ∼= ⊕ 4kj+1(c(t)) ∼= z 4 for j = 0, 1, where ⊕k means the direct sum of k copies. the commutative diagram also splits into two short exact sequences of k0 and k1-groups, by the splitting short exact sequence of c ∗-algebras. therefore, we obtain 0 → z4 → kj(c((⊔ 4 r) + × t)) → z → 0 for j = 0, 1. since extensions of groups by z also split, certainly known, we obtain that kj(c((⊔ 4 r)+× t)) ∼= z5 for j = 0, 1. hence we get theorem 3.1. let h2 = sl2(r)/sl2(z) = kan/kzaznz be the homogeneous space via the iwasawa decomposition. then h2 is homeomorphic to the product space (⊔ 4 r)+ × r × t, and kj(c0(h2)) ∼= z 5, (j = 0, 1). moreover, we obtain proposition 3.2. let k/kz = so(2)/so(2)z = kan/kzan be the homogeneous space of the compact group so(2). then k/kz is the compact space (⊔ 4 r)+, and k0(c(k/kz)) ∼= z and k1(c(k/kz)) ∼= z 4. proof. consider the following short exact sequence of c∗-algebras: 0 −−−−→ c0(⊔ 4 r) i −−−−→ c((⊔4r)+) q −−−−→ c −−−−→ 0. note that this extension of c∗-algebras splits. we then have the following six-term exact sequence of k-groups: k0(c0(⊔ 4 r)) i∗ −−−−→ k0(c((⊔ 4 r)+)) q∗ −−−−→ k0(c) x     y k1(c) q∗ ←−−−− k1(c((⊔ 4 r)+)) i∗ ←−−−− k1(c0(⊔ 4 r)), with kj(c0((⊔ 4 r))) ∼= ⊕ 4kj(c0(r)) ∼= ⊕ 4kj+1(c) for j = 0, 1. the commutative diagram also splits into two short exact sequences of k0 and k1-groups. therefore, we obtain that k0(c((⊔ 4 r)+)) ∼= z and k1(c((⊔ 4 r)+)) ∼= z4. remark. note that the quotient space n/nz is isomorphic to t as a group. thus, kj(c(n/nz)) ∼= z for j = 0, 1. furthermore, we have cubo 14, 2 (2012) k-theory for the c∗-algebras of continuous functions ... 187 proposition 3.3. the homogeneous space sl2(r)/k = an is homeomorphic to the product space r × t, and kj(c0(an)) ∼= z for j = 0, 1. proof. we have kj(c0(r × t)) ∼= kj+1(c(t)) ∼= z for j = 0, 1. notes. it is shown by natsume [2] that for c∗(sl2(z)) the full group c ∗-algebra of sl2(z), k0(c ∗(sl2(z))) ∼= z 8, k1(c ∗(sl2(z))) ∼= 0, and the same holds by replacing c∗(sl2(z)) with its reduced group c ∗-algebra of the regular representation of sl2(z). more precisely, since sl2(z) is isomorphic to the amalgam z4 ∗z2 z6 of cyclic groups with orders 2, 4, 6, we have c∗(sl2(z)) isomorphic to the amalgam c ∗(z4)∗c∗(z2) c ∗(z6) of their group c∗-algebras, so that kj(c ∗(z4) ∗c∗(z2) c ∗(z6)) ∼= (kj(c ∗(z4)) ⊕ kj(c ∗(z6)))/kj(c ∗(z2)) for j = 0, 1. in particular, k0(c ∗(sl2(z))) ∼= z 8 ∼= z10/z2. also, kj(c ∗(z4) ∗ c ∗(z6)) ∼= kj(c ∗(z4)) ⊕ kj(c ∗(z6)) for j = 0, 1, where c∗(z4)∗c ∗(z6) is the full free product of c ∗-algebras. more generally, for a∗b the full free product of c∗-algebras a and b, we have ([1]) kj(a ∗ b) ∼= kj(a) ⊕ kj(b), (j = 0, 1). corollary 1. we have k0(c0(h2)) ⊕ k1(c0(h2)) ∼= k0(c ∗ (z4) ∗ c ∗ (z6)) ⊕ k1(c ∗ (z4) ∗ c ∗ (z6)), as a group, but k0(c0(h2)) ⊕ k1(c0(h2)) 6∼= k0(c ∗(sl2(z))) ⊕ k1(c ∗(sl2(z))). remark. since 10 > 8, it may say to be possible that k-theory data of the homogeneous space c∗-algebra contains that of the group c∗-algebra of sl2(z). in fact, in the group non-isomorphic equation above, the right hand side can be a quotient of the left hand side. this picture might be extended to the more general setting. 188 takahiro sudo cubo 14, 2 (2012) 4 homogeneous spaces in sln(r) consider the following inclusion and its homogeneous space denoted as: 0 → sln(z) → sln(r), sln(r)/sln(z) ≡ hn. let sln(r) = kan be the iwasawa decomposition. more precisely, we have the following homeomorphism: sln(r) ≈ kan = so(n)annn, where an =        a1 0 ... 0 an     | πnj=1aj = 1, aj > 0    , nn =           1 b12 · · · b1n ... ... ... ... bn−1,n 0 1        | bi,j ∈ r, (i < j)    . it follows that sln(z) ≈ kzaznz = so(n)zan,znn,z, where so(n)z consists of all matrices of so(n) with components of integers, an,z of only the n-th identity matrix, and nn,z of all matrices of nn with components of integers. it follows from considering quotient spaces that the homogeneous space hn is homeomorphic to the following product space: hn ≈ (so(n)/so(n)z) × r n−1 × t (n−1)n 2 , where an ≈ r n−1 and nn/nn,z ≈ t (n−1)n 2 . recall that as a topological space, so(n)/so(n − 1) ≈ sn−1, where sn−1 is the n − 1 dimensional sphere. indeed, so(n) acts transitively on sn−1 by matrix multiplication, and the isotropy group for the n-th standard basis vector in sn−1 is so(n − 1), from which the homeomorphism is obtained. however, these quotient spaces do not split in general into the product spaces: so(n) ≈ so(n − 1) × sn−1, but this is certainly true if and only if there is a continuous section from sn−1 to so(n). this is just the cases where n = 4 or n = 8, a well-known, non-tirvial, important result in algebraic topology. note that what is necessary in what follows may be the isomorphisms in topological k-theory level: kj(so(n)) ∼= k j (so(n − 1) × sn−1) cubo 14, 2 (2012) k-theory for the c∗-algebras of continuous functions ... 189 (or mere replacements). we have shown that so(2)/so(2)z ≈ s 1/s1 z . if we assume the homeomorphisms for so(n), inductively we have so(n)/so(n)z ≈ (so(n − 1)/so(n − 1)z) × (s n−1/sn−1 z ), where sn−1 z means the set of all integral points in sn−1, and the equivalence relation on sn−1 by sn−1 z is defined as: for ξ, η ∈ sn−1, we have ξ ∼ η if and only if ξ = η, or ξ, η ∈ sn−1 z . therefore, we obtain so(n)/so(n)z ≈ (s1/sz) × · · · × (s n−1/sn−1 z ). however, this may not be true in general, but even in such a case, we may replace so(n)/so(n)z by the product space in the right hand side, as a reasonable candidate, and we continue. but what is necessary in what follows may be the isomorphisms in topological k-theory level: kj(so(n)/so(n)z) ∼= k j((so(n − 1)/so(n − 1)z) × (s n−1/sn−1 z )) (or mere replacements). we also have sn−1 z = {(±1, 0, · · · , 0), (0, ±1, 0, · · · , 0), · · · , (0, · · · , 0, ±1) ∈ rn}. hence we identify sn−1 z with ⊔nz2 the n-fold disjoint union of z2 = z/2z. therefore, we get sn−1/sn−1 z ≈ sn−1/ ⊔n z2. let c0(hn) be the c ∗-algebra of all continuous functions on hn vanishing at infinity. we compute its k-theory groups as follows. first of all, we have kj(c0(hn)) ∼= kj(c0((so(n)/so(n)z) × r n−1 × t (n−1)n 2 )) ∼= kj+n−1(c(so(n)/so(n)z) × t (n−1)n 2 )), by the bott periodicity, where j + n − 1 (mod 2). now let sn = so(n)/so(n)z and tn = t (n−1)n 2 . since c(sn × tn) ∼= c(sn) ⊗ c(tn) a c∗-tensor product, the künneth formula implies k0(c(sn × tn)) ∼= (k0(c(sn)) ⊗ k0(c(tn))) ⊕ (k1(c(sn)) ⊗ k1(c(tn))), k1(c(sn × tn)) ∼= (k0(c(sn)) ⊗ k1(c(tn))) ⊕ (k1(c(sn)) ⊗ k0(c(tn))). for j = 0, 1, we have kj(c(tn)) = kj(c(t (n−1)n 2 )) ∼= z 2 2−1(n−1)n−1 = z 2 2−1(n−2)(n+1) . let sk/sk z = vk for 1 ≤ k ≤ n − 1 and (s 1/s1 z ) × · · · × (sk/sk z ) = uk. since we have c((s1/sz) × · · · × (s n−1/sn−1 z )) ∼= c(s1/sz) ⊗ · · · ⊗ c(s n−1/sn−1 z ), 190 takahiro sudo cubo 14, 2 (2012) the künneth formula implies that, for instance, k0(c(u3)) ∼= ⊕(i1,i2,i3)∈i3ki1(c(v1)) ⊗ ki2(c(v2)) ⊗ ki3(c(v3)), k1(c(u3)) ∼= ⊕(j1,j2,j3)∈j3kj1(c(v1)) ⊗ kj2(c(v2)) ⊗ kj3(c(v3)), where i3 = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}, j3 = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)}, where note that for each tuple in i3, the number of 0 is 3 or 1 odd, while for each tuple in j3, the number of 0 is 2 or 0 even, and the cardinal numbers of i3 and j3 are computed as: |i3| = 3c3 + 3c1 = 1 + 3 = 2 2, |j3| = 3c2 + 3c0 = 3 + 1 = 2 2, where nck means the combination of k elements in n elements. as one more example, similarly, |i4| = 4c4 + 4c2 + 4c0 = 1 + 6 + 1 = 2 3, |j4| = 4c3 + 4c1 = 4 + 4 = 2 3. therefore, more generally, we have k0(c(uk)) ∼= ⊕(i1,··· ,ik)∈ikki1(c(v1)) ⊗ · · · ⊗ kik(c(vk)), k1(c(uk)) ∼= ⊕(j1,··· ,jk)∈jkkj1(c(v1)) ⊗ · · · ⊗ kjk(c(vk)), where if k is even, then |ik| = kck + kck−2 + · · · + kc0 = 2 k, |jk| = kck−1 + kck−3 + · · · + kc1 = 2 k. and if k is odd, then |ik| = kck + kck−2 + · · · + kc1 = 2 k, |jk| = kck−1 + kck−3 + · · · + kc0 = 2 k, and in both cases, ik and jk consist of tuples with elements 0 or 1 chosen accordingly to the above combinatorial sums. note that the quotient space vk−1 is just vk−1 = s k−1/ ⊔k z2 = (s k−1 \ (⊔kz2)) + ≡ v+k the one-point compactification v+k of the open subspace vk of s k−1obtained by removing points of ⊔nz2 from s k−1. consider the following short exact sequence of c∗-algebras: 0 −−−−→ c0(vk) i −−−−→ c(v+k ) q −−−−→ c −−−−→ 0. cubo 14, 2 (2012) k-theory for the c∗-algebras of continuous functions ... 191 note that this extension of c∗-algebras splits, clearly. we then have the following six-term exact sequence of k-groups: k0(c0(vk)) i∗ −−−−→ k0(c(v + k )) q∗ −−−−→ k0(c) x     y k1(c) q∗ ←−−−− k1(c(v + k )) i∗ ←−−−− k1(c0(vk)), and the commutative diagram also splits into two short exact sequences of k0 and k1-groups. it follows that k0(c(v + k )) ∼= k0(c0(vk)) ⊕ z, k1(c(v + k )) ∼= k1(c0(vk)). moreover consider the following short exact sequence of c∗-algebras: 0 −−−−→ c0(vk) i −−−−→ c(sk−1) q −−−−→ ⊕2kc −−−−→ 0 corresponding to attaching 2k points to 2k holes in vk to make s k−1. we then have the following six-term exact sequence of k-groups: k0(c0(vk)) i∗ −−−−→ k0(c(s k−1)) q∗ −−−−→ ⊕2kk0(c) x     y ⊕2kk1(c) q∗ ←−−−− k1(c(s k−1)) i∗ ←−−−− k1(c0(vk)). furthermore consider the following short exact sequence of c∗-algebras: 0 −−−−→ c0(r k−1) i −−−−→ c(sk−1) q −−−−→ c −−−−→ 0, where note that sk−1 ≈ (rn−1)+. note that this extension of c∗-algebras splits, clearly. we then have the following six-term exact sequence of k-groups: k0(c0(r k−1)) i∗ −−−−→ k0(c(s k−1)) q∗ −−−−→ k0(c) x     y k1(c) q∗ ←−−−− k1(c(s k−1)) i∗ ←−−−− k1(c0(r k−1)) and the commutative diagram also splits into two short exact sequences of k0 and k1-groups. it follows that for k ≥ 2, k0(c(s k−1 )) ∼= k0(c0(r k−1 )) ⊕ z ∼= { z if k even, z 2 if k odd; k1(c(s k−1)) ∼= k1(c0(r k−1)) ∼= { z if k even, 0 if k odd. 192 takahiro sudo cubo 14, 2 (2012) therefore, we obtain that if k is even, then k0(c0(vk)) i∗ −−−−→ z q∗ −−−−→ ⊕2kz x     y 0 q∗ ←−−−− z i∗ ←−−−− k1(c0(vk)) and if k is odd, then k0(c0(vk)) i∗ −−−−→ z2 q∗ −−−−→ ⊕2kz x     y 0 q∗ ←−−−− 0 i∗ ←−−−− k1(c0(vk)). in both cases, the k0-class corresponding to the unit of c(s k−1) is mapped injectively under the map q∗, while the k0-class corresonding to the bott projection in a matrix algebra over c(s k−1) for k odd is mapped to zero under q∗. it follows that if k is even, then k0(c(vk)) ∼= 0, while if k is odd, then k0(c(vk)) ∼= z. therefore, we obtain that if k is even, then k1(c0(vk)) ∼= z 2k, and if k is odd, then k1(c0(vk)) ∼= z 2k−1. hence we get k0(c(vk−1)) ∼= k0(c(v + k )) ∼= { z if k even, z 2 if k odd; k1(c(vk−1)) ∼= k1(c(v + k )) ∼= { z 2k if k even, z 2k−1 if k odd. note that the case where k = 2 is considered in the previous section. summing up the argument above, we obtain theorem 4.1. let hn = sln(r)/sln(z) = kan/kzaznz be the homogeneous space via the iwasawa decomposition. then hn is homeomorphic to the product space (so(n)/so(n)z)×r n−1× t (n−1)n 2 , and k0(c0(hn)) ∼= k1(c0(hn)) ∼= ⊕j=0,1(kj(c(so(n)/so(n)z)) ⊗ z 2 (n−2)(n+1)2−1 ). proof. if n is even, then k0(c0(hn)) ∼= k1(c(so(n)/so(n)z) ⊗ c(t (n−1)n 2 )) ∼= (k0(c(tn)) ⊗ z 2 (n−2)(n+1) 2 ) ⊕ k1(c(tn)) ⊗ z 2 (n−2)(n+1) 2 ), k1(c0(hn)) ∼= k0(c(so(n)/so(n)z) ⊗ c(t (n−1)n 2 )) ∼= (k0(c(tn)) ⊗ z 2 (n−2)(n+1) 2 ) ⊕ k1(c(tn)) ⊗ z 2 (n−2)(n+1) 2 ), cubo 14, 2 (2012) k-theory for the c∗-algebras of continuous functions ... 193 where tn = so(n)/so(n)z for short, and in particular, we get k0(c0(hn)) ∼= k1(c0(hn)). if n is odd, then we can deduce the same conclusions by the same calculation as above. remark. the results obtained above and below in k-theory might contain (some of) k-theory data for the (full or reduced) group c∗-algebra of sln(z) or the (full or reduced) free product c∗-algebra corresponding to the generators of sln(z). it is known that if n ≥ 3, then sln(z) is not an amalgam, but a certain multi-amalgam of subgroups, by soulé [4]. moreover, we obtain proposition 4.2. let k/kz = so(n)/so(n)z = kan/kzan be the homogeneous space of the compact group so(n). for convenience, as a candidate, we replace k/kz with the compact product space: (s1/s1 z ) × (s2/s2 z ) · · · × (sn−1/sn−1 z ), which is identified with (s1/ ⊔2 z2) × (s 2/ ⊔3 z2) × · · · × (s n−1/ ⊔n z2) ≈ (s1 \ ⊔2z2) + × (s2 \ ⊔3z2) + × · · · × (sn−1 \ ⊔nz2) +, or we may assume that we replace the topological k-theory of k/kz with that of the product space. then k0(c(k/kz)) ∼= ⊕(i1,i2,··· ,in−1)∈in−1(ki1(c(v1)) ⊗ · · · ⊗ kin−1(c(vn−1))), k1(c(k/kz)) ∼= ⊕(j1,j2,··· ,jn−1)∈jn−1(kj1(c(v1)) ⊗ · · · ⊗ kjn−1(c(vn−1))), with vk = s k/sk z , where if n is odd , then |in−1| = n−1cn−1 + n−1cn−3 + · · · + n−1c0 = 2 n−1, |jn−1| = n−1cn−2 + n−1cn−4 + · · · + n−1c1 = 2 n−1. and if n is even, then |in−1| = n−1cn−1 + n−1cn−3 + · · · + n−1c1 = 2 n−1, |jn−1| = n−1cn−2 + n−1cn−4 + · · · + n−1c0 = 2 n−1, and in both cases, in−1 and jn−1 consist of the tuples with elements 0 or 1 chosen accordingly to the above combinatorial sums. moreover, we obtain k0(c(vk−1)) ∼= { z if k even, z 2 if k odd; k1(c(vk−1)) ∼= { z 2k if k even, z 2k−1 if k odd. 194 takahiro sudo cubo 14, 2 (2012) remark. for example, as n = 5 we compute k0(c(v1)) ⊗ k1(c(v2)) ⊗ k1(c(v3)) ⊗ k0(c(v4)) ∼= z ⊗ z 3 ⊗ z6 ⊗ z2 ∼= z 3·6·2 = z36, where (0, 1, 1, 0) ∈ i4. note that the quotient space n/nz is homeomorphic to t (n−1)n2 −1 as a space. thus, kj(c(n/nz)) ∼= z 2 (n−2)(n+1)2−1 for j = 0, 1. furthermore, we have proposition 4.3. the homogeneous space sln(r)/k = an is homeomorphic to the product space r n−1 × t(n−1)n2 −1 , and kj(c0(an)) ∼= z 2 2−1(n−2)(n+1) for j = 0, 1. proof. we have kj(c0(r n−1 × t (n−1)n 2 )) ∼= kj+n−1(c(t (n−1)n 2 )) ∼= z 2 (n−2)(n+1) 2 for j = 0, 1. 5 nilpotent case recall that the discrete heisenberg group hz2n+1 of rank 2n + 1 is defined by hz2n+1 =        1 at c 0n 1n b 0 0tn 0     ∈ gln+2(z) | a, b ∈ z n, c ∈ z    where 1n is the n × n identity matrix, 0n is the zero in z n, a, b, 0n are column vectors, and x t means the transpose of x. the heisenberg lie group hr2n+1 with dimension 2n + 1 is defined by replacing z with r in the definition above. then we have the homogeneous space: hr2n+1/h z 2n+1 ≈ t 2n+1 as a space. let c∗(hz2n+1) be the group c ∗-algebra of hz2n+1. it is shown by the author [3] that for j = 0, 1, kj(c ∗ (hz2n+1)) ∼= z 3 n . it follows that cubo 14, 2 (2012) k-theory for the c∗-algebras of continuous functions ... 195 proposition 5.1. we have kj(c(h r 2n+1/h z 2n+1)) ∼= z 2 2n for j = 0, 1, but for n ≥ 1, kj(c(h r 2n+1/h z 2n+1)) 6 ∼= kj(c ∗ (hz2n+1)). proof. because 22n 6= 3n for n ≥ 1. remark. we have 4n > 3n, so that it may say to be possible that k-theory data of the homogeneous space c∗-algebra contains that of the group c∗-algebra. in fact, in the group non-isomorphic equation above, the right hand side can be a quotient of the left hand side. this picture might be extended to the more general setting. conjecture. let γ be a nilpotent discrete group with rank n. then we have rankzkj(c ∗(γ)) ≤ 2n−1 for j = 0, 1, where rankz(x) means the z-rank of x. remark. the equality holds if γ = zn and the estimate is ture if γ = hz2n+1 as checked above. it is certainly known that a discrete nilpotent group γ can be viewed as a subgroup of matrices, i.e. to be linear. also, it can be viewed as a successive semi-direct products by the abelian groups z kj of integers for some kj ≥ 1 (1 ≤ j ≤ n). in this case, γ is a subgroup of the connected, simply connected nilpotent lie group g obtained as a a successive semi-direct products by rkj, so that the homogeneous space g/γ is homeomorphic to: g/γ ≈ t ∑ n j=1 kj. our conjecture says that rankzkj(c ∗(γ)) ≤ 2−1+ ∑ n j=1 kj. acknowledgement. i would like to thank shuichi tsukuda and michishige tezuka for providing some useful (also in the future) information about spheres splitting in algebraic topology, received: july 2011. revised: december 2011. references [1] b. blackadar, k-theory for operator algebras, second edition, cambridge, (1998). [2] t. natsume, on k∗(c ∗(sl2(z))), j. operator theory 13 (1985), 103-118. 196 takahiro sudo cubo 14, 2 (2012) [3] t. sudo, k-theory of continuous fields of quantum tori, nihonkai math. j. 15 (2004), no. 2, 141-152. [4] c. soulé, the cohomology of sl3(z), topology, 17 (1978), 1-22. [5] n. e. wegge-olsen, k-theory and c∗-algebras, oxford univ. press, 1993. introduction abelian case homogeneous spaces in sl2(r) homogeneous spaces in sln(r) nilpotent case cubo a mathematical journal vol.15, no¯ 01, (171–189). march 2013 discrete almost periodic operators alexander pankov 1 morgan state university, department of mathematics, 1700 e. cold spring lane, baltimore, md 21251, usa. alexander.pankov@morgan.edu abstract this paper deals with discrete almost periodic linear operators in the space of bounded sequences. we study the invertibility of such operators in that space, as well as in the space of almost periodic sequences. one of main results is a discrete version of wellknown first favard theorem, and is based on the notion of the envelope of an almost periodic operator. another result is restricted to finite order operators. it characterizes the invertibility in therms of the operator in question only. resumen este trabajo trata de operadores lineales discretos casi periódicos en el espacio de las secuencias acotadas. estudiamos la invertibilidad de dichos operadores en ese espacio, aśı como en el espacio de secuencias casi periódicas. uno de los resultados principales es una versión discreta del conocido primer teorema de favard, y se basa en la noción de la envolvente de un operador casi periódico. otro resultado se restringe a los operadores de orden finito. se caracteriza la invertibilidad solamente en términos del operador en cuestión. keywords and phrases: almost periodic sequence, discrete operator, favard condition. 2010 ams mathematics subject classification: 39a24, 47b39 1dedicated to professor gaston m. n’guérékata on the occasion of his 60th birthday 172 alexander pankov cubo 15, 1 (2013) 1 introduction the theory of almost periodic differential equations has been initiated by j. favard in his pioneering work [5]. today the theory is well-developed not only for ordinary differential equations, but also for abstract evolution equations and partial differential equations. contemporary presentations of the theory can be found in many monographs and survey articles (see, e.g., [1, 3, 7, 9, 11, 13, 14] and references therein). difference equations constitute a natural counterpart of the theory of differential equations. we refer to [4] for a contemporary introductory presentation of the theory of difference equations. there is a number of papers that deal with almost periodic difference equations. most of them concern either special equations (see, e.g. [17]), or first order systems [6, 12]. in particular [12] contains certain discrete versions of first and second favard theorems. to the best of our knowledge, there are only few papers dedicated to general almost periodic linear difference equations, or, equivalently, almost periodic discrete linear operators (see [2, 15, 16] and references therein). in particular, in [15] certain discrete version of the first favard theorem is obtained (see corollary 5.2). this version requires coercivity estimate (10) and is not similar to typical favard type assumptions. thus, at the moment the theory of almost periodic difference equations is not completely parallel to the theory of almost periodic differential equations. the aim of the present paper is to fill, at list partially, the gap mentioned above. we accept the functional analytic point of view. this means that we study almost periodic operators in the space l∞ of bounded sequences with values in a finite dimensional space and their restrictions to the space ap of almost periodic sequences. we obtain certain criteria for such an operator a to have a bounded inverse operator. equivalently, the invertibility means that the equation ax = y has a unique solution x ∈ l∞ for every right hand side y ∈ l∞. one of our main result, theorem 5.1, is an exact analogue of the version of first favard theorem for differential equations obtained by e. mukhamadiev in [10] (see also [14]). the second result, theorem 6.2, is, in a sense , dual to theorem 5.1, but it holds for operators of finite order. this is an analogue of a result by m. krasnosel’skii, v. burd and y. kolesov [7]. the paper is organized as follows. section 2 is a quick reminder of basic facts about almost periodicity. in section 3 we discuss bounded linear operators in the space l∞. in particular, we introduce important concepts of c-convergence and c-continuity. the main result of the section is proposition 3.1 which shows that any c-continuous operator is of the form (1) (this is a result by v. slyusrchuk [16]). almost periodic operators are introduced in section 4. sections 5 and 6 contain our main results. in what follows, we consider elements of sequence spaces as functions on the set of integers z. we use the notation [·] to list the values of such a function. on the other hand, the notation {·} stands for the lists of elements of a set. cubo 15, 1 (2013) discrete almost periodic operators 173 2 almost periodic functions and sequences let e be a banach space, with the norm ‖·‖e, over real or complex numbers. we denote by cb(e) the space of bounded continuous functions on r with values in e. this is a banach space with the norm ‖f‖cb = sup t∈r ‖f(t)‖e . a function f ∈ cb(e) is almost periodic if the family {f(· + τ)}τ∈r of shifts is a precompact set in cb(e). almost periodic functions form a closed subspace ap(e) of cb(e), hence, a banach space. by l∞(e) we denote the space of all bounded two-sided sequences x = [x(n)]n∈z with values in e. this is a banach space endowed with the norm ‖x‖l∞ = sup n∈z ‖x(n)‖e . in what follows we also need the space of e-valued sequences l1(e), with the norm ‖x‖l1 = ∑ n∈z ‖x(n)‖e . in this paper we do not use other spaces lp. a sequence x = [x(n)]n∈z ∈ l ∞(e) is almost periodic if the set of its shifts {[x(· + q)]}q∈z is a precompact set in l∞(e). the set of all almost periodic sequences is a closed subspace ap(e) ⊂ l∞(e). hence, ap(e) is a banach space. it is convenient to introduce operators of translation tq, q ∈ z, acting in the space l ∞(e) by the formula (tqx)(n) = x(n + q) , n ∈ z . these are linear bounded operators. moreover, they are isometric operators, i.e., ‖tqx‖l∞ = ‖x‖l∞ , q ∈ z . the operators tq form a one-parameter discrete group of operators, i.e., (i) tq1+q2 = tq1tq2 , q1, q2 ∈ z, (ii) t0 = i, (iii) t−q = t −1 q , q ∈ z, where i stands for the identity operator. in terms of these operators the almost periodicity of a sequence x ∈ l∞ means that the set {tqx}q∈z is precompact in l ∞(e). the following simple statement is well-known (see, e.g., [3, theorem 1.27]) and clarifies a relation between almost periodic sequences and functions. 174 alexander pankov cubo 15, 1 (2013) proposition 2.1. the restriction operator r : cb(e) → l ∞(e) defined by f 7→ [f(n)]n∈z maps ap(e) onto ap(e). furthermore, there exists a linear operator j : e → cb(e), an extension operator, such that (jx)(n) = x(n) for all n ∈ z, ‖jx‖cb = ‖x‖l∞ for all x ∈ l ∞(e), and j(ap(e)) ⊂ ap(e). making use of proposition 2.1, one can transfer many results about almost periodic functions to almost periodic sequences. finally, we introduce a simple, but important, notion of periodization. given a positive integer j, let qj = {n ∈ z | j − 1 ≤ n ≤ j} . for any x ∈ l∞(e), its 2j-periodization is a 2j-periodic sequence, say xj = [xj(n)]n∈z, such that xj(n) = x(n) for all n ∈ qj. obviously, xj c → x and, hence, the space ap(e) is c-dense in l∞(e). 3 linear operators in l∞ in the rest of the paper e stands for a finite dimensional banach space. for sequences in the space l∞(e) one can introduces several kinds of convergence. in this paper we use the standard convergence with respect to the norm of l∞(e) and the so-called c-convergence. a sequence xk ∈ l ∞(e) c-converges to x ∈ l∞(e) (in symbols xk c → x) if the sequence xk is bounded in l ∞(e) and xk(n) → x(n) for all n ∈ z. in this case we also write x = clim xk. by l(l∞(e)) we denote the banach algebra of all bounded linear operators in l∞(e). an operator a ∈ l(l∞(e)) is c-continuous if for any sequence xk ∈ l ∞(e) such that xk c → x we have that axk c → ax. the set of all c-continuous operators is a closed subalgebra of the banach algebra l(l∞(e)) (see [2, proposition 1]). we denote this subalgebra by lc(l ∞(e)). we consider operators of the form (ax)(n) = ∑ m∈z a(n, m)x(m) , n ∈ z , (1) where a(n, m) ∈ l(e). the double sequence a = [a(n, m)]n,m∈z is called the kernel of a. given such a kernel a, we set ‖a‖ = sup n∈z ∑ m∈z ‖a(n, m)‖l(e) . (2) an alternative representation of operator (1) is (ax)(n) = ∑ k∈z a(n, k)x(n + k) , n ∈ z , (3) where a(n, k) = a(n, n + k) are called the coefficients of a. it is easily seen that, for double sequences a and a = [a(n, m)]n,m∈z, we have ‖a‖ = ‖a‖. such sequences can be considered as cubo 15, 1 (2013) discrete almost periodic operators 175 sequences indexed by n with values in the space of sequences indexed by m. from this point of view the norm defined by (2) is exactly the l∞(l1(l(e)))-norm. the nontrivial part of the next result goes back to [16]. since this paper is not available in english, we preset the proof here. proposition 3.1. a linear operator a in l∞(e) is a bounded c-continuous operator if and only if a is of the form (1) with ‖a‖ < ∞. moreover, in this case ‖a‖l(l∞(e)) = ‖a‖ . proof . suppose that ‖a‖ < ∞. then it is well-known, and easily seen, that ‖a‖l(l∞(e)) ≤ ‖a‖ . hence, a ∈ l(l∞(e)). to prove that a is c-continuous, suppose that xk c → 0. obviously, the sequence axk is bounded in l∞(e), and we need to show that (axk)(n) → 0 in e for every n ∈ z. for any positive n ∈ z ‖(axk)(n)‖e ≤   ∑ |m|≤n + ∑ |m|>n   ‖a(n, m)‖l(e)‖xk(m)‖e . since the sequence xk is bounded in l ∞(e) and ‖a‖ < ∞, choosing n large enough we can make the second sum in the right-hand side sufficiently small. next, since xk c → 0, the first sum is sufficiently small provided k is large enough. this proves the first statement. suppose that a ∈ lc(l ∞(e)). first, we define its kernel as follows. denote by jm : e → l ∞(e), m ∈ z, the operator defined by (jmu)(n) = { u if n = m 0 otherwise . the operator pn : l ∞(e) → e, n ∈ z, is defined by pnx = x(n). for all n, m ∈ z, we set a(n, m) = pnajm . obviously, a(n, m) ∈ l(e). now we prove (1), where the right-hand side converges in l∞(e) for each n ∈ z. indeed, given x ∈ l∞(e), we set xk(n) = { x(n) if |n| ≤ k 0 otherwise . it is easily seen that xk c → x. since a is c-continuous, then for every n ∈ z (axk)(n) = ∑ |m|≤k a(n, m)x(m) → (ax)(n) 176 alexander pankov cubo 15, 1 (2013) as required. let us prove that ‖a‖ ≤ ‖a‖l(l∞(e)) . for each k ∈ z, we consider an element xk ∈ l ∞(e) such that ‖xk(n)‖e = 1 and ‖a(k, n)xk(n)‖e = ‖a(k, n)‖l(e) for all n ∈ z. since dim e < ∞, such a sequence exists. then ∑ m∈z ‖a(k, m)‖l(e) = ‖(axk)(k)‖e ≤ ‖axk‖l∞ ≤ ‖a‖l(l∞) because ‖xk‖l∞(e) = 1. this implies the required. 2 proposition 3.1 and [2, proposition 3] imply immediately corollary 3.2. suppose that a ∈ lc(l ∞(e)) has a bounded inverse operator. then the inverse operator is c-continuous and, hence, is of the form (a−1x)(n) = ∑ m∈z g(n, m)x(n) , n ∈ z , with ‖g‖ = ‖a−1‖l(l∞(e)) < ∞. the kernel g in corollary 3.2 is often called the green function of the operator a−1. remark 3.3. under the assumption of corollary 3.2, suppose in addition that the kernel a satisfies ‖a(m, n)‖l(e) ≤ c (1 + |n − m|)α , m, n ∈ z , (4) with c > 0 and α > 2. then for every θ > 0 small enough there exists cθ > 0 such that the green function satisfies ‖g(m, n)‖l(e) ≤ cθ (1 + |n − m|)α−1−θ , m, n ∈ z . (5) furthermore, if ‖a(m, n)‖l(e) ≤ c exp(−δ|n − m|) , m, n ∈ z , (6) with c > 0 and δ > 0, then there exist c1 > 0 and ε > 0 such that ‖g(m, n)‖l(e) ≤ c1 exp(−ε|n − m|) , m, n ∈ z , (7) (see [2] and [14]). cubo 15, 1 (2013) discrete almost periodic operators 177 4 almost periodic operators we say that a ∈ l(l∞(e)) is an almost periodic operator if the sequence of operators [tqat−q]q∈z is an almost periodic sequence with values in l(l∞(e)). proposition 4.1. an operator a ∈ l(l∞(e)) is almost periodic if and only if the set {tqat−q}q∈z is precompact in l(l∞(e)). for the proof we refer to [2, proposition 6]. the envelope, or hull, h(a) of an almost periodic operator a ∈ l(l∞(e)) is the closure of the set {tqat−q}q∈z in the space l(l ∞(e)). this is a compact set. now we collect some properties of almost periodic operators obtained in [2]. proposition 4.2. suppose that a ∈ l(l∞(e)) is almost periodic operator. then the following statements hold: (a) a(ap(e)) ⊂ ap(e). (b) if a has a bounded inverse operator, then a−1 is an almost periodic operator and, hence, a|ap(e) has a bounded inverse operator in l(ap(e)). moreover, all operators in the envelope h(a) are invertible and h(a−1) = {ã−1 : ã ∈ h(a)} . (c) if, in addition, a is of the form (1), the kernel a satisfies (4) with c > 0 and α > 2, and if a|ap(e) has a bounded inverse operator in l(ap(e)), then the operator a has a bounded inverse operator in l(l∞(e)). remark 4.3. actually, under the assumptions of proposition 4.2(c) the operator a is c-continuous. proposition 4.4. suppose that a ∈ l(l∞(e)) is a c-continuous operator of the form (1). the following statements are equivalent: (i) the operator a is almost periodic. (ii) the kernel a is almost periodic along the diagonal with respect to norm (2), i.e., the sequence of kernels [a(· + q, · + q)]q∈z is an almost periodic sequence in with values in the space of kernels endowed with the norm ‖ · ‖. (iii) the sequence [a(n, ·)]n∈z of coefficients is an almost periodic sequence with values in l 1(l(e)). proof . a straightforward verification shows that [a(n+q, m+q)] is the kernel of the operator tqat−q. now the required follows immediately from proposition 3.1 and remarks after equation (3). 2 178 alexander pankov cubo 15, 1 (2013) remark 4.5. if an almost periodic operator a ∈ l(l∞(e)) is of the form (1), then every operator ã ∈ h(a) is of the same form. the kernel of ã is a limit point of the set {a(· + q, · + q)}q∈z in the space of kernels with respect to the norm ‖·‖, while the collection of the coefficients is the limit point of the set {a(· + q, ·)}q∈z in the space l 1(l(e)). 5 favard type theorem in this section we prove the following result of favard type for almost periodic operators in l∞(e). theorem 5.1. an almost periodic operator a ∈ lc(l ∞(e)) has a bounded inverse operator if and only if the following condition is satisfied: (f) every operator in the envelope h(a) is injective. proof . if a ∈ lc(l ∞(e)) is invertible, then, by proposition 4.2(b), all operators in the envelope are invertible, and (f) follows. now suppose that (f) is satisfied. to prove that the operator a has a bounded inverse, it is enough to show that for any y ∈ l∞(e) there exists x ∈ l∞(e) such that ax = y . (8) for any positive integer j, denote by xj = [xj(n)]n∈z the 2j-periodization of x. similarly, we denote by aj = [aj(n, k)]n,k∈z the 2j-periodization of a = [a(n, k)]n,k∈z with respect to the variable n. according to equation (3), aj generates an operator aj that belongs to lc(e). notice that 2j-periodic sequences form a finite dimensional subspace of l∞(e), and aj leaves that subspace invariant. we solve the equation ajxj = yj (9) in the subspace of 2j-periodic sequences provided j is large enough. since the problem is finite dimensional, it is sufficient to show that the associated homogeneous problem has only zero solution. assuming the contrary, one can find a sequence jl → ∞ such that ajlxjl = 0 for some xjl ∈ l ∞(e) with ‖xjl‖l∞ = 1. then there exists ql ∈ qjl such that ‖xjl(ql)‖e = 1. we put x̃l = tqlxjl and ãl = tqlajlt−ql. then ‖x̃l‖l∞ = 1, ‖x̃l(0)‖e = 1 and ãlx̃l = 0. passing to a subsequence, we can assume that x̃l c → x̃ and al = tqlat−ql → ã in l(l∞(e)). obviously, ‖x̃(0)‖e = 1 and ã ∈ h(a). given n ∈ z we have (ãx̃)(n) = [(ãx̃)(n) − (ãx̃l)(n)] + [(ãx̃l)(n) − (alx̃l)(n)] + (alx̃l)(n) . cubo 15, 1 (2013) discrete almost periodic operators 179 here the first term in the right-hand side tends to 0 since the operator ã is c-continuous, ‖(ãx̃l)(n) − (alx̃l)(n)‖e ≤ ‖ã − al‖l(l∞(e)) → 0 , and (alx̃l)(n) = [(tqlat−ql)tqlxl)](n) = (tqlaxjl)(n) = (tqlajlxjl)(n) = 0 provided |ql| ≥ |n|. hence, ãx̃ = 0, which contradicts to condition (f). thus, equation (9) has a unique 2j-periodic solution xj for all sufficiently large j. the sequence xj is bounded in the space l ∞(e). for if this is not so, we can find a subsequence xjl such that ‖xjl‖l∞ → ∞. set zl = xjl/‖xjl‖l∞ . then allzl = yjl/‖xjl‖l∞ . arguing exactly as above, we obtain that there is a nonzero x ∈ l∞(e) such that ãx = 0, and we arrive at contradiction to condition (f). now since the sequence xj is bounded in l ∞(e), then xj c → x along a subsequence. it is easy to see that ax = y. 2 the following result is obtained in [15]. corollary 5.2. let a ∈ lc(l ∞(e)) be an almost periodic operator. if there exists a constant c0 > 0 such that ‖ax‖l∞ ≥ c0‖x‖l∞ (10) for all x ∈ l∞(e), then the operator a is invertible in l∞(e). proof . since the operators tq are isometric, then, by the definition of the envelope, all operators in e(a) satisfy inequality (10). this implies condition (f), and we conclude. 2 by proposition 4.2(b), we obtain the following corollary 5.3. if an almost periodic operator a ∈ lc(l ∞(e)) satisfies condition (f), then the operator a|ap(e) has a bounded inverse operator in the space ap(e). furthermore, proposition 4.2(c) implies corollary 5.4. suppose that the kernel a of an almost periodic operator a ∈ lc(l ∞(e)) satisfies satisfies inequality (4) with c > 0 and α > 2. then the following statements are equivalent: (i) the operator a satisfies condition (f); 180 alexander pankov cubo 15, 1 (2013) (ii) the operator a has a bounded inverse operator in the space l∞(e); (iii) the operator a|ap(e) has a bounded inverse operator in the space ap(e). 6 operators of finite order now, under certain additional assumptions, we obtain yet another criterion for an almost periodic operator to be invertible. this result is, in a sense, dual to theorem 5.1. in this section we consider operators of finite order. these are of the form (ax)(n) = k2∑ k=k1 a(n, k)x(n + k) , n ∈ z , (11) where a(n, k1) and a(n, k2) are non-zero operators in e. the number k2 − k1 is called the order of a. in what follows we always suppose that the order of a is greater than zero. the kernel [a(n, m)] of a vanishes outside the strip {(n, m) : k1 ≤ m − n ≤ k2}. we impose the following assumptions: (a1) sup{‖a(n, k)‖l(e) : n ∈ z , k1 ≤ k ≤ k2} < ∞; (a2) for all n ∈ z the operators a(n, k1) and a(n, k2) are invertible in l(e), and there exists a constant c > 0 independent of n such that ‖a−1(n, k1)‖l(e) ≤ c and ‖a−1(n, k2)‖l(e) ≤ c . assumption (a1) is necessary and sufficient for an operator a of the form (11) to be a bounded linear operator in l∞(e). in this case, a is c-continuous automatically. assumption (a2) is natural because it is necessary for the existence of bounded inverse operator a−1. let us also mention that the operator a is almost periodic if and only if for any k, k1 ≤ k ≤ k2, the sequence [a(n, k)]n∈z is almost periodic. furthermore, the envelope h(a) of any almost periodic operator of finite order consists of operators of finite order. the following simple, but important, property is well-known. proposition 6.1. assume that an operator a of finite order ≥ 1 satisfies (a1) and (a2). then its null space {x ∈ l∞(e) | ax = 0} is finite dimensional. cubo 15, 1 (2013) discrete almost periodic operators 181 proof . let d ≥ 1 be the order of a. assumption (a2) implies immediately that the linear mapping from the null space into the space ed defined by x = [x(n)]n∈z 7→ (x(k1), x(k1 + 1), . . . , x(k2 − 1)) is one-to-one. 2 the main result of the section is the following. theorem 6.2. suppose that an operator a of the form (11) is almost periodic, of order d ≥ 1, and satisfies assumptions (a1) and (a2). then the following statements are equivalent: (i) the range of operator a contains ap(e); (ii) the operator a has an inverse operator in l(l∞(e)); (iii) the operator a|ap(e) has an inverse operator in l(ap(e)). proof . the equivalence of (ii) and (iii) follows from proposition 4.2. obviously, (ii) implies (i). now we prove that (i) implies (ii). assuming (i), we have to show that the equation ax = y (12) has a unique solution x ∈ l∞(e) for any y ∈ l∞(e). claim 1. for any y ∈ ap(e) there exists a solution x ∈ ap(e) of equation (12) such that ‖x‖l∞ ≤ c‖y‖l∞ , (13) with some constant c > 0 independent of y. denote by v the preimage of ap(u) under the operator a. since ap(e) is a closed subspace of l∞(e) and a is a bounded operator, v is a closed subspace as well, hence, a banach space. the operator a|v maps v onto ap(e). now the required is a particular case of a well-known result about linear operators from a banach space onto a banach space (for an excellent presentation see [8]). claim 2. for any y ∈ l∞(e) there exists a solution x ∈ l∞(e) of equation (12) that satisfies estimate (13). let yj be the 2j-periodization of y. obviously, ‖yj‖l∞ ≤ ‖y‖l∞. by claim 1, there exists an almost periodic solution xj of equation (12), with y replaced by yj, and ‖xj‖l∞ ≤ c‖yj‖l∞ ≤ c‖y‖l∞ . 182 alexander pankov cubo 15, 1 (2013) hence, along a subsequence, xjk c → x, and x is a solution of (12) that satisfies (13). claim 3. each operator ã ∈ h(a) maps l∞(e) onto l∞(e). let ã = lim tqjat−qj ∈ h(a) . by claim 2, given y ∈ l∞(e) there exists xj ∈ l ∞(e) such that axj = t−qjy and ‖xj‖l∞ ≤ c‖t−qjy‖l∞ = c‖y‖l∞ . setting x̃j = tqjxj, we have that tqjat−qjx̃j = y . (14) passing to a subsequence, we can suppose that there exists x ∈ l∞ such that x̃j c → x. passing to the limit in equation (14), we see that ãx = y. claim 4. if y ∈ ap(e), then every solution x ∈ l∞(e) of equation (12) is almost periodic. due to claim 1, it is enough to show that every solution of the homogeneous equation, i.e., with y = 0, is almost periodic. assume the contrary. then there exists x ∈ l∞ such that ax = 0 and x is not almost periodic, i.e., the family of shifts {tqx}q∈z is not precompact. then there exist ε0 > 0 and an infinite set of integers {qj} such that ‖xj − xi‖l∞ ≥ ε0 (15) for i 6= j, where xj = tqjx. without loss of generality, we can suppose that tqjat−qj → ã ∈ h(a) in the space l(l∞(e)). since ‖ãxj‖l∞ = ‖(ã − tqjat−qj)xj‖l∞ ≤ ‖ã − tqjat−qj‖l(l∞(e))‖x‖l∞ , then ãxj → 0 (16) in l∞(e) as j → ∞. let ṽ0 be the null space of the operator ã. by proposition 6.1, this is a finite dimensional subspace in l∞(e). hence, there exists a bounded projector p̃0 in l ∞(e) onto ṽ0. set p̃1 = i − p̃0 and ṽ1 = p̃1(l ∞(e)). obviously, ãp̃0xj = 0. hence, by (16), ãp̃1xj → 0 (17) in l∞(e) as j → ∞. cubo 15, 1 (2013) discrete almost periodic operators 183 the restriction ã |ṽ1 is one-to-one and, by claim 3, maps ṽ1 onto l ∞(e). hence, by (17), p̃1xj → 0 (18) in l∞(e) as j → ∞. by the triangle inequality, ‖p̃0xj − p̃0xi‖l∞ ≥ ‖xj − xi‖l∞ − ‖p̃1xj‖l∞ − ‖p̃1xi‖l∞ . now (15) and (18) imply that for any ε1 ∈ (0, ε0) ‖p̃0xj − p̃0xi‖l∞ ≥ ε1 , whenever both j and i are large enough. hence, the set {p̃0xj} is not a precompact set. on the other side, {p̃0xj} is a bounded subset of a finite dimensional space ṽ0. hence, it is precompact, and we arrive at a contradiction. claim 5. the null space v0 of the operator a is trivial. without loss of generality, we can assume that k2 = 1. the restriction operator r : l ∞(e) → ed is defined by r : x = [x(n)]n∈z 7→ (x(k1), x(k1 + 1), . . . , x(0)) . as we have mentioned in the proof of proposition 6.1, r maps v0 into e d in one-to-one manner. we set v0 = r(v0). this is a linear subspace of e d. choose any direct complement v1 to v0 in ed and set v1 = {x ∈ l ∞(e) : rx ∈ v1} . then v0 ⊕ v1 = l ∞(e) and a(v1) = l ∞(e). the operator a|v1 : v1 → l ∞(e) is one-to-one and onto and, hence, has a bounded inverse denoted by b : l∞(e) → v1 . now we prove that v0 = {0}. suppose that ax = 0, and x 6= 0. by claim 4, x is an almost periodic sequence. let θj = [θj(n)]n∈z be a (scalar valued) sequence defined by θj(n) =    0 if n ≤ 0 n if 1 ≤ n ≤ j j if n > j . consider the sequence xj = θj · x = [θj(n)x(n)]n∈z . 184 alexander pankov cubo 15, 1 (2013) obviously, xj ∈ l ∞(e). moreover, xj ∈ v1 because xj(n) = 0 if n ≤ 0. we have axj = θj · ax + zj = zj , (19) where zj = [zj(n)]n∈z with zj(n) = 1∑ k=k1 (θj(n + k) − θj(n))a(n, k)x(n + k) , n ∈ z . it is easily seen that |θj(n + k) − θj(n)| ≤ d for all n ∈ z and integer k ∈ [k1, 1]. hence, ‖zj‖l∞ is bounded above by a constant independent of j. since xj ∈ v1, we have that xj = bzj and, hence, ‖xj‖l∞ ≤ c, where c > 0 is independent on j. in particular, j‖x(j)‖e ≤ ‖xj‖l∞ ≤ c . hence, x(j) → 0 as j → ∞, which implies that x = 0 because, by claim 3, x is almost periodic. this completes the proof of the theorem. 2 combining theorem 6.2 and corollary 5.4, we obtain corollary 6.3. under the assumptions of theorem 6.2 the following statements are equivalent: (i) the operator a satisfies condition (f) of theorem 5.1; (ii) the operator a has a bounded inverse in l(l∞(e)); (iii) the operator a maps l∞(e) onto l∞(e); (iv) the restriction a|ap(e) has a bounded inverse in l(ap(e)); (v) the restriction a|ap(e) maps ap(e) onto ap(e). received: november 2012. revised: december 2012. references [1] l. amerio, g. prouse, almost periodic functions and functional equations, van nostrand, new york, 1971. 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() cubo a mathematical journal vol.17, no¯ 01, (41–64). march 2015 hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks and efficiency conditions for multiobjective fractional programming problems ram u. verma department of mathematics, texas state university, san marcos, tx 78666, usa. verma99@msn.com abstract the parametrically generalized sufficient efficiency conditions for multiobjective fractional programming based on the hybrid (φ, ψ, ρ, ζ, θ)−invexities are developed and then efficient solutions to the multiobjective fractional programming problems are established. plus, the obtained results on sufficient efficiency conditions are generalized to the case of the ǫ−efficient solutions. the results thus obtained generalize and unify a wider range of investigations on the theory and applications to the multiobjective fractional programming based on the hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks. resumen se desarrollan las condiciones de eficiencia suficiente generalizadas paramétricamente de programación multifraccional multiobjetivo basado en las invexidades-(φ, ψ, ρ, ζ, θ)− h́ıbridas y luego se establecen las soluciones eficientes a los problemas de programación fraccional multiobjetivo. además, los resultados obtenidos sobre condiciones de eficiencia suficiente se generalizan al caso de soluciones ǫ−-eficientes. los resultados obtenidos generalizan y unifican una amplia gama de investigaciones en la teoŕıa y aplicaciones de la programación fraccional multiobjetivo basado en el marco de trabajo de las invexidades-(φ, ψ, ρ, ζ, θ)−. keywords and phrases: generalized invexity, multiobjective fractional programming, efficient solutions, ǫ−efficient solutions, parametric sufficient efficiency conditions. 2010 ams mathematics subject classification: 90c32, 90c45. 42 ram u. verma cubo 17, 1 (2015) 1 introduction among recent developments on higher order generalized invexties and duality models for mathematical programming, we begin with the work of kawasaki [5] on some second order necessary conditions of the kuhn tucker type under new weaker constraint qualifications for twice continuously differentiable functions, while mishra and rueda [11] introduced higher order generalized invexity and duality models in mathematical programming. mangasarian [8] focused on the second order duality for a conventional nonlinear programming problem, where the approach is based on constructing a second order dual problem by taking linear and quadratic approximations of the objective and constraint functions for an arbitrary but fixed point leading to the wolfe dual model for the approximated problem, while letting the fixed point to vary. verma [24] introduced and studied the second order (ρ, η, θ)−invexities to the context of parametrically sufficient optimality conditions in semiinfinite discrete minimax fractional programming. zalmai and zhang [37] have established a set of efficiency conditions and a fairly large number of global nonparametric sufficient efficiency results under various frameworks for generalized (η, ρ)−invexity for the semiinfinite discrete minimax fractional programming. just recently, verma [22] investigated a general framework for a class of (ρ, η, θ)−invex functions to examine some parametric sufficient efficiency conditions for multiobjective fractional programming problems for weakly ǫ−efficient solutions. inspired by these research advances, we first introduce the hybrid (φ, ψ, ρ, ζ, θ)−invexities as well as the second order hybrid (φ, ψ, ρ, ζ, θ)−invexities, second, introduce some parametrically sufficient efficiency conditions for multiobjective fractional programming, and finally, explore the efficient solutions to multiobjective fractional programming problems. in addition, we generalize the obtained results based on the hybrid (φ, ψ, ρ, ζ, θ)−invexities regarding the efficient solutions to the multiobjective fractional programming problems to the case of the ǫ−efficient solutions to the multiobjective fractional programming problems. the results established in this communication, not only generalize (and unify) the results on general sufficient efficiency conditions for multiobjective fractional programming problems based on the hybrid invexity of functions, but also generalize second order invexity results in more general settings. there exists an enormous literature on higher order generalized invexity and duality models in mathematical programming. we consider, based on the generalized (φ, ψ, ρ, ζ, θ)−invexities of functions, the following multiobjective fractional programming problem: (p) minimize ( f1(x) g1(x) , f2(x) g2(x) , · · ·, fp(x) gp(x) ) subject to x ∈ q = {x ∈ x : hj(x) ≤ 0, j ∈ {1, 2, · · ·, m}}, where x is an open convex subset of ren (n-dimensional euclidean space), fi and gi for i ∈ {1, ···, p} and hj for j ∈ {1, · · ·, m} are real-valued functions defined on x such that fi(x) ≥ 0, gi(x) > 0 for i ∈ {1, · · ·, p} and for all x ∈ q. here q denotes the feasible set of (p). cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 43 next, we observe that problem (p) is equivalent to the nonfractional programming problem: (pλ) minimize ( f1(x) − λ1g1(x), · · ·, fp(x) − λpgp(x) ) subject to x ∈ q with λ = ( λ1, λ2, · · ·, λp ) = ( f1(x ∗) g1(x ∗) , f2(x ∗) g2(x ∗) , · · ·, fp(x ∗) gp(x ∗) ) , where x∗ is an efficient solution to (p). we all can agree that general mathematical programming problems offer a great opportunity for applications to other fields, for instance, applications to game theory, statistical analysis, engineering design (including design of control systems, design of earthquakes-resistant structures, digital filters, and electronic circuits), random graphs, boundary value problems, wavelet analysis, environmental protection planning, decision and management sciences, optimal control problems, continuum mechanics, robotics, and beyond. for more details on generalized efficiency and ǫ−efficiency results and applications, we recommend the reader [1 40]. this submission is organized as follows: the introductory section deals with a brief historical development for the multiobjective fractional mathematical programming, while emphasizing the roles of the generalized first (and second) order (φ, ψ, ρ, ζ, θ)−invex functions as well as the first (and second) order generalized (φ, ψ, ρ, ζ, θ)−invex functions. in section 2, the hybrid (φ, ψ, ρ, ζ, θ)−invex functions of higher orders are introduced, and section 3 deals with sufficient efficiency conditions leading to the solvability of the problem (p) using the hybrid (φ, ψ, ρ, ζ, θ)−invexities. in section 4, general remarks are presented. 2 hybrid invexities in this section, we introduce and develop some concepts and notations for the problem on hand. let x be an open convex subset of ren (n-dimensional euclidean space). let 〈·, ·〉 denote the inner product, and let z ∈ ren . suppose that f : x → re is a real-valued twice continuously differentiable function defined on x, and that ▽f(y) and ∇2f(y) denote, respectively, the gradient and hessian of f at y. recall a function ψ : ren → re is sublinear (super linear) if ψ(x + y) ≤ (≥)ψ(x) + ψ(y) for all x, y ∈ ren, and ψ(ax) = aψ(x) for all x ∈ ren and a ∈ re+ = [0, ∞). let x∗ ∈ x. definition 2.1. a twice differentiable function f : x → re is said to be hybrid (φ, ψ, ρ, ζ, θ)−invex at x∗ of second order if there exists a function φ : re → re such that for each x ∈ x, ρ : x×x → re, ψ : ren → re, ζ, θ : x × x → ren and z ∈ ren, φ ( f(x) − f(x∗) ) ≥ ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2. 44 ram u. verma cubo 17, 1 (2015) definition 2.2. a twice differentiable function f : x → re is said to be hybrid (φ, ψ, ρ, ζ, θ)−pseudoinvex at x∗ of second order if there exists a function φ : re → re and z ∈ ren such that for each x ∈ x, ρ : x × x → re, ψ : ren → re, and ζ, θ : x × x → ren, ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≥ 0 ⇒ φ ( f(x) − f(x∗) ) ≥ 0. definition 2.3. a twice differentiable function f : x → re is said to be strictly hybrid (φ, ψ, ρ, ζ, θ)− pseudo-invex at x∗ of second order if there exists a function φ : re → re and z ∈ ren such that for each x ∈ x, ρ : x × x → re, ψ : ren → re, and ζ, θ : x × x → ren, ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≥ 0 ⇒ φ ( f(x) − f(x∗) ) > 0. definition 2.4. a twice differentiable function f : x → re is said to be prestrictly hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ of second order if there if there exists a function φ : re → re and z ∈ ren such that for each x ∈ x, ρ : x × x → re, ψ : ren → re, and θ, ζ : x × x → ren, ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 > 0 ⇒ φ ( f(x) − f(x∗) ) ≥ 0. definition 2.5. a twice differentiable function f : x → re is said to be hybrid (φ, ψ, ρ, ζ, θ)−quasiinvex at x∗ of second order if there exists a function φ : re → re such that for each x ∈ x, ρ : x × x → re, ψ : ren → re, and θ, ζ : x × x → ren, φ ( f(x) − f(x∗) ) ≤ 0 ⇒ ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≤ 0. definition 2.6. a twice differentiable function f : x → re is said to be strictly hybrid (φ, ψ, ρ, ζ, θ)− quasi-invex at x∗ of second order if there exist a function φ : re → re, and z ∈ ren such that for each x ∈ x, ρ : x × x → re, ψ : ren → re, and θ, ζ : x × x → ren, φ ( f(x) − f(x∗) ) ≤ 0 ⇒ ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 < 0. cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 45 definition 2.7. a twice differentiable function f : x → re is said to be prestrictly hybrid (φ, ψ, ρ, ζ, θ)− quasi-invex at x∗ of second order if there exist a function φ : re → re, and z ∈ ren such that for each x ∈ x, ρ : x × x → re, ψ : ren → re, and θ, ζ : x × x → ren, φ ( f(x) − f(x∗) ) < 0 ⇒ ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≤ 0. definition 2.8. a point x∗ ∈ q is an efficient solution to (p) if there exists no x ∈ q such that fi(x) gi(x) ≤ fi(x ∗) gi(x ∗) ∀ i = 1, · · ·, p, fj(x) gj(x) < fj(x ∗) gj(x ∗) for some j ∈ {1, · · ·, p}. next to this context, we have the following auxiliary problem: (pλ̄) minimizex∈q(f1(x) − λ̄1g1(x), · · ·, fp(x) − λ̄pgp(x)), subject to x ∈ q, where λ̄i for i ∈ {1, · · ·, p} are parameters, and λ̄i = f(x ∗ ) gi(x ∗) . next, we introduce the efficiency solvability conditions for (pλ̄) problem. definition 2.9. a point x∗ ∈ q is an efficient solution to (pλ̄) if there does not exist an x ∈ q such that fi(x) − λ̄igi(x) ≤ fi(x ∗) − λ̄igi(x ∗) ∀ i = 1, · · ·, p, fj(x) − λ̄jgj(x) < fj(x ∗) − λ̄jgj(x ∗) for some j ∈ {1, · · ·, p}, where λ̄i = fi(x ∗ ) gi(x ∗) for i = 1, · · ·, p. next, we recall the following result (verma [24]) that is crucial to developing the results for the next section based on second order (φ, ψ, ρ, z, θ)−invexities. theorem 2.1. let x∗ ∈ f and λ∗ = max1≤i≤p fi(x ∗)/gi(x ∗), for each i ∈ p, let fi and gi be twice continuously differentiable at x∗, for each j ∈ q, let the function z → gj(z, t) be twice continuously differentiable at x∗ for all t ∈ tj, and for each k ∈ r, let the function z → hk(z, s) be twice continuously differentiable at x∗ for all s ∈ sk. if x ∗ is an efficient solution of (p), if the second 46 ram u. verma cubo 17, 1 (2015) order generalized guignard constraint qualification holds at x∗, and if for any critical direction y, the set cone ({ ( ∇gj(x ∗, t), 〈y, ∇2gj(x ∗, t)y〉 ) : t ∈ t̂j(x ∗), j ∈ q} ∪ {∇fi(x ∗) − λ∗i ∇gi(x ∗) : i ∈ p, i 6= i0}) + span({ ( ∇hk(x ∗, s), 〈y, ∇2hk(x ∗, s)y〉 ) : s ∈ sk, k ∈ r}), where t̂j(x ∗) ≡ {t ∈ tj : gj(x ∗, t) = 0}) is closed, then there exist u∗ ∈ u ≡ {u ∈ rp : u ≥ 0, ∑p i=1 ui = 1} and integers ν ∗ 0 and ν ∗, with 0 ≤ ν∗0 ≤ ν ∗ ≤ n + 1, such that there exist ν∗0 indices jm, with 1 ≤ jm ≤ q, together with ν ∗ 0 points tm ∈ t̂jm(x ∗), m ∈ ν∗0, ν ∗ − ν∗0 indices km, with 1 ≤ km ≤ r, together with ν ∗ − ν∗0 points sm ∈ skm for m ∈ ν ∗\ν∗0, and ν ∗ real numbers v∗m, with v ∗ m > 0 for m ∈ ν ∗ 0, with the property that p∑ i=1 u∗i [∇fi(x ∗) − λ∗(∇gi(x ∗)] + ν ∗ 0∑ m=1 v∗m[∇gjm(x ∗, tm) + ν ∗ ∑ m=ν∗ 0 +1 v∗m∇hk(x ∗, sm) = 0, (2.1) 〈y, [ p∑ i=1 u∗i [∇ 2fi(x ∗ ) − λ∗∇2gi(x ∗ )] + ν ∗ 0∑ m=1 v∗m∇ 2gjm(x ∗, tm) + ν ∗ ∑ m=ν∗ 0 +1 v∗m∇ 2hk(x ∗, sm) ] y〉 ≥ 0, (2.2) u∗i [fi(x ∗) − λ∗gi(x ∗)] = 0, i ∈ p, (2.3) where ν \ ν0 is the complement of the set ν0 relative to the set ν. 3 efficiency conditions for problem (p) this section deals with some parametrically sufficient efficiency conditions for problem (p) under the hybrid frameworks for (φ, ψ, ρ, ζ, θ)−invexities. we begin with real-valued functions ei(., x ∗, u∗) and bj(., v) defined by ei(x, x ∗, u∗) = ui[fi(x) − ( fi(x ∗) gi(x ∗) ) gi(x)], i ∈ {1, · · ·, p} and bj(., v) = vjhj(x), j = 1, · · ·, m. cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 47 theorem 3.1. let x∗ ∈ q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗ ) gi(x ∗) ≥ 0, gi(x ∗) > 0 and hj for j ∈ {1, · · ·, m} be twice continuously differentiable at x∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ rep : u > 0, σ p i=1 ui = 1} and v ∗ ∈ rem+ such that σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + σmj=1v ∗ j ▽ hj(x ∗) = 0, (3.1) 〈 ζ(x, x∗), [ p∑ i=1 u∗i [∇ 2fi(x ∗) − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)] + m∑ j=1 v∗j ∇ 2hj(x ∗) ] z 〉 ≥ 0, (3.2) and v∗j hj(x ∗) = 0, j ∈ {1, · · ·, m}. (3.3) suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): (i) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ with φ̄(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are hybrid (φ̃, ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for φ̃ increasing with φ̃(0) = 0, and ψ sublinear. (ii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are prestrictly hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (φ̃, ψ, ρ, ζ, θ)−quasiinvex at x∗ for φ̃ increasing with φ̃(0) = 0, and ψ sublinear. (iii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are prestrictly hybrid (φ, ψ, ρ, η, θ)−quasi-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (φ̃, ψ, ρ̄, ζ, θ)−quasiinvex at x∗ for φ̃ increasing with φ̃(0) = 0, and ψ sublinear. (iv) for each i ∈ {1, ···, p}, fi is hybrid (φ, ψ, ρ1, ζ, θ)−invex and −gi is hybrid (φ, ψ, ρ2, ζ, θ)−invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, hj(. , v∗) ∀ j ∈ {1, · · ·, m} is hybrid (φ̄, ψ, ρ3, ζ, θ)−quasiinvex at x∗ for φ̄ increasing with φ̄(0) = 0, ψ sublinear, and σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for ρ∗ = σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2) and for φ(x ∗) = fi(x ∗ ) gi(x ∗) . then x∗ is an efficient solution to (p). 48 ram u. verma cubo 17, 1 (2015) proof. if (i) holds, and if x ∈ q, then using the sublinearity of ψ, it follows from (3.1) and (3.2) that ψ (〈 σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗ )z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗ )z], ζ(x, x∗) 〉) + ψ ( 〈σmj=1v ∗ j ▽ hj(x ∗) + 1 2 ∇2hj(x ∗)z, ζ(x, x∗) 〉) ≥ 0. (3.4) since v∗ ≥ 0, x ∈ q and (3.3) holds, we have σmj=1v ∗ j hj(x) ≤ 0 = σ m j=1v ∗ j hj(x ∗ ), and in light of the hybrid (φ̃, ψ, ρ̄, ζ, θ)−quasi-invexity of bj(., v ∗) at x∗, and assumptions on φ̃, we find φ̃ ( σmj=1v ∗ j hj(x) − σ m j=1v ∗ j hj(x ∗) ) ≤ 0, which results in ψ ( 〈▽hj(x ∗) + 1 2 ∇2hj(x ∗)z, ζ(x, x∗)〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 ≤ 0. (3.5) it follows from (3.4) and (3.5) that ψ (〈 σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)z], ζ(x, x∗) 〉) ≥ ρ̄(x, x∗)‖θ(x, x∗)‖2 ≥ −ρ(x, x∗)‖θ(x, x∗)‖2. (3.6) since ρ(x, x∗) ≥ 0, applying the hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.6) and assumptions on φ, we have φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(g ∗) )gi(x)] − σ p i=1 u∗i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) )gi(x ∗ )] ) ≥ 0, which implies σ p i=1u ∗ i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)]) = 0. thus, we have σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ 0. (3.7) cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 49 since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) ) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) ) < 0 for some j ∈ {1, · · ·, p}. hence, x∗ is an efficient solution to (p). next, if (ii) holds, and if x ∈ q, then using the sublinearity of ψ, it follows from (3.1) and (3.2) that ψ ( 〈σ p i=1u ∗ i [▽fi(x ∗ ) − ( fi(x ∗) gi(x∗) ) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)z], ζ(x, x∗) 〉) + ψ ( 〈σmj=1v ∗ j ▽ hj(x ∗ ) + 1 2 ∇2hj(x ∗ )z, ζ(x, x∗) 〉) ≥ 0. (3.8) since v∗ ≥ 0, x ∈ q and (3.3) holds, we have σmj=1v ∗ j hj(x) ≤ 0 = σ m j=1v ∗ j hj(x ∗), which results (using assumptions on φ̃) in φ̃ ( σmj=1v ∗ j hj(x) − σ m j=1v ∗ j hj(x ∗) ) ≤ 0. now, in light of the strictly hybrid (φ̃, ψ, ρ̄, ζ, θ)−quasi-invexity of bj(., v ∗) at x∗, we find ψ ( 〈▽hj(x ∗) + 1 2 ∇2hj(x ∗)z, η(x, x∗)〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 < 0. (3.9) it follows from (3.8) and (3.9) that ψ ( 〈σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)z], ζ(x, x∗)〉 > ρ̄(x, x∗)‖θ(x, x∗)‖2 > −ρ(x, x∗)‖θ(x, x∗)‖2. (3.10) 50 ram u. verma cubo 17, 1 (2015) as a result, since ρ(x, x∗) ≥ 0, applying the prestrictly hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.10) and assumptions on φ, we have φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(g ∗) )gi(x)] − σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)] ) ≥ 0, which implies σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)]) = 0. thus, we have σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ 0. (3.11) since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) ) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) ) < 0 for some j ∈ {1, · · ·, p}. hence, x∗ is an efficient solution to (p). the proof applying (iii) is similar to that of (ii), and we just need to include the proof using (iv) as follows: since x ∈ q, it follows that hj(x) ≤ hj(x ∗), which implies φ̄ ( hj(x) − hj(x ∗) ) ≤ 0. then applying the hybrid (φ̄, ψ, ρ3, ζ, θ)−quasi-invexity of hj at x ∗ and v∗ ∈ rm+ , we have ψ ( 〈σmj=1v ∗ j ▽ hj(x ∗), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), σmj=1v ∗ j ∇ 2hj(x ∗)z 〉) ≤ −σmj=1v ∗ j ρ3‖θ(x, x ∗)‖2. since u∗ ≥ 0 and fi(x ∗ ) gi(x ∗) ≥ 0, it follows from the hybrid (φ, ψ, ρ3, ζ, θ)−invexity assumptions that cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 51 φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ) = φ ( σ p i=1 u∗i {[fi(x) − fi(x ∗)] − ( fi(x ∗) gi(x ∗) )[gi(x) − gi(x ∗)]} ) ≥ ψ ( σ p i=1 u∗i {〈▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗), ζ(x, x∗)〉} + 1 2 〈ζ(x, x∗), σ p i=1u ∗ i [∇ 2fi(x ∗ )z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗ )z〉] ) + σ p i=1 u∗i [ρ1 + φ(x ∗)ρ2]‖θ(x, x ∗)‖2 ≥ −ψ ( [ 〈σmj=1v ∗ j ▽ hj(x ∗), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), σmj=1v ∗ j ∇ 2hj(x ∗)z 〉 ] ) + σ p i=1 u∗i [ρ1 + φ(x ∗)ρ2]‖θ(x, x ∗)‖2 ≥ (σmj=1v ∗ j ρ3 + σ p i=1 u∗i [ρ1 + φ(x ∗ )ρ2])‖θ(x, x ∗ )‖2 = (σmj=1v ∗ j ρ3 + ρ ∗)‖θ(x, x∗)‖2 ≥ 0, where φ(x∗) = fi(x ∗ ) gi(x ∗) and ρ∗ = σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2). this implies that φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ) ≥ 0. theorem 3.2. let x∗ ∈ q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗ ) gi(x ∗) ≥ 0, gi(x ∗) > 0 and hj for j ∈ {1, · · ·, m} be continuously differentiable at x∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ rep : u > 0, σ p i=1 ui = 1} and v ∗ ∈ rem+ such that 〈 σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗ )] + σmj=1v ∗ j ▽ hj(x ∗ ), z) 〉 ≥ 0 (3.12) and v∗j hj(x ∗) = 0, j ∈ {1, · · ·, m}. (3.13) suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): 52 ram u. verma cubo 17, 1 (2015) (i) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are first-order hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are first-order hybrid (φ̄, ψ, ρ̄, ζ, θ)−quasiinvex at x∗ for φ̄ increasing with φ̄(0) = 0, and ψ sublinear. (ii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are first-order hybrid prestrictly (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are first-order strictly hybrid (φ̄, ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for φ̄ increasing with φ̄(0) = 0, and ψ sublinear. (iii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are first-order prestrictly hybrid (φ, ψ, ρ, ζ, θ)−quasi-invex at x∗ φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are first-order strictly hybrid (φ̄, ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for φ̄ increasing with φ̄(0) = 0, and ψ sublinear. and z (iv) for each i ∈ {1, · · ·, p}, fi is first-order hybrid (φ, ψ, ρ1, ζ, θ)−invex and −gi is first-order hybrid (φ, ψ, ρ2, ζ, θ)−invex at x ∗ for φ(a) ≥ 0 ⇒ a ≥ 0. hj(. , v∗) ∀ j ∈ {1, · · ·, m} is hybrid (φ̄, ψ, ρ̄3, ζ, θ)−quasi-invex at x ∗, and σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for φ̄ increasing with φ̄(0) = 0, ρ∗ = σ p i=1u ∗ i (ρ1 + φ(x ∗)ρ2) for φ(x ∗) = fi(x ∗ ) gi(x ∗) , and ψ sublineaer. then x∗ is an efficient solution to (p). proof. although the proof is similar to that of theorem 3.1), we include for the sake of the completeness. if we consider (i), then proceeding as in theorem 3.1 (and using the first-order hybrid (φ, ψ, ρ, ζ, θ)−invexity assumptions instead), we arrive at ψ ( 〈σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)], ζ(x, x∗)〉 ) ≥ ρ(x, x∗)‖θ(x, x∗)‖2. (3.14) since ρ(x, x∗) ≥ 0, applying the hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.14) and assumptions on φ, we have φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(g ∗) )gi(x)] − σ p i=1 u∗i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) )gi(x ∗ )] ) ≥ 0, which implies σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)]) = 0. cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 53 thus, we have σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ 0. (3.15) since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) ) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) ) < 0 for some j ∈ {1, · · ·, p}. hence, x∗ is an efficient solution to (p). theorem 3.3. let x∗ ∈ q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗ ) gi(x ∗) ≥ 0, gi(x ∗) > 0 and hj for j ∈ {1, · · ·, m} be twice continuously differentiable at x∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ rep : u > 0, σ p i=1 ui = 1} and v ∗ ∈ rem+ such that σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + σmj=1v ∗ j ▽ hj(x ∗) = 0 (3.16) 〈 ζ(x, x∗), [ p∑ i=1 u∗i [∇ 2fi(x ∗) − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)] + m∑ j=1 v∗j ∇ 2hj(x ∗) ] z 〉 ≥ 0, (3.17) and v∗j hj(x ∗) = 0, j ∈ {1, · · ·, m}. (3.18) suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): (i) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are hybrid (ρ, ζ, θ)−pseudo-invex at x∗, and bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are hybrid (ρ, ζ, θ)−quasi-invex at x∗. (ii) ei(. ; x ∗, u∗) ∀ i ∈ {1, ···, p} are prestrictly hybrid (ρ, ζ, θ)−pseudo-invex at x∗, and bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are hybrid (ρ, ζ, θ)−strictly-quasi-invex at x∗. (iii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are strictly hybrid (ρ, ζ, θ)−pseudo-invex at x∗, and bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (ρ, ζ, θ)−quasi-invex at x∗. (iv) for each i ∈ {1, · · ·, p}, fi is hybrid (ρ1, ζ, θ)−invex and −gi is (ρ2, ζ, θ)−invex at x ∗. hj(. , v ∗) ∀ j ∈ {1, · · ·, m} is hybrid (ρ3, ζ, θ)−quasi-invex at x ∗, and σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for ρ∗ = σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2) and for φ(x ∗) = fi(x ∗ ) gi(x ∗) . 54 ram u. verma cubo 17, 1 (2015) then x∗ is an efficient solution to (p). proof. the proof is similar to that of theorem 3.1 based on the second order hybrid (ρ, ζ, θ)− invexity assumptions. we observe that theorem 3.1 can be further generalized to the case of the ǫ−efficient conditions based on the hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks. as a matter of fact, we generalize the ǫ−efficient solvability conditions for problem (p) based on the work of verma [22], and kim, kim and lee [6], where they have investigated the ǫ−efficiency as well as the weak ǫ−efficiency conditions for multiobjective fractional programming problems under constraint qualifications. to the best of our knowledge, the results established in this communication (theorem 3.1 and theorem 3.4) generalize and unify most of the results on the multiobjective fractional programming to the context of the generalized invexities in the literature. we recall some auxiliary concepts (for the hybrid (φ, ψ, ρ, ζ, θ)−invexity) crucial to the problem on hand. definition 3.1. a point x∗ ∈ q is an ǫ−efficient solution to (p) if there does not exist an x ∈ q such that fi(x) gi(x) ≤ fi(x ∗) gi(x ∗) − ǫi ∀ i = 1, · · ·, p, fj(x) gj(x) < f(jx ∗) gj(x ∗) − ǫj for some j ∈ {1, · · ·, p}, where ǫi=(ǫ1, · · ·, ǫp) is with ǫi ≥ 0 for i = 1, · · ·, p. for ǫ = 0, definition 3.1 reduces to the case that x∗ ∈ q is an efficient solution to (p). next, we start with real-valued functions ei(., x ∗, u∗) and bj(., v), respectively, defined by ei(x, x ∗, u∗) = ui[fi(x) − ( fi(x ∗) gi(x ∗) − ǫi ) gi(x)], i ∈ {1, · · ·, p} and bj(., v) = vjhj(x), j = 1, · · ·, m. cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 55 theorem 3.4. let x∗ ∈ q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗) ≥ ǫigi(x ∗), gi(x ∗) > 0 and hj for j ∈ {1, · · ·, m} be twice continuously differentiable at x ∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ rep : u > 0, σ p i=1 ui = 1}, v ∗ ∈ rem+ and z ∈ re n such that σ p i=1 u∗i [∇fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi ) ▽ gi(x ∗ )] + σmj=1v ∗ j ▽ hj(x ∗ ) = 0, (3.19) 〈 ζ(x, x∗), [ p∑ i=1 u∗i [∇ 2fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi ) ∇2gi(x ∗ )] + m∑ j=1 v∗j ∇ 2hj(x ∗ ) ] z 〉 ≥ 0, (3.20) and v∗j hj(x ∗ ) = 0, j ∈ {1, · · ·, m}. (3.21) suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): (i) ei(. ; x ∗, u∗) ∀ i ∈ {1, ···, p} are hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v ∗) ∀ j ∈ {1, · · ·, m} arehybrid (φ̃, ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for φ̃ increasing with φ̃(0) = 0, and ψ sublinear. (ii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are prestrictly hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (φ, ψ, ρ, ζ, θ)−quasiinvex at x∗ for φ̄ increasing with φ̄(0) = 0, and ψ sublinear. (iii) ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are strictly hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (φ̄, ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for φ̄ increasing with φ̄(0) = 0, and ψ sublinear. (iv) for each i ∈ {1, · · ·, p}, fi is hybrid (φ, ψ, ρ1, ζ, θ)−invex and −gi is (φ, ψ, ρ2, ζ, θ)−invex at x∗ for φ(a) ≥ 0 ⇒ a ≥ 0, and hj(. , v∗) ∀ j ∈ {1, · · ·, m} is hybrid (φ̄, ψ, ρ3, ζ, θ)−quasiinvex at x∗ for φ̄ increasing with φ̄(0) = 0, ψ sublinear, and σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for ρ∗ = σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2), where φ(x ∗) = fi(x ∗ ) gi(x ∗) − ǫi. then x∗ is an ǫ−efficient solution to (p). proof. if (i) holds, and if x ∈ q, then it follows using the sublinearity of ψ from (3.1) and (3.2) 56 ram u. verma cubo 17, 1 (2015) that ψ ( 〈σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z], ζ(x, x∗) 〉) + ψ ( 〈σmj=1v ∗ j ▽ hj(x ∗ ) + 1 2 ∇2hj(x ∗ )z, ζ(x, x∗) 〉) ≥ 0. (3.22) since v∗ ≥ 0, x ∈ q and (3.3) holds, we have σmj=1v ∗ j hj(x) ≤ 0 = σ m j=1v ∗ j hj(x ∗), which implies σmj=1v ∗ j hj(x) − σ m j=1v ∗ j hj(x ∗) ≤ 0, so in light of the hybrid (φ̃, ψ, ρ̄, ζ, θ)−quasi-invexity of bj(., v ∗) at x∗, and assumptions on φ̃, it results in φ̃ ( σmj=1v ∗ j hj(x) − σ m j=1v ∗ j hj(x ∗ ) ) ≤ 0, which implies ψ ( 〈▽hj(x ∗), ζ(x, x∗)〉 + 1 2 〈ζ(x, x∗), ∇2hj(x ∗)z〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 ≤ 0. (3.23) it follows from (3.22) and (3.23) that ψ ( 〈σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗)], z〉 + 1 2 〈 ζ(x, x∗), p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z] 〉) ≥ ρ̄(x, x∗)‖θ(x, x∗)‖2 ≥ −ρ(x, x∗)‖θ(x, x∗)‖2. (3.24) as a result, since ρ(x, x∗) ≥ 0, applying the hybrid (φ, ψ, ρ, ζ, θ)− pseudo-invexity at x∗ to (3.24) and assumptions on φ, we have φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] −σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗)] ) ≥ 0, cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 57 which implies σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗)] ≥ σ p i=1u ∗ i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗ )] −σ p i=1 u∗i ǫigi(x ∗) = 0. thus, we have σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ 0. (3.25) since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ q such that ∑p i=1 fi(x)∑p i=1 gi(x) − ( fi(x ∗) gi(x ∗) − ǫi) ≤ 0 ∀ i = 1, · · ·, p, ∑p j=1 fj(x) ∑p j=1 gj(x) − ( fj(x ∗) gj(x ∗) − ǫj) < 0 for some j ∈ {1, · · ·, p}. hence, x∗ is an ǫ−efficient solution to (p). if (ii) holds, and if x ∈ q, then it follows from (3.1) and (3.2) that ψ (〈 σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z], ζ(x, x∗) 〉) + ψ (〈 σmj=1v ∗ j ▽ hj(x ∗ ) + 1 2 ∇2hj(x ∗ )z, ζ(x, x∗) 〉) ≥ 0. (3.26) since v∗ ≥ 0, x ∈ q and (3.3) holds, we have σmj=1v ∗ j hj(x) ≤ 0 = σ m j=1v ∗ j hj(x ∗), or σmj=1v ∗ j hj(x) − σ m j=1v ∗ j hj(x ∗ ) ≤ 0, 58 ram u. verma cubo 17, 1 (2015) which implies based on assumptions on φ̃ that φ̃ ( σmj=1v ∗ j hj(x) − σ m j=1v ∗ j hj(x ∗) ) ≤ 0. next, in light of the strict (φ̃, ψ, ρ̄, ζ, θ)−quasi-invexity of bj(., v ∗) at x∗ with φ̃ increasing and φ̃(0) = 0, we find ψ ( 〈▽hj(x ∗), ζ(x, x∗)〉 + 1 2 〈ζ(x, x∗), ∇2hj(x ∗)z〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 < 0. (3.27) it follows from (3.26) and (3.27) that ψ ( 〈σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗)], ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z] 〉) > ρ̄(x, x∗)‖θ(x, x∗)‖2 > −ρ(x, x∗)‖θ(x, x∗)‖2. (3.28) as a result, since ρ(x, x∗) ≥ 0, applying the prestrictly hybrid (φ, ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.28) and assumptions on φ, we have φ ( σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] − σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) ) − ǫi)gi(x ∗)] ) ≥ 0, which implies σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗)] ≥ σ p i=1 u∗i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗ )] − σ p i=1 u∗i ǫigi(x ∗ ) = 0. thus, we have σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ 0. (3.29) since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) − ǫi) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) − ǫj) < 0 for some j ∈ {1, · · ·, p}. cubo 17, 1 (2015) hybrid (φ, ψ, ρ, ζ, θ)−invexity frameworks . . . 59 hence, x∗ is an ǫ−efficient solution to (p). the proof applying (iii) is similar to that of (ii), and we just need to include the proof using (iv) as follows: since x ∈ q, it follows that hj(x) ≤ hj(x ∗). then applying the (φ̄, ψ, ρ3, ζ, θ)−quasiinvexity of hj at x ∗ and v∗ ∈ rm+ , we have ψ (〈 σmj=1v ∗ j ▽ hj(x ∗ ), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), σmj=1v ∗ j ∇ 2hj(x ∗ )z 〉) ≤ −σmj=1v ∗ j ρ3‖θ(x, x ∗)‖2. since u∗ ≥ 0 and fi(x ∗) ≥ ǫigi(x ∗), it follows from (φ, ψ, ρ3, ζ, θ)−invexity assumptions that φ ( σ p i=1u ∗ i [fi(x) − ( fi(x ∗) gi(x∗) − ǫi)gi(x)] ) = φ ( σ p i=1 u∗i {[fi(x) − fi(x ∗)] − ( fi(x ∗) gi(x ∗) − ǫi)[gi(x) − gi(x ∗)] + ǫigi(x ∗)} ) ≥ ψ ( σ p i=1 u∗i {〈▽fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗), ζ(x, x∗)〉} + 1 2 〈ζ(x, x∗), σ p i=1 u∗i [∇ 2fi(x ∗ )z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗ )z〉] ) + [ρ1 + φ(x ∗)ρ2]‖θ(x, x ∗)‖2 + σ p i=1 u∗i ǫigi(x ∗) ≥ −ψ ( [ 〈σmj=1v ∗ j ▽ hj(x ∗), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), σmj=1v ∗ j ∇ 2hj(x ∗)z 〉 ] ) + σ p i=1 u∗i [ρ1 + ( fi(x ∗) gi(x ∗) − ǫi)ρ2]‖θ(x, x ∗)‖2 + σ p i=1 u∗i ǫigi(x ∗) ≥ (σmj=1v ∗ j ρ3 + σ p i=1 u∗i [ρ1 + ( fi(x ∗) gi(x ∗) − ǫi)ρ2])‖θ(x, x ∗ )‖2 + σ p i=1 u∗i ǫigi(x ∗ ) = (σmj=1v ∗ j ρ3 + ρ ∗)‖θ(x, x∗)‖2 + σ p i=1 u∗i ǫigi(x ∗) ≥ (σmj=1v ∗ j ρ3 + ρ ∗)‖θ(x, x∗)‖2 ≥ 0. therefore, we have σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ 0. (3.30) 60 ram u. verma cubo 17, 1 (2015) thus, we conclude that there does not exist an x ∈ q such that ∑p i=1 fi(x)∑p i=1 gi(x) − ( fi(x ∗) gi(x ∗) − ǫi) ≤ 0 ∀ i = 1, · · ·, p, ∑p j=1 fj(x) ∑p j=1 gj(x) − ( fj(x ∗) gj(x ∗) − ǫj) < 0 for some j ∈ {1, · · ·, p}. hence, x∗ is an ǫ−efficient solution to (p). 4 concluding remarks we observe that the obtained results in this communication can be generalized to the case of multiobjective fractional programming with generalized hybrid invex functions of higher orders (including the exponential type generalized invexities), for instance, based on the work of mishra and rueda [11], mishra, laha and verma [13], and zalmai and zhang [37] to the case of the efficiency as well as to the ǫ−efficiency conditions relating to the minimax fractional programming problems involving generalized invex functions. received: may 2014. accepted: october 2014. references [1] a. ben-israel and b. mond, what is the invexity? journal of australian mathematical society ser. b 28 (1986), 1 9. 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[40] e. zeidler, nonlinear functional analysis and its applications iii, springer-verlag, new york, new york, 1985. introduction hybrid invexities efficiency conditions for problem (p) concluding remarks cubo a mathematical journal vol.15, no¯ 02, (79–88). june 2013 squares in euler triples from fibonacci and lucas numbers zvonko čerin university of zagreb, kopernikova 7, 10010 zagreb, croatia, europe, cerin@math.hr abstract in this paper we shall continue to study from [4], for k = −1 and k = 5, the infinite sequences of triples a = (f2n+1, f2n+3, f2n+5), b = (f2n+1, 5f2n+3, f2n+5), c = (l2n+1, l2n+3, l2n+5), d = (l2n+1, 5l2n+3, l2n+5) with the property that the product of any two different components of them increased by k are squares. the sequences a and b are built from the fibonacci numbers fn while the sequences c and d from the lucas numbers ln. we show some interesting properties of these sequences that give various methods how to get squares from them. resumen en este art́ıculo continuaremos el estudio de [4], para k = −1 y k = 5, las secuencias infinitas de tripletas a = (f2n+1, f2n+3, f2n+5), b = (f2n+1, 5f2n+3, f2n+5), c = (l2n+1, l2n+3, l2n+5), d = (l2n+1, 5l2n+3, l2n+5) con la propiedad que el producto de dos componentes diferentes que se aumenta en k son cuadrados. las secuencias a y b se construyen con los números de fibonacci fn mientras que las secuencias c y d se construyen con los números de lucas ln. mostramos algunas propiedades interesantes de estas secuencias que entregan muchos métodos de cómo conseguir los cuadrados de ellos. keywords and phrases: d(k)-triple, fibonacci numbers, lucas numbers, square, symmetric sum, alternating sum, product, component 2010 ams mathematics subject classification: 11b37, 11b39, 11d09. 80 zvonko čerin cubo 15, 2 (2013) 1 introduction for integers a, b and c, let us write a b ∼ c provided a + b = c2. for the triples a = (a,b,c), d = (d,e,f) and ã = (ã, b̃, c̃) the notation a d ∼ ã means that bc d ∼ ã, ca e ∼ b̃ and ab f ∼ c̃. when d = (k,k,k), let us write a k ∼ ã for a d ∼ ã. hence, a is the d(k)-triple (see [1]) if and only if there is a triple ã such that a k ∼ ã. in the paper [4] we constructed infinite sequences α = {α(n)}∞n=0 and β = {β(n)} ∞ n=0 of d(−1)triples and γ = {γ(n)}∞n=0 and δ = {δ(n)} ∞ n=0 of d(5)-triples. here, α(n) = a = (f2n+1, f2n+3, f2n+5), β(n) = b = (f2n+1, 5f2n+3, f2n+5) and γ(n) = c = (l2n+1, l2n+3, l2n+5), δ(n) = d = (l2n+1, 5l2n+3, l2n+5), where the fibonacci and lucas sequences of natural numbers fn and ln are defined by the recurrence relations f0 = 0, f1 = 1, fn = fn−1 + fn−2 for n > 2 and l0 = 2, l1 = 1, ln = ln−1 + ln−2 for n > 2. the numbers fk make the integer sequence a000045 from [6] while the numbers lk make a000032. the goal of this article is to further explore the properties of the sequences α, β, γ and δ. each member of these sequences is an euler d(−1)or d(5)-triple (see [2] and [3]) so that many of their properties follow from the properties of the general (pencils of) euler triples. it is therefore interesting to look for those properties in which at least two of the sequences appear. this paper presents several results of this kind giving many squares from the components, various sums and products of the sequences α, β, γ and δ. most of our theorems have also versions for the associated sequences α̃, β̃, γ̃ and δ̃, where α̃(n) = ã = (f2n+4,f2n+3,f2n+2), β̃(n) = b̃ = (l2n+4,f2n+3,l2n+2), γ̃(n) = c̃ = (l2n+4,l2n+3,l2n+2), δ̃(n) = d̃ = (5f2n+4,l2n+3, 5f2n+2) satisfy a −1 ∼ ã, b −1 ∼ b̃, c 5 ∼ c̃ and d 5 ∼ d̃. 2 squares from products of components the relations a −1 ∼ ã and c 5 ∼ c̃ imply that the components of a and c satisfy a2a3 −1 ∼ ã1 and c2c3 5 ∼ c̃1. our first theorem shows that the product a2a3c2c3 is in a similar relation with respect to 1. of course, the other products a3a1c3c1, a1a2c1c2 as well as b2b3d2d3, b3b1d3d1 and b1b2d1d2 exhibit a similar property. theorem 1. the following hold for the products of components: a2 a3 c2 c3 1 ∼ f4n+8, 1 5 b2 b3 d2 d3 9 ∼ l4n+8, a3 a1 c3 c1 9 ∼ f4n+6, b3 b1 d3 d1 9 ∼ f4n+6, a1 a2 c1 c2 1 ∼ f4n+4, 1 5 b1 b2 d1 d2 9 ∼ l4n+4. cubo 15, 2 (2013) squares in euler triples ... 81 proof. let ϕ = 1+ √ 5 2 and ψ = 1− √ 5 2 = − 1 ϕ . since fj = ϕ j −ψ j ϕ−ψ and lj = ϕ j + ψj, it follows that a2 = ϕ 2 n+3 −ψ 2 n+3 ϕ−ψ , a3 = ϕ 2 n+5 −ψ 2 n+5 ϕ−ψ and c2 = ϕ 2n+3 + ψ2n+3, c3 = ϕ 2n+5 + ψ2n+5. after the substitutions ψ = − 1 ϕ and m = ϕn, the sum of a2a3c2c3 and 1 becomes ϕ 16 (m 8 −ψ 16 ) 2 20m8 . however, this is precisely the square of f4n+8. this shows the first relation. the other relations have similar proofs. the version of the previous theorem for the sequences α̃, β̃, γ̃ and δ̃ is the following result. theorem 2. the products of components of ã, b̃, c̃ and d̃ satisfy: ã2ã3c̃2c̃3 1 ∼ f4n+5, b̃2b̃3d̃2d̃3 1 ∼ l4n+5, ã3ã1c̃3c̃1 1 ∼ f4n+6, 1 25 b̃3b̃1d̃3d̃1 1 ∼ f4n+6, ã1ã2c̃1c̃2 1 ∼ f4n+7, b̃1b̃2d̃1d̃2 1 ∼ l4n+7. proof. since ã2 = ϕ 2 n+3 −ψ 2 n+3 ϕ−ψ , ã3 = ϕ 2 n+2 −ψ 2 n+2 ϕ−ψ , c̃2 = ϕ 2n+3+ ψ2n+3 and c̃3 = ϕ 2n+2+ ψ2n+2, the sum of ã2ã3c̃2c̃3 and 1, after the substitutions ψ = − 1 ϕ and m = ϕn, becomes ϕ 10 (m 8 +ψ 10 ) 2 5m8 . however, the square of f4n+5 has the same value. this proves the first relation ã2ã3c̃2c̃3 1 ∼ f4n+5. the remaining five relations in this theorem have similar proofs. the same kind of relations hold also for the products of components from all four sequences α, β, γ and δ. theorem 3. the following relations for products of components hold: a2b3c2d3 1 ∼ f4n+8, 1 25 a3b2c3d2 1 ∼ f4n+8, a3b1c3d1 9 ∼ f4n+6, a1b3c1d3 9 ∼ f4n+6, 1 25 a1b2c1d2 1 ∼ f4n+4, a2b1c2d1 1 ∼ f4n+4. proof. since b3 = a3 and d3 = c3, the first relation is the consequence of the first relation in theorem 1. in order to prove the second relation, notice that b2 = 5a2 and d2 = 5c2 so that the multiplication of the identity behind the first relation in theorem 1 with 25 we conclude that the second relation holds. the other relations in this theorem have similar proofs. there is again the version of the previous theorem for the products of components from all four sequences α̃, β̃, γ̃ and δ̃. 82 zvonko čerin cubo 15, 2 (2013) theorem 4. the products of components of ã, b̃, c̃ and d̃ satisfy: ã2b̃3c̃2d̃3 1 ∼ l4n+5, ã3b̃2c̃3d̃2 1 ∼ f4n+5, ã3b̃1c̃3d̃1 9 ∼ l4n+6, ã1b̃3c̃1d̃3 9 ∼ l4n+6, ã1b̃2c̃1d̃2 1 ∼ f4n+7, ã2b̃1c̃2d̃1 1 ∼ l4n+7. proof. since ã2 = b̃2 and c̃2 = d̃2, the first, the second, the fifth and the sixth relations are the consequence of the second, the first, the fifth and the sixth relations in theorem 2. in order to prove the third relation, note that the components ã3, b̃1 and c̃3, d̃1 are ϕ 2 n+2 −ψ 2 n+2 ) ϕ−ψ , ϕ2n+4 + ψ2n+4, ϕ2n+2 + ψ2n+2 and 5(ϕ 2 n+4 −ψ 2 n+4 ) ϕ−ψ . it is now clear from the proof of theorem 1 that the sum of ã3b̃1c̃3d̃1 and 9 is precisely the square of l4n+6. this shows the third relation. the fourth relation has a similar proof. nice relationships of the same kind hold also for the products of components with other choices of indices. theorem 5. 1 5 a2 b2 c3 d3 0 ∼ f2n+3l2n+5, 1 5 a3 b3 c2 d2 0 ∼ f2n+5l2n+3, a3 b3 c1 d1 0 ∼ f2n+5l2n+1, a1 b1 c3 d3 0 ∼ f2n+1l2n+5, 1 5 a1 b1 c2 d2 0 ∼ f2n+1l2n+3, 1 5 a2 b2 c1 d1 0 ∼ f2n+3l2n+1. proof. since b2 = 5a2, a2 = f2n+3 and c3 = d3 = l2n+5, the product 1 5 a2 b2 c3 d3 is the square of f2n+3l2n+5. the other claims in this theorem have similar proofs. the version of the previous theorem for the products of components from all four associated sequences is the following result. theorem 6. the products of components of ã, b̃, c̃ and d̃ satisfy: ã2b̃2c̃3d̃3 5f4n+4 0 ∼ f2n+3, ã3b̃3c̃2d̃2 f4n+4 0 ∼ l2n+3, 1 5 ã3b̃3c̃1d̃1 1 ∼ f4n+6, 1 5 ã1b̃1c̃3d̃3 1 ∼ f4n+6, ã1b̃1c̃2d̃2 f4n+8 0 ∼ l2n+3, ã2b̃2c̃1d̃1 5f4n+8 0 ∼ f2n+3. proof. since ã2 = b̃2 = f2n+3, c̃3 = l2n+2, d̃3 = 5f2n+2, we see that the first relation clearly holds. the others in this theorem are proved similarly. cubo 15, 2 (2013) squares in euler triples ... 83 this time the pairs (a,d) and (b,c) have equal indices. theorem 7. the following hold for the products of components: 1 5 a2 b3 c3 d2 1 ∼ f4n+8, 1 5 a3 b2 c2 d3 1 ∼ f4n+8, a3 b1 c1 d3 9 ∼ f4n+6, a1 b3 c3 d1 9 ∼ f4n+6, 1 5 a1 b2 c2 d1 1 ∼ f4n+4, 1 5 a2 b1 c1 d2 1 ∼ f4n+4. proof. since a2 = f2n+3, b3 = f2n+5, c3 = l2n+5 and d2 = 5l2n+3, the sum of 1 5 a2 b3 c3 d2 and 1 is f4n+6f4n+10 + 1 = f 2 4n+8. the other claims in this theorem have similar proofs. once again the version of the previous theorem for the associated sequences includes interesting relations. theorem 8. the products of components of ã, b̃, c̃ and d̃ satisfy: ã2b̃3c̃3d̃2 f4n+6 0 ∼ l2n+2, ã3b̃2c̃2d̃3 5f4n+6 0 ∼ f2n+2, 1 5 ã3b̃1c̃1d̃3 0 ∼ f2n+2l2n+4, 1 5 ã1b̃3c̃3d̃1 0 ∼ f2n+4l2n+2, ã1b̃2c̃2d̃1 5f4n+6 0 ∼ f2n+4, ã2b̃1c̃1d̃2 f4n+6 0 ∼ l2n+4. proof. since ã2 = f2n+3 b̃3 = c̃3 = l2n+2, d̃2 = l2n+3 and f2n+3l2n+3 = f4n+6 we see that the first relation clearly holds. the other relations in this theorem are proved similarly. it is interesting that in some cases we can even mix components of the triples a, b, c, d and ã, b̃, c̃, d̃ as the relations ã2 b3 c3 d̃2 1 ∼ f4n+8, ã2 b̃2 c3 d3 0 ∼ f2n+3l2n+5 and ã2 b̃2 c1 d1 0 ∼ f2n+3l2n+1 show, but we do not see a general pattern here. 3 squares from symmetric sums let σ1, σ2, σ3 : z 3 → z be the basic symmetric functions defined for x=(a, b, c) by xσ1 = a + b + c, xσ2 = bc + ca + ab, xσ3 = abc. let σ ∗ 2, σ ∗ 1 : z 3 → z be defined by xσ∗ 2 = bc − ca + ab and xσ∗ 1 = a − b + c. note that xσ∗ 1 is the determinant of the 1 × 3 matrix [a,b,c] (see [5]). for the sums σ2 and σ ∗ 2 of the components the following relations are true. 84 zvonko čerin cubo 15, 2 (2013) theorem 9. the following is true for the sums σ2 of the components: aσ2cσ2 21 ∼ 4f4n+6, bσ2dσ2 69 ∼ 16f4n+6, ãσ2c̃σ2 5 ∼ 2f4n+7, b̃σ2d̃σ2 45 ∼ 10f4n+6. proof. since aσ2 = 1 5 (4l4n+6 + 13) and cσ2 = 4l4n+6 − 13, the sum aσ2cσ2 + 21 is 1 5 [(4l4n+6) 2 − 64] that we recognize as the square of 4f4n+6. this proves the first relation aσ2cσ2 21 ∼ 4f4n+6. the other relations in this theorem have similar proofs. the sums b̃σ∗ 2 and d̃σ∗ 2 have constant values −1 and 5. on the other hand, we have the following result. theorem 10. the following is true for the sums σ∗2 of the components: aσ∗ 2 cσ∗ 2 −3 ∼ 2f4n+6, bσ∗ 2 dσ∗ 2 −51 ∼ 14f4n+6, ãσ∗ 2 c̃σ∗ 2 5 ∼ 2f4n+6. proof. since bσ∗ 2 = 1 5 (14l4n+6 + 23) and dσ∗ 2 = 14l4n+6 − 23, the sum bσ∗ 2 dσ∗ 2 − 51 is the quotient 196(l 2 4n+6−4) 5 . it is now easy to check that this is the square of 14f4n+6. this proves the second relation bσ∗ 2 cσ2 −51 ∼ 14f4n+6. the other relations in this theorem have similar proofs. some similar relations where all four letters a, b, c and d appear make the following result. theorem 11. the following is true for the sums σ∗2 of the components: ãσ∗ 2 d̃σ∗ 2 + b̃σ∗ 2 c̃σ∗ 2 = 6, ãσ∗ 2 c̃σ∗ 2 + b̃σ∗ 2 d̃σ∗ 2 10 ∼ 2f4n+5, −ãσ∗ 2 b̃σ∗ 2 c̃σ∗ 2 d̃σ∗ 2 9 ∼ 2l4n+5. proof. since b̃σ∗ 2 = −1, ãσ∗ 2 = 1 5 (2l4n+5 + 3), c̃σ∗ 2 = 2l4n+5 − 3 and d̃σ∗ 2 = 5, it follows that ãσ∗ 2 d̃σ∗ 2 + b̃σ∗ 2 c̃σ∗ 2 = 6. the second and the third relations in this theorem have similar proofs. here are two relations which contains both sums σ2 and σ ∗ 2. theorem 12. the following is true for the sums σ2 and σ ∗ 2: 1 36 (aσ2dσ2 − bσ∗2cσ ∗ 2 ) 2 ∼ f4n+6, 3bσ∗ 2 cσ∗ 2 − aσ2dσ2 −74 ∼ 2l4n+6 + 6. proof. since the sums bσ∗ 2 , aσ2, cσ∗2 and dσ2 are equal 1 5 (14l4n+6 + 23), 1 5 (4l4n+6 + 13), 2l4n+6 + 1 and 16l4n+6 − 37, we infer that the sum 1 36 (aσ2dσ2 − bσ∗2cσ ∗ 2 ) + 2 is the square of f4n+6. the second relation in this theorem has analogous proof. in the next result we consider the products of the same components of the triples a, b, c and d and the product of their components. cubo 15, 2 (2013) squares in euler triples ... 85 theorem 13. the following relations hold: a1b1c1d1 0 ∼ f4n+2, a2b2c2d2 0 ∼ 5f4n+6, a3b3c3d3 0 ∼ f4n+10, aσ3bσ3cσ3dσ3 0 ∼ 5f4n+2 f4n+6 f4n+10. proof. since the product f2n+5l2n+5 is f4n+10, a3 = b3 = f2n+5 and c3 = d3 = l2n+5, it follows that a3b3c3d3 = (f2n+5 l2n+5) 2 = f24n+10. this proves the third relation. the other relations have similar proofs. the products of the same components of the triples ã, b̃, c̃ and d̃ and the product of their components appear in the following result. theorem 14. the following relations are true: 1 5 ã1b̃1c̃1d̃1 0 ∼ f4n+8, 1 5 ã3b̃3c̃3d̃3 0 ∼ f4n+4, ã2b̃2c̃2d̃2 0 ∼ f4n+6, ãσ3b̃σ3c̃σ3d̃σ3 0 ∼ 5f4n+8f4n+6f4n+4. proof. since ã3 = f2n+2, b̃3 = l2n+2, c̃3 = l2n+2, d̃3 = 5f2n+2 and f2n+2l2n+2 = f4n+4, it follows that the product 1 5 ã3b̃3c̃3d̃3 is the square of f4n+4. this proves the second relation. the other relations in this theorem have similar proofs. the products of the sums σ1 and σ ∗ of the components of the triples a, b, c and d show the same kind of relations. this is also true for the associated triples ã, b̃, c̃ and d̃. theorem 15. the following relations hold for the sums σ1 and σ ∗ 1: aσ1bσ1cσ1dσ1 0 ∼ 32f4n+6, 1 144 ãσ1b̃σ1c̃σ1d̃σ1 1 ∼ f4n+7, aσ∗ 1 bσ∗ 1 cσ∗ 1 dσ∗ 1 0 ∼ 4f4n+6, 1 64 ãσ∗ 1 b̃σ∗ 1 c̃σ∗ 1 d̃σ∗ 1 1 ∼ f4n+5. proof. the sums of the components aσ1, bσ1, cσ1 and dσ1 are equal 4f2n+3, 8f2n+3, 4l2n+3 and 8l2n+3. hence, the product aσ1bσ1cσ1dσ1 is the square of 32f4n+6 since f2n+3l2n+3 = f4n+6. this proves the above first relation. the other relations in this theorem have similar proofs. in the next result we combine the sums σ1 and σ ∗ 1 in each product. theorem 16. the following relations hold for the sums σ1 and σ ∗ 1: aσ1bσ∗1cσ1dσ ∗ 1 0 ∼ 8f4n+6, aσ∗ 1 bσ1cσ∗1dσ1 0 ∼ 16f4n+6, 1 24 ãσ1b̃σ∗1c̃σ ∗ 1 d̃σ1 1 ∼ 2f4n+6 + 1, 1 24 ãσ∗ 1 b̃σ1c̃σ1d̃σ∗1 1 ∼ 2f4n+6 − 1, 1 64 ãσ1b̃σ∗1c̃σ1d̃σ ∗ 1 1 ∼ f4n+7, 1 144 ãσ∗ 1 b̃σ1c̃σ∗1d̃σ1 1 ∼ f4n+5. 86 zvonko čerin cubo 15, 2 (2013) proof. the sums of the components aσ1, bσ∗1, cσ1 and dσ ∗ 1 are equal 4f2n+3, −2f2n+3, 4l2n+3 and −2l2n+3. the product aσ1bσ∗1cσ1dσ ∗ 1 is therefore the square of 8f4n+6 since f2n+3l2n+3 = f4n+6. this proves the first relation. the other relations in this theorem have analogous proofs. 4 squares from the sums of squares for a natural number k > 1, let the sums νk, ν ∗ k : z 3 → z of powers be defined for x = (a,b,c) by xνk = a k + bk + ck and xν∗ k = ak − bk + ck. we proceed with the version of the theorem 9 for the sums ν2 of the squares of components. theorem 17. the following relations are true for the sums ν2: 1 4 aν2cν2 −11 ∼ 4f4n+6, 1 4 bν2dν2 −59 ∼ 16f4n+6, 1 4 ãν2c̃ν2 −3 ∼ 2f4n+6, 1 4 b̃ν2d̃ν2 −27 ∼ 8f4n+6. proof. since aν2 and cν2 are 2 5 (4l4n+6 + 3) and 2(4l4n+6 − 3), the difference of 1 4 aν2cν2 and 11 is equal 16(l 2 4n+6−4) 5 . but, one can easily check that l 2 4n+6−4 5 = f24n+6 so that the above quotient is the square of 4f4n+6. this concludes the proof of the first relation. the other relations in this theorem have similar proofs. the next is the version of the theorem 10 for the alternating sums ν∗2 of the squares of components. theorem 18. the following relations are true for the sums ν∗2: 1 4 aν∗ 2 cν∗ 2 −7 ∼ 3f4n+6, 1 4 bν∗ 2 dν∗ 2 41 ∼ 9f4n+6, 1 4 ãν∗ 2 c̃ν∗ 2 1 ∼ f4n+6, 1 4 b̃ν∗ 2 d̃ν∗ 2 −23 ∼ 7f4n+6. proof. notice that the alternating sums of squares of components aν∗ 2 and cν∗ 2 are 2 5 (3l4n+6 + 1) and 2(3l4n+6 + 1). hence, the sum of 1 4 aν∗ 2 cν∗ 2 and −7 is equal to the following quotient 9(l 2 4n+6−4) 5 . this quotient is in fact the square of 3f4n+6. this proves the first relation. the remaining three relations in this theorem have similar proofs. certain sums of products of the sums ν∗2 of components show the same behavior. theorem 19. the following relations are true for the sums ν∗2: 1 8 (aν∗ 2 dν∗ 2 + bν∗ 2 cν∗ 2 ) 17 ∼ √ −27f4n+6, 1 8 (ãν∗ 2 d̃ν∗ 2 + b̃ν∗ 2 c̃ν∗ 2 ) −11 ∼ √ 7f4n+6. cubo 15, 2 (2013) squares in euler triples ... 87 proof. notice that the alternating sums of squares of components ãν∗ 2 , b̃ν∗ 2 , c̃ν∗ 2 and d̃ν∗ 2 are 2f2n+2f2n+4, 2 5 (7l4n+6 + 9), 2l2n+2l2n+4 and 2(7l4n+6 − 9). hence, the sum of 1 8 (ãν∗ 2 d̃ν∗ 2 + b̃ν∗ 2 c̃ν∗ 2 ) and −11 is equal to the square of √ 7f4n+6. this proves the second relation. the first relation has a similar proof. 5 squares from the products ⊙, ⊲ and ⊳ let us introduce three binary operations ⊙, ⊲ and ⊳ on the set z3 of triples of integers by the rules (a, b, c) ⊙ (u, v, w) = (au, bv, cw), (a, b, c) ⊲ (u, v, w) = (av, bw, cu), and (a, b, c) ⊳ (u, v, w) = (aw, bu, cv). this section contains four theorems which show that the operations ⊙, ⊲ and ⊳ are also the source of squares from components of the eight sequences. theorem 20. the following relations for the sequences a, b, c and d hold: (a ⊙ b)σ1(c ⊙ d)σ1 −76 ∼ 12f4n+6, (a ⊲ b)σ1(c ⊲ d)σ1 61 ∼ 4l4n+5 and (a ⊳ b)σ1(c ⊳ d)σ1 61 ∼ 4l4n+7. proof. since (a ⊲ b)σ1 = 4f4n+5 + 5 and (c ⊲ d)σ1 = 5(4f4n+5 − 5), it follows that the sum of (a ⊲ b)σ1(c ⊲ d)σ1 and 61 is the product 16(5f 2 4n+5 − 4), i. e., the square of 4l4n+5. this proves the second relation. the first and the third could be established similarly. theorem 21. the following relations for the sequences a, b, c and d hold: 1 4 (a ⊙ b)σ∗ 1 (c ⊙ d)σ∗ 1 1 ∼ f4n+6, (a ⊲ b)σ∗ 1 (c ⊲ d)σ∗ 1 69 ∼ 2f4n+2 and (a ⊳ b)σ∗ 1 (c ⊳ d)σ∗ 1 69 ∼ 2f4n+10. proof. since the sums (a ⊲ b)σ∗ 1 and (c ⊲ d)σ∗ 1 are 1 5 (2l4n+2 + 19) and 2l4n+2 − 19, it follows that the sum of (a ⊲ b)σ∗ 1 (c ⊲ d)σ∗ 1 and 69 is the square of 2f4n+2. this is the outline of the proof of the second relation. the similar proofs of the first and the third relation are left to the reader. theorem 22. the following relations for the triples ã, b̃, c̃ and d̃ hold: (ã ⊙ b̃)σ1(c̃ ⊙ d̃)σ1 36 ∼ 31f4n+3 + 7f4n, (ã ⊲ b̃)σ1(c̃ ⊲ d̃)σ1 −3 ∼ 2f4n+8, and (ã ⊳ b̃)σ1(c̃ ⊳ d̃)σ1 29 ∼ 4f4n+7. proof. since the sums (ã ⊳ b̃)σ1 and (c̃ ⊳ d̃)σ1 are 1 5 (2l4n+8 + 1) and 2l4n+8 − 1, it follows that the sum of (ã ⊳ b̃)σ1(c̃ ⊳ d̃)σ1 and −3 is the square of 2f4n+8. this is the outline of the proof of the third relation. the similar proofs of the first and the second relation are left to the reader. 88 zvonko čerin cubo 15, 2 (2013) theorem 23. the following relations hold for the triples ã, b̃, c̃ and d̃: (ã ⊙ b̃)σ∗ 1 (c̃ ⊙ d̃)σ∗ 1 36 ∼ 23f4n+3 + 5f4n, (ã ⊲ b̃)σ∗ 1 (c̃ ⊲ d̃)σ∗ 1 29 ∼ 4f4n+5 and (ã ⊳ b̃)σ∗ 1 (c̃ ⊳ d̃)σ∗ 1 −3 ∼ 2f4n+4. proof. since the sums (ã ⊳ b̃)σ∗ 1 and (c̃ ⊳ d̃)σ∗ 1 are −1 5 (2l4n+4 + 1) and 1 − 2l4n+4, it follows that the difference of (ã ⊳ b̃)σ∗ 1 (c̃ ⊳ d̃)σ∗ 1 and 3 is the square of 2f4n+4. this is the outline of the proof of the third relation. the similar proofs of the first and the second relation are left to the reader. received: march 2010. accepted: september 2012. references [1] e. brown, sets in which xy + k is always a square, mathematics of computation, 45 (1985), 613-620. [2] z. čerin, on pencils of euler triples, i, sarajevo journal of mathematics, 8 (1) (2012), 15–31. [3] z. čerin, on pencils of euler triples, ii, sarajevo journal of mathematics, 8 (2) (2012), 179-192. [4] z. čerin, on diophantine triples from fibonacci and lucas numbers, (preprint). [5] m. radić, a definition of determinant of rectangular matrix, glasnik mat. 1 (21) (1966), 17-22. [6] n. sloane, on-line encyclopedia of integer sequences, http://www.research. att.com/∼njas/sequences/. cubo a mathematical journal vol.16, no¯ 01, (81–93). march 2014 characterizations for certain analytic functions by series expansions with hadamard gaps a. el-sayed ahmed mathematics department, faculty of science, sohag university, 82524 sohag, egypt ahsayed80@hotmail.com a. kamal department of mathematics, faculty of science, port said university, port said, egypt alaa mohamed1@yahoo.com t.i. yassen department of mathematics, faculty of science,, el azhar university at assiut, assiut, egypt taha hmour@yahoo.com abstract in this paper we characterize qk,ω(p,q) functions by lacunary series under mild conditions posed on the weight functions k and ω, where qk,ω(p,q) is a space of analytic functions defined in the unit disk generalizing the well known analytic besov-type space. resumen en este art́ıculo caracterizamos las funciones qk,ω(p,q) por series lacunarias bajo condiciones medianas impuestas en las funciones de peso k y ω, donde qk,ω(p,q) es un espacio de funciones anaĺıticas definidas en el disco unitario generalizando el conocido espacio anaĺıtico del tipo besov. keywords and phrases: qk,ω(p,q)-type spaces, lacunary series. 2010 ams mathematics subject classification: 30b10, 30b50, 46e15. 82 a. el-sayed ahmed, a. kamal & t.i. yassen cubo 16, 1 (2014) 1 introduction let d = {z ∈ c : |z| < 1} be the open unit disk in the complex plane c, let h(d) denote the class of functions analytic in the unit disc d, while da(z) denotes the lebesgue area measure on the plane, normalized so that a(d) = 1. let the green’s function of d be defined as g(z,a) = log 1 |ϕa(z)| , where ϕa(z) = a−z 1−āz is the möbius transformation related to the point a ∈ d. for 0 < r < 1, let d(a,r) = {r ∈ d : |ϕa(z)| < r} be the pseudo-hyperbolic disk with center a ∈ d and radius r. definition 1.1. [17] let k : [0,∞) → [0,∞) be right-continuous and nondecreasing function, 0 < p < ∞,−2 < q < ∞ and for given reasonable function ω : (0,1] → (0,∞), an analytic function f in d is said to belong to the space qk,ω(p,q) if ||f||qk,ω(p,q) = sup a∈d ∫ d |f′(z)|p (1 − |z|)q ωp(1 − |z|) k(g(z,a))da(z) < ∞. in the past few decades both taylor and fourier series expansions for various classes of analytic function spaces where the studies are done by the help of hadamard gap class (see [1, 3, 10, 16] and others). it is well known that a lacunary series belongs to bmoa if and only if it is in the hardy space h2, (see [2]) for example. very recently, in [6, 7, 14] there are some characterizations for some classes of meromorphic functions by the coefficients of certain lacunary series expansions in the unit disk. on the other hand there are some studies of the same problem in clifford analysis (see [4, 5, 12, 13]). we assume throughout the paper that ∫1 0 (1 − r)q k ( log 1 r ) ωp ( log 1 r ) rdr < ∞. an important tool in the study of qk,ω(p,q) space is the auxiliary functions φk and ψω defined by φk(s) = sup 0 0, independent of t, such that ω1(t) ≤ cω2(t), k1(t) ≤ ck2(t) for all t. the notation ω1 & ω2, k1 & k2 is used in a similar fashion. when ω1 . ω2 . ω1, we write ω1 ≈ ω2. also for k1 . k2 . k1, we write k1 ≈ k2. 2 auxiliary lemmas in what follows we say f . g (for two functions f and g ) if there is a constant c such that f ≤ cg. we say f ≈ g (that is, f is comparable with g ) whenever g . f . g. in this section we prove several result about the weight function that are needed for subsequent sections and are of some independent interest. lemma 2.1. [20] if k satisfies condition (1), then the function k∗(t) = t ∫ ∞ t k(s) s2 ds (where, 0 < t < ∞), has the following properties : (a) k∗ is nondecreasing on (0,∞). (b) k∗(t)/t is nondecreasing on (0,∞). (c) k∗(t) ≥ k(t) for all t ∈ (0,∞). (d) k∗ . k on (0,1]. if k(t) = k(1) for t ≥ 1, then we also have (e) k∗(t) = k∗(1) = k(1) for t ≥ 1, so k∗ ≈ k on (0,∞). lemma 2.2. [20] if k satisfies condition (1), then we can find another non-negative weight function given by k∗(t) = t ∫ ∞ t k(s) s2 ds (where, 0 < t < ∞), such that qk,ω(p,q) = qk∗,ω(p,q) and that the new function k ∗ has the following properties : (a) k∗ is nondecreasing on (0,∞). (b) k∗(t)/t is nondecreasing on (0,∞). 84 a. el-sayed ahmed, a. kamal & t.i. yassen cubo 16, 1 (2014) (c) k∗(t) satisfies condition (1) (d) k∗(2t) ≈ k∗(t) on (0,∞). (e) k∗(t) ≈ k(t) on (0,1]. (f) k∗ is differentiable on (0,∞). (g) k∗ is concave on (0,∞). (h) k∗(t) = k∗(1) for t ≥ 1, lemma 2.3. [8] if ω satisfies condition (2), then the function ω∗(t) = t ∫1 t ω(s) s2 ds (where, 0 < t < 1), has the following properties : (a) ω∗ is nondecreasing on (0,1). (b) ω∗(t)/t is nondecreasing on (0,1). (c) ω∗(t) ≥ ω(t) for all t ∈ (0,1). (d) ω∗ . ω on (0,1). if ω(t) = ω(1) for t ≥ 1, then we also have (e) ω∗(t) = ω∗(1) = ω(1) for t ≥ 1, so ω∗ ≈ ω on (0,1). lemma 2.4. let α ≥ 1 and β > 0. if k satisfies (1) and ω satisfies (2), then ∫1 0 rα−1(log 1 r )β−1 k(log 1 r ) ωp(log 1 r ) dr ≈ ( 1 α )β k( 1 α ) ωp( 1 α ) proof. let i = ∫1 0 rα−1(log 1 r )β−1 k(log 1 r ) ωp(log 1 r ) dr. by the change variables we have i = ∫1 0 e−αttβ−1 k(t) ωp(t) dt. cubo 16, 1 (2014) characterizations for certain analytic functions by series . . . 85 we write i = i1 + i2 where i1 = ∫1 1 α e−αt tβ−1 k(t) ωp(t) dt, and i2 = ∫ 1 α 0 e−αt tβ−1 k(t) ωp(t) dt. by lemma 2.1 and lemma 2.2, we have i1 ≤ k( 1 α ) ( 1 α ) × ( 1 α ) ωp( 1 α ) ∫1 1 α e−αttβ−1 dt, = k( 1 α ) ωp( 1 α ) ∫1 1 α e−αttβ−1 dt. making the change of variables s = αt, we have i1 ≤ k( 1 α ) ωp( 1 α ) ∫α 1 e−ssβ−1 ( 1 α )β−1( 1 α ) ds = k( 1 α ) ωp( 1 α ) ( 1 α )β∫α 1 e−ssβ−1 ds. then i1 ≤ c(β) k( 1 α ) ωp( 1 α ) ( 1 α )β . since k(t) and ω(t) are non-decreasing, then by making the change of variables s = αt, we obtain i2 ≤ k( 1 α ) ωp( 1 α ) ∫ 1 α 0 e−αttβ−1 dt, and i2 ≤ k( 1 α ) ωp( 1 α ) ( 1 α )β∫1 0 e−ssβ−1 ds. then i2 ≤ c(β) k( 1 α ) ωp( 1 α ) ( 1 α )β . combining this with what was proved in the previous paragraph, we have ∫1 0 rα−1(log 1 r )β−1 k(log 1 r ) ωp(log 1 r ) dr ≤ c(β) k( 1 α ) ωp( 1 α ) ( 1 α )β , where c(β) is constant which only depends on (β). on the other hand, recall that k(t) and ω(t) are non-decreasing. then, i ≥ ∫1 1 α e−αttβ−1 k(t) ωp(t) dt ≥ k( 1 α ) ωp( 1 α ) ∫1 1 α e−αttβ−1 dt. 86 a. el-sayed ahmed, a. kamal & t.i. yassen cubo 16, 1 (2014) making the change of variables s = αt, we have i ≥ c(β) k( 1 α ) ωp( 1 α ) ( 1 α )β . then ∫1 0 rα−1(log 1 r ) β−1 k(log 1 r ) ωp(log 1 r ) dr ≈ c(β) k( 1 α ) ωp( 1 α ) ( 1 α )β . this completes the proof of the lemma. lemma 2.5. let 0 < γ ≤ 1 and η(r) = ∞∑ n=0 2nγr2 n , 0 ≤ r < 1, then η(r) ≤ 2γ(γ)(log 1 r )−γ. then above result can be found in [15]. lemma 2.6. [19] 1 2π ∫2π 0 g(reiθ,a)dθ =    log 1 |a| , 0 < r ≤ |a|, log 1 r , |a| < r ≤ 1. by jensen’s formula(see [11, 18]), we can directly obtain the above result. theorem 2.7. let k satisfy condition (1) and ω satisfy condition (2). suppose that f(x) = ∞∑ n=1 anx n, with an ≥ 0. for α > 0, p > 0, we have that ∫1 0 (1 − x)α−1(f(x))p k(log 1 x ) ωp(log 1 x ) dx ≈ ∞∑ n=0 2−nα tpn k( 1 2n ) ωp( 1 2n ) (3) where tn = ∑ k∈in ak, n ∈ n, in = {k : 2 n ≤ k < 2n+1; k ∈ n}. proof. let rn = 1 − 2 −n, n = 1,2, ..., then r2 n −1 n ≥ 1 e . by lemma 2.3, we have that ∫1 0 ( ∞∑ n=1 anr n)p (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ≥ ∫1 1 2 ( ∞∑ n=0 tnr 2 n+1 −1)p (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr & ∞∑ n=0 tpn (r 2 n+1 −1 n+1 ) p ∫rn+2 rn+1 (1 − r)α−1 k(log 1 rn+1 ) ωp(log 1 rn+1 ) dr & ∞∑ n=0 2−nα tpn k( 1 2n ) ωp( 1 2n ) . cubo 16, 1 (2014) characterizations for certain analytic functions by series . . . 87 the last inequality holds because of (log 1 r ) ≥ 1 − r. conversely, we first suppose that p > 1. let γ = min{1,α/p} and η(r) = ∞∑ k=0 2kγ r2k, 0 ≤ r < 1. then by jensen’s inequality (see [11, 18]), we have ( ∞∑ n=0 anr n )p ≤ ( ∞∑ n=0 tnr 2 n )p . (η(r))p−1 ∞∑ n=0 2nγ(1−p) r2n|tn| p. from lemma 2.2, lemma 2.3, lemma 2.4 and lemma 2.5, it follows that ∫1 0 ( ∞∑ n=1 anr n )p (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∫1 0 (2γ(γ))p−1(log 1 r )−γ(p−1) ( ∞∑ n=0 2nγ(1−p) r2n |tn| p(1 − r)α−1 ) k(log 1 r ) ωp(log 1 r ) dr ≤ 2nγ(1−p) ∞∑ n=0 |tn| p ∫1 0 r2n−1(log 1 r ) −γ(p−1)+α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ 2nγ(1−p) ∞∑ n=0 |tn| p( 1 2n )−γ(p−1)+α k(log 1 r ) ωp(log 1 r ) ≤ ∞∑ n=0 2−nα tpn k( 1 2n ) ωp( 1 2n ) . (4) denote r(1 − r)α−1 ≤ (log 1 r )α−1, 0 ≤ r < 1. secondly suppose that p = 1, by lemma 2.4, we obtain that ∫1 0 ( ∞∑ n=1 anr n ) (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ∫1 0 ( ∞∑ n=0 tnr 2 n ) (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∞∑ n=0 tn ∫1 0 ( r2 n −1 ) r(1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∞∑ n=0 tn ∫1 0 ( r2 n −1 ) (log 1 r )α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∞∑ n=0 2−nα tn k( 1 2n ) ωp( 1 2n ) . (5) 88 a. el-sayed ahmed, a. kamal & t.i. yassen cubo 16, 1 (2014) finally, if p < 1, we have ∫1 0 ( ∞∑ n=1 anr n )p (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ∫1 0 ( ∞∑ n=0 tnr 2 n )p (1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∞∑ n=0 tpn ∫1 0 ( rp2 n −1 ) r(1 − r)α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∞∑ n=0 tpn ∫1 0 ( rp2 n −1 ) (log 1 r )α−1 k(log 1 r ) ωp(log 1 r ) dr ≤ ∞∑ n=0 2−nα tpn k( 1 2n ) ωp( 1 2n ) (6) from (4), (5) and (6), we obtain the desired result and the proof is therefore established. 3 main results we prove that an analytic function f on the unit disk d with hadamard gaps, that is, f(z) = ∞∑ n=1 anz n satisfying nk+1 nk ≥ λ > 1 for all k ∈ n, belongs to the space qk,ω(p,q) with mild conditions on the weight functions k and ω if and only if ∞∑ k=0 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) < ∞, where 0 0, p > 0, we have ∫ d |f′(z)|p (1 − |z|)q ωp(1 − |z|) k(log 1 |z| )da(z) ≈ ∞∑ n=1 np−q−1 |an| p k( 1 n ) ωp( 1 n ) , (7) where tn = ∑ k∈in ak, n ∈ n, in = {k : 2 n ≤ k < 2n+1; k ∈∈ n}. proof. we write i(f) = ∫ d |f′(z)|p (1 − |z|)q ωp(1 − |z|) k(log 1 |z| )da(z). cubo 16, 1 (2014) characterizations for certain analytic functions by series . . . 89 integrating in polar coordinates we gets i(f) = ∫1 0 ∫2π 0 ( ∞∑ n=1 n |an|r n−1 )p (1 − r)q k(log 1 r ) ωp(log 1 r ) rdrdθ . ∫1 0 ( ∞∑ n=1 n |an|r n−1 )p (1 − r)q k(log 1 r ) ωp(log 1 r ) dr, by theorem 2.7, we obtain i(f) . ∞∑ n=0 2−n(q+1) tpn k( 1 2n ) ωp( 1 2n ) ≈ ∞∑ n=1 np−q−1 |an| p k( 1 n ) ωp( 1 n ) . the proof is therefore established. theorem 3.2. let 0 < p < ∞, −1 < q < ∞. let k satisfy condition (1) and ω satisfy condition (2). suppose that f(z) = ∞∑ k=1 ak z nk, has hadamard gaps, then f belongs to qk,ω(p,q) if and only if ∞∑ k=1 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) < ∞. (8) proof. suppose that f(z) = ∑ ∞ k=1 ak z nk is a lacunary series in qk,ω(p,q). without loss generality, we assume that nk ≥ 1. if f has hadamard gaps, then mp(r,f ′) ≈ m2(r,f ′) (see [21]), where mpp(r,f ′) = 1 2π ∫2π 0 |f′(reiθ)|pdθ. since f ∈ qk,ω(p,q), by theorem 2.7, we obtain ∞ > ∫ d |f′(z)|p (1 − |z|)q ωp(1 − |z|) k(log 1 |z| )da(z) = ∫1 0 mpp(r,f ′) (1 − r)q k(log 1 r ) ωp(log 1 r ) rdr, & ∫1 0 ( ∞∑ k=1 n2k |ak| 2r2(nk−1) ) p 2 (1 − r)q k(log 1 r ) ωp(log 1 r ) rdr, & ∞∑ k=1 2−k(q+1) t p 2 k k( 1 2k ) ωp( 1 2k ) , where tk = ∑ nj∈ik n2j |aj| 2. the taylor series of f has at most [logλ 2] + 1 terms aj z nj when nj ∈ ik for k ≥ 1. by hölder’s inequality, we note that t p 2 k & ∑ nj∈ik n p j |aj| p. then 90 a. el-sayed ahmed, a. kamal & t.i. yassen cubo 16, 1 (2014) ∞∑ k=0 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) . ∞∑ k=0 2−k(q+1) ( ∑ nj∈ik n p j |aj| p ) k( 1 2k ) ωp( 1 2k ) < ∞. next suppose that condition (8) holds, we write z = reiθ in polar form and observe that |f(z)| = ∞∑ k=1 |ak| r nk. since k and ω satisfies conditions (1) and (2) respectively, k is concave. then, by jensen’s inequality, lemma 2.2 and lemma 2.6, we deduce that ∫2π 0 k(g(reiθ,a))dθ ≤ k (∫2π 0 (g(reiθ,a))dθ ) ≤ k ( log 1 r ) . hence, ||f|| p k,ω = sup a∈d ∫ d |f′(z)|p (1 − |z|)q ωp(1 − |z|) k(log 1 |z| )da(z) ≤ sup a∈d ∫1 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q ωp(1 − r) × { 1 2π ∫2π 0 k(g(reiθ,a))dθ } rdr ≤ sup a∈d ∫1 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q ωp(1 − r) k ( log 1 r ) rdr by theorem 2.7,we obtain ||f||qk,ω(p,q) . ∞∑ n=0 2−n(q+1) tpn k( 1 2n ) ωp( 1 2n ) . ∞∑ k=1 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) < ∞. the proof is therefore established. theorem 3.3. let 0 0, then f belongs to qk,ω(p,q) if and only if f ∈ qk,ω,0(p,q). proof. since sufficiency is obvious because of qk,ω,0(p,q) ⊂ qk,ω(p,q). now we will prove necessity of theorem 3.3. suppose that the lacunary series f belongs to qk,ω(p,q). we must show that i(a) → 0 as |a| → 1 −, where i(a) = ∫ d |f′(z)|p (1 − |z|)q ωp(1 − |z|) k(log 1 |z| )da(z). cubo 16, 1 (2014) characterizations for certain analytic functions by series . . . 91 from the proof of theorem 3.2, we know that f ∈ qk,ω(p,q) implies that ∫1 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q ωp(log 1 r ) k ( log 1 r ) rdr < ∞. for any � < 0, there is a δ ∈ (0,1) such that ∫1 δ ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q k ( log 1 r ) ωp(log 1 r ) dr < �. we may as well assume that lim |a|→1− k(log 1 |a| ) = 0. if k satisfies the condition (1). then we choose a such that 1 > |a| > δ. by lemma 2.6, theorem 2.7 and theorem 3.2, we have ∫δ 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q ωp(1 − r) ∫2π 0 k(g(z,a))rdrdθ ≤ ∫δ 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q ωp(1 − r) k ( log 1 |a| ) dr = k ( log 1 |a| ) ∫δ 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q ωp(1 − r) dr ≤ k ( log 1 |a| ) k ( log 1 δ ) ∫δ 0 ( ∞∑ k=1 nk |ak|r (nk−1) )p (1 − r)q k ( log 1 r ) ωp(log 1 r ) dr ≤ k ( log 1 |a| ) k ( log 1 δ ) ∞∑ k=1 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) . since � is arbitrary, we conclude that i(a) → 0 as |a| → 1−. so f ∈ qk,ω,0(p,q) and the proof is complete. carefully checking the proof of theorems 3.2 and 3.3, we also obtain the following sufficient condition for f ∈ qk,ω,0(p,q) and hence in f ∈ qk,ω,0(p,q) in terms of taylor coefficients. theorem 3.4. let 0 < p < ∞ and −1 < q < ∞. let k satisfy condition (1) and ω satisfy condition (2). suppose that f(z) = ∞∑ k=0 ak z nk, has hadamard gaps, then f belongs to qk,ω,0(p,q) if and only if ∞∑ k=0 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) < ∞. (9) 4 conclusion from theorems 3.2, 3.3 and3.4, we can give the following theorem; 92 a. el-sayed ahmed, a. kamal & t.i. yassen cubo 16, 1 (2014) theorem 4.1. let 0 < p < ∞, −1 < q < ∞. let k satisfy condition (1) and ω satisfy condition (2). suppose that f(z) = ∞∑ k=0 ak z nk ∈ h(d) has hadamard gaps, then the following statements are equivalent: (i) f ∈ qk,ω(p,q); (ii) f ∈ qk,ω,0(p,q); (iii) ∑ ∞ k=0 n p−q−1 k |ak| p k( 1 nk ) ωp( 1 nk ) < ∞. received: april 2013. accepted: october 2013. references [1] k. l. avetisyan, sharp inclusions and lacunary series in mixed-norm spaces on the polydisc, complex variables and elliptic equations, 58(2)(2013), 185195. [2] a. baernstein ii, analytic functions of bounded mean oscillation, in aspects of contemporary complex analysis, academic press, london, (1980), 3-36. [3] j. s. choa, a property of series of holomorphic homogeneous polynomials with hadamard gaps, bull. aust. math. soc. 53(3)(1996), 479-484. [4] a. el-sayed ahmed, lacunary series in quaternion bp,q spaces, complex variables and elliptic equations, vol 54(7)(2009), 705-723. [5] a.el-sayed ahmed, lacunary series in weighted hyperholomorphic bp,q(g) spaces, numerical functional analysis and optimization, vol 32(1)(2011), 41-58. [6] a. el-sayed ahmed, meromorphic functions with hadamard gaps,to appear. [7] a. el-sayed ahmed and a. kamal, characterizations by gap series in meromorphic qp functions, to appear. [8] a. el-sayed ahmed and a. kamal, series expansions in some analytic function spaces, submitted. [9] m. essén, h. wulan, and j. xiao, several function theoretic characterizations of möbius invariant qk spaces, j. funct. anal. 230(2006), 78-115. [10] o. furdui, on a class of lacunary series in bmoa, j. math. anal. appl. 342(2)(2008), 773779. cubo 16, 1 (2014) characterizations for certain analytic functions by series . . . 93 [11] j. garnett, bounded analytic functions, academic press, new york, (1981). [12] k. gürlebeck and h. r. malonek, on strict inclusions of weighted dirichlet spaces of monogenic functions, bull. aust. math. soc. 64(2001), 33-50. [13] k. gürlebeck and a. el-sayed ahmed,on series expansions of hyperholomorphic bq functions, in tao qian et al. (eds), trends in mathematics advances in analysis and geometry, (basel/switzerland : birkhäuser verlarg publisher) (2004), 113-129. [14] a. kamal and a. el-sayed ahmed, a property of meromorphic functions with hadamard gaps, scientific research and essays, vol 8(15)(2013), 633-639. [15] m. mateljević and m. pavlović, lp-behavior of power series with positive coefficients and hardy spaces, proc. amer. math. soc, 87 (1983), 309 316. [16] j. miao, a property of analytic functions with hadamard gaps, bull. austral. math. soc. 45(1992), 105-112. [17] r. a. rashwan, a. el-sayed ahmed and alaa kamal, integral characterizations of weighted bloch spaces and qk,ω(p,q) spaces, mathematica 51(74)(1)(2009), 63-76. [18] w. rudin, real and complex analysis, megraw-hill, new york, (1987). [19] h. wulan and k. zhu, lacunary series in qk spaces, studia math, 178 (2007), 217-230. [20] h. wulan and k. zhu,lacunary series in qk spaces, journal of function spaces and applications, vol 6(3)(2008), 293-301. [21] a. zygmund, trigonometric series. volumes i and ii combined. 1. paperback ed. of the 2. ed. cambridge mathematical library. cambridge university press. xiv, (1988). trigonometric series, cambridge (1968). cubo a mathematical journal vol.15, no¯ 02, (53–63). june 2013 about cumulative idle time model of the message switching system s. minkevičius vu institute of mathematics and informatics, akademijos 4, 08663 vilnius, lithuania. vilnius university, faculty of mathematics and informatics, naugarduko 24, 03225 vilnius, lithuania. minkevicius.saulius@gmail.com abstract the purpose of this research in the queueing theory is the theorem about the law of the iterated logarithm in multiphase queueing systems and its application to the mathematical model of the message switching system. first the law of the iterated logarithm is proved for the cumulative idle time of a customer. finally we present an application of the proved theorem for the model of the message switching system. resumen el propósito de esta investigación en la teoŕıa de colas es el teorema sobre la ley de logaritmo iterado en sistemas multifase y su aplicación al modelo matemático del sistema de interruptores de mensajes. primero, la ley de logaritmo iterado se prueba para el tiempo ocioso acumulado de un cliente. finalmente presentamos una aplicación del teorema probado para el modelo de sistema de interruptores de mensajes. keywords and phrases: mathematical models of technical systems, queueing theory, multiphase queueing systems, a law of the iterated logarithm, cumulative idle time of a customer. 2010 ams mathematics subject classification: 60k25, 60g70, 60f17. 54 s. minkevičius cubo 15, 2 (2013) 1 introduction at first, the law of the iterated logarithm is considered by investigating the cumulative idle time of a customer in multiphase queueing systems. interest in the field of multiphase queueing systems has been stimulated by the theoretical values of the results as well as by their possible applications in information and computing systems, communication networks, and automated technological processes (see, for example, [20]). the methods of investigation of single-phase queueing systems are considered in [2], [3], etc. the asymptotic analysis of models of queueing systems in heavy traffic is of special interest (see, for example, [9], [10], [4], [5], etc.). the papers [11], [18] and others desribed the beginning of the investigation of diffusion approximation to queueing networks. intermediate models multiphase queueing systems are considered rarer due to serious technical difficulties (see, for example, book [7]). the works on cumulative idle time for the multiphase queueing systems and open jackson networks in heavy traffic are also sparse. in one of the first papers of this kind, [16] used numerical methods to study values of the mean of the cumulative idle time in single-server queues. [22] obtained limit theorems for the cumulative idle time in the systems gi/g/1 andm/g/1. [12] presented expressions for the cumulative idle time of a server in the gi/g/1 system. [19] found the laplace transform of the distribution of the cumulative idle time in a finite time interval for the gi/g/1 system. [8] conceived the laplace transform of the expected cumulative idle time in an m/g/1 queue. [17] considered the moderate-deviation behaviour of the cumulative idle time with single-server queues. these results complement the existing results on the heavy traffic behaviour of this process. [23] established functional central limit theorems for a cumulative idle time process in a fluid queue. these limit processes have discontinuous sample paths (e.g., to be a non-brownian stable process, or a more general levy process). let the cumulative idle time of a customer in the phases of a queueing system be unrestricted, the principle of service being “first come, first served”. all the random variables studied are defined on one basic probability space (ω, f, p). we present some definitions in the theory of metric spaces (see, for example, [1]). let c be a metric space consisting of real continuous functions in [0, 1] with a uniform metric ρ(x, y) = sup 0≤t≤1 |x(t) − y(t)|, x, y ∈ c . let d be a space of all real-valued right-continuous functions in [0,1] having left limits and endowed with the skorokhod topology induced by the metric d (under which d is complete and separable). also, note that d(x, y) ≤ ρ(x, y) for x, y ∈ d. in this paper, we will constantly use an analog of the theorem on converging together (see, for example, [6]): cubo 15, 2 (2013) about cumulative idle time model of the message switching system 55 theorem 1.1. let ε > 0 and xn, yn, x ∈ d. if p ( lim n→∞ d(x n , x) > ε ) = 0 and p ( lim n→∞ d(xn, yn) > ε ) = 0, then p ( lim n→∞ d(yn, x) > ε ) = 0. (1) 2 statement of the problem we investigate here a k-phase queue (i.e., after a customer has been served in the j-th phase of the queue, he is routed to the j + 1-th phase of the queue, and, after the service in the k-th phase of the queue, he leaves the queue). let us denote by tn the time of arrival of the n-th customer; by s (j) n – the service time of the n-th customer in the j-th phase; zn = tn+1 − tn; and by τj,n+j departure of the n-th customer from the j-th phase of the queue, j = 1, 2, · · · , k. let interarrival times (zn) at the multiphase queueing system and service times (s (j) n ) in each phase of the queue for j = 1, 2, · · · , k be mutually independent identically distributed random variables. next, denote by bij,n the idle time of the n-th customer in the j-th phase of the multiphase queue; f̂j,n = n∑ l=1 bij,l stands for a cummulative idle time of the n-th customer in the j-th phase of the multiphase queue, j = 1, 2, . . . , k. when j = 1, 2, . . . , k, let δj,n = { s (j) n−(j−1) − zn, if n ≥ k 0, if n < k. let us denote sj,n = ∑n−1 l=1 δj,l, s0,n ≡ 0, ŝj,n = sj−1,n − sj,n, xj,n = τj,n − tn, x0,n ≡ 0, x̂j,n+1 = xj,n − δj,n+1, x̂0,n ≡ 0, zj,n = x̂j,n − sj,n, αj = mδj,n, α0 ≡ 0, dzn = σ20, ds (j) n = σ2j , σ̃ 2 j = σ 2 0 + σ 2 j , s (0) n = zn, j = 1, 2, . . . , k, [x] as the integer part of number x. we assume that the following conditions are fulfilled: there exists a constant γ > 0 such that sup n≥1 m|s(j)n | 4+γ < ∞, j = 0, 1, 2, . . . , k (2) and αk < αk−1 < · · · < α1 < 0. (3) in this paper, we mostly use the equations presented in [13]: x̂j,n = max 0≤l≤n (x̂j−1,l − sj,l) + sj,n, x̂0,n ≡ 0, n ≥ k, j = 1, 2, . . . , k. (4) 56 s. minkevičius cubo 15, 2 (2013) 3 on the law of the iterated logarithm for the cumulative idle time of a customer first we investigate the law of the iterated logarithm for the cumulative idle time in multiphase queues. we prove such a theorem. theorem 3.1. if conditions (2) and (3) are fulfilled, then p ( lim n→∞ f̂j,n − (−αj) · n σ̃j · a(n) = 1 ) = p ( lim n→∞ f̂j,n − (−αj) · n σ̃j · a(n) = −1 ) = 1, j = 1, 2, . . . , k and a(n) = √ 2n ln ln n. proof. denote random functions d as follows f̂nj (t) = f̂j,[nt] − (−αj) · [nt] a(n) , ẑnj (t) = ẑj,[nt] − (−αj) · [nt] a(n) , ŝnj (t) = (−sj,[nt]) − (−αj) · [nt]√ n , j = 1, 2, . . . , k and 0 ≤ t ≤ 1. using a triangle inequality we see that, for each fixed ε > 0, p ( lim n→∞ d(f̂nj , ŝ n j ) > ε ) ≤ p ( lim n→∞ d(f̂nj , ẑ n j ) > ε 2 ) + p ( lim n→∞ d(ẑnj , ŝ n j ) > ε 2 ) ≤ p ( lim n→∞ ρ(f̂nj , ẑ n j ) > ε 2 ) + p ( lim n→∞ ρ(ẑnj , ŝ n j ) > ε 2 ) = p   lim n→∞ sup 0≤t≤1 |fj,[nt] − ẑj,[nt]| a(n) > ε 2   + p   lim n→∞ sup 0≤t≤1 |ẑj,[nt] − (−sj,[nt])| a(n) > ε 2   = p   lim n→∞ max 0≤l≤n |fj,l − ẑj,l| a(n) > ε 2   + p   lim n→∞ max 0≤l≤n |ẑj,l − (−sj,l)| a(n) > ε 2   , j = 1, 2, . . . , k. thus, we have for each fixed ε > 0, p ( lim n→∞ d(f̂nj , ŝ n j ) > ε ) ≤ p   lim n→∞ max 0≤l≤n |f̂j,l − ẑj,l| a(n) > ε 2  + p   lim n→∞ max 0≤l≤n |ẑj,l − (−sj,l)| a(n) > ε 2   , j = 1, 2, . . . , k. (5) cubo 15, 2 (2013) about cumulative idle time model of the message switching system 57 it is proved (see [14]) that, if conditions (3) are fulfilled, then, for each fixed ε > 0, p   lim n→∞ max 0≤l≤n |f̂j,l − ẑj,l| √ n > ε   = 0, j = 1, 2, . . . , k. using similar method as in [14] can be proven that, for each fixed ε > 0, p   lim n→∞ max 0≤l≤n |f̂j,l − ẑj,l| a(n) > ε   = 0, j = 1, 2, . . . , k. (6) so the first term in (5) converges to zero. we will prove that second term in (5) also converges to zero. using (4) we see that ẑj,n = max 0≤l≤n (x̂j−1,l − sj−1,l + sj−1,l − sj,l) = max 0≤l≤n (ẑj−1,l + sj,l), j = 1, 2, . . . , k. thus, ẑj,n = max 0≤l≤n (ẑj−1,l + sj,l), j = 1, 2, . . . , k, z0,· ≡ 0. (7) also we see that ẑj,n − j∑ i=1 ŝi,n ≥ ẑj−1,n + ŝj,n − j∑ i=1 ŝi,n = ẑj−1,n − j−1∑ i=1 ŝi,n ≥ · · · ≥ ẑ1,n − ŝ1,n = max 0≤l≤n (ŝ1,n) − ŝ1,n ≥ 0. so, ẑj,n − j∑ i=1 ŝi,n ≥ 0, j = 1, 2, . . . , k. (8) but ẑj,n ≤ max 0≤l≤n (ẑj−1,l) + max 0≤l≤n ŝj,l = ẑj−1,n + max 0≤l≤n ŝj,l ≤ · · · ≤ j∑ i=1 { max 0≤l≤n ŝi,l}. from it follows that ẑj,n ≤ j∑ i=1 { max 0≤l≤n ŝi,l}, j = 1, 2, . . . , k. (9) using (8) and (9) we get that 0 ≤ ẑj,n − j∑ i=1 ŝi,n ≤ j∑ i=1 { max 0≤l≤n ŝi,l − ŝi,n}, j = 1, 2, . . . , k. (10) 58 s. minkevičius cubo 15, 2 (2013) applying (9) we achieve for each fixed ε > 0, p      max 0≤l≤n |ẑj,l − j∑ i=1 ŝj,l| a(n) > ε      = p      max 0≤l≤n (ẑj,l − j∑ i=1 ŝj,l) a(n) > ε      ≤ p      j∑ i=1 max 0≤l≤n { max 0≤m≤l ŝi,m − ŝi,l} a(n) > ε      ≤ p      k∑ i=1 max 0≤l≤n { max 0≤m≤l ŝi,m − ŝi,l} a(n) > ε      ≤ k∑ i=1 p   max 0≤l≤n { max 0≤m≤l ŝi,m − ŝi,l} a(n) > ε k   = k∑ i=1 p   max 0≤l≤n { max 0≤m≤l (−ŝi,l−m)} a(n) > ε k   = k∑ i=1 p   max 0≤l≤n { max 0≤m≤l (−ŝi,m)} a(n) > ε k   ≤ k∑ i=1 p   max 0≤l≤n (−ŝi,l) a(n) > ε k   , j = 1, 2, . . . , k. (11) thus, we have that for each fixed ε > 0, p      max 0≤l≤n |ẑj,l − j∑ i=1 ŝj,l| a(n) > ε      ≤ k∑ i=1 p   max 0≤l≤n (−ŝi,l) a(n) > ε k   , j = 1, 2, . . . , k. (12) note (see, for example, [14]) that for each fixed ε > 0, p   lim n→∞ max 0≤l≤n (−ŝi,l) a(n) > ε   = 0, j = 1, 2, . . . , k, (13) if conditions (3) are fulfilled. using relation k∑ i=1 ŝi,n = −sj,n, j = 1, 2, . . . , k and (12) (13) we obtain that for each fixed ε > 0, p   lim n→∞ max 0≤l≤n |ẑj,l − (−sj,l)| a(n) > ε   = 0 j = 1, 2, . . . , k. (14) using the theorem on the law of the iterated logarithm for random functions ŝnj (t), j = 1, 2, . . . , k (see, for example, [21]) we achieve that p ( lim n→∞ (−sj,n) − (−αj) · n σ̃j · a(n) = 1 ) = 1 and p ( lim n→∞ (−sj,n) − (−αj) · n σ̃j · a(n) = −1 ) = 1, j = 1, 2, . . . , k. (15) cubo 15, 2 (2013) about cumulative idle time model of the message switching system 59 thus, applying (1), (5), (6), (14) and (15) we obtain that p ( lim n→∞ f̂j,n − (−αj) · n σ̃j · a(n) = 1 ) = 1 and p ( lim n→∞ f̂j,n − (−αj) · n σ̃j · a(n) = −1 ) = 1, j = 1, 2, . . . , k. (16) the proof of theorem 3.1 is complete. 4 on the model of switching facility in this part of the paper, we will present an application of the proved theorem a mathematical model of message switching system. as noted in the introduction, multiphase queueing systems are of special interest both in theory and in practical applications. such systems consist of several service nodes, and each arriving customer is served at each of the consecutively located node (frequently called phases). a typical example is provided by queueing systems with identical service. such systems are very important in applications, especially to message switching systems. in fact, in many comunication systems the transmission times of the customers do not vary in the delivery process. so, we investigate a message switching system which consists of k phases and in which s j n = sn, j = 1, 2, . . . , k (the service process is identical in phases of the system). let δn = { sn−k − zn, if n ≥ k 0, if n < k. also, let us note α = mδn, dzn = σ 2 0, dsn = σ 2, σ̃2 = σ20 + σ 2, f̂j,n = n∑ l=1 bij,l, j = 1, 2, . . . , k. we assume that the following conditions are fulfilled: there exists a constant γ > 0 such that sup n≥1 m|sn| 4+γ < ∞ (17) and α < 0. (18) similarly as in the proof of theorem 3.1, we present the following theorem on the law of the iterated logarithm for the cumulative idle time of a data packet in message switching systems. 60 s. minkevičius cubo 15, 2 (2013) theorem 4.1. if conditions (17) and (18) are fulfilled, then p ( lim n→∞ f̂j,n − (−α) · n σ̃ · a(n) = 1 ) = p ( lim n→∞ f̂j,n − (−α) · n σ̃ · a(n) = −1 ) = 1, j = 1, 2, . . . , k. we see that the cumulative idle time of data packet is the same in the all phases of system. 5 computing example we see that theorem 4.1 implies that for fixed ε > 0 there exists n(ε) such that for every n ≥ n(ε), with probability one (1 − ε) · σ̃ · a(n) − α · n ≤ f̂j,n ≤ (1 + ε) · σ̃ · a(n) − α · n, (19) where a(n) = √ 2n ln ln n, α = m(sn − zn) < 0, σ̃ 2 = dzn + dsn, j = 1, 2, . . . , k. from this we can obtain (1 − ε) · σ̃ · a(n) − α · n ≤ f̂j,n ≤ (1 + ε) · σ̃ · a(n) − α · n, |m(f̂j,n − (−α) · n) − {(1 − ε) · σ̃ · a(n)}| ≤ 2 · ε · σ̃ · a(n), |m ( f̂j,n − (−α) · n) σ̃ · a(n) ) − (1 + ε)| ≤ 2 · ε, j = 1, 2, . . . , k. (20) so from (20) we can get mf̂j,n ∼ (−α) · n + (1 + ε) · σ̃ · a(n), j = 1, 2, . . . , k. (21) mf̂j,n is average cumulative idle time of n-th message (time, which system is waiting for processing message until n-th message arrival to the system). we see from (21) that mf̂j,n consists of linear function and nonlinear slowly increasing function (1 + ε) · σ̃ · a(n), j = 1, 2, . . . , k. now we present a technical example from the computer network practice. assume that messages arrive at the computer v1 at the rate λ of 20 per hour during business hours. these messages are served at a rate µ of 25 per hour in the computer v1. after service in the computer v1 messages arrive at the second computer v2. also we note that messages are served at a rate µ of 25 per hour in the computer v2. so, messages is served in computers v1, v2,. . . ,vk, and after messages are served in computer vk, they leave computer network. so, mzn = 1/λ = 1/20 = 0.05, msn = 1/µ = 1/25 = 0.04, α = 0.04 − 0.05 = −0.01 < 0, dzn = 1/λ = 1/20 = 0.05, dsn = 1/µ = 1/25 = 0.04, σ̃ 2 = 41/104, σ̃ ∼ 0.064, ε = 0.001, n ≥ 100. cubo 15, 2 (2013) about cumulative idle time model of the message switching system 61 thus, mf̂j,n ∼ (−α) · n + (1 + ε) · σ̃ · a(n) = (0.01) · n + (0.064) · a(n), j = 1, 2, . . . , k. (22) from (22) we get mf̂j,n n = (0.01) + (0.064) · √ 2 ln ln n n , j = 1, 2, . . . , k. (23) now we present figure for mf̂j,n n , j = 1, 2, . . . , k, when 100 ≤ n ≤ 1000, ε = 0.001 (see (23) and table 1). time n mf̂j,n n , j = 1, 2, . . . , k 100 0.02118510415 200 0.01826415546 300 0.01689524794 400 0.01605525217 500 0.01547101209 600 0.01503369681 700 0.01469010032 800 0.01441067288 900 0.01417749453 1000 0.01397897294 table 1 summary of computing results. we see that when α = −0.01 < 0, computer network is busy 99 % of this time. corollary 5.1. average idle time of message system direcly depends of traffic coefficient α and time n and is the same in all phases of message system. received: september 2010. accepted: september 2012. references [1] billinsley p. (1968). convergence of probability measures. wiley, new york. [2] borovkov a. (1972). stochastic processes in queueing theory. nauka, moscow (in russian). [3] borovkov a. (1980). asymptotic methods in theory of queues. nauka, moscow (in russian). 62 s. minkevičius cubo 15, 2 (2013) [4] iglehart d.l., whitt w. (1970a). multiple channel queues in heavy traffic. i. advances in applied probability, 2, 150-177. [5] iglehart d.l., whitt w. (1970b). multiple channel queues in heavy traffic. ii. sequences, networks and batches. advances in applied probability, 2, 355-369. [6] iglehart d.l. (1971). multiple channel queues in heavy traffic. iv. law of the iterated logarithm. zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete, 17, 168-180. [7] karpelevich f.i., kreinin a.i. (1994). heavy traffic limits for multiphase queues. american mathematical society, providence. [8] kella o. (1992). concavity and reflected levy process. journal of applied probability, 29(1), 209-215. [9] kingman j. (1961). on queues in heavy traffic. j. r. statist. soc., 24, 383-392. [10] kingman j. (1962). the single server queue in heavy traffic. proc. camb. phil. soc., 57, 902-904. [11] kobyashi h. (1974). application of the diffusion approximation to queueing networks. journal of acm, 21, 316-328. [12] milch p., waggoner m. (1970). a random walk approach to a shutdown queueing system. siam j. appl. math., 19, 103-115. [13] minkevičius s. (1986). weak convergence in multiphase queues. lietuvos matematikos rinkinys, 26, 717-722 (in russian). [14] minkevičius s. (2005). on the full idle time in multiphase queues. lietuvos matematikos rinkinys (in russian, in appear). [15] minkevičius s, kulvietis g. (2004). a mathematical model of the message switching system. proceedings of the seventh international conference ”computer data analysis and modeling”, minsk, september 6-10, 2004, 2, 96-100. [16] pike m. (1963). some numerical results for the queueing system d/ek/1. j. r. statist. soc. ser. b, 25, 477-488. [17] puhalskii a. (1999). moderate deviations for queues in critical loading. queueing systems theory appl., 31(3-4), 359-392. [18] reiman m.i. (1984). open queueing networks in heavy traffic. mathematics of operations research, 9, 441-459. [19] ridel m. (1976). conditions for stationarity in a single server queueing system. zastos. mat., 15(1), 17-24. cubo 15, 2 (2013) about cumulative idle time model of the message switching system 63 [20] saati t., kerns k. (1971). analytic planning. organization of systems. mir, moscow (in russian). [21] strassen v. (1964). an invariance principle for the law of the iterated logarithm. zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete, 3, 211-226. [22] takacs l. (1974). occupation time problems in the theory of queues. in: lecture notes in economics and mathematical systems, 98, springer-verlag, berlin, heidelberg, new york, 91–131. [23] whitt w. (2000). limits for cumulative input processes to queues. probab. engrg. inform. sci., 14(2), 123-150. () cubo a mathematical journal vol.17, no¯ 01, (11–27). march 2015 periodic bvp for a class of nonlinear differential equation with a deviated argument and integrable impulses alka chadha and dwijendra n pandey department of mathematics, indian institute of technology roorkee, roorkee-247667 alkachaddha03@gmail.com, dwij.iitk@gmail.com abstract this paper deals with periodic bvp for integer/fractional order differential equations with a deviated argument and integrable impulses in arbitrary banach space x for which the impulses are not instantaneous. by utilizing fixed point theorems, we firstly establish the existence and uniqueness of the mild solution for the integer order differential system and secondly obtain the existence results for the mild solution to the fractional order differential system. also at the end, we present some examples to show the effectiveness of the discussed abstract theory. resumen este art́ıculo estudia las ecuaciones diferenciales de orden entero/fraccional con condiciones de frontera periódicas con un argumento desviado e impulsos integrables en espacios de banach arbitrarios x donde los pulsos no son instantáneos. utilizando teoremas de punto fijo, establecemos la existencia y unicidad de soluciones temperadas para los sistemas diferenciales de orden entero, y luego obtenemos resultados de existencia para soluciones temperadas del sistema diferencial de orden fraccional. además, presentamos un ejemplo para mostrar la efectividad de la teoŕıa abstracta discutida. keywords and phrases: deviating arguments, fixed point theorem, impulsive differential equation, periodic bvp, fractional calculus. 2010 ams mathematics subject classification: 34g20, 34k37, 34k45, 35r12, 45j05. 12 alka chadha & dwijendra n pandey cubo 17, 1 (2015) 1 introduction: in a few decades, impulsive differential equations have received much attention of researchers mainly due to its demonstrated applications in widespread fields of science and engineering. impulsive differential equations have played an important role in real world problems for describing a process which is characterized by the development of a sudden change in system’s state. such processes are investigated in various fields such as biology, physics, control theory, population dynamics, medicine and many others. impulsive differential equations are an appropriate model to hereditary phenomena for which a delay argument arises in modelling equations. for more details on impulsive differential equation, we refer to the monographs [1],[2] and papers [3]-[12] and references given therein. a differential equation with boundary conditions arise in many areas of applied sciences, for example, chemical engineering, blood flow problems, thermoelasticity, population models, underground water flow and many others. for more details on differential equation with integral boundary conditions, we refer to [10, 17, 18, 19, 23, 27] and references given therein. on the other hand, fractional calculus have many applications in various areas of sciences and engineering for example, fluid dynamics, like fractal theory, diffusion in porous media and fractional biological neurons, traffic flow, polymer rheology. the fractional differential equation is an important tool to describe nonlinear oscillation of earthquake. for more study on fractional calclus, we refer to books [13]-[16]. in this work, we consider the periodic boundary value problems for integer order nonlinear differential equations in a banach space x of the form with non-instantaneous integrable impulses u′(t) = f(t, u(t), u([h(u(t), t)])), t ∈ (sm, tm+1], m = 0, 1, 2, · · · , δ, (1.1) u(t) = ∫t tm gm(s, u(s))ds, t ∈ (tm, sm], m = 1, 2, · · · , δ, δ ∈ n (1.2) u(0) = u(t). (1.3) next, we consider the periodic boundary value problems for nonlinear fractional differential equations in a banach space x of the form with non-instantaneous integrable impulses cd q 0,tu(t) = i 2−q t f(t, u(t), u([h(u(t), t)])), t ∈ (sm, tm+1], 0 < q < 1, (1.4) u(t) = ∫t tm gm(s, u(s))ds, t ∈ (tm, sm], m = 0, 1, 2, · · · , δ, δ ∈ n, (1.5) u(0) = u(t), (1.6) where 0 < t < ∞, cd q 0,t represents the caputo fractional derivative of the order q with lower limit 0, 0 = t0 = s0 < t1 ≤ s1 ≤ t2 < · · · < tδ ≤ sδ ≤ tδ+1 = t are fixed numbers, gm : (tm, sm] × x → x, m = 1, · · · , δ. the nonlinear x-valued functions f and h are appropriate functions and satisfy some suitable conditions to be stated later. in this system of equations cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 13 (1.1)-(1.3) and (1.4)-(1.6), the impulses begin all of a sudden at the points ti and continue their proceeding on a finite interval [ti, si]. according to the authors in [4]-[5], there are many different inspirations for consideration of the problem (1.1)-(1.3) and (1.4)-(1.6). the hemodynamical equilibrium of a person is an example of such systems. one can prescribe some intravenous drugs (insulin) in the case of a decompensation (for example, low or high level of glucose). since the introduction of the drugs in the bloodstream and the consequent absorption of the body are successive and continuous processes, we can describe this situation as an impulsive action which start suddenly and stays active on a finite time interval. the organization of the paper is as follows: in section 2, we give some basic definitions, assumptions, lemmas and theorems as preliminaries which will be used for proving our main results. in section 3, we prove the existence of a mild solution to the problem (1.1)-(1.3) and problem (1.4)-(1.6). some examples are also presented at the end of the paper. 2 preliminaries and assumptions in this section, we discuss some basic definitions, preliminaries, theorem and lemmas which will be used for proving the required result. let (x, ‖ · ‖) be a banach space. let c(j; x), where j = [0, t] denotes the space of all continuous x-valued functions on interval j which is a banach space with the norm ‖ u‖c = supt∈j ‖ u(t)‖. the space of all bochner integrable functions u : (0, t) → x represented by l1((0, t); x), is a banach space with norm ‖ u‖1 = ∫t 0 ‖ u(t)‖dt. the br(x, x) denotes the closed ball with center at x and radius r in x. to study the impulsive differential equation, we introduce the following space pc([0, t]; x) = {u : [0, t] → x; u ∈ c((tj, tj+1]; x), j = 0, 1, · · · , m, and ∃ u(t+j ) and u(t − j ), j = 1, · · · , m exist with u(t − j ) = u(tj)}. it is clear that pc([0, t]; x) is a banach space with the norm ‖u‖pc = max t∈[0,t] ‖ u(t)‖. for a function u ∈ pc([0, t]; x) and j ∈ {0, 1, · · · , m}, we define the function ũj ∈ c([tj, tj+1]; x) such that ũj(t) = { u(t), for t ∈ (tj, tj+1], u(t+j ), for t = tj. (2.1) for b ⊂ pc([0, t]; x) and j ∈ {0, 1, · · · , m}, we have b̃j = {ũj : u ∈ b} and we have following accoli-arzelà type criteria. 14 alka chadha & dwijendra n pandey cubo 17, 1 (2015) lemma 2.1. [4]. a set b ⊂ pc([0, t]; x) is relatively compact in pc([0, t]; x) if and only if each set b̃j(j = 1, 2, · · · , m) is relatively compact in c([tj, tj+1], x)(j = 0, 1, · · · , m). now, we recall some basic definition. definition 2.1. the riemann-liouville fractional integral of f with order q defined by i q 0,tf(t) = 1 γ(q) ∫t 0 (t − s)q−1f(s)ds. (2.2) definition 2.2. the fractional derivative of function f : [0, ∞) → r in the riemann-liouville sense with order q is defined by d q 0,tf(t) = dn dtn 1 γ(n − q) ∫t 0 (t − s)n−q−1f(s)ds, t > 0, n − 1 < q < n. (2.3) definition 2.3. the fractional derivative of function f : [0, ∞) → r in the caputo sense of order q is defined by cd q 0,tf(t) = 1 γ(n − q) ∫t 0 (t − s)n−q−1fn(s)ds, (2.4) for n − 1 < q < n, n ∈ n, t > 0, with the following property: cd q 0,t(i q 0,tf(t)) = f(t) − n−1∑ k=1 tk k! fk(0). (2.5) before expressing and demonstrating the required main result, we present the following definition of mild solution to the system (1.1)-(1.3) and (1.4)-(1.6). lemma 2.2. for given continuous function f : [0, t] → x and gm ∈ c([tm, sm], x), a function u ∈ pc([0, t]; x) is a mild solution for the impulsive periodic boundary value problem u′(t) = f(t), t ∈ (sm, tm+1], m = 0, 1, 2, · · · , δ, δ ∈ n (2.6) u(t) = ∫t tm gm(s)ds, t ∈ (tm, sm], m = 1, 2, · · · , δ, (2.7) u(0) = u(t), (2.8) if and only if u(·) satisfies the following u(t) =    ∫sδ tδ gδ(s)ds + ∫t sδ f(s)ds + ∫t 0 f(s)ds, t ∈ [0, t1], ∫sm tm gm(s)ds + ∫t sm f(s)ds, t ∈ (sm, tm+1], ∫t tm gm(s)ds, t ∈ (tm, sm], (2.9) for each m = 1, · · · , δ. cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 15 lemma 2.3. for the continuous function f : [0, t] → x and gm ∈ c([tm, sm], x), a function u ∈ pc([0, t]; x) is said to be a mild solution for the system cd q 0,tu(t) = i 2−q t f(t), t ∈ (sm, tm+1], 0 < q < 1, (2.10) u(t) = ∫t tm gm(s)ds, t ∈ (tm, sm], m = 0, 1, 2, · · · , δ, δ ∈ n, (2.11) u(0) = u(t), (2.12) if and only if u(0) = u(t), u(t) = ∫t tm gm(t), ∀ t ∈ (tm, sm], m = 1, · · · , δ and u(·) satisfies the following integral equations u(t) =    ∫sδ tδ gδ(s)ds − ∫sδ 0 (sδ − s)f(s)ds + ∫t 0 (t − s)f(s)ds + ∫t 0 (t − s)f(s)ds, t ∈ [0, t1], ∫sm tm gm(s)ds − ∫sm 0 (sm − s)f(s)ds + ∫t 0 (t − s)f(s)ds, t ∈ (sm, tm+1], (2.13) for each m = 1, · · · , δ. further, we list the following assumption which will be used to establish the main result. assumptions on f, h and gm, (m = 1, · · · , δ) : (a1) the function f : [0, t] × x × x → x is continuous and there exist a positive constant lf and 0 < γ1 ≤ 1 such that ‖f(t1, u1, v1) − f(t2, u2, v2)‖ ≤ lf[|t1 − t2| γ1 + ‖u1 − u2‖x + ‖v1 − v2‖x], (2.14) for all (tj, uj, vj) ∈ [0, t] × x × x, j = 1, 2. (a2) h : x × [0, t] → [0, t] is continuous function and there exist positive constants lh and 0 < γ2 ≤ 1 such that |h(u1, t1) − h(u2, t2)| ≤ lh[‖u1 − u2‖x + |t1 − t2| γ2], (2.15) for each (uj, tj) ∈ x × [0, t], for j = 1, 2. (a3) gm : [0, t] × x → x, m = 1, 2, · · · , δ, are continuous functions and there exist constants lgm > 0 such that ‖ gm(t, x) − gm(t, y)‖ ≤ lgm‖ x − y‖, (2.16) ‖gm(t, u(t))‖ ≤ km, (2.17) for all (t, x), (s, y) ∈ [0, t] × x, u ∈ x and km > 0, m = 1, · · · , δ are constants. 16 alka chadha & dwijendra n pandey cubo 17, 1 (2015) 3 existence result in this section, we establish the existence of a mild solutions for the systems (1.1)-(1.3) and (1.4)(1.6) by using fixed point theorems. let y0 = pc(j; x) = {y ∈ pc(j; x) : y ∈ c((tm, tm+1], x), m = 0, 1, · · · , δ and y(t−m) = y(tm), y(t + m) exist}. (3.1) and y1 = {y ∈ y0 : ‖y(t) − y(s)‖ ≤ l|t − s|, ∀ t ∈ [tm, tm+1], m = 0, 1, · · · , δ}. (3.2) where l is an appropriate positive constant to be defined later. 3.1 integer order case theorem 3.1. we assume that assumptions (a1) − (a3) are satisfied. if θ = sup{ max m=1,··· ,δ [lgm(sm − tm) + lf(1 + lhl)(tm+1 − sm)], lgδ(sδ − tδ) + lf(1 + lhl)(t − sδ + t1)} < 1. (3.3) then, the system (1.1)-(1.3) has a unique mild solution on the interval j. proof. in order to transform the system (1.1)-(1.3) into a fixed point problem, we consider the map q : s → s defined by qu(t) =    ∫t tm gm(s, u(s))ds, t ∈ (tm, sm], m = 1, · · · , δ, ∫sδ tδ gδ(s, u(s))ds + ∫t sδ f(s, u(s), u([h(u(s), s)]))ds + ∫t 0 f(s, u(s), u([h(u(s), s)]))ds, t ∈ [0, t1], ∫sδ tδ gm(s, u(s))ds + ∫t sm f(s, u(s), u([h(u(s), s)]))ds, t ∈ (sm, tm+1], (3.4) where s = {u ∈ y0 ∩y1 : ‖u‖pc ≤ r}. clearly, s is a closed and bounded subset of y1 and complete metric space. it is not difficult to show that qu ∈ y0. now, it remains to show that qu ∈ y1. for u ∈ s and τ2, τ1 ∈ [0, t1] with τ1 < τ2, ‖qu(τ2) − qu(τ1)‖ ≤ ∫τ2 τ1 ‖f(s, u(s), u([h(u(s), s)]))‖ds, ≤ h|τ2 − τ1|. (3.5) where h = supt∈[0,t] ‖f(t, u(t), u([h(u(t), t)]))‖. similarly, τ2, τ1 ∈ (tm, sm], m = 1, · · · , δ ‖qu(τ2) − qu(τ1)‖ ≤ ‖ ∫τ2 tm gm(s, u(s))ds − ∫τ1 tm gm(s, u(s))ds‖ ≤ km|τ2 − τ1|, (3.6) cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 17 and for τ2, τ1 ∈ (sm, tm+1], m = 1, · · · , δ ‖qu(τ2) − qu(τ1)‖ ≤ h|τ2 − τ1|. (3.7) therefore, we conclude that qu ∈ y1 with suitable constant l = min{ max m=1,··· ,δ km, h}. now, we show that q(s) ⊆ s. for t ∈ [0, t1] and u ∈ s, we get ‖qu(t)‖ ≤ ‖ ∫sδ tδ gδ(s, u(s))ds‖ + ∫t sδ ‖f(s, u(s), u([h(u(s), s)]))‖ds + ∫t 0 ‖f(s, u(s), u([h(u(s), s)]))‖ds, ≤ kδ(sδ − tδ) + h(t − sδ + t1) ≤ kδt + ht. (3.8) for t ∈ [sm, tm+1], m = 1, · · · , δ, ‖qu(t)‖ ≤ km(sm − tm) + h(tm+1 − sm) ≤ kmt + ht, (3.9) and for t ∈ (sm, tm], we have that ‖qu(t)‖ ≤ kmt. we choose r = max[kδt +ht, sup m=1,··· ,δ {kmt + ht}]. thus, we get that q(s) ⊆ s. in the next step, we prove that q is a contraction map. for w1, w2 ∈ s and t ∈ [0, t1], we get ‖qw1(t) − qw2(t)‖ ≤ [lgδ(sδ − tδ) + lf(1 + lhl)(t − sδ + t1)] ×‖w1 − w2‖pc. (3.10) for t ∈ [sm, tm+1], m = 1, · · · , δ ‖qw1(t) − qw2(t)‖ ≤ [lgm(sm − tm) + lf(1 + lhl)(tm+1 − sm)]‖w1 − w2‖pc, ≤ max m=1,··· ,δ [lgm(sm − tm) + lf(1 + lhl)(tm+1 − sm)] ×‖w1 − w2‖pc, (3.11) and for t ∈ (tm, sm], we obtain that ‖qw1(t) − qw2(t)‖ ≤ max m=1,··· ,δ lgm(sm − tm) × ‖w1 − w2‖pc. (3.12) from the inequalities (3.10)-(3.12), we get ‖qw1 − qw2‖pc ≤ θ‖w1 − w2‖pc. (3.13) thus, by the inequality (3.3), we conclude that q is a contraction on s and there exists a unique fixed point u ∈ s of the map q. it is obvious that u is a mild solution for the system (1.1)-(1.3). our second existence result is based on krasnoselskii’s theorem. the statement of the theorem is given as: 18 alka chadha & dwijendra n pandey cubo 17, 1 (2015) theorem 3.2. let f ⊂ x be a closed convex and nonempty subset of x, where x is a banach space. let p1 and p2 be the operator such that (a) p1w1 + p2w2 ∈ f, whenever, w1, w2 ∈ f, (b) p1 is a contraction, (c) p2 is compact and continuous. then, the map p = p1 + p2 has a fixed point x ∈ f i.e., x = p1x + p2x. theorem 3.3. assume that (a1) − (a3) are satisfied. then, there exists a mild solution for the system (1.1)-(1.3) on j provided that ξ = max{kgm(sm − tm); m = 1, · · · , δ} < 1. (3.14) proof. we define the following operators q1 : s → s which is decomposition of operator q, by q1u(t) =    ∫sδ tδ gδ(s, u(s))ds, t ∈ [0, t1], ∫t sm gm(s, u(s))ds, t ∈ (tm, sm], m = 1, · · · , δ, ∫sm tm gm(sm, u(sm)), t ∈ (sm, tm+1] m = 1, · · · , δ. (3.15) and q2 : s → s by q2u(t) =    ∫t sδ f(s, u(s), u(h(u(s), s)))ds + ∫t 0 f(s, u(s), u(h(u(s), s)))ds, t ∈ [0, t1], 0, t ∈ (tm, sm], m = 1, · · · , δ, ∫t sm f(s, u(s), u(h(u(s), s)))ds, t ∈ (sm, tm+1] i = 1, · · · , δ. (3.16) we choose r such that max{ max m=1,··· ,δ (km + h)t, (kδ + h)t} < r. (3.17) consider br = {u ∈ y0 ∩ y1 : ‖u‖pc ≤ r}. (3.18) it is clear that the mappings q1 and q2 are well-defined. now, we show the result in several steps. step 1. for u, v ∈ br and t ∈ [0, t1], we have ‖(q1u + q2v)(t)‖ ≤ ‖ ∫sδ tδ gδ(s, u(s))ds‖ + ∫t sδ ‖f(s, v(s), v(h(v(s), s)))‖ds + ∫t 0 ‖f(s, v(s), v(h(v(s), s)))‖ds, ≤ kδ(sδ − tδ) + h[t − sδ − t1] ≤ kδt + ht. (3.19) for t ∈ (sm, tm+1], m = 1, · · · , δ, ‖(q1u + q2v)(t)‖ ≤ ‖ ∫sm tm gm(s, u(s))ds‖ + ∫t sm ‖f(s, u(s), u(h(u(s), s)))‖ds, ≤ km(sm − tm) + h(tm+1 − sm) ≤ (km + h)t, (3.20) cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 19 and for t ∈ [tm, sm], we have ‖(q1u + q2v)(t)‖ ≤ kmt. thus, by the choice of r, we get that ‖(q1u + q2v)‖pc ≤ r, for all t ∈ [0, t]. (3.21) hence, q1u + q2v ∈ br. step 2. we show that q1 is contraction map. for w1, w2 ∈ br and t ∈ [0, t1], ‖q1w1(t) − q2w2(t)‖ ≤ kgδ‖w1(sδ) − w2(sδ)‖ × |sδ − tδ| ≤ kgδ(sδ − tδ)‖w1 − w2‖pc. (3.22) for t ∈ (tm, sm], m = 1, · · · , δ, we get ‖q1w1(t) − q2w2(t)‖ ≤ kgm(sm − tm)‖w1 − w2‖pc, (3.23) and t ∈ (sm, tm+1], m = 1, · · · , δ ‖q1w1(t) − q2w2(t)‖ ≤ kgm(sm − tm)‖w1 − w2‖pc. (3.24) from the above inequalities, we conclude that ‖q1w1 − q2w2‖pc ≤ ξ‖w1 − w2‖pc, (3.25) which gives that q1 is a contraction. step 3. q2 is continuous map. let {zp} ∞ p=1 be a sequence such that zp → z ∈ br. for t ∈ [0, t1], ‖q2zp(t) − q2z(t)‖ ≤ ∫t sδ ‖f(s, zp(s), zp(h(zp(s), s))) − f(s, z(s), z(h(z(s), s)))‖ds + ∫t 0 ‖f(s, zp(s), zp(h(zp(s), s))) − f(s, z(s), z(h(z(s), s)))‖ds, by the continuity of f and h, we have that s ∈ [0, t] f(s, zp(s), zp(h(zp(s), s))) → f(s, z(s), z(h(z(s), s))), as p → ∞, (3.26) h(zp(s), s) → h(z(s), s), as p → ∞, (3.27) from the dominated convergence theorem, we get ‖q2zp − q2z‖pc → 0, as p → ∞, for t ∈ (tm, sm], m = 1, · · · , δ, ‖q2zp(t) − q2z(t)‖ = 0. similarly, for t ∈ (sm, tm+1], m = 1, · · · , δ ‖q2zp(t) − q2z(t)‖ ≤ ∫t sm ‖f(s, zp(s), zp(h(zp(s), s))) − f(s, z(s), z(h(z(s), s)))‖ds, 20 alka chadha & dwijendra n pandey cubo 17, 1 (2015) by the continuity of f, h and the dominated convergence theorem, we deduce that ‖q2zp − q2z‖pc → 0, as p → ∞. step 3. q2 is compact. since f is continuous map and ‖(q2u)(t)‖ ≤ 2ht < r. this implies that q2 is uniformly bounded on br. now, we show that q2 maps bounded set into equicontinuous set of br. for τ2 > τ1 ∈ [0, t1] and u ∈ br, we have ‖q2u(τ2) − q2u(τ1)‖ ≤ lf(τ2 − τ1). (3.28) for τ2 > τ1 ∈ (tm, sm], we have ‖q2u(τ2) − q2u(τ1)‖ = 0. for τ2 > τ1 ∈ (sm, tm+1], m = 1, · · · , δ and u ∈ br, we have ‖q2u(τ2) − q2u(τ1)‖ ≤ lf(τ2 − τ1). (3.29) thus, we conclude that ‖q2u(τ2) − q2u(τ1)‖ → 0 as τ2 → τ1. hence q2 is equicontinuous. by the steps (3) − (4) and arzela-ascoli theorem, we deduce that q2 : br → br is continuous and compact i.e. completely continuous. since q1 is contraction and q2 is completely continuous operator. thus, q = q1 + q2 has a fixed point by using krasnoselskiis fixed point theorem which is just a mild solution for the system (1.1)-(1.3). the proof of the theorem is finished. 3.2 fractional order case now, we obtain the existence results for the problem (1.4)-(1.6) via fixed points theorems, the first existence result of the mild solution for problem (1.4)-(1.6) is obtained by using banach fixed point theorem and second existence results is obtained by using krasnoselskii’s fixed point theorem. theorem 3.4. assume that hypotheses (a1) − (a3) are fulfilled and λ = sup{ max m=1,··· ,δ [lgm(sm − tm) + lf(1 + llh)(t 2 m+1 + s 2 m) 2 ], max m=1,··· .δ (sm − tm)lgm, lgδ(sδ − tδ) + lf(1 + llh)(t 2 + s2δ + t 2 1 2 } < 1. (3.30) then, the problem (1.4)-(1.6) has at least one mild solution on [0, t]. proof. we firstly define the operator q : s → s by (qu)(t) =    ∫sδ tδ gδ(s, u(s))ds − ∫sδ 0 (sδ − s)f(s, u(s), u([h(u(s), s)]))ds + ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds + ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds, t ∈ [0, t1], ∫t tm gm(s, u(s))ds, t ∈ (tm, sm], m = 1, · · · , δ, ∫sm tm gm(s, u(s))ds − ∫sm 0 (sm − s)f(s, u(s), u([h(u(s), s)]))ds + ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds, t ∈ (sm, tm+1], m = 1, · · · , δ. (3.31) cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 21 it is clear that qu ∈ y0. so it remains to show that qu ∈ y1. for u ∈ s and τ2, τ1 ∈ [0, t1] with τ1 ≤ τ2, we get ‖(qu)(τ2) − (qu)(τ1)‖ = ‖ ∫τ2 0 (τ2 − s)f(s, u(s), u([h(u(s), s)]))ds − ∫τ1 (τ1 − s)f(s, u(s), u([h(u(s), s)]))ds‖, ≤ ‖ ∫τ1 0 [(τ2 − s) − (τ1 − s)]f(s, u(s), u([h(u(s), s)]))ds‖ +‖ ∫τ2 τ1 (τ2 − s)f(s, u(s), u([h(u(s), s)]))ds‖, ≤ h(τ2 − τ1) 2 + h (τ2 − τ1) 2 2 , ≤ 2ht |τ2 − τ1|, (3.32) similarly, for τ2, τ1 ∈ (sm, tm+1], m = 1, · · · , δ, ‖(qu)(τ2) − (qu)(τ1)‖ ≤ ht |τ2 − τ1| (3.33) and for τ2, τ1 ∈ (tm, sm], ‖(qu)(τ2) − (qu)(τ1)‖ ≤ km|τ2 − τ1|. (3.34) thus, from (3.32)-(3.34), we conclude that qu ∈ y1 with l = min{ max m=1,··· ,δ km, 2ht, ht}. hence q is well defined on s. next we show that q(s) ⊆ s. for u ∈ s and t ∈ [0, t1], we get ‖qu(t)‖ ≤ ‖ ∫sδ tδ gδ(s, u(s))ds‖ + ∫sδ 0 (sδ − s)‖f(s, u(s), u([h(u(s), s)]))‖ds + ∫t 0 (t − s)‖f(s, u(s), u([h(u(s), s)]))‖ds + ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds, ≤ kδ(sδ − tδ) + h(t2 + s2δ + t 2 1) 2 ≤ kδt + 3ht2 2 . (3.35) similarly, for t ∈ (sm, tm+1], m = 1, · · · , δ, ‖qu(t)‖ ≤ kmt + t 2h, (3.36) and for t ∈ (tm, sm], we get ‖qu(t)‖ ≤ kmt. (3.37) we choose r = max{kδt + 3t2h 2 , supm=1,··· ,δ kmt +t 2h} such that ‖qu(t)‖ ≤ r, for all t ∈ [0, t]. we now show that q is a contraction map on s. for u∗, u∗∗ ∈ s and t ∈ [0, t1], we get 22 alka chadha & dwijendra n pandey cubo 17, 1 (2015) ‖(qu∗)(t) − (qu∗∗)(t)‖ ≤ ‖ ∫sδ tδ [gδ(s, u ∗ (s)) − gδ(s, u ∗∗ (s))]ds‖ + ∫sδ 0 (sδ − s)‖f(s, u ∗(s), u∗([h(u∗(s), s)])) − f(s, u∗∗(s), u∗∗([h(u∗∗(s), s)]))‖ds + ∫t 0 (t − s)‖f(s, u∗(s), u∗([h(u∗(s), s)])) − f(s, u∗∗(s), u∗∗([h(u∗∗(s), s)]))‖ds + ∫t 0 (t − s)‖f(s, u∗(s), u∗([h(u∗(s), s)])) − f(s, u∗∗(s), u∗∗([h(u∗∗(s), s)]))‖ds, ≤ [lgδ(sδ − tδ) + lf(1 + llh)(t 2 + s2δ + t 2 1 2 ]‖u∗ − u∗∗‖pc. (3.38) similarly, for t ∈ (sm, tm+1], m = 1, · · · , δ ‖(qu∗)(t) − (qu∗∗)(t)‖ ≤ ‖ ∫sm tm [gm(s, u ∗(s)) − gm(s, u ∗∗(s))]ds‖ + ∫sm 0 (sm − s)‖f(s, u ∗ (s), u∗([h(u∗(s), s)])) − f(s, u∗∗(s), u∗∗([h(u∗∗(s), s)]))‖ds + ∫t 0 (t − s)‖f(s, u∗(s), u∗([h(u∗(s), s)])) − f(s, u∗∗(s), u∗∗([h(u∗∗(s), s)]))‖ds, ≤ max m=1,··· ,δ [lgm(sm − tm) + lf(1 + llh)(t 2 m+1 + s 2 m) 2 ]‖u∗ − u∗∗‖pc, (3.39) and for t ∈ (tm, sm], we get ‖(qu∗)(t) − (qu∗∗)(t)‖ ≤ max m=1,··· .δ lgm(sm − tm)‖u ∗ − u∗∗‖pc. (3.40) from the inequalities (3.38)-(3.40), we obtain ‖(qu∗)(t) − (qu∗∗)(t)‖ ≤ λ‖u∗ − u∗∗‖pc. (3.41) thus, by the inequality (3.30), we conclude that q is a contraction on s i.e., there exists a unique fixed point of the map u ∈ s such that qu(t) = u(t) for all t ∈ [0, t]. hence problem (1.4)-(1.6) has a unique mild solution on [0, t]. theorem 3.5. assume that (a1)-(a3) are fulfilled and ξ = max{lgm|sm − tm|; m = 1, · · · , δ} < 1. (3.42) then, problem (1.4)-(1.6) has at least one mild solution on [0, t]. proof. we consider the operators q1 and q2 on bq,r = {u ∈ y0 ∩ y1 : ‖u‖pc ≤ r} defined by q1u(t) =    ∫t tm gm(s, u(s))ds, t ∈ (tm, sm], ∫sδ tδ gδ(s, u(s))ds, t ∈ [0, t1] ∫sm tm gm(s, u(s))ds, t ∈ (sm, tm+1], m = 1, · · · , δ, (3.43) cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 23 and q2u(t) =    ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds − ∫sδ 0 (sδ − s)f(s, u(s), u([h(u(s), s)]))ds + ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds, t ∈ [0, t1] 0, t ∈ (tm, sm], m = 1, · · · , δ, − ∫sm 0 (sm − s)f(s, u(s), u([h(u(s), s)]))ds + ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds, t ∈ (sm, tm+1], m = 1, · · · , δ. (3.44) where r is a positive constant such that max{ sup m=1,··· ,δ km(sm − tm) + h(t2m+1 + s 2 m) 2 , kδ(sδ − tδ) + h(t2 + s2δ + t 2 1) 2 } ≤ r. (3.45) for the purpose of convenience, we separate the proof into a few steps. step 1. we show that q1u + q2u ∈ bq,r for each u ∈ bq,r. for t ∈ [0, t1], we have ‖q1u(t) + q2u(t)‖ ≤ ‖ ∫sδ tδ gδ(s, u(s))ds‖ + ‖ ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds‖ +‖ ∫sδ 0 (sδ − s)f(s, u(s), u([h(u(s), s)]))ds‖ + ‖ ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds‖ ≤ kδ(sδ − tδ) + h(t2 + s2δ + t 2 1) 2 , (3.46) where h = supt∈[0,t] ‖f(t, u(t), u([h(u(t), t)]))‖. similarly, for t ∈ (sm, tm+1], m = 1, · · · , δ ‖q1u(t) + q2u(t)‖ ≤ ‖ ∫sm tm gm(tm, u(tm))‖ + ‖ ∫sm 0 (sm − s)f(s, u(s), u([h(u(s), s)]))ds‖ +‖ ∫t 0 (t − s)f(s, u(s), u([h(u(s), s)]))ds‖, ≤ km(sm − tm) + h(t2m+1 + s 2 m) 2 , (3.47) and for t ∈ (tm, sm], m = 1, · · · , δ, ‖q1u(t) + q2u(t)‖ ≤ km(sm − tm). (3.48) by inequality (3.45), we get ‖q1u(t) + q2u(t)‖ ≤ r for all t ∈ [0, t]. hence, q1u + q2u ∈ bq,r. step 2. the map q1 is contraction on bq,r. from the step 2 of theorem 3.3, we have that q1 is a contraction on bq,r. step 3. the map q2 is continuous on bq,r. let {up} ∞ p=1 be a sequence in bq,r such that limp→∞ up = u ∈ bq,r. for t ∈ (tm, sm], m = 24 alka chadha & dwijendra n pandey cubo 17, 1 (2015) 1, · · · , δ, it is obvious since q2up(t) = 0. for t ∈ [0, t1], we get ‖(q2up)(t) − (q2u)(t)‖ ≤ ∫t 0 (t − s)‖f(s, up(s), up([h(up(s), s)])) − f(s, u(s), u([h(u(s), s)]))‖ds + ∫sδ 0 (sδ − s)‖f(s, up(s), up([h(up(s), s)])) − f(s, u(s), u([h(u(s), s)]))‖ds + ∫t 0 (t − s)‖f(s, up(s), up([h(up(s), s)])) − f(s, u(s), u([h(u(s), s)]))ds‖ds, by the continuity of f and lebesgue dominated convergence theorem, we estimate ‖(q2up)(t) − (q2u)(t)‖ → 0, as p → ∞. (3.49) similarly, t ∈ (sm, tm+1], m = 1, · · · , δ, ‖(q2up)(t) − (q2u)(t)‖ ≤ ∫sm 0 (sm − s)‖f(s, up(s), up([h(up(s), s)])) − f(s, u(s), u([h(u(s), s)]))‖ds + ∫t 0 (t − s)‖f(s, up(s), up([h(up(s), s)])) − f(s, u(s), u([h(u(s), s)]))‖ds by the continuity of f and lebesgue dominated convergence theorem, we estimate ‖(q2up)(t) − (q2u)(t)‖ → 0, ∀ t ∈∈ (sm, tm+1] as p → ∞. (3.50) hence, q2 is continuous map on bq,r. step 4. q2 is compact. q2 is firstly uniformly bounded on bq,r, since ‖q2u‖pc ≤ r. we now prove that q2 maps bounded set into equicontinuous set of bq,r. for t ∈ (tm, sm], m = 1, · · · , δ, it is obvious. for τ2, τ1 ∈ [0, t1] with τ1 < τ2, we have ‖q2(τ2) − q2(τ1)‖ ≤ h(τ2 − τ1) 2 + h (τ2 − τ1) 2 2 . (3.51) for τ2, τ1 ∈ (sm, tm+1], m = 1, · · · , δ with τ2 > τ1, ‖q2(τ2) − q2(τ1)‖ ≤ h(τ2 − τ1) 2 + h (τ2 − τ1) 2 2 . (3.52) the right hand side of inequalities (3.51)-(3.52) tend to zero as τ2 → τ1. thus, q2(bq,r) is equicontinuous. by arzela-ascoli theorem, we conclude that q2 is completely continuous. therefore, from the krasnoselskiis fixed point theorem, we deduce that q = q1 + q2 has a fixed point which is just a mild solution for the problem (1.4)-(1.6). cubo 17, 1 (2015) periodic bvp for a class of nonlinear differential equation . . . 25 4 examples for illustrating the application of the theory, we consider the following examples. let us consider the following impulsive nonlinear cauchy problems with boundary conditions as u′(t)(or cd 1/2 0,t u(t)) = 1 3 + t3 [ |u(t)| 6(1 + |u(t)|) + |u(1 3 u(t))| 1 + |u(1 3 u(t))| ] or (i 3/2 0,t 1 3 + t3 [ |u(t)| 6(1 + |u(t)|) + |u(1 3 u(t))| 1 + |u(1 3 u(t))| ]), t ∈ (0, 1] ∪ (2, 3] (4.1) u(t) = ∫t 1 |u(s)| (9s + 1)(1 + |u(s)|) ds, t ∈ (1, 2], (4.2) u(0) = u(3), (4.3) where 0 = s0 < t1 = 1 < s1 = 2 < t2 = 3 and j = [0, 3] and u ∈ c 1([0, 3], [0, 3]). then, u ∈ cl([0, 3], [0, 3]). here cl([0, 3], [0, 3]) = {u ∈ c([0, 3], [0, 3]) : |u(t) − u(s)|l ≤ l|t − s|, ∀ t, s ∈ [0, 3]} (4.4) and f(t, u(t), u(h(u(t), t))) = 1 3 + t3 [ |u(t)| 6(1 + |u(t)|) + |u(1 3 u(t))| 1 + |u(1 3 u(t))| ], (4.5) g1(t, u(t)) = |u(t)| (9t + 1)(1 + |u(t)|) . (4.6) it is easy to show that f and g satisfy the following condition ‖f(t, u1, v1) − f(t, u2, v2)‖ ≤ lf[‖u1 − u2‖ + ‖v1 − v2‖l], u1, u2 ∈ [0, 3], (4.7) ‖g1(t, u1) − g1(t, u2)‖ ≤ 1 10 ‖u1 − u2‖, (4.8) ‖g1(t, u)‖ ≤ 1 9t + 1 = k1 ≤ 1 10 (4.9) thus all the assumptions of theorem 3.1/3.3 or 3.4/3.5 are fulfilled. hence, there exists a mild solution for the problem (4.1). acknowledgement. the authors would like to thank the referee for valuable comments and suggestions. the work of the first author is supported by the ugc (university grants commission, india) under grant no (6405 − 11 − 061) and indian institute of technology, roorkee. nt. received: december 2014. accepted: january 2015. 26 alka chadha & dwijendra n pandey cubo 17, 1 (2015) references [1] m. benchohra, j. henderson, s. k. ntouyas, impulsive differential equations and inclusions, contemporary mathematics and its applications, vol.2, hindawi publishing corporation, new york, 2006. 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[27] y. k. chang, a. anguraj, p. karthikeyan, existence results for initial value problems with integral conditions for impulsive fractional differential equations, j. fract. cal. appl., 2 (2011), 1-10. introduction: preliminaries and assumptions existence result integer order case fractional order case examples () cubo a mathematical journal vol.17, no¯ 01, (99–106). march 2015 semi open sets in bispaces amar kumar banerjee department of mathematics, burdwan university, burdwan-713104, w.b., india. akbanerjee1971@gmail.com pratap kumar saha behala college, kolkata,w.b., india. pratapsaha2@gmail.com abstract the notions of semi open sets in a topological space were introduced by n.levine in 1963. here we study the same using the idea of τ1(τ2) semi open sets with respect to τ2(τ1), pairwise semi open sets in a more general structure of a bispace and investigate how far several results as valid in a bitopological space are affected in bispaces. resumen las nociones de conjuntos semiabiertos en un espacio topológico se introdujeron por n. levine en 1963. aqúı estudiamos lo mismo usando la idea de conjuntos semiabiertos τ1(τ2) respecto de conjuntos abiertos semiabiertos dos a dos τ2(τ1), en una estructura más general de biespacio e investigamos cómo varios resultados válidos en un espacio bitopológico cambian en biespacios. keywords and phrases: bispaces, semi open sets, τ1 semi open sets with respect to τ2. 2010 ams mathematics subject classification: 54a05, 54e55, 54e99 100 amar kumar banerjee & pratap kumar saha cubo 17, 1 (2015) 1 introduction the notion of a σ space or simply a space was introduced by a.d.alexandroff [1] in 1940 generalising the idea of a topological space where only countable unions of open sets were taken to be open. in 2001 the idea of space was used by lahiri and das [8] to generalise the notion of a bitopological space to a bispace. n.levine [9] introduced the concept of semi open sets in a topological space in 1963 and this idea was generalised by s.bose [3] in the setting of a bitopological space (x, p, q) using the idea of p(q) semi open sets with respect to q(p) etc. later the same was studied in a space by lahiri and das [7] and they critically took the matter of generalisation in this setting. here we have studied the concept of τ1(τ2) semi open sets with respect to τ2(τ1) and some other properties in the setting of a bispace and have shown with typical examples how far several results as valid in[3] are affected in bispaces. also we have given a necessary and sufficient condition for a bispace to be a bitopological space in terms of τ1(τ2) semi open sets with respect to τ2(τ1). 2 preliminaries definition 2.1. [1] a set x is called an alexandroff space or simply a space if in it is chosen a system of subsets f satisfying the following axioms: (1) the intersection of a countable number of sets from f is a set in f. (2) the union of a finite number of sets from f is a set in f. (3) the void set φ is a set in f. (4) the whole set x is a set in f. . sets of f are called closed sets. their complementary sets are called open sets. it is clear that instead of closed set in the definition of the space, one may put open sets with subject to the condition of countable summability, finite intersectibility and the condition that x and φ should be open. the collection of all such open sets will sometimes be denoted by τ and the space by (x, τ). note that, in general τ is not a topology as can be easily seen by taking x = r, the set of real numbers and τ as the collection of all fσ sets in r. definition 2.2. [1] to every set m of (x, τ) we correlate its closure m = the intersection of all closed sets containing m. sometimes the closure of a set m will be denoted by τclm or simply clm when there is no confusion about τ. generally the closure of a set in a space is not a closed set. from the axioms, it easily follows that 1) m ∪ n = m ∪ n; 2) m ⊂ m ; 3) m = m ; 4) φ = φ. cubo 17, 1 (2015) semi open sets in bispaces 101 definition 2.3. [7] the interior of a set m in (x, τ) is defined as the union of all open sets contained in m and is denoted by τintm or intm when there is no confusion. definition 2.4. [6] a non empty set x on which are defined two arbitrary topologies p, q is called a bitopological space and denoted by (x, p, q). definition 2.5. [8] let x be a nonempty set. if τ1 and τ2 be two collections of subsets of x such that (x, τ1) and (x, τ2) are two spaces, then x is called a bispace and is denoted by (x, τ1, τ2). 3 pairwise semi open sets definition 3.1. (cf. definition 1[3] ): let (x, τ1, τ2) be a bispace. we say that a subset a of x is τ1 semi open with respect to τ2 (in short τ1 s.o.w.r.to τ2 ) if and only if there exists a τ1 open set o such that o ⊂ a ⊂ τ2clo. similarly a ⊂ x, is τ2 semi open with respect to τ1 (in short τ2 s.o.w.r.to τ1) if and only if there exists a τ2 open set o such that o ⊂ a ⊂ τ1clo. we say that a is pairwise semi open if and only if it is both τ1 s.o.w.r.to τ2 and τ2 s.o.w.r.to τ1. note that a τ1(τ2) open set is τ1(τ2) s.o.w.r.to τ2(τ1). throughout our discussion, (x, τ1, τ2) or simply x stands for a bispace, r stands for the set of real numbers, q for the set of rational numbers and n stands for the set of natural numbers and sets are always subsets of x unless otherwise stated. theorem 3.2. let (x, τ1, τ2) be a bispace. let a ⊂ x, and a is τ1 s.o.w.r.to τ2 then τ2cla = τ2cl(τ1inta). proof. let a is τ1 s.o.w.r.to τ2 then there exists a τ1 open set o such that o ⊂ a ⊂ τ2clo. also o ⊂ τ1inta. therefore, a ⊂ τ2clo ⊂ τ2cl(τ1inta) and hence τ2cla ⊂ τ2cl(τ2cl(τ1inta)) = τ2cl(τ1inta). also τ2cl(τ1inta) ⊂ τ2cla. therefore, τ2cla = τ2cl(τ1inta). corollary 3.3. if a is τ1 s.o.w.r.to τ2 and a 6= φ then τ1inta 6= φ. corollary 3.4. let a is τ1 s.o.w.r.to τ2 and a ⊂ b then a ⊂ τ2cl(τ1intb). if (x, τ1, τ2) is a bitopological space then converse part of the theorem 3.2 also holds which is seen in [3] . but in a bispace, this may not be true as shown below: example 3.5. let x = [0, 2] and {gi} be the collection of all countable subsets of irrational numbers in [0, 1]. let τ1 be the collection of all sets of the form gi ∪ { √ 2} together with x and φ, and τ2 be the collection of all sets gi together with x and φ. then (x, τ1, τ2) is a bispace. now consider a subset a = [0, 1] ∪ { √ 2} then τ2cla = x and τ1inta is set of all irrational numbers in [0, 1] together with √ 2 and hence τ2cl(τ1inta) = x. therefore, τ2cla = τ2cl(τ1inta). but for any τ1 102 amar kumar banerjee & pratap kumar saha cubo 17, 1 (2015) open set g(6= x, φ), τ2clg = g ∪ q1 ∪ [1, 2], where q1 is the set of all rational numbers in [0, 1]. clearly τ2clg does not contain a. therefore, there does not exist any τ1 open set g satisfying g ⊂ a ⊂ τ2clg. so a is not τ1 s.o.w.r.to τ2. however we observe in the following theorem that the converse part of theorem 3.2 holds under an additional condition. theorem 3.6. in a bispace (x, τ1, τ2), let τ2cla = τ2cl(τ1inta). then a is τ1 s.o.w.r.to τ2 for any subset a of x if the condition c1 is satisfied. c1 : arbitrary union of τ1 open sets is τ1 s.o.w.r.to τ2. proof. let o = τ1inta. then by the condition c1, o is τ1 s.o.w.r.to τ2. so there exists a τ1 open set g such that g ⊂ o ⊂ τ2clg. now since τ2clo = τ2cl(τ1inta) = τ2cla and o ⊂ τ2clg, it follows that τ2clo ⊂ τ2cl(τ2clg) = τ2clg and hence τ2cla ⊂ τ2clg. therefore, g ⊂ o ⊂ a ⊂ τ2cla ⊂ τ2clg and so a is τ1 s.o.w.r.to τ2. remark 3.7. we see in the example 3.8 below that there is a bispace which is not a bitopological space where the condition c1 holds good. example 3.8. let x = [0, 2], τ1 be the collection of all sets gi together with x and φ and τ2 be the collection of all sets fi together with x and φ where {gi} and {fi} are the collection of all countable subsets of irrational numbers in [0, 1] and [1, 2] respectively. then (x, τ1, τ2) is a bispace but not a bitopological space. now consider all τ1 open sets {gi}. then ∪gi is the set of all irrational numbers in [0, 1] which is not τ1 open. but since, for any τ1 open set gi, τ2clgi = [0, 1] ∪ q2 where q2 is set of all rational numbers in [1, 2]. it follows that gi ⊂ ∪gi ⊂ τ2clgi. this implies that ∪gi is τ1 s.o.w.r.to τ2 although it is not τ1 open. theorem 3.9. countable union of τ1 s.o.sets w.r.to τ2 is τ1 s.o.w.r.to τ2. proof. let {an : n ∈ n} be a countable collection of τ1 s.o.sets w.r.to τ2. then for each n ∈ n there exists a τ1 open set on such that on ⊂ an ⊂ τ2clon. this implies that ∪{on : n ∈ n} ⊂ ∪{an : n ∈ n} ⊂ ∪{τ2clon : n ∈ n} ⊂ τ2cl(∪{on : n ∈ n}) i.e, o ⊂ ∪{an : n ∈ n} ⊂ τ2clo, where o = ∪{on : n ∈ n}, a τ1 open set. hence ∪{an : n ∈ n} is τ1 s.o.w.r.to τ2. remark 3.10. in [3] it was proved that in a bitopological space arbitrary union of τ1 s.o.sets w.r.to τ2 is τ1 s.o.w.r.to τ2. but this may not be true in a bispace as shown in the example 3.11 below. example 3.11. consider x = r. let τ1 open sets are x, φ and all sets gi where {gi} is the collection of all countable subsets of irrational numbers in r and τ2 open sets are x, φ and all fσ sets in r. clearly τ2 closed sets are the gδ sets. so for any subset g in r we have τ2clg = g. therefore, for any set a which is τ1 s.o.w.r.to τ2 in x, there exists a τ1 open set gi such that gi ⊂ a ⊂ τ2clgi = gi. this implies that a = gi, i.e., a is τ1 open set. so τ1 open sets are the only τ1 s.o.sets w.r.to τ2. since the union of all τ1 open sets gi (gi 6= x) is precisely the set cubo 17, 1 (2015) semi open sets in bispaces 103 of all irrational numbers in r which is not τ1 open, it follows that arbitrary union of τ1 s.o.sets w.r.to τ2 may not be τ1 s.o.w.r.to τ2. however the additional condition c1 ensures the result in theorem 3.9 for arbitrary union. theorem 3.12. arbitrary union of τ1 s.o.sets w.r.to τ2 is τ1 s.o.w.r.to τ2 if and only if the condition c1 is satisfied. proof. assume first that the arbitrary union of τ1 s.o.sets w.r.to τ2 is τ1 s.o.w.r.to τ2. since every τ1 open set is τ1 s.o.w.r.to τ2, arbitrary union of τ1 open sets is τ1 s.o.w.r.to τ2, i.e., the condition c1 holds. next assume that the condition c1 holds. let {ai} be an arbitrary collection of τ1 s.o.sets w.r.to τ2 and a = ∪ai. for each i, there exists a τ1 open set gi such that gi ⊂ ai ⊂ τ2clgi. therefore, ∪gi ⊂ ∪ai = a ⊂ ∪τ2clgi ⊂ τ2cl(∪gi). since ∪gi, by assumption, is τ1 s.o.w.r.to τ2, there exists a τ1 open set g such that g ⊂ ∪gi ⊂ τ2clg. therefore, g ⊂ ∪gi ⊂ a ⊂ ∪τ2clgi ⊂ τ2cl(∪gi) ⊂ τ2clτ2clg = τ2clg. this proves that a is τ1 s.o.w.r.to τ2. theorem 3.13. let a be τ1 s.o.w.r.to τ2 in a bispace (x, τ1, τ2) and let a ⊂ b ⊂ τ2cla. then b is τ1 s.o.w.r.to τ2. proof. since a is τ1 s.o.w.r.to τ2, there exists a τ1 open set o such that o ⊂ a ⊂ τ2clo. therefore, o ⊂ a ⊂ b ⊂ τ2cla ⊂ τ2cl(τ2clo) = τ2clo and hence b is τ1 s.o.w.r.to τ2. theorem 3.14. let (x, τ1, τ2) be a bispace and let a ⊂ y ⊂ x. if a is pairwise s.o. in x, it is pairwise s.o. in y. proof. let a be τ1 s.o.w.r.to τ2. then there exists a τ1 open set o such that o ⊂ a ⊂ τ2clo. let oy = y ∩ o which is τ1 open in y. so oy = y ∩ o ⊂ y ∩ a ⊂ y ∩ τ2clo = τ2cloy in y. interchanging the role of τ1 and τ2 we get the result. we denote the class of all τ1 s.o.w.r.to τ2 by τ1s.o.(x)τ2. theorem 3.15. let b = {bα} be a collection of subsets of x such that (i) τ1 ⊂ b and (ii) b ∈ b and b ⊂ d ⊂ τ2clb imply d ∈ b, then τ1s.o.(x)τ2 ⊂ b. proof. let a ∈ τ1s.o.(x)τ2 , then there exists a τ1 open set o such that o ⊂ a ⊂ τ2clo. therefore, o ∈ b and o ⊂ a ⊂ τ2clo imply that a ∈ b and hence the result follows. we denote the set {τ1inta : a ∈ τ1s.o.(x)τ2} by τ1int(τ1s.o.(x)τ2 ). interchanging the role of τ1 and τ2 we may denote other such classes at our will. from the construction, it is obvious that τ1 ⊂ τ1int(τ1s.o.(x)τ2). but in general, τ1 may not be equal to τ1int(τ1s.o.(x)τ2) as shown in the following example. 104 amar kumar banerjee & pratap kumar saha cubo 17, 1 (2015) example 3.16. let (x, τ1, τ2) be the bispace as in example 3.8. now for any τ1 open set gi, τ2clgi = [0, 1] ∪ q2, where q2 is set of all rational numbers in [1, 2]. let a = [0, 1]. then for any τ1 open set gi, gi ⊂ a ⊂ [0, 1] ∪ q2 = τ2clgi. this implies that a is τ1 s.o.w.r.to τ2, i.e., a ∈ τ1s.o.(x)τ2. but τ1inta is the set of all irrational numbers in [0,1] which is not τ1 open. however equality τ1 = τ1int(τ1s.o.(x)τ2) holds if an additional condition holds. theorem 3.17. in a bispace (x, τ1, τ2), τ1 = τ1int(τ1s.o.(x)τ2) if and only if the condition c2 is satisfied. c2 : for any a ⊂ x which is τ1 s.o.w.r.to τ2, there exists a maximal τ1 open set o such that o ⊂ a ⊂ τ2clo. proof. first assume that τ1 = τ1int(τ1s.o.(x)τ2), and let a be any subset of x which is τ1 s.o.w.r.to τ2. then τ1inta ∈ τ1. also by theorem 3.2, a ⊂ τ2cl(τ1inta). again if g is any τ1 open set satisfying g ⊂ a ⊂ τ2clg, then g ⊂ τ1inta. hence τ1inta is the maximal τ1 open set contained in a such that τ1inta ⊂ a ⊂ τ2cl(τ1inta). taking o = τ1inta, we get o ⊂ a ⊂ τ2clo. conversely let a ∈ τ1s.o.(x)τ2. by the condition, there exists a maximal τ1 open set o such that o ⊂ a ⊂ τ2clo......(1). if possible let o 6= τ1inta. then there exists a τ1 open set g ⊂ a such that g is not contained in o . since o ∪ g is τ1 open set and o ∪ g ⊂ a ⊂ τ2clo ⊂ τ2cl(o ∪ g), this contradicts that o is maximal satisfying the condition (1). hence τ1inta = o and so τ1inta is a τ1 open set, i.e., τ1inta ∈ τ1. therefore τ1int(τ1s.o.(x)τ2 ) ⊂ τ1 and consequently τ1 = τ1int(τ1s.o.(x)τ2). remark 3.18. we see that there is a bispace which is not bitopological space where the condition c2 holds. for, consider the bispace (x, τ1, τ2) as in example 3.11 where the τ1 open sets are the only τ1 s.o.sets w.r.to τ2. so for any set a which τ1 s.o.w.r.to τ2 in x, there exists a maximal τ1 open set o(= a) such that o ⊂ a ⊂ τ2clo. we now give a necessary and sufficient condition in terms of semi open sets for a bispace to be a bitopological space. theorem 3.19. a bispace (x, τ1, τ2) is a bitopological space if and only if following condition holds: (i) arbitrary union of τ1(τ2) s.o.sets w.r.to τ2(τ1) is τ1(τ2) s.o.w.r.to τ2(τ1) (ii) τ1 = τ1int(τ1s.o.(x)τ2 ) and τ2 = τ2int(τ2s.o.(x)τ1). proof. if (x, τ1, τ2) is a bitopological space then (i) holds[3] . for (ii), let o ∈ τ1. then o ∈ τ1s.o.(x)τ2 and since o = τ1into, o ∈ τ1int(τ1s.o.(x)τ2). therefore, τ1 ⊂ τ1int(τ1s.o.(x)τ2 ). on the other hand, let o ∈ τ1int(τ1s.o.(x)τ2). then o = τ1inta for some a ∈ τ1s.o.(x)τ2 and hence o ∈ τ1. therefore, τ1int(τ1s.o.(x)τ2 ) ⊂ τ1. therefore, τ1 = τ1int(τ1s.o.(x)τ2). similarly we can prove that τ2 = τ2int(τ2s.o.(x)τ1). conversely, it suffices to show that an arbitrary union of τ1(τ2) open sets is τ1(τ2) open in cubo 17, 1 (2015) semi open sets in bispaces 105 (x, τ1, τ2). let {gi} be an arbitrary collection of τ1 open sets and g = ∪gi. each gi being τ1 open is τ1 s.o.w.r.to τ2. so by (i) g is τ1 s.o.w.r.to τ2. then by (ii) τ1intg is τ1 open. so τ1intg = g and similarly arbitrary union of τ2 open sets is τ2 open set and this proves the theorem. definition 3.20. (cf. [7] ): let (x, τ1, τ2) be a bispace. two non empty subsets a and b are said to be (i) pairwise weakly separated if there exist a τ1 open set u and a τ2 open set v such that a ⊂ u, b ⊂ v, a ∩ v = φ, b ∩ u = φ. (ii) pairwise strongly separated if there exists a τ1 open set u and a τ2 open set v such that a ⊂ u, b ⊂ v, u ∩ v = φ. definition 3.21. (cf.[7] ): a subset a in a bispace (x, τ1, τ2) is said to be pairwise connected if it can not be expressed as the unions of two pairwise weakly separated sets. remark 3.22. in [3] it is proved that in a bitopological space (x, τ1, τ2) if a = o ∪ b where (i) o 6= φ is τ1 open (ii) a is pairwise connected and (iii) b ′ τ 1 , the derived set of b w.r.to τ1 is empty, then a is τ1 s.o.w.r.to τ2. but this is not true in a bispace as shown in the following example. example 3.23. let x = ([0, 1] − q) ∪ { √ 2} and {gi} be the collection of all countable subsets of [0, 1] − q where q is the set of all rational numbers. let τ1 be the collection of all sets gi ∪ { √ 2} together with x and φ, and τ2 be the collection of all sets gi together with x and φ where gi ∈ {gi}. then (x, τ1, τ2) is a bispace. let a = ([0, 1 2 ] − q) ∪ { √ 2}. then a is pairwise connected, because if a = a1 ∪ a2, then at least one of a1 and a2 say a1 is uncountable and x is the only τ1 open set containing a1. let g be a nonempty countable subset of [0, 1 2 ] − q and o = g ∪ { √ 2}. then a = o ∪ (a − o) where o 6= φ, is τ1 open. also (a − o) ′ τ 1 , the set of all τ1 limit points of a − o is empty. indeed if α ∈ [0, 1] − q, then {α, √ 2} is a τ1 open set containing α satisfying {α, √ 2} ∩ ((a − o) − {α}) = φ. again if α = √ 2 then for any p ∈ g, {p, α} is a τ1 open set containing α satisfying {p, α} ∩ ((a − o) − {α}) = φ. thus no point of x can be a τ1 limit point of a − o. so all the conditions stated in the remark 3.22 above are satisfied, but a is not τ1 s.o.w.r.to τ2, because if gi ∪ { √ 2} is any τ1 open set contained in a, then τ2cl(gi ∪ { √ 2}) = gi ∪ { √ 2}. however the following theorem is true. theorem 3.24. let (x, τ1, τ2) be a bispace. if a = o ∪ b where (i) o(6= φ) is τ1 open (ii) a is pairwise connected (iii) there exists a τ2 closed set f1 ⊃ o such that b ∩ f1 ⊂ g ⊂ τ2clo for some τ1 open set g, then a is τ1 s.o.w.r.to τ2. proof. we show that b ⊂ τ2clo. if b 6⊂ τ2clo then there exists a τ2 closed set f ⊃ o such that b 6⊂ f. let b1 = b ∩ f, b2 = b − b1. then b2 ⊂ x − f and b2 6= φ. further, let b∗ 1 = b1 ∩ f1 and b∗∗1 = b1 ∩ (x − f1). then a = o ∪ b = o ∪ b1 ∪ b2 = (o ∪ b ∗ 1 ) ∪ (b∗∗ 1 ∪ b2). now b∗∗ 1 ∪ b2 ⊂ (x − f1) ∪ (x − f) = x − (f ∩ f1) = g2 (say), which is τ2 open. since b∗1 = 106 amar kumar banerjee & pratap kumar saha cubo 17, 1 (2015) b1 ∩f1 = b∩f∩f1 ⊂ b∩f1 ⊂ g, we have o∪b∗1 ⊂ o∪g = g1 (say), which is τ1 open set. since g ⊂ τ2clo ⊂ f∩f1 and g1 = o∪g ⊂ f∩f1, it follows that g1 ∩g2 = φ. this implies that o∪b∗1, b∗∗ 1 ∪ b2 are non empty strongly separated sets, a contradiction. hence o ⊂ a = o ∪ b ⊂ τ2clo and so a is τ1 s.o.w.r.to τ2. received: october 2013. accepted: november 2014. references [1] alexandroff a.d., additive set functions in abstract spaces, (a) mat.sb.(n.s), 8:50(1940), 307348.(english,russian summary). (b) ibid, 9:51(1941), 563-628, (english,russian summary) [2] biswas n., on semi open mapping in topological spaces, bull.cal.math.soc.(1969). [3] bose s., semi open sets, semi continuity and semi open mapping in bitopological spaces. bull.cal.math.soc. 73. 237-246 (1981). [4] das p. and rashid m.a., g∗-closed sets and a new separation axiom in alexandroff spaces,arch. math. (brno),30(2003),299-307. [5] das p. and rashid m.a., semi g∗-closed sets and a new separation axiom in the spaces, bulletine of the allahabad mathematical society, vol. 19, (2004), 87-98. [6] kelley j.c., bitopological spaces, proc.london.math.soc. 13,71(1963). [7] lahiri b.k. and das p., semi open set in a space, sains malaysiana 24(4) 1-11(1995). [8] lahiri b.k. and das p., certain bitopological concept in a bispace, soochow j.of math.vol.27.no.2,pp.175-185(2001). [9] levine n., semi open sets and semi continuity in topological spaces, amer.math.monthly 70,36(1963). [10] noiri takashi, remarks on semi open mapping, bull.cal.math.soc.65,197(1973). [11] pervin w.j., connectedness in bitopological spaces, ind.math.29,369(1967). [12] reilly i.l., on bitopological separation properties, nanta math.5,14-25(1972). [13] sikorski r., closure homomorphism and interior mapping, fund.math 41,12-20(1995). introduction preliminaries pairwise semi open sets cubo a mathematical journal vol.15, no¯ 03, (01–07). october 2013 an elementary study of a class of dynamic systems with single time delay akio matsumoto chuo university, department of economics, 742-1, higashi-nakano, hachioji, tokyo, 192-0393, japan. akiom@tamacc.chuo-u.ac.jp and ferenc szidarovszky university of pecs, department of applied mathematics, pecs, ifjusag u. 6, h-7624, pecs,hungary szidarka@gmail.com abstract a complete eigenvalue analysis is given for a certain class of dynamic systems with a single delay. the stability region is determined and it is demonstrated that there is only one stability switch. special cases from economics, biology and engineering illustrate the importance of such models. resumen un análisis completo de los autovalores se entrega para una clase de sistemas dinámicos con retardo simple. la región de estabilidad se determina y se demuestra que existe solamente un switch de estabilidad. casos especiales para economı́a, bioloǵıa e ingenieŕıa ilustran la importancia de los modelos mencionados. keywords and phrases: dynamic systems, time delay, stability. 2010 ams mathematics subject classification: 34k20, 37c75. 2 akio matsumoto & ferenc szidarovszky cubo 15, 3 (2013) 1 introduction in examining economic and engineering systems we often face with delayed data and delayed responses. in the case of fixed delays the system is described by a difference-differential equation and in the case of continuously distributed delay the model is a volterra-type integro-differential equation. delay models have many applications in engineering, biology and economics to name only the most important fields (hale (1979); cushing (1977); invernizzi and medio (1991)). without time delay the governing dynamic model is a system of ordinary differential equations, the asymptotical behavior of the solution trajectories can be examined by well established methods such as the usage of lyapunov functions and local linearization. if the system is linear, then local asymptotical stability implies global stability, and the spectrum is finite making the analytic investigation relatively simple. in the case of continuously distributed delay with gamma-density weighting functions, the spectrum remains finite, however for fixed delays the spectrum is usually infinite. in the case of linear systems with fixed delays the characteristic equation is an exponentialpolynomial equation. there is a large literature on delayed equations with one delay (see for example, hayes (1950) and burger (1956) for the earliest studies), however only very few studies are devoted to multiple delays (see for example, hale and huang (1993) and piotrowska (2007)). in this paper we will consider a special case of nonlinear dynamics with one delay and will present an elementary analysis of its spectrum which can lead to a complete understanding of its local asymptotical behavior. 2 practical examples consider first a monopoly where one firm produces a product and sells it to a homogeneous market. let x be the production output of the firm and p(x) = a − bx (a, b > 0) the price function. if the firm determines its production level based on a delayed price information, then it is a − bx(t − τ), where τ is the delay. assuming gradient adjustment process in the dynamics, in the absence of time delay the governing dynamic equation would be as follows: ẋ(t) = α(x(t))(a − c − 2bx(t)), (1) since the profit is given as ϕ = x(a − bx) − cx (2) where c is the firm’s marginal cost. in the presence of delay, equation (1) has to be modified as ẋ(t) = αx(t)(a − c − 2bx(t − τ)) (3) where we assume that α(x) = αx with a positive coefficient α. the only positive steady state of the system is x̄ = a − c 2b . cubo 15, 3 (2013) an elementary study of a class of dynamic systems . . . 3 in order to guarantee that this output level is positive, we have to assume that a > c. linearizing equation (3) around x̄ and introducing the new variable z = x − x̄ a single delay ode is obtained: ż(t) = −γz(t − τ) (4) where γ = α(a − c) > 0. consider next an electrical system with state feedback, where the feedback is delayed. assume the systems equation is linear: ẋ(t) = ax (t) +bu (t) (5) where x is the state and u is the input. let k be the feedback matrix and τ the delay. then the delayed feedback system can be written as ẋ(t) = ax (t) +b (u(t) +kx (t − τ)). (6) in the single-dimensonal case this equation reduces to the following: ẋ(t) = ax(t) + bkx(t − τ) + bu(t) in the special case of constant input, u(t) ≡ u0, and a = 0 introduce the new variable z = x+u0/k to have ż(t) = bkz(t − τ), (7) which has the same form as equation (4) with γ = −bk. models in population dynamics are often delayed equations, when reproduction is not instantenuous. assuming exponential growth rate, the model can be written as ẋ(t) = rx(t − τ) (8) where r is the reproduction rate and τ is the delay. notice that this equation also has the form as (4) with r = −γ. 3 spectrum analysis as usual, we look for the solution in the exponential form z(t) = eλtv, and substitute it into equation (4) to get λ + γe−λτ = 0. multiplying by τ and introducing the new variables ∆ = λτ and a = γτ, this equation is simplified as ∆ + ae−∆ = 0. (9) assume that ∆ = α + iβ is a complex root. then α + iβ + ae−α(cos β − i sin β) = 0. 4 akio matsumoto & ferenc szidarovszky cubo 15, 3 (2013) equating the real and imaginary parts with zero, α + ae−α cos β = 0 (10) and β − ae−α sin β = 0. (11) from (11), e−α = β a sin β , (12) if sin β 6= 0. if sin β = 0, then from (9), ∆ is real and therefore is the solution of the real equation ae−∆ = −∆. depending on the value of a, there is either no solution, or 1 or 2 negative solutions. so the real solutions (if exist) are negative. assume now that sin β 6= 0, then from (10) and (12), α + a β a sin β cos β = 0 showing that α = −β cot β. (13) without losing generality we may assume that β > 0, since if ∆ is a solution of equation (9), then its complex conjugate is also a solution. from (13) we see that the real part of ∆ is negative if and only if β ∈ (nπ, π 2 + nπ), n = 0, 1, 2, ... substituting (13) into relation (11), we get a single-variable equation for β : 1 a β = eβ cot β sin β. (14) let f(β) denote the right hand side of this equation. we will next examine the shape of the graph of this function. clearly lim β→0 f(β) = 0 and for n ≥ 1, lim β→nπ−0 f(β) = 0, since β cot β converges to −∞ as β tends to nπ from the left. similarly lim β→nπ+0 f(β) =    ∞ if n is even −∞ if n is odd since the value of k = β cot β tends to ∞ as β tends to nπ from the right and f(β) = ek 1 k β cos β. cubo 15, 3 (2013) an elementary study of a class of dynamic systems . . . 5 in addition, lim β→ π 2 +nπ f(β) =    1 if n is even −1 if n is odd, since k → 0 here. simple differentiation shows that f′(β) = cos βeβ cos β sin β + sin βeβ cos β sin β (cos β − β sin β) sin β − β cos2 β sin2 β = 1 sin β e β cos β sin β [sin 2β − β] (15) notice that there is a unique β∗ ∈ (0, π/2) such that sin 2β∗ = β∗, and for β < β∗, sin 2β > β and for β > β∗, sin 2β < β. therefore f′(β) > 0 if and only if either β ∈ (0, β∗) or β ∈ ((2k − 1)π, 2kπ), k = 1, 2, ... and f′(β) < 0 if and only if either β ∈ (β∗, π) or β ∈ (2kπ, (2k + 1)π), k = 1, 2, ... the graph of f(β) is shown in figure 1. the value of β is the intersection of this graph with the linear function β/a. assume first that a > π/2. then function β/a crosses the β = π/2 vertical line under one, so there is a root between π/2 and π. here the value of α is positive making the system unstable regardless of the other solutions. if a < π/2, then there is no intersection between π/2 and π, however depending on the value of a there is the possibility of solution between 0 and π/2, where α is negative. figure 1. shape of the graph of f(β) 6 akio matsumoto & ferenc szidarovszky cubo 15, 3 (2013) notice that in this case line β/a crosses the β = π/2 vertical line above one, so the other intersections with the graph of f(β) are in intervals (2kπ, π 2 +2kπ), k = 1, 2, ..., where the corresponding α value is negative. hence we have the following result. proposition 3.1. assume a > 0. then the system (9) is asymptotically stable if a < π 2 , and unstable if a > π 2 . consider next the case of a < 0 in equation (9). if sin β = 0, then equation (11) implies that β = 0, so ∆ is real, and solves equation ∆ = −ae−∆. (16) this equation always has a positive solution, so the system is always unstable. 4 stability switches assume again that a > 0. stability switches are usually examined by looking for pure complex eigenvalues ∆ = iβ (β > 0 since complex conjugate is also a solution). with α = 0, equations (10) and (11) are reduced to the more simple equations a cos β = 0 (17) and β − a sin β = 0 (18) from (17), β = π 2 + nπ, however from (18), sinβ has to be positive, so β = π 2 + 2nπ (n = 0, 1, 2, ...) are the stability switches with the corresponding values of a = β. in order to detect the direction of the stability switches we consider ∆ as the function of the bifurcation parameter a. implicitly differentiating equation (9) with respect to a gives ∆′ + e−∆ − ae−∆∆′ = 0 showing that ∆′ = e−∆ ae−∆ − 1 = ∆ a∆ + a = iβ iβa + a = iβ(a − iβa) a2β2 + a2 with real part β2/(aβ2+a) > 0 showing that the real part of the eigenvalue changes from negative to positive. notice that a = π/2 is the only stability switch, since for all a = π 2 + 2nπ (n ≥ 1) there is an eigenvalue with positive real part with β ∈ (π/2, π) so regardless what happens with the other eigenvalues the system is unstable anyway. at a = π 2 hopf bifurcation occurs giving the possibility of the birth of limit cycles. cubo 15, 3 (2013) an elementary study of a class of dynamic systems . . . 7 5 conclusions a special class of dynamic systems was examined where a single delay was present. based on elementary analysis the spectrum of the system was completely described and the stability region characterized. we proved that the system is asymptotically stable if a < π 2 and unstable if a > π 2 . if a = π 2 , then hopf bifurcation occurs. in the model of a monopolistic firm a = λτ, where τ is the delay and γ is the product of the marginal speed of adjustment and the difference of the maximum price and marginal cost. since both γ and τ are positive, the stability region is the domain between the positive branch of the hyperbola τ = π 2γ and the γ = 0 positive horizontal axis. similar interpretation can be given for the delay electrical systems and for the delay population dynamic models in the paper. we also demonstrated that there are infinitely many values of a which correspond to pure complex eigenvalues, however the smallest such value is the only stability switch. the case of multiple delays is much more complicated. it will be the subject of our future study. received: july 2011. accepted: october 2012. references [1] burger, e. (1956), on the stability of certain economic systems. econometrica, 24(4), 488-493. [2] cushing, j. m. (1977), integro-differential equations and delay models in population dynamics. springer-verlag, berlin/heidelberg/new york. [3] hale, j. (1979), nonlinear oscillations in equations with delays. in nonlinear oscillations in biology (k. c. hoppenstadt, ed.), lectures in applied mathematics, 17, 157-185. [4] hale, j. and w. huang (1993), global geometry of the stable regions for two delay differential equations. j. of math. analysis and appl., 178, 344-362. [5] hayes, n. d. (1950), roots of the transcendental equation associated with a certain difference-differential equation. j. of the london math. society, 25, 226-232. [6] invernizzi, s. and a. medio (1991), on lags and chaos in economic dynamic models. journal of math. econ., 20, 521-550. [7] piotrowska. m. (2007), a remark on the ode with two discrete delays. journal of math. analysis and appl., 329, 664-676. cubo a mathematical journal vol.14, no¯ 03, (167–190). october 2012 some generalized difference double sequence spaces defined by a sequence of orlicz-functions kuldip raj and sunil k. sharma school of mathematics, shri mata vaishno devi university, katra-182320, j&k, india email: kuldeepraj68@rediffmail.com, sunilksharma42@yahoo.co.in abstract in the present paper we introduce some generalized difference double sequence spaces defined by a sequence of orlicz-functions. we study some topological properties and some inclusion relations between these spaces. we also make an effort to study these properties over n-normed spaces. resumen en este art́ıculo introducimos algunos espacios de sucesiones doble-diferencia generalizadas definidas por una sucesión de funciones de orlicz. estudiamos algunas propiedades topológicas y algunas relaciones de inclusión entre estos espacios. además, hacemos un esfuerzo para estudiar estas propiedades en espacios n-normados. keywords and phrases: p-convergent, orlicz function, sequence spaces, paranorm space, nnormed space 2010 ams mathematics subject classification: primary 42b15; secondary 40c05 168 k. raj and s. k. sharma cubo 14, 3 (2012) 1 introduction and preliminaries the initial works on double sequences is found in bromwich [4]. later on, it was studied by hardy [6], moricz [17], moricz and rhoades [18], tripathy ([33], [34]), basarir and sonalcan [2] and many others. hardy[6] introduced the notion of regular convergence for double sequences. quite recently, zeltser [36] in her ph.d thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. mursaleen and edely [21] have recently introduced the statistical convergence and cauchy convergence for double sequences and given the relation between statistical convergent and strongly cesaro summable double sequences. nextly, mursaleen [19] and mursaleen and edely [22] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the m-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = (xmn) into one whose core is a subset of the m-core of x. more recently, altay and basar [1] have defined the spaces bs, bs(t), csp, csbp, csr and bv of double sequences consisting of all double series whose sequence of partial sums are in the spaces mu, mu(t), cp, cbp, cr and lu, respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces bs, bv, csbp and the β(v)-duals of the spaces csbp and csr of double series. now, recently basar and sever [3] have introduced the banach space lq of double sequences corresponding to the well known space ℓq of single sequences and examined some properties of the space lq. let w 2 denote the set of all double sequences of complex numbers. by the convergence of a double sequence we mean the convergence of the pringsheim sense i.e. a double sequence x = (xkl) has pringsheim limit l (denoted by p − lim x = l) provided that given ǫ > 0 there exists n ∈ n such that |xkl − l| < ǫ whenever k, l > n see [26]. we shall write more briefly as p-convergent. we shall denote the space of all p-convergent sequences by c2. the double sequence x = (xkl) is bounded if there exists a positive number m such that |xkl| < m for all k and l. let l2 ∞ the space of all bounded double sequence such that ||xkl||∞,2 = supkl |xkl| < ∞. for more details about double sequence spaces see ([30], [31],[32]) and references therein. the notion of difference sequence spaces was introduced by kızmaz [13], who studied the difference sequence spaces l∞(∆), c(∆) and c0(∆). the notion was further generalized by et. and çolak [5] by introducing the spaces l∞(∆ n), c(∆n) and c0(∆ n). let w be the space of all complex or real sequences x = (xk) and let m, s be non-negative integers, then for z = l∞, c, c0 we have sequence spaces z(∆ms ) = {x = (xk) ∈ w : (∆ m s xk) ∈ z}, where ∆ms x = (∆ m s xk) = (∆ m−1 s xk − ∆ m−1 s xk+1) and ∆ 0 sxk = xk for all k ∈ n, which is equivalent to the following binomial representation ∆ms xk = m∑ v=0 (−1)v ( m v ) xk+sv. taking s = 1, we get the spaces which were studied by et and çolak [5]. taking m = s = 1, we get the spaces which were introduced and studied by kızmaz [13]. cubo 14, 3 (2012) some generalized difference double sequence spaces ... 169 an orlicz function m : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex function such that m(0) = 0, m(x) > 0 for x > 0 and m(x) −→ ∞ as x −→ ∞. lindenstrauss and tzafriri [14] used the idea of orlicz function to define the following sequence space: lm = { x ∈ w : ∞∑ k=1 m ( |xk| ρ ) < ∞ } which is called as an orlicz sequence space. also lm is a banach space with the norm ||x|| = inf { ρ > 0 : ∞∑ k=1 m ( |xk| ρ ) ≤ 1 } . also, it was shown in [14] that every orlicz sequence space lm contains a subspace isomorphic to lp(p ≥ 1). the ∆2− condition is equivalent to m(lx) ≤ lm(x), for all l with 0 < l < 1. an orlicz function m can always be represented in the following integral form m(x) = ∫x 0 η(t)dt where η is known as the kernel of m, is right differentiable for t ≥ 0, η(0) = 0, η(t) > 0, η is non-decreasing and η(t) → ∞ as t → ∞. let x be a linear metric space. a function p : x → r is called paranorm, if (1) p(x) ≥ 0, for all x ∈ x, (2) p(−x) = p(x), for all x ∈ x, (3) p(x + y) ≤ p(x) + p(y), for all x, y ∈ x, (4) if (λn) is a sequence of scalars with λn → λ as n → ∞ and (xn) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λnxn − λx) → 0 as n → ∞. a paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (x, p) is called a total paranormed space. it is well known that the metric of any linear metric space is given by some total paranorm (see [35], theorem 10.4.2, p-183). for more details about sequence spaces see ([12], [15], [20], [23], [24], [25], [27]) and references therein. let m = (mk,l) be a sequence of orlicz functions,p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. let x be a seminormed space over the complex field c with the seminorm q. now we define the following classes of sequences in the present paper: c2(∆mn , m, u, p, q, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l − l ρ ))]pk,l = 0, for some ρ > 0, l and s ≥ 0 } , 170 k. raj and s. k. sharma cubo 14, 3 (2012) c20(∆ m n , m, u, p, q, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , m, u, p, q, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . if we take m(x) = x, we get c2(∆mn , u, p, q, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ q ( ∆mn xk,l − l ρ )]pk,l = 0, for some ρ > 0, l and s ≥ 0 } , c20(∆ m n , u, p, q, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ q ( ∆mn xk,l ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , u, p, q, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ q ( ∆mn xk,l ρ )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . if we take p = (pk,l) = 1, we get c2(∆mn , m, u, q, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l − l ρ ))] = 0, for some ρ > 0, l and s ≥ 0 } , c20(∆ m n , m, u, q, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))] = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , m, u, q, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))] < ∞, for some ρ > 0 and s ≥ 0 } . if we take m = n = 0 and q(x) = |x|, then we get new double sequence spaces as follows : c2(m, u, p, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( |xk,l − l| ρ )]pk,l = 0, cubo 14, 3 (2012) some generalized difference double sequence spaces ... 171 for some ρ > 0, l and s ≥ 0 } , c20(m, u, p, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( |xk,l| ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (m, u, p, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ mk,l ( |xk,l| ρ )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . if we take m = n = 1 and q(x) = |x|, then we get new double sequence spaces as follows : c2(∆mn , m, u, p, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( |∆xk,l − l| ρ )]pk,l = 0, for some ρ > 0, l and s ≥ 0 } , c20(∆ m n , m, u, p, s) = { x = (xk,l) ∈ w 2 : p − lim k,l (kl)−suk,l [ mk,l ( |∆xk,l| ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , m, u, p, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ mk,l ( |∆mn xk,l| ρ )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . the following inequality will be used throughout the paper. let p = (pk,l) be a double sequence of positive real numbers with 0 < pk,l ≤ sup k,l = h and let k = max{1, 2h−1}. then for the factorable sequences {ak,l} and {bk,l} in the complex plane, we have |ak,l + bk,l| pk,l ≤ k(|ak,l| pk,l + |bk,l| pk,l). (1.1) the main goal of this paper is to extend a few known results in the literature from single difference sequence spaces to double difference sequence spaces. we also make an effort to study some topological properties and inclusion relations between above defined sequence spaces. 172 k. raj and s. k. sharma cubo 14, 3 (2012) 2 main results theorem 2.1 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers, then the classes of sequences c20(∆ m n , m, u, p, q, s), c 2(∆mn , m, u, p, q, s) and l∞(∆ m n , m, u, p, q, s) are linear spaces over the field of complex numbers c. proof. let x = (xk,l), y = (yk,l) ∈ c 2 0(∆ m n , m, u, p, q, s) and α, β ∈ c. then there exist positive numbers ρ1 and ρ2 such that lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ1 ))]pk,l = 0, for some ρ1 > 0 and lim k,l (kl)−suk,l [ mk,l ( q ( ∆nmyk,l ρ2 ))]pk,l = 0, for some ρ2 > 0. let ρ3 = max(2|α|ρ1, 2|β|ρ2). since m = (mk,l) is non-decreasing convex function and so by using inequality (1.1), we have lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn (αxk,l + βyk,l) ρ3 ))]pk,l = lim k,l (kl)−suk,l [ mk,l ( q ( α∆mn xk,l ρ3 ) + q ( β∆mn yk,l ρ3 ))]pk,l ≤ k lim k,l 1 2pk,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ1 ))]pk,l + k lim k,l 1 2pk,l (kl)−suk,l [ mk,l ( q ( ∆mn yk,l ρ2 ))]pk,l ≤ k lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ1 ))]pk,l + k lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn yk,l ρ2 ))]pk,l = 0. so, αx + βy ∈ c20(∆ n m, m, u, p, q, s). hence c 2 0(∆ n m, m, u, p, q, s) is a linear space. similarly, we can prove that c2(∆nm, m, u, p, q, s) and l 2 ∞ (∆nm, m, u, p, q, s) are linear spaces. theorem 2.2 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. for z2 = l2 ∞ , c2 and c20, the spaces z 2(∆mn , m, u, p, q, s) are paranormed spaces, paranormed by g(x) = nm∑ k,l=1 q(xk,l) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q ( ∆mn xk,l ρ )) ≤ 1 } where h = max(1, sup k,l pk,l). cubo 14, 3 (2012) some generalized difference double sequence spaces ... 173 proof. clearly g(−x) = g(x), g(0) = 0. let (xk,l) and (yk,l) be any two sequences belong to any one of the spaces z2(∆nm, m, u, p, q, s), for z 2 = c20, c 2 and l2 ∞ . then, we get ρ1, ρ2 > 0 such that sup k,l (kl)−suk,lmk,l ( q ( ∆mn xk,l ρ1 )) ≤ 1 and sup k,l (kl)−suk,lmk,l ( q ( ∆mn yk,l) ρ2 ) ≤ 1. let ρ = ρ1 + ρ2. then by convexity of m = (mk,l), we have sup k,l (kl)−suk,lmk,l ( q ( ∆mn (xk,l + yk,l) ρ )) ≤ ( ρ1 ρ1 + ρ2 ) sup k,l (kl)−suk,lmk,l ( q ( ∆mn xk,l ρ1 )) + ( ρ2 ρ1 + ρ2 ) sup k,l (kl)−suk,lmk,l ( q ( ∆mn yk,l ρ2 )) ≤ 1. hence we have, g(x + y) = mn∑ k,l=1 q(xk,l + yk,l) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q ( ∆mn (xk,l + yk,l) ρ )) ≤ 1 } ≤ mn∑ k,l=1 q(xk,l) + inf { ρ pk,l h 1 : sup k,l (kl)−suk,lmk,l ( q ( ∆mn xk,l ρ1 )) ≤ 1 } + mn∑ k,l=1 q(yk,l) + inf { ρ pk,l h 2 : sup k,l (kl)−suk,lmk,l ( q ( ∆mn yk,l ρ2 )) ≤ 1 } . this implies that g(x + y) ≤ g(x) + g(y). the continuity of the scalar multiplication follows from the following inequality g(µx) = mn∑ k,l=1 q(µxk,l) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q ( ∆mn µxk,l ρ )) ≤ 1 } = |µ| mn∑ k,l=1 q(xk,l) + inf { (t|µ|) pk,l h : sup k,l (kl)−suk,lmk,l ( q ( ∆mn xk,l t )) ≤ 1 } , where t = ρ |µ| . hence the space z2(∆nm, m, u, p, q, s), for z 2 = c20, c 2 and l2 ∞ is a paranormed space, paranormed by g. theorem 2.3 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. 174 k. raj and s. k. sharma cubo 14, 3 (2012) for z2 = l2 ∞ , c2 and c20, the spaces z 2(∆mn , m, u, p, q, s) are complete paranormed spaces, paranormed by g(x) = nm∑ k,l=1 q(xk,l) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q ( ∆mn xk,l ρ )) ≤ 1 } , where h = max(1, sup k,l pk,l). proof. we prove the result for the space l2 ∞ (∆mn , m, u, p, q, s). let (x i k,l) be any cauchy sequence in l2 ∞ (∆mn , m, u, p, q, s). let ǫ > 0 be given and for t > 0, choose x0 be fixed such that uk,lmk,l ( tx0 2 ) ≥ 1, then there exists a positive integer n0 ∈ n such that g(x i k,l − x j k,l ) < ǫ x0t , for all i, j ≥ n0. using the definition of paranorm, we get mn∑ k,l=1 q(xik,l − x j k,l ) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q (∆mn (x i k,l − x j k,l ) ρ ))} < ǫ x0t , for all i, j ≥ n0 (2.1). hence we have, mn∑ k,l=1 q(xik,l − x j k,l) < ǫ, for all i, j ≥ n0. this implies that q(xik,l − x j k,l ) < ǫ, for all i, j ≥ n0 and 1 ≤ k ≤ nm. thus (xik,l) is a cauchy sequence in c for k, l = 1, 2, ...., nm. hence (x i k,l) is convergent in c for k, l = 1, 2, ...., nm. let lim i→∞ xik,l = xk,l, say for k, l = 1, 2, ...., nm. (2.2) again from equation (2.1) we have, inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q (∆mn (x i k,l − x j k,l ) ρ )) ≤ 1 } < ǫ, for all i, j ≥ n0. hence we get sup k,l (kl)−suk,lmk,l ( q (∆mn (x i k,l − x j k,l ) g(xi − xj) )) ≤ 1, for all i, j ≥ n0. it follows that (kl)−suk,lmk,l ( q ( ∆ m n (x i k,l−x j k,l ) g(xi−xj) )) ≤ 1, for each k, l ≥ 1 and for all i, j ≥ n0. for t > 0 with (kl)−suk,lmk,l( tx0 2 ) ≥ 1, we have (kl)−suk,lmk,l ( q (∆mn (x i k,l − x j k,l ) g(xi − xj) )) ≤ (kl)−suk,lmk,l( tx0 2 ). this implies that q(∆mn x i k,l − ∆ m n x j k,l ) < tx0 2 ǫ tx0 = ǫ 2 . cubo 14, 3 (2012) some generalized difference double sequence spaces ... 175 hence q(∆mn x i k,l) is a cauchy sequence in c for all k, l ∈ n. this implies that q(∆ m n x i k,l) is convergent in c for all k, l ∈ n. let lim i→∞ q(∆mn x i k,l) = yk,l for each k, l ∈ n. let k, l = 1, then we have lim i→∞ q(∆mn x i 1,1) = lim i→∞ m∑ v=0 (−1)v ( m v ) xi1+nv,1+mv = y1,1. (2.3) we have by equation (2.2) and equation (2.3) lim i→∞ ximn+1 = xmn+1, exists. proceeding in this way inductively, we have lim i→∞ xik,l = xk,l exists for each k, l ∈ n. now we have for all i, j ≥ n0, mn∑ k,l=1 q(xik,l − x j k,l ) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q (∆mn (x i k,l − x j k,l ) ρ )) ≤ 1 } < ǫ. this implies that lim j→∞ { mn∑ k,l=1 q(xik,l − x j k,l ) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q (∆mn (x i k,l − x j k,l) ρ )) ≤ 1 }} < ǫ, for all i ≥ n0. using the continuity of mk,l, we have mn∑ k,l=1 q(xik,l − xk,l) + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( q ( ∆mn x i k,l − ∆ m n xk,l ρ )) ≤ 1 } < ǫ, for all i ≥ n0. it follows that (x i −x) ∈ l2 ∞ (∆mn , m, u, p, q, s). since x i ∈ l2 ∞ (∆nm, m, u, p, q, s) and l2 ∞ (∆mn , m, u, p, q, s) is a linear space, so we have x = x i − (xi − x) ∈ l2 ∞ (∆nm, m, u, p, q, s). this completes the proof. similarly, we can prove that c2(∆nm, m, u, p, q, s) and c 2 0(∆ n m, m, u, p, q, s) are complete paranormed spaces in view of the above proof. theorem 2.4 let m ≥ 1, then for all 0 < i ≤ m, z2(∆in, m, u, p, q, s) ⊂ z 2(∆mn , m, u, p, q, s), where z2 = c2, c20 and l 2 ∞ . proof. we will prove it for only c20(∆ m−1 n , m, u, p, q, s). let x = (xk,l) ∈ c 2 0(∆ m−1 n , m, u, p, q, s). then p − lim k,l (kl)−suk,l [ mk,l ( q(∆m−1n xk,l) ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 (2.4) then from (2.4) we have p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))]pk,l+1 = 0, p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))]pk+1,l = 0 and p − lim k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))]pk+1,l+1 = 0. 176 k. raj and s. k. sharma cubo 14, 3 (2012) now for ∆mn x = (∆ m n xk,l) = (∆ m−1 n xk,l − ∆ m−1 n xk,l+1 − ∆ m−1 n xk+1,l + ∆ m−1 n xk+1,l+1), we have (kl)−suk,l [ mk,l ( q ( ∆ m n xk,l ρ ))]pk,l ≤ (kl)−suk,l [ mk,l ( q ( ∆m−1n xk,l ρ ) + q ( ∆m−1n xk,l+1 ρ ) + q ( ∆m−1n xk+1,l ρ ) + q ( ∆m−1n xk+1,l+1 ρ ))]pk,l ≤ k2(kl)−suk,l {[ mk,l ( q ( ∆m−1n xk,l ρ ))]pk,l + uk,l [ mk,l ( q ( ∆m−1n xk+1,l ρ ))]pk,l + uk,l [ m ( q ( ∆m−1n xk,l+1 ρ ))]pk,l + uk,l [ mk,l ( q ( ∆m−1n xk+1,l+1 ρ ))]pk,l } ≤ k2 {[ (kl)−suk,lmk,l ( q ( ∆m−1n xk,l ρ ))]pk,l + [ (kl)−suk,lmk,l ( q ( ∆m−1n xk+1,l ρ ))]pk+1,l + [ (kl)−suk,lmk,l ( q ( ∆m−1n xk,l+1 ρ ))]pk,l+1 + uk,l [ (kl)−smk,l ( q ( ∆m−1n xk+1,l+1 ρ ))]pk+1,l+1 } from this it follows that x = (xk,l) ∈ c 2 0(∆ m n , m, u, p, q, s) and hence c 2 0(∆ m−1 n , m, u, p, q, s) ⊂ c20(∆ m n , m, u, p, q, s). on applying the principle of induction, it follows that c 2 0(∆ i n, m, u, p, q, s) ⊂ c20(∆ m n , m, u, p, q, s) for i = 0, 1, 2, · · · , m − 1. similarly, we can prove the other cases. theorem 2.5 (a) if 0 < inf k,l pk,l ≤ pk,l < 1, then z 2(∆mn , m, u, p, q, s) ⊂ z 2(∆mn , m, u, q, s), (b) if 1 < pk,l ≤ sup k,l pk,l < ∞, then z 2(∆mn , m, u, q, s) ⊂ z 2(∆mn , m, u, p, q, s), where z2 = c2, c20 and l 2 ∞ . proof. (i) let x = (xk,l) ∈ l 2 ∞ (∆mn , m, u, p, q, s). since 0 < inf pk,l ≤ 1, we have sup k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))] ≤ sup k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))]pk,l , and hence x = (xk,l) ∈ l 2 ∞ (∆mn , m, u, p, q, s). (ii) let pk,l for each (k, l) and sup k,l pk,l < ∞. let x = (xk,l) ∈ l 2 ∞ (∆mn , m, u, q, s). then, for each 0 < ǫ < 1, there exists a positive integer n such that sup k,l (kl)−suk,l [ mk,l ( q ( ∆mn xk,l ρ ))] ≤ ǫ < 1, for all m, n ∈ n. this implies that sup k,l (kl)−suk,l [ mk,l ( q (∆mn xk,l ρ ))]pk,l ≤ sup k,l (kl)−suk,l [ mk,l ( q (∆mn xk,l ρ ))] . cubo 14, 3 (2012) some generalized difference double sequence spaces ... 177 thus x = (xk,l) ∈ l 2 ∞ (∆mn , m, u, p, q, s) and this completes the proof. theorem 2.6 let m′ = (m′k,l) and m ′′ = (m′′k,l) be two sequences of orlicz functions satisfying ∆2-condition. if β = lim t→∞ m′′k,l(t) t ≥ 1, then z2(∆mn , m ′, u, p, q, s) = z2(∆mn , m ′′◦m′, u, p, q, s), where z2 = c2, c20 and l 2 ∞ . proof. we prove it for z2 = c2 and the other cases will follows on applying similar techniques. let x = (xk,l) ∈ c 2(∆mn , m ′, u, p, q, s), then p − lim k,l (kl)−s [ m′k,l ( q ( ∆mn xk,l − l ρ ))]pk,l = 0. let 0 < ǫ < 1 and δ with 0 < δ < 1 such that m′′k,l(t) < ǫ for 0 ≤ t < δ. let yk,l = m ′ k,l ( q ( ∆mn xk,l − l ρ )) and consider [m′′k,l(yk,l)] pk,l = [m′′k,l(yk,l)] pk,l + [m′′k,l(yk,l)] pk,l (2.5) where the first term is over yk,l ≤ δ and the second is over yk,l > δ. from the first term in (2.5), we have (kl)−s[m′′k,l(yk,l)] pk,l < (kl)−s[m′′k,l(2)] h [(yk,l)] pk,l (2.6) on the other hand, we use the fact that yk,l < yk,l δ < 1 + yk,l δ . since (m′′k,l) for each k, l is non-decreasing and convex, it follows that m′′k,l(yk,l) < m ′′ k,l ( 1 + yk,l δ ) < 1 2 m′′k,l(2) + 1 2 m′′k,l( 2yk,l δ ). since (m′′k,l) for each k, l satisfies ∆2-condition, we have m′′k,l(yk,l) < 1 2 k yk,l δ m′′k,l(2) + 1 2 k yk,l δ m′′k,l(2) = k yk,l δ m′′k,l(2). hence, from the second term in (2.5), it follows that (kl)−s[m′′(yk,l)] pk,l ≤ max ( 1, (km′′(2)δ−1)h ) (kl)−s[(yk,l)] pk,l (2.7) by the inequalities (2.6) and (2.7), taking limit in the pringsheim sense, we have x = (xk,l) ∈ c2(∆mn , m ′′ ◦ m′, u, p, q, s). observe that in this part of the proof we did not need β ≥ 1. now, let β ≥ 1 and x = (xk,l) ∈ c 2(m′, ∆rn, u, q, p). then, we have m ′′ k,l(t) ≥ β(t) for all t ≥ 0. it follows that x = (xk,l) ∈ c 2(∆mn , m ′′ ◦ m′, u, p, q, s) implies x = (xk,l) ∈ c 2(∆mn , m ′, u, p, q, s). 178 k. raj and s. k. sharma cubo 14, 3 (2012) this implies c2(∆mn , m ′, u, p, q, s) = c2(∆mn , m ′′ ◦ m′, u, p, q, s). theorem 2.7 let m′ = (m′k,l) and m ′′ = (m′′k,l) be two sequences of orlicz functions, q, q1 and q2 be seminorms and s, s1 and s2 be positive real numbers. then (1) z2(∆mn , m ′, u, p, q, s) ∩ z2(∆mn , m ′′, u, p, q, s) ⊂ z2(∆mn , m ′ + m′′, u, p, q, s), (2) z2(∆mn , m, u, p, q1, s) ∩ z 2(∆mn , m, u, p, q2, s) ⊂ z 2(∆mn , m, u, p, q1 + q2, s), (3) if q1 is stronger than q2, then z 2(∆mn , m, u, p, q1, s) ⊂ z 2(∆mn , m, u, p, q2, s) (4) if s1 ≤ s2, then z 2(∆mn , m, u, p, q, s1) ⊂ z 2(∆mn , m, u, p, q, s2), where z2 = c2, c20 and l 2 ∞ . proof. (1) let x = (xk,l) ∈ c 2(∆mn , m ′, u, p, q, s) ∩ c2(∆m, m′′, u, p, q, s). then p − lim k,l (kl)−suk,l [ m′k,l ( q ( ∆mn xk,l − l ρ1 ))]pk,l = 0, for some ρ1 > 0, p − lim k,l (kl)−suk,l [ m′′k,l ( q ( ∆mn xk,l − l ρ2 ))]pk,l = 0, for some ρ2 > 0. let ρ = max(ρ1, ρ2). the result follows from the following inequality (kl)−s [ (m′ + m′′) ( q ( ∆ m n xk,l−l ρ ))]pk,l ≤ k { (kl)−suk,l [ m ′ ( q ( ∆mn xk,l − l ρ1 ))]pk,l + (kl)−suk,l [ m ′′ ( q ( ∆mn xk,l − l ρ2 ))]pk,l } . the proofs of (2), (3) and (4) follows by same pattern. theorem 2.8 for any sequence of orlicz functions, if q1 ≡ (equivalent to) q2, then z2(∆mn , m, u, p, q1, s) = z 2(∆mn , m, u, p, q2, s), where z 2 = c2, c20 and l 2 ∞ . proof.it is easy to prove so we omit the details. 3 some generalized difference double sequence spaces over n-normed spaces the concept of 2-normed spaces was initially developed by gähler[8] in the mid of 1960’s, while that of n-normed spaces one can see in misiak[16]. since then, many others have studied this concept and obtained various results, see gunawan ([9],[10]), gunawan and mashadi [11] and many others. cubo 14, 3 (2012) some generalized difference double sequence spaces ... 179 let n ∈ n and x be a linear space over the field k, where k is the field of real or complex numbers of dimension d, where d ≥ n ≥ 2. a real valued function ||·, · · · , ·|| on xn satisfying the following four conditions: (1) ||x1, x2, · · · , xn|| = 0 if and only if x1, x2, · · · , xn are linearly dependent in x; (2) ||x1, x2, · · · , xn|| is invariant under permutation; (3) ||αx1, x2, · · · , xn|| = |α| ||x1, x2, · · · , xn|| for any α ∈ k, and (4) ||x + x′, x2, · · · , xn|| ≤ ||x, x2, · · · , xn|| + ||x ′, x2, · · · , xn|| is called an n-norm on x and the pair (x, ||·, · · · , ·||) is called a n-normed space over the field k. for example, we may take x = rn being equipped with the n-norm ||x1, x2, · · · , xn||e = the volume of the n-dimensional parallelopiped spanned by the vectors x1, x2, · · · , xn which may be given explicitly by the formula ||x1, x2, · · · , xn||e = | det(xij)|, where xi = (xi1, xi2, · · · , xin) ∈ r n for each i = 1, 2, · · · , n and ||.||e denotes the euclidean norm. let (x, ||·, · · · , ·||) be an n-normed space of dimension d ≥ n ≥ 2 and {a1, a2, · · · , an} be linearly independent set in x. then the following function ||·, · · · , ·||∞ on x n−1 defined by ||x1, x2, · · · , xn−1||∞ = max{||x1, x2, · · · , xn−1, ai|| : i = 1, 2, · · · , n} defines an (n − 1)-norm on x with respect to {a1, a2, · · · , an}. a sequence (xk) in a n-normed space (x, ||·, · · · , ·||) is said to converge to some l ∈ x if lim k→∞ ||xk − l, z1, · · · , zn−1|| = 0 for every z1, · · · , zn−1 ∈ x. a sequence (xk) in a n-normed space (x, ||·, · · · , ·||) is said to be cauchy if lim k,p→∞ ||xk − xp, z1, · · · , zn−1|| = 0 for every z1, · · · , zn−1 ∈ x. if every cauchy sequence in x converges to some l ∈ x, then x is said to be complete with respect to the n-norm. any complete n-normed space is said to be n-banach space. for more details about sequence spaces see ([28], [29]) and references therein. let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of positive reals such that uk,l 6= 0 for all k, then we define the following sequences spaces in the present paper: c20(m, ∆ m n , p, u, s, ||·, · · · , ·||) = { x = (xk,l) ∈ w 2 : lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l = 0, 180 k. raj and s. k. sharma cubo 14, 3 (2012) for some ρ > 0 and s ≥ 0 } , c2(m, ∆mn , p, u, s, ||·, · · · , ·||) = { x = (xk,l) ∈ w 2 : lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| )]pk,l = 0, for some ρ > 0, l and s ≥ 0 } , and l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) = { x = (xk,l) ∈ w 2 : sup k,l≥1 (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . in this section of the present paper we shall study the topological properties and some interesting inclusion relation between the spaces c2(m, ∆mn , p, u, s, ||·, · · · , ·||), c 2 0(m, ∆ m n , p, u, s, ||·, · · · , ·||) and l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). theorem 3.1 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers, then the spaces c20(m, ∆ m n , p, u, s, ||·, · · · , ·||), c 2(m, ∆mn , p, u, s, ||·, · · · , ·||) and l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) are linear spaces. proof. let x = (xk,l), y = (yk,l) ∈ c 2 0(m, ∆ m n , p, u, s, ||·, · · · , ·||) and α, β ∈ c. then there exist positive number ρ1 and ρ2 such that lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )]pk,l = 0, for some ρ1 > 0 and lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )]pk,l = 0, for some ρ2 > 0. let ρ3 = max(2|α|ρ1, 2|β|ρ2). since m = (mk,l) is non-decreasing convex function and so by using inequality (1.1), we have cubo 14, 3 (2012) some generalized difference double sequence spaces ... 181 lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn (αxk,l + βyk,l)| ρ3 , z1, · · · , zn−1|| )]pk,l = lim k,l→∞ (kl)−suk,l [ mk,l ( || α∆mn xk,l ρ3 , z1, · · · , zn−1|| + || β∆mn yk,l ρ3 , z1, · · · , zn−1|| )]pk,l ≤ k lim k,l→∞ 1 2pk,l (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )]pk,l + k lim k,l→∞ 1 2pk,l (kl)−suk,l [ mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )]pk,l ≤ k lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )]pk,l + k lim k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )]pk,l = 0. so, αx + βy ∈ c20(m, ∆ n m, p, u, s, ||·, · · · , ·||). hence c 2 0(m, ∆ n m, p, u, s, ||·, · · · , ·||) is a linear space. similarly, we can prove that c2(m, ∆mn , p, u, s, ||·, · · · , ·||) and l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) are linear spaces. theorem 3.2 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. for z2 = l2 ∞ , c2 and c2o, the spaces z 2(m, ∆mn , p, u, s, ||·, · · · , ·||) are paranormed spaces, paranormed by g(x) = nm∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ) ≤ 1 } where h = max(1, sup k,l pk,l). proof. clearly g(−x) = g(x), g(0) = 0. let (xk) and (yk) be any two sequences belong to any one of the spaces z2(m, ∆mn , p, u, s, ||·, · · · , ·||), for z 2 = c20, c 2 and l2 ∞ . then, we get ρ1, ρ2 > 0 such that sup k,l (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )] ≤ 1 and sup k,l (kl)−suk,l [ mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )] ≤ 1. let ρ = ρ1 + ρ2. then by convexity of m = (mk,l), we have sup k,l (kl)−suk,l [ mk,l ( || ∆mn (xk,l + yk,l) ρ , z1, · · · , zn−1|| )] ≤ ( ρ1 ρ1 + ρ2 ) sup k,l (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )] + ( ρ2 ρ1 + ρ2 ) sup k,l (kl)−suk,l [ mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )] ≤ 1. 182 k. raj and s. k. sharma cubo 14, 3 (2012) hence we have, g(x + y) = mn∑ k,l=1 ||(xk,l + yk,l), z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,l [ mk,l ( || ∆mn (xk,l + yk,l) ρ , z1, · · · , zn−1|| )] ≤ 1 } ≤ mn∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { ρ pk,l h 1 : sup k,l (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )] ≤ 1 } + mn∑ k,l=1 ||yk,l, z1, · · · , zn−1|| + inf { ρ pk,l h 2 : sup k,l (kl)−suk,l [ mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )] ≤ 1 } . this implies that g(x + y) ≤ g(x) + g(y). the continuity of the scalar multiplication follows from the following inequality g(µx) = mn∑ k,l=1 ||µxk,l, z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,l [ mk,l ( || ∆mn µxk,l ρ , z1, · · · , zn−1|| )] ≤ 1 } = |µ| mn∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { (t|µ|) pk,l h : sup k,l (kl)−suk,l [ mk,l ( || ∆mn xk,l t , z1, · · · , zn−1|| )] ≤ 1 } , where t = ρ |µ| . hence the space z2(m, ∆mn , p, u, s, ||·, · · · , ·||), for z 2 = c20, c 2 and l2 ∞ is a paranormed space, paranormed by g. theorem 3.3 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. for z2 = l2 ∞ , c2 and c20, the spaces z 2(m, ∆mn , p, u, s, ||·, · · · , ·||) are complete paranormed spaces, paranormed by g(x) = nm∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ) ≤ 1 } , where h = max(1, sup k,l pk,l). proof. we prove the result for the space l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). let (x i) be any cauchy cubo 14, 3 (2012) some generalized difference double sequence spaces ... 183 sequence in l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). let ǫ > 0 be given and for t > 0, choose x0 be fixed such that uk,lmk,l ( tx0 2 ) ≥ 1, then there exists a positive integer n0 ∈ n such that g(x i k,l −x j k,l ) < ǫ x0t , for all i, j ≥ n0. using the definition of paranorm, we get mn∑ k,l=1 ||(xik,l−x j k,l ), z1, · · · , zn−1||+inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || ∆mn (x i k,l − x j k,l ) ρ , z1, · · · , zn−1|| )} < ǫ x0t , for all i, j ≥ n0 (3.1). hence we have, mn∑ k,l=1 ||(xik,l) − x j k,l ), z1, · · · , zn−1|| < ǫ, for all i, j ≥ no. this implies that ||(xik,l − x j k,l ), z1, · · · , zn−1|| < ǫ, for all i, j ≥ n0 and 1 ≤ k ≤ nm. thus (xik,l) is a cauchy sequence in c for k, l = 1, 2, ...., nm. hence (x i k,l) is convergent in c for k, l = 1, 2, ...., nm. let lim i→∞ xik,l = xk,l, say for k, l = 1, 2, ...., nm. (3.2) again from equation (3.1) we have, inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || ∆mn (x i k,l − x j k,l) ρ , z1, · · · , zn−1|| ) ≤ 1 } < ǫ, for all i, j ≥ n0. hence we get sup k,l (kl)−suk,lmk,l ( || ∆mn (x i k,l − x j k,l ) g(xi − xj) , z1, · · · , zn−1|| ) ≤ 1, for all i, j ≥ n0. it follows that (kl)−suk,lmk,l ( || ∆ m n (x i k,l−x j k,l ) g(xi−xj) , z1, · · · , zn−1|| ) ≤ 1, for each k, l ≥ 1 and for all i, j ≥ n0. for t > 0 with (kl)−suk,lmk,l( tx0 2 ) ≥ 1, we have (kl)−suk,lmk,l ( || ∆mn (x i k,l − x j k,l ) g(xi − xj) , z1, · · · , zn−1|| ) ≤ (kl)−suk,lmk,l( tx0 2 ). this implies that ||(∆mn x i k,l − ∆ m n x j k,l ), z1, · · · , zn−1|| < tx0 2 ǫ tx0 = ǫ 2 . hence (∆mn x i k,l) is a cauchy sequence in c for all k, l ∈ n. this implies that (∆ m n x i k,l) is convergent in c for all k, l ∈ n. let lim i→∞ ∆mn x i k,l = yk,l for each k, l ∈ n. let k, l = 1, then we have lim i→∞ ∆mn x i 1,1 = lim i→∞ n∑ v=0 (−1)v ( m v ) xi1+nv,mv+1 = y1. (3.3) 184 k. raj and s. k. sharma cubo 14, 3 (2012) we have by equation (3.2) and equation (3.3) lim i→∞ ximn+1 = xmn+1, exists. proceeding in this way inductively, we have lim i→∞ xik,l = xk,l exists for each k, l ∈ n. now we have for all i, j ≥ n0, mn∑ k,l=1 ||(xik,l − x j k,l ), z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || ∆mn (x i k,l − x j k,l ) ρ , z1, · · · , zn−1|| ) ≤ 1 } < ǫ. this implies that lim j→∞ { mn∑ k,l=1 ||(xik,l − x j k,l ), z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || ∆mn (x i k,l − x j k,l ) ρ , z1, · · · , zn−1|| ) ≤ 1 }} < ǫ, for all i ≥ n0. using the continuity of (mk,l), we have mn∑ k,l=1 ||(xik,l − xk,l), z1, · · · , zn−1|| + inf { ρ pk,l h : sup k,l (kl)−suk,lmk,l ( || (∆mn x i k,l − ∆ m n xk,l) ρ , z1, · · · , zn−1|| ) ≤ 1 } < ǫ, for all i ≥ n0. it follows that (x i − x) ∈ l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). since x i ∈ l2 ∞ (m, ∆nm, p, u, s, ||·, · · · , ·||) and l 2 ∞ (m, ∆nm, p, u, s, ||·, · · · , ·||) is a linear space, so we have x = xi − (xi − x) ∈ l2 ∞ (m, ∆nm, p, u, s, ||·, · · · , ·||). this completes the proof. similarly, we can prove that c2(m, ∆nm, p, u, ||·, · · · , ·||) and c 2 0(m, ∆ n m, p, u, ||·, · · · , ·||) are complete paranormed spaces in view of the above proof. theorem 3.4 if 0 < pk,l ≤ qk,l < ∞ for each k, l, then z 2(m, ∆mn , p, u, s, ||·, · · · , ·||) ⊆ z2(m, ∆mn , q, u, s, ||·, · · · , ·||), for z 2 = c20 and c 2. proof. let x = (xk,l) ∈ c 2(m, ∆mn , p, u, s, ||·, · · · , ·||). then there exists some ρ > 0 and l ∈ x such that lim k,l→∞ (kl)−suk,l ( mk,l ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| ))pk,l = 0. this implies that (kl)−suk,lmk,l ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| ) < ǫ (0 < ǫ < 1) for sufficiently large k, l. hence we get lim k,l→∞ (kl)−suk,l ( mk,l ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| ))qk,l ≤ lim k,l→∞ (kl)−suk,l ( mk,l ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| ))pk,l = 0. this implies that x = (xk,l) ∈ c 2(m, ∆mn , q, u, s, ||·, · · · , ·||). this completes the proof. similarly, we can prove for the case z2 = c20. cubo 14, 3 (2012) some generalized difference double sequence spaces ... 185 theorem 3.5 if m′ = (m′k,l) and m ′′ = (m′′k,l) be a sequence of orlicz functions. then (i) z2(m′, ∆mn , p, u, s, ||·, · · · , ·||) ⊆ z 2(m′′ ◦ m′, ∆mn , p, u, s, ||·, · · · , ·||), (ii) z2(m′, ∆mn , p, u, s, ||·, · · · , ·||) ∩ z 2(m′′, ∆mn , p, u, s, ||·, · · · , ·||) ⊆ z2(m′ + m′′, ∆mn , p, u, s, ||·, · · · , ·||), for z2 = l2 ∞ , c2 and c20. proof. (i) we prove this part for z2 = l2 ∞ and the rest of the cases will follow similarly. let (xk,l) ∈ l 2 ∞ (m′, ∆mn , p, u, s, ||·, · · · , ·||), then there exists 0 < u < ∞ such that (kl)−suk,l ( m′k,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l ≤ u, for all k, l ∈ n. let yk,l = (kl) −suk,lm ′ k,l ( || ∆ m n xk,l ρ , z1, · · · , zn−1|| ) . then yk,l ≤ u 1 pk,l ≤ v, say for all k, l ∈ n. hence we have ( (m′′k,l ◦ m ′ k,l) ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l = (m′′k,l(yk,l)) pk,l ≤ (m′′k,l(v)) pk,l < ∞, for all k, l ∈ n. hence sup k,l uk,l ( (m′′k,l◦m ′ k,l) ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l < ∞. thus x = (xk,l) ∈ l 2 ∞ (m′′◦ m′, ∆mn , p, u, s, ||·, · · · , ·||). (ii) we prove the result for the case z2 = c2 and the rest of the cases will follow similarly. let x = (xk,l) ∈ c 2(m′, ∆mn , p, u, s, ||·, · · · , ·||) ∩ c 2(m′′, ∆mn , p, u, s, ||·, · · · , ·||), then there exist some ρ1, ρ2 > 0 and l ∈ x such that lim k→∞ (kl)−suk,l ( m′k,l ( || ∆mn xk,l − l ρ1 , z1, · · · , zn−1|| ))pk,l = 0 and lim k→∞ (kl)−suk,l ( m′′k,l ( || ∆mn xk,l − l ρ2 , z1, · · · , zn−1|| ))pk,l = 0. let ρ = ρ1 + ρ2. then we have (kl)−suk,l ( (m′k,l + m ′′ k,l) ( || ∆ m n xk,l−l ρ , z1, · · · , zn−1|| ))pk,l ≤ k [( ρ1 ρ1 + ρ2 ) (kl)−suk,lm ′ k,l ( || ∆mn xk,l − l ρ1 , z1, · · · , zn−1|| )]pk,l + k [( ρ2 ρ1 + ρ2 ) (kl)−suk,lm ′′ k,l ( || ∆mn xk,l − l ρ2 , z1, · · · , zn−1|| )]pk,l . this implies that lim k→∞ (kl)−suk,l ( (m′k,l + m ′′ k,l) ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| ))pk,l = 0. 186 k. raj and s. k. sharma cubo 14, 3 (2012) thus x = (xk,l) ∈ c 2(m′ + m′′, ∆mm, p, u, s, ||·, · · · , ·||). this completes the proof. theorem 3.6 let m = (mk,l) be a sequence of orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers, then z2(m, ∆m−1n , p, u, s, ||·, · · · , ·||) ⊂ z 2(m, ∆mn , p, u, s, ||·, · · · , ·||) , for z 2 = l2 ∞ , c2 and c2o. proof. we prove the result for the case z2 = l2 ∞ and the rest of the cases will follow similarly. let x = (xk,l) ∈ l 2 ∞ (m, ∆m−1n , p, u, s, ||·, · · · , ·||). then we can have ρ > 0 such that (kl)−suk,l ( mk,l ( || ∆m−1n xk,l ρ , z1, · · · , zn−1|| ))pk,l < ∞, for all k ∈ n. (3.4) on considering 2ρ and using the convexity of (mk,l), we have (kl)−suk,lmk,l ( || ∆mn xk,l 2ρ , z1, · · · , zn−1|| ) ≤ 1 2 (kl)−suk,lmk,l ( || ∆m−1n xk,l ρ , z1, · · · , zn−1|| ) + 1 2 (kl)−suk,lmk,l ( || ∆m−1n xk+n,l+m ρ , z1, · · · , zn−1|| ) . hence we have (kl)−suk,l ( mk,l ( || ∆ m n xk,l 2ρ , z1, · · · , zn−1|| ))pk,l ≤ k { (kl)−suk,l ( 1 2 mk,l ( || ∆m−1n xk,l ρ , z1, · · · , zn−1|| ))pk,l + (kl)−suk,l ( 1 2 mk,l ( || ∆m−1n xk+n,l+m ρ , z1, · · · , zn−1|| ))pk,l } . then using equation (3.4), we have (kl)−suk,l ( mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l < ∞, for all k, l ∈ n. thus l2 ∞ (m, ∆m−1n , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). theorem 3.7 let m = (mk,l) be a sequence of orlicz functions. then c20(m, ∆ m n , p, u, s, ||·, · · · , ·||) ⊂ c 2 (m, ∆mn , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). proof. it is obvious that c20(m, ∆ m n , p, u, s, ||·, · · · , ·||) ⊂ c 2(m, ∆mn , p, u, s, ||·, · · · , ·||). we shall prove that c2(m, ∆mn , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). let x = (xk,l) ∈ c 2(m, ∆mn , p, u, s, ||·, · · · , ·||). then there exists some ρ > 0 and l ∈ x such that lim k,l→∞ (kl)−suk,l ( mk,l ( || ∆mn xk,l − l ρ , z1, · · · , zn−1|| ))pk,l = 0. cubo 14, 3 (2012) some generalized difference double sequence spaces ... 187 on taking ρ = 2ρ1, we have sup k,l (kl)−suk,l ( mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1 ))pk,l ≤ sup k,l k [ 1 2 (kl)−suk,l ( mk,l ( || ∆mn xk,l − l ρ1 , z1, · · · , zn−1|| ))]pk,l + sup k,l k [ 1 2 (kl)−suk,lmk,l ( || l ρ1 , z1, · · · , zn−1|| )]pk,l ≤ sup k,l k( 1 2 )pk,l(kl)−suk,l [ mk,l ( || ∆mn xk,l − l ρ1 , z1, · · · , zn−1|| )]pk,l + sup k,l k( 1 2 )pk,l(kl)−s max(1, uk,l ( mk,l ( || l ρ1 , z1, · · · , zn−1|| ))h) , where h = max(1, sup pk,l). thus we get x = (xk,l) ∈ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). hence c20(m, ∆ m n , p, u, s, ||·, · · · , ·||) ⊂ c 2 (m, ∆mn , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). theorem 3.8 the sequence space l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) is solid. proof. let x = (xk,l) ∈ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||), that is sup k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l < ∞. let (αk,l) be a sequence of scalars such that |αk,l| ≤ 1 for all k, l ∈ n. thus we have supk,l→∞(kl) −suk,l [ mk,l ( || αk,l∆ m n xk,l ρ , z1, · · · , zn−1|| )]pk,l ≤ sup k,l→∞ (kl)−suk,l [ mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l < ∞. this shows that (αk,lxk,l) ∈ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) for all sequences of scalars (αk,l) with |αk,l| ≤ 1 for all k, l ∈ n, whenever (xk,l) ∈ l 2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||). hence the space l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) is a solid sequence space. theorem 3.9 the sequence space l2 ∞ (m, ∆mn , p, u, s, ||·, · · · , ·||) is monotone. proof. the proof of the theorem is obvious and so we omit it. received: october 2011. revised: august 2012. 188 k. raj and s. k. sharma cubo 14, 3 (2012) references [1] b. altay and f. başar, some new spaces of double sequencs, j. math. anal. appl., 309 (2005), 70-90. 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[36] m. zeltser, investigation of double sequence spaces by soft and hard analytical methods, dissertationes mathematicae universitatis tartuensis 25, tartu university press, univ. of tartu, faculty of mathematics and computer science, tartu (2001). cubo a mathematical journal vol.14, no¯ 02, (91–109). june 2012 higher order terms for the quantum evolution of a wick observable within the hepp method sébastien breteaux irmar, umr-cnrs 6625, université de rennes 1, campus de beaulieu, 35042 rennes cedex, france. ens de cachan, antenne de bretagne, campus de ker lann, av. r. schuman, 35170 bruz, france. email: sebastien.breteaux@ens-cachan.org abstract the hepp method is the coherent state approach to the mean field dynamics for bosons or to the semiclassical propagation. a key point is the asymptotic evolution of wick observables under the evolution given by a time-dependent quadratic hamiltonian. this article provides a complete expansion with respect to the small parameter ε > 0 which makes sense within the infinite-dimensional setting and fits with finite-dimensional formulae. resumen el método de hepp describe en términos de estados coherentes la dinámica en campo medio de bosones o la propagación semiclásica. un punto clave es la evolución asintótica de observables de wick bajo la evolución dada por un hamiltoniano cuadrático dependiente del tiempo. este art́ıculo proporciona una expansión completa con respecto al parámetro pequeño ε > 0 válido en dimensión infinita y que corresponde a fórmulas en dimensión finita conocidas. keywords and phrases: mean field limit, semiclassical limit, coherent states, squeezed states. 2000 ams mathematics subject classification: 81r30, 35q40, 81s10, 81s30. 78 sébastien breteaux cubo 14, 2 (2012) 1 introduction in this article we derive two expansions with respect to a small parameter ε of quantum evolved wick observables under a time-dependent quadratic hamiltonian. the hepp method was introduced in [20] and then extended in [12, 13] in order to study the mean field dynamics of many bosons systems via a (squeezed) coherent states approach. the asymptotic analysis in the mean field limit is done with respect to a small parameter ε, where the number of particles is of order 1 ε . remember that the mean field dynamics is obtained as a classical hamiltonian dynamics which governs the evolution of the center z(t) of the gaussian state (squeezed coherent state). meanwhile the covariance of this gaussian as well as the control of the remainder term is determined by the evolution of a quadratic approximate hamiltonian around z(t). a key point in this method is the asymptotic analysis of the evolution of a wick quantized observable according to this quantum time-dependent quadratic hamiltonian. only a few results are clearly written about the remainder terms and some possible expansions in powers of ε, see the works of ginibre and velo [14, 15]. in the finite-dimensional case, entering into the semiclassical theory, accurate results have been given by combescure, ralston and robert in [6, 7, 8], hagedorn and joye in [17, 18, 19]. another viewpoint is used (in finite dimension) by paul and uribe in [25] to get approximate eigenvectors of semiclassical operators in terms of linear superpositions of coherent states. for the mean field infinite-dimensional setting some results have been proved in [16, 11, 28] with a different approach. we stick here with the hepp method with the presentation of [1] which puts the stress on the similarities and differences between the infinite-dimensional bosonic mean field problem and the finite-dimensional semiclassical analysis. nevertheless, in [1] the authors only considered the main order term although some of their formulae make possible complete expansions. in this article we derive two expansions of the quantum evolved wick observables which are equal term by term. two difficulties have to be solved : (1) unlike the time-independent finite-dimensional case, no mehler type explicit formula (see for example [22] or [9]) is available. a general time-dependent hamiltonian has no explicit dynamics. (2) in the infinite-dimensional framework the quantization of a linear symplectic transformation (a bogoliubov transformation) requires some care. useful references on this subject are [3] and [2]. its realization in the fock space relies on a hilbert-schmidt condition on the antilinear part connected with the shale theorem (see [29] and [26, 8, 5]). these things are well known but have to be considered accurately while writing complete expansions. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 79 two different methods, with apparently two different final formulae, will be used. a first one relies on a dyson expansion approach and provides the successive terms as time-dependent integrals. the second one uses the exact formulae for the finite-dimensional weyl quantization and after having made explicit the relationship between wick and weyl quantizations like in [4] or [1], the proper limit process with respect to the dimension is carried out. the outline of this article is the following. in section 2 we recall some facts and definitions about the fock space and wick quantization. we then present our main results in section 3 in theorems 3.1 and 3.2 and illustrate them by a simple example. section 4 and section 5 are devoted to the construction and properties of the classical and quantum evolution associated with a symmetric quadratic hamiltonian. section 7 and section 8 contain the proofs of our two expansion formulae. for the convenience of the reader we recall some facts about real-linear symplectomorphisms and symplectic fourier transform in the appendices. 2 wick calculus with polynomial observables 2.1 definitions we recall some definitions and results about wick quantization. more details can be found in [1]. in this paper (z,〈·, ·〉) denotes a separable hilbert space over c, the field of complex numbers. it is also a symplectic space with respect to the symplectic form σ(z1,z2) = ℑ 〈z1,z2〉. we use the physicists convention that all the scalar products over hilbert spaces are linear with respect to the right variable and antilinear with respect to the left variable. we denote by sm the symmetrization operator on ⊗m z (the completion for the natural hilbert scalar product of the algebraic tensor product ⊗m, alg z) defined by sm (z1 ⊗ · · · ⊗ zm) = 1 m! ∑ σ∈sm zσ1 ⊗ · · · ⊗ zσm , where the zj are vectors in z and sm denotes the set of the permutations of {1, . . . ,m}. we will use the notation z1 ∨ · · · ∨zm for sm (z1 ⊗ · · · ⊗ zm), and z∨m for z∨ · · · ∨z when the m terms of this product are equal to z. we call monomial of order (p,q) ∈ n2 a complex-valued application defined on z of the form b(z) = 〈 z∨q, b̃z∨p 〉 , with b̃ ∈ l (∨p z, ∨q z ) where ∨n z (or z∨n) denotes the hilbert completion of the n-fold symmetric tensor product, and for two banach spaces e and f, the space of continuous linear applications from e to f is denoted by l (e,f). we then write b ∈ pp,q (z). the total order of b is the integer m = p + q. the finite linear combinations of monomials are called polynomials. the set of all polynomials of this type is denoted by p (z). subsets of particular interest of p (z) are pm (z) and p≤m (z), the finite linear combinations of monomials of total order equal to m and not greater than m. 80 sébastien breteaux cubo 14, 2 (2012) the hilbert space h := ⊕ n∈n n∨ z is called the symmetric fock space associated with z, where tensor products and sum completions are made with respect to the natural hilbert scalar products inherited from z. we also consider the dense subspace hfin of h of states with a finite number of particles hfin := alg⊕ n∈n n∨ z , where the tensor products are completed but the sum is algebraic. the wick quantization of a monomial b ∈ pp,q (z) is the operator defined on hfin by its action on ∨n z as an element of l( ∨n z, ∨n+q−p z), bwick ∣∣∨ n z = 1[p,+∞) (n) √ n! (n + q − p) ! (n − p) ! ε p+q 2 ( b̃ ∨ i∨n−p z ) , where ix denotes the identity map on the space x and for aj ∈ l ( z∨pj,z∨qj ) , a1 ∨ a2 = sq1+q2a1 ⊗ a2sp1+p2. the wick quantization is extended by linearity to polynomials. we have a notion of derivative of a polynomial, first defined on the monomials and then extended by linearity. for b ∈ pp,q (z) and for any given z ∈ z, the operator ∂ j z̄∂ k zb(z) := p! (p − k) ! q! (q − j) ! (〈 z∨(q−j) ∣∣∣ ∨ i∨j z ) b̃ ( z∨(p−k) ∨ i∨k z ) (2.1) is an element of l (∨k z, ∨j z ) . we use the “bra” and “ket” notations of the physicists for vectors and forms in hilbert spaces. then we can define the poisson bracket of order k of two polynomials b1, b2, by {b1,b2} (k) = ∂kzb1.∂ k z̄b2 − ∂ k zb2.∂ k z̄b1 since, for any polynomial b, ∂kzb(z) is a k-form (on z) and ∂kz̄b(z) is a k-vector. remark 1. the product denoted by a dot in the definition of the poisson bracket is a c-bilinear duality-product between k-forms and k-vectors. as an example consider the polynomials b1 (z) = 〈 z∨3,ξ∨31 〉〈 η∨21 ,z ∨2 〉 and b2 (z) = 〈 z∨3,ξ∨32 〉 〈η2,z〉 . the poisson bracket of order 2 of b1 and b2 is {b1,b2} (2) (z) = 2 × 6 × 〈 z∨3,ξ∨31 〉〈 η∨21 ∨ z,ξ ∨3 2 〉 〈η2,z〉 − 0. 2.2 some examples of wick quantizations here is a quick review of the notations used for some useful examples of wick quantization. a vector of z is denoted by ξ, a is a bounded operator and z is the variable of the polynomials. in cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 81 the next table, the first column describes the polynomial and the second the corresponding wick quantization (as an operator on hfin). 〈z,az〉 ↔ dγ (a) |z| 2 ↔ n 〈z,ξ〉 ↔ a∗ (ξ) 〈ξ,z〉 ↔ a(ξ) √ 2ℜ 〈z,ξ〉 ↔ φ(ξ) the operator dγ (a) is the usual second quantization of an operator restricted to hfin multiplied by a factor ε. if a = iz we obtain n the usual number operator multiplied by a factor ε. the operators a, a∗ and φ are the usual annihilation, creation and field operators of quantum field theory with an additional √ ε factor. the real and imaginary parts of a complex number ζ are denoted by ℜζ and ℑζ. the field operators φ(ξ) are essentially self-adjoint and this enables us to define the (ε-dependent) weyl operators w (ξ) = eiφ(ξ) . 2.3 calculus here are some calculation rules for wick quantization of polynomials in p (z). the proofs can be found in [1]. proposition 2.1. for every polynomial b ∈ p (z), • bwick1 bwick2 = (∑min{p1,q2} k=0 εk k! ∂kzb1.∂ k z̄b2 )wick in hfin for any bi ∈ ppi,qi (z), • bwick is closable and the domain of the closure contains h0 = vect {w (z)ϕ,ϕ ∈ hfin, z ∈ z} , (we still denote by bwick the closure of bwick), • ( bwick )∗ = b̄wick on hfin (where the bar denotes the usual conjugation on complex numbers), • for any z0 in z, w (√ 2 iε z0 )∗ bwickw (√ 2 iε z0 ) = (b(z0 + z)) wick holds on h0 where b(z0 + ·) ∈ p (z). 3 main results and a simple example our two hypotheses are: 82 sébastien breteaux cubo 14, 2 (2012) h1 let (αt)t∈r be a one parameter family of self-adjoint operators on z defining a strongly continuous dynamical system uα(t,s). h1’ assume h1 and additionally that the dynamical system preserves a dense set d such that, for any ψ ∈ d, uα (·, ·)ψ belongs to c1 ( r2,z ) ∩ c0 ( r2,d ) . h2 let β be in c0 ( r; z∨2 ) , (βt defines a c-antilinear hilbert-schmidt operator by z 7→ (iz ∨ 〈z|)βt). with h1’ and h2, the classical flow associated with a family qt (z) = 〈z,αtz〉 + ℑ 〈 βt,z ∨2 〉 of quadratic polynomials is the solution ϕ(t,s) to the equation { i∂tϕ(t,0) [z] = ∂z̄qt (ϕ(t,0) [z]) ϕ(0,0) = iz (3.1) where ∂z̄qt (z) = αz + i(iz ∨ 〈z|β), written in a weak sense. although things are better visualized by writing a differential equation, the hypotheses h1 and h2 suffice to define the dynamical system ϕ(t,s). details about this point are given in section 4. actually ϕ(t,s) is a family of symplectomorphisms of (z,σ) which are naturally decomposed into their c-linear and c-antilinear parts: ϕ = l + a, l ∈ l (z) , aa∗ ∈ l1 (z) . see appendix a for more details about symplectomorphisms and this decomposition. similarly, the quantum flow associated with qt is the solution u(t,s) to { iε∂tu(t,0) = q wick t u(t,0) u(0,0) = ih . (3.2) the precise meaning of the solutions to this equation is specified in section 5. we are ready to state our two main results dealing with the evolution of a wick observable bwick, b ∈ p (z), under the quantum flow, that is to say the quantity u(0,t)bwicku(t,0). (we use the usual notation 〈n〉 = √ n2 + 1.) theorem 3.1. assume h1 and h2. let b ∈ p≤m (z) be a polynomial. then, for any time t ≥ 0, the formula u(0,t)bwicku(t,0) = ( b(0),t )wick + ⌊m/2⌋∑ k=1 ( ε 2 )k ˆ ∆kt ( b(k)t,s̄ k )wick ds̄k (3.3) holds as an equality of continuous operators from d ( 〈n〉m/2 ) to h, where • s̄k = (s1, . . . ,sk) ∈ rk+ and ∆kt = { s̄k ∈ rk+, ∑k j=1 sj ≤ t } , cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 83 • the polynomials b(k)t,s̄k are defined recursively by { b(0)t (z) = b(ϕ(t,0)z) b(k+1)t,s̄ k+1 = λsk+1b(k)t,s̄ k , with λsc = −i {c ◦ ϕ(0,s) ,qs}(2) ◦ ϕ(s,0) for any polynomial c. theorem 3.2. assume h1 and h2. let m ≥ 2 and b ∈ p≤m (z) a polynomial. then introducing • the vector vt ∈ ⊗2 z such that for all z1, z2 ∈ z, 〈z1 ⊗ z2,vt〉 = 〈z1,l∗ (t,0)a(t,0)z2〉 , • the operator on p (z) λtc(z) = tr [−2a∗ (t,0)a(t,0)∂z̄∂zc(z)] + 〈vt| .∂2z̄c(z) + ∂2zc(z) . |vt〉 , the formula u(0,t)bwicku(t,0) = ( e ε 2 λt (b ◦ ϕ(t,0)) )wick (3.4) holds as an equality of continuous operators from d ( 〈n〉m/2 ) to h. remark 2. the derivative ∂z̄∂zc(z) is in l (z) and tr denotes the trace on the subset of trace class operators of l (z). remark 3. for m ≥ 2 the operators λt and λt send pm (z) into pm−2 (z). remark 4. the exponential is intended in the sense e ε 2 λtb = ⌊m/2⌋∑ k=0 1 k! ( ε 2 λt )k b for a polynomial b in p≤m (z). example 1. to give an idea of the behavior of these formulae we apply them in the simplest (non trivial) possible situation, with z = c and qt (z) = ℑ ( z2 ) . as qt is time-independent the classical evolution equation is autonomous and thus we can write ϕ(t,s) = ϕ(t − s) and i∂tϕ(t)z = ∂z̄q(ϕ(t)z) = iϕ(t)z. the solution is ϕ(t)z = zcosht + z̄sinht. we can then compute both ˆ t 0 b(1)t,sds and λt (b ◦ ϕ(t)) . the first one is easily computed as ∂2zq(z) = −i, ∂ 2 z̄q(z) = i and, with c = b ◦ ϕ(t), −i {c ◦ ϕ(−s) ,q(z)}(2) = ( ∂2z + ∂ 2 z̄ ) (c ◦ ϕ(−s)) = [ cosh (−2s) ( ∂2z + ∂ 2 z̄ ) c +2sinh (−2s)∂z̄∂zc] ◦ ϕ(−s) 84 sébastien breteaux cubo 14, 2 (2012) and thus ˆ t 0 b(1)t,sds = ˆ t 0 ( cosh (−2s) ( ∂2z + ∂ 2 z̄ ) + 2sinh (−2s)∂z̄∂z ) ds(b ◦ ϕ(t)) = ( 1 2 sinh (2t) ( ∂2z + ∂ 2 z̄ ) + (1 − cosh (2t))∂z̄∂z ) (b ◦ ϕ(t)) . now we compute the second one. since l(t,0)z = l∗ (t,0)z = zcosht and a(t,0)z = a∗ (t,0)z = z̄sinht, we get vt = coshtsinht and then obtain directly λt = (1 − cosh (2t))∂z̄∂z + 1 2 sinh (2t) ( ∂2z + ∂ 2 z̄ ) . we thus obtain the same result with the two computations for the term of order 1 in ε. then we can show that ˆ ∆k t b(k)t,s̄ k ds̄k = 1 k! ( λt )k (b ◦ ϕ(t)) since ˆ ∆kt k∏ j=1 ( 2sinh (−2sj)∂z̄∂z + cosh (−2sj) ( ∂2z + ∂ 2 z̄ )) ds̄k = 1 k! ( (1 − cosh (−2t))∂z̄∂z − 1 2 sinh (−2t) ( ∂2z + ∂ 2 z̄ ))k because d ds [ (1 − cosh (−2s))∂z̄∂z − 1 2 sinh (−2s) ( ∂2z + ∂ 2 z̄ )] = 2sinh (−2s)∂z̄∂z + cosh (−2s) ( ∂2z + ∂ 2 z̄ ) . remark 5. since these two formulae will be proven independently and the identification of each term of order k in ε in the expansion of the symbol is clear, we carry out a computation only on the formal level for the convenience of the reader to show the link between the two formulae in the general case. we show (formally) that d ds λs = λs . then it is simple to show that ˆ s̄k∈∆kt λskλsk−1 · · ·λs1ds̄k = 1 k! ( λt )k as operators on p (z) once the case k = 2 is understood: 2 ˆ s̄2∈∆2t λs2λs1ds̄2 = ˆ t 0 ˆ s1 0 λs2λs1ds2ds1 + ˆ t 0 ˆ s2 0 λs2λs1ds1ds2 = ˆ t 0 λs1λs1ds1 + ˆ t 0 λs2λs2ds2 = ( λt )2 . cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 85 in this computation we have used that λ0 = 0 as a(0,0) = 0. we first give λs in a more explicit way. as ∂2z̄q = i |β〉 and ∂2zq = −i〈β| we first get λc = [ ∂2z ( c ◦ ϕ−1 ) . |β〉 + 〈β| .∂2z̄ ( c ◦ ϕ−1 )] ◦ ϕ with ϕ = ϕ(t,0) and omitting the time dependence everywhere. then with ϕ = l + a (and thus ϕ−1 = l∗ − a∗) and 〈z1,az2〉 = 〈z1 ⊗ z2,wa〉 we obtain λc(z) = ∂2zc(z) . ∣∣(l∗∨2 + a∗∨2 ) β 〉 + 〈( l∗∨2 + a∗∨2 ) β ∣∣ .∂2z̄c(z) −2 (〈( iz ⊗ ∂z̄∂zc(z)∗ l∗ ) β,wa 〉 + 〈wa,(iz ⊗ ∂z̄∂zc(z)l∗)β〉 ) . then we compute d ds λs in several steps. the linear and antilinear parts of the equation i∂sϕ(s,0)z = ∂z̄qs (ϕ(s,0)z) give ∂slz = −iαlz + (〈az| ∨ iz) |β〉 ∂saz = −iαaz + (〈lz| ∨ iz) |β〉 . we now show that ∂svs = ∣∣(l∗∨2 + a∗∨2 ) β 〉 , ∂s 〈z1 ⊗ z2,vs〉 = ∂s 〈lz1,az2〉 = 〈−iαlz1,az2〉 + 〈β,az2 ∨ az1〉 + 〈lz1,−iαaz2〉 + (〈lz2| ∨ 〈lz1|) |β〉 = 〈β,(a ∨ a) (z1 ∨ z2)〉 + 〈(l ∨ l) (z1 ∨ z2) ,β〉 = 〈 z1 ∨ z2, ( l∗∨ 2 + a∗∨ 2 ) β 〉 . and thus ∂s ( ∂2z. |v〉 + 〈v| .∂2z̄ ) = ∂2z. ∣∣(l∗∨2 + a∗∨2 ) β 〉 + 〈( l∗∨2 + a∗∨2 ) β ∣∣ .∂2z̄. we then show that ∂str [a ∗a∂z̄∂zc(z)] = 〈β,(iz ⊗ l∂z̄∂zc(z))wa〉 + 〈wa,(iz ⊗ ∂z̄∂zc(z)l∗)β〉 . we first observe that tr [a∗a∂z̄∂zc(z)] = 〈wa,(iz ⊗ ∂z̄∂zc(z))wa〉. a simple calculation using ∂saz = −iαaz + (〈lz| ∨ iz) |β〉 shows that ∂swa = (−iα ⊗ iz)wa + (iz ⊗ l∗)β and this immediately gives the result. 4 classical evolution of a wick polynomial under a quadratic evolution the adjoint of a c-antilinear operator is defined in appendix a. definition 4.1. a c-antilinear operator a on z is said of hilbert-schmidt class if ‖a‖la 2 (z) := ‖aa∗‖1/2l1(z) is finite, where ‖·‖l1(z) is the usual trace norm for c-linear operators. the set of hilbert-schmidt antilinear operators is denoted by la2 (z). 86 sébastien breteaux cubo 14, 2 (2012) let x (z) = l (z) + la2 (z) with norm ‖t‖x (z) = ‖l‖l(z) + ‖a‖la 2 (z) for t = l + a, where l and a are respectively c-linear and c-antilinear. the space x (z) is a banach algebra. remark 6. the norm ‖t‖x (z) is well defined as the decomposition t = l + a is unique (l = 1 2 (t − iti) and a = 1 2 (t + iti)). 4.1 construction of the classical flow without the α term let β ∈ c0 ( r; z∨2 ) and qt = ℑ 〈 βt,z ∨2 〉 . observe that ∂z̄q(t) (z) = i(iz ∨ 〈z|)βt and so (∂z̄qt)t is a continuous one parameter family of x (z), so that the theory of ordinary differential equations in banach algebras (see for example [21]) asserts that there exists a unique two parameters family ϕ(t2,t1) of elements of x (z) such that { i∂tϕ(t,0) = ∂z̄qt ϕ(t,0) ϕ(0,0) = iz , with ϕ of c1 class in both parameters such that for all r, s and t, ϕ(t,s)ϕ(s,r) = ϕ(t,r) . the classical flow ϕ(t,s) is a symplectomorphism with respect to the symplectic form σ(z1,z2) = ℑ 〈z1,z2〉. it can be checked deriving σ(ϕ(t,s)z1,ϕ(t,s)z2) with respect to t. 4.2 the strongly continuous dynamical system associated with (αt) we first state a proposition which is a direct consequence of theorem x.70 in [27] in the unitary case. this proposition provides a set of assumptions ensuring the existence of a strongly continuous dynamical system associated with a family (αt)t of self-adjoint operators. other more general situations can be considered as in [23, 24] for example. proposition 4.2. let (αt)t∈r be a family of self-adjoint operators on the hilbert space z satisfying the following conditions. (1) the αt have a common domain d (from which it follows by the closed graph theorem that c(t,s) = (αt − i) (αs − i) −1 is bounded). cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 87 (2) for each z ∈ z, (t − s)−1 c(t,s)z is uniformly strongly continuous and uniformly bounded in s and t for t 6= s lying in any fixed compact interval. (3) for each z ∈ z, c(t)z = limsրt (t − s)−1 c(t,s)z exists uniformly for t in each compact interval and c(t) is bounded and strongly continuous in t. the approximate propagator uk is defined by uk (t,s) = exp(− (t − s)iαj−1 k ) if j−1 k ≤ s ≤ t ≤ j k and uk (t,r) = uk (t,s)uk (s,r). then for all s, t in a compact interval and any z ∈ z, u(t,s)z = lim k→+∞ uk (t,s)z exists uniformly in s and t. further, if z ∈ d, then u(t,s)z is in d for all s, t and satisfies { i d dt u(t,s)z = αtu(t,s)z u(s,s)z = z . 4.3 construction of the classical flow with the α term assume h1 and h2. let ϕ̂ be the solution of { i∂tϕ̂(t,0) = ∂z̄q̂t ϕ̂(t,0) ϕ̂(0,0) = iz , with q̂t (z) = ℑ 〈 β̂t,z ∨2 〉 , β̂t = uα (t,0) ∗∨2 βt. what we call here the solution of { i∂tϕ(t,0) = ∂z̄qt ϕ(t,0) ϕ(0,0) = iz , (4.1) with qt = 〈z,αtz〉 + ℑ 〈 βt,z ∨2 〉 is ϕ(t,0) = uα (t,0) ◦ ϕ̂(t,0) . depending on the assumptions on (αt) it will be possible to precise if ϕ solves equation (4.1) in a usual sense (strongly, weakly, on some dense subset...). with the particular set of assumptions of theorem 4.2 we get that for all z1 ∈ d and z2 ∈ z, { i∂t 〈z1,ϕ(t,0)z2〉 = 〈αz1,ϕ(t,0)z2〉 + i〈z1 ∨ ϕ(t,0)z2,β〉 ϕ(0,0) = iz . 88 sébastien breteaux cubo 14, 2 (2012) 4.4 composition of a wick polynomial with the classical evolution the composition of a polynomial with the classical flow defines a time-dependent polynomial. definition 4.3. we define a norm on p (z) by ‖b‖p(z) = ∑ p, q ‖bp,q‖q←p where b = ∑ p, q bp,q is a polynomial with bp,q ∈ pp,q (z) and ‖bp,q‖q←p is a shorthand for ‖b̃p,q‖l(∨p z,∨q z). for a polynomial b in pm (z), we will sometimes write ‖b‖pm(z). proposition 4.4. let b ∈ pm (z) be a polynomial, and ϕ ∈ x (z). then b ◦ ϕ ∈ pm (z) and we have the estimate ‖b ◦ ϕ‖pm(z) ≤ ‖ϕ‖ m x (z) ‖b‖pm(z) . proof. the proof is essentially the same as in proposition 2.12 of [1]. 5 quantum evolution of a wick polynomial 5.1 without the α term definition 5.1. let β ∈ c0 ( r; z∨2 ) and qt (z) = ℑ 〈 βt,z ∨2 〉 . a family u(t,s) of unitary operators on h defined for s, t real is a solution of { i∂tu(t,0) = qwickt ε u(t,0) u(0,0) = ih (5.1) if (1) u(t,s) is strongly continuous in h with respect to s, t with u(s,s) = i, (2) u(t,r) = u(t,s)u(s,r), r ≤ s ≤ t, (3) i d dt u(t,s)y exists for almost every t (depending on s) and is equal to qwickt u(t,s)y, (4) iε d ds u(t,s)y = −u(t,s)qwicks y, y ∈ d (n + 1), 0 ≤ s ≤ t. this definition is made to fit the general framework of theorems 4.1 and 5.1 of [23]. more precisely we may check the following theorem. theorem 5.2. let β ∈ c0 ( r; z∨2 ) and qt (z) = ℑ 〈 βt,z ∨2 〉 . then the quantum flow equation (5.1) associated to the family 1 ε qt has a unique solution. this solution preserves the sets d(〈n〉k/2) for k ≥ 2. to establish this theorem we will use the following estimates. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 89 lemma 5.1. let β ∈ z∨2 and q(z) = ℑ 〈 β,z∨2 〉 . then, on hfin, and for k ≥ 1, qwick satisfies the estimates ∥∥qwick/εψ ∥∥ ≤ 3 2 ‖β‖z∨2 ‖(n/ε + 1)ψ‖ (5.2) and ± i [ qwick/ε,(n/ε + 1) k ] ≤ 3k √ 2‖β‖z∨2 (n/ε + 1) k . (5.3) the second estimate is in the sense of quadratic forms, for all ψ ∈ hfin, ± i (〈 1 ε qwickψ,(n/ε + 1) k ψ 〉 − 〈 (n/ε + 1) k ψ, 1 ε qwickψ 〉) ≤ 3 k √ 2 ‖β‖z∨2 〈 ψ,(n/ε + 1) k ψ 〉 . proof. the first estimate is a consequence of n + 2 ≤ 2(n + 1) associated to 2i ε qwick ∣∣ z∨n = √ n(n − 1) 〈β| ∨ i∨n−2 z − √ (n + 2) (n + 1) |β〉 ∨ i∨n z . for the second estimate, consider 2i ε 〈 ψ, [ (1 + n/ε) k ,qwick ] ψ 〉 . the first term of this commutator is ∑ n (n + 1) k (√ (n + 2) (n + 1) 〈 ψ(n) ∨ 〈β| ,ψ(n+2) 〉 − √ n(n − 1) 〈 ψ(n), |β〉 ∨ ψ(n−2) 〉) . then we deduce easily the second term and a reindexation gives the following form for the whole commutator: ∑ n [ (n + 1) k − ((n + 2) + 1) k ]√ (n + 2) (n + 1) × (〈 ψ(n) ∨ 〈β| ,ψ(n+2) 〉 + 〈 ψ(n+2), |β〉 ∨ ψ(n) 〉) . newton’s binomial formula and the inequalities ∑k−1 l=0 ( k l ) 2k−l ≤ 3k and (n + 1)l ≤ (n + 1)k−1 yield (n + 1) k − ((n + 2) + 1) k ≤ 3k (n + 1)k−1 . using also n + 2 ≤ 2(n + 1) to control √ (n + 2) (n + 1) we obtain ±i 〈 ψ, [ (1 + n/ε) k , qwick ε ] ψ 〉 ≤ 1 2 ∑ n 3k (n + 1) k−1 √ 2(n + 1) ∥∥∥ψ(n) ∥∥∥‖β‖z∨2 ∥∥∥ψ(n+2) ∥∥∥ . cauchy-schwarz’s inequality gives the claimed estimate. lemma 5.2. let β ∈ z∨2 and q(z) = ℑ 〈 β,z∨2 〉 . then qwick is essentially self-adjoint on hfin and its closure is essentially self-adjoint on any other core for n/ε+1. inequalities (5.2) and (5.3) still hold on d (n/ε + 1). 90 sébastien breteaux cubo 14, 2 (2012) we still denote by qwick this self-adjoint extension. proof. we apply the commutators theorem x.37 of [27] with the estimates of lemma 5.1 for k = 1. lemma 5.3. if a solution of the quantum flow equation (5.1) exists then it leaves q((n/ε + 1)k) = d((n/ε + 1)k/2) invariant for any integer k ≥ 2. in the time-independent case the estimate ‖u(t,0)‖l(d((n/ε+1)k/2)) ≤ exp ( 3k √ 2‖β‖ |t| ) holds. proof. from lemma 5.2, for any k ≥ 2, d((n/ε + 1)k/2) ⊂ d(qwick). we can adapt the proof of theorem 2 of [10] to the case of the quantization of a continuous one parameter family of quadratic polynomials with the estimates of lemma 5.1. proof of theorem 5.2. we use theorems 4.1 and 5.1 of [23] with the family of operators iq(t) wick /ε (here we directly consider the self-adjoint extension of qwickt /ε). we set y = d((n/ε + 1) k/2 ). (1) this family is stable in the sense that ‖ ∏k j=1 e −isjq(tj) wick/ε‖l(h) ≤ 1 (we actually have an equality here). (2) the space y is admissible for this family in the sense that for each t, (iqwickt /ε + λ) −1 leaves y invariant and ∥∥∥ ( iqwickt /ε + λ )−1∥∥∥ l(y) ≤ ( λ − 3k √ 2‖β‖ )−1 for ℜλ > 3k √ 2‖β‖. this is true because, as we have seen in lemma 5.3, (e−isq wick t /ε)s∈r leaves y invariant and, thanks to the estimate of the same lemma, we can apply the resolvent formula ( iqwickt /ε + λ )−1 = ˆ +∞ 0 e−λse−isq wick t /εds and obtain the desired estimate. (3) y ⊂ d ( qwickt /ε ) so that qwickt /ε ∈ l (y,h) for each t, and the map t → qwickt /ε ∈ l (y,h) is continuous. (4) y = d((n/ε + 1)k/2) is reflexive. theorems 4.1 and 5.1 of [23] thus apply and give the existence of an evolution operator. the preservation of the set d((n/ε + 1)k/2) comes from the application of lemma 5.3 to the solution of the time-dependent problem. to conclude it is then enough to observe that the domains d(〈n〉k/2) and d((n/ε + 1)k/2) are the same and have equivalent norms. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 91 5.2 with the α term assume h1 and h2. let û be the solution of { i∂tû(t,0) = q̂wickt ε û(t,0) û(0,0) = ih (5.4) with q̂t (z) = ℑ 〈 β̂t,z ∨2 〉 , β̂t = uα (t,0) ∗∨2 βt. what we call here the solution of { i∂tu(t,0) = qwickt ε u(t,0) u(0,0) = ih (5.5) with qt = 〈z,αtz〉 + ℑ 〈 βt,z ∨2 〉 is u(t,0) = γ (uα (t,0)) ◦ û(t,0) . 6 removal of the α part proposition 6.1. assume h1 and h2. suppose theorems 3.1 and 3.2 hold with a null one parameter family of self-adjoint operators on z, and β̂t = uα (t,0)∗∨2 βt. we denote with a hat the quantities associated with this solution. then theorems 3.1 and 3.2 hold. proof. for equation 3.3, we forget during the proof the (t,0) dependency in our notations and write ˆ ∆0t b(0)t,s̄ 0 ds̄0 instead of b(0),t. then u∗bwicku = û∗γ (u∗α)b wickγ (u)û = û∗ (b ◦ uα)wick û = ⌊ m2 ⌋∑ k=0 ( ε 2 )k ˆ ∆k t ( b̂ ◦ uα (k)t,s̄k )wick ds̄k where the b̂(k)t,s̄ k are defined recursively by { b̂(0)t (z) = b ◦ ϕ̂ b̂(k+1)t,s̄ k+1 = λ̂sk+1b̂(k)t,s̄ k with λ̂sc = −i { c ◦ ϕ̂(0,s) ,q̂s }(2) ◦ ϕ̂(s,0) for any polynomial c. thus it suffices to prove that b̂ ◦ uα (k)t,s̄k = b(k)t,s̄ k . 92 sébastien breteaux cubo 14, 2 (2012) this is clear for k = 0 as uα ◦ ϕ̂ = ϕ. then we observe that λ̂sc = −i { c ◦ ϕ̂−1,q̂ }(2) ◦ ϕ̂ = −i { c ◦ ϕ−1 ◦ uα,q̂ }(2) ◦ u−1α ◦ ϕ = −i { c ◦ ϕ−1,q }(2) ◦ ϕ where we used that ∂2z 〈z,αz〉 = 0, ∂2z̄ 〈z,αz〉 = 0 and βt = uα (t,0) ∨2 β̂t. we can thus restrict our proof to the case of a polynomial qt of the form qt (z) = ℑ 〈 βt,z ∨2 〉 with βt ∈ c0 ( r; z∨2 ) and no (αt) term. 7 a dyson type expansion formula for the wick symbol of the evolved quantum observable in this section we prove theorem 3.1. proof. we first prove that the formula, for c ∈ p≤m (z), u(0,s) (c ◦ ϕ(0,s))wick u(s,0) = cwick − iε 2 ˆ s 0 u(0,σ) {c ◦ ϕ(0,σ) ,qσ}(2)wick u(σ,0)dσ holds as an equality of continuous operators from d(〈n〉m/2) to h, with 〈n〉 = (n2 + 1)1/2. this is a consequence of the fact that the derivative of the left hand term as a function of s is −iε 2 u(0,s) {c ◦ ϕ(0,s) ,qs}(2)wick u(s,0) as it can be seen from the relation i∂σ (c ◦ ϕ(0,σ)) = −∂z (c ◦ ϕ(0,σ)) .∂z̄qσ + ∂zqσ.∂z̄ (c ◦ ϕ(0,σ)) and proposition 2.1. applying the previous formula with c = b(k)t,s̄ k we get recursively u(0,t)bwicku(t,0) = k−1∑ k=0 ( ε 2 )k ˆ s̄k∈∆kt ( b(k)t,s̄ k )wick ds̄k + ( ε 2 )k ˆ s̄k∈∆k t u(0,sk) ( b(k)t,s̄ k ◦ ϕ(0,sk) )wick u(sk,0)ds̄ k . this process gives a null remainder as soon as k > m/2 as for k ≤ ⌊m/2⌋, since the polynomial b(k)s̄ k is of total order m − 2k. 8 an exponential type expansion formula for the wick symbol of the evolved observable in this section we prove theorem 3.2. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 93 8.1 quantum evolution as a bogoliubov implementation some basic facts about symplectomorphisms are recalled in appendix a. definition 8.1. a symplectomorphism t is called implementable if and only if there exists a unitary operator u on h , called a bogoliubov implementer of t, such that ∀ξ ∈ z, u∗w (ξ)u = w (tξ) . proposition 8.2. assume αt ≡ 0 and h2. let qt = ℑ 〈 βt,z ∨2 〉 , ϕ(t,s) the associated classical evolution (see section 4) and u(t,s) the associated quantum evolution (see section 5). then for all t in r, u(t,0) is a bogoliubov implementer of −iϕ(0,t)i. remark 7. note that the symplectomorphism ϕ(0,t) is only r-linear and not c-linear in general, and thus −iϕ(0,t)i 6= ϕ(0,t). proof. we begin with a formal computation which will be justified further. it suffices to show that iε∂t [u(0,t)w (−iϕ(t,0)iξ)u(t,0)] = 0. computing this derivative and omitting the time and −iϕ(t,0)iξ dependencies in our notations, we get with u(t,0) = u u∗w { −w∗qwickw + qwick + w∗iε∂tw } u. then from proposition 2.10 (iii) in [1], the differential formula of weyl operators recalled in proposition 8.3 below and with ft = −iϕ(t,0)iξ it suffices to show that q ( z + iε√ 2 ft ) = q(z) + iε ( iε 2 ℑ 〈ft,∂tft〉 + i √ 2ℜ 〈∂tft,z〉 ) to get the result. this equality results from the expansion of q(z) = ℑ 〈 β,z∨2 〉 , recalling that i∂tϕ(t,0)ξ = ∂z̄q(ϕ(t,0)ξ), and observing that ∂z̄q(z) = i(〈z| ∨ iz) |β〉. we now need to clarify the meaning of this computation. it suffices to show that the quantity 〈φ,u(0,t)w (−iϕ(t,0)iξ)u(t,0)ψ〉 is constant for ψ, φ in d (n + 1). since this domain is preserved by the operators u(t,s), the weyl operators are weakly derivable on this domain (see next proposition), and u(t,s) is derivable on this domain, then we get the justification of the previous formal computation. proposition 8.3. let z, h be vectors in z, t be a real parameter and ϕ, ψ be in the domain of φ(h). then lim t→0 1 t (〈ϕ, [w (z + th) − w (z)]ψ〉) = 〈 ϕ,w (z) [ iφ(h) + iε 2 ℑ 〈z,h〉 + ] ψ 〉 = 〈 ϕ, [ iφ(h) − iε 2 ℑ 〈z,h〉 ] w (z)ψ 〉 . 94 sébastien breteaux cubo 14, 2 (2012) proof. for the first equality. the weyl commutation relations give 1 t 〈ϕ, [w(z + th) − w(z)]ψ〉 = 1 t 〈 w(−z)ϕ, [ e iε 2 ℑ〈z,th〉w(th) − iz ] ψ 〉 = 〈 w(−z)ϕ,e iε 2 ℑ〈z,th〉 1 t (w(th) − iz)ψ 〉 + 1 t ( e iε 2 ℑ〈z,th〉 − 1 ) 〈w(−z)ϕ,ψ〉 → t→0 〈 ϕ,w(z) [ iφ(h) + iε 2 ℑ 〈z,h〉 ] ψ 〉 . the convergence of the first term is due to the continuous one parameter group structure of w (th). the other equality is obtained in the same way. 8.2 action of bogoliubov transformations on wick symbols a theorem due to shale (see [29]) characterizes implementable symplectomorphisms. we quote here a version of this theorem fitting our needs. theorem 8.4 (shale, 1962). a symplectomorphism t is implementable if and only if the c-linear part of t∗t − id is trace class. we can now quote the main result of this part. theorem 8.5. let t = l + a with l c-linear and a c-antilinear, be an implementable symplectomorphism with a bogoliubov implementer u preserving d(〈n〉k/2) for any integer k≥ 2, then for any polynomial b in p≤m (z) with m ≥ 2, u∗bwicku = ( e ε 2 λ[t] [b(t∗·)] )wick (8.1) as an equality of continuous operators from d(〈n〉m/2) to h, with 〈n〉 = (n2 + 1)1/2, where • the exponential is a finite expansion whose rank depends on the degree of the polynomial b, • the operator λ [t] is defined on any polynomial c by λ [t]c(z) = tr [−2aa∗∂z̄∂zc(z)] + 〈v| .∂2z̄c(z) + ∂2zc(z) . |v〉 with v ∈ ⊗2 z the vector such that for all z1, z2 ∈ z, 〈z1 ⊗ z2,v〉 = 〈z1,la∗z2〉. in order to prove this result, we use intermediate steps. (1) we prove that u∗bweylu = b(t∗·)weyl in finite dimension. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 95 (2) we use the fourier transform and the formula bweyl = 1 (πε/2) d ( b ∗ e− |z|2 ε/2 )wick to get the result in finite dimension. (3) we extend the result to infinite dimension. 8.3 action of bogoliubov transformations on weyl quantizations of polynomials in finite dimension definition 8.6. in a finite-dimensional hilbert space z identified with cd, the symplectic fourier transform is defined by fσ [f] (z) = ˆ z e 2πiσ(z,z′)f(z′)l(dz′) where l denotes the lebesgue measure, and f is any schwartz tempered distribution. we associate with each polynomial b ∈ pp,q (z) a weyl observable by bweyl = ˆ z fσ [b] (z)w ( −i √ 2πz ) l(dz) . (8.2) this formula has a meaning as an equality of quadratic forms on s (z) since for any φ, ψ in s (z), z 7→ 〈 φ,w(−i √ 2πz)ψ 〉 and its derivative are continuous bounded functions and fσ [b] is made of derivatives of the delta function. proposition 8.7. let b ∈ p≤m (z) with m ≥ 2 be a polynomial on a finite-dimensional hilbert space z. let t be an implementable symplectomorphism with implementation u preserving the domain d(〈n〉m/2). then u∗bweylu = b(t∗·)weyl as a continuous operator from d(〈n〉m/2) to h. proof. we compute, in the sense of quadratic forms on s (z), u∗bweylu = ˆ fσ [b] (z)w ( − √ 2πtiz ) l(dz) = ˆ fσ [b] (t∗z)w ( −i √ 2πz ) l(dz) = ˆ fσ [b(t∗·)] (z)w ( −i √ 2πz ) l(dz) = b(t∗·)weyl where we made use of the relation ti = i(t∗) −1 , the volume preservation of t∗ in z seen as a r-vector space and the property of composition of a symplectic fourier transform by a symplectomorphism (see appendix c). the boundedness from d(〈n〉m/2) to h is deduced from the facts 96 sébastien breteaux cubo 14, 2 (2012) that the fourier transform of b involves only derivatives of the delta function of order smaller or equal to m and that a derivation of the weyl operator gives at worse a field factor which is controlled by 〈n〉1/2. 8.4 action of bogoliubov transformations on wick quantization of polynomials in finite dimension proposition 8.8. let b ∈ p≤m (z) with m ≥ 2 be a polynomial on a finite-dimensional hilbert space z. let t be an implementable symplectomorphism with implementation upreserving the domain d(〈n〉m/2). then u∗bwicku = ( e ε 2 λ[t] [b(t∗·)] )wick , (8.3) as a continuous operator from d(〈n〉m/2) to h, where λ [t] is defined as in theorem 8.5. proof. we search the polynomial c such that u∗bwicku = cwick. in finite dimension for polynomials we can use the well known deconvolution formula cwick = ( c ∗ 1 (πε/2) d e |z|2 ε/2 )weyl . by proposition 8.7 we boil down to search for a polynomial c such that ( b ∗ 1 (πε/2) d e |z|2 ε/2 ) (t∗·) = c ∗ 1 (πε/2) d e |z|2 ε/2 . using symplectic fourier transform (see appendix c) and its properties with respect to convolution, composition with symplectomorphisms and gaussians, we get fσc = [fσb(t∗·)] ×  fσ   e |z|2 ε/2 (πε/2) d  (t∗·)   ×  fσ   e − |z|2 ε/2 (πε/2) d     = e π2ε(|t ∗·|2−|·|2) 2 × fσb(t∗·) . writing t = l + a with l the c-linear and a the c-antilinear part of t we obtain |t∗z| 2 − |z| 2 = 〈l∗z,l∗z〉 + 〈a∗z,a∗z〉 + 〈l∗z,a∗z〉 + 〈a∗z,l∗z〉 − 〈z,z〉 = 〈z,ll∗z〉 + 〈z,aa∗z〉 + 〈la∗z,z〉 + 〈z,la∗z〉 − 〈z,z〉 = 〈z,2aa∗z〉 + 〈 v,z∨ 2 〉 + 〈 z2,v 〉 with v ∈ ⊗2 z the vector such that for all z1, z2 ∈ z, 〈z1 ⊗ z2,v〉 = 〈z1,la∗z2〉. by fourier transforming again, we get π2fσ [( |t∗·|2 − |·|2 ) × · ] fσc = tr [−2aa∗∂z̄∂zc(z)] + 〈v|∂2z̄c(z) + ∂2zc(z) |v〉 as the c-linear and c-antilinear parts behave differently under fourier transform (the c-linear part has a minus sign added, see appendix c). we then obtain the claimed result. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 97 8.5 extension to infinite dimension on a “cylindrical” class of polynomials theorem 8.9. let t̂ be symplectomorphism of the form t̂ = ecρ, with c a conjugation and ρ a positive, self-adjoint, hilbert-schmidt operator commuting with c. let (ξj)j∈n a hilbert basis in which ρ is diagonal. let πk be the orthogonal projection on the finite-dimensional space zk = vect({ξj}j≤k). then for any polynomial b in pm (z) with m ≥ 2 and any integer k û∗bwickk û = ( e ε 2 λ[t̂] [ bk ( t̂∗· )])wick as continuous operators from d(〈n〉 m 2 ) to h where bk (z) = b(πkz). proof. we first remark that, with q(z) = ℑ 〈cρz,z〉, e−iqwick/ε is a bogoliubov implementer of t̂ as it can be seen using proposition 8.2 and the hilbert-schmidt property of ρ. we define ρl = ρπl, t̂l = t̂πl and the operator ql (z) wick = ℑ 〈cρlz,z〉wick. we use the identification h = γs (zl)⊗ γs(z⊥l ) and observe that on γs (zl) ⊗ {ωz ⊥ l }, e−iq wick/ε = e−iq wick l /ε. for k ≤ l we obtain on γs (zl) ⊗ {ωz ⊥ l } û∗lb wick k ûl = ( e ε 2 λ[t̂l] [ bk ( t̂∗l· )])wick by proposition 8.8, with ûl = e −iqwickl /ε. but on this domain it is the same as û∗bwickk û = ( e ε 2 λ[t̂] [ bk ( t̂∗· )])wick with û = e−iq wick/ε. we thus get an equality on ∪lγs (zl), and by continuity of the involved operators from d(〈n〉 m 2 ) to h we get the expected result. we will first show that formula (8.1) apply in particular to a well chosen class of cylindrical polynomials, and then extend it by density to every polynomial. 8.6 extension to general polynomials we split the proof of formula (8.1) for general polynomials into several lemmata and propositions. lemma 8.1. let (ξj)j∈n be a hilbert basis of z, πm be the orthogonal projector on zm = vect({ξj}j≤m). let b be a polynomial in pp,q (z) and define bk = b(πk·). then (b̃k)k∈n is bounded and b̃ = w − lim j→∞ b̃k . to formulate more clearly some convergence results we need some extra definitions. 98 sébastien breteaux cubo 14, 2 (2012) definition 8.10. we define the spaces l∨p,q (z) = l ( z∨p,z∨q ) , l∨m = ⊕ p+q=m l∨p,q and l∨≤m = ⊕ m ′≤m l∨m ′ corresponding to pp,q (z), pm (z) and p≤m (z). let b = ∑ p,q bp,q be a polynomial, with bp,q ∈ p (z). we note b̃ = (b̃p,q) ∈ ⊕ p,q l∨p,q (z). the norm of b̃ = (b̃p,q) ∈ l∨≤m (z) is ‖b̃‖l∨ ≤m (z) = ∑ p,q ‖b̃p,q‖l(∨p z,∨q z) . a sequence (b̃k)k∈n of elements of l≤m (z) converges weakly to b̃ in l≤m (z) if b̃kp,q converges weakly to b̃p,q for every p and q as k → +∞. lemma 8.2. let t be an operator in x (z), (bk)k∈n and b be polynomials in pm (z) such that (b̃k)k∈n converges weakly to b̃. then bk (t·) and b(t·) are in pm (z) and b̃k (t·) converges weakly to b̃(t·). lemma 8.3. let t be an operator in x (z), (bk)k∈n and b be polynomials in pm (z) such that (b̃k)k∈n is bounded and converges weakly to b̃. then ( ˜e ε 2 λ[t]bk)k∈n converges weakly to ˜e ε 2 λ[t]b. proof. it is enough to show that weak convergence is preserved by the action of λ [t]. but, for any polynomial b, λ̃ [t]b = tr 1 [ (−2a∗a ⊗ iz∨q−1) b̃ ] + (〈v| ∨ iz∨q−2) b̃ + b̃(|v〉 ∨ iz∨p−2) , where tr1 is the partial trace on the first z subspace on the left and any direction on the right (so that if b̃ ∈ l∨p,q (z), then tr1[(−2a∗a ⊗ iz∨q−1)b̃] is in l∨p−1,q−1 (z)). with this formula the preservation of the weak convergence is clear. proposition 8.11. let b and (bk)k∈n be wick polynomials in pp,q (z) such that w − lim b̃k = b̃. then w − lim k (bk − b) wick 〈n〉− p+q 2 = 0. proposition 8.12. let b and (bk)k∈n be wick polynomials in pp,q (z) such that w − lim b̃k = b̃. let u be a unitary operator on the fock space h such that, for all k ≥ 2, 〈n〉 k 2 u〈n〉− k 2 is a bounded operator. then w − lim k u∗ (bk − b) wick u〈n〉− m ′ 2 = 0 with m′ = max (m,2), m = p + q. proposition 8.13. let t be an implementable symplectomorphism with bogoliubov implementer u. then for any polynomial b in p≤m (z), m ≥ 2, u∗bwicku = ( e ε 2 λ[t] [b(t∗·)] )wick as continuous operators from d(〈n〉 m 2 ) to h. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 99 proof. from the results 8.1 to 8.12 we deduce that proposition 8.13 holds for symplectomorphisms of the form t̂ = ecρ, with c a conjugation and ρ a positive, self-adjoint, hilbert-schmidt operator commuting with c. by theorem a.8 this assumption on the form of t̂ is not restrictive. indeed, if t = ut̂ with u unitary and û is a bogoliubov implementer for t̂, then u = ûγ (u∗) is a bogoliubov implementer for t and γ (u) ( e ε 2 λ[t̂] [ b ( t̂∗· )])wick γ (u∗) = ( e ε 2 λ[t] [b(t∗·)] )wick . indeed for any polynomial c, and operator ϕ in x (z), γ (ϕ)cwickγ (ϕ∗) = c(ϕ∗·)wick and λ [ t̂ ]k [ b ( t̂∗· )] (u∗·) = λ [ ut̂ ]k [ b ( t̂∗u∗· )] as can be checked using that l = ul̂ and a = uâ (l, l̂ and a, â denote respectively the c-linear and c-antilinear parts of t and t̂). this achieves the proof. 8.7 an evolution formula for the wick symbol we can now prove theorem 3.2. proof. we only need to apply propositions 8.2 and 8.13 with t = −iϕ(0,t)i = l∗ (t,0) +a∗ (t,0) (with ϕ(t,0) = l(t,0) + a(t,0)). we remark that for any symplectomorphism t, (−iti) ∗ = t−1 so that (−iϕ(0,t)i) ∗ = ϕ(t,0) and thus we get the result. 8.8 estimates we now give estimates for the different terms of the expansion of the symbol. proposition 8.14. let t = l + a be an implementable symplectomorphism with l c-linear and a c-antilinear. then the operator λ [t] defined on p (z) by λ [t]c(z) = tr [−2aa∗∂z̄∂zc] + 〈v|∂2z̄c(z) + ∂2zc(z) |v〉 , with v ∈ ⊗2 z the vector such that for all z1, z2 ∈ z, 〈z1 ⊗ z2,v〉 = 〈z1,la∗z2〉 is such that, for c in pm (z) ‖λ [t]c‖pm−2(z) ≤ 2‖t‖x (z) ‖a‖la2 (z) ‖c‖pm(z) . proof. we only have to remark that for any polynomial c in pp,q (z) the following estimates hold ‖tr [b∂z̄∂zc(z)]‖q−1←p−1 ≤ ‖b‖l1(z) ‖c‖q←p for any trace class operator b, and ∥∥〈v|∂2z̄c(z) ∥∥ q−2←p ≤ ‖v‖∨2 z ‖c‖q←p and that ‖v‖∨2 z = ‖la∗‖la 2 (z) ≤ ‖l‖l(z) ‖a‖la 2 (z). the same estimate holds for ∂ 2 zc(z) |v〉. 100 sébastien breteaux cubo 14, 2 (2012) we apply this result to the expression given in the theorem 3.2. proposition 8.15. let (qt)t be a continuous one parameter family of quadratic polynomials, ϕ the classical flow associated to (qt)t, and λ t the operator defined in theorem 3.2. then, for b in p≤m (z) ∥∥∥e ε 2 λt (b ◦ ϕ(t,0)) ∥∥∥ p(z) ≤ ‖b‖p(z) ‖ϕ(t,0)‖ m x (z) m∑ k=0 1 k! ( ε‖ϕ(t,0)‖x (z) ‖a(t,0)‖la 2 (z) )k where a is the c-antilinear part of ϕ. proof. it is enough to combine the propositions 4.4 and 8.14. remark 8. the norm ‖ϕ(t,0)‖x (z) is bigger than 1 as for any symplectic transformation t = l + a with l c-linear and a c-antilinear, l∗l = iz + a ∗a ≥ iz (see proposition a.4) and thus ‖t‖x (z) ≥ ‖l‖l(z) ≥ 1. appendices a r-linear symplectic transformations in this part we adapt and recall some results of [26] to fit our needs. let (z,〈·, ·〉) be a separable hilbert space over the complex numbers field c. the scalar products is linear with respect to the right variable and antilinear with respect to the left variable. we note autr (z) the group of r-linear continuous automorphisms on z. we define a symplectic form σ on z by σ(z1,z2) := ℑ 〈z1,z2〉 . definition a.1. a r-linear automorphism t is a symplectomorphism if it preserves the symplectic form, i.e. if ∀z1,z2 ∈ z, σ(tz1,tz2) = σ(z1,z2) . we note sp r (z) the set of symplectic transformations over the hilbert space z. it is a subgroup of autr (z). proposition a.2. a r-linear application t : z → z can be written as a sum of two applications respectively c-linear and c-antilinear in a unique way : t = t − iti 2 + t + iti 2 . definition a.3. let a be a (bounded) c-antilinear operator on the hilbert space z. we define its adjoint a∗ as the only antilinear operator such that ∀z1,z2 ∈ z, 〈z1,az2〉 = 〈z2,a∗z1〉 . cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 101 let t = l + a = z → z be a r-linear application with l c-linear and a c-antilinear. the adjoint t∗ of t is defined by t∗ = l∗ + a∗. proposition a.4. let t = l + a be a r-linear automorphism with l c-linear and a c-antilinear, then the following conditions are equivalent. (1) l + a is a symplectomorphism. (2) (l∗ − a∗) (l + a) = iz. (3) (l∗ + a∗) (l − a) = iz. (4) l∗l − a∗a = iz and l ∗a = a∗l. (5) l∗ − a∗ is a symplectomorphism. (6) l − a is a symplectomorphism. (7) ll∗ − aa∗ = iz and a ∗l = l∗a. proof. (1) ⇔ (2) let t = l + a a symplectomorphism, for all z1,z2 ∈ z, σ(z1,z2) = ℑ 〈z1,z2〉 = ℑ 〈(l + a)z1,tz2〉 = ℑ ( 〈z1,l∗tz2〉 + 〈z1,a∗tz2〉 ) = ℑ 〈z1,(l∗ − a∗)tz2〉 . replacing z1 by iz1 we get the same relation with a real part instead of an imaginary part and finally 〈z1, [(l∗ − a∗) (l + a) − iz]z2〉 = 0 and this in turn implies (l∗ − a∗) (l + a) = iz. we can reverse the order of these calculations in order to obtain the first equivalence. (2) ⇔ (3) the c-linearity and antilinearity properties of l and a give (l∗ − a∗) (l + a)i = i(l∗ + a∗) (l − a) so that we get the equivalent condition (3). ((2) and (3)) ⇔ (4) the sum and the difference of the equations of (2) and (3) give (4) and the sum and difference of the equations in (4) give (2) and (3). (1) ⇔ (5) from (1) and (3) we know that the inverse of a symplectomorphism t = l + a is t−1 = l∗−a∗ which is necessarily a symplectomorphism too, and thus (1) ⇒ (5). we get (5) ⇒ (1) exchanging t and t−1. (1) ⇔ (6) ⇔ (7) is easily deduced from the previous equivalences. 102 sébastien breteaux cubo 14, 2 (2012) proposition a.5. let t = l + a be a symplectomorphism with l c-linear and a c-antilinear, then l is invertible. proof. from proposition a.4 we get l∗l = iz + a ∗a ≥ iz and ll∗ = iz + aa∗ ≥ iz and thus l and l∗ are one to one. as l∗ is one to one we get ranl = (kerl∗) ⊥ = {0} ⊥ = z. it is now enough to show that the range of l is closed. pick a vector y ∈ z, there is a sequence (xn) ∈ zn such that lxn → y. the relation l∗l ≥ iz gives |lxm − lxn| ≥ |xn − xm|. the left hand part of the inequality goes to 0 for m,n → ∞, so that (xn) is a cauchy sequence and thus converges to a limit x. by continuity of l, lx = y and l is indeed one to one. definition a.6. an application c from z to z is a conjugation if and only if it satisfies the following conditions. (1) c is r-linear. (2) c2 = iz. (3) for all z1, z2 in z, 〈cz1,z2〉 = 〈cz2,z1〉. remark 9. it follows from the third condition in this definition that a conjugation is antilinear. one may define different conjugations on the same hilbert space over c (even for a one dimensional hilbert space). as an example one can consider a hilbert basis (ej) and define the application c : ∑ j αjej 7→ ∑ j αjej. definition a.7. let c be a conjugation on the hilbert space z. the real and imaginary parts of a vector z ∈ z (with respect to the conjugation c) are defined as ℜz := z + cz 2 and ℑz := z − cz 2i . they verify z = ℜz + iℑz. the space ec r := ℜz = ℑz is a subspace of z as r-vector space, 〈·, ·〉 restricted to ec r is a real scalar product and e = ec r ⊕ iec r . let f be a r-linear application on z, then we can define the applications from ec r to itself α : z 7→ ℜf(z) , γ : z 7→ ℜf(iz) , β : z 7→ ℑf(z) , δ : z 7→ ℑf(iz) . then, if a, b ∈ ec r , then f(a + ib) = α(a) + iβ(a) + γ(ib) + iδ(ib), and f can be represented as an application on ec r × ec r by the matrix ( α γ β δ ) . the following relations hold with the above sign if f is c-linear and with the below sign if f is c-antilinear: β = ∓γ and α = ±δ and f∗ is represented by the matrix ( αt ∓βt βt ±αt ) . cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 103 we want to show a reduction result for the symplectomorphisms in the spirit of the polar decomposition, in the case of an implementable symplectomorphism (see definition 8.1 and theorem 8.4). theorem a.8. let t be an implementable symplectomorphism. then t = uecρ where • u is a unitary operator, • c is a conjugation, • ρ is a hilbert-schmidt, self-adjoint, non-negative operator commuting with c. remark 10. the operator u is the unitary operator of the polar decomposition l = u |l| of the c-linear part of t. the conjugation c is a specific conjugation associated with l and will be constructed during the proof and ρ = arg cos |l|. proof. let us write t = l + a with l c-linear and a c-antilinear. with l = u |l| the polar decomposition of l we get t = u(|l| + u∗a) so that it is enough to show the two next lemmas. lemma a.1. let (e,〈·, ·〉) be a finite-dimensional hilbert space over c. let f : e → e be a c-antilinear application such that ff∗ = ie and f = f ∗ . then there exists an orthonormal basis (uj) of e such that ∀j, f(uj) = uj . proof. let us consider an arbitrary conjugation c0 on e and the ( α β β −α ) “matrix” of f (as a r-linear operator) on e = e c0 r ⊕ iec0 r identified with e c0 r × ec0 r . the matrix associated to f∗ is ( αt βt βt −αt ) so that the relation f = f∗ gives α = αt and β = βt . from ff∗ = ie we deduce α 2 + β2 = id and αβ = βα. we can thus diagonalize simultaneously α and β, and so in a convenient basis of e c0 r the matrix of f is of the form   ... 0 ... 0 λαj λ β j 0 ... 0 ... ... 0 ... 0 λ β j −λ α j 0 ... 0 ...   . 104 sébastien breteaux cubo 14, 2 (2012) we can thus confine ourself to the case of a space e of complex dimension 1 and of f with a matrix of the form ( α β β −α ) with α and β real numbers. we search a normalized vector z = ( cos θ sin θ ) = ( x y ) and a real λ such that f(z) = λz, i.e. λ( x y ) = ( αx+βy −αy+βx ) = ( x y −y x ) ( α β ) = √ α2 + β2 ( cos θ sin θ − sin θ cos θ )( cos φ sin φ ) = √ α2 + β2 ( cos(φ−θ) sin(φ−θ) ) so that if we choose θ such that φ − θ = θ we get the desired result with λ = √ α2 + β2. finally, from ff∗ = ie we deduce that λ = 1 and the result follows. lemma a.2. let t = l+a be an implementable symplectomorphism with l c-linear self-adjoint and positive, a c-antilinear. then l and a commute, there exist a conjugation c commuting with l and a such that ac is self-adjoint and non-negative and t = ecρ with ρ = arg coshl = arg sinh (ac) a hilbert-schmidt, non-negative and self-adjoint operator commuting with c. proof. as aa∗ ∈ l1 (z), aa∗ = ∑ j λ 2 j |ej〉〈ej|, with λj ∈ r and ∑ j λ 2 j < ∞, from l 2 = iz +aa ∗ we deduce l2 = ∑ j µ 2 j |ej〉〈ej| with µj = √ 1 + λ2j and thus l = ∑ j µj |ej〉〈ej|. from the equivalent characterizations of a symplectomorphism we get l2 − aa∗ = iz and l 2 − a∗a = iz multiplying the first equality on the right and the second on the left by a and computing the difference we get [ l2,a ] = 0. as l is self-adjoint and positive one can use the functional calculus and l = √ l2 to obtain [l,a] = 0. from [l,a] = 0, l = l∗ and the characterizations of a symplectomorphism, we also get al = la = l∗a = a∗l so that (a − a∗)l = 0 and from the invertibility of l one deduces a = a∗. the proper subspaces associated with l and ker (l − µiz), are thus stable by the action of a (and finite-dimensional). we also remark that on ker (l − µiz), aa ∗ = l2 − iz = (µ 2 − 1)iz, so that two cases are possible: µ = 1, thena = 0 or µ > 1, then 1√ µ2 − 1 a 1√ µ2 − 1 a∗ = iz . we apply lemma a.1 to the c-antilinear applications induced by the applications a/ √ µ2 − 1 on the hilbert spaces ker (l − µiz). this provides us with a hilbert basis (ej) of z which diagonalizes both l and a. we can also define a conjugation c (∑ j αjej ) = ∑ j αjej. this conjugation commutes with l and a, and ac is clearly a non-negative self-adjoint operator and so is necessarily √ aa∗. we finally get for every vector ej of the basis the relations lej = µjej and aej = λjej cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 105 with µ2j − λ 2 j = 1, and thus one can define ρj = arg coshµj (ρj = arg sinhλj as λj ≥ 0) and so we can define ρ = arg coshl = arg sinhac so that t = ecρ. b relations between weyl and wick symbols in finite dimension we want to use the relation between the weyl and wick symbols associated to a same wick polynomial in finite dimension, working with z = cr we have b = 1 (πε/2) r b̆ ∗ e − |z|2 ε/2 where b is the wick symbol and b̆ is the weyl symbol and bwick = b̆weyl. we want to get rid of the convolution and for this we use the fourier transform ff(x′) = 1 (2π) r ˆ r2r e−ix.x ′ f(x)dx where x,x′ ∈ r2r ∼= cr. the inverse fourier transform is then f−1f(x) = 1 (2π) r ˆ r2r eix.x ′ f(x′)dx′ . we can use the formulae f (f ∗ g) = (2π)r ff.fg f [ e−α |x|2 2 ] (x′) = 1 αr e− |x ′| 2 2α f−1 (x × ·) f = dx . we then obtain with m = 2n fb(z′) = (2π) r (πε/2) r f [ e − |z|2 ε/2 ] fb̆(z′) = ( 4 ε )r ( ε 4 )r e − ε 8 |z ′| 2 fb̆(z′) = e − ε 8 |z ′| 2 fb̆(z′) and b = f−1e− ε 8 |z ′| 2 fb̆ = e − ε 8 f−1|z′| 2f b̆ = e ε 2 ∂z.∂z̄b̆ 106 sébastien breteaux cubo 14, 2 (2012) using the fact that f−1 |z′|2 f = d2(x,ξ) = −4 × 1 2 (∂x − i∂ξ) . 1 2 (∂x + i∂ξ) = −4∂z.∂z̄ . it is clear that if b̆ is a polynomial in p≤m (z), then b is in this class of polynomials, as we can see deriving the convolution product. we want to show that the map p≤m (z) → p≤m (z) b̆ 7→ b = 1 (πε/2) n b̆ ∗ e − |z|2 ε/2 is bijective. as the dimension of z is finite, the dimension of p≤m (z) is finite and it is enough to show that this map is one to one. for this we want to justify that on the part of main degree this application is the identity. this is obvious from the following facts: • ∂qz̄∂ p zb = 1 (πε/2)r ∂ q z̄∂ p zb̆ ∗ e− |z|2 ε/2 • this application is the identity on the constants. thus we can also consider the reverse application that we will improperly note b̆ = e− ε 2 ∂z.∂z̄b. c symplectic fourier transform let us then consider the symplectic fourier transform on l2 ( cd; c ) ≡ l2 ( r2d ) with z = x + iy, defined by fσ (f) (z) = ˆ e i2πσ(z,z′)f(z′)l(dz′) with σ(z,z′) = ℑ 〈z,z′〉 = ℑ [〈x,x′〉 + 〈y,y′〉 + i〈x,y′〉 − i〈y,x′〉] and l denotes the lebesgue measure. we list here some properties of the symplectic fourier transform. (1) inverse. (fσ)−1 = fσ (2) convolution. fσ (f ∗ g) = fσf.fσg (3) composition with a symplectic transformation. let t be a symplectomorphism, then fσ [f(t·)] (z) = fσ [f] (tz) . cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 107 (4) gaussians. for a > 0, fσ [ e−a|·| 2 ] (z) = ( π a )d e−π 2 |z|2/a . (5) derivation. we consider the derivations ∂z = 1 2 (∂x − i∂y) and ∂z̄ = 1 2 (∂x + i∂y) then − 1 π ∂z.z0 = fσ (z̄.z0×) fσ and 1 π z̄0.∂z̄ = fσ (z̄0.z×) fσ . acknowledgment the author would like to thank francis nier and zied ammari for profitable discussions. received: october 2011. revised: november 2011. references [1] z. ammari and f. nier. mean field limit for bosons and infinite dimensional phase-space analysis. ann. henri poincaré,9(8):1503-1574, 2008. [2] j. c. baez, i. e. segal, and z.-f. zhou. introduction to algebraic and constructive quantum feld theory. princeton series in physics. princeton university press, princeton, nj, 1992. 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[26] l. polley, g. reents, and r. f. streater. some covariant representations of massless boson fields. j. phys. a, 14(9):2479-2488, 1981. cubo 14, 2 (2012) higher order terms for the quantum evolution of a wick ... 109 [27] m. reed and b. simon. methods of modern mathematical physics. ii. fourier analysis, selfadjointness. academic press [harcourt brace jovanovich publishers], new york, 1975. [28] i. rodnianski and b. schlein. quantum fuctuations and rate of convergence towards mean field dynamics. comm. math. phys., 291(1):31-61, 2009. [29] d. shale. linear symmetries of free boson fields. trans. amer. math. soc., 103:149-167, 1962. introduction wick calculus with polynomial observables definitions some examples of wick quantizations calculus main results and a simple example classical evolution of a wick polynomial under a quadratic evolution construction of the classical flow without the term the strongly continuous dynamical system associated with (t) construction of the classical flow with the term composition of a wick polynomial with the classical evolution quantum evolution of a wick polynomial without the term with the term removal of the part a dyson type expansion formula for the wick symbol of the evolved quantum observable an exponential type expansion formula for the wick symbol of the evolved observable quantum evolution as a bogoliubov implementation action of bogoliubov transformations on wick symbols action of bogoliubov transformations on weyl quantizations of polynomials in finite dimension action of bogoliubov transformations on wick quantization of polynomials in finite dimension extension to infinite dimension on a ``cylindrical'' class of polynomials extension to general polynomials an evolution formula for the wick symbol estimates r-linear symplectic transformations relations between weyl and wick symbols in finite dimension symplectic fourier transform cubo a mathematical journal vol.14, no¯ 03, (59–61). october 2012 erratum to “on the group of strong symplectic homeomorphisms” augustin banyaga department of mathematics, the pennsylvania state university, university park, pa 16802. email: banyaga@math.psu.edu, augustinbanyaga@gmail.com abstract we give a proof of the estimate (1.1) which is the main ingredient in the proof that the set ssympeo(m, ω) of strong symplectic homeomorphisms of a compact symplectic manifold (m, ω) forms a group [1]. resumen probamos la estimación (1.1) que es el principal elemento en la demostración que el conjuntos ssympeo(m, ω) de homeomorfismos simplécticos fuertes de una variedad simpléctica compacta (m, ω) genera un grupo [1]. keywords and phrases: c0-symplectic topology; strong symplectic homeomorphism. 2010 ams mathematics subject classification: 53d05; 53d35. 60 augustin banyaga cubo 14, 3 (2012) 1 erratum in the paper [1] mentioned in the title, the “constant e” at page 60 may be infinite ( so proposition 2 is meaningless). therefore, some of the estimates on pages 63 to 65 based on e, needed to show that ∫1 0 osc(vnt − v m t ) → 0 (1.1) as n, m → ∞ may not hold true. here is a direct proof of (1.1). first simplify the notations by writing khm for (µkt ) ∗ h m, h for h and omitting t. the function vn := vnt satisfies nh n − hn = dvn. fix a point ∗ in m and for each x ∈ m, pick an arbitrary curve γx from ∗ to x, then un(x) := ∫ γx (nhn − hn) = vn(x) − vn(∗). (the definition of un(x) is independent of the choice of the curve γx). hence osc(u n − um) = osc(vn − vm). since osc(f) ≤ 2|f|, where |.| is the uniform sup norm, we need to show that ∫1 0 | ∫ γx (nhn − hn) − (mhm − hm)|dt ≤ ∫1 0 | ∫ γx (nhn − mhm)|dt + ∫1 0 | ∫ γx (hn − hm)|dt, (1.2) goes to zero , when n, m are sufficiently large. the last integral tends to zero when n, m are large: indeed, ∫1 0 | ∫ γx (hn − hm)|dt = ∫1 0 | ∫1 0 (hn − hm)(γx(u))(γ ′ x(u)du)|dt| ≤ a ∫1 0 |hn − hm|dt, (1.3) where a = supu |γ ′ x(u)|. this goes to 0 since h n is a cauchy sequence. to prove that ∫1 0 | ∫ γx (nhn − mhm)|dt tends to zero when n, m → ∞, we write: | ∫ γx (nhn − mhm)| ≤ | ∫ γx (nhn − mhn)| + | ∫ γx (m(hn − hm) − n0(h n − hm))| + | ∫ γx (n0)(h n − hm)|, (1.4) for some large n0. cubo 14, 3 (2012) erratum to “on the group of ...” 61 the integral ∫1 0 | ∫ γx (n0)(h n − hm)|dt = ∫1 0 | ∫1 0 (hn − hm)(γn 0 (u))(dµ n0γ′x(u)du)|dt| ≤ b ∫1 0 |hn − hm|dt, (1.5) where b = supu|dµ n0γ′x(u)| goes to zero when n, m → ∞ since h n is a cauchy sequence and dµn0 is bounded. ( here γk = µ k(γx)). we now show that ∫ γx (nhn − mhn) = ∫ γn hn − ∫ γm hn tends to zero when n, m → ∞ let d0 be a distance induced by a riemmanian metric g and let r be its injectivity radius. for n, m large enough, supxd0(µ n t (x), µ m t (x)) ≤ r/2. it follows that there is a homotopy f : [0, 1] × m → m between µn and µm , i.e f(0, y) = µn(y) and f(1, y) = µm(y) and we may define f(s, y) to be the unique minimal geodesic v y mn(s) joining µ n(y) to µm(y). see [[3]] ( theorem 12.9). let �(s, u) =: {f(s, γn(u)), 0 ≤ s, u ≤ 1} since by stokes’ theorem, ∫ ∂� hn = 0, ∫ γn hn− ∫ γm hn = ∫ l hn− ∫ l′ hn where l, and l′ are the geodesics vxmn and v ∗ mn. the integral over l is bounded by sups|h n(vxmn(s)|d0(µ n t (x), µ m t (x)), because the speed of the geodesics l, l′ is bounded by d0(µ n t (x), µ m t (x)). this integral tends to zero when n, m → ∞ since hn is also bounded . analogously for the integral over l′. the same argument apply to hn − hm with the geodesics l, l′ replaced by vxmn0 and v ∗ mn0 . this finishes the proof of (1.1). remark : we will show in a forthcoming paper [2] that (1.1) is the main ingredient in the proof of the main theorem of [1]. acknowledgments i would like to thank mike usher to have pointed out to me that proposition 2 in [1] does not yield a finite constant e. received: january 2012. revised: january 2012. references [1] a. banyaga, on the group of strong symplectic homeomorphisms, cubo, vol 12, no 03 (2010), 49-69 [2] a. banyaga, on the group of strong symplectic homeomorphisms, ii, preprint [3] t. brocker and k. janich, introduction to differential topology, cambrigde university press, 1982 cubo a mathemati al journal vol.15, n o 03, (59�70). o tober 2013 on quasionformally �at and quasionformally semisymmetri generalized sasakian-spa e-forms d.g. prakasha department of mathemati s, karnatak university, dharwad-580 003 karnataka state, india. prakashadg�gmail. om h.g. nagaraja department of mathemati s central college campus, bangalore university, bangalore-560 001, india. hgnraj�yahoo. om abstract the obje t of the present paper is to study quasionformally �at and quasionformally semisymmetri generalized sasakian-spa e-forms. resumen el objeto del artí ulo a tual es estudiar formas de espa io sasakian uasionforma ionales planas y uasionforma ionales generalizadas semisimétri as. keywords and phrases: generalized sasakian-spa e-forms, quasionformally �at, quasionformally semisymmetri , einstein manifold, s alar urvature. 2010 ams mathemati s subje t classi� ation: 53c25, 53d15. 60 d.g. prakasha & h.g. nagaraja cubo 15, 3 (2013) 1 introdu tion the notion of generalized sasakian-spa e-forms was introdu ed and studied by alegre et al [1℄ with several examples. a generalized sasakian-spa e-form is an almost onta t metri manifold (m, φ, ξ, η, g) whose urvature tensor is given by r(x, y)z = f1{g(y, z)x − g(x, z)y} + f2{g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz} + f3{η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ} where f1, f2, f3 are di�erentiable fun tions on m and x, y, z are ve tor �elds on m. in su h ase we will write the manifold as m(f1, f2, f3). this kind of manifolds appears as natural generalization of the sasakian-spa e-forms by taking: f1 = c + 3 4 and f2 = f3 = c − 1 4 , where c denotes onstant φ-se tional urvature. the φ-se tional urvature of generalized sasakianspa e-forms m(f1, f2, f3) is f1 + 3f2. moreover, osymple ti spa e-forms and kenmotsu spa eforms are also parti ular ase of generalized sasakian-spa e-forms. in the re ent paper p. alegre and a. carriazo [2℄ studied onta t metri and trans-sasakian generalized sasakian-spa e-forms. generalized sasakian-spa e-forms have been studied by several authors, viz., [5,6,10,12℄. in riemannian geometry, many authors have studied urvature properties and to what extent they determined the manifold itself. two important urvature properties are �atness and symmetry. as a generalization of lo al symmetri spa e, the notion of semisymmetri spa e [13℄ is de�ned as r(x, y) · r = 0, where r(x, y) a ts on r as a derivation. in this onne tion, the onformal �atness and lo al symmetry of generalized sasakian-spa e-forms was studied in [10℄. also in [6℄, generalized sasakian-spa e-forms with vanishing proje tive urvature tensor and some symmetry properties have been onsidered. motivated by these fa ts, in this paper we study the �atness and semisymmetry property of generalized sasakian-spa e-form regarding the quasionformal urvature tensor. the notion of the quasionformal urvature tensor was given by yano and sawaki [14℄. a ording to them in a (2n + 1)-dimensional (n > 1) almost onta t metri manifold the quasionformal urvature tensor c̃ is de�ned by c̃(x, y)z = ar(x, y)z + b[s(y, z)x − s(x, z)y + g(y, z)qx − g(x, z)qy] − r 2n + 1 ( a 2n + 2b ) [g(y, z)x − g(x, z)y] (1) where a and b are onstants and r, s, q and r are the riemannian urvature tensor of type (1, 3), the ri i tensor of type (0, 2), the ri i operator de�ned by g(qx, y) = s(x, y) and the s alar cubo 15, 3 (2013) on quasionformally �at and quasionformally semisymmetri . . . 61 urvature of the manifold respe tively. if a = 1 and b = − 1 2n−1 , then (1) takes the form c(x, y)z = r(x, y)z − 1 2n − 1 [s(y, z)x − s(x, z)y + g(y, z)qx − g(x, z)qy] + r (2n)(2n − 1) [g(y, z)x − g(x, z)y] = c(x, y)z (2) where c is the onformal urvature tensor [8℄. thus onformal urvature tensor is a parti ular ase of quasionformal urvature tensor. a manifold (m, φ, ξ, η, g) shall be alled quasionformally �at if the quasionformal urvature tensor c̃ = 0. it is known that the quasionformally �at manifold is either onformally �at if a 6= 0 or einstein if a = 0 and b 6= 0 [3℄. if the manifold (m, φ, ξ, η, g) satis�es r(x, y).c̃ = 0, then the manifold is said to be quasionformally semisymmetri manifold. a manifold (m, φ, ξ, η, g) is said to be ri i symmetri if r · s = 0 holds on m [7℄. the lass of ri i semisymmetri manifolds in ludes the set of ri i symmetri manifolds (∇s = 0) as a proper subset. every semisymmetri manifold is ri i symmetri . the onverse is not true. in the present paper quasionformally �at and quasionformally semisymmetri generalized sasakian-spa e-forms are studied. the paper is organized as follows: se tion 2 of this paper ontains some preliminary results on generalized sasakian-spa e-forms. in se tion 3, we study quasionformally �at generalized sasakian-spa e-forms and obtain ne essary and su� ient onditions for a generlized sasakian-spa e-form to be quasionformally �at. also, we onsider quasionformally ri i tensor and quasionformally ri i symmetri generalized sasakian spa e-forms. in the next se tion, we deal with quasionformally semisymmetri generalized sasakian-spa e-forms and it is proved that a generalized sasakian-spa e-form is quasionformally semisymmetri if and only if the spa e-form is quasionformally �at and f1 = f3. 2 preliminaries an odd-dimensional manifold m2n+1 is said to admit an almost onta t stru ture (φ, ξ, η), if it arries a tensor �eld φ of type (1, 1), a ve tor �eld ξ and a 1-form η satisfying φ2 = −i + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0. (3) if g is a ompatible riemannian metri with (φ, ξ, η) su h that g(φx, φy) = g(x, y) − η(x)η(y) (4) or equivalently, g(x, ξ) = η(x), g(x, φy) = −g(φx, y) (5) for all ve tor �elds x, y on m2n+1, then m2n+1 be omes an almost onta t metri manifold with an almost onta t metri stru ture (φ, ξ, η, g). an almost onta t metri stru ture is alled a onta t metri stru ture if g(x, φy) = dη(x, y) 62 d.g. prakasha & h.g. nagaraja cubo 15, 3 (2013) an almost onta t metri manifold is sasakian if and only if (∇xφ)y = g(x, y)ξ − η(y)x (6) for all ve tor �elds x, y on m2n+1. for a (2n + 1)-dimensional generalized sasakian-spa e-form we have [1℄ r(x, y)z = f1{g(y, z)x − g(x, z)y} + f2{g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz} + f3{η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ}, (7) qx = (2nf1 + 3f2 − f3)x − (3f2 + (2n − 1)f3)η(x)ξ, (8) s(x, y) = (2nf1 + 3f2 − f3)g(x, y) − (3f2 + (2n − 1)f3)η(x)η(y), (9) r = 2n(2n + 1)f1 + 6nf2 − 4nf3 (10) for all ve tor �elds x, y, z. by virtue of equations(7) and (9), we have η(r(x, y)z) = (f1 − f3){g(y, z)η(x) − g(x, z)η(y)}, (11) r(x, y)ξ = (f1 − f3){η(y)x − η(x)y}, (12) r(ξ, x)y = (f1 − f3){g(x, y)ξ − η(y)x}, (13) s(x, ξ) = 2n(f1 − f3)η(x), (14) s(ξ, ξ) = 2n(f1 − f3). (15) the above results will be used in the next se tions. now we would like to re olle t some of the examples of generalized sasakian-spa e-forms. example 1: ( [11℄)a osymple ti -spa e-form, i.e., a osymple ti manifold with onstant φse tional urvature c, is a generalized sasakian-spa e-form with f1 = f2 = f3 = c/4. example 2:( [9℄)a kenmotsu-spa e-form, i.e., a kenmotsu manifolds with onstant φ-se tional urvature c, is a generalized sasakian-spa e-form with f1 = (c − 3)/4 and f2 = f3 = (c + 1)/4. example 3: ( [1℄)let n(f1, f2) be a generalized omplex-spa e-form. then, the warped produ t m = r ×f n, endowed with the almost onta t metri stru ture (φ, ξ, η, gf), is a generalized sasakian-spa e-form m(f1; f2; f3) with fun tions: f1 = (f1 ◦ π) − f ′2 f2 , f2 = f2 ◦ π f2 , f3 = (f1 ◦ π) − f ′2 f2 + f′′ f . in parti ular if n(c) is a omplex-spa e-form, we obtain the generalized sasakian-spa e-form m ( c − 4f′2 4f2 , c 4f2 , c − 4f′2 4f2 + f" f ) . hen e, the warped produ ts r ×f c n , r ×f cp n(4) and r ×f ch n(−4) are generalized sasakianspa e-forms. cubo 15, 3 (2013) on quasionformally �at and quasionformally semisymmetri . . . 63 example 4: r ×f c m is a generalized sasakian-spa e-form with f1 = − (f′)2 f2 , f2 = 0, f3 = − (f′1)2 f2 + f′′ f , where f = f(t). 3 quasionformally �at generalized sasakian-spa e-forms if the generalized sasakian-spa e-form m(f1, f2, f3) under onsideration is quasionformally �at, then we have from (1) r(x, y, z, w) = b a [s(x, z)g(y, w) − s(y, z)g(x, w) (16) +s(y, w)g(x, z) − s(x, w)g(y, z)] + r (2n + 1)a [ a 2n + 2b][g(y, z)g(x, w) − g(x, z)g(y, w)], where a and b are onstants and r(x, y, z, w) = g(r(x, y)z, w). now putting z = ξ in (16) and using (4), (12) and (14) we get (f1 − f3)[g(x, w)η(y) − g(y, w)η(x)] (17) = 2n(f1 − f3) b a [g(y, w)η(x) − g(x, w)η(y) +s(y, w)η(x) − s(x, w)η(y)] + r (2n + 1)a [ a 2n + 2b][g(x, w)η(y) − g(y, w)η(x)]. again putting x = ξ in (17) and using (4) and (14) it follows that s(y, w) = ag(y, w) + bη(y)η(w), (18) where a = [− a b (f1 − f3) − 2n(f1 − f3) + r (2n + 1)b ( a 2n + 2b)] (19) and b = [ a b (f1 − f3) + 4n(f1 − f3) − r (2n + 1)b ( a 2n + 2b)]. (20) here a+b = 2n(f1 −f3). in the equation (18) putting y = w = {ei}, where {ei} is an orthonormal basis of the tangent spa e at ea h point of the manifold and taking summation over i, 1 ≤ i ≤ 2n+1, we get r = (2n + 1)a + b. (21) now with the help of (19) and (20) the equation (21) gives [a + (2n − 1)b][ r 2n + 1 − 2n(f1 − f3)] = 0. (22) 64 d.g. prakasha & h.g. nagaraja cubo 15, 3 (2013) if a + (2n − 1)b = 0 and a 6= 0 6= b. then from (1) it follows that c̃(x, y)z = ac(x, y)z, (23) where c(x, y)z denotes the weyl onformal urvature tensor. but, under the onsideration c̃ = 0. so the quasionformal �atness and onformally �atness are equivalent. this implies that c = 0. if a + (2n − 1)b 6= 0 and a 6= 0. then from (22). r = 2n(2n + 1)(f1 − f3). (24) so by omparing (10) and (24) we have 3f2 + (2n − 1)f3 = 0. (25) by taking a ount of (25) in (9), we get s(x, y) = 2n(f1 − f3)g(x, y). (26) this shows that, m(f1, f2, f3) is an einstein. thus we state the following: theorem 1. a quasionformally �at generalized sasakian-spa e-from is either onformally �at or an einstein manifold with s alar urvature r = 2n(2n + 1)(f1 − f3). in the above theorem we have seen if a + (2n − 1)b = 0 and a 6= 0 6= b, then it follows that a quasionformally �at generalized sasakian-spa e-form is onformally �at. but, it is known that [10℄ a (2n + 1)-dimensional (n > 1) generalized sasakian-spa e-form m(f1, f2, f3) is onformally �at if and only if f2 = 0. so in this ase m(f1, f2, f3) is quasionformally �at if and only if f2 = 0. on the other hand, if a + (2n − 1)b 6= 0 and a 6= 0 then we have (24). by omparing the equations (10) and (24), one an get (25). conversely, suppose that (25) holds. then in view of (7), (9) and (25), we an write the equation (1) as c̃(x, y, z, w) = a 1 − 2n f2[g(y, z)g(x, w) − g(x, z)g(y, w)] +af2[g(x, φz)g(φy, w) − g(y, φz)g(φx, w) + 2g(x, φy)g(φz, w)] + 3a 1 − 2n f2[g(y, w)η(x)η(z) − g(x, w)η(y)η(z) +g(x, z)η(y)η(w) − g(y, z)η(x)η(w)], (27) where c̃(x, y, z, w) = g(c̃(x, y)z, w). repla ing x by φx and y by φy in (27) we get c̃(φx, φy, z, w) = a 1 − 2n f2[g(φy, z)g(φx, w) − g(φx, z)g(φy, w)] +af2[g(φx, φz)g(φ 2y, w) − g(φy, φz)g(φ2x, w) + 2g(φx, φ2y)g(φz, w)]. (28) cubo 15, 3 (2013) on quasionformally �at and quasionformally semisymmetri . . . 65 putting y = w = ei, where {ei} is an orthonormal basis of the tangent spa e at ea h point of the manifold, and taking summation over i, (1 ≤ i ≤ 2n + 1), we get 2n+1 ∑ i=1 c̃(φx, φei, z, ei) = a 2n − 1 f2g(φx, φz) + af2[−g(φx, φz)g(φei, φei) + 3g(φ 2x, φ2z)]. (29) again putting x = z = ei, where {ei} is an orthonormal basis of the tangent spa e at ea h point of the manifold, and taking summation over i, (1 ≤ i ≤ 2n + 1), we get after simpli� ation f2 = 0 with a 6= 0. then in view of (25), we get f3 = 0. therefore, we obtain from (7) that r(x, y)z = f1{g(y, z)x − g(x, z)y}. (30) from (30) we have s(x, y) = 2nf1g(x, y) and r = 2n(2n + 1)f1. hen e in view of (1), we have c̃(x, y)z = 0. this leads to the following: theorem 2. let m(f1, f2, f3) be a (2n+1)-dimensional (n > 1) generalized sasakian-spa e-form. then m(f1, f2, f3) is quasionformally �at if and only if one of the following statements is true: (i) a + (2n − 1)b = 0, a 6= 0 6= b and f2 = 0. (ii) a + (2n − 1)b 6= 0, a 6= 0 and 3f2 + (2n − 1)f3 = 0. in a (2n + 1)-dimensional (n > 1) manifold (m, φ, ξ, η, g), let {ei}, i = 1, 2, ..., 2n + 1 be a lo al orthonormal basis. then the quasionformal urvature tensor c̃(x, y)z de�ned as in (1), we an de�ne a symmetri tensor of type (0, 2) alled as quasionformal ri i tensor and whi h is denoted by s c̃ (x, y) = 2n+1 ∑ i=1 c̃(ei, x, y, ei), (31) where ∑ 2n+1 i=1 c̃(ei, x, y, ei) = ∑ 2n+1 i=1 g(c̃(ei, x)y, ei). from (31)and (1), we have s c̃ (x, y) = {a + (2n − 1)b}{s(x, y) − r 2n + 1 g(x, y)}. (32) we �rst assume that a (2n + 1)-dimensional generalized sasakian-spa e-form m(f1, f2, f3) is ri i semisymmetri . that is, (r(x, y).s)(z, w) = −s(r(x, y)z, w) − s(z, r(x, y)w) = 0. now, sin e the urvature tensor r of type (0, 4), de�ned by g(r(x, y)z, w) = r(x, y, z, w) 66 d.g. prakasha & h.g. nagaraja cubo 15, 3 (2013) is skew-symmetri where r is the urvature tensor of type (1, 3), we get from (33) and (11) by taking a ount that a + (2n − 1)b 6= 0 s c̃ (r(x, y)z, w) + s c̃ (z, r(x, y)w) = 0 (33) whi h implies that (r(x, y).s c̃ )(z, w) = 0. so the spa e-form m(f1, f2, f3) is quasionformally ri i semisymmetri . again, let us suppose that the spa e-form is quasionformally ri i semisymmetri , that is, r.s c̃ = 0 holds in m(f1, f2, f3). then (33) holds. now using (33), and the skew-symmetri properties of r we get after simpli� ation r.s = 0, whi h implies that the spa e-form is ri i semisymmetri . hen e the following theorem holds: theorem 3. a (2n + 1)-dimensional (n > 1) generalized sasakian-spa e-form m(f1, f2, f3) is ri i semisymmetri if and only if it is quasionformally ri i semisymmetri provided that a + (2n − 1)b 6= 0. 4 quasionformally semisymmetri generalized sasakianspa e-forms in this se tion we onsider a generalized sasakian-spa e-form m(f1, f2, f3) satisfying the ondition r(x, y) · c̃ = 0. (34) then we obtain from (1) by using (4), (12) and (14) η(c̃(x, y)z) = { (a + 2nb)(f1 − f3) − r (2n + 1) { a 2n + 2b } } [g(y, z)η(x) −g(x, z)η(y)] + b[s(y, z)η(x) − s(x, z)η(y)]. (35) on taking z = ξ in the equation (35), we get η(c̃(x, y)ξ) = 0. (36) again putting x = ξ in the equation (35), we have η(c̃(ξ, y)z) = { (a + 2nb)(f1 − f3) − r (2n + 1) { a 2n + 2b } } [g(y, z) −η(y)η(z)] + b[s(y, z) − 2n(f1 − f3)η(y)η(z)]. (37) in virtue of (34) we get r(x, y)c̃(u, v)w − c̃(r(x, y)u, v)w −c̃(u, r(x, y)v)w − c̃(u, v)r(x, y)w = 0. (38) cubo 15, 3 (2013) on quasionformally �at and quasionformally semisymmetri . . . 67 whi h implies that (f1 − f3){c̃(u, v, w, y) − η(y)η(c̃(u, v)w) +η(u)η(c̃(y, v)w) + η(v)η(c̃(u, y)w) +η(w)η(c̃(u, v)y) − g(y, u)η(c̃(ξ, v)w) −g(y, v)η(c̃(u, ξ)w) − g(y, w)η(c̃(u, v)ξ)} = 0. (39) putting u = y in (39) and with the help of (35) and (36) we get either f1 = f3 (40) or {c̃(y, v, w, y) + η(w)η(c̃(y, v)y) −g(y, y)η(c̃(ξ, v)w) − g(y, v)η(c̃(y, ξ)w)} = 0. (41) let {e1, e2, ..., e2n+1} is an orthonormal basis of the tangent spa e at ea h point of the manifold. putting y = ei in (41) and taking summation over i, (1 ≤ i ≤ 2n + 1), and using (35), (37) we get s(v, w) = a′g(v, w) + b′η(v)η(w)} (42) where a′ = 2n(a + 2nb)(f1 − f3) − rb a − b (43) and b′ = −2n(2n + 1)b(f1 − f3) + rb a − b . (44) here a′ + b′ = 2n(f1 − f3). now ontra ting (42) we get r = (2n + 1)a′ + b′. (45) by (43) and (44) the equation (45) gives (a + (2n − 1)b)(r − 2n(2n + 1)(f1 − f3)) = 0. (46) therefore, either a + (2n − 1)b = 0 or r = 2n(2n + 1)(f1 − f3). (47) from (43) and (47) we obtain a′ = 2n(f1 − f3). (48) by (44) and (47) we get b′ = 0. (49) so, from (42), (48) and (49) we have s(v, w) = 2n(f1 − f3)g(v, w). (50) 68 d.g. prakasha & h.g. nagaraja cubo 15, 3 (2013) therefore, m(f1, f2, f3) is an einstein manifold. now with the help of (47) and (50) the equations (35) and (37) imply that η(c̃(x, y)z) = 0 (51) and η(c̃(ξ, y)z) = 0 (52) respe tively. so using (36),(51) and (52) in (39) we get c̃(u, v, w, y) = 0. (53) therefore, by taking a ount of (40) and (53), we have either f1 = f3 or m(f1, f2, f3) is quasionformally �at. conversely, if f1 = f3 then from (13) r(ξ, u) = 0. then obviously the ondition r(ξ, u) · c̃ = 0, that is, quasionformally semisymmetri ondition is satis�ed. again if the spa e-form is quasionformally �at, then learly it is quasionformally semisymmetri . hen e we on lude the following: theorem 4. a (2n+1)-dimensional (n > 1) generalized sasakian-spa e-form is quasionformally semisymmetri if and only if either the spa e-form is quasionformally �at or f1 = f3. by ombining the theorem2 and theorem 4, we an state the following orollary: theorem 5. let m(f1, f2, f3) be a (2n+1)-dimensional (n > 1) generalized sasakian-spa e-form. then m(f1, f2, f3) is quasionformally semisymmetri if and only if f1 = f3 or one of the following statements is true: (i) a + (2n − 1)b = 0, a 6= 0 6= b and f2 = 0. (ii) a + (2n − 1)b 6= 0, a 6= 0 and 3f2 + (2n − 1)f3 = 0. it an be easily seen that ∇p = 0 implies r.p = 0. hen e by virtue of theorem 4 we get corollary 4.1. a (2n+1)-dimensional (n > 1) quasionformally symmetri generalized sasakianspa e-form is either quasionformally �at or f1 = f3. a riemannian manifold is said to be quasionformally re urrent if ∇p = a ⊗ p, where a is a non-zero 1-form. it an be easily shown that a quasionformally re urrent manifold satis�es r · p = 0. hen e we immediately get the following: corollary 4.2. a (2n+1)-dimensional (n > 1) quasionformally re urrent generalized sasakianspa e-form is either quasionformally �at or f1 = f3. in parti ular, for sasakian-spa e-form f1 = c+3 4 and f3 = c−1 4 . so, f1 6= f3. hen e we an have the following orollary: corollary 4.3. a (2n+1)-dimensional (n > 1) sasakian-spa e-form is quasionformally semisymmetri if and only if it is quasionformally �at. cubo 15, 3 (2013) on quasionformally �at and quasionformally semisymmetri . . . 69 remark: if we take f(t) = et in example 4, we have f1 = −1, f2 = 0 and f3 = 0. therefore, the ondition 3f2 + (2n − 1)f3 = 0 and f2 = 0 holds. hen e from theorem 2, generalized sasakianspa e-form r ×f c m with f(t) = et is quasionformally �at. similarly from theorem 5, generalized sasakian-spa e-form r ×f c m with f(t) = et is quasionformally semisymmetri . a knowledgement: the �rst author (dgp) is thankful to university grants commission, new delhi, india for �nan ial support in the form of major resear h proje t f. no. 39-30/2010 (sr), dated: 23-12-2010. re eived: may 2012. a epted: september 2013. referen es [1℄ p. alegre, d. blair and a. carriazo, generalized sasakian-spa e-forms, israel j. math. 14 (2004), 157-183. [2℄ p. alegre and a. carriazo, stru tures on generalized sasakian-spa e-form, di�erential geom. and its appli ation 26 (2008), 656-666. doi. 10.1016/j difgeo. [3℄ k. amur and y.b. maralabhavi, on quasionformally �at spa es, tensor (n.s.) 31 (2)(1977), 194-198. math so ., 45 (2)(2008), 313 319. [4℄ d.e. blair, conta t manifolds in riemannian geometry, le ture notes in mathemati s, vol. 509, springer-verlag, berlin, 1976. [5℄ u.c. de and a. sarkar, some results on generalized sasakian-spa e-forms, thai j. math. 8(1) (2010), 1-10. [6℄ u.c. de and a. sarkar, on the proje tive urvature tensor of generalized sasakian-spa eforms, quaestiones mathemati ae, 33(2)(2010), 245-252. [7℄ r. desz z, on the equivalan e of ri i-semisymmetry and semisymmetry, dept. math. agriultural univ. wro law, preprint no.64, 1998. [8℄ l.p. eisenhart, riemannian geometry, prin eton university press, prin eton, n. j., 1949. [9℄ k. kenmotsu, a lass of almost onta t riemannian manifolds, tohoku math.j., 24 (1972), 93-103. [10℄ u.k. kim, conformally �at generalized sasakian-spa e-forms and lo ally symmetri generalized sasakian-spa e-forms, note di matemeti a 26 (2006), 55-67. [11℄ l.d. ludden, submanifolds of osymple ti manifolds, j.di� geom. 4 (1970), 237-244. 70 d.g. prakasha & h.g. nagaraja cubo 15, 3 (2013) [12℄ d.g. prakasha, on generalized sasakian-spa e-forms with weylonformal urvature tensor, loba heviskii j. math., 33 (3) (2012), 223-228. [13℄ z.i. szabo, stru tures theorems on riemannian spa es satisfying r(x, y) · r = 0, i. the lo al version j. di�. geom. 17 (1982), 531-582. [14℄ k. yano and s. sawaki, riemannian manifolds admitting a onformal transformation group, j. di� geom. 2 (1968), 161-184. cubo a mathematical journal vol.14, no¯ 01, (21–27). march 2012 remarks on cotype and absolutely summing multilinear operators a. thiago bernardino ufrn/ceres, centro de ensino superior do seridó, rua joaquim gregório, s/n, 59300-000, caicórn, brazil, email: thiagobernardino@yahoo.com.br abstract in this short note we present some new results concerning cotype and absolutely summing multilinear operators, extending recent results from different authors. resumen en esta nota presentamos nuevos resultados sobre cotipo y suma absoluta de operadores multilineales, extendiendo resultados recientes de diferentes autores. keywords and phrases: absolutely p-summing multilinear operators, cotype. 2010 ams mathematics subject classification: 46g25, 47b10. 22 a. thiago bernardino cubo 14, 1 (2012) 1 introduction in this note the letters x1, ..., xn, x, y will denote banach spaces over the scalar field k = r or c. from now on the space of all continuous n-linear operators from x1 × · · · × xn to y will be denoted by l(x1, ..., xn; y). if 1 ≤ s < ∞, the symbol s∗ represents the conjugate of s. it will be convenient to adopt that s/∞ = 0 for any s > 0. for 1 ≤ q < ∞, we denote by `wq (x) the set {(xj) ∞ j=1 ⊂ x : supϕ∈bx∗ ∑ j |ϕ(xj)| q < ∞}. if 0 < p, q1, ..., qn < ∞ and 1 p ≤ 1 q1 + · · · + 1 qn , a multilinear operator t ∈ l(x1, ..., xn; y) is absolutely (p; q1, ..., qn)-summing if (t(x (1) j , ..., x (n) j ))∞j=1 ∈ `p(y) for every (x (k) j )∞j=1 ∈ `wqk(xk), k = 1, ..., n. in this case we write t ∈ πnp,q1,...,qn (x1, ..., xn; y). if q1 = · · · = qn = q, we write πnp,q (x1, ..., xn; y) instead of πnp,q,...,q (x1, ..., xn; y) . for details on the linear theory we refer to the excellent monograph [9] and for the multilinear theory we refer to [1, 7, 14] and references therein. this paper deals with the connection between cotype and absolutely summing multilinear operators; this line of investigation begins with [4] and was followed by several recent papers (we refer, for example, to [5, 6, 8, 11, 12, 13, 15, 16] and for a full panorama we mention [14]). the following result appears in [10, theorem 3 and remark 2] and [16, corollary 4.6] (see also [5, theorem 3.8 (ii)] for a particular case): theorem 1.1 (inclusion theorem). let x1, ..., xn be banach spaces with cotype s and n ≥ 2 be a positive integer: (i) if s = 2, then πnq;q(x1, ..., xn; y) j π n p;p(x1, ..., xn; y) holds true for 1 ≤ p ≤ q ≤ 2. (ii) if s > 2, then πnq;q(x1, ..., xn; y) j π n p;p(x1, ..., xn; y) holds true for 1 ≤ p ≤ q < s∗ < 2. as a consequence of results from [3] one can easily prove the following generalization of this result (see [2] for details): theorem 1.2. if x1 has cotype 2 and 1 ≤ p ≤ s ≤ 2, then πns;s,q,...,q(x1, ..., xn; y) j π n p;p,q,....,q(x1, ..., xn; y) cubo 14, 1 (2012) remarks on cotype absolutely summing multilinear operators 23 for all x2, ..., xn,y and all q ≥ 1. in particular πns;s(x1, ..., xn; y) j π n p;p,s,....,s(x1, ..., xn; y) j π n p;p(x1, ..., xn; y). (1.1) a similar result, mutatis mutandis, holds if xj (instead of x1) has cotype 2. in this note we remark that analogous results hold for other situations in which the spaces involved may have different cotypes and no space may have necessarily cotype 2. 2 results the following proposition can be found in [5]: proposition 2.1. let 1 ≤ p1, ..., pn, p, q1, ..., qn, q ≤ ∞ such that 1/t ≤ ∑nj=1 1/tj for t ∈ {p, q} . let 0 < θ < 1 and define 1 r = 1 − θ p + θ q and 1 rj = 1 − θ pj + θ qj for all j = 1, ..., n and let t ∈ l (x1, ..., xn; y) . then t ∈ πnp;p1,...,pn ∩ π n q;q1,...,qn implies t ∈ πnr;r1,...,rn, provided that for each j = 1, ..., n, one of the following conditions holds: (i) xj is an l∞-space; (ii) xj is of cotype 2 and 1 ≤ pj, qj ≤ 2; (iii) xj is of finite cotype sj > 2 and 1 ≤ pj, qj < s∗j ; (iv) pj = qj = rj. next lemma appears in [13, theorem 3.1] without proof. we present a proof for the sake of completeness: lemma 2.2. let s > 0. suppose that xj has cotype sj for all j = 1, ..., n and at least one of the sj is finite. if 1 s ≤ 1 s1 + . . . + 1 sn , then l (x1, ..., xn; y) = πns;b1,...,bn (x1, ..., xn; y) for bj = 1 if sj < ∞ and bj = ∞ if sj = ∞. 24 a. thiago bernardino cubo 14, 1 (2012) proof. let j1, ..., jk ∈ {1, ..., n} , k ≤ n such that sj1, ..., sjk are finite and sj = ∞ if j 6= j1, ..., jk. if ( x (jl) i )∞ i=1 ∈ `w1 (xjl) and (x (j) i )∞i=1 ∈ `∞(xj), j 6= jl, l = 1, ..., k, using generalized hölder inequality, we obtain( ∞∑ i=1 ∥∥∥t(x(1)i , ..., x(n)i )∥∥∥s ) 1 s ≤ ‖t‖ ( ∞∑ i=1 (∥∥∥x(1)i ∥∥∥ · · · ∥∥∥x(n)i ∥∥∥)s ) 1 s ≤ c ‖t‖ ( ∞∑ i=1 ∥∥∥x(j1)i ∥∥∥sj1 )1/sj1 · · · ( ∞∑ i=1 (∥∥∥x(jk)i ∥∥∥)sjk )1/sjk where c is such that n∏ j=1,j6=j1,...,jn ∥∥∥x(j)i ∥∥∥ ≤ c for all i. since xj has cotype sj, for j1, ..., jk, we have( ∞∑ i=1 ∥∥∥t(x(1)i , ..., x(n)i )∥∥∥s ) 1 s ≤ c ‖t‖ ( ∞∑ i=1 ∥∥∥x(j1)i ∥∥∥sj1 )1/sj1 · · · ( ∞∑ i=1 (∥∥∥x(jk)i ∥∥∥)sjk )1/sjk = c ‖t‖ k∏ t=1 ( ∞∑ i=1 ∥∥∥idxj t ( x (jt) i )∥∥∥sjt )1/sjt < ∞ and the result follows. the main result of this note is the following theorem. at first glance it seems to have too restrictive assumptions, but corollary 2.4 and example 2.5 will illustrate its usefulness: theorem 2.3. let k, n be natural numbers, n ≥ k ≥ 2 and xk+1, ..., xn, y be arbitrary banach spaces. if xj has finite cotype sj ≥ 2 for j = 1, ..., k, then πnp;p1,...,pk,q,...,q (x1, ..., xn; y) j π n r;r1,...,rk,q,...,q (x1, ..., xn; y) for any (q, θ) ∈ [1, ∞] × (0, 1), 1 ≤ pj ≤ 2 (when sj = 2), 1 ≤ pj < s∗j (when sj > 2) and s ∈ [1, ∞) so that 1 s ≤ 1 s1 + · · · + 1 sk , 1 r = 1 − θ s + θ p , 1 rj = 1 − θ 1 + θ pj , for all j = 1, ..., k. cubo 14, 1 (2012) remarks on cotype absolutely summing multilinear operators 25 proof. let t ∈ πnp;p1,...,pk,q,...,q (x1, ..., xn; y) . by the previous lemma, l (x1, ..., xn; y) = πns;1,...,1,∞,...,∞ (x1, ..., xn; y) , where 1 is repeated k times. a fortiori, we have l (x1, ..., xn; y) = πns;1,...,1,q,...,q (x1, ..., xn; y) . so, t ∈ πns;1,...,1,q,...,q (x1, ..., xn; y) ∩ π n p;p1,...,pk,q,...,q (x1, ..., xn; y) . from proposition 2.1 we get t ∈ πnr;r1,...,rk,q,...,q (x1, ..., xn; y) . corollary 2.4. let k, n be natural numbers, n ≥ k ≥ 2, xk+1, ..., xn, y be arbitrary banach spaces and q ∈ [1, ∞). if xj has finite cotype sj ≥ 2, j = 1, ..., k and 1 ≤ 1/s1 + · · · + 1/sk, then πnp;p1,...,pk,q,...,q (x1, ..., xn; y) j π n r;r1,...,rk,q,...,q (x1, ..., xn; y) , where pj = p and rj = r for all j = 1, ..., k, for all r so that 1 ≤ r < p < min s∗j if sj 6= 2 for some j = 1, ..., k, 1 ≤ r < p ≤ 2 if sj = 2 for all j = 1, ..., k. in particular πnp;p (x1, ..., xn; y) j π n r;r (x1, ..., xn; y) for all r so that 1 ≤ r < p < min s∗j if sj 6= 2 for some j = 1, ..., k, 1 ≤ r < p ≤ 2 if sj = 2 for all j = 1, ..., k. proof. since 1 ≤ 1/s1 + · · · + 1/sk, we can use s = 1 in the previous theorem. since p = pi and r = ri for all i = 1, ..., k and s = 1, we conclude that πnp;p,...,p,q,...,q (x1, ..., xn; y) j π n r;r,...,r,q,...,q (x1, ..., xn; y) . in fact, for any 1 ≤ r < p there is a θ ∈ (0, 1) so that 1 r = 1 − θ 1 + θ p and since p = pi and r = ri, the same θ ∈ (0, 1) satisfies 1 ri = 1 − θ 1 + θ pi . 26 a. thiago bernardino cubo 14, 1 (2012) choosing q = p, since r < p we have πnp;p (x1, ..., xn; y) j π n r;r,...,r,p,...,p (x1, ..., xn; y) j π n r;r (x1, ..., xn; y) . example 2.5. let x4, ..., xn, y be arbitrary banach spaces. then πnp;p,p,p,q,...,q (`3, `3, `3, x4, ..., xn; y) j π n r;r,r,r,q,...,q (`3, `3, `3, x4, , , ., xn; y) for all q ∈ [1, ∞) and 1 ≤ r < p < 3∗. in particular πnp;p (`3, `3, `3, x4, ..., xn; y) j πnr;r (`3, `3, `3, x4, , , ., xn; y) for all 1 ≤ r < p < 3∗. received: january 2011. revised: february 2011. references [1] r. alencar and m. c. matos, some classes of multilinear mappings between banach spaces, publicaciones del departamento de análisis matemático 12, universidad complutense madrid, (1989). [2] a. thiago bernardino and d. pellegrino, some remarks on absolutely summing multilinear operators, arxiv:1101.2119v2. [3] o. blasco, g. botelho, d. pellegrino and p. rueda, lifting summability properties for multilinear mappings, preprint. [4] g. botelho, cotype and absolutely summing multilinear mappings and homogeneous polynomials, proc. roy. irish acad sect. a 97 (1997), 145-153. [5] g. botelho, c. michels and d. pellegrino, complex interpolation and summability properties of multilinear operators, rev. matem. complut. 23 (2010), 139-161. [6] g. botelho, d. pellegrino and p. rueda, cotype and absolutely summing linear operators, mathematische zeitschrift, 267 (2011), 1–7. [7] e. çalışkan and d. m. pellegrino, on the multilinear generalizations of the concept of absolutely summing operators, rocky mount. j. math. 37 (2007), 1137-1154. [8] a. defant, d. popa and u. schwarting, coordenatewise multiple summing operators on banach spaces, j. funct. anal. 259 (2010), 220-242. cubo 14, 1 (2012) remarks on cotype absolutely summing multilinear operators 27 [9] j. diestel, h. jarchow and a. tonge, absolutely summing operators, cambridge university press, 1995. [10] h. junek, m.c. matos and d. pellegrino, inclusion theorems for absolutely summing holomorphic mappings, proc. amer. math. soc. 136 (2008), 3983-3991. [11] y. meléndez and a. tonge, polynomials and the pietsch domination theorem, proc. roy. irish acad sect. a 99 (1999), 195-212. [12] d. pellegrino, cotype and absolutely summing homogeneous polynomials in lp spaces, studia math. 157 (2003), 121-131. [13] d. pellegrino, cotype and nonlinear absolutely summing mappings, math. proc. roy. irish acad., 105a (2005), 75-91. [14] d. pellegrino and j. santos, absolutely summing operators: a panorama, quaestiones mathematicae 34 (2011), 447–478. [15] d. popa, reverse inclusions for multiple summing operators, j. math. anal. appl. 350 (2009), 360-368. [16] d. popa, multilinear variants of maurey and pietsch theorems and applications, j. math. anal. appl. 368 (2010) 157–168. () cubo a mathematical journal vol.13, no¯ 02, (73–84). june 2011 closure of pointed cones and maximum principle in hilbert spaces paolo d’alessandro math. department, third university of rome, email: dalex@mat.uniroma3.it abstract we prove, in a hilbert space setting, that all targets of the minimum norm optimal control problems reachable with inputs of minimum norm ρ are support points for the the set reachable by inputs with norm bounded by ρ. this amount to say that the maximum principle always holds in hilbert spaces. resumen en este art́ıculo se demuestra que, para el problema de control óptimo a un nivel mı́nimo en los espacios de hilbert, todos los estados alcanzables con un nivel mı́nimo de entrada de ρ son puntos de apoyo para el conjunto de estados alcanzables por la norma de entrada inferior o igual a ρ. esto es equivalente a decir que el principio máximo siempre es válido en los espacios de hilbert. keywords and phrases: linear control systems in hilbert spaces, norm optimal control, maximum principle. mathematics subject classification: 93e20, 93e25. 74 paolo d’alessandro cubo 13, 2 (2011) 1. introduction arguing in a hilbert space setting, suppose that the minimum norm of inputs, which steer the origin to a state ζ in a finite interval of time [0, γ ] is ρ. that the maximum principle holds, seen through the convex analysis optics, means that the target ζ is a support vector for the set rρ of all vectors reachable under an input of norm less than or equal to ρ. in other words the normal cone to rρ at ζ is non-trivial. in the literature vectors in the normal cone, which live in the dual space, are also called multipliers, a term that has a more general meaning. to verify that rρ has support at ζ, one might hope to apply to rρ and {ζ} the celebrated separation theorem for linear topological spaces, stating that, given two non-void convex sets a and b, and assuming that a has interior, they can be can be separated by a continuous linear functional if and only if b ∩ ai = φ. unfortunately, this application is not possible in the infinite dimensional case because it is not true in general that rρ has interior. it is very well known, and easy to show, that the set of support points s is contained and is dense in the set r∧ρ of target points reachable with minimum input norm ρ. this might suggest that some of the above targets are not support point. on the other hand the cited separation theorem above is indeed a sufficient condition, in view of the presiding hypothesis (one of the sets has interior), so that it leaves open the problem of determining if separation holds for all targets or if, instead, the dense subset of support points is proper. much attention (see [1]) has been devoted to more sophisticated banach space settings, obtaining a generalization of the maximum principle, thanks to the definition of a larger linear space of multipliers, which contains the dual space. similarly, this leaves to determine which multipliers are in the dual space, although sufficient conditions are known. for details as well as for accurate historical remarks and proper credits, reference can be made to the vast and outstanding work by fattorini, which covers a variety of banach space settings. recent work is, besides [1], [6] and [7]. the purpose here is to answer for hilbert spaces the question connected to the aforementioned density results: are there vectors of r∧ρ at which rρ has no support? the question remained open for quite a long time. and the answer is ”no”. the argument lean on a (very general) result of the theory of cones and on strict convexity of hilbert space norms. thus generalization are possible (although beyond the present purposes), but our technique cannot pass the barrier of the requirement of strict convexity of norms. more specifically, we will show that the tangent cone to rρ at any ζ ∈ r ∧ ρ is pointed. that some vector ζ ∈ r∧ρ is not a support point for rρ is equivalent to say that the polar cone to the tangent cone to rρ at ζ is trivial. this is in turn equivalent to say that the closure of the tangent cone is the whole space. that the closure of a pointed cone be the whole space is a rather counterintuitive proposition, and in fact we prove that this is not the case, in any linear topological space. cubo 13, 2 (2011) closure of pointed cones and maximum principle in hilbert spaces 75 from this fact it follows, as an immediate consequence, that all ζ ∈ r∧ρ are support points for rρ. no assumptions will be made on either separability or full control. more might be said in the separable case, but this is not dealt with here. with an apology, our exposition covers some well known basic facts and complements, in order to enhance readability. a more succinct exposition would result in a choppy and difficult to follow narration. 2. setting we refer to the cauchy problem with u ∈ l2([0, γ ], h1) as given in [2]. the abstract differential equation has the form: . x(t) = ax(t) + bu(t) where the spaces h1, h are real hilbert spaces, b is an operator (bounded linear transformation) h1 → h, a is the infinitesimal generator of a strongly continuous semigroup {t (t)} on h and x(0) = x is given. the equation is intended in weak sense and its unique solution is expressed by the formula of variation of constants: x(t) = t (t)x + ∫ [0,t] t (t − s)bu(s)ds which formally recurs also in other settings, like that of differential equations in banach spaces. in our specific case the integral is a pettis integral. the case h1 = h and b = i is referred to as full control case, but here we do not make any assumption in this respect. for simplicity, dealing with norm optimal control problems, we assume x = 0. the general case is a variant of this and can be obtained along the lines of [1]. it is assumed that we can reach a certain target vector ζ ∈ h at time γ > 0 or: ζ = ∫ [0,γ ] t (γ − s)bu(s)ds = lγ u and we look for the minimum norm input that does the job of reaching ζ. the linear transformation lγ : l2([0, γ ], h1) → h is well known to be continuous. this is all we will need in the sequel as to the properties of this operator. if the control steering the system from the origin at time 0 to ζ at time γ is unique, because the operator lγ is one to one, then, according to fattorini, who introduced this concept in sixties, the system is called rigid. fattorini constructed an example of rigid system showing that this phenomenon can actually occur (see e.g. [1] and references therein). in this case, characterizing the optimum control, becomes obviously pointless. we note that our arguments hold good even when lγ is one to one. in this case though, exploiting the fact that ζ is a support point, can only lead to retrieving the unique control that solves the reachability problem. 76 paolo d’alessandro cubo 13, 2 (2011) it is immediate to show that the minimum norm control exists and is unique, since this is a straightforward consequence of the projection theorem. 3. lemmata and definitions the environment is, unless otherwise stated, a real hilbert space h. the symbols l(c) and l−(c) denote, respectively, the linear and closed linear hull of the non-void set c, whereas co(c) and co−(c) denote, respectively, its conical and closed conical hull. given a convex cone c in h its polar cone is the (always closed) cone: cp = {y : (y, x) ≤ 0: ∀x ∈ c} definition 1. the tangent cone to c at y ∈ c is the cone co(−y + c), the normal cone to c at y is the polar of the tangent cone to c at y. consider a convex set c and ζ ∈ c. we say that ζ is a support point for c if there exists a nonzero vector n (which can be taken with unit norm) such that: (n, ζ) ≥ (n, z), ∀z ∈ c this is equivalent to say that c is contained in the closed half-space defined by the continuous linear functional (n, .): {y : (n, y) ≤ (n, ζ)} whose limiting (closed) hyperplane contains {ζ}. in still another equivalent terminology, this is equivalent to say that the tangent cone to c at ζ has a non-trivial polar cone (because n is in such cone). let n be an unit norm vector. the closed convex set: {x : a ≤ (n, x) ≤ b} (with b ≥ a) is called a sandwich. the number b − a is the thickness of the sandwich. if the thickness is zero, the sandwich is a closed hyperplane. the sandwich is symmetrical if it has the form {x : −a ≤ (n, x) ≤ a} for some a ≥ 0. the next lemma regards closed cones in hilbert spaces lemma 2. a closed cone in a hilbert space h is proper if and only if it is contained in a closed half-space. demostración. let c be a closed proper cone. then there is a singleton {y} disjoint from c. singletons are convex and compact and therefore the strong separation corollary 14.4 in [4] applies. the rest is immediate. cubo 13, 2 (2011) closure of pointed cones and maximum principle in hilbert spaces 77 we recall briefly some other relevant notions about cones in infinite dimensional hilbert spaces. incidentally, the literature on this topic tend to be either finite dimensional or infinite dimensional, but typically on the footsteps of the seminal work of choquet, aiming at extending the krein milman theorem in a measure theory setting. definition 3. a (convex) cone c is pointed if c ∩ −c = {0}. a cone c is blunt if l−(c) = h the following lemma well known. lemma 4. if a closed c cone is pointed, its polar cone is blunt, that is, l−(cp) = h. demostración. the proof is simple and based on elementary computations that, for brevity, are taken for granted here, but, on the other hand, are rather intuitive, since polarization is the analogous for cones of orthogonal complementation for subspaces. we can write: {0} = c ∩ −c = [cp + (−cp)]p = l(cp)p and, taking polars of the first and last cone the desired conclusion follows. if a pointed cone is not closed what can we say of its polar cone? notice that according to lemma 2, if the closure of the cone is proper, then the cone has a nontrivial polar cone. for the polar to be trivial instead, again in view of the same lemma, the closure of the cone must be the whole space, despite the fact that the cone is pointed. but we prove here that this cannot happen even in general, as stated by the following: theorem 5. the closure of a pointed cone in a linear topological space is a proper cone. demostración. suppose that it is not true, that is there is a pointed cone c in a linear topological space e, such that c− = e. consider a finite dimensional subspace f, which intersect c in a non trivial, necessarily pointed, cone. actually we can take instead of f, its subspace l(f ∩ c), without restriction of generality. for simplicity we leave the symbol f unchanged, and equip f with the relative topology. next notice that, as is well known, because f is the finite dimensional, the pointed convex cone υ = f ∩ c has interior. thus it can be separated by a continuous linear functional from the origin and therefore it is contained in a closed semi-space. it follows that the closure of υ in f is contained in a closed half-space and therefore is a proper cone. but by theorem 1.16 in [5], such closure is c− ∩ f. by the initial assumption c− ∩ f = e ∩ f = f. this is a contradiction and therefore the proof is finished. 4. existence and uniqueness in l2(h1) assume that for some ζ ∈ h, ∃uζ such that x = ζ = lγ (uζ). the set of all u satisfying ζ = lγ (u) is given by uζ + n (lγ ). this is a closed affine space, because lγ is continuous. if n (lγ ) is trivial, the unique uζ solving ζ = lγ (uζ) is already optimum. optimization in this case 78 paolo d’alessandro cubo 13, 2 (2011) is pointless, but the arguments below hold good anyway. to obtain the minimum norm solution we can apply the projection theorem and project the origin on this closed convex set. moreover, we know, from the celebrated projection theorem for hilbert spaces, that this projection exists and is unique and hence the minimum norm solution always exists and is unique. we call this unique minimum norm control uo. we can put: ‖uo‖l2(h1) = ρ which is the optimum value of the norm. in particular we can say that the optimum control belongs to the closed sphere sρ of radius ρ, around the origin in l2([0, γ ], h1) (briefly l2(h1)): uo ∈ sρ and so: ζ ∈ lγ (sρ) ⊂ r(lγ ) it is immediate to verify that: lγ (sρ) = {z : mı́n{‖u‖ : lγ (u) = z} ≤ ρ}} for notational simplicity we put lγ (sρ) = rρ. 5. reachability in this section we recast a few well known facts of reachability theory. we noted that ζ ∈ lγ (l2(h1)) = r(lγ ) = rγ . this is the reachable set at time γ and is a linear subspace of h, which is in general not closed. however, we can always argue in the hilbert space lγ (l2(h1)) − and thus assume, without restriction of generality, that rγ is a dense linear subspace in h. this is equivalent to n (l ∗ γ ) = {0}. to streamline the exposition this assumption will be in force thoroughly. note that l∗γ is the map x → b ∗t ∗(γ − .)x. thus x ∈ n (l ∗ γ ) if and only if the continuous function b∗t ∗(γ − .)x is identically zero. next notice that: n (l ∗ γ ) = n (l γ l∗γ ) for n (l ∗ γ ) ⊂ n (l γ l∗γ ) is obvious and, on the other end: lγ l ∗ γ x = 0 ⇒ (x, lγ l ∗ γ x) = ‖l ∗ γ x‖ 2 = 0 therefore, under the present hypothesis that n (l ∗ γ )⊥ = r− γ = h, the selfadjoint operator gγ = lγ l ∗ γ cubo 13, 2 (2011) closure of pointed cones and maximum principle in hilbert spaces 79 has a trivial kernel, and so it is one to one. moreover, obviously r(gγ ) ⊂ r(lγ ) = rγ . indeed, r(gγ ) − = r− γ , for: r(gγ ) − = n (gγ ) ⊥ = n (l ∗ γ )⊥ = r−γ thus both r(gγ ) and r(lγ ) are dense, so that for any x ∈ r(gγ ) and ε > 0 there is an y ∈ r(lγ ) such that ‖x − y‖ ≤ ε and vice-versa. notice that, by a change of variables, ∀x ∈ h: gγ x = γ∫ 0 t (σ)bb∗t ∗(σ)xdσ we claim that we can define the integral as a riemann integral. in fact we can prove that the integrand is a continuous function. to this purpose first note that the function bb∗t ∗(σ)x is continuous. then, to show that the integrand function is continuous, apply the exponential growth property of the semigroup and following well known: theorem 6. consider a set t ⊂ lc(h,h) and suppose that it is bounded in the norm operator topology. then the evaluation map is jointly continuous in t ×h where t is equipped with the (relativized) pointwise topology. if the state y is reachable and y ∈ r(gγ ) , y = gγ x has solution in x, so that y = γ∫ 0 t (σ)bb∗t ∗(σ)dσx = γ∫ 0 t (γ − τ)bw(τ)dτ where w(τ) = b∗t ∗(γ − τ)x. in this case, there is a smooth control that solves the problem. 6. the quasi-topology of rρ and main theorem there are a number of interesting properties of rρ = lt (sρ), which depend both on the environment (hilbert space), on continuity of lt and on the fact that the closed sphere sρ is convex and weakly compact. in particular recall that s − s (strong-strong) continuity is equivalent to w − w (weak-weak) continuity and, therefore, lt is w − w continuous. the set rρ is obviously: 80 paolo d’alessandro cubo 13, 2 (2011) convex (as image under an operator of a convex set) with 0 ∈ rρ, and symmetrical. weakly compact, as image of a weakly compact set under a w − w continuous linear transformation weakly closed + convex and hence strongly closed bounded (as image under an operator of a bounded set) not a convex body in general. we now describe what we mean with the quasi-topology of rρ, adding to the above list the fact that rρ has a ”quasi-interior”, at whose points it is densely radial and circled, and a quasiboundary, which is the complement in rρ of the quasi-interior. these terms are justified because this quasi-topology can be realized as an actual topology, using the topology introduced by fattorini for rρ, based on the norm: p(z) = ı́nf{‖u‖ : lγ u = z} this concept is of primary importance in more general and complex banach space settings, but it will not be used here. definition 7. the set of all vectors x in rρ, such that the minimum norm control to reach x has norm ρ, is called quasi-boundary of rρ and denoted by r ∧ ρ . the set of all vectors x in rρ, such that the minimum norm control to reach x has norm ρ′ < ρ, is called quasi-interior of rρ and denoted by r∨ρ . it is well known that no point of the quasi-interior can be a support point for rρ. consider the origin, which belongs to r∨ρ . the set rρ is densely radial at the origin. in fact consider any point z ∈ rγ and let η be the minimum norm of the unique control u that reaches z. if η ≤ ρ the whole segment [0 : z] ⊂ lγ (sρ). otherwise take the point z ′ reached by the control ρ η u, and [0 : z′] ⊂ lγ (sρ). observe that ∀z ∈ rγ is positively proportional to a vector in lγ (sρ). in other words: ∪{αlγ (sρ) : α > 0} = rγ moreover it is clear that αlγ (sρ) ⊂ lγ (sρ) for any positive α ≤ 1, so that lγ (sρ) is also circled. on the other hand, if for ζ ∈ lγ (sρ) the minimum norm of the corresponding control to reach ζ is ρ, by a very well known argument, see e.g. [1], (1 + ε)ζ /∈ lγ (sρ) for ∀ε > 0. all the points of the quasi-interior have the same dense radiality property as the origin. in fact, if ξ ∈ r∨ρ , ξ 6= 0, then the minimum norm of the control that steers the origin to the state to ξ is some ρ′ with 0 < ρ′ < ρ. let uξ be the corresponding minimum norm control. radiality in the direction of the origin or of ξ itself is obvious. next consider any z ∈ rγ with z 6= 0 and z not proportional to ξ, and let uz be corresponding the minimum norm control. it will be ‖uz‖ = γ > 0. then it is immediate that ξ + αz ∈ rρ for any α, such that: 0 < α ≤ ρ−ρ ′ γ . the role of the dense radiality is emphasized by the proof of the anticipated well known result: cubo 13, 2 (2011) closure of pointed cones and maximum principle in hilbert spaces 81 theorem 8. no point of r∨ρ can be a support point of rρ. demostración. suppose that a point y ∈ r∨ρ is a support point. then the projection of some vector z /∈ rρ on rρ is y. by the projection theorem we have: (z − y, x − y) ≤ 0, ∀x ∈ rρ if equality holds for all x ∈ rρ then rρ and hence also rγ would be contained in a closed hyperplane, contradicting that rγ is dense in the space. so ∃x ∈ rρ such that (z − y, x − y) < 0. because of the radiality property at y we can take in lieu of x, a state w = y + α(y − x) ∈ rρ for some α > 0. but then (z − y, w − y) = α(z − y, y − x) > 0 contradicting the projection theorem. at this point we know that any support point is in r∧ρ . it is well known that support points are dense in the quasi-boundary of rρ (e.g.[2]). to show this, one may use the following sequence of support points converging to an arbitrary ζ ∈ r∧ρ . for any positive integer i, (1 + 1 i )ζ /∈ rρ. because the projection prρ on rρ is continuous, the sequence {prρ ((1 + 1 i )ζ)} converges strongly to ζ and obviously, because the points in the sequence are projections, they are all support points (and hence also lie on the quasi-boundary of rρ). however, more is true for arbitrary vectors of the quasi-boundary of rρ in the present hilbert space setting, as we show in the next: theorem 9. all vectors in r∧ρ are extreme. no other point of rρ can be extreme demostración. suppose that for the quasi-boundary vector ζ it is true that ζ = ζ1+ζ2 2 with ζ1 and ζ2 in rρ, and let ρ1 ≤ ρ be the norm of the minimum norm control u1 that steers the system to ζ1 and ρ2 ≤ ρ be the norm of the minimum norm control u2 that steers the system to ζ2. the control u1+u2 2 has norm strictly less than ρ, because the norm of a hilbert space is strictly convex, and steers the system to ζ. but this contradicts that ζ ∈ r∧ρ , and so the proof of the first statement is finished. points of the quasi-interior cannot be extreme in view of the specific dense radiality property we have illustrated. therefore we are done. corollary 10. the tangent cone to rρ at any ζ ∈ r ∧ ρ is pointed demostración. obviously if it were not so it would be contradicted that ζ is an extreme point. at this point, putting together this corollary with lemmata 5 and 2 we have established the following main theorem 11. any point ζ ∈ r∧ρ is a support point of rρ. in other words ∀ζ ∈ r∧ρ , ∃n ∈ h ∗ = h, ‖n‖ = 1 82 paolo d’alessandro cubo 13, 2 (2011) such that (n, ζ) ≥ (n, z), ∀z ∈ c given this result we may expect all the more a large normal fan for rρ. indeed the normal fan is the whole space, but this fact is independent on the main theorem, as we illustrate in the next section. 7. the set of support points and the normal fan in this section we collect some, mostly well known, facts about the set of support points s (which we now know to be the same as r∧ρ ) and the normal fan. at each support point ζ the tangent cone has a non-trivial polar cone, also called the normal cone at ζ. the union of these cones is the normal fan of rρ. naturally, it is often convenient to normalize vectors in the normal fan. suppose ζ is a support point of rρ. using the pairing in h and the cbs inequality, we have a well known expression for ζ. let n a unit norm vector in the normal cone at ζ. then it must be for any y ∈ rρ: (n, y − ζ) = (n, lγ uy − lγ uζ) ≤ 0 or (n, lγ uy) ≤ (n, lγ uζ) or (l∗γ n, uy) ≤ (l ∗ γ n, uζ) so that, by the cbs inequality, the optimal control corresponding to ζ has the expression: uζ = ρ l∗γ n ‖l∗ γ n‖ , a.e. and: ζ = lγ ρ ‖l∗ γ n‖ l∗γ n = ρ ‖l∗ γ n‖ gγ n on the other hand, for an arbitrary unit norm vector n consider the linear program: máx{(n, z) : z ∈ rρ} the existence of solutions of this linear program derive from the fact that rρ is convex and weakly compact and the functional is continuous (and hence also weakly continuous). let ζ be a solution, then for any y ∈ rρ: (n, y − ζ) ≤ 0 so that ζ is a support point and has the above expression. cubo 13, 2 (2011) closure of pointed cones and maximum principle in hilbert spaces 83 thus s = {ζ : ζ = ρ ‖l∗ γ n‖ gγ n, ‖n‖ = 1} notice that because gγ is one-to-one, to each support ζ there corresponds a unique normal n. in other words, all normal cones are rays. we collect these facts and more in the following: theorem 12. the normal fan of rρ, less the origin, is the whole space h\{0}. all normal cones at support vectors ζ are rays, under the correspondence between unit normal vectors n and support points ζ: ζ = ρ ‖l∗ γ n‖ gγ n the correspondence between unit vectors in the normal fan and the corresponding support points is one-to-one. if ζ is a support point {ζ} is an exposed face of rρ. equivalently , for any pair normal-support point it is true that: (n, y − ζ) < 0, ∀y ∈ rρ consequently (since 0 ∈ rρ) the value of the functional (n, .) at ζ is always positive. demostración. it remains to be proved the last statement. suppose a ζ ∈ rρ is a support point, so that it is necessarily a quasi-boundary and extreme point. suppose that for some normal n there is z such that: (n, z − ζ) = 0 then z is a support point as well with normal n. in fact, ∀y ∈ rρ: (n, y − z) = (n, y − ζ) + (n, ζ − z) ≤ 0 therefore z is a support point and hence it is an quasi-boundary point, so that the minimum norm control that steers the system to z has norm ρ. denote by uz the minimum norm control to reach z and by uζ the minimum norm control to reach ζ. by the same argument above all points in [ζ : z] are support points with normal n. in fact for 0 < α < 1, ∀y ∈ rρ: (n, y − (ζ + α(z − ζ))) ≤ 0 but the point ζ+z 2 can be reached by the control uζ+uz 2 and ‖ uζ+uz 2 ‖ < ρ. this means that ζ+z 2 is a quasi-interior point, which is a contradiction and the proof is finished. returning to the linear program: máx{(n, z) : z ∈ rρ} call the maximum m, then m = ρ ‖l∗ γ n‖ (n, gγ n) = ρ(n, gγ n) 1/2 = ρ‖l∗γ n‖ 1/2 84 paolo d’alessandro cubo 13, 2 (2011) recall that m > 0. because rρ is symmetric, it is contained in the symmetrical sandwich: rρ ⊂ {y : −m ≤ (n, y) ≤ m} and the limiting hyperplanes of the sandwich meet rρ only in the two points ζ and −ζ. also the sandwich cannot degenerate to a hyperplane, because its thickness is 2m > 0. received: october 2009. revised: january 2010. referencias [1] h.o. fattorini, infinite dimensional linear control systems, elsevier, amsterdam 2005. [2] a.v. balakrishnan, applied functional analysis,springer-verlag berlin-heidelberg-new york 1976. [3] p.r. halmos, a hilbert space problem book van nostrand, new york, 1967. [4] j.l. kelley and i. namioka, linear topological spaces, springer, new york, 1963 [5] j.l. kelley, general topology, springer, new york, 1955. [6] h.o. fattorini smoothness of the costate and the target in the time and norm optimal problems optimization, vol.55, no.2, 2006, 19-36 [7] h.o. fattorini regular and strongly regular time and norm optimal controls to appear introduction setting lemmata and definitions existence and uniqueness in l2(h1) reachability the quasi-topology of r and main theorem the set of support points and the normal fan cubo a mathematical journal vol.16, no¯ 01, (09–20). march 2014 viscosity approximation methods with a sequence of contractions koji aoyama† department of economics, chiba university, yayoi-cho, inage-ku, chiba-shi, chiba 263-8522, japan. aoyama@le.chiba-u.ac.jp yasunori kimura‡ department of information science, toho university, miyama, funabashi, chiba 274-8510, japan. yasunori@is.sci.toho-u.ac.jp abstract the aim of this paper is to prove that, in an appropriate setting, every iterative sequence generated by the viscosity approximation method with a sequence of contractions is convergent whenever so is every iterative sequence generated by the halpern type iterative method. then, using our results, we show some convergence theorems for variational inequality problems, zero point problems, and fixed point problems. resumen la meta de este art́ıculo es probar en un marco de trabajo adecuado que cada sucesión iterativa generada por el método de aproximación de viscosidad con una sucesión cualquiera de contracciones es convergente como lo es cada sucesión iterativa generada por el método iterativo del tipo halpern. aśı, usando nuestro resultado mostramos algunos teoremas de convergencia para problemas de desigualdades variacionales, problemas de punto cero y problemas de punto fijo. keywords and phrases: viscosity approximation method, nonexpansive mapping, fixed point, hybrid steepest descent method. 2010 ams mathematics subject classification: 47h09, 47j20, 47h10. 10 koji aoyama & yasunori kimura cubo 16, 1 (2014) 1 introduction let c be a nonempty closed convex subset of a hilbert space. this paper is devoted to the study of strong convergence of a sequence {yn} in c defined by an arbitrary point y1 ∈ c and yn+1 = λnfn(yn) + (1 − λn)tnyn (1.1) for n ∈ n, where λn is a real number in [0, 1], fn is a contraction on c, and tn is a nonexpansive mapping on c for n ∈ n. in particular, our main interest is the relationship between convergence of such a sequence {yn} and a sequence {xn} defined by an arbitrary point x1 ∈ c and xn+1 = λnu + (1 − λn)tnxn (1.2) for n ∈ n, where u is a point in c. in §3, using the technique developed in [21], we prove that their convergence are equivalent under some assumptions. then, as applications of our convergence results in §3, we discuss strong convergence of the sequences generated by the hybrid steepest descent method [30] and we give another proof of iemoto and takahashi’s theorem [17] in §4. moreover, we show one generalization of ceng, petruşel, and yao’s theorem [13] in §5. the iterative method defined by (1.1) is based on the viscosity approximation method due to moudafi [19]. he considered the fixed point problem of a single nonexpansive mapping and proved strong convergence of sequences generated by the viscosity approximation methods; see also xu [28] and suzuki [21]. the iterative method defined by (1.2) is called the halpern type iterative method; see halpern [16], wittmann [25], and shioji and takahashi [20]; see also [1,5,7]. 2 preliminaries throughout the present paper, h denotes a real hilbert space with the inner product 〈 · , · 〉 and the norm ‖ · ‖, c a nonempty closed convex subset of h, i the identity mapping on h, and n the set of positive integers. a mapping s: c → h is said to be lipschitzian if there exists a constant η ≥ 0 such that ‖sx − sy‖ ≤ η ‖x − y‖ for all x, y ∈ c. in this case, s is called an η-lipschitzian mapping. in particular, an η-lipschitzian mapping is said to be nonexpansive if η = 1; an η-lipschitzian mapping is said to be an η-contraction if 0 ≤ η < 1. it is known that fix(s) is closed and convex if s: c → h is nonexpansive, where fix(s) denotes the set of fixed points of s. the metric projection of h onto c is denoted by pc and we know that pc is nonexpansive. we also know the following; see [22]. lemma 2.1. let x ∈ h and z ∈ c. then z = pc(x) if and only if 〈y − z, x − z〉 ≤ 0 for all y ∈ c. let {sn} be a sequence of nonexpansive mappings of c into h. we say that {sn} satisfies the condition (z) if every weak cluster point of {xn} is a common fixed point of {sn} whenever {xn} cubo 16, 1 (2014) viscosity approximation methods with a sequence . . . 11 is a bounded sequence in c and xn − snxn → 0; see [1, 3, 8–11]. we say that {sn} satisfies the condition (r) if lim n→∞ sup y∈d ‖sn+1y − sny‖ = 0 for every nonempty bounded subset d of c; see [1,5]. we say that {sn} is stable on a nonempty subset d of c if {snz : n ∈ n} is a singleton for every z ∈ d. we need the following lemmas: lemma 2.2. let c1 and c2 be nonempty closed convex subsets of h, {sn} a sequence of nonexpansive mappings of c1 into h, and {tn} a sequence of nonexpansive mappings of c2 into h. suppose that {sn} and {tn} satisfy the condition (r), c1 ⊃ tn(c2) for every n ∈ n, and {tn} has a common fixed point. then {sntn} satisfies the condition (r). proof. let d be a nonempty bounded subset of c2. then it is clear that each sntn is nonexpansive and ‖sn+1tn+1y − sntny‖ ≤ ‖sn+1tn+1y − sntn+1y‖ + ‖sntn+1y − sntny‖ ≤ ‖sn+1tn+1y − sntn+1y‖ + ‖tn+1y − tny‖ (2.1) for all y ∈ d and n ∈ n. let z be a common fixed point of {tn}. then it is obvious that ‖tny‖ ≤ ‖tny − tnz‖ + ‖z‖ ≤ ‖y − z‖ + ‖z‖ for all y ∈ d and n ∈ n. this shows that d′ = {tny : n ∈ n, y ∈ d} is a bounded subset of c1. since {sn} and {tn} satisfy the condition (r), it follows from (2.1) that sup y∈d ‖sn+1tn+1y − sntny‖ ≤ sup y ′∈d′ ‖sn+1y ′ − sny ′‖ + sup y∈d ‖tn+1y − tny‖ → 0. therefore, {sntn} satisfies the condition (r). lemma 2.3. let {sn} be a sequence of nonexpansive mappings of c into h and {γn} a sequence in [0, 1] such that γn+1 − γn → 0. suppose that {sn} satisfies the condition (r) and {sn} has a common fixed point. then {γni + (1 − γn)sn} satisfies the condition (r). proof. set un = γni + (1 − γn)sn for n ∈ n. let d be a nonempty bounded subset of c. then it is clear that each un is nonexpansive and ‖un+1y − uny‖ ≤ |γn+1 − γn| ‖y − sny‖ + |1 − γn+1| ‖sn+1y − sny‖ ≤ |γn+1 − γn| ‖y − sny‖ + ‖sn+1y − sny‖ (2.2) for all y ∈ d and n ∈ n. let z be a common fixed point of {sn}. then it is obvious that ‖y − sny‖ ≤ ‖y − z‖ + ‖snz − sny‖ ≤ 2 ‖y − z‖ (2.3) 12 koji aoyama & yasunori kimura cubo 16, 1 (2014) for all y ∈ d and n ∈ n. since {sn} satisfies the condition (r) and γn+1 − γn → 0, it follows from (2.2) and (2.3) that sup y∈d ‖un+1y − uny‖ ≤ 2 |γn+1 − γn| ‖y − z‖ + sup y∈d ‖sn+1y − sny‖ → 0. therefore, {un} satisfies the condition (r). a set-valued mapping a of h into h, which is denoted by a ⊂ h×h, is said to be a monotone operator if 〈x − y, x′ − y′〉 ≥ 0 for all (x, x′), (y, y′) ∈ a. a monotone operator a ⊂ h × h is said to be maximal if a = b whenever b ⊂ h × h is a monotone operator such that a ⊂ b. let a ⊂ h × h be a maximal monotone operator. it is known that (i + ρa)−1 is a single-valued mapping of h onto dom(a) = {x ∈ h : ax 6= ∅} for all ρ > 0. such a mapping (i + ρa)−1 is called the resolvent of a and denoted by jρ. it is also known that the resolvent jρ is nonexpansive and fix(jρ) = a −10 = {x ∈ h : ax 3 0}; see [22] for more details. a mapping a : h → h is said to be strongly monotone if there is a constant κ > 0 such that 〈x − y, ax − ay〉 ≥ κ ‖x − y‖ 2 for all x, y ∈ h. in this case, a is called a κ-strongly monotone mapping. the following lemma is well known; see, for example, [4]. lemma 2.4. let κ and η be positive real numbers such that η2 < 2κ. let f be a nonempty closed convex subset of h and a : h → h a κ-strongly monotone and η-lipschitzian mapping. then the following hold: (1) κ ≤ η, 0 ≤ 1 − 2κ + η2 < 1 and i − a is a θ-contraction, where θ = √ 1 − 2κ + η2. (2) there exists a unique point z ∈ f such that 〈y − z, az〉 ≥ 0 for all y ∈ f, and moreover, z is the unique fixed point of pf(i − a). the following lemma is well known; see [7,18,24,26,27]. lemma 2.5. let {�n} be a sequence of nonnegative real numbers, {γn} a sequence of real numbers, and {λn} a sequence in [0, 1]. suppose that �n+1 ≤ (1 − λn)�n + λnγn for every n ∈ n, lim supn→∞ γn ≤ 0, and ∑ ∞ n=1 λn = ∞. then �n → 0. 3 viscosity approximation method with a sequence of contractions in this section, we deal with the viscosity approximation method due to moudafi [19] in order to find a common fixed point of a sequence of nonexpansive mappings. in particular, we focus on the viscosity approximation method with a sequence of contractions. we first investigate the relationship between this method and the halpern type iterative method (theorem 3.1). then, by using known results (theorems 3.3 and 3.5), we show convergence theorems (theorems 3.4 and 3.6). cubo 16, 1 (2014) viscosity approximation methods with a sequence . . . 13 using the technique in [21], we can prove the following: theorem 3.1. let h be a hilbert space, c a nonempty closed convex subset of h, {tn} a sequence of nonexpansive self-mappings of c, f a nonempty closed convex subset of c, θ a nonnegative real number with θ < 1, and {λn} a sequence in [0, 1] such that ∑ ∞ n=1 λn = ∞. then the following are equivalent: (1) for any (x, u) ∈ c × c, the sequence {xn} defined by x1 = x and xn+1 = λnu + (1 − λn)tnxn (3.1) for n ∈ n converges strongly to pf(u). (2) for any y ∈ c and any sequence {fn} of θ-contractions on c which is stable on f, the sequence {yn} defined by y1 = y and yn+1 = λnfn(yn) + (1 − λn)tnyn (3.2) for n ∈ n converges strongly to w, where w is the unique fixed point of pf ◦ f1. proof. we first show that (1) implies (2). let {fn} be a sequence of θ-contractions on c which is stable on f, w the fixed point of a contraction pf ◦ f1, and y ∈ c. let {xn} be a sequence defined by x1 = y and xn+1 = λnf1(w) + (1 − λn)tnxn for n ∈ n. then xn → pf ( f1(w) ) = w by (1). since tn is nonexpansive and fn is a θ-contraction, it follows from f1(w) = fn(w) that ‖xn+1 − yn+1‖ = ∥ ∥(1 − λn)(tnxn − tnyn) + λn ( f1(w) − fn(yn) ) ∥ ∥ ≤ (1 − λn) ‖tnxn − tnyn‖ + λn ‖fn(w) − fn(yn)‖ ≤ (1 − λn) ‖xn − yn‖ + λnθ ‖w − yn‖ ≤ (1 − λn) ‖xn − yn‖ + λnθ(‖w − xn‖ + ‖xn − yn‖) ≤ ( 1 − (1 − θ)λn ) ‖xn − yn‖ + (1 − θ)λn θ 1 − θ ‖xn − w‖ for every n ∈ n. since ∑ ∞ n=1 (1 − θ)λn = ∞ and xn → w, lemma 2.5 shows that xn − yn → 0. therefore, we conclude that yn → w. we next show that (2) implies (1). let (x, u) ∈ c × c be given. for each n ∈ n, let fn be a mapping defined by fn(z) = u for z ∈ c. then, obviously, each fn is a 0-contraction and {fn} is stable on f. thus it follows from (2) that {xn} converges strongly to w = pf ( f1(w) ) = pf(u). remark 3.2. it is easy to check that theorem 3.1 holds even if h is a banach space under appropriate conditions. we know the following result; see [2,7] and see also [3,11]. 14 koji aoyama & yasunori kimura cubo 16, 1 (2014) theorem 3.3. let h be a hilbert space, c a nonempty closed convex subset of h, {tn} a sequence of nonexpansive self-mappings of c with a common fixed point, f the set of common fixed points of {tn}, and {λn} a sequence in [0, 1] such that λn → 0, ∞∑ n=1 λn = ∞, and ∞∑ n=1 |λn+1 − λn| < ∞. (3.3) suppose that {tn} satisfies the condition (z) and ∞∑ n=1 sup{‖tn+1y − tny‖ : y ∈ d} < ∞ for every nonempty bounded subset d of c. let x and u be points in c and {xn} a sequence defined by x1 = x and (3.1) for n ∈ n. then {xn} converges strongly to pf(u). using theorems 3.1 and 3.3, we obtain the following: theorem 3.4. let h, c, {tn}, f, and {λn} be the same as in theorem 3.3. let θ be a nonnegative real number with θ < 1 and {fn} a sequence of θ-contractions on c which is stable on f. let y be a point in c and {yn} a sequence defined by y1 = y and (3.2) for n ∈ n. then {yn} converges strongly to w, where w is the unique fixed point of pf ◦ f1. we also know the following result; see [1,5]. theorem 3.5. let h be a hilbert space, c a nonempty closed convex subset of h, {sn} a sequence of nonexpansive self-mappings of c with a common fixed point, f the set of common fixed points of {sn}. let {λn} and {βn} be sequences in [0, 1] such that λn → 0, ∞∑ n=1 λn = ∞, and 0 < lim inf n→∞ βn ≤ lim sup n→∞ βn < 1. suppose that {sn} satisfies the conditions (z) and (r). let x and u be points in c and {xn} a sequence defined by x1 = x and xn+1 = λnu + (1 − λn) ( (1 − βn)xn + βnsnxn ) for n ∈ n. then {xn} converges strongly to pf(u). using theorems 3.1 and 3.5, we also obtain the following: theorem 3.6. let h, c, {sn}, f, {λn}, and {βn} be the same as in theorem 3.5. let θ be a nonnegative real number with θ < 1 and {fn} a sequence of θ-contractions on c which is stable on f. let y be a point in c and {yn} a sequence defined by y1 = y and yn+1 = λnfn(yn) + (1 − λn) ( (1 − βn)yn + βnsnyn ) for n ∈ n. then {yn} converges strongly to w, where w is the unique fixed point of pf ◦ f1. cubo 16, 1 (2014) viscosity approximation methods with a sequence . . . 15 proof. set tn = (1 − βn)i + βnsn for n ∈ n. then it is clear that each tn is nonexpansive and yn+1 = λnfn(yn) + (1 − λn)tnyn for n ∈ n. let x and u be points in c and {xn} a sequence defined by x1 = x and xn+1 = λnu + (1 − λn)tnxn for n ∈ n. then it follows from theorem 3.5 that xn → pf(u). therefore, theorem 3.1 implies the conclusion. 4 convergence theorems by the hybrid steepest descent method in this section, we deal with the variational inequality problem over the set of common fixed points of a sequence of nonexpansive mappings; see problem 4.1 below. then we prove some strong convergence theorems by the hybrid steepest descent method introduced by yamada [30]. we know many results by using the hybrid steepest descent method; see [2,14,17,29,31]. problem 4.1. let h be a hilbert space, {tn} a sequence of nonexpansive self-mappings of h with a common fixed point, f the set of common fixed points of {tn}, and a : h → h a κ-strongly monotone and η-lipschitzian mapping, where κ and η are positive real numbers such that η2 < 2κ. then find z ∈ f such that 〈y − z, az〉 ≥ 0 for all y ∈ f. remark 4.2. the assumption that η2 < 2κ in problem 4.1 is not restrictive. indeed, suppose that a κ-strongly monotone and η-lipschitzian mapping a is given. let us choose a positive constant µ such that µ < 2κ/η2, and define κ′ = µκ and η′ = µη. then it is easy to verify that (η′)2 < 2κ′, µa is κ′-strongly monotone and η′-lipschitzian, and moreover, 〈y − z, az〉 ≥ 0 is equivalent to 〈y − z, µaz〉 ≥ 0 for every y, z ∈ h. as a consequence of theorem 3.1, we can obtain the following theorem, which shows that every sequence generated by the hybrid steepest descent method for problem 4.1 is convergent whenever so is every sequence generated by the halpern type iterative method for the sequence of nonexpansive mappings. theorem 4.3. let h, {tn}, f, κ, η, and a be the same as in problem 4.1. let {λn} be a sequence in [0, 1] such that ∑ ∞ n=1 λn = ∞. suppose that for any (x, u) ∈ h × h, the sequence {xn} defined by x1 = x and xn+1 = λnu + (1 − λn)tnxn (4.1) for n ∈ n converges strongly to pf(u). let y be a point in h and {yn} a sequence defined by y1 = y and yn+1 = (i − λna)tnyn (4.2) for n ∈ n. then {yn} converges strongly to the unique solution of problem 4.1. 16 koji aoyama & yasunori kimura cubo 16, 1 (2014) proof. set fn = (i − a)tn for n ∈ n. since tn is nonexpansive, fn is a θ-contraction by lemma 2.4, where θ = √ 1 − 2κ + η2. by the definition of {yn}, it is clear that yn+1 = λn(i − a)tnxn + (1 − λn)tnyn = λnfn(yn) + (1 − λn)tnyn for every n ∈ n. it is also clear that {fn} is stable on f. thus theorem 3.1 implies that {yn} converges strongly to w = pf ( (i − a)t1w ) = pf(i − a)w, which is the unique solution of problem 4.1 by lemma 2.4. using theorem 3.6 and other known results, we also obtain the following: theorem 4.4 (iemoto and takahashi [17, theorem 3.1]). let h, {tn}, f, κ, η, and a be the same as in problem 4.1. let {λn} be a sequence in [0, 1] such that λn → 0 and ∞∑ n=1 λn = ∞ and {γn} a sequence in [a, b], where 0 < a ≤ b < 1. for each n ∈ n and k ∈ {1, 2, . . . , n + 1}, let un,k be a mapping defined by un,k = { i if k = n + 1; un,k = (1 − γk)i + γktkun,k+1 if k ∈ {1, 2, . . . , n}. let y be a point in h and {yn} a sequence defined by y1 = y and yn+1 = (i − λna)un,1yn (4.3) for n ∈ n. then {yn} converges strongly to the unique solution of problem 4.1. proof. set fn = (i − a)un,1 and sn = t1un,2 for n ∈ n. then it is obvious from (4.3) that yn+1 = λn(i − a)un,1yn + (1 − λn)un,1yn = λnfn(yn) + (1 − λn) ( (1 − γ1)yn + γ1snyn ) for every n ∈ n. it is known that fix(sn) = fix(un,1) = n ⋂ k=1 fix(tk) by [23, lemma 3.2]; see also [9, lemma 4.2]. hence we have ∞ ⋂ n=1 fix(sn) = ∞ ⋂ n=1 fix(un,1) = ∞ ⋂ n=1 n ⋂ k=1 fix(tk) = f and thus fn(z) = (i − a)un,1z = (i − a)z for all z ∈ f. therefore, {fn} is stable on f. since un,1 is nonexpansive, fn is a θ-contraction by lemma 2.4, where θ = √ 1 − 2κ + η2. it is also known that {sn} satisfies the conditions (z) and (r); see [3, 5, 9, 11]. therefore, theorem 3.6 implies that {yn} converges strongly to w = pf ( f1(w) ) = pf(i − a)w, which is the unique solution of problem 4.1 by lemma 2.4. cubo 16, 1 (2014) viscosity approximation methods with a sequence . . . 17 5 zero point problems and fixed point problems motivated by ceng, petruşel, and yao [13], we consider the problem of finding a common solution of the zero point problem for a maximal monotone operator and the fixed point problems for nonexpansive mappings. then, by using theorem 3.6, we prove the following strong convergence theorem, which is a generalization of [13, theorem 3.1]; see remark 5.2 below. theorem 5.1. let h be a hilbert space, c a nonempty closed convex subset of h, {tn} a sequence of nonexpansive self-mappings of c, a ⊂ h × h a maximal monotone operator with dom(a) ⊂ c, θ a nonnegative real number with θ < 1, and f a θ-contraction on c. let {αn}, {βn}, and {γn} be sequences in [0, 1) such that αn → 0, ∑ ∞ n=1 αn = ∞, 0 < lim infn→∞ βn ≤ supn βn < 1, αn + βn ≤ 1 for every n ∈ n, 0 < lim infn→∞ γn ≤ supn γn < 1, and γn+1 − γn → 0. let {ρn} be a sequnece of positive real numbers such that infn ρn > 0 and ρn+1 − ρn → 0. suppose that f = ⋂ ∞ n=1 fix(tn) ∩ a −10 is nonempty and lim n→∞ sup y∈d ‖tny − tmtny‖ = 0 and lim n→∞ sup y∈d ‖tn+1y − tmtny‖ = 0 (5.1) for any m ∈ n and nonempty bounded subset d of c. let y be a point in c and {yn} a sequence defined by y1 = y and yn+1 = αnf(vnxn) + (1 − αn − βn)xn + βntnvnxn (5.2) for n ∈ n, where vn = γni + (1 − γn)tnjρn and jρn is the resolvent of a. then {yn} converges strongly to the unique fixed point of pf ◦ f. proof. since γn 6= 1 and fix(tn) ∩ fix(jρn) = fix(tn) ∩ a −10 is nonempty, it follows from [8, corollary 3.9] and [9, corollary 3.6] that fix(vn) = fix(tnjρn) = fix(tn) ∩ fix(jρn) = fix(tn) ∩ a −10 and fix(tnvn) = fix(tn) ∩ fix(vn) = fix(vn) for every n ∈ n. therefore, we have ∞ ⋂ n=1 fix(tnvn) = ∞ ⋂ n=1 fix(vn) = ∞ ⋂ n=1 fix(tn) ∩ a −10 = f 6= ∅. (5.3) it is clear that each vn is nonexpansive and thus f ◦ vn is a θ-contraction for every n ∈ n. since f(vnz) = f(z) for all z ∈ f by (5.3), we see that {f ◦ vn} is stable on f. we next show that {tnvn} satisfies the condition (r). let d be a nonempty bounded subset of c. by (5.1), we have lim n→∞ sup y∈d ‖tn+1y − tny‖ ≤ lim n→∞ sup y∈d ‖tn+1y − t1tny‖ + lim n→∞ sup y∈d ‖t1tny − tny‖ = 0 18 koji aoyama & yasunori kimura cubo 16, 1 (2014) and hence {tn} satisfies the condition (r). since {jρn} satisfies the condition (r) by [5, example 4.2], lemma 2.2 shows that {tnjρn} satisfies the condition (r). thus lemma 2.3 implies that {vn} satisfies the condition (r). therefore, it follows from lemma 2.2 that {tnvn} satisfies the condition (r). we next show that {tnvn} satisfies the condition (z). let {xn} be a bounded sequence in c such that xn − tnvnxn → 0 and {xni} a subsequence of {xn} such that xni ⇀ z. it is enough to show that z ∈ f. it follows from [8, theorem 3.10] that xn − tnxn → 0 and xn − vnxn → 0. let d be a nonempty bounded subset of c such that xn ∈ d for all n ∈ n. for fixed m ∈ n, it follows from (5.1) and xn − tnxn → 0 that ‖xn − tmxn‖ ≤ ‖xn − tnxn‖ + ‖tnxn − tmtnxn‖ + ‖tmtnxn − tmxn‖ ≤ 2 ‖xn − tnxn‖ + sup y∈d ‖tny − tmtny‖ → 0 as n → 0. thus, by the demiclosedness [15, p.109] of i − tm, z ∈ fix(tm) and hence z ∈ ⋂ ∞ n=1 fix(tn). on the other hand, xn−vnxn → 0 and [9, corollary 3.2] imply that xn−tnjρnxn → 0 and hence xn − jρnxn → 0 by [8, theorem 3.10]. thus z ∈ a −10 because {jρn} satisfies the condition (z); see [8, lemma 5.1], [10, lemma 2.1], and [12, lemma 2.4]. consequently, we conclude that z ∈ f. finally, by assumption, it is obvious that yn+1 = αnf(vnxn) + (1 − αn) ( ( 1 − βn 1 − αn ) xn + βn 1 − αn tnvnxn ) for every n ∈ n and 0 < lim inf n→∞ βn 1 − αn ≤ lim sup n→∞ βn 1 − αn < 1. thus theorem 3.6 implies the conclusion. remark 5.2. ceng, petruşel, and yao [13] considered an equilibrium problem for a real-valued function φ defined on c×c and they adopted the resolvent of φ in [13, theorem 3.1]. according to [6], such an equilibrium problem can be regarded as a zero point problem for a maximal monotone operator a ⊂ h × h and we know that the resolvent φ is equivalent to that of a. thus theorem 5.1 is a generalization of [13, theorem 3.1]. acknowledgment. the authors are supported by grant-in-aid for scientific research no. 22540175 from japan society for the promotion of science. received: march 2012. accepted: march 2013. cubo 16, 1 (2014) viscosity approximation methods with a sequence . . . 19 references [1] k. aoyama, an iterative method for fixed point problems for sequences of nonexpansive mappings, fixed point theory and its applications, yokohama publ., yokohama, 2010, pp. 1–7. 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() cubo a mathematical journal vol.17, no¯ 02, (73–87). june 2015 on an anisotropic allen-cahn system alain miranville laboratoire de mathématiques et applications, université de poitiers, umr cnrs 7348 sp2mi, boulevard marie et pierre curie téléport 2, f-86962 chasseneuil futuroscope cedex, france alain.miranville@math.univ-poitiers.fr abstract our aim in this paper is to prove the existence and uniqueness of solutions for an allen-cahn type system based on a modification of the ginzburg-landau free energy proposed in [11]. in particular, the free energy contains an additional term called willmore regularization and takes into account anisotropy effects. resumen nuestro propósito en este trabajo es probar la existencia y unicidad de soluciones para un sistema de tipo allen-cahn basados en una modificación de la enerǵıa libre ginzburg-landau propuesta en [11]. en particular, la enerǵıa libre contiene un término adicional llamado regularización de willmore y considera efectos de anisotroṕıa. keywords and phrases: allen-cahn equation, willmore regularization, anisotropy effects, wellposedness 2010 ams mathematics subject classification: 35b45, 35k55 74 alain miranville cubo 17, 2 (2015) 1 introduction the allen-cahn equation, ∂u ∂t − ∆u + f(u) = 0, (1.1) where u is the order parameter and f(s) = s3 −s, describes important processes related with phase separation in binary alloys, namely, the ordering of atoms in a lattice (see [1]). this equation is obtained by considering the ginzburg-landau free energy, ψgl = ∫ ω ( 1 2 |∇u|2 + f(u)) dx, (1.2) where ω is the domain occupied by the material and, typically, f(s) = 1 4 (s2 − 1)2. assuming a relaxation dynamics, i.e., writing ∂u ∂t = − dψgl du , (1.3) where d du denotes a variational derivative, we obtain (1.1). in [11] (see also [2]), the authors introduced the following modification of the ginzburg-landau free energy: ψagl = ∫ ω (δ( ∇u |∇u| )( 1 2 |∇u|2 + f(u)) + β 2 ω2) dx, β > 0, (1.4) ω = −∆u + f(u), (1.5) where g(u) = 1 2 ω2 is called nonlinear willmore regularization, β is a small regularization parameter and the function δ accounts for anisotropy effects. the willmore regularization is relevant, e.g., in determining the equilibrium shape of a crystal in its own liquid matrix, when anisotropy effects are strong. indeed, in that case, the equilibrium interface may not be a smooth curve, but may present facets and corners with slope discontinuities (see, e.g., [9]), which can lead to an ill-posed problem and requires regularization. the allen-cahn equation associated with (1.4) has been studied in [6] in the particular cases δ ≡ 1 (isotropic case) and δ ≡ −1 (in that case, ψagl is also called functionalized cahn-hilliard energy in [8]). in particular, well-posedness results have been obtained. the cahn-hilliard equation associated with (1.4) (obtained by writing ∂u ∂t = ∆dψagl du ) has been studied in [4], again, in the isotropic case δ ≡ 1; we also refer the reader to [2] and [12] for numerical studies. in this paper, we actually consider the following regularization of ψagl: cubo 17, 2 (2015) on an anisotropic allen-cahn system 75 ψragl = ∫ ω (δ( ∇u (ǫ + |∇u| 2 ) 1 2 )( 1 2 |∇u|2 + f(u)) + β 2 ω2) dx, ǫ > 0. (1.6) we have, in that case and formally, dψragl = ∫ ω (δ( ∇u (ǫ + |∇u| 2 ) 1 2 )(∇u · ∇du + f(u)du) + βωf′(u)du − βω∆du) dx + ∫ ω δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇du (ǫ + |∇u| 2 ) 1 2 ( 1 2 |∇u|2 + f(u)) dx − ∫ ω δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u ∇u.∇du (ǫ + |∇u| 2 ) 3 2 ( 1 2 |∇u|2 + f(u)) dx, where δ′ denotes the differential (gradient) of δ. therefore, assuming proper boundary conditions, dψragl du = −div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (1.7) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( f(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u f(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω. assuming again a relaxation dynamics, we finally obtain the following regularized anisotropic allen-cahn system: ∂u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (1.8) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( f(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u f(u)∇u (ǫ + |∇u| 2 ) 3 2 ) 76 alain miranville cubo 17, 2 (2015) +βωf′(u) − β∆ω = 0, ω = −∆u + f(u). (1.9) we proved in [5] the existence and uniqueness of solutions to (1.8)-(1.9), but only in one space dimension, due to a lack of regularity on ∂u ∂t . thus, in order to handle the problem in higher space dimensions, we consider in this paper the following further regularized allen-cahn system: ∂u ∂t − α ∂∆u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (1.10) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( f(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u f(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω = 0, α > 0, ω = −∆u + f(u). (1.11) a term of the form −α∂∆u ∂t appears in generalizations of the allen-cahn equation proposed in [3], based on a separate balance law for internal microforces, i.e., for interactions at a microscopic level (we can note that the derivation proposed in [3] is strongly based on the usual ginzburg-landau free energy; it would thus be interesting to go back to the arguments in [3] and see whether/how they can be adapted to a more general free energy). such a regularization is also similar to the viscous cahn-hilliard equation proposed in [7]. actually, the approach in [3], applied to the cahn-hilliard equation, allows to recover the viscous cahn-hilliard equation. we prove, in the next sections, the existence and uniqueness of solutions to (1.10)-(1.11). it is important to note however that our estimates are not independent of ǫ, so that we cannot pass to the limit as ǫ goes to 0. this is not surprising, as the problem formally obtained by taking ǫ = 0 cannot correspond to the (allen-cahn) problem associated with the free energy (1.4) (see also [2] and [11]). actually, this is related with a proper functional setting for the limit problem and, more precisely, for the allen-cahn system associated with (1.4). this will be studied elsewhere. cubo 17, 2 (2015) on an anisotropic allen-cahn system 77 2 setting of the problem we consider the following initial and boundary value problem (for simplicity, we take α = β = 1): ∂u ∂t − ∂∆u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (2.1) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( f(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u f(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +ωf′(u) − ∆ω = 0, ω = −∆u + f(u), (2.2) u = ω = 0 on γ, (2.3) u|t=0 = u0, (2.4) in a bounded and regular domain ω ⊂ rn, n = 1, 2 or 3, with boundary γ. as far as the nonlinear terms δ and f are concerned, we assume that δ is of class c2, (2.5) f is of class c2, f(0) = 0, f′ ≥ −c0, c0 ≥ 0, (2.6) sf′(s)f(s) − f(s)2 ≥ −c1, c1 ≥ 0, s ∈ r, (2.7) sf′′(s) ≥ −c2, c2 ≥ 0, s ∈ r, (2.8) 78 alain miranville cubo 17, 2 (2015) |f(s)| ≤ σf(s)2 + cσ, ∀σ > 0, s ∈ r, (2.9) |f(s)| ≤ c3(|s| p + 1), c3 > 0, p ≥ 0 if n = 1 or 2, p ∈ [0, 7] if n = 3, (2.10) where f is an antiderivative of f. these assumptions are satisfied by polynomials of the form f(s) = ∑q i=1 ais i, q ≥ 3 odd (q ≤ 5 when n = 3), aq > 0, and, in particular, by the usual cubic nonlinear term f(s) = s 3 − s. we denote by ((·, ·)) the usual l2-scalar product, with associated norm ‖ · ‖, and we denote by ‖ · ‖x the norm in the banach space x. throughout the paper, the same letter c (and, sometimes, c′) denotes constants which may vary from line to line. similarly, the same letter q denotes monotone increasing (with respect to each argument) functions which may vary from line to line. 3 a priori estimates we multiply (2.1) by u and have, integrating over ω and by parts and owing to (2.2), 1 2 d dt (‖u‖2 + ‖∇u‖2) + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇u)) + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), u)) (3.1) + 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇u (ǫ + |∇u| 2 ) 1 2 )) + ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), f(u)∇u (ǫ + |∇u| 2 ) 1 2 )) − 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|4 (ǫ + |∇u| 2 ) 3 2 )) − ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, f(u)|∇u|2 (ǫ + |∇u| 2 ) 3 2 )) +‖ω‖2 + ∫ ω (uf′(u)f(u) − f(u)2) dx + ((uf′′(u)∇u, ∇u)) = 0. we note that |s| (ǫ + |s|2) 1 2 ≤ 1, s ∈ rn, so that cubo 17, 2 (2015) on an anisotropic allen-cahn system 79 |δ( ∇u (ǫ + |∇u| 2 ) 1 2 )| ≤ c, |δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )| ≤ c′. therefore, |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇u))| ≤ c‖∇u‖2, (3.2) |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), u))| ≤ σ‖f(u)‖2 + cσ‖∇u‖ 2, ∀σ > 0, (3.3) |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇u (ǫ + |∇u| 2 ) 1 2 ))| ≤ c‖∇u‖2 (3.4) ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), f(u)∇u (ǫ + |∇u| 2 ) 1 2 )) ≤ c ∫ ω |f(u)| dx (3.5) ≤ (owing to (2.9)) ≤ σ‖f(u)‖2 + cσ, ∀σ > 0, |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|4 (ǫ + |∇u| 2 ) 3 2 ))| ≤ c‖∇u‖2 (3.6) and, as above, |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, f(u)|∇u|2 (ǫ + |∇u| 2 ) 3 2 ))| ≤ c ∫ ω |f(u)| dx ≤ σ‖f(u)‖2 + cσ, ∀σ > 0. (3.7) it thus follows from (2.7)-(2.8) and (3.1)-(3.7) that d dt (‖u‖2 + ‖∇u‖2) + 2‖ω‖2 ≤ c‖∇u‖2 + σ‖f(u)‖2 + cσ, ∀σ > 0. (3.8) we then note that, owing to (2.6), ‖ω‖2 ≥ ‖∆u‖2 + ‖f(u)‖2 − 2c0‖∇u‖ 2 (3.9) and it follows from (3.8)-(3.9) that, taking σ = 1, 80 alain miranville cubo 17, 2 (2015) d dt (‖u‖2 + ‖∇u‖2) + c(‖u‖2 h2(ω) + ‖f(u)‖2) ≤ c′(‖u‖2 h1(ω) + 1), c > 0. (3.10) we then multiply (2.1) by ∂u ∂t and obtain, owing to (2.2), ‖ ∂u ∂t ‖2 + ‖∇ ∂u ∂t ‖2 + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇ ∂u ∂t )) + ((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), ∂u ∂t )) (3.11) + 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 )) + ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), f(u)∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 )) − 1 2 ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|2∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 )) − ((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, f(u)∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 )) + 1 2 d dt ‖ω‖2 = 0. we have, proceeding as above, |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u, ∇ ∂u ∂t ))| + |((δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u), ∂u ∂t ))| (3.12) + 1 2 |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), |∇u|2∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 ))| + 1 2 |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, |∇u|2∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 ))| ≤ c(‖∇u‖ + ‖f(u)‖)‖∇ ∂u ∂t ‖ ≤ 1 2 ‖∇ ∂u ∂t ‖2 + c(‖∇u‖2 + ‖f(u)‖2). furthermore, for the most difficult case n = 3 and p = 7 and owing to (2.10) and agmon’s inequality (see, e.g., [10]), |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ), f(u)∇∂u ∂t (ǫ + |∇u| 2 ) 1 2 ))| + |((δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u, f(u)∇u · ∇∂u ∂t (ǫ + |∇u| 2 ) 3 2 ))| (3.13) ≤ cǫ− 1 2 ∫ ω |f(u)||∇ ∂u ∂t | dx ≤ 1 2 ‖∇ ∂u ∂t ‖2 + c′ǫ−1 ∫ ω (|u|14 + 1) dx cubo 17, 2 (2015) on an anisotropic allen-cahn system 81 ≤ 1 2 ‖∇ ∂u ∂t ‖2 + cǫ−1(‖u‖8l∞(ω) + 1)(‖u‖ 6 l6(ω) + 1) ≤ 1 2 ‖∇ ∂u ∂t ‖2 + cǫ−1(‖u‖4 h1(ω) ‖u‖4 h2(ω) + 1)(‖u‖6 h1(ω) + 1) ≤ 1 2 ‖∇ ∂u ∂t ‖2 + cǫ−1(‖u‖10 h1(ω) + 1)(‖u‖4 h2(ω) + 1). we thus deduce from (3.11)-(3.13) that d dt ‖ω‖2 + ‖ ∂u ∂t ‖2h1(ω) ≤ cǫ −1(‖u‖10h1(ω) + 1)(‖u‖ 4 h2(ω) + 1). (3.14) recalling that ‖ω‖2 ≥ ‖∆u‖2 + ‖f(u)‖2 − 2c0‖∇u‖ 2, we finally deduce that d dt ‖ω‖2 + ‖ ∂u ∂t ‖2h1(ω) ≤ cǫ −1(‖u‖10h1(ω) + 1)(‖u‖ 2 h2(ω) + 1)(‖ω‖ 2 + ‖u‖2h1(ω) + 1). (3.15) we now multiply (2.1) by −∆u and have, owing to (2.2), 1 2 d dt (‖∇u‖2 + ‖∆u‖2) + ((divϕ1(∇u), ∆u)) − ((ϕ2(∇u)f(u), ∆u)) (3.16) +((divϕ3(∇u), ∆u)) + ((div(f(u)ϕ4(∇u)), ∆u)) −((divϕ5(∇u), ∆u)) − ((div(f(u)ϕ6(∇u)), ∆u)) −((ωf′(u), ∆u)) + ((∆f(u), ∆u)) + ‖∇∆u‖2 = 0, where ϕ1(s) = δ( s (ǫ + |s|2) 1 2 )s, ϕ2(s) = δ( s (ǫ + |s|2) 1 2 ), ϕ3(s) = 1 2 |s|2 (ǫ + |s|2) 1 2 δ′( s (ǫ + |s|2) 1 2 ), 82 alain miranville cubo 17, 2 (2015) ϕ4(s) = 1 (ǫ + |s|2) 1 2 δ′( s (ǫ + |s|2) 1 2 ), ϕ5(s) = 1 2 δ′( s (ǫ + |s|2) 1 2 ) · s |s|2s (ǫ + |s|2) 3 2 , ϕ6(s) = δ ′( s (ǫ + |s|2) 1 2 ) · s s (ǫ + |s|2) 3 2 . noting that divϕi(∇u) = ϕ ′ i(∇u) · ∇∇u, i = 1, 3, 5, div(f(u)ϕi(∇u)) = f(u)ϕ ′ i(∇u) · ∇∇u + f(u)ϕi(∇u) · ∇u, i = 4, 6, it follows from the continuous embedding h2(ω) ⊂ c(ω) and (2.2) that |((divϕ1(∇u), ∆u))| + |((ϕ2(∇u)f(u), ∆u))| +|((divϕ3(∇u), ∆u))| + |((div(f(u)ϕ4(∇u)), ∆u))| +|((divϕ5(∇u), ∆u))| + |((div(f(u)ϕ6(∇u)), ∆u))| +|((ωf′(u), ∆u)) + ((∆f(u), ∆u))| ≤ q(ǫ−1, ‖u‖h2(ω)) (here, we have used the facts that the ϕ′i’s are bounded and that 1 ǫ+|s|2 ≤ ǫ−1 and |s| 2 ǫ+|s|2 ≤ 1), hence, d dt (‖∇u‖2 + ‖∆u‖2) + c‖u‖2h3(ω) ≤ q(ǫ −1, ‖u‖h2(ω)), c > 0. (3.17) we finally multiply (2.1) by −∆∂u ∂t and find, owing to (2.2), ‖∇ ∂u ∂t ‖2 + ‖∆ ∂u ∂t ‖2 + ((divϕ1(∇u), ∆ ∂u ∂t )) − ((ϕ2(∇u)f(u), ∆ ∂u ∂t )) +((divϕ3(∇u), ∆ ∂u ∂t )) + ((div(f(u)ϕ4(∇u)), ∆ ∂u ∂t )) −((divϕ5(∇u), ∆ ∂u ∂t )) − ((div(f(u)ϕ6(∇u)), ∆ ∂u ∂t )) cubo 17, 2 (2015) on an anisotropic allen-cahn system 83 −((ωf′(u), ∆ ∂u ∂t )) + ((∆f(u), ∆ ∂u ∂t )) + 1 2 d dt ‖∇∆u‖2 = 0, which yields, proceeding as above, d dt ‖∇∆u‖2 + c‖ ∂u ∂t ‖2h2(ω) ≤ q(ǫ −1, ‖u‖h2(ω)), c > 0. (3.18) 4 existence and uniqueness of solutions we have the theorem 1. we assume that u0 ∈ h 2(ω) ∩ h10(ω). then, (2.1)-(2.4) possesses a unique solution u such that u ∈ l∞(0, t ; h2(ω) ∩ h10(ω)) ∩ l 2(0, t ; h3(ω)) and ∂u ∂t ∈ l2(0, t ; h10(ω)), ∀t > 0. furthermore, if u0 ∈ h 3(ω)∩h10(ω), then u ∈ l ∞(0, t ; h3(ω)∩h10(ω)) and ∂u ∂t ∈ l2(0, t ; h2(ω)∩ h10(ω)), ∀t > 0. proof. a) existence: the proof of existence is based on the a priori estimates derived in the previous section and, e.g., a classical galerkin scheme. in particular, we first deduce from (3.10) that u ∈ l∞(0, t ; h10(ω)) ∩ l 2(0, t ; h2(ω)), ∀t > 0. having this, it follows from (3.15) that u ∈ l∞(0, t ; h2(ω)) and ∂u ∂t ∈ l2(0, t ; h10(ω)). the only difficulty here is to pass to the limit in the nonlinear terms when considering galerkin approximations. more precisely, we have, owing to classical aubin-lions compactness results, a sequence um of solutions to approximated problems such that um → u in l ∞ (0, t ; h2(ω)) weak star, l2(0, t ; h1(ω)) and a.e.. we consider, for instance, the passage to the limit in the term f(um)ϕ4(∇um) (the other terms can be handled similarly or are simpler to treat). we have |f(um)ϕ4(∇um) − f(u)ϕ4(∇u)| ≤ |f(um)(ϕ4(∇um) − ϕ4(∇u))| + |(f(um) − f(u))ϕ4(∇u)|, so that, proceeding as in the previous section (using, in particular, the fact that ϕ′4 is bounded), ‖f(um)ϕ4(∇um) − f(u)ϕ4(∇u)‖ ≤ q(ǫ −1, ‖um‖h2(ω), ‖u‖h2(ω))‖um − u‖h1(ω), 84 alain miranville cubo 17, 2 (2015) hence the convergence in l2(0, t ; l2(ω)). furthermore, noting that ωm (defined as in (2.2)) converges to ω in l∞(0, t ; l2(ω)) weak star and f′(um) converges to f ′(u) in l2(0, t ; h1(ω)), we easily see that ωmf ′(um) converges to ωf ′(u) in l 3 2 (0, t ; l 3 2 (ω)) weak. b) uniqueness: let u1 and u2 be two solutions to (2.1)-(2.3) (ω1 and ω2 being defined as in (2.2)) with initial data u0,1 and u0,2, respectively. we set u = u1 − u2, ω = ω1 − ω2, u0 = u0,1 − u0,2 and have ∂u ∂t − ∂∆u ∂t − div(ϕ1(∇u1) − ϕ1(∇u2)) + ϕ2(∇u1)f(u1) − ϕ2(∇u2)f(u2) (4.1) −div(ϕ3(∇u1) − ϕ3(∇u2)) − div(f(u1)ϕ4(∇u1) − f(u2)ϕ4(∇u2)) +div(ϕ5(∇u1) − ϕ5(∇u2)) + div(f(u1)ϕ6(∇u1) − f(u2)ϕ6(∇u2)) +ω1f ′(u1) − ω2f ′(u2) − ∆ω = 0, ω = −∆u + f(u1) − f(u2), (4.2) u = ω = 0 on γ, (4.3) u|t=0 = u0. (4.4) we multiply (4.1) by u and obtain, owing to (4.2), 1 2 d dt (‖u‖2 + ‖∇u‖2) + ((ϕ1(∇u1) − ϕ1(∇u2), ∇u)) (4.5) +((ϕ2(∇u1)f(u1) − ϕ2(∇u2)f(u2), u)) + ((ϕ3(∇u1) − ϕ3(∇u2), ∇u)) +((f(u1)ϕ4(∇u1) − f(u2)ϕ4(∇u2)), ∇u)) − ((ϕ5(∇u1) − ϕ5(∇u2), ∇u)) −((f(u1)ϕ6(∇u1) − f(u2)ϕ6(∇u2)), ∇u)) + ((ω1f ′ (u1) − ω2f ′ (u2), u)) cubo 17, 2 (2015) on an anisotropic allen-cahn system 85 +((f(u1) − f(u2), u)) + ‖∆u‖ 2 = 0. we have, for instance (again, the other terms can be handled similarly or are easier to treat) and proceeding as in the previous section, |((f(u1)ϕ4(∇u1) − f(u2)ϕ4(∇u2)), ∇u))| ≤ |((f(u1)(ϕ4(∇u1) − ϕ4(∇u2)), ∇u))| +|(((f(u1) − f(u2))ϕ4(∇u2), ∇u))| ≤ q(ǫ −1, t, ‖u0,1‖h2(ω), ‖u0,2‖h2(ω))‖∇u‖ 2. furthermore, owing to (4.2) and a proper interpolation inequality, |((ω1f ′(u1) − ω2f ′(u2), u))| ≤ |((ωf ′(u1), u))| + |((ω2(f ′u1) − f ′(u2)), u))| ≤ q(t, ‖u0,1‖h2(ω), ‖u0,2‖h2(ω))‖∆u‖‖u‖. we thus find an inequality of the form d dt (‖u‖2 + ‖∇u‖2) ≤ q(ǫ−1, t, ‖u0,1‖h2(ω), ‖u0,2‖h2(ω))‖∇u‖ 2, hence the uniqueness, as well as the continuous dependence with respect to the initial data in the h1-norm. remark 4.1. the viscous cahn-hilliard system associated with the free energy (1.6) reads ∂u ∂t − α ∂∆u ∂t − ∆[−div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (4.6) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( f(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u f(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω] = 0, α > 0, 86 alain miranville cubo 17, 2 (2015) ω = −∆u + f(u). (4.7) taking, for simplicity, dirichlet boundary conditions, u = ∆u = ω = ∆ω = 0 on γ, we can rewrite (4.6) equivalently as (−∆)−1 ∂u ∂t + α ∂u ∂t − div(δ( ∇u (ǫ + |∇u| 2 ) 1 2 )∇u) + δ( ∇u (ǫ + |∇u| 2 ) 1 2 )f(u) (4.8) − 1 2 div( |∇u|2 (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) − div( f(u) (ǫ + |∇u| 2 ) 1 2 δ′( ∇u (ǫ + |∇u| 2 ) 1 2 )) + 1 2 div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u |∇u|2∇u (ǫ + |∇u| 2 ) 3 2 ) + div(δ′( ∇u (ǫ + |∇u| 2 ) 1 2 ) · ∇u f(u)∇u (ǫ + |∇u| 2 ) 3 2 ) +βωf′(u) − β∆ω = 0. even though (4.8) bears some resemblance with (2.1), we have less regularity on ∂u ∂t and thus cannot proceed as above to prove the existence and uniqueness of solutions. we can however prove the well-posedness in one space dimension (see [5]). received: may 2013. accepted: may 2013. references [1] s.m. allen and j.w. cahn, a microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, acta metall. 27 (1979), 1085–1095. [2] f. chen and j. shen, efficient energy stable schemes with spectral discretization in space for anisotropic cahn-hilliard systems, commun. comput. phys., to appear. [3] m. gurtin, generalized ginzburg-landau and cahn-hilliard equations based on a microforce balance, phys. d 92 (1996), 178–192. [4] a. miranville, asymptotic behavior of a sixth-order cahn-hilliard system, submitted. [5] a. miranville, existence of solutions for a one-dimensional allen-cahn equation, submitted. cubo 17, 2 (2015) on an anisotropic allen-cahn system 87 [6] a. miranville and r. quintanilla, a generalization of the allen-cahn equation, submitted. [7] a. novick-cohen, on the viscous cahn-hilliard equation, in material instabilities in continuum and related problems, j.m. ball ed., oxford university press, oxford, 329–342, 1988. [8] k. promislow and h. zhang, critical points of functionalized lagrangians, discrete cont. dyn. system 33 (2013), 1231–1246. [9] j.e. taylor and j.w. cahn, diffuse interfaces with sharp corners and facets: phase-field models with strongly anisotropic surfaces, phys. d 112 (1998), 381–411. [10] r. temam, infinite-dimensional dynamical systems in mechanics and physics, second edition, applied mathematical sciences, vol. 68, springer-verlag, new york, 1997. [11] s. torabi, j. lowengrub, a. voigt and s. wise, a new phase-field model for strongly anisotropic systems, proc. r. soc. a 465 (2009), 1337–1359. [12] s.m. wise, c. wang and j.s. lowengrub, solving the regularized, strongly anisotropic cahnhilliard equation by an adaptative nonlinear multigrid method, j. comput. phys. 226 (2007), 414–446. introduction setting of the problem a priori estimates existence and uniqueness of solutions () cubo a mathematical journal vol.16, no¯ 03, (67–85). october 2014 on the supersingular loci of quaternionic siegel space oliver bültel mathematische fakultät, universität duisburg-essen, thea-leymann strasse 9, 45127 essen, germany. oliver.bueltel@uni-due.de abstract the paper studies the supersingular locus of the characteristic p moduli space of principally polarized abelian 8-folds that are equipped with an action of a maximal order in a quaternion algebra, that is non-split at ∞ but split at p. the main result is that its irreducible components are fermat surfaces of degree p + 1. resumen el art́ıculo estudia el lugar supersingular del espacio módulo caracteŕıstico p de abeliano polarizado 8-veces principal que son equipados con una acción de un orden maximal en un álgebra quaterniona, que es no-divisible en ∞, pero se divide en p. el principal resultados es que los componentes irreducibles son superficies de fermat de grado p+1. keywords and phrases: supersingular abelian varieties, shimura varieties, orthogonal groups. 2010 ams mathematics subject classification: 14l05, 14k10. 68 oliver bültel cubo 16, 3 (2014) 1 introduction let p be a prime number. in [9] oort and li give a description of the supersingular locus sg,1 of ag,1 × fp, the fibre over p of the siegel modular variety of principally polarized abelian g-folds. among their results are that sg,1 has hg(p,1) irreducible components if g is odd and hg(1,p) if g is even, and all of these components have dimension [g2/4]. in this paper we study the supersingular locus of certain pel-moduli spaces sk of type d h 4 , see body of text for a more precise explanation. these moduli spaces are associated to groups g that are twists of go(8). in the complex analytic context there exist uniformisations by quaternionic siegel half-spaces, these are tube domains of the shape h = {x + iy|x,y ∈ mat2(h),x t,ι = x,yt,ι = y > 0}, (1) where h is the non-split quaternion algebra over r, and ι is the standard involution. in the algebraic context sk is a 6-dimensional variety parameterizing abelian 8-folds with a particular kind of additional structure, and on a mild assumption on the level structure this variety is smooth. for every prime of good reduction we introduce the usual integral model for this shimura variety, and we move on to exhibit the geometry of each individual irreducible component of the supersingular locus of the mod p-reduction. our main result says that these components are fermat surfaces. this comes as a surprise, because for a more general shimura variety, the structure of the supersingular locus is usually quite complicated and might not even be smooth, for example this happens in the case of s3,1 the 2-dimensional space of supersingular principally polarized abelian 3-folds, cf. [12, paragraph(4)], [14, proposition 2.4], [9, example(9.4)] or [15] for a very precise exposition. we round off the discussion by turning to the non-supersingular points also, we prove that their p-divisible groups do not have parameters, which is somewhat the exact opposite to their behaviour on the supersingular locus. this too seems unusual, as one sees from the non-supersingular principally polarized abelian 3-folds. these results were already applied in [2] to obtain an eichler-shimura congruence relation for sk and for its shimura divisors. i am indebted to the referee, and there remains the pleasant task of thanking c.kaiser, prof. m.rapoport, and prof. t.wedhorn for remarks on the topic and especially prof. r.taylor for much good advice and prof. f.oort for some email exchanges. the paper is organized as follows: section 2 focuses on local aspects, section 3 on global ones. in subsections 2.1/ 3.1 we sum up definitions and conventions. in subsections 2.2 we explain techniques needed to understand supersingular dieudonné modules. we apply these techniques to supersingular dieudonné modules with the particular additional structure under consideration in the subsections 2.32.4. in subsection 2.5 a result on the non-supersingular locus is obtained (corollary 2.7). cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 69 2 structure theorems on dieudonné modules with pairing 2.1 notions and notations we continue to fix a prime p. if k is a perfect field of characteristic p, then one denotes by w(k) and k(k) the witt ring and fraction field thereof. unless otherwise said, k will be assumed to be algebraically closed. the absolute frobenius x 7→ xp induces automorphisms x 7→ fx on w(k) and k(k) which again will be referred to as absolute frobenii. recall that a dieudonné module is a finitely generated, torsion free w(k)-module together with a f-linear endomorphism f and a f−1-linear endomorphism v that satisfies fv = vf = p. if one tensorizes with q one obtains the isocrystal of m, this is a finite dimensional k(k)-vector space together with a f-linear bijection f. dieudonné modules are called isogenous if they give rise to isomorphic isocrystals. by a pairing on a dieudonné module m one understands a w(k)-bilinear map φ : m × m → w(k) which satisfies φ(x,fy) = fφ(vx,y). when thinking of m as the co-variant dieudonné module of a p-divisible group a over k, this means that φ gives rise to a morphism from a to the serre-dual of a. dieudonné modules with pairings (m,φ) and (m′,φ′) are called isometric if there exists an isomorphism from m to m′ taking φ to φ′. the dimension, dimk(m/vm) of a dieudonné module with non-degenerate pairing is equal to the codimension dimk(m/fm), the rank of m is necessarily even. a pairing is called antisymmetric if φ(x,y) = −φ(y,x) and symmetric if φ(x,y) = φ(y,x). in either of these cases we denote {x ∈ m ⊗ q|φ(m,x) ⊂ w(k)} by mt, if m = mt we say that φ is perfect. in [13, definition(3.1)] the crucial notion of crystalline discriminant of a non-degenerate symmetric pairing is introduced: say the underlying dieudonné module has rank 2n, and choose a k(k)-basis x1, . . . ,x2n with the additional property fx1 ∧ · · · ∧ fx2n = p nx1 ∧ · · · ∧ x2n. the determinant det(φ(xi,xj)), regarded as an element in k(fp) ×/(k(fp) ×)2, can be checked to be independent of the choice of basis and is called the crystalline discriminant crisdisc(m,φ). it only depends on the isogeny class of m allowing one to also write crisdisc(m ⊗ q,φ) for crisdisc(m,φ). when fixing once and for all an element t ∈ w(fp2) × with tσ = −t, the target group k(fp) ×/(k(fp) ×)2 can be identified with {1,t2,p,pt2} if p is odd and with {±1,±t2,±p,±pt2} if p = 2, notice that the kernel of the forgetful map from k(fp) ×/(k(fp) ×)2 to k(k)×/(k(k)×)2 consists of {1,t2}. notice also that the image of crisdisc(m,φ) in the group k(k)×/(k(k)×)2 is the discriminant of a symmetric pairing in the usual sense of linear algebra, hence is independent of the structure of m as a dieudonné module. the following characterization of crystalline discriminants within {1,t2} will be useful, see [13, corollary(3.5)] for a proof: fact 1 (ogus). assume that φ is a non-degenerate symmetric pairing on a dieudonné module m of rank 2n. assume also that there exists a φ-isotropic k(k)-subspace a ⊂ m ⊗ q of dimension n. then crisdisc(m,φ) = (−1)nt2 dimk(k)(a+fa/a), in particular, if a is an isocrystal, then crisdisc(m,φ) = (−1)n. 70 oliver bültel cubo 16, 3 (2014) recall that the grassmannian of n-dimensional isotropic subspaces of the k(k)-vector space m ⊗ q has two connected components. two such spaces a1 and a2 lie in the same component if and only if the integer dimk(k)(a1 + a2/a1) is even, we will say that a1 and a2 have the same parity if this is the case. thus by the above fact (−1)n crisdisc(m,φ) is the trivial element of k(fp) ×/(k(fp) ×)2 if and only if the bijection f does not change the parity of the maximal isotropic subspaces. if p is odd and if φ is a perfect form on m, then one can deduce a further formulation: pick a maximal isotropic k-subspace a ⊂ m = m/pm complementary to vm. lift it to w(k) to obtain a maximal isotropic w(k) submodule a of m (the grassmannian is smooth). observe that fa = fm to conclude that (mod 2): dimk(a + fa/a) ≡ dimk(a + vm/a) + dimk(vm + fm/vm) ≡ dimk(m/vm + fm) (mod 2). the integer dimk(m/vm + fm) is called the oort invariant of m and denoted a(m). thus, we have derived the consequence, implicitely stated in [11, section 5.3]: fact 2 (moonen). assume that φ is a perfect symmetric pairing on the dieudonné module m of rank 2n. if p is odd, then crisdisc(m,φ) = (−1)nt2a(m). the dieudonné module m is called superspecial if it satisfies fm = vm, i.e. if rankw(k)(m) = 2a(m). superspecial dieudonné modules may conveniently be described in terms of their skeletons, these are the w(fp2)-submodules defined by m̃ = {x ∈ m|fx = vx}. we write ob for the ring extension of w(fp2), obtained by adjoining an indeterminate σ subject to the relations σ 2 = p and σa = faσ, it operates in a self-explanatory way on m̃. as remarked in [8] the assignment m 7→ m̃ sets up an equivalence of the category of superspecial dieudonné modules with the category of finitely generated torsion free ob-modules. we also write b for ob ⊗ q, it is the unique non-split quaternion algebra over k(fp). observe that ob is the maximal order of b. we let mb be the maximal ideal of ob, one has ob/mb ∼= fp2. we need to put pairings into the picture as follows. if φ is a pairing on a superspecial dieudonné module m, then one considers a ob-valued pairing on m̃ defined by: φ(x,y) = φ(x,σty) − φ(x,y)σt (2) this is ob-sesquilinear, i.e. satisfies φ(ux,vy) = uφ(x,y)v ι, for u,v ∈ ob. the involution ι is the standard one, mapping a+bσ to fa−bσ. conversely any ob-sesquilinear form arises from a pairing on m in the way described, φ is non-degenerate/perfect if and only if φ is. unless otherwise said we assume from now on that φ is symmetric, in terms of φ this means cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 71 φ(y,x)ι = −φ(x,y) for all x,y ∈ m̃. the ob-module m̃ with form φ is called hyperbolic if on a suitable ob-basis e1, . . . ,en/2,f1, . . . ,fn/2 of m̃ one has φ(ei,ej) = φ(fi,fj) = 0 , φ(ei,fj) = wδi,j (3) for some non-zero w ∈ ob, uniquely determined only up to multiplication by o × b . it turns out that w = −σrt is a very convenient choice as the values of the corresponding form φ will then read: φ(ei,fj) = { 0 r ≡ 0 (mod 2) pr−1/2δi,j r ≡ 1 (mod 2) and φ(ei,ffj) = { pr/2δi,j r ≡ 0 (mod 2) 0 r ≡ 1 (mod 2) and φ(ei,ej) = φ(fi,fj) = 0. equivalently, (m,φ) is hyperbolic if and only if the dieudonné module m allows a decomposition into a direct sum of dieudonné modules a and b with φ(a,a) = φ(b,b) = 0, and mt = f−rm, so that φ identifies the dual of a with f−rb. 2.2 results of oort and li a dieudonné module is called supersingular if it is isogenous to a superspecial one, or equivalently if all its newton slopes are equal to 1/2. this section is primarily concerned with supersingular dieudonné modules, so recall some of the techniques which are usually applied to them: if m is supersingular it has a biggest superspecial sub-module s0(m) which one can construct as s0(m) = m̃ ⊗w(f p2 ) w(k). dually there is s 0(m), the smallest superspecial module containing m, see [10, chapter iii.2] for proofs of this. the following facts on the relation of the lattices s0(m) ⊂ m ⊂ s 0(m) are basic to the study of supersingular dieudonné modules. the first of them can be found in [8, corollary(1.7)], along with more information on the functors s0 and s 0. for the other two facts we refer the reader to [8, lemma(1.5/1.6)] (or [9, fact(5.8)]) and [8, 1.10(i)] (or [9, chapter(12.2)]): fact 3 (li). let m be a supersingular dieudonné module of rank 2g over w(k). then one has fg−1s0(m) = ∑ i+j=g−1 f ivjm. it follows that fg−1s0(m) ⊂ s0(m), in particular the length of the w(k)-module s0(m)/s0(m) is bounded by g(g − 1) and equality is acquired if and only if a(m) = 1. fact 4 (li). let n be a superspecial dieudonné module of rank 2g over w(k). let x be an element of n. then one has s0(w(k)[f,v]x) = n if and only if the elements fg−1x,fg−2vx,. . . ,fvg−2x,vg−1x form a basis of the k-vector space fg−1n/fgn. moreover, an element with this property exists. 72 oliver bültel cubo 16, 3 (2014) fact 5 (li). let m be a supersingular dieudonné module of rank 2g over w(k). for a non-negative integer i let si = dimk(m ∩ f is0(m)/m ∩ fi+1s0(m)). then one has si ≤ si+1 and equality holds if and only if si = g. the work [9] studies supersingular dieudonné modules which are equipped with a perfect anti-symmetric form ψ. following their method we notice that we have to incorporate additional structure which by the morita-equivalence of subsection 3.2 leads to dieudonné modules m of rank 8 equipped with a symmetric form φ. analogous to [9, proposition(6.1)] we need to analyze the restriction of φ to n = s0(m), or more generally, a classification of non-degenerate symmetric forms on superspecial dieudonné modules: theorem 2.1. let k be an algebraically closed field of characteristic p 6= 2 and let n be a superspecial dieudonné module of rank 2n over w(k), which is equipped with a non-degenerate symmetric pairing φ. then n contains dieudonné modules ni of rank 2, with φ(ni,nj) = 0, for i 6= j, and n = ⊕n i=1 ni. moreover, each ni has a w(k)-basis consisting of elements xi,fxi = vxi = yi such that one of the two cases: (i) φ(xi,xi) = φ(yi,yi) = 0, and φ(xi,yi) = p ni, (ii) φ(xi,yi) = 0, φ(xi,xi) = ǫip ni, and φ(yi,yi) = ǫ σ i p ni+1, holds for some integers ni and some elements ǫi ∈ w(fp2) × which are unique up to multiplication by elements in (w(fp2) ×)2. moreover, the cristalline discriminant can be computed from this decomposition as crisdisc(ni,φ) = { −t2 (ni,φ) of type (i) pt2ǫiǫ σ i (ni,φ) is of type (ii) , and crisdisc(n,φ) = ∏n i=1 crisdisc(ni,φ). proof. the skeleton construction descends n to a w(fp2)-dieudonné module ñ which at the same time is a ob-module. as in (2) we consider the ob-valued sesquilinear form φ and diagonalize it as follows: let x0 ∈ ñ be an element with φ(x0,x0) of mb-adic valuation as small as possible, i.e. such that φ(x,x) ∈ mrb = obφ(x0,x0) for all x ∈ ñ. by the usual polarization process it follows that φ(x,y)−φ(x,y)ι ∈ mrb, and also φ(x,y)+φ(x,y) ι ∈ mrb by replacing tx for x. consequently φ(ñ,ñ) ⊂ mrb. therefore we obtain an orthogonal direct sum ñ = (ñ ∩ (bx0) ⊥) ⊕ obx0, as any x ∈ ñ has φ(x,x0)φ(x0,x0) −1 = α ∈ ob which allows to write x as a sum of αx0 ∈ ñ and x − αx0 ∈ ñ ∩ (bx0) ⊥. having obtained a decomposition ñ = ⊕n i=1 ñi we search for basis elements x̃i ∈ ñi with φ(x̃i, x̃i) manageable: in ñi ⊗ q one can certainly find elements x̃i with φ(x̃i, x̃i) ∈ w(fp2) × ∪ fw(fp2) × for example by [16, chapter 10, theorem(3.6.(i))]. observe that the mb-adic valuation of φ(x̃i, x̃i) must be congruent modulo 2 to ri = lengthob ñ t i/ñi. hence after adjusting the x̃i’s by multiplying cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 73 them by fri/2, if ri is even, and by f (ri−1)/2, if ri is odd, one gets generators of the ob-modules ñi on which the sesquilinear form takes values in f riw(fp2) ×. it is clear how to obtain the desired basis x1, . . . ,xn,y1, . . . ,yn from these generators. if ri is even ni will be of type (i) with ni = ri/2, and if ri is odd then ni will be of the type (ii) with ni = (ri − 1)/2. remark 2.2. suppose n is a superspecial dieudonné module of rank 2 with a symmetric form φ. then one checks from the above classification that (n,φ) is isometric to (n,−φ). it follows that n⊕2, the orthogonal direct sum of two copies of n, is hyperbolic. one checks this by using the sesquilinear form (2) as φ((u1 + u2,u1 − u2),(v1 + v2,v1 − v2)) = (u1 + u2)w(v1 + v2) ι − (u1 − u2)w(v1 − v2) ι = (2u1w)v ι 2 − u2(2v1w) ι. (cf. [9, remark(6.1)] for the analog in the anti-symmetric setting) for later use we note an immediate corollary: corollary 2.3. let (n1,φ1) and (n2,φ2) be supersingular dieudonné modules of rank two, equipped with symmetric pairings. there exists an isometry between them if and only if the following holds: lengthw(k) n t 1/n1 = lengthw(k) n t 2/n2 crisdisc(n1,φ1) = crisdisc(n2,φ2). consequently for any non negative integer n, there is only one isometry class of rank two supersingular dieudonné modules with pairing (n,φ) where lengthw(k) n t/n = 2n. there are two such classes of modules with pairing where lengthw(k) n t/n = 2n + 1. 2.3 classification of symmetric dieudonné modules this section is the core of the work, we give a classification of dieudonné modules with the additional structure of interest. theorem 2.4. let m be a supersingular dieudonné module over w(k) with perfect symmetric pairing φ. assume that: rankw(k) m = 8 crisdisc(m,φ) = 1. consider s0(m) = n, the smallest superspecial dieudonné lattice in m ⊗ q, which contains m. choose a decomposition n = ⊕4 i=1 ni with properties as granted by theorem 2.1, and with s0(m) = n t = ⊕4 i=1 f rini for integers r1 ≤ r2 ≤ r3 ≤ r4. then (r1,r2,r3,r4) is one of (i) (0,0,0,0) (ii) (1,1,1,1) 74 oliver bültel cubo 16, 3 (2014) (iii) (0,2,2,2) (iv) (2,2,2,2), moreover, there exists a superspecial dieudonné lattice q, which contains fm and satisfies (a) qt = q (b) dimk(m/m ∩ q) = dimk(q/m ∩ q) = 1. if m is of the form (iii) or (iv), then the superspecial dieudonné lattice q, satisfying (a) and (b) is unique. proof. for the proof we need two auxiliary lemmas: lemma 2.5. let the assumptions on m be as in the above theorem, then there exist two different indices i1 and i2 such that ni1 and ni2 are isometric. proof. if an even integer r occurs twice amongst the various ri’s one is done, and if an odd integer r occurs three times one is done as well, use the pigeon hole principle and corollary 2.3. the condition on the discriminant forces the number of indices i with ri odd to be even. this means that one is left with checking the lemma for the ri-quadruples (0,1,2,3), (0,2,3,3), (0,1,1,2), and (1,1,3,3). the three quadruples with r1 = 0 do not arise, because otherwise m would be an orthogonal direct sum of n1 and some supersingular dieudonné module m ′ of rank 6 and equipped with a perfect symmetric form φ′. applying fact 2 to m′ would give that m′ has oort invariant 1 or 3, as crisdisc(m′) = crisdisc(n1) = −t 2. fact 3 applied to m′ would further imply that the elementary divisors of s0(m′)/s0(m ′) are either all 0 or all equal to 2. hence the elementary divisors of s0(m)/s0(m) would be (0,0,0,0) or (0,2,2,2). it remains to do the (r1,r2,r3,r4) = (1,1,3,3)-case. assume that no two of the n ′ is were isometric. this would lead to a basis xi,fxi = vxi = yi with φ(x1,x1) = p −1, φ(y1,y1) = 1 φ(x2,x2) = ǫp −1, φ(y2,y2) = ǫ σ φ(x3,x3) = p −2, φ(y3,y3) = p −1 φ(x4,x4) = ǫp −2, φ(y4,y4) = ǫ σp−1, other products = 0, and with ǫ some non-square in w(fp2) ×. the module m has to contain an element of the form α1x1 +α2x2 + α3x3 + α4x4 + β3y3 +β4y4 such that βi,αi ∈ w(k) but not both of α3 and α4 in pw(k). as φ(x,x) = p−1(α21 + ǫα 2 2 + β 2 3 + ǫ σβ24) + p −2(α23 + ǫα 2 4) cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 75 one has α23 + ǫα 2 4 ≡ 0 (mod p), but as φ(x,f2x) = α1α σ2 1 + ǫα2α σ2 2 + β3β σ2 3 + ǫ σβ4β σ2 4 + p −1(α3α σ2 3 + ǫα4α σ2 4 ) one has α p2+1 3 +ǫα p2+1 4 ≡ 0 (mod p) as well. as ǫ is a non-square in w(fp2) ×, one has ǫ p2−1 2 ≡ −1 (mod p), so that we derive the contradiction α p2+1 3 ≡ (−ǫα 2 4) p2+1 2 ≡ ǫα p2+1 4 (mod p). lemma 2.6. with the same notation as in the theorem ri ≤ 2 for all indices i. proof. observe that the lemma would be immediate if one of the ri was zero. so we can assume 0 < ri for all indices i. pick two indices i 6= j with ri = rj = r and crisdisc(ni) = crisdisc(nj), according to the previous lemma such indices will exist. say (i, j) = (1,2) after relabeling, and write according to remark 2.2 n1⊕n2 = a⊕b, with φ(a,a) = φ(b,b) = 0 and a×f rb → w(k) a perfect pairing. consider along the lines of [9, proposition(6.3)] a w(k)-module m′ which is the image of (b⊕n3⊕n4)∩m under the projection map b⊕n3⊕n4 → n3⊕n4. m ′ inherits a perfect form and is indeed canonically isomorphic to the sub-quotient (b⊥ ∩ m)/(b ∩ m) of m. one has crisdisc(m′) = 1 because m′ is isogenous to n3 ⊕n4. by fact 2 it follows that m ′ is superspecial. furthermore the proof of [9, proposition(6.3)] shows that fn3 ⊕ fn4 ⊂ m ′ ⊂ n3 ⊕ n4. for convenience of the reader we reproduce the argument in loc.cit.: pick an element in m of the form x = e + f + n3 + n4 with e ∈ ã, f ∈ b, n3 ∈ n3, n4 ∈ n4 and s0(m) = s0(w(k)[f,v]x), it exists due to fact 4. the elements f3x, f2vx, fv2x, v3x will then form a basis of the k-vector space f3n/f4n so that f3x−f2vx, f2vx−fv2x, fv2x−v3x is a basis of f3(b⊕n3 ⊕n4)/f 4(b⊕ n3 ⊕ n4). it follows that s0(w(k)[f,v](f − v)x) = f(b ⊕ n3 ⊕ n4), but (f − v)x ∈ m ∩ (b ⊕ n3 ⊕ n4) which projects surjectively onto m ′. as s0 is a functor in supersingular dieudonné modules fn3 ⊕fn4 will be contained in s 0(m′) = m′, and consequently fn3 ⊕ fn4 ⊂ m ′ = m′t ⊂ f−1nt3 ⊕ f −1nt4 = f r3−1n3 ⊕ f r4−1n4 i.e. r3,r4 ≤ 2. however, r3 ≡ r4 (mod 2), as crisdisc(n3) = crisdisc(n4). therefore r3 = r4, as r3,r4 ∈ {1,2}. now, note that this does indeed imply that n3 is isometric to n4. in order to find that r1,r2 ≤ 2 also, we redo the whole argument, with the roles of n1 and n2 being replaced by n3 and n4. return to proof of theorem 76 oliver bültel cubo 16, 3 (2014) we move on to investigate the set of possible quadruples (r1,r2,r3,r4). if one of the numbers in that sequence is 0, then fact 2 shows that we must have either (0,0,0,0) or (0,2,2,2). for the remaining cases (2,2,2,2), (1,1,1,1) and (1,1,2,2) are conceivable. we show that (1,1,2,2) can not arise: assume we had a dieudonné module m with (r1,r2,r3,r4) = (1,1,2,2). it would follow that one had crisdisc(n3) = crisdisc(n4) by corollary 2.3, and so would crisdisc(n1) = crisdisc(n2). by applying remark 2.2 to both n1 ⊕ n2 and n3 ⊕ n4 one obtains a basis of n consisting of say e1, e2, f1, f2, fe1 = ve1, fe2 = ve2, ff1 = vf1, ff2 = vf2 and with the only non-zero products being given by φ(fe1,ff1) = 1 φ(e1,f1) = φ(e2,ff2) = φ(f2,fe2) = p −1. as f−1nt is superspecial one has m 6⊂ f−1nt, so that m contains an element of the form x = α1e1 + β1f1 + α2e2 + β2f2 + α3fe2 + β3ff2, with all α1, . . . ,β3 ∈ w(k) and at least one of α2 and β2 a unit. from fx ∈ α σ 2fe2 + β σ 2ff2 + n t and φ(m,m) ⊂ w(k) one infers φ(x,fx) ∈ p−1(ασ2β2 + β σ 2α2) + w(k), which means that α σ 2β2 + β σ 2α2 ≡ 0 (mod p). as we may alter the elements α1, . . . ,β3 by any element in pw(k) we can actually assume that α σ 2β2 + β σ 2α2 = 0, but then the dieudonné module w(k)fx + nt = w(k)(ασ2fe2 + β σ 2ff2) + n t is superspecial contradicting s0(m) = n t. having done the first assertion of the theorem we now focus on the existence of q. if m is of the form (i), then use remark 2.2 to write n1 ⊕n2 as direct sum of two isotropic dieudonné modules a and b, between which there is the duality that is induced from the pairing on n. then one finds that q = f−1a ⊕ fb ⊕ n3 ⊕ n4 is a superspecial dieudonné lattice that does the job. similarly for the (ii)-case: write n = a1 ⊕ a2 ⊕ b1 ⊕ b2 with isotropic ai and bi, this time equipped with a canonical isomorphism ati ∼= fbi. the superspecial lattices fa1 ⊕ fa2 ⊕ b1 ⊕ b2 a1 ⊕ fa2 ⊕ fb1 ⊕ b2 both satisfy qt = q, and one of them satisfies property (b) as well. in the (iii)-case property (a) forces to look at q = n1 ⊕ ⊕4 i=2 fni, whereas q = fn in the (iv)-case. we have to show that this module does indeed satisfy (b), to this end observe that the numbers dimk m/m ∩ q and dimk m ∩ q/m ∩ fq are nonzero and sum up to 4, it thus suffices to see that the first of them is strictly smaller than the second. in the (iv)-case this is the content of fact 5. in the (iii)-case apply fact 5 to the orthogonal complement of n1 in m, which is a dieudonné module of rank 6 with perfect symmetric form. 2.4 moduli of symmetric dieudonné modules we consider the graded fp-algebra r := fp[a1,a2,b1,b2]/( ∑2 i=1 aib p i +bia p i ), and its associated projective variety x1 := projr, which is smooth of relative dimension 2. let y1 denote the affine cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 77 chart determined by a1 6= 0, it is the spectrum of r(a1) ∼= fp[a2,b1,b2]/(b1 +b p 1 +a2b p 2 +b2a p 2), where a2 := a2 a1 , b1 := b1 a1 , and b2 := b2 a1 . let α2,β1,β2 ∈ w(r(a1)) be lifts of a2,b1,b2 with β1 + fβ1 + α2 fβ2 + β2 fα2 = 0. let t(a1) be the w(r(a1))-module ⊕4 i=1 w(r(a1))ti, l(a1) be the w(r(a1))-module ⊕4 i=1 w(r(a1))li, and m(a1) be l(a1) ⊕ t(a1). putting: f(t1) = l1 f(t2) = l2 + (β2 − f2β2)t1 f(t3) = l3 + ( f 2 α2 − α2)t2 + ( f 2 β2 − β2)t4 f(t4) = l4 + (α2 − f2α2)t1 v−1(l1) = t1 v−1(l2) = t2 v−1(l3) = t3 v−1(l4) = t4 and using the formula v−1(vαx) = αf(x) defines the structure of a display ([17]) on m(a1), which moreover has the normal decomposition l(a1) ⊕t(a1). one checks that a pairing is given on m(a1) by φ(li, lj) = φ(ti,tj) = 0, φ(li,tj) = δ|i−j|,2. let also n = ln ⊕tn be the display obtained from the formulas f(ti) = li, v −1(li) = ti and with pairing defined analogously. putting: ǫ(t1) = pt3 ǫ(t2) = l2 − fβ2l3 ǫ(t3) = t1 + fα2t2 + fβ1t3 + fβ2t4 ǫ(t4) = − fα2l3 + l4 ǫ(l1) = pl3 ǫ(l2) = pt2 − pβ2t3 ǫ(l3) = l1 + α2l2 + β1l3 + β2l4 ǫ(l4) = −pα2t3 + pt4 defines an embedding of displays ǫ(a1) : m(a1) →֒ n ×fp y1, satisfying pφ(x,y) = φ(ǫ(x),ǫ(y)). neither m(a1) nor ǫ(a1) depend on the choice of the lifts α2,β1,β2, which can be checked upon passage to the perfection r perf (a1) (here notice that r(a1) → r perf (a1) is flat, because r(a1) is regular). moreover, the natural action of the kleinian group on x1 gives rise to analogous subdisplays of the constant display n regarded over each of the translates {a2 6= 0}, {b1, 6= 0}, and {b2 6= 0}, which in turn gives rise to an inclusion ǫ : m →֒ n ×fp x1, of sheaves of displays with respect to the zariski topology of x1. this is because the closed points can be used to check the cocycle condition. however, notice that there does not exist a global normal decomposition for m. 78 oliver bültel cubo 16, 3 (2014) 2.5 miscellaneous the study of families of dieudonné modules with our additional structure within a given isogeny class is meaningful not just for the supersingular one. recall that every isogeny class of dieudonné modules can be written as a direct sum of certain simple ones. these are parameterized by pairs of coprime non-negative integers a and b and denoted by ga,b, see [10] for details. the isogeny class ga,b contains usually more than one dieudonné module except if a or b is equal to 1, in which case we are allowed to speak of “the” dieudonné module of type ga,b. we have the following result: corollary 2.7. let m be a non-supersingular dieudonné module over w(k) that is equipped with a perfect symmetric pairing φ. assume that: rankw(k) m = 8 crisdisc(m,φ) = 1. then m is an orthogonal direct sum ⊕ i mi where for each of the (mi,φ) one of the following alternatives hold: (i.n) (mi,φ) can be written as a ⊕ b with mutually dual isotropic dieudonné modules a and b, which lie in the isogeny classes g1,n and gn,1 for some n ∈ {0,1,2,3}. (ii) (mi,φ) is supersingular of rank 2 and the perfect pairing thereon is the one described by part (i) of theorem 2.1. (iii) (mi,φ) is supersingular of rank 4, and the pairing is such that s0(mi) decomposes into the two dieudonné modules with pairings described by part (ii) of theorem 2.1. moreover, the only combinations which occur are: • 4 × (i.0) • 2 × (i.0) ⊕ (i.1) • (i.0) ⊕ (ii) ⊕ (iii) • (i.0) ⊕ (i.2) • (i.3) proof. we consider the canonical decomposition of m = m0 ⊕m ′ ⊕m1 into the étale-local, locallocal, local-étale parts. the assertion of the corollary has solely something to do with m′ which is of some even rank equal to 8 − 2f and has crisdisc(m′) = (−1)f, here f is the p-rank of m. as m′ is also self-dual it can have only one of the following isogeny types: cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 79 (1) 3 × g1,1 (2) 2 × g1,1 (3) g1,2 ⊕ g2,1 (4) g1,3 ⊕ g3,1 if m′ has the above isogeny types 3., or 4. we deduce from [7, paragraph(16), satz(3)] and a(m′) = 2 that m′ is a direct sum of two dieudonné modules a and b, each with oort invariant equal to one. the assertion on the pairing is then immediate as neither a nor b is selfdual. if m′ has isogeny type 2g1,1 it must be superspecial. then use theorem 2.1 in conjunction with remark 2.2 to check that m′ has the shape a ⊕ b with isotropic a and b. in the case in which the isogeny type of m′ is 3g1,1, we have to work a bit harder: first consider a diagonalization of s0(m′) = n = ⊕3 i=1 ni with s0(m ′) = nt = frini. an analysis as in the proof of lemma 2.5 yields that (r1,r2,r3) = (0,1,1), therefore the orthogonal direct summand (n1,φ) has a complement with perfect form, say m′′, its oort invariant is 1. therefore crisdisc(m′′) = t2. as r2 = r3 = 1 this implies that crisdisc(n2,φ), and crisdisc(n3,φ), are the two numbers p, and pt2, which is what we wanted. 3 the shimura variety skp 3.1 further notation before we proceed we want to introduce the input data for our pel-moduli problem: fix once and for all a quaternion algebra b over q and write r for the set of places at which b is non-split. assume that ∞ ∈ r, i.e. that br is definite. let p be a prime which is not in {2} ∪ r and choose a maximal z(p)-order ob ⊂ b, together with an isomorphism κp : zp ⊗ ob ∼= mat2(zp). the standard involution b 7→ bι = tr(b) − b preserves ob and is positive. let v be a left b-module of rank 4 with non-degenerate alternating pairing satisfying (bv,w) = (v,bιw). for simplicity we require that the skew-hermitian b-module v is hyperbolic in the following sense: we want it to have a b-basis e1,e2,f1,f2 such that ( ∑2 i=1 aiei +bifi, ∑2 i=1 a ′ iei + b′ifi) = trb/q( ∑2 i=1 aib ′ι i − bia ′ι i ) for all ai,bi,a ′ i,b ′ i ∈ b. set further λ0 = ⊕2 i=1 obei ⊕ obfi, it is a self-dual ob-invariant z(p)-lattice in v. let g/q be the reductive group of all b-linear symplectic similitudes of v. this group is a form of go(8). write kp ⊂ g(qp) for the hyperspecial subgroup consisting of group elements that preserve λ0 and let k p ⊂ g(a∞,p) be an arbitrary compact open subgroup. finally we specify a particular ∗-homomorphism h0 : c → endb(vr) by the rule h0(i)( ∑2 i=1 aiei+ bifi) = ∑2 i=1 biei − aifi, and r-linear extension. the reflex field of (g,h0) is equal to q. now, for every connected scheme skp/z(p) with a geometric base point s we consider the set of z(p)-isogeny classes of quadruples (a,λ, ı,η) with: 80 oliver bültel cubo 16, 3 (2014) (m1) a is a 8-dimensional abelian scheme over s up to prime-to-p isogeny (m2) λ : a → at is a z× (p) -class of prime-to-p polarizations of a (m3) ı : ob → end(a) ⊗ z(p) is a homomorphism satisfying ı(b ι) = ı(b)∗, here ∗ is the rosati involution associated to λ (m4) η is a π1(s,s)-invariant k p-orbit of ob-linear isomorphisms η : v ⊗ a ∞,p ∼= h1(as,a ∞,p) which are compatible with the alternating form up to scalars. by geometric invariant theory this functor is representable by a quasi-projective z(p)-scheme skp . moreover, the deformation theory of grothendieck-messing shows that s is smooth of relative dimension 6 over z(p), cf. [6, chapter 5]. see also [6, chapter 8] for the complex uniformizations of skp(c). finally, let us write ssikp (resp. s sp kp ) for the subsets skp × f ac p whose sets of geometric points consist of those quadruples (a,λ, ı,η) where d(a[p∞]) is supersingular (resp. superspecial), here d(g) denotes the (covariant) dieudonné module of a p-divisible group g over a perfect field. notice that we always have crisdisc(d(g),φ) = 1, by [1]. 3.2 morita equivalence let us write g∗ for the serre-dual of a p-divisible group g = ⋃ l g[p l] over some base scheme s. we will say that g is polarized (resp. anti-polarized) if it is endowed with an isomorphism φ to its dual which satisfies φ = −φ∗ (resp. φ = φ∗). in particular, consider the anti-polarized p-divisible groups g1 := bt (m) and g0 := bt (n), where m and n are as in section 2.4. the emdedding ǫ : m →֒ n ×fp x1 gives rise to a canonical isogeny ǫ : g1 → g0 ×fp x1 satisfying ǫ ∗ ◦ ǫ = p idg1 and ǫ ◦ ǫ∗ = p id g0×fp x1 , notice also that ker(ǫ) ⊂ g1[p] and ker(ǫ ∗) ⊂ g0[p] ×fp x1 are finite, flat, maximal isotropic subgroup schemes of order p4. if an isomorphism zp ⊗ob κp → mat2(zp) is fixed once and for all, one obtains a morita-equivalence (g,φ) 7→ (g⊕2, ( 0 φ −φ 0 ) ) from the category of anti-polarized p-divisible groups to the category of polarized p-divisible groups with rosati-invariant ob-action. in this manner one obtains an anti-polarized p-divisible group (g,φ) from every s-valued point on skp, say represented by (a,λ, ı,η), by the requirement (a[p∞],ψλ) ∼= (g ⊕2, ( 0 φ −φ 0 ) ), where ψλ : a[p ∞] → a[p∞]∗ is the p-adic weil-pairing, which is induced from the polarization λ : a → at. if s is the spectrum of a perfect field of characteric p, we always have crisdisc(d(g),φ) = cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 81 1, by [1]. we next want to define a family of morphisms cx,ηp : x1 × f ac p → skp × f ac p (4) which are indexed by superspecial facp -points x = (a,λ, ı,η), equipped with the following additional datum: by a frame for x we mean an isomorphism ηp : g0 ×fp f ac p → g, where (g,φ) corresponds to x ∈ skp(f ac p ) by the above morita-equivalence while (g0,φ0) is the previously exhibited antipolarized p-divisible group. let us consider the abelian variety which is defined by the exact sequence: 0 → ηp(ker(ǫ ∗))⊕2 → a et → a1 → 0, the isotropicity and the ob-invariance of ηp(ker(ǫ ∗))⊕2 give rise to a canonical z× (p) -class of prime-to-p polarizations λ1 : a1 → a t 1, together with a rosati-invariant operation ı1 : ob → end(a1) ⊗ z(p) and level structure η1, each gotten by transport of structure. finally one sees that the quadruple x1 = (a1,λ1, ı1,η1) thus obtained constitutes a x1 × f ac p -valued point, whose classifying morphism we define to be (4). it is easy to see that the image of cx,ηp is a closed subset, whose geometric points consist of exactly those quadruples (a1,λ1, ı1,η1) which allow an ob-linear isogeny e : a1 → a, wich is compatible with the level structure and satisfies pλ1 = e t ◦ λ ◦ e. remark 3.1. fix (a,λ, ı,η) = x ∈ s sp kp (f ac p ). notice, that we have just shown, that the zariskiclosed subset cx,ηp(x1 × f ac p ) := s sp x,kp does not dependent on the choice of frame. 3.3 description of ssikp now, we would like to investigate whether or not cx,ηp is a closed immersion, the next lemma is a step towards this direction: lemma 3.2. let x and ηp be as above, then cx,ηp induces an injection on the tangentspaces to each geometric point u ∈ x1(k), where k is an arbitrary algebraically closed field of characteristic p. proof. recall that every k-display p of dimension d and codimension c allows structural equations: f(tj) = d∑ i=1 ui,jti + c∑ i=1 ui+d,jli v−1(lj) = d∑ i=1 ui,j+dti + c∑ i=1 ui+d,j+dli for some display-matrix     u1,1 . . . u1,c+d ... ... ... uc+d,1 . . . uc+d,c+d     = u ∈ gl(c + d,w(k)), 82 oliver bültel cubo 16, 3 (2014) where t1, . . . ,td, l1, . . . , lc ∈ p, and t1 +q,.. . ,td +q ∈ p/q are bases. let l and t be the w(k)submodules of p that are generated by l1, . . . , lc and t1, . . . ,td, and write j := homw(k)(l,t). due to the technique of norman-oort the isomorphism classes of infinitesimal deformations of p over the ring of dual numbers kd := k[s]/(s 2) are parameterized by the elements in j ⊗w(k) k = homk(q/pp,p/q), in fact each deformation may be described explicitly as follows: pick a tangent direction n ∈ j ⊗w(k) k, say with d × c-matrix representation     n1,1 . . . n1,c ... ... ... nd,1 . . . nd,c     (with respect to the two bases above). write w(skd) for the kernel of the natural map from w(kd) to w(k), and choose elements ñi,j ∈ w(skd) whose 0-th witt coordinate is equal to the dual number sni,j. then ũ :=              1 . . . 0 ñ1,1 . . . ñ1,c ... ... ... ... ... ... 0 . . . 1 ñd,1 . . . ñd,c 0 . . . 0 1 . . . 0 ... ... ... ... ... ... 0 . . . 0 0 . . . 1              u ∈ gl(c + d,w(kd)) displays an infinitesimal deformation of p, that corresponds to the tangent direction n, in particular it is the trivial deformation if and only of n = 0. now let (x1 : x2 : y1 : y2) be the homogeneous coordinates of u ∈ x1(k), and fix one of its non-zero tangent directions u′ ∈ x1(kd). to finish the proof of the lemma we only have to show that the associated kd-display mu′ is a non-trivial infinitesimal deformation (of mu, i.e. the special fiber of mu′). of course we can assume (x1 : x2 : y1 : y2) = (1 : x2 : y1 : y2) from the start, so let (1 : x2 + sa : y1 − s(ay p 2 − bx p 2) : y2 + sb) be the homogeneous coordinates of u ′, where (a,b) ∈ k2 − {(0,0)}. now recall from section 2.4 that the restriction of m to the affine chart spec fp[a2,b1,b2]/(b1 +b p 1 +a2b p 2 +b2a p 2) ⊂ x1 has already a normal decomposition and is explicitly displayed in an extremely convenient way, namely by means of the matrix u = ( h e e 0 ) , where e denotes the identity matrix, and where the (so-called ‘hasse-witt’) matrix h is given by:       0 β2 − f2β2 0 α2 − f2α2 0 0 f 2 α2 − α2 0 0 0 0 0 0 0 f 2 β2 − β2 0       , for certain α2,β1,β2 ∈ w(fp[a2,b1,b2]/(b1 + b p 1 + a2b p 2 + b2a p 2)). now consider the skdvalued witt-vectors α := u′(α2) − u(α2) and β := u ′(β2) − u(β2), in fact it is easy to see that cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 83 u′(β1)−u(β1) = −(αu(β2) σ +βu(α2) σ), because α and β are killed by f. moreover, the 0th wittcoordinates of α and β are just sa and sb. it follows immediately that u′(u) = ( e ñ 0 e ) u(u), with ñ being the deformation matrix:       0 β 0 α 0 0 −α 0 0 0 0 0 0 0 −β 0       , whose matrix of 0th witt-components is clearly nonvanishing. as a consequence of theorem 2.4 we have: ssikp = ⋃ x∈s sp kp ssix,kp, and s sp kp is a finite set of closed points. it follows from this (or from grothendieck’s specialization theorem [3, p.149]), that ssikp is zariski closed. our aim is to describe s si kp together with its induced reduced subscheme structure. let us fix x ∈ s sp kp, which classifies some quadruple (a,λ, ı,η), and let ∗ denote the rosati-involution on the q-algebra end0b(a). let us write ix/q for the group scheme which represents the functor c 7→ {g ∈ (end0b(a) ⊗ c) ×|ggt ∈ c×}. (5) every full level structure η : v ⊗ a∞,p ∼= h1(a,a ∞,p) yields an isomorphism i × a∞,p ∼= → g × a∞,p;γ 7→ η−1γη. notice that the preimage of kp under the above isomorphism depends only on the kp-orbit of η, and hence we can define k p x := ηk pη−1 for any η ∈ η, this is again a compact open subgroup of ix(a ∞,p). consider the compact set k̃p := {γ ∈ i(qp)|γ,γ −1 ∈ p−1zp ⊗ endb(a)}, and let us say that kp is superneat for x if and only if ix(q) ∩ k̃p × k p x = {1}. the left-hand side is always a finite group, because ix is anisotropic. in particular k p will always contain some a compact open subgroup which is superneat for every x ∈ ssikp lemma 3.3. if kp is superneat for x, then (4) is a closed immersion. proof. a morphism from a proper facp -variety to a separated one is a closed immersion if and only if it radicial and injective on the tangent spaces to all facp -valued points, this is elementary and can be proved along the lines of [4, lemma 7.4.]. in view of lemma 3.2 it suffices to check that (4) is indeed injective on geometric points. suppose it wasn’t. then there existed skp (k) ∋ x1 = (a1,λ1, ı1,η1) which lies in the image of (4) in two different ways. according to the thoughts at the end of subsection 3.2, this means that there existed two degree-p8-isogenies e,e′ : a1 → a each of which induce the additional structures λ1, ı1, η1 from the additional structures λ, ı, η on a. it follows immediately that ida 6= e ′ ◦ e−1 is in contradiction to kp being superneat for x. 84 oliver bültel cubo 16, 3 (2014) received: october 2012. accepted: march 2013. references [1] bültel, o., 1999, rational points on some pel-stacks, manuscripta math. volume 99, p.395-410 [2] bültel, o., 2002, the congruence relation in the non-pel case, j. reine angew. math. volume 544, p.133-159 [3] grothendieck, a., 1970, groupes de barsotti-tate et cristaux de dieudonné, sém. math. sup. volume 45, presses de l‘univ. de montreal [4] hartshorne, r., 1977, algebraic geometry, graduate texts in mathematics volume 52, springer verlag, new york [5] katsura, t. and oort, f., 1987, families of supersingular abelian surfaces, compos. math. volume 62, p.107-167 [6] kottwitz, r., 1992, points on some shimura varieties over finite fields, journal of the ams volume 5, p.373-444 [7] kraft, h., 1975, kommutative algebraische p-gruppen, sonderforschungsbereich theoretische mathematik, universität bonn [8] li, k-z., 1989, classification of supersingular abelian varieties, math. ann. band 283, p.333351 [9] li, k-z. and oort, f., 1998, moduli of supersingular abelian varieties, lecture notes in mathematics volume 1680 [10] manin, y., 1963, the theory of commutative formal groups over fields of positive characteristic, russ. math. surveys volume 18, p.1-80 [11] moonen, b., 2001, group schemes with additional structures and weyl group cosets, in: moduli of abelian varieties, (eds. faber, van der geer, oort), progress in mathematics volume 195, p.255-298 [12] oda, t. and oort, f., 1977, supersingular abelian varieties, intl. sympos. on algebraic geometry, kyoto (ed. nagata), p.595-621 [13] ogus, a., 1989, absolute hodge cycles and crystalline cohomology, in: hodge cycles, motives, and shimura varieties (eds. deligne, milne, ogus, shih) lecture notes in mathematics volume 900, p.357-414 [14] oort, f., 1991, hyperelliptic supersingular curves, in: arithmetic algebraic geometry, texel 1989 (eds. van der geer, oort, steenbrink), progress in mathematics volume 89, p.247-284 cubo 16, 3 (2014) on the supersingular loci of quaternionic siegel space 85 [15] richartz, m., 1998, klassifikation von selbstdualen dieudonnégittern in einem dreidimensionalen polarisierten supersingulärem isokristall, bonn thesis [16] scharlau, w., 1985, quadratic and hermitian forms, grundlehren der mathematischen wissenschaften 270 [17] zink, t., 2002, the display of a formal p-divisible group, in: cohomologies p-adiques et applications arithmétiques (i) (eds. berthelot, fontaine, illusie, kato, rapoport) astérisque volume 278, p.127-248 introduction structure theorems on dieudonné modules with pairing notions and notations results of oort and li classification of symmetric dieudonné modules moduli of symmetric dieudonné modules miscellaneous the shimura variety skp further notation morita equivalence description of skpsi cubo a mathematical journal vol.14, no¯ 03, (129–142). october 2012 existence of deviating fractional differential equation rabha w. ibrahim institute of mathematical sciences university malaya, 50603 kuala lumpur, malaysia email: rabhaibrahim@yahoo.com abstract in this paper we shall establish sufficient conditions for the existence of solutions of a class of fractional differential equation (cauchy type ) and its solvability in a subset of the banach space. the main tool used in our study is the non-expansive operator technique. the non integer case is taken in sense of riemann-liouville fractional operators. applications are illustrated. resumen en este art́ıculo establecemos condiciones suficientes para la existencia de soluciones de una clase de ecuaciones diferenciales fraccionales (del tipo cauchy) y su solubilidad en un subconjunto de un espacio de banach. la principal herramienta utilizada en nuestro estudio es la técnica del operador no expansivo. el caso no entero se escoge en el sentido de operadores fraccionales riemann-liouville. además, se ilustran aplicaciones. keywords and phrases: fractional calculus; fractional differential equation; cauchy equation; riemann-liouville fractional operators; volterra integral equation; non-expansive mapping; iterative differential equation 2010 ams mathematics subject classification: 34a12. 130 rabha w. ibrahim cubo 14, 3 (2012) 1 introduction fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) has importance in several widely diverse areas of mathematical physical and engineering sciences. it generalized the ideas of integer order differentiation and nfold integration. fractional derivatives introduce an excellent instrument for the description of general properties of various materials and processes. this is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. the advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of properties of gases, liquids and rocks, and in many other fields [1-5]. the class of fractional differential equations of various types plays important roles and tools not only in mathematics but also in physics, control systems, dynamical systems and engineering to create the mathematical modeling of many physical phenomena. naturally, such equations required to be solved. many studies on fractional calculus and fractional differential equations, involving different operators such as riemann-liouville operators, erdlyi-kober operators, weylriesz operators, caputo operators and grnwald-letnikov operators, have appeared during the past three decades. the existence of positive solution and multi-positive solutions for nonlinear fractional differential equation are established and studied [6-8]. moreover, by using the concepts of the subordination and superordination of analytic functions, the existence of analytic solutions for fractional differential equations in complex domain are suggested and posed in [9,10]. our aim in this paper is to consider the existence and uniqueness of nonlinear cauchy problems of fractional order in sense of riemann-liouville operators. also, two theorems in the analytic continuation of solutions are studied. in the fractional cauchy problems, we replace the first order time derivative by a fractional derivative. fractional cauchy problems are useful in physics. recently, the author studied the the fractional cauchy problems in complex domain [11]. one of the most frequently used tools in the theory of fractional calculus is furnished by the riemann-liouville operators (see[6-8]). the riemann-liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. moreover, this operator possesses advantages of fast convergence, higher stability and higher accuracy to derive different types of numerical algorithms. definition 1.1. the fractional (arbitrary) order integral of the function f of order α > 0 is defined by iαaf(t) = ∫t a (t − τ)α−1 γ(α) f(τ)dτ. cubo 14, 3 (2012) existence of deviating fractional differential equation 131 when a = 0, we write iαaf(t) = f(t) ∗ φα(t), where (∗) denoted the convolution product (see [7]), φα(t) = t α−1 γ(α) , t > 0 and φα(t) = 0, t ≤ 0 and φα → δ(t) as α → 0 where δ(t) is the delta function. definition 1.2. the fractional (arbitrary) order derivative of the function f of order 0 ≤ α < 1 is defined by dαaf(t) = d dt ∫t a (t − τ)−α γ(1 − α) f(τ)dτ = d dt i1−αa f(t). remark 1.1. from definition 1.1 and definition 1.2, we have dαtµ = γ(µ + 1) γ(µ − α + 1) tµ−α, µ > −1; 0 < α < 1 and iαtµ = γ(µ + 1) γ(µ + α + 1) tµ+α, µ > −1; α > 0. 2 preliminaries we extract here the basic theory of non-expansive mappings in order to offer the notions and results that will be needed in the next sections of the paper. let (x, d) be a metric space. a mapping p : x → x is said to be an ν-contraction if there exists ν ∈ [0, 1) such that d(px, py) ≤ νd(x, y), ∀ x, y ∈ x. in the case where ν = 1 the mapping p is said to be non expansive. let k be a nonempty subset of a real normed linear space e and p : k → k be a map. in this setting, p is non-expansive if ‖px − py‖ ≤ ‖x − y‖ ∀x, y ∈ k. the following result is a fixed point theorem for non expansive mappings, due to browder, ghode and kirk, see e.g. [12]: theorem 2.1. let k be a nonempty closed convex and bounded subset of a uniformly banach space e. then any non expansive mapping p : k → k has at least a fixed point. definition 2.1. let k be a convex subset of a normed linear space e and let p : k → k be a self-mapping. given an x0 ∈ k and a real number λ ∈ [0, 1], the sequence xn defined by the formula xn+1 = (1 − λ)xn + λpxn, n = 0, 1, 2, ... is usually called krasnoselskij iteration or krasnoselskij-mann iteration. 132 rabha w. ibrahim cubo 14, 3 (2012) definition 2.2. let k be a convex subset of a normed linear space e and let p : k → k be a self-mapping. given an x0 ∈ k and a real number λn ∈ [0, 1], the sequence xn defined by the formula xn+1 = (1 − λn)xn + λnpxn, n = 0, 1, 2, ... is usually called mann iteration. edelstein [13] proved that strict convexity of e suffices for the krasnoselskij iteration converge to a fixed point of p. while, egri and rus [14] proved that for any subset of e, the mann iteration converge to a fixed point of p when p is a non-expansive mapping. we need the following results, which can be found in [15]: lemma 2.1. let k be a convex and compact subset of a banach space e and let p : k → k be a non-expansive mapping. if the mann iteration process xn satisfies the assumptions (a) xn ∈ k for all positive integers n, (b) 0 ≤ λn ≤ b < 1 for all positive integers n, (c) ∑ ∞ n=0 λn = ∞. then xn converges strongly to a fixed point of p. lemma 2.2. let k be a closed bounded convex subset of a real normed space e and p : k → k be a non-expansive mapping. if i − p maps closed bounded subset of e into closed subset of e and xn is the mann iteration, with λn satisfying assumptions (a)-(c) in lemma 2.1, then xn converges strongly to a fixed point of p in k. 3 existence theorems and approximation of solutions for most of the differential and integral equations with deviating arguments that appear in recent literature, the deviation of the argument usually involves only the time itself. however, another case, in which the deviating arguments depend on both the state variable u and the time t, is of importance in theory and practice. equations of the form u′(t) = f ( t, u(u(t)) ) are called iterative differential equations. these equations are important in the study of infection models and are related to the study of the motion of charged particles with retarded interaction (see [16-18]). in this section, we establish the existence and uniqueness results for the fractional differential equation dαu(t) = f ( t, u(t), u(u(t)) ) (3.1) cubo 14, 3 (2012) existence of deviating fractional differential equation 133 with initial condition u(0) = u0, where t, u0 ∈ j := [0, t] and f ∈ c(j × j × j, j). for t ∈ j denote mt = max{t, t − t} and cl,α = {u : |u(t1) − u(t2)| ≤ l γ(α + 1) |t1 − t2| α, ∀t1, t2 ∈ j}, l > 0. it is clear that cl,α is a nonempty convex and compact subset of the banach space ( c[j], ‖.‖ ) , where ‖x‖ = supt∈j |x(t)|. theorem 3.1. assume that the following conditions are satisfied for the initial value problem (1): (a1) f ∈ c[j × j × j, j]; (a2) ∃ℓ > 0 : |f(t, u1, u2) − f(t, v1, v2)| ≤ ℓ[|u1 − v1| + |u2 − v2|], ∀ t, ui, vi, i = 1, 2 ∈ j; (a3) if l is the lipschitz constant such that |u(t1) − u(t2)| ≤ lγ(α+1) |t1 − t2| α, then m = max{|f(t, u, v)| : (t, u, v) ∈ j × j × j} ≤ l 2 ; (a4) one of the following conditions holds: (a) m t α γ(α+1) ≤ mu0, where mu0 = max{u0, t − u0}; (b) u0 = 0, m t α γ(α+1) ≤ t − u0, f(t, u, v) ≥ 0, ∀t, u, v ∈ j; (c) u0 = t, m t α γ(α+1) ≤ u0, f(t, u, v) ≥ 0, ∀t, u, v ∈ j. if (l̃ + 2)tαℓ γ(α + 1) ≤ 1, (3.2) then there exists at least one solution of problem (1) in cl,α which can be approximated by the krasnoselskij iteration un+1 = (1 − λ)un + λu0 + λ ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ, where λ ∈ (0, 1) and u1 ∈ cl,α is arbitrary. proof. consider the integral operator p : cl,α → c(j) pu(t) = u0 + ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ, t ∈ j, u ∈ cl,α. 134 rabha w. ibrahim cubo 14, 3 (2012) our aim is show that p has a fixed point in cl,α. we proceed to apply schauder fixed point theorem or banach fixed point theorem. first we show that cl,α is invariant set with respect to p, i.e. t(cl,α) ⊂ cl,α. in virtue of condition (a4a) and for all t ∈ j, u ∈ cl,α we have |pu(t)| ≤ |u0| + | ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ| ≤ |u0| + m tα γ(α + 1) ≤ t and |pu(t)| ≥ |u0| − | ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ| ≥ |u0| − m tα γ(α + 1) ≥ u0 − mu0 ≥ 0. thus (pu)(t) ∈ j, t ∈ j. in the similar manner of (a4a), we treat the cases (a4b) and (a4c). now for every t1, t2 ∈ j, by (a3), we obtain |(pu)(t1) − (pu)(t2)| = | ∫t1 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ − ∫t2 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ| ≤ m |t α 1 − t α 2 + 2(t1 − t2) α| γ(α + 1) ≤ 2m |t1 − t2| α γ(α + 1) ≤ l |t1 − t2| α γ(α + 1) . hence (pu) ∈ cl,α whenever u ∈ cl,α. therefore, p : cl,α → cl,α (i. e., p is a self-mapping of cl,α). let u, v ∈ cl,α and t ∈ j, by employing (a2) we have cubo 14, 3 (2012) existence of deviating fractional differential equation 135 |(pu)(t) − (pv)(t)| = | ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ − ∫t 0 (t − τ)α−1 γ(α) f(τ, v(τ), v(v(τ)))dτ| ≤ ∫t 0 (t − τ)α−1 γ(α) |f(τ, u(τ), u(u(τ))) − f(τ, v(τ), v(v(τ)))|dτ ≤ ∫t 0 (t − τ)α−1 γ(α) [|u(τ) − v(τ)| + |u(u(τ)) − v(v(τ))|]dτ ≤ ∫t 0 (t − τ)α−1 γ(α) max[|u(τ) − v(τ)| + |u(u(τ)) − u(v(τ)) + u(v(τ)) − v(v(t))|]dτ ≤ tαℓ γ(α + 1) [l̃‖u − v‖ + 2‖u − v‖] ≤ (l̃ + 2)t αℓ γ(α + 1) ‖u − v‖, where 1 ≤ ‖u − v‖ ≤ t and l̃ := max l γ(α + 1) , α ∈ (0, 1]. now, by taking the supremum in the last assertion, we get ‖pu) − (pv)‖ ≤ (l̃ + 2)t αℓ γ(α + 1) ‖u − v‖. if (l̃+2)t α ℓ γ(α+1) < 1, then p is a contraction mapping and hence in view of banach fixed point theorem, eq. (1) has a unique solution. now if (l̃ + 2)tαℓ γ(α + 1) = 1 then p is non-expansive and, hence, continuous; thus schauder fixed point theorem implies that eq. (1) has a solution in cl,α. finally, in view of lemmas 2.1 and 2.2, we obtain the second part of the theorem. next we establish the solution of eq. (1) in a subset of cl,α defined by cl,α,δ = { u ∈ cl,α : u(t) ≤ δtα γ(α + 1) , ∀t ∈ j } , δ ∈ (0, 1). it is clear that cl,α,δ is non-empty, convex and compact subset in c[j]. theorem 3.2. assume that the following conditions are satisfied: (a5) u0 ≤ δt α 0 2γ(α+1) t0(6= 0) ∈ j; 136 rabha w. ibrahim cubo 14, 3 (2012) (a6) if l is the lipschitz constant such that |u(t1)−u(t2)| ≤ lγ(α+1) |t1 −t2| α, then m ≤ min{δ 2 , l 2 }; (a7) there exists a τ > 0 such that τ > − ln(1−δ) δ(t−t0) , t 6= t0 and tα−1ℓ γ(α)τ ( 1 δ + l̃ + 1) max{eτt0 − 1, 1 − eτ(t0−t)} ≤ 1 (3.3) if (a2), (a4) hold then there exists at least one solution of problem (1) in cl,α,δ which can be approximated by the krasnoselskij iteration un+1 = (1 − λ)un + λu0 + λ ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ, where λ ∈ (0, 1) and u1 ∈ cl,α,δ is arbitrary. proof. we assume the banach space c[j] endowed with bieleckis norm given by the formula ‖u‖b = max t∈j |u(t)|e−s(t−t0), s > 0, t > t0 (t, s, t0 ∈ j = [0, t], t < ∞). let p be defined as in the proof of theorem 3.1. by assumptions (a2), (a4), and (a6), it follows that p(cl,α,δ) ⊂ cl,α,δ. now we prove that cl,α,δ is an invariant set with respect to the operator p. indeed, if u ∈ cl,α,δ and t ∈ j then in view of (a5) and (a6), we have pu(t) ≤ u0 + m tα γ(α + 1) = u0 + m (tα − tα0 ) + t α 0 γ(α + 1) ≤ δtα0 2γ(α + 1) + δtα 2γ(α + 1) − δtα0 2γ(α + 1) + δtα0 2γ(α + 1) ≤ δt α γ(α + 1) , t > t0, that is pu ∈ cl,α,δ. let u, v ∈ cl,α,δ and t ∈ j, we have cubo 14, 3 (2012) existence of deviating fractional differential equation 137 |(pu)(t) − (pv)(t)| = | ∫t 0 (t − τ)α−1 γ(α) f(τ, u(τ), u(u(τ)))dτ − ∫t 0 (t − τ)α−1 γ(α) f(τ, v(τ), v(v(τ)))dτ| ≤ t α−1ℓ γ(α) ∣∣∣ ∫t 0 ( |u(τ) − v(τ)| + l̃|u(τ) − v(τ)| + |u(v(τ)) − v(v(τ))| ) dτ ∣∣∣ ≤ t α−1ℓ γ(α) (∣∣∣ ∫t 0 (l̃ + 1)es(τ−t0)dτ ∣∣∣ + ∣∣∣ ∫t 0 es(δτ−t0)dτ ∣∣∣ ) ‖u − v‖b ≤ tα−1ℓ γ(α) (∣∣∣ (l̃ + 1) s (es(t−t0) − 1) ∣∣∣ + 1 δs ∣∣∣es(δt−t0) − es(δt0−t0) ∣∣∣ ) ‖u − v‖b. this yields |(pu)(t) − (pv)(t)|e−s(τ−t0) ≤ t α−1ℓ sγ(α) ( (l̃ + 1) ∣∣∣1 − e−s(t−t0) ∣∣∣ + 1 δ ∣∣∣es(δ−1)t − es(δt0−t) ∣∣∣ ) ‖u − v‖b := l(t)‖u − v‖b where l(t) is a continuous function. then there exists a constant l̂ > 0 such that max t∈j |l(t)| ≤ l̂. thus we have ‖pu − pv‖b ≤ l̂‖u − v‖b, which shows that p is lipschitzian, hence continuous. by schauders fixed point theorem it follows that t has at least one fixed point which is actually a solution of the initial value problem (1). we proceed to show that p is non-expansive function. the function g(t) = 1 − e−s(t−t0), s > 0, t > t0 is strictly increasing on j and g(t0) = 0; furthermore, max t∈j g(t) = max{eτ̃t0 − 1, 1 − eτ̂(t0−t)}. similarly for the function h(t) = es(δ−1)t − es(δt0−t) then h′(t) = ses(δ−1)t[(δ − 1) + esδ(t−t0)]. now the function k(t) = (δ − 1) + esδ(t−t0) 138 rabha w. ibrahim cubo 14, 3 (2012) is strictly decreasing on j; hence, k(t) ≥ k(t) = (δ − 1) + esδ(t−t0). (3.4) for δ ∈ (0, 1) and t 6= t0 then by the assumption (a7) there exists a τ > 0 such that τ > − ln(1 − δ) δ(t − t0) , t 6= t0 which implies that k(t) > 0 and hence h is strictly increasing on j. if we put s = τ we have max t∈j |h(t)| = max { |1 − esδt0 |, |es(δ−1)t − es(δt0−t)| } . but since δ ∈ (0, 1) thus we get |es(δ−1)t − es(δt0−t)| = es(δ−1)t |1 − esδ(t0−t)| ≤ 1 − esδ(t0−t) for sufficient s, δ, t and t0. moreover, we have |1 − e sδt0 | ≤ esδt0 − 1. consequently, we receive l(t) ≤ max { esδt0 − 1, 1 − esδ(t0−t) } tα−1ℓ sγ(α) ( 1 δ + l̃ + 1). this shows that p is non-expansive. similar argument holds when t = t0 in eq. (4) we have k(t) = δ > 0 hence h is strictly increasing on j. finally, one can use lemmas 2.1 and 2.2 to obtain the second part of the theorem. this completes the proof. example 3.1. consider the following initial value problem associated to an fractional iterative differential equation    d0.5u(t) = −1 3 + 1 4 u(t) + 1 4 u(u(t)), t ∈ [0, 1] u(0) = 1 3 (3.5) where u ∈ c1([0, 1], [0, 1]). we are focused in the solutions u ∈ c1([0, 1], [0, 1]) belonging to the set c1,0.5 = {u : |u(t1) − u(t2)| ≤ 1 γ(3 2 ) |t1 − t2| 0.5, ∀t1, t2 ∈ [0, 1]} = {u : |u(t1) − u(t2)| ≤ 1 0.886 √ |t1 − t2|, ∀t1, t2 ∈ [0, 1]} = {u : |u(t1) − u(t2)| ≤ 1.1 √ |t1 − t2|, ∀t1, t2 ∈ [0, 1]}. cubo 14, 3 (2012) existence of deviating fractional differential equation 139 to satisfy (a4a), we have m ≤ l 2 ≃ 1 2 , m 1 3 = max{1 3 , 2 3 } = 2 3 = 0.666 and m tα γ(α + 1) = 1 2 × 1 0.886 = 0.56 < 0.666. hence (a4a) is satisfied. the function f(t) = −1 3 + 1 4 (u + v), v := u(u(t)), is lipschitzian with the lipschitz constant ℓ = 1 4 . this shows that (l̃ + 2)tαℓ γ(α + 1) = 3.1 × 0.25 0.886 = 0.874 < 1. therefore, by theorem 3.1 we obtain information on the existence and approximation of the solutions of the initial value problem (5). if we consider the function f(t) = −1 3 + 286 1000 (u + v) in example 3.1, then we obtain (l̃ + 2)tαℓ γ(α + 1) = 3.1 × 0.286 0.886 ≃ 1. therefore, again by theorem 3.1 we pose the existence and approximation of the solutions of the initial value problem (5). again, we consider the problem (5) on the interval [3 4 , 1] for ℓ = 0.015, where u ∈ c1([3 4 , 1], [3 4 , 1]). we are interested in the solutions u ∈ c1([3 4 , 1], [3 4 , 1]) belonging to the set c1, 1 2 , 3 4 = { u ∈ c1, 1 2 : u(t) ≤ δt α γ(α + 1) , ∀t ∈ j } , δ ∈ (0, 1)} = {u : u(t) ≤ 3 4 t 1 2 γ(3 2 ) , ∀t ∈ [3 4 , 1]} = {u : u(t) ≤ 0.846 √ t, t ∈ [3 4 , 1]}. our aim is to satisfy the assumptions of theorem 3.2. (a2) and (a4) are valid. since u0 = 1 3 and t0 = 3 4 we have u0 ≤ δtα0 2γ(α + 1) =⇒ 1 3 < 3 8 ; hence (a5) is satisfied. moreover, a computation gives m ≤ min{ δ 2 , l 2 } = min{ δ 2 , l 2 } = { 3 8 , 1 2 } = 3 8 thus (a6) is satisfied. now we proceed to satisfy (a7); since − ln(1 − δ) δ(t − t0) = − ln 1 4 × 16 3 = 6.933 140 rabha w. ibrahim cubo 14, 3 (2012) and max{eτt0 − 1, 1 − eτ(t0−t)} = max{189.5, .826} then for τ = 7 we impose tα−1ℓ γ(α)τ ( 1 δ + l̃ + 1) max{eτt0 − 1, 1 − eτ(t0−t)} = 0.15 37.17 max{189.5, .826} = 0.758 < 1. hence in view of theorem 3.2, problem (5) has a solution in the set c1, 1 2 , 3 4 . note that when ℓ = 0.025 we obtain tα−1ℓ γ(α)τ ( 1 δ + l̃ + 1) max{eτt0 − 1, 1 − eτ(t0−t)} = 0.25 37.17 max{189.5, .826} = 1.137 > 1. thus problem (5) hasn’t a solution in c1, 1 2 , 3 4 . while, for ℓ ≃ .02, implies tα−1ℓ γ(α)τ ( 1 δ + l̃ + 1) max{eτt0 − 1, 1 − eτ(t0−t)} ≃ 1; therefore, in virtue of theorem 3.2, eq.(5) has a solution. moreover, we can observe that problem (5) hasn’t a solution on the set c1, 1 2 , 1 2 over the interval [1 2 , 1] : c1, 1 2 , 1 2 = { u ∈ c1, 1 2 : u(t) ≤ δt α γ(α + 1) , ∀t ∈ j } , δ ∈ (0, 1)} = {u : u(t) ≤ 1 2 t 1 2 γ(3 2 ) , ∀t ∈ [1 2 , 1]} = {u : u(t) ≤ 0.564 √ t, t ∈ [1 2 , 1]}. for u0 = 1 3 , t0 = 1 2 , α = 1 2 , δ = 1 2 , a calculation poses u0 ≤ δtα0 2γ(α + 1) =⇒ 1 3 > 0.35 1.772 ; therefore, condition (a5) dose not satisfy. finally, problem (5) hasn’t a solution on the set c1, 1 2 , 1 2 over the interval [3 4 , 1] : c1, 1 2 , 1 2 = { u ∈ c1, 1 2 : u(t) ≤ δt α γ(α + 1) , ∀t ∈ j } , δ ∈ (0, 1)} = {u : u(t) ≤ 1 2 t 1 2 γ(3 2 ) , ∀t ∈ [3 4 , 1]} = {u : u(t) ≤ 0.5 √ t, t ∈ [3 4 , 1]}. cubo 14, 3 (2012) existence of deviating fractional differential equation 141 for u0 = 1 3 , t0 = 3 4 , α = 1 2 , δ = 1 2 , a calculation yields u0 ≤ δtα0 2γ(α + 1) =⇒ 1 3 > 1 4 ; therefore, condition (a5) dose not satisfy. as such iterative fractional differential equations are used to generalize the model infective disease processes, pattern formation in the plane, and are important in investigations of dynamical systems, future works will be also devoted to them. received: november 2011. revised: august 2012. references [1] r. w. ibrahim , s. momani, on the existence and uniqueness of solutions of a class of fractional differential equations, j. math. anal. appl. 334 (2007) 1-10. [2] s. m. momani, r. w. ibrahim, on a fractional integral equation of periodic functions involving weyl-riesz operator in banach algebras, j. math. anal. appl. 339 (2008) 1210-1219. [3] r. a. el-nabulsi, the fractional calculus of variations from extended erdelyi-kober operatorint. j. mod. phys. b23(16)(2009) 3349-3361. [4] z. wei, w. dong, j. che, periodic boundary value problems for fractional differential equations involving riemann-liouville fractional derivative, nonlinear analysis: theory, methods and applications, 73(10) (2010) 3232-3238. [5] b. ahmad, s. k. ntouyas, a. alsaedi, new existence results for nonlinear fractional differential equations with three-point integral boundary conditions, advances in difference equations, volume 2011, article id 107384, 11 pages. [6] j. sabatier, o. p. agrawal, j. a. tenreiro machado, advance in fractional calculus: theoretical developments and applications in physics and engineering, springer, 2007. 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[18] v. berinde, existence and approximation of solutions of some first order iterative differential equations, miskolc math. notes,vol. 11 (2010) pp. 1326. cubo a mathematical journal vol.19, no¯ 01, (79–87). march 2017 three dimensional f-kenmotsu manifold satisfying certain curvature conditions venkatesha and divyashree g. department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india. vensmath@gmail.com, gdivyashree9@gmail.com abstract the purpose of the present paper is to study pseudosymmetry conditions on f-kenmotsu manifolds. resumen el propósito del presente art́ıculo es estudiar condiciones de pseudosimetŕıa en variedades f-kenmotsu. keywords and phrases: f-kenmotsu manifold, cyclic parallel ricci tensor, almost pseudo ricci symmetry, pseudosymmetry, ricci pseudosymmetry, ricci generalized pseudosymmetry. 2010 ams mathematics subject classification: 47a63; 47a99. 80 venkatesha and divyashree g. cubo 19, 1 (2017) 1 introduction let mn be an almost contact manifold with an almost contact metric structure (φ,ξ,η, g) [1]. we denote by φ, the fundamental 2-form of mn i.e., φ(x, y) = g(x,φy) for any vector fields x, y ∈ χ(mn), where χ(mn) being the lie algebra of differentiable vector fields on mn. furthermore, we recollect the following definitions [1, 3, 8]. the manifold mn and its structure (φ,ξ,η, g) is said to be: i) normal if the almost complex structure defined on the product manifold mn ×r is integrable (equivalently, [φ,φ] + 2dη ⊗ ξ = 0), ii) almost cosymplectic if dη = 0 and dφ = 0, iii) cosymplectic if it is normal and almost cosymplectic (equivalently, ∇φ = 0, where ∇ is covariant differentiation with respect to the levi-civita connection). the manifold mn is called locally conformal almost cosymplectic (respectively, locally conformal cosymplectic) if mn has an open covering {ut} endowed with differentiable functions σt : ui −→ r such that over each ut the almost contact metric structure (φt,ξt,ηt, gt) defined by φt = φ, ξt = e σtξ, ηt = e −σtη, gt = e −2σtg is almost cosymplectic (respectively, locally conformal cosymplectic). normal locally conformal almost cosymplectic manifold were studied by olszak and rosca [7]. an almost contact metric manifold is said to be f-kenmotsu if it is normal and locally conformal almost cosymplectic. the same type of manifold was also studied by yildiz et al. [9] using the projective curvature tensor. olszak and rosca [7] also gave a geometric interpretation of fkenmotsu manifolds and studied some curvature restrictions. among others, they proved that a ricci symmetric f-kenmotsu manifold is an einstein manifold. our work is structured in the following way: after introduction, we have given some basic equations of f-kenmotsu manifold in section 2. section 3 deals with the study of 3-dimensional fkenmotsu manifold with cyclic parallel ricci tensor. and we study almost pseudo ricci symmetric, pseudosymmetric, ricci pseudosymmetric and ricci generalized pseudosymmetric 3-dimensional fkenmotsu manifolds in sections 4, 5, 6 and 7, respectively. 2 f-kenmotsu manifolds let mn be a smooth (2n + 1)-dimensional manifold endowed with an almost contact metric structure (φ,ξ,η, g) which satisfy φ2 = −id + η ⊗ ξ, η(ξ) = 1, η · φ = 0, (2.1) cubo 19, 1 (2017) three dimensional f-kenmotsu manifold satisfying certain . . . 81 φξ = 0, η(x) = g(x,ξ), g(φx,φy) = g(x, y) − η(x)η(y), (2.2) for any vector fields x, y ∈ χ(mn) where id is the identity of the tangent bundle tmn, φ is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form and g is a riemannian metric. we say that (mn,φ,ξ,η, g) is an f-kenmotsu manifold if the levi-civita connection ∇ of φ satisfies the condition [6] (∇xφ)(y) = f[g(φx, y)ξ − η(y)φx], (2.3) where f ∈ c∞(mn) is strictly positive and df ∧ η = 0. if f = 0, then the manifold is cosymplectic [5]. an f-kenmotsu manifold is called regular if f2 + f′ ̸= 0 where f′ = ξf. in an f-kenmotsu manifold, from (2.3) we have ∇xξ = f[x − η(x)ξ]. (2.4) the condition df ∧ η = 0 holds if dim mn ≥ 5 but it does not hold if dim mn = 3 [7]. (∇xη)(y) = f[g(x, y) − η(x)η(y)]. (2.5) in a 3-dimensional riemannian manifold, we have r(x, y)z = g(y, z)qx − g(x, z)qy + s(y, z)x − s(x, z)y (2.6) − r 2 {g(y, z)x − g(x, z)y}. in a 3-dimensional f-kenmotsu manifold, we see that [7] r(x, y)z = ( r 2 + 2f2 + 2f′)(x ∧ y)z − ( r 2 + 3f2 + 3f′){η(x)(ξ ∧ y)z (2.7) +η(y)(x ∧ ξ)z}, s(x, y) = ( r 2 + f2 + f′)g(x, y) − ( r 2 + 3f2 + 3f′)η(x)η(y), (2.8) where r, s, q and r are the riemannian curvature tensor, the ricci tensor, the ricci operator and the scalar curvature, respectively. now from(2.7), we have the following: r(x, y)ξ = −(f2 + f′)[η(y)x − η(x)y], (2.9) r(ξ, y)z = −(f2 + f′)[g(y, z)ξ − η(z)y], (2.10) η(r(x, y)z) = −(f2 + f′)[g(y, z)η(x) − g(x, z)η(y)]. (2.11) and from (2.8), we get s(x,ξ) = −2(f2 + f′)η(x), (2.12) and qξ = −2(f2 + f′)ξ. (2.13) 82 venkatesha and divyashree g. cubo 19, 1 (2017) 3 3-dimensional f-kenmotsu manifold with cyclic parallel ricci tensor suppose the manifold mn under consideration satisfies the cyclic parallel ricci tensor condition [4]. then we have (∇xs)(y, z) + (∇ys)(z, x) + (∇zs)(x, y) = 0, (3.1) for all x, y, z ∈ χ(mn). from the above equation, it is seen that r is constant. and we have (∇xs)(y, z) + (∇ys)(z, x) + (∇zs)(x, y) = −( r 2 + 3f2 + 3f′)[(∇xη)(y)η(z) (3.2) +η(y)(∇xη)(z) + (∇yη)(z)η(x) +η(z)(∇yη)(x) + (∇zη)(x)η(y) +η(x)(∇zη)(y)]. from (3.1) and (3.2), we get ( r 2 + 3f2 + 3f′)[(∇xη)(y)η(z) + η(y)(∇xη)(z) + (∇yη)(z)η(x) (3.3) +η(z)(∇yη)(x) + (∇zη)(x)η(y) + η(x)(∇zη)(y)] = 0. using (2.5) in (3.3), we get ( r 2 + 3f2 + 3f′)[g(x, y)η(z) + g(x, z)η(y) + g(y, z)η(x) + g(y, x)η(z) (3.4) +g(z, x)η(y) + g(z, y)η(x) − 6η(x)η(y)η(z)] = 0, since f ̸= 0. on substituting x = y = ei in (3.4), where ei is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, 1 ≤ i ≤ 3, which gives 4{ r 2 + 3f2 + 3f′}η(z) = 0. (3.5) hence, we get η(z) = 0, which is a contradiction. therefore, from (3.5) we have r = −6(f2 + f′). (3.6) conversely, if r = −6(f2 + f′) then from (3.2), we obtain (∇xs)(y, z) + (∇ys)(z, x) + (∇zs)(x, y) = 0. (3.7) from the above discussions we have the following: theorem 3.1. a 3-dimensional f-kenmotsu manifold satisfies cyclic parallel ricci tensor if and only if the scalar curvature r = −6(f2 + f′), provided f ̸= 0. cubo 19, 1 (2017) three dimensional f-kenmotsu manifold satisfying certain . . . 83 4 almost pseudo ricci symmetric 3-dimensional f-kenmotsu manifold satisfying cyclic ricci tensor chaki and kawaguchi [2] introduced the concept of almost pseudo ricci symmetric manifolds as an extended class of pseudo symmetric manifolds. a riemannian manifold (mn, g) is called an almost pseudo ricci symmetric manifold (aprs)n, if its ricci tensor s of type (0, 2) is not identically zero and satisfies the following condition (∇us)(v, w) = [a(u) + b(u)]s(v, w) + a(v)s(u, w) + a(w)s(u, v), (4.1) where a and b are two non-zero 1-forms defined by a(u) = g(u, p1), b(u) = g(u, p2). (4.2) by taking the cyclic sum of (4.1), we see that (∇us)(v, w) + (∇vs)(w, u) + (∇ws)(u, v) = [3a(u) + b(u)]s(v, w) (4.3) +[3a(v) + b(v)]s(u, w) + [3a(w) + b(w)]s(u, v). let mn admit a cyclic ricci tensor, then (4.3) becomes [3a(u) + b(u)]s(v, w) + [3a(v) + b(v)]s(u, w) + (4.4) [3a(w) + b(w)]s(u, v) = 0. replacing w by ξ in the above equation and using (2.12) and (4.2), we get −{2(f2 + f′)}[3a(u) + b(u)]η(v) − {2(f2 + f′)}[3a(v) + b(v)]η(u) (4.5) +[3η(p1) + η(p2)]s(u, v) = 0. in (4.5), substituting v = ξ and using (2.12) and (4.2), we have −{2(f2 + f′)}[3a(u) + b(u)] − 4{2(f2 + f′)}[3η(p1) + η(p2)]η(u) = 0. (4.6) again treating u by ξ and using (4.2) in (4.6), we obtain {f2 + f′}[3η(p1) + η(p2)] = 0, (4.7) which implies [3η(p1) + η(p2)] = 0, (4.8) since {f2 + f′} ̸= 0. from (4.8) and (4.6), it follows that 3a(u) + b(u) = 0. (4.9) thus, we can state: theorem 4.1. there is no almost pseudo ricci symmetric 3-dimensional f-kenmotsu manifold admitting cyclic ricci tensor, unless 3a + b vanishes everywhere. 84 venkatesha and divyashree g. cubo 19, 1 (2017) 5 pseudosymmetric 3-dimensional f-kenmotsu manifold let mn be an pseudosymmetric 3-dimensional f-kenmotsu manifold. then we have, (r(x, y) · r)(u, v)w = frq(g, r)(u, v, w; x, y), (5.1) for all x, y, u, v, w ∈ χ(mn). from the above relation it follows that r(x, y)r(u, v)w − r(r(x, y)u, v)w − r(u, r(x, y)v)w (5.2) −r(u, v)r(x, y)w = fr[(x ∧g y)r(u, v)w − r((x ∧g y)u, v)w −r(u, (x ∧g y)v)w − r(u, v)(x ∧g y)w], where (x ∧g y)z = g(y, z)x − g(x, z)y. (5.3) substituting x by ξ and using (2.10) and (5.3), (5.2) yields [(f2 + f′) + fr]{g(y, r(u, v)w)ξ − η(r(u, v)w)y − g(y, u)r(ξ, v)w (5.4) +η(u)r(y, v)w − g(y, v)r(u,ξ)w + η(v)r(u, y)w − g(y, w)r(u, v)ξ +η(w)r(u, v)y} = 0. taking inner product of (5.4) with ξ, we get [(f2 + f′) + fr]{r(u, v, w, y) − η(y)η(r(u, v)w) − g(y, u)η(r(ξ, v)w) (5.5) +η(u)η(r(y, v)w) − g(y, v)η(r(u,ξ)w) + η(v)η(r(u, y)w) −g(y, w)η(r(u, v)ξ) + η(w)η(r(u, v)y)} = 0. by using (2.11), (5.5) becomes [(f2 + f′) + fr]{r(u, v, w, y) − (f 2 + f′)[−g(v, w)η(y)η(u) (5.6) +g(u, w)η(y)η(v) − g(y, u)g(v, w) + g(y, u)η(v)η(w) + g(v, w)η(u)η(y) −g(y, w)η(u)η(v) − g(y, v)η(w)η(u) + g(y, v)g(u, w) + g(y, w)η(v)η(u) −g(u, w)η(y)η(v) + g(v, y)η(w)η(u) − g(u, y)η(v)η(w)]} = 0. contracting the above equation, we obtain [(f2 + f′) + fr]{s(v, w) + 2(f 2 + f′)g(v, w)} = 0. (5.7) the above equation can hold only if either (i) (f2 + f′) = −fr, or cubo 19, 1 (2017) three dimensional f-kenmotsu manifold satisfying certain . . . 85 (ii) s(v, w) = αg(v, w), where α = −2(f2 + f′). this leads to the following: theorem 5.1. a 3-dimensional pseudosymmetric f-kenmotsu manifold with never vanishing function {(f2 + f′) = −fr} is an einstein manifold. 6 ricci pseudosymmetric 3-dimensional f-kenmotsu manifold suppose (mn, g) be a 3-dimensional ricci pseudosymmetric f-kenmotsu manifold. then we have, (r(x, y) · s)(u, v) = fsq(g, s)(u, v; x, y), (6.1) for all x, y, u, v, w ∈ χ(mn). from the above relation it follows that (r(x, y) · s)(u, v) = fs((x ∧g y) · s)(u, v), or −s(r(x, y)u, v) − s(u, r(x, y)v) = f[−g(y, u)s(x, v) + g(x, u)s(y, v) (6.2) −g(y, v)s(u, x) + g(x, v)s(u, y)]. replacing x and u by ξ and using (2.1), (2.10) and (2.12) in the above equation, we get [(f2 + f′) + fs]{s(y, v) + 2(f 2 + f′)g(y, v)} = 0, (6.3) which follows that either [(f2 + f′) + fs] = 0 or s(y, v) = αg(y, v), (6.4) where α = −2(f2 + f′). thus we can state: theorem 6.1. if a 3-dimensional f-kenmotsu manifold mn is ricci pseudosymmetric with restrictions x = u = ξ, then either [(f2 + f′) + fs] = 0 or the manifold is an einstein manifold. 7 ricci generalized pseudosymmetric 3-dimensional f-kenmotsu manifold consider a ricci generalized pseudosymmetric 3-dimensional f-kenmotsu manifold. then we have (r(x, y) · r)(u, v)w = f((x ∧s y) · r)(u, v)w, (7.1) 86 venkatesha and divyashree g. cubo 19, 1 (2017) for all x, y, u, v, w ∈ χ(mn). we can write the above form as r(x, y)r(u, v)w − r(r(x, y)u, v)w − r(u, r(x, y)v)w (7.2) −r(u, v)r(x, y)w = f[s(y, r(u, v)w)x − s(x, r(u, v)w)y −s(y, u)r(x, v)w + s(x, u)r(y, v)w − s(y, v)r(u, x)w +s(x, v)r(u, y)w − s(y, w)r(u, v)x + s(x, w)r(u, v)y]. on substituting x = u = ξ and using (2.10) and (2.12), (7.2) reduces to −(f2 + f′)[(f2 + f′)g(v, w)y + r(y, v)w − (f2 + f′)g(y, w)v] (7.3) = f[(f2 + f′)s(y, v)η(w)ξ − 2(f2 + f′)2g(v, w)y − 2(f2 + f′)r(y, v)w +2(f2 + f′)2g(y, w)η(v)ξ + (f2 + f′)s(y, w)η(v)ξ − (f2 + f′)s(y, w)v +2(f2 + f′)2g(v, y)η(w)ξ]. taking inner product of the above equation with z, we get −(f2 + f′)[(f2 + f′)g(v, w)g(y, z) + g(r(y, v)w, z) − (f2 + f′)g(y, w)g(v, z)] (7.4) = f[(f2 + f′)s(y, v)η(w)η(z) − 2(f2 + f′)2g(v, w)g(y, z) −2(f2 + f′)g(r(y, v)w, z) + 2(f2 + f′)2g(y, w)η(v)η(z) +(f2 + f′)s(y, w)η(v)η(z) − (f2 + f′)s(y, w)g(v, z) +2(f2 + f′)2g(v, y)η(w)η(z)]. contracting (7.4) and simplifying gives (f2 + f′)(3f − 1)[s(y, z) + 2(f2 + f′)g(y, z)] = 0, (7.5) which means that either (f2 + f′)(3f − 1) = 0 or s(y, z) = αg(y, z), where α = −2(f2 + f′). hence we can state the following: theorem 7.1. if a 3-dimensional f-kenmotsu manifold is ricci generalized pseudosymmetric then either (i) (f2 + f′)(3f − 1) = 0, or (ii) it is an einstein manifold. acknowledgement: the second author is thankful to ugc for financial support in the form of rajiv gandhi national fellowship (f1-17.1/2015-16/rgnf-2015-17-sc-kar-26367). references [1] d.e blair, contact manifolds in riemannian geometry, lecture notes in mathematics, 509, springer-verlag, berlin-new york, 1976. cubo 19, 1 (2017) three dimensional f-kenmotsu manifold satisfying certain . . . 87 [2] m.c. chaki and t. kawaguchi, on almost pseudo ricci symmetric manifolds, tensor n. s. 68, 10-14, 2007. [3] goldberg s. i. and yano k, integrability of almost cosymplectic structures, pacific j. math., 31, 373-382, 1969. [4] a. gray, two classes of riemannian manifolds, geom. dedicata. 7, 259-280, 1978. [5] janssens d and vanhecke l, almost contact structures and curvature tensors, kodai math. j. 4(1), 1-27, 1981. [6] olszak z, locally conformal almost cosymplectic manifolds, colloq. math. 57, 73-87, 1989. [7] olszak z and rosca r, normal locally conformal almost cosymplectic manifolds, publ. math. debrecen. 39, 315-323, 1991. [8] sasaki s. and hatakeyama y, on differentiable manifolds with certain structures which are closely related to almost contact structures ii, tohoku math. j., 13, 281-294, 1961. [9] yildiz a, de u.c. and turan m, on 3-dimensional fkenmotsu manifolds and ricci solitons, ukrainian math. j., 65(5), 684-693, 2013. cubo a mathematical journal vol.14, no¯ 03, (85–101). october 2012 a common fixed point theorem in g-metric spaces s.k.mohanta and srikanta mohanta department of mathematics, west bengal state university, barasat, 24 pargans (north), west bengal, kolkata 700126, email: smwbes@yahoo.in abstract we prove a common fixed point theorem for a pair of self mappings in complete gmetric spaces. our result will improve and supplement some recent results in the setting of g-metric spaces. resumen probamos un teorema de punto fijo genérico para un par de auto-aplicaciones en espacios g-métricos completos. nuestro resultado mejorará y complementará algunos de los resultados recientes en el marco de los espacios g-métricos. keywords and phrases: g-metric space, g-cauchy sequence, g-continuity, common fixed point. 2010 ams mathematics subject classification: 54h25, 47h10. 86 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) 1 introduction the study of metric fixed point theory has been at the centre of vigorous activity and it has a wide range of applications in applied mathematics and sciences. over the past two decades a considerable amount of research work for the development of fixed point theory have executed by several authors. different generalizations of the usual notion of a metric space have been proposed by gähler [4, 5] and by dhage [2, 3]. unfortunately, it was found that most of the results claimed by dhage are invalid. these errors were pointed out by mustafa and sims in [13]. they also introduced a more appropriate concept of generalized metric space called g-metric space [9] and developed a new fixed point theory for various mappings in this new structure. our aim in this study is to prove a common fixed point theorem in a complete g-metric space. this theorem generalizes the fixed point results of [1], [10] and [11]. 2 preliminaries in this section, we present some basic definitions and results for g-metric spaces which will be needed in the sequel. throughout this paper we denote by n the set of positive integers. definition 2.1. (see[9]) let x be a nonempty set, and let g : x × x × x → r+ be a function satisfying the following axioms: (g1) g(x, y, z) = 0 if x = y = z, (g2) 0 < g(x, x, y), for all x, y ∈ x, with x 6= y, (g3) g(x, x, y) ≤ g(x, y, z), for all x, y, z ∈ x, with z 6= y, (g4) g(x, y, z) = g(x, z, y) = g(y, z, x) = · · · (symmetry in all three variables), (g5) g(x, y, z) ≤ g(x, a, a) + g(a, y, z), for all x, y, z, a ∈ x, (rectangle inequality). then the function g is called a generalized metric , or, more specifically a g-metric on x, and the pair (x, g) is called a g-metric space. proposition 2.1. (see[9]) let (x, g) be a g-metric space. then for any x, y, z, and a ∈ x, it follows that (1) if g(x, y, z) = 0 then x = y = z, cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 87 (2) g(x, y, z) ≤ g(x, x, y) + g(x, x, z), (3) g(x, y, y) ≤ 2g(y, x, x), (4) g(x, y, z) ≤ g(x, a, z) + g(a, y, z), (5) g(x, y, z) ≤ 2 3 (g(x, y, a) + g(x, a, z) + g(a, y, z)), (6) g(x, y, z) ≤ g(x, a, a) + g(y, a, a) + g(z, a, a). definition 2.2. (see[9]) let (x, g) be a g-metric space, let (xn) be a sequence of points of x, we say that (xn) is g-convergent to x if lim n,m→∞ g(x, xn, xm) = 0; that is, for any ǫ > 0, there exists n0 ∈ n such that g(x, xn, xm) < ǫ, for all n, m ≥ n0. we call x as the limit of the sequence (xn) and write xn −→ x. proposition 2.2. (see[9]) let (x, g) be a g-metric space, then the following are equivalent. (1) (xn) is g − convergent to x. (2) g(xn, xn, x) → 0, as n → ∞. (3) g(xn, x, x) → 0, as n → ∞. (4) g(xn, xm, x) → 0, as n, m → ∞. definition 2.3. (see[9]) let (x, g) be a g-metric space, a sequence (xn) is called g-cauchy if given ǫ > 0, there is n0 ∈ n such that g(xn, xm, xl) < ǫ, for all n, m, l ≥ n0; that is, if g(xn, xm, xl) → 0 as n, m, l → ∞. definition 2.4. (see[9]) let (x, g) and (x ′ , g ′ ) be g-metric spaces and let f : (x, g) → (x ′ , g ′ ) be a function, then f is said to be g-continuous at a point a ∈ x if given ǫ > 0, there exists δ > 0 such that x, y ∈ x; g(a, x, y) < δ implies g ′ (f(a), f(x), f(y)) < ǫ. a function f is g-continuous on x if and only if it is g-continuous at all a ∈ x. proposition 2.3. (see[9]) let (x, g) and (x ′ , g ′ ) be g-metric spaces, then a function f : x → x ′ is g-continuous at a point x ∈ x if and only if it is g-sequentially continuous at x; that is, whenever (xn) is g-convergent to x, (f(xn)) is g-convergent to f(x). proposition 2.4. (see[9]) let (x, g) be a g-metric space, then the function g(x, y, z) is jointly continuous in all three of its variables. proposition 2.5. (see[9]) every g-metric space (x, g) will define a metric space (x, dg) by dg(x, y) = g(x, y, y) + g(y, x, x), for all x, y ∈ x. 88 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) definition 2.5. (see[9]) a g-metric space (x, g) is said to be g-complete (or a complete g-metric space) if every g-cauchy sequence in (x, g) is g-convergent in (x, g). proposition 2.6. (see[9]) a g-metric space (x, g) is g-complete if and only if (x, dg) is a complete metric space. 3 main results theorem 3.1. let (x, g) be a complete g-metric space, and let t1, t2 be mappings from x into itself satisfying max    g(t1(x), t2(t1(x)), t2(t1(x))), g(t2(x), t1(t2(x)), t1(t2(x)))    ≤ r min    g(x, t1(x), t1(x)), g(x, t2(x), t2(x))    (3.1) for every x ∈ x, where 0 ≤ r < 1 and that inf [g(x, y, y) + min {g(x, t1(x), t1(x)), g(x, t2(x), t2(x))} : x ∈ x] > 0 for every y ∈ x with y is not a common fixed point of t1 and t2. then t1 and t2 have a common fixed point in x. proof. let x0 ∈ x be arbitrary and define a sequence (xn) by xn = t1(xn−1), if n is odd = t2(xn−1), if n is even. then for any odd positive integer n ∈ n, we have g(xn, xn+1, xn+1) = g(t1(xn−1), t2(xn), t2(xn)) = g(t1(xn−1), t2(t1(xn−1)), t2(t1(xn−1))) ≤ max    g(t1(xn−1), t2(t1(xn−1)), t2(t1(xn−1))), g(t2(xn−1), t1(t2(xn−1)), t1(t2(xn−1)))    ≤ r min    g(xn−1, t1(xn−1), t1(xn−1)), g(xn−1, t2(xn−1), t2(xn−1))    , by (3.1) ≤ r g(xn−1, t1(xn−1), t1(xn−1)) = r g(xn−1, xn, xn). cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 89 if n is even, then by (3.1), we have g(xn, xn+1, xn+1) = g(t2(xn−1), t1(xn), t1(xn)) = g(t2(xn−1), t1(t2(xn−1)), t1(t2(xn−1))) ≤ max    g(t1(xn−1), t2(t1(xn−1)), t2(t1(xn−1))), g(t2(xn−1), t1(t2(xn−1)), t1(t2(xn−1)))    ≤ r min    g(xn−1, t1(xn−1), t1(xn−1)), g(xn−1, t2(xn−1), t2(xn−1))    ≤ r g(xn−1, t2(xn−1), t2(xn−1)) = r g(xn−1, xn, xn). thus for any positive integer n, it must be the case that g(xn, xn+1, xn+1) ≤ r g(xn−1, xn, xn). (3.2) by repeated application of (3.2), we obtain g(xn, xn+1, xn+1) ≤ r n g(x0, x1, x1). (3.3) then, for all n, m ∈ n, n < m, we have by repeated use of the rectangle inequality and (3.3) that g(xn, xm, xm) ≤ g(xn, xn+1, xn+1) + g(xn+1, xn+2, xn+2) +g(xn+2, xn+3, xn+3) + · · · + g(xm−1, xm, xm) ≤ ( rn + rn+1 + · · · + rm−1 ) g(x0, x1, x1) ≤ rn 1 − r g(x0, x1, x1). then, lim g(xn, xm, xm) = 0, as n, m → ∞, since lim r n 1−r g(x0, x1, x1) = 0, as n, m → ∞. for n, m, l ∈ n, (g5) implies that g(xn, xm, xl) ≤ g(xn, xm, xm) + g(xl, xm, xm), taking limit as n, m, l → ∞, we get g(xn, xm, xl) → 0. so (xn) is a g-cauchy sequence. by completeness of (x, g), there exists u ∈ x such that (xn) is g-convergent to u. let n ∈ n be fixed. then, since (xm) g-converges to u and g is continuous on its variables, we have g(xn, u, u) = lim m→∞ g(xn, xm, xm) ≤ rn 1 − r g(x0, x1, x1). 90 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) assume that u is not a common fixed point of t1 and t2. then, by hypothesis, we have 0 < inf [ g(x, u, u) + min {g(x, t1(x), t1(x)), g(x, t2(x), t2(x))} : x ∈ x] ≤ inf [ g(xn, u, u) + min { g(xn, t1(xn), t1(xn)), g(xn, t2(xn), t2(xn))} : n ∈ n] ≤ inf [ rn 1 − r g(x0, x1, x1) + g(xn, xn+1, xn+1) : n ∈ n ] ≤ inf [ rn 1 − r g(x0, x1, x1) + r n g(x0, x1, x1) : n ∈ n ] = 0, which is a contradiction. therefore, u is a common fixed point of t1 and t2. theorem 3.2. let (x, g) be a complete g-metric space, and let t1, t2 be mappings from x into itself satisfying max    g(t1(x), t1(x), t2(t1(x))), g(t2(x), t2(x), t1(t2(x)))    ≤ r min    g(x, x, t1(x)), g(x, x, t2(x))    (3.4) for every x ∈ x, where 0 ≤ r < 1 and that inf [g(x, x, y) + min {g(x, x, t1(x)), g(x, x, t2(x))} : x ∈ x] > 0 for every y ∈ x with y is not a common fixed point of t1 and t2. then t1 and t2 have a common fixed point in x. proof. let x0 ∈ x be arbitrary and define a sequence (xn) by xn = t1(xn−1), if n is odd = t2(xn−1), if n is even. then by the argument similar to that used in theorem 3.1, we have for any positive integer n, g(xn, xn, xn+1) ≤ r n g(x0, x0, x1). (3.5) then, for all n, m ∈ n, n < m, we have by repeated use of the rectangle inequality and (3.5) that g(xm, xn, xn) ≤ g(xm, xm−1, xm−1) + g(xm−1, xm−2, xm−2) +g(xm−2, xm−3, xm−3) + · · · + g(xn+1, xn, xn) ≤ ( rn + rn+1 + · · · + rm−1 ) g(x0, x0, x1) ≤ rn 1 − r g(x0, x0, x1). cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 91 thus (xn) becomes a g-cauchy sequence. by completeness of (x, g), there exists u ∈ x such that (xn) is g-convergent to u. let n ∈ n be fixed. then since (xm) g-converges to u and g is continuous on its variables, we have g(xn, xn, u) = lim m→∞ g(xn, xn, xm) ≤ rn 1 − r g(x0, x0, x1). the argument similar to that used in the proof of theorem 3.1 establishes that u is a common fixed point of t1 and t2. combining theorem 3.1 and theorem 3.2, we state the following theorem: theorem 3.3. let (x, g) be a complete g-metric space, and let t1, t2 be mappings from x into itself satisfying one of the following conditions: max    g(t1(x), t2(t1(x)), t2(t1(x))), g(t2(x), t1(t2(x)), t1(t2(x)))    ≤ r min    g(x, t1(x), t1(x)), g(x, t2(x), t2(x))    for every x ∈ x, where 0 ≤ r < 1 and that inf [g(x, y, y) + min {g(x, t1(x), t1(x)), g(x, t2(x), t2(x))} : x ∈ x] > 0 for every y ∈ x with y is not a common fixed point of t1 and t2. or max    g(t1(x), t1(x), t2(t1(x))), g(t2(x), t2(x), t1(t2(x)))    ≤ r min    g(x, x, t1(x)), g(x, x, t2(x))    for every x ∈ x, where 0 ≤ r < 1 and that inf [g(x, x, y) + min {g(x, x, t1(x)), g(x, x, t2(x))} : x ∈ x] > 0 for every y ∈ x with y is not a common fixed point of t1 and t2. then t1 and t2 have a common fixed point in x. as an application of theorem 3.3, we have the following corollary. 92 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) corollary 1. let (x, g) be a complete g-metric space, and let t : x → x be a mapping satisfying one of the following conditions: g(t(x), t2(x), t2(x)) ≤ r g(x, t(x), t(x)) (3.6) for every x ∈ x, where 0 ≤ r < 1 and that inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] > 0 (3.7) for every y ∈ x with y 6= t(y). or g(t(x), t(x), t2(x)) ≤ r g(x, x, t(x)) (3.8) for every x ∈ x, where 0 ≤ r < 1 and that inf [g(x, x, y) + g(x, x, t(x)) : x ∈ x] > 0 (3.9) for every y ∈ x with y 6= t(y). then t has a fixed point in x. proof. take t1 = t2 = t in theorem 3.3. we now supplement corollary 1 by examination of condition (3.6)(or, (3.8)) and condition (3.7)(or, (3.9)) in respect of their independence. in fact, we furnish example 3.1 and example 3.2 below to show that these two conditions are independent in the sense that corollary 1 shall fall through by dropping one in favour of the other. example 3.1. let x = {0} ∪ { 1 2n : n ≥ 1 } . define g : x × x × x → r+ by g(x, y, z) = 1 4 | x − y | + 1 4 | y − z | + 1 4 | z − x |, for all x, y, z ∈ x. then (x, g) is a complete g-metric space. define t : x → x by t(0) = 1 2 and t ( 1 2n ) = 1 2n+1 for n ≥ 1.clearly, t has got no fixed point in x. also, it is easy to check that g(t(x), t2(x), t2(x)) = 1 2 g(x, t(x), t(x)) for every x ∈ x. thus, condition (3.6) in corollary 1 is satisfied. however, t(y) 6= y for all y ∈ x and so inf {g(x, y, y) + g(x, t(x), t(x)) : x, y ∈ x with y 6= t(y)} cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 93 = inf {g(x, y, y) + g(x, t(x), t(x)) : x, y ∈ x} = inf { 1 2 | x − y | + 1 2 | x − t(x) |: x, y ∈ x } = 0. thus condition (3.7) in corollary 1 does not hold. similarly, we can verify that condition (3.8) in corollary 1 is also satisfied but condition (3.9) fails. clearly, the conclusion of corollary 1 is not valid. example 3.2. take x = {0, 1} ∪ [2, ∞). define g : x × x × x → r+ by g(x, y, z) = 1 4 | x − y | + 1 4 | y − z | + 1 4 | z − x |, for all x, y, z ∈ x. then (x, g) is a complete g-metric space. define t : x → x by t(x) = 0, for x 6= 0 = 1, for x = 0. clearly, t possesses no fixed point in x. since t(y) 6= y for all y ∈ x, we have inf {g(x, y, y) + g(x, t(x), t(x)) : x, y ∈ x with y 6= t(y)} = inf {g(x, y, y) + g(x, t(x), t(x)) : x, y ∈ x} = inf { 1 2 | x − y | + 1 2 | x − t(x) |: x, y ∈ x } > 0. thus condition (3.7) in corollary 1 is satisfied. however, for x = 0, we have g(t(x), t2(x), t2(x)) = 1 2 | t(x) − t2(x) |= 1 2 > r g(x, t(x), t(x)) for any r ∈ [0, 1). this shows that condition (3.6) in corollary 1 does not hold. similarly, we can check that condition (3.9) in corollary 1 is also satisfied but condition (3.8) fails. obviously, corollary 1 is invalid in this case. as an application of corollary 1, we have the following results. 94 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) corollary 2. let (x, g) be a complete g-metric space, and let t : x → x be a g-continuous mapping satisfying one of the following conditions: g(t(x), t2(x), t2(x)) ≤ r g(x, t(x), t(x)) (3.10) or g(t(x), t(x), t2(x)) ≤ r g(x, x, t(x)) (3.11) for every x ∈ x, where 0 ≤ r < 1. then t has a fixed point in x. proof. suppose that t satisfies condition (3.10) for every x ∈ x. assume that there exists y ∈ x with y 6= t(y) and inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] = 0. then there exists a sequence (xn) in x such that lim n→∞ {g(xn, y, y) + g(xn, t(xn), t(xn))} = 0, which implies that, g(xn, y, y) → 0 and g(xn, t(xn), t(xn)) → 0 as n → ∞. so, by proposition 2.2, the sequence (xn) is g-convergent to y. but by (g5), we have g(t(xn), y, y) ≤ g(t(xn), xn, xn) + g(xn, y, y) ≤ 2 g(xn, t(xn), t(xn)) + g(xn, y, y) → 0 as n → ∞. again, by proposition 2.2, the sequence (t(xn)) is g-convergent to y. so g-continuity of t implies that, (t2(xn)) g-converges to t(y). then by use of the rectangle inequality and (3.10) that g(xn, t 2(xn), t 2(xn)) ≤ g(xn, t(xn), t(xn)) + g(t(xn), t 2(xn), t 2(xn)) ≤ g(xn, t(xn), t(xn)) + r g(xn, t(xn), t(xn)). taking the limit as n → ∞, and using the fact that the function g is continuous on its variables, we have g(y, t(y), t(y)) ≤ g(y, y, y) + r g(y, y, y) = 0, cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 95 which implies that, y = t(y). this is a contradiction. hence, if y 6= t(y), then inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] > 0. if t satisfies condition (3.11), then using the same methods as above one can prove that inf [g(x, x, y) + g(x, x, t(x)) : x ∈ x] > 0. so, using corollary 1, we have the desired result. the following corollary is a generalization of the result [[1], theorem2.1]. corollary 3. let (x, g) be a complete g-metric space, and let t : x → x be a mapping satisfying one of the following conditions: g(t(x), t(y), t(z)) ≤ k max    g(x, y, z), g(x, t(x), t(x)), g(y, t(y), t(y)), g(z, t(z), t(z)), g(x,t(y),t(y))+g(z,t(x),t(x)) 2 , g(x,t(y),t(y))+g(y,t(x),t(x)) 2 , g(y,t(z),t(z))+g(z,t(y),t(y)) 2 , g(x,t(z),t(z))+g(z,t(x),t(x)) 2    (3.12) or g(t(x), t(y), t(z)) ≤ k max    g(x, y, z), g(x, x, t(x)), g(y, y, t(y)), g(z, z, t(z)), g(x,x,t(y))+g(z,z,t(x)) 2 , g(x,x,t(y))+g(y,y,t(x)) 2 , g(y,y,t(z))+g(z,z,t(y)) 2 , g(x,x,t(z))+g(z,z,t(x)) 2    (3.13) 96 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) for all x, y, z ∈ x, where 0 ≤ k < 1. then t has a unique fixed point (say u) in x and t is g-continuous at u. proof. suppose that t satisfies condition (3.12) for all x, y, z ∈ x. then replacing y and z by t(x), we obtain from (3.12) and using (g5) that g(t(x), t2(x), t2(x)) ≤ k max    g(x, t(x), t(x)), g(x, t(x), t(x)), g(t(x), t2(x), t2(x)), g(t(x), t2(x), t2(x)), g(x,t 2 (x),t 2 (x))+g(t(x),t(x),t(x)) 2 , g(x,t 2 (x),t 2 (x))+g(t(x),t(x),t(x)) 2 , g(t(x),t 2 (x),t 2 (x))+g(t(x),t 2 (x),t 2 (x)) 2 , g(x,t 2 (x),t 2 (x))+g(t(x),t(x),t(x)) 2    ≤ k max    g(x, t(x), t(x)), g(t(x), t2(x), t2(x)), g(x,t(x),t(x))+g(t(x),t 2 (x),t 2 (x)) 2    (3.14) without loss of generality we may assume that t(x) 6= t2(x). for, otherwise, t has a fixed point. so, (3.14) leads to the following cases, (1) g(t(x), t2(x), t2(x)) ≤ k g(x,t(x),t(x))+g(t(x),t 2 (x),t 2 (x)) 2 , (2) g(t(x), t2(x), t2(x)) ≤ k g(x, t(x), t(x)). in the first case, we have g(t(x), t2(x), t2(x)) ≤ k 2 − k g(x, t(x), t(x)). put r = k 2−k . then 0 ≤ r < 1. thus, in each case we must have g(t(x), t2(x), t2(x)) ≤ r g(x, t(x), t(x)) for every x ∈ x, where 0 ≤ r < 1. assume that there exists y ∈ x with y 6= t(y) and inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] = 0. cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 97 proceeding exactly the same way as in the proof of corollary 2, there exists a sequence (xn) in x such that (xn) is g-convergent to y and (t(xn)) is g-convergent to y. now applying (3.12), we have g(t(xn), t(y), t(y)) ≤ k max    g(xn, y, y), g(xn, t(xn), t(xn)), g(y, t(y), t(y)), g(y, t(y), t(y)), g(xn,t(y),t(y))+g(y,t(xn),t(xn)) 2 , g(xn,t(y),t(y))+g(y,t(xn),t(xn)) 2 , g(y,t(y),t(y))+g(y,t(y),t(y)) 2 , g(xn,t(y),t(y))+g(y,t(xn),t(xn)) 2    . taking the limit as n → ∞, and using the fact that the function g is continuous on its variables, we obtain g(y, t(y), t(y)) ≤ k g(y, t(y), t(y)), which is a contradiction. hence, if y 6= t(y), then inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] > 0. now corollary 1 applies to obtain a fixed point (say u) of t. the proof using (3.13) is similar. uniqueness of u and g-continuity of t at u may be verified in the usual way by using any one of condition (3.12) and condition (3.13) that t satisfies. remark 1. we see that special cases of corollary 3 are theorem 2.1 of [1], theorem 2.1 of [11] and theorems 2.1, and 2.4 of [10]. the following corollary is the result [[1], theorem 2.2]. corollary 4. let (x, g) be a complete g-metric space, and let t : x → x be a mapping satisfying one of the following conditions: g(t(x), t(y), t(z)) ≤ k max    g(x, y, z), g(x, t(x), t(x)), g(y, t(y), t(y)), g(x, t(y), t(y)), g(y, t(x), t(x)), g(z, t(z), t(z))    (3.15) 98 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) or g(t(x), t(y), t(z)) ≤ k max    g(x, y, z), g(x, x, t(x)), g(y, y, t(y)), g(x, x, t(y)), g(y, y, t(x)), g(z, z, t(z))    (3.16) for all x, y, z ∈ x, where 0 ≤ k < 1. then t has a unique fixed point (say u) in x and t is g-continuous at u. proof. suppose that t satisfies condition (3.15) for all x, y, z ∈ x. then replacing z by x ; y and x by t(x) in (3.15), we have g(t2(x), t2(x), t(x)) ≤ k max    g(t(x), t(x), x), g(t(x), t2(x), t2(x)), g(t(x), t2(x), t2(x)), g(t(x), t2(x), t2(x)), g(t(x), t2(x), t2(x)), g(x, t(x), t(x))    ≤ k max { g(x, t(x), t(x)), g(t(x), t2(x), t2(x)) } . without loss of generality we may assume that t(x) 6= t2(x). for, otherwise, t has a fixed point. so, it must be the case that, g(t(x), t2(x), t2(x)) ≤ k g(x, t(x), t(x)) for every x ∈ x, where 0 ≤ k < 1. by the same argument used in the proof of corollary 3, we see that if y 6= t(y), then inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] > 0. now corollary 1 applies to obtain a fixed point (say u) of t. the proof using (3.16) is similar. uniqueness of u and g-continuity of t at u are obtained by the same argument used in corollary 3. the following corollary is the result [[11], theorem2.9]. corollary 5. let (x, g) be a complete g-metric space, and let t : x → x be a mapping satisfying one of the following conditions: g(t(x), t(y), t(y)) ≤ a {g(x, t(y), t(y)) + g(y, t(x), t(x))} (3.17) cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 99 or g(t(x), t(y), t(y)) ≤ a {g(x, x, t(y)) + g(y, y, t(x))} (3.18) for all x, y ∈ x, where 0 ≤ a < 1 2 . then t has a unique fixed point (say u) in x and t is g-continuous at u. proof. suppose that t satisfies condition (3.17) for all x, y ∈ x. then replacing y by t(x) in (3.17), we have g(t(x), t2(x), t2(x)) ≤ a { g(x, t2(x), t2(x)) + g(t(x), t(x), t(x)) } ≤ a { g(x, t(x), t(x)) + g(t(x), t2(x), t2(x)) } , by (g5). so, it must be the case that, g(t(x), t2(x), t2(x)) ≤ a 1 − a g(x, t(x), t(x)). put r = a 1−a . then 0 ≤ r < 1 since 0 ≤ a < 1 2 . thus, g(t(x), t2(x), t2(x)) ≤ r g(x, t(x), t(x)). for every x ∈ x, where 0 ≤ r < 1. assume that there exists y ∈ x with y 6= t(y) and inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] = 0. as in the proof of corollary 2, there exists a sequence (xn) in x such that (xn) is g-convergent to y and (t(xn)) is g-convergent to y. now using (3.17), we have g(t(xn), t(y), t(y)) ≤ a {g(xn, t(y), t(y)) + g(y, t(xn), t(xn))} . taking the limit as n → ∞, and using the fact that the function g is continuous on its variables, we have g(y, t(y), t(y)) ≤ a {g(y, t(y), t(y)) + g(y, y, y)} = a g(y, t(y), t(y)), which is a contradiction. hence, if y 6= t(y), then inf [g(x, y, y) + g(x, t(x), t(x)) : x ∈ x] > 0. now applying corollary 1, we obtain a fixed point (say u) of t. the proof using (3.18) is similar. uniqueness of u and g-continuity of t at u are obtained by the same argument used above. received: may 2011. revised: june 2012. 100 s.k.mohanta and srikanta mohanta cubo 14, 3 (2012) references [1] r.chugh, t.kadian, a.rani, and b.e.rhoades, ”property p in g-metric spaces,” fixed point theory and applications, vol. 2010, article id 401684, 12 pages, 2010. [2] b.c.dhage, ”generalised metric spaces and mappings with fixed point,” bulletin of the calcutta mathematical society, vol.84, no. 4, pp. 329-336, 1992. [3] b.c.dhage, ”generalised metric spaces and topological structurei,” analele stiintifice ale universitǎtii ”al.i.cuza” din iasi. serie nouǎ. matematicǎ, vol.46, no. 1, pp. 3-24, 2000. [4] s.gähler, ”2-metrische räume und ihre topologische struktur,” mathematische nachrichten, vol.26, pp. 115-148, 1963. [5] s.gähler, ”zur geometric 2-metrische räume,” revue roumaine de mathématiques pures et appliquées, vol.40, pp. 664-669, 1966. [6] k.s.ha,y.j.cho, and a.white, ”strictly convex and strictly 2-convex 2-normed spaces,” mathematica japonica, vol.33, no. 3, pp. 375-384, 1988. [7] osamu kada, tomonari suzuki and wataru takahashi, ”nonconvex minimization theorems and fixed point theorems in complete metric spaces,” math. japonica , vol.44, no. 2, pp. 381-391, 1996,. [8] sushanta kumar mohanta, ”property p of ćirić operators in g-metric spaces,” international j. of math. sci. and engg. appls., vol. 5, no. ii, pp. 353-367, 2011. [9] z.mustafa and b.sims, ”a new approach to generalized metric spaces,” journal of nonlinear and convex analysis, vol. 7, no. 2, pp. 289-297, 2006. [10] z.mustafa and b.sims, ”fixed point theorems for contractive mappings in complete g-metric spaces,” fixed point theory and applications, vol. 2009, article id 917175, 10 pages, 2009. [11] z.mustafa, h.obiedat, and f. awawdeh, ”some fixed point theorem for mapping on complete g-metric spaces,” fixed point theory and applications, vol. 2008, article id 189870, 12 pages, 2008. [12] z.mustafa, w.shatanawi, and m.bataineh, ”existence of fixed point results in g-metric spaces,” international journal of mathematics and mathematical sciences, vol. 2009, article id 283028, 10 pages, 2009. [13] z.mustafa and b.sims, ”some remarks concerning d-metric spaces,” in proceedings of the international conference on fixed point theory and applications, pp. 189-198, valencia, spain, july 2004. [14] z.mustafa, a new structure for generalized metric spaces-with applications to fixed point theory, ph.d. thesis, the university of newcastle, callaghan, australia, 2005. cubo 14, 3 (2012) a common fixed point theorem in g-metric spaces ... 101 [15] z.mustafa and h. obiedat, ”a fixed points theorem of reich in g-metric spaces,” cubo a mathematics journal, vol. 12, no. 01, pp. 83-93, 2010. [16] z.mustafa, f. awawdeh and w.shatanawi, ”fixed point theorem for expansive mappings in g-metric spaces,” int. j. contemp. math. sciences, vol. 5, no. 50, pp. 2463-2472, 2010. [17] s.v.r.naidu, k.p.r.rao, and n.srinivasa rao, ”on the concept of balls in a d-metric space,” international journal of mathematics and mathematical sciences, no. 1, pp. 133-141, 2005. [18] w.shatanawi,”fixed point theory for contractive mappings satisfying φ-maps in g-metric spaces,” fixed point theory and applications, vol. 2010, article id 181650, 9 pages, 2010. cubo a mathematical journal vol.16, no¯ 02, (111–119). june 2014 on the uniform asymptotic stability to certain first order neutral differential equations cemil tunç department of mathematics, faculty of science, yüzüncü yıl university, 65080, van, turkey cemtunc@yahoo.com abstract in this paper, the uniform asymptotic stability of the zero solution of a kind of neutral differential equations is discussed. based on the lyapunov functional approach, a new stability criterion is derived, which is delay dependent on two positive constants. the result to be obtained here extends and generalizes the existing ones in the literature. resumen en este art́ıculo se discute la estabilidad asintótica uniforme de la solución cero de un tipo de ecuaciones diferenciales neutrales. basados en la técnica de funcional de lyapunov, se deriva un nuevo criterio de estabilidad, el cual es dependiente de atrasos basados en dos constantes positivas. el resultado que se obtiene extiende y generaliza los encontrados en la literatura. keywords and phrases: neutral differential equation; first order, uniform asymptotic stability; lyapunov functional. 2010 ams mathematics subject classification: 34k20, 34k40. 112 cemil tunç cubo 16, 2 (2014) 1 introduction in 2009, nam and phat [4] considered the first order neutral differential equation d dt [x(t) + px(t − τ)] = −ax(t) + b tanh x(t − σ), t ≥ 0, (1.1) where a, τ and σ are positive constants, σ ≥ τ, b and p are real numbers with |p| < 1. using a lyapunov functional, the authors established some sufficient conditions for the zero solution of eq. (1.1) to be uniformly asymptotically stable. by this work, nam and phat [4] established an improved criterion for the uniform asymptotic stability of the zero solution of eq. (1.1). in this paper, instead of eq. (1.1), we consider the first order neutral differential equation of the form d dt [x(t) + px(t − τ)] = −f(x(t))x(t) − g(x(t − τ))x(t) + b tanh x(t − σ), t ≥ 0, (1.2) where f and g are continuous functions on ℜ = (−∞, ∞), τ and σ are positive constants with σ ≥ τ, b and p are real numbers, |p| < 1. for each solution of eq. (1.2), we assume the initial condition x(t) = φ(t), t ∈ [−σ, 0], φ ∈ c([−σ, 0], ℜ). a primary purpose of this paper is to study the uniform asymptotic stability of the zero solution of eq. (1.2). motivated by nam and phat [4], we obtain some sufficient conditions which guarantee the uniform asymptotic stability of the zero solution of eq. (1.2). it is clear that the equation discussed by nam and phat [4], eq. (1.1), is a special case of our equation, eq. (1.2). that is, our equation, eq. (1.2), includes and extends the equation discussed in [4]. by this paper, we generalize and extend the result obtained by nam and phat [4]. finally, in paticular, one can refer to the papers of agarwal and grace [1], park [5], sun and wang [6], tunç [7], tunç and sirma [9] and the references listed in these sources to see some recent contributions focused on the topic of this paper. 2 description of problem before introducing our main result, we give some basic information relative to the topic. lemma 2.1. ([4]). assume that s ∈ ℜn×n is a symmetric positive definite matrix. then for every q ∈ ℜn×n, 2 < qy, x >≤< qs−1qt x, x > + < sy, y >, ∀x, y ∈ ℜn. cubo 16, 2 (2014) on the uniform asymptotic stability to certain first order neutral . . . 113 lemma 2.2. ([2]). for any symmetric positive definite matrix m ∈ ℜn×n, scalar σ ≥ 0 and vector function w : [0, σ] → ℜn such that the integrations concerned are well defined, then ⎛ ⎝ σ∫ 0 w(s)ds ⎞ ⎠ t m ⎛ ⎝ σ∫ 0 w(s)ds ⎞ ⎠ ≤ σ σ∫ 0 wt (s)mw(s)ds. definition ([3]). suppose ω ⊆ ℜ × c is open, f : ω → ℜn, d : ω → ℜn are given continuous functions with d atomic at zero (see hale and lunel [[3], pp.53]. the relation d dt d(t, xt) = f(t, xt) is called the neutral functional differential equation. theorem 2.1. ([3]). suppose d is a stable operator, f : ℜ × c → ℜn, f : ℜ × (bounded sets of c) into bounded sets of ℜn and suppose u(s), v(s), and w(s) are continuous, nonnegative, and non-decreasing with u(s), v(s) > 0 for s ̸= 0, and u(0) = v(0) = 0. if there is a continuous function v : ℜ × c → ℜn such that u(|dφ|) ≤ v(t, φ) ≤ v(|φ|) and v̇(t, φ) ≤ −w(|dφ|), then the solution x = 0 of d dt d(t, xt) = f(t, xt) are uniformly stable. if u(s) → ∞ as s → ∞, then the solutions of d dt d(t, xt) = f(t, xt) are uniformly bounded. if w(s) > 0 for s > 0, then the solution x = 0 of d dt d(t, xt) = f(t, xt) is uniformly asymptotically stable. the same conclusion holds if the upper bound of v̇(t, φ) is given by −w(|φ(0)| . for a real number α, eq. (1.2) can be rewritten in the form d dt [x(t) + px(t − τ) + α t∫ t−τ x(s)ds + b t−τ∫ t−σ tanh x(s)ds] = {α − [f(x(t)) + g(x(t − τ))]}x(t) − αx(t − τ) + b tanh x(t − τ), t ≥ 0. (2.1) define the operators ([4]): d1(xt) = x(t) + px(t − τ) + α t∫ t−τ x(s)ds + b t−τ∫ t−σ tanh x(s)ds and d2(xt) = x(t) + px(t − τ). our main result is the following theorem. 114 cemil tunç cubo 16, 2 (2014) theorem 2.2. in addition to the all the basic assumptions imposed to the functions f and g that appear in eq. (1.2), we assume that there exist a constant α with 0 < |α| < 1 and positive constants β, γ, η, θ, a1 and a2 such that the following conditions hold: f(x) ≥ a1, g(x) ≥ a2, a = a1 + a2, ω = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ω11 ω12 b ω14 ω15 ∗ −β − 2pα pb −α −bα ∗ ∗ θ(σ − τ)2 − η b b2 ∗ ∗ ∗ −γ 0 ∗ ∗ ∗ ∗ −θ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, (2.2) where ω11 = 2(α − a) + β + γτ 2 + η, ω12 = p(α − f(x) − g(x(t − τ))) − α, ω14 = α − f(x(t)) − g(x(t − τ)), ω15 = b{α − f(x(t)) − g(x(t − τ))}, then, the zero solution of eq. (1.2) is uniformly asymptotically stable. remark. here the symbol * represents the elements below the main diagonal of the matrix ω in (4). proof. since ω < 0, there is a positive constant δ such that the following inequality holds: ω1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ω11 ω12 b ω14 ω15 ∗ −β − 2pα pb −α −bα ∗ ∗ θ(σ − τ)2 − η + δ b b2 ∗ ∗ ∗ −γ 0 ∗ ∗ ∗ ∗ −θ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0. we use the lyapunov functional ([4]) defined by v = v1 + v2 + v3 + v4 + v5 + v6 + v7, cubo 16, 2 (2014) on the uniform asymptotic stability to certain first order neutral . . . 115 where v1 = d t 1 (xt)d1(xt), v2 = β t∫ t−τ x2(s)ds, v3 = γτ t∫ t−τ (τ − t + s)(αx(s))2ds, v4 = θ(σ − τ) t−τ∫ t−σ (s − t + σ) tanh2 x(s)ds, v5 = η t∫ t−τ tanh2 x(s)ds, v6 = δ t−τ∫ t−σ tanh2 x(s)ds, (δ is a positive constant), v7 = εd 2 2(xt), ε is a positive constant that will be chosen later. let v0 = v1 +v2 +v3 +v4 +v5 +v6. the time derivative of the functional v0 along solutions of eq. (2.1) is given by d dt v0 = 2 ⎡ ⎣x(t) + px(t − τ) + α t∫ t−τ x(s)ds + b t−τ∫ t−σ tanh x(s)ds ⎤ ⎦ t ×[{α − [f(x(t)) + g(x(t − τ))]}x(t) − αx(t − τ) + b tanh x(t − τ)] +βx2(t) − βx2(t − τ) + γτ2α2x2(t) − γτ t∫ t−τ (αx(s))2ds +θ(σ − τ)2 tanh2 x(t − τ) − θ(σ − τ) t−τ∫ t−σ tanh2 x(s)ds +η tanh2 x(t) − η tanh2 x(t − τ) + δ tanh2 x(t − τ) − δ tanh2 x(t − σ). 116 cemil tunç cubo 16, 2 (2014) using lemma 2.2 and applying the estimate tanh2 x(t) ≤ x2(t), we obtain d dt v0 ≤ 2 ⎡ ⎣x(t) + px(t − τ) + α t∫ t−τ x(s)ds + b t−τ∫ t−σ tanh x(s)ds ⎤ ⎦ t ×[{α − [f(x(t)) + g(x(t − τ))]}x(t) − αx(t − τ) + b tanh x(t − τ)] +βx2(t) − βx2(t − τ) + γτ2α2x2(t) − γ ⎛ ⎝ t∫ t−τ αx(s)ds ⎞ ⎠ 2 +θ(σ − τ)2 tanh2 x(t − τ) − θ ⎛ ⎝ t−τ∫ t−σ tanh x(s)ds ⎞ ⎠ 2 +ηx2(t) − η tanh2 x(t − τ) + δ tanh2 x(t − τ) − δ tanh2 x(t − σ). choosing α t∫ t−τ x(s)ds = u(t), t−τ∫ t−σ tanh x(s)ds = v(t) and using the assumptions f(x) ≥ a1 , g(x) ≥ a2 and the estimate 0 < |α| < 1, it follows that d dt v0 ≤ [2{α − (a1 + a2)} + β + γτ 2 + η]x2(t) +[−2α + 2p(α − f(x(t)) − g(x(t − τ)))]x(t)x(t − τ) +2bx(t) tanh x(t − τ) + 2{α − f(x(t)) − g(x(t − τ))}x(t)u(t) +2b{α − f(x(t)) − g(x(t − τ))}x(t)v(t) −(2pα + β2)x2(t − τ) + 2pbx(t − τ) tanh x(t − τ) − 2αx(t − τ)u(t) −2αbx(t − τ)v(t) + [θ(σ − τ)2 − η + δ] tanh2 x(t − τ) − δ tanh2 x(t − σ) +2b tanh x(t − τ)u(t) + 2b2 tanh x(t − τ)v(t) − γu2(t) − θv2(t) = [2(α − a) + β + γτ2 + η]x2(t) +[−2α + 2p(α − f(x(t)) − g(x(t − τ)))]x(t)x(t − τ) +2bx(t) tanh x(t − τ) + 2(α − f(x(t)) − g(x(t − τ)))x(t)u(t) +2b(α − f(x(t)) − g(x(t − τ)))x(t)v(t) −(2pα + β2)x2(t − τ) + 2pbx(t − τ) tanh x(t − τ) − 2αx(t − τ)u(t) −2αbx(t − τ)v(t) + [θ(σ − τ)2 − η + δ] tanh2 x(t − τ) − δ tanh2 x(t − σ) +2b tanh x(t − τ)u(t) + 2b2 tanh x(t − τ)v(t) − γu2(t) − θv2(t). since the terms on the right hand side of d dt v0 represent a specific quadratic form, we can rearrange the terms given on the right hand side of d dt v0 as the following: d dt v0 ≤ ζ t (t) ω2ζ(t), cubo 16, 2 (2014) on the uniform asymptotic stability to certain first order neutral . . . 117 where ω2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ω11 ω12 b ω14 ω15 0 ∗ −β − 2pα pb −α −bα 0 ∗ ∗ θ(σ − τ)2 − η + δ b b2 0 ∗ ∗ ∗ −γ 0 0 ∗ ∗ ∗ ∗ −θ 0 ∗ ∗ ∗ ∗ ∗ −δ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . ζt (t) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) x(t − τ) tanh x(t − τ) u(t) v(t) tanh x(t − σ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ t , ζ(t) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) x(t − τ) tanh x(t − τ) u(t) v(t) tanh x(t − σ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . making use of the assumption ω < 0, we have d dt v0 ≤ ζ t (t)ω2ζ(t) < 0. therefore, there exists a positive constant λ such that d dt v0 ≤ −λ{∥x(t)∥ 2 + ∥x(t − τ)∥ 2 + ∥tanh x(t − τ)∥ 2 + ∥u(t)∥ 2 + ∥v(t)∥ 2 + ∥tanh x(t − σ)∥ 2 }. by a straightforward calculation for the time derivative of v7, it follows that d dt v7 = 2ε[x(t) + px(t − τ)] [−f(x(t))x(t) − g(x(t − τ))x(t) + b tanh x(t − σ)] = −2εf(x(t))x2(t) − 2εg(x(t − σ))x2(t) + 2bεx(t) tanh x(t − σ) −2εpf(x(t))x(t)x(t − τ) − 2εpg(x(t − τ))x(t)x(t − τ) +2εpbx(t − τ)) tanh x(t − σ). taking into account the assumptions f(x) ≥ a1 , g(x) ≥ a2, a = a1 + a2, and lemma 2.1, we find d dt v7 ≤ −2ε(a1 + a2)x 2(t) + 2bεx(t) tanh x(t − σ) +2εp{f(x(t)) + g(x(t − τ))} |x(t)| |x(t − τ)| +2εpbx(t − τ)) tanh x(t − σ) ≤ −2εax2(t) + 2bεx(t) tanh x(t − σ) + 2εap |x(t)| |x(t − τ)| +2εpbx(t − τ)) tanh x(t − σ) ≤ ε{−2a + |b| 2 + |ap| 2 }x2(t) + ε{1 + |bp| 2 }x2(t − τ) + 2ε tanh2 x(t − σ). 118 cemil tunç cubo 16, 2 (2014) let us choose the constant ε as ε = { 2−1λ min{(1 + |bp| 2 )−1, 2−1}, if − 2a + |b| 2 + |ap| 2 ≤ 0 2−1λ min{(1 + |b| 2 + |ap| 2 )−1, (1 + |bp| 2 )−1, 2−1}, if − 2a + |b| 2 + |ap| 2 > 0. hence, we can obtain d dt v < − λ 2 ∥x(t)∥ 2 . further, we have v ≥ εd2 2 (xt), and d2(xt) is stable due to |p| < 1. the proof of theorem (2.2) is now completed. 3 conclusion a non-linear neutral differential equation of the first order is considered. we establish certain sufficient conditions which guarantee that the zero solution of this equation is uniformly asymptotically stable. in proving our main result, we employ the lyapunov functional approach as a basic tool. this paper has a contribution to the subject in the literature, and it may be useful for researchers work on the qualitative behaviors of solutions. received: december 2012. revised: may 2013. references [1] agarwal, r. p.; grace, s. r., asymptotic stability of certain neutral differential equations, math. comput. modelling, 31 (2000), no. 8-9, 9–15. 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[9] tunç, c.; sirma, a., stability analysis of a class of generalized neutral equations, j. comput. anal. appl., 12 (2010), no. 4, 754–759. c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2601 .dvi cubo a mathematical journal vol.13, no¯ 01, (1–10). march 2011 on strongly α-i-open sets and a new mapping r.devi, a.selvakumar, m.parimala department of mathematics, kongunadu arts and science college, coimbatore 641 029, tamilnadu, india. email: rdevicbe@yahoo.com and s.jafari college of vestsjaelland syd, herrestraede 11,4200 slagelse, denmark. abstract in this paper, we introduce the notion of strongly α-i-open sets in ideal topological spaces and investigate some of their properties. further we study the continuous functions for the above set and derive the some of their properties. resumen en este trabajo, se introduce la noción del gran conjunto α-i-abierto ideal en espacios topológicos y se investigan algunas de sus propiedades. además se estudian las funciones continuas para el conjunto y parte de sus propiedades. keywords: α-i-open set, strongly α-i-open set and bi set. mathematics subject classification: 54a05,54d10,54f65,54g05. 2 r.devi, s.jafari, a.selvakumar and m.parimala cubo 13, 1 (2011) 1 introduction the notion of α-open sets was introduced and investigated by njastad [16]. by using α-open sets. mashhour et al. [14] defined and studied α-continuity and α-openness in topological spaces. ideals in topological spaces have been considered since 1930. this topic has won its importance by the paper of vaidyanathaswamy [19]. in 2002, hatir and noiri [6] have introduced the notion of α-i-continuous functions and used it to obtain a decomposition of continuity. the notion of bi sets introduced by hatir and noiri [6] and provided a decomposition of continuity. in this paper, we introduce strongly α-i-open sets and establish a decomposition of continuity. in 1990, jankovic and hamlett [9] introduced the notion of i-open sets in ideal topological spaces. an ideal is defined as a non-empty collection i of subsets of x satisfying the following two conditions. (1) if a ∈ i and b ⊂ a, then b ∈ i. (2) if a ∈ i and b ∈ i, then a ∪ b ∈ i. an ideal topological space is a topological space (x, τ ) with an ideal i on x and it is denoted by (x, τ, i). for a subset a ⊂ x, a∗(i) = {x ∈ x : u ∩ a /∈ i for each neighbourhood u of x} is called the local function of a with respect to i and τ [9]. we simply write a∗ instead of a∗(i) to be brief. for every ideal topological space (x, τ, i), there exists a topology τ ∗(i), finer than τ , generated by β(i, τ ) = {u − i : u ∈ τ and i ∈ i}, but in general β(i, τ ) is not always a topology [9]. additionally, cl∗(a) = a ∪ a∗ defines a kuratowski closure operator for τ ∗(i). given a space (x, τ, i) and a ⊂ x, a is called i-open if a ⊂ int(a∗) and a subset k is called i-closed if its complement is i-open [8,9]. 2 preliminaries first we will recall some definitions used in sequel. definition 2.1. a subset a of an ideal topological space (x, τ, i) is said to be 1. α-i-open [6] (resp. α-open [16]) if a ⊂ int(cl∗(int(a))) (resp. a ⊂ int(cl(int(a)))), 2. semi-i-open [6] (resp. semi-open [12]) if a ⊂ cl∗(int(a)) (resp. a ⊂ cl(int(a))), 3. pre-i-open [1] (resp. pre-open [13]) if a ⊂ int(cl∗(a)) (resp. a ⊂ int(cl(a))), 4. b-i-open [3] (resp. b-open [2]) if a ⊂ int(cl∗(a)) ∪ cl∗(int(a)) (resp. a ⊂ int(cl(a)) ∪ cl(int(a)), 5. t-i-set [6] (resp. t-set [18]) if int(cl∗(a)) = int(a) (resp. int(cl(a)) = int(a)), 6. bi -set [6] if a = u ∩ v , where u ∈ τ and v is a t-i-set, 7. ci -set [6] if a = u ∩ v , where u ∈ τ and int(cl ∗(int(v ))) = int(v ), 8. ai -set [11] if a = u ∩ v , where u ∈ τ and v = (int(v )) ∗, cubo 13, 1 (2011) on strongly α-i-open sets and a new mapping 3 9. strongly pre-i-open [17] if a is pre-i-open as well as a ci -set, 10. strongly b-i-open [4] if a is b-i-open as well as a ci -set and 11. i-locally closed set [5] if a = u ∩ v , where u ∈ τ and v = v ∗. definition 2.2. a subset a of an ideal topological space (x, τ, i) is said to be i-nowhere dense if int(cl∗(a)) = φ. observe that if a is rare, then a t-i-set (resp. t-set) is i-nowhere dense (resp. nowhere dense). recall that a set a of x is rare if it has no interior points. also notice that if a is rare, then b-i-open sets and b-open sets are pre-i-open and preopen, respectively. definition 2.3. a function f : (x, τ, i) → (y, σ) is said to be 1. semi-i-continuous [6] if for every v ∈ σ, f −1(v ) is semi-i-open, 2. pre-i-continuous [5] if for every v ∈ σ, f −1(v ) is pre-i-open, 3. b-i-continuous [3] if for every v ∈ σ, f −1(v ) is b-i-open, 4. ai -continuous [11] if for every v ∈ σ, f −1(v ) is ai -set, 5. bi -continuous [6] if for every v ∈ σ, f −1(v ) is bi -set. 6. i-locally continuous [5] if for every v ∈ σ, f −1(v ) is i-locally closed, 7. strongly pre-i-continuous [17] if for every v ∈ σ, f −1(v ) is strongly pre-i-open and 8. strongly b-i-continuous [4] if for every v ∈ σ, f −1(v ) is strongly b-i-open. 3 stongly α-i-open sets definition 3.1. a subset a of an ideal space (x, τ, i) is said to be strongly α-i-open set if a is b-i-open as well as a bi -set. the family of all strongly α-i-open sets in (x, τ, i) is denoted by s-αio(x, τ ) or s-αio(x). for a subset a of (x, τ, i), intsα(a)= ⋃ {u ⊂ a, u ∈ s-αio(x, τ )}. clearly τ ⊂ s-αio(x) ⊂ αio(x). the following examples 3.2 and 3.3 show that these inclusions are not reversible. example 3.2. let x = {a, b, c, d}, τ = {x, φ, {a}, {c}, {a, c}} and i = {φ, {a}}. if a = {c}, then a∗ = {b, c, d} and so int(cl∗(int(a))) = int(cl∗({c})) = int({b, c, d}) = {c} = a. therefore a is α-i-open. since x is the only open set containing a, a = x ∩ a is the only possibility to write a as the intersection with x. since int(cl∗(a)) = int(cl∗({c})) = int({b, c, d}) = int(a). this shows that a is a bi -set and hence a is strongly α-i-open set, but a is not i-open. this shows 4 r.devi, s.jafari, a.selvakumar and m.parimala cubo 13, 1 (2011) the existence of non trivial strongly α-i-open sets. example 3.3. let x = {a, b, c, d}, τ = {x, φ, {a}, {c}, {a, c}} and i = {φ, {a}}. if a = {a, b, c}, then a∗ = {a, c, d} and so int(cl∗(int(a))) = int(cl∗({a, c})) = int(x) = x ⊃ a. therefore a is α-i-open. since x is the only open set containing a, a = x ∩ a is the only possibility to write a as the intersection with x. since int(cl∗(a)) = int(cl∗({a, b, c})) = int(x) = x 6= int(a), a is not a bi -set and hence a is not a strongly α-i-open set. the following example shows that α-i-open sets and bi sets are independent concepts. example 3.4. consider the ideal space (x, τ, i) of example 3.3. (a) if a = {a, b, d}, then int(a) = {a} and int(cl∗(int(a))) = int(cl∗({a})) = int({a}) does not contains a. therefore a is not a α-i-open set. but int(cl∗(a)) = int(cl∗({a, b, d})) = int({a, b, d}) = {a} = int(a) and a = x ∩ a. therefore a is a bi -set. (b) if b = {a, b, c}, then int(b) = {a, c} and so int(cl∗(int(b))) = int(cl∗({a, c})) = x ⊃ b. therefore b is α-i-open set. but b is not a bi -set, since int(cl ∗(b)) = int(x) = x 6= int(b). theorem 3.5. every strongly α-i-open set is strongly pre-i-open. proof. it follows from the fact that every α-i-open set is pre-i-open and let a be a bi set. then a = u ∩v , where u ∈ τ and v is a t-i-set. then int(v ) = int(cl∗(v )) ⊃ int(cl∗(int(v ))) ⊃ int(v ) and hence int(v ) = int(cl∗(int(v ))). this shows that a is a ci -set. therefore a is strongly prei-open set. the converse of the above theorem need not be true by the following example. example 3.6. consider r, the set of all real numbers with the usual topology and the ideal if consisting of all finite subsets of r. if a = q, the set of all rational numbers, then a ∗ = r. since int(cl∗(a)) = r ⊃ a, a is pre-i-open. since a = r ∩ a where r is open and int(cl∗(int(a))) = φ = int(a), it follows that a is strongly pre-i-open but a is not strongly α-i-open, since int(cl∗(int(a))) = φ does not contains a. theorem 3.7. every strongly α-i-open set is strongly b-i-open. proof. it follows from theorem 3.5. and [4, theorem 3.7]. the converse of the above theorem need not be true by the following example. example 3.8. let x = {a, b, c}, τ = {x, φ, {a}, {b}, {a, b}} and i = {φ, {b}}. then a = {a, c} is strongly b-i-open, but it is not strongly α-i-open. for int(cl∗(a)) ∪ cl∗(int(a)) = int({a, c}∗ ∪ {a, c}) ∪ cl∗({a}) = {a, c} ⊃ a. therefore, a is b-i-open. since x is the only open set containing a, a = x ∩ a is the only possibility to write a as the intersection with x. since int(cl∗(int(a))) = int(cl∗({a})) = int({a, c}) = int(a) and hence a is strongly b-i-open set. since int(cl∗(int(a))) = int(cl∗({a})) = int({a, c}) = {a} is not contains a. hence a is not cubo 13, 1 (2011) on strongly α-i-open sets and a new mapping 5 strongly α-i-open. proposition 3.9. let (x, τ, i) be an ideal topological space. a subset a of x is i-locally closed set if a is both open and ai -set. proof. let a be an open and ai -set, then a = g ∩ v , where g ∈ τ and v = (int(v )) ∗ = v ∗. this shows that a is i-locally closed set. observe that if v is rare, then a is empty. the following theorem gives a characterization of open sets in terms of strongly α-i-open sets and ai -sets. theorem 3.10. if (x, τ, i) is an ideal topological space. for a subset a of x, the following conditions are equivalent. (a) a is open. (b) a is open, strongly α-i-open and ai -set. (c) a is strongly α-i-open and i-locally closed set. (d) a is strongly pre-i-open and i-locally closed set. (e) a is strongly pre-i-open and ai -set. proof. (a) ⇒ (b) is obvious. (b) ⇒ (c) it follows from proposition 3.9. (c) ⇒ (d) it follows from theorem 3.5. (d) ⇒ (e) if a is i-locally closed, then a = g ∩ a∗ for some open set g. since a ⊂ a∗, by [17, lemma 2.5] a∗ = cl∗(a). now a ⊂ int(cl∗(a)) = int(a∗) and so a∗ ⊂ (int(a∗))∗ ⊂ (a∗)∗ ⊂ a∗. therefore a∗ = (int(a∗))∗ which implies that a is an ai set. (e) ⇒ (a) suppose a is strongly pre-i-open and ai -set. a ⊂ int(cl∗(a)) = int(cl∗(u ∩ v )) where u is open and v = (int(v ))∗. by [10, theorem 2.1.] a ⊂ u ∩ (int(cl∗(v ))) ⊂ u ∩ int(v ∗) ⊂ u ∩ int(int(v ))∗ ⊂ u ∩ int(cl∗(int(v ))) ⊂ u ∩ int(v ) = int(a) 6 r.devi, s.jafari, a.selvakumar and m.parimala cubo 13, 1 (2011) theorem 3.11. let (x, τ, i) be an ideal topological space. a subset a of (x, τ, i) is pre-i-open and bi -set if a is strongly α-i-open. proof. let a be strongly α-i-open set. since every α-i-open set is pre-i-open, then a is pre-iopen and bi -set. theorem 3.12. let (x, τ, i) be an ideal topological space. a subset a of (x, τ, i) is strongly α-i-open if and only if it is semi-i-open, pre-i-open and bi -set. proof. necessity. it follows from the fact that every α-i-open set is semi-i-open and pre-i-open. sufficiency. let a be semi-i-open, pre-i-open and bi -set. then, we have a ⊂ int(cl ∗(a)) ⊂ int(cl∗(cl∗(int(a)))) = int(cl∗(int(a))). this shows that a is α-i-open set and also a is bi -set. therefore a is a strongly α-i-open set. 4 strongly α-i-ccontinuous maps definition 4.1. a mapping f : (x, τ, i) → (y, σ) is said to be strongly α-i-continuous if for every v ∈ σ, f −1(v ) is strongly α-i-open. theorem 4.2. every strongly α-i-continuous map is strongly pre-i-continuous. proof. it follows from theorem 3.5. theorem 4.3. every strongly α-i-continuous map is strongly b-i-continuous. proof. it follows from theorem 3.7. theorem 4.4. let f : (x, τ, i) → (y, σ) be any mapping. then f is i-locally continuous map if it is both continuous and ai -continuous. proof. it follows from proposition 3.9. theorem 4.5. let f : (x, τ, i) → (y, σ) be any mapping. then the following conditions are equivalent. (a) f is continuous. (b) f is continuous, strongly α-i-continuous and ai -continuous. (c) f is strongly α-i-continuous and i-locally continuous. (d) f is strongly pre-i-continuous and i-locally continuous. (e) f is strongly pre-i-continuous and ai -continuous. proof. it follows from theorem 3.10. cubo 13, 1 (2011) on strongly α-i-open sets and a new mapping 7 theorem 4.6. let f : (x, τ, i) → (y, σ) be any mapping. then f is pre-i-continuous and bi -continuous if f is strongly α-i-continuous. proof. it follows from theorem 3.11. theorem 4.7. let f : (x, τ, i) → (y, σ) be any mapping. then f is strongly α-i-continuous if and only if it is semi-i-continuous, pre-i-continuous and bi -continuous. proof. it follows from theorem 3.12. definition 4.8. a mapping f : (x, τ, i) → (y, σ, i) is said to be strongly α-i-irresolute if f −1(v ) is strongly α-i-open in x for every strongly α-i-open set v of y . theorem 4.9. let f : (x, τ, i) → (y, σ) and g : (y, σ) → (z, η) be mappings. then the composition g ◦f : x → z is strongly α-i-continuous if g is continuous and f is strongly α-i-continuous. proof. let w be any open subset of z. since g is continuous, g−1(w ) is open in y . since f is strongly α-i-continuous, then (g ◦ f )−1(w ) = f −1(g−1(w )) is strongly α-i-open in x and hence g ◦ f is strongly α-i-continuous. theorem 4.10. let f : (x, τ, i1) → (y, σ, i2) and g : (y, σ, i2) → (z, η, i3) be mappings. then the composition g ◦ f : x → z is strongly α-i-continuous if g is strongly α-i-continuous and f is strongly α-i-irresolute. proof. let w be any open subset of z. since g is strongly α-i-continuous, g−1(w ) is strongly α-i-open in y . since f is strongly α-i-irresolute, then (g ◦ f )−1(w ) = f −1(g−1(w )) is strongly α-i-open in x and hence g ◦ f is strongly α-i-continuous. theorem 4.11. let f : (x, τ, i1) → (y, σ, i2) and g : (y, σ, i2) → (z, η, i3) be mappings. then the composition g ◦ f : x → z is strongly α-i-irresolute if both f and g are strongly α-i-irresolute. proof. let w be any strongly α-i-open subset of z. since g is strongly α-i-irresolute, g−1(w ) is strongly α-i-open in y . since f is strongly α-i-irresolute, then (g ◦ f )−1(w ) = f −1(g−1(w )) is strongly α-i-open in x and hence g ◦ f is strongly α-i-irresolute. definition 4.12. [15] let a be a subset of a space (x, τ ) then the set ∩{u ∈ τ : a ⊂ u} is called the kernel of a and denoted by ker(a). lemma 4.13. [7] let a be a subset of a space (x, τ ), then (a) x ∈ ker(a) if and only if a ∩ f 6= φ for any closed subset f of x with x ∈ f ; (b) a ⊂ ker(a) and a = ker(a) if a is open in x; (c) if a ⊂ b, then ker(a) ⊂ ker(b). definition 4.14. let n be a subset of a space (x, τ, i) and x ∈ x. then n is called strongly α-i-neighbourhood of x, if there exists a strongly α-i-open set u containing x such that u ⊂ n . 8 r.devi, s.jafari, a.selvakumar and m.parimala cubo 13, 1 (2011) theorem 4.15. the following statements are equivalent for a mapping f : (x, τ, i) → (y, σ). 1. f is strongly α-i-continuous. 2. for each x ∈ x and each open set v in y with f (x) ∈ v , there exists a strongly α-i-open set u containing x such that f (u ) ⊂ v . 3. for each x ∈ x and each open set v in y with f (x) ∈ v , f −1(v ) is a strongly α-ineighbourhood of x. proof. (1) ⇒ (2) let x ∈ x and v be an open set in y such that f (x) ∈ v . since f is strongly α-i-continuous, f −1(v ) is a strongly α-i-open containing x. set u = f −1(v ). then we have f (u ) ⊂ v . (2) ⇒ (3) let v be an open set in y and let f (x) ∈ v . then by (2), there exists a strongly α-i-open set u containing x such that f (u ) ⊂ v . so x ∈ u ⊂ f −1(v ). hence f −1(v ) is a strongly α-i-neighbourhood of x. (3) ⇒ (1) let v be an open set in y and let f (x) ∈ v then by (3), f −1(v ) is a strongly α-ineighbourhood of x. thus for each x ∈ f −1(v ) there exists a strongly α-i-open set ux containing x such that x ∈ ux ⊂ f −1(v ). hence f −1(v ) ⊂ ∪x∈f −1(v )ux so f −1(v ) ∈ s-αio(x). theorem 4.16. the following mappings are equivalent for a mapping f : (x, τ, i) → (y, σ). 1. f is strongly α-i-continuous. 2. for every subset a of x, f (intsαi(a)) ⊂ ker(f (a)). 3. for every subset b of y , intsαi(f −1(b)) ⊂ f −1(ker(b)). proof. (1) ⇒ (2) let a be any subset of x. suppose that y /∈ ker(f (a)). then by lemma 4.13. there exists a closed subset f of y such that y ∈ f and f (a)∩f = φ. thus we have a∩f −1(f ) = φ and (intsα(i(a))) ∩ f −1(f ) = φ. therefore, we obtain f (intsα(i(a))) ∩ f = φ and y /∈ f (intsαi(a)). this implies that f (intsαi(a)) ⊂ ker(a). (2) ⇒ (3) let b be any subset of y by (2) and lemma 4.13., we have f (intsαi(f −1(b))) ⊂ ker(f (f −1(b))) ⊂ ker(b) and intsαi(f −1(b)) ⊂ f −1(ker(b)). (3) ⇒ (1) let v be an open set of y . then by lemma 4.13. and (3), we have intsαi(f −1(v )) ⊂ f −1(ker(v )) = f −1(v ) and intsαi(f −1(v )) = f −1(v ). this implies f −1(v ) is strongly α-i-open. received: april 2009. revised: august 2009. cubo 13, 1 (2011) on strongly α-i-open sets and a new mapping 9 references [1] m.e. abd el-monsef, e.f.lashien and a.a. nasef, on i-open sets and i-continuous mappings, kyungpook mathematical journal, vol. 32, no. 1 (1992), 21-30. 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() cubo a mathematical journal vol.17, no¯ 02, (01–14). june 2015 reproducing inversion formulas for the dunkl-wigner transforms fethi soltani 1 department of mathematics, faculty of science, jazan university, p.o.box 277, jazan 45142, saudi arabia, fethisoltani10@yahoo.com abstract we define and study the fourier-wigner transform associated with the dunkl operators, and we prove for this transform a reproducing inversion formulas and a plancherel formula. next, we introduce and study the extremal functions associated to the dunklwigner transform. resumen definimos y estudiamos la transformada de fourier-wigner asociada a los operadores de dunkl, y probamos una fórmula de inversion y una formula de plancherel para esta transformada. luego introducimos y estudiamos las funciones extramales asociadas a la transformada de dunkl-wigner. keywords and phrases: dunkl transform; dunkl-wigner transform; inversion formulas; extremal functions. 2010 ams mathematics subject classification: 42b10; 44a20; 46f12. 1author partially supported by the dgrst research project lr11es11 and cmcu program 10g/1503 2 fethi soltani cubo 17, 2 (2015) 1 introduction in this paper, we consider rd with the euclidean inner product 〈., .〉 and norm |y| := √ 〈y, y〉. for α ∈ rd\{0}, let σα be the reflection in the hyperplane hα ⊂ rd orthogonal to α: σαy := y − 2〈α, y〉 |α|2 α. a finite set re ⊂ rd\{0} is called a root system, if re ∩r.α = {−α, α} and σα re = re for all α ∈ re. we assume that it is normalized by |α|2 = 2 for all α ∈ re. for a root system re, the reflections σα, α ∈ re, generate a finite group g. the coxeter group g is a subgroup of the orthogonal group o(d). all reflections in g, correspond to suitable pairs of roots. for a given β ∈ rd\ ⋃ α∈re hα, we fix the positive subsystem re+ := {α ∈ re : 〈α, β〉 > 0}. then for each α ∈ re either α ∈ re+ or −α ∈ re+. let k : re → c be a multiplicity function on re (a function which is constant on the orbits under the action of g). as an abbreviation, we introduce the index γ = γk := ∑ α∈re+ k(α). throughout this paper, we will assume that k(α) ≥ 0 for all α ∈ re. moreover, let wk denote the weight function wk(y) := ∏ α∈re+ |〈α, y〉| 2k(α), for all y ∈ rd, which is g-invariant and homogeneous of degree 2γ. let ck be the mehta-type constant given by ck := ( ∫ rd e−|y| 2/2wk(y)dy) −1. we denote by µk the measure on r d given by dµk(y) := ckwk(y)dy; and by l p(µk), 1 ≤ p ≤ ∞, the space of measurable functions f on rd, such that ‖f‖lp(µk) := ( ∫ rd |f(y)|pdµk(y) )1/p < ∞, 1 ≤ p < ∞, ‖f‖l∞(µk) := ess sup y∈rd |f(y)| < ∞, and by l p rad(µk) the subspace of l p(µk) consisting of radial functions. for f ∈ l1(µk) the dunkl transform of f is defined (see [3]) by fk(f)(x) := ∫ rd ek(−ix, y)f(y)dµk(y), x ∈ rd, where ek(−ix, y) denotes the dunkl kernel. (for more details see the next section.) the dunkl translation operators τx, x ∈ rd, [18] are defined on l2(µk) by fk(τxf)(y) = ek(ix, y)fk(f)(y), y ∈ rd. let g ∈ l2rad(µk). the dunkl-wigner transform vg is the mapping defined for f ∈ l2(µk) by vg(f)(x, y) := ∫ rd f(t)τxgk,y(−t)dµk(t), cubo 17, 2 (2015) reproducing inversion formulas for the dunkl-wigner transforms 3 where gk,y(z) := fk ( √ τy|fk(g)|2 ) (z). we study some of its properties, and we prove reproducing inversion formulas for this transform. next, building on the ideas of matsuura et al. [6], saitoh [11, 13] and yamada et al. [20], and using the theory of reproducing kernels [10], we give best approximation of the mapping vg on the sobolev-dunkl spaces hs(µk). more precisely, for all λ > 0, h ∈ l2(µk ⊗ µk), the infimum inf f∈hs(µk) { λ‖f‖2hs(µk) + ‖h − vg(f)‖ 2 l2(µk⊗µk) } , is attained at one function f∗λ,h, called the extremal function, and given by f∗λ,h(y) = ∫ rd ∫ rd ek(iy, z) √ τt|fk(g)|2(z)fk(h(., t))(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(t)dµk(z). in the dunkl setting, the extremal functions are studied in several directions [14, 15, 16, 17]. in the classical case, the fourier-wigner transforms are studied by weyl [21] and wong [22]. in the bessel-kingman hypergroups, these operators are studied by dachraoui [1]. this paper is organized as follows. in section 2, we recall some properties of harmonic analysis for the dunkl operators. next, we define the fourier-wigner transform vg in the dunkl setting, and we have established for it a reproducing inversion formulas. in section 3, we introduce and study the extremal functions associated to the dunkl-wigner transform vg. 2 the dunkl-wigner transform the dunkl operators dj; j = 1, ..., d, on rd associated with the finite reflection group g and multiplicity function k are given, for a function f of class c1 on rd, by djf(y) := ∂ ∂yj f(y) + ∑ α∈re+ k(α)αj f(y) − f(σαy) 〈α, y〉 . for y ∈ rd, the initial problem dju(., y)(x) = yju(x, y), j = 1, ..., d, with u(0, y) = 1 admits a unique analytic solution on rd, which will be denoted by ek(x, y) and called dunkl kernel [2, 4]. this kernel has a unique analytic extension to cd × cd (see [7]). the dunkl kernel has the laplace-type representation [8] ek(x, y) = ∫ rd e〈y,z〉dγx(z), x ∈ rd, y ∈ cd, (2.1) where 〈y, z〉 := ∑d i=1 yizi and γx is a probability measure on r d, such that supp(γx) ⊂ {z ∈ rd : |z| ≤ |x|}. in our case, |ek(ix, y)| ≤ 1, x, y ∈ rd. (2.2) 4 fethi soltani cubo 17, 2 (2015) the dunkl kernel gives rise to an integral transform, which is called dunkl transform on rd, and was introduced by dunkl in [3], where already many basic properties were established. dunkl’s results were completed and extended later by de jeu [4]. the dunkl transform of a function f in l1(µk), is defined by fk(f)(x) := ∫ rd ek(−ix, y)f(y)dµk(y), x ∈ rd. we notice that f0 agrees with the fourier transform f that is given by f(f)(x) := (2π)−d/2 ∫ rd e−i〈x,y〉f(y)dy, x ∈ rd. some of the properties of dunkl transform fk are collected bellow (see [3, 4]). theorem 2.1. (i) l1 − l∞-boundedness. for all f ∈ l1(µk), fk(f) ∈ l∞(µk), and ‖fk(f)‖l∞(µk) ≤ ‖f‖l1(µk). (ii) inversion theorem. let f ∈ l1(µk), such that fk(f) ∈ l1(µk). then f(x) = f(fk(f))(−x), a.e. x ∈ rd. (iii) plancherel theorem. the dunkl transform fk extends uniquely to an isometric isomorphism of l2(µk) onto itself. in particular, we have ‖f‖l2(µk) = ‖fk(f)‖l2(µk). (iv) parseval theorem. for f, g ∈ l2(µk), we have 〈f, g〉l2(µk) = 〈fk(f), fk(g)〉l2(µk). the dunkl transform fk allows us to define a generalized translation operators on l2(µk) by setting fk(τxf)(y) = ek(ix, y)fk(f)(y), y ∈ rd. (2.3) it is the definition of thangavelu and xu given in [18]. it plays the role of the ordinary translation τxf = f(x + .) in r d, since the euclidean fourier transform satisfies f(τxf)(y) = eixyf(f)(y). note that from (2.2) and theorem 2.1 (iii), the definition (2.3) makes sense, and ‖τxf‖l2(µk) ≤ ‖f‖l2(µk), f ∈ l 2 (µk). (2.4) cubo 17, 2 (2015) reproducing inversion formulas for the dunkl-wigner transforms 5 rösler [9] introduced the dunkl translation operators for radial functions. if f are radial functions, f(x) = f(|x|), then τxf(y) = ∫ rd f ( √ |x|2 + |y|2 + 2〈y, z〉 ) dγx(z); x, y ∈ rd, where γx is the representing measure given by (2.1). this formula allows us to establish the following results [18, 19]. proposition 2.2. (i) for all p ∈ [1, 2] and for all x ∈ rd, the dunkl translation τx : lprad(µk) → lp(µk) is a bounded operator, and for f ∈ lprad(µk), we have ‖τxf‖lp(µk) ≤ ‖f‖lprad(µk). (ii) let f ∈ l1rad(µk). then, for all x ∈ rd, we have ∫ rd τxf(y)dµk(y) = ∫ rd f(y)dµk(y). the dunkl convolution product ∗k of two functions f and g in l2(µk) is defined by f ∗k g(x) := ∫ rd τxf(−y)g(y)dµk(y), x ∈ rd. (2.5) we notice that ∗k generalizes the convolution ∗ that is given by f ∗ g(x) := (2π)−d/2 ∫ rd f(x − y)g(y)dy, x ∈ rd. the proposition 2.2 allows us to establish the following properties for the dunkl convolution on rd (see [18]). proposition 2.3. (i) assume that p ∈ [1, 2] and q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r . then the map (f, g) → f ∗k g extends to a continuous map from lprad(µk) × lq(µk) to lr(µk), and ‖f ∗k g‖lr(µk) ≤ ‖f‖lprad(µk)‖g‖lq(µk). (ii) for all f ∈ l1rad(µk) and g ∈ l2(µk), we have fk(f ∗k g) = fk(f) fk(g). (iii) let f ∈ l2rad(µk) and g ∈ l2(µk). then f∗k g belongs to l2(µk) if and only if fk(f)fk(g) belongs to l2(µk), and fk(f ∗k g) = fk(f)fk(g), in the l2(µk) − case. 6 fethi soltani cubo 17, 2 (2015) (iv) let f ∈ l2rad(µk) and g ∈ l2(µk). then ∫ rd |f ∗ g(x)|2dµk(x) = ∫ rd |fk(f)(z)|2|fk(g)(z)|2dµk(z), where both sides are finite or infinite. let g ∈ l2rad(µk) and y ∈ rd. the modulation of g by y is the function gk,y defined by gk,y(z) := fk ( √ τy|fk(g)|2 ) (z), z ∈ rd. thus, ‖gk,y‖l2(µk) = ‖g‖l2rad(µk). (2.6) let g ∈ l2rad(µk). the fourier-wigner transform associated to the dunkl operators, is the mapping vg defined for f ∈ l2(µk) by vg(f)(x, y) := ∫ rd f(t)τxgk,y(−t)dµk(t), x, y ∈ rd. (2.7) proposition 2.4. let (f, g) ∈ l2(µk) × l2rad(µk). (i) vg(f)(x, y) = gk,y ∗k f(x). (ii) vg(f)(x, y) = ∫ rd ek(ix, z)fk(f)(z) √ τy|fk(g)|2(z)dµk(z). (iii) the function vg(f) belongs to l ∞(µk ⊗ µk), and ‖vg(f)‖l∞(µk⊗µk) ≤ ‖f‖l2(µk)‖g‖l2rad(µk). proof. (i) follows from (2.5), (2.7) and the fact that τxgk,y(−t) = τxgk,y(−t). (ii) by theorem 2.1 (iv) and (2.3) we have vg(f)(x, y) = ∫ rd ek(ix, z)fk(f)(z)fk(gk,y)(−z)dµk(z). we obtain the result from the fact that fk(gk,y)(−z) = fk(gk,y)(z) = √ τy|fk(g)|2(z). (iii) follows from (2.7), by using hölder’s inequality, (2.4) and (2.6). ✷ theorem 2.5. let g ∈ l2rad(µk). (i) plancherel formula: for every f ∈ l2(µk), we have ‖vg(f)‖l2(µk⊗µk) = ‖g‖l2rad(µk)‖f‖l2(µk). cubo 17, 2 (2015) reproducing inversion formulas for the dunkl-wigner transforms 7 (ii) parseval formula: for every f, h ∈ l2(µk), we have 〈vg(f), vg(h)〉l2(µk⊗µk) = ‖g‖ 2 l2 rad (µk) 〈f, h〉l2(µk). (iii) inversion formula: for all f ∈ l1 ∩ l2(µk) such that fk(f) ∈ l1(µk), we have f(z) = 1 ‖g‖2 l2 rad (µk) ∫ rd ∫ rd vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y). proof. (i) from theorem 2.1 (iii), proposition 2.2 (ii), proposition 2.3 (iv) and proposition 2.4 (i), we obtain ∫ rd ∫ rd |vg(f)(x, y)| 2dµk(x)dµk(y) = ∫ rd ∫ rd |gk,y ∗k f(x)|2dµk(x)dµk(y) = ∫ rd ∫ rd |fk(gk,y)(z)|2|fk(f)(z)|2dµk(z)dµk(y) = ∫ rd ∫ rd τy|fk(g)|2(z)|fk(f)(z)|2dµk(z)dµk(y) = ‖g‖2 l2 rad (µk) ∫ rd |fk(f)(z)|2dµk(z). (ii) follows from (i) by polarization. (iii) from theorem 2.1 (iv), proposition 2.3 (ii) and (iii), we have ∫ rd ∫ rd vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y) = ∫ rd ∫ rd τy|fk(g)|2(t)fk(f)(t)ek(iz, t)dµk(t)dµk(y). then, by fubini’s theorem, theorem 2.1 (ii) and proposition 2.2 (ii) we deduce that ∫ rd ∫ rd vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y) = ‖g‖2l2 rad (µk) ∫ rd fk(f)(t)ek(iz, t)dµk(t) = ‖g‖2 l2 rad (µk) f(z). ✷ in the following we establish reproducing inversion formula of calderón’s type for the dunklwigner transform on rd. theorem 2.6. let ∆ = ∏d j=1[aj, bj], −∞ < aj < bj < ∞; and let g ∈ l2rad(µk) such that fk(g) ∈ l∞(µk). then, for f ∈ l2(µk), the function f∆ given by f∆(z) = 1 ‖g‖l2 rad (µk) ∫ ∆ ∫ rd vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y), 8 fethi soltani cubo 17, 2 (2015) belongs to l2(µk) and satisfies lim aj→−∞ bj→+∞ ‖f∆ − f‖l2(µk) = 0. (2.8) proof. from theorem 2.1 (iii), proposition 2.3 (iv) and proposition 2.4 (i), we have f∆(z) = 1 ‖g‖2 l2 rad (µk) ∫ ∆ ∫ rd τy|fk(g)|2(t)fk(f)(t)ek(iz, t)dµk(t)dµk(y). by fubini’s theorem we get f∆(z) = ∫ rd k∆(t)fk(f)(t)ek(iz, t)dµk(t). (2.9) where k∆(t) = 1 ‖g‖2 l2 rad (µk) ∫ ∆ τy|fk(g)|2(t)dµk(y). it is easily to see that ‖k∆‖l∞(µk) ≤ 1. on the other hand, by hölder’s inequality, we deduce that |k∆(t)| 2 ≤ µk(∆) ‖g‖4 l2 rad (µk) ∫ ∆ |τy|fk(g)|2(t)|2dµk(y). hence, by (2.4) we find ‖k∆‖2l2(µk) ≤ (µk(∆)) 2 ‖g‖4 l2 rad (µk) ∫ rd |fk(g)(t)|4dµk(t) ≤ (µk(∆)) 2‖fk(g)‖2l∞(µk) ‖g‖2 l2 rad (µk) . thus k∆ ∈ l∞ ∩ l2(µk). therefore and by (2.9) we obtain fk(f∆)(t) = k∆(t)fk(f)(t). from this relation and theorem 2.1 (iii), it follows that f∆ ∈ l2(µk) and ‖f∆ − f‖2l2(µk) = ∫ rd |fk(f)(t)|2(1 − k∆(t))2dµk(t). but by proposition 2.2 (ii) we have lim aj→−∞ bj→+∞ k∆(t) = 1, for all t ∈ rd, and |fk(f)(t)|2(1 − k∆(t))2 ≤ |fk(f)(t)|2, for all t ∈ rd. so, the relation (2.8) follows from the dominated convergence theorem. ✷ cubo 17, 2 (2015) reproducing inversion formulas for the dunkl-wigner transforms 9 3 extremal functions for the mapping vg let s ≥ 0. we define the sobolev-dunkl space of order s, that will be denoted hs(µk), as the set of all f ∈ l2(µk) such that (1 + |z|2)s/2fk(f) ∈ l2(µk). the space hs(µk) provided with the inner product 〈f, g〉hs(µk) = ∫ rd (1 + |z|2)sfk(f)(z)fk(g)(z)dµk(z), and the norm ‖f‖hs(µk) = [∫ rd (1 + |z|2)s|fk(f)(z)|2dµk(z) ]1/2 . the space hs(µk) satisfies the following properties. (a) h0(µk) = l 2(µk). (b) for all s > 0, the space hs(µk) is continuously contained in l 2(µk) and ‖f‖l2(µk) ≤ ‖f‖hs(µk). (c) for all s, t > 0, such that t > s, the space ht(µk) is continuously contained in h s(µk) and ‖f‖hs(µk) ≤ ‖f‖ht(µk). (d) the space hs(µk), s ≥ 0 provided with the inner product 〈., .〉hs(µk) is a hilbert space. remark 3.1. for s > γ + d/2, the function y → (1 + |z|2)−s/2 belongs to l2(µk). hence for all f ∈ hs(µk), we have ‖fk(f)‖l2(µk) ≤ ‖f‖hs(µk), and by hölder’s inequality ‖fk(f)‖l1(µk) ≤ [∫ rd dµk(z) (1 + |z|2)s ]1/2 ‖f‖hs(µk) . then the function fk(f) belongs to l1 ∩ l2(µk), and therefore f(x) = ∫ rd ek(ix, z)fk(f)(z)dµk(z), a.e. x ∈ rd. let λ > 0. we denote by 〈., .〉λ,hs(µk) the inner product defined on the space hs(µk) by 〈f, h〉λ,hs(µk) := λ〈f, h〉hs(µk) + 〈vg(f), vg(h)〉l2(µk⊗µk) , and the norm ‖f‖λ,hs(µk) := √ 〈f, f〉λ,hs(µk) . in the next we suppose that g ∈ l2rad(µk). by theorem 2.5 (ii), the inner product 〈., .〉λ,hs(µk) can be written 〈f, h〉λ,hs(µk) = λ〈f, h〉hs(µk) + ‖g‖ 2 l2 rad (µk) 〈f, h〉l2(µk) . (3.1) theorem 3.2. let λ > 0 and s > γ+d/2 and let g ∈ l2rad(µk). the space (hs(µk), 〈., .〉λ,hs(µk)) has the reproducing kernel ks(x, y) = ∫ rd ek(ix, z)ek(−iy, z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(z), (3.2) 10 fethi soltani cubo 17, 2 (2015) that is (i) for all y ∈ rd, the function x → ks(x, y) belongs to hs(µk). (ii) the reproducing property: for all f ∈ hs(µk) and y ∈ rd, 〈f, ks(., y)〉λ,hs(µk) = f(y). proof. (i) let y ∈ rd. from (2.2), the function φy : z → ek(−iy,z)λ(1+|z|2)s+‖g‖2 l2 rad (µk) belongs to l1 ∩ l2(µk). then, the function ks is well defined and by theorem 2.1 (ii), we have ks(x, y) = f−1k (φy)(x), x ∈ r d. from theorem 2.1 (iii), it follows that ks(., y) belongs to l 2(µk), and we have fk(ks(., y))(z) = ek(−iy, z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) , z ∈ rd. (3.3) then by (2.2), we obtain |fk(ks(., y))(z)| ≤ 1 λ(1 + |z|2)s , and ‖ks(., y)‖2hs(µk) ≤ 1 λ2 ∫ rd dµk(z) (1 + |z|2)s < ∞. this proves that for all y ∈ rd the function ks(., y) belongs to hs(µk). (ii) let f ∈ hs(µk) and y ∈ rd. from (3.1) and (3.3), we have 〈f, ks(., y)〉λ,hs(µk) = ∫ rd ek(iy, z)fk(f)(z)dµk(z), and from remark 3.1, we obtain the reproducing property: 〈f, ks(., y)〉λ,hs(µk) = f(y). this completes the proof of the theorem. ✷ the main result of this subsection can then be stated as follows. theorem 3.3. let s > γ + d/2 and g ∈ l2rad(µk). for any h ∈ l2(µk ⊗ µk) and for any λ > 0, there exists a unique function f∗λ,g, where the infimum inf f∈hs(µk) { λ‖f‖2hs(µk) + ‖h − vg(f)‖ 2 l2(µk⊗µk) } (3.4) is attained. moreover, the extremal function f∗λ,h is given by f∗λ,h(y) = ∫ rd ∫ rd h(x, t)qs(x, y, t)dµk(t)dµk(x), cubo 17, 2 (2015) reproducing inversion formulas for the dunkl-wigner transforms 11 where qs(x, y, t) = ∫ rd ek(−ix, z)ek(iy, z) √ τt|fk(g)|2(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(z). proof. the existence and unicity of the extremal function f∗λ,h satisfying (3.4) is given by kimeldorf and wahba [5], matsuura et al. [6] and saitoh [12]. especially, f∗λ,h is given by the reproducing kernel of hs(µk) with ‖.‖λ,hs(µk) norm as f∗λ,h(y) = 〈h, vg(ks(., y))〉l2(µk⊗µk), (3.5) where ks is the kernel given by (3.2). but by proposition 2.4 (ii) and (3.3), we have vg(ks(., y))(x, t) = ∫ rd ek(ix, z)fk(ks(., y))(z) √ τt|fk(g)|2(z)dµk(z) = ∫ rd ek(ix, z)ek(−iy, z) √ τt|fk(g)|2(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(z). this clearly yields the result. ✷ theorem 3.4. let s > γ + d/2 and g ∈ l2rad(µk). for any h ∈ l2(µk ⊗ µk) and for any λ > 0, we have (i) |f∗λ,h(y)| ≤ ‖h‖ l2(µk⊗µk) 2 √ λ [∫ rd dµk(z) (1 + |z|2)s ]1/2 . (ii) ‖f∗λ,h‖2l2(µk) ≤ 1 4λ ∫ rd ∫ rd |h(x, t)|2e(|x| 2 +|t|2)/2dµk(t)dµk(x). proof. (i) from (3.5) and theorem 2.5 (i), we have |f∗λ,h(y)| ≤ ‖h‖l2(µk⊗µk)‖vg(ks(., y))‖l2(µk⊗µk) ≤ ‖h‖l2(µk⊗µk)‖g‖l2rad(µk)‖ks(., y)‖l2(µk). then, by theorem 2.1 (iii) and (3.3), we deduce that |f∗λ,g(y)| ≤ ‖h‖l2(µk⊗µk)‖g‖l2rad(µk)‖fk(ks(., y))‖l2(µk) ≤ ‖h‖l2(µk⊗µk)‖g‖l2rad(µk) [ ∫ rd dµk(z) [λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) ]2 ]1/2 . using the fact that [ λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) ]2 ≥ 4λ(1 + |z|2)s‖g‖2 l2 rad (µk) , we obtain the result. (ii) we write f∗λ,h(y) = ∫ rd ∫ rd e−(|x| 2 +|t|2)/4e(|x| 2 +|t|2)/4h(x, t)qs(x, y, t)dµk(t)dµk(x). applying hölder’s inequality, we obtain |f∗λ,h(y)| 2 ≤ ∫ rd ∫ rd |h(x, t)|2e(|x| 2 +|t|2)/2 |qs(x, y, t)| 2dµk(t)dµk(x). 12 fethi soltani cubo 17, 2 (2015) thus and from fubini-tonnelli’s theorem, we get ‖f∗λ,h‖2l2(µk) ≤ ∫ rd ∫ rd |h(x, t)|2e(|x| 2 +|t|2)/2‖qs(x, ., t)‖2l2(µk)dµk(t)dµk(x). the function z → ek(−ix,z) √ τt|fk(g)|2(z) λ(1+|z|2)s+‖g‖2 l2 rad (µk) belongs to l1 ∩ l2(µk), then by theorem 2.1 (ii), we get qs(x, y, t) = f−1k ( ek(−ix, z) √ τt|fk(g)|2(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) ) (y). thus, by theorem 2.1 (iii) we deduce that ‖qs(x, ., t)‖2l2(µk) = ∫ rd |fk(qs(x, ., t))(z)|2dµk(z) ≤ ∫ rd τt|fk(g)|2(z)dµk(z) [λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) ]2 . then ‖q(x, ., t)‖2l2(µk) ≤ 1 4λ‖g‖2 l2 rad (µk) ∫ rd τt|fk(g)|2(z)dµk(z) ≤ 1 4λ . from this inequality we deduce the result. ✷ theorem 3.5. let s > γ + d/2 and g ∈ l2rad(µk). for any h ∈ l2(µk ⊗ µk) and for any λ > 0, we have (i) f∗λ,h(y) = ∫ rd ∫ rd ek(iy, z) √ τt|fk(g)|2(z)fk(h(., t))(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(t)dµk(z). (ii) fk(f∗λ,h)(z) = ∫ rd √ τt|fk(g)|2(z)fk(h(., t))(z)dµk(t) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) . (iii) ‖f∗λ,h‖hs(µk) ≤ 1 2 √ λ ‖h‖l2(µk⊗µk). proof. (i) from theorem 3.3 and fubini’s theorem, we have f∗λ,h(y) = ∫ rd ∫ rd ek(iy, z) √ τt|fk(g)|2(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) [∫ rd h(x, t)ek(−ix, z)dµk(x) ] dµk(t)dµk(z) = ∫ rd ∫ rd ek(iy, z) √ τt|fk(g)|2(z)fk(h(., t))(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(t)dµk(z). (ii) the function z → ∫ rd √ τt|fk(g)|2(z)fk(h(., t))(z)dµk(t) λ(1+|z|2)s+‖g‖2 l2 rad (µk) belongs to l1 ∩ l2(µk). then cubo 17, 2 (2015) reproducing inversion formulas for the dunkl-wigner transforms 13 by theorem 2.1 (ii) and (iii), it follows that f∗λ,h belongs to l 2(µk), and fk(f∗λ,h)(z) = ∫ rd √ τt|fk(g)|2(z)fk(h(., t))(z)dµk(t) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) . (iii) from (ii), hölder’s inequality and (2.6) we have |fk(f∗λ,h)(z)|2 ≤ ‖g‖2 l2 rad (µk) [λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) ]2 ∫ rd |fk(h(., t))(z)|2dµk(t). thus, ‖f∗λ,h‖2hs(µk) ≤ ∫ rd (1 + |z|2)s‖g‖2 l2 rad (µk) [λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) ]2 [∫ rd |fk(h(., t))(z)|2dµk(t) ] dµk(z) ≤ 1 4λ ∫ rd [∫ rd |fk(h(., t))(z)|2dµk(t) ] dµk(z) = 1 4λ ‖h‖2l2(µk⊗µk), which ends the proof. ✷ theorem 3.6. let s > γ + d/2 and g ∈ l2rad(µk). for any h ∈ l2(µk ⊗ µk) and for any λ > 0, we have vg(f ∗ λ,h)(x, y) = ∫ rd ∫ rd ek(ix, z) √ τt|fk(g)|2(z)τy|fk(g)|2(z)fk(h(., t))(z) λ(1 + |z|2)s + ‖g‖2 l2 rad (µk) dµk(t)dµk(z). proof. from proposition 2.4 (ii), we have vg(f ∗ λ,h)(x, y) = ∫ rd ek(ix, z)fk(f∗λ,h)(z) √ τy|fk(g)|2(z)dµk(z). then by theorem 3.5 (ii), we obtain the result. ✷ received: january 2015. accepted: april 2015. references [1] a. dachraoui, weyl-bessel transforms, j. comput. appl. math. 133 (2001) 263–276 [2] c.f. dunkl, integral kernels with reflection group invariance, canad. j. math. 43 (1991) 1213– 1227. 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[22] m.w. wong, weyl transforms, universitext, springer, new york, 1998. introduction the dunkl-wigner transform extremal functions for the mapping vg cubo a mathematical journal vol.14, no¯ 01, (111–117). march 2012 more on approximate operators philip j. maher mathematics and statistics group, middlesex university, hendon campus, the burrough, london nw4 4 bt, united kingdom. email: p.maher@mdx.ac.uk and mohammad sal moslehian department of pure mathematics, centre of excellence in analysis on algebraic structures, (ceaas), ferdowsi university of mashhad, p.o. box 1159, mashhad 91775, iran. email: moslehian@ferdowsi.um.ac.ir, moslehian@member.ams.org abstract this note is a continuation of the work on (p, �)–approximate operators studied by mirzavaziri, miura and moslehian. [4]. we investigate approximate partial isometries and approximate generalized inverses. we also prove that if t is an invertible contraction satisfying ‖tt∗t − t‖ < � < 2 3 √ 3 . then there exists a partial isometry v such that ‖t − v‖ < k� for k > 0. resumen esta trabajo es una continuación del trabajo sobre operadores (p, �)–aproximados estudiados por mirzavaziri, miura y moslehian [4]. investigamos isometŕıas parciales 112 philip j. maher and mohammad sal moslehian cubo 14, 1 (2012) aproximadas e inversas aproximadas generalizadas. también probamos que si t es una contracción invertible que satisface ‖tt∗t − t‖ < � < 2 3 √ 3 entonces existe una isometŕıa parcial v tal que ‖t − v‖ < k� para k > 0. keywords and phrases: hilbert space; approximation; unitary; partial isometry; polar decomposition; (p, �)-approximate operator 2010 ams mathematics subject classification: primary 47a55; secondary 39b52. 1 introduction this note is a continuation of the work on (p, �)–approximate operators and operator approximation studied in [4]. mirzavaziri et al investigated (p, �)–approximate (co) isometries and (p, �)– approximate unitaries. for example, a (p, �)–approximate isometry is defined as an operator t in l(h) for which ‖ [t∗t − i] f‖ ≤ �‖f‖p (1.1) where p is a real number and � a fixed positive number. they also proved, for example, the following result on unitary approximation: if to each 0 < � < 1 an operator t in l(h) satisfies ‖t∗t − i‖ ≤ � and ‖tt∗ − i‖ ≤ � there corresponds a unitary operator u such that ‖t − u‖ < �. in section 2 we investigate approximate partial isometries and approximate generalized inverses. in section 3 we investigate operator approximation. we prove (theorem 3.2 below) that an invertible contraction t satisfying ‖tt∗t − t‖ < � < 2 3 √ 3 can be approximated by a partial isometry. recall that a contraction t in l(h) is an operator such that ‖t‖ ≤ 1. recall that the polar decomposition of an operator t says that t can be expressed uniquely as t = u|t |, provided keru = ker|t |, where u is a partial isometry. by definition, a partial isometry u is a isometric on (keru)+; and |t | denotes the positive square root of t∗t. 2 approximate operators in (1.1) (the example of a (p, �)–approximate isometry) there is no question of letting � → 0; for otherwise, the subject would collapse into triviality. for fixed � the upshot of this section is that the (p, �)–approximate operators considered here coincide with their ordinary (exact) counterparts provided p 6= 1. in the cases studied here the operator t we are concerned with must satisfy an operator equation of the form f(t, t∗, t−) = 0 cubo 14, 1 (2012) more on approximate operators 113 (where t− is a generalized inverse of t: see example 2.2 below). our results hinge on the following lemma. lemma 2.1. let p be a real number such that p 6= 1 and let � > 0. if ‖f(t, t∗, t−)‖ ≤ �‖f‖p (2.1) then ‖f(t, t∗, t−)‖ = 0. proof. in (2.1) substitute rf for f where r > 0. then, by the linearity of t, ‖f(t, t∗, t−)‖ ≤ �rp−1‖f‖p. (2.2) if p < 1 so that rp−1 = r−k where k > 0 then �rp−1‖f‖p = �‖f‖p rk → 0 as r → ∞. if p > 1 then �rp−1‖f‖p → 0 as r → 0. example 2.2. (partial isometries). there is the following (equivalent) algebraic definition of a partial isometry: t is a partial isometry if t = tt∗t [3, problem 127, corollary 3]. given a real number p and � > 0, a (p, �)–approximate partial isometry is an operator t in b(h) for which ‖ [tt∗t − t] f‖ ≤ ‖f‖p. let f(t, t∗) = tt∗t − t; if p 6= 1 then, by lemma 2.1, f(t, t∗) = 0 i.e. t is an (exact) partial isometry. counterexample 2.3. this shows that the condition p 6= 1 in lemma 2.1 cannot be dropped. let t =   0 0 √ � 0 √ � 0 √ � 0 0   ∈ m3(c). then, for 1 < � < 2, ‖ [tt∗t − t] f‖ = |� − 1| √ � ‖f‖ < √ � ‖f‖ < � ‖f‖ yet t is not a partial isometry since ‖tf‖ = √ � ‖f‖ for all f in h. a (p, �)–approximate normal partial isometry is an operator t in b(h) for which∥∥[t∗t2 − t] f∥∥ ≤ �‖f‖p (a) and ‖ [ t2t∗ − t ] f‖ ≤ �‖f‖p (b) 114 philip j. maher and mohammad sal moslehian cubo 14, 1 (2012) for given � > 0 and a real number p. let f1(t, t ∗) = t∗t2 − t and f2(t, t ∗) = t2t∗ − t; then if p 6= 1 lemma 2.1 applied to f1 and f2 yields (a) t∗t2 = t and (b) t2t∗ = t. therefore, from (a), t∗t2t∗ = tt∗ and, from (b), t∗t2t∗ = t∗t. thus, t is normal and hence by, say (a), tt∗t = t. example 2.4. (generalized inverses). an operator t− is said to be a generalized inverse of the operator t if tt−t = i. an operator t in b(h) has a generalized inverse if and only if rant is closed [7, p. 261]. for an operator t, with closed range, its moore – penrose inverse t+ has range rant+ = (kert)⊥ and satisfies tt+t = t (i) t+tt+ = t+ (ii) (tt+)∗ = tt+ (iii) (t+t)∗ = t+t (iv)   (mp) and, further, t+ is uniquely determined by these properties. if an operator t− satisfies properties (i), (iii) [(i), (iv)] it will be called a (i), (iii) [(i), (iv)] inverse of t. a (p, �)–approximate generalized inverse of t is an operator t− in b(h) for which ‖[tt−t − t]f‖ ≤ �‖f‖p for � > 0 and real p. let f1(t, t +) = tt+t − t, f2(t, t +) = t+tt+ − t+, f3(t, t +) = (tt+)∗ − tt∗ − t and f4(t, t +) = (t+t)∗ − t+t; then a (p, �)–approximate moore–penrose inverse pf t is an operator t+ in b(h) for which ‖fi(t < t+)f‖ ≤ �‖f‖p for i = 1, . . . , 4 and for � > 0 and real p. let f(t, t−) = tt−t − t; then if p 6= 1, by lemma 2.1, f(t, t−) = 0 i.e. t− is a (exact) generalized inverse of t; and, for p 6= 1, applying lemma 2.1 successively to f1, f2, f3 and f4 yields f1 = f2 = f3 = f4 = 0 i.e. t + satisfies (mp) so that t+ is the (exact) moore – penrose inverse of t. counterexample 2.5. again, we cannot drop the condition p 6= 1 in lemma 2.1. take t = �s where s =  12 12 1 2 1 2   and 0 < � ≤ 1. let t ′ = t. then ‖[tt ′t − t]f‖ = |�3 − �|‖sf‖ ≤ �|�2 − 1|‖s‖‖f‖ = �|�2 − 1|‖f‖ ≤ �‖f‖ yet t ′ = � [ 1 2 1 2 1 2 1 2 ] is not a generalized inverse of t (except, as can be verified, if � = 1) for, e.g., if � = 1 2 then tt ′t = 1 4 t. cubo 14, 1 (2012) more on approximate operators 115 does the algebraic structure of approximate operators mirror that of their exact counterparts? for approximate isometries the answer is “ yes”. the product of two (exact) isometries is an (exact) isometry. the same is true for approximate isometries. proposition 2.6. the product of two (p, �)–approximate isometries is a (p, �′)–approximate isometry. proof. for p 6= 1, by lemma 2.1, a (p, �)–approximate isometry is an (exact) isometry. therefore, we need to prove this result in the case of p = 1. accordingly, let t1 and t2 be two approximate isometries such that ‖[t∗1 t1 − i]f‖ ≤ �1‖f‖ and ‖[t ∗ 2 t2 − i]f‖ ≤ �2‖f‖ for �1 > 0, �2 > 0 for all f in h. assertion: if ‖[t∗t − i]f‖ ≤ �‖f‖ for � > 0 and for all f in h then ‖t‖2 ≤ � + 1. proof of assertion: ‖t∗tf‖ = ‖[t∗t − i]f + f‖ ≤ ‖[t∗t − i]f‖ + ‖f‖ ≤ (� + 1)‖f‖ whence the result ‖t‖2 = ‖t∗t‖ ≤ � + 1 follows by taking supremum over unit vectors. now, ‖[(t1t2)∗(t1t2) − i]f‖ = ‖[t∗2 (t ∗ 1 t1 − i)t2 − i + t ∗ 2 t2]f‖ ≤ ‖t∗2 ‖‖[t ∗ 1 t1 − i]t2f‖ + ‖[t ∗ 2 t2 − i]f‖ ≤ ‖t∗2 ‖�1‖t2f‖ + �2‖f‖ ≤ ((�2 + 1)�1 + �2)‖f‖ = (�1 + �1�2 + �2)‖f‖. we cannot expect a similar result about product of approximate partial isometries since it is not true that the product of two (exact) partial isometries is an (exact) partial isometry. 3 approximating contractions we need the following lemma. lemma 3.1. let t ≤ 0, ‖t‖ ≤ 1 and ‖t3 − t‖ < � < 2 3 √ 3 . then there is a self–adjoint partial isometry s such that ‖t − s‖ < k� for a certain constant k > 0. proof. the conditions t ≤ 0, ‖t‖ ≤ 1 imply that sp(t) ⊆ [−1, 0]. let δ1, δ2(δ1 < δ2) be the solutions of polynomial equation t3 − t = � in [−1, 0]. then |t3 − t| < � for all sp(t), whence t ∈ sp(t) ⊆ [−1, δ1] ∪ [δ2, 0]. 116 philip j. maher and mohammad sal moslehian cubo 14, 1 (2012) t 6 2 3 √ 3 s = t3−t −1√ 3 δ1 δ2 s = � s therefore, ϕ(t) = { −1 t ∈ [−1, δ1] 0 t ∈ [δ2, 0] is a continuous function on sp(t). using the functional calculus, we observe that s = ϕ(t) satisfies s∗ = s and ss∗s = s and ‖t − s‖ = sup t∈sp(t) |ϕ(t) − t| = max{1 + δ1, |δ2|} < k�, for certain k > 0. now we are ready to proof our next result. theorem 3.2. let t be an invertible contraction and let ‖tt∗t − t‖ < � < 2 3 √ 3 . then there exists a partial isometry v such that ‖t − v‖ < k� for a certain constant k > 0. proof. let t = u|t | be the polar decomposition of t. it is known that u is unitary, since t is invertible. then ‖|t |3 − |t |‖ = ‖u|t |t∗t − u|t |‖ = ‖tt∗t − t‖ < � since the operator norm is unitarily invariant in the sense that ‖vxw‖ = ‖x‖ for all arbitrary operators x and all unitaries v, w in b(h). utilizing lemma 3.1 for −|t | we get a self-adjoint partial isometry s such that ‖|t | − s‖ < k� for a certain positive number k. hence ‖t − us‖ = ‖u|t | − us‖ = ‖|t | − s‖ < k� since us(us)∗us = us, the operator us turns into a partial isometry v. if t acts on a finite dimensional hilbert space h, then the partial isometry u appeared in the polar decomposition of t is a unitary. so the proof of theorem 3.2 above follows the following fact. cubo 14, 1 (2012) more on approximate operators 117 corollary 3.3. let a be an m × m contractive matrix such that ‖aa∗a − a‖ < � < 2 3 √ 3 . then there exists a partial isometry v such that ‖a − v‖ < k� for a certain constant k > 0. acknowledgement. the second author was supported by a grant from ferdowsi university of mashhad (no, mp89163mos). received: june 2011. revised: august 2011. references [1] s. aaronson, algorithms for boolean function query properties, siam j. comput. 32 (2003), no. 5, 1140–1157. 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[7] a. e. taylor and d. c. lay, introduction to functional analysis, 2nd ed, john wiley and sons, new york, 1980. cubo a mathematical journal vol.14, no¯ 01, (55–79). march 2012 bounded and periodic solutions of integral equations t. a. burton northwest research institute 732 caroline street, port angeles, wa 98362 email: taburton@olypen.com and bo zhang department of mathematics and computer science fayetteville state university fayetteville, nc 28301 email: bzhang@uncfsu.edu abstract in this paper we introduce a new method for obtaining boundedness of solutions of integral equations. from the integral equation we form an integrodifferential equation by computing x′ + kx to which we apply a liapunov functional. this can be far more effective than the usual technique of differentiating the equation. the qualitative properties derived from that equation are then transferred to a majorizing function for the integral equation. schaefer’s fixed point theorem is used to conclude that there is a periodic solution. three kinds of integral equations are studied and they are shown to be related through two examples. resumen en este art́ıculo presentamos un nuevo método para obtener acotación de soluciones de ecuaciones integrales. a partir de la ecuación integral, formamos una ecuación ı́ntegro56 t. a. burton and bo zhang cubo 14, 1 (2012) diferencial calculando x′ + kx mediante la aplicación de un funcional de liapunov. ello puede resultar bastante más efectivo que la técnica usual de diferenciación de la ecuación. las propiedades cualitativas derivadas de la ecuación son entonces transferidas a la función mayorante para la ecuación integral. el teorema del punto fijo de schaefer es usado para concluir que hay una solución periódica. se estudia tres tipos de ecuaciones integrales y se muestra que ellas están relacionadas a través de dos ejemplos. keywords and phrases: integral equations, boundedness, periodic solutions, liapunov functions. 2010 ams mathematics subject classification: 45d05, 45d20, 45m15. 1 introduction in this paper we consider several nonlinear scalar integral equations of the form x(t) = a(t) − ∫t α(t) c(t, s)g(x(s))ds (1.1) where α(t) may be zero, −∞, or t − h for some constant h > 0. in each of the problems the kernel need not be convex, but the assumption is that there is a constant k > 0 with d(t, s) := ct(t, s) + kc(t, s) (1.2) convex. for existence theory see [3], [5], [15]. in 1928 volterra [17] noted that many physical problems were being modeled by integral and integrodifferential equations with convex kernels. such kernels are natural representations of fading memory. today we see such models in problems in biology, neural networks, viscoelasticity, nuclear reactors, and many other places. see [4]–[8],[11], [14], [17], [19]–[20] for work on integral equations with convex kernels. in addition to the natural fading memory, by 1963 there arose another good reason to try to formulate problems with such kernels. in that same paper volterra had suggested that there might be constructed a liapunov functional which would yield very precise qualitative properties of solutions, and that it would admit arbitrarily large kernels. this is in marked contrast to so much of the theory which leaves the investigator strapped with draconian conditions such as∫t 0 |c(t, s)|ds ≤ γ < 1. in 1963 levin [10] followed volterra’s suggestion and constructed a liapunov functional for the convolution case (see also [13]) and in 1968 [12] he constructed (1.5) below for x′(t) = − ∫t 0 c(t, s)g(x(s))ds, xg(x) > 0 if x 6= 0 (1.3) where c(t, s) is convex: c(t, s) ≥ 0, cs(t, s) ≥ 0, cst(t, s) ≤ 0, ct(t, s) ≤ 0. (1.4) cubo 14, 1 (2012) bounded and periodic solutions of integral equations 57 his functional has the form v(t)=2 ∫x 0 g(s)ds + ∫t 0 cs(t, s) (∫t s g(x(u))du )2 ds + c(t, 0) (∫t 0 g(x(u))du )2 (1.5) with derivative along solutions of (1.3) satisfying v ′(t) = ∫t 0 cst(t, s) (∫t s g(x(u))du )2 ds + ct(t, 0) (∫t 0 g(x(u))du )2 ≤ 0 (1.6) much work followed in that same vein. see zhang [19]–[20] for systems. in 1992 (see [1] and [2]) we constructed parallel liapunov functions for (1.1), taking into account the various forms of α(t). here, also, one finds much work following in the same vein. long before the 1992 work appeared, investigators had differentiated (1.1) to obtain an integrodifferential equation to which they could apply liapunov’s direct method. miller [15] formally starts his chapter 6 with such a presentation. fruitful as that approach has been, it also has significant difficulties. a more refined approach was introduced in chapter 9 of [5] and that has been used with success in a number of subsequent papers, each of which features a new advantage to the technique. in each case, the idea is to form x′ + kx from (1.1). for α = 0 we have x′ + kx = a′(t) + ka(t) − c(t, t)g(x(t)) − ∫t 0 d(t, s)g(x(s))ds (1.7) with d defined in (1.2). there are six important observations. (i) x′ + kx is a uniformly asymptotically stable operator for k > 0. (ii) if c(t, t) ≥ 0 and if xg(x) > 0 for x 6= 0, then x′ + kx + c(t, t)g(x) is an operator of the same, but stronger, type. (iii) if c and ct differ in sign then d(t, s) is smaller than the larger of the two terms. (see [5] and [9].) (iv) under general conditions if c is convex and k is large then d is convex, while the kernel for x′ alone has lost its convexity. (see [6].) (v) if c(t, s) is not convex while d(t, s) is, then the combined equation (1.7) is the right form to apply liapunov functionals. (see example 3.1.) (vi) the utility of a liapunov functional often depends on the separation of its derivative into a difference, say |p(t)| − |h(x)|. using c alone, that can require strong conditions on g, but when using d there is a natural separation [6]. to be fair, one should ask if something has been lost. it has, and it introduces a new problem. the liapunov functional which we constructed in 1992 for (1.1) works with a more general g(t, x), but the levin liapunov functional which we will use on the x′ + kx equation needs g(x). it would be so interesting to extend levin’s liapunov functional to g(t, x) for (1.3). 58 t. a. burton and bo zhang cubo 14, 1 (2012) that is the background and we now move along with some new problems. the first of which is to use a combination of the two liapunov functionals and the systems (1.1) and (1.7). first we prove the existence of a periodic solution when α = ∞. we then study the case of α = 0 proving boundedness properties. finally, we take α = t − h and prove both boundedness and periodicity. let r+ = [0, ∞), r = (−∞, ∞), and c(x, y) denote the space of continuous functions φ : x → y. we also denote by (pt , ‖ · ‖) the banach space of continuous t-periodic functions φ : r → r with the supremum norm. for the existence of periodic solutions, we apply schaefer’s fixed point theorem (see below) with f(x) being the right-hand side of (1.1) so that if f has a fixed point, then this fixed point is a periodic solution of (1.1). theorem 1.1 (schaefer [16]). let (p, ‖ · ‖) be a normed space, f a continuous mapping of p into p which is compact on each bounded subset of p. then either (i) the equation φ = λfφ has a solution for λ = 1, or (ii) the set of all such solutions φ, for 0 < λ < 1, is unbounded. 2 boundedness and periodicity we consider the equation x(t) = λ[a(t) − ∫t −∞ c(t, s)g(x(s))ds], 0 ≤ λ ≤ 1 (2.1) where a : r → r, c : r × r → r, g : r → r are all continuous. suppose that there is a positive constant k so that d(t, s) := ct(t, s) + kc(t, s) is convex. (2.2) we first want to show that there exists a constant γ > 0 such that |x(t)| ≤ γ whenever x is a t-periodic solution of (2.1) for all 0 ≤ λ ≤ 1. we then show the existence of a t-periodic solution of (2.1) for λ = 1 by applying schaefer’s fixed point theorem. our main assumptions are that there is a t > 0 and j > 0 such that a(t + t) = a(t), c(t + t, s + t) = c(t, s) (2.3) for all s ≤ t with a′ continuous and sup 0≤t≤t ∫t −∞ d(t, s)(t − s)ds ≤ j (2.4) and that d satisfies d(t, s) ≥ 0, ds(t, s) ≥ 0, dst(t, s) ≤ 0. (2.5) cubo 14, 1 (2012) bounded and periodic solutions of integral equations 59 we differentiate (2.1) and form x′ + kx = λ[a′ + ka − c(t, t)g(x) − ∫t −∞ d(t, s)g(x(s))ds]. (2.6) now, we define the liapunov functional v(t) = 2 ∫x(t) 0 g(s)ds + λ ∫t −∞ ds(t, s) (∫t s g(x(v))dv )2 ds (2.7) for x ∈ (pt , ‖ · ‖). theorem 2.1. suppose that (2.3), (2.4), (2.5), (2.9) and (2.10) hold. if x(t) is a t-periodic solution of (2.1), then the derivative of v along that solution satisfies v ′(t) ≤ 2λg(x)[a′(t) + ka(t)] − 2g(x)[kx + λc(t, t)g(x)]. (2.8) if there is an l > 0 with xg(x) ≥ 0 for |x| ≥ l (2.9) and if, in addition, there is a µ > 0 with g(x)[kx + c(t, t)g(x)] ≥ µg2(x) (2.10) for |x| ≥ l, then there is an m > 0 with v′(t) ≤ −|g(x)| + m. (2.11) proof. we first define some constants to simplify notation. integrating by parts, we obtain∫t b ds(t, s)(t − s) 2ds = d(t, s)(t − s)2 ∣∣∣t b + 2 ∫t b d(t, s)(t − s)ds = −d(t, b)(t − b)2 + 2 ∫t b d(t, s)(t − s)ds for each b < t. since d(t, s) ≥ 0 and ds(t, s) ≥ 0, letting b → −∞, we see that∫t −∞ds(t, s)(t − s) 2ds + lim s→−∞ d(t, s)(t − s)2 = 2 ∫t −∞d(t, s)(t − s)ds ≤ 2j. (2.12) observe also that d(t, b)(t − b)2 ≤ 2 ∫t b d(t, s)(t − s)ds ≤ 2j 60 t. a. burton and bo zhang cubo 14, 1 (2012) for all b ≤ t. this then implies that d(t, s)(t − s) ≤ 2j/(t − s) for all s < t, and so we arrive at lim s→−∞(t − s)d(t, s) = 0 for fixed t (2.13) and obtain ∫t −∞ ds(t, s)ds = limb→−∞[d(t, t) − d(t, b)] ≤ sup0≤t≤t d(t, t) =: b. (2.14) now let x be a t-periodic solution of (2.1) and v(t) be defined in (2.7). it follows from (2.12) that v(t) is well-defined and t-periodic. we then find v′(t) = 2g(x)x′(t) + λ ∫t −∞ dst(t, s) (∫t s g(x(v))dv )2 ds +2λg(x) ∫t −∞ ds(t, s) ∫t s g(x(v))dvds. integration of the last term by parts and use of (2.13) in the lower limit for that periodic solution yields ∫t −∞ ds(t, s) ∫t s g(x(v))dvds = ∫t −∞ d(t, s)g(x(s))ds. since dst(t, s) ≤ 0, the second term of v′ is not positive, and thus, if we use (2.6), we obtain v′(t) ≤ 2g(x) [ −kx + λ(a′ + ka) − λc(t, t)g(x) − λ ∫t −∞ d(t, s)g(x(s))ds ] +2λg(x) ∫t −∞ d(t, s)g(x(s))ds = 2λg(x)[a′(t) + ka(t)] − 2g(x)[kx + λc(t, t)g(x)] verifying (2.8). next we choose n > 1 so large that −µ(n − 1) < c∗ = min{c(t, t) : 0 ≤ t ≤ t}, where µ > 0 cubo 14, 1 (2012) bounded and periodic solutions of integral equations 61 is defined in (2.10). if |x| ≥ l, then xg(x) ≥ 0, and by (2.10), we obtain for |x(t)| ≥ l that v′(t) ≤ 2λ|g(x)|[‖a′‖ + k‖a‖] − 2g(x)[kx + λc(t, t)g(x)] ≤ 2|g(x)|[‖a′‖ + k‖a‖] − 2k 1 n xg(x) −2λ ( 1 − 1 n ) g(x)[kx + c(t, t)g(x)] − 2λ 1 n c(t, t)g2(x) ≤ 2|g(x)|[‖a′‖ + k‖a‖] − 2k 1 n xg(x) −2λ ( 1 − 1 n ) µg2(x) − 2λ 1 n c(t, t)g2(x) ≤ −2|g(x)| [ 1 n k|x| − (‖a′‖ + k‖a‖) ] . we may assume that l ≥ n(‖a′‖ + k‖a‖ + 1)/k. thus, if |x(t)| ≥ l, then v′(t) ≤ −|g(x(t))|. it is clear that v′(t) ≤ m for 0 ≤ |x(t)| ≤ l, where m = 2gl[‖a′‖ + k‖a‖] + 2gl[kl + c∗gl] with gl = sup{|g(x)| : |x| ≤ l} and c∗ = sup{|c(t, t)| : 0 ≤ t ≤ t}, and hence, v′(t) ≤ −|g(x(t))| + m for all t ≥ 0. this completes the proof. to establish an a priori bound for all possible t-periodic solutions of (2.1), we assume that lim s→−∞(t − s)c(t, s) = 0 and ∫t −∞ |cs(t, s)|(t − s)ds ≤ j1 (2.15) for j1 > 0. theorem 2.2. suppose that (2.3)-(2.5), (2.9)-(2.10) and (2.15) hold. then there exists a constant γ > 0 such that ‖x‖ < γ whenever x is a t-periodic solution of (2.1). proof. let x be a t-periodic solution of (2.1) and v(t) be defined in (2.7). then (2.11) holds. since v(t) is t-periodic, v(t) has a global maximum at q ∈ [0, t] and, hence, at tn = q + nt. so for s ≤ tn, we have 0 ≤ v(tn) − v(s) ≤ − ∫tn s |g(x(v))|dv + m(tn − s). and so ∫tn s |g(x(v))|dv ≤ m(tn − s). 62 t. a. burton and bo zhang cubo 14, 1 (2012) then (x(t) − λa(t))2 has a global maximum at hn := tn + p, where 0 ≤ p ≤ t, and for s ≤ hn we have ∫hn s |g(x(v))|dv ≤ ∫tn+1 s |g(x(v))|dv ≤ m(tn+1 − s). it follows from (2.1) that ( x(hn) − λa(hn) )2 ≤ (∫hn −∞ c(hn, s)g(x(s))ds )2 = ( −c(hn, s) ∫hn s g(x(v))dvds ∣∣∣hn −∞ + ∫hn −∞ cs(hn, s) ∫hn s g(x(v))dvds )2 = (∫hn −∞ cs(hn, s) ∫hn s g(x(v))dvds )2 ≤ (∫hn −∞ |cs(hn, s)| ∫tn+1 s |g(x(v))|dvds )2 ≤ m2 (∫hn −∞ |cs(hn, s)|(hn + t − s)ds )2 . since ∫t −∞ |cs(t, s)|ds is t-periodic, we see from (2.15) that sup 0≤t≤t ∫t −∞ |cs(t, s)|ds ≤ j0 (2.16) for j0 > 0, and hence, ( x(hn) − λa(hn) )2 ≤ m2 (tj0 + j1) 2 . noticing that m is a function of l, we find that |x(hn)| < ‖a‖ + m (tj0 + j1) + 1 := γ. this implies that ‖x‖ < γ whenever x is a t-periodic solution of (2.1) for 0 ≤ λ ≤ 1, and the proof is complete. we now define a mapping f on pt by f(φ)(t) = a(t) − ∫t −∞ c(t, s)g(φ(s))ds for φ ∈ pt . (2.17) cubo 14, 1 (2012) bounded and periodic solutions of integral equations 63 theorem 2.3. if (2.3)-(2.5), (2.9)-(2,10) and (2.15) hold, then (2.1) has a t-periodic solution for λ = 1. proof. it is clear that f(φ) ∈ pt . we show that f is continuous on pt and is compact on each bounded subset of pt . if φ̃, φ ∈ pt , then |f(φ)(t) − f(φ̃)(t)| = ∣∣∣∣ ∫t −∞ c(t, s)g(φ(s))ds − ∫t −∞ c(t, s)g(φ̃(s))ds ∣∣∣∣ (2.18) = ∣∣∣∣ ∫t −∞ cs(t, s) (∫t s g(φ(v))dv − ∫t s g(φ̃(v))dv ) ds ∣∣∣∣ . since g is uniformly continuous on {x ∈ r : |x| ≤ ‖φ̃‖ + 1}, for any ε > 0, there exists 0 < δ < 1 such that ‖φ − φ̃‖ < δ implies |g(φ(s)) − g(φ̃(s))| < ε for all s ∈ [0, t]. it follows from (2.18) that ‖f(φ) − f(φ̃)‖ ≤ j1ε. thus, f is continuous on pt . we now show that f is compact on each bounded subset of pt . let η > 0 and define γ = {f(φ) : φ ∈ pt , ‖φ‖ ≤ η}. (2.19) since d dt f(φ)(t) = a′(t)−c(t, t)g(φ(t)) − ∫t −∞ ct(t, s)g(φ(s))ds = a′(t)−c(t, t)g(φ(t)) − ∫t −∞ d(t, s)g(φ(s))ds + k ∫t −∞ c(t, s)g(φ(s))ds = a′(t)−c(t, t)g(φ(t)) − ∫t −∞d(t, s)g(φ(s))ds + k ∫t −∞cs(t, s) ∫t s g(φ(v))dvds we have∣∣∣∣ ddtf(φ)(t) ∣∣∣∣ ≤ ‖a′‖ + gη sup 0≤t≤t ( |c(t, t)| + ∫t −∞ d(t, s)ds + k ∫t −∞ |cs(t, s)|(t − s)ds ) where gη = {|g(x)| : |x| ≤ η}, and thus, γ is equi-continuous. the uniform boundedness of γ follows from the inequality |f(φ)(t)| ≤ ‖a‖ + ∫t −∞ |cs(t, s)| ∫t s |g(φ(v))|dv ≤ ‖a‖ + j1gη for all φ ∈ γ. so, by the ascoli-arzela theorem, γ lies in a compact subset of pt . by combining schaefer’s theorem with theorem 2.2, we see that f has a fixed point which is a t-periodic solution of (2.1) for λ = 1. this completes the proof. corollary 2.1. suppose that (2.3)-(2.5) hold. if there is an l > 0 and µ > 0 with xg(x) ≥ 0 for |x| ≥ l and c(t, t) ≥ µ, (2.20) 64 t. a. burton and bo zhang cubo 14, 1 (2012) then there is an m > 0 with v′(t) ≤ −|g(x)| + m (2.21) whenever x is a t-periodic solution of (2.1). if, in addition, (2.15) is satisfied, then (2.1) has a t-periodic solution for λ = 1. 3 boundedness we turn now to x(t) = a(t) − ∫t 0 c(t, s)g(x(s))ds (3.1) where a : r+ → r, c : r+ × r+ → r, g : r → r are all continuous with a, a′ bounded. the project here is to show that solutions of (3.1) are bounded. we define d(t, s) := ct(t, s) + kc(t, s) (3.2) for a constant k > 0. our main assumptions are that d(t, s) is convex for t ≥ s ≥ 0 (3.3) and there exists b > 0 with c(t, t) ≥ −b and ∫t 0 d(t, s)(t − s)ds ≤ b (3.4) for all t ≥ 0. we differentiate (3.1) and form x′ + kx = [a′ + ka − c(t, t)g(x) − ∫t 0 d(t, s)g(x(s))ds]. (3.5) now, we define the liapunov functional v(t) = 2g(x(t))+ ∫t 0 ds(t, s) (∫t s g(x(u))du )2 ds+d(t, 0) (∫t 0 g(x(u))du )2 (3.6) for x ∈ c(r+, r), where g(x) = ∫x 0 g(s)ds. theorem 3.1. suppose that d(t, s) is convex and c(t, t) ≥ −b for a constant b > 0. if x(t) is a solution of (3.1), then the derivative of v along that solution satisfies v ′(t) ≤ 2g(x)[a′(t) + ka(t)] − 2g(x)[kx + c(t, t)g(x)]. (3.7) if there is an l > 0 with xg(x) ≥ 0 for |x| ≥ l (3.8) cubo 14, 1 (2012) bounded and periodic solutions of integral equations 65 and if, in addition, there is a µ > 0 with g(x)[kx + c(t, t)g(x)] ≥ µg2(x) (3.9) for |x| ≥ l, then there is an m > 0 with v′(t) ≤ −|g(x)| + m. (3.10) proof. we first observe that if x is a solution of (3.1), then x is also a solution of (3.5). now let x be a solution (3.1) and v(t) be defined in (3.6). we then find v′(t) = 2g(x)x′(t) + ∫t 0 dst(t, s) (∫t s g(x(u))du )2 ds +2g(x(t)) ∫t 0 ds(t, s) ∫t s g(x(u))duds + dt(t, 0) (∫t 0 g(x(u))du )2 +2d(t, 0)g(x(t)) ∫t 0 g(x(u))du. integrate the third to last term by parts to obtain 2g(x(t)) ∫t 0 ds(t, s) ∫t s g(x(u))duds = 2g(x(t)) [ d(t, s) ∫t s g(x(u))du ∣∣∣s=t s=0 + ∫t 0 d(t, s)g(x(s))ds ] = 2g(x(t)) [ −d(t, 0) ∫t 0 g(x(s))ds + ∫t 0 d(t, s)g(x(s))ds ] . (3.11) cancel terms, use the sign conditions, and use (3.5) in the process to unite the liapunov functional and the equation to obtain v′(t) ≤ 2g(x) [ −kx + (a′ + ka) − c(t, t)g(x) − ∫t 0 d(t, s)g(x(s))ds ] +2g(x) ∫t 0 d(t, s)g(x(s))ds = 2g(x)[a′(t) + ka(t)] − 2g(x)[kx + c(t, t)g(x)] verifying (3.7). now we assume that (3.8) and (3.9) hold. we may choose n > 1 so large that µ(n − 1) > b, where b and µ are defined in (3.4) and (3.9), respectively. if |x| ≥ l, then xg(x) ≥ 0, and by (3.9), 66 t. a. burton and bo zhang cubo 14, 1 (2012) we obtain for |x(t)| ≥ l that v′(t) ≤ 2|g(x)|[|a′| + k|a|] − 2g(x)[kx + c(t, t)g(x)] = 2|g(x)|[|a′| + k|a|] − 2k 1 n xg(x) −2 ( 1 − 1 n ) g(x)[kx + c(t, t)g(x)] − 2 1 n c(t, t)g2(x) ≤ 2|g(x)|[|a′| + k|a|] − 2k 1 n xg(x) −2 ( 1 − 1 n ) µg2(x) + 2 1 n bg2(x) ≤ −2|g(x)| [ 1 n k|x| − (|a′(t)| + k|a(t)|) ] . we may assume that l ≥ n[ sup t≥0 (|a′(t)| + k|a(t)|) + 1 ] / k. thus, if |x(t)| ≥ l, then v′(t) ≤ −|g(x(t))|. since −c(t, t) ≤ b, it is clear that v′(t) ≤ m for 0 ≤ |x(t)| ≤ l, where the constant m > 0 is a function of l, and hence, v′(t) ≤ −|g(x(t))| + m for all t ≥ 0. this completes the proof. relations related to (3.9) and (3.10) are found in [4]. to establish the boundedness of solutions, we assume that there is a b1 > 0 with |c(t, 0)| t ≤ b1 and ∫t 0 |cs(t, s)|(t − s + 1)ds ≤ b1 (3.12) for t ≥ 0. we also observe that∫t 0 ds(t, s)(t − s) 2ds = d(t, s)(t − s)2 ∣∣∣t 0 + 2 ∫t 0 d(t, s)(t − s)ds = −d(t, 0) t2 + 2 ∫t 0 d(t, s)(t − s)ds. by (3.4), we now have∫t 0 ds(t, s)(t − s) 2ds + d(t, 0) t2 = 2 ∫t 0 d(t, s)(t − s)ds ≤ 2b (3.13) for all t ≥ 0. cubo 14, 1 (2012) bounded and periodic solutions of integral equations 67 theorem 3.2. if (3.3)-(3.4), (3.8)-(3.9), and (3.12) hold, then any solution of (3.1) is bounded. proof. let x be a solution of (3.1) and v(t) be defined in (3.6). then v(t) is bounded below and satisfies (3.10). we now show that v(t) is bounded above. if v(t) is unbounded, then there exists a sequence {tn} ↑ ∞ with v(tn) → ∞ as n → ∞ and v(tn) ≥ v(s) for 0 ≤ s ≤ tn. it then follows from (3.10) that 0 ≤ v(tn) − v(s) ≤ − ∫tn s |g(x(u))|du + m(tn − s). this implies that ∫tn s |g(x(u))|du ≤ m(tn − s). (3.14) applying (3.14) to v(tn) and taking into account (3.13), we find that v(tn) ≤ 2g(x(tn)) + m2 [∫tn 0 ds(tn, s)(tn − s) 2ds + d(tn, 0) t 2 ] ≤ 2g(x(tn)) + 2bm2. (3.15) we now use (3.1), (3.12), and (3.14) to obtain (x(tn) − a(tn)) 2 = (∫tn 0 c(tn, s)g(x(s))ds )2 = ( −c(tn, s) ∫tn s g(x(u))du ∣∣∣s=tn s=0 + ∫tn 0 cs(tn, s) ∫tn s g(x(u))duds )2 = ( c(tn, 0) ∫tn 0 g(x(s))ds + ∫tn 0 cs(tn, s) ∫tn s g(x(u))duds )2 ≤ ( |c(tn, 0)|m tn + m ∫tn 0 |cs(tn, s)|(tn − s)ds )2 ≤ m2(2b1)2. this implies that |x(tn)| ≤ sup s≥0 |a(s)| + 2b1m := b2 (3.16) and that |g(x(tn))| ≤ b3 for a b3 > 0. we now find that v(tn) ≤ 2g(x(tn)) + 2bm2 ≤ 2b3 + 2bm2 := b4, 68 t. a. burton and bo zhang cubo 14, 1 (2012) a contradiction. thus, v(t) is bounded. in fact, we have 2g(x(t)) ≤ v(t) ≤ 2g(x(0)) + b4 (3.17) and hence |v(t)| ≤ k for all t ≥ 0 (3.18) where k := 2|η| + 2|g(x(0))| + b4 where η = inf{g(u) : u ∈ r}. we also observe that |c(t, 0)| ≤ b5 for a b5 > 0 whenever (3.12) holds. we now integrate (3.10) from s to t and use (3.18) to obtain∫t s |g(x(u))|du ≤ v(s) − v(t) + m(t − s) ≤ 2k + m(t − s). (3.19) applying (3.19) to (3.1) we find |x(t)| ≤ |a(t)| + ∣∣∣∣ ∫t 0 c(t, s)g(x(s))ds ∣∣∣∣ ≤ |a(t)| + ∣∣∣∣c(t, 0) ∫t 0 g(x(s))ds + ∫t 0 cs(t, s) ∫t s g(x(u))duds ∣∣∣∣ ≤ |a(t)| + |c(t, 0)|(mt + 2k) + ∫t 0 |cs(t, s)|[m(t − s) + 2k]ds ≤ sup s≥0 |a(s)| + b1m + 2kb5 + b1(m + 2k). this implies that x is bounded. the proof is complete. corollary 3.1. suppose that (3.3)-(3.4) hold. if there is an l > 0 and µ > 0 with xg(x) ≥ 0 for |x| ≥ l and c(t, t) ≥ µ, (3.20) then there is an m > 0 with v′(t) ≤ −|g(x)| + m (3.21) whenever x is a solution of (3.1). if, in addition, (3.12) holds, then v(t) satisfies (3.17) and any solution of (3.1) is bounded. remark 3.1. inequalities related to (3.17) and (3.21) are of fundamental importance in the study of boundedness and periodic solutions in differential equations by liapunov’s direct method (see burton [3] and yoshizawa [18]). not only are these practical inequalities with many applications, but such combined relations are directly linked to the right-hand side of the equations, and hence, cubo 14, 1 (2012) bounded and periodic solutions of integral equations 69 much of the qualitative properties of solutions can be derived by taking full advantage of the liapunov functions. the following example shows that if c(t, s) is not convex while d(t, s) is, then the combined equation (1.7) is the right form to apply liapunov functionals. example 3.1. we consider the equation x(t) = a(t) − ∫t 0 c(t, s)g(x(s))ds (3.22) where a : r+ → r and g : r → r are continuous with a, a′ bounded, and c(t, s) = c(t − s) = −e−(t−s+3) 2 for t ≥ s ≥ 0. it is clear that c(t, s) is not convex (even not positive). if we choose k = 4, then d(t, s) = ct(t, s) + kc(t, s) = 2(t − s + 1)e −(t−s+3)2. a straightforward calculation shows that d(t, s) is convex and (3.4) holds. we also see that c(t, s) satisfies (3.12). thus, if there exist constants l > 0 and µ > 0 with xg(x) ≥ 0 for |x| ≥ l and g(x)[kx + c(t, t)g(x)] = g(x)[4x − e−9g(x)] ≥ µg2(x) for |x| ≥ l, then any solution of (3.22) is bounded by theorem 3.2. 4 a truncated equation and unification we consider the finite delay equation x(t) = a(t) − ∫t t−h c(t, s)g(x(s))ds (4.1) in which h > 0 is a constant, a : r+ → r, c : r+ × [−h, ∞) → r, g : r → r are all continuous with a, a′ bounded. we write d(t, s) := ct(t, s) + kc(t, s) (4.2) for a positive constant k and assume that d(t, s) is convex: d(t, s) ≥ 0, ds(t, s) ≥ 0, dst(t, s) ≤ 0, dt(t, s) ≤ 0 (4.3) for t ≥ s ≥ −h and that c(t, t − h) = 0, ct(t, t − h) = 0, c(t, t) ≥ −b (4.4) 70 t. a. burton and bo zhang cubo 14, 1 (2012) for all t ≥ 0 and a constant b > 0, where ct(t, t−h) is the partial derivative of c(t, s) with respect the first variable for s = t − h. before we get too far into the work, it is interesting to point out classical forms for c. let c(t, s) = c(t − s) = (−1)n(t − s − h)n, n > 2. not only does it satisfy (4.4), but it is a convex kernel for 0 ≤ s ≤ t ≤ h. moreover, if we let c(t) = 0 for t > h then that kernel will satisfy our work in both sections 2 and 3. in section 3 something very interesting happens. in the linear case we have x(t) = a(t) − ∫t 0 c(t − s)x(s)ds, an equation about which there is a very straightforward theory. however, for t ≥ h it becomes x(t) = a(t) − ∫t t−h c(t − s)x(s)ds and that belongs to a class of far more complex structure. we differentiate (4.1) and take into account (4.4) to form x′ + kx = [ a′ + ka − c(t, t)g(x) − ∫t t−h d(t, s)g(x(s))ds ] . (4.5) now, we define the liapunov functional v(t) = 2g(x(t)) + ∫t t−h ds(t, s) (∫t s g(x(u))du )2 ds (4.6) for x ∈ c([−h, ∞), r), where g(x) = ∫x 0 g(s)ds. theorem 4.1. suppose that (4.3) and (4.4) hold. if x(t) is a solution of (4.1), then the derivative of v along that solution satisfies v ′(t) ≤ 2g(x)[a′(t) + ka(t)] − 2g(x)[kx + c(t, t)g(x)]. (4.7) if there is an l > 0 with xg(x) ≥ 0 for |x| ≥ l (4.8) and if, in addition, there is a µ > 0 with g(x)[kx + c(t, t)g(x)] ≥ µg2(x) (4.9) for |x| ≥ l, then there is an m > 0 with v′(t) ≤ −|g(x)| + m (4.10) cubo 14, 1 (2012) bounded and periodic solutions of integral equations 71 for all t ≥ 0. proof. we first observe that if x is a solution of (4.1), then x is also a solution of (4.5). now let x be a solution (4.1) and v(t) be defined in (4.6). we then find that v′(t) = 2g(x)x′(t) + ∫t t−h dst(t, s) (∫t s g(x(u))du )2 ds −ds(t, t − h) (∫t t−h g(x(u))du )2 + 2g(x(t)) ∫t t−h ds(t, s) ∫t s g(x(u))duds. integration of the last term by parts and use of (4.4) yield 2g(x(t)) ∫t t−h ds(t, s) ∫t s g(x(u))duds = 2g(x(t)) [ d(t, s) ∫t s g(x(u))du ∣∣∣s=t s=t−h + ∫t t−h d(t, s)g(x(s))ds ] = 2g(x(t)) ∫t t−h d(t, s)g(x(s))ds. (4.11) since dst(t, s) ≤ 0 and ds(t, t − h) ≥ 0, the middle two terms of v′ are not positive, and if we use (4.5) and (4.11), we obtain v′(t) ≤ 2g(x) [ −kx + (a′ + ka) − c(t, t)g(x) − ∫t t−h d(t, s)g(x(s))ds ] +2g(x) ∫t t−h d(t, s)g(x(s))ds = 2g(x)[a′(t) + ka(t)] − 2g(x)[kx + c(t, t)g(x)] verifying (4.7). now we assume that (4.8) and (4.9) hold. we may choose n > 1 so large that µ(n−1) > −b, where b and µ are defined in (4.4) and (4.9), respectively. if |x| ≥ l, then xg(x) ≥ 0, and by (4.9), 72 t. a. burton and bo zhang cubo 14, 1 (2012) we obtain for |x(t)| ≥ l that v′(t) ≤ 2|g(x)|[|a′| + k|a|] − 2g(x)[kx + c(t, t)g(x)] = 2|g(x)|[|a′| + k|a|] − 2k 1 n xg(x) −2 ( 1 − 1 n ) g(x)[kx + c(t, t)g(x)] − 2 1 n c(t, t)g2(x) ≤ 2|g(x)|[|a′| + k|a|] − 2k 1 n xg(x) −2 ( 1 − 1 n ) µg2(x) + 2 1 n bg2(x) ≤ −2|g(x)| [ 1 n k|x| − (|a′(t)| + k|a(t)|) ] . we may assume that l ≥ n[ sup t≥0 (|a′(t)| + k|a(t)|) + 1 ] / k. thus, if |x(t)| ≥ l, then v′(t) ≤ −|g(x(t))|. since −c(t, t) ≤ b, it is clear that v′(t) ≤ m for 0 ≤ |x(t)| ≤ l, where the constant m > 0 is a function of l, and hence, v′(t) ≤ −|g(x(t))| + m for all t ≥ 0. this completes the proof. to establish boundedness of solutions, we assume that there is a b1 > 0 with d(t, t) ≤ b1 and ∫t t−h |cs(t, s)|ds ≤ b1 (4.12) for t ≥ 0. we then see that∫t t−h ds(t, s)ds = d(t, t) ≤ b1 and |c(t, t)| ≤ ∫t t−h |cs(t, s)|ds ≤ b1. theorem 4.2. if (4.3)-(4.4), (4.8)-(4.9), and (4.12) hold, then any solution of (4.1) is bounded. proof. let x be a solution of (4.1) and v(t) be defined in (4.6). then v(t) is bounded below and satisfies (4.10). we now show that v(t) is bounded above. if v(t) is unbounded, then there exists a sequence {tn} ↑ ∞ with v(tn) → ∞ as n → ∞ and v(tn) ≥ v(s) for 0 ≤ s ≤ tn. cubo 14, 1 (2012) bounded and periodic solutions of integral equations 73 it then follows from (4.10) that 0 ≤ v(tn) − v(s) ≤ − ∫tn s |g(x(u))|du + m(tn − s). this implies that ∫tn s |g(x(u))|du ≤ m(tn − s) (4.13) and, in particular, that ∫tn s |g(x(u))|du ≤ m(tn − s) ≤ hm (4.14) for all tn − h ≤ s ≤ tn. applying (4.14) to (4.1), we see that (x(tn)−a(tn)) 2 = (∫tn tn−h c(tn, s)g(x(s))ds )2 = ( −c(tn, s) ∫tn s g(x(s))ds ∣∣∣s=tn s=tn−h + ∫tn tn−h cs(tn, s) ∫tn s g(x(u))duds )2 = (∫tn tn−h cs(tn, s) ∫tn s g(x(u))duds )2 ≤ ( hm ∫tn tn−h |cs(tn, s)|ds )2 ≤ h2b21m 2. this implies that |x(tn)| ≤ sup s≥0 |a(s)| + hb1m := b2 (4.15) and that |g(x(tn))| ≤ b3 for a b3 > 0. we now arrive at v(tn) = 2g(x(tn)) + ∫tn tn−h ds(tn, s) (∫tn s g(x(u))du )2 ds ≤ 2g(x(tn)) + h2m2 ∫tn tn−h ds(tn, s)ds ≤ 2b3 + b1h2m2 := b4, (4.16) a contradiction. thus, v(t) is bounded. in fact, we have 2g(x(t)) ≤ v(t) ≤ max{v(0), b4} and hence |v(t)| ≤ k for all t ≥ 0 (4.17) 74 t. a. burton and bo zhang cubo 14, 1 (2012) where k := 2|η| + 2|g(x(0))| + b4 with η = inf{g(u) : u ∈ r}. we now integrate (4.10) from s to t and use (4.17) to obtain∫t s |g(x(u))|du ≤ v(s) − v(t) + m(t − s) ≤ 2k + m(t − s) and hence ∫t s |g(x(u))|du ≤ 2k + hm for tn − h ≤ s ≤ tn. (4.18) applying (4.18) to (4.1) we find that |x(t)| ≤ |a(t)| + ∣∣∣∣ ∫t t−h c(t, s)g(x(s))ds ∣∣∣∣ ≤ |a(t)| + ∣∣∣∣c(t, t − h) ∫t t−h g(x(s))ds+ ∫t t−h cs(t, s) ∫t s g(x(u))duds ∣∣∣∣ ≤ |a(t)| + ∫t t−h |cs(t, s)|[2k + hm]ds ≤ sup s≥0 |a(s)| + b1(2k + hm). (4.19) this implies that x is bounded. the proof is complete. we now consider the existence of periodic solutions of (4.1). we assume that a : r → r, c : r × r → r, and g : r → r are continuous and that there is a t > 0 with a(t + t) = a(t), c(t + t, s + t) = c(t, s) (4.20) for all t ≥ s. if (4.20) holds, then c(t, t) and ∫t t−h |cs(t, s)|ds are t-periodic, and so there are b and b1 with c(t, t) ≥ −b, ∫t t−h |cs(t, s)|ds ≤ b1; then part of (4.4) and (4.12) are satisfied. we define a companion of (4.1) by x(t) = λ [ a(t) − ∫t t−h c(t, s)g(x(s))ds ] , 0 ≤ λ ≤ 1 (4.21) for t ∈ r and form a differential equation x′ + kx = λ [ a′ + ka − c(t, t)g(x) − ∫t t−h d(t, s)g(x(s))ds ] . (4.22) cubo 14, 1 (2012) bounded and periodic solutions of integral equations 75 to obtain an a priori bound for all t-periodic solutions of (4.21), we define v1(t) = 2g(x(t)) + λ ∫t t−h ds(t, s) (∫t s g(x(u))du )2 ds (4.23) for t ∈ r and x ∈ (pt , ‖ · ‖). theorem 4.3. if (4.3)-(4.4), (4.8)-(4.9), and (4.20) hold for t ≥ s, then (4.1) has a t-periodic solution. proof. let x be a t-periodic solution of (4.21) and v1(t) be defined in (4.23). then we have v′1(t) ≤ −|g(x)| + m (4.24) for t ≥ 0 and for an m > 0 independent of x and λ. since v1(t) is t-periodic, v1(t) has a global maximum at q ∈ [0, t], and hence, at tn = q + nt. we then have 0 ≤ v1(tn) − v1(s) ≤ − ∫tn s |g(x(u))|du + m(tn − s) for all s ≤ tn. an argument similar to that of (4.13)-(4.16) shows that v1(tn) ≤ b4 with b4 defined just after (4.16). observing that v1(0) ≤ v1(tn) ≤ b4, we see that |v1(t)| ≤ k with k = 2|η| + b4, where η = inf{g(u) : u ∈ r}. we then follow the argument in (4.19) to arrive at |x(t)| < sup s≥0 |a(s)| + b1(2k + hm)) + 1 := b ∗ (4.25) for all t ∈ r. this implies that ‖x‖ < b∗ whenever x is a t-periodic solution of (4.21) for 0 ≤ λ ≤ 1. define a mapping f on pt by f(φ)(t) = a(t) − ∫t t−h c(t, s)g(φ(s))ds (4.26) for each φ ∈ pt . it is clear that f(φ) ∈ pt . we will show that f is continuous on pt and is compact on each bounded subset of pt . if φ̃, φ ∈ pt , then |f(φ)(t) − f(φ̃)(t)| = ∣∣∣∣ ∫t t−h c(t, s)g(φ(s))ds − ∫t t−h c(t, s)g(φ̃(s))ds ∣∣∣∣ (4.27) = ∣∣∣∣ ∫t t−h cs(t, s) (∫t s g(φ(v))dv − ∫t s g(φ̃(v))dv ) ds ∣∣∣∣ . 76 t. a. burton and bo zhang cubo 14, 1 (2012) since g is uniformly continuous on {x ∈ r : |x| ≤ ‖φ̃‖+1}, then for any ε > 0, there exists 0 < δ < 1 such that ‖φ − φ̃‖ < δ implies |g(φ(s)) − g(φ̃(s))| < ε for all s ∈ [0, t]. it follows from (4.27) that ‖f(φ) − f(φ̃)‖ ≤ hb1ε. thus, f is continuous on pt . we now show that f is compact on each bounded subset of pt . let η > 0 and define γ = {f(φ) : φ ∈ pt , ‖φ‖ ≤ η}. (4.28) observe that d dt f(φ)(t) = a′(t) − c(t, t)g(φ(t)) − ∫t t−h ct(t, s)g(φ(s))ds = a′(t)−c(t, t)g(φ(t))− ∫t t−h d(t, s)g(φ(s))ds + k ∫t t−h c(t, s)g(φ(s))ds = a′(t)−c(t, t)g(φ(t))− ∫t t−h d(t, s)g(φ(s))ds + k ∫t t−h cs(t, s) ∫t s g(φ(v))dvds and that | d dt f(φ)(t)| ≤ ‖a′‖ + g∗ sup 0≤t≤t [ |c(t, t)| + ∫t t−h d(t, s)ds + k ∫t t−h |cs(t, s)|(t − s)ds ] ≤ ‖a′‖ + g∗[ sup 0≤t≤t |c(t, t)| + hb1 + hkb1] where g∗ = sup{|g(u)| : |u| ≤ η}; thus, γ is equi-continuous. the uniform boundedness of γ follows from the inequality |f(φ)(t)| ≤ ‖a‖ + ∫t t−h |cs(t, s)| ∫t s |g(φ(v))|dv ≤ ‖a‖ + hb1g∗ for all φ ∈ γ. so, by the ascoli-arzela theorem, γ lies in a compact subset of pt . by schaefer’s theorem, we see f has a fixed point which is a t-periodic solution of (4.1). the proof is complete. we now give two examples which show a connection between this section and sections 2 and 3. example 1. consider the scalar equation x(t) = a(t) − ∫t −∞ c(t − s)g(x(s))ds (4.29) where c(t) = (−1)n(t − h)n, 0 ≤ t ≤ h, n = 3, 4, .. (4.30) = 0, t ≥ h cubo 14, 1 (2012) bounded and periodic solutions of integral equations 77 for some h > 0. it is readily verified that c′′ is continuous for 0 ≤ t < ∞ and that c(t − s) is convex. moreover, using two changes of variable we find that x(t) = a(t) − ∫∞ 0 c(u)g(x(t − u))du (4.31) = a(t) − ∫h 0 c(u)g(x(t − u))du = a(t) − ∫t t−h c(t − s)g(x(s))ds and c(h) = c′(h) = c′′(h) = 0. all of the work in sections 2 and 4 hold for this equation. example 2. consider x(t) = a(t) − ∫t 0 c(t − s)g(x(s))ds (4.32) with solution φ on [0, h] where c satisfies (4.30). then for t ≥ h we have x(t) = a(t) − ∫t t−h c(t − s)g(x(s))ds, t ≥ h, (4.33) with initial function φ on [0, h]. theorem 4.4 let dg(x) dx be continuous, let x(t) be the unique solution of (4.32), and let y(t) be any continuous solution of (4.33). suppose that there is an l > 0 with dg(x) dx ≥ l. then z(t) := x(t) − y(t) ∈ l2[h, ∞). if, in addition, there is an m > 0 with dg(x) dx ≤ m, then z(t) → 0 as t → ∞. proof. we have for t ≥ h that z(t) = − ∫t t−h c(t − s)[g(x(s) − g(y(s))]ds (4.34) = − ∫t t−h c(t − s) dg(ξ(s)) dx z(s)ds (4.35) where ξ(s) is between x(s) and y(s). define a liapunov functional by v(t) = ∫t t−h cs(t − s) (∫t s [g(x(u)) − g(y(u))]du )2 ds (4.36) with derivative satisfying v ′(t) ≤ −2[g(x(t)) − g(y(t))][x(t) − y(t)] = −2 dg(ξ(t)) dx z2(t) ≤ −2lz2(t). (4.37) 78 t. a. burton and bo zhang cubo 14, 1 (2012) this yields the first conclusion. with the last assumption, note that |z(t)| ≤ m ∫t t−h |c(t − s)||z(s)|ds (4.38) ≤ m √∫t t−h c2(t − s)ds ∫t t−h z2(s)ds (4.39) ≤ m √ h2n+1 ∫t t−h z2(s)ds (4.40) and ∫t t−h z2(s)ds → 0 as t → ∞. (4.41) under the conditions here, with c defined by (4.30) we see that the solutions of the equations in sections 2, 3, and 4 all converge to the same function both pointwise and in l2. received: april 2011. revised: may 2011. references [1] burton, t. a., examples of lyapunov functionals for non-differentiated equations, proc. first world congress of nonlinear analysts, 1992. v. lakshmikantham, ed., walter de gruyter, new york, (1996) 1203–1214. [2] burton, t. a., boundedness and periodicity in integral and integro-differential equations, diff. eq. dynamical systems 1(1993) 161–172. [3] burton, t. a., volterra integral and differential equations, elsevier, amsterdam, 2005 [4] burton, t. a., integral equations, periodicity, and fixed points, fixed point theory 9 (2008) 47–65. [5] burton, t. a., liapunov functionals for integral equations, trafford, victoria, b. c., canada, 2008. (www.trafford.com/08-1365) [6] burton, t. a., liapunov functionals, convex kernels, and strategy, nonlinear dynamics and systems theory 10(4)(2010) 325-337. [7] burton, t. a., six integral equations and a flexible liapunov functional, trudy instituta matematiki i mekhaniki uro ran 16(5)(2010) 241-252. [8] burton, t. a. and dwiggins, d. p., resolvents, integral equations, limit sets, mathematica bohemica 135(2010) 337-354. cubo 14, 1 (2012) bounded and periodic solutions of integral equations 79 [9] burton, t. a. and haddock, john r., qualitative properties of solutions of integral equations, nonlinear analysis 71 (2009) 5712–5723. [10] levin, j. j., the asymptotic behavior of a volterra equation, proc. amer. math. soc. 14(1963) 434–451. [11] levin, j. j., the qualitative behavior of a nonlinear volterra equation, proc. amer. math. soc. 16 (1965), 711–718. [12] levin, j. j., a nonlinear volterra equation not of convolution type, j. differential equations 4 (1968), 176–186. [13] levin, j. j. and nohel, j. a. , note on a nonlinear volterra equation, proc. amer. math. soc. 14 (1963), 924-929. [14] londen, stig-olof., on the solutions of a nonlinear volterra equation, j. math. anal. appl. 39 (1972), 564-573. [15] miller, richard k., nonlinear volterra integral equations, benjamin, new york, 1971. [16] schaefer, h., über die methode der a priori schranken, math. ann. 129(1955), 415–416. [17] volterra, v., sur la théorie mathématique des phénomès héréditaires, j. math. pur. appl. 7 (1928) 249–298. [18] yoshizawa t., stability theory by liapunov’s second method, math. soc. japan, tokyo, 1966. [19] zhang, b., boundedness and global attractivity of solutions for a system of nonlinear integral equations, cubo: a mathematical journal 11 (2009) 41–53. [20] zhang, b., liapunov functionals and periodicity in a system of nonlinear integral equations, electronic journal of qualitative theory of differential equations, spec. ed. i, 2009 no. 1, 1–15. cubo a mathematical journal vol.15, no¯ 02, (65–69). june 2013 a note on modifications of rg-closed sets in topological spaces takashi noiri 2949-1 shiokita-cho, hinagu, yatsushiro-shi, kumamoto-ken, 869-5142 japan. t.noiri@nifty.com valeriu popa department of mathematics, university of bacǎu,, 600 115 bacǎu,, romania, vpopa@ub.ro abstract we point out that a certain modification of regular generalized closed sets due to palaniappan and rao [15] means nothing to the family of semi-open sets. resumen destacamos que una modificación de conjuntos cerrados regulares generalizados debido a palaniappan and rao [15] no tiene importancia para la familia de conjuntos semiabiertos. keywords and phrases: g-closed, rg-closed. 2010 ams mathematics subject classification: 54a05. 66 takashi noiri and valeriu popa cubo 15, 2 (2013) 1 introduction in 1970, levine [11] introduced the notion of generalized closed (briefly g-closed) sets in topological spaces and showed that compactness, locally compactness, countably compactness, paracompactness, and normality etc are all g-closed hereditary. and also he introduced a separation axiom called t1/2 between t1 and t0. since then, many modifications of g-closed sets are introduced and investigated. among them, dontchev and ganster [5] introduced the notion of t3/4-spaces which are situated between t1 and t1/2 and showed that the digital line or the khalimsky line [9] (z, κ) lies between t1 and t3/4. as a modification of g-closed sets, regular generalized closed sets are introduced and investigated by palaniappan and rao [15]. as the further modification of g-closed sets, gnanambal [7] introduced the notion of generalized preregular closed sets. the purpose of this note is to present some remarks concerning modifications of regular generalized closed sets. 2 preliminaries let (x, τ) be a topological space and a a subset of x. the closure of a and the interior of a are denoted by cl(a) and int(a), respectively. we recall some generalized open sets in topological spaces. definition 2.1. let (x, τ) be a topological space. a subset a of x is said to be (1) α-open [14] if a ⊂ int(cl(int(a))), (2) semi-open [10] if a ⊂ cl(int(a)), (3) preopen [12] if a ⊂ int(cl(a)), (4) semi-preopen [2] or β-open [1] if a ⊂ cl(int(cl(a))), (5) b-open [3] if a ⊂ int(cl(a)) ∪ cl(int(a)), (6) regular open if a = int(cl(a)). the family of all α-open (resp. semi-open, preopen, semi-preopen, b-open, regular open) sets in (x, τ) is denoted by τα (resp. so(x), po(x), spo(x), bo(x), ro(x, τ)). for generalizations of open sets defined above, the following relations are well known: diagram i open ⇒ α-open ⇒ preopen ⇓ ⇓ semi-open ⇒ b-open ⇒ semi-preopen definition 2.2. let (x, τ) be a topological space. a subset a of x is said to be α-closed [13] (resp. semi-closed [4], preclosed [12], semi-preclosed [2], b-closed [3]) if the complement of a is α-open (resp. semi-open, preopen, semi-preopen, b-open). cubo 15, 2 (2013) a note on modifications of rg-closed sets in topological spaces 67 definition 2.3. let (x, τ) be a topological space and a a subset of x. the intersection of all α-closed (resp. semi-closed, preclosed, semi-preclosed, b-closed) sets of x containing a is called the α-closure [13] (resp. semi-closure [4], preclosure [6], semi-preclosure [2], b-closure [3]) of a and is denoted by αcl(a) (resp. scl(a), pcl(a), spcl(a), bcl(a)). definition 2.4. let (x, τ) be a topological space. a subset a of x is said to be (1) generalized closed (briefly g-closed) [11] if cl(a) ⊂ u whenever a ⊂ u and u ∈ τ, (2) regular generalized closed (briefly rg-closed) [15] if cl(a) ⊂ u whenever a ⊂ u and u ∈ ro(x, τ), (3) generalized preregular closed (briefly gpr-closed) [7] if pcl(a) ⊂ u whenever a ⊂ u and u ∈ ro(x, τ). for generalizations of closed sets defined above, the following relations are well known: diagram ii closed ⇒ g-closed ⇒ rg-closed ⇒ gpr-closed 3 modifications of rg-closed sets first we shall define a modification of rg-closed sets. definition 3.1. let (x, τ) be a topological space. a subset a of x is said to be regular generalized α-closed (briefly rgα-closed) if αcl(a) ⊂ u whenever a ⊂ u and u ∈ ro(x, τ). lemma 3.2. if a is a subset of (x, τ), then τα-int(τα-cl(a)) = int(cl(a)). proof. this is shown in corollary 2.4 of [8]. lemma 3.3. let v be a subset of a topological space (x, τ). then v ∈ ro(x, τ) if and only if v ∈ ro(x, τα). proof. this is an immediate consequence of lemma 3.2. theorem 3.4. a subset a of a topological space (x, τ) is rgα-closed in (x, τ) if and only if a is rg-closed in the topological space (x, τα). proof. necessity. suppose that a is rgα-closed in (x, τ). let a ⊂ v and v ∈ ro(x, τα). by lemma 3.3, v ∈ ro(x, τ) and we have τα-cl(a) = αcl(a) ⊂ v. therefore, a is rg-closed in (x, τα). sufficiency. suppose that a is rg-closed in (x, τα). let a ⊂ v and v ∈ ro(x, τ). by lemma 3.3, v ∈ ro(x, τα) and hence αcl(a) = τα-cl(a) ⊂ v. therefore, a is rgα-closed in (x, τ). 68 takashi noiri and valeriu popa cubo 15, 2 (2013) remark 3.5. it turns out that, by therem 3.4, we can not obtain the essential notion even if we replace cl(a) in definition 2.4(2) with αcl(a). next, we try to replace cl(a) in definition 2.4(2) with scl(a). lemma 3.6. let (x, τ) be a topological space. if a ⊂ v and v ∈ ro(x, τ), then scl(a) ⊂ v. proof. let a ⊂ v and v ∈ ro(x, τ). then we have scl(a) ⊂ scl(v) = v ∪ int(cl(v)) = v and hence scl(a) ⊂ v. remark 3.7. (1) lemma 3.6 shows that in case so(x, τ) the condition ”scl(a) ⊂ v whenever a ⊂ v and v ∈ ro(x, τ)” does not define a subset like regualr generalized semi-closed sets. (2) by diagram i, so(x) ⊂ bo(x) ⊂ spo(x) and hence spcl(a) ⊂ bcl(a) ⊂ scl(a) for any subset a of x. therefore, we can not obtain any notions even if we replace cl(a) in definitiion 2.4(2) with scl(a), bcl(a) or spcl(a). received: october 2010. accepted: september 2012. references [1] m. e. abd el-monsef, s. n. el-deep and r. a. mahmoud, β-open sets and β-continuous mappings,bull. fac. sci. assiut univ., 12 (1983), 77–90. [2] d. andrijević,semi-preopen sets,mat. vesnik, 38 (1986), 24–32. [3] d. andrijević,on b-open sets,mat. vesnik, 48 (1996), 59–64. [4] s. g. crossley and s. k. hildebrand, semi-closure, texas j. sci., 22 (1971), 99–112. [5] j. dontchev and m. ganster, on δ-generalized closed sets and t3/4-spaces, mem. fac. sci. kochi univ. ser. a math., 17 (1996), 15–31. [6] s. n. el-deeb, i. a. hasanein, a. s. mashhour and t. noiri, on p-regular spaces, bull. math. soc. sci. math. r. s. roumanie, 27(75) (1983), 311–315. [7] y. gnanambal,on generalized preregular closed sets in topological spaces,indian j. pure appl. math., 28 (1997), 351–360. [8] d. s. janković, a note on mappings of extremally disconnected spaces, acta math. hungar., 46 (1985), 83–92. [9] e. d. khalimsky, r. kopperman and p. r. meyer,computer graphics and connected topologies on finite ordered sets,topology appl., 36 (1990), 1–17. cubo 15, 2 (2013) a note on modifications of rg-closed sets in topological spaces 69 [10] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly, 70 (1963), 36–41. [11] n. levine,generalized closed sets in topology,rend. circ. mat. palermo (2), 19 (1970), 89–96. [12] a. s. mashhour, m. e. abd el-monsef and s. n. el-deep, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt, 53 (1982), 47–53. [13] a. s. mashhour, i. a. hasanein and s. n. el-deeb, α-continuous and α-open mappings,acta math. hungar., 41 (1983), 213–218. [14] o. nj̊astad, on some classes of nearly open sets,pacific j. math., 15 (1965), 961–970. [15] n. palaniappan and k. c. rao,regular generalized closed sets,kyungpook math. j., 33 (1993), 211–219. cubo a mathematical journal vol.15, no¯ 02, (21–31). june 2013 nonnegative solutions of quasilinear elliptic problems with sublinear indefinite nonlinearity1 weihui wang a and zuodong yang a,b a institute of mathematics, school of mathematical sciences, nanjing normal university, jiangsu nanjing 210046, china. b college of zhongbei, nanjing normal university, jiangsu nanjing 210046, china. 335348332@qq.com zdyang jin@263.net abstract we study the existence, nonexistence and multiplicity of nonnegative solutions for the quasilinear elliptic problem { − △p u = a(x)u q + λb(x)ur, in ω u = 0, on ∂ω where ω is a bounded domain in rn, λ > 0 is a parameter, △p = div(|∇u| p−2∇u) is the p−laplace operator of u, 1 < p < n, 0 < q < p − 1 < r ≤ p∗ − 1, a(x), b(x) are bounded functions, the coefficient b(x) is assumed to be nonnegative and a(x) is allowed to change sign. the results of the semilinear equations are extended to the quasilinear problem. 1project supported by the national natural science foundation of china(grant no.11171092). project supported by the natural science foundation of the jiangsu higher education institutions of china (grant no.08kjb110005) 22 weihui wang and zuodong yang cubo 15, 2 (2013) resumen estudiamos la existencia, no existencia y multiplicidad de soluciones no negativas del problema eĺıptico cuasi-lineal { − △p u = a(x)u q + λb(x)ur, in ω u = 0, on ∂ω donde ω es un dominio acotado en rn, λ > 0 es un parámetro, △p = div(|∇u| p−2∇u) es el operador p−laplaciano de u , 1 < p < n, 0 < q < p − 1 < r ≤ p∗ − 1, a(x), b(x) son funciones acotadas, el coeficiente b(x) se supone que es no negativo y a(x) se le permite cambiar de signo. los resultados de las ecuaciones semilineales se extienden a el problema cuasi-lineal. keywords and phrases: nonnegative solutions; quasilinear elliptic problems; sublinear indefinite nonlinearity; existence and nonexistence. 2010 ams mathematics subject classification: 35j50, 35j55, 35j60. cubo 15, 2 (2013) nonnegative solutions of quasilinear elliptic problems ... 23 1 introduction let us consider the problem { − △p u = a(x)u q + λb(x)ur, in ω u = 0, on ∂ω (pλ) where ω ⊂ rn is a smooth bounded domain, λ > 0, 1 < p < n, 0 < q < p − 1 < r ≤ p∗ − 1, p∗ = np n−p , b(x) ≥ 0, a(x) change its sign, △p = div(|∇u| p−2∇u) is the p−laplace operator of u. equations of the above form are mathematical models occuring in studies of the p-laplace equation, generalized reaction-diffusion theory([7]), non-newtonian fluid theory, and the turbulent flow of a gas in porous medium([8]). in the non-newtonian fluid theory, the quantity p is characteristic of the medium. media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. if p = 2, they are newtonian fluids. recently, a.v.lair and a.mohammed in [11] considered the existence and nonexistence of positive entire large solutions of the semilinear elliptic equation △u = p(x)uα + q(x)uβ, 0 < α ≤ β. francisco in [1] considered a sublinear indefinite nonlinearity problem of the form { −△u = a(x)uq + λb(x)up, in ω u = 0, on ∂ω where ω is a smooth bounded domain in rn, λ ∈ r, 0 < q < 1 < p < r ≤ 2∗ − 1, b(x) ≥ 0, a(x) change its sign. for more results we refer the reader to the works [12-15] and the references therein. in recent years, the existence and uniqueness of the positive solutions for the single quasilinear elliptic equation with eigenvalue problems { div(|∇u|p−2∇u) + λf(u) = 0 in ω, u(x) = 0 ∂ω, (1.1) with λ > 0, p > 1, ω ⊂ rn, n ≥ 2 have been studied by many authors, see [16-23] and the references therein. when f is strictly increasing on r+, f(0) = 0, lims→0+ f(s)/s p−1 = 0 and f(s) ≤ α1 + α2s µ, 0 < µ 0, it was shown in [16] that there exist at least two positive solutions for eqs (1.1) when λ is sufficiently large. if lims→0+ inf f(s)/s p−1 > 0, f(0) = 0 and the monotonicity hypothesis (f(s)/sp−1)′ < 0 holds for all s > 0. it was also shown in [17] that problem (1.1) has a unique positive large solution and at least one positive small solution when λ is large if f is nondecreasing; there exist α1, α2 > 0 such that f(s) ≤ α1 + α2s β, 0 < β 0 with y ≥ t such that (f(s)/sp−1)′ > 0 for s ∈ (0, t) 24 weihui wang and zuodong yang cubo 15, 2 (2013) and (f(s)/sp−1)′ < 0 for s > y. yang and xu in [10] established the existence for quasilinear elliptic equation    −△pu = a(x)(u m + λun), x ∈ rn u > 0, x ∈ rn u → 0, |x| → ∞ (1.2) where 0 < m 0 such that (1.2) has a positive solution for 0 < λ < λ∗. the quasilinear elliptic equations when a(x) ≡ b(x) ≡ 1 was considered in [2], although here under some restrictions on the p, q in the critical case r = p∗−1. problems of local ”superlinearrity” and ”sublinearity” for the p− laplace problem was considered in [3]. a class of quasilinear elliptic equations are study in [4]. for more results we refer the reader to the works [5-6] and the references therein. motivated by the results of the above papers. in this paper, we consider the quasilinear elliptic equations (pλ). we modify the method developed francisco odair de paiva in [1] and extend the results a quasilinear elliptic equation (pλ), and complement results in [2-4, 10]. the paper is organized as follows. in section 2, we recall some facts that will be needed in the paper, and give the main results. in section 3, we give the proofs of the main results in this paper. 2 main results and preliminary let us first consider the following parameterized elliptic problems    −△pu = a(x)u q + λb(x)ur, in ω u ≥ 0, in ω u = 0, on ∂ω (qλ) where ω is a bounded domain in rn, λ > 0 is a parameter, 1 < p < n, 0 < q < p − 1 < r ≤ p∗ − 1, a(x), b(x) are bounded functions, the coefficient b(x) is assumed to be nonnegative and a(x) is allowed to change sign. because that a(x) changes sign in ω, so the maximum principal is not applicable. then, define fλ(u) = 1 p ∫ ω |∇u|p − 1 q + 1 ∫ ω a(x)(u+)q+1 − λ r + 1 ∫ ω b(x)(u+)r+1, u ∈ w 1,p 0 (ω) we know that fλ(u) is well define in w 1,p 0 (ω) and is of c10(ω) cubo 15, 2 (2013) nonnegative solutions of quasilinear elliptic problems ... 25 definition 2.1. we call u ∈ w 1,p 0 (ω) is a weak solution of (qλ), if u is a critical points of fλ(u). throughout this paper, we always suppose that (h1) there exist λ > 0,a smooth subdomain b1 ∈ ω + a , m(x) ∈ l ∞(b1) with m(x) ≥ 0, m(x) 6≡ 0, µ > λ1(b1, m(x)) such that a(x)sq + λb(x)sr ≥ µm(x)sp−1 for a.e.x ∈ b1 and all s ≥ 0; here λ1(b1, m(x)) denotes the principal eigenvalue of −△p on w 1,p 0 (b1) for the weight m(x). (h2) for any λ > 0, there exists a smooth subdomain b2 ⊂ ω + a ,s1 > 0 and θ1 > λ1(b2), such that a(x)sq + λb(x)sr ≥ θ1s p−1 for a.e. x ∈ b2, and all s ∈ [0, s1]; here λ1(b2) denotes the principal eigenvalue of −△p on w 1,p 0 (ω) (f1) a(x), b(x) ∈ l ∞(ω), and ωa = {x ∈ ω : a(x) ≥ 0}, ω + a = {x ∈ ω : a(x) > 0} ω−a = {x ∈ ω : a(x) < 0}, ω + b = {x ∈ ω : b(x) > 0} are nonempty; (f2) ω + a is open, |ω − a | > 0 and ω + a ⋂ ω−a = ∅; (f3) int(ω + b ) 6= ∅ and b ≥ 0; (f4) ω + a ⊂ ω + b and ω+a ⊂ ω; (f5) int(ωa) = ⋃k 1 ui, ui connected, and ui ⋂ ω+a 6= ∅. as a consequence of assumption (f5), by the maximum principle, if u is a solution of (qλ) such that u is nontrivial in the components of ωa, then u > 0 in int(ωa) ⊃ ω + a . definition 2.2. if u is a weak solution of (qλ) and u(x) > 0, a.e.x ∈ ω + a , then u ∈ w 1,p 0 (ω) is a solution of (1.1). let λ∗ = sup{λ > 0; (1.1) has a solution}. by a modification of the method given in [1], we obtain the following main results. theorem 2.1. let 0 < q < p − 1 < r ≤ p∗ − 1. assume that (f1) − (f5) hold, then there exists λ∗ ∈ (0, ∞) such that (1) for all λ ∈ (0, λ∗), problem (pλ) has at least one weak solutions; (2) for λ = λ∗, problem (pλ) has at least one solution; 26 weihui wang and zuodong yang cubo 15, 2 (2013) (3) for all λ > λ∗, problem (pλ) has no solution. theorem 2.2. let 0 < q 0 such that for 0 < λ ≤ λ0, problem (pλ) has a solution. proof. let e be the unique positive solution of { −△pe = 1, in ω; e = 0, on ∂ω. since 0 < q < p−1 < r, we can find λ0 > 0 such that for all 0 < λ ≤ λ0 there exists m = m(λ) > 0 satisfying mp−1 ≥ mq‖a‖∞‖e‖ q ∞ + λmr‖b‖∞‖e‖ r ∞ . as a consequence, the function me satisfies −△p(me) = m p−1 ≥ mq‖a‖∞‖e‖ q ∞ + λmr‖b‖∞‖e‖ r ∞ . hence me is a supersolution of (pλ). then let u = me, we have that u is a supersolution for (qλ). moreover 0 is a solution of (qλ), so let u = 0 is a subsolution for (qλ). it follows form the sub-supersolution argument as in [5] or [6] that (qλ) has a nonnegative solution in a = {u ∈ w 1,p 0 : 0 ≤ u(x) ≤ me a.e.x ∈ ω}. then let c = infa fλ, fλ(u) = 1 p ∫ ω |∇u|p − 1 q + 1 ∫ ω a(x)(u+)q+1 − λ r + 1 ∫ ω b(x)(u+)r+1, u ∈ w 1,p 0 (ω), there exist uλ ∈ a such that c = infa fλ(uλ) and uλ is a solution of (qλ). also uλ solves (pλ) if uλ > 0 a.e.x ∈ ω + a . by contradiction, suppose that uλ ≡ 0 a.e.x ∈ ω + a , let ϕ ∈ c ∞ c (ω + a ) be nonnegative and nontrivial, then for sufficiently small s > 0, uλ + sϕ ∈ a fλ(uλ + sϕ) = fλ(uλ) + fλ(sϕ) = fλ(uλ) + sp p ‖ϕ‖p − sq+1 q + 1 ∫ ω a(x)ϕq+1 − λsr+1 r + 1 ∫ ω b(x)ϕr+1 then we have fλ(uλ + sϕ) < fλ(uλ), if s > 0 is small enough, however this contradicts that the infimum c = inf fλ is achieve at uλ . so uλ > 0 a.e.x ∈ ω + a and is a solution of (pλ). lemma 3.2. (pλ) has a solution for all λ ∈ (0, λ ∗). cubo 15, 2 (2013) nonnegative solutions of quasilinear elliptic problems ... 27 proof. given λ < λ∗, let u λ be a solution of (p λ ), with λ < λ < λ∗. then − △p uλ = a(x)u q λ + λb(x)ur λ ≥ a(x)u q λ + λb(x)ur λ , which u λ is a supersolution for (pλ). consider a = {u ∈ w 1,p 0 : 0 ≤ u ≤ u λ }, there exist uλ ∈ a such that fλ(uλ) = infa fλ, and uλ is a solution of (qλ), as the proof of lemma 3.1, uλ is also the solution of (pλ). lemma 3.3. let λ∗ = sup{λ > 0 : (pλ) has a solution}, then 0 < λ ∗ < ∞. proof. under the assume (h1), suppose that when λ > 0, (pλ) has a solution uλ ∈ w 1,p 0 (ω) ⋂ l∞(ω). consider the eigenvalue problem with weight { − △p v = µm(x)|v| p−2, in b1; v = 0, on ∂b1. since by (h1), we have ∫ b1 | ▽ uλ| p−2 ▽ uλ ▽ ϕ = ∫ b1 (a(x)u q λ + λb(x)urλ)ϕ ≥ µ ∫ b1 m(x)u p−1 λ ϕ for all ϕ ∈ c∞c (ω), ϕ ≥ 0. this show that uλ is an supersolution of (eµ). furthermore, ǫϕ1 is a subsolution of (eµ), and ǫϕ1 ≤ uλ for ǫ small enough. ∫ b1 | ▽ (ǫϕ1)| p−2 ▽ (ǫϕ1) ▽ ϕ = λ1 ∫ b1 m(x)(ǫϕ1) p−1ϕ < µ ∫ b1 m(x)(ǫϕ1) p−1ϕ for ϕ ∈ c∞c (ω), ϕ ≥ 0; ϕ1 is a positive eigenfunction associated to λ1(b1, m(x)). then (eλ) has a solution v with ǫϕ1 ≤ v ≤ uλ,in particular v ≥ 0, v 6≡ 0. for above that µ is a principal eigenvalue of −△p u on b for the weight m(x). this is contradiction with µ > λ1(b1, m(x)), and consequently λ∗ < +∞, moreover we can also obtain λ∗ > 0 to the lemma 4.1, so, λ∗ ∈ (0, ∞). hence, when λ > λ∗, problem (pλ) has no solution. lemma 3.4. for λ = λ∗, problem (pλ) has at least one solution. proof. for the definition of λ∗, let λn be a sequence such that λn −→ λ ∗ with 0 < λn < λ ∗,λn increasing, let un be a solution of pλn with fλn(un) < 0 and f ′ λn (un) = 0. we obtain fλn(un) + f ′ λn (un) · un ≤ c‖un‖, where fλn(un) = 1 p ∫ ω |∇un| p − 1 q + 1 ∫ ω a(x)(u+n) q+1 − λn r + 1 ∫ ω b(x)(u+n) r+1, f ′ λn (un) · un = ∫ ω |∇un| p − ∫ ω a(x)(u+n) q+1 − λn ∫ ω b(x)(u+n) r+1 so by theorem 1.2.1 of [9], we have ( 1 p + 1)‖un‖ p ≤ c‖un‖ q+1 + c. 28 weihui wang and zuodong yang cubo 15, 2 (2013) it shows that un is bounded in w 1,p 0 , we have, for a subsequence, un −→ u ∗ in c1(ω), hence u∗ solves (qλ) in ω. u ∗ is a solution of (pλ)) if u ∗ 6≡ 0 in ω+a . assume by contradiction u ∗ ≡ 0 in ω+a . under the assume (h2), we have ∫ b2 | ▽ un| p−2 ▽ un ▽ ϕ = ∫ b2 (a(x)uqn + λnb(x)u r)ϕ ≥ θ1 ∫ b2 up−1n ϕ for n sufficiently large(so that 0 ≤ un(x) ≤ s1 on b2,which is possible since un −→ 0 uniformly). so that un is a supersolution for the problem { − △p v = θ1|v| p−2v, in b2; v = 0, on ∂b2. moreover, since θ1 > λ1, let uε = εϕ1. we have − △p (uε) = λ1u p−1 ε < θ1u p−1 ε and εϕ1 ≤ un on b2, for (ε > 0 sufficiently small). it shows that the existence of a solution v of (eθ1) with εϕ1 ≤ v ≤ un. this is a contradiction with θ1 > λ1 in assume (h2). so, u ∗ 6≡ 0 in ω+a and is a solution of (pλ). proof of theorem 2.2. from the lemma 3.2, we have obtained uλ is a local minimizer of fλ(u) and is a solution of (pλ). in this section, we hope to find the second solution of the form v = uλ + u, by the moutnain pass theorem,where u is a nonnegative solution of { − △p (uλ + u) = a(x)(uλ + u +)q + λb(x)(uλ + u +)r, in ω; u = 0, on ∂ω. u ∈ w 1,p 0 (ω), and u ≥ 0. then, uλ + u is a second solution of (pλ). define the associated functional iλ(u) = 1 p ∫ ω |∇(uλ + u)| p − ∫ ω hλ(x, u) hλ(x, u) = gλ(x, uλ + u +) − gλ(x, uλ) − gλ(x, uλ)u +; gλ(x, u) = ∫ ω gλ(x, u)du; gλ(x, u) = a(x)u q + λb(x)ur. then, it follows that iλ(u) = 1 p ∫ ω |∇(uλ + u)| p − 1 q + 1 ∫ ω a(x)[(uλ + u +)q+1 − u q+1 λ − (q + 1)u q λ u+] − λ r + 1 ∫ ω b(x)[(uλ + u +)r+1 − ur+1λ − (r + 1)u r λu +] (i) let u+ ∈ w 1,p 0 (ω+a ), and for ‖u +‖ sufficiently small, we have iλ(u) ≥ 1 p ∫ ω |∇(uλ + u)| p − 1 p ∫ |ω∇(uλ + u +)|p + 1 p ∫ ω |∇uλ| p + ∫ ω gλ(x, uλ)u + cubo 15, 2 (2013) nonnegative solutions of quasilinear elliptic problems ... 29 then, iλ(u) ≥ 1 p ∫ ω |∇uλ| p + ∫ ω gλ(x, uλ)u + ≥ 1 p ∫ ω |∇uλ| p = iλ(0) (ii) let v1 ∈ w 1,p 0 (ω+ b ), v1 ≥ 0, v1 6≡ 0,such that ∫ ω b(x)vr+1 1 > 0. we have, for large s iλ(sv1) = 1 p ∫ ω |∇(uλ + sv1)| p − 1 q + 1 ∫ ω a(x)[(uλ + sv1) q+1 − u q+1 λ − (q + 1)u q λ sv1] − λ r + 1 ∫ ω b(x)[(uλ + sv1) r+1 − ur+1λ − (r + 1)u r λsv1] = sp p ∫ ω |∇( uλ s + v1)| p − sq+1 q + 1 ∫ ω a(x)[( uλ s + v1) q+1 − ( uλ s )q+1 − (q + 1)u q λ v1 sq ] − λsr+1 r + 1 ∫ ω b(x)[( uλ s + v1) r+1 − ( uλ s )r+1 − (r + 1)urλv1 sr ] = o(sp) − λsr+1 r + 1 ∫ ω b(x)vr+11 −→ −∞ as s −→ ∞. (iii) we now prove iλ(u) satisfies the (ps) condition in w 1,p 0 (ω). indeed, if uk is a (ps) sequence, i.e. iλ(uk) −→ c, i ′ λ(uk) −→ 0. then, for p < θ < r + 1,εk −→ 0, and some constant c, we have, θiλ(uk) − i ′ λ(uk) · uk ≤ c + εk‖uk‖ where ‖uk‖ denotes the w 1,p 0 (ω) norm ( ∫ ω |∇u|p) 1 p . ( θ p − 1)‖uk‖ p ≤ ( θ q + 1 − 1) ∫ ω a(x)u q+1 k + λ( θ r + 1 − 1) ∫ ω b(x)ur+1k + c + εk‖uk‖ ( θ p − 1)‖uk‖ p + λ(1 − θ r + 1 ) ∫ ω b(x)ur+1k ≤ ( θ q + 1 − 1) ∫ ω a(x)u q+1 k + c + εk‖uk‖ by a(x), b(x) is bounded in ω, we obtain, ( θ p − 1)‖uk‖ p + c2λ(1 − θ r + 1 )‖uk‖ r+1 ≤ c1( θ q + 1 − 1)‖uk‖ q+1 + c + εk‖uk‖ since q + 1 < p < r + 1, this implies that the sequence (uk) be bounded in w 1,p 0 (ω). thus, from (i)-(iii), iλ satisfies the assumptions of the mountain pass theorem,i.e.iλ has a nontrivial critical point. this concludes the proof of theorem 2.2. received: february 2012. accepted: october 2012. 30 weihui wang and zuodong yang cubo 15, 2 (2013) references [1] francisco odair de paiva, nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, j.funct.anal. 261(2011) 2569-2586. 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[23] zuodong yang, existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation , j.comput.appl.math.197(2006),355-364. cubo a mathemati al journal vol.15, n o 03, (105�122). o tober 2013 euler's onstant, new lasses of sequen es and estimates alina sînt m rian department of mathemati s, te hni al university of cluj-napo a, str. memorandumului nr. 28, 400114 cluj-napo a, romania. alina.sintamarian�math.ut luj.ro abstract we give two lasses of sequen es with the argument of the logarithmi term modi�ed and also with some additional terms besides those in the de�nition sequen e, and that onverge qui kly to γ(a) = lim n→∞ (∑n k=1 1 a+k−1 − ln a+n−1 a ) , where a ∈ (0,+∞). we present the pattern in forming these sequen es, expressing the oe� ients that appear with the bernoulli numbers. also, we obtain estimates ontaining best onstants for ∑n k=1 1 k − 1 24(n+1/2)2 − ln ( n + 1 2 − 7 960(n+1/2)3 ) −γ and γ− (∑n k=1 1 k − 1 24(n+1/2)2 + 7 960(n+1/2)4 − ln ( n + 1 2 + 31 8064(n+1/2)5 )) , where γ = γ(1) is the euler's onstant. resumen mostramos dos lases de se uen ias on el argumento del término logarítmi o modi�ado y también on algunos términos adi ionales además de los de�nidos en la se uenia y que onvergen rápidamente a γ(a) = lim n→∞ (∑n k=1 1 a+k−1 − ln a+n−1 a ) , donde a ∈ (0,+∞). presentamos el patrón que forma las se uen ias expresando los oe� ientes que apare en en los números de bernoulli. además, obtenemos estima iones que ontienen las mejores onstantes para ∑n k=1 1 k − 1 24(n+1/2)2 −ln ( n + 1 2 − 7 960(n+1/2)3 ) −γ y γ − (∑n k=1 1 k − 1 24(n+1/2)2 + 7 960(n+1/2)4 − ln ( n + 1 2 + 31 8064(n+1/2)5 )) , donde γ = γ(1) es la onstante de euler. keywords and phrases: sequen e, onvergen e, approximation, euler's onstant, bernoulli number, estimate. 2010 ams mathemati s subje t classi� ation: 11y60, 11b68, 40a05, 41a44, 33b15. 106 alina sînt m rian cubo 15, 3 (2013) 1 introdu tion let hn = ∑n k=1 1/k be the nth harmoni number and let dn = hn − lnn. euler's onstant γ = limn→∞ dn is one of the most important onstants in mathemati s and is also the topi of many papers in the literature. this omes as a on�rmation of what leonhard euler said about γ, namely that it is �worthy of serious onsideration� ( [10, pp. xx, 51℄). it is well-known (see [19℄, [20℄, [2℄, [5℄) that 1 2n + 2γ−1 1−γ ≤ dn − γ < 1 2n + 1 3 , n ∈ n, the numbers 2γ−1 1−γ and 1 3 being the best onstants with this property, i.e. 2γ−1 1−γ annot be repla ed by a smaller one and 1 3 annot be repla ed by a larger one, so that the above-mentioned inequalities to hold for all n ∈ n. having in view that limn→∞ n(dn −γ) = 1/2, one an say that the sequen e (dn)n∈n onverges to γ very slowly. in order to in rease the speed of onvergen e to γ, d. w. detemple [7℄ modi�ed the argument of the logarithmi term from dn, onsidering the sequen e (rn)n∈n de�ned by rn = hn − ln(n+1/2), and he proved that 1 24(n+1)2 < rn −γ < 1 24n2 , n ∈ n. sequen es with higher rate of onvergen e to γ an be also obtained by subtra ting a rational term from dn: it is shown (see [21℄) that limn→∞ n 2(γ−dn +1/(2n)) = 1/12. in our paper we shall try to ombine these two methods, modifying the argument of the logarithmi term, and subtra ting and adding terms in the de�nition sequen e, to obtain qui ker onvergen es to a generalization of euler's onstant. also, we shall provide estimates regarding euler's onstant γ and this is the reason why further on we remind some of the estimates related to γ and ontaining best onstants, that have been obtained in the literature: 1 24(n+a1) 2 ≤ rn − γ < 1 24(n+b1) 2 , n ∈ n ([3℄); 1 12n2+a2 < γ − ( dn − 1 2n ) ≤ 1 12n2+b2 , n ∈ n ([9℄); 7 960(n+a3) 4 ≤ γ − ( hn − ln ( n + 1 2 ) − 1 24(n+1/2)2 ) < 7 960(n+b3) 4 , n ∈ n ([4℄); 17 3840(n+a4) 5 ≤ hn − ln ( n + 1 2 + 1 24n − 1 48n2 + 23 5760n3 ) − γ < 17 3840(n+b4) 5 , n ∈ n ([6℄), with a1 = 1/ √ 24(1 − γ − ln(3/2)) − 1 and b1 = 1/2; a2 = 6/5 and b2 = 2(7 − 12γ)/(2γ − 1); a3 = 1/ 4 √ 960/7(ln(3/2) + γ − 53/54) − 1 and b3 = 1/2; a4 = 1/ 5 √ 3840/17(1 − γ − ln(8783/5760)) − 1 and b4 = 3305/12852, where ai and bi are the best onstants with the property that the orresponding inequalities hold for all n ∈ n, i ∈ {1,2,3,4}. as we anti ipated earlier, in the present paper we shall investigate a generalization of euler's onstant, namely the limit γ(a) of the sequen e (yn(a))n∈n de�ned by (see [11, p. 453℄, [16℄, [17℄, cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 107 [18℄) yn(a) = n∑ k=1 1 a + k − 1 − ln a + n − 1 a , where a ∈ (0,+∞). obviously, γ(1) = γ. numerous results related to γ(a) an be found, for example, in [16℄, [17℄, [18℄, [14℄, [12℄. in se tion 2 we give two lasses of sequen es with the argument of the logarithmi term modi�ed and also with some additional terms besides those in the de�nition sequen e (yn(a))n∈n, and that onverge qui kly to γ(a). we present the pattern in forming these sequen es, expressing the oe� ients that appear with the bernoulli numbers. the two lasses of sequen es have as base the sequen es (αn,2(a))n∈n and (αn,3(a))n∈n de�ned by αn,2(a) = n∑ k=1 1 a + k − 1 − 1 24 ( a + n − 1 2 )2 − ln ( a + n − 1 2 a − 7 960a ( a + n − 1 2 )3 ) , αn,3(a) = n∑ k=1 1 a + k − 1 − 1 24 ( a + n − 1 2 )2 + 7 960 ( a + n − 1 2 )4 − ln ( a + n − 1 2 a + 31 8064a ( a + n − 1 2 )5 ) . in se tion 3 we prove estimates ontaining best onstants for αn,2(1) −γ and γ −αn,3(1), n ∈ n. the following lemma, whi h we shall need in our proofs, was given by c. morti i [13, lemma℄ and is a onsequen e of the the stolz-cesàro theorem, the 0/0 ase [8, theorem b.2, p. 265℄. lemma 1.1. let (xn)n∈n be a onvergent sequen e of real numbers and x ∗ = lim n→∞ xn. we suppose that there exists α ∈ r, α > 1, su h that lim n→∞ nα(xn − xn+1) = l ∈ r. then there exists the limit lim n→∞ nα−1(xn − x ∗ ) = l α − 1 . also, re all that the digamma fun tion ψ is the logarithmi derivative of the gamma fun tion, i.e. ψ(x) = γ ′(x) γ(x) , x ∈ (0,+∞). it is known that ( [1, p. 258℄, [15, p. 337℄) ψ(n + 1) = −γ + hn, n ∈ n. (1) from the re urren e formula ( [1, p. 258℄) ψ(x + 1) = ψ(x) + 1 x , x ∈ (0,+∞), 108 alina sînt m rian cubo 15, 3 (2013) and the asymptoti formula ( [1, p. 259℄) ψ(x) ∼ lnx − 1 2x − 1 12x2 + 1 120x4 − 1 252x6 + 1 240x8 − 1 132x10 + · · · (x → ∞), one obtains ψ(x + 1) ∼ lnx + 1 2x − 1 12x2 + 1 120x4 − 1 252x6 + 1 240x8 − 1 132x10 + · · · (x → ∞). (2) 2 sequen es that onverge to γ(a) theorem 2.1. let a ∈ (0,+∞) and let γ(a) be the limit of the sequen e (yn(a))n∈n from introdu tion. we onsider the sequen es (αn,2(a))n∈n and (βn,2(a))n∈n de�ned by αn,2(a) = n∑ k=1 1 a + k − 1 − 1 24 ( a + n − 1 2 )2 − ln ( a + n − 1 2 a − 7 960a ( a + n − 1 2 )3 ) , βn,2(a) = αn,2(a) − 31 8064 ( a + n − 1 2 )6 . then: (i) lim n→∞ n6(αn,2(a) − γ(a)) = 31 8064 ; (ii) lim n→∞ n8(γ(a) − βn,2(a)) = 7571 1843200 . proof. (i) set εn := 1 a+n , n ∈ n. sin e ±1 2 εn ∈ (−1,1), − 1 2 εn − 7 960 · ε 4 n (1− 12 εn) 3 ∈ (−1,1] and 1 2 εn − 7 960 · ε 4 n (1+ 12 εn) 3 ∈ (−1,1], for every n ∈ n, using the series expansion ( [11, pp. 171�179℄) we obtain αn,2(a) − αn+1,2(a) = −εn − 1 24 · ε2n ( 1 − 1 2 εn )2 + 1 24 · ε2n ( 1 + 1 2 εn )2 − ln ( 1 − 1 2 εn − 7 960 · ε4n ( 1 − 1 2 εn )3 ) + ln ( 1 + 1 2 εn − 7 960 · ε4n ( 1 + 1 2 εn )3 ) = 31 1344 ε7n + 4829 230400 ε9n + 2913 225280 ε11n + 20456239 2875392000 ε13n + o(ε 15 n ). it follows that lim n→∞ n7(αn,2(a) − αn+1,2(a)) = 31 1344 . now, a ording to lemma 1.1, we get lim n→∞ n6(αn,2(a) − γ(a)) = 31 8064 . cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 109 (ii) we are able to write that βn+1,2(a) − βn,2(a) = αn+1,2(a) − αn,2(a) − 31 8064 · ε6n (1 + 1 2 εn) 6 + 31 8064 · ε6n (1 − 1 2 εn) 6 = 7571 230400 ε9n + 10727 225280 ε11n + o(ε 13 n ). therefore lim n→∞ n9(βn+1,2(a) − βn,2(a)) = 7571 230400 , and based on lemma 1.1 we obtain lim n→∞ n8(γ(a) − βn,2(a)) = 7571 1843200 . also, onsidering the sequen e in ea h of the following parts and using similar arguments as in theorem 2.1, we get the indi ated limit: δn,2(a) = βn,2(a) + 7571 1843200 ( a + n − 1 2 )8 , n ∈ n, lim n→∞ n10(δn,2(a) − γ(a)) = 511 67584 ; ηn,2(a) = δn,2(a) − 511 67584 ( a + n − 1 2 )10 , n ∈ n, lim n→∞ n12(γ(a) − ηn,2(a)) = 5092085987 241532928000 ; θn,2(a) = ηn,2(a) + 5092085987 241532928000 ( a + n − 1 2 )12 , n ∈ n, lim n→∞ n14(θn,2(a) − γ(a)) = 8191 98304 ; λn,2(a) = θn,2(a) − 8191 98304 ( a + n − 1 2 )14 , n ∈ n, lim n→∞ n16(γ(a) − λn,2(a)) = 25599939583183 57755566080000 ; µn,2(a) = λn,2(a) + 25599939583183 57755566080000 ( a + n − 1 2 )16 , n ∈ n, lim n→∞ n18(µn,2(a) − γ(a)) = 5749691557 1882718208 . 110 alina sînt m rian cubo 15, 3 (2013) we remark the pattern in forming the sequen es from theorem 2.1 and those mentioned above. for example, the general term of the sequen e (µn,2(a))n∈n an be written in the form µn,2(a) = n∑ k=1 1 a + k − 1 − 1 2 · b2 2 · 1 ( a + n − 1 2 )2 − ln ( a + n − 1 2 a + 23 − 1 23 · b4 4 · 1 a ( a + n − 1 2 )3 ) − 8∑ k=3 ck,2 ( a + n − 1 2 )2k , with ck,2 =    22k−1 − 1 22k−1 · b2k 2k , if k = 2p + 1,p ∈ n, 22k−1 − 1 22k−1 · b2k 2k + 2 k ( − 23 − 1 23 · b4 4 ) k 2 , if k = 2p + 2,p ∈ n, where b2k is the 2kth bernoulli number. related to this remark, see also [16, remark 3.4℄, [18, p. 71, remark 2.1.3; pp. 100, 101, remark 3.1.6℄. for euler's onstant γ = 0.5772156649. . . we obtain, for example: α2,2(1) = 0.5772292855. . .; α3,2(1) = 0.5772175963. . .; β2,2(1) = 0.5772135395. . .; β3,2(1) = 0.5772155051. . .; δ2,2(1) = 0.5772162314. . .; δ3,2(1) = 0.5772156875. . .; η2,2(1) = 0.5772154386. . .; η3,2(1) = 0.5772156600. . .; θ2,2(1) = 0.5772157923. . .; θ3,2(1) = 0.5772156663. . .; λ2,2(1) = 0.5772155686. . .; λ3,2(1) = 0.5772156643. . .; µ2,2(1) = 0.5772157590. . .; µ3,2(1) = 0.5772156651. . .. as an be seen, λ3,2(1) is a urate to nine de imal pla es in approximating γ. theorem 2.2. let a ∈ (0,+∞) and let γ(a) be the limit of the sequen e (yn(a))n∈n from introdu tion. we onsider the sequen es (αn,3(a))n∈n, (βn,3(a))n∈n and (δn,3(a))n∈n de�ned by αn,3(a) = n∑ k=1 1 a + k − 1 − 1 24 ( a + n − 1 2 )2 + 7 960 ( a + n − 1 2 )4 − ln ( a + n − 1 2 a + 31 8064a ( a + n − 1 2 )5 ) , βn,3(a) = αn,3(a) + 127 30720 ( a + n − 1 2 )8 , δn,3(a) = βn,3(a) − 511 67584 ( a + n − 1 2 )10 . then: (i) lim n→∞ n8(γ(a) − αn,3(a)) = 127 30720 ; cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 111 (ii) lim n→∞ n10(βn,3(a) − γ(a)) = 511 67584 ; (iii) lim n→∞ n12(γ(a) − δn,3(a)) = 178161637 8453652480 . proof. (i) set εn := 1 a+n , n ∈ n. sin e ±1 2 εn ∈ (−1,1), 1 2 εn + 31 8064 · ε 6 n (1+ 12 εn) 5 ∈ (−1,1] and −1 2 εn + 31 8064 · ε 6 n (1− 12 εn) 5 ∈ (−1,1], for every n ∈ n, using the series expansion ( [11, pp. 171�179℄) we obtain αn+1,3(a) − αn,3(a) = εn − 1 24 · ε2n ( 1 + 1 2 εn )2 + 1 24 · ε2n ( 1 − 1 2 εn )2 + 7 960 · ε4n ( 1 + 1 2 εn )4 − 7 960 · ε4n ( 1 − 1 2 εn )4 − ln ( 1 + 1 2 εn + 31 8064 · ε6n ( 1 + 1 2 εn )5 ) + ln ( 1 − 1 2 εn + 31 8064 · ε6n ( 1 − 1 2 εn )5 ) = 127 3840 ε9n + 409 8448 ε11n + 5873471 140894208 ε13n + 2502391 92897280 ε15n + 2826605 210567168 ε17n + 33340423721 8302787297280 ε19n + o(ε 21 n ). consequently, lim n→∞ n9(αn+1,3(a) − αn,3(a)) = 127 3840 , and from this, based on lemma 1.1, we get lim n→∞ n8(γ(a) − αn,3(a)) = 127 30720 . (ii) we have βn,3(a) − βn+1,3(a) = αn,3(a) − αn+1,3(a) + 127 30720 · ε8n (1 − 1 2 εn) 8 − 127 30720 · ε8n (1 + 1 2 εn) 8 = 2555 33792 ε11n + 114794663 704471040 ε13n + 18092183 92897280 ε15n + 36074257 210567168 ε17n + o(ε 19 n ). thus lim n→∞ n11(βn,3(a) − βn+1,3(a)) = 2555 33792 , and applying lemma 1.1, it follows that lim n→∞ n10(βn,3(a) − γ(a)) = 511 67584 . 112 alina sînt m rian cubo 15, 3 (2013) (iii) we an write that δn+1,3(a) − δn,3(a) = βn+1,3(a) − βn,3(a) − 511 67584 · ε10n (1 + 1 2 εn) 10 + 511 67584 · ε10n (1 − 1 2 εn) 10 = 178161637 704471040 ε13n + 69794707 92897280 ε15n + o(ε 17 n ). hen e lim n→∞ n13(δn+1,3(a) − δn,3(a)) = 178161637 704471040 . this, along with lemma 1.1, gives lim n→∞ n12(γ(a) − δn,3(a)) = 178161637 8453652480 . also, onsidering the sequen e in ea h of the following parts and using similar arguments as in theorem 2.2, we get the indi ated limit: ηn,3(a) = δn,3(a) + 178161637 8453652480 ( a + n − 1 2 )12 , n ∈ n, lim n→∞ n14(ηn,3(a) − γ(a)) = 8191 98304 ; θn,3(a) = ηn,3(a) − 8191 98304 ( a + n − 1 2 )14 , n ∈ n, lim n→∞ n16(γ(a) − θn,3(a)) = 118518239 267386880 ; λn,3(a) = θn,3(a) + 118518239 267386880 ( a + n − 1 2 )16 , n ∈ n, lim n→∞ n18(λn,3(a) − γ(a)) = 91282102592903 29890034270208 ; µn,3(a) = λn,3(a) − 91282102592903 29890034270208 ( a + n − 1 2 )18 , n ∈ n, lim n→∞ n20(γ(a) − µn,3(a)) = 91546277357 3460300800 . we remark the pattern in forming the sequen es from theorem 2.2 and those mentioned above. for example, the general term of the sequen e (µn,3(a))n∈n an be written in the form µn,3(a) = n∑ k=1 1 a + k − 1 − 1 2 · b2 2 · 1 ( a + n − 1 2 )2 − 23 − 1 23 · b4 4 · 1 ( a + n − 1 2 )4 − ln ( a + n − 1 2 a + 25 − 1 25 · b6 6 · 1 a ( a + n − 1 2 )5 ) − 9∑ k=4 ck,3 ( a + n − 1 2 )2k , cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 113 with ck,3 =    22k−1 − 1 22k−1 · b2k 2k , if k = 3p + 1,p ∈ n, 22k−1 − 1 22k−1 · b2k 2k , if k = 3p + 2,p ∈ n, 22k−1 − 1 22k−1 · b2k 2k + 3 k ( − 25 − 1 25 · b6 6 ) k 3 , if k = 3p + 3,p ∈ n, where b2k is the 2kth bernoulli number. related to this remark, see also [16, remark 3.4℄, [18, p. 71, remark 2.1.3; pp. 100, 101, remark 3.1.6℄. for euler's onstant γ = 0.5772156649. . . we obtain, for example: α2,3(1) = 0.5772135222. . .; α3,3(1) = 0.5772155039. . .; β2,3(1) = 0.5772162315. . .; β3,3(1) = 0.5772156875. . .; δ2,3(1) = 0.5772154387. . .; δ3,3(1) = 0.5772156601. . .; η2,3(1) = 0.5772157923. . .; η3,3(1) = 0.5772156663. . .; θ2,3(1) = 0.5772155686. . .; θ3,3(1) = 0.5772156643. . .; λ2,3(1) = 0.5772157590. . .; λ3,3(1) = 0.5772156651. . .; µ2,3(1) = 0.5772155491. . .; µ3,3(1) = 0.5772156647. . .. as an be seen, θ3,3(1) and µ3,3(1) are a urate to nine de imal pla es in approximating γ. con luding, the following remark an be made. let a ∈ (0,+∞) and q ∈ n \ {1}. let n0 = min { n ∈ n ∣ ∣ ∣ ∣ a + n − 1 2 + 2 2q−1 −1 22q−1 · b2q 2q · 1 (a+n− 12) 2q−1 > 0 } . we onsider the sequen e (αn,q(a))n≥n0 de�ned by αn,q(a) = n∑ k=1 1 a + k − 1 − q−1∑ k=1 22k−1 − 1 22k−1 · b2k 2k · 1 ( a + n − 1 2 )2k − ln ( a + n − 1 2 a + 22q−1 − 1 22q−1 · b2q 2q · 1 a ( a + n − 1 2 )2q−1 ) , for every n ∈ n, n ≥ n0. clearly, lim n→∞ αn,q(a) = γ(a). based on the sequen e (αn,q(a))n≥n0, a lass of sequen es onvergent to γ(a) an be onsidered, namely {(αn,q,r(a))n≥n0|r ∈ n,r ≥ q+1}, where αn,q,r(a) = αn,q(a) − r∑ k=q+1 ck,q ( a + n − 1 2 )2k , for every n ∈ n, n ≥ n0, with ck,q =    22k−1 − 1 22k−1 · b2k 2k , if k ∈ {qp + 1,qp + 2, . . . ,qp + q − 1},p ∈ n, 22k−1 − 1 22k−1 · b2k 2k + q k ( − 22q−1 − 1 22q−1 · b2q 2q ) k q , if k = qp + q,p ∈ n. 114 alina sînt m rian cubo 15, 3 (2013) in this se tion we have obtained that the sequen e (αn,q(a))n∈n onverges to γ(a) as n −(2q+2) and that the sequen e (αn,q,r(a))n∈n onverges to γ(a) as n −(2r+2) , for q ∈ {2,3} and r ∈ {q + 1,q + 2,q + 3,q + 4,q + 5,q + 6}. 3 best bounds let (αn)n∈n be the sequen e de�ned by αn = αn,2(1). in part (i) of theorem 2.1 we have proved that lim n→∞ n6(αn − γ) = 31 8064 . (3) proposition 3.1. we have γ < αn+1 < αn, for every n ∈ n. proof. we have αn+1 − αn = 1 n + 1 − 1 24 ( n + 3 2 )2 + 1 24 ( n + 1 2 )2 − ln 960 ( n + 3 2 )4 − 7 ( n + 3 2 )3 + ln 960 ( n + 1 2 )4 − 7 ( n + 1 2 )3 . considering the fun tion h : [1,+∞) → r, de�ned by h(x) = 1 x + 1 − 1 24 ( x + 3 2 )2 + 1 24 ( x + 1 2 )2 − ln 960 ( x + 3 2 )4 − 7 ( x + 3 2 )3 + ln 960 ( x + 1 2 )4 − 7 ( x + 1 2 )3 , and di�erentiating it, we obtain that h′(x) = − 1 (x + 1)2 + 1 12 ( x + 3 2 )3 − 1 12 ( x + 1 2 )3 − 3840 ( x + 3 2 )3 960 ( x + 3 2 )4 − 7 + 3 x + 3 2 + 3840 ( x + 1 2 )3 960 ( x + 1 2 )4 − 7 − 3 x + 1 2 = (28569600x8 + 228556800x7 + 783330048x6 + 1500185088x5 + 1754428416x4 +1282873344x3 + 573399572x2 + 143606824x + 15502847) /[3(x + 1)2(2x + 1)3(2x + 3)3(960x4 + 1920x3 + 1440x2 + 480x + 53) ×(960x4 + 5760x3 + 12960x2 + 12960x + 4853)] > 0, for every x ∈ [1,+∞). it follows that the fun tion h is stri tly in reasing on [1,+∞). also, one an observe that limx→∞ h(x) = 0. these imply that h(x) < 0, for every x ∈ [1,+∞). therefore αn+1 − αn < 0, for every n ∈ n, i.e. the sequen e (αn)n∈n is stri tly de reasing. be ause limn→∞ αn = γ, we on lude that γ < αn+1 < αn, for every n ∈ n. now we give our �rst main result of this se tion. cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 115 theorem 3.2. let c = 6 √ 31 8064( 5354 −ln 4853 3240 −γ) . we have 31 8064(n + c − 1)6 ≤ αn − γ < 31 8064 ( n + 1 2 )6 , for every n ∈ n. moreover, the onstants c − 1 and 1 2 are the best possible with this property. proof. note that h is the fun tion from the proof of proposition 3.1. let (un)n∈n be the sequen e de�ned by un = αn − 31 8064(n + c − 1)6 . we have un+1 − un = αn+1 − αn − 31 8064(n + c)6 + 31 8064(n + c − 1)6 . we onsider the fun tion f : [1,+∞) → r de�ned by f(x) = h(x) − 31 8064(x + c)6 + 31 8064(x + c − 1)6 . di�erentiating, we get that f′(x) = h′(x) + 31 1344(x + c)7 − 31 1344(x + c − 1)7 = [(x − 2) 20∑ k=0 akx k + a]/[1344(x + 1)2(2x + 1)3(2x + 3)3 ×(960x4 + 1920x3 + 1440x2 + 480x + 53) ×(960x4 + 5760x3 + 12960x2 + 12960x + 4853)(x + c)7(x + c − 1)7]. one an verify that ai > 0, i ∈ {0,1, . . . ,20} and a > 0. it follows that f ′(x) > 0, for every x ∈ [2,+∞). hen e, the fun tion f is stri tly in reasing on [2,+∞). also, one an see that limx→∞ f(x) = 0. from these we obtain that f(x) < 0, for every x ∈ [2,+∞). so, un+1−un < 0, for every n ≥ 2, i.e. the sequen e (un)n≥2 is stri tly de reasing. having in view that limn→∞ un = γ, we are able to write that γ < un, for every n ≥ 2. consequently, 31 8064(n + c − 1)6 ≤ αn − γ, for every n ∈ n, and the onstant c − 1 is the best possible with this property (the equality holds only when n = 1). let (vn)n∈n be the sequen e de�ned by vn = αn − 31 8064 ( n + 1 2 )6 . 116 alina sînt m rian cubo 15, 3 (2013) then vn+1 − vn = αn+1 − αn − 31 8064 ( n + 3 2 )6 + 31 8064 ( n + 1 2 )6 . di�erentiating the fun tion g : [1,+∞) → r, de�ned by g(x) = h(x) − 31 8064 ( x + 3 2 )6 + 31 8064 ( x + 1 2 )6 , we obtain that g′(x) = h′(x) + 31 1344 ( x + 3 2 )7 − 31 1344 ( x + 1 2 )7 = −(93776707584x14 + 1312873906176x13 + 8441879298048x12 +33033108455424x11 + 87842644390912x10 + 167855050098688x9 +237559279782912x8 + 252802412814336x7 + 203105932312256x6 +122459215673472x5 + 54452798252624x4 + 17269355301696x3 +3675508601216x2 + 465873090688x + 26069935939) /[21(x + 1)2(2x + 1)7(2x + 3)7(960x4 + 1920x3 + 1440x2 + 480x + 53) ×(960x4 + 5760x3 + 12960x2 + 12960x + 4853)]. thus g′(x) < 0, for every x ∈ [1,+∞). hereby, the fun tion g is stri tly de reasing on [1,+∞). clearly, limx→∞ g(x) = 0. these yield g(x) > 0, for every x ∈ [1,+∞). then vn+1 − vn > 0, for every n ∈ n, whi h means that the sequen e (vn)n∈n is stri tly in reasing. sin e limn→∞ vn = γ, it follows that vn < γ, for every n ∈ n. we an therefore write that αn − γ < 31 8064 ( n + 1 2 )6 , (4) for every n ∈ n. it remains to prove that the onstant 1 2 is the best possible with the property that the above inequality (4) holds for every n ∈ n, and this an be a hieved as follows. we have just proved that 6 √ 31 8064(αn − γ) − n > 1 2 , n ∈ n. (5) cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 117 using (1) and (2), we get that αn − γ = hn − 1 24 ( n + 1 2 )2 − ln ( n + 1 2 − 7 960 ( n + 1 2 )3 ) − γ = ψ(n + 1) − 1 24 ( n + 1 2 )2 − ln ( n + 1 2 − 7 960 ( n + 1 2 )3 ) = 1 2n − 1 12n2 + 1 120n4 − 1 252n6 + 1 240n8 + o ( 1 n10 ) − 1 24n2 ( 1 + 1 2n )2 − ln ( 1 + 1 2n − 7 960n4 ( 1 + 1 2n )3 ) = 31 8064n6 − 31 2688n7 + o ( 1 n8 ) . (6) let an = 6 √ 31 8064n6(αn−γ) , n ∈ n. clearly, lim n→∞ an = 1, having in view (3). then, based on (6), we have 6 √ 31 8064(αn − γ) − n = n(an − 1) = n ∑5 k=0 akn ( 1 8064 31 n6(αn − γ) − 1 ) = n ∑5 k=0 akn ( 1 1 − 3 n + o ( 1 n2 ) − 1 ) = 1 ∑5 k=0 akn · 3 + o ( 1 n ) 1 − 3 n + o ( 1 n2 ) → 1 6 · 3 = 1 2 (n → ∞). (7) indeed, from (5) and (7) we obtain that 1 2 is the best onstant with the property that inequality (4) holds for every n ∈ n, and now the proof is omplete. let (�αn)n∈n be the sequen e de�ned by �αn = αn,3(1). in part (i) of theorem 2.2 we have proved that lim n→∞ n8(γ − �αn) = 127 30720 . (8) proposition 3.3. we have �αn < �αn+1 < γ, for every n ∈ n. proof. we have �αn+1 − �αn = 1 n + 1 − 1 24 ( n + 3 2 )2 + 1 24 ( n + 1 2 )2 + 7 960 ( n + 3 2 )4 − 7 960 ( n + 1 2 )4 − ln 8064 ( n + 3 2 )6 + 31 ( n + 3 2 )5 + ln 8064 ( n + 1 2 )6 + 31 ( n + 1 2 )5 . 118 alina sînt m rian cubo 15, 3 (2013) considering the fun tion �h : [1,+∞) → r, de�ned by �h(x) = 1 x + 1 − 1 24 ( x + 3 2 )2 + 1 24 ( x + 1 2 )2 + 7 960 ( x + 3 2 )4 − 7 960 ( x + 1 2 )4 − ln 8064 ( x + 3 2 )6 + 31 ( x + 3 2 )5 + ln 8064 ( x + 1 2 )6 + 31 ( x + 1 2 )5 , and di�erentiating it, we obtain that �h′(x) = − 1 (x + 1)2 + 1 12 ( x + 3 2 )3 − 1 12 ( x + 1 2 )3 − 7 240 ( x + 3 2 )5 + 7 240 ( x + 1 2 )5 − 48384 ( x + 3 2 )5 8064 ( x + 3 2 )6 + 31 + 5 x + 3 2 + 48384 ( x + 1 2 )5 8064 ( x + 1 2 )6 + 31 − 5 x + 1 2 = −(297308454912x14 + 4162318368768x13 + 26769400971264x12 +104792256479232x11 + 278852137150464x10 + 533369371889664x9 +755912773435392x8 + 806006485057536x7 + 649383667564032x6 +393129697342464x5 + 175861357984144x4 + 56282483209792x3 +12150419739472x2 + 1576326994464x + 91873672505) /[15(x + 1)2(2x + 1)5(2x + 3)5 ×(8064x6 + 24192x5 + 30240x4 + 20160x3 + 7560x2 + 1512x + 157) ×(8064x6 + 72576x5 + 272160x4 + 544320x3 +612360x2 + 367416x + 91885)] < 0, for every x ∈ [1,+∞). it follows that the fun tion �h is stri tly de reasing on [1,+∞). also, one an observe that limx→∞ �h(x) = 0. these imply that �h(x) > 0, for every x ∈ [1,+∞). therefore �αn+1 − �αn > 0, for every n ∈ n, i.e. the sequen e (�αn)n∈n is stri tly in reasing. be ause limn→∞ �αn = γ, we on lude that �αn < �αn+1 < γ, for every n ∈ n. now we give our se ond main result of this se tion. theorem 3.4. let �c = 8 √ 127 30720(γ− 47774860 −ln 91885 61236) . we have 127 30720(n + �c − 1)8 ≤ γ − �αn < 127 30720 ( n + 1 2 )8 , for every n ∈ n. moreover, the onstants �c − 1 and 1 2 are the best possible with this property. proof. note that �h is the fun tion from the proof of proposition 3.3. let (�un)n∈n be the sequen e de�ned by �un = �αn + 127 30720(n + �c − 1)8 . cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 119 we have �un+1 − �un = �αn+1 − �αn + 127 30720(n + �c)8 − 127 30720(n + �c − 1)8 . we onsider the fun tion �f : [1,+∞) → r de�ned by �f(x) = �h(x) + 127 30720(x + �c)8 − 127 30720(x + �c − 1)8 . di�erentiating, we get that �f′(x) = �h′(x) − 127 3840(x + �c)9 + 127 3840(x + �c − 1)9 = −[(x − 2) 30∑ k=0 �akx k + �a]/[3840(x + 1)2(2x + 1)5(2x + 3)5 ×(8064x6 + 24192x5 + 30240x4 + 20160x3 + 7560x2 + 1512x + 157) ×(8064x6 + 72576x5 + 272160x4 + 544320x3 + 612360x2 +367416x + 91885)(x + �c)9(x + �c − 1)9]. one an verify that �ai > 0, i ∈ {0,1, . . . ,30} and �a > 0. it follows that �f ′(x) < 0, for every x ∈ [2,+∞). hen e, the fun tion �f is stri tly de reasing on [2,+∞). also, one an see that limx→∞ �f(x) = 0. from these we obtain that �f(x) > 0, for every x ∈ [2,+∞). so, �un+1−�un > 0, for every n ≥ 2, i.e. the sequen e (�un)n≥2 is stri tly in reasing. having in view that limn→∞ �un = γ, we are able to write that �un < γ, for every n ≥ 2. consequently, 127 30720(n + �c − 1)8 ≤ γ − �αn, for every n ∈ n, and the onstant �c − 1 is the best possible with this property (the equality holds only when n = 1). let (�vn)n∈n be the sequen e de�ned by �vn = �αn + 127 30720 ( n + 1 2 )8 . then �vn+1 − �vn = �αn+1 − �αn + 127 30720 ( n + 3 2 )8 − 127 30720 ( n + 1 2 )8 . di�erentiating the fun tion �g : [1,+∞) → r, de�ned by �g(x) = �h(x) + 127 30720 ( x + 3 2 )8 − 127 30720 ( x + 1 2 )8 , 120 alina sînt m rian cubo 15, 3 (2013) we obtain that �g′(x) = �h′(x) − 127 3840 ( x + 3 2 )9 + 127 3840 ( x + 1 2 )9 = (212667885158400x20 + 4253357703168000x19 + 40151062789226496x18 +237836352044924928x17 + 991344446628691968x16 + 3090222937974767616x15 +7473501796931665920x14 + 14356208148056506368x13 + 22242021354079121408x12 +28060052688951263232x11 + 28976739296839394304x10 + 24531604126817085440x9 +16993882015446322432x8 + 9580270969643116544x7 + 4353524933287391360x6 +1571495149494432512x5 + 440878222795013392x4 + 93004870693412928x3 +13980891645509980x2 + 1352763769145912x + 64676820697555) /[15(x + 1)2(2x + 1)9(2x + 3)9 ×(8064x6 + 24192x5 + 30240x4 + 20160x3 + 7560x2 + 1512x + 157) ×(8064x6 + 72576x5 + 272160x4 + 544320x3 + 612360x2 + 367416x + 91885)]. thus �g′(x) > 0, for every x ∈ [1,+∞). hereby, the fun tion �g is stri tly in reasing on [1,+∞). clearly, limx→∞ �g(x) = 0. these yield �g(x) < 0, for every x ∈ [1,+∞). then �vn+1 − �vn < 0, for every n ∈ n, whi h means that the sequen e (�vn)n∈n is stri tly de reasing. sin e limn→∞ �vn = γ, it follows that γ < �vn, for every n ∈ n. we an therefore write that γ − �αn < 127 30720 ( n + 1 2 )8 , (9) for every n ∈ n. it remains to prove that the onstant 1 2 is the best possible with the property that the above inequality (9) holds for every n ∈ n, and this an be a hieved as follows. we have just proved that 8 √ 127 30720(γ − �αn) − n > 1 2 , n ∈ n. (10) using (1) and (2), we get that γ − �αn = γ − hn + 1 24 ( n + 1 2 )2 − 7 960 ( n + 1 2 )4 + ln ( n + 1 2 + 31 8064 ( n + 1 2 )5 ) = −ψ(n + 1) + 1 24 ( n + 1 2 )2 − 7 960 ( n + 1 2 )4 + ln ( n + 1 2 + 31 8064 ( n + 1 2 )5 ) = − 1 2n + 1 12n2 − 1 120n4 + 1 252n6 − 1 240n8 + 1 132n10 + o ( 1 n12 ) + 1 24n2 ( 1 + 1 2n )2 − 7 960n4 ( 1 + 1 2n )4 + ln ( 1 + 1 2n + 31 8064n6 ( 1 + 1 2n )5 ) = 127 30720n8 − 127 7680n9 + o ( 1 n10 ) . (11) cubo 15, 3 (2013) euler's onstant, new lasses of sequen es and estimates 121 let �an = 8 √ 127 30720n8(γ−�αn) , n ∈ n. clearly, lim n→∞ �an = 1, having in view (8). then, based on (11), we have 8 √ 127 30720(γ − �αn) − n = n(�an − 1) = n ∑7 k=0 �akn ( 1 30720 127 n8(γ − �αn) − 1 ) = n ∑7 k=0 �akn ( 1 1 − 4 n + o ( 1 n2 ) − 1 ) = 1 ∑7 k=0 �akn · 4 + o ( 1 n ) 1 − 4 n + o ( 1 n2 ) → 1 8 · 4 = 1 2 (n → ∞). (12) combining (10) and (12) we obtain that 1 2 is the best onstant with the property that inequality (9) holds for every n ∈ n, and now the proof is omplete. re eived: september 2012. a epted: september 2013. referen es [1℄ m. abramowitz, i. a. stegun, handbook of mathemati al fun tions with formulas, graphs, and mathemati al tables, national bureau of standards applied mathemati s series 55, washington, 1964. [2℄ h. alzer, inequalities for the gamma and polygamma fun tions, abh. math. semin. univ. hamb. 68, 1998, 363�372. [3℄ c.-p. chen, inequalities for the euler�mas heroni onstant, appl. math. lett. 23 (2), 2010, 161�164. [4℄ c.-p. chen, monotoni ity properties of fun tions related to the psi fun tion, appl. math. comput. 217 (7), 2010, 2905�2911. [5℄ c.-p. chen, f. qi, the best lower and upper bounds of harmoni sequen e, rgmia 6 (2), 2003, 303�308. [6℄ c.-p. chen, c. morti i, new sequen e onverging towards the euler�mas heroni onstant, comput. math. appl. 64 (4), 2012, 391�398. 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[15℄ i. m. rîji , i. s. grad³tein, tabele de integrale. sume, serii ³i produse (tables of integrals. sums, series and produ ts), editura tehni , bu ure³ti, 1955. [16℄ a. sînt m rian, a generalization of euler's onstant, numer. algorithms 46 (2), 2007, 141� 151. [17℄ a. sînt m rian, some inequalities regarding a generalization of euler's onstant, j. inequal. pure appl. math. 9 (2), 2008, 7 pp., arti le 46. [18℄ a. sînt m rian, a generalization of euler's constant, editura mediamira, cluj-napo a, 2008. [19℄ l. tóth, problem e 3432, amer. math. monthly 98 (3), 1991, 264. [20℄ l. tóth, problem e 3432 (solution), amer. math. monthly 99 (7), 1992, 684�685. [21℄ a. vernes u, o nou onvergenµ a elerat tre onstanta lui euler (a new a elerated onvergen e towards euler's onstant), gazeta matemati , seria a, 17 (96) (4), 1999, 273� 278. cubo a mathemati al journal vol.15, n o 03, (31�44). o tober 2013 coin iden e and ommon �xed point theorems in non-ar himedean menger pm-spa es sunny chauhan r.h. government postgraduate college, kashipur-244713, (u.s. nagar), uttarakhand, india. sun.gkv�gmail. om b. d. pant government degree college, champawat-262523, uttarakhand, india. badridatt.pant�gmail. om mohammad imdad department of mathemati s, aligarh muslim university, aligarh 202 002, india. mhimdad�yahoo. o.in abstract the obje t of this work is to point out a falla y in the proof of theorem 1 ontained in the re ent paper of khan et al. [jordan j. math. stat. (jjms) 5(2) (2012), 137�150℄ proved in non-ar himedean menger pm-spa e by using the notions of subompatibility and sub-sequential ontinuity. we show that the results of khan et al. [jordan j. math. stat. (jjms) 5(2) (2012), 137�150℄ an be re overed in two ways. further, we establish some illustrative examples to show the validity of the main results. our results improve a multitude of relevant �xed point theorems of the existing literature. resumen el objetivo de este trabajo es señalar una fala ia en la demostra ión del teorema 1 ontenido en un artí ulo re iente de khan et al. [jordan j. math. stat. (jjms) 5(2) (2012), 137�150℄ probado en un espa io-pm no-arquimedeano menger usando no iones de ontinuidad sub ompatible y sub se uen ial. mostramos que el resultado de khan et al. [jordan j. math. stat. (jjms) 5(2) (2012), 137�150℄ puede re uperarse de dos maneras. además, estable emos algunos ejemplos ilustrativos que muestran la validez de los resultados prin ipales. nuestro resultado mejora una gran antidad de teoremas de punto �jo importantes existentes en la literatura. keywords and phrases: t-norm, ompatible mappings, re ipro al ontinuity, sub ompatible mappings, subsequential ontinuity. 2010 ams mathemati s subje t classi� ation: 47h10, 54h25. 32 sunny chauhan, b. d. pant & mohammad imdad cubo 15, 3 (2013) 1 introdu tion istr tes u and criv t [19℄ introdu ed the on ept of non-ar himedean probablisti metri spa es (brie�y, n.a. pm-spa es) in 1974. in this sequen e, istr tes u [16,17℄ obtained some �xed point theorems on n.a. menger pm-spa es and generalized the results of sehgal and bharu ha-reid [32℄ (see [18,20℄). further, had�zi¢ [13℄ improved the results of istr tes u [16,17℄. in 1987, singh and pant [33℄ introdu ed the notion of weakly ommuting mappings on n.a. menger pm-spa es and proved some ommon �xed point theorems. dimri and pant [10℄ studied the appli ation of n.a. menger pm-spa es to produ t spa es. in 1997, cho et al. [8℄ introdu ed the on epts of ompatible mappings and ompatible mappings of type (a) in n.a. menger pmspa es and obtained some �xed point theorems for these mappings. most of the ommon �xed point theorems for ontra tion mappings invariably require a ompatibility ondition besides assuming ontinuity of at least one of the mappings. sin e then, pant [27℄ noti ed these riteria for �xed points of ontra tion mappings and introdu ed a new ontinuity ondition, known as re ipro al ontinuity and obtained a ommon �xed point theorem by using the ompatibility in metri spa es. he also showed that in the setting of ommon �xed point theorems for ompatible mappings satisfying ontra tion onditions, the notion of re ipro al ontinuity is weaker than the ontinuity of one of the mappings. jung k and rhoades [21℄ weakened the notion of ompatible mappings by introdu ing weakly ompatible mappings and proved ommon �xed point theorems without any requirement of ontinuity of the involved mappings. in 2009, kutuk u and sharma [26℄ introdu ed the on ept of ompatible mappings of type (a-1) and type (a-2) in n.a. menger pm-spa es and showed that they are equivalent to ompatible mappings under ertain onditions. many mathemati ians proved several ommon �xed point theorems in non-ar himedean menger pm-spa es using di�erent ontra tive onditions (see [4,6,9,22�24,34℄). in 2008, al-thaga� and shahzad [1℄ introdu ed the on ept of o asionally weakly ompatible (shortly, ow ) mappings in metri spa es. bouhadjera and godet-thobie [2℄ weakened the on ept of o asionally weak ompatibility and reipro al ontinuity in the form of subompatibility and sub-sequential ontinuity respe tively and proved some interesting results with these on epts in metri spa es. re ently, imdad et al. [14℄ showed that the results ontained in [2℄ an easily re overed by repla ing subompatibility with ompatibility or sub-sequential ontinuity with re ipro ally ontinuity (also see [3,5,12℄). in this paper, we prove ommon �xed point theorems for two pairs of self mappings by using the notions of ompatibility and sub-sequentially ontinuity (alternately subompatibility and re ipro ally ontinuity) in n.a. menger pm-spa es. some examples are also derived to support our results. 2 preliminaries de�nition 2.1. [31℄ a triangular norm t (brie�y, t-norm) is a binary operation on the unit interval [0,1℄ su h that for all a, b, c, d ∈ [0, 1] and the following onditions are satis�ed: cubo 15, 3 (2013) coin iden e and ommon �xed point theorems in n.a. menger . . . 33 (1) t (a, 1) = a; (2) t (a, b) = t (b, a); (3) t (a, b) ≤ t (c, d), whenever a ≤ c and b ≤ d; (4) t (a, t (b, c)) = t (t (a, b), c). de�nition 2.2. [31℄ a mapping f : r → r+ is said to be a distribution fun tion if it is nonde reasing and left ontinuous with inf{f(t) : t ∈ r} = 0 and sup{f(t) : t ∈ r} = 1. we shall denote by im, the set of all distribution fun tions whereas h stands for spe i� distribution fun tion (also known as heaviside fun tion) de�ned as h(t) = { 0, if t ≤ 0; 1, if t > 0. if x is a non-empty set, f : x × x → im is alled a probabilisti distan e on x and f(x, y) is usually denoted by fx,y. de�nition 2.3. [17,19℄ the ordered pair (x, f) is said to be an n.a. pm-spa e if x is a nonempty set and f is a probabilisti distan e satisfying the following onditions: for all x, y, z ∈ x and t, t1, t2 > 0, (1) fx,y(t) = 1 ⇔ x = y; (2) fx,y(t) = fy,x(t); (3) if fx,y(t1) = 1 and fy,z(t2) = 1 then fx,z(max{t1, t2}) = 1. the ordered triplet (x, f, t ) is alled an n.a. menger pm-spa e if (x, f) is an n.a. pmspa e, t is a t-norm and the following inequality holds: fx,z(max{t1, t2}) ≥ t (fx,y(t1), fy,z(t2)) , for all x, y, z ∈ x and t1, t2 > 0. example 2.4. let x be any set with at least two elements. if we de�ne fx,x(t) = 1 for all x ∈ x, t > 0 and fx,y(t) = { 0, if t ≤ 1; 1, if t > 1, where x, y ∈ x, x 6= y, then (x, f, t ) is an n.a. menger pm-spa e with t (a, b) = min{a, b} or (ab) for all a, b ∈ [0, 1]. 34 sunny chauhan, b. d. pant & mohammad imdad cubo 15, 3 (2013) example 2.5. let x = r be the set of real numbers equipped with metri de�ned by d(x, y) =| x−y | and fx,y(t) = { t t+|x−y| , if t > 0; 0, if t = 0. then (x, f, t ) is an n.a. menger pm-spa e with t as ontinuous t-norm satisfying t (a, b) = min{a, b} or ab for all a, b ∈ [0, 1]. de�nition 2.6. [8℄ an n.a. menger pm-spa e (x, f, t ) is said to be of type (c)g if there exists a g ∈ ω su h that g(fx,z(t)) ≤ g(fx,y(t)) + g(fy,z(t)), for all x, y, z ∈ x, t ≥ 0, where ω = {g | g : [0, 1] → [0, ∞) is ontinuous with g(1) = 0 i� t = 1}. de�nition 2.7. [8℄ an n.a. menger pm-spa e (x, f, t ) is said to be of type (d)g if there exists a g ∈ ω su h that g(t (t1, t2)) ≤ g(t1) + g(t2), for all t1, t2 ∈ [0, 1]. remark 2.8. [8℄ if an n.a. menger pm-spa e (x, f, t ) is of type (d)g, then (x, f, t ) is of type (c)g. on the other hand, (x, f, t ) is an n.a. menger pm-spa e su h that t (a, b) ≥ max{a+b−1, 0} for all a, b ∈ [0, 1], then (x, f, t ) is of type (d)g for g ∈ ω de�ned by g(t) = 1−t, t ∈ [0, 1]. throughout this paper (x, f, t ) is an n.a. menger pm-spa e with a ontinuous stri tly in reasing t-norm t . let φ : [0, ∞) → [0, ∞) be a fun tion satisfying the ondition (φ): φ is upper semiontinuous from the right and φ(t) < t for t > 0. lemma 2.9. [8℄ if a fun tion φ : [0, ∞) → [0, ∞) satis�es the ondition (φ) then we have: (1) for all t ≥ 0, limn→∞ φ n(t) = 0, where φn(t) is the nth iteration of φ(t). (2) if {tn} is a non-de reasing sequen e of real numbers and tn+1 ≤ φ(tn) where n = 1, 2, . . . then limn→∞ tn = 0. in parti ular, if t ≤ φ(t), for ea h t ≥ 0 then t = 0. de�nition 2.10. [8℄ a pair (a, s) of self mappings de�ned on an n.a. menger pm-spa e (x, f, t ) is said to be ompatible if and only if fasxn,saxn(t) → 1 for all t > 0, whenever {xn} is a sequen e in x su h that axn, sxn → z for some z ∈ x as n → ∞. cubo 15, 3 (2013) coin iden e and ommon �xed point theorems in n.a. menger . . . 35 de�nition 2.11. a pair (a, s) of self mappings de�ned on an n.a. menger pm-spa e (x, f, t ) satis�es the (e.a) property, if there exists a sequen e {xn} su h that lim n→∞ axn = lim n→∞ sxn = z, for some z ∈ x. de�nition 2.12. [29℄ a pair (a, s) of self mappings of a non-empty set x is said to be weakly ompatible (or oin identally ommuting) if they ommute at their oin iden e points, i.e. if az = sz for some z ∈ x, then asz = saz. it is easy to see that two ompatible mappings are weakly ompatible but onverse is not true. de�nition 2.13. [21℄ a pair (a, s) of self mappings of a non-empty set x is ow i� there is a point x ∈ x whi h is a oin iden e point of a and s at whi h a and s ommute. in an interesting note, �ori et al. [11℄ showed that the notion of ow redu es to weak ompatibility in the presen e of a unique point of oin iden e (or a unique ommon �xed point) of the given pair of single valued mappings. thus, no generalization an be obtained by repla ing weak ompatibility with ow . inspired by bouhadjera and godet-thobie [2℄, we de�ne the notion of subompatible mappings in n.a. menger pm-spa e as follows: de�nition 2.14. a pair (a, s) of self mappings de�ned on an n.a. menger pm-spa e (x, f, t ) is said to be sub ompatible i� there exists a sequen e {xn} su h that lim n→∞ axn = lim n→∞ sxn = z, for some z ∈ x and limn→∞ fasxn,saxn(t) = 1, for all t > 0. remark 2.15. two ow mappings are subompatible, however the onverse is not true in general (see [3, example 1.2℄). remark 2.16. a pair of subompatible mapping satis�es the (e.a) property. obviously, ompatible mappings whi h satisfy the (e.a) property are subompatible but the onverse statement does not hold in general (see [30, example 2.3℄). de�nition 2.17. a pair (a, s) of self mappings de�ned on an n.a. menger pm-spa e (x, f, t ) is alled re ipro ally ontinuous if for a sequen e {xn} in x, limn→∞ asxn = az and limn→∞ saxn = sz, whenever lim n→∞ axn = lim n→∞ sxn = z, for some z ∈ x. 36 sunny chauhan, b. d. pant & mohammad imdad cubo 15, 3 (2013) remark 2.18. if two self mappings a and b are ontinuous, then they are obviously re ipro ally ontinuous but onverse is not true. moreover, in the setting of ommon �xed point theorems for ompatible pair of self mappings satisfying ontra tive onditions, ontinuity of one of the mappings implies their re ipro al ontinuity but not onversely (see [27℄). now we de�ne the notion of sub-sequentially ontinuous mappings in n.a. menger pm-spa e due to bouhadjera and godet-thobie [2℄: de�nition 2.19. a pair of self mappings (a, s) de�ned on an n.a. menger pm-spa e (x, f, t ) is alled sub-sequentially ontinuous i� there exists a sequen e {xn} in x su h that, lim n→∞ axn = lim n→∞ sxn = z, for some z ∈ x and limn→∞ asxn = az and limn→∞ saxn = sz. remark 2.20. one an easily he k that if two self-mappings a and s are both ontinuous, hen e also re ipro ally ontinuous mappings but a and s are not sub-sequentially ontinuous (see [28, example 1℄). 3 results in 2012, khan et al. [25℄ proved the following ommon �xed point theorem for two pairs of subompatible as well as sub-sequentially ontinuous mappings in n.a. menger pm-spa e. theorem 3.1. [25, theorem 1℄ let a, b, s and t be four self mappings of an n.a. menger pmspa e (x, f, t ). if the pairs (a, s) and (b, t) are subompatible and sub-sequentially ontinuous, then (1) a and s have a oin iden e point, (2) b and t have a oin iden e point. further, if g(fax,by(t)) ≤ φ ( max { g(fsx,ty(t)), g(fsx,ax(t)), g(fty,by(t)), g(fsx,by(t)), g(fty,ax(t)) }) , (1) holds for all x, y ∈ x, t > 0, φ ∈ φ and g : [0, 1] → [0, ∞) is ontinuous and stri tly de reasing with g(1) = 0 and g(0) < ∞. then a, b, s and t have a unique ommon �xed point in x. unfortunately, theorem 3.1 is not true in its present form. to substantiate this viewpoint, we refer to imdad et al. [15, example 0.1℄ wherein it an be easily seen that involved mappings do not have a oin iden e or ommon �xed point in the underlying spa e. cubo 15, 3 (2013) coin iden e and ommon �xed point theorems in n.a. menger . . . 37 motivated by a re ent note of imdad et al. [14℄, the on lusions of theorem 3.1 remain valid if we repla e ompatibility with subompatibility and sub-sequential ontinuity with re ipro al ontinuity. however, theorem 3.1 an be orre ted in two ways under more general onditions as follows: theorem 3.2. let a, b, s and t be self mappings of an n.a. menger pm-spa e (x, f, t ). if the pairs (a, s) and (b, t) are ompatible and sub-sequentially ontinuous, then (1) the pair (a, s) has a oin iden e point, (2) the pair (b, t) has a oin iden e point. (3) there exists φ ∈ φ su h that g(fax,by(t)) ≤ φ ( max { g(fsx,ty(t)), g(fsx,ax(t)), g(fty,by(t)), 1 2 (g(fsx,by(t)) + g(fty,ax(t))) }) , (2) holds for all x, y ∈ x, t > 0 and g ∈ ω. then a, b, s and t have a unique ommon �xed point in x. proof. sin e the pair (a, s) (also (b, t)) is sub-sequentially ontinuous and ompatible mappings, therefore there exists a sequen e {xn} in x su h that lim n→∞ axn = lim n→∞ sxn = z, (3) for some z ∈ x, and lim n→∞ fasxn,saxn(t) = faz,sz(t) = 1, for all t > 0 then az = sz, whereas in respe t of the pair (b, t), there exists a sequen e {yn} in x su h that lim n→∞ byn = lim n→∞ tyn = w, (4) for some w ∈ x, and lim n→∞ fbtyn,tbyn(t) = fbw,tw(t) = 1, for all t > 0 then bw = tw. hen e z is a oin iden e point of the pair (a, s) whereas w is a oin iden e point of the pair (b, t). now we show that z = w. on using inequality (2) with x = xn, y = yn, we get g(faxn,byn(t)) ≤ φ ( max { g(fsxn,tyn(t)), g(fsxn,axn(t)), g(ftyn,byn(t)), 1 2 (g(fsxn,byn(t)) + g(ftyn,axn(t))) }) , 38 sunny chauhan, b. d. pant & mohammad imdad cubo 15, 3 (2013) passing to limit as n → ∞, we get g(fz,w(t)) ≤ φ ( max { g(fz,w(t)), g(fz,z(t)), g(fw,w(t)), 1 2 (g(fz,w(t)) + g(fw,z(t))) }) , = φ ( max { g(fz,w(t)), g(1), g(1), 1 2 (g(fz,w(t)) + g(fz,w(t))) }) = φ (max{g(fz,w(t)), 0, 0, g(fz,w(t))}) = φ (g(fz,w(t))) . owing lemma 2.9, we have z = w. we assert that az = z. on using (2) with x = z and y = yn, we get g(faz,byn(t)) ≤ φ ( max { g(fsz,tyn(t)), g(fsz,az(t)), g(ftyn,byn(t)), 1 2 (g(fsz,byn(t)) + g(ftyn,az(t))) }) , passing to limit as n → ∞, we get g(faz,z(t)) ≤ φ ( max { g(faz,z(t)), g(faz,az(t)), g(fz,z(t)), 1 2 (g(faz,z(t)) + g(fz,az(t))) }) , = φ ( max { g(faz,z(t)), g(1), g(1), 1 2 (g(faz,z(t)) + g(fz,az(t))) }) = φ (max {g(faz,z(t)), 0, 0, g(faz,z(t))}) = φ (g(faz,z(t))) . on employing lemma 2.9, we have z = az. therefore az = z = sz and hen e z is a ommon �xed point of (a, s). now we show that z is a ommon �xed point of (b, t). on using (2) with x = xn and y = z, we get g(faxn,bz(t)) ≤ φ ( max { g(fsxn,tz(t)), g(fsxn,axn(t)), g(ftz,bz(t)), 1 2 (g(fsxn,bz(t)) + g(ftz,axn(t))) }) , passing to limit as n → ∞, we get g(fz,bz(t)) ≤ φ ( max { g(fz,bz(t)), g(fz,z(t)), g(fbz,bz(t)), 1 2 (g(fz,bz(t)) + g(fbz,z(t))) }) , = φ ( max { g(fz,bz(t)), g(1), g(1), 1 2 (g(fz,bz(t)) + g(fbz,z(t))) }) = φ (max{g(fz,bz(t)), 0, 0, g(fz,bz(t))}) = φ (g(fz,bz(t))) . in view of lemma 2.9, we have z = bz. therefore bz = z = tz. thus we on lude that z is a ommon �xed point of a, b, s and t. the uniqueness of ommon �xed point is an easy onsequen e of inequality (2). cubo 15, 3 (2013) coin iden e and ommon �xed point theorems in n.a. menger . . . 39 theorem 3.3. let a, b, s and t be self mappings of an n.a. menger pm-spa e (x, f, t ). if the pairs (a, s) and (b, t) are subompatible and re ipro ally ontinuous, then (1) the pair (a, s) has a oin iden e point, (2) the pair (b, t) has a oin iden e point. (3) further, the mappings a, b, s and t have a unique ommon �xed point in x provided the involved mappings satisfy the inequality (2) of theorem 3.2. proof. sin e the pair (a, s) (also (b, t)) is subompatible and re ipro ally ontinuous, therefore there exists a sequen es {xn} in x su h that lim n→∞ axn = lim n→∞ sxn = z, for some z ∈ x, and lim n→∞ fasxn,saxn(t) = lim n→∞ faz,sz(t) = 1, for all t > 0, whereas in respe t of the pair (b, t), there exists a sequen e {yn} in x with lim n→∞ byn = lim n→∞ tyn = w, for some w ∈ x, and lim n→∞ fbtxn,tbxn(t) = lim n→∞ fbz,tz(t) = 1, for all t > 0. therefore, az = sz and bw = tw, i.e., z is a oin iden e point of the pair (a, s) whereas w is a oin iden e point of the pair (b, t). the rest of the proof an be ompleted on the lines of theorem 3.2. remark 3.4. the on lusions of theorem 3.2 and theorem 3.3 remain true if we repla e the inequality (2) by one of the following: g(fax,by(t)) ≤ φ (maxg(fsx,ty(t)), g(fsx,ax(t)), g(fty,by(t)), g(fsx,by(t))) , (5) for all x, y ∈ x, t > 0, where g ∈ ω and φ satis�es the ondition (φ). or, g(fax,by(t)) ≤ φ (max g(fsx,ty(t)), g(fsx,ax(t)), g(fty,by(t))) , (6) for all x, y ∈ x, t > 0, where g ∈ ω and φ satis�es the ondition (φ). 40 sunny chauhan, b. d. pant & mohammad imdad cubo 15, 3 (2013) or, g(fax,by(t)) ≤ φ     max    g(fsx,ty(t)) + g(fsx,ax(t)) + g(fty,by(t)), g(fsx,ax(t)) + g(fsx,by(t)), g(fax,ty(t)) + +g(fty,by(t))        , (7) for all x, y ∈ x, t > 0, where g ∈ ω and φ satis�es the ondition (φ). remark 3.5. theorem 3.2 and theorem 3.3 (also in view of remark 3.4) improve the results of rao and ramudu [29, theorem 14℄, khan and sumitra [23, theorem 2℄ and kutuk u and sharma [26, theorem 1℄. by hoosing a, b, s and t suitably, we an drive a multitude of ommon �xed point theorems for a pair or triod of mappings. as a sample, we outline the following natural result for a pair of self mappings. corollary 3.6. let a and s be self mappings of an n.a. menger pm-spa e (x, f, t ). if the pair (a, s) is ompatible and sub-sequentially ontinuous (alternately subompatible and re ipro ally ontinuous), then (1) the pair (a, s) has a oin iden e point. (2) there exists φ ∈ φ su h that, g(fax,ay(t)) ≤ φ ( max { g(fsx,sy(t)), g(fsx,ax(t)), g(fsy,ay(t)), 1 2 (g(fsx,ay(t)) + g(fsy,ax(t))) }) , (8) holds for all x, y ∈ x, t > 0 and g ∈ ω. then a and s have a unique ommon �xed point in x. remark 3.7. the results similar to corollary 3.6 an also be outlined in respe t of inequalities (5)-(7). now we give some illustrative examples. example 3.8. let (x, d) be a metri spa e with the usual metri d where x = [0, ∞) and (x, f, t ) be the indu ed n.a. menger pm-spa e with g(t) = 1−t for all t ∈ [0, 1], and fx,y(t) = h(t−d(x, y)) for all x, y ∈ x and all t > 0 and t (a, b) = min{a, b} for all a, b ∈ [0, 1]. set a = b and s = t. de�ne the self mappings a and s on x by a(x) = { x 4 , if x ∈ [0, 1]; 5x − 4, if x ∈ (1, ∞). s(x) = { x 5 , if x ∈ [0, 1]; 4x − 3, if x ∈ (1, ∞). cubo 15, 3 (2013) coin iden e and ommon �xed point theorems in n.a. menger . . . 41 consider a sequen e {xn} = { 1 n } n∈n in x. then lim n→∞ a(xn) = lim n→∞ ( 1 4n ) = 0 = lim n→∞ ( 1 5n ) = lim n→∞ s(xn). next, lim n→∞ as(xn) = lim n→∞ a ( 1 5n ) = lim n→∞ ( 1 20n ) = 0 = a(0), lim n→∞ sa(xn) = lim n→∞ s ( 1 4n ) = lim n→∞ ( 1 20n ) = 0 = s(0), and lim n→∞ fasxn,saxn(t) = 1, for all t > 0. consider another sequen e {xn} = { 1 + 1 n } n∈n in x. then lim n→∞ a(xn) = lim n→∞ ( 5 + 5 n − 4 ) = 1 = lim n→∞ ( 4 + 4 n − 3 ) = lim n→∞ s(xn). also, lim n→∞ as(xn) = lim n→∞ a ( 1 + 4 n ) = lim n→∞ ( 5 + 20 n − 4 ) = 1 6= a(1), lim n→∞ sa(xn) = lim n→∞ s ( 1 + 5 n ) = lim n→∞ ( 4 + 20 n − 3 ) = 1 6= s(1), but limn→∞ fasxn,saxn(t) = 1. thus, the pair (a, s) is ompatible as well as sub-sequentially ontinuous but not re ipro ally ontinuous. therefore all the onditions of corollary 3.6 are satis�ed. here, 0 is a oin iden e as well as unique ommon �xed point of the pair (a, s). it is noted that this example annot be overed by those �xed point theorems whi h involve ompatibility and re ipro al ontinuity both or by involving onditions on ompleteness (or losedness) of underlying spa e (or subspa es). also, in this example neither x is omplete nor any subspa e a(x) = [ 0, 1 4 ] ∪ (1, ∞) and s(x) = [ 0, 1 5 ] ∪ (1, ∞) are losed. it is noted that this example annot be overed by those �xed point theorems whi h involve ompatibility and re ipro al ontinuity both. example 3.9. in the setting of example 3.8, de�ne x = r (set of real numbers) and the self mappings a and s on x by a(x) = { x 4 , if x ∈ (−∞, 1); 5x − 4, if x ∈ [1, ∞). s(x) = { x + 3, if x ∈ (−∞, 1); 4x − 3, if x ∈ [1, ∞). consider a sequen e {xn} = { 1 + 1 n } n∈n in x. then lim n→∞ a(xn) = lim n→∞ ( 5 + 5 n − 4 ) = 1 = lim n→∞ ( 4 + 4 n − 3 ) = lim n→∞ s(xn). 42 sunny chauhan, b. d. pant & mohammad imdad cubo 15, 3 (2013) also, lim n→∞ as(xn) = lim n→∞ a ( 1 + 4 n ) = lim n→∞ ( 5 + 20 n − 4 ) = 1 = a(1), lim n→∞ sa(xn) = lim n→∞ s ( 1 + 5 n ) = lim n→∞ ( 4 + 20 n − 3 ) = 1 = s(1), and lim n→∞ fasxn,saxn(t) = 1, for all t > 0. consider another sequen e {xn} = { 1 n − 4 } n∈n in x. then lim n→∞ a(xn) = lim n→∞ ( 1 4n − 1 ) = −1 = lim n→∞ ( 1 n − 4 + 3 ) = lim n→∞ s(xn). next, lim n→∞ as(xn) = lim n→∞ a ( 1 n − 1 ) = lim n→∞ ( 1 4n − 1 4 ) = − 1 4 = a(−1), lim n→∞ sa(xn) = lim n→∞ s ( 1 4n − 1 ) = lim n→∞ ( 1 4n − 1 + 3 ) = 2 = s(−1), and limn→∞ fasxn,saxn(t) 6= 1. thus, the pair (a, s) is re ipro ally ontinuous as well as subompatible but not ompatible. therefore all the onditions of corollary 3.6 are satis�ed. thus 1 is a oin iden e as well as unique ommon �xed point of the pair (a, s). it is also noted that this example too annot be overed by those �xed point theorems whi h involve ompatibility and re ipro al ontinuity both. re eived: june 2012. a epted: september 2013. referen es [1℄ m.a. al-thaga� and n. shahzad, generalized i-nonexpansive selfmaps and invariant approximations, a ta math. sin. 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[34℄ s.l. singh, b.d. pant and s. chauhan, fixed point theorems in non-ar himedean menger pm-spa es, j. nonlinear anal. optim. theory appl. 3(2) (2012), 153�160. cubo a mathematical journal vol.16, no¯ 02, (135–148). june 2014 coupled coincidence points for generalized (ψ, ϕ)-pair mappings in ordered cone metric spaces sushanta kumar mohanta department of mathematics, west bengal state university, barasat, 24 parganas (north), kolkata 700126, west bengal, india. smwbes@yahoo.in abstract the existence of coupled coincidence points for mappings satisfying generalized contractive conditions related to ψ and ϕ-maps in an ordered cone metric space is proved. our results extend and generalize some well-known comparable results in the existing literature. resumen se prueba la existencia de puntos coincidentes acoplados para aplicaciones que satisfacen las condiciones de contractividad generalizada relacionada a las aplicaciones ψ y ϕ en un espacio métrico cono ordenados. nuestro resultado extiende y generaliza algunos resultados comparables conocidos en la literatura. keywords and phrases: cone metric space, ψ-map, ϕ-map, coupled coincidence point. 2010 ams mathematics subject classification: 54h25, 47h10. 136 sushanta kumar mohanta cubo 16, 2 (2014) 1 introduction fixed point theory plays a major role in mathematics because of its applications in many important areas such as optimization, mathematical models, nonlinear and adaptive control systems. over the past two decades a considerable amount of research work for the development of metric fixed point theory have executed by numerous mathematicians. the fixed points for certain mappings in ordered metric spaces has been studied by ran and reurings [16]. in [11] nieto and lópez extended the result of ran and reurings [16] for nondecreasing mappings and applied their results to obtain a unique solution for a first order differential equation. the existence of coupled fixed points in partially ordered metric spaces was first investigated by bhaskar and laksmikantham [3]. so far, many mathematicians have studied coupled fixed point results for mappings under various contractive conditions in different metric spaces. in 2007, huang and zhang [5] introduced the concept of cone metric spaces and proved some important fixed point theorems. afterwards, sabetghadam and masiha [17] obtained some fixed point results for generalized ϕ-pair mappings in cone metric spaces. the purpose of this paper is to obtain sufficient conditions for existence of coupled coincidence points for mappings satisfying generalized contractive conditions related to ψ and ϕ -maps in ordered cone metric spaces. 2 preliminaries in this section we need to recall some basic notations, definitions, and necessary results from existing literature. definition 1. [3] let (x, ⊑) be a partially ordered set and f : x × x → x be a self-map. one can say that f has the mixed monotone property if f(x, y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for all x1, x2 ∈ x, x1 ⊑ x2 implies f(x1, y) ⊑ f(x2, y) for any y ∈ x, and for all y1, y2 ∈ x, y1 ⊒ y2 implies f(x, y1) ⊑ f(x, y2) for any x ∈ x. definition 2. [4] let (x, ⊑) be a partially ordered set and f : x × x → x and g : x → x be two self-mappings. f has the mixed g-monotone property if f is monotone g-nondecreasing in its first argument and is monotone g-nonincreasing in its second argument, that is, for all x1, x2 ∈ x, gx1 ⊑ gx2 implies f(x1, y) ⊑ f(x2, y) for any y ∈ x, and for all y1, y2 ∈ x, gy1 ⊑ gy2 implies f(x, y1) ⊒ f(x, y2) for any x ∈ x. definition 3. [3] an element (x, y) ∈ x × x is called a coupled fixed point of the mapping f : x × x → x if x = f(x, y) and y = f(y, x). definition 4. [8] an element (x, y) ∈ x × x is called (i) a coupled coincidence point of the mappings f : x × x → x and g : x → x if gx = f(x, y) and gy = f(y, x), cubo 16, 2 (2014) coupled coincidence points for generalized (ψ, ϕ)-pair . . . 137 (ii) a common coupled fixed point of the mappings f : x×x → x and g : x → x if x = gx = f(x, y) and y = gy = f(y, x). definition 5. [4] let x be a nonempty set. one can say that the mappings f : x × x → x and g : x → x are commutative if g(f(x, y)) = f(gx, gy), for all x, y ∈ x. let e be a real banach space and θ denote the zero element in e. a cone p is a subset of e such that (i) p is closed, nonempty and p ̸= {θ}; (ii) a, b ∈ r, a, b ≥ 0, x, y ∈ p ⇒ ax + by ∈ p; (iii) p ∩ (−p) = {θ}. for any cone p ⊆ e, we can define a partial ordering ≼ on e with respect to p by x ≼ y if and only if y − x ∈ p. we shall write x ≺ y (equivalently, y ≻ x) if x ≼ y and x ̸= y, while x ≪ y will stand for y − x ∈ int(p), where int(p) denotes the interior of p. the cone p is called normal if there is a number k > 0 such that for all x, y ∈ e, θ ≼ x ≼ y implies ∥x∥ ≤ k ∥y∥. the least positive number satisfying the above inequality is called the normal constant of p. rezapour and hamlbarani [13] proved that there are no normal cones with normal constant k < 1. definition 6. [2] let p be a cone. a nondecreasing mapping ϕ : p → p is called a ϕ-map if (ϕ1) ϕ(θ) = θ and θ ≺ ϕ(w) ≺ w for w ∈ p \ {θ}, (ϕ2) w − ϕ(w) ∈ int(p) for every w ∈ int(p), (ϕ3) lim n→∞ ϕn(w) = θ for every w ∈ p \ {θ}. definition 7. [17] let p be a cone and let (wn) be a sequence in p. one says that wn → θ if for every ϵ ∈ p with θ ≪ ϵ there exists n0 ∈ n such that wn ≪ ϵ for all n ≥ n0. a nondecreasing mapping ψ : p → p is called a ψ-map if (ψ1)ψ(w) = θ if and only if w = θ, (ψ2) for every wn ∈ p, wn → θ if and only if ψ(wn) → θ, (ψ3) for every w1, w2 ∈ p, ψ(w1 + w2) ≼ ψ(w1) + ψ(w2). definition 8. [5] let x be a nonempty set. suppose the mapping d : x × x → e satisfies (i) θ ≼ d(x, y) for all x, y ∈ x and d(x, y) = θ if and only if x = y ; (ii) d(x, y) = d(y, x) for all x, y ∈ x; 138 sushanta kumar mohanta cubo 16, 2 (2014) (iii) d(x, y) ≼ d(x, z) + d(z, y) for all x, y, z ∈ x. then d is called a cone metric on x, and (x, d) is called a cone metric space. definition 9. [5] let (x, d) be a cone metric space. let (xn) be a sequence in x and x ∈ x. if for every c ∈ e with θ ≪ c there is a natural number n0 such that for all n > n0, d(xn, x) ≪ c, then (xn) is said to be convergent and (xn) converges to x, and x is the limit of (xn). we denote this by lim n→∞ xn = x or xn → x (n → ∞). definition 10. [5] let (x, d) be a cone metric space, (xn) be a sequence in x. if for any c ∈ e with θ ≪ c, there is a natural number n0 such that for all n, m > n0, d(xn, xm) ≪ c, then (xn) is called a cauchy sequence in x. definition 11. [5] let (x, d) be a cone metric space, if every cauchy sequence is convergent in x, then x is called a complete cone metric space. lemma 1. [19] every cone metric space (x, d) is a topological space. for c ≫ θ, c ∈ e, x ∈ x let b(x, c) = {y ∈ x : d(y, x) ≪ c} and β = {b(x, c) : x ∈ x, c ≫ θ}. then τc = {u ⊆ x : ∀x ∈ u, ∃b ∈ β, x ∈ b ⊆ u} is a topology on x. definition 12. [19] let (x, d) be a cone metric space. a map t : (x, d) → (x, d) is called sequentially continuous if xn ∈ x, xn → x implies txn → tx. lemma 2. [19] let (x, d) be a cone metric space, and t : (x, d) → (x, d) be any map. then, t is continuous if and only if t is sequentially continuous. lemma 3. [14] let e be a real banach space with a cone p. then (i) if a ≪ b and b ≪ c, then a ≪ c. (ii) if a ≼ b and b ≪ c, then a ≪ c. lemma 4. [5] let e be a real banach space with cone p. then one has the following. (i) if θ ≪ c, then there exists δ > 0 such that ∥b∥ < δ implies b ≪ c. (ii) if an, bn are sequences in e such that an → a, bn → b and an ≼ bn for all n ≥ 1, then a ≼ b. proposition 1. [6] if e is a real banach space with cone p and if a ≼ λa where a ∈ p and 0 ≤ λ < 1 then a = θ. 3 main results in this section we always suppose that e is a real banach space, p is a cone in e with int(p) ̸= ∅ and ≼ is the partial ordering on e with respect to p. also, we mean by ϕ the ϕ-map and by ψ the ψ-map, unless otherwise stated. now, we state and prove our main results. cubo 16, 2 (2014) coupled coincidence points for generalized (ψ, ϕ)-pair . . . 139 theorem 1. let (x, ⊑) be a partially ordered set and (x, d) be a complete cone metric space. suppose f : x × x → x and g : x → x be two continuous and commuting functions with f(x × x) ⊆ g(x). let f satisfy mixed g-monotone property and ψ(d(f(x, y), f(u, v)) + d(f(y, x), f(v, u))) ≼ ϕ(ψ(d(gx, gu) + d(gy, gv))) (1) for all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. proof. let x0, y0 be such that gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0. since f(x × x) ⊆ g(x), we can choose x1, y1 ∈ x such that gx1 = f(x0, y0) and gy1 = f(y0, x0). continuing this process one can construct sequences (xn) and (yn) in x such that gxn+1 = f(xn, yn) and gyn+1 = f(yn, xn) for all n ≥ 0. we shall show that gxn ⊑ gxn+1 and gyn ⊒ gyn+1 (2) for all n ≥ 0. we shall use the mathematical induction. for n = 0, (2) follows by the choice of x0 and y0. suppose now (2) holds for n = k, k ≥ 0. then gxk ⊑ gxk+1 and gyk ⊒ gyk+1. mixed g-monotonicity of f now implies that gxk+1 = f(xk, yk) ⊑ f(xk+1, yk) ⊑ f(xk+1, yk+1) = gxk+2. similarly, we have gyk+1 ⊒ gyk+2. thus (2) follows for k + 1. hence, by the mathematical induction we conclude that (2) holds for n ≥ 0. now for all n ∈ n, ψ(d(gxn, gxn+1) + d(gyn, gyn+1)) = ψ ⎛ ⎜ ⎜ ⎝ d(f(xn−1, yn−1), f(xn, yn)) +d(f(yn−1, xn−1), f(yn, xn)) ⎞ ⎟ ⎟ ⎠ ≼ ϕ(ψ(d(gxn−1, gxn) + d(gyn−1, gyn))) ≼ ϕ2(ψ(d(gxn−2, gxn−1) + d(gyn−2, gyn−1))) · · · ≼ ϕn(ψ(d(gx0, gx1) + d(gy0, gy1))). let ϵ ∈ int(p), then by (ϕ2), ϵ0 = ϵ − ϕ(ϵ) ∈ int(p). by (ϕ3), lim n→∞ ϕn(ψ(d(gx0, gx1) + d(gy0, gy1))) = θ. 140 sushanta kumar mohanta cubo 16, 2 (2014) so, there exists n0 ∈ n such that for all m ≥ n0, ψ(d(gxm, gxm+1) + d(gym, gym+1)) ≪ ϵ − ϕ(ϵ). we show that ψ(d(gxm, gxn+1) + d(gym, gyn+1)) ≪ ϵ, (3) for a fixed m ≥ n0 and n ≥ m. clearly, this holds for n = m. we now suppose that (3) holds for some n ≥ m. then by using (ψ3) and condition (1), we obtain ψ(d(gxm, gxn+2) + d(gym, gyn+2)) ≼ ψ ⎛ ⎜ ⎜ ⎝ d(gxm, gxm+1) + d(gxm+1, gxn+2) +d(gym, gym+1) + d(gym+1, gyn+2) ⎞ ⎟ ⎟ ⎠ ≼ ψ(d(gxm, gxm+1) + d(gym, gym+1)) +ψ(d(gxm+1, gxn+2) + d(gym+1, gyn+2)) ≼ ψ(d(gxm, gxm+1) + d(gym, gym+1)) +ϕ(ψ(d(gxm, gxn+1) + d(gym, gyn+1))) ≪ ϵ − ϕ(ϵ) + ϕ(ϵ) = ϵ. therefore, by induction (3) holds. since ψ is nondecreasing, it follows from (3) that ψ(d(gxm, gxn+1)) ≼ ψ(d(gxm, gxn+1) + d(gym, gyn+1)) ≪ ϵ for a fixed m ≥ n0 and n ≥ m. similarly, ψ(d(gym, gyn+1)) ≪ ϵ for a fixed m ≥ n0 and n ≥ m. therefore, by using (ψ2) we deduce that (gxn) and (gyn) are cauchy sequences in x. since x is complete, there exist x∗, y∗ ∈ x such that gxn → x ∗ and gyn → y ∗ as n → ∞. by continuity of g we get lim n→∞ ggxn = gx ∗ and lim n→∞ ggyn = gy ∗. commutativity of f and g now implies that ggxn = g(f(xn−1, yn−1)) = f(gxn−1, gyn−1) for all n ∈ n and ggyn = g(f(yn−1, xn−1)) = f(gyn−1, gxn−1) for all n ∈ n. since f is continuous, gx∗ = lim n→∞ ggxn = lim n→∞ f(gxn−1, gyn−1) = f( lim n→∞ gxn−1, lim n→∞ gyn−1) = f(x∗, y∗) cubo 16, 2 (2014) coupled coincidence points for generalized (ψ, ϕ)-pair . . . 141 and gy∗ = lim n→∞ ggyn = lim n→∞ f(gyn−1, gxn−1) = f( lim n→∞ gyn−1, lim n→∞ gxn−1) = f(y∗, x∗). thus, f and g have a coupled coincidence point. if we let ψ be the identity map in theorem 1, then we have the following corollary. corolary 1. let (x, ⊑) be a partially ordered set and (x, d) be a complete cone metric space. suppose f : x × x → x and g : x → x be two continuous and commuting functions with f(x × x) ⊆ g(x). let f satisfy mixed g-monotone property and d(f(x, y), f(u, v)) + d(f(y, x), f(v, u)) ≼ ϕ(d(gx, gu) + d(gy, gv)) for all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. corolary 2. let (x, ⊑) be a partially ordered set and (x, d) be a complete cone metric space. suppose f : x × x → x and g : x → x be two continuous and commuting functions with f(x × x) ⊆ g(x). let f satisfy mixed g-monotone property and d(f(x, y), f(u, v)) + d(f(y, x), f(v, u)) ≼ k(d(gx, gu) + d(gy, gv)) for some k ∈ [0, 1) and all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. proof. the proof can be obtained from theorem 1 by taking ψ = i, the identity map and ϕ(x) = kx, where k ∈ [0, 1) is a constant. the following corollary is a generalization of the result [[3], theorem 2.1]. corolary 3. let (x, ⊑) be a partially ordered set and (x, d) be a complete cone metric space. suppose f : x × x → x and g : x → x be two continuous and commuting functions with f(x × x) ⊆ g(x). let f satisfy mixed g-monotone property and d(f(x, y), f(u, v)) ≼ ad(gx, gu) + bd(gy, gv) (4) for some a, b ∈ [0, 1) with a + b < 1 and all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. 142 sushanta kumar mohanta cubo 16, 2 (2014) proof. let x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). using (4), we have d(f(x, y), f(u, v)) ≼ ad(gx, gu) + bd(gy, gv) and d(f(y, x), f(v, u)) ≼ ad(gy, gv) + bd(gx, gu). therefore, d(f(x, y), f(u, v)) + d(f(y, x), f(v, u)) ≼ (a + b)(d(gx, gu) + d(gy, gv)). the result follows from corollary 2. theorem 2. let (x, ⊑) be a partially ordered set and (x, d) be a cone metric space. suppose f : x × x → x and g : x → x be two functions such that f(x × x) ⊆ g(x) and (g(x), d) is a complete subspace of x. let f satisfy mixed g-monotone property and ψ(d(f(x, y), f(u, v)) + d(f(y, x), f(v, u))) ≼ ϕ(ψ(d(gx, gu) + d(gy, gv))) for all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). suppose x has the following property: (i) if a nondecreasing sequence (xn) → x, then xn ⊑ x for all n. (ii) if a nonincreasing sequence (yn) → y, then y ⊑ yn for all n. if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. proof. consider cauchy sequences (gxn) and (gyn) as in the proof of theorem 1. since (g(x), d) is complete, there exist x∗, y∗ ∈ x such that gxn → gx ∗ and gyn → gy ∗. it is to be noted that the sequence (gxn) is nondecreasing and converges to gx ∗. by given condition (i) we have, therefore, gxn ⊑ gx ∗ for all n ≥ 0 and similarly gyn ⊒ gy ∗ for all n ≥ 0. by (ψ2), for θ ≪ c, one can choose a natural number n0 such that ψ(d(gxn, gx ∗)) ≪ c 4 and ψ(d(gyn, gy ∗)) ≪ c 4 for all n ≥ n0. cubo 16, 2 (2014) coupled coincidence points for generalized (ψ, ϕ)-pair . . . 143 then, ψ ⎛ ⎜ ⎜ ⎝ d(f(x∗, y∗), gx∗) +d(f(y∗, x∗), gy∗) ⎞ ⎟ ⎟ ⎠ ≼ ψ ⎛ ⎜ ⎜ ⎝ d(f(x∗, y∗), gxn+1) + d(gxn+1, gx ∗) +d(f(y∗, x∗), gyn+1) + d(gyn+1, gy ∗) ⎞ ⎟ ⎟ ⎠ ≼ ψ(d(gxn+1, gx ∗) + d(gyn+1, gy ∗)) +ψ ⎛ ⎜ ⎜ ⎝ d(f(x∗, y∗), f(xn, yn)) +d(f(y∗, x∗), f(yn, xn)) ⎞ ⎟ ⎟ ⎠ ≼ ψ(d(gxn+1, gx ∗)) + ψ(d(gyn+1, gy ∗)) +ϕ(ψ(d(gxn, gx ∗) + d(gyn, gy ∗))) ≺ ψ(d(gxn+1, gx ∗)) + ψ(d(gyn+1, gy ∗)) +ψ(d(gxn, gx ∗) + d(gyn, gy ∗)) ≼ ψ(d(gxn+1, gx ∗)) + ψ(d(gyn+1, gy ∗)) +ψ(d(gxn, gx ∗)) + ψ(d(gyn, gy ∗)) ≪ c 4 + c 4 + c 4 + c 4 = c. so, c i − ψ(d(f(x∗, y∗), gx∗) + d(f(y∗, x∗), gy∗)) ∈ p, for all i ≥ 1. since c i → θ as i → ∞ and p is closed, −ψ(d(f(x∗, y∗), gx∗) + d(f(y∗, x∗), gy∗)) ∈ p. but p ∩ (−p) = θ gives that ψ(d(f(x∗, y∗), gx∗) + d(f(y∗, x∗), gy∗)) = θ. by (ψ1), we get d(f(x∗, y∗), gx∗) + d(f(y∗, x∗), gy∗) = θ. this shows that d(f(x∗, y∗), gx∗) = d(f(y∗, x∗), gy∗) = θ and so f(x∗, y∗) = gx∗, f(y∗, x∗) = gy∗. thus, f and g have a coupled coincidence point. if we let ψ be the identity map in theorem 2, then we have the following corollary. corolary 4. let (x, ⊑) be a partially ordered set and (x, d) be a cone metric space. suppose f : x × x → x and g : x → x be two functions such that f(x × x) ⊆ g(x) and (g(x), d) is a complete subspace of x. let f satisfy mixed g-monotone property and d(f(x, y), f(u, v)) + d(f(y, x), f(v, u)) ≼ ϕ(d(gx, gu) + d(gy, gv)) for all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). suppose x has the following property: (i) if a nondecreasing sequence (xn) → x, then xn ⊑ x for all n. (ii) if a nonincreasing sequence (yn) → y, then y ⊑ yn for all n. if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. 144 sushanta kumar mohanta cubo 16, 2 (2014) corolary 5. let (x, ⊑) be a partially ordered set and (x, d) be a cone metric space. suppose f : x × x → x and g : x → x be two functions such that f(x × x) ⊆ g(x) and (g(x), d) is a complete subspace of x. let f satisfy mixed g-monotone property and d(f(x, y), f(u, v)) + d(f(y, x), f(v, u)) ≼ k(d(gx, gu) + d(gy, gv)) for some k ∈ [0, 1) and all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). suppose x has the following property: (i) if a nondecreasing sequence (xn) → x, then xn ⊑ x for all n. (ii) if a nonincreasing sequence (yn) → y, then y ⊑ yn for all n. if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. proof. the proof can be obtained from theorem 2 by taking ψ = i, the identity map and ϕ(x) = kx, where k ∈ [0, 1) is a constant. the following corollary is a generalization of the result [[3], theorem 2.2]. corolary 6. let (x, ⊑) be a partially ordered set and (x, d) be a cone metric space. suppose f : x × x → x and g : x → x be two functions such that f(x × x) ⊆ g(x) and (g(x), d) is a complete subspace of x. let f satisfy mixed g-monotone property and d(f(x, y), f(u, v)) ≼ ad(gx, gu) + bd(gy, gv) for some a, b ∈ [0, 1) with a + b < 1 and all x, y, u, v ∈ x with (gx ⊑ gu) and (gy ⊒ gv) or (gx ⊒ gu) and (gy ⊑ gv). suppose x has the following property: (i) if a nondecreasing sequence (xn) → x, then xn ⊑ x for all n. (ii) if a nonincreasing sequence (yn) → y, then y ⊑ yn for all n. if there exist x0, y0 ∈ x satisfying gx0 ⊑ f(x0, y0) and f(y0, x0) ⊑ gy0, then f and g have a coupled coincidence point. proof. the proof follows from theorem 2 by an argument similar to that used in corollary 3. theorem 3. in addition to hypothesis of either theorem 1 or theorem 2, suppose that any two elements of g(x) are comparable and g is one-one. then f and g have a coupled coincidence point of the form (x∗, x∗) for some x∗ ∈ x. proof. we first note that the set of coupled coincidence points of f and g is nonempty. we will show that if (x∗, y∗) is a coupled coincidence point of f and g, then x∗ = y∗. since the elements of g(x) are comparable, we may assume that gx∗ ⊑ gy∗. suppose that d(gx∗, gy∗) ̸= θ. then, by using (ϕ1) we have ψ(d(gx∗, gy∗) + d(gy∗, gx∗)) = ψ(d(f(x∗, y∗), f(y∗, x∗)) + d(f(y∗, x∗), f(x∗, y∗))) ≼ ϕ(ψ(d(gx∗, gy∗) + d(gy∗, gx∗))) ≺ ψ(d(gx∗, gy∗) + d(gy∗, gx∗)), cubo 16, 2 (2014) coupled coincidence points for generalized (ψ, ϕ)-pair . . . 145 a contradiction. therefore, d(gx∗, gy∗) = θ which gives that gx∗ = gy∗. since g is one-one, it follows that x∗ = y∗. we conclude with an example. example 1. let e = r2, the euclidean plane and p = {(x, x) ∈ r2 : x ≥ 0} a cone in e. let x = [0, ∞) with the usual ordering and define d : x × x → e by d(x, y) = (| x − y |, | x − y |) for all x, y ∈ x. then (x, d) is a partially ordered complete cone metric space. define f : x×x → x as follows: f(x, y) = ⎧ ⎪⎪⎨ ⎪⎪⎩ x−y 6 , if x ≥ y 0, if x < y, for all x, y ∈ x and g : x → x with gx = x 3 for all x ∈ x. then f(x × x) ⊆ g(x) = x and f satisfy mixed g-monotone property. also f and g are continuous and commuting, g(0) ≤ f(0, 1) and g(1) ≥ f(1, 0). let ψ,ϕ : p → p be defined by ψ(x, x) = (x 2 , x 2 ) and ϕ(x, x) = (3x 4 , 3x 4 ). let x, y, u, v ∈ x be such that gx ≤ gu and gy ≥ gv. now, we have case-i (y > x and v > u). then ψ(d(f(x, y), f(u, v)) + d(f(y, x), f(v, u))) = ψ ( d(0, 0) + d ( y − x 6 , v − u 6 )) = ψ ( | y − x − v + u | 6 , | y − x − v + u | 6 ) = ( | y − x − v + u | 12 , | y − x − v + u | 12 ) ≼ ( | x − u | 12 + | y − v | 12 , | x − u | 12 + | y − v | 12 ) . (5) again, ϕ(ψ(d(gx, gu) + d(gy, gv))) = ϕ ( ψ ( d ( x 3 , u 3 ) + d ( y 3 , v 3 ))) = ϕ ( ψ (( | x − u | 3 , | x − u | 3 ) + ( | y − v | 3 , | y − v | 3 ))) = ( 3 | x − u | 24 + 3 | y − v | 24 , 3 | x − u | 24 + 3 | y − v | 24 ) . (6) 146 sushanta kumar mohanta cubo 16, 2 (2014) it follows from conditions (5) and (6) that ψ(d(f(x, y), f(u, v)) + d(f(y, x), f(v, u))) ≼ ϕ(ψ(d(gx, gu) + d(gy, gv))). case-ii (y > x and u ≥ v). then ψ(d(f(x, y), f(u, v)) + d(f(y, x), f(v, u))) = ψ ( d ( 0, u − v 6 ) + d ( y − x 6 , 0 )) = ψ (( u − v 6 , u − v 6 ) + ( y − x 6 , y − x 6 )) = ( u − v + y − x 12 , u − v + y − x 12 ) ≼ ( | x − u | 12 + | y − v | 12 , | x − u | 12 + | y − v | 12 ) ≺ ( 3 | x − u | 24 + 3 | y − v | 24 , 3 | x − u | 24 + 3 | y − v | 24 ) = ϕ(ψ(d(gx, gu) + d(gy, gv))). case-iii (x ≥ y and u ≥ v). then ψ(d(f(x, y), f(u, v)) + d(f(y, x), f(v, u))) = ψ ( d ( x − y 6 , u − v 6 ) + d(0, 0) ) = ( | x − y − u + v | 12 , | x − y − u + v | 12 ) ≼ ( | x − u | 12 + | y − v | 12 , | x − u | 12 + | y − v | 12 ) ≼ ( 3 | x − u | 24 + 3 | y − v | 24 , 3 | x − u | 24 + 3 | y − v | 24 ) = ϕ(ψ(d(gx, gu) + d(gy, gv))). the case x ≥ y and v > u is not possible. as gx ≤ gu and gy ≥ gv, it follows that x ≤ u and y ≥ v. so, v ≤ y ≤ x ≤ u when x ≥ y. thus, we have all the conditions of theorem 1. moreover, (0, 0) is the coupled coincidence point of f and g. received: november 2013. revised: march 2014. cubo 16, 2 (2014) coupled coincidence points for generalized (ψ, ϕ)-pair . . . 147 references [1] m. abbas and g. jungck, common fixed point results for noncommuting mappings without continuity in cone metric spaces, j. math. anal. appl., 341, 2008, 416-420. 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[20] s. wang, b. guo, distance in cone metric spaces and common fixed point theorems, applied mathematics letters, 24, 2011, 1735-1739. cubo a mathematical journal vol.15, no¯ 01, (33–47). march 2013 on fokker-planck and linearized korteweg-de vries type equations with complex spatial variables 1 ciprian g. gal florida international university, department of mathematics, dm 435b, miami, florida 33199 cgal@fiu.edu sorin g. gal department of mathematics and computer science, university of oradea, str. universitatii no. 1 410087, romania galso@uoradea.ro abstract in the present work, we construct solutions to a fokker-planck type equation with real time variable and complex spatial variable, and prove some properties. the equations are obtained from the complexification of the spatial variable by two different methods. firstly, one complexifies the spatial variable in the corresponding convolution integral in the solution, by replacing the usual sum of variables (translation) by an exponential product (rotation). secondly, one complexifies the spatial variable directly in the corresponding evolution equation and then one searches for analytic solutions. these methods are also applied to a linear evolution equation related to the korteweg-de vries equation. resumen en este trabajo construimos soluciones de una ecuación tipo fokker-planck con variable de tiempo real y variable espacial compleja. las ecuaciones se obtienen de la complejización de la variable espacial por dos métodos diferentes. primero, se complejiza la variable espacial en la integral de convolución respectiva en la solución reemplazando la suma usual de las variables (traslaciones) por un producto de exponenciales (rotación). luego, se complejiza la variable espacial directamente en la respectiva la ecuación de evolución y se busca por las soluciones anaĺıticas. estos métodos también se aplican a la ecuación de evolución lineal relacionada a la ecuación korteweg-de vries. keywords and phrases: fokker-planck equation, korteweg-de vries equation, complex convolution integrals, complex spatial variables. 2010 ams mathematics subject classification: 47d03, 47d06, 47d60. 1dedicated to professor gaston n’guerekata on the occasion of his 60th birthday. 34 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) 1 introduction let us consider the following initial value problem: { ∂u ∂t (t, x) = α(t)∂ 2 u ∂x2 (t, x) + β(t)xu(t, x), (t, x) ∈ r+ × r, u(0, x) = f(x), x ∈ r, (1) where α ∈ c ([0, +∞), r+) and β ∈ c ([0, +∞), r) . using the exponential operator method, it is shown in [6] that u(t, x) = ea(t)+d(t)[c(t)d(t)+b(t)+x] 2 √ πc(t) ∫∞ −∞ e−[u+b(t)+2c(t)d(t)] 2 /(4c(t))f(x − u)du, (2) is a solution of (1) provided that the integral in (2) converges, with a(t), b(t), c(t), d(t) depending on α(t), β(t), and given in [6, (54)] as follows: c(t) = ∫t 0 α(u)du, d(t) = ∫t 0 β(u)du, (3) b(t) = −2 ∫t 0 [ β(u) ∫u 0 α(s)ds ] du, a(t) = 2 ∫t 0 β(u) {∫u 0 [ β(s) ∫s 0 α(v)dv ] ds } du. note that by the assumptions on α, β, the functions a, b, c and d are differentiable for every t > 0, and that c(t) > 0, for all t > 0. for β(t) ≡ 0 and α(t) ≡ c (c =constant), we recapture the initial value problem for the classical heat equation. for β (t) 6= 0, α (t) 6= 0, the main equation in (1) is known as a fokker-planck type equation. in the second part of this article, we’ll devote our attention to the ”linearized” korteweg-de vries equation with real time variable and complex spatial variable. indeed, let us consider the well-known korteweg-de vries equation ∂u ∂t + αu ∂u ∂x = ∂3u ∂x3 , (4) where α ∈ r (see, e.g., widder [7]), and the related linear problem { ∂u ∂t (t, x) = ∂ 3 u ∂x3 (t, x), (t, x) ∈ r+ × r, limtց0 u(t, x) = f(x), x ∈ r. (5) for problem (5), the following is known to hold. theorem 1.1. ([7, theorem 4]) let f : r → r be continuous and of bounded variation, such that it satisfies the conditions: (i) the integral f(s) = ∫∞ −∞ e−sx f (x)dx cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 35 converges absolutely for s ∈ c with re(s) = c; (ii) let ∫∞ −∞ |f (c + iτ)|dτ < ∞ setting u(t, x) := ∫∞ −∞ k(x − y, t)f(y)dy = ∫∞ −∞ k(u, t)f(x − u)du, (6) where k(u, t) := 1 π ∫∞ 0 cos(uτ − tτ3)dτ = 1 (3t)1/3 ai ( −u (3t)1/3 ) , (7) the function u (t) satisfies (5). above, ai(u) := 1 π ∫∞ 0 cos(uτ + τ3/3)dτ is also called the airy function. moreover, it is well-known (see, e.g., widder [7, (5.1)]) that ∂k ∂t (x, t) = ∂3k ∂x3 (x, t), t > 0, x ∈ r. (8) it is natural to ask what happens if in the above equations we complexify the spatial variable and keep the time variable real? we shall proceed as follows. the complexification of the spatial variable in the above mentioned equations is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding formula for the solutions in (2) and (5), respectively, by replacing in the integral the usual sum of variables (translation) by an exponential product(rotation) and looking for solutions in a disk dr of radius r > 1. this method yields solutions that satisfy differential equations similar to (2) and (5). secondly, one directly complexifies the spatial variable in the corresponding evolution equations, and then one searches for analytic and non-analytic solutions for the resulting equation. the topic was already developed in detail for complex heat and laplace equations in [1, 2], for complex wave and telegraph equations in [3] and for complex schrödinger type equations in [4]. 2 generalized heat type equations with complex spatial variable let r ≥ 1 and let us now consider the open disk dr = {z ∈ c; |z| < r} and a(dr) = {f : dr → c; f is analytic on dr, continuous on dr}, endowed with the uniform norm ‖f‖r = sup{|f(z)|; z ∈ dr}. is well-known that (a(dr), ‖ · ‖r) is a banach space. if f ∈ a(dr), then we have f(z) = ∞∑ k=0 akz k, for all z ∈ dr. finally, ω1(f; δ)dr denotes the modulus of continuity, ω1 (f ; δ)dr = sup{|f (u) − f (v)|; |u − v | ≤ δ, u, v ∈ dr}. 36 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) concerning the system (1), we will first complexify the solution in (2) as follows. for f ∈ a(dr) and t > 0, let us replace x and the translation x−u in (2) by z and the rotation ze−iu, respectively, and consider the complex integral gt(f)(z) = ea(t)+d(t)[c(t)d(t)+b(t)+z] 2 √ πc(t) ∫∞ −∞ e−[u+b(t)+2c(t)d(t)] 2 /(4c(t))f(ze−iu)du, z ∈ dr. (9) the first goal of this section is to prove some properties of the complex integral (9). theorem 2.1. let r > 1 and f ∈ a(dr). (i) for all t > 0, gt(f) ∈ a(dr), namely gt(f) is continuous on dr, is analytic in dr, and the following holds: gt(f)(z) = e φ(t,z) ∞∑ k=0 akdk(t)z k, z ∈ dr, (10) where φ(t, z) = a(t) + d(t)[c(t)d(t) + b(t) + z] and, for all k ≥ 0, dk(t) := e −k 2 c(t)+ikg(t), g(t) := b(t) + 2c(t)d(t). (ii) setting wt(f)(z) := e −φ(t,z)gt(f)(z) = 1 2 √ πc(t) ∫∞ −∞ f(ze−iu)e−[u+g(t)] 2 /(4c(t))du, the following estimate holds: |wt(f)(z) − f(z)| ≤ (r + 1) [ 1 + 2√ π + |g(t)| 2 √ c(t) ] ω1(f; √ c(t)) dr , for all z ∈ dr, t > 0. (iii) the operator wt is contractive, that is, ‖wt(f )‖dr ≤ ‖f ‖dr , for all t > 0, f ∈ a(dr). (iv) let u(t, ϕ) := ea(t)d(t)[c(t)d(t)+b(t)+ϕ] ∞∑ k=0 akdk(t)z k, for every z 6= 0, z = re−iϕ such that r ∈ (0, r) , ϕ ∈ (0, 2π]. then, u(t, ϕ) satisfies ∂u ∂t (t, ϕ) = α(t) ∂2u ∂ϕ2 (t, ϕ) + β(t)ϕu(t, ϕ), (11) for (t, z) ∈ r+ × dr\ {0} , z = reiϕ, r ∈ (0, r), and u(0, ϕ) = f(reiϕ), ϕ ∈ (0, 2π], 0 < r ≤ r, f ∈ a(dr). (12) cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 37 proof. (i) for fixed z ∈ dr, we have f(ze−iu) = ∞∑ k=0 ake −ikuzk. (13) since |ake −iku| = |ak|, for all u ∈ r, and since ∞∑ k=0 akz k is absolutely convergent, it follows that ∞∑ k=0 ake −ikuzk is uniformly convergent with respect to u ∈ r. thus, on account of (13), we can integrate in (9) term by term. this yields gt(f)(z) = e φ(z) ∞∑ k=0 ak [ 1 2 √ πc(t) ∫+∞ −∞ e−iku · e−[u+g(t)] 2 /(4c(t))du ] zk == eφ(z) ∞∑ k=0 ak [ 1 2 √ πc(t) ∫+∞ −∞ (cos[k(v − g(t)] − i sin[k(v − g(t)])e−v 2 /(4c(t))dv ] zk := eφ(z)[i1 − ii2]. since sin(kv)e−v 2 /(4c(t)) is odd as function of v, we have i1 = ∞∑ k=0 ak [ cos(kg(t)) 1 2 √ πc(t) ∫+∞ −∞ cos(kv)e−v 2 /(4c(t))dv ] zk = ∞∑ k=0 ak[cos(kg(t)) · e−k 2 c(t)]zk and i2 = − ∞∑ k=0 ak[sin(kg(t))e −k 2 c(t)]zk. hence, these calculations give the formula (10) and prove the analyticity of gt(f)(z), as function of z ∈ dr. to prove the continuity in dr, it suffices to prove the continuity in z, of the function ht(f)(z) := 1 2 √ πc(t) ∫∞ −∞ e−[u+g(t)] 2 /(4c(t))f(ze−iu)du. to this end, let z0, zn ∈ dr be such that limn→∞ zn = z0. we get |ht(f)(zn) − ht(f)(z0)| ≤ 1 2 √ πc(t) ∫+∞ −∞ |f(zne −iu) − f(z0e −iu)|e−[u+g(t)] 2 /(4c(t)) du (14) ≤ 1 2 √ πc(t) ∫+∞ −∞ ω1(f; |zne −iu − z0e −iu|) dr e−[u+g(t)] 2 /(4c(t)) du = 1 2 √ πc(t) ∫+∞ −∞ ω1(f; |zn − z0|)dre −[u+g(t)] 2 /(4c(t)) du = ω1(f; |zn − z0|)dr. 38 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) passing to the limit as n → ∞, the continuity of ht(f)(z) at z0 ∈ dr is a consequence of (14) since f is continuous on dr. (ii) first, note that we can also write wt(f) (z) = 1 2 √ πc(t) ∫+∞ −∞ e−v 2 /(4c(t))f(ze−i(v−g(t)))dv. a simple calculation gives |wt(f)(z) − f(z)| ≤ 1 2 √ πc ∫+∞ −∞ |f(ze−i(u−g)) − f(z)|e−u 2 /(4c) du (15) ≤ r + 1 2 √ πc ∫∞ −∞ ω1(f; |1 − e −i(u−g)|) dr e−u 2 /(4c) du = r + 1 2 √ πt ∫+∞ −∞ ω1 ( f; 2 ∣∣∣∣sin u − g 2 ∣∣∣∣ ) dr e−u 2 /(4c) du ≤ r + 1 2 √ πt ∫+∞ −∞ ω1(f; |u − g|)dre −u 2 /(4c) du ≤ r + 1 2 √ πt ∫+∞ −∞ ω1(f; √ c) dr ( |u − g|√ c(t) + 1 ) e−u 2 /(4c) du ≤ (r + 1) [ ω1(f; √ c) dr + ω1(f; √ c) dr√ c2 √ πc ∫∞ 0 2ue−u 2 /(4c) du ] + (r + 1) [ |a| · ω1(f; √ c) dr√ c2 √ πc ∫∞ 0 e−u 2 /(4c) du ] . since ∫∞ 0 2ue−u 2 /(4c(t))du = 4c (t), we infer |wt(f)(z) − f(z)| ≤ ω1(f; √ c) dr [ 1 + 2√ π + |g| 2 √ c ] (r + 1). this proves (ii). (iii) since 1 2 √ πc ∫+∞ −∞ e−u 2 /(4c)du = 1, we also have |wt(f)(z)| ≤ 1 2 √ πc ∫+∞ −∞ |f(ze−i(u−g))|e−u 2 /(4c)du ≤ ‖f‖ dr , z ∈ dr, which proves the claim. (iv) let f ∈ a(dr), and z ∈ dr, z = reiϕ, 0 < r < r. set b(t, ϕ) := a(t) + d(t)[c(t)d(t) + b(t) + ϕ], ak(t) := −k 2c(t) + ikg(t); by (i), we have dk(t) = e ak(t), and we can write u(t, ϕ) = eb(t,ϕ) ∞∑ k=0 ake ak(t)rkekiϕ. (16) cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 39 consequently, since the series representation (10) for gt(f)(z) is uniformly convergent in any compact disk included in dr, it follows that u(t, ϕ) can be differentiated term by term with respect to t and ϕ. therefore, simple calculations show that u, given by (16), satisfies. we emphasize again that in (11) we must take z 6= 0 simply because z = 0 has no polar representation, that is, z = 0 cannot be represented as function of ϕ. finally, it is also easy to check that u(t, ϕ) satisfies (12) since a(0) = b(0) = c(0) = d(0) = 0 on account of (3). this completes the proof of the theorem. remark 2.2. in theorem 2.1-(ii), we have limt→0 c(t) = 0. therefore, there exists a sufficiently small δ0 > 0 such that 0 < c(t) < 1, for all t ∈ [0, δ0). this implies that |g (t)| / √ c (t) ≤ |b (t)| /c (t) + 2 |d (t)| , (17) for all t ∈ (0, δ0) . exploiting (3) once more again it is easy to show that |b (t)| /c (t) ≤ 2β0t, for all t ∈ (0, δ0), where β0 = ‖β‖[0,δ0]. this together with the inequality (17) yields limt→0 |g(t)|/ √ c(t) = 0. as a consequence, cf. the estimate of theorem 2.1-(ii), it also follows that limtց0 wt(f)(z) = f(z), for all z ∈ dr. in what follows, the system (1) is complexified, by replacing x ∈ r with z ∈ c directly in the equations. more precisely, our goal is to study the following initial value problem { ∂u ∂t (t, z) = α(t)∂ 2 u ∂z2 (t, z) + β(t)zu(t, z), u(0, z) = f(z). (18) we will first consider the case when f is analytic. our first goal is to search for analytic solutions u(t, z), as functions of z, for any t > 0. first, we need some basic notations. for r > 0, define the strip sr = {z = x + iy ∈ c; x ∈ r, |y| ≤ r} and a(sr) = {f : sr → c; f is analytic in sr}, ( i.e., f is analytic in a domain that contains sr). next, let mr be the set of all f ∈ a (sr) such that there exists g ∈ l1 (r+) ∩ l∞ (r+) with the property that |f′ (z)| ≤ g (|z|), as |z| goes to infinity. finally, let u(t, z) := eφ(t,z) 2 √ πc(t) ∫∞ −∞ f(z − ξ)e−[ξ+g(t)] 2 /(4c(t))dξ, (t, z) ∈ r+ × sr, where c(t), φ(t, z), and g(t) are defined in the statement of theorem 2.1-(i). theorem 2.3. for each f ∈ mr we have u = u(t, ·) ∈ a (sr) , for any t > 0. moreover, u(t, z) solves (18) for (t, z) ∈ r+ × sr. 40 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) proof. if f ∈mr, there exists m > 0 such that sup{|f′(z)|; z ∈ sr} = ‖f′‖sr ≤ m. by the mean value theorem in complex analysis, f is uniformly continuous in sr, and that |f(z − ξ)| ≤ |f(0)| + ‖f′‖sr · |z − ξ| ≤ |f(0)| + m|z − ξ|. the latter implies that the integral i(t, z) := 1 2 √ πc(t) ∫∞ −∞ f(z − ξ)e−[ξ+g(t)] 2 /(4c(t))dξ exists and is absolutely convergent in c, and that i(t, z) is differentiable with respect to any z ∈ sr, with ∂zi(t, z) = ∫∞ −∞ f′(z − ξ)e−[ξ+g(t)] 2 /(4c(t))dξ. the analicity of φ(t, z) with respect to z ∈ sr implies that u = u(t, ·) also belongs to a (sr), for any t > 0. analogous calculations to those performed in (15) yield |i(t, z) − f(z)| ≤ ω1(f; |g(t)|)sr + ω1(f; √ c (t))sr 1 2 √ πc (t) ∫∞ −∞ ( |u| (c (t)) −1/2 + 1 ) e−u 2 /(4c(t))du = ω1(f; |g(t)|)sr + ( 1 + 2√ π ) ω1(f; √ c(t))sr. taking now into account the remark 2.2, and the uniform continuity of f on sr, it easily follows that limtց0 i(t, z) = f(z), for all z ∈ sr. this together with the fact that φ(0, z) = 1 yields u(0, z) = limtց0 u(t, z) = f(z), for all z ∈ sr, i.e., u satisfies the initial condition of (18). it remains to show that u also solves the main equation of (18). to this end, define f(t, z) := ∂u ∂t (t, z) − α(t) ∂2u ∂z2 (t, z) − β(t)zu(t, z), for all (t, z) ∈ r+ × sr, and recall that u(0, z) = f(z), z ∈ sr. for each t > 0, f (t, ·) is analytic in sr. taking now z = x ∈ r in all the equations of (18), we can now apply known theory to deduce that u (t, z) = u (t, x) also solves (1). hence, f(t, x) = 0, for all (t, x) ∈ r+ × r. the identity theorem for holomorphic functions (in a domain that contains sr) implies that we must also have f(t, z) = 0, (t, z) ∈ r+ × sr. this finishes the proof of the theorem. 3 linearized korteweg-de vries type equations for r > 1, let us define the open disk dr := {z ∈ c : |z| < r}. next we endow the local convex space a∗(dr) := {f : dr → c : f is analytic on dr}, cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 41 with the countable family of seminorms ‖f‖n := sup{|f(z)| : z ∈ drn}, rn ր r, rn ≥ 1, and metric d(f, g) := ∞∑ n=0 1 2n ‖f − g‖n 1 + ‖f − g‖n . then a∗(dr) is a fréchet space. let f ∈ a∗(dr) such that f(z) = ∞∑ k=0 akz k, for all z ∈ dr. we consider the integral operator tk (t) (f) (z) := ∫∞ −∞ k(u, t)f(ze−iu)du, z ∈ dr, (19) with k(u, t) given by (7). evidently, since k(−u, t) 6= k(u, t), we can naturally introduce another complex integral by t̃k (t) (f) (z) := ∫∞ −∞ k(−u, t)f(zeiu)du, z ∈ dr. (20) the first goal of this section is to prove some properties for (19) and (20). theorem 3.1. let r > 1, f ∈ a∗(dr), that is, f(z) = ∞∑ k=0 akz k, for all z ∈ dr. let tk (·) (f) and t̃k (·) (f) be as in (19) and (20), respectively. (i) for all t ≥ 0, as functions of z, we have tk (·) (f) (z) ∈ a∗(dr), t̃k (·) (f) (z) ∈ a∗(dr), and there hold tk (t) (f) (z) = ∞∑ k=0 akdk(t)z k and t̃k (t) (f) (z) = ∞∑ k=0 akbk(t)z k, z ∈ dr, (21) where for all k ≥ 0, dk(t) = e itk 3 and bk(t) = e −itk 3 . moreover, tk (0) (f) = t̃k (0) (f) = f. (ii) for all z ∈ dr with 1 ≤ r < r and t ∈ r+, the following estimate holds: |tk (t) (f) (z) − f(z)| ≤ t2 2 ∞∑ k=0 |ak|k 6rk + |t| ∞∑ k=0 |ak|k 3rk, where ∞∑ k=0 |ak|k 6rk < ∞, since f(6) ∈ a∗(dr). 42 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) (iii) for all z ∈ dr with 1 ≤ r < r and t, s ∈ r+, there holds: |tk (t) (f) (z) − tk (s) (f) (z) | ≤ 2 ∞∑ k=0 |ak|k 3rk|t − s| and |t̃k (t) (f) (z) − t̃k (s) (f) (z) | ≤ 2 ∞∑ k=0 |ak|k 3rk|t − s|. (iv) the families {tk (t)}t≥0 and { t̃k (t) } t≥0 are (c0)-semigroups of linear operators, locally equicontinuous (that is, equicontinuous for t ∈ [0, a], for some a > 0) on a∗(dr). for each f ∈ a∗(dr), the corresponding cauchy problems { ∂u ∂t + ∂ 3 u ∂ϕ3 = 0, (t, z) ∈ r+ × dr\ {0} , u(0, z) = f(z), z ∈ dr (22) and { ∂w ∂t = ∂ 3 w ∂ϕ3 , (t, z) ∈ r+ × dr\ {0} , u(0, z) = f(z), z ∈ dr (23) are well-posed, with solutions given by u (t) = tk (t) (f) ∈ c∞ (r+; a∗(dr)) , w (t) = t̃k (t) (f) ∈ c∞ (r+; a∗(dr)) , respectively. proof. we will prove the above statements only for the family {tk (t)}t≥0 (the proof for { t̃k (t) } t≥0 is the same). (i) let f(z) = ∞∑ k=0 akz k, for all z ∈ dr. for fixed z ∈ dr, we get f(ze−iu) = ∞∑ k=0 ake −ikuzk. since |ake −iku| = |ak|, for all u ∈ r, and since ∞∑ k=0 akz k is absolutely convergent, the series ∞∑ k=0 ake −ikuzk is also uniformly convergent with respect to u ∈ r. therefore, the latter can be integrated term by term. using (19), we deduce tk (t) (f) (z) = ∞∑ k=0 ak {∫∞ −∞ [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] e−ikudu } zk = ∞∑ k=0 akdk(t)z k, cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 43 where dk(t) = ∫∞ −∞ [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] cos(ku)du − i ∫∞ −∞ [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] sin(ku)du = ∫0 −∞ [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] cos(ku)du + ∫∞ 0 [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] cos(ku)du − i ∫0 −∞ [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] sin(ku)du + ∫∞ 0 [ 1 π ∫+∞ 0 cos(uτ − tτ3)dτ ] sin(ku)du = 1 π ∫∞ 0 [∫∞ 0 ( cos(uτ + tτ3) + cos(uτ − tτ3) ) dτ ] cos(ku)du + i π {∫∞ 0 [∫∞ 0 ( cos(uτ + tτ3) − cos(uτ − tτ3) ) dτ ] sin(ku)du } . on the other hand, it is well-known that the fourier transform of the airy’s function is given by (see, e.g., [5, p. 87, table 4.2]) ∫∞ −∞ [ 1 π ∫∞ 0 cos(uτ + τ3/3)dτ ] e−ikωudu = ei(ωk) 3 /3 =: i1 + i2, where i1 := 1 π ∫∞ 0 [∫∞ 0 ( cos(τ3/3 + uτ) + cos(τ3/3 − uτ) ) dτ ] cos(kωu)du, i2 := i π {∫∞ 0 [∫∞ 0 ( cos(τ3/3 + uτ) − cos(τ3/3 − uτ) ) dτ ] sin(kωu)du } . now, by a change of variable τ = (3t)1/3η, and then another u = v (3t)1/3 , (t > 0 is a fixed parameter), simple calculations yield i1 + i2 = 1 π ∫∞ 0 [∫∞ 0 (cos(tη3 + vη) + cos(tη3 − vη))dη ] cos(kωv/(3t)1/3)dv + i π ∫∞ 0 [∫∞ 0 (cos(tη3 + vη) − cos(tη3 − vη))dη ] sin(kωv/(3t)1/3)dv = ei(ωk) 3 /3. choosing ω = (3t)1/3, and taking into account that i1 + i2 = dk(t), we easily arrive at dk(t) = e itk 3 . 44 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) this proves the analyticity of tk (·) (f) (z) , as function of z ∈ dr. finally, the relation tk (0) = i is immediate in view of (21). (ii) let |z| ≤ r. we obtain |tk (t) (f) (z) − f(z)| = ∣∣∣∣∣ ∞∑ k=0 akz k[eik 3 t − 1] ∣∣∣∣∣ = ∣∣∣∣∣ ∞∑ k=0 akz k[−2 sin2(k3t/2) + i sin(k3t/2) cos(k3t/2)] ∣∣∣∣∣ ≤ ∞∑ k=0 |ak|r k|2 sin2(k3t/2)| + ∞∑ k=0 |ak|r k|2 sin(k3t/2)| ≤ t2 2 ∞∑ k=0 |ak|k 6rk + |t| ∞∑ k=0 |ak|k 3rk, since | sin(x)| ≤ |x|, for all x ∈ r. (iii) we have |tk (t) (f) (z) − tk (s) (f) (z)| = ∣∣∣∣∣ ∞∑ k=0 akz k[eik 3 t − eik 3 s] ∣∣∣∣∣ = ∞∑ k=0 [cos(k3t) − cos(k3s) + i(sin(k3t) − sin(k3s))] ≤ 4 ∞∑ k=0 |ak|r k ∣∣∣∣sin ( k3(t − s) 2 )∣∣∣∣ ≤ 2 ∞∑ k=0 |ak|k 3rk|t − s|. (iv) from (i), it is immediate that tk (t + s) = tk (t) tk (s) , for all t, s ∈ r+. the strong continuity of tk (t) follows from (iii). we can argue as in the proof of theorem 6.2.1 to deduce the first part of the statement in (iv). let f ∈ a∗(dr). we can compute the generators of the semigroups tk (t) and t̃k (t), t ∈ r+, respectively, as follows: ( d dt tk (t) (f) (z) ) |t=0 = i ∞∑ k=0 k3ake ik 3 tzk = − ∂3 ∂ϕ3 tk (t) (f) (z) and ( d dt t̃k (t) (f) (z) ) |t=0 = −i ∞∑ k=0 k3ake −ik 3 tzk = ∂3 ∂ϕ3 d dt t̃k (t) f (z) , for all z = reiϕ ∈ dr\ {0} . the proof of the theorem is complete. cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 45 in what follows, the linearized korteweg-de vries equation from (5) is complexified by replacing x ∈ r with z ∈ ω ⊂ c directly in the equations. more precisely, we aim to study the following initial value problem    ∂u ∂t = ∂ 3 u ∂z3 , (t, z) ∈ r+ × ω, lim tց0 u(t, z) = f(z), z ∈ ω. (24) we look for classical solutions of (24) which belong to the class: u ∈ c1 (r+; a◦ (sr)) , (25) where a◦(sr) := {f : sr → c : f is analytic in sr}, i.e., f is analytic in a domain that contains the closure sr of sr := {z = x + iy ∈ c : x ∈ r, |y| < r}. theorem 3.2. let f ∈ a◦(sr) such that the following are satisfied: (i) f′ is bounded on sr; (ii) for all |y| ≤ r, the integral f(s, y) = ∫+∞ −∞ e−sxf(x + iy)dx is absolutely convergent for s = c1 + iτ; (iii) for all |y| ≤ r, the integral g(s, y) = ∫+∞ −∞ e−sxf′(x + iy)dx is absolutely convergent for s = c1 + iτ; (iv) for all |y| ≤ r, ∫+∞ −∞ |f(c1 + iτ, y)|dτ < +∞. setting tkv (t) (f) (z) := ∫+∞ −∞ k(u, t)f(z − u)du, (t, z) ∈ r+ × sr, there holds u (t) = tkv (t) (f) ∈ c∞ (r+; a◦(sr)) , and tkv (t) (f) solves the initial value problem (24). proof. let f ∈ a◦(sr) and decompose f(z) = u(x, y) + iv(x, y), with u and v having continuous partial derivatives of first order. moreover, u, v satisfy the cauchy-riemann conditions at any (x, y) ∈ sr. we can write tkv (t) (f) (z) = ∫+∞ −∞ k(u, t)u(x − u, y)du + i ∫+∞ −∞ k(u, t)v(x − u, y)du := t1(t, x, y) + it2(t, x, y). step 1. let z = x + iy ∈ sr. since f′(z) = ∂u ∂x (x, y) + i ∂v ∂x (x, y), (26) 46 ciprian g. gal and sorin g. gal cubo 15, 1 (2013) we note that f′′′(z) = ∂3u ∂x3 (x, y) + i ∂3v ∂x3 (x, y). we claim that for any fixed |y| ≤ r, conditions (i)-(ii) in the statement of theorem 7.1.1, are fulfilled for t1 and t2, with respect to (t, x) ∈ r+ × r. as a consequence,    ∂t1 ∂t (t, x, y) = ∂ 3 t1 ∂x3 (t, x, y), (t, x, y) ∈ r+ × sr, lim tց0 t1(t, x, y) = u(x, y), (x, y) ∈ sr.    ∂t2 ∂t (t, x, y) = ∂ 3 t2 ∂x3 (t, x, y), (t, x, y) ∈ r+ × sr, lim tց0 t2(t, x, y) = v(x, y), (x, y) ∈ sr, and, therefore, tkv (t) (f) solves (24). indeed, by the first hypothesis above (see (i)), there exists m > 0 such that sup{|f′(z)| : z ∈ sr} = ‖f′‖sr ≤ m. now, from (26), ∂u ∂x (x, y) and ∂v ∂x (x, y) are bounded on sr. therefore, for any fixed |y| ≤ r, u(xy) and v(x, y) are continuous and of bounded variation with respect to x ∈ r. setting f1(s, y) = ∫+∞ −∞ e−sxu(x, y)dx, f2(s, y) = ∫+∞ −∞ e−sxv(x, y)dx, clearly, f(s, y) = f1(s, y) + if2(s, y), and by virtue of (ii), both f1(s, y) and f2(s, y) are absolutely convergent, for a fixed (but otherwise arbitrary) |y| ≤ r. furthermore, in view of (iv), we deduce ∫+∞ −∞ |f1(c1 + iτ, y)|dτ < +∞, ∫+∞ −∞ |f2(c1 + iτ, y)|dτ < +∞. therefore, we can apply theorem 1.1 to the functions t1 and t2, respectively. this yields the above claim. step 2. clearly, tkv (t) (f) belongs to the class (25) for any f ∈ a◦(sr). indeed, the fact that ∂xt1(·, x, y) and ∂xt2(·, x, y) are continuous on sr was already proved in step 1. the existence and continuity of the partial derivatives ∂yt1(·, x, y), ∂yt2(·, x, y) follow from condition (iii). finally, the functions ti (·, x, y) , i = 1, 2, also satisfy the cauchy-riemann equations since u, v do. the proof is finished. � remark 3.1. a simple example of boundary data in (24) that satisfies all the hypotheses of theorem 3.2 is f(z) = e−z 2 . in this case, one can prove that f(s, y) = √ πes 2 /4 cos(sy). received: october 2012. revised: february 2013. cubo 15, 1 (2013) on fokker-planck and linearized korteweg-de vries ... 47 references [1] c. g. gal, s. g. gal and j. a. goldstein, evolution equations with real time variable and complex spatial variables, complex variables and elliptic equations, 53 (2008), no. 8, 753– 774. [2] c. g. gal, s. g. gal and j. a. goldstein, higher order heat and laplace type equations with real time variable and complex spatial variable, complex variables and elliptic equations, 55 (2010), no. 4, 357–373. [3] c. g. gal, s. g. gal and j. a. goldstein, wave and telegraph equations with real time variable and complex spatial variables, complex variables and elliptic equations, 57 (2012), 91–109 [4] c. g. gal, s. g. gal and j. a. goldstein, schrödinger type equations with real time variable and complex spatial variables, complex variables and elliptic equations, 58 (2013), 415-430. [5] o. vallée and m. suarez, airy functions and applications to physics, imperial college press, new jersey, london, singapore, 2004. [6] k.v. zhukovsky and g. dattoli, evolution of non-spreading airy wavepackets in time dependent linear potentials, applied mathematics and computation, 217(2011), 7966-7974. [7] d.v. widder, the airy transform, amer. math. monthly, 86(1979), no. 4, 271-277. () cubo a mathematical journal vol.13, no¯ 02, (127–137). june 2011 module amenability for banach modules d. ebrahimi bagha department of mathematics, islamic azad university, central tehran branch, tehran, iran. email: d.ebrahimibagha@iauctb.ac.ir and m. amini department of mathematics, tarbiat modares university, p.o.box 14115-175, tehran, iran. email: mamini@modares.ac.ir abstract we study the module amenability of banach modules. this is a natural generalization of johnson’s amenability of banach algebras. as an example we show that for a discrete abelian group g, ℓp(g) is amenable as an ℓ1(g)-module if and only if g is amenable, where ℓ1(g) is a banach algebra with pointwise multiplication. resumen se estudia el módulo de receptividad de los módulos de banach. esta es una generalización natural de la receptividad de johnson de las álgebras de banach. como ejemplo se muestra que para un grupo abeliano discreto g ℓp(g) es receptivo como un g ℓp(g)módulo, si y sólo si g es receptivo, donde ℓ1(g) es un álgebra de banach con producto punto. 128 d. ebrahimi bagha, m. amini cubo 13, 2 (2011) keywords and phrases: banach modules, module amenability, weak module amenability, semigroup algebra, inverse semigroup. mathematics subject classification: 43a07, 46h25 1. introduction the concept of amenability for banach algebras was introduced by b.e. johnson in [j]. the main example in [j] asserts that the group algebra l1(g) of a locally compact group g is amenable if and only if g is amenable. this is far from true for semigroups. if s is a discrete inverse semigroup with the set of idempotents es, ℓ 1(s) is amenable if and only if es is finite and all the maximal subgroups of s are amenable [dn]. for an arbitrary discrete semigroup s, ℓ1(s) is amenable if and only if the minimum ideal of s exists and is an amenable group and s has a principal series whose corresponding quotients are regular rees matrix semigroups of special form [dls, 10.12].this failure is partly due to the fact that ℓ1(s) is equipped with two (related) algebraic structures. it is a banach algebra and a banach module over ℓ1(es). this consideration was the motivation of the second named author to study the concept of module amenability for banach algebras which have an extra banach module structure (with compatible actions) in [a]. in particular it is shown in [a] that for an inverse semigroup s, ℓ1(s) is module amenable as a banach module over ℓ1(es) if and only if s is amenable. the authors introduced the concept of weak module amenability in [ae] and showed that for a commutative inverse semigroup s, ℓ1(s) is always weak module amenable as a banach module over ℓ1(es). the present paper investigates module amenability from a different angle. there are many examples of banach modules which do not have any natural algebra structure. one example is lp(g) which is a left banach l1(g)-module, for a locally compact group g [d, 3.3.19]. as another example of this sort, one may consider a banach algebraic bundle over a locally compact group g [fd]. then the fibers on elements of g are banach modules over the fiber on the identity. crossed products of banach algebra by groups are special cases of banach algebraic bundles. the theory of module amenability developed in [a] does not cover these examples. there is one thing in common in these examples and that is the existence of a module homomorphism from the banach module to the underlying banach algebra. for instance in the case of crossed products, x is a banach algebra, g is a topological group, and xg = x × {g}, for g ∈ g, and {xg} is a banach algebraic bundle over g. in this case we have a module homomorphism ∆g : xg → xe which sends (x,g) to (x,e), where e is the identity of g. also if g is a compact group and f ∈ lq(g), then one has the module homomorphism ∆f : l p(g) → l1(g) which sends g to f ∗ g. in this paper, the concept of module amenability (more precisely ∆-amenability) is defined for a banach module e over a banach algebra a with a given module homomorphism ∆ : e → a. the next section gives the basic properties of module amenability and in particular establishes the equivalence of this concept with the existence of module virtual (approximate) diagonals in an appropriate sense. section 3 covers the weak ∆-amenability. a few examples are discussed in the cubo 13, 2 (2011) module amenability for banach modules 129 last section. 2. module amenability let a be a banach algebra and e be a banach space with a left a-module structure such that, for some m > 0, ‖a.x‖ ≤ m‖a‖ ‖x‖ (a ∈ a,x ∈ e), then e is called a left banach a-module. right and two-sided banach a-modules are defined similarly. throughout this section e is a banach a-bimodule and ∆ : e → a is a bounded banach a-bimodule homomorphism. definition 2.1. let x be a banach a-bimodule. a bounded linear map d : a → x is called a module derivation (or more specifically a ∆-derivation) if d(∆(a.x)) = a.d(∆(x)) + d(a).∆(x), d(∆(x.a)) = d(∆(x)).a + ∆(x).d(a), for each a ∈ a and x ∈ e. also d is called inner (or ∆-inner) if there is f ∈ x such that d(∆(x)) = f.∆(x) − ∆(x).f =: df(∆(x)) (x ∈ e). when ∆ has a dense range, df extends uniquely to a ∆-derivation from a to x. definition 2.2. a bimodule e is called module amenable (or more specifically ∆-amenable as a a-bimodule) if for each banach a-bimodule x, all ∆-derivations from a to x∗ are ∆-inner. it is clear that a is a-module amenable (with ∆ = id) if and only if it is amenable as a banach algebra. a right bounded approximate identity of e is a bounded net {aα} in a such that for each x ∈ e, ‖∆(x).aα −∆(x)‖ → 0, as α → ∞. the left and two-sided approximate identities are defined similarly. proposition 2.3. if e is module amenable, then e has a bounded approximate identity. proof consider the double conjugate space a∗∗ as a banach a-module with 〈f.a,f〉 = 〈f,a.f〉,〈a.f,b〉 = f(ba),a.f = 0 (a,b ∈ a,f ∈ a∗,f ∈ a∗∗). then the canonical embedding d : a → a∗∗ is a module derivation, hence d = df on ∆(e), for some f ∈ a∗∗. choose a net {aα} in a which is w ∗-convergent to f in a∗∗. clearly {aα} is a left bounded approximate identity of e. right and two sided approximate identities now could be constructed similar to the classical case [d]. definition 2.4. a banach a-module x is called right ∆-essential if for each x ∈ x there is a ∈ ∆(e) and y ∈ x such that x = y.a. the left ∆-essential and (two sided) ∆-essential modules are defined similarly. 130 d. ebrahimi bagha, m. amini cubo 13, 2 (2011) the following two results are proved as in the classical case [j]. we just include the proof of lemma 2.5(i), as it involves a variation of the cohen factorization theorem. lemma 2.5. (i) if ∆ has a dense range and e has a (right) bounded approximate identity, then e is module amenable iff for each (right) ∆-essential banach a-bimodule x, all ∆-derivations from a to x∗ are ∆-inner. (ii) if e and e ′ are banach a-modules with module homomorphisms ∆ and ∆ ′ and θ : e → e ′ is a bounded module map with dense range such that ∆ ′ ◦ θ = ∆, then ∆-amenability of e implies ∆ ′ -amenability of e ′ . (iii) if j is a closed submodule of e and j∆ is the closed ideal of a generated by ∆(j), and q : a → a/j∆ and q̃ : e → e/j are the corresponding quotient maps, then e is ∆ amenable whenever j is ∆|j-amenable and e/j is ∆̃-amenable, where ∆̃ : e/j → a/j∆ is the unique a/j∆module map with ∆̃ ◦ q̃ = q ◦ ∆. proof we prove part (i) as promised. we just need to check the necessity. let {aα} ⊆ a be a right bounded approximate identity for e. let x be a banach a-bimodule. consider tα : x ∗ → x∗ defined by t(f) = aα.f, for f ∈ x ∗, where a.f(x) = f(x.a), for a ∈ a,x ∈ x. since {aα} is bounded in a, {tα} is bounded in b(x ∗). hence it has a w∗-cluster point t. we may assume that tα → t in w∗-topology. for each e ∈ e,x ∈ x,f ∈ x∗, we have 〈x,∆(e),tf〉 = ĺım α 〈x,∆(e),tαf〉 = ĺım α 〈x,∆(e),aα.f〉 = ĺım α 〈x,∆(e)aα,f〉 = 〈x,∆(e),f〉. hence t −i : x∗ → (x.∆(e))⊥ is a bounded projection and we have the admissible short exact sequence 0 → (x.∆(e))⊥ → x∗ → (x.∆(e))∗ → 0 of banach a-bimodules. but ∆(e).(x/(x.∆(e))) = 0 and ∆ has a dense range, hence each bounded ∆-derivation d1 : a → (x.∆(e)) ⊥ = (x/(x.∆(e)))∗ is zero. on the other hand, each bounded ∆-derivation d2 : a → (x.∆(e)) ∗ is ∆-inner, by assumption. therefore each bounded ∆-derivation d : a → (x.∆(e))∗ is ∆-inner, and we are done. lemma 2.6. assume that a and b are banach algebras, j is a closed ideal of a, e is a banach a-module, and ∆ : e → a is an a-module homomorphism. (i) if f is a banach a-module and φ : e → f is an a-module homomorphism with dense range, then ∆-amenability of e implies ∆ ◦ φ-amenability of f. (ii) if ψ : a → b is a banach algebra epimorphism with e.ker(ψ) = ker(ψ).e = {0}, and e is considered as a b-module via b.x := a.x, x.b := x.a (b ∈ b,x ∈ e), cubo 13, 2 (2011) module amenability for banach modules 131 where a ∈ a on the right hand side is any element with b = ψ(a). then ∆-amenability of e, as an a-module, implies ψ ◦ ∆-amenability of e as a b-module. (iii) in (ii), if b = a/j, ψ : a → a/j is the quotient map, and e.j = j.e = {0}, then ∆-amenability of e, as an a-module, implies ψ ◦ ∆-amenability of e as a a/j-module. (iv) if i is a closed ideal of a, e ′ is the closed submodule of e generated by ie, and ∆ ′ : e ′ → i is the restriction of ∆ : e → a, then e ′ is ∆ ′ -amenable whenever e is ∆-amenable and e ′ has a bounded approximate identity. proposition 2.7. if i is a closed ideal of a which contains a bounded approximate identity (of itself), e is a banach a-bimodule with module homomorphism ∆ : e → a, and x is an essential banach i-module, then x is (canonically) a banach a-module and each ∆|i-derivation d : i → x ∗ uniquely extends to a ∆-derivation d̃ : a → x∗ which is continuous with respect to the strict topology of a (induced by i) and w∗-topology of x∗. proof each x ∈ x decomposes (not uniquely) as x = a.y, for some a ∈ i and y ∈ x. it is easy to see that x is a left banach a-module under the action b.x = ba.y (a ∈ i,b ∈ a,x,y ∈ x,x = a.y). this is well defined, as i has a bounded approximate identity. define d̃ : a → x∗ by d̃(b) = w∗ĺım α (d(beα) − b.d(eα)), where {eα} is a bounded approximate identity of i. now beα → b strictly, for each b ∈ a. hence, given b ∈ a and e ∈ e, we have d̃(∆(b.e)) = d̃(b∆(e)) = w∗ĺım α w∗ĺım β d(beα∆(e)eβ) = w∗ĺım α w∗ĺım β [beαd(∆(e)eβ) + d(beα).∆(e)eβ] = bd̃(∆(e)) + d̃(b).∆(e). hence d̃ is a ∆-derivation. the rest of the proof is similar to [ru, 2.1.6]. proposition 2.8. if ∆ : e → a has a dense range, then ∆-amenability of e is equivalent to amenability of a. proof if e is ∆-amenable, then each derivation d : a → x∗, where x is a banach a-module, is a module derivation and so inner on ∆(e). by continuity, d is inner on a. conversely each module derivation d : a → x∗ is a derivation. indeed, given b ∈ a, there is a sequence {xn} ⊆ e such that ∆(xn) → b, and so d(ab) = ĺım n d(a∆(xn)) = ĺım n (d(a).∆(xn) + a.d(∆(xn))) = d(a).b + a.d(b), for each a ∈ a. hence, if a is amenable, then e is ∆-amenable. 132 d. ebrahimi bagha, m. amini cubo 13, 2 (2011) definition 2.9. let π : a⊗̂a → a be the continuous lift of the multiplication map of a to the projective tensor product a⊗̂a. a module approximate diagonal of e is a bounded net {eα} in a⊗̂a such that ‖eα.∆(x) − ∆(x).eα‖ → 0, ‖π(eα).∆(x) − ∆(x)‖ → 0 (x ∈ e), as α → ∞. a module virtual diagonal of e is an element m in (a⊗̂a)∗∗ such that m.∆(x) − ∆(x).m = 0, π∗∗(m).∆(x) − ∆(x) = 0 (x ∈ e). it is clear that if e has a module virtual diagonal, then a contains a bounded approximate identity. theorem 2.10. consider the following assertions. (i) e is module amenable, (ii) e has a module virtual diagonal, (iii) e has a module approximate diagonal. we have (i) → (ii) ↔ (iii). if moreover ∆ has a dense range, all the assertions are equivalent. proof (i) → (ii). by proposition 2.3, we may choose a bounded approximate identity {eα} for e. we may assume that {eα ⊗ eα} is w ∗-convergent to a point p ∈ (a⊗̂a)∗∗. then for each x ∈ e and f ∈ a∗, 〈π∗∗((dp ◦ ∆)(x),f〉 = w ∗ĺım α 〈π∗(f).∆(x) − ∆(x).π∗(f),eα ⊗ eα〉 = w∗ĺım α f(∆(x.eα)eα − eα∆(eα.x)) = 0. hence im(dp ◦ ∆) ⊆ ker(π ∗∗). now ker(π∗∗) is isometrically isomorphic to x∗, where x = (∆(e)⊗̂a)∗∗/im(π∗)⊥, so by assumption, there is q ∈ ker(π∗∗) with dp ◦ ∆ = dq ◦ ∆. it is easy to see that m := p − q is a module virtual diagonal for e. (ii) → (iii). let m be a module virtual diagonal and let {eα} be a net in a⊗̂a which w ∗clusters to m. then clearly eα.∆(x) − ∆(x).eα → 0, ∆◦π(eα).∆(x) − ∆(x) → 0 as α → ∞ for each x ∈ e in the w∗-topology of (a⊗̂a)∗∗. a standard argument based on mazur’s theorem shows that the same holds in the norm topology for a net consisting of appropriate convex combinations of elements of {eα}. (iii) → (ii). just take any w∗-cluster point. (iii) → (i). now assume that ∆ has a dense range. let {mα} be a module approximate diagonal for e with w∗-cluster point m, then {π(mα)} is a bounded approximate identity for e. by lemma 2.5(i), it is enough to show that for each essential a-module y, all module derivation d from a to y∗ are inner. each y ∈ y could be regarded as a bounded linear functional ŷ on a⊗̂a via 〈ŷ,b ⊗ a〉 := 〈b.d(a),y〉 (a,b ∈ a). cubo 13, 2 (2011) module amenability for banach modules 133 then for each x,x ′ ∈ e, a ∈ a, and y ∈ y 〈(y.∆(x) − ∆(x).y)̂,∆(x ′ ) ⊗ a〉 = 〈ŷ.∆(x) − ∆(x).ŷ,∆(x ′ ) ⊗ a〉 + 〈∆(x ′ )a.d ◦ ∆(x),y〉. it follows that 〈(y.∆(x) − ∆(x).y)̂,m〉 = 〈ŷ.∆(x) − ∆(x).ŷ,m〉 + 〈π(m).d ◦ ∆(x),y〉, for each m ∈ a⊗̂a. if we identify m with an element of y∗ with m(y) = 〈ŷ,m〉, for y ∈ y, then 〈dm ◦ ∆(x),y〉 = w ∗ĺım α 〈y.∆(x) − ∆(x).y,mα〉 = 〈m,ŷ.∆(x) − ∆(x).ŷ〉 + w∗ĺım α 〈π(mα).d ◦ ∆(x),y〉. now in the last equation, the first term is zero, as m is a module virtual diagonal, and the second term is easily seen to be equal to 〈d◦∆(x),y〉, using the fact that y = z.∆(x ′ ), for some z ∈ y and x ′ ∈ e. therefore d = dm on ∆(e), as required. 3. weak module amenability in this section we study weak module amenability of banach modules. all over this section e is a commutative banach a-module (that is a.x = x.a, for each a ∈ a,x ∈ e) and ∆ : e → a is a bounded banach a-module homomorphism. a banach a-module x is called ∆-commutative (or more specifically ∆(e)-commutative) if a.x = x.a (a ∈ ∆(e),x ∈ x). definition 3.1. e is called weak module amenable (or more specifically weak ∆-amenable as an a-module) if each ∆-derivation from a to ∆(e)∗ is inner on ∆(e). clearly a is weak a-module amenable (with ∆ = id) if and only if it is weakly amenable as a banach algebra. the following result could be proved as in the classical case. proposition 3.2. (i) if e ′ is a commutative a-module and φ : e ′ → e is a module homomorphism with dense range, and e is weak ∆-amenable then e ′ is weak ∆ ◦ φ-amenable. (ii) if i is a closed ideal of a with ie = ei = {0} and q : a → a/i is the quotient map, then e is weak q ◦ ∆-amenable as an a/i-module if it is ∆-amenable as an a-bimodule. proposition 3.3. if e is weak ∆-amenable, then the closed linear span f of a∆(e) is dense in ∆(e). proof if not, there is a nonzero bounded linear functional λ in ∆(e)∗ which vanishes on f. by hahn-banach theorem λ extends to an element of a∗, which we still denote by λ. define d : a → ∆(e)∗ by d(a) = λ(a)λ. this is a module derivation which is not inner, a contradiction. now if a is a (commutative) banach algebra with maximal ideal space ma and φ ∈ ma, then c is a banach a-module with respect to the module action a.z = z.a = φ(a)z (a ∈ a,z ∈ c), 134 d. ebrahimi bagha, m. amini cubo 13, 2 (2011) which is denoted by cφ. each module derivation d : a → cφ is called a module point derivation (at φ). clearly when a commutative banach a-module e is ∆-weak amenable, all module point derivations vanish on ∆(e). this holds in general. proposition 3.4. if e is weak ∆-amenable, there is no nonzero point derivation on a. proof let d : a → cφ be a nonzero module point derivation. let ψ be the restriction of φ to ∆(e) and define d : a → ∆(e)∗ by d(a) = d(a)ψ. then d is a ∆-derivation and so d = dλ on ∆(e), for some λ ∈ ∆(e) ∗. choose e,f ∈ e so that ψ(∆(e)) = 1, ψ(∆(f)) = 0, and d(∆(f)) = 1. then for a = ∆(e) + (1−d(∆(e)))∆(f), we have ψ(a) = d(a) = 1, hence d(a)(a) = 1, a contradiction. theorem 3.5. if e is a commutative a-module and there is a ∆-commutative a-module x and a module derivation d0 : a → x which is not identically zero on ∆(e), then there is a nonzero module derivation d : a → ∆(e)∗. proof we consider two cases. first assume that a∆(e) is not dense in ∆(e). by hahn-banach theorem, there is a nonzero functional λ ∈ (∆(e))∗ whose kernel contains a∆(e). extend λ to an element of a∗ (still denoted by λ) and define d : a → ∆(e)∗ by d(a) = λ(a)λ (a ∈ a), next consider the case where a∆(e) is dense in ∆(e). we know that there is a ∆-commutative a-module x and a module derivation d0 : a → x which is not identically zero on ∆(e). choose a ∈ a, e ∈ e, and λ ∈ x∗ such that d0(a∆(e)) 6= 0 and λ(d0(a∆(e))) 6= 0. define d : a → ∆(e) ∗ by 〈d(a),∆(e)〉 = λ(∆(e).d0(a)) (e ∈ e,a ∈ a). in both cases d is a nonzero module derivation. 4. examples in this sections we give three examples in which strong and weak module amenability of some banach modules are demonstrated. example 4.1. let s be an inverse semigroup and es be the commutative sub-semigroup of idempotents in s. then a = ℓ1(es) is a commutative banach algebra and e = ℓ 1(s) is a commutative banach a-bimodule with the module actions δe.δx = δx.δe = δex (e ∈ es,x ∈ s). also there is a surjective module homomorphism ∆ : ℓ1(s) → ℓ1(es) defined by ∆(δx) = δxx∗ (x ∈ s). cubo 13, 2 (2011) module amenability for banach modules 135 we show that ℓ1(s) is always module weakly amenable. if d : ℓ1(es) → ℓ ∞ (es) is a ∆-derivation, then for each e ∈ es, d(δe) = d(δee∗ ) = d(∆(δe)) = d(∆(δe.δe)) = ∆(δe).d(δe) + δe.d(∆(δe)) = 2δed(δe). applying the same formula to the right hand side, δe.d(δe) = 2δe.(δe.d(δe)) = 2(δe ∗ δe).d(δe)) = 2δe.d(δe), hence d(δe) = δe.d(δe) = 0. example 4.2. in the above example, if ℓ1(s) is ∆-amenable, then ℓ1(es) is amenable (proposition 2.8). hence es is finite, ℓ 1(s) is weakly amenable, and it has a bounded approximate identity [dn]. example 4.3. it is well known that the disk algebra a(d) is non amenable [bd]. a(d) is a cmodule with respect to the scalar product. now evaluation at zero defines a module epimorphism ∆ : a(d) → c and a(d) is ∆-amenable. example 4.4. if a is an amenable banach algebra, the canonical map π : a⊗̂a → a is an amodule epimorphism (it is surjective, since a has a bounded approximate identity) and a⊗̂a is π-amenable. example 4.5. for a locally compact group g, l1(g) is a closed two sided ideal in m(g), so we can consider it as a banach m(g) module. now if g is a non discrete amenable group, m(g) is not amenable [dgh] but l1(g) is i-amenable, where i : l1(g) → m(g) is the canonical injection. example 4.6. let 1 < p < ∞ and 1 p + 1 q = 1. then ℓ1 is a banach algebra and ℓp is a banach ℓ1-bimodule, both with respect to pointwise multiplication. also each f ∈ ℓq defines a module homomorphism ∆f : ℓ p → ℓ1 by ∆f(g) = g ∗ f. if f = ∑∞ k=−∞ 1 k δk, then ∆f has dense range(as its range contains all finitely supported element) and ℓp is ∆f-amenable, by proposition 2.8. this example could also be stated for any discrete group g, where ℓp(g) is considered as a banach ℓ1(g)-bimodule. same is true for lp(g) with convolution, when g is a compact group [d, 3.3.19]. in this case we have the module homomorphism ∆f : l p(g) → l1(g) defined by ∆f(g) = g ∗ f, where f ∈ lq(g). if ∆f has a dense range, then l p(g) is ∆f -amenable. this is always the case when g is an abelian compact group. we illustrate this for g = t. the same proof basically works for arbitrary abelian compact groups as well. take f = ∑∞ k=−∞ 1 k e2πikt ∈ lq(t) (which is basically the fourier transform of the above function f used in the discrete case). then, for each g ∈ lp(g), ∆f(g) = ∞∑ k=−∞ 1 k ĝ(k)e2πikt, where ĝ ∈ c0 is the fourier transform of g. in particular, range of ∆f includes all trigonometric functions which are dense in l1(g). example 4.7. if g is a discrete group (with identity e) which acts on a c∗-algebra a, then a× {e} could be identified with a and a × {g} is a banach a-module under a.(b,g) = ((g.a)b,g) , (b,g).a = (ba,g) (a,b ∈ a,g ∈ g), 136 d. ebrahimi bagha, m. amini cubo 13, 2 (2011) and there is a natural surjective module homomorphism ∆g : a × {g} → a which sends (a,g) to a. the crossed product c∗-algebra a ⋊ g is nuclear iff a is nuclear and g is amenable [ro] iff g is amenable and modules a × {g} are ∆g-amenable, for each g ∈ g. example 4.8. if a is a banach algebra such that a∗ ⊆ a and a∗ is a dense subspace of a, then a∗ is a banach a-bimodule (with canonical arens actions) and ∆ = id : a∗ → a has dense range. therefore a∗ is ∆-amenable as a banach a-bimodule iff a is amenable as a banach algebra. there are many examples of this type. if g is a compact group, then the fourier algebra a(g) is dense in the group c∗-algebra c∗(g). indeed a(g) ⊆ c(g) ⊆ l1(g) ⊆ c∗(g), and each space in this chain is dense in the subsequent space (with respect to the norm of the bigger space). but the norms of the last three spaces satisfy ‖.‖c∗(g) ≤ ‖.‖1 ≤ ‖.‖∞ [ey]. hence a(g) is dense in c∗(g). also, since g is compact, a(g) = b(g) ≃ c∗(g)∗, where b(g) is the fourier-stieltjes algebra [ey]. but c∗(g) is amenable when g is compact [ru]. hence a(g) is idamenable in this case. this becomes more interesting when we recall that there are compact groups for which the fourier algebra a(g) is not amenable [j2]. another example is ℓ1 which is dense in c0. it follows that ℓ 1 ≃ c∗0 is id-amenable as a c0-bimodule. finally, for a compact group g, l1(g) is an amenable banach algebra with convolution, and so l1(g) is id-amenable as a banach l1(g)-bimodule. received: december 2009. revised: april 2010. referencias [a] m. amini, module amenability for semigroup algebras, semigroup forum 69 (2004) 243-254. [ae] m. amini and d. ebrahimi bagha, weak module amenability for semigroup algebras, semigroup forum 71 (2005) 18-26. [bd] f. f. bonsall and j. duncan, complete normed algebras, springer-verlag, new york, 1973. [d] h.g. dales, banach algebras and automatic continuity, london math. soc. monographs, volume 24, clarendon press, oxford, 2000. [dgh] h.g. dales, f. ghahramani and a. ya. helemskii, the amenability of measure algebras, j. london math. soc. (2) 66 (2002), no. 1, 213-226. [dls] h.g. dales, a.t.m. lau and d. strauss, banach algebras on semigroups and their compactification, to appear in memoirs amer. math. soc. [dn] j. duncan and i. namioka, amenability of inverse semigroups and their semigroup algebras, proceedings of the royal society of edinburgh 80a (1978), 309-321. cubo 13, 2 (2011) module amenability for banach modules 137 [ey] p. eymard, l’algebre de fourier d’un groupe localement compact, bull. soc. math. france 92 (1964), 181-236. [fd] j. m. g. fell and r. s. doran, representations of ∗-algebras, locally compact groups, and banach ∗-algebraic bundles, pure and applied mathematics, vols. 125 & 126, academic press inc., boston, ma, 1988. [j] b.e. johnson, cohomology in banach algebras. memoirs of the american mathematical society, no. 127, american mathematical society, providence, 1972. [j2] b.e. johnson,non-amenability of the fourier algebra of a compact group, j. london math. soc. (2), 50 (1994), 361-374. [ro] j. rosenberg, amenability of crossed products of c∗-algebras, comm. math. phys. 57 (1977), no. 2, 187-191. [ru] v. runde, lectures on amenability, lecture notes in mathematics 1774, springer-verlag, berlin, 2002. introduction module amenability weak module amenability examples cubo a mathematical journal vol.14, no¯ 01, (01–07). march 2012 univariate right fractional ostrowski inequalities george a. anastassiou university of memphis, department of mathematical sciences, memphis, tn 38152, u.s.a. email: ganastss@memphis.edu abstract very general univariate right caputo fractional ostrowski inequalities are presented. one of them is proved sharp and attained. estimates are with respect to ‖·‖ p , 1 ≤ p ≤ ∞. resumen se presenta de manera muy general desigualdades univariadas derechas de caputo fraccionarias de ostrowski. se prueba que una de ellas es aguda las estimaciones con respecto a ‖·‖ p , 1 ≤ p ≤ ∞. keywords and phrases: ostrowski inequality, right caputo fractional derivative. 2010 ams mathematics subject classification: 26a33, 26d10, 26d15. 2 george a. anastassiou cubo 14, 1 (2012) 1 introduction in 1938, a. ostrowski [7] proved the following important inequality: theorem 1. let f : [a,b] → r be continuous on [a,b] and differentiable on (a,b) whose derivative f′ : (a,b) → r is bounded on (a,b), i.e., ‖f′‖∞ := sup t∈(a,b) |f′ (t)| < +∞. then ∣∣∣∣∣ 1b − a ∫b a f(t)dt − f(x) ∣∣∣∣∣ ≤ [ 1 4 + ( x − a+b 2 )2 (b − a) 2 ] · (b − a)‖f′‖∞ , (1) for any x ∈ [a,b]. the constant 1 4 is the best possible. since then there has been a lot of activity around these inequalities with important applications to numerical analysis and probability. this paper is greatly motivated and inspired also by the following result. theorem 2. (see [1]) let f ∈ cn+1 ([a,b]), n ∈ n and x ∈ [a,b] be fixed, such that f(k) (x) = 0, k = 1, ...,n. then it holds∣∣∣∣∣ 1b − a ∫b a f(y)dy − f(x) ∣∣∣∣∣ ≤ ∥∥f(n+1)∥∥∞ (n + 2) ! · ( (x − a) n+2 + (b − x) n+2 b − a ) . (2) inequality (2) is sharp. in particular, when n is odd is attained by f∗ (y) := (y − x) n+1 · (b − a), while when n is even the optimal function is f(y) := |y − x| n+α · (b − a) , α > 1. clearly inequality (2) generalizes inequality (1) for higher order derivatives of f. also in [2], see chapters 24-26, we presented a complete theory of left fractional ostrowski inequalities. 2 main results we need definition 3. ([3], [4], [5], [6], [8]) let f ∈ l1 ([a,b]), α > 0. the right riemann-liouville fractional operator of order α by iαb−f(x) = 1 γ (α) ∫b x (j − x) α−1 f(j)dj, (3) ∀ x ∈ [a,b], where γ is the gamma function. we set i0b− := i (the identity operator). definition 4. ([3], [4], [5], [6], [8]) let f ∈ acm ([a,b]) (f(m−1) is in ac([a,b])), m ∈ n, m = dαe, α > 0 (d·e the ceiling of the number). we define the right caputo fractional derivative of order α > 0, by dαb−f(x) = (−1) m γ (m − α) ∫b x (j − x) m−α−1 f(m) (j)dj, ∀ x ≤ b. (4) cubo 14, 1 (2012) univariate right fractional ostrowski inequalities 3 if α = m ∈ n, then dmb−f(x) = (−1) m f(m) (x) , ∀ x ∈ [a,b] . if x > b we define dαb−f(x) = 0. we also need theorem 5. ([3]) let f ∈ acm ([a,b]), x ∈ [a,b], α > 0, m = dαe. then f(x) = m−1∑ k=0 f(k) (b) k! (x − b) k + 1 γ (α) ∫b x (j − x) α−1 dαb−f(j)dj, (5) the right caputo fractional taylor formula with integral remainder. we present theorem 6. let α > 0, m = dαe, f ∈ acm ([a,b]). assume f(k) (b) = 0, k = 1, ...,m − 1, and dαb−f ∈ l∞ ([a,b]). then∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ ≤ ∥∥dαb−f∥∥∞,[a,b] γ (α + 2) (b − a) α . (6) proof. let x ∈ [a,b]. we have f(x) − f(b) = 1 γ (α) ∫b x (j − x) α−1 dαb−f(j)dj. then |f(x) − f(b)| ≤ 1 γ (α) ∫b x (j − x) α−1 ∣∣dαb−f(j)∣∣dj ≤ 1 γ (α) (∫b x (j − x) α−1 dj )∥∥dαb−f∥∥∞,[a,b] = 1 γ (α) ( (j − x) α α ∣∣∣∣b x )∥∥dαb−f∥∥∞,[a,b] = 1 γ (α + 1) (b − x) α ∥∥dαb−f∥∥∞,[a,b] . therefore |f(x) − f(b)| ≤ (b − x) α γ (α + 1) ∥∥dαb−f∥∥∞,[a,b] , ∀ x ∈ [a,b] . hence it holds ∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ = ∣∣∣∣∣ 1b − a ∫b a (f(x) − f(b))dx ∣∣∣∣∣ ≤ 1 b − a ∫b a |f(x) − f(b)|dx ≤ 1 b − a ∫b a (b − x) α γ (α + 1) ∥∥dαb−f∥∥∞,[a,b] dx 4 george a. anastassiou cubo 14, 1 (2012) = ∥∥dαb−f∥∥∞,[a,b] (b − a)γ (α + 1) ∫b a (b − x) α dx = ∥∥dαb−f∥∥∞,[a,b] (b − a)γ (α + 1)  −   (b − x)α+1 α + 1 ∣∣∣∣∣ b a     = ∥∥dαb−f∥∥∞,[a,b] (b − a)γ (α + 1) (−1) ( 0 − (b − a) α+1 α + 1 ) = ∥∥dαb−f∥∥∞,[a,b] (b − a)γ (α + 2) · (b − a)α+1 = ∥∥dαb−f∥∥∞,[a,b] · (b − a)α γ (α + 2) , proving the claim. we also give theorem 7. let α ≥ 1, m = dαe, f ∈ acm ([a,b]). assume that f(k) (b) = 0, k = 1, ...,m−1, and dαb−f ∈ l1 ([a,b]). then∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ ≤ ∥∥dαb−f∥∥l1([a,b]) γ (α + 1) (b − a) α−1 . (7) proof. we have again |f(x) − f(b)| ≤ 1 γ (α) ∫b x (j − x) α−1 ∣∣dαb−f(j)∣∣dj ≤ 1 γ (α) (b − x) α−1 ∫b x ∣∣dαb−f(j)∣∣dj ≤ 1 γ (α) (b − x) α−1 ∥∥dαb−f∥∥l1([a,b]) . hence |f(x) − f(b)| ≤ ∥∥dαb−f∥∥l1([a,b]) γ (α) (b − x) α−1 , ∀ x ∈ [a,b] . therefore ∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ ≤ 1b − a ∫b a |f(x) − f(b)|dx ≤ 1 b − a ∫b a ∥∥dαb−f∥∥l1([a,b]) γ (α) (b − x) α−1 dx = ∥∥dαb−f∥∥l1([a,b]) (b − a)γ (α) ∫b a (b − x) α−1 dx = ∥∥dαb−f∥∥l1([a,b]) (b − a)γ (α) (b − x) α α = ∥∥dαb−f∥∥l1([a,b]) γ (α + 1) (b − x) α−1 , proving the claim. cubo 14, 1 (2012) univariate right fractional ostrowski inequalities 5 we continue with theorem 8. let p,q > 1 : 1 p + 1 q = 1, α > 1 − 1 p , m = dαe, f ∈ acm ([a,b]). assume that f(k) (b) = 0, k = 1, ...,m − 1, and dαb−f ∈ lq ([a,b]). then∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ ≤ ∥∥dαb−f∥∥lq([a,b]) γ (α) (p(α − 1) + 1) 1 p ( α + 1 p ) (b − a)α−1+ 1p . (8) proof. we have again |f(x) − f(b)| ≤ 1 γ (α) ∫b x (j − x) α−1 ∣∣dαb−f(j)∣∣dj ≤ 1 γ (α) (∫b x (j − x) p(α−1) dj )1 p (∫b x ∣∣dαb−f(j)∣∣q dj )1 q ≤ 1 γ (α) (b − x) (α−1)+ 1 p (p(α − 1) + 1) 1 p (∫b x ∣∣dαb−f(j)∣∣q dj )1 q ≤ 1 γ (α) (b − x) (α−1)+ 1 p (p(α − 1) + 1) 1 p ∥∥dαb−f∥∥lq([a,b]) . therefore |f(x) − f(b)| ≤ ∥∥dαb−f∥∥lq([a,b]) γ (α) (p(α − 1) + 1) 1 p (b − x) α−1+ 1 p , ∀ x ∈ [a,b] . hence ∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ ≤ 1b − a ∫b a |f(x) − f(b)|dx ≤ ∥∥dαb−f∥∥lq([a,b]) (b − a)γ (α) (p(α − 1) + 1) 1 p ∫b a (b − x) α−1+ 1 p dx = ∥∥dαb−f∥∥lq([a,b]) γ (α) (p(α − 1) + 1) 1 p (b − a) α−1+ 1 p( α + 1 p ) . corollary 9. let α > 1 2 , m = dαe, f ∈ acm ([a,b]). assume f(k) (b) = 0, k = 1, ...,m − 1, dαb−f ∈ l2 ([a,b]). then∣∣∣∣∣ 1b − a ∫b a f(x)dx − f(b) ∣∣∣∣∣ ≤ ∥∥dαb−f∥∥l2([a,b]) γ (α) (√ 2α − 1 )( α + 1 2 ) (b − a)α− 12 . (9) we finish with 6 george a. anastassiou cubo 14, 1 (2012) proposition 10. inequality (6) is sharp, namely it is attained by f(x) = (b − x) α , α > 0, α /∈ n, x ∈ [a,b] . proof. notice that (b − x) α ∈ acm ([a,b]). we see that f′ (x) = −α(b − x) α−1 , f′′ (x) = (−1) 2 α(α − 1) (b − x) α−2 , ..., f(m−1) (x) = (−1) m−1 α(α − 1) (α − 2) ...(α − m + 2) (b − x) α−m+1 , and f(m) (x) = (−1) m α(α − 1) (α − 2) ...(α − m + 2) (α − m + 1) (b − x) α−m . thus dαb−f(x) = (−1) 2m γ (m − α) α(α − 1) ...(α − m + 1) ∫b x (j − x) m−α−1 (b − j) α−m dj = α(α − 1) ...(α − m + 1) γ (m − α) ∫b x (b − j) (α−m+1)−1 (j − x) (m−α)−1 dj = α(α − 1) ...(α − m + 1) γ (m − α) γ (α − m + 1)γ (m − α) γ (1) = α(α − 1) ...(α − m + 1)γ (α − m + 1) = γ (α + 1) . that is dαb−f(x) = γ (α + 1) , ∀ x ∈ [a,b] . also we see that f(k) (b) = 0, k = 0,1, ...,m−1, and dαb−f ∈ l∞ ([a,b]). so f fulfills all assumptions. next we see r.h.s.(6) = γ (α + 1) γ (α + 2) (b − a) α = (b − a) α (α + 1) . l.h.s.(6) = 1 b − a ∫b a (b − x) α dx = 1 b − a (b − a) α+1 (α + 1) = (b − a) α α + 1 , proving attainability and sharpness of (6). received: november 2010. revised: march 2011. cubo 14, 1 (2012) univariate right fractional ostrowski inequalities 7 references [1] g.a. anastassiou, ostrowski type inequalities, proc. ams 123 (1995), 3775-3781. [2] g.a. anastassiou, fractional differentiation inequalities, research monograph, springer, new york, 2009. [3] g.a. anastassiou, on right fractional calculus, chaos, solitons and fractals, 42 (2009), 365376. [4] a.m.a. el-sayed, m. gaber, on the finite caputo and finite riesz derivatives, electronic journal of theoretical physics, vol. 3, no. 12 (2006), 81-95. [5] g.s. frederico, d.f.m. torres, fractional optimal control in the sense of caputo and the fractional noether’s theorem, international mathematical forum, vol. 3, no. 10 (2008), 479493. [6] r. gorenflo, f. mainardi, essentials of fractional calculus, 2000, maphysto center, http://www.maphysto.dk/oldpages/events/levycac2000/ mainardinotes/fm2k0a.ps. [7] a. ostrowski, über die absolutabweichung einer differentiebaren funcktion von ihrem integralmittelwert, comment. math. helv., 10 (1938), 226-227. [8] s.g. samko, a.a. kilbas, o.i. marichev, fractional integrals and derivatives, theory and applications, (gordon and breach, amsterdam, 1993) [english translation from the russian, integrals and derivatives of fractional order and some of their applications (nauka i tekhnika, minsk, 1987)]. introduction main results () cubo a mathematical journal vol.17, no¯ 03, (53–70). october 2015 gronwall-bellman type integral inequalities and applications to global uniform asymptotic stability mekki hammi and mohamed ali hammami university of sfax, faculty of sciences of sfax, department of mathematics, route soukra, bp 1171, 3000 sfax, tunisia, mohamedali.hammami@fss.rnu.tn abstract in this paper, some new nonlinear generalized gronwall-bellman-type integral inequalities are established. these inequalities can be used as handy tools to research stability problems of perturbed dynamic systems. as applications, based on these new established inequalities, some new results of practical uniform stability are also given. a numerical example is presented to illustrate the validity of the main results. resumen en este art́ıculo, establecemos algunas desigualdades integrales nolineales nuevas de tipo gronwall-bellman. estas desigualdades pueden ser usadas como herramientas utiles para estudiar problemas de estabilidad de sistemas dinámicos perturbados. como aplicaciones, basados en las nuevas desigualdades establecidas, también damos algunos resultados nuevos de estabilidad uniforme prácticos. un ejemplo numérico es presentado para ilustrar la validez de los resultados principales. keywords and phrases: gronwall-bellman inequality, perturbed systems, stability. 2010 ams mathematics subject classification: 26d15, 26d20, 34a40, 34h15. 54 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) 1 introduction in 1919, t.h. gronwall [6] proved a remarkable inequality which has attracted and continues to attract considerable attention in the literature. in the qualitative theory of differential, the gronwall type inequalities of one variable for the real functions play a very important role. the first use of the gronwall inequality to establish boundedness and uniqueness is due to r. bellman [1] . gronwall-bellman inequality, which is usually proved in elementary differential equations using continuity arguments, is an important tool in the study of of qualitative behavior of solutions of differential and stability. the problem of stability analysis of nonlinear time-varying systems has attracted the attention of several researchers and has produced a vast body of important results (see [2]-[15] and the references therein). in this paper, we present a new generalization of the gronwallbellman lemma. this new generalization can develop a simple command to exponentially stabilize a large class of nonlinear systems. in this paper, some new nonlinear generalized gronwall-bellman-type integral inequalities are given. as applications, we give some new classes of time-varying perturbed systems which are globally uniformly practically asymptotically stable. moreover, we give an example to illustrate the applicability of the results. 2 definitions and notations we consider the following system : ẋ(t) = f(t, x(t)), x(t0) = x0, (1) where t ∈ r+ is the time and x ∈ rn is the state. definition 1. (uniform boundedness). a solution of (1) is said to be globally uniformly bounded if for every α > 0 there exists c = c(α) such that, for all t0 ≥ 0, ‖ x0 ‖≤ α ⇒‖ x(t) ‖≤ c, ∀t ≥ t0. let r ≥ 0 and br = {x ∈ rn/ ‖ x ‖≤ r}. first, we give the definition of uniform stability and uniform attractivity of br. definition 2. (uniform stability of br). i. br is uniformly stable if for all ε > r, there exists δ = δ(ε) > 0 such that for all t0 ≥ 0, ‖ x0 ‖≤ δ ⇒‖ x(t) ‖≤ ε, ∀t ≥ t0. ii. br is globally uniformly stable if it is uniformly stable and the solutions of system (1) are globally uniformly bounded. cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 55 definition 3. (uniform attractivity of br). br is globally uniformly attractive if for all ε > r and c, there exists t(ε, c) > 0 such that for all t0 ≥ 0, ‖ x(t) ‖≤ ε, ∀t ≥ t0 + t(ε, c), ‖ x0 ‖≤ c the system (1) is globally uniformly practically asymptotically stable if there exists r ≥ 0 such that br is globally uniformly stable and globally uniformly attractive. definition 4. a continuous function α : [0, a) → [0, +∞) is said to belong to class k if it is strictly increasing and α(0) = 0. it is said to belong to class k∞ if a = +∞ and α(r) → +∞ as r → +∞. definition 5. a continuous function β : [0, a) × [0, +∞) → [0, +∞) is said to belong to class kl if, for each fixed s, the mapping β(r, s) belongs to class k with respect to r and, for each r, the mapping β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → +∞. proposition 1. if there exists a class k-function α, and a constant r > 0 such that, given any initial state x0, the solution satisfies ‖ x(t) ‖≤ α(‖ x0 ‖) + r ∀t ≥ t0, then the system (1) is globally uniformly practically stable. proposition 2. if there exist a class kl-function β, a constant r > 0 such that, given any initial state x0, the solution satisfies ‖ x(t) ‖≤ β(‖ x0 ‖, t − t0) + r ∀t ≥ t0, then the system (1) is globally uniformly practically asymptotically stable. the next definition concerns a special case of practical global uniform asymptotic stability, namely, if the class kl in the above proposition consists of functions β(r, s) = kre−γs. definition 6. br is globally uniformly exponentially stable if there exist γ > 0 and k ≥ 0 such that for all t0 ∈ r+ and x0 ∈ rn, ‖ x(t) ‖≤ k ‖ x0 ‖ exp(−γ(t − t0)) + r ∀t ≥ t0. system (1) is globally practically uniformly exponentially stable if there exist r > 0 such that br is globally uniformly exponentially stable. 3 basic results lemma 1. let u, v and w nonnegative piecewise continuous functions on [0, +∞) for which the inequality u(t) ≤ c + ∫t a (uv + w) ∀t ≥ a (2) 56 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) holds, where a and c are nonnegative constants. then, u(t) ≤ ce ∫t a v + re ∫t a (v + w r ) ∀t ≥ a, ∀r > 0. (3) proof it follows from (2) and the classic inequality ex > x + 1 ∀x > 0 that for all r > 0 and t ≥ a 0 ≤ u(t) < (c + re ∫ t a w r ) + ∫t a uv (4) which implies that u(t) c + re ∫ t a w r + ∫t a uv ≤ 1. since v ≥ 0, we obtain u(t)v(t) + w(t)e ∫ t a w r c + re ∫ t a w r + ∫t a uv ≤ v(t) + w(t)e ∫ t a w r c + re ∫ t a w r (5) then we take f(t) = ∫t a v + log(c + re ∫ t a w r ) − log(c + re ∫ t a w r + ∫t a uv) ∀t ≥ a. it is clear that f is defined, continuous and piecewise continuously differentiable on [a, +∞). consequently, we get for all b > a, a sequence {a0, ..., an} of [a, b] verifying f′(t) = v(t) + w(t)e ∫ t a w r c + re ∫ t a w r − w(t)e ∫ t a w r + u(t)v(t) c + re ∫ t a w r + ∫t a uv ∀t ∈ [a, b] − {a0, ..., an}. by using the inequality (5), we obtain f′(t) ≥ 0. thus, f is increasing on the intervals [a, a0), ...(an, b]. since f is continuous on [a, b], then f is increasing on [a, b]. consequently, we get f(b) ≥ f(a) however, f(a) = 0 which implies that f(b) ≥ 0 ∀b ≥ a. consequently log(c + re ∫ t a w r + ∫t a uv) ≤ ∫t a v + log(c + re ∫ t a w r ) ∀t ≥ a cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 57 hence c + re ∫ t a w r + ∫t a uv ≤ (c + re ∫ t a w r )e ∫ t a v by using the inequality (4), we have finally u(t) ≤ ce ∫ t a v + re ∫ t a (v+ w r ). lemma 2. let φ ∈ lp(r+, r+) where p ∈ [1, +∞]. we denote by ‖ φ ‖p the p-norm of φ. then, for all t0 ≥ 0, s ≥ 0 and t ≥ t0 ∫t t0 φ ≤ n + l(t − t0) (6) where n = ∫s 0 φ + ms p and l = p−1 p ms with ms =‖ φ|[s,+∞) ‖p . proof we first consider the case p ∈ (1, +∞). by using hölder inequality to the function φ, we obtain for all t ≥ t0 : ∫t t0 φ ≤ ( ∫t t0 φp) 1 p ( ∫t t0 1) p−1 p ≤ (t − t0) p−1 p ( ∫+∞ t0 φp) 1 p . we put f(x) = 1 p + p − 1 p x − x p−1 p ∀x > 0 then, f is differentiable on (0, +∞) and verifying f′(x) = p − 1 p (1 − x − 1 p ). hence, f is decreasing on (0, 1] and increasing on [1, +∞). since f(1) = 0, we conclude that f is positive on (0, +∞) which means that x p−1 p ≤ 1 p + p − 1 p x ∀x > 0 consequently, we have (t − t0) p−1 p ≤ 1 p + p − 1 p (t − t0) ∀t ≥ t0 then 0 ≤ ∫t t0 φ ≤ mt0[ 1 p + p − 1 p (t − t0)] (7) where mt0 =‖ φ|[t0,+∞) ‖p . this inequality holds also for p ∈ {1, +∞}. now, for all t0 ≥ 0, s ≥ 0 and t ≥ t0, we distingue three cases : 58 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) • s ≤ t0 ≤ t in view of (7), we obtain ∫t t0 φ ≤ mt0[ 1 p + p − 1 p (t − t0)] ≤ ms p + p − 1 p (t − t0)ms. now, since ∫s 0 φ ≥ 0, we obtain ∫t t0 φ ≤ ( ∫s 0 φ + ms p ) + p − 1 p (t − t0)ms. • t0 < s ≤ t we can write by using (7) ∫t t0 φ ≤ ∫s t0 φ + ∫t s φ ≤ ∫s 0 φ + [ 1 p + p − 1 p (t − s)]ms. then ∫t t0 φ ≤ ∫s 0 φ + ms p + p − 1 p (t − s)ms however, s ∈ (t0, t] then ∫t t0 φ ≤ ( ∫s 0 φ + ms p ) + p − 1 p (t − t0)ms. • t0 ≤ t < s it is clear that ∫t t0 φ ≤ ∫s 0 φ ≤ ( ∫s 0 φ + ms p ) + p − 1 p (t − t0)ms. then the lemma is proved. lemma 3. consider the differential system ẋ(t) = a(t)x(t) + h(t, x(t)) (8) where : i. a is an n× n matrix whose entries are all real-valued piecewise-continuous functions of t ∈ r+. ii. the function h is defined on r+ × rn, piecewise continuous in t, and locally lipshitz in x. cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 59 iii. there exist φ and ε piecewise continuous functions, positives and verifying ‖ h(t, x) ‖≤ φ(t) ‖ x ‖ +ε(t) ∀t ∈ r+. (9) then, for all (t0, x0) ∈ r+×rn, there exist a unique maximal solution x of (8) such that x(t0) = x0. moreover, x is defined on [t0, +∞). proof it is clear that the system (8) can be written ẋ(t) = f(t, x(t)) where f(t, x) = a(t)x + h(t, x). the function f is piecewise continuous in t and locally lipshitz in x, then we have : for all (t0, x0) ∈ r+ × rn, there exist a unique maximal solution x of (8) such that x(t0) = x0. we will prove that x is defined on [t0, +∞). supposed that is not true, then there exist tmax ∈ (t0, +∞) such that x is defined on [t0, tmax). then, for all t ∈ [t0, tmax) ‖ ẋ(t) ‖≤ (m1 + m2) ‖ x(t) ‖ +m3 where m1 = sup [t0,tmax] ‖ a(t) ‖, m2 = sup [t0,tmax] ‖ φ(t) ‖ and m3 = sup [t0,tmax] ‖ ε(t) ‖ . it is clear that m1, m2 and m3 ∈ r+, therefore ‖ ∫t t0 ẋ(s)ds ‖ ≤ ∫t t0 ‖ ẋ(s) ‖ ds ≤ ∫t t0 [(m1 + m2) ‖ x(s) ‖ +m3]ds then ‖ x(t) ‖≤‖ x(t0) ‖ + ∫t t0 [(m1 + m2) ‖ x(s) ‖ +m3]ds by using the lemma 1, we obtain for all t ∈ [t0, tmax) ‖ x(t) ‖ ≤ ‖ x(t0) ‖ e ∫ t t0 (m1+m2)ds + e ∫ t t0 (m1+m2+m3)ds ≤ m4 60 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) where m4 =‖ x(t0) ‖ e(m1+m2)tmax + e(m1+m2+m3)tmax. consequently, x remains within the compact bm4, which is impossible. so, we conclude that tmax = +∞. theorem 1. consider the following time-varying : ẋ(t) = a(t)x(t) + h(t, x(t)) (10) where : (1) a is an n × n matrix whose entries are all real-valued piecewise-continuous functions of t ∈ r+. (2) the transition matrix for the system ẋ = a(t)x satisfies : ‖ r(t, s) ‖≤ ke−γ(t−s) ∀(t, s) ∈ r2+ (11) for some k > 0 and γ > 0. (3) the function h is defined on r+ × rn, piecewise continuous in t, and locally lipshitz in x. (4) there exist φ and ε piecewise continuous functions, positives and verifying ‖ h(t, x) ‖≤ φ(t) ‖ x ‖ +ε(t) ∀t ∈ r+. (12) (5) φ ∈ lp(r+, r+) for some p ∈ [1, +∞). (6) there exist a constant m > 0 such that ε(t) ≤ me−γt. (13) then for all (t0, x0) ∈ r+ × rn, the maximal solution x of (10) such that x(t0) = x0, is verifying : i. the function x is defined on [t0, +∞). ii. for all t ≥ t0 ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + ne−θt where n, l > 0 and δ, θ ∈ (0, γ]. cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 61 proof of theorem 1 i. by using the lemma 3, we proved that the system (10) has a unique maximal solution x such that x(t0) = x0. moreover, x is defined on [t0, +∞). ii. we can write the solution x of (10) as x(t) = r(t, t0)x(t0) + ∫t t0 r(t, s)h(s, x(s))ds where r(t, t0) is the transition matrix of the system ẋ = a(t)x. further, we have : ‖ x(t) ‖ ≤ ‖ r(t, t0) ‖‖ x(t0) ‖ + ∫t t0 ‖ r(t, s) ‖‖ h(s, x(s)) ‖ ds ≤ ke−γ(t−t0) ‖ x0 ‖ + ∫t t0 ke−γ(t−s) ‖ h(s, x(s)) ‖ ds. from the inequalities (11) and (12), we deduce that u(t) ≤ ku(t0) + ∫t t0 [kφ(s)u(s) + keγsε(s)]ds where u(t) = eγt ‖ x(t) ‖ . now by the lemma 1, we get u(t) ≤ ku(t0)e ∫t t0 kφ + re ∫t t0 [kφ(s) + keγsε(s) r ]ds ∀t ≥ t0, ∀r > 0 since ‖ x(t) ‖= e−γtu(t) we obtain the estimation ‖ x(t) ‖≤ k ‖ x0 ‖ e ∫t t0 kφ − γ(t − t0) + re ∫t t0 [kφ(s) + keγsε(s) r ]ds − γt . (14) let us denote m = sup t≥0 [eγtε(t)] and ms = ( ∫+∞ s φp) 1 p we deduce from the assumptions 5 and 6, that m, ms ∈ r+ 62 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) it follows that ∫t t0 keγsε(s) r ds ≤ km r t ∀t ≥ t0. (15) moreover φ ∈ lp(r+, r+), then ∫+∞ t φp −−−−→ t→+∞ 0 and so there exist s ≥ 0 such that ms < γ k p p − 1 . by using the lemma 2, we find for all t ≥ t0 ∫t t0 φ ≤ ∫s 0 φ + ms p + ms p − 1 p (t − t0) (16) from (15) and (16), we get : ∫t t0 kφ − γ(t − t0) ≤ k( ∫s 0 φ + ms p ) + [kms p − 1 p − γ](t − t0) and ∫t t0 [kφ(s) + keγsε(s) r ]ds − γt ≤ [−γ + kms p − 1 p + km r ]t + k( ∫s 0 φ + ms p ). thus, (14) yields ‖ x(t) ‖≤ kek( ∫ s 0 φ+ ms p ) ‖ x0 ‖ e−[γ−kms p−1 p ](t−t0) + re −[γ−kms p−1 p − km r ]t+k( ∫ s 0 φ+ ms p ) . taking r > m γ k − p−1 p ms l = ke k( ms p + ∫ s 0 φ) n = re k( ms p + ∫ s 0 φ) = r k l δ = γ − k p − 1 p ms ∈ (0, γ] θ = γ − k p − 1 p ms − km r ∈ (0, δ). finally, we obtain ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + ne−θt ∀t ≥ t0. corollary 1. under the same assumptions of theorem 1, we get ∀r > 0, ∀t ≥ t0, ∀x0 ∈ rn \ br : ‖ x(t) ‖≤ p ‖ x0 ‖ e−θ(t−t0) where p > 0 and θ ∈ (0, γ). cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 63 proof due to theorem 1, we have ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + ne−θt ∀t ≥ t0. let r > 0, then for all x0 ∈ rn \ br ‖ x(t) ‖ ≤ l ‖ x0 ‖ e−δ(t−t0) + n r re−θ(t−t0) ≤ (l + n r ) ‖ x0 ‖ e−θ(t−t0). taking p = l + n r > 0, we obtain ‖ x(t) ‖≤ p ‖ x0 ‖ e−θ(t−t0). remark 1. take limit as r → m γ k − p−1 p ms in theorem 1, we obtain ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + n ∀t ≥ t0 (17) with n = m γ k − p−1 p ms e k( ms p + ∫s 0 φ) . in particular, if we choose p = 1, we find ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + n ∀t ≥ t0 (18) with l = kek‖φ‖1 and n = km γ ek‖φ‖1. the estimation (17) and (18) implies that the system (10) is globally uniformly practically asymptotically stable in the sense that the ball bn is globally uniformly asymptotically stable. theorem 2. consider the following time-varying : ẋ(t) = a(t)x(t) + h(t, x(t)) (19) where : (1) a is an n × n matrix whose entries are all real-valued piecewise-continuous functions of t ∈ r+. 64 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) (2) the transition matrix for the system ẋ = a(t)x satisfies : ‖ r(t, s) ‖≤ ke−γ(t−s) ∀(t, s) ∈ r2+ (20) for some k > 0 and γ > 0. (3) the function h is defined on r+ × rn, piecewise continuous in t, and locally lipshitz in x. (4) there exist φ and ε piecewise continuous functions, positives and verifying ‖ h(t, x) ‖≤ φ(t) ‖ x ‖ +ε(t) ∀t ∈ r+. (21) (5) sup [s,+∞) φ < γ k for some s ∈ [0, +∞). (6) there exist a constant m > 0 such that ε(t) ≤ me−γt. (22) then for all (t0, x0) ∈ r+ × rn, the maximal solution x of (10) such that x(t0) = x0, is verifying : i. the function x is defined on [t0, +∞). ii. for all t ≥ t0 ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + ne−θt where n, l > 0 and δ, θ ∈ (0, γ]. proof of theorem 2 i. by using the lemma 3, we proved that the system (19) has a unique maximal solution x such that x(t0) = x0. moreover, x is defined on [t0, +∞). ii. similar to the proof of theorem 1, it can be shown that : ‖ x(t) ‖≤ k ‖ x0 ‖ e ∫t t0 kφ − γ(t − t0) + re ∫t t0 [kφ(s) + keγsε(s) r ]ds − γt ∀t ≥ t0, ∀r > 0. let us denote m = sup t≥0 [eγtε(t)] ∈ r+ it follows that ∫t t0 keγsε(s) r ds ≤ km r t ∀t ≥ t0. hence, there exist s ∈ r+ such that sup [s,+∞) φ < γ k cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 65 then we can apply the lemma 2, we deduce that ∫t t0 φ ≤ ∫s 0 φ + ( sup [s,+∞) φ)(t − t0) ∀t ≥ t0 consequently, we obtain ‖ x(t) ‖≤ ke k ∫s 0 φ ‖ x0 ‖ e −[γ − k sup [s,+∞) φ](t − t0) + re −[γ − k sup [s,+∞) φ − km r ]t + k ∫s 0 φ . taking r > m γ k − sup [s,+∞) φ l = kek ∫ s 0 φ n = rek ∫ s 0 φ) = r k l δ = γ − k sup [s,+∞) φ ∈ (0, γ] θ = γ − k sup [s,+∞) φ − km r ∈ (0, δ). finally, we obtain ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) + ne−θt ∀t ≥ t0. corollary 2. under the assumptions (1),(2),(3),(4) and (6) of theorem 2, and by replacing the condition (5) by (5′) : φ(t) −−−−→ t→+∞ 0 then, we obtain the same consequences of theorem 2. proof since lim t→+∞ φ(t) = 0 , then there exist s ≥ 0 such that ∀t ≥ s φ(t) ≤ γ 2k therefore sup [s,+∞) φ < γ k . thus, we can apply theorem 2 to prove the result. remark 2. it is clear that if we choose ε = 0 in theorem 1 or 2, we obtain due to m = 0 : θ = δ = γ − k p − 1 p ms 66 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) l = ke k( ms p + ∫ s 0 φ) n = re k( ms p + ∫ s 0 φ) ∀r > 0 as r → 0+, we get the classic result: ‖ x(t) ‖≤ l ‖ x0 ‖ e−δ(t−t0) ∀t ≥ t0. we can see that the claim of the theorem 1 is true by examining a specific example, where a solution of the scalar equation can be found. example 1. consider the stability of following system :    ẋ1 = −x1 − tx2 + 1 (1 + t2)2 x21 1 + √ x2 1 + x2 2 + e−2t 1 + x2 1 ẋ2 = tx1 − x2 + t (1 + t2)2 x22 1 + √ x21 + x 2 2 (23) which can be writing as ẋ = a(t)x + h(t, x) where x = ( x1 x2 ) , a(t) = ( −1 −t t −1 ) and h(t, x) = ( h1(t, x) h2(t, x) ) with    h1(t, x) = 1 (1 + t2)2 x21 1 + √ x21 + x 2 2 + e−2t 1 + x2 1 h2(t, x) = t (1 + t2)2 x22 1 + √ x2 1 + x2 2 it is clear that the system ẋ = a(t)x cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 67 is globally uniformly asymptotically stable. indeed, the transition matrix r(t, t0) satisfies : r(t, t0) = e (t−t0)a = e−(t−t0) ( cos t − sin t sin t cos t ) thus, we obtain ‖ r(t, t0) ‖= ke−γ(t−t0) with γ = k = 1 and ‖ ‖ represents the euclidean norm. on the other hand, ‖ h(t, x) ‖ = h21(t, x) + h22(t, x) ≤ 1 (1 + t2)3 (x21 + x 2 2) + 2e −2t. by using the classic inequality √ a2 + b2 ≤ a + b ∀a, b ≥ 0 we get ‖ h(t, x) ‖≤ φ(t) ‖ x(t) ‖ +ε(t) ∀t ≥ 0 where φ(t) = 1 (1 + t2) 3 2 and ε(t) = √ 2e−t. it is easy to verify that φ and ε are continuous, positive and bounded on [0, +∞), in particular φ ∈ lp(r+, r+) ∀p ∈ [1, +∞]. to estimate ‖ φ ‖p, we use the inequality : φp(t) ≤ φ(t) ∀t ≥ 0 since ‖ φ ‖∞= 1, then ∫+∞ 0 φp ≤ ∫+∞ 0 φ however ∫+∞ 0 φ = 1, then ‖ φ ‖p≤ 1 ∀p ≥ 1. consequently ‖ φ ‖p< pp−1 ∀p ≥ 1, and we can apply theorem 1 to prove the following results : • ∀(t0, x0) ∈ r+ × r2, there exist a unique maximal solution x of (8) such that x(t0) = x0. moreover, x is defined on [t0, +∞). • ∀t ≥ t0, ∀p ≥ 1 ‖ x(t) ‖≤ e 1 p ‖ x0 ‖ e− 1 p (t−t0) + 2 √ 2e − 1 2p t+ 1 p 68 mekki hammi & mohamed ali hammami cubo 17, 3 (2015) by choosing r = 2 √ 2p. in particular ‖ x(t) ‖≤ e ‖ x0 ‖ e−(t−t0) + 2 √ 2e. (24) the estimation (24) implies that the system (23) is globally uniformly practically asymptotically stable in the sense that the ball b 2 √ 2e is globally uniformly asymptotically stable. 0 5 10 15 20 25 30 35 40 45 50 −0.5 0 0.5 1 1.5 2 time (s) x 1 figure 1: time evolution of the state x1(t) of system (23) cubo 17, 3 (2015) gronwall-bellman type integral inequalities and applications . . . 69 0 5 10 15 20 25 30 35 40 45 50 −0.5 0 0.5 1 1.5 2 time (s) x 2 figure 2: time evolution of the state x2(t) of system (23) received: september 2013. accepted: february 2015. references [1] r. bellman, stability theory of differential equations, mcgraw hill, new york, (1953). [2] a. benabdallah, m. dlala and m.a. hammami. a new lyapunov function for stability of timevarying nonlinear perturbed systems, systems and control letters 56, (2007) 179-187 . 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[15] v. i. zubov, methods of a.m. lyapunov and their application, p. noordhoff, groningen, the netherlands, (1964). introduction definitions and notations basic results () cubo a mathematical journal vol.17, no¯ 02, (97–122). june 2015 spacetime singularity, singular bounds and compactness for solutions of the poisson’s equation carlos cesar aranda blue angel navire research laboratory, rue eddy 113 gatineau, qc, canada carloscesar.aranda@gmail.com abstract a black hole is a spacetime region in whose interior lies a structure known as a spacetime singularity whose scientific description is profoundly elusive, and which depends upon the still missing theory of quantum gravity. using the classical weak comparison principle we are able to obtain new bounds, compactness results and concentration phenomena in the theory of newtonian potentials of distributions with compact support which gives a suitable mathematical theory of spacetime singularity. we derive a rigorous renormalization of the newtonian gravity law using nonlinear functional analysis and we have a solid set of astronomical observations supporting our new equation. this general setting introduces a new kind of ill posed problem with a very simple physical interpretation. resumen un hoyo negro es una región espacio-temporal en cuyo interior hay una estructura llamada singularidad espacio-temporal cuya descripción cient́ıfica es dif́ıcil de encontrar, y que depende de la aún inexistente teoŕıa de la gravedad cuántica. usando el clásico principio de comparación débil, aqúı probamos nuevas cotas, resultados de compacidad y fenómenos de concentración en la teoŕıa de potenciales newtonianos de distribuciones de soporte compacto, que dan una teoŕıa matemática adecuada de la singularidad espacio-temporal. derivamos una rigurosa renormalización de la ley de gravitación newtoniana usando análisis funcional no lineal y tenemos un contundente conjunto de datos de observaciones astronómicas que apoyan nuestra nueva ecuación. este marco general introduce una nueva forma de problema mal-puesto con una interpretación f́ısica muy simple. keywords and phrases: black hole, spacetime singularity, quantum field theory, newtonian potentials, elliptic equations, compact imbedding, sobolev’s spaces. 2010 ams mathematics subject classification: 35j25, 35j60, 35j75. 98 carlos cesar aranda cubo 17, 2 (2015) 1 introduction. in [2, 3], the authors introduced a new concentration phenomena for the poisson’s equation using techniques from nonlinear functional analysis. in this article we are concerned with several simple consequences of this new concentration of compactness results. using the classical theory of newtonian potentials of distributions with compact support we are able to derive concentration of compactness for newtonian potentials with singular behaviour. for a review of this topic see [10]. newtonian potentials are useful in the description of gravity fields of celestial bodies [5, 20, 21]. today black holes in gravity theory and astronomy plays a central role [8, 12, 15, 27, 30, 31, 33, 40, 46]. the interior of a black hole is usually called spacetime singularity [7, 31]. in [3] we obtain the existence of a sequence {pj} ∞ j=1 ∈ c 2(ω) for any ω bounded domain in rn such that − limj→∞ ∆pj = ∞ uniformly on ω and 0 < pj(x) ≤ pj+1(x) ≤ cte for all x ∈ ω. this sequence proof that it is possible to do rigorous treatment of divergence to infinite in the frame of newtonian potentials extending rigorous quantum field theories on large scale [37, 38]. black holes are complicated real objects, and our newtonian equation is a mathematical object but we have astronomical observations of supermassive black holes given a concrete physical significance to this new theoretical frame [27]. this is a remarkable fact in gravity theory [13, 41]. lemma 1.1 (lemma 1 page 277 [11]). let ω be an open set of rn, f ∈ d′(ω) and u a solution (in the sense of distribution) of poisson’s equation ∆u = f on ω. then for every bounded open set ω1 with ω1 ⊂ ω there exists f1 ∈ e ′ the space of distributions on rn with compact support, such that f1 = f on ω, u = the newtonian potential of f1 on ω1. (1) therefore for ω1 ⊂ ω and our sequence {pj} ∞ j=1 there exists a sequence {f1,j} ∞ j=1 ∈ e ′ the space of distributions on rn with compact support, such that f1,j = ∆pj on ω, pj = the newtonian potential of f1,j on ω1. (2) we have associated to each pair {pj, ω1} a gravitational newtonian potential defined on all r n. using this lemma we have a very simple newtonian interpretation of the interior of a black hole. we have a set of solid astronomic observations supporting our new equation. cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 99 black hole name solar mass (sun=1) references holmberg 15a 170.000.000.000 [23] s5 0014+813 40.000.000.000 [16] sdss j085543.40-001517.7 25.000.000.000 [47] apm 08279+5255 23.000.000.000 [35] ngc 4889 21.000.000.000 [25] central black hole of phoenix cluster 20.000.000.000 [26] sdss j07451.78+734336.1 19.500.000.000 [47] oj 285 primary 18.000.000.000 [39] sdss j08019.69+373047.3 15.140.000.000 [47] sdss j115954.33+20192.1 14.120.000.000 [47] sdss j075303.34+423130.8 13.500.000.000 [47] sdss j081855.77+095848.0 12.000.000.000 [47] sdss j0100+2802 12.000.000.000 [44] sdss j082535.19+512706.3 11.220.000.000 [47] sdss j013127.34-0321000.1 11.000.000.000 ♦ central black hole of rx j1532.9+3021 10.000.000.000 ♦ qso b2126-158 10.000.000.000 [16] sdss j015741.57-010629.6 9.800.000.00 [47] ngc 3842 9.700.000.000 [25] sdss j2330301.45-093930.7 9.120.000.000 [47] sdss j075819.70+202300.9 7.800.000.000 [47] sdss j080956.02+50200.9 6.450.000.000 [47] sdss j0142114.75+0023224.2 6.310.000.000 [47] messier 87 6.300.000.000 ♦ qso b0746+254 5.000.000.000 [16] qso b2149-306 5.000.000.000 [16] ngc 1277 5.000.000.000 ♦ sdss j090033.50+421547.0 4.700.000.000 [47] messier 60 4.500.000.000 ♦ sdss j011521.20+152453.3 4.100.000.000 [47] qso b0222+185 4.000.000.000 [16] hercules a (3c 348) 4.000.000.000 ♦ sdss j213023.61+122252.0 3.500.000.000 [47] 100 carlos cesar aranda cubo 17, 2 (2015) j173352.23+540030.4 3.400.000.000 [47] sdss j025021.76-075749.9 3.100.000.000 [47] sdss j030341.04-002321.9 3.000.000.000 [47] qso b0836+710 3.000.000.000 [16] sdss j224956.08+000218.0 2.630.000.000 [47] sdss j030449.85-000813.4 2.400.000.000 [47] sdss j234625.66-001600.4 2.240.000.000 [47] ulas j1120+0641 2.000.000.000 ♦ qso 0537-286 2.000.000.000 [16] ngc 3115 2.000.000.000 ♦ q0906+6930 2.000.000.000 ♦ qso b0805+614 1.500.000.000 [16] messier 84 1.500.000.000 ♦ qso b225155+2217 1.000.000.000 [16] qso b1210+330 1.000.000.000 [16] ngc 6166 1.000.000.000 ♦ cygnus a 1.000.000.000 ♦ sombrero galaxy 1.000.000.000 ♦ markarian 501 900.000.000-3.400.000.000 ♦ pg 1426+015 467.740.000 [28] 3c 273 550.000.000 [28] messier 49 560.000.000 ♦ pg 0804+761 190.550.000 [28] pg 1617+175 275.420.000 [28] pg 1700 + 518 60.260.000 [28] ngc 4261 400.000.000 ♦ ngc 1275 340.000.000 ♦ 3c 390.3 338.840.000 [28] ii zwicky 136 144.540.000 [28] pg 0052+251 218.780.000 [28] messier 59 270.000.000 ♦ pg 1411+442 79.430.000 [28] markarian 876 240.000.000 [28] andromeda galaxy 230.000.000 ♦ pg 0953+414 182.000.000 [28] pg 0026+129 53.700.000 [28] fairall 9 79.430.000 [28] cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 101 markarian 1095 182.000.000 [28] messier 105 200.000.000 ♦ markarian 509 57.550.000 [28] oj 287 secondary 100.000.000 [39] rx j124236.9-1111935 100.000.000 ♦ messier 85 100.000.000 ♦ ngc 5548 123.000.000 ♦ pg 1221+143 40.740.000 [28] messier 88 80.000.000 ♦ messier 81 70.000.000 ♦ markarian 771 75.860.000 [28] messier 58 70.000.000 ♦ pg 0844+349 21.380.000 [28] centaurus a 55.000.000 ♦ markarian 79 52.500.000 [28] messier 96 48.000.000 ♦ markarian 817 43.650.000 [28] ngc 3227 38.900.000 [28] ngc 4151 primary 40.000.000 ♦ 3c 120 22.900.000 ♦ markarian 279 41.700.000 ♦ ngc 3516 23.000.000 ♦ ngc 863 17.700.00 ♦ messier 82 30.000.000 ♦ messier 108 24.000.000 ♦ ngc 3783 9.300.000 ♦ markarian 110 5.620.000 ♦ markarian 335 6.310.000 ♦ ngc 4151 secondary 10.000.000 ♦ ngc 7469 6.460.000 ♦ ic 4329 a 5.010.000 ♦ ngc 4593 8.130.000 ♦ messier 61 5.000.000 ♦ messier 32 1.500.000-5.000.000 ♦ sagittarius a* 4.100.000 ♦ ngc 4051 1.300.000 ♦ ♦ only main 102 carlos cesar aranda cubo 17, 2 (2015) references are provided. our function p = limj→∞ pj is very ’rough’ from the point of view of regularity on sobolev’s spaces in opposition of the poisson’s equation ∆u = 0 on ω. theorem 1.2 (weyl page 118[10]). let ω ⊂ rn be open and u ∈ l1loc(ω) satisfy ∫ ω u∆v = 0 for all v ∈ c∞0 (ω). then u ∈ c ∞(ω) and ∆u = 0. a simpler use of green’s identity allow us to imply the discontinuity at infinitum of the functional g : c2(ω) × c2(ω) → r, g(u, v) = ∫ ω (v∆u − u∆v) dx − ∫ ∂ω ( v ∂u ∂n − u ∂v ∂n ) dγ, (3) where ω is a bounded domain with c1 boundary ∂ω. the functional g for a fixed pair (u, v) ∈ c2(ω) × c2(ω) in a smooth bounded domain ω ⊂ rn satisfies the green’s identity g(u, v) = 0 (for a proof of green’s formula see page 20 [29]). this equality is a consequence of the divergence theorem, (for a proof of the divergence theorem stated by e. heinz see page 46 [36], also page 570 [11]). let us to remember several classical results: theorem 1.3. [theorem 6.6 [18]] let ω be a c2,α domain in rn and let u ∈ c2,α(ω) be a solution of the equation lu = n∑ i,j=1 aijuxi,xj + n∑ i=1 biuxi + cu = f, (4) where f ∈ cα(ω) and the coefficients of l satisfy, for positive constants λ, λ n∑ i,j=1 aijξiξj ≥ λ | ξ | 2, for all x ∈ ω, ξ ∈ rn, (5) | ai,j |0,α;ω, | bi |0,α;ω, | ci |0,α;ω≤ λ. (6) let ϕ ∈ c2,α(ω) and suppose u = ϕ on ∂ω. then | u |2,α;ω≤ c{| u |0,ω + | ϕ |2,α;ω + | f |0,α;ω}, (7) where c = c(n, α, λ, λ, ω). lu = di ( aij(x)dju + b i(x)u ) + ci(x)diu + d(x)u, (8) l∗u = di ( aijdju − c iu ) − bidiu + du, (9) aij(x)ξiξj ≥ λ | ξ | 2, for all x ∈ ω, ξ ∈ rn, (10) ∑ | aij(x) |2≤ λ2, (11) λ−2 ∑ ( | bi(x) |2 + | ci(x) |2 ) + λ−1 | d(x) |≤ v2, (12) cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 103 theorem 1.4. [theorem 9.11 [18]] let ω be an open set in rn and w2,p(ω)∩lp(ω), 1 < p < ∞, a strong solution of the equation lu = f in ω where the coefficients of l satisfy for positive constants λ, λ aij ∈ c0(ω), bi, c ∈ l∞(ω), f ∈ lp(ω); (13) aijξiξj ≥ λ | ξ | 2 for all ξ ∈ rn, (14) | ai,j |, | bi |, | c |≤ λ, (15) where i, j = 1, . . . , n. then for any domain ω′ ⊂⊂ ω, ‖ u ‖2,p;ω′≤ c (‖ u ‖p;ω + ‖ f ‖p;ω) , (16) where c depends on n, p, λ, λ, ω′, ω and the moduli of continuity of the coefficients aij on ω′. now the existence of our concentrating sequence of non negative, bounded functions {pj} ∞ j=1 is a compactness property not established and noncontradictory with the statements in theorems 1.3 and 1.4. this is a singular property that using very elementary techniques, allow us to obtain new bounds and several interesting results related to the newtonian potential in the theory of distributions [11]. moreover we built a sequence fj : [a, b] → [0,∞) satisfying limj→∞ f′′j (x) = −∞ in measure, cte ≥ fj+1 ≥ fj ≥ 0 and each fj non increasing on [a, b]. by considering fj(x1, x2, . . . , xn) = fj(x1) we obtain a sequence for bounded smooth domains in r n, n ≥ 2 with similar properties. 2 preliminaries. the results of this section are contained in [3]. for the sake of the readability, we stated and prove it here again. lemma 2.1. let b(0, r) be a ball of radius r > 0 in rn, n > 2. consider the singular nonlinear elliptic equation −∆pj = hj(pj) in b(0, r) pj = 0 on ∂b(0, r). (17) where hj : (0,∞) → (0,∞) is locally hölder continuous function hj(s) = { s−j if 0 < s < 1, s−1 if s ≥ 1. then the next properties holds: (i) the sequence {pj} ∞ j=1 ∈ c 2(b(0, r)) ∩ c(b(0, r)) are radial functions with ∂p ∂r < 0. (ii) the sequence {pj} ∞ j=1 satisfies pj ≤ pj+1. (iii) the sequence {hj(pj)} ∞ j=1 satisfies hj(pj) ≤ hj+1(pj+1). (iv) the sequence {pj} ∞ j=1 satisfies w ≤ pj ≤ e, where −∆v = v −1 in b(0, r), v = 0 on ∂b(0, r), 104 carlos cesar aranda cubo 17, 2 (2015) −∆e = e−1 in b(0, r), e = 1 on ∂b(0, r) and −∆w = e−1 in b(0, r), w = 0 on ∂b(0, r). proof. the enunciate (i) is a consequence of classical results on radial symmetry. the points (ii) and (iii) have been stated at [2, 3] and the point (iv) is proved in [1]. theorem 2.2 ([3]). let b(0, r) ⊂ rn, a ball of radius r, with n ≥ 3. then there exists a sequence of radial, nonnegative and bounded functions {pj} ∞ j=1 and 0 ≤ r0 < r such that − lim j→∞ ∆pj = ∞ on a(r0, r), (18) where a(r0, r) is the annulus of external radius r and internal radius r0. moreover pj ∈ c∞(a(r0, r)) and pj ≤ pj+1. proof. our proof is a reductio ad absurdum procedure. let us to consider p = lim j→∞ pj. (19) now if there exists no sequence such that it is stated in our theorem 2.2, then lim rրr p(r) ≥ 1, (20) because if limrրr p(r) < 1 for a all nonnegative and small enough ǫ, there exist δ > 0 such that p(r) ≤ 1 − ǫ for all r ∈ (r − δ, r). therefore pj(r) ≤ p(r) ≤ 1 − ǫ for all r ∈ (r − δ, r) and − limj→∞ ∆pj = limj→∞ hj(pj) = limj→(pj) −j ≥ limj→∞(1 − ǫ) −j = ∞ on a(r − δ, r). (21) similarly if there exists no sequence satisfying our theorem 2.2 then lim j→∞ hj(pj(r)) < ∞ for all r ∈ [0, r), (22) because if lim j→∞ hj(pj(r)) = ∞ for r0 ∈ [0, r), (23) from hj(pj(r)) ≥ hj(pj(r0)) for all | x |= r ∈ [r0, r), (24) we deduce − limj→∞ ∆pj = limj→∞ hj(pj) ≥ limj→∞ hj(pj(r0)) = ∞ on a(r0, r). (25) cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 105 contradiction so 22 holds and we obtain that p ∈ c1(b(0, r)), because using theorem 9.11 page 235 in [18] we derive ‖ pj ‖h2,p(ω′′) ≤ c(n, p, ω ′, ω′′) { ‖ pj ‖lp(ω′) + ‖ hj(pj) ‖lp(ω′) } ≤ c(n, p, ω′, ω′′) { ‖ e ‖lp(ω′) + ‖ limj→∞ hj(pj(r3)) ‖lp(ω′) } (26) where ω′ ⊂ ω′′, ω′′ ⊂ b(0, r), p > n and r3 = supx∈ω′ | x |. according to theorem 7.26 page 171 [18], we have the bounds ‖ pj ‖ c 1,1− n p (ω′′) ≤‖ pj ‖h2,p(ω′′) . (27) therefore p ∈ c1(b(0, r)) and pj → p in c1,αloc(b(0, r) for 0 < α < 1 − n p . one more time if there exist no sequence satisfying the conclusions of theorem 2.2 we imply p : [0, r) → [0,∞) is a strictly nonincreasing radial function because if there exist 0 ≤ r1 < r2 < r with p(r1) = p(r2) and the fact of being p nonincreasing implies −∆p = 0 on the annulus a(r1, r2). using ‖ pj ‖c1,α(a(r1,r2)≤ c and a nonnegative test function ϕ with support contained in a(r1, r2): 0 = ∫ a(r1,r2) ∇p · ∇ϕdx = limj→∞ ∫ a(r1,r2) ∇pj · ∇ϕdx = ∫ a(r1,r2) hj(p1)ϕdx ≥ ∫ a(r1,r2) h1(p1)ϕdx > 0. (28) contradiction. so from the negation of the conclusion of theorem 2.2 we derive that the function p : [0, r) → r satisfies p ∈ c1(0, r), p is strictly nonincreasing in (0, r) and limrրr p ≥ 1 and therefore p(r) > 1 for all r ∈ [0, r). (29) finally we are ready to finish the proof of theorem 2.2. independently of the hypothesis in the reductio ad absurdum, there exists 0 < r0 < r such that h1(p1(r0)) > h1(1) = 1 and therefore using (iii) of lemma 2.1 1 < h1(p1(r0)) ≤ hj(pj(r0)) for all j ≥ 1. (30) but limj→∞ pj(r0) = p(r0) > 1 and therefore for j big enough pj(r0) > 1 and hj(pj(r0)) = h1(pj(r0)) < h1(1) = 1. (31) therefore 1 < h1(p1(r0)) ≤ hj(pj(r0)) < 1, (32) for j big enough. in page 291 [11] it is stated that for every function f ∈ cm,α(ω), 0 < α < 1, m ≥ 1 the solutions of poisson’s equation ∆u = f on ω are of class cm+2,α on ω, and so pj are of class c∞(a(r, r)). this end the proof. 106 carlos cesar aranda cubo 17, 2 (2015) the next lemma is new lemma 2.3. the sequence {hj(pj)} ∞ j=1 is unbounded in c α loc(a(r, r)). proof. in theorem 4.6 page 60 [18] it is stated that: let ω be a domain in rn and let u ∈ c2(ω), f ∈ cα(ω) satisfy poisson’s equation ∆u = f. then u ∈ c2,α(ω) and for any two concentric balls b1 = b(x0, r), b2 = b(x0, 2r) ⊂⊂ ω we have | u |′2,α;b1≤ c(| u |0;b2 +r 2 | f |′0,α;b2). (33) therefore limj→∞ | hj(pj) | ′ 0,α;b2 = ∞. 3 the construction of the sequence of functions fj. in [19] it is stated that [page xviii, [19]] when we refer to a set f as a fractal, therefore, we will typically have the following in mind. (i) f has a fine structure, i. e. detail on arbitray small scales. (ii) f is too irregular to be described in traditional geometrical language, both locally and globally. (iii) often f has some form of self-similarity, perhaps approximate or statistical. (iv) usually, the ‘fractal dimension’ of f (defined in some way) is greater than its topological dimension. (v) in most cases of interest f is defined in a very simple way, perhaps recursively. moreover [page xxii, [19]] the highly intricate structure of the julia set illustrated in figure 0.6 stems from the single quadratic function f(z) = z2 + c for a suitable constant c. although the set is not strictly self-similar in the sense that the cantor’s set and von koch curve are, it is ‘quasi-self-similar’ in that arbitrarily small portions of the set can be magnified an distorted smoothly to coincide with a large part of the set. cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 107 let [0, a] be a bounded interval in r. let us to consider, the infinite sequence of linear functions h(x) = a − x, h0(x) = a + 1 − x, h1(x) = a + 1 + 1 2 − x, · · · hj(x) = a + j∑ n=0 1 2n − x. we introduce a infinite sequence of functions defined on [0, a], p(x) = p(x; s; k) = −t(x − s k a)2 + cs for all x ∈ [a s k , a s + 1 k ], s = 0, 1, 2, . . . k − 1, p0(x) = p0(x; s; k0) = −t0(x − s k0 a)2 + c0,s for all x ∈ [a s k0 , a s + 1 k0 ], s = 0, 1, 2, . . . k0 − 1, p1(x) = p1(x; s; k1) = −t1(x − s k1 a)2 + c1,s for all x ∈ [a s k1 , a s + 1 k1 ], s = 0, 1, 2, . . . k1 − 1, . . . , pj(x) = pj(x; s; kj) = −tj(x − s kj a)2 + cj,s for all x ∈ [a s kj , a s + 1 kj ], s = 0, 1, 2, . . . kj − 1. where the sequence c0 = 1, c0,0 = 1 + 1, c1,0 = 1 + 1 + 1 2 , . . . , cj,0 = 1 + j∑ s=0 1 2j . satisfies cs = p ( s k ; s; k ) for all s = 1, . . . , k − 1, (34) c0,s = p0 ( s k ; s; k0 ) for all s = 1, . . . , k0 − 1, (35) c1,s = p1 ( s k ; s; k1 ) for all s = 1, . . . , k1 − 1, (36) · · · , (37) cj,s = pj ( s kj ; s; kj ) for all s = 1, . . . , kj − 1. (38) 108 carlos cesar aranda cubo 17, 2 (2015) we have the association: {cs} ∞ s=0 ←→ h ←→ p, {c0,s} ∞ s=0 ←→ h0 ←→ p0, {c1,s} ∞ s=0 ←→ h1 ←→ p1, . . . , {cj,s} ∞ s=0 ←→ hj ←→ pj. the choice of the sequence of non negative numbers k ∪ {kj} ∞ j=1 determines t ∪ {tj} ∞ j=1. we keep k such that the equation p(x; 0, k) = h0(x), (39) has no solutions. similarly we divide this first interval [0, a k ] in k0 intervals and setting k0 = kk0 such that the equation p0(x; 0, k) = h1(x), (40) has no solution. now we complete the procedure by induction. therefore lim j→∞ tj = ∞. (41) it is follow that the non decreasing, bounded, sequence of functions {p}∞j=0 defined on [0, a] has second derivative defined almost everywhere and p′′j (x) = −2tj. using the rolle’s theorem for the functions pj(·; s; kj), hj(·) in the interval [ sa kj , (s+1)a kj ], we deduce the existence of xj,s ∈ ( sa kj , (s+1)a kj ) such that p′j(xj,s; s; kj) = h ′ j(xj,s) = −1, (42) where xj,s = 1 2tj + sa kj . (43) similarly for x̃j,s = 1 20tj + sa kj , (44) we have p′j(x̃j,s; s; kj) = − 1 10 . (45) for j big enough, let us to consider the sequence of intervals (sa kj − δj, sa kj + δj) ⊂ [0, a] for s = 1, 2, . . . , kj − 1 where δj = min{ 1 200tj , a kjj 2 , a kj − 1 2tj 10 }. (46) cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 109 by smoothing each pj on ( sa kj − δj, sa kj + δj), we obtain the desired sequence {fj} ∞ j=1 ∈ c ∞[a, b]. we point that c∞ extension is a nontrivial task, see for example page 136 [18] or the nikolskii’s extension method page 69 [29] and the calderon’s extension method page 72 [29]. therefore we give a full description of our procedure. for the smoothing method we use the functions s(x; a; b) = x − ∫x 0 g(t)dt, (47) where g(x) = 0 for x < a − δ, g(x) = 1 for a < x < b, g(x) = 0 for x > b + δ, 0 < δ ∈ r, and g ∈ c∞(r) (see the c∞ urysohn lemma page 245, [14]). it it follows that s(x; a; b) = x for x < a − δ, s(x; a; b) = x − c for x > b + δ and s′(x; a; b) ≥ 0. taking a suitable composition with functions s(·; a; b) we accomplish with the smoothing procedure and moreover f′j(x) ≤ 0. the sequence {∆fj} ∞ j=1 converges in measure to −∞. note that the set of numbers {xj,s, x̃j,s} are numerable and dense in [0, a]. therefore by 42 and 45 we imply if fj(x1, x2, . . . , xn) = fj(x1) then lim j→∞ ‖ fj ‖c1,α(b(x,r))= ∞, (48) for any ball b(x, r) with center at x with radius r and b(x, r) ⊂ ω. it follows that lim j→∞ ‖ fj ‖c2,α(b(x,r))= ∞. (49) let us to consider the function iaj(x) = { 1 if x ∈ aj, 0 if x /∈ aj, (50) where aj = {x ∈ ( sa kj + δj, (s+1)a kj − δj) | s = 0, 1, 2, . . . kj − 1}. therefore ∫a 0 f′′j iajdx = tj kj−1∑ s=0 ( a kj − 2δj ) = tja − 2 kj−1∑ s=0 δj ≥ tja − 2 a j2 . so limj→∞ ∫a 0 f′′j iajdx = ∞ and lim j→∞ ‖ f′′j ‖lp[0,a]= ∞ for all 1 ≤ p ≤ ∞. (51) 4 a primer analysis. 4.1 a new kind of ill-posed problem. the concept of a well-posed problem of mathematical physics was introduced by j. hadamard. the solution of any quantitative problem usually ends in a equation z = r(u) where u is the 110 carlos cesar aranda cubo 17, 2 (2015) initial data and z is the solution, r : u → z, u and z are metric spaces with distances ρu and ρz respectively. the problem of determining the solution z in the space z from the initial data u in the space u is said to be well-posed on the pair of metric space (z, u) if the following three conditions are satisfied: (i) for every element u ∈ u there exists a solution z in the space z. (ii) the solution is unique. (iii) for every positive number ǫ > 0 there exists a positive number δ such that ρu(u1, u2) ≤ δ implies ρz(z1, z2) ≤ ǫ, where z1 = s(u1), z2 = s(u2). problems that do not satisfy them are said ill-posed. the sequence {pj} ∞ j=1 is a new kind of illposed problem related to sobolev’s spaces or even for the laplacian operator in the the context of distributions. given ω a bounded domain in rn and u ∈ w1,1(ω) whose laplacian is a bounded measure µ on ω, we call normal derivative in the sense of distributions of u on ∂ω the distribution v1 defined on rn by 〈v1, ϕ〉 = ∫ ω ϕdµ + ∫ ω ∇ϕ · ∇udx, ϕ ∈ d(rn) (52) the distribution v1 defined by 52 is of compact support in ∂ω, if suppϕ ∩ ∂ω = ∅ then ϕiω and ∇(ϕiω) = (∇ϕ)iω. then by the definition of laplacian in the sense of distributions ∫ ω ϕdµ = 〈∆u, ϕiω〉 = − ∫ ∇(ϕiω) · ∇udx = − ∫ ω ∇ϕ · ∇udx. (53) proposition 4.1 (page 500 [11]). let ω be a regular bounded open set with boundary of class w2,∞, µ a bounded radon measure and v1 a radon measure on ∂ω. let us to consider the neumann problem u ∈ w1,1(ω), ∆u = µ in d ′(ω), (54) v1 is the normal derivative on ∂ω of u. (55) (i) there exists a solution of 54,55 if and only if ∫ ∂ω dv1 = ∫ ω dµ. (ii) if ∫ ∂ω dv1 = ∫ ω dµ then the solution of 54, 55 is defined to whitin an additive constant and u ∈ w1,p(ω) for all 1 ≤ p < n n−1 . our sequence {∆pj} ∞ j=1 is non bounded in l ∞(ω) therefore the limit p is outside of the scope of application of proposition 4.1. cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 111 4.1.1 green’s identities. we show the nature of the discontinuity of the the sequence {pj} ∞ j=1 in sobolev’s spaces and in the context of distribution theory. from (page 17 [18] or [6]) ∫ ω (∆u) vdx = ∫ ∂ω ∂u ∂n vdγ − ∫ ω ∇u · ∇vdx for all u ∈ c2(ω), for all v ∈ c1(ω), (56) we calculate lim j→∞ ∫ ω ∇pj · ∇vdx = ∞ for all non negative v ∈ c 1(ω) with v = 0 on ∂ω. (57) taking v ≡ 1 in 56, we derive lim j→∞ ∫ ∂ω ∂pj ∂n dγ = −∞. (58) now we use the second green’s identity ∫ ω (v∆u − u∆v) dx = ∫ ∂ω ( v ∂u ∂n − u ∂v ∂n ) dγ for all u, v ∈ c2(ω), (59) it follows lim j→∞ ∫ ∂ω ( v ∂pj ∂n − pj ∂v ∂n ) dγ = −∞ for all nonnegative v ∈ c2(ω). (60) therefore for non negative v ∈ c2(ω) with v = 0 on ∂ω, we deduce ∫ ω ((v + ǫ)∆pj − pj∆v) dx = ǫ ∫ ∂ω ∂pj ∂n dγ − ∫ ∂ω pj ∂v ∂n dγ, (61) ∫ ω (v∆pj − pj∆v) dx = − ∫ ∂ω pj ∂v ∂n dγ, (62) and letting j →∞ we demonstrate the discontinuity at infinitum of the functional 3. theorem 4.2 (theorem 4.11 page 85 [29]). let ω ∈ rn a bounded domain with smooth boundary; if 1 ≤ p < n put 1 q = 1 p − 1 n−1 p−1 p ; if p = n, put q ≥ 1. there exists a unique mapping z ∈ [w2,p(ω) → w1,q(∂ω)] such that u ∈ c∞(ω) =⇒ zu = u. the second green’s identity 59 is valid on w2,2(ω), it follows from 58 and theorem 4.2 that our sequence {pj} ∞ j=1 is unbounded in w 1,2 n−1 n−2 (∂ω). from 57 we imply lim j→∞ ‖ pj ‖w1,p(ω)= ∞ for all 1 ≤ p ≤ ∞, (63) and lim j→∞ ‖ pj ‖w2,p(ω)= ∞ for all 1 ≤ p ≤ ∞. (64) therefore the sequence is unbounded in the domain of definition of the trace operator stated in theorem 4.2. similar considerations are implied easily from 112 carlos cesar aranda cubo 17, 2 (2015) theorem 4.3 (page 5 [29]). let ω be a bounded domain with lipschitzian boundary. then there exists a uniquely defined, linear an continuous mapping t : wk,2(ω) → l2(∂ω) such that for x ∈ ∂ω and v ∈ c∞(ω), it is defined by t(v)(x) = v(x) theorem 4.4 (page 135 [36]). we consider a solution u = u(x) ∈ c2(ω) of poisson’s differential equation ∆u(x) = f(x), x ∈ ω in the domain ω ⊂ rn, n ≥ 3. for each ball br(a) ⊂⊂ ω then we have the identity u(a) = 1 rn−1ωn ∫ |x−a|=r u(x)dσ − 1 (n − 2)ωn ∫ |x−a|≤r ( | x − a |2−n −r2−n ) f(x)dx (65) the same discontinuity at infinitum appears in the context of singular phenomena in nonlinear elliptic problems [1, 2, 9, 17, 34]. ∫ ω ((v + ǫ)∆uj − uj∆v) dx = ǫ ∫ ∂ω ∂uj ∂n dγ − ∫ ∂ω uj ∂v ∂n dγ, (66) ∫ ω (v∆uj − uj∆v) dx = − ∫ ∂ω uj ∂v ∂n dγ, (67) where uj solves the problem −∆uj = u −γ j in ω, (68) uj = 1 j on ∂ω, (69) with γ > 1. moreover limj→∞ uj = u ∈ c 2(ω) ∩ c0(ω) and ∫ ω u−γdx = ∞, showing the same kind of discontinuity at infinitum. this discontinuity property is an interesting example in the friedrichs method of extension of semibounded operators to self-adjoint operators (page 228 [4], see also page 205 [24]). 4.1.2 integration by parts. in the one dimension if u, v ∈ w1,p(i) with 1 ≤ p ≤ ∞, then ∫y x u′v = u(x)v(x)−u(y)v(x)− ∫x y uv′ for all x, y ∈ i but even if a distribution has second distributional derivative the integration by parts is not true. for example for fh = ∫x 0 1 h i[−h,0](t)dt, we have: ∫y x [f′h]ϕ ′ = [f′h(y)]ϕ(y) − [f ′ h(x)]ϕ(x) − ∫y x [f′′h]ϕ = 0 for all ϕ ∈ c ∞ 0 (x, y)∫y x [f′h]ϕ ′ = [fh(y)]ϕ ′(y) − [fh(x)]ϕ ′(x) − ∫y x [fh]ϕ ′′ = ϕ ′(−h)−ϕ′(0) −h for all ϕ ∈ c∞0 (x, y). (70) taking a radial ϕ ∈ d(a(r, r)) we get ∫ a(r,r) ∇pj · ∇ϕdx = ∫ a(r,r) ∂pj ∂r xi r ϕ ∂r xi r dx = ∫ a(r,r) ∂pj ∂r ∂ϕ ∂r rdx = ∫r r rn−1dr ∫ sn−1 ∂pj ∂r ∂ϕ ∂r rdω = ∫r r ∂pj ∂r ∂ϕ ∂r rndr ∫ sn−1 dω. (71) cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 113 so lim j→∞ ∫r r ∂pj ∂r ∂ϕ ∂r dr = ∞ (72) and the integration by parts rule not hold in the limit because ∫r r ∂pj ∂r ∂ϕ ∂r dr = pj(r) ∂ϕ ∂r (r) − pj(r) ∂ϕ ∂r (r) − ∫r r pj ∂2ϕ ∂r2 dr. (73) also the cantor’s function not satisfies the integration by part rule (this function is monotone and it has zero derivative almost everywhere). 4.1.3 a detour with monotone functions. let ω be a domain in rn, p a line verifying p ∩ ω is a nonempty set. a function defined almost everywhere in ω is said absolutely continuous on the line p if it is continuous on each closed interval of p ∩ ω. theorem 4.5 (page 55 [29]). suppose u ∈ l1loc(ω) and ∂u ∂xi ∈ lp(ω), p ≥ 1. this function changed on a set of measure zero is absolutely continuous on almost all lines parallel to axis xi. let us denote by [ ∂u ∂xi ] the usual derivative and by ∂u ∂xi the distribution derivative. then we have almost everywhere [ ∂u ∂xi ] = ∂u ∂xi . conversely, if u ∈ l1loc(ω) is absolutely continuous on almost all lines parallel to the axis xi with [ ∂u ∂xi ] ∈ lp(ω), then we have ∂u ∂xi = [ ∂u ∂xi ] . by the lebesgue’s differentiation theorem the function p has derivative almost everywhere with respect the radius. varpj = supr≤r0 0 and limh→0+ ‖ [ dfh dt ] ‖lp[−1,1]= ∞ for all 1 < p ≤ ∞. moreover for all nonnegative test function ϕ ∈ c∞0 (−1, 1), we have lim h→0+ ∫1 −1 [f′h(x)] ϕ ′(x)dx = lim h→0+ ∫0 −h ϕ′(x) h dx = lim h→0+ ϕ(−h) − ϕ(0) −h = (−1)(dδ0)ϕ. (81) therefore if we define the distribution λh(ϕ) = ∫1 −1 [f′h(x)] ϕ ′(x)dx then limh→0+ λh = −(dδ0) where dδ0 is the distributional derivative of dirac’s δ distribution and it is well known that distribution has not weak derivative. the space of functions of pointwise bounded variation admits discontinuous functions and therefore both topologies on the same set c∞(r, r) produce completely different objects in metrics and associated functionals. our functions the sequence {pj} ∞ j=1 is bounded in w1,1(a(r, r)) but is it not ensured the strong or weak convergence. the function p has derivative [ ∂p ∂r ] almost everywhere on (r, r) because is a monotone function and moreover we have p = pac + pc + pj, where pac is an absolutely continuous function, pj is continuous and singular, and pj is the jump function of p. theorem 4.6 (page 3 [22]). let i ⊂ r be an interval and let u : i → r be a monotone function. then u has as most countable many discontinuity points. conversely, given a countable set e ⊂ r, there exists a monotone function u : r → r whose set of discontinuity points is exactly e. so by theorem 4.5 the function u has derivative [u′] but no weak derivative if for example e is dense on i. 4.2 the solid mean value. despite the difficulty posed by the discontinuity of green’s identities on the sequence {pj} ∞ j=1 we can obtain several properties. if a function u is absolutely continuous on the interval (a, b) page 225 [22], then u(x) − 1 b − a ∫b a u(t)dt = 1 b − a [∫x a (t − a)u′(t)dt − ∫b x (b − t)u′(t)dt ] . (82) using the lebesgue’s dominated convergence theorem, we obtain a one dimensional solid mean average identity f(x) − 1 b − a ∫b a f(t)dt = lim j→∞ { 1 b − a [∫x a (t − a)f′j(t)dt − ∫b x (b − t)f′j(t)dt ]} . (83) cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 115 now we recall the extension to poisson’s equation of the solid mean value for laplace’s equation in [18] chapter 4: let v ∈ c2(ω) ∩ c0(ω) satisfy −∆v = f then for any ball b = br(y), we have v(y) = 1 | b | ∫ b vdx + 1 nωn ∫ b f(x)θ(r, r)dx, r =| x − y |, (84) where θ(r, r) = 1 n − 2 ( r2−n − r2−n ) − 1 2rn ( r2 − r2 ) , (85) for n > 2 and θ(r, r) = log ( r r ) − 1 2 ( 1 − r2 r2 ) , (86) for n = 2, where ωn is the volume of the unit ball in r n. we deduce that if p(y) = limj→∞pj(y), using the lebesgue’s dominated convergence theorem then p(y) − 1 | b | ∫ b pdx = lim j→∞ 1 nωn ∫ b hj(pj(x))θ(r, r)dx, (87) r =| x − y | . (88) this elementary result involves several strong indeterminations. lemma 4.7 (lemma 3.1.1 page 113 [45]). let u ∈ w1,p[b(x0, r)], p ≥ 1, where x0 ∈ r n and r > 1. let 0 < δ < r. then ∫ b(x0,r) u(y)dy rn − ∫ b(x0,δ) u(y)dy δn = ∫ b(x0,r) [∇u(y)·(y−x0)]dy nrn − ∫ b(x0,δ) [∇u(y)·(y−x0)]dy nδn . (89) the lebesgue’s dominated convergence theorem implies ∫ b(x0,r) p(y)dy rn − ∫ b(x0,δ) p(y)dy δn = limj→∞ {∫ b(x0,r) [∇pj(y)·(y−x0)]dy nrn − ∫ b(x0,δ) [∇pj(y)·(y−x0)]dy nδn } . (90) 4.3 newtonian potentials. the theory of newtonian potentials for distributions with compact support are defined on ω ⊂ r n, n ≥ 1. we shall make use of the following results: proposition 4.8 (proposition 5 page 281 [11]). let ω be a regular bounded open set and let u ∈ c2(ω) ∩ c1n(ω) with ∆u ∈ l 1(ω). then u = u0 + u1 + u2 on ω, (91) 116 carlos cesar aranda cubo 17, 2 (2015) where u0, u1, u2 are the newtonian potentials of the distributions f0, f1, f2 on r n defined by 〈f0, ζ〉 = ∫ ω ζ∆udx, (92) 〈f1, ζ〉 = ∫ ∂ω ζ ( − ∂u ∂n ) dγ, (93) 〈f2, ζ〉 = ∫ ∂ω ∂ζ ∂n udγ. (94) we note that f0, f1, f2 ∈ e ′: f0 is an integrable function on r n with support contained in ω, f1 is a measure on r n with support contained in ∂ω, f2 is a distribution of order 1 on r n with support contained in ∂ω. we say that u1 is the simple layer (respectively double layer) potential defined by the function − ∂u ∂n (respectively u) continuous on ∂ω. we apply this results to our sequence {pj} ∞ j=1. using proposition 4.8, we have pj = p0,j + p1,j + p2,j on ω, (95) where p0,j, p1,j, p2,j are the newtonian potentials of the distributions f0,j, f1,j, f2,j. therefore ∆p0,j = f0,j on r n, (96) ∆p1,j = f1,j on r n, (97) ∆p2,j = f2,j on r n. (98) we obtain lim j→∞ 〈f0,j, ζ〉 = −∞, (99) lim j→∞ 〈f1,j, 1〉 = ∫ ∂ω 1 ( − ∂pj ∂n ) dγ = ∞, simple layer potentials, (100) lim j→∞ 〈f2,j, ζ〉 = ∫ ∂ω ∂ζ ∂n ( lim j→∞ pj ) dγ, double layer potentials. (101) proposition 4.9 (proposition 2 page 278 [11]). let f ∈ e ′ and let u be the newtonian potential of f. then for every multi-index α ∈ n ∂αu ∂xα (x) = 〈f, 1〉 + o ( 1 | x |n+|α|−1 ) when | x |→∞. (102) in particular if n ≥ 3, lim |x|→∞ u(x) = 0, (103) if n ≥ 2, lim |x|→∞ ∇u(x) = 0. (104) the last proposition is useful in the description of the sequence of newtonian potentials {f1,j} ∞ j=1, {f0,j} ∞ j=1, {f1,j} ∞ j=1 and {f2,j} ∞ j=1. cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 117 4.4 the spherical average. for u in 56 with {0} ∈ ω ⊂ rn, n ≥ 2, let u be the spherical average of u, i.e., u(r) = 1 ωnr n−1 ∫ |x|=r u(x)dγx. (105) with the change of variable x → y, we have u(r) = 1 ωn ∫ |y|=1 u(ry)dγy, (106) and du dr = 1 ωn ∫ |y|=1 ∇u(ry) · ydγy. (107) hence du dr = 1 ωn ∫ |y|=1 ∂u ∂r (ry)dγy = 1 ωnr n−1 ∫ |x|=r ∂u ∂r (x)dγx, (108) that is du dr = 1 ωnr n−1 ∫ b(0,r) ∆u(x)dx. (109) therefore from theorem 2.2 it follows that lim j→∞ dpj dr = −∞. (110) 5 statement and proof of the main results. our main theorem states the concentration of compactness. it can be regarded as a classical counterpart of helly’s selection theorem in the space of functions of bounded point variations (theorem 2.35 page 59 [22]). theorem 5.1. let ω be a bounded domain in rn, n ≥ 3 and any sequence of functions {uj} ∞ j=1 in c2(ω) ∩ c0(ω) satisfying −∆pj ≥ −∆uj in ω, (111) pj ≥ uj on ∂ω. (112) then there exist a constant c depending only on the sequence {p}∞j=1, such that uj ≤ c for all j = 1, . . . ,∞. proof. it is a simple consequence of theorem 3.3 page 33 in [18]. 118 carlos cesar aranda cubo 17, 2 (2015) theorem 5.2 (strong concentration of compactness for newtonian potentials). let ω be a bounded smooth domain in rn, n > 2. then there exist a sequence {f1,j} ∞ j=1 ∈ e ′ the space of distributions on rn with compact support such that: (i) f1,j = ∆pj on ω in the sense of the distributions. (ii) the sequence of functions {pj} ∞ j=1 ∈ c ∞(ω) is non decreasing and bounded. (iii) limj→∞ ∆pj = −∞ uniformly on ω. (iv) pj is the newtonian potential of f1,j on ω. (v) the simple layer potential a f1,j ∈ e ′ associated to pj satisfies limj→∞〈f1,j, 1〉 = ∫ ∂ω 1 ( − ∂pj ∂n ) dγ = ∞. (vi) limj→∞ ‖ ∆pj ‖ ′ cα(b(x0,r)) = ∞ for all b(x0, r) ⊂⊂ ω. (vii) solid mean value property. p(y) − 1 | b | ∫ b pdx = lim j→∞ (−1) nωn ∫ b ∆pj(x)θ(r, r)dx, (113) r =| x − y | . (114) where θ(r, r) = 1 n − 2 ( r2−n − r2−n ) − 1 2rn ( r2 − r2 ) , (115) for n > 2, where ωn is the volume of the unit ball in r n. (viii) for n > 2 we have ∫ b(x0,r) p(y)dy rn − ∫ b(x0,δ) p(y)dy δn = limj→∞ {∫ b(x0,r) [∇pj(y)·(y−x0)]dy nrn − ∫ b(x0,δ) [∇pj(y)·(y−x0)]dy nδn } . (116) (ix) the spherical average pj satisfy lim j→∞ dpj dr = −∞. (117) proof. this theorem is a collection of results stated in the a primer analysis section. theorem 5.3 (weak concentration of compactness for newtonian potentials). let ω be a bounded smooth domain in rn, n ≥ 1. then there exist a sequence {f1,j} ∞ j=1 ∈ e ′ the space of distributions on rn with compact support such that: (i) f1,j = ∆fj on ω in the sense of the distributions. (ii) the sequence of functions {fj} ∞ j=1 ∈ c ∞(ω) is non decreasing and bounded. (iii) limj→∞ ∆fj = −∞ in measure on ω. (iv) fj is the newtonian potential of f1,j on ω. (v) limj→∞ ‖ f ‖c1,α(b(x0,r))= ∞ for all b(x0, r) ⊂⊂ ω. cubo 17, 2 (2015) spacetime singularity, singular bounds and compactness . . . 119 (vi) one dimensional mean value property. f(x) − 1 b − a ∫b a f(t)dt = lim j→∞ { 1 b − a [∫x a (t − a)f′j(t)dt − ∫b x (b − t)f′j(t)dt ]} . (118) (vii) solid mean value property. f(y) − 1 | b | ∫ b fdx = lim j→∞ (−1) nωn ∫ b ∆fj(x)θ(r, r)dx, (119) r =| x − y | . (120) where θ(r, r) = 1 n − 2 ( r2−n − r2−n ) − 1 2rn ( r2 − r2 ) , (121) for n > 2 and θ(r, r) = log ( r r ) − 1 2 ( 1 − r2 r2 ) , (122) for n = 2, where ωn is the volume of the unit ball in r n. (viii) for n > 2 ∫ b(x0,r) f(y)dy rn − ∫ b(x0,δ) f(y)dy δn = limj→∞ {∫ b(x0,r) [∇fj(y)·(y−x0)]dy nrn − ∫ b(x0,δ) [∇fj(y)·(y−x0)]dy nδn } . (123) proof. this theorem is a collection of results stated in the a primer analysis section. received: july 2014. accepted: april 2015. references [1] c. c. aranda and t. godoy. existence and multiplicity of positive solutions for a singular problem associated to the p-laplacian operator. electronic journal of differential equations, vol. 2004(2004), no. 132, pp. 1-15. [2] c. c. aranda. bounds and compactness for solutions of second-order elliptic equations. electronic journal of differential equations, vol. 2012 (2012), no. 134, pp. 1-5. 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[47] w. zuo, x. b. wu, x. fan, r. green, r. wang and f. bian. black hole mass estimates and rapid growth of supermassive black holes in luminous z ∼ 3.5 quasars. preprint (2014). introduction. preliminaries. the construction of the sequence of functions fj. a primer analysis. a new kind of ill-posed problem. green's identities. integration by parts. a detour with monotone functions. the solid mean value. newtonian potentials. the spherical average. statement and proof of the main results. cubo a mathematical journal vol.14, no¯ 02, (61–80). june 2012 bicomplex numbers and their elementary functions m.e. luna-elizarrarás, m. shapiro departamento de matemáticas, e.s.f.m del i.p.n., méxico. and d.c. struppa1, a. vajiac schmid college of science and technology, chapman university, orange california, 1 email: struppa@chapman.edu abstract in this paper we introduce the algebra of bicomplex numbers as a generalization of the field of complex numbers. we describe how to define elementary functions in such an algebra (polynomials, exponential functions, and trigonometric functions) as well as their inverse functions (roots, logarithms, inverse trigonometric functions). our goal is to show that a function theory on bicomplex numbers is, in some sense, a better generalization of the theory of holomorphic functions of one variable, than the classical theory of holomorphic functions in two complex variables. resumen en este art́ıculo introducimos el álgebra de números bicomplejos como una generalización del campo de números complejos. describimos cómo definir funciones elementales en tales álgebras (polinomios y funciones exponenciales y trigonométricas) aśı como sus funciones inversas (ráıces, logaritmos, funciones trigonométricas inversas). nuestro objetivo es mostrar que una teoŕıa de funciones sobre números bicomplejos es, en cierto sentido, una mejor generalización de la teoŕıa de funciones holomorfas de una variable compleja, que la teoŕıa de funciones holomorfas en dos variables complejas. keywords and phrases: bicomplex numbers, elementary functions 2010 ams mathematics subject classification: 30g35 62 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) 1 introduction consider two c1 functions u,v from r2 = {(x1,x2) : x1 ∈ r,x2 ∈ r} to r. it is well known that if these two functions satisfy the so-called cauchy-riemann system ∂u ∂x1 = ∂v ∂x2 ∂u ∂x2 = − ∂v ∂x1 then the function f(x1 + ix2) := u(x1,x2) + iv(x1,x2) admits complex derivative (i.e. the limit limh→0 f(z+h)−f(z) h exists), and is what we call a holomorphic function. this observation is the key point of the theory of one complex variable, and shows that the entire theory relies on considering pairs of differentiable functions connected by a simple system of linear, constant coefficients, first order, partial differential equations. there are different ways to attempt to generalize this observation to the case of more pairs of real variables. for example, if we consider two such pairs defined on two independent sets of variables (i.e. a map defined on r4 with values in r4), one can consider quaternion valued functions of a quaternionic variable, and a very interesting theory of holomorphicity was developed by fueter [3] (though others like moisil [4] and mosil-teodorescu [5] introduced similar ideas before him). another way to generalize this observation consists in looking at maps ~f = (f1,f2) from c 2 to c2, and to ask that each component f1,f2 be holomorphic in both variables in c 2, without assuming any additional relationship between them. though both generalizations are important, and give rise to large and interesting theories, we believe that there is another even more appropriate generalization, which so far has not received enough attention. to this purpose, we propose to complexify the cauchy-riemann system and to apply it to pairs of holomorphic functions u,v from c2 = {(z1,z2) : z1 ∈ c,z2 ∈ c} to c, so that the pair (u,v) can be interpreted as a map of c2 to itself. it is then natural to ask whether it makes any sense to consider pairs (u,v) for which the following system is satisfied: ∂u ∂z1 = ∂v ∂z2 ∂u ∂z2 = − ∂v ∂z1 . formally, we have replaced r by c, and differentiability in the real sense by holomorphicity. does this have any implications on the pair (u,v)? as it turns out, it is possible to give a very interesting interpretation of this complexified cauchy-riemann system, if we endow the pair (z1,z2) with a special algebraic structure. instead of considering (z1,z2) as a point in c 2 we now consider, in analogy with what we did in the case of r2, a new space whose elements are of the form z = z1 + jz2, where j is a new imaginary unit (i.e. j 2 = −1), which commutes with the original imaginary unit i. this creates a new algebra, the algebra of bicomplex numbers, and as we will cubo 14, 2 (2012) bicomplex numbers and their elementary functions 63 show in the next sections, such an algebra enjoys most of the properties one would expect from a good generalization of the field of complex numbers. there is another equally interesting way of introducing bicomplex numbers. we recall, for example, that complex numbers are important because they allow the factorization of the positively definite real quadratic form x21 + x 2 2 = (x1 + ix2)(x1 − ix2), which defines the geometry of the plane and is the symbol of the 2−dimensional laplace operator. one may therefore ask whether it is possible to factor the complex analog of the form, i.e. the c−valued quadratic form z21 + z 2 2, which is the symbol of the 2−dimensional complex laplace operator. a trivial answer is to express such form as a product of two linear complex valued factors as z21 + z 2 2 = (z1 + iz2)(z1 − iz2). these two factorizations may appear superficially similar, but in fact there is a deep difference between them. the factorization of x21 + x 2 2 is realized through real 2−dimensional factors, while the factorization of z21 +z 2 2 is realized through complex 1−dimensional factors. one may therefore ask whether it is possible to factor the complex quadratic form through two factors which have complex dimension 2. in order to do so, we consider a distributive, and associative (but not necessarily commutative) complex algebra over c2, and we assume we have two elements a,b in this algebra, such that z21 + z 2 2 = (z1 + az2)(z1 + bz2) = z 2 1 + az2z1 + z1bz2 + az2bz2. (1.1) this implies immediately that, for every z1,z2 we must have az2z1 + z1bz2 = 0, (1.2) which, for z1 = 1, gives az2 + bz2 = 0, and therefore a = −b. by substituting in (1.2) we obtain az2z1 − z1az2 = 0, which, for z2 = 1, gives az1 = z1a. this shows that a is not a complex number, but it commutes with every complex number. by inserting these results in (1.1) we obtain z21 + z 2 2 = z 2 1 − a 2z22, i.e. a2 = −1. once again, we have arrived to a new structure, which requires a to be a second imaginary unit, which we will call j in the sequel, that commutes with the initial imaginary unit i. 64 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) the algebra which one obtains is the bicomplex algebra. in this paper we show how to introduce elementary functions, such as polynomials, exponentials, trigonometric functions, in this algebra, as well as their inverses (something that, incidentally, is not possible in the case of quaternions). we will show how these elementary functions enjoy properties that are very similar to those enjoyed by their complex counterparts. in addition, as we will indicate below, this algebraic structure will allow us to show that any pair of holomorphic functions that satisfies the complexified cauchyriemann system admits derivative in the sense of bicomplex numbers. 2 the bicomplex numbers we gave, in the introduction, a couple of justifications for the introduction of the notion of bicomplex numbers. it is however also possible to arrive to bicomplex numbers by means of purely algebraic considerations. for example, if in a complex number a + ib we replace the real numbers a and b by complex numbers z1 = a1 + ia2 and z2 = b1 + ib2, then we get just another complex number: z1 + iz2 = (a1 + ia2) + i (b1 + ib2) = (a1 − b2) + i (a2 + b1) . if we want to obtain a new type of number, then we must use another imaginary unit, say j, with j2 = −1, and set z1 + jz2 = (a1 + ia2) + j (b1 + ib2) , which gives a new object, outside the field of complex numbers. if we want to be able to operate with these new numbers, we need to define the product of the two imaginary units. this was a problem that was solved by hamilton by requiring that they anticommute, and his solution led to the introduction of quaternions. hamilton’s decision was influenced by many considerations, including the desire to obtain a field, which of course the quaternions form (a skew field). but one could explore what happens if we assume that the two new imaginary units commute. in this case we obtain a new, and lesser known object, the algebra of bicomplex numbers. the set bc of bicomplex numbers is therefore defined as follows: bc = {z1 + jz2 ∣ ∣z1,z2 ∈ c}, where i and j are commuting imaginary units, i.e. ij = ji, i2 = j2 = −1, and c is the set of complex numbers with the imaginary unit i. thus bicomplex numbers are “complex numbers with complex coefficients”, which explains the name of bicomplex, and in what follows we will try to emphazise the deep similarities between the properties of complex and bicomplex numbers. we should probably point out that bicomplex numbers were apparently first introduced in 1892 by segre, [12], that the origin of their function theory is due to the italian school of scorza-dragoni ([13], [14], [15]), and that a first theory of differentiability in bc was developed by price in [7]. cubo 14, 2 (2012) bicomplex numbers and their elementary functions 65 subsequently, other authors have developed further the study of these objects, [2], [6], [8], [9] . a key role, in this evolution, has been played by john ryan, who was probably the first to understand the importance of complex clifford algebras (of which bc is the simplest example, and the only commutative one), and to highlight their role in analysis, [10], [11]. a bicomplex number can be written in cartesian form as z = z1 + jz2 or, at least as long as z2 1 + z2 2 6= 0, in trigonometric form as z = z1 + jz2 = √ z2 1 + z2 2   z1 √ z2 1 + z2 2 + j z2 √ z2 1 + z2 2   = √ z2 1 + z2 2 (cosθ + j sinθ) , (2.1) where the complex number θ is a solution of the system cos(θ) = z1 √ z2 1 + z2 2 , sin(θ) = z2 √ z2 1 + z2 2 . (2.2) since for the particular case of real z1 and z2, √ z2 1 + z2 2 is the modulus of a complex number and θ is its argument, the complex number √ z2 1 + z2 2 is called the complex modulus of the bicomplex number z, denoted by |z|c, and θ is called the complex argument of z, denoted by argc(z). these names can be given a deeper justification, but this is beyond the scope of this article. it can be shown by elementary complex analysis that the apparent ambiguities in formula (2.1) can always be resolved by choosing either the principal or the secondary branch of the complex square root √ z2 1 + z2 2 . either way formula (2.1) is well-defined. now the addition and the multiplication of bicomplex numbers are introduced in a natural way: given z1 = z11 + jz12 and z2 = z21 + jz22 in bc, then z1 + z2 := (z11 + z21) + j(z12 + z22). (2.3) and z1 · z2 := (z11 + jz12)(z21 + jz22) = (z11z21 − z12z22) + j(z11z22 + z21z12). (2.4) it is a simple exercise left to the reader to verify the following proposition 1. (bc,+, ·) is a commutative ring, i.e. (1) the addition is associative, commutative, with identity element 0 = 0+j0, and all bicomplex numbers have an additive inverse. this is to say that (bc,+) is an abelian group. (2) the multiplication is associative, commutative, with identity element 1 = 1 + j0. 66 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) (3) the multiplication is distributive with respect to the addition, i.e. for any z,z1,z2 ∈ bc, we have: z(z1 + z2) = zz1 + zz2. (2.5) remark 2. the process described above has allowed us to endow the complex linear space c2 with a structure of a commutative complex algebra. note that quaternions form only a real, not a complex algebra. on the other hand, quaternions are a (skew) field, while we will soon show that not all bicomplex numbers have a multiplicative inverse. 2.1 bicomplex conjugation given a bicomplex number z = z1 + jz2, its (bicomplex) conjugate is defined by z† := z1 − jz2. we immediately notice that z · z† = z21 + z22 ∈ c. (2.6) this last equality is only apparently similar to the corresponding identity for complex numbers; however the quadratic form in (2.6) takes complex values (rather than real non-negative values), and this implicates significant differences with the complex situation. in particular it implies that a bicomplex number z = z1 + jz2 is invertible if and only if z · z† = z21 + z22 6= 0. (2.7) in this case, it is easy to verify that the inverse of z is given by z−1 = z† z2 1 + z2 2 . if both z1 and z2 are non-zero but the sum z 2 1 +z 2 2 = 0, then the corresponding bicomplex number z = z1 + jz2 is a zero divisor. in fact all zero divisors z = z1 + jz2 in bc are characterized by the equations z21 = −z 2 2, i.e. z1 = ±iz2. thus all zero divisors are of the form: z = λ(1 ± ij), for any λ ∈ c \ {0}. the following proposition is a simple exercise. proposition 3. the bicomplex numbers e := 1 + ij 2 and e† := 1 − ij 2 cubo 14, 2 (2012) bicomplex numbers and their elementary functions 67 are zero divisors, which are linearly independent in the c-linear space c2, and satisfy the identities: e + e† = 1, e − e† = ij, e · e† = 0, e2 = e, e†2 = e†. the next property has no analog for complex numbers, and it exemplifies one of the interesting peculiarities of the bicomplex setting. for any bicomplex number z = z1 +jz2 ∈ bc one can write z = αe + βe†, (2.8) where α = z1 −iz2 and β = z1 +iz2 are uniquely defined complex numbers. formula (2.8) is called the idempotent representation of z. this shows that the set {e,e†} is another basis for the complex space bc, and writing z as a pair (z1,z2) in c 2, one has the transition formula     z1 z2     =      1 2 1 2 − 1 2i 1 2i          α β     . (2.9) note that this new basis is orthogonal with respect to the euclidean inner product in c2 which is given for (z1,w1) and (z2,w2) by 〈(z1,w1),(z2,w2)〉c2 := z1z2 + w1w2. since e = ( 1 2 , i 2 ) and e† = ( 1 2 ,− i 2 ) in c2, we have 〈e,e†〉c2 = 0, and 〈e,e〉c2 = 〈e†,e†〉c2 = 1 2 , so that we have an orthogonal but not orthonormal basis for c2. the following result shows the importance of the idempotent representation of bicomplex numbers in all algebraic operations. proposition 4. the addition and multiplication of bicomplex numbers can be realized “term-byterm” in the idempotent representation. specifically, if z1 = αe + βe † and z2 = γe + δe † are two bicomplex numbers, then z1 + z2 = (α + γ) e + (β + δ) e †, z1 · z2 = αγe + βδe†, zn1 = α n e + βn e†. 68 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) moreover, the inverse of an invertible bicomplex number z = αe + βe† is given by z−1 = α−1e + β−1 e†, where α−1 and β−1 are the complex multiplicative inverses of α and β, respectively. 3 bicomplex derivatives 3.1 let f : u ⊂ bc → bc be a bicomplex function. there is a definition for the derivative of a bicomplex function (see e.g. [7]) which looks quite similar to its complex counterpart. definition 5. the derivative of the function f at a point z0 ∈ u is the limit, if it exists, f′(z0) := lim z→z0 f(z) − f(z0) z − z0 , (3.1) for z in the domain of f such that z − z0 is an invertible bicomplex number. it is never emphasized in the literature that this limit is tacitly taken in the usual euclidean topology of c2 (we call this the usual euclidean convergence in bc), which seems not to be the one generated by the natural structure of bc (see for instance (2.7)). we will show below that there is another, equivalent approach to the limit of a bicomplex function which employs the specific algebraic structure of bicomplex numbers. 3.2 let zn = αne + βne † for n ≥ 1, be a sequence of bicomplex numbers. definition 6. the sequence {zn}n≥1 is said to converge component-wise if the sequences of complex numbers {αn} and {βn} are convergent in the complex plane to complex numbers α0 and β0. we then write that zn → z0 := α0e + β0e †, and we say that zn has limit z0. formula (2.9) shows the equivalence between the usual euclidean convergence and the componentwise convergence defined above. moreover, the euclidean convergence is the general definition of convergence in c2, but the component-wise convergence expresses the relation between the topology and the algebraic structure of bc. 3.3 component-wise convergence of sequences allows us to define the notion of component-wise limits of bicomplex functions. cubo 14, 2 (2012) bicomplex numbers and their elementary functions 69 consider a bicomplex function f = g+jh, f : u ⊂ bc → bc, and its idempotent representation f = ue +ve†. if for any sequence {zn}n≥1 component-wise convergent to z0, the complex number sequences {u(zn)} and {v(zn)} are convergent in c to λ and µ respectively, then the function f has a (usual) limit as z → z0 (with respect to the canonical topology in c 2), and we have: lim z→z0 f(z) = λe + µe†. a description of continuity of a bicomplex function which is compatible with the algebraic structure of bc follows immediately. we mentioned in the introduction that bicomplex numbers are the appropriate setting to consider a complexification of the cauchy-riemann equations. that this is the case is demonstrated by the following important result: theorem 7. let u be an open set in bc, whose variable we indicate with z = z1 + jz2 and let f : u → bc be such that f = u+ jv ∈ c1(u). then f admits bicomplex derivative f′ if and only if: (1) u and v are complex holomorphic in z1 and z2 (2) ∂u ∂z1 = ∂v ∂z2 and ∂u ∂z2 = − ∂v ∂z1 on u. 4 bicomplex polynomials let p(z) = n∑ k=0 akz k be a bicomplex polynomial of degree n, with z = z1 + jz2 = αe + βe †, and bicomplex coefficients ak = γke + δke †, for k = 0. . .n. then zk = αke + βke† and we can rewrite p(z) as p(z) = n∑ k=0 ( γkα k ) e + n∑ k=0 ( δkβ k ) e † =: φ(α)e + ψ(β)e†. if we denote the set of distinct roots of φ and ψ by s1 and s2, and if we denote by s the set of distinct roots of the polynomial p, it is easy to see that s = s1e + s2e†, so that the structure of the zero-set of a bicomplex polynomial p(z) of degree n is fully described by the following three cases: (1) if both polynomials φ and ψ are of degree at least one, and if s1 = {α1, . . . ,αk} and s2 = {β1, . . . ,βℓ}, then the set of distinct roots of p is given by s = {zs,t = αse + βte† ∣ ∣s = 1, . . . ,k, t = 1, . . . , ℓ} . 70 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) (2) if φ ≡ 0, then s1 = c and s2 = {β1, . . . ,βℓ}, with ℓ ≤ n. then s = {zt = λe + βte† ∣ ∣λ ∈ c, t = 1, . . . , ℓ} . similarly, if ψ ≡ 0, then s2 = c and s1 = {α1, . . . ,αk}, where k ≤ n. then s = {zs = αse + λe† ∣ ∣λ ∈ c, s = 1, . . . ,k} . (3) if all the coefficients ak with the exception of a0 = γ0e + δ0e † are complex multiples of e (respectively of e†), but a0 has δ0 6= 0 (respectively γ0 6= 0), then p has no roots. we now discuss a few examples, to give a flavor for computations in bc. first, consider the polynomial p(z) = ( 1 2 + j i 2 ) z5 + (−(1 + 4i) + 2j(2 − i))z4 + ((−11 + 6i) − j (12 + 11i))z3 + (( 29 2 + 13i ) + j ( −13 + 47 2 i )) z2 + (( 13 2 − 17i ) + j ( 17 + 13 2 i )) z − ( 11 2 + i ) + j ( 1 − 11 2 i ) . the corresponding complex polynomials are: φ(α) = α5 − (3 + 8i)α4 + 2(−11 + 9i)α3 + 2(19 + 13i)α2 + (13 − 34i)α − (11 + 2i) , ψ(β) = β4 − 6iβ3 − 9β2. their distinct roots are s1 = {i,1 + 2i} and s2 = {0,3i}. then p has the following four roots: s = { 1 2 i − 1 2 j, 2i + j, 1 + 2i 2 + j −2 + i 2 , 1 + 5i 2 + j 1 + i 2 } . as another example, consider the polynomial p(z) = (1 + ji)z2 − (i − j) . the associated complex polynomials are: φ(α) = 2(α2 − i), ψ(β) ≡ 0. the null set of p is s = { ± (√ 2 2 + i √ 2 2 ) e + λe† ∣ ∣λ ∈ c } . slightly adjusting the previous example, i.e. taking ψ(β) ≡ 2, we get the polynomial p(z) = (1 + ji)z2 + (1 − i) + j (1 − i) , cubo 14, 2 (2012) bicomplex numbers and their elementary functions 71 which has no roots. it is also important to note that a bicomplex polynomial may not have a unique factorization into linear polynomials. for example, the polynomial p(z) = z3 − 1 has 9 solutions. indeed, the associated complex polynomials are φ(α) = α3 − 1, φ(β) = β3 − 1. the set of zeros of φ and ψ are, respectively: s1 = { α1 = 1,α2 = − 1 2 + i √ 3 2 ,α3 = − 1 2 − i √ 3 2 } s2 = { β1 = 1,β2 = − 1 2 + i √ 3 2 ,β3 = − 1 2 − i √ 3 2 } then the set of solutions of p is s = { zkl = αke + βℓe † ∣ ∣k,ℓ = 1. . .3 } , and we have at least two distinct factorizations: z3 − 1 = (z − 1) ( z + 1 2 − √ 3 2 i )( z + 1 2 + √ 3 2 i ) and z3 − 1 = (z − 1) ( z + 1 2 − j √ 3 2 )( z + 1 2 + j √ 3 2 ) . it is therefore clear from what we have indicated that bicomplex polynomials do not satisfy the fundamental theorem of algebra in its original form. at the same time, the following is true and summarizes the comments above. theorem 8 (analogue of the fundamental theorem of algebra for bicomplex polynomials). consider a bicomplex polynomial p(z) = n∑ k=0 akz k. if all the coefficients ak with the exception of the free term a0 = γ0e + δ0e † are complex multiple of e (respectively of e†), but a0 has δ0 6= 0 (respectively γ0 6= 0), then p has no roots. in all other cases, p has at least one root. corollary 9. assume that a bicomplex polynomial p of degree n ≥ 1 has at least one root. then: (1) if at least one of the coefficients ak, for k = 1. . .n, is invertible, then p has at most n 2 distinct roots. (2) if all coefficients are complex multiples of e (respectively e†) then p has infinitely many roots. note that zeros of bicomplex polynomials were originally investigated in [6]. 72 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) 5 the exponential function in bicomplex numbers in this section, we are going to introduce the exponential function of a bicomplex variable. our approach is based on the following theorem, whose proof requires the usage of minimal mathematical tools. theorem 10. let z = z1 + jz2 be any bicomplex number. then the sequence zn := ( 1 + z n )n is convergent. proof. the computation below proves that the sequence is component-wise convergent. set as before z = αe + βe†. then ( 1 + z n )n = ( 1 + α n e + β n e † )n = ( e + e† + α n e + β n e † )n = ( ( 1 + α n ) e + ( 1 + β n ) e † )n = ( 1 + α n )n e + ( 1 + β n )n e † . by taking the limit as n → ∞, and relying on the fact that the corresponding sequences of complex numbers ( 1 + α n )n and ( 1 + β n )n are convergent to the complex exponentials eα and eβ, respectively, we get that the limit of the right-hand-side exists and lim n→∞ ( 1 + z n )n = lim n→∞ ( ( 1 + α n )n e + ( 1 + β n )n e † ) = eαe + eβe† = 1 2 (eα + eβ) + j i 2 (eα − eβ) = 1 2 (ez1−iz2 + ez1+iz2) + j i 2 (ez1−iz2 − ez1+iz2) = ez1 ( 1 2 (e−iz2 + eiz2) + j i 2 (e−iz2 − eiz2) ) = ez1 (cos(z2) + j sin(z2)) . (5.1) this concludes our proof. clearly, the theorem justifies the following definition. cubo 14, 2 (2012) bicomplex numbers and their elementary functions 73 definition 11. we set ez := lim n→∞ ( 1 + z n )n = ez1 (cos(z2) + j sin(z2)) . one observes a marvelous similarity with the definitions of the euler number e and with the exponential functions in real and complex numbers. we pass now to the properties of this newly introduced bicomplex exponential function. • first we note that the bicomplex exponential is an extension to bc of the complex exponential function: indeed, for z = z1 + j0 ∈ c, we have that ez = ez1 (cos(0) + j sin(0)) = ez1, which is the usual complex exponential function. • note that ez1 is the complex modulus of the bicomplex number ez, and z2 is the complex argument of the same bicomplex number ez. the reader may find it instructive to compare this fact with what happens in the complex case. • for z = 0 = 0e + 0e†, we have: e0 = 1e + 1e† = 1. • for any bicomplex number z, the exponential ez is invertible. this is because ez = ez1−iz2e + ez1+iz2e† and the exponential terms ez1−iz2 and ez1+iz2 are complex exponential functions, so they are never zero. the inverse multiplicative of ez is e−z = e−(z1−iz2)e + e−(z1+iz2)e† = e−z1 (cos(z2) − j sin(z2)) . thus, the range of the bicomplex exponential function does not contain neither the zero nor any zero divisors. • two curious facts. for e = 1 · e + 0 · e†, and e† = 0 · e + 1 · e†, we have: ee = e · e + 1 · e† = e12 ( cos ( i 2 ) + j sin ( i 2 )) = e 1 2 ( cosh ( 1 2 ) + ji sinh ( 1 2 )) . similarly: ee † = 1 · e + e · e† = e12 ( cos ( i 2 ) − j sin ( i 2 )) = e 1 2 ( cosh ( 1 2 ) − ji sinh ( 1 2 )) . 74 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) • due to the commutativity of the multiplication in bc, we can show that for any z1 = z11 + jz12 and z2 = z21 + jz22 in bc, the following formula holds: ez1ez2 = ez1+z2. (5.2) indeed, we have: ez1ez2 = (ez11 (cos(z12) + j sin(z12))) (e z21 (cos(z22) + j sin(z22))) = ez11ez21 ((cos(z12) cos(z22) − sin(z12) sin(z22)) +j(sin(z12) cos(z22) + sin(z22) cos(z12))) = ez11+z21 (cos(z12 + z22) + j sin(z12 + z22)) = e z1+z2 . this equality means that the exponential function is a homomorphism from the additive group of bicomplex numbers into the multiplicative group of invertible bicomplex numbers. • in the case z = 0 + jz2, we have: ez = ejz2 = cos(z2) + j sin(z2). • the complex formula eiπ+1 = 0 remains valid for bicomplex numbers, but it is complemented with its mirror image ejπ + 1 = 0. • for any z = αe+βe† ∈ bc, and any invertible bicomplex number w = γe+δe†, i.e. γδ 6= 0, the equation ez = w is equivalent to the system eα = γ and eβ = δ. because γδ 6= 0, it follows that there is always a solution. • we leave as an exercise to the reader to verify that the bicomplex derivative of ez is still ez. • recalling that the complex exponential function and the complex trigonometric functions are periodic, we obtain that ez = ez1 (cos(z2) + j sin(z2)) = ez1+2πim (cos(z2 + 2πn) + j sin(z2 + 2πn)) = ez+2π(mi+nj), for m and n integer numbers. thus the bicomplex exponential function is periodic with bicomplex periods 2π(mi + nj). one can prove that these are the only periods. 6 trigonometric functions of a bicomplex variable adding and subtracting the formulas ejz2 = cos(z2) + j sin(z2) and e −jz2 = cos(z2) − j sin(z2), for any z2 ∈ c, we express the complex cosine and sine via the bicomplex exponential: cubo 14, 2 (2012) bicomplex numbers and their elementary functions 75 cosz2 = ejz2 + e−jz2 2 , sinz2 = ejz2 − e−jz2 2j . (6.1) thus we are in a position to introduce the bicomplex sine and cosine functions which are direct extensions of their complex antecedents. definition 12. let z = z1 + jz2 ∈ bc. we define the bicomplex cosine and sine functions of a bicomplex variable as follows: cosz := ejz + e−jz 2 , sinz := ejz − e−jz 2j . (6.2) given z = z1 + jz2 = αe + βe † ∈ bc, the properties of the bicomplex exponential bring us immediately to the idempotent representation of cosz and sinz: cosz = cos(α)e + cos(β)e† , sinz = sin(α)e + sin(β)e† . (6.3) in terms of the components of the cartesian representation, one gets: cosz = cos(z1 − iz2)e + cos(z1 + iz2)e † . since for a complex variable z the following formulas hold: cosh(z) = cos(iz), sinh(z) = −i sin(iz), we obtain that cosz = cosh(z2) cos(z1) − j sinh(z2) sin(z1). we continue with a description of some basic properties of the trigonometric bicomplex functions. • since the complex sine and cosine functions are periodic with principal period 2π, then taking z = αe +βe† and setting zk,ℓ = (α+2kπ)e + (β+2ℓπ)e † for arbitrary integers k,ℓ we have: cos(zk,ℓ) = cos(z), sin(zk,ℓ) = sin(z). thus the real number (2π)e + (2π)e† = 2π remains the principal period of both bicomplex sine and cosine functions. 76 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) • from (6.3), the equation cosz = 0 is equivalent to the equations in complex variables α and β: cos(α) = 0, cos(β) = 0. the solutions are α = π 2 + kπ, and β = π 2 + ℓπ, for k,ℓ ∈ z. note that α and β are never 0, so the bicomplex solutions z to cosz = 0 are always invertible. in the {1,j} basis, we get the general solution to cosz = 0 as z = z1 + jz2 = ((1 + k + ℓ) + j i(k − ℓ)) π 2 . (6.4) • similarly, the equation sinz = 0 is equivalent to sin(α) = 0, sin(β) = 0. the solutions are α = kπ, and β = ℓπ, for k,l ∈ z. note that there are non-invertible solutions for sinz = 0, e.g. for α = 0, i.e. k = 0, and β 6= 0. in the {1,j} basis, we get the general solution for sinz = 0 as z = z1 + jz2 = (k + ℓ + j i(k − ℓ)) π 2 . • the component-wise formulas (6.3) guarantee that the usual trigonometric identities are true, e.g., the sums and differences of angle formulas, the double angle identities, etc. for example: sin2 z + cos2 z = (sin2(α) + cos2(α))e + (sin2(β) + cos2(β))e† = 1. • it turns out that both functions have the derivatives which extend directly their complex antecedents, i.e. (cosz)′ = − sinz, (6.5) (sinz)′ = cosz. (6.6) 7 bicomplex radicals in this and the next section we begin the study of inverse functions in bc. we start by looking at the equation zn = w, where z = z1 + jz2 = αe + βe †, and w = w1 + jw2 = ae + be †. this system is equivalent to the following two complex equations in variables α and β: αn = a, βn = b. if w is invertible, i.e. ab 6= 0, each complex equation has n distinct complex solutions, and the equations are independent of each other. denote these solutions by ak ∈ n √ a and bℓ ∈ n √ b, respectively. therefore the bicomplex equation zn = w has, in general, n2 solutions given by the bicomplex numbers zkℓ = ake + bℓe † = ak + bℓ 2 + j bℓ − ak 2i cubo 14, 2 (2012) bicomplex numbers and their elementary functions 77 for all k,ℓ = 1. . .n. we define the n-th root of w to be the set of all of these solutions, n √ w := {zkℓ}. note that if we start with formula (2.1) for the bicomplex number w = w1 + jw2, i.e. w = |w|c(cosθ + j sinθ) where |w|c = √ w2 1 + w2 2 is the complex modulus of w, and θ is the complex argument of w, then the solutions zkℓ of the equation z n = w have complex modulus n √ |w|c, which is a set of n complex numbers, and arguments θ + 2ℓπ n , for ℓ = 1. . .n. in conclusion, we find again that there are n2 bicomplex n − th roots, and more precisely n √ w = { n √ |w|c(cos θ + 2ℓπ n + j sin θ + 2ℓπ n ) : ℓ ∈ {0,1, . . . ,n − 1}}. if w = ae + be† is a zero divisor then exactly one of the complex numbers a or b is zero, so the bicomplex equation zn = w has exactly n solutions, all of them zero divisors. obviously if w = 0 there is only one solution, z = 0, to the equation zn = 0. 8 the bicomplex logarithmic function in this section we define the notion of the logarithm of a bicomplex number. in the complex case, we look for the solutions of ez = w, where z and w 6= 0 are complex numbers. if ln(|w|) is the real logarithm of the positive number |w|, and arg(w) is the principal argument of w, then the complex logarithm is defined as the set log(w) := {ln |w| + i(arg(w) + 2mπ) : m ∈ z} and its m−th branch is defined by logm(w) := ln |w| + i(arg(w) + 2mπ). (8.1) we will finally denote by log(w) the principal branch of log(w), i.e. the branch for m = 0. similarly, we will denote by arg(w) the principal argument, so that arg(w) := {arg(w) + 2mπ ∣ ∣m ∈ z}. we note, in this respect, that our notation differs a bit from other more frequently used, but we believe our convention will be useful in discussing the bicomplex case. we pass now to our task to define the logarithm of a bicomplex number. take a bicomplex number z = z1 + jz2, and an invertible bicomplex number w = w1 + jw2. we study the solutions to the bicomplex equation ez = w. recall again from (2.1) that w = |w|c(cosθ + j sinθ). 78 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) then ez1 = |w|c, z2 = θ + 2kπ for any k ∈ z. the first equation implies that z1 is in the set log|w|c, i.e. for every m ∈ z the complex number z1 = logm |w|c, is a solution of the first equation. in the idempotent representation, taking γ = w1 − iw2 and δ = w1 + iw2, then w 2 1 + w2 2 = γδ. then the complex modulus of w is |w|c = √ w2 1 + w2 2 = √ γδ, and logm |w|c = logm √ γδ. (8.2) one can show that θ = i logℓ √ γ δ (8.3) for any ℓ ∈ z. the (principal) complex argument argc(w) is given by the formula above for ℓ = 0. finally we obtain z2 = θ + 2kπ = argc(w) + 2nπ, where n = k + ℓ is an integer number. set argc(w) := {argc(w) + 2nπ ∣ ∣n ∈ z} = ilog √ γ δ . these facts motivate the following definition: definition 13. the bicomplex logarithm is defined by: log(w) := log|w|c + j argc(w), which is an infinite set of bicomplex numbers. the (m,n)-th branch of the bicomplex logarithm of w is given by: logm,n(w) := logm |w|c + j(argc(w) + 2nπ) for m and n integer numbers. in the idempotent representation w = γe + δe†, it turns out that log(w) = log(γ)e + log(δ)e†. (8.4) as in the case of the complex logarithm, the formula (8.4) has to be interpreted in the sense that both terms of the right-hand-side represent the infinite sets of complex numbers multiplied by e and e†, respectively. this formula is obtained immediately by studying the equation ez = w in its idempotent form. in more detail, if z = αe + βe†, then ez = w is equivalent to the two complex equations eα = γ, eβ = δ, cubo 14, 2 (2012) bicomplex numbers and their elementary functions 79 which have as solutions the complex logarithms log(γ) and log(δ), respectively. in the cartesian basis, formula (8.4) agrees with definition 13: log(w) = 1 2 log(γδ) + 1 2 ji log γ δ = log √ γδ + ji log √ γ δ = log|w|c + j argc(w). we state below some properties of the bicomplex logarithm. • the bicomplex logarithm is not defined for zero-divisors, as the bicomplex exponential w = ez is always invertible. • if z = z1 + jz2 is an invertible bicomplex number, if m,n ∈ z, then: elogm,n(z) = elogm |z|c+j argc(z)+2nπj = elogm |z|cej argc(z) = |z|c(cos(argc(z)) + j sin(argc(z))) = z • for z = 1 = 1 + j0, we have: logm,n(1) = 0 + 2mπi + 2nπj for all m,n ∈ z. • for z1 and z2 two invertible bicomplex numbers, the following formula holds log(z1z2) = log(z1) + log(z2). (8.5) the inverses of the bicomplex trigonometric functions are defined in complete analogy with the complex case, as we have already properly defined the notions of bicomplex exponential, logarithm, and square root. for example, the inverse of the bicomplex cosine function is obtained by solving the equation cos(z) = ejz + e−jz 2 = w. this is a quadratic equation in ejz with roots ejz = w ± √ w2 − 1. therefore, for m,n ∈ z, arccos(w) := −j logm,n(w ± √ w2 − 1) = ±j logm,n(w + √ w2 − 1). received: may 2011. revised: october 2011. 80 m.e. luna-elizarrarás, m. shapiro, d.c. struppa and a. vajiac cubo 14, 2 (2012) references [1] ahlfors, l., complex analysis, mcgraw hill, new york, 1966. [2] charak, k.s., rochon, d., sharma, n., normal families of bicomplex holomorphic functions, arxiv:0806.4403v1 (2008). [3] fueter, r., analytische funktionen einer quaternionen variablen, comm. math. helv. 4 (1932), 9-20. 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[11] ryan, j., c2 extensions of analytic functions defined in the complex plane, adv. in applied clifford algebras 11 (2001), 137–145. [12] segre, c., le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, math. ann. 40 (1892), 413–467. [13] scorza dragoni, g., sulle funzioni olomorfe di una variabile bicomplessa, reale accad. d’italia, mem. classe sci. nat. fis. mat. 5 (1934), 597–665. [14] spampinato, n., estensione nel campo bicomplesso di due teoremi, del levi-civita e del severi, per le funzioni olomorfe di due variabili bicomplesse i, ii, reale accad. naz. lincei 22 (1935), 38–43, 96–102. [15] spampinato, n., sulla rappresentazione di funzioni di variabile bicomplessa totalmente derivabili, ann. mat. pura appli. 14 (1936), 305–325. introduction the bicomplex numbers bicomplex conjugation bicomplex derivatives bicomplex polynomials the exponential function in bicomplex numbers trigonometric functions of a bicomplex variable bicomplex radicals the bicomplex logarithmic function cubo a mathematical journal vol.15, no¯ 02, (89–103). june 2013 numerical solution of singular and non singular integral equations m.h.saleh and d.sh.mohammed zagazig university, mathematics department, faculty of science, zagazig, egypt. doaamohammedshokry@yahoo.com abstract this paper is devoted to study the approximate solution of singular and non singular integral equations by means of chebyshev polynomial and shifted chebyshev polynomial. some examples are presented to illustrate the mothed. resumen este art́ıculo se dedica al estudio de la solución aproximada de ecuaciones integrales singulares y no singulares por medio de polinomios de chebyshev con o sin corrimiento. se presentan algunos ejemplos para ilustrarlo. keywords and phrases: linear hypersingular integral equations, nonlinear integral equations, chebyshev polynomial, shifted chebyshev polynomial, approximate solution. 2010 ams mathematics subject classification: 45e05; 65r20; 78w23. 90 m.h.saleh and d.sh.mohammed cubo 15, 2 (2013) introduction singular integral equations are usually difficult to solve analytically so it required to obtain the approximate solution [7,8]. chebyshev polynomials are of great importance in many areas of mathematics particalarly approximation theory see ([1-3], [9-13] ). in this paper we analyze the numerical solution of singular and non singular integral equations by using chebyshev polynomial and shifted chebyshev polynomial. this paper consists of two parts i and ii. in part i, we study the approximate solution of hypersingular integral equations by means of chebyshev polynomial. in part ii, we study the approximate solution of nonlinear integral equations by means of shifted chebyshev polynomial. part i: approximate solution of hypersingular integral equations 1. formulation of the problem consider the following hypersingular integral equation: = ∫1 −1 k(x, t) (t − x)2 ϕ(t) dt + ∫1 −1 l(x, t) ϕ(t) dt = f(x) , − 1 ≤ x ≤ 1 (1.1) where k(x, t) , l(x, t) and f(x) are given real-valued continuous functions defined on the set [−1, 1] × [−1, 1], [−1, 1] × [−1, 1] and [−1, 1] respectively and ϕ(t) is unknown function satisfy the following condition ϕ(±1) = 0. the simplest hypersingular integral equation of the form (1.1) given by the following form: = ∫1 −1 ϕ(t) (t − x)2 dt = f(x) , (1.2) where the finite-part integral in (1.2) can be defined as the derivative of a cauchy principle value integral as = ∫1 −1 ϕ(t) (t − x)2 dt = d dx ∫1 −1 ϕ(t) t − x dt. (1.3) cubo 15, 2 (2013) numerical solution of singular and non singular integral equations 91 thus equation (1.2) can be written in the following form d dx ∫1 −1 ϕ(t) t − x dt = f(x) . (1.4) integrating both sides of equation (1.4) with respect to x , we obtain ∫1 −1 ϕ(t) t − x dt = f(x) , (1.5) where f(x) = ∫ f(x) dx . (1.6) the exact solution of equation (1.5) given by the following form: ϕ(x) = − √ 1 − x2 π2 ∫1 −1 f(t) √ 1 − t2(t − x) dt . (1.7) the solution exists if and only if the function f(x) satisfies the following condition : ∫1 −1 f(t) √ 1 − t2 dt = 0 . (1.8) 2. the approximate solution in this section we shall study the approximate solution of the hypersingular integral equation (1.1). the unknown function ϕ(x) satisfying the condition ϕ(±1) = 0, can be represented by the following form: ϕ(x) = √ 1 − x2 φ(x), − 1 ≤ x ≤ 1 (2.1) where φ(x) is a well-behaved function of x in the interval x ∈ [−1, 1] . let the unknown function φ(x) be approximated by means of a polynomial of degree n as the following : φ(x) ≈ n ∑ j=0 cj x j , (2.2) where cj (j = 0, 1, 2, ..., n) are unknown constants. 92 m.h.saleh and d.sh.mohammed cubo 15, 2 (2013) substituting from form (2.1) and (2.2) into equation (1.1) we obtain: n ∑ j=0 cj [= ∫1 −1 √ 1 − t2 k(x, t) tj (t − x)2 dt + ∫1 −1 √ 1 − t2 l(x, t) tj dt] = f(x) , − 1 ≤ x ≤ 1. (2.3) by using the following ” chebyshev approximations” to the kernels k(x, t) and l(x, t) k(x, t) ≈ ∑m p=0 kp(x) t p , l(x, t) ≈ ∑s q=0 lq(x) t q ,            (2.4) with known functions kp(x) and lq(x) , then (2.3) takes n ∑ j=0 cj [ m ∑ p=0 kp(x) up+j (x) + s ∑ q=0 lq(x) γq+j ] = f(x) , − 1 ≤ x ≤ 1 (2.5) where up+j (x) = = ∫1 −1 √ 1 − t2 tp+j (t − x)2 dt (2.6) and γq+j = ∫1 −1 √ 1 − t2 tq+j dt . (2.7) using the zeros xk of the chebyshev polynomial tn+1 (x) into equation (2.5) we obtain the following system of linear equations with (n + 1) of the unknown constants cj (j = 0, 1, 2, ..., n) n ∑ j=0 cj αj(xk) = f(xk) , (2.8) where xk = cos ( (2k − 1) 2(n + 1) π) , k = 1, 2, 3, ..., n + 1 (2.9) and αj(xk) = m ∑ p=0 kp(xk) up+j (xk) + s ∑ q=0 lq(xk) γq+j . (2.10) cubo 15, 2 (2013) numerical solution of singular and non singular integral equations 93 by solving the system of linear equations (2.8) we obtain the unknown constants cj and substituting into (2.1) and (2.2) we obtain the approximate solution of equation (1.1). 3. numerical examples in this section we shall give two examples to illustrate the above results . example 3.1 consider the following hypersingular integral equation = ∫1 −1 ϕ(t) (t − x)2 dt + ∫1 −1 x t ϕ(t) dt = x + x3 , − 1 ≤ x ≤ 1 . (3.1) equation (3.1) can be written in the following form: = ∫1 −1 ϕ(t) (t − x)2 dt = (1 + µ) x + x3 , (3.2) where µ = − ∫1 −1 t ϕ(t) dt . according to (1.3), (1.5), (1.6) and (1.7) it easy to show that the exact solution of equation (3.2) given by: ϕ(x) = − 1 40 π √ 1 − x2 [10 x3 + 27 x ] . (3.3) now, we study the approximate solution of equation (3.1). since k(x, t) = 1 and l(x, t) = x t , then we have k0(x) = 1 , k1(x) = k2(x) = ... = km(x) = 0 l1(x) = x , l0(x) = l2(x) = ... = ls(x) = 0 .            (3.4) 94 m.h.saleh and d.sh.mohammed cubo 15, 2 (2013) substituting from (3.4) into (2.10) we obtain: αj(xk) = uj (xk) + xk γj+1 , (j = 0, 1, 2, ..., n) (3.5) by using the following formula = ∫1 −1 √ 1 − t2 uj(t) (t − x)2 dt = −π (j + 1) uj(x) , (3.7) where uj(x) is chebyshev polynomial of the second kind. from (2.6) and (3.7) it easy to show that u0(x) = −π , u1(x) = −2π x, u2(x) = −π (3x 2 − 1 2 ), u3(x) = −π (4x 3 − x) , u4(x) = −π (5x 4 − 3 2 x2 − 1 8 ) , u5(x) = −π (6x 5 − 2x3 − 1 4 x) , ... (3.8) from (2.7) we have γ0 = π 2 , γ2 = π 8 , γ4 = π 16 , γ6 = 5π 128 , γ8 = 7π 256 , γ1 = 0 , γ3 = 0 , γ5 = 0 , γ7 = 0 , γ9 = 0 . (3.9) substituting from (3.8) and (3.9) into (2.10) and take n = 5 , we obtain the following system of linear equations: −π [c0 + 15 8 c1 xk + c2 (3x 2 k − 1 2 ) + c3 (4x 3 k − 17 16 xk) + c4 (5x 4 k − 3 2 x2k − 1 8 ) + c5 (6x 5 k − 2x 3 k − 37 128 xk)] = xk + x 3 k , (k = 1, 2, ..., 6). (3.10) solving system (3.10) by using the zeros xk of chebyshev polynomial tn+1(x) , we obtain the values of the constants as follows: { c0 = c2 = c4 = c5 = 0 , c1 = −27 40π and c3 = −1 4π } . (3.11) cubo 15, 2 (2013) numerical solution of singular and non singular integral equations 95 substituting from (3.11) into (2.1) and (2.2) we obtain the approximate solution of equation (3.1) which is the same as the exact solution which given by (3.3). example 3.2 consider the following hypersingular integral equation = ∫1 −1 ϕ(t) (t − x)2 dt + ∫1 −1 (2x t + 4x3 t3) ϕ(t) dt = 4x3 − 2x + 1 . − 1 ≤ x ≤ 1 (3.12) equation (3.12) can be written in the following form: = ∫1 −1 ϕ(t) (t − x)2 dt = 2(µ1 − 1) x + 4(µ2 + 1)x 3 + 1 , (3.13) µ1 = − ∫1 −1 t ϕ(t) dt and µ2 = − ∫1 −1 t3 ϕ(t) dt . (3.14) it is easy to show that the exact solution of equation (3.13) given by: ϕ(x) = − 1 π √ 1 − x2 [ 832 825 x3 − 136 275 x + 1 ] . (3.15) similarly as in example 3.1 we study the approximate solution of equation (3.12). since k(x, t) = 1 and l(x, t) = 2x t + 4x3 t3 , then we have k0(x) = 1 , k1(x) = k2(x) = ... = km(x) = 0 l1(x) = 2x , l3(x) = 4x 3 , l0(x) = l2(x) = ... = ls(x) = 0 .            (3.16) substituting from (3.16) into (2.10) we obtain: αj(xk) = uj (xk) + 2xk γj+1 + 4x 3 k γj+3 . (j = 0, 1, 2, ..., n) (3.17) 96 m.h.saleh and d.sh.mohammed cubo 15, 2 (2013) substituting from (3.17) into (2.8) we obtain the following system of linear equations: n ∑ j=0 cj (uj (xk) + 2xk γj+1 + 4x 3 k γj+3 ) = 4x 2 k − 2xk + 1. (j = 0, 1, 2, ..., n) (3.18) substituting from (3.8) and (3.9) into (3.18) and take n = 5 , we obtain the following system of linear equations: −π [c0 + c1 ( 7 4 xk − 1 4 x3k) + c2 (3x 2 k − 1 2 ) + c3 ( 123 32 x3k − 9 8 xk) + c4 (5x 4 k − 3 2 x2k − 1 8 ) + c5 (6x 5 k − 135 64 x3k − 21 64 xk)] = 4x 3 k − 2xk + 1 , (k = 1, 2, ..., 6). (3.19) solving system (3.19) by using the zeros xk of chebyshev polynomial tn+1(x) , we obtain the values of the constants as follows: { c2 = c4 = c5 = 0 , c0 = − 1 π , c1 = −136 275π and c3 = −832 825π } . (3.20) substituting from (3.20) into (2.1) and (2.2) we obtain the approximate solution of equation (3.12) which is the same as the exact solution which is given by (3.15). part ii: approximate solution of nonlinear integral equations in this part we transform the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown chebyshev coefficients. 1. formulation of the problem consider the following nonlinear integral equation: φ(x) = f(x) + λ ∫1 0 k(x, t) [φ(t)]2 dt , (1.1) cubo 15, 2 (2013) numerical solution of singular and non singular integral equations 97 where f(x) , k(x, t) are given functions, λ is a real parameter and φ(x) is unknown function. the unknown function φ(x) can be represented by truncated chebyshev series as follows: φ(x) = n ∑ j=0 ′ a∗j t ∗ j (x) , 0 ≤ x ≤ 1 (1.2) where t∗j (x) denoted the shifted chebyshev polynomial of the first kind, a ∗ j are the unknown chebyshev coefficients, ∑ ′ is a sum whose first term is halved and n is any positive integer. suppose that the solution φ(x) of equation (1.1) and k(x, t) can be expressed as a truncated chebyshev series. then (1.2) can be written in the following form φ(x) = t∗(x) a∗ , (1.3) where t∗(x) = [ t∗0 (x) t ∗ 1 (x) ... t ∗ n(x)] , a ∗ = [ a∗0 2 a∗1 ... a ∗ n] t , and the function [φ(t)]2 can be written in the following matrix form [1] [φ(t)]2 = t∗(t) b ∗ , (1.4) where t∗(t) = [t∗0 (x) t ∗ 1 (x) ... t ∗ 2n(x)] , b ∗ = [ b∗0 2 b∗1 ... b ∗ 2n] t , and the elements b∗i consists of a ∗ i and a ∗ −i = a ∗ i as follows: b∗i =                    (a ∗ i/2 ) 2 2 + ∑n−i/2 r=1 (a ∗ i 2 −r ) (a∗i 2 +r ) for even i ∑n− i−1 2 r=1 (a ∗ i+1 2 −r ) (a∗i−1 2 +r ) for odd i. 98 m.h.saleh and d.sh.mohammed cubo 15, 2 (2013) now, k(x, t) can be expanded by chebyshev series as follows: k(xi, t) = n ∑ r=0 ′′ kr(xi) t ∗ r (t) , where ∑ ′′ denotes a sum with first and last terms halved, xi are the chebyshev collocation points defined by xi = 1 2 [ 1 + cos ( iπ n )] , i = 0, 1, ..., n (1.5) and chebyshev coefficients kr(xi) are determined by the following relation: kr(xi) = 2 n n ∑ j=0 ′′ k(xi, tj) t ∗ r (tj) , ti = 1 2 [ 1 + cos ( jπ n )] which is given by [5 ]. then the matrix representation of k(xi, t) given by k(xi, t) = k(xi) t ∗(t) t . (1.6) where k(xi) = [ k0(xi) 2 k1(xi) ... kn−1(xi) kn(xi) 2 ] . 2. solution of nonlinear integral equation our aim in this section to find the chebyshev coefficients of (1.2), that is the matrix a∗. by substituting from chebyshev collocation points defined by (1.5) into equation (1.1) we obtain a matrix equation of the form φ = f + λ i , (2.1) where i(x) denotes the integral part of equation (1.1) and cubo 15, 2 (2013) numerical solution of singular and non singular integral equations 99 φ =                    φ(x0) φ(x1) . . . φ(xn)                    , f =                    f(x0) f(x1) . . . f(xn)                    , i =                    i(x0) i(x1) . . . i(xn)                    , t∗ =                    t∗(x0) t∗(x1) . . . t∗(xn)                    . when we substitute from chebyshev collocation points (1.5) into (1.3), the matrix φ becomes φ = t∗ a∗ . (2.2) substituting from (1.4) and (1.6) in i(xi) for i = 0, 1, ..., n , i = 0, 1, ..., 2n and using the following relation [6 ], z = ∫1 0 t∗(t) t t∗(t) dt = [ ∫1 0 t∗i (t) t ∗ j (t) dt ] = 1 2 [zij] , where zij =                  1 1−(i+j)2 + 1 1−(i−j)2 for even i+j 0 for odd i+j , we obtain i(xi) = k(xi) z b ∗ . (2.3) therefore, we obtain the matrix i in terms of chebyshev coefficients matrix in the following form: i = k z b∗ , (2.4) 100 m.h.saleh and d.sh.mohammed cubo 15, 2 (2013) where k = [ k(x0) k(x1) ... k(xn)] t . now, by using the relation (2.2) and (2.4), the integral equation (1.1) transform to a matrix equation which is given by: t∗ a∗ − λ k z b∗ = f . (2.5) the matrix equation (2.5) corresponds to a system of (n + 1) nonlinear algebraic equations with (n + 1) unknown chebyshev coefficients. thus the unknown coefficients a∗j can be computed from this equation and substituting from these coefficients into (1.2) we obtain the approximate solution. 3. numerical examples in this section we shall give two examples to illustrate the above results . example 3.1 consider the following nonlinear integral equation φ(x) = x − 1 3 + ∫1 0 [φ(t)]2 dt . (3.1) from (3.1) we have f(x) = x − 1 3 , k(x, t) = 1 and λ = 1 . for n = 2 , the chebyshev collocation points on [0, 1] can be found from (1.5) as x0 = 1 , x1 = 1 2 , x2 = 0 and the matrix equation corresponds to the integral equation (3.1) given by t∗ a∗ − k z b∗ = f , (3.2) where cubo 15, 2 (2013) numerical solution of singular and non singular integral equations 101 t∗ =            1 1 1 1 0 −1 1 −1 1            , f =            2 3 1 6 −1 3            , z =            1 0 −1 3 0 − 1 15 0 1 3 0 −1 5 0 −1 3 0 7 15 0 − 19 105            k =            1 0 0 1 0 0 1 0 0            , a∗ =            a∗0/2 a∗1 a∗2            , b∗ =                       1 2 ( a ∗ 0 2 2 + a∗1 2 + a∗2 2 ) a∗0 a ∗ 1 + a ∗ 1 a ∗ 2 a ∗ 1 2 2 + a∗0 a ∗ 2 a∗1 a ∗ 2 a ∗ 2 2 2                       . substituting from these matrices into (3.2), we obtain a system of nonlinear algebraic equations, the solution of this given by: (a∗0 = 1 , a ∗ 1 = 1 2 , a∗2 = 0) . substituting from these values into (1.2) when n = 2 we obtain the approximate solution φ(x) = x , which is the exact solution. example 3.2 consider the following nonlinear integral equation φ(x) = x2 − x 6 − 1 + ∫1 0 x t [φ(t)]2 dt . 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[13] tweed j., john r. st. and dunn m. h.; algorithms for the numerical solution of a finitepart integral equation. applied mathematics letters; 12 (1999) 3-6. cubo a mathematical journal vol.15, no¯ 01, (01–12). march 2013 weak solutions of fractional order pettis integral inclusions with multiple time delay in banach spaces mouffak benchohra université de sidi bel-abbès laboratoire de mathématiques, bp 89, 22000 sidi bel-abbès, algérie benchohra@univ-sba.dz fatima-zohra mostefai université de saida département de mathématiques, bp 138 cité ennasr, 20000, saida, algérie f.z.mostefai@gmail.com abstract we study the existence of weak solutions for nonlinear integral inclusion with multiple time delay. the main result of the paper is based on the fixed point theorem of mönch type and the technique of measure of weak noncompactness. resumen estudiamos la existencia de soluciones débiles de la inclusión integral no lineal con retardos temporales múltiples. el resultado principal del art́ıculo se basa en el teorema de punto fijo de tipo mönch y la técnica de medida de la no-compacidad débil keywords and phrases: hyperbolic differential inclusion, measure of weak noncompactness, left sided mixed pettis integral, weak solution, banach space. 2010 ams mathematics subject classification: 26a33, 35h10, 35d30 2 mouffak benchohra and fatima-zohra mostefai cubo 15, 1 (2013) 1 introduction fractional differential equations have been of great interest recently. it is due to the development of the theory of fractional calculus itself and by application of such constructions in various fields of science and engineering such as control theory, physics, mechanics, electrochemistry, porous media, etc. there are many works discussing the solvability of nonlinear fractional differential equations and inclusions, see the monographs of abbas et al. [2], kilbas et al. [14], lakshmikantham et al. [15], podlubny [18], tarasov [20], the papers of agarwal et al. [3, 4, 5], benchohra et al. [7, 8], kilbas and marzan [13], salem [19], vityuk and golushkov [21], and the references therein. in [12], r. w. ibrahim and h. a. jalab studied the existence of solutions of the following fractional integral inclusion u(t) − m∑ i=1 bi(t)u(t − τi) ∈ i αf(t, u(t)); if t ∈ [0, t], (1) where τi < t ∈ [0, t], bi : [0, t] → r, i = 1 . . . , n are continuous functions, and f : [0, t]×r → p(r) is a given multivalued map. in [1], abbas and benchohra considered the following fractional integral equation with delay u(x, y) = m∑ i=1 gi(x, y)u(x − ξi, y − µi) + i r θf(x, y, u(x, y)); if (x, y) ∈ j := [0, a] × [0, b], (2) u(x, y) = φ(x, y); if (x, y) ∈ j̃ := [−ξ, a] × [−µ, b]\(0, a] × (0, b], (3) where a, b > 0, θ = (0, 0), ξi, µi ≥ 0; i = 1 . . . , m, ξ = maxi=1...,m{ξi}, µ = maxi=1...,m{µi}, i r θ is the left-sided mixed riemann-liouville integral of order r = (r1, r2) ∈ (0, ∞) × (0, ∞), f : j × rn → rn, gi : j → r; i = 1 . . . m are given continuous functions, and φ : j̃ → rn is a given continuous function such that φ(x, 0) = m∑ i=1 gi(x, 0)φ(x − ξi, −µi); x ∈ [0, a], and φ(0, y) = m∑ i=1 gi(0, y)φ(−ξi, y − µi); y ∈ [0, b]. motivated by the above papers, in this paper, we consider the following fractional integral inclusion with multiple time delay: u(x, y) − m∑ i=1 gi(x, y)u(x − ξi, y − µi) ∈ i α θ f(x, y, u(x, y)); (x, y) ∈ ja × jb. (4) u(x, y) = ψ(x, y); (x, y) ∈ j̃ = [−ξ, a] × [−µ, b]\(0, a] × (0, b], (5) cubo 15, 1 (2013) weak solutions of fractional order pettis integral inclusions ... 3 where ja = [0, a], jb = [0, b] for a, b > 0, θ = (0, 0), ξ = maxi=1...m{ξi}, µ = maxi=1...m{µi}, i α θ is the left sided mixed pettis integral of order α, α = (α1, α2) ∈ (0, ∞) × (0, ∞), f : ja × jb × e → p(e) is a multivalued map (p(e) is the family of all nonempty subsets of e), gi : ja × jb → r; i = 1, . . . m are given continuous functions, and ψ : j̃ → e is a given continuous function such that ψ(0, y) = m∑ i=1 gi(0, y)ψ(−ξi, y − µi); y ∈ [0, b], and ψ(x, 0) = m∑ i=1 gi(x, 0)ψ(x − ξi, −µi); x ∈ [0, a]. e is a banach space with norm ‖.‖. our result is based on fixed point theorem of mönch type and the technique of measure of weak noncompactness. let us mention that other tools like the nonlinear alternative of leray-schauder type, the banach fixed point theorem and schauder’s fixed point theorem, such have been used to analyze the above problem in the scalar case [1, 2]. the present results complement and extend those considered in the scalar case. 2 preliminaries in this section, we introduce the notation, definitions, and preliminary facts that will be used in the remainder of this survey paper. let r denote the real line and let ja = [0, a] and jb = [0, b] be two closed and bounded intervals in r for some real numbers a > 0 and b > 0. throughout the paper, e is a banach space with norm ‖.‖ and dual e∗. also (e, w) = (e, σ(e, e∗)) denotes the space e with its weak topology. we take c(ja × jb, e) to be the banach space of continuous functions u : ja × jb → e, with the usual supremum norm ‖u‖ ∞ = sup{‖u(x, y)‖, (x, y) ∈ ja × jb}. definition 2.1. [17] the function x : ja × jb → e is said to be pettis integrable on ja × jb if and only if there is an element xi×j ∈ e corresponding to each i×j ⊂ ja ×jb (i and j are measurable), such that ϕ(xi×j) = ∫ i ∫ j ϕ(x(s, t))dsdt for all ϕ ∈ e∗ where the integral on the right is assumed to exist in the sense of lebesgue (by definition, xi×j = ∫ i ∫ j x(s, t)dsdt). we let l1(ja × jb, e) denote the banach space of measurable functions u : ja × jb → e that are pettis integrable, equipped with the norm ‖u‖l1 = ∫a 0 ∫b 0 ‖u(x, y)‖dxdy. let p(e) is the family of all nonempty subsets of e. a multivalued map g : e → p(e) has convex (closed) valued if g(x) is convex (closed) for all x ∈ e. we say that g is bounded on bounded sets if g(b) is bounded in e for each bounded set 4 mouffak benchohra and fatima-zohra mostefai cubo 15, 1 (2013) b of e (i.e. supx∈b{sup{‖y‖ : y ∈ g(x)}} < ∞). the map g is upper semicontinuous (u.s.c) on e if for each x0 ∈ e, the set g(x0) is a nonempty closed subset of e and for each open set n of e containing g(x0) there exists an open neighborhood m of x0 such that g(m) ⊆ n. the mapping g has a fixed point if there exists x ∈ e such that x ∈ g(x). in what follows pcl(e) = {y ∈ p(e) : y is closed}, pb(e) = {y ∈ p(e) : y is bounded}, pcp(e) = {y ∈ p(e) : y is compact}, and pcp,cv(e) = {y ∈ p(e) : y is compact and convex}. a multivalued map g : ja × jb → pcl(e) is said to be measurable if for each ω ∈ e the function (x, y) → d(ω, g(x, y)) = inf{|ω − υ| : υ ∈ g(x, y)} is measurable. for more details on multivalued maps see the books of aubin and cellina [6], deimling [10]. definition 2.2. a function h : e → e is said to be weakly sequentially continuous if h takes each weakly convergent sequence in e to weakly convergent sequence in e (ie for any (xn)n in e with xn → x in (e, w), h(xn) → h((x)) in (e, w)). definition 2.3. a function f : q → pcl,cv(q) has weakly sequentially closed graph if for any sequence (xn, yn) ∈ q × q, where yn ∈ f(xn) for n ∈ {1, 2, ...},and where both xn → x in (e, ω) and yn → y in (e, ω) then y ∈ f(x). proposition 2.4. [17, 11] if x(.) is pettis integrable and h(.) is a measurable and essentially bounded real-valued function, then x(.)h(.) is pettis integrable. definition 2.5. [9] let e be a banach space, ωe the bounded subsets of e and b1 the unit ball of e. the de blasi measure of weak noncompactness is the map β : ωe → [0, ∞) defined by β(x) = inf{ǫ > 0 : there exists a weakly compact subset ω of e : x ⊂ ǫb1 + ω} properties: de blasi measure of noncompactness satisfies some properties (a) a ⊂ b ⇒ β(a) ≤ β(b), (b) β(a) = 0 ⇔ a is relatively compact, (c) β(a ∪ b) = max{β(a), β(b)}, (d) β(a ω ) = β(a), (a ω denotes the weak closure of a), (e) β(a + b) ≤ β(a) + β(b), (f) β(λa) = |λ|β(a), (g) β(conv(a)) = β(a), (h) β(∪|λ|≤hλa = hβ(a). the following result follows directly from the hahn-banach theorem. proposition 2.6. let e be a normed space with x0 6= 0 then there exists ϕ ∈ e ∗ with ‖ϕ‖ = 1 and ϕ(x0) = ‖x0‖. cubo 15, 1 (2013) weak solutions of fractional order pettis integral inclusions ... 5 for a given set v of functions v : ja × jb → e let us denote by v(x, y) = {v(x, y), v ∈ v}, (x, y) ∈ ja × jb and v(ja × jb) = {v(x, y) : v ∈ v, (x, y) ∈ ja × jb}. for completeness, we recall the definition of the fractional pettis-integral of order α > 0. let α1, α2 > 0 and α = (α1, α2). for h ∈ l 1(ja × jb, e), the expression (iα0 h)(x, y) = 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1h(s, t)dsdt where the sign " ∫ " denotes the pettis integral and γ(.) is the euler gamma function, is called the left sided mixed pettis integral of order α. for our purpose we will need the following fixed point theorem. theorem 2.7. [16] let e be a banach space with q a nonempty, bounded, closed, convex and equicontinuous subset of metrizable locally convex vector space c(j, e) such that 0 ∈ q. suppose that t : q → pcl,cv(q) has weakly-sequentially closed graph. if the implication v = conv({0} ∪ t(v)) ⇒ v is relatively weakly compact, (6) holds for every subset v ⊂ q, then the operator t has a fixed point. 3 main results we first define what we mean by solution of the problem (4)-(5). definition 3.1. a function u ∈ c(j, e) is said to be solution of problem (4)-(5) if there exists a function v ∈ l1(ja × jb, e) with v(x, y) ∈ f(x, y, u(x, y)) and such that u(x, y) = m∑ i=1 gi(x, y)u(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1v(s, t)dsdt and the function u satisfies condition (5) on j̃. for any u ∈ c(ja × jb, e), we define the set sf,u = {v ∈ l 1(ja × jb, e), v(x, y) ∈ f(x, y, u(x, y)), (x, y) ∈ ja × jb} this is known as the set of selection function. set g = max i=1...m { sup (x,y)∈ja×jb |gi(x, y)|}. we are now in the position to state and prove our existence result for the problem (4)-(5). we first list the following hypotheses. 6 mouffak benchohra and fatima-zohra mostefai cubo 15, 1 (2013) (h1) f : ja × jb × e → pcp,cl,cv(e), has weakly sequentially closed graph. (h2) for each u ∈ c(ja × jb, e), there exists a measurable function v : ja × jb → e with v(x, y) ∈ f(x, y, u(x, y)) a.e. on ja × jb and v is pettis integrable on ja × jb. (h3) there exists p ∈ l∞(ja × jb, r+) such that ‖f(x, y, u)‖p = sup{‖v‖ : v ∈ f(x, y, u)} ≤ p(x, y), for (x, y) ∈ ja × jb and each u ∈ e. (h4) there exists a number r > 0 such that p∗aα1bα2 γ(α1 + 1)γ(α2 + 1)(1 − mg) < r, (7) where p∗ = ‖p‖ ∞ . (h5) let r0 > 0 be arbitrary (but fixed). for any ǫ > 0 and for any subset x ⊂ br0, there exists a closed subset iǫ ⊂ ja × jb such that µ(ja × jb\iǫ) < ǫ and β(f(t × x)) ≤ sup (x,y)∈t p(x, y)β(x), for each closed subset t of iǫ, where µ denotes the lebesgue measure in r 2. the main result in this paper reads as follows. theorem 3.2. assume that assumptions (h1)-(h5) hold. if mg + p∗aα1bα2 γ(α1 + 1)γ(α2 + 1) < 1, (8) then problem(4)-(5) has at least one solution on j. proof. to transform problem (4)-(5) into a fixed point problem, we define a multivalued map ω : c(j, e) → pcl(c(j, e)) as ω(u) = {h ∈ c(j, e) such that h(x, y) =    ψ(x, y) if (x, y) ∈ j̃, ∑m i=1 gi(x, y)u(x − ξi, y − µi) if υ ∈ sf,u, + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt; (x, y) ∈ ja × jb. where ψ(·, ·) is the function defined by (5). now, we prove that ω satisfies all the assumptions of the theorem 2.7 and thus ω has a fixed point which is a solution of problem (4)-(5). cubo 15, 1 (2013) weak solutions of fractional order pettis integral inclusions ... 7 first notice that, for all u ∈ c(j, e), there exists a pettis integral υ : ja × jb → e such that υ(x, y) ∈ f(x, y, u(x, y)) for a.e. (x, y) ∈ ja × jb (assumption (h2)) then ϕ(υ(x, y)) ∈ l 1(ja × jb) for any ϕ ∈ e∗. from the definition of the integral of fractional order we have iαϕ(υ(x, y, )) = ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1 γ(α1)γ(α2) ϕ(υ(s, t))dsdt = ∫x 0 ∫y 0 ϕ ( (x − s)α1−1(y − t)α2−1 γ(α1)γ(α2) υ(s, t) ) dsdt exists for almost every (x, y) ∈ ja ×jb and is an element from l 1(ja ×jb), that is, for almost every (x, y) ∈ ja × jb, s ∈ (0, x), t ∈ (0, y) the measurable function ϕ ( (x − s)α1−1(y − t)α2−1 γ(α1)γ(α2) υ(s, t) ) = (x − s)α1−1(y − t)α2−1 γ(α1)γ(α2) ϕ(υ(s, t)) is lebesgue integrable, hence the function (s, t) → (x−s) α 1 −1 (y−t) α 2 −1 γ(α1)γ(α2) υ(s, t) is pettis integrable on ja × jb, and thus the operator ω is well defined. let r > 0 and consider the set q = {u ∈ c(j, e) : ‖u‖ ∞ ≤ r and ‖u(x2, y2) − u(x1, y1)‖ ≤ r m∑ i=1 |gi(x2, y2) − gi(x1, y1)| + p∗ γ(α1 + 1)γ(α2 + 1) [x α1 2 y α2 2 − x α1 1 y α2 1 ]; for (x1, y1), (x2, y2) ∈ ja × jb} clearly, the subset q is closed, bounded, convex and equicontinuous subset of a metrisable locally convex vector space c(j, e). the remainder of the proof will be given in four steps. step 1: ω(u) is convex for each u ∈ q. for that, let 0 < λ < 1, h1, h2 ∈ ω(u), obviously if (x, y) ∈ j̃ then λh1(x, y)+(1−λ)h2(x, y) ∈ ω(u). now if (x, y) ∈ ja × jb, then there exists υ1, υ2 ∈ sf,u such that hi(x, y) = m∑ i=1 gi(x, y)u(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υi(s, t)dsdt; i = 1, 2, then for each (x, y) ∈ ja × jb we have (λh1 + (1 − λ)h2)(x, y) = m∑ i=1 gi(x, y)u(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1(λυ1(s, t) + (1 − λ)υ2(s, t))dsdt. 8 mouffak benchohra and fatima-zohra mostefai cubo 15, 1 (2013) since sf,u is convex ( because f has convex values), it follows that λh1 + (1 − λ)h2 ∈ ω(u). step 2: ω maps q into q. to see this, take h ∈ ωq. then there exists u ∈ q with h ∈ ωu. and there exists υ : ja × jb → e pettis integrable with υ(x, y) ∈ f(x, y, u(x, y)) h(x, y) =    ψ(x, y) if (x, y) ∈ j̃, ∑m i=1 gi(x, y)u(x − ξi, y − µi) if υ ∈ sf,u, + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt; (x, y) ∈ ja × jb. we can consider that h(x, y) 6= 0 and by proposition 2.6 there exists ϕ ∈ e∗ with ‖ϕ‖ = 1 and ϕ(h(x, y)) = ‖h(x, y)‖ for (x, y) ∈ ja × jb, we have ‖h(x, y)‖ = ϕ(h(x, y)) = ϕ ( m∑ i=1 gi(x, y)u(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt ) = ϕ ( m∑ i=1 gi(x, y)u(x − ξi, y − µi) ) + ϕ ( 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt ) ≤ mgr + p∗ γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1dsdt ≤ mgr + p∗aα1bα2 γ(α1 + 1)γ(α2 + 1) ≤ r. on the other hand, for (x, y) ∈ j̃, we have ‖h(x, y)‖ = ϕ(h(x, y)) ≤ r. next, suppose that (x1, y1), (x2, y2) ∈ ja × jb with x1 < x2 and y1 < y2, and let h ∈ ωu, so h(x1, y1) − h(x2, y2) 6= 0. then there exists ϕ ∈ e ∗ such that ‖h(x1, y1) − h(x2, y2)‖ = ϕ(h(x1, y1) − h(x2, y2)), and ‖ϕ‖ = 1. thus cubo 15, 1 (2013) weak solutions of fractional order pettis integral inclusions ... 9 ‖h(x2, y2) − h(x1, y1)‖ = ϕ( m∑ i=1 gi(x2, y2)u(x2 − ξi, y2 − µi) + 1 γ(α1)γ(α2) ∫x2 0 ∫y2 0 (x2 − s) α1−1(y2 − t) α2−1υ(s, t)dsdt − m∑ i=1 gi(x1, y1)u(x1 − ξi, y1 − µi) + 1 γ(α1)γ(α2) ∫x1 0 ∫y1 0 (x1 − s) α1−1(y1 − t) α2−1υ(s, t)dsdt) = ϕ( m∑ i=1 gi(x2, y2)u(x2 − ξi, y2 − µi) − m∑ i=1 gi(x1, y1)u(x1 − ξi, y1 − µi)) +ϕ( 1 γ(α1)γ(α2) ∫x2 x1 ∫y2 y1 (x2 − s) α1−1(y2 − t) α2−1υ(s, t)dsdt + 1 γ(α1)γ(α2) ∫x1 0 ∫y1 0 [(x2 − s) α1−1(y2 − t) α2−1 − (x1 − s) α1−1(y1 − t) α2−1] ×υ(s, t)dsdt + 1 γ(α1)γ(α2) ∫x1 0 ∫y2 y1 (x2 − s) α1−1(y2 − t) α2−1υ(s, t)dsdt + 1 γ(α1)γ(α2) ∫x2 x1 ∫y1 0 (x2 − s) α1−1(y2 − t) α2−1υ(s, t)dsdt) ≤ m∑ i=1 ‖gi(x2, y2)u(x2 − ξi, y2 − µi) − gi(x1, y1)u(x1 − ξi, y1 − µi)‖ + p∗ γ(α1)γ(α2) ∫x1 0 ∫y1 0 [(x2 − s) α1−1(y2 − t) α2−1 − (x1 − s) α1−1(y1 − t) α2−1]dsdt + p∗ γ(α1)γ(α2) ∫x2 x1 ∫y2 y1 (x2 − s) α1−1(y2 − t) α2−1dsdt + p∗ γ(α1)γ(α2) ∫x1 0 ∫y2 y1 (x2 − s) α1−1(y2 − t) α2−1dsdt + p∗ γ(α1)γ(α2) ∫x2 x1 ∫y1 0 (x2 − s) α1−1(y2 − t) α2−1dsdt ≤ r m∑ i=1 ‖gi(x2, y2) − gi(x1, y1)‖ + p∗ γ(α1 + 1)γ(α2 + 1) [x α1 2 y α2 2 − x α1 1 y α2 1 ]. this implies that h ∈ q, hence ωq ⊂ q step 3: ω has weakly sequentially closed graph. let (un, wn) be a sequence in q×q with un(x, y) → u(x, y) in (e, w) for each (x, y) ∈ ja×jb, wn(x, y) → w(x, y) in (e, w) for each (x, y) ∈ ja ×jb and wn ∈ ω(un) for n ∈ {1, 2, . . .}. we show that w ∈ ω(u). since wn ∈ ω(un), there exists υn ∈ sf,un such that wn(x, y) = m∑ i=1 gi(x, y)un(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υn(s, t)dsdt. 10 mouffak benchohra and fatima-zohra mostefai cubo 15, 1 (2013) we show that there exists υ ∈ sf,u such that w(x, y) = m∑ i=1 gi(x, y)u(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt. since f(·, ·, ·) has compact values, there exists a subsequence υnm ∈ sf,un such that υnm is pettis integrable and υnm(x, y) ∈ f(x, y, un(x, y)) a.e.(x, y) ∈ ja × jb and υnm(·, ·) → υ(·, ·) in (e, w) as m → ∞. as f(x, y, ·) has weakly sequentially closed graph, υ(x, y) ∈ f(x, y, u(x, y)). then lebesgue dominated convergence theorem for pettis integral implies that ϕ(wn(x, y)) → ϕ ( m∑ i=1 gi(x, y)u(x − ξi, y − µi) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt ) i.e. wn(x, y) → ωu(x, y) in (e, w). since this holds, for each (x, y) ∈ ja × jb, we have w ∈ ωu. step 4: the implication (6) holds. let v be a subset of q such that v = conv(ω(v)∪{0}). obviously v(x, y) ⊂ conv(ω(v(x, y))∪ {0}), ∀(x, y) ∈ j. further, as v is bounded and equicontinuous, the function (x, y) → υ(x, y) = β(v(x, y)) is continuous on j. if (x, y) ∈ j̃ then ωv(x, y) = {ωu(x, y) : u ∈ v} = {ψ(x, y) : (x, y) ∈ j̃}. and since ψ is continuous on [−ξ, 0] × [−µ, 0], the set {ψ(x, y), (x, y) ∈ [−ξ, 0] × [−µ, 0]} ⊂ e is compact. now by (h3) and the properties of the measure β, for any (x, y) ∈ ja × jb, we have υ(x, y) ≤ β((ωv)(x, y) ∪ {0}) ≤ β((ωv)(x, y)) ≤ β{ωu(x, y) : u ∈ v} ≤ β {∑m i=1 gi(x, y)u(x − ξi, y − µi); u ∈ v } +β { 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt; υ(x, y) ∈ f(x, y, u), u ∈ v } ≤ ∑m i=1 β({gi(x, y)u(x − ξi, y − µi); u ∈ v}) cubo 15, 1 (2013) weak solutions of fractional order pettis integral inclusions ... 11 + 1 γ(α1)γ(α2) β {∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1υ(s, t)dsdt; υ(x, y) ∈ f(x, y, u), u ∈ v } ≤ ∑m i=1 |gi(x, y)|β(v(x, y)) + 1 γ(α1)γ(α2) ∫x 0 ∫y 0 (x − s)α1−1(y − t)α2−1p(s, t)β(v(s, t))dsdt ≤ mg‖υ‖ ∞ + p ∗ a α 1 b α 2 γ(α1+1)γ(α2+1) ‖υ‖ ∞ in particular, ‖υ‖ ∞ ≤ ‖υ‖ ∞ ( mg + p∗aα1bα2 γ(α1 + 1)γ(α2 + 1) ) . by (8) it follows that ‖υ‖ ∞ = 0, that is υ(x, y) = β(v(x, y)) = 0 for each (x, y) ∈ j and then v is weakly relatively compact in c(j, e). applying now theorem 2.7 we conclude that t has a fixed point which is a solution of problem (4)-(5). 2 received: october 2012. revised: february 2013. references [1] s. abbas and m. benchohra, fractional order riemann-liouville integral equations with multiple time delay, appl. math. e-notes 12 (2012), 79-87. 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[21] a. n. vityuk and a. v. golushkov, existence of solutions of systems of partial differential equations of fractional order, nonlinear oscil. 7 (3) (2004), 318-325. cubo a mathematical journal vol.19, no¯ 01, (01–15). march 2017 almost ω-continuous functions defined by ω-open sets due to arhangel’skĭı e. rosas 1 , c. carpintero2, m. salas2, j. sanabria2 and l. vásquez 1 department of mathematics, universidad de oriente, núcleo de sucre, cumaná, venezuela. departamento de ciencias naturales y exactas, universidad de la costa barranquilla, colombia. 2 department of mathematics, universidad de oriente, núcleo de sucre, cumaná, venezuela. ennisrafael@gmail.com, carpintero.carlos@gmail.com, salasbrown@gmail.com, jesanabri@gmail.com, eligiovm85@gmail.com abstract in this paper, we apply the notion of ω-open sets due to arhangel’skĭı [1] to present and study a new class of functions called almost ω-continuous functions. relationships between this new class and other classes of functions are established. resumen en este art́ıculo, aplicamos la noción de ω-conjuntos abiertos dada por arhangel’skĭı [1] para presentar y estudiar una nueva clase de funciones llamadas funciones casi ωcontinuas. establecemos relaciones entre esta nueva clase y otras clases de funciones. keywords and phrases: ω-open sets, almost ω-continuous functions. 2010 ams mathematics subject classification: 54d10. 2 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) 1 introduction and preliminaries generalized open sets play a very important role in general topology and they are now the research topics of many topologist worldwide. indeed, a significant theme in general topology and real analysis concerns the variously modified forms of continuity and separation axioms, by utilizing generalized closed sets. recently, as generalization of closed sets, the notion of β-closed sets were introduced and studied by noiri et al. [12] and the notion of ω-closed sets were introduced and studied by hdeib [8]. let (x,τ) be a topological space and let a be a subset of x. we denote the closure and the interior of a by cl(a) and int(a), respectively. a point x ∈ x is called a condensation point of a if for each u ∈ τ with x ∈ u, the set u ∩ a is uncountable. a subset a is said to be ω-closed [8] if it contains all its condensation points. it is well known that a subset w of a space (x,τ) is ω-open[8] if and only if for each x ∈ w, there exists u ∈ τ such that x ∈ u and u \ w is countable. other notion of ω-closed sets were introduced and studied by arhangel’skĭi [1]. a subset a of x is called ω-closed [1], if cl(b) ⊂ a whenever b ⊂ a and b is a countable set. the complement of an ω-closed set is said to be an ω-open set [1]. in the sequel, we will use the ω-closed and ω-open sets in the sense of [1]. in the case that, we use the ω-closed and ω-open sets in the sense of [8], this will be explicitly stated. the family of all ω-open subsets of a topological space (x,τ) forms a topology on x which is finer than τ. the set of all ω-open sets of (x,τ) is denoted by ωo(x). the set of all ω-open sets of (x,τ) containing a point x ∈ x is denoted by ωo(x, x). the intersection of all ω-closed sets containing a is called the ω-closure of a and is denoted by ωcl(a). the ω-interior of a is defined by the union of all ω-open sets contained in a and is denoted by ωint(a). a point x ∈ x is called a θ-cluster point of a if cl(v) ∩ a ̸= ∅ for every open set v of x containing x. the set of all θ-cluster points of a is called the θ-closure of a and is denoted by clθ(a). if a = clθ(a), then a is said to be θ-closed. the complement of a θ-closed set is said to be a θ-open set. the union of all θ-open sets contained in a is called the θ-interior of a and is denoted by intθ(a). it follows from [17] that the collection of all θ-open sets in a topological space (x,τ) forms a topology on x which is coarser than τ and is denoted by τθ. a subset a of x is said to be regular open [16] if a = int(cl(a)). a subset a of x is said to be δ-open [17] if it is the union of regular open sets of x. the complement of a regular open (resp. δ-open) set is called regular closed (resp. δ-closed). the intersection of all δ-closed sets of (x,τ) containing a is called the δ-closure [17] of a and is denoted by clδ(a). a subset a of a topological space (x,τ) is said to be β-open [2] (resp. semiopen [10], preopen [11]) if a ⊂ cl(int(cl(a))) (resp. a ⊂ cl(int(a)), a ⊂ int(cl(a))). the complement of a semiopen (resp. preopen, β-open) set is called a semiclosed (resp. preclosed, β-closed) set. the set of all regular open (resp. regular closed, δ-open, δ-closed, β-open, preopen, semiclosed, preclosed, β-closed) sets of (x,τ) is denoted by ro(x) (resp. rc(x), δo(x), δc(x), βo(x), po(x), sc(x), pc(x), βc(x)). the intersection of all semiclosed sets of (x,τ) containing a is called the semiclosure [5] of a and is denoted by scl(a). in this article, using the notions of ω-open sets given in [1], we introduce and study a new class of functions called almost ω-continuous functions. the connections between these functions and other existing well-known related functions are investigated. cubo 19, 1 (2017) almost ω-continuous functions 3 the following two examples shows that the notions of ω-open set in sense of [1] and ω-open set in sense of [8] are independent. that means, the topologies τω generated by the ω-open sets in the sense of [1] and [8] are different. example 1.1. let x = r with the usual topology. then a = r \ q is an ω-open set in the sense of [8], but a is not an ω-open set in the sense of [1]. example 1.2. consider the topology of the countable complement on x = r. then a = {1} is an ω-open set in the sense of [1], but a is not an ω-open set in the sense of [8]. definition 1.3. a topological space (x,τ) is said to be: (1) ω-t1 (resp. r-t1 [7]) if for each pair of distinct points x and y of x, there exist ω-open (resp. regular open) sets u and v such that x ∈ u, y /∈ u and x /∈ v, y ∈ v. (2) ω-t2 (resp. r-t2 [7]) if for each pair of distinct points x and y of x, there exist ω-open (resp. regular open) sets u and v such that x ∈ u, y ∈ v and u ∩ v = ∅. lemma 1.4. let (x,τ) be a space and let a be a subset of x. the following statements are true: (1) a ∈ po(x) if and only if scl(a) = int(cl(a)) [9]. (2) a ∈ βo(x) if and only if cl(a) is regular closed [3]. definition 1.5. a function f : (x,τ) → (y,σ) is said to be: (1) ω-continuous [6] if f−1(v) is ω-open in x for every open set v of y. (2) almost continuous [15] if f−1(v) is open in x for every regular open set v of y. (3) r-map [4] if f−1(v) is regular open in x for every regular open set v of y. (4) weakly ω-continuous [6] if for each point x ∈ x and each open subset v in y containing f(x), there exists u ∈ ωo(x, x) such that f(u) ⊂ cl(v). the proof of the following lemma is a direct consequence of definition 1.5(1). lemma 1.6. a function f : (x,τ) → (y,σ) is ω-continuous if and only if f−1(v) ∈ ωc(x) for every closed set v of y. 2 almost ω-continuous functions definition 2.1. a function f : (x,τ) → (y,σ) is said to be almost ω-continuous if for each point x ∈ x and each open subset v of y containing f(x), there exists u ∈ ωo(x, x) such that f(u) ⊂ int(cl(v)). 4 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) theorem 2.2. for a function f : (x,τ) → (y,σ), the following statements are equivalent: (1) f is almost ω-continuous, (2) f−1(v) ∈ ωo(x) for every v ∈ ro(y), (3) f−1(f) ∈ ωc(x) for every f ∈ rc(y), (4) f(ωcl(a)) ⊂ clδ(f(a)) for every subset a of x, (5) ωcl(f−1(b)) ⊂ f−1(clδ(b)) for every subset b of y, (6) f−1(f) ∈ ωc(x) for every f ∈ δc(y), (7) f−1(v) ∈ ωo(x) for every v ∈ δo(y). proof. (1) ⇒ (2) suppose that v ∈ ro(y) and let x ∈ f−1(v), then f(x) ∈ v. since v is an open set and f is an almost ω-continuous function, there exists u ∈ ωo(x, x) such that f(u) ⊂ int(cl(v)) = v. thus x ∈ u ⊂ f−1(f(u)) ⊂ f−1(v) and hence, we obtain that f−1(v) ∈ ωo(x). (2) ⇒ (3) let f ∈ rc(y), then y \ f ∈ ro(y). by hypothesis, f−1(y \ f) ∈ ωo(x) and since f−1(y \ f) = x \ f−1(f), we have x \ f−1(f) ∈ ωo(x). therefore f−1(f) ∈ ωc(x). (3) ⇒ (4) suppose that k is a δ-closed set in y containing f(a). observe that k = clδ(k) = ! {f : k ⊂ f and f ∈ rc(y)} and so f−1(k) = ! {f−1(f) : k ⊂ f and f ∈ rc(y)}. now, by part (3), we have that f−1(k) ∈ ωc(x) and a ⊂ f−1(k). hence ωcl(a) ⊂ f−1(k), and it follows that f(ωcl(a)) ⊂ k. since this is true for any δ-closed set k containing f(a), we have f(ωcl(a)) ⊂ clδ(f(a)). (4) ⇒ (5) let b be a subset of y, then f−1(b) is a subset of x. by part (4), f(ωcl(f−1(b))) ⊂ clδ(f(f −1(b))) ⊂ clδ(b) and so, ωcl(f −1(b)) ⊂ f−1(f(ωcl(f−1(b)))) ⊂ f−1(clδ(b)). (5) ⇒ (6) suppose that f ∈ δc(y), then ωcl(f−1(f)) ⊂ f−1(clδ(f)) = f −1(f). in consequence, ωcl(f−1(f)) = f−1(f) and hence f−1(f) ∈ ωc(x). (6) ⇒ (7) let v ∈ δo(y), then y \ v ∈ δc(y). by hypothesis, f−1(y \ v) ∈ ωc(x) and since f−1(y \ v) = x \ f−1(v), we have x \ f−1(v) ∈ ωc(x). therefore f−1(v) ∈ ωo(x). (7) ⇒ (1) let x ∈ x and let v any open set in y such that f(x) ∈ v. put w = int(cl(v)) and u = f−1(w). since cl(v) is a closed set in y, we have w = int(cl(v)) ∈ δo(y) and by part (7), u = f−1(w) ∈ ωo(x). now, f(x) ∈ v = int(v) ⊂ int(cl(v)) = w, it follows that x ∈ f−1(w) = u and f(u) = f(f−1(w)) ⊂ w = int(cl(v)). we note that nour [13], has also defined a type of function which he calls almost ω-continuous. but this definition is given by using the ω-open sets in sense of [8]. the following example shows that the notions of almost ω-continuous function in the sense of this paper and almost ω-continuous function in the sense of [13], are independent. cubo 19, 1 (2017) almost ω-continuous functions 5 example 2.3. consider x = r with the countable complement topology τc and y = r with the discrete topology τd. then, the function f : (x,τc) → (y,τd) defined as f(x) = x, is almost ω-continuous in the sense of this paper, but f is not almost ω-continuous in the sense [13]. example 2.4. let x = r with the topology τ = {∅, r, q} and y = {a, b, c} with the topology σ = {∅, y, {a}, {b}, {a, b}}. consider the function f : (x,τ) → (y,σ) defined as follows: f(x) = { a, if x ∈ q b, if x /∈ q. then, f is an almost ω-continuous function in the sense [13], but f is not almost ω-continuous in the sense of this paper. proposition 2.5. every almost ω-continuous function is weakly ω-continuous. proof. let x ∈ x and let v an open subset of y such that f(x) ∈ v. since f is an almost ω-continuous function, there exists u ∈ ωo(x) such that x ∈ u and f(u) ⊂ int(cl(v)) ⊂ cl(v). therefore, f is a weakly ω-continuous function. the following examples show that the converse of proposition 2.5 is not true in general. example 2.6. consider the function f in example 2.4. it is easy to see that f is weakly ωcontinuous but is not almost ω-continuous. example 2.7. let x = r with the topology τ = {∅, r, r \ q} and y = {a, b, c} with the topology σ = {∅, y, {a}, {b}, {a, b}}. take a ⊂ q and define the function f : (x,τ) → (y,σ) as follows: f(x) = ⎧ ⎪⎪⎨ ⎪⎪⎩ a, if x ∈ q \ a. c, if x ∈ r \ q. b, if x ∈ a. then, f is weakly ω-continuous, but f is not almost ω-continuous in in the sense [13]. theorem 2.8. for a function f : (x,τ) → (y,σ), the following statements are equivalent: (1) f is almost ω-continuous, (2) for each x ∈ x and each open set v of y containing f(x) there exists u ∈ ωo(x, x) such that f(u) ⊂ scl(v), (3) for each x ∈ x and each regular open set v of y containing f(x) there exists u ∈ ωo(x, x) such that f(u) ⊂ v, (4) for each x ∈ x and each δ-open set v of y containing f(x) there exists u ∈ ωo(x, x) such that f(u) ⊂ v. 6 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) proof. (1)⇒(2): let x ∈ x and v be an open set of y containing f(x). by part (1), there exists u ∈ ωo(x, x) such that f(u) ⊂ int(cl(v)). since v is a preopen set, then by lemma 1.4, f(u) ⊂ scl(v). (2)⇒(3): let x ∈ x and v be a regular open set of y containing f(x). then v is an open set of y containing f(x). by part (2), there exists u ∈ ωo(x, x) such that f(u) ⊂ scl(v). since v is a preopen set, then by lemma 1.4, f(u) ⊂ int(cl(v)) = v. (3)⇒(4). let x ∈ x and v be a δ-open set of y containing f(x). then, there exists an open set g containing f(x) such that g ⊂ int(cl(g)) ⊂ v. since int(cl(g)) is a regular open set of y containing f(x), by part (3), there exists u ∈ ωo(x, x) such that f(u) ⊂ int(cl(g)) ⊂ v. (4)⇒(1). let x ∈ x and v be an open set of y containing f(x). then int(cl(v)) is a δ-open set of y containing f(x). by part (4), there exists u ∈ ωo(x, x) such that f(u) ⊂ int(cl(v)). therefore, f is almost ω-continuous. theorem 2.9. for a function f : (x,τ) → (y,σ), the following statements are equivalent: (1) f is almost ω-continuous, (2) f−1(int(cl(v))) ∈ ωo(x) for every open set v of y, (3) f−1(cl(int(f))) ∈ ωc(x) for every closed set f of y. proof. (1)⇒(2): let v be an open set in y. we have to show that f−1(int(cl(v))) is an ω-open set in x. let x ∈ f−1(int(cl(v))). then f(x) ∈ int(cl(v)) and int(cl(v)) is a regular open set in y. since f is almost ω-continuous, there exists u ∈ ωo(x, x) such that f(u) ⊂ int(cl(v)). these implies that x ∈ u ⊂ f−1(int(cl(v))), in consequence, f−1(int(cl(v))) is ω-open set in x. (2)⇒(3): let f be a closed set of y. then y \ f is an open set of y. by part (2), we have f−1(int(cl(y \ f)))) is ω-open set in x and as f−1(int(cl(y \ f))) = f−1(int(y \ int(f))) = f−1(y \ cl(int(f))) = x \ f−1(int(cl(f))) then f−1(int(cl(f))) is an ω-closed set in x. (3)⇒(1): let f be a regular closed set of y. then f is a closed set of y. by part (3), f−1(cl(int(f))) is an ω-closed set in x. since f is a regular closed set, then f−1(cl(int(f))) = f−1(f). therefore, f−1(f) is an ω-closed set in x. by theorem 2.2, f is an almost ω-continuous function. theorem 2.10. a function f : (x,ωo(x)) → (y,σ) is almost ω-continuous if and only if it is almost continuous. proof. this is an immediate consequence of theorem 2.2. theorem 2.11. for a function f : (x,τ) → (y,σ), the following statements are equivalent: (1) f is almost ω-continuous, (2) ωcl(f−1(v)) ⊂ f−1(cl(v)) for every v ∈ βo(y), (3) f−1(int(f)) ⊂ ω int(f−1(f)) for every f ∈ βc(y), cubo 19, 1 (2017) almost ω-continuous functions 7 (4) f−1(int(f)) ⊂ ω int(f−1(f)) for every f ∈ sc(y), (5) ωcl(f−1(v)) ⊂ f−1(cl(v)) for every v ∈ so(y), (6) f−1(v) ⊂ ω int(f−1(int(cl(v)))) for every v ∈ po(y). proof. (1)⇒ (2): let v be any β-open set of y. since cl(v) ∈ rc(y), by theorem 2.2, f−1(cl(v)) is ω-closed in x and f−1(v) ⊂ f−1(cl(v)). therefore, ωcl(f−1(v)) ⊂ f−1(cl(v)). (2)⇒(3): let f be any β-closed set of y. then y\f is β-open set of y. by part (2), ωcl(f−1(y\f)) ⊂ f−1(cl(y \f)) and ωcl(x\f−1(f)) ⊂ f−1(y \int(f)) and hence, x\ωint(f−1(f)) ⊂ x\f−1(int(f)). therefore, f−1(int(f)) ⊂ ωint(f−1(f)). (3)⇒(4): this is obvious since sc(y) ⊂ βc(y). (4)⇒(5): let v be any semiopen set of y. then y \ v is a semiclosed set in y. by part (4), f−1(int(y \ v)) ⊂ ω int(f−1(y \ v)) and f−1(y \ cl(v)) ⊂ ω int(x \ f−1(v)) and hence, x \ f−1(cl(v)) ⊂ x \ ωcl(f−1(v)). therefore, ωcl(f−1(v)) ⊂ f−1(cl(v)). (5)⇒(1): let k ∈ rc(y). then k ∈ so(y) and by part (5), ωcl(f−1(k)) ⊂ f−1(cl(k)) = f−1(k). therefore, f−1(k) is ω-closed in x and hence f is almost ω-continuous by theorem 2.2. (1)⇒(6): let v be any preopen set of y. since int(cl(v)) ∈ ro(y), by theorem 2.2, we have f−1(int(cl(v))) ∈ ωo(x) and hence f−1(v) ⊂ f−1(int(cl(v))) = ω int(f−1(int(cl(v)))). (6)⇒(1): let v be any regular open set of y. since v ∈ po(y), f−1(v) ⊂ ω int(f−1(int(cl(v)))) = ω int(f−1(v)) and hence f−1(v) ∈ ωo(x). it follows from theorem 2.2, that f is almost ωcontinuous. as a direct consequence of theorem 2.11, we obtain the following two corollaries corollary 2.12. for a function f : (x,τ) → (y,σ), the following statements are equivalent: (1) f is almost ω-continuous, (2) f−1(v) ⊂ ω int(f−1(scl(v))) for each preopen set v of y, (3) ωcl(f−1(cl(int(f)))) ⊂ f−1(f) for each preclosed set f of y, (4) ωcl(f−1(sint(f))) ⊂ f−1(f) for each preclosed set f of y. corollary 2.13. for a function f : (x,τ) → (y,σ), the following statements are equivalent: (1) f is almost ω-continuous, (2) for each neighborhood v of f(x), x ∈ ω int(f−1(scl(v))), (3) for each neighborhood v of f(x), x ∈ ω int(f−1(int(cl(v)))). 8 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) theorem 2.14. let f : (x,τ) → (y,σ) be an almost ω-continuous function and let v be any open subset of y. if x ∈ ωcl(f−1(v)) \ f−1(v), then f(x) ∈ ωcl(v). proof. let x ∈ x such that x ∈ ωcl(f−1(v)) \ f−1(v) and suppose f(x) /∈ ωcl(v). then there exists an ω-open set h containing f(x) such that h ∩ v = ∅. then cl(h) ∩ v = ∅ implies int(cl(h)) ∩ v = ∅ and int(cl(h)) is a regular open set. since f is almost ω-continuous, there exists an ω-open set u in x containing x such that f(u) ⊂ int(cl(h)). therefore, f(u) ∩ v = ∅. however, since x ∈ ωcl(f−1(v), u ∩ f−1(v) ̸= ∅ for every ω-open set u in x containing x, so that f(u) ∩ v ̸= ∅. we have a contradiction. it follows that f(x) ∈ ωcl(v). recall that the family of all ω-open subsets of a topological space (x,τ) forms a topology on x finer than τ. from this fact we obtain immediately the following result. lemma 2.15. let a and b be subsets of a topological space (x,τ). if a ∈ ωo(x) and b ∈ τ, then a ∩ b ∈ ωo(b). proof. since τ is a topology, then the induced topology on b, denoted by τb is {k ∩ b : k ∈ τ}. let x ∈ a ∩ b, then x ∈ a and x ∈ b. since a ∈ ωo(x), there exists u ∈ τ with x ∈ u and u \ a is countable. since u ∩ b is an open subset in τb with x ∈ u ∩ b, follows that (u ∩ b) \ (a ∩ b) = (u \ a) ∩ b. since u \ a is countable, (u ∩ b) \ (a ∩ b) is countable, in consequence, a ∩ b ∈ ωo(b). theorem 2.16. let f : (x,τ) → (y,σ) be an almost ω-continuous function and a ⊂ x. if a ∈ τ, then f|a : (a,τa) → (y,σ) is almost ω-continuous. proof. it follows from lemma 2.15. theorem 2.17. let f : (x,τ) → (y,σ) be a function and u = {ui : i ∈ i} be an open cover of x. if f|ui is almost ω-continuous for each i ∈ i, then f is almost ω-continuous. proof. suppose that v is a regular open set of y. since f|ui is almost ω-continuous for each i ∈ i, it follows that (f|ui) −1(v) is ω-open in ui. since f −1(v) = x ∩ f−1(v) = ( " i∈i ui) ∩ f −1(v) = " i∈i (ui ∩ f −1(v)) = " i∈i (f|ui) −1(v), then f−1(v) ∈ ωo(x), which means that f is almost ω-continuous. definition 2.18. let (x,τ) be a topological space. a filter base λ is said to be: (1) ω-convergent to a point x in x if for every u ∈ ωo(x, x), there exists b ∈ λ such that b ⊂ u. (2) r-convergent to a point x in x if for every regular open set u of x containing x, there exists b ∈ λ such that b ⊂ u. theorem 2.19. if f : (x,τ) → (y,σ) is an almost ω-continuous function, then for each point x ∈ x and each filter base λ in x ω-converging to x, the filter base f(λ) is r-convergent to f(x). cubo 19, 1 (2017) almost ω-continuous functions 9 proof. let x ∈ x and λ be any filter base in x, ω-converging to x. by theorem 2.8, for any regular open set v of (y,σ) containing f(x), there exists u ∈ ωo(x, x) such that f(u) ⊂ v. since λ is ω-converging to x, there exists b ∈ λ such that b ⊂ u. this means that f(b) ⊂ v and hence the filter base f(λ) is r-convergent to f(x). definition 2.20. a net (x λ ) is said to be ω-convergent to a point x if for every ω-open set v containing x, there exists an index λ0 such that for λ ≥ λ0, xλ ∈ v. theorem 2.21. if f : (x,τ) → (y,σ) is an almost ω-continuous function, then for each point x ∈ x and each net (x λ ) which is ω-convergent to x, the net f((x λ )) is r-convergent to f(x). proof. the proof is similar to that of theorem 2.19. theorem 2.22. if f : (x,τ) → (y,σ) is an almost ω-continuous injective function and (y,σ) is r-t1, then (x,τ) is ω-t1. proof. suppose that (y,σ) is r-t1. for any distinct points x and y in x, using the injectivity of f, f(x) ̸= f(y) and then, there exist regular open sets v and w such that f(x) ∈ v, f(y) /∈ v, f(x) /∈ w and f(y) ∈ w. since f is almost ω-continuous, f−1(v) and f−1(w) are ω-open subsets of (x,τ) such that x ∈ f−1(v), y /∈ f−1(v), x /∈ f−1(w) and y ∈ f−1(w). this shows that (x,τ) is ω-t1. theorem 2.23. if f : (x,τ) → (y,σ) is an almost ω-continuous injective function and (y,σ) is r-t2, then (x,τ) is ω-t2. proof. for any pair of distinct points x and y in x, using the injectivity of f, f(x) ̸= f(y) and then, there exist disjoint regular open sets u and v in y such that f(x) ∈ u and f(y) ∈ v. since f is almost ω-continuous, f−1(u) and f−1(v) are ω-open sets in x containing x and y, respectively. therefore, f−1(u) ∩ f−1(v) = ∅ because u ∩ v = ∅. this shows that (x,τ) is ω-t2. theorem 2.24. if f : (x,τ) → (y,σ) is an almost continuous function and g : (x,τ) → (y,σ) is an almost ω-continuous function and y is a r-t2-space, then the set e = {x ∈ x : f(x) = g(x)} is an ω-closed set in (x,τ). proof. if x ∈ x \ e, then it follows that f(x) ̸= g(x). since y is r-t2, there exist disjoint regular open sets v and w of y such that f(x) ∈ v and g(x) ∈ w. since f is almost continuous and g is almost ω-continuous, then f−1(v) is open and g−1(w) is ω-open in x with x ∈ f−1(v) and x ∈ g−1(w). put a = f−1(v) ∩ g−1(w). since ωo(x) is a topology on x finer than τ, we have a is ω-open in x. therefore, f(a) ∩ g(a) = ∅ and it follows that x /∈ ωcl(e). this shows that e is ω-closed in x. definition 2.25. a function f : (x,τ) → (y,σ) is said to be: (1) ω-irresolute if f−1(v) is ω-open in x for every ω-open set v of y. 10 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) (2) faintly ω-continuous if for each point x ∈ x and each θ-open set v of y containing f(x), there exists u ∈ ωo(x, x) such that f(u) ⊂ v. theorem 2.26. a function f : (x,τ) → (y,σ) is faintly ω-continuous if and only if f−1(v) ∈ ωo(x) for every θ-open set v of y. proof. suppose that f is faintly ω-continuous. let v be a θ-open set of y and x ∈ f−1(v). since f(x) ∈ v and f is faintly ω-continuous, there exists u ∈ ωo(x, x) such that f(u) ⊂ v. it follows that x ∈ u ⊂ f−1(v). hence f−1(v) is ω-open in x. conversely, let x ∈ x and v be a θ-open set of y containing f(x), by hypothesis f−1(v) is an ω-open set containing x. take u = f−1(v), then f(u) ⊂ v. this shows that f is faintly ω-continuous. as a direct consequence of the theorem 2.26, we obtain the following corollary. corollary 2.27. a function f : (x,τ) → (y,σ) is faintly ω-continuous if and only if f−1(v) ∈ ωc(x) for every θ-closed set v of y. theorem 2.28. the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) hold for the following properties of a function f : (x,τ) → (y,σ): (1) f is ω-continuous. (2) f−1(clδ(b)) is ω-closed in x for every subset b of y. (3) f is almost ω-continuous. (4) f is weakly ω-continuous. (5) f is faintly ω-continuous. if, in addition, y is regular, then the five properties are equivalent of one another. proof. (1) ⇒ (2): since clδ(b) is closed in y for every subset b of y, by theorem 2.2, f −1(clδ(b)) is ω-closed in x. (2) ⇒ (3): for any subset b of y, f−1(clδ(b)) is ω-closed in x and hence we have ωcl(f −1(b)) ⊂ ωcl(f−1(clδ(b)) = f −1(clδ(b)). it follows from theorem 2.2 that f is almost ω-continuous. (3) ⇒ (4): this is obvious. (4) ⇒ (5): let f be any θ-closed set of y. since f is closed, it follows from lemma 1.6 that, f−1(f) is ω-closed in x and hence, by theorem 2.26, f is faintly ω-continuous. suppose that y is regular. we prove that (5) ⇒ (1). let v be any open set of y. since y is regular, v is θ-open in y. by the faintly ω-continuity of f, f−1(v) is ω-open in x. therefore, f is ω-continuous. definition 2.29. a function f : (x,τ) → (y,σ) is said to be ω-preopen if f(u) ∈ po(y) for every ω-open set u of x. cubo 19, 1 (2017) almost ω-continuous functions 11 theorem 2.30. if a function f : (x,τ) → (y,σ) is ω-preopen and weakly ω-continuous, then f is almost ω-continuous. proof. let x ∈ x and let v be an open set of y containing f(x). since f is weakly ω-continuous, there exists u ∈ ωo(x, x) such that f(u) ⊂ cl(v). since f is ω-preopen, f(u) ⊂ int(cl(f(u))) ⊂ int(cl(v)) and hence f is almost ω-continuous. theorem 2.31. let f : (x,τ) → (y,σ) and g : (y,σ) → (z,η) be functions. then the composition g ◦ f : (x,τ) → (z,η) is almost ω-continuous if f and g satisfy one of the following conditions: (1) f is almost ω-continuous and g is r-map. (2) f is ω-irresolute and g is almost ω-continuous. (3) f is ω-continuous and g is almost continuous. proof. (1) follows from theorem 2.2 and definition 1.5. (2) follows from theorem 2.2 and definition 2.25. (3) follows from theorem 2.2 and definition 1.5. theorem 2.32. if f : x → ∏ i∈i yi is almost ω-continuous function then pi ◦ f : x → yi is almost ω-continuous for each i ∈ i, where pi is the projection of ∏ i∈i yi onto yi. proof. let v be a regular open set of yi. since pi is continuous open, it is an r-map and hence p−1 i (v) is regular open in ∏ i∈i yi, it follows that f −1(p−1 i (v)) = (pi ◦ f) −1(v) ∈ ωo(x). this shows that pi ◦ f is almost ω-continuous for each i ∈ i. definition 2.33. a topological space (x,τ) is said to be almost regular [14] if for any regular closed set f of x and any point x ∈ x \ f there exist disjoint open sets u and v such that x ∈ u and f ⊂ v. theorem 2.34. if f : (x,τ) → (y,σ) is a weakly ω-continuous function and y is almost regular, then f is almost ω-continuous. proof. let x ∈ x and let v be an open set of y containing f(x). by the almost regularity of y, there exists a regular open set g of y such that f(x) ∈ g ⊂ cl(g) ⊂ int(cl(v)) [14, theorem 2.2]. since f is weakly ω-continuous, there exists u ∈ ωo(x, x) such that f(u) ⊂ cl(g) ⊂ int(cl(v)). therefore, f is almost ω-continuous. definition 2.35. an ω-frontier of a subset a of (x,τ), denoted by ωfr(a), is defined by ωfr(a) = ωcl(a) ∩ ωcl(x \ a). 12 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) theorem 2.36. the set of all points x ∈ x in which a function f : (x,τ) → (y,σ) is not almost ω-continuous is identical with the union of ω-frontier of the inverse images of regular open sets containing f(x). proof. suppose that f is not almost ω-continuous at x ∈ x. then there exists a regular open set v of y containing f(x) such that u ∩ (x \ f−1(v)) ̸= ∅ for every u ∈ ωo(x, x). therefore, x ∈ ωcl(x \ f−1(v)) = x \ ω int(f−1(v)) and x ∈ f−1(v). thus, x ∈ ωfr(f−1(u)). conversely, suppose that f is almost ω-continuous at x ∈ x and let v be a regular open set of y containing f(x). then there exists u ∈ ωo(x, x) such that u ⊂ f−1(v). that is x ∈ ω int(f−1(v)). therefore, x ∈ x \ ωfr(f−1(v)). definition 2.37. a function f : (x,τ) → (y,σ) is said to be complementary almost ω-continuous if for each regular open set v of y, f−1(fr(v)) is ω-closed in x, where fr(v) denotes the frontier of v. theorem 2.38. if f : (x,τ) → (y,σ) is weakly ω-continuous and complementary almost ωcontinuous, then f is almost ω-continuous. proof. let x ∈ x and v be a regular open set of y containing f(x). then f(x) ∈ y \ fr(v) and hence x ∈ x \ f−1(fr(v)). since f is weakly ω-continuous there exists g ∈ ωo(x, x) such that f(g) ⊂ cl(v). put u = g ∩(x\f−1(fr(v))). then u ∈ ωo(x, x) and f(u) ⊂ f(g)∩(y \fr(v)) ⊂ cl(v) ∩ (y \ fr(v)) = v. this shows that f is almost ω-continuous. theorem 2.39. if f : (x,τ) → (y,σ) is almost ω-continuous, g : (x,τ) → (y,σ) is weakly ωcontinuous and y is hausdorff, then the set {x ∈ x : f(x) = g(x)} is ω-closed in (x,τ). proof. let a = {x ∈ x : f(x) = g(x)} and x ∈ x \ a. then f(x) ̸= g(x). since (y,σ) is hausdorff, there exist open sets v and w of y such that f(x) ∈ v, g(x) ∈ w and v ∩ w = ∅, hence int(cl(v)) ∩ cl(w) = ∅. since f is almost ω-continuous, there exists g ∈ ωo(x, x) such that f(g) ⊂ int(cl(v)). since g is weakly ω-continuous, there exists h ∈ ωo(x) such that g(h) ⊂ cl(w). now put u = g ∩ h, then u ∈ ωo(x, x) and f(u) ∩ g(u) ⊂ int(cl(v)) ∩ cl(w) = ∅. therefore, we obtain u ∩ a = ∅ and hence a is ω-closed in x. theorem 2.40. suppose that the product of two ω-open sets is ω-open. if f1 : (x1,τ) → (y,σ) is weakly ω-continuous, f2 : (x2,τ) → (y,σ) is almost ω-continuous and (y,σ) is hausdorff, then the set {(x1, x2) ∈ x1 × x2 : f1(x1) = f2(x2)} is ω-closed in x1 × x2. proof. let a = {(x1, x2) ∈ x1 × x2 : f(x1) = f(x2)}. if (x1, x2) ∈ (x1 × x2) \ a, then we have f(x1) ̸= f(x2). since (y,σ) is hausdorff, there exist disjoint open sets v1 and v2 in y such that f(x1) ∈ v1 and f(x2) ∈ v2 and cl(v1)∩int(cl(v2)) = ∅. since f1 (resp. f2) is weakly ω-continuous (resp. almost ω-continuous), there exists u1 ∈ ωo(x1, x1) such that f(u1) ⊂ cl(v1) (resp. u2 ∈ ωo(x2, x2) such that f(ωcl(u1)) ⊂ int(cl(v2))). thus, we obtain (x1, x2) ∈ u1 × u2 ⊂ x1 × x2 \ a. therefore, (x1 × x2) \ a is ω-open and so a is ω-closed in x1 × x2. cubo 19, 1 (2017) almost ω-continuous functions 13 theorem 2.41. if g : (x,τ) → (y,σ) is almost ω-continuous and s is a δ-closed subset of x × y, then px(s ∩ g(g)) is ω-closed in x, where px represents the projection of x × y onto x and g(g) denotes the graph of g. proof. let s be a δ-closed set of x × y and x ∈ ωcl(px(s ∩ g(g))). let u be an open set of x containing x and v an open set of y containing g(x). since g is almost ω-continuous, we have x ∈ g−1(v) ⊂ ω int(g−1(int(cl(v)))) and u ∩ ω int(g−1(int(cl(v)))) ∈ ωo(x, x). since x ∈ ωcl(px(s ∩ g(g))), (u ∩ ω int(g −1(int(cl(v))))) ∩ px(s ∩ g(g)) contains some point u of x. this implies that (u, g(u)) ∈ s and g(u) ∈ int(cl(v)). thus, we have ∅ ̸= (u × int(cl(v))) ∩ s ⊂ int(cl(u × v)) ∩ s and hence (x, g(x)) ∈ clδ(s). since s is δ-closed, (x, g(x)) ∈ px(s ∩ g(g)) and x ∈ px(s ∩ g(g)). then px(s ∩ g(g)) is ω-closed. corollary 2.42. if f : (x,τ) → (y,σ) has a δ-closed graph and g : (x,τ) → (y,σ) is almost ω-continuous, then the set {x ∈ x : f(x) = g(x)} is ω-closed in x. proof. since g(f) is δ-closed and px(g(f) ∩ g(g)) = {x ∈ x : f(x) = g(x)} it follows from theorem 2.41, that {x ∈ x : f(x) = g(x)} is ω-closed in x. theorem 2.43. if for each pair of points x1, x2 distinct in a topological space (x,τ) there exists a function f on (x,τ) into a hausdorff space (y,σ) such that f(x1) ̸= f(x2), f is weakly ω-continuous at x1 and f is almost ω-continuous at x2, then x is ω-t2. proof. since (y,σ) is hausdorff, for each pair of point x1, x2 distinct, there exist disjoint open sets v1 and v2 of y containing f(x1) and f(x2), respectively; hence cl(v1) ∩ int(cl(v2)) = ∅. since f is weakly ω-continuous at x1, there exists u1 ∈ ωo(x, x1) such that f(u1) ⊂ cl(v1). since f is almost ω-continuous at x2, there exists u2 ∈ ωo(x, x2) such that f(u2) ⊂ int(cl(v2)). therefore, we obtain u1 ∩ u2 = ∅. this shows that (x,τ) is ω-t2. definition 2.44. a function f : (x,τ) → (y,σ) is said to have an ω-strongly closed graph if for each (x, y) ∈ (x × y) \ g(f), there exists an ω-open subset u of x and an open subset v of y such that (u × cl(v)) ∩ g(f) = ∅. as a consequence of definition 2.44, we obtain easily the following lemma. lemma 2.45. a function f : (x,τ) → (y,σ) has ω-strongly closed graph g(f) if and only if for each (x, y) ∈ (x × y) \ g(f) there exists an ω-open set u and an open set v containing x and y, respectively such that f(u) ∩ cl(v) = ∅. theorem 2.46. if f : (x,τ) → (y,σ) is an almost ω-continuous function and (y,σ) is hausdorff, then f has an ω-strongly closed graph. proof. let (x, y) ∈ x × y such that y ̸= f(x). since (y,σ) is hausdorff, there exist open sets v and w of y containing f(x) and y, respectively, such that v ∩ w = ∅. then f(x) ∈ y \ cl(w) and 14 e. rosas, c. carpintero, m. salas, j. sanabria, l. vásquez cubo 19, 1 (2017) y \cl(w) is regular open in y. since f is almost ω-continuous function, there exists u ∈ ωo(x, x) such that f(u) ⊂ y \ cl(w) and hence f(u) ∩ cl(w) = ∅. therefore, by lemma 2.45 f has an ω-strongly closed graph. the following corollary is an immediate consequence of lemma 2.45, as we can see. corollary 2.47. if f : (x,τ) → (y,σ) is an ω-continuous function and (y,σ) is hausdorff, then f has an ω-strongly closed graph. proof. let (x, y) ∈ x × y such that y ̸= f(x). since (y,σ) is hausdorff, there exist open sets v and w of y containing f(x) and y, respectively such that v ∩ w = ∅. then f(x) ∈ y \ cl(w) and y \ cl(w) is an open set in y. since f is weakly ω-continuous function, there exists u ∈ ωo(x, x) such that u ⊂ f−1(y \ cl(w)) and then f(u) ⊂ y \ cl(w), hence f(u) ∩ cl(w) = ∅. therefore, by lemma 2.45 f has an ω-strongly closed graph. references [1] a. v. arhangel’skĭı, bicompacta that satisfy the suslin condition hereditarily. tightness and free 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[10] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly, 70 (1963), 36-41. cubo 19, 1 (2017) almost ω-continuous functions 15 [11] a. s. mashhour, m. e. abd el-monsef and s. n. el-deep, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt, 53 (1982), 47-53. [12] t. noiri and v. popa, on almost β-continuous functions, acta math. hungar., 79 (4) (1998), 329-339. [13] t. m. nour, almost ω-continuous functions, european j. sci. res., 8 (1) (2005), 43-47. [14] m. k. singal and s. p. arya, on almost regular spaces, glasnik mat., 4 (24) (1969), 89-99. [15] m. k. singal and a. r. singal, almost-continuous mappings, yokohama math. j., 16 (1968), 63-73. [16] m. stone, applications of the theory of boolean rings to general topology, trans. amer. math. soc., 41 (1937), 374-381. [17] n. v. veličko, h-closed topological spaces, amer. math. soc. transl., 78 (1968), 103-118. cubo a mathematical journal vol.16, no¯ 01, (63–72). march 2014 a common fixed point theorem for pairs of mappings in cone metric spaces i̇lker s. ahi̇n and mustafa telci̇ department of mathematics, faculty of science, trakya university, 22030 edirne, turkey isahin@trakya.edu.tr, mtelci@trakya.edu.tr abstract in this paper, we present a common fixed point theorem in complete cone metric spaces which is a generalization of the theorem in [6]. this result also generalizes some theorems given in [4] and [9]. resumen en este art́ıculo presentamos un teorema de punto de fijo común en espacios métricos cono completos, el cual es una generalización del teorema en [6]. también este resultado generaliza algunos teoremas en [4] y [9]. keywords and phrases: cone metric space; complete cone metric space; fixed point 2010 ams mathematics subject classification: 47h10, 54h25. 64 i̇lker s. ahin & mustafa telci cubo 16, 1 (2014) 1 introduction in [4], guang and xian reintroduced the concept of a cone metric space ( known earlier as k-metric space, see [12]), replacing the set of real numbers by an ordered banach space and proved some fixed point theorems for mapping satisfying various contractive conditions. recently, rezapour and hamlbarani [9] generalized some results of [4] by omitting the assumption of normality in the results. also many authors proved some fixed point theorems for contractive type mappings in cone metric spaces (see [1, 2, 3, 5, 7, 8, 10, 11]). the main purpose of this paper is to present a common fixed point theorem for mappings in complete cone metric spaces. 2 preliminaries throughout this paper, we denote by n the set of positive integers and by r the set of real numbers. definition 2.1. let e be a real banach space and p be a subset of e. p is called a cone if and only if: (i) p is closed, nonempty and p 6= {0}, (ii) a, b ∈ r, a, b ≥ 0, x, y ∈ p implies ax + by ∈ p, (iii) x ∈ p and −x ∈ p implies x = 0. given a cone p ⊆ e, we define a partial ordering ≤ with respect to p by x ≤ y if and only if y − x ∈ p. we shall write x < y if x ≤ y and x 6= y, and x � y if y − x ∈ intp, where intp is the interior of p. the cone p is called normal if there is a number k > 0 such that for all x, y ∈ e, 0 ≤ x ≤ y implies ‖x‖ ≤ k‖y‖. the least positive number satisfying the above is then called the normal constant of p. lemma 2.1. ([13]) let e be a real banach space with a cone p. then: (i) if x ≤ y and 0 ≤ a ≤ b, then ax ≤ by for x, y ∈ p, (ii) if x ≤ y and u ≤ v, then x + u ≤ y + v, cubo 16, 1 (2014) a common fixed point theorem for pairs . . . 65 (iii) if xn ≤ yn for each n ∈ n, and limn→∞ xn = x, limn→∞ yn = y then x ≤ y. lemma 2.2. ([10]) if p is a cone, x ∈ p, α ∈ r, 0 ≤ α < 1, and x ≤ αx, then x = 0. in the following definition, we suppose that e is a real banach space, p is a cone in e with intp 6= ∅ and ≤ is partial ordering with respect to p. definition 2.2. let x be a non-empty set. suppose the mapping d : x × x → e satisfies: (d1) 0 ≤ d(x, y) for all x, y ∈ x and d(x, y) = 0 if and only if x = y, (d2) d(x, y) = d(y, x) for all x, y ∈ x, (d3) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ x. then d is called a cone metric on x and (x, d) is called a cone metric space. this definition is more general than that of a metric space. example 2.1. let e = r2, p = {(x, y) ∈ e : x, y ≥ 0} ⊂ r2, x = r2 and d : x × x → e defined by d(x, y) = d((x1, x2), (y1, y2)) = ( max{|x1 − y1|, |x2 − y2|}, α max{|x1 − y1|, |x2 − y2|}), where α ≥ 0 is a constant. then (x, d) is a cone metric space. 3 definitions and lemmas in this section we shall give some definitions and lemmas. definition 3.1.([4]) let (x, d) be a cone metric space. a sequence {xn} in x is said to be: (a) a convergent sequence if for every c ∈ e with 0 � c, there is n ∈ n such that for all n ≥ n, d(xn, x) � c for some fixed x in x. we denote this by limn→∞ xn = x or xn → x, n → ∞. (b) a cauchy sequence if for every c ∈ e with 0 � c, there is n ∈ n such that for all n, m ≥ n, d(xn, xm) � c. a cone metric space (x, d) is said to be complete if every cauchy sequence is convergent in x. the following lemma was recently proved in ([3]), by omitting the normality condition. lemma 3.1. let (x, d) be a cone metric space. if {xn} is a convergent sequence in x, then the limit of {xn} is unique. 66 i̇lker s. ahin & mustafa telci cubo 16, 1 (2014) the proof of the following lemma is straighforward and is omitted. lemma 3.2. let (x, d) be a cone metric space, {xn} be a sequence in x. if {xn} converges to x and {xnk} is any subsequence of {xn}, then {xnk} converges to x. lemma 3.3. ([10]) let (x, d) be a cone metric space and {xn} be a sequence in x. if there exists a sequence {an} in r with an > 0 for all n ∈ n and ∑ an < ∞, which satisfies d(xn+1, xn) ≤ anm for all n ∈ n and for some m ∈ e with m ≥ 0, then {xn} is a cauchy sequence in (x, d). definition 3.2. ([11]) let e and f be reel banach spaces and p and q be cones on e and f, respectively. let (x, d) and (y, ρ) be cone metric spaces, where d : x × x → e and ρ : y × y → f. a function f : x → y is said to be continuous at x0 ∈ x, if for every c ∈ f with 0 � c, there exists b ∈ e with 0 � b such that, ρ(f(x), f(x0)) � c whenever x ∈ x and d(x, x0) � b. if f is continuous at every point of x, then it is said to be continuous on x. lemma 3.4. ([11]) let (x, d) and (y, ρ) be cone metric spaces as in definition 3.2. a function f : x → y is continuous at a point x0 ∈ x if and only if whenever a sequence {xn} in x converges to x0, the sequence {f(xn)} converges to f(x0). 4 main result the following common fixed point theorem was proved in [6]. theorem 4.1. let (x, d) be a complete metric space and let f and g be two continuous selfmappings of x. if there are positive numbers α < 1 and β < 1 such that, for all x, y ∈ x, d(fgx, gy) ≤ αd(x, gy) (1) and d(gfx, fy) ≤ βd(x, fy), (2) then f and g have a unique common fixed point. we now prove the following common fixed point theorem in complete cone metric spaces: theorem 4.2. let (x, d) be a complete cone metric space and p be a cone. let f and g be self-mappings of x satisfying the following inequalities d(fgx, gx) ≤ ad(x, gx), (3) d(gfx, fx) ≤ bd(x, fx) (4) cubo 16, 1 (2014) a common fixed point theorem for pairs . . . 67 for all x in x, where a, b < 1. if either f or g is continuous, then f and g have a common fixed point. proof. let x0 be an arbitrary point in x and define the sequence {xn} inductively by x2n+1 = fx2n, x2n+2 = gx2n+1 for n = 0, 1, 2, . . .. note that if xn = xn+1 for some n, then xn is a fixed point of f and g. indeed, if x2n = x2n+1 for some n ≥ 0, then x2n is a fixed point of f. on the other hand, we have from inequality (4) that d(x2n+2, x2n+1) = d(gx2n+1, fx2n) = d(gfx2n, fx2n) ≤ bd(x2n, fx2n) = bd(x2n, x2n+1) = 0 which implies −d(x2n+1, x2n+2) ∈ p. also we have d(x2n+1, x2n+2) ∈ p. hence d(x2n+1, x2n+2) = 0 and so x2n+1 = x2n+2. thus, x2n is a common fixed point of f and g. if x2n+1 = x2n+2 for some n ≥ 0, similarly, by using inequality (3) leads to x2n+1 is a common fixed point of f and g. now we suppose that xn 6= xn+1 for all n. using inequality (4), we have d(x2n+2, x2n+1) = d(gx2n+1, fx2n) = d(gfx2n, fx2n) ≤ bd(x2n, fx2n) = bd(x2n, x2n+1). (5) similarly, using inequality (3) we have d(x2n+1, x2n) = d(fx2n, gx2n−1) = d(fgx2n−1, gx2n−1) ≤ ad(x2n−1, gx2n−1) = ad(x2n−1, x2n). (6) suppose that α = max{a, b}. then from inequalities (5) and (6) we have d(x2n+1, x2n+2) ≤ αd(x2n, x2n+1) and d(x2n, x2n+1) ≤ αd(x2n−1, x2n). thus, we obtain d(xn+1, xn+2) ≤ αd(xn, xn+1) for n = 0, 1, 2, . . . and it follows that d(xn, xn+1) ≤ α nd(x0, x1). for n = 1, 2, 3 . . .. since ∑ ∞ n=0 αn < ∞, it follows from lemma 3.3 that {xn} is a cauchy sequence in the complete cone metric space (x, d) and so has a limit z in x. 68 i̇lker s. ahin & mustafa telci cubo 16, 1 (2014) now we consider that f is continuous. since x2n+1 = fx2n, it follows from lemma 3.4 that z = lim n→∞ x2n+1 = lim n→∞ fx2n = fz and so z is a fixed point of f. using inequality (4) we have d(gz, z) = d(gfz, fz) ≤ bd(z, fz) = bd(z, z) = 0 which implies −d(gz, z) ∈ p. also we have d(gz, z) ∈ p. hence d(gz, z) = 0 and so gz = z. we have therefore proved that z is a common fixed point of f and g. similarly, considering the continuity of g, it can be seen that f and g have a common fixed point and this completes the proof. putting f = g and k = max{a, b} in theorem 4.2, we get corollary 4.1. let (x, d) be a complete cone metric space and p be a cone. let f be a self-mapping of x satisfying the following inequality d(f2x, fx) ≤ kd(fx, x) (7) for all x in x, where k < 1. if f is continuous, then f has a fixed point. putting e = r, p = {x ∈ r : x ≥ 0} ⊂ r and d : x × x → r in theorem 4.2 and corollary 4.1, then we obtain the following corollaries. corollary 4.2. let (x, d) be a complete metric space and let f and g be self-mappings of x satisfying the following inequalities d(fgx, gx) ≤ ad(x, gx), (8) d(gfx, fx) ≤ bd(x, fx) (9) for all x in x, where a, b < 1. if either f or g is continuous, then f and g have a common fixed point. corollary 4.3. let (x, d) be a complete metric space and let f be a self-mapping of x satisfying the following inequality d(f2x, fx) ≤ kd(fx, x) for all x in x, where k < 1. if f is continuous, then f has a fixed point. cubo 16, 1 (2014) a common fixed point theorem for pairs . . . 69 we now illustrate theorem 4.2 by the following example. example 4.1. let e = r2, p = {(x, y) ∈ e : x, y ≥ 0} ⊂ r2, x = r and the mapping d : x × x → e defined by d(x, y) = ( |x − y|, |x − y|). then (x, d) is a complete cone metric space. define the self-mappings f, g : x → x by fx = { 0 if x ≤ 1 x/4 if x > 1 and gx = 1 4 x for all x in x. if x ≤ 1, then we have d(fgx, gx) = d ( 0, x 4 ) = ( |x| 4 , |x| 4 ) = 1 3 ( ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ , ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ ) = d ( x, x 4 ) = 1 3 d(x, gx) and d(gfx, fx) = (0, 0) ≤ 1 3 (|x|, |x|) = 1 3 d(x, 0) = 1 3 d(x, fx). if 1 < x ≤ 4, then we have d(fgx, gx) = d ( 0, x 4 ) = ( |x| 4 , |x| 4 ) = 1 3 ( ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ , ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ ) = d ( x, x 4 ) = 1 3 d(x, gx) and d(gfx, fx) = d ( x 16 , x 4 ) = ( 3|x| 16 , 3|x| 16 ) ≤ 1 3 ( ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ , ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ ) = 1 3 d ( x, x 4 ) = 1 3 d(x, fx). if x > 4, then we have d(fgx, gx) = d ( x 16 , x 4 ) = ( 3|x| 16 , 3|x| 16 ) ≤ 1 3 ( ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ , ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ ) = 1 3 d ( x, x 4 ) = 1 3 d(x, gx) and d(gfx, fx) = d ( x 16 , x 4 ) = ( 3|x| 16 , 3|x| 16 ) ≤ 1 3 ( ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ , ∣ ∣ ∣ x − x 4 ∣ ∣ ∣ ) = 1 3 d ( x, x 4 ) = 1 3 d(x, fx). 70 i̇lker s. ahin & mustafa telci cubo 16, 1 (2014) thus, inequalities (3) and (4) are satisfied and also x = 0 is a common fixed point of f and g. remark 4.1. inequalities (1) and (2) of theorem 4.1 obviously imply inequalities (8) and (9) of corollary 4.2. in general, inequalities (8) and (9) do not imply inequalities (1) and (2). example 4.2. let x = r and d(x, y) = |x − y|. define the self-mappings f, g : x → x by fx = 1 2 x and gx = 1 4 x for all x in x. then f and g satisfy inequalities (8) and (9). but, for x = 0 and y ∈ r (y 6= 0) we get d(fg(0), g(y)) = 1 4 |y| 6≤ α 1 4 |y| = αd(0, g(y)), d(gf(0), f(y)) = 1 2 |y| 6≤ β 1 2 |y| = βd(0, f(y)) where 0 ≤ α, β < 1. therefore inequalities (1) and (2) are not satisfied. the following theorems were proved in [4]. theorem 4.3. let (x, d) be a complete cone metric space, p be a normal cone with normal constant k. suppose the mapping t : x → x satisfies the contractive condition d(tx, ty) ≤ kd(x, y), (10) for all x, y ∈ x, where k ∈ [0, 1) is a constant. then t has a unique fixed point in x. theorem 4.4. let (x, d) be a complete cone metric space, p be a normal cone with normal constant k. suppose the mapping t : x → x satisfies the contractive condition d(tx, ty) ≤ k(d(tx, x) + d(ty, y)), (11) for all x, y ∈ x, where k ∈ [0, 1 2 ) is a constant. then, t has a unique fixed point in x. theorem 4.5. let (x, d) be a complete cone metric space , p be a normal cone with normal constant k. suppose the mapping t : x → x satisfies the contractive condition d(tx, ty) ≤ k(d(tx, y) + d(x, ty)), (12) for all x, y ∈ x, where k ∈ [0, 1 2 ) is a constant. then, t has a unique fixed point in x. note that rezapour and hamlbarani also proved these theorems by omitting the normality condition, see [9]. remark 4.2. inequalities (10), (11) and (12) obviously imply inequality (7) of corollary 4.1. in general, this inequality do not imply inequalities (10), (11) and (12). thus, it is obvious that cubo 16, 1 (2014) a common fixed point theorem for pairs . . . 71 corollary 4.1 that is a generalization of theorem 4.3. if t is continuous in theorem 4.4 and theorem 4.5, then corollary 4.1 is also a generalization of theorem 4.4 and theorem 4.5. example 4.3. let e = r2, p = {(x, y) ∈ e : x, y ≥ 0} ⊂ r2, x = r and the mapping d : x × x → e defined by d(x, y) = ( |x − y|, |x − y|). define f : x → x by fx = { 0 if x ≤ 0 x if x > 0 for all x in x. then we have, d(f2x, fx) = d(0, 0) = (0, 0) = kd(fx, x) for x ≤ 0 and d(f2x, fx) = d(x, x) = (0, 0) = kd(fx, x) for x > 0 where k ∈ [0, 1). thus, inequality (7) is satisfied and also each x ∈ [0, ∞) is a fixed point of f. now let x > 0, y > 0 and x 6= y. then inequalities (10), (11) and (12) are not satisfied. in fact, if inequality (10) hold for x > 0 and y > 0 (x 6= y) where 0 ≤ k < 1, then we have d(fx, fy) = d(x, y) = (|x − y|, |x − y|) ≤ kd(x, y) = k(|x − y|, |x − y|), and so 1 < k. this is a contradiction because of 0 ≤ k < 1. if inequality (11) hold for x > 0 and y > 0 (x 6= y) where 0 ≤ k < 1 2 , then we have d(fx, fy) = d(x, y) = (|x − y|, |x − y|) ≤ k(d(fx, x) + d(fy, y)) = k(d(x, x) + d(y, y)) = k(0, 0), and so this is a contradiction. if inequality (12) hold for x > 0 and y > 0 (x 6= y) where 0 ≤ k < 1 2 , then we have d(fx, fy) = d(x, y) = (|x − y|, |x − y|) ≤ k(d(fx, y) + d(fy, x)) = k(d(x, y) + d(y, x)) = 2k(|x − y|, |x − y|), and so 1 2 ≤ k. this is a contradiction because of 0 ≤ k < 1 2 . received: september 2011. accepted: november 2012. 72 i̇lker s. ahin & mustafa telci cubo 16, 1 (2014) references [1] abbas, m. and jungck, g., common fixed point results for noncommuting mappings without continuity in cone metric spaces, j. math. anal. appl., 341 (2008), 416-420. [2] abdeljawad, t., karapinar, e. and taş, k, common fixed point theorems in cone banach spaces , hacet. j. math. stat., 40 (2011), 211-217. [3] di bari, c. and vetro p., ϕpairs and common fixed points in cone metric spaces, rend. circ. mat. palermo, 57 (2008), 279-285. [4] huang long-guang and zhang xian., cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl., 332 (2007), 1468-1476. [5] ilić, d. and rakočević, v., common fixed points for maps on cone metric space, j. math. anal. appl., 341 (2008), 876-882. [6] iseki, k., common fixed point theorem of contraction mappings, math. sem. notes kobe univ., 2 (1974), 7-10. [7] janković, s., kadelburg, z. and radenović, s., on cone metric spaces: a survey, nonlinear analysis, 74 (2011), 2591-2601. [8] raja, p. and vaezpour, s. m., some extensions of banach’s contraction principle in complete cone metric spaces, fixed point theory and applications, article id 768294 (2008), 11 pages. [9] rezapour, sh. and hamlbarani, r., some notes on the paper ”cone metric spaces and fixed point theorems of contractive mappings”, j. math. anal. appl., 345 (2008), 719-724. [10] s.ahi̇n, i̇. and telci̇, m., fixed points of contractive mappings on complete cone metric spaces, hacet. j. math. stat., 38 (2009), 59-67. [11] s.ahi̇n, i̇. and telci̇, m., a theorem on common fixed points of expansion type mappings in cone metric spaces, an. şt. univ. ovidius constanţa 18(1) (2010), 329-336. [12] zabrejko, p. p., k-metric and k-normed spaces: a survay, collect. math., 48 (4-6) (1997) 825-859. [13] zeidler, e., nonlinear functional analysis and its applications, volume i, fixed point theorems, springer-verlag, 1993. cubo a mathematical journal vol.14, no¯ 02, (01–13). june 2012 the ǫ−optimality conditions for multiple objective fractional programming problems for generalized (ρ, η)−invexity of higher order ram u. verma international publications, 3400 s brahma blvd, suite 31b, kingsville, texas 78363, usa email: verma99@msn.com abstract motivated by the recent investigations in literature, a general framework for a class of (ρ, η)−invex n-set functions of higher order is introduced, and then some results on the ǫ−optimality conditions for multiple objective fractional subset programming are explored. the obtained results are general in nature, while generalize and unify results on generalized invexity as well as on generalized invexity of higher order to the context of multiple fractional programming. resumen motivado por investigaciones recientes en la literatura, se introduce un marco general para una clase de funciones (ρ, η)−invex n-set de orden superior y se exploran algunos resultados sobre condiciones de épsilon-optimalidad para objetivos múltiples fraccionales de subconjuntos de programación. los resultados obtenidos son de naturaleza general, dado que generalizan y unifican resultados sobre invexity generalizada e invexity generalizada de orden superior en el contexto de la programación múltiple fraccionaria. keywords and phrases: generalized invexity of higher order, multiple objective fractional subset programming, ǫ−efficient solution, semi-parametric sufficient ǫ−optimality conditions. 2010 ams mathematics subject classification: 49j40, 90c25. 2 ram u. verma cubo 14, 2 (2012) 1 introduction recently, kim et al. [9] investigated some results based on ǫ−optimality conditions for multiple objective fractional optimization problems. they used both approaches of parametric as well as non-parametric sufficient conditions to achieving an equivalence between them. motivated by these developments, we examine some ǫ−optimality conditions for multiple objective fractional programming problems based on a generalized (ρ, η)−invexity for higher order [1,7,8] of n-set functions, more specifically, results on parametric and semi-parametric sufficient ǫ−efficiency conditions for multiobjective fractional subset programming. more recently, mishra et al. [11] published some results on optimality conditions for multiple objective fractional subset programming with invex and related non-convex n-set functions (also studied by verma [13,15] and zalmai [16]) to the case of parametric and semi-parametric sufficient efficiency conditions for a multiobjective fractional subset programming problem. jeyakumar et al. [5,6] and kim et al. [9] investigated some results on ǫ−optimality conditions for multiobjective fractional programming problems. we present using the generalized invexity of higher order for differentiable functions, the following multiple objective fractional subset programming problem: (p) minimize ( f1(s) g1(s) , f2(s) g2(s) , ..., fp(s) gp(s) ) subject to hj(s) ≤ 0 for j ∈ {1, ..., m}, s ∈ q = {x ∈ x : hj(x) ≤ 0, j ∈ {1, ..., m}}, where x is an open convex subset of ℜn (n-dimensional euclidean space), fi, gi, i ∈ {1, ..., p} and hj for j ∈ {1, ..., m} are real-valued functions defined on x and gi(s) > 0 for each i ∈ {1, ..., p} and for all s ∈ x. next, we observe that problem (p) is equivalent to the parametric multiobjective non-fractional programming problem: (pλ) minimize (f1(s) − λ1g1(s), ..., fp(s) − λpgp(s)), where λi, i = 1, 2, ..., p are parameters, and s ∈ q. mishra et al. [11] investigated several parametric and semi-parametric sufficient conditions for the multiobjective fractional subset programming problems based on generalized invexity assumptions. moreover, these results are also applicable to other classes of problems with multiple, fractional, and conventional objective functions. furthermore, among other results, the obtained results generalize the recent results on generalized cubo 14, 2 (2012) the ǫ−optimality conditions ... 3 invexity to the case of the generalized invexity of higher order m ≥ 1 relating to the case of semiparametric sufficient ǫ−efficiency conditions for the multiobjective fractional subset programming problems. for more details, we refer the reader [1–17]. 2 generalized invexities of higher order in this section, we develop some concepts and notations for the problem on hand. let x be an open convex subset of ℜn (n-dimensional euclidean space). let 〈·, ·〉 the inner product, and let η : x × x → ℜn be a vector-valued function. suppose that ▽f(x)denotes the gradient of f at x defined by ▽f(x) = ( ∂f(x) ∂x1 , ..., ∂f(x) ∂xn ), where f : x → ℜn is real-valued function on x. next, we recall the notions of the generalized invexity. let s, s∗ ∈ x, let the function f : x → ℜn with components fi for i ∈ {1, ..., n}, be differentiable at s ∗. definition 2.1. a differentiable function f : x → ℜn is said to be (ρ, η)−invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, f(s) − f(s∗) ≥ 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m, where m ≥ 1 is an integer. definition 2.2. a differentiable function f : x → ℜn is said to be (ρ, η)−strictly-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, f(s) − f(s∗) > 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m, where m ≥ 1 is an integer. definition 2.3. a differentiable function f : x → ℜn is said to be (ρ, η)−quasi-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, f(s) ≤ f(s∗) ⇒ 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m ≤ 0, where m ≥ 1 is an integer. definition 2.4. a differentiable function f : x → ℜn is said to be (ρ, η)−prestrictly-quasi-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each 4 ram u. verma cubo 14, 2 (2012) s∗ ∈ x, and ρ > 0, f(s) < f(s∗) ⇒ 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m ≤ 0, where m ≥ 1 is an integer. definition 2.5. a differentiable function f : x → ℜn is said to be (ρ, η)−pseudo-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m ≥ 0 ⇒ f(s) ≥ f(s∗), where m ≥ 1 is an integer. definition 2.6. a differentiable function f : x → ℜn is said to be (ρ, η)−strictly-pseudo-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m ≥ 0 ⇒ f(s) > f(s∗), where m ≥ 1 is an integer. definition 2.7. a differentiable function f : x → ℜn is said to be (ρ, η)−prestrictly-pseudo-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m > 0 ⇒ f(s) ≥ f(s∗), where m ≥ 1 is an integer. definition 2.8. a differentiable function f : x → ℜn is said to be (ρ, η)−strictly-quasi-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, f(s) ≤ f(s∗) ⇒ 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m < 0, where m ≥ 1 is an integer. definition 2.9. a differentiable function f : x → ℜn is said to be (ρ, η)−prestrictly-quasi-invex of higher order at s∗ if there exists a vector-valued function η : x × x → ℜn such that for each s∗ ∈ x, and ρ > 0, f(s) < f(s∗) ⇒ 〈▽f(s∗), η(s, s∗)〉 + ρ‖s − s∗‖m ≤ 0, cubo 14, 2 (2012) the ǫ−optimality conditions ... 5 where m ≥ 1 is an integer. now we introduce the generalized ǫ−solvability conditions for (p) and (pλ) problems as follows: s∗ is a generalized ǫ−efficient solution to (p) if there does not exist an s ∈ q such that fi(s) gi(s) ≤ fi(s ∗) gi(s ∗) − ǫi(s ∗) ∀ i = 1, .., p, fj(s) gj(s) < fj(s ∗) gj(s ∗) − ǫj(s ∗) for some j ∈ {1, .., p}, where ǫi,ǫj : ℜ n → ℜ are with ǫ(s∗) = (ǫ1(s ∗), ..., ǫp(s ∗)), ǫi ≥ 0 for i=1,...,p. for ǫ = ǫ(s∗), (p) reduces to kim et al. [9], and for ǫ = 0, it reduces to the case that s∗ ∈ q is an efficient solution to (p) if there exists no s ∈ q such that fi(s) gi(s) ≤ fi(s ∗) gi(s ∗) ∀ i = 1, ..., p. to this context, based on mishra et al. [11], we consider the following auxiliary problem: (pλ) minimizes∈q(f1(s) − λ1g1(s), ..., fp(s) − λpgp(s)), where λi for i ∈ {1, ..., p} are parameters. next, we introduce the generalized ǫ−solvability conditions for (pλ) problem as follows: s∗ is a generalized ǫ−efficient solution to (pλ) if there does not exist an s ∈ q such that fi(s) − λigi(s) ≤ fi(s ∗) − λigi(s ∗) − ǭi(s ∗) ∀ i = 1, .., p, fj(s) − λjgj(s) < fj(s ∗ ) − λjgj(s ∗ ) − ǭj(s ∗ ) for some j ∈ {1, .., p}, where λi = fi(s ∗ ) gi(s ∗) − ǫi, ǭi(s ∗) = ǫi(s ∗)gi(s ∗), and ǭj(s ∗) = ǫj(s ∗)gj(s ∗), where ǫi,ǫj : ℜ n → ℜ are with ǫ(s∗) = (ǫ1(s ∗), ..., ǫp(s ∗)), ǫi ≥ 0 for i=1,...,p. for ǫ = ǫ(s∗), (p) reduces to kim et al. [9], and for ǫ = 0, it reduces to the case that is an efficient solution to (p) if there exists no s ∈ ξ such that ( f1(s) g1(s) , f2(s) g2(s) , ..., fp(s) gp(s) ) ≤ ( f1(s ∗) g1(s ∗) , f2(s ∗) g2(s ∗) , ..., fp(s ∗) gp(s ∗) ). lemma 2.1. [9] let s∗ ∈ q = {x ∈ x : hj(x) ≤ 0 for i = 1, ..., m}, where hj : x → ℜ is a real-valued function on x. then the following statements are mutually equivalent: (i) s∗ is a generalized ǫ(s∗)−efficient solution to (p). 6 ram u. verma cubo 14, 2 (2012) (ii) s∗ is a generalized ǫ∗(s∗)−solution to (pλ), where λ = ( f1(s ∗) g1(s ∗) − ǫ1(s ∗ ), ..., fp(s ∗) gp(s ∗) − ǫp(s ∗ )) and ǫ∗(s∗) = (ǫ1(s ∗)g1(s ∗), ..., ǫp(s ∗)gp(s ∗)). lemma 2.2. [15] let s∗ ∈ q={x ∈ x : hj(x) ≤ 0 for i = 1, ..., m}, where hj : x → ℜ is a real-valued function on x. then the following statements are mutually equivalent: (i) s∗ is a generalized ǫ(s∗)−efficient solution to (p). (ii) there exists s ∈ q such that σ p i=1 [fi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗ ))gi(s)] ≥ 0. lemma 2.3. [15] let s∗ ∈ q = {x ∈ x : hj(x) ≤ 0 for i = 1, ..., m}, where hj : x → ℜ is a real-valued function on x. then the following statements are mutually equivalent: (i) s∗ is a generalized ǫ(s∗)−efficient solution to (pλ). (ii) there exists s ∈ q such that σ p i=1 [fi(s)−( fi(s ∗) gi(s) −ǫi(s ∗))gi(s)] ≥ σ p i=1 [fi(s ∗)−( fi(s ∗) gi(s ∗) −ǫi(s ∗))gi(s ∗)]−σ p i=1 ǫi(s ∗)gi(s ∗). 3 parametric sufficient ǫ− optimality conditions this section deals with some parametric sufficient ǫ− optimality conditions for problem (p) under the generalized frameworks for generalized (ρ, η)−invexity of higher order m ≥ 1. we begin with real-valued functions ai(.; λ, u) and bj(., v) defined by ai(.; λ, u) = ui[fi(s) − λigi(s)] for i = 1, ..., p, and for fixed λ, u and v and bj(., v) = vjhj(s), j = 1, ..., m. theorem 3.1. let s∗ ∈ q = {s ∈ x : hj(s) ≤ 0 for j ∈ {1, ..., m}, the feasible set of (p). let fi, gi, i ∈ {1, ..., p}, and hj, j ∈ {1, ..., m}, be differentiable at s ∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ ℜp : u > 0, σ p i=1 ui = 1} and v ∗ ∈ rm+ such that 〈σ p i=1 u∗i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )] + σmj=1v ∗ j ▽ hj(s ∗ ), η(s, s∗)〉 ≥ 0 ∀s ∈ q, (3.1) cubo 14, 2 (2012) the ǫ−optimality conditions ... 7 fi(s ∗) − λ∗i gi(s ∗) = 0 for i ∈ {1, ..., p}, (3.2) v∗j hj(s ∗ ) = 0 for j ∈ {1, ..., m}, (3.3) where λ∗i = ( fi(s ∗ ) gi(s ∗) − ǫi(s ∗)). suppose, in addition, that any one of the following assumptions holds: (i) ai(.; λ ∗, u∗) (∀i = 1, ..., p) are(ρ, η)−pseudo-invex at s∗ of higher order and bj(., v ∗) ∀j ∈ {1, ..., m} are (ρ, η)−quasi-invex at s∗ of higher order. (ii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at s∗ of higher order and bj(., v ∗) ∀j ∈ {1, ..., m} are strictly-quasi-invex at s∗ of higher order. (iii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-quasi-invex at s∗ of higher order and bj(., v ∗) ∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at s∗ of higher order. then s∗ is an ǫ−efficient solution to (p). proof. if (i) holds, and if s∗ ∈ q, then it follows from (3.1) that 〈σ p i=1 u∗i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )], η(s, s∗〉 + 〈σmj=1v ∗ j ▽ hj(s ∗), η(s, s∗)〉 ≥ 0 ∀s ∈ q. (3.4) since v∗ ≥ 0, s ∈ q and (3.3) holds, we have σmj=1v ∗ j hj(s) ≤ 0 = σ m j=1v ∗ j hj(s ∗), and in light of the (ρ, η)−quasi-invexity of bj(., v ∗) at s∗, we arrive at 〈σmj=1v ∗ j ▽ hj(s ∗), η(s, s∗)〉 ≤ −ρ‖s − s∗‖m. (3.5) it follows from (3.4) and (3.5) that 〈σ p i=1 u∗i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )], η(s, s∗〉 ≥ ρ‖s − s∗‖m. (3.6) this further implies 〈σ p i=1 u∗i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )], η(s, s∗〉 ≥ −ρ‖s − s∗‖m. (3.7) next, applying the (ρ, η)−pseudo-invexity at s∗ to (3.6), we have σ p i=1 u∗i [fi(s) − λ ∗ i gi(s)] ≥ σ p i=1 u∗i [fi(s ∗ ) − λ∗i gi(s ∗ )], 8 ram u. verma cubo 14, 2 (2012) that is equivalent to σ p i=1 u∗i [fi(s) − λ ∗ i gi(s)] ≥ σ p i=1 u∗i [fi(s ∗) − λ∗i gi(s ∗)] − σ p i=1 u∗i ǫi(s ∗)gi(s ∗) = 0, where λ∗i = ( fi(s ∗ ) gi(s ∗) − ǫi(s ∗)). thus, we have σ p i=1 u∗i [fi(s) − λ ∗ i gi(s)] ≥ 0. (3.8) since u∗i > 0 for each i ∈ {1, ..., p}, we conclude that there does not exist an s ∈ q such that fi(s) gi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗)) ≤ 0 ∀ i = 1, ..., p, fj(s) gj(s) − ( fj(s ∗) gj(s ∗) − ǫj(s ∗ )) < 0 ∀ j ∈ {1, ..., p}. hence, s∗ is an ǫ−efficient solution to (p). similar proofs hold for (ii) and (iii). when m=2, we have theorem 3.2. let s∗ ∈ q = {s ∈ x : hj(s) ≤ 0 for j ∈ {1, ..., m}, the feasible set of (p). let fi, gi, i ∈ {1, ..., p}, and hj, j ∈ {1, ..., m}, be differentiable at s ∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ ℜp : u > 0, σ p i=1ui = 1} and v ∗ ∈ rm+ such that 〈σ p i=1 u∗i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )] + σmj=1v ∗ j ▽ hj(s ∗ ), η(s, s∗)〉 ≥ 0 ∀s ∈ q, (3.9) fi(s ∗) − λ∗i gi(s ∗) = 0 for i ∈ {1, ..., p}, (3.10) v∗j hj(s ∗ ) = 0 for j ∈ {1, ..., m}, (3.11) where λ∗i = ( fi(s ∗ ) gi(s ∗) − ǫi(s ∗)). suppose, in addition, that any one of the following assumptions holds: (i) ai(.; λ ∗, u∗) (∀i = 1, ..., p) are (ρ, η)−pseudo-invex at s∗ and bj(., v ∗) ∀j ∈ {1, ..., m} are (ρ, η)−quasi-invex at s∗. (ii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at s∗ and bj(., v ∗) ∀j ∈ {1, ..., m} are strictly-quasi-invex at s∗. cubo 14, 2 (2012) the ǫ−optimality conditions ... 9 (iii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-quasi-invex at s∗ and bj(., v ∗) ∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at s∗. then s∗ is an ǫ−efficient solution to (p). for ǫ = 0, we have theorem 3.3. let s∗ ∈ q, let fi, gi, i ∈ {1, ..., p}, and hj, j ∈ {1, ..., m}, be differentiable at s∗ ∈ λ, and let there exist u∗ ∈ u = {u ∈ ℜp : u > 0, σ p i=1 ui = 1} and v ∗ ∈ rm+ such that 〈σ p i=1u ∗ i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )] + σmj=1v ∗ j ▽ hj(s ∗ ), η(s, s∗)〉 + ρ‖s − s∗‖2 ≥ 0 ∀s ∈ λn, (3.12) fi(s ∗) − λ∗i gi(s ∗) = 0 for i ∈ {1, ..., p}, (3.13) v∗j hj(s ∗) = 0 for j ∈ {1, ..., m}. (3.14) suppose, in addition, that any one of the following assumptions holds: (i) ai(.; λ ∗, u∗) (∀i = 1, ..., p) are (ρ, η)−pseudo-invex at s∗ of higher order and bj(., v ∗) (∀j ∈ {1, ..., m} are (ρ, η)−quasi-invex at s∗ of order. (ii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at s∗ of higher order and bj(., v ∗) (∀j ∈ {1, ..., m} are(ρ, η)−strictly-quasi-invex at s∗ of higher order. (iii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-quasi-invex at s∗ of higher order and bj(., v ∗) (∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at s∗ of higher order. then s∗ is an efficient solution to (p). theorem 3.4. ([11], theorem 3.1) let s∗ ∈ q, let fi, gi, i ∈ {1, ..., p}, and hj, j ∈ {1, ..., m}, be differentiable at s∗ ∈ q, and let there exist u∗ ∈ u = {u ∈ ℜp : u > 0, σ p i=1ui = 1} and v ∗ ∈ rm+ such that 〈σ p i=1u ∗ i [▽fi(s ∗ ) − λ∗i ▽ gi(s ∗ )] + σmj=1v ∗ j ▽ hj(s ∗ ), η(s, s∗)〉 + ρ‖s − s∗‖2 ≥ 0 ∀s ∈ λn, (3.15) 10 ram u. verma cubo 14, 2 (2012) fi(s ∗) − λ∗i gi(s ∗) = 0 for i ∈ {1, ..., p}, (3.16) v∗j hj(s ∗ ) = 0 for j ∈ {1, ..., m}. (3.17) suppose, in addition, that any one of the following assumptions holds: (i) ai(.; λ ∗, u∗) (∀i = 1, ..., p) arepseudo-invex at s∗ and bj(., v∗) (∀j ∈ {1, ..., m} are quasiinvex at s∗. (ii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are prestrictly-pseudo-invex at s∗ and bj(., v ∗) (∀j ∈ {1, ..., m} are strictly-quasi-invex at s∗. (iii) ai(.; λ ∗, u∗) (∀i ∈ {1, ..., p} are prestrictly-quasi-invex at s∗ and bj(., v ∗) (∀j ∈ {1, ..., m} are strictly-pseudo-invex at s∗. then s∗ is an efficient solution to (p). 4 semi-parametric sufficient ǫ− optimality conditions this section deals with some semi-parametric sufficient ǫ− optimality conditions for problem (p) under the generalized frameworks for generalized invexity. we start with real-valued functions ei(., s ∗, u∗), bj(., v), and hi(., s ∗, u∗, v∗) defined by ei(s, s ∗, u∗) = ui[fi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s)] for i ∈ {1, ..., p}, li(s, s ∗, u∗, v∗) = u∗i [fi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s)] + σj∈j0v ∗ j hj(s) for i ∈ {1, ..., p}, and bj(., v) = vjhj(s), j = 1, ..., m. theorem 4.1. let s∗ ∈ q = {s ∈ x : hj(s) ≤ 0 for j ∈ {1, ..., m}, the feasible set of (p). let fi, gi, i ∈ {1, ..., p}, and hj, j ∈ {1, ..., m}, be differentiable at s ∗ ∈ q, and let there exist cubo 14, 2 (2012) the ǫ−optimality conditions ... 11 u∗ ∈ u = {u ∈ ℜp : u > 0, σ p i=1 ui = 1} and v ∗ ∈ rm+ such that 〈σ p i=1 u∗i [▽fi(s ∗) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗)) ▽ gi(s ∗)] + σj∈j0v ∗ j ▽ hj(s ∗), η(s, s∗)〉 ≥ 0 , (4.1) and v∗j hj(s ∗) = 0 for j ∈ {1, ..., m}. (4.2) suppose, in addition, that any one of the following assumptions holds: (i) ei(.; s ∗, u∗) (∀i = 1, ..., p) are (ρ, η)−pseudo-invex at s∗ of higher order and bj(., v ∗) (∀j ∈ {1, ..., m} are (ρ, η)−quasi-invex at s∗ of higher order. (ii) ei(.; s ∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at s∗ of higher order and bj(., v ∗) (∀j ∈ {1, ..., m} are(ρ, η)−strictly-quasi-invex at s∗ of higher order. (iii) ei(.; s ∗, u∗) (∀i ∈ {1, ..., p} are(ρ, η)−prestrictly-quasi-invex at s∗ of higher order and bj(., v ∗) (∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at s∗ of higher order. then s∗ is an ǫ−efficient solution to (p). proof. if (i) holds, and if s ∈ q, then it follows from (4.1) that 〈σ p i=1 u∗i [▽fi(s ∗ ) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗ )) ▽ gi(s ∗ )], η(s, s∗〉 + 〈σmj=1v ∗ j ▽ hj(s ∗), η(s, s∗)〉 ≥ 0 ∀s ∈ λn. (4.3) since v∗ ≥ 0, s ∈ q and (4.2) holds, we have σmj=1v ∗ j hj(s) ≤ 0 = σ m j=1v ∗ j hj(s ∗), and in light of the (ρ, η)−quasi-invexity of bj(., v ∗) at s∗, we arrive at 〈σmj=1v ∗ j ▽ hj(s ∗), η(s, s∗)〉 ≤ −ρ‖s − s∗‖m. (4.4) it follows from (4.3) and (4.4) that 〈σ p i=1 u∗i [▽fi(s ∗) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗)) ▽ gi(s ∗)], η(s, s∗〉 ≥ ρ‖s − s∗‖m. (4.5) next, applying the (ρ, η)−pseudo-invexity at s∗ to (4.5), we have σ p i=1 u∗i [fi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s)] ≥ σ p i=1 u∗i [fi(s ∗) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s ∗)], 12 ram u. verma cubo 14, 2 (2012) that is equivalent to σ p i=1 u∗i [fi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s)] ≥ σ p i=1 u∗i [fi(s ∗) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s ∗)] − σ p i=1 u∗i ǫigi(s ∗) = 0. thus, we have σ p i=1 u∗i [fi(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗))gi(s)] ≥ 0. (4.6) since u∗i > 0 for each i ∈ {1, ..., p}, we conclude that there does not exist an s ∈ q such that f1(s) g1(s) − ( fi(s ∗) gi(s ∗) − ǫi(s ∗)) ≤ 0 ∀ i = 1, ..., p, fj(s) gj(s) − ( fj(s ∗) gj(s ∗) − ǫj(s ∗)) < 0 ∀ j ∈ {1, ..., p}. hence, s∗ is an ǫ−efficient solution to (p). similar proofs hold for (ii) and (iii). received: august 2011. revised: august 2011. references [1] k.d. bae and d.s. kim, optimality and duality theorems in nonsmooth multiobjective optimization, fixed point theory & applications 2011,2011:42 doi:10.1186/1687-1812-2011-42 [2] l. caiping and y. xinmin, generalized (ρ, θ, η)−invariant monotonicity and generalized (ρ, θ, η)−invexity of non-differentiable functions, journal of inequalities and applications vol. 2009(2009), article id # 393940, 16 pages. [3] h.w. corley, optimization theory for n-set functions, journal of mathematical analysis and applications 127(1987), 193-205. 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[9] m.h. kim, g.s. kim and g.m. lee, on ǫ−optimality conditions for multiobjective fractional optimization problems, fpta 2011:6 doi:10.1186/1687-1812-2011-6. [10] l.j. lin, on the optimality conditions for vector-valued n-set functions, journal of mathematical analysis and applications 161(1991), 367–387. [11] s.k. mishra, m. jaiswal, and pankaj, optimality conditions for multiple objective fractional subset programming with invex an related non-convex functions, communications on applied nonlinear analysis 17(3)(2010), 89–101. [12] s.k. mishra, s.y. wang and k.k. lai, generalized convexity and vector optimization, nonconvex optimization and its applications, vol. 19, springer-verlag, 2009. [13] r.u. verma, general parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized (ρ, η, a)−invexity, numerical algebra, control & optimization, in press. 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[17] e. zeidler, nonlinear functional analysis and its applications iii, springer-verlag, new york, new york, 1985. introduction generalized invexities of higher order parametric sufficient optimality conditions semi-parametric sufficient optimality conditions cubo a mathematical journal vol.16, no¯ 01, (95–104). march 2014 lp local uncertainty inequality for the sturm-liouville transform fethi soltani 1 department of mathematics, faculty of science, jazan university, p.o.box 114, jazan, kingdom of saudi arabia. fethisoltani10@yahoo.com abstract in this paper, we give analogues of local uncertainty inequality for the sturm-liouville transform on [0, ∞[. a generalization of donoho-stark’s uncertainty principle is obtained for this transform. resumen en este art́ıculo entregamos resultados análogos de una desigualdad de incertidumbre local de la transformada sturm-liouville en [0, ∞[. una generalización del principio de incertidumbre de donoho-stark se obtiene de esta transformación. keywords and phrases: sturm-liouville transform; local uncertainty principle; donoho-stark’s uncertainty principle. 2010 ams mathematics subject classification: 42b10; 44a20; 46g12. 1author partially supported by dgrst project 04/ur/15-02 and cmcu program 10g 1503 96 fethi soltani cubo 16, 1 (2014) 1 introduction we consider the second-order differential operator defined on ]0, ∞[ by ∆u := u′′ + a′ a u′ + ρ2u, where a is a nonnegative function satisfying certain conditions and ρ is a nonnegative real number. this operator plays an important role in analysis. for example, many special functions (orthogonal polynomials) are eigenfunctions of an operator of ∆ type. the radial part of the beltrami-laplacian in a symmetric space is also of ∆ type. many aspects of such operators have been studied [2, 10, 17, 18, 19]. in particular, the two references [2, 17] investigate standard constructions of harmonic analysis, such as translation operators, convolution product, and fourier transform, in connection with ∆. many uncertainty principles have already been proved for the sturm-liouville operarator ∆, namely by rösler and voit [14] who established an uncertainty principle for hankel transforms. bouattour and trimèche [1] proved a beurling’s theorem for the sturm-liouville transform. daher et al. [3, 4, 5] give some related versions of the uncertainty principle for the sturm-liouville transform (hardy’s theorem and miyachi’s theorem). ma [9] proved a heisenberg uncertainty principle for the sturm-liouville transform. building on the ideas of faris [7] and price [12, 13], we show a local uncertainty principle for the sturm-liouville transform f. more precisely, we will show the following result. if 1 < p ≤ 2, q = p/(p − 1) and 0 < a < (2α + 2)/q, there is a constant k(a) such that for every f ∈ lp(µ) and every measurable subset e ⊂ [0, ∞[ such that 0 < ν(e) < ∞, (∫ e |f(f)(λ)|qdν(λ) )1/q ≤ k(a) ( ν(e) ) a 2α+2 ‖xaf‖lp(µ), (1.1) where µ is the measure given by dµ(x) := a(x)dx, and ν is the plancherel measure associated to f. (for more details see the next section.) this inequality generalizes the local uncertainty principle for the hankel transform given by ghobber et al. [8] and omri [11]. we shall use the local uncertainty principle (1.1); and building on the techniques of donoho and stark [6], we show a continuous-time principles for the lp theory, when 1 < p ≤ 2. this paper is organized as follows. in section 2 we list some basic properties of the sturmliouville transform f (plancherel theorem, inversion formula,...). in section 3 we show a local uncertainty principle for the sturm-liouville f. the section 4 is devoted to donoho-stark’s uncertainty principle for the sturm-liouville transform f in the lp theory, when 1 < p ≤ 2. cubo 16, 1 (2014) lp local uncertainty inequality for the sturm-liouville transform 97 2 the sturm-liouville transform f we consider the second-order differential operator ∆ defined on ]0, ∞[ by ∆u := u′′ + a′ a u′ + ρ2u, where ρ is a nonnegative real number and a(x) := x2α+1b(x), α > −1/2, for b a positive, even, infinitely differentiable function on r such that b(0) = 1. moreover we assume that a and b satisfy the following conditions: (i) a is increasing and lim x→∞ a(x) = ∞. (ii) a′ a is decreasing and lim x→∞ a′(x) a(x) = 2ρ. (iii) there exists a constant δ > 0 such that a′(x) a(x) = 2ρ + d(x) exp(−δx) if ρ > 0, a′(x) a(x) = 2α + 1 x + d(x) exp(−δx) if ρ = 0, where d is an infinitely differentiable function on ]0, ∞[, bounded and with bounded derivatives on all intervals [x0, ∞[, for x0 > 0. this operator was studied in [2, 10, 17], and the following results have been established: (i) for all λ ∈ c, the equation { ∆u = −λ2u u(0) = 1, u′(0) = 0 admits a unique solution, denoted by ϕλ, with the following properties: • for x ≥ 0, the function λ → ϕλ(x) is analytic on c; • for λ ∈ c, the function x → ϕλ(x) is even and infinitely differentiable on r; • for all λ, x ∈ r, |ϕλ(x)| ≤ 1. (2.1) (ii) for nonzero λ ∈ c, the equation ∆u = −λ2u has a solution φλ satisfying φλ(x) = 1 √ a(x) exp(iλx)v(x, λ), with limx→∞ v(x, λ) = 1. consequently there exists a function (spectral function) λ 7→ c(λ), 98 fethi soltani cubo 16, 1 (2014) such that ϕλ = c(λ)φλ + c(−λ)φ−λ for nonzero λ ∈ c. moreover there exist positive constants k1, k2 and k such that k1|λ| 2α+1 ≤ |c(λ)|−2 ≤ k2|λ| 2α+1 for all λ such that imλ ≤ 0 and |λ| ≥ k. notation 2.1. we denote by • µ the measure defined on [0, ∞[ by dµ(x) := a(x)dx; and by lp(µ), 1 ≤ p ≤ ∞, the space of measurable functions f on [0, ∞[, such that ‖f‖lp(µ) := (∫ ∞ 0 |f(x)|pdµ(x) )1/p < ∞, 1 ≤ p < ∞, ‖f‖l∞(µ) := ess sup x∈[0,∞[ |f(x)| < ∞; • ν the measure defined on [0, ∞[ by dν(λ) := dλ 2π|c(λ)|2 ; and by lp(ν), 1 ≤ p ≤ ∞, the space of measurable functions f on [0, ∞[, such that ‖f‖lp(ν) < ∞. the fourier transform associated with the operator ∆ is defined on l1(µ) by f(f)(λ) := ∫ ∞ 0 ϕλ(x)f(x)dµ(x) for λ ∈ r. some of the properties of the fourier transform f are collected bellow (see [2, 10, 17, 18]). theorem 2.2. (i) l1 − l∞-boundedness. for all f ∈ l1(µ), f(f) ∈ l∞(ν) and ‖f(f)‖l∞(ν) ≤ ‖f‖l1(µ). (2.2) (ii) inversion theorem. let f ∈ l1(µ), such that f(f) ∈ l1(ν). then f(x) = ∫ ∞ 0 ϕλ(x)f(f)(λ)dν(λ), a.e. x ∈ [0, ∞[. (2.3) (iii) plancherel theorem. the fourier transform f extends uniquely to an isometric isomorphism of l2(µ) onto l2(ν). in particular, ‖f‖l2(µ) = ‖f(f)‖l2(ν). (2.4) using relations (2.2) and (2.4) with marcinkiewicz’s interpolation theorem [15, 16], we deduce that for every 1 ≤ p ≤ 2, and for every f ∈ lp(µ), the function f(f) belongs to the space lq(ν), q = p/(p − 1), and ‖f(f)‖lq(ν) ≤ ‖f‖lp(µ). (2.5) cubo 16, 1 (2014) lp local uncertainty inequality for the sturm-liouville transform 99 3 lp local uncertainty inequality this section is devoted to establish a local uncertainty principle for the sturm-liouville transform f, more precisely, we will show the following theorem. theorem 3.1. if 1 < p ≤ 2, q = p/(p − 1) and 0 < a < (2α + 2)/q, then for all f ∈ lp(µ) and all measurable subset e ⊂ [0, ∞[ such that 0 < ν(e) < ∞, (∫ e |f(f)(λ)|qdν(λ) )1/q ≤ k(a) ( ν(e) ) a 2α+2 ‖xaf‖lp(µ), k(a) = ( qa )− qa 2α+2 ( 2α + 2 − qa ) (q−1)a 2α+2 [ 1 + qa 2α + 2 − qa ( sup x∈[0,r0] b(x) )1/q ] , where r0 = ( qa ) q 2α+2 ( 2α + 2 − qa ) 1−q 2α+2 ( ν(e) )− 1 2α+2 . proof. for r > 0, denote by χe, χ[0,r[ and χ[r,∞[ the characteristic functions. let f ∈ lp(µ), 1 0, ‖f(f)χe‖lq(ν) ≤ ‖f(fχ[0,r[)χe‖lq(ν) + ‖f(fχ[r,∞[)χe‖lq(ν) ≤ ( ν(e) )1/q ‖f(fχ[0,r[)‖l∞(ν) + ‖f(fχ[r,∞[)‖lq(ν); hence it follows from (2.2) and (2.5) that ‖f(f)χe‖lq(ν) ≤ ( ν(e) )1/q ‖fχ[0,r[‖l1(µ) + ‖fχ[r,∞[‖lp(µ). (3.1) on the other hand, by hölder’s inequality, ‖fχ[0,r[‖l1(µ) ≤ ‖x −aχ[0,r[‖lq(µ)‖x af‖lp(µ). by hypothesis a < (2α + 2)/q, ‖x−aχ[0,r[‖lq(µ) ≤ r−a+(2α+2)/q (2α + 2 − qa)1/q ( sup x∈[0,r] b(x) )1/q , and therefore, ‖fχ[0,r[‖l1(µ) ≤ r−a+(2α+2)/q (2α + 2 − qa)1/q ( sup x∈[0,r] b(x) )1/q ‖xaf‖lp(µ). (3.2) moreover, ‖fχ[r,∞[‖lp(µ) ≤ ‖x −aχ[r,∞[‖l∞(µ)‖x af‖lp(µ) ≤ r −a‖xaf‖lp(µ). (3.3) combining the relations (3.1), (3.2) and (3.3), we deduce that ‖f(f)χe‖lq(ν) ≤ [ r−a + ( ν(e) )1/q r−a+(2α+2)/q (2α + 2 − qa)1/q ( sup x∈[0,r] b(x) )1/q ] ‖xaf‖lp(µ). 100 fethi soltani cubo 16, 1 (2014) we choose r = r0 = ( qa ) q 2α+2 ( 2α + 2 − qa ) 1−q 2α+2 ( ν(e) )− 1 2α+2 , we obtain the desired inequality. 2 remark 3.2. (i) the local uncertainty principle for the sturm-liouville transform f generalizes the local uncertainty principle for the hankel transform (see [8, 11]). (ii) if 1 < p ≤ 2 and 0 < a < (2α + 2)/q, where q = p/(p − 1), then for every f ∈ lp(µ), sup e⊂[0,∞[, 0<ν(e)<∞ [ ( ν(e) )− a 2α+2 ‖f(f)χe‖lq(ν) ] ≤ k(a)‖xaf‖lp(µ). the left hand side is known to be an equivalent norm of f(f) in the lorentz-space lpa,q(ν), where pa = q(2α + 2) 2α + 2 − qa . 4 lp donoho-stark uncertainty principle let t and e be measurable subsets of [0, ∞[. we introduce the time-limiting operator pt by pt f := fχt , (4.1) and, we introduce the partial sum operator se by f(sef) = f(f)χe. (4.2) lemma 4.1. if ν(e) < ∞ and f ∈ lp(µ), 1 ≤ p ≤ 2, sef(x) = ∫ e ϕλ(x)f(f)(λ)dν(λ). proof. let f ∈ lp(µ), 1 ≤ p ≤ 2 and let q = p/(p − 1). then by (2.1), hölder’s inequality and (2.5), ‖f(f)χe‖l1(ν) = ∫ e |f(f)(λ)|dν(λ) ≤ ( ν(e) )1/p ‖f(f)‖lq(ν) ≤ ( ν(e) )1/p ‖f‖lp(µ), and ‖f(f)χe‖l2(ν) = (∫ e |f(f)(λ)|2dν(λ) )1/2 ≤ ( ν(e) ) q−2 2q ‖f(f)‖lq(ν) ≤ ( ν(e) ) q−2 2q ‖f‖lp(µ). cubo 16, 1 (2014) lp local uncertainty inequality for the sturm-liouville transform 101 thus f(f)χe ∈ l 1(µ) ∩ l2(µ) and by (4.2), sef = f −1 ( f(f)χe ) . this combined with (2.3) gives the result. 2 let t and e be measurable subsets of [0, ∞[. we say that a function f ∈ lp(µ), 1 ≤ p ≤ 2, is ε-concentrated to t in lp(µ)-norm, if there is a measurable function g(t) vanishing outside t such that ‖f − g‖lp(µ) ≤ ε‖f‖lp(µ). similarly, we say that f(f) is ε-concentrated to e in l q(ν)-norm, q = p/(p−1), if there is a function h(λ) vanishing outside e with ‖f(f)−h‖lq(ν) ≤ ε‖f(f)‖lq(ν). if f is εt -concentrated to t in l p(µ)-norm (g being the vanishing function) then by (4.1), ‖f − pt f‖lp(µ) = (∫ [0,∞[\t |f(t)|pdµ(t) )1/p ≤ ‖f − g‖lp(µ) ≤ εt ‖f‖lp(µ) (4.3) and therefore f is εt -concentrated to t in l p(µ)-norm if and only if ‖f − pt f‖lp(µ) ≤ εt ‖f‖lp(µ). from (4.2) it follows as for pt that f(f) is εe-concentrated to e in l q(ν)-norm, q = p/(p−1), if and only if ‖f(f) − f(sef)‖lq(ν) ≤ εe‖f(f)‖lq(µ). (4.4) let bp(e), 1 ≤ p ≤ 2, be the set of functions f ∈ l p(µ) that are bandlimited to e (i.e. f ∈ bp(e) implies sef = f). the spaces bp(e) satisfy the following property. lemma 4.2. let t and e be measurable subsets of [0, ∞[ such that 0 < ν(e) < ∞. for f ∈ bp(e), 1 < p ≤ 2 and 0 < a < (2α + 2)(1 − 1 p ), ‖ptf‖lp(µ) ≤ k(a) ( µ(t) )1/p( ν(e) ) 1 p + a 2α+2 ‖xaf‖lp(µ), where k(a) is the constant given by theorem 3.1. proof. if µ(t) = ∞, the inequality is clear. assume that µ(t) < ∞. for f ∈ bp(e), 1 < p ≤ 2, from lemma 4.1, f(t) = ∫ e ϕλ(t)f(f)(λ)dν(λ), and by (2.1), hölder’s inequality and theorem 3.1, |f(t)| ≤ ( ν(e) )1/p (∫ e |f(f)(λ)|qdν(λ) )1/q ≤ k(a) ( ν(e) ) 1 p + a 2α+2 ‖xaf‖lp(µ), 102 fethi soltani cubo 16, 1 (2014) where q = p/(p − 1). hence, ‖pt f‖lp(µ) = (∫ t |f(t)|pdµ(t) )1/p ≤ k(a) ( µ(t) )1/p( ν(e) ) 1 p + a 2α+2 ‖xaf‖lp(µ), which yields the result. 2 it is useful to have uncertainty principle for the lp(µ)-norm. theorem 4.3. let t and e be measurable subsets of [0, ∞[ such that 0 < ν(e) < ∞; and let f ∈ bp(e), 1 < p ≤ 2 and 0 < a < (2α + 2)(1 − 1 p ). if f is εt -concentrated to t, then ‖f‖lp(µ) ≤ k(a) 1 − εt ( µ(t) )1/p( ν(e) ) 1 p + a 2α+2 ‖xaf‖lp(µ). proof. let f ∈ bp(e), 1 < p ≤ 2. since f is εt -concentrated to t in l p(µ)-norm, then by (4.3) and lemma 4.2, ‖f‖lp(µ) ≤ εt ‖f‖lp(µ) + ‖pt f‖lp(µ) ≤ εt ‖f‖lp(µ) + k(a) ( µ(t) )1/p( ν(e) ) 1 p + a 2α+2 ‖xaf‖lp(µ). thus, (1 − εt )‖f‖lp(µ) ≤ k(a) ( µ(t) )1/p( ν(e) ) 1 p + a 2α+2 ‖xaf‖lp(µ), which gives the result. 2 another uncertainty principle for the lp(µ) theory is obtained. theorem 4.4. let e be measurable subset of [0, ∞[ such that 0 < ν(e) < ∞; and let f ∈ lp(µ), 1 < p ≤ 2 and 0 < a < (2α + 2)(1 − 1 p ). if f(f) is εe-concentrated to e in l q(ν)-norm, q = p/(p − 1), then ‖f(f)‖lq(ν) ≤ k(a) 1 − εe ( ν(e) ) a 2α+2 ‖xaf‖lp(µ). proof. let f ∈ lp(µ), 1 < p ≤ 2. since f(f) is εe-concentrated to e in l q(ν)-norm, q = p/(p−1), then by (4.4) and theorem 3.1, ‖f(f)‖lq(ν) ≤ εe‖f(f)‖lq(ν) + (∫ e |f(f)(λ)|qdν(λ) )1/q ≤ εe‖f(f)‖lq(ν) + k(a) ( ν(e) ) a 2α+2 ‖xaf‖lp(µ). thus, (1 − εe)‖f(f)‖lq(ν) ≤ k(a) ( ν(e) ) a 2α+2 ‖xaf‖lp(µ), which proves the result. 2 received: march 2013. accepted: september 2013. cubo 16, 1 (2014) lp local uncertainty inequality for the sturm-liouville transform 103 references [1] l. bouattour and k. trimèche, beurling-hörmander’s theorem for the chébli-trimèche transform, glob. j. pure appl. math. 1(3) (2005) 342–357 [2] h. chébli, théorème de paley-wiener associé à un opérateur différentiel singulier sur (0, ∞), j. math. pures appl. 58(1) (1979) 1–19. 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() cubo a mathematical journal vol.13, no¯ 03, (141–152). october 2011 on weak concircular symmetries of trans-sasakian manifolds shyamal kumar hui nikhil banga sikshan mahavidyalaya, bishnupur, bankura – 722 122, west bengal, india. email: shyamal hui@yahoo.co.in abstract the object of the present paper is to study weakly concircular symmetric and weakly concircular ricci symmetric trans-sasakian manifolds. resumen el objeto del presente trabajo es el estudio de variedades simétricas débilmente concirculares y variedades simétricas trans-sasakian débilmente concircular de ricci. keywords. weakly symmetric manifold, weakly concircular symmetric manifold, weakly ricci symmetric manifold, concircular ricci tensor, weakly concircular ricci symmetric manifold, αsasakian manifold, β-kenmotsu manifold, trans-sasakian manifold. mathematics subject classification: 53c15, 53c25. 142 shyamal kumar hui cubo 13, 3 (2011) 1 introduction the notion of weakly symmetric manifolds was introduced by tamássy and binh [9]. a non-flat riemannian manifold (mn, g) (n > 2) is called a weakly symmetric manifold if its curvature tensor r of type (0, 4) satisfies the condition (∇xr)(y, z, u, v) = a(x)r(y, z, u, v) + b(y)r(x, z, u, v) (1.1) + h(z)r(y, x, u, v) + d(u)r(y, z, x, v) + e(v)r(y, z, u, x) for all vector fields x, y, z, u, v ∈ χ(mn); χ(m) being the lie algebra of smooth vector fields of m, where a, b, h, d and e are 1-forms (not simultaneously zero) and ∇ denotes the operator of covariant differentiation with respect to the riemannian metric g. the 1-forms are called the associated 1-forms of the manifold and an n-dimensional manifold of this kind is denoted by (ws)n. in 1999 de and bandyopadhyay [3] studied a (ws)n and proved that in such a manifold the associated 1-forms b = h and d = e. hence (1.1) reduces to the following: (∇xr)(y, z, u, v) = a(x)r(y, z, u, v) + b(y)r(x, z, u, v) (1.2) + b(z)r(y, x, u, v) + d(u)r(y, z, x, v) + d(v)r(y, z, u, x). a transformation of an n-dimensional riemannian manifold m, which transforms every geodesic circle of m into a geodesic circle, is called a concircular transformation [11]. the interesting invariant of a concircular transformation is the concircular curvature tensor c̃, which is defined by [11] c̃(y, z, u, v) = r(y, z, u, v) − r n(n − 1) [ g(z, u)g(y, v) − g(y, u)g(z, v) ] , (1.3) where r is the scalar curvature of the manifold. recently shaikh and hui [7] introduced the notion of weakly concircular symmetric manifolds. a riemannian manifold (mn, g)(n > 2) is called weakly concircular symmetric manifold if its concircular curvature tensor c̃ of type (0, 4) is not identically zero and satisfies the condition (∇xc̃)(y, z, u, v) = a(x)c̃(y, z, u, v) + b(y)c̃(x, z, u, v) (1.4) + h(z)c̃(y, x, u, v) + d(u)c̃(y, z, x, v) + e(v)c̃(y, z, u, x) for all vector fields x, y, z, u, v ∈ χ(mn), where a, b, h, d and e are 1-forms (not simultaneously zero) an n-dimensional manifold of this kind is denoted by (wc̃s)n. also it is shown that [7], in a (wc̃s)n the associated 1-forms b = h and d = e, and hence the defining condition (1.4) of a (wc̃s)n reduces to the following form: (∇xc̃)(y, z, u, v) = a(x)c̃(y, z, u, v) + b(y)c̃(x, z, u, v) (1.5) + b(z)c̃(y, x, u, v) + d(u)c̃(y, z, x, v) + d(v)c̃(y, z, u, x), cubo 13, 3 (2011) on weak concircular symmetries . . . 143 where a, b and d are 1-forms (not simultaneously zero). again tamássy and binh [10] introduced the notion of weakly ricci symmetric manifolds. a riemannian manifold (mn, g) (n > 2) is called weakly ricci symmetric manifold if its ricci tensor s of type (0, 2) is not identically zero and satisfies the condition (∇xs)(y, z) = a(x)s(y, z) + b(y)s(x, z) + d(z)s(y, x), (1.6) where a, b and d are three non-zero 1-forms, called the associated 1-forms of the manifold, and ∇ denotes the operator of covariant differentiation with respect to the metric tensor g. such an n-dimensional manifold is denoted by (wrs)n. let {ei : i = 1, 2, · · · , n} be an orthonormal basis of the tangent space at each point of the manifold and let p(y, v) = n∑ i=1 c̃(y, ei, ei, v), (1.7) then from (1.3), we get p(y, v) = s(y, v) − r n g(y, v). (1.8) the tensor p is called the concircular ricci symmetric tensor [4], which is a symmetric tensor of type (0, 2). in [4] de and ghosh introduced the notion of weakly concircular ricci symmetric manifolds. a riemannian manifold (mn, g)(n > 2) is called weakly concircular ricci symmetric manifold [4] if its concircular ricci tensor p of type (0, 2) is not identically zero and satisfies the condition (∇xp)(y, z) = a(x)p(y, z) + b(y)p(x, z) + d(z)p(y, x), (1.9) where a, b and d are three 1-forms (not simultaneously zero). in [5] oubiña introduced the notion of trans-sasakian manifolds which contains both the class of sasakian and cosympletic structures, and are closely related to the locally conformal kähler manifolds. a trans-sasakian manifold of type (0, 0), (α, 0) and (0, β) are the cosympletic, α-sasakian and β-kenmotsu manifold respectively. in particular, if α = 1, β = 0; and α = 0, β = 1, then a trans-sasakian manifold reduces to a sasakian and kenmotsu manifold respectively. thus transsasakian structures provide a large class of generalized quasi-sasakian structures. tamássy and binh [10] studied weakly symmetric and weakly ricci symmetric sasakian manifolds and proved that in such a manifold the sum of the associated 1-forms vanishes everywhere. again özgür [6] studied weakly symmetric and weakly ricci symmetric kenmotsu manifolds and proved that in such a manifold the sum of the associated 1-forms is zero everywhere and hence such a manifold does not exist unless the sum of the associated 1-forms is everywhere zero. the object of the present paper is to study weakly concircular symmetric and weakly concircular ricci symmetric trans-sasakian manifolds. section 2 deals with preliminaries of trans-sasakian manifolds. recently shaikh and hui [8] studied weakly symmetric and weakly ricci symmetric trans-sasakian manifolds and proved that the sum of the associated 1-forms of a weakly symmetric and also of a weakly ricci symmetric trans-sasakian manifold of non-vanishing ξ-sectional curvature are non-zero everywhere and hence such two structure exists, provided that the manifold is 144 shyamal kumar hui cubo 13, 3 (2011) of non-vanishing ξ-sectional curvature. however, in section 3 of the paper we have obtained all the 1-forms of a weakly concircular symmetric trans-sasakian manifold and hence such a structure exist always. again in section 4 we study weakly concircular ricci symmetric trans-sasakian manifolds and obtained all the 1-forms of a weakly concircular ricci symmetric trans-sasakian manifold and consequently such a structure is always exist. also it is proved that the sum of the associated 1-forms of a weakly concircular ricci symmetric trans-sasakian manifold is non-vanishing everywhere. 2 trans-sasakian manifolds a (2n + 1)-dimensional smooth manifold m is said to be an almost contact metric manifold [1] if it admits an (1, 1) tensor field φ, a vector field ξ, an 1-form η and a riemannian metric g, which satisfy φξ = 0, η(φx) = 0, φ2x = −x + η(x)ξ, (2.1) g(φx, y) = −g(x, φy), η(x) = g(x, ξ), η(ξ) = 1, (2.2) g(φx, φy) = g(x, y) − η(x)η(y) (2.3) for all vector fields x, y on m. an almost contact metric manifold m2n+1(φ, ξ, η, g) is said to be trans-sasakian manifold [5] if (m × r, j, g) belongs to the class w4 of the hermitian manifolds, where j is the almost complex structure on m × r defined by j ( z, f d dt ) = ( φz − fξ, η(z) d dt ) for any vector field z on m and smooth function f on m × r and g is the product metric on m × r. this may be stated by the condition [2] (∇xφ)(y) = α{g(x, y)ξ − η(y)x} + β{g(φx, y)ξ − η(y)φx}, (2.4) where α, β are smooth functions on m and such a structure is said to be the trans-sasakian structure of type (α, β). from (2.4) it follows that ∇xξ = −αφx + β{x − η(x)ξ}, (2.5) (∇xη)(y) = −αg(φx, y) + βg(φx, φy). (2.6) in a trans-sasakian manifold m2n+1(φ, ξ, η, g), the following relations hold: r(x, y)ξ = (α2 − β2)[η(y)x − η(x)y] − (xα)φy − (xβ)φ2(y) (2.7) + 2αβ[η(y)φx − η(x)φy] + (yα)φx + (yβ)φ2(x), cubo 13, 3 (2011) on weak concircular symmetries . . . 145 η(r(x, y)z) = (α2 − β2)[g(y, z)η(x) − g(x, z)η(y)] (2.8) − 2αβ[g(φx, z)η(y) − g(φy, z)η(x)] − (yα)g(φx, z) − (xβ){g(y, z) − η(y)η(z)} + (xα)g(φy, z) + (yβ){g(x, z) − η(z)η(x)}, s(x, ξ) = [2n(α2 − β2) − (ξβ)]η(x) − ((φx)α) − (2n − 1)(xβ), (2.9) r(ξ, x)ξ = (α2 − β2 − ξβ)[η(x)ξ − x], (2.10) s(ξ, ξ) = 2n(α2 − β2 − ξβ), (2.11) (ξα) + 2αβ = 0, (2.12) qξ = [2n(α2 − β2) − (ξβ)]ξ + φ(gradα) − (2n − 1)(gradβ), (2.13) where r is the curvature tensor of type (1, 3) of the manifold and q is the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the ricci tensor s, that is, g(qx, y) = s(x, y) for any vector fields x, y on m. 3 weakly concircular symmetric trans-sasakian manifolds definition 3.1. a trans-sasakian manifold (m2n+1, g)(n > 1) is said to be weakly concircular symmetric if its concircular curvature tensor c̃ of type (0, 4) satisfies (1.5). setting y = v = ei in (1.5) and taking summation over i, 1 ≤ i ≤ 2n + 1, we get (∇xs)(z, u) − dr(x) n g(z, u) (3.1) = a(x) [ s(z, u) − r n g(z, u) ] + b(z) [ s(x, u) − r n g(x, u) ] +d(u) [ s(x, z) − r n g(x, z) ] + b(r(x, z)u) + d(r(x, u)z) − r n(n − 1) [ {b(x) + d(x)}g(z, u) − b(z)g(x, u) − d(u)g(z, x) ] . plugging x = z = u = ξ in (3.1) and then using (2.7) and (2.11), we obtain a(ξ) + b(ξ) + d(ξ) = 2n2{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) 2n2{α2 − (ξβ) − β2} − r . (3.2) this leads to the following: theorem 3.1. in a weakly concircular symmetric trans-sasakian manifold (m2n+1, g) (n > 1), the relation (3.2) holds. 146 shyamal kumar hui cubo 13, 3 (2011) next, substituting x and z by ξ in (3.1) and then using (2.7) and (2.12) we obtain (∇ξs)(ξ, u) − dr(ξ) n η(u) (3.3) = [ a(ξ) + b(ξ) ] [ s(u, ξ) − r n η(u) ] + d(u) [ (2n − 1){α2 − (ξβ) −β2} − n − 2 n(n − 1) r ] + [ α 2 − (ξβ) − β2 − r n(n − 1) ] η(u)d(ξ). from (2.9), we have (∇ξs)(ξ, u) = ∇ξs(ξ, u) − s(∇ξξ, u) − s(ξ, ∇ξu) (3.4) = ∇ξs(ξ, u) − s(ξ, ∇ξu) = [2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ))]η(u) −(2n − 1)(u(ξβ)) − (φu(ξα)). by virtue of (3.3) and (3.4) we obtain from (3.2) that d(u) = [2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ]η(u) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r (3.5) − (2n − 1)(u(ξβ)) + (φu(ξα)) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r + d(ξ) [ (2n − 1){(α2 − β2)η(u) − (uβ)} − ((φu)α) − n−2 n(n−1) rη(u) (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ][(2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r] [ {2n(α2 − β2) − (ξβ) − r n }η(u) − (2n − 1)(uβ) − ((φu)α) ] . next, setting x = u = ξ in (3.1) and proceeding in a similar manner as above, we get b(z) = [2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ]η(z) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r (3.6) − (2n − 1)(z(ξβ)) + (φz(ξα)) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r + b(ξ) [ (2n − 1){(α2 − β2)η(z) − (zβ)} − ((φz)α) − n−2 n(n−1) rη(z) (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ][(2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r] [ {2n(α2 − β2) − (ξβ) − r n }η(z) − (2n − 1)(zβ) − ((φz)α) ] . cubo 13, 3 (2011) on weak concircular symmetries . . . 147 again, setting z = u = ξ in (3.1), we get (∇xs)(ξ, ξ) − dr(x) n = a(x) [ s(ξ, ξ) − r n ] + [b(ξ) + d(ξ)][s(x, ξ) (3.7) − n − 2 n(n − 1) rη(x)] + b(r(x, ξ)ξ) + d(r(x, ξ)ξ) − r n(n − 1) [b(x) + d(x)] = [2n{α2 − (ξβ) − β2} − r n ]a(x) + [b(ξ) + d(ξ)] [ s(x, ξ) − { n − 2 n(n − 1) r + α2 − (ξβ) − β2}η(x) ] + [b(x) + d(x)] [ α 2 − (ξβ) − β2 − r n(n − 1) ] . now we have (∇xs)(ξ, ξ) = ∇xs(ξ, ξ) − 2s(∇xξ, ξ), which yields by using (2.5) and (2.9) that (∇xs)(ξ, ξ) = 2n[2α(xα) − 2β(xβ) − (x(ξβ))] (3.8) + 2α[(xα) − η(x)(ξα) − (2n − 1)((φx)β)] + 2β[((φx)α) + (2n − 1){(xβ) − (ξβ)η(x)}]. in view of (3.8), (3.7) yields [ 2n{α 2 − (ξβ) − β2} − r n ] a(x) + [ α 2 − (ξβ) − β2 − r n(n − 1) ][ b(x) + d(x) ] (3.9) = 2n [ 2α(xα) − 2β(xβ) − (x(ξβ)) ] + 2α [ (xα) − η(x)(ξα) −(2n − 1)((φx)β) ] + 2β [ ((φx)α) + (2n − 1){(xβ) − (ξβ)η(x)} ] − dr(x) n − {b(ξ) + d(ξ)} [ {(2n − 1)(α2 − β2) − n − 2 n(n − 1) r}η(x) − ((φx)α) − (2n − 1)(xβ) ] . 148 shyamal kumar hui cubo 13, 3 (2011) using (3.5) and (3.6) in (3.9), we obtain [2n{α2 − (ξβ) − β2} − r n ]a(x) (3.10) = 2n[2α(xα) − 2β(xβ) − (x(ξβ))] + 2α[(xα) − η(x)(ξα) −(2n − 1)((φx)β)] + 2β[((φx)α) + (2n − 1){(xβ) − (ξβ)η(x)}] − dr(x) n − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ {(2n − 1)(α2 −β2) − n − 2 n(n − 1) r}η(x) − ((φx)α) − (2n − 1)(xβ) ] +a(ξ) 2n{α2 − (ξβ) − β2} − r n (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ {(2n − 1)(α2 −β2) − n − 2 n(n − 1) r}η(x) − ((φx)α) − (2n − 1)(xβ) ] − 2{α2 − (ξβ) − β2 − r n(n−1) } (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ 2n{2α(ξα) −2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(x) + 2{α2 − (ξβ) − β2 − r n(n−1) } (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ (2n − 1)(x(ξβ)) + (φx(ξα)) ] + 2{α2 − (ξβ) − β2 − r n(n−1) }[2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n ] [2n{α2 − (ξβ) − β2} − r n ][(2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r] [ {2n(α2 − β2) − (ξβ) − r n }η(x) − (2n − 1)(xβ) − ((φx)α) ] . this leads to the following: theorem 3.2. in a weakly concircular symmetric trans-sasakian manifold (m2n+1, g) (n > 1), the associated 1-forms d, b and a are given by (3.5), (3.6) and (3.10) respectively. 4 weakly concircular ricci symmetric trans-sasakian manifolds definition 4.1. a trans-sasakian manifold (m2n+1, g)(n > 1) is said to be weakly concircular ricci symmetric if its concircular ricci tensor p of type (0, 2) satisfies (1.9). cubo 13, 3 (2011) on weak concircular symmetries . . . 149 in view of (1.8), (1.9) yields (∇xs)(y, z) − dr(x) n g(y, z) = a(x) [ s(y, z) − r n g(y, z) ] (4.1) + b(y) [ s(x, z) − r n g(x, z) ] + d(z) [ s(x, y) − r n g(x, y) ] . setting x = y = z = ξ in (4.1), we get the relation (3.2) and hence we can state the following: theorem 4.1. in a weakly concircular ricci symmetric trans-sasakian manifold (m2n+1, g) (n > 1), the relation (3.2) holds. next, substituting x and y by ξ in (4.1), we obtain (∇ξs)(ξ, z) − dr(ξ) n η(z) = [a(ξ) + b(ξ)] [ s(ξ, z) (4.2) − r n η(z) ] + d(z) [ s(ξ, ξ) − r n ] . using (3.2) and (3.4) in (4.2), we get d(z) = [ 2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(z) 2n[α2 − (ξβ) − β2] − r n (4.3) − (2n − 1)(z(ξβ)) + (φz(ξα)) 2n[α2 − (ξβ) − β2] − r n + d(ξ) [ 2n{(α2 − β2) − (ξβ) − r n }η(z) − ((φz)α) − (2n − 1)(zβ) 2n{α2 − (ξβ) − β2} − r n ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) − (ξβ) − r n }η(z) − (2n − 1)(zβ) − ((φz)α) ] for all z. again putting x = z = ξ in (4.1) and proceeding in a similar manner as above we get b(y) = [ 2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(y) 2n[α2 − (ξβ) − β2] − r n (4.4) − (2n − 1)(y(ξβ)) + (φy(ξα)) 2n[α2 − (ξβ) − β2] − r n + b(ξ) [ 2n{(α2 − β2) − (ξβ) − r n }η(y) − ((φy)α) − (2n − 1)(yβ) 2n{α2 − (ξβ) − β2} − r n ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) − (ξβ) − r n }η(y) − (2n − 1)(yβ) − ((φy)α) ] for all y. 150 shyamal kumar hui cubo 13, 3 (2011) again, setting y = z = ξ in (4.1) and using (2.9) and (2.11), we get (∇xs)(ξ, ξ) − dr(x) n = [ 2n{α 2 − (ξβ) − β2} − r n ] a(x) (4.5) + [b(ξ) + d(ξ)] [ {2n(α2 − β2) − (ξβ)}η(x) − ((φx)α) − (2n − 1)(xβ) ] . using (3.2) and (3.8) in (4.5), we get a(x) = 2n[2α(xα) − 2β(xβ) − (x(ξβ))] 2n{α2 − (ξβ) − β2} − r n (4.6) + 2α[(xα) − η(x)(ξα) − (2n − 1)((φx)β)] 2n{α2 − (ξβ) − β2} − r n + 2β[((φx)α) + (2n − 1){(xβ) − (ξβ)η(x)}] 2n{α2 − (ξβ) − β2} − r n + a(ξ) [ {2n(α2 − β2) − (ξβ) − r n }η(x) − ((φx)α) − (2n − 1)(xβ) 2n{α2 − (ξβ) − β2} − r n ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) − (ξβ) − r n }η(x) − (2n − 1)(xβ) − ((φx)α) ] for all x. this leads to the following: theorem 4.2. in a weakly concircular ricci symmetric trans-sasakian manifold (m2n+1, g) (n > 1), the associated 1-forms d, b and a are given by (4.3), (4.4) and (4.6) respectively. adding (4.3), (4.4) and (4.6) and using (3.2), we get a(x) + b(x) + d(x) (4.7) = 2n[2α(xα) − 2β(xβ) − (x(ξβ))] 2n{α2 − (ξβ) − β2} − r n + 2α[(xα) − η(x)(ξα) − (2n − 1)((φx)β)] 2n{α2 − (ξβ) − β2} − r n + 2β[((φx)α) + (2n − 1){(xβ) − (ξβ)η(x)}] 2n{α2 − (ξβ) − β2} − r n 2 [ 2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(x) 2n{α2 − (ξβ) − β2} − r n − (2n − 1)(x(ξβ)) + (φx(ξα)) n{α2 − (ξβ) − β2} − r 2n − 2 [ 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n ] [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) −(ξβ) − r n }η(x) − (2n − 1)(xβ) − ((φx)α) ] cubo 13, 3 (2011) on weak concircular symmetries . . . 151 for any vector field x. this leads to the following: theorem 4.3. in a weakly concircular ricci symmetric trans-sasakian manifold (m2n+1, g) (n > 1), the sum of the associated 1-forms is given by (4.7). in particular, if φ(grad α) = grad β, then (ξβ) = 0 and hence the relation (4.7) reduces to the following form a(x) + b(x) + d(x) (4.8) = 2n[2α(xα) − 2β(xβ)] 2n(α2 − β2) − r n + 2α{(xα) − η(x)(ξα) − (2n − 1)((φx)β)} 2n(α2 − β2) − r n + 2β{((φx)α) + (2n − 1)(xβ)} + 2{4nα(ξα) − dr(ξ) n }η(x) − 2(φx(ξα)) 2n(α2 − β2) − r n − 2[4nα(ξα) − dr(ξ) n ] [2n(α2 − β2) − r n ]2 [ {2n(α2 − β2) − r n }η(x) − ((φx)α) − (2n − 1)(xβ) ] . for any vector field x. this leads to the following: corollary 4.1. if a weakly concircular ricci symmetric trans-sasakian manifold (m2n+1, g) (n > 1) satisfies the condition φ(grad α) = grad β, then the sum of the associated 1-forms is given by (4.8). if β = 0 and α = 1, then (4.7) yields a(x) + b(x) + d(x) = 0 for all x and hence we can state the following: corollary 4.2. there is no weakly concircular ricci symmetric sasakian manifold m2n+1(n > 1), unless the sum of the 1-forms is everywhere zero. corollary 4.3. if an α-sasakian manifold is weakly concircular ricci symmetric, then the sum of the 1-forms, i.e., a + b + d is given by a(x) + b(x) + d(x) = 2α[(2n + 1)(xα) − η(x)(ξα)] − 2(φx(ξα)) 2nα2 − r n + 2[4nα(ξα) − dr(ξ) n ]((φx)α) (2nα2 − r n )2 . again, if α = 0 and β = 1, then (4.7) yields a(x) + b(x) + d(x) = 0 for all x. this leads to the following: corollary 4.4. there is no weakly concircular ricci symmetric kenmotsu manifold m2n+1(n > 1), unless the sum of the 1-forms is everywhere zero. corollary 4.5. if a β-kenmotsu manifold is weakly concircular ricci symmetric, then the sum of 152 shyamal kumar hui cubo 13, 3 (2011) the 1-forms, i.e., a + b + d is given by a(x) + b(x) + d(x) = 2n{2β(xβ) + (x(ξβ))} − 2(2n − 1)β{(xβ) − (ξβ)η(x)} 2n{(ξβ) + β2} + r n + 2[{4nβ(ξβ) + (ξ(ξβ) + dr(ξ) n }η(x) + (2n − 1)(x(ξβ))] 2n{(ξβ) + β2} + r n − 2[2n{2β(ξβ) + (ξ(ξβ))} + dr(ξ) n ][{2nβ2 + (ξβ) + r n }η(x) + (2n − 1)(xβ)] [2n{(ξβ) + β2} + r n ]2 . received: april 2010. revised: september 2010. references [1] blair, d. e., contact manifolds in riemannian geometry, lecture notes in math. 509, springer-verlag, 1976. [2] blair, d. e. and oubina, j. a., conformal and related changes of metric on the product of two almost contact metric manifolds, publ. math. debrecen, 34 (1990), 199–207. [3] de, u. c. and bandyopadhyay, s., on weakly symmetric riemannian spaces, publ. math. debrecen, 54/3-4 (1999), 377-381. [4] de, u. c. and ghosh, g. c., on weakly concircular ricci symmetric manifolds, south east asian j. math. and math. sci., 3(2) (2005), 9–15. [5] oubina, j. a., new class of almost contact metric manifolds, publ. math. debrecen, 32 (1985), 187–193. [6] özgür, c., on weakly symmetric kenmotsu manifolds, diff. geom.-dynamical systems, 8 (2006), 204–209. [7] shaikh, a. a. and hui, s. k., on weakly concircular symmetric manifolds, ann. sti. ale univ., “al. i. cuza”, din iasi, lv, f.1 (2009), 167–186. [8] shaikh, a. a. and hui, s. k., on weak symmetries of trans-sasakian manifolds, proc. estonian acad. sci., 58(4) (2009), 213–223. [9] tamássy, l. and binh, t. q., on weakly symmetric and weakly projective symmetric rimannian manifolds, coll. math. soc., j. bolyai, 56 (1989), 663–670. [10] tamássy, l. and binh, t. q., on weak symmetries of einstein and sasakian manifolds, tensor n. s., 53 (1993), 140–148. [11] yano, k., concircular geometry i, concircular transformations, proc. imp. acad. tokyo, 16 (1940), 195–200. introduction trans-sasakian manifolds weakly concircular symmetric trans-sasakian manifolds weakly concircular ricci symmetric trans-sasakian manifolds cubo a mathematical journal vol.18, no¯ 01, (01–14). december 2016 uniqueness of meromorphic functions sharing a set in annuli renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai department of mathematics, karnatak university, dharwad 580003, india renukadyavanal@gmail.com, ashwinimhmaths@gmail.com, madhuramathai@gmail.com abstract the purpose of this article is to investigate the uniqueness of meromorphic functions sharing a set with counting multiplicity and also with weight 1 in annuli. resumen el propósito de este art́ıculo es investigar la unicidad de funciones meromorfas compartiendo un conjunto contando multiplicidad y también con peso 1 en un anillo. keywords and phrases: annuli, meromorphic, sharing set, uniqueness. 2010 ams mathematics subject classification: 30d35, 39a05. 2 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) 1 introduction and main results we assume the reader is familiar with standard results and notations of nevanlinna’s theory of meromorphic functions [4],[10],[11]. the purpose of this paper is to study the uniqueness of meromorphic functions in doubly connected domains of complex plane c. by the doubly connected mapping theorem [1], each doubly connected domain is conformally equivalent to the annulus {z : r < |z| < r}, 0 ≤ r < r ≤ +∞. we consider only two cases: r = 0, r = +∞ simultaneously and 0 < r < r < +∞. in the latter case, the homothety z "−→ z√ rr reduces the given domain to the annulus a = { z : 1 r0 < |z| < r0 } , where r0 = ! r r . thus, in two cases every annulus is invariant with respect to the inversion z "−→ 1 z . hence, we consider the uniqueness of mermorphic functions in the annulus a = {z : 1 r0 < |z| < r0}, where 1 < r0 ≤ +∞. we denote by s the subset of distinct elements in c = c ∪ {∞}. for a meromorphic function f in a, we define ea(s, f) = ∪a∈s{z ∈ a : fa(z) = 0, counting multiplicities}, e a (s, f) = ∪a∈s{z ∈ a : fa(z) = 0, ignoring multiplicities}, where fa(z) = f(z) − a if a ∈ c. we also define e a 1(s, f) = ∪a∈s{z ∈ a : all the simple zeros of fa(z)}. for any constant a, we say that f and g share a counting multiplicity(cm), provided that f − a and g − a have the same zeros with same multiplicities. similarly, we say that f and g share a ignoring multiplicity(im), provided that f − a and g − a have the same zeros ignoring multiplicities. in 2009, cao et al.[3] obtained an analog of nevanlinna’s famous five -value theorem as follows. theorem 1.1. let f1 and f2 be two transcendental or admissible meromorphic functions on the annulus a = { z : 1 r0 < |z| < r0 } , where 1 < r0 ≤ +∞. let aj(j = 1, 2, 3, 4, 5) be five distinct complex numbers in c. if f1, f2 share aj im for j = 1, 2, 3, 4, 5, then f1(z) ≡ f2(z). in 2012, cao and deng [2] investigated the uniqueness of two meromorphic functions in a sharing two or three finite sets and obtained the following theorems. theorem 1.2. let f and g be two admissible meromorphic functions in the annulus a. put s1 = {0}, s2 = {∞}, and s3 = {w : p(w) = 0}, where p(w) = awn + n(n − 1)w2 + 2n(n − 2)bw − (n − 1)(n − 2)b2, where n ≥ 5 is an integer and a and b are two non-zero complex numbers satisfying abn−2 ̸= 1, 2. if e(s2, f) = e(s2, g) and e(sj, f) = e(sj, g)(j = 1, 3), then f ≡ g. theorem 1.3. let f and g be two admissible meromorphic functions in the annulus a. put s1 = {∞} and s2 = {w : p(w) = 0}, where p(w) is stated as in theorem 1.2, and a and b are two non-zero complex numbers satisfying abn−2 ̸= 2, n ≥ 8 is an integer. if e(s1, f) = e(s1, g) and e(s2, f) = e(s2, g), then f ≡ g. cubo 18, 1 (2016) uniqueness of meromorphic functions sharing a set in annuli 3 in 2013, xu and wu [8] obtained the following theorems. theorem 1.4. let f and g be two admissible meromorphic functions in the annulus a. let s = {w ∈ a : p1(w) = 0}, where p1(w) = (n−1)(n−2) 2 wn − n(n − 2)wn−1 + n(n−1) 2 wn−2 − c and c(̸= 0, 1) is a complex number. if ea(s, f) = ea(s, g) and n is an integer satisfying ≥ 11, then f ≡ g. theorem 1.5. let f and g be two admissible meromorphic functions in the annulus a. if n is an integer ≥ 7, ea(s, f) = ea(s, g) and θ0(∞, f) > 3 4 , θ0(∞, g) > 3 4 , where s is defined as in theorem 1.4, then f ≡ g. theorem 1.6. let f and g be two admissible meromorphic functions in the annulus a. if ea1 (s, f) = e a 1 (s, g) and n is an integer ≥ 15, where s is defined as in theorem 1.4, then f ≡ g. theorem 1.7. let f and g be two admissible meromorphic functions in the annulus a. if n is an integer ≥ 9, ea1 (s, f) = e a 1(s, g) and θ0(∞, f) > 5 6 , θ0(∞, g) > 5 6 , where s is defined as in theorem 1.4, then f ≡ g. the main purpose of this paper is to investigate the uniqueness of meromorphic functions sharing a set s = {w ∈ a : p(w) = 0}, where p(w) = wn + αwn−m + β, α and β are two non-zero constants different from the sets considered by cao and deng [2] and x.y.xu and z.t.wu [8]. the set considered in this paper is more general, as set varies by varying value of m and constants α and β for a fixed n, where as the set in [2] and [8] are specific for a fixed n. to prove the main results, we follow the techniques used by x.y.xu and z.t.wu [8] till the conclusion part and using different technique, conclusion part is effectively proved as the sharing set s is different from the set considered in [8]. theorem 1.8. let f and g be two admissible meromorphic functions in the annulus a. let n > 2m + 8 and m ≥ 2 be integers with n and m having no common factors. let s = {w ∈ a : p(w) = 0}, where p(w) = wn + αwn−m + β, α and β are two non-zero constants such that the algebraic equation wn + αwn−m + β = 0 has no multiple roots. if ea(s, f) = ea(s, g), then f ≡ g. as inspired by the proof of the theorem 1.5, theorem 1.6 and theorem 1.7, we proved the following theorems. theorem 1.9. let f and g be two admissible meromorphic functions in the annulus a. let n > 2m + 5 and m ≥ 2 be integers. if ea(s, f) = ea(s, g) and θ0(∞, f) > 3 4 , θ0(∞, g) > 3 4 , where s is defined as in theorem 1.8, then f ≡ g. theorem 1.10. let f and g be two admissible meromorphic functions in the annulus a. let n > 2m + 12 and m ≥ 2 be integers. if ea1 (s, f) = e a 1 (s, g), where s is defined as in theorem 1.8, then f ≡ g. theorem 1.11. let f and g be two admissible meromorphic functions in the annulus a. let n > 2m+7 and m ≥ 2 be integers. if ea1 (s, f) = e a 1(s, g) and θ0(∞, f) > 5 6 , θ0(∞, g) > 5 6 , where s is defined as in theorem 1.8, then f ≡ g. 4 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) basic notations in the nevanlinna theory on annuli let f be a meromorphic function in c. we recall the classical notations of the nevanlinna theory as follows: n(r, f) = ∫r 0 n(t, f) − n(0, f) t dt + n(0, f) log r, m(r, f) = 1 2π ∫2π 0 log+ |f(reiθ)|dθ, t(r, f) = m(r, f) + n(r, f) where log+ x = max{log x, 0} and n(t, f) is the counting function of poles of the function f in {z : |z| ≤ t}.the following are the notations and basic results of nevanlinna theory on annuli a = {z : 1 r0 < |z| < r0} for 1 < r < r0 ≤ +∞, which can be found in [5] and [6]. let n1(r, f) = ∫1 1 r n1(t, f) t dt, n2(r, f) = ∫r 1 n2(t, f) t dt where n1(t, f) and n2(t, f) are the counting function of poles of the function f in {z : t < |z| ≤ 1} and {z : 1 < |z| ≤ t}, respectively. let n0(r, f) = n1(r, f) + n2(r, f) , m0(r, f) = m(r, f) + m " 1 r , f # and n0(r, f) = n1(r, f) + n2(r, f) = ∫1 1 r n1(t, f) t dt + ∫r 1 n2(t, f) t dt where n1(t, f) and n2(t, f) are the reduced counting function of poles of the function f in {z : t < |z| ≤ 1} and {z : 1 < |z| ≤ t}, respectively. finally, we define the nevanlinna characteristic of f on the annulus a by t0(r, f) = m0(r, f) − 2m(1, f) + n0(r, f) in addition, we have n (2) 0 (r, f) = n0(r, f) + n (2 0 (r, f), n (2 0 (r, f) = n(r, f) − n 1) 0 (r, f) where n 1) (r, f) is the reduced counting function which only counts simple zeros of the function f. and the nevanlinna characteristic of f has the following properties. (i)t0(r, f) = t0 $ r, 1 f % , (ii)max{t0(r, f1 · f2), t0(r, f1 f2 ), t0(r, f1 + f2)} ≤ t0(r, f1) + t0(r, f2) + o(1) by above properties, the first fundamental theorem on the annulus a is immediately obtained as follows. cubo 18, 1 (2016) uniqueness of meromorphic functions sharing a set in annuli 5 first fundamental theorem in annuli : let f be a non-constant meromorphic function on the annulus a = {z : 1 r0 < |z| < r0}. for 1 < r < r0 ≤ +∞, we have t0 " r, 1 f − a # = t0(r, f) + o(1) for every fixed a ∈ c. definition 1. for every a ∈ c, the reduced deficiency is given by θ0(a, f) = 1 − lim sup r→∞ n0 $ r, 1 f−a % t0(r, f) . definition 2. let f be a meromorphic function on the annulus a = {z : 1 r0 < |z| < r0}, where 1 < r0 ≤ +∞. the function f is called an admissible meromorphic function on the annulus a provided that lim sup r→∞ t0(r, f) log r = ∞, 1 < r < r0 = +∞, or lim sup r→r0 t0(r, f) − log (r0 − r) = ∞, 1 < r < r0 < +∞, respectively. khrystiyanyn and kondratyuk[5] obtained the following lemma on the logarithmic derivative on the annulus a. lemma on the logarithmic derivative : let f be a non-constant meromorphic function on the annulus a = {z : 1 r0 < |z| < r0}, where r0 ≤ +∞ and let λ > 0. then, (i) if r0 = +∞, then m0 & r, f ′ f ' = o(log(rt0(r, f))), for r ∈ (1, +∞) except for the set ∆r such that ∫ ∆r rλ−1dr < +∞; (ii) if r0 < +∞, then m0 & r, f ′ f ' = o & log & t0(r,f) r0−r '' , for r ∈ (1, r0) except for the set ∆ ′ r such that ∫ ∆′ r dr (r0−r) λ−1 < +∞. 2 some lemmas for the proof of our main results, we need the following lemmas. lemma 2.1. ([3]) let f be a non-constant meromorphic function on the annulus a = {z : 1 r0 < |z| < r0}, where 1 < r0 ≤ +∞. let a1, a2, · · · , aq be q distinct complex numbers in the extended complex plane c. then, (q − 2)t0(r, f) < q∑ j=1 n0 " r, 1 f − aj # + s(r, f) 6 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) (1) if r0 = +∞, then s(r, f) = o(log(rt0(r, f))), for r ∈ (1, +∞) except for the set ∆r such that ∫ ∆r rλ−1dr < +∞. (2) if r0 < +∞, then s(r, f) = o & log & t0(r,f) r0−r '' , for r ∈ (1, r0) except for the set ∆ ′ r such that ∫ ∆′ r dr (r0−r)λ−1 < +∞. lemma 2.2. ([2]) let f be a non-constant meromorphic function in a, q1(f) and q2(f) be two mutually prime polynomials in f with degree m and n, respectively. then, t0 " r, q1(f) q2(f) # = max{m, n}t0(r, f) + s(r, f), where s(r, f) is defined as in lemma 2.1. lemma 2.3. ([8]) let f be a non-constant meromorphic function in a. then, n0 " r, 1 f′ # ≤ n0 " r, 1 f # + n0(r, f) + s(r, f) + o(1), where s(r, f) is defined as in lemma 2.1. lemma 2.4. ([8]) let f and g be admissible meromorphic functions in a satisfying ea(f, 0) = ea(g, 0) and c1, c2, · · · , cq be q ≥ 2 distinct non-zero complex numbers. if lim sup r→∞,r∈i 3n0(r, f) + ∑q j=1 n (2) 0 & r, 1 f−cj ' + n0 $ r, 1 f′ % t0(r, f) < q lim sup r→∞,r∈i 3n0(r, g) + ∑q j=1 n (2) 0 & r, 1 g−cj ' + n0 $ r, 1 g′ % t0(r, g) < q where n (2) 0 (r, ∗) = n0(r, ∗) + n (2 0 (r, ∗), n (2 0 (r, ∗) = n0(r, ∗) − n 1) 0 (r, ∗) and i is some set of r of infinite linear measure, then f = ag + b cg + d where a, b, c, d ∈ c are constants with ad − bc ̸= 0. lemma 2.5. ([8]) let f and g be admissible meromorphic functions in a satisfying ea1(f, 0) = ea1 (g, 0) and let c1, c2, · · · , cq be q ≥ 2 distinct non-zero complex numbers. if lim sup r→∞,r∈i 3n0(r, f) + ∑q j=1 n (2) 0 & r, 1 f−cj ' + n0 $ r, 1 f′ % + 2n (2 0 $ r, 1 f % t0(r, f) < q lim sup r→∞,r∈i 3n0(r, g) + ∑q j=1 n (2) 0 & r, 1 g−cj ' + n0 $ r, 1 g′ % + 2n (2 0 $ r, 1 g % t0(r, g) < q cubo 18, 1 (2016) uniqueness of meromorphic functions sharing a set in annuli 7 where n (2) 0 (r, ∗), n (2 0 (r, ∗) and i are stated as in lemma 2.4, then f = ag + b cg + d , where a, b, c, d ∈ c are constants with ad − bc ̸= 0. lemma 2.6. [6] let f be a non-constant meromophic function on a = {z : 0 < |z| < ∞}. let {av} be any finite collection of complex numbers possibly including ∞. then, q∑ v=1 θ0 (av) ≤ 2. 3 proof of theorem 1.1 if p(w) = wn + αwn−m + β, we can get that p(0) = β = c1(̸= 0) and p(1) = 1 + α + β = c2(̸= 0), and p ′(w) = wn−m−1 [nwm + α(n − m)] (3.1) p(w) − c1 = w n−mq1(w) (3.2) where q1(w) is a polynomial of degree m and q1(0) ̸= 0. p(w) − c2 = (w − 1)q2(w) (3.3) where q2(w) is a polynomial of degree n − 1 and q2(1) ̸= 0. from (3.2) and (3.3), we notice that q1(w)and q2(w) have only simple zeros. let f = p(f) and g = p(g). since ea(s, f) = ea(s, g), we get that f and g share the value 0 cm. from (3.2) and (3.3), we have n (2) 0 " r, 1 f − c1 # = n0 " r, 1 f − c1 # + n (2 0 " r, 1 f − c1 # ≤ 2n0 " r, 1 f # + m∑ i=1 n0 " r, 1 f − ai # = (m + 2)t0(r, f) + s(r) (3.4) where ai(i = 1, 2, · · · , m) are the zeros of q1(w) in a and s(r) = o{t0(r)}, t0(r) = max{t0(r, f), t0(r, g)}. and n (2) 0 " r, 1 f − c2 # = n0 " r, 1 f − c2 # + n (2 0 " r, 1 f − c2 # ≤ n0 " r, 1 f − 1 # + n−1∑ j=1 n0 " r, 1 f − bj # 8 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) = nt0(r, f) + s(r) (3.5) where bj(j = 1, 2, · · · , n − 1) are the zeros of q2(w) in a. from (3.1), we obtain n0 " r, 1 f′ # = n0 " r, 1 fn−m−1[nfm + α(n − m)]f′ # ≤ n0 " r, 1 fn−m−1 # + n0 " r, 1 nfm + α(n − m) # + n0 " r, 1 f′ # + s(r) (3.6) using lemma 2.3, (3.6) reduces to n0 " r, 1 f′ # ≤ n0 " r, 1 f # + n0 " r, 1 nfm + α(n − m) # + n0 " r, 1 f # + n0(r, f) + s(r) ≤ (m + 3)t0(r, f) + s(r) (3.7) from lemma 2.2, we get t0(r, f) = nt0(r, f) + s(r) (3.8) using (3.4) − (3.8) in lemma 2.4 and since n > 2m + 8, we get lim sup r→∞,r∈i 3n0(r, f) + ∑2 j=1 n (2) 0 & r, 1 f−cj ' + n0 $ r, 1 f′ % t0(r, f) ≤ n + 2m + 8 n < 2 (3.9) similarly, we obtain lim sup r→∞,r∈i 3n0(r, g) + ∑2 j=1 n (2) 0 & r, 1 g−cj ' + n0 $ r, 1 g′ % t0(r, g) < 2 (3.10) thus, by lemma 2.4, we have f ≡ ag+b cg+d , where, a, b, c, d ∈ c and ad − bc ̸= 0. since ea(s, f) is non-empty and ea(s, f) = ea(s, g), we have b = 0, a ̸= 0. hence, f = ag cg + d = g ag + b (3.11) where a = c a and b = d a . now, we consider the following two cases: case 1: suppose a ̸= 0. from (3.11), we notice that every zero of p(g) + b a in a has multiplicity n. cubo 18, 1 (2016) uniqueness of meromorphic functions sharing a set in annuli 9 next, the case 1 is followed by three following subcases: subcase 1: suppose b a = −c1. from (3.2), we have p(g) + b a = gn−m(gm + α) = gn−m(g − a1) · · · (g − am) (3.12) where ai(i = 1, 2, · · · , m) are non-zero distinct roots of g. it follows that every zero of g in a has multiplicity at least m and every zero of g−ai in a has multiplicity of at least n. then by lemma 2.1, we have (m − 1)t0(r, g) ≤ n0 " r, 1 g # + n0 " r, 1 g − a1 # + · · · + n0 " r, 1 g − am # + s(r, g) ≤ 1 m n0 " r, 1 g # + 1 n n0 " r, 1 g − a1 # + · · · + 1 n n0 " r, 1 g − am # + s(r, g) ≤ " 1 m + m n # t0(r, g) + s(r) (3.13) since m ≥ 2 and n > 2m + 8, we arrive at a contradiction. subcase 2: suppose b a = −c2. from (3.3), we have p(g) + b a = (g − 1)(g − b1)(g − b2) · · · (g − bn−1) (3.14) where bj ̸= 0, 1 are distinct values. for j = 1, 2, · · · , n − 1, consider θ(bj, f) = 1 − lim sup r→∞ n0(r, bj, f) t0(r, f) > 1 2 (3.15) we can see that p(g) + b a has n values satisfying the above inequality. thus, by lemma 2.6, we get a contradiction. subcase 3: suppose b a ̸= −c1, −c2. by using the same argument as in subcase 1 or subcase 2, we get a contradiction. case 2: suppose a = 0. if b ̸= 1, then from (3.11), we have f = g b ; that is p(f) = 1 b p(g) (3.16) from (3.2) and (3.16), we have p(f) − c1 b = 1 b gn−m(g − a1)(g − a2) · · · (g − am) (3.17) since c1 b ̸= c1, from (3.14), it follows that p(f) − c1 b has at least n distinct zeros e1, e2, · · · , en. 10 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) then by applying lemma 2.1, we have (n − 2)t0(r, f) ≤ n∑ i=1 n0 " r, 1 f − ei # + s(r) ≤ n0 " r, 1 g # + n0 " r, 1 g − a1 # + n0 " r, 1 g − a2 # + · · · + n0 " r, 1 g − am # + s(r) ≤ (m + 1)t0(r, g) + s(r) (3.18) by applying lemma 2.4 to (3.16) and from (3.18) and since n > 2m + 8 and m ≥ 2, we arrive at a contradiction. thus, we get a = 0 and b = 1, that is p(f) = p(g) ⇒ fn + αfn−m = gn + αgn−m (3.19) we set h = f g , we substitute f = hg in (3.19), it follows that gn−m[gm(hn − 1) + α(hn−m − 1)] = 0 (3.20) if h is a constant. we have from (3.20) that hn − 1 = 0 and hn−m − 1 = 0, which implies h = 1 and hence f ≡ g. if h is not a constant, then suppose fn ̸≡ gn now consider, gm = −α (hn−m − 1) hn − 1 (3.21) gm = −α(hn−m−1 + hn−m−2 + · · · + 1) (hn−1 + hn−2 + · · · + 1) (3.22) gm = −α(h − vn−m−1)(h − vn−m−2) · · · (h − v) (h − un−1)(h − un−2) · · · (h − u) (3.23) where v = exp((2πi)/(n − m)) and u = exp((2πi)/n). since n and m have no common factors, we have vj ̸= uk(j = 1, 2, · · · , n − m − 1; k = 1, 2, · · · , n − 1). suppose that zk is a zero of h − u k of order pk. from (3.23), we have pk ≥ m. thus n0 " r, 1 h − uk # ≤ 1 m n0 " r, 1 h − uk # ≤ 1 2 t0(r, h) + o(1) (3.24) cubo 18, 1 (2016) uniqueness of meromorphic functions sharing a set in annuli 11 by lemma 2.1 and from (3.24), we obtain (n − 3)t0(r, h) < n−1∑ k=1 n0 " r, 1 h − uk # + s(r, h) ≤ n − 1 2 t0(r, h) + s(r, h) where s(r, h) is defined as in lemma 2.1. since n > 2m + 8 and m ≥ 2, we arrive at a contradiction and since n and m have no common factors, we get f ≡ g. this completes the proof of theorem 1.8. proof of theorem 1.9 : since θ0(∞, f) > 3 4 and θ0(∞, g) > 3 4 it follows that lim sup r→∞ n0(r, f) t0(r, f) < 1 4 , lim sup r→∞ n0(r, g) t0(r, g) < 1 4 (3.25) by applying (3.25), from lemma 2.4 and since n > 2m + 5, we deduce lim sup r→∞,r∈i 3n0(r, f) + ∑2 j=1 n (2) 0 & r, 1 f−cj ' + n0 $ r, 1 f′ % t0(r, f) (3.26) < 4 n lim sup r→∞,r∈i n0(r, f) t0(r, f) + lim sup r→∞,r∈i (n + 2m + 4)t0(r, f) nt0(r, f) < 2 (3.27) similarly, we get lim sup r→∞,r∈i 3n0(r, g) + ∑2 j=1 n (2) 0 & r, 1 g−cj ' + n0 $ r, 1 g′ % t0(r, g) < 2 (3.28) then, from lemma 2.4, we have f = ag+b cg+d , where a, b, c, d ∈ c and ad − bc ̸= 0. thus, by using the same argument as that in theorem 1.8, we can prove the conclusion of theorem 1.9. proof of theorem 1.10 : since ea1 (s, f) = e a 1(s, g), we have e a 1(f, 0) = e a 1 (g, 0) from (3.1)-(3.3), we can get n (2 0 " r, 1 f # = n∑ i=1 n0 " r, 1 f − di # ≤ n0 " r, 1 f′ # (3.29) 12 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) where di(i = 1, 2, · · · , n) are the distinct zeros of p(w). and from (3.6), (3.29) and by lemma 2.3, we have n0 " r, 1 f′ # + 2n (2 0 " r, 1 f # ≤ n0 " r, 1 f # + n0 " r, 1 nfm + α(n − m) # + 3n0 " r, 1 f # + 3n0(r, f) + s(r) ≤ (4 + m)t0(r, f) + 3n0(r, f) + s(r) (3.30) since n > 2m + 12, m ≥ 2, t0(r, f) = nt0(r, f) + s(r) and using equations (3.4), (3.5) and (3.30), we deduce lim sup r→∞,r∈i 3n0(r, f) + ∑2 j=1 n (2) 0 & r, 1 f−cj ' + n0 $ r, 1 f′ % + 2n (2 0 $ r, 1 f % t0(r, f) ≤ n + 2m + 12 n < 2 (3.31) similarly, we get lim sup r→∞,r∈i 3n0(r, g) + ∑2 j=1 n (2) 0 & r, 1 g−cj ' + n0 $ r, 1 g′ % + 2n (2 0 $ r, 1 g % t0(r, g) < 2 (3.32) thus, from lemma 2.5, we have f = ag+b cg+d , where a, b, c, d ∈ c and ad − bc ̸= 0. hence, by using the same argument as that in theorem 1.8, we can prove the conclusion of theorem 1.10. proof of theorem 1.11 : since θ0(∞, f) > 5 6 and θ0(∞, g) > 5 6 , it follows that lim sup r→∞ n0(r, f) t0(r, f) < 1 6 , lim sup r→∞ n0(r, g) t0(r, g) < 1 6 (3.33) by lemma 2.5, (3.31)(3.33) and since n > 2m + 7, we deduce lim sup r→∞,r∈i 3n0(r, f) + ∑2 j=1 n (2) 0 & r, 1 f−cj ' − n0 $ r, 1 f′ % + 2n (2 0 $ r, 1 f % t0(r, f) (3.34) cubo 18, 1 (2016) uniqueness of meromorphic functions sharing a set in annuli 13 < 6 n lim sup r→∞,r∈i n0(r, f) t0(r, f) + lim sup r→∞,r∈i (n + 2m + 6)t0(r, f) nt0(r, f) < 2 (3.35) similarly, we get lim sup r→∞,r∈i 3n0(r, g) + ∑2 j=1 n (2) 0 & r, 1 g−cj ' + n0 $ r, 1 g′ % + 2n (2 0 $ r, 1 g % t0(r, g) < 2 (3.36) then, from lemma 2.5, we have f = ag+b cg+d , where a, b, c, d ∈ c and ad − bc ̸= 0. thus, by using the same argument as that in theorem 1.8, we can prove the conclusion of theorem 1.11. remark: the method used in this paper to prove the conclusion part of main results can be applied to the sets of zeros of polynomials containing only three terms including constant term, but not for more than three terms. acknowledgements: the authors are grateful to the referee for his/her keen observations, comments and valuable suggestions towards the improvement of the present paper. first author is supported by ugc sap drsiii with ref. no. f.510/3/drs-iii/2016(sap-i) dated: 29th feb. 2016. second author and third author were supported by ugc’s research fellowship in science for meritorious students, ugc, new delhi. ref. no.f.7101/2007(bsr) and ref. no. ku/sch/ugc-upe/2014-15/894. references [1] s.axler, harmonic functions from a complex analysis view point, amer. math. monthly 93 (1986), 246-258. [2] t.b.cao, z.s.deng, on the uniqueness of meromorphic functions that share three or two finite sets on annuli, proceedings of the indian academy of mathematical sciences, 122(2)(2012), 203-220. [3] t.b.cao, h.x.yi, h.y.xu, on the multiple values and uniqueness of meromorphic functions on annuli, comp. math. appl. 58(7)(2009), 1457-1465. [4] w.k.hayman, meromorphic functions, claredon press, oxford, 1964. [5] a.ya.khrystiyanyn, a.a.kondratyuk, on the nevanlinna theory for meromorphic functions on annuli-i, mathematychin studii, 23(2005), 19-30. [6] a.ya.khrystiyanyn, a.a.kondratyuk, on the nevanlinna theory for meromorphic functions on annuli-ii, mathematychin studii, 24(2005), 57-68. 14 renukadevi s. dyavanal, ashwini m. hattikal, madhura m. mathai cubo 18, 1 (2016) [7] w.c.lin, h.x.yi, uniqueness theorems for meromorphic functions that share three sets, complex variables, 48(4)(2003), 315-327. [8] h.y.xu, z.j.wu, the shared set and uniqueness of meromorphic functions on annuli, abst. appl. anal. 2013 (2013), 10 pages(article id 758318). [9] c.c.yang, x.h.hua, uniqueness and value sharing of meromorphic functions, ann. acad. sci. fenn. math. 22(1997), 395-406. [10] c.c.yang, h.x.yi, uniqueness theory of meromorphic functions, kluwer academic publishers, dordrecht,2003; chinese original: science press, beijing, 1995. [11] l.yang, value distribution theory, springer-verlag berlin, 1993. cubo a mathematical journal vol.15, no¯ 01, (131–149). march 2013 planar pseudo-almost limit cycles and applications to solitary waves bourama toni virginia state university, department of mathematics & computer science, petersburg va 23806. btoni@vsu.edu abstract we investigate the existence of pseudo-almost limit cycles, a new class of non-periodicity at the interface of the theories of limit cycles and pseudo-almost periodicity. we determine the conditions of existence for several systems including some pseudo-almost periodic perturbations of the harmonic oscillator and the renowned liénard systems. we apply to derive the existence of pseudo-almost periodic solitary waves by perturbing first then transforming some hyperbolic and parabolic partial differential equations to liénard-type equations. included also are open questions on the co-existence of limit cycles and strictly pseudo-almost periodic limit cycles partitioning the phase space, and the existence of isochronous pseudo-almost limit cycles. resumen investigamos la existencia de ciclos seudo-casi ĺımites, una nueva clase de no-periodicidad en la interfaz de las teoŕıas de ciclos ĺımites y seudo-casi periodicidad. determinamos condiciones de existencia de muchos sistemas, incluyendo algunas perturbaciones seudocasi periódicas del oscilador armónico y los sistemas de liénard. aplicamos las condiciones para derivar la existencia de ondas solitarias seudo-cuasi periódicas, primero perturbando y luego transformando algunas ecuaciones diferenciales parciales hiperbólicas y parabólicas a ecuaciones del tipo liénard. también se incluyen preguntas abiertas sobre la co-existencia de ciclos ĺımite y estrictamente pseudo-casi periódicos ciclos ĺımite de partición del espacio de fases, y la existencia de isócrono pseudo-casi ciclos ĺımite . keywords and phrases: limit cycles. almost and pseudo-almost periodic orbits. periodic waves. isochronous systems and isochrons. liénard systems. hyperbolic and parabolic equations. 2010 ams mathematics subject classification: 34c05, 34c07, 34c27, 34k14 132 bourama toni cubo 15, 1 (2013) 1 introduction limit cycles are used to model the dynamical state of self-sustained oscillations found very often in biology, chemistry, mechanics, electronics, fluid dynamics, etc. see for example [2, 16, 18, 26]. they often arise in many physical systems around a state at which energy generation and dissipation balance. one of the most important limit cycles of our lives is the heartbeat. a spectacular example is the tacoma narrows bridge1. and its 1940 dramatic collapse, where the limit cycle drew its energy from the wind and involved torsional oscillations of the roadbed. in robotics the stable gait to which the repeated dynamic walking pattern converges is modeled as a stable limit cycle, stability easily lost to even small disturbances, evidence of a narrow basin of attracting of the limit cycle. planar limit cycles were defined by poincaré2 in the famous paper mémoire sur les courbes définies par une équation différentielle [22], using his so-called method of sections. however much attention in this century has been drawn to the determination of the number, amplitude and configuration of limit cycles in a general nonlinear system, which is still an unsolved problem. this is part of the so-called hilbert’s 16th problem3. a weakened version4 by arnold called the tangential hilbert’s problem, concerns the bound on the number of limit cycles which can bifurcate from a first-order perturbation of a hamiltonian system.[3, 9, 13, 14, 17] the possibility of a limit cycle on a plane or a two-dimensional manifold is restricted to nonlinear dynamical systems, due to the fact that, for linear systems, kx(t) is also a solution for any constant k if x(t) is a solution. therefore the phase space will contain an infinite number of closed trajectories encircling the origin, with none of them isolated. conservative and gradient systems do not have limit cycles, but these systems may exhibit almost or pseudo-almost limit cycles. the most common techniques for predicting the absence or existence of periodicity and limit cycles include the index theory, dulac’s criterion, poincaré-bendixson test, perturbation and bifurcation theory, configuration of limit cycles, the toroidal principle. these concepts and related examples could be found in [2, 5, 6, 9, 10, 13, 18, 25]. the nonlinear character of isolated periodic oscillations renders their detection and construction challenging. in mechanical terms the appraisal of the regions of the phase plane where energy loss and energy gain occur might reveal a limit cycle. let us emphasize that even though in most studies periodicity has been illustrated more frequently, almost and pseudo-almost periodic oscillations or waves actually occur much more 1a wealth of information including historical and anecdotal facts could be found in http://en.wikipedia.org/wiki/tacoma-narrows-bridge(1940) 2jules henri poincaré has excelled in all fields of knowledge and is often described as a polymath or the last universalist. the famous poincaré conjecture named after him was finally solved in 2002-2003 by grigori perelman who turned down the related prize of $1,000,000! 3determine the maximum number and relative positions of limit cycles in polynomial vector fields of degree n. stated in 1900, it was only in 1987 that ecalle and ilyashenko proved independently the finiteness of that number using the compactification of the phase space to poincaré disk 4the number of limit cycles in a small perturbation of a polynomial hamiltonian system is given by the number of zeroes of abelian integrals at least far from polycycles. cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 133 frequently than periodic ones. for instance, in the simplest model of harmonic oscillator or mathematical pendulum, as well as for the one-dimensional wave equation, diverse kinds of oscillatory trajectories can be displayed, both periodic and more generally non-periodic. the theory of almost periodic functions introduced by h. bohr [6] in the 1920s and extended to pseudo-almost periodicity5 by zhang [27] in the 1990s is also connected with problems in differential equations, stability theory, dynamical systems, partial differential equations or equations in banach spaces. there are several results concerning the existence and uniqueness of almost and pseudoalmost periodic solutions for first-order differential equations, e.g., in [7, 11, 12, 15, 20, 21, 23, 24, 27]. but the authors usually derived their results from the existence of bounded solutions. we extend the theory of limit cycles and pseudo-almost periodicity to that of pseudo-almost limit cycles, isolated pseudo-almost periodic orbits, and we investigate in the current and future work the usual questions of conditions of existence and uniqueness, stability, bifurcation and perturbation, the coexistence of limit cycles and strictly pseudo-almost limit cycles. we also introduce the idea of isochronous pseudo-almost limit cycles and pseudo-almost isochrons6. section 2 overviews the theory of limit cycles recalling the definitions and presenting some classic and concrete examples relevant to our study. in section 3, we develop the concept of pseudoalmost limit cycle, its properties, several illustrative examples including the so-called linear pseudocenter, and existence theorems in the case of the well-known liénard systems. section 4 shows the applications of the existence theorems for liénard systems to obtain pseudo-almost periodic solitary waves for some hyperbolic and parabolic partial differential equations. finally in section 5 we discuss some directions for future research, and state some open problems, defining in the process the concept of isochronous pseudo-almost limit cycles and pseudo-almost isochrons. 2 preliminary definitions and examples let the multi-dimensional space rn represents all the possible states of a system modeling nonlinear phenomena. the dynamics of the system are determined by the values in rn in terms of the time. that is to say we define an evolution map or flow φ, smooth on the smooth manifold rn : φ : rn × r −→ rn, (2.1) such that φ(x,t) = y indicates that the state x ∈ rn evolved into the state y ∈ rn after t units of time, together with the usual flow properties φ(x,0) = x, φ(x,t1 + t2) = φ(φ(x,t1),t2). (2.2) 5any pseudo-almost periodic function is also a besicovitch almost periodic function 6the development of the concept of isochrons and the recognition of their significance is due to winfree (1980) 134 bourama toni cubo 15, 1 (2013) the flow φ then determines a vector field x (conversely as well) such that, for x ∈ m x(x) := ∂φ ∂t (x,0). (2.3) the orbit or trajectory of the flow through x ∈ rn is given by: o(x) := {φx(t) := φ(x,t)|t ∈ r}. (2.4) definition 2.1. the orbit γ = o(x) based at x is called a limit cycle if there is a neighborhood n of γ such that γ is the only periodic orbit contained in n . the limit cycle7 is stable (unstable) if ω(s) = γ (α(s) = γ) for any s ∈ n , that is, γ is the ω−limit set (α− limit set) of any point in n . in other words, the limit cycle, isolated periodic orbit of some period τ, is stable (resp. unstable) if it has a neighborhood n such that, for some distance function d on rn, d(φ(y,t),γ) −→ 0, as t → ∞ (resp. t → −∞), for any y ∈ n . note that the phase ϕ = t t0 of a limit cycle of period t0 refers to the relative position on the orbit, which is measured by the elapsed time (modulo the period) to go from a reference point to the current position on the limit cycle. the most common illustrative examples are from the perturbations of the linear center or linear isochrone. 2.1 linear center and its perturbations 2.1.1 poincaré oscillator the linear center or linear isochrone8 ẋ = −y, ẏ = x, (2.5) where the origin of the plane is surrounded by a continuum of periodic orbits (not isolated) given by x2 + y2 = c > 0, is perturbed into the following system, in polar coordinates (r,θ) ṙ = r(1 − r), θ̇ = 1 (2.6) the circle r = 1 is a 2π−periodic orbit and is unique. it is therefore a limit cycle. moreover r is a monotone function on each orbit (ṙ > 0 inside and < 0 outside) so that all non constant orbits tend towards the limit cycle which is therefore stable.9[2, 18].10 7a limit cycle actually controls the behavior of neighboring orbits, attracting/repelling on both sides, or attracting on one side and repelling on the other 8the term isochrone refers to the fact that all the periodic orbits in the continuum have the same constant period normalized to 2π. 9the poincaré’s oscillator has been considered a model of biological oscillations, in particular with respect to the effects of periodic stimulation of cardiac oscillators 10the isochrons here are radial lines from which the trajectories evolve to equal phase cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 135 2.1.2 limit cycles annulus the linear center could also be perturbed into a system to generate several limit cycles as in the following example. the c∞−system ẋ = −y + xp(x,y), ẏ = x + yp(x,y), (2.7) where p(x,y) = sin( 1 x2 + y2 )e − 1 x2+y2 , has an infinite number of limit cycles γn : x 2 + y2 = 1 nπ , n ∈ n (2.8) accumulating at the origin.[9] 2.1.3 remarks the linear center is a continuum of periodic orbits encircling a critical point. the perturbation in examples 1 and 2 has in fact destroyed these orbits to give birth to respectively a unique limit cycle in example 1, and an accumulating family of limit cycles in example 2. we will see below that a time-dependent pseudo-almost perturbation could lead to the emergence of the so-called pseudo-almost limit cycles. 3 pseudo-almost limit cycles 3.1 introductory concepts let c(r × ω,rn), ω ⊂ rn open, be the banach space of bounded continuous functions φ(t,x) endowed with the norm ||φ|| = supt∈r,x∈ω|φ(t,x)|. the set c(r × ω,rn) is a subset of the more general space lb(r × ω,rn) of all lebesgue measurable and bounded functions. definition 3.1. a function f in lb(r × ω,rn) is said to be ergodic if for every compact subset k ⊂ ω the mean defined by m(f) := lim t→∞ 1 2t ∫t −t f(t,x)dt, (3.1) exists uniformly for x ∈ k. we say that the function has a vanishing mean if m(f) = 0. let e(r × ω,rn) denote the space of all ergodic functions on r×ω. note in passing that not all uniformly continuous bounded 136 bourama toni cubo 15, 1 (2013) functions on r are ergodic. for instance the function f(t) = {1 − t2, for |t| < 1, and sin(log( 1 t2 )), for |t| ≥ 1, } (3.2) is uniformly continuous in r, but not ergodic. in the space l(r × ω,rn) of all lebesgue measurable functions on r × ω, we consider next the following subspace l0 of all functions φ : r × ω → rn such that ∀x ∈ ω, φ̃(.) := φ(.,x) is lebesgue measurable on r with m(|φ̃|) = 0, and m(|φ|) = 0. for example the function φ(t) = t| sinπt|t n , n > 6, (3.3) is unbounded, lebesgue measurable with vanishing mean m. the unbounded and discontinuous function φ(t) := { √ n, n ≤ t ≤ n + 1/n, and 0, otherwise} (3.4) is also an element of l0. indeed we have limt→∞ 12t ∫t −t |φ(t)|dt = limn→∞ 1 n ∑n k=1 1√ k = 0. definition 3.2. the orbit o(x0) based at x0 as defined above is called a pseudo-almost limit cycle if it is isolated, and more importantly if the function φ(.) := φx0(.) : r −→ rn defining the orbit is pseudo-almost periodic in the following sense: ∀ǫ > 0, ∃δ = δ(ǫ) > 0, a relatively dense subset dǫ ⊂ r, a subset cǫ ⊂ r, such that: (1) for m the lebesgue measure on r, lim t→∞ m(cǫ ∩ [−t,t]) 2t = 0, (cǫ is called an ergodic zero set), (3.5) (2) let tτφ denotes the translate of φ by τ, that is, (tτφ(t)) := φ(t + τ). then ||(tτφ)(t) − φ(t)|| < ǫ, τ ∈ dǫ, t,t + τ ∈ r − cǫ, (3.6) (3) finally |t1 − t2| < δ =⇒ ||φ(t1) − φ(t2)|| < ǫ, t1,t2 ∈ r − cǫ. (3.7) denote pa the space of pseudo-almost periodic functions. these functions satisfy the following properties widely available in the relevant literature. [11, 12, 27] 3.1.1 some properties of pseudo-almost periodicity we first give an equivalent definition of a pseudo-almost periodic function, in particular in the space c(r×ω,rn), with the restriction of l0 to the space e0 containing all functions φ ∈ c(r×ω,rn) cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 137 such that lim t→∞ 1 2t ∫t −t |φ(t,x)|dt = 0, (3.8) uniformly in x ∈ ω. definition 3.3. a function f : r × ω −→ rn is called pseudo-almost periodic in t uniformly on compact subsets k of ω if it has a unique decomposition in the form f(t,x) = a(t,x) + e(t,x), (3.9) where the component a is almost periodic, and the component e ∈ e ⊂ l0 is called the ergodic perturbation of f. recall that a is almost periodic if it satisfies the so-called bohr’s property. that is: ∀ǫ > 0 ∃l = l(ǫ) such that any interval (t,t + l) ⊂ r contains a number τǫ, the ǫ−almost period or ǫ−translation number, such that: ||f(t + τ,x) − f(t,x)|| < ǫ, t ∈ r,x ∈ ω. (3.10) we have the following properties relevant to our study and details could be found in zhang [27] and also in [11, 12]. (1) for f ∈ pa, the range f(r,k) := {f(t,x)|t ∈ r,x ∈ k} is bounded for every bounded subset k ⊂ ω. (2) the function f(t, .) ∈ pa is uniformly continuous in each bounded subset of ω uniformly in t. (3) when the ergodic zero set cǫ = ∅, the space pa coincides with the space ap of almost periodic functions. (4) if both functions f and its derivative f′ are pseudo-almost periodic, with f = a + e and f′ = a′ + e′, where a and a′ in pa and e and e′ in l0, then the functions a and e are differentiable with a′ = a and e′ = e. (5) the space pa is convolution invariant with the space l1(r) of integrable functions on r. 3.1.2 illustrative examples we present some by now classic examples of pseudo-almost periodic functions. see also [12, 27]. we include here their graphics. (1) example 1 the function φ1(t) = sint + sin √ 2t + e−|t| 1 + t2 (3.11) 138 bourama toni cubo 15, 1 (2013) has the almost periodic component a(t) = sint+ sin √ 2t, and the ergodic perturbation e(t) = e −|t| 1+t2 . we represent it along with its components in figure 1. figure 1: φ1(t) = sint + sin √ 2t + e −|t| 1+t2 (2) example 2 we have also the function φω(t) = i1(t) + i2(t), ω 6= 0, (3.12) with the almost periodic component i1(t) = ∫ ∞ −∞ h(t − s)(sins + sin √ 2s)ds, h ∈ l1(r) (3.13) and the ergodic component i2(t) = ∫ ∞ −∞ h(t − s) s2 + ω2 ds (3.14) we take h(t) = t2, in l1(r), ω = 1 to illustrate in figure 2. figure 2: φω(t) = i1(t) + i2(t) cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 139 3.2 existence of pseudo-almost limit cycles first note that a periodic or almost periodic function is also pseudo-almost periodic with a zero ergodic perturbation. consequently a limit cycle is also an almost or a pseudo-almost limit cycle, but not inversely to make the distinction, we will call strictly pseudo-almost limit cycles those pseudo-almost limit cycles that are not limit cycles. we start with the case of the linear pseudo-almost center. 3.2.1 linear pseudo-almost center: an example let p(t) ∈ pa(r,c) be a complex-valued pseudo-almost periodic function defined on the real numbers, and consider the differential equation (see also [11]) ẋ(t) = −αx(t) + p(t), α > 0. (3.15) define a kernel k(t) = { 0, for t < 0 e−αt, for t ≥ 0 } (3.16) we have k ∈ l1(r,c). thus the convolution xα(t) = (k ∗ p)(t) = e−αt ∫t −∞ eαsp(s)ds (3.17) is also in pa(r,c), for every α > 0. indeed the space pa is convolution invariant with l1. the equation being linear, it results in the existence of a continuum of parameterized pseudo-almost periodic solutions which we called linear pseudo-almost center. therefore these solutions are not isolated, and are not pseudo-almost limit cycles. a graphical representation for the case k(t) = t2, p(t) = sint + sin √ 2t, α = 1,2,3,4 is given in figure 3. figure 3: xα(t) = (k ∗ p)(t) = e−αt ∫t −∞ eαsp(s)ds. k(t) = t2, p(t) = sint + sin √ 2t. 140 bourama toni cubo 15, 1 (2013) 3.2.2 pseudo-almost periodic perturbations of the harmonic oscillator consider the forced oscillations of the harmonic oscillator given by ẍ(t) + x(t) = f(t) (3.18) where the forcing term is f(t) = − sin √ 2t + t2(t2 + 4) (t2 + 1)3 (3.19) or equivalently, for ẋ = y ẋ = y, ẏ = −x + f(t) (3.19a) clearly the function explicitly given by x(t) = sint + sin √ 2t + 1 t2 + 1 (3.20) is the unique solution of the equation and it is one of the classic examples of pseudo-almost periodic function that is not periodic. see also [11]. therefore we obtain an explicit example of pseudoalmost limit cycle. figure 4 gives the phase portrait of (3.19a) and the graph of the pseudo-almost periodic function in (3.20). figure 4 ẍ(t) + x(t) = − sin √ 2t + t 2 (t 2 +4) (t2+1)3 we further illustrate the theory of pseudo-almost limit cycles with the well-known liénard systems. 3.3 liénard pseudo-almost limit cycles liénard equation, which also generalizes the famous van der pol oscillator, is ubiquitous in the study of nonlinear systems. consider the one-parameter family of forced liénard systems ẍ + f(x)ẋ + g(x) = µh(t), (3.21) cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 141 or equivalently ẋ = y − f(x), ẏ = −g(x) + µh(t), (3.22) where f, g, are two functions generally nonlinear, continuous and differentiable from r to r, and h is a time-dependent continuous functions on r, µ ≥ 0 a small real parameter, and f(x) := ∫x 0 f(s)ds. for the homogeneous liénard systems at µ = 0 we recall the following classical result. see more details in, e.g., [5, 10, 18] theorem 3.4. if the homogeneous liénard systems satisfy the following conditions: (1) f(x) is continuous, even and f(0) < 0. (2) g(x) is locally lipschitz, odd, and such that xg(x) > 0 for x 6= 0. (3) f(x) has a unique positive zero at x = b, and it increases at ∞ for x > b. then there exists a unique stable limit cycle. therefore this theorem provides conditions under which there exists, for the unperturbed liénard systems, a unique limit cycle, isolated periodic orbit controlling the behavior of neighboring trajectories. we next show that we could subject some classes of liénard systems to perturbations that, in fact, destroy the limit cycles to give birth to strictly pseudo-almost limit cycles under suitable conditions. we study system (3.21) or its equivalent form (3.22) under the following additional assumptions: l1 : f(x) > 0, in r, with f(x)sgnx → ∞ as |x| → ∞. l2 : xg(x) > 0 for x 6= 0, g(x) → ∞ as |x| → ∞, with g(x) := ∫x 0 g(s)ds. l3 : |h(t)| ≤ k, and |h(t)| ≤ k, with h(t) = ∫t 0 h(s)ds, t ∈ r, and k a positive constant. l4 : g ′(x) > 0, and g′′(x) exists and is bounded. it is known that, under such assumptions, for 0 < µ ≪ 1, there exists in the xy-plane a region r bounded by a regular simple curve (c1 except possibly at a finite number of points) such that: (1) for every solution γ(t) = (x(t),y(t)) of system (3.21) there is a value t0 such that γ(t0) ∈ r. (2) if, for a value t0 of t, we have γ(t0) ∈ r, then we have also γ(t) ∈ r, for t ≥ t0. that is, solutions entering the set cannot leave it for increasing time. moreover the region r depends only on the functions f(x), g(x), h(t), the parameter µ and the constant k. equivalently, the region r may be described by the inequalities |x(t)| ≤ x0 |ẋ(t)| ≤ v0, for a solution x(t) of the equation (3.22), and where x0 and v0 are constants independent of µ. see, 142 bourama toni cubo 15, 1 (2013) for example, [7, 15, 21, 23]. in other words, under the above conditions the solutions ultimately settle in a c1−bounded region r in r2. actually we obtain lemma 3.5. assume the conditions l1, . . .,l4. let γ(t) = (x(t),y(t)) be a solution of the system, and γ̃(t) = (x̃(t), ỹ(t)) either another solution of the system or a solution of an associated system with a sufficiently small perturbation h̄(t) of the forcing term h(t). then we have lim t→∞ |γ̃(t) − γ(t)| = 0, (3.23) moreover there exists a unique solution x(t) for all t ∈ r. proof. let γ(t) = (x(t),y(t)) a solution of the system, and γ̃(t) = (x̃(t), ỹ(t)) either another solution of the system or a solution of an associated system with a sufficiently small perturbation h̄(t) of the forcing term h(t). lim t→∞ |γ̃(t) − γ(t)| = 0, is equivalent to lim t→∞ |x̃(t) − x(t)| = 0 = lim t→∞ |ỹ(t) − y(t)|. (3.24). upon the change of variables u(t) = x̃(t) − x(t), v(t) = x̃(t) − y(t), we obtain the system u̇(t) = v(t) − ϕ(t)u(t) v̇(t) = −ψ(t)u(t) + µ∆h(t), (3.25) where ϕ(t) = f(x2) − f(x1) x2 − x1 , ψ(t) = g(x2) − g(x1) x2 − x1 . (3.26) note that the function f, g′ and g′′ are bounded on the compact region r. for sufficiently small values of the parameter µ ≪ 1, we can construct a lyapunov-type quadratic form v(t,u,v) = ψ(t)u2 + v2 − 2cuv, (3.27) with c > 0 chosen small enough for v(t,u,v) to be positive definite such that v(t,u,v) ≥ c(u2 + v2), (3.28) c a positive constant, and such that v̇(t,u,v) + cv(t,u,v) < 0. (3.29) actually we have dv dt (t,u,v) = −2(ϕψ − ψ̇ − 2cψ)u2 − 2cv2 + 2cϕuv, (3.30) yielding ṽ(t,u,v) := v̇(t,u,v) + cv(t,u,v) = −(2ϕψ − ψ̇ − 3cψ)u2 + 2c(ϕ − c)uv − cv2. (3.31) cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 143 the quadratic form ṽ(t,u,v) can be made negative definite by taking the constant c such that c < 2ϕψ − ψ̇ 3ψ , c(3ψ + (ϕ − c)2) < 2ϕψ − ψ̇, (3.32) which entails v̇(t,u,v) < v(t0)e −c(t−t0). (3.33) therefore v(t) → 0 as t → ∞, implying that u → 0 and v → 0. the constant c is appropriately chosen so that, when |∆h(t)| = |h̃(t) − h(t)| → 0, we can make v(t) → 0 for t → ∞. that is, the solutions of the system of the perturbed forcing term ultimately converge to the solutions of the original system. next let γ(t) = (x(t),y(t)) be one of these solutions which settled in r for t ≥ t0. we then define the sequence of solutions γn(t) = γ(t + n) = (xn(t),yn(t)), t ≥ t0 − n. the sequence is therefore equicontinuous and uniformly bounded. consequently we can extract a subsequence γnk(t) converging uniformly to a solution γ̄(t) = (x̄(t), ȳ(t)) lying completely in r for all t ∈ r. (limn→∞(t0 + n,∞) = (−∞,∞)). and of course γ̄(t) is unique. 3.3.1 remarks indeed the proof of the theorem actually accomplishes the followings: the solutions of the system associated to the perturbed forcing term ultimately converge to the solutions of the original system; moreover, under the assumptions above, only one solution of the system settles in the bounded region r for all time in r.; as we show below, that single solution will be of the same nature as the forcing term, when it becomes pseudo-almost periodic. in a previous work, [24] the case of the pseudo-almost periodic forcing was presented as a corollary to that of almost periodic forcing; here we present a more elegant and self-contained proof drawing from the above definitions of pseudo-almost periodicity, definitions not used in the cited work. we state and prove theorem 3.6. assume the forcing term h(t) is a pseudo-almost periodic function. then under the conditions l1, . . .,l4, the forced liénard system exhibits a unique asymptotically stable pseudoalmost limit cycle. proof. the proof is based on the previous lemma, including the existence of a unique solution enclosed in r for all time. first assuming the forcing term h(t) is pseudo-almost periodic entails from the definition above that, for any arbitrary ǫ, there exists δ = δ(ǫ), an ǫ−pseudo-almost period τ ∈ dǫ, a relatively dense set in r such that ‖h(t + τ) − h(t)‖ < ǫ, t,t + τ ∈ r − cǫ (3.34) 144 bourama toni cubo 15, 1 (2013) and |t1 − t2| < δ =⇒ ||h(t1) − h(t2)|| < ǫ, t1,t2 ∈ r − cǫ, (3.35) where cǫ is the ergodic zero set defined above. for such an ǫ− pseudo-almost period consider the unique solution γ̄(t) given in the previous lemma that settles in r for all time t ∈ (−∞,∞), and the associated function γ̄(t + τ) = (x̄(t + τ), ȳ(t + τ)). this function is readily a solution of the following system (eτ) ẋ = y − f(x, ẏ = −g(x) + µh(t + τ), (3.36) take h(t + τ) as a sufficiently small perturbation of h(t) as above. therefore, according to the previous propositions, the solutions γ̄(t) and γ̄(t + τ) converge. thus we obtain ‖γ̄(t + τ) − γ̄(t)‖ < ǫ, t,t + τ ∈ r − cǫ. (3.37) moreover we also have, for t1,t2 ∈ r − cǫ, |γ̄(t2) − γ̄(t1)| ≤ |t2 − t1|supr| ˙̄γ|, (3.38) which ensures the existence of δ such that |t1 − t2| < δ =⇒ ||γ̄(t1) − γ̄(t2)|| < ǫ, t1,t2 ∈ r − cǫ, (3.39) therefore we conclude that the unique solution γ̄(t) is pseudo-almost periodic. moreover, from the previous lemma, all other solutions of the system that ultimately settle in r converge to this unique pseudo-almost periodic solution γ̄(t) ∈ r. therefore the system has a unique (isolated) almost periodic solution to which any other solution unwinds in the c1−bounded set r. it is a stable pseudo-almost limit cycle as defined above. hence the claim. 4 pseudo-almost periodic waves the importance of liénard systems among nonlinear systems also comes from the fact that several systems can be transformed into liénard systems and solved. [1, 19]. we present next some partial differential equations solvable first by reducing them to some liénard-type equations, then by applying the previous theorems. 4.1 hyperbolic pseudo-almost periodic wave consider systems described by the time-perturbed nonlinear hyperbolic equation utt = uxx + f0(u)ux + g0(u) + p(t) (h) cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 145 the search of special solutions of the form u(x,t) = y(x + λt), λ ∈ r (4.1) defining the wave with speed v = |λ|, yields the liénard-type equation (1 − λ2)ÿ + f0(y)ẏ + g0(y) = −p(t) (4.2) define f(y) = f0(y) 1−λ2 , g(y) = g0(y) 1−λ2) , and h(t) = −p(t) 1−λ2 . the functions f0 and g0 are continuously differentiable chosen together with the speed v = |λ| of the waves u(t,x) such that the function f, g, and h satisfy the assumptions l1, . . .,l4. obviously assuming p(t) pseudo-almost periodic implies h(t) is pseudo-almost periodic. therefore we conclude under these assumptions theorem 4.1. for a pseudo-almost periodic perturbation p(t), the nonlinear hyperbolic equation (h) has a pseudo-almost periodic solitary wave u(x,t) = y(x + λt), where y(x) is the unique pseudo-almost limit cycle of the perturbed liénard-type equation (4.2). proof. the proof is immediate and is adapted from theorem (3.6). we next consider a parabolic partial differential equation describing a reaction-diffusion model. 4.2 parabolic pseudo-almost periodic wave: a reaction-diffusion model consider now the time-perturbed parabolic equation describing a reaction-diffusion model ut = uxx + f0(u)ux + g0(u) + p(t) (rd) looking again for special solutions of the form (4.1) leads to the liénard-type equation ÿ + [f0(y) − λ]ẏ + g0(y) = 0 (4.3) as in the previous case we set f(y) = f0(y) − λ, g(y) = g0(y), and h(t) = −p(t). the functions f0 and g0 are continuously differentiable and determined together with the speed |λ| of the waves u(t,x) such that the function f, g, and h satisfy the assumptions l1, . . .,l4. again assuming p(t) pseudo-almost periodic implies h(t) is also pseudo-almost periodic. we therefore obtain the equivalent theorems of existence of pseudo-almost solitary waves to the reaction-diffusion equation as functions of the corresponding liénard pseudo-almost limit cycles. that is, theorem 4.2. for a pseudo-almost periodic perturbation p(t), the nonlinear parabolic equation(rd) has a pseudo-almost periodic solitary wave u(x,t) = y(x + λt), where y(x) is the unique pseudoalmost limit cycle of the perturbed liénard-type equation (4.3). 146 bourama toni cubo 15, 1 (2013) 5 outlook and open problems arnold in [3] states une trajectoire fermée nondégénérée ne disparait pas par une petite déformation du système, mais se déforme légèrement. donc le système des trajectoires est structurellement stable dans le voisinage de la trajectoire fermée générique that is, periodic orbits do not just disappear under small perturbation, but they may be slightly deformed, due to the fact that the system of trajectories is structurally stable in the neighborhood of a periodic orbit. many forced systems such as the liénard ones are actually small perturbations of systems having periodic orbits (limit cycles) in their unperturbed form, and many results do imply the disappearance of these orbits upon perturbation. the appearing of pseudo-almost periodic solutions could result from the deformation/bifurcation of existing orbits. therefore one must investigate the relation between the “new” pseudo-almost periodic solutions appearing upon perturbation and the periodic-type orbits of the unperturbed system, including the question in the following open problem 1. (1) open problem 1: co-existence of limit cycles and strictly pseudo-almost limit cycles for parameterized systems, including the above liénard systems, investigate conditions under which co-exist limit cycles and strictly almost or pseudo-almost limit cycles partitioning the phase space. (2) open problem 2: isochronous pseudo-almost limit cycles let γ be a strictly pseudo-almost limit cycle of a flow φ on rn. a point x1 in r n has asymptotic phase with respect to γ if there is a point x0 ∈ γ such that limt−→±∞ |φt(x1) − φt(x0)| = 0. we say that x1 is in phase with x0. it is well known that a hyperbolic limit cycle has some neighborhood where every point has asymptotic phase with respect to the limit cycle, due to the existence of invariant foliation.[8] similar question needs to be addressed as well in case of strictly pseudo-almost limit cycles. definition 5.1. a strictly pseudo-almost limit cycle is said to be isochronous if there is a neighborhood of γ in which every point is in phase with a point on γ. in the case of limit cycles, we have, for instance, the following examples. system ṙ = − 1 3 (r − 1)4e|r−1| −3 , θ̇ = 2π (5.1) has a nonhyperbolic limit cycle at the unit cycle with period 1, attracting for r > 1. the asymptotic phase of any point (r0,θ0) in its neighborhood is (1,θ0). the limit cycle is therefore isochronous. for more details see [8]. it would be interesting to: cubo 15, 1 (2013) planar pseudo-almost limit cycles and applications ... 147 (a) perturb system (5.1), in particular in the angle variable, and study the conditions of appearance of strictly pseudo-almost limit cycles. (b) investigate the conditions of existence of isochronous strictly pseudo-almost limit cycles, in particular for the forced liénard systems. (c) investigate the bifurcation of pseudo-almost limit cycles from an isochronous period annulus, as in [25] (3) open problem 3: pseudo-almost isochrons as above, we further define: definition 5.2. given x0 ∈ γ where γ is a strictly pseudo-almost limit cycle, a pseudoalmost isochron i(x0) based at x0 is the set of all point x ∈ rn in phase with x0. as in the case of limit cycles we conjecture the existence of pseudo-almost isochrons, and that they will foliate the neighborhood of pseudo-almost limit cycles. their determination is definitely an interesting but difficult question of research. one line of attack might be similar to guckenheimer and winfree investigation of isochrons of limit cycles. 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(1) the function ω1 is continuous at t = 0 if and only if f is uniformly continuous on r. so that here ω1 (f, t) → ω1 (f, 0) = 0, as t → 0. it also holds ω1 (f, λt) ≤ (λ + 1) ω1 (f, t) , λ ≥ 0. (2) clearly ω1 (f, t) is finite for each t ≥ 0. in [1] we studied extensively the convergence to the unit operator of various integral singular operators. here we define the discrete analogs of these operators next, and we study their uniform convergence to the unit operator with rates. let 0 < ξ ≤ 1, such that ξ → 0+, x ∈ r; 1 ξ ≥ 1. i) we define the discrete picard operators: ( p∗ξf ) (x) := ∑ ∞ ν=−∞ f (x + ν) e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ . (3) ii) we define the discrete gauss-weierstrass operators: ( w∗ξf ) (x) := ∑ ∞ ν=−∞ f (x + ν) e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ . (4) iii) we define the general discrete poisson-cauchy operators: let α ∈ n, β > 1 α ; ( m∗ξf ) (x) := ∑ ∞ ν=−∞ f (x + ν) ( ν2α + ξ2α )−β ∑ ∞ ν=−∞ (ν2α + ξ2α) −β . (5) iv) we define the basic discrete convolution operators: let ϕ : r → r, with ‖ϕ‖ ∞ := sup x∈r |ϕ (x)| ≤ k, k > 0, β ∈ n − {1}; ( θ∗ξf ) (x) := f (x) + ∑ ν∈z−{0} f (x + ν) ( ϕ( νξ ) ν ξ )2β 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ν ξ )2β . (6) cubo 15, 1 (2013) approximation by discrete singular operators 99 v) we define the general discrete convolution operators: let ϕ : r → r+ with ϕ (x) ≤ ax2β, ∀ x ∈ r, β ∈ n − {1}, a > 0; ( t∗ξf ) (x) := f (x) + ∑ ν∈z−{0} f (x + ν) ϕ( νξ ) ( νξ ) 4β 1 + ∑ ν∈z−{0} ϕ( νξ ) ( νξ ) 4β . (7) the above operators, as we will see, are well defined and are linear, positive, and bounded when ‖f‖ ∞ := sup x∈r |f (x)| < ∞. furthermore p∗ξ (1) = w ∗ ξ (1) = m ∗ ξ (1) = θ ∗ ξ (1) = t ∗ ξ (1) = 1, (8) with ∥ ∥p∗ξ ∥ ∥ = ∥ ∥w∗ξ ∥ ∥ = ∥ ∥m∗ξ ∥ ∥ = ∥ ∥θ∗ξ ∥ ∥ = ∥ ∥t∗ξ ∥ ∥ = 1, (9) on continuous bounded functions. in this article we are motivated by [3]. 2 main results all here as in preliminaries earlier. we start with the basic approximation properties of discrete picard operators. we present theorem 2.1. it holds ∥ ∥p∗ξf − f ∥ ∥ ∞ ≤   1 + 2e− 1 ξ ( 2ξ + 2 + 1 ξ ) 1 + 2ξe− 1 ξ  ω1 (f, ξ) . (10) the constant in the right hand side of (10) converges to 1 as ξ → 0+. so that p∗ξ → i (unit operator), uniformly with rates, as ξ → 0 + . proof. we will use a lot ∞∑ ν=1 1 ν2 = π2 6 (euler, 1741). we see that −∞∑ ν=−1 e− |ν| ξ = ∞∑ ν=1 e− ν ξ < ∞∑ ν=1 1 ν2 = π2 6 , it converges. 100 george a. anastassiou cubo 15, 1 (2013) thus ∞∑ ν=−∞ e− |ν| ξ = 2 ∞∑ ν=1 e− ν ξ + 1 < π2 3 + 1. (11) using [4] we obtain ∞∑ ν=1 e− ν ξ − e− 1 ξ ≤ ∫ ∞ 1 e− ν ξ dν ≤ ∞∑ ν=1 e− ν ξ . (12) hence 2 ∫ ∞ 1 e− ν ξ dν + 1 ≤ 2 ∞∑ ν=1 e− ν ξ + 1 = ∞∑ ν=−∞ e− |ν| ξ . (13) thus 0 < 1 ∑ ∞ ν=−∞ e− |ν| ξ ≤ 1 2 ∫ ∞ 1 e− ν ξ dν + 1 = 1 2ξe− 1 ξ + 1 → 1, as ξ → 0 + . (14) we need to prove that g (ν) = νe− ν ξ is decreasing for ν ≥ 1. indeed we have that g′ (ν) = e− ν ξ ( 1 − ν ξ ) ≤ 0, by ξ ≤ 1 ≤ ν. so that, again by [4], we get that 1 + 2 ∞∑ ν=1 ( 1 + ν ξ ) e− ν ξ ≤ 1 + 2 [∫ ∞ 1 ( 1 + ν ξ ) e− ν ξ dν + ( 1 + 1 ξ ) e− 1 ξ ] =: (∗) (15) using integration by parts we have ∫ ∞ 1 ξ xe−xdx = −e−x (x + 1) |∞1 ξ = e− 1 ξ ( 1 ξ + 1 ) . (16) hence we get ∫ ∞ 1 ( 1 + ν ξ ) e− ν ξ dν = ∫ ∞ 1 e− ν ξ dν + ∫ ∞ 1 ν ξ e− ν ξ dν = ξe− 1 ξ + ξ ∫ ∞ 1 ξ xe−xdx = e− 1 ξ (2ξ + 1) . (17) therefore (∗) = 1 + 2 [ e− 1 ξ (2ξ + 1) + ( 1 + 1 ξ ) e− 1 ξ ] = 1 + 2e− 1 ξ ( 2ξ + 2 + 1 ξ ) . (18) consequently we have found that ∞∑ ν=−∞ ( 1 + |ν| ξ ) e− |ν| ξ = 1 + 2 ∞∑ ν=1 ( 1 + ν ξ ) e− ν ξ ≤ 1 + 2e− 1 ξ ( 2ξ + 2 + 1 ξ ) (finite) → 1, as ξ → 0 + . (19) cubo 15, 1 (2013) approximation by discrete singular operators 101 finally we observe ( p∗ξf ) (x) − f (x) = ∑ ∞ ν=−∞ (f (x + ν) − f (x)) e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ . (20) so that ∣ ∣ ( p∗ξf ) (x) − f (x) ∣ ∣ ≤ ∑ ∞ ν=−∞ |f (x + ν) − f (x)| e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ (21) ≤ ∑ ∞ ν=−∞ ω1 (f, |ν|) e − |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ = ∑ ∞ ν=−∞ ω1 ( f, ξ |ν| ξ ) e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ (by (2)) ≤ ω1 (f, ξ)   ∑ ∞ ν=−∞ ( 1 + |ν| ξ ) e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ   (22) (by (14), (19)) ≤ ω1 (f, ξ) ( 1 + 2e− 1 ξ ( 2ξ + 2 + 1 ξ )) ( 2ξe− 1 ξ + 1 ) . (23) we notice that ( 1 + 2e− 1 ξ ( 2ξ + 2 + 1 ξ )) ( 2ξe− 1 ξ + 1 ) → 1, as ξ → 0 + . we have proved ∣ ∣ ( p∗ξf ) (x) − f (x) ∣ ∣ ≤   1 + 2e− 1 ξ ( 2ξ + 2 + 1 ξ ) 1 + 2ξe− 1 ξ  ω1 (f, ξ) , (24) ∀ x ∈ r. the proof now is completed. next we prove preservation of global smoothness of p∗ξ. theorem 2.2. it holds ω1 ( p∗ξf, δ ) ≤ ω1 (f, δ) , ∀ δ > 0. (25) inequality (25) is sharp, namely it is attained by f (x) = identity (x) = x. proof. we see that ( p∗ξf ) (x) − ( p∗ξf ) (y) = ∑ ∞ ν=−∞ (f (x + ν) − f (y + ν)) e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ . (26) 102 george a. anastassiou cubo 15, 1 (2013) hence ∣ ∣ ( p∗ξf ) (x) − ( p∗ξf ) (y) ∣ ∣ ≤ ∑ ∞ ν=−∞ |f (x + ν) − f (y + ν)| e− |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ ≤ ∑ ∞ ν=−∞ ω1 (f, |x − y|) e − |ν| ξ ∑ ∞ ν=−∞ e− |ν| ξ = ω1 (f, |x − y|) . (27) so that for any x, y ∈ r : |x − y| < δ we get (25). if f = id, then trivially we get ( p∗ξid ) (x) − ( p∗ξid ) (y) = x − y = id (x) − id (y) , (28) thus (25) is attained. next we study the approximation properties of discrete gauss-weierstrass operators. theorem 2.3. let f ∈ cu (r), 0 < ξ ≤ 1. then ∥ ∥w∗ξf − f ∥ ∥ ∞ ≤ c (ξ) ω1 ( f, √ ξ ) , (29) where c (ξ) :=  1 +   e− 1 ξ (√ ξ + 2 + 2√ ξ ) √ πξ ( 1 − erf ( 1√ ξ )) + 1     . (30) we have lim ξ→0+ c (ξ) = 1, and by lim ξ→0+ ω1 ( f, √ ξ ) = 0, we get w∗ξ → i uniformly with rates, as ξ → 0 + . proof. we notice easily that −∞∑ ν=−1 e− ν2 ξ = ∞∑ ν=1 e− ν2 ξ < ∞∑ ν=1 1 ν2 = π2 6 < ∞. (31) so we can write ∞∑ ν=−∞ e− ν2 ξ = 2 ∞∑ ν=1 e− ν2 ξ + 1 < π2 3 + 1. (32) since e− ν2 ξ is positive, continuous and decreasing, by [4], we get ∞∑ ν=1 e− ν2 ξ − e− 1 ξ ≤ ∫ ∞ 1 e− ν2 ξ dν ≤ ∞∑ ν=1 e− ν2 ξ . (33) so that 2 ∫ ∞ 1 e− ν2 ξ dν + 1 ≤ 2 ∞∑ ν=1 e− ν2 ξ + 1 = ∞∑ ν=−∞ e− ν2 ξ , (34) cubo 15, 1 (2013) approximation by discrete singular operators 103 and 0 < 1 ∑ ∞ ν=−∞ e− ν2 ξ ≤ 1 2 ∫ ∞ 1 e− ν2 ξ dν + 1 . (35) we know that ∫ ∞ 0 e−t 2 dt = √ π 2 , and erf (x) := 2√ π ∫x 0 e−t 2 dt, with erf (∞) = 1. hence 2 ∫ ∞ 1 e− ν2 ξ dν + 1 = 2 √ ξ ∫ ∞ 1 e − ( ν √ ξ ) 2 d ( ν√ ξ ) + 1 = (36) 2 √ ξ ∫ ∞ 1 √ ξ e−θ 2 dθ + 1 = 2 √ ξ [∫ ∞ 0 e−θ 2 dθ − ∫ 1 √ ξ 0 e−θ 2 dθ ] + 1 = 2 √ ξ [√ π 2 − √ π 2 erf ( 1√ ξ )] + 1 = √ πξ ( 1 − erf ( 1√ ξ )) + 1. (37) therefore 2 ∫ ∞ 1 e− ν2 ξ dν + 1 = √ πξ ( 1 − erf ( 1√ ξ )) + 1 → 1, as ξ → 0 + . (38) so we got that 0 < 1 ∑ ∞ ν=−∞ e− ν2 ξ ≤ 1 √ πξ ( 1 − erf ( 1√ ξ )) + 1 → 1, as ξ → 0 + . (39) next we prove that g (ν) = νe− ν2 ξ is decreasing for ν ≥ 1. indeed we have g′ (ν) = e− ν2 ξ ( 1 − 2ν 2 ξ ) ≤ 0, iff 1 − 2ν 2 ξ ≤ 0, iff ξ ≤ 2ν2, which is true. so that we have (by [4]) ∞∑ ν=1 ( 1 + ν√ ξ ) e− ν2 ξ ≤ ∫ ∞ 1 ( 1 + ν√ ξ ) e− ν2 ξ dν + ( 1 + 1√ ξ ) e− 1 ξ = (40) ∫ ∞ 1 e− ν2 ξ dν + ∫ ∞ 1 ν√ ξ e− ν2 ξ dν + e− 1 ξ + e− 1 ξ √ ξ = √ πξ 2 ( 1 − erf ( 1√ ξ )) + √ ξ 2 ∫ ∞ 1 e− ν2 ξ d ( ν2 ξ ) + e− 1 ξ + e− 1 ξ √ ξ = √ πξ 2 ( 1 − erf ( 1√ ξ )) + √ ξ 2 ∫ ∞ 1 ξ e−xdx + e− 1 ξ + e− 1 ξ √ ξ = √ πξ 2 ( 1 − erf ( 1√ ξ )) + √ ξ 2 e− 1 ξ + e− 1 ξ + e− 1 ξ √ ξ . (41) that is ∞∑ ν=1 ( 1 + ν√ ξ ) e− ν2 ξ ≤ √ πξ 2 ( 1 − erf ( 1√ ξ )) + e− 1 ξ ( √ ξ 2 + 1 + 1√ ξ ) (42) (finite) → 0, as ξ → 0 + . 104 george a. anastassiou cubo 15, 1 (2013) since ∞∑ ν=−∞ ( 1 + |ν|√ ξ ) e− ν2 ξ = 2 ∞∑ ν=1 ( 1 + ν√ ξ ) e− ν2 ξ + 1 < ∞, (43) we find ∞∑ ν=−∞ ( 1 + |ν|√ ξ ) e− ν2 ξ ≤ √ πξ ( 1 − erf ( 1√ ξ )) + e− 1 ξ ( √ ξ + 2 + 2√ ξ ) + 1 (44) (is finite) → 1, as ξ → 0 + . next we observe that ( w∗ξf ) (x) − f (x) = ∑ ∞ ν=−∞ (f (x + ν) − f (x)) e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ . (45) thus ∣ ∣ ( w∗ξf ) (x) − f (x) ∣ ∣ ≤ ∑ ∞ ν=−∞ |f (x + ν) − f (x)| e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ ≤ (46) ∑ ∞ ν=−∞ ω1 (f, |ν|) e − ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ = ∑ ∞ ν=−∞ ω1 ( f, √ ξ |ν|√ ξ ) e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ ≤ (47) ω1 ( f, √ ξ )∑ ∞ ν=−∞ ( 1 + |ν|√ ξ ) e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ (48) (by (39), (44)) ≤ ω1 ( f, √ ξ )   √ πξ ( 1 − erf ( 1√ ξ )) + e− 1 ξ (√ ξ + 2 + 2√ ξ ) + 1 √ πξ ( 1 − erf ( 1√ ξ )) + 1   = (49) ω1 ( f, √ ξ )  1 + e− 1 ξ (√ ξ + 2 + 2√ ξ ) √ πξ ( 1 − erf ( 1√ ξ )) + 1   . so we have proved that ∣ ∣ ( w∗ξf ) (x) − f (x) ∣ ∣ ≤ ω1 ( f, √ ξ )  1 + e− 1 ξ (√ ξ + 2 + 2√ ξ ) √ πξ ( 1 − erf ( 1√ ξ )) + 1   , (50) ∀ x ∈ r, any 0 < ξ ≤ 1. the constant in the last inequality converges to 1, as ξ → 0 + . the proof of the theorem is completed. it follows the global smoothness preservation property of w∗ξ. cubo 15, 1 (2013) approximation by discrete singular operators 105 theorem 2.4. it holds ω1 ( w∗ξf, δ ) ≤ ω1 (f, δ) , ∀ δ > 0. (51) inequality (51) is sharp, attained by f (x) = id (x) = x. proof. we see that ( w∗ξf ) (x) − ( w∗ξf ) (y) = ∑ ∞ ν=−∞ (f (x + ν) − f (y + ν)) e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ , ∀ x, y ∈ r. (52) hence ∣ ∣ ( w∗ξf ) (x) − ( w∗ξf ) (y) ∣ ∣ ≤ ∑ ∞ ν=−∞ |f (x + ν) − f (y + ν)| e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ ≤ ω1 (f, |x − y|)   ∑ ∞ ν=−∞ e− ν2 ξ ∑ ∞ ν=−∞ e− ν2 ξ   = ω1 (f, |x − y|) , ∀ x, y ∈ r, (53) proving (51). sharpness is obvious. next we study the approximation properties of general discrete poisson-cauchy operators. theorem 2.5. let f ∈ cu (r), 0 < ξ ≤ 1. then ∥ ∥m∗ξf − f ∥ ∥ ≤ d (ξ) ω1 (f, ξ) , (54) where d (ξ) := [ 1 + 4ξ2αβ ( αβ 2αβ − 1 ) + ξ2αβ−1 ( 2αβ − 1 αβ − 1 )] . (55) we have lim ξ→0+ d (ξ) = 1, and by lim ξ→0+ ω1 (f, ξ) = 0, we get m ∗ ξ → i uniformly with rates, as ξ → 0 + . proof. here 0 < ξ ≤ 1, α ∈ n, β > 1 α , x ∈ r. by [5], p. 397, formula 595, we have ∫ ∞ 0 1 (t2α + ξ2α) β dt = γ ( 1 2α ) γ ( β − 1 2α ) 2γ (β) αξ2αβ−1 . (56) clearly ( ν2α + ξ2α )−β is decreasing, continuous and positive for ν ∈ [1, ∞). hence by [4], we get 0 < ∞∑ ν=1 ( ν2α + ξ2α )−β ≤ ( 1 + ξ2α )−β + ∫ ∞ 1 ( ν2α + ξ2α )−β dν ≤ (57) ( 1 + ξ2α )−β + ∫ ∞ 0 ( ν2α + ξ2α )−β dν = ( 1 + ξ2α )−β + γ ( 1 2α ) γ ( β − 1 2α ) 2γ (β) αξ2αβ−1 < ∞, ∀ ξ ∈ (0, 1]. 106 george a. anastassiou cubo 15, 1 (2013) consequently we find convergence of 0 < s1 := ∞∑ ν=−∞ ( ν2α + ξ2α )−β = ξ−2αβ + 2 ∞∑ ν=1 ( ν2α + ξ2α )−β ≤ ξ−2αβ + 2 ( 1 + ξ2α )−β + γ ( 1 2α ) γ ( β − 1 2α ) γ (β) αξ2αβ−1 < ∞, ∀ ξ ∈ (0, 1]. (58) similarly we have ∞∑ ν=1 ( ν2α + ξ2α )−β ≥ ∫ ∞ 1 ( ν2α + ξ2α )−β dν, (59) and ∞∑ ν=−∞ ( ν2α + ξ2α )−β ≥ ξ−2αβ + 2 ∫ ∞ 1 ( ν2α + ξ2α )−β dν. (60) that is 0 < 1 ∑ ∞ ν=−∞ (ν2α + ξ2α) −β ≤ 1 ξ−2αβ + 2 ∫ ∞ 1 (ν2α + ξ2α) −β dν < ξ2αβ. (61) that 0 < 1 s1 < ξ2αβ → 0, as ξ → 0 + . (62) hence lim ξ→0+ 1 s1 = 0. (63) call g (ν) := ν ( ν2α + ξ2α )−β , ν ∈ [1, ∞). we have that g′ (ν) = ( ν2α + ξ2α )−β [ 1 − ( 2αβν2α ν2α + ξ2α )] ≤ 0, (64) iff 1 − ( 2αβν 2α ν2α+ξ2α ) ≤ 0, iff ν2α + ξ2α ≤ 2αβν2α, iff ξ2α ≤ ν2α (2αβ − 1), which is true because 2αβ − 1 ≥ 1 and ν2α (2αβ − 1) ≥ 1 ≥ ξ2α. that is g is decreasing, positive and continuous on [1, ∞). hence ( 1 + ν ξ ) ( ν2α + ξ2α )−β is decreasing, positive and continuous on [1, ∞). thus again by [4] we derive ∞∑ ν=1 ( 1 + ν ξ ) ( ν2α + ξ2α )−β ≤ (65) ( 1 + 1 ξ ) ( 1 + ξ2α )−β + ∫ ∞ 1 ( 1 + ν ξ ) ( ν2α + ξ2α )−β dν. we further notice that ∫ ∞ 1 ( 1 + ν ξ ) ( ν2α + ξ2α )−β dν = ∫ ∞ 1 ( ν2α + ξ2α )−β dν + 1 ξ ∫ ∞ 1 ν ( ν2α + ξ2α )−β dν < cubo 15, 1 (2013) approximation by discrete singular operators 107 ∫ ∞ 1 ν−2αβdν + 1 ξ ∫ ∞ 1 ν−2αβ+1dν = ( 1 2αβ − 1 ) + (66) ( 1 2ξ (αβ − 1) ) < ∞, ∀ ξ ∈ (0, 1]. so that ∞∑ ν=1 ( 1 + ν ξ ) ( ν2α + ξ2α )−β < (67) ( 1 + 1 ξ ) ( 1 + ξ2α )−β + ( 1 2αβ − 1 ) + ( 1 2ξ (αβ − 1) ) < ∞, ∀ ξ ∈ (0, 1]. consequently we obtain 0 < s2 := ∞∑ ν=−∞ ( 1 + |ν| ξ ) ( ν2α + ξ2α )−β = (68) ξ−2αβ + 2 ∞∑ ν=1 ( 1 + ν ξ ) ( ν2α + ξ2α )−β < ξ−2αβ + 2 ( 1 + 1 ξ ) ( 1 + ξ2α )−β + ( 2 2αβ − 1 ) + ( 1 ξ (αβ − 1) ) < 1 ξ2αβ + 2 ( 1 + 1 ξ ) + ( 2 2αβ − 1 ) + ( 1 ξ (αβ − 1) ) =: ϕ (ξ) . so that 0 < s2 < ϕ (ξ) < ∞, ∀ ξ ∈ (0, 1], (69) and 0 < s2 s1 (62) < ξ2αβϕ (ξ) = 1 + 2ξ2αβ ( 1 + 1 2αβ − 1 ) + ξ2αβ−1 ( 2 + 1 αβ − 1 ) = [ 1 + 4ξ2αβ ( αβ 2αβ − 1 ) + ξ2αβ−1 ( 2αβ − 1 αβ − 1 )] → 1, as ξ → 0 + . (70) hence 0 < lim ξ→0+ s2 s1 < 1. (71) finally we have that m∗ξ (f, x) − f (x) = ∑ ∞ ν=−∞ (f (x + ν) − f (x)) ( ν2α + ξ2α )−β ∑ ∞ ν=−∞ (ν2α + ξ2α) −β , (72) and ∣ ∣m∗ξ (f, x) − f (x) ∣ ∣ ≤ ∑ ∞ ν=−∞ |f (x + ν) − f (x)| ( ν2α + ξ2α )−β ∑ ∞ ν=−∞ (ν2α + ξ2α) −β ≤ (73) 108 george a. anastassiou cubo 15, 1 (2013) ∑ ∞ ν=−∞ ω1 ( f, ξ |ν| ξ ) ( ν2α + ξ2α )−β ∑ ∞ ν=−∞ (ν2α + ξ2α) −β ≤ ω1 (f, ξ)   ∑ ∞ ν=−∞ ( 1 + |ν| ξ ) ( ν2α + ξ2α )−β ∑ ∞ ν=−∞ (ν2α + ξ2α) −β   = ( s2 s1 ) ω1 (f, ξ) (70) ≤ (74) ≤ [ 1 + 4ξ2αβ ( αβ 2αβ − 1 ) + ξ2αβ−1 ( 2αβ − 1 αβ − 1 )] ω1 (f, ξ) . we have derived ∣ ∣m∗ξ (f, x) − f (x) ∣ ∣ ≤ [ 1 + 4ξ2αβ ( αβ 2αβ − 1 ) + ξ2αβ−1 ( 2αβ − 1 αβ − 1 )] ω1 (f, ξ) , (75) ∀ x ∈ r, ∀ ξ ∈ (0, 1], proving the claim. it follows the global smoothness preservation property of m∗ξ. theorem 2.6. it holds ω1 ( m∗ξf, δ ) ≤ ω1 (f, δ) , ∀ δ > 0. (76) inequality (76) is sharp, attained by f (x) = id (x) = x. proof. similar to the proof of theorem 2.4. we continue with theorem 2.7. it holds ∥ ∥θ∗ξf − f ∥ ∥ ∞ ≤ ( 2 3 π2k2β ) ξ2β−1ω1 (f, ξ) → 0, as ξ → 0 + . (77) proof. here we use a lot ∞∑ ν=1 1 ν2 = π2 6 (euler, 1741). (78) we have θ∗ξ (f, x) − f (x) = ξ2β ∑ ν∈z−{0} (f (x + ν) − f (x)) ( ϕ( νξ ) ν )2β 1 + ξ2β ∑ ν∈z−{0} ( ϕ( νξ ) ν )2β . (79) hence ∣ ∣θ∗ξ (f, x) − f (x) ∣ ∣ ≤ ξ2β ∑ ν∈z−{0} |f (x + ν) − f (x)| ( ϕ( νξ ) ν )2β 1 + ξ2β ∑ ν∈z−{0} ( ϕ( νξ ) ν )2β cubo 15, 1 (2013) approximation by discrete singular operators 109 ≤ ξ2β ∑ ν∈z−{0} ω1 ( f, ξ |ν| ξ ) ( ϕ( νξ ) ν )2β 1 + ξ2β ∑ ν∈z−{0} ( ϕ( νξ ) ν )2β (80) ≤ ξ2βω1 (f, ξ) ∑ ν∈z−{0} ( 1 + |ν| ξ ) ( ϕ( νξ ) ν )2β 1 + ξ2β ∑ ν∈z−{0} ( ϕ( νξ ) ν )2β (81) ≤ ξ2βω1 (f, ξ) k 2β (∑ ν∈z−{0} ( 1 + |ν| ξ ) 1 ν2β ) 1 + ξ2β ∑ ν∈z−{0} ( ϕ( νξ ) ν )2β =: (∗) . we observe that 0 ≤ s2 := ∑ ν∈z−{0}   ϕ ( ν ξ ) ν   2β ≤ k2β ∑ ν∈z−{0} 1 ν2β = 2k2β ∞∑ ν=1 1 ν2β < 2k2β ∞∑ ν=1 1 ν2 = k2βπ2 3 . (82) thus 0 ≤ s2 ≤ k2βπ2 3 . (83) so that 1 ≤ 1 + ξ2βs2 ≤ 1 + ξ2βk2βπ2 3 < ∞, ∀ ξ > 0. (84) that is 0 < 1 1 + ξ2βs2 ≤ 1, (85) with lim ξ→0+ 1 1 + ξ2βs2 = 1. (86) consequently it holds (∗) ≤ 2ξ2βω1 (f, ξ) k2β ( ∞∑ ν=1 ( 1 + ν ξ ) 1 ν2β ) = 2ξ2βω1 (f, ξ) k 2β ( ∞∑ ν=1 1 ν2β + 1 ξ ∞∑ ν=1 1 ν2β−1 ) ≤ 2ξ2βω1 (f, ξ) k2β ( ∞∑ ν=1 1 ν2 + 1 ξ ∞∑ ν=1 1 ν2 ) = ξ2βπ2 3 ω1 (f, ξ) k 2β ( 1 + 1 ξ ) ≤ 2ξ2β−1π2 3 k2βω1 (f, ξ) . (87) 110 george a. anastassiou cubo 15, 1 (2013) we have proved that ∣ ∣θ∗ξ (f, x) − f (x) ∣ ∣ ≤ ( 2 3 π2k2β ) ξ2β−1ω1 (f, ξ) → 0, as ξ → 0 + . (88) the proof is completed. example 2.8. in theorem 2.7 we can take ϕ to be sine, cosine with k = 1. theorem 2.9. it holds ω1 ( θ∗ξf, δ ) ≤ ω1 (f, δ) , ∀ δ > 0. (89) inequality (89) is attained by f = id. we finish by studying t∗ξ. theorem 2.10. it holds ∥ ∥t∗ξf − f ∥ ∥ ∞ ≤ ( 2π2a 3 ) ξ2β−1ω1 (f, ξ) → 0, as ξ → 0 + . (90) theorem 2.11. it holds ω1 ( t∗ξf, δ ) ≤ ω1 (f, δ) , ∀ δ > 0. (91) inequality (91) is attained by f = id. proof. of theorem 2.10. we have t∗ξ (f, x) − f (x) = ∑ ν∈z−{0} (f (x + ν) − f (x)) ( ϕ( νξ ) ( νξ ) 4β ) 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) . (92) thus ∣ ∣t∗ξ (f, x) − f (x) ∣ ∣ ≤ ∑ ν∈z−{0} |f (x + ν) − f (x)| ( ϕ( νξ ) ( νξ ) 4β ) 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) ≤ ∑ ν∈z−{0} ω1 ( f, ξ |ν| ξ ) ( ϕ( νξ ) ( νξ ) 4β ) 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) (93) ≤ ω1 (f, ξ) ∑ ν∈z−{0} ( 1 + |ν| ξ ) ( ϕ( νξ ) ( νξ ) 4β ) 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) (94) cubo 15, 1 (2013) approximation by discrete singular operators 111 ≤ ω1 (f, ξ) ∑ ν∈z−{0} ( 1 + |ν| ξ ) a ( νξ ) 2β 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) = aω1 (f, ξ) ξ 2β ∑ ν∈z−{0} ( 1 + |ν| ξ ) 1 ν2β 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) (95) = 2aω1 (f, ξ) ξ 2β ∑ ∞ ν=1 ( 1 + ν ξ ) 1 ν2β 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) = 2aω1 (f, ξ) ξ 2β [∑ ∞ ν=1 1 ν2β + 1 ξ ∑ ∞ ν=1 1 ν2β−1 ] 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β ) (96) ≤ 2aω1 (f, ξ) ξ 2β [ π 2 6 + 1 ξ π 2 6 ] ( 1 + ∑ ν∈z−{0} ( ϕ( νξ ) ( νξ ) 4β )) ≤ ( 2π 2 3 a ) ω1 (f, ξ) ξ 2β−1 1 + ∑ ν∈z−{0} ϕ( νξ ) ( νξ ) 4β =: (∗) . (97) we see that 1 + ∑ ν∈z−{0} ϕ ( ν ξ ) ( ν ξ )4β ≤ 1 + a ∑ ν∈z−{0} 1 ( ν ξ )2β (98) = 1 + aξ2β ∑ ν∈z−{0} 1 ν2β = 1 + 2aξ2β ∞∑ ν=1 1 ν2β < 1 + 2aξ2β ( ∞∑ ν=1 1 ν2 ) = 1 + aξ2βπ2 3 < ∞, ∀ ξ > 0. (99) that is 1 ≤ 1 + ∑ ν∈z−{0} ϕ ( ν ξ ) ( ν ξ )4β < 1 + aπ2ξ2β 3 < ∞, ∀ ξ > 0. (100) also 1 + aπ 2 ξ 2β 3 → 1, as ξ → 0+, that is lim ξ→0+    1 + ∑ ν∈z−{0} ϕ ( ν ξ ) ( ν ξ )4β    = 1. (101) furthermore we have 0 < 1 1 + ∑ ν∈z−{0} ϕ( νξ ) ( νξ ) 4β ≤ 1. (102) 112 george a. anastassiou cubo 15, 1 (2013) hence (∗) ≤ ( 2π2 3 a ) ω1 (f, ξ) ξ 2β−1. (103) we have proved ∣ ∣t∗ξ (f, x) − f (x) ∣ ∣ ≤ ( 2π2a 3 ) ξ2β−1ω1 (f, ξ) → 0, as ξ → 0+, ∀ x ∈ r, (104) proving the claim. note 2.12. all estimates of this article are also true for f ∈ cb (r), continuous and bounded functions on r. however the convergences fail if f is not uniformly continuous. received: october 2012. revised: march 2013. references [1] george anastassiou and razvan mezei, approximation by singular integrals, to appear, cambridge scientific publishers, cambridge, u.k, 2013. [2] r.a. devore and g.g. lorentz, constructive approximation, springer-verlag, n.y., heildelberg, 1993. [3] i. favard, sur les multiplicateurs d’interpolation, j. math. pures appl., ix, 23(1944), 219-247. [4] f. smarandache, a triple inequality with series and improper integrals, arxiv. org/ftp/math/papers/0605/0605027.pdf, 2006. [5] d. zwillinger, crc standard mathematical tables and formulae, 30th edition, chapman & hall/crc, boca raton, 1995. cubo a mathematical journal vol.14, no¯ 03, (103–113). october 2012 an elementary study of a class of dynamic systems with two time delays akio matsumoto 1 department of economics, chuo university, 742-1, higashi-nakano, hachioji, tokyo, 192-0393, japan email: akiom@tamacc.chuo-u.ac.jp ferenc szidarovszky department of systems and industrial engineering, university of arizona, tucson, 85721-0020, usa. email: szidar@sie.arizona.edu abstract an elementary analysis is developed to determine the stability region of a certain class of ordinary differential equations with two delays. our analysis is based on determining stability switches first where an eigenvalue is pure complex, and then checking the conditions for stability loss or stability gain. in the case of both stability losses and stability gains hopf bifurcation occurs giving the possibility of the birth of limit cycles. resumen se realiza un análisis básico para determinar la estabilidad de la región de una cierta clase de ecuaciones diferenciales ordinaras con dos retrasos. nuestro análisis se basa en la determinación de switches de estabilidad, en primer lugar cuando un autovalor es complejo puro, y luego revisando las condiciones para la pérdida o ganancia de estabilidad. en el caso de ambas pérdidas de estabilidad y ganancias de estabilidad, se obtiene la bifurcación de hopf dando la posibilidad del nacimiento de ciclos ĺımites. keywords and phrases: dynamic systems, time delays, stabiliy analysis. 2010 ams mathematics subject classification: 34k20, 37c75 1 the authors highly appreciate financial supports from the japan society for the promotion of science (grantin-aid for scientific research (c) 24530202) and chuo university (grant for special research and joint research grant 0981). 104 akio matsumoto and ferenc szidarovszky cubo 14, 3 (2012) 1 introduction dynamic models with time delays have many applications in many fields of quantitative sciences (see for example, cushing (1977) and invernizzi and medio (1991)). the case of a single delay is well established in the literature (hayes (1950) and burger (1956)), however the presence of multiple delays makes analysis much more complicated. sufficient and necessary conditions were derived for several classes of models giving a complete description of the stability region (hale (1979), hale and huang (1993) and piotrowska (2007)). in this paper a special class of dynamic systems is considered which are governed by delay differential equations with two delays. it is well known (hayes (1950) and cooke and grossman (1982)) that stability can be lost or gained on a curve of stability switches, where an eigenvalue is pure complex. we will therefore determine these curves and then by bifurcation analysis characterize those segments where stability is gained or lost. in this way the stability region can be completely described. this paper is the continuation of our previous work (matsumoto and szidarovszky (2011)) where an elementary analysis was presented with a single delay. the paper is organized in the following way. section 2 determines the curves where stability switches are possible and characterizes those segments where stability is lost or gained in the nonsymmetric cases. section 3 discusses the symmetric case and section 4 concludes the paper. 2 stability switches and stability region we will examine the asymptotical stability of the delay differential equation ẋ(t) + ax(t − τ1) + bx(t − τ2) = 0 (2.1) where a and b are positive constants. the characteristic equation can be obtained by looking for the solution in the exponential form αeλt. by substitution, αλeλt + aαeλ(t−τ1) + bαeλ(t−τ2) = 0 or λ + ae−λτ1 + be−λτ2 = 0. (2.2) introduce the new variables ω = a a + b , 1 − ω = b a + b , λ̄ = λ a + b γ1 = τ1(a + b) and γ2 = τ2(a + b) to reduce equation (2.2) to the following: λ̄ + ωe−λ̄γ1 + (1 − ω)e−λ̄γ2 = 0. (2.3) cubo 14, 3 (2012) an elementary study of a class of dynamic systems ... 105 because of symmetry we can assume that ω ≥ 1/2.in order to find the stability region in the (γ1, γ2) plane we will first characterize the cases when an eigenvalue is pure complex, that is, when λ̄ = iυ. we can assume that υ > 0, since if λ̄ is an eigenvalue, its complex conjugate is also an eigenvalue. substituting λ̄ = iυ into equation (2.3) we have ιυ + ωe−iυγ1 + (1 − ω)e−iυγ2 = 0. in the special case of γ1 = 0, the equation becomes ιυ + ω + (1 − ω)e−iυγ2 = 0. the real and imaginary parts imply that ω + (1 − ω) cos(υγ2) = 0 υ − (1 − ω) sin(υγ2) = 0. we can assume first ω > 1/2, so from the first equation cos(υγ2) = − ω 1 − ω < −1 so no stability switch is possible. if ω = 1/2, then cos(υγ2) = −1 implying that sin(υγ2) = 0 and so υ = 0 showing that there is no pure complex root. hence for γ1 = 0 the system is asymptotically stable with all γ2 ≥ 0. assume now that γ1 > 0, γ2 ≥ 0. the real and imaginary parts give two equations: ω cos(υγ1) + (1 − ω) cos(υγ2) = 0 (2.4) and υ − ω sin(υγ1) − (1 − ω) sin(υγ2) = 0. (2.5) we consider the case of ω > 1/2 first and the symmetric case of ω = 1/2 will be discussed later. introduce the variables x = sin(υγ1) and y = sin(υγ2), then (2.4) implies that ω2(1 − x2) = (1 − ω)2(1 − y2) or − ω2x2 + (1 − ω)2y2 = 1 − 2ω. (2.6) from (2.5), υ − ωx − (1 − ω)y = 0 106 akio matsumoto and ferenc szidarovszky cubo 14, 3 (2012) implying that y = υ − ωx 1 − ω (2.7) combining (2.6) and (2.7) yields −ω2x2 + (1 − ω)2 ( υ − ωx 1 − ω )2 = 1 − 2ω from which we can conclude that x = υ2 + 2ω − 1 2υω (2.8) and then from (2.7), y = υ2 − 2ω + 1 2υ(1 − ω) . (2.9) equations (2.8) and (2.9) provide a parameterized curve in the (γ1, γ2) plane: sin(υγ1) = υ2 + 2ω − 1 2υω and sin(υγ2) = υ2 − 2ω + 1 2υ(1 − ω) . (2.10) in order to guarantee feasibility we have to satisfy − 1 ≤ υ2 + 2ω − 1 2υω ≤ 1 (2.11) and − 1 ≤ υ2 − 2ω + 1 2υ(1 − ω) ≤ 1. (2.12) simple calculation shows that with ω > 1/2 these relations hold if and only if 2ω − 1 ≤ υ ≤ 1. from (2.10) we have four cases for γ1 and γ2, since γ1 = 1 υ { sin−1 ( υ2 + 2ω − 1 2υω ) + 2kπ } or γ1 = 1 υ { π − sin−1 ( υ2 + 2ω − 1 2υω ) + 2kπ } (k = 0, 1, 2, ...) and similarly γ2 = 1 υ { sin−1 ( υ2 − 2ω + 1 2υ(1 − ω) ) + 2nπ } or γ2 = 1 υ { π − sin−1 ( υ2 − 2ω + 1 2υ(1 − ω) ) + 2nπ } (n = 0, 1, 2, ...). cubo 14, 3 (2012) an elementary study of a class of dynamic systems ... 107 however from (2.4) we can see that cos(υγ1) and cos(υγ2) must have different signs, so we have only two possibilities: l1(k, n) :              γ1 = 1 υ { sin−1 ( υ2 + 2ω − 1 2υω ) + 2kπ } γ2 = 1 υ { π − sin−1 ( υ2 − 2ω + 1 2υ(1 − ω) ) + 2nπ } (2.13) and l2(k, n) :              γ1 = 1 υ { π − sin−1 ( υ2 + 2ω − 1 2υω ) + 2kπ } γ2 = 1 υ { sin−1 ( υ2 − 2ω + 1 2υ(1 − ω) ) + 2nπ } (2.14) for each υ ∈ [2ω − 1, 1] these equations determine the values of γ1 and γ2. at the initial point υ = 2ω − 1, we have υ2 + 2ω − 1 2υω = 1 and υ2 − 2ω + 1 2υ(1 − ω) = −1 and if υ = 1, then υ2 + 2ω − 1 2υω = 1 and υ2 − 2ω + 1 2υ(1 − ω) = 1. therefore the starting point and end point of l1(k, n) are given as γs1 = 1 2ω − 1 ( π 2 + 2kπ ) , γs2 = 1 2ω − 1 ( 3π 2 + 2nπ ) and γe1 = π 2 + 2kπ and γe2 = π 2 + 2nπ. similarly, the starting and end points of l2(k, n) are as follows: γs1 = 1 2ω − 1 ( π 2 + 2kπ ) , γs2 = 1 2ω − 1 ( − π 2 + 2nπ ) and γe1 = π 2 + 2kπ and γe2 = π 2 + 2nπ. figure 1 illustrates the loci l1(k, n) and l2(k, n) of the corresponding points (γ1, γ2), when υ increases from 2ω − 1 to unity. the parameter value ω = 0.8 is selected. the red curves show l1(0, n) and the blue curves show l2(0, n) with n = 0, 1, 2, .... notice that γ s 2 is infeasible at n = 0 and only the segment of l2(0, 0) between υ = √ 2ω − 1 and υ = 1 is feasible. with fixed value of k, l1(k, n) and l2(k, n) have the same end point, however the starting point of l1(k, n) is the same as that of l2(k, n + 1). therefore the segments l1(k, n) and l2(k, n) with fixed k form a continuous curve with n = 0, 1, 2, .... 108 akio matsumoto and ferenc szidarovszky cubo 14, 3 (2012) l1h0,0l l1h0,1l l1h0,2l l2h0,0l l2h0,1l l2h0,2l e0 e1 e2 s0 s1 γ1 m γ1 m γ1 3π 2 h2ω-1l 7π 2 h2ω-1l γ2 figure 1. partition curve in the (γ1, γ2) plane with fixing k = 0. consider first the segment l1(k, n). since ( υ2 − 2ω + 1 ) /(2υ(1 − ω)) is strictly increasing in υ, γ2 is strictly decreasing in υ. by differentiation and substitution of equation (2.4), we have ∂γ1 ∂υ ∣ ∣ ∣ ∣ l1 = − 1 υ2 ( sin−1 ( υ 2 +2ω−1 2υω ) + 2kπ ) + 1 υ √ 1− ( υ 2 +2ω−1 2υω ) 2 2υ(2υω)−(υ 2 +2ω−1)2ω 22υ2ω2 = − 1 υ2 υγ1 + 1 υ cos(υγ1) υ2 − 2ω + 1 2υ2ω = − 1 υ2 (υγ1 + tan(υγ2)) . (2.15) consider next segment l2(k, n), similarly to (2.15) we can shown that ∂γ1 ∂υ ∣ ∣ ∣ ∣ l2 = − 1 υ2 (υγ1 + tan(υγ2)) which is the same as in l1(k, n), since from (2.14), cos(υγ1) < 0. similarly ∂γ2 ∂υ ∣ ∣ ∣ ∣ l2 = − 1 υ2 (υγ2 + tan(υγ1)) (2.16) where we used again equation (2.4). in order to visualize the curves l1(k, n) and l2(k, n), we change the coordinates (γ1, γ2) to (υγ1, υγ2) to get the transformed segments: ℓ1(k, n) :              υγ1 = sin −1 ( υ2 + 2ω − 1 2υω ) + 2kπ υγ2 = π − sin −1 ( υ2 − 2ω + 1 2υ(1 − ω) ) + 2nπ (2.17) cubo 14, 3 (2012) an elementary study of a class of dynamic systems ... 109 and ℓ2(k, n) :              υγ1 = π − sin −1 ( υ2 + 2ω − 1 2υω ) + 2kπ υγ2 = sin −1 ( υ2 − 2ω + 1 2υ(1 − ω) ) + 2nπ (2.18) they also form a continuous curve with each fixed value of k, and they are periodic in both directions υγ1 and υγ2. figure 2 shows them with k = 0 where the curves ℓ1(0, n) are shown in red color while the curves ℓ2(0, n) with blue color. l1h0,0l l1h0,1l l2h0,0l l2h0,1l π�2xm x m νγ1 π 2 0 π 2 π 3 π 2 2 π 5 π 2 3 π 7 π 2 νγ2 figure 2. partition curve in the (υγ1, υγ2) plane with fixing k = 0 we will next examine the directions of the stability switches on the different segments of the curves l1(k, n) and l2(k, n). we fix the value of γ2 and select γ1 as the bifurcation parameter, so the eigenvalues are functions of γ1 : λ̄ = λ(γ1). by differentiating the characteristic equation (2.3) implicitly with respect to γ1 we have dλ̄ dγ1 + ωe−λ̄γ1(− dλ̄ dγ1 γ1 − λ̄) + (1 − ω)e −λ̄γ2 ( − dλ̄ dγ1 γ2 ) = 0 implying that dλ̄ dγ1 = λ̄ωe−λ̄γ1 1 − ωγ1e −λ̄γ1 − (1 − ω)γ2e −λ̄γ2 (2.19) from (2.3) we have (1 − ω)e−λ̄γ2 = −λ̄ − ωe−λ̄γ1, so dλ̄ dγ1 = λ̄ωe−λ̄γ1 1 + λ̄γ2 + ω(γ2 − γ1)e −λ̄γ1 110 akio matsumoto and ferenc szidarovszky cubo 14, 3 (2012) if λ̄ = ιυ, then dλ̄ dγ1 = iυω(cos(υγ1) − i sin(υγ1)) 1 + iυγ2 + ω(γ2 − γ1)(cos(υγ1) − i sin(υγ1)) and the real part of this expression has the same sign as υω sin(υγ1)[1 + ω(γ2 − γ1) cos(υγ1)] + υω cos(υγ1)[υγ2 − ω(γ2 − γ1) sin(υγ1)] = υω [sin(υγ1) + υγ2 cos(υγ1)] hence re ( dλ̄ dγ1 ) r 0 if and only if sin(υγ1) + υγ2 cos(υγ1) r 0 consider first the case of crossing any segment l1(k, n) from the left. here υγ1 ∈ (0, π/2], so both sin(υγ1) and cos(υγ2) are positive. hence stability is lost everywhere on any segment of l1(k, n). consider the case when crossing the segments of l2(k, n) from the left. here υγ1 ∈ [π/2, π], so cos(υγ1) < 0. combining (2.16) and the conditions for the sign of re[dλ̄/dγ1], we have that re ( dλ̄ dγ1 ) r 0 if and only if ∂γ2 ∂υ r 0. that is, stability is lost when γ2 increases in υ and stability is gained when γ2 decreases in υ. we can also show that at any intercept with l1(k, n) or l2(k, n) the complex root is single. otherwise λ = iυ would satisfy both equations λ + ωe−λγ1 + (1 − ω)e−λγ2 = 0 and 1 − ωγ1e −λγ1 − (1 − ω)γ2e −λγ2 = 0, from which we have e−λγ1 = 1 + λγ2 (γ1 − γ2)ω and e−λγ2 = −1 − λγ1 (γ1 − γ2)(1 − ω) . by substituting λ = iυ and comparing the real and imaginary parts yield sin(υγ1) + υγ2 cos(υγ1) = sin(υγ2) + υγ1 cos(υγ2) = 0. therefore this intercept is at an extremum in υ of a segment l1(k, n) and also at an extremum of a segment l2(k̄, n̄) which is impossible. for each γ2 > 0, define m(γ2) = min γ1 {(γ1, γ2) ∈ l1(k, n) ∪ l2(k, n), k, n ≥ 0} (2.20) at γ1 = 0 the system is asymptotically stable with all γ2 > 0. with fixed value of γ2 by increasing the value of γ1 the first intercept with m(γ2) should be a stability loss, since there is no stability switch before. then by increasing the value of γ1 further, the next intercept is either a stability cubo 14, 3 (2012) an elementary study of a class of dynamic systems ... 111 gain or a stability loss. in the first case the equilibrium becomes asymptotically stable. in the second case the equilibrium remains unstable, which will not change even if the next intercept is an stability gain, since the real part of only one eigenvalue becomes negative. consider next a point (γ∗ 1 , γ∗ 2 ) with γ∗ 1 , γ∗ 2 > 0 which is not located on any curve l1(k, n) or l2(k, n), and consider the horizontal line γ2 = γ ∗ 2 and its segment with γ1 ∈ (0, γ∗1). if it has no stability switch, then the equilibrium is asymptotically stable. this is the case even if the number of stability losses equals the number of stability gains, otherwise the equilibrium is unstable. the stability region is shown as the shaded region in figure 1. notice that this is the same result which was obtained earlier by hale and huang (1993) by using different approach. 3 the symmetric case assume next that ω = 1/2. then equations (2.4) and (2.5) imply that cos(υγ1) + cos(υγ2) = 0 υ − 1 2 (sin(υγ1) + sin(υγ2)) = 0 and the curves l1(k, n) and l2(k, n) are simplified as follows: l1(k, n) :          γ1 = 1 υ ( sin−1(υ) + 2kπ ) γ2 = 1 υ ( π − sin−1(υ) + 2nπ ) (3.1) and l2(k, n) :          γ1 = 1 υ ( π − sin−1(υ) + 2kπ ) γ2 = 1 υ ( sin−1(υ) + 2nπ ) (3.2) which are shown in figure 3. the same argument as shown above for the nonsymmetric case can be applied here as well to show that stability region is left of l1(0, 0) and below l2(0, 0), where the shape of the stability region differs from that of the nonsymmetric case. it is illustrated in figure 3 by the shaded domain. 112 akio matsumoto and ferenc szidarovszky cubo 14, 3 (2012) l1h0,0l l1h0,1l l2h0,0l l2h0,1l π�2xm x m νγ1 π 2 0 π 2 π 3 π 2 2 π 5 π 2 3 π 7 π 2 νγ2 figure 3. partition curve in the (γ1, γ2) plane with ω = 1 2 notice that at each segment of ℓ2(k, n) there are at most two intercepts with the υγ2 = − tan(υγ1) curve, so the same holds for l2(k, n). at every other point re[dλ̄/dγ1] 6= 0, so at these points hopf bifurcation occurs giving the possibility of the birth of limit cycles. 4 conclusions ordinary differential equation were examined with two delays. after finding the possible stability switches, their curves were determined. hopf bifurcation was used to find segments with stability losses and stability gains. the boundary of the stability region are the γ2 = 0, γ1 = 0 and a continuous curve consisting of certain portions of the segments l1(0, n) and l2(0, n). all other points on the curves l1(k, n) and l2(k, n) for k ≥ 1 do not lead to actual stability switches, since the system is already unstable. received: july 2011. revised: june 2012. references [1] burger, e. (1956), on the stability of certain economic systems. econometrica, 24(4), 488-493. [2] cooke, k. l. and z. grossman (1982), discrete delay, distributed delay and stability switches. j. of math. analysis and appl., 86, 592-627. [3] cushing, j. m. (1977), integro-differential equations and delay models in population dynamics. springer-verlag, berlin/heidelberg/new york. [4] hayes, n. d. (1950), roots of the transcendental equation associated with a certain difference-differential equation. j. of the london math. society, 25, 226-232. cubo 14, 3 (2012) an elementary study of a class of dynamic systems ... 113 [5] hale, j. (1979), nonlinear oscillations in equations with delays. in nonlinear oscillations in biology (k. c. hoppenstadt, ed.). lectures in applied mathematics, 17, 157-185. [6] hale, j. and w. huang (1993), global geometry of the stable regions for two delay differential equations. j. of math. analysis and appl., 178, 344-362. [7] invernizzi, s. and a. medio (1991), on lags and chaos in economic dynamic models. journal of math. econ., 20, 521-550. [8] matsumoto, a. and f. szidarovszky (2011), an elementary study of a class of dynamic systems with single delay. mimeo: dp 161 (http://www2.chuo-u.ac.jp/keizaiken/discuss.htm), institute of economic research, chuo university, tokyo, japan. [9] piotrowska, m. (2007), a remark on the ode with two discrete delays. journal of math. analysis and appl., 329, 664-676. () cubo a mathematical journal vol.17, no¯ 01, (65–73). march 2015 spline left fractional monotone approximation involving left fractional differential operators george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract let f ∈ cs ([−1, 1]), s∈ n and l∗ be a linear left fractional differential operator such that l∗ (f) ≥ 0 on [0, 1]. then there exists a sequence qn, n ∈ n of polynomial splines with equally spaced knots of given fixed order such that l∗ (qn) ≥ 0 on [0, 1]. furthermore f is approximated with rates fractionally and simultaneously by qn in the uniform norm. this constrained fractional approximation on [−1, 1] is given via inequalities invoving a higher modulus of smoothness of f(s). resumen sea f ∈ cs ([−1, 1]), s∈ n y l∗ un operador diferencial fraccionario lineal izquierdo tal que l∗ (f) ≥ 0 en [0, 1].. entonces, existe una sucesión qn, n ∈ n de splines polinomiales con nodos equiespaciados de un orden fijo dado tal que l∗ (qn) ≥ 0 en [0, 1]. además, f se aproxima con velocidades fraccionales y simultáneamente por qn en la norma uniforme. esta aproximación fraccional restringida a [−1, 1] se encuentra por medio de desigualdades que involucran un módulo alto de suavidad de f(s). keywords and phrases: monotone approximation, caputo fractional derivative, fractional linear differential operator, modulus of smoothness, splines. 2010 ams mathematics subject classification: 26a33, 41a15, 41a17, 41a25, 41a28, 41a29, 41a99. 66 george a. anastassiou cubo 17, 1 (2015) 1 introduction let [a, b] ⊂ r and for n ≥ 1 consider the partition ∆n with points xin = a + i ( b−a n ) , i = 0, 1, ..., n. hence ∆n ≡ max1≤i≤n (xin − xi−1,n) = b−a n . let sm (∆n) be the space of polynomial splines of order m > 0 with simple knots at the points xin, i = 1, ..., n − 1. then there exists a linear operator qn : qn ≡ qn (f), mapping b [a, b]: the space of bounded real valued functions f on [a, b], into sm (∆n) (see [4], p. 224, theorem 6.18). from the same reference [4], p. 227, corollary 6.21, we get corolary 1. let 1 ≤ σ ≤ m, n ≥ 1. then for all f ∈ cσ−1 [a, b] ; r = 0, ..., σ − 1, ∥ ∥ ∥ f(r) − q(r)n ∥ ∥ ∥ ∞ ≤ c1 ( b − a n )σ−r−1 ωm−σ+1 ( f(σ−1), b − a n ) , (1) where c1 depends only on m, c1 = c1 (m) . by denoting c2 = c1 max0≤r≤σ−1 (b − a) σ−r−1 we obtain lemma 1. ([1]) let 1 ≤ σ ≤ m, n ≥ 1. then for all f ∈ cσ−1 [a, b]; r = 0, ..., σ − 1, ∥ ∥ ∥ f(r) − q(r)n ∥ ∥ ∥ ∞ ≤ c2 nσ−r−1 ωm−σ+1 ( f(σ−1), b − a n ) , (2) where c2 depends only on m, σ and b − a. here ωm−σ+1 is the usual modulus of smoothness of order m − σ + 1. we are motivated by theorem 1. ([1]) let h, k, σ, m be integers, 0 ≤ h ≤ k ≤ σ − 1, σ ≤ m and let f ∈ cσ−1 [a, b]. let αj (x) ∈ b [a, b], j = h, h + 1, ..., k and suppose that αh (x) ≥ α > 0 or αh (x) ≤ β < 0 for all x ∈ [a, b] . take the linear differential operator l = k∑ j=h αj (x) [ dj dxj ] (3) and assume, throughout [a, b], l (f) ≥ 0. (4) then, for every integer n ≥ 1, there is a polynomial spline function qn (x) of order m with simple knots at { a + i ( b−a n ) , i = 1, ..., n − 1 } such that l (qn) ≥ 0 throughout [a, b] and ∥ ∥ ∥ f(r) − q(r)n ∥ ∥ ∥ ∞ ≤ c nσ−k−1 ωm−σ+1 ( f(σ−1), b − a n ) , 0 ≤ r ≤ h. (5) cubo 17, 1 (2015) spline left fractional monotone approximation involving left . . . 67 moreover, we find ∥ ∥ ∥ f(r) − q(r)n ∥ ∥ ∥ ∞ ≤ c nσ−r−1 ωm−σ+1 ( f(σ−1), b − a n ) , h + 1 ≤ r ≤ σ − 1, (6) where c is a constant independent of f and n. it depends only on m, σ, l, a, b. next we specialize on the case of a = −1, b = 1. that is working on [−1, 1] . by lemma 1 we get lemma 2. let 1 ≤ σ ≤ m, n ≥ 1. then for all f ∈ cσ−1 ([−1, 1]); j = 0, 1, ..., σ − 1, ∥ ∥ ∥ f(j) − q(j)n ∥ ∥ ∥ ∞ ≤ c2 nσ−j−1 ωm−σ+1 ( f(σ−1), 2 n ) , (7) where c2 := c2 (m, σ) := c1 (m) 2 σ−1. since ωm−σ+1 ( f(σ−1), 2 n ) ≤ 2m−σ+1ωm−σ+1 ( f(σ−1), 1 n ) (8) (see [2], p. 45), we get lemma 3. let 1 ≤ σ ≤ m, n ≥ 1. then for all f ∈ cσ−1 ([−1, 1]); j = 0, 1, ..., σ − 1, ∥ ∥ ∥ f(j) − q(j)n ∥ ∥ ∥ ∞ ≤ c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (9) where c∗2 := c ∗ 2 (m, σ) := c1 (m) 2 m. we use a lot in this article lemma 3. in this article we extend theorem 1 over [−1, 1] to the fractional level. indeed here l is replaced by l∗, a linear left caputo fractional differential operator. now the monotonicity property is only true on the critical interval [0, 1]. simultaneous fractional convergence remains true on all of [−1, 1] . we make definition 1. ([3], p. 50) let α > 0 and ⌈α⌉ = m, (⌈·⌉ ceiling of the number). consider f ∈ cm ([−1, 1]). we define the left caputo fractional derivative of f of order α as follows: ( dα∗−1f ) (x) = 1 γ (m − α) ∫x −1 (x − t) m−α−1 f(m) (t) dt, (10) for any x ∈ [−1, 1], where γ is the gamma function. we set d0∗−1f (x) = f (x) , dm∗−1f (x) = f (m) (x) , ∀ x ∈ [−1, 1] . (11) 68 george a. anastassiou cubo 17, 1 (2015) 2 main result theorem 2. let h, k, σ, m be integers, 1 ≤ σ ≤ m, n ∈ n, with 0 ≤ h ≤ k ≤ σ − 2 and let f ∈ cσ−1 ([−1, 1]), with f(σ−1) having modulus of smoothness ωm−σ+1 ( f(σ−1), δ ) there, δ > 0. let αj (x), j = h, h+1, ..., k be real functions, defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [0, 1]. let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < ... < ασ−2 ≤ σ−2. here d αj ∗−1 f stands for the left caputo fractional derivative of f of order αj anchored at −1. consider the linear left fractional differential operator l∗ := k∑ j=h αj (x) [ d αj ∗−1 ] (12) and suppose, throughout [0, 1], l∗ (f) ≥ 0. then, for every integer n ≥ 1, there exists a polynomial spline function qn (x) of order m > 0 with simple knots at { −1 + i 2 n , i = 1, ..., n − 1 } such that l∗ (qn) ≥ 0 throughout [0, 1] , and sup −1≤x≤1 ∣ ∣ ( d αj ∗−1f ) (x) − ( d αj ∗−1qn ) (x) ∣ ∣ ≤ 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (13) j = h + 1, ..., σ − 2. set lj :≡ sup x∈[−1,1] ∣ ∣α−1h (x) αj (x) ∣ ∣ , h ≤ j ≤ k. (14) when j = 1, ..., h we derive max −1≤x≤1 ∣ ∣ ( d αj ∗−1 f ) (x) − ( d αj ∗−1 qn ) (x) ∣ ∣ ≤ c∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) · [( k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) )( h−j∑ λ=0 2h−αj−λ λ!γ (h − αj − λ + 1) ) + 2j−αj γ (j − αj + 1) ] . (15) finally it holds sup −1≤x≤1 |f (x) − qn (x)| ≤ c∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) + 1 ] . (16) proof. set α0 = 0, thus ⌈α0⌉ = 0. we have ⌈αj⌉ = j, j = 1, ..., σ − 2. let qn as in lemma 5. cubo 17, 1 (2015) spline left fractional monotone approximation involving left . . . 69 we notice that (x ∈ [−1, 1]) ∣ ∣ ( d αj ∗−1f ) (x) − ( d αj ∗−1qn ) (x) ∣ ∣ = 1 γ (j − αj) ∣ ∣ ∣ ∣ ∫x −1 (x − t) j−αj−1 f(j) (t) dt − ∫x −1 (x − t) j−αj−1 q(j)n (t) dt ∣ ∣ ∣ ∣ = (17) 1 γ (j − αj) ∣ ∣ ∣ ∣ ∫x −1 (x − t) j−αj−1 ( f(j) (t) − q(j)n (t) ) dt ∣ ∣ ∣ ∣ ≤ 1 γ (j − αj) ∫x −1 (x − t) j−αj−1 ∣ ∣ ∣ f(j) (t) − q(j)n (t) ∣ ∣ ∣ dt (9) ≤ (18) 1 γ (j − αj) (∫x −1 (x − t) j−αj−1 dt ) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = 1 γ (j − αj) (x + 1) j−αj (j − αj) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = (19) (x + 1) j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) ≤ 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) . (20) hence ∥ ∥d αj ∗−1 f − d αj ∗−1 qn ∥ ∥ ∞,[−1,1] ≤ 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (21) j = 0, 1, ..., σ − 2. we set ρn := c ∗ 2ωm−σ+1 ( f(σ−1), 1 n )   k∑ j=h lj 2j−αj γ (j − αj + 1) n σ−j−1  . (22) i. suppose, throughout [0, 1], αh (x) ≥ α > 0. let qn (x), x ∈ [−1, 1], the polynomial spline of order m > 0 with simple knots at the points xin, i = 1, ..., n − 1, on [−1, 1] (xin = −1 + i 2 n , i = 0, 1, ..., n, here ∆n = 2 n ), so that max −1≤x≤1 ∣ ∣ ∣ ∣ d αj ∗−1 ( f (x) + ρn xh h! ) − ( d αj ∗−1 qn ) (x) ∣ ∣ ∣ ∣ ≤ 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (23) j = 0, 1, ..., σ − 2. when j = h + 1, ..., σ − 2, then max −1≤x≤1 ∣ ∣ ( d αj ∗−1 f ) (x) − ( d αj ∗−1 qn ) (x) ∣ ∣ ≤ 70 george a. anastassiou cubo 17, 1 (2015) 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (24) proving (13). for j = 1, ..., h we find that d αj ∗−1 ( xh h! ) = h−j∑ λ=0 (−1) λ (x + 1) h−αj−λ λ!γ (h − αj − λ + 1) . (25) therefore we get from (23) max −1≤x≤1 ∣ ∣ ∣ ∣ ∣ ( d αj ∗−1 f ) (x) + ρn ( h−j∑ λ=0 (−1) λ (x + 1) h−αj−λ λ!γ (h − αj − λ + 1) ) − ( d αj ∗−1 qn ) (x) ∣ ∣ ∣ ∣ ∣ ≤ (26) 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , j = 1, ..., h. therefore we get for j = 1, ..., h, that max −1≤x≤1 ∣ ∣ ( d αj ∗−1f ) (x) − ( d αj ∗−1qn ) (x) ∣ ∣ ≤ (27) ρn ( h−j∑ λ=0 2h−αj−λ λ!γ (h − αj − λ + 1) ) + 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = c∗2ωm−σ+1 ( f(σ−1), 1 n )   k∑ j=h l j 2 j−α j γ ( j − α j + 1 ) nσ−j−1   · ( h−j∑ λ=0 2h−αj−λ λ!γ (h − αj − λ + 1) ) + 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = c∗2ωm−σ+1 ( f(σ−1), 1 n )     k∑ j=h l j 2 j−α j γ ( j − α j + 1 ) 1 nσ−j−1   · (28) ( h−j∑ λ=0 2h−αj−λ λ!γ (h − αj − λ + 1) ) + 2j−αj γ (j − αj + 1) 1 nσ−j−1 ] ≤ c∗2ωm−σ+1 ( f(σ−1), 1 n ) 1 nσ−k−1     k∑ j=h l j 2j−αj γ ( j − α j + 1 )   · (29) ( h−j∑ λ=0 2h−αj−λ λ!γ (h − αj − λ + 1) ) + 2j−αj γ (j − αj + 1) ] . cubo 17, 1 (2015) spline left fractional monotone approximation involving left . . . 71 hence for j = 1, ..., h we derived (15): max −1≤x≤1 ∣ ∣ ( d αj ∗−1 f ) (x) − ( d αj ∗−1 qn ) (x) ∣ ∣ ≤ c∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) · [( k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) )( h−j∑ λ=0 2h−αj−λ λ!γ (h − αj − λ + 1) ) + 2j−αj γ (j − αj + 1) ] . (30) when j = 0 from (23) we obtain max −1≤x≤1 ∣ ∣ ∣ ∣ f (x) + ρn xh h! − qn (x) ∣ ∣ ∣ ∣ ≤ c∗2 nσ−1 ωm−σ+1 ( f(σ−1), 1 n ) . (31) and max −1≤x≤1 |f (x) − qn (x)| ≤ ρn h! + c∗2 nσ−1 ωm−σ+1 ( f(σ−1), 1 n ) = (32) c∗2 h! ωm−σ+1 ( f(σ−1), 1 n ) ( k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) n σ−τ−1 ) + c∗2 nσ−1 ωm−σ+1 ( f(σ−1), 1 n ) = c∗2ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) n σ−τ−1 + 1 nσ−1 ] ≤ (33) c∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) + 1 ] . proving max −1≤x≤1 |f (x) − qn (x)| ≤ c∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ γ (τ − ατ + 1) + 1 ] , (34) so that (16) is established. also if 0 ≤ x ≤ 1, then α−1h (x) l ∗ (qn (x)) = α −1 h (x) l ∗ (f (x)) + ρn (x + 1) h−αh γ (h − αh + 1) + (35) k∑ j=h α−1 h (x) αj (x) [ d αj ∗−1 qn (x) − d αj ∗−1 f (x) − ρn h! d αj ∗−1 xh ] (23) ≥ ρn (x + 1) h−αh γ (h − αh + 1) −   k∑ j=h lj 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n )   = 72 george a. anastassiou cubo 17, 1 (2015) ρn (x + 1) h−αh γ (h − αh + 1) − ρn = ρn [ (x + 1) h−αh γ (h − αh + 1) − 1 ] = ρn [ (x + 1) h−αh − γ (h − αh + 1) γ (h − αh + 1) ] ≥ ρn [ 1 − γ (h − αh + 1) γ (h − αh + 1) ] ≥ 0. (36) explanation: we know that γ (1) = 1, γ (2) = 1, and γ is convex and positive on (0, ∞) . here 0 ≤ h − αh < 1 and 1 ≤ h − αh + 1 < 2. thus γ (h − αh + 1) ≤ 1 and 1 − γ (h − αh + 1) ≥ 0. hence l∗ (qn (x)) ≥ 0, x ∈ [0, 1] . ii. suppose on [0, 1] that αh (x) ≤ β < 0. let qn (x), x ∈ [−1, 1], be the polynomial spline of order m > 0, (as before), so that max −1≤x≤1 ∣ ∣ ∣ ∣ d αj ∗−1 ( f (x) − ρn xh h! ) − ( d αj ∗−1 qn ) (x) ∣ ∣ ∣ ∣ ≤ 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (37) j = 0, 1, ..., σ − 2. similarly as before we obtain again inequalities of convergence (13), (15) and (16). also if 0 ≤ x ≤ 1, then α−1h (x) l ∗ (qn (x)) = α −1 h (x) l ∗ (f (x)) − ρn (x + 1) h−αh γ (h − αh + 1) + (38) k∑ j=h α−1h (x) αj (x) [ d αj ∗−1 qn (x) − d αj ∗−1 f (x) + ρn h! ( d αj ∗−1 xh ) ] (37) ≤ − ρn (x + 1) h−αh γ (h − αh + 1) + k∑ j=h lj 2j−αj γ (j − αj + 1) c∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = (39) ρn ( 1 − (x + 1) h−αh γ (h − αh + 1) ) = ρn ( γ (h − αh + 1) − (x + 1) h−αh γ (h − αh + 1) ) ≤ (40) ρn ( 1 − (x + 1) h−αh γ (h − αh + 1) ) ≤ 0, and hence again l∗ (qn (x)) ≥ 0, x ∈ [0, 1] . received: may 2014. accepted: october 2014. cubo 17, 1 (2015) spline left fractional monotone approximation involving left . . . 73 references [1] g.a. anastassiou, spline monotone approximation with linear differential operators, approximation theory and its applications, vol. 5 (4) (1989), 61-67. [2] r. de vore, g. lorentz, constructive approximation, springer-verlag, heidelberg, new york, 1993. [3] k. diethelm, the analysis of fractional differential equations, lecture notes in mathematics, vol. 2004, 1st edition, springer, new york, heidelberg, 2010. [4] l.l. schumaker, spline functions: basic theory, john wiley and sons, inc., new york, 1981. introduction main result cubo a mathematical journal vol.18, no¯ 01, (27–45). december 2016 on generalized closed sets in generalized topological spaces b. k. tyagi1, harsh v. s. chauhan2 1 department of mathematics, atmaram sanatan dharma college, university of delhi, new delhi-110021, india. 2 department of mathematics, university of delhi, new delhi-110007, india brijkishore.tyagi@gmail.com, harsh.chauhan111@gmail.com abstract in this paper, we introduce several types of generalized closed sets in generalized topological spaces (gtss). their interrelationships are investigated and several characterizations of µ-t0, µ-t1, µ-t1/2, µ-regular, µ-normal gtss and extremally µ-disconnected gtss are obtained. resumen en este art́ıculo introducimos varios tipos de conjuntos cerrados generalizados en espacios topológicos generalizados (gtss). sus interrelaciones son investigadas y varias caracterizaciones de gtss µ-t0, µ-t1, µ-t1/2, µ-regulares, µ-normales y extremalmente µ-disconexos son obtenidas. keywords and phrases: generalized topological spaces, generalized closed sets, extremally µ-disconnectedness, separation axioms. 2010 ams mathematics subject classification: 54a05, 54d15. 28 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) 1 introduction several types of generalized closed sets are investigated in the literature of topological spaces [3, 5, 6, 7, 16, 18, 19, 20, 23, 24, 26, 27, 28, 29, 30, 35, 38, 37, 39, 43, 44, 48, 49]. their relationship with one another is shown by a diagram in benchalli et al. [4] and dontchev [17]. using the concept of generalized closed sets, several separation axioms [17, 21] are introduced which are found to be useful in the study of digital topology (digital line) [25]. cao et al. [9] obtained several characterizations of extremally disconnectedness in terms of generalized closed sets. the purpose of this paper is to show that these diagrams can be obtained in the setting of generalized topological spaces (gtss) introduced by császár [11]. let x be a set and p(x) be the power set of x. a subset µ of p(x) is called generalized topology (gt) on x if µ is closed under arbitrary unions and in that case (x, µ) is called a generalized topological space (gts). the elements of µ are called µ-open sets and their complements are called µ-closed sets. the closure of a, denoted by cµa, is the intersection of µ-closed sets containing a. the interior of a, denoted by iµa, is the union of µ-open sets contained in a. in a gts (x, µ), we define mµ = ∪{u : u ∈ µ}. a gts (x, µ) is called strong if mµ = x. the notions of various generalized closed sets depend on several types of stronger or weaker forms of open sets, for example, regular open set [44], semi open set [26], preopen set [31], semi preopen set [2], α-open set [36], θ-open set [50], δ-open set [50], π-open set [20] etc. all these notions are extended to the setting of generalized topological spaces. the concept of µ-t1/2 gts depends in turn on the concept of a generalized closed set. we explore the relationship of generalized closed sets with several separation axioms, µ-t0, µ-t1, µ-t1/2, µ-regularity, and µ-normality [32, 33]. a concept of extremally µ-disconnectedness was introduced in [46]; a gts (x, µ) is extremallay µ-disconnected if cµu ∩ mµ ∈ µ for every u ∈ µ. it may be remarked that in strong gts, this notion concide with the notion of extremally disconnectedness in császár [12]. several characterizations of extremally µ-disconnectedness in terms of generalized closed sets are obtained. section 2 contains preliminaries. in section 3, we introduce various notions of generalized closed sets and obtain several implications among them. section 4 contains characterizations of µ-t0, µ-t1 and µ-t1/2 gtss. in section 5, we study the characterization of µ-regularity and µ-normality. section 6 obtains some characterizations of extremally µ-disconnected gtss. 2 preliminaries let (x, µ) be a gts and a ⊆ x. ac denotes the complement of a in x. the collection of all µ-closed sets in x is denoted by ω. cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 29 theorem 2.1. let (x, µ) be a gts and a, b ⊆ x. then the following statements hold. (i) x ∈ cµa if and only if x ∈ u ∈ µ implies u ∩ a ̸= ∅. (ii) cµa = cµ(a ∩ mµ). (iii) cµa = x − iµ(x − a). (iv) if u, v ∈ µ and u ∩ v = ∅ then cµu ∩ v = ∅ and u ∩ cµv = ∅. (v) mµ − cµa = x − cµa. (vi) iµa = iµ(a ∩ mµ). (vii) iµ(cµa − a) = ∅. (viii) cµ and iµ are monotone: a ⊆ b implies cµa ⊆ cµb (respectively iµa ⊆ iµb), idempotent cµcµa = cµa (respectively iµiµa = iµa), cµ is enhancing (a ⊆ cµa), iµ is contracting (iµa ⊆ a). proof. (vii). if x ∈ iµ(cµa − a) then there exists a u ∈ µ such that x ∈ u ⊆ cµa − a. then x ∈ u ⊆ cµa and u ∩ a = ∅. now x ∈ u ⊆ cµa implies u ∩ a ̸= ∅, a contradiction. let (x, µ) be a gts and y ⊆ x. then the collection µy = {u ∩ y : u ∈ µ} is a gt on y and (y, µy) is called a generalized subspace of (x, µ). it may be noted that cµy a = cµa ∩ y for any a ⊆ y. thus, a set a ⊆ y is µy-closed if and only if it is the intersection with y of a µ-closed set. definition 2.2. a subset a of a gts (x, µ) is called (i) µ-regular open (or roµ-open) if iµcµa = a. (ii) µ-semi open (or sµ-open) if a ⊆ cµiµa ∩ mµ. (iii) µ-preopen (or pµ-open ) if a ⊆ iµcµa. (iv) µ-α-open (or αµ-open) if a ⊆ iµcµiµa. (v) µ-semi preopen (or spµ-open) if a ⊆ cµiµcµa ∩ mµ. (vi) µ-θ-closed (or θµ-closed) [34] if a = γθa, where γθ(a) = {x ∈ x : cµg ∩ mµ ∩ a ̸= ∅ for all g ∈ µ, x ∈ g}. the complement of a θµ-closed set is called µ-θ-open (θµ-open). (vii) µ-δ-closed (or δµ-closed) [12] if a = cδa, where cδa = {x ∈ x : iµcµu ∩ a ̸= ∅ for u ∈ µ and x ∈ u}. the complement of a δµ-closed set is called µ-δ-open (δµ-open). 30 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) (viii) µ-π-open (or πµ-open) if a is the union of finitely many µ-regular open sets. (ix) µ-regular semi open (or rsµ-open) if there exists a µ-regular open set u such that u ⊆ a ⊆ cµu ∩ mµ. the collections of all µ-( ) sets in (i) to (ix) of the above definitions are denoted by roµ, sµ, pµ, αµ, spµ, θµ, δµ, πµ, rsµ respectively. the complements of the sets in the above definitions are named similarly by replacing the word “open” by “closed”, for example µ-semi closed (or sµclosed) for the complement of a sµ-open set and vice-versa. it follows using theorem 2.1, a subset a of gts (x, µ) is a regular µ-closed (or roµ-closed) if and only if cµiµa = a; a set a is µ-semi open if and only if cµa = cµiµa and a ⊆ mµ. a is sµ-closed if and only if iµcµa ⊆ a and x − mµ ⊆ a; a is pµ-closed if and only if cµiµa ⊆ a; a is αµ-closed if and only if cµiµcµa ⊆ a; a is spµ-closed if iµcµiµa ⊆ a and x − mµ ⊆ a. for any set a, cµiµcµa is αµ-closed. also if a ∈ rsµ then a ∈ sµ but not conversely. theorem 2.3. [46] for a gts (x, µ), θµ, αµ, sµ, pµ and spµ are gtss and (i) θµ ⊆ µ ⊆ αµ ⊆ sµ ⊆ spµ, (ii) αµ ⊆ pµ ⊆ spµ. theorem 2.4. a is αµ-open if and only if a ∈ sµ ∩ pµ. proof. if a ⊆ iµcµiµa then a ⊆ cµiµa, a ⊆ mµ and a ⊆ iµcµa. so a ∈ sµ ∩ pµ. conversely, let a ∈ sµ ∩ pµ. then a ⊆ cµiµa ∩ mµ. therefore, cµa ⊆ cµiµa. also a ⊆ iµcµa. therefore, a ⊆ iµcµiµa a subset a of a gts (x, µ) is µ-nowhere dense if iµcµa = ∅. lemma 2.5. let x be a point in a gts (x, µ). then {x} is µ-nowhere dense or pµ-open. proof. suppose {x} is not µ-nowhere dense. then iµcµ{x} ̸= ∅. then x ∈ iµcµ{x}. so {x} ⊆ iµcµ{x}. lemma 2.6. if {x} is µ-nowhere dense in a gts (x, µ) then {x} ∪ (x − mµ) is αµ-closed. proof. cµiµcµ({x} ∪ (x − mµ)) = cµiµcµ{x} = cµ∅ = x − mµ. so cµiµcµ({x} ∪ (x − mµ)) ⊆ {x} ∪ (x − mµ). lemma 2.7. for a subset a containing x − mµ, csµa = a ∪ iµcµa. proof. since csµa is sµ-closed, iµcµ(csµ a) ⊆ csµa. on the other hand iµcµ(a ∪ iµcµa) ⊆ iµcµcµa = iµcµa. therefore, iµcµ(a ∪ iµcµa) ⊆ a ∪ iµcµa. since x − mµ ⊆ a, a ∪ iµcµa is sµclosed. cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 31 lemma 2.8. for a subset a, cαµa = a ∪ cµiµcµa. proof. since cαµa is αµ-closed, cµiµcµcαµa ⊆ cαµa. therefore, a ∪ cµiµcµa ⊆ cαµa. on the other hand cµiµcµ(a ∪ cµiµcµa) ⊆ cµiµcµcµa = cµiµcµa ⊆ a ∪ cµiµcµa. thus, a ∪ cµiµcµa is αµ-closed set containing a. lemma 2.9. for a subset a, a ∪ iµcµiµa ⊆ cspµa. proof. iµcµiµa ⊆ iµcµiµ(cspµ )a ⊆ cspµa, since cspµa is spµ-closed. 3 various type of generalized closed sets definition 3.1. let (x, µ) be a gts. a subset a of x containing x − mµ is called (i) a µ-generalized closed (or gµ-closed) set if cµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ µ. the complement of a gµ-closed set is called µ-generalized open (or gµ-open). the set of all gµ-open sets is denoted by gµ. (ii) a µ-semi generalized closed (or sgµ-closed) set if csµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ sµ. (iii) a µ-generalized semi closed (or gsµ-closed) set if csµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ µ. (iv) a µ-generalized α-closed (or gαµ-closed) set if cαµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ αµ. (v) a µα-generalized closed (or αgµ-closed) set if cαµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ µ. (vi) a µ-generalized semi preclosed (or gspµ-closed) set if cspµ a ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ µ. (vii) a µ-regular generalized closed (or rgµ-closed) set if cµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ roµ. (viii) a µ-generalized preclosed (or gpµ-closed) set if cpµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ µ. (ix) a µ-generalized preregular closed (or gprµ-closed) set if cpµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ roµ. (x) a µ-θ-generalized closed (or θgµ-closed) set if γθµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ µ. (xi) a µ-δ-generalized closed (or δgµ-closed) set if cδµa ∩ mµ ⊆ u whenever a ⊆ u ∈ µ. (xii) a µ-weakly generalized closed (or wgµ-closed) set if cµiµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ µ. (xiii) a µ-strongly generalized closed (or gµ ∗-closed) set if cµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ gµ. (xiv) a µ-π-generalized closed (or πgµ-closed) set if cµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ πµ. 32 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) (xv) a µ-weakly closed (or wµ-closed) set if cµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ sµ. (xvi) a µ-mildly generalized closed (or mgµ-closed) set if cµiµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ gµ. (xvii) a µ-semi-weakly generalized closed (or swgµ-closed) set if cµiµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ sµ. (xviii) a µ-regular weakly generalized closed (or rwgµ-closed) set if cµiµa ∩ mµ ⊆ u whenever a ∩ mµ ⊆ u ∈ roµ. (xix) a µ-regular generalized w-closed (or rwµ-closed) set if cµa∩mµ ⊆ u whenever a∩mµ ⊆ u ∈ rsµ. lemma 3.2. (i) a ∈ gµ then a ⊆ mµ. (ii) µ⊆ gµ. proof. (i) since x − mµ is contained in a generalized closed set a, the complement of a is contained in mµ. (ii) let a ∈ µ and (x − a) ∩ mµ ⊆ u ∈ µ. then cµ(x − a) ∩ mµ = (x − a) ∩ mµ ⊆ u. theorem 3.3. a subset a of gts (x, µ) is gµ-closed if and only if for any µ-closed set f such that f ∩ mµ ⊆ cµa − a implies f ∩ mµ = ∅. proof. let f be a µ-closed set such that f∩mµ ⊆ cµa−a . then a∩mµ ⊆ f c ∈ µ. since a is gµclosed, cµa ∩ mµ ⊆ f c. that is, f ∩ mµ ⊆ (cµa) c. therefore, f ∩ mµ ⊆ (cµa − a) ∩ (cµa) c = ∅. conversely, let a ∩ mµ ⊆ u ∈ µ and if cµa ∩ mµ is not contained in u then cµa ∩ mµ ∩ u c ̸= ∅. since cµa ∩ u c is µ-closed and cµa ∩ u c ∩ mµ ⊆ cµa − a, a contradiction. theorem 3.4. if a gµ-closed subset a of a gts (x, µ) be such that cµa − (a ∩ mµ) is µ-closed then a is µ-closed. proof. let a be a gµ-closed set such that cµa − (a ∩ mµ) is µ-closed. then cµa − (a ∩ mµ) is µ-closed subset of itself. since cµa − (a ∩ mµ) is gµ-closed subset of itself, by theorem 3.3 (cµa−(a∩mµ))∩mµy , where y = cµa−(a∩mµ), is empty. since mµy = (cµa−(a∩mµ))∩mµ, a is µ-closed. theorem 3.5. if a is a gµ-closed set and a ⊆ b ⊆ cµa then b is gµ-closed. proof. let b ∩ mµ ⊆ u ∈ µ. since a is gµ-closed and a ∩ mµ ⊆ u, cµa ∩ mµ ⊆ u. then cµb ∩ mµ ⊆ cµa ∩ mµ ⊆ u. theorem 3.6. in a gts (x, µ), µ= ω if and only if (x, µ) is strong and every subset of x is gµ-closed. cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 33 proof. if µ= ω then obviously (x, µ) is strong. now if a ⊆ u ∈ µ then cµa ⊆ cµu = u since u ∈ µ. conversely, let u ∈ µ. since u is gµ-closed, cµu ⊆ u. thus, u is µ-closed. on the other hand if f ∈ ω then fc ∈ µ. since µ ⊆ ω, f ∈ µ. theorem 3.7. a subset a of mµ of a gts (x, µ) is gµ-open if and only if f ∩ mµ ⊆ iµa whenever f is µ-closed and f ∩ mµ ⊆ a. proof. let a be a gµ-open set and f be a µ-closed set such that f ∩ mµ ⊆ a. then x − a ⊆ x − (f ∩ mµ). since (x−a)∩mµ ⊆ (x − (f ∩ mµ)) ∩ mµ = x−f and x−a is gµ-closed, cµ(x− a)∩mµ ⊆ x − f. then (x−iµa)∩mµ ⊆ x − f. that is, f∩mµ ⊆ (x − (x − iµa) ∩ mµ) ∩ mµ = iµa. conversely, let a ⊆ mµ and (x − a) ∩ mµ ⊆ u ∈ µ. then x − u ⊆ x − ((x − a) ∩ mµ). so (x − u) ∩ mµ ⊆ a. then (x − u) ∩ mµ ⊆ iµa. so x − iµa ⊆ x − ((x − u) ∩ mµ). therefore, cµ(x − a) ∩ mµ ⊆ u. thus, a is gµ-open. theorem 3.8. a set a in gts (x, µ) is gµ-open if and only if iµa∪(a c ∩ mµ) ⊆ u ∈ µ implies u = mµ. proof. let a be a gµ-open set and iµa ∪ (a c ∩ mµ) ⊆ u ∈ µ. then u c ⊆ (iµa) c ∩ (a ∪ mcµ) = cµ(x − a) ∩ (a ∪ m c µ). therefore, u c ∩ mµ ⊆ (cµ(x − a) ∩ mµ) ∩ a = cµ(x − a) − (x − a). then by theorem 3.3 uc ∩ mµ = ∅, that is, u = mµ. conversely, let f be a µ-closed set such that f ∩ mµ ⊆ a. then iµa ∪ (a c ∩ mµ) ⊆ iµa ∪ f c ∈ µ. by the assumption, iµa ∪ f c = mµ, that is, f ∩ mµ ⊆ iµa. now apply theorem 3.7. theorem 3.9. a subset a of a gts (x, µ) is gµ-closed if and only if cµa − a is gµ-open. proof. suppose that a is gµ-closed and f∩mµ ⊆ cµa − a, where f is a µ-closed set. by theorem 3.3 f ∩ mµ = ∅. so f ∩ mµ ⊆ iµ(cµa − a). therefore, cµa − a is gµopen by theorem 3.7. conversely, assume that x − mµ ⊆ a and a ∩ mµ ⊆ u ∈ µ. now cµa ∩ u c ∩ mµ ⊆ cµa ∩ (mµ − a) = cµa − a. by theorem 3.7 cµa ∩ u c ∩ mµ ⊆ iµ(cµa − a) = ∅. thus, cµa ∩ mµ ⊆ u and a is gµ-closed. the following diagram extends to the setting of gtss the corresponding diagram of benchalli and wali [4] and dontchev [17]. 34 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) for examples showing independence a ! b in the above diagram see [4]. theorem 3.10. let (x, µ) be a gts and a ⊆ x. then the following statements hold. (i) µ-closed ⇒ αµ-closed ⇒ sµ-closed ⇒ spµ-closed. (ii) αµ-closed ⇒ pµ-closed ⇒ spµ-closed. (iii) µ-closed ⇒ gµ-closed ⇒ rgµclosed. (iv) gµ-closed ⇒ αgµ-closed ⇒ gsµ-closed ⇒ gspµ-closed. (v) αµ-closed ⇒ gαµ-closed ⇒ αgµ-closed. (vi) sµ-closed ⇒ sgµ-closed ⇒ gspµ-closed. (vii) sgµ-closed ⇒ gsµ-closed. (viii) spµ-closed ⇒ gspµ-closed. cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 35 (ix) pµ-closed ⇒ gspµ-closed. (x) αgµ-closed ⇒ gspµ-closed. (xi) gαµ-closed ⇒ gsµ-closed. proof. (i) let a be µ-closed set. then cµa = a. therefore, iµcµa = iµa ⊆ a. thus, cµiµcµa ⊆ cµa = a. now let a be a αµ-closed set. then iµcµa ⊆ cµiµcµa ⊆ a. now let a be a sµ-closed set. then iµcµa ⊆ a and x − mµ ⊆ a. therefore, iµcµiµa ⊆ iµcµa ⊂ a. this proves (i). the proofs of other parts also follow easily. theorem 3.11. (i) every sgµ-closed sets is spµ-closed. (ii) every gαµ-closed set is pµ-closed. proof. (i) let a be a sgµ-closed set and x ∈ cspµa ∩ mµ. then {x} is pµ-open or µ-nowhere dense. if {x} is pµ-open then by theorem 2.3, {x} is spµ-open. since x ∈ spµa ∩ mµ, {x}∩a ̸= ∅. therefore, x ∈ a. if {x} is µ-nowhere dense then {x} ∪ (x − mµ) is αµ-closed and hence sµ-closed. therefore, the complement b = mµ − {x} is sµ-open. assume that x /∈ a, then a ∩ mµ ⊆ b. since a is sgµ-closed, and cspµ a ⊆ csµa. cspµa ∩ mµ ⊆ b. hence x /∈ cspµa ∩ mµ. by contradiction x ∈ a. thus, a is spµ-closed. (ii) let a be a gαµ-closed set. let x ∈ cpµa ∩ mµ. if {x} is pµ-open, then {x} ∩ a ̸= ∅. so that x ∈ a. if {x} is µ-nowhere dense and does not meet a then {x} ∪ (x − mµ) is αµ-closed. then b = mµ − {x} is αµ-open and a ∩ mµ ⊆ b. since a is gαµ-closed, cαµa ∩ mµ ⊆ b. therefore, x /∈ cαµa ∩ mµ, a contradiction. thus, x ∈ a and a is pµ-closed. the following theorem also covers some immediate implications. theorem 3.12. for a set in a gts (x, µ), the following statements hold. (i) πµ-closed ⇒ δµ-closed. (ii) θµ-closed ⇒ θgµ-closed. (iii) πµ-closed ⇒ πgµ-closed. (iv) δµ-closed ⇒ δgµ-closed. (v) µ-closed ⇒ gµ ∗-closed. (vi) µ-closed ⇒ wµ-closed. 36 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) (vii) gµ ∗-closed ⇒ mgµ-closed. (viii) gµ ∗-closed ⇒ gµ-closed. (ix) gµ-closed ⇒ wgµ-closed. (x) rgµ-closed ⇒ gprµ-closed. (xi) gpµ-closed ⇒ gprµ-closed. (xii) αgµ-closed ⇒ gpµ-closed. (xiii) wµ-closed ⇒ rwµ-closed. (xiv) rwµ-closed ⇒ rgµ-closed. (xv) rwµ-closed ⇒ rwgµ-closed. proof. (i) let a be a πµ-closed set. then there are µ-regular closed sets ri, r2, .....rn such that a = !n i=1ri. let x ∈ x − a = ∪ n i=1ri c. then x ∈ ri c for some i and iµcµri c ∩ a = ri c ∩ a = ∅. so x /∈ cδµa. the proofs of other parts are also easy and left to the reader. 4 µ-t0, µ-t1 and µ-t1/2 generalized topological spaces definition 4.1. a gts (x, µ) is said to be (i) µ-t0 if x, y ∈ mµ, x ̸= y implies the existence of a µ-open set containing precisely one of x and y. (ii) [32] µ-t1 if x, y ∈ mµ, x ̸= y implies the existence of µ-open sets u1 and u2 such that x ∈ u1 and y /∈ u1 and y ∈ u2 and x /∈ u2. (iii) µ-t1/2 if every gµ-closed set is µ-closed. easy examples of gt-spaces which are not strong and having the properties of the above separation axioms may be provided. for example, let r be the set of real numbers and x, y, x ̸= y be any two real numbers. then µ = {∅, {x}, {x, y}} is a gt which is not strong and has the property of µ-t0 but not µ-t1. it is obvious that µ-t1 implies µ-t0. also (x, µ) is µ-t0 if and only if for each x, y ∈ mµ, cµ({x}) = cµ({y}) implies x = y. theorem 4.2. if a gts (x, µ) is µ-t1/2 then it is µ-t0. cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 37 proof. suppose that (x, µ) is not a µ-t0 space. then there exist distinct points x and y in mµ such that cµ({x}) = cµ({y}). let a = cµ({x})∩{x} c . we show that a is gµ-closed but not µ-closed. x − mµ ⊆ a. let a ∩ mµ ⊆ v ∈ µ. since a ⊆ cµ({x}), cµa ∩ mµ ⊆ cµ({x}) ∩ mµ. thus, we show that cµ({x}) ∩ mµ ⊆ v. since cµ({x}) ∩ {x} c ∩ mµ ⊆ v, it is enough to show that x ∈ v. if x is not in v then y ∈ v and y ∈ cµ({y}) = cµ({x}) ⊆ v c as vc is a µ-closed set containing the set {x}. thus, y ∈ v ∩ vc, a contradiction. now if x ∈ u ∈ µ then u ∩ a ⊇ {y} ̸= ∅, and hence x ∈ cµa. but x is not in a and thus, a is not a µ-closed set. theorem 4.3. if a gts (x, µ) is µ-t1 then for each x ∈ x, a = {x} ∪ (x − mµ) is µ-closed. proof. let y ∈ cµa ∩ mµ and y ̸= x. then y ∈ cµ(a ∩ mµ) ∩ mµ = cµ({x}) ∩ mµ. then y ∈ cµ({x}). so y ∈ u ∈ µ implies x ∈ u which is against our hypothesis. so cµa ∩ mµ = {x}, that is, cµa = a. theorem 4.4. if a gts (x, µ) is µ-t1 then it is µ-t1/2. proof. let a be a subset of x which is not µ-closed. if x−mµ is not contained in a, then a is not gµ-closed. so let x − mµ ⊆ a. since a is not µ-closed, cµa − a is non empty. let x ∈ cµa − a. by theorem 4.3 {x} ∪ (x − mµ) is µ-closed. as ({x} ∪ (x − mµ)) ∩ mµ = {x} ⊆ cµa − a, by theorem 3.3 a is not gµ-closed. definition 4.5. a gts (x, µ) is said to be µ-symmetric if for each x, y ∈ mµ, x ∈ cµ({y}) implies y ∈ cµ({x}). theorem 4.6. a gts (x, µ) is µ-symmetric if and only if {x} ∪ (x − mµ) is gµ-closed for each x ∈ x. proof. let a = {x} ∪ (x − mµ) and a ∩ mµ ⊆ u ∈ µ. if a ∩ mµ = ∅ then cµa = cµ(a ∩ mµ) = cµ∅ = x − mµ. so cµa ∩ mµ ⊆ u. otherwise cµa ∩ mµ = cµ(a ∩ mµ) ∩ mµ = cµ({x}) ∩ mµ. if cµ({x}) ∩ mµ " u then assume that y ∈ cµ({x}) ∩ uc ∩ mµ. since (x, µ) is µ-symmetric, x ∈ cµ({y}). since x ∈ u, y ∈ u, then y ∈ u ∩ u c, a contradiction. conversely, let for each x ∈ x, {x}∪(x − mµ) is gµ-closed. let x, y ∈ mµ, x ∈ cµ({y}) and y /∈ cµ({x}). then y ∈ (cµ({x})) c. let a = {y} ∪ (x − mµ). then a is gµ-closed and a ∩ mµ = {y} ⊆ (cµ({x})) c. so cµa ∩ mµ = (cµ({y})) ∩ mµ ⊆ (cµ({x})) c. then x ∈ (cµ({y})) ∩ mµ ⊆ (cµ({x})) c, a contradiction. corollary 4.7. if a gts (x, µ) is µ-t1 then it is µ-symmetric. proof. the proof follows from theorem 4.3, theorem 4.6 and lemma 3.2. theorem 4.8. a gts (x, µ) is µ-symmetric and µ-t0 if and if only (x, µ) is µ-t1. 38 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) proof. if (x, µ) is µ-t1 then by corollary 4.7 (x, µ) is µ-symmetric and obviously µ-t0. conversely, let (x, µ) be µ-symmetric and µ-t0. let x, y ∈ mµ and x ̸= y. then by µ-t0 property there exists a u ∈ µ such that x ∈ u ⊆ ({y})c. then x is not in cµ({y}). since (x, µ) is µ-symmetric, y is not in cµ({x}). then there exists v = (cµ({x})) c such that y ∈ v and x /∈ v. theorem 4.9. if (x, µ) is µ-symmetric then (x, µ) is µ-t0 if and only if (x, µ) is µ-t1/2 if and only if (x, µ) is µ-t1. proof. the proof follows from theorems 4.8, 4.4 and 4.2. theorem 4.10. a gts (x, µ) is µ-t1/2 if and only if for each x ∈ x, either {x} is µ-open or {x} ∪ (x − mµ) is µ-closed. proof. suppose x is µ-t1/2 and for some x ∈ x, {x} ∪ (x − mµ) is not µ-closed. then mµ is the only µ-open set containing mµ − {x}. therefore, (mµ − {x}) ∪ (x − mµ) is gµ-closed. so it is µ-closed. thus, {x} is µ-open. conversely, let a be a gµ-closed set with x ∈ cµa ∩ mµ and x /∈ a. if {x} is µ-open then ∅ ̸= {x} ∩ a. thus, x ∈ a. otherwise {x} ∪ (x − mµ) is µ-closed. then({x} ∪ (x − mµ)) ∩ mµ = {x} ⊆ cµa − a. then by theorem 3.3 {x} = ∅, a contradiction. thus, x ∈ a and so a is µ-closed. theorem 4.11. for a gts (x, µ), the following statements are equivalent. (i) x is µ-t1/2. (ii) every αgµ-closed set is αµ-closed. proof. (i) ⇒ (ii). let a be a αgµ-closed set and x ∈ cαµa∩mµ. if {x} is µ-open then {x} ∈ αµ so that {x} ∩ a ̸= ∅. thus, x ∈ a. otherwise {x} ∪ (x − mµ) is µ-closed. let x /∈ a. then mµ − {x} is µ-open and a ∩ mµ ⊆ mµ − {x}. since a is αgµ-closed, cαµa ∩ mµ ⊆ mµ − {x}. therefore, x /∈ cαµa ∩ mµ, a contradiction. thus, x ∈ a and a is αµ-closed. (ii) ⇒ (i). if some set {x} ∪ (x − mµ) is not µ-closed then x ∈ mµ and mµ − {x} is not µopen. then (mµ − {x}) ∪ (x − mµ) is trivially αgµ-closed. by (ii), (mµ − {x}) ∪ (x − mµ) is αµ-closed. so {x} is αµ-open. since a non-empty αµ-open set contains a non-empty µ-open set, {x} is µ-open. this shows that (x, µ) is µ-t1/2. 5 µ-regular and µ-normal generalized topological spaces definition 5.1. [33] a gts (x, µ) is said to be µ-regular if for each µ-closed set f of x not containing x ∈ x there exist disjoint µ-open subsets u and v of x such that x ∈ u and f∩mµ ⊆ v. theorems 5.2, 5.4, and 5.5 generalize the corresponding results in roy [40]. cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 39 theorem 5.2. for a gts (x, µ), the following statements are equivalent. (i) x is µ-regular. (ii) x ∈ u ∈ µ implies that there exists v ∈ µ such that x ∈ v ⊆ cµv ∩ mµ ⊆ u. (iii) for each µ-closed set f, f = ∩{cµv : f ∩ mµ ⊆ v ∈ µ}. (iv) for each subset a of x and each u ∈ µ with a ∩ u ̸= ∅ there exists a v ∈ µ such that a ∩ v ̸= ∅ and cµv ∩ mµ ⊆ u. (v) for each non-empty set a ⊆ x and each µ-closed set f with a ∩ f = ∅ there exist u, v ∈ µ such that a ∩ v ̸= ∅, f ∩ mµ ⊆ u and u ∩ v = ∅. (vi) for each µ-closed set f and x /∈ f there exist u ∈ µ and a gµ-open set v such that x ∈ u, f ∩ mµ ⊆ v and u ∩ v = ∅. (vii) for each non-empty a ⊆ x and each µ-closed set f with a ∩ f = ∅ there exist a u ∈ µ and a gµ-open set v such that a ∩ u ̸= ∅, f ∩ mµ ⊆ v and u ∩ v = ∅. (viii) for each µ-closed set f of x, f = ∩{cµv : f ∩ mµ ⊆ v and v is gµ-open}. proof. (i)⇔ (ii) [32]. (ii)⇒ (iii). suppose x /∈ f. then by (ii) there exists a v ∈ µ such that x ∈ v ⊆ cµv ∩mµ ⊆ x − f. then f ∩ mµ ⊆ (x − (cµv ∩ mµ)) ∩ mµ = x − cµv = w ∈ µ. since cµw ∩ v = ∅, (iii) follows. (iii) ⇒ (iv). x ∈ a ∩ u implies that x /∈ x − u. by (iii) there exists a w ∈ µ such that (x − u) ∩ mµ ⊆ w and x /∈ cµw. let v = x − cµw then x ∈ v ∩ a and v ⊆ x − w. thus, cµv ⊆ x − w. therefore, cµv ∩ mµ ⊆ (x − w) ∩ mµ ⊆ (x − ((x − u) ∩ mµ)) ∩ mµ = u. (iv) ⇒ (v). a ∩ (x − f) ̸= ∅. by (iv) there exists a µ-open set v such that a ∩ v ̸= ∅ and cµv ∩mµ ⊆ x − f. let w = x−cµv. then f∩mµ ⊆ ⊆ (x − (cµv ∩ mµ)) ∩ mµ = x−cµv = w and w ∩ v = ∅. (v) ⇒ (i). let f be a µ-closed set not containing x. by (v) there exist disjoint µ-open sets u and v such that x ∈ u and f ∩ mµ ⊆ v. (i) ⇒ (vi). follows from lemma 3.2. (vi) ⇒ (vii). note that a ⊆ mµ. since a is non-empty and a ∩ f = ∅ there exists a point x ∈ a such that x /∈ f. by (vi) there exist a u ∈ µ and a gµ-open set v such that x ∈ u, f ∩ mµ ⊆ v and u ∩ v = ∅. then u ∩ a ̸= ∅. (vii) ⇒ (i). let x /∈ f, where f is a µ-closed set. then {x} ∩ f = ∅. by (vii) there exist a u ∈ µ and a gµ-open set v such that x ∈ u, f∩mµ ⊆ v and u∩v = ∅. now f∩mµ ⊆ iµv by theorem 3.7. (iii) ⇒ (viii). we have f ⊆ ∩{cµv : f ∩ mµ ⊆ v and v is gµ-open} ⊆ ∩{cµv : f ∩ mµ ⊆ v ∈ µ} = f. (viii) ⇒ (i). let f be a µ-closed set such that x /∈ f. then by (viii) there exists gµ-open set w such that f ∩ mµ ⊆ w and x /∈ cµw. since f is µ-closed, w is gµ-open and f ∩ mµ ⊆ w, by theorem 3.7, f ∩ mµ ⊆ iµw. 40 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) definition 5.3. [32] a gts (x, µ) is µ-normal if for any pair of µ-closed sets a and b such that a ∩ b ∩ mµ = ∅ there exist disjoint µ-open sets u and v such that a ∩ mµ ⊆ u and b ∩ mµ ⊆ v. theorem 5.4. for a gts (x, µ), the following statements are equivalent. (i) x is µ-normal. (ii) for any µ-closed set a and µ-open set u such that a∩mµ ⊆ u there is a µ-open set v such that a ∩ mµ ⊆ v ⊆ cµv ∩ mµ ⊆ u. proof. let a be a µ-closed set such that a ∩ mµ ⊆ u ∈ µ. then b = x − u is µ-closed and a ∩ b ∩ mµ is empty. then by (i) there exist disjoint µ-open sets v and w such that a ∩ mµ ⊆ v and b∩mµ ⊆ w. then a∩mµ ⊆ v ⊆ cµv ∩ mµ ⊆ (x − w) ∩ mµ ⊆ (x − (b ∩ mµ)) ∩ mµ = u. conversely, assume that a and b be µ-closed sets such that a∩b∩mµ = ∅. then u = x−b is µopen and a∩mµ ⊆ u. by (ii) there exists a µ-open set v such that a∩mµ ⊆ v ⊆ cµv ∩ mµ ⊆ u. let w = x − cµv. since cµv ∩ b ∩ mµ = ∅, b ∩ mµ ⊆ x − cµv = w. theorem 5.5. in a gts (x, µ), the following statements are equivalent. (i) x is µ-normal. (ii) for any pair of µ-closed sets a and b such that a ∩ b ∩ mµ = ∅ then there exist disjoint gµ-open sets u and v such that a ∩ mµ ⊆ u and b ∩ mµ ⊆ v. (iii) for every µ-closed set a and µ-open set u such that a ∩ mµ ⊆ u there exists a gµ-open set v such that a ∩ mµ ⊆ v ⊆ cµv ∩ mµ ⊆ u. (iv) for every µ-closed set a and every gµ-open set u containing a ∩ mµ there exists a µ-open set v such that a ∩ mµ ⊆ v ⊆ cµv ∩ mµ ⊆ u. (v) for every gµ-closed set a and every µ-open set u containing a ∩ mµ there exists a µ-open set v such that cµa ∩ mµ ⊆ v ⊆ cµv ∩ mµ ⊆ u. proof. (i) ⇒ (ii). follows from lemma 3.2. (ii) ⇒ (iii). assume that b = x − u. by (ii) there exist disjoint gµ-open sets v and w such that a ∩ mµ ⊆ v and b ∩ mµ ⊆ w. since b ∩ mµ ⊆ w, (x − u) ∩ mµ ⊆ w. therefore, (x−w)∩mµ ⊆ (x − (x − u) ∩ mµ) ∩ mµ = u. since x−w is gµ-closed, cµ(x−w)∩mµ ⊆ u. since cµv ∩ mµ ⊆ cµ(x − w) ∩ mµ, the implication is established. (iii) ⇒ (iv). by theorem 3.7 a ∩ mµ ⊆ iµu. then by (iii) there exists a gµ-open set v such that a ∩ mµ ⊆ v ⊆ cµv ∩ mµ ⊆ iµu. by theorem 3.7 a ∩ mµ ⊆ iµv ⊆ cµ(iµv) ∩ mµ ⊆ cµv ∩ mµ ⊆ u. (iv) ⇒ (v). let a be a gµ-closed set and a ∩ mµ ⊆ u ∈ µ. then cµa ∩ mµ ⊆ u. by (iv) there exists a µ-open set v such that cµa ∩ mµ ⊆ v ⊆ cµv ∩ mµ ⊆ u. (v) ⇒ (i). let a and b be µ-closed sets such that a∩b∩mµ = ∅. then a∩mµ ⊆ x − b ∈ µ. by cubo 18, 1 (2016) on generalized closed sets in generalized topological spaces. 41 (v) there exists a µ-open set v such that cµa∩mµ ⊆ v ⊆ cµv ∩ mµ ⊆ x−b. thus, a∩mµ ⊆ v and b ⊆ x − (cµv ∩ mµ). therefore, b∩mµ ⊆ (x − (cµv ∩ mµ)) ∩ mµ = x−cµv = w ∈ µ. 6 extremally µ-disconnectedness theorem 6.1. for a gts (x, µ), the following statements are equivalent. (i) (x, µ) is extremally µ-disconnected. (ii) every spµ-closed set is pµ-closed. (iii) every sgµ-closed set is pµ-closed. (iv) every sµ-closed set is pµ-closed. (v) every sµ-closed set is αµ-closed. (vi) every sµ-closed set is gαµ-closed. proof. (i) ⇒ (ii). let a be a spµ-closed set. then by lemma 2.9 iµcµiµa ⊆ a. since x is extremally µ-disconnected, cµiµa∩mµ = iµ(cµiµa∩mµ) ⊆ iµcµiµa. therefore, cµiµa∩mµ ⊆ a. since x − mµ ⊆ a, cµiµa ⊆ a. (ii) ⇒ (iii). is theorem 3.11(i). (iii) ⇒ (iv). since a sµ-closed set is sgµclosed, the result follows. (iv) ⇒ (v). follows from theorem 2.4. (v) ⇒ (vi). follows from theorem 3.10(v). (vi) ⇒ (i). let u be a µ-open set. we need to show that iµ(cµu ∩ mµ) = cµu ∩ mµ. now iµ(cµu ∩ mµ) = iµcµu. since iµcµu ⊆ cµu ∩ mµ, we prove the inclusion cµu ∩ mµ ⊆ iµcµu. let a = iµcµu ∪ x − mµ. now iµcµa = iµcµiµcµu = iµcµu ⊆ a. so a is sµ-closed. by our assumption a is gαµ-closed. since iµcµiµ(iµcµu) = iµcµu, iµcµu is αµ-open. since a ∩ mµ = iµcµu ∈ αµ and a is gαµ-closed, cαµa ∩ mµ ⊆ iµcµu ⊆ a. thus, cαµa ⊆ a. on the other hand cαµa = a ∪ cµiµcµa implies cµiµcµa ⊆ a. therefore, cµiµcµa ∩ mµ ⊆ a ∩ mµ = iµcµa, which implies that a is µ-closed. now u ⊆ iµcµu. then cµu ⊆ cµiµcµu = cµa = a. therefore, cµu ∩ mµ ⊆ iµcµu. future scope: this paper may be useful in the study of digital topology since generalized closed sets and t1/2 separation axiom have already proved their utility in that area. 42 b. k. tyagi, harsh v. s. chauhan cubo 18, 1 (2016) 7 acknowledgements the second author acknowledges the fellowship grant of university grant commission, india. the authors are very grateful to anonymous referee for his observations which improved the paper. references [1] m.e. abd el-monsef, s.n. el-deep and r.a. mahmoud, β-open sets and βcontinuous mappings, bull. fac. sci. assiut univ. 12 (1983), 77–90. 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[51] ge xun, ge ying, µ-separations in generalized topological spaces, appl. math. j. chiness univ. 25(2) (2010), 243–252. cubo a mathematical journal vol.16, no¯ 02, (01–31). june 2014 pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory toka diagana howard university, 2441 6th street n.w., washington, d.c. 20059, usa. tdiagana@howard.edu khalil ezzinbi université cadi ayyad, faculté des sciences semlalia, département de mathématiques, bp 2390, marrakesh, maroc mohsen miraoui institut supérieur des etudes technologiques de kairouan, rakkada-3191 kairouan, tunisie. abstract motivated by the recent works by the first and the second named authors, in this paper we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively, doubly-weighted pseudo-almost automorphy) using theoretical measure theory. basic properties of these new spaces are studied. to illustrate our work, we study, under acquistapace–terreni conditions and exponential dichotomy, the existence of (µ,ν)pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to some nonautonomous partial evolution equations in banach spaces. a few illustrative examples will be discussed at the end of the paper. resumen motivado por los trabajos recientes del primer y segundo autor, en este art́ıculo introducimos la noción de seudo-casi periodicidad con doble peso (seudo-casi automorf́ıa con doble peso respectivamente) usando teoŕıa de la medida. se estudian las propiedades básicas de estos espacios nuevos. para ilustrar nuestro trabajo, bajo las condiciones de acquistapace-terreni y dicotomı́a exponencial estudiamos la existencia de soluciones (respectivamente, (µ,ν) seudo-casi periódicas (µ,ν) seudo-casi automórficas) para algunas ecuaciones parciales de evolución autónomas en espacios de banach. algunos ejemplos ilustrativos se discutirán al final del art́ıculo. keywords and phrases: evolution family; exponential dichotomy; acquistapace–terreni conditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomous equation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy; (µ,ν)-pseudo-almost periodicity; (µ,ν)-pseudo-almost automorphy; neutral systems; positive measure. 2010 ams mathematics subject classification: 34c27; 34k14; 34k30; 35b15; 43a60; 47d06; 28axx; 58d25; 65j08. 2 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) 1 introduction motivated by the recent works by ezzinbi et al. [12, 13] and diagana [30], in this paper we make extensive use of theoretical measure theory to introduce and study the concept of doubly-weighted pseudo almost periodicity (respectively, doubly-weighted pseudo almost automorphy). obviously, these new notions generalize all the different notions of weighted pseudo-almost periodicity (respectively, weighted pseudo-almost automorphy) recently introduced in the literature. in contrast with [12, 13], here the idea consists of using two positive measures instead of one. doing so will provide us a larger and richer class of weighted ergodic spaces. basic properties of these new functions will be studied including their translation invariance and compositions etc. to illustrate our study, we study the existence of (µ,ν)-pseudo-almost periodic (respectively, (µ,ν)-pseudo-almost automorphic) solutions to the following nonautonomous differential equations, d dt u(t) = a(t)u(t) + f(t, u(t)), t ∈ r, (1.1) and d dt ! u(t) − g(t, u(t)) " = a(t) ! u(t) − g(t, u(t)) " + f(t, u(t)), t ∈ r, (1.2) where a(t) : d(a(t)) ⊂ x #→ x for t ∈ r is a family of closed linear operators on a banach space x, satisfying the well-known acquistapace–terreni conditions, and f, g : r × x #→ x are jointly continuous functions satisfying some additional conditions. one should indicate that the autonomous case, i.e., a(t) = a for all t ∈ r, and the periodic case, that is, a(t + θ) = a(t) for some θ > 0, have been extensively studied, see [8, 10, 40, 41, 53, 56] for the almost periodic case and [18, 22, 39, 42, 50, 51] for the almost automorphic case. recently, diagana [24, 25, 26, 32] studied the existence and uniqueness of weighted pseudo-almost periodic and weighted pseudoalmost automorphic solutions to some classes of nonautonomous partial evolution equations of type eq. (1.1). similarly, in diagana [33], the existence of pseudo-almost periodic solutions to eq. (1.2) has been studied in the particular case when g = 0. in this paper it goes back to studying the existence of doubly-weighted pseudo-almost periodic (respectively, doubly-weighted pseudo-almost automorphic) solutions in the general case as outlined above using theoretical measure theory. the existence and uniqueness of almost periodic, almost automorphic, pseudo-almost periodic and pseudo-almost automorphic solutions is one of the most attractive topics in the qualitative theory of ordinary or functional differential equations due to applications in the physical sciences, mathematical biology, and control theory. the concept of almost automorphy, which was introduced by bochner [15], is an important generalization of the classical almost periodicity in the sense of bohr. for basic results on almost periodic and almost automorphic functions we refer the reader to [7, 59, 61], where the authors give an important overview about their applications to differential equations. in recent years, the existence of almost periodic, pseudo-almost periodic, almost automorphic, and pseudo-almost automorphic solutions to different kinds of differential equations have been extensively investigated by many people, see, e.g., cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 3 [3, 4, 5, 16, 17, 19, 20, 30, 23, 24, 32, 33, 34, 35, 36, 37, 39, 43, 44, 45, 46, 48, 57, 58, 60] and the references therein. the concept of weighted pseudo-almost periodicity, which was introduced by diagana [25, 26, 27, 29] is a natural generalization of the classical pseudo-almost periodicity due to zhang [59, 60, 61]. a few years later, blot et al. [11], introduced the concept of weighted pseudo-almost automorphy as a generalization of weighted pseudo-almost periodicity. more recently, ezzinbi et al. [12, 13] presented a new approach to study weighted pseudo-almost periodic and weighted pseudo-almost automorphic functions using theoretical measure theory, which turns out to be more general than diagana’s approach. let us explain the meaning of this notion as introduced by ezzinbi et al.et al. [12, 13]. let µ be a positive measure on r. we say that a continuous function f : r #→ x is µ-pseudo-almost periodic (respectively, µ-pseudo almost automorphic) if f = g + ϕ, where g is almost periodic (respectively, almost automorphic) and ϕ is ergodic with respect to the measure µ in the sense that lim r→∞ 1 µ(qr) ∫ qr ∥ϕ(s)∥dµ(s) = 0, where qr := [−r, r] and µ(qr) := ∫ qr dµ(t). one can observe that a ρ-weighted pseudo almost automorphic function is µ-pseudo almost automorphic, where the measure µ is absolutely continuous with respect to the lebesgue measure and its radonnikodym derivative is ρ, dµ(t) dt = ρ(t). here we generalize the above-mentioned notion of µ-pseudo-almost periodicity. fix two positive measures µ,ν in r. we say that a function f : r #→ x is (µ,ν)-pseudo-almost periodic (respectively, (µ,ν)-pseudo-almost automorphic) if f = g + ϕ, where g is almost periodic (respectively, almost automorphic) and ϕ is (µ,ν)-ergodic in the sense that lim r→∞ 1 ν(qr) ∫ qr ∥ϕ(s)∥dµ(s) = 0, clearly, the (µ, µ)-pseudo-almost periodicity coincides with the µ-pseudo-almost periodicity. more generally, the (µ,ν)-pseudo-almost periodicity coincides with the µ-pseudo-almost periodicity when the measures µ and ν are equivalent. in this paper, we introduce and study properties of (µ,ν)-pseudo-almost periodic functions and make use of these new functions to study the existence and uniqueness of (µ,ν)-pseudo-almost periodic (respectively, (µ,ν)pseudo-almost automorphic) solutions of the nonautonomous partial evolution equations eq. (1.1) and eq. (1.2) in a banach space. 4 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) the organization of this paper is as follows. in section 2, we recall some definitions and lemmas of (µ,ν)-pseudo almost periodic functions, (µ,ν)-pseudo-almost automorphic functions, and the basic notations of evolution family and exponential dichotomy. in section 3, we study the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to both eq. (1.1) and eq. (1.2). in section 4, we give some examples to illustrate our abstract results. 2 preliminaries 2.1 (µ,ν)-pseudo-almost periodic and (µ,ν)-pseudo-almost automorphic functions let (x, ∥ · ∥), (y, ∥ · ∥) be two banach spaces and let bc(r, x) (respectively, bc(r × y, x)) be the space of bounded continuous functions f : r −→ x (respectively, jointly bounded continuous functions f : r × y −→ x). obviously, the space bc(r, x) equipped with the super norm ∥f∥∞ := sup t∈r ∥f(t)∥ is a banach space. let b(x, y) denote the banach spaces of all bounded linear operator from x into y equipped with natural topology with b(x, x) = b(x). definition 2.1. [21] a continuous function f : r #→ x is said to be almost periodic if for every ε > 0 there exists a positive number l(ε) such that every interval of length l(ε) contains a number τ such that ∥f(t + τ) − f(t)∥ < ε for t ∈ r. let ap(r, x) denote the collection of almost periodic functions from r to x. it can be easily shown that (ap(r, x), ∥ · ∥∞) is a banach space. definition 2.2. [38] a jointly continuous function f : r × y #→ x is said to be almost periodic in t uniformly for y ∈ y, if for every ε > 0, and any compact subset k of y, there exists a positive number l(ε) such that every interval of length l(ε) contains a number τ such that ∥f(t + τ, y) − f(t, y)∥ < ε for (t, y) ∈ r × k. we denote the set of such functions as apu(r × y, x). let µ,ν ∈ m. if f : r #→ x is a bounded continuous function, we define its doubly-weighted mean, if the limit exists, by m(f, µ,ν) := lim r→∞ 1 ν(qr) ∫ qr f(t)dµ(t). cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 5 it is well-known that if f ∈ ap(r, x), then its mean defined by m(f) := lim r→∞ 1 2r ∫ qr f(t)dt exists [15]. consequently, for every λ ∈ r, the following limit a(f,λ) := lim r→∞ 1 2r ∫ qr f(t)e−iλtdt exists and is called the bohr transform of f. it is well-known that a(f,λ) is nonzero at most at countably many points [15]. the set defined by σb(f) := { λ ∈ r : a(f,λ) ̸= 0 } is called the bohr spectrum of f [47]. theorem 2.3. [47] let f ∈ ap(r, x). then for every ε > 0 there exists a trigonometric polynomial pε(t) = n∑ k=1 ake iλkt where ak ∈ x and λk ∈ σb(f) such that ∥f(t) − pε(t)∥ < ε for all t ∈ r. theorem 2.4. let µ,ν ∈ m and suppose that lim r→∞ µ(qr) ν(qr) = θµν. if f : r #→ x is an almost periodic function such that lim r→∞ # # # # # 1 ν(qr) ∫ qr eiλtdµ(t) # # # # # = 0 (2.1) for all 0 ̸= λ ∈ σb(f), then the doubly-weighted mean of f, m(f, µ,ν) = lim t→∞ 1 ν(qt ) ∫ qt f(t)dµ(t) exists. furthermore, m(f, µ,ν) = θµνm(f). proof. the proof of this theorem was given in [30] in the case of measures of the form ρ(t)dt. for the sake of completeness we reproduce it here for positive measures. if f is a trigonometric polynomial, say, f(t) = ∑n k=0 ake iλkt where ak ∈ x − {0} and λk ∈ r for k = 1, 2, ..., n, then σb(f) = {λk : k = 1, 2, ..., n}. moreover, 1 ν(qr) ∫ qr f(t)dµ(t) = a0 µ(qr) ν(qr) + 1 ν(qr) ∫ qr $ n∑ k=1 ake iλkt % dµ(t) = a0 µ(qr) ν(qr) + n∑ k=1 ak $ 1 ν(qr) ∫ qr eiλktdµ(t) % 6 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) and hence & & & & 1 ν(qr) ∫ qr f(t)dµ(t) − a0 µ(qr) ν(qr) & & & & ≤ n∑ k=1 ∥ak∥ # # # 1 ν(qr) ∫ qr eiλktdµ(t) # # # which by eq. (2.1) yields & & & & 1 ν(qr) ∫ qr f(t)dµ(t) − a0θµν & & & & → 0 as r → ∞ and therefore m(f, µ,ν) = a0θµν = θµνm(f). if in the finite sequence of λk there exist λnk = 0 for k = 1, 2, ...l with am ∈ x − {0} for all m ̸= nk (k = 1, 2, ..., l), it can be easily shown that m(f, µ,ν) = θµν l∑ k=1 ank = θµνm(f). now if f : r #→ x is an arbitrary almost periodic function, then for every ε > 0 there exists a trigonometric polynomial (theorem 2.3) pε defined by pε(t) = n∑ k=1 ake iλkt where ak ∈ x and λk ∈ σb(f) such that ∥f(t) − pε(t)∥ < ε (2.2) for all t ∈ r. proceeding as in bohr [15] it follows that there exists r0 such that for all r1, r2 > r0, & & & 1 ν(qr1) ∫ qr1 pε(t)dµ(t) − 1 ν(qr2) ∫ qr2 pε(t)dµ(t) & & & = θµν & & & m(pε) − m(pε) & & & = 0 < ε. in view of the above it follows that for all r1, r2 > r0, & & & 1 ν(qr1) ∫ qr1 f(t)dµ(t) − 1 ν(qr2) ∫ qr2 pε(t)dµ(t) & & & ≤ 1 ν(qr1) ∫ qr1 ∥f(t) − pε(t)∥dµ(t) + & & & 1 ν(qr1) ∫ qr1 pε(t)dµ(t) − 1 ν(qr2) ∫ qr2 pε(t)dµ(t) & & & < ε. now for all r > r0, & & & 1 ν(qr) ∫ qr f(t)dµ(t) − 1 ν(qr) ∫ qr pε(t)dµ(t) & & & < ε and hence m(f, µ,ν) = m(pε, µ,ν) = θµνm(pε) = θµνm(f). the proof is complete. cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 7 definition 2.5. [51] a continuous function f : r → x is called almost automorphic if for every sequence (σn)n∈n there exists a subsequence (sn)n∈n ⊂ (σn)n∈n such that lim n,m→∞ f(t + sn − sm) = f(t) for each t ∈ r. equivalently, g(t) := lim n→∞ f(t + sn) and f(t) = lim n→∞ g(t − sn) are well defined for each t ∈ r. let aa(r, x) denote the collection of all almost automorphic functions from r to x. it can be easily shown that (aa(r, x), ∥.∥∞) is a banach space. definition 2.6. [13] a function f : r × x → y is said to be almost automorphic in t uniformly with respect to x in x if the following two conditions hold: (i) for all x ∈ x, f(., x) ∈ aa(r, y), (ii) f is uniformly continuous on each compact set k in x with respect to the second variable x, namely, for each compact set k in x, for all ε > 0, there exists δ > 0 such that for all x1, x2 ∈ k, one has ∥x1 − x2∥ ≤ δ ⇒ sup t∈r ∥f(t, x1) − f(t, x2)∥ ≤ ε. denote by aau(r × x, y) the set of all such functions. remark 2.7. [13] note that in the above limit the function g is just measurable. if the convergence in both limits is uniform in t ∈ r, then f is almost periodic. the concept of almost automorphy is then larger than almost periodicity. if f is almost automorphic, then its range is relatively compact, thus bounded in norm. example 2.8. [49] let k : r → r be such that k(t) = sin ! 1 2 + cos(t) + cos( √ 2t) " , t ∈ r. then k is almost automorphic, but it is not uniformly continuous on r. then, it is not almost periodic. in what follows, we introduce a new concept of ergodicity, which will generalize those given in [12] and [29, 31]. let b denote the lebesque σ-field of r and let m be the set of all positive measures µ on b satisfying µ(r) = +∞ and µ([a, b]) < ∞, for all a, b ∈ r (a ≤ b). definition 2.9. [12] let µ,ν ∈ m. the measures µ and ν are said to be equivalent there exist constants c0, c1 > 0 and a bounded interval ω ⊂ r (eventually ∅) such that c0ν(a) ≤ µ(a) ≤ c1ν(a) for all a ∈ b satisfying a ∩ ω = ∅. 8 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) we introduce the following new space. definition 2.10. let µ,ν ∈ m. a bounded continuous function f : r → x is said to be (µ,ν)ergodic if lim r→∞ 1 ν(qr) ∫ qr ∥f(s)∥dµ(s) = 0. we then denote the collection of all such functions by e(r, x, µ,ν). we are now ready to introduce the notion of (µ,ν)-pseudo-almost periodicity (respectively, (µ,ν)-pseudo-almost automorphy) for two positive measures µ,ν ∈ m. definition 2.11. let µ,ν ∈ m. a continuous function f : r → x is said to be (µ,ν)-pseudo almost periodic if it can be written in the form f = g + h, where g ∈ ap(r, x) and h ∈ e(r, x, µ,ν). the collection of such functions is denoted by pap(r, x, µ,ν). definition 2.12. let µ,ν ∈ m. a continuous function f : r → x is said to be (µ,ν)-pseudo almost automorphic if it can be written in the form f = g + h, where g ∈ aa(r, x) and h ∈ e(r, x, µ,ν). the collection of such functions will be denoted by paa(r, x, µ,ν). we formulate the following hypotheses. (m.1) let µ,ν ∈ m such that lim sup r→∞ µ(qr) ν(qr) < ∞. (2.3) (m.2) for all τ ∈ r, there exist β > 0 and a bounded interval i such that µ({a + τ : a ∈ a}) ≤ βµ(a) when a ∈ b satisfies a ∩ i = ∅. theorem 2.13. let µ,ν ∈ m satisfy (m.2). then the spaces pap(r, x, µ,ν) and paa(r, x, µ,ν) are translation invariants. proof. we show that e(r, x, µ,ν) is translation invariant. let f ∈ e(r, x, µ,ν), we will show that t #→ f(t + s) belongs to e(r, x, µ,ν) for each s ∈ r. cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 9 indeed, letting µs = µ({t + s : t ∈ a}) for a ∈ b it follows from (m.2) that µ and µs are equivalent (see [12]). now 1 ν(qr) ∫ qr ∥f(t + s)∥dµ(t) = ν(qr+|s|) ν(qr) . 1 ν(qr+|s|) ∫ qr ∥f(t + s)∥dµ(t) = ν(qr+|s|) ν(qr) . 1 ν(qr+|s|) ∫ qr+|s| ∥f(t)∥dµ−s(t) ≤ ν(qr+|s|) ν(qr) . cst. ν(qr+|s|) ∫ qr+|s| ∥f(t)∥dµ(t). since ν satisfies (m.2) and f ∈ e(r, x, µ,ν), we have lim r→∞ 1 ν(qr) ∫ qr ∥f(t + s)∥dµ(t) = 0. therefore, e(r, x, µ,ν) is translation invariant. since ap(r, x) and aa(r, x) are translation invariants, then pap(r, x, µ,ν) and paa(r, x, µ,ν) are translation invariants. theorem 2.14. let µ,ν ∈ m satisfy (m.1), then (e(r, x, µ,ν), ∥.∥∞) is a banach space. proof. it is clear that (e(r, x, µ,ν) is a vector subspace of bc(r, x). to complete the proof, it is enough to prove that (e(r, x, µ,ν) is closed in bc(r, x). if (fn)n be a sequence in (e(r, x, µ,ν) such that lim n→∞ fn = f uniformly in r. from ν(r) = ∞, it follows ν(qr) > 0 for r sufficiently large. using the inequality ∫ qr ∥f(t)∥dµ(t) ≤ ∫ qr ∥f(t) − fn(t)∥dµ(t) + ∫ qr ∥fn(t)∥dµ(t) we deduce that 1 ν(qr) ∫ qr ∥f(t)∥dµ(t) ≤ µ(qr) ν(qr) ∥f − fn∥∞ + 1 ν(qr) ∫ qr ∥fn(t)∥dµ(t), then from (m.1) we have lim sup r→∞ 1 ν(qr) ∫ qr ∥f(t)∥dµ(t) ≤ cst.∥f − fn∥∞ for all n ∈ n. since lim n→∞ ∥f − fn∥∞ = 0, we deduce that lim r→∞ 1 ν(qr) ∫ qr ∥f(t)∥dµ(t) = 0. 10 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) lemma 2.15. [13] let g ∈ aa(r, x) and ε > 0 be given. then there exist s1, ..., sm ∈ r such that r = i=1 ' m (si + cε), where cε := {t ∈ r : ∥g(t) − g(0)∥ < ε}. theorem 2.16. let µ,ν ∈ m and f ∈ paa(r, x, µ,ν) be such that f = g + φ, where g ∈ aa(r, x) and φ ∈ e(r, x, µ,ν). if paa(r, x, µ,ν) is translation invariant, then {g(t); t ∈ r} ⊂ {f(t); t ∈ r}, (the closure of the range of f). (2.4) proof. the proof is similar to the one given in [13]. indeed, if we assume that (2.4) does not hold, then there exists t0 ∈ r such that g(t0) is not in {f(t); t ∈ r}. since the spaces aa(r, x) and e(r, x, µ,ν) are translation invariants, we can assume that t0 = 0, then there exists ε > 0 such that ∥f(t) − g(0)∥ > 2ε for all t ∈ r. then we have ∥φ(t)∥ = ∥f(t) − g(t)∥ ≥ ∥f(t) − g(0)∥ − ∥g(t) − g(0)∥ ≥ ε for all t ∈ cε. therefore, ∥φ(t − si)∥ ≥ ε, for all i ∈ {1, ..., m}, and t ∈ si + cε. let φ be the function defined by φ(t) := i=m∑ i=1 ∥φ(t − si)∥. from lemma 2.15, we deduce that ∥φ(t)∥ ≥ ε for all t ∈ r. (2.5) since e(r, x, µ,ν) is translation invariant, then [t → φ(t − si)] ∈ e(r, x, µ,ν) for all i ∈ {1, ..., m}, then φ ∈ e(r, x, µ,ν) which is a contradiction. consequently (2.4) holds. theorem 2.17. let µ,ν ∈ m satisfy (m.2), then the decomposition of a (µ,ν)-pseudo almost automorphic function in the form f = g + h, where g ∈ aa(r, x) and h ∈ e(r, x, µ,ν), is unique. proof. suppose that f = g1 + φ1 = g2 + φ2, where g1, g2 ∈ aa(r, x) and φ1,φ2 ∈ e(r, x, µ,ν). then 0 = (g1 − g2) + (φ1 − φ2) ∈ paa(r, x, µ,ν) where g1 − g2 ∈ aa(r, x) and φ1 − φ2 ∈ e(r, x, µ,ν). from theorem 2.16, we obtain (g1 − g2)(r) ⊂ {0}, therefore we have g1 = g2 and φ1 = φ2. from theorem 2.17, we deduce cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 11 theorem 2.18. let µ,ν ∈ m satisfy (m.2), then the decomposition of a (µ,ν)-pseudo almost periodic function in the form f = g + h, where g ∈ ap(r, x) and h ∈ e(r, x, µ,ν), is unique. theorem 2.19. let µ,ν ∈ m satisfy (m.1) and (m.2). then, the spaces (pap(r, x, µ,ν), ∥.∥∞) and (paa(r, x, µ,ν), ∥.∥∞) are banach spaces. proof. the proof is similar to the one given in [13], in fact we assume that (fn)n is a cauchy sequence in paa(r, x, µ,ν). we have fn = gn +φn where gn ∈ aa(r, x) and φn ∈ e(r, x, µ,ν). from theorem 2.16 we see that ∥gn − gm∥∞ ≤ ∥fn − fm∥∞, therefore (gn)n is a cauchy sequence in the banach space (aa(r, x), ∥.∥∞). so, φn = fn − gn is also a cauchy sequence in the banach space e((r, x, µ,ν), ∥.∥∞). then we have limn→∞ gn = g ∈ aa(r, x) and limn→∞ φn = φ ∈ e(r, x, µ,ν). finally we have lim n→∞ fn = g + φ ∈ paa(r, x, µ,ν). the proof for pap(r, x, µ,ν) is similar to that of paa(r, x, µ,ν). definition 2.20. let µ,ν ∈ m. a continuous function f : r × y → x is said to be (µ,ν)-ergodic in t uniformly with respect to y ∈ y if the following conditions are true (i) for all y ∈ y, f(., y) ∈ e(r, x, µ,ν). (ii) f is uniformly continuous on each compact set k in y with respect to the second variable y. the collection of such function is denoted by eu(r × y, x, µ,ν). definition 2.21. let µ,ν ∈ m. a continuous function f : r × y → x is said to be (µ,ν)-pseudo almost periodic if is written in the form f = g + h, where g ∈ apu(r × y, x) and h ∈ eu(r × y, x, µ,ν). the collection of such functions is denoted by papu(r × y, x, µ,ν). theorem 2.22. let µ, ν ∈ m and i be a bounded interval (eventually i = ø). assume that (m1) and f ∈ bc(r, x). then the following assertions are equivalent: (i) f ∈ e(r, x, µ,ν). (ii) lim r→∞ 1 ν(qr \ i) ∫ qr\i ∥f(t)∥dµ(t) = 0. (iii) for any ε > 0, lim r→∞ µ({t ∈ qr \ i : ∥f(t)∥ > ε}) ν({qr \ i) = 0. proof. the proof is similar to the one given in [13], in fact we have (i) ⇔ (ii) : denote by a = ν(i), b = ∫ i ∥f(t)∥dµ(t) and c = µ(i). since the interval i is bounded 12 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) and the function f is bounded and continuous, then a, b and c are finite. for r > 0 such that i ⊂ qr and ν(qr \ i) > 0, we have 1 ν(qr \ i) ∫ qr\i ∥f(t)∥dµ(t) = 1 ν(qr) − a ! ∫ qr ∥f(t)∥dµ(t) − b " = ν(qr) ν(qr) − a ! 1 ν(qr) ∫ qr ∥f(t)∥dµ(t) − b ν(qr) " . since ν(r) = ∞, we deduce that (ii) is equivalent to (i). (iii) ⇒ (ii) denote by aεr and b ε r the following sets aεr = {t ∈ qr \ i : ∥f(t)∥ > ε} and b ε r = {t ∈ qr \ i : ∥f(t)∥ ≤ ε}. assume that (iii) holds, that is lim r→∞ µ(aεr) ν(qr \ i) = 0. from the following equality ∫ qr\i ∥f(t)∥dµ(t) = ∫ aεr ∥f(t)∥dµ(t) + ∫ bεr ∥f(t)∥dµ(t), and (m.1), we deduce for r large enough that , 1 ν(qr \ i) ∫ qr\i ∥f(t)∥dµ(t) ≤ ∥f∥∞ µ(aεr) ν(qr \ i) + µ(bεr) ν(qr \ i) ε ≤ ∥f∥∞ µ(aεr) ν(qr \ i) + µ(qr \ i) ν(qr \ i) ε = ∥f∥∞ µ(aεr) ν(qr \ i) + µ(qr) − c ν(qr) − a ε = ∥f∥∞ µ(aεr) ν(qr \ i) + µ(qr) ν(qr) 1 − c µ(qr) 1 − a ν(qr) ε ≤ ∥f∥∞ µ(aεr) ν(qr \ i) + cst. 1 − c µ(qr) 1 − a ν(qr) ε. since µ(r) = ν(r) = ∞, then for all ε > 0 we have lim sup r→∞ 1 ν(qr \ i) ∫ qr\i ∥f(t)∥dµ(t) ≤ cst.ε, cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 13 consequently (ii) holds. (ii) ⇒ (iii) assume that (ii) holds. from the following inequality: 1 ν(qr \ i) ∫ qr\i ∥f(t)∥dµ(t) ≥ 1 ν(qr \ i) ∫ aεr ∥f(t)∥dµ(t) ≥ ε µ(aεr) ν(qr \ i) , for r sufficiently large, we obtain (iii). theorem 2.23. [12] let f ∈ apu(r×x, y) and h ∈ ap(r, x). then [t #−→ f(t, h(t))] ∈ ap(r, y). proposition 2.24. [12] let f : r × x → y be a continuous function. then f ∈ apu(r × x, y) if and only if the two following conditions hold: (i) for all x ∈ x, f(., x) ∈ ap(r, y), (ii) f is uniformly continuous on each compact set k in x with respect to the second variable x, namely, for each compact set k in x, for all ε > 0, there exists δ > 0 such that for all x1, x2 ∈ k, one has ∥x1 − x2∥ ≤ δ ⇒ sup t∈r ∥f(t, x1) − f(t, x2)∥ ≤ ε. the proof of our result of composition of (µ,ν)-pseudo-almost periodic functions is based on the following lemma due to schwartz [54]. lemma 2.25. if ψ ∈ c(x, y), then for each compact set k in x and all ε > 0, there exists δ > 0, such that for any x1, x2 ∈ x, one has x1 ∈ k and ∥x1 − x2∥ ≤ δ ⇒ ∥ψ(x1) − ψ(x2)∥ ≤ ε. theorem 2.26. let µ, ν ∈ m, f ∈ papu(r × x, y, µ,ν) and h ∈ pap(r, y, µ,ν). assume that (m.1) and the following hypothesis holds: (c) for all bounded subset b of y, f is bounded on r × b. then t #−→ f(t, h(t)) ∈ pap(r, y, µ,ν). proof. the function [t #→ f(t, h(t))] is continuous and by hypothesis (c), it is bounded. since h ∈ pap(r, x, µ,ν), we can write h = h1 + h2, where h1 ∈ ap(r, x) and h2 is (µ,ν)-ergodic. since f ∈ papu(r × x, y, µ,ν), we have f = f1 + f2 14 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) where f1 ∈ apu(r × x, y) and f2 ∈ e(r × x, y, µ,ν). the function f can be written in the form f(t, h(t)) = f1(t, h1(t)) + [f(t, h(t)) − f(t, h1(t))] + [f(t, h1(t)) − f1(t, h1(t))] = f1(t, h1(t)) + [f(t, h(t)) − f(t, h1(t))] + f2(t, h1(t)). from theorem 2.23, we have [t #−→ f1(t, h1(t))] ∈ ap(r, y). denote by k the closure of the range of h1: k = ¯{h1(t) : t ∈ r} . since h1 is almost periodic, k is a compact subset of x. denote by φ the function defined by φ : x → pap(r, y, µ,ν) x #→ f(., x) since f ∈ papu(r × x, y, µ,ν) ), by using proposition 2.24, we deduce that the restriction of φ on all compact k of x, is uniformly continuous, which is equivalent to saying that the function φ is continuous on x. from lemma 2.25 applied to φ, we deduce that for given ε > 0, there exists δ > 0 such that, for all t ∈ r, ξ1 and ξ2 ∈ x, one has ξ1 ∈ k and ∥ξ1 − ξ2∥ ≤ δ ⇒ ∥f(t,ξ1) − f(t,ξ2)∥ ≤ ε. since h(t) = h1(t) + h2(t) and h1(t) ∈ k, we have t ∈ r and ∥h2(t)∥ ≤ δ ⇒ ∥f(t, h(t)) − f(t, h2(t))∥ ≤ ε, therefore, we have µ{t ∈ qr : ∥f(t, h(t)) − f(t, h1(t))∥ > ε} ν(qr) ≤ µ{t ∈ qr : ∥h2(t)∥ > δ} ν(qr) . since h2 is (µ,ν)-ergodic, theorem 2.22 yields that for the above-mentioned δ we have lim r→∞ µ{t ∈ qr : ∥h2(t)∥ > δ} ν(qr) = 0, then we obtain lim r→∞ µ{t ∈ qr : ∥f(t, h(t)) − f(t, h1(t))∥ > ε} ν(qr) = 0. by theorem 2.22 we have t #→ f(t, h(t))−f(t, h1(t)) is (µ,ν)-ergodic. now to complete the proof, it is enough to prove that t #→ f2(t, h1(t)) is (µ,ν)-ergodic. since f2 is uniformly continuous on the compact set k = ¯{h1(t) : t ∈ r} with respect to the second variable x, we deduce that for given ε > 0, there exists δ > 0 such that, for all t ∈ r, ξ1 and ξ2 ∈ k, one has ∥ξ1 − ξ2∥ ≤ δ ⇒ ∥f2(t,ξ1) − f2(t,ξ2)∥ ≤ ε. then, there exist n(ε) and {xi} n(ε) i=1 ⊂ k, such that k ⊂ n(ε) ' i=1 b(xi,δ), cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 15 and then ∥f2(t, h1(t))∥ ≤ ε + n(ε)∑ i=1 ∥f2(t, xi)∥. since ∀i ∈ {1, ..., n(ε)}, lim r→+∞ 1 ν(qr) ∫ qr ∥f2(t, xi)∥dµ(t) = 0, we deduce that ∀ε > 0, lim sup r→∞ 1 ν(qr) ∫ qr ∥f2(t, h1(t))∥dµ(t) ≤ ε, that implies lim r→∞ 1 ν(qr) ∫ qr ∥f2(t, h1(t))∥dµ(t) = 0, then t #→ f2(t, h1(t)) is (µ,ν)-ergodic and the theorem is proved. definition 2.27. let µ,ν ∈ m. a continuous function f : r × y → x is said to be (µ,ν)-pseudo almost automorphic if is written in the form f = g + h, where g ∈ aau(r × y, x) and h ∈ eu(r × y, x, µ,ν). the collection of such functions is denoted by paau(r × y, x, µ,ν). theorem 2.28. let µ,ν ∈ m, f ∈ paau(r × x, y, µ,ν) and h ∈ paa(r, y, µ,ν). assume that, for all bounded subset b of y, f is bounded on r × b. then t #−→ f(t, h(t)) ∈ paa(r, x, µ,ν). proof. the proof for paa(r, y, µ,ν) is similar to that of pap(r, y, µ,ν). 2.2 evolution families and exponential dichotomy (h0) a family of closed linear operators a(t) for t ∈ r on x with domain d(a(t)) (possibly not densely defined) is said to satisfy the so-called acquistapace–terreni conditions, if there exist constants ω ∈ r, θ ∈ (π 2 ,π), k, l ≥ 0 and µ0,ν0 ∈ (0, 1] with µ0 + ν0 > 1 such that σθ ∪ {0} ⊂ ρ(a(t) − ω) ∋ λ, ∥r(λ, a(t) − ω)∥ ≤ k 1 + |λ| (2.6) and ∥(a(t) − ω)r(λ, a(t) − ω) [r(ω, a(t)) − r(ω, a(s))]∥ ≤ l |t − s|µ0 |λ|ν0 (2.7) for t, s ∈ r, λ ∈ σθ := {λ ∈ c \ {0} : | argλ| ≤ θ}. 16 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) note that in the particular case when a(t) has a constant domain d = d(a(t)), it is well-known on [6] that condition (2.7) can be replaced with the following one: there exist constants l and 0 < γ ≤ 1 such that ∥(a(t) − a(s))r(ω, a(r))∥ ≤ l|t − s|γ, for all, s, t, r ∈ r. (2.8) for a given family of linear operators a(t), the existence of an evolution family associated with it is not always guaranteed. however, if a(t) satisfies acquistapace–terreni, then there exists a unique evolution family(see[1, 2, 52]) u = {u(t, s) : t, s ∈ r, t ≥ s} on x associated with a(t) such that u(t, s)x ⊆ d(a(t)) for all t, s ∈ r with t ≥ s, and, (1) u(t, r)u(r, s) = u(t, s) and u(s, s) = i for all t ≥ r ≥ s and t, r, s ∈ r; (2) the map (t, s) → u(t, s)x is continuous for all x ∈ x, t ≥ s and t, s ∈ r; (3) u(·, s) ∈ c1((s, ∞), b(x)), ∂u ∂t (t, s) = a(t)u(t, s) and & & & a(t)ku(t, s) & & & ≤ k (t − s)−k for 0 < t − s ≤ 1, k = 0, 1. definition 2.29. an evolution family (u(t, s))t≥s on a banach space x is x is called hyperbolic (or has exponential dichotomy) if there exist projections p(t), t ∈ r, uniformly bounded and strongly continuous in t, and constant n ≥ 1, δ > 0 such that (1) u(t, s)p(s) = p(t)u(t, s) for t ≥ s; (2) the restriction uq(t, s) : q(s)x → q(t)x of u(t, s) is invertible for t, s ∈ r and we set uq(t, s) = u(s, t) −1; (3) ∥u(t, s)p(s)∥ ≤ ne−δ(t−s) (2.9) and ∥uq(s, t)q(t)∥ ≤ ne−δ(t−s) (2.10) for t ≥ s and t, s ∈ r, were q(t) := i − p(t). 3 existence results to study the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to equation (1.1), in addition to above, we also assume that the next assumption holds: cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 17 (h1) the evolution family u generated by a(.) has an exponential dichotomy with constants n ≥ 1, δ > 0 and dichotomy projections p(t). we recall from [48, 55], the following sufficient conditions to fulfill the assumption (h1). (h1.1) let (a(t), d(a(t)))t∈r be generators of analytic semigroups on x of the same type. suppose that d(a(t)) = d(a(0)), a(t) is invertible, supt,s∈r ∥a(t)a(s)−1∥ is finite, and ∥a(t)a(s)−1 − i∥ ≤ l0|t − s|µ1 for t, s ∈ r and constants l0 ≥ 0 and 0 < µ1 ≤ 1. (h1.2) the semigroups (eτa(t))τ≥0, t ∈ r, are hyperbolic with projection pt and constants n,δ > 0. moreover, let ∥a(t)eτa(t)pt∥ ≤ ψ(τ) and ∥a(t)eτaq(t)qt∥ ≤ ψ(−τ) for τ > 0 and a function ψ such that r ∋ s → ϕ(s) := |s|µψ(s) is integrable with l0∥ϕ∥l1(r) < 1. now, we first introduce the definition of the mild solution to eq. (1.1). definition 3.1. a continuous function u : r #→ x is called a bounded mild solution of equation (1.1) if: u(t) = u(t, s)u(s) + ∫t s u(t,τ)f(τ, u(τ))dτ, t ≥ s, t, s ∈ r. (3.1) theorem 3.2. assume that (h0) and (h1) hold. if there exists 0 < kf < δ 2n such that ∥f(t, u) − f(t, v)∥ ≤ kf∥u − v∥, for all u, v ∈ x and t ∈ r, then the equation (1.1) has a unique bounded mild solution u : r #→ x given by u(t) = ∫ r γ(t, s)f(s, u(s))ds, t ∈ r, where the operator family γ(t, s), called green’s function corresponding to u and p(.), is given by { γ(t, s) = u(t, s)p(s), t ≥ s, t, s ∈ r γ(t, s) = −uq(t, s)q(s), t < s, t, s ∈ r. proof. if one supposes u(t) = ∫ r γ(t,τ)f(τ, u(τ))dτ, t ∈ r, 18 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) thus we have u(t) = ∫t −∞ u(t,τ)p(τ)f(τ, u(τ))dτ − ∫+∞ t uq(t,τ)q(τ)f(τ, u(τ))dτ, for all t ∈ r. for t = s, one obtains u(s) = ∫s −∞ u(s,τ)p(τ)f(τ, u(τ))dτ − ∫+∞ s uq(s,τ)q(τ)f(τ, u(τ))dτ, and u(t, s)u(s) = ∫s −∞ u(t,τ)p(τ)f(τ, u(τ))dτ − ∫+∞ s uq(t,τ)q(τ)f(τ, u(τ))dτ. now, we have u(t) − u(t, s)u(s) = ∫t s u(t,τ)f(τ, u(τ))dτ. then, u(t) = ∫ r γ(t,τ)f(τ, u(τ))dτ checks equation (3.1). consider the nonlinear operator k defined on x by ku(t) = ∫ r γ(t,τ)f(τ, u(τ))dτ, t ∈ r. to complete the proof, one has to show that k is a contraction map on x. from assumption (h1), there exist two constant n ≥ 1 and δ > 0 such that ∥γ(t, s)∥ ≤ ne−δ|t−s| for all t, s ∈ r if u, v ∈ x, then one has ∥kv − ku∥∞ < 2nkf δ ∥v − u∥∞, and k is a contraction map on x. therefore, k has unique fixed point in x, that is, there exists unique u ∈ x such that ku = u. therefore, eq.(1.1) has unique mild solution. denote by γ1 and γ2 the nonlinear integral operators defined by, (γ1u)(t) := ∫t −∞ u(t, s)p(s)f(s, u(s))ds, and (γ2u)(t) := ∫+∞ t uq(t, s)q(s)f(s, u(s))ds. in the rest of the paper, we fix µ,ν ∈ m satisfy (m1) and (m2). cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 19 3.1 existence of (µ,ν)-pseudo-almost periodic solutions in addition to the previous assumptions, we require the following additional ones: (h2) r(ω, a(·)) ∈ ap(r, l(x)). (h3) we suppose f : r × x #→ x belongs to pap(r × x, x, µ,ν) and there exists kf > 0 such that ∥f(t, u) − f(t, v)∥ ≤ kf∥u − v∥, for all u, v ∈ x and t ∈ r. the following lemma plays an important role to prove main results of this work. lemma 3.3. [48] assume that (h0), (h1)and (h2) hold. then r → γ(t + r, s + r) belongs to ap(r, l(x)) for t, s ∈ r, where we may take the same pseudo periods for t, s with |t − s| ≥ h > 0. if f ∈ ap(r, l(x)), then the unique bounded mild solution u(t) = ∫ r γ(t, s)f(s)ds of the following equation u′(t) = a(t)u(t) + f(t), t ∈ r, is almost periodic. lemma 3.4. under assumptions (h0)–(h3), then the integral operators γ1 and γ2 defined above map pap(r, x, µ,ν) into itself. proof. let u ∈ pap(r, x, µ,ν). setting h(t) = f(t, u(t)), using assumption (h3) and theorem 2.26 it follows that h ∈ pap(r, x, µ). now write h = ψ1 + ψ2 where ψ1 ∈ ap(r, x) and ψ2 ∈ e(r, x, µ,ν). that is, γ1h = ξ(ψ1) + ξ(ψ2) where ξψ1(t) := ∫t −∞ u(t, s)p(s)ψ1(s)ds, and ξψ2(t) := ∫t −∞ u(t, s)p(s)ψ2(s)ds. from lemma 3.3, we have ξ(ψ1) ∈ ap(r, x). to complete the proof, we will prove that ξ(ψ2) ∈ e(r, x, µ,ν). now, let r > 0. again from eq. (2.9), we have 1 ν(qr) ∫ qr ∥(ξψ2)(t)∥dµ(t) ≤ 1 ν(qr) ∫ qr ∫+∞ 0 ∥u(t, t − s)p(t − s)ψ2(t − s)∥dsdµ(t) ≤ n ν(qr) ∫ qr ∫+∞ 0 e−δs∥ψ2(t − s)∥dsdµ(t) ≤ n ∫+∞ 0 e−δs ( 1 ν(qr) ∫ qr ∥ψ2(t − s)∥dµ(t) ) ds. 20 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) since µ and ν satisfy (m2), then from theorem 2.13, we have t #→ ψ2(t − s) ∈ e(r, x, µ,ν) for every s ∈ r. by the lebesgue’s dominated convergence theorem, we have lim r→∞ 1 ν(qr) ∫ qr ∥(ξψ2)(t)∥dµ(t) = 0. the proof for γ2u(·) is similar to that of γ1u(·) except that one makes use of equation (2.10) instead of equation (2.9). theorem 3.5. under assumptions (h0)—(h3), then eq. (1.1) has a unique (µ,ν)-pseudo almost periodic mild solution whenever kf is small enough. proof. consider the nonlinear operator m defined on pap(r, x, µ,ν) by mu(t) = ∫t −∞ u(t, s)p(s)f(s, u(s))ds − ∫+∞ t uq(t, s)q(s)f(s, u(s))ds, for all t ∈ r. in view of lemma 3.4, it follows that m maps pap(r, x, µ,ν) into itself. to complete the proof one has to show that m is a contraction map on pap(r, x, µ,ν). let u, v ∈ pap(r, x, µ,ν). firstly, we have ∥γ1(v)(t) − γ1(u)(t)∥ ≤ ∫t −∞ ∥u(t, s)p(s) [f(s, v(s)) − f(s, u(s))] ∥ds ≤ nkf ∫t −∞ e−δ(t−s)∥v(s) − u(s)∥ds ≤ nkfδ−1∥v − u∥∞. next, we have ∥γ2(v)(t) − γ2(u)(t)∥ ≤ ∫+∞ t ∥uq(t, s)q(s) [f(s, v(s)) − f(s, u(s))] ∥ds ≤ n ∫+∞ t eδ(t−s)∥f(s, v(s)) − f(s, u(s))∥ds ≤ nkf ∫+∞ t eδ(t−s)∥v(s) − u(s)∥ds ≤ nkf∥v − u∥∞ ∫+∞ t eδ(t−s)ds = nkfδ −1∥v − u∥∞. finally, combining previous approximations it follows that ∥mv − mu∥∞ < 2nkfδ−1∥v − u∥∞. thus if kf is small enough, that is, kf < δ(2n) −1, then m is a contraction map on pap(r, x, µ,ν). therefore, m has unique fixed point in pap(r, x, µ,ν), that is, there exists unique function u satisfying mu = u, which is the unique (µ,ν)-pseudo almost periodic mild solution to eq. (1.1). cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 21 3.2 existence of (µ,ν)-pseudo-almost automorphic solutions in this section we consider the following conditions: (h’2) r(ω, a(·)) ∈ aa(r, l(x)). (h’3) we suppose f : r × x #→ x belongs to paa(r × x, x, µ,ν) and there exists kf > 0 such that ∥f(t, u) − f(t, v)∥ ≤ kf∥u − v∥∞, for all u, v ∈ x and t ∈ r. lemma 3.6. [9] assume that (h0), (h1)and (h’2) hold. let a sequence (s′l)l∈n ∈ r there is a subsequence (sl)l∈n such that for every h > 0 ∥γ(t + sl − sk, s + sl − sk) − γ(t, s)∥ → 0, k, l → ∞ for |t − s| ≥ h. lemma 3.7. under assumptions (h0), (h1) , (h’2) and (h’3), then the integral operators γ1 and γ2 defined above map paa(r, x, µ,ν) into itself. proof. let u ∈ paa(r, x, µ,ν). setting g(t) = f(t, u(t)), using assumption (h’3) and theorem 2.28 it follows that g ∈ paa(r, x, µ,ν). now write g = u1 + u2 where u1 ∈ aa(r, x) and u2 ∈ e(r, x, µ,ν). that is, γ1g = su1 + su2, where su1(t) := ∫t −∞ u(t, s)p(s)u1(s)ds, and su2(t) := ∫t −∞ u(t, s)p(s)u2(s)ds. from eq. (2.9), we obtain ∥su1(t)∥ ≤ nδ−1∥u1∥∞ and ∥su2(t)∥ ≤ nδ−1∥u2∥∞ for all t ∈ r. then su1, su2 ∈ bc(r, x). now, we prove that su1 ∈ aa(r, x). since u1 ∈ aa(r, x), then for every sequence (τ′n)n∈n ∈ r there exists a subsequence (τn)n∈n such that v1(t) := lim n→∞ u1(t + τn), (3.2) is well defined for each t ∈ r, and lim n→∞ v1(t − τn) = u1(t), (3.3) for each t ∈ r. set m(t) = ∫t −∞ u(t, s)p(s)u1(s)ds and n(t) = ∫t −∞ u(t, s)p(s)v1(s)ds, t ∈ r. 22 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) now, we have m(t + τn) − n(t) = ∫t+τn −∞ u(t + τn, s)p(s)u1(s)ds − ∫t −∞ u(t, s)p(s)v1(s)ds = ∫t −∞ u(t + τn, s + τn)p(s + τn)u1(s + τn)ds − ∫t −∞ u(t, s)p(s)v1(s)ds = ∫t −∞ u(t + τn, s + τn)p(s + τn) $ u1(s + τn) − v1(s) % ds + ∫t −∞ $ u(t + τn, s + τn)p(s + τn) − u(t, s)p(s) % v1(s)ds. using eq. (2.9), eq. (3.2) and the lebesgue’s dominated convergence theorem, it follows that ∥ ∫t −∞ u(t + τn, s + τn)p(s + τn) $ u1(s + τn) − v1(s) % ds∥ → 0 as n → ∞, t ∈ r. (3.4) similarly, using lemma 3.6 it follows that ∥ ∫t −∞ $ u(t + τn, s + τn)p(s + τn) − u(t, s)p(s) % v1(s)ds∥ → 0 as n → ∞, t ∈ r. (3.5) therefore, we have n(t) := lim n→∞ m(t + τn), t ∈ r. (3.6) using similar ideas as the previous ones, then m(t) := lim n→∞ n(t − τn), t ∈ r. (3.7) therefore, su1 ∈ aa(r, x). finally, by using the same stages that the proof of lemma 3.4 one obtains su2 ∈ e(r, x, µ,ν). the proof for γ2u(·) is similar to that of γ1u(·) except that one makes use of equation (2.10) instead of equation 2.9. theorem 3.8. under assumptions (h0), (h1) , (h’2) and (h’3), then eq. (1.1) has a unique (µ,ν)-pseudo almost automorphic mild solution whenever kf is small enough. proof. the proof for theorem 3.8 is similar to that of theorem 3.5 except that one makes use of lemma 3.7 instead of lemma 3.4. cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 23 3.3 neutral systems in this subsection, we establish the existence and uniqueness of (µ,ν)-pseudo-almost periodic (respectively, (µ,ν)-pseudo-almost automorphic) solutions for the nonautonomous neutral partial evolution equation (1.2). for that, we need the following assumptions: (h4) we suppose g : r × x #→ x belongs to pap(r × x, x, µ,ν) and there exists kg > 0 such that ∥g(t, u) − g(t, v)∥ ≤ kg∥u − v∥, for all u, v ∈ x and t ∈ r. (h’4) g : r × x #→ x belongs to paa(r × x, x, µ,ν) and there exists kg > 0 such that ∥g(t, u) − g(t, v)∥ ≤ kg∥u − v∥, for all u, v ∈ x and t ∈ r. definition 3.9. a function v : r #→ x is said to be a bounded mild solution to equation (1.2) and we have: v(t) = g(t, v(t)) + ∫t −∞ u(t, s)p(s)f(s, v(s))ds − ∫+∞ t uq(t, s)q(s)f(s, v(s))ds for all t ∈ r. theorem 3.10. if assumptions (h0), (h1), (h2), (h3) and (h4) hold and ! kg+2nkfδ −1 " < 1, then eq. (1.2) has a unique (µ,ν)-pseudo almost periodic mild solution. proof. we consider the nonlinear operator w defined on pap(r, x, µ) by wv(t) = g(t, v(t)) + ∫t −∞ u(t, s)p(s)f(s, v(s))ds − ∫+∞ t uq(t, s)q(s)f(s, v(s))ds for all t ∈ r. from (h4), theorem (2.26), and lemma 3.4 it follows that w maps pap(r, x, µ,ν) into itself. to complete the proof we need to show that w is a contraction map on pap(r, x, µ,ν). for that, letting u, v ∈ pap(r, x, µ,ν), we obtain ∥wv − wu∥∞ ≤ ! kg + 2nkfδ −1 " ∥v − u∥∞, which yields w is a contraction map on pap(r, x, µ,ν). therefore, w has unique fixed point in pap(r, x, µ,ν). therefore, eq.(1.2), has unique (µ,ν)-pseudo-almost periodic mild solution. theorem 3.11. if assumptions (h0), (h1), (h’2), (h’3) and (h’4) hold and ! kg+2nkfδ −1 " < 1, then eq. (1.2) has a unique (µ,ν)-pseudo almost automorphic mild solution. 24 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) proof. similarly the proof of theorem 3.10, we can show, by using the assumption (h’4), theorem 2.28 and lemma 3.7, that the eq. (1.2) has a unique (µ,ν)-pseudo almost automorphic mild solution. 4 examples example 4.1. let x = l2([0, 1]) be equipped with its natural topology. in order to illustrate theorem 3.5, we consider the following one-dimensional heat equation with dirichlet conditions, ∂ ∂t $ v(t, x) % = $ ∂2 ∂x2 + ξ ! sin(at) + sin(bt) "% v(t, x) + f(t, v(t, x)), on r × (0, 1) v(t, 0) = v(t, 1) = 0, t ∈ r, (4.1) where the coefficient ξ ∈]0, 1 2 [, the constants a, b ∈ r with a b /∈ q, and the forcing term f : r×x #→ x is continuous function. in order to write eq.(4.1) in the abstract form eq.(1.1), we consider the linear operator a : d(a) ⊂ x −→ x, given by d(a) = h2(0, 1) ∩ h10(0, 1) and au = u ′′ for u ∈ d(a). it is well known that a is the infinitesimal generator of an exponentially stable c0-semigroup * t(t) + t≥0 such that ∥t(t)∥ ≤ e−π 2t for t ≥ 0. define a family of linear operator a(t) as follows: ⎧ ⎪⎪⎨ ⎪⎪⎩ d(a(t)) = d(a) = h2[0, 1] ∩ h10[0, 1] a(t)v = $ a + ξ ! sin(at) + sin(bt) "% v, v ∈ d(a). obviously, d(a(t)) = d(a). furthermore, ∥a(t) − a(s)∥ = ∥ξ ! sin(at) − sin(as) + sin(bt) − sin(bs) " ∥ ≤ ξ ! |a| + |b| " |t − s|, for all s, t ∈ r and hence (h0) holds. consequently, a(t) generates an evolution family, which we denote by u(t, s)t≥s and which satisfies u(t, s)v = t(t − s) exp $ ∫t s ξ ! sin(aτ) + sin(bτ) " dτ % v. since ∥u(t, s)∥ ≤ e−(π 2−1)(t−s) for t ≥ s and t, s ∈ r, it follows that (h1) holds with n = 1, δ = π2 − 1 > 0. and since t #→ sin(at) + sin(bt) is almost periodic, then r(ω, a(·)) ∈ ap(r, l(x)) and so (h2) holds. cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 25 let f : r × x #→ x be the mapping defined by f(t,ϕ)(x) = f(t,ϕ(x)) for x ∈ [0, 1], and let y : r → x be the function defined by y(t) = v(t, .), for t ∈ r. then eq. (4.1) takes the abstract form, d dt , y(t) = a(t)y(t) + f(t, y(t)), t ∈ r. (4.2) let µ = ν and suppose that its radon-nikodym derivative is given by ρ(t) = { et if t ≤ 0, 1 if t > 0. then from [12], µ ∈ m satisfies (m1) and (m2). if we assume that f is µ-pseudo almost periodic in t ∈ r uniformly in u ∈ x and is globally lipschitz with respect to the second argument in the following sense: there exists kf > 0 such that & & & f(t, u) − f(t, v) & & & ≤ kf & & & u − v & & & for all t ∈ r and u, v ∈ x, then f satisfies (h3). we deduce all assumptions (h0),(h1),(h2),(h3), (m.1) and (m.2) of theorem 3.5 are satisfied and thus equation (4.1) has a unique (µ, µ)-pseudo almost periodic solution whenever kf is small enough (kf < π2−1 2 ). to illustrate the result in theorem 3.11, we consider the following equation ∂ ∂t $ v(t, x) − g1(t, v(t, x)) % = ! ∂2 ∂x2 + ξ(sin(at) + sin(bt)) "$ v(t, x) − g1(t, v(t, x)) % +f1(t, v(t, x)), on r × (0, 1) v(t, 0) = v(t, 1) = 0, t ∈ r, (4.3) where the coefficient ξ ∈ (0, 1 2 ) , a, b ∈ r and a b /∈ q, f1, g1 : r × x #→ x are given by f1(t, x) = x sin 1 2 + cos t + cos √ 2t + max k∈z {e−(t±k 2)2} cos x, t ∈ r , x ∈ x, g1(t, x) = x 4 sin 1 2 + sin t + sin √ 2t + 1 4 max k∈z {e−(t±k 2)2} cosx, t ∈ r , x ∈ x. clearly, f1, g1 ∈ paa(r × x, x, µ, µ) and satisfies the lipschitz condition in theorem 3.11 with n = 1, δ = π2 − 1, kf1 = 2 and kg1 = 1 2 . by theorem 3.11, the evolution equation (4.3) has a unique (µ, µ)-pseudo almost automorphic solution, with µ being the measure defined in the example above. example 4.2. let ω ⊂ rn (n ≥ 1) be an open bounded subset with regular boundary γ = ∂ω and let x = l2(ω) equipped with its natural topology ∥ · ∥2. 26 toka diagana, khalil ezzinbi & mohsen miraoui cubo 16, 2 (2014) we study the existence of (µ,ν)-pseudo-almost automorphic solutions to the n-dimensional heat equation ⎧ ⎪⎪⎨ ⎪⎪⎩ ∂ϕ ∂t = a(t, x)∆ϕ + g(t,ϕ), in r × ω ϕ = 0, on r × γ (4.4) where a : r × ω #→ r is almost automorphic, and g : r × l2(ω) #→ l2(ω) is (µ,ν)-pseudo almost automorphic function. define the linear operator appearing in eq. (4.4) as follows: a(t)u = a(t, x)∆u for all u ∈ d(a(t)) = h10(ω) ∩ h 2(ω), where a : r×ω #→ r, in addition of being almost automorphic satisfies the following assumptions: (h.1) inf t∈r,x∈ω a(t, x) = m0 > 0, and (h.2) there exists l > 0 and 0 < µ ≤ 1 such that |a(t, x) − a(s, x)| ≤ l|s − t|µ for all t, s ∈ r uniformly in x ∈ ω. (h.3) sup t∈r,x∈ω a(t, x) < ∞. (h.4) g is µ-pseudo-almost periodic in t ∈ r uniformly in u ∈ l2(ω) and satisfying globally lipschitz with respect to the second argument in the following sense: there exists kg > 0 such that ∥g(t, u) − g(t, v)∥2 ≤ kg∥u − v∥2 for all t ∈ r and u, v ∈ l2(ω), a classical example of a function a satisfying the above-mentioned assumptions is for instance a(t, x) = 3 + sin(t) + sin( √ 2t) + l(x), for x ∈ ω and t ∈ r, where l : ω #−→ r+, continuous and bounded on ω. under previous assumptions, it is clear that the operators a(t) defined above are invertible and satisfy acquistapace–terreni conditions. moreover, it can be easily shown that r ! ω, a(·, x)∆ " ϕ = 1 a(·, x) r ! ω a(·, x) ,∆ " ϕ ∈ aa(r, l2(ω)) for each ϕ ∈ l2(ω) with & & & r ! ω, a∆ " & & & b(l2(ω)) ≤ const. |ω| . cubo 16, 2 (2014) pseudo-almost periodic and pseudo-almost automorphic . . . 27 let f : r × l2(ω) #→ l2(ω) be the mapping defined by f(t,φ)(x) = g(t,φ(x)) for x ∈ ω, and z : r → l2(ω) be the function defined by z(t) = ϕ(t, .), for t ∈ r. then the eq.(4.4) takes the abstract form d dt , z(t) = a(t)z(t) + f(t, z(t)), t ∈ r, (4.5) furthermore, all assumptions (h0), (h1), (h’2), (h’3), (m1) and (m2) of theorem 3.8 are fulfilled. then, the evolution equation (4.4) has a unique (µ,ν)-pseudo almost automorphic solution whenever kg is small enough, with µ,ν being arbitrary positive measures of m satisfying (m.1) and (m.2). received: january 2014. revised: april 2014. references [1] p. acquistapace, f. flandoli, b. terreni, initial boundary value problems and optimal control for nonautonomous parabolic systems, siam journal on control and optimization, 29, (1991), 89-118. 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[61] c.y. zhang, pseudo almost periodic solutions of some differential equations ii, journal of mathematical analysis and applications, 192, (1995), 543-561. cubo a mathematical journal vol.14, no¯ 03, (63–69). october 2012 uniformly boundedness of a class of non-linear differential equations of third order with multiple deviating arguments cemil tunç, hilmi ergören department of mathematics, faculty of sciences, yüzüncü yıl university, 65080, van, turkey email: cemtunc@yahoo.com, hergoren@yahoo.com abstract this paper deals with a certain third-order non-linear differential equation with multiple deviating arguments. some sufficient conditions are set up for all solutions and their derivatives to be uniformly bounded. resumen en este art́ıculo se estudia un tipo de ecuaciones diferenciales no lineales de tercer orden con argumentos de desviación múltiple. se establecen algunas condiciones suficientes para que todas las soluciones y sus derivadas sean uniformemente acotadas. keywords and phrases: non-linear differential equation; third order; multiple deviating arguments; bounded solutions. 2010 ams mathematics subject classification: 34c25; 34k13; 34k25. 64 cemil tunç and hilmi ergören cubo 14, 3 (2012) 1 introduction consider the following third order nonlinear differential equation with multiple deviating arguments x ′′′ (t) + f1(t, x(t))x ′′ (t) + f2(t, x(t))x ′ (t) + g0(t, x(t)) + n∑ i=1 gi(t, x(t − τi(t)) = p(t), (1.1) where f1, f2 and gi(i = 0, 1, 2, ..., n) are continuous functions on r + × r, τi(t) ≥ 0 (i = 1, 2, ..., n) and p(t) are bounded continuous functions on r and r+ = [0, +∞), respectively. define y(t) = dx(t) dt +α1x(t) and z(t) = dy(t) dt +α2y(t), where α1 and α2 are some constants. then, we can transform (1.1) into the following system dx(t) dt = −α1x(t) + y(t), dy(t) dt = −α2y(t) + z(t), dz(t) dt = −(f1(t, x(t)) − α1 − α2)z(t) +((f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α 2 2 )y(t) +((α1 − f1(t, x(t)))α 2 1 + f2(t, x(t))α1)x(t) − g0(t, x(t)) − n∑ i=1 gi(t, x(t − τi(t)) + p(t). (1.2) in applied science some practical problems are associated with higher-order nonlinear differential equations, such as nonlinear oscillations (afuwape et al.[1], andres[2] and fridedrichs[3]), electronic theory (rauch[4]), biological models and other models (cronin[5] and gopalsamy[6]). just as above, in the past few decades, the study for third order differential equations has been paid attention by many scholars. many results relative to the stability, boundedness of solutions of third order differential equations with delays or without delays have been obtained (see li[7], murakami[8], ademola et al.[9], tunç and ergören[10], tunç[11−13] and references therein). however, to the best of our knowledge, no authors have considered the boundedness of solutions of third order differential equations with multiple deviating arguments in non-liapunov sense, in spite of the fact that some authors (see afuwape and castellanos[14], gao and liu[15], and yu and zhao[16]) have obtained some results for the third order ones with a deviating argument and second order ones with multiple deviating arguments. thus, it is worthwhile to continue to the investigation of the boundedness of solutions of (1.1) in this case. the main objective of this paper is to study the uniformly boundedness of solutions of (1.1). we will establish some sufficient conditions satisfying the solutions of (1.1) to be uniformly bounded. our result is new and complement to previously known results. in particular, an example is also given to illustrate the effectiveness of the new result. cubo 14, 3 (2012) uniformly boundedness of a class of non-linear ... 65 2 definitions and assumptions we assume that h = max 1≤i≤n { sup t∈r τi(t) } ≥ 0. let c ([−h, 0], r) denote the space of continuous functions φ : [−h, 0] → r with the supremum norm ‖ .‖. it is known in burton[17], hale[18] and kuang[19] that there exists a solution of (1.2) on an interval [0, t) satisfying the initial condition and (1.1) on [0, t) for gi(i = 0, 1, 2, ..., n), φ, f1, f2, p and τi(t)(i = 1, 2, ..., n) continuous, given a continuous initial function φ ∈ c ([−h, 0], r) and a vector (y0, z0) ∈ r 2. if the solution remains bounded, then t = +∞. we denote such a solution by (x(t), y(t), z(t)) = (x(t, φ, y0, z0), y(t, φ, y0, z0), z(t, φ, y0, z0)), where y(s) = y(0) and z(s) = z(0) for all s ∈ [−h, 0]. then, it follows that (x(t), y(t), z(t)) can be defined on [−h, +∞). definition. solutions of (1.2) are called uniformly bounded (ub) if for each b1 > 0 there is a b2 > 0 such that (φ, y0, z0) ∈ c ([−h, 0], r) × r 2 and ‖φ‖ + ‖y0‖ + ‖z0‖ ≤ b1 imply that |x(t, φ, y0, z0)| + |y(t, φ, y0, z0)| + |z(t, φ, y0, z0)| ≤ b2 for all t ∈ r +. in this work, we also assume that the following conditions hold: i) ∣ ∣((α1 − f1(t, u))α 2 1 + f2(t, u)α1)u − g0(t, u) ∣ ∣ ≤ l0 |u| + q0 for all u ∈ r and t ≥ 0, ii) |g1(t, u)| ≤ l1 |u| + q1, |g2(t, u)| ≤ l2 |u| + q2, ..., |gn(t, u)| ≤ ln |u| + qn for all u ∈ r and t ≥ 0, iii) α3 = inf t≥0 (f1(t, u) − α1 − α2) −sup t≥0 ∣ ∣(f1(t, u) − α1)(α1 + α2) − f2(t, u) − α 2 2 ∣ ∣ > n∑ i=0 li, where α1 > 1, α2 > 1, α3 > 0 are some constants, and li and qi (i = 0, 1, 2, ..., n) are nonnegative constants. 3 main result theorem 1. suppose (i)-(iii) hold. then, solutions of (1.2) are uniformly bounded. proof. let (x(t), y(t), z(t)) = (x(t, φ, y0, z0), y(t, φ, y0, z0), z(t, φ, y0, z0)) be a solution of system (1.2) defined on [0, t). we may assume that t = +∞ since the following estimates give a priori bound on (x(t), y(t), z(t)) . calculating the upper right derivative of |x(s)| , |y(s)| and |z(s)| , in view of (i) − (iii), we have d+(|x(s)|)s=t = sgn(x(t)){−α1x(t) + y(t)} ≤ −α1 |x(t)| + |y(t)| , (3.1) d+(|y(s)|)s=t = sgn(y(t)){−α2y(t) + z(t)} ≤ −α2 |y(t)| + |z(t)| (3.2) 66 cemil tunç and hilmi ergören cubo 14, 3 (2012) and d+(|z(s)|)s=t = sgn(z(t)){−(f1(t, x(t)) − α1 − α2)z(t) +((f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α 2 2 )y(t) +((α1 − f1(t, x(t)))α 2 1 + f2(t, x(t))α1)x(t) − g0(t, x(t)) − n∑ i=1 gi(t, x(t − τi(t)) + p(t)} ≤ −inf t≥0 (f1(t, x(t)) − α1 − α2) |z(t)| +sup t≥0 ∣ ∣(f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α 2 2 ∣ ∣ |y(t)| +l0 |x(t)| + n∑ i=1 li |x(t − τi(t))| + n∑ i=0 qi + |p(t)| . (3.3) let m(t) = max −h≤s≤t {max {|x(s)| , |y(s)| , |z(s)|}} , (3.4) where y(s) = y(0), z(s) = z(0) for all −h ≤ s ≤ 0. it is clear that max {|x(t)| , |y(t)| , |z(t)|} ≤ m(t) and m(t) is non-decreasing for t ≥ −h. now, we consider the following two cases: case i): m(t) > max {|x(t)| , |y(t)| , |z(t)|} (3.5) for all t ≥ 0, then we claim that m(t) ≡ m(0) (3.6) is a constant for all t ≥ 0. by contrapositive, assume (3.6) does not hold, then, there exists t1 > 0 such that m(t1) > m(0). here max {|x(t)| , |y(t)| , |z(t)|} ≤ m(0) for all −h ≤ t ≤ 0 and there exists β ∈ (0, t1) such that max {|x(β)| , |y(β)| , |z(β)|} = m(t1) ≥ m(β) which contradicts (3.5). this contradiction implies that (3.6) holds. it follows that there exists t2 > 0 such that max {|x(t)| , |y(t)| , |z(t)|} ≤ m(t) = m(0) for all t ≥ t2. case ii): there is a point t0 ≥ 0 such that m(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|}. let η = min { α1 − 1, α2 − 1, α3 − n∑ i=0 li } > 0 and θ = n∑ i=0 qi + sup t∈r+ |p(t)| + 1 be constants, where t ≥ 0. then, if m(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|} = |x(t0)|, then we obtain d+(|x(s)|)s=t0 ≤ −α1 |x(t0)| + |y(t0)| ≤ (−α1 + 1)m(t0) < −ηm(t0) + θ. (3.7) cubo 14, 3 (2012) uniformly boundedness of a class of non-linear ... 67 if m(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|} = |y(t0)|, then we have d+(|y(s)|)s=t0 ≤ −α2 |y(t)| + |z(t)| ≤ (−α2 + 1)m(t0) < −ηm(t0) + θ. (3.8) if m(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|} = |z(t0)|, then we get d+(|z(s)|)s=t0 ≤ −inf t≥0 (f1(t, x(t)) − α1 − α2) |z(t0)| +sup t≥0 ∣ ∣(f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α 2 2 ∣ ∣ |y(t0)| +l0 |x(t0)| + n∑ i=1 li |x(t0 − τi(t0))| + n∑ i=0 qi + |p(t)| ≤ ( n∑ i=0 li − α3)m(t0) + n∑ i=0 qi + |p(t)| < −ηm(t0) + θ. (3.9) in addition, if m(t0) ≥ θ η , then (3.7), (3.8) and (3.9) imply that m(t) is strictly decreasing in a small neighborhood (t0, t0 + δ0). this contradicts that m(t) is non-decreasing. therefore, m(t0) < θ η and max {|x(t0)| , |y(t0)| , |z(t0)|} < θ η . (3.10) for ∀t > t0, by the same approach used in the proof of (3.10), we have max {|x(t)| , |y(t)| , |z(t)|} < θ η , if m(t) = max {|x(t)| , |y(t)| , |z(t)|} . on the other hand, if m(t) > max {|x(t)| , |y(t)| , |z(t)|} , t > t0, we can choose t0 ≤ t3 < t such that m(t3) = max {|x(t3)| , |y(t3)| , |z(t3)|} < θ η and m(s) > max {|x(s)| , |y(s)| , |z(s)|} for all s ∈ (t3, t]. using a similar argument as in the proof of case (i), we can show that m(s) ≡ m(t3) is a constant, for all s ∈ (t3, t], which implies max {|x(t)| , |y(t)| , |z(t)|} < m(t) = m(t3) = max {|x(t3)| , |y(t3)| , |z(t3)|} < θ η . to sum up, the solutions of (1.2) are uniformly bounded. the proof is complete. 4 an example consider the following equation 68 cemil tunç and hilmi ergören cubo 14, 3 (2012) x ′′′ (t) + (11 − 1 1 + t + x2(t) )x ′′ (t) + (31 − 4 1 + t + x2(t) )x ′ (t) + (26 − 1 1 + t + x2(t) )x(t) + 1 1 + t + x2(t) x(t − |sin t|) + (sin t) sin x(t − e|sin t|) = 1 1 + t2 . (4.1) setting y(t) = dx(t) dt + 2x(t) and z(t) = dy(t) dt + 2y(t), we can transform (4.1) into dx(t) dt = −2x(t) + y(t), dy(t) dt = −2y(t) + z(t), dz(t) dt = −(7 − 1 1 + t + x2(t) )z(t) + y(t) + 1 1 + t + x2(t) x(t)) (4.2) − 1 1 + t + x2(t) x(t − |sin t|) − (sin t) sin x(t − e|sin t|) + 1 1 + t2 . then, we can satisfy the following assumptions: i) ∣ ∣((α1 − f1(t, u))α 2 1 + f2(t, u)α1)u − g0(t, u) ∣ ∣ = ∣ ∣ ∣ ∣ 1 1 + t + u2 u ∣ ∣ ∣ ∣ ≤ l0 |u| + q0, ii) ∣ ∣ ∣ ∣ 1 1 + t + u2 u ∣ ∣ ∣ ∣ ≤ l1 |u| + q1and |sin t sin u| ≤ l2 |u| + q2 , iii) α3 = inf t≥0 (7− 1 1 + t + u2 )−1 > 2∑ i=0 li by taking suitable li and qi such as l0 = l1 = l2 = 1 for appropriate qi (i = 0, 1, 2). hence, all solutions of the system (4.2) are uniformly bounded. received: april 2011. revised: february 2012. references [1] afuwape, a.u., omari, p., and zanolin, f. nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary problems. j. math. anal. appl. 143, 35-56 (1989). 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() cubo a mathematical journal vol.17, no¯ 01, (29–40). march 2015 maps preserving fredholm or semi-fredholm elements relative to some ideal mohadeseh rostamani department of pure mathematics, ferdowsi university of mashhad, mashhad, iran. mohadeseh.rostamani@gmail.com shirin hejazian department of pure mathematics, ferdowsi university of mashhad, tusi mathematical research group (tmrg), mashhad, iran. hejazian@um.ac.ir abstract we consider the calkin algebra cr(a) and the fredholm theory in a banach algebra a, relative to some fixed ideal f of a. our aim is to study linear maps between unital banach algebras a and b which are surjective up to the inessential elements relative to f, and preserve fredholm or semi-fredholm elements in both directions or equivalently different relatively essential spectral sets such as essential spectrum, left or right essential spectrum, the boundary of essential spectrum or the full essential spectrum. we characterize such mappings when one of cr(a) or cr(b) is commutative and also investigate similar problems when a is assumed to be a unital c∗-algebra of real rank zero and b is an arbitrary banach algebra. resumen consideramos el álgebra de calkin cr(a) y la teoŕıa de fredholm en un álgebra de banach a relativa a algún ideal fijo f de a. nuestra meta es estudiar aplicaciones lineales entre álgebras de banach unitales a y b las cuales son sobrejectivas salvo los elementos no esenciales relativos a f y preservan los elementos de fredholm o semi-fredholm en ambas direcciones o equivalentemente conjuntos espectrales esenciales relativos diferentes tales como el espectro esencial izquierdo o derecho, la frontera del espectro esencial o el espectro esencial completo. caracterizamos dichas aplicaciones cuando uno de los cr(a) o cr(b) es conmutativo e investigamos problemas similares cuando a se asume que es una c∗-álgebra unital de rango real cero y b es una álgebra de banach cualquiera. keywords and phrases: linear preservers, fredholm element, semi-fredholm element, inessential ideal, relative calkin algebra. 2010 ams mathematics subject classification: 47b49, 47a10. 30 mohadeseh rostamani & shirin hejazian cubo 17, 1 (2015) 1 introduction let a and b be algebras, and let p be a property. we say that a linear map θ : a → b (i) preserves property p, if for each a ∈ a, θ(a) has property p whenever a has property p; (ii) preserves property p in both directions, if for each a ∈ a, θ(a) has property p if and only if a has this property. in operator theory, one of the most active research areas is linear preservers. the main problems in this subject concern with characterizing those linear maps that leave certain properties invariant. in the context of operator algebras, some of the most well known preserving problems deal with characterizing linear maps on the algebra of all bounded linear operators acting on a banach space which preserve fredholm, semi-fredholm or generalized invertible operators. new contributions to the study of linear preserver problems in l(h), the algebra of all bounded linear operators on an infinite dimensional complex hilbert space h, have been recently made by several authors, see [3, 4, 5, 12, 13, 14]. in [11], kim and park investigated linear maps φ on a unital c∗-algebra a of real rank zero that are surjective up to some fixed closed ideal i and π(a) is invertible in a/i if and only if π(φ(a)) is invertible in a/i, where π : a → a/i is the canonical quotient map. in this paper we consider fredholm theory in a banach algebra a relative to some fixed ideal f, c.f. [16], and investigate linear preservers between banach algebras. in section 2 we give some preliminaries concerning fredholm theory in banach algebras and the ideal of inessential elements relative to a fixed ideal. we will also consider the relative calkin algebra of a banach algebra with respect to some fixed ideal. section 3 is devoted to investigating linear maps between two unital banach algebras a and b which are surjective up to inessential elements and preserve any of the sets of fredholm, left semi-fredholm, right semi-fredholm or semi-fredholm elements in both directions (equivalently, preserves different spectral sets). we characterize such mappings whenever one of the relative calkin algebras in a or b is assumed to be commutative. in section 4 we consider similar linear preservers between unital c*-algebras or from a unital c*-algebra of real rank zero to an arbitrary unital banach algebra. in each case we show that a linear map which is surjective up to inessential ideal and preserves a certain essential spectral set, induces a continuous jordan isomorphism between the corresponding relative calkin algebras. 2 preliminaries throughout this section a is a unital banach algebra with unit element 1 and f is a fixed two sided ideal of a. the elements of f are called finite elements of a. set i(a) = k(h(f)) := ⋂ {p ∈ πa; f ⊆ p}, where πa denotes the set of all primitive ideals of a. by [15, theorem 4.3], i(a) = {a ∈ a; a + f ∈ rad(a/f)}. cubo 17, 1 (2015) maps preserving fredholm or semi-fredholm elements . . . 31 clearly i(a) is a closed two sided ideal of a containing f. we call i(a) the inessential ideal of a (relative to f), and the elements of i(a) are called the inessential elements relative to f. definition 2.1. an element a ∈ a is said to be left (resp. right) semi-fredholm (relative to f), if it is left (resp. right) invertible modulo f, and a is called fredholm if it is invertible modulo f. an element a ∈ a is a semi-fredholm or an atkinson element if it is either left or right semi-fredholm. we denote by φl(a) (resp. φr(a)) the set of all left (resp. right) semi-fredholm elements of a. also we denote by φ(a) := φl(a) ∩ φr(a) and ψ(a) := φl(a) ∪ φr(a) the sets of fredholm and semi-fredholm elements of a, respectively. by [2, ba.2.4] an element a ∈ a is left (resp. right) invertible modulo f if and only if it is left (resp. right) invertible modulo i(a). since i(a) is a closed ideal it is easy to see that φ(a),φl(a) and φr(a) are open multiplicative semigroups of a and adding i(a) to each of these classes leaves it stable. for more about fredholm theory in banach algebras see [2, 15, 16]. definition 2.2. the quotient algebra a/i(a), denoted by cr(a), is called the relative calkin algebra of a with respect to f. the relative calkin algebra of a is semisimple; see [16, theorem 3.2 (vi)]. we recall that if a has minimal left ideals the smallest left ideal containing all of them is called the left socle of a. the right socle is similarly defined in terms of right ideals. if a has both minimal left and minimal right ideals, and if the left socle coincides with the right socle, it is called the socle of a denoted by soc(a). in this case we say for brevity that soc(a) exists. if a has no minimal left or minimal right ideals we put soc(a) = {0}. clearly the socle, if it exists, is a non-zero ideal of a. if a is a semiprime algebra which has minimal left ideals, then soc(a) exists (see [2, ba.3.3]). when a is a semisimple banach algebra, as a special case in definition 2.2, we may assume f to be the socle of a. in this case the relative calkin algebra coincides with the generalized calkin algebra as in [4]. for x,y ∈ a let x ◦ y = x + y − xy denote the quasi product of x and y. set q(a) := {x ∈ a; there exist u,v ∈ a such that u ◦ x,x ◦ v ∈ f}. by [16, theorem 3.2 (viii)], i(a) is the largest ideal of a contained in q(a). lemma 2.3. let a be a unital banach algebra. then i(a) = {x ∈ a; u + x ∈ φ(a) for all u ∈ φ(a)} = {x ∈ a; u + x ∈ φl(a) for all u ∈ φl(a)} = {x ∈ a; u + x ∈ φr(a) for all u ∈ φr(a)} = {x ∈ a; u + x ∈ ψ(a) for all u ∈ ψ(a)}. 32 mohadeseh rostamani & shirin hejazian cubo 17, 1 (2015) proof. the first equality is [16, theorem 3.5]. we prove the second equality; the others are proved similarly. let j = {x ∈ a;u + x ∈ φl(a) for all u ∈ φl(a)}. clearly i(a) ⊆ j. a modification of the proof of [16, theorem 3.5] shows that j is an ideal of a. we show that j ⊆ q(a). let x ∈ j, then 1 − x ∈ φl(a). thus there exists y ∈ a such that (1 − y)(1 − x) − 1 ∈ f. since f ⊆ j we have y ∈ j. therefore there exists z ∈ a such that (1−z)(1−y) −1 ∈ f. this shows that 1−y+f is both left and right invertible in a/f with right inverse 1−x+f. it follows that 1−x+f is invertible in a/f and so x ∈ q(a), therefore j ⊆ q(a). now [16, theorem 3.2 (viii)] implies that j = i(a). in [4, lemma 3.2] a different method has been applied to prove the same equalities as in lemma 2.3 when a is assumed to be semisimple and f is assumed to be the socle of a. however, it should be emphasized that semisimplicity has been used just to guarantee the existence of the socle. for any a ∈ a σ(a) := {λ ∈ c : a − λ1 is not invertible}, σl(a) := {λ ∈ c : a − λ1 is not left invertible}, σr(a) := {λ ∈ c : a − λ1 is not right invertible}, denote the spectrum, the left spectrum and the right spectrum of a, respectively. also ∂σ(a) and r(a) are the boundary of the spectrum and the spectral radius of a, respectively. we recall that for every compact set k ⊆ c, the polynomial convex hull ηk, is defined by ηk = {z ∈ c : |p(z)| ≤ sup ξ∈k |p(ξ)|, for each polynomial p}. for an element a in a banach algebra, the polynomial convex hull ησ(x) of σ(x) is called the full spectrum of a. let πa : a → cr(a) denote the canonical quotient map. for a ∈ a, σe(a) := σ(πa(a)), σle(a) := σl(πa(a)), σre(a) := σr(πa(a)), ησe(a) := ησ(πa(a)) and re(a) := r(πa(a)) are called, respectively, the essential spectrum, the left essential spectrum, the right essential spectrum, the full essential spectrum and the essential spectral radius of a relative to f. let λ(.) denote any of the spectral sets σe(.),σle(.), σre(.),σle(.) ∩ σre(.),∂σe(.) and ησe(.). a linear map θ : a → b is said to be λ(.)-preserving if λ(θ(x)) = λ(x) for all x ∈ a, and θ is called essentially spectrally bounded if there exists a positive constant m such that re(θ(x)) ≤ mre(x) for all x ∈ a. if re(θ(x)) = re(x) for all x ∈ a, θ is called an essential spectral isometry. throughout this paper we will always suppose the following. assumption a. we assume that a and b are unital banach algebras and denote by 1 the unit element both in a and b. we consider arbitrary fixed ideals f and f′ in a and b, respectively. the inessential ideals in a and b relative to f and f′ are denoted by i(a) and i(b), respectively. cr(.) cubo 17, 1 (2015) maps preserving fredholm or semi-fredholm elements . . . 33 denotes the relative calkin algebra in the corresponding banach algebra. note that when we use the terms fredholm, left semi-fredholm, right semi-fredholm or semi-fredholm, we always mean relative to the specific fixed ideal considered in the corresponding algebra. for a linear map θ : a → b, n(θ) will denote the null space of θ. if θ(1) = 1, θ is called unital. we say that θ is surjective up to inessential ideal if b = θ(a) + i(b). if θ(a2) = θ(a)2 for all a ∈ a (equivalently, θ(ab + ba) = θ(a)θ(b) + θ(b)θ(a) for all a,b ∈ a), then θ is said to be a jordan homomorphism. a jordan isomorphism is a bijective jordan homomorphism. 3 fredholm and semi-fredholm preservers we consider assumption a and begin with the following lemma. note that there is no assumption of semisimplicity neither for a nor for b. the proof of the following lemma is a modification of a part of the proof of [3, theorem 1.1]. we give the proof for the sake of convenience. lemma 3.1. if the linear map θ : a → b is surjective up to the inessential ideal and essentially spectrally bounded then θ(i(a)) ⊆ i(b). proof. assume that there is a positive constant m such that re(θ(a)) ≤ mre(a) for all a ∈ a. take u ∈ i(a) and let b ∈ b be an arbitrary element. since θ is surjective up to the inessential ideal, there exist x ∈ a and y ∈ i(b) such that b = θ(x) + y. for every λ ∈ c, we have r(λπb(θ(u)) + πb(b)) = r(πb(λθ(u) + b)) = re(λθ(u) + b) = re(θ(λu + x) + y) = re(θ(λu + x)) ≤ mre(λu + x) = mre(x). the function λ 7→ r(λπb(θ(u)) + πb(b)) is subharmonic on c, and so by liouville’s theorem r(πb(θ(u)) + πb(b)) = r(πb(b)). zemanek’s characterization of the radical (see [1, theorem 5.3.1]), implies that πb(θ(u)) ∈ rad(cr(b)) = {0} and thus θ(u) ∈ i(b). lemma 3.2. let the linear map θ : a → b be surjective up to the inessential ideal. if θ preserves any of the sets of fredholm, left semi-fredholm, right semi-fredholm or semi-fredholm elements in both directions, then (i) θ(i(a)) ⊆ i(b); (ii) n(θ) ⊆ i(a). proof. (i) if θ(1) = 1 (mod i(b)), then re(θ(a)) = re(a) for all a ∈ a and the result follows by lemma 3.1. otherwise, let s(.) denote any of the sets of fredholm, left semi-fredholm, right semifredholm or semi-fredholm elements in the corresponding banach algebra. let x ∈ i(a). then we show that θ(x) ∈ i(b). suppose u ∈ s(b). since θ is surjective up to the inessential ideal i(b), 34 mohadeseh rostamani & shirin hejazian cubo 17, 1 (2015) there exist u′ ∈ a and x′ ∈ i(b) such that u = θ(u′)+x′. it follows that θ(u′) = u−x′ ∈ s(b) and so u′ ∈ s(a). now u+θ(x) = θ(u′)+x′ +θ(x) = θ(u′ +x)+x′ ∈ s(b). thus, for every u ∈ s(b), u + θ(x) ∈ s(b). lemma 2.3 implies that θ(x) ∈ i(b), which establishes that θ(i(a)) ⊆ i(b). (ii) let s(.) be as above. if x ∈ n(θ) and u ∈ s(a), then θ(x + u) = θ(u) ∈ s(b). thus, for all u ∈ s(a), x + u ∈ s(a) and by lemma 2.3, x ∈ i(a). remark 3.3. by [1, corollary 5.2.3] a semisimple banach algebra a is commutative if and only if the spectral radius is subadditive on a, that is, there exists m > 0 such that r(x + y) ≤ m(r(x) + r(y)) (x,y ∈ a). now, let a and b be semisimple banach algebras and let θ : a → b be a surjective linear map which preserves spectral radius. it is easy to see that commutativity of either of a or b, implies commutativity of the other one. theorem 3.4. let at least one of cr(a) or cr(b) be commutative. suppose that θ : a → b is a linear map which is surjective up to the inessential ideal. then the following assertions are equivalent. (i) θ preserves the set of fredholm elements in both directions; (ii) θ(i(a)) ⊆ i(b) and the induced map θ̂ : cr(a) → cr(b), θ̂ ◦ πa = πb ◦ θ, is a continuous isomorphism multiplied by an invertible element in cr(b). proof. (i) ⇒ (ii). by lemma 3.2, θ(i(a)) ⊆ i(b)); thus θ induces a linear map θ̂ : cr(a) → cr(b) such that θ̂ ◦ πa = πb ◦ θ. clearly θ̂ is surjective. since θ(1) is a fredholm element, πb(θ(1)) is invertible in cr(b). take s ∈ b such that πb(s) = πb(θ(1)) −1 in cr(b). consider ψ := lπb(s) ◦ θ̂ : cr(a) → cr(b), where the linear mapping lπb(s) is the left multiplication by πb(s) in cr(b). then ψ(πa(x)) = πb(sθ(x)) for all x ∈ a. now it is easy to see that ψ is surjective, ψ(πa(1)) = πb(1) and πa(x) ∈ inv(cr(a)) ⇐⇒ x ∈ φ(a) ⇐⇒ θ(x) ∈ φ(b) ⇐⇒ πb(θ(x)) ∈ inv(cr(b)) ⇐⇒ πb(s)πb(θ(x)) ∈ inv(cr(b)) ⇐⇒ ψ(πa(x)) ∈ inv(cr(b)). thus ψ is unital and preserves the set of invertible elements in both directions and so it preserves the spectrum. the banach algebras cr(a) and cr(b) are semisimple, so by remark 3.3 both cr(a) and cr(b) are commutative. it follows from [1, theorem 4.1.17] that ψ is an isomorphism which is continuous by semisimplicity of cr(b). the assertion (ii) ⇒ (i) is obvious. we recall that a c∗-algebra a is said to be of real rank zero, if the set of all invertible selfadjoint elements of a is dense in the set of selfadjoint elements. suppose that a is a c*-algebra of real rank zero and b is a semisimple banach algebra. let i(a) and i(b) denote the inessential ideals cubo 17, 1 (2015) maps preserving fredholm or semi-fredholm elements . . . 35 relative to soc(a) and soc(b), respectively. it is proved in [4, corollary 3.5] that if θ : a → b is a linear map which is surjective up to i(b) and if θ preserves any of the sets φl(a), φr(a) or ψ(a) in both directions, then θ(i(a)) ⊆ i(b) and the induced map θ̂ : c(a) → c(b) is a continuous jordan isomorphism multiplied by an invertible element in c(b). here we prove a similar result in a different setting. proposition 3.5. let a, b satisfy assumption a. suppose that θ : a → b is a linear map which is surjective up to the inessential ideal. if θ preserves any of the sets of left semi-fredholm, right semifredholm or semi-fredholm elements in both directions, then θ(i(a)) ⊆ i(b) and θ preserves the set of fredholm elements in both directions. moreover, if one of cr(a) or cr(b) is commutative, then the induced map θ̂ : cr(a) → cr(b) is a continuous isomorphism multiplied by an invertible element in cr(b). proof. by assumption and lemma 3.2, θ̂ : cr(a) → cr(b) is well defined. since cr(a),cr(b) are semisimple, [4, theorem 2.2] implies that θ̂ preserves invertibility in both directions. therefore θ preserves the set of fredholm elements in both directions. the last assertion follows from theorem 3.4. we will need the following theorem in the sequel. we recall that a linear map θ : a → b preserves idempotents if for each idempotent e in a, θ(e) is an idempotent in b. theorem 3.6. [10, corollary 2.3] let a and b be two semisimple banach algebras and let λ(.) denote any of the spectral sets σ(.),σl(.),σr(.),σl(.)∩σr(.),∂σ(.) and ησ(.). suppose that θ : a → b is a surjective linear map. if λ(θ(x)) = λ(x) for every x ∈ a, then θ is a bijective linear map preserving idempotents. in the above theorem, since λ(θ(x)) = λ(x) for one of the spectral sets, we have θ(1) = 1. remark 3.7. let k ⊆ c be compact. it is well known that ∂η(k) is the boundary of the unbounded component of c \ k which is called the outer boundary of k. thus max{|z| : z ∈ k} = max{|z| : z ∈ ∂η(k)}. theorem 3.8. let cr(a) or cr(b) be commutative. suppose that θ : a → b is a linear map which is surjective up to the inessential ideal. the following assertions are equivalent. (i) θ preserves the set of fredholm elements in both directions and θ(1) = 1(mod i(b)). (ii) θ(i(a)) ⊆ i(b) and the induced map θ̂ : cr(a) → cr(b), θ̂ ◦ πa = πb ◦ θ, is a continuous unital isomorphism. (iii) θ is σe-preserving; (iv) θ is σle-preserving; (v) θ is σre-preserving; 36 mohadeseh rostamani & shirin hejazian cubo 17, 1 (2015) (vi) θ is (σle ∩ σre)-preserving; (vii) θ preserves the set of left semi-fredholm elements in both directions and θ(1) = 1(mod i(b)). (viii) θ preserves the set of right semi-fredholm elements in both directions and θ(1) = 1(mod i(b)). (ix) θ preserves the set of semi-fredholm elements in both directions and θ(1) = 1(mod i(b)). (x) θ is ∂σe-preserving; (xi) θ is ησe-preserving. (xii) θ is an essential spectral isometry and θ(1) = 1(mod i(b)). proof. (i) ⇒ (ii). since θ(1) = 1 (mod i(b)), we have the result from theorem 3.4. (ii) ⇒ (iii). for every x ∈ a, we have σe(θ(x)) = σ(πb(θ(x))) = σ(θ̂(πa(x))) = σ(πa(x)) = σe(x), and the result holds. (iii) ⇒ (i). since θ is σe-preserving, it preserves fredholm elements in both directions. lemma 3.2 implies that θ(i(a)) ⊆ i(b) and by the hypothesis the induced map θ̂ : cr(a) → cr(b) is spectrum preserving. it follows from theorem 3.6 that θ̂ is unital, that is θ(1) − 1 ∈ i(b). it is easy to see that (ii) implies all other assertions. (iv) ⇒ (vii). if θ is σle-preserving then θ preserves the set of left semi-fredholm elements in both directions. by lemma 3.2, θ̂ is well defined and by the hypothesis it preserves the left spectrum. so by theorem 3.6, θ̂ is unital. the inverse implication is obvious. the assertions (v) ⇐⇒ (viii), (vi) ⇐⇒ (ix) are proved similarly. (vii) ⇒ (i). if θ preserves the set of left semi-fredholm elements in both directions then by proposition 3.5, θ preserves the set of fredholm elements in both directions. the same reasoning gives (viii) ⇒ (i) and (ix) ⇒ (i). if each of assertions (x),(xi) and (xii) holds then by lemma 3.1, θ̂ is well defined and from theorem 3.6, θ̂ is bijective, unital and also a spectral isometry. thus both cr(a) and cr(b) are commutative. now from [1, theorem 4.1.17], θ̂ is an isomorphism which is continuous by semisimplicity of c(b) and (ii) follows. 4 essential spectral preservers on c∗-algebras let a be a banach algebra. we recall that an element a ∈ a is called regular if there exists b ∈ a such that a = aba and b = bab; b is said to be a generalized inverse for a. note that if a and b in a satisfy a = aba, then a is regular and b′ = bab is a generalized inverse for a. this shows, in particular, that the generalized inverse of a regular element is not unique. for an element a in a banach algebra a, the regular set of a denoted by reg(a) is the set of all λ ∈ c such that there cubo 17, 1 (2015) maps preserving fredholm or semi-fredholm elements . . . 37 exists a neighborhood uλ of λ, and an analytic function f : uλ → a, such that f(µ) is a generalized inverse of a − µ1 for all µ ∈ uλ. the generalized spectrum (also called saphar spectrum) of a is given by σg(a) := c \ reg(a). the conorm or the reduced minimum modulus of a ∈ a is given by γ(a) := { inf{‖ax‖ : dist(x,ker(la)) ≥ 1}, if a 6= 0 ∞, if a = 0, (4.1) where la denotes the operator x 7→ ax on a. if f is a fixed ideal in a then the essential conorm and the generalized essential spectrum of a relative to f is given by γe(a) := γ(πa(a)) and σge(a) := σg(πa(a)), respectively. here πa denotes the canonical quotient map from a onto the relative calkin algebra cr(a). it is easy to see that if a is a c ∗-algebra then σge(a) = {λ ∈ c : limµ→λ γe(a − µ) = 0}, and that ∂σe(a) ⊆ σge(a) ⊆ σe(a). it is clear from the definition of γ(.) that, if θ preserves essential conorm that is, γe(θ(a)) = γe(a) for all a ∈ a, and θ(1) = 1 (mod i(b)), then σge(θ(a)) = σge(a) for all a ∈ a. we recall that a unital c∗-algebra a is said to be finite if a∗a = 1 implies that aa∗ = 1. it is easy to see that in a finite c∗-algebra every one-sided invertible element is invertible. c∗-algebras with finite traces and those with dense subset of invertible elements are examples of finite c∗algebras. the quotient of a finite c∗ algebra a need not be finite. a unital c∗ algebra a is said to be residually finite if every quotient of a is finite; [6, v.2.1.3]. note that, in this section a and b as banach algebras, satisfy assumption a. theorem 4.1. let a and b be unital c∗-algebras and let θ : a → b be a ∗-preserving linear map which is surjective up to the inessential ideal. then in the following assertions, (i) − (iv) are equivalent and (v) − (vii) imply (ii). moreover, if a and b are residually finite c∗-algebras, then all of the following conditions are equivalent. (i) θ is σe-preserving; (ii) θ(i(a)) ⊆ i(b) and the induced map θ̂ : cr(a) → cr(b), is an isometric jordan isomorphism; (iii) θ preserves fredholm elements in both directions and θ(1) = 1(mod i(b)); (iv) γe(θ(a)) = γe(a) for all a ∈ a and θ(1) = 1(mod i(b)). (v) θ is σle-preserving; (vi) θ is σre-preserving; (vii) θ is (σle ∩ σre)-preserving. proof. (i) ⇒ (iii). since θ is σe-preserving, it preserves fredholm elements in both directions and it follows from lemma 3.2 that θ̂ : cr(a) → cr(b) is well defined and is unital by theorem 3.6. (iii) ⇒ (ii). it follows from lemma 3.2 that θ̂ is well defined and by the assumption it is unital 38 mohadeseh rostamani & shirin hejazian cubo 17, 1 (2015) and preserves invertibility in both directions. it is easy to see that θ̂ is ∗-preserving. therefore, θ̂ is a jordan isomorphism by [9] and is an isometry by [8, lemma 4.3]. (ii) ⇒ (i) is obvious. (iv) ⇒ (ii). if (iv) holds then σge(θ(a)) = σge(a) for all a ∈ a and thus θ is an essential spectral isometry and so θ̂ is well defined by lemma 3.1. moreover, θ̂ is injective and preserves the conorm. thus by [8, theorem 3.1], θ̂ is an isometric jordan isomorphism. (ii) ⇒ (iv) follows easily. now suppose that (v) holds. then θ preserves the set of left semi-fredholm elements in both directions. lemma 3.2 implies that θ̂ is well defined. by theorem 3.6, θ̂ is unital and bijective. θ̂ preserves left invertibility in both directions and so preserves invertibility in both directions by [4, theorem 2.2]. therefore θ̂(inv(cr(a))) = inv(cr(b)). it is easy to see that θ̂ is ∗-preserving. therefore, θ̂ is a jordan isomorphism by [9] and is an isometry by [8, lemma 4.3]. (vi) ⇒ (ii) and (vii) ⇒ (ii) are proved in a similar way. finally, if a and b are residually finite c∗-algebras, then cr(a) and cr(b) are finite and since a jordan isomorphism is spectrum preserving it is easy to see that (ii) implies each of the assertions (v) − (vii). theorem 4.2. let a be a unital c∗-algebra of real rank zero and b a banach algebra. suppose that the linear map θ : a → b is surjective up to the inessential ideal. consider the following assertions. then (i) − (v) are equivalent and (vi) − (viii) imply (v). (i) θ is σe-preserving; (ii) θ is ∂σe-preserving; (iii) θ is ησe-preserving; (iv) θ preserves fredholm elements in both directions and θ(1) = 1(mod i(b)). (v) θ(i(a)) ⊆ i(b) and the induced map θ̂ : cr(a) → cr(b) is a continuous jordan isomorphism; (vi) θ is σle-preserving; (vii) θ is σre-preserving; (viii) θ is (σle ∩ σre)-preserving; proof. it is obvious that (i) ⇒ (ii),(i) ⇒ (iii),(i) ⇔ (iv) and (v) implies each of the statements (i) − (iv) . if each of the statements (i) − (iii) holds then re(θ(x)) = re(x) and lemma 3.1 implies that θ(i(a)) ⊆ i(b). thus the induced map θ̂ : cr(a) → cr(b) is a spectral isometry and so is injective. by theorem 3.6, θ̂ is unital and preserves idempotents. let p and q be orthogonal projections in a. then, p + q is a projection and θ̂(p) + θ̂(q) = ( θ̂(p) + θ̂(q) )2 = θ̂(p) + θ̂(p)θ̂(q) + θ̂(q)θ̂(p) + θ̂(q). cubo 17, 1 (2015) maps preserving fredholm or semi-fredholm elements . . . 39 it follows that θ̂(p)θ̂(q)+θ̂(q)θ̂(p) = 0 and hence θ̂(p)θ̂(q) = θ̂(p)θ̂(q)θ̂(p) = θ̂(q)θ̂(p). therefore, θ̂(p) and θ̂(q) are orthogonal idempotents. let a = ∑n j=1 λjpj be a linear combination of mutually orthogonal projections p1, ...,pn ∈ a. then, θ̂(a2) = θ̂ ( n∑ j=1 λ2j pj ) = n∑ j=1 λ2j θ̂pj = ( θ̂(a) )2 for θ̂(p1), ..., θ̂(pn) are mutually orthogonal idempotents. since every selfadjoint element in a is the norm limit of finite linear combinations of mutually orthogonal projections and θ̂ is continuous, θ̂(a2) = (θ̂(a))2 for every selfadjoint element a in a. replacing a by a + b in this identity yields θ̂(ab + ba) = θ̂(a)θ̂(b) + θ̂(b)θ̂(a) for all a,b ∈ asa. suppose a = a1 + ia2 with a1,a2 ∈ asa. by the above argument θ̂(a2) = θ̂ ( a21 + i(a1a2 + a2a1) − a 2 2 ) = θ̂(a1) 2 + i ( θ̂(a1)θ̂(a2) + θ̂(a2)θ̂(a1) ) − θ̂(a2) 2 = (θ̂(a))2. this proves that θ̂ is a jordan isomprphism and (v) holds. (vi) ⇒ (v). since ∂σe(θ(x)) ⊆ σle(θ(x)) for all x ∈ a, we have re(θ(x)) = max { |λ| : λ ∈ σle(θ(x)) } = max { |λ| : λ ∈ σle(x) } = re(x). lemma 3.1 implies that θ(i(a)) ⊆ i(b). the induced map θ̂ : cr(a) → cr(b) is σl-preserving and by theorem 3.6, θ̂ is a bijective unital linear map preserving idempotents and hence is a jordan isomorphism. (vii) ⇒ (v) and (viii) ⇒ (v) are proved similarly. the following corollary is a direct consequence of theorem 4.2 and [7, corollary 5.3]. corollary 4.3. let a be a c∗-algebra of real rank zero and b a c∗-algebra. for a surjective up to inessential ideal linear map θ : a → b such that θ(1) = 1(mod i(b)), the conditions (i) − (v) in the above theorem are equivalent to the following conditions. (i) there exist α,β > 0 such that αγe(a) ≤ γe(θ(a)) ≤ βγe(a), for all a ∈ a. (ii) σge(θ(a)) = σge(a), for all a ∈ a. acknowledgement. the authors would like to thank the referee for valuable comments and suggestions. received: november 2012. accepted: march 2013. references [1] b. aupetit, a primer on spectral theory, springer-verlag, new york, 1991. 40 mohadeseh rostamani & shirin hejazian cubo 17, 1 (2015) [2] b. a. barnes, g. j. murphy, m. r. f. smyth and t. t. west, riesz and fredholm theory in banach algebras, research notes in math. 67, pitman, boston, ma, 1982. [3] m. bendaoud and a. bourhim, essentially spectrally bounded linear maps, proc. amer. math. soc. 137 (2009) 3329-3334. [4] m. bendaoud, a. bourhim, m. burgos and m. sarih, linear maps preserving fredholm and atkinson elements of c*-algebras, linear multilinear algebra, 57 (2009) 823-838. [5] m. bendaoud, a. bourhim and m. sarih, linear maps preserving the essential spectral radius, linear alg. appl. 428 (2008) 1041-1045. [6] b. blackadar, operator algebras: theory of c*-algebras and von neumann algebras, springer-verlag, berlin heidelberg, 2006. [7] a. bourhim and m. burgos, linear maps preserving regularity in c*-algebras, illinois j. math. 53 (2009), 899-914. [8] a. bourhim, m. burgos and v. shulman, linear maps preserving the minimum and reduced minimum moduli, j. funct. anal. 258 (2010) 50-66 [9] m-d. choi, d. hadwin, e. nordgren, h. radjavi and p. rosenthal, on positive linear maps preserving invertibility, j. funct. anal. 59 (1984) 462-469 [10] j. l. cui and j. c. hou, linear maps between banach algebras compressing certain spectral functions, rocky mountain j. math. 34 (2004) 565-585. [11] s. o. kim and c. park, linear maps on c∗-algebras preserving the set of operators that are invertible in a/i. canad. math. bull. 54 (1) (2011) 141-146. [12] m. mbekhta, linear maps preserving the set of fredholm operators, proc. amer. math. soc. 135 (2007) 3613-3619. 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[16] m. r. f. smyth, fredholm theory in banach algebras, banach center publications, 8 (1982) 403-414. introduction preliminaries fredholm and semi-fredholm preservers essential spectral preservers on c*-algebras cubo a mathematical journal vol.15, no¯ 03, (09–18). october 2013 approximating a solution of an equilibrium problem by viscosity iteration involving a nonexpansive semigroup binayak s. choudhury & subhajit kundu bengal engineering and science university, shibpur department of mathematics, p.o.: b. garden, shibpur, howrah 711103, west bengal, india. binayak12@yahoo.co.in, subhajit.math@gmail.com abstract in this paper we have defined a new iteration in order to solve an equilibrium problem in hilbert spaces. the iteration we have introduced is a viscosity type iteration and involves a semigroup of nonexpansive operators. we have established that depending on some control conditions, our iteration strongly converges to a solution of the equilibrium problem. resumen en este art́ıculo hemos definido una iteración nueva para resolver un problema de equilibrio en espacios de hilbert. la iteración que introducimos es de tipo viscoso e involucra un semigrupo de operadores no expansivos. hemos establecido que dependiendo de las condiciones de control, nuestra iteración converge fuertemente a una solución de un problema de equilibrio. keywords and phrases: equilibrium problem, nonexpansive semigroup, viscosity iteration, fixed point, weak convergence, hilbert space. 2010 ams mathematics subject classification: 46c05, 47h10, 91b50. 10 binayak s. choudhury & subhajit kundu cubo 15, 3 (2013) 1 introduction and mathematical preliminaries the equilibrium problem we consider in this paper is formulated in the framework of real hilbert spaces. this problem is a generalization of several problems in physics, optimization and economics. references [3, 9] give a good account of this feature. there are several iterative methods for obtaining solutions of this equilibrium problem in hilbert spaces and also in the more general settings of banach spaces [19, 10, 22]. a particular category of these iterations is viscosity iteration which was first developed by moudafi [15] to obtain fixed points of nonexpansive mappings. viscosity iterations have been used for solving equilibrium problems in works noted in [20, 16]. semigroup of nonexpansive operators have been considered in the context of constructing fixed point iteration in banach and hilbert spaces [8, 21, 1, 6, 5, 13, 7, 11, 12, 17]. the purpose of this paper is to use nonexpansive semigroups in a viscosity iteration scheme in order to construct a two step iteration for approximating a solution of an equilibrium problem in real hilbert spaces. precisely, we have shown that under suitable choices of the control conditions, our iteration strongly converges to solution of the equilibrium problem. let h be a hilbert space and c be a nonempty closed convex subset of h. a mapping t : c → c is said to be a nonexpansive mapping if for all x, y ∈ c ‖tx − ty‖ ≤ ‖x − y‖. (1.1) a mapping f : c→ c is said to be a θ contraction if for each x, y ∈ c, ‖fx − fy‖ ≤ θ‖x − y‖ when 0 < θ < 1. (1.2) for any x ∈ h, the metric projection pc from h into c is defined as pcx = {z ∈ c : ‖z − x‖ = inf y∈c ‖y − x‖}. (1.3) obviously, ‖x − pcx‖ ≤ ‖x − y‖. it is well known that pc is a firmly nonexpansive mapping from h onto c, that is, ‖pcx − pcy‖ 2 ≤ 〈pcx − pcy, x − y〉 for all x, y ∈ h. pc is also nonexpansive mapping from h onto c. the set of fixed point of an operator t from h to h is denoted by fix(t), that is, fix(t)={x ∈ h : tx = x}. a family s = (t(s)) s≥ 0 is a nonexpansive semigroup on h if it satisfies the following conditions: (a1) t(0)x = x for all x ∈ h, (a2) t(s + t) = t(s)t(t) for all s, t ≥ 0, (a3) ‖t(s)x − t(s)y‖ ≤ ‖x − y‖ for all x, y ∈ h and s ≥ 0, (a4) for all x ∈ h, s → t(s)x is continuous. a sequence {xn} of elements of a banach space x is said to converge weakly to an element x ∈ x if f(xk) → f(x) as k → ∞ for all f ∈ x ′ where f is a continuous linear functional from x to r or c where r is the set of real numbers and c is the set of complex number, and x′ is the dual of x. a sequence {xn} is said to have a weak limit point l if there exists a subsequence {xnk} of {xn} which converges weakly to l. cubo 15, 3 (2013) approximating a solution of an equilibrium problem . . . 11 we denote wω(xn) as the set of all weak limit point of {xn} and ws(xn) as the set of all strong limit point of {xn} . we denote the set of fixed point of t(s) by fix(t(s)). the set of all common fixed points of s is denoted by fix(s). so fix(s)= ∩s≥0fix(t(s)) . baillon proved the following nonlinear ergodic theorem: theorem 1.1. [1] if t is a nonexpansive mapping from c into itself such that fix(t) 6= φ and x ∈ c, then 1 n n−1∑ k=0 tkx converges weakly to a fixed point of t. later baillon and brezis proved the following theorem for semigroup of nonexpansive operator: theorem 1.2. [2] if s = (t(s)) s≥ 0 is a nonexpansive semigroup on c, then { 1 t ∫t 0 t(s)xtds}t>0, t ∈ (0, 1), s ∈ r+, where r+ is the set of positive real numbers, converges weakly to a common fixed point of s. let f : c×c→ r be a bifunction where r is the set of real numbers. the equilibrium problem is to find some x ∈ c such that f(x, y) ≥ 0, for all y ∈ c . (1.4) the set of solutions of (1.4) is denoted by ep(f), that is, ep(f) = {x ∈ c : f(x, y) ≥ 0 for all y ∈ c}. in the equilibrium problem for the bifunction f from c×c→r , we assume that f satisfies following conditions: (c1) f(x, x) = 0 for all x ∈ c, (c2) f is monotone, that is, f(x, y) + f(y, x) ≤ 0, (c3) for each x, y, z ∈ c, lim t→0+ f(tz + (1 − t)x, y) ≤ f(x, y), (c4) for each x ∈ c, y → f(x, y) is convex and lower semicontinuous. lemma 1.1. [9] let c be a nonempty closed convex subset of h and let f be a bifunction from c×c into r satisfying conditions (c1)(c4). then for any r > 0 and x ∈ h there exists z ∈ c such that f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0 for all y ∈ c. further, if trx = {z ∈ c : f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0 for all y ∈ c} then the following hold: (1) tr is single valued, (2) tr is firmly nonexpansive, that is, for any x, y ∈ h ‖trx − try‖ 2 ≤ 〈trx − try, x − y〉, (3) fix(tr) = ep(f), (4) ep(f) is closed and convex. lemma 1.2. [4] let c be a nonempty closed convex subset of a real hilbert space h. given z ∈ h and x ∈ c, the inequality 〈x − z, y − x〉 ≥ 0, for all y ∈ c holds if and only if x = pcz, where pc denotes the metric projection from h onto c. lemma 1.3. [14] let {an},{bn} and {cn} be three nonnegative real sequences satisfying an+1 ≤ (1 − λn)an + bn + cn, n ≥ n0, where n0 is some nonnegative integer, λn ∈ [0,1] , ∞∑ n=1 λn=∞ , bn=o(λn) , and ∞∑ n=1 cn < ∞. 12 binayak s. choudhury & subhajit kundu cubo 15, 3 (2013) then an → 0 as n → ∞. lemma 1.4. [18] let c be a nonempty bounded closed convex subset of a hilbert space h and let (t(s)) s≥ 0 be a nonexpansive semigroup on c. then for every h ≥ 0, lim t→∞ sup x∈c ‖1 t ∫t 0 t(s)xds − t(h) 1 t ∫t 0 t(s)xds‖= 0. lemma 1.5. [4] let x be a uniformly convex banach space, c be a nonempty closed convex subset of x and t : c → x be a nonexpansive mapping. then, the mapping (i − t) is demiclosed on c, that is, if {xn} is weakly convergent to x and {(i − t)xn} is strongly convergent to y, then (i − t)x = y. lemma 1.6. [8] let us suppose (c1)-(c4) hold. let x, y ∈ h, r1, r2 > 0. then ‖tr2y − tr1x‖ ≤ ‖y − x‖ + | r2−r1 r2 |‖tr2y − y‖. lemma 1.7. [23] let {sn} be a sequence of nonnegative real numbers satisfying sn+1 ≤ (1 − γn)sn + σn + δn, for all n ≥ 0, where {γn} is a sequence in (0,1) and {σn}, {δn} are sequences of real numbers such that (a) lim n→∞ γn = 0 and ∞∑ n=0 γn = ∞, (b) lim sup n→∞ σn γn ≤ 0, (c) δn ≥ 0 and ∞∑ n=1 δn < ∞. then {sn} converges to zero. the following lemma is a well known result of functional analysis. lemma 1.8. let x be a reflexive banach space. then every bounded sequence in x has a weakly convergent subsequence. 2 main result theorem 2.1. let s = (t(s)) s≥ 0 be a nonexpansive semigroup on a real hilbert space h. let f : h → h be a θ-contraction, with 0 < θ < 1. let f : h × h → r be a mapping satisfying hypothesis (c1)-(c4). assume that fix(s) ∩ ep(f) 6= φ. let x0 ∈ h, {zn} ⊂ h and {xn} ⊂ h be the sequences generated by    xn+1 = βnxn + (1 − βn) 1 sn ∫sn 0 t(s)ynds, yn = αnf(xn) + (1 − αn)zn, f(zn, y) + 1 rn 〈y − zn, zn − xn〉 ≥ 0, for all y ∈ h where {αn}, {βn}, {sn} and {rn} satisfy the following conditions. cubo 15, 3 (2013) approximating a solution of an equilibrium problem . . . 13 (i) αn ∈ [0, 1], lim n→∞ αn= 0, ∞∑ n=1 αn = ∞ and ∞∑ n=1 |αn − αn−1| < ∞, (ii) lim n→∞ sn = ∞ and lim n→∞ | sn − sn−1| sn 1 αn = 0, (iii) lim inf n→∞ rn > 0, ∞∑ n=0 |rn+1 − rn| < ∞, (iv) 0 < βn ≤ d < 1, lim n→∞ βn = 0, ∞∑ n=0 | βn+1 − βn| < ∞. then {xn} converges strongly to a point p ∈ fix(s) ∩ ep(f). proof. here pfix(s)∩ ep(f) f is a mapping of h into fix(s) ∩ ep(f) ⊂ h such that ‖pfix(s)∩ ep(f)f(x) − pfix(s)∩ ep(f)f(y)‖ ≤ ‖f(x) − f(y)‖ ≤ θ‖x − y‖. therefore pfix(s)∩ ep(f)f is a contractive mapping and hence, by banach’s contraction principle, there exists a unique element p ∈ fix(s) ∩ ep(f) such that p = pfix(s)∩ ep(f)f(p). now, for this p ∈ fix(s) ∩ ep(f), n ≥ 0, we have, ‖ xn+1 − p ‖ = ‖ βnxn + (1 − βn) 1 sn ∫sn 0 t(s)ynds − p ‖ = ‖ βnxn + (1 − βn) 1 sn ∫sn 0 t(s)ynds − 1 sn ∫sn 0 t(s)p ds ‖ = ‖ βn(xn − p) + (1 − βn) 1 sn ∫sn 0 (t(s)yn − t(s)p)ds ‖ ≤ βn‖ xn − p ‖ + (1 − βn)‖ yn − p ‖. now, ‖ yn − p ‖ = ‖ αnf(xn) + (1 − αn)zn − p ‖ ≤ αn‖f(xn) − f(p)‖ + αn‖f(p) − p‖ + (1 − αn)‖ zn − p ‖. since by lemma 1.1 we have zn = trnxn, p = trnp it follows that for all n ≥ 0, ‖ zn − p ‖ = ‖trnxn − trnp‖ ≤ ‖ xn − p ‖. therefore, for all n ≥ 0, ‖ yn − p ‖ ≤ αnθ‖ xn − p ‖ + αn‖ f(p) − p ‖ + (1 − αn)‖ xn − p ‖ = {1 − αn(1 − θ)}‖ xn − p ‖ + αn(1 − θ) ‖ f(p)−p ‖ 1−θ . therefore, for all n ≥ 0, ‖ yn − p ‖ ≤ max { ‖ xn − p ‖, ‖ f(p)−p ‖ 1−θ }. therefore, for all n ≥ 0, {yn} is bounded. so { zn } and {f(xn)} are also bounded. hence for all n ≥ 0, ‖ xn+1 − p ‖ ≤ max { ‖ xn − p ‖, ‖ f(p)−p ‖ 1−θ }. proceeding in the same way we get for all n ≥ 0, ‖ xn+1 − p ‖ ≤ max { ‖ x0 − p ‖, ‖ f(p)−p ‖ 1−θ }. therefore, { xn } is bounded. again, for all n ≥ 0, we have, xn+1 = βnxn + (1 − βn)un where un= 1 sn ∫sn 0 t(s)ynds. again, for all n ≥ 0, ‖un − p‖ = ‖ 1 sn ∫sn 0 t(s)ynds − 1 sn ∫sn 0 t(s)p ds‖ ≤ ‖yn − p‖ ≤max{ ‖ xn − p ‖, ‖ f(p)−p ‖ 1−θ }. so {un} is also bounded. 14 binayak s. choudhury & subhajit kundu cubo 15, 3 (2013) now, for all n ≥ 0, xn+1 − xn =βnxn + (1 − βn)un − βn−1xn−1 − (1 − βn−1)un−1. therefore, for all n ≥ 0, ‖xn+1 − xn‖ =‖(1 − βn)(un − un−1) − (βn − βn−1)un−1 + βn(xn − xn−1) + (βn − βn−1)xn−1‖ ≤ (1 − βn)‖un − un−1‖ + |βn − βn−1|{ ‖ un−1 ‖ + ‖ xn−1 ‖ } + βn‖ xn − xn−1 ‖. (2.1) now, for all n ≥ 0, ‖ un − un−1 ‖ =‖ 1 sn ∫sn 0 t(s)ynds − 1 sn−1 ∫sn−1 0 t(s)yn−1ds ‖ =‖ 1 sn ∫sn 0 (t(s)yn − t(s)yn−1)ds + ( 1 sn − 1 sn−1 ) ∫sn−1 0 t(s)yn−1ds + 1 sn ∫sn sn−1 t(s)yn−1ds ‖. if p ∈ fix(s) where s is the nonexpansive semigroup, then for all n ≥ 0, we have ‖ un − un−1 ‖ = ‖ 1 sn ∫sn 0 (t(s)yn − t(s)yn−1)ds + ( 1 sn − 1 sn−1 ) ∫sn−1 0 (t(s)yn−1 − t(s)p)ds + 1 sn ∫sn sn−1 t(s)yn−1 − t(s)pds ‖ ≤ ‖ yn − yn−1 ‖ + ( 2|sn−sn−1| sn )‖ yn−1 − p ‖. (2.2) now, for all n ≥ 0, ‖ yn − yn−1 ‖ = ‖ αnf(xn) + (1 − αn)zn − αn−1f(xn−1) − (1 − αn−1)zn−1 ‖ =‖αn(f(xn) − f(xn−1)) + (αn − αn−1)(f(xn−1) − zn−1) + (1 − αn)(zn − zn−1) ‖. therefore, for all n ≥ 0, ‖ yn − yn−1 ‖ ≤ αn‖ f(xn) − f(xn−1)‖ + |αn − αn−1|‖ f(xn−1) − zn−1 ‖ + (1 − αn)‖zn − zn−1 ‖. (2.3) again, for all n ≥ 0, ‖zn − zn−1‖ ≤ ‖xn − xn−1‖ + |rn−rn−1| rn ‖zn − xn‖ (using lemma 1.6) we have lim inf n→∞ rn > 0. therefore there exists b > 0 such that rn > b for large n ∈ n where n is the set of positive integers. then, for all n ≥ 0, ‖zn − zn−1‖ ≤ ‖xn − xn−1‖ + |rn−rn−1| b ‖zn − xn‖. (2.4) using (2.3), (2.4) in (2.2), we get, for all n ≥ 0, ‖un − un−1‖ ≤ αnθ‖xn − xn−1‖ + |αn − αn−1|‖f(xn−1) − zn−1‖ + (1 − αn)‖xn − xn−1‖ + (1 − αn) |rn−rn−1| b ‖zn − xn‖ + 2|sn−sn−1| sn ‖yn−1 − p ‖. (2.5) using (2.5) in (2.1) we get, for all n ≥ 0, ‖xn+1 − xn‖ ≤ (1−βn)αnθ‖xn −xn−1‖+(1−βn)|αn −αn−1|‖f(xn−1)−zn−1‖+(1−βn)(1−αn)‖xn −xn−1‖ +(1 − βn)(1 − αn) |rn−rn−1| b ‖zn − xn‖ + (1 − βn) 2|sn−sn−1| sn ‖yn−1 − p ‖ + |βn − βn−1|{ ‖ un−1 ‖ + ‖ xn−1 ‖ }+βn‖ xn − xn−1 ‖ ={(1−βn)αnθ+(1−βn)(1−αn)+βn}‖xn −xn−1‖+(1−βn)|αn −αn−1|‖f(xn−1)−zn−1‖+(1− βn)(1−αn) |rn−rn−1| b ‖zn −xn‖+(1−βn) 2|sn−sn−1| sn ‖yn−1 −p ‖+ |βn −βn−1|{ ‖ un−1 ‖+‖ xn−1 ‖ } = {1 − αn(1 − θ)(1 − βn)}‖ xn − xn−1 ‖ + (1 − βn)|αn − αn−1|‖f(xn−1) − zn−1‖ + (1 − βn)(1 − αn) |rn−rn−1| b ‖zn − xn‖ + (1 − βn) 2|sn−sn−1| sn ‖yn−1 − p ‖ + |βn − βn−1|{ ‖ un−1 ‖ + ‖ xn−1 ‖ } ≤ {1 − αn(1 − θ)(1 − d)}‖ xn − xn−1 ‖ + |αn − αn−1|‖f(xn−1) − zn−1‖ + |rn−rn−1| b ‖yn − xn‖ + cubo 15, 3 (2013) approximating a solution of an equilibrium problem . . . 15 2|sn−sn−1| sn ‖yn−1 − p ‖ + |βn − βn−1|{ ‖ un−1 ‖ + ‖ xn−1 ‖ } let m= max{ sup n∈n ‖f(xn−1) − zn−1‖, sup n∈n ‖zn − xn‖, sup n∈n ‖yn−1 − p‖, sup n∈n (‖(un−1‖ + ‖xn−1‖)}. therefore, for all n ≥ 0, ‖xn+1 − xn‖ ≤ {1 − αn(1 − θ)(1 − d)}‖ xn − xn−1 ‖ + m[|αn − αn−1| + |rn−rn−1| b + 2|sn−sn−1| sn + |βn − βn−1|]. let γn = αn(1 − θ)(1 − d), σn = 2m |sn−sn−1| sn , δn = m[|αn − αn−1| + |rn−rn−1| b + |βn − βn−1|]. using the lemma 1.7 we get lim n→∞ ‖xn+1 − xn‖ = 0. now, ‖ yn − zn ‖ = ‖ αnf(xn) + (1 − αn)zn − zn ‖ = αn‖ f(xn) − zn‖ → 0 as n → ∞. (2.6) again, for all n ≥ 0, ‖ xn − un ‖ =‖ βn−1xn−1 + (1 − βn−1)un−1 − un ‖ ≤ ‖ un − un−1‖ + βn−1‖ xn−1 − un−1‖. since βn → 0 and ‖ un − un−1‖ → 0 as n → ∞, we have, ‖ xn − un ‖ → 0 as n → ∞. (2.7) also, for all n ≥ 0, ‖ zn − p ‖ 2 = ‖trnxn − trnp ‖ 2 ≤ 〈trnxn − trnp, xn − p〉 ( by lemma 1.1) =〈zn − p, xn − p〉 =1 2 [‖ zn − p ‖ 2 + ‖ xn − p ‖ 2 − ‖ xn − zn ‖ 2]. therefore, for all n ≥ 0, ‖ zn − p ‖ 2 ≤ ‖ xn − p ‖ 2 − ‖ xn − zn ‖ 2. (2.8) now, for all n ≥ 0, ‖ xn+1 − p ‖ 2 =‖ βnxn + (1 − βn)un − p ‖ 2 ≤ βn‖ xn − p ‖ 2 + (1 − βn)‖ un − p ‖ 2 (2.9) also for all n ≥ 0, ‖ yn − p ‖ 2 =‖ αnf(xn) + (1 − αn)zn − p ‖ 2 ≤ αn‖f(xn) − p‖ 2 + (1 − αn)‖ zn − p ‖ 2 (2.10) using (2.8) and (2.10) in (2.9), for all n ≥ 0, we get ‖ xn+1 − p ‖ 2 ≤ βn‖ xn − p ‖ 2 + (1 − βn)[αn‖f(xn) − p‖ 2 + (1 − αn)‖ zn − p ‖ 2] (since ‖ un − p ‖ ≤ ‖ yn − p ‖) ≤ βn‖ xn − p ‖ 2 + (1 − βn)αn‖f(xn) − p‖ 2 + (1 − βn)(1 − αn)‖ zn − p ‖ 2 ≤ ‖ xn − p ‖ 2 + αn‖ f(xn) − p ‖ 2 − (1 − βn)‖ xn − zn ‖ 2 [by (2.8)] therefore, (1 − βn)‖ xn − zn ‖ 2 ≤ ‖ xn − p ‖ 2 − ‖ xn+1 − p ‖ 2 + αn‖ f(xn) − p ‖ 2 ≤ {‖ xn − p ‖ + ‖ xn+1 − p ‖}‖ xn − xn+1 ‖ + αn‖ f(xn) − p ‖ 2 therefore, ‖ xn − zn ‖ → 0 as n → ∞. (2.11) again ‖ zn − un ‖ ≤ ‖ zn − xn ‖ + ‖ xn − un ‖ by (2.7) and (2.11) we have ‖ zn − un ‖ → 0 as n → ∞. (2.12) by (2.6), (2.7), (2.11) and (2.12) we can say that one of the sequences {xn}, {un}, {zn}, {yn} converge if and only if the other three converge to the same limit. by (2.6), (2.7), (2.11) and (2.12) we have 16 binayak s. choudhury & subhajit kundu cubo 15, 3 (2013) ωw(xn)=ωw(un)= ωw(zn) =ωw(yn), ωs(xn) = ωs(un)=ωs(zn)=ωs(yn). (2.13) now we have, p = pfix(s)∩ ep(f)f(p). we shall prove that lim sup n→∞ 〈f(p) − p, yn − p〉 ≤ 0. we take a subsequence {yni} of { yn} such that lim sup n→∞ 〈f(p) − p, yn − p〉 = lim i→∞ 〈f(p) − p, yni − p〉. (2.14) since {yni} is bounded and the hilbert space h is reflexive, by lemma 1.8, there exists a subsequence {yni k } of {yni} which converges weakly to x ∗. then x∗ is also a weak limit of {xn}. let v0=pfix(s)∩ ep(f)x0. since {xn} is a bounded sequence, there exists k such that b(v0, k) contains {xn}. moreover, b(v0, k) is t(s)-invariant for every s ≥ 0. therefore, we can assume that (t(s))s≥ 0 is a nonexpansive semigroup on b(v0, k). so by (2.13), x ∗ ∈ ωw(un)= ωw(zn). then, from lemma 1.4, we have , for every h ≥ 0, lim n→∞ ‖ 1 sn ∫sn 0 t(s)ynds−t(h) 1 sn ∫sn 0 t(s)ynds‖= lim n→∞ ‖un− t(h)un‖ = 0. therefore from lemma 1.5 , we have x ∗ ∈ fix(s). next we prove that x∗ ∈ ep(f). let {xni k } be a subsequence of {xni} such that xni k ⇀ x∗. from (2.11) we can say that zk ⇀ x ∗. moreover, by (c2) we obtain (1/rk)〈y − zk, zk − xk〉 ≥ f(y, zk), for all y ∈ h. by condition (c4), for fixed x ∈ h, the function f(x, .) is lower semicontinuous and convex and thus is weakly lower semicontinuous. since zk ⇀ x, by (2.11) and the fact that lim inf n→∞ rn = b > 0, we get (zk − xk)/rk → 0. letting k → ∞, we have, f(y, x∗) ≤ lim inf k→∞ f(y, zk) ≤ 0, for all y ∈ h. replacing y by yt where yt = ty + (1 − t)x ∗, t ∈ [0, 1] and using (c1) and (c4), we get 0=f(yt, yt) ≤ tf(yt, y) + (1 − t)f(yt, x ∗) ≤ f(yt, y). therefore, f(ty+(1−t)x∗, y) ≥ 0, t ∈ [0, 1], y ∈ h. letting t → 0+ and using (c3), we conclude that f(x∗, y) ≥ 0, y ∈ h. therefore, x∗ ∈ ep(f). since x∗ ∈ fix(s) ∩ ep(f), from lemma 1.2, we have, lim n→∞ 〈f(p) − p, yn − p〉 = lim i→∞ 〈f(p) − p, yni − p〉 (by using(2.14)) =〈f(p) − p, x∗ − p〉 ≤ 0. now for p ∈ fix(s) ∩ ep(f), for all n ≥ 0, we have, ‖xn+1 − p‖ 2 = ‖βn(xn − p) + (1 − βn)(un − p) ‖ 2 ≤ βn‖xn − p‖ 2 + (1 − βn)‖un − p‖ 2 ≤ βn‖xn − p‖ 2 + (1 − βn)‖yn − p‖ 2 ≤ βn‖xn − p‖ 2 + (1 − βn){(1 − αn) 2‖zn − p‖ 2 + 2αn〈f(xn) − p, yn − p〉} ≤ βn‖xn −p‖ 2 +(1−βn)(1−2αn +α 2 n)‖xn −p‖ 2 +2αn(1−βn)〈f(xn)−p, yn −p〉 ≤ (1 − 2(1 − βn)αn)‖xn − p‖ 2 + α2n‖xn − p‖ 2 + 2αn(1 − βn){〈f(xn) − f(p), yn − p〉 + 〈f(p) − p, yn − p〉} ≤ (1 − 2(1 − βn)αn)‖xn − p‖ 2 + α2nm0 + 2(1 − βn)αn(θ‖ xn − p ‖.‖ yn − p ‖ + ηn) ≤ (1−2(1−βn)αn)‖xn−p‖ 2+α2nm0 +(1−βn)θαn(‖xn−p‖ 2+‖yn−p‖ 2)+2αnηn where ηn=max{〈f(p) − p, yn − p〉, 0} and m0 =sup n≥0 {‖xn − p‖ 2 + ‖f(xn) − p‖ 2}. now, for all n ≥ 0, ‖yn − p‖ 2 ≤ αn‖f(xn) − p ‖ 2 + (1 − αn)‖zn − p ‖ 2 cubo 15, 3 (2013) approximating a solution of an equilibrium problem . . . 17 ≤ αn‖f(xn) − p ‖ 2 + (1 − αn)‖xn − p ‖ 2 ≤ αnm0 + ‖xn − p ‖ 2. therefore, for all n ≥ 0, ‖xn+1 − p ‖ 2 ≤ (1 − 2(1 − βn)αn)‖xn − p ‖ 2 + α2nm0 + (1 − βn)θαn(2‖xn − p ‖ 2 + αnm0) + 2αnηn ≤ (1 − 2(1 − βn)(1 − θ)αn)‖xn − p‖ 2 + α2nm0 + θα 2 nm0 + 2αnηn ≤ (1 − 2(1 − d)(1 − θ)αn)‖xn − p‖ 2 + (α2nm0 + θα 2 nm0 + 2αnηn) therefore, by lemma 1.3, we get xn → p as n → ∞. received: november 2012. accepted: september 2013. references [1] baillon, j. b., un theorème de type ergodique pour les contractions non linèaires dans un espace de hilbert, c.r. acad. sci. paris sèr., 280 (1975), no. a-b , 1511-1514. 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(2), 66 (2002), no. 1, 240-256. cubo a mathematical journal vol.14, no¯ 01, (119–125). march 2012 majorization for certain classes of analytic functions defined by a new operator e. a. eljamal and m. darus school of mathematical sciences, faculty of science and technology, universiti kebangsaan malaysia, bangi 43600 selangor d. ehsan, malaysia. email: n-ebtisam@yahoo.com , maslina@ukm.my abstract in the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by a new operator. moreover, we pointed out some new and known consequences of our main result. resumen en el presente art́ıculo, investigamos las propiedades de mayorización para ciertas clases de funciones anaĺıticas multivalentes definidas por un nuevo operador. además, resaltamos algunas consecuencias -nuevas y conocidasde nuestro resultado princresultado. keywords and phrases: majorization properties, multivalent functions, ruscheweyh derivative operator, hadamard product. 2010 ams mathematics subject classification: 30c45. 120 e. a. eljamal and m. darus cubo 14, 1 (2012) 1 introduction let f and g be analytic in the open unit disk u = {z : z ∈ c, |z| < 1}. we say that f is majorized by g in u and write f(z) � g(z) (z ∈ u) (1.1) if there exists a function ϕ, analytic in u such that |ϕ(z)| ≤ 1 and f(z) = ϕ(z)g(z) (z ∈ u). (1.2) it maybe noted here that (1.1) is closely related to the concept of quasi-subordination between analytic functions. let ap denote the class of functions of the form f(z) = zp + ∞∑ k=p+1 akz k,(p ∈ n = {1,2, ...}), (1.3) which are analytic and multivalent in the open unit disk u. in particular, if p = 1, thena1 = a. for functions fj ∈ ap given by fj(z) = z p + ∞∑ k=p+1 ak,jz k,(j = 1,2;p ∈ n), (1.4) we define the hadamard product or convolution of two functions f1 and f2 by f1 ∗ f2(z) = zp + ∞∑ k=p+1 ak1,ak2z k = (f2 ∗ f1)(z). (1.5) . definition 1.1. let the function f be in the class ap. ruscheweyh derivative operator is given by rn = zp + ∞∑ k=p+1 c(k,n)akz k. (1.6) next we define the following differential operator, d0 = f(z) = zp + ∞∑ k=p+1 akz k d1n,λ1,λ2,p = d 0f(z) p − pλ1 + λ2(k − p) p + λ2(k − p) + (d0f(z))′ zλ1 p + λ2(k − p) = zp + σ∞k=p+1 [ p + (λ1 + λ2)(k − p) p + λ2(k − p) ] akz k, and d2n,λ1,λ2,p = d 1 n,λ1,λ2,p f(z) p − pλ1 + λ2(k − p) p + λ2(k − p) + (d1n,λ1,λ2,pf(z)) ′ zλ1 p + λ2(k − p) cubo 14, 1 (2012) majorization for certain classes of analytic functions . . . 121 = zp + σ∞k=p+1 [ p + (λ1 + λ2)(k − p) p + λ2(k − p) ]2 akzk. in general, dmn,λ1,λ2,pf(z) = d(d n−1f(z) = zp + σ∞k=p+1 [ p + (λ1 + λ2)(k − p) p + λ2(k − p) ]m akz k (1.7) where (m,n ∈ n0 = n ∪ {0},λ2 ≥ λ1 ≥ 0). by applying convolution product on (1.6) and (1.7) we have the following operator dmn,λ1,λ2,pf(z) = z p + σ∞k=p+1 [ p + (λ1 + λ2)(k − p) p + λ2(k − p) ]m c(k,n)akz k, (1.8) where c(k,n) = γ(k+n) γ(k) . moreover, for m,n ∈ n0, λ2 ≥ λ1 ≥ 0 (p + λ2(k − p))d m,n λ1,λ2,p f(z) = (p + λ2(k − p) − pλ1)d m,n λ1,λ2,p f(z) + λ1z(d m,n λ1,λ2,p f(z))′ (1.9) special cases of this operator include: • the ruscheweyh derivative operator in the case d0,n0,0,1f(z) ≡ r n [6], • the salagean derivative operator in the case dm,01,0,1f(z) ≡ d m ≡ sn [2], • the generalized salagean derivative operator introduced by al-oboudi in the case dm,0λ1,0,1f(z) ≡ dmλ1[1], • the generalized ruscheweyh derivative operator in the case d1,nλ1,0,1f(z) ≡ d λ1 n [3], and • the generalized al-shaqsi and darus derivative operator in the casedm,nλ1,0,1f(z) ≡ d m,λ1 n [4]. to further our work, we need to define a class of functions as follows: definition 1.2. a function f ∈ ap is said to be in the class s m,p,j λ1,λ2,n [a,b,γ] of p-valent functions of complex order γ 6= 0 in u if and only if{ 1 + 1 γ ( z(dm,nλ1,λ2,pf(z)) (j+1) (dm,n λ1,λ2,p f(z))(j) − p + j )} ≺ 1 + az 1 + bz . (1.10) (z ∈ u,p ∈ n, j ∈ n0 = n ∪ {0},γ ∈ c − {0},λ2 ≥ λ1 ≥ 0). clearly, we have the following relationships: (i) s0,1,00,0,0[1,−1,γ] = s(γ) 122 e. a. eljamal and m. darus cubo 14, 1 (2012) (ii) s0,1,10,0,0[1,−1,γ] = k(γ) (iii) s0,1,00,0,0[1,−1,1 − α] = s ∗ for 0 < α < 1. the classes s(γ) and k(γ) are said to be classes of starlike and convex of complex order γ 6= 0 in u and s∗(α) denote the class of starlike functions of order α in u. a majorization problem for the class s(γ) has been investigated by altintas e.tal [5] and for the class s∗=s∗(0) has been investigated by macgregor [7]. in the present paper, we investigate a majorization problem for the class s m,p,j λ1,λ2,α [a,b,γ]. 2 majorization problem for the class s m,p,j λ1,λ2,n [a,b,γ] theorem 2.1. let the function f ∈ ap and suppose that g ∈ s m,p,j λ1,λ2,n [a,b,γ]. if (dm,nλ1,λ2,pf(z)) (j) is majorized by (dm,nλ1,λ2,pg(z)) (j) in u, then∣∣∣(dm+1,nλ1,λ2,pf(z))(j)∣∣∣ ≤ ∣∣∣(dm,nλ1,λ2,pg(z))(j)∣∣∣ for |z| ≤ r0, (2.1) where r0 = r0(p,γ,λ1,λ2,a,b) is the smallest positive root of the equation r3 ∣∣∣∣γ(a − b) − ( p + λ2(k − p) λ1 ) b ∣∣∣∣ − [ p + λ2(k − p) λ1 + 2|b| ] r2− [∣∣∣∣γ(a − b) − (p + λ2(k − p)λ1 )b ∣∣∣∣ + 2 ] r + ( p + λ2(k − p) λ1 ) = 0, (2.2) (−1 ≤ b < a ≤ 1;p ∈ n;γ ∈ c − {0}). proof. since g ∈ sm,p,j λ1,λ2,n [a,b,γ] we find from (1.10) that 1 + 1 γ ( z(dm,nλ1,λ2,pg(z)) (j+1) (dm,n λ1,λ2,p g(z))(j) − p + j ) = 1 + aw(z) 1 + bw(z) (2.3) (γ ∈ c − 0,j,p ∈ n and p > j), where w is analytic in u with w(0) = 0 and |w(z)| < z (z ∈ u). from (2.3) we get z(dm,nλ1,λ2,pg(z)) (j+1) (dm,n λ1,λ2,p g(z))(j) = (p − j) + [γ(a − b) + (p − j)b]w(z) 1 + bw(z) (2.4) and z(dm,nλ1,λ2,pf(z)) (j+1) = (p + λ2(k − p) λ1 )(dm+1,nλ1,λ2,pf(z)) (j)+ cubo 14, 1 (2012) majorization for certain classes of analytic functions . . . 123 (p − j − λ2(k − p) λ1 )(dm,nλ1,λ2,pf(z)) (j). (2.5) by virtue of (2.4) and (2.5) we get ∣∣∣(dm,nλ1,λ2,pg(z))(j)∣∣∣ ≤ p+λ2(k−p) λ1 [1 + |b|z|] ( p+λ2(k−p) λ1 )|γ(a − b) − ( p+λ2(k−p) λ1 )|b|z| |(dm+1,nλ1,λ2,pg(z)) (j)|. (2.6) next, since (dm,nλ1,λ2,pf(z)) (j) is majorized by (dm,nλ1,λ2,pg(z)) (j) in the unit disk u, we have from (1.2) that (dm,nλ1,λ2,pf(z)) (j) = ϕ(z)(dm,nλ1,λ2,pg(z)) (j). differentiating it with respect to z and multiplying by z we get z(dm,nλ1,λ2,pf(z)) (j+1) = zϕ′(z)(dm,nλ1,λ2,pg(z)) (j) + zϕ(z)(dm,nλ1,λ2,pg(z)) (j+1). now by using (2.5) in the above equation, it yields (dm,nλ1,λ2,pf(z)) (j) = zϕ′(z)(dm,nλ1,λ2,pg(z)) (j) p+λ2(k−p) λ1 + ϕ(z)(dm,nλ1,λ2,pg(z)) (j) (2.7) thus, by noting that ϕ ∈ ω satisfies the inequality (see, e.g. nehari [8]) |ϕ′(z)| ≤ 1 − |ϕ(z)|2 1 − |z|2 (z ∈ u) (2.8) and using (2.6) and (2.8) in (2.7), we get∣∣∣(dm+1,nλ1,λ2,pf(z))(j)∣∣∣ ≤[ |ϕ(z)| + 1 − |ϕ(z)|2 1 − |z|2 |z|(1 + |b||z|) p+λ2(k−p) λ1 − |γ(a − b) − ( p+λ2(k−p) λ1 )|b||z ]∣∣∣(dm,nλ1,λ2,pg(z))(j+1)∣∣∣ (2.9) which upon setting |z| = r and |ϕ(z)| = ρ (0 ≤ ρ ≤ 1) leads us to the inequality ∣∣∣(dm+1,nλ1,λ2,pf(z))(j)∣∣∣ ≤ φ(ρ) (1 − r2)( p+λ2(k−p) λ1 ) − |γ(a − b) − ( p+λ2(k−p) λ1 )b|r ∣∣∣(dm+1,nλ1,λ2,pg(z))(j)∣∣∣ (2.10) where φ(ρ) = −r(1 + |b|)ρ2 + (1 − r2)[ ( p + λ2(k − p) λ1 ) − |γ(a − b) + ( p + λ2(k − p) λ1 )b|r) ] ρ + r(1 + |b|r) (2.11) 124 e. a. eljamal and m. darus cubo 14, 1 (2012) takes its maximum value at ρ = 1 with r1 = r1(p,γ,λ1,λ2,a,b) for r1(p,γ,λ1,λ2,a,b) is the smallest positive root of equation (2.2). furthermore, if 0 ≤ ρ ≤ r1(p,γ,λ1,λ2,a,b), then function ψ(ρ) defined by ψ(ρ) = −σ(1 + |b|σ)ρ2 + (1 − σ2)[ ( p + λ2(k − 1) λ1 ) − |γ(a − b) + ( p + λ2(k − p) λ1 )b|σ) ] ρ + σ(1 + |b|σ) (2.12) is seen to be an increasing function on the interval 0 ≤ ρ ≤ 1 so that ψ(ρ) ≤ ψ(1) = (1 − σ2)( p + λ2(k − p) λ1 ) − |γ(a − b) + ( p + λ2(k − p) λ1 )b|σ) (2.13) 0 ≤ ρ ≤ 1; (0 ≤ σ ≤ r1(p,γ,λ1,λ2,a,b)). hence upon setting ρ = 1 in (2.13) we conclude that (2.1) of theorem 2.1 holds true for |z| ≤ r1(p,γ,λ1,λ2,a,b) where r1(p,γ,λ1,λ2,a,b) is the smallest positive root of equation (2.2). this completes the proof of the theorem 2.1. setting p = 1, m = 0, a = 1,b = −1 and j = 0 in theorem 2.1 we get corollary 2.1. let the function f ∈ a be analytic in the open unit disk uand suppose that g ∈ s0,1,00,0,0[1,−1,γ] = s(γ). if f(z) is majorized by g(z) in u, then |f′(z)| ≤ |g′(z)| (|z| < r3) where r3 = r3(γ) = 3 + |2γ − 1| − √ 9 + 2|2γ − 1| + |2γ − 1|2 2|2γ − 1| . this is a known result obtained by altintas[5]. for γ = 1, the above corollary reduces to the following result: corollary 2.2. let the function f(z) ∈ a be analytic univalent in the open unit disk u and suppose that g ∈ s∗ = s∗(0). if f is majorized by g in u, then |f′(z)| ≤ |g′(z)| (|z| ≤ 2 − √ 3) which is a known result obtained by macgregor [7]. some other work related to the class defined by (1.3) can be seen in [9] and of course elsewhere. in fact, recently ibrahim [10] used the concept of majorization to find solutions of fractional differential equations in the unit disk. acknowledgement the work presented here was supported by ukm-st-06-frgs0244-2010. received: april 2011. revised: june 2011. cubo 14, 1 (2012) majorization for certain classes of analytic functions . . . 125 references [1] f. m. al-oboudi, on univalent functions defined by a generalized salagean operator, internat. j. math. math. sci., 27(2004), 1429-1436. [2] g. salagean, subclasses of univalent functions, lecture in math. springer verlag, berlin, 1013(1983), 362-372. [3] k. al-shaqsi and m. darus, on univalent functions with respect to k-symmetric points defined by a generalization ruscheweyh derivative operators, jour. anal. appl., 7(2009), 53-61. [4] m. darus and k. al-shaqsi, differential sandwich theorems with generalised derivative operator, int. j. comput. math. sci., (22)(2008), 75-78. [5] o. altintas, ö.özkan and h. m. srivastava, majorization by starlike functions of complex order, complex var. 46(2001), 207-218. [6] st. ruscheweyh, new certain for univalent functions, proc. amer.math. soc. 49(1975),109-115. [7] t. h. macgregor, majorization by univalent functions, duke math. j. 34(1967), 95-102. [8] z. nehari, confformal mapping, macgraw-hill book company, new york,toronto and london (1955). [9] m. darus and r. w. ibrahim, multivalent functions based on a linear operator, miskolc mathematical notes, 11(1) (2010), 43-52. [10] r. w. ibrahim, existence and uniqueness of holomorphic solutions for fractional cauchy problem, j. math. anal. appl., 380 (2011), 232-240. () cubo a mathematical journal vol.17, no¯ 02, (49–71). june 2015 on a type of volterra integral equation in the space of continuous functions with bounded variation valued in banach spaces hugo leiva & jesús matute 1 dpto. de matemáticas, universidad de los andes, la hechicera. mérida 5101. venezuela. hleiva@ula.ve jmatute@ula.ve nelson merentes & josé sánchez. escuela de matemáticas, universidad central de venezuela, caracas. venezuela. nmerucv@gmail.com, casanay085@hotmail.com abstract in this paper we prove existence and uniqueness of the solutions for a kind of volterra equation, with an initial condition, in the space of the continuous functions with bounded variation which take values in an arbitrary banach space. then we give a parameters variation formula for the solutions of certain kind of linear integral equation. finally, we prove exact controllability of a particular integral equation using that formula. moreover, under certain condition, we find a formula for a control steering of a type of system which is studied in the current work, from an initial state to a final one in a prescribed time. resumen en este trabajo probamos existencia y unicidad de las soluciones para una ecuación de volterra, con condición inicial, en el espacio de funciones continuas con variación acotada y valores en un espacio de banach arbitrario. damos una formula de variación de parámetros para las soluciones de cierta clase de ecuación lineal integral. finalmente probamos la controlabilidad exacta de una ecuación integral particular usando esa formula. más aún, bajo cierta condición, encontramos una formula para una dirección de control de un tipo de sistema que se estudia en el presente trabajo, desde un estado inicial a uno final en un tiempo prescrito. keywords and phrases: existence and uniqueness of solutions of integral equations in banach spaces; continuous functions; bounded variation norm; parameters variation formula; controllability. 2010 ams mathematics subject classification: 26b30, 34a12, 45d99, 45n05. 1corresponding author 50 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) 1 introduction in the course of the last decades, many authors have studied integral equations in the banach spaces. some examples of this kind of work are [1], [10], [13], [17] and [22]. simultaneously, such as it has been done in [7], [9] and [21], the study of the solutions of the integral equations has considered various spaces of bounded variation functions. even more, there are works that combine these two trends in the field of the integral equations, such as it is done in [2], [8], [11] and [15]. the present paper follows this tendency, since in section 3 we prove the existence and uniqueness of the solution of the nonlinear problem with initial condition    x(t) − x(a) = a(g)(t) + λ ∫t a k(t, s)f(x(s))ds, x(a) = x0 ∈ x, in the space of continuous functions of bounded variation defined on the interval [a, b] with values in a normed space x, endowed with the norm |x|1 := |x(a)| + ∨ (x), where ∨ (x) is the bounded variation of the function x, a is a convenient function defined on such functions space and the symbol of integral is referred to the riemann integral on any banach space x. furthermore, k(t, s) is a continuous linear operator from x in x for each (t, s) belonging to a convenient subset in [a, b] × [a, b], f : x −→ x and where the expression k(t, s)f(x(s)) in the integral part means that the linear operator k(t, s) is evaluated in f(x(s)). an important part of our considerations is the representation of the solutions. there are known many formulas for the solutions of several kind of integral equations in the space of the real continuous functions defined on set of the real numbers, such as we can find in [20]. in the case of abstract banach spaces, we find in [16] a parameters variation formula for the volterra-stieltjest integral equations in the space of regulated functions. in the same way, in section 4, we give a formula of this type for the solution of linear problem    x(t) − x(a) = ∫t a b(g)(s)ds + λ ∫t a k(t, s)x(s)ds, x(a) = x0 ∈ x, where b : f −→ f is an adequate function, f denotes the space of continuous functions of bounded variation defined from the interval [a, b] into any fixed normed space x and the expression k(t, s)x(s) in the integral part means that the continuous linear operator k(t, s) : x −→ x is evaluated in x(s). another interesting question is referred to the controllability of such equations. in papers like [3], [4] and [5] is studied the controllability of some types of integral equations, and most recently in [2], [13] and [19], is studied the controllability of hammerstein or volterra integral equation on banach or hilbert spaces. in this setting, in section 5, we prove the existence of an exact control of the linear system    x(t) − x(a) = ∫t a b(g)(s)ds + λ ∫t a k(t, s)x(s)ds, x(a) = x0 ∈ x, cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 51 and then, in section 6, we verify the controllability of nonlinear system    x(t) − x(a) = ∫t a b(g)(s)ds + λ ∫t a t(s)f(x(s))ds, x(a) = x0 ∈ x, under assumption that f is a globally lipschitz function and such that t(s) : x −→ x is a bounded linear operator for each s ∈ [a, b]. the study of the above two systems was motivated by the work [13], since they are particular forms of the integral equation which is studied in it. in section 2, we have gathered the definitions and properties of riemann integral and functions of bounded variations in normed spaces which are used in this work. 2 preliminares in this paper, we use the symbol x to denote any banach space endowed with a norm | · |. now, let us recall the following definition. definition 2.1. [18] let us fix an interval i := [a, b] and consider a function y : i −→ x. let π := {a = t0 < t1 < · · · < tn = b} be a partition of interval i. we define ∨ (y, π) by ∨ (y, π) = n∑ i=1 ∣∣ y(ti) − y(tti−1) ∣∣. we call the least upper bound of the set of all possible sums ∨ (y, π) the total variation of the function y(t) on the interval [a, b] and we denote it by ∨ (y). if ∨ (y) < ∞, then we say that the function y(t) is of bounded variation on [a, b]. also, we recall two basic properties of functions of bounded variation in the subsequent proposition. proposition 2.1. if x and y are of bounded variation, then 1. ∨ (x + y) ≤ ∨ (x) + ∨ (y) and 2. ∨ (αy) = | α | ∨ (y) for each real number α ∈ r. notation 2.1. we denote the vector space of functions of bounded variation on [a, b] by bv[a, b]. the following proposition will be useful later. proposition 2.2. if y ∈ bv[a, b], then y(t) is bounded. furthermore, sup s∈i | y(s) | ≤ | y(a) | + ∨ (y). 52 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) definition 2.2. we define the set of functions f by f := c[a, b] ⋂ bv[a, b], where c[a, b] := { x : i −→ x : x is continuous } . proposition 2.3. the vector space f endowed with the norm |x|1 := |x(a)| + ∨ (x) is a banach space. now, let us recall the definitions of the riemann integral for the functions of one and two real variables with values in a normed space and some of their properties, which we shall use in the following sections. definition 2.3. [18] let us consider a function x : [a, b] −→ x. we denote a partition π := {a = t0 < t1 < · · · < tn = b}, together with the set of real numbers τi ∈ [ti−1, ti] for i = 1, . . . , n , by p and put |p| := max{|ti − ti−1| : i = 1, . . . , n}. we define the riemann sum sp by sp := n∑ i=1 ( ti − ti−1) x(τi). moreover, we say that the riemann integral is i ∈ x if for each real number ǫ > 0 there exists δ > 0 such that when |p| < δ, then | i − sp | < ǫ. in this case the element i ∈ x is called the riemann integral of the function x(t) and is denoted by ∫b a x(t)dt. proposition 2.4. [18] using the definition of riemann integral one can easily verify the following properties: 1. ∫b a x(t)dt = − ∫a b x(t)dt, provided that one of integrals exists. 2. ∫b a x(t)dt = ∫c a x(t)dt + ∫b c x(t)dt , a < c < b provided that the integral on the left member exists. cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 53 3. if x(t) = x0 ∈ x for all t ∈ [a, b], then ∫b a x0 = (b − a)x0. 4. if x : [a, b] −→ x is continuous, then the riemann integral ∫b a x(t)dt exists. 5. if x : [a, b] −→ x is continuous, then ∣∣∣∣∣ ∫b a x(t)dt ∣∣∣∣∣ ≤ ∫b a | x(t) | dt. definition 2.4. let us consider a function f : [a, b]× [a, b] −→ x. we denote two partitions π1 := {a = σ0 < σ1 < · · · < σn = b}, together with the set of real numbers αi ∈ [σi−1, σi] for i = 1, . . . , n and π2 := {a = s0 < s1 < · · · < sm = b}, together with the set of real numbers βj ∈ [sj−1, sj] for j = 1, . . . , m, by p put and |p| := max{|σi − σi−1| : i = 1, . . . , n} + max{|sj − sj−1| : j = 1, . . . , m}. we define the riemann sum sp by the expression sp := n∑ i=1 m∑ j=1 ( σi − σi−1 ) ( sj − sj−1 ) f(αi, βj) . moreover, we say that the riemann integral is i ∈ x if for each real number ǫ > 0 there exists δ > 0 such that when |p| < δ, then | i − sp | < ǫ. in this case the element i ∈ x is called the riemann integral of the function f and is denoted by ∫ ∫ f dσds. theorem 2.1. let us assume that there exist the integrals ∫b a f(σ, s)ds for each σ ∈ [a, b] and ∫b a [∫b a f(σ, s)ds ] dσ. if there exists ∫∫ f dσds, then ∫ ∫ f dσds = ∫b a [∫b a f(σ, s)ds ] dσ. 3 existence and uniqueness of the solutions of the nonlinear problem let us assume that f : x −→ x is globally lispchitz with lipschitz constant l ≥ 0. we denote by the letter l the set of continuous linear operator acting from x in x and given (t, s) ∈ ∆ := 54 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) {(t, s) ∈ r2 : a ≤ s ≤ t ≤ b}, we shall suppose that k(t, s) ∈ l. in this section we prove the existence and uniqueness of the solution for the problem    x(t) − x(a) = a(g)(t) + λ ∫t a k(t, s)f(x(s))ds, x(a) = x0 ∈ x, in the banach space f, where a is a function from f in f, such that a(g)(a) is equal to the null vector of x for each g ∈ f, the integral symbol is referred to the riemann integral in x and the expression k(t, s)f(x(s)) in the integral part means that the linear operator k(t, s) is evaluated in f(x(s)). example 3.1. a pair of examples of the function f : x −→ x are f(x) := lx + x0 and f(x) := l sin ‖x‖x0, where l is a real number and x0 is any fixed element belonging to x. example 3.2. two examples of the above function a : f −→ f are a(g)(t) := g(t) − g(a) and a(g)(t) := ∫t a b(g)(s)ds, where b is any function from f into f. definition 3.1. a function k : ∆ −→ l is uniformly lipschitz in the first variable, if there exists l̂ > 0 such that ∥∥ k(t, s) − k(τ, s) ∥∥ ≤ l̂ | t − τ | for all pairs (t, s) and (τ, s) belonging to the set ∆, where ∆ is defined by ∆ := { (t, s) ∈ r2 : a ≤ s ≤ t ≤ b } and the symbol ‖ · ‖ is referred to the usual operator norm in the space l. assumption 3.1. we suppose that: 1. the function k : ∆ −→ l is uniformly lipschitz in the first variable, 2. k(t, ·) : [a, t] −→ l is continuous for each t ∈ (a, b] and 3. sup s∈[a,b] ‖k(s, s)‖ < ∞. example 3.3. let t : x −→ x be any fixed bounded linear operator different than the null operator in l. an example of the function k : ∆ −→ l which is mentioned in the above assumption is k(t, s) := q(t, s)t, where q : [a, b] × [a, b] −→ r is a continuous function such that ∂q ∂t is continuous. the following proposition is a straightforward consequence from above assumption 3.1. proposition 3.1. the function k is bounded. cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 55 definition 3.2. given x ∈ f, we define f(x) : [a, b] −→ x by f(x)(t) := ∫t a k(t, s)f(x(s))ds. now, we shall prove a pair of propositions about the function f. proposition 3.2. if x ∈ f, then f(x) ∈ f. proof. let us note that ∣∣f(x)(t + h) − f(x)(t) ∣∣ = ∣∣∣∣ ∫t+h a k(t + h, s)f(x(s))ds − ∫t a k(t, s)f(x(s))ds ∣∣∣∣ ≤ ∫t a ∥∥k(t + h, s) − k(t, s) ∥∥ · ∣∣f(x(s)) ∣∣ds + ∫t+h t ∥∥k(t + h, s) ∥∥ · ∣∣f(x(s)) ∣∣ds ≤ l̂ · |h| · max s∈[a,b] ∣∣f(x(s)) ∣∣ ∫t a ds + sup (t,s)∈∆ ∥∥k(t, s) ∥∥ · max s∈[a,b] ∣∣f(x(s)) ∣∣ ∫t+h t ds ≤ { l̂ · (b − a) · max s∈[a,b] ∣∣f(x(s)) ∣∣ds + sup (t,s)∈∆ ∥∥k(t, s) ∥∥ · max s∈[a,b] ∣∣f(x(s)) ∣∣ } |h| . from the above inequality we can deduce that the function f is continuous. now we shall convince ourselves that if x ∈ f, then f(x) ∈ bv. let π := { a = t0 < t1 < · · · < tn = b} be a partition of interval [a, b] and observe that n∑ k=1 | f(x)(tk) − f(x)(tk−1) | = n∑ k=1 ∣∣∣∣∣ ∫tk a k(tk, s)f(x(s))ds − ∫tk−1 a k(tk−1, s)f(x(s))ds ∣∣∣∣∣ = n∑ k=1 ∣∣∣∣∣ ∫tk−1 a k(tk, s)f(x(s))ds − ∫tk−1 a k(tk−1, s)f(x(s))ds + ∫tk tk−1 k(tk, s)f(x(s))ds ∣∣∣∣∣ ≤ n∑ k=1 ∫tk−1 a ‖k(tk, s) − k(tk−1, s)‖ | f(x(s)) | ds + n∑ k=1 ∫tk tk−1 ‖k(tk, s)‖ | f(x(s)) | ds ≤ max s∈[a,b] | f(x(s)) | { l̂(b − a) n∑ k=1 (tk − tk−1) + sup (t,s)∈∆ ‖k(t, s)‖ n∑ k=1 (tk − tk−1) } = max s∈[a,b] | f(x(s)) | { l̂(b − a) + sup (t,s)∈∆ ‖k(t, s)‖ } (b − a) . ✷ 56 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) proposition 3.3. there exists a constant c > 0 such that ∣∣f(x) − f(y) ∣∣ 1 ≤ c ∣∣x − y ∣∣ 1 , for each pair of functions x, y ∈ f. proof. first, let us note that |f(x) − f(y)|1 = ∨ (f(x) − f(y)). now, if ∏ := { a = t0 < t1 < · · · < tn = b } is any partition of interval [a, b], then we obtain that n∑ i=1 |(f(x) − f(y))(ti) − (f(x) − f(y))(ti−1)| = n∑ i=1 ∣∣∣∣∣ ∫ti a k(ti, s) [ f(x(s)) − f(y(s)) ] ds − ∫ti−1 a k(ti−1, s) [ f(x(s)) − f(y(s)) ] ds ∣∣∣∣∣ ≤ n∑ i=1 ∫ti−1 a ‖k(ti, s) − k(ti−1, s)‖ · | f(x(s)) − f(y(s)) | ds + n∑ i=1 ∫ti ti−1 ‖k(ti, s)‖ · | f(x(s)) − f(y(s)) | ds ≤ l|x − y|1 [ n∑ i=1 ∫ti−1 a l̂(ti − ti−1)ds + sup (t,s)∈∆ ‖k(t, s)‖ n∑ i=1 (ti − ti−1) ] ≤ l (b − a) [ l̂(b − a) + sup (t,s)∈∆ | k(t, s) | ]∣∣x − y ∣∣ 1 . this proof ends when we realize that one adequate constant is c := l (b − a) [ l̂(b − a) + sup (t,s)∈∆ | k(t, s) | ] . ✷ theorem 3.1. let x0 be any element belonging to x. given g ∈ f, there is a real number ρ > 0 such that for each fixed real number λ with |λ| ≤ ρ, there exists a unique solution xg ∈ f for the initial value problem    x(t) − x(a) = a(g)(t) + λ ∫t a k(t, s)f(x(s))ds, x(a) = x0. proof. given x, g ∈ f, x0 ∈ x and λ ∈ r, we define g(x) : [a, b] −→ x ; cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 57 g(x)(t) := x0 + a(g)(t) + λf(x)(t). observe that if x ∈ f, then g(x) ∈ f. moreover, note that |g(x) − g(y)|1 = | λ | |f(x) − f(y)|1. from this equality, together with proposition 3.3, we ensure the existence of a real number ρ > 0 such that the function gg,λ,x0 : f −→ f; gg,λ,x0(x) := g(x) is a contraction for each λ ∈ r with |λ| ≤ ρ. the banach’s fixed point theorem implies the existence of an unique function xg ∈ f, which is the fixed point of the function gg,λ,x0 : f −→ f. but this indicates that the function xg is a solution of the above initial value problem. ✷ remark 3.1. the number ρ > 0 in theorem 3.1 does not depend on the function g ∈ f. 4 a formula for the solution of the linear problem in this section we find a parameters variation formula for the solutions of the linear problem    x(t) − x(a) = ∫t a b(g)(s)ds + λ ∫t a k(t, s)x(s)ds, x(a) = x0 ∈ x, (1) in the banach space f, where b : f −→ f. let us begin this part with two definitions. definition 4.1. we say that the function u : ∆ −→ l belongs to the set h, if u is continuous and uniformly lispchitz in the first variable. remark 4.1. if u ∈ h, then u holds each one of the properties in assumption 3.1. definition 4.2. we define the function ‖ · ‖1 : h −→ r by ‖ u ‖1:= ‖ u ‖11 + ‖ u ‖12 , where ‖ u ‖11 := max (t,s)∈∆ ‖u(t, s)‖ and ‖ u ‖12 := sup s∈[a,b), t 6=τ ‖u(t, s) − u(τ, s)‖ | t − τ | , such that (t, s), (τ, s) ∈ ∆. theorem 4.1. the real vectorial space h endowed with the above function ‖·‖1 is a banach space. now, we define an integral operator on the banach space h. 58 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) definition 4.3. let k be a fixed element belonging to h. given u ∈ h, we define f(u) : ∆ −→ l by f(u)(t, s) := ∫t s k(t, σ)u(σ, s)dσ, where k(t, σ)u(σ, s) is referred to the composition of the linear operators u(σ, s) and k(t, σ). remark 4.2. the operator f in above definition 4.3. is different than function f in definition 3.2. moreover, the integral symbol in above definition 4.3. is referred to the riemann integral in the space of the linear continuous operator l := l(x, x). the following theorem will play an important role in this section. theorem 4.2. the above operator f is a well defined bounded linear operator from h into h. proof. the proof will be given by claims. claim 1: the operator f is well defined, since the integral part [s, t] ∋ σ 7→ k(t, σ)u(σ, s) ∈ l is a continuous function and the riemann integral is unique. claim 2: if u ∈ h, then f(u) is continuous. claim 3: if u ∈ h, then the function f(u) is uniformly lispchitz in the first variable. to prove this claim, let us suppose that s ∈ [a, b). without lost of generality, we assume that a ≤ s ≤ τ < t < b and observe that ∥∥f(u)(t, s) − f(u)(τ, s) ∥∥ | t − τ | = ∥∥∥∥ ∫t s k(t, σ)u(σ, s)dσ − ∫τ s k(τ, σ)u(σ, s)dσ ∥∥∥∥ | t − τ | ≤ max (t,s)∈∆ ‖u(t, s)‖ · sup s∈[a,b) ∥∥k(t, s) − k(τ, s) ∥∥ | t − τ | ∫τ s dσ + 1 | t − τ | max (t,s)∈∆ ‖u(t, s)‖ · max (t,s)∈∆ ‖k(t, s)‖ ∫t τ dσ ≤ ‖ u ‖11 (‖ k ‖11 + (b − a) ‖ k ‖12). claim 4: the function f is a bounded linear transformation from h into h. since it is easy to verify that the function f is a linear transformation from h into h, then only we prove that such operator is bounded. firstly, note that max (t,s)∈∆ ∥∥f(u)(t, s) ∥∥ ≤ (b − a) max (t,σ)∈∆ ∥∥k(t, σ) ∥∥ · max (σ,s)∈∆ ∥∥u(σ, s) ∥∥. from here, together with the inequality in claim 3, we can conclude that the linear operator f is bounded. cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 59 ✷ now, we are ready to present and prove one important result of this section. theorem 4.3. let k be any fixed function belonging to h. there is a real number ρ > 0 such that if | λ | < ρ, then there exists a unique function r ∈ h, which satisfies the equality r(t, s) = i + λ ∫t s k(t, σ)r(σ, s)dσ for each pair of real numbers t and s such that a ≤ s ≤ t ≤ b, where the symbol i denote the identity operator i : x −→ x. proof. the proof follows from the fact that f : h −→ h is a bounded linear operator and applying the banach’s fixed point theorem to the operator t λ : h −→ h, which is given by t λ(u) := i + λf(u). ✷ remark 4.3. in above theorem 4.3, the function r could depend on the real number λ and r(t, t) = i for each t ∈ [a, b]. the function r in theorems 4.3 will allow us to find a representation of the solution of the linear problem (1) by means of two functions u and v, which are defined below. definition 4.4. given x0 ∈ x, we define u : [a, b] −→ x by u(t) := r(t, a)x0. lemma 4.1. the above function u(t) belongs to the space f. lemma 4.2. let us fix λ ∈ r such that | λ | < ρ. given any x0 ∈ x, we have that u(t) = x0 + λ ∫t a k(t, σ)u(σ)dσ, for each real number t ∈ [a, b]. proof. evaluate both of the members of the equality of theorem 4.3 in x0 with s = a. ✷ now we define another function v. 60 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) definition 4.5. we define v : [a, b] −→ x by v(t) := ∫t a r(t, s)b(g)(s)ds. lemma 4.3. the function v in definition 4.5 belongs to f. we need the following technical lemma in order to prove below lemma 4.5. lemma 4.4. let us fix t ∈ [a, b]. if k ∈ h, then ∫t a [ ∫t s k(t, σ)r(σ, s)dσ ] b(g)(s)ds = ∫t a [ k(t, σ) ∫σ a r(σ, s)b(g)(s)ds ] dσ. proof. lemma 4.4 can be deduced from theorem 2.1. ✷ lemma 4.5. if k ∈ h, then v(t) = ∫t a b(g)(s)ds + λ ∫t a k(t, σ)v(σ)dσ. proof. from lemma 4.4 and theorem 4.3, we have that v(t) − λ ∫t a k(t, σ)v(σ)dσ = ∫t a r(t, s)b(g)(s)ds − λ ∫t a k(t, σ) [∫σ a r(σ, s)b(g)(s)ds ] dσ = ∫t a r(t, s)b(g)(s)ds − λ ∫t a [∫t s k(t, σ)r(σ, s)dσ ] b(g)(s)ds = ∫t a r(t, s)b(g)(s)ds − ∫t a [ r(t, s) − i ] b(g)(s)ds = ∫t a b(g)(s)ds. ✷ as a consequence of lemmas 4.2 and 4.5, we obtain the following important result of this work. cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 61 theorem 4.4. let us suppose that | λ |< ρ, k ∈ h and y0 ∈ x. if b is any fixed function from f into f and g ∈ f, then a solution yg ∈ f for the problem with initial condition    y(t) − y(a) = ∫t a b(g)(s)ds + λ ∫t a k(t, s)y(s)ds, y(a) = y0 can be expressed by yg(t) = r(t, a)y0 + ∫t a r(t, s)b(g)(s)ds, where r is the function which was found in theorem 4.3. remark 4.4. the number ρ > 0 in above theorem 4.4 does not depend on the functions b or g. moreover, if this real number ρ is small enough, then the mentioned solution yg is unique. 5 controllability of the linear integral equation anew, we consider the linear problem which was studied in section 4    y(t) − y(a) = ∫t a b(g)(s)ds + λ ∫t a k(t, s)y(s)ds, y(a) = y0, (2) where k ∈ h, b : f −→ f and g ∈ f. definition 5.1. we say that the system (2) is exactly controllable on the interval [a, b], if for each pair of elements y0 and y1 belonging to x, there exists a function g ∈ f such that the corresponding solution y ∈ f of the problem (2) verify that y(a) = y0 and y(b) = y1. assumption 5.1. let us consider a function b : f −→ f, which is not necessarily a linear operator. from now on, we assume that the function b is surjective and λ is such as in theorem 4.4 statement. example 5.1. an example of a function b : f −→ f, such as in the above assumption, is defined by b(g) := γg + g0, where γ is a real number different than zero and g0 is a fixed function belonging to f. definition 5.2. we define the controller map g : f −→ x by g(g) = ∫b a r(b, s)b(g)(s)ds. as a consequence of theorem 4.4, we have the following proposition. proposition 5.1. the system (2) is exactly controllable on [a, b] if, and only if, rang(g) = x. 62 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) theorem 5.1. if the function b is surjective, then the system    x(t) − x(a) = ∫t a b(g)(s) + λ ∫t a k(t, s)x(s)ds, x(a) = x0 is exactly controllable on the interval [a, b]. proof. from the foregoing proposition 5.1, it is enough to prove that rang(g) = x. in order to show this, let us recall that the function r(b, ·) : [a, b] −→ l is continuous by theorem 4.3 and definition 4.1. from this and remark 4.1., there is a real number δ > 0 such that∥∥ i−r(b, s) ∥∥ < 1 4 for all s ∈ i = (b−δ, b] ⊂ [a, b]. moreover, there exists a continuous function α : [a, b] −→ r such that: 1. α(s) = 0 if s ∈ [a , b − δ], 2. 0 < α(s) ≤ 1 δ if s ∈ ( b − δ , b], 3. ∫b b−δ (1 δ − α(s))ds < 1 4 , 4. 0 < ∫b b−δ α(s)ds, 5. ∨ (α) < ∞. now, we define the function h : x −→ f by h(x)(s) := α(s)x. furthermore, we consider the linear operator t : x −→ x which is defined by t(x) := ∫b a r(b, s)h(x)(s) ds = ∫b a r(b, s)α(s)x ds. let us prove that ‖ i − t ‖ < 1. to this end, observe that ∣∣∣∣∣ i(x) − t(x) ∣∣∣∣∣ = ∣∣∣∣∣ x − ∫b a r(b, s) ( α(s)x ) ds ∣∣∣∣∣ = ∣∣∣∣∣ 1 δ ∫b b−δ xds − ∫b b−δ α(s)r(b, s)(x)ds ∣∣∣∣∣ ≤ ∣∣∣∣∣ 1 δ ∫b b−δ xds − ∫b b−δ α(s)xds ∣∣∣∣∣ + ∣∣∣∣∣ ∫b b−δ α(s)xds − ∫b b−δ α(s)r(b, s)(x)ds ∣∣∣∣∣ = ∣∣∣∣∣ ∫b b−δ ( 1 δ − α(s) ) xds ∣∣∣∣∣ + ∣∣∣∣∣ ∫b b−δ α(s) ( i − r(b, s) ) (x)ds ∣∣∣∣∣ ≤ ∫b b−δ ( 1 δ − α(s) ) |x|ds + ∫b b−δ |α(s)| ∥∥i − r(b, s) ∥∥ |x|ds = [∫b b−δ ( 1 δ − α(s) ) ds + ∫b b−δ |α(s)| ∥∥i − r(b, s) ∥∥ ds ] |x| cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 63 ≤ [ 1 4 + ∫b b−δ 1 δ 1 4 ds ] |x| = [ 1 4 + 1 4 ] |x| = 1 2 |x|. we have just proved that ( t − i ) is bounded. now, let us recall the following theorem. theorem.[14] let v : x −→ x be a bounded linear operator on a banach space x such that ‖ v ‖< 1. then the inverse (i − v)−1 exists on x and is bounded. from above two paragraphs, we can infer that t−1 : x −→ x exists and is bounded. since we have assumed that b is surjective, then for a given x ∈ x there exists a function gx ∈ f such that b(gx) = h(t −1(x)). now observe that g(gx) = ∫b a r(b, s)b(gx)(s)ds = ∫b a r(b, s)h(t−1(x))ds = ∫b a r(b, s)α(s)t−1(x)ds = t(t−1(x)) = x. from here we conclude that rang(g) = x. ✷ theorem 5.2. the function b : f −→ f admits a right inverse λ : f −→ f, i.e. b ◦ λ = i and there exists a control gy0,y1 steering the system (2) from the initial state y0 to a final state y1 which is given by gy0,y1 := λ ◦ h ◦ t −1 ( y1 − r(b, a)y0 ) ∈ f. proof. recall that any surjective function admits a right inverse. now observe that g(gy0,y1) = ∫b a r(b, s)b(gy0,y1)(s)ds = ∫b a r(b, s)b ◦ λ ( h ◦ t−1 ( y1 − r(b, a)y0 )) (s)ds = ∫b a r(b, s) ( h ◦ t−1 ( y1 − r(b, a)y0 )) (s)ds = y1 − r(b, a)y0. therefore, y1 = r(b, a)y0 + ∫b a r(b, s)b(gy0,y1)(s)ds. ✷ 64 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) remark 5.1.[6] if b : f −→ f is a surjective continuous linear operator and ker(b) admits a complement, then b has a right inverse λ : f −→ f, that is to say, b ◦ λ = i, such that λ is a continuous linear operator. definition 5.3. let λ : f −→ f be a right inverse of b, i.e. b◦λ = i. we define γ : x −→ f by γ(x) := λ◦h◦t−1 ( x−r(b, a)x0 ) , where t−1 and h are defined in the proof of above theorem 5.1. remark 5.2. in above theorem 5.2, we proved that ( g ◦ γ ) (x) = x − r(b, a)x0 for each x ∈ x 6 controllability of a type of nonlinear integral equation in this section we prove the controllability of nonlinear system    x(t) − x(a) = ∫t a b(g)(s)ds + λ ∫t a t(s)f(x(s))ds, x(a) = x0, (3) but before to do this, we shall prove that the following semilinear system is controllable,    x(t) − x(a) = ∫t a b(g)(s)ds + λ ∫t a t(s)x(s)ds + λ ∫t a t(s)f(x(s))ds, x(a) = x0. (4) to this end, we shall consider a particular integral equation under the following assumption. assumption 6.1. the function t : [a, b] −→ l is lipschitz on [a, b]. remark 6.1. the function k : ∆ −→ l, such that k(t, s) := t(s), belongs to the banach space h which was defined at the beginning in section 4. theorem 6.1. there exists ρ > 0 such that for any λ with |λ| ≤ ρ and g ∈ f, the problem (4) admits only one solution xg ∈ f, which simultaneously is a solution for the integral equation xg(t) = r(t, a)x0 + ∫t a r(t, s)b(g)(s)ds + λ ∫t a r(t, s)t(s)f(xg(s))ds, (5) where r is the function mentioned in theorem 4.4. proof. the existence of xg and ρ is a straightforward consequence of theorem 3.1 with a(g)(t) := ∫t a b(g)(s)ds, k(t, s) := t(s) and the function ( i + f ) instead of f. now, in order to prove the above equality (5), given a function g ∈ f, we define the function bg : [a, b] −→ x by bg(s) := b(g)(s) + λt(s)f(xg(s)), cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 65 where |λ| < ρ and xg is the function just mentioned. observe that the function xg ∈ f is a solution in f of the problem with initial value    y(t) − y(a) = ∫t a bg(s)ds + λ ∫t a k(t, s)y(s)ds, y(a) = x0 ∈ x, where k(t, s) := t(s). from theorem 4.4, we get xg(t) = r(t, a)x0 + ∫t a r(t, s)bg(s)ds, which indicates that xg(t) = r(t, a)x0 + ∫t a r(t, s)b(g)(s)ds + λ ∫t a r(t, s)t(s)f(xg(s))ds. ✷ we continue this section with a pair of definitions, and then we prove a set of technical propositions, which we shall use in order to verify that above semilinear problem (4) is controllable. definition 6.1. we define the controller map gf : f −→ x for the system (4) by gf(g) = r(b, a)x0 + ∫b a r(b, s)b(g)(s)ds + λ ∫b a r(b, s)t(s)f(xg(s))ds. = gr(g) + h(g) , such that gr : f −→ x is defined by gr(g) = r(b, a)x0 + ∫b a r(b, s)b(g)(s)ds = r(b, a)x0 + g(g), where g is the controller map in definition 5.2. and h : f −→ x is defined by h(g) = λ ∫b a r(b, s)t(s)f(xg(s))ds. now, let us give the definition of controllability for the nonlinear system (4). definition 6.2. we say that the system (4) is exactly controllable on the interval [a, b], if for each pair of elements x0 and x1 belonging to x, there exists a function g ∈ f and a solution xg ∈ f of the system (4) which could depend on g, such that xg(a) = x0 and xg(b) = x1. proposition 6.1. the system (4) is exactly controllable on [a, b] if, and only if, rang(gf) = x. 66 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) assumption 6.2. in this section we assume that there exists c ≥ 0 such that |b(u)−b(v)|1 ≤ c |u − v|1 for all pair u, v ∈ f. proposition 6.2. let us fix any two functions u, v ∈ f. if we define the function t 7→ ∫t a [ b(u)(s) − b(v)(s) ] ds, then there is a constant c > 0 such that ∣∣∣∣∣ ∫t a [ b(u)(s) − b(v)(s) ] ds ∣∣∣∣∣ 1 ≤ c (b − a) ∣∣ u − v ∣∣ 1 for each pair of elements u, v belonging to f. proof. this proposition is a consequence of proposition 2.4 and assumption 6.2. ✷ proposition 6.3. there exists a constant c > 0 such that |h(u) − h(v)| ≤ |λ| c |xu − xv|1 for each pair of elements u, v belonging to f. proposition 6.4. let us fix any two functions u, v ∈ f. if we define the function t 7→ ∫t a t(s) [ f(xu(s)) − f(xv(s)) ] ds, then there is a constant c > 0 such that ∣∣∣∣∣ ∫t a t(s) [ f(xu(s)) − f(xv(s)) ] ds ∣∣∣∣∣ 1 ≤ c ∣∣ xu − xv ∣∣ 1 for each pair of elements u, v belonging to f. proof. observe that for any partition { a = t0 < t1 < · · · < tn = b } of interval [a, b], we have that n∑ i=1 ∣∣∣∣∣ ∫ti a t(s) [ f(xu(s)) − f(xv(s)) ] − ∫ti−1 a t(s) [ f(xu(s)) − f(xv(s)) ] ds ∣∣∣∣∣ = n∑ i=1 ∣∣∣∣∣ ∫ti ti−1 t(s) [ f(xu(s)) − f(xv(s)) ] ds ∣∣∣∣∣ cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 67 ≤ n∑ i=1 ∫ti ti−1 ∥∥t(s) ∥∥ · ∣∣f(xu(s)) − f(xv(s)) ∣∣ ds ≤ n∑ i=1 l max s∈[a,b] ∥∥t(s) ∥∥ · max s∈[a,b] ∣∣xu(s) − xv(s) ∣∣ ∫ti ti−1 ds = l max s∈[a,b] ∥∥t(s) ∥∥ · max s∈[a,b] ∣∣xu(s) − xv(s) ∣∣ n∑ i=1 ∫ti ti−1 ds = l max s∈[a,b] ∥∥t(s) ∥∥ · max s∈[a,b] ∣∣xu(s) − xv(s) ∣∣ (b − a) ≤ l(b − a) max s∈[a,b] ∥∥t(s) ∥∥ ∣∣xu − xv ∣∣ 1 . ✷ proposition 6.5. there exist two constants c1 > 0 and c2 > 0, such that if |λ| < 1 c2 , then ∣∣xu − xv ∣∣ 1 ≤ c1 1 − |λ|c2 ∣∣u − v ∣∣ 1 for each pairs of functions u, v ∈ f. proof. since xu and xv denote the solutions for the system (4) for u and v respectively, we have that xu(t) − xv(t) = ∫t a ( b(u)(s) − b(v)(s) ) ds + λ ∫t a t(s) [ xu(s) − xv(s) ] ds + λ ∫t a t(s) [ f(xu(s)) − f(xv(s)) ] ds. now observe that ∣∣xu − xv ∣∣ 1 ≤ ∣∣∣ ∫t a ( b(u)(s) − b(v)(s) ) ds ∣∣∣ 1 + |λ| [ ∣∣∣ ∫t a t(s) [ xu(s) − xv(s) ] ds ∣∣∣ 1 + ∣∣∣ ∫t a t(s) [ f(xu(s)) − f(xv(s)) ] ds ∣∣∣ 1 ] . if we use once proposition 6.2 and twice proposition 6.4, we obtain ∣∣xu − xv ∣∣ 1 ≤ c(b − a) ∣∣u − v ∣∣ 1 + |λ| ( c̃ + ĉ )∣∣ xu − xv ∣∣ 1 . therefore ( 1 − |λ| ( c̃ + ĉ )) ∣∣xu − xv ∣∣ 1 ≤ c(b − a) ∣∣u − v ∣∣ 1 . ✷ 68 hugo leiva, jesús matute, nelson merentes & josé sánchez. cubo 17, 2 (2015) proposition 6.6. there exists a constant c > 0 such that |h(u) − h(v)| ≤ |λ| c |u − v|1 for each pair of functions u, v ∈ f. proof. this is a consequence of propositions 6.3 and 6.5. ✷ let us consider a further result before we can prove the controllability of system (4). theorem 6.2. [12] let z be a banach space and s : z −→ z a lipschitz function with a lipschitz constant l < 1 and consider ĝ(z) := z+s(z). then ĝ is a homeomorphism whose inverse is a lipschitz function with a lipschitz constant (1 − l)−1. now we shall prove the exact controllability of the system (4). theorem 6.3. let us assume that λ : f −→ f is a right inverse of b, i.e. b ◦ λ = i, such that λ is a lipschitz function. there exists a real number ρ > 0 such that if |λ| ≤ ρ, then the system (4) is exactly controllable on [a, b]. proof. in view of proposition 6.1 it is enough to prove that rang(gf) = x. to this end, we consider the operator g : x → x, which is defined by g := gf ◦ γ, where γ is the function in definition 5.3. now, from the remark 5.2. is obtained that gr ◦ γ(x) = i(x) = x. then, g(x) = x + s(x), where s := h ◦ γ. moreover, due to the proposition 6.6 and the definition 5.3., we can assure that ∣∣ s(x) − s(y) ∣∣ ≤ |λ| · c· ∣∣x − y ∣∣ for some fixed real number c ≥ 0 and all pair of elements x, y ∈ x. if the real number λ is small enough, then s is a lipschitz function with a lipschitz constant κ < 1. then, by the theorem 6.2, the function g : x −→ x is a homeomorphisms. hence we obtain that the function g is surjective , which implies that rang(gf) = x. therefore, the system (4) is exactly controllable on the interval [a, b]. ✷ under the assumptions made in the statement of above theorem 6.3, we obtain the following two theorems. theorem 6.4. let i : x −→ x be the identity function i(x) := x. the operator γ̂ : x −→ f defined by γ̂(x) = ( γ ◦ ( i + s )−1 ) (x) is a right inverse of the nonlinear operator gf, i.e. , cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 69 gf ◦ γ̂ = i. moreover, a control g ∈ f steering the system (4) from an initial state x0 to a final state x1 on [a, b] is given by g(t) = γ̂( x1 )(t) = ( γ ◦ ( i + s )−1)( x1 ) (t) . proof. this theorem is a consequence from the proof of the above theorem 6.3. ✷ before to prove the controllability of the system (3), let us consider one more definition. definition 6.3. we say that the system (3) is exactly controllable on the interval [a, b], if for each pair of real numbers x0 and x1 there exists a function g ∈ f and a solution x ∈ f of the system (3), which could depend on g, such that x(a) = x0 and x(b) = x1. now we are ready to prove the exact controllability of the nonlinear system (3). theorem 6.5. there exists a real number ρ > 0 such that if |λ| ≤ ρ, then the system (3) is exactly controllable on the interval [a, b]. proof. putting f(x) = x + ( f(x) − x ) = x + f̂(x), we obtain that f̂ := f(x) − x has the same properties as f. therefore, the system (3) can be written as follows    x(t) − x(a) = g(t) − g(a) + λ ∫t a t(s)x(s)ds + λ ∫t a t(s)f̂(x(s))ds, x(a) = x0. then, applying theorem 6.3 to this system, we obtain the result which we were looking for. ✷ acknowledgements. this work has been supported by bcv and cdcht-ula-c-aa received: september 2013. accepted: february 2015. references [1] g. astorga y l. barbanti; ecuaciones de evolución como ecuaciones integrales, revista de la facultad de ingenieŕıa. www.ingenieria.uda.cl . 22 (2008) 46-51. 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[17] o. a. i̇lhan; solvability of some integral equations in banach space and their applications to the theory of viscoelasticity, abstract and applied analysis, volume 2012, article id 717969, 13 pages. cubo 17, 2 (2015) on a type of volterra integral equation in the space . . . 71 [18] g. e. ladas and v. lakshmikantham; differential equations in abstract spaces, academic press, 1972. [19] m. h. noori, h. r. erfabian, a. v. kamyad; a new approach for a class of optimal control problems of volterra integral equations, intelligent control and automation, 2011, 2, 121-125. [20] a. d. polyanin and a. v. manzhirov; handbook of integral equations, second edition, chapman hall/crc, c©2008. [21] s̆. schwabik, m. tvrdý, o. vejvoda; differential and integral equations. boundary value problems and adjoints, d. reidel publishing co., dordrechtboston, mass.london, 1979. [22] g. f. webb; asymtotic stability in the α-norm for an abstract nonlinear volterra integral equation, stability of dynamical systems, theory and applications, volume 28, chapter 19, lectures notes in pure and applied mathematics, edited by john r. graef, marcel dekker inc., new york and basel. introduction preliminares existence and uniqueness of the solutions of the nonlinear problem a formula for the solution of the linear problem controllability of the linear integral equation controllability of a type of nonlinear integral equation cubo a mathematical journal vol.16, no¯ 01, (21–35). march 2014 weighted pseudo almost automorphic solutions of fractional functional differential equations syed abbas school of basic sciences, indian institute of technology mandi, mandi, h.p. 175001, india. sabbas.iitk@gmail.com, abbas@iitmandi.ac.in abstract in this paper we discuss the existence of weighted pseudo almost automorphic solution of fractional order functional differential equations. using the fixed point theorem we establish existence and uniqueness of solution to the problem under consideration. the results obtained extend the theory of almost automorphic solutions to a more general class of weighted pseudo almost automorphic solutions. these extensions allow to treat infinite dimensional dynamics such as fractional wave and heat equation which are presented in the paper. at the end we give several example to illustrate the analytical findings. resumen en este art́ıculo discutimos la existencia de una solución seudo casi automórfica con peso de ecuaciones diferenciales funcionales de orden fraccional. usando el teorema del punto fijo, establecemos la existencia y unicidad de la solución del problema en estudio. los resultados obtenidos extienden la teoŕıa de soluciones casi automórficas a clases más generales de soluciones seudo casi automórficas con peso. estas extensiones permiten estudiar dinámicas infinito-dimensional como la onda fraccionaria y la ecuación del calor, las cuales se presentan en este art́ıculo. al final, mostramos varios ejemplos para ilustrar los resultados anaĺıticos obtenidos. keywords and phrases: fractional differential equation, fixed point theorem, almost automorphic functions, abstract differential equations. 2010 ams mathematics subject classification: 34k40, 34k14. 22 syed abbas cubo 16, 1 (2014) 1 introduction in this work we consider the following functional differential equations of fractional order α ∈ (1,2), dαt u(t) = au(t) + d α−1 t f(t,u(t),ut), t ∈ r, u(t) = φ(t), t ∈ (−∞,0], ut(θ) = u(t + θ), θ ∈ (−∞,0], (1) where f : r × x × x → x, φ ∈ c0((−∞,0],r) and a : d(a) ⊂ x → x is a linear densely defined operator of sectorial type on a complex banach space x. with motivation coming from a wide range of engineering and physical applications, fractional differential equations have recently attracted great attention of mathematicians and scientists. this kind of equations is a generalization of ordinary differential equations to arbitrary non integer orders. the origin of fractional calculus goes back to newton and leibniz in the seventieth century. it is widely and efficiently used to describe many phenomena arising in engineering, physics, economy and science. recent investigations have shown that many physical systems can be represented more accurately through fractional derivative formulation [36]. fractional differential equations find numerous applications in the field of viscoelasticity, feedback amplifiers, electrical circuits, electro analytical chemistry, fractional multipoles, neuron modelling encompassing different branches of physics, chemistry and biological sciences [39]. many physical processes appear to exhibit fractional order behavior that may vary with time or space. the fractional calculus has allowed the operations of integration and differentiation to any fractional order. the order may take on any real or imaginary value. the existence and uniqueness of solutions to fractional differential equations have been shown by many authors [1, 2, 4, 5, 7, 11, 12, 16, 9, 10, 17, 18, 23, 37, 28, 31, 32, 37, 39]. agarwal et. al. [6] have shown the existence of weighted pseudo almost periodic solutions of semilinear fractional differential equations. since bohr [15] introduced the concept of almost periodic functions, there have been many important generalizations of this functions in the past few decades. the generalization includes pseudo almost periodic functions [41], where the function can be decomposable in two part. these functions are further generalized to weighted pseudo almost periodic function by diagana, where the weighted mean of the second component is zero [20]. another direction of generalization is almost automorphic functions introduced by bochner [14]. the pseudo almost automorphic functions are natural generalization of almost automorphic functions [14] and introduced by liang et. al. [33]. these functions are further generalized by blot et.al. [13] and named weighted pseudo almost automorphic. the authors in [13] have proved very important properties of these functions including composition theorem and completeness property. the study of weighted pseudo almost automorphic solutions of various kind of differential equations are very new and an attractive area of research. for more details on theory and applications of these functions we refer to [13] and references therein. the existence and uniqueness of almost automorphic and pseudo almost automorphic solutions have been established by many authors, for example [3, 13, 26] and references therein. cubo 16, 1 (2014) weighted pseudo almost automorphic solutions of fractional . . . 23 the problem considered in this work is motivated by the work of claudio cuevas and carlos lizama [17] work in which they have considered the following fractional differential equations dαt u(t) = au(t) + d α−1 t f(t,u(t)), t ∈ r, (2) and proved the existence of almost automorphic solutions under certain assumptions. in this paper we discuss existence and uniqueness of weighted pseudo almost automorphic solutions of problem (1). the concept of stepanov like pseudo almost periodicity is introduced by diagana [21, 22], which is a generalization of pseudo almost periodicity. further stepanov like almost automorphy has been introduced by n’guerekata and pankov [29]. 2 preliminaries denote b(x) be the banach space of all linear and bounded operators on x endowed with the norm ‖ · ‖b(x) and c = c(r,x) the set of all continuous functions from r to x. let u the collection of all positive integrable functions ρ : r → r. for each ρ ∈ u define m(r,ρ) = ∫r −r ρ(s)ds. denote u∞ : the set of all ρ ∈ u such that limr→∞ m(r,ρ) = ∞ ub : the set of all bounded ρ ∈ u∞ such that inft∈r ρ(t) > 0. now we state the definitions of weighted almost automorphic functions. definition 2.1. a continuous function f : r → x is called almost automorphic if for every real sequence (sn), there exists a subsequence (snk) such that g(t) = lim n→∞ f(t + snk) is well defined for each t ∈ r and lim n→∞ g(t − snk) = f(t) for each t ∈ r. the set of all almost automorphic functions from r to x are denoted by aa(x). the set of all almost automorphic functions from r to x are denoted by aa(x) and it is a banach space equipped with the sup norm ‖f‖∞ = sup t∈r ‖f(t)‖. definition 2.2. a continuous function f : r ×x → r is called almost automorphic in t uniformly for x in compact subsets of x if for every compact subset k of x and every real sequence (sn), there exists a subsequence (snk) such that g(t,x) = lim n→∞ f(t + snk,x) 24 syed abbas cubo 16, 1 (2014) is well defined for each t ∈ r, x ∈ k and lim n→∞ g(t − snk,x) = f(t,x) for each t ∈ r, x ∈ k. denote by aa(r × x) the set of all such functions. we denote by aa0(x) = { f ∈ bc(r,x) : lim r→∞ 1 m(r,ρ) ∫r −r ρ(ξ)‖f(ξ)‖dξ = 0 } , and by aa0(r × x × x,x) the set of all continuous functions f : r × x × x → x such that f(.,u,φ) ∈ aa0(x) and lim r→∞ 1 m(r,ρ) ∫r −r ρ(ξ)‖f(ξ,u,φ)‖dξ = 0, uniformly in (u,φ) ∈ x × x. definition 2.3. a mapping f ∈ bc(r,x) is called weighted pseudo almost automorphic if it can be written as f = f1 + f2, where f1 ∈ aa(x) and f2 ∈ aa0(x). the functions f1 and f2 are called the almost automorphic and the weighted ergodic perturbation components of f respectively. the set of all such functions will be denoted by paa(x). remark 2.4. a classical example of pseudo almost automorphic function is f(t) = sin 1 2 + cost + cos √ 2t + 1 1 + t2 . t ∈ r. one can easily see that this function is not almost periodic. example: consider the function f(t) = sin 1 2 + cost + cos √ 2t + eαt it is well known that the function sin 1 2+cos t+cos √ 2t is almost automorphic. now consider the weight function ρ defined by ρ(t) = 1 t < 0 and ρ(t) = e−βt t ≥ 0 for some β > 0. it is easy to verify that m(r,ρ) = ∫r −r ρ(t)dt = ∫0 −r ρ(t)dt + ∫r 0 ρ(t)dt = r + 1 − e−βr β . thus limr→∞ m(r,ρ) = ∞ which implies that ρ ∈ u∞. further ∫r −r eαtρ(t)dt = ∫0 −r eαtdt + ∫r 0 eαte−βtdt = 1 − e−αr α + ∫r 0 e(α−β)tdt = 1 − e−αr α + e(α−β)r − 1 α − β . (3) cubo 16, 1 (2014) weighted pseudo almost automorphic solutions of fractional . . . 25 thus for α ≤ β, we have lim r→∞ 1 m(r,ρ) ∫r −r eαtρ(t)dt = 0. hence eαt ∈ paa0(r,ρ) and so f(t) ∈ wpaa(r). it is also interesting to note that lim r→∞ 1 2r ∫r −r eαtdt = lim r→∞ eαr − e−αr 2αr = ∞. this implies that f(t) does not belongs to paa(r), the space of all pseudo almost automorphic functions. definition 2.5. a continuous mapping f : r × x × x → x is called weighted pseudo almost automorphic in t ∈ r uniformly in (x,φ) ∈ x × x if it can be written as f = f1 + f2, where f1 ∈ aa(r × x × x,x) and f2 ∈ aa0(r × x × x,x). we denote the set of all weighted pseudo almost automorphic functions f : r × x × x → x by wpaa(r × x × x). the following theorems are from [13]. theorem 2.6. the decomposition of a weighted pseudo almost automorphic function is unique for any ρ ∈ ub. theorem 2.7. let wpaa(r,ρ) 3 f = g + φ where ρ ∈ u∞ and assume that f(t,u) is uniformly continuous in any bounded subset k of x uniformly in t ∈ r and g(t,u) is uniformly continuous in any bounded subset k of x uniformly in t ∈ r. then if u ∈ wpaa(r,ρ), implies f(·,u(·)) ∈ wpaa(r,ρ). the above theorem holds if both functions f,g are lipschitz continuous in u uniformly in t ∈ r. the weight one functions that is ρ = 1, are called pseudo almost automorphic. 3 weighted pseudo almost automorphic solutions assumptions: let us consider the the following assumptions: (a1) the function f : r × x × x → x is weighted pseudo almost automorphic with respect to t uniformly in (u,φ) ∈ x × x, and there exists 0 < l < 1, such that ‖f(t,u,φ) − f(t,v,ψ)‖ ≤ l(‖u − v‖ + ‖φ − ψ‖. (a2) the function f is bounded. lemma 3.1. let {s(t)}t>0 ⊂ b(x) be a strongly continuous family of bounded and linear operators such that ‖s(t)‖ ≤ φ(t) for almost all t ∈ r+ with φ ∈ l1(r+). if f : r → x is a weighted pseudo almost automorphic function then ∫t −∞ s(t − s)f(s)ds ∈ wpaa(x). 26 syed abbas cubo 16, 1 (2014) a closed and linear operator a is said to be sectorial of type ω and angle θ if there exists 0 < θ < π 2 ,m > 0 and ω ∈ r such that its resolvent exists outside the sector ω + sθ := {ω + λ : λ ∈ c, |arg(−λ)| < θ}, and ‖(λ − a)−1‖ ≤ m |λ − ω| , λ 6∈ ω + sθ. sectorial operators are well studied in the literature. for a recent reference including several examples and properties we refer the reader to [30]. note that an operator a is sectorial of type ω if and only if ωi − a is sectorial of type 0. the equation 1 can be thought as a limiting case of the following equation v′(t) = ∫t 0 (t − s)α−2 γ(α − 1) av(s)ds + f(t,u(t),ut), t ≥ 0, vt(θ) = φ(t),t ∈ (−∞,0), (4) in the sense that the solutions are asymptotic to each other as t → ∞. if we consider that the operator a is sectorial of type ω with θ ∈ [0,π(1 − α 2 )), then problem 4 is well posed [19]. thus we can use variation of parameter formulae to get v(t) = sα(t)u0 + ∫t 0 sα(t − s)f(s,u(s),us)ds, t ≥ 0, where sα(t) = 1 2πi ∫ γ eλtλα−1(λαi − a)−1dλ, t ≥ 0, where the path γ lies outside the sector ω+sθ. if sα(t) is integrable then the solution is given by u(t) = ∫t −∞ sα(t − s)f(s,u(s),us)ds. now one can easily see that v(t) − u(t) = sα(t)u0 − ∫ ∞ t sα(s)f(t − s,u(t − s),ut−s). hence for f ∈ lp(r+ × x × x,x), p ∈ [1,∞) we have v(t) − u(t) → 0 as t → ∞. definition 3.2. a function u : r → x is said to be a mild solution to 1 if the function sα(t − s)f(s,u(s),us) is integrable on (−∞,t) for each t ∈ r and u(t) = ∫t −∞ sα(t − s)f(s,u(s),us)ds, for each t ∈ r. cubo 16, 1 (2014) weighted pseudo almost automorphic solutions of fractional . . . 27 recently, cuesta in [19], theorem-1, has proved that if a is a sectorial operator of type ω < 0 for some m > 0 and θ ∈ [0,π(1 − α 2 )), then there exists c > 0 such that ‖sα(t)‖ ≤ cm 1 + |ω|tα for t ≥ 0. also the following relation [17], theorem-3.4 holds, ∫ ∞ 0 dt 1 + |ω|tα = |ω| −1 α π αsin π α for α ∈ (1,2). define the operator fu(t) = ∫t −∞ sα(t − s)f(s,u(s),us)ds, t ∈ r. first thing we observe about the operator f is boundedness and continuity. indeed, ‖fu‖ ≤ ∫t −∞ ‖sα(t − s)‖ × ‖f(s,u(s),us)‖ds ≤ ∫ ∞ 0 ‖sα(s)‖‖f(t − s,u(t − s),ut−s)‖ds ≤ cm ∫ ∞ 0 1 1 + |ω|sα ‖f(t − s,u(t − s),ut−s)‖ds ≤ cm‖f‖ ∫ ∞ 0 1 1 + |ω|sα ds = cm‖f‖ω− 1α π αsin π α (5) thus f is bounded. further, we have ‖fu(t + h) − fu(t)‖ = ∥ ∥ ∥ ∫t+h −∞ sα(t + h − s)f(s,u(s),us)ds − ∫t −∞ sα(t − s)f(s,u(s),us)ds ∥ ∥ ∥ ≤ ∫t −∞ ‖sα(t − s)‖ × ‖f(s + h,u(s + h),us+h) − f(s,u(s),us)‖ds ≤ ∫ ∞ 0 ‖sα(s)‖ × ‖f(t − s + h,u(t − s + h),ut−s+h) − f(t − s,u(t − s),ut−s)‖ds ≤ cmsup t∈r ‖f(t − s + h,u(t − s + h),ut−s+h) − f(t − s,u(t − s),ut−s)‖ × ∫ ∞ 0 1 1 + |ω|sα ds = cmω− 1 α π αsin π α sup t∈r ‖f(t − s + h,u(t − s + h),ut−s+h) − f(t − s,u(t − s),ut−s)‖, (6) 28 syed abbas cubo 16, 1 (2014) which goes to zero as h → 0 and hence f is continuous. it is easy to see that the operator f maps wpaa(x) to wpaa(x), which we represent in the form of a lemma as follows. lemma 3.3. the operator f maps wpaa(x) to wpaa(x) if f ∈ wpaa(x). proof: as f ∈ wpaa(x), we can decompose it into two part f1 ∈ aa(x) and f2 ∈ aa0(x). now define the operators f1u(t) = ∫t −∞ sα(t − s)f1(s,u(s),us)ds, t ∈ r and f2u(t) = ∫t −∞ sα(t − s)f2(s,u(s),us)ds, t ∈ r. also for every sequence tn there exists a subsequence tnk such that f1(t + tnk,u,ψ) → g1(t,u,ψ), g1(t − tnk,u,ψ) → f1(t,u,ψ), u,ψ ∈ d, where d is a compact subset of x × x. f1u(t + tnk) = ∫t+tn k −∞ sα(t + tnk − s)f1(s,u(s),us)ds = ∫t −∞ sα(t − s)f1(s + tnk,u(s + tnk),u(s+tn k ))ds → ∫t −∞ sα(t − s)g1(s,u(s),us)ds = (f∗u)(t). (7) thus (f1u)(t + tnk) → (f ∗u)(t). similarly one can get (f∗u)(t − tnk) → (f1u)(t). now we need to show lim r→∞ 1 m(r,ρ) ∫r −r ∫t −∞ ρ(s)|sα(t − s)|‖f2(s,u(s),us)‖dsdt = 0. consider 1 m(r,ρ) ∫r −r ∫t −∞ ρ(s)‖sα(t − s)‖‖f2(s,u(s),us)‖dsdt ≤ i1(r) + i2(r), where i1(r) = 1 m(r,ρ) ∫r −r dt ∫t −r ρ(s)‖sα(t − s)‖‖f2(s,u(s),us)‖ds cubo 16, 1 (2014) weighted pseudo almost automorphic solutions of fractional . . . 29 and i2(r) = 1 m(r,ρ) ∫r −r dt ∫−r −∞ ρ(s)‖sα(t − s)‖‖f2(s,u(s),us)‖ds. thus we have i1(r) ≤ 1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ ∫r s ‖sα(t − ξ)‖dt ≤ 1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ ∫r−s 0 ‖sα(t)‖dt ≤ 1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ ∫ ∞ 0 ‖sα(t)‖dt ≤ cm m(r,ρ) ∫r −r ρξ‖f2(ξ,u(ξ),uξ)‖dξ |ω| −1 α π αsin π α ≤ m1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ, (8) for some positive constant m1. the above calculations imply that lim r→∞ i1(r) = 0 as f2 ∈ aa0(r × x × x). now consider i2(r) ≤ 1 m(r,ρ) ∫r −r dt ∫ ∞ t+r ρ(t − s)‖sα(s)‖‖f2(t − s,u(t − s),ut−s)‖ds ≤ 1 m(r,ρ) ∫r −r dt ∫ ∞ 2r ρ(t − s)‖sα(s)‖‖f2(t − s,u(t − s),ut−s)‖ds ≤ ‖f2‖∞ ∫ ∞ 2r ‖sα(s)‖ds. (9) from the above analysis we get lim r→∞ i2(r) = 0. thus we have lim r→∞ 1 2r ∫r −r ‖f2(u)(t)‖dt = 0. hence the result is proved. theorem 3.4. problem (1) has a unique solution in wpaa(x) under the assumption (a1) provided that 2l|ω| −1 α π αsin π α < 1. 30 syed abbas cubo 16, 1 (2014) proof: in order to prove that the operator f has a fixed point, consider ‖fu1(t) − fu2(t)‖ ≤ ∫t −∞ ‖sα(t − s)‖‖f(s,u1(s),u1s) − f(s,u2(s),u2s)‖ds ≤ l ∫t −∞ ‖sα(t − s)‖ ( ‖u1(s) − u2(s)‖ + ‖u1s − u1s‖b(x) ) ds ≤ 2l‖u1 − u2‖∞ ∫ ∞ 0 ‖sα(t)‖dt. (10) thus for 2l ∫ ∞ 0 |sα(t)|dt < 1, the problem (1) has an unique solution. we have mentioned that ∫ ∞ 0 1 1 + |ω|tα = |ω| −1 α π αsin π α for α ∈ (1,2). thus the above condition reduces to 2cml|ω| −1 α π α sin π α < 1. remark 3.5. one can easily show that for f stepanov almost automorphic, the problem (1) has a unique stepanov almost automorphic solutions under the same condition as in both theorems. remark 3.6. it is to note that for differential equation du(t) dt = au(t) + f(t,u(t)), t ∈ r u(0) = u0, (11) where a generates an exponentially stable c0 semigroup {t(t)}t≥0, we can conclude that, if f if lipschitz continuous, bounded and weyl almost automorphic or weyl pseudo almost automorphic, then there exists a unique weyl almost automorphic or weyl pseudo almost automorphic solution accordingly of the problem provided that lfm1 δ < 1, where ‖t(t)‖ ≤ m1e−δt for some m ≥ 1 and δ > 0. 4 examples example-1: consider the following fractional order partial differential equation for α ∈ (1,2), ∂αu(t,x) ∂tα = ∂2u(t,x) ∂x2 + ∂α−1 ∂tα−1 (g(t,u(t,x),u(t − τ,x))), τ > 0, t ∈ r, x ∈ (0,π) u(t,0) = u(t,π) = 0, t ∈ r, u(t,x) = φ(t,x) t ∈ [−τ,0], (12) where g is a weighted pseudo almost automorphic function in t. also assume that g satisfies lipschitz condition in both variable with lipschitz constant lg. using the transformation u(t)x = cubo 16, 1 (2014) weighted pseudo almost automorphic solutions of fractional . . . 31 u(t,x) and a = ∂ 2 ∂x2 with d(a) = { u ∈ l2((0,π),r), u ′ ∈ l2((0,π),r), u ′′ ∈ l2((0,π),r),u(0) = u(π) = 0 } , the above equation can be transform into dαu(t) dtα = au(t) + dα−1 dtα−1 g(t,u(t),ut(−τ)), (13) t ∈ r and u(t) = φ(t) t ∈ [−τ,0]. it is to note that a generates an analytic semigroup {t(t) : t ≥ 0 on x, where x = l2((0,π),r). further a has discrete spectrum with eigenvalues of the form −k2;k ∈ n, and corresponding normalized eigenfunctions given by zk(x) = ( 2π) 1 2 sin(kx). as a is analytic, let us assume that it is sectorial of type ω1 and let the following relation holds 2lg|ω1| −1 α π αsin π α < 1. thus under all the required assumption on g, the existence of weighted almost automorphic solutions is ensured accordingly. example-2: consider the following fractional order delay relaxation oscillation equation for α ∈ (1,2), ∂αu(t,x) ∂tα = ∂2u(t,x) ∂x2 − pu(t,x) + ∂α−1 ∂tα−1 (f(t,u(t,x),u(t − τ,x))), τ > 0, t ∈ r, x ∈ (0,π) u(t,0) = u(t,π) = 0, t ∈ r, u(t,x) = φ(t,x) t ∈ [−τ,0], (14) where p > 0 and f is a weighted pseudo almost automorphic function in t. also assume that f satisfies lipschitz condition in both variable with lipschitz constant lf. using the transformation u(t)x = u(t,x) and define au = ∂ 2 u ∂x2 − pu, u ∈ d(a), where d(a) = { u ∈ l2((0,π),c), u ′ ∈ l2((0,π),c), u ′′ ∈ l2((0,π),c),u(0) = u(π) = 0 } , the above equation can be transform into dαu(t) dtα = au(t) + dα−1 dtα−1 g(t,u(t),ut(−τ)), (15) t ∈ r and u(t) = φ(t) t ∈ [−τ,0]. it is to note that a generates an analytic semigroup {t(t) : t ≥ 0 on x, where x = l2((0,π),r). hence pi−a is sectorial of type ω = −p < 0. further a has discrete spectrum with eigenvalues of the form −k2;k ∈ n, and corresponding normalized eigenfunctions given by zk(x) = ( 2 π ) 1 2 sin(kx). as a is analytic. let us assume that 2lf|ω| −1 α π αsin π α < 1. 32 syed abbas cubo 16, 1 (2014) thus under all the required assumption on f, the existence of weighted almost automorphic solutions is ensured accordingly. example-3: consider the following abstract differential equations of fractional order over a complex banach space (x,‖ · ‖), dαu(t) dtα = au(t) + dα−1 dtα−1 (g(t,u(t)) + ku(t)), (16) t ∈ r, where ku(t) = ∫t −∞ k(t − s)u(s)ds and a : d(a) ⊂ x → x is a linear densely defined operator of sectorial type on a complex banach space x. we assume that g is weighted pseudo almost automorphic in t uniformly in u and k satisfy |k(t)| ≤ ce−bt for some c,b > 0. for u weighted pseudo almost automorphic, it is not difficult to see that ku(t) is also weighted pseudo almost automorphic. let us assume that g satisfy lipschitz condition with lipschitz constant lg. now for u1,u2 ∈ x, consider ‖g(t,u1(t)) − g(t,u2(t))‖ + ‖ku1(t) − ku2(t)‖ ≤ lg‖u1 − u2‖ + ∫t −∞ |k(t − s)||u1(s) − u2(s)|ds ≤ lg‖u1 − u2‖ + ‖u1 − u2‖ ∫ ∞ 0 |k(s)|ds ≤ lg‖u1 − u2‖ + c b ‖u1 − u2‖ ≤ ( lg + c b ) ‖u1 − u2‖. (17) thus we have ‖g(t,u1) − g(t,u2)‖ + ‖ku1 − ku2‖ ≤ ( lg + c b ) ‖u1 − u2‖. considering t − s = s1 we have ku(t) = ∫ ∞ 0 k(s1)u(t + s1) = ∫ ∞ 0 k(s)ut(s). thus if we take g1(t,u(t),ut) = g(t,u(t)) + ku(t), the above equation is similar to (1). from the above analysis, one can easily see that g1 satisfies lipschitz condition with lipschitz constant lg + c b . further assume that a is sectorial of type ω2 and the following condition hold 2(lg + c b )|ω2| −1 α π αsin π α < 1. one can easily see that for u ∈ wpaa(x),ku(t) ∈ wpaa(x). thus we can apply our result to ensure the existence of weighted almost automorphic solution for g weighted almost automorphic. received: november 2012. accepted: may 2013. cubo 16, 1 (2014) weighted pseudo almost automorphic solutions of fractional . . . 33 references [1] abbas, s., existence of solutions to fractional order ordinary and delay differential equations and applications, electron. j. diff. equ., vol. 2011 (2011), no. 09, pp. 1-11. 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() cubo a mathematical journal vol.13, no¯ 03, (69–89). october 2011 lightlike geometry of leaves in indefinite kenmotsu manifolds fortuné massamba department of mathematics, university of botswana, private bag 0022 gaborone, botswana. email: massfort@yahoo.fr, massambaf@mopipi.ub.bw abstract in this paper, we study some aspects of the geometry of leaves of integrable distributions of lightlike hypersurfaces in indefinite kenmotsu manifolds, tangent to the structure vector field. theorems on parallel vector field, killing distribution, geodesibility of lightlike hypersurfaces are obtained. some characterization theorems of leaves of integrable distributions are given. we prove that there exists a distribution, subset of the screen distribution, in which, under the integrability condition, any totally contact umbilical leaf is an extrinsic sphere. resumen en este trabajo se estudian algunos aspectos de la geometŕıa de las hojas de distribuciones integrables de las hipersuperficies luminosas en variedades de kenmotsu indefinidas, tangentes a la estructura de un campo vectorial. se obtienen algunos teoremas sobre el campo de vectores paralelo, distribución de killing y geodesbilidad de hipersuperficies liminosas. se dan algunos teoremas de caracterización de las hojas de las distribuciones integrables. se demuestra que existe una distribucin, subconjunto de 70 fortuné massamba cubo 13, 3 (2011) la distribucin de la pantalla, en la que, bajo la condición de integrabilidad, cualquier contacto con la hoja umbilical es una esfera extŕınseca. keywords: lightlike hypersurface; indefinite kenmotsu manifold; screen distribution. mathematics subject classification: 53c15, 53c25, 53c40, 53c50. 1 introduction several authors have studied some properties of kenmotsu manifolds. in [7], for instance, the authors partially classified the kenmotsu manifolds and considered manifolds admitting the transformation which keeps the riemannian curvature tensor and ricci tensor invariant. the present paper aims to investigate the geometry of lightlike hypersurfaces of indefinite kenmotsu manifolds, tangent to the structure vector field, with specific attention to the geometry of leaves of its integrable distributions. as is well known, the geometry of lightlike submanifolds [4] are different because of the fact that their normal vector bundle intersects with the tangent bundle. thus, the study becomes more difficult and strikingly different from the study of non-degenerate submanifolds. this means that one cannot use, in the usual way, the classical submanifold theory to define any induced object on a lightlike submanifold. to deal with this anomaly, the lightlike submanifolds were introduced and presented in a book by duggal and bejancu [4]. they introduced a non-degenerate screen distribution to construct a nonintersecting lightlike transversal vector bundle of the tangent bundle. several authors have studied lightlike hypersurfaces of semi-riemannian manifolds (see [5] and many more references therein). physically, lightlike hypersurfaces are interesting in general relativity since they produce models of different types of horizons. on the latter, the relationship between killing and geodesic notions is well specified. in [4], the authors discussed the cauchy riemann (cr) lightlike submanifolds of indefinite käehler manifolds in ([4], chapter 6) and proved that, in a totally umbilical real lightlike hypersurface of an indefinite käehler space form, the nonzero mean curvature vector satisfies partial differential equations which imply that the nonzero mean curvature vector is not parallel. the usual terminology says that such an umbilical lightlike submanifold is not an extrinsic sphere (see [3] for more details). as the notion of totally umbilical submanifolds of kaehlerian manifolds corresponds to that of totally contact umbilical submanifolds of sasakian manifolds [10], the author in [13] showed that, in a totally contact umbilical lightlike hypersurface of an indefinite sasakian space form, the nonzero mean curvature vector also is not parallel. but in [15] it is proved that any totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface in an indefinite sasakian space form is an extrinsic sphere. considering the mentioned notions above and those given in [6], [11], [12] and [14] on lightlike cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 71 hypersurfaces of indefinite sasakian manifolds, similar research is needed for the geometry of leaves of distribution of lightlike hypersurfaces in indefinite kenmotsu manifolds. it is important to note that kenmotsu manifolds are different from sasakian manifolds. the paper is organized as follows. in section 2, we recall some basic definitions for indefinite kenmotsu manifolds and lightlike hypersurfaces of semi-riemannian manifolds. in section 3, we give a decomposition of almost contact metric of lightlike hypersurfaces in indefinite kenmotsu manifolds, tangent to the structure vector field, supported by an example, as well as theorems on lie derivatives and parallel vector field. in section 4, we investigate the geometry of leaves of integrable distributions of lightlike hypersurfaces. we prove that, if a leaf m′ of the integrable distribution d0 ⊥ 〈ξ〉 is totally geodesic, then φ(tm ⊥) ⊕ φ(n(tm)) is a killing distribution. moreover, if the second fundamental form of m′ is parallel with respect to the levi-civita connection ∇′, then m′ is totally geodesic (theorem 4.3). a characterization of a leaf of d0 ⊥ 〈ξ〉 is given (theorem 4.4). we show that any totally contact umbilical leaf of an integrable distribution d̂ of a lightlike hypersurface is an extrinsic sphere (theorem 4.5). 2 preliminaries let m be a (2n + 1)-dimensional manifold endowed with an almost contact structure (φ, ξ, η), i.e. φ is a tensor field of type (1, 1), ξ is a vector field, and η is a 1-form satisfying φ 2 = −i + η ⊗ ξ, η(ξ) = 1, η ◦ φ = 0 and φξ = 0. (2.1) then (φ, ξ, η, g) is called an indefinite almost contact metric structure on m if (φ, ξ, η) is an almost contact structure on m and g is a semi-riemannian metric on m such that, for any vector field x, y on m (see [2] for riemannian case) η(x) = g(ξ, x), g(φ x, φ y) = g(x, y) − η(x) η(y). (2.2) if, moreover, (∇ x φ)y = g(φ x, y)ξ − η(y)φ x, where ∇ is the levi-civita connection for the semi-riemannian metric g, we call m an indefinite kenmotsu manifold [9]. a plane section σ in tpm is called a φ-section if it is spanned by x and φ x, where x is a unit tangent vector field orthogonal to ξ. the sectional curvature of a φ-section σ is called a φ-sectional curvature. if a kenmotsu manifold m has constant φ-sectional curvature c, then, by virtue of the proposition 12 in [9], the curvature tensor r of m is given by r(x, y)z = c − 3 4 { g(y, z)x − g(x, z)y } + c + 1 4 { η(x)η(z)y −η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ + g(φ y, z)φ x − g(φ x, z)φ y − 2g(φ x, y)φ z } , x, y, z ∈ γ (t m). (2.3) a kenmotsu manifold m of constant φ-sectional curvature c will be called kenmotsu space form and denoted m(c). 72 fortuné massamba cubo 13, 3 (2011) let (m, g) be a (2n + 1)-dimensional semi-riemannian manifold with index s, 0 < s < 2n + 1 and let (m, g) be a hypersurface of m, with g = g|m. m is a lightlike hypersurface of m if g is of constant rank 2n − 1 and the normal bundle tm⊥, defined as tm ⊥ = ⋃ p∈m { yp ∈ tpm : gp(xp, yp) = 0, ∀ xp ∈ tpm } , (2.4) is a distribution of rank 1 on m [4]. a complementary bundle of tm⊥ in tm is a rank 2n − 1 non-degenerate distribution over m. it is called a screen distribution and is often denoted by s(tm). in general, s(tm) is not canonical (thus it is not unique). a lightlike hypersurface endowed with a specific screen distribution is denoted by the triple (m, g, s(tm)). as tm⊥ lies in the tangent bundle, the following result has an important role in studying the geometry of a lightlike hypersurface. theorem 2.1. [4] let (m, g, s(tm)) be a lightlike hypersurface of (m, g). then, there exists a unique vector bundle n(tm) of rank 1 over m such that for any non-zero section e of tm⊥ on a coordinate neighborhood u ⊂ m, there exists a unique section n of n(tm) on u satisfying g(n, e) = 1 and g(n, n) = g(n, w) = 0, ∀ w ∈ γ (s(tm)|u ). throughout the paper, all manifolds are supposed to be paracompact and smooth. we denote γ (e) the set of smooth sections of the vector bundle e. also by ⊥ and ⊕ we denote the orthogonal and nonorthogonal direct sum of two vector bundles. by theorem 2.1 we may write down the following decompositions tm = s(tm) ⊥ tm⊥, t m = tm ⊕ n(tm) = s(tm) ⊥ (tm⊥ ⊕ n(tm)). (2.5) let ∇ be the levi-civita connection on (m, g), then by using the second decomposition of (2.5) and considering a normalizing pair {e, n} as in theorem 2.1, we have the gauss and weingarten formulae in the form, ∇xy = ∇xy + h(x, y), and ∇xv = −av x + ∇ ⊥ x v, (2.6) for any x, y ∈ γ (tm|u ), v ∈ γ (n(tm)), where ∇xy, av x ∈ γ (tm) and h(x, y), ∇ ⊥ x v ∈ γ (n(tm)). ∇ is an induced symmetric linear connection on m, ∇⊥ is a linear connection on the vector bundle n(tm), h is a γ (n(tm))-valued symmetric bilinear form and av is the shape operator of m concerning v. equivalently, consider a normalizing pair {e, n} as in theorem 2.1. then (2.6) takes the form, ∇xy = ∇xy + b(x, y) n and ∇xn = −anx + τ(x)n, (2.7) for any x, y ∈ γ (tm|u ), where b, an, τ and ∇ are called the local second fundamental form, the local shape operator, the transversal differential 1-form and the induced linear torsion free cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 73 connection, respectively, on tm|u . it is important to mention that b is independent of the choice of screen distribution, in fact, from (2.7), we obtain b(x, y) = g(∇xy, e) and τ(x) = g(∇ ⊥ x n, e). let p be the projection morphism of tm on s(tm) with respect to the orthogonal decomposition of tm. we have, ∇xpy = ∇ ∗ xpy + c(x, py)e, and ∇xe = −a ∗ ex − τ(x)e, (2.8) where ∇∗xpy and a ∗ ex belong to γ (s(tm)). c, a ∗ e and ∇ ∗ are called the local second fundamental form, the local shape operator and the induced connection on s(tm). the induced linear connection ∇ is not a metric connection and we have, for any x, y ∈ γ (tm|u ), (∇xg)(y, z) = b(x, y)θ(z) + b(x, z)θ(y), (2.9) where θ is a differential 1-form locally defined on m by θ(·) := g(n, ·). also, we have, g(a∗ex, py) = b(x, py), g(a∗ex, n) = 0, b(x, e) = 0. using (2.7), the curvature tensor fields r and r of m and m, respectively, are related as r(x, y)z = r(x, y)z + b(x, z)any − b(y, z)anx + {(∇xb)(y, z) − (∇y b)(x, z) + τ(x)b(y, z) − τ(y)b(x, z)} n, (2.10) where (∇xb)(y, z) = x.b(y, z) − b(∇xy, z) − b(y, ∇xz). (2.11) 3 lightlike hypersurfaces of indefinite kenmotsu manifolds let (m, φ, ξ, η, g) be an indefinite kenmotsu manifold and (m, g) be a lightlike hypersurface of (m, g), tangent to the structure vector field ξ (ξ ∈ tm). if e is a local section of tm⊥, then g(φe, e) = 0, and φe is tangent to m. thus φ(tm⊥) is a distribution on m of rank 1 such that φ(tm⊥) ∩ tm⊥ = {0} . this enables us to choose a screen distribution s(tm) such that it contains φ(tm⊥) as a vector subbundle. if we consider a local section n of n(tm), since g(φ n, e) = −g(n, φ e) = 0, φ e belongs to s(tm). since g(φ n, n) = 0, φ n ∈ γ (s(tm)). from (2.1), we have g(φ n, φe) = 1. therefore, φ(tm⊥) ⊕ φ(n(tm)) is a non-degenerate vector subbundle of s(tm) of rank 2. if m is tangent to the structure vector field ξ, then, we may choose s(tm) so that ξ belongs to s(tm). using this and g(φe, ξ) = g(φn, ξ) = 0, there exists a non-degenerate distribution d0 of rank 2n − 4 on m such that s(tm) = { φ(tm⊥) ⊕ φ(n(tm)) } ⊥ d0 ⊥< ξ >, (3.1) where 〈ξ〉 is the distribution spanned by ξ, that is, 〈ξ〉 = span{ξ}. it is easy to check that the distribution d0 is invariant under φ, i.e. φ(d0) = d0. example 3.1. we consider the 7-dimensional manifold m 7 = { x ∈ r7 : x7 6= 0 } , where x = (x1, x2, ..., x7) are the standard coordinates in r 7. the vector fields e1 = x7 ∂ ∂x1 , e2 = x7 ∂ ∂x2 , e3 = x7 ∂ ∂x3 , e4 = x7 ∂ ∂x4 , e5 = −x7 ∂ ∂x5 , e6 = −x7 ∂ ∂x6 , e7 = −x7 ∂ ∂x7 are linearly independent at each 74 fortuné massamba cubo 13, 3 (2011) point of m 7 . let g be the semi-riemannian metric defined by g(ei, ej) = 0, ∀ i 6= j, i, j = 1, 2, ..., 7 and g(ek, ek) = 1, ∀ k = 1, 2, 3, 4, 7, g(em, em) = −1, ∀ m = 5, 6. let η be the 1-form defined by η(·) = g(·, e7). let φ be the (1, 1) tensor field defined by φe1 = −e2, φe2 = e1, φe3 = −e4, φe4 = e3, φe5 = −e6, φe6 = e5, φe7 = 0. then, using the linearity of φ and g, we have φ 2 x = −x + η(x)e7, g(φ x, φ y) = g(x, y) − η(x)η(y), for any x, y ∈ γ (t m 7 ). thus, for e7 = ξ, (φ, ξ, η, g) defines an almost contact metric structure on m 7 . let ∇ be the levicivita connection with respect to the metric g. then, we have [ei, e7] = ei, ∀ i = 1, 2, ..., 6 and [ei, ej] = 0, ∀ i 6= j, i, j = 1, 2, ..., 6. the metric connection ∇ of the metric g is given by 2g(∇ x y, z) = x(g(y, z)) + y(g(z, x)) − z(g(x, y)) − g(x, [y, z]) −g(y, [x, z]) + g(z, [x, y]), which is known as koszul’s formula. using this formula, the non-vanishing covariant derivatives are given by ∇e1 e1 = −e7, ∇e2 e2 = −e7, ∇e3 e3 = −e7, ∇e4 e4 = −e7, ∇e5 e5 = e7, ∇e6 e6 = e7, ∇e1 e7 = e1, ∇e2 e7 = e2, ∇e3 e7 = e3, ∇e4 e7 = e4, ∇e5 e7 = e5, ∇e6 e7 = e6. from these relations, it follows that the manifold m 7 satisfies ∇ x ξ = x − η(x)ξ. hence, m 7 is indefinite kenmotsu manifold. we now define a hypersurface m of (m 7 , φ, ξ, η, g) as m = {x ∈ m 7 : x5 = x2}. thus, the tangent space tm is spanned by {ui}1≤i≤6, where u1 = e1, u2 = e2 − e5, u3 = e3, u4 = e4, u5 = e6, u6 = ξ and the 1-dimensional distribution tm ⊥ of rank 1 is spanned by e, where e = e2 − e5. it follows that tm ⊥ ⊂ tm. then m is a 6-dimensional lightlike hypersurface of m 7 . also, the transversal bundle n(tm) is spanned by n = 1 2 (e2 + e5) . using the almost contact structure of m 7 and (3.1), d0 is spanned by { f, φf } , where f = u3, φf = −u4 and the distributions 〈ξ〉, φ(tm⊥) and φ(n(tm)) are spanned, respectively, by ξ, φe = u1 + u5 and φn = 1 2 (u1 − u5). hence, m is a lightlike hypersurface of m 7 . moreover, from (2.5) and (3.1) we obtain the decompositions tm = { φ(tm⊥) ⊕ φ(n(tm)) } ⊥ d0 ⊥< ξ >⊥ tm ⊥ , (3.2) t m = { φ(tm⊥) ⊕ φ(n(tm)) } ⊥ d0 ⊥< ξ >⊥ (tm ⊥ ⊕ n(tm)). (3.3) now, we consider the distributions on m, d := tm⊥ ⊥ φ(tm⊥) ⊥ d0, d ′ := φ(n(tm)). then d is invariant under φ and tm = d ⊕ d′ ⊥ 〈ξ〉. (3.4) let us consider the local lightlike vector fields u := − φ n, v := − φ e. then, from (3.4), any x ∈ γ (tm) is written as x = rx + qx + η(x)ξ, qx = u(x) u, where r and q are the projection morphisms of tm into d and d′, respectively, and u is a differential 1-form locally defined on m by u(·) := g(v, ·). applying φ to x and using (2.1), one obtains φ x = φ x + u(x) n, where φ is a tensor field of type (1, 1) defined on m by φ x := φ rx. also, we obtain φ 2 x = − x + η(x)ξ + u(x) u, (3.5) ∇xξ = x − η(x)ξ, ∀ x ∈ γ (tm). (3.6) cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 75 for the sake of future use, we have the following identities: for any x, y ∈ γ (tm), b(x, ξ) = 0, (3.7) c(x, ξ) = θ(x), (3.8) b(x, u) = c(x, v) (3.9) (∇xu)y = −b(x, φ y) − u(y)τ(x) − η(y)u(x), (3.10) (∇xφ)y = g(φx, y)ξ − η(y)φx − b(x, y)u + u(y)anx. (3.11) proposition 3.1. let m be a lightlike hypersurface of an indefinite kenmotsu manifold m with ξ ∈ tm. the lie derivative of g with respect to the vector field v is given by, for any x, y ∈ γ (tm), (lv g)(x, y) = x(u(y)) + y(u(x)) + u([x, y]) − 2u(∇xy). (3.12) proof: the proof follows by direct calculation. the relation (3.12) can be rewritten as, for any x, y ∈ γ (tm), (lv g)(x, y) = (∇xu)y + (∇y u)x. (3.13) as the geometry of a lightlike hypersurface depends on the chosen screen distribution, it is important to investigate the relationship between geometrical objects induced by two screen distributions. we ask the following question: is the lie derivative lv (3.12) independent of the choice of a screen distribution s(tm)? the answer is negative because of the differential 1-form τ which appears in the relation (3.10) and which is not unique. indeed, we prove the following with respect to a change in s(tm). note that the 1-dimensional distribution tm⊥ is independent of the choice of a screen distribution and hence so is also v := −φe (e ∈ tm⊥). suppose a screen s(tm) changes to another screen s(tm)′. following are the transformation equations due to this change (see details in [4], page 87). w ′ i = 2n−1∑ j=1 w j i (wj − ǫjcje), n ′ = n − 1 2 { 2n−1∑ i=1 ǫi(ci) 2 }e + w, τ ′(x) = τ(x) + b(x, w), ∇′xy = ∇xy + b(x, y){ 1 2 ( 2n−1∑ i=1 ǫi(ci) 2)e − w}, (3.14) where w = ∑2n−1 i=1 ciwi, {wi} and {w ′ i } are the local orthonormal bases of s(tm) and s(tm) ′ with respective transversal sections n and n′ for the same null section e. here ci and w j i are smooth functions on u and {ǫ1, ..., ǫ2n−1} is the signature of the basis {w1, ..., w2n−1}. the lie 76 fortuné massamba cubo 13, 3 (2011) derivatives lv and l ′ v of the screen distributions s(tm) and s(tm) ′, respectively, are related through the relation [12]: (l′v g)(x, y) = (∇ ′ xu)y + (∇ ′ y u)x = −b(x, φ y) − u(y)τ′(x) − η(y)u(x) − b(y, φ x) −u(x)τ′(y) − η(x)u(y) = −b(x, φ y) − u(y) (τ(x) + b(x, w)) − η(y)u(x) −b(y, φ x) − u(x) (τ(y) + b(y, w)) − η(x)u(y) = (lv g)(x, y) − u(x)b(y, w) − u(y)b(x, w). (3.15) the lie derivative lv is unique, that is, lv is independent of s(tm), if and only if, the second fundamental form h (or equivalently b) of m vanishes identically on m. if a (2n + 1)-dimensional kenmotsu manifold m has a constant φ-sectional curvature c, then the ricci tensor ric and the scalar curvature r are given by [9] ric = 1 2 (n(c − 3) + c + 1) g − 1 2 (n + 1)(c + 1)η ⊗ η. (3.16) this means that m is η-einstein. since m is kenmotsu and η-einstein, by corollary 9 in [9], m is an einstein one and consequently, c + 1 = 0, that is, c = −1. so, the ricci tensor (3.16) becomes ric = −2ng and the scalar curvature is given by r = −2n(2n + 1). thus, if a kenmotsu manifold m is a space form, then it is einstein and c = −1. let m be a lightlike hypersurface of an indefinite kenmotsu space form m(c) with ξ ∈ tm. using (3.7) and the fact that b(., e) = 0, the local second fundamental form b of m can be written as, for any x, y ∈ γ (tm), b(x, y) = 2n−4∑ i=1 b(x, fi) g(fi, fi) g(y, fi) + v(y)u(a ∗ ex) + u(y)v(a ∗ ex), (3.17) where {fi}1≤i≤2n−4 is an orthogonal basis of d0 and g(fi, fi) 6= 0. this means that m is not totally geodesic in general. so, only some privileged conditions on distributions could allow m to be totally geodesic. next we study some classes of lightlike hypersurfaces m of m(c), tangent to the structure vector field ξ, which are totally geodesic. let us consider the pair {e, n} on u ⊂ m (see theorem 2.1) and by using (2.10), we obtain (∇xb)(y, z) − (∇y b)(x, z) = τ(y)b(x, z) − τ(x)b(y, z). (3.18) theorem 3.1. let m be a lightlike hypersurface of an indefinite kenmotsu space form m(c), with ξ ∈ tm. then, the lie derivative of the second fundamental form b with respect to ξ is given by (lξb)(x, y) = (1 − τ(ξ))b(x, y), ∀ x, y ∈ γ (tm). (3.19) cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 77 moreover, if τ(ξ) 6= 1, then ξ is a killing vector field with respect to the second fundamental form b if and only if m is totally geodesic. proof: using (2.11), we obtain (∇ξb)(x, y) = (lξb)(x, y) − 2b(x, y). (3.20) likewise, using again (2.11), we have (∇xb)(ξ, y) = −b(x, y). (3.21) subtracting (3.20) and (3.21), we obtain (∇ξb)(x, y) − (∇xb)(ξ, y) = (lξb)(x, y) − b(x, y). (3.22) from (3.18) and after calculations, the left hand side of (3.22) becomes (∇ξb)(x, y) − (∇xb)(ξ, y) = −τ(ξ)b(x, y). (3.23) the expressions (3.22) and (3.23) implies (lξb)(x, y) = (1 − τ(ξ))b(x, y). the last assertion is obvious by definitions of killing distribution and totally geodesic submanifold. as an example to the last part of the theorem 3.1, we have a lightlike hypersurface of an indefinite kenmotsu space form, tangent to the structure vector field ξ, with parallel vector field u or v. in fact, when the vector field u or v is parallel, the differential 1-form τ vanishes on m and consequently, the equivalence of the theorem 3.1 holds. the second fundamental form h = b ⊗ n of m is said to be parallel if (∇xh)(y, z) = 0, ∀ x, y, z ∈ γ (tm). that is, (∇xb)(y, z) = −τ(x)b(y, z). (3.24) theorem 3.2. let m be a lightlike hypersurface of an indefinite kenmotsu space form m(c) with ξ ∈ tm. if the second fundamental form h of m is parallel, then m is totally geodesic. proof: suppose that the second fundamental form h of m is parallel. using (3.24), we obtain (∇ξb)(x, y) = −τ(ξ)b(x, y). (3.25) from (2.11) and using (3.19), the left hand side of (3.25) becomes (∇ξb)(x, y) = (lξb)(x, y) − 2b(x, y) = −(1 + τ(ξ))b(x, y). (3.26) from the expressions (3.25) and (3.26) we complete the proof. this means that any parallel lightlike hypersurface m of an indefinite kenmotsu manifold m admits a metric connection. 78 fortuné massamba cubo 13, 3 (2011) the covariant derivative of the second fundamental form h depends on ∇, n and τ which depend on the choice of the screen vector bundle. using equations (3.14), the covariant derivatives ∇ of h = b⊗n and ∇′ of h′ = b⊗n′ in the screen distributions s(tm) and s(tm)′, respectively, are related as follows: for any x, y, z ∈ γ (tm), g((∇′xh ′)(y, z), e) = g((∇xh)(y, z), e) + l(x,y)z, with l(x,y)z = b(x, y)b(z, w) + b(x, z)b(y, w) + b(y, z)b(x, w). it is easy to check that the parallelism of h is independent of the screen distribution s(tm) (∇′h′ ≡ ∇h) if and only the second fundamental form b of m vanishes identically on m. we note that the theorem 3.2 arises when the local second fundamental form b of m is also parallel. therefore, the theorem 3.2 generates some lightlike geometric aspects on any parallel lightlike hypersurface of an indefinite kenmotsu manifold by using the theorem 2.2 in [4]. from (2.3) and (2.10), a direct calculation shows that (∇xc)(y, pz) − (∇y c)(x, pz) + τ(y)c(x, pz) − τ(x)c(y, pz) = g(x, pz)θ(y) − g(y, pz)θ(x). (3.27) lemma 3.1. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm. then, the covariant derivative of v and the lie derivative of g with respect to the vector field u are given, respectively, by, for any x, y ∈ γ (tm), (∇xv)y = −c(x, φy) − v(x)η(y) + τ(x)v(y), (3.28) (lug)(x, y) = x(v(y)) + y(v(x)) + v([x, y]) − 2v(∇xy), (3.29) where v(·) := g(u, ·). proof: the proof of (3.28) and (3.29) follows from direct calculations. the lie derivative (3.29) can be written in terms of the second fundamental form c of s(tm) using the relation v(∇xy) = c(x, φy) + η(y)v(x), ∀ x, y ∈ γ (tm). (3.30) example 3.2. let m be a hypersurface of m 7 defined in the example 3.1. the tangent space tm is spanned by {ui}1≤i≤6, where u1 = e1, u2 = e2 − e5, u3 = e3, u4 = e4, u5 = e6, u6 = ξ and the 1-dimensional distribution tm⊥ of rank 1 is spanned by e, where e = e2 − e5. also, the transversal bundle n(tm) is spanned by n = 1 2 (e2 + e5) . it follows that tm ⊥ ⊂ tm. then m is a 6-dimensional lightlike hypersurface of m 7 having a local quasi-orthogonal field of frames {u1, u2 = e, u3, u4, u5, u6 = ξ, n} along m. denote by ∇ the levi-civita connection on m 7 . then, by straightforward calculations, we obtain ∇xn = 0, ∀ x ∈ γ (tm). using these equations above, the differential 1-form τ vanishes i.e. τ(x) = 0, for any x ∈ γ (tm). so, from the gauss and weingarten formulae we have anx = 0, a ∗ ex = 0 and ∇xe = 0, ∀ x ∈ γ (tm). therefore, by cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 79 theorem 2.2 and proposition 2.7 in [4] pages 88-89, the lightlike hypersurface m of m 7 is totally geodesic and its distribution is parallel. the non-vanishing components of the lie derivatives (3.12) and (3.29) are given by lv g(u1, ξ) = lv g(ξ, u1) = 1, lv g(u5, ξ) = lv g(ξ, u5) = −1, lv g(u, ξ) = lv g(ξ, u) = −1, lug(v, ξ) = lug(ξ, v) = −1, lug(u1, ξ) = lug(ξ, u1) = 1 2 , lug(u5, ξ) = lug(ξ, u5) = − 1 2 . 4 lightlike geometry of leaves in indefinite kenmotsu manifolds let m be a lightlike hypersurface of an indefinite kenmotsu space form m(c) with ξ ∈ tm. from the differential geometry of lightlike hypersurfaces, we recall the following desirable property for lightlike geometry. it is known that lightlike submanifolds whose screen distribution is integrable have interesting properties. now, we study the geometry of leaves of integrable distributions with specific attention to leaves of screen distribution s(tm), the distributions d, d0, d0 ⊥ 〈ξ〉 and{ φ(tm⊥) ⊕ φ(n(tm)) } ⊥ d0. by theorem 2.3 in [4] page 89, the screen distribution s(tm) of m is integrable if and only if the second fundamental form of s(tm) is symmetric on γ (s(tm)). however, for any x, y ∈ γ (d ⊥ 〈ξ〉), u([x, y]) = b(x, φy) − b(φx, y). so, it is very easy to see that the distribution d ⊥ 〈ξ〉 is integrable if and only if b(x, φy) = b(φx, y). theorem 4.1. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu space form m(c) with ξ ∈ tm such that the distribution d ⊥ 〈ξ〉 is integrable. then, m is d ⊥ 〈ξ〉-totally geodesic if and only if φ(tm⊥) is a d ⊥ 〈ξ〉-killing distribution. proof: since d ⊥ 〈ξ〉 is integrable, using (3.10) and (3.12), one obtains, (lv g)(x, y) = −b(x, φy) − b(φx, y) = −2b(x, φy), x, y ∈ γ (d ⊥ 〈ξ〉). using (3.7) and the fact that φ(d ⊥ 〈ξ〉) = d, we complete the proof. note that the theorem 4.1 also holds when the distribution d ⊥ 〈ξ〉 is replaced by d. example 4.1. consider the lightlike hypersurface m of m 7 defined in the example 3.2. since m is totally geodesic, so it is obviously d ⊥ 〈ξ〉-totally geodesic. since the only nonvanishing brackets on the distribution d ⊥ 〈ξ〉 are [v, ξ] = v, [e, ξ] = e, [f, ξ] = f and [φf, ξ] = φf, it is easy to check that the distribution d ⊥ 〈ξ〉 is integrable and (lv g)(x, y) = −2b(x, φy) = 0, x, y ∈ γ (d ⊥ 〈ξ〉), that is, φ(tm⊥) is a d ⊥ 〈ξ〉-killing distribution. proposition 4.1. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu space form m(c) with ξ ∈ tm. if the screen distribution s(tm) is integrable, then, (lξc)(x, py) = τ(ξ)c(x, py), x, y ∈ γ (tm). (4.1) 80 fortuné massamba cubo 13, 3 (2011) proof: if the screen distribution s(tm) of a lightlike hypersurface m is integrable, then, from (3.27) and using (3.8), we have, for any x, y ∈ γ (tm), (∇ξc)(x, py) − (∇xc)(ξ, py) = η(py)θ(x) + τ(ξ)c(x, py). (4.2) on the other hand, using (3.8), we have (∇ξc)(x, py) = ξ(c(x, py)) − c(∇ξx, py) − c(x, ∇ξ(py)) = (lξc)(x, py) − 2c(x, py) + η(py)θ(x), (4.3) and (∇xc)(ξ, py) = x(c(ξ, py)) − c(∇xξ, py) − c(ξ, ∇xpy) = −2c(x, py). (4.4) putting (4.3) and (4.4) together in (4.2), we obtain (4.1). let us assume that the screen distribution s(tm) of m is integrable and let m′ be a leaf of s(tm). then, using (2.7) and (2.8), we obtain, for any x, y ∈ γ (tm′), ∇xy = ∇ ∗ xy + c(x, y)e + b(x, y)n = ∇ ′ xy + h ′(x, y), (4.5) where ∇′ and h′ are, respectively, the levi-civita connection and second fundamental form of m′ in m. thus, for any x, y ∈ γ (tm′), h ′(x, y) = c(x, y)e + b(x, y)n. (4.6) in the sequel, we need the following lemma. lemma 4.1. let (m, g, s(tm)) be a screen integrable lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm and m′ be a leaf of s(tm). then, ∇′xξ = x − η(x)ξ, (4.7) ∇′xu = −v(x)ξ − v(anx)e − v(a ∗ ex)n + φ(anx) + τ(x)u, (4.8) ∇′xv = −u(x)ξ − u(anx)e − u(a ∗ ex)n + φ(a ∗ ex) − τ(x)v, (4.9) for any x ∈ γ (tm′). proof: from a straightforward calculation we complete the proof. it is well known that the second fundamental form and the shape operators of a non-degenerate hypersurface (in general, submanifold) are related by means of the metric tensor field. contrary to this, we see from (2.8), in the case of lightlike hypersurfaces, the second fundamental forms on m and their screen distribution s(tm) are related to their respective shape operators an and a∗e. as the shape operator is an information tool in studying the geometry of submanifolds, their studying turns out very important. for instance, in [5] a class of lightlike hypersurfaces whose shape operators are the same as the one of their screen distribution up to a conformal non zero cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 81 smooth factor in f(m) was considered. that work gave a way to generate, under some geometric conditions, an integrable canonical screen (see [5] for more details). next, we study these operators and give their implications in lightlike hypersurface of indefinite kenmotsu manifolds with ξ ∈ tm. proposition 4.2. let (m, g, s(tm)) be a screen integrable lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm and m′ be a leaf of s(tm). then we have (i) the vector field u is parallel with respect to the levi-civita connection ∇′ on m′ if and only if anx = u(anx)u, ∀ x ∈ γ (tm ′), v and τ vanish on m′. (ii) the vector field v is parallel with respect to the levi-civita connection ∇′ on m′ if and only if a ∗ ex = v(a ∗ ex)v, ∀ x ∈ γ (tm ′), u and τ vanishes on m′. proof: (i) suppose u is parallel with respect to the levi-civita connection ∇′ on m′. then, by using (4.8), we have, for any x ∈ γ (tm′), φ(anx) = v(x)ξ + v(anx)e + v(a ∗ ex)n − τ(x)u. (4.10) since φ(anx) = φ(anx) + u(anx)n, by using (3.9), we obtain φ(anx) = v(x)ξ + v(anx)e − τ(x)u. (4.11) apply φ to (4.11) and using (3.5) and the fact that φu = 0, we obtain anx = η(anx)ξ + u(anx)u + v(anx)v = u(anx)u + v(anx)v, (4.12) since θ(x) = 0, for any x ∈ γ (tm′). putting (4.12) in (4.8) and using (3.9), one obtains v(x)ξ − τ(x)u = 0 which is equivalent to v(x) = 0 and τ(x) = 0. since anx ∈ γ (tm ′), then (4.12) is reduced to anx = u(anx)u. the converse is obvious. in the similar way, by using (4.9) the assertion (ii) follows. corollary 4.1. (to proposition 4.2) let (m, g, s(tm)) be a screen integrable lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm and m′ be a leaf of s(tm) such u and v are parallel with respect to the levi-civita connection ∇′ on m′. then, the type number t′(x) of m′ (with x ∈ m′) satisfies t′(x) ≤ 1. 82 fortuné massamba cubo 13, 3 (2011) proof: the proof follows from proposition 4.2. let w be an element of φ(tm⊥) ⊕ φ(n(tm)) which is a non-degenerate vector subbundle of s(tm) of rank 2. then there exist non-zero functions a and b such that k = av + bu. (4.13) it is easy to check that a = v(k) and b = u(k). let κ be a 1-form locally defined by κ(·) = g(k, ·). lemma 4.2. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm. then, the covariant derivative of κ and the lie derivative of g with respect to the vector field k are given, respectively, by (∇xκ)y = −v(k)b(x, φy) − u(k)c(x, φy) − κ(x)η(y), (4.14) (lkg)(x, y) = x(κ(y)) + y(κ(x)) + κ([x, y]) − 2κ(∇xy), (4.15) for any x, y ∈ γ (tm). proof: using (3.10) and (3.28), we obtain, for any x, y ∈ γ (tm), (∇xκ)y = −v(k)b(x, φy) − u(k)c(x, φy) − κ(x)η(y) (4.16) which proves (4.14) and (4.15) follows from a direct calculation. from (3.30), one obtains, for any x, y ∈ γ (tm), κ(∇xy) = v(k)b(x, φy) + u(k)c(x, φy) + κ(x)η(y). (4.17) lemma 4.3. let (m, g, s(tm)) be a screen integrable lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm and m′ be a leaf of s(tm). then, for any, x, y ∈ γ (tm′), κ(∇′xy) = −κ(φh ′(x, φy)) + κ(x)η(y), (4.18) κ([x, y]) = κ(φh′(φx, y) − φh′(x, φy)) + κ(x)η(y) − κ(y)η(x). (4.19) proof: using (4.5) and (4.6), we obtain, for any x, y ∈ γ (tm′), κ(∇′xy) = g(k, ∇ ′ xy) = v(k)u(∇xy) + u(k)v(∇xy) = v(k)b(x, φy) + u(k)c(x, φy) + κ(x)η(y) = −κ(φh′(x, φy)) + κ(x)η(y) and κ([x, y]) = κ(∇′xy) − κ(∇ ′ y x) = −κ(φh ′(x, φy) − φh′(y, φx)) + κ(x)η(y) −κ(y)η(x), which completes the proof. cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 83 theorem 4.2. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm. then, the distribution d0 ⊥ 〈ξ〉 is integrable if and only if c(φx, y) = c(x, φy), b(φx, y) = b(x, φy), (4.20) and c(x, y) = c(y, x), ∀ x, y ∈ γ (d0 ⊥ 〈ξ〉). (4.21) proof: the proof follows from a direct calculation. note that when the distribution d0 is integrable, the relations (4.20) and (4.21) are satisfied and vice versa. theorem 4.3. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm. suppose the distribution d0 ⊥ 〈ξ〉 is integrable. let m ′ be a leaf of d0 ⊥ 〈ξ〉. then (i) if m′ is totally geodesic in m, then m′ is auto-parallel with respect to the levi-civita connection ∇′ in m and φ(tm⊥) ⊕ φ(n(tm)) is a killing distribution on m′. (ii) if m′ is parallel with respect to the levi-civita connection ∇′ in m, then m′ is totally geodesic. proof: (i) writing y ∈ γ (d0 ⊥ 〈ξ〉) as y = ∑2n−4 i=1 g(y,fi) g(fi,fi) fi + η(y)ξ, where g(fi, fi) 6= 0 and {fi}1≤i≤2n−4 an orthogonal basis of d0. so, it is easy to check that, for any x, y ∈ γ (tm ′), h′(x, φy) = ∑2n−4 i=1 g(y,fi) g(fi,fi) h′(x, φfi). if m ′ is totally geodesic, then, for any x, y ∈ γ (d0 ⊥ 〈ξ〉), h′(x, y) = 0. in particular h′(x, φy) = ∑ i g(y,fi) g(fi,fi) h′(x, φfi) = 0. the auto-parallelism of m ′ follows from (4.18). using (4.15), (4.18), (4.19) and the fact that κ(x) = 0, ∀ x ∈ γ (d0 ⊥ 〈ξ〉), we obtain (lkg)(x, y) = 0. so φ(tm ⊥) ⊕ φ(n(tm)) is a killing distribution on m′. (ii) if m′ is parallel with respect to the connection in m, then, for any x, y, z ∈ γ (tm′), (∇′xh ′)(y, z) = 0. so, (∇′xc)(y, z) − c(y, z)τ(x) = 0 and (∇ ′ xb)(y, z) + b(y, z)τ(x) = 0. using (3.7) and (3.19), since d0 ⊥ 〈ξ〉 integrable, for z = ξ, we have, 0 = (∇′ξb)(x, y) + τ(ξ)b(x, y) = −b(x, y). (4.22) also, using (4.1), we obtain, for any x, y ∈ γ (tm′), 0 = (∇′ξc)(x, y) − τ(ξ)c(x, y) = −2c(x, y). (4.23) from (4.22) and (4.23), we get h′(x, y) = 0 which completes the proof. note that, the lie derivative (4.15) can be expressed in functions of lie derivatives (3.12) and (3.29) as, for any x, y ∈ γ (tm), (lkg)(x, y) = x(v(k))u(y) + y(v(k))u(x) + x(u(k))v(y) + y(u(k))v(x) +v(k)(lv g)(x, y) + u(k)(lug)(x, y). (4.24) 84 fortuné massamba cubo 13, 3 (2011) theorem 4.4. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm. suppose the distribution d0 ⊥ 〈ξ〉 is integrable. let m ′ be a leaf of d0 ⊥ 〈ξ〉. then, the following assertions are equivalent: (i) m′ is totally geodesic in m, (ii) a∗ex and anx ∈ γ (φ(tm ⊥) ⊕ φ(n(tm))), ∀ x ∈ γ (tm′), (iii) φ(tm⊥) ⊕ φ(n(tm)) is a killing distribution on m′, (iv) φ(tm⊥) and φ(n(tm)) are killing distribution on m′. proof: the equivalence of (i) and (ii) follows from direct calculations. using the relation (4.24), we obtain the equivalence of (iii) and (iv). next we prove the equivalence of (i) and (iii). using the fact that κ vanishes on m′ and the relation (4.17), and since d0 ⊥ 〈ξ〉 is integrable, (4.15) becomes, for any x, y ∈ γ (tm′), (lkg)(x, y) = −v(k) {b(x, φy) + b(φx, y)} − u(k) {c(x, φy) + c(φx, y)} = −2κ(φh′(x, φy)) (4.25) suppose m′ is totally geodesic in m. then, h′(x, y) = 0. in particular h′(x, φy) = 0, since φ(d0 ⊥ 〈ξ〉) ⊂ d0. therefore (lkg)(x, y) = 0, i.e. φ(tm ⊥) ⊕ φ(n(tm)) is a killing distribution on m′. the converse is obvious by (4.25). now, we deal with the geometry of the mean curvature vector of a leaf of a integrable distribution of a lightlike hypersurface m of an indefinite kenmotsu space form m(c). first of all, a submanifold m is said to be totally umbilical lightlike hypersurface of a semi-riemannian manifold m if the local second fundemental form b of m satisfies b(x, y) = ρg(x, y), ∀x, y ∈ γ (tm) (4.26) where ρ is a smooth function on u ⊂ m. if m is totally umbilical lightlike hypersurface of a semi-riemannian manifold m, then, we have b(x, y) = ρg(x, y), for any x, y ∈ γ (tm), which implies, by using (3.7), that 0 = b(ξ, ξ) = ρ. hence m is totally geodesic. it follows from this that a kenmotsu m(c) does not admit any non-totally geodesic, totally umbilical lightlike hypersurface. from this point of view, bejancu [1] considered the concept of totally contact umbilical semi-invariant submanifolds. the notion of totally contact umbilical submanifolds was first defined by kon [10]. it is now important to investigate the parallelism of the nonzero mean curvature vector by regarding the effect of the totally contact umbilical condition on the geometry of lightlike submanifolds in kenmotsu manifolds case. as it was done in case of lightlike hypersurfaces of indefinite sasakian manifolds [15], the terminology of extrinsic sphere [3] will be used in case of totally contact geodesic submanifolds. we say that a totally contact umbilical submanifold is an extrinsic cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 85 sphere when it has parallel non zero mean curvature vector [3]. in [15], the author showed that if m is a totally contact umbilical lightlike hypersurface of m(c) with ξ ∈ tm, that is, the second fundamental form h of m satisfies h(x, y) = {g(x, y) − η(x)η(y)} h + η(x)h(y, ξ) + η(y)h(x, ξ), (4.27) where h = λ n normal vector field and λ is a smooth function on u ⊂ m, then λ satisfies the partial differential equations [16] e(λ) + λτ(e) − λ2 = 0, ξ(λ) + λ(τ(ξ) + 1) = 0, (4.28) and px(λ) + λτ(px) = 0, px 6= ξ, ∀ x ∈ γ (tm). (4.29) some of these equations are similar to those of the generic submanifold of indefinite sasakian manifolds case given in [13]. from the equations (4.28) and (4.29), the geometry of the mean curvature vector h of m is discussed. from (4.28) and (4.29), we have ∇⊥e h = g(h, e) 2n, ∇⊥ξ h = −g(h, e)n and ∇ ⊥ pxh = 0, px 6= ξ, ∀x ∈ γ (tm). this means that h is not parallel on m. now, we prove that there exists a distribution which is a subset of the screen distribution s(tm) in which, under the integrability condition, any totally contact umbilical leaf has a parallel nonzero mean curvature. note that any totally contact umbilical leaf of an integrable screen distribution of a lightlike hypersurface of an indefinite kenmotsu space form cannot be an extrinsic sphere [15]. let us consider the following distribution d̂ = { φ(tm⊥) ⊕ φ(n(tm)) } ⊥ d0 (4.30) so that the tangent space of m is written tm = d̂ ⊥ 〈ξ〉 ⊥ tm⊥. (4.31) now, referring to the decomposition (4.31), for any x ∈ γ (tm), y ∈ γ (d̂), we have ∇xy = ∇̂xy + ĥ(x, y), (4.32) where ∇̂ is a linear connection on the bundle d̂ and ĥ : γ (tm) × γ (d̂) −→ γ (〈ξ〉 ⊥ tm⊥) is f(m)-bilinear. let u ⊂ m be a coordinate neighbourhood as fixed in theorem 2.1. then, using (4.31), (4.32) can be locally rewritten in the following way: ∇xy = ∇̂xy + g(∇xy, ξ)ξ + g(∇xy, n)e = ∇̂xy − g(x, y)ξ + c(x, y)e, (4.33) for any x ∈ γ (tm), y ∈ γ (d̂|u ) and the local expression of ĥ is ĥ(x, y) = −g(x, y)ξ + c(x, y)e. (4.34) 86 fortuné massamba cubo 13, 3 (2011) lemma 4.4. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu manifold (m, g) with ξ ∈ tm. then d̂ is integrable if and only if c(x, y) = c(y, x), ∀ x, y ∈ γ (d̂). proof: the proof follows by direct calculation, using (4.33). theorem 4.5. let (m, g, s(tm)) be a lightlike hypersurface of an indefinite kenmotsu space form (m(c), g) with ξ ∈ tm such that the distribution d̂ is integrable. suppose any leaf m′ of d̂ is totally contact umbilical immersed in m as non-degenerate submanifold. then, m′ is an extrinsic sphere. proof: by combining the first equations of (2.7) and (2.8), we obtain ∇xy = ∇̂xy − g(x, y)ξ + c(x, y)e + b(x, y)n = ∇̂′xy + ĥ ′(x, y), ∀ x, y ∈ γ (tm′), (4.35) where ∇̂′ and ĥ′ are the levi-civita connection and second fundamental form of m′ in m. denote by h′ the mean curvature vector of m′. as n(tm) ⊕ tm⊥ is the normal bundle of m′, there exist smooth functions λ and ρ such that h′ = λe + ρn. since m′ is totally contact umbilical immersed in m we have ĥ ′(x, y) = (g(x, y) − η(x)η(y)) h′ + η(x)ĥ′(y, ξ) + η(y)ĥ′(x, ξ). (4.36) since ĥ′(x, ξ) = 0, for any x ∈ γ (tm′), from (4.35) we obtain ∇xy = ∇̂ ′ xy + (g(x, y) − η(x)η(y)) h ′ (4.37) which implies ∇x∇y z = ∇̂ ′ x∇̂ ′ y z + { g(x, ∇̂′y z) − η(x)η(∇̂ ′ y z) } h ′ + { g(∇̂′xy, z) + g(y, ∇̂ ′ xz) − η(z)g(x, y) + 2η(x)η(y)η(z) − η(z)η(∇̂′xy) − η(y)g(x, z) − η(y)η(∇̂ ′ xz) } h ′ + {g(y, z) − η(y)η(z)} ∇xh ′ . (4.38) since d̂ is integrable, θ([x, y]) = 0, for any x, y ∈ γ (tm′) and we have ∇[x,y]z = ∇̂ ′ [x,y]z + {g([x, y], z) − η([x, y])η(z)} h ′ . (4.39) from (4.38), (4.39) and (4.7)-(4.9), after calculations, we obtain r(x, y)z = r̂′(x, y)z + {g(y, z)η(x) − g(x, z)η(y)} h′ + {g(y, z) − η(y)η(z)} ∇xh ′ − {g(x, z) − η(x)η(z)} ∇y h ′ , (4.40) cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 87 where r̂′ is the curvature tensor field of m′. consequently, g(r(x, y)z, e) = {g(y, z)η(x) − g(x, z)η(y)} g(h′, e) + {g(y, z) − η(y)η(z)} g(∇xh ′ , e) − {g(x, z) − η(x)η(z)} g(∇y h ′ , e), (4.41) g(r(x, y)z, n) = {g(y, z)η(x) − g(x, z)η(y)} g(h′, n) + {g(y, z) − η(y)η(z)} g(∇xh ′ , n) − {g(x, z) − η(x)η(z)} g(∇y h ′ , n). (4.42) from (4.41) and using (2.3), we obtain 0 = {g(y, z)η(x) − g(x, z)η(y)} g(h′, e) + {g(y, z) − η(y)η(z)} g(∇xh ′ , e) − {g(x, z) − η(x)η(z)} g(∇y h ′ , e). (4.43) now, since x, y, z ∈ γ (tm′), the relation (4.43) is reduced as have g(∇xh ′ , e)y = g(∇y h ′ , e)x. (4.44) likewise, from (4.42) and (2.3), we have g(∇xh ′ , n)y = g(∇y h ′ , n)x. (4.45) now suppose that there exists a vector field x0 on some neighborhood of m ′ such that g(∇x0 h ′, e) 6= 0 and g(∇x0 h ′, n) 6= 0 at some point p in the neighborhood. from (4.44) and (4.45) it follows that all vectors of the fibre tm′ are collinear with x0|p. this contradicts dim tm ′ > 1. this implies g(∇xh ′, e) = 0 and g(∇xh ′, n) = 0, ∀ x ∈ γ (tm′). these lead, respectively, to g(∇̂′⊥x h ′, e) = 0 and g(∇̂′⊥x h ′, n) = 0, where ∇̂′⊥ is a linear connection on n(tm) ⊕ tm⊥ defined by ∇̂′⊥x e = ∇ ∗⊥ x e = −τ(x)e and ∇̂ ′⊥ x n = ∇ ⊥ x n = τ(x)n, which completes the proof. we discuss here the effect of the change of the screen distribution (3.14) on all the results above. the lie derivative (3.29) depends on c and v which are not unique and their change can be seen as follows. denote by ω the dual 1-form of w = ∑2n−1 i=1 ciwi with respect to the induced metric g of m, that is ω(·) = g(·, w). let p and p ′ be projections of tm on s(tm) and s(tm)′, respectively with respect to the orthogonal decomposition of tm. so, any vector field x on m can be written as x = px + θ(x)e = p ′x + θ′(x)e, where θ′(x) = g(x, n′). then, using (3.14) we have p ′x = px − ω(x)e and c′(x, p ′y) = c′(x, py), ∀ x, y ∈ γ (tm). the relationship between the second fundamental forms c and c′ of the screen distribution s(tm) and s(tm)′, respectively, is given by c ′(x, py) = c(x, py) − 1 2 ω(∇xpy + b(x, y)w). (4.46) all equations above depending only on the local second fundamental form c (making equations non unique) are independent of the screen distribution s(tm) if and only if ω(∇xpy + b(x, y)w) = 0, ∀ x, y ∈ γ (tm). 88 fortuné massamba cubo 13, 3 (2011) the equations (3.29) and (4.15) also are not unique as they depend on c, θ and τ which depend on the choice of a screen vector bundle. the lie derivatives l(·) and l ′ (·) of the screen distributions s(tm) and s(tm)′, respectively, are related through the relations: (l′u ′ g)(x, y) = (lug)(x, y) + 1 2 ω(∇{xpφy} + (b(x, φy) + b(φx, y))w) + 1 2 {η(x)ω(−2φy + u(y)w) + η(y)ω(−2φx + u(x)w)} − 1 2 {τ(x)ω(−2φy + u(y)w) + τ(y)ω(−2φx + u(x)w)} + vx(y) − 1 2 {b(x, w)ω(−2φy + u(y)w) + b(y, w)ω(−2φx + u(x)w)} , (l′k ′ g)(x, y) = (lkg)(x, y) + (v ′(k′) − v(k)) ( (lv g)(x, y) + u(x(τ+η)(y)) ) + (u(k′) − u(k)) ( (lug)(x, y) + v(x(η−τ)(y)) ) + η(x(κ−κ ′)(y)) + 1 2 u(k′)ω(∇{xpφy} + (b(x, φy) + b(φx, y))w), where fx(y) = f(x)b(y, w) + f(y)b(x, w), ∇{xpy} = ∇xpy + ∇y px, v ′(x) = v(x) − 1 2 κ(−2φx + u(x)w) and xf(y) = xf(y) + yf(x), f denoting a 1-form. received: august 2009. revised: august 2010. references [1] a. bejancu, umbilical semi-invariant submanifolds of a sasakian manifold, tensor n. s., 37 (1982), 203-213. [2] d. e. blair, riemannian geometry of contact and symplectic manifolds, progress in mathematics 203. birkhauser boston, inc., boston, ma, 2002. [3] m. dajczer et al., submanifolds and isometric immersions, mathematics lecture series 13. publish or perish, inc., houston, texas, 1990. [4] k. l. duggal and a. bejancu, lightlike submanifolds of semi-riemannian manifolds and applications, mathematics and its applications. kluwer publishers, 1996. [5] k. l. duggal and d. h. jin, null curves and hypersurfaces of semi-riemannian manifolds, world scientific publishing co. pte. ltd. 2007. [6] k. l. duggal and b. sahin, lightlike submanifolds of indefinite sasakian manifolds, internat. j. math. math. sci., vol. 2007, article id 57585, 21 pages. [7] j-b. jun, u. c. de and g. pathak, on kenmotsu manifolds, j. korean math. soc., 42 (3) (2005), 435-445. [8] d. janssens and l. vanhecke, almost contact structures and curvature tensors, kodai math. j., 4 (1981), 1-27. cubo 13, 3 (2011) lightlike geometry of leaves in indefinite . . . 89 [9] k. kenmotsu, a class of almost contact riemannian manifolds, tohoku math. j., 24 (1972), 93-103. [10] m. kon, remarks on anti-invariant submanifold of a sasakian manifold, tensor, n. s., 30 (1976), 239-246. [11] f. massamba, a note on umbilical lightlike hypersurfaces of indefinite sasakian manifolds, int. j. contemp. math. sciences, 2, 32 (2007), 1557-1568. [12] f. massamba, lightlike hypersurfaces of indefinite sasakian manifolds with parallel symmetric bilinear forms, differ. geom. dyn. syst., 10 (2008), 226-234. [13] f. massamba, totally contact umbilical lightlike hypersurfaces of indefinite sasakian manifolds, kodai math. j., 31 (2008), 338-358. [14] f. massamba, screen integrable lightlike hypersurfaces of indefinite sasakian manifolds, mediterr. j. math., 6 (2009), 27-46. [15] f. massamba, on lightlike geometry in indefinite kenmotsu manifolds, to appear in mathematica slovaca. [16] f. massamba, on semi-parallel lightlike hypersurfaces of indefinite kenmotsu manifolds, j. geom, 95 (2009), 73-89. introduction preliminaries lightlike hypersurfaces of indefinite kenmotsu manifolds lightlike geometry of leaves in indefinite kenmotsu manifolds references () cubo a mathematical journal vol.16, no¯ 03, (87–96). october 2014 computing the resolvent of composite operators abdellatif moudafi l.s.i.s, aix marseille université, u.f.r sciences, domaine universitaire de saint-jérôme. avenue escadrille normandie-niemen, 13397 marseille cedex 20, abdellatif.moudafi@lsis.org abstract based in a very recent paper by micchelli et al. [8], we present an algorithmic approach for computing the resolvent of composite operators: the composition of a monotone operator and a continuous linear mapping. the proposed algorithm can be used, for example, for solving problems arising in image processing and traffic equilibrium. furthermore, our algorithm gives an alternative to dykstra-like method for evaluating the resolvent of the sum of two maximal monotone operators. resumen basados en un art́ıculo reciente de micchelli et al. [8], presentamos una manera algoŕıtmica para calcular la resolvente de operadores compuestos: la composición de un operador monótono y una aplicación lineal continua. el algoritmo propuesto puede usarse, por ejemplo, para resolver problemas que aparecen en procesamiento de imágenes y equilibrio de tránsito. además, nuestro algoritmo entrega una alternativa a métodos tipo dykstra para evaluar al resolvente de la suma de dos operadores monótonos maximales. keywords and phrases: 49j53, 65k10, 49m37, 90c25. 2010 ams mathematics subject classification: maximal monotone operators, km-algorithm, yosida regularization, douglas-rachford algorithm, dykstra-like method. 88 abdellatif moudafi cubo 16, 3 (2014) 1 introduction and preliminaries to begin with, let us recall the following concepts which are of common use in the context of convex and nonlinear analysis, see for example takahashi [13] or [1]. throughout, h is a real hilbert space, 〈·, ·〉 denotes the associated scalar product and ‖ · ‖ stands for the corresponding norm. an operator with domain dom(t) and range r(t) is said to be monotone if 〈u − v,x − y〉 ≥ 0 whenever u ∈ t(x),v ∈ t(y). it is said to be maximal monotone if, in addition, its graph, gpht := {(x,y) ∈ h × h : y ∈ t(x)}, is not properly contained in the graph of any other monotone operator. it is well-known that for each x ∈ h and λ > 0 there is a unique z ∈ h such that x ∈ (i + λt)z. (1) the single-valued operator jtλ := (i + λt) −1 is called the resolvent of t of parameter λ. it is a nonexpansive mapping which is everywhere defined and is related to its yosida approximate, namely tλ(x) := x−j t λ (x) λ , by the relation tλ(x) ∈ t(j t λ(x)). (2) the latter is lipschitz continuous with constant 1 λ . recall also that the inverse t−1 of t is the operator defined by x ∈ t−1(y) if and only if t(x). now, let a : h1 → h2 be a continuous linear operator with adjoint a∗, t : h2 → h2 a maximal monotone operator, h1 and h2 being two hilbert spaces. it is easily checked that the composite operator a∗ta is monotone. this kind of operator appears, for example, in partial differential equations in divergence form and in signal and image processing. without further conditions, however, it may fail to be maximal monotone, see for sufficient conditions [12] or [13]. now, let us state the two following key facts: fact 1: a∗ta is maximal monotone, see for instance [12]-corollary 4.4, if 0 ∈ ri(r(a) − domt), where ri stands for the relative interior of a set. the krasnoselski-mann algorithm is a widely used method for solving fixed-point problems. this algorithm generates from an arbitrary initial guess v0 ∈ c (a closed convex set), a sequence (yk) by the recursive formula yk+1 = (1 − αk)yk + αkq(yk), k ∈ in, where (αk) is a sequence in [0,1]. fact 2: it is known that for a nonexpansive mapping q, the krasnoselski-mann’s algorithm converges weakly to a fixed point of q provided that the sequence (αk) is such that ∑ ∞ k=0 αk(1 − αk) = +∞ and the underling space is a hilbert space, see for example [2]. given x ∈ h1, we will focus our attention in this paper on the following problem compute y := ja ∗ ta 1 (x). (3) cubo 16, 3 (2014) computing the resolvent of composite operators 89 the problem (1.3) was studied in the case where t = ∂φ because it arises in many applications in signal processing and image denoising, see for example, [8], [9] and references therein. more precisely, if φ : h2 → ir ∪ {+∞} is a convex and lower semicontinuous, and a : h1 → h2 is a continuous and linear mapping , then the composition φ◦a is also convex and lower semicontinuous. furthermore, by the chain rule of convex analysis, a∗∂φa ⊂ ∂(φ ◦ a), where equality holds whenever the constraint qualification 0 ∈ ri(r(a) − domφ) is satisfied, see for instance [12], (here domφ = {x ∈ h2;φ(x) < +∞} and ∂φ(x) means the subdifferential of φ at x and is equal to the set {u ∈ h2 : φ(y) ≥ φ(x) + 〈u,y − x〉 for all y ∈ h2} as usual). in this context (1.3) reduces to the problem of evaluating the proximity operator, proxφ◦ax := argminu{φ ◦ a(u) + 1 2 ‖u − x‖2}, (4) of φ ◦ a at x. this arises, for instance, in studying the l1/tv image denoising model that involves the total-variation regularization term which is non-differentiable. this causes algorithmic difficulties for its numerical treatment. to overcome the difficulties, the authors formulate in [8] the total-variation as a composition of a convex function (the l1-norm or the l2norm) and the first order difference operator, and then express the solution of the model in terms of the proximity operator of the composition, by identifying the solution as fixed point of a nonlinear mapping expressed in terms of the proximity operator of the l1-norm or the l2-norm, each of which is explicitly given. our aim here is to extend their analysis to evaluate the resolvent of the main operator a∗ta. to begin with, let us recall that m. fukushima [6] proved that if a ◦ a∗ is an isomorphism, then the operator a∗ta is maximal monotone. whereas, in finite dimensional setting, s. robinson [13] observed that we may avoid this condition provided that r(a) ∩ ri(domt) 6= ∅. moreover, ja ∗ ta λ (x) = x − λu with u = (t −1 + λaa∗)−1(ax). however, the above formula is difficult to evaluate in practice since it involves t−1. to overcome this difficulty, m. fukushima [6] proposed an alternative computation of y = ja ∗ ta λ (x): given x ∈ h1 (i) find the unique solution z of 0 ∈ 1 λ (aa∗)−1(z − ax) + tz, (ii) compute u, by u = 1 λ (aa∗)−1(ax − z) (iii) compute y = x − λa∗u. let us remark the potential difficulties in implementing the above algorithm lie in the fact that it involves the inverse operator aa∗. the evaluation of such a mapping is in general expensive. in the sequel we will follow another approach developed in [8] in a convex optimization context. 90 abdellatif moudafi cubo 16, 3 (2014) 2 fixed-point approach under the assumption that we can explicitly compute the resolvent operator of t and by assuming that a∗ta is a maximal monotone operator, our aim is to develop an algorithm for evaluating the resolvent of a∗ta. remember that from (1.1), for each x ∈ h1 there is a unique z := j a ∗ ta 1 x ∈ h1 such that x ∈ (i + a∗ta)z. our main is to provide a constructive method to compute it. from (1.2), we clearly have ja ∗ ta 1 (x) ∈ x − a ∗t(a(ja ∗ ta 1 x). (5) this combined with the fact that y ∈ a∗ta(x) ⇔ x = ja ∗ ta 1 (x + y) enable us to establish a relationship between the resolvent of a∗ta and that of t. to that end, we define the following affine transformation for a fixed x ∈ h1 at y ∈ h2 by fy := ax + (i − λaa∗)y and the operator q := (i − jt1/λ) ◦ f. (6) theorem 2.1. if t is a maximal monotone operator, a a continuous linear operator and λ > 0 then ja ∗ ta 1 x = x − λa ∗y if and only if y is a fixed point of q. proof. according to (2.1), we can write ja ∗ ta 1 x = x − λa ∗y, where y ∈ 1 λ t(aja ∗ ta 1 x). thus y ∈ 1 λ t(a(x − λa∗y)). using, for instance, the fact that y ∈ t(x) ⇔ x = jt1(x + y), we deduce ax − λaa∗y = jt1/λ(ax + (i − λaa ∗)y), that is y is a fixed-point of q. conversely, if y is a fixed-point of q then ax − λaa∗y = jt1/λ(ax + (i − λaa ∗)y), definition of the resolvent yields λy ∈ ta(x − λa∗y), thus λa∗y ∈ a∗ta(x − λa∗y), and by (2.1) we obtain ja ∗ ta 1 x = x − λa ∗y. this completes the proof. cubo 16, 3 (2014) computing the resolvent of composite operators 91 hence, it suffices to find a fixed-point y of q and the resolvent of t is then equal to x−λa∗y. now, by assuming ‖i−λaa∗‖ ≤ 1 and having in mind that i−jt 1/λ is nonexpansive, we easily derive: lemma 2.2. if t is a maximal monotone operator and a a continuous linear mapping and λ > 0 such that ‖i − λaa∗‖ ≤ 1, then the operator q is nonexpansive. to find a fixed point y∞ of the operator q, we can use, for example, the km-algorithm, namely yk+1 = αkyk + (1 − αk)q(yk), (7) and then set ja ∗ ta 1 x = x − λa ∗y∞. using fact 2, we derive the following result. corollary 2.3. under assumptions of lemma 1.2, for any αk ∈ [0,1] satisfying ∑ k αk(1−αk) = +∞, the sequence (xk) generated by (2.3) weakly converges to a fixed-point of q. 3 applications 3.1 image denoising l1/tv models can be cast into the following general form 0 ∈ λs(x) + a∗tax, (8) where s and t are two maximal monotone operator and a a linear continuous mapping. for example, given a noisy image x which was contaminated by impulsive noise, we consider a denoised image of x as a minimizer of the following l1/tv model min{λ‖u − x‖1 + ‖u‖tv}, (9) where ‖ · ‖1 represents the l1−norm, ‖ · ‖tv denotes the total-variation, and λ is the regularization parameter balancing the fidelity term‖u · −x‖1 and the regularization term ‖ · ‖tv. by rewriting ‖u‖tv = φ ◦ a, with φ a convex lower-semicontinuous function and a a real matrix and by using the chain rule, we obtain the following optimality condition of (3.2) 0 ∈ λ∂‖u − x‖1 + a t∂φa(u), which is nothing else than (3.1) with s = ∂‖ · −x‖1 and t = ∂φ. it is well accepted that the l1-norm fidelity term can effectively suppress the effect of outliers that may contaminate a given image, and is therefore particularly suitable for handling non-gaussian additive noise. the l1/tv model (3.1) has many distinctive and desirable features. for the 92 abdellatif moudafi cubo 16, 3 (2014) anisotropic total variation, φ is expressed with ‖ · ‖1 while for the isotropic total variation, φ is expressed with ‖ · ‖2. thus, we can obtain their proximity mappings in closed-forms. indeed, for x ∈ irm and λ > 0 prox1 λ ‖·‖1 (x) = (prox1 λ |·|(x), · · ·,prox1 λ |·|(x)) with prox1 λ |·|(xi) = max(|xi| − 1 λ ,0)sign(xi), while prox1 λ ‖·‖2 (x) = max(‖x‖2 − 1 λ ,0) x ‖x‖2 . since in section 2, we developed a method for computing the resolvent of the operator a∗ta, to solve (3.1), we can make use of any splitting algorithm for finding the zero of the sum of two maximal monotone operators, for instance that of passty which consists in the picard iteration for the composition of the resolvents of λs and a∗ta. but, it is well known that we only have ergodic convergence. the algorithm which is of common use in this type of context is the so-called douglas-rachford (dr) proposed in its initial form by lions and mercier [7] and that takes here the following form { xk = j a ∗ ta γ (yk) (by a loop which uses the method introduced in section 2); yk+1 = yk + αk(j s λγ(2xk − yk) − yk); (10) thanks to the well-known convergence result for dr-algorithm, we derive that the sequence (yk) converges weakly to a fixed-point y∞ of (2ja ∗ ta γ − i) ◦ (2j s λγ −i) and j s λγy ∞ solves (1.3) provided that ∑ k αk(2 − αk) = +∞ and αk < 2. moreover, if dimh < +∞, then the sequence (xk) converges to a solution of (3.1). 3.2 resolvent of a sum of two operators let x ∈ h, let t1 and t2 be two maximal monotone operators from h to h. to compute the resolvent of the sum of t1 and t2 at x, bauschke and combettes [4] proposed the following dykstralike method: and set    x0 = x, p0 = 0 q0 = 0 and { yk = j t2 1 (xk + pk); pk+1 = xk + pk − yk, and { xk+1 = j t1 1 (yk + qk); qk+1 = yk + qk − xk+1. (11) they proved that if x ∈ r(id+t1 +t2), then both the sequences (xk) and (yk) strongly converges to jt1+t2 1 x. our aim is to propose an alternative approach for computing the resolvent of a monotone operator which can be decomposed as a sum of two maximal monotone operators, such that their individual resolvents can be implemented easily. both methods we present will proceed by splitting in the sense that, at each iteration, they employ these resolvents separately. cubo 16, 3 (2014) computing the resolvent of composite operators 93 first, note that if t1 and t2 are set-valued mappings from h to h, their pointwise sum can be expressed in the composite form a∗ta; by defining h = h × h;ax = (x,x) and t(x1,x2) = t1(x1) × t2(x2). indeed, then a∗(y1,y2) = y1 + y2 and so a ∗ta(x) = t1(x) + t2(x). this fact will allow us to give alternative algorithms to compute the resolvent operator of the sum of two maximal monotone operators relying on the fixed-point approach developed in section 2. it is easily seen that finding a fixed point y = (y1,y2) of the operator q, defined by (2.2), amounts to solving the following system { y1 = 1 λ (i − j t1 1 )(x − λy2); y2 = 1 λ (i − j t2 1 )(x − λy1), (12) note that the operator q̃(y1,y2) := { 1 λ (i − j t1 1 )(x − λy2); 1 λ (i − j t2 1 )(x − λy1), is nonexpansive for all λ > 0. thus, we can use the algorithm (yk+11 ,y k+1 2 ) = αk(y k 1,y k 2) + (1 − αk)q̃(y k 1,y k 2), (13) to find a fixed-point (y∞1 ,y ∞ 2 ) and then we deduce the resolvent of the sum of t1 and t2. indeed, we have the following result: proposition 3.1. let t1,t2 be two maximal monotone operators, then for any αk ∈ [0,1] satisfying∑ k αk(1 − αk) = +∞, the sequence (y k 1,y k 2) generated by (3.6) weakly converges to a fixed-point (y∞1 ,y ∞ 2 ). furthermore, we have j t1+t2 1 (x) = x − λa∗(y∞1 ,y ∞ 2 ) = x − λ(y ∞ 1 + y ∞ 2 ). we would like to emphasize that we can also use a von neumann-like alternating algorithm for solving system (3.5). indeed, the latter is equivalent to { λy1 = (i − j t1 1 )(x − (i − j t2 1 )(x − λy1)); λy2 = (i − j t2 1 )(x − λy1). (14) by defining ã(y) = −a(−y) for a given operator a, a simple computation gives jã1 (y) = −j a 1 (−y) and j a(·−x) 1 (y) = x + ja1 (y − x). hence, by setting a := t−1 1 , b := t−1 2 , v1 := λy1 and v2 := λy2, and according to the fact that i − ja1 = j a −1 1 , we finally obtain { v1 = j a 1 ◦ j b̃(·−x) 1 (v1); v2 = j b 1 ◦ j ã(·−x) 1 (v2). (15) 94 abdellatif moudafi cubo 16, 3 (2014) this suggests to consider the following alternating resolvent method: { v01 ∈ h and ṽ k 1 = j b̃(·−x) 1 (vk1),v k+1 1 = ja1 (ṽ k 1); v02 ∈ h and ṽ k 2 = j ã(·−x) 1 (vk2),v k+1 2 = ja1 (ṽ k 2). (16) from [3]-theorem 3.3, we deduce: proposition 3.2. let t1,t2 be two maximal monotone operators, then the sequence (v k 1,v k 2) generated by (3.9) weakly converges to a solution (v∞1 ,v ∞ 2 ) of (3.8) provided that the latter exists. moreover, t he resolvent of the sum of t1 and t2 at x is then given by j t1+t2 1 (x) = x − λa∗(y∞1 ,y ∞ 2 ) = x − λ(y∞1 + y ∞ 2 ). to conclude, it is worth mentioning that in the case of convex optimization, the problem under consideration in this section amounts to evaluating the proximity operator of the sum of two proper, lower semicontinuous convex functions φ,ψ : h → ir ∪ {+∞}, namely: given x, compute y := proxφ+ψ(x). (17) in this context, the resolvent operator is nothing else than the proximity operator and according to the fact that (i − proxφ) = proxφ∗, system (3.5) reduces to { y1 = 1 λ (i − proxφ)(x − λy2) = 1 λ proxφ∗(x − λy2); y2 = 1 λ (i − proxψ)(x − λy1) = 1 λ proxψ∗(x − λy1), (18) where φ∗,ψ∗ stand for the fenchel conjugate of the functions φ and ψ. remember that the fenchel conjugate of a given function f is defined as f∗(x) = supu{〈x,u〉 − f(u)}. finally, we would like to emphasize that we can also consider as an application the traffic equilibrium problem consider in [6]. the advantage of our approach is that it does not require the inversion of the operator aa∗. roughly speaking, see [6], the traffic equilibrium problem can be written as (3.1) with λ = 1, s = ∂δc, namely 0 ∈ a∗tax + ∂δc(x), (19) s = ∂δc the partial differential of the indicator function of a polyhedral convex set, t a cost function (not assumed to be single-valued as in [6], but we suppose that its resolvent operator can be computed efficiently) and a satisfying aa∗ = νi for ν ∈ in. (20) the latter is also satisfied in image restoration problems by the so-called tight frames operators. it is well known that in this case the resolvent operator of ∂δc is exactly the projection onto c (algorithms for computing projection which do not require any particular hypothesis on the input cubo 16, 3 (2014) computing the resolvent of composite operators 95 polyhedral sets are available) and a simple calculation shows that the method developed in section 2 gives ja ∗ ta 1 (x) = x − a ∗tν(ax). the latter formula is in fact valid for any positive real number ν and generalizes a formula by combettes and pesquet [5] obtained in the case where t is the subdifferential of a proper lowersemicontinuous function and was used in many applications in image restoration. more precisely, relying on a qualification condition and by taking into account (1.4), we clearly have proxφ◦ax = x − a ∗ (∂φ)ν(ax) = x − ν −1a∗(i − proxνφ)ax. now, an application of passty’s algorithm gives xk+1 = pc(xk − 1 ν a∗(i − jtν)axk), (21) which is nothing else than an extension of the celebrated cq-algorithm of byrne, see [10]. [10]theorem 3.1 assures: proposition 3.3. let t be a given maximal monotone operator and c a nonempty closed convex set, the sequence f (xk) generated by (3.14) converges to a solution of the traffic equilibrium problem (3.12) if ν > l, with l being the spectral radius of the operator a∗a. while applying the douglas-rachford’s algorithm yields to { xk = yk − 1 ν a∗(i − jtν)ayk; yk+1 = (1 − αk)yk + αkpc(yk − 2 ν a∗(i − jtν)ayk). (22) which looks like a relaxed version of algorithm (3.13). since, we are in a finite dimensional setting, we obtain proposition 3.4. let t be a given maximal monotone operator and c a nonempty closed convex set, , then for any αk ∈]0,2[ satisfying ∑ k αk(2 − αk) = +∞, the sequence (xk) generated by (3.15) converges to a solution of the equilibrium problem (3.12). received: april 2014. accepted: august 2014. references [1] h. attouch, a. moudafi, h. riahi, quantitative stability analysis for maximal monotone operators and semi-groups of contractions, journal nonlinear analysis, theory, methods & applications, vol. 21, (1993), 697-723. [2] h. h. bauschke and p. l. combettes, a weak-to-strong convergence principle for fejérmonotone methods in hilbert spaces, mathematics of operations research, 26, no. 2 (2001), 248-264. 96 abdellatif moudafi cubo 16, 3 (2014) [3] h.h. bauschke, p.l. combettes, and s. reich: the asymptotic behavior of the composition of two resolvents, nonlinear analysis: theory, methods, and applications 60, (2005) 283-301. [4] h.h. bauschke and p.l. combettes: a dykstra-like algorithm for two monotone operators, pacific journal of optimization 4, (2008) 383-391. [5] p. l. combettes and j.-c. pesquet, a douglas-rachford splitting approach to nonsmooth convex variational signal recovery, ieee j. selected topics signal process. 1, (2007) 564574. [6] m. fukushima,the primal douglas-rachford splitting algorithm for a class of monotone operators with application to thetraffic equilibrium problem, mathematical programming, vol.72 (1996) 1-15. [7] p.-l. lions, b. mercier, splitting algorithms for the sum of two nonlinear operators, siam j. numer. anal. 16 (1979) 964-979. [8] ch. a. micchelli, l. chen and y. xu, proximity algorithms for image models:denoising inverse problems (2011) doi:10.1088/0266-5611/27/4/045009. [9] ch. a. micchelli, l. shen, y.xu, x. zeng, proximity algorithms for the l1/tv image denoising model, adv comput math 38 (2013) 401-426. [10] a. moudafi, m. théra, finding the zero for the sum of two maximal monotone operators, journal optimization theory & applications, vol. 94, n2, (1997), 425-448. [11] a. moudafi, split monotone variational inclusions, journal of optimization, theory and applications 150 (2011), p. 275-283. [12] t. pennanen, dualization of generalized equations of maximal monotone type, siam j. optim. 10 (2000) 809-835. [13] s.m. robinson, composition duality and maximal monotonicity, math. programing ser. a 85 (1999) 1-13. [14] w. takahashi, nonlinear functional analysis, yokohama publishers, yokohama, 2000. introduction and preliminaries fixed-point approach applications image denoising resolvent of a sum of two operators cubo a mathematical journal vol.14, no¯ 01, (81–91). march 2012 special recurrent transformation in an npr-finsler space anjali goswami department of mathematics jagannath gupta institute of engineering and technology sitapura, jaipur, india email: dranjaligoswami@rediffmail.com abstract in this paper, an infinitesimal transformation x̄i = xi + �vi (xj), where the vector vi is recurrent has been considered in an nprfinsler space. such transformation is being called special recurrent transformation if the recurrence vector of the nprfinsler space is lie invariant. besides different properties of such transformation, the conditions for such transformation to be curvature collineation and an affine motion have been obtained. resumen en este art́ıculo se considera una transformación infinitesimal x̄i = xi +�vi (xj), donde el vector vi es recurrente, en un espacio nprfinsler. tal transformación se dice transformación recurrente especial si el vector recurrente del espacio nprfinsler es lie invariante. además se han obtenido diferentes propiedades de dicha transformación y las condiciones para que ésta sea una colineación de curvatura y una moción af́ın. keywords and phrases: npr-finsler space, recurrent vector fields, special recurrent transformation, curvature collineation, affine motion. 2010 ams mathematics subject classification: 53b40. 82 anjali goswami cubo 14, 1 (2012) 1 introduction let an n-dimensional finsler space fn be equipped with fundamental metric function f(x k, ẋk), metric tensor gij and berwald connection g i jk. covariant derivative of any tensor with respect to berwald connection is given by [6] bk t i j = ∂kt i j − (∂̇rt i j)g r k h ẋ h + trjg i kr − t i rg r j k (1.1) where ∂k ≡ ∂∂ xk and ∂̇r ≡ ∂ ∂ ẋr . the commutation formulae for the operators bk and ∂̇k are given by ∂̇jbkt i h − bk∂̇jt i h = t r hg i jkr − t i rg r j kh, (1.2) bjbkt i h − bkbjt i h = t r hh i jkr − t i rh r j kh − (∂̇rt i h)h r j k, (1.3) where gijkh = ∂̇hg i jk, (1.4) hijkh = ∂jg i kh + g i hrjg r k + g i rjg r kh − j/k (1.5) and hijk = h i jkhẋ h. (1.6) the symbol -j/k means the subtraction of the earlier terms after interchanging j and k. the tensor gijkh is symmetric in its lower indices and satisfies gijkhẋ h = gijhkẋ h = gihjkẋ h = 0 (1.7) while the berwald curvature tensor hijkh satisfies (a) hijkh = −h i kjh, (b) h i jkh = ∂̇hh i jk. (1.8) the berwald deviation tensor hij is defined by (a)hij = h i jkẋ k, (b) hijk = 1/3∂̇kh i j − j/k. (1.9) pandey[2] proved that the relation between the normal projective curvature tensor nijkh defined by yano [7] and the berwald curvature tensor hijkh is given by nijkh = h i jkh − ẋi n + 1 ∂̇hh r j kr, (1.10) nrj kr = h r j kr. (1.11) the relation between the tensors nijkh and h i jk is given by nijkhẋ h = hijk. (1.12) cubo 14, 1 (2012) special recurrent transformation in an npr-finsler space 83 2 an npr-finsler space an npr-finsler space was defined by p. n. pandey [2] in 1980. it is a finsler space whose normal projective curvature tensor nij k h satisfies bm n i j k h = λm n i j k h, (2.1) where λm is a covariant vector called recurrence vector. this vector is atmost a point function, i.e. independent of the directional arguments. it was observed by p. n. pandey [2] that the tensors hij k and h i j are recurrent in npr-finsler space. thus in an npr-finsler space, we have (a) bm h i j k = λm h i j k, (b) bm h i j = λm h i j. (2.2) however, an npr-finsler space is not necessarily a recurrent finsler space. also, a recurrent finsler sapce is not necessarily an npr-finsler space. in another paper, p.n. pandey [4] established the following identities: λm n i j k h + λj n i k m h + λk n i m j h = 0, (2.3) λm h i j k h + λj h i k m h + λk h i m j h = 0, (2.4) λm h i j k + λj h i k m + λk h i m j = 0, (2.5) he further proved that in such space, the second bianchi identity splits into the following identities: bm h i j k h + bj h i k m h + bk h i m j h = 0, (2.6) hrj k g i m h r + h r k m g i j h r + h r m j g i k h r = 0. (2.7) contracting the indices in (2.2b) and using hii = (n − 1)h, we get bm h = λm h. (2.8) differentiating (2.8) covariantly with respect to xh and taking skew-symmetric part, we have (bh bm − bm bh)h = ah m h (2.9) where ahm = bh λm − bm λh. using (1.3) in (2.9), we have − ∂̇r hh r h m = ah m h, (2.10) which after further covariant differentiation gives − (bk∂̇r h)h r h = (bk ah m)h. (2.11) using the commutation formula (1.2) and the equation (2.10), we get bk ah m = λk ah m (2.12) 84 anjali goswami cubo 14, 1 (2012) provided h is non-vanishing. if we multiply (2.10) with λk and take skew-symmetric part, we find λk ah m + λh am k + λm ak h = 0 (2.13) provided h 6= 0. thus, we find that the recurrence vector λm of an npr-finsler space satisfies (2.12) and (2.13) provided h 6= 0. in view of the commutation formula given by (1.2), we get ∂̇j bm λk − bm ∂̇jλk = −λr g r j m k which due to the fact that the recurrence vector is independent of ẋi, gives ∂̇j bm λk = −λr g r j m k. (2.14) taking skew-symmetric part of (2.14), we get ∂̇j am k = 0. (2.15) now ∂̇j bk ah m − bk ∂̇j ah m = −ar m g r jkh l − ah r g r j k m (2.16) which, in view of (2.12) and (2.15), gives ar m g r j k h + ah r g r j k m = 0. (2.17) 3 a recurrent vector field in an npr-finsler space a vector field vi is called recurrent if it satisfies bk v i = µk v i. (3.1) differentiating (3.1) covariantly with respect to xj and using the commutation formula (1.3), we get hijkh v h = µjk v i (3.2) where µjk = bj µk − bk µj. the tensor µjk may or may not vanish. let us consider the case when µjk 6= 0. from (1.10) and (3.2), we find( nijkh + ẋi n + 1 ∂̇h n r jkr ) vh = µjkv i. (3.3) differentiating (3.3) covariantly with respect to xm, and using (2.1) and (3.1), we have( λm n i jkh + ẋi n + 1 bm∂̇h n r jkr ) vh = vi bmµjk, (3.4) which in view of (1.2), gives λm ( nijkh + ẋi n + 1 ∂̇h n r jkr ) vh + ẋi n + 1 ( nrskrg s hmj + n r jsrg s hmk ) vh = vibmµjk. (3.5) cubo 14, 1 (2012) special recurrent transformation in an npr-finsler space 85 from (3.3) and (3.5), we get ( λm µjk − bm µjk ) vi + ẋi n + 1 vh ( nrskr g s hmj + n r jsr g s hmk ) = 0. (3.6) transvecting (3.6) by yi and using yi ẋ i = f2, we get( λm µjk − bm µjk ) yi v i + f 2 n+1 vh ( nrskr g s hmj + n r jsr g s hmk ) = 0 which implies vh n + 1 ( nrskr g s hmj + n r jsr g s hmk ) = 1 f2 ( bm µjk − λm µjk ) yi v i . (3.7) using (3.7) in (3.6), we get( λm µjk − bm µjk ) vi − lilr v r ( λm µjk − bm µjk ) = 0 (3.8) where li = ẋi/f and lr = yr/f. (3.8) may be rewritten as( λm µjk − bm µjk ) ( vi − li lr v r ) = 0. this implies at least one of the conditions (a) bm µjk = λm µjk, (b) v i = li lr v r. (3.9) suppose that the condition (3.9 b)holds. then the partial differentiation with respect to ẋh gives 0 = (∂̇hl i)lrv r + li(∂̇hlr)v r. (3.10) using ∂̇hl i = 1 f (δih − l ilh) and ∂̇hlr = 1 f (ghr − lhlr) in (3.10),we find 0 = (δih − l ilh)lrv r + li(ghr − lhlr)v r. contracting the indices i and h and using δii = n and l rlr = 1, we get (n − 1)lrv r = 0. this implies lrv r = 0 for n 6= 1. in view of lrvr = 0, (3.9 b) gives vi = 0, a contradiction. therefore (3.9b) can not be true. hence, we have (3.9a). from (2.4) and (3.2), we may deduce λm µjk + λj µkm + λk µmj = 0. (3.11) this leads to: theorem 3.1. in an npr-finsler space admitting a recurrent vector field vi given by (3.1), the tensor µjk either vanishes identically or is recurrent and satisfies the identity (3.11). differentiating (3.1) partially with respect to ẋj and using the commutation formula (1.2), we get gijkr v r = (∂̇j µk)v i. (3.12) transvecting (2.17) by vj ẋm and using (3.12), we get arm v r ẋm ∂̇k µh = 0. (3.13) 86 anjali goswami cubo 14, 1 (2012) this gives at least one of the following conditions: (a) arm v r ẋm = 0, (b) ∂̇k µh = 0. (3.14) if (3.14a) holds, then its partial drivatives with respect to ẋk gives ark v r = 0. (3.15) transvecting (2.13) by vk and using (3.15), we find λk v k ahm = 0. (3.16) since ahm 6= 0, we have λk v k = 0. (3.17) thus we have theorem 3.2. in an npr-fnsler space admitting a recurrent vector field vi characterized by (3.1), we have at least one of the conditions (3.14b) and (3.17). suppose (3.14b) holds, then we have ∂̇jbk µm = −µr g r jkm. (3.18) taking skew-symmetric part of (3.18) with respect to the indices k and m, we get ∂̇jµkm = 0. (3.19) differentiating (3.19) covariantly with respect to xh and using commutation formula exhibitted by (1.2) and the equation (3.9a), we find µrm g r kjh + µkr g r mjh = 0. 4 a special recurrent transformation an infinitesimal transformation x̄i = xi + �vi(xj) (4.1) where vi is a covariant vector field and � is an infinitesimal constant, is called a special recurrent transformation if the vector field vi is recurrent and the transformation does not deform the recurrence vector λm of the npr-finsler space, i.e. if the vector field v i satisfies (3.1) and £λm = 0 (4.2) where £ is the operator of lie differentiation with respect to the infinitesimal transformation (4.1). the necessary and sufficient condition for (4.1) to be an affine motion is given by £gijk = 0. (4.3) cubo 14, 1 (2012) special recurrent transformation in an npr-finsler space 87 since every affine motion is a curvature collination, (4.3) implies £hijkh = 0. (4.4) operating (1.10) by the operator £ and using (4.4), we get £nijkh = − ẋi n + 1 £ ∂̇hh r jkr, (4.5) since the operators £ and ∂̇h are commutative, (4.5) becomes £n i jkh = − ẋi n+1 ∂̇h £ h r jkr which in view of (4.4), gives £nijkh = 0. (4.6) let us consider an npr-finsler space admitting an affine motion. then we have (2.1), (4.3), (4.4) and (4.6). operating (2.1) by the operator £ and using (4.6), we have £bm n i jkh = (£λm)n i jkh. (4.7) in view of the commutation formula £bk t i j − bk£t i j = t r j £g i rk − t i r £g r jk − (∂̇r t i j ) £g r ks ẋ s (4.8) and equations (4.3) and (4.6), the equation (4.7) gives (4.2) for nijkh 6= 0. thus, we obsereve that every affine motion generated by a recurrent vector field in an npr-finsler space is a special recurrent transformation. now, we wish to discuss its converse problem. let us consider a special recurrent transformation (4.1) in an npr-finsler space. this transformation is characterized by (3.1) and (4.2). in view of theorem (3.2), we have at least one of the equations (3.14b) and (3.17). if (3.14b) does not hold, we must have (3.17), i.e. l = λr v r = 0. we shall divide the special recurrent transformations in two classes according as l 6= 0 and l = 0. a special recurrent transformation is called of first kind if l 6= 0 while it is called of second kind if l = 0. let us consider a special recurrent transformation of the first kind. for such transformation l 6= 0. therefore in view of theorem (3.2), the vector field µk must be a point function, i.e. ∂̇j µk = 0. expanding the left hand side of equation (4.2) with the help of the formula £t ij = v r br t i j − t r j br v i + t ir bj v r + (∂̇rt i j ) bs v r ẋs, (4.9) we get vr br λm + lµm = 0. (4.10) also bm l = bm (λr v r) = vr bm λr + lµm . (4.11) using (4.10) in (4.11), we have vr ark + bm l = 0. (4.12) 88 anjali goswami cubo 14, 1 (2012) differentiating (2.3) covariantly with respect to xp and using (2.1), we have (bpλm)n i jkh + (bpλj)n i kmh + (bpλk)n i mjh = 0. (4.13) transvecting (4.13) by vp and using (4.10), we get µmn i jkh + µjn i kmh + µkn i mjh = 0. (4.14) differentiating (2.11) and (2.13) covariantly with respect to xp and then multiplying by vp, we get (vp bp λk)ahm + (v p bp λh)amk + (v p bp λm)akh = 0, and (vp bp λk)µhm + (v p bp λh)µmk + (v p bp λm)µkh = 0, which imply µkahm + µhamk + µmakh = 0 (4.15) and µkµhm + µhµmk + µmµkh = 0 (4.16) since l 6= 0. this proves the following: theorem 4.1. an npr-finsler space admitting a special recurrent transformation admits the identities (4.14), (4.15) and (4.16) provided l 6= 0. the commutation formula for the operators £ and bk in case of the recurrence vector λm is given by £bkλm − bk£λm = −λr£g r mk, which, in view of (4.2), gives £bkλm = −λr£g r mk. (4.17) taking skew-symmetric part of (4.17), we get £amk = 0. (4.18) transvecting (4.14) by ẋh and using (1.12), we get µmh i jk + µjh i km + µkh i mj = 0. (4.19) now £hijk = lh i jk + µh i jkrv r − µrh r jkv i + µjh i rkv r + µkh i jrv r. transvecting (4.19) by vm and using (3.2) in the above equation, we get £hijk = (l + µmv m)hijk + (µµjk − µrh r jk)v i. this shows that £hijk = 0 if l + µmv m = 0 and µµjk − µrh r jk = 0. (4.20) we know that £hijk = 0 is equivalent to £h i jkh = 0. therefore we have: cubo 14, 1 (2012) special recurrent transformation in an npr-finsler space 89 theorem 4.2. a special recurrent transformation of the first kind is a curvature collineation if (4.20) holds. the lie derivative of gijk is given by £gijk = bj bkv i + himjkv m + gijkrbsv rẋs, (4.21) which in the present case is given by £gijk = (bjµk + µjµk)v i + himjkv m , (4.22) for gijkrv r = ∂̇jµkv i = 0. differentiating (2.4) covariantly with respect to xp and transvecting by vp, we ge (vpbpλm)h i jkh + (v pbpλj)h i kmh + (v pbpλk)h i mjh = 0. using (4.10) in it, we find µmh i jkh + µjh i km + µkh i mj = 0 (4.23) for l 6= 0. transvecting (2.4) and (4.23) by vm and adding, we get (λk + µk)h i mjhv m − (λj + µj)h i mkhv m = 0. from this we may conclude himjhv m = φ(λj + µj)x i h. (4.24) for some tensor xih. therefore £gijk = (bjµk + µjµk)v i + φ(λj + µj)x i k. (4.25) from this we find that the special recurrent transformation is affine motion if (bj µk + µjµk)v i = −φ(λj + µj)x i k. now we consider a special recurrent transformation of the second kind (l = 0). transvecting (2.5) by vm and using l = λmv m = 0, we get λjh i kmv m + λkh i mjv m = 0. this is possible only when himkv m = λkx i (4.27) for some vector field xi. since yih i jk = 0, yix i = 0. £hijk, in view of (2.2), (3.1) and (3.17), becomes £hijk = µh i jkrv r − hrjkµrv i + µjh i rkv r + µkh i jrv r (4.28) where µ = µkẋ k. using (3.2) and (4.9) in (4.10), we get £hijk = (µµjk − µrh r jk)v i + (µjλk − µkλj)x i. (4.29) 90 anjali goswami cubo 14, 1 (2012) this shows that £hijk = 0 if (a) µrh r jk = µµjk (b) µj = ψλj, (4.30) where ψ is a scalar. also £hijk = 0 if and only if £h i jkh = 0. this leads to theorem 4.3. a special recurrent transformation of the second kind in an npr-finsler space is a curvature collineation if (4.30) holds. in view of (4.21), we have £gijk = (bjµk + µjµk + µ∂̇jµk)v i + himjkv m (4.31) which gives £gijk = (bjµk + µjµk + µ∂̇jµk)v i + λjx i k (4.32) where xik = ∂̇kx i. this shows that a special recurrent transformation of the second kind is an affine motion if (bjµk + µjµk + µ∂̇jµk)v i = −λjx i k. (4.33) transvecting this equation by ẋk, we get (bjµk + µjµk)ẋ kvi = −λjx i. (4.34) transvecting this equation by yi, we have (bjµk + µjµk)ẋ k = 0 (4.35) for yiv i 6= 0 and yixi = 0. using (4.35) in (4.34), we get xi = 0. therefore xik = 0. using xik = 0 in equation (4.33), we get bjµk + µjµk + µ∂̇jµk = 0. (4.36) thus (4.33) implies (4.36). conversely if (4.36) holds, its skew symmetric part gives µjk = bjµk − bkµj = 0. (4.37) using this in (3.2) we get hijkhv h = 0, which implies himjkv m = 0. therefore xik = 0. hence we conclude: theorem 4.4. a special recurrent transformation of the second kind in an npr-finsler space is an affine motion if bjµk + µjµk + µ∂̇jµk = 0. received: september 2010. revised: may 2011. cubo 14, 1 (2012) special recurrent transformation in an npr-finsler space 91 references [1] pandey, p. n., a recurrent finsler manifold admitting special transformations, progress of mathematics, 13 (1979), 85-98. [2] pandey, p. n., on npr-finsler manifold, ann. fac. kinshasha, 6 (1980), 65-77. [3] pandey, p. n., affine motion in a recurrent finsler manifold, ann. fac. kinshasha, 6 (1980), 51-63. [4] pandey, p. n., some identities in an npr-finsler manifold, proc. nat. acad. sci. (india), 51 (1981), 105-109. [5] pandey, p. n., certain types of affine motion in a finsler manifold, colloquium mathematicum, 49 (1985), 243-252. [6] rund, h., the differential geometry of finsler spaces, springer-verlag, berlin, 1959. [7] yano, k., the theory of lie derivatives and its applications, north holland publ. co., amsterdam, 1957. introduction an npr-finsler space a recurrent vector field in an npr-finsler space a special recurrent transformation () cubo a mathematical journal vol.17, no¯ 03, (15–41). october 2015 degenerate k-regularized (c1,c2)-existence and uniqueness families marko kostić 1 faculty of technical sciences, university of novi sad, trg d. obradovića 6, 21125 novi sad, serbia. marco.s@verat.net abstract in this paper, we consider various classes of degenerate k-regularized (c1,c2)-existence and uniqueness families. the main purpose of the paper is to report how the techniques established in a joint paper of c.-g. li, m. li and the author [32] can be successfully applied in the analysis of a wide class of abstract degenerate multi-term fractional differential equations with caputo derivatives. resumen en este art́ıculo, consideramos varias clases de familias k-regularizadas (c1,c2)-de existencia y unicidad. el principal objetivo de este trabajo es mostrar como las técnicas establecidas en un trabajo conjunto de c.-g. li, m. li y el autor [27], pueden ser aplicadas satisfactoriamente en el análisis de una clase amplia de ecuaciones fracionarias multi-término degeneradas con derivadas de caputo. keywords and phrases: abstract multi-term fractional differential equations, degenerate differential equations, fractional calculus, mittag-leffler functions, caputo time-fractional derivatives. 2010 ams mathematics subject classification: 47d06, 47d60, 47d62, 47d99. 1the author is partially supported by grant 174024 of ministry of science and technological development, republic of serbia. 16 marko kostić cubo 17, 3 (2015) 1 introduction and preliminaries during the past three decades, considerable interest in fractional calculus and fractional differential equations has been stimulated due to their numerous applications in engineering, physics, chemistry, biology and other sciences. basic information about fractional calculus and non-degenerate fractional differential equations can be obtained by consulting [6], [15], [23]-[25], [42]-[44] and the references cited therein. for the basic source of information on the abstract degenerate differential equations, we refer the reader to [1], [3], [5], [7], [12], [17], [35], [40]-[41], [45]-[49] and [52]-[53]. the theory of abstract degenerate (multi-term) fractional differential equations is at its beginning stage and we can freely say that it is a still-undeveloped subject. the most important qualitative properties of abstract degenerate (multi-term) fractional differential equations have been recently considered in the papers [19]-[20], [22], [27]-[30] and [34]. the existence and uniqueness of solutions of the cauchy and showalter problems for a class of degenerate fractional evolution systems have been analyzed by v. e. fedorov and a. debbouche in [19], while the necessary and sufficient conditions for the relative p-boundedness of a pair of operators have been obtained by v. e. fedorov and d. m. gordievskikh in [20]. in [27]-[28], the author has investigated degenerate volterra integro-differential equations in locally convex spaces, as well as the generation of degenerate fractional resolvent operator families associated with abstract differential operators and the generation of various classes of exponentially equicontinuous k-regularized c-resolvent propagation families associated with the degenerate multi-term problem (1.1) below. the hypercyclic and topologically mixing properties of degenerate multi-term fractional differential equations with caputo derivatives have been analyzed in [29]-[30]. among many other things, in a joint research study with v. e. fedorov [22], the author has analyzed the existence and uniqueness of regularized solutions for a class of abstract degenerate multi-term fractional differential equations with caputo derivatives. the abstract degenerate multi-term fractional differential equations with classical riemann-liouville fractional derivatives have been recently investigated by the author in [34], following the methods used in [33] and this paper. the main subject under our consideration is the following degenerate multi-term problem: bd αn t u(t) + n−1∑ i=1 aid αi t u(t) = ad α t u(t) + f(t), t ≥ 0; u(j)(0) = uj, j = 0, · · ·,⌈αn⌉ − 1, (1.1) where n ∈ n \ {1}, a, b and a1, · · ·,an−1 are closed linear operators on a sequentially complete locally convex space x, 0 ≤ α1 < · · · < αn, 0 ≤ α < αn, f(t) is an x-valued function, and d α t denotes the caputo fractional derivative of order α ([6], [25]). define an := b, a0 := a, m := ⌈α⌉, α0 := α and mi := ⌈αi⌉, i ∈ n 0 n, where nn := {1,2, · · ·,n} and n 0 n := nn ∪ {0}. as mentioned in the abstract, the main purpose of this paper is to reconsider the various notions of non-degenerate k-regularized (c1,c2)-existence and uniqueness families introduced in the paper [32], whose organization is very similar to that of this paper. without any doubt, this cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 17 causes the expositority of our paper in a certain sense. on the other hand, we will not be in wrong if we say that our paper proposes an important theoretical novelty method capable of seeking of solutions of some very atypical degenerate differential equations in lp-spaces. in this place, it is also worth noting that we initiate the analysis of existence of local solutions of abstract degenerate differential equations in this paper; furthermore, we provide generalizations of [36, theorem 2.3, theorem 3.1] for degenerate multi-term problems, and successfully apply the obtained theoretical results in the analysis of some very interesting degenerate differential equations. before explaining the notation used in the paper, we would like to note that it is quite questionable whether there exists any other significant reference which treats the existence and uniqueness of various types of automorphic solutions to abstract degenerate multi-term fractional differential equations (cf. [2], [8]-[9] and [13]-[14] for some results in the non-degenerate case). unless specifed otherwise, we assume that x is a hausdorff sequentially complete locally convex space over the field of complex numbers, sclcs for short. if y is also an sclcs over the same field of scalars as x, then we denote by l(y,x) the space consisting of all continuous linear mappings from y into x; l(x) ≡ l(x,x). by ⊛x (⊛, if there is no risk for confusion), we denote the fundamental system of seminorms which defines the topology of x. the fundamental system of seminorms which defines the topology on y is denoted by ⊛y. the symbol i denotes the identity operator on x. let 0 < τ ≤ ∞. a strongly continuous operator family (w(t))t∈[0,τ) ⊆ l(y,x) is said to be locally equicontinuous iff, for every t ∈ (0,τ) and for every p ∈ ⊛x, there exist qp ∈ ⊛y and cp > 0 such that p(w(t)y) ≤ cpqp(y), y ∈ y, t ∈ [0,t]; the notions of equicontinuity of (w(t))t∈[0,τ) and the exponential equicontinuity of (w(t))t≥0 are defined similarly. notice that (w(t))t∈[0,τ) is automatically locally equicontinuous in case that the space y is barreled ([39]). by b we denote the family consisting of all bounded subsets of y. define pb(t) := supy∈b p(ty), p ∈ ⊛x, b ∈ b, t ∈ l(y,x). then pb(·) is a seminorm on l(y,x) and the system (pb)(p,b)∈⊛x×b induces the hausdorff locally convex topology on l(y,x). if x is a banach space, then we denote by ‖x‖ the norm of an element x ∈ x. suppose that a is a closed linear operator acting on x. then we denote the domain, kernel space and range of a by d(a), n(a) and r(a), respectively. since no confusion seems likely, we will identify a with its graph. set pa(x) := p(x) +p(ax), x ∈ d(a), p ∈ ⊛. then the calibration (pa)p∈⊛ induces the hausdorff sequentially complete locally convex topology on d(a); we denote this space simply by [d(a)]. if v is a general topological vector space, then a function f : ω → v, where ω is an open non-empty subset of c, is said to be analytic if it is locally expressible in a neighborhood of any point z ∈ ω by a uniformly convergent power series with coefficients in v. we refer the reader to [4], [25, section 1.1] and references cited there for the basic information about vector-valued analytic functions. in our approach the space x is sequentially complete, so that the analyticity of a mapping f : ω → x is equivalent with its weak analyticity. it is said that a function f : [0,∞) → e is locally hölder continuous with the exponent r ∈ (0,1] iff for each p ∈ ⊛ and t > 0 there exists m ≥ 1 such that p(f(t) − f(s)) ≤ m|t − s|r, provided 0 ≤ t,s ≤ t. sometimes we use the following condition on a scalar-valued function k(·): 18 marko kostić cubo 17, 3 (2015) (p1) k(·) is laplace transformable, i.e., it is locally integrable on [0,∞) and there exists β ∈ r so that k̃(λ) := l(k)(λ) := lim b→∞ ∫b 0 e−λtk(t)dt := ∫ ∞ 0 e−λtk(t)dt exists for all λ ∈ c with reλ > β. put abs(k) :=inf{reλ : k̃(λ) exists}, and denote by l−1 the inverse laplace transform. we say that a function h(·) belongs to the class lt − e iff there exists a function f ∈ c([0,∞) : e) such that for each p ∈ ⊛ there exists mp > 0 satisfying p(f(t)) ≤ mpe at, t ≥ 0 and h(λ) = (lf)(λ),λ > a. the reader may consult [4], [51, chapter 1] and [25, section 1.2] for the basic properties of vector-valued laplace transform. given θ ∈ (0,π] in advance, define σθ := {λ ∈ c : λ 6= 0, | arg(λ)| < θ}. further on, ⌈β⌉ := inf{n ∈ z : β ≤ n} (β ∈ r). a scalar-valued function k ∈ l1loc[(0,τ)) is said to be a kernel on [0,τ) iff for any scalar-valued continuous function t 7→ u(t), t ∈ [0,τ), the preassumption ∫t 0 k(t − s)u(s)ds = 0, t ∈ [0,τ) implies u(t) = 0, t ∈ [0,τ). if τ < ∞, then the titchmarsh– foiaş theorem (see e.g. [24, theorem 3.4.40]) states that the function k(t) is a kernel on [0,τ) iff 0 ∈ supp(k); on the other hand, if τ = ∞ and k 6= 0 in l1loc([0,∞)), then it is well known that the function k(t) is automatically a kernel on [0,∞). the gamma function is denoted by γ(·) and the principal branch is always used to take the powers; the convolution like mapping ∗ is given by f ∗ g(t) := ∫t 0 f(t − s)g(s)ds. set gζ(t) := t ζ−1/γ(ζ), 0ζ := 0 (ζ > 0, t > 0) and g0(t) := the dirac δ-distribution. if f : [0,∞) → x is a continuous function, then we set g0 ∗ f ≡ f. the reader may consult [43, definition 4.5, p. 96] for the notion of a completely positive function on [0,∞) (cf. also [37, remark 3.6, (3.3)]). denote by sα,p(rn) the fractional sobolev space of order α (cf. [38, definition 12.3.1, p. 297]). let ζ > 0. then the caputo fractional derivative dζtu ([6], [25]) is defined for those functions u ∈ c⌈ζ⌉−1([0,∞) : e) for which g⌈ζ⌉−ζ ∗ (u − ∑⌈ζ⌉−1 j=0 u (j)(0)gj+1) ∈ c ⌈ζ⌉([0,∞) : e), by d ζ tu(t) := d⌈ζ⌉ dt⌈ζ⌉ [ g⌈ζ⌉−ζ ∗ ( u − ⌈ζ⌉−1∑ j=0 u(j)(0)gj+1 )] . if the caputo fractional derivative dζtu(t) exists, then for each number ν ∈ (0,ζ) the caputo fractional derivative dνt u(t) exists, as well, and the following equality holds: d ν t u(t) = ( gζ−ν ∗ d ζ tu(·) ) (t) + ⌈ζ⌉−1∑ j=⌈ν⌉ u(j)(0)gj+1−ν(t), t ≥ 0. (1.2) the mittag-leffler function eβ,γ(z) (β > 0, γ ∈ r) is defined by eβ,γ(z) := ∞∑ k=0 zk γ(βk + γ) , z ∈ c. cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 19 in this place, we assume that 1/γ(βk + γ) = 0 if βk + γ ∈ −n0. set, for short, eβ(z) := eβ,1(z), z ∈ c. let β ∈ (0,1). then the wright function φβ(·) is defined by φβ(t) := l −1 ( eβ(−λ) ) (t), t ≥ 0. for further information about the mittag-leffler and wright functions, cf. [6], [25] and references cited there. 2 degenerate k-regularized (c1,c2)-existence and uniqueness propagation families for (1.1) we start this section by recalling that n ∈ n \ {1}, 0 ≤ α1 < · · · < αn, 0 ≤ α < αn, as well as that a, b and a1, · · ·,an−1 are closed linear operators acting on x. further on, an = b, a0 = a, m = ⌈α⌉, α0 = α and mi = ⌈αi⌉, i ∈ n 0 n. set di := {j ∈ nn−1 : mj − 1 ≥ i} (i ∈ n 0 mn−1 ). let t > 0 and f ∈ c([0,t] : e). by a strong solution of problem (1.1) on the interval [0,t] we mean any continuous function t 7→ u(t), t ∈ [0,t] satisfying that the term aid αi t u(t) is well-defined and continuous on [0,t] (i ∈ n0n), as well as that (1.1) holds identically on [0,t]. convoluting both sides of (1.1) with gαn(t), we get that: b [ u(·)− mn−1∑ k=0 ukgk+1 ( · ) ] + n−1∑ j=1 gαn−αj ∗ aj [ u(·) − mj−1∑ k=0 ukgk+1 ( · ) ] = gαn−α ∗ a [ u(·) − m−1∑ k=0 ukgk+1 ( · ) ] + ( gαn ∗ f ) (·), t ∈ [0,t]. (2.1) by a mild solution of (1.1) on [0,t] we mean any continuous x-valued function t 7→ u(t), t ∈ [0,t] satisfying b [ u(·)− mn−1∑ k=0 ukgk+1 ( · ) ] + n−1∑ j=1 aj ( gαn−αj ∗ [ u(·) − mj−1∑ k=0 ukgk+1 ( · ) ]) = a ( gαn−α ∗ [ u(·) − m−1∑ k=0 ukgk+1 ( · ) ]) + ( gαn ∗ f ) (·), t ∈ [0,t]. consider the following inhomogeneous integral equation: bu(t) + n−1∑ j=1 ( gαn−αj ∗ aju ) (t) = f(t) + ( gαn−α ∗ au ) (t), t ∈ [0,t]. (2.2) similarly to the above, we say that a function u ∈ c([0,t] : e) is: (i) a strong solution of (2.2) iff aju ∈ c([0,t] : e), j ∈ n 0 n−1 and (2.2) holds for every t ∈ [0,t]. 20 marko kostić cubo 17, 3 (2015) (ii) a mild solution of (2.2) iff (gαn−αj ∗ u)(t) ∈ d(aj), t ∈ [0,t], j ∈ n 0 n−1 and bu(t) + n−1∑ j=1 aj ( gαn−αj ∗ u ) (t) = f(t) + a ( gαn−α ∗ u ) (t), t ∈ [0,t]. a mild (strong) solution of problem (1.1), resp. (2.2), on [0,∞) is defined analogously. we will be interested in the following notions. definition 2.1. (cf. [32, definition 2.2] for the case b = i) suppose 0 < τ ≤ ∞, k ∈ c([0,τ)), c, c1, c2 ∈ l(x), c and c2 are injective. (i) a sequence ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)) of strongly continuous operator families in l(x, [d(b)]) is called a (local, if τ < ∞) k-regularized c1-existence propagation family for (1.1) iff, for every i = 0, · · ·,mn − 1, the following holds: b [ ri(·)x − ( k ∗ gi ) (·)c1x ] + ∑ j∈di aj [ gαn−αj ∗ ( ri(·)x − ( k ∗ gi ) (·)c1x )] + ∑ j∈nn−1\di aj ( gαn−αj ∗ ri ) (·)x =    a ( gαn−α ∗ ri ) (·)x, m − 1 < i, x ∈ x, a [ gαn−α ∗ ( ri(·)x − ( k ∗ gi ) (·)c1x )] (·), m − 1 ≥ i, x ∈ x. (ii) a sequence ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)) of strongly continuous operator families in l(x) is called a (local, if τ < ∞) k-regularized c2-uniqueness propagation family for (1.1) iff [ ri(·)bx − ( k ∗ gi ) (·)c2bx ] + ∑ j∈di gαn−αj ∗ [ ri(·)ajx − ( k ∗ gi ) (·)c2ajx ] + ∑ j∈nn−1\di ( gαn−αj ∗ ri(·)ajx ) (·) =    ( gαn−α ∗ ri(·)ax ) (·), m − 1 < i, gαn−α ∗ [ ri(·)ax − ( k ∗ gi ) (·)c2ax ] (·), m − 1 ≥ i, for any x ∈ ⋂ 0≤j≤n d(aj) and i ∈ n 0 mn−1 . (iii) a sequence ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)) of strongly continuous operator families in l(x) is called a (local, if τ < ∞) k-regularized c-resolvent propagation family for (1.1), in short k-regularized c-propagation family for (1.1), iff ((r0(t))t∈[0,τ), ···,(rmn−1(t))t∈[0,τ)) is a k-regularized c-uniqueness propagation family for (1.1), and for every t ∈ [0,τ), i ∈ n0mn−1 and j ∈ n0n, one has ri(t)aj ⊆ ajri(t), ri(t)c = cri(t) and caj ⊆ ajc. cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 21 in case k(t) = gζ+1(t), where ζ ≥ 0, it is also said that ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)) is a ζ-times integrated c1-existence propagation family for (1.1); 0-times integrated c1-existence propagation family for (1.1) is simply called c1-existence propagation family for (1.1). for a kregularized c1-existence propagation family ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)), it is said that is locally equicontinuous (exponentially equicontinuous) iff each single operator family (r0(t))t∈[0,τ) ⊆ l(x, [d(b)]), · · ·, (rmn−1(t))t∈[0,τ) ⊆ l(x, [d(b)]) is; ((r0(t))t≥0, · · ·,(rmn−1(t))t≥0) is said to be an exponentially equicontinuous k-regularized c1existence propagation family for problem (1.1), of angle α ∈ (0,π/2], iff the following holds: (a) for every x ∈ e and i ∈ n0mn−1, the mappings t 7→ ri(t)x, t > 0 and t 7→ bri(t)x, t > 0 can be analytically extended to the sector σα; since no confusion seems likely, we shall denote these extensions by the same symbols. (b) for every x ∈ e, β ∈ (0,α) and i ∈ n0mn−1, one has limz→0,z∈σβ ri(z)x = ri(0)x and limz→0,z∈σβ bri(z)x = bri(0)x. (c) for every β ∈ (0,α) and i ∈ n0mn−1, there exists ωβ ≥ max(0,abs(k)) (ωβ = 0) such that the family {e−ωβzri(z) : z ∈ σβ} ⊆ l(e, [d(b)]) is equicontinuous. the above terminological agreements and abbreviations can be also understood for the classes of kregularized c2-uniqueness propagation families for (1.1) and k-regularized c-resolvent propagation families for (1.1). the reader with a little experience can simply state a few noteworthy facts about the existence and uniqueness of solutions of mild (strong) solutions of problem (2.2) provided that there exists a k-regularized c1-existence propagation family for problem (1.1) (k-regularized c2-uniqueness propagation family for problem (1.1)); because of that, the corresponding discussion is omitted. the proof of following extension of [32, proposition 2.3] is omitted, too. proposition 2.2. let i ∈ n0mn−1, and let ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)) be a locally equicontinuous k-regularized c1-existence propagation family for (1.1). if ri(t)aj ⊆ ajri(t) (j ∈ n0n, t ∈ [0,τ)), ri(t)c1 = c1ri(t) (t ∈ [0,τ)), c1 is injective, k(t) is a kernel on [0,τ) and c1aj ⊆ ajc1 (j ∈ n 0 n), then the following holds: (i) the equality ri(t)ri(s) = ri(s)ri(t), 0 ≤ t, s < τ (2.3) holds, provided that m − 1 < i and that the condition (⋄) the assumption bf(t)+ ∑ j∈di aj(gαn−αj ∗f)(t) = 0, t ∈ [0,τ) for some f ∈ c([0,τ) : e), implies f(t) = 0, t ∈ [0,τ), holds. (ii) the equality (2.3) holds provided that m − 1 ≥ i, nn−1 \ di 6= ∅, and that the condition 22 marko kostić cubo 17, 3 (2015) (⋄⋄) if ∑ j∈nn−1\di aj(gαn−αj ∗f)(t) = 0, t ∈ [0,τ), for some f ∈ c([0,τ) : e), then f(t) = 0, t ∈ [0,τ), holds. the assertions of [32, proposition 2.5, proposition 2.6] can be reformulated for degenerate multi-term problems. this is also the case with the assertion of generalized variation of parameters formula [32, proposition 2.8]: theorem 2.3. let c2 ∈ l(x) be injective. suppose that ((r0(t))t∈[0,τ), ···,(rmn−1(t))t∈[0,τ)) is a locally equicontinuous k-regularized c2-uniqueness propagation family for (1.1), t ∈ (0,τ) and f ∈ c([0,t] : x). then the following holds: (i) if m − 1 < i, then any strong solution u(t) of (2.2) satisfies the equality: ( ri ∗ f ) (t) = ( k ∗ gi ∗ c2bu ) (t) + ∑ j∈di ( gαn−αj+i ∗ k ∗ c2aju ) (t), for any t ∈ [0,t]. therefore, there is at most one strong (mild) solution for (2.2), provided that k(t) is a kernel on [0,τ) and (⋄) holds. (ii) if m − 1 ≥ i, then any strong solution u(t) of (2.2) satisfies the equality: ( ri ∗ f ) (t) = − ∑ j∈nn−1\di ( gαn−αj+i ∗ k ∗ c2aju ) (t), t ∈ [0,t]. therefore, there is at most one strong (mild) solution for (2.2), provided that k(t) is a kernel on [0,τ), nn−1 \ di 6= ∅ and (⋄⋄) holds. as explained in [25, section 2.10], the notion of a k-regularized c-resolvent propagation family is probably the best theoretical concept for the investigation of integral solutions of non-degenerate abstract time-fractional equation (1.1) with aj ∈ l(e), 1 ≤ j ≤ n − 1. if aj /∈ l(e) for some j ∈ nn−1, then the vector-valued laplace transform cannot be so easily applied, which certainly implies that there exist some limitations to this class of propagation families. a similar problem appears in the analysis od degenerate multi-term fractional differential equation (1.1) and, because of that, we will leave the problem of restating [32, theorem 2.9(i), theorem 2.10-theorem 2.12] in our new framework to the reader’s own exploration. in contrast to the above, it is very simple to reformulate the assertion of [32, theorem 2.9(ii)] to degenerate equations, without imposing any additional barriers at: theorem 2.4. suppose k(t) satisfies (p1), ω ≥ max(0,abs(k)), (ri(t))t≥0 is strongly continuous, and the family {e−ωtri(t) : t ≥ 0} ⊆ l(x) is equicontinuous (0 ≤ i ≤ mn − 1). let c2 ∈ l(x) be injective. then ((r0(t))t≥0, ···,(rmn−1(t))t≥0) is a global k-regularized c2-uniqueness propagation family for (1.1) iff, for every λ ∈ c with reλ > ω, and for every x ∈ ⋂ 0≤j≤n d(aj), the following cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 23 equality holds: ∞∫ 0 e−λt [ ri(t)bx − ( k ∗ gi ) (t)c2bx ] dt + ∑ j∈di λαj−αn ∞∫ 0 e−λt [ ri(t)x − ( k ∗ gi ) (t)c2ajx ] dt + ∑ j∈nn−1\di λαj−αn ∞∫ 0 e−λtri(t)ajxdt =    λα−αn ∞∫ 0 e−λtri(t)axdt, m − 1 < i, λα−αn ∞∫ 0 e−λt [ ri(t)ax − ( k ∗ gi ) (t)c2ax ] dt, m − 1 ≥ i. now we would like to present an intriguing example of a local k-regularized i-resolvent propagation family for (1.1): example 2.5. (cf. [32, example 5.2] for non-degenerate case) suppose 1 ≤ p ≤ ∞, e := lp(r), m : r → c is measurable, aj ∈ l ∞(r), (ajf)(x) := aj(x)f(x), x ∈ r, f ∈ e (1 ≤ j ≤ n), (af)(x) := m(x)f(x), x ∈ r, with maximal domain, and α = 0. assume s ∈ (1,2), δ = 1/s, mp = p! s and kδ(t) = l −1(exp(−λδ))(t), t ≥ 0. denote by m(t) the associated function of sequence (mp) (cf. [24, section 1.3] for more details) and put λ ′ α′,β′,γ ′ := {λ ∈ c : reλ ≥ γ′−1m(α′λ) + β′}, α′, β′, γ′ > 0. clearly, there exists a constant cs > 0 such that m(λ) ≤ cs|λ| 1/s, λ ∈ c. assume that the following condition holds (ch): for every τ > 0, there exist α′ > 0, β′ > 0 and d > 0 such that τ ≤ cos( δπ 2 ) cs(α′)1/s and ∣∣∣∣∣ n∑ j=1 λαj−αaj(x) − m(x) ∣∣∣∣∣ ≥ d, x ∈ r, λ ∈ λα ′,β′,1. notice that the above condition holds provided n = 2, α2 = 2, α1 = 1, c1 ∈ l ∞(r), |c1(x)| ≥ d1 > 0 for a.e. x ∈ r, a2(x) ∈ l ∞(r), a2(x) = 0, x ∈ (−1,1), a1(x) = a2(x)c1(x) and m(x) = 1 4 c21(x)a2(x) − 1 16 c41(x)a2(x) − a2(x), x ∈ r (cf. [32, (5.7)]), and that the validity of condition (ch) does not imply, in general, the essential boundedness of function m(·) or the injectivity of the operator b. we will prove that there exists a global (not exponentially bounded, in general) kδ-regularized i-resolvent propagation family ((r0(t))t≥0, · · ·,(rmn−1(t))t≥0) for (1.1). clearly, it suffices to show that, for every τ > 0, there exists a local kδ-regularized i-resolvent propagation family for (1.1) on [0,τ). suppose τ > 0 is given in advance, and α′ > 0, β′ > 0 and d > 0 satisfy (ch), with this τ. let γ denote the upwards oriented boundary of ultra-logarithmic region 24 marko kostić cubo 17, 3 (2015) λα′,β′,1. put, for every t ∈ [0,τ), f ∈ e and x ∈ r, ( ri(t)f ) (x) := 1 2πi ∫ γ eλt−λ δ [ λαn−α−ian(x) + ∑ j∈di λαj−α−iaj(x) ] f(x) λαn−αan(x) + n−1∑ j=1 λαj−αaj(x) − m(x) dλ. then the analysis contained in [32, example 5.2] shows that ((r0(t))t∈[0,τ), · · ·,(rmn−1(t))t∈[0,τ)) is a local kδ-regularized i-resolvent propagation family for (1.1), as well as that, for every compact set k ⊆ [0,∞), there exists hk > 0 such that sup t∈k,p∈n0,i∈n 0 mn−1 ∥∥∥hpk dp dtp ri(t) ∥∥∥ p!s < ∞. we can similarly consider the existence of local k1/2-regularized i-resolvent propagation families for (1.1) which obey slight modifications of the properties stated above with s = 2, and with the operators aj not belonging to the space l(e) for some indexes j ∈ nn. furthermore, we can similarly construct some relevant examples of local k-regularized i-resolvent propagation families for (1.1) in certain classes of fréchet function spaces. 3 degenerate k-regularized (c1,c2)-existence and uniqueness families for (1.1) in this section, we investigate the class of degenerate k-regularized (c1,c2)-existence and uniqueness families for (1.1). recall that di = {j ∈ nn−1 : mj − 1 ≥ i} (i ∈ n 0 mn−1 ), as well as that a, b and a1, · · ·,an−1 are closed linear operators acting on x. by y we denote another sclcs over the same field of scalars as x. in the following definition, we will generalize the notion introduced in our previous joint research with c.-g. li and m. li (cf. [32, definition 3.1], [31], r. delaubenfels [10]-[11], and t.-j. xiao-j. liang [54] for some other known concepts in the case that b = i). definition 3.1. suppose 0 < τ ≤ ∞, k ∈ c([0,τ)), c1 ∈ l(y,x), and c2 ∈ l(x) is injective. (i) a strongly continuous operator family (e(t))t∈[0,τ) ⊆ l(y,x) is said to be a (local, if τ < ∞) k-regularized c1-existence family for (1.1) iff, for every y ∈ y, the following holds: e(·)y ∈ cmn−1([0,τ) : [d(b)]), e(i)(0)y = 0 for every i ∈ n0 with i < mn − 1, aj(gαn−αj ∗ e(mn−1))(·)y ∈ c([0,τ) : x) for 0 ≤ j ≤ n, and be(mn−1)(t)y + n−1∑ j=1 aj ( gαn−αj ∗ e (mn−1) ) (t)y − a ( gαn−α ∗ e (mn−1) ) (t)y = k(t)c1y, (3.1) for any t ∈ [0,τ). cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 25 (ii) a strongly continuous operator family (u(t))t∈[0,τ) ⊆ l(x) is said to be a (local, if τ < ∞) k-regularized c2-uniqueness family for (1.1) iff, for every τ ∈ [0,τ) and x ∈ ⋂ 0≤j≤n d(aj), the following holds: u(t)bx + n−1∑ j=1 ( gαn−αj ∗ u(·)ajx ) (t) − ( gαn−α ∗ u(·)ax ) (t)y = ( k ∗ gmn−1 ) (t)c2x. (3.2) (iii) a strongly continuous family ((e(t))t∈[0,τ),(u(t))t∈[0,τ)) ⊆ l(y,x)×l(x) is said to be a (local, if τ < ∞) k-regularized (c1,c2)-existence and uniqueness family for (1.1) iff (e(t))t∈[0,τ) is a k-regularized c1-existence family for (1.1), and (u(t))t∈[0,τ) is a k-regularized c2uniqueness family for (1.1). (iv) suppose y = x and c = c1 = c2. then a strongly continuous operator family (r(t))t∈[0,τ) ⊆ l(x) is said to be a (local, if τ < ∞) k-regularized c-resolvent family for (1.1) iff (r(t))t∈[0,τ) is a k-regularized c-uniqueness family for (1.1), r(t)aj ⊆ ajr(t), for 0 ≤ j ≤ n and t ∈ [0,τ), as well as r(t)c = cr(t), t ∈ [0,τ), and caj ⊆ ajc, for 0 ≤ j ≤ n. if k(t) = gζ+1(t), where ζ ≥ 0, then it is also said that (e(t))t∈[0,τ) is a ζ-times integrated c1-existence family for (1.1); 0-times integrated c1-existence family for (1.1) is also said to be a c1-existence family for (1.1). a similar notion can be introduced for all other classes of uniqueness and resolvent families introduced in definition 3.1. albeit the choice of an sclcs space y different from x can produce a larger set of initial data for which the abstract cauchy problem (1.1) has a strong solution (see e.g. [54, example 2.5]), in our furher work the most important case will be that in which y = x. keeping in mind that the operators a, b, a1, · · ·,an−1 are closed, we can integrate the both sides of (3.1) sufficiently many times in order to see that: be(l)(t)y + n−1∑ j=1 aj ( gαn−αj ∗ e (l) ) (t)y − a ( gαn−α ∗ e (l) ) (t)y = ( k ∗ gmn−1−l ) (t)c1y, (3.3) for any t ∈ [0,τ), y ∈ y and l ∈ n0mn−1. proposition 3.2. suppose that ((e(t))t∈[0,τ),(u(t))t∈[0,τ)) is a k-regularized (c1,c2)-existence and uniqueness family for (1.1), and let (u(t))t∈[0,τ) be locally equicontinuous. then c2e(t)y = u(t)c1y, t ∈ [0,τ), y ∈ y. proof. the proof of proposition is almost the same as the corresponding proof of [32, proposition 3.2]. observe only that we can always assume, without loss of generality, that the number α is less than or equal to α1. 26 marko kostić cubo 17, 3 (2015) definition 3.3. (cf. [32, definition 3.3]) suppose 0 ≤ i ≤ mn − 1. then we define d ′ i := {j ∈ n 0 n−1 : mj − 1 ≥ i}, d ′′ i := n 0 n−1 \ d ′ i and di := { ui ∈ ⋂ j∈d′′ i d(aj) : ajui ∈ r(c1), j ∈ d ′′ i } . it is not so predictable that the assertion of [32, theorem 3.4] continues to hold in degenerate case without any terminological changes, and that the operator b does not appear in the definition of set di, for which it is well known that represents, in non-degenerate case, the set which consists of all initial values for which the homogeneous counterpart of abstract cauchy problem (1.1), with b = i and uj = 0, j ∈ n 0 mn−1 \ {i}, has a strong solution (provided that there exists a c1-existence family for (1.1)). it is also worth nothing that we do not use the injectiveness of the operator b in (ii): theorem 3.4. (i) suppose (e(t))t∈[0,τ) is a c1-existence family for (1.1), t ∈ (0,τ), and ui ∈ di for 0 ≤ i ≤ mn − 1. then the function u(t) = mn−1∑ i=0 uigi+1(t) − mn−1∑ i=0 ∑ j∈nn−1\di ( gαn−αj ∗ e (mn−1−i) ) (t)vi,j + mn−1∑ i=m ( gαn−α ∗ e (mn−1−i) ) (t)vi,0, 0 ≤ t ≤ t, is a strong solution of the problem (1.1) on [0,t], with f(t) ≡ 0, where vi,j ∈ y satisfy ajui = c1vi,j for 0 ≤ j ≤ n − 1. (ii) suppose (u(t))t∈[0,τ) is a locally equicontinuous k-regularized c2-uniqueness family for (1.1), t ∈ (0,τ) and 0 ∈ supp(k). then there exists at most one strong (mild) solution of (1.1) on [0,t], with ui = 0, i ∈ n 0 mn−1 . proof. we will provide all the relevant details for the sake of completeness. making use of (3.3), cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 27 it can be easily verified that: b [ u(·) − mn−1 ∑ i=0 uigi+1 ( · ) ] + n−1 ∑ j=1 aj ( gαn−αj ∗ [ u(·) − mj−1 ∑ i=0 uigi+1 ( · ) ]) = − mn−1 ∑ i=0 ∑ j∈nn−1\di ( gαn−αj ∗ be (mn−1−i) ) (·)vi,j + mn−1 ∑ i=m ( gαn−α ∗ be (mn−1−i) ) (·)vi,0 + n−1 ∑ j=1 aj ( gαn−αj ∗ { mn−1 ∑ i=mj gi+1 ( · ) ui − mn−1 ∑ i=0 ∑ l∈nn−1\di ( gαn−αl ∗ e (mn−1−i) ) (·)vi,l + mn−1 ∑ i=m ( gαn−α ∗ e (mn−1−i) ) (·)vi,0 }) = − mn−1 ∑ i=0 ∑ j∈nn−1\di ( gαn−αj ∗ be (mn−1−i) ) (·)vi,j + mn−1 ∑ i=m ( gαn−α ∗ be (mn−1−i) ) (·)vi,0 + n−1 ∑ j=1 mn−1 ∑ i=mj c1vi,jgαn−αj+i+1 ( · ) − mn−1 ∑ i=0 ∑ l∈nn−1\di gαn−αl ∗ [ −be (mn−1−i) ( · ) vi,l + a ( gαn−α ∗ e (mn−1−i) ) (·)vi,l + gi+1 ( · ) c1vi,l ] + mn−1 ∑ i=m gαn−α ∗ [ −be (mn−1−i) ( · ) vi,0 + a ( gαn−α ∗ r (mn−1−i) ) (·)vi,0 + gi+1 ( · ) c1vi,0 ] = gαn−α ∗ a [ u(·) − m−1 ∑ i=0 uigi+1 ( · ) ] , since n−1 ∑ j=1 mn−1 ∑ i=mj c1vi,jgαn−αj+i+1(·) = mn−1 ∑ i=0 ∑ j∈nn−1\di c1vi,jgαn−αj+i+1(·). this implies that u(t) is a mild solution of (1.1) on [0,t]. in order to complete the proof of (i), it suffices to show that dαnt u(t) ∈ c([0,t] : x) and aid αi t u ∈ c([0,t] : x) for all i ∈ n 0 n. towards this end, notice that the partial integration implies that, for every t ∈ [0,t], gmn−αn ∗ [ u(·) − mn−1 ∑ i=0 uigi+1(·) ] (t) = mn−1 ∑ i=m ( gmn−α+i ∗ e (mn−1) ) (t)vi,0 − mn−1 ∑ i=0 ∑ j∈nn−1\di ( gmn−αj+i ∗ e (mn−1) ) (t)vi,j. 28 marko kostić cubo 17, 3 (2015) therefore, dαnt u ∈ c([0,t] : x) and, for every t ∈ [0,t], d αn t u(t) = dmn dtmn { gmn−αn ∗ [ u(·) − mn−1 ∑ i=0 uigi+1(·) ] (t) } = mn−1 ∑ i=m ( gi−α ∗ e (mn−1) ) (t)vi,0 − mn−1 ∑ i=0 ∑ j∈nn−1\di ( gi−αj ∗ e (mn−1) ) (t)vi,j, (3.4) whence we may directly conclude that bdαnt u ∈ c([0,t] : x). suppose, for the time being, i ∈ n0n−1. then aiuj ∈ r(c1) for j ≥ mi. moreover, the inequality l ≥ αj holds provided 0 ≤ l ≤ mn −1 and j ∈ nn−1 \dl, and aj(gαn−αj ∗e (mn−1))(·)y ∈ c([0,t] : x) for 0 ≤ j ≤ n−1 and y ∈ y. using (1.2) and (3.4), it is not difficult to prove that: aid αi t u(·) = mn−1 ∑ j=mi gj+1−αi(·)aiuj − mn−1 ∑ l=0 ∑ j∈nn−1\dl [ gl−αj ∗ ai ( gαn−αi ∗ e (mn−1) )] (·)vl,j + mn−1 ∑ l=m [ gl−α ∗ ai ( gαn−αi ∗ e (mn−1) )] (·)vl,0 ∈ c([0,t] : x), finishing the proof of (i). the second part of theorem can be proved as follows. suppose u(t) is a strong solution of (1.1) on [0,t], with ui = 0, i ∈ n 0 mn−1 . making use of (3.2) and the equality t ∫ 0 t−s ∫ 0 gαn−αj(r)u(t − s − r)aju(s)drds = t ∫ 0 s ∫ 0 gαn−αj(r)u(t − s)aju(s − r)drds, holding for any t ∈ [0,t] and j ∈ n0n−1, we have that ( ub ∗ u ) (t) = ( k ∗ gmn−1c2 ∗ u ) (t) + t∫ 0 t−s∫ 0 [ gαn−αj(r)u(t − s − r)aju(s) − gαn−α(r)u(t − s − r)au(s) ] drds = ( k ∗ gmn−1c2 ∗ u ) (t) + ( u ∗ bu ) (t), t ∈ [0,t]. therefore, (k ∗ gmn−1c2 ∗ u)(t) = 0, t ∈ [0,t] and u(t) = 0, t ∈ [0,t]. the standard proof of following theorem is omitted. theorem 3.5. suppose k(t) satisfies (p1), (e(t))t≥0 ⊆ l(y,x), (u(t))t≥0 ⊆ l(x), ω ≥ max(0, abs(k)), c1 ∈ l(y,x) and c2 ∈ l(x) is injective. set pλ := b + ∑n−1 j=1 λ αj−αnaj − λ α−αna, reλ > 0. (i) (a) let (e(t))t≥0 be a k-regularized c1-existence family for (1.1), let the family {e −ωte(t) : t ≥ 0} be equicontinuous, and let the family {e−ωtaj(gαn−αj ∗ e)(t) : t ≥ 0} be equicontinuous (0 ≤ j ≤ n). then the following holds: pλ ∞∫ 0 e−λte(t)ydt = k̃(λ)λ1−mnc1y, y ∈ y, reλ > ω. cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 29 (b) let the operator pλ be injective for every λ > ω with k̃(λ) 6= 0. suppose, additionally, that there exist strongly continuous operator families (w(t))t≥0 ⊆ l(y,x) and (wj(t))t≥0 ⊆ l(y,x) such that {e −ωtw(t) : t ≥ 0} and {e−ωtwj(t) : t ≥ 0} are equicontinuous (0 ≤ j ≤ n) as well as that: ∞∫ 0 e−λtw(t)ydt = k̃(λ)p−1λ c1y and ∞∫ 0 e−λtwj(t)ydt = k̃(λ)λ αj−αnajp −1 λ c1y, for every λ > ω with k̃(λ) 6= 0, y ∈ y and j ∈ n0n. then there exists a k-regularized c1existence family for (1.1), denoted by (e(t))t≥0. furthermore, e (mn−1)(t)y = w(t)y, t ≥ 0, y ∈ y and aj(gαn−αj ∗ e (mn−1))(t)y = wj(t)y, t ≥ 0, y ∈ y, j ∈ n 0 n−1. (ii) suppose (u(t))t≥0 is strongly continuous and the operator family {e−ωtu(t) : t ≥ 0} is equicontinuous. then (u(t))t≥0 is a k-regularized c2-uniqueness family for (1.1) iff, for every x ∈ ⋂n j=0 d(aj), the following holds: ∞∫ 0 e−λtu(t)pλxdt = k̃(λ)λ 1−mnc2x, reλ > ω. the assertion of [32, theorem 3.7], concerning the inhomogeneous cauchy problem (1.1), can be stated for degenerate equations without any terminological changes, as well: theorem 3.6. suppose (e(t))t∈[0,τ) is a locally equicontinuous c1-existence family for (1.1), t ∈ (0,τ), and ui ∈ di for 0 ≤ i ≤ mn − 1. let f ∈ c([0,t] : x), let g ∈ c([0,t] : y) satisfy c1g(t) = f(t), t ∈ [0,t], and let g ∈ c([0,t] : y) satisfy (gαn−mn+1 ∗g)(t) = (g1 ∗g)(t), t ∈ [0,t]. then the function u(t) = mn−1∑ i=0 uigi+1(t) − mn−1∑ i=0 ∑ j∈nn−1\di ( gαn−αj ∗ e (mn−1−i) ) (t)vi,j + mn−1∑ i=m ( gαn−α ∗ e (mn−1−i) ) (t)vi,0 + t∫ 0 e(t − s)g(s)ds, 0 ≤ t ≤ t, (3.5) is a mild solution of the problem (2.1) on [0,t], where vi,j ∈ y satisfy ajui = c1vi,j for 0 ≤ j ≤ n − 1. if, additionally, g ∈ c1([0,t] : y) and (e(mn−1)(t))t∈[0,τ) ⊆ l(y,x) is locally equicontinuous, then the solution u(t), given by (3.5), is a strong solution of (1.1) on [0,t]. 30 marko kostić cubo 17, 3 (2015) contrary to the assertion of [32, theorem 3.7], the final conclusions of [32, remark 3.8] cannot be proved for degenerate equations without imposing some additional conditions. details can be left to the interested reader. concerning the action of subordination principles, we can state the following analogue of [32, theorem 4.1] for degenerate multi-term problems (the final conclusions of [32, remark 4.2] can be restated in our new setting, as well). theorem 3.7. suppose c1 ∈ l(y,x), c2 ∈ l(x) is injective and γ ∈ (0,1). (i) let ω ≥ max(0,abs(k)), and let the assumptions of theorem 3.5(i)-(b) hold. put wγ(t) := ∞∫ 0 t−γφγ ( t−γs ) w(s)yds, t > 0, y ∈ y and wγ(0) := w(0). (3.6) define, for every j ∈ n0n and t ≥ 0, wj,γ(t) by replacing w(t) in (3.6) with wj(t). suppose that there exist a number ν > 0 and a continuous kernel kγ(t) on [0,∞) satisfying (p1) and k̃γ(λ) = λ γ−1k̃(λγ), λ > ν. then there exists an exponentially equicontinuous kγ-regularized c1-existence family (eγ(t))t≥0 for (1.1), with αj replaced by αjγ therein (0 ≤ j ≤ n). furthermore, the family {(1 + t⌈αnγ⌉−2)−1e−ω 1/γteγ(t) : t ≥ 0} is equicontinuous. (ii) suppose (u(t))t≥0 is a k-regularized c2-uniqueness family for (1.1), and the family {e −ωtu(t) : t ≥ 0} is equicontinuous. define, for every t ≥ 0, uγ(t) by replacing w(t) in (3.6) with u(t). suppose that there exist a number ν > 0 and a continuous kernel kγ(t) on [0,∞) satisfying (p1) and k̃γ(λ) = λ γ(2−mn)−2+⌈αnγ⌉k̃(λγ), λ > ν. then there exists a kγ-regularized c2uniqueness family for (1.1), with αj replaced by αjγ therein (0 ≤ j ≤ n). furthermore, the family {e−ω 1/γtuγ(t) : t ≥ 0} is equicontinuous. of importance is the following abstract degenerate volterra equation: bu(t) = f(t) + n−1∑ j=0 ( aj ∗ aju ) (t), t ∈ [0,τ), (3.7) where 0 < τ ≤ ∞, f ∈ c([0,τ) : x), a0, · · ·,an−1 ∈ l 1 loc([0,τ)), and a = a0, · · ·,an−1,b are closed linear operators on x. we define the notion of a mild (strong) solution of problem (3.7) in the same way as it has been done before for the problem (2.2). the following definition plays a crucial role in our investigation of problem (3.7). definition 3.8. (cf. [32, definition 4.3] for the case b = i) suppose 0 < τ ≤ ∞, k ∈ c([0,τ)), c1 ∈ l(y,x), and c2 ∈ l(x) is injective. (i) a strongly continuous operator family (e(t))t∈[0,τ) ⊆ l(y, [d(b)]) is said to be a (local, if τ < ∞) k-regularized c1-existence family for (3.7) iff be(t)y = k(t)c1y + n−1∑ j=0 aj ( aj ∗ e ) (t)y, t ∈ [0,τ), y ∈ y. cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 31 (ii) a strongly continuous operator family (u(t))t∈[0,τ) ⊆ l(x) is said to be a (local, if τ < ∞) k-regularized c2-uniqueness family for (3.7) iff u(t)bx = k(t)c2x + n−1∑ j=0 ( aj ∗ aju ) (t)x, t ∈ [0,τ), x ∈ n⋂ j=0 d(aj). as in non-degenerate case, we have the following: (i) suppose (e(t))t∈[0,τ) is a k-regularized c1-existence family for (3.7). then, for every y ∈ y, the function u(t) = e(t)y, t ∈ [0,τ) is a mild solution of (3.7) with f(t) = k(t)c1y, t ∈ [0,τ). (ii) let (u(t))t∈[0,τ) be a locally equicontinuous k-regularized c2-uniqueness family for (3.7). then there exists at most one mild (strong) solution of (3.7). the most important structural properties of k-regularized c1-existence families for (3.7) and k-regularized c2-uniqueness families for (3.7) are stated in the following analogue of theorem 3.5. theorem 3.9. suppose that k(t) and a0(t), · · ·,an−1(t) satisfy (p1), (e(t))t≥0 ⊆ l(y,x), (u(t))t≥0 ⊆ l(x), ω ≥ max(0,abs(k),abs(a0), · · ·,abs(an−1)), c1 ∈ l(y,x) and c2 ∈ l(x) is injective. set pλ := b − ∑n−1 j=0 ãj(λ)aj, reλ > ω. (i) (a) let (e(t))t≥0 be a k-regularized c1-existence family for (3.7), let the family {e −ωte(t) : t ≥ 0} ⊆ l(y, [d(b)]) be equicontinuous, and let the family {e−ωtaj(aj ∗e)(t) : t ≥ 0} ⊆ l(y,x) be equicontinuous (0 ≤ j ≤ n − 1). then the following holds: pλ ∞∫ 0 e−λte(t)ydt = k̃(λ)c1y, y ∈ y, reλ > ω. (b) let the operator pλ be injective for every λ > ω with k̃(λ) 6= 0. suppose, additionally, that there exist strongly continuous operator families (e(t))t≥0 ⊆ l(y,x), (eb(t))t≥0 ⊆ l(y,x), and (ej(t))t≥0 ⊆ l(y,x) such that the operator families {e −ωte(t) : t ≥ 0}, {e−ωteb(t) : t ≥ 0}, and {e −ωtej(t) : t ≥ 0} are equicontinuous (0 ≤ j ≤ n − 1) as well as that: ∞ ∫ 0 e −λt e(t)ydt = k̃(λ)p −1 λ c1y, ∞ ∫ 0 e −λt eb(t)ydt = k̃(λ)bp −1 λ c1y and ∞ ∫ 0 e −λt ej(t)ydt = k̃(λ)ãj(λ)ajp −1 λ c1y, for every λ > ω with k̃(λ) 6= 0, y ∈ y and j ∈ n0n−1. then (e(t))t≥0 is a k-regularized c1-existence family for (3.7). furthermore, be(t)y = eb(t)y, t ≥ 0, y ∈ y and aj(aj ∗ e)(t)y = ej(t)y, t ≥ 0, y ∈ y, j ∈ n 0 n−1. 32 marko kostić cubo 17, 3 (2015) (ii) suppose (u(t))t≥0 is strongly continuous and the operator family {e−ωtu(t) : t ≥ 0} ⊆ l(x) is equicontinuous. then (u(t))t≥0 is a k-regularized c2uniqueness family for (3.7) iff, for every x ∈ ⋂n j=0 d(aj), the following holds: ∞∫ 0 e−λtu(t)pλxdt = k̃(λ)c2x, reλ > ω. theorem 3.10. (subordination principle) (i) suppose that the requirements of theorem 3.9(i)-(b) hold. let c(t) be completely positive, let c(t), k(t), k1(t), a0(t), · · ·,an−1(t) and b0(t), · · ·,bn−1(t) satisfy (p1), and let ω0 > 0 be such that, for every λ > ω0 with c̃(λ) 6= 0 and k̃(1/c̃(λ)) 6= 0, the following holds: ãj(1/c̃(λ)) = b̃j(λ), j ∈ n 0 n−1 and k̃1(λ) = 1 λc̃(λ) k̃(1/c̃(λ)). (3.8) then for each r ∈ (0,1] there exists a locally hölder continuous (with exponent r), exponentially equicontinuous (k1 ∗ gr)-regularized c1-existence family for bu(t) = f(t) + n−1∑ j=0 ( bj ∗ aju ) (t), t ∈ [0,τ). (3.9) (ii) suppose that the requirements of theorem 3.9(ii) hold. let c(t) be completely positive, let c(t), k(t), k1(t) a0(t), ···,an−1(t) and b0(t), ···,bn−1(t) satisfy (p1), and let ω0 > 0 be such that, for every λ > ω0 with c̃(λ) 6= 0 and k̃(1/c̃(λ)) 6= 0, (3.8) holds. then for each r ∈ (0,1] there exists a locally hölder continuous (with exponent r), exponentially equicontinuous (k1 ∗ gr)regularized c2-uniqueness family for (3.9). the interested reader may try to transfer the final conclusions of [36, theorem 2.1, theorem 2.2, remark 2.1, proposition 2.1] to degenerate multi-term fractional differential equations. in order to do the same with the perturbation result [36, theorem 2.3], we need to introduce the following notion. definition 3.11. a strongly continuous operator family (u(t))t∈[0,τ) ⊆ l(x) is said to be a (local, if τ < ∞) (k,c2)-uniqueness family for (1.1) iff, for every t ∈ [0,τ) and x ∈ ⋂ 0≤j≤n d(aj), the following holds: u(t)bx + n−1∑ j=1 ( gαn−αj ∗ u(·)ajx ) (t) − ( gαn−α ∗ u(·)ax ) (t)x = k(t)c2x. then it is clear that for any strongly continuous operator family (u(t))t∈[0,τ) the following equivalence relation holds: (u(t))t∈[0,τ) is a (local) (k ∗ gmn−1,c2)-uniqueness family for (1.1) iff (u(t))t∈[0,τ) is a (local) k-regularized c2-uniqueness family for (1.1). cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 33 consider now the perturbed equation: bd αn t u(t) + n−1∑ i=1 (ai + fi)d αi t u(t) = (a + f)d α t u(t) + f(t), t ≥ 0; u(j)(0) = uj, j = 0, · · ·,⌈αn⌉ − 1, (3.10) where fi ∈ l(x) for 0 ≤ i ≤ n − 1 and f0 ≡ f. a similar line of reasoning as in the proof of [36, theorem 2.3] shows that the following result about the c-wellposedness of problem (3.10) holds good (observe that the employed method is based on the arguments contained in the proof of [43, theorem 6.1], which does not work any longer if we replace the term bdαnt u(t), in (3.10), with (b + fn)d αn t u(t)): theorem 3.12. (i) suppose y = x, (e(t))t∈[0,τ) ⊆ l(x) is a (local) c1-existence family for (1.1), ej ∈ l(x) and fj = c1ej (j ∈ n 0 n−1). suppose that the following conditions hold: (a) for every p ∈ ⊛x and for every t ∈ (0,τ), there exists cp,t > 0 such that p ( e(mn−1)(t)x ) ≤ cp,tp(x), x ∈ x, t ∈ [0,t]. (b) for every p ∈ ⊛x, there exists cp > 0 such that p ( ejx ) ≤ cpp(x), j ∈ n 0 n−1, x ∈ x. (c) αn − αn−1 ≥ 1 and αn − α ≥ 1. then there exists a (local) c1-existence propagation family (r(t))t∈[0,τ) for (3.10). if τ = ∞ and if, for every p ∈ ⊛x, there exist m ≥ 1 and ω ≥ 0 such that p ( e(mn−1)(t)x ) ≤ meωtp(x), t ≥ 0, x ∈ x, (3.11) respectively (3.11) and p ( be(mn−1)(t)x ) ≤ meωtp(x), t ≥ 0, x ∈ x, (3.12) then (r(t))t≥0 is exponentially equicontinuous, and moreover, (r(t))t≥0 also satisfies the condition (3.11), repectively (3.11) and (3.12), with possibly different numbers m ≥ 1 and ω > 0. (ii) suppose y = x, (u(t))t∈[0,τ) ⊆ l(x) is a (local) (1,c2)-uniqueness family for (1.1), ej ∈ l(e) and fj = ejc2 (j ∈ n 0 n−1). suppose that (b)-(c) hold, and that (a) holds with (e(mn−1)(t))t∈[0,τ) replaced by (u(t))t∈[0,τ) therein. then there exists a (local) (1,c2)uniqueness family (w(t))t∈[0,τ) for (3.10). if τ = ∞ and if, for every p ∈ ⊛x, there exist m ≥ 1 and ω ≥ 0 such that (3.11) holds, then (w(t))t≥0 is exponentially equicontinuous, and moreover, (w(t))t≥0 also satisfies the condition (3.11), with possibly different numbers m ≥ 1 and ω > 0. 34 marko kostić cubo 17, 3 (2015) for some other results concerning perturbation properties of abstract degenerate differential equations, one may refer e.g. to [18], [21] and [50]. concerning the existence of strong solutions of (1.1), we can prove the following slight extension of [36, theorem 3.1]; this result can be viewed of some independent interest. theorem 3.13. suppose a, b, a1, · · ·, an−1 are closed linear operators on x, ω > 0, l(x) ∋ c is injective and u0, · · ·,umn−1 ∈ x. set pλ := λ αn−αb + ∑n−1 j=1 λ αj−αaj − a, λ ∈ c \ {0}. let the following conditions hold: (i) the operator pλ is injective for λ > ω and d(p −1 λ c) = x, λ > ω. (ii) if 0 ≤ j ≤ n, 0 ≤ k ≤ mn − 1, m − 1 < k, 1 ≤ l ≤ n, ml − 1 ≥ k and λ > ω, then cuk ∈ d(p −1 λ al), aj { λαj [ λαn−α−k−1p−1λ bcuk + ∑ l∈dk λαl−α−k−1p−1λ alcuk ] − mj−1∑ l=0 δklλ αj−1−lcuk } ∈ lt − x (3.13) and λαn [ λαn−α−k−1p−1λ bcuk + ∑ l∈dk λαl−α−k−1p−1λ alcuk ] − λαn−1−kcuk ∈ lt − x. (3.14) (iii) if 0 ≤ j ≤ n, 0 ≤ k ≤ mn − 1, m − 1 ≥ k, nn−1 \ dk 6= ∅, s ∈ nn−1 \ dk and λ > ω, then cuk ∈ d(as), ∑ l∈nn−1\dk λαl−α−k−1alcuk ∈ d(p −1 λ ), aj { λαj [ λ−k−1cuk−p −1 λ ∑ l∈nn−1\dk λαl−α−k−1alcuk ] − mj−1∑ l=0 δklλ αj−1−lcuk } ∈ lt − x (3.15) and λαn [ λ−k−1cuk−p −1 λ ∑ l∈nn−1\dk λαl−α−k−1alcuk ] − λαn−1−kcuk ∈ lt − x. (3.16) then the abstract cauchy problem (1.1) has a strong solution, with uk replaced by cuk (0 ≤ k ≤ mn − 1). cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 35 remark 3.14. let 0 ≤ k ≤ mn − 1 and m − 1 < k. then theorem 3.13 continues to hold if we replace the term λαn−α−k−1p−1λ bcuk + ∑ l∈dk λαl−α−k−1p−1λ alcuk i.e., the laplace transform of uk(t), in (3.13)-(3.14) by λ−k−1cuk − ∑ l∈nn−1\dk λαl−α−k−1p−1λ alcuk + λ −k−1p−1λ acuk; in this case, it is necessary to assume that cuk ∈ d(p −1 λ al), provided 0 ≤ l ≤ n − 1, k > ml − 1 and λ > ω. let us also observe that a similar modification can be made in the case 0 ≤ k ≤ mn −1 and m − 1 ≥ k. strictly speaking, one can replace the term λ−k−1cuk − p −1 λ ∑ l∈nn−1\dk λαl−α−k−1alcuk i.e., the laplace transform of uk(t), in (3.15)-(3.16) by λαn−α−k−1p−1λ bcuk + ∑ l∈dk λαl−α−k−1p−1λ alcuk − λ −k−1p−1λ acuk; in this case, one has to assume that cuk ∈ d(p −1 λ al), provided 0 ≤ l ≤ n, ml −1 ≥ k and λ > ω. now we would like to illustrate the obtained results with some examples. example 3.15. suppose 1 ≤ p < ∞, ∅ 6= ω ⊆ rn is an open bounded domain with smooth boundary, and x := lp(ω). consider the equation (α − ∆)utt = β∆ut + ∆u + ∫t 0 g(t − s)∆u(s,x)ds = 0, t > 0, x ∈ ω; u(0,x) = φ(x), ut(0,x) = ψ(x), (3.17) where g ∈ l1loc([0,∞)) satisfies (p1), α > 0 and β ∈ r \ {0}. as explained by m. v. falaleev and s. s. orlov in [16], the equation (3.17) appears in the models of nonlinear viscoelasticity provided n = 3. integrating (3.17) twice with the respect to the time-variable t, we obtain the associated integral equation (α − ∆)u(t) = (α + (β − 1)∆)φ(x) + t(α − ∆)ψ + β∆ ( g1 ∗ u ) (t) + ∆ ( g2 ∗ u ) + ∆ ( g2 ∗ g ∗ u ) (t), (3.18) which is of the form (3.7) with b := α − ∆, a2 := β∆, a1 = a0 := ∆ (acting with the dirichlet boundary conditions) and a2(t) := g1(t), a1(t) := g2(t), a0(t) := (g2 ∗ g)(t). then pλ = λ2 + βλ + g̃(λ) + 1 λ2 [ αλ2 λ2 + βλ + g̃(λ) + 1 − ∆ ] . 36 marko kostić cubo 17, 3 (2015) we assume that g(t) is of the following form: g(t) = l∑ j=0 cjgβj (t) + f(t), t > 0, where l ∈ n, cj ∈ c (0 ≤ j ≤ l), 0 < β1 < · · · < βl < 1 and the function f(t) satisfies the requirements of [26, theorem 3.4(i)-(a)] with α = π/2 and ω > 0 sufficiently large. using the resolvent equation and the fact that the operator ∆ generates a bounded analytic c0-semigroup of angle π/2, it can be simply verified that 1 λ p −1 λ ∈ lt − l(x), 1 λ bp −1 λ ∈ lt − l(x) and ãj(λ) λ p −1 λ ∈ lt − l(x), j = 0,1,2. this implies by theorem 3.9 that there exists an exponentially bounded i-existence family (e(t))t≥0 for (3.18), satisfying additionally that for each f ∈ x the mappings t 7→ e(t)f, t > 0, t 7→ be(t)f, t > 0 and t 7→ aj(aj ∗ e)(t)f, t > 0 can be analytically extended to the sector σπ/2; furthermore, (e(t))t≥0 is an exponentailly bounded i-uniqueness family for (3.18). this implies that for each φ, ψ ∈ w2,p(ω) ∩ w 1,p 0 (ω), there exists a unique strong solution u(t) = e(t)(α + (β − 1)∆)φ(x) + ∫t 0 e(s)(α − ∆)ψds of the integral equation (3.18), and that u(t) can be analytically extended to the sector σπ/2. example 3.16. suppose 1 < p < ∞, x := lp(rn), 1/2 < γ ≤ 1, q ∈ n\{1}, p1(x) = ∑ |η|≤q aηx η, p2(x) = −1 − |x| 2 (x ∈ rn), p1(x) is positive, σ ≥ 0, the estimate ∣∣∣∣∣d η ( p1(x) p2(x) )∣∣∣∣∣ ≤ cη ( 1 + |x| )|η|(σ−1) , x ∈ rn holds for each multi-index η ∈ nn0 with |η| > 0, v2 ≥ 0 and for each η ∈ n n 0 there exists mη > 0 such that |dη(p2(x) −1)| ≤ mη(1 + |x|) |η|(v2−1), x ∈ rn. set a2 := ∆ − i, a0f := ∑ |η|≤q aηd ηf with maximal distributional domain, where dη ≡ (−i)|η|fη, and c1 := (i − ∆) − n 2 | 1 p − 1 2 | max(σ,v2). let ei ∈ l(x) and fi = c1ei (i = 0,1). then we know (cf. [27]-[28]) that λ(λ 2a2 − a0) −1c1 ∈ lt −l(x) and λa2(λ 2a2 −a0) −1c1 ∈ lt − l(x), which implies by theorem 3.9(i)-(b) that there exists an exponentially bounded c1-existence family (e(t))t≥0 for the following degenerate second order cauchy problem: { ( ∆ − i ) utt(t,x) = ∑ |η|≤q aηd ηu(t,x), u(0,x) = u0(x) = φ(x), ut(0,x) = u1(x) = ψ(x), obeying the properties (3.11)-(3.12) stated in the formulation of theorem 3.12. applying theorem 3.12, we get there exists an exponentially bounded c1-existence family (r(t))t≥0 for the following degenerate second order cauchy problem: (p) :    ( ∆ − i ) utt(t,x) + f1ut(t,x) = (∑ |η|≤q aηd η + f0 ) u(t,x), u(0,x) = u0(x) = φ(x), ut(0,x) = u1(x) = ψ(x). cubo 17, 3 (2015) degenerate k-regularized (c1,c2)-existence . . . 37 then theorem 3.4(i) shows that there exists a strong solution of problem (p) provided that φ, ψ ∈ sq+n| 1 p − 1 2 | max(σ,v2),p(r n ), ( a0 + f ) φ ∈ sn| 1 p − 1 2 | max(σ,v2),p(r n ), f1ψ ∈ s n| 1 p − 1 2 | max(σ,v2),p(r n ) and ( a0 + f ) ψ ∈ sn| 1 p − 1 2 | max(σ,v2),p(r n ). if we denote by u(t,x), resp. v(t,x), the corresponding strong solution of problem (p) with the initial values φ(x) and ψ(x) ≡ 0, resp. φ(x) ≡ 0 and ψ(x), then one can simply verify that the function u(t,x) := ∞ ∫ 0 t −γ φγ ( st −γ ) u(s,x)ds + t ∫ 0 g1−γ(t − s) ∞ ∫ 0 s −γ φγ ( rs −γ ) v(r,x)drds, is a strong solution of the following integral equation a2 [ u(t,x) − φ(x) − tψ(x) ] + f1 ∫t 0 gγ(t − s) [ u(s,x) − φ(x) ] ds = ∫t 0 g2γ(t − s) ( a0 + f ) u(s,x)ds, t ≥ 0, x ∈ rn; (3.19) furthermore, the function t 7→ u(t, ·) ∈ x can be analytically extended to the sector σ( 1 γ −1) π 2 . in the present situation, we can only prove that there is at most one strong solution of the integral equation (3.19) provided that p = 2. speaking-matter-of-factly, suppose that u(t,x) is a strong solution of (3.19) with φ(x) ≡ ψ(x) ≡ 0. then a−12 ∈ l(x), c1 = i and the function v(t,x) := a2u(t,x) is a strong solution of the following non-degenerate integral equation u(t,x) + ∫t 0 gγ(t − s)f1a −1 2 u(s,x)ds = ∫t 0 g2γ(t − s) ( a0a −1 2 + fa −1 2 ) u(s,x)ds, t ≥ 0, x ∈ rn. (3.20) since λ(λ2 − a0a −1 2 ) −1 = λa2(λ 2a2 − a0) −1 ∈ lt − l(x), the operator a0a −1 2 generates a cosine operator function and we can apply theorem 3.12(ii) in order to see that there exists an exponentially bounded (1,i)-uniqueness family for (3.20), with the meaning clear. this proves the claimed assertion on the uniqueness of strong solutions of problem (3.19). in general case p 6= 2, it is not clear how we can prove that there is at most one strong solution of the integral equation (3.19) without assuming that f1 and f take some specific forms. acknowledgments. the author would like to convey a special vote of thanks to prof. v. e. fedorov, chelyabinsk state university (russia), and prof. r. ponce, universidad de talca (chile), for many stimulating discussions and valuable suggestions. received: june 2014. accepted: march 2015. 38 marko kostić cubo 17, 3 (2015) references [1] n. h. abdelaziz and f. neubrander, degenerate abstract cauchy problems, in: seminar notes in functional analysis and pde, louisiana state university 1991/1992. 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[54] t.-j. xiao and j. liang, higher order abstract cauchy problems: their existence and uniqueness families, j. lond. math. soc. 67 (2003), 149–164. introduction and preliminaries degenerate k-regularized (c1,c2)-existence and uniqueness propagation families for (1.1) degenerate k-regularized (c1,c2)-existence and uniqueness families for (1.1) cubo a mathematical journal vol.16, no¯ 02, (149–160). june 2014 diagana space and the gas absorption model najja s. al-islam department of mathematics, medgar evers college of the city university of new york, 1650 bedford ave., brooklyn, n.y. 11225 usa nalislam@mec.cuny.edu abstract poorkarimi and wiener established the existence of almost periodic solutions to a class of nonlinear hyperbolic partial differential equations with delay. al-islam then generalized the results of poorkarimi and weiner to the pseudo-almost periodic setting. in this paper, the results of al-islam will be extended to the space of weighted pseudo almost periodic functions, also known as diagana space. the class of nonlinear hyperbolic partial differential equations of poorkarimi and wiener represents a mathematical model for the dynamics of gas absorption. resumen poorkarimi y wiener establecieron la existencia de soluciones casi periódicas de una clase de ecuaciones diferenciales parciales hiperbólicas no lineales con retraso. luego, al-islam generalizó los resultados de poorkarimi y wiener al caso seudo-cuasi periódico. en este art́ıculo los resultados de al-islam se extenderán al espacio de funciones seudocuasi periódicas con peso, también conocido como espacio de diagana. la clase de ecuaciones diferenciales parciales hiperbólicas no lineales de poorkarimi y wiener representa un modelo matemático de la dinámica de absorción de gas. keywords and phrases: almost periodic solution, weighted pseudo-almost periodic, gas absorption. 2010 ams mathematics subject classification: 35b10; 35b10; 35j60; 35l70 150 najja s. al-islam cubo 16, 2 (2014) 1 introduction let l > 0. in poorkarimi and wiener [22], under some reasonable assumptions, the existence of both periodic and almost periodic solutions to the nonlinear hyperbolic second-order partial differential equation with delay given by ⎧ ⎪⎪⎨ ⎪⎪⎩ uxt(x, t) + a(x, t)ux(x, t) = c(x, t, u(x, ⌊t⌋)) u(0, t) = ϕ(t) (1.1) where a : [0, l] × r $→ r, c : [0, l] × r × r $→ r, and ϕ : r $→ r are periodic (respectively, almost periodic) functions in the variable t and ⌊t⌋ denotes the greatest integer function: ⌊t⌋ := n for n ≤ t < n + 1 for an integer n, was established. extensive use of similar assumptions as in [22] and [3] will be used to extend the above-mentioned existence results to the weighted pseudo-almost periodic setting. eq.(1.1) is of great interest, being that it represents a mathematical model for the dynamics of gas absorption. further details of the gas absorption model, eq.(1.1), can also be seen in [22]. the existence of almost periodic, asymptotically almost periodic, almost automorphic [21], pseudo almost periodic [11], and more recently, weighted pseudo-almost periodic solutions to differential equations are among the most attractive topics in the qualitative theory of differential equations due to their applications in physics, mathematical biology, along with other areas of science and engineering. the concept of pseudo almost periodicity was first introduced by zhang [23, 25, 24] and generalizes the almost periodicity of bohr. more details on the concept of pseudo almost periodicity as well as its applications to differential equations, functional differential, and partial differential equations can be easily found in the literature, especially in [1, 2, 4, 11, 7, 9, 10] and the references therein. the more recent generalization of zhang almost periodicity is the weighted pseudo almost periodicity of diagana. the text that follows this introduction shows, as well as, compares the properties of the zhang and diagana almost periodic spaces. following the comparisons of the zhang and diagana almost periodic spaces, the results contained in [3] will be generalized in the setting of diagana space. for more details on diagana space, the reader should refer to [6, 14, 20]. 2 weighted pseudo almost periodic functions let (bc(r), ∥ · ∥∞) denote the banach space of all bounded continuous functions ϕ : r $→ r endowed with the sup norm defined by ∥ϕ∥∞ := sup t∈r |ϕ(t)|. cubo 16, 2 (2014) diagana space and the gas absorption model 151 definition 2.1. [5, 11, 21] a continuous function g : r $→ r is called (bohr) almost periodic if for each ε > 0, there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the following property |g(t + τ) − g(t)| < ε for each t ∈ r. the number τ above is then called an ε-translation number of g, and the collection of those almost periodic functions will be denoted as ap(r). although the concept of almost periodicity is a natural generalization of the classical periodicity, there are almost periodic functions that are not periodic. a classical example of an almost periodic function that is not periodic is the function defined by: g(t) = sin t + sin( √ 7t) for each t ∈ r. more details on properties of almost periodic functions can be found in the literature by corduneanu [5], diagana [11], n’guérékata [21] and the references therein. let u be the collection of all functions w, weights, such that w(t) > 0 for almost each t ∈ r and w ∈ l1loc(r). also, for each w ∈ u and r > 0, µ(r, w) := ∫r −r w(t)dt. from the collection of weights, u, we define two subcollections of u as: u∞ := {w ∈ u : lim r→∞ µ(r, w) = ∞ and lim inf t→∞ w(t) > 0}, uw := {w ∈ u∞ : w is bounded} one can see that the subcollections defined above can be written as: uw ⊂ u∞ ⊂ u define pap0(r) := { φ ∈ bc(r) : lim r→∞ 1 2r ∫r −r |φ(σ)|dσ = 0 } . definition 2.2. [11, 23] a function f ∈ bc(r) is called pseudo almost periodic if it can be expressed as f = g + ϕ, where g ∈ ap(r) and ϕ ∈ pap0(r). the collection of such functions will be denoted by pap(r). note that the functions g and ϕ appearing in definition 2.2 are respectively called the almost periodic and the ergodic perturbation components of f. furthermore, the decomposition in definition 2.2 is unique [23, 25, 24]. 152 najja s. al-islam cubo 16, 2 (2014) we now equip pap(r) the collection of all pseudo almost periodic functions from r into r with the sup norm. it is not really hard to see that (pap(r), ∥ · ∥∞) is a closed subspace of bc(r) and hence is a banach space. an example of a pseudo almost periodic function is the function f defined by f(t) = sin t + sin t √ 2 + e−|t| for each t ∈ r. the core of the construction of the weighted pseudo almost periodic space, is the enrichment of the space of ergodic perturbations, pap0(r). that is, for w ∈ u∞, the weighted ergodic space is defined by: pap0(r, w) := { φ ∈ bc(r) : lim r→∞ 1 µ(r, w) ∫r −r |φ(σ)|w(σ)dσ = 0 } . definition 2.3. [14]a function f ∈ bc(r) is called weighted pseudo almost periodic if it can be expressed as f = g + ϕ, where g ∈ ap(r) and ϕ ∈ pap0(r, w). the collection of the functions defined above is diagana space, and will be denoted as dw(r). an example of a function f ∈ dw(r) is the function f(t) = cos t + cos √ 2t + 1 1 + t2 , where w(t) = 1 + t2 for each t ∈ r. lemma 2.4. ap(r) ⊂ pap(r) ⊂ dw(r). in [19], it was shown that the decomposition f = g+ϕ, where g ∈ ap(r) and ϕ ∈ pap0(r, w) is not unique. hence, one cannot define dw(r), equipped with the sup norm, to be a banach space, despite ap(r) and pap0(r, w) being closed subspaces with respect to the sup norm. therefore, with the possibility of being able to construct countably many decompositions of any weighted pseudo almost-periodic function, written as: {gn + φn, n ∈ n}, had to be resolved to ensure the criterion of completeness for a banach space. to resolve this dilemma, in [20] another norm, which will be known as the w-norm in this writing, was constructed and defined as follows: ||f||w := inf n∈n (||gn|| + ||φn||) = inf n∈n ! sup t∈r ||gn(t)|| + sup t∈r ||φn(t)|| " . || · ||w is undoubtedly a norm on dw(r). theorem 2.5. dw(r) is a banach space under the norm || · ||w. cubo 16, 2 (2014) diagana space and the gas absorption model 153 proof. the proof of the theorem can be found in [20]. let w∞ be the set of all functions w ∈ u∞ where there exists a measurable set k ⊂ r such that for each τ ∈ r, lim sup |t|→+∞, t∈k w(t + τ) w(t) := inf m>0 # sup |t|>m,t∈k w(t + τ) w(t) $ < ∞ and lim r→+∞ ∫ kτ r w(t)dt µ(r, w) = 0, where kτr = [−r, r] \ k + τ. lemma 2.6. [17] let w ∈ w∞ and f ∈ dw(r) and if g is its almost periodic component, then g(r) ⊂ f(r). therefore, ||f||∞ ≥ ||g||∞ ≥ inf t∈r |g(t)| ≥ inf t∈r |f(t)|. proof. the proof of the lemma can be found in [17]. theorem 2.7. if (fn)n∈n ⊂ dw(r) is a sequence which converges uniformly with respect to the w-norm to some f : r $→ r, then f is necessarily a weighted pseudo-almost periodic function. proof. write fn = gn + ϕn where (gn)n ⊂ ap(r) and (ϕn)n∈n ⊂ pap0(r, w). suppose ||fn − f||w → 0 as n → ∞ for some function f : r $→ r. of course, f ∈ bc(r), as a uniform limit of a sequence of bounded continuous functions. so to complete the proof, it needs to be shown that f ∈ dw(r). for that, notice that by using lemma 2.6, it follows that ∥gn − gm∥w ≤ ∥fn − fm∥w for all n, m ∈ n. now letting n, m → ∞ in the previous inequality it follows that lim n,m→∞ ∥gn − gm∥w ≤ lim n,m→∞ ∥fn − fm∥w = 0, and hence (gn)n∈n ⊂ ap(r) is a cauchy sequence. since (ap(r), ∥ · ∥∞) is a banach space, it follows that there exists g ∈ ap(r) such that ∥gn − g∥∞ → 0 as n → ∞. 154 najja s. al-islam cubo 16, 2 (2014) now, fn − gn = φn → ϕ := f − g uniformly with respect to the w-norm as n → ∞. thus, writing ϕ = (ϕ − ϕn) + ϕn, it follows that: 1 µ(r, w) ∫r −r |φ(σ)|w(σ)dσ ≤ ∥φn − φ∥w + 1 µ(r, w) ∫r −r |φn(σ)|w(σ)dσ. let r → ∞ in the previous inequality, then lim r→∞ 1 µ(r, w) ∫r −r |φ(σ)|w(σ)dσ ≤ ∥φn − φ∥w + lim r→∞ 1 µ(r, w) ∫r −r |φn(σ)|w(σ)dσ = ∥φn − φ∥w. letting n → ∞ in the previous inequality, then 0 ≤ lim r→∞ 1 µ(r, w) ∫r −r |φ(σ)|w(σ)dσ ≤ lim n→∞ ∥φn − φ∥w = 0, and hence lim r→∞ 1 µ(r, w) ∫r −r |φ(σ)|w(σ)dσ = 0. therefore, f = g + ϕ ∈ dw(r). more details on properties of weighted pseudo-almost periodic functions can be found in the literature, especially in diagana [14]. 3 existence of weighted pseudo almost periodic solutions throughout the rest of the paper, it is assumed that the function a : [0, l] × r $→ r satisfies the following: inf x∈[0,l], t∈r a(x, t) := m > 0. using the previous assumption, the initial value problem, eq.(1.1), has a unique bounded solution, which can be explicitly given by: u(x, t) = ϕ(t) + ∫x 0 ∫t −∞ exp { − ∫t τ a(ξ, θ)dθ } c(ξ, τ, u(ξ, ⌊τ⌋))dτdξ (3.1) for each (x, t) ∈ [0, l] × r. before exploring the existence and uniqueness of a weighted pseudo-almost periodic solution to eq.(1.1), consider the existence of a weighted pseudo-almost periodic solution to the first-order partial differential equation cubo 16, 2 (2014) diagana space and the gas absorption model 155 ∂v ∂t (x, t) + a(x, t)v(x, t) = f(x, t), (3.2) for each (x, t) ∈ [0, l] × r. lemma 3.1. assume t $→ f(x, t) is weighted pseudo-almost periodic uniformly in x ∈ [0, l], t $→ a(x, t) is almost periodic uniformly in x ∈ [0, l]. then eq. (4) has a unique weighted pseudoalmost periodic solution, which can be explicitly expressed by v(x, t) = ∫t −∞ exp { − ∫t τ a(x, θ)dθ } f(x, τ)dτ. (3.3) moreover, v satisfies the following a priori inequality ∥v∥w ≤ 1 m ∥f∥w where the w-norm is taken in both x ∈ [0, l] and t ∈ r. proof. it is clear that the only bounded solution to eq.(4) is given by eq.(3.3). now from the weighted pseudo almost periodicity of t $→ f(x, t) it follows that there exist two functions g and h with t $→ g(x, t) ∈ ap(r) for each x ∈ [0, l] and t $→ h(x, t) ∈ pap0(r, w) for each x ∈ [0, l] such that f = g + h. consequently, v(x, t) = vg(x, t) + vh(x, t) for x ∈ [0, l] and t ∈ r, where vg(x, t) = ∫t −∞ exp { − ∫t τ a(x, θ)dθ } g(x, τ)dτ, and vh(x, t) = ∫t −∞ exp { − ∫t τ a(x, θ)dθ } h(x, τ)dτ. thus to complete the proof we must show that t $→ vg(x, t) belongs to ap(r) and that t $→ vh(x, t) belongs to pap0(r, w) uniformly in x ∈ [0, l]. the almost periodicity of t $→ vg(x, t) (x ∈ [0, l]) was established in [22]. thus, it remains to show that t $→ vh(x, t) belongs to pap0(r, w) uniformly in x ∈ [0, l]. now for r > 0, 156 najja s. al-islam cubo 16, 2 (2014) 1 µ(r, w) ∫r −r % % % % % ∫t −∞ e− ∫ t τ a(x,θ)dθh(x, τ)w(τ)dτ % % % % % w dt ≤ 1 µ(r, w) ∫r −r ∫t −∞ e− ∫ t τ a(x,θ)dθ % %h(x, τ)dτ % % w w(τ)dt ≤ 1 µ(r, w) ∫r −r &∫t −∞ e−m(t−τ) % %h(x, τ) % % w dτ ' w(τ)dt = 1 µ(r, w) ∫r −r &∫+∞ 0 e−ms % %h(x, t − s) % % w w(s)ds ' dt = ∫+∞ 0 e−ms & 1 µ(r, w) ∫r −r % %h(x, t − s) % % w w(t)dt ' ds, by letting s = t − τ (ds = −dτ). for any w ∈ w∞, it was shown in [17] that dw(r) is translation invariant with respect to the time variable t ∈ r. therefore, it follows that t $→ h(x, t − s) belongs to pap0(r, w) uniformly in x ∈ [0, l]. that is, lim r→∞ 1 µ(r, w) ∫r −r % %h(x, t − s) % % w w(t)dt = 0 uniformly in x ∈ [0, l]. using the lebesgue dominated convergence theorem completes the proof. theorem 3.2. assume t $→ a(x, t) is almost periodic and the functions t $→ ϕ(t), t $→ c(x, t, u(x, ⌊t⌋) are weighted pseudo-almost periodic uniformly in x ∈ [0, l]. additionally, assume c(x, t, u(x, ⌊t⌋) satisfies the lipschitz condition, that is, there exists k > 0 such that % %c(x, t, u(x, ⌊t⌋)) − c(x, t, v(x, ⌊t⌋)) % % w ≤ k % %u(x, ⌊t⌋) − v(x, ⌊t⌋) % % w for all x ∈ [0, l] and t ∈ r. then eq.(1.1) has a unique weighted pseudo-almost periodic solution. proof. our proof follows along the same line as that given in [22] with the appropriate modifications. indeed, for the first approximation, let u0(x, t) ≡ 0. the next approximation is u1(x, t) = ϕ(t) + ∫x 0 ∫t −∞ exp { − ∫t τ a(ξ, θ)dθ } c(ξ, τ, 0)dτdξ. now since v(x, t) = ∂ ∂x u1(x, t), then from ∂v ∂t (x, t) + a(x, t)v(x, t) = c(x, t, 0) cubo 16, 2 (2014) diagana space and the gas absorption model 157 and by lemma 3.1 the weighted pseudo-almost periodicity of v(x, t) is obtained. now u1(x, t) = ϕ(t) + ∫x 0 v(ξ, t)dξ. and hence t $→ u1(x, t) is pseudo almost periodic in t uniformly with respect to x ∈ [0, l] and u2(x, t) = ϕ(t) + ∫x 0 ∫t −∞ exp { − ∫t τ a(ξ, θ)dθ } c(ξ, τ, u1(τ, ξ))dτdξ. the relation ṽ(x, t) = ∂ ∂x u2(x, t) and the equation ∂ṽ ∂t (x, t) + a(x, t)ṽ(x, t) = c(x, t, u1(x, t)) yields the weighted pseudo-almost periodicity of u2(x, t) = ϕ(t) + ∫x 0 ṽ(ξ, t)dξ. this shows that all successive approximations un(t) are weighted pseudo-almost periodic functions in t uniformly in x ∈ [0, l]. therefore, lim n→∞ un(x, t) = u(x, t) is a weighted pseudo-almost periodic function in t uniformly with respect to x ∈ [0, l], by using theorem 2.7. 4 example to illustrate the main result of this paper(theorem 3.2), consider the following nonlinear hyperbolic second-order partial differential equation ⎧ ⎪⎪⎨ ⎪⎪⎩ uxt(x, t) + ! p(x) + sin t " ux(x, t) = h(t) sin(u(x, ⌊t⌋)) u(0, t) = 0 = ϕ(t) (4.1) where p(x) = n∑ k=0 akx k for x ∈ [0, 1] is a polynomial of degree n with real coefficients, h(t) = sin t + sin t √ 2 + w(t) sin t where w(t) = { 1, t ∈ [0, ∞), e−t 2 , t ∈ (−∞, 0), 158 najja s. al-islam cubo 16, 2 (2014) a(x, t) = p(x) + sin t, c(x, t, u(x, t)) = h(t) sin(u(x, ⌊t⌋)), ϕ(t) = 0 for all x ∈ [0, 1] and t ∈ r. suppose inf x∈[0,1],t∈r a(x, t) = a0 − 1, where a0 > 1. then all the assumptions of theorem 3.2 are fulfilled and therefore the next theorem holds. theorem 4.1. under previous assumptions, the hyperbolic partial differential equation eq. 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() cubo a mathematical journal vol.13, no¯ 03, (17–48). october 2011 applications and lipschitz results of approximation by smooth picard and gauss-weierstrass type singular integrals razvan a. mezei the university of memphis, department of mathematical sciences, memphis, tn 38152, u.s.a. email: rmezei@memphis.edu abstract we continue our studies in higher order uniform convergence with rates and in lp convergence with rates. namely, in this article we establish some lipschitz type results for the smooth picard type singular integral operators and for the smooth gaussweierstrass type singular integral operators. resumen continuamos nuestros estudios sobre convergencia uniforme de orden superior con radios y sobre convergencia lp con radios. concretamente, en este art́ıculo establecemos algunos resultados de tipo lipschitz para operadores integrales suves del tipo picard singulares y para operadores integrales singulares de tipo gauss-weierstrass. keywords: smooth picard type singular integral, smooth gauss-weierstrass type singular integral, modulus of smoothness, rate of convergence, lp convergence, higher order uniform convergence with rates, sharp inequality, lipschitz functions. mathematics subject classification: 26a15, 26d15, 41a17, 41a35, 41a60, 41a80. 18 razvan a. mezei cubo 13, 3 (2011) 1. introduction we are motivated by [1], [2], [3] and [4]. we denote by lp, 1 ≤ p < ∞, the classes of functions f (x) , integrable in −∞ < x < ∞ with the norm ‖f‖ p = [∫ ∞ −∞ |f (u)| p du ] 1 p . (1.1) the picard singular integral pξ(f; x) corresponding to the function f (x) , is defined as follows pξ(f; x) = 1 2ξ ∫ ∞ −∞ f(x + y)e−|y|/ξdy, for all x ∈ r, ξ > 0. (1.2) the gauss weierstrass singular integral wξ(f; x) corresponding to the function f (x) , is defined as follows wξ(f; x) = 1√ πξ ∫ ∞ −∞ f(x + y)e−y 2 /ξ dy, for all x ∈ r, ξ > 0. (1.3) 2. convergence with rates of smooth picard singular integral operators in the next we deal with the following smooth picard singular integral operators pr,ξ(f; x) defined as follows. for r ∈ n and n ∈ z+ we set αj =    (−1)r−j ( r j ) j −n , j = 1, . . . , r, 1 − r∑ j=1 (−1)r−j ( r j ) j −n , j = 0, (2.1) that is r∑ j=0 αj = 1. let f : r → r be lebesgue measurable, we define for x ∈ r, ξ > 0 the lebesgue integral pr,ξ(f; x) := 1 2ξ ∫ ∞ −∞   r∑ j=0 αjf(x + jt)  e −|t|/ξ dt. (2.2) we assume that pr,ξ(f; x) ∈ r for all x ∈ r. we mention the useful here formula ∫ ∞ 0 t k e −t/ξ dt = γ (k + 1) ξk+1, k > −1. (2.3) cubo 13, 3 (2011) applications and lipschitz results . . . 19 we need to introduce δk := r∑ j=1 αjj k , k = 1, . . . , n ∈ n. (2.4) denote by ⌊·⌋ the integral part. we give a special related result. proposition 1. let f be defined as above in this section. it holds that |p2,ξ(f; x) − f(x)| ≤ 1 ξ ∫ ∞ 0 (∫ |t| 0 ω2(f ′ , w)dw ) e −t/ξ dt. (2.5) proof. in theorem 1 of [1] we use n = 1, r = 2. � we also present the lipschitz type result corresponding to the theorem 1 of [1]. theorem 2. let f be defined as above in this section, with n ∈ n. furthermore we assume the following lipschitz condition: ωr ( f(n), δ ) ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then it holds that ∣ ∣ ∣ ∣ ∣ ∣ pr,ξ(f; x) − f(x) − ⌊ n 2 ⌋ ∑ m=1 f (2m)(x)δ2mξ 2m ∣ ∣ ∣ ∣ ∣ ∣ ≤ kγ (γ + r) ξn+r+γ−1. (2.6) in l.h.s.(2.6) the sum collapses when n = 1. proof. as in the proof of theorem 1, of [1], we get again that pr,ξ(f; x) − f(x) = n∑ k=1 f(k)(x) k! δk 1 2ξ (∫ ∞ −∞ t k e −|t|/ξ dt ) + r∗n, (2.7) where r∗n := 1 2ξ ∫ ∞ −∞ rn(0, t)e−|t|/ξdt, (2.8) with rn(0, t) := ∫ t 0 (t − w)n−1 (n − 1)! τ(w)dw, (2.9) and τ(w) := r∑ j=0 αjj n f (n)(x + jw) − δnf (n)(x). also we get |rn(0, t)| ≤ ∫ |t| 0 (|t| − w)n−1 (n − 1)! ωr(f (n) , w)dw. (2.10) 20 razvan a. mezei cubo 13, 3 (2011) using the lipschitz type condition we obtain |rn(0, t)| ≤ ∫ |t| 0 (|t| − w)n−1 (n − 1)! kw r−1+γ dw = k|t|n+r+γ−2 (n − 1)! ∫ |t| 0 ( 1 − w |t| )n−1 ( w |t| )r−1+γ dw = k|t|n+r+γ−1 (n − 1)! ∫ 1 0 (1 − y) n−1 y r−1+γ dy = k|t|n+r+γ−1γ (γ + r) γ (n + γ + r) . (2.11) then, by (2.3), we obtain |r∗n| ≤ 1 2ξ ∫ ∞ −∞ k|t|n+r+γ−1γ (γ + r) γ (n + γ + r) e −|t|/ξ dt = k 2ξ γ (γ + r) γ (n + γ + r) ∫ ∞ −∞ |t| n+r+γ−1 e −|t|/ξ dt = k ξ γ (γ + r) γ (n + γ + r) ∫ ∞ 0 t n+r+γ−1 e −t/ξ dt (2.3) = kγ (γ + r) ξn+r+γ−1. (2.12) we also notice that pr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δk 1 2ξ (∫ ∞ −∞ t k e −|t|/ξ dt ) = pr,ξ(f; x) − f(x) − ⌊ n 2 ⌋ ∑ m=1 f (2m)(x)δ2mξ 2m = r∗n. (2.13) by (2.12) and (2.13) we complete the proof of the theorem. � corollary 3. let f be defined as above in this section. furthermore we assume the following lipschitz condition ω2 (f ′, δ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |p2,ξ(f; x) − f(x)| ≤ kγ (γ + 2) ξ2+γ. (2.14) proof. in theorem 2 we use n = 1, r = 2. � for the case n = 0 we have theorem 4. let f be defined as above in this section, with n = 0. furthermore we assume the following lipschitz condition: ωr (f, δ) ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. it holds that |pr,ξ(f; x) − f(x)| ≤ kγ (r + γ) ξr+γ−1. (2.15) cubo 13, 3 (2011) applications and lipschitz results . . . 21 proof. as in the proof of corollary 1, of [1], with n = 0, using the lipschitz type condition, we get that |pr,ξ(f; x) − f(x)| ≤ 1 ξ ∫ ∞ 0 ωr(f, t)e −t/ξ dt ≤ 1 ξ ∫ ∞ 0 kt r−1+γ e −t/ξ dt (2.3) = kγ (r + γ) ξr+γ−1 (2.16) this completes the proof of theorem 4. � corollary 5. let f be defined as above in this section, with n = 0. furthermore we assume the following lipschitz condition: ω2 (f, δ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |p2,ξ(f; x) − f(x)| ≤ kγ (2 + γ) ξγ+1. (2.17) proof. in theorem 4 we use r = 2. � in the next we consider f ∈ cn(r), n ≥ 2 even and the simple smooth singular operator of symmetric convolution type pξ(f, x0) := 1 2ξ ∫ ∞ −∞ f(x0 + y)e −|y|/ξ dy, for all x0 ∈ r, ξ > 0. (2.18) that is pξ(f; x0) = 1 2ξ ∫ ∞ 0 ( f(x0 + y) + f(x0 − y) ) e −y/ξ dy, for all x0 ∈ r, ξ > 0. (2.19) we assume that f is such that pξ(f; x0) ∈ r, ∀x0 ∈ r, ∀ξ > 0 and ω2(f(n), h) < ∞, h > 0. note that p1,ξ = pξ and if pξ(f; x0) ∈ r then pr,ξ(f; x0) ∈ r. proposition 6. assume ω2(f, h) < ∞, h > 0. furthermore we assume the following lipschitz condition: ω2 (f, δ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖pξ(f) − f‖∞ ≤ kγ (2 + γ) 2 ξ γ+1 . (2.20) proof. using proposition 1 of [1] we obtain |pξ(f; x0) − f(x0)| ≤ 1 2ξ ∫ ∞ 0 ω2(f, y)e −y/ξ dy ≤ 1 2ξ ∫ ∞ 0 ky 1+γ e −y/ξ dy (2.3) = kγ (2 + γ) 2 ξ γ+1 , (2.21) 22 razvan a. mezei cubo 13, 3 (2011) proving the claim of the proposition. � let k2(x0) := pξ(f; x0) − f(x0) − n/2∑ ρ=1 f (2ρ)(x0)ξ 2ρ . (2.22) we give theorem 7. let f ∈ cn(r), n even, pξ(f) real valued. furthermore we assume the following lipschitz condition: ω2 ( f(n), δ ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |k2(x0)| ≤ kγ (n + γ + 2) 2n! ξ n+γ+1 . (2.23) proof. using theorem 6 of [1] we obtain |k2(x0)| ≤ 1 2ξn! ∫ ∞ 0 ω2(f (n) , y)yne−y/ξdy ≤ 1 2ξn! ∫ ∞ 0 ky 1+γ y n e −y/ξ dy (2.3) = kγ (n + γ + 2) 2n! ξ n+γ+1 , (2.24) proving the claim of the theorem. � in particular we have corollary 8. let f ∈ c4(r) such that pξ(f) is real valued. furthermore we assume the following lipschitz condition: ω2 ( f(4), δ ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |k2(x0)| ≤ kγ (γ + 6) 48 ξ γ+5 . (2.25) proof. in theorem 7 we use n = 4. � we also give corollary 9. let f ∈ c2(r), such that ω2(f ′′ , |y|) ≤ 2a|y|γ, 0 < γ ≤ 2, a > 0. then for x0 ∈ r we have ∣ ∣pξ(f; x0) − f(x0) − f ′′(x0)ξ 2 ∣ ∣ ≤ γ (α + 1)aξγ+2. (2.26) cubo 13, 3 (2011) applications and lipschitz results . . . 23 inequality (2.16 ) is sharp, namely it is attained at x0 = 0 by f∗(y) = a|y|γ+2 (γ + 1)(γ + 2) . proof. in theorem 7 of [1] we use n = 2. � we also give corollary 10. assume that ω2(f, ξ) < ∞ and n = 0. then ‖p2,ξ(f) − f‖∞ ≤ 5ω2(f, ξ), (2.27) and as ξ → 0, p2,ξ u → i with rates. proof. by formula (37) of [1] with r = 2. � next let k1 := ∥ ∥ ∥ ∥ ∥ ∥ pr,ξ(f; x) − f(x) − ⌊n/2⌋∑ m=1 [ f (2m)(x)δ2mξ 2m ] ∥ ∥ ∥ ∥ ∥ ∥ ∞ ,x . (2.28) we present corollary 11. assuming f ∈ c2(r) and ω2(f′′, ξ) < ∞, ξ > 0 we have k1 = ∥ ∥p2,ξ(f; x) − f(x) − f ′′(x)δ2ξ 2 ∥ ∥ ∞ ,x ≤ 21 4 ξ 2 ω2(f ′′ , ξ). (2.29) that is as ξ → 0 we get p2,ξ → i, pointwise with rates, given that ‖f′′‖∞ < ∞. proof. in theorem 11 of [1] we use r = n = 2. � we also present corollary 12. assuming f ∈ c2(r) and ω2(f′′, ξ) < ∞, ξ > 0 we have ‖k2(x)‖∞ ,x = ∥ ∥pξ(f; x0) − f(x0) − f ′′(x0)ξ 2 ∥ ∥ ∞ ,x ≤ 21 8 ξ 2 ω2(f ′′ , ξ). (2.30) that is as ξ → 0 we get pξ → i, pointwise with rates, given that ‖f′′‖∞ < ∞. proof. in theorem 12 of [1] we use n = 2. � 24 razvan a. mezei cubo 13, 3 (2011) 3. lp convergence with rates of smooth picard singular integral operators for r ∈ n and n ∈ z+ we let αj as in (2.1). let f ∈ cn(r) and f(n) ∈ lp(r), 1 ≤ p < ∞, we define for x ∈ r, ξ > 0 the lebesgue integral pr,ξ(f; x) as in (2.2). we need the rth lp-modulus of smoothness ωr(f (n) , h)p := sup |t|≤h ‖∆rtf(n)(x)‖p,x, h > 0, (3.1) where ∆ r tf (n)(x) := r∑ j=0 (−1)r−j ( r j ) f (n)(x + jt), (3.2) here we have that ωr(f (n), h)p < ∞, h > 0. we need to introduce δk’s as in (2.4). we define ∆(x) := pr,ξ(f; x) − f(x) − ⌊n/2⌋∑ m=1 f (2m)(x)δ2mξ 2m . (3.3) we have the following results. corollary 13. let n ∈ n and the rest as above in this section. then ‖∆(x)‖2 ≤ √ 2τξn √ (2r + 1)(4n − 2)(n − 1)! ωr(f (n) , ξ)2, (3.4) where 0 < τ := [∫ ∞ 0 (1 + u)2r+1u2n−1e−udu − (2n − 1)! ] < ∞. (3.5) hence as ξ → 0 we obtain ‖∆(x)‖2 → 0. if additionally f(2m) ∈ l2(r), m = 1, 2, . . . , ⌊ n 2 ⌋ then ‖pr,ξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 1 of [2], we place p = q = 2. � corollary 14. let f be as above in this section. in particular, for n = 1, we have ‖pr,ξ(f; ·) − f‖2 ≤ √ τξ √ (2r + 1) ωr(f ′ , ξ)2, (3.6) cubo 13, 3 (2011) applications and lipschitz results . . . 25 where 0 < τ := [∫ ∞ 0 (1 + u)2r+1ue−udu − 1 ] < ∞. (3.7) hence as ξ → 0 we obtain ‖pr,ξ(f; ·) − f‖2 → 0. proof. in theorem 1 of [2], we place p = q = 2, n = 1. � corollary 15. let f be as above in this section and n = 2. then ‖pr,ξ(f; x) − f(x) − f′′(x)δ2ξ2‖2 ≤ √ 2τξ2 √ 6(2r + 1) ωr(f ′′ , ξ)2, (3.8) where 0 < τ := [∫ ∞ 0 (1 + u)2r+1u3e−udu − 6 ] < ∞. (3.9) hence as ξ → 0 we obtain ‖∆(x)‖2 → 0. if additionally f′′ ∈ l2(r), then ‖pr,ξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 1 of [2], we place p = q = n = 2. � next we present the lipschitz type result corresponding to theorem 1 of [2]. theorem 16. let p, q > 1 such that 1 p + 1 q = 1, n ∈ n, and the rest as above in this section. furthermore we assume the following lipschitz condition: ωr ( f(n), δ ) p ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖∆(x)‖p ≤ (γ (p (r − 1 + γ + n) + 1)) 1 p 2(r+γ+n)k [ (n − 1)!q 1 q p r− 1 q +γ+n (q(n − 1) + 1) 1 q (p (r − 1 + γ) + 1) 1 p ]ξ (r−1+γ+n) . (3.10) hence as ξ → 0 we obtain ‖∆(x)‖p → 0. if additionally f(2m) ∈ lp(r), m = 1, 2, . . . , ⌊ n 2 ⌋ then ‖pr,ξ(f) − f‖p → 0, as ξ → 0. proof. as in the proof of theorem 1, [2], we get again i : = ∫ ∞ −∞ |∆(x)|pdx ≤ c1 (∫ ∞ −∞ ((∫ |t| 0 ωr(f (n) , w)ppdw ) |t| np−1 e −|pt|/2ξ ) dt ) , (3.11) where c1 := 2p−2 ξqp−1((n − 1)!)p(q(n − 1) + 1)p/q . (3.12) 26 razvan a. mezei cubo 13, 3 (2011) using the lipschitz condition, we obtain i ≤ c1 (∫ ∞ −∞ (∫ |t| 0 ( kw r−1+γ )p dw ) |t| np−1 e −p|t|/2ξ dt ) = c1k p (p (r − 1 + γ) + 1) (∫ ∞ −∞ |t| p(r−1+γ+n) e −p|t|/2ξ dt ) = 2c1k p (p (r − 1 + γ) + 1) (∫ ∞ 0 t p(r−1+γ+n) e −pt/2ξ dt ) = 2c1k p (p (r − 1 + γ) + 1) ( 2 p )p(r−1+γ+n)+1 (∫ ∞ 0 z p(r−1+γ+n) e −z/ξ dz ) (2.3) = 2c1k pγ (p (r − 1 + γ + n) + 1) (p (r − 1 + γ) + 1) ( 2 p )p(r−1+γ+n)+1 ξ p(r−1+γ+n)+1 . (3.13) thus we obtain i ≤ γ (p (r − 1 + γ + n) + 1) qp−1((n − 1)!)p(q(n − 1) + 1)p/qpp(r−1+γ+n)+1 2p(r+γ+n)kp (p (r − 1 + γ) + 1) ξ p(r−1+γ+n) . (3.14) that is finishing the proof of the theorem. � in particular we have corollary 17. let f such that the following lipschitz condition holds: ω7 ( f(4), δ ) 2 ≤ kδ6+γ, k > 0, 0 < γ ≤ 1, for any δ > 0, and the rest as above in this section. then ‖∆(x)‖2 ≤ k 6 √ (γ (2γ + 21)) 7 (2γ + 13) ξ (γ+10) . (3.15) hence as ξ → 0 we obtain ‖∆(x)‖2 → 0. if additionally f(2m) ∈ l2(r), m = 1, 2, then ‖p7,ξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 16 we place p = q = 2, n = 4, and r = 7. � the counterpart of theorem 16 follows, case of p = 1. theorem 18. let f ∈ cn(r) and f(n) ∈ l1(r), n ∈ n. furthermore we assume the following lipschitz condition: ωr ( f(n), δ ) 1 ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖∆(x)‖1 ≤ k (n − 1)! (r + γ) γ (r + γ + n) ξr+γ+n−1. (3.16) hence as ξ → 0 we obtain ‖∆(x)‖1 → 0. cubo 13, 3 (2011) applications and lipschitz results . . . 27 if additionally f(2m) ∈ l1(r), m = 1, 2, . . . , ⌊ n 2 ⌋ then ‖pr,ξ(f) − f‖1 → 0, as ξ → 0. proof. as in the proof of theorem 2 of [2] we get ‖∆(x)‖1 ≤ 1 2ξ(n − 1)! (∫ ∞ −∞ (∫ |t| 0 ωr(f (n) , w)1dw ) |t| n−1 e −|t|/ξ dt ) . (3.17) consequently we have ‖∆(x)‖1 ≤ 1 2ξ(n − 1)! (∫ ∞ −∞ (∫ |t| 0 kw r−1+γ dw ) |t| n−1 e −|t|/ξ dt ) (3.18) = k 2ξ(n − 1)! (∫ ∞ −∞ ( |t|r+γ r + γ ) |t| n−1 e −|t|/ξ dt ) = k 2ξ(n − 1)! (r + γ) (∫ ∞ −∞ |t| r+γ+n−1 e −|t|/ξ dt ) = k ξ(n − 1)! (r + γ) (∫ ∞ 0 t r+γ+n−1 e −t/ξ dt ) (2.3) = k (n − 1)! (r + γ) γ (r + γ + n) ξr+γ+n−1, (3.19) proving (3.16). � corollary 19. let f ∈ c2(r) and f′′ ∈ l1(r). furthermore we assume the following lipschitz condition: ω2 (f ′′, δ) 1 ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖∆(x)‖1 ≤ k (2 + γ) γ (4 + γ) ξγ+3. (3.20) hence as ξ → 0 we obtain ‖∆(x)‖1 → 0. if additionally f′′ ∈ l1(r),then ‖p2,ξ(f) − f‖1 → 0, as ξ → 0. proof. in theorem 18 we place n = r = 2. � next, when n = 0 we get proposition 20. let r ∈ n and the rest as above. then ‖pr,ξ(f) − f‖2 ≤ θ1/2ωr(f, ξ)2, (3.21) where 0 < θ := ∫ ∞ 0 (1 + x)2re−xdx < ∞. (3.22) hence as ξ → 0 we obtain pr,ξ → unit operator i in the l2 norm. 28 razvan a. mezei cubo 13, 3 (2011) proof. in the proof of proposition 1 of [2] we use p = q = 2. � we continue with proposition 21. let p, q > 1 such that 1 p + 1 q = 1 and the rest as above. furthermore we assume the following lipschitz condition: ωr (f, δ)p ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖pr,ξ(f) − f‖p ≤ p √ γ (p (r − 1 + γ) + 1) k q1/q 2(r+γ)ξ(r+γ−1) p( r−1+γ+ 1 p ) . (3.23) hence as ξ → 0 we obtain pr,ξ → unit operator i in the lp norm, p > 1. proof. as in the proof of proposition 1 of [2] we find ∫ ∞ −∞ |pr,ξ(f; x) − f(x)| p dx ≤ 1 2p−1ξp ( 4ξ q )p/q (∫ ∞ 0 ωr(f, t) p pe −pt/(2ξ) dt ) ≤ 1 2p−1ξp ( 4ξ q )p/q (∫ ∞ 0 ( kt r−1+γ )p e −pt/(2ξ) dt ) (2.3) = kp qp−1 γ (p (r − 1 + γ) + 1) 2p(r+γ)ξ(r−1+γ)p p(p(r+γ−1)+1) . (3.24) we have established the claim of the proposition. � corollary 22. let f such that the following lipschitz condition holds: ω4 (f, δ)2 ≤ kδ3+γ, k > 0, 0 < γ ≤ 1, for any δ > 0, and the rest as above in this section. then ‖p4,ξ(f) − f‖2 ≤ √ γ (2γ + 7)kξ(3+γ). (3.25) hence as ξ → 0 we obtain p4,ξ → unit operator i in the l2 norm. proof. in proposition 21 we place p = q = 2 and r = 4. � in general, for the l1 case, n = 0 we have proposition 23. it holds ‖p2,ξf − f‖1 ≤ 5ω2(f, ξ)1. (3.26) hence as ξ → 0 we get p2,ξ → i in the l1 norm. proof. in the proof of proposition 2 of [2] we use r = 2. � proposition 24. we assume the following lipschitz condition: ωr (f, δ)1 ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖pr,ξf − f‖1 ≤ kγ (r + γ) ξr−1+γ. (3.27) cubo 13, 3 (2011) applications and lipschitz results . . . 29 hence as ξ → 0 we get pr,ξ → i in the l1 norm. proof. as in the proof of proposition 2 of [2] we get ∫ ∞ −∞ |pr,ξ(f; x) − f(x)| dx ≤ 1 ξ ∫ ∞ 0 ωr(f, t)1e −t/ξ dt ≤ k ξ ∫ ∞ 0 t r−1+γ e −t/ξ dt = kγ (r + γ) ξr−1+γ, (3.28) proving the claim. � corollary 25. assume the following lipschitz condition: ω2 (f, δ)1 ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖p2,ξf − f‖1 ≤ kγ (2 + γ) ξ1+γ. (3.29) hence as ξ → 0 we get p2,ξ → i in the l1 norm. proof. in proposition 24 we place r = 2. � in the next we consider f ∈ cn(r) and f(n) ∈ lp(r), n = 0 or n ≥ 2 even, 1 ≤ p < ∞ and the similar smooth singular operator of symmetric convolution type pξ(f; x) = 1 2ξ ∫ ∞ −∞ f(x + y)e−|y|/ξdy, for all x ∈ r, ξ > 0. (3.30) denote k(x) := pξ(f; x) − f(x) − n/2∑ ρ=1 f (2ρ)(x)ξ2ρ. (3.31) we give theorem 26. let n ≥ 2 even and the rest as above. then ‖k(x)‖2 ≤ ( √ τ̃ 20(2n − 1) ) ξn (n − 1)! ω2(f (n) , ξ)2, (3.32) where 0 < τ̃ = (∫ ∞ 0 (1 + x)5x2n−1e−xdx − (2n − 1)! ) < ∞. (3.33) hence as ξ → 0 we get ‖k(x)‖2 → 0. if additionally f(2m) ∈ l2(r), m = 1, 2, . . . , n2 then ‖pξ(f) − f‖2 → 0, as ξ → 0. proof. in the proof of theorem 3 of [2] we use p = q = 2. � 30 razvan a. mezei cubo 13, 3 (2011) it follows a lipschitz type approximation result. theorem 27. let p, q > 1 such that 1 p + 1 q = 1, n ≥ 2 even and the rest as above. furthermore we assume the following lipschitz condition: ω2 ( f(n), δ ) p ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖k(x)‖p ≤ ( 2 p )(γ+n+1) k [γ (p (γ + n + 1) + 1)] 1/p (n − 1)!q1/qp1/p(q(n − 1) + 1)1/q [p (γ + 1) + 1] 1/p ξ γ+n+1 . (3.34) hence as ξ → 0 we get ‖k(x)‖p → 0. if additionally f(2m) ∈ lp(r), m = 1, 2, . . . , n2 then ‖pξ(f) − f‖p → 0, as ξ → 0. proof. as in the proof of theorem 3, of [2] we find ∫ ∞ −∞ |k(x)|pdx ≤ c2 (∫ ∞ 0 (∫ y 0 ω2(f (n) , t)ppdt ) y pn−1 e −py/(2ξ) dy ) ≤ kpc2 (∫ ∞ 0 ( yp(γ+1)+1 p (γ + 1) + 1 ) y pn−1 e −py/(2ξ) dy ) = kpc2 p (γ + 1) + 1 ( 2 p )p(γ+n+1)+1 (∫ ∞ 0 z p(γ+n+1) e −z/ξ dz ) (2.3) = kpc2γ (p (γ + n + 1) + 1) p (γ + 1) + 1 ( 2 p )p(γ+n+1)+1 ξ p(γ+n+1)+1 . (3.35) where here we denoted c2 := 1 2ξqp/q((n − 1)!)p(q(n − 1) + 1)p/q . (3.36) we have established the claim of the theorem. � corollary 28. assume the following lipschitz condition: ω2 (f ′′, δ) 2 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0, and the rest as above in this section.then ‖k(x)‖2 ≤ √ γ (2γ + 7) 6γ + 9 k 2 ξ γ+3 . (3.37) hence as ξ → 0 we get ‖k(x)‖2 → 0. if additionally f′′ ∈ l2(r), then ‖pξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 27 we place p = q = n = 2. � theorem 29. let f ∈ c2(r) and f′′ ∈ l1(r). here k(x) = pξ(f; x) − f(x) − f′′(x)ξ2. then ‖k(x)‖1 ≤ 8ω2(f′′, ξ)1ξ2. (3.38) hence as ξ → 0 we obtain ‖k(x)‖1 → 0. cubo 13, 3 (2011) applications and lipschitz results . . . 31 also ‖pξ(f) − f‖1 → 0, as ξ → 0. proof. in the proof of theorem 4 of [2] we use n = 2. � the lipschitz case of p = 1 follows. theorem 30. let f ∈ cn(r) and f(n) ∈ l1(r), n ≥ 2 even. furthermore we assume the following lipschitz condition: ω2 ( f(n), δ ) 1 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖k(x)‖1 ≤ γ (γ + n + 2) k 2(n − 1)! (γ + 2) ξ γ+n+1 . (3.39) hence as ξ → 0 we obtain ‖k(x)‖1 → 0. if additionally f(2m) ∈ l1(r), m = 1, 2, . . . , n2 then ‖pξ(f) − f‖1 → 0, as ξ → 0. proof. as in the proof of theorem 4 of [2] we have ‖k(x)‖1 ≤ 1 2ξ (∫ ∞ 0 (∫ y 0 ω2(f (n) , t)1dt ) yn−1 (n − 1)! e −y/ξ dy ) ≤ 1 2ξ (∫ ∞ 0 (∫ y 0 kt γ+1 dt ) yn−1 (n − 1)! e −y/ξ dy ) = k 2ξ(n − 1)! (γ + 2) (∫ ∞ 0 y γ+n+1 e −y/ξ dy ) (2.3) = γ (γ + n + 2) k 2(n − 1)! (γ + 2) ξ γ+n+1 . (3.40) we have proved the claim of the theorem. � corollary 31. let f ∈ c6(r) and f(6) ∈ l1(r). furthermore we assume the following lipschitz condition: ω2 ( f(6), δ ) 1 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖k(x)‖1 ≤ γ (γ + 8) k 240 (γ + 2) ξ γ+7 . (3.41) hence as ξ → 0 we obtain ‖k(x)‖1 → 0. if additionally f(2m) ∈ l1(r), m = 1, 2, 3 then ‖pξ(f) − f‖1 → 0, as ξ → 0. proof. in theorem 30 we place n = 6. � the case of n = 0 follows. proposition 32. let f as above in this section. then ‖pξ(f) − f‖2 ≤ √ 65 2 ω2(f, ξ)2. (3.42) hence as ξ → 0 we obtain pξ → i in the l2 norm. 32 razvan a. mezei cubo 13, 3 (2011) proof. in the proof of proposition 3 of [2] we use p = q = 2. � the related lipschitz case for n = 0 comes next. proposition 33. let p, q > 1 such that 1 p + 1 q = 1 and the rest as above. furthermore we assume the following lipschitz condition: ω2 (f, δ)p ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖pξ(f) − f‖p ≤ ( 2 p )1+γ [γ ((1 + γ) p + 1)] 1/p k q1/qp1/p ξ 1+γ . (3.43) hence as ξ → 0 we obtain pξ → i in the lp norm, p > 1. proof. as in the proof of proposition 3 of [2] we get ∫ ∞ −∞ |pξ(f; x) − f(x)| p dx ≤ 1 2ξqp/q (∫ ∞ 0 ω2(f, y) p pe −py/(2ξ) dy ) ≤ kp 2ξqp/q (∫ ∞ 0 y (1+γ)p e −py/(2ξ) dy ) (2.3) = kp qp/qp ( 2 p )(1+γ)p γ ((1 + γ) p + 1) ξ(1+γ)p. (3.44) the proof of the claim is now completed. � a particular example follows corollary 34. let f as above in this section. furthermore we assume the following lipschitz condition: ω2 (f, δ)2 ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖pξ(f) − f‖2 ≤ k 2 √ γ (3 + 2γ)ξ1+γ. (3.45) hence as ξ → 0 we obtain pξ → i in the l2 norm. proof. in proposition 33 we place p = q = 2. � it follows the lipschitz type result proposition 35. assume the following lipschitz condition: ω2 (f, δ)1 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0. it holds, ‖pξf − f‖1 ≤ k 2 γ (γ + 2) ξγ+1. (3.46) hence as ξ → 0 we get pξ → i in the l1 norm. proof. as in the proof of proposition 4 of [2] we derive ∫ ∞ −∞ |pξ(f; x) − f(x)|dx ≤ 1 2ξ ∫ ∞ 0 ω2(f, y)1e −y/ξ dy ≤ 1 2ξ ∫ ∞ 0 ky γ+1 e −y/ξ dy (2.3) = k 2 γ (γ + 2) ξγ+1, (3.47) cubo 13, 3 (2011) applications and lipschitz results . . . 33 proving the claim. � 4. convergence with rates of smooth gauss weierstrass singular integral operators in the next we deal with the following smooth gauss weierstrass singular integral operators wr,ξ(f; x) defined as follows. for r ∈ n and n ∈ z+ we set αj’s as in (2.1). let f : r → r be lebesgue measurable, we define for x ∈ r, ξ > 0 the lebesgue integral wr,ξ(f; x) := 1√ πξ ∫ ∞ −∞   r∑ j=0 αjf(x + jt)  e −t 2 /ξ dt. (4.1) we assume that wr,ξ(f; x) ∈ r for all x ∈ r. we mention the useful here formula ∫ ∞ 0 t k e −t 2 /ξ dt = 1 2 γ ( k + 1 2 ) ξ k+1 2 , for any k > −1. (4.2) we also need to introduce δk’s as in (2.4). proposition 36. let f ∈ c1(r) be defined as above in this section, and assume that w2,ξ(f; x) ∈ r for all x ∈ r. then |w2,ξ(f; x) − f(x)| ≤ 2√ πξ ∫ ∞ 0 (∫ |t| 0 ω2(f ′ , w)dw ) e − t 2 ξ dt. (4.3) proof. in theorem 1 of [3] we use n = 1, r = 2. � we present the lipschitz type result corresponding to the theorem 1 of [3]. theorem 37. let f ∈ cn(r), n ∈ z+ and assume that wr,ξ(f; x) ∈ r for all x ∈ r. furthermore we assume the following lipschitz condition: ωr ( f(n), δ ) ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then it holds that ∣ ∣ ∣ ∣ ∣ ∣ wr,ξ(f; x) − f(x) − ⌊n/2⌋∑ m=1 f (2m)(x)δ2m 1 m! ( ξ 4 )m ∣ ∣ ∣ ∣ ∣ ∣ ≤ k√ π γ (γ + r) γ (n + γ + r) γ ( n + r + γ 2 ) ξ n+r+γ−1 2 . (4.4) 34 razvan a. mezei cubo 13, 3 (2011) in l.h.s.(4.4) the sum collapses when n = 1. proof. as in the proof of theorem 1, of [3], we get again that wr,ξ(f; x) − f(x) = n∑ k=1 f(k)(x) k! δk 1√ πξ (∫ ∞ −∞ t k e − t 2 ξ dt ) + r∗n, (4.5) where r∗n := 1√ πξ ∫ ∞ −∞ rn(0, t)e− t2 ξ dt, (4.6) with rn(0, t) := ∫ t 0 (t − w)n−1 (n − 1)! τ(w)dw, (4.7) and τ(w) := r∑ j=0 αjj n f (n)(x + jw) − δnf (n)(x). also we get |rn(0, t)| ≤ ∫ |t| 0 (|t| − w)n−1 (n − 1)! ωr(f (n) , w)dw. (4.8) using the lipschitz type condition we obtain again |rn(0, t)| ≤ k|t|n+r+γ−1γ (γ + r) γ (n + γ + r) , (4.9) and, using (4.2), we obtain |r∗n| ≤ 1√ πξ ∫ ∞ −∞ k|t|n+r+γ−1γ (γ + r) γ (n + γ + r) e − t 2 ξ dt = k√ πξ γ (γ + r) γ (n + γ + r) ∫ ∞ −∞ |t| n+r+γ−1 e − t 2 ξ dt = 2k√ πξ γ (γ + r) γ (n + γ + r) ∫ ∞ 0 t n+r+γ−1 e − t 2 ξ dt (4.2) = k√ π γ (γ + r) γ (n + γ + r) γ ( n + r + γ 2 ) ξ n+r+γ−1 2 . (4.10) we notice also that wr,ξ(f; x) − f(x) − n∑ k=1 f(k)(x) k! δk 1√ πξ (∫ ∞ −∞ t k e − t 2 ξ dt ) = wr,ξ(f; x) − f(x) − ⌊ n 2 ⌋ ∑ m=1 [ f(2m)(x) (2m)! √ π δ2mγ ( 2m + 1 2 ) ξ m ] = r∗n. (4.11) cubo 13, 3 (2011) applications and lipschitz results . . . 35 furthermore we have that 1 (2m)! √ π γ ( 2m + 1 2 ) = = 1 (2m) · (2m − 1) · ... · 3 · 2 · 1 · 1√ π · 2m − 1 2 · 2m − 3 2 · ... · 3 2 · 1 2 γ ( 1 2 ) = 1 m! ( 1 4 )m . (4.12) by (4.10), (4.11) and (4.12) we complete the proof of the theorem. � corollary 38. let f ∈ c1(r), and assume that w2,ξ(f; x) ∈ r for all x ∈ r. furthermore we assume the following lipschitz condition: ω2 (f ′, δ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |w2,ξ(f; x) − f(x)| ≤ k (γ + 2) √ π γ ( 3 + γ 2 ) ξ 2+γ 2 . (4.13) proof. in theorem 37 we use n = 1, r = 2. � for the case n = 0 we have theorem 39. let f be defined as above in this section, with n = 0. furthermore we assume the following lipschitz condition: ωr (f, δ) ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. it holds that |wr,ξ(f; x) − f(x)| ≤ k√ π γ ( r + γ 2 ) ξ r+γ−1 2 . (4.14) proof. as in the proof of corollary 1, of [3], with n = 0, using the lipschitz type condition, we get that |wr,ξ(f; x) − f(x)| ≤ 2√ πξ ∫ ∞ 0 ωr(f, t)e − t 2 ξ dt ≤ 2√ πξ ∫ ∞ 0 kt r−1+γ e − t 2 ξ dt (4.2) = k√ π γ ( r + γ 2 ) ξ r+γ−1 2 . (4.15) this completes the proof of theorem 39. � corollary 40. let f be defined as above in this section, with n = 0. furthermore we assume the following lipschitz condition: ω2 (f, δ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |w2,ξ(f; x) − f(x)| ≤ k√ π γ ( 2 + γ 2 ) ξ γ+1 2 . (4.16) proof. in theorem 39 we use r = 2. � 36 razvan a. mezei cubo 13, 3 (2011) in the next we consider f ∈ cn(r), n ≥ 2 even and the simple smooth singular operator of symmetric convolution type wξ(f, x0) := 1√ πξ ∫ ∞ −∞ f(x0 + y)e −y 2 /ξ dy, for all x0 ∈ r, ξ > 0. (4.17) that is wξ(f; x0) = 1√ πξ ∫ ∞ 0 (f(x0 + y) + f(x0 − y)) e −y 2 /ξ dy, for all x0 ∈ r, ξ > 0. (4.18) we assume that f is such that wξ(f; x0) ∈ r, ∀x0 ∈ r, ∀ξ > 0 and ω2(f(n), h) < ∞, h > 0. note that w1,ξ = wξ and if wξ(f; x0) ∈ r then wr,ξ(f; x0) ∈ r. proposition 41. assume f ∈ cn(r), ω2(f, h) < ∞, h > 0. furthermore we assume the following lipschitz condition: ω2 (f, δ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖wξ(f) − f‖∞ ≤ k 2 √ π γ ( 2 + γ 2 ) ξ γ+1 2 . (4.19) proof. using proposition 1 of [3] we obtain |wξ(f; x0) − f(x0)| ≤ 1√ πξ ∫ ∞ 0 ω2(f, y)e −y 2 /ξ dy ≤ 1√ πξ ∫ ∞ 0 ky 1+γ e −y 2 /ξ dy (4.2) = k 2 √ π γ ( 2 + γ 2 ) ξ γ+1 2 , (4.20) proving the claim of the proposition. � define the quantity k2(x0) := wξ(f; x0) − f(x0) − n/2∑ ρ=1 f (2ρ)(x0) 1 ρ! ( ξ 4 )ρ . (4.21) we give theorem 42. let f ∈ cn(r), n even, wξ(f) real valued. furthermore we assume the following lipschitz condition: ω2 ( f(n), δ ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |k2(x0)| ≤ k n!2 √ π γ ( n + γ + 2 2 ) ξ n+γ+1 2 . (4.22) cubo 13, 3 (2011) applications and lipschitz results . . . 37 proof. using theorem 6 of [3] we obtain |k2(x0)| ≤ 1 n! √ πξ ∫ ∞ 0 ω2(f (n) , y)yne−y 2 /ξ dy ≤ 1 n! √ πξ ∫ ∞ 0 ky 1+γ y n e −y 2 /ξ dy (4.2) = k n!2 √ π γ ( n + γ + 2 2 ) ξ n+γ+1 2 , (4.23) proving the claim of the theorem. � in particular we have corollary 43. let f ∈ c4(r) such that wξ(f) is real valued. furthermore we assume the following lipschitz condition: ω2 ( f(4), δ ) ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then |k2(x0)| ≤ k 48 √ π γ ( γ + 6 2 ) ξ γ+5 2 . (4.24) proof. in theorem 42 we use n = 4. � we also give corollary 44. let f ∈ c2(r), such that ω2(f ′′ , |y|) ≤ 2a|y|γ, 0 < γ ≤ 2, a > 0. then for x0 ∈ r we have ∣ ∣ ∣ ∣ wξ(f; x0) − f(x0) − f′′(x0)ξ 4 ∣ ∣ ∣ ∣ ≤ a (γ + 1)(γ + 2) √ π γ ( 3 + γ 2 ) ξ 2+γ 2 . (4.25) inequality (4.25) is sharp, namely it is attained at x0 = 0 by f∗(y) = a|y|γ+2 (γ + 1)(γ + 2) . proof. in theorem 7 of [3] we use n = 2. � we also give corollary 45. assume that ω2(f, ξ) < ∞ and n = 0. then ‖w2,ξ(f) − f‖∞ ≤ [ 2√ π + 3 2 ] ω2(f, √ ξ). (4.26) 38 razvan a. mezei cubo 13, 3 (2011) and as ξ → 0, w2,ξ u → i with rates. proof. by formula (37) of [3] with r = 2. � define the quantity k1 := ∥ ∥ ∥ ∥ ∥ ∥ wr,ξ(f; x) − f(x) − ⌊n/2⌋∑ m=1 f (2m)(x)δ2m 1 m! ( ξ 4 )m ∥ ∥ ∥ ∥ ∥ ∥ ∞ ,x . (4.27) we present corollary 46. assuming f ∈ c2(r) and ω2(f′′, ξ) < ∞, ξ > 0 we have k1 = ∥ ∥ ∥ ∥ w2,ξ(f; x) − f(x) − f ′′(x)δ2 ξ 4 ∥ ∥ ∥ ∥ ∞ ,x ≤ { 1 3 √ π + 5 16 } ω2(f ′′ , √ ξ)ξ. (4.28) that is as ξ → 0 we get w2,ξ → i, pointwise with rates, given that ‖f′′‖∞ < ∞. proof. in theorem 11 of [3] we use r = n = 2. � we also present corollary 47. assuming f ∈ c2(r) and ω2(f′′, ξ) < ∞, ξ > 0 we have ∥ ∥k2(x) ∥ ∥ ∞ ,x = ∥ ∥ ∥ ∥ wξ(f; x0) − f(x0) − f ′′(x0) ξ 4 ∥ ∥ ∥ ∥ ∞ ,x ≤ { 1 6 √ π + 5 32 } ω2(f ′′ , √ ξ)ξ. (4.29) that is as ξ → 0 we get wξ → i, pointwise with rates, given that ‖f′′‖∞ < ∞. proof. in theorem 12 of [3] we use n = 2. � 5. lp convergence with rates of smooth gauss weierstrass singular integral operators for r ∈ n and n ∈ z+ we let αj as in (2.1). cubo 13, 3 (2011) applications and lipschitz results . . . 39 let f ∈ cn(r) and f(n) ∈ lp(r), 1 ≤ p < ∞, we define for x ∈ r, ξ > 0 the lebesgue integral wr,ξ(f; x) as in (4.1). the rth lp-modulus of smoothness ωr(f (n), h)p was defined in (3.1). here we have that ωr(f (n), h)p < ∞, h > 0. the δk’s were introduced in (2.4). we define ∆(x) := wr,ξ(f; x) − f(x) − ⌊n/2⌋∑ m=1 f (2m)(x)δ2m 1 m! ( ξ 4 )m . (5.1) we have the following results. corollary 48. let n ∈ n and the rest as above in this section. then ‖∆(x)‖2 ≤ √ 2τξ n 2 (n − 1)! 4 √ π √ (2r + 1) (2n − 1) ωr(f (n) , √ ξ)2, (5.2) where 0 < τ := [∫ ∞ 0 (1 + u) 2r+1 u 2n−1 e −u 2 du − ∫ ∞ 0 u 2n−1 e −u 2 du ] < ∞. (5.3) hence as ξ → 0 we obtain ‖∆(x)‖2 → 0. if additionally f(2m) ∈ l2(r), m = 1, 2, . . . , ⌊ n 2 ⌋ then ‖wr,ξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 1 of [4], we place p = q = 2. � corollary 49. let f be as above in this section. in particular, for n = 1, we have ‖wr,ξ(f; ·) − f‖2 ≤ √ 2τ 4 √ π √ (2r + 1) √ ξωr(f ′ , √ ξ)2, (5.4) where 0 < τ := [∫ ∞ 0 (1 + u) 2r+1 ue −u 2 du − 1 2 ] < ∞. (5.5) hence as ξ → 0 we obtain ‖wr,ξ(f; ·) − f‖2 → 0. proof. in theorem 1 of [4], we place p = q = 2, n = 1. � corollary 50. let f be as above in this section and n = 2. then ‖wr,ξ(f; x) − f(x) − f′′(x)δ2 4 ξ‖2 ≤ √ 2τ 4 √ π √ 3 (2r + 1) ξωr(f ′′ , √ ξ)2, (5.6) 40 razvan a. mezei cubo 13, 3 (2011) where 0 < τ := [∫ ∞ 0 (1 + u) 2r+1 u 3 e −u 2 du − 1 2 ] < ∞. (5.7) hence as ξ → 0 we obtain ‖∆(x)‖2 → 0. if additionally f′′ ∈ l2(r), then ‖wr,ξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 1 of [4], we place p = q = n = 2. � next we present the lipschitz type result corresponding to theorem 1 of [4]. theorem 51. let p, q > 1 such that 1 p + 1 q = 1, n ∈ n, and the rest as above in this section. furthermore we assume the following lipschitz condition: ωr ( f(n), δ ) p ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖∆(x)‖p ≤ ( γ ( p(r−1+γ+n)+1 2 )) 1 p 2 (r+γ+n) 2 kξ (r−1+γ+n) 2 [ (n − 1)!p r− 1 q +γ+n 2 q 1 2q π 1 2p (q(n − 1) + 1) 1 q (p (r − 1 + γ) + 1) 1 p ]. (5.8) hence as ξ → 0 we obtain ‖∆(x)‖p → 0. if additionally f(2m) ∈ lp(r), m = 1, 2, . . . , ⌊ n 2 ⌋ then ‖wr,ξ(f) − f‖p → 0, as ξ → 0. proof. as in the proof of theorem 1, [4], we get again i := ∫ ∞ −∞ |∆(x)|pdx ≤ c1 (∫ ∞ −∞ (∫ |t| 0 ωr(f (n) , w)ppdw ) |t| np−1 e − pt2 2ξ dt ) , (5.9) where c1 := 2 p−1 2 q p−1 2 √ πξ((n − 1)!)p(q(n − 1) + 1)p/q . (5.10) using the lipschitz condition, we obtain i ≤ c1 (∫ ∞ −∞ (∫ |t| 0 ( kw r−1+γ )p dw ) |t| np−1 e − pt2 2ξ dt ) = c1k p (p (r − 1 + γ) + 1) (∫ ∞ −∞ |t| p(r−1+γ+n) e − pt2 2ξ dt ) = 2c1k p (p (r − 1 + γ) + 1) (∫ ∞ 0 t p(r−1+γ+n) e − pt2 2ξ dt ) (4.2) = c1k pγ ( p(r−1+γ+n)+1 2 ) (p (r − 1 + γ) + 1) ( 2 p ) p(r−1+γ+n)+1 2 ξ p(r−1+γ+n)+1 2 . (5.11) cubo 13, 3 (2011) applications and lipschitz results . . . 41 thus we obtain i ≤ kp2 p(r+γ+n) 2 γ ( p(r−1+γ+n)+1 2 ) ξ p(r−1+γ+n) 2 q p−1 2 √ π((n − 1)!)p(q(n − 1) + 1)p/q (p (r − 1 + γ) + 1) p p(r−1+γ+n)+1 2 . (5.12) that is finishing the proof of the theorem. � in particular we have corollary 52. let f such that the following lipschitz condition holds: ω7 ( f(4), δ ) 2 ≤ kδ6+γ, k > 0, 0 < γ ≤ 1, for any δ > 0, and the rest as above in this section. then ‖∆(x)‖2 ≤ k 6 √ √ √ √ γ ( 2γ+21 2 ) 7 √ π (2γ + 13) ξ (γ+10) 2 . (5.13) hence as ξ → 0 we obtain ‖∆(x)‖2 → 0. if additionally f(2m) ∈ l2(r), m = 1, 2, then ‖w7,ξ(f) − f‖2 → 0, as ξ → 0. proof. in theorem 51 we place p = q = 2, n = 4, and r = 7. � the counterpart of theorem 51 follows, case of p = 1. theorem 53. let f ∈ cn(r) and f(n) ∈ l1(r), n ∈ n. furthermore we assume the following lipschitz condition: ωr ( f(n), δ ) 1 ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖∆(x)‖1 ≤ k (n − 1)! (r + γ) √ π γ ( r + γ + n 2 ) ξ r+γ+n−1 2 . (5.14) hence as ξ → 0 we obtain ‖∆(x)‖1 → 0. if additionally f(2m) ∈ l1(r), m = 1, 2, . . . , ⌊ n 2 ⌋ then ‖wr,ξ(f) − f‖1 → 0, as ξ → 0. proof. as in the proof of theorem 2, [4] we get ‖∆(x)‖1 ≤ 1 (n − 1)! √ πξ (∫ ∞ −∞ (∫ |t| 0 ωr(f (n) , w)1dw ) |t| n−1 e −t 2 /ξ dt ) . (5.15) consequently we have 42 razvan a. mezei cubo 13, 3 (2011) ‖∆(x)‖1 ≤ 1 (n − 1)! √ πξ (∫ ∞ −∞ (∫ |t| 0 kw r−1+γ dw ) |t| n−1 e −t 2 /ξ dt ) = k (n − 1)! √ πξ (∫ ∞ −∞ ( |t|r+γ r + γ ) |t| n−1 e −t 2 /ξ dt ) = k (n − 1)! (r + γ) √ πξ (∫ ∞ −∞ |t| r+γ+n−1 e −t 2 /ξ dt ) = 2k (n − 1)! (r + γ) √ πξ (∫ ∞ 0 t r+γ+n−1 e −t 2 /ξ dt ) (4.2) = k (n − 1)! (r + γ) √ πξ γ ( r + γ + n 2 ) ξ r+γ+n 2 . (5.16) we have gotten that ‖∆(x)‖1 ≤ k (n − 1)! (r + γ) √ π γ ( r + γ + n 2 ) ξ r+γ+n−1 2 . (5.17) hence the validity of (5.14). � corollary 54. let f ∈ c2(r) and f′′ ∈ l1(r). furthermore we assume the following lipschitz condition: ω2 (f ′′, δ) 1 ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖∆(x)‖1 ≤ k (2 + γ) √ π γ ( 4 + γ 2 ) ξ γ+3 2 . (5.18) hence as ξ → 0 we obtain ‖∆(x)‖1 → 0. also we get ‖w2,ξ(f) − f‖1 → 0, as ξ → 0. proof. in theorem 53 we place n = r = 2. � next, when n = 0 we get proposition 55. let r ∈ n and the rest as above. then ‖wr,ξ(f) − f‖2 ≤ 2 3 4 θ 1 2 q 1 4 π 1 4 ωr(f, √ ξ)2, (5.19) where 0 < θ := ∫ ∞ 0 (1 + t) 2r e −t 2 dt < ∞. (5.20) hence as ξ → 0 we obtain wr,ξ → unit operator i in the l2 norm, p > 1. proof. in the proof of proposition 1 of [4] we use p = q = 2. � we continue with cubo 13, 3 (2011) applications and lipschitz results . . . 43 proposition 56. let p, q > 1 such that 1 p + 1 q = 1 and the rest as above. furthermore we assume the following lipschitz condition: ωr (f, δ)p ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖wr,ξ(f) − f‖p ≤ p √ γ ( p(r − 1 + γ) + 1 2 )( 2 p ) r+γ 2 ( p q ) 1 2q k p √ π ξ (r−1+γ) 2 . (5.21) hence as ξ → 0 we obtain wr,ξ → unit operator i in the lp norm, p > 1. proof. as in the proof of proposition 1 of [4] we find ∫ ∞ −∞ |wr,ξ(f; x) − f(x)| p dx ≤ 2 (πξ) p 2 ( 2πξ q ) p 2q ∫ ∞ 0 ωr(f, t) p pe − pt2 2ξ dt ≤ 2kp (πξ) p 2 ( 2πξ q ) p 2q ∫ ∞ 0 t p(r−1+γ) e − pt2 2ξ dt (4.2) = kp π p 2 ( 2π q ) p 2q ( 2 p ) p(r−1+γ)+1 2 γ ( p(r − 1 + γ) + 1 2 ) ξ p(r−1+γ) 2 . (5.22) we have established the claim of the proposition. � corollary 57. let f such that the following lipschitz condition holds: ω4 (f, δ)2 ≤ kδ3+γ, k > 0, 0 < γ ≤ 1, for any δ > 0, and the rest as above in this section. then ‖w4,ξ(f) − f‖2 ≤ √ γ ( 2γ + 7 2 ) k√ π ξ (3+γ) 2 . (5.23) hence as ξ → 0 we obtain w4,ξ → unit operator i in the l2 norm. proof. in proposition 56 we place p = q = 2 and r = 4. � in the l1 case, n = 0 we have proposition 58. it holds ‖w2,ξf − f‖1 ≤ ( 2√ π + 3 2 ) ω2(f, √ ξ)1. (5.24) hence as ξ → 0 we get w2,ξ → i in the l1 norm. proof. in the proof of proposition 2 of [4] we use r = 2. � proposition 59. we assume the following lipschitz condition: ωr (f, δ)1 ≤ kδr−1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖wr,ξf − f‖1 ≤ k√ π γ ( r + γ 2 ) ξ r−1+γ 2 . (5.25) 44 razvan a. mezei cubo 13, 3 (2011) hence as ξ → 0 we get wr,ξ → i in the l1 norm. proof. as in the proof of proposition 2 of [4] we get ∫ ∞ −∞ |wr,ξ(f; x) − f(x)| dx ≤ 1√ πξ ∫ ∞ −∞ ωr(f, |t|)1e −t 2 /ξ dt ≤ 1√ πξ ∫ ∞ −∞ k|t| r−1+γ e −t 2 /ξ dt = 2k√ πξ ∫ ∞ 0 t r−1+γ e −t 2 /ξ dt (4.2) = k√ π γ ( r + γ 2 ) ξ r−1+γ 2 . (5.26) we have proved the claim of the proposition. � corollary 60. assume the following lipschitz condition: ω2 (f, δ)1 ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖w2,ξf − f‖1 ≤ k√ π γ ( 2 + γ 2 ) ξ 1+γ 2 . (5.27) hence as ξ → 0 we get w2,ξ → i in the l1 norm. proof. in proposition 59 we place r = 2. � in the next we consider f ∈ cn(r) and f(n) ∈ lp(r), n = 0 or n ≥ 2 even, 1 ≤ p < ∞ and the similar smooth singular operator of symmetric convolution type wξ(f; x) = 1√ πξ ∫ ∞ −∞ f(x + y)e−y 2 /ξ dy, for all x ∈ r, ξ > 0. (5.28) denote k(x) := wξ(f; x) − f(x) − n/2∑ ρ=1 f(2ρ)(x) ρ! · ( ξ 4 )ρ . (5.29) we give theorem 61. let n ≥ 2 even and the rest as above. then ‖k(x)‖2 ≤ √ τ̃ 10 √ π (2n − 1) ξ n 2 (n − 1)! ω2(f (n) , √ ξ)2, (5.30) where 0 < τ̃ = ∫ ∞ 0 ( (1 + u) 5 − 1 ) u 2n−1 e −u 2 du < ∞. (5.31) hence as ξ → 0 we get ‖k(x)‖2 → 0. cubo 13, 3 (2011) applications and lipschitz results . . . 45 if additionally f(2m) ∈ l2(r), m = 1, 2, . . . , n2 then ‖wξ(f) − f‖2 → 0, as ξ → 0. proof. in the proof of theorem 3 of [4] we use p = q = 2. � it follows a lipschitz type approximation result. theorem 62. let p, q > 1 such that 1 p + 1 q = 1, n ≥ 2 even and the rest as above. furthermore we assume the following lipschitz condition: ω2 ( f(n), δ ) p ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0.then ‖k(x)‖p ≤ k [ γ ( p(γ+n+1)+1 2 )] 1 p √ 2π 1 2p (n − 1)!p 1 2p q 1 2q [q(n − 1) + 1] 1 q [p (γ + 1) + 1] 1 p ( 2 p ) (γ+n+1) 2 ξ (γ+n+1) 2 . (5.32) hence as ξ → 0 we get ‖k(x)‖p → 0. if additionally f(2m) ∈ lp(r), m = 1, 2, . . . , n2 then ‖wξ(f) − f‖p → 0, as ξ → 0. proof. as in the proof of theorem 3, of [4] we find ∫ ∞ −∞ |k(x)|pdx ≤ c2 (∫ ∞ 0 (∫ y 0 ω2(f (n) , t)ppdt ) y pn−1 e − py2 2ξ dy ) ≤ kpc2 (∫ ∞ 0 ( yp(γ+1)+1 p (γ + 1) + 1 ) y pn−1 e − py2 2ξ dy ) = kpc2 p (γ + 1) + 1 (∫ ∞ 0 y p(γ+n+1) e − py2 2ξ dy ) (4.2) = kpc2 p (γ + 1) + 1 ( 2 p ) p(γ+n+1)+1 2 · 1 2 γ ( p (γ + n + 1) + 1 2 ) ξ p(γ+n+1)+1 2 . (5.33) where here we denoted c2 := 1 2 p 2q q p 2q (q(n − 1) + 1)p/q ((n − 1)!) p √ πξ . (5.34) we have established the claim of the theorem. � corollary 63. assume the following lipschitz condition: ω2 (f ′′, δ) 2 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0, and the rest as above in this section.then ‖k(x)‖2 ≤ √ √ √ √ [ γ ( 2γ+7 2 )] √ π [6γ + 9] k 2 ξ (γ+3) 2 . (5.35) hence as ξ → 0 we get ‖k(x)‖2 → 0. if additionally f′′ ∈ l2(r), then ‖wξ(f) − f‖2 → 0, as ξ → 0. 46 razvan a. mezei cubo 13, 3 (2011) proof. in theorem 62 we place p = q = n = 2. � theorem 64. let f ∈ c2(r) and f′′ ∈ l1(r). here k(x) = wξ(f; x) − f(x) − f ′′ (x) 4 ξ. then ‖k(x)‖1 ≤ ( 1 2 √ π + 3 8 ) ω2(f ′′ , √ ξ)1ξ. (5.36) hence as ξ → 0 we obtain ‖k(x)‖1 → 0. also ‖wξ(f) − f‖1 → 0, as ξ → 0. proof. in the proof of theorem 4 of [4] we use n = 2. � the lipschitz case of p = 1 follows. theorem 65. let f ∈ cn(r) and f(n) ∈ l1(r), n ≥ 2 even. furthermore we assume the following lipschitz condition: ω2 ( f(n), δ ) 1 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖k(x)‖1 ≤ γ ( γ+n+2 2 ) k (n − 1)! (γ + 2) 2 √ π ξ γ+n+1 2 . (5.37) hence as ξ → 0 we obtain ‖k(x)‖1 → 0. if additionally f(2m) ∈ l1(r), m = 1, 2, . . . , n2 then ‖wξ(f) − f‖1 → 0, as ξ → 0. proof. as in the proof of theorem 4 of [4] we have ‖k(x)‖1 ≤ 1√ πξ ∫ ∞ 0 ((∫ y 0 ω2(f (n) , t)1dt ) yn−1 (n − 1)! e −y 2 /ξ ) dy ≤ 1√ πξ ∫ ∞ 0 ((∫ y 0 kt γ+1 dt ) yn−1 (n − 1)! e −y 2 /ξ ) dy = k (n − 1)! (γ + 2) √ πξ ∫ ∞ 0 ( y γ+n+1 e −y 2 /ξ ) dy (4.2) = γ ( γ+n+2 2 ) k (n − 1)! (γ + 2) 2 √ π ξ γ+n+1 2 . (5.38) we have proved the claim of the theorem. � corollary 66. let f ∈ c6(r) and f(6) ∈ l1(r). furthermore we assume the following lipschitz condition: ω2 ( f(6), δ ) 1 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖k(x)‖1 ≤ γ ( γ+8 2 ) k 240 (γ + 2) √ π ξ γ+7 2 . (5.39) hence as ξ → 0 we obtain ‖k(x)‖1 → 0. if additionally f(2m) ∈ l1(r), m = 1, 2, 3 then ‖wξ(f) − f‖1 → 0, as ξ → 0. cubo 13, 3 (2011) applications and lipschitz results . . . 47 proof. in theorem 65 we place n = 6. � the case of n = 0 follows. proposition 67. let f as above in this section. then ‖wξ(f) − f‖2 ≤ √ 2√ π + 19 16 ω2(f, √ ξ)2. (5.40) hence as ξ → 0 we obtain wξ → i in the l2 norm. proof. in the proof of proposition 3 of [4] we use p = q = 2. � the related lipschitz case for n = 0 comes next. proposition 68. let p, q > 1 such that 1 p + 1 q = 1 and the rest as above. furthermore we assume the following lipschitz condition: ω2 (f, δ)p ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖wξ(f) − f‖p ≤ ( 2 p ) (1+γ) 2 [ γ ( (1+γ)p+1 2 )] 1 p k π 1 2p p 1 2p q 1 2q √ 2 ξ (1+γ) 2 . (5.41) hence as ξ → 0 we obtain wξ → i in the lp norm, p > 1. proof. as in the proof of proposition 3 of [4] we get ∫ ∞ −∞ |wξ(f; x) − f(x)| p dx ≤ 1 √ πξ (2q) p 2q ∫ ∞ 0 ω2(f, y) p pe −py2 2ξ dy ≤ 1 √ πξ (2q) p 2q ∫ ∞ 0 ( ky 1+γ )p e −py2 2ξ dy (4.2) = kp √ π (2q) p 2q ( 2 p ) (1+γ)p+1 2 1 2 γ ( (1 + γ) p + 1 2 ) ξ (1+γ)p 2 . (5.42) the proof of the claim is now completed. � a particular example follows corollary 69. let f as above in this section. furthermore we assume the following lipschitz condition: ω2 (f, δ)2 ≤ kδ1+γ, k > 0, 0 < γ ≤ 1, for any δ > 0. then ‖wξ(f) − f‖2 ≤ k 2 √ √ √ √ γ ( 3+2γ 2 ) √ π ξ (1+γ) 2 . (5.43) hence as ξ → 0 we obtain wξ → i in the l2 norm. proof. in proposition 68 we place p = q = 2. � 48 razvan a. mezei cubo 13, 3 (2011) we finish with the lipschitz type result proposition 70. assume the following lipschitz condition: ω2 (f, δ)1 ≤ kδγ+1, k > 0, 0 < γ ≤ 1, for any δ > 0. it holds, ‖wξf − f‖1 ≤ k 2 √ π γ ( γ + 2 2 ) ξ γ+1 2 . (5.44) hence as ξ → 0 we get wξ → i in the l1 norm. proof. as in the proof of proposition 4 of [4] we derive ∫ ∞ −∞ |wξ(f; x) − f(x)|dx ≤ 1√ πξ ∫ ∞ 0 ω2(f, y)1e −y 2 /ξ dy ≤ 1√ πξ ∫ ∞ 0 ky γ+1 e −y 2 /ξ dy (4.2) = k 2 √ π γ ( γ + 2 2 ) ξ γ+1 2 . (5.45) we have established the claim. � received: september 2009. revised: july 2010. references [1] george a. anastassiou, ”basic convergence with rates of smooth picard singular operators”, j. comput. anal. appl., 8 (2006), 313–334. [2] george a. anastassiou, ”lp convergence with rates of smooth picard singular operators”, differential & difference equations and applications, hindawi publ. corp., new york, (2006), 31–45. [3] george a. anastassiou, razvan a. mezei, “uniform convergence with rates of smooth gauss-weierstrass singular integral operators”, applicable analysis, 88:7 (2009), 1015 — 1037. [4] george a. anastassiou, razvan a. mezei, “lp convergence with rates of smooth gaussweierstrass singular operators”, nonlinear studies, accepted 2009. introduction convergence with rates of smooth picard singular integral operators lp convergence with rates of smooth picard singular integral operators convergence with rates of smooth gauss weierstrass singular integral operators lp convergence with rates of smooth gauss weierstrass singular integral operators cubo a mathemati al journal vol.15, n o 03, (45�50). o tober 2013 on entralizers of standard operator algebras with involution maja fo²ner, benjamin mar en fa ulty of logisti s, university of maribor, mariborska esta 7 3000 celje slovenia, maja.fosner�fl.uni-mb.si, benjamin.mar en�fl.uni-mb.si nej �irovnik fa ulty of natural s ien es and mathemati s, university of maribor, koro²ka esta 160 2000 maribor slovenia. nej .sirovnik�uni-mb.si abstract the purpose of this paper is to prove the following result. let x be a omplex hilbert spa e, let l(x) be the algebra of all bounded linear operators on x and let a(x) ⊂ l(x) be a standard operator algebra, whi h is losed under the adjoint operation. let t : a(x) → l(x) be a linear mapping satisfying the relation 2t(aa∗a) = t(a)a∗a + aa∗t(a) for all a ∈ a(x). in this ase t is of the form t(a) = λa for all a ∈ a(x), where λ is some �xed omplex number. resumen el propósito de este artí ulo es probar el siguiente resultado. sea x un espa io de hilbert omplejo, sea l(x) el álgebra de todos los operadores lineales a otados sobre x y sea a(x) ⊂ l(x) la álgebra de operadores lási a, la ual es errada bajo la opera ión adjunto. sea t : a(x) → l(x) una apli a ión lineal satisfa iendo la rela ión 2t(aa∗a) = t(a)a∗a+aa∗t(a) para todo a ∈ a(x). en este aso, t es de la forma t(a) = λa para todo a ∈ a(x), donde λ es un número omplejo �jo. keywords and phrases: ring, ring with involution, prime ring, semiprime ring, bana h spa e, hilbert spa e, standard operator algebra, h∗-algebra, left (right) entralizer, two-sided entralizer. 2010 ams mathemati s subje t classi� ation: 16n60, 46b99, 39b42. 46 maja fo²ner, benjamin mar en & nej �irovnik cubo 15, 3 (2013) this resear h has been motivated by the work of vukman, kosi-ulbl [5℄ and zalar [13℄. throughout, r will represent an asso iative ring with enter z(r). given an integer n ≥ 2, a ring r is said to be n-torsion free if for x ∈ r, nx = 0 implies x = 0. an additive mapping x 7→ x∗ on a ring r is alled involution if (xy)∗ = y∗x∗ and x∗∗ = x hold for all pairs x, y ∈ r. a ring equipped with an involution is alled a ring with involution or ∗ -ring. re all that a ring r is prime if for a, b ∈ r, arb = (0) implies that either a = 0 or b = 0, and is semiprime in ase ara = (0) implies a = 0. we denote by qr and c the martindale right ring of quotients and the extended entroid of a semiprime ring r, respe tively. for the explanation of qr and c we refer the reader to [2℄. an additive mapping t : r → r is alled a left entralizer in ase t(xy) = t(x)y holds for all pairs x, y ∈ r. in ase r has the identity element, t : r → r is a left entralizer i� t is of the form t(x) = ax for all x ∈ r, where a is some �xed element of r. for a semiprime ring r all left entralizers are of the form t(x) = qx for all x ∈ r, where q ∈ qr is some �xed element (see chapter 2 in [2℄). an additive mapping t : r → r is alled a left jordan entralizer in ase t(x2) = t(x)x holds for all x ∈ r. the de�nition of right entralizer and right jordan entralizer should be self-explanatory. we all t : r → r a two-sided entralizer in ase t is both a left and a right entralizer. in ase t : r → r is a two-sided entralizer, where r is a semiprime ring with extended entroid c, then t is of the form t(x) = λx for all x ∈ r, where λ ∈ c is some �xed element (see theorem 2.3.2 in [2℄). zalar [13℄ has proved that any left (right) jordan entralizer on a semiprime ring is a left (right) entralizer. let us re all that a semisimple h∗-algebra is a omplex semisimple bana h∗-algebra whose norm is a hilbert spa e norm su h that (x, yz∗) = (xz, y) = (z, x∗y) is ful�lled for all x, y, z ∈ a. for basi fa ts on erning h∗-algebras we refer to [1℄. vukman [10℄ has proved that in ase there exists an additive mapping t : r → r, where r is a 2-torsion free semiprime ring satisfying the relation 2t(x2) = t(x)x + xt(x) for all x ∈ r, then t is a two-sided entralizer. kosi-ulbl and vukman [9℄ have proved the following result. let a be a semisimple h∗−algebra and let t : a → a be an additive mapping su h that 2t(xn+1) = t(x)xn + xnt(x) holds for all x ∈ r and some �xed integer n ≥ 1. in this ase t is a two-sided entralizer. re ently, benkovi£, eremita and vukman [3℄ have onsidered the relation we have just mentioned above in prime rings with suitable hara teristi restri tions. kosi-ulbl and vukman [9℄ have proved that in ase there exists an additive mapping t : r → r, where r is a 2-torsion free semiprime ∗-ring, satisfying the relation t(xx∗) = t(x)x∗ (t(xx∗) = xt(x∗)) for all x ∈ r, then t is a left (right) entralizer. for results on erning entralizers on rings and algebras we refer to [4�13℄, where further referen es an be found. let x be a real or omplex bana h spa e and let l(x) and f(x) denote the algebra of all bounded linear operators on x and the ideal of all �nite rank operators in l(x), respe tively. an algebra a(x) ⊂ l(x) is said to be standard in ase f(x) ⊂ a(x). let us point out that any standard operator algebra is prime, whi h is a onsequen e of a hahn-bana h theorem. in ase x is a real or omplex hilbert spa e, we denote by a∗ the adjoint operator of a ∈ l(x). we denote cubo 15, 3 (2013) on entralizers of standard operator algebras with involution 47 by x∗ the dual spa e of a real or omplex bana h spa e x. vukman and kosi-ulbl [5℄ have proved the following result. theorem 0.1. let r be a 2-torsion free semiprime ring and let t : r → r be an additive mapping. suppose that 2t(xyx) = t(x)yx + xyt(x) (1) holds for all x, y ∈ r. in this ase t is a two-sided entralizer. in ase we have a ∗ -ring, we obtain, after putting y = x∗ in the relation (1), the relation 2t(xx∗x) = t(x)x∗x + xx∗t(x). it is our aim in this paper to prove the following result, whi h is related to the above relation. theorem 0.2. let x be a omplex hilbert spa e and let a(x) be a standard operator algebra, whi h is losed under the adjoint operation. suppose t : a(x) → l(x) is a linear mapping satisfying the relation 2t(aa∗a) = t(a)a∗a + aa∗t(a) (2) for all a ∈ a(x). in this ase t is of the form t(a) = λa, where λ is a �xed omplex number. proof. let us �rst onsider the restri tion of t on f(x). let a be from f(x) (in this ase we have a∗ ∈ f(x)). let p ∈ f(x) be a self-adjoint proje tion with the property ap = pa = a (we also have a∗p = pa∗ = a∗). putting p for a in (2) we obtain 2t(p) = t(p)p + pt(p). left multipli ation by p in the above relation gives pt(p) = pt(p)p. similarly, right multipli ation by p in the above relation leads to t(p)p = pt(p)p. therefore t(p) = t(p)p = pt(p) = pt(p)p. (3) putting a + p for a in the relation (2) we obtain 2t(a2) + 2t(aa∗ + a∗a) + 4t(a) + 2t(a∗) = = t(a)(a + a∗) + t(a)p + t(p)a∗a + t(p)(a + a∗)+ + (a + a∗)t(a) + pt(a) + aa∗t(p) + (a + a∗)t(p). putting −a for a in the above relation and omparing the relation so obtained with the above relation gives 2t(a2) + 2t(aa∗ + a∗a) = = t(a)(a + a∗) + t(p)a∗a + (a + a∗)t(a) + aa∗t(p) (4) 48 maja fo²ner, benjamin mar en & nej �irovnik cubo 15, 3 (2013) and 4t(a) + 2t(a∗) = = t(a)p + pt(a) + t(p)(a + a∗) + (a + a∗)t(p). (5) so far we have not used the assumption of the theorem that x is a omplex hilbert spa e. putting ia for a in the relations (4) and (5) and omparing the relations so obtained with the above relations, respe tively, we obtain 2t(a2) = t(a)a + at(a), (6) 4t(a) = t(a)p + pt(a) + t(p)a + at(p). (7) putting a∗ for a in the relation (5) gives 4t(a∗) + 2t(a) = = t(a∗)p + pt(a∗) + t(p)(a + a∗) + (a + a∗)t(p). putting ia for a in the above relation and omparing the relation so obtained with the above relation leads to 2t(a) = t(p)a + at(p). comparing the above relation and (7), we obtain 2t(a) = t(a)p + pt(a). (8) right (left) multipli ation by p in the above relation gives t(a)p = pt(a)p and pt(a) = pt(a)p, respe tively. hen e, pt(a) = t(a)p, whi h redu es the relation (8) to t(a) = t(a)p. from the above relation one an on lude that t maps f(x) into itself. we therefore have a linear mapping t : f(x) → f(x) satisfying the relation (6) for all a ∈ f(x). sin e f(x) is prime, one an on lude, a ording to theorem 1 in [10℄ that t is a two-sided entralizer on f(x). we intend to prove that there exists an operator c ∈ l(x), su h that t(a) = ca (9) for all a ∈ f(x). for any �xed x ∈ x and f ∈ x∗ we denote by x ⊗ f an operator from f(x) de�ned by (x ⊗ f)y = f(y)x, y ∈ x. for any a ∈ l(x) we have a(x ⊗ f) = (ax) ⊗ f. now let us hoose su h f and y that f(y) = 1 and de�ne cx = t(x ⊗ f)y. obviously, c is linear and applying the fa t that t is a left entralizer on f(x), we obtain (ca)x = c(ax) = t((ax) ⊗ f)y = t(a(x ⊗ f))y = t(a)(x ⊗ f)y = t(a)x for any x ∈ x. we therefore have t(a) = ca for any a ∈ f(x). as t is a right entralizer on f(x), we obtain c(ab) = t(ab) = at(b) = acb. we therefore have [a, c]b = 0 for any cubo 15, 3 (2013) on entralizers of standard operator algebras with involution 49 a, b ∈ f(x), when e it follows that [a, c] = 0 for any a ∈ f(x). using losed graph theorem one an easily prove that c is ontinuous. sin e c ommutes with all operators from f(x), we an on lude that cx = λx holds for any x ∈ x and some �xed omplex number λ, whi h gives together with the relation (9) that t is of the form t(a) = λa (10) for any a ∈ f(x) and some �xed omplex number λ. it remains to prove that the relation (10) holds on a(x) as well. let us introdu e t1 : a(x) → l(x) by t1(a) = λa and onsider t0 = t −t1. the mapping t0 is, obviously, additive and satis�es the relation (2). besides, t0 vanishes on f(x). it is our aim to show that t0 vanishes on a(x) as well. let a ∈ a(x), let p ∈ f(x) be a onedimensional self-adjoint proje tion and s = a + pap − (ap + pa). su h s an also be written in the form s = (i − p)a(i − p), where i denotes the identity operator on x. sin e s − a ∈ f(x), we have t0(s) = t0(a). it is easy to see that sp = ps = 0. by the relation (2) we have t0(s)s ∗s + ss∗t0(s) = = 2t0(ss ∗s) = = 2t0((s + p)(s + p) ∗(s + p)) = = t0(s + p)(s + p) ∗(s + p) + (s + p)(s + p)∗t0(s + p) = t0(s)s ∗s + t0(s)p + ss ∗t0(s) + pt0(s). we therefore have t0(s)p + pt0(s) = 0. considering t0(s) = t0(a) in the above relation, we obtain t0(a)p + pt0(a) = 0. (11) multipli ation from both sides by p in the above relation leads to pt0(a)p = 0. right multipli ation by p in the relation (11) and onsidering the above relation gives t0(a)p = 0. sin e p is an arbitrary one-dimensional self-adjoint proje tion, it follows from the above relation that t0(a) = 0 for all a ∈ a(x), whi h ompletes the proof of the theorem. we on lude the paper with the following onje ture. conje ture 0.3. let r be a semiprime ∗-ring with suitable torsion restri tions and let t : r → r be an additive mapping satisfying the relation 2t(xx∗x) = t(x)x∗x + xx∗t(x) for all x ∈ r. in this ase t is a two-sided entralizer. re eived: april 2013. a epted: september 2013. 50 maja fo²ner, benjamin mar en & nej �irovnik cubo 15, 3 (2013) referen es [1℄ w. ambrose: stru ture theorems for a spe ial lass of bana h algebras, trans. amer. math. so . 57 (1945), 364-386. [2℄ k. i. beidar, w. s. martindale 3rd, a. v. mikhalev: rings with generalized identities, mar el dekker, in ., new york, (1996). [3℄ d. benkovi£, d. eremita, j. vukman: a hara terization of the entroid of a prime ring, studia s i. math. hungar. 45 (3) (2008), 379-394. [4℄ i. kosi-ulbl, j. vukman: an equation related to entralizers in semiprime rings, glas. mat. 38 (58) (2003), 253-261. [5℄ i. kosi-ulbl, j. vukman: on entralizers of semiprime rings, aequationes math. 66 (2003), 277-283. [6℄ i. kosi-ulbl, j. vukman: on ertain equations satis�ed by entralizers in rings, internat. math. j. 5 (2004), 437-456. [7℄ i. kosi-ulbl, j. vukman: centralizers on rings and algebras, bull. austral. math. so . 71 (2005), 225-234. [8℄ i. kosi-ulbl, j. vukman: a remark on a paper of l. molnár, publ. math. debre en. 67 (2005), 419-421. [9℄ i. kosi-ulbl, j. vukman: on entralizers of standard operator algebras and semisimple h∗algebras, a ta math. hungar. 110 (3) (2006), 217-223. [10℄ j. vukman: an identity related to entralizers in semiprime rings, comment. math. univ. carol. 40 (1999), 447-456. [11℄ j. vukman: centralizers of semiprime rings, comment. math. univ. carol. 42 (2001), 237245. [12℄ j. vukman: identities related to derivations and entralizers on standard operator algebras, taiwan. j. math. vol. 11 (2007), 255-265. [13℄ b. zalar: on entralizers of semiprime rings, comment. math. univ. carol. 32 (1991), 609614. c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2602 .dvi cubo a mathematical journal vol.13, no¯ 01, (11–24). march 2011 weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space hiroko manaka1 and wataru takahashi2 department of mathematical and computing sciences, tokyo institute of technology, ohokayama, meguroku, tokyo 152-8552, japan. email: hiroko.manaka@is.titech.ac.jp email: wataru@is.titech.ac.jp abstract let c be a closed convex subset of a real hilbert space h. let t be a nonspreading mapping of c into itself, let a be an α-inverse strongly monotone mapping of c into h and let b be a maximal monotone operator on h such that the domain of b is included in c. we introduce an iterative sequence of finding a point of f (t )∩(a+b)−10, where f (t ) is the set of fixed points of t and (a + b)−10 is the set of zero points of a + b. then, we obtain the main result which is related to the weak convergence of the sequence. using this result, we get a weak convergence theorem for finding a common fixed point of a nonspreading mapping and a nonexpansive mapping in a hilbert space. further, we consider the problem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonspreading mapping. resumen sea c un subconjunto convexo cerrado de un espacio real de hilbert h. sea t una asignación de c en śı mismo, sea a una asignación monótona α-inversa de c en h y 12 hiroko manaka and wataru takahashi cubo 13, 1 (2011) sea b un operador monotono máximal en h tal que el dominio de b está incluido en c. se introduce una secuencia iterativa para encontrar un punto de f (t ) ∩ (a + b)−10, donde f (t ) es el conjunto de puntos fijos de t y (a + b)−10 es el conjunto de los puntos cero de a + b. entonces, se obtiene el resultado principal que se relaciona con la convergencia débil de la secuencia. utilizando este resultado, obtenemos un teorema de convergencia para encontrar un punto común de una asignación fija y una asignación en un espacio de hilbert. además, consideramos el problema para encontrar un elemento común del conjunto de soluciones de un problema de equilibrio y el conjunto de puntos fijos de una asignación. keywords: nonspreading mapping, maximal monotone operator, inverse strongly-monotone mapping, fixed point, iteration procedure. mathematics subject classification: 46c05. 1 introduction let h be a real hilbert space with inner product 〈·, ·〉 and induced norm ‖·‖ and let c be a nonempty closed convex subset of h. for a constant α > 0, the mapping a : c → h is said to be α-inverse strongly monotone if for any x, y ∈ c, 〈x − y, ax − ay〉 ≥ α ‖ax − ay‖ 2 . it is well-known that an α-inverse strongly monotone mapping is also lipschitz continuous with a lipschitz constant 1 α . let s be a mapping of c into itself. we denote by f (s) the set of fixed points of s. a mapping s of c into itself is nonexpansive if ‖su − sv‖ ≤ ‖u − v‖, ∀u, v ∈ c. if s : c → c is a nonexpansive mapping, then i − s is 1 2 -inverse strongly monotone, where i is the identity mapping on h; see, for instance, [18]. a mapping s of c into itself is nonspreading if 2‖su − sv‖2 ≤ ‖su − v‖2 + ‖sv − u‖2, ∀u, v ∈ c; see [6, 7]. a multi-valued mapping b ⊂ h × h is said to be monotone if 〈x − y, u − v〉 ≥ 0 for all x, y ∈ h, u ∈ bx and v ∈ by. a monotone operator b on h is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on h. recently, in the case when s : c → c is a nonexpansive mapping, a : c → h is an α-inverse strongly monotone mapping and b ⊂ h × h is a maximal monotone operator, takahashi, takahashi and toyoda [15] proved a strong convergence theorem for finding a point of f (s) ∩ (a + b)−10, where f (s) is the set of fixed points of s and (a + b)−10 is the set of zero points of a + b. cubo 13, 1 (2011) weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space 13 in this paper, motivated by takahashi, takahashi and toyoda [15], we introduce an iteration sequence of finding a common point of the set f (s) of fixed points of a nonspreading mapping s and the set (a+b)−10 of zero points of a+b, where a : c → h is an α-inverse strongly monotone mapping and b ⊂ h × h is a maximal monotone operator. then, we prove a weak convergence theorem. using this result, we get a weak convergence theorem for finding a common fixed point of a nonspreading mapping and a nonexpansive mapping in a hilbert space. further, we obtain a weak convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonspreading mapping. 2 preliminaries throughout this paper, let n be the set of positive integers and let h be a real hilbert space with inner product 〈 · , · 〉 and norm ‖ · ‖. a hilbert space satisfies opial’s condition [10], that is, lim inf n→∞ ‖xn − u‖ < lim inf n→∞ ‖xn − v‖ if xn ⇀ u and u 6= v; see [10]. let c be a nonempty closed convex subset of a hilbert space h. the nearest point projection of h onto c is denoted by pc , that is, ‖x − pc x‖ ≤ ‖x − y‖ for all x ∈ h and y ∈ c. such pc is called the metric projection of h onto c. we know that the metric projection pc is firmly nonexpansive, i.e., ‖pc x − pc y‖ 2 ≤ 〈pc x − pc y, x − y〉 for all x, y ∈ h. further 〈x − pc x, y − pc x〉 ≤ 0 holds for all x ∈ h and y ∈ c; see, for instance, [16]. let α > 0 be a given constant. a mapping a : c → h is said to be α-inverse strongly monotone if 〈x − y, ax − ay〉 ≥ α ‖ax − ay‖ 2 for all x, y ∈ c. we have that ‖ax − ay‖ ≤ (1/α) ‖x − y‖ for all x, y ∈ c if a is α-inverse strongly monotone. let b be a mapping of h into 2h . the effective domain of b is denoted by d(b), that is, d(b) = {x ∈ h : bx 6= ∅}. a multi-valued mapping b is said to be a monotone operator on h if 〈x − y, u − v〉 ≥ 0 for all x, y ∈ d(b), u ∈ bx, and v ∈ by. a monotone operator b on h is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on h. for a maximal monotone operator b on h and r > 0, we may define a single-valued operator jr = (i + rb) −1 : h → d(b), which is called the resolvent of b for r > 0. let b be a maximal monotone operator on h and let b−10 = {x ∈ h : 0 ∈ bx}. it is known that the resolvent jr is firmly nonexpansive and b −10 = f (jr) for all r > 0. we give the crucial lemmas in order to prove the main theorem. lemma 2.1 ([12]). let h be a real hilbert space, let {αn} be a sequence of real numbers such that 0 < a ≤ αn ≤ b < 1 for all n ∈ n and let {vn} and {wn} be sequences in h such that for some c, lim supn→∞ ‖vn‖ ≤ c, lim supn→∞ ‖wn‖ ≤ c and lim supn→∞ ‖αnvn + (1 − αn)wn‖ = c. then limn→∞ ‖vn − wn‖ = 0. 14 hiroko manaka and wataru takahashi cubo 13, 1 (2011) lemma 2.2 ([19]). let h be a hilbert space and let s be a nonempty closed convex subset of h. let {xn} be a sequence in h. if ‖xn+1 − x‖ ≤ ‖xn − x‖ for all n ∈ n and x ∈ s, then {ps (xn)} converges strongly to some z ∈ s, where ps stands for the metric projection on h onto s. using opial’s theorem [10], we can also prove the following lemma; see, for instance, [18]. lemma 2.3. let h be a hilbert space and let {xn} be a sequence in h such that there exists a nonempty subset s ⊂ hsatisfying (i) and (ii): (i) for every x∗ ∈ s, limn→∞ ‖xn − x ∗‖ exists: (ii) if a subsequence {xnj } ⊂ {xn} converges weakly to x ∗, then x∗ ∈ s. then there exists x0 ∈ s such that xn ⇀ x0. let c be a nonempty closed convex subset of a real hilbert space h, let f : c × c → r be a bifunction and let a : c → h be a nonlinear mapping. then, we consider the following equilibrium problem [8]: find z ∈ c such that f (z, y) + 〈az, y − z〉 ≥ 0, ∀y ∈ c. (2.1) the set of such z ∈ c is denoted by ep (f, a), i.e., ep (f, a) = {z ∈ c : f (z, y) + 〈az, y − z〉 ≥ 0, ∀y ∈ c}. in the case of a ≡ 0, ep (f, a) is denoted by ep (f ). in the case of f ≡ 0, ep (f, a) is also denoted by v i(c, a). for solving the equilibrium problem, let us assume that the bifunction f satisfies the following conditions: (a1) f (x, x) = 0 for all x ∈ c; (a2) f is monotone, i.e., f (x, y) + f (y, x) ≤ 0 for all x, y ∈ c; (a3) for all x, y, z ∈ c, lim sup t↓0 f (tz + (1 − t)x, y) ≤ f (x, y); (a4) f (x, ·) is convex and lower semicontinuous for all x ∈ c. we know the following lemmas; see, for instance, [1] and [2]. lemma 2.4 ([1]). let c be a nonempty closed convex subset of h, let f be a bifunction from c × c to r satisfying (a1)-(a4) and let r > 0 and x ∈ h. then, there exists z ∈ c such that f (z, y) + 1 r 〈y − z, z − x〉 ≥ 0 for all y ∈ c. cubo 13, 1 (2011) weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space 15 lemma 2.5 ([2]). for r > 0 and x ∈ h, define the resolvent tr : h → c of f for r > 0 as follows: trx = { z ∈ c : f (z, y) + 1 r 〈y − z, z − x〉 ≥ 0, ∀y ∈ c } for all x ∈ h. then, the following hold: (i) tr is single-valued; (ii) tr is firmly nonexpansive, i.e., for all x, y ∈ h, ‖trx − try‖ 2 ≤ 〈trx − try, x − y〉; (iii) f (tr) = ep (f ); (iv) ep (f ) is closed and convex. 3 main result let h be a real hilbert space and let c be a nonempty closed convex subset of h. then, a mapping s of c into itself is nonspreading if 2‖su − sv‖2 ≤ ‖su − v‖2 + ‖sv − u‖2, ∀u, v ∈ c; see [6, 7]. we know from [6, 7, 3] that if the bifunction f : c × c → r satisfies the conditions (a1), (a2), (a3) and (a4), then for any r > 0, tr is a nonspreading mapping of c into itself. further, we can give the following example of nonspreading mappings in a hilbert space. let h be a real hilbert space; see [4]. set e = {x ∈ h : ‖x‖ ≤ 1}, d = {x ∈ h : ‖x‖ ≤ 2} and c = {x ∈ h : ‖x‖ ≤ 3}. define a mapping s : c → c as follows: sx { 0, x ∈ d, pe x, x /∈ d. then, this mapping s is not nonexpansive but nonspreading because it is not continuous. this implies that the class of nonexpansive mappings does not contain the class of nonspreading mappings. now, we can prove a weak convergence theorem. before proving it, we give the following lemma. lemma 3.1. let h be a real hilbert space and let c be a nonempty closed convex subset of h. let α > 0. let a be an α-inverse strongly monotone mapping of c into h and let b be a maximal monotone operator on h such that the domain of b is included in c. let jλ = (i + λb) −1 be the resolvent of b for any λ > 0. then, the following hold: (i) if u, v ∈ (a + b)−10, then au = av; 16 hiroko manaka and wataru takahashi cubo 13, 1 (2011) (ii) for any λ > 0, u ∈ (a + b)−1(0) if and only if u = jλ(i − λa)u. proof. (i) if u, v ∈ (a + b)−1(0), then 0 ∈ au + bu and 0 ∈ av + bv. then, we have −au ∈ bu and −av ∈ bv. since b is monotone, we have 〈u − v, −au − (−av)〉 ≥ 0. on the other hand, since a is α-inverse strongly monotone, we have 〈u − v, au − av〉 ≥ ‖au − av‖2. so, we have 〈u − v, −au − (−av)〉 = 0 and hence au = av. (ii) for any λ > 0, we have that u = jλ(i − λa)u ⇔ u − λau ∈ u + λbu ⇔ 0 ∈ λau + λbu ⇔ 0 ∈ au + bu ⇔ u ∈ (a + b)−1(0). this completes the proof. now, we can prove the main theorem. theorem 3.1. let c be a nonempty convex closed subset of a real hilbert space h, let a : c → h be α-inverse strongly monotone, let b : d(b) ⊂ c → 2h be maximal monotone, let jλ = (i + λb) −1 be the resolvent of b for any λ > 0, and let t : c → c be a nonspreading mapping. assume that f (t ) ∩ (a + b)−1(0) 6= ∅. for any x = x1 ∈ c, define xn+1 = βnxn + (1 − βn)t (jλn (i − λna)xn), ∀n ∈ n, where {βn} and {λn} satisfy the following conditions (∗): 0 < c ≤ βn ≤ d < 1 and 0 < a ≤ λn ≤ b < 2α. (∗) then, xn ⇀ z0 ∈ f (t ) ∩ (a + b) −1(0), where z0 = limn→∞ pf (t )∩(a+b)−1(0)(xn). proof. set e = f (t ) ∩ (a + b)−1(0). let yn = jλn (i − λna)xn for all n ∈ n and let z ∈ e. since z = jλn (i − λna)z from lemma 3.1 and a is α-inverse strongly monotone, we have that ‖yn − z‖ 2 = ‖jλn (i − λna)xn − jλn (i − λna)z‖ 2 (3.1) ≤ ‖xn − λnaxn − z + λnaz‖ 2 = ‖xn − z‖ 2 − 2λn〈xn − z, axn − az〉 + λ 2 n ‖axn − az‖ 2 ≤ ‖xn − z‖ 2 − 2λnα ‖axn − az‖ 2 + λ2n ‖axn − az‖ 2 = ‖xn − z‖ 2 + λn(λn − 2α) ‖axn − az‖ 2 . from (∗), we have that ‖yn − z‖ 2 ≤ ‖xn − z‖ 2 , ∀n ∈ n cubo 13, 1 (2011) weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space 17 and hence ‖xn+1 − z‖ = ‖βnxn + (1 − βn)t yn − z‖ ≤ βn ‖xn − z‖ + (1 − βn) ‖t yn − z‖ ≤ βn ‖xn − z‖ + (1 − βn) ‖yn − z‖ ≤ ‖xn − z‖ . this means that the condition (i) of lemma 2.3 holds for s = e. we also obtain that limn→∞ ‖xn − z‖ exists. thus, {xn}, {axn}, {yn} and {t yn} are bounded. by the inequality (2), ‖xn+1 − z‖ 2 ≤ βn ‖xn − z‖ 2 + (1 − βn) ‖yn − z‖ 2 ≤ βn ‖xn − z‖ 2 + (1 − βn){‖xn − z‖ 2 + λn(λn − 2α) ‖axn − ax‖ 2 } ≤ ‖xn − z‖ 2 + λn(λn − 2α)(1 − βn) ‖axn − az‖ 2 . thus we have 0 ≤ (1 − d)a(2α − d) ‖axn − az‖ 2 ≤ ‖xn − z‖ 2 − ‖xn+1 − z‖ 2 → 0, as n → ∞. this means that lim n→∞ ‖axn − az‖ = 0. (3.2) on the other hand, since jλn is firmly nonexpansive, we have that ‖yn − z‖ 2 = ‖jλn (i − λna)xn − jλn (i − λna)z‖ 2 ≤〈yn − z, (i − λna)xn − (i − λna)z〉 = 1 2 {‖yn − z‖ 2 + ‖(i − λna)xn − (i − λna)z‖ 2 − ‖yn − z − (i − λna)xn + (i − λna)z‖ 2 } = 1 2 {‖yn − z‖ 2 + ‖xn − z‖ 2 − ‖yn − z − (i − λna)xn + (i − λna)z‖ 2 } = 1 2 {‖yn − z‖ 2 + ‖xn − z‖ 2 − ‖yn − xn‖ 2 − 2λn〈yn − xn, axn − az〉 − λn 2 ‖axn − az‖ 2 }. therefore we have ‖yn − z‖ 2 ≤ ‖xn − z‖ 2 − ‖yn − xn‖ 2 − 2λn〈yn − xn, axn − az〉 − λn 2 ‖axn − az‖ 2 18 hiroko manaka and wataru takahashi cubo 13, 1 (2011) and hence ‖xn+1 − z‖ 2 ≤βn ‖xn − z‖ 2 + (1 − βn) ‖t yn − z‖ 2 ≤βn ‖xn − z‖ 2 + (1 − βn) ‖yn − z‖ 2 ≤βn ‖xn − z‖ 2 + (1 − βn){‖xn − z‖ 2 − ‖yn − xn‖ 2 − 2λn〈yn − xn, axn − az〉 − λn 2 ‖axn − az‖ 2 } ≤ ‖xn − z‖ 2 − (1 − d) ‖yn − xn‖ 2 − λn 2(1 − βn) ‖axn − az‖ 2 − 2λn(1 − βn)〈yn − xn, axn − az〉. this means that (1 − d) ‖yn − xn‖ 2 ≤ ‖xn − z‖ 2 − ‖xn+1 − z‖ 2 + ‖axn − az‖{2b(1 − c) ‖yn − xn‖ + b 2(1 − c) ‖axn − az‖}. since {yn} and {xn} are bounded, limn→∞ ‖axn − az‖ = 0 and limn→∞ ‖xn − z‖ exists, we have lim n→∞ ‖yn − xn‖ = 0. since a is lipschitz continuous, we also have lim n→∞ ‖ayn − axn‖ = 0. let x∗ be a weak cluster point of {xn}. first, we prove that x ∗ ∈ (a + b)−1(0). since yn = jλn (i − λna)xn, we have that yn = (i + λnb) −1(i − λna)xn ⇔ (i − λna)xn ∈ (i + λnb)yn = yn + λnbyn ⇔ xn − yn − λnaxn ∈ λnbyn ⇔ 1 λn (xn − yn − λnaxn) ∈ byn. since b is monotone, we have that for (u, v) ∈ b, 〈 yn − u, 1 λn (xn − yn − λnaxn) − v 〉 ≥ 0 and hence 〈yn − u, xn − yn − λn(axn + v)〉 ≥ 0. suppose that a subsequence {xnj } ⊂ {xn} satisfies xnj ⇀ x ∗. then, since a is α-inverse strongly monotone and axn → az by (3), 〈 xnj − x ∗, axnj − ax ∗ 〉 ≥ α ∥ ∥axnj − ax ∗ ∥ ∥ 2 cubo 13, 1 (2011) weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space 19 implies that axnj → ax ∗ as j → ∞. moreover, since limn→∞ ‖yn − xn‖ = 0 implies ynj ⇀ x ∗, we have lim j→∞ 〈 ynj − u, xnj − ynj − λnj (axnj + v) 〉 ≥ 0 and hence 〈x∗ − u, −ax∗ − v〉 ≥ 0. since b is maximal monotone, (−ax∗) ∈ bx∗. that is, x∗ ∈ (a + b)−1(0). next, we show x∗ ∈ f (t ). putting c = limn→∞ ‖xn − z‖, we have lim sup n→∞ ‖t yn − z‖ = lim sup n→∞ ‖t yn − t z‖ ≤ lim sup n→∞ ‖yn − z‖ ≤ lim sup n→∞ ‖xn − z‖ ≤ c. on the other hand, we have lim n→∞ ‖xn+1 − z‖ = lim n→∞ ‖βnxn + (1 − βn)t yn − z‖ = c. from lemma 2.1, we have lim n→∞ ‖(xn − z) − (t yn − z)‖ = lim n→∞ ‖xn − t yn‖ = 0. (3.3) we have also ‖yn − t yn‖ ≤ ‖yn − xn‖ + ‖xn − t yn‖. hence, we have lim n→∞ ‖yn − t yn‖ = 0. since xnj ⇀ x ∗ and xn − yn → 0, we have ynj ⇀ x ∗. now we shall show that t x∗ = x∗. since t is nonspreading, we have 0 ≤(‖t yn − x ∗‖ 2 − ‖t yn − t x ∗‖ 2 ) + (‖t x∗ − yn‖ 2 − ‖t yn − t x ∗‖ 2 ) =2 〈t yn, t x ∗ − x∗〉 + ‖x∗‖ 2 − ‖t x∗‖ 2 + 2 〈t yn − yn, t x ∗〉 + ‖yn‖ 2 − ‖t yn‖ 2 ≤ 2 〈t yn − yn, t x ∗ − x∗〉 + 2 〈yn, t x ∗ − x∗〉 + ‖x∗‖ 2 − ‖t x∗‖ 2 + 2 〈t yn − yn, t x ∗〉 + (‖yn‖ + ‖t yn‖)(‖yn − t yn‖). thus, we have that for all j ∈ n, 0 ≤2 〈 t ynj − ynj , t x ∗ − x∗ 〉 + 2 〈 ynj , t x ∗ − x∗ 〉 + ‖x∗‖ 2 − ‖t x∗‖ 2 + 2 〈 t ynj − ynj , t x ∗ 〉 + ( ∥ ∥ynj ∥ ∥ + ∥ ∥t ynj ∥ ∥)( ∥ ∥ynj − t ynj ∥ ∥). 20 hiroko manaka and wataru takahashi cubo 13, 1 (2011) since limn→∞ ∥ ∥t ynj − ynj ∥ ∥ = 0 and ynj ⇀ x ∗ as j → ∞, the above inequality implies that 0 ≤2 〈x∗, t x∗ − x∗〉 + ‖x∗‖ 2 − ‖t x∗‖ 2 =2 〈x∗, t x∗〉 − ‖x∗‖ 2 − ‖t x∗‖ 2 = − ‖x∗ − t x∗‖ 2 . so, we have t x∗ = x∗, i.e., x∗ ∈ f (t ). therefore we obtain that x∗ ∈ e = f (t ) ∩ (a + b)−1(0). this implies that the condition (ii) of lemma 2.3 holds for s = e. we also know that limn→∞ ‖xn − z‖ exists for z ∈ s = e. so, we have from lemma 2.3 that there exists z∗ ∈ e such that xn ⇀ z ∗ as n → ∞. moreover, since for any z ∈ s = e, ‖xn+1 − z‖ ≤ ‖xn − z‖ , ∀n ∈ n, by lemma 2.2 there exists some z0 ∈ s such that ps (xn) → z0. the property of metric projection implies that 〈z∗ − ps (xn), xn − ps (xn)〉 ≤ 0. therefore, we have 〈z∗ − z0, z ∗ − z0〉 = ‖z ∗ − z0‖ 2 ≤ 0. this means that z∗ = z0, i.e., xn ⇀ z ∗ = limn→∞ pe (xn). 4 applications let h be a hilbert space and let f be a proper lower semicontinuous convex function of h into (−∞, ∞]. then the subdifferential ∂f of f is defined as follows: ∂f (x) = {z ∈ h : f (x) + 〈z, y − x〉 ≤ f (y), ∀y ∈ h} for all x ∈ h. by rockafellar [11], it is shown that ∂f is maximal monotone. let c be a nonempty closed convex subset of h and let ic be the indicator function of c, i.e., ic (x) { 0, if x ∈ c, ∞, if x 6∈ c. further, for any u ∈ c, we also define the normal cone nc (u) of c at u as follows; nc (u) = {z ∈ h : 〈z, y − u〉 ≤ 0, ∀y ∈ c}. then ic : h → (−∞, ∞] is a proper lower semicontinuous convex function on h and ∂ic is a maximal monotone operator. let jλx = (i + λ∂ic ) −1x for λ > 0 and x ∈ h. since ∂ic (x) = {z ∈ h : ic (x) + 〈z, y − x〉 ≤ ic (y), ∀y ∈ h} = {z ∈ h : 〈z, y − x〉 ≤ 0, ∀y ∈ c} = nc (x) cubo 13, 1 (2011) weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space 21 for x ∈ c, we have u = jλx ⇔ (i + λ∂ic ) −1x = u ⇔ x ∈ u + λ∂ic (u) ⇔ x ∈ u + λnc (u) ⇔ x − u ∈ λnc (u) ⇔ 〈x − u, y − u〉 ≤ 0, ∀y ∈ c ⇔ pc (x) = u. similarly, we have that for x ∈ c, x ∈ (a + ∂ic ) −1(0) ⇔ 〈−ax, y − x〉 ≤ 0, ∀y ∈ c ⇔ x ∈ v i(a, c). thus, putting b = ∂ic , we have jλn = pc for any n ∈ n. thus, we have the following theorem from theorem 3.1. theorem 4.1. let c be a nonempty closed convex subset of a real hilbert space h, let a be an α-inverse strongly monotone mapping of c into h and let t : c → c be a nonspreading mapping. assume f (t ) ∩ (a + ∂ic ) −1(0) = f (t ) ∩ v i(a, c) 6= ∅. define a sequence {xn} in c as follows: x = x1 ∈ c and xn+1 = βnxn + (1 − βn)t (pc (i − λna)xn) for all n ∈ n, where the sequences {βn} and {λn} satisfy the condition (∗): 0 < c ≤ βn ≤ d < 1 and 0 < a ≤ λn ≤ b < 2α. (∗) then, xn ⇀ z0 ∈ f (t ) ∩ v i(a, c) and z0 = limn→∞ pf (t )∩v i(a,c)(xn). let s : c → c be nonexpansive. then, i − s is 1 2 -inverse strongly monotone. so, we obtain the following result. theorem 4.2. let c be a nonempty closed convex subset of a real hilbert space h, let s : c → c be a nonexpansive mapping and let t : c → c be a nonspreading mapping. assume that f (t ) ∩ f (s) 6= ∅. let x = x1 ∈ c and define xn+1 = βnxn + (1 − βn)t ((1 − λn)xn + λnsxn) for all n ∈ n , where {λn} and {βn} satisfy the condition (∗): 0 < c ≤ βn ≤ d < 1 and 0 < a ≤ λn ≤ b < 1. (∗) then, xn ⇀ z0 ∈ f (t ) ∩ f (s) and z0 = limn→∞ pf (t )∩f (s)(xn). 22 hiroko manaka and wataru takahashi cubo 13, 1 (2011) proof. put a = i − s. then we have pc (xn − λnaxn) = pc (xn − λn(i − s)xn) = pc ((1 − λn)xn + λnsxn) = (1 − λn)xn + λnsxn. for u ∈ c, we have su ∈ c and u ∈ (a + ∂ic ) −1(0) ⇔ 0 ∈ au + nc (u) ⇔ su − u ∈ nc (u) ⇔ 〈su − u, v − u〉 ≤ 0, ∀v ∈ c ⇔ pc (su) = u ⇔ su = u. thus, we obtain (a + ∂ic ) −1(0) = v i(a, c) = f (s). so, by theorem 4.1 we have the desired result. next, we deal with the equilibrium problem with nonspreading mappings in a hilbert space. takahashi, takahashi and toyoda [15] showed the following. theorem 4.3 ([15]). let c be a nonempty closed convex subset of a hibert space h and let f : c × c → r be a bifunction satisfying the conditions (a1)-(a4). define af as follows: af (x) { {z ∈ h : f (x, y) ≥ 〈y − x, z〉 , ∀y ∈ c}, if x ∈ c, ∅, if x 6∈ c. then, ep (f ) = a−1 f (0) and af is maximal monotone with the domain of af in c. furthermore, tr(x) = (i + raf ) −1(x), ∀r > 0. we obtain the following theorem from theorem 3.1. theorem 4.4. let c be a nonempty closed convex subset of a real hilbert space h, let f : c ×c → r satisfy the conditions (a1)-(a4) and let tλ be the resolvent of f for λ > 0. let s : c → c be a nonspreading mapping. assume that f (t ) ∩ ep (f ) 6= ∅. for x = x1 ∈ c, define xn+1 = βnxn + (1 − βn)stλn xn, ∀n ∈ n, where {βn} and {λn} satisfy the following conditions: 0 < c ≤ βn ≤ d < 1, 0 < a ≤ λn ≤ b < ∞. then, xn ⇀ z0 ∈ f (t ) ∩ ep (f ) and z0 = limn→∞ pf (s)∩ep (f )(xn). cubo 13, 1 (2011) weak convergence theorems for maximal monotone operators with nonspreading mappings in a hilbert space 23 proof. suppose a = 0. then, we have that 〈x − y, ax − ay〉 ≥ α ‖ax − ay‖ 2 = 0, ∀α ∈ r. so, we can choose α = ∞ in theorem 3.1. since tλn = (i + λnaf ) −1 is the resolvent of af and af is maximal monotone, theorem 3.1 implies that xn ⇀ z0 ∈ f (t ) ∩ a −1 f (0). moreover, we know a−1 f (0) = ep (f ). so, we have the desired result. received: june 2009. revised: september 2009. references [1] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, math. student 63 (1994), 123–145. 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[22] h. k. xu, viscosity approximation methods for nonexpansive mappings, j. math. anal. appl. 298 (2004), 279–291. cubo a mathemati al journal vol.15, n o 03, (123�132). o tober 2013 k-theory for the group c∗-algebras of nilpotent dis rete groups takahiro sudo department of mathemati al s ien es, fa ulty of s ien e, university of the ryukyus, senbaru 1, nishihara, okinawa 903-0213, japan. sudo�math.u-ryukyu.a .jp abstract we study the k-theory groups for the group c∗-algebras of nilpotent dis rete groups, mainly, without torsion. we determine the k-theory lass generators for the k-theory groups by using generalized bott proje tions. resumen estudiamos los grupos de la k-teoría para el grupo de álgebras c∗ de grupos dis retos nilpotentes prin ipalmente sin torsión. determinamos los generadores de la lase de k-teoría para los grupos de la k-teoría usando proye iones generalizadas de bott. keywords and phrases: group c*-algebra, k-theory, nilpotent dis rete group, bott proje tion. 2010 ams mathemati s subje t classi� ation: 46l05, 46l80, 19k14. 124 takahiro sudo cubo 15, 3 (2013) 1 introdu tion the k-theory groups for the group c∗-algebra of the dis rete heisenberg nilpotent group are omputed in the paper [1℄ of anderson and pas hke by determining the k-thoery lass generators for the k-theory groups by using the bott proje tion on the two dimensional torus. the ktheory groups for the group c∗-algebras of the generalized dis rete heisenberg nilpotent groups are omputed in the paper [3℄ of the author by determining the k-theoy lass generators for the k-theory groups by using generalized bott proje tions on the higher dimensional torus de�ned in [3℄. in this paper, based on those results in the typi al ase of two-step, nilpotent dis rete groups, we study the k-theory groups for the group c∗-algebras of general, nilpotent dis rete groups, mainly, without torsion, and it is found out that we an determine the k-theory lass generators for the k-theory groups by using the generalized bott proje tions. moreover, several onsequen es of this main result are also obtained. notation. we denote by c(x) the c∗-algebra of all ontinuous, omplex-valued fun tions on a ompa t hausdor� spa e x. denote by c∗(g) the (full or redu ed) group c∗-algebra of a nilpotent, dis rete group g (that is amenable). note that c∗(g) is generated by unitaries that orrespond to generators of g. denote by k0(a) and k1(a) the k0-group and the k1-group of a c ∗ -algebra a respe tively (see [2℄). 2 finitely generated nilpotent dis rete ase re all that a k-dimensional non ommutative torus denoted by tkθ is the universal c ∗ -algebra generated by k unitaries uj (1 ≤ j ≤ k) with the relations uiuj = e 2πiθij ujui for i 6= j and θij ∈ r and θ = (θij) ∈ mk(r) a k × k skew adjoint matrix over the �eld r of real numbers with θii = 0 and θji = −θij (i 6= j). lemma 2.1. let g be a �nitely generated, two-step nilpotent dis rete group without torsion and with z its enter and c∗(g) be the group c∗-algebra of g. then c∗(g) an be viewed as a ontinuous �eld c∗-algebra over the dual group z∧ of z with �bers given by non ommutative tori t n−k θλ with the relations by θλ varing over elements λ ∈ z ∧ , where z∧ is an ordinary torus tk by pontrjagin duality theorem with 1 ≤ k = rank(z) the rank of z, and n = rank(g). proof. this is ertainly known and may follow from the same way as done by [1℄ in the ase of g the dis rete heisenberg group of rank 3. indeed, note that sin e g/z is ommutative, the ommutator subgroup [g, g] of g is ontained in z. as a fa t of the unitaty representation theory for g, that is identi�ed with the representation cubo 15, 3 (2013) k-theory for the group c∗-algebras of nilpotent dis rete groups 125 theory of c∗(g), any element λ in z∧ indu es an irredu ible indu ed representation πλ of g and of c∗(g) and any element of [g, g] is mapped to a omplex number in the one-torus t, so that the image of c∗(g) under πλ is a non ommutive torus t n−k θλ with θλ asso iated to λ. sin e elements λ ∈ z∧ = tk vary ontinuously on z∧, the norms of πλ([ui, uj]) for ui, uj unitary generators of c∗(g) orresponding to generators of g also vary ontinuously, to make a ontinuous �eld c∗-algebra over z∧ with �bers non ommutative tori tn−k θλ . as the main result we obtain theorem 1. let g be a �nitely generated, nilpotent dis rete group without torsion and c∗(g) be the group c∗-algebra of g. then the k-theory lass generators in the k0-group k0(c ∗(g)) are given by the lass of the identity of c∗(g) and the lasses of generalized bott proje tions ombinatori ly orresponding to abelian subalgebras of c∗(g) that orrespond to even subsets of mutually ommuting generators, even numbered, in the set of generators of g. moreover, the k-theory lass generators in the k1-group k1(c ∗(g)) are given by the lass of unitary generators of c∗(g) that orrespond to ea h of generators of g, or orrespond both to generators of g and to the generalized bott proje tions, ea h of whi h is obtained ombinatori ly from both the generalized bott proje tion and ea h of generators of g whi h is not involved in the generalized bott proje tion. the statement above an be understood pre isely by helpful examples and remark below the following proof. proof. re all that under the assumption on g, the group g is isomorphi to a su essive semi-dire t produ t of z the group of integers: g ∼= z ⋊ z ⋊ · · · ⋊ z rossed by z rank(g) − 1 times, with rank(g) the rank of g. then c∗(g) ∼= c ∗(z) ⋊ z ⋊ · · · ⋊ z a su essive rossed produ t c∗-algebra by z, and c∗(z) ∼= c(t) by the fourier transform. set rank(g) = n. let u = {g1, · · · , gn} be the set of generators of g. note that sin e g is dis rete, the generators of g an be identi�ed with orresponding unitary generators of c∗(g) (via the left regular, or universal representation on the orresponding hilbert spa e sin e g is amenable). suppose that v is an even subset of u with some mutually ommuting generators of g. denote by c∗(v) the c∗-algebra generated by elements of v. then c∗(v) is an abelian subalgebra of c∗(g) and is isomorphi to c(t|v|) the c∗-algebra of all ontinuous, omplex-valued fun tions on the |v|-dimensional torus t|v|, where |v| is the ardinality of v. we assign su h even 126 takahiro sudo cubo 15, 3 (2013) subset v ea h to the generalized bott proje tion pv in m2(c(t |v|)) the 2 × 2 matrix algebra over c(t|v|), involving all elements of v. see the remark below for the de�nition of pv. it follows that k0(c(t |v|)) an be embedded in k0(c ∗(g)) anoni ally. therefore, the k0group lass [pv] an be viewed in k0(c ∗(g)). it also follows that if v 6= v ′ even subsets in u, then [pv] 6= [pv ′], i.e., pv is not equivalent to pv ′. indeed, if pv is equivalent to pv ′, then we an dedu e a ontradition, by observing that the oordinates of t |v| orresponding to v are di�erent from those of t |v ′ | of v ′. if g is ommutative, then g ∼= zn, and c∗(g) ∼= c(tn) by the fourier transform, and it is shown by [3℄ that the k0-group lasses of generalized bott proje tions on the even dimensional tori t 2k (2 ≤ 2k ≤ n) ombinatori ly in tn and the lass of the identity generate all lasses in k0(c(t n)). by the lemma above, if g is a �nitely generated, two-step nilpotent dis rete group without torsion, then c∗(g) an be viewed as a ontinuous �eld c∗-algebra over the dual group z∧ of the enter z of g with �bers given by non ommutative tori, that are su essive rossed produ t c∗-algebras by z, generated by unitaries orresponding to generators of g not in z, where their relations vary over z∧. it is also shown by [3℄ that even in this ase, the same holds as in the ommutative ase. indeed, the lass of the identity and the lasses of generalized bott proje tions in m2(c ∗(g)) generate all lasses in k0(c ∗(g)), be ause it is noti ed in [3℄ that the lasses of the genearalized rie�el proje tions de�ned in [3℄ and the lass of the identity generate all lasses in the k0-group of a �ber, a non ommutative torus, and the lasses of the genearalized rie�el proje tions an not ontribute to a lass of k0(c ∗(g)) sin e those proje tions are not ontinuous over z∧. therefore, a proje tion for a lass of k0(c ∗(g)) an not involve the generalized rie�el proje tions in �bers. we now onsider the general ase by indu tion. suppose that the theorem on k0 is true when rank(g) ≤ n. let rank(g) = n + 1. let [p] ∈ k0(c ∗(g)) for a proje tion p in a matrix algebra over c∗(g). if p is generated by k unitaries orresponding to k generators of g with k ≤ n, then p is ontained in the group c∗-algebra c∗(h) of a nilpotent subgroup h of g generated by v the set of the k generators of g, that is c∗(h) = c∗(v) ⊂ c∗(g). by indu tion hypothesis, the lass [p] is spanned by the lass of the identity and the lasses of generalized bott proje tions in m2(c ∗(h)). we now assume that the proje tion p involves all elements of u. we also may assume that g is not two-step nilpotent. therefore, the quotient group g/z is not ommutative and nilpotent. there is a quotient map q from c∗(g) to c∗(g/z) and is extended to their matrix algebras. then q(p) is a proje tion that involves all generators of g/z. but by indu tion, and sin e g/z is non ommutative, the k0-group lasses of k0(c ∗(g/z)) an not involve all generators of g/z. this is a ontradi tion. hen e, there is no su h proje tion p. in fa t, this redu tion an be ontinued until that p is ontained in an abelian subalgebra of c∗(g) that is generated by unitaries orresponding to a set of mutually ommuting generators of g cubo 15, 3 (2013) k-theory for the group c∗-algebras of nilpotent dis rete groups 127 the k1-group ase for c ∗(g) is treated similarly as in the k0-group ase above. indeed, when g is ommutative, it is shown by [3℄ that the k1-group k1(c ∗(g)) an be generated by the lasses represented by either unitary generators of c∗(g) orresponding to generators of g or the unitaires that ombinatori ly orrespond to both generalized bott proje tions and ea h of unitary generators of c∗(g) orresponding to generators of g. see the remark below for the de�nition of the unitaries. moreover, even in the ase of g two-step nilpotent, the same holds for k1(c ∗(g)). and the general ase an be proved by the same way as in the proof for that ase of k0(c ∗(g)). in fa t, the onstru tion of generators of k1(c ∗(g)) an be made by bije tively orreponding to the generators of k0(c ∗(g)) onstru ted, in a suitable and ombinatori way (see the examples below). remark. re all from [3℄ (or [1℄ originally) that the bott proje tion p in m2(c(t 2)) is de�ned as a proje tion-valued fun tion from t 2 to m2(c): p(w, z) = ad(u(w, z)) ( 1 0 0 0 ) ∈ m2(c), (w, z) ∈ t 2, where u(w, z) = y(t, z)∗ with w = e2πit ∈ t for t ∈ [0, 1] and y(t, z) = exp ( iπt 2 k(z) ) exp ( iπt 2 s ) k(z) = ( 0 z z 0 ) , s = k(1). moreover, the generalized bott proje tion qk in m2(c(t 2k)) is de�ned in [3℄ by a proje tion-valued fun tion from t 2k to m2(c): qk(z1, · · · , z2k) = ad(u1(z1, z2))ad(u2(z3, z4)) · · · ad(uk(z2k−1, z2k)) ( 1 0 0 0 ) where uj(·, ·) = u(·, ·) for 1 ≤ j ≤ k. furthermore, the unitary vk in m2(c(t 2k+1)) obtained from the generalized bott proje tion qk and a unitary generator u of c ∗(g) orresponding to a generator of g is de�ned in [3℄ by vk = ( 1 0 0 1 ) + (u − 1) ⊗ qk ∈ m2(c(t 2k+1 )). example 2.2. if g = zn, then c∗(g) ∼= c(tn) by the fourier transform, and k∗(c(t n)) ∼= z2 n−1 for ∗ = 0, 1 ( [4℄). note that the generators of k0(c(t n)) are given by the lass of the identity and the lasses of generalized bott proje tions de�ned as above and the generators of k1(c(t n)) are given by the lasses of unitary generators of c∗(zn) orresponding to generators of zn and the 128 takahiro sudo cubo 15, 3 (2013) lasses of the unitaries asso iated to both generalized bott proje tions and the unitary generators of c∗(zn) de�ned as above (see [3℄). more pre isely, when n = 4, the generators of k0(c ∗(z4)) ∼= z8 is given by the following lasses: [1], [p12], [p13], [p14], [p23], [p24], [p34], [q1234], where 1 is the identity of c∗(z4) and ea h pij over t 4 is identi�ed with the bott proje tion over t 2 that orresponds to i, j oordinates in t4, and q1234 is the generalized bott proje tion over t 4 . also, the generators of k1(c ∗(z4)) ∼= z8 is given by the following lasses: [u1], [u2], [u3], [u4], [v123], [v124], [v134], [v234], where ea h uj is the unitary generator of c ∗(z4) orresponding to generators of z4 and ea h unitary vijk in m2(c ∗(z4)) is obtained by pij and uk. note that vijk may be obtained from either pjk and ui, or pik and uj. example 2.3. let g be the dis rete heisenberg group of rank 3: g =        1 a c 0 1 b 0 0 1     | a, b, c ∈ z    . then z = z and g/z ∼= z2. also, c∗(g) is viewed as a ontinuous �led c∗-algebra over t = z∧ with �bers non ommutative 2-tori t2θλ. it is omputed by [1℄ (and also [3℄) that k0(c ∗ (g)) ∼= z 3, k1(c ∗ (g)) ∼= z 3, and the generators of k0(c ∗(g)) is given by the lass of the identity of c∗(g) and two lasses of the bott proje tions over t 2 , where their domains are di�erent in the sense as one t 2 = t×z∧ with the �rst fa tor t orresponding to one of two generators of the �bers and the other t 2 = t × z∧ with the �rst fa tor t orresponding to the other of two generators of the �bers, and the generators of k1(c ∗(g)) is given by two lasses of unitary generators of c∗(g) orresponding both to generators of g and to one of two bott proje tions and the lass of the unitary of m2(c ∗(g)) obtained from both the hosen bott proje tion and the rest of unitary generators of c∗(g) orresponding to generators of g. namely, k0(c ∗(g)) ∼= 〈[1], [p13], [p23]〉, k1(c ∗(g)) ∼= 〈[u1], [u3], [v123]〉, where the equations mean that the left hand sides are generated by the lasses in the bra kets in the right hand sides, and the third oordinate in t 3 orresponds to z∧ and the unitary v123 is cubo 15, 3 (2013) k-theory for the group c∗-algebras of nilpotent dis rete groups 129 obtained from the bott proje tion p13 and u2. note that the above set of generators of k1(c ∗(g)) may be repla ed with {[u2], [u3], [v ′ 123]}, where v ′ 123 is obtained from the bott proje tion p23 and u1. example 2.4. let g × g be the dire t produ t of g the dis rete heisenberg nilpotent group of rank 3. then c∗(g×g) ∼= c∗(g)⊗c∗(g) the tensor produ t of c∗(g). sin e kj(c ∗(g)) (j = 0, 1) are torsion free, the künneth theorem in k-theory for c∗-algebras (see [2℄) implies that k0(c ∗(g × g)) ∼= [k0(c ∗(g)) ⊗ k0(c ∗(g))] ⊕ [k1(c ∗(g)) ⊗ k1(c ∗(g))] ∼= [z 3 ⊗ z3] ⊕ [z3 ⊗ z3] ∼= z 18, k1(c ∗ (g × g)) ∼= [k0(c ∗ (g)) ⊗ k1(c ∗ (g))] ⊕ [k1(c ∗ (g)) ⊗ k0(c ∗ (g))] ∼= [z 3 ⊗ z3] ⊕ [z3 ⊗ z3] ∼= z 18. our theorem tells us that k0(c ∗ (g × g)) ∼= 〈[1], [p13], [p23], [p46], [p56], [p14], [p15], [p16], [p24], [p25], [p26], [p34], [p35], [p36], [q1346], [q1356], [q2346], [q2356]〉, where the subindi es 1, 2, 3 orrespond to the unitary generators uj of c ∗(g)⊗c and the subindi es 4, 5, 6 orrespond to the unitary generators uj of c⊗c ∗(g) and both subindi es 3 and 6 orresponds to the enter z of g. also, k1(c ∗(g × g)) ∼= 〈[u1], [u3], [v123], [u4], [u6], [v456], [v(p14, u2)], [v(p15, u2)], [v(p16, u2)], [v(p24, u3)], [v(p25, u3)], [v(p26, u3)], [v(p34, u5)], [v(p35, u6)], [v(p36, u4)], [v(q1346, u2)], [v(q1356, u2)], [v(q2346, u1)]〉, where ea h v(pij, uk) means the unitary obtained from pij and uk and ea h v(qijkl, um) means the unitary obtained from qijkl and um. note that the unions of subindei es su h as (1, 2, 4) of (14, 2) and (1, 2, 5) of (15, 2) are taken only on e among ombinations of (i, j, k) with i < j < k. also, the hoi e of adding um to either pij or qijkl may be di�erent to make the same set of unions of subindi es, and the set of generators of k0(c ∗(g × g)) orresponds to the set of generators of k1(c ∗(g × g)) bije tively. corollary 2.5. if g is a �nitely generated, dis rete nilpotent group without torsion, then k0(c ∗(g)) ∼= k1(c ∗(g)). example 2.6. the isomorphism in the orollary above does not hold if g has torsion. indeed, if g = zn = z/nz (n ≥ 2) a y li group, then c ∗(g) ∼= cn, so that k0(c ∗(g)) ∼= zn but k1(c ∗(g)) ∼= 0. 130 takahiro sudo cubo 15, 3 (2013) corollary 2.7. if g is a �nitely generated, dis rete nilpotent group without torsion, then the k-theory groups k0(c ∗(g)) and k1(c ∗(g)) are torsion free. proof. this follows from the onstru tion of the generators of k0(c ∗(g)) and k1(c ∗(g)) obtained in the theorem above. remark. possibly, in the last orollary, the group g may have torsion. example 2.8. we onsider a version of the dis rete heisenberg nilpotent group with torsion (see [1℄ or [3℄ for the dis rete heisenberg nilpotent group). let g = z22 ⋊α z2 be a semi-dire t produ t of the produ t group z 2 2 of the y i group z2 = z/2z by an a tion of z2 de�ned by αt(b + tc, c) for b, c, t ∈ z2. then the group c ∗ -algebra c∗(g) is isomorphi to the rossed produ t c(z∧2 × z ∧ 2 ) ⋊α∧ z2 via the fourier transform, where the dual a tion α ∧ of z2 on the produ t spa e of the dual group z ∧ 2 ∼= z2 is de�ned by α ∧ t (z, w) = (z, z tw) for z, w ∈ z∧2 via the duality α∧t (ϕz,w)(b, c) = ϕz,w(b + tc, c) = z b+tcwc = ϕz,ztw(b, c) where ϕz,w ∈ z ∧ 2 ×z ∧ 2 indenti�ed with (z, w) ( f. [5℄). we then obtain the following de omposition: c(z∧2 × z ∧ 2 ) ⋊α∧ z2 ∼= [c ⊗ (c 2 ⋊α∧ z2)] ⊕ [c ⊗ (c 2 ⋊α∧ z2)] ∼= [c 2 ⊗ c∗(z2)] ⊕ [m2(c)] ∼= [c 2 ⊗ c2] ⊕ m2(c) ∼= c 4 ⊕ m2(c) where the a tion α∧ on c2 in the �rst dire t summand is trivial and that in the se ond is the shift. therefore, k0(c ∗ (g)) ∼= z 5, but k1(c ∗ (g)) ∼= 0. hen e the k-theory groups are torsion free. in this ase, the dire t sum fa tor z 4 in z 5 = k0(c ∗(g)) omes from c4 in c∗(g) whi h is a maximal abelian subalgebra of c∗(g) but the other dire t sum fa tor z in z5 = k0(c ∗(g)) omes from m2(c) in c ∗(g) whi h is a non ommutative subalgebra of c∗(g). therefore, the ase with torsion is ertainly di�erent from the torsion free ase onsidered above, but the nilpotent ase with torsion is just the same as the abelian ase with torsion as in the example above, in the k-theory level. corollary 2.9. if g is a �nitely generated, dis rete nilpotent group without torsion, then both k0(c ∗(g)) and k1(c ∗(g)) are isomorphi to a �nitely generated, free abelian group, i.e., zm for some positive integer m. cubo 15, 3 (2013) k-theory for the group c∗-algebras of nilpotent dis rete groups 131 3 in�nitely generated ase we assume that g is a ountable dis rete group. theorem 2. let g be an in�nitely generated, nilpotent dis rete group without torsion. then both k0(c ∗(g)) and k1(c ∗(g)) of the group c∗-algebra c∗(g) are isomorphi to an indu tive limit of �nitely generated free abelian groups: k0(c ∗ (g)) ∼= k1(c ∗ (g)) ∼= lim −→ z mn, for some positive integers mn with mn < mn+1, where the onne ting maps z mn → zmn+1 are inje tive. therefore, k0(c ∗(g)) ∼= k1(c ∗(g)) ∼= ⊕ ∞ z, whi h is the in�nite dire t sum of z, as a group. proof. let u = {g1, g2, · · · } be an in�nite set of generators of g and set un = {g1, g2, . . . , gn}, where gn+1 is not generated by g1, · · · , gn. let c ∗(un) denote the c ∗ -algebra generated by the elements of c∗(g) that orrespond to the elements of un. then c ∗(un) is a c ∗ -subalgebra of c∗(g). there is the anoni al in lusion in from c ∗(un) → c ∗(un+1). it follows that c ∗(g) is an indu tive limit of the c∗-subalgebras c∗(un) under the in lusions in. by ontinuity of k-theory, we have kj(c ∗(g)) ∼= lim −→ kj(c ∗(un)) for j = 0, 1. by the theorem in the previous se tion, we see that both kj(c ∗(un)) for j = 0, 1 are isomorphi to z mn for some positive integer mn and mn ≤ mn+1 and also that there is a anoni al in lusion from kj(c ∗(un)) ∼= z mn to kj(c ∗(un+1)) ∼= z mn+1 . we need to he k that mn 6= mn+1 for ea h n. note that the group hn generated by elements of un an be written as a su essive semi-dire t produ t by z: hn ∼= z ⋊ z ⋊ · · · ⋊ z rossed by z n − 1 times. then hn+1 ∼= hn ⋊ z. it follows that the a tion of z on hn an not be non-trivial on every generator of hn. be ause, if non-trivial, hn+1 is not nilpotent (but solvable). indeed, then there is no enter in hn+1, a ontradi tion to the nilpotentness of hn+1. therefore, there is a generator of hn su h that the a tion of z is trivial on it. therefore, we an onstru t a new bott proje tion from these ommuting elements of hn+1, and not from hn. it follows that mn < mn+1. example 3.1. if g is an in�nitely generated abelian dis rete group, then c∗(g) is isomorphi to an indu tive limit of c(tn) with the anoni al in lusion from c(tn) to c(tn+1). then kj(c ∗ (g)) ∼= lim −→ kj(c(t n )) ∼= lim −→ z 2 n−1 ∼= ⊕ ∞ z 132 takahiro sudo cubo 15, 3 (2013) for j = 0, 1. if g is an indu tive limit of the produ t groups πnz2 with the anoni al in lusion from π n z2 to πn+1z2, then g is ommutative and in�nitely generated and has torsion. then c∗(g) ∼= lim −→ c∗(πnz2) ∼= lim−→ ⊗nc∗(z2) ∼= lim −→ ⊗nc2 ∼= lim −→ c 2 n ∼= ⊕ ∞ c where the last side means the in�nite dire t sum of c, so that kj(c ∗ (g)) ∼= kj(⊕ ∞ c) ∼= ⊕ ∞kj(c) ∼= { ⊕∞z if j = 0, 0 if j = 1. re eived: mar h 2012. a epted: september 2013. referen es [1℄ j. anderson and w. pas hke, the rotation algebra, houston j. math. 15 (1989), 1-26. [2℄ b. bla kadar, k-theory for operator algebras, se ond edition, cambridge, (1998). [3℄ t. sudo, k-theory of ontinuous �elds of quantum tori, nihonkai math. j. 15 (2004), 141152. [4℄ n. e. wegge-olsen, k-theory and c∗-algebras, oxford univ. press (1993). [5℄ d. p. williams, crossed produ ts of c∗-algebras, surv 134, amer. math. so . (2007). cubo a mathematical journal vol.15, no¯ 02, (33–42). june 2013 on weak concircular symmetries of lorentzian concircular structure manifolds dhruwa narain d.d.u.gorakhpur university, department of mathematics & statistics, gorakhpur, india. profdndubey@yahoo.co.in sunil yadav alwar institute of engineering & technology, department of mathematics, matsya industrial area, alwar-301030, rajasthan, india. prof sky16@yahoo.com abstract the object of the present paper is to study weakly concircular symmetric, weakly concircular ricci symmetric and special weakly concircular ricci symmetric lorentzian concircular structure manifolds. resumen el objetivo del presente art́ıculo es estudiar las variedades de estructura simétricas concirculares débiles, las simétricas ricci concirculares débiles y concirculares lorentzianas simétricas ricci concirculares débiles especiales. keywords and phrases: weakly concircular symmetric manifold, weakly concircular ricci symmetric manifold, concircular ricci tensor, special weakly concircular ricci symmetric and lorentzian concircular structure manifold. 2010 ams mathematics subject classification: 53c10, 53c15, 53c25. 34 dhruwa narain and sunil yadav cubo 15, 2 (2013) 1 introduction the notion of weakly symmetric manifolds was introduced by tamassy and binh [8].a non-flat riemannian manifold (mn, g) (n > 2) is called weakly symmetric manifold if its curvature tensor r of type (0, 4) satisfies the condition (∇xr)(y, z, u, v) = a(x)r(y, z, u, v) + b(y)r(x, z, u, v) +h(z)r(y, x, u, v) + d(u)r(y, z, x, v) + e(v)r(y, z, u, x) , (1.1) for all vector fields x, y, z, u, v ∈ χ(mn), χ(m) being the lie-algebra of the smooth vector fields of m, where a, b, h, d and e are 1−forms (not simultaneously zero) and ∇ denote the operator of the covariant differentiations with respect to riemannian metric g. the 1−forms are called the associated 1−forms of the manifold and n−dimensional manifold of this kind is denoted by (ws)n. in 1999, de and bandyopadhyay [2] studied a (ws)n and prove that in such a manifold the associated 1–forms b = h and d = e. hence from (1.1) reduces to the following: (∇xr)(y, z, u, v) = a(x)r(y, z, u, v) + b(y)r(x, z, u, v) +b(z)r(y, x, u, v) + d(u)r(y, z, x, v) + d(v)r(y, z, u, x). (1.2) a transformation of n–dimensional riemannian manifold m, which transform every geodesic circle of m into a geodesic circle, is called a concircular transformation [11].the intersecting invariant of a concircular transformation is the concircular curvature tensor c̃ which is defined by [11]. c̃ (y, z, u, v) = r (y, z, u, v) − k n (n − 1) [ g (z, u)g(y, v) − g(y, u) g(z, v)] , (1.3) where k is the scalar curvature of the manifold. recently shaikh and hui [5] introduced the notion of weakly concircular symmetric manifolds. a riemannian manifold is called weakly concircular symmetric manifold if its concircular curvature tensor c̃ of type (0, 4) is not identically zero and satisfies the condition ( ∇xc̃ ) (y, z, u, v) = a (x) c̃(y, z, u, v) + b (y) c̃(x, z, u, v) +h(z)c̃(y, x, u, v) + d (u) c̃(y, z, x, v) + e (v) c̃(y, z, u, x) , (1.4) for all vector fields x, y, z, u, v ∈ χ(mn) where a, b, h, d and e are 1–form (not simultaneously zero) an n–dimensional manifold of this kind is denoted by ( w c̃ s ) n .also it is known that [5], in a ( wc̃s ) n the associated 1–forms b = h and d = e, and hence the defining the condition (1.4) of a ( w, c̃s ) n reduces to the following form: ( ∇xc̃ ) (y, z, u, v) = a (x) c̃(y, z, u, v) + b (y) c̃(x, z, u, v) +b (z) c̃(y, x, u, v) + d (u) c̃(y, z, x, v) + d (v) c̃(y, z, u, x) , (1.5) cubo 15, 2 (2013) on weak concircular symmetries of lorentzian concircular . . . 35 where a, b and d are 1-forms (not simultaneously zero). again tamassy and binh [9] introduced the notion of weakly ricci symmetric manifolds. a riemannian manifold (mn, g), (n > 2) is called weakly ricci symmetric manifold if its ricci tensor s of type(0, 2) is not identically zero and satisfies the condition: (∇xs) (y, z) = a(x)s(y, z) + b(y)s(x, z) + d(z)s(y, x), (1.6) where a, b and d are three non-zero 1–forms called the associate 1-forms of the manifold, and ∇ is the operator of covariant differentiation with respect to metric g. such n–dimensional manifold is denoted by (w r s) n. if a = b = d then is called pseudo ricci symmetric. let{ei : i = 1, 2, ......n} be an orthonormal basis of the tangent space at each point of the manifold and let s̃ (y, v) = n∑ i=1 c̃ (y, ei, ei, y) then from (1.3), we have s̃ (y, v) = s (y, v) − k n g (y, v), (1.7) the tensor s̃ is called the concircular ricci symmetric tensor which is symmetric tensor of type (0, 2). in [1] de and ghose introduced the notion of weakly concircular ricci symmetric manifolds. a riemannian manifold (mn, g), (n > 2) is called weakly concircular ricci symmetric manifolds [1] if its concircular ricci tensor s̃ of type (0, 2) is not identically zero satisfies the condition: ( ∇xs̃ ) (y, z) = a (x) s̃(y, z) + b (y) s̃(x, z) + d (z) s̃(y, x), (1.8) wherea, b and d are three 1-form (not simultaneously zero).if a = b = d then mn is called pseudo concircular ricci symmetric. a riemannian manifold is called special weakly ricci symmetric manifold if (∇xs) (y, z) = 2a(x)s(y, z) + a(y)s(x, z) + a(z)s(y, x), (1.9) where a is a 1–form and is defined by a(x) = g(x, ρ). (1.10) where ρ is the associated vector field. motivated by above studied we define and syudy special weakly concircular ricci symmetric manifold. an n−dimensional riemannian manifold is called special weakly concircular ricci symmetric manifolds. if ( ∇xs̃ ) (y, z) = 2 a (x) s̃(y, z) + a (y) s̃(x, z) + a (z) s̃(y, x). (1.11) where a is a 1–form and is defined by (1.10). 36 dhruwa narain and sunil yadav cubo 15, 2 (2013) an (2n + 1)-dimensional lorentzian manifold m is smooth connected para contact hausdorff manifold with lorentzian metric g, that is, m admits a smooth symmetric tensor field g of type (0, 2) such that for each point p ∈ m, the tensor gp : tpm × tpm → r is a non degenerate inner product of signature (−, +, .....+)where tpmdenotes the tangent space of m at p and r is the real number space. in a lorentzian manifold (m, g) a vector field ρ defined by g(x, ρ) = a(x) for any vector field x ∈ χ(m) is said to be concircular vector field [5] if (∇xa)(y) = α [g(x, y) + ω(x)a(y)] where α is a non zero scalar function, a is a 1-form and ω is a closed 1-form. let m be a lorentzian manifold admitting a unit time like concircular vector field ξ, called the characteristic vector field of the manifold. then we have g(ξ, ξ) = −1, (1.12) since ξ is the unit concircular vector field, there exist a non zero 1-form η such that g(x, ξ) = η(x), (1.13) the equation (1.13) of the following form holds: (∇xη)(y) = α [g(x, y) + η(x)η(y)] (α 6= 0), (1.14) for all vector field x, y, where ∇denotes the operator of covariant differentiation with respect to lorentzian metric g and α is a non zero scalar function satisfying (∇xα) = (xα) = ρη(x), (1.15) where ρ being a scalar function. if we put φx = 1 α ∇xξ, (1.16) then from (1.14) and (1.16), we have φ2x = x + η(x)ξ, (1.17) from which it follows that φ is a symmetric (1, 1)-tensor. thus the lorentzian manifold m together with unit time like concircular vector field ξ, it’s associate 1-form η and (1, 1)–tensor field φ is said to be lorentzian concircular structure manifolds (briefly (lcs)2n+1-manifold) [6]. in particular if α = 1, then the manifold becomes lp-sasakian structure of matsumoto [3]. cubo 15, 2 (2013) on weak concircular symmetries of lorentzian concircular . . . 37 2 lorentzian concircular structure manifolds a differentiable manifold m of dimension (2n+1) is called (lcs)2n+1-manifold if it admits a (1, 1) -tensor φ, a contravarient vector field ξ, a covariant vector field η and a lorentzian metric g which satisfy the following η(ξ) = −1, (2.1) φ2 = i + η ⊗ ξ, (2.2) g(φx, φy) = g(x, y) + η(x)η(y), (2.3) g(x, ξ) = η(x), (2.4) φξ = 0 , η(φx) = 0, (2.5) for all x, y ∈ tm. also in a (lcs)2n+1-manifold the following relations are satisfied [7]. η(r(x, y)z) = (α2 − ρ) [g(y, z)η(x) − g(x, z)η(y)] , (2.6) r(x, y)ξ = (α2 − ρ) [η(y)x − η(x)y] , (2.7) r(ξ, x)y = (α2 − ρ) [g(x, y)ξ − η(y)x] , (2.8) r(ξ, x)ξ = (α2 − ρ) [η(x)ξ + x] , (2.9) (∇xφ)(y) = α [g(x, y)ξ + 2η(x)η(y)ξ + η(y)x] , (2.10) s(x, ξ) = 2n(α2 − ρ)η(x), (2.11) s(φx, φy) = s(x, y) + 2n(α2 − ρ)η(x)η(y), (2.12) definition 2.1 a lorentzian concircular structure manifold is said to be η–einstein if the ricci operator q satisfies q = aid + bη ⊗ ξ, where a and b are smooth functions on the manifolds, in particular if b = 0, then m is an einstein manifold. 38 dhruwa narain and sunil yadav cubo 15, 2 (2013) 3 main results definition 3.1a lorentzian concircular structure manifold(m2n+1, g) (n > 1) is said to be weakly concircular symmetric if its concircular curvature tensor c̃ of type (0, 4) satisfies (1.5) substituting y = v = ei in (1.5) and taking summation over i , 1 ≤ i ≤ 2n + 1, we get (∇xs) (z, u) − dκ(x) n g(z, u) = a(x) [ s(z, u) − κ n g(z, u) ] + b(z) [s(x, u) − κ n g(x, u) + d(u) [ s(x, z) − κ n g(x, z) ] + b(r(x, z)u) + d(r(x, u)z) − κ n(n−1) [{b(x) + d(x) g(z, u) − b(z)g(x, u) − d(u)g(z, x)] (3.1) again setting x = z = u = ξ in (3.1) and using (2.7)and (2.11), we have a(ξ) + b(ξ) + d(ξ) = dk(ξ) k − 2n2(α2 − ρ) (3.2) this leads to the following result. theorem 3.1.in a weakly concircular symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the relation (3.2) holds. corollary 3.1 in a weakly concircular symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the sum of 1-forms a, b and d is zero everywhere if and only if the scalar curvature κ of the manifold is constant. next, putting x and z by ξ in (3.1) and using (2.4), (2.7)and (2.11) we obtain (∇xs) (ξ, u) − dκ(ξ) n η(u) = [a(ξ) + b(ξ)] { 2n(α2 − ρ) − κ n } η(u)+ [ κ n − 2n(α2 − ρ) − κ n(n−1) + 1 ] d(u) + d(ξ) [ (α2 − ρ) − κ n(n−1) ] η(u). (3.3) also from (2.11), we have (∇ξs) (ξ, u) = 0. (3.4) in view of (3.2) and (3.4), equation (3.3) reduces to d(u) = [ k + (n − 1) { k − 2n2(α2 − ρ) } −k + (n − 1) {k − 2n2(α2 − ρ)} ] d(ξ) η(u). (3.5) next setting x = u = ξ in (3.1) and proceeding in the similar manner as above, we have b(z) = [ k − (n − 1) { k − 2n2(α2 − ρ) } −k + (n − 1) {k − 2n2(α2 − ρ)} ] b(ξ) η(z), (3.6) cubo 15, 2 (2013) on weak concircular symmetries of lorentzian concircular . . . 39 again, substituting z = u = ξ (3.1), we obtain (∇xs) (ξ, ξ) + dκ(x) n = a(x) [ κ n − 2n(α2 − ρ) ] + [ κ n(n−1) − (α2 − ρ) ] {b(x) + d(x) + (b(ξ) + d(ξ))η(x)} + [ 2n(α2 − ρ) − κ n ] {b(ξ) + d(ξ)} η(x) (3.7) on the other hand we have (∇ξs) (ξ, ξ) = ∇xs(ξ, ξ) − 2s(∇xξ, ξ), which yield by using (1.16) and (2.1) that. (∇ξs) (ξ, ξ) = −2n(α 2 − ρ)ξ, (3.8) in view of (3.7) and (3.8), we get a(x) = [ dk(x)−2n 2 (α 2 −ρ)ξ k−2n2(α2−ρ) ] − [ k (n−1){(k−2n2(α2−ρ)} ] {b(x) + d(x)} − {d(ξ) + b(ξ)}η(x) − { dk(ξ) k−2n2(α2−ρ) − a(ξ) } η(x) (3.9) this leads to the following result. theorem 3.2. in a weakly concircular symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the associated 1-forms d, b and a are given by (3.5) (3.6) and (3.9) respectively. definition 3.2 a lorentzian concircular structure manifold (m2n+1, g) (n > 1) is said to be weakly concircular ricci symmetric if its concircular ricci tensor s of type (0, 2) satisfies (1.8). in view of (1.8) and (1.9) yield (∇xs) (y, z) − dk(x) n g(y, z) = a(x) [ s(y, z) − k n g(y, z) ] + b(y) [ s(x, z) − k n g(x, z) ] + d(z) [ s(x, y) − k n g(x, y) ] (3.10) setting x = y = z = ξ in above we get the relation (3.2). hence we can state the following theorem 3.3. in a weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the relations (3.2) holds corollary 3.2. in a weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the sum of 1-forms a, b and d is zero everywhere and only if the scalar curvature k of the manifold is constant. now, taking x and y by ξ in (3.10), we have 40 dhruwa narain and sunil yadav cubo 15, 2 (2013) (∇ξs) (ξ, z) − dκ(x) n η(z) = {a(ξ) + b(ξ)} + d(z) [ s(ξ, ξ) + κ n ] (3.11) in view of (3.2) and (3.4), equation (3.11) yields. d(z) = −dk(ξ) k − 2n2(α2 − ρ) η(z) + [ dk(ξ) k − 2n2(α2 − ρ) − d(ξ) ] η(z), (3.12) again putting x = z = ξ in (3.11) and proceeding in a similar manner as above, we get b(y) = −dk(ξ) k − 2n2(α2 − ρ) η(y) + [ dk(ξ) k − 2n2(α2 − ρ) − b(ξ) ] η(y), (3.13) a(x) = −2n2(α2 − ρ)ξη(x) k − 2n2(α2 − ρ) + dk(x) k − 2n2(α2 − ρ) + [ dk(ξ) k − 2n2(α2 − ρ) − a(ξ) ] η(x), (3.14) this leads to the following result. theorem 3.4. in a weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the associated 1-form d, b and a are given by (3.12) (3.13) and (3.14) respectively adding equations (3.12) (3. 13) and (3.14), using (3.3) we obtain a(x) + b(x) + d(x) = dk(x) − 2n2(α2 − ρ)ξ k − 2n2(α2 − ρ) (3.15) for any vector field x . this leads to the following result. theorem 3.5. in a weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the sum of the associated 1-form a, b and d is given by (3.15) corollary 3.3 there is no weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g)(n > 1) unless the sum of the 1-forms is everywhere zero if dk(x) = 2n2(α2− ρ)ξ. also taking cyclic sum of (1.11), we get ( ∇xs̃ ) (y, z) + ( ∇ys̃ ) (z, y) + ( ∇zs̃ ) (x, y) = 4 a (x) s̃(y, z) +a (y) s̃(x, z) + a (z) s̃(y, x), (3.16) let m2n+1 admits a cyclic ricci tensor. then (3.16) reduces to cubo 15, 2 (2013) on weak concircular symmetries of lorentzian concircular . . . 41 a (x) s̃(y, z) + a (y) s̃(x, z) + a (z) s̃(y, x) = 0. taking z = ξ in above and then using (1.7), (1.10) and (2.11), we obtain [ 2n2(α2 − ρ) − κ n ] {a(x)η(y) + a(y)η(x)} + η(ρ)s(x, y) = 0. (3.17) again taking z = ξ in (3.17), we get 2η(ρ)η(x) = a(x) (3.18) taking x = ξ in (3.18)and using (1.7), we yields η(ρ) = 0. (3.19) in view of (3.18) and (3.19), we get a(x) = 0, ∀x. this leads to the following result. theorem 3.6.if a special weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) admits cyclic ricci tensor then the 1-form a must vanishes. finally for einstein manifold (∇xs) (y, z) = 0 and s(y, z) = ag(y, z). then (1.7) and (1.11), we get − dκ(x) n g(y, z) = 2a(x) [( a − κ n ) g(y, z) ] + a(y) [( a − κ n ) g(x, z) ] +a(z) [( a − κ n ) g(x, y) ] , (3.20) plugging z = x = y = ξ in (3.20), we obtain that 4 η (ρ) (an − κ) = dκ (ξ) which implies that if κ is constant then η(ρ) = 0,that is a(y) = 0, ∀y. therefore we state the results theorem 3.7. a special weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1)can not einstein manifold if the scalar curvature of the manifold is constant. corollary 3.4.in a special weakly concircular ricci symmetric lorentzian concircular structure manifold (m2n+1, g) (n > 1) the 1-form a is given by [ a (ξ) = d k (ξ) − 2n2 (α2 − ρ) 2 { k − 2n2 (α2 − ρ) + n } . ] 42 dhruwa narain and sunil yadav cubo 15, 2 (2013) received: january 2012. accepted: october 2012. references [1] de,u.c. and ghose,g.c.,on weakly concircular ricci symmetric manifolds, south east assian j. math. and math. sci. 3(2)(2005), 9-15. 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() cubo a mathematical journal vol.16, no¯ 03, (37–53). october 2014 existence results for some neutral partial functional differential equations of fractional order with state-dependent delay mouffak benchohra laboratory of mathematics, university of sidi bel abbes, po box 89, 22000 sidi bel abbes, algeria. benchohra@yahoo.com omar bennihi département de mathématiques et informatique, université de saida, 20000, saida, algérie. obennihidz@yahoo.fr khalil ezzinbi département de mathématiques, faculté des sciences, semlalia, b.p.2390, marrakech, morocco. ezzinbi@ucam.ac.ma abstract in this paper we provide sufficient conditions for the existence and uniqueness of mild solutions for a class of neutral partial functional differential equations of fractional order with state-dependent delay. the nonlinear alternative of frigon-granas type for contractions maps in fréchet spaces combined with α-resolvent family is the main tool in our analysis. resumen en este art́ıculo entregamos condiciones suficientes para la existencia y unicidad de soluciones mild para una clase de ecuaciones diferenciales funcionales parciales neutrales de orden fraccionario con retraso dependiente del estado. la alternativa no lineal de tipo frigon-granas para contracciones en espacios de féchet combinados con familias α-resolvente es la herramienta principal en nuestro análisis. keywords and phrases: neutral functional differential equations; riemman-liouville fractional derivative; mild solution; delay; fixed point; α-resolvent; fréchet space. 2010 ams mathematics subject classification: 26a33; 34a08; 34k37. 38 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) 1 introduction in recent years fractional calculus has found many applications in physics, mechanics, chemistry, porous media, viscoelasticity, electrochemistry, electromagnetism, engineering, control, etc. [11, 24, 34, 37]. recent developments of differential and integral equations of fractional order are reported in the books by abbas et al. [1], baleanu et al. [11], kilbas et al. [27], lakshmikantham et al. [28], and the references therein. in this work, we discuss the existence of the unique mild solution defined on the semi-infinite positive real interval [0,+∞) for a class of neutral partial functional differential evolution equations of fractional order with state dependent delay. complicated situations in which the delay depends on the unknown functions have been proposed in modeling in recent years. these equations are frequently called equations with state-dependent delay. existence results and among other things were derived recently from functional differential equations when the solution is depending on the delay. we refer the reader to the papers by adimy and ezzinbi [2], adimy et al. [3], [4], agarwal et al. [5, 6], aissani and benchohra [7], ait dads and ezzinbi [8], dos santos et al [17], and hernandez et al. [23]. over the past several years it has become apparent that equations with state-dependent delay arise also in several areas such as in classical electrodynamics [18], in population models [12], in models of commodity price fluctuations [13, 31], and in models of blood cell productions [32]. recently li and peng [30] studied a class of abstract homogeneous fractional evolution equation. baghli et al. [10], have proved global existence and uniqueness results for an initial value problem for functional differential equations of first order with state-dependent delay. functional differential equations involving the riemann-liouville fractional derivative were considered by benchohra et al. [14]. n’guérékata and mophou [33] studied semi linear neutral fractional functional evolution equations with infinite delay using the notion of α resolvent family. several works was published on the existence of mild solutions for this type of problems using different approaches and techniques like the approach of probability density function given by el borai [19] and developed by zhou and jiao [41]. one can see also the work by darwish and ntouyas [15]. recently velusamy et al [38] studied the same problem using approach and technics based on the nonlinear alternative of leray-schauder. motivated by the above papers, in this paper we studied the existence and uniqueness of solutions for neutral partial functional differential equations of fractional order with state-dependent delay in a real banach space (e, |.|) when the delay is infinite. our contribution is to introduce a new approach based on the notion of semi norms in fréchet spaces. in particular, we consider the following initial value problem dα[x(t) − g(t,xρ(t,xt))] = ax(t) + f(t,xρ(t,xt)),a.e. t ∈ [0,+∞),0 < α < 1, (1) x0 = ϕ, ϕ ∈ b, (2) where a : d(a) ⊂ e → e is the infinitesimal generator of an α-resolvent family (tα(t))t≥0 defined on a real banach space e, dα is understood here in the riemann-liouville sense, f : j × b → e, ρ : j × b → r and g : j × b → e are appropriate given functions and satisfy some conditions that will be specified later, ϕ belongs to an abstract space denoted b and called phase space with ϕ(0) − g(0,ϕ) = 0. for any function x defined on (−∞,+∞) and any t ∈ j, we denote by xt the cubo 16, 3 (2014) existence results for some neutral partial functional . . . 39 element of b defined by xt(θ) = x(t + θ), θ ∈ (−∞,0]. the function xt represents the history of the state from −∞ up to the present time t. our approach is based on the nonlinear alternative of leray-schauder type due to frigon and granas [21]. these results can be considered as a contribution to this emerging field. this paper is arranged as follows. in section 2, some preliminary results are introduced. the main results are presented in section 3, and in section 4, an example is given to illustrate the abstract theory. 2 preliminaries in this section, we collect a few auxiliary results which will be needed in the sequel. let b > 0. by c([0,b];e) we denote the banach space of continuous functions from [0,b] into e, normed by ‖x‖∞ = sup t∈[0,b] |x(t)|. b(e) is the space of bounded linear operators from e into e, with the usual supremum norm ‖n‖b(e) = sup{|n(x)| : |x| = 1}. a measurable function x : [0,b] → e is bochner integrable if and only if |x| is lebesgue integrable. let l1([0,b],e) denotes the banach space of measurable functions x : [0,b] → e which are bochner integrable normed by ‖x‖l1 = ∫b 0 |x(t)|dt. (for the bochner integral properties, see the classical monograph of yosida[39]). definition 2.1. [14] the fractional primitive of order α > 0 of a function h : r+ → r is defined by iα0 h(t) := 1 γ(α) ∫t 0 (t − s)α−1h(s)ds, (3) provided the right side exist point wise on r+. γ is the gamma function. definition 2.2. [14] the fractional derivative of order α > 0 of a function h : r+ → r is defined as follow dα0h(t) := 1 γ(1 − α) d dt ∫t 0 (t − s)−αh(s)ds = d dt i1−α0 h(t). (4) definition 2.3. [33] the laplace transform of a function f ∈ l1loc(r +,e) is defined by f̂(λ) := ∫ ∞ 0 e−λtf(t)dt, re(λ) > ω, if the integral is absolutely convergent for re(λ) > ω. in order to defined the mild solution of the considered problem, we recall the following definition 40 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) definition 2.4. [33] let a be a closed and linear operator with domain d(a) defined on a banach space e. we call a the generator of an α-resolvent family or solution operator if there exists ω > 0 and a strongly continuous function tα : r + → l(e) such that {λ : re(λ) > ω} ⊂ ρ(a), and (λα − a)−1x = ∫ ∞ 0 e−λttα(t)xdt, re(λ) > ω and x ∈ e. tα(t) is called the solution operator generated by a. proposition 2.5. [29] let tα(t) ∈ l(e) be the solution operator with generator a. then the following conditions are satisfied: (1) tα(t) is strongly continuous for t ≥ 0 and tα(0) = i. (2) tα(t)d(a) ⊂ d(a) and atα(t)x = tα(t)ax for all x ∈ d(a), t ≥ 0. (3) for every x ∈ d(a) and t ≥ 0, one has tα(t)x = x + ∫t 0 (t − s)α−1 γ(α) atα(s)xds. (4) let x ∈ d(a). then ∫t 0 (t − s)α−1 γ(α) tα(s)xds ∈ d(a), and tα(t)x = x + a ∫t 0 (t − s)α−1 γ(α) tα(s)xds remark 2.6. the concept of a solution operator, as defined above, is closely related to the concept of a resolvent family (see prüss)[35]. because of the uniqueness of the laplace transform, in the border case α = 1, the family tα(t) corresponds to the c0semigroup (see [20]), whereas in the case α = 2 a solution operator corresponds to the concept of cosine family (see [9]). for more details on α-resolvent family, we refer to [33] and the references therein. we will define the phase space b axiomatically, using ideas and notations developed by hale and kato [22]. (see also kapper and chappacher [26] and schumacher [36]). more precisely, b will denote the vector space of functions defined from (−∞,0] into e endowed with a semi norm denoted ‖.‖b and such that the following axioms hold. • (a1) if x : (−∞,b) → e, is continuous on [0,b] and x0 ∈ b, then for t ∈ [0,b) the following conditions hold cubo 16, 3 (2014) existence results for some neutral partial functional . . . 41 – (i) xt ∈ b – (ii)‖xt‖b ≤ k(t) sup{|x(s)| : 0 ≤ s ≤ t} + m(t)‖x0‖b, – (iii)|x(t)| ≤ h‖xt‖b where h ≥ 0 is a constant, k : [0,b) → [0,+∞), m : [0,+∞) → [0,+∞) with k continuous and m locally bounded and h, k and m are independent of x(.). • (a2) for the function x in (a1), the function t → xt is a b-valued continuous function on [0,b]. • (a3) the space b is complete. denote kb = sup{k(t) : t ∈ [0,b]} and mb = sup{m(t) : t ∈ [0,b]}. remark 2.7. (1) [(iii)] is equivalent to |φ(0)| ≤ h‖φ‖b for every φ ∈ b. (2) since ‖·‖b is a semi norm, two elements φ,ψ ∈ b can verify ‖φ−ψ‖b = 0 without necessarily φ(θ) = ψ(θ) for all θ ≤ 0. (3) from the equivalence of in the first remark, we can see that for all φ,ψ ∈ b such that ‖φ − ψ‖b = 0, we necessarily have that φ(0) = ψ(0). we now indicate some examples of phase spaces. for other details we refer, for instance to the book by hino et al. [25]. example 2.8. let: bc the space of bounded continuous functions defined from (−∞,0] to e; buc the space of bounded uniformly continuous functions defined from (−∞,0] to e; c∞ := {φ ∈ bc : limθ→−∞ φ(θ) exist in e} ; c0 := {φ ∈ bc : limθ→−∞ φ(θ) = 0} , endowed with the uniform norm ‖φ‖ = sup{|φ(θ)| : θ ≤ 0}. we have that the spaces buc, c∞ and c0 satisfy conditions (a1) − (a3). however, bc satisfies (a1),(a3) but (a2) is not satisfied. example 2.9. the spaces cg, ucg, c ∞ g and c 0 g. let g be a positive continuous function on (−∞,0]. we define: cg := { φ ∈ c((−∞,0],e) : φ(θ) g(θ) is bounded on (−∞,0] } ; 42 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) c0g := { φ ∈ cg : limθ→−∞ φ(θ) g(θ) = 0 } , endowed with the uniform norm ‖φ‖ = sup { |φ(θ)| g(θ) : θ ≤ 0 } . then we have that the spaces cg and c 0 g satisfy conditions (a1)−(a3). we consider the following condition on the function g. (g1) for all a > 0, sup0≤t≤a sup { g(t+θ) g(θ) : −∞ < θ ≤ −t } < ∞. they satisfy conditions (a1) and (a2) if (g1) holds. example 2.10. the space cγ. for any real constant γ, we define the functional space cγ by cγ := { φ ∈ c((−∞,0],e) : lim θ→−∞ eγθφ(θ) exists in e } endowed with the following norm ‖φ‖ = sup{eγθ|φ(θ)| : θ ≤ 0}. then cγ satisfies axioms (a1) − (a3). let e = (e,‖.‖n) be a fréchet space with a family of semi-norms {‖.‖n}n∈n. we say that x is bounded if for every n ∈ n, there exists mn > 0 such that ‖x‖n ≤ mn for all x ∈ x. to e we associate a sequence of banach spaces {(en,‖ · ‖n)} as follows: for every n ∈ n, we consider the equivalence relation ∼n defined by : x ∼n y if and only if ‖x − y‖n = 0 for x,y ∈ e. we denote en = (e| ∼n ,‖ · ‖n) the quotient space, and we set (e n,‖ · ‖n) the completion of e n with respect to ‖ · ‖n. to every x ⊂ e, we associate a sequence {x n} of subsets xn ⊂ en as follows: for every x ∈ e, we denote [x]n the equivalence class of x in e n and we define xn = {[x]n : x ∈ x}. we denote xn, intn(x n) and ∂nx n, respectively, the closure, the interior and the boundary of xn with respect to ‖ · ‖n in e n. we assume that the family of semi-norms {‖.‖n}n∈n verifies: ‖x‖1 ≤ ‖x‖2 ≤ ‖x‖3 ≤ . . . for every x ∈ x. the following definition is the appropriate concept of contraction in e. definition 2.11. [21] a function f : e → e is said to be a contraction if for every n ∈ n there exists kn ∈ [0,1) such that: ‖f(x) − f(y)‖n ≤ kn‖x − y‖n for all x,y ∈ e. cubo 16, 3 (2014) existence results for some neutral partial functional . . . 43 the corresponding nonlinear alternative result is as follows: theorem 2.12. (nonlinear alternative) [21]. let e be a fréchet space and x be a closed subset of e such that 0 ∈ x and let n : x → e be a contraction map such that n(x) is bounded. then one of the following statements holds: (s1) n has a unique fixed point in e. (s2) there exist 0 ≤ λ < 1, n ∈ n and x ∈ ∂nx n such that ‖x − λn(x)‖n = 0. 3 main results before starting and proving the main results, let us give the definition of mild solution to the neutral partial evolution problem (1)-(2). throughout this work, the function f : j × b → e will be continuous. definition 3.1. a function x is said to be a mild solution of (1)-(2) if x satisfies x(t) =    ϕ(t), t ∈ (−∞,0], g(t,xρ(t,xt)) + t∫ 0 tα(t − s)a(s)g(s,xρ(s,xs))ds + t∫ 0 tα(t − s)f(s,xρ(s,xs))ds, t ∈ j. (5) set r(ρ−) = {ρ(s,ϕ) : (s,ϕ) ∈ j × b, ρ(s,ϕ) ≤ 0}. we always assume that ρ : j × b → r is continuous. let m̂ be such that m̂ = supt∈j |tα(t)| then ‖tα(t)‖b(e) ≤ m̂, t ∈ j. additionally, we will need to introduce the following hypotheses which are assumed thereafter: (hϕ) the function t → ϕt is continuous from r(ρ −) into b and there exists a continuous and bounded function lϕ : r(ρ−) → (0,∞) such that ‖ϕt‖b ≤ l ϕ(t)‖ϕ‖b for every t ∈ r(ρ −) remark 3.2. the condition (hϕ), is frequently verified by continuous and bounded functions. for more details, see for instance ([25], proposition 7.1.1). (h1) there exist a function p ∈ l 1 loc(j,r +) and a continuous nondecreasing function ψ : [0,+∞) → (0,∞) such that |f(t,u)| ≤ p(t)ψ(‖u‖b) for a.e. t ∈ j and each u ∈ b. 44 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) (h2) for all n > 0, there exists ln ∈ l 1 loc(j,r +) such that: |f(t,u) − f(t,v)| ≤ ln(t)‖u − v‖b for all t ∈ [0,n] and u,v ∈ b. (h3) there exists a constant m̄0 > 0 such that ‖a−1(t)‖b(e) ≤ m̄0 for all t ∈ j. (h4) there exists a constant l∗ > 0 such that |a(s) g(s,ϕ) − a(s̄) g(s̄, ϕ̄)| ≤ l∗ (|s − s̄| + ‖ϕ − ϕ̄‖b) for all s, s̄ ∈ j and ϕ,ϕ̄ ∈ b. consider the following space b+∞ = {x : r → e : x|[0,b] continuous for b > 0 and x0 ∈ b}, where x|[0,b] is the restriction of x to the real compact interval [0,b]. let us fix r > 1. for every n ∈ n, we define in b+∞ the semi norms by: ‖x‖n := sup{e −rl ∗ n (t)|x(t)| : t ∈ [0,n]} where l∗n(t) = t∫ 0 ln(s)ds, ln(t) = knm̂ln(t) and ln is the function given in (h2). then b+∞ is a fréchet space with those semi norms family ‖.‖n. lemma 3.3. [23], (lemma 2.4) if x : (−∞,b] → e is a function such that x0 = ϕ, then ‖xs‖b ≤ (mb + l ϕ)‖ϕ‖b + kb sup{|x(θ)|, θ ∈ [0,max(0,s)]}, s ∈ r(ρ −) ∪ j, where lϕ = supt∈r(ρ−) l ϕ(t). theorem 3.4. suppose the hypothesis (hϕ) and (h1) − (h4) are satisfied and moreover for each n ∈ n ∫+∞ δn ds s + ψ(s) > m̂kn 1 − m̄0lkn ∫n 0 max(l,p(s))ds (6) with δn = cn + knl m̄0(1 + m̂) + m̂n + m̄0[cn + m̂‖ϕ‖b] 1 − m̄0lkn , and cn = (mn + l ϕ + knm̂h)‖ϕ‖b, then the problem (1) − (2) has a unique mild solution. cubo 16, 3 (2014) existence results for some neutral partial functional . . . 45 proof. consider the operator n : b+∞ → b+∞ defined by : n(x)(t) =    ϕ(t), if t ≤ 0; g(t,xρ(t,xt)) + ∫t 0 tα(t − s)a(s)g(s,xρ(s,xs))ds + ∫t 0 tα(t − s)f(s,xρ(s,xs))ds, if t ∈ j. (7) then, fixed points of the operator n are mild solutions of the problem (1) − (2). for ϕ ∈ b, we consider the function x(.) : r → e defined as bellow by y(t) =    ϕ(t), if t ≤ 0; 0, if t ∈ j. then y0 = ϕ. for each function z ∈ b+∞ with z(0) = 0, we consider the function z̄ by z̄(t) =    0, if t ≤ 0; z(t), if t ∈ j. if x(·) satisfies (3.1), we decompose it as x(t) = z(t) + y(t), t ≥ 0, which implies xt = zt + yt, for every t ∈ j and the function z(·) satisfies z0 = 0 and for t ∈ j, we get z(t) = g(t,zρ(t,zt+yt) + yρ(t,zt+yt)) + ∫t 0 tα(t − s)a(s)g(s,zρ(s,zs+ys) + yρ(s,zs+ys))ds + ∫t 0 tα(t − s)f(s,zρ(s,zs+ys) + yρ(s,zs+ys))ds. define the operator f : b0+∞ → b 0 +∞ by : f(z)(t) = g(t,zρ(t,zs+ys) + yρ(t,zs+ys)) + ∫t 0 tα(t − s)a(s)g(s,zρ(s,zs+ys) + yρ(s,zs+ys))ds + ∫t 0 tα(t − s)f(s,zρ(s,zs+ys) + yρ(s,zs+ys))ds. (8) obviously the operator n has a fixed point is equivalent to f has one, so it turns to prove that f has a fixed point. let z ∈ b0+∞ be such that z = λf(z) for some λ ∈ [0,1). then, using (h1)−(h4), 46 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) we have for each t ∈ [0,n] |z(t)| ≤ |g(t,zρ(t,zt+yt) + yρ(t,zt+yt))| + | ∫t 0 tα(t − s)a(s)g(s,zρ(s,zs+ys) + yρ(s,zs+ys))ds| + | ∫t 0 tα(t − s)f(s,zρ(s,zs+ys) + yρ(s,zs+ys))ds| ≤ ‖a−1(t)‖b(e)‖a(t) g(t,zρ(t,zt+yt) + yρ(t,zt+yt))‖ + ∫t 0 ‖tα(t − s)‖b(e)‖a(s) g(s,zρ(s,zs+ys) + yρ(s,zs+ys))‖ds + ∫t 0 ‖tα(t − s)‖b(e)|f(s,zρ(s,zs+ys) + yρ(s,zs+ys))|ds ≤ m̄0l(‖zρ(t,zs+ys) + yρ(t,zs+ys)‖b + 1) + m̂m̄0l(‖ϕ‖b + 1) + m̂ ∫t 0 l(‖zρ(s,zs+ys) + yρ(s,zs+ys)‖b + 1)ds + m̂ ∫t 0 p(s)ψ(‖zρ(s,zs+ys) + yρ(s,zs+ys)‖b)ds ≤ m̄0l‖zρ(t,zt+yt) + yρ(t,zt+yt)‖b + m̄0l(1 + m̂) + m̂ln + m̂m̄0l‖ϕ‖b + m̂l ∫t 0 ‖zρ(s,zs+ys) + yρ(s,zs+ys)‖bds + m̂ ∫t 0 p(s)ψ(‖zρ(s,zs+ys) + yρ(s,zs+ys)‖b)ds. using the assumption a1, we get ‖zρ(t,zs+ys) + yρ(t,zs+ys)‖b ≤ ‖zρ(t,zs+ys)‖b + ‖yρ(t,zs+ys)‖b ≤ k(s)|z(s)| + m(s)‖z0‖b + k(s)|y(s)| + m(s)‖y0‖b ≤ kn|z(s)| + mnm|ϕ(0)| + mn‖ϕ‖b ≤ kn|z(s)| + mnmh‖ϕ‖b + mn‖ϕ‖b ≤ kn|z(s)| + (knmh + mn)‖ϕ‖b. set cn = (knmh + mn)‖ϕ‖b we obtain |z(t)| ≤ m̄0l(kn|z(t)| + cn) + m̄0l(1 + m̂) + m̂ln + m̂m̄0l‖ϕ‖b + m̂l ∫t 0 (kn|z(s)| + cn)ds + m̂ ∫t 0 p(s)ψ(kn|z(s)| + cn)ds ≤ m̄0lkn|z(t)| + m̄0l(1 + m̂) + m̂ln + m̄0lcn + m̂m̄0l‖ϕ‖b + m̂l ∫t 0 (kn|z(s)| + cn)ds + m̂ ∫t 0 p(s)ψ(kn|z(s)| + cn)ds. then cubo 16, 3 (2014) existence results for some neutral partial functional . . . 47 (1 − m̄0lkn)|z(t)| ≤ l(m̄0(1 + m̂) + m̂n + m̄0cn + m̂m̄0‖ϕ‖b) + m̂l ∫t 0 (kn|z(s)| + cn)ds + m̂ ∫t 0 p(s)ψ(kn|z(s)| + cn)ds. set δn := cn + lkn 1 − m̄0lkn [m̄0(1 + m̂) + m̂n + m̄0cn + m̂m̄0‖ϕ‖b]. thus kn|z(t)| + cn ≤ δn + m̂lkn 1 − m̄0lkn ∫t 0 (kn|z(s)| + cn)ds + m̂kn 1 − m̄0lkn ∫t 0 p(s)ψ(kn|z(s)| + cn)ds. we consider the function µ defined by µ(t) := sup{ kn|z(s)| + cn : 0 ≤ s ≤ t }, 0 ≤ t < +∞. let t⋆ ∈ [0,t] be such that µ(t⋆) = kn|z(t ⋆)| + cn. by the previous inequality, we have µ(t) ≤ δn + m̂kn 1 − m̄0lkn [∫t 0 lµ(s)ds + ∫t 0 p(s)ψ(µ(s))ds ] for t ∈ [0,n]. let us take the right-hand side of the above inequality as v(t). then, we have µ(t) ≤ v(t) for all t ∈ [0,n]. from the definition of v, we have v(0) = δn and v′(t) = m̂kn 1 − m̄0lkn [lµ(t) + p(t)ψ(µ(t))] a.e. t ∈ [0,n]. using the nondecreasing character of ψ, we get v′(t) ≤ m̂kn 1 − m̄0lkn [ l v(t) + p(t) ψ(v(t))] a.e. t ∈ [0,n]. using the condition (6), this implies that for each t ∈ [0,n], we have ∫v(t) δn ds s + ψ(s) ≤ m̂kn 1 − m̄0lkn ∫t 0 max(l,p(s))ds ≤ m̂kn 1 − m̄0lkn ∫n 0 max(l,p(s))ds < ∫+∞ δn ds s + ψ(s) . 48 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) thus, for every t ∈ [0,n], there exists a constant λn such that v(t) ≤ λn and hence µ(t) ≤ λn. since ‖z‖n ≤ µ(t), we have ‖z‖n ≤ λn. now, we shall show that f : z → b0+∞ is a contraction operator. indeed, consider z, z̄ ∈ z, thus for each t ∈ [0,n] and n ∈ n |f(z)(t) − f(z̄)(t)| ≤ |g(t,zρ(t,zt+yt) + yρ(t,zt+yt)) − g(t, z̄ρ(t,zt+yt) + yρ(t,zt+yt))| + ∫t 0 ‖tα(t − s)‖b(e)|a(s)[g(s,zρ(s,zs+ys) + yρ(s,zs+ys)) − g(s, z̄ρ(s,zs+ys) + yρ(s,zs+ys))]|ds + ∫t 0 ‖tα(t − s)‖b(e)|f(s,zρ(s,zs+ys) + yρ(s,zs+ys)) − f(s, z̄ρ(s,zs+ys) + yρ(s,zs+ys))|ds ≤ ‖a−1(t)‖b(e) |a(t)g(t,zρ(t,zt+yt) + yρ(t,zt+yt)) − a(t)g(t, z̄ρ(t,zt+yt) + yρ(t,zt+yt))| + ∫t 0 m̂ |a(s)g(s,zρ(s,zs+ys) + yρ(s,zs+ys)) − a(s)g(s, z̄ρ(s,zs+ys) + yρ(s,zs+ys))|ds + ∫t 0 m̂ |f(s,zρ(s,zs+ys) + yρ(s,zs+ys)) − f(s, z̄ρ(s,zs+ys) + yρ(s,zs+ys))|ds ≤ m̄0l⋆‖zρ(t,zt+yt) − z̄ρ(t,zt+yt)‖b + ∫t 0 m̂l⋆‖zρ(s,zs+ys) − z̄ρ(s,zs+ys)‖bds + ∫t 0 m̂ln(s)‖zρ(s,zs+ys) − z̄ρ(s,zs+ys)‖bds ≤ m̄0l⋆‖zρ(t,zt+yt) − z̄ρ(t,zt+yt)‖b + ∫t 0 m̂[l⋆ + ln(s)]‖zρ(s,zs+ys) − z̄ρ(s,zs+ys)‖bds. since ‖zρ(t,zt+yt)‖b ≤ kn|z(t)| + cn we obtain |f(z)(t) − f(z̄)(t)| ≤ m̄0l∗kn|z(t) − z̄(t)| + ∫t 0 m̂[l∗ + ln(s)]kn|z(s) − z(s)|ds. let us take here l̄n(t) = m̂kn[l∗ + ln(t)] for the family semi norm {‖ · ‖n}n∈n, then |f(z)(t) − f(z̄)(t)| ≤ m̄0l∗kn|z(t) − z̄(t)| + ∫t 0 l̄n(s) |z(s) − z̄(s)|ds ≤ [m̄0l∗kne τl ∗ n (t)][e−τl ∗ n (t)|z(t) − z̄(t)|] + ∫t 0 [̄ln(s)e τl ∗ n (s)][e−τl ∗ n (s)|z(s) − z̄(s)|]ds ≤ m̄0l∗kne τ l ∗ n (t)‖z − z̄‖n + ∫t 0 [ eτl ∗ n (s) τ ]′ ds‖z − z̄‖n ≤ [m̄0l∗kn + 1 τ ]eτl ∗ n (t)‖z − z̄‖n. therefore, ‖f(z) − f(z̄)‖n ≤ [m̄0l∗kn + 1 τ ]‖z − z̄‖n. cubo 16, 3 (2014) existence results for some neutral partial functional . . . 49 so, for an appropriate choice of l∗ and τ such that m̄0l∗kn + 1 τ < 1, the operator f is a contraction for all n ∈ n. from the choice of z there is no z ∈ ∂zn such that z = λ f(z) for some λ ∈ (0,1). then the statement s2 in theorem 2.12 does not hold. a consequence of the nonlinear alternative of frigon and granas shows that the statement s1 holds. we deduce that the operator f has a unique fixed-point z⋆. then x⋆(t) = z⋆(t) + y⋆(t), t ∈ (−∞,+∞) is a fixed point of the operator n, which is the unique mild solution of the problem (1) − (2). 4 example to illustrate our results, we give an example example 4.1. consider the neutral evolution equation    ∂ α ∂tα [u(t,ξ) − ∫0 −∞ a3(s − t)u(s − ρ1(t)ρ2( ∫π 0 a2(θ)|u(t,θ)| 2dθ),ξ)ds] = ∂ 2 u(t,ξ) ∂ξ2 + a0(t,ξ)u(t,ξ) + ∫0 −∞ a1(s − t)u(s − ρ1(t)ρ2( ∫π 0 a2(θ)|u(t,θ)| 2dθ),ξ)ds, t ≥ 0, ξ ∈ [0,π], v(t,0) = v(t,π) = 0, t ≥ 0, v(θ,ξ) = v0(θ,ξ), −∞ < θ ≤ 0, ξ ∈ [0,π], (9) to represent this system in the abstract form (1)-(2), we choose the space e = l2([0,π],r) and the operator a : d(a) ⊂ e → e is given by aω = ω′′ with domain d(a) := {ω ∈ e : ω′′ ∈ e,ω(0) = ω(π) = 0}. it is well known that a is an infinitesimal generator of an α-resolvent family (tα(t))t≥0 on e. furthermore, a has discrete spectrum with eigenvalues −n2, with n ∈ n and corresponding normalized eigenfunctions given by zn(ξ) = √ 2 π sin(nξ). in addition, {zn : n ∈ n} is an orthonormal basis of e. and tα(t)x = ∞∑ n=1 exp−n 2 t(x,zn)zn for x ∈ e and t ≥ 0. theorem 4.2. let b = buc(r;e) and ϕ ∈ b. assume that condition (hϕ) holds, ρi : [0,∞) → [0,∞), i = 1,2, are continuous and the functions ai : r → r are continuous for i = 1,2,3. then there exists a unique mild solution of (9). 50 m. benchohra, o. bennihi & k. ezzinbi cubo 16, 3 (2014) proof. from the assumptions of the above theorem, we have that f(t,ψ)(ξ) = ∫0 −∞ a1(s)ψ(s,ξ)ds, g(t,ψ)(ξ) = ∫0 −∞ a3(s)ψ(s,ξ)ds, ρ(s,ψ) = s − ρ1(s)ρ2 (∫π 0 a2(θ)|ψ(0,ξ)| 2dθ ) , are well defined functions, which permit to transform system (9) into the abstract system (1)−(2). moreover, the function f is bounded linear operator. now, the existence of a mild solutions can be deduced from a direct application of theorem 2.12. we have the following result. corollary 4.3. let ϕ ∈ b be continuous and bounded. then the problem (1) − (2) has a unique mild solution on (−∞,+∞). acknowledgement: the second author would like to express his thanks to prof. k. ezzinbi for his invitation and the warm hospitality. received: march 2014. accepted: september 2014. references [1] s. abbas, m. benchohra and g.m. n’guérékata, topics in 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[41] y. zhou and f. jiao, nonlocal cauchy problem for fractional evolution equations, nonlinear anal.: real world appl 11 (2010), 4465-4475. introduction preliminaries main results example cubo a mathemati al journal vol.15, n o 03, (71�87). o tober 2013 convergen e theorems for generalized asymptoti ally quasi-nonexpansive mappings in one metri spa es g. s. saluja department of mathemati s and information te hnology, govt. nagarjuna p.g. college of s ien e, raipur 492010 (c.g.), india. saluja_1963�rediffmail. om, saluja1963�gmail. om abstract the purpose of this paper is to study an ishikawa type iteration pro ess with errors to approximate the ommon �xed point of two generalized asymptoti ally quasinonexpansive mappings in the framework of one metri spa es. our results extend and generalize many known results from the existing literature. resumen el propósito de este artí ulo es estudiar el pro eso de itera ión del tipo ishikawa on errores para aproximar el puto �jo omún de dos apli a iones uasi-expansivas asintóti amente generalizadas en el mar o de espa ios métri os óni os. nuestro resultado extiende y generaliza mu hos resultados de la literatura existente. keywords and phrases: generalized asymptoti ally quasi-nonexpansive mapping, ommon �xed point, ishikawa type iteration pro ess with errors, one metri spa e, normal and non-normal one. 2010 ams mathemati s subje t classi� ation: 47h10, 54h25. 72 g. s. saluja cubo 15, 3 (2013) 1 introdu tion and preliminaries the well-known bana h ontra tion prin iple and its several generalization in the setting of metri spa es play a entral role for solving many problems of nonlinear analysis. for example, see [2,6,7,20,21℄. in 1980, rzepe ki [23℄ introdu ed a generalized metri by repla ing the set of real numbers with normal one of the bana h spa e. in 1987, lin [17℄ introdu ed the notion of k-metri spa es by repla ing the set of real numbers with one in the metri fun tion. zabrejko [33℄ studied new revised version of the �xed point theory in k-metri and k-normed linear spa es by repla ing an ordered bana h spa e instead of the set of real numbers, as the o-domain for a metri . ordered normed spa es and ones have appli ations in applied mathemati s, for instan e, in using newton's approximations method [28,33℄, and in optimization theory [7℄. re ently, huang and zhang [10℄ used the notion of one metri spa es as a generalization of metri spa es. they have repla ed the real numbers (as the o-domain of a "metri ") by an ordered bana h spa e. the authors des ribed the onvergen e in one metri spa es and introdu ed their ompleteness. then they proved some �xed point theorems for ontra tive single-valued mappings in su h spa es. in their theorems one is normal. for more �xed point results in one metri spa es, see [1,3,11,24,25,32℄. most re ently, duki et al. [8℄ studied an ishikawa type iteration pro ess with errors for two uniformly quasi-lips hitzian mappings in omplete onvex one metri spa es and they gave a ne essary and su� ient ondition to approximate the ommon �xed point for said mappings. their results extended and generalized many known results from the literature. the main goal of this paper is to study an ishikawa type iteration pro ess with errors for two generalized asymptoti ally quasi-nonexpansive mappings in the setting of omplete onvex one metri spa es and also give a ne essary and su� ient ondition to approximate the ommon �xed point for said mappings. the results presented in this paper extend and generalize many known results from the literature. consistent with [7℄ and [10℄, the following de�nitions and results will be needed in the sequel. let e be a real bana h spa e. a subset p of e is alled a one whenever the following onditions hold: (c1) p is losed, nonempty and p 6= {0}; cubo 15, 3 (2013) convergen e theorems for generalized . . . 73 (c2) a, b ∈ r, a, b ≥ 0 and x, y ∈ p imply ax + by ∈ p; (c3) p ∩ (−p) = {0}. given a one p ⊂ e, we de�ne a partial ordering � with respe t to p by x � y if and only if y − x ∈ p. we shall write x ≺ y to indi ate that x � y but x 6= y, while x ≪ y will stand for y − x ∈ intp (interior of p). if intp 6= ∅ then p is alled a solid one (see [28℄). there exist two kinds of ones: normal (with the normal onstant k) and non-normal ones [7℄). let e be a real bana h spa e, p ⊂ e a one and � partial ordering de�ned by p. then p is alled normal if there is a number k > 0 su h that for all x, y ∈ p, 0 � x � y imply ‖x‖ ≤ k ‖y‖ , (1) or equivalently, if (∀n) xn � yn � zn and lim n→∞ xn = lim n→∞ zn = x imply lim n→∞ yn = x. (2) the least positive number k satisfying (1) is alled the normal onstant of p. it is lear that k ≥ 1. example 1.1. (see [28℄) let e = c1 r [0, 1] with ‖x‖ = ‖x‖∞ + ‖x ′‖∞ on p = {x ∈ e : x(t) ≥ 0}. this one is not normal. consider, for example, xn(t) = t n n and yn(t) = 1 n . then 0 � xn � yn, and limn→∞ yn = 0, but ‖xn‖ = maxt∈[0,1] | t n n | + maxt∈[0,1] |t n−1| = 1 n + 1 > 1; hen e xn does not onverge to zero. it follows by (2) that p is a non-normal one. de�nition 1.1. (see [10,33℄) let x be a nonempty set. suppose that the mapping d: x×x → e satis�es: (d1) 0 � d(x, y) for all x, y ∈ x and d(x, y) = 0 if and only if x = y; (d2) d(x, y) = d(y, x) for all x, y ∈ x; (d3) d(x, y) � d(x, z) + d(z, y) x, y, z ∈ x. 74 g. s. saluja cubo 15, 3 (2013) then d is alled a one metri [10℄ or k-metri [33℄ on x and (x, d) is alled a one metri [10℄ or k-metri spa e [33℄ (we shall use the �rst term). the on ept of a one metri spa e is more general than that of a metri spa e, be ause ea h metri spa e is a one metri spa e where e = r and p = [0, +∞). example 1.2. (see [10℄) let e = r2, p = {(x, y) ∈ r2 : x ≥ 0, y ≥ 0}, x = r and d: x×x → e de�ned by d(x, y) = (|x − y|, α|x − y|), where α ≥ 0 is a onstant. then (x, d) is a one metri spa e with normal one p where k = 1. example 1.3. (see [24℄) let e = ℓ2, p = {{xn}n≥1 ∈ e : xn ≥ 0, for all n}, (x, ρ) a metri spa e, and d: x × x → e de�ned by d(x, y) = {ρ(x, y)/2n}n≥1. then (x, d) is a one metri spa e. clearly, the above examples show that the lass of one metri spa es ontains the lass of metri spa es. de�nition 1.2. (see [10℄) let (x, d) be a one metri spa e. we say that {xn} is: (i) a cau hy sequen e if for every ε in e with 0 ≪ ε, then there is an n su h that for all n, m > n, d(xn, xm) ≪ ε; (ii) a onvergent sequen e if for every ε in e with 0 ≪ ε, then there is an n su h that for all n > n, d(xn, x) ≪ ε for some �xed x in x. a one metri spa e x is said to be omplete if every cau hy sequen e in x is onvergent in x. let us re all ( [10℄) that if p is a normal solid one, then xn ∈ x is a cau hy sequen e if and only if ‖d(xn, xm)‖ → 0 as n, m → ∞. further, xn ∈ x onverges to x ∈ x if and only if ‖d(xn, x)‖ → 0 as n → ∞. in the sequel we assume that e is a real bana h spa e and that p is a normal solid one in e, that is, normal one with intp 6= ∅. the last assumption is ne essary in order to obtain reasonable results onne ted with onvergen e and ontinuity. the partial ordering indu ed by the one p will be denoted by �. cubo 15, 3 (2013) convergen e theorems for generalized . . . 75 2 convexity in one metri spa e let (x, d) be a one metri spa e with solid one p and t : x → x a given mapping. let f(t) denote the set of �xed points of t. de�nition 2.1. (1) the mapping t is said to be nonexpansive if d(tx, ty) � d(x, y) (3) for all x, y ∈ x. (2) the mapping t is said to be quasi-nonexpansive if f(t) 6= ∅ and d(tx, p) � d(x, p) (4) for all x ∈ x and p ∈ f(t). (3) the mapping t is said to be asymptoti ally nonexpansive if there exists a sequen e {rn} ∈ [0, ∞) with rn → 0 as n → ∞ su h that d(tnx, tny) � (1 + rn)d(x, y) (5) for all x, y ∈ x. (4) the mapping t is said to be asymptoti ally quasi-nonexpansive if f(t) 6= ∅ and there exists a sequen e {rn} ∈ [0, ∞) with rn → 0 as n → ∞ su h that d(tnx, p) � (1 + rn)d(x, p) (6) for all x ∈ x and p ∈ f(t). (5) the mapping t is said to be generalized asymptoti ally quasi-nonexpansive [12℄ if f(t) 6= ∅ and there exist two sequen es of real numbers {rn} and {sn} ∈ [0, ∞) with rn → 0 and sn → 0 as n → ∞ su h that d(tnx, p) � (1 + rn)d(x, p) + sn, (7) 76 g. s. saluja cubo 15, 3 (2013) for all x ∈ x and p ∈ f(t). (6) the mapping t is said to be uniformly l-lips hitzian if there exists a onstant l > 0 su h that d(tnx, tny) � l d(x, y), (8) for all x, y ∈ x. remark 2.1. (i) it is lear that the nonexpansive mappings with the nonempty �xed point set f(t) are quasi-nonexpansive. (ii) the linear quasi-nonexpansive mappings are nonexpansive, but it is easily seen that there exist nonlinear ontinuous quasi-nonexpansive mappings whi h are not nonexpansive; for example, de�ne t(x) = (x/2)sin(1/x) for all x 6= 0 and t(0) = 0 in r. (iii) it is obvious that if t is nonexpansive, then it is asymptoti ally nonexpansive with the onstant sequen e {1}. (iv) if t is asymptoti ally nonexpansive, then it is uniformly lips hitzian with the uniform lips hitz onstant l = sup{1 + rn : n ≥ 1}. however, the onverse of this laim is not true. (v) if in de�nition (5), sn = 0 for all n ∈ n, then t be omes asymptoti ally quasi-nonexpansive, and hen e the lass of generalized asymptoti ally quasi-nonexpansive maps in ludes the lass of asymptoti ally quasi-nonexpansive maps. in re ent years, asymptoti ally nonexpansive mappings, asymptoti ally nonexpansive type mappings, asymptoti ally quasi-nonexpansive mappings and asymptoti ally quasi-nonexpansive type mappings have been studied extensively in the setting of onvex metri spa es (see e.g. [5,9,14�16,18,19,27℄). in 1970, takahashi [26℄ introdu ed the on ept of onvexity in a metri spa e and the properties of the spa e. de�nition 2.2. (see [26℄) let (x, d) be a metri spa e and i = [0, 1]. a mapping w : x × x × i → x is said to be a onvex stru ture on x if for ea h (x, y, λ) ∈ x × x × i and u ∈ x, d(u, w(x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y). cubo 15, 3 (2013) convergen e theorems for generalized . . . 77 x together with a onvex stru ture w is alled a onvex metri spa e, denoted by (x, d, w). a nonempty subset k of x is said to be onvex if w(x, y, λ) ∈ k for all (x, y, λ) ∈ k × k × i. remark 2.2. every normed spa e is a onvex metri spa e, where a onvex stru ture w(x, y, z; α, β, γ) = αx + βy + γz, for all x, y, z ∈ e and α, β, γ ∈ i with α + β + γ = 1. in fa t, d(u, w(x, y, z; α, β, γ)) = ‖u − (αx + βy + γz)‖ ≤ α ‖u − x‖ + β ‖u − y‖ + γ ‖u − z‖ = αd(u, x) + βd(u, y) + γd(u, z), ∀u ∈ x. but there exists some onvex metri spa es whi h an not be embedded into normed spa e. now we de�ne the following: de�nition 2.3. let (x, d) be a one metri spa e and i = [0, 1]. a mapping w : x×x×i → x is said to be a onvex stru ture on x if for any (x, y, λ) ∈ x × x × i and u ∈ x, the following inequality holds: d(u, w(x, y, λ)) � λd(u, x) + (1 − λ)d(u, y). if (x, d) be a one metri spa e with a onvex stru ture w, then (x, d) is alled a onvex abstra t metri spa e or onvex one metri spa e (see also [13℄, [22℄). moreover, a nonempty subset c of x is said to be onvex if w(x, y, λ) ∈ c, for all (x, y, λ) ∈ c × c × i. de�nition 2.4. let (x, d) be a one metri spa e, i = [0, 1] and {an}, {bn}, {cn} are real sequen es in [0,1℄ with an + bn + cn = 1. a mapping w : x 3 × i3 → x is said to be a onvex stru ture on x if for any (x, y, z, an, bn, cn) ∈ x 3 × i3 and u ∈ x, the following inequality holds: d(u, w(x, y, z, an, bn, cn)) � and(u, x) + bnd(u, y) + cnd(u, z). if (x, d) be a one metri spa e with a onvex stru ture w, then (x, d) is alled a generalized onvex one metri spa e. moreover, a nonempty subset c of x is said to be onvex if w(x, y, z, an, bn, cn) ∈ c, for all (x, y, z, an, bn, cn) ∈ c 3 × i3. remark 2.3. if e = r, p = [0, +∞), ‖.‖ = |.|, then (x, d) is a onvex metri spa e, i.e., generalized onvex metri spa e as in [30℄. 78 g. s. saluja cubo 15, 3 (2013) example 2.1. let (x, d) be a one metri spa e as in example (1.2)(1). if w(x, y, λ) =: λx + (1 − λ)y, then (x, d) is a one metri spa e. hen e, this notion is more general than that of a onvex metri spa e. de�nition 2.5. let (x, d) be a one metri spa e with a onvex stru ture w : x3 × i3 → x, s, t : x → x be two generalized asymptoti ally quasi-nonexpansive mappings with sequen es of real numbers {rn} and {sn} ∈ [0, ∞) su h that rn → 0 and sn → 0 as n → ∞ and {an}, {bn}, {cn}, {a′n}, {b ′ n}, {c ′ n} are six sequen es in [0,1℄ with an + bn + cn = a ′ n + b ′ n + c ′ n = 1, n = 1, 2, . . . . for any given x1 ∈ x, de�ne a sequen e {xn} as follows: yn = w(xn, s nxn, vn, a ′ n, b ′ n, c ′ n), xn+1 = w(xn, t nyn, un, an, bn, cn), (9) where {un}, {vn} are two sequen es in x satisfying the following ondition: for any nonnegative integers n, m, 1 ≤ n < m, if δ(anm) > 0, then max n≤i,j≤m { ‖d(x, y)‖ : x ∈ {ui, vi}, y ∈ {xj, yj, tyj, sxj, uj, vj} } < δ(anm), (10) where anm = {xi, yi, tyi, sxi, ui, vi : n ≤ i ≤ m}, δ(anm) = sup x,y∈anm ‖d(x, y)‖ . then {xn} is alled the ishikawa type iteration pro ess with errors for two generalized asymptoti ally quasi-nonexpansive mappings s and t in onvex one metri spa e (x, d). remark 2.4. note that some iteration pro esses onsidered in [9,14,19,27℄ an be obtained from the above pro ess (9) as spe ial ases by hoosing suitable spa es and mappings. in the sequel, we shall need the following lemma. lemma 2.1. (see [19℄) let {an}, {bn} and {αn} be sequen es of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + αn)an + bn, n ≥ 1. if ∑∞ n=1 bn < ∞ and ∑∞ n=1 αn < ∞. then (a) limn→∞ an exists. (b) if lim infn→∞ an = 0, then limn→∞ an = 0. cubo 15, 3 (2013) convergen e theorems for generalized . . . 79 3 main results now we give our main results of this paper. lemma 3.1. let c be a nonempty losed onvex subset of a omplete onvex one metri spa e x, s, t : c → c be two generalized asymptoti ally quasi-nonexpansive mappings with sequen es {rn} and {sn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞ and ∑∞ n=1 sn < ∞. assume that f = f(s) ∩ f(t) 6= ∅. let {xn} be the ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. then (i) there exists a onstant ve tor v ∈ p\ {0} su h that ‖d(xn+1, p)‖ ≤ k. (1 + an). ‖d(xn, p)‖ + k. bn + k. ‖v‖ . cn, where an = r 2 n + 2rn, bn = (2 + rn)sn and cn = (1 + rn)d(un, p) + d(vn, p), for all n ∈ n and for all p ∈ f, where k is the normal onstant of a one p; (ii) there exists a real onstant m > 0 su h that ‖d(xn+m, p)‖ ≤ k. m. ‖d(xn, p)‖ + k. m. n+m−1∑ j=n bj + k. m. ‖v‖ . n+m−1∑ j=n cj, for all n, m ∈ n and for all p ∈ f, where k is the normal onstant of a one p. proof. for any p ∈ f, from (7) and (9), we have d(xn+1, p) = d(w(xn, t nyn, un, an, bn, cn), p) � and(xn, p) + bnd(t nyn, p) + cnd(un, p) � and(xn, p) + bn[(1 + rn)d(yn, p) + sn] + cnd(un, p) = and(xn, p) + bn(1 + rn)d(yn, p) + bnsn + cnd(un, p) � and(xn, p) + bn(1 + rn)d(yn, p) + sn + cnd(un, p) (11) and d(yn, p) = d(w(xn, s nxn, vn, a ′ n, b ′ n, c ′ n), p) � a′nd(xn, p) + b ′ nd(s nxn, p) + c ′ nd(vn, p) � a′nd(xn, p) + b ′ n[(1 + rn)d(xn, p) + sn] + c ′ nd(vn, p) � (a′n + b ′ n)(1 + rn)d(xn, p) + b ′ nsn + c ′ nd(vn, p) = (1 − c′n)(1 + rn)d(xn, p) + b ′ nsn + c ′ nd(vn, p) � (1 + rn)d(xn, p) + sn + c ′ nd(vn, p). (12) 80 g. s. saluja cubo 15, 3 (2013) substituting (12) into (11), it an be obtained that d(xn+1, p) � and(xn, p) + bn(1 + rn)[(1 + rn)d(xn, p) + sn +c′nd(vn, p)] + sn + cnd(un, p) � (an + bn)(1 + rn) 2d(xn, p) + bn(1 + rn)sn + sn +bnc ′ n(1 + rn)d(vn, p) + cnd(un, p) = (1 − cn)(1 + rn) 2d(xn, p) + bn(1 + rn)sn + sn +bnc ′ n(1 + rn)d(vn, p) + cnd(un, p) � (1 + rn) 2d(xn, p) + (2 + rn)sn + cnd(un, p) +c′n(1 + rn)d(vn, p) � (1 + an)d(xn, p) + bn + vcn, (13) where an = r 2 n + 2rn, bn = (2 + rn)sn, cn = cn + c ′ n and v = (1 + rn)d(vn, p) + d(un, p). now, (i) follows from (1), where k is a normal onstant of the one p. (ii) it is well known that 1 + x ≤ ex for all x ≥ 0. using this, it follows from on lusion (i) that for all n, m ∈ n and p ∈ f, we have d(xn+m, p) � (1 + an+m−1)d(xn+m−1, p) + bn+m−1 + v.cn+m−1 � ean+m−1d(xn+m−1, p) + bn+m−1 + v.cn+m−1 � ean+m−1[ean+m−2d(xn+m−2, p) + bn+m−2 +v.cn+m−2] + bn+m−1 + v.cn+m−1 � ean+m−1+an+m−2d(xn+m−2, p) + e an+m−1+an+m−2. [bn+m−1 + bn+m−1] + e an+m−1+an+m−2. [cn+m−1 + cn+m−1].v . . . � m. d(xn, p) + m. n+m−1∑ j=n bj + m. ( n+m−1∑ j=n cj ) . v, (14) where m = e ∑ ∞ j=1 aj . further, (ii) follows from (1), be ause p is a normal one with the normal onstant k. theorem 3.1. let c be a nonempty losed onvex subset of a omplete onvex one metri spa e x, s, t : c → c be two generalized asymptoti ally quasi-nonexpansive mappings with sequen es {rn} and {sn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞ and ∑∞ n=1 sn < ∞. assume that f = f(s)∩f(t) 6= ∅. let {xn} be the ishikawa type iteration pro ess with errors de�ned by (9) {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. then {xn} onverges to a ommon �xed cubo 15, 3 (2013) convergen e theorems for generalized . . . 81 point of s and t if and only if lim infn→∞ ‖d(xn, f)‖ = 0, where ‖d(x, f)‖ = inf{‖d(x, q)‖ : q ∈ f}. proof. the ne essity of ondition is obvious. thus, we will only prove the su� ien y. from lemma 3.1(i), we have ‖d(xn+1, p)‖ ≤ k. (1 + an). ‖d(xn, p)‖ + k. bn + k. ‖v‖ . cn, where an = r 2 n + 2rn, bn = (2 + rn)sn, cn = cn + c ′ n and v = (1 + rn)d(un, p) + d(vn, p) with∑∞ n=1 an < ∞, ∑∞ n=1 bn < ∞ and ∑∞ n=1 cn < ∞. sin e ∑∞ n=1 an < ∞, ∑∞ n=1 bn < ∞ and ∑∞ n=1 cn < ∞, it follows from lemma 2.1 that limn→∞ ‖d(xn, f)‖ exists. a ording to the hypothesis, lim infn→∞ ‖d(xn, f)‖ = 0, hen e we have that limn→∞ ‖d(xn, f)‖ = 0. next, we show that {xn} is a cau hy sequen e. let ε > 0 be given. there exists an integer n0 su h that for all n > n0, we have ‖d(xn, f)‖ < ε 6k2m , ∞∑ n=n0+1 bn < ε 6k2m , and ∞∑ n=n0+1 cn < ε 6k2 ‖v‖ m . in parti ular, there exists a q ∈ f and an integer n1 > n0 su h that ‖d(xn1, q)‖ < ε 6k2m . it follows from lemma 3.1(ii) that when n > n1, we get ‖d(xn+m, q)‖ = ∥ ∥d ( xn1+(n+m−n1), q ) ∥ ∥ ≤ k. m. . ‖d(xn1, q)‖ + k. m. ( n+m−1∑ j=n1 bj ) +k. m. ‖v‖ . ( n+m−1∑ j=n1 cj ) (15) and 82 g. s. saluja cubo 15, 3 (2013) ‖d(xn, q)‖ = ∥ ∥d ( xn1+(n−n1), q ) ∥ ∥ ≤ k. m. ‖d(xn1, q)‖ + k. m. ( n−1∑ j=n1 bj ) +k. m. ‖v‖ . ( n−1∑ j=n1 cj ) . (16) therefore from (1), (15) and (16), we obtain that ‖d(xn+m, xn)‖ ≤ k. ‖d(xn+m, q) + d(q, xn)‖ ≤ k. ‖d(xn+m, q)‖ + k. ‖d(q, xn)‖ ≤ 2k2. m. ‖d(xn1, q)‖ + 2k 2. m. ( n+m−1∑ j=n1 bj + n−1∑ j=n1 bj ) +2k2. m. ‖v‖ . ( n+m−1∑ j=n1 cj + n−1∑ j=n1 cj ) ≤ 2k2. m. ‖d(xn1, q)‖ + 2k 2. m. ( n+m−1∑ j=n1 bj ) +2k2. m. ‖v‖ . ( n+m−1∑ j=n1 cj ) ≤ 2k2. m. ε 6k2m + 2k2. m. ε 6k2m +2k2. m. ‖v‖ . ε 6k2 ‖v‖ m = ε. (17) hen e {xn} is a cau hy sequen e in losed onvex subset c of a omplete one metri spa e x. therefore, it must be onvergent to a point in c. suppose limn→∞ xn = p. we will prove that p ∈ f. for a given ε > 0, there exists an integer n2 su h that for all n ≥ n2, we have ‖d(xn, p)‖ < ε 2k(2 + r1) and ‖d(xn, f)‖ < ε 4k(2 + r1) (18) in parti ular, there exists a p1 ∈ f and an integer n3 > n2 su h that ‖d(xn3, p1)‖ < ε 2k(2 + r1) (19) cubo 15, 3 (2013) convergen e theorems for generalized . . . 83 then, we have d(tp, p) � d(tp, p1) + d(p1, xn3) + d(xn3, p) � (1 + r1)d(p, p1) + s1 + d(p1, xn3) + d(xn3, p) � (1 + r1)d(p, p1) + d(p1, xn3) + d(xn3, p) � (1 + r1)d(p, xn3) + (1 + r1)d(xn3, p1) +d(p1, xn3) + d(xn3, p) = (2 + r1)d(xn3, p) + (2 + r1)d(xn3, p1) (20) now using (1), (18) and (19), we obtain ‖d(tp, p)‖ ≤ k(2 + r1) ‖d(xn3, p)‖ + k(2 + r1) ‖d(xn3, p1)‖ < k(2 + r1). ε 2k(2 + r1) + k(2 + r1). ε 2k(2 + r1) = ε. (21) similarly, we an also have ‖d(sp, p)‖ < ε. sin e ε is arbitrary, it follows that d(tp, p) = d(sp, p) = 0, that is, p is a ommon �xed point of s and t. this ompletes the proof of theorem 3.1. we dedu e some results from theorem 3.1 as follows. corollary 3.1. let c be a nonempty losed onvex subset of a omplete onvex one metri spa e x, s, t : c → c be asymptoti ally quasi-nonexpansive mappings with sequen e {rn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞. assume that f = f(s) ∩ f(t) 6= ∅. let {xn} be the ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. then {xn} onverges to a ommon �xed point of s and t if and only if lim infn→∞ ‖d(xn, f)‖ = 0, where ‖d(x, f)‖ = inf{‖d(x, q)‖ : q ∈ f}. proof. it follows from theorem 3.1 with sn = 0 for all n ≥ 1. corollary 3.2. let c be a nonempty losed onvex subset of a omplete onvex one metri spa e x, s, t : c → c be uniformly quasi-lips hitzian mappings with l > 0. assume that f = f(s) ∩ f(t) 6= ∅. let {xn} be the ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. then {xn} onverges to a ommon �xed point of s and t if and only if lim infn→∞ ‖d(xn, f)‖ = 0, where ‖d(x, f)‖ = inf{‖d(x, q)‖ : q ∈ f}. proof. sin e {rn} ∈ [0, ∞) with rn → 0 as n → ∞, then there exists l > 0 su h that l = sup{1 + rn : n ≥ 1}. in this ase s and t are uniformly quasi-lips hitzian mappings with 84 g. s. saluja cubo 15, 3 (2013) l > 0. hen e, corollary 3.2 an be proven by corollary 3.1. corollary 3.3. let c be a nonempty losed onvex subset of a omplete onvex one metri spa e x, s, t : c → c be asymptoti ally nonexpansive mappings with sequen e {rn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞. assume that f = f(s) ∩ f(t) 6= ∅. let {xn} be the ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. then {xn} onverges to a ommon �xed point of s and t if and only if lim infn→∞ ‖d(xn, f)‖ = 0, where ‖d(x, f)‖ = inf{‖d(x, q)‖ : q ∈ f}. proof. it is lear that an asymptoti ally nonexpansive mapping must be asymptoti ally quasinonexpansive. therefore, corollary 3.3 an be proven by corollary 3.1. corollary 3.4. in corollary 3.1 by setting e = r, p = [0, ∞), d(x, y) = |x−y|, x, y ∈ r (that is ‖.‖ = |.|), we get lemma 2 and theorem 1, 2, 3 of [29℄. corollary 3.5. if we set in corollary 3.1 e = r, p = [0, ∞), d(x, y) = |x − y|, x, y ∈ r (that is ‖.‖ = |.|), we obtain the main result of [4℄, theorem 2.1 and 2.2 of [30℄ and theorem 2.1 and corollary 2.3 of [31℄. remark 3.1. our results extend the orresponding results of duki et al. [8℄ to the ase of more general lass of uniformly quasi-lips hitzian mappings onsidered in this paper. remark 3.2. our results also extend, improve and generalize many known results from the existing literature. example 3.1. let e be the real line with the usual norm |.| and k = [0, 1]. de�ne s, t : k → k by t(x) = x 2 sin ( 1 x ) , x ∈ [0, 1], and s(x) = x 3 , x ∈ [0, 1], for x ∈ k. obviously s(0) = 0 and t(0) = 0. hen e, f = f(s) ∩ f(t) = {0}, that is, 0 is a ommon �xed point of s and t. now we he k that s and t are generalized asymptoti ally quasinonexpansive mappings. in fa t, if x ∈ [0, 1] and p = 0 ∈ [0, 1], then |t(x) − p| = |t(x) − 0| = |(x/2) sin (1/x) − 0| = |(x/2) sin (1/x)| cubo 15, 3 (2013) convergen e theorems for generalized . . . 85 ≤ |x/2| ≤ |x| = |x − 0| = |x − p|, that is, |t(x) − p| ≤ |x − p|. thus, t is quasi-nonexpansive. it follows that t is asymptoti ally quasi-nonexpansive with the onstant sequen e {kn} = {1} for ea h n ≥ 1 and hen e it is generalized asymptoti ally quasinonexpansive mapping with onstant sequen es {kn} = {1} and {sn} = {0} for ea h n ≥ 1. similarly, we an show that s is also generalized asymptoti ally quasi-nonexpansive mapping with onstant sequen es {kn} = {1} and {sn} = {0} for ea h n ≥ 1. but the onverse does not hold in general. re eived: mar h 2012. a epted: february 2013. referen es [1℄ m. abbas and b.e. rhoades, fixed and periodi point results in one metri spa es, appl. math. lett. 22(4)(2009), 511-515. 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() cubo a mathematical journal vol.17, no¯ 02, (15–30). june 2015 an other uncertainty principle for the hankel transform chirine chettaoui département de mathématiques et d’informatique, institut national des sciences appliquées et de thechnologie, centre urbain nord bp 676 1080 tunis cedex, tunisia. chirine.chettaoui@insat.rnu.tn abstract we use the hausdorff-young inequality for the hankel transform to prove the uncertainly principle in terms of entropy. next, we show that we can obtain the heisenbergpauli-weyl inequality related to the same hankel transform. resumen usamos la desigualdad de hausdorff-young para la transformada de hankel para probar el principio de incertidumbre en términos de la entroṕıa. además probamos que podemos obtener la desigualdad de heisenberg-pauli-weyl relacionada con la misma transformada de hankel. keywords and phrases: uncertainty principle, hausdorff-young inequality, entropy, hankel transform 2010 ams mathematics subject classification: 43a32, 42b25. 16 chirine chettaoui cubo 17, 2 (2015) 1 introduction: the uncertainly principles play an import role in harmonic analysis. they state that a function f and its fourier transform f̂ can not be simultaneously sharply localized in the sense that it is impossible for a nonzero function and its fourier transform to be simultaneously small. many mathematical formulations of this fact can be found in [2, 5, 6, 11, 16, 17]. for a probability density function f on rn, the entropy of f is defined according to [18] by e(f) = − ∫ rn f(x) ln(f(x))dx. the entropy e(f) is closely related to quantum mechanics [4] and constitutes one of the important way to measure the concentration of f. the uncertainly principle in terms of entropy consists to compare the entropy of |f|2 with that of |f̂|2. a first result has been given in [13], where hirschman established a weak version of this uncertainly principle by showing that for every square integrable function f on rn with respect to the lebesgue measure, such that ||f||2 = 1, we have e(|f|2) + e(|f̂|2) > 0. later, in [1, 2], beckner proved the following stronger inequality, that is for every square integrable function f on rn; ||f||2 = 1, e(|f|2) + e(|f̂|2) > n(1 − ln2). the last inequality constituted a very powerful result which implies in particular the well known heisenberg-pauli-weyl uncertainly principle [17]. in this paper, we consider the singular differential operator defined on ]0,+∞[ by ℓα = d2 dr2 + 2α + 1 r d dr = 1 r2α+1 d dr [r2α+1 d dr ]; α > −1 2 . the hankel transform associated with the operator ℓα is defined by hα(f)(λ) = ∫+∞ o f(r)jα(λr)dµα(r), where . dµα(r) is the measure defined on [0,+∞[ by dµα(r) = r2α+1dr 2α γ(α + 1) . . jα is the modified bessel-function that will be defined in the second section . many harmonic analysis results have been establish for the hankel transform hα [14, 19, 20]. also, many uncertainly principles have been proved for the transform hα [17, 21]. cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 17 our investigation in this work consists to establish the uncertainly principle in terms of entropy for the hankel transform hα. for a nonnegative measurable function f on [0,+∞[, the entropy of f is defined by eµα(f) = − ∫+∞ 0 f(r) ln(f(r))dµα(r). then using the hausdorff-young inequality for hα [9], we establish the main result of this work. . let f ∈ l2(dµα); ||f||2,µα = 1 such that ∫+∞ o |f(r)|2 ∣∣ ln(|f(r)|) ∣∣dµα(r) < +∞, and ∫+∞ o ∣∣hα(f)(λ) ∣∣2 ∣∣∣ ln ( |hα(f)(λ)| )∣∣∣dµα(λ) < +∞. then eµα ( |f|2 ) + eµα (∣∣hα(f) ∣∣2 ) > (2α + 1)(1 − ln2), where lp(dµα); p ∈ [1,+∞], is the lebesgue space of measurable functions on [0,+∞[ such that ||f||p,µα < +∞, with ||f||p,µα =    ( ∫+∞ 0 ∣∣f(r) ∣∣pdµα(r) ) 1 p , if p ∈ [1,+∞[, ess sup r∈ [0,+∞[ ∣∣f(r) ∣∣, if p = +∞. using this result, we prove that we can derive the heisenberg -pauli-weyl inequality for hα, that is . for every f ∈ l2(dµα); we have ||rf||2,µα||λhα(f)||2,µα > (α + 1)||f|| 2 2,µα . 2 the hankel operator in this section, we recall some harmonic analysis results related to the convolution product and the fourier transform associated with hankel operator. let ℓα be the bessel operator defined on ]0 + ∞[ by ℓαu = d2 dr2 u + 2α + 1 r du dr . 18 chirine chettaoui cubo 17, 2 (2015) then, for every λ ∈ c, the following system    ℓαu(x) = −λ 2u(x), u(0) = 1, u′(0) = 0, admits a unique solution given by jα(λ.), where jα(z) = 2αγ(α + 1) zα jα(z) (2.1) = γ(α + 1) +∞∑ k=0 (−1)k k!γ(α + k + 1) (z 2 )2k , with jα is the bessel function of first kind and index α [7, 8, 15, 22]. the modified bessel function jα satisfies the following properties for every α > −1 2 , the modified bessel function jα has the mehler integral representation, for every z ∈ c, jα(z) =    2γ(α + 1) √ πγ(α + 1 2 ) ∫1 0 (1 − t2)α− 1 2 cosztdt, if α > −1 2 , cosz, if α = −1 2 . consequently, for every k ∈ n and z ∈ c; we have ∣∣j(k)α (z) ∣∣ 6 exp ( |imz| ) . (2.2) the eigenfunction jα(λ.) satisfies the following product formula [22], for all r,s ∈ [0,+∞[ jα(λr)jα(λs) =    γ(α + 1) √ πγ(α + 1 2 ) ∫π 0 jα ( λ √ r2 + s2 + 2rscosθ ) sin2α θdθ; if α > −1 2 , jα ( λ(r + s) ) + jα ( λ ( (r − s) )) 2 , if α = −1 2 . (2.3) the previous product formula allows us to define the hankel translation operator and the convolution product related to the operator ℓα as follows definition 2.1. i) for every r ∈ [0,+∞[, the hankel translation operator ταr is defined on lp(dµα); p ∈ [1,+∞], by ταr f(s) =    γ(α + 1) √ πγ(α + 1 2 ) ∫π 0 f (√ r2 + s2 + 2rscosθ ) sin2α θdθ; if α > −1 2 , f(r + s) + f(|r − s|) 2 , if α = −1 2 . cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 19 ii) the convolution product of f,g ∈ l1(dµα) is defined for every r ∈ [0,+∞[, by f ∗α g(r) = ∫+∞ o ταr (f)(s)g(s)dµα(s). (2.4) then the product formula (2.3) can be written ταr (jα(λ.))(s) = jα(λr)jα(λs). (2.5) we have the properties proposition 2.2. i. for every f ∈ lp(dµα); 1 6 p 6 +∞, and for every r ∈ [0,+∞[, the function ταr (f) belongs to l p(dµα) and we have ∣∣∣∣ταr (f) ∣∣∣∣ p,µα 6 ||f||p,µα. (2.6) ii. for f, g ∈ l1(dµα), the function f ∗α g belongs to l1(dµα); the convolution product is commutative, associative and we have ||f ∗α g||1,µα 6 ||f||1,µα||g||1,µα. (2.7) moreover, if 1 6 p,q,r 6 +∞ are such that 1 r = 1 p + 1 q − 1 and if f ∈ lp(dµα),g ∈ lq(dµα), then the function f ∗α g belongs to lr(dµα), and we have the young’s inequality ||f ∗α g||r,µα 6 ||f||p,µα||g||q,µα. (2.8) iii. for every f ∈ l1(dµα), and r ∈ [0,+∞[ the function ταr (f) belongs to l1(dµα) and we have ∫+∞ o ταr (f)(s)dµα(s) = ∫+∞ o f(r)dµα(r). (2.9) we denoted by . c∗,0(r) the space of even continuous functions f on r such that lim |r|→+∞ f(r) = 0. . se(r) the space of even infinitely differentiable functions on r; rapidly decreasing together with all their derivatives. now, we shall define the hankel transform and we give the most important properties. definition 2.3. the hankel transform hα is defined on l 1(dµα) by ∀λ ∈ r; hα(f)(λ) = ∫+∞ o f(r)jα(λr)dµα(r), (2.10) where jα is the modified bessel function defined by the relation(2.1). 20 chirine chettaoui cubo 17, 2 (2015) proposition 2.4. i. for every f ∈ l1(dµα), the function hα(f) belongs to the space c∗,0(r) and ∣∣∣∣hα(f) ∣∣∣∣ ∞,µα 6 ||f||1,µα. (2.11) ii. for every f ∈ l1(dµα) and r ∈ [0,+∞[, hα(τ α r (f))(λ) = jα(λr)hα(f)(λ). (2.12) iii. for all f,g ∈ l1(dµα), hα(f ∗α g)(λ) = hα(f)(λ)hα(g)(λ). (2.13) theorem 2.5. (inversion formula) let f ∈ l1(dµα) such that hα(f) ∈ l1(dµα), then for almost every r ∈ [0,+∞[, we have f(r) = ∫+∞ o hα(f)(λ)jα(λr)dµα(λ) = hα ( hα(f) ) (r). (2.14) theorem 2.6. (plancherel) the hankel transform hα can be extented to an isometric isomorphism from l2(dµα) onto itself. in particular, for all f and g ∈ l2(dµα), we have (parseval equality) ∫+∞ o f(r)g(r)dµα(r) = ∫+∞ o hα(f)(λ)hα(g)(λ)dµα(λ). (2.15) proposition 2.7. i. let f ∈ l1(dµα)and g ∈ l2(dµα); by the relation (2.8), the function f∗αg belongs to l2(dµα), moreover hα(f ∗α g)(λ) = hα(f)(λ)hα(g)(λ). (2.16) ii. for all f and g ∈ l2(dµα), the function f ∗α g belongs to c∗,0(r) and we have f ∗α g = hα ( hα(f).hα(g) ) . (2.17) iii. the hankel transform hα is a topological isomorphism from se(r) onto itself and we have h −1 α = hα. (2.18) iv. for every f ∈ s(r) and g ∈ l2(dµα), we have hα(fg)(λ) = hα(f)(λ) ∗α hα(g)(λ). (2.19) definition 2.8. the gaussian kernel associated with the hankel operator is defined by ∀t > 0, gt(r) = e −r2 2t2 t2α+2 . (2.20) thus, one can easily see that the family (gt)t>0 is an approximation of the identity, in particular, for every f ∈ l2(dµα) we have lim t→0+ ∣∣∣∣gt ∗α f − f ∣∣∣∣ 2,µα = 0. (2.21) cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 21 3 uncertainty principle in terms of entropy for the hankel transform this section contains the main result of this paper that is the uncertainty principle in terms of entropy for the hankel transform hα. we start this section by the following hausdorff-young inequality. theorem 3.1. [9] let p ∈ ]1,2], for every f ∈ lp(dµα), the function hα(f) belongs to lp ′ (dµα); p ′ = p p−1 , and we have ||hα(f)||p′,µα 6 ap,α||f||p,µα, (3.1) where ap,α is the babenko-beckner constant, ap,α = ( p 1 p ( p p−1 ) p−1 p )α+1 . lemma 3.2. let x be a positive real number. for every p ∈ [1,2[, x2 − x 6 xp − x2 p − 2 6 x2 lnx. (3.2) proof. let x > 0 and let us put g(p) = xp − x2 p − 2 . g is differentiable on [1,2[ and we have g′(p) = h(p) (p − 2)2 , where h(p) = (p − 2) ln(x).xp − xp + x2. on the other hand, ∀p ∈ [1,2[; h′(p) = (p − 2)xp(ln(x))2 < 0 and h(2) = 0. thus, for every p ∈ [1,2], h(p) > 0 and the function g is increasing on [1,2[, hence g(1) 6 g(p) 6 lim p→2− g(p). in the following, we shall prove the uncertainty principle in terms of entropy for f ∈ l1(dµα)∩ l2(dµα) such that ||f||2,µα = 1. 22 chirine chettaoui cubo 17, 2 (2015) theorem 3.3. let f ∈ l1(dµα) ∩ l2(dµα); ||f||2,µα = 1, such that ∫ ∞ 0 |f(r)|2 ∣∣ ln(|f(r)|) ∣∣ dµα(r) < +∞, and ∫ ∞ 0 |h (f)(λ)|2 ∣∣ ln(|h (f)(λ)|) ∣∣ dµα(λ) < +∞. then, eµα(|f| 2) + eµα(|hα(f)| 2) > (2α + 2)(1 − ln2). (3.3) proof. let f ∈ l1(dµα) ∩ l2(dµα); ||f||2,µα = 1 and let ϕ be the function defined on ]1,2] by ϕ(p) = ∫+∞ o |hα(f)(λ)| p p−1 dµα(λ) − ( 1 p 1 p ( p p−1 ) p−1 p ) p(α+1) p−1 ( ∫+∞ o |f(x)|pdµα(x) ) 1 p−1 . then, by relation (3.1), ∀p ∈]1,2]; ϕ(p) 6 0. on the other hand, theorem 2.6 means that ϕ(2) = 0. this implies that dϕ dp (2−) > 0. (3.4) since f ∈ l1(dµα)∩l2(dµα), then by a standard interpolation argument, f belongs to lp(dµα); p ∈ [1,2]. using lemma 3.2, the hypothesis and lebesgue dominated convergence theorem, we deduce that d dp [ ∫+∞ o |f(r)|pdµα(r) ] (2−) = ∫+∞ o lim p→2− |f(r)|p − |f(r)|2 p − 2 dµα(r). (3.5) thus d dp [ ∫+∞ o |f(r)|pdµα(r) ] (2−) = 1 2 ∫+∞ o ln |f(r)|2|f(r)|2dµα(r)]. (3.6) now, since f ∈ l1(dµα) ∩ l2(dµα), by using again the lebesgue dominated convergence theorem, we get lim p→2 ∫+∞ o |f(r)|pdµα(r) = ∫+∞ o |f(r)|2dµα(r) = 1. (3.7) combining (3.6) and (3.7), we get d dp [( ∫+∞ o |f(r)|pdµα(r) ) 1 p−1 ] (2−) = − 1 2 eµα(|f| 2 ). (3.8) cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 23 as the same way, one can see that d dp [ ∫+∞ o |hα(f)(λ)| p p−1 dµα(λ) ] (2−) = − 1 2 eµα(|hα(f)| 2). (3.9) finally, basic calculations show that d dp [ ( 1 p 1 p ( p p−1 ) p−1 p ) p p−1 (α+1) ] (2−) = (α + 1)(1 − ln2). (3.10) thus according to relations(3.8), (3.9) and (3.10), it follows that dϕ dp (2−) = 1 2 eµα(|f| 2) + 1 2 eµα(|hα(f)| 2) − (α + 1)(1 − ln2). (3.11) the proof is complete by using (3.4). lemma 3.4. let f be a measurable function on [0,+∞[ and let ω : [0,+∞[7−→ [0,+∞[ be a convex function such that ω(|f|) belongs to l1(dµα). let (fk)k∈n be a sequence of nonnegative measurable functions on r+ such that for every k ∈ n; ||fk||1,µα = 1 and the sequence (fk ∗α f)k∈n converges pointwise to f. then, for every k ∈ n, the function ω(|fk ∗α f| belongs to l1(dµα), and we have lim k→+∞ ∫+∞ o ω(|fk ∗α f|)(r)dµα(r) = ∫+∞ o ω(|f(r)|)dµα(r). (3.12) proof. we have ||ω ◦ |f|||1,µα = ∫+∞ o lim inf k→+∞ ω(|fk ∗α f|)(r)dµα(r), (3.13) and by using fatou’s lemma, we get ||ω ◦ |f|||1,α 6 lim inf p→+∞ ∫+∞ o ω(|fk ∗α f(r)|)dµα(r). (3.14) conversely, according to relation (2.9), we have ∀k ∈ n; ∀λ ∈ r+, ∫+∞ o ταλ (fk)(r)dµα(r) = ||fk||1,µα = 1, (3.15) which means that for every λ ∈ r+, ταλ (fk)(r)dµα(r) is a probability measure on r+. therefore, by using jensen’s inequality for convex functions, we get 24 chirine chettaoui cubo 17, 2 (2015) ∀r ∈ r+, ω(|fk ∗α f(r)|) = ω(| ∫+∞ o f(x)ταr (fk)(x)dµα(x)) 6 ω( ∫+∞ o |f(x)ταr (fk)(x)|dµα(x) 6 ∫+∞ o ω(|f(x)|)ταr (fk)(x)dµα(x) = fk ∗α ( ω ◦ |f| ) (r). (3.16) in particular, ω ◦ |fk ∗α f| ∈ l1(dµα). hence by relations (2.9) and (3.16), we deduce that lim sup k→+∞ ∫+∞ o ω(|fk ∗α f(r)|)dµα(r) 6 lim sup k→+∞ ∫+∞ o fk ∗α ( ω ◦ |f(r)| ) dµα(r) = lim sup k→+∞ ||fk ∗α ( ω ◦ |f| ) ||1,α 6 ||ω ◦ |f|||1,α. (3.17) the relations (3.14) and (3.17) show that lim k→+∞ ∫+∞ o ω(|fk ∗α f|)(r)dµα(r) = ∫+∞ o ω(|f(r)|)dµα(r). now, e are able to prove the main result. theorem 3.5. let f ∈ l2(dµα); ||f||2,µα = 1, such that ∫ ∞ 0 |f(r)|2 ∣∣ ln(|f(r)|) ∣∣ dµα(r) < +∞, and ∫ ∞ 0 |hα(f)(λ)| 2 ∣∣ ln(|hα(f)(λ)|) ∣∣ dµα(λ) < +∞. then, we have eµα(|f| 2 ) + eµα(|hα(f)| 2 ) > (2α + 2)(1 − ln2). (3.18) proof. the main idea of this proof is to combine theorem 3.3 with the standard density argument. indeed, we will show that for every f ∈ l2(dµα); there exists a sequence (fk)k∈n ∈ l1(dµα) ∩ l2(dµα) such that lim k→+∞ ||fk||2,µα = ||f||2,µα, (3.19) cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 25 lim k→+∞ eµα(|fk| 2) = eµα(|f| 2), (3.20) lim k→+∞ eα(|hα(fk)| 2) = eα(|hα(f)| 2). (3.21) let (hk)k∈n be the sequence of functions defined by hk(r) = 2 α+1k2α+2e−k 2 r 2 = g 1 k √ 2 (r). (3.22) then, by relation(2.21), we have ∀f ∈ l2(dµα); lim k→+∞ ||hk ∗α f − f||2,µα = 0. (3.23) furthermore, according to weber’s formula [15], we know that for all k ∈ n∗, α > −1 2 , ∫+∞ o e−k 2 r 2 jα(xr)r 2α+1dr = γ(α + 1) e − x 2 4k2 2k2α+2 . (3.24) hence, by relation (3.24), we deduce that h −1 α (hk)(λ) = 2k2α+2 γ(α + 1) ∫+∞ o e−k 2 r 2 jα(λr)r 2α+1dµα(r) = e − λ 2 4k2 . (3.25) let (ψk)k∈ n be the sequence of functions defined on r+ by ψk(λ) = e − λ 2 4k2 = h −1α (hk)(λ). (3.26) let f ∈ l2(dµα); ||f||2,µα = 1, then according to relation(3.23), we have lim k→+∞ ||hα(ψk) ∗α hα(f) − hα(f)||2,µα = 0. in particular, there is a subsequence (ψσ(k))k∈n such that hα(ψσ(k)) ∗α hα(f) = hσ(k) ∗α hα(f) converges pointwise to hα(f). let (fk)k∈n be the sequence of measurable functions on r+ defined by fk = ψσ(k)f. (3.27) since ψσ(k) ∈ l2(dµα) ∩ c∗,0(r+), then fk belongs to the space l1(dµα) ∩ l2(dµα). replacing f by fk ||fk||2,µα in theorem 3.3 and using the fact that ||f||2,µα = ||hα(f)||2,µα; f ∈ l 2 (dµα), 26 chirine chettaoui cubo 17, 2 (2015) we deduce that − ∫+∞ o ln ( |fk(r)| 2 ) |fk(r)| 2dµα(r) − ∫+∞ o ln ( |hα(fk)(λ)| 2 ) |hα(fk)(λ)| 2dµα(λ) (3.28) > (2α + 2)(1 − ln2)||fk|| 2 2,µα − 2||fk|| 2 2,α ln ( ||fk|| 2 2,µα ) . (3.29) now, by using the lebesgue dominated convergence theorem, we have lim k→+∞ ||fk||2,µα = ||f||2,µα. (3.30) on the other hand, one can see that for all k ∈ n, and for almost every r ∈ [0,+∞[, we have ln |fk(r)| 2 |fk(r)| 2 6 c|f(r)|2 + ln |f(r)|2|f(r)|2, (3.31) hence, using again the lebesgue dominated convergence theorem, we get − lim k→+∞ ∫+∞ o ln ( |fk(r)| 2 ) |fk(r)| 2dµα(r) = eµα(|f| 2). (3.32) now, let us show that lim k→+∞ eµα(|hα(fk)| 2 ) = eµα(hα(f)| 2 ). for this, we denote by ω1 ,ω2 the functions defined on r by ω1(t)= { t2 ln |t|, if |t| > 1 0, if |t| 6 1, and ω2(t)=    2t2, if |t| > 1 −t2 ln |t| + 2t2, if |t| 6 1 ,t 6= 0 0, if t = 0. then ω1 and ω2 are both nonnegative and convex, moreover; we have ∀t > 0, t2 ln |t| = ω1(t) − ω2(t) + 2t2. (3.33) from the hypothesis, for each i ∈ {1,2}, the function ωi(|hα(f)|) belongs to l1(dµα). now, from proposition 2.7 iv), for every k ∈ n∗; we have hα(fk) = hσ(k) ∗α hα(f) and we know that hσ(k) ∗α hα(f) converges pointwise to hα(f) and ||hσ(k)||1,µα = 1. so, the hypothesis of lemma 3.4 are satisfies and we get lim k→+∞ ∫+∞ o ωi(|hα(fk)|)(r)dµα(r) = ∫+∞ o ωi(|hα(f)|)(r)dµα(r), (3.34) and therefore, by relations(3.30) and (3.33) we get lim k→+∞ ∫+∞ o ln |hα(fk)| 2|hα(fk)(r)| 2dµα(r) = eµα(|hα(f)| 2). (3.35) the proof is complete by combining relations (3.29), (3.30), (3.32)and(3.35). cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 27 4 heisenberg-pauli-weyl inequality for the hankel transorm lemma 4.1. let f ∈ l2(dµα) such that ||f||2,µα = 1.then, for every t > 0, ∫+∞ o |f(r)|2 ln ( |f(r)|2 gt(r) ) dµα(r) > 0, (4.1) where gt(r)is given by(2.20). proof. since the function ω(t) = t lnt is convex on ]0,+∞[, and dνα(r) = gt(r)dµα(r) is a probability measure on ]0,+∞[ then, applying jensen’s inequality, we get ∫+∞ o |f(r)|2 ln( |f(r)|2 gt(r) )dµα(r) = ∫+∞ o ω( |f(r)|2 gt(r) )dνα(r) > ω( ∫+∞ o |f(r)|2 gt(r) dνα(r)) = ω(||f||22,µα) = ω(1) = 0. theorem 4.2. (heisenberg-pauli-weyl inequality) let f ∈ l2(dµα), then ||rf||2,µα||λhα(f)||2,µα > (α + 1)||f|| 2 2,µα . (4.2) proof. let h ∈ l2(dµα); ||h||2,µα = 1. from lemma 4.1, we get ∫+∞ o [ |h(r)|2 ln(|h(r)|2) − |h(r)|2 ln(|gt(r)|) ] dµα(r) > 0. (4.3) then, eµα(|h| 2) 6 ln ( t2α+2 ) + 1 2t2 ∫+∞ o |h(r)|2r2dµα(r). (4.4) since ||hα(h)||2,µα = ||h||2,µα = 1, we get also eµα(|hα(h)| 2) 6 ln ( t2α+2 ) + 1 2t2 ∫+∞ o |hα(h)(λ)| 2λ2dµα(λ), (4.5) adding (4.4) and (4.6), it follows that ||rh||22,µα + ||λhα(h)|| 2 2,µα > 2t2 [ eµα(|h| 2 ) + eµα(|hα(h)| 2 ) − 2 ln(t2α+2) ] . 28 chirine chettaoui cubo 17, 2 (2015) applying theorem 3.5, we obtain ||rh||22,µα + ||λhα(h)|| 2 2,µα > 2t2 [ (2α + 2)(1 − 2 ln2) − 2(2α + 2) lnt ] = 2t2(2α + 2)(1 − ln2t2). in particular, for t = 1√ 2 ; we deduce that for every h ∈ l2(dµα);||h||2,µα = 1, ||rh||22,µα + ||λhα(h)|| 2 2,µα > 2α + 2. (4.6) let f ∈ l2(dµα), replacing h by f ||f||2 2,µα in relation (4.6), we claim that for every f ∈ l2(dµα), ||rf||22,µα + ||λhα(f)|| 2 2,µα > (2α + 2)||f||22,µα. (4.7) on the other hand, for f ∈ l2(dµα) and t > 0, we denote by ft the dilated of f defined by ft(r) = f(tr), then, ft belongs to l 2(dµα) and we have ||ft|| 2 2,µα = ∫+∞ o |ft(r)| 2dµα(r) = 1 t2α+2 ∫+∞ o |f(r)|2dµα(r) = 1 t2α+2 ||f||22,µα. (4.8) moreover ||rft|| 2 2,µα = ∫+∞ o r2|ft(r)| 2dµα(r) = 1 t2α+4 ||rf||22,µα, (4.9) and ||λhα(ft)|| 2 2,µα = ∫+∞ o λ2|hα(ft)(λ)| 2dµα(λ) (4.10) hα(ft)(λ) = ∫+∞ o ft(x)jα(λx)dµα(x) = 1 t2α+2 hα(f)( λ t ). (4.11) then ||λhα(ft)|| 2 2,µα = 1 t2α ||λhα(f)|| 2 2,µα . (4.12) cubo 17, 2 (2015) an other uncertainty principle for the hankel transform 29 now, we assume that ||rf||2,µα and ||λhα(f)||2,µα are both non zero and finite, hence the same holds for ft , t > 0 and from relation (4.7), we have ||rft|| 2 2,µα + ||λhα(ft)|| 2 2,µα > (2α + 2)||ft|| 2 2,µα . (4.13) then, by relations (4.8), (4.9) and (4.12), we get for every t > 0 1 t2α+4 ||rf||22,µα + 1 t2α ||λhα(f)|| 2 2,µα > (2α + 2) 1 t2α+2 ||f||22,µα, or 1 t2 ||rf||22,µα + t 2||λhα(f)|| 2 2,µα > (2α + 2)||f||22,µα. in particular for t = t0 = √ ||rf||2,µα ||λhα(f)||2,µα . we obtain ||λhα(f)||2,µα||rf||2,µα > (α + 1)||f|| 2 2,µα . received: june 2014. accepted: march 2015. references [1] w. beckner, inequalities in fourier analysis, ann. of math., 102 (1975) 159-182. [2] w. beckner, pitt’s inequality and the uncertainty principle, proc. amer. math. soc., (1995) 1897-1905. [3] a. beurling, the collected works of arne beurling, birkhäuser, vol.1-2, boston 1989. [4] i. bialynicki-birula, entropic uncertainty relations in quantum mechanics, quantum probability and applications ii, eds. l. accardi and w. von waldenfels, lecture notes in mathematics, 1136 (1985) 90-103. [5] a. bonami, b. demange, and p. jaming, hermite functions and uncertainty priciples for the fourier and the widowed fourier transforms, rev. mat. iberoamericana, 19 (2003) 23–55. [6] m.g. cowling and j. f. price, generalizations of heisenberg inequality in harmonic analysis, (cortona, 1982), lecture notes in math., 992 (1983) 443–449. [7] a. erdélyi et al., tables of integral transforms, mc graw-hill book compagny., vol.2, new york 1954. [8] a. erdélyi , asymptotic expansions, dover publications, new-york 1956. 30 chirine chettaoui cubo 17, 2 (2015) [9] a. fitouhi, inégalité de babenko et inégalité logarithmique de sobolev pour l’opérateur de bessel, c. r. acad. sci. paris, t. 305, srie i, (1987) 877-880. [10] g. b. folland, real analysis modern thechniques and their applications, pure and applied mathematics, john wiley and sons, new york 1984. [11] g. b. folland and a. sitaram, the uncertainty principle: a mathematical survey, j. fourier anal. appl., 3 (1997) 207–238. [12] g. h. hardy, a theorem concerning fourier transform, j. london. math. soc., 8 (1933) 227231. [13] i. i. hirschman, a note on entropy, amer. j. math., 79, 1 (1957) 152-156. [14] i. i. hirschman, variation diminishing hankel transforms, j. anal. math., vol. 8, no. 01, (1960) 307-336. [15] n. n. lebedev, special functions and their applications, dover publications, new-york 1972. [16] g. w. morgan, a note on fourier transforms, j. london. math. soc., 9 (1934) 178–192. [17] m. rösler and m. voit, an uncertaintly principle for the hankel transforms, american mathematical society, 127 (1) (1999), 183-194. [18] c. shanon, a mathematical theory of communication, bell system tech. j., 27 (1948/1949), 379-423 and 623-656. [19] k. trimèche, transformation intégrale de weyl et théorème de paley-wiener associés à un opérateur différentiel singulier sur (0,+∞), j. math. pures appl., 60 (1981) 51–98. [20] k. trimèche, inversion of the lions transmutation operator using genaralized wavelets, appl. comput. harmon. anal., 4 (1997) 97–112. [21] vu kim tuan, uncertainty principle for the hankel transform. integral transforms spec. funct. vol. 18, issue 5 (2007), 369-381. [22] g. n. watson, a treatise on the theory of bessel functions, cambridge univ. press., 2nd ed., cambridge 1959. introduction: the hankel operator uncertainty principle in terms of entropy for the hankel transform heisenberg-pauli-weyl inequality for the hankel transorm c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2609 .dvi cubo a mathematical journal vol.13, no¯ 01, (137–147). march 2011 strong convergence of an implicit iteration process for a finite family of strictly asymptotically pseudocontractive mappings gurucharan singh saluja department of mathematics & information technology, govt. nagarjuna p.g. college of science, raipur (c.g.). email: saluja 1963@rediffmail.com and hemant kumar nashine department of mathematics, disha institute of management and technology, satya vihar, vidhansabha-chandrakhuri marg mandir hasaud, raipur-492101(chhattisgarh), india. email: hnashine@rediffmail.com, nashine 09@rediffmail.com abstract in this paper, we establish the strong convergence theorems for a finite family of kstrictly asymptotically pseudo-contractive mappings in the framework of hilbert spaces. our results improve and extend the corresponding results of liu [5] and many others. resumen en este trabajo, hemos establecido los teoremas de convergencia para una familia finita de asignaciones de k-estrictamente asintticamente pseudo-contraccin en el marco de los espacios de hilbert. nuestros resultados mejoran y amplan los resultados correspondientes de liu [5] y muchos otros. 138 gurucharan singh saluja and hemant kumar nashine cubo 13, 1 (2011) keywords: strictly asymptotically pseudo-contractive mapping, implicit iteration scheme, common fixed point, strong convergence, hilbert space. ams subject classification: 47h09, 47h10. 1 introduction let h be a real hilbert space with the scalar product and norm denoted by the symbols 〈., .〉 and ‖ . ‖ respectively, and c be a closed convex subset of h. let t be a (possibly) nonlinear mapping from c into c. we now consider the following classes: (1) t is contractive, i.e., there exists a constant k < 1 such that ‖t x − t y‖ ≤ k ‖x − y‖ , (1.1) for all x, y ∈ c. (2) t is nonexpansive, i.e., ‖t x − t y‖ ≤ ‖x − y‖ , (1.2) for all x, y ∈ c. (3) t is uniformly l-lipschitzian, i.e., if there exists a constant l > 0 such that ‖t nx − t ny‖ ≤ l ‖x − y‖ , (1.3) for all x, y ∈ c and n ∈ n. (4) t is pseudo-contractive, i.e., 〈t x − t y, j(x − y)〉 ≤ ‖x − y‖ 2 , (1.4) for all x, y ∈ c. (5) t is strictly pseudo-contractive, i.e., there exists a constant k ∈ [0, 1) such that ‖t x − t y‖ 2 ≤ ‖x − y‖ 2 + k ‖(x − t x) − (y − t y)‖ 2 , (1.5) cubo 13, 1 (2011) strong convergence of an implicit iteration process for a finite family of strictly asymptotically pseudocontractive mappings 139 for all x, y ∈ c. (6) t is asymptotically nonexpansive [3], i.e., if there exists a sequence {rn} ⊂ [0, ∞) with limn→∞ rn = 0 such that ‖t nx − t ny‖ ≤ (1 + rn) ‖x − y‖ , (1.6) for all x, y ∈ c and n ∈ n. (7) t is k-strictly asymptotically pseudo-contractive [6], i.e., if there exists a sequence {rn} ⊂ [0, ∞) with limn→∞ rn = 0 such that ‖t nx − t ny‖ 2 ≤ (1 + rn) 2 ‖x − y‖ 2 +k ‖(x − t nx) − (y − t ny)‖ 2 (1.7) for some k ∈ [0, 1) for all x, y ∈ c and n ∈ n. remark 1.1 [6]: if t is k-strictly asymptotically pseudo-contractive mapping, then it is uniformly l-lipschitzian, but the converse does not hold. concerning the convergence problem of iterative sequences for strictly pseudocontractive mappings has been studied by several authors (see, e.g., [2, 4, 7, 11, 12]). concerning the class of strictly asymptotically pseudocontractive mappings, liu [5] proved the following result in hilbert space: theorem 1.1(liu [5]): let h be a real hilbert space, let c be a nonempty closed convex and bounded subset of h, and let t : c → c be a completely continuous uniformly l-lipschitzian (λ, {kn})-strictly asymptotically pseudocontractive mapping such that ∑∞ n=1(k 2 n − 1) < ∞. let {αn} ⊂ (0, 1) be a sequence satisfying the following condition: 0 < ǫ ≤ αn ≤ 1 − λ − ǫ ∀ n ≥ 1 and some ǫ > 0. then, the sequence {xn} generated from an arbitrary x1 ∈ c by xn+1 = (1 − αn)xn + αnt n xn, ∀ n ≥ 1 (1.8) converges strongly to a fixed point of t . 140 gurucharan singh saluja and hemant kumar nashine cubo 13, 1 (2011) in 2001, xu and ori [12] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a hilbert space h. let c be a nonempty subset of h. let t1, t2, . . . , tn be self-mappings of c and suppose that f = ∩ni=1f (ti) 6= ∅, the set of common fixed points of ti, i = 1, 2, . . . , n . an implicit iteration process for a finite family of nonexpansive mappings is defined as follows, with {tn} a real sequence in (0, 1), x0 ∈ c: x1 = t1x0 + (1 − t1)t1x1, x2 = t2x1 + (1 − t2)t2x2, ... xn = tn xn−1 + (1 − tn )tn xn , xn+1 = tn+1xn + (1 − tn+1)t1xn+1, ... which can be written in the following compact form: xn = tnxn−1 + (1 − tn)tnxn, n ≥ 1 (1.9) where tk = tk mod n . (here the mod n function takes values in {1, 2, . . . , n}). and they proved the weak convergence of the process (1.9). very recently, acedo and xu [1] still in the framework of hilbert spaces introduced the following cyclic algorithm. let c be a closed convex subset of a hilbert space h and let {ti} n−1 i=0 be n k-strict pseudocontractions on c such that f = ⋂n−1 i=0 f (ti) 6= ∅. let x0 ∈ c and let {αn} be a sequence in (0, 1). the cyclic algorithm generates a sequence {xn} ∞ n=1 in the following way: x1 = α0x0 + (1 − α0)t0x0, x2 = α1x1 + (1 − α1)t1x1, ... xn = αn−1xn−1 + (1 − αn−1)tn−1xn−1, xn+1 = αn xn + (1 − αn )t0xn , ... in general, {xn+1} is defined by xn+1 = αnxn + (1 − αn)t[n]xn, (1.10) cubo 13, 1 (2011) strong convergence of an implicit iteration process for a finite family of strictly asymptotically pseudocontractive mappings 141 where t[n] = ti with i = n (mod n ), 0 ≤ i ≤ n −1. they also proved a weak convergence theorem for k-strict pseudo-contractions in hilbert spaces by cyclic algorithm (1.10). more precisely, they obtained the following theorem: theorem ax [1]: let c be a closed convex subset of a hilbert space h. let n ≥ 1 be an integer. let for each 0 ≤ i ≤ n − 1, ti : c → c be a ki-strict pseudo-contraction for some 0 ≤ ki < 1. let k = max{ki : 1 ≤ i ≤ n}. assume the common fixed point the set ⋂n−1 i=0 f (ti) of {ti} n−1 i=0 is nonempty. given x0 ∈ c, let {xn} ∞ n=0 be the sequence generated by the cyclic algorithm (1.10). assume that the control sequence {αn} is chosen so that k + ǫ < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). then {xn} converges weakly to a common fixed point of the family {ti} n−1 i=0 . motivated by xu and ori [12], acedo and xu [1] and some others we introduce and study the following: let c be a closed convex subset of a hilbert space h and let {ti} n−1 i=0 be n k-strictly asymptotically pseudo-contractions on c such that f = ⋂n−1 i=0 f (ti) 6= ∅. let x0 ∈ c and let {αn} be a sequence in (0, 1). the implicit iteration scheme generates a sequence {xn} ∞ n=0 in the following way: x1 = α0x0 + (1 − α0)t0x0, x2 = α1x1 + (1 − α1)t1x1, ... xn = αn−1xn−1 + (1 − αn−1)tn−1xn−1, xn+1 = αn xn + (1 − αn )t 2 0 x0, ... x2n = α2n−1x2n−1 + (1 − α2n−1)t 2 n−1x2n−1, x2n+1 = α2n x2n + (1 − α2n )t 3 0 x0, ... in general, {xn} is defined by xn+1 = αnxn + (1 − αn)t s [n]xn, (1.11) where t s [n] = t s n (mod n) = t si with n = (s − 1)n + i and i ∈ i = {0, 1, . . . , n − 1}. the aim of this paper is to establish strong convergence theorems of implicit iteration process (1.11) for a finite family of k-strictly asymptotically pseudo-contraction mappings in hilbert 142 gurucharan singh saluja and hemant kumar nashine cubo 13, 1 (2011) spaces. our results extend the corresponding results of liu [5] and many others. in the sequel, we will need the following lemmas. lemma 1.1: let h be a real hilbert space. there holds the following identities: (i) ‖x − y‖ 2 = ‖x‖ 2 − ‖y‖ 2 − 2〈x − y, y〉 ∀ x, y ∈ h. (ii) ‖tx + (1 − t)y‖ 2 = t ‖x‖ 2 + (1 − t) ‖y‖ 2 − t(1 − t) ‖x − y‖ 2 , ∀ t ∈ [0, 1], ∀ x, y ∈ h. (iii) if {xn} be a sequence in h weakly converges to z, then lim sup n→∞ ‖xn − y‖ 2 = lim sup n→∞ ‖xn − z‖ 2 + ‖z − y‖ 2 ∀y ∈ h. lemma 1.2 [9]: let {an} ∞ n=1, {βn} ∞ n=1 and {rn} ∞ n=1 be sequences of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + rn)an + βn, n ≥ 1. if ∑∞ n=1 rn < ∞ and ∑∞ n=1 βn < ∞, then limn→∞ an exists. if in addition {an} ∞ n=1 has a subsequence which converges strongly to zero, then limn→∞ an = 0. 2 main results theorem 2.1: let c be a closed convex subset of a hilbert space h. let n ≥ 1 be an integer. let for each 0 ≤ i ≤ n − 1, ti : c → c be n ki-strictly asymptotically pseudo-contraction mappings for some 0 ≤ ki < 1 and ∑∞ n=1 rn < ∞. let k = max{ki : 0 ≤ i ≤ n − 1} and rn = max{rni : 0 ≤ i ≤ n − 1}. assume that f = ⋂n−1 i=0 f (ti) 6= ∅. given x0 ∈ c, let {xn} ∞ n=0 be the sequence generated by an implicit iteration scheme (1.11). assume that the control sequence {αn} is chosen so that k < αn < 1 for all n and ∑∞ n=0(αn − k)(1 − αn) = ∞. then the iterative sequence {xn} has the following properties: cubo 13, 1 (2011) strong convergence of an implicit iteration process for a finite family of strictly asymptotically pseudocontractive mappings 143 (1) limn→∞ ‖xn − p‖ exists for each p ∈ f , (2) limn→∞ d(xn, f ) exists, (3) lim infn→∞ ∥ ∥ ∥ xn − t s [n] xn ∥ ∥ ∥ = 0, (4) the sequence {xn} ∞ n=0 converges strongly to a common fixed point p ∈ f if and only if lim inf n→∞ d(xn, f ) = 0. proof: we divide the proof of theorem 2.1 into three steps. (i) first, we proof the conclusions (1)and (2). for any p ∈ f , it follows from (1.11) and lemma 1.1(ii), we note that ‖xn+1 − p‖ 2 = ∥ ∥ ∥ αnxn + (1 − αn)t s [n]xn − p ∥ ∥ ∥ (2.1) = ∥ ∥ ∥ αn(xn − p) + (1 − αn)(t s [n]xn − p) ∥ ∥ ∥ ≤ αn ‖xn − p‖ 2 + (1 − αn) ∥ ∥ ∥ t s [n]xn − p ∥ ∥ ∥ 2 −αn(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 ≤ αn ‖xn − p‖ 2 + (1 − αn)[(1 + rn) 2 ‖xn − p‖ 2 +k ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 ] − αn(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 ≤ [αn(1 + rn) 2 + (1 − αn)(1 + rn) 2] ‖xn − p‖ 2 −(αn − k)(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 ≤ (1 + rn) 2 ‖xn − p‖ 2 − (αn − k)(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 ≤ (1 + dn) ‖xn − p‖ 2 − (αn − k)(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 where dn = r 2 n + 2rn, since ∑∞ n=1 rn < ∞ thus ∑∞ n=1 dn < ∞ and since k < αn < 1, we get ‖xn+1 − p‖ 2 ≤ (1 + dn) ‖xn − p‖ 2 (2.2) and therefore 144 gurucharan singh saluja and hemant kumar nashine cubo 13, 1 (2011) ‖xn+1 − p‖ ≤ (1 + dn) 1/2 ‖xn − p‖ . (2.3) since ∑∞ n=1 dn < ∞, it follows from lemma 1.2, we know that limn→∞ ‖xn − p‖ exists for each p ∈ f . so that there exists k > 0 such that ‖xn − p‖ ≤ k for all n ≥ 1. consequently, we obtain from (2.3) that ‖xn+1 − p‖ ≤ (1 + dn) 1/2 ‖xn − p‖ ≤ (1 + dn) ‖xn − p‖ ≤ ‖xn − p‖ + kdn. (2.4) it follows from (2.4) that d(xn+1, f ) ≤ (1 + dn)d(xn, f ), ∀ n ≥ 1 (2.5) so that it again follows from lemma 1.2 that limn→∞ d(xn, f ) exists. the conclusions (1)and (2) are proved. (ii) the proof of conclusion (3). it follows from (2.1) that ‖xn+1 − p‖ 2 ≤ (1 + dn) ‖xn − p‖ 2 (2.6) −(αn − k)(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 where dn = r 2 n + 2rn, since ∑∞ n=1 rn < ∞ thus ∑∞ n=1 dn < ∞ and since k < αn < 1, we get ‖xn+1 − p‖ 2 ≤ (1 + dn) ‖xn − p‖ 2 (2.7) that means the sequence {‖xn − p‖} is decreasing. now, since ∑∞ n=1 dn < ∞ it follows that ∏∞ i=1(1 + di) < ∞, from (2.6), we have ∞ ∑ n=0 (αn − k)(1 − αn) ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ 2 ≤ ∞ ∏ i=1 (1 + di) ‖x0 − p‖ 2 (2.8) < ∞. cubo 13, 1 (2011) strong convergence of an implicit iteration process for a finite family of strictly asymptotically pseudocontractive mappings 145 since ∑∞ n=0(αn − k)(1 − αn) = ∞, (2.8) implies that lim inf n→∞ ∥ ∥ ∥ xn − t s [n]xn ∥ ∥ ∥ = 0. (2.9) (iv) next, we prove the conclusion (4). necessity if {xn} converges strongly to some point p ∈ f , then from 0 ≤ d(xn, f ) ≤ ‖xn − p‖ → 0 as n → ∞, we have lim inf n→∞ d(xn, f ) = 0. (2.10) sufficiency if lim infn→∞ d(xn, f ) = 0, it follows from the conclusion (2) that limn→∞ d(xn, f ) = 0. next, we prove that {xn} is a cauchy sequence in c. in fact, since for any x > 0, 1 + x ≤ exp(x), therefore, for any m, n ≥ 1 and for given p ∈ f , from (2.4), we have ‖xn+m − p‖ ≤ (1 + dn+m−1) ‖xn+m−1 − p‖ ≤ edn+m−1 ‖xn+m−1 − p‖ ≤ edn+m−1 [edn+m−2 ‖xn+m−2 − p‖] ≤ e{dn+m−1+dn+m−2} ‖xn+m−2 − p‖ ≤ . . . ≤ e ∑ n+m−1 j=n dj ‖xn − p‖ ≤ k′ ‖xn − p‖ < ∞ (2.11) where k′ = e ∑ ∞ j=1 dj < ∞. since lim n→∞ d(xn, f ) = 0, (2.12) for any given ǫ > 0, there exists a positive integer n1 such that 146 gurucharan singh saluja and hemant kumar nashine cubo 13, 1 (2011) d(xn, f ) < ǫ 2(k′ + 1) , ∀ n ≥ n1. (2.13) hence, there exists p1 ∈ f such that ‖xn − p1‖ < ǫ (k′ + 1) ∀ n ≥ n1. (2.14) consequently, for any n ≥ n1 and m ≥ 1, from (2.11), we have ‖xn+m − xn‖ ≤ ‖xn+m − p1‖ + ‖xn − p1‖ ≤ k′ ‖xn − p1‖ + ‖xn − p1‖ ≤ (k′ + 1) ‖xn − p1‖ < (k′ + 1). ǫ (k′ + 1) = ǫ. this implies that {xn} is a cauchy sequence in c. let xn → x ∗ ∈ c. since lim infn→∞ d(xn, f ) = 0, and so d(x∗, f ) = 0. again since {ti} n−1 i=0 is a finite family of k-strictly asymptotically pseudocontractive mappings, by remark 1.1 of [6], it is a finite family of uniformly lipschitzian mappings. hence, the set f of common fixed points of {ti} n−1 i=0 is closed and so x ∗ ∈ f . thus the sequence {xn} converges strongly to a common fixed point of the family {ti} n−1 i=0 . this completes the proof. theorem 2.2: let c be a closed convex compact subset of a hilbert space h. let n ≥ 1 be an integer. let for each 0 ≤ i ≤ n − 1, ti : c → c be n ki-strictly asymptotically pseudocontraction mappings for some 0 ≤ ki < 1 and ∑∞ n=1 rn < ∞. let k = max{ki : 0 ≤ i ≤ n −1} and rn = max{rni : 0 ≤ i ≤ n − 1}. assume that f = ⋂n−1 i=0 f (ti) 6= ∅. given x0 ∈ c, let {xn} ∞ n=0 be the sequence generated by an implicit iteration scheme (1.11). assume that the control sequence {αn} is chosen so that k < αn < 1 for all n. then {xn} converges strongly to a common fixed point of the family {ti} n−1 i=0 . proof: we only conclude the difference. by compactness of c this immediately implies that there is a subsequence {xnj } of {xn} which converges to a common fixed point of {ti} n−1 i=0 , say, p. combining (2.3) with lemma 1.2, we have limn→∞ ‖xn − p‖ = 0. thus {xn} converges strongly to a common fixed point of the family {ti} n−1 i=0 . this completes the proof. remark 2.1 our results extend and improve the corresponding results of liu [5] and we also extend the iteration process (1.8) of [5] to an implicit iteration process for a finite family of cubo 13, 1 (2011) strong convergence of an implicit iteration process for a finite family of strictly asymptotically pseudocontractive mappings 147 mappings. received: june 2009. revised: november 2009. references [1] g.l. acedo and h.k. xu, iterative methods for strict pseudo-contractions in hilbert spaces, nonlinear anal. 67(2007), 2258-2271. [2] f.e. browder and w.v. ptryshyn, construction of fixed points of nonlinear mappings in hilbert spaces, j. math. anal. appl. 20(1967), 197-228. [3] k. goebel and w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35(1972), 171-174. [4] f. gu, the new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings, j. math. anal. appl. 329(2) (2007), 766-776. [5] q. liu, convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, nonlinear anal. 26(1996), 1835-1842. [6] m.o. osilike, iterative approximation of fixed points of asymptotically demicontractive mappings, indian j. pure appl. math. 29(12), december 1998, 1291-1300. [7] m.o. osilike, implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, j. math. anal. appl. 294(1)(2004), 73-81. [8] m.o. osilike and a. udomene, demiclosedness principle and convergence results for strictly pseudocontractive mappings of browder-petryshyn type, j. math. anal. appl. 256(2001), 431-445. [9] m.o. osilike, s.c. aniagbosor and b.g. akuchu, fixed points of asymptotically demicontractive mappings in arbitrary banach spaces, panam. math. j. 12(2002), 77-78. [10] s. reich, weak convergence theorems for nonexpansive mappings in banach spaces, j. math. anal. appl. 67(1979), 274-276. [11] y. su and s. li, composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, j. math. anal. appl. 320(2)(2006), 882-891. [12] h.k. xu and r.g. ori, an implicit iteration process for nonexpansive mappings, numer. funct. anal. optim. 22(2001), 767-773. cubo a mathematical journal vol.14, no¯ 01, (29–47). march 2012 spectral shift function for slowly varying perturbation of periodic schrödinger operators. mouez dimassi univ. paris 13, laga, (umr cnrs 7539), f-93430 villetaneuse, france, email: dimassi@math.univ-paris13.fr and maher zerzeri univ. paris 13, laga, (umr cnrs 7539), f-93430 villetaneuse, france, email: zerzeri@math.univ-paris13.fr abstract in this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic schrödinger operators. we give a weak and pointwise asymptotic expansions in powers of h of the derivative of the spectral shift function corresponding to the pair ( p(h) = p0 + ϕ(hx),p0 = −∆ + v(x) ) , where ϕ(x) ∈ c∞(rn,r) is a decreasing function, o(|x|−δ) for some δ > n and h is a small positive parameter. here the potential v is real, smooth and periodic with respect to a lattice γ in rn. to prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption theorem for p(h). 30 m. dimassi and m. zerzeri cubo 14, 1 (2012) resumen en este art́ıculo estudiamos la expansión asintótica de la función shift espectral para perturbaciones de variación lenta de operadores periódicos de schrödinger. proporcionamos una expansión débil y puntual en potencias de h de la derivada de la función shift espectral que corresponde al par ( p(h) = p0 + ϕ(hx),p0 = −∆ + v(x) ) , donde ϕ(x) ∈ c∞(rn,r) es una función decreciente, o(|x|−δ) para algún δ > n y h un parámetro positivo pequeño. aqúı el potencial v es real, suave y periódico con respecto a un ret́ıculo γ en rn. para demostrar la expansión asintótica puntual de la función shift espectral establecemos un teorema de absorción ĺımite para p(h). keywords and phrases: periodic schrödinger operator, spectral shift function, asymptotic expansions, limiting absorption theorem. 2010 ams mathematics subject classification: 81q10 (35p20 47a55 47n50 81q15) 1 introduction the aim of this paper is to give an asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic schrödinger operator: p(h) = p0 + ϕ(hx), h > 0, (1.1) p0 = −∆x + v(x), here v is a real-valued, c∞ function and periodic with respect to a lattice γ of rn. the hamiltonian p(h) describes the quantum motion of an electron in a crystal placed in an external field. there are many works devoted to the spectral properties of this model, see [1, 3, 5, 8, 9, 10, 12, 13, 15, 16, 18, 21, 36]. we assume that ϕ ∈ c∞(rn; r) and satisfies the following estimate: for all α ∈ nn, there exists cα > 0 such that |∂αxϕ(x)| ≤ cα(1 + |x|) −δ−|α|, ∀x ∈ rn, with δ > n . (1.2) the operators p0,p(h) are self-adjoint on h 2(rn). under the assumption (1.2) we show in theorem 2.2 below that the operator [ f(p(h))−f(p0) ] belongs to the trace class for all f ∈ c∞0 (r). following the general setup we define the spectral shift function, ssf, ξh(µ) := ξ(µ;p(h),p0) related to the pair (p(h),p0) by tr [ f(p(h)) − f(p0) ] = −〈ξ′h(·),f(·)〉 = ∫ r ξh(µ)f ′(µ)dµ, ∀f ∈ c∞0 (r). (1.3) by this formula ξh is defined modulo a constant but for the analysis of the derivative ξ ′ h(µ) this is not important. see [22] and [2]. cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 31 the ssf may be considered as a generalization of the eigenvalues counting function. the notion of ssf was first singled out by the outstanding theoretical physicist i-m.lifshits in his investigations in the solid state theory, in 1952, see [23]. it was brought into mathematical use in m-g. krĕın’s famous paper [22], where the precise statement of the problem was given and explicit representation of the ssf in term of the perturbation determinant was obtained. the work of m-g. krĕın’s on the ssf has been described in detail in [2]. for more details about the interpretation of ssf we refer to the survey by d. robert [32] and to the monograph by d-r. yafaev [42, chapter 8]. in the case where v = 0, the asymptotic behavior of the ssf of the schrödinger operator has been intensively studied in different aspects (see [6, 17, 24, 25, 30, 31, 34] and the references given there). in the semi-classical regime (i.e. h(h) = −h2∆x + ϕ(x),(h ↘ 0)) the weyl type asymptotics of ξh(·) with sharp remainder estimate has been obtained (see [30, 31, 34, 35]). on the other hand, if an energy µ > 0 is non-trapping for the classical hamiltonian p(x,ζ) = |ζ|2 + ϕ(x) (i.e. for all (x,ζ) ∈ p−1{µ}, |expthp(x,ζ)| → ∞ when t → ∞) a complete asymptotic expansion in powers of h of ξ′h(µ) has been obtained (see [30, 31, 34, 35]). similar results are well-known for the ssf at high energy (see [4, 6, 25, 26, 29]). there are only few works treating the ssf in perturbed periodic schrödinger operator. see [3], [10] and also [16]. in [10] the connection between the resonances of p(h) and the ssf associated to the pair ( p(h),p0 ) were studied. under the assumption that ϕ is analytic in some conic complex neighborhood of the real axis and that p(h) has no resonances in a small complex neighborhood of some interval i the first author obtained a full asymptotic expansion in powers of h of the derivative of ssf: ξ′(µ;h) ∼ ∞∑ j=0 bj(µ)h j−n, h ↘ 0, (1.4) uniformly with respect to µ ∈ i. nevertheless, there is a lot of examples of perturbed periodic schrödinger operator that the perturbation ϕ does not satisfies the analyticity assumption. in this paper, we improve the result of [10] concerning the behavior of the derivative of ssf by removing the analyticity assumption on the potential ϕ. our proof is based on a limiting absorption principle and some arguments due to d. robert and h. tamura, see [33] and [35]. for v 6= 0, the limiting absorption theorem is new (see theorem 3.11). they are in harmony with the physical intuition which argues that, when h sufficiently small, the main effect of the periodic potential v consists in changing the dispersion relation from the free kinetic |k|2 to the modified kinetic energy λp(k) given by the pth band. by the method of effective hamiltonian spectral problems of p(h) can be reduced to similar problem of system of h−pseudodifferential operators (see [9] and also [12]). using a well-known results on h−pseudodifferential calculus we get the asymptotic (1.4) in the sense of distributions. if the values of the principal term of the effective hamiltonian are contained in non-trapping energy region we prove a limiting absorption principle for p(h) (theorem 3.11) and we get a pointwise 32 m. dimassi and m. zerzeri cubo 14, 1 (2012) asymptotic expansion for the derivative of the spectral shift function. the paper is organized as follows: in the next section, we recall some well-known results concerning the spectra of a periodic schrödinger operator (subsection 2.1) and we state the assumptions and the results precisely (subsection 2.2). we give an outline of the proofs in subsection 2.3. section 3 is devoted to the proofs. roughly, we introduce a class of symbols and the corresponding h-weyl operators (subsection 3.1). in the subsection 3.2 we recall the effective hamiltonian method and we give a representation of the derivative of the spectral shift function, denoted by ζ′h(·). the proof of the weak asymptotic expansion of ξ ′ h is given in subsection 3.3. we establish a limiting absorption principle for p(h) in the subsection 3.4 at last, the pointwise asymptotic expansion of ξ′h is proved in subsection 3.5. 2 statements 2.1 preliminaries let γ = n ⊕ i=1 zei be a lattice generated by some basis (e1,e2, · · · ,en) of rn. the dual lattice γ∗ is given by γ∗ := {γ∗ ∈ rn; 〈γ|γ∗〉 ∈ 2πz, ∀γ ∈ γ}. a fundamental domain of γ (resp. γ∗) is denoted by e (resp. e∗). if we identify opposite edges of e (resp. e∗) then it becomes a flat torus denoted by t = rn/γ (resp. t∗ = rn/γ∗). let v be a real-valued potential, c∞ and γ−periodic. for k ∈ rn, we define the operator p(k) on l2(t) by p(k) := (dy +k)2 +v(y). the operator p(k) is a semi-bounded self-adjoint with k-independent domain h2(t). since the resolvent of p(k) is compact, p(k) has a complete set of (normalized) eigenfunctions φn(·,k) ∈ h2(t), n ∈ n, called bloch functions. the corresponding eigenvalues accumulate at infinity and we enumerate them according to their multiplicities, λ1(k) ≤ λ2(k) ≤ · · · . the operator p(k) satisfies the identity e−iy·γ ∗ p(k)eiy·γ ∗ = p(k + γ∗), ∀γ∗ ∈ γ∗, then for every p ≥ 1, the function k 7→ λp(k) is γ∗−periodic. ordinary perturbation theory shows that λp(k) are continuous functions of k for any fixed p, and λp(k) is even an analytic function of k near any point k0 ∈ t∗ where λp(k0) is a simple eigenvalue of p(k0). the function λp(k) is called the band function and the closed intervals λp := λp(t∗) are called bands. see [27], [39] and also [37, 38]. consider the self-adjoint operator on l2(rn) with domain h2(rn): p0 = −∆x + v(x), where ∆x = n∑ j=1 ∂2 ∂x2 j . (2.1) the spectrum of p0 is absolutely continuous (see [41]) and consists of the bands λp, p = 1,2, · · · . indeed, σ(p0) = σac(p0) = ∪ p≥1 λp. see also [40]. definition 2.1. let µ ∈ r and f(µ) = { k ∈ t∗; µ ∈ σ ( p(k) )} the corresponding fermi-surface. cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 33 a) we will say that µ ∈ σ(p0) is a simple energy level if and only if µ is a simple eigenvalue of p(k), for every k ∈ f(µ). b) assume that µ is a simple energy level of p0 and let λ(k) be the unique eigenvalue defined on a neighborhood of f(µ) such that λ(k) = µ, for all k ∈ f(µ). we say that µ is a non-critical energy of p0 if dkλ(k) 6= 0 for all k ∈ f(µ). note that in one dimension case f(µ) is just a finite set of points. now, let us recall some well-known facts about the density of states associated with p0, see [40]. the density of states measure ρ is defined as follows: ρ(µ) := 1 (2π)n ∑ p≥1 ∫ {k∈e∗; λp(k)≤µ} dk. (2.2) since the spectrum of p0 is absolutely continuous, the measure ρ is absolutely continuous with respect to the lebesgue measure dµ. therefore the density of states, dρ de (e), of p0 is locally integrable. 2.2 results we now consider the perturbed periodic schrödinger operator: p(h) := p0 + ϕ(hx), h ↘ 0, (2.3) where ϕ ∈ c∞(rn; r) and satisfies: (a1) there exists δ > 0 such that ∀α ∈ nn, ∃cα > 0 s.t.∣∣∂αxϕ(x)∣∣ ≤ cα(1 + |x|)−δ−|α| uniformly on x ∈ rn. the operator p(h) is self-adjoint, semi-bounded on l2(rn) with domain h2(rn). the assumption (a1) and the perturbation theory (weyl theorem) give: σess ( p(h) ) = σess(p0) = σ(p0) = ⋃ p≥1 λp. (2.4) recall that σess(a), the essential spectrum of a, is defined by σess(a) = σ(a) \ σdisc(a), where σdisc(a) is the set of isolated eigenvalues of a with finite multiplicity. here a is an unbounded operator on a hilbert space. our first theorem in this section concerns the weak asymptotic of ζ′h(µ). let i =]a,b[⊂ r. theorem 2.2 (weak asymptotic). assume (a1) with δ > n . for f ∈ c∞0 (i), the operator[ f(p(h)) − f(p0) ] is of trace class and tr [ f(p(h)) − f(p0) ] ∼ h−n +∞∑ j=0 aj(f)h j, when h ↘ 0, (2.5) 34 m. dimassi and m. zerzeri cubo 14, 1 (2012) with a0(f) = (2π) −n ∑ p≥1 ∫ rnx ∫ e∗ [ f ( λp(k) + ϕ(x) ) − f ( λp(k) )] dkdx. (2.6) the coefficients f → aj(f) are distributions of finite order ≤ j + 1. moreover, if µ is a non-critical energy of p0 for all µ ∈ i, then aj(f) = −〈γj(·),f〉, for all f ∈ c∞0 (i). here γj(µ) are smooth functions of µ ∈ i. in particular, γ0(µ) = d dµ [∫ rnx { ρ ( µ ) − ρ ( µ − ϕ(x) )} dx ] . (2.7) the proof of theorem 2.2 is contained in subsection 3.3. let [a,b] ⊂ r. assume that: (a2) for all µ ∈ [a,b], µ is a non-critical energy of p0. for all µ ∈ [a,b], let λ(k) be the unique eigenvalue defined on a neighborhood of f(µ) such that λ(k) = µ. we assume that for all (k,r) ∈ t∗ × rn such that µ = λ(k) + ϕ(r) ∈ σ(p0) ∩ [a,b], µ is a simple energy level, and that: (a3) |∇λ(k)|2 − r∇ϕ(r)∆λ(k) > 0, for all (k,r) s.t. λ(k) + ϕ(r) ∈ [a,b]. remark. note that the assumption (a2) is fulfilled in the bottom of the spectrum of p0. moreover, assuming (a2) the hypothesis (a3) is satisfied if ‖ϕ‖∞ +‖x∇ϕ‖∞ << 1, (see [27], [37, 38]). our main result concerning the derivative of the spectral shift function is the following. theorem 2.3 (pointwise asymptotic). assume (a2), (a3) and (a1) with δ > n . then the following asymptotic expansion holds: ζ′h(µ) ∼ h −n ∑ j≥0 γj(µ)h j as h ↘ 0, (2.8) uniformly for µ ∈ [a,b]. the coefficients ( γj(µ) ) j≥0 are given in theorem 2.2. furthermore, this expansion has derivate in µ to any order. remark 2.4. theorems 2.2 and 2.3 still true also in the case when the potential ϕ(x) depend on h, i.e. ϕ(x,h) = ϕ(x) + hϕ1(x) + h 2ϕ2(x) · · · in sδ(1). see the next section for the definition of sδ(1). (subsection 3.1). 2.3 outline of the proofs let q1(h) = q w 1 (x,hdx), q0(h) = q w 0 (x,hdx) be two h-pseudodifferential operators such that qj(h) = q ∗ j (h), (j = 0,1), and (q1(h) + i) −1 − (q0(h) + i) −1 cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 35 is an operator of trace class. in this case, theorem 2.2 is well-known see [28] and the references therein. on the other hand, in the case of non-trapping geometrie, the asymptotic follows from the results of robert-tamura (see [33, 34, 35] and also [31]). the main ingredient in the roberttamura method is the limiting absorption theorem and the construction of a long-time parametrix for the time-dependent equation( hdt − q w j (x,hdx) ) uj(t) = 0, uj(0) = i. in our case p(h) = −∆ + v(x) + ϕ(hx) is not an h-pseudodifferential operator. in fact, when (h ↘ 0), there are two spatial scales in equation (1.1). the first one of the order of the linear dimension γ of the periodicity cell and the second one of order γ h on which the perturbation of the potential varies appreciably. to remedy this we reduce the study of p(h), to the one of a system of h-pseudodifferential operators. more precisely, following [7, 9, 12, 15, 18], we can reduce the spectral study of p(h) near any fixed energy z to the study of a finite system of h-pseudodifferential operators, e−+(z,h), acting on l 2(t∗n; cn). in general, for the reduced problem, the dependence on the spectral parameter is non-linear. however, in the case of simple band (see assumption (a2)) we show that e−+(z,h) = z − ( λ(k) + ϕ(r) + hk1(k,r) + h 2k2(k,r;z,h) ) , where k1 ∈ sδ+1(t∗ × rn) and k2(·;z,h) ∈ sδ+2(t∗ × rn), holomorphic with respect to z in a small complex neighborhood ω of a bounded interval i. (see 3.24). now, considering s ∈ ω∩ r as a parameter and assuming (a3), we can apply the robert-tamura approach to the hamiltonian bs(k,−hdk;h) := λ(k) + ϕ(−hdk) + hk w 1 (k,−hdk) + h 2gw(k,−hdk;s,h) where g satisfies the same properties as k2, and we obtain the theorem 2.3. here we use the following crucial argument: the assumption (a3) implies that the interval i is a non-trapping region of the classical hamiltonian associated to bs for all s in the compact set ω∩r. in fact, r·ki(k,r) ∈ sδ(t∗×rn) ⊂ s0(t∗×rn), (i = 1,2) then the corresponding operators are bounded uniformly for s ∈ ω ∩ r and moreover, the principal symbol of bs does not depend on s. 3 proofs 3.1 definitions and notations let h be a hilbert space. the scalar product in h will be denoted by 〈·, ·〉. the set of linear bounded operators from h1 to h2 is denoted by l(h1,h2). for (m,n) ∈ r × n we denote by sm(t∗ × rn; mn(c)) the space of p ∈ c∞(r2nk,r; mn(c)), γ∗-periodic with respect to k, such that for all α and β in nn there exists cα,β > 0 such that ‖∂αr ∂ β k p(k,r)‖mn(c) ≤ cα,β〈r〉 −m−|α|, 〈r〉 = ( 1 + |r|2 )1 2 , (3.1) where mn(c) is the set of n×n-matrices. in the special case when n = 1 (i.e., p is real valued), we will write sm(t∗ × rn) instead of sm(t∗ × rn; m1(c)). 36 m. dimassi and m. zerzeri cubo 14, 1 (2012) if p depends on a semi-classical parameter h ∈]0,h0] and possibly on other parameters as well, we require (3.1) to hold uniformly with respect to these parameters. for h dependent symbols, we say that p(k,r;h) has an asymptotic expansion in powers of h, and we write p(k,r;h) ∼ ∞∑ j=0 pj(k,r)h j, if for every n ∈ n, h−(n+1) ( p − n∑ j=0 pjh j ) ∈ sm ( t∗ × rn; mn(c) ) . for p ∈ sm(t∗ × rn; mn(c)), the h-weyl operator p = pw(k,hdk;h) = opwh (p) is defined by: pw(k,hdk;h)u(k) = (2πh) −n ∫ ∫ e i h (k−y)rp( k + y 2 ,r;h)u(y)dydr. here dk = 1 i ∂ ∂k . 3.2 effective hamiltonian in this subsection, we recall the effective hamiltonian method. more precisely, we will construct a suitable auxiliary (so-called grushin) problem associated with the operator ( p(h) − z ) for z in a small complex neighborhood of i, where i = [a,b] ⊂ r is some bounded interval. the reader can find more details and the proofs of the results of this subsection in [20] (see also [11, 12, 15]). for the reader convenience, let us point out the main change in our situation and fix the notations. denote by tγ the distribution in s ′(r2n) defined by tγ (x,y) = 1 vol(e)hn ∑ β∗∈γ∗ ei(x−hy) β∗ h . we recall that e is a fundamental domain of γ. for m ∈ n, put lm := {u(x)tγ (x,y); ∂αxu ∈ l 2(rn), ∀α, |α| ≤ m}. it was shown in [11, chapter 13, proposition 13.5], that the operator p(h) acting on l2(rn) with domain h2(rn) is unitary equivalent to p1(h) := ( dy + hdx )2 + v(y) + ϕ(x), (3.2) acting on l0 with domain l2, and the following proposition holds. proposition 3.1. assume (a1). there exist n ∈ n, a complex neighborhood ω of i, and a bounded operator r+ in l ( l0;l2(t∗; cn) ) such that for all z ∈ ω and 0 < h < h0 small enough, the operator p1(z,h) := ( p1(h) − z r∗+ r+ 0 ) : l2 × l2(t∗; cn) → l0 × l2(t∗; cn), (3.3) is bijective with bounded two-sided inverse e1(z,h) := ( e1(z,h) e1,+(z,h) e1,−(z,h) e1,−+(z,h) ) . (3.4) cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 37 here e1,−+ := e w 1,−+(k,−hdk;z,h) is an h−pseudodifferential operator with symbol e1,−+(k,r;z,h) ∼ ∑ l≥0 el1,−+(k,r;z)h l, ∀0 < h < h0, (3.5) in s0 ( t∗ × rn; l(cn,cn) ) . remark 3.2. (1) we denote by p0(z,h) and e0(z,h) := ( e0(z,h) e0,+(z,h) e0,−(z,h) e0,−+(z,h) ) the operators given by proposition 3.1 when ϕ = 0. (2) note that, r+ depends only on the non-perturbed periodic schrödinger operator p0. see [15, proposition 2.1] and [11, chapter 13]. therefore, we may take the same r+ for p1(z,h) and p0(z,h). the following well-known formulas are a consequence of proposition 3.1 (see also [20]), for j = 0,1. ( pj(h) − z )−1 = ej(z,h) − ej,+(z,h)ej,−+(z,h) −1ej,−(z,h), (3.6) ej,−+(z,h) −1 = −r+ ( pj(h) − z )−1 r∗+ , (3.7) and ∂zej,−+(z,h) = ej,−(z,h)ej,+(z,h). (3.8) here p0(h) := ( dy + hdx )2 + v(y). we observe that pj(z,h)∗ = pj(z,h), which implies that ej(z,h)∗ = ej(z,h). from this, we deduce the following identity: ej,−+(z,h) ∗ = ej,−+(z,h), j = 0,1. (3.9) in the following, we write [aj] 1 j=0 = a1 − a0. lemma 3.3. we have [ ej,+(z,h) ]1 j=0 = e1(z,h)ϕ(r)e0,+(z,h), (3.10) [ ej,−(z,h) ]1 j=0 = e0,−(z,h)ϕ(r)e1(z,h), (3.11) and [ ej,−+(z,h) ]1 j=0 = e1,−(z,h)ϕ(r)e0,+(z,h). (3.12) in particular, if (a1) is satisfied then [ ej,−+ ( k,r;z,h )]1 j=0 ∈ sδ ( t∗k × r n r ; mn(c n) ) . (3.13) 38 m. dimassi and m. zerzeri cubo 14, 1 (2012) proof. identities (3.10)-(3.12) follow from the first resolvent equation[ ej(z,h) ]1 j=0 = e1(z,h) [ p0(z,h) − p1(z,h) ] e0(z,h) = −e0(z,h) [ p1(z,h) − p0(z,h) ] e1(z,h) and the fact that [ pj(z,h) ]1 j=0 = ( ϕ(r) 0 0 0 ) . formula (3.13) is a simple consequence of (3.12) and standard h-pseudodifferential calculus. lemma 3.4. assume (a1) with δ > n , the operator ϕ(r)e0,+(z,h) : l 2(t∗; cn) → l2(rn), (3.14) and e0,−(z,h)ϕ(r) : l 2(rn) → l2(t∗; cn), (3.15) are of trace class. proof. since ( e0,−(z,h)ϕ(r) )∗ = ϕ(r)e0,+(z,h) it suffice to prove 3.14. without any loss of generality, we may assume that n = 1. consider the operator a = ( id − h2∆t∗ )− δ 2 on l2(t∗; c). set b = ϕ(r)e0,+(z,h), c = b∗b and d = a−1ca−1. since ϕ ∈ sδ(r2n) and e0,+(k,r;z,h) ∈ s0, a standard result of h-pseudodifferential calculus shows that d ∈ s0(t∗k × r n r ). therefore, d extends to a bounded operator from l 2(t∗; c) into l2(t∗; c), see [11, chapter 13]. combining this with the fact that c is positif, we get: 0 ≤ c = ada ≤ ‖d‖a2, which implies 0 ≤ c 1 2 ≤ √ ‖d‖a. since δ > n then a : l2(t∗; c) → l2(t∗; c) is of trace class and the lemma follows from the above inequality. remark 3.5. notice that if p ∈ sδ(t∗ × rn; mn(c)) with δ > n, then the operator pw(k,hdk) is a trace class, see [11]. proposition 3.6. assume (a1) with δ > n . for z ∈ ω such that =(z) 6= 0, the operator [ ej,+(z,h)ej,−+(z,h) −1ej,−(z,h) ]1 j=0 cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 39 is of trace class from l2(rn) to l2(rn) and tr ([ ej,+(z,h)ej,−+(z,h) −1ej,−(z,h) ]1 j=0 ) = tr ([ ej,−+(z,h) −1∂zej,−+(z,h) ]1 j=0 ) . (3.16) here the operator in the right member of (3.16) is defined on l2(t∗; cn). proof. let z ∈ ω such that =(z) 6= 0, we have the following identity: [ ej,+(z,h)ej,−+(z,h) −1ej,−(z,h) ]1 j=0 = (3.17)[( [ej,+(z,h)] 1 j=0 ) e1,−+(z,h) −1e1,−(z,h) ] +[ e0,+(z,h)e0,−+(z,h) −1 ( [ej,−(z,h)] 1 j=0 )] −[ e0,+(z,h)e1,−+(z,h) −1 ( [ej,−+(z,h)] 1 j=0 ) e0,−+(z,h) −1e1,−(z,h) ] . according to lemmas 3.3 and 3.4, all the term of the right member in the last equality are of trace class. using the cyclicity of the trace and identity (3.8), we obtain the proposition. using again the cyclicity of the trace in (3.17) and the identity (3.8) we obtain tr ([ ej,−+(z,h) −1∂zej,−+(z,h) ]1 j=0 ) = tr ([ ∂zej,−+(z,h)ej,−+(z,h) −1 ]1 j=0 ) . (3.18) the main result in this subsection is proposition 3.7. assume (a1) with δ > n . let ψ ∈ c∞0 (r) and let ψ̃ be an almost analytic extension of ψ. then the operator [ψ(p(h)) − ψ(p0)] is of trace class as an operator from l 2(rn) to l2(rn) and tr[ψ(p(h)) − ψ(p0)] = tr[ψ(p1(h)) − ψ(p0(h))] = (3.19) − 1 π ∫ c ∂ψ̃(z)tr ([ ej,−+(z) −1∂zej,−+ ]1 j=0 ) l(dz). here ∂ = ∂ ∂z and l(dz) = dxdy denotes the lebesgue measure on c. recall that ψ̃ ∈ c∞0 (c) is an almost analytic extension of ψ, i.e. ψ̃|r = ψ and ∂ψ̃ = o(|=(z)|n) for all n ∈ n. we refer to [11] for the existence of ψ̃. proof. by helffer-sjöstrand formula (see [19]), we have ψ(p1(h)) − ψ(p0(h)) = − 1 π ∫ c ∂ψ̃(z) [ (z − p1(h))−1 − (z − p0(h))−1 ] l(dz). 40 m. dimassi and m. zerzeri cubo 14, 1 (2012) combining this with (3.6), we obtain ψ(p1(h)) − ψ(p0(h)) = 1 π ∫ c ∂ψ̃(z) [ ej(z,h)] 1 j=0 l(dz) (3.20) − 1 π ∫ c ∂ψ̃(z) [ ej,+(z,h)ej,−+(z,h) −1ej,−(z,h) ]1 j=0 l(dz). since ej(z,h), j = 0,1 is holomorphic in a neighborhood of supp(ψ̃), the first term in the right member of (3.20) vanishes. consequently, ψ(p1(h)) − ψ(p0(h)) = − 1 π ∫ c ∂ψ̃(z) [ ej,+(z,h)ej,−+(z,h) −1ej,−(z,h) ]1 j=0 l(dz). using proposition 3.6, we conclude that [ψ(p1(h))−ψ(p0(h))] is of trace class and applying (3.16), we obtain the second equality of (3.19). the first equality follows from the fact that p1(h) (resp. p0(h)) is unitary equivalent to p(h) (resp. p0). now, we recall a representation of the derivative of the spectral shift function in term of the effective hamiltonian ej,−+(z,h) (see [10, lemma 1]). let i ⊂ r be some bounded interval and ω be the complex neighborhood of i given by the proposition 3.1. put ω± = ω ∩ {z ∈ c; ±=(z) > 0}. we introduce the functions e±(z) = tr ([ ej,−+(z,h) −1∂zej,−+(z,h) ]1 j=0 ) , for z ∈ ω±. since ej,−+(z,h) is holomorphic on z, we deduce from (3.9) that ∂zej,−+(z,h) ∗ = ∂zej,−+(z,h). using the fact that tr(a) = tr(a∗) with (3.9), (3.18) and the above equality we obtain e+(z) = e−(z), for all z ∈ ω+. (3.21) lemma 3.8. [10, lemma 1] assume (a1) with δ > n . in d ′(i), we have ζ′h(µ) = lim �↘0 1 2πi [ e+(µ + i�) − e+(µ + i�) ] . (3.22) for the reader convenience we give the proof of the lemma. proof. let ψ ∈ c∞0 (i). in the previous proposition, we proved that − 〈ζ′h(·),ψ〉 = tr[ψ(p(h)) − ψ(p0)] = − 1 π ∫ c ∂ψ̃(z)tr ([ ej,−+(z) −1∂zej,−+ ]1 j=0 ) l(dz). since e±(z) = o ( h−n|=(z)|−2 ) and ∂ψ̃(z) = o ( |=(z)|2 ) , the integral in the identity above converge. thus the r.h.s. of the previous equality can be written as − 〈ζ′h(·),ψ〉 = lim �↘0 − 1 π [∫ =(z)>0 ∂ψ̃(z)e+(z + i�)l(dz) + ∫ =(z)<0 ∂ψ̃(z)e−(z − i�)l(dz) ] . cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 41 since e±(z ± i�) is holomorphic in ω±, green’s formula then gives − 〈ζ′h(·),ψ〉 = lim �↘0 i 2π ∫ r ψ(µ) [ e+(µ + i�) − e−(µ − i�) ] dµ. (3.23) thus lemma 3.8 follows from (3.21) and (3.23). in the following we will use the result concerning the expression of the effective hamiltonian ej,−+(z,h), j = 0,1 given in [7] (see also [10]). in fact, under the assumptions (a1)-(a2), the two leading terms of the symbol e1,−+(k,r;z,h) are computed in [7, section 4, formulas (4.5)-(4.7)], it was shown that: e1,−+(k,r;z,h) = z − ( λ(k) + ϕ(r) + hk1(k,r) + h 2k2(k,r;z,h) ) , (3.24) where k1 ∈ sδ+1(t∗ × rn) and k2(·;z,h) ∈ sδ+2(t∗ × rn), holomorphic with respect to z in ω. note that e0,−+(k,r;z) = z − λ(k), k ∈ t∗, z ∈ ω. (3.25) from now on, we consider the h-pseudodifferential operator hw(k,−hdk;h) with the following symbol: h(k,r;h) = λ(k) + ϕ(r) + hk1(k,r). (3.26) remark that this operator is z-independent. corollary 3.9. under assumptions (a1) with δ > n and (a2), there exists g(k,r;z,h) ∼∞∑ j=0 gj(k,r;z)h j in sδ+2(t∗ × rn) such that for µ ∈ i and h small enough, we have: ζ′h(µ) = lim �↘0 1 2πi [ tr ( (z − bµ) −1 − ( z − λ(k) )−1)]z=µ+i� z=µ−i� , (3.27) where bµ := h w(k,−hdk;h) + h 2gw(k,−hdk,µ;h). here h(k,r;h) is given by 3.26. proof. identity (3.24) gives ∂ze1,−+(k,r;z,h) = 1+h 2∂zk2(k,r;z,h) and since ∂zk2 ∈ sδ+2(t∗ × rn) ⊂ s0(t∗ × rn) it follows from the calderon-vaillancourt’s theorem and the beal’s characterization (see [11]) that the corresponding operator ∂ze1,−+(z,h) is invertible for h small enough and his inverse is given by [ ∂ze1,−+(z) ]−1 = i + h2r(z), where r(z) is an h-pseudodifferential operator with symbol satisfying the same properties as k2. combining this with (3.24) and using the composition formula of h-pseudodifferential operators we see that there exists g ∼ ∑∞ j=0 gj(k,r;z)h j in sδ+2(t∗ × rn) such that( ∂ze1,−+(z,h) )−1 e1,−+(z,h) = z − h w(k,−hdk;h) + h 2gw(k,−hdk;z,h), which together with (3.25), lemma 3.8 and the holomorphy of g on z give the corollary. 42 m. dimassi and m. zerzeri cubo 14, 1 (2012) 3.3 proof of the weak asymptotic expansion of ξ′h(·) let i =]a,b[∈ r and f ∈ c∞0 (i). the proof of theorem 2.2 is a simple consequence of proposition 3.7 (with ψ = f) and symbolic calculus. here, we only give an outline of the proof. for the details, we refer to [7]. fix � in ]0, 1 2 [. the integral (3.19) over {z ∈ c; |=z| ≤ h�} is o(h∞), since ∂f̃(z) = o(|=z|∞) and ∥∥∥[ej,−+(z)−1∂zej,−+(z)]1 j=0 ∥∥∥ tr = o ( |=z|−1 ) . on the other hand, [ ej,−+(z,h) −1∂zej,−+(z,h) ]1 j=0 has an asymptotic expansion in powers of h uniformly for z in {z ∈ suppf̃; |=(z)| ≥ h�} (see [7]). therefore, as in [11, theorem 13.28], we have − 1 π ∫ c ∂f̃(z)tr ([ ej,−+(z) −1∂zej,−+(z) ]1 j=0 ) l(dz) ∼ h−n ∑ j≥0 ajh j, (h ↘ 0) with a0 = (2π) −n ∑ p≥1 ∫ rnx (∫ e∗ [ f ( λp(k) + ϕ(x) ) − f ( λp(k) )] dk ) dx. note that, the sum in the last equality is finite, since lim p→+∞ λp(k) = +∞ and ϕ is bounded. the coefficient aj is a finite sum of term of the form ∫ ∫ cl(x,k)f (l) ( b(x,k) ) dxdk, where cl depends on ϕ and their derivatives and b(x,k) ∈ { λp(k),λp(k) + ϕ(x) } , which complete the first part of the theorem 2.2. see [11, chapter 8, identity (8.16)]. if µ is a non-critical energy of p0 for all µ ∈ i. then d ( λp(k) ) 6= 0 and d ( λp(k) + ϕ(x) ) 6= 0 for all k ∈ f(µ). we recall that f(µ) is the fermi surface. therefore, aj(f) = −〈γj(·),f〉, for all f ∈ c∞0 (i) and γj(µ) are smooth functions of µ ∈ i, in particular, γ0(µ) = ddµ [∫ rnx ρ ( µ ) − ρ ( µ − ϕ(x) ) dx ] , which complete the proof of the theorem 2.2. � 3.4 limiting absorption theorem in this subsection we establish a limiting absorption principle for the operator p(h), see theorem 3.11 below. we start by the following lemma, let [a,b] ∈ r and ω given by proposition 3.1. lemma 3.10. assume that (a1), (a2) and (a3) are satisfied on [a,b]. then, for l ∈ n∗, there exists h0(l) > 0 small enough such that for all 0 < h < h0(l):∥∥ < hdk >−α ej,−+(k,−hdk,µ ± i0;h)−l < hdk >−α ∥∥ = o(h−l), (j = 0,1), (3.28) for all α > l− 1 2 and uniformly for µ ∈ [a,b], where ‖·‖ denotes the operator norm when considered as an operator from l2(t∗k) into itself. proof. recall that ω is a small complex neighborhood of [a,b] such that the proposition 3.1 holds. for s ∈ ω∩ r, we consider the h-pseudodifferential operator p̃s(k,−hdk;h) := hw(k,−hdk;h) + cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 43 h2kw2 ((k,−hdk;s,h) with the following symbol: p̃s(k,r;h) = h(k,r;h) + h 2k2(k,r;s,h), where h(k,r;h) given by (3.26) and k2(·;z,h) ∈ sδ+2(t∗ × rn), holomorphic with respect to z in ω. put a = opwh ( − ∇λ(k) · r ) . since k2 ∈ sδ+2(t∗ × rn) ⊂ s0(t∗ × rn), it follows from the h-pseudodifferential calculus and the calderon-vaillancourt’s theorem that[ p̃s,a ] = opwh (∣∣∇kλ(k)∣∣2 − ( − r∇rϕ(−r)) · ∆λ(k)) + o(h), in l(l2(t∗)), uniformly for s ∈ ω ∩ r. here [·, ·] denotes the commutator. the assumption (a3) and the gärding inequality (see [11, chapter 8]) imply that for f ∈ c∞0 (ω ∩ r) there exists c > 0 such that f(p̃s) [ p̃s,a ] f(p̃s) ≥ cf(p̃s)2 for h small enough and uniformly for s ∈ ω ∩ r. now applying [14, theorem 1], we get, for all l ∈ n∗,∥∥ < hdk >−α (µ ± i0 − p̃s)−l < hdk >−α ∥∥ = o(h−l), for all α > l − 1 2 , (3.29) uniformly for µ ∈ [a,b] and s ∈ ω ∩ r. take s = µ in (3.29) we obtain the estimation (3.28) for j = 1. theorem 3.11 (limiting absorption theorem). with the same assumptions as lemma 3.10. one has, for l ∈ n∗, there exists h0(l) > 0 small enough such that for all 0 < h < h0(l):∥∥ < hx >−α (p(h) − µ ± i0)−l < hx >−α ∥∥ l ( l2(rnx ) ) = o(h−l), (3.30) for all α > l − 1 2 and uniformly for µ ∈ [a,b]. proof. recall that the operator p(h) acting on l2(rn) with domain h2(rn) is unitary equivalent to p1(h) := ( dy + hdx )2 + v(y) + ϕ(x), acting on l0 with domain l2, where lm := {u(x)tγ (x,y); ∂αxu ∈ l 2(rn), ∀α, |α| ≤ m} for m ∈ n and tγ is a distribution in s ′(r2n) defined by tγ (x,y) = 1 vol(e)hn ∑ β∗∈γ∗ ei(x−hy) β∗ h . here e is a fundamental domain of γ. then we will prove (3.30) for p1(h). it follows from identity (3.6) that: 〈hy〉−α ( p1(h) − z )−1〈hy〉−α = 〈hy〉−αe1(z,h)〈hy〉−α − 〈hy〉−αe1,+(z,h)〈hdk〉α· · 〈hdk〉−αe1,−+(z,h)−1〈hdk〉−α · 〈hdk〉αe1,−(z,h)〈hy〉−α. 44 m. dimassi and m. zerzeri cubo 14, 1 (2012) since e1(z,h) is holomorphic then the first term of the r.h.s is bounded. on the other hand, as in proposition 3.1, we prove that ( 〈hy〉−α ( p1(h) − z ) 〈hy〉α 〈hy〉−αr∗+〈hdk〉α 〈hdk〉−αr+〈hy〉α 0 ) = ( 〈hy〉−α 0 0 〈hdk〉−α )( p1(h) − z r∗+ r+ 0 )( 〈hy〉α 0 0 〈hdk〉α ) is well-defined as a bounded operator from l2 × l2(t∗; cn) to l0 × l2(t∗; cn) and is bijective with bounded two-sided inverse given by: ( 〈hy〉−αe1(z,h)〈hy〉α 〈hy〉−αe1,+(z,h)〈hdk〉α 〈hdk〉−αe1,−(z,h)〈hy〉α 〈hdk〉−αe1,−+(z,h)〈hdk〉α ) = ( 〈hy〉−α 0 0 〈hdk〉−α )( e1(z,h) e1,+(z,h) e1,−(z,h) e1,−+(z,h) )( 〈hy〉α 0 0 〈hdk〉α ) . then 〈hy〉−αe1,+(z,h)〈hdk〉α and 〈hdk〉−αe1,−(z,h)〈hy〉α are well-defined and bounded. therefore combining this with lemma 3.10 we obtain theorem 3.11 for l = 1. with the same arguments we obtain the result for l ≥ 2. remark 3.12. a simple consequence of theorem 3.11 is that p(h) has no embedded eigenvalues in [a,b]. 3.5 proof of the pointwise asymptotic expansion of ξ′h(·) recall that ω is a small complex neighborhood of [a,b] such that the proposition 3.1 holds. for s ∈ ω ∩ r, we consider the h-pseudodifferential operator bs(k,−hdk;h) := hw(k,−hdk;h) + h2gw(k,−hdk;s,h) with the following symbol: bs(k,r;h) = h(k,r;h) + h 2g(k,r;s,h), where h(k,r;h) given by (3.26) and g(·;z,h) ∈ sδ+2(t∗ × rn), holomorphic with respect to z in ω (see corollary 3.9). clearly, the principal symbol of bs (i.e. b 0 s = λ(k) + ϕ(r)) is independent on s. moreover, the assumption (a3) implies that [a,b] is a non-trapping region for the classical hamiltonian bs. now, by constructing a long-time parametrix for the time-dependent equation ( hdt − bs ) u(t) = 0, u(0) = i, cubo 14, 1 (2012) ssf for perturbed periodic schrödinger operators 45 we can apply the robert-tamura method [33, 34, 35] (see also [31, remark 6.1]) to prove that[ tr ( (z − bs) −1 − (z − λ(k))−1 )]z=µ+i0 z=µ−i0 , has a complete asymptotic expansion in powers of h uniformly for µ ∈ [a,b] and s ∈ ω ∩ r. remembering (3.27) and take s = µ ∈ [a,b] ⊂ ω we obtain (2.8). received: february 2011. revised: march 2011. references [1] s. alama, p-a. deift, and r. hempel, eigenvalue branches of the schrödinger operator, comm. math. phys. 121 (1989), no. 2, 291–321. 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[42] d. yafaev, scattering theory: some old and new problems, scattering theory: some old and new problems, lecture notes in math., vol. 1737, springer, berlin, 2000, pp. xvi+169 pp. c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2603 .dvi cubo a mathematical journal vol.13, no¯ 01, (25–43). march 2011 evolutionary method of construction of solutions of polynomials and related generalized dynamics robert m. yamaleev facultad de estudios superiores, universidad nacional autonoma de mexico, cuautitlán izcalli, campo 1, c.p.54740, méxico. joint institute for nuclear research, lit, dubna, russia. email: iamaleev@servidor.unam.mx abstract invariant theory as a study of properties of polynomials under translational transformations is developed. class of polynomials with congruent set of eigenvalues is introduced. evolution equations for eigenvalues and coefficients remaining the polynomial within proper class of polynomials are formulated. the connection with equations for hyper-elliptic weierstrass and hyper-elliptic jacobian functions is found. algorithm of calculation of eigenvalues of the polynomials based on the evolution process is elaborated. elements of the generalized dynamics with n-order characteristic polynomials are built. resumen la teoŕıa de invariantes es un estudio de las propiedades de los polinomios que se desarrolla en las transformaciones de traslación. se introduce una clase de polinomios congruentes con un conjunto de valores propios. se formulan ecuaciones de evolución de los valores propios y los coeficientes del polinomio restante dentro de la clase adecuada 26 robert m. yamaleev cubo 13, 1 (2011) de los polinomios. se encuentra la conexión con las ecuaciones de weierstrass hipereĺıpticas y funciones jacobiano hiper-eĺıptica. son elaborados algoritmos de cálculo de valores propios de los polinomios basado en el proceso de evolución. keywords: nonspreading mapping, maximal monotone operator, inverse strongly-monotone mapping, fixed point, iteration procedure mathematics subject classification: 12yxx 1 introduction the problem of construction of solutions of the polynomial equations as a certain functions of the coefficients is one of the oldest mathematical problems. e.galois and h.abel had proved that the polynomial higher than fourth order, in general, does not admit a presentation of solutions by radicals [3]. this rigorous mathematical theory directed mathematicians to look for the other ways of solutions. in particularly, the eigenvalues of the polynomials can be expressed in analytical way as certain functions of the coefficients [2], [7]. ch.hermite was first who found an elegant expression of eigenvalues of the quintic equation by modular functions [6]. the theory of elliptic functions originally was related with the problem of finding of eigenvalues of the cubic polynomial. in fact, the eigenvalues of the cubic equation admit presentation by weierstrass elliptic functions at the periods [12]. it is clear, however, that for a search of analytical solutions of the n > 5degree polynomials one needs of mathematical tools beyond the elliptic functions. in this context as hopeful tools one may consider the theories of hyper-elliptic functions [9] and multi-complex algebras [8], [13]. the main purpose of the present paper is to construct an algorithm for calculation roots of the polynomials. for that purpose, firstly, we study invariant properties of the polynomials with respect to simultaneous translations of the roots. secondly, we build the system of evolution equations for the coefficients transforming the given polynomial into the polynomial with one trivial solution. the evolution process is directed in a such way that remain the original polynomial within proper class where the roots of the polynomials form congruent set of values. the coefficients of the polynomial with respect to the parameter of evolution are solutions of the cauchy problem for ordinary differential equations. as soon as the cauchy problem is resolved, the eigenvalues of initial polynomial are found simply by simultaneous translations of the set of eigenvalues of the final polynomial. since the final polynomial possesses with one trivial solution, the degree of the polynomial is reduced from n to (n − 1) which simplifies the process of solution. if solution of the obtained polynomial still is a difficult problem, then the method can be applied again in order to reduce the degree of the obtained polynomial. this process can be continued up till linear equation. the inverse process is fulfilled simply by simultaneous translations of the roots from the solution of the linear equation up till solutions of n-degree polynomial. it is shown, the evolution equations for the coefficients can be identified with equations for weierstrass-type cubo 13, 1 (2011) evolutionary method of construction of solutions of polynomials and related generalized dynamics 27 hyper-elliptic functions, whereas the evolution equations for the roots are given by equations for jacobian-type hyper-elliptic functions. furthermore, it is demonstrated a link between evolution equations for the coefficients of the polynomials and the classical dynamic equations. in n-order generalized mechanics the innerand outer-momenta are inter-related as roots and coefficients of the characteristic n-degree polynomial. in particular n = 2 case, we return to well-known equations of the relativistic dynamics closely related with quadratic polynomial of the mass-shell equation. besides the introduction the paper contains the following sections. in section 2, the equations of evolution for the coefficients of n-degree polynomial are formulated. in section 3, the algorithm of finding of eigenvalues of the n-degree polynomial is built. in section 4, some peculiarities of the cubic equation is explored. in section 5, the elements of the relativistic dynamics based on quadratic characteristic polynomial is presented. in section 6 elements of the generalized dynamics related with n-degree characteristic polynomial is outlined. 2 evolution equations for eigenvalues and coefficients of ndegree polynomial if f is a field and q1, ..., qn are algebraically independent over f , the polynomial p(x) = n∏ i=1 (x − qi) is referred to as generic polynomial over f of degree n. the polynomial equation of n-degree over field f is written in the form p(x) := xn + n−1∑ k=1 (−)k(n − k + 1)pkxn−k + (−)np 2 = 0, (2.1) where the coefficients pk ∈ f (q1, ..., qn). in this paper we shall restrict our attention only to polynomials with real coefficients and with simple roots. the signs at the coefficients in eq.(2.1) are changed from term to term which allows in vièta formulae to keep only the positive signs. the expressions at the coefficients are included for a convenience and have no real bearing on the theory. the mapping from the set of eigenvalues onto the set of coefficients is given by vièta formulae (a) np1 = n∑ i=1 qi, (b) p 2 = n∏ i=1 qi, (c) pk = ∑ 1≤r1<... c, we suppose that there exist a strictly positive integer m such that sup x∈rd |dkxd k ′ ξ a(x, ξ)| ≤ c|ξ| 2m−k ′ (2) we suppose that the symbol is elliptic: inf x∈rd |a(x, ξ)| ≥ c|ξ|2m (3) we put by standard theory on pseudo-differential operators ([3], [4], [5]) l̂f(x) = ∫ rd a(x, ξ)f̂(ξ)dξ (4) cubo 15, 1 (2013) a girsanov formula associated to a big order pseudo-differential ... 115 where ξ → f̂(ξ) is the fourier transform of x → f(x).he can be extended continuously on the space of smooth functions with bounded derivatives at each order. we suppose because later we consider girsanov type formulas that l1 = 0. hypothesis: we suppose that −l is positive essentially self-adjoint on l2(rd). l generates a contraction semigroup exp[tl] on l2(rd). by elliptic theory, exp[tl]f(x) = ∫ rd f(y)µt(x, dy) (5) where µt(x, dy) is a measure on r d (but not a probability measure). we consider an operator l1 on l2(rd) and we suppose that it is a pseudodifferential operator of order strictly smaller than 2m − 1 of the type (2) and (4). he can be extended continuously on the space of smooth functions with bounded derivatives at each order. we suppose because later we consider girsanov type formulas that l11 = 0. we consider the pseudo-differential operator densely defined on l2(rd × r) − ltot = −l − l1 ∂ ∂u + (−1)m ∂2m ∂u2m (6) by elliptic theory, it generates a semi group exp[tltot]on l2(rd × r) (but not a contraction semigroup due to the perturbation term l1 ∂ ∂u in the total operator ltot). the main remark is that if f depends only on u l1 ∂ ∂u f = 0! by elliptic theory exp[tltot]f(x, u) = ∫ rd×r f(y, v)µtott (x, u, dy, dv) (7) where µtot is a measure on rd × r (but not a probability measure). we consider the operator densely defined on l2(rd) − lper = −l − l1 (8) by elliptic theory, it generates a semi-group on l2(rd) (but not a contraction semi-group due to the perturbation term l1). by elliptic theory, it generates a semi-group on l2(rd) (but not a contraction semi-group due to the perturbation term l1). by elliptic theory, exp[tlper]f(x) = ∫ rd f(y)µ per t (x, dy) (9) where µ per t (x, dy) is a measure on r d (but not a probability measure). we get theorem 2.1. (girsanov): we have if f is continuous with compact support and if we consider the doleans-dade exponential exp[u + (−1)mt] = g(u, t) exp[tlper]f(x) = exp[tltot][f(.)g(., t)](x, 0) (10) 116 rémi léandre cubo 15, 1 (2013) proof:let us begin by doing formal computations. ∂ ∂u commute with ltot. therefore ltot exp[tltot][f(.)g(., t)](x, u) = lexp[tltot][f(.)g(., t)](x, u)+ l1exp[tltot][f(.) ∂ ∂v g(., t)](x, u) + exp[tltot][f(.)(−1)m+1 ∂2m ∂v2m g(., t)](x, u) = a1 + a2 + a3 (11) the term a3 is boring. this explain that we introduce exp[(−1) mt] in the doleans-dade exponential in order to remove it. namely we consider linear semi-groups such that exp[tltot][f(.)g(., t)](x, u) = exp[tltot][f(.) exp[.]](x, 0) exp[(−1)mt] (12) therefore a3 disappears and ∂ ∂t exp[tltot][f(.)g(., t)](x, 0) = lper exp[tltot][f(.)g(., t)](x, 0) (13) the only problem in this formal comutation is that u → exp[u] is not bounded!. but if f is with compact support continuous | exp[tltot][f(.) exp[.]](x, 0)| ≤ ∫ rd×r |f(y)| exp[v]|µtott |(x, u, dy, dv) ≤ ( ∫ rd |f(y)|2|µt|(x, dy)) 1/2( ∫ r exp[2u]|νt|(0, dv)) 1/2 (14) in (14), νt(u, dv) represents the semi group associated to l2m = (−1) m+1 ∂ 2m ∂u2m . by [1], this semi-group has an heat-kernel bounded by ct−1/4mg2m,a( |u−v| t1/4m ) (a > 0) where g2m,a(u) = exp[−au 2m/2m−1] (15) this inequality justifies the formal considerations above! ♦ received: november 2012. revised: february 2013. references [1] davies b.: unifomly elliptic operators with measurable coefficients. j. funct. ana. 132, 141– 169. 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() cubo a mathematical journal vol.13, no¯ 02, (139–149). june 2011 fractional order differential inclusions via the topological transversality method mouffak benchohra1 and naima hamidi2 laboratoire de mathématiques, université de sidi bel abbès, b.p. 89, sidi bel-abbès, 22000. email: benchohra@univ-sba.dz email: hamidi.naima@yahoo.fr abstract the aim of this paper is to present new results on the existence of solutions for a class of boundary value problem for differential inclusions involving the caputo fractional derivative. our approach is based on the topological transversality method. resumen el objetivo de este trabajo es presentar nuevos resultados sobre la existencia de soluciones para una clase de problemas de contorno para inclusiones diferenciales derivados de la participación de caputo fraccionada. nuestro enfoque se basa en el método de la transversalidad topológica. keywords and phrases: fractional differential inclusions; fixed point, caputo fractional derivative, existence, topological transversality theorem. mathematics subject classification: 26a33, 26a42, 34a60, 34b15. 140 mouffak benchohra and naima hamidi cubo 13, 2 (2011) 1 introduction this paper deals with the existence of solutions for boundary value problems (bvp for short) for fractional order differential inclusion of the form c d α y(t) ∈ f(t,y(t)), t ∈ j := [0,t], 1 < α ≤ 2, (1.1) y(0) = y0, y(t) = yt (1.2) where cdα is the caputo fractional derivative, f : j× irn → p(irn) is a carathéodory multifunction, y0,yt ∈ ir n. here p(irn) denotes the family of all nonempty subsets of p(irn). differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [11, 16, 20, 24, 27, 28]). there has been a significant development in fractional differential equations and inclusions in recent years; see the monographs of kilbas et al [21], lakshmikantham et al. [22], miller and ross [25], podlubny [28], samko et al [29] and the survey by agarwal et al. [1], benchohra et al. [5, 6, 7], chang and nieto [10], diethelm et al [11, 12], ouahab [26], yu and gao [30] and zhang [31] and the references therein. very recently, in [4, 8] the authors studied the existence and uniqueness of solutions of some classes of functional differential equations with infinite delay and fractional order, and in [3] a class of perturbed functional differential equations involving the caputo fractional derivative has been considered. these papers have relied on different methods such as banach fixed point theorem, schaefer’s theorem, leray-schauder nonlinear alternative. in this paper we use a powerful method due to granas [17] to prove the existence of solution to bvp (1.1)-(1.2). granas’ method is commonly known as topological transversality and relies on the idea of an essential map. the method has been highly useful for proving existence of solutions for initial and boundary value problem for integer order differential equations, see for example [9, 14, 18, 19]. this paper is organized as follows: in section 2 we introduce some backgrounds on fractional calculus and the topological transversality theorem. in section 3 we present our main results and an illustrative example will be presented in section 4. this paper initiates the application of the topological transversality method to boundary value problems for fractional order differential inclusions. 2 preliminaries we now introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. we denote by ‖y‖ the norm of any element y ∈ irn. cubo 13, 2 (2011) fractional order differential inclusions via the topological . . . 141 c(j,ir n ) is the banach space of all continuous functions from j into irn with the usual norm |y|∞ = sup {|y(t)| : 0 ≤ t ≤ t} . ac1(j,ir n ) is the space of differentiable functions y : j → irn, whose first derivative, y′ is absolutely continuous. l1(j,ir n ) denote the banach space of functions y : j −→ irn that are lebesgue integrable with the norm ‖y‖l1 = ∫ t 0 ‖y(t)‖dt. 2.1 some properties of fractional calculus definition 1. ([21, 28]). given an interval [a,b] of ir. the fractional (arbitrary) order integral of the function h ∈ l1([a,b], r) of order α ∈ r+ is defined by i α ah(t) = ∫ t a (t − s)α−1 γ(α) h(s)ds, where γ is the gamma function. when a = 0, we write iαh(t) = [h ∗ ϕα](t), where ϕα(t) = tα−1 γ(α) for t > 0, and ϕα(t) = 0 for t ≤ 0, and ϕα → δ(t) as α → 0, where δ is the delta function. definition 2. ([21]). for a given function h on the interval [a,b], the caputo fractional-order derivative of h, is defined by (cdαa+h)(t) = 1 γ(n − α) ∫ t a (t − s)n−α−1h(n)(s)ds, where n = [α] + 1. lemma 3. (lemma 2.22 [21]). let α > 0, then the differential equation c d α h(t) = 0 has solutions h(t) = c0 + c1t + c2t 2 + . . . + cn−1t n−1 ,ci ∈ ir, i = 0,1,2, . . . ,n − 1, n = [α] + 1. lemma 4. (lemma 2.22 [21]). let α > 0, then i α c d α h(t) = h(t) + c0 + c1t + c2t 2 + . . . + cn−1t n−1 , for arbitrary ci ∈ ir, i = 0,1,2, . . . ,n − 1, n = [α] + 1. 142 mouffak benchohra and naima hamidi cubo 13, 2 (2011) 2.2 set-valued maps. let x and y be banach spaces. a set-valued map g : x → p(y) is said to be compact if g(x) = ¯ ⊔ {g(y); y ∈ x} is compact. g has convex (closed, compact) values if g(y) is convex(closed, compact) for every y ∈ x. g is bounded on bounded subsets of x if g(b) is bounded in y for every bounded subset b of x. a set-valued map g is upper semicontinuous (usc for short) at z0 ∈ x if for every open set o containing gz0, there exists a neighborhood v of z0 such that g(v) ⊂ o. g is usc on x if it is usc at every point of x if g is nonempty and compact-valued then g is usc if and only if g has a closed graph. the set of all bounded closed convex and nonempty subsets of x is denoted by bcc(x). a closed valued set-valued map g : j → p(x) is measurable if for each y ∈ x, the function t 7→ dist(y,g(t)) is measurable on j. if x ⊂ y, g has a fixed point if there exists y ∈ x such that y ∈ gy. also, ‖g(y)‖p = sup{|x|; x ∈ g(y)}. definition 5. a multivalued map f : j × irn → p(irn) is said to be l1-carathéodory if (i) t 7−→ f(t,y) is measurable for each x ∈ irn; (ii) y 7−→ f(t,y) is upper semicontinuous for almost each t ∈ j; (iii) for each q > 0, there exists ϕq ∈ l 1(j,ir+) such that ‖f(t,y)‖p = sup{‖v‖ : v ∈ f(t,y)} ≤ ϕq(t) for all ‖y‖ ≤ q and for a.e. t ∈ [0,1]. for each y ∈ c(j,irn), define the set of selections of f by s 1 f,y = {v ∈ l 1(j,ir n ) : v(t) ∈ f(t,y(t)) a.e. t ∈ j}, denotes the set of selections of f. remark 6. note that for an l1-carathéodory multifunction f : j × irn → p(irn) the set s1f,y is not empty (see [23]). for more details on set-valued maps we refer to [2, 13]. 2.3 topological transversality theory. let x be a banach space, c a convex subset of x and u an open subset of c. k∂u(ū,p(c)) denotes the set of all set-valued maps g : u → p(c) which are compact, usc with closed convex values and have no fixed points on ∂u (i.e., u ∈ gu for all u ∈ ∂u). a compact homotopy is a set-valued map h : [0,1] × ū → p(c) which is compact, usc with closed convex values. if u ∈ h(λ,u) for every λ ∈ [0,1], u ∈ ∂u, h is said to be fixed point free on ∂u. two set-valued maps f,g ∈ k∂u(ū,p(c)) are called homotopic in k∂u(ū,p(c)) if there exists a compact homotopy h : [0,1]×ū → p(c) which is fixed point free on ∂u and such that h(0,.) = f and h(1,.) = g. the function g ∈ k∂u(ū,p(c)) is called essential if every f ∈ k∂u(ū,p(c)) such that g |∂u= f |∂u, has a fixed point. otherwise g is called inessential. cubo 13, 2 (2011) fractional order differential inclusions via the topological . . . 143 theorem 7. [17] let g : ū → p(c) be the constant set-valued map g(u) ≡ u0. then, if u0 ∈ u, g is essential. theorem 8. (topological transversality theorem) [17]. let f,g be two homotopic maps in k∂u(ū,p(c))). then f is essential if and only if g is essential. for further details of the topological transversality theory we refer the reader to [18]. 3 main results in this section, we are concerned with the existence of solutions for the problem (1.1)-(1.2). consider the following spaces ac 1 b(j,ir n ) = {y ∈ ac1(j,irn); y(0) = y0, y(t) = yt }, ac 1,α(j,ir n ) = {y ∈ ac1b(j,ir n ); ∫ t 0 | c d α y(t)|dt < ∞}. ac1,α(j,ir n ) is a banach space with norm ‖y‖ac1,α = max{‖y‖∞,‖y ′‖∞,‖ c d α y‖l1 }. for the existence of solutions for the problem (1.1)-(1.2), we have the following result which is useful in what follows. definition 9. a function y ∈ ac1,α(j,irn) is said a solution to bvp (1.1)-(1.2) if there exists a function v ∈ l1(j,ir) with v(t) ∈ f(t,y(t)), for a.e. t ∈ j, such that cdαy(t) = v(t), a.e t ∈ j, 1 < α ≤ 2, and the function y satisfies condition (1.2). let h : j → irn be continuous, and consider the linear fractional order differential equation c d α y(t) = h(t), t ∈ j, 1 < α ≤ 2. (3.1) we shall refer to (3.1)-(1.2) as (lp). for the existence of solutions for the problem (1.1)-(1.2), we have the following result which is useful in what follows. lemma 10. let 1 < α ≤ 2 and let h : j → irn be continuous. a function y is a solution of the fractional integral equation y(t) = 1 γ(α) ∫ t 0 g(t,s)h(s)ds + y0 + (yt − y0)t t , (3.2) if and only if y is a solution of (lp), where g(t,s) is the green’s function defined by g(t,s) = { (t − s)α−1 − t(t −s) α−1 t , 0 ≤ s ≤ t ≤ t, −t(t −s) α−1 t , 0 ≤ t ≤ s ≤ t. (3.3) 144 mouffak benchohra and naima hamidi cubo 13, 2 (2011) proof. assume y satisfies (3.1), then lemma 4 implies that y(t) = c0 + c1t + 1 γ(α) ∫ t 0 (t − s)α−1h(s)ds. from (1.2), a simple calculation gives c0 = y0, and c1 = − 1 tγ(α) ∫ t 0 (t − s)α−1h(s)ds + yt − y0 t . hence we get equation (3.2). inversely, it is clear that if y satisfies equation (3.2), then equations (3.1)-(1.2) hold. our main result is theorem 11. assume the following hypotheses hold: (a1) the function f : j × ir n → bcc(irn) is a l1-carathéodory multi-valued map; (a2) there exist a function p ∈ l 1(j,ir+), and a continuous nondecreasing function ψ : [0, ∞) −→ (0, ∞), such that ‖f(t,y)‖p ≤ p(t)ψ(‖y‖) for each (t,y) ∈ j × ir n ; (a3) lim sup r→ +∞ r ψ(r) = +∞. then, the fractional bvp (1.1)-(1.2) has a least one solution on j. proof. this proof will be given in several steps. step 1: consider the set-valued operator f : c(j,irn) → p(l1(j,irn)) defined by (fy)(t) = f(t,y(t)). f is well defined, upper semicontinuous, with convex values and sends bounded subsets of c(j,irn) into bounded subsets of l1(j,irn). in fact, we have fy := {u : j → irn, measurable u(t) ∈ f(t,y(t)), a.e. t ∈ j}. let z ∈ c(j,irn). and u ∈ fz. then ‖u(t)‖ ≤ p(t)ψ(‖z(t)‖) ≤ p(t)ψ(‖z‖0). hence ‖u‖l1 ≤ k0 := ‖p‖l1ψ(‖z‖0). this shows that f is well defined. it is clear that f is convex valued. cubo 13, 2 (2011) fractional order differential inclusions via the topological . . . 145 now, let b be a bounded subset of c(j,irn). then, there exists k > 0 such that ‖u‖0 ≤ k for u ∈ b. so, for w ∈ fu we have ‖w‖l1 ≤ k1, where k1 = ‖p‖l1ψ(k). also, we can argue as in [15] to show that f is usc. step 2: a priori bounds on solutions. we shall show that if y be a possible solution of (1.1)-(1.2), then there exists a positive constant r∗, independent of y, such that ‖y‖ac1,α ≤ r ∗ . let y be a possible solution of (1.1)-(1.2), by lemma 10, there exits v ∈ s1f,y such that, for each t ∈ j y(t) = 1 γ(α) ∫ t 0 g(t,s)v(s)ds − ( t t − 1 ) y0 + 1 t yt, (3.4) where g is given by (3.3). let g0 := sup{‖g(t,s)‖; (t,s) ∈ j × j}, p0 = sup{p(t) : t ∈ j}. hence for t ∈ j ‖y(t)‖ ≤ 1 γ(α) ∫ t 0 g(t,s)‖v(s)‖ds − ( t t − 1 ) ‖y0‖ + 1 t ‖yt ‖ ≤ g0 γ(α) ∫ t 0 p(s)ψ(‖y(s)‖)ds + ∣ ∣ ∣ ∣ ( t t − 1 ) ∣ ∣ ∣ ∣ ‖y0‖ + 1 t ‖yt ‖ ≤ g0 γ(α) ∫ t 0 p(s)ψ(‖y(s)‖)ds + ‖y0‖ + 1 t ‖yt ‖. since ψ is nondecreasing we have ‖y‖∞ ≤ g0ψ(‖y‖∞ )p0t γ(α) + ‖y0‖ + 1 t ‖yt ‖. thus ‖y‖∞ ψ(‖y‖∞ ) ≤ g0p0t γ(α) + ‖y0‖ ψ(‖y‖∞ ) + ‖yt ‖ tψ(‖y‖∞ ) = r̃ so ‖y‖∞ ψ(‖y‖∞ ) ≤ r̃. (3.5) now, the condition ψ in (a3) shows that there exists r ∗ 1 > 0 such that for all r > r ∗ 1 r ψ(r) > r̃. (3.6) comparing these last two inequalities (3.5) and (3.6) we see that r0 ≤ r ∗ 1. consequently, we obtain ‖y(t)‖ ≤ r∗1 for all t ∈ j. 146 mouffak benchohra and naima hamidi cubo 13, 2 (2011) from (3.4) we have for each t ∈ j y ′(t) = 1 γ(α) ∫ t 0 ∂g(t,s) ∂t f(s,y(s))ds − y0 t , (3.7) using a similar argument as before we can show that there exists r∗2 > 0 such that ‖y′(t)‖ ≤ r∗2 for all t ∈ j. (3.8) now from (1.1) and (a1) we have ∫ t 0 ‖cdαy(t)‖dt ≤ ψ(r∗1) ∫ t 0 p(s)ds := r∗3. (3.9) hence ‖y‖ac1,α ≤ max{r ∗ 1,r ∗ 2,r ∗ 3} := r ∗ . step 3: existence of solutions. for 0 ≤ λ ≤ 1 consider the one-parameter family of problems cdαy(t) ∈ λf(t,y(t)), a.e. t ∈ j, 1 < α ≤ 2, (1λ) y(0) = λy0, y(t) = λyt (2λ) which reduces to (1.1)-(1.2) for λ = 1. for 0 ≤ λ ≤ 1, we define the operator fλ : c(j,ir n ) → p(l1(j,irn)) by (fλy)(t) = λf(t,y(t)). step 1 shows that fλ is usc, has convex values and sends bounded subsets of c(j,ir n ) into bounded subsets of l1(j,irn) and if y is a solution of (1λ)−(2λ) for some λ ∈ [0,1], then ‖y‖ac1,α ≤ r∗, where r∗ does not depend on λ. for λ ∈ [0,1], we define the operators j : ac1,α(j,irn) → c(j,irn) by (jy)(t) = y(t), l : ac1,α(j,ir n ) → l1(j,irn) by (ly)(t) =c dαy(t). it is clear that j is continuous and completely continuous and l is linear, continuous and has a bounded inverse denoted by l−1. let v := {y ∈ ac1,α(j,irn); ‖y‖ac1,α < r ∗ + 1}. define a map h : [0,1] × v → ac1,α(j,irn) by h(λ,y) = (l−1 ◦ fλ ◦ j )(y). cubo 13, 2 (2011) fractional order differential inclusions via the topological . . . 147 we can show that the fixed points of h(λ, ·) are solutions of (1λ) − (2λ). moreover, h is a compact homotopy between h(0, ·) ≡ 0 and h(1, ·). in fact, h is compact since j is completely continuous, fλ is continuous and l −1 is continuous. since solutions of (1λ) satisfy ‖y‖ac1,α ≤ r ∗ we see that h(λ,.) has no fixed points on ∂v. now h(0, ·) is essential by theorem 7. hence by theorem 8, h(1, ·) is essential. this implies that l−1 ◦ f1 ◦ j has a fixed point which is a solution to problem (1.1)-(1.2). 4 an example as an application of our results we consider the following boundary value problem c d α y(t) ∈ f(t,y), t ∈ j := [0,1], 1 < α ≤ 2, (4.1) y(0) = 1, y(1) = 2, (4.2) where cdα is the caputo fractional derivative. set f(t,y) = {v ∈ ir : f1(t,y) ≤ v ≤ f2(t,y)}, where f1,f2 : j × ir → ir are measurable in t. we assume that for each t ∈ j, f1(t, ·) is lower semi-continuous (i.e, the set {y ∈ ir : f1(t,y) > µ} is open for each µ ∈ ir), and assume that for each t ∈ j, f2(t, ·) is upper semi-continuous (i.e the set {y ∈ ir : f2(t,y) < µ} is open for each µ ∈ ir). assume that there exists p ∈ l1(j,ir+) and δ ∈ (0,1) such that max(|f1(t,y)|, |f2(t,y)|) ≤ p(t)|y| δ , t ∈ j, and all y ∈ ir. it is clear that f is compact and convex valued, and it is upper semi-continuous (see [13]). since assumptions (a1) − (a3) hold, theorem 11 implies that the bvp (4.1)-(4.2) has at least one solution. received: november 2009. revised: july 2010. references [1] r.p agarwal, m. benchohra and s. hamani, a survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, acta. appl. math. 109 (3) (2010), 973-1033. 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[31] s. zhang, positive solutions for boundary-value problems of nonlinear fractional diffrential equations, electron. j. differential equations 2006, no. 36, 12 pp. introduction preliminaries some properties of fractional calculus set-valued maps. topological transversality theory. main results an example cubo a mathematical journal vol.14, no¯ 02, (111–152). june 2012 on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| e.a. grove department of mathematics, university of rhode island, kingston, rhode island, 02881-0816, usa email: grove@math.uri.edu and e. lapierre 2 department of mathematics, johnson and wales university, providence, rhode island 02903, usa. email: elapierre@jwu.edu and w. tikjha faculty of science and technology, pibulsongkram rajabhat university, muang district, phitsanuloke, 65000, thailand email: wirot tik@yahoo.com abstract in this paper we consider the system of piecewise linear difference equations in the title, where the initial conditions x0 and y0 are real numbers. we show that there exists a unique equilibrium solution and exactly two prime period-3 solutions, and that except for the unique equilibrium solution, every solution of the system is eventually one of the two prime period-3 solutions. resumen en este art́ıculo consideramos el sistema de ecuaciones en diferencia lineales por partes indicado en el t́ıtulo, donde las condiciones iniciales x0 e y0 son números reales. demostramos que existe una única solución de equilibrio y exactamente dos soluciones de peŕıodo 3-primo, y que exceptuando la solución única de equilibrio, toda solución del sistema es eventualmente una de las dos soluciones de periodo 3-primo. keywords and phrases: periodic solution; systems of piecewise linear difference equations 2010 ams mathematics subject classification: 39a10, 65q10. 2 corresponding author. 112 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) 1 introduction in this paper we consider the system of piecewise linear difference equations        xn+1 = |xn| − yn − 1 , n = 0, 1, . . . yn+1 = xn + |yn| (1.1) where the initial conditions x0 and y0 are arbitrary real numbers. we show that every solution of system(1.1) is either (from the beginning) the unique equilibrium point (x̄, ȳ) = ( − 2 5 , − 1 5 ) or else is eventually one of the following period-3 cycles: p 1 3 =          x0 = 0 , y0 = −1 x1 = 0 , y1 = 1 x2 = −2 , y2 = 1          or p2 3 =            x0 = 0 , y0 = − 1 3 x1 = − 2 3 , y1 = 1 3 x2 = − 2 3 , y2 = − 1 3            . this study of system(1.1) was motivated by devaney’s celebrated gingerbreadman map        xn+1 = |xn| − yn + 1 , n = 0, 1, . . . . yn+1 = xn see ref. [1, 2, 3, 4]. we believe that the methods and techniques used in this paper will be useful in discovering the global behavior of similar piecewise linear systems of the form        xn+1 = |xn| + ayn + b , n = 0, 1, 2... yn+1 = xn + c |yn| + d for another system of this form see [5]. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 113 2 the global behavior of the solutions of system(1.1) set l1 = {(x, y) : x ≥ 0, y = 0} l2 = {(x, y) : x = 0, y ≥ 0} l3 = {(x, y) : x ≤ 0, y = 0} l4 = {(x, y) : x = 0, y ≤ 0} q1 = {(x, y) : x > 0, y > 0} q2 = {(x, y) : x < 0, y > 0} q3 = {(x, y) : x < 0, y < 0} q4 = {(x, y) : x > 0, y < 0}. theorem 1. let {(xn, yn)} ∞ n=0 be a solution of system(1.1) with (x0, y0) ∈ r 2. then either {(xn, yn)} ∞ n=0 is the unique equilibrium (x̄, ȳ), or else there exists a non-negative integer n ≥ 0 such that the solution {(xn, yn)} ∞ n=n of system(1.1) is either the prime period-3 cycle p 1 3 or the prime period-3 cycle p2 3 . the proof of theorem 1 is a direct consequence of the following lemmas. lemma 2. suppose there exists a non-negative integer n ≥ 0 such that yn = −xn − 1 and yn ≥ 0. then (xn+1, yn+1) = (0, −1), and so {(xn, yn)} ∞ n=n+1 is the period-3 cycle p 1 3 . proof. note that xn = −yn − 1 ≤ −1, and so xn+1 = |xn| − yn − 1 = −xn − (−xn − 1) − 1 = 0 yn+1 = xn + |yn| = xn + (−xn − 1) = −1. the proof is complete. lemma 3. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ l2. then {(xn, yn)} ∞ n=n+2 is the period-3 cycle p 1 3 . 114 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) proof. we have xn+1 = |xn| − yn − 1 = 0 − yn − 1 = −yn − 1 < 0 yn+1 = xn + |yn| = 0 + yn = yn ≥ 0 and so it follows by lemma 2 that {(xn, yn)} ∞ n=n+2 is the period-3 cycle p 1 3 . lemma 4. suppose there exists a non-negative integer n ≥ 0 such that xn = 0 and yn < −1. then (1) xn+3 = 2yn + 2 < 0. (2) if − 3 2 ≤ yn < −1, then yn+3 = −2yn − 3 ≤ 0. (3) if yn < − 3 2 , then {(xn, yn)} ∞ n=n+4 is the period-3 cycle p 1 3 . proof. we have xn+1 = |xn| − yn − 1 = −yn − 1 > 0 yn+1 = xn + |yn| = −yn > 0 xn+2 = |xn+1| − yn+1 − 1 = −2 yn+2 = xn+1 + |yn+1| = −2yn − 1 > 0 xn+3 = |xn+2| − yn+2 − 1 = 2yn + 2 < 0 yn+3 = xn+2 + |yn+2| = −2yn − 3. if − 3 2 ≤ yn < −1, then yn+3 = −2yn − 3 ≤ 0. if yn < − 3 2 , then yn+3 = −2yn − 3 > 0 and so by lemma 2 {(xn, yn)} ∞ n=n+4 is the period-3 cycle p 1 3 . the proof is complete. lemma 5. suppose there exists a non-negative integer n ≥ 0 such that xn = 0 and −1 < yn ≤ 0. then (1) if − 1 4 < yn ≤ 0, then {(xn, yn)} ∞ n=n+5 is the period-3 cycle p 1 3 . (2) if − 1 2 < yn ≤ − 1 4 , then xn+5 = 8yn + 2, yn+5 = −8yn − 3, and xn+6 = 0. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 115 (3) if −1 < yn ≤ − 1 2 , then {(xn, yn)} ∞ n=n+6 is the period-3 cycle p 1 3 . proof. we have xn+1 = |xn| − yn − 1 = −yn − 1 < 0 yn+1 = xn + |yn| = −yn ≥ 0 xn+2 = |xn+1| − yn+1 − 1 = 2yn ≤ 0 yn+2 = xn+1 + |yn+1| = −2yn − 1 xn+3 = |xn+2| − yn+2 − 1 = 0. if − 1 4 < yn ≤ 0, then yn+2 < 0 and yn+3 = xn+2 + |yn+2| = 4yn +1 > 0. it follows by lemma 3 that {(xn, yn)} ∞ n=n+5 is the period-3 cycle p 1 3 , and so statement 1 is true. if − 1 2 < yn ≤ − 1 4 , then yn+2 < 0 and yn+3 = xn+2 + |yn+2| = 4yn + 1 ≤ 0 xn+4 = |xn+3| − yn+3 − 1 = −4yn − 2 < 0 yn+4 = xn+3 + |yn+3| = −4yn − 1 ≥ 0 xn+5 = |xn+4| − yn+4 − 1 = 8yn + 2 ≤ 0 yn+5 = xn+4 + |yn+4| = −8yn − 3 xn+6 = |xn+5| − yn+5 − 1 = 0 and so statement 2 is true. if −1 < yn ≤ − 1 2 , then yn+6 = xn+5 + |yn+5| = −1 and so {(xn, yn)} ∞ n=n+6 is the period-3 cycle p1 3 . the proof is complete. lemma 6. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ l4. then the following five statements are true: 116 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) (1) suppose − 1 3 < yn ≤ 0. then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 . (2) suppose yn = − 1 3 . then {(xn, yn)} ∞ n=n is the period-3 cycle p 2 3 . (3) suppose − 4 3 < yn < − 1 3 . then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 . (4) suppose yn = − 4 3 . then {(xn, yn)} ∞ n=n+3 is the period-3 cycle p 2 3 . (5) suppose yn < − 4 3 . then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 . proof. we have xn = 0 and yn ≤ 0. (1) suppose − 1 3 < yn ≤ 0. note that by statement 1 of lemma 5, that if − 1 4 < yn ≤ 0, then {(xn, yn)} ∞ n=n+5 is the period-3 cycle p 1 3 . so suppose − 1 3 < yn ≤ − 1 4 . for each integer n ≥ 1, let an = −22n + 1 3 · 22n . observe that − 1 4 = a1 > a2 > a3 > . . . > − 1 3 and lim n→∞ an = − 1 3 . thus there exists a unique integer k ≥ 1 such that yn ∈ (ak+1, ak]. we first consider the case k = 1; that is, yn ∈ ( − 5 16 , −1 4 ] . it follows from statement 2 of lemma 5 that xn+5 = 8yn + 2 ≤ 0, yn+5 = −8yn − 3 < 0, and xn+6 = 0. thus yn+6 = xn+5 + |yn+5| = 16yn + 5 > 0, and so by lemma 3 we have {(xn, yn)} ∞ n=n+8 is the period-3 cycle p 1 3 . hence without loss of generality, we may assume k ≥ 2. for each integer m ≥ 1, let p(m) be the following statement: xn+3m+3 = 0 yn+3m+3 = 2 2m+2yn + 22m+2 − 1 3 ≤ 0. claim: p(m) is true for 1 ≤ m ≤ k − 1. the proof of the claim will be by induction on m. we shall first show that p(1) is true. recall that xn = 0 and yn ∈ (ak+1, ak] ⊂ ( −1 3 , − 5 16 ] , and so by statement 2 of lemma 5 cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 117 we have xn+5 = 8yn + 2 < 0 and yn+5 = −8yn − 3 < 0. then xn+3(1)+3 = 0 yn+3(1)+3 = 16yn + 5 = 2 2(1)+2yn + 22(1)+2 − 1 3 ≤ 0 and so p(1) is true. thus if k = 2, then we have shown that for 1 ≤ m ≤ k − 1, p(m) is true. it remains to consider the case k ≥ 3. so assume that k ≥ 3. let m be an integer such that 1 ≤ m ≤ k−2, and suppose p(m) is true. we shall show that p(m+1) is true. since p(m) is true we know xn+3m+3 = 0 yn+3m+3 = 2 2m+2yn + 22m+2 − 1 3 ≤ 0. recall that yn ∈ (ak+1, ak] = ( −22(k+1) + 1 3 · 22(k+1) , −22k + 1 3 · 22k ] . then xn+3m+4 = |xn+3m+3| − yn+3m+3 − 1 = −2 2m+2yn − ( 22m+2 − 1 3 ) − 1. note that xn+3m+4 = −yn+3m+3 − 1. in particular, xn+3m+4 = −2 2m+2yn − ( 22m+2 − 1 3 ) − 1 < −22m+2 ( −22(k+1) + 1 3 · 22(k+1) ) − ( 22m+2 − 1 3 ) − 1 = 22m+2k+4 3 · 22k+2 − 22m+2 3 · 22k+2 − 22m+2 3 + 1 3 − 1 = − 22m−2k 3 − 2 3 < 0 and yn+3m+4 = xn+3m+3 + |yn+3m+3| = 0 + |yn+3m+3| = −yn+3m+3 ≥ 0. 118 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) thus xn+3m+5 = |xn+3m+4| − yn+3m+4 − 1 = yn+3m+3 + 1 − (−yn+3m+3) − 1 = 2yn+3m+3 ≤ 0 and yn+3m+5 = xn+3m+4 + |yn+3m+4| = −yn+3m+3 − 1 + (−yn+3m+3) = −2yn+3m+3 − 1. in particular, yn+3m+5 = −2 ( 22m+2yn + 22m+2 − 1 3 ) − 1 < −2 [ 22m+2 ( −22(k+1) + 1 3 · 22(k+1) ) + 22m+2 − 1 3 ] − 1 = 22m+2k+5 3 · 22k+2 − 22m+3 3 · 22k+2 − 22m+3 3 + 2 3 − 1 = − 22m−2k+1 3 − 1 3 < 0. finally, xn+3(m+1)+3 = xn+3m+6 = |xn+3m+5| − yn+3m+5 − 1 = −2yn+3m+3 − (−2yn+3m+3 − 1) − 1 = 0 cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 119 and yn+3(m+1)+3 = yn+3m+6 = xn+3m+5 + |yn+3m+5| = 2yn+3m+3 + 2yn+3m+3 + 1 = 4yn+3m+3 + 1 = 22 ( 22m+2yn + 22m+2 − 1 3 ) + 1 = 22m+4yn + 22m+4 − 4 3 + 1 = 22(m+1)+2yn + 22(m+1)+2 − 1 3 . in particular, yn+3(m+1)+3 ≤ 2 2(m+1)+2 ( −22k + 1 3 · 22k ) + 22(m+1)+2 − 1 3 = − 22m+2k+4 3 · 22k + 22m+4 3 · 22k + 22m+4 3 − 1 3 = − 1 3 ( 1 − 22m−2k+4 ) ≤ 0 and so p(m + 1) is true. thus the proof of the claim is complete. that is, p(m) is true for 1 ≤ m ≤ k − 1. specifically, p(k − 1) is true, and so xn+3(k−1)+3 = xn+3k = 0 yn+3(k−1)+3 = yn+3k = 2 2kyn + 22k − 1 3 < 0. note that 22k ( −22k+2 + 1 3 · 22k+2 ) + 22k − 1 3 < yn+3k ≤ 2 2k ( −22k + 1 3 · 22k ) + 22k − 1 3 . so as 120 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) 22k ( −22k+2 + 1 3 · 22k+2 ) + 22k − 1 3 = −24k+2 3 · 22k+2 + 22k 3 · 22k+2 + 22k 3 − 1 3 = 1 3 ( 1 22 − 1 ) = − 1 4 and 22k ( −22k + 1 3 · 22k ) + 22k − 1 3 = −22k + 1 3 + 22k − 1 3 = 0 we have − 1 4 < yn+3k ≤ 0 and so it follows from statement 1 of lemma 5 that {(xn, yn)} ∞ n=n+3k+5 is the period-3 cycle p1 3 . (2) suppose yn = − 1 3 . note that (0, −1 3 ) ∈ p1 3 and so {(xn, yn)} ∞ n=n is the period-3 cycle p 1 3 . (3) suppose −4 3 < yn ≤ − 1 3 . we shall first consider the case where −4 3 < yn ≤ −1. so suppose −4 3 < yn ≤ −1. for each integer n ≥ 0, let bn = −22n+2 + 1 3 · 22n . observe that −1 = b0 > b1 > b2 > . . . > − 4 3 and lim n→∞ bn = − 4 3 . thus there exists a unique integer k ≥ 1 such that yn ∈ (bk, bk−1]. we first consider the case k = 1; that is, yn ∈ ( −5 4 , −1 ] . note that if yn = −1 then (xn, yn) = (0, −1) and {(xn, yn)} ∞ n=n is the period-3 cycle p 1 3 . so assume yn ∈ ( −5 4 , −1 ) . by statements 1 and 2 of lemma 4, we have xn+3 = 2yn + 2 < 0 and yn+3 = −2yn − 3 ≤ 0. then xn+4 = |xn+3| − yn+3 − 1 = 0 yn+4 = xn+3 + |yn+3| = 4yn + 5 > 0 cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 121 and so it follows by lemma 3 that {(xn, yn)} ∞ n=n+6 is the period-3 cycle p 1 3 . hence without loss of generality, we may assume k ≥ 2. for each integer m ≥ 1, let q(m) be the following statement: xn+3m+1 = 0 yn+3m+1 = 2 2myn + 22m+2 − 1 3 ≤ 0. claim: q(m) is true for 1 ≤ m ≤ k − 1. the proof of the claim will be by induction on m. we shall first show that q(1) is true. recall that xn = 0 and yn ∈ (bk, bk−1] ⊂ ( −4 3 , −5 4 ] , and so by statements 1 and 2 of lemma 4 we have xn+3 = 2yn + 2 < 0 yn+3 = −2yn − 3 < 0 xn+3(1)+1 = |xn+3| − yn+3 − 1 = 0 yn+3(1)+1 = xn+3 + |yn+3| = 4yn + 5 ≤ 0 = 22(1)yn + 22(1)+2 − 1 3 ≤ 0 and so q(1) is true. thus if k = 2, then we have shown that for 1 ≤ m ≤ k − 1, q(m) is true. it remains to consider the case k ≥ 3. so assume that k ≥ 3. let m be an integer such that 1 ≤ m ≤ k−2, and suppose q(m) is true. we shall show that q(m+1) is true. since q(m) is true we know xn+3m+1 = 0 yn+3m+1 = 2 2myn + 22m+2 − 1 3 ≤ 0 and so xn+3m+2 = |xn+3m+1| − yn+3m+1 − 1 = 0 − yn+3m+1 − 1. recall that yn ∈ (bk, bk−1] = ( −22k+2 + 1 3 · 22k , −22k + 1 3 · 22k−2 ] . 122 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) in particular, xn+3m+2 = −2 2myn − ( 22m+2 − 1 3 ) − 1 < −22m ( −22k+2 + 1 3 · 22k ) − ( 22m+2 − 1 3 ) − 1 = 22k+2m+2 3 · 22k − 22m 3 · 22k − 22m+2 3 + 1 3 − 1 = − 1 3 ( 22m−2k+2 + 2 ) < 0 and yn+3m+2 = xn+3m+1 + |yn+3m+1| = 0 − yn+3m+1 ≥ 0. hence xn+3m+3 = |xn+3m+2| − yn+3m+2 − 1 = yn+3m+1 + 1 − (−yn+3m+1) − 1 = 2yn+3m+1 ≤ 0 and yn+3m+3 = xn+3m+2 + |yn+3m+2| = −yn+3m+1 − 1 + (−yn+3m+1) = −2yn+3m+1 − 1. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 123 in particular, yn+3m+3 = −2 [ 22myn + 22m+2 − 1 3 ] − 1 < −2 [ 22m ( −22k+2 + 1 3 · 22k ) + 22m+2 − 1 3 ] − 1 = 22k+2m+3 3 · 22k − 22m+1 3 · 22k − 22m+3 3 + 2 3 − 1 = − 1 3 ( 22m−2k+1 + 1 ) < 0. finally, xn+3(m+1)+1 = xn+3m+4 = |xn+3m+3| − yn+3m+1 − 1 = −2yn+3m+1 − (−2yn+3m+1 − 1) − 1 = 0 and yn+3(m+1)+1 = yn+3m+4 = xn+3m+3 + |yn+3m+3| = 2yn+3m+1 + 2yn+3m+1 + 1 = 4yn+3m+1 + 1 = 22(m+1)yn + 22(m+1)+2 − 1 3 . in particular, yn+3(m+1)+1 ≤ 2 2m+2 ( −22k + 1 3 · 22k−2 ) + 22m+4 − 1 3 124 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) = − 22k+2m+2 3 · 22k−2 + 22m+2 3 · 22k−2 + 22m+4 3 − 1 3 = 1 3 ( 22m−2k+4 − 1 ) ≤ 0 and so q(m + 1) is true. thus the proof of the claim is complete. that is, q(m) is true for 1 ≤ m ≤ k − 1. specifically, q(k − 1) is true, and so xn+3(k−1)+1 = 0 yn+3(k−1)+1 = 2 2(k−1)yn + 22(k−1)+2 − 1 3 ≤ 0. note that 0 ≥ yn+3(k−1)+1 > 2 2(k−1) ( −22k+2 + 1 3 · 22k ) + 22k − 1 3 = − 24k 3 · 22k + 22k−2 3 · 22k + 22k 3 − 1 3 = 1 3 ( 1 4 − 1 ) = − 1 4 and so it follows by statement 1 of lemma 5 that {(xn, yn)} ∞ n=n+3k+3 is the period-3 cycle p1 3 . suppose −1 < yn < − 1 2 . by statement 3 of lemma 5 we have {(xn, yn)} ∞ n=n+3 is the period-3 cycle p1 3 . to complete the proof of statement 3 we shall now suppose that −1 2 ≤ yn < − 1 3 . for each integer n ≥ 1, let αn = −22n−1 − 1 3 · 22n−1 . observe that − 1 2 = α1 < α2 < α3 < . . . < − 1 3 and lim n→∞ αn = − 1 3 . cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 125 thus there exists a unique integer k ≥ 1 such that yn ∈ [αk, αk+1). we first consider the case k = 1; that is, yn ∈ [ −1 2 , −3 8 ) . by statement 2 of lemma 5 we have xn+5 = 8yn + 2 ≤ 0, yn+5 = −8yn − 3 > 0, and so it follows by lemma 2 that {(xn, yn)} ∞ n=n+6 is the period-3 cycle p1 3 . without loss of generality we may assume k ≥ 2. for each integer m ≥ 1, let r(m) be the following statement: xn+3m+2 = 2 2m+1yn + 22m+1 − 2 3 < 0 yn+3m+2 = −2 2m+1yn − ( 22m+1 + 1 3 ) ≤ 0. claim: r(m) is true for 1 ≤ m ≤ k − 1. the proof of the claim will be by induction on m. we shall first show that r(1) is true. recall that xn = 0 and yn ∈ [αk, αk+1) ⊂ [ −3 8 , −1 3 ) , and so it follows from statement 2 of lemma 5 that xn+3(1)+2 = 8yn + 2 = 2 2(1)+1yn + 22(1)+1 − 2 3 < 0 yn+3(1)+2 = −8yn − 3 = −2 2(1)+1yn − ( 22(1)+1 + 1 3 ) ≤ 0 and so r(1) is true. thus if k = 2, then we have shown that for 1 ≤ m ≤ k − 1, r(m) is true. it remains to consider the case k ≥ 3. so assume that k ≥ 3. let m be an integer such that 1 ≤ m ≤ k−2, and suppose r(m) is true. we shall show that r(m+1) is true. since r(m) is true we know xn+3m+2 = 2 2m+1yn + 22m+1 − 2 3 < 0 yn+3m+2 = −2 2m+1yn − ( 22m+1 + 1 3 ) ≤ 0. then xn+3m+3 = |xn+3m+2| − yn+3m+2 − 1 = −22m+1yn − 22m+1 − 2 3 − ( −22m+1yn − 22m+1 + 1 3 ) − 1 = 0 126 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) yn+3m+3 = xn+3m+2 + |yn+3m+2| = 22m+1yn + 22m+1 − 2 3 + 22m+1yn + 22m+1 + 1 3 = 22m+2yn + 22m+2 − 1 3 . recall that yn ∈ [αk, αk+1) = [ −22k−1 − 1 3 · 22k−1 , −22(k+1)−1 − 1 3 · 22(k+1)−1 ) . in particular, yn+3m+3 < 2 2m+2 ( −22(k+1)−1 − 1 3 · 22(k+1)−1 ) + 22m+2 − 1 3 = − 22k+2m+3 3 · 22k+1 − 22m+2 3 · 22k+1 + 22m+2 3 − 1 3 = − 1 3 ( 1 + 22m−2k+1 ) < 0. then xn+3m+4 = |xn+3m+3| − yn+3m+3 − 1 = 0 − yn+3m+3 − 1 = −yn+3m+3 − 1. in particular, xn+3m+4 = −2 2m+2yn − 22m+2 − 1 3 − 1 ≤ −22m+2 ( −22k−1 − 1 3 · 22k−1 ) − ( 22m+2 − 1 3 ) − 1 = 22m+2k+1 3 · 22k−1 + 22m+2 3 · 22k−1 − 22m+2 3 + 1 3 − 1 = − 2 3 ( 1 − 22m−2k+2 ) < 0. hence yn+3m+4 = xn+3m+3 + |yn+3m+3| = 0 + (−yn+3m+3) = −yn+3m+3 > 0. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 127 finally, xn+3(m+1)+2 = xn+3m+5 = |xn+3m+4| − yn+3m+4 − 1 = yn+3m+3 + 1 − (−yn+3m+3) − 1 = 2yn+3m+3 < 0 = 22(m+1)+1yn + 22(m+1)+1 − 2 3 < 0 and yn+3(m+1)+2 = yn+3m+5 = xn+3m+4 + |yn+3m+4| = −yn+3m+3 − 1 + (−yn+3m+3) = −2yn+3m+3 − 1 = −22(m+1)+1yn − ( 22(m+1)+1 + 1 3 ) . in particular, yn+3(m+1)+2 ≤ −2 2m+3 ( −22k−1 − 1 3 · 22k−1 ) − ( 22m+3 + 1 3 ) = 22m+2k+2 3 · 22k−1 + 22m+3 3 · 22k−1 − 22m+3 3 − 1 3 = 1 3 ( 22m−2k+4 − 1 ) ≤ 0 and so r(m + 1) is true. thus the proof of the claim is complete. that is, r(m) is true for 1 ≤ m ≤ k − 1. specifically, r(k − 1) is true, and so xn+3(k−1)+2 = 2 2(k−1)+1yn + 22(k−1)+1 − 2 3 < 0 yn+3(k−1)+2 = −2 2(k−1)+1yn − ( 22(k−1)+1 + 1 3 ) ≤ 0. 128 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) then xn+3k = xn+3(k−1)+3 = |xn+3(k−1)+2| − yn+3(k−1)+2 − 1 = 0 and yn+3k = yn+3(k−1)+3 = xn+3(k−1)+2 + |yn+3(k−1)+2| = 22kyn + 22k − 1 3 . note that 22k ( −22k−1 − 1 3 · 22k−1 ) + 22k − 1 3 ≤ yn+3k < 2 2k ( −22k+1 − 1 3 · 22k+1 ) + 22k − 1 3 . so as 22k ( −22k−1 − 1 3 · 22k−1 ) + 22k − 1 3 = −24k−1 3 · 22k−1 + 22k 3 · 22k−1 + 22k 3 − 1 3 = − 2 3 − 1 3 = −1 and 22k ( −22k+1 − 1 3 · 22k+1 ) + 22k − 1 3 = −24k+1 3 · 22k+1 + 22k 3 − 1 3 = − 1 6 − 1 3 = − 1 2 we have −1 ≤ yn+3k < − 1 2 and so it follows by statement 3 of lemma 5 and the fact (0, −1) ∈ p1 3 that the solution {(xn, yn)} ∞ n=n+3k+3 is the period-3 cycle p 1 3 . (4) suppose yn = − 4 3 . by direct computations we have (xn+3, yn+3) = (− 2 3 , −1 3 ) ∈ p2 3 , and so {(xn, yn)} ∞ n=n+3 is the period-3 cycle p 2 3 . cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 129 (5) suppose yn < − 4 3 . first consider the case −3 2 ≤ yn < − 4 3 . for each integer n ≥ 0, let βn = −22n+3 − 1 3 · 22n+1 . observe that − 3 2 = β0 < β1 < β2 < . . . < − 4 3 and lim n→∞ βn = − 4 3 . thus there exists a unique integer k ≥ 1 such that yn ∈ [βk−1, βk). we first consider the case k = 1; that is, yn ∈ [ −3 2 , −11 8 ) . by statements 1 and 2 of lemma 4 we have xn+3 = 2yn + 2 < 0 yn+3 = −2yn − 3 ≤ 0 and so xn+4 = |xn+3| − yn+3 − 1 = 0 yn+4 = xn+3 + |yn+3| = 4yn + 5 < 0. in particular, −1 ≤ yn+4 < − 1 2 . it follows by statement 3 of lemma 5 that the solution {(xn, yn)} ∞ n=n+7 is the period-3 cycle p 1 3 . thus without loss of generality, we may assume that k ≥ 2. for each integer m ≥ 1, let s(m) be the following statement: xn+3m+3 = 2 2m+1yn + 22m+3 − 2 3 < 0 yn+3m+3 = −2 2m+1yn − ( 22m+3 − 2 3 ) − 1 ≤ 0. claim: s(m) is true for 1 ≤ m ≤ k − 1. the proof of the claim will be by induction on m. we shall first show that s(1) is true. recall that xn = 0 and yn ∈ [βk−1, βk) ⊂ [ −11 8 , −4 3 ) , and so by statements 1 and 2 of lemma 4 we have xn+3 = 2yn + 2 < 0 130 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) yn+3 = −2yn − 3 < 0 xn+4 = |xn+3| − yn+3 − 1 = 0 yn+4 = xn+3 + |yn+3| = 4yn + 5 < 0 xn+5 = |xn+4| − yn+4 − 1 = −4yn − 6 < 0 yn+5 = xn+4 + |yn+4| = −4yn − 5 > 0. finally, xn+3(1)+3 = xn+6 = |xn+5| − yn+5 − 1 = 8yn + 10 < 0 yn+3(1)+3 = yn+6 = xn+5 + |yn+5| = −8yn − 11 ≤ 0. it follows that s(1) is true. thus if k = 2, then we have shown that for 1 ≤ m ≤ k − 1, s(m) is true. it remains to consider the case k ≥ 3. so assume that k ≥ 3. let m be an integer such that 1 ≤ m ≤ k−2, and suppose s(m) is true. we shall show that s(m+1) is true. since s(m) is true, we know xn+3m+3 = 2 2m+1yn + 22m+3 − 2 3 < 0 yn+3m+3 = −2 2m+1yn − ( 22m+3 − 2 3 ) − 1 ≤ 0. note that yn+3m+3 = −xn+3m+3 − 1, and so −1 ≤ xn+3m+3 < 0. thus xn+3m+4 = |xn+3m+3| − yn+3m+3 − 1 = −xn+3m+3 − (−xn+3m+3 − 1) − 1 = 0 cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 131 and yn+3m+4 = xn+3m+3 + |yn+3m+3| = xn+3m+3 + xn+3m+3 + 1 = 2xn+3m+3 + 1. recall that yn ∈ [βk−1, βk) = [ −22(k−1)+3 − 1 3 · 22(k−1)+1 , −22k+3 − 1 3 · 22k+1 ) . in particular, yn+3m+4 = 2 [ 22m+1yn + 22m+3 − 2 3 ] + 1 < 2 [ 22m+1 ( −22k+3 − 1 3 · 22k+1 ) + 22m+3 − 2 3 ] + 1 = − 22k+2m+5 3 · 22k+1 − 22m+2 3 · 22k+1 + 22m+4 3 − 1 3 = − 1 3 ( 22m−2k+1 + 1 ) < 0. also note that −1 < xn+3m+3 < − 1 2 . thus xn+3m+5 = |xn+3m+4| − yn+3m+4 − 1 = 0 − (2xn+3m+3 + 1) − 1 = −2xn+3m+3 − 2 < 0 132 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) and yn+3m+5 = xn+3m+4 + |yn+3m+4| = 0 + (−2xn+3m+3 − 1) = −2xn+3m+3 − 1 > 0. finally, xn+3(m+1)+3 = xn+3m+6 = |xn+3m+5| − yn+3m+5 − 1 = 2xn+3m+3 + 2 − (−2xn+3m+3 − 1) − 1 = 4xn+3m+3 + 2 < 0 = 4 [ 22m+1yn + ( 22m+3 − 2 3 )] + 2 < 0 = 22(m+1)+1yn + ( 22(m+1)+3 − 2 3 ) + 2 < 0 and yn+3(m+1)+3 = yn+3m+6 = xn+3m+5 + |yn+3m+5| = −2xn+3m+3 − 2 + (−2xn+3m+3 − 1) = −4xn+3m+3 − 3 = −4 [ 22m+1yn + ( 22m+3 − 2 3 )] − 3 = −22(m+1)+1yn − ( 22(m+1)+3 − 2 3 ) − 1. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 133 in particular, yn+3m+6 ≤ −4 [ 22m+1 ( −22(k−1)+3 − 1 3 · 22(k−1)+1 ) + 22m+3 − 2 3 ] − 3 = 22k+2m+4 3 · 22k−1 + 22m+3 3 · 22k−1 − 22m+5 3 − 1 3 = 1 3 ( 22m−2k+4 − 1 ) < 0 and so s(m + 1) is true. thus the proof of the claim is complete. that is, s(m) is true for 1 ≤ m ≤ k − 1. specifically, s(k − 1) is true, and so xn+3(k−1)+3 = xn+3k = 2 2k−1yn + 22k+1 − 2 3 < 0 yn+3(k−1)+3 = yn+3k = −2 2k−1yn − ( 22k+1 − 2 3 ) − 1 < 0. note that yn+3k = −xn+3k − 1. thus xn+3k+1 = |xn+3k| − yn+3k − 1 = −xn+3k − (−xn+3k − 1) − 1 = 0 and yn+3k+1 = xn+3k + |yn+3k| = xn+3k + xn+3k + 1 = 2xn+3k + 1 = 2 ( 22k−1yn + 22k+1 − 2 3 ) + 1. 134 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) note that 2 [ 22k−1 ( −22(k−1)+3 − 1 3 · 22(k−1)+1 ) + 22k+1 − 2 3 ] + 1 ≤ yn+3k+1 < 2 [ 22k−1 ( −22k+3 − 1 3 · 22k+1 ) + 22k+1 − 2 3 ] + 1. so as 2 [ 22k−1 ( −22(k−1)+3 − 1 3 · 22(k−1)+1 ) + 22k+1 − 2 3 ] + 1 = −24k+1 3 · 22k−1 − 22k 3 · 22k−1 + 22k+2 3 − 1 3 = − 1 3 (2 + 1) = −1 and 2 [ 22k−1 ( −22k+3 − 1 3 · 22k+1 ) + 22k+1 − 2 3 ] + 1 = −22k+3 3 · 2 − 1 6 + 22k+2 3 − 1 3 = − 1 6 − 1 3 = − 1 2 we have −1 ≤ yn+3k+1 < − 1 2 and hence it follows from case 3 of this lemma and the fact that (0, −1) ∈ p1 3 that the solution {(xn, yn)} ∞ n=n+3k+5 is eventually the period-3 cycle p 1 3 . finally, suppose yn < − 3 2 . then by statement 3 of lemma 4 the solution {(xn, yn)} ∞ n=n+4 is the period-3 cycle p1 3 . lemma 7. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ q1. then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 135 proof. we have xn+1 = |xn| − yn − 1 = xn − yn − 1 yn+1 = xn + |yn| = xn + yn > 0. if xn+1 ≥ 0 then xn+2 = |xn+1| − yn+1 − 1 = −2yn − 2 < 0 yn+2 = xn+1 + |yn+1| = 2xn − 1 > 0 xn+3 = |xn+2| − yn+2 − 1 = −2xn + 2yn + 2 ≤ 0 yn+3 = xn+2 + |yn+2| = 2xn − 2yn − 3 xn+4 = |xn+3| − yn+3 − 1 = 0 and so (xn+4, yn+4) ∈ l2 ∪ l4. by lemmas 3 and 6, the solution {(xn, yn)} ∞ n=n is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . if xn+1 < 0 then xn+2 = |xn+1| − yn+1 − 1 = −2xn < 0 yn+2 = xn+1 + |yn+1| = 2xn − 1 xn+3 = |xn+2| − yn+2 − 1 = 0 and so (xn+3, yn+3) ∈ l2 ∪ l4. by lemmas 3 and 6, the solution {(xn, yn)} ∞ n=n is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . lemma 8. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ q2. then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . proof. we have xn+1 = |xn| − yn − 1 = −xn − yn − 1 yn+1 = xn + |yn| = xn + yn. 136 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) case 1: suppose yn+1 ≥ 0. then by lemma 2, the solution {(xn, yn)} ∞ n=n+2 is the period-3 cycle p1 3 . case 2: suppose yn+1 < 0 and xn+1 ≤ 0. then xn+2 = |xn+1| − yn+1 − 1 = 0 and so (xn+2, yn+2) ∈ l2 ∪l4. by lemmas 3 and 6, the solution {(xn, yn)} ∞ n=n is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . case 3: suppose yn+1 < 0 and xn+1 > 0. then xn+2 = |xn+1| − yn+1 − 1 = −2xn − 2yn − 2 > 0 yn+2 = xn+1 + |yn+1| = −2xn − 2yn − 1 > 0 xn+3 = |xn+2| − yn+2 − 1 = −2 yn+3 = xn+2 + |yn+2| = −4xn − 4yn − 3 > 0 xn+4 = |xn+3| − yn+3 − 1 = 4xn + 4yn + 4 < 0 yn+4 = xn+3 + |yn+3| = −4xn − 4yn − 5 xn+5 = |xn+4| − yn+4 − 1 = 0 and so (xn+5, yn+5) ∈ l2 ∪ l4. by lemmas 3 and 6, the solution {(xn, yn)} ∞ n=n is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . lemma 9. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ q4. then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . proof. we have xn+1 = |xn| − yn − 1 = xn − yn − 1 yn+1 = xn + |yn| = xn − yn > 0 case 1: suppose xn+1 > 0. then (xn+1, yn+1) ∈ q1 and so by lemma 7, the solution {(xn, yn)} ∞ n=n+2 is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . case 2: suppose xn+1 = 0. then (xn+1, yn+1) ∈ l2 and so by lemma 3, the solution {(xn, yn)} ∞ n=n+4 is the period-3 cycle p1 3 . cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 137 case 3: suppose xn+1 < 0. then (xn+1, yn+1) ∈ q2 and so by lemma 8, the solution {(xn, yn)} ∞ n=n+1 is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . lemma 10. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ l1. then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 or p2 3 . proof. we have xn+1 = |xn| − yn − 1 = xn − 1 yn+1 = xn + |yn| = xn case 1: suppose xn = 0. then (xn+1, yn+1) = (−1, 0), and so (xn+2, yn+2) = (0, −1). hence the solution {(xn, yn)} ∞ n=n+2 is the period-3 cycle p 1 3 . case 2: suppose 0 < xn ≤ 1. then xn+1 ≤ 0 and yn+1 > 0. thus (xn+1, yn+1) ∈ q2 ∪ l2, and hence by lemmas 3 and 8, the solution {(xn, yn)} ∞ n=n+1 is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . case 3: suppose xn > 1. then xn+1 > 0 and yn+1 > 0. thus (xn+1, yn+1) ∈ q1 and by lemma 7, the solution {(xn, yn)} ∞ n=n+1 is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . lemma 11. suppose there exists a non-negative integer n ≥ 0 such that (xn, yn) ∈ l3. then {(xn, yn)} ∞ n=n is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . proof. we have xn+1 = |xn| − yn − 1 = −xn − 1 yn+1 = xn + |yn| = xn < 0. case 1: suppose −1 < xn ≤ 0. then xn+2 = |xn+1|−yn+1 −1 = 0, and so (xn+2, yn+2) ∈ l2 ∪l4. it follows by lemmas 3 and 6, that the solution {(xn, yn)} ∞ n=n+2 is eventually the period-3 cycle p1 3 or the period-3 cycle p2 3 . case 2: suppose xn = −1. then (xn+1, yn+1) = (0, −1) ∈ p 1 3 , and so the solution {(xn, yn)} ∞ n=n+1 is the period-3 cycle p1 3 . case 3: suppose xn < −1. then (xn+1, yn+1) ∈ q4 ∪ l1. it follows by lemmas 9 and 10, the solution {(xn, yn)} ∞ n=n+2 is eventually the period-3 cycle p 1 3 or the period-3 cycle p2 3 . to complete the proof of theorem 2.1 it remains to consider the case where the initial condition (x0, y0) ∈ q3. 138 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) lemma 12. suppose (x0, y0) ∈ q3. then {(xn, yn)} ∞ n=0 is the unique equilibrium solution (x̄, ȳ) = ( −2 5 , −1 5 ) , or else is eventually either the period-3 cycle p1 3 or the period-3 cycle p2 3 . proof. if (x0, y0) = ( −2 5 , −1 5 ) , then the solution {(xn, yn)} ∞ n=0 is the equilibrium. so suppose (x0, y0) ∈ q3 \ {( −2 5 , −1 5 )} . it suffices to show that there exists an integer n ≥ 0 such that {(xn, yn)} ∞ n=n is either the period-3 cycle p 1 3 or the period-3 cycle p2 3 . for the sake of contradiction, assume that it is false that there exists an integer n ≥ 0 such that {(xn, yn)} ∞ n=n is either the period-3 cycle p 1 3 or the period-3 cycle p2 3 . it follows from the previous lemmas that xn < 0 and yn < 0 for every integer n ≥ 0. case 1: suppose x0 ≤ −2 and y0 < 0. then x1 = |x0| − y0 − 1 = −x0 − y0 − 1 > 0 which is a contradiction, and the proof is complete. case 2: suppose −2 < x0 < 0 and y0 ≤ −1. then x1 = |x0| − y0 − 1 = −x0 − y0 − 1 > 0 which is a contradiction, and the proof is complete. case 3: it remains to consider the case (x0, y0) ∈ (−2, 0) × (−1, 0). for each integer n ≥ 0, let an = −24n−2 − 1 5 · 24n−3 , bn = −24n + 1 5 · 24n−1 , cn = −24n−2 − 1 5 · 24n−2 , dn = −24n + 1 5 · 24n and dn = 24n − 1 5 . observe that −2 = a0 < a1 < a2 < . . . < − 2 5 and lim n→∞ an = − 2 5 0 = b0 > b1 > b2 > . . . > − 2 5 and lim n→∞ bn = − 2 5 −1 = c0 < c1 < c2 < . . . < − 1 5 and lim n→∞ cn = − 1 5 0 = d0 > d1 > d2 > . . . > − 1 5 and lim n→∞ dn = − 1 5 . cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 139 there exists a unique integer k ≥ 0 such that (x0, y0) ∈ [ak, bk] × [ck, dk] \ [ak+1, bk+1] × [ck+1, dk+1]. we first consider the case k = 0; that is, (x0, y0) ∈ [−2, 0] × [−1, 0] \ [− 1 2 , −3 8 ] × [−1 4 , − 3 16 ]. note that by lemmas 6 and 11, and by case 1 and case 2 of this lemma, we know that the solution {(xn, yn)} ∞ n=0 is eventually either the period-3 cycle p 1 3 or the period-3 cycle p2 3 when (x0, y0) is an element of the outer boundaries of [−2, 0] × [−1, 0]. recall by assumption that xn < 0 and yn < 0 for every integer n ≥ 0. so suppose (x0, y0) ∈ (−2, 0) × (−1, 0) \ [ − 1 2 , − 3 8 ] × [ − 1 4 , − 3 16 ] . then x1 = |x0| − y0 − 1 = −x0 − y0 − 1 y1 = x0 + |y0| = x0 − y0 x2 = |x1| − y1 − 1 = (x0 + y0 + 1) − (x0 − y0) − 1 = 2y0 y2 = x1 + |y1| = (−x0 − y0 − 1) + (−x0 + y0) = −2x0 − 1. if −2 < x0 < − 1 2 , then y2 > 0 which is a contradiction. thus −1 2 ≤ x0 < 0. then x3 = |x2| − y2 − 1 = (−2y0) − (−2x0 − 1) − 1 = 2x0 − 2y0 y3 = x2 + |y2| = (2y0) + (2x0 + 1) = 2x0 + 2y0 + 1 x4 = |x3| − y3 − 1 = (−2x0 + 2y0) − (2x0 + 2y0 + 1) − 1 = −4x0 − 2 y4 = x3 + |y3| = (2x0 − 2y0) + (−2x0 − 2y0 − 1) = −4y0 − 1. if −1 < y0 < − 1 4 , then y4 > 0 which is a contradiction. hence −1 4 ≤ y0 < 0. then x5 = |x4| − y4 − 1 = (4x0 + 2) − (−4y0 − 1) − 1 = 4x0 + 4y0 + 2 y5 = x4 + |y4| = (−4x0 − 2) + (4y0 + 1) = −4x0 + 4y0 − 1 x6 = |x5| − y5 − 1 = (−4x0 − 4y0 − 2) − (−4x0 + 4y0 − 1) − 1 = −8y0 − 2 y6 = x5 + |y5| = (4x0 + 4y0 + 2) + (4x0 − 4y0 + 1) = 8x0 + 3. 140 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) if −3 8 < x0 < 0, then y6 > 0 which is a contradiction. hence −1 2 < x0 ≤ − 3 8 . thus x7 = |x6| − y6 − 1 = (8y0 + 2) − (8x0 + 3) − 1 = −8x0 + 8y0 − 2 y7 = x6 + |y6| = (−8y0 − 2) + (−8x0 − 3) = −8x0 − 8y0 − 5 x8 = |x7| − y7 − 1 = (8x0 − 8y0 + 2) − (−8x0 − 8y0 − 5) − 1 = 16x0 + 6 y8 = x7 + |y7| = (−8x0 + 8y0 − 2) + (8x0 + 8y0 + 5) = 16y0 + 3 > 0, which is a contradiction. thus the case k = 0 is complete. next consider the case k ≥ 1. recall that xn < 0 and yn < 0 for all n ≥ 0. for each integer m such that 0 ≤ m ≤ k − 1, let p(m) be the following proposition: x8m+1 = −2 4mx0 − 2 4my0 − 3dm − 1 y8m+1 = 2 4mx0 − 2 4my0 + dm x8m+2 = 2 4m+1y0 + 2dm y8m+2 = −2 4m+1x0 − 4dm − 1 x8m+3 = 2 4m+1x0 − 2 4m+1y0 + 2dm y8m+3 = 2 4m+1x0 + 2 4m+1y0 + 6dm + 1 x8m+4 = −2 4m+2x0 − 8dm − 2 y8m+4 = −2 4m+2y0 − 4dm − 1 x8m+5 = 2 4m+2x0 + 2 4m+2y0 + 12dm + 2 y8m+5 = −2 4m+2x0 + 2 4m+2y0 − 4dm − 1 x8m+6 = −2 4m+3y0 − 8dm − 2 y8m+6 = 2 4m+3x0 + 16dm + 3 cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 141 x8m+7 = −2 4m+3x0 + 2 4m+3y0 − 8dm − 2 y8m+7 = −2 4m+3x0 − 2 4m+3y0 − 24dm − 5 x8m+8 = 2 4m+4x0 + 32dm + 6 y8m+8 = 2 4m+4x0 + 16dm + 3. claim: p(m) is true for 0 ≤ m ≤ k − 1. the proof of the claim will be by induction on m. we shall first show that p(0) is true. x8(0)+1 = −x0 − y0 − 1 = −2 4(0)x0 − 2 4(0)y0 − 3d0 − 1 y8(0)+1 = x0 − y0 = 2 4(0)x0 − 2 4(0)y0 − d0 x8(0)+2 = 2y0 = 2 4(0)+1y0 + 2d0 y8(0)+2 = −2x0 − 1 = −2 4(0)+1x0 − 4d0 − 1 x8(0)+3 = 2x0 − 2y0 = 2 4(0)+1x0 − 2 4(0)+1y0 + 2d0 y8(0)+3 = 2x0 + 2y0 + 1 = 2 4(0)+1x0 + 2 4(0)+1y0 + 6d0 + 1 x8(0)+4 = −4x0 − 2 = −2 4(0)+2x0 − 8d0 − 2 y8(0)+4 = −4y0 − 1 = −2 4(0)+2x0 − 4d0 − 1 x8(0)+5 = 4x0 + 4y0 + 2 = 2 4(0)+1x0 − 2 4(0)+2y0 + 12d0 + 2 y8(0)+5 = −4x0 + 4y0 − 1 = −2 4(0)+2x0 + 2 4(0)+2y0 − 4d0 − 1 x8(0)+6 = −8x0 − 2 = −2 4(0)+3x0 − 8d0 − 2 y8(0)+6 = 8y0 + 3 = 2 4(0)+3x0 + 16d0 + 3 x8(0)+7 = −8x0 + 8y0 − 2 = −2 4(0)+3x0 + 2 4(0)+3y0 − 8d0 − 2 y8(0)+7 = −8x0 − 8y0 − 5 = −2 4(0)+3x0 − 2 4(0)+3y0 − 24d0 + 5 x8(0)+8 = 16x0 + 6 = 2 4(0)+4x0 + 32d0 + 6 y8(0)+8 = 16y0 + 3 = 2 4(0)+4x0 + 16d0 + 3 and so p(0) is true. thus if k = 1, then we have shown that for 0 ≤ m ≤ k − 1, p(m) is true. it remains to consider the case k ≥ 2. so assume that k ≥ 2. suppose that m is an integer such that 0 ≤ m ≤ k − 2, and that p(m) is true. we shall show that p(m + 1) is true. 142 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) since p(m) is true, we know x8m+8 = 2 4m+4x0 + 32dm + 6 y8m+8 = 2 4m+4x0 + 16dm + 3. hence x8(m+1)+1 = x8m+9 = |x8m+8| − y8m+8 − 1 = −(24m+4x0 + 32dm + 6) − (2 4m+4y0 + 16dm + 3) − 1 = −24m+4x0 − 2 4m+4y0 − 48dm − 10 = −24m+4x0 − 2 4m+4y0 − 48 ( 24m − 1 5 ) − 10 = −24(m+1)x0 − 2 4(m+1)y0 − 3 ( 24(m+1) − 1 5 ) − 1 = −24(m+1)x0 − 2 4(m+1)y0 − 3dm+1 − 1 and y8(m+1)+1 = y8m+9 = x8m+8 + |y8m+8| = 24m+4x0 + 32dm + 6 + (−2 4m+4y0 − 16dm − 3) = 24m+4x0 − 2 4m+4y0 + 16dm + 3 = 24(m+1)x0 − 2 4(m+1)y0 + 16 ( 24m − 1 5 ) + 3 = 24(m+1)x0 − 2 4(m+1)y0 + dm+1. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 143 thus x8(m+1)+2 = x8m+10 = |x8m+9| − y8m+9 − 1 = −(−24(m+1)x0 − 2 4(m+1)y0 − 3dm+1 − 1) −(24(m+1)x0 − 2 4(m+1)y0 + dm+1) − 1 = 24(m+1)+1y0 + 2dm+1 and y8(m+1)+2 = y8m+10 = x8m+9 + |y8m+9| = −24(m+1)x0 − 2 4(m+1)y0 − 3dm+1 − 1 + (−2 4(m+1)x0 + 2 4(m+1)y0 − dm+1) = −24(m+1)+1x0 − 4dm+1 − 1. then x8(m+1)+3 = x8m+11 = |x8m+10| − y8m+10 − 1 = −24(m+1)+1y0 − 2dm+1 + 2 4(m+1)+1x0 + 4dm+1 + 1 − 1 = 24(m+1)+1x0 − 2 4(m+1)+1y0 + 2dm+1 and y8(m+1)+3 = y8m+11 = x8m+10 + |y8m+10| = 24(m+1)+1y0 + 2dm+1 + 2 4(m+1)+1x0 + 4dm+1 + 1 = 24(m+1)+1x0 + 2 4(m+1)+1y0 + 6dm+1 + 1. 144 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) hence x8(m+1)+4 = x8m+12 = |x8m+11| − y8m+11 − 1 = −24(m+1)+1x0 + 2 4(m+1)+1y0 − 2dm+1 − 2 4(m+1)+1x0 −24(m+1)+1y0 − 6dm+1 − 2 = −24(m+1)+2x0 − 8dm+1 − 2 and y8(m+1)+4 = y8m+12 = x8m+11 + |y8m+11| = 24(m+1)+1x0 − 2 4(m+1)+1y0 + 2dm+1 − 2 4(m+1)+1x0 −24(m+1)+1y0 − 6dm+1 − 1 = −24(m+1)+2y0 − 4dm+1 − 1. thus x8(m+1)+5 = x8m+13 = |x8m+12| − y8m+12 − 1 = 24(m+1)+2x0 + 8dm+1 + 2 + 2 4(m+1)+2y0 + 4dm+1 + 1 − 1 = 24(m+1)+2x0 + 2 4(m+1)+2y0 + 12dm+1 + 2 and y8(m+1)+5 = y8m+13 = x8m+12 + |y8m+12| = −24(m+1)+2x0 − 8dm+1 − 2 + 2 4(m+1)+2y0 + 4dm+1 + 1 = −24(m+1)+2x0 + 2 4(m+1)+2y0 − 4dm+1 − 1. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 145 hence x8(m+1)+6 = x8m+14 = |x8m+13| − y8m+13 − 1 = −24(m+1)+2x0 − 2 4(m+1)+2y0 − 12dm+1 − 2 +24(m+1)+2x0 − 2 4(m+1)+2y0 + 4dm+1 + 1 − 1 = −24(m+1)+3y0 − 8dm+1 − 2 and y8(m+1)+6 = y8m+14 = x8m+13 + |y8m+13| = 24(m+1)+2x0 + 2 4(m+1)+2y0 + 12dm+1 + 2 + 2 4(m+1)+2x0 −24(m+1)+2y0 + 4dm+1 + 1 = 24(m+1)+3x0 + 16dm+1 + 3. then x8(m+1)+7 = x8m+15 = |x8m+14| − y8m+14 − 1 = 24(m+1)+3y0 + 8dm+1 + 2 − 2 4(m+1)+3x0 − 16dm+1 − 3 − 1 = −24(m+1)+3x0 + 2 4(m+1)+3y0 − 8dm+1 − 2 and y8(m+1)+7 = y8m+15 = x8m+14 + |y8m+14| = −24(m+1)+3y0 − 8dm+1 − 2 − 2 4(m+1)+3x0 − 16dm+1 − 3 = −24(m+1)+3x0 − 2 4(m+1)+3y0 − 24dm+1 − 5. 146 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) thus x8(m+1)+8 = x8m+16 = |x8m+15| − y8m+15 − 1 = 24(m+1)+3x0 − 2 4(m+1)+3y0 + 8dm+1 + 2 +24(m+1)+3x0 + 2 4(m+1)+3y0 + 24dm+1 + 5 − 1 = 24(m+1)+4x0 + 32dm+1 + 6 and y8(m+1)+8 = y8m+16 = x8m+15 + |y8m+15| = −24(m+1)+3x0 + 2 4(m+1)+3y0 − 8dm+1 − 2 +24(m+1)+3x0 + 2 4(m+1)+3y0 + 24dm+1 + 5 = 24(m+1)+4y0 + 16dm+1 + 3 and so p(m + 1) is true. thus the proof of the claim is complete. that is, p(m) is true for 0 ≤ m ≤ k − 1. in particular, p(k − 1) is true. thus x8k = x8(k−1)+8 = 2 4(k−1)+4x0 + 32dk−1 + 6 and y8k = y8(k−1)+8 = 2 4(k−1)+4y0 + 16dk−1 + 3. cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 147 hence x8k+1 = |x8k| − y8k − 1 = −24kx0 − 32dk−1 − 6 − 2 4ky0 − 16dk−1 − 3 − 1 = −24kx0 − 2 4ky0 − 48dk−1 − 10 = −24kx0 − 2 4ky0 − 48 ( 24(k−1) − 1 5 ) − 10 = −24kx0 − 2 4ky0 − 3 · 24k 5 + 3 5 − 1 = −24kx0 − 2 4ky0 − 3dk − 1 and y8k+1 = x8k + |y8k| = 24kx0 + 32dk−1 + 6 − 2 4ky0 − 16dk−1 − 3 = 24kx0 − 2 4ky0 + 16dk−1 + 3 = 24kx0 − 2 4ky0 + 16 ( 24(k−1) − 1 5 ) + 3 = 24kx0 − 2 4ky0 + 24k 5 − 24 5 + 3 = 24kx0 − 2 4ky0 + dk. hence x8k+2 = |x8k+1| − y8k+1 − 1 = 24kx0 + 2 4ky0 + 3dk + 1 − 2 4kx0 + 2 4ky0 − dk − 1 = 24k+1y0 + 2dk 148 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) and y8k+2 = x8k+1 + |y8k+1| = −24kx0 − 2 4ky0 − 3dk − 1 − 2 4kx0 + 2 4ky0 − dk = −24k+1x0 − 4dk − 1. recall that (x0, y0) ∈ [ak, bk] × [ck, dk] \ [ak+1, bk+1] × [ck+1, dk+1] = [ −24k−2 − 1 5 · 24k−3 , −24k + 1 5 · 24k−1 ] × [ −24k−2 − 1 5 · 24k−2 , −24k + 1 5 · 24k ] \ [ −24(k+1)−2 − 1 5 · 24(k+1)−3 , −24(k+1) + 1 5 · 24(k+1)−1 ] × [ −24(k+1)−2 − 1 5 · 24(k+1)−2 , −24(k+1) + 1 5 · 24(k+1) ] . suppose (x0, y0) ∈ [ak, ak+1) × [ck, dk]. hence y8k+2 > −2 4k+1 (ak+1) − 4dk − 1 = −24k+1 ( −24(k+1)−2 − 1 5 · 24(k+1)−3 ) − 4dk − 1 = 28k+3 5 · 24k+1 + 24(k+1) − 1 5 · 24(k+1)−3 − 24(k+2) 5 + 4 5 − 1 = 0 which is a contradiction. next suppose (x0, y0) ∈ [ak+1, bk] × [ck, ck+1). then x8k+3 = |x8k+2| − y8k+2 − 1 = −24k+1y0 − 2dk + 2 4k+1x0 + 4dk + 1 − 1 = 24k+1x0 − 2 4k+1y0 + 2dk cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 149 and y8k+3 = x8k+2 + |y8k+2| = 24k+1y0 + 2dk + 2 4k+1x0 + 4dk + 1 = 24k+1x0 + 2 4k+1y0 + 6dk + 1. hence x8k+4 = |x8k+3| − y8k+3 − 1 = −24k+1x0 + 2 4k+1y0 − 2dk − 2 4k+1x0 − 2 4k+1y0 − 6dk − 1 − 1 = −24k+2x0 − 8dk − 2 and y8k+4 = x8k+3 + |y8k+3| = 24k+1x0 − 2 4k+1y0 + 2dk − 2 4k+1x0 − 2 4k+1y0 − 6dk − 1 = −24k+2y0 − 4dk − 1. recall that (x0, y0) ∈ [ak+1, bk] × [ck, ck+1). thus y8k+4 > −2 4k+2(ck+1) − 4dk − 1 = −24k+2 ( −24k+2 − 1 5 · 24k+2 ) − 4 ( 24k − 1 5 ) − 1 = 28k+4 5 · 24k+2 + 24k+2 5 · 24k+2 − 24k+2 5 + 4 5 − 1 = 0 which is a contradiction. now suppose that (x0, y0) ∈ (bk+1, bk] × [ck+1, dk]. 150 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) hence x8k+5 = |x8k+4| − y8k+4 − 1 = 24k+2x0 + 8dk + 2 + 2 4k+2y0 + 4dk + 1 − 1 = 24k+2x0 + 2 4k+2y0 + 12dk + 2 and y8k+5 = x8k+4 + |y8k+4| = −24k+2x0 − 8dk − 2 + 2 4k+2y0 + 4dk + 1 = −24k+2x0 + 2 4k+2y0 − 4dk − 1. then x8k+6 = |x8k+5| − y8k+5 − 1 = −24k+2x0 − 2 4k+2y0 − 12dk − 2 + 2 4k+2x0 − 2 4k+2y0 + 4dk + 1 − 1 = −24k+3y0 − 8dk − 2 and y8k+6 = x8k+5 + |y8k+5| = 24k+2x0 + 2 4k+2y0 + 12dk + 2 + 2 4k+2x0 − 2 4k+2y0 + 4dk + 1 = 24k+3x0 + 16dk + 3. recall that (x0, y0) ∈ (bk+1, bk] × [ck+1, dk] . thus y8k+6 > 2 4k+3 (bk+1) + 16 ( 24k − 1 5 ) + 3 = 24k+3 ( −24(k+1) + 1 5 · 24(k+1)−1 ) + 16 ( 24k − 1 5 ) + 3 cubo 14, 2 (2012) on the global behavior of xn+1 = |xn| − yn − 1 and yn+1 = xn + |yn| 151 = −24k+4 5 + 1 5 + 24k+4 5 − 16 5 + 3 = 0 which is a contradiction. finally, suppose (x0, y0) ∈ [ak+1, bk+1] × (dk+1, dk]. thus x8k+7 = |x8k+6| − y8k+6 − 1 = 24k+3y0 + 8dk + 2 − 2 4k+3x0 − 16dk − 3 − 1 = −24k+3x0 + 2 4k+3y0 − 8dk − 2 and y8k+7 = x8k+6 + |y8k+6| = −24k+3y0 − 8dk − 2 − 2 4k+3x0 − 16dk − 3 = −24k+3x0 − 2 4k+3y0 − 24dk − 5. hence x8k+8 = |x8k+7| − y8k+7 − 1 = 24k+3x0 − 2 4k+3y0 + 8dk + 2 + 2 4k+3x0 + 2 4k+3y0 + 24dk + 5 − 1 = 24k+3x0 + 32dk + 6 and y8k+8 = x8k+7 + |y8k+7| = −24k+3x0 + 2 4k+3y0 − 8dk − 2 + 2 4k+3x0 + 2 4k+3y0 + 24dk + 5 = 24k+4y0 + 16dk + 3. recall that (x0, y0) ∈ [ak+1, bk+1] × (dk+1, dk]. 152 e.a. grove, e. lapierre and w. tikjha cubo 14, 2 (2012) thus y8k+8 > 2 4k+4 (dk+1) + 16 ( 24k − 1 5 ) + 3 > 24k+4 ( −24(k+1) + 1 5 · 24(k+1) ) + 16 ( 24k − 1 5 ) + 3 = − 24k+4 5 + 1 5 + 24k+4 5 − 16 5 + 3 = 0 which is a contradiction. the proof is complete. received: november 2011. revised: november 2011. references [1] e. camouzis, and g. ladas, dynamics of third-order rational difference equations 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[2] r.l. devaney, a piecewise linear model of the the zones of instability of an area-preserving map, physica 10d (1984), 387-393. [3] m.r.s. kulenovic, and o. merino, discrete dynamical systems and difference equations with mathematica, chapman & hall/crc, new york, 2002. [4] h.o. peitgen and d. saupe, (eds.) the science of fractal images, springer-verlog, new york, 1991. [5] w. tikjha, y. lenbury, and e. g. lapierre, on the global character of the system of piecewise linear difference equations xn+1 = |xn|−yn −1 and yn+1 = xn − |yn|, advances in difference equations, volume 2010 (2010), article id 573281. introduction the global behavior of the solutions of system(1.1) () cubo a mathematical journal vol.13, no¯ 02, (1–35). june 2011 homogeneous besov spaces associated with the spherical mean operator l.t.rachdi and a.rouz department of mathematics, faculty of sciences of tunis, 2092 el manar 2 tunis, tunisia. email: ahlemrouz@yahoo.fr abstract we define and study homogeneous besov spaces associated with the spherical mean operator. we establish some results of completeness, continuous embeddings and density of subspaces. next, we define a discrete equivalent norm on this space and we give other properties. resumen definimos y estudiamos los espacios homogneos besov asociados con el operador esférico medio. se establecen algunos resultados de la exhaustividad, de inclusiones continuas y de la densidad de subespacios. a continuación, se define una norma equivalente discreta en este espacio y se dan otras propiedades. keywords and phrases:: spherical mean operator, besov space, banach space, fourier transform. mathematics subject classification: 46e35 , 44a35. 2 l.t.rachdi and a.rouz cubo 13, 2 (2011) 1 introduction for a continuous function f on r × rn, even with respect to the first variable, the spherical mean operator r is defined as r(f)(r,x) = ∫ sn f(rη,x + rξ)dσn(η,ξ); (r,x) ∈ r × rn, where sn is the unit sphere, i.e. sn = {(η,ξ) ∈ r × rn ; η2 + |ξ|2 = 1} and σn is the surface measure on sn normalized to have total measure one. the dual of the spherical mean operator tr is defined by t r(g)(r,x) = γ ( n+1 2 ) π n+1 2 ∫ rn g( √ r2 + |x − y|2,y)dy, where dy is the lebesgue measure on rn. the spherical mean operator r and its dual tr play an important role and have many applications, for example, in image processing of so-called synthetic aperture radar (sar) data [14, 15], or in the linearized inverse scattering problem in acoustics [9]. many aspects of such operator have been studied [1, 3, 6, 18, 21]. in particular, in [18] the first author with the others associated to the spherical mean operator the fourier transform defined by ∀(µ,λ) ∈ γ, f (f)(µ,λ) = ∫ rn ∫ +∞ 0 f(r,x) ϕµ,λ(r,x) dνn(r,x), where • ϕµ,λ is the function defined by ∀(r,x) ∈ r × rn, ϕµ,λ(r,x) = r ( cos(µ.)e−i〈λ|.〉 ) (r,x). • νn is the measure defined on [0, +∞[ × rn, by dνn(r,x) = 1 2 n−1 2 γ(n+1 2 )(2π) n 2 rn dr ⊗ dx. • γ is the set given by γ = r × rn ∪ {(iµ,λ); (µ,λ) ∈ r × rn, |µ| 6 |λ|} . they have constructed the harmonic analysis related to the fourier transform f (inversion formula, schwartz theorem, paley-wiener theorem, plancherel theorem). there are many ways to define besov spaces [4, 5, 13, 16, 20, 23]. it is well known that besov spaces can be defined for instance in terms of convolutions f ∗ φt with different kinds of smooth functions φ and that can be also described by means of differences △xf [10, 11, 22]. cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 3 in this work, we define and study a class of homogeneous besov spaces connected with the spherical mean operator r. more precisely, let φ be a smooth function on r×rn, even with respect to the first variable. for all p,q ∈ [1, +∞] and γ ∈ r, we define the besov space bγ,φp,q ( [0, +∞[ × r n ) to be the space of tempered distributions f on r × rn, even with respect to the first variable such that f = ∫ +∞ 0 f ∗ φt ∗ φt dt t , where ∗ is the convolution product associated with the spherical mean operator and φt; t > 0 is the dilated function of φ defined by ∀(r,x) ∈ [0, +∞[ × rn, φt(r,x) = 1 t2n+1 φ( r t , x t ) (see definition 10 below). the space b γ,φ p,q ( [0, +∞[ × rn ) is equipped firstly with the norm mγ,φp,q (f) =    (∫ +∞ 0 (‖f ∗ φt‖p,νn tγ )qdt t ) 1 q , if 1 6 q < +∞; esssup t>0 ‖f ∗ φt‖p,νn tγ , if q = +∞. with ‖f ∗ φt‖p,νn =    (∫ rn ∫ +∞ 0 |f ∗ φt(r,x)|p dνn(r,x) ) 1 p , if p ∈ [1, +∞[ ; esssup (r,x)∈[0,+∞ [×rn |f ∗ φt(r,x)|, if p = +∞. 4 l.t.rachdi and a.rouz cubo 13, 2 (2011) then we have established the coming results • the besov space bγ,φp,q ( [0, +∞[ × rn ) is independent of the choice of the function φ and will be denoted by b γ p,q ( [0, +∞[ × rn ) . this means that for all smooth functions φ and ψ, there exists a positive constant cφ,ψ such that ∀f ∈ bγ,φp,q ( [0, +∞[ × rn ) , mγ,φp,q (f) 6 cφ,ψ m γ,ψ p,q (f). • the space bγp,q ( [0, +∞[ × rn ) is homogeneous with degree equal to (2n + 1)/p − γ − 2n − 1, that is for all f ∈ bγp,q ( [0, +∞[ × rn ) and t > 0, the distribution ft belongs to the space b γ p,q ( [0, +∞[×rn ) and we have mγ,φp,q (ft) = t 2n+1 p −γ−2n−1 mγ,φp,q (f). • the besov space is a banach one when γ < (2n + 1)/p. we have also proved some continuous embeddings and density of subspaces. next, we define the following discrete norm on the space b γ p,q ( [0, +∞[ × rn ) by setting nγ,φp,q (f) =    ( ∑ k∈z (‖f ∗ φ2k‖p,νn 2kγ )q ) 1 q , if 1 6 q < +∞; esssup k∈z ‖f ∗ φ2k‖p,νn 2kγ , if q = +∞. we show that this norm defines the same topology as the norm m γ,φ p,q . we prove that this space is homogeneous in a weaker sense when equipped with the norm n γ,φ p,q , that is there exist two positive constants c1 and c2 such that for all f ∈ bγp,q ( [0, +∞[ × rn ) and t > 0 c1 t 2n+1 p −2n−1−γ nγ,φp,q (f) 6 n γ,φ p,q (ft) 6 c2 t 2n+1 p −2n−1−γ nγ,φp,q (f). finally, we establish some new continuous embedding. 2 fourier transform associated with the spherical mean operator in this section, we recall some harmonic analysis results related to the fourier transform associated with the spherical mean operator. let ϕµ,λ, (µ,λ) ∈ c × cn, be the function defined by ∀(r,x) ∈ r × rn, ϕµ,λ(r,x) = r ( cos(µ.)e−i〈λ|.〉 ) (r,x). it’s well known ([18, 21]) that cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 5 i. the function ϕµ,λ is given by ∀(r,x) ∈ r × rn, ϕµ,λ(r,x)e−i〈λ|x〉j(n−1)/2(r √ µ2 + λ2 1 + . . . + λ2n), where j(n−1)/2 is the modified bessel function defined by jn−1 2 (s) = 2 n−1 2 γ ( n + 1 2 )jn−1 2 (s) s n−1 2 = γ ( n + 1 2 ) +∞∑ k=0 (−1)k k!γ (α + k + 1) ( s 2 )2k , and j(n−1)/2 is the bessel function of first kind and index (n − 1)/2 [7, 8, 17, 27]. ii. for all (µ,λ) ∈ c × cn, ϕµ,λ is the unique infinitely differentiable function on r × rn, even with respect to the first variable, satisfying    dju(r,x1, ...,xn) = −iλju(r,x1, ...,xn), 1 6 j 6 n, ξu(r,x1, ...,xn) = −µ 2u(r,x1, ...,xn), u(0,...,0) = 1, ∂u ∂r (0,x1, ...,xn) = 0, ∀(x1, ...,xn) ∈ rn. where dj = ∂ ∂xj ; 1 6 j 6 n, and ξ = ∂2 ∂r2 + n r ∂ ∂r − n∑ j=1 d2j . (2.1) iii. the function ϕµ,λ is bounded on r × rn if, and only if (µ,λ) belongs to the set γ given by γ = r × rn ∪ {(iµ,λ); (µ,λ) ∈ r × rn, |µ| 6 |λ|} . (2.2) in this case, we have sup (r,x)∈r×rn |ϕµ,λ(r,x)| = 1. we denote by • lp(dνn), p ∈ [1, +∞] , the space of measurable functions f on [0, +∞[ × rn, such that ‖f‖ p,νn =    (∫ rn ∫ +∞ 0 |f(r,x)| p dνn(r,x) ) 1 p < +∞, if p ∈ [1, +∞[ ; esssup (r,x)∈[0,+∞ [×rn |f(r,x)| < +∞, if p = +∞, where νn is the measure defined in the introduction. • γ+ the subset of γ given by γ+ = [0, +∞[ × rn ∪ {(iµ,λ); (µ,λ) ∈ r × rn, 0 6 µ 6 |λ|} . 6 l.t.rachdi and a.rouz cubo 13, 2 (2011) • bγ+ the σ-algebra on γ+ defined by bγ+ = θ−1(b[0,+∞ [×rn ), where θ is the bijective function defined on γ+ by θ(µ,λ) = ( √ µ2 + λ2,λ). (2.3) • γn the measure defined on γ+ by γn(a) = νn(θ(a)); a ∈ bγ+. • lp(dγn), p ∈ [1, +∞] , the space of measurable functions on γ+ satisfying ‖f‖ p,γn < +∞. then we have the coming properties proposition 1. i) for all non negative measurable function f on γ+ (respectively integrable on γ+ with respect to the measure dγn), we have ∫ ∫ γ+ f(µ,λ)dγn(µ,λ) 1 2 n−1 2 γ(n+1 2 )(2π) n 2 { ∫ rn ∫ +∞ 0 f(µ,λ) ( µ2 + |λ|2 )n−1 2 µdµdλ + ∫ rn ∫ |λ| 0 f(iµ,λ) ( |λ|2 − µ2 )n−1 2 µdµdλ } . ii) for all non negative measurable function g on [0, +∞[×rn (respectively integrable on [0, +∞[× r n with respect to the measure dνn), the function g◦θ is measurable positive on γ+ (respectively integrable on γ+ with respect to the measure dγn) and we have ∫ rn ∫ ∞ 0 g(r,x)dνn(r,x) = ∫ ∫ γ+ g ◦ θ(µ,λ)dγn(µ,λ). in the following, we shall define the translation operator and the convolution product associated with the spherical mean operator. for this, we use the product formula for the function ϕµ,λ, for all (r,x), (s,y) ∈ r × rn, we have ϕµ,λ(r,x)ϕµ,λ(s,y) γ(n+1 2 ) √ π γ ( n 2 ) ∫π 0 ϕµ,λ (√ r2 + s2 + 2rs cos θ,x + y ) sinn−1(θ)dθ (2.4) definition 2. i) for all (r,x) ∈ [0, +∞[×rn, the translation operator τ(r,x) associated with the spherical mean operator is defined on lp(dνn), p ∈ [1, +∞] , by τ(r,x)(f)(s,y) = γ(n+1 2 ) √ π γ ( n 2 ) ∫π 0 f (√ r2 + s2 + 2rs cos θ,x + y ) sinn−1(θ)dθ. cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 7 ii) the convolution product of f,g ∈ l1(dνn) is defined by ∀(r,x) ∈ [0, +∞[ × rn, f ∗ g(r,x) = ∫ rn ∫ +∞ 0 f(s,y)τ(r,−x)(ǧ)(s,y)dνn(s,y), where ǧ(s,y) = g(s, −y). we have the following properties • for all (r,x), (s,y) ∈ [0, +∞[ × rn, the relation (2.4) can be written τ(r,x)(ϕµ,λ)(s,y)ϕµ,λ(r,x) ϕµ,λ(s,y). (2.5) • if f ∈ lp(dνn), 1 6 p 6 +∞, then for all (s,y) ∈ [0, +∞[ × rn, the function τ(s,y)(f) belongs to lp(dνn) and we have ∥∥τ(s,y)(f) ∥∥ p,νn 6 ‖f‖ p,νn . (2.6) • let p, q, r ∈ [1, +∞] such that 1 r = 1 p + 1 q − 1. then for all f ∈ lp(dνn) and g ∈ lq(dνn), the function f ∗ g belongs to lr(dνn) and we have ‖f ∗ g‖ r,νn 6 ‖f‖ p,νn ‖g‖ q,νn . (2.7) now, we will define the fourier transform f connected with the spherical mean operator and we recall some properties that we need in the next section. definition 3. the fourier transform associated with the spherical mean operator is defined on l1(dνn) by ∀(µ,λ) ∈ γ, f (f)(µ,λ) = ∫ rn ∫ +∞ 0 f(r,x) ϕµ,λ(r,x) dνn(r,x), where γ is the set defined by the relation (2.2). the fourier transform f satisfies the properties • for every f in l1(dνn) and (r,x) ∈ [0, +∞[ × rn, we have ∀(µ,λ) ∈ γ, f ( τ(r,−x)(f) ) (µ,λ)ϕµ,λ(r,x)f (f)(µ,λ). (2.8) • for all f,g ∈ l1(dνn), we have ∀(µ,λ) ∈ γ, f (f ∗ g) (µ,λ) = f (f)(µ,λ)f (g)(µ,λ). (2.9) • for all f ∈ l1(dνn), we have ∀(µ,λ) ∈ γ, f (f) (µ,λ)f̃ (f) ◦ θ(µ,λ), (2.10) 8 l.t.rachdi and a.rouz cubo 13, 2 (2011) where ∀(µ,λ) ∈ r × rn, f̃ (f) (µ,λ)= ∫ rn ∫ +∞ 0 f(r,x)jn−1 2 (rµ)e−i〈λ|x〉 dνn(r,x) (2.11) and θ is the function defined by the relation (2.3). theorem 4. (inversion formula for f ) let f ∈ l1(dνn) such that the function f (f) belongs to l1(dγn), then for almost every (r,x) ∈ [0, +∞[ × rn, we have f(r,x) = ∫ ∫ γ+ f (f)(µ,λ) ϕµ,λ(r,x) dγn(µ,λ). we denote by • e∗ (r × rn) the space of infinitely differentiable functions on r × rn, even with respect to the first variable. • s∗ (r × rn) the subspace of e∗ (r × rn) consisting of functions rapidly decreasing together with all their derivatives. • s∗ (γ) the space of functions f : γ −→ c infinitely differentiable, even with respect to the first variable and rapidly decreasing together with all their derivatives, i.e ∀k1,k2 ∈ n, ∀α ∈ nn, sup (µ,λ)∈γ ( 1 + µ2 + 2|λ|2 )k1 ∣∣∣ ( ∂ ∂µ )k2 dαλf(µ,λ) ∣∣∣ < +∞, where ∂f ∂µ (µ,λ)    ∂ ∂r (f(r,λ)) , if µ = r ∈ r 1 i ∂ ∂t (f(it,λ)) , if µ = it, |t| 6 |λ| and dαλ = ( ∂ ∂λ1 )α1 . . . ( ∂ ∂λn )αn . • s′∗ (r × rn) and s ′ ∗(γ) are respectively the topological dual spaces of s∗ (r × rn) and s∗(γ). each of these spaces is equipped with its usual topology. theorem 5. (schwartz theorem)[2, 18] i) the fourier transform f is a topological isomorphism from s∗(r × rn) onto s∗(γ). the inverse mapping is given by ∀(r,x) ∈ r × rn, f −1(f)(r,x) = ∫ ∫ γ+ f(µ,λ) ϕµ,λ(r,x) dγn(µ,λ). (2.12) ii) (plancherel formula) for all f,g ∈ s∗(r × rn), we have ∫ +∞ 0 ∫ rn f(r,x) g(r,x) dνn(r,x) = ∫ ∫ γ+ f (f)(µ,λ) f (g)(µ,λ) dγn(µ,λ). cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 9 in particular ‖f (f)‖ 2,γn ‖f‖ 2,νn . theorem 6. (plancherel theorem) the fourier transform f can be extended to an isometric isomorphism from l2(dνn) onto l 2(dγn). for t ∈ s′∗(r × rn), we put 〈f (t),ϕ〉 = 〈t, f −1(ϕ)〉; ϕ ∈ s∗(γ). (2.13) then from theorem 5, we get the following result corollary 7. the transform f defined by the relation (2.13) is a topological isomorphism from s ′ ∗(r × rn) onto s ′ ∗(γ). proposition 8. i) let f ∈ e∗ (r × rn) , f slowly increasing and let g ∈ s∗(r × rn). then the function f ∗ g belongs to the space e∗ (r × rn) . ii) for all f ∈ s∗(r × rn) and t ∈ s ′ ∗(r × rn). the function t ∗ f defined by ∀(r,x) ∈ r × rn, t ∗ f(r,x) = 〈t,τ(r,−x)(f̌)〉 belongs to the space e∗ (r × rn) and is slowly increasing. moreover, we have f ( tt∗f ) = f (f̌)f (t). 3 besov spaces this section contains the main result of this paper. indeed, we define and study a class of besov spaces b γ,φ p,q ( [0, +∞[×rn ) , where φ is a smooth function. we show that this space is independant of the choice of φ and is a banach space for γ < (2n+1)/p. next, we prove that b γ,φ p,q ( [0, +∞[ × r n ) is an homogeneous space with degree equal to (2n + 1)/p − γ − 2n − 1. lemma 9. let a, b, a1, b1 be real numbers such that 0 < a1 < a 0. ii) ∀(µ,λ) ∈ γ ; a2 6 µ2 + 2|λ|2 6 b2, f (ψ)(µ,λ) = c where c is a positive constante. iii) f (ψ)(µ,λ) = 0 if µ2 + 2|λ|2 > b21 or µ 2 + 2|λ|2 < a21. 10 l.t.rachdi and a.rouz cubo 13, 2 (2011) iv) for all (µ,λ) ∈ γ \ {(0,0)}, ∫ +∞ 0 ( f (ψ)(tµ,tλ) )2dt t = 1. proof. from uryshon’s lemma, there exists an infinitely differentiable function ω on r such that • ∀t ∈ r; 0 6 ω(t) 6 1. • ∀t ∈ [a,b]; ω(t) = 1. • supp(ω) ⊂]a1,b1[. let g be the function defined on r × rn by g(r,x) = ω (√ r2 + |x|2 ) (∫ +∞ 0 (ω(t))2 dt t )1 2 , then the function g belongs to the space s∗(r × rn). since, the transform f̃ defined by the relation (2.11) is a topological isomorphism from the space s∗(r × rn) onto itself [24, 25], then there exists ψ ∈ s∗(r × rn) such that f̃ (ψ) = g. thus, by the relation (2.10), we deduce that the function ψ satisfies the hypothesis of the lemma. we denote by • d∗(γ) the space of real infinitely differentiable functions g on γ, even with respect to the first variable such that, there exist two positive real numbers 0 < a b2. • s∗,0(r × rn) the subspace of s∗(r × rn) consisting of functions f such that f (f) belongs to the space d∗(γ). • s1∗,0(r × rn) the subspace of s∗,0(r × rn) formed by the functions f such that ∀(µ,λ) ∈ γ \ {(0,0)}, ∫ +∞ 0 ( f (f)(tµ,tλ) )2dt t = 1. (3.1) these functions are known as wavelets on [0, +∞[ × rn [19, 26]. cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 11 • lp(dt t ); p ∈ [1, +∞], the space of measurable functions on ]0, +∞[ such that ∥∥f ∥∥ lp( dt t ) =    (∫ +∞ 0 ∣∣f(t) ∣∣pdt t ) 1 p < +∞, 1 6 p < +∞; esssup t>0 ∣∣f(t) ∣∣ < +∞, p = +∞. • ⋆ the convolution product defined on the group ( ]0, +∞[, . ) by f ⋆ g(s) = ∫ +∞ 0 f(t) g( s t ) dt t . (3.2) • for all measurable function φ on [0, +∞[ × rn, the dilated φt; t > 0 of φ is defined by ∀(r,x) ∈ [0, +∞[ × rn, φt(r,x) = 1 t2n+1 φ( r t , x t ). then we have the following properties • let p, q, r ∈ [1, +∞] such that 1 p + 1 q = 1 + 1 r . then for all f ∈ lp(dt t ) and g ∈ lq(dt t ), the function f ⋆ g belongs to lr(dt t ) and we have ∥∥f ⋆ g ∥∥ lr( dt t ) 6 ‖f‖lp( dt t )‖g‖lq( dt t ). (3.3) • for every φ ∈ lp(dνn); p ∈ [1, +∞] , the function φt belongs to lp(dνn) and we have ∥∥φt ∥∥ p,νn = t − 2n+1 p ′ ∥∥φ ∥∥ p,νn , (3.4) where p ′ = p/(p − 1). • for all φ ∈ l1(dνn) and for every (µ,λ) ∈ γ, f (φt)(µ,λ) = f (φ)(tµ,tλ). (3.5) definition 10. let p, q ∈ [1, +∞] , γ ∈ r and φ ∈ s1∗,0(r × rn). we define the besov space b γ,φ p,q ( [0, +∞[ × rn ) to be the space of tempered distributions f on r × rn, even with respect to the first variable and satisfying • for all t > 0, the function f ∗ φt belongs to the space lp(dνn). • the function t 7−→ ‖f ∗ φt‖p,νn tγ belongs to the space lq(dt t ). • the integral (r,x) 7−→ ∫ +∞ 0 f ∗ φt ∗ φt(r,x) dt t 12 l.t.rachdi and a.rouz cubo 13, 2 (2011) is convergent in s ′ ∗(r × rn) and f = ∫ +∞ 0 f ∗ φt ∗ φt dt t . (3.6) the space b γ,φ p,q ( [0, +∞[ × rn ) is equipped with the norm mγ,φp,q (f) =    (∫ +∞ 0 (‖f ∗ φt‖p,νn tγ )qdt t ) 1 q , if 1 6 q < +∞; esssup t>0 ‖f ∗ φt‖p,νn tγ , if q = +∞. lemma 11. let ψ ∈ s∗(r × rn) and let φ ∈ s∗,0(r × rn). then for all k ∈ n, there exists φk ∈ s∗,0(r × rn) such that ψ ∗ φt = t2k ( ∆kψ ) ∗ (φk)t, where ∆ is the differential operator defined by ∆ = − ( ∂2 ∂r2 + n r ∂ ∂r + n∑ j=1 ( ∂ ∂xj )2) . moreover, for all p ∈ [1, +∞] ∥∥ψ ∗ φt ∥∥ p,νn 6 t2k ∥∥∆kψ ∥∥ p,νn ∥∥φk ∥∥ 1,νn (3.7) and ∥∥ψ ∗ φt ∥∥ p,νn 6 t − 2n+1 p ′ ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p,νn . (3.8) proof. the operator ∆ is continuous from s∗(r × rn) into itself and for all f ∈ s∗(r × rn), we have f ( ∆f ) (µ,λ) = ( µ2 + 2|λ|2 ) f (f)(µ,λ). (3.9) let ψ ∈ s∗(r × rn) and let φ ∈ s∗,0(r × rn). from the relations (2.9) and (3.5), we get f ( ψ ∗ φt ) (µ,λ) = f (ψ)(µ,λ) f (φ)(tµ,tλ) = t2 ( µ2 + 2|λ|2 ) f (ψ)(µ,λ) f (φ)(tµ,tλ) t2 ( µ2 + 2|λ|2 ), and from the equality (3.9), we obtain f ( ψ ∗ φt ) (µ,λ) = t2f (∆ψ)(µ,λ) f (φ)(tµ,tλ) t2 ( µ2 + 2|λ|2 ). (3.10) cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 13 since, the function φ belongs to the space s∗,0(r × rn) then the function (µ,λ) 7−→ f (φ)(µ,λ) µ2 + 2|λ|2 belongs to the space s∗(γ) and from theorem 5, there exists φ1 ∈ s∗(r × rn) such that f (φ1)(µ,λ) = f (φ)(µ,λ) µ2 + 2|λ|2 . in particular, φ1 lies in s∗,0(r × rn) and the relation (3.10) leads to f ( ψ ∗ φt ) (µ,λ) = t2f (∆ψ)(µ,λ) f ( (φ1)t ) (µ,λ), which implies that ψ ∗ φt = t2 (∆ψ) ∗ (φ1)t. by induction, for all k ∈ n∗, there exists φk ∈ s∗,0(r × rn) verifying ψ ∗ φt = t2k (∆kψ) ∗ (φk)t. (3.11) on the other hand, for every t > 0 and by the relation (3.4), we get ∥∥ψ ∗ φt ∥∥ p,νn 6 ‖ψ‖1,νn ‖φt‖p,νn = t − 2n+1 p ′ ‖ψ‖1,νn ‖φ‖p,νn as the same way and using the relation (3.11), it follows that ∥∥ψ ∗ φt ∥∥ p,νn 6 t2k ‖∆kψ‖p,νn‖φk‖1,νn. proposition 12. let φ ∈ s1∗,0(r × rn). i) for all f ∈ l2(dνn) we have f = ∫ +∞ 0 f ∗ φt ∗ φt dt t ; in l2(dνn). ii) let γ ∈ r; γ < (2n + 1)/p and f ∈ s′∗(r × rn) such that for all t > 0, the function f ∗ φt belongs to lp(dνn) and the function t 7−→ ‖f ∗ φt‖p,νn tγ belongs to the space lq(dt t ). then the integral ∫ +∞ 0 f ∗ φt ∗ φt dt t converges in s ′ ∗(r × rn). 14 l.t.rachdi and a.rouz cubo 13, 2 (2011) proof. i) let f ∈ l2(dνn) and let fa,b(f) be the function defined by ∀(r,x) ∈ [0, +∞[ × rn, fa,b(f)(r,x) = ∫b a f ∗ φt ∗ φt(r,x) dt t ; 0 < a < b. the function fa,b(f) is well defined and by the relation (3.4) we have ∣∣fa,b(f)(r,x) ∣∣ 6 ∫b a ‖f‖2,νn ‖φt ∗ φt‖2,νn dt t 6 ‖f‖2,νn ∫b a ‖φt‖1,νn ‖φt‖2,νn dt t 6 ‖f‖2,νn ‖φ‖1,νn ‖φ‖2,νn ∫b a t− 2n+1 2 −1dt < +∞. moreover, the function fa,b(f) belongs to l 2(dνn). indeed by minkowski’s inequality [12] and the relation (3.4) we get ∥∥fa,b(f) ∥∥ 2,νn 6 ∫b a ∥∥f ∗ φt ∗ φt ∥∥ 2,νn dt t 6 ∫b a ‖f‖2,νn ‖φt‖21,νn dt t = ‖f‖2,νn ‖φ‖21,νn log( b a ) < +∞. on the other hand, by fubini’s theorem and the relation (3.5), we have f ( fa,b(f) ) (µ,λ) = f (f)(µ,λ) ∫b a ( f (φ)(tµ,tλ) )2dt t . thus, by the plancherel theorem ∥∥f − fa,b(f) ∥∥2 2,νn = ∥∥f (f) − f (fa,b(f)) ∥∥2 2,γn = ∫ ∫ γ+ ∣∣f (f)(µ,λ) ∣∣2 ∣∣∣1 − ∫b a ( f (φ)(tµ,tλ) )2dt t ∣∣∣dγn(µ,λ). using the fact that ∫ +∞ 0 ( f (φ)(tµ,tλ) )2dt t = 1, we have ∀(µ,λ) ∈ γ\{(0,0)}, ∣∣∣1 − ∫b a ( f (φ)(tµ,tλ) )2dt t ∣∣∣ 6 1 and applying the dominated convergence theorem, we deduce that lim a→0+ b→ +∞ ∥∥f − fa,b(f) ∥∥ 2,νn = 0. cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 15 ii) let f be in s ′ ∗(r × rn) satisfying the hypothesis, then the function fa,b(f) defined above is bounded on r × rn. in fact ∣∣fa,b(f)(r,x) ∣∣ 6 ∫b a ∥∥f ∗ φt ∥∥ p,νn ∥∥φt ∥∥ p ′ ,νn dt t = ∥∥φ ∥∥ p ′ ,νn ∫b a ∥∥f ∗ φt ∥∥ p,νn tγ t − 2n+1 p +γ dt t 6 ∥∥φ ∥∥ p ′ ,νn [∫b a (∥∥f ∗ φt ∥∥ p,νn tγ )qdt t ] 1 q [∫b a t (− 2n+1 p +γ) q ′ dt t ] 1 q ′ < +∞, where q ′ is the conjugate exponent of q. thus for all a, b ∈ r; b > a > 0, the function fa,b(f) defines an element of s ′ ∗(r × rn). let ψ ∈ s∗(r × rn), by fubini’s theorem, we have 〈fa,b(f),ψ〉 = ∫ +∞ 0 ∫ rn { ∫b a f ∗ φt ∗ φt(r,x) ψ(r,x) dt t } dνn(r,x) = ∫b a { ∫ +∞ 0 ∫ rn f ∗ φt ∗ φt(r,x) ψ(r,x) dνn(r,x) } dt t = ∫b a { ∫ +∞ 0 ∫ rn ψ(r,x) [∫ +∞ 0 ∫ rn f ∗ φt(s,y) τ(r,−x)(φ̌t)(s,y) dνn(s,y) ] dνn(r,x) } dt t = ∫b a { ∫ +∞ 0 ∫ rn f ∗ φt(s,y) [∫ +∞ 0 ∫ rn ψ(r,x) τ(s,−y)(φt)(r,x) dνn(r,x) ] dνn(s,y) } dt t = ∫b a [∫ +∞ 0 ∫ rn f ∗ φt(s,y) φ̌t ∗ ψ(s,y) dνn(s,y) ] dt t . however, ∫ +∞ 0 [∫ +∞ 0 ∫ rn ∣∣f ∗ φt(s,y) ∣∣ ∣∣φ̌t ∗ ψ(s,y) ∣∣ dνn(s,y) ] dt t 6 ∫ +∞ 0 ∥∥f ∗ φt ∥∥ p,νn ∥∥φ̌t ∗ ψ ∥∥ p ′ ,νn dt t 6 ∫1 0 ∥∥f ∗ φt ∥∥ p,νn ∥∥φ̌t ∗ ψ ∥∥ p ′ ,νn dt t + ∫ +∞ 1 ∥∥f ∗ φt ∥∥ p,νn ∥∥φ̌t ∗ ψ ∥∥ p ′ ,νn dt t . 16 l.t.rachdi and a.rouz cubo 13, 2 (2011) using the relations (3.7) and (3.8), we get ∫ +∞ 0 { ∫ +∞ 0 ∫ rn ∣∣f ∗ φt(s,y) ∣∣ ∣∣φ̌t ∗ ψ(s,y) ∣∣ dνn(s,y) } dt t 6 ∥∥∆kψ ∥∥ p ′ ,νn ∥∥φk ∥∥ 1,νn ∫1 0 t2k ∥∥f ∗ φt ∥∥ p,νn dt t + ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p ′ ,νn ∫ +∞ 1 t − 2n+1 p ∥∥f ∗ φt ∥∥ p,νn dt t ∥∥∆kψ ∥∥ p ′ ,νn ∥∥φk ∥∥ 1,νn ∫ +∞ 0 t2k+γ 1[0,1](t) ∥∥f ∗ φt ∥∥ p,νn tγ dt t + ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p ′ ,νn ∫ +∞ 0 t − 2n+1 p +γ 1[1,+∞ [(t) ∥∥f ∗ φt ∥∥ p,νn tγ dt t . let k be sufficiently large. using the hypothesis γ < (2n + 1)/p and applying hölder’s inequality, we obtain ∫ +∞ 0 { ∫ +∞ 0 ∫ rn ∣∣f ∗ φt(s,y) ∣∣ ∣∣φ̌t ∗ ψ(s,y) ∣∣ dνn(s,y) } dt t 6 ∥∥∆kψ ∥∥ p ′ ,νn ∥∥φk ∥∥ 1,νn ∥∥∥t2k+γ 1[0,1] ∥∥∥ lq ′ ( dt t ) ∥∥∥ ∥∥f ∗ φt ∥∥ p,νn tγ ∥∥∥ lq( dt t ) + ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p ′ ,νn ∥∥∥t− 2n+1 p +γ 1[1,+∞ [ ∥∥∥ lq ′ ( dt t ) ∥∥∥ ∥∥f ∗ φt ∥∥ p,νn tγ ∥∥∥ lq( dt t ) < +∞. this shows that for all ψ ∈ s∗(r × rn), lim a→0+ b→ +∞ 〈fa,b(f),ψ〉 exists and lim a→0+ b→ +∞ 〈fa,b(f),ψ〉 = ∫ +∞ 0 ∫ +∞ 0 ∫ rn f ∗ φt(s,y) φ̌t ∗ ψ(s,y) dνn(s,y) dt t . this means that the integral ∫ +∞ 0 f ∗ φt ∗ φt dt t converges in s ′ ∗(r × rn). lemma 13. 1) let f ∈ bγ,φp,q ( [0, +∞[ × rn ) . then i) for all ψ ∈ s∗(r × rn), we have f ∗ ψ = ∫ ∞ 0 f ∗ φt ∗ φt ∗ ψ dt t . ii) for all ψ ∈ s1∗,0(r × rn), f = ∫ +∞ 0 f ∗ ψρ ∗ ψρ dρ ρ . cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 17 2) for all g ∈ s∗(r × rn) and for all ψ ∈ s1∗,0(r × rn), we have ∫ +∞ 0 g ∗ ψρ ∗ ψρ dρ ρ = g. proof. 1) let f ∈ bγ,φp,q ( [0, +∞[ × rn ) . i) for every ψ ∈ s∗(r × rn), we have f ∗ ψ(r,x) = 〈f,τ(r,−x)ψ̌〉 = lim a→0+ b→ +∞ 〈 ∫b a f ∗ φt ∗ φt dt t ,τ(r,−x)ψ̌〉 = lim a→0+ b→ +∞ ∫ +∞ 0 ∫ rn (∫b a f ∗ φt ∗ φt(s,y) dt t ) τ(r,−x)ψ̌(s,y) dνn(s,y), and by fubini’s theorem, we obtain f ∗ ψ(r,x) = lim a→0+ b→ +∞ ∫b a (∫ +∞ 0 ∫ rn f ∗ φt ∗ φt(s,y) τ(r,−x)ψ̌(s,y) dνn(s,y) ) dt t = lim a→0+ b→ +∞ ∫b a f ∗ φt ∗ φt ∗ ψ(r,x) dt t = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ψ(r,x) dt t . ii) let ψ ∈ s1∗,0(r × rn). for all positive real number ρ, we have ψρ ∗ ψρ = (ψ ∗ ψ)ρ. (3.12) applying i) we get f ∗ ψρ ∗ ψρ = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t . now, let a1, a2, b1, b2 be positive real numbers such that f (φ)(µ,λ) = 0 if µ2 + 2|λ|2 < a21 or µ 2 + 2|λ|2 > b21 and f (ψ)(µ,λ) = 0 if µ2 + 2|λ|2 < a22 or µ 2 + 2|λ|2 > b22 then f (φt)(µ,λ)f (ψρ)(µ,λ) = 0 if t ρ /∈ [ a1 b2 , b1 a2 ] = [α, β] , and consequently, by the relation (2.9) and theorem 4 φt ∗ ψρ = 0 if t ρ /∈ [α, β] . (3.13) 18 l.t.rachdi and a.rouz cubo 13, 2 (2011) thus, f ∗ ψρ ∗ ψρ = ∫ρβ ρα f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t . so for all a, b ∈ r; 0 < a < b, ∫b a f ∗ ψρ ∗ ψρ dρ ρ = ∫b a (∫ρβ ρα f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t ) dρ ρ . by fubini’s theorem, we get ∫b a f ∗ ψρ ∗ ψρ dρ ρ = ∫bβ aα (∫ t α t β f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dρ ρ ) dt t . (3.14) on the other hand, we have ∫ t α t β f ∗ φt ∗ φt ∗ ψρ ∗ ψρ(r,x) dρ ρ = ∫ t α t β (∫ +∞ 0 ∫ rn φt ∗ ψρ ∗ ψρ(s,y) τ(r,−x) ˇ(f ∗ φt)(s,y)dνn(s,y) ) dρ ρ = ∫ +∞ 0 ∫ rn τ(r,−x)( ˇf ∗ φt)(s,y) (∫ t α t β φt ∗ ψρ ∗ ψρ(s,y) dρ ρ ) dνn(s,y) = ∫ +∞ 0 ∫ rn τ(r,−x)( ˇf ∗ φt)(s,y) (∫ +∞ 0 φt ∗ ψρ ∗ ψρ(s,y) dρ ρ ) dνn(s,y). however by i) of proposition 12, it follows that ∫ t α t β f ∗ φt ∗ φt ∗ ψρ ∗ ψρ(r,x) dρ ρ = ∫ +∞ 0 ∫ rn φt(s,y) τ(r,−x)( ˇf ∗ φt)(s,y) dνn(s,y) = f ∗ φt ∗ φt(r,x). replacing in the equality (3.14), we obtain ∫b a f ∗ ψρ ∗ ψρ dρ ρ = ∫bβ aα f ∗ φt ∗ φt dt t . 2) we know that for all g ∈ s∗(r × rn) and ψ ∈ s1∗,0(r × rn), the function g ∗ ψρ ∗ ψρ belongs to the space s∗(r × rn). by theorem 4 and the relation (3.5), we have g ∗ ψρ ∗ ψρ(r,x) = ∫ ∫ γ+ f (g)(µ,λ) ( f (ψ)(ρµ,ρλ) )2 ϕµ,λ(r,x) dγn(µ,λ), then ∫ +∞ 0 g ∗ ψρ ∗ ψρ(r,x) dρ ρ ∫ ∫ γ+ f (g)(µ,λ)ϕµ,λ(r,x) [∫ +∞ 0 ( f (ψ)(ρµ,ρλ) )2 dρ ρ ] dγn(µ,λ), cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 19 and by the relation (3.1) and theorem 4, we get ∫ +∞ 0 g ∗ ψρ ∗ ψρ(r,x) dρ ρ = ∫ ∫ γ+ f (g)(µ,λ) ϕµ,λ(r,x) dγn(µ,λ) = g(r,x). theorem 14. let p, q ∈ [1, +∞] and γ ∈ r, the space bγ,φp,q ( [0, +∞[ × rn ) is independent of the choice of the function φ in s1∗,0 ( r × rn ) and will be denoted by b γ p,q ( [0, +∞[ × rn ) . proof. let f ∈ bγ,φp,q ( [0, +∞[ × rn ) and let ψ ∈ s1∗,0 ( r × rn ) . from lemma 13 and the relation (3.13), we have f ∗ ψρ = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ψρ dt t = ∫ρβ ρα f ∗ φt ∗ φt ∗ ψρ dt t = ∫β α f ∗ φρs ∗ φρs ∗ ψρ ds s . thus, from minkowski’s inequality and the relations (2.7) and (3.4), we get ∥∥f ∗ ψρ ∥∥ p,νn 6 ∫β α ∥∥f ∗ φρs ∗ ψρ ∗ φρs ∥∥ p,νn ds s 6 ∫β α ∥∥f ∗ φρs ∥∥ p,νn ∥∥ψρ ∗ φρs ∥∥ 1,νn ds s 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∫β α ∥∥f ∗ φρs ∥∥ p,νn ds s , (3.15) and by hölder’s inequality, it follows that ∥∥f ∗ ψρ ∥∥ p,νn 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn (∫β α (∥∥f ∗ φρs ∥∥ p,νn (ρs)γ )qds s ) 1 q (∫β α ( ρs )γ q′ ds s ) 1 q ′ 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn mγ,φp,q (f)ρ γ (∫β α sγ q ′ ds s ) 1 q ′ < +∞, where q ′ is the conjugate exponent of q. now, by the relation (3.15), we have ∥∥f ∗ ψρ ∥∥ p,νn ργ 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∫β α sγ ∥∥f ∗ φρs ∥∥ p,νn (ρs)γ ds s 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∫ 1 α 1 β t−γ ∥∥f ∗ φρ t ∥∥ p,νn ( ρ t )γ dt t = ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn [ t−γ1[ 1β, 1 α ] ⋆ (∥∥f ∗ φt ∥∥ p,νn tγ )] (ρ), 20 l.t.rachdi and a.rouz cubo 13, 2 (2011) where ⋆ is the convolution product defined on ]0, +∞[ by the relation (3.2). by the relation (3.3), we obtain mγ,ψp,q (f) 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∥∥∥t−γ1[ 1β, 1α ] ∥∥∥ l1( dt t ) mγ,φp,q (f) < +∞, and the proof is complete if we take into account lemma 13. proposition 15. let p, q ∈ [1, +∞] and γ ∈ r. the besov space b γ p,q ( [0, +∞[ × rn ) is homogeneous of degree equal to (2n + 1)/p − γ − 2n − 1, that is for every f ∈ bγp,q ( [0, +∞[ × rn ) and t > 0, the distribution ft belongs to the space b γ p,q ( [0, +∞[×rn ) and we have mγ,φp,q (ft) = t 2n+1 p −γ−2n−1 mγ,φp,q (f), where 〈ft,ϕ〉 = 〈f, 1 t2n+1 ϕ1 t 〉; ϕ ∈ s∗(r × rn). proof. let φ ∈ s1∗,0(r × rn), we have ft ∗ φρ(r,x) = 〈ft,τ(r,−x)(φ̌ρ)〉 = 〈f, 1 t2n+1 ( τ(r,−x)(φ̌ρ) ) 1 t 〉. however, 1 t2n+1 ( τ(r,−x)(φ̌ρ) ) 1 t (s,y) = τ(r,−x)(φ̌ρ)(ts,ty) = 1 t2n+1 τ( r t ,− x t )(φ̌ρ t )(s,y) consequently, ft ∗ φρ(r,x) = 〈f, 1 t2n+1 τ( r t ,− x t )(φ̌ρ t )〉 = ( f ∗ φρ t ) t (r,x). (3.16) hence, from the relation (3.4), we get ∥∥ft ∗ φρ ∥∥ p,νn = t − 2n+1 p ′ ∥∥f ∗ φρ t ∥∥ p,νn , this shows that for all ρ > 0, the function ft ∗ φρ belongs to lp(dνn) and we have ∥∥∥ ∥∥ft ∗ φρ ∥∥ p,νn ργ ∥∥∥ q lq( dρ ρ ) = t − 2n+1 p ′ q ∫ +∞ 0 (∥∥f ∗ φρ t ∥∥ p,νn ργ )q dρ ρ = t − 2n+1 p ′ q t−γ q ∫ +∞ 0 (∥∥f ∗ φs ∥∥ p,νn sγ )q ds s = t −q ( 2n+1 p ′ +γ )[ mγ,φp,q (f) ]q , cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 21 which proves that the function ρ 7−→ ∥∥ft ∗ φρ ∥∥ p,νn ργ belongs to the space lq(dρ ρ ) and that mγ,φp,q (ft) = t 2n+1 p −γ−2n−1 mγ,φp,q (f). on the other hand, from the relations (3.12) and (3.16), we have ∫ +∞ 0 ft ∗ φρ ∗ φρ(r,x) dρ ρ 1 t2n+1 ∫ +∞ 0 f ∗ ( φ ∗ φ ) s ( r t , x t ) ds s = 1 t2n+1 ∫ +∞ 0 f ∗ φs ∗ φs( r t , x t ) ds s , and from the relation (3.6), it follows ∫ +∞ 0 ft ∗ φρ ∗ φρ(r,x) dρ ρ 1 t2n+1 f( r t , x t ) = ft(r,x). this completes the proof. proposition 16. let p, q ∈ [1, +∞] and γ ∈ r. the space b γ p,q ( [0, +∞[ × rn ) ∩ e∗ ( r × rn ) is dense in b γ p,q ( [0, +∞[ × rn ) . proof. let f ∈ bγp,q ( [0, +∞[ × rn ) and φ ∈ s1∗,0(r × rn). for all t > 0, the function (r,x) 7−→ f ∗ φt(r,x) = 〈f,τ(r,−x)(φ̌t)〉 belongs to the space e∗ ( r × rn ) and is slowly increasing. from i) of proposition 8, we deduce that the function f ∗ φt ∗ φt belongs to the space e∗ ( r × rn ) . thus, from derivative’s theorem it follows that for all k ∈ n∗; the function fk(r,x) = ∫k 1 k f ∗ φt ∗ φt(r,x) dt t is infinitely differentiable on r × rn, even with respect to the first variable. on the other hand, let ψ ∈ s1∗,0(r × rn), by fubini’s theorem, we have fk ∗ ψρ = ∫k 1 k f ∗ φt ∗ φt ∗ ψρ dt t . and by the same way as the proof of theorem 14, we deduce that for all ρ > 0, the function fk ∗ ψρ belongs to lp(dνn) and that the function ρ 7−→ ∥∥fk ∗ ψρ ∥∥ p,νn ργ 22 l.t.rachdi and a.rouz cubo 13, 2 (2011) belongs to lq(dρ ρ ). again, by fubini’s theorem, for all ψ ∈ s1∗,0(r × rn), ∫ +∞ 0 fk ∗ ψρ ∗ ψρ dρ ρ = ∫ +∞ 0 (∫k 1 k f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t ) dρ ρ = ∫k 1 k (∫ +∞ 0 f ∗ ψρ ∗ ψρ ∗ φt ∗ φt dρ ρ ) dt t , and by lemma 13 and theorem 14, we obtain ∫ +∞ 0 fk ∗ ψρ ∗ ψρ dρ ρ = ∫k 1 k f ∗ φt ∗ φt dt t = fk. this shows that for all k ∈ n∗, the function fk belongs to the space b γ p,q ( [0, +∞[ × rn ) ∩ e∗ ( r × rn ) . moreover, for every ϕ ∈ s1∗,0(r × rn), we have fk ∗ ϕρ = ∫k 1 k f ∗ φt ∗ φt ∗ ϕρ dt t , and by i) of lemma 13, we get f ∗ ϕρ = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ϕρ dt t , thus, ( f − fk ) ∗ ϕρ = ∫ 1 k 0 f ∗ φt ∗ φt ∗ ϕρ dt t + ∫ +∞ k f ∗ φt ∗ φt ∗ ϕρ dt t = ∫ [0, 1k ]∪[k,+∞ [ f ∗ φt ∗ φt ∗ ϕρ dt t . now using the relation (3.13), we obtain ( f − fk ) ∗ ϕρ = ∫ ( [0, 1kρ ]∪[ k ρ ,+∞ [ ) ∩[α,β] f ∗ φρs ∗ φρs ∗ ϕρ ds s = ∫ +∞ 0 1( [0, 1kρ ]∪[ k ρ ,+∞ [ ) ∩[α,β] (s) f ∗ φρs ∗ φρs ∗ ϕρ ds s . now minkowski’s inequality leads to ∥∥(f − fk ) ∗ ϕρ ∥∥ p,νn 6 ∫β α 1( [0, 1kρ ]∪[ k ρ ,+∞ [ )(s) ∥∥f ∗ φρs ∗ φρs ∗ ϕρ ∥∥ p,νn ds s 6 ∫β α 1( [0, 1kρ ]∪[ k ρ ,+∞ [ )(s) ∥∥f ∗ φρs ∥∥ p,νn ∥∥φρs ∗ ϕρ ∥∥ 1,νn ds s 6 ‖φ‖1,νn ‖ϕ‖1,νn ∫β α 1( [0, 1kρ ]∪[ k ρ ,+∞ [ )(s) ∥∥f ∗ φρs ∥∥ p,νn ds s . cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 23 consequently; ∥∥(f − fk ) ∗ ϕρ ∥∥ p,νn ργ 6 ‖φ‖1,νn ‖ϕ‖1,νn ∫β α 1( [0, 1k ]∪[k,+∞ [ )(ρs) ∥∥f ∗ φρs ∥∥ p,νn ργ ds s 6 ‖φ‖1,νn ‖ϕ‖1,νn ∫ 1 α 1 β 1( [0, 1k ]∪[k,+∞ [ )( ρ t ) t−γ ∥∥f ∗ φρ t ∥∥ p,νn( ρ t )γ dt t = ‖φ‖1,νn ‖ϕ‖1,νn ( t−γ 1[ 1β, 1 α ] ⋆ ∥∥f ∗ φt ∥∥ p,νn tγ 1( [0, 1k ]∪[k,+∞ [ ) ) (ρ). thus, by the relation (3.3), we obtain mγ,ϕp,q (fk − f) 6 ‖φ‖1,νn ‖ϕ‖1,νn ∥∥t−γ 1[ 1 β , 1 α ] ∥∥ l1( dt t ) × [∫ 1 k 0 (∥∥f ∗ φt ∥∥ p,νn tγ )q dt t + ∫ +∞ k (∥∥f ∗ φt ∥∥ p,νn tγ )q dt t ] 1 q . so, lim k→ +∞ mγ,ϕp,q (fk − f) = 0 because ∫ +∞ 0 (∥∥f ∗ φt ∥∥ p,νn tγ )q dt t < +∞ and the proof is complete. we denote by lq ( ]0, +∞[ , lp(dνn), dt t ) the space of measurable functions g on ]0, +∞[ × [0, +∞[ × rn such that for all t > 0, the function g(t, (., .)) belongs to the space lp(dνn) and the function t 7−→ ∥∥g(t, (., .)) ∥∥ p,νn belongs to lq(dt t ). this space is equipped with the norm ∥∥g ∥∥ lq ( ]0,+∞ [, lp(dνn), dt t ) = (∫ +∞ 0 ∥∥g(t, (., .)) ∥∥q p,νn dt t ) 1 q . then we have lemma 17. let p,q ∈ [1, +∞] and let γ < (2n + 1)/p. for all φ ∈ s1∗,0(r × rn), the mapping f defined by f(g)(r,x) = ∫ +∞ 0 tγ g ( t, (., .) ) ∗ φt(r,x) dt t is continuous from lq ( ]0, +∞[ , lp(dνn), dt t ) into b γ p,q ( [0, +∞[ × rn ) . 24 l.t.rachdi and a.rouz cubo 13, 2 (2011) proof. let φ ∈ s1∗,0(r × rn) and g ∈ lq ( ]0, +∞[ , lp(dνn), dt t ) . • let a, b be real numbers such that b > a > 0 and f (φ)(µ,λ) = 0 if µ2 + 2|λ|2 < a2 or µ2 + 2|λ|2 > b2. let ψ ∈ s∗(r × rn) such that f (ψ)(µ,λ) = 1 if a2 6 µ2 + 2|λ|2 6 b2, then from the relation (2.9), we deduce that for every t > 0 ψt ∗ φt = φt. (3.17) for every k ∈ n∗, the function f(g)k defined by f(g)k(r,x) = ∫k 1 k tγ g(t, (., .)) ∗ φt(r,x) dt t . is bounded on r × rn. in fact, from the relations (2.7) and (3.4), we deduce that for all (r,x) ∈ r × rn, ∣∣f(g)k(r,x) ∣∣ 6 ∫k 1 k tγ ∥∥g(t, (., .)) ∥∥ p,νn ∥∥φt ∥∥ p ′ ,νn dt t 6 ∥∥φ ∥∥ p ′ ,νn ∫k 1 k t γ − 2n+1 p ∥∥g ( t, (., .) )∥∥ p,νn dt t 6 ∥∥φ ∥∥ p ′ ,νn ∥∥g ∥∥ lq ( ]0,+∞ [, lp(dνn), dt t ) [∫k 1 k t (γ − 2n+1 p ) q ′ dt t ] 1 q ′ < +∞. thus, for all k ∈ n∗ the function f(g)k defines a tempered distribution on r × rn, even with respect to the first variable. moreover, for all h ∈ s∗(r × rn), we have 〈f(g)k,h〉 = ∫k 1 k tγ [∫ +∞ 0 ∫ rn h(r,x) g(t, (., .)) ∗ φt(r,x) dνn(r,x) ] dt t = ∫k 1 k tγ 〈g(t, (., .)) ∗ φt,h〉 dt t , and by the relation (3.17), it follows that 〈f(g)k,h〉 = ∫k 1 k tγ 〈g(t, (., .)) ∗ φt ∗ ψt,h〉 dt t = ∫k 1 k tγ 〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 dt t . (3.18) cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 25 however, ∫ +∞ 0 tγ ∣∣〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 ∣∣ dt t 6 ∫ +∞ 0 tγ ∥∥g(t, (., .)) ∗ φt ∥∥ ∞,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t 6 ∫ +∞ 0 tγ ∥∥g(t, (., .)) ∥∥ p,νn ∥∥φt ∥∥ p ′ ,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t = ∥∥φ ∥∥ p ′ ,νn ∫ +∞ 0 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t = ∥∥φ ∥∥ p ′ ,νn { ∫1 0 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t + ∫ +∞ 1 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t } applying the relations (3.7) and (3.8), we get ∫ +∞ 0 tγ ∣∣〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 ∣∣ dt t 6 ∥∥φ ∥∥ p ′ ,νn ‖∆kh‖1,νn ‖ψ̌k‖1,νn ∫1 0 t 2k+γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn dt t + ∥∥φ ∥∥ p ′ ,νn ‖h‖1,νn ‖ψ̌‖1,νn ∫ +∞ 1 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn dt t ; and by hölder’s inequality, we have ∫ +∞ 0 tγ ∣∣〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 ∣∣ dt t 6 ∥∥φ ∥∥ p ′ ,νn ∥∥g ∥∥ lq ( ]0,+∞ [, lp(dνn), dt t ) { ‖∆kh‖1,νn ‖ψ̌k‖1,νn (∫1 0 t (2k+γ − 2n+1 p ) q ′ dt t ) 1 q ′ + ‖h‖1,νn ‖ψ̌‖1,νn (∫ +∞ 1 t (γ − 2n+1 p ) q ′ dt t ) 1 q ′ } < +∞. the last inequality together with the relation (3.18) show that for all h ∈ s∗(r × rn), lim k→ +∞ 〈f(g)k,h〉 exists and lim k→ +∞ 〈f(g)k,h〉 = ∫ +∞ 0 tγ 〈g(t, (., .)) ∗ φt,h〉 dt t . consequently, the function f(g)(r,x) = ∫ +∞ 0 tγ g ( t, (., .) ) ∗ φt(r,x) dt t defines an element of s ′ ∗(r × rn). 26 l.t.rachdi and a.rouz cubo 13, 2 (2011) • let ϕ ∈ s1∗,0(r × rn), we have f(g) ∗ ϕρ(r,x) = 〈f(g),τ(r,−x)ϕ̌ρ〉 = lim k→ +∞ 〈f(g)k,τ(r,−x)ϕ̌ρ〉 = lim k→ +∞ f(g)k ∗ ϕρ(r,x) = lim k→ +∞ ∫k 1 k tγ g(t, (., .)) ∗ φt ∗ ϕρ(r,x) dt t . however, the relation (3.13) implies ∫ +∞ 0 tγ ∣∣g(t, (., .)) ∗ φt ∗ ϕρ(r,x) ∣∣ dt t ∫ρβ ρα tγ ∣∣g(t, (., .)) ∗ φt ∗ ϕρ(r,x) ∣∣ dt t 6 ∫ρβ ρα tγ ∥∥g(t, (., .)) ∥∥ p,νn ∥∥φt ∗ ϕρ ∥∥ p ′ ,νn dt t 6 ‖φ‖p′,νn‖ϕ‖1,νn ∫ρβ ρα t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn dt t , 6 ‖φ‖p′,νn ‖ϕ‖1,νn (∫ρβ ρα t (γ − 2n+1 p ) q ′ dt t ) 1 q ′ ∥∥g ∥∥ lq ( ]0,+∞ [, lp(dνn), dt t ) < +∞. thus, f(g) ∗ ϕρ(r,x) = ∫ +∞ 0 tγ g(t, (., .)) ∗ φt ∗ ϕρ(r,x) dt t = ∫β α (ρs)γ g(ρs, (., .)) ∗ φρs ∗ ϕρ(r,x) ds s . (3.19) by minkowski’s inequality, we obtain ∥∥f(g) ∗ ϕρ ∥∥ p,νn 6 ∫β α (ρs)γ ∥∥g(ρs, (., .)) ∗ φρs ∗ ϕρ ∥∥ p,νn ds s 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∫β α (ρs)γ ∥∥g(ρs, (., .)) ∥∥ p,νn ds s < +∞ and ∥∥f(g) ∗ ϕρ ∥∥ p,νn ργ 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∫β α sγ ∥∥g(ρs, (., .)) ∥∥ p,νn ds s = ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∫ 1 α 1 β t−γ ∥∥g( ρ t , (., .)) ∥∥ p,νn dt t = ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ( t−γ1[ 1β, 1 α ] ⋆ ∥∥g(t, (., .)) ∥∥ p,νn ) (ρ), cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 27 and by the relation (3.3) it follows that ∥∥∥ ∥∥f(g) ∗ ϕρ ∥∥ p,νn ργ ∥∥∥ lq( dρ ρ ) 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∥∥∥t−γ1[ 1β, 1α ] ∥∥∥ l1( dt t ) ∥∥g ∥∥ lq ( ]0,+∞ [,lp(dνn), dt t ) < +∞. (3.20) • let ϕ ∈ s1∗,0(r × rn), from the relation (3.19), we have f(g) ∗ ϕρ(r,x) = ∫β α (ρs)γ g(ρs, (., .)) ∗ φρs ∗ ϕρ(r,x) ds s , and by fubini’s theorem, we get f(g) ∗ ϕρ ∗ ϕρ(r,x) = ∫β α (ρs)γ g(ρs, (., .)) ∗ φρs ∗ ϕρ ∗ ϕρ(r,x) ds s = ∫ρβ ρα tγ g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dt t . thus, ∫k 1 k f(g) ∗ ϕρ ∗ ϕρ(r,x) dρ ρ ∫k 1 k [∫ρβ ρα tγ g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dt t ] dρ ρ = ∫βk α k tγ [∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ ] dt t . (3.21) however, ∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫ t α t β [∫ +∞ 0 ∫ rn τ(r,−x)ǧ(t, (., .))(s,y) φt ∗ ϕρ ∗ ϕρ(s,y) dνn(s,y) ] dρ ρ . again, by fubini’s theorem, we have ∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫ +∞ 0 ∫ rn τ(r,−x)ǧ(t, (., .))(s,y) [∫ +∞ 0 φt ∗ ϕρ ∗ ϕρ(s,y) dρ ρ ] dνn(s,y). applying 2) of lemma 13, we obtain ∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫ +∞ 0 ∫ rn τ(r,−x)ǧ(t, (., .))(s,y) φt(s,y) dνn(s,y) = g(t, (., .)) ∗ φt(r,x). 28 l.t.rachdi and a.rouz cubo 13, 2 (2011) replacing in the equality (3.21), it follows that ∫k 1 k f(g) ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫βk α k tγ g(t, (., .)) ∗ φt(r,x) dt t . hence, ∫ +∞ 0 f(g) ∗ ϕρ ∗ ϕρ dρ ρ = f(g). this shows that the function f(g) belongs to the space b γ p,q ( [0, +∞[×rn ) and from the inequality (3.20), we have mγ,ϕp,q ( f(g) ) 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∥∥∥t−γ1[ 1β, 1α ] ∥∥∥ l1( dt t ) ∥∥g ∥∥ lq ( ]0,+∞ [, lp(dνn), dt t ) which means that the mapping f is continuous from lq ( ]0, +∞[ , lp(dνn), dt t ) into b γ p,q ( [0, +∞[ × rn ) . theorem 18. let p,q ∈ [1, +∞] and let γ ∈ r, γ < (2n+1)/p. then the besov space bγp,q ( [0, +∞[× r n ) is a banach one. proof. let φ ∈ s1∗,0(r × rn). we define the mapping g on the space b γ p,q ( [0, +∞[ × rn ) by setting g(f)(t, (r,x)) = f ∗ φt(r,x) tγ . the mapping g is continuous from b γ p,q ( [0, +∞[ × rn ) into lq ( ]0, +∞[ , lp(dνn), dt t ) and we have ∥∥g(f) ∥∥ lq ( ]0,+∞ [, lp(dνn), dt t ) = mγ,φp,q (f). (3.22) moreover, for all f ∈ bγp,q ( [0, +∞[ × rn ) , we have f ◦ g(f)(r,x) = ∫ +∞ 0 tγ g(f)(t, (., .)) ∗ φt(r,x) dt t = ∫ +∞ 0 tγ f ∗ φt ∗ φt(r,x) tγ dt t = ∫ +∞ 0 f ∗ φt ∗ φt(r,x) dt t and by ii) of lemma 13, we get f ◦ g(f) = f. this equality shows that g ( b γ p,q ( [0, +∞[ × rn )) ker ( g ◦ f − id (lq ( ]0,+∞ [, lp(dνn), dt t ) ) ) . cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 29 in particular, g ( b γ p,q ( [0, +∞[ × rn )) is a closed subspace of lq ( ]0, +∞[ , lp(dνn), dt t ) . let (fk)k∈n be a cauchy sequence in b γ p,q ( [0, +∞[ × rn ) . from the relation (3.22), the sequence ( g(fk) ) k is a cauchy’s one in lq ( ]0, +∞[ , lp(dνn), dt t ) . since g ( b γ p,q ( [0, +∞[×rn )) is a closed subspace of lq ( ]0, +∞[ , lp(dνn), dt t ) , then there exists a function f in b γ p,q ( [0, +∞[ × rn ) such that lim k→ +∞ g(fk) = g(f) in l q ( ]0, +∞[ , lp(dνn), dt t ) . again by the relation (3.22), lim k→ +∞ fk = f in b γ p,q ( [0, +∞[ × rn ) . proposition 19. i) let q ∈ [1, +∞] , p1, p2 ∈ [1, +∞] ; p1 < p2 and let γ1, γ2 ∈ r such that 2n + 1 p1 − γ1 = 2n + 1 p2 − γ2. (3.23) then b γ1 p1,q ( [0, +∞[ × rn ) →֒ bγ2p2,q ( [0, +∞[ × rn ) . ii) for all p ∈ [1, +∞] , b 0 p,1 ( [0, +∞[ × rn ) →֒ lp(dνn). proof. i) let p1, p2, γ1, γ2, q be real numbers satisfying the hypothesis. let p3 be an exponent such that 1 p1 + 1 p3 = 1 + 1 p2 . (3.24) finally, let f ∈ bγ1p1,q ( [0, +∞[ × rn ) and φ ∈ s1∗,0 ( r × rn ) such that f (φ)(µ,λ) = 0 if µ2 + 2|λ|2 > b2 or µ2 + 2|λ|2 < a2. let us take ψ ∈ s∗ ( r × rn ) satisfying ∀(µ,λ) ∈ γ ; a2 6 µ2 + 2|λ|2 6 b2, f (ψ)(µ,λ) = 1. then for all t > 0, we have φt ∗ ψt = φt and mγ2,φp2,q (f) = (∫ +∞ 0 (‖f ∗ φt‖p2,νn tγ2 )qdt t ) 1 q = (∫ +∞ 0 (‖f ∗ φt ∗ ψt‖p2,νn tγ2 )qdt t ) 1 q . 30 l.t.rachdi and a.rouz cubo 13, 2 (2011) by the relations (2.7), (3.4), (3.23) and (3.24) we get mγ2,φp2,q (f) 6 ‖ψ‖p3,νn [∫ +∞ 0 (‖f ∗ φt‖p1,νn tγ1 )qdt t ] 1 q 6 ‖ψ‖p3,νn mγ1,φp1,q (f). this shows that the space b γ1 p1,q ( [0, +∞[×rn ) is contained in b γ2 p2,q ( [0, +∞[×rn ) and that the canonical injection is continuous from b γ1 p1,q ( [0, +∞[ × rn ) into the space b γ2 p2,q ( [0, +∞[ × rn ) . ii) let f ∈ b0p,1 ( [0, +∞[ × rn ) ; p ∈ [1, +∞] . from ii) of lemma 13, we have f = ∫ +∞ 0 f ∗ φt ∗ φt dt t ; φ ∈ s1∗,0 ( r × rn ) thus, ‖f‖p,νn 6 ∫ +∞ 0 ∥∥f ∗ φt ∗ φt ∥∥ p,νn dt t 6 ‖φ‖1,νn m 0,φ p,1 (f). this completes the proof. in the following, we shall define a discrete norm on the besov space b γ p,q ( [0, +∞[ × rn ) and we will prove that it is equivalent to the norm m γ,φ p,q ; φ ∈ s1∗,0 ( r × rn ) . more precisely, we have theorem 20. let p, q ∈ [1, +∞], γ ∈ r. let a, b be real numbers such that 0 < a < b and φ ∈ s∗,0 ( r × rn ) verifying f (φ)(µ,λ) = 1 if a2 6 µ2 + 2|λ|2 6 b2. then the mapping n γ,φ p,q defined by nγ,φp,q (f) =    ( ∑ k∈z (‖f ∗ φ2k‖p,νn 2kγ )q ) 1 q , if 1 6 q < +∞; esssup k∈z ‖f ∗ φ2k‖p,νn 2kγ , if q = +∞ is a norm on the besov space b γ p,q ( [0, +∞[ × rn ) which defines the same topology as the norm m γ,ψ p,q ; ψ ∈ s1∗,0 ( r × rn ) . proof. • from lemma 9, there exists ψ ∈ s1∗,0 ( r × rn ) such that f (ψ)(µ,λ) = 0 if µ2 + 2|λ|2 < a2 or µ2 + 2|λ|2 > b2. cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 31 then for all s ∈ [1,2] and k ∈ z, we have f (ψ)(2ksµ,2ksλ) = f (ψ)(2ksµ,2ksλ) f (φ)(2kµ,2kλ) which leads to ψ2ks = ψ2ks ∗ φ2k and therefore, for all f ∈ bγp,q ( [0, +∞[ × rn ) f ∗ ψ2ks = f ∗ φ2k ∗ ψ2ks. (3.25) then for all q ∈ [1, +∞[ mγ,ψp,q (f) = (∫ +∞ 0 (‖f ∗ ψt‖p,νn tγ )q dt t ) 1 q = ( ∑ k∈z ∫2k+1 2k (‖f ∗ ψt‖p,νn tγ )q dt t ) 1 q = ( ∑ k∈z ∫2 1 (‖f ∗ ψ2ks‖p,νn (2ks)γ )q ds s ) 1 q . using the relations (2.7), (3.4) and (3.25), we obtain mγ,ψp,q (f) 6 ‖ψ‖1,νn [ ∑ k∈z (‖f ∗ φ2k‖p,νn 2kγ )q ∫2 1 ds sγq+1 ] 1 q = ‖ψ‖1,νn (1 − 2−qγ qγ ) 1 q nγ,φp,q (f). on the other hand, for q = +∞ and again by the relation (3.25), we deduce that for all k ∈ z and s ∈ [1,2] ‖f ∗ ψ2ks‖p,νn (2ks)γ 6 (1 + 2−γ) ‖ψ‖1,νn ‖f ∗ φ2k‖p,νn 2kγ . consequently, for all k ∈ z and t ∈ [2k,2k+1] ‖f ∗ ψt‖p,νn tγ 6 (1 + 2−γ) ‖ψ‖1,νn nγ,φp,∞ (f), which shows that mγ,ψp,∞ (f) 6 (1 + 2 −γ) ‖ψ‖1,νn nγ,φp,∞ (f). • let a1, b1 be two real numbers; 0 < a1 < a b21. from lemma 9, there exists ψ ∈ s1∗,0 ( r × rn ) such that f (ψ)(µ,λ) = c, for all (µ,λ) ∈ γ ; a21 6 µ2 + 2|λ|2 6 4b21 32 l.t.rachdi and a.rouz cubo 13, 2 (2011) where c is a positive constant. then for all k ∈ z and s ∈ [1,2], cf (φ)(2kµ,2kλ) = f (φ)(2kµ,2kλ) f (ψ)(2kµs,2kλs) so, c.φ2k = φ2k ∗ ψ2ks. hence, for all f ∈ bγp,q ( [0, +∞[×rn ) c f ∗ φ2k = f ∗ ψ2ks ∗ φ2k (3.26) and c ‖f ∗ φ2k‖p,νn 2kγ 6 (1 + 2γ) ‖φ‖1,νn ‖f ∗ ψ2ks‖p,νn (2ks)γ . integrating over [1,2] with respect to the measure ds s , we get for all q ∈ [1, +∞[, (‖f ∗ φ2k‖p,νn 2kγ )q 6 ( (1 + 2γ) ‖φ‖1,νn )q cq log 2 ∫2k+1 2k (‖f ∗ ψt‖p,νn tγ )qdt t which leads to nγ,φp,q (f) 6 1 c (log 2)− 1 q (1 + 2γ) ‖φ‖1,νn mγ,ψp,q (f). on the other hand, for q = +∞ and using the relation (3.26), we deduce that for all k ∈ z ‖f ∗ φ2k‖p,νn 2kγ 6 (1 + 2γ) c ‖φ‖1,νn mγ,ψp,∞ (f), which implies that nγ,φp,∞ (f) 6 (1 + 2γ) c ‖φ‖1,νn mγ,ψp,∞ (f). this completes the proof of theorem. remark 21. 1) from theorem 14 and theorem 20, we deduce that the besov space b γ p,q ( [0, +∞[× r n ) is independent of the choice of the function φ ∈ s∗,0 ( r × rn ) , when it is endowed with the norm n γ,φ p,q . from proposition 15 and theorem 20, we deduce the following proposition proposition 22. the besov space b γ p,q ( [0, +∞[ × rn ) is homogeneous in a weaker sense when equipped with the norm n γ,φ p,q , that is there exist c1, c2 > 0 such that for all f ∈ bγp,q ( [0, +∞[ × r n ) and t > 0 c1 t 2n+1 p −2n−1−γ nγ,φp,q (f) 6 n γ,φ p,q (ft) 6 c2 t 2n+1 p −2n−1−γ nγ,φp,q (f). proposition 23. let p ∈ [1, +∞] and γ ∈ r. then for all q1, q2 ∈ [1, +∞] ; q1 6 q2, we have the continuous embedding b γ p,q1 ( [0, +∞[ × rn ) →֒ bγp,q2 ( [0, +∞[×rn ) . cubo 13, 2 (2011) homogeneous besov spaces associated with the spherical . . . 33 proof. let f ∈ bγp,q1 ( [0, +∞[ × rn ) and φ ∈ s∗,0 ( r × rn ) . since ∑ k∈z (‖f ∗ φ2k‖p,νn 2kγ )q1 < +∞ then, nγ,φp,∞ (f) < +∞ and we have nγ,φp,q2 (f) 6 ( nγ,φp,∞ (f) )1− q1 q2 ( nγ,φp,q1 (f) )q1 q2 . however, nγ,φp,∞ (f) 6 n γ,φ p,q1 (f) and consequently, nγ,φp,q2 (f) 6 n γ,φ p,q1 (f). received: october 2009. revised: december 2009. references [1] l. e. andersson, on the determination of a function from spherical averages, siam j. math. anal 19 (1988), 214–234. 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[27] g. n. watson, a treatise on the theory of bessel functions, 2nd ed. cambridge univ. press. london/new-york, (1966). introduction fourier transform associated with the spherical mean operator besov spaces cubo a mathematical journal vol.19, no¯ 01, (89–110). march 2017 weighted pseudo almost periodic solutions for fractional order stochastic impulsive differential equations vikram singh and dwijendra n pandey department of mathematics, indian institute of technology roorkee roorkee-247667, india. vikramiitr1@gmail.com, dwij.iitk@gmail.com abstract in this paper, we deal with the existence and uniqueness of piecewise square mean weighted pseudo almost periodic solutions for a class of fractional order stochastic impulsive differential equations. the working tools are based on fixed point technique, fractional power operators and stochastic analysis; methods and theory are adopted from deterministic fractional systems. in addition, an example is given to illustrate the theory. resumen en este art́ıculo estudiamos la existencia y unicidad de soluciones pseudo casi periódicas con pesos promedio cuadrado a trozos para una clase de ecuaciones diferenciales estocásticas impulsivas de orden fraccional. las herramientas de trabajo están basadas en la técnica de punto fijo, operadores de potencia fraccional y análisis estocástico; los métodos y teoŕıa están adaptados a partir de sistemas fraccionales deterministas. adicionalmente, damos un ejemplo para ilustrar la teoŕıa. keywords and phrases: fractional stochastic impulsive differential equation, square-mean piecewise weighted pseudo almost periodicity, analytic semigroup, fractional power operator. 2010 ams mathematics subject classification: 26a33, 34a37, 34c27, 34g20, 35r12, 35r60, 43a60. 90 vikram singh and dwijendra n pandey cubo 19, 1 (2017) 1 introduction in recent years, fractional differential equations have been gaining considerable attention of many scientists and mathematicians because of their demonstrated applications in widespread fields of science and engineering. since noises or stochastic perturbations are unavoidable and omnipresent in nature as well as in man-made systems, so we have to move from deterministic models to stochastic models. stochastic differential equations play an important role in formulation and analysis of fluctuations in stock market prices, asset prices, population modeling, control engineering, and chemical engineering [12, 20] ect. motivated by these facts many researchers are showing great interest to establish an appropriate system to investigate qualitative properties such as existence, uniqueness, controllability and stability of these physical processes with the help of fractional calculus, stochastic analysis and fixed point theorems. for more details, we refer to [1, 3, 10, 11, 16, 19, 28, 29] and references therein. on the other hand, the study of differential equations with impulsive effect constitutes a useful and important field of research due to a lot of applications. in particular, differential equations with impulsive effects arise in various deterministic and stochastic processes which appear in chemical technology, physics, medicine and economics ect. the fractional differential equations involving impulsive effects came out as a natural description of observed phenomena. for more details see [5, 13, 14, 21, 22, 24] and the references therein. the concept of pseudo almost periodic solutions introduced by zhang [25, 26] is a natural and good generalization of the classical almost periodic functions. further, diagana investigated weighted pseudo almost periodic solutions in [8]. moreover, the authors investigated piecewise almost periodic solutions in [22], piecewise square mean almost periodic solutions in [11], pseudo almost periodic solutions in [5, 27] for impulsive differential equations. recently, zhinan [23] analyzed piecewise weighted pseudo almost periodic functions, which was more tricky and changeable than those of the classical functions. many authors have been made important contributions in study of almost periodic functions and its generalizations, one can see [6, 11, 13, 14, 22, 23, 24] and the references therein. however, piecewise square mean weighted pseudo almost periodic mild solutions for the fractional order stochastic impulsive differential equations, is an untreated topic in the literature and this fact is the motivation of the present work. in this paper, we are interested to investigate the existence and uniqueness of piecewise square mean weighted pseudo almost periodic mild solution for the following fractional order stochastic impulsive differential system cdαy(t) + ay(t) = g(t, y(t)) + f ! t, y(t), ∫ t −∞ k(t − s)g(s, y(s))ds " dw(t) dt , t0 < t ̸= ti, t ∈ r, (1.1) y(t+ i ) = y(t− i ) + gi(y(ti)), i ∈ z, (1.2) y(t0) = y0, (1.3) where the state y(·) take values in l2(p, h), h is a separable real hilbert space; cdα,α ∈ (0, 1) cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 91 symbolizes the caputo fractional derivative of order α; −a : d(a) ⊂ l2(p, h) → l2(p, h), is the infinitesimal generator of an analytic semigroup of exponentially bounded linear operator {s(t)}t≥0; {w(t) : t ≥ 0} is a k-valued wiener process, k is another separable hilbert space; g, f, gi are some suitable functions will be mention later; δ(·) is dirac’s delta function and k ∈ l1(r) with |k(t)| ≤ cke −bt, b, ck > 0. the rest of this paper is organized as follows: in section 2, we define some fundamental results about the notion of piecewise square mean weighted pseudo almost periodic functions. section 3 is devoted to the main results ensuring the existence and uniqueness of mild solutions of (1.1)−(1.3) via fractional power of operator and fixed point technique. at last, we will provide an example to show the feasibility of the theory discussed in this paper. 2 preliminaries let l(k, h) denote the collection of all bounded linear operators form k to h. for convenience, without confusion we will employ the same notation ∥.∥ to denote the norms in h, k and l(k, h) and ⟨·, ·⟩ for inner product in h and k. let (ω, f, {ft}t≥0, p) be a complete probability space equipped with a normal filtration {ft}t≥0 satisfying the usual conditions(i.e right continuous and {f0} containing all p-null sets). suppose {w(t) : t ≥ 0} is a k-valued wiener process with a finite nuclear covariance operator q ≥ 0 denote tr(q) = ∑ ∞ k=1 #λk = #λ < ∞ with qek = $λkek, where ek are complete orthonormal basis of k. in fact, w(t) = ∑ ∞ k=1 % #λkwk(t)ek, here {wk(t)} ∞ k=1 are mutually independent one dimensional standard wiener process. we consider that ft = {w(s) : 0 ≤ s ≤ t} is the σ algebra generated by w. assume that l0 2 = l2(q 1 2 k, h) represent the space of all hilbert schmidt operators from q 1 2 k to h with inner product ⟨φ,ψ⟩ = tr[φqψ∗]. for more details we refer to the book by da prato and zabczyk [7]. let the collection of all strongly measurable, square integrable h valued random variables be denoted by l2(p, h) which a banach space endowed with the norm ∥x(·)∥l2 = (e∥x(·)∥ 2) 1 2 , where e(·) represents the expectations with measure p. moreover l2f0(p, h) denote the collection of all f0 measurable, h valued random variable y(0). let ω be a subspace of l2(p, h) and e be a compact set of ω. assume that r, n, z, and c represent the sets of real number, natural number, integers and complex numbers respectively. for a being a linear operator on l2(p, h), d(a), r(a) and ρ(a) stands for domain, range and resolvent of a, repectively. let b = {ti : ti ∈ r, ti < ti+1, i ∈ z} be the set of all strictly increasing and unbounded sequences. for {ti : i ∈ z} ∈ b, let pc(r, l 2(p, h)) denote the space of all piecewise stochastically continuous processes y : r → l2(p, h) such that y(t) is stochastically continuous at t for any t /∈ b, y(t− i ), y(t+ i ) exists and y(t− i ) = y(ti) for all i ∈ r. in particular, the space pc(r × ω, l2(p, h)) consists of all piecewise stochastically continuous processes y : r × ω → l2(p, h) such that for any x ∈ ω,, y(t, ·) ∈ pc(r, l2(p, h)) and for any t ∈ r, y(t, ·) is stochastically continuous at x ∈ ω. 92 vikram singh and dwijendra n pandey cubo 19, 1 (2017) 2.1 fractional calculus and fractional power operator following [16, 18]) we recall some definitions and basic results of fractional calculus. definition 1. the riemann-liouville fractional integral of a function g ∈ l1 loc (r+, r) with the lower limit zero of order α > 0 is defined by jαg(t) = 1 γ(α) ∫ t 0 (t − ξ)α−1g(ξ)dξ, t > 0, and j0g(t) := g(t). this fractional integral satisfies the properties jα ◦ jb = jα+b for b > 0. definition 2. the riemann-liouville fractional derivative of a function g ∈ l1 loc (r+, r) with the lower limit zero of order α > 0, n − 1 < α < n, n ∈ n is given by dαg(t) = 1 γ(n − α) dn dtn ∫t 0 (t − ξ)n−α−1g(ξ)dξ, moreover d0g(t) = g(t) and dαjαg(t) = g(t) for t > 0. definition 3. the caputo fractional derivative of a function g : [0, ∞) → r with the lower limit 0 of order α > 0 is given by cdαg(t) = dα ! g(t) − n−1∑ k=0 tk k! g(k)(0) " , t > 0, n − 1 < α < n. remark 1. (i) if g(t) ∈ cn([0, ∞)), then cdαg(t) = 1 γ(n − α) ∫t 0 (t − ξ)n−α−1 dn dξn g(ξ)dξ, where n − 1 < α < n, n ∈ n. (ii) if g is an abstract function with values in h, then integral defined in definition 1 and 2 are taken in bochner’s sense. if −a generates an analytic semigroup s(t) in l2(p, h) and 0 ∈ ρ(a), then for σ > 0, we can define fractional power a−σ of the operator a by a−σ = 1 γ(σ) ∫ ∞ 0 tσ−1s(t)dt where a−σ is bijective, bounded and aσ = (a−σ)−1, σ > 0 a closed linear operator on d(aσ) such that d(aσ) = r(a−σ). moreover d(aσ) is dense in l2(p, h) and the expression ∥y∥σ = ∥a σy∥, y ∈ d(aσ) defines a norm on d(aσ). let us denote by l2(p, hσ) the banach space d(aσ) with norm ∥.∥σ. the following properties are well recognized. lemma 2.1. [17] let a be an infinitesimal generator of an analytic semigroup s(t) and 0 ∈ ρ(a). then cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 93 (i) s(t) : l2(p, h) → d(aσ), for σ ≥ 0, and t > 0. (ii) for every y ∈ d(aσ), we have s(t)aσy = aσs(t)y. (iii) the operator aσs(t) is bounded and ∥aσs(t)∥ ≤ mσt −σe−λt, mσ, t,λ > 0. (2.1) (iv) for y ∈ d(aσ), and 0 < σ ≤ 1, we have ∥s(t)y − y∥ ≤ cσt σ∥aσy∥, cσ > 0. (2.2) 2.2 square-mean piecewise weighted pseudo almost periodic function now we define square-mean piecewise weighted pseudo almost periodic function and explore its properties definition 4. a stochastic process y : r → l2(p, h) is said to be stochastically continuous for s ∈ r if limt→s e∥y(t) − y(s)∥ 2 = 0. definition 5. a stochastically continuous process y : r → l2(p, h) is said to be square mean almost periodic if for ever ϵ > 0, there exists a l(ϵ) > 0 such that every interval l of length l(ϵ) > 0 contains a number τ with the property e∥y(t + τ) − y(t)∥2 < ϵ for all t ∈ r. definition 6. a sequence zi : z → l 2(p, h) is said to be square-mean almost periodic sequence if for ever ϵ > 0, there exists a l(ϵ) > 0 such that every p ∈ z there is at least one number k in [p, p+l], with the property e∥zi+k −zi∥ 2 < ϵ for all i ∈ z. we denote the set of all such processes by ap(z, l2(p, h)). remark 2. let {zi} ∈ ap(z, l 2(p, h)), then {zi : i ∈ z} is stochastically bounded. let wd denote the collection of all functions (weights) ρm : z → (0, +∞), m ∈ z. for ρm ∈ wd and m ∈ z + = {m ∈ z, m ≥ 0}, set µ(m,ρ) := m∑ k=−m ρm. denote wd,∞ := {ρ ∈ wd : lim m→∞ (m,ρ) = ∞}. for ρ ∈ wd,∞, we define papρ(z, l 2(p, h)) = { zm ∈ l ∞(z, l2(p, h)) : lim m→∞ 1 µ(m,ρ) m∑ k=−m e∥zm∥ 2ρm = 0 } (2.3) definition 7. let ρ ∈ wd,∞. a sequence {zi}i∈z ∈ l ∞(z, l2(p, h)) is called square mean discrete weighted pseudo almost periodic if zi = ai + bi, where ai ∈ ap(z, l 2(p, h)) and bi ∈ papρ(z, l 2(p, h)). the set of all such functions denoted by wpapρ(z, l 2(p, h)). definition 8. a stochastic process y ∈ pc(r, l2(p, h)) is said to be square-mean piecewise almost periodic if: 94 vikram singh and dwijendra n pandey cubo 19, 1 (2017) (i) the set of all sequences {tj i : t j i := ti+j − ti, ti ∈ b, i, j ∈ z} are equipotentially almost periodic i.e. for every ϵ > 0 there exists a relatively dϵ ⊂ r of ϵ periods common for all sequences {t j i }. (ii) for any ϵ > 0, there exists a δ > 0 such that if the points s and t are in the same interval of continuity of y(t) and |t − s| < δ, then e∥y(t) − y(s)∥2 < ϵ. (iii) for any ϵ > 0, there exists a relatively dense set rϵ of r such that if τ ∈ rϵ, then e∥y(t + τ) − y(t)∥2 < ϵ, with the condition |t − ti| > ϵ, i ∈ z. we denote by app(r, l2(p, h)) the space of all square-mean piecewise almost periodic processes. we denote by upc(r, l2(p, h)) the space of all stochastic processes such that y satisfy the condition (ii) in definition 8 and y ∈ pc(r, l2(p, h)). definition 9. [6] for {ti} ∈ b, i ∈ z, the function f(t, y) ∈ pc(r × ω, l 2(p, h)) is called squaremean piecewise almost periodic in t ∈ r and uniformly on e ⊆ ω, {f(·, y) : y ∈ e} is uniformly bounded, and for every ϵ > 0 there exists a relatively compact set rϵ of r, such that e∥f(t+τ, y)− f(t, y)∥2 < ϵ, for all y ∈ e, t ∈ r and τ ∈ rϵ with |t − ti| > ϵ, i ∈ z. the set of all such processes is denoted by app(r × ω, l2(p, h)). lemma 2.2. [13] let f ∈ app(r, l2(p, h)), {zi : i ∈ z} is square mean almost periodic sequence in l2(p, h) and {t j i : i, j ∈ z} is equipotentially almost periodic. then for each ϵ > 0 there exist relatively dense sets rϵ of r and zϵ of z such that the following conditions hold: (i) e∥f(t + τ) − f(t)∥2 < ϵ for all τ ∈ rϵ, t ∈ r, |t − ti| > ϵ, i ∈ z. (ii) e∥zi+p − zi∥ 2 < ϵ for all p ∈ zϵ, and i ∈ z. (iii) for any τ ∈ rϵ there exists at least a number p ∈ zϵ such that |t p i − τ| < ϵ, i ∈ z. next, we introduce the concept of piecewise square mean weighted pseudo almost periodic functions and explore its properties. let w be the collections of all positive and locally integrable functions ρ : r → (0, ∞). for each ρ ∈ w and γ > 0, set µ(γ,ρ) := ∫γ −γ ρ(t)dt. define w∞ := {ρ ∈ w : lim γ→∞ µ(γ,ρ) = ∞}, wb := {ρ ∈ w∞ : ρ is bounded and inf t∈r ρ(t) > 0}. it is clear that wb ⊂ w∞ ⊂ w. definition 10. let ρ1,ρ2 ∈ w∞. ρ1 is said to be equivalent to ρ2 (i.e. ρ1 ∼ ρ2) if ρ1 ρ2 ∈ wb. cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 95 it is clear that ‘‘ ∼ " binary equivalence relation on w∞. for a given weight ρ ∈ w∞, the equivalence class is denoted by cl(ρ) := {ρ ∗ ∈ w∞ : ρ ∼ ρ ∗}. moreover w∞ = ∪ρ∈w∞cl(ρ). for ρ ∈ w∞, we define paaρ(r, l 2(p, h)) := { f ∈ pc(r, l2(p, h)) : lim γ→∞ 1 µ(γ,ρ) ∫ γ −γ e∥f(t)∥2ρ(t)dt = 0 } . similarly paaρ(r × ω, l 2(p, h)) := { f ∈ pc(r × ω, l2(p, h)) : lim γ→∞ 1 µ(γ,ρ) ∫ γ −γ e∥f(t, y)∥2ρ(t)dt = 0 uniformly in y ∈ e } . definition 11. a function f ∈ pc(r, l2(p, h)) is called piecewise square mean weighted pseudo almost periodic if it has a decomposition of the form f = φ+ ψ, where φ ∈ app(r, l2(p, h)) and ψ ∈ papρ(r, l 2(p, h)). the set of all such functions denoted by wpapρ(r, l 2(p, h)). definition 12. a function f ∈ pc(r × ω, l2(p, h)) is called piecewise square mean weighted pseudo almost periodic if it has a decomposition of the form f = φ + ψ, where φ ∈ app(r × ω, l2(p, h)) and ψ ∈ papρ(r×ω, l 2(p, h)). the set of all such functions denoted by wpapρ(r× ω, l2(p, h)). for ρ ∈ w∞ and τ ∈ r define ρ τ by ρτ(t) = ρ(t + τ) for all t ∈ r. define wt = {ρ ∈ w∞ : ρ ∼ ρ τ for each t ∈ r}. it is clear that wt contains many of weights, such as 1, e t and 1 + |t|n with n ∈ n among others. remark 3. (i) for ρ ∈ wt , papρ(r, l 2(p, h)) is a translation invariant set of pc(r, l2(p, h)). (ii) it is easy to see that wpapρ(r, l 2(p, h))(resp., wpapρ(r × ω, l 2(p, h))) are banach spaces with sup norm. similar as the proof of lemma 2.5 in [9], we have the following result. lemma 2.3. let {fn}n∈n be a sequence of functions in wpapρ(r, l 2(p, h)). if fn converge uniformly to f, then f ∈ wpapρ(r, l 2(p, h)). similar as the proof of [14] the following composition theorems hold for piecewise square mean weighted pseudo almost periodic functions. theorem 2.1. let f(t, y, z) ∈ wpapρ(r × ω × ω, l 2(p, h)),ξ,χ ∈ wpapρ(r, l 2(p, h)) and r(ξ) × r(ξ) ⊂ ω × ω. assume that there exists a number lf > 0 such that e∥f(t, y1, z1)−f(t, y2, z2)∥ 2 ≤ lf.(e∥y1 −y2∥ 2 +e∥z1 −z2∥ 2), for all t ∈ r, yi, zi ∈ ω, i = 1, 2, then f(·,ξ(·),χ(·)) ∈ wpapρ(r, l 2(p, h)). 96 vikram singh and dwijendra n pandey cubo 19, 1 (2017) theorem 2.2. let {ii(y) : i ∈ z} for any y ∈ ω be a piecewise square mean weighted pseudo almost periodic sequence. assuming that there exists a constant l0 > 0 such that e∥ii(x) − ii(y)∥ 2 ≤ l0.e∥x − y∥ 2, for all x, y ∈ ω, i ∈ z. if ξ ∈ wpapρ(r, l 2(p, h)) ∩ upc(r, l2(p, h)) such that r(ξ) ⊂ ω, then ii(ξ(ti)) is piecewise square mean weighted pseudo almost periodic. lemma 2.4. [21] assume that {t j i : i, j ∈ z} are equipotentially almost periodic sequences, then for each p > 0 there exists a positive integer n0 such that each interval of length p has no more than n0 elements of the sequence {ti} and n(s, t) ≤ n0(t − s) + n0, where n(t, s) denotes the number of the points ti in the interval [t, s]. 3 main results in this section, we establish piecewise square mean weighted pseudo almost periodic mild solution to the fractional order stochastic impulsive differential system (1.1)-(1.3). in formulation of the system (1.1)-(1.3), we consider the following assumptions: (h1) the collection of sequences {t j i : i, j ∈ z} is equipotentially almost periodic and there exists θ > 0 such that infi τ 1 i = θ. (h2) −a is the infinitesimal generator of an analytic semigroup s(t), t ≥ 0, on l 2(p, h). (h3) for ρ ∈ wt , g ∈ wpapρ(r × l 2(p, hσ), l 2(p, h)) and there exists a lg > 0, 0 < η < 1, such that e∥g(t1, u1) − g(t2, u2)∥ 2 ≤ lg(|t1 − t2| η + e∥u1 − u2∥ 2 σ ), for each (ti, ui) ∈ r × l 2(p, hσ) i = 1, 2. (h4) for ρ ∈ wt , g ∈ wpapρ(r × l 2(p, hσ), l 2(p, h)) and there exists lf > 0, 0 < η < 1, such that e∥g(t1, u1) − g(t2, u2)∥ 2 ≤ lg(|t1 − t2| η + e∥u1 − u2∥ 2 σ ), for each (ti, ui) ∈ r × l 2(p, hσ), i = 1, 2. (h5) for ρ ∈ wt , f ∈ wpapρ(r×l 2(p, hσ)×l 2(p, hσ), l 2(p, l0 2 )) and there exists lf > 0, 0 < η < 1, such that e∥f(t, u1, v1) − f(t, u2, u2)∥ 2 ≤ lf(|t1 − t2| η + e∥u1 − u2∥ 2 σ + e∥v1 − v2∥ 2 σ ), for each (ti, ui, vi) ∈ r × l 2(p, hσ) × l 2(p, hσ) i = 1, 2. cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 97 (h6) {gi(y) : k ∈ z} is piecewise square mean weighted pseudo almost periodic sequence uniformly y ∈ ω and that there exists a constant lg > 0 such that e∥gi(x) − gi(y)∥ 2 ≤ lg.e∥x − y∥ 2 σ, for all x, y ∈ l 2(p, hσ). (h7) for any l1, l2 > 0, denote cf := supt∈r,∥u∥∞ 0 such that 3m2σ & 4c2gin 2 0 ! 1 mσ 0 + 1 eλ − 1 "2 + c2g γ2(1 − σ) λ2(1−σ) + c2fn0 γ(1 − 2σ) λ(2−2σ) ' ≤ r0. now, we define the mild solutions for the system (1.1) − (1.3). definition 13. a stochastic process y ∈ pc(j, l2(p, h)), j ⊂ r is a mild solution of the system (1.1) -(1.3), if (i) y0 ∈ l 2 f0 (p, h). (ii) y(t) ∈ l2(p, h) has càdlàg path on t ∈ j a.s., and satisfies the following integral equation y(t) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ i(t − t0)y0 + ∫t t0 (t − s)α−1j (t − s)g(s, y(s))ds + ∫ t t0 (t − s)α−1j (t − s)f(s, y(s), ∫ s −∞ k(s − ξ)g(ξ, y(ξ))dξ)dw(s), t ∈ [t0, t1]; i(t − t0)y0 + i(t − t1)y1 + ∫t t0 (t − s)α−1j (t − s)g(s, y(s))ds + ∫ t t0 (t − s)α−1j (t − s)f(s, y(s), ∫ s −∞ k(s − ξ)g(ξ, y(ξ))dξ)dw(s), t ∈ (t1, t2]; ... i(t − t0)y0 + k∑ i=1 i(t − ti)gi(y(ti)) + ∫ t t0 (t − s)α−1j (t − s)g(s, y(s))ds + ∫ t t0 (t − s)α−1j (t − s)f(s, y(s), ∫ s −∞ k(s − ξ)g(ξ, y(ξ))dξ)dw(s), t ∈ (tk, tk+1], (3.1) where i(t) = ∫ ∞ 0 nα(θ)s(t αθ)dθ, j (t) = α ∫ ∞ 0 θnα(θ)s(t αθ)dθ, and for θ ∈ (0, ∞) nα(θ) = 1 α θ−1− 1 α ωα(θ − 1 α ) ≥ 0, ωα(θ) = 1 π ∞∑ n=1 (−1)n−1θ−nα−1 γ(nα + 1) n! sin(nπα), 98 vikram singh and dwijendra n pandey cubo 19, 1 (2017) nα denote the probability density function on (0, ∞) such that nα(θ) ≥ 0, θ ∈ (0, ∞) and ∫ ∞ 0 nα(θ)dθ = 1. noth that when (h2) holds, we observe that if y(t) is stochastically bounded solution of the system (1.1) − (1.3) on r, then the mild solution (3.1) take the following form as t0 → −∞. y(t) = ∑ ti 0, there exists a relatively dense set rϵ of r formed by ϵ-periods of ψ1. for τ ∈ rϵ, t ∈ r, |t−ti| > ϵ, i ∈ z, we have ∥ψ1(t + τ) − ψ1(t)∥ < ϵ. hence for t ∈ r, |t − ti| > ϵ, i ∈ z, we get e∥θ1(t + τ) − θ1(t)∥ 2 =e ( ( ( ( ∫t+τ −∞ k(t + τ − s)ψ1(s)ds − ∫t −∞ k(t − s)ψ1(s)ds ( ( ( ( 2 ≤e ( ( ( ( ∫ t −∞ k(t − s)[ψ1(s + τ) − ψ1(s)]ds ( ( ( ( 2 ≤c2k ∫ t −∞ e−2b(t−s)e∥ψ1(s + τ) − ψ1(s)∥ 2ds < c2k 2b ϵ, which implies that θ1 ∈ ap p (r, l2(p, h)). cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 99 next we show that θ2 ∈ papρ(r, l 2(p, h)). in fact for γ > 0, we have 1 µ(γ,ρ) ∫ γ −γ e∥θ2(t)∥ 2ρ(t)dt = 1 µ(γ,ρ) ∫ γ −γ e ( ( ( ( ∫ t −∞ k(t − s)ψ2(s)ds ( ( ( ( 2 ρ(t)dt = 1 µ(γ,ρ) ∫ γ −γ e ( ( ( ( ∫ ∞ 0 k(s)ψ2(t − s)ds ( ( ( ( 2 ρ(t)dt ≤ c2k µ(γ,ρ) ∫γ −γ ∫ ∞ 0 e−2b(s)ρ(t)e∥ψ2(t − s)∥ 2dsdt ≤c2k ∫ ∞ 0 e−2b(s)λγ(s)ds, where λγ(s) = 1 µ(γ,ρ) ∫ γ −γ ρ(t)e∥ψ2(t − s)∥ 2dt. since ψ2(s) ∈ papρ(r, l 2(p, h)), ρ ∈ wt , this implies that ψ2(· − s) ∈ papρ(r, l 2(p, h)) for each s ∈ r by remark 3. hence lim γ→∞ λγ(s) = 0 for all s ∈ r. now, by lebesgue dominated convergence theorem, we have θ2 ∈ papρ(r, l 2(p, h)). lemma 3.2. assume that (h1) − (h2) hold, if φ(t) ∈ wpapρ(r, l 2(p, l0 2 )), then λφ(t) = ∫t −∞ aσ(t − s)α−1j (t − s)φ(s)dw(s) ∈ wpapρ(r, l 2(p, h)). proof. since φ(t) ∈ wpapρ(r, l 2(p, l0 2 )) and ∥φ∥∞ := supt∈r(e∥φ(t)∥ 2) 1 2 < ∞. now, using ito’s isometry property of stochastic integral and lemma 2.1 , we get e∥λφ(t)∥ 2 =e ( ( ( ( ∫t −∞ aσ(t − s)α−1j (t − s)φ(s)dw(s) ( ( ( ( 2 ≤α2e ( ( ( ( ∫ t −∞ ∫ ∞ 0 θ(t − s)α−1nα(θ)a σs((t − s)αθ)φ(s)dθdw(s) ( ( ( ( 2 ≤α2 &∫ t −∞ ∫ ∞ 0 e∥θ(t − s)α−1nα(θ)a σs((t − s)αθ)φ(s)∥2dθds ' ≤α2m2σ ∫t −∞ ∫ ∞ 0 θ2(1−σ)n 2α(θ)(t − s) 2(α−ασ−1)e2λθ(t−s) α e∥φ(s)∥2dθds ≤α2m2 σ ∥φ∥2 ∫ ∞ 0 n 2 α (θ) ∫ ∞ 0 θ2(1−σ)ξ2(α−ασ−1)e2λθξ α dξdθ. since n 2α(θ) ∈ l 1(r+), then by calculating we get (see [11]) α2 ∫ ∞ 0 n 2α(θ) ∫ ∞ 0 θ2(1−σ)ξ2(α−ασ−1)e2λθξ α dξdθ = n0 γ(1 − 2σ) λ(2−2σ) (3.3) where n0 = supθ≥0 n 2 α (θ). then e∥λφ(t)∥ 2 ≤ m2 σ ∥φ∥2n0 γ(1 − 2σ) λ(2−2σ) , 100 vikram singh and dwijendra n pandey cubo 19, 1 (2017) this implies that λφ is well defined. now let φ = φ1 + φ2, with φ1 ∈ ap p(r, l2(p, h)) and φ2 ∈ papρ(r, l 2(p, h)), then λφ(t) = ∫t −∞ aσ(t − s)α−1j (t − s)φ1(s)dw(s) + ∫t −∞ aσ(t − s)α−1j (t − s)φ2(s)dw(s) :=λφ1(t) + λφ2(t). it is easy to check that λφ1 ∈ upc(r, l 2(p, h)). since φ1 ∈ ap p(r, l2(p, h)), for ϵ > 0, there exists a relatively dense set rϵ of r such that e∥φ1(t+τ)−φ1(t)∥ 2 < ϵ for τ ∈ rϵ, t ∈ r, |t−ti| > ϵ, i ∈ z, note that #w(s) := w(s+τ)−w(s), s ∈ r, is also a brownian motion with same distribution as w(s). now for t ∈ r, |t − ti| > ϵ, i ∈ z, using lemma 2.1 and ito’s isometry property of stochastic integral, we have e∥λφ1(t + τ) − λφ1(t)∥ 2 =e ( ( ( ( ∫ t −∞ (t − s)α−1aσjα(t − s)[φ1(s + τ) − φ1(s)]d#w(s) ( ( ( ( 2 ≤α2e ( ( ( ( ∫ t −∞ ∫ ∞ 0 θ(t − s)α−1nα(θ)a σs((t − s)αθ)[φ1(s + τ) − φ1(s)]dθd#w(s) ( ( ( ( 2 ≤α2m2 σ ∫ t −∞ ∫ ∞ 0 θ2(1−σ)n 2 α (θ)(t − s)2(α−σα−1)e−2λθ(t−s) α e∥[φ1(s + τ) − φ1(s)]∥ 2dθds <ϵα2m2σ ∫t −∞ ∫ ∞ 0 θ2(1−σ)n 2α(θ)(t − s) 2(α−σα−1)e−2λθ(t−s) α dθds. ≤ϵα2m2σ ∫ ∞ 0 n 2α(θ) ∫ ∞ 0 θ2(1−σ)ξ2(α−σα−1)e−2λθξ α dξdθ ≤m2 σ n0 γ(1 − 2σ) λ(2−2σ) ϵ, that is λφ1 ∈ ap p(r, l2(p, h)). next we show that λφ2 ∈ papρ(r, l 2(p, h)). in fact for γ > 0, we have 1 µ(γ,ρ) ∫ γ −γ e∥λφ2(t)∥ 2ρ(t)dt = 1 µ(γ,ρ) ∫ γ −γ e ( ( ( ( ∫ t −∞ aσ(t − s)α−1j (t − s)φ2(s)dw(s) ( ( ( ( 2 ρ(t)dt ≤ 1 µ(γ,ρ) ∫γ −γ ∫t −∞ ∥aσ(t − s)α−1j (t − s)∥2e∥φ2(s)∥ 2dsρ(t)dt ≤ 1 µ(γ,ρ) ∫ γ −γ ∫ ∞ 0 ∥ξα−1aσj (ξ)∥2e∥φ2(t − ξ)∥ 2dξρ(t)dt. cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 101 similar as previous calculation, we have ∫ ∞ 0 ∥ξα−1aσj (ξ)∥2e∥φ2(t − ξ)∥ 2dξ ≤α2m2 σ ∫ ∞ 0 n 2 α (θ) ∫ ∞ 0 θ2(1−σ)ξ2(α−σα−1)e−2λθξ α e∥φ2(t − ξ)∥ 2dξdθ. now, we have 1 µ(γ,ρ) ∫γ −γ e∥λ2y(t)∥ 2ρ(t)dt ≤α2m2σ ∫ ∞ 0 n 2α(θ) ∫ ∞ 0 θ2(1−σ)ξ2(α−σα−1)e−2λθξ α tγ(t)dξdθ. where tγ(t) = 1 µ(γ,ρ) ∫γ −γ e∥φ2(t − ξ)∥ 2ρ(t)dt. (3.4) since φ2(s) ∈ papρ(r, l 2(p, h)), ρ ∈ wt and translation invariant, this implies that φ2(· − s) ∈ papρ(r, l 2(p, h)) for each s ∈ r by remark 3. hence γ→∞tγ(t) = 0 for all s ∈ r. now, by lebesgue dominated convergence theorem, we have λφ2 ∈ papρ(r, l 2(p, h)). theorem 3.1. assume the conditions (h1) − (h7) are satisfy, if ∆ := 3m2σ & 4lgn 2 0 ! 1 mσ 0 + 1 eλ − 1 "2 + lg γ2(1 − σ) λ2(1−σ) + lfn0 ! 1 + lgc 2 k 2b " γ(1 − 2σ) λ(2−2σ) ' < 1, then the system (1.1) − (1.3) admits a unique mild solution in wpapρ(r, l 2(p, h)). proof. let m := {y ∈ wpapρ(r, l 2(p, h)) with discontinuity of first type at ti, i ∈ z satisfying e∥y∥2 ≤ r0, r0 > 0}. obviously, m is a closet subspace of wpapρ(r, l 2(p, h)). define an operator q in m by (qy)(t) = ∑ ti 0 there exists relative dense set rϵ of real numbers and zϵ of integers, such that for ti < t < ti+1, τ ∈ rϵ, p ∈ zϵ, |t − ti| > ϵ, |t − ti+1| > ϵ, i ∈ z, we have t + τ > ti + ϵ + τ > ti+p, ti+p+1 > ti+1 + τ − ϵ > t + τ, that is, ti+p < t + τ < ti+p+1, then using cauchy-schwarz inequality we have e∥υ1(t + τ) − υ1(t)∥ 2 ≤e ( ( ( ( ∑ ti 0, we have 1 µ(γ,ρ) ∫γ −γ e∥y2(t − η)∥ 2ρ(t)dt = 1 µ(γ,ρ) ∫γ−η −γ−η e∥y2(t)∥ 2ρ(t + η)dt = µ(γ + η,ρ) µ(γ,ρ) × 1 µ(γ + η,ρ) ∫ γ−η −γ−η e∥y2(t)∥ 2 ρ(t + η) ρ(t) ρ(t)dt. cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 107 since ρ ∈ wt , then there exists a0 > 0 such that ρ(t+η) ρ(t) ≤ a0, ρ(t−η) ρ(t) ≤ a0,. for γ > η µ(γ + η,ρ) = ∫γ−η −γ−η ρ(t)dt + ∫γ+η γ−η ρ(t)dt ≤ ∫γ−η −γ−η ρ(t)dt + ∫γ+η −γ+η ρ(t)dt = ∫γ −γ ρ(t − η) ρ(t) ρ(t)dt + ∫γ −γ ρ(t + η) ρ(t) ρ(t)dt ≤ 2a0µ(γ,ρ). then by y2 ∈ papρ(r, l 2(p, h)), we get 1 µ(γ,ρ) ∫γ −γ e∥y2(t − η)∥ 2ρ(t)dt ≤ 2a2 0 µ(γ + η,ρ) ∫γ+η −γ−η e∥y2(t)∥ 2ρ(t)dt → 0, as γ → ∞. hence y(t − η) ∈ wpapρ(r, l 2(p, h)) for η ∈ r+. thus the results of theorem 3.1 holds for the system (3.12) − (3.14). 4 example now,we present an example, which do not aim at generality but indicate how our abstract result can be applied to concrete problem. consider the stochastic fractional differential equation with impulsive effects ∂α ∂tα z(t, x) − ∂2 ∂x2 z(t, x) =g(t, x, z(t, x)) + f ! t, x, z(t, x), ∫ t ∞ e−2(t−s)g(s, x, z(t, x)ds " dw(t) dt , t ∈ r, (4.1) z(t+ i , x) =z(t− i , x) + λi(z(ti, x)), i ∈ z, x ∈ (0, 1), (4.2) z(t, 0) =z(t, 1) = 0, (4.3) where ti = i + 1 4 | sin 3i + sin √ 3i| and assume that λi ∈ wpapρ(z, l 2(p, h)), ρ ∈ wt . note that {t j i }, i, j ∈ z are equipotentially almost periodic and κ = infi∈z(ti+1 − ti) > 0, for more details see [14, 21, 24]. note that w(t) represents a standard wiener process on a complete probability space (ω, f, {ft}t≥0, p), where {ft}t≥0 is sigma algebra generated by {w(u) − w(v) : u, v ≤ t}. let h = (l2([0, 1], ∥ · ∥l2) be a hilbert space. now define the operator ay(ξ) := −y′′(ξ), ξ ∈ (0, 1), y ∈ d(a), where d(a) := {h2 ∩ h10 : y ′′ ∈ h}. then, a is the infinitesimal generator of analytic semigroup s(t) on h. now, we have zn(t) = (2) 1 2 sin nπt, n = 1, 2, 3, ..., are the eigenfunction of a corresponding to the eigenvalues nπ. for σ = 1 4 denote d(a 1 4 ) by l2(p, h 1 4 ) is a banach space equipped with the norm ∥y∥ 1 4 = ∥a 1 4 y∥, y ∈ d(a 1 4 ). 108 vikram singh and dwijendra n pandey cubo 19, 1 (2017) let y(t)x = z(t, x), t ∈ r, x ∈ [0, 1], then f ! t, y(t), ∫ t −∞ k(t − s)g(s, y(s))ds " = f ! t, x, z(t, x), ∫ t ∞ e−2(t−s)g(s, x, z(t, x)ds " . now the system (4.1)-(4.3) can be reformulated in the abstract form of the system (1.1)-(1.3). since gi = λi, then (h6) holds with lg = supi∈z ∥λi∥. we have the following result. theorem 4.1. assume that the following assumptions hold: (i) for ρ ∈ wt , g ∈ wpapρ(r × l 2(p, h 1 4 ), l2(p, h)) and there exists a lg > 0, 0 < η < 1 such that e∥g(t1, u1) − g(t2, u2)∥ 2 ≤ lge(|t2 − t1| η + ∥u1 − u2∥ 2 1 2 , for all, (ti, ui) ∈ r × l 2(p, h 1 4 ), i = 1, 2. (ii) for ρ ∈ wt , g ∈ wpapρ(r × l 2(p, h 1 4 ), l2(p, h)) and there exists lg > 0, 0 < η < 1 such that e∥g(t1, u1) − g(t2, u2)∥ 2 ≤ lg(|t2 − t1| η + e∥u1 − u2∥ 2 1 2 ), for each (ti, ui) ∈ r × l 2(p, h 1 4 ), i = 1, 2. (iii) for ρ ∈ wt , f ∈ wpapρ(r × l 2(p, h 1 4 ) × l2(p, h 1 4 ), l2(p, l2 0 )) and there exists a lf > 0 such that e∥f(t, u1, v1) − f(t, u2, u2)∥ 2 ≤ lf(|t2 − t1| ηe∥u1 − u2∥ 2 1 2 + e∥v1 − v2∥ 2 1 2 ), for each (ti, ui, vi) ∈ r × l 2(p, h 1 4 ) × l2(p, h 1 4 ), i = 1, 2. let us choose the constants mσ = 1,λ = 9, lg = 1, lf = 1, lg = 1, lg = 1 2 and n0 = 1 4 , then we have ∆ := 3m2σ & 4lgn 2 0 ! 1 mσ 0 + 1 eλ − 1 "2 + lg γ2(3 4 ) λ 3 2 + lfn0 ! 1 + lg 4 "√ π λ 3 2 ' = 0.69 < 1, this implies that the system (4.1)−(4.3) has a unique piecewise square mean weighted pseudo almost periodic solution. acknowledgment the authors would like to thank the editor and the reviewers for their valuable comments and suggestions. the work of the first author is supported by the “ministry of human resource and development, india under grant number:mhr-02-23-200-44”. cubo 19, 1 (2017) weighted pseudo almost periodic solutions for fractional order . . . 109 references [1] abbas s., benchohra m. and n’guérékata g.m.; topics in fractional differential equations, developments in mathematics, springer, new york (2012). 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[29] zhou y. and jiao f.; existence of mild solutions for fractional neutral evolution equations, comput. math. appl. 59 (2010), 1063–1077. cubo a mathematical journal vol.14, no¯ 03, (143–166). october 2012 weak and strong convergence theorems of a multistep iteration to a common fixed point of a family of nonself asymptotically nonexpansive mappings in banach spaces shrabani banerjee and binayak s.choudhury department of mathematics, bengal engineering and science university, shibpur, howrah-711103, india. email: banerjee.shrabani@yahoo.com, binayak12@yahoo.co.in abstract in this paper we have defined a multistep iterative scheme with errors involving a family of asymptotically nonexpansive nonself mappings in banach spaces. a retraction has been used in the construction of the iteration. we prove here weak and strong convergences of the iteration to common fixed points of the family of asymptotically nonexpansive nonself mappings. we have used several concepts of banach space geometry. our results improve and extend some recent results. resumen en este art́ıculo definimos un esquema de multi paso iterativo con errores que involucran una familia de aplicaciones no expansivas y no auto asintóticamente en espacios de banach. una retracción se ha usado en la construcción de la iteración. probamos convergencias débiles y fuertes de las iteraciones a puntos fijos clásicos de la familia de aplicaciones no expansivas no auto asintóticamente. hemos usado varios conceptos de geometŕıa en espacios de banach. nuestro resultado mejora y extiende algunos resultados recientes. keywords and phrases: modified multistep iterative process with errors; nonself asymptotically nonexpansive mapping; retraction; opial’s condition; uniformly convex banach space; common fixed point; kadec-klee property; condition (b); weak and strong convergence. 2010 ams mathematics subject classification: 47h10 144 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) 1 introduction let k be a nonempty subset of real normed space e. a self mapping t : k → k is called nonexpansive if ‖tx − ty‖ ≤ ‖x − y‖, for all x, y ∈ k a self mapping t : k → k is called asymptotically nonexpansive if there exists a sequence {kn} ⊂ [1, ∞) with limn→∞ kn = 1 such that ‖tnx − tny‖ ≤ kn‖x − y‖ for all x, y ∈ k and n ≥ 1. (1.1) t is said to be uniformly l-lipschitzian if there exists a constant l > 0 such that ‖tnx − tny‖ ≤ l‖x − y‖ for all x, y ∈ k and n ≥ 1. (1.2) the class of asymptotically nonexpansive mappings was introduced by goebel and kirk[7] in 1972 as a generalization of the class of nonexpansive self mappings. they proved that if t is a selfmap on k where k is a nonempty closed convex subset of a real uniformly convex banach space, then t has a fixed point. fixed point iterative processes for asymptotically nonexpansive self-mappings on convex subsets of banach spaces have been studied extensively by many authors. since t remains a selfmapping of a nonempty closed convex subset k of a banach space e, the well known mann[11] and ishikawa[8] iterative processes are well defined. if however the domain k of t is a proper subset of e (and it is the case of several applications) and t maps k into e, then the iteration processes of mann and ishikawa and their modifications fail to be well defined. to overcome this problem chidume et al.[2] introduced the concept of nonself asymptotically nonexpansive mappings in 2003 as a generalization of asymptotically nonexpansive self mappings. a subset k of e is said to be a retract of e if there exists a continuous mapping p : e → k such that px = x for all x ∈ k. every closed convex subset of a uniformly convex banach space is a retract. a mapping p : e → e is said to be a retraction if p2 = p. it follows that if a map p is a retraction then py = y for all y in the range of p. the nonself asymptotically nonexpansive mapping is defined as follows: definition 1.1. ([2]) let e be a real normed linear space, k be a nonempty subset of e and p : e → k be the nonexpansive retraction of e onto k. let t : k → e be a non-self mapping. t is said to be a non-self asymptotically nonexpansive mapping if there exists a sequence {kn} ⊂ [1, ∞) with kn → 1 as n → ∞ such that the following inequality holds: ‖t(pt)n−1x − t(pt)n−1y‖ ≤ kn‖x − y‖, for all x, y ∈ k and n ≥ 1. (1.3) t is said to be uniformly l-lipschitzian if there exists a constant l > 0 such that ‖t(pt)n−1x − t(pt)n−1y‖ ≤ l‖x − y‖, for all x, y ∈ k and n ≥ 1. (1.4) if t is a self map, then p becomes the identity map so that (1.3) and (1.4) coinside with (1.1) and (1.2) respectively. cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 145 chidume et al.[2] introduced and studied the weak and strong convergences of the following iterative process { x1 ∈ k xn+1 = p((1 − αn)xn + αnt(pt) n−1xn) (1.5) where {αn} is a appropriate real sequence in [0, 1]. if t is a self map, then p becomes the identity map so that (1.5) reduces to the mann-type iteration scheme[11]. then wang[19] used a similar scheme to prove the weak and strong convergence theorems for a pair of non-self asymptotically nonexpansive mappings which is given by    x1 ∈ k xn+1 = p((1 − αn)xn + αnt1(pt1) n−1yn) yn = p((1 − βn)xn + βnt2(pt2) n−1xn), n ≥ 1. (1.6) if t is a self map, then p becomes the identity map so that (1.6) reduces to the ishikawa-like iteration scheme without errors [9] involving two asymptotically nonexpansive self mappings. after that chidume and bashir ali [3] introduced a new iteration process for approximating common fixed points for finite families of nonself asymptotically nonexpansive mappings which is defined as follows:    x1 ∈ k xn+1 = p[(1 − α1n)xn + α1nt1(pt1) n−1yn+r−2] yn+r−2 = p[(1 − α2n)xn + α2nt2(pt2) n−1yn+r−3] · · · · yn = p[(1 − αmn)xn + αmntm(ptm) n−1xn] (1.7) very recently yang [20] introduced and studied a modified multistep iteration for a finite family of nonself asymptotically nonexpensive mappings and discuss their convergences which is defined as follows. 146 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) for a given x1 ∈ k and n ≥ 1, compute the iterative sequences {xn}, {yn}, ....., {yn+r−2} defined by    yn = p[(1 − anr)xn + anrtr(ptr) n−1xn] yn+1 = p[(1 − an(r−1) − bn(r−1))xn + an(r−1)tr−1(ptr−1) n−1yn +bn(r−1)tr−1(ptr−1) n−1xn] yn+2 = p[(1 − an(r−2) − bn(r−2))xn + an(r−2)tr−2(ptr−2) n−1yn+1 +bn(r−2)tr−2(ptr−2) n−1yn] . . . . yn+r−2 = p[(1 − an2 − bn2)xn + an2t2(pt2) n−1yn+r−3 +bn2t2(pt2) n−1yn+r−4] xn+1 = p[(1 − an1 − bn1)xn + an1t1(pt1) n−1yn+r−2 +bn1t1(pt1) n−1yn+r−3] (1.8) where {ani}, {bni}, {1 − ani − bni} are appropriate real sequences in [0, 1] for i ∈ i where i = {1, 2, ....., r}. motivated by these facts we have introduced and studied a new type of multistep iterative process with errors which is defined as follows: let e be a normed space, k be a nonempty convex subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with limn→∞ k i n = 1 for i ∈ i. then for a given x1 ∈ k and n ≥ 1, compute the iterative sequences {xn}, {yn}, ....., {yn+r−2} defined by    yn = p[(1 − a 1 nr − bnr)xn + a 1 nrtr(ptr) n−1xn + bnrunr] yn+1 = p[(1 − a 1 n(r−1) − a2 n(r−1) − bn(r−1))xn + a 1 n(r−1) tr−1(ptr−1) n−1yn +a2 n(r−1) tr−1(ptr−1) n−1xn + bn(r−1)un(r−1)] yn+2 = p[(1 − a 1 n(r−2) − a2 n(r−2) − a3 n(r−2) − bn(r−2))xn + a 1 n(r−2) tr−2(ptr−2) n−1yn+1 +a2 n(r−2) tr−2(ptr−2) n−1yn + a 3 n(r−2) tr−2(ptr−2) n−1xn + bn(r−2)un(r−2)] . . . . yn+r−2 = p[(1 − a 1 n2 − a 2 n2 − ..... − a r−1 n2 − bn2)xn + a 1 n2t2(pt2) n−1yn+r−3 +a2n2t2(pt2) n−1yn+r−4 + ..... + a r−1 n2 t2(pt2) n−1xn + bn2un2] xn+1 = p[(1 − a 1 n1 − a 2 n1 − ..... − a r n1 − bn1)xn + a 1 n1t1(pt1) n−1yn+r−2 +a2n1t1(pt1) n−1yn+r−3 + ..... + a r n1t1(pt1) n−1xn + bn1un1] (1.9) where {aknj}, {bnj}, {1 − ∑r−j+1 k=1 aknj − bnj} are appropriate real sequences in [0, 1] for j ∈ i and k ∈ {1, ..., r − j + 1} and {unj} are bounded sequences in k for j ∈ i. the iterative sequence cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 147 (1.9) is called the new modified multistep iteration for a finite family of nonself asymptotically nonexpansive mappings. the iterative sequence (1.9) can be written as in the compact form yn+r−j = p[(1 − r−j+1∑ k=1 aknj − bnj)xn + r−j∑ k=1 aknjtj(ptj) n−1yn+r−j−k + a r−j+1 nj tj(ptj) n−1xn + bnjunj] where j ∈ i and xn+1 = yn+r−1. as an illustration, for r = 3, (1.9) reduces to the new modified three-step iteration with errors:    yn = p[(1 − a 1 n3 − bn3)xn + a 1 n3t3(pt3) n−1xn + bn3un3] yn+1 = p[(1 − a 1 n2 − a 2 n2 − bn2)xn + a 1 n2t2(pt2) n−1yn +a2n2t2(pt2) n−1xn + bn2un2] xn+1 = p[(1 − a 1 n1 − a 2 n1 − a 3 n1 − bn1)xn + a 1 n1t1(pt1) n−1yn+1 +a2n1t1(pt1) n−1yn + a 3 n1t1(pt1) n−1xn + bn1un1] (1.10) where {aknj}, {bnj}, {1 − ∑3−j+1 k=1 aknj − bnj} are appropriate real sequences in [0, 1] for j ∈ {1, 2, 3} and k ∈ {1, ..., 3 − j + 1} and {unj} are bounded sequences in k for j ∈ {1, 2, 3}. for aknj = 0 for all j ∈ {1, 2, ...., r−2} and k ∈ {3, 4, ..., r−j+1} and bnj = 0 for all j ∈ i, (1.9) reduces to the iteration (1.8). again if aknj = 0 for all j ∈ {1, 2, ...., r − 2, r − 1} and k = {2, 3, 4, ..., r − j + 1} and bnj = 0 for all j ∈ i, then (1.9) reduces to the iteration (1.7). next we recall the following definitions and results. let e be a real normed linear space. the modulus of convexity of e is a function δe : (0, 2] → [0, 1] defined by δe(ǫ) = inf{1 − ‖ 1 2 (x + y)‖ : ‖x‖ = 1, ‖y‖ = 1, ǫ = ‖x − y‖} . e is called uniformly convex if and only if δe(ǫ) > 0 for all ǫ ∈ (0, 2]. the norm of e is said to be frèchet differentiable if for each x ∈ e with ‖x‖ = 1 the limit limt→0 ‖x+ty‖−‖x‖ t exists and is attained uniformly for y with ‖y‖ = 1 and in this case it has been shown that in [18] that < h, j(x) > + 1 2 ‖x‖2 ≤ 1 2 ‖x + h‖2 ≤< h, j(x) > + 1 2 ‖x‖2 + b(‖h‖) (1.11) for all x, h ∈ e where j is the frèchet derivative of the functional ‖.‖2 at x ∈ e, < ., . > is the pairing between e and e⋆ and b is a function defined on [0, ∞) such that limt→0 b(t) t = 0. a banach space e is said to satisfy opial’s condition [12] if xn ⇀ x and x 6= y imply lim sup n→∞ ‖xn − x‖ < lim sup n→∞ ‖xn − y‖ . a banach space e is said to satisfy kadec-klee property, if for every sequence {xn} ∈ e, xn ⇀ x and ‖xn‖ → ‖x‖ together imply that xn → x as n → ∞. there are uniformly convex banach spaces which have neither a frèchet differentiable norm nor satisfy opial’s property but their dual does have the kadec-klee property (see [6],[10]). 148 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) lemma 1.1. ([18], lemma1) let {an}, {bn} and {δn} be sequences of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + δn)an + bn, ∀n ≥ 1. if ∑ ∞ n=1 δn < ∞ and ∑ ∞ n=1 bn < ∞, then (i) limn→∞ an exists, (ii) limn→∞ an = 0 whenever lim infn→∞ an = 0. lemma 1.2. ([21], theorem2) let p > 1 and r > 0 be two fixed real numbers. then a banach space e is uniformly convex if and only if there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that ‖λx + (1 − λ)y‖p ≤ λ‖x‖p + (1 − λ)‖y‖p − ωp(λ)g(‖x − y‖) for all x, y ∈ br(0) = {x ∈ e : ‖x‖ ≤ r} and λ ∈ [0, 1] where ωp(λ) = λ p(1 − λ) + λ(1 − λ)p. lemma 1.3. ([5], lemma1.4) let e be a uniformly convex banach space and br = {x ∈ e : ‖x‖ ≤ r}, r > 0. then there exist a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞), g(0) = 0 such that ‖λx + βy + γz‖2 ≤ λ‖x‖2 + β‖y‖2 + γ‖z‖2 − λβg(‖x − y‖) for all x, y, z ∈ br and all λ, β, γ ∈ [0, 1] with λ + β + γ = 1. by using lemma 1.2 and lemma 1.3 we can easily prove the following lemma: lemma 1.4. let e be a uniformly convex banach space and br = {x ∈ e : ‖x‖ ≤ r}, r > 0. then there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞), g(0) = 0 such that ‖λ1x1 + λ2x2 + ... + λnxn‖ 2 ≤ λ1‖x1‖ 2 + λ2‖x2‖ 2 + ... + λn‖xn‖ 2 − λ1λ2g(‖x1 − x2‖) for all xi ∈ br and all λi ∈ [0, 1] for all i = 1, 2, ..., n with ∑n i=1 λi = 1. proof: the lemma is true for n = 2 since for n = 2 using lemma 1.2 we get ‖λ1x1 + λ2x2‖ 2 ≤ λ1‖x1‖ 2 + λ2‖x2‖ 2 − ω2(λ1)g(‖x1 − x2‖) where ω2(λ1) = λ 2 1(1 − λ1) + λ1(1 − λ1) 2 = λ21λ2 + λ1λ 2 2 = λ1λ2(λ1 + λ2) = λ1λ2. also by lemma 1.3 we see that this lemma is true for n = 3. now let the lemma is true for n = m. now ‖λ1x1 + λ2x2 + ... + λmxm + λm+1xm+1‖ 2 = ‖λ1x1 + λ2x2 + ... + λm−1xm−1 + (1 − λ1 − λ2 − .... − λm−1)( λm 1 − λ1 − λ2 − .... − λm−1 xm + λm+1 1 − λ1 − λ2 − .... − λm−1 xm+1)‖ 2. cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 149 by using the above inequality, ‖λ1x1 + λ2x2 + ... + λmxm + λm+1xm+1‖ 2 = ‖λ1x1 + λ2x2 + ... + λm−1xm−1 + (1 − λ1 − λ2 − .... − λm−1)( λm 1 − λ1 − λ2 − .... − λm−1 xm + λm+1 1 − λ1 − λ2 − .... − λm−1 xm+1)‖ 2 ≤ λ1‖x1‖ 2 + λ2‖x2‖ 2 + ... + λm−1‖xm−1‖ 2 + (1 − λ1 − λ2 − .... − λm−1)‖ λm 1 − λ1 − λ2 − .... − λm−1 xm + λm+1 1 − λ1 − λ2 − .... − λm−1 xm+1‖ 2 −λ1λ2g(‖x1 − x2‖) . since λ1 + λ2 + ... + λm + λm+1 = 1, so λm 1 − λ1 − λ2 − .... − λm−1 + λm+1 1 − λ1 − λ2 − .... − λm−1 = λm + λm+1 1 − λ1 − λ2 − .... − λm−1 = 1 . then from lemma 1.2 we get that ‖ λm 1 − λ1 − λ2 − .... − λm−1 xm + λm+1 1 − λ1 − λ2 − .... − λm−1 xm+1‖ 2 ≤ λm 1 − λ1 − λ2 − .... − λm−1 ‖xm‖ 2 + λm+1 1 − λ1 − λ2 − .... − λm−1 ‖xm+1‖ 2 −ω2( λm 1 − λ1 − λ2 − .... − λm−1 )g(‖x1 − x2‖) . now, ω2( λm 1 − λ1 − λ2 − .... − λm−1 ) = ( λm 1 − λ1 − λ2 − .... − λm−1 )2(1 − λm 1 − λ1 − λ2 − .... − λm−1 ) + λm 1 − λ1 − λ2 − .... − λm−1 (1 − λm 1 − λ1 − λ2 − .... − λm−1 )2 = λm 1 − λ1 − λ2 − .... − λm−1 . λm+1 1 − λ1 − λ2 − .... − λm−1 ( λm 1 − λ1 − λ2 − .... − λm−1 + λm+1 1 − λ1 − λ2 − .... − λm−1 ) = λmλm+1 (1 − λ1 − λ2 − .... − λm−1)2 ≥ 0 . therefore from above we have ‖ λm 1 − λ1 − λ2 − .... − λm−1 xm + λm+1 1 − λ1 − λ2 − .... − λm−1 xm+1‖ 2 ≤ λm 1 − λ1 − λ2 − .... − λm−1 ‖xm‖ 2 + λm+1 1 − λ1 − λ2 − .... − λm−1 ‖xm+1‖ 2 . 150 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) so finally we get that ‖λ1x1 + λ2x2 + ... + λmxm + λm+1xm+1‖ 2 = ‖λ1x1 + λ2x2 + ... + λm−1xm−1 + (1 − λ1 − λ2 − .... − λm−1)( λm 1 − λ1 − λ2 − .... − λm−1 xm + λm+1 1 − λ1 − λ2 − .... − λm−1 xm+1)‖ 2 ≤ λ1‖x1‖ 2 + λ2‖x2‖ 2 + ... + λm−1‖xm−1‖ 2 + (1 − λ1 − λ2 − .... − λm−1)( λm 1 − λ1 − λ2 − .... − λm−1 ‖xm‖ 2 + λm+1 1 − λ1 − λ2 − .... − λm−1 ‖xm+1‖ 2 ) − λ1λ2g(‖x1 − x2‖) = λ1‖x1‖ 2 + λ2‖x2‖ 2 + ... + λm−1‖xm−1‖ 2 + λm‖xm‖ 2 + λm+1‖xm+1‖ 2 − λ1λ2g(‖x1 − x2‖) . hence the lemma is true for n = m + 1. thus, by induction, the lemma is true for all n ≥ 2. this completes the proof of the lemma. lemma 1.5. ([2], theorem3.4) let e be a real uniformly banach space and k be a nonempty closed convex subset of e and t : k → e be asymptotically nonexpansive mapping with a sequence {kn} ⊂ [1, ∞) with limn→∞ kn = 1. then i − t is demiclosed at zero, i.e. if {xn} is a sequence in k which converges weakly to x and if the sequence {xn − txn} converges strongly to zero, then x − tx = 0. lemma 1.6. ([10], theorem2) let e be a real reflexive banach space such that e⋆ has the kadecklee property. let {xn} be a bounded sequence in e and x ⋆, y⋆ ∈ ww(xn)(weak w-limit set of {xn}). suppose limn→∞ ‖txn + (1 − t)x ⋆ − y⋆‖ exists for all t ∈ [0, 1]. then x⋆ = y⋆. lemma 1.7. ([1]) let e be a uniformly convex banach space k be a nonempty bounded closed convex subset of e. then there exists a strictly increasing continuous convex function φ : [0, ∞) → [0, ∞) with φ(0) = 0 such that for any lipschitzian mapping t : k → e with the lipschitz constant l ≥ 1 and for any x, y ∈ k and t ∈ [0, 1] the following inequality holds: ‖t(tx + (1 − t)y) − (ttx + (1 − t)ty)‖ ≤ lφ−1(‖x − y‖ − l−1‖tx − ty‖) the purpose of this paper is to introduce a new modified multi step iteration with errors for approximating common fixed points for finite families of nonself asymptotically nonexpansive mappings. we prove some strong and weak convergence theorems in real uniformly convex banach spaces. more precisely we prove convergence theorems in a uniformly convex banach space which satisfy opial’s condition or have frèchet differentiable norm or whose duals have the kadec-klee property. our results generalize some recent results. 2 main results we begin this section with the following lemmas. cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 151 lemma 2.1. let e be a real normed space and k be a nonempty subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with ∑ ∞ n=1 bni < ∞ for i ∈ i. if f = ⋂r i=1 f(ti) 6= ∅, then limn→∞ ‖xn − q‖ exists for all q ∈ f. proof: let q ∈ f. for each n ≥ 1, let kn = max {k 1 n, k 2 n, .........., k r n} so that {kn} ⊂ [1, ∞) with ∑ ∞ n=1 (kn − 1) < ∞. since {uni} are bounded sequences in k for i ∈ i, let m = sup n≥1,i=1,2,...,r‖uni − q‖. from (1.9) we get ‖yn − q‖ = ‖p((1 − a 1 nr − bnr)xn + a 1 nrtr(ptr) n−1xn + bnrunr) − pq‖ ≤ ‖(1 − a1nr − bnr)(xn − q) + a 1 nr(tr(ptr) n−1xn − q) + bnr(unr − q)‖ ≤ (1 − a1nr − bnr)‖xn − q‖ + a 1 nr‖tr(ptr) n−1xn − q‖ + bnr‖unr − q‖ ≤ (1 − a1nr)‖xn − q‖ + a 1 nrkn‖xn − q‖ + bnrm ≤ kn‖xn − q‖ + bnrm = kn‖xn − q‖ + σ 1 n (2.1) where σ1n = bnrm. by the given condition we get that ∑ ∞ n=1 σ1n < ∞. also from (1.9) and (2.1) we have ‖yn+1 − q‖ = ‖p((1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))xn + a 1 n(r−1)tr−1(ptr−1) n−1yn + a2n(r−1)tr−1(ptr−1) n−1xn + bn(r−1)un(r−1)) − pq‖ ≤ ‖(1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))(xn − q) + a 1 n(r−1)(tr−1(ptr−1) n−1yn − q) +a2n(r−1)(tr−1(ptr−1) n−1xn − q) + bn(r−1)(un(r−1) − q)‖ ≤ (1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))‖xn − q‖ + a 1 n(r−1)‖tr−1(ptr−1) n−1yn − q‖ +a2n(r−1)‖tr−1(ptr−1) n−1xn − q‖ + bn(r−1)‖un(r−1) − q‖ ≤ (1 − a1n(r−1) − a 2 n(r−1))‖xn − q‖ + a 1 n(r−1)kn‖yn − q‖ +a2n(r−1)kn‖xn − q‖ + bn(r−1)m ≤ (1 − a1n(r−1) − a 2 n(r−1))‖xn − q‖ + a 1 n(r−1)kn[kn‖xn − q‖ + bnrm] +a2n(r−1)kn‖xn − q‖ + bn(r−1)m ≤ [1 + a1n(r−1)(k 2 n − 1) + a 2 n(r−1)(kn − 1)]‖xn − q‖ + a 1 n(r−1)knbnrm + bn(r−1)m ≤ k2n‖xn − q‖ + knbnrm + bn(r−1)m = k2n‖xn − q‖ + σ 2 n (2.2) where σ2n = knbnrm + bn(r−1)m. by the given condition we get that ∑ ∞ n=1 σ2n < ∞. also from 152 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) (1.9) and (2.2) we have ‖yn+2 − q‖ = ‖p((1 − a1n(r−2) − a 2 n(r−2) − a 3 n(r−2) − bn(r−2))xn + a 1 n(r−2)tr−2(ptr−2) n−1yn+1 +a2n(r−2)tr−2(ptr−2) n−1yn + a 3 n(r−2)tr−2(ptr−2) n−1xn + bn(r−2)un(r−2)) − pq‖ ≤ ‖(1 − a1n(r−2) − a 2 n(r−2) − a 3 n(r−2) − bn(r−2))(xn − q) + a 1 n(r−2)(tr−2(ptr−2) n−1yn+1 − q) +a2n(r−2)(tr−2(ptr−2) n−1yn − q) + a 3 n(r−2)(tr−2(ptr−2) n−1xn − q) +bn(r−2)(un(r−2) − q)‖ ≤ (1 − a1n(r−2) − a 2 n(r−2) − a 3 n(r−2) − bn(r−2))‖xn − q‖ + a 1 n(r−2)‖tr−2(ptr−2) n−1yn+1 − q‖ +a2n(r−2)‖tr−2(ptr−2) n−1yn − q‖ + a 3 n(r−2)‖tr−2(ptr−2) n−1xn − q‖ +bn(r−2)‖un(r−2) − q‖ ≤ (1 − a1n(r−2) − a 2 n(r−2) − a 3 n(r−2))‖xn − q‖ + a 1 n(r−2)kn‖yn+1 − q‖ +a2n(r−2)kn‖yn − q‖ + a 3 n(r−2)kn‖xn − q‖ + bn(r−2)m ≤ (1 − a1n(r−2) − a 2 n(r−2) − a 3 n(r−2))‖xn − q‖ + a 1 n(r−2)kn[k 2 n‖xn − q‖ +knbnrm + bn(r−1)m] + a 2 n(r−2)kn[kn‖xn − q‖ + bnrm] +a3n(r−2)kn‖xn − q‖ + bn(r−2)m ≤ k3n‖xn − q‖ + σ 3 n (2.3) where σ3n = k 2 nbnrm + knbnrm + knbn(r−1)m + bn(r−2)m. by the given condition we get that∑ ∞ n=1 σ3n < ∞. in general after (j + 1) steps we get ‖yn+j − q‖ ≤ k j+1 n ‖xn − q‖ + σ j+1 n (2.4) for j = 0, 1, ..., r − 2 and {σ j+1 n } is a nonnegative real sequence such that ∑ ∞ n=1 σ j+1 n < ∞ for j = 0, 1, ..., r − 2. therefore it follows from (1.9) and (2.4) that ‖xn+1 − q‖ ≤ k r n‖xn − q‖ + σ r n = [1 + (k r n − 1)]‖xn − q‖ + σ r n (2.5) where {σrn} is a nonnegative real sequence such that ∑ ∞ n=1 σrn < ∞. since 0 ≤ t r −1 ≤ rtr−1(t−1) for all t ≥ 1, so 0 ≤ krn − 1 ≤ rk r−1 n (kn − 1). since ∑ ∞ n=1 (kn − 1) < ∞ so {kn} is bounded, kn ∈ [1, m ′] for some m′ > 0. so ∑ ∞ n=1 (krn−1) < ∞. thus by lemma 1.1 we get limn→∞ ‖xn−q‖ exists for all q ∈ f. ♦ lemma 2.2. let e be a uniformly convex banach space and k be a nonempty closed convex subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with ∑ ∞ n=1 bni < ∞ for i ∈ i. if f = ⋂r i=1 f(ti) 6= ∅, then the following results hold (1) if lim infn→∞ a 1 nk > 0, for all k < r and 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr + bnr) < 1 then limn→∞ ‖tr(ptr) n−1xn − xn‖ = 0. cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 153 (2) if 0 < lim infn→∞ a 1 n1 ≤ lim supn→∞( ∑r k=1 akn1+bn1) < 1 then limn→∞ ‖t1(pt1) n−1yn+r−2− xn‖ = 0. (3) if lim infn→∞ a 1 nk > 0, for all k < j and 0 < lim infn→∞ a 1 nj ≤ lim supn→∞( ∑r−j+1 m=1 amnj + bnj) < 1, then limn→∞ ‖tj(ptj) n−1yn+r−j−1 − xn‖ = 0 for j = 2, 3, ....., r − 1. proof: let q ∈ f. by lemma 2.1 we have that limn→∞ ‖xn − q‖ exists for all q ∈ f. so {xn −q} is bounded in k. since {kn} and {σ j+1 n } are bounded so it follows from (2.4) that {yn+j −q} are bounded for j = 0, 1, ..., r − 2. since tj is a nonself asymptotically nonexpansive mapping, we have ‖tj(ptj) n−1yn+r−j−1 − q‖ ≤ k j n‖yn+r−j−1 − q‖ for j = 1, ..., r−1. therefore the sequences {tj(ptj) n−1yn+r−j−1 −q} are bounded for j = 1, ..., r−1. therefore there exists d > 0 such that k ⊆ bd. from (1.9) and lemma 1.3 we get ‖yn − q‖ 2 = ‖p((1 − a1nr − bnr)xn + a 1 nrtr(ptr) n−1xn + bnrunr) − pq‖ 2 ≤ ‖(1 − a1nr − bnr)(xn − q) + a 1 nr(tr(ptr) n−1xn − q) + bnr(unr − q)‖ 2 ≤ (1 − a1nr − bnr)‖xn − q‖ 2 + a1nr‖tr(ptr) n−1xn − q‖ 2 + bnr‖unr − q‖ 2 − (1 − a1nr − bnr)a 1 nrg1(‖tr(ptr) n−1xn − xn‖) ≤ (1 − a1nr)‖xn − q‖ 2 + a1nrk 2 n‖xn − q‖ 2 + bnrm 2 − (1 − a1nr − bnr)a 1 nrg1(‖tr(ptr) n−1xn − xn‖) ≤ k2n‖xn − q‖ 2 + µ1n − a 1 nr(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) (2.6) 154 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) where µ1n = bnrm 2 so that ∑ ∞ n=1 µ1n < ∞. from (1.9) and (2.6) and from lemma 1.4 we get ‖yn+1 − q‖ 2 = ‖p[(1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))xn + a 1 n(r−1)tr−1(ptr−1) n−1yn + a2n(r−1)tr−1(ptr−1) n−1xn + bn(r−1)un(r−1)] − pq‖ 2 ≤ ‖(1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))(xn − q) + a 1 n(r−1)(tr−1(ptr−1) n−1yn − q) +a2n(r−1)(tr−1(ptr−1) n−1xn − q) + bn(r−1)(un(r−1) − q)‖ 2 ≤ (1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))‖xn − q‖ 2 + a1n(r−1)‖tr−1(ptr−1) n−1yn − q‖ 2 +a2n(r−1)‖tr−1(ptr−1) n−1xn − q‖ 2 + bn(r−1)‖un(r−1) − q‖ 2 −a1n(r−1)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) ≤ (1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))‖xn − q‖ 2 + a1n(r−1)k 2 n‖yn − q‖ 2 +a2n(r−1)k 2 n‖xn − q‖ 2 + bn(r−1)m 2 −a1n(r−1)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) ≤ (1 − a1n(r−1) − a 2 n(r−1) − bn(r−1))‖xn − q‖ 2 + a1n(r−1)k 2 n[k 2 n‖xn − q‖ 2 + µ1n −a1nr(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖)] + a 2 n(r−1)k 2 n‖xn − q‖ 2 +bn(r−1)m 2 −a1n(r−1)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) ≤ k4n‖xn − q‖ 2 + k2nµ 1 n + bn(r−1)m 2 −a1n(r−1)a 1 nr(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) −a1n(r−1)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) ≤ k4n‖xn − q‖ 2 + µ2n −a1n(r−1)a 1 nr(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) −a1n(r−1)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) (2.7) where µ2n = k 2 nµ 1 n + bn(r−1)m 2, so that ∑ ∞ n=1 µ2n < ∞. proceeding in this way we have ‖yn+j − q‖ 2 ≤ k2(j+1)n ‖xn − q‖ 2 + µ(j+1)n − ( r∏ i=r−j a1ni)(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) − ( r−1∏ i=r−j a1ni)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) − .... − a1n(r−j)(1 − j+1∑ k=1 akn(r−j) − bn(r−j))gj+1(‖tr−j(ptr−j) n−1yn+j−1 − xn‖) (2.8) for j = 1, 2, ..., r − 2 and {µ (j+1) n } is a nonnegative real sequence such that ∑ ∞ n=1 µ (j+1) n < ∞. thus cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 155 we get ‖xn+1 − q‖ 2 ≤ k2rn ‖xn − q‖ 2 + µrn − ( r∏ i=1 a1ni)(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) − ( r−1∏ i=1 a1ni)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(||tr−1(ptr−1) n−1yn − xn||) −.... − a1n1a 1 n2(1 − r−1∑ k=1 akn2 − bn2)gr−1(‖t2(pt2) n−1yn+r−3 − xn‖) −a1n1(1 − r∑ k=1 akn1 − bn1)gr(‖t1(pt1) n−1yn+r−2 − xn‖) (2.9) where {µrn} is a nonnegative real sequence such that ∑ ∞ n=1 µrn < ∞. since {kn} is bounded so there exists m1 > 0 such that kn ∈ [1, m1] for all n ≥ 1. hence k 2r n − 1 ≤ 2rk 2r−1 n (kn − 1) ≤ 2rm2r−1 1 (kn − 1) holds for all n ≥ 1. so ∑ ∞ n=1 (kn − 1) < ∞ implies that ∑ ∞ n=1 (k2rn − 1) < ∞. therefore from (2.9) we get ‖xn+1 − q‖ 2 ≤ ‖xn − q‖ 2 + (k2rn − 1)‖xn − q‖ 2 + µrn − ( r∏ i=1 a1ni)(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) − ( r−1∏ i=1 a1ni)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) −.... − a1n1a 1 n2(1 − r−1∑ k=1 akn2 − bn2)gr−1(‖t2(pt2) n−1yn+r−3 − xn‖) −a1n1(1 − r∑ k=1 akn1 − bn1)gr(‖t1(pt1) n−1yn+r−2 − xn‖) ≤ ‖xn − q‖ 2 + 2rm2r−11 (kn − 1)d 2 + µrn − ( r∏ i=1 a1ni)(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) − ( r−1∏ i=1 a1ni)(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))g2(‖tr−1(ptr−1) n−1yn − xn‖) −.... − a1n1a 1 n2(1 − r−1∑ k=1 akn2 − bn2)gr−1(‖t2(pt2) n−1yn+r−3 − xn‖) −a1n1(1 − r∑ k=1 akn1 − bn1)gr(‖t1(pt1) n−1yn+r−2 − xn‖) . (2.10) 156 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) from (2.10) we get ( r∏ i=1 a1ni)(1 − a 1 nr − bnr)g1(‖tr(ptr) n−1xn − xn‖) ≤ ‖xn − q‖ 2 − ‖xn+1 − q‖ 2 + 2rm2r−1 1 (kn − 1)d 2 + µrn (2.11) and ( j∏ i=1 a1ni)(1 − r−j+1∑ k=1 aknj − bnj)gr−j+1(‖tj(ptj) n−1yn+r−j−1 − xn‖) ≤ ‖xn − q‖ 2 − ‖xn+1 − q‖ 2 + 2rm2r−11 (kn − 1)d 2 + µrn (2.12) for j = 1, 2, ..., r−1. if lim infn→∞ a 1 ni > 0, for all i < r and 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr+ bnr), then there exists a positive integer n0 and η, η ′ ∈ (0, 1) such that 0 < η < a1ni(i ∈ i), a1nr + bnr < η ′ < 1, for all n ≥ n0. thus from (2.11) we get ηr(1 − η′)g1(‖tr(ptr) n−1xn − xn‖) ≤ ‖xn − q‖ 2 − ‖xn+1 − q‖ 2 + 2rm2r−11 (kn − 1)d 2 +µrn, for all n ≥ n0 . this implies that ∞∑ n=n0 g1(‖tr(ptr) n−1xn − xn‖) ≤ 1 ηr(1 − η′) (‖xn0 − q‖ 2 + 2rm2r−1 1 d2 ∞∑ n=n0 (kn − 1) + ∞∑ n=n0 µrn) < ∞ which further implies that limn→∞ g1(‖tr(ptr) n−1xn − xn‖) = 0. since g1 is strictly increasing and continuous with g1(0) = 0, so limn→∞ ‖tr(ptr) n−1xn − xn‖ = 0. similarly from (2.12) using the fact that gr−j+1 is strictly increasing and continuous with gr−j+1(0) = 0 we get limn→∞ ‖tj(ptj) n−1yn+r−j−1 − xn‖ = 0 for j = 1, 2, ....., r − 1. ♦ lemma 2.3. let e be a uniformly convex banach space and k be a nonempty closed convex subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with ∑ ∞ n=1 bni < ∞ for i ∈ i. if f = ⋂r i=1 f(ti) 6= ∅ and (1) 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr + bnr) < 1 (2) 0 < lim infn→∞ a 1 nj ≤ lim supn→∞( ∑r−j+1 m=1 amnj + bnj) < 1, then limn→∞ ‖xn − tixn‖ = 0 for i ∈ i. proof: by lemma 2.2 we get lim n→∞ ‖tr(ptr) n−1xn − xn‖ = 0 and lim n→∞ ‖tj(ptj) n−1yn+r−j−1 − xn‖ = 0 for j = 1, 2, ....., r − 1. (2.13) cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 157 since p is nonexpansive mapping so from (1.9) together with (2.13) we have ‖yn − xn‖ ≤ a 1 nr‖tr(ptr) n−1xn − xn‖ + bnr‖unr − xn‖ ≤ ‖tr(ptr) n−1xn − xn‖ + bnr‖unr − xn‖ → 0 as n → ∞. (2.14) since tr−1 is nonself asymptotically nonexpansive mapping, from (2.13) and (2.14) we get ‖tr−1(ptr−1) n−1xn − xn‖ ≤ ‖tr−1(ptr−1) n−1xn − tr−1(ptr−1) n−1yn‖ + ‖tr−1(ptr−1) n−1yn − xn‖ ≤ kn‖yn − xn‖ + ‖tr−1(ptr−1) n−1yn − xn‖ → 0 as n → ∞. (2.15) again from (1.9), (2.13) and (2.15) it follows that ‖yn+1 − xn‖ ≤ a 1 n(r−1)‖tr−1(ptr−1) n−1yn − xn‖ + a 2 n(r−1)‖tr−1(ptr−1) n−1xn − xn‖ +bn(r−1)‖un(r−1) − xn‖ → 0 as n → ∞. (2.16) from (2.16) and (2.13) we have that ‖tr−2(ptr−2) n−1xn − xn‖ ≤ ‖tr−2(ptr−2) n−1xn − tr−2(ptr−2) n−1yn+1‖ + ‖tr−2(ptr−2) n−1yn+1 − xn‖ ≤ kn‖yn+1 − xn‖ + ‖tr−2(ptr−2) n−1yn+1 − xn‖ → 0 as n → ∞. (2.17) also from (2.17) and (2.14) we have that ‖tr−2(ptr−2) n−1yn − xn‖ ≤ ‖tr−2(ptr−2) n−1yn − tr−2(ptr−2) n−1xn‖ + ‖tr−2(ptr−2) n−1xn − xn‖ ≤ kn‖yn − xn‖ + ‖tr−2(ptr−2) n−1xn − xn‖ → 0 as n → ∞ . (2.18) continuing in this way we have that lim n→∞ ‖tj(ptj) n−1yn+r−j−2 − xn‖ = 0 for j = 1, 2, ....., r − 2. (2.19) again from (1.9), (2.13), (2.17) and (2.18) it follows that ‖yn+2 − xn‖ ≤ a 1 n(r−2)‖tr−2(ptr−2) n−1yn+1 − xn‖ + a 2 n(r−2)‖tr−2(ptr−2) n−1yn − xn‖ +a3n(r−2)‖tr−2(ptr−2) n−1xn − xn‖ + bn(r−2)‖un(r−2) − xn‖ → 0 as n → ∞. (2.20) 158 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) from (2.20) and (2.13) we have that ‖tr−3(ptr−3) n−1xn − xn‖ ≤ ‖tr−3(ptr−3) n−1xn − tr−3(ptr−3) n−1yn+2‖ + ‖tr−3(ptr−3) n−1yn+2 − xn‖ ≤ kn‖yn+2 − xn‖ + ‖tr−3(ptr−3) n−1yn+2 − xn‖ → 0 as n → ∞. (2.21) thus from (2.21) and (2.14) it follows that ‖tr−3(ptr−3) n−1yn − xn‖ ≤ ‖tr−3(ptr−3) n−1yn − tr−3(ptr−3) n−1xn‖ + ‖tr−3(ptr−3) n−1xn − xn‖ ≤ kn‖yn − xn‖ + ‖tr−3(ptr−3) n−1xn − xn‖ → 0 as n → ∞ . continuing in this way we have that lim n→∞ ‖tj(ptj) n−1yn+r−j−3 − xn‖ = 0 for j = 1, 2, ....., r − 3. continuing in this way after a finite steps we have that lim n→∞ ‖ti(pti) n−1xn − xn‖ = 0, for i ∈ i, and lim n→∞ ‖tj(ptj) n−1yn+r−j−k − xn‖ = 0 for j = 1, 2, ....., r − k. (2.22) from (1.9), (2.22) we have that ‖xn+1 − xn‖ = ‖p[(1 − r∑ k=1 akn1 − bn1)xn + r−1∑ k=1 akn1t1(pt1) n−1yn+r−1−k + arn1t1(pt1) n−1xn + bn1un1] − xn‖ ≤ r−1∑ k=1 akn1‖t1(pt1) n−1yn+r−1−k − xn‖ + a r n1‖t1(pt1) n−1xn − xn‖ +bn1‖un1 − xn‖ → 0 as n → ∞. (2.23) since every nonself asymptotically nonexpansive mapping uniformly l-lipschitzian, so from (2.22) and (2.23) we get ‖ti(pti) n−2xn − xn‖ ≤ ‖ti(pti) n−2xn − ti(pti) n−2xn−1‖ + ‖ti(pti) n−2xn−1 − xn−1‖ + ‖xn−1 − xn‖ ≤ (1 + l)‖xn − xn−1‖ + ‖ti(pti) n−2xn−1 − xn−1‖ → 0 as n → ∞. (2.24) now from (2.22) and (2.24) it follows that ‖xn − tixn‖ ≤ ‖xn − ti(pti) n−1xn‖ + ‖ti(pti) n−1xn − tixn‖ ≤ ‖xn − ti(pti) n−1xn‖ + l‖ti(pti) n−2xn − xn‖ → 0 as n → ∞ . cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 159 thus we have that limn→∞ ‖xn − tixn‖ = 0 for i ∈ i.♦ lemma 2.4. let e be a uniformly convex banach space and k be a nonempty closed convex subset of e which is also a nonexpansive retract of e which has a frèchet differentiable norm. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with∑ ∞ n=1 bni < ∞ for i ∈ i. if f = ⋂r i=1 f(ti) 6= ∅ and (1) 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr + bnr) < 1 (2) 0 < lim infn→∞ a 1 nj ≤ lim supn→∞( ∑r−j+1 m=1 amnj + bnj) < 1, then for any p1, p2 ∈ f, limn→∞ < xn, j(p1 − p2) > exists. in particular limn→∞ = 0 for all p, q ∈ ww(xn). proof: since e has frèchet differentiable norm, taking x = p1 − p2 with p1 6= p2 and h = t(xn − p1) in the inequality (1.11) we get that t < xn − p1, j(p1 − p2) > + 1 2 ‖p1 − p2‖ 2 ≤ 1 2 ‖txn + (1 − t)p1 − p2‖ 2 ≤ t < xn − p1, j(p1 − p2) > + 1 2 ‖p1 − p2‖ 2 + b(t‖xn − p1‖) . (2.25) again p1 ∈ f, so by lemma 2.1 we have that limn→∞ ‖xn − p1‖ exists. let sup{‖xn − p1‖ : n ∈ n} ≤ m′ for some m′ > 0. thus from (2.25) we get 1 2 ‖p1 − p2‖ 2 + lim sup n→∞ t < xn − p1, j(p1 − p2) >≤ 1 2 lim n→∞ ‖txn + (1 − t)p1 − p2‖ 2 ≤ 1 2 ‖p1 − p2‖ 2 + b(tm′) + lim inf n→∞ t < xn − p1, j(p1 − p2) > ⇒ lim sup n→∞ < xn − p1, j(p1 − p2) >≤ lim inf n→∞ < xn − p1, j(p1 − p2) > + b(tm′) tm′ m′ ⇒ lim n→∞ < xn − p1, j(p1 − p2) > exists as t → 0. in particular limn→∞ = 0 for all p, q ∈ ww(xn). lemma 2.5. let e be a uniformly convex banach space and k be a nonempty closed convex subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with ∑ ∞ n=1 bni < ∞ for i ∈ i. if f = ⋂r i=1 f(ti) 6= ∅ and (1) 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr + bnr) < 1 (2) 0 < lim infn→∞ a 1 nj ≤ lim supn→∞( ∑r−j+1 m=1 amnj + bnj) < 1, then limn→∞ ‖txn + (1 − t)p1 − p2‖ exists for all p1, p2 ∈ f. 160 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) proof: let dn(t) = ‖txn + (1 − t)p1 − p2‖ for all t ∈ [0, 1] and p1, p2 ∈ f. then limn→∞ dn(0) = ‖p1 − p2‖ exists and limn→∞ dn(1) = ‖xn − p2‖ exists by lemma 2.1. define qn : k → e by qnx = p[(1 − a 1 n1 − a 2 n1 − ..... − a r n1 − bn1)x + a 1 n1t1(pt1) n−1xr−2 + a2n1t1(pt1) n−1xr−3 + ..... + a r n1t1(pt1) n−1x + bn1un1] xr−2 = p[(1 − a 1 n2 − a 2 n2 − ..... − a r−1 n2 − bn2)x + a 1 n2t2(pt2) n−1xr−3 + a2n2t2(pt2) n−1xr−4 + ..... + a r−1 n2 t2(pt2) n−1x + bn2un2] . . . . x2 = p[(1 − a 1 n(r−2) − a 2 n(r−2) − a 3 n(r−2) − bn(r−2))x + a 1 n(r−2)tr−2(ptr−2) n−1x1 +a2n(r−2)tr−2(ptr−2) n−1x0 + a 3 n(r−2)tr−2(ptr−2) n−1x + bn(r−2)un(r−2)] x1 = p[(1 − a 1 n(r−1) − a 2 n(r−1) − bn(r−1))x + a 1 n(r−1)tr−1(ptr−1) n−1x0 + a2n(r−1)tr−1(ptr−1) n−1x + bn(r−1)un(r−1)] x0 = p[(1 − a 1 nr − bnr)x + a 1 nrtr(ptr) n−1x + bnrunr] for all x ∈ k. thus for all x, z ∈ k ‖x0 − z0‖ ≤ (1 − a 1 nr − bnr)‖x − z‖ + a 1 nr‖tr(ptr) n−1x − tr(ptr) n−1z‖ ≤ (1 − a1nr − bnr)‖x − z‖ + a 1 nrkn‖x − z‖ ≤ kn‖x − z‖ . proceeding in this way we get ‖qnx − qnz‖ ≤ k r n‖x − z‖ = [1 + (k r n − 1)]‖x − z‖ . set sn,m = qn+m−1qn+m−2........qn, m ≥ 1 and bn,m = ‖sn,m(txn + (1 − t)p1) − (txn+m + (1 − t)p1)‖. then ‖sn,mx − sn,my‖ ≤ ( n+m−1∏ j=n krj )‖x − y‖ = hnmr‖x − y‖ where hnmr = ( ∏n+m−1 j=n krj ) for n ≥ 1, sn,mxn = xn+m and sn,mp = p for all p ∈ f. from the facts ∑ ∞ n=1 (kn−1) < ∞ and 0 ≤ tr−1 ≤ rtr−1(t−1) for all t ≥ 1 we have that ∑ ∞ n=1 (krn−1) < ∞ cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 161 which in turn implies that hnmr → 1 as n, m → ∞. also we have that sn,m is lipschitzian with the lipschitz constant hnmr. by lemma 1.7 we have bn,m ≤ hnmrφ −1(‖xn − p1‖ − h −1 nmr‖sn,mxn − sn,mp1‖) = hnmrφ −1(‖xn − p1‖ − h −1 nmr‖xn+m − p1‖) . now, dn+m(t) = ‖txn+m + (1 − t)p1 − p2‖ ≤ bn,m + ‖sn,m(txn + (1 − t)p1) − p2‖ = bn,m + ‖sn,m(txn + (1 − t)p1) − sn,mp2‖ ≤ bn,m + hnmr‖txn + (1 − t)p1 − p2‖ = bn,m + hnmrdn(t) . it then follows from lemma 2.1 that the sequence {bn,m} converges uniformly to 0 as n → ∞ for all m ≥ 1. thus from above we get lim sup n→∞ dn(t) ≤ φ −1(0) + lim inf n→∞ dn(t) = lim inf n→∞ dn(t) . this completes the proof. theorem 1. let e be a uniformly convex banach space and k be a nonempty closed convex subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with ∑ ∞ n=1 bni < ∞ for i ∈ i. let f = ⋂r i=1 f(ti) 6= ∅ and (1) 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr + bnr) < 1 (2) 0 < lim infn→∞ a 1 nj ≤ lim supn→∞( ∑r−j+1 m=1 amnj + bnj) < 1. assume that any one of the following conditions holds: (1)e satisfies opial’s property (2)e has a frechet differentiable norm (3)e⋆ has the kadec-klee property then {xn} converges weakly to some common fixed point of {ti}, i ∈ i. proof: since f 6= ∅, so let q ∈ f. then by lemma 2.1 limn→∞ ‖xn − q‖ exists and so {xn} is bounded. since e be a uniformly convex banach space so {xn} has a subsequence {xnj} which is weakly convergent to p ∈ k (say). from lemma 2.3 we get limn→∞ ‖xn − tixn‖ = 0 for i ∈ i. by lemma 1.5 we have ti is demiclosed at 0 so p ∈ f(ti) for all i ∈ i. then p ∈ f. if possible let {xn} has another subsequence {xnk} which converges weakly to another point q ∈ k. then by similar 162 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) argument as above we have that q ∈ f(t). let (1) hold. then by opial’s property we have ‖xn − p‖ = lim sup j→∞ ‖xnj − p‖ < lim sup j→∞ ‖xnj − q‖ = lim n→∞ ‖xn − q‖ = lim sup k→∞ ‖xnk − q‖ < lim sup k→∞ ‖xnk − p‖ = lim n→∞ ‖xn − p‖ a contradiction. so p = q. let (2) hold. then from lemma 2.4 we get limn→∞ = 0 for all p, q ∈ ww(xn) and p1, p2 ∈ f. since p, q ∈ f also so from above we get = 0, that is, ‖p − q‖2 = 0 which implies that p = q. let (3) hold. then from lemma 2.5 we get limn→∞ ‖txn + (1 − t)p − q‖ exists, so by lemma 1.6 we have that p = q. so {xn} converges weakly to some common fixed point of {ti}, i ∈ i. this completes the proof of the theorem. ♦ condition(a)[14] a mapping t : k → e with nonempty fixed point set f(t) in k satisfies condition (a) if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(m) > 0 for all m ∈ (0, ∞) such that f(d(x, f(t))) ≤ ‖x − tx‖ for all x ∈ k . a finite family of mappings ti : k → e, for all i = 1, 2, 3, ..., r with nonempty fixed point set f = ⋂r i=1 f(ti) 6= ∅ satisfies (i) condition(a)[4] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(m) > 0 for all m ∈ (0, ∞) such that f(d(x, f)) ≤ 1 r ( r∑ i=1 ‖x − tix‖) for all x ∈ k (ii) condition(b)[4] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(m) > 0 for all m ∈ (0, ∞) such that f(d(x, f)) ≤ max 1≤i≤r {‖x − tix‖} for all x ∈ k (iii) condition(c)[4] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(m) > 0 for all m ∈ (0, ∞) such that at least one of the ti’s satisfies condition(a). clearly if ti = t, for all i = 1, 2, 3, ..., r, then condition(a) reduces to condition(a). also condition(b) reduces to condition(a) if all but one of ti’s are identities. also it contains condition(a). furthermore condition(c) and condition(b) are equivalent. tan and xu [18] pointed out that the condition(a) is weaker than the compactness of k. it is well known that every continuous and demicompact mapping must satisfy condition(a) [14]. since every completely continuous mapping is continuous and demicompact so it must satisfy condition(a). also condition(b) contains cubo 14, 3 (2012) weak and strong convergence theorems of a multistep ... 163 condition(a) therefore to study the strong convergence of the iterative sequence {xn} be defined by (1.9) we use condition(b) instead of the complete continuity of the mappings {t1, t2, ...., tr} and condition(a). theorem 2. let e be a uniformly convex banach space and k be a nonempty closed convex subset of e which is also a nonexpansive retract of e. let ti : k → e(i ∈ i = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with ∑ ∞ n=1 (kin − 1) < ∞ for i ∈ i. let {xn} be defined by (1.9) with ∑ ∞ n=1 bni < ∞ for i ∈ i. let f = ⋂r i=1 f(ti) 6= ∅ and (1) 0 < lim infn→∞ a 1 nr ≤ lim supn→∞(a 1 nr + bnr) < 1 (2) 0 < lim infn→∞ a 1 nj ≤ lim supn→∞( ∑r−j+1 m=1 amnj + bnj) < 1. if the family of mappings {t1, t2, ...., tr} satisfy condition(b), then {xn} converges strongly to some common fixed point of {t1, t2, ...., tr}. proof: let q ∈ f then by lemma 2.1 limn→∞ ‖xn − q‖ exists. let limn→∞ ‖xn − q‖ = a, for some a ≥ 0. let a > 0. now from (2.5) we get ‖xn+1 − q‖ ≤ [1 + (k r n − 1)]‖xn − q‖ + σ r n = [1 + δn]‖xn − q‖ + σ r n (2.26) where {σrn} is a nonnegative real sequence such that ∑ ∞ n=1 σrn < ∞ and δn = k r n − 1 such that∑ ∞ n=1 δn < ∞. so d(xn+1, f) ≤ (1 + δn)d(xn, f) + σ r n . by lemma 1.1 we have that limn→∞ d(xn, f) exists. by condition (b) and lemma 2.3 we get, lim n→∞ f(d(xn, f)) = 0. since f : [0, ∞) → [0, ∞) is a nondecreasing function with f(0) = 0 so we have limn→∞ d(xn, f) = 0. since limn→∞ ‖xn −q‖ exists, it follows that {‖xn −q‖} is bounded, so there exists m′′ > 0 such that ‖xn − q‖ ≤ m ′′. from (2.26) we get ‖xn+1 − q‖ ≤ ‖xn − q‖ + δnm ′′ + σrn = ‖xn − q‖ + θn where θn = δnm ′′ + σrn. now ∑ ∞ n=1 θn < ∞. now for any m > 1 we have that ‖xn+m − q‖ ≤ ‖xn+m−1 − q‖ + θn+m−1 ≤ ‖xn+m−2 − q‖ + θn+m−2 + θn+m−1 · · · · · · · · · · · · · · · · · · · · · · · · · · · ≤ ‖xn − q‖ + n+m−1∑ k=n θk . 164 shrabani banerjee and binayak s.choudhury cubo 14, 3 (2012) since ∑ ∞ n=1 θn < ∞ and limn→∞ d(xn, f) = 0, there exists n1 ∈ n such that for all n ≥ n1 we have d(xn, f) < ǫ 3 and ∑ ∞ n=n1 θn < ǫ 6 . therefore there exists x ∈ f such that ||xn1 − x|| < ǫ 3 . therefore we have ||xn+m − xn|| ≤ ||xn+m − x|| + ||xn − x|| < ||xn1 − x|| + n+m−1∑ k=n1 θk + ||xn1 − x|| + n−1∑ k=n1 θk < ǫ 3 + ǫ 6 + ǫ 3 + ǫ 6 = ǫ (2.27) , .nonumber (2.28) hence {xn} is a cauchy sequence. since e is complete so xn → p ∈ e, so for given ǫ > 0 there exists n1 ∈ n such that for all n ≥ n1, ‖xn − p‖ ≤ ǫ 2(1+k1) . again since limn→∞ d(xn, f) = 0, so for given ǫ > 0 there exists n2 ∈ n such that for all n ≥ n2(≥ n1), d(xn, f) < ǫ 2(1+k1) . so there exists p ∈ f such that ‖xn2 − p‖ ≤ ǫ 2(1+k1) . therefore ‖p − tip‖ = ‖p − xn2 + xn2 − p + p − tip‖ ≤ ‖p − xn2‖ + ‖xn2 − p‖ + ‖p − tip‖ = ‖p − xn2‖ + ‖xn2 − p‖ + ‖tip − tip‖ ≤ ‖p − xn2‖ + ‖xn2 − p‖ + k1‖p − p‖ ≤ (1 + k1)‖p − xn2‖ + (1 + k1)‖xn2 − p‖ ≤ (1 + k1) ǫ 2(1 + k1) + (1 + k1) ǫ 2(1 + k1) = ǫ . since ǫ is arbitrary so we have tip = p for all i ∈ i. so p ∈ f(ti) for all i ∈ i. thus p ∈ f. hence {xn} converges strongly to some common fixed point of {t1, t2, ...., tr}. remark 2.6. theorem 1 and theorem 2 extends and generalize theorem 2.1 and 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[21] h.k.xu, inequalities in banach spaces with applications, nonlinear anal. 16(1991) 1127-1138. paperbc.dvi cubo a mathematical journal vol.12, no¯ 01, (1–6). march 2010 characterization of the banzhaf value using a consistency axiom joss erick sánchez pérez facultad de economı́a, uaslp, av. pintores s/n, col. b. del estado 78213, san luis potośı, méxico email : joss.sanchez@uaslp.mx abstract one of the important properties characterizing cooperative game solutions is consistency. this notion establishes connections between the solution vectors of a cooperative game and those of its reduced game. the last one is obtained from the initial game by removing one or more players and by giving them payoffs according to a specific principle. consistency of a solution means that the restriction of a solution payoff vector of the initial game to any coalition belongs to the solution set of the corresponding reduced game. in this paper, we show that the banzhaf value is consistent with respect to a suitable reduced game. we also show that consistency together with standardness for two-person games characterize the banzhaf value. resumen una de las propiedades importantes caracterizando soluciones de juegos cooperativos es consistencia. esta noción establece conexiones entre los vectores solución de un juego cooperativo y aquellos de su juego reducido. este último es obtenido a partir del juego inicial al remover uno o más jugadores y asignándoles pagos de acuerdo a un principio espećıfico. consistencia de una solución significa que la restricción de una solución del juego inicial a cualquier coalición, pertenece al conjunto de soluciones del correspondiente juego reducido. en este art́ıculo, mostramos que consistencia junto con estandarización para juegos de dos personas caracterizan el valor de banzhaf. 2 joss erick sánchez pérez cubo 12, 1 (2010) key words and phrases: games in characteristic function form, banzhaf value, consistency. math. subj. class.: 91a12. 1 introduction the mathematical approach to a proposed solution is to examine a number of its (elementary) properties and, if possible, to provide a minimal number of properties which fully characterize the solution. in their well known paper, hart and mas-colell (1989) characterized the shapley value for the class of tu cooperative games by means of the following axioms1: i) consistency. ii) standardness for two-person games. a cooperative game is always described by a finite player set as well as a real-valued ”characteristic function” on the collection of subsets of the player set. a so-called reduced game is deducible from a given cooperative game by removing one or more players on the understanding that the removed players will be paid according to a specific principle (e.g., a proposed payoff vector). the remaining players form the player set of the reduced game: the characteristic function of which is composed of the original characteristic function, the proposed payoff vector, and / or the solution in question. the consistency property for the solution states that if all the players are supposed to be paid according to a payoff vector in the solution set of the original game, then the players of the reduced game can achieve the correspondig payoff vector in the solution set of the reduced game. in other words, there is no inconsistency in what the players of the reduced game can achieve, in either the original game or the reduced game. this paper presents, using a standard notion of consistency, that the banzhaf value is the unique consistent extension which ”divides the surplus equally” in two-person games. 2 preliminaries by an n-person game in characteristic function form (or a tu cooperative game), in what follows just a game, we mean a pair (n,v), where n = {1, ...,n} is a finite set of players and v is a function v : 2n → r with the property that v(φ) = 0 (2n denotes the set of subsets of n). we usually refer to subsets s of n as coalitions and to the number v(s) as the worth of s. a game (n,v) is superadditive if v(s∪t ) ≥ v(s)+v(t ) for all s,t ⊂ n, and it is subadditive if the inequality holds in the other direction. there are several interpretations for (n,v), it depends on what people want to model. for instance, if the game is superadditive, v(s) means the maximal amount the players 1the precise definitions will be provided in section 3. cubo 12, 1 (2010) characterization of the banzhaf value ... 3 in s can get if they decide to play together. while the game is subadditive, v(s) usually means the joint cost that players in s have to pay to get a service if they hired the service together. additionally, we will denote the cardinality of a set by its corresponding lower-case letter, for instance n = |n|, s = |s|, t = |t |, and so on. we denote by gn the set of all games with a fixed set of players n, i.e., gn = {v : 2n → r | v(φ) = 0}. a solution ϕ : gn → rn in gn is a rule to divide the common gain or cost among the players in n. let γn be the set of solutions in gn. given v,w ∈ gn and λ ∈ r, we define the sum and the product v + w and λv in gn in the usual form, i.e., (v + w)(s) = v(s) + w(s) and (λv)(s) = λv(s) respectively. in a similar manner, for ϕ,ψ ∈ γn and λ ∈ r, we define the sum and the product ϕ + ψ and λϕ in gn by (ϕ + ψ)(v + w) = ϕ(v) + ψ(w) and (λϕ)(v) = λϕ(v) it is easy to verify that gn and γn are vector spaces with these operations. next, we define some axioms which are common in game theory. axiom 1 (linearity). the solution ϕ is linear if ϕ(v + w) = ϕ(v) + ϕ(w) and ϕ(cv) = cϕ(v), for all v,w ∈ gn and c ∈ r. let us consider the group sn = {θ : n → n | θ is bijective}, the group of permutations of the set of players. for every θ ∈ sn and v ∈ g n we define another game (n,θ · v) as follows θ · v(s) = v(θ−1s) axiom 2 (symmetry). the solution ϕ is said to be symmetric if and only if ϕ(θ ·v) = θ ·ϕ(v) for every θ ∈ sn and v ∈ g n . the player i is said to be a dummy player in the game (n,v) if v(s ∪ {i}) = v(s) for every s ⊆ n. the dummy axiom requests that every dummy player in (n,v) gets a zero payoff. axiom 3 (dummy). if the player i is a dummy player in (n,v) then ϕi(v) = 0. for every coalition t ⊆ n we obtain a new game vt where the set of players t is considered as one player; we denote this player by t́ . the space of players for vt is n\t ∪ {t́} and is defined by: vt (s) = v(s) and vt (s ∪ {t́}) = v(s ∪ t ) where s ⊆ n\t . the reduction axiom says that unification of any two players has to be profitable (or at least never harmful). 4 joss erick sánchez pérez cubo 12, 1 (2010) axiom 4 (reduction). the solution ϕ satisfies the reduction axiom if and only if ϕi(v) + ϕj(v) ≤ ϕt́ (vt ) for every coalition t = {i,j} of two players. theorem 5 (lehrer, 1988). there exists a unique solution ϕ that satisfies linearity, symmetry, dummy and reduction axioms. furthermore it is given by ϕi(v) = 1 2n−1 ∑ s⊆n\{i} [v(s ∪ {i}) − v(s)] 3 consistency and the main result this section is devoted to a characterization of the banzhaf value by means of an internal consistency property. intuitively speaking, let ϕ be a function that associates a payoff to every player in every game and suppose that one player leaves a game with some payoff. reduced games describe what games are played between the remaining players, i.e. what is earned by coalitions of the remaining players after one player has left the game. then ϕ is said to be consistent if, when it is applied to any reduced game, it yields the same payoffs as in the original game. definition 6. a value ϕ : gn → rn is consistent with respect to the reduced game (n\{i},v ϕ n\{i} ) if, for n\{i} ⊂ n and all v ∈ gn, one has ϕj(n\{i},v ϕ n\{i} ) = ϕj(n,v) for all j ∈ n\{i} the various consistency requirements differ in the precise definition of the reduced game (i.e., exactly how the players outside are being paid off). it is well known that different solutions are consistent with respect to different reduced games. next, we will see that the banzhaf value is consistent according to the following reduced game concept: definition 7. given a solution ϕ : gn → rn in gn, a game (n,v) (with n ≥ 3). the reduced game (n\{k},v ϕ n\{k} ) is defined by v ϕ n\{k} (s) =        1 2 [v(n) + v(n\{k}) − v({k})] if s = n\{k} 1 2 [v(s) − v(n\s)] if φ 6= s ⊂ n\{k} 0 if s = φ (1) the first result is as follows: proposition 8. the banzhaf value is a consistent solution function. proof. we must apply this value to the reduced game (n\{k},v ϕ n\{k} ). we note that player j in (n\{k},v ϕ n\{k} ) gets: ϕj(n\{k},v ϕ n\{k} ) = 1 2n−2 ∑ s⊆n\{j,k} [ v ϕ n\{k} (s ∪ {j}) − v ϕ n\{k} (s) ] cubo 12, 1 (2010) characterization of the banzhaf value ... 5 then, using the reduced game concept (1) in the last expression, we obtain: 2n−2 · ϕj(n\{k},v ϕ n\{k} ) = ∑ s⊆n\{j,k} [ v ϕ n\{k} (s ∪ {j}) − v ϕ n\{k} (s) ] = ∑ φ 6=s⊂n\{j,k} [v(s ∪ {j}) − v(n\(s ∪ {j})) − v(s) + v(n\s)] +v({j}) − v(n\{j}) + v(n) + v(n\{k}) − v(n\{j,k}) + v({j,k}) = 1 2 ∑ s⊆n\{j,k} [v(s ∪ {j}) − v(s) + v(n\s) − v(n\(s ∪ {j}))] we observe that in the sum, the worth v(n\s) corresponds to the worths of coalitions containing {j,k}; while the worth v(n\s ∪ {j}) corresponds to the worths of coalitions containing {k} but not {j}. hence, we can apply the change: ϕj(n\{k},v ϕ n\{k} ) = 1 2n−1    ∑ s⊆n\{j,k} [v(s ∪ {j}) − v(s)] + ∑ s⊆n\{j},s∋{k} [v(s ∪ {j}) − v(s)]    = 1 2n−1 ∑ s⊆n\{j} [v(s ∪ {j}) − v(s)] = ϕj(n,v) next, we will show that the property of consistency almost characterizes the banzhaf value. ”almost” refers to the ”initial conditions”, namely, the behavior of the solution for two-person games. the next theorem establishes the main result of this work, we show that consistency together with standardness for two-person games characterize the banzhaf value. this characterization is equivalent to that of the shapley value given by hart and mas-colell (1989) where the only difference rests upon the reduced game concept underlying the notion of consistency. first, we recall that a solution ϕ is standard for two-person games if ϕi({i,j},v) = v({i}) + 1 2 [v({i,j}) − v({i}) − v({j})] for all i 6= j and all v. thus, the surplus [v({i,j}) − v({i}) − v({j})] is equally divided among the two players. theorem 9. let ϕ : gn → rn in gn be a solution function. then: (i) ϕ is consistent with respect to the reduced game(1); and (ii) ϕ is standard for two-person games; if and only if ϕ is the banzhaf value. 6 joss erick sánchez pérez cubo 12, 1 (2010) proof. one direction is immediate, recall proposition 8 and the fact that: ϕi({i,j},v) = 1 2 ∑ s⊆{j} [v(s ∪ {i}) − v(s)] = v({i}) + 1 2 [v({i,j}) − v({i}) − v({j})] for the other direction, the proof is by induction on the number of players. assume ϕ,ψ : gn → rn satisfy (i) and (ii). we claim first that ϕ = ψ for two-person games, since for k ∈ {i,j}: ϕk({i,j},v) = v({k}) + 1 2 [v({i,j}) − v({i}) − v({j})] = 1 2 ∑ s∋k s⊆{i,j} [v(s) − v({i,j}\s)] = ψk({i,j},v) now, suppose that ϕ and ψ coincide for all the games with n− 1 players. the two-person reduced games: ({i,j},v ϕ {i,j} ) and ({i,j},v ψ {i,j} ) coincide since, v ϕ {i,j} (s) = v ψ {i,j} (s) for any s ⊂ {i,j}. then ϕi(v ϕ {i,j} ) = ψi(v ψ {i,j} ) by standarness for two-person games, and ϕi(n,v) = ϕi({i,j},v ϕ {i,j} ) = ψi({i,j},v ψ {i,j} ) = ψi(n,v) by consistency. in a similar manner, ϕj(n,v) = ψj(n,v). applying this to any pair of players, one gets ϕi(n,v) = ψi(n,v) ∀i ∈ n. received: april, 2008. revised: september, 2009. references [1] hart, s. and mas-colell, a., potencial, value, and consistency, econometrica, 57 (1989), 589–614. [2] lehrer, e., an axiomatization of the banzhaf value, international journal of game theory, 17 (1988), 89–99. [3] owen, g., consistency in values, sixth spanish meeting on game theory and practice, (2003), 15–30. () cubo a mathematical journal vol.13, no¯ 02, (85–117). june 2011 differential forms versus multi-vector functions in hermitean clifford analysis f. brackx, h. de schepper and v. souček ghent university faculty of engineering department of mathematical analysis gent, belgium email: fb@cage.ugent.be and f. brackx, h. de schepper and v. souček charles university faculty of mathematics and physics praha, czech republic abstract similarities are shown between the algebras of complex differential forms and of complex clifford algebra-valued multi-vector functions in an open region of euclidean space of even dimension. resumen se presentan las similitudes entre las álgebras de formas diferenciales complejas y de las funciones de álgebras de clifford complejas con valores de múltiples vectores aplicados en una región abierta del espacio euclidiano de dimensión par. 86 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) keywords and phrases: complex differential forms, multi-vector functions, hermitean clifford analysis. mathematics subject classification: 30g35 1 introduction usually clifford analysis is understood to be the study of the solutions of the dirac equation for functions defined on the (anti-)euclidean vector space r0,m and taking values in the corresponding clifford algebra r0,m. it thus offers a proper analogue to the cauchy-riemann equations for holomorphic functions in the complex plane. for a thorough study of the so-called monogenic functions of clifford analysis we refer to the standard textbooks [5, 15, 17, 18]. the symmetry group of the dirac equation is either so(m) or spin(m), according to the definition of the group action on the values taken by the functions under consideration. if these values are in the clifford algebra with left multiplication, the symmetry group is spin(m), which then usually is realized inside the clifford algebra. in the case of functions with values in the clifford algebra with both side action, it is more natural to identify the dirac equation with the equation (d + d∗)f = 0, and to identify the space of values, in casu the clifford algebra, as a vector space, with the grassmann algebra of rm. this grassmann algebra may then be decomposed into the direct sum of its homogeneous parts, which is a decomposition into irreducible parts under the action of so(m). in this framework it was shown (see [13]) that, on the polynomial level, the space of monogenic functions can be split into a direct sum of solutions of the hodge-de rham equations for homogeneous differential forms. this entails a finer structure of the space of monogenic functions, which manifests itself explicitly in a finer form of the corresponding fischer decomposition (see [14]). an important ingredient in the latter approach is the translation of spaces and operators from the language of multivector functions with values in a clifford algebra to the language of real differential forms, as was described in detail in [6]. let us give a very brief overview. on the one hand we have the cartan algebra ∧ (g) of smooth real differential forms in an open subset g of euclidean space rm, endowed with exterior multiplication. a fundamental operator on ∧ (g) is the exterior derivative d with its important property that for any differential form ω, d2ω = d(dω) = 0. introducing the hodge co-derivative d∗ leads to the differential operator d = d + d∗, by means of which the so-called ”harmonic” r-forms (0 < r < m) are characterized as smooth differential rforms ωr satisfying dωr = 0. on the other hand we have the algebra e(g) of smooth multi-vector functions in g. multi-vector functions arise in a natural way when considering functions defined in g and taking values in the universal real clifford algebra r0,m constructed over r 0,m, i.e. rm equipped with an anti-euclidean metric. if rr0,m (0 ≤ r ≤ m) denotes the space of r-vectors, then the clifford algebra r0,m is precisely the associative algebra r0,m = ⊕m r=0 r r 0,m, and an r-vector function fr is a map fr : g → rr0,m. a fundamental operator on the space of smooth multi-vector cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 87 functions, is the rotation-invariant dirac operator ∂x, by means of which the so-called monogenic functions are characterized as the smooth functions f satisfying ∂xf = 0, as already mentioned above. the spaces of smooth differential forms and of smooth multi-vector functions were shown to be isomorphic in a natural way: a smooth r-form is identified with a smooth r-vector function, and the action of the differential operator d + d∗ on the space ∧r (g) of smooth r-forms, is identified with the action of the dirac operator ∂x on the space er(g) of smooth r-vector functions. also other correspondences were studied in detail in [6]. when the dimension is taken to be even (m = 2n), one can make the framework of clifford analysis closer to complex analysis by introducing on r2n a complex structure j. the symmetry group then reduces to the subgroup u(n) ⊂ so(2n) preserving the chosen complex structure j. this is the basic setting for so-called hermitean clifford analysis, which recently has emerged as a new and successful branch of clifford analysis, offering yet a refinement of the euclidean case. the functions studied are defined in open regions of cn and take their values in the complex clifford algebra c2n. more particularly hermitean clifford analysis focusses on the simultaneous null solutions, called hermitean (or h–) monogenic functions, of two hermitean dirac operators ∂z and ∂z† . a systematic development of this function theory, including the invariance properties with respect to the underlying lie groups and lie algebras, is still in full progress, see e.g. [9, 1, 2, 7, 8, 3, 4, 23, 12]. part of this program also concerns the study of the finer structure induced on the space of monogenic functions by the choice of the complex structure j. when studying the dirac equation for functions with values in a clifford algebra, it is well known that the clifford algebra can be split into the direct sum of a number of isomorphic copies of the basic spinor representation. accordingly, the set of equations will split into a number of independent subsets of equations for functions with values in the various copies of spinor space. it is a trivial observation that all these subsystems are equivalent to each other and their solutions will have the same properties, whence, without any loss of generality, we can restrict the study to functions with values in the space of spinors (or half-spinors in even dimension). in the standard situation, this space of values cannot be split further since they are already irreducible under the (left) action of the spin(m) group. however, after having fixed the complex structure j, the symmetry group is reduced, as explained above, and the spinor space decomposes further into smaller pieces. if it is realized in a standard way as the grassmann algebra over the maximal isotropic subspace in c2n, then this splitting is just the splitting into homogeneous components; for details see e.g. [2]. our final aim is to understand the finer structure of the space of monogenic functions induced by this splitting. a first step towards that goal is to establish a scheme for the translation of spaces and operators between the language of complex clifford algebra and the language of complex differential forms. in fact this is the complex analogue of the translation in the euclidean situation 88 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) mentioned above, see [6]. the purpose of the underlying paper is precisely to describe in a rather formal, yet detailed, way the similarities between complex differential forms in open regions of cn on the one hand and multivector functions in the hermitean clifford analysis setting on the other. crucial to this description is the detailed analysis of the structure of complex clifford algebra as carried out in [10]. the hermitean dirac operators ∂z and ∂z† and the associated operators ∂z•, ∂z∧, ∂z†• and ∂z† ∧, originating by splitting the clifford or geometric product into its “inner” or “dot” and “outer” or “wedge” parts, are identified with well-known differential operators for complex differential forms on kählerian manifolds in cn. however it should be emphasized that, in this paper, we restrict ourselves to the flat kählerian metric on cn with fundamental form ω = i 2 ∂∂|z|2. the more general approach of hermitean clifford analysis on complex hermitean manifolds and its comparison with complex analysis on kählerian manifolds is the subject of the forthcoming paper [11]. the paper is structured as follows. sections 2 and 3 are introductory, fixing our definitions and notations. an identification of all differential operators and forms under consideration in both pictures is described in section 4. the relation to the operators which are standard in kählerian geometry is clarified in section 5. the last section adds some remarks on the hodge operator. 2 multi-vector functions: preliminaries in this section we recall some basic notions and results from clifford algebra and clifford analysis. the construction of the universal real clifford algebra is well-known; for an in-depth study we refer the reader to e.g. [22]. here we restrict ourselves to a schematic approach. let r0,m be the real vector space rm (m ≥ 1) endowed with a non-degenerate symmetric bilinear form b of signature (0, m), and let (e1, ..., em) be an associated orthonormal basis, i.e. b(ei, ej) = { −1 if i = j 0 if i 6= j (1 ≤ i, j ≤ m) then the anti-euclidean metric on r0,m is induced by the scalar product 〈ei, ej〉 = −b(ei, ej) = δij, 1 ≤ i, j ≤ m we first introduce the anti-symmetric outer product by the rules ei ∧ ei = 0, 1 ≤ i ≤ m ei ∧ ej + ej ∧ ei = 0, 1 ≤ i 6= j ≤ m and for each a = {i1, i2, ..., ir} ⊂ m = {1, ..., m}, with 1 ≤ i1 < i2 < ... < ir ≤ m, i.e. ordered in the natural way, we put ea = ei1 ∧ ei2 ∧ ... ∧ eir cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 89 while e∅ = 1. then for each r = 0, 1, ..., m, the set {ea : a ⊂ m and |a| = r} is a basis for the space rr0,m of so-called r-vectors. next, we introduce the inner product ei • ej = −〈ei, ej〉 = b(ei, ej) = −δij, 1 ≤ i, j ≤ m leading to the so-called geometric product of vectors in the clifford algebra: eiej = ei • ej + ei ∧ ej, 1 ≤ i, j ≤ m the respective definitions of the inner, the outer and the geometric product are then extended to r-vectors as follows: for the inner product, we have ej • ea = ej • (ei1 ∧ ... ∧ eir ) = r∑ k=1 (−1)kδjik ea\{ik} with ea\{ik} = ei1 ∧ ... ∧ eik−1 ∧ [eik ∧] eik+1 ∧ ... ∧ eir while for the outer product    ej ∧ ea = ej ∧ (ei1 ∧ ... ∧ eir ) = ej ∧ ei1 ∧ ... ∧ eir , if j /∈ a ej ∧ ea = 0, if j ∈ a and finally, for the geometric product (or product for short) ejea = ej • ea + ej ∧ ea finally, these definitions are linearly extended to the whole of the clifford algebra r0,m, which is the associative algebra r0,m = m ⊕ r=0 r r 0,m if [ · ]r : r0,m → rr0,m denotes the projection operator from r0,m onto r r 0,m, then each clifford number a ∈ r0,m may be written as a = m∑ r=0 [a]r note that in particular for a 1-vector u and an r-vector vr, one has u vr = u • vr + u ∧ vr with u • vr = [u vr]r−1 = 1 2 ( u vr − (−1) rvr u ) u ∧ vr = [u vr]r+1 = 1 2 ( u vr + (−1) rvr u ) 90 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) usually r and rm are identified with r00,m and r 1 0,m respectively. an element x = (x1, . . . , xm) ∈ r m is thus identified with the 1-vector x = ∑m j=1 xjej. now let g be an open region in rm. a smooth r-vector function fr is a map fr : g → r r 0,m, x 7→ ∑ |a|=r fr,a(x) ea where for each a, fr,a is a smooth real valued function in g. we denote by er(g) the space of smooth r-vector functions in g, and we put e(g) = m ⊕ r=0 er(g) the projection operator from e(g) onto er(g) is denoted by [ . ]r. a fundamental operator in clifford analysis is the so-called dirac operator, a first order vector valued differential operator given by ∂x = m∑ j=1 ej ∂xj since the multiplication in the clifford algebra is non-commutative, operators can act from the left or from the right on a function. for the dirac operator and a function f = ∑ a eafa ∈ e(g), these actions are given by ∂xf = m∑ j=1 ∑ a ejea ∂xj fa and f∂x = m∑ j=1 ∑ a eaej ∂xj fa a function f ∈ e(g) is called left (resp. right) monogenic in g if and only if it satisfies in g the equation ∂xf = 0 (resp. f∂x = 0). restricting the dirac operator ∂x to the space er(g), we find for an r-vector function fr that ∂xfr and fr∂x split into an (r − 1)-vector part and an (r + 1)-vector part: ∂xfr = m∑ j=1 ej ∂xj fr = m∑ j=1 ej • ∂xj fr + m∑ j=1 ej ∧ ∂xj fr fr∂x = m∑ j=1 ∂xj fr ej = m∑ j=1 ∂xj fr • ej + m∑ j=1 ∂xj fr ∧ ej cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 91 it readily follows that [ ∂xfr ] r−1 = m∑ j=1 ej • ∂xj fr = (−1) r+1 m∑ j=1 ∂xj fr • ej = (−1) r+1 [ fr∂x ] r−1 [ ∂xfr ] r+1 = m∑ j=1 ej ∧ ∂xj fr = (−1) r m∑ j=1 ∂xj fr ∧ ej = (−1) r [ fr∂x ] r+1 usually one introduces the notations ∂x • fr = [∂xfr]r−1, ∂x ∧ fr = [∂xfr]r+1 fr • ∂x = [fr∂x]r−1, fr ∧ ∂x = [fr∂x]r+1 the action of the dirac operator ∂x on er(g) thus gives rise to two auxiliary differential operators ∂x• : er(g) → er−1(g); fr 7→ (∂x• )fr = ∂x • fr = [∂xfr]r−1 ∂x∧ : er(g) → er+1(g); fr 7→ (∂x∧)fr = ∂x ∧ fr = [∂xfr]r+1 for which it holds that ∂x = ∂x • + ∂x∧ symbolically these operators may be written as (∂x•) = m∑ j=1 (ej • )∂xj (∂x∧) = m∑ j=1 (ej ∧)∂xj their action on er(g) is two-fold in the sense that they act on the multi-vector by means of the inner and outer product with basis vectors, and at the same time on the function coefficients by partial differentiation. we thus have that, for a smooth r-vector function fr, the notions of left monogenicity and right monogenicity coincide, and moreover that fr is left as well as right monogenic in g if and only if in g ∂xfr = 0 ⇐⇒ fr∂x = 0 ⇐⇒ { ∂x • fr = 0 ∂x ∧ fr = 0 as the dirac operator ∂x factorizes the laplace operator, viz ∂2x = ∂x • ∂x + ∂x ∧ ∂x = ∂x • ∂x = −〈∂x, ∂x〉 = −∆m a monogenic function in g is also harmonic in g, but the converse clearly is not true. as moreover (∂x•) 2 = (∂x∧) 2 = 0 92 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) we have that −∆m = (∂x • +∂x∧) 2 = ∂x • ∂x ∧ +∂x ∧ ∂x• the two second order differential operators ( ∂x • ∂x∧ ) and ( ∂x ∧ ∂x• ) arising above being scalar operators in the sense that they keep the order of the multi-vector function invariant. however the function coefficients, while being differentiated, are interchanged w.r.t. the basis multi-vectors. when allowing for complex constants and moreover taking the dimension to be even: m = 2n, the same generators (e1, . . . , en, en+1, . . . , e2n) produce the complex clifford algebra c2n, which is the complexification of the real clifford algebra r0,2n, i.e. c2n = r0,2n ⊕ i r0,2n. any complex clifford number λ ∈ c2n may thus be written as λ = a + ib, a, b ∈ r0,2n, an observation leading to the definition of the hermitean conjugation λ† = (a + ib)† = a − ib, where the bar notation stands for the usual clifford conjugation in r0,2n, i.e. the main anti–involution for which ej = −ej, j = 1, . . . , 2n. this hermitean conjugation also leads to a hermitean inner product and its associated norm on c2n given by (λ, µ) = [λ †µ]0 and |λ| = √ [λ†λ]0 = ( ∑ a |λa| 2)1/2. this is the framework for so–called hermitean clifford analysis, a refinement of euclidean clifford analysis. an elegant way of introducing this setting consists in considering a so–called complex structure, i.e. a specific so(2n; r)–element j for which it holds that j2 = −1 (see [1, 2]). here, j is chosen to act upon the generators e1, . . . , e2n of the clifford algebra as j[ej] = −en+j and j[en+j] = ej, j = 1, . . . , n with j one may associate two projection operators 1 2 (1 ± ij) which produce the main objects of the hermitean setting by acting upon the corresponding objects in the euclidean framework. first of all, the so–called witt basis elements (fj, f † j )nj=1 for c2n are obtained through the action of ± 1 2 (1 ± ij) on the original orthogonal basis: fj = 1 2 (1 + ij)[ej] = 1 2 (ej − i en+j), j = 1, . . . , n f † j = − 1 2 (1 − ij)[ej] = − 1 2 (ej + i en+j), j = 1, . . . , n the witt basis elements satisfy the grassmann identities fjfk + fkfj = f † j f † k + f † k f † j = 0 , j, k = 1, . . . , n including their isotropy: f2j = f † j 2 = 0, j = 1, . . . , n, as well as the duality identities fjf † k + f † k fj = δjk , j, k = 1, . . . , n the witt basis of the complex clifford algebra c2n is then obtained, in much the same way as is done for the basis of the real clifford algebra, by taking all possible products of witt basis vectors. cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 93 introducing the inner and outer products for the witt basis vectors we have, see also [9], fj • fk = f † j • f † k = 0, j, k = 1, . . . , n fj • f † k = f † j • fk = 1 2 δjk, j, k = 1, . . . , n and fj ∧ fk = −fk ∧ fj, j, k = 1, . . . , n f † j ∧ f † k = −f † k ∧ f † j , j, k = 1, . . . , n eventually yielding fjfk = fj • fk + fj ∧ fk = fj ∧ fk, j, k = 1, . . . , n f † j f † k = f † j • f † k + f † j ∧ f † k = f † j ∧ f † k , j, k = 1, . . . , n fjf † k = fj • f † k + fj ∧ f † k = 1 2 δjk + fj ∧ f † k , j, k = 1, . . . , n this leads to the grassmann structure of the complex clifford algebra c2n ∼= n ⊕ p=0 n ⊕ q=0 ∧p,q 2n where ∧p,q 2n = span c { f † j1 ∧ . . . ∧ f † jp ∧ fk1 ∧ . . . ∧ fkq |j1 < j2 < . . . < jp, k1 < k2 < . . . < kq } a vector (x1, . . . , x2n) in r 0,2n is now denoted by (x1, . . . , xn, y1, . . . , yn) and is identified with the clifford vector x = ∑n j=1 (ej xj + en+j yj); the action of the complex structure j on x yields the twisted vector x| = j[x] = n∑ j=1 (ej yj − en+j xj) note that x and x| anti-commute, since they are orthogonal w.r.t. the standard euclidean scalar product; more precisely they satisfy the following properties. lemma 2.1. one has (i) x • x| = 0 (ii) x ∧ x| = ∑ j 6=k xjyk(ejek − en+ken+j) − ∑ j,k ejen+k(xjxk + yjyk) (iii) x| ∧ x = ∑ j 6=k xjyk(ekej − en+jen+k) − ∑ j,k en+kej(xjxk + yjyk) (iv) x x| + x| x = x ∧ x| + x| ∧ x = 0 94 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) the actions of the projection operators on the clifford vector x then produce the mutually hermitean conjugate hermitean clifford variables z and z†, i.e. z = 1 2 (1 + ij)[x] = 1 2 (x + i x|) z† = − 1 2 (1 − ij)[x] = − 1 2 (x − i x|) which may also be rewritten in terms of the witt basis elements as z = n∑ j=1 fj zj and z † = (z)† = n∑ j=1 f † j zcj where n complex variables zj = xj + iyj have been introduced, with complex conjugates z c j = xj − iyj, j = 1, . . . , n. finally, the hermitean dirac operators ∂z and ∂z† are obtained from the euclidean dirac operator ∂x: ∂z† = 1 4 (1 + ij)[∂x] = 1 4 (∂x + i ∂x|) ∂z = − 1 4 (1 − ij)[∂x] = − 1 4 (∂x − i ∂x|) where also the so–called twisted dirac operator arises: ∂x| = j[∂x] = n∑ j=1 ( ej ∂yj − en+j ∂xj ) as for ∂x, a notion of (twisted) monogenicity may be associated in a natural way to ∂x| as well. passing to the witt basis, the hermitean dirac operators are expressed as ∂z = n∑ j=1 f † j ∂zj and ∂z† = (∂z) † = n∑ j=1 fj ∂zc j involving the classical cauchy–riemann operators ∂zc j = 1 2 (∂xj +i∂yj ) and their complex conjugates ∂zj = 1 2 (∂xj − i∂yj ) in the complex zj-planes, j = 1, . . . , n. as a consequence of the isotropy of the witt basis vectors, the hermitean vector variables and dirac operators are isotropic, i.e. (z)2 = (z†)2 = 0 and (∂z) 2 = (∂z† ) 2 = 0 whence the laplacian ∆2n = −∂ 2 x = −∂ 2 x| allows for the decomposition ∆2n = 4(∂z∂z† + ∂z† ∂z) = 4(∂z + ∂z† ) 2 while also (z + z†)2 = z z† + z†z = |z|2 = |z†|2 = |x|2 = |x||2 the central notion in hermitean clifford analysis is that of hermitean monogenicity. a continuously differentiable function g on an open region g of r2n ∼= cn with values in the complex cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 95 clifford algebra c2n is called (left) hermitean monogenic (or h–monogenic for short) in g if and only if it simultaneously is ∂x– and ∂x|–monogenic in g, i.e. it satisfies in g the system ∂x g = 0 = ∂x| g which is equivalent with the system ∂z g = 0 = ∂z† g now the multivector functions in the hermitean clifford analysis setting are smooth functions defined in an open region g of r2n ∼= cn and taking their values in the grassmann subspaces ∧p,q 2n . they thus take the form fp,q(x1, . . . , xn, y1, . . . , yn) = ∑ φj1...jpk1...kq f † j1 ∧ . . . ∧ f † jp ∧ fk1 ∧ . . . ∧ fkq where the scalar functions φj1...jpk1...kq are assumed to be smooth functions in g. the space of these multivector functions is denoted by ep,q(g), and we have er(g) = ⊕ p+q=r ep,q(g) similarly to what was done for the euclidean dirac operator ∂x (and holds for ∂x| as well), also the hermitean dirac operators may be split into their scalar or ”dot” part and their bivector or ”wedge” part, leading to ∂z∧ = n∑ i=1 ∂zi f † i ∧ ∂z• = n∑ i=1 ∂zi f † i • ∂z† ∧ = n∑ i=1 ∂zc i fi ∧ ∂z†• = n∑ i=1 ∂zc i fi• for which it thus holds that ∂z ∧ + ∂z• = ∂z, ∂z† ∧ + ∂z†• = ∂z† these operators have a two-fold action on ep,q(g) in the sense that they act on the multi-vector by means of the inner and outer product with witt basis vectors, and at the same time on the function coefficients by partial differentiation. they enjoy the following properties, which can be obtained through direct calculation. property 2.2. the hermitean dirac dot and wedge operators are interrelated by complex conjugation as follows: 96 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) (i) (∂z∧) c = −∂z† ∧ (ii) (∂z•) c = −∂z†• property 2.3. the hermitean dirac dot and wedge operators act as follows on the spaces ep,q: (i) ∂z∧ : e p,q −→ ep+1,q (ii) ∂z• : e p,q −→ ep,q−1 (iii) ∂z† ∧ : e p,q −→ ep,q+1 (iv) ∂z†• : e p,q −→ ep−1,q property 2.4. the hermitean dirac dot and wedge operators are isotropic: (i) (∂z∧) 2 = (∂z•) 2 = (∂z† ∧) 2 = (∂z†•) 2 = 0 and they show the following anticommutation relations: (ii) (∂z∧)(∂z•) + (∂z•)(∂z∧) = 0 (iii) (∂z† ∧)(∂z†•) + (∂z†•)(∂z† ∧) = 0 (iv) (∂z∧)(∂z† ∧) + (∂z† ∧)(∂z∧) = 0 (v) (∂z•)(∂z†•) + (∂z†•)(∂z•) = 0 property 2.5. composition of the hermitean dirac dot and wedge operators yields the following actions on the spaces ep,q: (i) (∂z∧)(∂z•) = −(∂z∧)(∂z•) = (∂z•)(∂z∧) : e p,q −→ ep+1,q−1 (ii) (∂z† ∧)(∂z†•) = −(∂z† ∧)(∂z†•) = (∂z†•)(∂z† ∧) : e p,q −→ ep−1,q+1 (iii) (∂z∧)(∂z† ∧) = −(∂z† ∧)(∂z∧) : e p,q −→ ep+1,q+1 (iv) (∂z•)(∂z†•) = −(∂z†•)(∂z•) : e p,q −→ ep−1,q−1 property 2.6. the hermitean dirac dot and wedge operators establish a decomposition of the laplacian in the following ways: (i) (∂z∧)(∂z†•) + (∂z†•)(∂z∧) = 1 8 ∆2n (ii) (∂z† ∧)(∂z•) + (∂z•)(∂z† ∧) = 1 8 ∆2n cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 97 property 2.7. the hermitean dirac dot and wedge operators establish decompositions of the corresponding euclidean ones as follows: (i) (∂z† ∧) − (∂z∧) = 1 2 ∂x∧, (∂z†•) − (∂z•) = 1 2 ∂x• (ii) (∂z† ∧) + (∂z∧) = i 2 ∂x|∧, (∂z†•) + (∂z•) = i 2 ∂x|• whence they also decompose the actual euclidean dirac operators as follows: (iii) (∂z† ∧) − (∂z∧) + (∂z†•) − (∂z•) = 1 2 ∂x (iv) (∂z† ∧) + (∂z∧) + (∂z†•) + (∂z•) = i 2 ∂x| now, let us come back for a moment to the notion of hermitean monogenicity for multivector functions. a multivector function fp,q is h-monogenic if and only if simultaneously ∂zf p,q = (∂z • +∂z∧)f p,q = 0 and ∂z† f p,q = (∂z† • +∂z† ∧)f p,q = 0, which, due to property 2.3, is equivalent with the system { ∂z • f p,q = 0, ∂z ∧ f p,q = 0, ∂z† • f p,q = 0, ∂z† ∧ f p,q = 0 } in view of property 2.7 we then obtain the following remarkable result. proposition 2.8. for a multivector function fp,q the notions of ∂x-monogenicity, ∂x|-monogenicity and hermitean monogenicity coincide. remark 2.9. obviously the system of equations describing hermitean monogenicity will take particular forms according to the values of the functions considered. in [2] we have shown e.g. that, if the function takes its values in the subspace of spinor space corresponding to minimal or maximal degree of homogeneity, then hermitean monogenicity reduces to (anti-)holomorphy for a function of several complex variables. in that sense proposition 2.8 now reveals that one particular grassmann cell ∧p,q 2n can not be considered as an appropriate value space to study hermitean monogenicity, since in that case it coincides with euclidean monogenicity. it remains an interesting problem to discover appropriate value spaces in order to see the hermitean monogenicity system reduce to a significant system of differential equations. to that end we have investigated in [10] how the complex clifford algebra c2n decomposes into subspaces leading to exact sequences for the multiplicative action of the witt basis vectors. 3 differential forms: preliminaries there exists a vast literature on differential forms; in particular we refer to e.g. [19, 24] for real differential forms and to [20, 21] for complex differential forms. here we will only recall the basic concepts needed. 98 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) let rm be endowed with the standard euclidean metric. denoting by ∧r r m the space of alternating (or skew multilinear) real valued r-forms (0 ≤ r ≤ m), the grassmann algebra or exterior algebra over rm is the graded associative algebra ∧ r m = m ⊕ r=0 ∧r r m endowed with the exterior multiplication. a basis for ∧r r m is obtained as follows. let {dx1, ..., dxm} be a basis for the dual space (rm)∗ of rm. if, as before, the set a = {i1, . . . , ir} ⊂ m = {0, 1, ..., m} is ordered in the natural way, put dxa = dxi1 ∧ dxi2 ∧ ... ∧ dxir and dx∅ = 1. then for each r = 0, 1, ..., m, the set {dxa : a ⊂ m and |a| = r} is a basis for ∧r r m. note that in particular dxi ∧ dxi = 0, i = 0, 1, . . . , m and dxi ∧ dxj + dxj ∧ dxi = 0, 0 ≤ i 6= j ≤ m a smooth r-form in an open region g of rm is a map ωr : g → ∧r r m, x 7→ ∑ |a|=r ωra(x1, . . . , xm) dx a where, for each a, ωra is a smooth real valued function in g. we denote by ∧r (g) the space of smooth r-forms in g and we put ∧ (g) = m ⊕ r=0 ∧r (g) the projection operator from ∧ (g) onto ∧r (g) is denoted by [ · ]r. a fundamental linear operator on the space of smooth forms is the exterior derivative d. it is first defined as d : ∧r (g) → ∧r+1 (g) (r < m), by ωr = ∑ |a|=r ωra dx a 7−→ dωr = ∑ a ∑ j ∂xj ω r a dx j ∧ dxa a definition which is then extended to ∧ (g) by linearity. a second fundamental linear operator on the space of smooth forms is the hodge co-derivative d∗. for a = {ii, ..., ir} ⊂ m we denote dxa\{ij} = dxi1 ∧ ... ∧ dxij−1 ∧ [dxij ∧] dxij+1 ∧ ... ∧ dxir and in a first step we put d∗(ωradx a) = r∑ j=1 (−1)j ∂xij ω r a dx a\{ij} cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 99 then d∗ is defined as d∗ : ∧r (g) → ∧r−1 (g) (r > 0), by ωr = ∑ |a|=r ωra dx a 7−→ d∗(ωr) = ∑ |a|=r d∗(ωra dx a) and this definition again is extended to the whole of ∧ (g) by linearity. a smooth r-form ωr in g is called closed if and only if dωr = 0; it is called co-closed if and only if d∗ωr = 0; and it is called harmonic (in the sense of hodge) when it is at the same time closed and co-closed. introducing the operator d = d + d∗, a necessary and sufficient condition for a smooth r-form ωr in g to be harmonic thus reads dωr = (d + d∗)ωr = 0 ⇐⇒ { dωr = 0 d∗ωr = 0 (∗) the system (∗) is called the hodge-de rham system. note that if ωr is harmonic in an open region g of rm then automatically ωr satisfies ∆mω r = 0 in g, since d2 = (d + d∗)2 = d d∗ + d∗ d = −∆m the converse, however, is not true. the action of the operators d and d∗ on differential forms is two-fold in the sense that they act on the form itself as well as on the function coefficients by partial differentiation. in order to make this double action explicit we introduce the following symbolic notations for the operators d and d∗: d = m∑ j=1 (dxj∧) ∂xj d∗ = m∑ j=1 (dxj •) ∂xj with dxj • dxa = dxj • (dxi1 ∧ ... ∧ dxir ) = r∑ k=1 (−1)k δjik dx a\{ik} in this last action we recognize the contraction operators ∂xj ⌋, j = 1, ...m, given by ∂xj ⌋dx a = ∂xj ⌋ ( dxi1 ∧ ... ∧ dxir ) = r∑ k=1 (−1)k−1δjik dx a\{ik} acting only on the basis elements of the differential form, and not on the function coefficients. apparently the contraction operator ∂xj ⌋ coincides with the ”inner product”-operator dx j • up to a minus sign: ∂xj ⌋ = (−dx j •), j = 0, 1, ..., m 100 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) however bear in mind that contractions are more fundamental than dot products. indeed, they can be introduced independently of a scalar product, and their behaviour is invariant under diffeomorphisms, which is not the case for the dot product. we then indeed have for the operators d and d∗   m∑ j=1 (dxj∧)∂xj     ∑ |a|=r ωra dx a   = ∑ |a|=r m∑ j=1 (∂xj ω r a) dx j ∧ dxa = dωr   m∑ j=1 (dxj •)∂xj     ∑ |a|=r ωra dx a   = ∑ |a|=r r∑ k=1 (−1)k(∂xik ω r a) dx a\{ik} = d∗ωr at this moment we make the transition from the euclidean to the hermitean clifford setting, which, as above, is established by the introduction of the complex structure j, forcing the dimension to be even: m = 2n. we may now also consider a twisted exterior derivative d| and a twisted co-derivative d∗|, satisfying the following identities. property 3.1. it holds that (i) dd| + d|d = 0 = d∗d|∗ + d|∗d∗ = 0 (ii) dd|∗ + d|∗d = 0 = d∗d| + d|d∗ = 0 appropriate complex linear combinations of these real operators will give rise to complex exterior derivatives and co-derivatives, but we will first consider the traditional complex differential forms in cn or in an open region g of cn. we call ∧p,q (g) the space of complex differential forms of bidegree (p, q) in g; it contains elements ωp,q of the form ωp,q = ∑ |j|=p ∑ |k|=q ωj,k(z, z †) dzj ∧ dz c k where ωk,l(z1, . . . , zn, z c 1, . . . , z c n) are smooth functions in g and dzj = dzj1 ∧ . . . ∧ dzjp , j1 < j2 < . . . < jp dzck = dz c k1 ∧ . . . ∧ dzckq , k1 < k2 < . . . < kp the traditional derivatives in this setting are ∂, ∂c, ∂∗ and ∂∗c. they are defined as follows on a complex differential form of bidegree (p, q), definition which is then extended by linearity to an cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 101 arbitrary complex differential form: ∂ωp,q = ∑ |j|=p ∑ |k|=q ∂ωj,k ∧ dzj ∧ dz c k ∂cωp,q = ∑ |j|=p ∑ |k|=q ∂cωj,k ∧ dzj ∧ dz c k ∂∗ωp,q = ∑ |j|=p ∑ |k|=q ∂∗ (ωj,kdzj ∧ dz c k) ∂∗cωp,q = ∑ |j|=p ∑ |k|=q ∂∗c (ωj,kdzj ∧ dz c k) with ∂ωj,k = n∑ i=1 (∂zi ωj,k) dzi ∂cωj,k = n∑ i=1 (∂zc i ωj,k) dz c i ∂∗ (ωj,kdzj ∧ dz c k) = n∑ i=1 (∂zc i ωj,k) dz c i • (dzj ∧ dz c k) ∂∗c (ωj,kdzj ∧ dz c k) = n∑ i=1 (∂zi ωj,k) dzi • (dzj ∧ dz c k) here we have introduced, for j = 1, . . . , n, the not commonly used operators dzj• and dz c j •, which, via their euclidean counterparts, are in fact complex contraction operators. we have indeed, for all j = 1, . . . , n, that dzj• = (dxj + idyj)• = dxj • +idyj• = − ( ∂xj⌋ + i∂yj⌋ ) = −2∂zc j ⌋ dzcj • = (dxj − idyj)• = dxj • −idyj• = − ( ∂xj⌋ − i∂yj⌋ ) = −2∂zj ⌋ the four complex derivatives may thus be written symbolically as ∂ = n∑ i=1 ∂zi dzi ∧ ∂c = n∑ i=1 ∂zc i dzci ∧ ∂∗ = n∑ i=1 ∂zc i dzci • ∂∗c = n∑ i=1 ∂zi dzi• where it is explicitly shown that ∂ and ∂c act with a wedge product and ∂∗ and ∂∗c with a dot product or contraction. they enjoy the following properties. 102 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) property 3.2. the complex derivatives ∂, ∂c, ∂∗ and ∂∗c act as follows on the spaces ∧p,q (g) of complex differential forms of bidegree (p, q) in g: (i) ∂ : ∧p,q (g) −→ ∧p+1,q (g) (ii) ∂c : ∧p,q (g) −→ ∧p,q+1 (g) (iii) ∂∗ : ∧p,q (g) −→ ∧p−1,q (g) (iv) ∂∗c : ∧p,q (g) −→ ∧p,q−1 (g) property 3.3. the complex derivatives ∂, ∂c, ∂∗ and ∂∗c satisfy the kähler identities (i) ∂∂∗c + ∂∗c∂ = 0 = ∂∗∂c + ∂c∂∗ (ii) ∂∂c + ∂c∂ = 0 = ∂∗∂∗c + ∂∗c∂∗ (iii) ∂∂∗ + ∂∗∂ = − 1 2 ∆2n = ∂ c∂∗c + ∂∗c∂c in a very similar way as the hermitean variables and dirac operators are linked to their euclidean counterparts, the kählerian derivatives ∂, ∂c, ∂∗ and ∂∗c are linked to the exterior derivative and co-derivative and their twisted analogues. property 3.4. it holds that (i) ∂c + ∂ = d, ∂c − ∂ = id| (ii) ∂∗ + ∂∗c = d∗, ∂∗ − ∂∗c = id|∗ whence we may also write (iii) ∂c = 1 2 (d + id|), ∂ = 1 2 (d − id|) (iv) ∂∗ = 1 2 (d∗ + id|∗), ∂∗c = 1 2 (d∗ − id|∗) 4 differential forms and multi-vector functions: an identification in [6] it is shown how the world of real differential forms in an open region g of rm and the world of clifford algebra valued multi-vector functions in g may be naturally identified. the fundamental identification, adapted to the hermitean setting, reads ei ←→ dx i, en+i ←→ dy i, i = 1, . . . , n resulting in the identifications listed in table 1. note that we have listed here only a few of these identifications; for more details we refer the reader to [6]. cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 103 d = n∑ i=1 (dxi∧)∂xi + (dy i ∧)∂yi ∂x∧ = n∑ i=1 (ei∧)∂xi + (en+i∧)∂yi d∗ = n∑ i=0 (dxi •)∂xi + (dy i •)∂yi ∂x• = n∑ i=1 (ei •)∂xi + (en+i •)∂yi d| = n∑ i=1 (dxi∧)∂yi − (dy i ∧)∂xi ∂x|∧ = n∑ i=1 (ei∧)∂yi − (en+i∧)∂xi d|∗ = n∑ i=0 (dxi •)∂yi − (dy i •)∂xi ∂x|• = n∑ i=1 (ei •)∂yi − (en+i •)∂xi table 1: identification of the euclidean dirac operators this identification is now further developed in the hermitean setting. for the witt basis vectors one explicitly obtains the identifications f † j ∧ = − 1 2 (ej + ien+j)∧ = − 1 2 (ej ∧ +ien+j∧) ←→ − 1 2 (dxj ∧ +idyj∧) = − 1 2 (dzj∧) f † j • = − 1 2 (ej + ien+j)• = − 1 2 (ej • +ien+j• ) ←→ − 1 2 (dxj • +idyj•) = − 1 2 (dzj•) fj∧ = 1 2 (ej − ien+j)∧ = 1 2 (ej ∧ −ien+j∧) ←→ 1 2 (dxj ∧ −idyj∧) = 1 2 (dzcj ∧) and fj• = 1 2 (ej − ien+j)• = 1 2 (ej • −ien+j•) ←→ 1 2 (dxj • −idyj•) = 1 2 (dzcj •) listed in table 2. the so-called inflation operator, denoted ·⌉, is introduced below. f † j ∧ − 1 2 (dzj∧) = ∂zc j ⌉ f † j • − 1 2 (dzj•) = ∂zc j ⌋ fj∧ 1 2 (dzcj ∧) = −∂zj ⌉ fj• 1 2 (dzcj •) = −∂zj ⌋ table 2: identification of the witt basis vectors 104 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) in the same order of ideas one explicitly obtains for the hermitean dirac operators ∂z† ∧ = n∑ j=1 ∂zc j fj∧ ←→ n∑ j=1 ∂zc j 1 2 (dzcj ∧) = 1 2 (∂c∧) ∂z†• = n∑ j=1 ∂zc j fj• ←→ n∑ j=1 ∂zc j 1 2 (dzcj •) = 1 2 (∂∗•) ∂z∧ = n∑ j=1 ∂zj f † j ∧ ←→ n∑ j=1 ∂zj (− 1 2 )(dzj∧) = (− 1 2 )(∂∧) ∂z• = n∑ j=1 ∂zj f † j • ←→ n∑ j=1 ∂zj (− 1 2 )(dzj•) = (− 1 2 )(∂∗c•) as summarized in table 3. ∂z† ∧ 1 2 (∂c∧) ∂z†• 1 2 (∂∗•) ∂z∧ − 1 2 (∂∧) ∂z• − 1 2 (∂∗c•) table 3: identification of the hermitean dirac operators through these identifications it becomes clear that the properties of the hermitean dirac operators on multivector functions listed in section 2 and those of the kählerian differential operators on complex differential forms listed in section 3 are two faces of the same coin. this also implies that it suffices to prove a property in only one of these two worlds, automatically gaining the similar property in the other. to give an example, proposition 2.12 is transposed as follows. proposition 4.1. a (p, q)-form ωp,q ∈ ∧p,q (g) is harmonic in an open region g of cn, i.e. it satisfies the hodge-de rham system {dωp,q = 0, d∗ωp,q = 0}, if and only if in g it is hermitean monogenic, i.e. it satisfies the system {∂ωp,q = 0, ∂cωp,q = 0, ∂∗ωp,q = 0, ∂∗cωp,q = 0}, which implies that for a (p, q)-form ωp,q ∈ ∧p,q (g) the notions harmonic, twisted harmonic and hermitean monogenic coincide. another nice illustration of this identification is procured by the euler operators. the hermitean euler operators ez = n∑ j=1 zj ∂zj = 2 z • ∂z e†z = n∑ j=1 zcj ∂zcj = 2 z † • ∂z† cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 105 have shown their importance in constructing the fischer decomposition of homogeneous polynomials in terms of hermitean monogenic polynomials and the corresponding howe dual pair (see [16, 7]). they have a natural close connection with the traditional euclidean euler operators, since ez = 1 2 ex + i 2 x • ∂x| e†z = 1 2 ex + i 2 x| • ∂x whence ez + e † z = n∑ j=1 ( xj ∂xj + yj ∂yj ) = ex = ex| = −x • ∂x = −x| • ∂x| ez − e † z = i n∑ j=1 ( −xj ∂yj + yj ∂xj ) = i x • ∂x| = −i x| • ∂x it thus becomes clear that the hermitean euler operators are mutually complex conjugated scalar operators; note that they have the same expression in both worlds. in the world of differential forms we now focus on the contraction operators associated to the hermitean euler operators. to that end recall that we tend to denote contraction of a differential form by means of a ”dot”, more specifically ∂xα⌋ = −dxα•, yielding ∂x ⌋ = m∑ α=1 eα ∂xα⌋ = − m∑ α=1 eα dxα• = −dx• and also ∂z ⌋ = n∑ j=1 f † j ∂zj ⌋ = n∑ j=1 f † j (− 1 2 dzcj •) = − 1 2 dz† • ∂z† ⌋ = n∑ j=1 fj ∂zc j ⌋ = n∑ j=1 fj (− 1 2 dzj•) = − 1 2 dz • for the contracted hermitean euler operators we then obtain ez ⌋ = n∑ j=1 zj ∂zj ⌋ = (− 1 2 ) n∑ j=1 zj (dz c j •) or ez ⌋ = 2z • ∂z⌋ = −z • dz † • e†z ⌋ = n∑ j=1 zcj ∂zcj ⌋ = (− 1 2 ) n∑ j=1 zcj (dzj•) or e † z⌋ = 2z † • ∂z†⌋ = −z † • dz• we could as well, for symmetry’s sake, have introduced a so-called inflation operator (see [6]), denoted by a ”wedge”, i.e. ∂xα ⌉ = −dxα∧, yielding ∂x⌉ = m∑ α=1 eα ∂xα ⌉ = − m∑ α=1 eα dxα∧ = −dx∧ 106 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) and ∂z⌉ = n∑ j=1 f † j ∂zj ⌉ = n∑ j=1 f † j (− 1 2 dzcj ∧) = − 1 2 dz† ∧ ∂z†⌉ = n∑ j=1 fj ∂zc j ⌉ = n∑ j=1 fj (− 1 2 dzj∧) = − 1 2 dz ∧ note that the above notations ∂zj ⌉ and ∂zcj ⌉ where already used in table 2. this leads to ez ⌉ = n∑ j=1 zj ∂zj ⌉ = (− 1 2 ) n∑ j=1 zj (dz c j ∧) or ez⌉ = 2z • ∂z⌉ = −z • dz † ∧ e†z ⌉ = n∑ j=1 zcj ∂zcj ⌉ = (− 1 2 ) n∑ j=1 zcj (dzj∧) or e † z⌉ = 2z † • ∂z†⌉ = −z † • dz∧ these four contracted and inflated hermitean euler operators enjoy the properties summarized in the following two propositions. proposition 4.2. one has (i) ez⌋ + e † z⌋ = ex⌋ = x • dx• (ii) ez⌉ + e † z⌉ = ex⌉ = x • dx∧ (iii) ez⌋ − e † z⌋ = ix| • dx• = −ix • dx|• (iv) ez⌉ − e † z⌉ = ix| • dx∧ = −ix • dx|∧ (v) ez⌋ + ez⌉ = (− 1 2 ) n∑ j=1 zj dz c j = −z • dz † (vi) e†z⌋ + e † z⌉ = (− 1 2 ) n∑ j=1 zcj dzj = −z † • dz proposition 4.3. one has (i) ( ez⌋ + e † z⌋ )2 = ( ex⌋ )2 = 0 (ii) ( ez⌉ + e † z⌉ )2 = ( ex⌉ )2 = 0 (iii) (ez⌋ + ez⌉) 2 = 0 (iv) ( e † z⌋ + e † z⌉ )2 = 0 (v) (ez⌋ + ez⌉) ( e † z⌋ + e † z⌉ ) + ( e † z⌋ + e † z⌉ ) (ez⌋ + ez⌉) = −|z 2| cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 107 (vi) ( ez⌋ + e † z⌋ )( ez⌉ + e † z⌉ ) + ( ez⌉ + e † z⌉ )( ez⌋ + e † z⌋ ) = −|z2| these properties may be proven by direct calculation, but things become more transparent after identification in the multivector setting; to that end we look at the analogues of the operators involved, given by ez ⌋ = (− 1 2 ) n∑ j=1 zj (dz c j • ) ←→ (− 1 2 ) n∑ j=1 zj (2fj•) = −z • e†z ⌋ = (− 1 2 ) n∑ j=1 zcj (dzj• ) ←→ (− 1 2 ) n∑ j=1 zcj (−2f † j •) = z† • ez ⌉ = (− 1 2 ) n∑ j=1 zj (dz c j ∧) ←→ (− 1 2 ) n∑ j=1 zj (2fj∧) = −z ∧ e†z ⌉ = (− 1 2 ) n∑ j=1 zcj (dzj∧) ←→ (− 1 2 ) n∑ j=1 zcj (−2f † j ∧) = z†∧ propositions 4.2 and 4.3 now take a rather trivial form and are easily proven (see also [6]), as may be observed from their reformulation in the propositions below. proposition 4.4. one has (i) (−z•) + (z†•) = −x• (ii) (−z∧) + (z†∧) = −x∧ (iii) (−z•) − (z†•) = −ix|• (iv) (−z∧) − (z†∧) = −ix|∧ (v) (−z•) + (−z∧) = −z (vi) (z†•) + (z†∧) = z† proposition 4.5. one has (i) (−x•)(x•) = 0 (ii) (−x∧)(x∧) = 0 (iii) (−z • −z∧) 2 = (−z) 2 = 0 (iv) ( z† • +z†∧ )2 = ( z† )2 = 0 (v) (−z) ( z† ) + ( z† ) (−z) = −|z2| (vi) ( −z • +z†• )( −z ∧ +z†∧ ) + ( −z ∧ +z†∧ )( −z • +z†• ) = −|z2| 108 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) in the same order of ideas, starting from the operators d and d∗, we introduce the contraction and inflation operators d⌋ = m∑ j=1 (dxj∧) ∂xj ⌋ = m∑ j=1 (dxj∧)(−dxj •) d∗⌉ = m∑ j=1 (dxj •) ∂xj ⌉ = m∑ j=1 (dxj •)(−dxj∧) the operators d⌋ and d∗⌉ have er(ω) as an eigenspace since d⌋ωr = r ωr and d∗⌉ωr = (m − r) ωr in other words: they measure the order of a differential form. they are sometimes called fermionic euler operators. in the clifford analysis setting they read ∂x∧⌋ = m∑ j=1 (ej∧)(−ej •) and ∂x•⌉ = m∑ j=1 (ej •)(−ej∧) for which it indeed holds that ∂x∧⌋fr = r fr and ∂x•⌉fr = (m − r) fr note that d⌉, d∗⌋, ∂x∧⌉ and ∂x•⌋ are zero operators. the same can be done now with the kählerian derivatives, leading to ∂⌋ = (− 1 2 ) n∑ j=1 dzj ∧ dz c j • ∂c⌋ = (− 1 2 ) n∑ j=1 dzcj ∧ dzj • ∂∗⌋ = (− 1 2 ) n∑ j=1 dzcj • dzj • ∂∗c⌋ = (− 1 2 ) n∑ j=1 dzj • dz c j • and ∂⌉ = (− 1 2 ) n∑ j=1 dzj ∧ dz c j ∧ ∂c⌉ = (− 1 2 ) n∑ j=1 dzcj ∧ dzj ∧ ∂∗⌉ = (− 1 2 ) n∑ j=1 dzcj • dzj ∧ ∂∗c⌉ = (− 1 2 ) n∑ j=1 dzj • dz c j ∧ cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 109 while their hermitean multivector analogues are given by ∂z∧⌋ = − n∑ j=1 f † j ∧ fj • ∂z† ∧⌋ = n∑ j=1 fj ∧ f † j • ∂z†•⌋ = n∑ j=1 fj • f † j • ∂z•⌋ = − n∑ j=1 f † j • fj • and by ∂z∧⌉ = − n∑ j=1 f † j ∧ fj ∧ ∂z† ∧⌉ = n∑ j=1 fj ∧ f † j ∧ ∂z†•⌉ = n∑ j=1 fj • f † j ∧ ∂z•⌉ = − n∑ j=1 f † j • fj ∧ the spaces ep,q of smooth vector functions of bidegree (p, q) are eigenspaces of the operators ∂z∧⌋, ∂z† ∧⌋, ∂z†•⌋ and ∂z•⌋. more precisely we have the following. proposition 4.6. for fp,q ∈ ep,q, one has (i) ( ∂z∧⌋ ) fp,q = ( − p 2 ) fp,q (ii) ( ∂z† ∧⌋ ) fp,q = ( q 2 ) fp,q (iii) ( ∂z†•⌉ ) fp,q = ( n − p 2 ) fp,q (iv) ( ∂z•⌉ ) fp,q = ( −n + q 2 ) fp,q note that, by similarity, the same eigenvalue equations hold for the operators ∂⌋, ∂c⌋, ∂∗⌉ and ∂∗c⌉. moreover observe that the eigenvalue equations for the operators d⌋ or ∂x∧⌋ and d ∗⌉ or 110 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) ∂x•⌉ are refined by the ones of proposition 4.6, and may be recovered from them: ( ∂x∧⌋ ) fp,q = 2 ( ∂z† ∧⌋ − ∂z∧⌋ ) fp,q = 2 ( q 2 + p 2 ) fp,q = (p + q)fp,q ( ∂x•⌉ ) fp,q = 2 ( ∂z†•⌉ − ∂z• ⌉ ) fp,q = 2 ( n − p 2 − −n + q 2 ) fp,q = (2n − (p + q)) fp,q furthermore, it may be verified that ∂x∧⌉ and ∂x•⌋ indeed are zero operators: ∂x∧⌉ = 2 ( ∂z† ∧⌉ − ∂z∧⌉ ) = 2   n∑ j=1 fj ∧ f † j ∧ + n∑ j=1 f † j ∧ fj∧   = 0 ∂x•⌋ = 2 ( ∂z†•⌋ − ∂z• ⌋ ) = 2   n∑ j=1 fj • f † j • + n∑ j=1 f † j • fj•   = 0 finally, also the original expressions for ∂x∧⌋ and ∂x•⌉ as obtained in [6] may be recovered: ∂x∧⌋ = 2 ( ∂z† ∧⌋ − ∂z∧⌋ ) = 2   n∑ j=1 fj ∧ f † j • + n∑ j=1 f † j ∧ fj•   = − n∑ j=1 ej ∧ ej • + en+j ∧ en+j• = − 2n∑ α=1 eα ∧ eα • ∂x•⌉ = 2 ( ∂z†•⌉ − ∂z• ⌉ ) = 2   n∑ j=1 fj • f † j ∧ + n∑ j=1 f † j • fj∧   = − n∑ j=1 ej • ej ∧ + en+j • en+j∧ = − 2n∑ α=1 eα • eα∧ we shall encounter the operators ∂z∧⌉ and ∂z•⌋ again in the next section in a different context. 5 the kählerian metric we will now use known results from kählerian geometry, however restricted to the flat kählerian manifold cn, and transpose them to obtain results, not yet known in the hermitean clifford analysis setting. our guides are [21, 20]. each kählerian metric induces a fundamental form ω, which is a 2-form derived from the corresponding kähler potential u, i.e. ω ∧ = i 2 ∂∂cu the potential of the flat metric or the canonical hermitean metric is given by u = 1 2 |z|2 = 1 2 |z†|2 = 1 2 n∑ i=1 ziz c i = z • z † = 1 2 (zz† + z†z) = 1 2 |x|2 = 1 2 |x||2 cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 111 yielding the flat fundamental form ω ∧ = i 2 (−2∂z∧)(2∂z† ∧)|z| 2 = (−2i)   n∑ j=1 ∂zj f † j ∧   ( n∑ k=1 ∂zc k fk∧ ) |z|2 = (−2i) n∑ j=1 f † j ∧ fj∧ = 2i∂z∧⌉ or, in terms of the original basis vectors, ω ∧ = n∑ j=1 ej ∧ en+j∧ introducing the so-called spin-euler operator, which is a parabivector valued multiplicative constant, i.e. the sum of a scalar and a bivector, β = n∑ j=1 f † j fj = n∑ j=1 ( f † j • fj + f † j ∧ fj ) = n 2 + n∑ j=1 f † j ∧ fj we find that the fundamental form appears as the bivector part of that spin-euler operator, meaning that we may write β = n 2 + i 2 ω its complex conjugate is then given by βc = n∑ j=1 fjf † j = n∑ j=1 ( fj • f † j + fj ∧ f † j ) = n 2 − n∑ j=1 f † j ∧ fj = n 2 − i 2 ω usually, one also introduces the associated fundamental form ω = 1 2i ω = n 2 − β = βc − n 2 = n∑ j=1 fj ∧ f † j for the sake of completeness we recall the following intertwining relations of the spin-euler operator and its complex conjugate with the witt basis vectors; for more of these intertwining relations we refer to [9]. proposition 5.1. one has (i) [fk, β] = fk, [f † k , β] = −f † k (ii) [f † k , βc] = f † k , [fk, β c] = −fk an important operator in kähler geometry is the so-called l-operator, which is defined by means of the fundamental form. 112 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) definition 5.2. the l-operator is defined as l : ep,q −→ ep+1,q+1 : fp,q 7→ ω ∧ fp,q, where, explicitly ω ∧ fp,q = n∑ j=1 ej ∧ en+j ∧ f p,q = (−2i) n∑ j=1 f † j ∧ fj ∧ f p,q = 2i∂z∧⌉f p,q the l-operator enjoys the properties listed in the proposition below (see also [21]). proposition 5.3. one has (i) [l, ∂x∧] = 0, [l, ∂x•] = −∂x|∧ and also (ii) [l, ∂z† ∧] = 0, [l, ∂z∧] = 0 (iii) [l, ∂z†•] = i∂z† ∧, [l, ∂z•] = −i∂z∧ the counterpart of the l-operator is the λ-operator. definition 5.4. the λ-operator is defined as λ : ep,q −→ ep−1,q−1 : fp,q 7→ λfp,q, where, explicitly λfp,q = n∑ j=1 ej • en+j • f p,q = (−2i) n∑ j=1 f † j • fj • f p,q = 2i∂z•⌋f p,q it shows the following properties (see also [21]). proposition 5.5. one has (i) [λ, ∂x∧] = −∂x|•, [λ, ∂x•] = 0 and also (ii) [λ, ∂z† ∧] = i∂z†•, [λ, ∂z∧] = −i∂z• (iii) [λ, ∂z†•] = 0, [λ, ∂z•] = 0 a rather tedious computation leads to the commutator of the l and λ operators. proposition 5.6. one has [l , λ] fp,q = (n − p − q)fp,q finally, putting for an arbitrary multivector function f ≡ n∑ p=0 n∑ q=0 fp,q: h [f] = n∑ p=0 n∑ q=0 (n − p − q)fp,q = ( n − 2 ( ∂z† ∧⌋ − ∂z∧⌋ )) f we obtain the following relations (see also [20]). cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 113 proposition 5.7. one has (i) [l , λ] = h (ii) [h , λ] = 2λ (iii) [h , l] = −2l meaning that the operators (l, λ, h) generate the lie algebra slc(2) 6 the hodge “star”-operator the hodge ∗-operator for smooth real differential forms in rm may be defined as follows (see e.g. [19, 24]). definition 6.1. let {j1, . . . , jr} ∪ {jr+1, . . . , jm} = {1, . . . , m} and {j1, . . . , jr} ∩ {jr+1, . . . , jm} = ∅, with j1 < . . . < jr. then ∗ (dxj1 ∧ · · · ∧ dxjr ) = σ dxjr+1 ∧ · · · dxjm where σ is the signature of the permutation (jr+1, . . . , jm, j1, . . . , jr). it constitutes an isomorphism ∗ : ∧r −→ ∧m−r , its inverse being given by ∗−1 = (−1)r(m−r)∗ which implies that ∗2 = (−1)r(m−r) by means of this ∗-operator the hodge co-derivative d∗ may be expressed in terms of the derivative d as d∗ ωr = (−1)r ∗ d ∗−1 ωr = (−1)r(m+1−r) ∗ d ∗ ωr in the actual case of even dimension (m = 2n) we find that the hodge star-operator is an isomorphism ∗ : ∧p,q −→ ∧n−q,n−p for which ∗−1 = (−1)(p+q) 2 ∗ and thus ∗2 = (−1)(p+q) 2 . for the hodge derivative and co-derivative we then obtain d∗ = ∗ d ∗ and d = ∗ d∗ ∗ and similarly for the twisted versions d|∗ = ∗ d| ∗ and d| = ∗ d∗| ∗ simply applying the conversion rules of the foregoing section we obtain the counterparts of these relations in the setting of multivector functions, involving then the dirac operators. proposition 6.2. one has 114 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) (i) ∂x• = ∗ ( ∂x∧ ) ∗ and ∂x∧ = ∗ ( ∂x• ) ∗ (ii) ∂x|• = ∗ ( ∂x|∧ ) ∗ and ∂x|∧ = ∗ ( ∂x|• ) ∗ and also (iii) ∂z• = ∗ ( ∂z∧ ) ∗ and ∂z∧ = ∗ ( ∂z• ) ∗ (iv) ∂z†• = ∗ ( ∂z† ∧ ) ∗ and ∂z† ∧ = ∗ ( ∂z†• ) ∗ clearly, we may convert proposition 6.2 back to the differential form setting. proposition 6.3. one has (i) ∂∗c• = ∗ (∂∧) ∗ and ∂∧ = ∗ (∂∗c•) ∗ (ii) ∂∗• = ∗ (∂c∧) ∗ and ∂c∧ = ∗ (∂∗•) ∗ of course, it is also possible to express the hodge star operator in rm directly in the clifford algebra setting; definition 6.1 is then converted as follows (see [6]): ∗(ej1 · · · ejr ) = (−1) r(r+1) 2 em ej1 · · · ejr where em is the so-called pseudoscalar given by em = e1 · · · em, of which the square equals e2m = (−1) m(m+1) 2 . it follows that for 1-vectors the ∗-operation reduces to a multiplication from the left by −em. also ∗ 1 = em and ∗ em = 1. in the hermitean case with even dimension m = 2n, let us compute em in terms of the witt basis vectors. we consecutively obtain em = 2n∏ α=1 eα = n∏ j=1 (fj − f † j ) n∏ j=1 i(fj + f † j ) = in (−1) n(n−1) 2 n∏ j=1 (fj − f † j )(fj + f † j ) = in (−1) n(n−1) 2 n∏ j=1 (fjf † j − f † j fj) = 2n in (−1) n(n−1) 2 f1 ∧ f † 1 ∧ f2 ∧ f † 2 ∧ · · · ∧ fn ∧ f † n showing that the pseudoscalar has bidegree (n, n). as an example we have, for m = 4, n = 2, that the images under the ∗-operation of the euclidean basis vectors are 3-vectors given by ∗ e1 = −e2e3e4, ∗ e2 = e1e3e4, ∗ e3 = −e1e2e4, ∗ e4 = e1e2e3 cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 115 the witt basis vectors f1 and f2, of bidegree (0, 1), transform into (1, 2)-multivectors: ∗ f1 = −2 f1 ∧ f2 ∧ f † 2 ∗ f2 = −2 f1 ∧ f † 1 ∧ f2 while f † 1 and f † 2 , of bidegree (1, 0), transform into (2, 1)-multivectors: ∗ f † 1 = 2 f † 1 ∧ f2 ∧ f † 2 ∗ f † 2 = 2 f1 ∧ f2 ∧ f † 2 7 afterword in the previous sections we established and illustrated a ”natural” isomorphism between on the one hand the algebra of complex differential forms (extended with the hodge star operator and the inner product or dot product) with the underlying structure of a grassmann algebra, and on the other hand the algebra of multi-vector functions in hermitean clifford analysis with the underlying structure of a complex clifford algebra. the hermitean dirac operators, underlying the notion of hermitean monogenicity, may well be identified with the kählerian derivatives for complex differential forms, one of which is the famous ∂ operator from several complex variables theory. it should be emphasized, as was done from the beginning, that only differential forms in cn or in open regions thereof were considered, and that actually hermitean clifford analysis was developed only in flat space cn. as was also mentioned hermitean clifford analysis on curved kählerian manifolds is the subject of the forthcoming paper [11]. finally this paper is by no means a plea for substituting hermitean multivector functions for complex differential forms. both worlds, how convincing the similarities might be, have their own interest en properties; this paper intended to illustrate the very close connections between hermitean clifford analysis and complex analysis and the benefits obtained from exchanging knowledge between both. received: december 2009. revised: april 2010. references [1] f. brackx , j. bureš, h. de schepper, d. eelbode, f. sommen and v. souček, fundaments of hermitean clifford analysis. part i: complex structure, compl. anal. oper. theory 1(3), 2007, 341–365. [2] f. brackx, j. bureš, h. de schepper, d. eelbode, f. sommen and v. souček, fundaments of hermitean clifford analysis. part ii: splitting of h-monogenic equations, complex var. elliptic eq. 52(10-11), 2007, 1063–1079. [3] f. brackx, b. de knock and h. de schepper, a matrix hilbert transform in hermitean clifford analysis, j. math. anal. appl. 344 (2008), 1068–1078. 116 f. brackx, h. de schepper and v. souček cubo 13, 2 (2011) [4] f. brackx, b. de knock, h. de schepper and f. sommen, on cauchy and martinelli– bochner integral formulae in hermitean clifford analysis, bull. braz. math. soc. new series 40(3), 2009, 395-416. [5] f. brackx, r. delanghe and f. sommen, clifford analysis, pitman publishers, bostonlondon-melbourne, 1982. [6] f. brackx, r. delanghe and f. sommen, differential forms and/or multivector functions, cubo 7(2), 2005, 139-169. [7] f. brackx, h. de schepper, d. eelbode and v. souček, the howe dual pair in hermitean clifford analysis, accepted for publication in rev. mat. iberoam. 26(2), 2010, 449479. [8] f. brackx, h. de schepper and f. sommen, a theoretical framework for wavelet analysis in a hermitean clifford setting, comm. pure appl. anal. 6(3), 2007, 549-567. [9] f. brackx, h. de schepper and f. sommen, the hermitian clifford analysis toolbox, appl. clifford algebras 18(3-4), 2008, 451–487. [10] f. brackx, h. de schepper and v. souček, on the structure of complex clifford algebra adv. appl. clifford alg. doi: 10.1007/s0006-010-0270-4. [11] f. brackx, h. de schepper and v. souček, hermitean clifford analysis on kählerian manifolds (in preparation). [12] a. damiano, d. eelbode and i. sabadini, invariant syzygies for the hermitian dirac operator, math. zeitschrift 262, 2009, 929-945. [13] r. delanghe, r. lávička and v. souček, on polynomial solutions of generalized moisilthéodoresco systems and hodge-de rham systems (arxiv: 0908.0842). [14] r. delanghe, r. lávička and v. souček, the fischer decomposition for hodge-de rham systems in euclidean spaces (arxiv: 1012.4994). [15] r. delanghe, f. sommen and v. souček, clifford algebra and spinor–valued functions, kluwer academic publishers, dordrecht, 1992. [16] d. eelbode, stirling numbers and spin-euler polynomials, exp. math. 16(1) (2007), 55-66. [17] j. gilbert and m. murray, clifford algebras and dirac operators in harmonic analysis, cambridge university press, cambridge, 1991. [18] k. gürlebeck, k. habetha and w. sprößig, holomorphic functions in the plane and n-dimensional space, birkhäuser verlag, basel, 2008. [19] k. maurin, analysis, part ii, d. reidel publishing company, dordrecht boston london, pwn-polish scientific publishers, warszawa, 1980. cubo 13, 2 (2011) differential forms versus multi-vector functions . . . 117 [20] m. l. michelsohn, clifford and spinor cohomology of kähler manifolds, american journal of mathematics 102(6) (1980), 1083-1146. [21] a. moroianu, lectures on kähler geometry, london mathematical society student texts 69, cambridge university press (cambridge, 2007). [22] i. r. porteous, topological geometry, van nostrand reinhold company, london new york toronto melbourne, 1969. [23] i. sabadini and f. sommen, hermitian clifford analysis and resolutions, math. meth. appl. sci. 25 (16–18) (2002), 1395–1414. [24] c. von westenholz, differential forms in mathematical physics, stud. math. appl., vol 3, north-holland, amsterdam, 1978. introduction multi-vector functions: preliminaries differential forms: preliminaries differential forms and multi-vector functions: an identification the kählerian metric the hodge ``star"-operator afterword c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2606 .dvi cubo a mathematical journal vol.13, no¯ 01, (73–101). march 2011 existence of pseudo almost automorphic solutions to a nonautonomous heat equation toka diagana department of mathematics, howard university, 2441 6th street nw, washington, dc 20005 usa. email: tokadiag@gmail.com abstract in this paper, upon making some suitable assumptions, we obtain the existence of pseudo-almost automorphic solutions to a nonautonomous heat equation with gradient coefficients. resumen en este trabajo, al hacer algunos supuestos adecuados, se obtiene la existencia de pseudo soluciones automorfas a una ecuacin del calor no autnoma con coeficientes degradados. keywords: pseudo almost periodicity; almost automorphic; pseudo almost automorphic; sppseudo almost automorphic; sp-almost automorphic; sp-pseudo almost periodic; acquistapace and terreni conditions; intermediate space; exponential dichotomy. ams subject classification: 43a60; 34g20. 1 introduction fix p > 1 and α,β ∈ r with 0 ≤ α < β < 1. this paper is concerned with the existence of pseudoalmost automorphic solutions to the the class of abstract nonautonomous differential equations given by d dt [ u(t) + f ( t,bu(t) )] = a(t)u(t) + g ( t,cu(t) ) , t ∈ r, (1.1) 74 toka diagana cubo 13, 1 (2011) where a(t) for t ∈ r is a family of closed linear operators on their domains d(a(t)) satisfying the well-known acquistapace-terreni conditions, b,c are (possibly unbounded) linear operators, and f : r × x 7→ xtβ and g : r × x 7→ x are s p-pseudo-almost automorphic functions in t ∈ r uniformly in the second variable. in view of above, there exists an evolution family u = {u(t,s)}t≥s associated with the family of operators a(t). assuming that u = {u(t,s)}t≥s is exponentially dichotomic (hyperbolic) and under some additional assumptions, it will be shown that eq. (1.1) has a unique pseudo-almost automorphic solution. it is worth mentioning that the main result of this paper (theorem 5.5) generalizes, to some extent, most of known results on (pseudo) almost automorphic solutions to autonomous and nonautonomous differential equations, especially those in [9, 22, 48]. let ω ⊂ rn (n ≥ 1) be a bounded subset with regular boundary γ = ∂ω and let x = l2(ω) be the space of square integrable functions equipped with its natural topology. to illustrate our main result, we study the existence of pseudo-almost automorphic solutions to the nonautonomous heat equation with gradient coefficients given by    ∂ ∂t [ ϕ + f ( t,∇ϕ )] = a(t,x)∆ϕ + g ( t,∇ϕ ) , in r × ω ϕ = 0, on r × γ (1.2) where the diffusion coefficient a : r × ω 7→ r, and f,g : r × x1/2 7→ l 2(ω) are sp-pseudo-almost automorphic functions. the concept of pseudo-almost automorphy, which is the central tool here, was introduced in the literature a few years ago by liang et al. [34, 47] and is a generalization of both the classical almost automorphy due to bochner [8] and that of pseudo almost periodicity due to zhang [18]. such a concept has recently generated several developments and extensions. for the most recent developments, we refer the reader to [17, 25, 34, 47, 35]. more recently, in diagana [17], the concept of sp-pseudo almost automorphy (or stepanov-like pseudo almost automorphy) was introduced, which in fact is a natural generalization of the notion of pseudo almost automorphy. in this paper, we make extensive use of the concept of sp-pseudo almost automorphy combined with intermediate space techniques to investigate the existence of pseudo-almost automorphic solutions eq. (1.1) and then to the n-dimensional heat equation eq. (1.2). the literature related to those intermediate spaces is very extensive, in particular, we refer the reader to the excellent book by lunardi [33], which contains a comprehensive presentation on that topic and related issues. existence results related to almost periodic, asymptotically almost periodic, pseudo almost periodic and almost automorphic solutions to eq. (1.1) in the autonomous case have recently been established in [19], [20], [22], [23], [28], [29], and [30], respectively. though to the best of our knowledge, the existence of pseudo almost automorphic solutions to eq. (1.1) in the case when the coefficients f,g are sp-pseudo almost automorphic is an untreated original problem, which cubo 13, 1 (2011) pseudo almost automorphic solutions 75 constitutes one of the main motivations of the present paper. 2 preliminaries let (x,‖ · ‖), (y,‖ · ‖y) be two banach spaces. let bc(r, x) (respectively, bc(r × y, x)) denote the collection of all x-valued bounded continuous functions (respectively, the class of jointly bounded continuous functions f : r×y 7→ x). the space bc(r, x) equipped with the sup norm ‖ · ‖∞ is a banach space. furthermore, c(r, y) (respectively, c(r × y, x)) denotes the class of continuous functions from r into y (respectively, the class of jointly continuous functions f : r × y 7→ x). the notation b(x, y) stands for the banach space of bounded linear operators from x into y equipped with its natural topology; in particular, this is simply denoted b(x) whenever x = y. definition 2.1. [41] the bochner transform fb(t,s), t ∈ r, s ∈ [0, 1] of a function f : r 7→ x is defined by fb(t,s) := f(t + s). remark 2.2. (i) a function ϕ(t,s), t ∈ r, s ∈ [0, 1], is the bochner transform of a certain function f, ϕ(t,s) = fb(t,s) , if and only if ϕ(t + τ,s−τ) = ϕ(s,t) for all t ∈ r, s ∈ [0, 1] and τ ∈ [s− 1,s]. (ii) note that if f = h + ϕ, then fb = hb + ϕb. moreover, (λf)b = λfb for each scalar λ. definition 2.3. the bochner transform f b(t,s,u), t ∈ r, s ∈ [0, 1], u ∈ x of a function f(t,u) on r × x, with values in x, is defined by f b(t,s,u) := f(t + s,u) for each u ∈ x. definition 2.4. let p ∈ [1,∞). the space bsp(x) of all stepanov bounded functions, with the exponent p, consists of all measurable functions f : r 7→ x such that fb belongs to l∞ ( r; lp((0, 1), x) ) . this is a banach space with the norm ‖f‖sp := ‖f b‖l∞(r,lp) = sup t∈r (∫ t+1 t ‖f(τ)‖p dτ )1/p . 2.1 sp-pseudo almost periodicity definition 2.5. a function f ∈ c(r, x) is called (bohr) almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that ‖f(t + τ) − f(t)‖ < ε for each t ∈ r. the number τ above is called an ε-translation number of f, and the collection of all such functions will be denoted ap(x). 76 toka diagana cubo 13, 1 (2011) definition 2.6. a function f ∈ c(r × y, x) is called (bohr) almost periodic in t ∈ r uniformly in y ∈ k where k ⊂ y is any compact subset k ⊂ y if for each ε > 0 there exists l(ε) such that every interval of length l(ε) contains a number τ with the property that ‖f(t + τ,y) − f(t,y)‖ < ε for each t ∈ r, y ∈ k. the collection of those functions is denoted by ap(r × y). define the classes of functions pap0(x) and pap0(r × x) respectively as follows: pap0(x) := { u ∈ bc(r, x) : lim t →∞ 1 2t ∫ t −t ‖u(s)‖ds = 0 } , and pap0(r × y) is the collection of all functions f ∈ bc(r × y, x) such that lim t →∞ 1 2t ∫ t −t ‖f(t,u)‖dt = 0 uniformly in u ∈ y. definition 2.7. [18] a function f ∈ bc(r, x) is called pseudo almost periodic if it can be expressed as f = h + ϕ, where h ∈ ap(x) and ϕ ∈ pap0(x). the collection of such functions will be denoted by pap(x). definition 2.8. [18] a function f ∈ c(r × y, x) is said to be pseudo almost periodic if it can be expressed as f = g + φ, where g ∈ ap(r × y) and φ ∈ pap0(r × y). the collection of such functions will be denoted by pap(r × y). define aa0(r × y) as the collection of all functions f ∈ bc(r × y, x) such that lim t →∞ 1 2t ∫ t −t ‖f(t,u)‖dt = 0 uniformly in u ∈ k, where k ⊂ y is any bounded subset. obviously, pap0(r × y) ⊂ aa0(r × y). a weaker version of definition 2.8 is the following: definition 2.9. a function f ∈ c(r × y, x) is said to be b-pseudo almost periodic if it can be expressed as f = g + φ, where g ∈ ap(r × y) and φ ∈ aa0(r × y). the collection of such functions will be denoted by bpap(r × y). cubo 13, 1 (2011) pseudo almost automorphic solutions 77 definition 2.10. [15, 16] a function f ∈ bsp(x) is called sp-pseudo almost periodic (or stepanovlike pseudo almost periodic) if it can be expressed as f = h+ϕ, where hb ∈ ap ( lp((0, 1), x) ) and ϕb ∈ pap0 ( lp((0, 1), x) ) . the collection of such functions will be denoted by pap p(x). in other words, a function f ∈ l p loc(r, x) is said to be s p-pseudo almost periodic if its bochner transform fb : r → lp((0, 1), x) is pseudo almost periodic in the sense that there exist two functions h,ϕ : r 7→ x such that f = h + ϕ, where hb ∈ ap(lp((0, 1), x)) and ϕb ∈ pap0(l p((0, 1), x)). to define the notion of sp-pseudo almost automorphy for functions of the form f : r×y 7→ y, we need to define the sp-pseudo almost periodicity for these functions as follows: definition 2.11. a function f : r × y 7→ x, (t,u) 7→ f(t,u) with f(·,u) ∈ l p loc(r, x) for each u ∈ x, is said to be sp-pseudo almost periodic if there exist two functions h, φ : r × y 7→ x such that f = h + φ, where hb ∈ ap(r × lp((0, 1), x)) and φb ∈ aa0(r × l p((0, 1), x)). the collection of those sp-pseudo almost periodic functions f : r × y 7→ x will be denoted pap p(r × y). 2.2 sp-almost automorphy the notion of sp-almost automorphy is a new notion due to n’guérékata and pankov [40]. definition 2.12. (bochner) a function f ∈ c(r, x) is said to be almost automorphic if for every sequence of real numbers (s′n)n∈n, there exists a subsequence (sn)n∈n such that g(t) := lim n→∞ f(t + sn) is well defined for each t ∈ r, and lim n→∞ g(t − sn) = f(t) for each t ∈ r. remark 2.13. the function g in definition 2.12 is measurable, but not necessarily continuous. moreover, if g is continuous, then f is uniformly continuous. if the convergence above is uniform in t ∈ r, then f is almost periodic. denote by aa(x) the collection of all almost automorphic functions r → x. note that aa(x) equipped with the sup norm, ‖·‖∞, turns out to be a banach space. we will denote by aau(x) the closed subspace of all functions f ∈ aa(x) with g ∈ c(r, x). equivalently, f ∈ aau(x) if and only if f is almost automorphic and the convergence in definition 2.12 are uniform on compact intervals, i.e. in the fréchet space c(r, x). indeed, if f is almost automorphic, then, its range is relatively compact. obviously, the following inclusions hold: ap(x) ⊂ aau(x) ⊂ aa(x) ⊂ bc(x). 78 toka diagana cubo 13, 1 (2011) definition 2.14. [40] the space asp(x) of stepanov-like almost automorphic functions (or spalmost automorphic) consists of all f ∈ bsp(x) such that fb ∈ aa ( lp(0, 1; x) ) . that is, a function f ∈ l p loc(r; x) is said to be s p-almost automorphic if its bochner transform fb : r → lp(0, 1; x) is almost automorphic in the sense that for every sequence of real numbers (s′n)n∈n, there exists a subsequence (sn)n∈n and a function g ∈ l p loc(r; x) such that [∫ t+1 t ‖f(sn + s) − g(s)‖ pds ]1/p → 0, and [∫ t+1 t ‖g(s − sn) − f(s)‖ pds ]1/p → 0 as n → ∞ pointwise on r. remark 2.15. it is clear that if 1 ≤ p < q < ∞ and f ∈ l q loc(r; x) is s q-almost automorphic, then f is sp-almost automorphic. also if f ∈ aa(x), then f is sp-almost automorphic for any 1 ≤ p < ∞. moreover, it is clear that f ∈ aau(x) if and only if f b ∈ aa(l∞(0, 1; x)). thus, aau(x) can be considered as as ∞(x). definition 2.16. a function f : r × y 7→ x, (t,u) 7→ f(t,u) with f(·,u) ∈ l p loc(r; x) for each u ∈ y, is said to be sp-almost automorphic in t ∈ r uniformly in u ∈ y if t 7→ f(t,u) is sp-almost automorphic for each u ∈ y, that is, for every sequence of real numbers (s′n)n∈n, there exists a subsequence (sn)n∈n and a function g(·,u) ∈ l p loc(r; x) such that [∫ t+1 t ‖f(sn + s,u) − g(s,u)‖ pds ]1/p → 0, and [∫ t+1 t ‖g(s − sn,u) − f(s,u)‖ pds ]1/p → 0 as n → ∞ pointwise on r for each u ∈ y. the collection of those sp-almost automorphic functions f : r × y 7→ x will be denoted by asp(r × y). 2.3 pseudo almost automorphy the notion of pseudo almost automorphy is a new notion due to liang, xiao and zhang [47, 35]. definition 2.17. a function f ∈ c(r, x) is called pseudo almost automorphic if it can be expressed as f = h + ϕ, where h ∈ aa(x) and ϕ ∈ pap0(x). the collection of such functions will be denoted by paa(x). obviously, the following inclusions hold: ap(x) ⊂ pap(x) ⊂ paa(x) and ap(x) ⊂ aa(x) ⊂ paa(x). cubo 13, 1 (2011) pseudo almost automorphic solutions 79 definition 2.18. a function f ∈ c(r × y, x) is said to be pseudo almost automorphic if it can be expressed as f = g + φ, where g ∈ aa(r × y) and ϕ ∈ aa0(r × y). the collection of such functions will be denoted by paa(r × y). a substantial result is the next theorem, which is due to liang et al. [47]. theorem 2.19. [47] the space paa(x) equipped with the sup norm ‖ · ‖∞ is a banach space. we also have the following composition result. theorem 2.20. [47] if f : r × y 7→ x belongs to paa(r × y) and if x 7→ f(t,x) is uniformly continuous on any bounded subset k of y for each t ∈ r, then the function defined by h(t) = f(t,ϕ(t)) belongs to paa(x) provided ϕ ∈ paa(y). 3 sp-pseudo almost automorphy this section is devoted to the notion of sp-pseudo almost automorphy. such a concept is completely new and is due to diagana [17]. definition 3.1. [17] a function f ∈ bsp(x) is called sp-pseudo almost automorphic (or stepanovlike pseudo almost automorphic) if it can be expressed as f = h + ϕ, where hb ∈ aa ( lp((0, 1), x) ) and ϕb ∈ pap0 ( lp((0, 1), x) ) . the collection of such functions will be denoted by paap(x). clearly, a function f ∈ l p loc(r, x) is said to be s p-pseudo almost automorphic if its bochner transform fb : r → lp((0, 1), x) is pseudo almost automorphic in the sense that there exist two functions h,ϕ : r 7→ x such that f = h + ϕ, where hb ∈ aa(lp((0, 1), x)) and ϕb ∈ pap0(l p((0, 1), x)). theorem 3.2. [17] if f ∈ paa(x), then f ∈ paap(x) for each 1 ≤ p < ∞. in other words, paa(x) ⊂ paap(x). obviously, the following inclusions hold: ap(x) ⊂ pap(x) ⊂ paa(x) ⊂ paap(x), ap(x) ⊂ aa(x) ⊂ paa(x) ⊂ paap(x). theorem 3.3. [17] the space paap(x) equipped with the norm ‖ · ‖sp is a banach space. 80 toka diagana cubo 13, 1 (2011) definition 3.4. a function f : r × y 7→ x, (t,u) 7→ f(t,u) with f(·,u) ∈ lp(r, x) for each u ∈ y, is said to be sp-pseudo almost automorphic if there exists two functions h, φ : r×y 7→ x such that f = h + φ, where hb ∈ aa(r × lp((0, 1), x)) and φb ∈ aa0(r × l p((0, 1), x)). the collection of those sp-pseudo almost automorphic functions will be denoted by paap(r × y). we have the following composition theorems. theorem 3.5. let f : r × x 7→ x be a sp-pseudo almost automorphic function. suppose that f(t,u) is lipschitzian in u ∈ x uniformly in t ∈ r, that is there exists l > 0 such ‖f(t,u) − f(t,v)‖ ≤ l.‖u − v‖ (3.1) for all t ∈ r, (u,v) ∈ x × x. if φ ∈ paap(x), then γ : r → x defined by γ(·) := f(·,φ(·)) belongs to paap(x). proof. let f = h + φ, where hb ∈ aa(r × lp((0, 1), x)) and φb ∈ aa0(r × l p((0, 1), x)). similarly, let φ = φ1 + φ2, where φ b 1 ∈ aa(l p((0, 1), x)) and φb2 ∈ pap0(l p((0, 1), x)), that is, lim t →∞ 1 2t ∫ t −t (∫ t+1 t ‖ϕ2(σ)‖ pdσ )1/p dt = 0 (3.2) for all t ∈ r. it is obvious to see that f b(·,φ(·)) : r 7→ lp((0, 1), x). now decompose f b as follows f b(·,φ(·)) = hb(·,φ1(·)) + f b(·,φ(·)) − hb(·,φ1(·)) = hb(·,φ1(·)) + f b(·,φ(·)) − f b(·,φ1(·)) + φ b(·,φ1(·)). using the theorem of composition of almost automorphic functions, it is easy to see that hb(·,φ1(·)) ∈ aa(l p((0, 1), x)). now, set gb(·) := f b(·,φ(·)) − f b(·,φ1(·)). clearly, gb(·) ∈ pap0(l p((0, 1), x)). indeed, we have ∫ t+1 t ‖g(σ)‖pdσ = ∫ t+1 t ‖f(σ,φ(σ)) − f(σ,φ1(σ))‖ pdσ ≤ lp ∫ t+1 t ‖φ(σ) − φ1(σ)‖ pdσ = lp ∫ t+1 t ‖φ2(σ)‖ pdσ, cubo 13, 1 (2011) pseudo almost automorphic solutions 81 and hence for t > 0, 1 2t ∫ t −t (∫ t+1 t ‖g(σ)‖pdσ )1/p dt ≤ l 2t ∫ t −t (∫ t+1 t ‖φ2(σ)‖ pdσ )1/p dt. now using eq. (3.2), it follows that lim t →∞ 1 2t ∫ t −t (∫ t+1 t ‖g(σ)‖pdσ )1/p dt = 0. using the theorem of composition of functions of pap(lp((0, 1), x)) (see [18]) it is easy to see that φb(·,φ1(·)) ∈ pap0(l p((0, 1), x)). theorem 3.6. let f = h + φ : r × x 7→ x be a sp-pseudo almost automorphic function, where hb ∈ aa(r × lp((0, 1), x)) and φb ∈ aa0(r × l p((0, 1), x)). suppose that f(t,u) and φ(t,x) are uniformly continuous in every bounded subset k ⊂ x uniformly for t ∈ r. if g ∈ paap(x), then γ : r → x defined by γ(·) := f(·,g(·)) belongs to paap(x). proof. let f = h + φ, where hb ∈ aa(r × lp((0, 1), x)) and φb ∈ aa0(r × l p((0, 1), x)). similarly, let g = φ1 + φ2, where φ b 1 ∈ aa(l p((0, 1), x)) and φb2 ∈ pap0(l p((0, 1), x)). it is obvious to see that f b(·,g(·)) : r 7→ lp((0, 1), x). now decompose f b as follows f b(·,g(·)) = hb(·,φ1(·)) + f b(·,g(·)) − hb(·,φ1(·)) = hb(·,φ1(·)) + f b(·,g(·)) − f b(·,φ1(·)) + φ b(·,φ1(·)). using the theorem of composition of almost automorphic functions, it is easy to see that hb(·,φ1(·)) ∈ aa(l p((0, 1), x)). now, set gb(·) := f b(·,g(·)) − f b(·,φ1(·)). we claim that gb(·) ∈ pap0(l p((0, 1), x)). first of all, note that the uniformly continuity of f on bounded subsets k ⊂ x yields the uniform continuity of its bohr transform f b on bounded subsets of x. since both g,φ1 are bounded functions it follows that there exists k ⊂ x a bounded subset such that g(σ),φ1(σ) ∈ k for each σ ∈ r. now from the uniform continuity of f b on bounded subsets of x, it obviously follows that f b is uniformly continuous on k uniformly for each t ∈ r. therefore for every ε > 0 there exists δ > 0 such that for all x,y ∈ k with ‖x − y ‖ < δ yield ‖f b(σ,x) − f b(σ,y )‖ < ε for all σ ∈ r. using the proof of the composition theorem [47, theorem 2.4] (applied to f b) it follows lim t →∞ 1 2t ∫ t −t (∫ t+1 t ‖g(σ)‖pdσ )1/p dt = 0. using the theorem of composition [47, theorem 2.4] for functions of pap0(l p((0, 1), x)) it is easy to see that φb(·,φ1(·)) ∈ pap0(l p((0, 1), x)). 82 toka diagana cubo 13, 1 (2011) 4 evolution families this section is devoted to some preliminary results needed in the sequel. we basically use the same setting as in [7] with slight adjustments. throughout the rest of this paper, (x,‖ · ‖) stands for a banach space, a(t) for t ∈ r is a family of closed linear operators on d = d(a(t)), which is independent of t, satisfying the so-called acquistapace and terreni conditions (hypothesis (h.1)). moreover, the operators a(t) are not necessarily densely defined. the linear operators b,c are (possibly unbounded) defined on x such that a(t) + b + c is not trivial for each t ∈ r. the functions, f : r × x 7→ xtβ (0 < α < β < 1), g : r × x 7→ x are respectively jointly continuous satisfying some additional assumptions. if l is a linear operator on the banach space x, then: • d(l) stands for its domain; • ρ(l) stands for its resolvent; • σ(l) stands for its spectrum; • n(l) stands for its null-space or kernel; and • r(l) stands for its range. moreover, one sets r(λ,l) := (λi − l)−1 for all λ ∈ ρ(a). furthermore, we set q = i − p for a projection p . hypothesis (h.1). the family of closed linear operators a(t) for t ∈ r on x with domain d(a(t)) (possibly not densely defined) satisfy the so-called acquistapace-terreni conditions, that is, there exist constants ω ∈ r, θ ∈ ( π 2 ,π ) , l > 0 and µ,ν ∈ (0, 1] with µ + ν > 1 such that σθ ∪ { 0 } ⊂ ρ ( a(t) − ω ) ∋ λ, ∥∥∥r ( λ,a(t) − ω )∥∥∥ ≤ k 1 + |λ| for all t ∈ r, (4.1) and ∥∥∥ ( a(t) − ω ) r ( λ,a(t) − ω )[ r ( ω,a(t) ) − r ( ω,a(s) )]∥∥∥ ≤ l |t − s|µ |λ|−ν (4.2) for t,s ∈ r, λ ∈ σθ := { λ ∈ c \ {0} : | arg λ| ≤ θ } . note that in the particular case when a(t) has a constant domain d = d(a(t)), it is wellknown [4, 42] that eq. (4.2) can be replaced with the following: there exist constants l and 0 < µ ≤ 1 such that ∥∥∥ ( a(t) − a(s) ) r ( ω,a(r) )∥∥∥ ≤ l|t − s|µ, s,t,r ∈ r. cubo 13, 1 (2011) pseudo almost automorphic solutions 83 it should mentioned that (h.1) was introduced in the literature by acquistapace and terreni in [2, 3] for ω = 0. among other things, the acquistapace-terreni conditions ensure that there exists a unique evolution family u = {u(t,s) : t,s ∈ r such that t ≥ s} on x associated with a(t) such that u(t,s)x ⊆ d(a(t)) for all t,s ∈ r with t ≥ s, and (a) u(t,s)u(s,r) = u(t,r) for t,s ∈ r such that t ≥ s ≥ r; (b) u(t,t) = i for t ∈ r where i is the identity operator of x; (c) (t,s) 7→ u(t,s) ∈ b(x) is continuous for t > s; it should also be mentioned that the above-mentioned proprieties were mainly established in [1, theorem 2.3] and [50, theorem 2.1], see also [3, 49]. one says that an evolution family u has an exponential dichotomy (or is hyperbolic) if there are projections p(t) (t ∈ r) that are uniformly bounded and strongly continuous in t and constants δ > 0 and n ≥ 1 such that (f) u(t,s)p(s) = p(t)u(t,s); (g) the restriction uq(t,s) : q(s)x → q(t)x of u(t,s) is invertible (we then set ũq(s,t) := uq(t,s) −1); and (h) ∥∥∥u(t,s)p(s) ∥∥∥ ≤ ne−δ(t−s) and ∥∥∥ũq(s,t)q(t) ∥∥∥ ≤ ne−δ(t−s) for t ≥ s and t,s ∈ r. according to [44], the following sufficient conditions are required for a(t) to have exponential dichotomy. (i) let (a(t),d(t))t∈r be generators of analytic semigroups on x of the same type. suppose that d(a(t)) ≡ d(a(0)), a(t) is invertible, sup t,s∈r ∥∥∥a(t)a(s)−1 ∥∥∥ is finite, and ∥∥∥a(t)a(s)−1 − i ∥∥∥ ≤ l0|t − s|µ for t,s ∈ r and constants l0 ≥ 0 and 0 < µ ≤ 1. (j) the semigroups (eτ a(t))τ≥0, t ∈ r, are hyperbolic with projection pt and constants n,δ > 0. moreover, let ∥∥∥a(t)eτ a(t)pt ∥∥∥ ≤ ψ(τ) 84 toka diagana cubo 13, 1 (2011) and ∥∥∥a(t)eτ aq(t)qt ∥∥∥ ≤ ψ(−τ) for τ > 0 and a function ψ such that r ∋ s 7→ ϕ(s) := |s|µψ(s) is integrable with l0‖ϕ‖l1(r) < 1. this setting requires some estimates related to u(t,s). for that, we introduce the interpolation spaces for a(t). we refer the reader to the following excellent books [4], [24], and [33] for proofs and further information on theses interpolation spaces. let a be a sectorial operator on x (in assumption (h.1), replace a(t) with a) and let α ∈ (0, 1). define the real interpolation space x a α := { x ∈ x : ∥∥∥x ∥∥∥ a α := supr>0 ∥∥∥rα ( a − ω ) r ( r,a − ω ) x ∥∥∥ < ∞ } , which, by the way, is a banach space when endowed with the norm ∥∥∥ · ∥∥∥ a α . for convenience we further write x a 0 := x, ∥∥∥x ∥∥∥ a 0 := ∥∥∥x ∥∥∥, xa1 := d(a) and ∥∥∥x ∥∥∥ a 1 := ∥∥∥(ω − a)x ∥∥∥. moreover, let x̂a := d(a) of x. in particular, we have the following continuous embedding d(a) →֒ xaβ →֒ d((ω − a) α) →֒ xaα →֒ x̂ a →֒ x, (4.3) for all 0 < α < β < 1, where the fractional powers are defined in the usual way. in general, d(a) is not dense in the spaces xaα and x. however, we have the following continuous injection x a β →֒ d(a) ‖·‖aα (4.4) for 0 < α < β < 1. given the family of linear operators a(t) for t ∈ r, satisfying (h.1), we set x t α := x a(t) α , x̂ t := x̂a(t) for 0 ≤ α ≤ 1 and t ∈ r, with the corresponding norms. then the embedding in eq. (4.3) holds with constants independent of t ∈ r. these interpolation spaces are of class jα ([33, definition 1.1.1 ]) and hence there is a constant c(α) such that ∥∥∥y ∥∥∥ t α ≤ c(α) ∥∥∥y ∥∥∥ 1−α∥∥∥a(t)y ∥∥∥ α , y ∈ d(a(t)). (4.5) we have the following fundamental estimates for the evolution family u. its proof was given in [7] though for the sake of clarity, we reproduce it here. cubo 13, 1 (2011) pseudo almost automorphic solutions 85 proposition 4.1. for x ∈ x, 0 ≤ α ≤ 1 and t > s, the following hold: (i) there is a constant c(α), such that ∥∥∥u(t,s)p(s)x ∥∥∥ t α ≤ c(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥. (4.6) (ii) there is a constant m(α), such that ∥∥∥ũq(s,t)q(t)x ∥∥∥ s α ≤ m(α)e−δ(t−s) ∥∥∥x ∥∥∥. (4.7) proof. (i) using (4.5) we obtain ∥∥∥u(t,s)p(s)x ∥∥∥ t α ≤ c(α) ∥∥∥u(t,s)p(s)x ∥∥∥ 1−α∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α ≤ c(α) ∥∥∥u(t,s)p(s)x ∥∥∥ 1−α∥∥∥a(t)u(t,t − 1)u(t − 1,s)p(s)x ∥∥∥ α ≤ c(α) ∥∥∥u(t,s)p(s)x ∥∥∥ 1−α∥∥∥a(t)u(t,t − 1) ∥∥∥ α∥∥∥u(t − 1,s)p(s)x ∥∥∥ α ≤ c(α)n′ e−δ(t−s)(1−α)e−δ(t−s−1)α ∥∥∥x ∥∥∥ ≤ c(α)(t − s)−αe− δ 2 (t−s)(t − s)αe− δ 2 (t−s) ∥∥∥x ∥∥∥ for t − s ≥ 1 and x ∈ x. since (t − s)αe− δ 2 (t−s) → 0 as t → ∞ it easily follows that ∥∥∥u(t,s)p(s)x ∥∥∥ t α ≤ c(α)(t − s)−αe− δ 2 (t−s) ∥∥∥x ∥∥∥. if 0 < t − s ≤ 1, we have ∥∥∥u(t,s)p(s)x ∥∥∥ t α ≤ c(α) ∥∥∥u(t,s)p(s)x ∥∥∥ 1−α∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α ≤ c(α) ∥∥∥u(t,s)p(s)x ∥∥∥ 1−α∥∥∥a(t)u(t, t + s 2 )u( t + s 2 ,s)p(s)x ∥∥∥ α ≤ c(α) ∥∥∥u(t,s)p(s)x ∥∥∥ 1−α∥∥∥a(t)u(t, t + s 2 ) ∥∥∥ α∥∥∥u( t + s 2 ,s)p(s)x ∥∥∥ α ≤ c(α)ne−δ(t−s)(1−α)2α(t − s)−αe− δα 2 (t−s) ∥∥∥x ∥∥∥ ≤ c(α)ne− δ 2 (t−s)(1−α)2α(t − s)−αe− δα 2 (t−s) ∥∥∥x ∥∥∥ ≤ c(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥, and hence ∥∥∥u(t,s)p(s)x ∥∥∥ t α ≤ c(α)(t − s)−αe− δ 2 (t−s) ∥∥∥x ∥∥∥ for t > s. 86 toka diagana cubo 13, 1 (2011) (ii) ∥∥∥ũq(s,t)q(t)x ∥∥∥ s α ≤ c(α) ∥∥∥ũq(s,t)q(t)x ∥∥∥ 1−α∥∥∥a(s)ũq(s,t)q(t)x ∥∥∥ α ≤ c(α) ∥∥∥ũq(s,t)q(t)x ∥∥∥ 1−α∥∥∥a(s)q(s)ũq(s,t)q(t)x ∥∥∥ α ≤ c(α) ∥∥∥ũq(s,t)q(t)x ∥∥∥ 1−α∥∥∥a(s)q(s) ∥∥∥ α∥∥∥ũq(s,t)q(t)x ∥∥∥ α ≤ c(α)ne−δ(t−s)(1−α) ∥∥∥a(s)q(s) ∥∥∥ α e−δ(t−s)α ∥∥∥x ∥∥∥ ≤ m(α)e−δ(t−s) ∥∥∥x ∥∥∥. in the last inequality we have used that ∥∥∥a(s)q(s) ∥∥∥ ≤ c for some constant c ≥ 0, see e.g. [46, proposition 3.18]. in addition to above, we also need the following assumptions: hypothesis (h.2). the evolution family u generated by a(·) has an exponential dichotomy with constants n,δ > 0 and dichotomy projections p(t) for t ∈ r. moreover, 0 ∈ ρ(a(t)) for each t ∈ r and the following holds sup t,s∈r ∥∥∥a(s)a−1(t) ∥∥∥ b(xα,x) < c0. (4.8) remark 4.2. note that eq. (4.8) is satisfied in many cases in the literature. in particular, it holds when a(t) = d(t)a where a : d(a) ⊂ x 7→ x is any closed linear operator such that 0 ∈ ρ(a) and d : r 7→ r with inf t∈r |d(t)| > 0 and sup t∈r |d(t)| < ∞. hypothesis (h.3). there exists 0 ≤ α < β < 1 such that x t α = xα and x t β = xβ for all t ∈ r, with uniform equivalent norms. if 0 ≤ α < β < 1, then we let k(α) and c′ denote respectively the bounds of the embedding xβ →֒ xα and xα →֒ x, that is, ∥∥∥u ∥∥∥ α ≤ k(α) ∥∥∥u ∥∥∥ β for each u ∈ xβ and ∥∥∥u ∥∥∥ ≤ c′ ∥∥∥u ∥∥∥ α for each u ∈ xα cubo 13, 1 (2011) pseudo almost automorphic solutions 87 5 main results to study the existence and uniqueness of pseudo almost automorphic solutions to eq. (1.1) we first introduce the notion of bounded solution. definition 5.1. a function u : r 7→ xα is said to be a bounded solution to eq. (1.1) provided that the function s → a(s)u(t,s)p(s)f(s,bu(s)) is integrable on (−∞, t), and the function s → a(s)u(t,s)q(s)f(s,bu(s)) is integrable on (t,∞) for each t ∈ r, and u(t) = −f(t,bu(t)) − ∫ t −∞ a(s)u(t,s)p(s)f(s,bu(s))ds + ∫ ∞ t a(s)u(t,s)q(s)f(s,bu(s))ds + ∫ t −∞ u(t,s)p(s)g(s,cu(s))ds − ∫ ∞ t u(t,s)q(s)g(s,cu(s))ds for each ∀t ∈ r. throughout the rest of the paper we denote by γ1, γ2, γ3, and γ4, the nonlinear integral operators defined by (γ1u)(t) := ∫ t −∞ a(s)u(t,s)p(s)f(s,bu(s))ds, (γ2u)(t) := ∫ ∞ t a(s)u(t,s)q(s)f(s,bu(s))ds, (γ3u)(t) := ∫ t −∞ u(t,s)p(s)g(s,cu(s))ds, (γ4u)(t) := ∫ ∞ t u(t,s)q(s)g(s,cu(s))ds. in this paper we suppose that the linear operators b,c : xα 7→ x are bounded and set ̟ := max (∥∥∥b ∥∥∥ b(xα,x) , ∥∥∥c ∥∥∥ b(xα,x) ) . to study eq. (1.1), in addition to the previous assumptions, we require that p > 1, 1 p + 1 q = 1, and that the following additional assumptions hold: (h.4) r(ω,a(·))u ∈ aa(xα) for each u ∈ x. moreover, for any sequence of real numbers (τ ′ n)n∈n there exist a subsequence (τn)n∈n and g(·, ·) such that g(t,s)p(s)u = lim n→∞ a(s + τn)u(t + τn,s + τn)p(s + τn)u and a(s)u(t,s)p(s)u = lim n→∞ g(t − τn,s − τn)p(s − τn)u for all t,s ∈ r and u ∈ xα. 88 toka diagana cubo 13, 1 (2011) (h.5) let 0 ≤ α < β < 1, and f : r×x 7→ xβ , g : r×x 7→ x are s p-pseudo almost automorphic. moreover, the functions f,g are uniformly lipschitz with respect to the second argument in the following sense: there exists k > 0 such that ∥∥∥f(t,u) − f(t,v) ∥∥∥ β ≤ k ∥∥∥u − v ∥∥∥, and ∥∥∥g(t,u) − g(t,v) ∥∥∥ ≤ k ∥∥∥u − v ∥∥∥ for all u,v ∈ x and t ∈ r. the proof of our main result requires the following key technical lemma. lemma 5.2. under assumptions (h.1)—(h.3), then there exist constant m(α,β),n(α) > 0 such that ∥∥∥a(s)ũq(t,s)q(s)x ∥∥∥ α ≤ m(α,β)eδ(s−t) ∥∥∥x ∥∥∥ β for t ≤ s, (5.1) ∥∥∥a(s)u(t,s)p(s)x ∥∥∥ α ≤ n(α)(t − s)−αe− δ 2 (t−s) ∥∥∥x ∥∥∥ β , for t > s. (5.2) proof. let x ∈ xβ . since the restriction of a(s) to r(q(s)) is a bounded linear operator it follows that ∥∥∥a(s)ũq(t,s)q(s)x ∥∥∥ α ≤ ck(α) ∥∥∥ũq(t,s)q(s)x ∥∥∥ β ≤ ck(α)m(β)eδ(s−t) ∥∥∥x ∥∥∥ ≤ m(α,β)eδ(s−t) ∥∥∥x ∥∥∥ β for t ≤ s by using eq. (4.7). similarly, for each x ∈ xβ , using eq. (4.8), we obtain ∥∥∥a(s)u(t,s)p(s)x ∥∥∥ α = ∥∥∥a(s)a(t)−1a(t)u(t,s)p(s)x ∥∥∥ α ≤ ∥∥∥a(s)a(t)−1 ∥∥∥ b(xα,x) ∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α ≤ c0 ∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α for t ≥ s. first of all, note that ∥∥∥a(t)u(t,s) ∥∥∥ ≤ k(t − s)−1 for all t,s such that 0 < t − s ≤ 1. now, let t − s ≥ 1. then, using eq. (4.6), we obtain cubo 13, 1 (2011) pseudo almost automorphic solutions 89 ∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α = ∥∥∥a(t)u(t,t − 1)u(t − 1,s)p(s)x ∥∥∥ α ≤ ∥∥∥a(t)u(t,t − 1) ∥∥∥ b(xα,x) ∥∥∥u(t − 1,s)p(s)x ∥∥∥ α ≤ kc(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥ ≤ kk′c(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥ α ≤ kk′k(α)c(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥ β ≤ n′(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥ β . now, let 0 < t − s ≤ 1. again, using eq. (4.6), we obtain ∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α = ∥∥∥a(t)u(t, t + s 2 )u( t + s 2 ,s)p(s)x ∥∥∥ α ≤ ∥∥∥a(t)u(t, t + s 2 ) ∥∥∥ b(xα,x) ∥∥∥u( t + s 2 ,s)p(s)x ∥∥∥ α ≤ kc(α)e− δ 4 (t−s)2α(t − s)−α‖x‖ ≤ kk′c(α)e− δ 4 (t−s)2α(t − s)−α ∥∥∥x ∥∥∥ α ≤ kk′k(α)c(α)e− δ 4 (t−s)2α(t − s)−α ∥∥∥x ∥∥∥ β ≤ n′′(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥ β . therefore, ∥∥∥a(t)u(t,s)p(s)x ∥∥∥ α ≤ n(α)e− δ 2 (t−s)(t − s)−α ∥∥∥x ∥∥∥ β for all t,s ∈ r with t ≥ s. lemma 5.3. under assumptions (h.1)—(h.5) and if n(α,q,δ) := ∞∑ n=1 [∫ n n−1 e−q δ 2 ss−qαds ]1/q < ∞, then the integral operators γ3 and γ4 defined above map paa(xα) into itself. proof. let u ∈ paap(xα). since c ∈ b(xα, x) then cu(·) ∈ paa p(x). setting h(t) = g(t,c(u(t)) and using the theorem of composition of sp-pseudo almost automorphic functions (theorem 3.5) it follows that h ∈ paap(x). now write h = φ+ζ where φb belongs to aa(lp(0, 1), xα)) and ζb ∈ pap0(l p(0, 1), xα)). define for all n = 1, 2, ..., the sequence of integral operators 90 toka diagana cubo 13, 1 (2011) φn(t) = ∫ n n−1 u(t,t − s)p(t − s)φ(t − s)ds, and ψn(t) = ∫ n n−1 u(t,t − s)p(t − s)ζ(t − s)ds for each t ∈ r. letting r = t − s it follows that φn(t) = ∫ t−n+1 t−n u(t,r)p(r)φ(r)dr, and hence from the hölder’s inequality and the estimate eq. (4.6) it follows that ∥∥∥φn(t) ∥∥∥ α ≤ ∫ t−n+1 t−n c(α)e− δ 2 (t−r)(t − r)−α ∥∥∥φ(r) ∥∥∥dr ≤ c′ ∫ t−n+1 t−n c(α)e− δ 2 (t−r)(t − r)−α ∥∥∥φ(r) ∥∥∥ α dr ≤ c′c(α) [∫ n n−1 e−q δ 2 ss−qαds ]1/q∥∥∥φ ∥∥∥ sp . from the assumption that n(α,q,δ) is finite we then deduce from weirstrass theorem that the series d(t) := ∞∑ n=1 φn(t) is uniformly convergent on r. moreover, d ∈ c(r, xα) and ∥∥∥d(t) ∥∥∥ α ≤ ∞∑ n=1 ∥∥∥φn(t) ∥∥∥ α ≤ c′c(α)n(α,q,δ) ∥∥∥φ ∥∥∥ sp for all t ∈ r. let us show that φn ∈ aa(xα) for each n = 1, 2, 3, ... indeed, since φ ∈ as p(xα), for every sequence of real numbers (τ′n)n∈n there exist a subsequence (τnk )k∈n and a function φ̂ such that ∫ t+1 t ∥∥∥φ̂(s) − φ(s + τnk ) ∥∥∥ p α ds → 0 and ∫ t+1 t ∥∥∥φ̂(s − τnk ) − φ(s) ∥∥∥ p α ds → 0 as k → ∞ pointwise in r. define for all n = 1, 2, 3, ..., the sequence of integral operators φ̂n(t) = ∫ n n−1 u(t,t − s)p(t − s)φ̂(t − s)ds for all t ∈ r. now cubo 13, 1 (2011) pseudo almost automorphic solutions 91 φ(t + τnk ) − φ̂(t) = ∫ n n−1 u(t,t + τnk − s)p(t + τnk − s)φ(t + τnk − s)ds − ∫ n n−1 u(t,t − s)p(t − s)φ̂(t − s)ds = ∫ n n−1 u(t,t + τnk − s)p(t + τnk − s)φ(t + τnk − s)ds + ∫ n n−1 u(t,t + τnk − s)p(t + τnk − s)φ̂(t − s)ds − ∫ n n−1 u(t,t + τnk − s)p(t + τnk − s)φ̂(t − s)ds − ∫ n n−1 u(t,t − s)p(t − s)φ̂(t − s)ds = ∫ n n−1 u(t,t + τnk − s)p(t + τnk − s) [ φ(t + τnk − s) − φ̂(t − s) ] ds + ∫ n n−1 [ u(t,t + τnk − s)p(t + τnk − s) − u(t,t − s)p(t − s) ] φ̂(t − s)ds. using lebesgue dominated convergence theorem, one can easily see that ∥∥∥ ∫ n n−1 u(t,t + τnk − s)p(t + τnk − s) [ φ(t + τnk − s) − φ̂(t − s) ] ds ∥∥∥ α → 0 as k → ∞, t ∈ r. similarly, using [9, proposition 3.3] it follows that ∥∥∥ ∫ n n−1 [ u(t,t + τnk − s)p(t + τnk − s) − u(t,t − s)p(t − s) ] φ̂(t − s)ds ∥∥∥ α → 0 as k → ∞, t ∈ r. thus φ̂n(t) = lim k→∞ φn(t + τnk ), t ∈ r. similarly, one can easily see that φn(t) = lim k→∞ φ̂n(t − τnk ) for all t ∈ r and n = 1, 2, 3, ... therefore the sequence φn ∈ aa(xα) for each n = 1, 2, ... and hence its uniform limit d ∈ aa(xα). to complete the proof for γ3, we have to show that ψn ∈ pap0(xα). letting r = t − s it follows that φn(t) = ∫ t−n+1 t−n u(t,r)p(r)ζ(r)dr, 92 toka diagana cubo 13, 1 (2011) and hence from the hölder’s inequality and the estimate eq. (4.6) it follows that ∥∥∥φn(t) ∥∥∥ α ≤ ∫ t−n+1 t−n c(α)e− δ 2 (t−r)(t − r)−α ∥∥∥ζ(r) ∥∥∥dr ≤ c′ ∫ t−n+1 t−n c(α)e− δ 2 (t−r)(t − r)−α ∥∥∥ζ(r) ∥∥∥ α dr ≤ c′c(α) [∫ n n−1 e−q δ 2 ss−qαds ]1/q[∫ t−n+1 t−n ∥∥∥ζ(s) ∥∥∥ p α ds ]1/p , and hence ψn ∈ pap0(xα) as ζ b ∈ pap0(l p((0, 1), xα). from the assumption that n(α,q,δ) is finite we then deduce from weirstrass theorem that the series e(t) := ∞∑ n=1 ψn(t) is uniformly convergent on r. moreover, e ∈ c(r, xα) and ∥∥∥e(t) ∥∥∥ α ≤ ∞∑ n=1 ∥∥∥φn(t) ∥∥∥ α ≤ c′c(α)n(α,q,δ) ∥∥∥φ ∥∥∥ sp for all t ∈ r. consequently, the uniform limit e(t) = ∞∑ n=1 ψn(t) belongs to pap0(xα). therefore γ3 maps paa(xα) into itself. the proof for γ4 is similar to that of γ3 except that we make use of approximate eq. (4.7) rather than eq. (4.6). lemma 5.4. under assumptions (h.1)—(h.5) and if n(α,q,δ) := ∞∑ n=1 [∫ n n−1 e−q δ 2 ss−qαds ]1/q < ∞, then the integral operators γ1 and γ2 defined above map paa(xα) into itself. proof. let u ∈ paap(xα). since b ∈ b(xα, x) then bu(·) ∈ paa p(x). setting h(t) = f(t,bu(t)) and using the theorem of composition of sp-pseudo almost automorphic functions (theorem 3.5) it follows that h ∈ paap(x). now write h = φ+ζ where φb belongs to aa(lp(0, 1), xα)) and ζb ∈ pap0(l p(0, 1), xα)). define for all n = 1, 2, ..., the sequence of integral operators φn(t) = ∫ n n−1 a(t − s)u(t,t − s)p(t − s)φ(t − s)ds and ψn(t) = ∫ n n−1 a(t − s)u(t,t − s)p(t − s)ζ(t − s)ds cubo 13, 1 (2011) pseudo almost automorphic solutions 93 for each t ∈ r. letting r = t − s it follows that φn(t) = ∫ t−n+1 t−n a(r)u(t,r)p(r)φ(r)dr, and hence from the hölder’s inequality and the estimate eq. (5.2) it follows that ∥∥∥φn(t) ∥∥∥ α ≤ ∫ t−n+1 t−n n(α)e− δ 2 (t−r)(t − r)−α ∥∥∥φ(r) ∥∥∥ β dr ≤ c′ ∫ t−n+1 t−n n(α)e− δ 2 (t−r)(t − r)−α ∥∥∥φ(r) ∥∥∥ β dr ≤ c′n(α) [∫ n n−1 e−q δ 2 ss−qαds ]1/q∥∥∥φ ∥∥∥ sp . from the assumption that n(α,q,δ) is finite we then deduce from weirstrass theorem that the series d(t) := ∞∑ n=1 φn(t) is uniformly convergent on r. moreover, d ∈ c(r, xα) and ∥∥∥d(t) ∥∥∥ α ≤ ∞∑ n=1 ∥∥∥φn(t) ∥∥∥ α ≤ c′n(α)n(α,q,δ) ∥∥∥φ ∥∥∥ sp for all t ∈ r. let us show that φ ∈ aa(xα). indeed, since φ ∈ as p(xα), for every sequence of real numbers (τ′n)n∈n there exist a subsequence (τnk )k∈n and a function φ̂ such that ∫ t+1 t ∥∥∥φ̂(s) − φ(s + τnk ) ∥∥∥ p α ds → 0 and ∫ t+1 t ∥∥∥φ̂(s − τnk ) − φ(s) ∥∥∥ p α ds → 0 as k → ∞ pointwise in r. define for all n = 1, 2, 3, ..., the sequence of integral operators φ̂n(t) = ∫ n n−1 a(t − s)u(t,t − s)p(t − s)φ̂(t − s)ds for all t ∈ r. now φ(t + τnk ) − φ̂(t) = ∫ n n−1 a(t + τnk − s)u(t,t + τnk − s)p(t + τnk − s)φ(t + τnk − s)ds − ∫ n n−1 a(t − s)u(t,t − s)p(t − s)φ̂(t − s)ds = i1k,n(t) + i 2 k,n(t), 94 toka diagana cubo 13, 1 (2011) where i1k,n(t) = ∫ n n−1 a(t + τnk − s)u(t,t + τnk − s)p(t + τnk − s) [ φ(t + τnk − s) − φ̂(t − s) ] ds and i2k,n(t) = ∫ n n−1 [ a(t+τnk −s)u(t,t+τnk −s)p(t+τnk −s) −a(t−s)u(t,t−s)p(t−s) ] φ̂(t−s)ds. using lebesgue dominated convergence theorem, one can easily see that ∥∥∥ ∫ n n−1 a(t + τnk − s)u(t,t + τnk − s)p(t + τnk − s) [ φ(t + τnk − s) − φ̂(t − s) ] ds ∥∥∥ α → 0 as k → ∞ for each t ∈ r. similarly, using (h.4) it follows that ∥∥∥ ∫ n n−1 [ a(t + τnk −s)u(t,t + τnk −s)p(t + τnk −s) −a(t−s)u(t,t−s)p(t−s) ] φ̂(t−s)ds ∥∥∥ α → 0 as k → ∞ for each t ∈ r. thus φ̂n(t) = lim k→∞ φn(t + τnk ), t ∈ r. similarly, one can easily see that φn(t) = lim k→∞ φ̂n(t − τnk ) for all t ∈ r and n = 1, 2, 3, ... therefore the sequence φn ∈ aa(xα) for each n = 1, 2, ... and hence its uniform limit e ∈ aa(xα). to complete the proof for γ1, we have to show that ψn ∈ pap0(xα). letting r = t − s it follows that φn(t) = ∫ t−n+1 t−n a(r)u(t,r)p(r)ζ(r)dr, and hence from the hölder’s inequality and the estimate eq. (4.6) it follows that ∥∥∥φn(t) ∥∥∥ α ≤ ∫ t−n+1 t−n n(α)e− δ 2 (t−r)(t − r)−α ∥∥∥ζ(r) ∥∥∥dr ≤ c′ ∫ t−n+1 t−n n(α)e− δ 2 (t−r)(t − r)−α ∥∥∥ζ(r) ∥∥∥ α dr ≤ c′n(α) [∫ n n−1 e−q δ 2 ss−qαds ]1/q[∫ t−n+1 t−n ∥∥∥ζ(s) ∥∥∥ p α ds ]1/p , cubo 13, 1 (2011) pseudo almost automorphic solutions 95 and hence ψn ∈ pap0(xα) as ζ b ∈ pap0(l p((0, 1), xα). from the assumption that n(α,q,δ) is finite we then deduce from weirstrass theorem that the series e(t) := ∞∑ n=1 ψn(t) is uniformly convergent on r. moreover, e ∈ c(r, xα) and ∥∥∥e(t) ∥∥∥ α ≤ ∞∑ n=1 ∥∥∥φn(t) ∥∥∥ α ≤ c′n(α)n(α,q,δ) ∥∥∥φ ∥∥∥ sp for all t ∈ r. consequently, the uniform limit e(t) = ∞∑ n=1 ψn(t) belongs to pap0(xα). therefore γ1 maps paa(xα) into itself. the proof for γ2u(·) is similar to that of γ1u(·) except that one makes use of eq. (5.1) rather than eq. (5.2). theorem 5.5. suppose assumptions (h.1)—(h.5) hold. moreover, suppose n(α,q,δ) := ∞∑ n=1 [∫ n n−1 e−q δ 2 ss−qαds ]1/q < ∞. then the evolution equation (1.1) has a unique pseudo-almost automorphic mild solution whenever k is small enough. proof. consider the nonlinear operator m defined on paa(xα) by mu(t) = −f(t,bu(t)) − ∫ t −∞ a(s)u(t,s)p(s)f(s,bu(s))ds + ∫ ∞ t a(s)u(t,s)q(s)f(s,bu(s))ds + ∫ t −∞ u(t,s)p(s)g(s,cu(s))ds − ∫ ∞ t u(t,s)q(s)g(s,cu(s))ds for each t ∈ r. as we have previously seen, for every u ∈ paa(xα), f(·,bu(·)) ∈ paa(xβ ) ⊂ paa(xα). in view of lemma 5.3 and lemma 5.4, it follows that m maps paa(xα) into itself. to complete the proof one has to show that m has a unique fixed-point. let v,w ∈ paa(xα). it is not difficult to see that ∥∥∥mv − mw ∥∥∥ ∞,α ≤ kθ . ∥∥∥v − w ∥∥∥ ∞,α , where θ := ̟ [ k(α) + δ−1 ( m(α,β) + m(α) ) + ( n(α) + c(α) ) 21−α γ(1 − α)δα−1 ] , and hence if k is small enough, then eq. (1.1) has a unique solution, which obviously is its only pseudo-almost automorphic solution. 96 toka diagana cubo 13, 1 (2011) example 5.6. let ω ⊂ rn (n ≥ 1) be an open bounded subset with regular boundary γ = ∂ω and let x = l2(ω) equipped with its natural topology ‖ · ‖l2(ω). define the linear operator appearing in eq. (1.2) as follows: a(t)u = a(t,x)∆u for all u ∈ d(a(t)) = h10(ω) ∩ h 2(ω), where a : r × ω 7→ r is a jointly continuous function, almost automorphic and satisfies the following assumptions: (h.6) inf t∈r,x∈ω a(t,x) = m0 > 0, and (h.7) there exists l > 0 and 0 < µ ≤ 1 such that |a(t,x) − a(s,x)| ≤ l|s − t|µ for all t,s ∈ r uniformly in x ∈ ω. first of all, note that in view of the above, sup t∈r,x∈ω a(t,x) < ∞. also, a classical example of a function a satisfying the above-mentioned assumptions is for instance aγ (t,x) = 3 + sin |x|t + sin γ|x|t, where |x| = (x21 + ... + x 2 n ) 1/2 for each x = (x1,x2, ...,xn ) ∈ ω and γ ∈ r \ q. under previous assumptions, it is clear that the operators a(t) defined above are invertible and satisfy acquistapace-terreni conditions. moreover, it can be easily shown that r ( ω,a(·,x)∆ ) ϕ = 1 a(·,x) r ( ω a(·,x) , ∆ ) ϕ ∈ aa(x1/2) for each ϕ ∈ l2(ω) with ∥∥∥r ( ω,a∆ )∥∥∥ b(l2(ω)) ≤ const. |ω| . furthermore, assumptions (h.1)—(h.4) are fulfilled. we require the following assumption: (h.8) let 1 2 < β < 1, and f,g : r × x1/2 7→ xβ be s p-pseudo-almost automorphic functions in t ∈ r uniformly in u ∈ x1/2. moreover, the functions f,g are uniformly lipschitz with respect to the second argument in the following sense: there exists k′ > 0 such that ∥∥∥f(t,ϕ) − f(t,ψ) ∥∥∥ β ≤ k′ ∥∥∥ϕ − ψ ∥∥∥ l2(ω) , and ∥∥∥g(t,ϕ) − g(t,ψ) ∥∥∥ l2(ω) ≤ k′ ∥∥∥ϕ − ψ ∥∥∥ l2(ω) for all ϕ,ψ ∈ l2(ω) and t ∈ r. cubo 13, 1 (2011) pseudo almost automorphic solutions 97 we have theorem 5.7. under previous assumptions including (h.6)-(h.8), then eq. 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[51] s. zaidman, topics in abstract differential equations, pitman research notes in mathematics ser. ii john wiley and sons, new york, 1994–1995. cubo a mathematical journal vol.15, no¯ 01, (151–158). march 2013 on the poisson’s equation −∆u = ∞. carlos cesar aranda 1 blue angel navire research laboratory, rue eddy 113 gatineau qc canada. carloscesar.aranda@gmail.com abstract let ω ⊂ rn be a bounded domain. we proof the existence of a bounded solution of the poisson’s equation −∆u = ∞ on ω. resumen sea ω ⊂ rn un dominio acotado. probamos la existencia de una solución acotada para la ecuación de poisson −∆u = ∞ en ω. keywords and phrases: newtonian potential; nonlinear analysis; celestial mechanics 2010 ams mathematics subject classification: 35j25, 35j60. 1dedicated to professor gaston m n’guérékata on the occasion of his 60th birthday. 152 carlos c. aranda cubo 15, 1 (2013) 1 introduction. in [19] it is stated that le mouvement d’un corps libre consiste dans le mouvement de translation de son centre de gravite et dans le changement de sa position autour de ce point. la recherche du mouvument du centre de gravité se réduit à déterminer le mouvement d’un point sollicité par des forces donnés; et, relativament aux corps célestes, ces forces sont le résultat des attractions de spheroides dont la figure est supposée connu. soient dm une molécule d’un sphéroide; x′, y′, z′ les trois coordennées orthogonales de cette molécule; dm sera de la forme ξdx′dy′dz′, ξ étant fontion de x′, y′, z′. soient encore x, y, z les coordonnées d’un point attir’e, on aura v = ∫ ξdx′dy′dz′ √ (x′ − x)2 + (y′ − y)2 + (x′ − y)2 (1) cette intégrale étant prise relativementà toute l’étendue du sphéroide. ses limites étant indépendantes de x, y, z ainsi que les variables x′, y′, z′, il est clair qu’en differential l’expression de v par rapport àx, y, z il suffira, dans cette différentiation, d’avoir égard au radical que renferme cette expression, et alors il est facile de voir que l’on a 0 = ∂2v ∂x2 + ∂2v ∂y2 + ∂2v ∂y2 . (2) in modern interpretation of potential v of mass distributions, we have v(x, y, z) = ∫ g ξ(x′, y′, z′)dx′dy′dz′ √ (x′ − x)2 + (y′ − y)2 + (x′ − y)2 . (3) where ξ(x′, y′, z′) is the density of a mass distribution in the space x′, y′, z′. then ∇v furnishes the gravity field force and −∆v = 0 on r3 − g. in 1813 poisson found that for a ball g the following equation is valid in the case of constant density ξ(x, y, z′) = ρ −∆u = 4πρ on g poisson’s equation. therefore a natural question is: there exists a solution for poisson’s equation with ρ = ∞?. that kind of solution will be related to gravity potential of bodies with infinite density or black holes. the authors are not aware of a previous result deducing the existence of black holes using newton gravity theory or the gravity potential inside of a black hole. the equation − ∆u = up, (4) for p a nonnegative real number and u > 0in a ball of radius r in r3, with dirichlet boundary conditions was introduced by lane [18] for modelling both the temperature and the density of cubo 15, 1 (2013) on the poisson’s equation −∆u = ∞. 153 mass on the surface of the sun. today the problem (4) is named lane-emden-fowler equation. it was used first in the mid-19th century in the study of internal structure of stars mainly by chandrasekhar [4, 7, 9]. singular lane-emden-fowler equations (p < 0) has been considered in a remarkable pioneering paper by fulks and maybe [10]. eddigton [6] proposed the equation − ∆u = exp(2u) 1+ | x |2 in r3, (5) in order to represent the gravitational potential u of a globular cluster of stars. matukuma [20] introduced the equation − ∆u = ur 1+ | x |2 in r3, (6) where u is the gravity potential, ρ = (2π)−1(1+ | x |2)−1ur is the density and ∫ r3 ρdx is the total mass to study the gravitational potential u of a globular cluster of stars. for the same problem hénon [15] suggested − ∆u =| x |l ur in ω ⊂ r3. (7) black holes solutions means that the gravitational potential of the cluster behaves like 1 r (r =| x |) near the center. peebles [16, 17] gives for the first time a derivation of the steady state distribution of the star near a massive collapsed object. the question of the existence of black hole in a globular cluster is still open (1995). core collapse does occur, for instance using hubble space telescope, bendinelli et.al. [2] reported the first detection of a collapsed core globular cluster in m31. on may 25, 1994 astronomers at nasa headquarters announced the hubble space telescope finding of a supermassive black hole in the heart of the giant galaxy m87, more than 50 million light-years. the equation − ∆ 1 | x − x0 | = 4πδ(x − x0) in r 3, has a deep insight because relate the formulation of the laplace operator and the dirac δ function in a weak sense. the laplace operator with point interaction in r3 given by −∆ + αδ, α ∈ r has been widely study for your applications in quantum physics (see for expample [11]) and in seismic imaging [3]. our purpose in this paper is to give a classical interpretation to the equation − ∆u = ∞ in ω ⊂ rn. (8) we define: 154 carlos c. aranda cubo 15, 1 (2013) definition 1.1. the equation (8) has a classical solution if there exist two non decreasing sequences of functions {uj} ∞ j=1 ∈ c(ω) ∩ c 2(ω) and {fj} ∞ j=1 such that −∆uj = fj in ω, and limj→∞ fj(x) = ∞ for all x ∈ ω and limj→∞ uj(x) = u(x) < ∞ for all x ∈ ω. our main result in this article is as follows. theorem 1.2. let ω be a bounded domain in rn, n ≥ 3. then the problem −∆u = ∞ in ω, (9) has a non negative classical solution u. under the authors knowledge this is the first compactness result dealing with infinite on a non trivial domain (see for example [21] first chapter: direct methods in the calculus of variations). similarly the theory of generalized functions not allow solutions to this kind of problem because every distribution is locally a newtonian potential: theorem 1.3. (page 277 [5]) let ω be an open set of rn, f ∈ d′(ω) and u a solution (in the sense of distributions) of poisson’s equation ∆u = f on ω . then for every bounded open set ω1 with ω1 ⊂ ω there exists f1 ∈ e ′ the space of distributions on rn with compact support, such that f1 = f on ω and u = the newtonian potential of f1 on ω1. moreover if we study this problem using a weak formulation in sobolev’s spaces, the georginash-moser theory cannot be used to derive any comparable compacity result [14]. we will use a non linear singular elliptic approach as in [1, 8, 13] to obtain the result. our strategy is study the auxiliary problem −∆uǫ,m = gm(uǫ) in ω, uǫ,m = ǫ on ∂ω, where gm : (0, ∞) → (0, ∞), m = 1, . . . ∞ is non increasing locally hölder continuous function singular at the origin with the properties gm(s) = g(s) for all s ≥ 1 and limm→∞ gm(s) = ∞ for all s ∈ (0, 1), m = 1, . . . , ∞ and g : (0, ∞) → (0, ∞) is strictly non increasing locally hölder continuous function singular at the origin. our result 1.2 is obtained letting limm→∞,ǫ→0+ uǫ,m. this limit by definition has not weak derivatives of first or second order. cubo 15, 1 (2013) on the poisson’s equation −∆u = ∞. 155 2 auxiliary results theorem 2.1 ([1]). let ω be a smooth bounded domain in rn, n ≥ 3, g : (0, ∞) → (0, ∞) is non increasing locally hölder continuous function (that may be singular at the origin). then the problem −∆uǫ = g(uǫ) in ω, uǫ = ǫ on ∂ω, has a unique positive solution u ∈ c(ω) ∩ c2(ω) for ǫ ≥ 0. moreover uǫ2 ≥ uǫ1 for ǫ2 ≥ ǫ1. we consider the the auxiliary problem −∆um = gm(um) in ω, um = 0 on ∂ω, (10) lemma 2.2. let um be a solution of the equation (10). then um+j ≥ um. proof. suppose that there exists x0 ∈ ω such that um(x0) > um+j(x0). therefore for τ > 0 small enough we have the inequality um(x0) > τ + um+j(x0). then by continuity in ω of the function f(x) = um(x)−τ−um+j(x) there exist a non empty open set ωτ such that f(x) > 0 for all x ∈ ωτ and f = 0 on ∂ωτ. using that um(x) > τ + um+j(x) for all x ∈ ωτ, we deduce gm(um(x)) ≤ gm+j(um(x)) ≤ gm+j(τ + um+j(x)) ≤ gm+j(um+j(x)) for all x ∈ ωτ. then −∆um ≤ −∆(um+j + τ) in ωτ, um = um+j + τ on ∂ωτ. and we obtain um ≤ um+j + τ in ωτ (theorem 3.3 [14]) a contradiction. lemma 2.3. let um be a solution of the equation (10). then gm+j(um+j(x)) ≥ gm(um(x)). proof. suppose that there exists x0 ∈ ω such that gm(um(x0)) > gm+j(um+j(x0)). then by continuity in ω of the function h(x) = gm(um(x)) − gm+j(um+j(x)), there exists ω̂ ⊂ ω such that h(x) > 0 in ω̂ and h(x) = 0 on ∂ω̂ − ∆um ≥ −∆um+j in ω̂, um = um+j on ∂ω̂. we imply um ≥ um+j in ω̂ (theorem 3.3 [14]). therefore gm(um(x)) ≤ gm(um+j(x)) ≤ gm+j(um+j(x)) for all x ∈ ω̂. a contradiction. remark 2.4. in the proof of lemmas 2.2 and 2.3 it is assumed only that gm is a non increasing continuous function. 156 carlos c. aranda cubo 15, 1 (2013) 3 proof proof of theorem 1.2. let us consider the problem −∆v = g(v) in ω, v = 0 on ∂ω. we introduce the equations −∆e = g(e) in ω, e = 1 on ∂ω. −∆w = g(e) in ω, w = 0 on ∂ω. using v ≤ e (see lemma 2.3 and 2.6 in [1]), we infer −∆w = g(e) ≤ g(v) = −∆v in ω, w = 0 = v on ∂ω. then w ≤ v in ω . setting g0 = g and using the auxiliary results with the new sequence {gj} ∞ j=0, we conclude that w ≤ um ≤ e for m = 1, . . . ∞. using lemma 2.2, we infer the existence of limm→∞ um(x) = u(x) for all x ∈ ω. we restrict ourselves to the situation ω = b1(0) where b1(0) is the ball of radius 1 with center at 0. applying the main result of [12] we infer that um is a radial function with ∂um ∂r < 0. therefore u is also a radial non increasing function. we proceed by contradiction, suppose that lim m→∞ gm(um(x)) < ∞ for all 0 ≤‖ x ‖< 1. our first implication is that the function u is strictly non increasing, because if exists (r1, r2) with r2 < 1, and u(r1) = u(r2). then −∆u = 0 on the annulus a(r1, r2). using theorem 9.11 page 235 in [14], we deduce ‖ um ‖h2,p(ω′)≤ c(n, p, ω ′, a(r1, r2))(‖ um ‖lp(a(r1,r2)) + ‖ g(um) ‖lp(a(r1,r2))) ≤ c(n, p, ω′, a(r1, r2))(‖ e ‖lp(a(r1,r2)) + ‖ lim sup m→∞ gm(um(r2)) ‖lp(a(r1,r2))), for all p > n, therefore by sobolev’s embedding theorem (theo. 7.26 [14]) we deduce ‖ um ‖c1,α(ω′)≤ c. we use a non negative test function ϕ with support contained in ω′: 0 = ∫ ω′ ∇u · ∇ϕdx = lim m→∞ ∫ ω′ ∇um · ∇ϕdx = ∫ ω′ gm(um)ϕdx ≥ ∫ ω′ g0(u0)ϕdx > 0. cubo 15, 1 (2013) on the poisson’s equation −∆u = ∞. 157 contradiction, therefore we deduce that u is a strictly non increasing function. moreover using again estimates in theorems 9.11 and 9.12 in [14] we have u ∈ c1,αloc(b1(0)). by assumption lim supr→1 u(r) ≥ 1, therefore u(r) > 1 for 0 ≤ r < 1. by construction there exists 0 < r0 < 1 such that g0(u0(r0)) > g0(1). using lemma 2.3, we derive g0(u0(r0)) ≤ gm(um(r0)). but limm→∞ um(r0) = u(r0) > 1 and therefore for m big enough um(r0) > 1. moreover gm(um(r0)) = g0(um(r0)) < g0(1) because g0 is strictly non increasing. contradiction. it is follows that there exists 0 ≤ r1 < 1 such that lim m→∞ gm(um(r1)) = ∞. now, because um is a radial non increasing function, we infer that gm(um(r1)) ≤ gm(um(r)) for all r1 < r < 1. so lim m→∞ gm(um(r)) = ∞ for all r1 ≤ r < 1. now for ω a bounded domain in rn, n ≥ 3 consider the transformation um( a+x r ). this end the proof. received: october 2012. revised: february 2013. references [1] c. c. aranda and t. godoy. existence and multiplicity of positive solutions for a singular problem associated to the p-laplacian operator. electron. j. diff. eqns., vol. 2004(2004), no. 132, pp. 1-15 [2] o. bendinelli, c. cacciari, s. djorgorski, l. federici, er. ferraro, fi fusi pecci, g. parmeggiani, n. weir and e zaratti, “the first detection of a collapsed core globular cluster in m31”. astrophy. i., 409 (1993), li7-li9. [3] n. bleistein, mathematics of modelling, migration and inversion with gaussian beams. center for wave phenomena, colorado school of mines, 2009. [4] s. chandrasekhar, an introduction to the study of stellar structure. dover, new york, 1957. [5] r. dautray and j. l. lions mathematical analysis and numerical methods for science and technology. vol. 1 physical origins and classical methods. springer verlag 1990. [6] a.s. eddington, the dynamics of a globular stellar system. monthly notices roy. ast. soc., 75 (1915), 366-376. 158 carlos c. aranda cubo 15, 1 (2013) [7] v. r. emden. gaskugeln. teubner, leipzig, 1907. [8] j. hérnandez and f. j. mancebo. singular elliptic and parabolic equations. in handbook of differential equations (ed. m. chipot and p. quittner), vol 3 (elsevier 2006) [9] r. h. fowler. further studies of emden’s and similar differential equations. q. j. math. (oxford series) 2 (1931), 259-288. [10] w. fulks and j. s. maybe. a singular nonlinear equation. osaka j. math. 12 (1960), 1-19. [11] p. kurasov singular and superlinear perturbation hilbert space methods. contemporary mathematics vol. 340, 2004. [12] b. gidas, wei ming ni and l. nirenberg. symmetry and related properties via the maximum principle. comm. math. phys. volume 68 number 3(1979), 209-243. [13] m. ghergu and v. rădulescu. singular elliptic problems: bifurcation & asymptotic analysis. oxford lecture series in mathematics and its applications, 2008. [14] david gilbarg and neil s. trudinger elliptic partial differential equations of second order. classics in mathematics reprint of 1998 edition springer [15] m. hénon. numerical experiments on the stability of spherical stellar systems. astronomy and astrophysics 24 (1973), 229-238. [16] p.i.e. peebles,black holes are where you find them. gen. rel. grav., 3 (1972), 63-82. [17] p.i.e. peebles,star distribution near a collapsed object. ast. i., 178 (1972), 371-375. [18] j. homer lane. on the theoretical temperature of the sun under the hypothesis of a gaseous mass maintining its volume by its internal heat and depending on the law of gases known to terrestrial experiment. amer. j. sci, 2d ser. 50 (1869), 57-74. [19] p. s. de laplace memoire sur les mouvements des corps celestes autout de leurs centre de gravite, ouvres completes tome douzieme. [20] t. matukuma. sur la dynamique des amas globulaires stellaires. proc. imp. acad. 6 (1930), 133-136. [21] m. struwe. variational methods: applications to nonlinear partial differential equations and hamiltonian systems. springer 2000. cubo a mathematical journal vol.14, no¯ 03, (01–39). october 2012 fundamentals of scattering theory and resonances in quantum mechanics peter d. hislop department of mathematics, university of kentucky, lexington, kentucky 40506-0027, usa email: hislop@ms.uky.edu abstract we present the basics of two-body quantum-mechanical scattering theory and the theory of quantum resonances. the wave operators and s-matrix are constructed for smooth, compactly-supported potential perturbations of the laplacian. the meromorphic continuation of the cut-off resolvent is proved for the same family of schrödinger operators. quantum resonances are defined as the poles of the meromorphic continuation of the cut-off resolvent. these are shown to be the same as the poles of the meromorphically continued s-matrix. the basic problems of the existence of resonances and estimates on the resonance counting function are described and recent results are presented. resumen presentamos los conceptos básicos de la teoŕıa de dispersión cuanto-mecánica de dos cuerpos y la teoŕıa de resonancias cuánticas. el operador de ondas y la matriz s se construyen para perturbaciones del potencial suaves y de soporte compacto del laplaciano. la continuación meromórfica de la resolvente truncada se prueba para la misma familia de operadores de schrdinger. las resonancias cuánticas se definen como los polos de la continuación meromórifca de la resolvente truncada. se muestra que ellas son las mismas que los polos de la matriz s continuada meromórficamente. los problemas básicos de la existencia de resonancias y las estimaciones de la función de conteo de la resonancia se describen y resultados recientes se presentan. keywords and phrases: scattering theory, resonances, schrödinger equation, wave operators, quantum mechanics 2010 ams mathematics subject classification: 35j10, 35p25, 35q40,47a40, 47a55, 81u05, 81u20 2 peter d. hislop cubo 14, 3 (2012) 1 introduction: schrödinger operators the purpose of these notes is to present the necessary background and the current state-of-theart concerning quantum resonances for schrödinger operators in a simple, but nontrivial, setting. the unperturbed hamiltonian h0 = −∆ is the laplacian on l 2(rd). in quantum mechanics, the schrödinger operator or hamiltonian h0 represents the kinetic energy operator of a free quantum particle. many interactions are represented by a potential v that is a real-valued function with v ∈ l∞0 (r d), the essentially bounded functions of compact support. occasionally, we need the potential to have some derivatives and this will be indicated. if, for example, the potential v ∈ c∞0 (rd), then all the results mentioned here hold true. the perturbed hamiltonian is hv = −∆ + v. a fundamental property shared by both hamiltonians is self-adjointness. the unperturbed hamiltonian h0 is self-adjoint on its natural domain h 2(rd), the sobolev space of order two, which is dense in l2(rd). the self-adjoint operator h0 is the generator of a one-parameter stronglycontinuous unitary group t ∈ r → u0(t) = e−ih0t. the potential v is relatively h0-bounded with relative bound zero. by the kato-rellich theorem [14, theorem 13.5], the perturbed operator hv is self-adjoint on the same domain h 2(rd). this self-adjoint operator generates a one-parameter strongly-continuous unitary group t ∈ r → uv(t) = e −ihv t. the unitary groups u0(t) and uv(t) provide solutions to the initial value problem for the schrödinger operator in l2(rd). for example, the solution to i ∂ψ(t) ∂t = hvψ(t), ψ(0) = ψ0 ∈ h2(rd), (1.1) is formally given by ψ(t) = uv(t)ψ0. in this way, the unitary group uv(t) provides the timeevolution of the initial state ψ0. scattering theory seeks to provide a description of the perturbed time-evolution uv(t) in terms of the simpler (as we will show below) time-evolution u0(t). although we will work on the hilbert space l2(rd), much of scattering theory can be formulated in a more abstract setting. consequently, we will often write h for a general hilbert space. suppose we take a state f ∈ h and consider the interacting time-evolution uv(t)f. what is the behavior of uv(t)f as t → ±∞? there is one exactly solvable case, although, as we will see, it is not too interesting. suppose that f is an eigenfunction of hv with eigenvalue e so that f satisfies the eigenvalue equation hvf = ef. then, the time evolution is rather simple since uv(t)f = e −itef, as is easily verified by differentiation. we do not expect this simple oscillating state to be approximated by the free dynamics so we should eliminate these states from our consideration. let hcont(hv) be the closed subspace of h orthogonal to the span of all the eigenfunctions of hv. we will call these states the scattering states of hv. given f ∈ hcont(hv), can we find a state f+ ∈ h so that as time runs to plus infinity, the state uv(t)f looks approximately like the free time-evolved state u0(t)f+? in particular, we ask if given f ∈ hcont(hv), does there cubo 14, 3 (2012) scattering theory and resonances ... 3 exist a state f+ ∈ h so that uv(t)f − u0(t)f+ → 0, as t → +∞. (1.2) when it is possible to find such a vector f+, we have a simpler description of the dynamics uv(t) generated by hv in terms of the free dynamics u0(t) generated by h0. we can also pose the question concerning the existence of a state f− so that (1.2) holds for t → −∞ with f− replacing f+. we understand (1.2) to mean convergence as a vector in h, that is lim t→+∞ ‖uv(t)f − u0(t)f+‖h = 0. (1.3) note that if f+ is an eigenfunction of h0 with eigenvalue e, that is h0f+ = ef+, then u0(t)f+ = e−itef+, we would not expect the limit (1.3) to exist. hence, we want f+ to be a state with nontrivial free time evolution. this means that we want f+ to be a scattering state for h0, that is, f+ ∈ hcont(h0). for our specific example, h0 = −∆, there are no eigenfunctions so hcont(h0) = h. because the operators u0(t) and uv(t) are unitary, the limit in (1.3) is equivalent to lim t→+∞ ‖f − uv(t)∗u0(t)f+‖h = 0. (1.4) since h0 = −∆ has no eigenvalues and only continuous spectrum, we expect that the limit lim t→+∞ uv(t) ∗u0(t)f+ = f, (1.5) if it exists, should exist for all states f+ ∈ h. similarly, we might expect that the limit lim t→−∞ uv(t) ∗u0(t)f− = f, (1.6) exists for all f ∈ h. we will prove in section 2 that these limits do exist and define bounded operators ω±(hv,h0) on h called the wave operators for the pair (h0,hv). if we consider the original problem: given f ∈ hcont(hv), find f± so that the limit in (1.2), and the similar limit for t → −∞, it might seem strange that we consider ω±(hv,h0) rather than the limit of the operators in the other order, namely, u0(t) ∗uv(t) on the scattering states of hv. as we will see, it is much more difficult to prove the existence of the latter limit. let us consider, however, the inner product (g,ω±(hv,h0)f) for g in the range of the wave operator ω±(hv,h0). using the definition and unitarity of the time evolution groups, we have (g,ω±(hv,h0)f) = lim t→±∞ (g,uv(t) ∗u0(t)f) = lim t→±∞ (u0(t) ∗uv(t)g,f) = (ω±(hv,h0) ∗g,f). (1.7) since this holds for all f ∈ h, it follows that for g ∈ ran ω±(hv,h0), lim t→±∞ u0(t) ∗uv(t)g = ω±(hv,h0) ∗g. (1.8) 4 peter d. hislop cubo 14, 3 (2012) comparing this to (1.3), it is clear that we obtain the desired states by f± = ω±(hv,h0) ∗f. as we will see in proposition 4, the existence of the strong limits of u0(t) ∗uv(t) on the scattering states of hv as t → ±∞ is related to asymptotic completeness. the existence of the wave operators ω±(hv,h0) allow us to define states f± for any scattering state f ∈ hcont(hv). the map s : f− → f+ plays an important role in scattering theory. this map is called the s-operator for the pair (h0,hv). two technical remarks. 1) the subspace of scattering states hcont(hv) is technically the absolutely continuous spectral subspace of hv (see section 8.1). the unperturbed operator h0 = −∆ has spectrum equal to the half-line [0,∞) and is purely absolutely continuous. in our setting, the perturbed operator hv has only absolutely continuous spectrum and possibly eigenvalues. in general, it is a difficult task to prove the absence of singular continuous spectrum. there is an orthogonal spectral projector econt(hv) so that hcont(hv) = econt(hv)h. we will use either notation interchangeably. 2) the type of convergence described in (1.5) and (1.6) is called strong convergence of operators. we say that a sequence of bounded operators an on h converges strongly to a ∈ b(h) if for all f ∈ h, we have limn→∞ anf = af. 2 fundamentals of two-body scattering theory the basic objects of scattering theory are the wave operators and the scattering operator. the crucial property of the wave operators ω±(hv,h0) is called asymptotic completeness. this condition guarantees the unitarity of the scattering operator. on the level of spectral theory, asymptotic completeness means that the restrictions of the operators h0 and hv to their absolutely continuous subspaces are unitarily equivalent. from this viewpoint, scattering theory is a tool for studying the absolutely continuous spectral components of the pair (h0,hv) of self-adjoint operators. the theory has been developed to a very abstract level and the reader is referred to the references for further details (for example, [32, 45]). 2.1 wave operators another way to write (1.4) is lim t→∞ uv(t) ∗u0(t)f+ = f, (2.1) so one of our first tasks is to ask whether the limit on the left side of (2.1) exists. proposition 1. suppose that the real-valued potential v ∈ l∞0 (rd) and that d ≥ 3. for any f ∈ h, the limit lim t→∞ uv(t) ∗u0(t)f (2.2) exists. this limit defines a bounded linear transformation ω+(hv,h0) with ‖ω+(hv,h0)‖ = 1. cubo 14, 3 (2012) scattering theory and resonances ... 5 the linear operator ω+(hv,h0) is called a wave operator. we can also consider the limit in (2.1) as time runs to minus infinity. we introduce another wave operator ω−(hv,h0) defined by s − lim t→−∞ uv(t) ∗u0(t) ≡ ω−(hv,h0), (2.3) when the strong limit exists. of course, we can introduce another pair of wave operators by interchanging the order of hv and h0. we will consider these wave operators ω±(h0,hv) in section 2.3 when we discuss asymptotic completeness. we will see that it is much more difficult to prove the existence of these wave operators. we prove proposition 1 using the classic cook-hack method (see, for example, [31, section xi.4]). in the following proof, we drop the hamiltonians from the notation for the wave operators and simply write ω± for the wave operators ω±(hv,h0). proof. 1. the proof of proposition 1 relies on an explicit estimate for the free propagation given by u0(t). for any f ∈ l1(rd) ∩ l2(rd), and for t 6= 0, we have ‖u0(t)f‖∞ ≤ cd td/2 ‖f‖1. (2.4) this estimate is proved (see [1, lemma 3.12]) using an explicit formula for u0(t)f, t 6= 0. for any f ∈ l1(rd) ∩ l2(rd), we have (u0(t)f)(x) = ( 1 4πit )d/2 ∫ rd ei|x−y| 2/(4t) f(y) ddy. (2.5) this representation is based on the fact that the fourier transform (see (3.4) and (3.4)) of the action of the free propagation group is (f(u0(t)f))(k) = e −i|k|2t(ff)(k). (2.6) formally, formula (2.5) is obtained by computing the inverse fourier transform. this involves a singular integral: ∫ rd eik·(x−y)e−i|k| 2t ddk. (2.7) this integral can be done by first regularizing the integrand by replacing t by t − iǫ, for ǫ > 0. this results in a gaussian function of k, and the fourier transform is explicitly computable. it is also a gaussian function. one can then take ǫ → 0 and recover the formula (2.5) since f ∈ l2(rd) ∩ l1(rd) guarantees convergence of the integral. 2. given this result (2.4), we proceed as follows. let us define ω(t) by ω(t) ≡ uv(t)∗u0(t). (2.8) from this definition, we compute for any f ∈ l1(rd) ∩ l2(rd) (ω(t) − 1)f = ∫t 0 d ds uv(s) ∗u0(s)f ds = i ∫t 0 uv(s) ∗vu0(s)f ds. (2.9) 6 peter d. hislop cubo 14, 3 (2012) since u0(t) maps l 2(rd) to itself and v ∈ l∞0 (rd), the integral on the right is well-defined. to prove the existence of the limit, consider 0 << t1 < t2 and note that from (2.9) and the estimate (2.4), we have ∥ ∥ ∥ ∥ ∫t2 t1 uv(s) ∗vu0(s)f ds ∥ ∥ ∥ ∥ ≤ ‖v‖l2(rd) ∫t2 t1 ‖u0(s)f‖l∞(rd) ds ≤ cd‖v‖l2(rd) ‖f‖1 ∫t2 t1 s−d/2 ds ≤ c̃d‖v‖l2(rd)‖f‖1(t 1−d/2 1 − t 1−d/2 2 ). (2.10) it follows that for d ≥ 3, we have the bound ‖(ω(t2) − ω(t1)f‖ ≤ c̃d‖v‖‖f‖1(t1−d/21 − t 1−d/2 2 ). (2.11) consequently, for any sequence tn → ∞, the sequence of vectors ω(tn)f is a norm-convergent cauchy sequence so limt→∞ ω(t)f ≡ f̃+ exists. we must show that the map f ∈ l1(rd)∩l2(rd) → f̃+ defines a linear bounded operator. since ‖ω(tn)f‖ ≤ ‖f‖l2(rd), for any tn, it follows that ‖f̃+‖ ≤ ‖f‖. this defines ω+ : f → f̃+ on a dense domain l1(rd) ∩ l2(rd). a densely-defined bounded linear operator can be extended to h without increasing the norm. finally, one verifies that s − limt→∞ ω(t) = ω+ by approximating any g ∈ h by a sequence in l1(rd) ∩ l2(rd) and using a triangle inequality argument. the simplicity of this proof relies on the estimate (2.4) for the group u0(t). it is more difficult to consider the strong limit of u0(t) ∗uv(t) since no general formula is available for uv(t)f. 2.2 properties of wave operators the wave operators ω± are bounded operators on h with ‖ω±‖ = 1. they satisfy a number of important properties. first, they are partial isometries in the sense that e± ≡ ω∗±ω± are orthogonal projections. in our case, e± = i, the identity operator on h. in the general case, the operator e± is the projection onto the continuous subspace of h0. for any f,g ∈ h, we have (ω±f,ω±g) = (f,e±g) = (e±f,e±g), (2.12) so that ‖ω±f‖ = ‖e±f‖. (2.13) it follows that ω± are isometries on e±h and that the kernel of ω± is (1 − e±)h. we have that ω±e± = ω±. the subspaces of h given by e±h are called the initial spaces of the partial isometries ω±. second, the adjoints ω∗± are partial isometries. since (ω ∗ ±) ∗ω∗± = ω±ω ∗ ±, the operator f± ≡ ω±ω∗± satisfies f2± = ω±(ω∗±ω±)ω∗± = ω±e±ω∗± = f±, and in a similar manner f∗± = f±, cubo 14, 3 (2012) scattering theory and resonances ... 7 so f± are orthogonal projections. it follows that f±ω ∗ ± = ω ∗ ± and that ‖ω∗±f‖ = ‖f±f‖. one can show that f± are the orthogonal projections onto the closed ranges of the wave operators ran ω± = f±h. the subspaces f±h are called the final subspaces of the partial isometries ω±. proposition 2. the wave operators satisfy the following intertwining relations: ω±u0(t) = uv(t)ω± u0(t)ω ∗ ± = ω ∗ ±uv(t). (2.14) proof. these relations follow from the existence of the wave operators and the simple properties of the unitary evolution groups. for any f ∈ h, we have uv(t)ω+f = lim s→∞ uv(t)uv (s) ∗u0(s)f = lim s→∞ [uv(s − t) ∗u0(s − t)]u0(t)f = lim u→∞ [uv(u) ∗u0(u)]u0(t)f = ω+u0(t)f, (2.15) proving the first intertwining relation. the second is proven in the same manner. 2.3 asymptotic completeness the existence of the wave operators ω±(hv,h0) means the existence of a orthogonal projectors onto the initial space e± ≡ ω±(hv,h0)∗ω±(hv,h0) = i and final subspaces f± ≡ ω±(hv,h0)ω±(hv,h0) ∗ that are the ranges of the wave operators ω±(hv,h0). the range of the wave operators must be contained in the continuous spectral subspace of hv. definition 3. the pair of self-adjoint operators (h0,hv) is said to be asymptotically complete if f−h = f+h = econt(hv)h, that is, if ran ω− = ran ω+ = econt(hv)h. in our situation, with h0 = −∆, the spectrum of h0 is purely absolutely continuous and econt(h0)h = h. in particular, e± = 1h. also, neither operator h0 nor hv has singular continuous spectrum. in more general situations, one needs to prove that the perturbed operator hv has no singular continuous spectrum. in these more general cases, the subspace hcont(hv) must be taken as the absolutely continuous spectral subspace. one can also consider wave operators ω±(h0,hv) defined by switching the order of the unitary operators in (2.2): ω±(h0,hv) ≡ s − lim t→±∞ u0(−t)uv(t)econt(hv). (2.16) at first sight, it would seem that the existence of these wave operators would be equivalent to the existence of ω±(hv,h0). however, we have no explicit control over the dynamics generated by hv such as formula (2.5). consequently, it is difficult to use the cook-hack method to prove the existence of the wave operators ω±(h0,hv). in fact, the existence of the wave operators ω±(h0,hv) is equivalent to asymptotic completeness. 8 peter d. hislop cubo 14, 3 (2012) proposition 4. suppose that the wave operators ω±(hv,h0) exist. then the pair of operators (h0,hv) are asymptotically complete if and only if the wave operators ω±(h0,hv) exist. proof. 1. suppose that both sets of wave operators exist. then, we know that the projection econt(hv) = ω±(hv,hv). but, we have uv(−t)uv(t) = uv(−t)u0(t) · u0(−t)uv(t), (2.17) from which it follows that ω±(hv,hv) = ω±(hv,h0)ω∓(h0,hv). (2.18) this implies that hcont(hv) ⊂ ran ω±(hv,h0). since the existence of ω±(hv,h0) means that ran ω±(hv,h0) ⊂ hcont(hv), these two inclusions mean that ran ω+(hv,h0) = ω−(hv,h0) = hcont(hv). 2. to prove the other implication, we assume that the wave operators ω±(hv,h0) exist and are asymptotically complete. then, for any φ ∈ hcont(hv), there exists a ψ ∈ h so that φ = ω+(hv,h0)ψ. this means that u0(t)ψ − uv(t)φ converges to zero as t → +∞. by unitarity of the operator u0(t), this means that limt→+∞ u0(−t)uv(t)φ = ψ for all φ ∈ hcont(hv). this implies the existence of ω+(h0,hv). the proof of the existence of the other wave operator is analogous. we now turn to proving the existence of the wave operators ω±(h0,hv). many methods have been developed over the years in order to do this. the classic result of birman [31, theorem xi.10] is perhaps the simplest to apply to our simple two-body situation. there are more elegant and far-reaching methods. the enss method, in particular, is based on a beautiful phase-space analysis of the scattering process. a thorough account of the enss method may be found in perry’s book [27]. perry combined the enss method with the melin transform in [26] to present a new, clear, and short proof of asymptotic completeness for two-body systems more general than those considered here. finally, the problem of asymptotic completeness for n-body schrödinger operators with short-range, two-body potentials, was solved by sigal and soffer [38]. they developed a very useful technique of local decay estimates. in preparation, we recall that a bounded operator k is in the trace class if the following condition is satisfied. the singular values of a compact operator a are given by µj(a) = √ λj(a ∗a), where {λj(b)} are the eigenvalues of b. we say that k is in the trace class if ∑ j µj(k) < ∞. we say that k is in the hilbert-schmidt class if ∑ j µj(k) 2 < ∞. we refer to [29] or [39] for details concerning the von neumann-schatten trace ideals of bounded of operators. theorem 5. let v ∈ l∞0 (rd) be a real-valued potential and d ≥ 3. then the pair (h0,hv)is asymptotically complete. proof. 1. by proposition 4, it suffices to prove that ω±(h0,hv) exist since we know from proposition 1 that the wave operators ω±(hv,h0) exist. for any interval i ⊂ r and self-adjoint operator cubo 14, 3 (2012) scattering theory and resonances ... 9 a, let ei(a) denote the spectral projection for a and the interval i. in the first step, we note that ei(h0)vei(hv),ei(hv)vei(h0) ∈ i1. (2.19) the trace class property of these operators is easily demonstrated by proving that |v|1/2r0(i) k is a hilbert-schmidt operator for k > d/2 and noting that ei(h0)r0(i) −k is a bounded operator. 2. next, we need the following result called pearson’s theorem in [31, theorem xi.7]. let a > 0 and define the bounded operator ja ≡ e(−a,a)(h0)e(−a,a)(hv). the trace class property (2.19) means that h0ja − jahv ∈ i1. the main result of [31, theorem xi.7] is that s − lim t→±∞ u0(t) ∗jauv(t)econt(hv) (2.20) exists. let 0 < a0 < a and choose φ ∈ e(−a0,a0)(hv)econt(hv)h. we then have u0(t) ∗e(−a,a)(h0)uv(t)φ = u0(t) ∗jauv(t)φ, (2.21) so by (2.20), the strong limit of the term on the left in (2.21) exists. 3. we can now write the expression that gives the wave operator acting on any φ ∈ e(−a0,a0)(hv)econt(hv)h: u0(t) ∗uv(t)φ = u0(t) ∗[e(−a,a)(h0) + er\(−a,a)(h0)]uv(t)φ. (2.22) since the strong limit of the first term on the right in (2.22) exists by (2.21), it suffices to prove that lim a→∞ { sup t∈r ‖u0(t)∗er\(−a,a)(h0)uv(t)φ‖ } = 0. (2.23) once this is proven, we can first take a → ∞ and then a0 → ∞ so that the limit in (2.22) holds for any φ ∈ econt(hv)h. 4. to prove (2.23), we need some estimates. let f(s) = s2 + 1 ≥ 1. the fact that v is relatively h0-bounded means that ‖f(hv)f(h0)−1‖ < c1 < ∞. (2.24) next, recall that φ ∈ e(−a0,a0)(hv)h, for 0 < a0 < a, so that ‖f(hv)uv(t)φ‖ ≤ sup |s|≤a0 f(s) = a20 + 1 < ∞. (2.25) finally, since f is invertible, we have ‖f(h0)−1er\(−a,a)(h0) ≤ [ inf |s|≥a0 f(s) ]−1 = (a2 + 1)−1. (2.26) note that this vanishes as a → ∞. 10 peter d. hislop cubo 14, 3 (2012) 5. returning to (2.23), we write the norm as ‖u0(t)∗er\(−a,a)(h0)uv(t)φ‖ ≤ ‖u0(t)∗ · f(h0)−1er\(−a,a)(h0) · f(h0)f(hv)−1 · f(hv)uv(t)φ‖ ≤ ‖f(h0)−1er\(−a,a)(h0)‖ ‖f(h0)f(hv)−1‖ ‖f(hv)uv(t)φ‖ ≤ c1(a20 + 1)(a2 + 1)−1, (2.27) independently of t. taking a → ∞ proves (2.23). the asymptotic completeness of (h0,hv) means that the absolutely continuous parts of each operator are unitarily equivalent. recall that our condition on the real-valued potential v ∈ l∞0 (rd) means that v(h0 + i)−1 is compact. by weyl’s theorem (see, for example, [14, theorem 14.6]), the essential spectrum of hv is the same as the essential spectrum of h0 that is [0,∞). hence, the perturbation can add at most a discrete set of isolated eigenvalues with finite multiplicities. the property of asymptotic completeness goes beyond this and establishes the unitary equivalence of the absolutely continuous components. 3 the scattering operator the existence of the wave operators ω±(hv,h0) guarantees the existence of the asymptotic states f±. for any f ∈ ran ω±(hv,h0) ⊂ econt(hv)h, we have f± = ω±(hv,h0)∗f. the s-operator maps f− to f+. it is a bounded operator on l 2(rd). furthermore, the s-operator commutes with the free time evolution u0(t). this allows for a reduction of the s-operator to a family of operators s(λ) defined on l2(sd−1) called the s-matrix. 3.1 basic properties of the s-operator an important use of the wave operators is the construction of the s-operator on h. for any f ∈ ran ω±(hv,h0), we have from section 2.1 that f± = ω±(hv,h0)∗f, or, for example, f = ω−f−. as a result, we can compute a formula for the map f− → f+ in terms of the wave operators: sf− = f+ = ω ∗ +f = ω ∗ +ω−f−. consequently, the s-operator is defined as the bounded operator s ≡ ω∗+ω− : h → h. (3.1) proposition 6. suppose ran ω− ⊂ ran ω+. then, the scattering operator is a partial isometry on l2(rd). proof. to prove this, we need to show that s∗s is an orthogonal projection. this follows from the properties of the wave operators: s∗s = (ω∗+ω−) ∗ (ω∗+ω−) = ω ∗ −[ω+ω ∗ +]ω− = ω ∗ −f+ω−. (3.2) cubo 14, 3 (2012) scattering theory and resonances ... 11 since we assume that ran ω− ⊂ f+h, we have f+ω− = ω−, so from (3.2), s∗s = f−, an orthogonal projection. if h0 = −∆, this operator f− is the identity operator on h. since ran s ⊂ ran ω∗+ ⊂ econt(h0)h, we have that s : econt(h0)h → econt(h0)h. an essential property of the s-operator is that it commutes with the free time evolution, as stated in the following proposition. proposition 7. the s-operator commutes with the free time evolution: [s,u0(t)] ≡ su0(t) − u0(t)s = 0. consequently, the s-operator satisfies e0(i)s = se0(i), where e0(i) is the spectral projector for h0 and any lebesgue measurable i ⊂ r. proof. this follows from the definition s = ω∗+ω− and the intertwining properties (2.14) of the wave operators. we compute: su0(t) = ω ∗ +uv(t)ω− = (uv(−t)ω+) ∗ω− = (ω+u0(−t)) ∗ω− = u0(t)s. (3.3) it follows from proposition 7 that for a wide class of reasonable functions φ, we have the general result sφ(h0) = φ(h0)s. the key property of the equality of the ranges of the wave operators (part of asymptotic completeness) has important consequences for the s-operator. theorem 8. suppose that for a pair of self-adjoint operators (h0,hv), we have ran ω−(hv,h0) = ran ω+(hv,h0). then, the s-operator is a unitary operator on l 2(rd). to prove the unitarity of the s-operator, we recall from (3.2) that, in general, s∗s = ω∗−f+ω−. if ran ω− = ran ω+, we have f+ω− = ω−. furthermore, under our hypotheses, we have ω∗−ω− = 1l2(rd), so that s ∗s = 1. as for ss∗, a similar calculation gives ss∗ = ω∗+f−ω+. it could happen that ran ω+ is strictly larger that ran ω−. in this case, the kernel of ss ∗ is nontrivial and consists of any element of ran ω+ orthogonal to ran ω−. in this case, ss ∗ is not invertible. our condition that ran ω− = ran ω+ eliminates this possibility and we find ss∗ = ω∗+f−ω+ = ω ∗ +ω+ = 1. hence, the s-operator s is invertible and s −1 = s∗. 3.2 the s-matrix because the s-operator commutes with spectral family for h0, both operators admit a simultaneous spectral decomposition. this is achieved with the fourier transform. we define the fourier transform of f ∈ s(rd) by (ff)(k) ≡ (2π)−d/2 ∫ rd e−ik·xf(x) ddx. (3.4) 12 peter d. hislop cubo 14, 3 (2012) the inverse fourier transform is defined, for any g ∈ s(rd), by (f−1g)(x) ≡ (2π)−d/2 ∫ rd eik·xg(k) ddk. (3.5) the fourier transform extends to a unitary map on l2(rd). note that for h0 = −∆, and f ∈ s(rd), we have (f(h0f))(k) = |k| 2(ff)(k). (3.6) it is convenient to write k = λω ∈ rd, where λ ∈ [0,∞) and ω ∈ sd−1. with this decomposition a function f(k) may be viewed as a function on sd−1 parameterized by λ ∈ [0,∞). we need a family of maps from l2(rd) → l2(sd−1) parameterized by the energy λ. these maps e±(λ) can be defined via the fourier transform (3.4). for λ ∈ r, and any f ∈ s(rd), we define (e±(λ)f)(ω) ≡ (2π)−d/2 ∫ rd e±iλx·ωf(x) ddx, ω ∈ sd−1. (3.7) the transpose of these maps, te±(λ) : l 2(sd−1) → l2(rd). the formula for the s-matrix involves the resolvent rv(λ) ≡ (hv − λ2) of hv. we will study the resolvent in detail in section 4. provided ℑλ2 6= 0 and −λ2 is not an eigenvalue of hv, the resolvent rv(λ) is a bounded operator. we need to understand the behavior of vrv(λ + iǫ)v, for λ ∈ r, in the limit as ǫ → 0. that this limit exists as a compact operator is part of the limiting absorption principle that is discussed in section 4.1. we will write vrv(λ + i0)v for this limit. recall from section 3.2 that the singular values of a compact operator a are given by µj(a) = √ λj(a ∗a), where {λj(b)} are the eigenvalues of b, and that k is in the trace class if ∑ j µj(k) < ∞. theorem 9. assume that the pair (h0,hv) is asymptotically complete with h0 = −∆. then the s-matrix is the unitary family of operators s(λ), for λ ∈ r, on l2(sd−1) given by s(λ) = 1l2(sd−1) − πiλ d−2e−(λ)(v − vrv(λ + i0)v) te+(λ) = 1l2(§d−1) − a(λ). (3.8) the operator a(λ) is the scattering amplitude. it is given by a(λ) ≡ −πiλd−2e−(λ)(v − vrv(λ + i0)v)te+(λ), (3.9) and is in the trace class. we can also express the s-matrix in terms of localization operators in the case the support of v is compact. we assume that suppv ⊂ b(0,r1). we choose two other length scales so that 0 < r1 < r2 < r3 < ∞. let 0 ≤ χj ∈ c20(rd) have the property that χjv = v and suppχ2 ⊂ b(0,r2) and suppχ3 ⊂ b(0,r3). finally, let w(φ) denote the commutator w(φ) ≡ [−∆,φ], for any φ ∈ c2(rd). the following representation is due to petkov and zworski [28]. theorem 10. let v ∈ c20(rd) and consider the s-matrix s(λ), λ ∈ r, as a unitary operator on l2(sd−1). then, the s-matrix has the form s(λ) = 1l2(sd−1) + a(λ), λ ∈ r, (3.10) cubo 14, 3 (2012) scattering theory and resonances ... 13 where a(λ) is in the trace class. explicitly, the scattering amplitude a(λ) has the form a(λ) = cdλ d−2e−(λ)w(χ2)rv(λ)w(χ1) te+(λ), (3.11) where the constant cd = −i(2π) −d2(1−d)/2. 4 the resolvent and resonances we now switch our perspective and return to the study of the resolvent of the schrödinger operator hv. we will connect these results with the s-matrix in section 4.5. we recall from section 2 that the spectrum of a self-adjoint operator a, denoted by σ(a), is a closed subset of the real line. the discrete spectrum of a, denoted σdisc(a), is the subset of the spectrum consisting of all isolated eigenvalues with finite multiplicity. the complement of the spectrum is called the resolvent set of a, denoted by ρ(a) ≡ c\σ(a). the resolvent of a self-adjoint operator a is defined, for any z ∈ ρ(a), as the bounded operator ra(z) = (a − z)−1. it is a bounded operator-valued analytic function on ρ(a). this means that about any point z0 ∈ ρ(a), the resolvent ra(z) has a norm convergent power series of the form ra(z) = ∞∑ j=0 aj(z − z0) j, (4.1) for bounded operators aj depending on z0. we note that for a self-adjoint operator a, the set c\r is always in the resolvent set. for a schrödinger operator hv = −∆+v, we reparameterized the spectrum by setting z = λ 2 and write rhv (z) = rv(λ). under this change of energy parameter, the spectrum in the complex λ-plane is the union of the line ℑλ = 0 and at most finitely-many points of the form iλj on the positive imaginary axis λj > 0. these points correspond to the negative eigenvalues of hv so that z = −λ2j ∈ σdisc(a). let χv ∈ c∞0 (r) be a compactly-supported function so that χvv = v. we are most concerned with the properties of the localized resolvent rv(λ) ≡ χvrv(λ)χv. the operator-valued function rv(λ) is defined for ℑλ > 0 and λ 6= iλj, with λj > 0 and −λ2j an eigenvalue of hv. we would like to find the largest region in the complex λ-plane on which rv(λ) can be defined. 4.1 limiting absorption principle one might first ask if the bounded operator rv(λ) has a limit as ℑλ → 0, from ℑλ > 0. that is, does the boundary-value of this operator-valued meromorphic function exist as a bounded operator for λ ∈ r? because of the weight functions χv the answer to this question is yes. in more general settings, this result is part of what is referred to as the limiting absorption principle (lap). the lap plays an important role in scattering theory. 14 peter d. hislop cubo 14, 3 (2012) theorem 11. the meromorphic bounded operator-valued function rv(λ) on the open set ρ+(hv) ≡ {λ ∈ c | ℑλ > 0,−λ2 6∈ σdisc(hv)} admits continuous boundary values rv(λ) for λ ∈ r, except possibly at λ = 0. that is, limǫ→0+ rv(λ + iǫ) exists for all λ ∈ r\{0}, and is a bounded, continuous operator-valued function on that set. the proof of this is given for more general potentials and n-body schrödinger operators in, for example, [9, chapter 4]. the key ingredient is a local commutator estimate called the mourre estimate, due to e. mourre [22]. let a = (1/2)(x · ∇ + ∇ ·x) be the generator of the unitary group implementing the dilations x → eθx, for θ ∈ r, on l2(rd). one formally computes the following commutator, assuming ∇v exists: [hv,a] = 2h0 − x · ∇v = 2hv − (2v + x · ∇v). (4.2) let i ⊂ r be a closed interval. let ev(i) be the projector for hv and the interval i. we conjugate the commutator in (4.2) by this spectral projector: ev(i)[hv,a]ev(i) = 2ev(i)hvev(i) − k(v,i), (4.3) where k(v,i) ≡ ev(i)(2v + x · ∇v)ev(i) is a compact, self-adjoint operator due to the properties of v. we now assume that there are no eigenvalues of hv in the interval i. for i ⊂ r+, this means that there are no positive eigenvalues of hv. in our situation, this is true (see [9, chapter 4]). then, the spectral theorem implies that s−lim|i|→0 ev(i) = 0. since k(v,i) is a compact operator and k(v,i) = k(v,i)ev(i), it follows that lim|i|→0 ‖k(v,i)‖ = 0. furthermore, if i = [e1,e2], then 2e(i)hve(i) ≥ 2e1. given any ǫ > 0, we choose i so that |i| is so small that ‖k(v,i)‖ ≤ ǫ. consequently, the commutator on the left in (2.9) is strictly nonnegative and bounded below: ev(i)[hv,a]ev(i) ≥ (2e1 − ǫ)ev(i) ≥ 0, |i| = e2 − e1 sufficiently small. (4.4) one of the main results of mourre theory is that for any interval i for which a positive commutator estimate of the form (4.4) holds, the boundary value of the weighted resolvent exists. more precisely, for any α > 1, one has lim ǫ→0+ { sup e∈i ‖(a2 + 1)−α/2(hv − e − iǫ)−1(a2 + 1)−α/2‖ } < ∞. (4.5) this technical estimate is the heart of the lap. estimate (4.5) is proved using a differential inequality-type argument. in our case, the function χv serves as the weight for the resolvent. one also proves that the limit in (4.5) is continuous in e ∈ i. if there are no embedded eigenvalues, as in our case, this holds for all e > 0. let us summarize what we have proved so far. the cut-off resolvent rv(λ) is meromorphic on c + with poles having finite-rank residues at at most finitely-many values iλj, with λj > 0 such that −λ2j is an eigenvalue of hv. using the lap, we can extend the cut-off resolvent rv(λ) onto the real axis as a bounded operator rv(λ), for λ ∈ r\{0}. this extension is continuous in λ. hence, the cut-off resolvent is meromorphic on c+ and continuous on c+\{0}. cubo 14, 3 (2012) scattering theory and resonances ... 15 4.2 analytic continuation of the cut-off resolvent of h0 our cut-off resolvent rv(λ) is meromorphic on c+ and continuous on the real axis, except possibly at zero. it is now natural to ask if the operator has a meromorphic extension to the entire complex λ-plane as a bounded operator. we first consider the simpler case when v = 0. in this case, let χ ∈ c∞0 (rd) be any compactly-supported cut-off function and consider the compact operator r0(λ) ≡ χr0(λ)χ. we mention that the kernel of this operator is known explicitly: r0(λ)(x,y) = i 4 χ(x) ( λ 2π|x − y| )(d−2)/2 h (1) (d−2)/2 (λ|x − y|)χ(y), (4.6) where h (1) j (s) is the hankel function of the first kind with index j. we remark that the lap is not necessary in order to construct an analytic continuation of the free cut-off resolvent r0(λ). an alternate and very nice method, based on the explicit formula (4.6), is presented in vodev’s review article [44]. we are tempted to define the continuation r̃0(λ) of r0(λ) for ℑλ < 0 as the operator χr0(−λ)χ since, if λ ∈ c−, then −λ ∈ c+ and χr0(−λ)χ is well defined away from σdisc(hv). clearly, r̃0(λ) ≡ χr0(−λ)χ for ℑλ < 0 is a meromorphic function in c−. the problem with this extension is that the two functions r0(λ) and r̃0(λ) do not match up on the real axis. in order to understand this, recall that in the z-plane, the resolvent (h0 − z) −1 is analytic on c\[0,∞). for λ0 > 0 and ǫ > 0, we are interested in the discontinuity of the resolvent across the positive z-axis at the point λ20 > 0. we can measure this by computing the following limit of the difference of the resolvents from above and below the point λ20 > 0: (h0 − (λ 2 0 + iǫ)) −1 − (h0 − (λ 2 0 − iǫ)) −1, (4.7) as ǫ → 0. the point z+ = λ 2 0 + iǫ has two square roots in the λ-plane. let λ̃0 ≡ √ λ40 + ǫ 2. for z+, let θ be the angle in the first quadrant so that 0 ≤ θ < π/2. then, the the two square roots are ±λ̃0[cos(θ/2) + isin(θ/2)]. the positive square root lies in c+ in the λ-plane so we work with this root λ+(ǫ) ≡ λ̃0[cos(θ/2) + isin(θ/2)]. similarly, the point z− = λ20 − iǫ has two square roots ±λ̃0[cos(θ/2)−isin(θ/2)]. note that because z− lies in the fourth quadrant, the imaginary part is negative. we choose the negative square root of z− because it lies in c + and call it λ−(ǫ). finally, for ǫ small, we may write λ±(ǫ) = ±λ0 + iǫ ∈ c+. consequently, the jump discontinuity in (4.7) across r+ in the z-plane corresponds to studying lim ǫ→0 [r0(λ0 + iǫ) − r0(−λ0 + iǫ)], (4.8) in the λ-plane. both terms in (4.8) are well-defined since the points ±λ+iǫ have positive imaginary parts ǫ > 0. we will compute the limit as ǫ → 0 in (4.8) and show that it is nonzero. furthermore, we will see that the limit extends to an analytic function on c. this is the term that must be added to 16 peter d. hislop cubo 14, 3 (2012) r0(−λ), for ℑλ < 0, in order to obtain a function that is continuous (and actually analytic) across ℑλ = 0. we follow a calculation in [19, sections 1.5-1.6]. for f ∈ c∞0 (rd) and ℑλ > 0, we have (r0(λ)f)(x) = (2π) −d/2 ∫ rd eiξ·x (ff)(ξ) (ξ2 − λ2) ddξ. (4.9) the fourier transform ff is a schwartz function so it decays rapidly in |ξ| (see, for example, [30, section ix.1, theorem ix.1]). since ℑλ > 0, this guarantees that the integral in (4.9) is absolutely convergent. switching to polar coordinates ξ = ρω, with ρ ≥ 0 and ω ∈ sd−1, we obtain for the integral ∫ rd eiξ·x (ff)(ξ) (ξ2 − λ2) ddξ = ∫ sd−1 dω ∫ ∞ 0 dρ eiρω·x ρn−1(ff)(ρω) (ρ2 − λ2) . (4.10) in order to compute r0(λ0 + iǫ), we deform the ρ-contour into the lower-half complex ρplane in a small, counter-clockwise oriented semicircle centered at λ0. the fourier transform ff extends to an analytic function (see, for example [30, section ix.3]) so there is no difficulty with this. similarly, in order to compute r0(−λ0 + iǫ), we note that this is the same as computing the integral in (4.10) with λ = λ0 − iǫ. this allows us to deform the ρ-integral into the upper-half complex ρ-plane and integrate around a small, clockwise semicircle centered at λ0. subtracting the two terms as in (4.8), we obtain r0(λ0 + iǫ) − r0(−λ0 + iǫ) = ∫ sd−1 dω ∫ γ(λ0) dρ eiρω·x ρd−1(ff)(ρω) (ρ2 − (λ0 + iǫ) 2) (4.11) where γ(λ0) is a counter-clockwise oriented circle about λ0 > 0. evaluating the integral by the residue theorem, we obtain for λ0 > 0, lim ǫ→0 [(r0(λ0 + iǫ) − r0(−λ0 + iǫ))f)(x)] = i 2 λd−20 (2π)(d−1)/2 ∫ sd−1 dω (ff)(λ0ω) e iλ0ω·x. (4.12) we define the kernel m(λ;x,y) by m(λ;x,y) ≡ i 2 1 (2π)d−1/2 ∫ sd−1 dω eiλω·(x−y). (4.13) undoing the fourier transform in (4.12), we can write the limit in (4.12) as lim ǫ→0 [(r0(λ0 + iǫ) − r0(−λ0 + iǫ))f)(x)] = λ d−2 0 ∫ rd m(λ0;x,y)f(y) d dy. (4.14) because the integration is over a compact set, the sphere, the kernel m(λ;x,y) extends to an analytic function on c. furthermore, recalling that we have compactly supported cut-off functions, the localized kernel m(λ;x,y) ≡ χ(x)m(λ;x,y)χ(y), (4.15) is square integrable for any λ ∈ c. hence, the operator m(λ) is an analytic, operator-valued function on c with values in the hilbert-schmidt class of operators (see [29, section vi.6]). cubo 14, 3 (2012) scattering theory and resonances ... 17 we can now define an extension r̃0(λ) of the cut-off resolvent r0(λ) from ℑλ > 0 to c−\(−∞,0] by r̃0(λ) ≡ χr̃0(λ)χ = χr0(−λ)χ + λd−2χm(λ)χ, ℑλ < 0. (4.16) we then have for λ > 0, lim ǫ→0 χr̃0(λ − iǫ)χ = lim ǫ→0 [χr0(−λ + iǫ)χ + λ d−2χm(λ − iǫ)χ] = χr0(λ)χ, (4.17) and thus we have continuity across the positive λ half-axis. it can be checked that this actually gives analyticity in a neighborhood of r\(−∞,0]. as for the open negative real axis (−∞,0), we note that m(−λ) = m(λ) since the sphere is invariant under the antipodal map ω → −ω. a similar analysis can be performed for d ≥ 2 even. we summarize the main results on the analytic continuation for the free cut-off resolvent. proposition 12. suppose that the dimension d ≥ 3 is odd. the cut-off resolvent χr0(λ)χ of the lapalcian admits an analytic continuation as a compact operator-valued function to the entire complex plane. in the case d = 1, there is an isolated pole of order one at λ = 0. when the dimension d ≥ 4 is even, the cut-off resolvent admits an analytic continuation as a compact operator-valued function to the infinite-sheeted riemann surface of the logarithm λ. in the case d = 2, there is a logarithmic singularity at λ = 0. 4.3 meromorphic continuation of the cut-off resolvent of hv we can use proposition 12 and the second resolvent formula to obtain a meromorphic continuation of the resolvent rv(λ). first, we write the second resolvent equation for λ ∈ c+, rv(λ) = r0(λ) − rv(λ)vr0(λ). (4.18) conjugating this equation by the cut-off function χv and using the fact that χvv = v, we obtain rv(λ) = χr0(λ)χ − rv(λ)vχr0(λ)χ. (4.19) solving this for rv(λ), we obtain rv(λ)(1 + vχvr0(λ)χv) = χvr0(λ)χv. (4.20) we use this equality in order to construct the meromorphic continuation of rv(λ). the right side of (4.20) has an analytic continuation as does the second factor on the left. we need to prove that this factor (1 + vχvr0(λ)χv) has a continuation that is boundedly invertible, at least away from a discrete set of λ. recall that an operator of the form 1 + k, for a bounded operator k, is boundedly invertible if, for example, ‖k‖ < 1. the inverse is constructed as a norm convergent geometric series. 18 peter d. hislop cubo 14, 3 (2012) there is another sufficient condition for invertibility. if the operator k is compact, then the fredholm alternative theorem [14, theorem 9.12] states that either k has an eigenvalue −1, and consequently, the operator 1 + k is not injective, or 1 + k is boundedly invertible. it follows from section 4.2 that the operator k(λ) = vχvr0(λ)χv in our expression (4.20) extends to a compact operator-valued analytic function. in this setting, the analytic fredholm theorem [29, theorem vi.14] is most useful. theorem 13. suppose that k(λ) is a compact operator-valued analytic function on a open connected set ω ⊂ c. then, either the operator 1+k(λ) is not invertible for any λ ∈ ω, or else it is boundedly invertible on ω except possibly on a discrete set d of points having no accumulation point in ω. the operator is meromorphic on ω\d at those points, the inverse has a residue that is a finite-rank operator. this theorem tells us that 1+k(λ), the first factor on the right of (4.20), is boundedly invertible for λ ∈ c except at a discrete set of points. since we know that rχ(λ) is invertible for ℑλ > 0, except for a finite number of points on the positive imaginary axis corresponding to eigenvalues, it also follows from (4.20) that the discrete set of points at which 1 + k(λ) fails to be invertible lies in c− if d is odd, or on λ\c+ if d is even. consequently, the analytic fredholm theorem allows us to establish the existence of a meromorphic extension of rv(λ). proposition 14. let v ∈ c20(rd) be a real-valued potential and let χv ∈ c∞0 (rd) be any function such that χvv = v. then the cut-off resolvent rv(λ) admits a meromorphic extension to c if d is odd and to λ if d is even. the poles have finite-rank residues. 4.4 resonances of hv having constructed the meromorphic continuation of the cut-off resolvent rv(λ), we can now define the resonances of hv. definition 15. let v ∈ c20(rd) be a real-valued potential. the resonances of hv are the poles of the meromorphic continuation of the compact operator rv(λ) occurring in c− for d odd, or on λ\c+ for d even. this definition can also be extended to complex-valued potentials. the residues of the extension of rv(λ) at the poles are finite rank operators. if λ0 ∈ c− is a resonance, then a resonance state ψλ0 ∈ h corresponding to λ0 is a solution to (1 + vχvr0(λ0)χv)ψλ0 = 0. (4.21) the poles are independent of the cut-off function used provided it has compact support and satisfies χv = v. cubo 14, 3 (2012) scattering theory and resonances ... 19 4.5 meromorphic continuation of the s-matrix the meromorphic continuation of the cut-off resolvent rv(λ) permits us to mermorphically continue the s-matrix s(λ) as a bounded operator on l2(sd) from λ ∈ c+ to all of c or λ depending on the parity of d. this follows from formula (3.8) of theorem 9. because of the compactness of the support of v, the operators e±(λ), and their transposes, admit analytic continuations. this property, together with the continuation properties of rv(λ) and formula (3.8), establish the meromorphic continuation of s(λ). for complex λ, the s-matrix is no longer unitary. the relation s(λ)s(λ)∗ = 1, however, does continue to hold for λ ∈ c (or λ). theorem 16. the s-matrix s(λ) admits a mermorphic continuation to c if d is odd, or to the riemann surface λ, if d is even, with poles precisely at the resonances of hv. the order of the poles are the same as the order of the poles for hv and the residues at these poles have the same finite rank. for the schrödinger operator hv, the resonances may be defined as the poles of the meromorphic continuation of the cut-off resolvent rv(λ), or in terms of the meromorphic continuation of the s-matrix s(λ). from formula (3.8), it follows that the poles of the meromorphic continuation of the s-matrix are included in the poles of the continuation of the resolvent. it is not always true that the scattering poles, defined via the s-matrix, are the same as the resolvent poles. a striking example where the scattering poles differ from the resolvent poles occurs for hyperbolic spaces. however, in the schrödinger operator case considered here, these are the same. a proof is given by shenk and thoe [37]. 5 resonances: existence and the counting function the resonance set rv for a schrödinger operator was defined in definition 15 as the poles of the meromorphic continuation of the cut-off resolvent rv(λ) to c for d ≥ 3 odd or to the riemann surface λ for d ≥ 4 even, together with their multiplicities. there are two basic questions that arise: (1) existence: do resonances exist for schrödinger operators hv with our class of potentials? (2) counting: how many resonances exist? 5.1 existence of resonances there are many different proofs of the existence of resonances for various quantum mechanical systems. resonances are considered as almost bound states or long-lived states that eventually decay to spatial infinity. to understand this physical description, let us consider the time evolution of a resonance state ψ0 corresponding to a resonance energy z0 = e0−iγ (in the z-parametrization), with γ > 0. a resonance state ψ0 solves the partial differential equation hvψ0 = z0ψ0 and is 20 peter d. hislop cubo 14, 3 (2012) purely outgoing. since v has compact support, the function ψ0 satisfies −∆ψ0 = z0ψ0 for |x| large enough. the outgoing condition means that the radial behavior of a component of ψ0 with angular momentum ℓ ≥ 0 is a hankel function of the first kind h(1) (d−2)/2+ℓ ( √ z0|x|). such a function ψ0 grows exponentially as |x| → ∞ and is not in h. we can formally compute uv(t)ψ0 by expressing the time evolution group as an integral of the resolvent over the energy uv(t)ψ0 = −1 2πi ∫ r e−iterv(e) de. (5.1) performing a deformation of the contour to capture the pole of the resolvent at z0 and applying the residue theorem, one finds that the time evolution behaves like e−iz0tψ0 = e −γte−ie0tψ0. the factor e−ite0ψ0 has an oscillatory time evolution similar to that of a bound state with energy e0, whereas the factor e −tγ is an exponentially decaying amplitude. the lifetime of the state is τ = γ−1. this is, roughly, the time it takes the amplitude to decay to e−1 times its original size. as noted above, there is no such state ψ0 ∈ h corresponding to a resonance z0 in the sense that hvψ0 = z0ψ0. since hv is self-adjoint and z0 has a nonzero imaginary part, the solutions of this eigenvalue equation are not in h. there are, however, approximate resonance states obtained by truncating such ψ0 to bounded regions, say k ⊂ rd. the truncated state χkψ0 ∈ h has an approximate time evolution like e−γte−ie0tχkψ0 showing that the amplitude of the resonance state in the bounded region k decays exponentially to zero. a typical situation for which resonances are expected to exist is the hydrogen atom hamiltonian hv = −∆ − |x| −1 acting on l2(r3), perturbed by an external constant electric field vpert(x) = ex1 in the x1-direction. the total stark hydrogen schrödinger operator is hv(e) = −∆ − |x|−1 + ex1. when e = 0, the spectrum of hv consists of an infinite sequence of eigenvalues en = −1/4n 2 plus the half line [0,∞). when e is turned on, the spectrum of hv(e) is purely absolutely continuous and equal to exactly the entire real line. there are no eigenvalues! it is expected that the bound states en of the hydrogen atom have become resonances for e 6= 0. these finite-lifetime states are observed in the laboratory. these resonances, in the zparametrization, have their real parts close to the eigenvalues en. their imaginary parts are exponentially small behaving like e−α/e. this means their lifetime is very long. the proof of the existence of these resonances for the the stark hydrogen hamiltonian was given by herbst [13] in 1979. the method of proof is perturbative in that the electric field strength is assumed to be very small. more generally, there are various models for which one can prove the existence of resonances using the smallness of some parameter. the semiclassical approximation is the most common regime. the quantum hamiltonian is written as hv(h) = −h 2∆ + v0 + v1 and h is considered as a small parameter. for a discussion of resonances in the semiclassical regime, see, for example, [14, chapters 20–23]. for more information on the semiclassical approximation for eigenvalues, eigenfunctions and resonances, see, for example, the monographs [10, 18, 33]. cubo 14, 3 (2012) scattering theory and resonances ... 21 if we inquire about the existence of resonances for the models studied here, hv = −∆ + v, with v ∈ c∞0 (rd), with no parameters, the proof is much harder and requires different techniques. melrose [19] gave perhaps the first proof of the existence of infinity many resonances for such hv. the proof holds for smooth, real-valued, compactly-supported potentials v ∈ c∞0 (rd), for d ≥ 3 odd. the proof requires two ingredients that will be presented here without proof. 5.1.1 small time expansion of the wave trace the wave group wv(t) associated with the schrödinger operator hv is defined as follows. let ∂t denote the partial derivative ∂/∂t. consider the wave equation associated with hv: (∂2t − hv)u = 0, u(t = 0) = u0, ∂tu(t = 0) = u1. (5.2) the solution can be expressed in terms of the initial conditions (u0,u1). the time evolution occurs on a direct sum of two hilbert spaces hfe = {(u0,u1) | ∫ [|∇u0|2 + |u1|2] < ∞}. this is the space of finite energy solutions. in two-by-two matrix notation, the time evolution acts as wv(t) ( u0 u1 ) = ( u ∂tu ) (5.3) the infinitesimal generator of the wave group wv(t) is the two-by-two matrix-valued operator av ≡ ( 0 1 hv 0 ) . (5.4) the evolution group wv(t) is unitary on hfe. similarly, the free wave group w0(t) is generated by a0 that is expressed as in (5.4) with v = 0. if hv ≡ −∆ + v ≥ 0, then this operator can be diagonalized. the diagonal form is ( √ hv 0 0 − √ hv ) . (5.5) in this case, the wave group wv(t) can be considered as two separate unitary groups e ±i √ hv t each acting on a single component hilbert space. the basic fact that we need is that the map t ∈ r → tr[wv(t) − w0(t)] is a distribution. this means that for any ρ(t), a smooth, compactly-supported function, the integral ∫ r dt ρ(t)tr[wv(t) − w0(t)] (5.6) is finite and bounded above by an appropriate sum of semi-norms of ρ. the distribution has a singularity at t = 0 and the behavior of the distribution at t = 0 has been well-studied. for d ≥ 3 22 peter d. hislop cubo 14, 3 (2012) odd, the wave trace has the following expansion as t → 0: tr[wv(t) − w0(t)] = (d−1)/2∑ j=1 wj(v)(−i) d−1−2jδ(d−1−2j)(t) + n∑ j=(d+1)/2 wj(v)|t| 2j−d + rn(t), (5.7) where the remainder rn(t) ∈ c2n−d(r). the first sum consists of derivatives of the delta function δ(t) at zero. we recall that for any smooth function f, these distributions are defined as 〈δj,f〉 = (−1)jf(j)(0). the second part of the sum consists of distributions that are polynomial in t. the coefficients wj(v) are integrals of the potential v and its derivatives. these are often called the ‘heat invariants’. the first three are: w1(v) = c1,d ∫ rd v(x) ddx w2(v) = c2,d ∫ rd v2(x) ddx w3(v) = c3,d ∫ rd (v3(x) + |∇v(x)|2) ddx, (5.8) where the constants cj,d are nonzero and depend only on the dimension d. for some insight as to why the trace in (5.7) exists, note that for ρ ∈ c∞0 (r), a formal calculation gives ∫ r ρ(t)tr[wv(t) − w0(t)] dt = tr( (fρ)(av ) − (fρ)(a0)). (5.9) the fourier transform fρ is a smooth, rapidly decreasing function. because v has compact support, the difference (fρ)(av )−(fρ)(a0) is in the trace class. this follows from the fact that the difference of the resolvents rv(z) k − r0(z) k is in the trace class for ℑz 6= 0 and k > d/2. 5.1.2 poisson formula the key formula that links the resonances with the trace of the difference of the wave groups is the poisson formula. in our context it was proved by melrose [20]. it is named this because of the analogy with the classical poisson summation formula. let f ∈ c∞(rd) be a schwarz function meaning that the function and all of its derivatives decay faster that 〈‖x‖〉−n, for any n ∈ n. the classical poisson summation formula states that ∑ k∈zd f(x + k) = ∑ k∈zd (ff)(k)e2πix·k. (5.10) the poisson formula for the wave group states that tr[wv(t) − w0(t)] = ∑ ξ∈rv m(ξ)ei|t|ξ, t 6= 0, (5.11) cubo 14, 3 (2012) scattering theory and resonances ... 23 where m(ξ) is the algebraic multiplicity of the resonance ξ. this multiplicity is defined as the rank of the residue of the resolvent at the pole ξ or, equivalently, by the rank of the contour integral: m(ξ) = rank (∫ γξ r(s) ds ) , (5.12) where γξ is a small contour enclosing only the pole ξ of the resolvent. it is important to note that the poisson formula (5.11) is not valid at t = 0. 5.1.3 melrose’s proof of the existence of resonances melrose [19, section 4.3] observed that the poisson formula (5.11) and the trace formula (5.7) can be used together to prove the existence of infinitely many resonances for schrödinger operators. theorem 17. let us suppose that d ≥ 3 is odd and that v ∈ c∞0 (rd; r). suppose also that wj(v) 6= 0 for some j ≥ (d + 1)/2. then hv has infinitely many resonances. in particular, for d = 3, since w2(v) = c2 ∫ v2(x) dx, for a positive constant c2 > 0, if v ∈ c∞0 (r3; r) is nonzero, then hv has an infinite number of resonances. proof. 1. suppose that hv has no resonances. then the right side of the poisson formula (5.11) is zero. on the other hand, it follows from the small time expansion (5.7) and the assumption that wj(v) 6= 0 for some j ≥ (d + 1)/2 that for t > 0 the right side of the expansion (5.7) is nonzero. note that for t > 0 all the contributions from the delta functions vanish. hence we obtain a contradiction. consequently, there must be at least one resonance. 2. if there are only finitely-many resonances, then the sum on the right in (5.11) is finite and the formula can be extended to t = 0. in particular, at t = 0, it is a finite positive number greater than or equal to the number of resonances. on the other hand, looking at the trace formula (5.7), if only one or more of the coefficients wj(v) 6= 0 for j > (d+1)/2, then the trace is zero at t = 0 (the coefficients of the derivatives of the delta functions being zero), so we get a contradiction. hence, at least one of the coefficients of a delta function term is nonzero. then the trace formula indicates that the distribution tr[wv(t) − w0(t)] is not continuous at t = 0 whereas the poisson formula indicates that it is continuous through t = 0, and we again obtain a contradiction. consequently, there must be an infinite number of resonances. we remark that in the even dimensional case for d ≥ 4, sá barreto and tang [36] proved the existence of at least one resonance for a real-valued, compactly-supported, smooth nontrivial potential. having settled the question of existence, we now turn to counting the number of resonances. 24 peter d. hislop cubo 14, 3 (2012) 5.2 the one-dimensional case: zworski’s asymptotics as with many problems, the one-dimensional case is special since many techniques of ordinary differential equations can be used. the most complete result on resonances for hv = −d 2/dx2 +v on l2(r) with a compactly-supported potential was proven by zworski [46]. theorem 18. let v ∈ l∞0 (r). then the number of resonances nv(r) with modulus less that r > 0 satisfies: nv(r) = 2 π ( sup x,y∈supp v |x − y| ) r + o(r). (5.13) there are extensions of this result to a class of super-exponentially decaying potentials due to r. froese [11]. we will not comment further on the one-dimensional case. 5.3 estimates on the number of resonances: upper bounds the resonance counting function counts the number of poles, including multiplicities, of the meromorphic continuation of the cut-off resolvent in c− for d odd, and on λ for d even. we will concentrate on the odd d-dimensional case, although we will give comments on the even dimensional case too. for any r > 0, we define nv(r) as nv(r) = #{j | λj(v) satisfies |λj(v)| ≤ r and ℑλj(v) < 0}. (5.14) this function is monotone increasing in r. it is the analogue of the eigenvalue counting function nm(r) studied by weyl to count the number of eigenvalues of the laplace-beltrami operator on a compact riemannian manifold m with size less than r > 0. the weyl upper bound on the eigenvalue counting function is nm(r) ≤ cdvol(m)〈r〉d, (5.15) where 〈r〉 = √ 1 + r2. it is natural to ask if the resonance counting function nv(r) satisfies a similar upper bound. since melrose’s early work [21], many people have established upper bounds on nv(r) with increasing optimality. zworski [49] presents a good survey of the state-of-the-art up to 1994. the optimal upper bound, having the same polynomial behavior as weyl’s eigenfunction counting function (5.15), was achieved by zworski [47]. a significant simplification of the proof was given by vodev [41]. theorem 19. for d ≥ 3 odd, the resonance counting function nv(r) satisfies nv(r) ≤ c(d,v)〈r〉d, (5.16) for a constant 0 < c(d,v) < ∞ depending on d and v. a sketch of the proof of this theorem will be given following the beautiful exposition of zworski [49, section 5], using vodev’s simplification [41]. one basic idea of the proof is to find a suitable cubo 14, 3 (2012) scattering theory and resonances ... 25 analytic or meromorphic function that has zeros exactly at the resonances. suppose h(λ) is one such function analytic on c. then one can count the number of zeros using jensen’s formula. this formula relates the number of zeros of h to growth properties of h. if a circle of radius r > 0 crosses no zero of h, if h(0) 6= 0, and if ak are the zeros of h inside the circle, then jensen’s formula states that nh(r)∑ k=1 log ( r |ak| ) = 1 2π ∫2π 0 log |h(reiθ)| dθ − log |h(0)|. (5.17) if we only sum over those zeros inside the circle of radius r/2, we have that log(r/|ak|) ≥ log2, so that nh(r/2)[log2] ≤ 1 2π ∫2π 0 | log |h(reiθ)|| dθ + | log |h(0)||. (5.18) this inequality shows that it suffices to bound h on circles of radius 2r in order to count the number of zeros inside the circle of radius r > 0. we will use some inequalities for singular values, the proofs of which can be found in [39]. proof. 1. the first observation is that the operator (vr0(λ)χv) d+1 is in the trace class for ℑλ ≥ 0. consequently, the following determinant is well-defined: h(λ) ≡ det(1 − (vr0(λ)χv)d+1). (5.19) this function is analytic on c+ with at most a finite number of zeros corresponding to the eigenvalues of hv. it follows from section 4.3 that this function has an analytic continuation to all of c. furthermore, the zeros of this function for ℑλ < 0 include with the resonances of hv that are given as the zeros of the analytic continuation of 1 + vr0(λ)χv according to (4.20). the problem, then, is to count the number of zeros of the analytic function h(λ) inside a ball of radius r > 0 in c. by jensen’s inequality (5.18), it suffices to obtain a growth estimate on h of the form |h(λ)| ≤ c1ec2|λ| d . (5.20) 2. we first estimate h in the half space ℑλ ≥ 0 using the fact that v has compact support contained inside of a bounded region ω. let −∆ω ≥ 0 denote the dirichlet laplacian on ω. by weyl’s bound (5.15), the jth eigenvalue λj(ω) of −∆ω grows like λj(ω) ∼ j 2/d. furthermore, we have ∆ωv = ∆v. using these ideas and the simple inequality for the singular values µj(ab) ≤ ‖a‖µj(b), we have µj(χvr0(λ)χv) = µj((−∆ω + 1) −1/2(−∆ω + 1) 1/2χvr0(λ)χv) ≤ ‖(−∆ω + 1)1/2χvr0(λ)χv‖ µj((−∆ω + 1)−1/2) ≤ cj−1/d. (5.21) it is important to note that the operator χvr0(λ)χv : l 2(rd) → h1(rd) is bounded uniformly in λ, for ℑλ ≥ 0. consequently, the norm ‖(−∆ω + 1)1/2χvr0(λ)χv‖ is bounded uniformly in λ in the upper half space. upon squaring this norm, the operator −∆ω can be replaced by −∆ 26 peter d. hislop cubo 14, 3 (2012) because of the support of χv. since µm+k−1(ab) ≤ µk(a)µm(b), we have µ2j−1(a2) ≤ µj(a)2, and consequently, for all large j µj((χvr0(λ)χv) d+1) ≤ cj−(d+1)/d. (5.22) it follows that |h(λ)| ≤ c for ℑλ ≥ 0. 3. for ℑλ < 0, we make use of the following formula from scattering theory used already in section 4.2. for λ ∈ r, we have χv(r0(λ) − r0(−λ))χv = cd(λ d−2) teχ(λ)eχ(λ), (5.23) where eχ(λ) : l 2(rd) → l2(sd−1) is given by (eχ(λ)g)(ω) ≡ ∫ rd eiλω·xχv(x)g(x) d dx, (5.24) and teχ(λ) denotes the transpose of this operator. this formula can be extended to all of c. we compute the singular values of the operator on the left in (5.23): µj(χv(r0(λ) − r0(−λ))χv) ≤ c|λ|d−2ec2|λ|µj(eχ(λ)). (5.25) since eχ(λ) ∗ : l2(sd−1) → l2(rd), the operator eχ(λ) ∗eχ(λ) : l 2(sd−1) → l2(sd−1). this is a crucial observation since the operator acts on a d − 1 dimensional space. without this reduction, one obtains an upper bound but with exponent d + 1 rather than the optimal exponent d. in a manner similar to (5.21), we compute for any m > 0, µj(eχ(λ)) ≤ µj((−∆sd−1 + 1)−m(−∆sd−1 + 1)meχ(λ)) ≤ ‖(−∆sd−1 + 1)meχ(λ))‖l2(sd−1) µj((−∆sd−1 + 1)−m) ≤ cm(2m)!j−2m/(d−1)ec|λ|. (5.26) this follows from the explicit formula for the kernel of eχ(λ)), eχ(λ)(ω,x) = e −iλω·xχv(x). (5.27) in particular, the factor (2m)! comes from differentiating the exponential factor. using stirling’s formula for the factorial, we obtain from (5.25)–(5.26) µj(χv(r0(λ) − r0(−λ))χv) ≤ |λ|d−2ec2|λ|cm(2m + 1)2m+(1/2)(j−1/(d−1))2m. (5.28) we now optimize over the free parameter m by choosing m ∼ j−1/(d−1). as a result, we obtain µj(χv(r0(λ) − r0(−λ))χv) ≤ ec|λ|e−cj 1/(d−1) . (5.29) 4. we now combine (5.21) with (5.29). for this, we need fan’s inequality for singular values [39, theorem 1.7] that states that µn+m+1(a + b) ≤ µm+1(a) + µn+1(b). (5.30) cubo 14, 3 (2012) scattering theory and resonances ... 27 for ℑλ < 0, we write µj(χvr0(λ)χv) = µj([χv(r0(λ) − r0(−λ))χv] + χvr0(−λ)χv). (5.31) applying fan’s inequality (5.30) to the right side of (5.31), we find that for ℑλ ≤ 0, the singular values satisfy µj(χvr0(λ)χv) ≤ ec|λ|e−cj 1/(d−1) + cj−1/d. (5.32) taking the (d + 1)st power of the operators, as in (5.22), we find µj((χvr0(λ)χv) d+1 ) ≤ ec|λ|e−cj 1/(d−1) + cj−(d+1)/d, (5.33) for a constant c > 0. as j → ∞, the first term dominates until j ∼ [|λ|d−1], where [·] denotes the integer part. we then use the weyl estimate for the determinant (see [39]), factorize the product using the first estimate in (5.33) for j ≤ [d|λ|d−1], to obtain |h(λ)| ≤ | det(1 + (vr0(λ)χv)d+1)| ≤ π∞j=1(1 + µj((vr0(λ)χv)d+1)) ≤ ( π [d|λ|d−1] j=1 e c|λ| ) ( πj≥[d|λ|d−1](1 + c2j −(d+1)/d) ) ≤ cec|λ| d . (5.34) this establishes (5.20) so by jensen’s inequality (5.18) we obtain the optimal upper bound on the resonance counting function. upper bounds for super-exponentially decaying potentials in d ≥ 3 odd dimensions were proved by r. froese [12]. there are fewer results in even dimensions. we refer to [7] for a discussion and the papers [15, 42, 43]. 5.4 estimates on the number of resonances: lower bounds one might hope to have a lower bound on the number of resonances of the form nv(r) ≥ cdrd. (5.35) this is known to hold in two cases. the first case is zworski’s result for d = 1. the second is for a class of spherically symmetric potentials in dimension d ≥ 3 odd. zworski proved that if v(r) has the property that v ′(a) 6= 0, where a > 0 is the radius of the support of v, then an asymptotic expansion holds for the number of resonances: nv(r) = cda drd + o(rd), d ≥ 3 and odd. (5.36) in general, for v ∈ l∞0 (rd) (or, even v ∈ c∞0 (rd)), there is presently no known proof of the optimal lower bound (5.35). there are some partial results for d ≥ 3 odd. these include nonoptimal lower bounds, estimates on the number of purely imaginary poles for potentials with fixed sign, and counterexamples made from certain complex potentials. 28 peter d. hislop cubo 14, 3 (2012) 5.4.1 nonoptimal lower bounds for the case of d ≥ 3 odd, the first quantitative lower bounds for the resonance counting function for nontrivial, smooth, real-valued v ∈ c∞0 (rd), not of fixed sign, were proved in [2]. in particular, it was proved there that lim sup r→∞ nv(r) r(logr)−p = ∞, (5.37) for all p > 1. for the same family of potentials, sá barreto [34] improved this to lim sup r→∞ nv(r) r > 0. (5.38) we mention that, in particular, all these lower bounds require the potential to be smooth. concerning lower bounds in the even dimensional case for d ≥ 4, sá barreto [35] studied the resonance counting function nsab(r) defined to be the number of resonances λj with 1/r < |λj| < r and | argλj| < logr. as r → ∞, this region in the riemann surface λ opens like logr. sá barreto proved that for even d ≥ 4, lim sup r→∞ nsab(r) (logr)(log logr)−p = ∞, (5.39) for all p > 1. 5.4.2 purely imaginary poles lax and phillips [17] noticed that for odd dimensions d ≥ 3, the wave operator associated with exterior obstacle scattering has an infinite number of purely imaginary resonances. they remarked that their proof held for schrödinger operators with nonnegative, compactly-supported, nontrivial potentials. vasy [40] used their method to prove that a schrödinger operator hv with a compactlysupported, bounded, real-valued potential with fixed sign (either positive or negative) has an infinite number of purely imaginary resonances. these resonances are poles of the meromorphic continuation of the resolvent of the form −iµj(v), with µj(v) > 0. in the z = λ 2 plane, these are located on the second riemann sheet of the square root function. furthermore, vasy is able to count these poles and prove the following lower bound nv(r) ≥ cdrd−1. (5.40) this is not an optimal lower bound on the total number of resonances. recently, the author and t. j. christiansen [7] proved that in even dimension there are no purely imaginary resonances on any sheet for hv with bounded, positive, real-valued potentials with compact support. 5.4.3 complex potentials most surprisingly, christiansen [3] gave examples of compactly supported, bounded complex-valued potentials having no resonances in any dimension d ≥ 2! this result, while interesting in its own cubo 14, 3 (2012) scattering theory and resonances ... 29 right, means that any technique that provides a result of the type (5.35) must be sensitive to whether the potential is realor complex-valued. 6 maximal order of growth is generic for the resonance counting function there is one general result that is a weak form of (5.35) due to the author and t. j. christiansen [5]. this result states that for almost all potentials v ∈ l∞0 (k), for a compact subset k ⊂ rd, realor complex-valued, the lower bound holds in the following sense as determined by the order of growth of the resonance counting function nv(r). definition 20. the order of growth of the monotone increasing function nv(r) is defined by ρv ≡ lim r→∞ lognv(r) logr , (6.1) if the limit exists and is finite. because of the upper bound (5.16), the order of growth of the resonance counting function is bounded from above as ρv ≤ d. we say that the order of growth is maximal for a potential v if ρv = d. by “almost all potentials” referred to above, we mean that the set of potentials in l∞0 (k), for a fixed compact subset k ⊂ rd with nonempty interior, for which the resonance counting function has maximal order of growth, is a dense gδ-set. recall that a gδ-set is a countable intersection of open sets. one sometimes says that a property that holds for all elements in a dense gδ-set is generic. (added in proof: for some recent developments, see dinh and vu arxiv:1207.4273v1.) 6.1 generic behavior: odd dimensions the basic theorem on generic behavior is the following. theorem 21. [5] let d ≥ 3 be odd and k ⊂ rd be a fixed, compact set with nonempty interior. let mf(k) ⊂ l∞0 (k), for f = r or f = c, be the set of all real-valued, respectively, complex-valued potentials in l∞0 (k) such that the resonance counting function nv(r) has maximal order of growth. then, the set mf(k) is a dense gδ set for f = r or f = c. this holds for both real-valued and complex-valued potentials. by [3], we know there are complex potentials with zero order of growth. an interesting open question is whether there exist real-valued potentials in l∞0 (r d) for which the resonance counting function has less than maximal order of growth. the proof of this theorem uses the s-matrix and its continuation to the entire complex plane. in section 3, we defined the scattering matrix for the pair h0 = −∆ and hv = h0 + v. the s-matrix s(λ), acting on l2(sd−1), is the bounded operator defined in (3.8). in the case that v is 30 peter d. hislop cubo 14, 3 (2012) real-valued, this is a unitary operator for λ ∈ r. under the assumption that supp v is compact, the scattering amplitude a(λ) : l2(sd−1) → l2(sd−1), defined in (3.9), is a trace class operator. hence, the function fv(λ) ≡ dets(λ), (6.2) is well-defined, at least for ℑλ > 0 sufficiently large. what are the meromorphic properties of fv(λ)? as proved in theorem 16, the s-matrix has a meromorphic continuation to the entire complex plane with finitely many poles for ℑλ > 0 corresponding to eigenvalues of hv, and poles in ℑλ < 0 corresponding to resonances. we recall that if ℑλ0 ≥ c0〈‖v‖l∞〉, the multiplicity of λ0, as a zero of detsv(λ), and of −λ0, as a pole of the cut-off resolvent rv(λ), coincide. consequently, the function fv(λ) is holomorphic for ℑλ > c0〈‖v‖l∞〉, and well-defined for ℑλ ≥ 0 with finitely many poles corresponding to the eigenvalues of hv. hence, the problem of estimating the number of zeros of fv(λ) in the upper half plane is the same as estimating the number of resonances in the lower half plane. the estimates on fv(λ) are facilitated in the odd dimensional case by the well-known representation of fv(λ) in terms of canonical products. let g(λ;p) be defined for integer p ≥ 1, by g(λ;p) = (1 − λ)eλ+λ 2/2+···+λp/p, (6.3) and define p(λ) = πλj∈rv ,λj 6=0 g(λ/λj;d − 1). (6.4) then the function fv(λ) may be written as fv(λ) = αe ig(λ)p(−λ) p(λ) , (6.5) for some constant α > 0 and where g(λ) is a polynomial of order at most d. careful study of the scattering matrix and the upper bound of theorem 19 may be used to show that fv(λ) is of order at most d in the half-plane ℑλ > c0〈‖v‖∞〉, see [48]. we consider a fixed compact set k ⊂ rd with nonempty interior. let m(k) be the subset of potentials in l∞0 (k) having a resonance counting function with maximal order of growth. we can separately consider realor complex-valued potentials. the proof of theorem 21 requires that we prove 1) that m(k) is a gδ-set, and 2) that m(k) is dense in l∞0 (k). the proof that m(k) is a gδ-set uses standard estimates on the s-matrix as in [5]. for n,m,j ∈ n with j > 2n+1, and for q > 0, we define sets of potentials a(n,m,q,j) ⊂ l∞0 (k) by a(n,m,q,j) ≡ {v ∈ l∞0 (k) | ‖v‖l∞ ≤ n, log | det(sv(λ))| ≤ m|λ|q, for ℑλ ≥ 2n + 1 and |λ| ≤ j} (6.6) one proves that these sets are closed. more importantly, we can use these sets to characterize the set of potentials having a resonance counting function with an order of growth strictly less that d. cubo 14, 3 (2012) scattering theory and resonances ... 31 for this, we define sets b(n,m,q) by b(n,m,q) ≡ ⋂ j≥2n+1 a(n,m,q,j). (6.7) one proves that if nv(r) has order of growth strictly less than d, then one can find (n,m,ℓ) ∈ n3 so that v ∈ b(n,m,d − 1/ℓ). since the sets a(n,m,j,q) are closed, so are the sets b(n,m,j). one notes that ∪(n,m,j)∈n3b(n,m,j) is an fσ set. the final step of the proof is to show that m(k) is the complement of this set. it follows that m(k) is a gδ-set. the proof of the density of m(k) is more involved and relies on machinery from complex analysis as developed in [4]. the basic idea is to consider a wider family of potentials v(x;z) holomorphic in the complex variable z ∈ ω ⊂ c, for some open subset ω. the construction of the s-matrix goes through for these complex-valued potentials. the key result is that if for some z0 ∈ ω the order of growth ρv(z0) for nv(z0) is equal to d, then there is a pluripolar subset e ⊂ ω so that the order of growth for all potentials v(z), with z ∈ ω\e, is equal to d. a pluripolar set is very small, in particular, the lebesgue measure of e ∩ r is zero. how do we know there is a potential for which nv(r) has maximal order of growth? for d ≥ 3 odd, we can use the result of zworski [47]. as mentioned in section 5.4, zworski proved the an asymptotic expansion for nv(r) for a class of radially symmetric potentials with compact support. let v0 be one of these potentials so that v0 ∈ m(k). to prove the density of m(k) in l∞0 (rd), we take any v1 ∈ l∞0 (k) and form v(z) = zv0 + (1 − z)v1. this is a holomorphic function of z for z ∈ ω = c. we apply the result described above to this family of holomorphic potentials. in particular, for z0 = 1, we have v(z0) = v0 and ρv(z0) = d by zworski’s result. let e ⊂ c be the pluripolar set so that for z ∈ c\e, the order of growth ρv(z) = d. since the lebesgue measure of e ∩ r is zero, we can find z ∈ r\(e ∩ r), with |z| as small as desired, for which ρv(z) = d. so, given ǫ > 0, we take z̃ ∈ r\(e ∩ r) so that |z̃| ≤ ǫ(1 + ‖v1‖l∞ + ‖v0‖l∞)−1. with this choice, we have ‖v1 − v(z̃)‖l∞ ≤ |z̃| (‖v1‖l∞‖v0‖l∞) ≤ ǫ. (6.8) this proves the density of m(k) in l∞0 (rd). note that we can take v0 real-valued and so v(z̃) is real-valued. we remark that the representation (6.5) is not available in the even dimensional case. 6.2 generic behavior: even dimensions we now summarize the corresponding results in the even dimensional case. let χv ∈ c∞0 (rd) be a smooth, compactly-supported function satisfying χvv = v, and denote the resolvent of hv by rv(λ) = (hv −λ 2)−1. in the even-dimensional case, the operator-valued function χvrv(λ)χv has a meromorphic continuation to the infinitely-sheeted riemann surface of the logarithm λ. we denote by λm the m th open sheet consisting of z ∈ λ with mπ < argz < (m + 1)π. the physical sheet corresponds to λ0 and it is identified with the upper half complex plane. we denote the number 32 peter d. hislop cubo 14, 3 (2012) of the poles nv,m(r) of the meromorphic continuation of the truncated resolvent χvrv(λ)χv on each sheet λm, counted with multiplicity, and with modulus at most r > 0. the order of growth of the resonance counting function nv,m(r) for hv on the m th-sheet is defined by ρv,m ≡ lim sup r→∞ lognv,m(r) logr . (6.9) it is known that ρv,m ≤ d for d ≥ 2 even [41, 42]. as in the odd dimensional case, it is proved that generically (in the sense of baire typical) the resonance counting function has the maximal order of growth d on each non-physical sheet. theorem 22. let d ≥ 2 be even, and let k ⊂ rd be a fixed, compact set with nonempty interior. let mf(k) ⊂ l∞0 (k), for f = r or f = c, be the set of all real-valued, respectively, complexvalued potentials in l∞0 (k) such that the resonance counting functions nv,m(r), for m ∈ z\{0}, have maximal order of growth. then, the set mf(k) is a dense gδ set for f = r or f = c. this theorem states that for a generic family of real or complex-valued potentials in l∞0 (k), the order of growth of the resonance counting function is maximal on each sheet, ρv,m = d, for m ∈ z\{0}. this implies that there are generically infinitely many resonances on each nonphysical sheet. there are two challenges in proving theorem 22. the first is to construct a function whose analytic extension to the mth-sheet λm has zeros at the resonances of hv. this function will substitute for (6.2). the second problem is prove a lower bound (5.35) for some potential in l∞0 (k) in even dimensions. to resolve the first problem, we use the following key identity, that follows from (4.16) and the formulas for the meromorphic continuation of hankel functions (see [6, section 6] or [23, chapter 7]), relating the free resolvent on λm to that on λ0, for any m ∈ z, r0(e imπλ) = r0(λ) − m(d)t(λ), where m(d) = { m mod 2 d odd m d even. (6.10) the operator t(λ) on l2(rd) has a schwartz kernel t(λ;x,y) = iπ(2π)−dλd−2 ∫ sd−1 eiλ(x−y)·ωdω, (6.11) see [19, section 1.6]. this operator is related to m(λ) introduced in section 4.2 in (4.13) (see also (5.24)). we note that for any χ ∈ c∞0 (rd), χt(λ)χ is a holomorphic trace-class operator for λ ∈ c. the operator t(λ) has a kernel proportional to |x−y|(−d+2)/2j(d−2)/2(λ|x−y|) when d is odd, and to |x − y|(−d+2)/2n(d−2)/2(λ|x − y|) when d is even. the different behavior of the free resolvent for d odd or even is encoded in (6.10). by the second resolvent formula (4.20), the poles of rv(λ) with multiplicity, correspond to the zeros of i+vr0(λ)χv. we can reduce the analysis of the zeros of the continuation of i+vr0(λ)χv cubo 14, 3 (2012) scattering theory and resonances ... 33 to λm to the analysis of zeros of a related operator on λ0 using (6.10). if 0 < argλ < π and m ∈ z, then eimπλ ∈ λm, and i + vr0(e imπλ)χ = i + v(r0(λ) − mt(λ))χv = (i + vr0(λ)χv)(i − m(i + vr0(λ)χv) −1vt(λ)χv). for any fixed v ∈ l∞0 (rd), there are only finitely many poles of (i+vr0(λ)χv)−1 with 0 < argλ < π. thus fv,m(λ) = det(i − m(i + vr0(λ)χv) −1vt(λ)χv) (6.12) is a holomorphic function of λ when 0 < argλ < π and |λ| > c0〈‖v‖l∞〉. moreover, with at most a finite number of exceptions, the zeros of fv,m(λ), with 0 < argλ < π correspond, with multiplicity, to the poles of rv(λ) with mπ < argλ < (m + 1)π. henceforth, we will consider the function fv,m(λ), for m ∈ z∗ ≡ z\{0}, on λ0. for d odd, we are only interested in m = −1. in this case, the zeros of fv,−1(λ), for λ ∈ λ0, correspond to the resonances. this provides an alternative to the s-matrix formalism, as presented in section 6.1, for estimating the resonance counting function in the odd dimensional case. the second problem in even dimensions is to prove that there are some potentials in l∞0 (k) for which the resonance counting function has the correct lower bound on each sheet. this is done by an explicit calculation. we prove (5.35) in even dimensions d ≥ 2 for schrödinger operators hv with radial potentials v(x) = v0χbr(0)(x), with v0 > 0, using separation of variables and uniform asymptotics of bessel and hankel functions due to olver [23, 24, 25]. this method can also be used in odd dimensions as an alternative to [47] thus providing examples as required in section 6.1. 7 topics not covered and some literature this notes focussed on perturbations of the laplacian on rd by real-valued, smooth, compactly supported potentials. this is just one family of examples where resonances arise. other topics concerning resonances include: (1) complex-spectral deformation method and resonances (2) obstacle scattering (3) resonance free regions (4) resonances for the wave equation (5) resonances for elastic media (6) resonances for manifolds hyperbolic at infinity (7) semiclassical theory of resonances 34 peter d. hislop cubo 14, 3 (2012) (8) description of resonance wave functions (9) approximate exponential decay of resonance states (10) local energy decay estimates there are some reviews on resonances that cover many aspects of the theory in this list. these reviews include: (1) the long discussion by m. zworski [49] that covers the complex scaling method developed by sjöstrand and zworski (inspired by the baslev-combes method) and its applications. (2) g. vodev has written an expository article in cubo [44]. many aspects of resonances for elastic bodies and obstacle scattering are described there. (3) the proof of the generic properties of the resonance counting function for even and odd dimensions is described in christiansen and hislop [8], an expository summary written for les journées edp 2008 evian available on the arxiv and from cedram. (4) text book versions of spectral deformation and quantum resonances, with an emphasis on the semiclassical regime, can be found in [9] and [14]. finally, for a lighter and intuitive discussion of resonances, the reader is referred to zworski’s article resonances in physics and geometry that appeared in the notices of the american mathematical society [50]. 7.1 acknowledgements these notes are an extended version of lectures on scattering theory and resonances given as a mini-course during the penn state-göttingen international summer school in mathematics at the pennsylvania state university in august 2010. i would like to thank the organizers juan gil, thomas krainer, gerardo mendoza, and ingo witt for the invitation to present this mini-course. i would like to thank gerardo mendoza and peter a. perry for some useful discussions. i also thank tanya christiansen for our enjoyable collaboration. i was partially supported by nsf grant dms 0803379 during the time this work was done. 8 appendix: assorted results two groups of results that are related to material in the text are summarized here. the first is a synopsis of the spectral theory of linear self-adjoint operators. the second is an analysis of the time evolution of states lying in various spectral subspaces of a self-adjoint operator. detailed discussions of these topics may be found in the reed-simon series [29]-[32], for example, and many other texts. cubo 14, 3 (2012) scattering theory and resonances ... 35 8.1 spectral theory let a be a self-adjoint operator on a separable hilbert space h. then, there is a direct sum decomposition h = hac(a)⊕hsc(a)⊕hpp(a) into three orthogonal subspaces that are a-invariant in that a : d(a) ∩ hx(a) → hx(a) for x = ac,sc,pp. the pure point subspace hpp(a) is the closure of the span of all the eigenfunctions of a. the continuous subspace hcon(a) ≡ hac(a) ⊕ hsc(a) is the orthogonal complement of hpp(a). for most schrödinger operators hv = −∆ + v, one has hsc(hv) = ∅. the proof of the absence of singular continuous spectrum is one of the main applications of the mourre estimate, see the discussion in section 4.1, [9, chapter 4], and the original paper [22]. as the names suggest, there is a measure associated with a self-adjoint operator and this measure has a lebesgue decomposition into pure point and continuous measures. the continuous measure admits a decomposition relative to lebesgue measure into a singular continuous and absolutely continuous parts. 8.2 the rage theorem the rage theorem (ruelle, amrein, georgescu, enss) (see, for example, [9, section 5.4]) is a general result about the averaged time evolution of states in the continuous subspace hcont(a) of a self-adjoint operator a. theorem 23. let a be a self-adjoint operator and φ ∈ hcont(a), where hcont(a) is the continuous spectral subspace of a. suppose that c is a bounded operator and that c(a + i)−1 is compact. then, we have lim t→∞ 1 2t ∫t −t ‖ce−itaφ‖ dt = 0. (8.1) furthermore, if φ ∈ h satisfies (8.1), then φ ∈ hcont(a). let a = hv be a schrödinger operator of the type considered here, and c = χbr(0), the characteristic function on a ball of radius r > 0 centered at the origin. the rage theorem (8.1) states that a state, initially localized near the origin, and in the continuous spectral subspace of hv, will eventually leave this neighborhood of the origin in this time-averaged sense. the continuous spectral subspace hcont(hv) has a further decomposition into the singular and absolutely continuous subspaces. it is the possible recurrent behavior of states in the singular continuous subspace that requires the time averaging in (8.1). corollary 24. let hv be a self-adjoint operator on l 2(rd). let φ ∈ hac(hv), where hac(hv) is the absolutely continuous spectral subspace of hv. let χk be the characteristic function for a compact subset k ⊂ rd. then, we have lim t→∞ ‖χkuv(t)φ‖ = 0. (8.2) as one might expect, if φ is a finite linear combination of eigenfunctions, then the state χkuv(t)φ remains localized for all time. indeed, for any ǫ > 0, there is a compact subset kǫ ⊂ rd so that ‖χrd\kǫuv(t)φ‖ < ǫ, for all t. 36 peter d. hislop cubo 14, 3 (2012) received: november 2010. revised: enero 2012. references [1] w. o. amrein, j. m. jauch, k. b. sinha, scattering theory in quantum mechanics. reading, ma: w. a. benjamin, inc., 1977. [2] t. christiansen, some lower bounds on the number of resonances in euclidean scattering, math. res. lett. 6 (1999), no. 2, 203–211. [3] t. christiansen, schrödinger operators with complex-valued potentials and no resonances, duke math. journal 133, no. 2 (2006), 313-323. [4] t. christiansen, several complex variables and the distribution of resonances for potential scattering, commun. math. phys 259 (2005), 711-728. 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[44] g. vodev, resonances in euclidean scattering, cubo matemática educacional 3 no. 1, enero 2001, 319–360. [45] d. yafaev, mathematical scattering theory, providence, ri: ams, 2000. [46] m. zworski, distribution of poles for scattering on the real line, j. funct. anal. 73 (1987), 277–296. cubo 14, 3 (2012) scattering theory and resonances ... 39 [47] m. zworski, sharp polynomial bounds on the number of scattering poles of radial potentials, j. funct. anal. 82 (1989), 370–403. [48] m. zworski, poisson formulae for resonances, séminaire sur les equations aux dérivées partielles, 1996–1997, exp. no. xiii, 14 pp., ecole polytech., palaiseau, 1997. [49] m. zworski, counting scattering poles, in: spectral and scattering theory (sanda, 1992), 301–331, lectures in pure and appl. math. 161, new york: dekker, 1994. [50] m. zworski, resonances in physics and geometry, notices amer. math. soc. 46, no. 3 (1999), 319–328. inversesemigrouplap.dvi cubo a mathematical journal vol.12, no¯ 03, (83–97). october 2010 the semigroup and the inverse of the laplacian on the heisenberg group1 aparajita dasgupta department of mathematics, indian institute of science, bangalore–560012, india email: adgupta@math.iisc.ernet.in and m.w. wong department of mathematics and statistics, york university, 4700 keele street, toronto, ontario m3j 1p3, canada email: mwwong@mathstat.yorku.ca abstract by decomposing the laplacian on the heisenberg group into a family of parametrized partial differential operators l̃τ,τ ∈ r \ {0}, and using parametrized fourier-wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by l̃τ, and the inverse of l̃τ. using these formulas and estimates, we obtain sobolev estimates for the one-parameter semigroup and the inverse of the laplacian. 1this research has been supported by the natural sciences and engineering research council of canada. 84 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) resumen mediante descomposición del laplaceano sobre el grupo de heisenberg en una familia de operadores diferenciales parciales parametrizados l̃τ,τ ∈ r\{0}, y usando transformada de fourier-wigner parametrizada, damos fórmulas y estimativas para la continuidad fuerte del semigrupo generado por l̃τ, y la inversa de l̃τ. usando esas fórmulas y estimativas obtenemos estimativas de sobolev para el semigrupo a un parámetro y la inversa del laplaceano. key words and phrases: heisenberg group, laplacian, parametrized partial differential operators, hermite functions, fourier-wigner transforms, heat equation, one parameter semigroup, inverse of laplacian, sobolev spaces. math. subj. class.: 47f05, 47g30, 35j70. 1 the laplacian on the heisenberg group if we identify r2 with the complex plane c via r 2 ∋ (x, y) ↔ z = x + i y ∈ c and let h = c×r, then h becomes a non-commutative group when equipped with the multiplication · given by (z, t) · (w, s) = ( z + w, t + s + 1 4 [z, w] ) , (z, t), (w, s) ∈ h, where [z, w] is the symplectic form of z and w defined by [z, w] = 2 im(zw). in fact, h is a unimodular lie group on which the haar measure is just the ordinary lebesgue measure d z dt. let h be the lie algebra of left-invariant vector fields on h. a basis for h is then given by x , y and t, where x = ∂ ∂x + 1 2 y ∂ ∂t , y = ∂ ∂y − 1 2 x ∂ ∂t , cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 85 and t = ∂ ∂t . the laplacian ∆h on h is defined by ∆h = −(x 2 + y 2 + t 2). a simple computation gives ∆h = −∆− 1 4 (x 2 + y2) ∂2 ∂t2 + ( x ∂ ∂y − y ∂ ∂x ) ∂ ∂t − ∂2 ∂t2 , where ∆= ∂2 ∂x2 + ∂2 ∂y2 . let g be the riemannian metric on r3 given by g(x, y, t) =   1 0 y/2 0 1 −x/2 y/2 −x/2 1 4 (x2 + y2)   for all (x, y, t) ∈ r3. then ∆h is also given by −∆h = 1 √ det g ∑ 1≤ j,k≤3 ∂j ( √ det g g j,k∂k), where ∂1 = ∂/∂x, ∂2 = ∂/∂y, ∂3 = ∂/∂t. since the symbol σ(∆h) of ∆h is given by σ(∆h)(x, y, t;ξ,η,τ) = ( ξ+ 1 2 yτ )2 + ( η− 1 2 xτ )2 +τ2 for all (x, y, t) and (ξ,η,τ) in r3, it is easy to see that ∆h is an elliptic partial differential operator on r3 but not globally elliptic in the sense of shubin [11]. let us recall that ∆h is globally elliptic if there exist positive constants c and r such that |σ(∆h)(x, y, t;ξ,η,τ)| ≥ c ( 1 +|x|+|y|+|t|+|ξ|+|η|+|τ| )2 whenever |x|+|y|+|t|+|ξ|+|η|+|τ| ≥ r. the aim of this paper is to give new estimates for the strongly continuous one-parameter semigroup e−u∆h , u > 0, generated by ∆h and the inverse ∆−1h of ∆h. more precisely, we use the sobolev spaces l2s (h), s ∈ r, as in [1, 2] to estimate ‖e −u∆h f ‖ l2s (h) , u > 0, in terms of ‖f ‖l2 (h) for all f in l2(h), and to give an estimate for ‖e−u∆h f ‖l2(h) in terms of ‖f ‖l2s (h). these sobolev spaces are also used to estimate ||∆−1 h f || l2 s+2 (h) in terms of ||f || l2s (h) for all f in l2s (h). 86 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) the function f on h× (0,∞) given by f(z, t, u) = (e−u∆h f )(z, t), (z, t) ∈ h, u > 0, is in fact the solution of the initial value problem    ∂f ∂u (z, t, u) = −(∆hf)(z, t, u), f(z, t, 0) = f (z, t), (z, t) ∈ h, u > 0, (z, t) ∈ h, for the laplacian ∆h. using the same techniques as in [1], we get for all f ∈ l2(h) and u > 0, (e −u∆h f )(z, t) = (2π)−1/2 ˆ ∞ −∞ e −itτ (e −ul̃τ f τ)(z) dτ, (z, t) ∈ h, (1.1) where l̃τ, τ ∈ r \ {0}, is given by l̃τ = −∆+ 1 4 (x 2 + y2)τ2 − i ( x ∂ ∂y − y ∂ ∂x ) τ+τ2 and f τ is the function on c given by f τ (z) = (2π)−1/2 ˆ ∞ −∞ e itτ f (z, t) dt, z ∈ c, provided that the integral exists. in fact, f τ(z) is the inverse fourier transform of f (z, t) with respect to t evaluated at τ. in this paper, the nonzero parameter τ can be looked at as planck’s constant. to obtain the estimates in this paper, we use formulas for e−ul̃τ and l̃−1τ in terms of the τweyl transforms and the τ-fourier–wigner transforms of hermite functions, τ ∈ r\ {0}, which we recall in, respectively, section 2 and section 3. the l2-boundedness and the hilbert– schimdt property of τ-weyl transforms are instrumental in obtaining the estimates. basic information on the classical fourier–wigner transforms, wigner transforms and weyl transforms can be found in [13] among others. in section 2, we introduce the τ-weyl transforms and prove results on the l2-boundedness and the hilbert–schmidt property of the τ-weyl transforms. the τ-fourier–wigner transforms of hermite functions are recalled in section 3. a formula for e−ul̃τ f , u > 0, for every function f in l2(c) and an estimate for ‖e−ul̃τ f ‖l2 (c), u > 0, in terms of ‖f ‖l p (c), 1 ≤ p ≤ 2, are given in section 4. this formula gives a formula for e−u∆h , u > 0, immediately using the inverse fourier transform as indicated by (1.1). in section 5, we use the family l2s (h), s ∈ r, of sobolev spaces with respect to the center of the heisenberg group as in [1, 2] to obtain sobolev estimates for e−u∆h f , u > 0, in terms of ‖f ‖l2 (h), and sobolev estimates for cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 87 ‖e−u∆h f ‖l2(h), u > 0, in terms of the sobolev norms ‖f ‖l2s (h) of f in l 2 s (h). in section 6, we obtain a formula for l̃−1τ and estimates for l̃ −1 τ which are then used to estimate ∆ −1 h . in section 7, estimates for ‖∆−1 h f ‖l2 s+2 (h) in terms of ‖f ‖l2s (h) for all f in l 2 s (h) are given. we end this section by putting in perspectives the results in this paper. while the semigroup and the inverse can be studied in the framework of functional analysis as explained in [3, 4, 5, 8, 9, 16], the results and methods in this paper are based on explicit formulas in hard analysis and are related to the works in [1, 2, 6, 7, 10, 12, 14, 15]. 2 τ-weyl transforms let f and g be functions in l2(r). then for τ in r\{0}, the τ-fourier–wigner transform vτ( f , g) is defined by vτ( f , g)(q, p) = (2π)−1/2|τ|1/2 ˆ ∞ −∞ e iτq y f ( y + p 2 ) g ( y − p 2 ) d y for all q and p in r. in fact, vτ( f , g)(q, p) = |τ|1/2v ( f , g)(τq, p), q, p ∈ r, where v ( f , g) is the classical fourier–wigner transform of f and g. a proof can be found in [1]. it can be proved that vτ( f , g) is a function in l 2(c) and we have the moyal identity stating that ‖vτ( f , g)‖l2 (c) = ‖f ‖l2 (r)‖g‖l2 (r), τ ∈ r \ {0}. (2.1) we define the τ-wigner transform wτ( f , g) of f and g by wτ( f , g) = vτ( f , g)∧. (2.2) then we have the following connection of the τ-wigner transform with the usual wigner transform. theorem 2.1. let τ ∈ r \ {0}. then for all functions f and g in l2(r), wτ( f , g)(x,ξ) = |τ|−1/2w ( f , g)(x/τ,ξ), x,ξ ∈ r, where w ( f , g) is the classical wigner transform of f and g. it is obvious that wτ( f , g) = wτ( g, f ), f , g ∈ l2(r). (2.3) 88 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) let σ ∈ l p(c), 1 ≤ p ≤ ∞. then for all τ in r\{0} and all functions f in the schwartz space s (r) on r, we define wτσ f to be the tempered distribution on r by (w τ σ f , g) = (2π) −1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ(x,ξ)wτ( f , g)(x,ξ) dx dξ (2.4) for all g in s (r), where (f, g) is defined by (f, g) = ˆ rn f(z)g(z) d z for all measurable functions f and g on rn, provided that the integral exists. we call wτσ the τ-weyl transform associated to the symbol σ. it is easy to see that if σ is a symbol in the schwartz space s (c) on c, then wτσ f is a function in s (r) for all f in s (r). we have the following estimate for the norm of the weyl transform wτ σ̂ in terms of the l p norm of the symbol σ when σ ∈ l p(c), 1 ≤ p ≤ 2. theorem 2.2. let σ ∈ l p(c), 1 ≤ p ≤ 2. then wτ σ̂ : l2(r) → l2(r) is a bounded linear operator and ‖wτσ̂‖∗ ≤ (2π) −1/p|τ|−(1/2)+(1/p)‖σ‖l p (c), where ‖wτ σ̂ ‖∗ is the operator norm of wτσ̂ : l 2(r) → l2(r). proof let f and g be functions in s (r). then (w τ σ̂ f , g) = (2π) −1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ̂(x,ξ)wτ( f , g)(x,ξ) dx dξ = (2π)−1|τ|−1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ̂(x,ξ)w ( f , g)(x/τ,ξ) dx dξ = (2π)−1|τ|1/2 ˆ ∞ −∞ ˆ ∞ ∞ σ̂(τx,ξ)w ( f , g)(x,ξ) dx dξ. but σ̂(τx,ξ) = |τ|−1σ̂1/τ(x,ξ), x,ξ ∈ r, where σ1/τ is the dilation of σ with respect to the first variable by the amount 1/τ. more precisely, σ1/τ(q, p) = σ(q/τ, p), q, p ∈ r. so, (w τ σ̂ f , g) = (2π) −1/2|τ|−1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ̂1/τ(x,ξ)w ( f , g)(x,ξ) dx dξ = |τ|−1/2(wσ̂1/τ f , g), cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 89 where wσ̂1/τ is the classical weyl transform with symbol σ̂1/τ. thus, it follows from theorem 21.1 in [14] that wτ σ̂ : l2(r) → l2(r) is a bounded linear operator and ‖wτσ̂‖∗ ≤ |τ| −1/2 (2π) −1/p‖σ1/τ‖l p (c) = (2π)−1/p|τ|−(1/2)+(1/p)‖σ‖l p (c). � we have the following result for the hilbert–schmidt norm of the weyl transform wτ σ̂ in terms of the l2 norm of the symbol σ when σ ∈ l2(c). theorem 2.3. let σ ∈ l2(c). then wτ σ̂ : l2(r) → l2(r) is a hilbert–schmidt operator and ‖wτσ̂‖hs = (2π) −1/2‖σ‖l2 (c), where ‖wτ σ̂ ‖hs is the hilbert–schmidt norm of wτσ̂ : l 2(r) → l2(r). proof let f and g be functions in s (r). then (w τ σ̂ f , g) = (2π) −1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ̂(x,ξ)wτ( f , g)(x,ξ) dx dξ = (2π)−1/2|τ|−1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ̂(x,ξ)w ( f , g)(x/τ,ξ) dx dξ = (2π)−1/2|τ|1/2 ˆ ∞ −∞ ˆ ∞ ∞ σ̂(τx,ξ)w ( f , g)(x,ξ) dx dξ. but σ̂(τx,ξ) = |τ|−1/2σ̂1/τ(x,ξ), x,ξ ∈ r, where σ1/τ is the dilation of σ with respect to the first variable by the amount 1/τ, i.e., σ1/τ(q, p) = σ(q/τ, p), q, p ∈ r. so, (w τ σ̂ f , g) = (2π) −1|τ|−1/2 ˆ ∞ −∞ ˆ ∞ −∞ σ̂1/τ(x,ξ)w ( f , g)(x,ξ) dx dξ = |τ|−1/2(wσ̂1/τ f , g), where wσ̂1/τ is the classical weyl transform with symbol σ̂1/τ. thus, it follows from theorem 7.5 in [13] that wτ σ̂ : l2(r) → l2(r) is a hilbert–schmidt operator and ‖wτσ̂‖hs = |τ| −1/2‖wσ̂1/τ ‖hs = (2π)−1/2|τ|−1/2‖σ1/τ‖l2 (c) = (2π)−1/2‖σ‖l2 (c). � 90 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) 3 fourier–wigner transforms of hermite functions for τ ∈ r \ {0} and for k = 0, 1, 2, . . . , we define eτ k to be the function on r by e τ k (x) = |τ|1/4 e k ( √ |τ|x), x ∈ r. here, e k is the hermite function of order k defined by e k(x) = 1 (2k k! p π)1/2 e −x2/2 hk (x), x ∈ r, where hk is the hermite polynomial of degree k given by hk(x) = (−1)k ex 2/2 ( d dx )k (e −x2 ), x ∈ r. for j, k = 0, 1, 2, . . . , we define eτ j,k on r2 by e τ j,k = vτ(eτj , e τ k ). the following theorem gives the connection of {eτ j,k : j, k = 0, 1, 2, . . . } with {e j,k : j, k = 0, 1, 2, . . . }, where e j,k = v (e j , e k ), j, k = 0, 1, 2, . . . . a proof can be found in [1]. theorem 3.1. for τ ∈ r \ {0} and for j, k = 0, 1, 2, . . . , e τ j,k (q, p) = |τ|1/2 e j,k ( τ p |τ| q, √ |τ|p ) , q, p ∈ r. theorem 3.2. {eτ j,k : j, k = 0, 1, 2, . . . } forms an orthonormal basis for l2(r2). theorem 3.2 follows from theorem 3.1 and theorem 21.2 in [13] to the effect that {e j,k : j, k = 0, 1, 2, . . . } is an orthonormal basis for l2(r2). theorem 3.3. for j, k = 0, 1, 2, . . . , l̃τ e τ j,k = (2k + 1 +|τ|)|τ|eτ j,k . theorem 3.3 can be proved using theorem 3.1, theorem 3.3 in [2] and theorem 22.2 in [13] telling us that for j, k = 0, 1, 2, . . . , e j,k is an eigenfunction of l1 corresponding to the eigenvalue 2k + 1 and the fact that, l̃τ = lτ +τ2. cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 91 4 a formula and an estimate for e−ul̃τ , u > 0 let τ ∈ r \ {0}. then a formula for e−ul̃τ , u > 0, is given by the following theorem. theorem 4.1. let f ∈ l2(c). then for u > 0, e −ul̃τ f = (2π)1/2 ∞∑ k=0 e −(2k+1+|τ|)|τ|u vτ(w τ f̂ e τ k , e τ k ), where the convergence of the series is understood to be in l2(c). proof let f ∈ l2(c). then from theorem 3.3 we have for u > 0 e −ul̃τ f = ∞∑ k=0 ∞∑ j=0 e −(2k+1+|τ|)|τ|u ( f , e τ j,k )e τ j,k = e−|τ| 2 u e −ulτ f , (4.1) where the series is convergent in l2(c). now, using the formula for e−ulτ f in [2] and (4.1), we get e −ul̃τ f = (2π)1/2 ∞∑ k=0 e −(2k+1+|τ|)|τ|u vτ(w τ f̂ e τ k , e τ k ) for all f in l2(c) and u > 0. � for all τ in r \ {0}, we have the following estimate for the l2 norm of e−ul̃τ f , u > 0, in terms of the l p norm of f . theorem 4.2. let τ ∈ r \ {0}. then for all functions f in l p(c), 1 ≤ p ≤ 2, ‖e−ul̃τ f ‖l2(c) ≤ (2π) −(1/p)+(1/2)|τ|−(1/2)+(1/p) e−τ 2 u 1 2 sinh(|τ|u) ‖f ‖l p (c). proof by theorem 4.1, the moyal identity (2.1) and the fact that ‖eτ k ‖l2 (r) = 1, k = 0, 1, 2, . . . , we get ‖e−ul̃τ f ‖l2(c) ≤ (2π) 1/2 e −(|τ|+|τ|2 )u ∞∑ k=0 e −2k|τ|u‖wτ f̂ e τ k ‖l2 (r), u > 0. (4.2) applying theorem 2.2 to (4.2), we get ‖e−ul̃τ f ‖l2 (c) ≤ (2π)−(1/p)+(1/2)|τ|−(1/2)+(1/p) e−(|τ|+|τ| 2 )u ( ∞∑ k=0 e −2k|τ|u ) ‖f ‖l p (c) = (2π)−(1/p)+(1/2)|τ|−(1/2)+(1/p) e−|τ| 2 u 1 2 sinh(|τ|u) ‖f ‖l p (c), as asserted. � 92 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) 5 sobolev estimates for e−∆h, u > 0 let s ∈ r. then we define l2s (h) to be the set of all tempered distributions f in s ′ (h) such that f τ(z) is a measurable function and ˆ c ˆ ∞ −∞ |τ|2s|f τ(z)|2dτ d z < ∞. for every f in l2s (h), we define the norm ‖f ‖l2s (h) by ‖f ‖2 l2s (h) = ˆ c ˆ ∞ −∞ |τ|2s|f τ(z)|2 dτ d z. then it can be shown easily that l2s (h) is an inner product space in which the inner product ( , ) l2s (h) is given by ( f , g) l2s (h) = ˆ c ˆ ∞ −∞ |τ|2s f τ(z) gτ(z) dτ d z for all f and g in l2s (h). theorem 5.1. let s ≥ 1. then for u > 0, e−u∆h : l2(h) → l2s (h) is a bounded linear operator and ‖e−u∆h f ‖ l2s (h) ≤ cs 2us ‖f ‖l2 (h), f ∈ l 2 (h), where cs = sup τ∈r\{0} (|τ|s/sinh|τ|). proof let u > 0 and f ∈ l2(h). then by (1.1), fubini’s theorem, plancherel’s theorem and theorem 4.2 with p = 2, ‖e−u∆h f ‖2 l2s (h) = ˆ c ˆ ∞ −∞ |τ|2s|(e−u∆h f )τ(z)|2dτ d z = ˆ ∞ −∞ |τ|2s (ˆ c |(e−u∆h f )τ(z)|2d z ) dτ = ˆ ∞ −∞ |τ|2s (ˆ c |(e−ul̃τ f τ)(z)|2d z ) dτ = ˆ ∞ −∞ |τ|2s‖e−ul̃τ f τ‖2 l2(c) dτ ≤ 1 4 ( ˆ ∞ −∞ e−2τ 2 u|τ|2s sinh2(|τ|u) ‖f τ‖2 l2 (c) dτ ) ≤ 1 4 ˆ ∞ −∞ |τ|2s sinh2(|τ|u) (ˆ c |f τ(z)|2d z ) dτ cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 93 = 1 4 u2s+1 ˆ ∞ −∞ |τ|2s sinh2(|τ|u) (ˆ c | f̌ (z,τ/u)|2 d z ) dτ, where f̌ is the inverse fourier transform of f with respect to t. so, using a simple change of variable and letting cs = sup τ∈r\{0} (|τ|2s/sinh2|τ|), we get ‖e−u∆h f ‖2 l2s (h) ≤ cs 4u2s ˆ ∞ −∞ (ˆ c | f̌ (z,τ)|2 d z ) dτ = cs 4u2s ‖f ‖2 l2 (h) and this completes the proof. � the following result complements theorem 5.1. theorem 5.2. let s ≤ −1. then for u > 0, e−u∆h : l2s (h) → l 2(h) is a bounded linear operator and ‖e−u∆h f ‖l2(h) ≤ c−s 2u−s ‖f ‖ l2s (h) , f ∈ l2s (h), where c−s = sup τ∈{0} (|τ|−ssinh|τ|). the proof of theorem 5.2 is very similar to that of theorem 5.1 and is hence omitted. 6 two formulas and an estimate for l̃−1τ let τ ∈ r \ {0}. then a formula for l−1τ is given by the following theorem. theorem 6.1. let f ∈ l2(c). then l̃ −1 τ f = (2π) 1/2 ∞∑ k=0 1 (2k + 1 +|τ|)|τ| vτ(w τ f̂ e τ k , e τ k ), where the convergence of the series is understood to be in l2(c). proof let f ∈ l2(c). then l̃ −1 τ f = ∞∑ k=0 ∞∑ j=0 1 (2k + 1 +|τ|)|τ| ( f , e τ j,k )e τ j,k , (6.1) where the series is convergent in l2(c). now, by plancherel’s theorem and (2.2)–(2.4), ( f , e τ j,k ) = ˆ c f (z)vτ(e τ j , eτ k )(z) d z = ˆ c f̂ (ζ)vτ(e τ j , eτ k )∧(ζ) dζ 94 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) = ˆ c f̂ (ζ)wτ(e τ j , eτ k )(ζ) dζ = (2π)1/2(w f̂ e τ k , e τ j ) (6.2) for j, k = 0, 1, 2, . . . . similarly, for j, k = 0, 1, 2, . . . , and g in l2(c), we get (e τ j,k , g) = ( g, eτ j,k ) = (2π)1/2(wτ ĝ eτ k , eτ j ) = (2π)1/2(eτ j , w τ ĝ e τ k ). (6.3) so, by (6.1)–(6.3), fubini’s theorem and parseval’s identity, (l̃ −1 τ f , g) = 2π ∞∑ k=0 1 (2k + 1 +|τ|)|τ| ∞∑ j=0 (w τ f̂ e τ k , e τ j )(e τ j , w τ ĝ e τ k ) = 2π ∞∑ k=0 1 (2k + 1 +|τ|)|τ| (w τ f̂ e τ k , w τ ĝ e τ k ). (6.4) by plancherel’s theorem and (2.2)–(2.4), (w τ f̂ e τ k , w τ ĝ e τ k ) = (2π)−1/2 ˆ c ĝ(z)wτ(e τ k , wτ f̂ eτ k )(z) d z = (2π)−1/2 ˆ c wτ(w τ f̂ e τ k , e τ k )(z) ĝ(z) d z = (2π)−1/2 ˆ c vτ(w τ f̂ e τ k , e τ k )(z) g(z) d z (6.5) for k = 0, 1, 2, . . . . thus, by (6.4), (6.5) and fubini’s theorem, (l̃ −1 τ f , g) = (2π) 1/2 ∞∑ k=0 1 (2k + 1 +|τ|)|τ| (vτ(w τ f̂ e τ k , e τ k ), g) = (2π)1/2 ( ∞∑ k=0 1 (2k + 1 +|τ|)|τ| vτ(w τ f̂ e τ k , e τ k ), g ) (6.6) for all f and g in l2(c). thus, by (6.6), l̃ −1 τ f = (2π) 1/2 ∞∑ k=0 1 (2k + 1 +|τ|)|τ| vτ(w τ f̂ e τ k , e τ k ) for all f in l2(c). � the formula (6.4) is an important formula in its own right and we upgrade it to the status of a theorem. theorem 6.2. for all τ ∈ r \ {0}, the inverse l̃−1τ of the parametrized partial differential operators l̃τ is given by (l̃ −1 τ f , g) = 2π ∞∑ k=0 1 (2k + 1 +|τ|)|τ| (w τ f̂ e τ k , w τ ĝ e τ k ), f , g ∈ l2(c). cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 95 for all τ in r\ {0}, we have the following estimate for the l2 norm of l̃−1τ f in terms of the l2 norm of f . theorem 6.3. let τ ∈ r \ {0}. then for all functions f in l2(c), ‖l̃−1τ f ‖l2 (c) ≤ |τ| −2‖f ‖l2 (c). proof let f and g be functions in l2(r). then by theorems 2.3 and 6.2, |(l̃−1τ f , g)| ≤ 2π 1 |τ|2 ∞∑ k=0 |(wτ f̂ e τ k , w τ ĝ e τ k )| ≤ 2π 1 |τ|2 ‖wτ f̂ ‖hs‖wτĝ ‖hs = 1 |τ|2 ‖f ‖l2 (c)‖g‖l2 (c) and this completes the proof. � 7 sobolev estimates for ∆−1 h we have the following simple result giving the connection of ∆−1 h with l̃−1τ , τ ∈ r \ {0}, which can be proved easily using the elementary properties of the fourier transform and the fourier inversion formula. theorem 7.1. let f ∈ l2(h). then (∆ −1 h f )(z, t) = (2π)−1/2 ˆ ∞ −∞ e −itτ (l̃ −1 τ f τ )(z) dτ, (z, t)∈ h. we can now give the following theorem, which can be seen as another manifestation of the ellipticity of ∆h. theorem 7.2. let s ∈ r. then ∆−1 h : l2s (h) → l 2 s+2(h) and ‖∆−1 h f ‖l2 s+2 (h) ≤ ‖f ‖l2s (h), f ∈ l 2 s (h). proof by fubini’s theorem, plancherel’s theorem, theorems 6.3 and 7.1, ‖∆−1 h f ‖2 l2 s+2 (h) = ˆ c ˆ ∞ −∞ |τ|2(s+2)|(∆−1 h f ) τ (z)|2dτ d z = ˆ ∞ −∞ |τ|2(s+2) (ˆ c |(∆−1 h f ) τ (z)|2d z ) dτ 96 aparajita dasgupta & m.w. wong cubo 12, 3 (2010) = ˆ ∞ −∞ |τ|2(s+2) (ˆ c |(l̃−1τ f τ )(z)|2d z ) dτ = ˆ ∞ −∞ |τ|2(s+2)‖l̃−1τ f τ‖2 l2 (c) dτ ≤ ˆ ∞ −∞ |τ|2s‖f τ‖2 l2 (c) dτ = ˆ ∞ −∞ |τ|2s (ˆ c |f τ(z)|2d z ) dτ = ˆ c ˆ ∞ −∞ |τ|2s|f τ(z)|2dτ d z = ‖f ‖2 l2s (h) , as asserted. � references [1] dasgupta, a and wong, m.w., weyl transforms and the inverse of the sub-laplacian on the heisenberg group, in pseudo-differential operators: partial differential equations and time-frequency analysis, fields institute communications, 52, american mathematical society, 2007, 27–36. [2] dasgupta, a. and wong, m.w., weyl transforms and the heat equation for the sublaplacian on the heisenberg group, in new developments in pseudo-differential operators, operator theory: advances and applications, 189, birkhäuser, 2009, 33–42. [3] davies, e.b., one-parameter semigroups, academic press, 1980. [4] davies, e.b., linear operators and their spectra, cambridge university press, 2007. [5] friedman, a., partial differential equations, holt, reinhart and winston, 1969. [6] furutani, k., the heat kernel and the spectrum of a class of manifolds, comm. partial differential equations, 21 (1996), 423–438. [7] gaveau, b., principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, acta math., 139 (1977), 95–153. [8] hille, e. and phillips, r.s., functional analysis and semi-groups, third printing of revised version of 1957, american mathematical society, 1974. [9] iancu, g.m. and wong, m.w., global solutions of semilinear heat equations in hilbert spaces, abstr. appl. anal., 1 (1996), 263–276. cubo 12, 3 (2010) the semigroup and the inverse of the laplacian ... 97 [10] kim, j. and wong, m.w., positive definite temperature functions on the heisenberg group, appl. anal., 85 (2006), 987–1000. [11] shubin, m.a., pseudodifferential operators and spectral theory, springer-verlag, 1987. [12] stein, e.m., harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, princeton university press, 1993. [13] wong, m.w., weyl transforms, springer-verlag, 1998. [14] wong, m.w., weyl transforms, the heat kernel and green function of a degenerate elliptic operator, ann. global anal. geom., 28 (2005), 271–283. [15] wong, m.w., the heat equation for the hermite operator on the heisenberg group, hokkaido math. j., 34 (2005), 393–404. [16] yosida, k., functional analysis, reprint of sixth edition, springer-verlag, 1995. cubo a mathematical journal vol.18, no¯ 01, (69–88). december 2016 parametrised databases of surfaces based on teichmüller theory armando rodado amaris1, gina lusares2 1 departamento de ciencias exactas, universidad de los lagos, av. fuchslocher 1305, osorno, chile 2 departamento de estadistica, universidad de valparaiso, blanco 951, valparaiso, chile armando.rodado@ulagos.cl, gina.lusares@postgrado.uv.cl abstract we propose a new framework to build databases of surfaces with rich mathematical structure. our approach is based on ideas that come from teichmüller and moduli space of closed riemann surfaces theory, and the problem of finding a canonical and explicit cell decomposition of these spaces. databases built using our approach will have a graphical underlying structure, which can be built from a single graph by contraction and expansion moves. resumen proponemos un nuevo marco teórico para construir bases de datos de superficies con rica estructura matemática. nuestro enfoque está basado en ideas que vienen de teoŕıa de espacios de teichmüller y espacios módulares de superficies de riemann cerradas, y el problema de encontrar una descomposición celular canónica y expĺıcita de estos espacios. las bases de datos construidas usando nuestro enfoque tendrán una estructura gráfica subyacente, la que se puede construir a partir de un solo grafo por movimientos de expansión y contracción. keywords and phrases: database of surfaces design teichmüller space moduli space of riemann surfaces canonical cell decomposition of riemann surfaces teichmüller surfaces descriptor. 2010 ams mathematics subject classification: 70 armando rodado amaris, gina lusares cubo 18, 1 (2016) 1 introduction the idea of surface deformation in real life is common and increasingly important for sciences and industry. for example, it is known that in the automotive industry around 90% of the time a new surface design start with the smooth modification of a prototype surface [1]. a process that mirrors the exploration of the possible paths in the teichmüller or moduli space of riemann surfaces. this suggests the need of encouraging the application of these theories, which already have been applied for instance to neurosciences [24, 25], but it is still basically unexploited. a systematic study of surfaces, its parametrisation and classification are tasks of real importance in applied sciences and industrial applications. formal studies on spaces of surfaces can be traced back to riemann and to the work of teichmüller [7, 11, 18]. nowadays moduli and teichmüller spaces theories are deep and active branches of mathematics with many connections to many other fields. hyperbolic geometry [16, 19] plays an important role in the understanding of riemann surfaces because by the classical uniformization problem each riemann surfaces s of genus g ≥ 2 can be represented as a quotient of h/g, where h is the hyperbolic upper half plane and g is a discrete group of mobiüs transformations keeping h invariant. the quest to find a canonical and explicit cell decomposition of the moduli space of closed riemann surfaces lead to combinatorial structures and circle pattern systems. indeed, a combinatorial analysis shows the existence of a family of graphs that contains all possible graphs corresponding to the 1-skeleton of the voronoi cells, determined by the characteristic set w of s; and circle pattern systems of equations come from the study of realization problems of graphs. in addition, circle patterns systems lead to a polytope pg complex which can be viewed as a parametrisation of tg for the genus two case [2], and more generally for the hyperelliptic locus of the teichmüller and moduli spaces of closed riemann surfaces for genus g ≥ 2. the development of applications of moduli spaces of riemann surfaces theory can be facilitated by assigning a computable combinatorial structure to each surface s on the moduli space, mg, or the teichmüller space, tg, where g ≥ 2. combinatorial structures can be based on w, a characteristic set of points on s, which usually is the set of weierstrass points on s, because it carries a lot of information about a riemann surfaces as shown on [2, 12, 10, 14, 15]. however, for specific application to choose a different set could be sensible. since a surface s embedded in a 3-dimensional space has a conformal structure and, if it has a negative euler characteristic, it can be provided of a hyperbolic metric. [24, 25], we can define a new surface descriptor from the embedding of s on the poincare disk by following the next steps: (1) compute the uniformization metric of the surface s (2) get a conformal model s̄ of s defined as a quotient of the poincaré disc d by a suitable fuchsian group cubo 18, 1 (2016) surface databases based on teichmüller theory 71 (3) compute the set w of weierstrass points of s̄ (4) assign the voronoi diagram, a(s), determined by w to s (5) assign a circle intersection angle θi to each edge i of a(s) (6) define the descriptor dθ(s) for the surface s by dθ(s) = (a(s), θ = (θi)i). for any genus dθ(s) depends only on the teichmüller class of s up to labelling, and circle pattern coordinates -intersection anglescan be translated to teichmüller coordinates [2]. the technical procedures that are needed to build dθ(s) for the first and second steps are well known [24, 25]. the location of the weierstrass points for the genus two case are described on [10, 14, 15]. to find the locations of the weierstrass points of a riemann surfaces is in general challenging, for this reason we mainly study the descriptor dθ(s) for genus 2 surfaces. our ideas show a pathway for the implementation of the general case. dθ(s) for genus 2 surfaces can be simplified and associated to a linear system, whose solutions are on a polytopes complex. this will allow us to approach surfaces descriptions on a marked polytope complexeach polytope marked by a graph. we propose the polytope complex that arise as a natural structure to support database of surfaces. in the sequel, we will describe the theory of graphs associated to riemann surfaces based on weierstrass points, the theory of linear systems associated to graphs, explain the construction of a polytope complex based on the previous ideas, and describe our proposal for databases designs based on teichmüller theory. 2 graphs associated to riemann surfaces for each riemann surface s with its hyperbolic metric, the 1-skeleton of the voronoi cell decomposition determined by the weierstrass points of s is a unique graph, â(s), which is a subset of s and depends only of the class of s in the teichmüller space, tg, and in the moduli space mg, for any genus g [2]. determining which graphs are associated to riemann surfaces based on weierstrass points in general is a difficult problem. we restrict ourselves to the hyperelliptic case, which allow us to find all graphs associated to riemann surfaces computationally by considering a similar problem on the 2-dimensional sphere. indeed, the hyperelliptic involution, τ, of s induces an action on s such that s/τ is the two dimensional hyperbolic sphere. s/τ has exactly 2g + 2 cone points which has measure π. then, τ projects â(s) into a graph on s/τ that we denote by a(s). by a standard lifting procedure [2] we can recover s from the marked sphere s/τ. this shows the existence of a one to one correspondence between set of hyperelliptic surfaces of a given genus and a set of graphs, assigning s to a(s). 72 armando rodado amaris, gina lusares cubo 18, 1 (2016) figure 1: the weierstrass point on a hyperelliptic surface s of genus g are the 2g+2 fixed points of its unique hyperelliptic involution. here, we represent the genus 2 case. definition 2.1. the graph, a(s), associated to a hyperelliptic riemann surface (s, τ) of genus g is the image on s/τ under τ of the voronoi graph determined by the weierstrass fixed points of s. proposition 2.1. if s is a hyperelliptic hyperbolic riemann surface of genus g, its associated graph a(s) satisfies the following properties: (1) g is connected (2) g does not have monogons (3) g divides s2 into 2g + 2 regions (4) all vertices of g have valence ≥ 3. the above properties of a(s) motivates the following definition of the family of ce(g) graphs. definition 2.2. a ce(g) graph is a connected graph g with vertices of valence greater than two that can be embedded in the sphere s2, determining 2g + 2 regions and having no monogons. if the valence of each vertex of g is 3, then we say that the graph is generic. 2.1 generic graphs: the genus two case to find all the possible generic graphs in the genus two case, we take advantage of the fact that all generic graphs are connected by whitehead moves and do not have any monogons. we find 20 generic graphs g1, g2, . . . g20 which are subset of the two dimensional sphere, 17 of which are non-isomorphic, see figure 3. counting the number of sides on each of the faces of these graphs, we always get six numbers which are arrange in a non-increasing order. this gives a natural labelling to each generic graph, cubo 18, 1 (2016) surface databases based on teichmüller theory 73 figure 2: a whitehead move on the red edge on graph g1, contracts the edge to a point, as shown on the middle graph, followed by an edge expansion as shown on the right graph. the result is the new graph which is represented on the right. called the face labelling, which is unique except for the generic graphs g11 and g12. however, this exception is not really important because g11 is not associated to a riemann surface of genus two. on figure 4, the combinatorics of genus two generic graphs is represented by a graph whose nodes are all possible generic graphs. we join graph gi with gj by an edge if there is a whitehead move transforming gi into gj. 2.2 stratification of ce(g) an immediate consequence of the euler characteristic formula of the sphere is that any generic graph that belong to ce(g) has 4g vertices, 6g edges and by definition 2g + 2 faces. then, the family ce(g) of graphs is stratified in 4g − 1 levels. indeed, since the number of faces in all ce(g) graphs is 2g + 2, and each vertex is of valence not smaller than 3 then 3v ≤ 2e. hence, by the euler characteristic formula of the sphere v ≤ 4g. the maximum number of vertices that a ce(g) graph can have is 4g and the minimum number is 2, because no monogons are allowed. thus, ce(g) has 4g−1 levels, if we define the k-th stratum of ce(g) as the set of all graphs in ce(g) that have exactly 4g − (k − 1) vertices. the members of the first stratum of ce(g) are all cubic graphs and is easy to verify that graphs at this level has 2g + 2 faces and 6g edges. this stratum of ce(g) also has great importance since any graph of ce(g) in a higher strata can be obtained by contraction moves starting from a graph of the first strata. the fact that cubic graphs are connected by a sequence of whitehead moves [3], allow us to view ce(g) as generated by any of its cubic graphs by sequences of whitehead moves and contraction 74 armando rodado amaris, gina lusares cubo 18, 1 (2016) figure 3: genus two generic graphs and their face labelling-l[f1f2f3f4f5f6]. moves. in particular, any graph in ce(2) can be connected to the graphs which are illustrated on figure 3. for another equivalent point of view, any stratum of ce(g) can be obtained inductively by contracting each graph of the previous strata by a single contraction move on one of its edges in such way that no monogon is created. furthermore, if we define a generalised whitehead move on a graph as the graph obtained by contracting one edge and expanding a new edge in such way that cubo 18, 1 (2016) surface databases based on teichmüller theory 75 no monogons are created, we have that two graphs at a fixed strata are connected by a sequence of generalised whitehead moves. finally, to complete a view of how graphs are connected on ce(g), its most contracted graph can generate all graphs by a sequence of contraction and expansions moves. then, ce(g) is analogous to a universe that can be generated by transforming any of its graphs by contraction or expansion moves from a single graph. 3 generic graphs on the teichmüller space informally, the teichmülller of riemann surfaces of genus g, tg is a space whose elements are classes of marked surfaces, and paths on tg can be viewed as deformation of a surface. this intuitive idea suggests that tg or an equivalent space could be an ideal space to model the deformation of real surfaces. for a background on teichmülller theory see [18]. next, we list a few facts about teichmülller theory, (1) let s be a fixed riemann surface of genus g. a marked surface is pair (r, [f]), where r is a riemann surface, [f] is the homotopy class of a homeomorphism f : s → r. two marked surfaces (s, [f]) and (s′, [f′]) are equivalent if there is a conformal map g : r → r′ such that [g ◦ f] = [f′]. (2) the teichmüller space is the set of marked classes. (3) the teichmüller space tg,p has a natural topology, which makes it homeomorphic to an open set of r6g−6+2p, where g and p are the genus and the number of punctures of each surface in a class of tg,p. on this space, we will only consider the case when p = 0. (4) the teichmüller space tg,p can be parametrised by fenchel-nielsen coordinates [18] in addition to the above, if g is a cubic graph associated to s, then g can be associated to an open subset of tg, with its teichmüller metric. the fenchel-nielsen coordinates of tg allow us to see it as a 6(g − 1) real dimensional manifold, and informally we could imagine a deformation of s by slightly changing its fenchel-nielsen coordinates which cannot change the graph g because all its vertices are trivalent and then g is stable under small perturbations. the following properties which are proved in [2] establish the relationship between generic graphs and the teichmüller space. proposition 3.1. if τ is a hyperelliptic involution of a hyperelliptic riemann surface r and h : r → r′ is an isometry, then r′ is hyperelliptic and its hyperelliptic involution is τ′ = h ◦ τ ◦ h−1. proposition 3.2. the associated graphs corresponding to two equivalent marked surfaces (r, [f]), (r′, [f′]) are equal. 76 armando rodado amaris, gina lusares cubo 18, 1 (2016) denote by õg, the set of all points in the teichmüller space with associated graph g, where g is a graph embedded in s2. proposition 3.3. if g is a cubic graph, then õg is an open set of the teichmüller topology of tg. 4 linear systems associated to graphs 4.1 delaunay realization problem if a graph g on the 2-sphere is the associated graph to a hyperelliptic surface s of genus g, then g is the boundary of a cell decomposition of the 2-sphere which is s/τ, with 2g + 2 faces, each one containing an interior point with cone angles equal to π. by lifting s/τ to its two fold branched covering space s can be recovered. the lifting of s/τ is standard. however, for a given graph g, the problem of finding a hyperbolic metric on the sphere with 2g + 2 π-cone angles whose associated graph is g is not trivial. we call this problem the delaunay realization problem (drp). a solution to the drp defined by g can be identified with a unique hyperbolic surface s and also with a collection of circles with a set of intersection angles. this connects the realizability problem that we have described with the theory of circle patterns, which can be traced back to koebe’s work (1936), e.m. andreev’s work (1970) and thurston [13], and has had great impact in many fields including conformal mapping, complex analysis, teichmüller theory, brain mapping, random walks, tilings, minimal surfaces and integrable systems, numerical analysis, metric spaces and more [21]. 4.2 circle patterns and quotients given a closed riemann surface s, and a cell decomposition {ci}i∈i of s, which might have cone singularities at a vertex or at the center of a cell, we say that a circle pattern is a configuration of disks {di}i∈i, where the boundary of each di contains all the vertices of ci and no vertex of the cell decomposition is in the interior of any di. in a circle pattern, for each edge e of the cell decomposition, two circles have as intersection the extremes of e. let ŝ be a hyperelliptic riemann surface with metric d̃ and hyperelliptic involution τ. the quotient space s = ŝ/τ is also a metric space with the quotient metric d. we are interested in delaunay circle patterns where the vertices of the circle pattern are the fixed points of the hyperelliptic involution. we will project a circle pattern on ŝ to a circle pattern of the quotient s/τ. proposition 4.1. a circle pattern of a cell decomposition of a hyperelliptic riemann surface s of genus g is projected to a circle pattern of the sphere by the hyperelliptic involution of s. cubo 18, 1 (2016) surface databases based on teichmüller theory 77 to study circle patterns, several variational approaches have been introduced and several circle packing results were proved using different functionals [23]. figure 4: above, we show the 10 realizable generic graphs with their groups of rigid symmetries. note that whitehead moves on red edges are prohibited. 78 armando rodado amaris, gina lusares cubo 18, 1 (2016) 4.3 delaunay circle patterns, voronoi cells and duality a set of points f = {p1, p2, . . . , pn} on a riemann surface s determines a voronoi decomposition v of s: this is a cell decomposition determined from the points pi ∈ f by taking the sets of points closest to pi, for each i. the open 2-cells are sets of form vi := {p ∈ s : d(p, pi) < d(p, pj) ∀j ∈ ii} where ii = {1, 2, . . . , n} − {i}. thus, s = ∪iv̄i and g = ∪i(v̄i − vi) is a graph whose edges are geodesic segments. the dual cell decomposition v ′ of v is by definition constructed by joining two vertices p1 ∈ f and p2 ∈ f by a geodesic segment, one for each common edge of the corresponding voronoi cells v1 and v2. the collection v ′ is itself a cell decomposition of s with the property that each of its cells is inscribed in a unique circle: the collection of such circles is a delaunay circle pattern for v ′. conversely, if we start with a cell decomposition v ′ of s which has a delaunay circle pattern, by joining the centers of adjacent circles we get a voronoi cell decomposition of s whose centers correspond to the vertices of v ′. a delaunay decomposition of a constant curvature surface is a cellular decomposition such that the boundary of each face is a geodesic polygon which is inscribed in a circular disc, and these discs have no vertices in their interiors. the poincare-dual decomposition of a delaunay decomposition with the centers of the circles as vertices and geodesics edges is a voronoi cell decomposition. a delaunay type circle pattern is the circle pattern formed by the circles of a delaunay decomposition. we will allow the surface to have cone-like singularities in the hyperbolic metric at the vertices of delauney decomposition, and centers of the circles: a k-cone cell is a hyperbolic polygon with interior cone point obtained from a collection of hyperbolic polygons glued together cyclically around a common vertex, by isometric identification of edges, such that the sum of angles at the vertex is k. a cellular decomposition of a surface with n-cone singularities is a collection {cj} of cone kj-cone cells such that each side of each cell has been glued to a unique side of another cell (possibly the same), by hyperbolic isometry. from a delaunay type circle pattern, one may obtain the following data: • a cell decomposition σ of a 2-dimensional manifold. • for each edge e of σ, the exterior (respectively interior) intersection angle θe (respectively θe ∗ := π − θe). thus, 0 < θe, θ ∗ e < π. • for each face σ, the cone angle φf > 0 of the cone-like singularity at the center of the circle corresponding to f. if there is no cone-like singularity at the center, then φf = 2π. cubo 18, 1 (2016) surface databases based on teichmüller theory 79 note that the cone angle θv at a vertex v of σ is determined by the intersection angles θe: θv = σθe (1) where the sum is over all edges e around v. next, we present theorem 1.8 (ii) on [20] and [5] for the case of a closed oriented surface, the main tool that we use on this paper. theorem 4.1 (springborn). let σ be a cell decomposition of a closed oriented surface. suppose the interior intersection angles are prescribed by a function θ∗ ∈ (0, π)e0 on the set e0 of edges. let φ ∈ (0, ∞)f be a function on the set f of faces, which prescribe, the cone angle corresponding to a face. a hyperbolic circle pattern corresponding to this data exists if and only if the following condition is satisfied: if f′ ⊆ f is any nonempty set of faces and e′ ⊆ e0 is the set of edges which are incident with any face f ∈ f′, then σf∈f′φf < σe∈e′2θe ∗, (2) if it exist, the circle pattern is unique up to hyperbolic isometry. the above observations can be integrated into a system of linear inequalities l(g, σ) to solve the delaunay realization problem for a delaunay graph g′ embedded in s2 with edge-labeling σ, dual to g, as a consequence of springborn theorem. below l(g, σ) is defined : l(g, σ) = ⎧ ⎪⎨ ⎪⎩ 2πq(i) < σ12j=12p(i, j)θ ∗ j , for each i ∈ {1, 2, . . . , 255} σk∈jv(π − θk ∗) = π, for v = 1, 2, . . . , 6 0 < θj ∗ < π, for j = 1, 2, . . . , 12 (3) where jv is the set of edges incident with vertex v, q(i) is the number of faces in the subset i of faces, and p(i, j) is the characteristic function, which is 1 if the edge j belongs to the subset i and 0 otherwise. if l(g, σ) has a solution then, by springborn theorem there is a hyperbolic circle pattern inducing a delaunay cell decomposition isomorphic to g′. hence, the two-fold covering space of the sphere that realize one of the solutions is a delaunay triangulated riemann surface of genus two whose dual graph projects to g′, solving the delaunay problem. the above system can be solved using commercial computer programs that work even with thousand of constraints [4]. however, we do not need to do so for the genus two case because we can reduce the linear system that we obtain substantially, which help us to understand the structure of the solutions. 80 armando rodado amaris, gina lusares cubo 18, 1 (2016) 5 solutions for the genus 2 case to approach the problem of finding which of the 20 generic graphs that we found for the genus two case are realizable, we used the package convex [6] which solves linear systems using symbolic algebra and allows the computation of exact solution. solving the reduced system, with only angle equalities and the angle constraints of the type 0 ≤ θ ≤ 1, we obtained that there are at most 10 generic graphs which are realizable. then, we showed that the face constraint of the systems associated to generic graphs for the genus two case are consequence of the equations and angle constraints of the systems, which allow us to claim that there are exactly 10 realizable generic graphs associated to riemann surfaces of genus two. proposition 5.1. there are at most ten genus two generic delaunay realizable graphs. as an additional conclusion from our computation, we can say that the face labelling of generic graphs is unique, e.g. delaunay realizable generic graphs are uniquely determined by their labelling. then, it is convenient to relabel the realizable generic graphs by ascending lexicographic ordering as d1, d2, . . . d10, table 1. generic delaunay realizable graphs face labels d1 444444 d2 554433 d3 555522 d4 654432 d5 663333 d6 664422 d7 754332 d8 773322 d9 854322 d10 943332 table 1: generic delaunay realizable graphs 5.1 independence of face constraints for generic linear systems in this section, we will prove that all the 10 delaunay realizable generic graphs are determined by six angle inequalities and six linear equations, corresponding to an angle equality for each vertex of a dual graph: from these all face inequalities follow, and are thus redundant. this radically improves our ability to understand these polytopes and corresponding realization solutions. each polytope is obtained as a cube cut by hyperplanes in r6. cubo 18, 1 (2016) surface databases based on teichmüller theory 81 our main result on this section is that for each delaunay realizable generic graph the angle constraints can be deduced from angle constraints. in other words, each system l(di, σ) is face constraint independent. 5.2 face inequalities for more than 3 triangles if g is a generic graph then its dual g◦ determines a triangulation t of s2 that has 255 non-empty subsets of triangles, since t has 8 triangles. let ci, i = 1, 2, . . . , 255 be the collection of non-empty subsets of t and let ei be the set of labels of edges which belong to ci. the face constraint for the linear program l(g, σ) corresponding to ci is q(i) < 12∑ j=1 p(i, j)θ∗j (4) where θj ∗ is the normalized exterior angle for the labeled edge j (we divided each angle by π to get 0 < θj ∗ < 1), and q(i) = card(ci). recall that p(i, j) = 1 when the edge labeled j ∈ ei belongs to one of the triangles in the subset ci, where ei ⊂ {1, . . . , 12}. from the above constraint, we can say that q(i) < 12∑ j=1 p(i, j)θ∗j = 12∑ j=1 θ∗j − ∑ j̸∈ei θ∗j the following proposition reduces drastically the number of constraint that we need to consider [2]. proposition 5.2. let g be a delaunay realizable generic graph with labelling σ corresponding to a hyperelliptic riemann surface of genus g. then, the system l(g, σ) associated to g is independent of all face constraints corresponding to subsets of triangles with more than 2g − 1 triangles. an immediate consequence of the above proposition is that in the genus two case, polytopes associated to a delaunay realizable graph can only depend on face constraints corresponding to subsets with one, two or three triangles. corollary 5.1. let g be a delaunay realizable generic graph with labelling σ corresponding to a hyperelliptic riemann surface of genus 2. then, the system l associated to g is independent of all face constraints corresponding to subsets of triangles with more than 3 triangles. the solutions of the angles systems associated to all generic graphs who are feasible are given on table 2. proposition 5.3. the linear system l(di, σ) for i = 1, 2, . . . , 10 is independent of its face constraints. 82 armando rodado amaris, gina lusares cubo 18, 1 (2016) 1 2 3 4 5 6 7 8 9 10 11 12 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 p1 1 1 1 1 1 1 1 0 1 0 1 0 + + + + p2 1 1 1 1 1 1 0 1 0 1 1 0 + + + + + + + p3 1 1 0 1 1 1 1 1 1 1 0 1 + + + + + + p4 0 1 1 0 1 1 1 1 1 1 0 1 + p5 1 0 1 1 1 0 1 1 1 0 1 1 + + p6 0 1 1 1 0 1 1 1 0 1 1 1 + p7 1 1 0 1 0 1 1 0 1 1 1 1 + + + + + + + + + p8 1 0 1 0 1 1 0 1 1 1 1 1 + + p9 0 1 0 1 1 1 0 1 1 1 1 1 + + + + + + + + + p10 1 1 0 1 1 1 0 1 1 0 1 1 + + + + + + p11 1 2 1 1 1 1 1 2 1 1 1 2 1 2 1 2 1 2 + p12 1 1 1 2 1 2 1 1 1 2 1 2 1 1 1 2 1 2 + + + + + + + + + + p13 1 2 1 2 1 1 2 1 2 1 1 1 2 1 1 2 1 1 + p14 1 1 2 1 2 1 1 2 1 2 1 2 1 1 2 1 1 1 + + + p15 1 1 2 1 2 1 1 1 1 2 1 2 1 1 1 2 1 2 + + + + + + + p16 1 1 1 2 1 2 1 1 2 1 2 1 2 1 1 1 1 2 + + + + p17 1 1 2 1 2 1 1 1 2 1 2 1 2 1 1 1 1 2 + + + + p18 1 1 0 1 1 1 1 2 1 2 1 2 1 1 2 1 + + + p19 1 1 0 1 1 1 2 1 2 1 2 1 2 1 1 1 + + p20 1 1 1 2 1 2 1 2 1 0 1 1 1 1 1 2 + p21 1 1 2 1 2 1 1 2 1 0 1 1 1 1 1 2 + p22 1 1 0 1 1 2 1 0 1 1 2 1 1 1 + table 2: on this table, the 22 vertices p1, p2, . . . p22, of the solutions of all angle systems for generic graphs of genus two is presented. each vertex pi is a point in r 12, whose coordinates are on columns 2 to 13. the entry corresponding to the vertex pi and the column dj is filled with + if the vertex pi is a vertex of the polytope of solutions corresponding to the generic graph dj. cubo 18, 1 (2016) surface databases based on teichmüller theory 83 figure 5: the solutions, φ, of top linear systems, l(g), are modified to get solutions for the bottom graphs system by adding ±δ to φ solutions of the original graph which are on the circuit and assigning δ to the new red edge. proof. by corollary 1, we only need to prove the statement for all face constraints which corresponds to sets with 1, 2 or 3 triangles. in addition, to check that this linear system is independent of any given linear constraint, one can just verify that the 22 vertices given on table 2 satisfy the constraints, which in our case can be done by simple inspection. 5.3 independence of face constraints for general genus two ce-graphs proposition 5.4. the solution of the system l(g) is face independent for any delaunay realizable graph g corresponding to a riemann surface of genus two. the basic idea to prove the above proposition is to use induction on the number k of contractions that are needed to obtain g from a generic graph and notice that if g is a delunay realisable graph which is obtained by k + 1 contraction moves then there exist a solution φi of the system l(g) such that one of the vertices of g can be expanded to obtain a graph g′ which is closer to the generic level and the solution of φi can be modified to obtain a solution of g ′ which is as close as we want to φi. see figure 5 and figure 6. 84 armando rodado amaris, gina lusares cubo 18, 1 (2016) figure 6: the solution, φ, of top linear system l(g) is modified to get solutions for the bottom graph system by adding ±δ to φ solutions of the original graph which are on the circuit and assigning δ to the new two red edges. 6 compactification of t2, mg and polytope complexes figure 4 not only shows how the generic delaunay realizable graphs are connected by whitehead moves, but the connections among the polytope complex p2, which have been described on table 2. from the geometric point of view, p2 is a covering of teichmüller space t2 from which the moduli space of riemann surfaces m2 can be obtained as the interior of the quotient space determined by the groups of symmetries shown on figure 4. polytope complexes, as the one described for genus 2, also exist for the hyperelliptic locus of any genus g ≥ 3. let pg be the polytope complex pg obtained for g ≥ 2. the mathematical theory relating the polytope complex pg, tg and mg is not yet completely developed. however, there is no reason why we could not take advantage of this structure for applications. for a background on polytopes see [8, 26]. 7 polytope complexes for indexing databases it is desirable to have indexing systems for databases of surfaces which mirror the internal structure of the space where the potential members to be included in the database belongs to. this can be done for surfaces of genus two base on the polytope complex p2 whose vertices are given on table cubo 18, 1 (2016) surface databases based on teichmüller theory 85 2. the internal structure of the teichmüller space for riemann surfaces of genus two is given by the polytope complex p2 and the knowledge of its structure which is represented on figure 4. then, if an open database is build on top of p2, a person who have classified a surface s ⋆ of genus two, using the descriptor dθ, could include s ⋆ in the database and annotate the database with new additional features. in addition, since the hyperbolic surface that is associated to dθ⋆, where θ⋆ is determined by the circle pattern angles corresponding to s⋆, we can build and support both new surface knowledge and the applications of surfaces theory. we think that the design of the structure of a database for surfaces of genus g could be enhanced by building on sound combinatorics and hyperbolic geometry, and propose to include the next steps in its design: (1) find cg, the combinatorics of the hyperelliptic locus of tg by generating all non-isomorphic generic graphs (2) construct the polytope complex pg by solving the linear programs of the form l(g) (3) determine the map of the database structure, using the generic realizable graphs as vertices, and connecting these vertices by edges corresponding to whitehead moves (4) finally, a new surface s entering the database would be indexed by the descriptor dθ(s). in addition to the above, to construct the deeper layers of cg, we could build all non-isomorphic graphs which can be obtained by contraction moves from a graph on the generic layer of cg. however, graphs on deeper layer of c(g)-non-genericcan be viewed as well as elements of the polytope cg. 8 conclusion we have shown a framework to develop databases based on polytope complexes which arise by considering the combinatorics and geometry of the teichmüller space for closed surfaces of a given genus. databases supported on the mathematical structures that we have described are desirable because they can be used to study deformation of surfaces in real applications, and supported by rich mathematical results generations of mathematicians have developed. however, our description is not complete because we do not know a canonical cell decomposition of tg for general g. then, further theoretical and computational tools are needed in this area. in particular, research on the applications of canonical decomposition of the moduli space of riemann surfaces which have punctures or boundaries [17] for the development of databases is desirable. we hope to encourage collaboration between researchers with different backgrounds by building database of surfaces having the indexing system that we have proposed. teichmüller and moduli 86 armando rodado amaris, gina lusares cubo 18, 1 (2016) space theories are in the heart of surfaces theory, modelling and its applications. databases of surfaces which mirror the amazing structure of mg and tg should be one of the tools that support the increasingly complex study of surfaces. references [1] f. albat and r. müller: free-form surface construction in a commercial cad/cam system, mathematical of surfaces xi: 11th ima international conference proceedings, loughborough, uk, september 2005. 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[8] b. grünbaum: convex polytopes, pure and applied mathematics, vol xvi, interscience publishers, 1967. [9] t. kuusalo and m. näätänen: geometric uniformization in genus 2, annales academiae scientiarum fennicae , series a.i. mathematica vol 20, 1995, 401–418. cubo 18, 1 (2016) surface databases based on teichmüller theory 87 [10] t. kuusalo and m. näätänen: weierstrass points of extremal surfaces in genus 2, http://www.math.jyu.fi/research/pspdf/231.pdf [11] j. harris and i. morrison: moduli of curves, graduate texts in mathematics, springer-verlag, 1998. [12] a. hass and p. susskind: the geometry of the hyperelliptic involutions in genus two, proceeding of the american mathematical society, vol 105, 1, january 1989. [13] a. marden and b. rodin: on thurston’s formulation and proof of andreev’s theorem, in computational methods and function theory, volume1435 of lecture notes in mathematics, pages 103–115. springer-verlag, 1990. 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[22] w. p. thurston: three-dimensional geometry and topology, i, princeton 1997. [23] y. c. de verdieré: une principe variationnel pour les empilements de cercles, invent. math. 104, 655–669, 1991. [24] wang, yalin and dai, wei and gu, xianfeng and chan, tony f. and yau, shing-tung and toga, arthur w. and thompson, paul m: teichmuller shape space theory and its application to brain morphometry, medical image computing and computer-assisted intervention – miccai 2009: 12th international conference, london, uk, proceedings, part ii, 133–140, september 20-24, 2009, . [25] w. zeng, r. shi, y. wang and x. gu. : teichmuller shape descriptor and its application to alzheimer’s disease study, international journal of computer vision (ijcv), 105(2):155-170, 2013. [26] g. ziegler: lectures on polytopes, springer-verlag, vol 152, 1994. cubo a mathematical journal vol.14, no¯ 02, (81–90). june 2012 an immediate derivation of maximum principle in banach spaces, assuming reflexive input and state spaces. paolo d’alessandro department of mathematics, third university of rome. email: dalex@mat.uniroma3.it abstract we consider a standard setting for the norm optimal problem in banach spaces and show that with a simple argument which invokes some appropriately selected powerful general theorems for banach spaces a straightforward derivation of the maximum principle is obtained. resumen consideramos una formulación estándar para el problema de norma optimal en espacios de banach y mostramos que con un argumento simple que invoca algunos fuertes teoremas generales de la teoŕıa de espacios de banach elegidos apropiadamente se deriva directamente el principio del máximo. keywords and phrases: linear control systems in banach spaces, norm optimal control, support of closed convex sets 2010 ams mathematics subject classification: 49k20. 82 paolo d’alessandro cubo 14, 2 (2012) 1 introduction we consider an optimal control setting proposed by fattorini [1], where the control functions are in the space l ∞ ([0, γ], eu), and eu is a real banach space. to simplify the matter we make the assumption that eu and state space e (another real banach space) be a reflexive banach spaces (but, anyway, l ∞ ([0, γ], eu) is not reflexive). the spaces eu and e are not assumed to be separable. as explained later on, the hypothesis on eu has a justification, in that it is a possible way to make the setting to work, which requires l ∞ ([0, γ], eu) to be the adjoint of another space. the hypothesis on e make it easier on the semigroup front. essentially this is one out of the many settings covered in [1], but handled with a different technique. targets states for which the maximum principle may hold must be support points of the set of states reachable under bounded norm. applying the results in [8] we show that a weath of support points exists. then, suitably blending certain ingredients and, more precisely: the properties of the setting, the celebrated bishop phelps support theorem, the determination and characterization of the radial kernel of the bounded norm reachable set, and, finally, an important technical lemma by fattorini, we obtain a very simple derivation of the maximum principle for a dense set of targets and functionals in e∗. in addition, we geometrically characterize the set of targets for which the principle holds, along with the conditions, which decide whether the principle is only necessary or necessary and sufficient. naturally, all this simplicity is made possible by the profound and powerful results we invoke along the way. but, on the other hand, this is what powerful theorems are mainly useful for: make life easier. there is clearly the need to investigate the connections of our analysis with recent research work on maximum principle, mainly by fattorini. we comment briefly on this in the conclusions, touching upon the open problems it poses. 2 unconstrained and constrained reachable sets consider the variation of constants formula: x(t) = t(t)x + ∫ [0,t] t(t − σ)bu(σ)ds where t ∈ [0, γ], {t(t)}, is a c0 semigroup on a real banach space e, b: eu → e is an operator on the real banach space eu, u ∈ l∞([0, γ], eu), and the integral is a bochner integral. the case eu = e and b = i is referred to as full control case, but here we do not make the full control assumption. we assume both e and eu reflexive. these hypotheses will be in force throughout the paper. for simplicity, dealing with norm optimal control problems, we assume x = 0. the general cubo 14, 2 (2012) an immediate derivation of maximum principle in banach spaces ... 83 case where the system starts from a non-zero state x is dealt with considering the target ζ − t(γ)x in place of ζ (see [1]). it is assumed that we can reach a certain target vector ζ ∈ e at fixed time γ > 0 or: ζ = ∫ [0,γ] t(γ − σ)bu(σ)ds = lγ u and we look for the minimum norm input that does the job of reaching ζ. the relevant linear transformation lγ : l∞([0, γ], eu) → e is well known to be continuous. on the other hand, in our setting (see [1]): l ∞ ([0, γ], eu) = (l1([0, γ], e ∗ u)) ∗ remark 1. as stated by [1], if we had put eu = f, and had taken x such that f = x ∗ then for l ∞ ([0, γ], f) = (l1([0, γ], x)) ∗ to hold, barring separability assumptions, it remains to assume reflexivity of x. for simplicity, we have taken directly eu reflexive which implies e ∗ u reflexive. thus our hypothesis on value space eu is justified, as long as we are interested in a setting where the above duality relation on time function spaces holds. reflexivity of e provides instead a simplification in terms of semigroup theory: in fact, in this case, the adjoint semigroup is c0, and we are dispensed to invoke the phillips dual, as in the general case (again [1]). this setting has the advantage that we find ourselves on the dual side, where the situation is much more favorable. while in general, in view of the celebrated james’ theorem, the unit ball of a banach space is not weakly compact, in the dual, thanks to the banach-alaoglu theorem ([6]), the unit ball is weak∗ compact. this circumstance is obviously useful if we can prove that the operator lγ is weak ∗ to weak continuous. we begin recalling that the assumed strong measurability of u(.) implies weak measurability ([7]). henceforth we use the symbol < ., . > to denote the canonical pairing functionals. if different pair of spaces are involved we either use a suffixes, or leave distinctions to the context when we feel it is safe to do so. proposition 2. the operator lγ is weak ∗ to weak continuous. proof. consider any y ∈ e∗. recall that under our hypothesis that e be reflexive {t∗(.)} is also a c0 semigroup. thus g(.) = b∗t∗(γ − .)y ∈ c([0, γ], e∗u) ⊂ l1([0, γ].e ∗ u) in particular it is weakly measurable. next recall that continuous linear functionals and transformation can go in and out the bochner integral ([7]), and that, since e is reflexive {t∗(.)} is a c0. semigroup. so, consider a weak∗ convergent net {uα} → u in l∞([0, γ], eu) and write: < y, ∫ [0,γ] t(γ − σ)buα(σ)ds >= 84 paolo d’alessandro cubo 14, 2 (2012) ∫ [0,γ] < b∗t∗(γ − σ)y, uα(σ) > ds = < g(.), uα(.) >l1l∞ thus the net {< g(.), uα(.) >l1l∞} converges and the proof is finished. this result implies that rρ = lγ (bρ), the image under lγ of the unit ball bρ of l∞([0, γ], eu), is weakly compact and hence weakly closed. moreover, since rρ is convex, it is also strongly closed. 3 properties of bounded norm reachable set. let rγ = r(lγ ), which is the reachable set in the interval [0, γ]. we may assume, without restriction of generality that r− γ = e. if this where not the case it suffices to consider r− γ in lieu of e. generality is not restricted because a closed subspace of a reflexive banach space is reflexive. for convenience we summarize the relevant properties of the constrained reachable set lγ (bρ) = rρ. • convex and circled (and hence also symmetric). in particular 0 ∈ rρ. • l(rρ) = rγ • weakly compact and both weakly and strongly closed. • in general it has no interior. in special cases rρ might well have interior, but in what follows we assume r i ρ = φ. we add to this list a further important property. first we state the following: definition 3. we define: r∨ρ = {z : z ∈ rρ, inf{‖u‖, lγ (u) = z} < ρ} and put: r∧ρ = rρ\r ∨ ρ proposition 4. r∨ρ is the radial kernel of rρ in rγ. if ζ ∈ r ∧ ρ then ∃ũ such that lγ (ũ) = ζ, ‖ũ‖ = ρ = min{‖u‖ : lγ (u) = ζ}. a necessary condition for a state ζ to be a support point of rρ is that ζ ∈ r∧ρ . cubo 14, 2 (2012) an immediate derivation of maximum principle in banach spaces ... 85 proof. clearly 0 ∈ r∨ρ . if w 6= 0 and w ∈ rγ , ∃u 6= 0 s.t. lγ (u) = w. if 0 ≤ ε < ρ, then the non-zero state: lγ (ε u ‖u‖ ) ∈ rρ thus rρ is radial at zero in rρ. next, if ξ 6= 0, ξ ∈ r ∨ ρ , then ∃u s.t.‖u‖ = ρ ′, 0 < ρ′ < ρ and lγ (u) = ξ . then for an arbitrary z 6= ξ, 0 6= z ∈ rγ , let lγ (uz) = z − ξ and ‖uz‖ = γ 6= 0. the state ξ+ α(z − ξ), with α ≤ ρ−ρ ′ γ , is reachable by the control u + αuz, whose norm is less or equal to ρ, by the triangle inequality. thus ξ+ β(z − ξ) ∈ rρ for 0 ≤ β ≤ α and rρ is radial at ξ. next ζ ∈ r∧ρ implies inf{‖u‖, lγ (u) = ζ} ≥ ρ but, also, ζ ∈ rρ implies ∃u s.t.‖u‖ ≤ ρ and so the second statement is proved. the just proved radiality property prevents any state in r∨ρ to be a support point of rρ. in fact suppose that for some ζ ∈ r∨ρ there exists a continuous linear functional f, such that < f, rρ >≤< f, ζ >. if it were < f, ζ >= 0 then < f, rρ >= {0} =< f, l(rρ) >=< f, rγ >. but the fact that rγ is contained in a closed hyperplane contradicts the fact that rγ is dense. if < f, ζ >= α > 0, then because by radiality, for some ε > 0, ξ = (1 + ε)ζ ∈ rρ, we can write < f, ξ >>< f, ζ >, contradicting separation. this concludes the proof. remark 5. if rρ had interior then r ∨ ρ = r i ρ and b(rρ) = r ∧ ρ , and all points of r ∧ ρ would be support points. if not then, since rρ is closed, b(rρ) = rρ, but this proposition tell us that we can find support points only in the proper subset r∧ρ . in this case we may view the above partition as a quasi-topological decomposition in which r∨ρ plays the role of quasi-interior and r ∧ ρ as a quasi-boundary. the results on the support problem given in [8] hold good here because they are surely true for hausdorff complete locally convex spaces. we summarize the argument here. first notice that the tangent cone to a convex set at an extreme point is always pointed. theorem 5 in [8] states that the closure of a pointed cone in a linear topological space is a proper cone. in the same paper lemma 2 states that a closed proper cone is contained in a closed semispace; the statement is made for hilbert spaces but it is obvious from it simple proof that it is indeed valid for hausdorff complete locally convex spaces. it follows that all extreme points of a convex set are support points. next, by the krein-milman theorem, the set ex(rρ) of extreme points of rρ is non-void (and generates rρ by closed convex extension). clearly ex(rρ) ⊂ r ∧ ρ , because there cannot be radiality in an extreme point. as recalled, all points of ex(rρ) are support points. let us call the set of all support points sρ. we collect this remarks in the following: proposition 6. ex(rρ) ⊂ sρ ⊂ r ∧ ρ . in particular sρ 6= φ. 4 support for void interior convex sets in a nutshell the maximum principle is about showing that a target is a support point for rρ and then translating the supporting condition in a pointwise in time condition for the optimal control. 86 paolo d’alessandro cubo 14, 2 (2012) the support condition is a special separation condition (separation between a singleton in a set and the set itself). one classical way to provide support points for a set is to invoke a topological separation theorem, which states that, in a linear topological space, given two convex sets a and b and assuming that a has interior, there is a continuous linear functional separating a and b iff b ∩ ai = φ. one obviously uses this theorem taking b = {ζ} with ζ ∈ b(a) thereby obtaining a support point for a−. the derivation of this separation theorem in [6] is essentially pre-topological and based on the theory of cones. despite the ”iff” we are in presence of a masked sufficient condition, because of the presiding hypothesis that ai 6= φ. thus we cannot exclude the possibility of finding support points for void interior sets. if the dimension is finite, then every convex set has (relative) interior, so that the application of this separation theorem to find support targets is direct and general. in infinite dimension, as already mentioned, rρ has no interior in general. we see various possible techniques to overcome this hurdle. the first consists in re-topologizing rρ in such a way that it has interior in the new topology. this is the technique introduced by fattorini (see [1]), who has developed a very complete and advanced theory for both norm and time optimality. because the new topology is stronger, in the larger dual space, singular functionals appear (non-zero functionals that are zero on the domain of the infinitesimal generator). the second possibility (developed in [8]) consisted in introducing and applying a support theorem for extreme points of convex sets (like all other separation/support theorems, this too is based on theory of cones). this technique works well in a hilbert space setting. in the present setting it allows us to exhibit an already large set of support points via the krein-milman theorem. but it would be nice to tell more about the structure of the set of support points of rρ. to this effect we apply a further tool: the celebrated bishop-phelps theorem for closed sets in banach spaces (see [9]). we recall the relevant part of the bishop-phelps theorem: theorem 7. a closed convex set of a real banach space has a non-void set of support points which is dense in its boundary. remark 8. some authors have shown that the corresponding statement for complex banach spaces does not hold (see e.g. [10]). this is completely irrelevant here. basically the idea of the proof of the bishop phelps theorem is to observe that the cone generated by a translated ball not containing the origin is a pointed cone with interior. if we can place, by translation, the apex of this cone on a point of the convex set in such a way that the apex is the only point in their intersection, then the cone is separated from the convex set. note the ingenuous swap of roles: here it is the cone in charge of insuring that at least one of the convex sets has non-void interior, so that the convex set, for which support is sought, is allowed to have cubo 14, 2 (2012) an immediate derivation of maximum principle in banach spaces ... 87 void interior. but such separation and the fact that the second set is a cone, imply that the point in question is a support point for the convex set. 5 norm optimality putting together the bishop-phelps theorem and proposition 4, we obtain the following result: theorem 9. the set sρ of all support points for rρ, which we proved to be non-void and contained in r∧ρ , is dense in r ∧ ρ . for all points ζ ∈ sρ, the minimum norm of controls that steer the system from 0 (at t = 0) to ζ (at t = γ) exists and is ρ. the proof of this theorem is contained in the previous analysis and can be omitted. notice that if the interior of rρ is void (as we are assuming) then sρ is dense in the whole rρ and hence also in the set r∧ρ into which is contained. if the interior is non-void, then sρ is dense in b(rρ), which, in such case, coincides with r∧ρ . let’s write down the support condition for ζ ∈ sρ. there exists a continuous linear functional f ∈ e∗ such that: < f, ζ >≥< f, z > , ∀z ∈ rρ the same condition holds, however, for all ξ ∈ rρ such that: < f, ξ >=< f, ζ > define ω = {ξ : ξ ∈ rρ, < f, ξ >=< f, ζ >} note that ω is a closed convex set, as a matter of facts it is a closed exposed face of rρ and obviously: ω ⊂ sρ it is well possible that ω = {ζ}, in which case ζ is an exposed extreme point. but it may happen, as well, that ω is a proper superset of {ζ}. next, to move back to the control space, we begin recalling the following well known proposition ([6]): proposition 10. let t be a linear transformation e → g and c be a convex subset of e. if a is a face of t(c) then t−1(a) ∩ c is a face of c. it follows that l−1 γ (ω) ∩ bρ is a closed face of bρ. it is also an exposed face because we can retrieve it as a face of bρ, generated by the support functional g = l∗γ f = b ∗t∗(γ − .)f ∈ c([0, γ], e∗u) ⊂ l1([0, γ], e ∗ u) that ”pushes back” the support functional f ∈ e∗. 88 paolo d’alessandro cubo 14, 2 (2012) in fact for each ξ ∈ ω, for all the corresponding uξ ∈ l −1 γ (ω) ∩ bρ it must be (in the next formula there are two different pairing functionals, but we leave unchanged the symbol): < f, ξ >=< f, lγ uξ >=< l ∗ γ f, uξ >≥< f, lγ u >=< l ∗ γ f, u > , ∀u ∈ bρ if instead u ∈ bρ\l −1 γ (ω) ∩ bρ: < g, uξ >>< g, u > thus we have proved that the functional g is a support functional for bρ at all points of l −1 γ (ω)∩bρ and this set is a closed and exposed face of bρ. remark 11. a consequence of james’ theorem insures that, for a non-reflexive banach spaces there exist some continuous linear functionals, that do not attain their supremum on the unit ball (but, of course, by the separation theorem, there is also a profusion of continuous linear functionals that do attain their supremum on the unit ball). clearly g is is not one of those pathological continuous linear functionals, because we have proved that it attains its supremum on bρ. if we use the above condition to characterize uξ, we have a necessary condition for the optimum controls corresponding to the target ζ. the condition is not sifficient if ω\ {ζ} 6= φ. if, by the contrary, no other target is involved, or ω = {ζ}, the condition becomes sufficient (independently of the fact that the optimum control be unique). reall that ω = {ζ}means that ζ is an exposed extreme point. the proof is implicit in the above discussion illustrating the correspondence between the two exposed faces of rρ and bρ. we register this fact in the following: theorem 12. the condition on u: < l∗γ f, u >= max {< l ∗ γ f, u > : u ∈ bρ} is necessary for a control u to be the norm optimal for the target ζ the condition becomes also sufficient if ω = {ζ}, or, equivalently, if ζ is an exposed extreme point. 6 maximum principle to state the maximum principle, we need to recast the support condition in an equivalent condition on the optimal input, which characterizes the input pointwise in time. given the reflexivity assumptions in force, the function l∗γ f(.) = b ∗t∗(γ − .)f is continuous thus borel and weakly measurable. also continuity of the norm implies that ‖l∗ γ f(.)‖e∗ is a continuous and thus measurable function. the support condition for the optimum control u yields: < l∗γ f, u >l 1 l∞ = ∫ [0,γ] < l∗γ f, u >e∗ueu (σ)dσ = cubo 14, 2 (2012) an immediate derivation of maximum principle in banach spaces ... 89 = max    ∫ [0,γ] < l∗γ f, u > (σ)dσ: u ∈ bρ    ≤ ρ ∫ [0,γ] ‖l∗γ f‖e∗u(σ)dσ = = ∫ [0,γ] max{< l∗γ f, v >: ‖v‖eu ≤ ρ}(σ)dσ < ∞ on the other hand, by lemma 2.2.10 in [1], the inequality can be substituted by equality (indeed fattorini proved this for arbitrary banach spaces) and so the support condition implies (and clearly is implied by): < b∗t∗(γ − σ)f, u(σ) >= max{< b∗t∗(γ − σ)f, v >: ‖v‖eu ≤ ρ} a.e. for σ ∈ [0, γ]. this characterization of the optimal control is the maximum principle. it yields a set of optimal controls. in view of the equivalence with the support condition, our consideration on whether necessity or necessity and sufficiency prevail hold good as well for the maximum principle. thus if ζ is a an exposed extreme points all controls defined by the principle are optimal (or the condition is necessary and sufficient). otherwise we can only say that the optimal controls are among those functions satysfying this condition (the condition is necessary). 7 conclusions the following considerations are inspired by the cited work by fattorini, including a private communication. one open problem is the relationship between the dense subset of targets for which the maximum principle holds for functionals in e∗, that we have shown to exists, and the domain of a. in [1] the maximum principle is proved for targets in d(a) using functionals in a linear space z larger than e∗. on the other hand [3] shows that, in general, if the target is in d(a), then the functional in e∗does not always exist. he also conjectures that this implication may fail even under the assumption that the semigroup is selfadjoint and the state space is a hilbert space, albeit the implication has been found to hold for the left translation semigroup and e = l2([0, ∞)) ([4]). thus he puts the question of finally determining a condition, stronger than target ζ ∈ d(a), ensuring that the functionals appearing in the maximum principle exists in e∗. this question is of course crucial but it is open at the moment. in some cases this whole issue is known to be connected the other important aspect of regularity of optimal control (e.g. [2]). a further motivation for orientating research toward these open problems. 90 paolo d’alessandro cubo 14, 2 (2012) received: october 2010. revised: october 2011. references [1] h.o. fattorini ”infinite dimensional linear control systems”, elsevier, amsterdam 2005. [2] h.o. fattorini, ”strong regularity of time and norm optimal controls”, submitted [3] h.o. fattorini, ”linear control systems in sequence spaces” functional analysis and evolution equations, the gunter lumer volume 2007, pp 273-290. [4] h.o. fattorini, ”regular and strongly regular time and norm optimal controls” submitted. [5] h.o. fattorini, private communication. [6] j.l. kelley and i. namioka, linear topological spaces, springer, new york, 1963 [7] k. yosida, ”functional analysis”, springer-verlag, new york, 1974. [8] p. d’alessandro, ”closure of pointed cones and maximum principle in hilbert spaces”, cubo a mathematical journal, vol.13, no.2 (73-84), june 2011. [9] r.r. phelps, ”support cones in banach spaces and their application”, adv. in math. 13 (1974), 1-19. [10] w.b. johnson, j. lindenstrauss eds, ”handbook of the geometry of banach spaces”, vol. 1, elsevier, amsterdam, 2001. introduction unconstrained and constrained reachable sets properties of bounded norm reachable set. support for void interior convex sets norm optimality maximum principle conclusions cubo a mathematical journal vol.18, no¯ 01, (47–57). december 2016 s-paracompactness modulo an ideal josé sanabria1, ennis rosas1, neelamegarajan rajesh2, carlos carpintero1, amalia gómez1 1 departamento de matemáticas, universidad de oriente, cumaná, venezuela. 2 department of mathematics, rajah serfoji govt. college, thanjavur-613005, tamilnadu, india. jesanabri@gmail.com, ennisrafael@gmail.com, nrajesh topology@yahoo.co.in, carpintero.carlos@gmail.com, amaliagomez1304@gmail.com abstract the notion of s-paracompactness modulo an ideal was introduced and studied in [15]. in this paper, we introduce and investigate the notion of αs-paracompact subset modulo an ideal which is a generalization of the notions of αs-paracompact set [1] and αparacompact set modulo an ideal [7]. resumen la noción de s-paracompacidad módulo un ideal fue introducida y estudiada en [15]. en este art́ıculo, introducimos e investigamos la noción de un subconjunto αs-paracompacto módulo un ideal, que es una generalización de las nociones de conjunto αs-paracompacto [1] y conjunto α-paracompacto módulo un ideal [7]. keywords and phrases: semi-open, ideal, s-paracompact. research partially suported by consejo de investigación udo. 2010 ams mathematics subject classification: 54a05, 54d20. 48 j. sanabria, e. rosas, n. rajesh, c. carpintero and a. gómez cubo 18, 1 (2016) 1 introduction the concept of α-paracompact subset modulo an ideal was defined and investigated by ergun and noiri [7]. the notions of s-paracompact spaces and αs-paracompact subsets were introduced in 2006 by al-zoubi [1] and also have been studied by li and song [13]. very recently, sanabria, rosas, carpintero, salas and garćıa [15] have introduced and investigated the concept of s-paracompact space with respect to an ideal as a generalization of the s-paracompact spaces. in this paper, we introduce the notion of αs-paracompact subset modulo an ideal which is a generalization of both αs-paracompact subset [1] and α-paracompact subset modulo an ideal. 2 preliminaries throughout this paper, (x, τ) always means a topological space on which no separation axioms are assumed unless explicitly stated. if a is a subset of (x, τ), we denote the closure of a and the interior of a by cl(a) and int(a), respectively. also, we denote by ℘(x) the class of all subset of x. a subset a of (x, τ) is said to be semi-open [11] (resp. semi-preopen [2]) if a ⊂ cl(int(a)) (resp. a ⊂ cl(int(cl(a)))). the complement of a semi-open set is called a semi-closed set. the semiclosure of a, denoted by scl(a), is defined by the intersection of all semi-closed sets containing a. the collection of all semi-open sets of a topological space (x, τ) is denoted by so(x, τ). a collection v of subsets of a space (x, τ) is said to be locally finite, if for each x ∈ x there exists ux ∈ τ containing x and ux intersects at most finitely many members of v. a space (x, τ) is said to be paracompact (resp. s-paracompact [1]), if every open cover of x has a locally finite open (resp. semi-open) refinement which covers to x (we do not require a refinement to be a cover). lemma 2.1. let (x, τ) be a space. then, the following properties hold: (1) if (a, τa) is a subspace of (x, τ), b ⊆ a and b ∈ so(x, τ), then b ∈ so(a, τa) [11]. (2) if a ∈ τ and b ∈ so(x, τ), then a ∩ b ∈ so(x, τ) [4]. (3) if (a, τa) is an open subspace of (x, τ), b ⊆ a and b ∈ so(a, τa), then b ∈ so(x, τ) [5]. an ideal i on a nonempty set x is a nonempty collection of subset of x which satisfies the following two properties: (1) a ∈ i and b ⊂ a implies b ∈ i; (2) a ∈ i and b ∈ i implies a ∪ b ∈ i. in this paper, the triplet (x, τ, i) denote a topological space (x, τ) together with an ideal i on x and will simply called a space. given a space (x, τ, i), a set operator (.)⋆ : ℘(x) → ℘(x), called the local function [10] of a with respect to τ and i, is defined as follows: for a ⊂ x, cubo 18, 1 (2016) s-paracompactness modulo an ideal 49 a⋆(i, τ) = {x ∈ x : u ∩ a /∈ i for every u ∈ τ(x)}, where τ(x) = {u ∈ τ : x ∈ u}. when there is no chance for confusion, we will simply write a⋆ for a⋆(i, τ). in general, x⋆ is a proper subset of x. the hypothesis x = x⋆ is equivalent to the hypothesis τ ∩ i = ∅. according to [14], we call the ideals which satisfy this hypothesis τ-boundary ideals. note that cl⋆(a) = a ∪ a⋆ defines a kuratowski closure for a topology τ⋆(i), finer than τ. a basis β(i, τ) for τ⋆(i) can be described as follows: β(i, τ) = {v \ j : v ∈ τ and j ∈ i}. when there is no chance for confusion, we will simply write τ⋆ for τ⋆(i) and β for β(i, τ). in the sequel, the ideal of nowhere dense (resp. meager) subsets of (x, τ) is denoted by n (resp. m). 3 αs-paracompactness modulo an ideal in this section, we shall introduce and study the αs-paracompact subsets modulo an ideal i, which is a natural generalization of αs-paracompact subsets. first recall some notions of paracompactness. definition 3.1. a subset a of a space (x, τ) is said to be α-paracompact [3] (resp. α-almost paracompact [9]) if for any open cover u of a, there exists a locally finite collection v of open sets such that v refines u and a ⊂ ⋃ {v : v ∈ v} (resp. a ⊂ ⋃ {cl(v) : v ∈ v}). a space (x, τ) is said to be paracompact (resp. almost-paracompact) if x is α-paracompact (resp. α-almost paracompact). definition 3.2. a subset a of a space (x, τ, i) is said to be α-paracompact modulo i [7] (briefly α-paracompact (mod i)), if for any open cover u of a, there exist i ∈ i and a locally finite collection v of open sets such that v refines u and a ⊂ ⋃ {v : v ∈ v} ∪ i. a space (x, τ, i) is said to be i-paracompact or paracompact with respect to i [16], if x is αparacompact modulo i. in the present, it is called paracompact modulo i (or briefly paracompact (mod i)). definition 3.3. a subset a of a space (x, τ) is said to be αs-paracompact [1] if for any open cover u of a, there exists a locally finite collection v of open sets such that v refines u and a ⊂ ⋃ {v : v ∈ v}. a space (x, τ) is said to be s-paracompact if x is αs-paracompact. now, we give the definition of αs-paracompact subset modulo an ideal i. definition 3.4. a subset a of a space (x, τ, i) is said to be αs-paracompact modulo i (briefly αs-paracompact (mod i)), if for any open cover u of a, there exist i ∈ i and a locally finite collection v of semi-open sets such that v refines u and a ⊂ ⋃ {v : v ∈ v} ∪ i. a space (x, τ, i) is said to be i-s-paracompact or s-paracompact with respect to i [15], if x is αs-paracompact modulo i. in the present, it is called s-paracompact modulo i (or briefly s-paracompact (mod i)). we say that a is s-paracompact (mod i) if (a, τ a , i a ) is s-paracompact (mod i a ) as a subspace, where τ a is the relative topology induced on a by τ and i a = {i∩a : i ∈ i}. 50 j. sanabria, e. rosas, n. rajesh, c. carpintero and a. gómez cubo 18, 1 (2016) proposition 3.1. let a be a subset of a space (x, τ) and i an ideal on (x, τ). then, the following properties hold: (1) if a is α-paracompact (mod i), then a is αs-paracompact (mod i). (2) every i ∈ i is an αs-paracompact (mod i). (3) (x, τ, i) is s-paracompact (mod i) if there exists i ∈ i such that x − i is αs-paracompact (mod i). (4) a is αs-paracompact if and only if it is αs-paracompact (mod {∅}). proof. (1) follows from the fact that every open set is semi-open. (2) suppose that there exists i ∈ i such that i is not αs-paracompact (mod i). then, there exists an open cover u of i such that i ̸⊂ ⋃ {v : v ∈ v} ∪ j for every j ∈ i and every locally finite collection v which refines u. this is a contradiction, because i ∈ i and i ⊂ ⋃ {v : v ∈ v} ∪ i. (3) suppose that there exists i ∈ i such that x − i is αs-paracompact (mod i) and let u be an open cover of x. then, u is an open cover of x − i and hence there exist j ∈ i and a locally finite collection v of semi-open sets such that v refines u and x − i ⊂ ⋃ {v : v ∈ v} ∪ j. thus, x = (x − i) ∪ i ⊂ ⋃ {v : v ∈ v} ∪ (j ∪ i) and as j ∪ i ∈ i, we have (x, τ, i) is s-paracompact (mod i). (4) it is obvious. now, we give some comments related with the proposition 3.1. remark 3.1. according to proposition 3.1(1), every α-paracompact (mod i) (resp. αsparacompact) subset is αs-paracompact (mod i), and from this point of view, the notion of αs-paracompact (mod i) subset is a natural generalization of the notion of α-paracompact (mod i) (resp. αs-paracompact) subset. on the other hand, in example 2.11 of [13], it is shows that there exists a semiregular hausdorff space x and a regular closed subset m of x such that m is an αs-paracompact (mod {∅}) subset of x, but m is not α-paracompact (mod {∅}). thus, the converse of proposition 3.1(1) in general is not true. proposition 3.2. let a be a subset of a space (x, τ) and i an ideal on (x, τ). then, the following properties hold: (1) if a is a semi-open and αs-paracompact (mod i) set and i is τ-boundary, then a is α-almost paracompact. (2) a semi-preopen set a is αs-paracompact (mod n) if and only if it is α-almost paracompact. proof. (1) let u be any open cover of a. then there exist i ∈ i and a locally finite collection v = {vλ : λ ∈ λ} of semi-open sets such that v refines u and a ⊂ ⋃ {vλ : λ ∈ λ} ∪ i. since a is cubo 18, 1 (2016) s-paracompactness modulo an ideal 51 semi-open, a ⊂ cl(int(a)) and as i is τ-boundary, int(i) = ∅. now, by the locally finiteness of v, the collection v′ = {int(vλ) : λ ∈ λ} is also locally finite, it follows that a ⊂ cl(int(a)) ⊂ cl ( int ( ⋃ λ∈λ vλ ∪ i )) ⊂ cl ( int ( ⋃ λ∈λ cl(int(vλ)) ∪ i )) = cl ( int ( cl ( ⋃ λ∈λ int(vλ) ) ∪ i )) = cl ( int ( cl ( ⋃ λ∈λ int(vλ) ) ∪ int(i) )) = cl ( int ( cl ( ⋃ λ∈λ int(vλ) ))) ⊂ cl ( ⋃ λ∈λ int(vλ) ) = ⋃ λ∈λ cl(int(vλ)). if wλ = int(vλ), then a ⊂ ⋃ λ∈λ cl(wλ). observe that wλ is open for each λ ∈ λ and wλ ⊂ vλ ⊂ u for some u ∈ u, hence w = {wλ : λ ∈ λ} is a locally finite open refinement of u. therefore, a is α-almost paracompact. (2) similar to the proof of (1), if a is semi-preopen, then a ⊂ cl(int(cl(a))) ⊂ cl ( int ( cl ( ⋃ λ∈λ vλ ∪ i ))) = cl ( int ( cl ( ⋃ λ∈λ vλ ) ∪ cl(i) )) = cl ( int ( cl ( ⋃ λ∈λ vλ ) ∪ int(cl(i)) )) = cl ( int ( cl ( ⋃ λ∈λ vλ ))) ⊂ cl ( int ( cl ( ⋃ λ∈λ cl(int(vλ)) ))) = cl ( int ( cl ( ⋃ λ∈λ int(vλ) ))) ⊂ cl ( ⋃ λ∈λ int(vλ ) = ⋃ λ∈λ cl(int(vλ)). 52 j. sanabria, e. rosas, n. rajesh, c. carpintero and a. gómez cubo 18, 1 (2016) therefore, the proof follows. as a consequence of proposition 3.2, we obtain the following result. corollary 3.1. (sanabria et al. [15]) let i be an ideal on a space (x, τ). then, the following properties hold: (1) if i is τ-boundary and (x, τ) is s-paracompact (mod i), then (x, τ) is almost-paracompact. (2) (x, τ) is s-paracompact (mod n) if and only if it is almost-paracompact. theorem 3.1. if every open subset of a space (x, τ, i) is αs-paracompact (mod i), then every subspace of (x, τ, i) is s-paracompact (mod i). proof. suppose that a is any subspace of (x, τ, i) and let u = {uµ : µ ∈ ∆} be a τa-open cover of a. for every µ ∈ ∆ there exists vµ ∈ τ such that uµ = vµ ∩a. put v = ⋃ {vµ : µ ∈ ∆}, then v ∈ τ and v = {vµ : µ ∈ ∆} is a τ-open cover of v. by hypothesis, there exist i ∈ i and a τ-locally finite collection w = {wλ : λ ∈ λ} of τ-semi-open sets such that w refines v and v ⊂ ⋃ {wλ : λ ∈ λ}∪i. then, we have a = ⋃ µ∈∆ uµ = ⋃ µ∈∆ (vµ ∩ a) = ⎛ ⎝ ⋃ µ∈∆ vµ ⎞ ⎠ ∩ a = v ∩ a ⊂ ( ⋃ λ∈λ wλ ∪ i ) ∩ a = ⋃ λ∈λ (wλ ∩ a) ∪ ia, where ia = i ∩ a ∈ ia. if x ∈ a, then there exists gx ∈ τ containing x such that wλ ∩ gx = ∅ for all λ ̸= λ1, λ2, . . . , λn and so (wλ ∩ gx) ∩ a = ∅ for all λ ̸= λ1, λ2, . . . , λn. it follows that (wλ ∩ a) ∩ (gx ∩ a) = ∅ for all λ ̸= λ1, λ2, . . . , λn and hence, the collection h = {wλ ∩ a : λ ∈ λ} is τ a -locally finite. if wλ ∩ a ∈ h, then wλ ∈ w and since w refines v, wλ ⊆ vµ for some vµ ∈ v, which implies that wλ ∩ a ⊂ vµ ∩ a = uµ ∈ u. therefore, h refines u. this shows that h = {wλ ∩ a : λ ∈ λ} is a τa-locally finite collection of τa-semi-open sets which refines u such that a ⊂ ⋃ {h : h ∈ h} ∪ ia. thus, every subspace of (x, τ, i) is s-paracompact (mod i). the following result is an immediate consequence of theorem 3.2. corollary 3.2. if every open subset of a space (x, τ, i) is αs-paracompact (mod i), then (x, τ, i) is s-paracompact (mod i). recall that a subset a of a space (x, τ) is said to be g-closed [12] if cl(a) ⊂ u whenever a ⊂ u and u ∈ τ. theorem 3.2. if (x, τ, i) is s-paracompact (mod i) and a is a g-closed subset of x, then a is αs-paracompact (mod i). cubo 18, 1 (2016) s-paracompactness modulo an ideal 53 proof. suppose that a is a g-closed subset of an s-paracompact (mod i) space (x, τ, i). let u = {uµ : µ ∈ ∆} be an open cover of a. since a is g-closed and a ⊂ ⋃ {uµ : µ ∈ ∆}, then scl(a) ⊂ ⋃ {uµ : µ ∈ ∆}. for each x /∈ cl(a) there exists a τ-open set gx containing x such that a ∩ gx = ∅. put u ′ = {uµ : µ ∈ ∆} ∪ {gx : x /∈ cl(a)}. then u ′ is an open cover of the s-paracompact (mod i) space x and so, there exist i ∈ i and a locally finite collection v = {vλ : λ ∈ λ} of semi-open sets such that v refines u and x = ⋃ {vλ : λ ∈ λ} ∪ i. for each λ ∈ λ, either vλ ⊂ uµ(λ) for some µ(λ) ∈ ∆ or vλ ⊂ gx(λ) for some x(λ) /∈ cl(a). now, put λ0 = {λ ∈ λ : vλ ⊂ uβ(λ)}. then v ′ = {vλ : λ ∈ λ0} is a collection of semi-open sets which is locally finite and refines u. also, x − ⋃ λ∈λ0 vλ = ( ⋃ λ∈λ vλ ∪ i ) − ⋃ λ∈λ0 vλ = ⋃ λ/∈λ0 vλ ∪ i ⊂ ⋃ λ/∈λ0 gx(λ) ∪ i ⊂ (x − a) ∪ i = x − (a − i), which implies a − i ⊂ ⋃ λ∈λ0 vλ and hence a ⊂ ⋃ λ∈λ0 vλ ∪ i. this shows that a is αs-paracompact (mod i). theorem 3.3. let (x, τ, i) be a space. then, the following properties hold: (1) if a is an open αs-paracompact (mod i) subset of (x, τ, i), then a is s-paracompact (mod i). (2) if a is a clopen subset of (x, τ, i), then a is αs-paracompact (mod i) if and only if it is s-paracompact (mod i). proof. (1) let a be an open αs-paracompact (mod i) subset of (x, τ, i). let u = {uµ : µ ∈ ∆} be a τ a -open cover of a. since a is τ-open, we have u is a τ-open cover of a and hence, there exist i ∈ i and a τ-locally finite collection v = {vλ : λ ∈ λ} of τ-semi-open sets which refines u such that a ⊂ ⋃ {vλ : λ ∈ λ} ∪ i. it follows that a ⊂ ⋃ {vλ ∩ a : λ ∈ λ} ∪ (i ∩ a) and so, the collection va = {vλ ∩ a : λ ∈ λ} is a τa-locally finite τa-semi-open refinement of u and is an ia-cover of a. therefore, a is s-paracompact (mod i). (2) if a is a clopen and αs-paracompact (mod i) subset of (x, τ, i), then from (1) we obtain that a is is s-paracompact (mod i). conversely, let u = {uµ : µ ∈ ∆} be a τ-open cover of a. the collection v = {a ∩ uµ : µ ∈ ∆} is a τa-open cover of the s-paracompact (mod i) subspace (a, τ a , i a ) and hence, there exist ia ∈ ia and a τa-locally finite τa-semi-open refinement w = {wλ : λ ∈ λ} of v such that a = ⋃ {wλ : λ ∈ λ} ∪ ia. it is easy to see that w refines u and by lemma 2.1(3), we have that wλ ∈ so(x, τ) for each λ ∈ λ. to show w = {wλ : λ ∈ λ} is τ-locally finite, let x ∈ x. si x ∈ a, then there exists ox ∈ τa ⊂ τ containing x such that ox intersects at most finitely many members of w. otherwise x \ a is a τ-open set containing x which intersects no member of w. therefore, w is τ-locally finite and such that 54 j. sanabria, e. rosas, n. rajesh, c. carpintero and a. gómez cubo 18, 1 (2016) a = ⋃ {wλ : λ ∈ λ} ∪ ia ⊂ ⋃ {wλ : λ ∈ λ} ∪ i for some i ∈ i. thus, a is αs-paracompact (mod i). as a consequence of theorem 3.3, we obtain the following result. corollary 3.3. every clopen subspace of a s-paracompact (mod i) space is s-paracompact (mod i). lemma 3.1. let a be a subset of a space (x, τ, i). if every open cover of a has a locally finite closed refinement v such that a ⊂ ⋃ {v : v ∈ v} ∪ i for some i ∈ i, then v has a locally finite open refinement w such that a ⊂ ⋃ {w : w ∈ w} ∪ i. proof. let u be an open cover of a. by hypothesis, there exist i ∈ i and a locally finite closed refinement v = {vλ : λ ∈ λ} of u such that a ⊂ ⋃ {vλ : λ ∈ λ} ∪ i. for each x ∈ a, there exists an open set gx containing x such that gx intersects at most finitely many members of v. note that the collection g = {gx : x ∈ a} is an open cover of a and again by hypothesis, there exist j ∈ i and a locally finite closed refinement h = {hµ : µ ∈ ∆} of g such that a ⊂ ⋃ {hµ : µ ∈ ∆} ∪ j. now, as {hµ : hµ ∩ vλ = ∅} ⊂ h, then the collection {hµ : hµ ∩ vλ = ∅} is locally finite and ⋃ {hµ : hµ ∩ vλ = ∅} = ⋃ {cl(hµ) : hµ ∩ vλ = ∅} = cl( ⋃ {hµ : hµ ∩ vλ = ∅}), it follows that oλ = x − ⋃ {hµ : hµ ∩ vλ = ∅} is an open set and vλ ⊂ oλ, for each λ ∈ λ. for each µ ∈ ∆ and λ ∈ λ, we have hµ ∩ oλ ̸= ∅ ⇐⇒ hµ ∩ vλ ̸= ∅. (∗) since v refines u, for every λ ∈ λ there exists u(λ) ∈ u such that vλ ⊂ u(λ). put wλ = oλ∩u(λ), then the collection w = {wλ : λ ∈ λ} is an open refinement of u. furthermore, if x ∈ a there exists an open set dx such that dx intersects at most finitely many members of h, it follows from (∗) that w is locally finite. also, a ⊂ ⋃ {vλ : λ ∈ λ} ∪ i ⊂ ⋃ {oλ ∩ u(λ) : λ ∈ λ} ∪ i = a ⊂ ⋃ {wλ : λ ∈ λ} ∪ i. the following theorem shows that, in the presence of the axiom of regularity, the notions of α-paracompact (mod i) and αs-paracompact (mod i) subsets are equivalent. theorem 3.4. let i be an ideal on a regular space (x, τ) and a be a subset of x. then, a is α-paracompact (mod i) if and only if it is αs-paracompact (mod i). proof. necessity is obvious from the definitions. to show sufficiency, assume a is an αs-paracompact (mod i) subset of (x, τ, i) and let u = {uµ : µ ∈ ∆} be an open cover of a. for each x ∈ a, there exists µ(x) ∈ ∆ such that x ∈ uµ(x) and since (x, τ, i) is a regular space, there exists an open set vx such that x ∈ vx ⊂ cl(vx) ⊂ uµ(x). thus, v = {vx : x ∈ a} is an open cover of a and because a is αs-paracompact (mod i), there exist i ∈ i and a locally finite semi-open refinement w = {wλ : λ ∈ λ} of v such that a ⊂ ⋃ {wλ : λ ∈ λ} ∪ i. since w refines v, then for each λ ∈ λ there exists x(λ) ∈ x such that wλ ⊂ vx(λ) and so, wλ ⊂ cl(wλ) ⊂ cl(vx(λ)) ⊂ uµ(x(λ)). obviously the collection {cl(wλ) : λ ∈ λ} is a locally finite closed refinement of u such that cubo 18, 1 (2016) s-paracompactness modulo an ideal 55 a ⊂ ⋃ {cl(wλ) : λ ∈ λ} ∪ i. by lemma 3.1, the open cover u of a has a locally finite open refinement h such that a ⊂ ⋃ {h : h ∈ h} ∪ i. therefore, a is an α-paracompact (mod i) subset of (x, τ, i). proposition 3.3. if a is an αs-paracompact (mod i) subset of a space (x, τ, i) and b is a subset of x with ∂(b) ∈ i, then a ∩ cl(b) is αs-paracompact (mod i). proof. let u be an open cover of a ∩ cl(b). then u′ = u ∪ {x − cl(b)} is an open cover of a and so, there exist i ∈ i and a locally finite semi-open refinement v = {vλ : λ ∈ λ} of u ′ such that a ⊂ ⋃ {vλ : λ ∈ λ} ∪ i. then, ∂(cl(b)) ⊂ ∂(b) ∈ i and a ∩ cl(b) ⊂ ⋃ λ∈λ vλ ∩ int(cl(b)) ∪ j, where j = [( ⋃ {vλ : λ ∈ λ})∩∂(cl(b))]∪(i∩cl(b)) ∈ i. thus, the collection v ′ = {vλ ∩int(cl(b)) : λ ∈ λ} is a locally finite semi-open refinement of u such that a ∩ cl(b) ⊂ ⋃ {v : v ∈ v′} ∪ j. therefore, a ∩ cl(b) is αs-paracompact (mod i). the following result follows from proposition 3.3 and the fact that the topological frontier of a semi-open (resp. semi-closed) set is nowhere dense. corollary 3.4. if a is an αs-paracompact (mod n) subset of a space (x, τ, i) and b is either semi-open or semi-closed, then a ∩ cl(b) is αs-paracompact (mod n). remark 3.2. if {vλ : λ ∈ λ} is a locally finite collection of subsets of a space (x, τ), then the collection {∂(vλ) : λ ∈ λ} is locally finite. according to [7], if i is an ideal on a space (x, τ) and f is the collection of all closed sets of (x, τ), then the collection {a ⊂ x : cl(a) ∈ i} is an ideal contained in i. the ideal generated by the collection of whole closed sets in i is denoted by ⟨i ∩ f⟩. it is clear that ⟨i ∩ f⟩ = {a ⊂ x : cl(a) ∈ i}. proposition 3.4. let a be a subset of a space (x, τ, i). if a is αs-paracompact (mod ⟨i ∩ f⟩) and n ⊂ i, then cl(a) is αs-paracompact (mod i). proof. let u be an open cover of cl(a). by hypothesis, there exist ia ∈ ⟨i ∩f⟩ and a locally finite collection v = {vλ : λ ∈ λ} of semi-open sets such that v refines u and a ⊂ ⋃ {vλ : λ ∈ λ} ∪ ia. then, cl(a) ⊂ ⋃ λ∈λ cl(vλ) ∪ cl(ia) = ( ⋃ λ∈λ vλ ) ∪ ( ⋃ λ∈λ ∂(vλ) ) ∪ cl(ia). by remark 3.2, the collection {∂(vλ) : λ ∈ λ} is locally finite and ∂(vλ) ∈ n for each λ ∈ λ. thus, by [6, lemma 2.1], we have ⋃ {∂(vλ) : λ ∈ λ} ∈ n ⊂ i. put i = ⋃ {∂(vλ) : λ ∈ λ} ∪ cl(ia), then i ∈ i and cl(a) ⊂ ⋃ λ∈λ vλ ∪ i. therefore, cl(a) is αs-paracompact (mod i). 56 j. sanabria, e. rosas, n. rajesh, c. carpintero and a. gómez cubo 18, 1 (2016) since n is the ideal of nowhere dense subsets of (x, τ), a ∈ n if and only if cl(a) ∈ n . in the case that i = n , then ⟨i ∩ f⟩ = n . the following corollary is a direct consequence of proposition 3.4. corollary 3.5. if a is an αs-paracompact (mod n) subset of a space (x, τ, i) , then cl(a) is αs-paracompact (mod n). lemma 3.2. [7] if {aλ : λ ∈ λ} is a locally finite collection of meager sets of a space (x, τ), then ⋃ {aλ : λ ∈ λ} is meager. theorem 3.5. if {aλ : λ ∈ λ} is a locally finite collection of αs-paracompact (mod m) subsets of a space (x, τ), then ⋃ {aλ : λ ∈ λ} is αs-paracompact (mod m). proof. let u be an open cover of ⋃ {aλ : λ ∈ λ} and put uλ = {u ∈ u : u ∩ aλ ̸= ∅} for each λ ∈ λ. by the hypothesis, there exist mλ ∈ m and a locally finite collection vλ of semi-open sets such that vλ refines uλ and aλ ⊂ ⋃ {v : v ∈ vλ} ∪ mλ. then, we have aλ ⊂ ⋃ v∈vλ (v ∩ int(cl(aλ))) ∪ ⋃ v∈vλ (v ∩ ∂(cl(aλ))) ∪ mλ. for each v ∈ vλ and each λ ∈ λ, v ∩∂(cl(aλ)) is nowhere dense and the collection {v ∩∂(cl(aλ)) : v ∈ vλ, λ ∈ λ} is locally finite, so by [6, lemma 2.1], the union of all elements of {v ∩ ∂(cl(aλ)) : v ∈ vλ, λ ∈ λ} is a nowhere dense set. by lemma 3.2, we obtain ⋃ {mλ : λ ∈ λ} ∈ m and m = ⋃ λ∈λ ⋃ v∈vλ v ∩ ∂(cl(aλ)) ∪ ⋃ λ∈λ mλ ∈ m. now, the collection {v ∩ int(cl(aλ)) : v ∈ vλ, λ ∈ λ} of semi-open sets is locally finite and refines u and also ⋃ λ∈λ aλ ⊂ ⋃ λ∈λ ⋃ v∈vλ v ∩ int(cl(aλ)) ∪ m. therefore, ⋃ {aλ : λ ∈ λ} is αs-paracompact (mod m). references [1] k. y. al-zoubi: s-paracompact spaces, acta. math. hungar. 110 (1-2) (2006), 165-174. [2] d. andrijević: semi-preopen sets, mat. vesnik 38 (1986), 24-32. [3] c. e. aull, α-paracompact subsets, proc. second prague topological symp. 1966, acad. pub. house czechoslovak acad. sci., prague (1967), 45-51. [4] s. g. crossley, s. k. hildebrand: semi-closure, texas j. sci. 22 (1971), 99-112. [5] s. g. crossley, s. k. hildebrand: semi-topological properties, fund. math. 74 (1972), 233-254. cubo 18, 1 (2016) s-paracompactness modulo an ideal 57 [6] n. ergun, t. noiri: on α∗-paracompactness subsets, bull. math. soc. sci. math. roumanie 36 (84) (1992), 259-268. [7] n. ergun, t. noiri: paracompactness modulo an ideal, math. japonica 42 (1) (1995), 15-24. [8] t. r. hamlett, d. rose, d. janković: paracompactness with respect to an ideal, internat. j. math. & math. sci. 20 (3) (1997), 433-442. [9] i. kovačević: locally almost paracompact spaces, review of research, faculty of science, univ. novi sad 10 (1980), 85-91. [10] k. kuratowski: topologie i, warszawa, 1933. 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() cubo a mathematical journal vol.16, no¯ 01, (75–84). march 2015 continuity via λsi-open sets josé sanabria, edumer acosta, ennis rosas and carlos carpintero1 departamento de matemáticas, núcleo de sucre, universidad de oriente, avenida universidad, cerro colorado, cumaná, estado sucre, venezuela jesanabri@gmail.com, edumeracostab@gmail.com, ennisrafael@gmail.com, carpintero.carlos@gmail.com abstract sanabria, rosas and carpintero [7] introduced the notions of λsi-sets and λ s i-closed sets using ideals on topological spaces. given an ideal i on a topological space (x, τ), a subset a ⊂ x is said to be λsi-closed if a = u ∩ f where u is a λ s i-set and f is a τ⋆-closed set. in this work we use sets that are complements of λsi-closed sets, which are called λsi-open, to characterize new variants of continuity namely λ s i-continuous, quasi-λsi-continuous y λ s i-irresolute functions. resumen sanabria, rosas y carpintero [7] introdujeron las nociones de conjuntos λsi y conjuntos λsi-cerrados usando ideales sobre espacios topológicos. dado un ideal i sobre un espacio topológico (x, τ), un subconjunto a ⊂ x se llama λsi-cerrado si a = u ∩ f donde u es un λsi-conjunto y f es un conjunto τ ⋆-cerrado . en este trabajo usamos conjuntos que son complementos de conjuntos λsi-cerrado, los cuales son llamados λ s i-abiertos, para caracterizar nuevas variantes de continuidad, denominadas, funciones λsi-continuas y funciones λsi-irresolutas. keywords and phrases: local function, λsi-open sets, λ s i-irresolute functions. 2010 ams mathematics subject classification: 54c08, 54d05. 1research partially suported by consejo de investigación udo 76 josé sanabria, edumer acosta, ennis rosas and carlos carpintero cubo 17, 1 (2015) 1 introduction the theory of ideal on topological spaces has been the subject of many studies in recent years. it was the works of hamlet and jankovic [5], abd el-monsef, lashien and nasef [1] and hatir and noiri [2] which motivated the research in applying topological ideals to generalize the most basic properties in general topology. in 2002, hatir and noiri [2] introduced the notion of semii-open sets in topological spaces. also, hatir and noiri [3] investigated semi-i-open sets and semi-i-continuous functions. quite recently, sanabria, rosas and carpintero [7] have introduced the notions of λsi-sets and λ s i-closed sets to obtain characterizations of two low separation axioms, namely semi-i-t1 and semi-i-t1/2 spaces. in this article we introduce the notion of λ s i-open sets in order to characterize new variants of continuity in ideal topological spaces. 2 preliminaries throughout this paper, p(x), cl(a) and int(a) denote the power set of x, the closure of a and the interior of a, respectively. an ideal i on a topological space (x, τ) is a nonempty collection of subsets of x which satisfies the following two properties: (1) a ∈ i and b ⊂ a implies b ∈ i; (2) a ∈ i and b ∈ i implies a ∪ b ∈ i. a topological space (x, τ) with an ideal i on x is called an ideal topological space and is denoted by (x, τ, i). given an ideal topological space (x, τ, i), a set operator (.)⋆ : p(x) → p(x), called a local function [6] of a with respect to τ and i, is defined as follows: for a ⊂ x, a⋆(i, τ) = {x ∈ x : u∩a /∈ ifor every u ∈ τ(x)}, where τ(x) = {u ∈ τ : x ∈ u}. when there is no chance for confusion, we will simply write a⋆ for a⋆(i, τ). in general, x⋆ is a proper subset of x. a kuratowski closure operator cl⋆(.) for a topology τ⋆(i, τ), called the ⋆-topology, finer than τ, is defined by cl⋆(a) = a∪a⋆(i, τ) [5]. for any ideal topological space (x, τ, i), the collection β(i, τ) = {v \ j : v ∈ τ and j ∈ i} is a basis for τ⋆(i, τ). according to the above, in this article, we denote by τ⋆ to topology τ⋆(i, τ) generated by cl⋆, that is, τ⋆ = {u ∈ p(x) : cl⋆(x − u) = x − u}. the elements of τ⋆ are called τ⋆-open and the complement of a τ⋆-open is called τ⋆-closed. it is well known that a subset a of an ideal topological space (x, τ, i) is τ⋆-closed if and only if a⋆ ⊂ a [5]. definition 2.1. a subset a of an ideal topological space (x, τ, i) is said to be semi-i-open [2] if a ⊂ cl ⋆ (int(a)). the complement of a semi-i-open set is said to be semi-i-closed. the family of all semi-i-open sets of an ideal topological space (x, τ, i) is denoted by sio(x, τ). the following three notions has been introduced by sanabria et al. [7]. definition 2.2. let a be a subset of an ideal topological space (x, τ, i). a subset λsi(a) is defined as follows: λsi(a) = ∩{u : a ⊂ u, u ∈ sio(x, τ)}. cubo 17, 1 (2015) continuity via λsi-open sets 77 definition 2.3. let (x, τ, i) an ideal topological space. a subset a of x is said to be: (1) λsi-set if a = λ s i(a). (2) λsi-closed if a = u ∩ f, where u is a λ s i-set and f is an τ ⋆-closed set. since each open set is semi-i-open, by propositions 3.1(3) and 4.1 of [7], we have the following implications: open =⇒ semi-i-open =⇒ λsi-set =⇒ λ s i-closed. lemma 2.1. (sanabria, rosas and carpintero [7]) for an ideal topological space (x, τ, i), we take τλ s i = {a : a is a λsi-set of (x, τ, i)}. then the pair (x, τ λ s i ) is an alexandroff space. remark 2.1. according to lemma 2.1, a subset a of an ideal topological space (x, τ, i) is open in (x, τλ s i ), if a is a λsi-set of (x, τ, i). definition 2.4. a subset a of an ideal topological space (x, τ, i) is called λsi-open if x \ a is a λsi-closed set. in the sequel, the ideal topological space (x, τ, i) is simply denoted by x. next we present some results related with λsi-open sets. lemma 2.2. every τ⋆-open set is λsi-open. proof. this follows from proposition 4.1 of [7]. lemma 2.3. let {bα : α ∈ ∆} be a family of subsets of x. if bα is λ s i-open for each α ∈ ∆, then⋃ {bα : α ∈ ∆} is λ s i-open. proof. the proof is an immediate consequence from proposition 4.2 of [7]. 3 new variants of continuity in this section we use the notions of open, λsi-open and τ ⋆-open sets in order to introduce new forms of continuous functions called λsi-continuous, quasi-λ s i-continuous and λ s i-irresolute. we study the relationships between these classes of functions and also obtain some properties and characterizations of them. definition 3.1. a function f : (x, τ, i) → (y, σ, j) is said to be semi-i-irresolute [4], if f−1(v) is a semi-i-open set in (x, τ, i) for each semi-j-open set of (y, σ, j). theorem 3.1. if a function f : (x, τ, i) → (y, σ, j) is semi-i-irresolute, then f : (x, τλ s i ) → (y, σλ s j ) is continuous. 78 josé sanabria, edumer acosta, ennis rosas and carlos carpintero cubo 17, 1 (2015) proof. let v be any λsj-set of (y, σ, j), that is v ∈ σ λ s j , then v = λsj(v) = ∩{w : v ⊂ w and w is semi-j-open in (y, σ, j)}. since f is semi-i-irresolute, f−1(w) is a semi-i-open set in (x, τ, i) for each w, hence we have λsi(f −1 (v)) = ∩{u : f−1(v) ⊂ u and u ∈ sio(x, τ)} ⊂ ∩{f−1(w) : f−1(v) ⊂ f−1(w) and w ∈ sjo(y, σ)} = f−1(v). on the other hand, always we have f−1(v) ⊂ λsi(f −1(v)) and so f−1(v) = λsi(f −1(v)). therefore, f−1(v) ∈ τλ s i and f : (x, τλ s i ) → (y, σλ s j ) is continuous. definition 3.2. a function f : (x, τ, i) → (y, σ, j) is called: (1) λsi-continuous, if f −1(v) is a λsi-open set in (x, τ, i) for each open set v of (y, σ, j). (2) quasi-λsi-continuous, if f −1(v) is a λsi-open set in (x, τ, i) for each σ ⋆-open set v of (y, σ, j). (3) λsi-irresolute, if f −1(v) is a λsi-open set in (x, τ, i) for each λ s j-open set v of (y, σ, j). theorem 3.2. if f : (x, τ, i) → (y, σ, j) is λsi-irresolute function, then f is quasi-λ s i-continuous. proof. let v be a σ⋆-open set of (y, σ, j), then by lemma 2.2, we have v is a λsj-open set of (y, σ, j) and since f is λsi-irresolute, f −1(v) is a λsi-open set of (x, τ, i). therefore, f is quasi-λ s icontinuous. the following example shows a function quasi-λsi-continuous which is not λ s i-irresolute. example 3.1. let x = {a, b, c}, τ = {∅, {a, c}, x}, σ = {∅, {a}, {a, b}, {a, c}, x}, i = {∅, {c}} and j = {∅, {b}}. the collection of the λsi-open sets of (x, τ, i) is {∅, {a, b}, {a, c}, {a}, {b}, x}, the collection of the σ⋆-open sets of (x, σ, j) is {∅, {a}, {a, c}, {a, b}, x} and the collection of the λsj-open sets of (x, σ, j) is {{∅}, {a}, {b}, {c}, {a, c}, {b, c}, x}. the identity function f : (x, τ, i) → (x, σ, j) is quasi-λsicontinuous, but is not λsi-irresolute, since f −1({b, c}) = {b, c} and f−1({c}) = {c} are not λsi-open sets. theorem 3.3. if f : (x, τ, i) → (y, σ, j) is quasi-λsi-continuous function, then f is λ s i-continuous. proof. let v be an open set of (y, σ, j), then v is σ⋆-open set of (y, σ, j) and since f is quasi-λsicontinuous, f−1(v) is a λsi-open set of (x, τ, i). this shows that f is λ s i-continuous. the following example shows a function λsi-continuous which is not quasi-λ s i-continuous. example 3.2. let x = {a, b, c}, τ = {∅, {a, c}, x}, σ = {∅, {a}, {b}, {a, b}, x}, i = {∅, {c}} and j = {∅, {a}}. the collection of the λsi-open sets of (x, τ, i) is {∅, {a, b}, {a, c}, {a}, {b}, x} and the collection of σ⋆-open sets of (x, σ, j) is {∅, {a}, {b, c}, {a, b}, {b}, x}. the identity function f : (x, τ, i) → (x, σ, j) is λsi-continuous, but is not quasi-λ s i-continuous, because f −1({b, c}) = {b, c} is not a λsi-open set. cubo 17, 1 (2015) continuity via λsi-open sets 79 corollary 3.1. if f : (x, τ, i) → (y, σ, j) is a λsi-irresolute function, then f is λ s i-continuous. proof. this is an immediate consequence of theorems 3.2 and 3.3. by the above results, we have the following diagram and none of these implications is reversible: λsi-irresolute =⇒ quasi-λ s i-continuous =⇒ λ s i-continuous. proposition 3.1. let f : (x, τ, i) → (y, σ, j) and g : (y, σ, j) → (z, θ, k) be two functions, where i, j, k are ideals on x, y, z respectively. then: (1) g ◦ f is λsi-irresolute, if f is λ s i-irresolute and g is λ s j-irresolute. (2) g ◦ f is λsi-continuous, if f is λ s i-irresolute and g is λ s j-continuous. (3) g ◦ f is λsi-continuous, if f is λ s i-continuous and g is continuous. (4) g ◦ f is quasi-λsi-continuous, if f is λ s i-irresolute and g is quasi-λ s j-continuous. proof. (1) let v be a λsk-open set in (z, θ, k). since g is λ s j-irresolute, then g −1(v) is a λsj-open set in (y, σ, j), using that f is λsi-irresolute, we obtain that f −1(g−1(v)) is a λsi-open set in (x, τ, i). but (g ◦ f)−1(v) = (f−1 ◦ g−1)(v) = f−1(g−1(v)) and hence, (g ◦ f)−1(v) is a λsi-open set in (x, τ, i). this shows that g ◦ f is λsi-irresolute. the proofs of (2), (3) and (4) are similar to the case (1). in the next three theorems, we characterize λsi-continuous, quasi-λ s i-continuous and λ s i-irresolute functions, respectively. theorem 3.4. for a function f : (x, τ, i) → (y, σ), the following statements are equivalent: (1) f is λsi-continuous. (2) f−1(b) is a λsi-closed set in (x, τ, i) for each closed set b in (y, σ). (3) for each x ∈ x and each open set v in (y, σ) containing f(x) there exists a λsi-open set u in (x, τ, i) containing x such that f(u) ⊂ v. proof. (1)⇒(2) let b be any closed set in (y, σ), then v = y\b is an open set in (y, σ) and since f is λsi-continuous, f −1(v) is a λsi-open subset in (x, τ, i), but f −1(v) = f−1(y\b) = f−1(y)\f−1(b) = x \ f−1(b) and hence, f−1(b) is a λsi-closed set in (x, τ, i). (2) ⇒ (1) let v be any open set in (y, σ), then b = y \v is a closed set in (y, σ). by hypothesis, we have f−1(b) is a λsi-closed set in (x, τ, i), but f −1(b) = f−1(y \ v) = f−1(y) \ f−1(v) = x \ f−1(v) and so, f−1(v) is a λsi-open set in (x, τ, i). this shows that f is λ s i-continuous. (1) ⇒ (3) let x ∈ x and v any open set in (y, σ) such that f(x) ∈ v, then x ∈ f−1(v) and since f is a λsi-continuous function, f −1(v) is a λsi-open set in (x, τ, i). if u = f −1(v), then u is a λsi-open 80 josé sanabria, edumer acosta, ennis rosas and carlos carpintero cubo 17, 1 (2015) set in (x, τ, i) containing x such that f(u) = f(f−1(v)) ⊂ v. (3) ⇒ (1) let v be any open set in (y, σ) and x ∈ f−1(v), then f(x) ∈ v and by (3) there exists a λsi-open set ux in (x, τ, i) such that x ∈ ux and f(ux) ⊂ v. thus, x ∈ ux ⊂ f −1(f(ux)) ⊂ f −1(v) and hence f−1(v) = ⋃ {ux : x ∈ f −1(v)}. by lemma 2.3, we have f−1(v) is a λsi-open set in (x, τ, i) and so f is λsi-continuous. theorem 3.5. for a function f : (x, τ, i) → (y, σ, j), the following statements are equivalent: (1) f is quasi-λsi-continuous. (2) f−1(b) is a λsi-closed set in (x, τ, i) for each σ ⋆-closed set b in (y, σ, j). (3) for each x ∈ x and each σ⋆-open set v in (y, σ, j) containing f(x) there exists a λsi-open set u in (x, τ, i) containing x such that f(u) ⊂ v. proof. the proof is similar to theorem 3.4. theorem 3.6. for a function f : (x, τ, i) → (y, σ, j), the following statements are equivalent: (1) f is λsi-irresolute. (2) f−1(b) is a λsi-closed set in (x, τ, i) for each λ s j-closet set b in (y, σ, j). (3) for each x ∈ x and each λsj-open set v in (y, σ, j) containing f(x) there exists a λ s i-open set u in (x, τ, i) containing x such that f(u) ⊂ v. proof. the proof is similar to theorem 3.4. 4 λsi-compactness and λ s i-connectedness in this section, new notions of compactness and connectedness are introduced in terms of λsi-open sets and semi-i-open sets, in order to study their behavior under the direct images of the new forms of continuity defined in the previous section. definition 4.1. an ideal topological space (x, τ, i) is said to be: (1) λsi-compact if every cover of x by λ s i-open sets has a finite subcover. (2) τ⋆-compact if every cover of x by τ⋆-open sets has a finite subcover. (3) semi-i-compact if every cover of x by semi-i-open sets has a finite subcover. theorem 4.1. let (x, τ, i) be an ideal topological space, the following properties hold: cubo 17, 1 (2015) continuity via λsi-open sets 81 (1) (x, τ, i) is λsi-compact if and only if for every collection {aα : α ∈ ∆} of λ s i-closed sets in (x, τ, i) satisfying ⋂ {aα : α ∈ ∆} = ∅, there is a finite subcollection aα1, aα2, . . . , aαn with⋂ {aαk : k = 1, . . . , n} = ∅. (2) (x, τ, i) is τ⋆-compact if and only if for every collection {aα : α ∈ ∆} of τ ⋆-closed sets in (x, τ, i) satisfying ⋂ {aα : α ∈ ∆} = ∅, there is a finite subcollection aα1, aα2, . . . , aαn with⋂ {aαk : k = 1, . . . , n} = ∅. (3) (x, τ, i) is semi-i-compact if and only if for every collection {aα : α ∈ ∆} of semi-i-closed sets in (x, τ, i) satisfying ⋂ {aα : α ∈ ∆} = ∅, there is a finite subcollection aα1, aα2, . . . , aαn with⋂ {aαk : k = 1, . . . , n} = ∅. proof. (1) let {aα : α ∈ ∆} be a collection of λ s i-closed sets such that ⋂ {aα : α ∈ ∆} = ∅, then {x − aα : α ∈ ∆} is a collection of λ s i-open sets such that x = x − ∅ = x − ⋂ {aα : α ∈ ∆} = ⋃ {x − aα : α ∈ ∆}, that is, {x − aα : α ∈ ∆} is a cover of x by λ s i-open sets. since (x, τ, i) is λ s i-compact, there exists a finite subcollection x − aα1, x − aα2, . . . , x − aαn such that x = ⋃ {x − aαk : k = 1, . . . , n} = x − ⋂ {aαk : k = 1, . . . , n}. this shows that ⋂ {aαk : k = 1, . . . , n} = ∅. conversely, suppose that {uα : α ∈ ∆} is a cover of x by λsi-open sets, then {x − uα : α ∈ ∆} is a collection of λ s i-closed sets such that ⋂ {x − uα : α ∈ ∆} = x − ⋃ {uα : α ∈ ∆} = x − x = ∅. by hypothesis, there exists a finite subcollection x − uα1, x − uα2, . . . , x − uαn such that ⋂ {x − uαk : k = 1, . . . , n} = ∅. follows x = x − ∅ = x − ⋂ {x − uαk : k = 1, . . . , n} = x − (x − ⋃ {uαk : k = 1, . . . , n}) = ⋃ {uαk : k = 1, . . . , n}. this shows that (x, τ, i) is λsi-compact. the proofs of (2) and (3) are similar to the case (1). theorem 4.2. let (x, τ, i) be an ideal topological space, the following properties hold: (1) if (x, τλ s i ) is compact, then (x, τ, i) is semi-i-compact. (2) if (x, τ, i) is λsi-compact, then (x, τ, i) is τ ⋆-compact. (3) if (x, τ, i) is λsi-compact, then (x, τ, i) is compact. proof. (1) let {uα : α ∈ ∆} any cover of x by semi-i-open sets, since every α ∈ ∆, uα is a λ s i-set and hence, uα ∈ τ λ s i for each α ∈ ∆. since (x, τλ s i ) is compact, there exists a finite subset ∆0 of ∆ such that x = ⋃ {uα : α ∈ ∆0}. this shows that (x, τ) is semi-i-compact. (2) let {fα : α ∈ ∆} be a collection of τ ⋆-closed sets of x such that ⋂ {fα : α ∈ ∆} = ∅. since every τ⋆-closed set is λsi-closed, then {fα : α ∈ ∆} is a collection of λ s i-closed sets and (x, τ, i) is λsi-compact. by theorem 4.1(1), there exists a finite subset ∆0 of ∆ such that ⋂ {fα : α ∈ ∆0} = ∅ 82 josé sanabria, edumer acosta, ennis rosas and carlos carpintero cubo 17, 1 (2015) and by theorem 4.1(2), we conclude that (x, τ, i) is τ⋆-compact. (3) follows from (2) and the fact that every τ⋆-compact space is compact. theorem 4.3. if f : (x, τ, i) → (y, σ, j) is a surjective function, the following properties hold: (1) if f is λsi-irresolute and (x, τ, i) is λ s i-compact, then (y, σ, j) is λ s j-compact. (2) if f is semi-i-irresolute and (x, τ, i) is semi-i-compact, then (y, σ, j) is semi-j-compact. (3) if f is quasi-λsi-continuous and (x, τ, i) is λ s i-compact, then (y, σ, j) is σ ⋆-compact. (4) if f is λsi-continuous and (x, τ, i) is λ s i-compact, then (y, σ, j) is compact. proof. (1) let {vα : α ∈ ∆} be a cover of y by λ s j-open sets. since f is λ s i-irresolute, {f −1(vα) : α ∈ ∆} is a cover of x by λsi-open sets and by the λ s i-compactnes of (x, τ, i), there exists a finite subset ∆0 of ∆ such that x = ⋃ {f−1(vα) : α ∈ ∆0}. since f is surjective, then y = f(x) = f( ⋃ {f−1(vα) : α ∈ ∆0}) = ⋃ {f(f−1(vα)) : α ∈ ∆0} = {vα : α ∈ ∆0} and this shows that (y, θ, j) is λ s j-compact. the proofs of (2), (3) and (4) are similar to case (1). definition 4.2. an ideal topological space (x, τ, i) is said to be: (1) λsi-connected if x cannot be written as a disjoint union of two nonempty λ s i-open sets. (2) τ⋆-connected if x cannot be written as a disjoint union of two nonempty τ⋆-open sets. (3) semi-i-connected if x cannot be written as a disjoint union of two nonempty semi-i-open sets. theorem 4.4. let (x, τ, i) be an ideal topological space, the following properties hold: (1) if (x, τλ s i ) is connected, then (x, τ; i) is semi-i-connected. (2) if (x, τ, i) is λsi-connected, then (x, τ, i) is τ ⋆-connected. (3) if (x, τ, i) is λsi-connected, then (x, τ, i) is connected. proof. (1) suppose that (x, τ, i) is not semi-i-connected, then there exist non-empty semi-i-open sets a and b such that a∩b = ∅ and a∪b = x. by proposition 3.1(3) of [7], a and b are λsi-sets and hence, (x, τλ s i ) is not connected. (2) suppose that (x, τ, i) is not τ⋆-connected, then there exist non-empty τ⋆-open sets a and b such that a ∩ b = ∅ and a ∪ b = x. by lemma 2.2, we have a and b are λsi-open sets and so, (x, τ, i) is not λsi-connected. (3) follows from (2) and the fact that every τ⋆-connected space is connected. theorem 4.5. for an ideal topological space (x, τ, i), the following statements are equivalent: (1) (x, τ, i) is λsi-connected. cubo 17, 1 (2015) continuity via λsi-open sets 83 (2) ∅ and x are the only subsets of x which are both λsi-open and λ s i-closed. (3) every λsi-continuous function of x into a discrete space y with at least two points, is a constant function. proof. (1)⇒(2) let v be a subset of x which is both λsi-open and λ s i-closed, then x − v is both λsi-open and λ s i-closed, so x = v ∪ (x − v). since (x, τ, i) is λ s i-connected, then one of those sets is ∅. therefore, v = ∅ or v = x. (2)⇒(1) suppose that (x, τ, i) is not λsi-connected and let x = u ∪ v, where u and v are disjoint nonempty λsi-open sets in (x, τ, i), then u = x−v is both λ s i-open and λ s i-closed. by hypothesis, u = ∅ or u = x, which is a contradiction. therefore, (x, τ, i) is λsi-connected. (2)⇒(3) let f : (x, τ, i) → y be a λsi-continuous function, where y is a topological space with the discrete topology and contains at least two points, then x can be cover by a collection of sets which are both λsi-open and λ s i-closed of the form {f −1(y) : y ∈ y}, from these, we conclude that there exists a y0 ∈ y such that f −1({y0}) = x and so, f is a constant function. (3)⇒(2) let w be a subset of (x, τ, i) which is both λsi-open and λ s i-closed. suppose that w 6= ∅ and let f : (x, τ, i) → y be the function defined by f(w) = {y1} and f(x − w) = {y2} for y1, y2 ∈ y and y1 6= y2. then f is λ s i-continuous, since the inverse image de each open set in y is λ s i-open in x. hence, by (3), f must be a constant function. it follows that x = w. theorem 4.6. if f : (x, τ, i) → (y, σ, j) is a surjective function, the following properties hold: (1) if f is a λsi-irresolute and (x, τ, i) is λ s i-connected, then (y, σ, j) is λ s j-connected. (2) if f is a semi-i-irresolute function and (x, τ, i) is semi-i-connected, then (y, σ, j) is semi-jconnected. (3) if f is a quasi-λsi-continuous function and (x, τ, i) is λ s i-connected, then (y, σ, j) is σ ⋆-connected. (4) if f is a λsi-continuous function and (x, τ, i) is λ s i-connected, then (y, σ) is connected. proof. (1) suppose that (y, σ, j) is not λsj-connected, then there exist nonempty λ s j-open sets h, g in (y, σ, j) such that g ∩ h = ∅ and g ∪ h = y. hence, we have f−1(g) ∩ f−1(h) = ∅, f−1(g) ∪ f−1(h) = x and moreover, f−1(g) and f−1(h) are nonempty λsi-open sets in (x, τ, i). this shows that (x, τ, i) is not λsi-connected. the proofs of (2), (3) and (4) are similar to case (1). open problems. the theorems 4.2 and 4.4 have been proved using the fact that every semii-open set is λsi-open and that every τ ⋆-open set is λsi-open. but until today, we dont have any contra example in order to shows that the converse of such theorems are not true. in that sense we write the following questions. (1) does there exists an ideal topological space (x, τ, i) which is semi-i-compact (resp. semi-iconnected) but (x, τλ s i ) is not a compact (resp. connected) space.? 84 josé sanabria, edumer acosta, ennis rosas and carlos carpintero cubo 17, 1 (2015) (2) does there exists an ideal topological space (x, τ, i) which is τ⋆-compact (resp. τ⋆-connected) but (x, τ) is not λsi-compact space (resp. λ s i-connected space.)? received: january 2014. accepted: january 2015. references [1] m. e. abd el-monsef, e. f. lashien, a. a. nasef: on i-open sets and i-continuous functions, kyungpook math. j. 32 (1992), 21-30. [2] e. hatir, t. noiri: on descompositions of continuity via idealization, acta. math. hungar. 96 (4) (2002), 341-349. [3] e. hatir, t. noiri: on semi-i-open sets and semi-i-continuous functions, acta math. hungar. 107 (4) (2005), 345-353. [4] e. hatir, t. noiri: on hausdorff spaces via ideals and semi-i-irresolute functions, eur. j. pure. appl. math. 2 (2) (2009), 172-181. [5] d. jankovic, t. r. hamlett: new topologies from old via ideals, amer. math. monthly 97 (1990), 295-310. [6] k. kuratowski: topology, vol. i, academic press, new york-london, 1966. [7] j. sanabria, e. rosas, c. carpintero: on λsi-sets and the related notions in ideal topological spaces, math. slovaca 63 (6) (2013), 1403-1411. introduction preliminaries new variants of continuity si-compactness and si-connectedness articulo 12.dvi cubo a mathematical journal vol.12, no¯ 02, (189–197). june 2010 fischer decomposition by inframonogenic functions helmuth r. malonek1, dixan peña peña2 department of mathematics, aveiro university, 3810-193 aveiro, portugal 1email: hrmalon@ua.pt 2email: dixanpena@ua.pt, dixanpena@gmail.com and frank sommen department of mathematical analysis, ghent university, 9000 gent, belgium email: fs@cage.ugent.be abstract let ∂x denote the dirac operator in r m. in this paper, we present a refinement of the biharmonic functions and at the same time an extension of the monogenic functions by considering the equation ∂xf ∂x = 0. the solutions of this “sandwich” equation, which we call inframonogenic functions, are used to obtain a new fischer decomposition for homogeneous polynomials in rm. resumen denotemos por ∂x el operador de dirac en r m. en este art́ıculo, nosotros presentamos un refinamiento de las funciones biarmónicas y al mismo tiempo una extensión de las funciones monogénicas considerando la ecuación ∂xf ∂x = 0. las soluciones de esta ecuación tipo “sándwich”, las cuales llamaremos inframonogénicas, son utilizadas para obtener una nueva descomposición de fischer para polinomios homogéneos en rm. 190 helmuth r. malonek, dixan peña peña and frank sommen cubo 12, 2 (2010) key words and phrases: inframonogenic functions; fischer decomposition. mathematics subject classification: 30g35; 31b30; 35g05. 1 introduction let r0,m be the 2 m-dimensional real clifford algebra constructed over the orthonormal basis (e1, . . . , em) of the euclidean space rm (see [6]). the multiplication in r0,m is determined by the relations ej ek + ekej = −2δjk and a general element of r0,m is of the form a = ∑ a aaea, aa ∈ r, where for a = {j1, . . . , jk} ⊂ {1, . . . , m}, j1 < · · · < jk, ea = ej1 . . . ejk . for the empty set ∅, we put e∅ = 1, the latter being the identity element. notice that any a ∈ r0,m may also be written as a = ∑m k=0[a]k where [a]k is the projection of a on r (k) 0,m. here r (k) 0,m denotes the subspace of k-vectors defined by r (k) 0,m = { a ∈ r0,m : a = ∑ |a|=k aaea, aa ∈ r } . in particular, r (1) 0,m and r (0) 0,m ⊕ r (1) 0,m are called, respectively, the space of vectors and paravectors in r0,m. observe that r m+1 may be naturally identified with r (0) 0,m ⊕ r (1) 0,m by associating to any element (x0, x1, . . . , xm) ∈ r m+1 the paravector x = x0 + x = x0 + ∑m j=1 xj ej . conjugation in r0,m is given by a = ∑ a aaea, ea = (−1) |a|(|a|+1) 2 ea. one easily checks that ab = ba for any a, b ∈ r0,m. moreover, by means of the conjugation a norm |a| may be defined for each a ∈ r0,m by putting |a|2 = [aa]0 = ∑ a a2a. the r0,m-valued solutions f (x) of ∂xf (x) = 0, with ∂x = ∑m j=1 ej ∂xj being the dirac operator, are called left monogenic functions (see [4, 8]). the same name is used for null-solutions of the operator ∂x = ∂x0 + ∂x which is also called generalized cauchy-riemann operator. in view of the non-commutativity of r0,m a notion of right monogenicity may be defined in a similar way by letting act the dirac operator or the generalized cauchy-riemann operator from the right. functions that are both left and right monogenic are called two-sided monogenic. one can also consider the null-solutions of ∂kx and ∂ k x (k ∈ n) which gives rise to the so-called k-monogenic functions (see e.g. [2, 3, 15]). it is worth pointing out that ∂x and ∂x factorize the laplace operator in the sense that ∆x = m ∑ j=1 ∂2xj = −∂ 2 x, ∆x = ∂ 2 x0 + ∆x = ∂x∂x = ∂x∂x. let us now introduce the main object of this paper. cubo 12, 2 (2010) fischer decomposition by inframonogenic functions 191 definition 1.1. let ω be an open set of rm (resp. rm+1). an r0,m-valued function f ∈ c 2(ω) will be called an inframonogenic function in ω if and only if it fulfills in ω the “sandwich” equation ∂xf ∂x = 0 (resp. ∂xf ∂x = 0). here we list some motivations for studying these functions. 1. if a function f is inframonogenic in ω ⊂ rm and takes values in r, then f is harmonic in ω. 2. the left and right monogenic functions are also inframonogenic. 3. if a function f is inframonogenic in ω ⊂ rm, then it satisfies in ω the overdetermined system ∂3xf = 0 = f ∂ 3 x. in other words, f is a two-sided 3-monogenic function. 4. every inframonogenic function f ∈ c4(ω) is biharmonic, i.e. it satisfies in ω the equation ∆2xf = 0 (see e.g. [1, 11, 13, 16]). the aim of this paper is to present some simple facts about the inframonogenic functions (section 2) and establish a fischer decomposition in this setting (section 3). 2 inframonogenic functions: simple facts it is clear that the product of two inframonogenic functions is in general not inframonogenic, even if one of the factors is a constant. proposition 2.1. assume that f is an inframonogenic function in ω ⊂ rm such that ej f (resp. f ej ) is also inframonogenic in ω for each j = 1, . . . , m. then f is of the form f (x) = cx + m (x), where c is a constant and m a right (resp. left) monogenic function in ω. proof. the proposition easily follows from the equalities ∂x ( ej f (x) ) ∂x = −2∂xj f (x)∂x − ej ( ∂xf (x)∂x ) , ∂x ( f (x)ej ) ∂x = −2∂xj ∂xf (x) − ( ∂xf (x)∂x ) ej , (1) j = 1, . . . , m. � for a vector x and a k-vector yk, the inner and outer product between x and yk are defined by (see [8]) x • yk = { [xyk]k−1 for k ≥ 1 0 for k = 0 and x ∧ yk = [xyk]k+1 . in a similar way yk • x and yk ∧ x are defined. we thus have that xyk = x • yk + x ∧ yk, ykx = yk • x + yk ∧ x, 192 helmuth r. malonek, dixan peña peña and frank sommen cubo 12, 2 (2010) where also x • yk = (−1) k−1yk • x, x ∧ yk = (−1) kyk ∧ x. let us now consider a k-vector valued function fk which is inframonogenic in the open set ω ⊂ r m. this is equivalent to say that fk satisfies in ω the system        ∂x • (∂x • fk) = 0 ∂x ∧ (∂x • fk) − ∂x • (∂x ∧ fk) = 0 ∂x ∧ (∂x ∧ fk) = 0. in particular, for m = 2 and k = 1, a vector-valued function f = f1e1 + f2e2 is inframonogenic if and only if { ∂x1x1 f1 − ∂x2x2 f1 + 2∂x1x2 f2 = 0 ∂x1x1 f2 − ∂x2x2 f2 − 2∂x1x2 f1 = 0. we now try to find particular solutions of the previous system of the form f1(x1, x2) = α(x1) cos(nx2), f2(x1, x2) = β(x1) sin(nx2). it easily follows that α and β must fulfill the system α′′ + n2α + 2nβ′ = 0 β′′ + n2β + 2nα′ = 0. solving this system, we get f1(x1, x2) = ( (c1 + c2x1) exp(nx1) + (c3 + c4x1) exp(−nx1) ) cos(nx2), (2) f2(x1, x2) = ( (c3 + c4x1) exp(−nx1) − (c1 + c2x1) exp(nx1) ) sin(nx2). (3) therefore, we can assert that the vector-valued function f (x1, x2) = ( (c1 + c2x1) exp(nx1) + (c3 + c4x1) exp(−nx1) ) cos(nx2)e1 + ( (c3 + c4x1) exp(−nx1) − (c1 + c2x1) exp(nx1) ) sin(nx2)e2, cj , n ∈ r, is inframonogenic in r2. note that if c1 = c3 and c2 = c4, then f1(x1, x2) = 2(c1 + c2x1) cosh(nx1) cos(nx2), f2(x1, x2) = −2(c1 + c2x1) sinh(nx1) sin(nx2). since the functions (2) and (3) are harmonic in r2 if and only if c2 = c4 = 0, we can also claim that not every inframonogenic function is harmonic. here is a simple technique for constructing inframonogenic functions from two-sided monogenic functions. cubo 12, 2 (2010) fischer decomposition by inframonogenic functions 193 proposition 2.2. let f (x) be a two-sided monogenic function in ω ⊂ rm. then xf (x) and f (x)x are inframonogenic functions in ω. proof. it is easily seen that ( xf (x) ) ∂x = m ∑ j=1 ∂xj ( xf (x) ) ej = x ( f (x)∂x ) + m ∑ j=1 ej f (x)ej = m ∑ j=1 ejf (x)ej . we thus get ∂x ( xf (x) ) ∂x = − m ∑ j=1 ej ( ∂xf (x) ) ej − 2f (x)∂x = 0. in the same fashion we can prove that f (x)x is inframonogenic. � we must remark that the functions in the previous proposition are also harmonic. this may be proved using the following equalities ∆x ( xf (x) ) = 2∂xf (x) + x ( ∆xf (x) ) , (4) ∆x ( f (x)x ) = 2f (x)∂x + ( ∆xf (x) ) x, (5) and the fact that every monogenic function is harmonic. at this point it is important to notice that an r0,m-valued harmonic function is in general not inframonogenic. take for instance h(x)ej , h(x) being an r-valued harmonic function. if we assume that h(x)ej is also inframonogenic, then from (1) it may be concluded that ∂xh(x) does not depend on xj . clearly, this condition is not fulfilled for every harmonic function. we can easily characterize the functions that are both harmonic and inframonogenic. indeed, suppose that h(x) is a harmonic function in a star-like domain ω ⊂ rm. by the almansi decomposition (see [12, 15]), we have that h(x) admits a decomposition of the form h(x) = f1(x) + xf2(x), where f1(x) and f2(x) are left monogenic functions in ω. it is easy to check that ∂xh(x) = −mf2(x) − 2exf2(x), ex = ∑m j=1 xj ∂xj being the euler operator. thus h(x) is also inframonogenic in ω if and only if mf2(x) + 2exf2(x) is right monogenic in ω. in particular, if h(x) is a harmonic and inframonogenic homogeneous polynomial of degree k, then f1(x) is a left monogenic homogeneous polynomial of degree k while f2(x) is a two-sided monogenic homogeneous polynomial of degree k − 1. the following proposition provides alternative characterizations for the case of k-vector valued functions. proposition 2.3. suppose that fk is a harmonic (resp. inframonogenic) k-vector valued function in ω ⊂ rm such that 2k 6= m. then fk is also inframonogenic (resp. harmonic) if and only if one of the following assertions is satisfied: (i) fk(x)x is left 3-monogenic in ω; 194 helmuth r. malonek, dixan peña peña and frank sommen cubo 12, 2 (2010) (ii) xfk(x) is right 3-monogenic in ω; (iii) xfk(x)x is biharmonic in ω. proof. we first note that ejeaej = { (−1)|a|ea for j ∈ a, (−1)|a|+1ea for j /∈ a, which clearly yields ∑m j=1 ej eaej = (−1) |a|(2|a|−m)ea. it thus follows that for every k-vector valued function fk, m ∑ j=1 ej fkej = (−1) k(2k − m)fk. using the previous equality together with (4) and (5), we obtain ∂x∆x ( fk(x)x ) = 2∂xfk(x)∂x + ( ∂x∆xfk(x) ) x + (−1)k(2k − m)∆xfk, ∆x ( xfk(x) ) ∂x = 2∂xfk(x)∂x + x ( ∆xfk(x)∂x ) + (−1)k(2k − m)∆xfk, ∆2x ( xfk(x)x ) = 4 ( 2∂xfk(x)∂x + (−1) k(2k − m)∆xfk + ( ∂x∆xfk(x) ) x + x ( ∆xfk(x)∂x ) ) + x ( ∆2xfk(x) ) x. the proof now follows easily. � before ending the section, we would like to make two remarks. first, note that if m even, then a m/2-vector valued function fm/2(x) is inframonogenic if and only if fm/2(x) and fm/2(x)x are left 3-monogenic, or equivalently, fm/2(x) and xfm/2(x) are right 3-monogenic. finally, for m odd the previous proposition remains valid for r0,m-valued functions. 3 fischer decomposition the classical fischer decomposition provides a decomposition of arbitrary homogeneous polynomials in rm in terms of harmonic homogeneous polynomials. in this section we will derive a similar decomposition but in terms of inframonogenic homogeneous polynomials. for other generalizations of the fischer decomposition we refer the reader to [5, 7, 8, 9, 10, 12, 14, 17, 18]. let p(k) (k ∈ n0) denote the set of all r0,m-valued homogeneous polynomials of degree k in r m. it contains the important subspace i(k) consisting of all inframonogenic homogeneous polynomials of degree k. an an inner product may be defined in p(k) by setting 〈pk(x), qk(x)〉k = [ pk(∂x) qk(x) ] 0 , pk(x), qk(x) ∈ p(k), pk(∂x) is the differential operator obtained by replacing in pk(x) each variable xj by ∂xj and taking conjugation. cubo 12, 2 (2010) fischer decomposition by inframonogenic functions 195 from the obvious equalities [eja b]0 = −[aej b]0, [aej b]0 = −[abej]0, a, b ∈ r0,m, we easily obtain 〈xpk−1(x), qk(x)〉k = − 〈 pk−1(x), ∂xqk(x) 〉 k−1 , 〈pk−1(x)x, qk(x)〉k = − 〈 pk−1(x), qk(x)∂x 〉 k−1 , with pk−1(x) ∈ p(k − 1) and qk(x) ∈ p(k). hence for pk−2(x) ∈ p(k − 2) and qk(x) ∈ p(k), we deduce that 〈xpk−2(x)x, qk(x)〉k = 〈 pk−2(x), ∂xqk(x)∂x 〉 k−2 . (6) theorem 3.1 (fischer decomposition). for k ≥ 2 the following decomposition holds: p(k) = i(k) ⊕ xp(k − 2)x. moreover, the subspaces i(k) and xp(k − 2)x are orthogonal w.r.t. the inner product 〈 , 〉k. proof. the proof of this theorem will be carried out in a similar way to that given in [8] for the case of monogenic functions. as p(k) = xp(k − 2)x ⊕ (xp(k − 2)x) ⊥ it is sufficient to show that i(k) = (xp(k − 2)x) ⊥ . take pk(x) ∈ (xp(k − 2)x) ⊥ . then for all qk−2(x) ∈ p(k − 2) it holds 〈 qk−2(x), ∂xpk(x)∂x 〉 k−2 = 0, where we have used (6). in particular, for qk−2(x) = ∂xpk(x)∂x we get that ∂xpk(x)∂x = 0 or pk(x) ∈ i(k). therefore (xp(k − 2)x) ⊥ ⊂ i(k). conversely, let pk(x) ∈ i(k). then for each qk−2(x) ∈ p(k − 2), 〈xqk−2(x)x, pk(x)〉k = 〈 qk−2(x), ∂xpk(x)∂x 〉 k−2 = 0, whence pk(x) ∈ (xp(k − 2)x) ⊥ . � by recursive application of the previous theorem we get: corollary 3.1 (complete fischer decomposition). if k ≥ 2, then p(k) = [k/2] ⊕ s=0 xsi(k − 2s)xs. acknowledgement. d. peña peña was supported by a post-doctoral grant of fundação para a ciência e a tecnologia, portugal (grant number: sfrh/bpd/45260/2008). received: march 2009. revised: may 2009. 196 helmuth r. malonek, dixan peña peña and frank sommen cubo 12, 2 (2010) references [1] s. bock and k. gürlebeck, on a spatial generalization of the kolosov-muskhelishvili formulae, math. methods appl. sci. 32 (2009), no. 2, 223–240. 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[14] g. ren and h. r. malonek, almansi decomposition for dunkl-helmholtz operators, wavelet analysis and applications, 35–42, appl. numer. harmon. anal., birkhäuser, basel, 2007. [15] j. ryan, basic clifford analysis, cubo mat. educ. 2 (2000), 226–256. [16] l. sobrero, theorie der ebenen elastizität unter benutzung eines systems hyperkomplexer zahlen, hamburg. math. einzelschriften, leipzig, 1934. [17] f. sommen, monogenic functions of higher spin, z. anal. anwendungen 15 (1996), no. 2, 279– 282. cubo 12, 2 (2010) fischer decomposition by inframonogenic functions 197 [18] f. sommen and n. van acker, functions of two vector variables, adv. appl. clifford algebr. 4 (1994), no. 1, 65–72. bengse1$09$.dvi cubo a mathematical journal vol.12, no¯ 03, (35–48). october 2010 measure of noncompactness and nondensely defined semilinear functional differential equations with fractional order mouffak benchohra1 laboratoire de mathématiques, université de sidi bel-abbès, b.p. 89, 22000, sidi bel-abbès, algérie email: benchohra@univ-sba.dz gaston m. n’guérékata department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore m.d. 21252, usa email: gaston.n’guerekata@morgan.edu and djamila seba département de mathématiques, université de boumerdès, avenue de l’indépendance, 35000 boumerdès, algérie email: djam_seba@yahoo.fr abstract this paper is devoted to study the existence of integral solutions for a nondensely defined semilinear functional differential equations involving the riemann-liouville derivative in 1corresponding author 36 m. benchohra, gaston m. n’guérékata & d. seba cubo 12, 3 (2010) a banach space. the arguments are based upon mönch’s fixed point theorem and the technique of measures of noncompactness. resumen este artículo es dedicado al estudio de existencia de soluciones integrales para ecuaciones diferenciales funcionales semilineales envolviendo la derivada de riemann-liouville en un espacio de banach. los argumentos se basan en un teorema de punto fijo de mönch y la técnica de no compacidad. key words and phrases: partial differential equations, fractional derivative, fractional integral, fixed point, semigroups, integral solutions, finite delay, measure of noncompactness, fixed point, banach space. math. subj. class.: 34g20, 34g25, 26a33, 26a42. 1 introduction the theory of functional differential equations has emerged as an important branch of nonlinear analysis. it is worthwhile mentioning that several important problems of the theory of ordinary and delay differential equations lead to investigations of functional differential equations of various types, see the books of hale and verduyn lunel [22], kolmanovskii and myshkis [26], wu [42], and the references therein. on the other hand the theory of fractional differential equations is also intensively studied and finds numerous applications in describing real world problems (see for instance the monographs of lakshmikantham et al. [27], kilbas et al. [25], miller and ross [31], podlubny [39], samko et al. [40], and the papers of agarwal et al. [1], benchohra et al. [11, 12], chang and nieto [14], diethelm et al. [16], furati and tatar [17, 18], gaul et al. [19], glockle and nonnenmacher [20], lakshmikantham and devi [28], mainardi [29], metzler et al. [30], n’guérékata et al [33, 34, 35], and the references therein). jaradat et al. [23], studied the existence and uniqueness of mild solutions for a class of initial value problem for a semilinear integrodifferential equation involving the caputo’s fractional derivative. in this paper we will examine the following semilinear functional differential equation of fractional order d r y(t) = a y(t) + f (t, yt ), t ∈ j = [0, b], r > 0 (1) y(t) = φ(t), t ∈ [−ρ, 0], (2) where d r is the standard riemann-liouville fractional derivative, f : j × c([−ρ, 0], e) → e is a given function, a : d(a) ⊂ e → e is a nondensely defined closed linear operator on e. cubo 12, 3 (2010) semilinear fractional differential equations 37 φ : [−ρ, 0] → e a given continuous function with φ(0) = 0 and (e,| · |) a banach space. for any function y defined on [−ρ, b] and any t ∈ j we denote by yt the element of c([−ρ, 0], e) defined by yt(θ) = y(t +θ), θ ∈ [−ρ, 0]. here yt(·) represents the history of the state from time t −ρ, up to the present time t. let us mention that the functional differential equation of the type (1) was investigated, in the case when a generates a c0−semigroup, in a lot of papers and developed with the help of various tools of fixed-point theory see, for instance belmekki et al. [8, 9, 10]. the principal goal of this paper is to extend such results to the case when the operator a is nondensely defined and satisfies the hille-yosida condition, and to initiate the application of the technique of measures of noncompactness to investigate the problem of the existence of integral solutions for (1)–(2). especially that technique combined with an appropriate fixed point theorem has proved to be a very useful tool in the study of the existence of solutions for several types of integral and differential equations; see for example alvàrez [3], banas̀ et al. [5, 6, 7], benchohra et al. [13], guo et al. [21], mönch [32], mönch and von harten [37], and szufla [41]. 2 preliminaries in this section we collect some definitions, notations and results needed in the sequel. at first, we recall the definition of riemann-liouville fractional primitive and fractional derivative. denote by c(j, e) the banach space of continuous functions j → e, with the usual supremum norm ‖y‖∞ = sup{|y(t)|, t ∈ j}. for ψ ∈ c([−ρ, 0], e) the norm of ψ is defined by ‖ψ‖c = sup{|ψ(θ)|, θ ∈ [−ρ, 0]}. b(e) denotes the banach space of all bounded linear operators from e into e, with norm ‖n‖b(e) = su p{|n( y)| : |y| = 1}. let l1(j, e) be the banach space of measurable functions y : j → e which are bochner integrable, equipped with the norm ‖y‖l1 = ˆ j |y(t)|dt. let l∞(j, e) be the banach space of measurable functions y : j → e which are bounded, equipped with the norm ‖y‖l∞ = inf{c > 0 : ‖y(t)‖ ≤ c, a.e. t ∈ j}. 38 m. benchohra, gaston m. n’guérékata & d. seba cubo 12, 3 (2010) for a given set v of functions v : [−ρ, b] → e, let us denote by v (t) = {v(t) : v ∈ v }, t ∈ [−ρ, b] and v (j) = {v(t) : v ∈ v , t ∈ [−ρ, b]}. definition 2.1. ([25, 39]). the riemann-liouville fractional primitive of order r > 0 of a function h : (0, b] → e is defined by i r 0 h(t) = 1 γ(r) ˆ t 0 (t − s)r−1 h(s)ds, provided the right side is pointwise defined on (0, b], and where γ is the gamma function. definition 2.2. ([25, 39]). the riemann-liouville fractional derivative of order r ∈ (0, 1] of a continuous function h : (0, b] → e is defined by dr h(t) dtr = 1 γ(1 − r) d dt ˆ t 0 (t − s)−r h(s)ds = d dt i 1−r 0 h(t). definition 2.3. a map f : j × c([−ρ, 0], e) → e is said to be carathéodory if (i) t 7−→ f (t, u) is measurable for each u ∈ c([−ρ, 0], e); (ii) u 7−→ f(t, u) is continuous for almost each t ∈ j. for completeness we gather some definitions and basic facts of integrated semigroups, and operators satisfying hille-yosida condition. definition 2.4. [4]. let e be a banach space. an integrated semigroup is a family of operators (s(t))t≥0 of bounded linear operators s(t) on e with the following properties: (i) s(0) = 0; (ii) t → s(t) is strongly continuous; (iii) s(s)s(t) = ˆ s 0 (s(t +τ) − s(τ))dτ, for all t, s ≥ 0. definition 2.5. [24]. an operator a is called a generator of an integrated semigroup if there exists ω ∈ r such that (ω,∞) ⊂ ρ0(a) (ρ0(a), is the resolvent set of a) and there exists a strongly continuous exponentially bounded family (s(t))t≥0 of bounded operators such that s(0) = 0 and r(λ, a) := (λi − a)−1 = λ ˆ ∞ 0 e −λt s(t)dt exists for all λ with λ > ω. cubo 12, 3 (2010) semilinear fractional differential equations 39 proposition 2.1. [4]. let a be the generator of an integrated semigroup (s(t))t≥0. then for all x ∈ e and t ≥ 0, ˆ t 0 s(s)xds ∈ d(a) and s(t)x = a ˆ t 0 s(s)xds + tx. definition 2.6. we say that the linear operator a satisfies the hille-yosida condition if there exists m ≥ 0 and ω ∈ r such that (ω,∞) ⊂ ρ0(a) and sup{(λ−ω)n|(λi − a)−n| : n ∈ in, λ > ω} ≤ m. definition 2.7. [24]. (i) an integrated semigroup (s(t))t≥0 is called locally lipschitz continuous if, for all τ > 0, there exists a constant l such that |s(t) − s(s)| ≤ l|t − s|, t, s ∈ [0,τ]. (ii) an integrated semigroup (s(t))t≥0 is called non degenerate if s(t)x = 0, for all t ≥ 0, implies that x = 0. theorem 2.1. [24]. the following assertions are equivalent: (i) a is the generator of a non degenerate, locally lipschitz continuous integrated semigroup; (ii) a satisfies the hille-yosida condition. if a is the generator of an integrated semigroup (s(t))t≥0 which is locally lipschitz, then from [4], s(·)x is continuously differentiable if and only if x ∈ d(a) and (s′(t))t≥0 is a c0−semigroup on d(a). let (s(t))t≥0 be the integrated semigroup generated by a. we note that, if a satisfies the hille-yosida condition, then ‖s′(t)‖b(e) ≤ m e ωt, t ≥ 0, where m and ω are the constants considered in the hille-yosida condition. now let us recall some fundamental facts of the notion of kuratowski measure of noncompactness. definition 2.8. ([6]) let e be a banach space and ωe the bounded subsets of e. the kuratowski measure of noncompactness is the map α : ωe → [0,∞] defined by α(b) = inf{ǫ > 0 : b ⊆ ∪n i=1bi and d iam(bi ) ≤ ǫ}; here b ∈ ωe . 40 m. benchohra, gaston m. n’guérékata & d. seba cubo 12, 3 (2010) properties: the kuratowski measure of noncompactness satisfies the following properties (for more details see [6]). (a) α(b) = 0 ⇔ b is compact (b is relatively compact). (b) α(b) = α(b). (c) a ⊂ b ⇒ α(a) ≤ α(b). (d) α(a + b) ≤ α(a) +α(b) (e) α(cb) = |c|α(b); c ∈ ir. (f) α(convb) = α(b). theorem 2.2. ([2, 32]) let d be a bounded, closed and convex subset of a banach space such that 0 ∈ d, and let n be a continuous mapping of d into itself. if the implication v = conv n(v ) or v = n(v ) ∪ {0} ⇒ α(v ) = 0 holds for every subset v of d, then n has a fixed point. lemma 2.1. ([41]) let d be a bounded, closed and convex subset of the banach space c(j, e), g a continuous function on j × j and f a function from j × c([−ρ, 0], e) → e which satisfies the carathéodory conditions and there exists p ∈ l1(j, ir+) such that for each t ∈ j and each bounded set b ⊂ c([−ρ, 0], e) we have lim k→0+ α( f (jt,k × b)) ≤ p(t)α(b); here jt,k = [t − k, t] ∩ j. if v is an equicontinuous subset of d, then α ({ ˆ j g(s, t) f (s, ys )ds : y ∈ v }) ≤ ˆ j ‖g(t, s)‖p(s)α(v (s))ds. 3 main results we start with the following principal assumption and the definition of integral solutions to the problem (1)-(2). (h1) a satisfies the hille-yosida condition. definition 3.1. we say that a continuous function y : [−ρ, b] → e is an integral solution of problem (1)-(2) if cubo 12, 3 (2010) semilinear fractional differential equations 41 (i) ˆ t 0 (t − s)r−1 y(s)ds ∈ d(a) for t ∈ j, (ii) y(t) = φ(t), t ∈ [−ρ, 0], and (iii) y(t) = 1 γ(r) a ˆ t 0 (t − s)r−1 y(s)ds + 1 γ(r) ˆ t 0 (t − s)r−1 f (s, ys )ds, t ∈ j. from the definition it follows that y(t) ∈ d(a),∀ t ≥ 0. moreover, y satisfies the following variation of constants formula: y(t) = 1 γ(r) d dt ˆ t 0 s(t − s)(t − s)r−1 f (s, ys )ds, t ≥ 0. (3) let bλ = λr(λ, a), then for all x ∈ d(a), bλx 7→ x as λ 7→ ∞. we notice also that, if y satisfies (3), then y(t) = lim λ→∞ 1 γ(r) ˆ t 0 s ′(t − s)(t − s)r−1 bλ f (s, ys )ds, t ≥ 0. without lost of generality, we will assume that w > 0. let us list some conditions on the functions involved in the problem (1)-(2). (h2) the operator s′(t) is compact in d(a) whenever t > 0 and ‖s ′(t)‖b(e) ≤ m e ωt, t ∈ j. (h3) f : j × c([−ρ, 0], e) → e is of carathéodory. (h4) there exists a function p ∈ l∞(j, ir+) such that |f (t, u)| ≤ p(t)(‖u‖c + 1), for a.e. t ∈ j, and each u ∈ c([−ρ, 0], e). (h5) for almost each t ∈ j and each bounded set b ⊂ c([−ρ, 0], e) we have lim h→0+ α( f (jt,h × b)) ≤ p(t)α(b); here jt,h = [t − h, t] ∩ j. (h6) assume mbr p∗ eωb γ(r + 1) < 1. let p∗ = ‖p‖l∞ . our main result reads as follows theorem 3.1. assume that assumptions (h1) − (h6) hold. then the the problem (1)-(2) has at least one integral solution. 42 m. benchohra, gaston m. n’guérékata & d. seba cubo 12, 3 (2010) proof. we shall reduce the existence of solutions of (1)-(2) to a fixed point problem. consider the operator n : c([−ρ, b], e) → c([−ρ, b], e) defined by n( y)(t) =      φ(t), t ∈ [−ρ, 0], 1 γ(r) d dt ˆ t 0 (t − s)r−1 s(t − s) f (s, ys )ds, t ∈ [0, b]. let r0 > 0 be such that r0 ≥ m p∗br eωb γ(r + 1) − mbr p∗ eωb , and consider the set dr0 = { y ∈ c([−ρ, b], e) : ‖y‖∞ ≤ r0}. clearly, the subset dr0 is closed, bounded and convex. we shall show that n satisfies the assumptions of theorem 2.2. the proof will be given in three steps. step 1: n is continuous. let us consider a sequence { yn} such that yn → y in c([−ρ, b], e). then for each t ∈ j |n( yn)(t) − n( y)(t)| = ∣ ∣ ∣ ∣ 1 γ(r) d dt ˆ t 0 (t − s)r−1 s(t − s)[ f (s, yns ) − f (s, ys )]ds ∣ ∣ ∣ ∣ ≤ m eωt γ(r) ˆ t 0 e −ωs(t − s)r−1|f (s, yns ) − f (s, ys )|ds ≤ m eωb γ(r) ˆ t 0 (t − s)r−1|f (s, yns ) − f (s, ys )|ds. let µ > 0 be such that ‖yn‖∞ ≤ µ, ‖y‖∞ ≤ µ. by (h4) we have |(t − s)r−1 [ f (s, yns ) − f (s, ys )]| ≤ 2 p ∗(µ+ 1)(t − s)r−1 ∈ l1(j, ir+). since f is a carathéodory function, the lebesgue dominated convergence theorem implies that ‖n( yn) − n( y)‖∞ → 0 as n → ∞. step 2: n maps dr0 into itself. for each y ∈ dr0 , by (h4) and (h6) we have for each t ∈ j |n( y)(t)| = ∣ ∣ ∣ ∣ 1 γ(r) d dt ˆ t 0 (t − s)r−1 s(t − s) f (s, ys )ds ∣ ∣ ∣ ∣ cubo 12, 3 (2010) semilinear fractional differential equations 43 ≤ mbr p∗ eωb(r0 + 1) γ(r + 1) ≤ r0 step 3: n(dr0 ) is bounded and equicontinuous. by step 2, it is obvious that n(dr0 ) ⊂ dr0 is bounded. for the equicontinuity of n(dr0 ). let τ1, τ2 ∈ j, τ1 < τ2, thus if ǫ > 0 and ǫ ≤ τ1 ≤ τ2 we have for any y ∈ dr0 ; |n( y)(τ2) − n( y)(τ1)| = ∣ ∣ ∣ ∣ lim λ 7→∞ 1 γ(r) ˆ τ2 0 (τ2 − s) r−1 s ′(τ2 − s)bλ f (s, ys )ds − lim λ 7→∞ 1 γ(r) ˆ τ1 0 (τ1 − s) r−1 s ′(τ1 − s)bλ f (s, ys )ds ∣ ∣ ∣ ∣ ≤ m p ∗(r0 + 1) ( 1 γ(r) ˆ τ1−ǫ 0 [(τ2 − s) r−1 − (τ1 − s) r−1]ds + ‖s ′(τ2 −τ1 +ǫ) − s ′(ǫ)‖b(e) { 1 γ(r) ˆ τ1−ǫ 0 (τ2 − s) r−1 ds } + 1 γ(r) ˆ τ1 τ1−ǫ [(τ2 − s) r−1 − (τ1 − s) r−1]ds + ‖s ′(τ2 −τ1) − i‖b(e) { 1 γ(r) ˆ τ1 τ1−ǫ (τ2 − s) r−1 ds } + 1 γ(r) ˆ τ2 τ1 (τ2 − s) r−1 ds ) . as τ1 → τ2 and ǫ sufficiently small, the right-hand side of the above inequality tends to zero, since s′(t) is a strongly continuous operator and the compactness of s′(t) for t > 0 implies the continuity in the uniform operator topology (see [38]). now let v be a subset of dr0 such that v ⊂ conv(n(v ) ∪ {0}). v is bounded and equicontinuous and therefore the function v → v(t) = α(v (t)) is continuous on [−ρ, b]. by (h5), lemmas 2.1 and the properties of the measure α we have for each t ∈ [−ρ, b] v(t) ≤ α(n(v )(t) ∪ {0}) ≤ α(n(v )(t)) ≤ lim λ 7→∞ 1 γ(r) ˆ t 0 (t − s)r−1 s′(t − s)bλ p(s)α(v (s))ds 44 m. benchohra, gaston m. n’guérékata & d. seba cubo 12, 3 (2010) ≤ m p∗ eωt γ(r) ˆ t 0 (t − s)r−1 v(s)ds ≤ ‖v‖∞ mbr p∗ eωb γ(r + 1) . this means that ‖v‖∞ ( 1 − mbr p∗ eωb γ(r + 1) ) ≤ 0. by (h6) it follows that ‖v‖∞ = 0, that is v(t) = 0 for each t ∈ [−ρ, b], and then v (t) is relatively compact in e. in view of the ascoli-arzelà theorem, v is relatively compact in dr0 . applying now theorem 2.2 we conclude that n has a fixed point which is an integral solution for the problem (1)-(2). � 4 an example as an application of our results we consider the following fractional time partial functional differential equation of the form ∂α ∂tα z(t, x) = ∂2 ∂x2 z(t, x) + q(t, z(t − r, x)), x ∈ [0,π], t ∈ [0, 1], α ∈ (0, 1], (4) z(t, 0) = z(t,π) = 0, t ∈ [0, 1] (5) z(t, x) = ϕ(t, x), t ∈ [−r, 0], x ∈ [0,π], (6) where r > 0, ϕ : [−r, 0] × [0,π] → ir is continuous and q : [0, 1] × ir → ir is a given function. let y(t)(x) = z(t, x), t ∈ j, x ∈ [0,π], φ(θ)(x) = ϕ(θ, x), θ ∈ [−r, 0], x ∈ [0,π], f(t,φ)(x) = q(t,ϕ(θ, x)), θ ∈ [−r, 0], x ∈ [0,π]. we choose e = c([0,π]; ir) endowed with the uniform topology and consider the operator a : d(a) ⊂ e → e defined by: d(a) = { y ∈ c2([0,π], ir) : y(0) = y(π) = 0} a y = y′′. it is well known (see [15]) that the operator a satisfies the hille-yosida condition with (0,+∞) ⊂ ρ(a), ‖(λi − a)−1‖ ≤ 1 λ for λ > 0, and d(a) = { y ∈ e; y(0) = y(π) = 0} 6= e. cubo 12, 3 (2010) semilinear fractional differential equations 45 it follows that a generates an integrated semigroup (s(t))t≥0 and ‖s ′(t)‖ ≤ e−µt for t ≥ 0 and for some constant µ > 0. we can show that problem (1)-(2) is an abstract formulation of problem (4)-(6). assume that the function q satisfies the following conditions (i) the function q : j × ir → ir is of carathéodory. 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[42] wu, j., theory and applications of partial functional differential equations, springerverlag, new york, 1996. dhage_multi_09.dvi cubo a mathematical journal vol.12, no¯ 03, (139–151). october 2010 some generalizations of mulit-valued version of schauder’s fixed point theorem with applications bapurao c. dhage kasubai, gurukul colony, ahmedpur – 413515, distr. latur, maharashtra, india email: bcdhage@yahoo.co.in abstract in this article, a generalization of a kakutani-fan fixed point theorem for multi-valued mappings in banach spaces is proved under weaker upper semi-continuity condition and it is further applied to derive a generalized version of krasnoselskii’s fixed point theorem and some nonlinear alternatives of leray-schauder type for multi-valued closed mappings in banach spaces. resumen en este artículo probamos una generalización para el teorema del punto fijo de kakutanifan para aplicaciones multi-valuadas en espacios de banach, bajo condición de semi-continuidad superior debil. este resultado es aplicado para obtener una versión generalizada del teorema del punto fijo krasnoselskii y algunas alternativas de tipo leray-schauder para aplicaciones multi-valuadas cerradas en espacios de banach. key words and phrases: multi-valued mappings, fixed point theorem, nonlinear alternative. math. subj. class.: 47h10. 140 bapurao c. dhage cubo 12, 3 (2010) 1 introduction throughout this paper, unless otherwise mentioned, let e be a banach space and let p (e) denote the class of all subsets of e. denote p p(e) = {a ⊂ e | a is non-empty and has a property p}. here, p may be the property p = closed (in short cl), or p = compact (in short cp), or p = convex (in short cv), or p = bounded (in short bd) etc. thus, p bd (e), p cl (e), p cv(e), p c p (e), p cl,bd (e), p c p,cv (e) denote the classes of all bounded, closed, convex, compact, closed-bounded and compact-convex subsets of e respectively. similarly, p cl,cv,bd (e) and p r c p (e) denote respectively the classes of closed, convex and bounded and relatively compact subsets of e. a correspondence q : e → p p(e) is called a multi-valued operator or multi-valued mapping on e into e. a point u ∈ e is called a fixed point of q if u ∈ qu. for the sake of convenience, we denote q(a) = ⋃ x∈a t x for all subsets a of e. let e1 and e2 be two banach spaces and let q : e1 → p p (e2) be a multi-valued operator. then for any non-empty subset a of e2, define q + (a) = {x ∈ e1 | t x ⊂ a}, q − (a) = {x ∈ e1 | t x ∩ a 6= ;}, and q −1 (a) = {x ∈ e1 | ∪xt x = a}. definition 1.1. a multi-valued operator q : e1 → p p(e2) is called upper semi-continuous (resp. lower semi-continuous and continuous) if q+(u) (resp. q−(u) and q−1(u)) is open set in e1 for every open subset u of e2. in what follows,we confine ourselves only to the fixed point theory related to upper semicontinuous multi-valued mappings in banach spaces. the first fixed point theorem in this direction is due to kakutani-fan [11] which is as follows. theorem 1.1. let k be a compact subset of a banach space e and let q : e → p c p,cv (e) be an upper semi-continuous multi-valued operator. then q has a fixed point. note that following are the main three ingredients for the above theorem 1.1. (i) the domain space e, cubo 12, 3 (2010) multi-valued fixed point theory 141 (ii) the domain set k , and (iii) the nature of the multi-valued operator q. theorem 1.1 has been extended in the literature by generalizing or modifying the above three hypotheses with the same conclusion. in the following discussion, we do not change the hypothesis on the domain space, and thus keep us in the practical applicability of the so obtained fixed point theorem to other areas of mathematics. however, the generalizations of the above theorem 1.1 with change of domain space may be found in the works of browder-fan [11] and himmelberg [9] etc. a first generalization of theorem 1.1 is due to bohnenblust-karlin as given in petruşel [12]. theorem 1.2 (bohnenblust-karlin). let x be a closed convex and bounded subset of a banach algebra e and let q : x → p c p,cv (x ) be a upper semi-continuous multi-valued operator with a relatively compact range. then q has a fixed point. a multi-valued map q : x → p c p (x ) is called compact if q(x ) is a compact subset of x . q is called totally bounded if for any bounded subset a of x , q(a) = ⋃ x∈a q x is a totally bounded subset of x . it is clear that every compact multi-valued operator is totally bounded, but the converse may not be true. however, these two notions are equivalent on a bounded subset of x . finally, q is called completely continuous if it is upper semi-continuous and totally bounded on x . the upper semi-continuity is further weakened to closed graph operators as follows. if q : e1 → e2 is a multi-valued operator, then the graph gr(q) of the operator q is defined by gr(q) = {(x, y) ∈ e1 × e2 | y ∈ t x}. the graph gr(q) of the operator q is said to be closed if {(xn, yn)} be a sequence in gr(q) such that (xn, yn) → (x, y), then we have that (x, y) ∈ gr(q). definition 1.2. a multi-valued operator q : e1 → p cl (e2) is called closed if it has a closed graph in e1 × e2. the following result concerning the upper semi-continuity of multi-valued mappings in banach spaces is very much useful in the study of multi-valued analysis. the details appears in deimling [5]. lemma 1.1. a multi-valued operator q : e1 → p cl (e2) is upper semi-continuous if and only if it is closed and has compact range. 142 bapurao c. dhage cubo 12, 3 (2010) theorem 1.3 (o’regan [13]). let x be a closed convex and bounded subset of a banach algebra e and let q : x → p c p,cv (x ) be a compact and closed multi-valued operator. then q has a fixed point. the compactness of q in theorem 1.3 is further weakened to condensing operators with the help of measure of noncompactness in the banach space e. the kuratowskii measure α and the ball or hausdorff measure β of noncompactness of a bounded set in the banach space e are the functions α,β : p bd (e) → r + defined by α(a) = inf { r > 0 : a ⊂ ∪ n i=1 ai , diam(si ) ≤ r ∀ i } , (1.1) and β(a) = inf { r > 0 | a ⊂ n ⋃ i=1 br (xi) for some xi ∈ x } (1.2) for all a ∈ p bd (e), where diam (ai ) = sup{‖x − x‖ : x, y ∈ ai } and br (xi ) are the open balls centered at xi of radius r. definition 1.3. a multi-valued operator q : e → p cl,bd (e) is called β-condensing if for all bounded sets a in e, q(a) is bounded and β(q(a)) < β(a) for β(a) > 0 . theorem 1.4. let x be a closed convex and bounded subset of a banach space e and let q : x → p cl,cv(x ) be a upper semi-continuous and β-condensing multi-valued operator. then q has a fixed point. in this article, we generalize theorem 1.1 by weakening the upper semi-continuity as well as compactness of the multi-valued operator q in a banach space e and discuss some of its applications. 2 fixed point theory a function dh : p p (e) × p p(e) → r + defined by dh (a, b) = max { sup a∈a d(a, b) , sup b∈b d(b, a) } (2.1) satisfies all the conditions of a metric on p p (e) and is called a hausdorff-pompeiu metric on e, where d(a, b) = inf {‖a − b‖ : b ∈ b}. it is known that the hyperspace ( p cl (e), dh ) is a complete metric space. the axiomatic way of defining the measures of noncompactness has been adopted in several papers in the literature. see akhmerov et al. [2], banas and goebel [3], and the cubo 12, 3 (2010) multi-valued fixed point theory 143 references given therein. in this paper, we define the measure of noncompactness in a banach space on the lines of dhage [6] which is slightly different manner from that given in the above monographs. definition 2.1. a sequence {an} of non-empty sets in p p(e) is said to converge to a set a, called the limiting set, if dh (an, a) → 0 as n → ∞. definition 2.2. a mapping µ : p p (e) → r + is continuous if for any sequence {an } in p p(e), we have that dh (an, a) → 0 implies |µ(an) −µ(a)| → 0 as n → ∞. definition 2.3. a mapping µ : p p(e) → r + is called nondecreasing if a, b ∈ p p(e) are any two sets with a ⊆ b, then µ(a) ≤ µ(b), where ⊆ is a order relation by inclusion in p p(e). now we are equipped with the necessary details to define the measures of noncompactness of a bounded subset of the banach space e. definition 2.4. a function µ : p cl,bd (e) → r + is called a measure of noncompactness if it satisfies (µ1) ; 6= µ −1(0) ⊂ p r c p (e), (µ2) µ(a) = µ(a), where a denotes the closure of a, (µ3) µ(conv a) = µ(a), where conv a denotes the convex hull of a, (µ4) µ is nondecreasing, and (µ5) if {an } is a decreasing sequence of sets in p cl,bd (e) satisfying lim n→∞ µ(an) = 0, then the limiting set a∞ = lim n→∞ an is non-empty. note that the functions α and β defined by (1.1) and (1.2) satisfy the conditions (µ1) through (µ5). hence α and β are the measures of noncompactness on e. moreover, they are locally lipschitz and hence are locally continuous on p cl,bd (e). some nice properties of α and β have been discussed in akhmerov et al. [2] and banas and goebel [3]. we remark that if (µ4) holds, then a∞ ∈ p r c p (e). to see this, let limn→∞ µ(an) = 0. as a∞ ⊆ an for each n = 0, 1, 2, ...; by the monotonicity of µ, we obtain µ(a∞) ≤ lim n→∞ an = lim n→∞ µ(an) = 0. hence, by assumption (µ1), we get a∞ is nonempty and a∞ ∈ p r c p (e). a measure µ is called complete or full if the kernel of µ consists of all possible relatively compact subsets of e. next, a measure µ is called sublinear if it satisfies 144 bapurao c. dhage cubo 12, 3 (2010) (µ6) µ(λa) = |λ|µ(a) for λ ∈ r, and (µ7) µ(a + b) ≤ µ(a) +µ(b) for a, b ∈ p cl,bd (e). there do exist the sublinear measures of noncompactness in banach spaces e. indeed, the measures α and β of noncompactness defined by (1.1) and (1.2) are sublinear on e. now we prove a fixed point theorem for the mappings in banach spaces involving the measures of noncompactness. before going to the main results, we give a useful definition. definition 2.5. a multi-valued mapping q : e → p cl,bd (e) is called d-set-lipschitz if there exists a continuous nondecreasing function ψ : r+ → r+ such that µ(q(a)) ≤ ψ(µ(a)) for all a ∈ p cl,bd (e) with q(a) ∈ p cl,bd (e), where ψ(0) = 0. sometimes we call the function ψ to be a d-function of q on e. in the special case, when ψ(r) = kr, k > 0, q is called a k-set-lipschitz mapping and if k < 1, then q is called a k-set-contraction on e. further, if ψ(r) < r for r > 0, then q is called a nonlinear d-set-contraction on e. we need the following lemma in the sequel. lemma 2.1 (dhage [8]). if ψ is a d-function with ψ(r) < r for r > 0, then lim n→∞ ψ n (t) = 0 for all t ∈ [0,∞). theorem 2.1. let x be a non-empty, closed, convex and bounded subset of a banach space e and let q : x → p cl,cv(x ) be a closed and nonlinear d-set-contraction. then q has a fixed point. proof. define a sequence {x n} of sets in p cl,bd (e) by x0 = x , x n+1 = conv q(x n), n = 0, 1, ... clearly, x0 ⊃ x1 ⊃ ··· ⊃ x n ⊃ x n+1 ··· . and so, {x n} is a decreasing sequence of subsets of e. since µ(x n+1) = µ ( conv q(x n) ) = µ(q(x n)) ≤ ψ(µ(x n)) for all n = 0, 1, 2, . . ., we have µ(x n+1) ≤ ψ n (µ(x0)). therefore lim sup n→∞ µ(x n+1) ≤ lim sup n→∞ ψ n (µ(x0)) = 0. cubo 12, 3 (2010) multi-valued fixed point theory 145 from the monotonicity of µ it follows that lim n→∞ x n = x∞ is a compact subset of e. as x n+1 ⊂ x n and q : x n → x n for all n = 0, 2, . . ., we have x∞ = lim n→∞ x n = ∞ ⋂ n=1 x n 6= ; is a convex subset of e and q : x∞ → p c p,cv (x∞) which is upper-semi-continuous in view of lemma 1.1. now the desired conclusion follows by an application of theorem 1.1 to the operator q on x∞. this completes the proof. remark 2.1. the fixed point set fix(q) of the multi-valued operator q in above theorem 2.1 is compact. in fact if µ(fix(q)) > 0, then from nonlinear d-set-contraction it follows that µ(fix(q)) = µ(q(fix(q))) ≤ ψ(µ(fix(q))) which is a contradiction since ψ(r) < r for r > 0. as a consequence of theorem 2.1 we obtain a fixed point theorem of darbo [3] type for linear set-contractions, corollary 2.1. let x be a closed, convex and bounded subset of a banach space e and let q : x → p cl,cv (x ) be a closed and k-set-contraction. then q has a fixed point. before stating the generalization of theorem 2.1of sadovskii [14] type, we give a useful definition. definition 2.6. a multi-valued mapping q : e → p (e) is called µ-condensing if for any bounded subset a of e, q(a) is bounded and µ(q(a)) < µ(a) for µ(a) > 0. theorem 2.2. let x be a nonempty, closed, convex and bounded subset of a banach space e and let q : x → p cl,cv (x ) be a closed and µ-condensing mapping. then q has a fixed point. thus, we have a one way implication that sadovskii’s type theorem ⇒ theorem 2.1 ⇒ darbo’s type theorem. however, it is rather difficult to find the operators satisfying the conditions on banach spaces given in sadovskii’s type fixed point theorem. 3 applications 3.1 hybrid fixed point theory first, we derive a krasnoselskii type fixed point theorem for the sum of two multi-valued mappings in banach spaces. before stating this result, we need the following definition. definition 3.1. a multi-valued mapping q : e → p cl,cv (e) is said to be nonlinear d-contraction if there is a d-function ψ such that dh (q x, q y) ≤ ψ(d(x, y)) 146 bapurao c. dhage cubo 12, 3 (2010) for all x, y ∈ e, where ψ(r) < r. theorem 3.1. let x be a closed, convex and bounded subset of a banach space e and let µ be a sublinear measure of noncompactness in it. let s, t : x → p cl,cv (e) be two operators such that (a) s is closed and nonlinear d-set-contraction, (b) t is compact and closed, and (c) sx + t x ⊂ x for all x ∈ x . then the operator inclusion x ∈ sx + t x has a solution and the set of all solutions is compact in e. proof. define a mapping q : x → p cl,cv (x ) by q x = sx + t x. (3.1) we show that q satisfies all the conditions of theorem 2.1. obviously, by hypothesis (c), q defines a mapping q : x → p cl,cv(x ). since s and t are closed, the sum q = s + t is also closed on x . as hypothesis (a) holds, there is a d-function ψ such that ψ(r) < r for r > 0. further, let a be a non-empty subset of x .then a bounded and q(a) ⊆ x and q(a) ⊆ s(a) + t(a), and hence q(a) is bounded. by sublinearity of µ, we obtain µ(q(a)) ≤ µ(s(a)) +µ(t(a)) ≤ ψ(µ(a)) where, ψ(r) < r for r > 0. this shows that q is a nonlinear d-set-contraction on x into itself. now an application of theorem 2.1 yields that q has a fixed point. consequently, the operator equation x ∈ sx + t x has a solution. this completes the proof. the following lemma is obvious and the proof may be found in the monographs of deimling [5] and hu and papageorgiou [10]. lemma 3.1. if q : e → p c p,cv (e) is nonlinear contraction. then for any bounded subset a of e with q(a) bounded, we have β(q(a)) ≤ ψ(β(a)), where β is a ball measure of noncompactness in e defined by (1.2). theorem 3.2. let x be a closed, convex and bounded subset of a banach space e and let s, t : x → p c p,cv (e) be two multi-valued operators such that cubo 12, 3 (2010) multi-valued fixed point theory 147 (a) s is a nonlinear d-contraction, (b) t is compact and closed, and (c) sx + t x ⊂ x for all x ∈ x . then the operator inclusion x ∈ sx + t x has a solution and the set of all solutions is compact in e. proof. since s is nonlinear d-contraction, it is closed on x and there is a d-function ψ of s on x with the properties that ψ(r) < r for r > 0. again from lemma 3.1, it follows that it is a also nonlinear d-set-contraction with respect to the hausdorff measure of noncompactness β and with a d-function ψ on x . now the desired conclusion follows by a direct application of theorem 2.1. 3.2 nonlinear alternative the following nonlinear alternative for multi-valued mappings in banach spaces is wellknown in the literature. theorem 3.3 (o’regan [13]). let u be a open bounded subset of a banach space e with 0 ∈ u and let q : u → p cl,cv(e) be a compact and closed multi-valued operator. then either (i) the operator inclusion x ∈ q x has a solution in u, or (ii) there is an element u ∈ ∂u such that λu ∈ qu for some λ > 1, where ∂u is the boundary of u in e. a generalization of above theorem 3.4 is theorem 3.4. let u be a open bounded subset of a banach space e with 0 ∈ u and let q : u → p cl,cv(e) be a µ-condensing and closed multi-valued operator. then either (i) the operator inclusion x ∈ q x has a solution in u and the set of all solutions is compact in e, or (ii) there is an element u ∈ ∂u such that λu ∈ qu for some λ > 1, where ∂u is the boundary of u in e. proof. the proof is similar to that given for theorem 3.3 in o’regan [13]( see also agarwal et al. [1]) and now the conclusion follows by an application of theorem 3.2. 148 bapurao c. dhage cubo 12, 3 (2010) as a consequence of theorem 3.4, we obtain corollary 3.1. let br (0) be a open ball in a banach space e centered at origin 0 ∈ e of radius r and let q : br (0) → p cl,cv(e) be a µ-condensing and closed multi-valued operator. then either (i) the operator inclusion x ∈ q x has a solution in br (0) and the set of all solutions is compact in e, or (ii) there is an element u ∈ e such that ‖u‖ = r satisfying λu ∈ qu for some λ > 1. corollary 3.2. let e be a banach space and let q : e → p cl,cv(e) be a µ-condensing and closed multi-valued operator. then, either (i) the operator inclusion x ∈ q x has a solution and the set of all solutions is compact in e, or (ii) the set e = {u ∈ e | λu ∈ qu} is in unbounded for some λ > 1. the above corollary 3.1 includes the following fixed point result due to martelli [10] which has been used by several authors in the literature for proving the existence theorems for differential and integral inclusions. corollary 3.3. let e be a banach space and let q : e → p cl,cv (e) be a upper semi-continuous and α-condensing( or β-condensing) multi-valued operator. then, either (i) the operator inclusion x ∈ q x has a solution in x , or (ii) the set e = {u ∈ e | λu ∈ qu} is in unbounded for some λ > 1. similarly, we can apply theorem 3.4 to obtain the following nonlinear alternatives for sum of the two multi-valued operators in banach spaces. theorem 3.5. let u be a open bounded subset of a banach space e with 0 ∈ u and let s, t : u → p cl,cv(e) be two multi-valued operators such that (a) s is closed and nonlinear d-set-contraction, and (b) t is compact and closed. then, either (i) the operator inclusion x ∈ sx + t x has a solution in u and the set of all solutions is compact in e, or cubo 12, 3 (2010) multi-valued fixed point theory 149 (ii) there is an element u ∈ ∂u such that λu ∈ su + t u for some λ > 1, where ∂u is the boundary of u in e. theorem 3.6. let u be a open bounded subset of a banach space e with 0 ∈ u and let s, t : u → p c p,cv (e) be two multi-valued operators such that (a) s is nonlinear d-contraction, and (b) t is compact and closed. then, either (i) the operator inclusion x ∈ sx + t x has a solution in u and the set of all solutions is compact in e, or (ii) there is an element u ∈ ∂u such that λu ∈ su + t u for some λ > 1, where ∂u is the boundary of u in e. corollary 3.4. let br (0) be a open ball in a banach space e centered at origin 0 ∈ e of radius r and let s, t : b r (0) → p c p,cv (e) be two multi-valued operators such that (a) s is nonlinear d-contraction, and (b) t is compact and closed. then, either (i) the operator inclusion x ∈ sx + t x has a solution in br (0) and the set of all solutions is compact in e, or (ii) there is an element u ∈ e such that ‖u‖ = r satisfying λu ∈ su + t u for some λ > 1. corollary 3.5. let e be a banach space e and let s, t : e → p c p,cv (e) be two multi-valued operators such that (a) s is nonlinear d-contraction, and (b) t is compact and closed. then, either (i) the operator inclusion x ∈ sx + t x has a solution and the set of all solutions is compact in e, or 150 bapurao c. dhage cubo 12, 3 (2010) (ii) the set e = {u ∈ e | λu ∈ su + t u} is in unbounded for some λ > 1. remark 3.1. note that our theorem 3.6 and corollary 3.4 improve the hybrid fixed point theorems for multi-valued mappings proved in dhage [6, 7] under weaker upper semi-continuity conditions. 4 the conclusion finally, while concluding, we remark that the multi-valued fixed point theorems of this paper have some nice applications to differential and integral inclusions for proving the existence as well as some characterizations of solutions such as global and local asymptotic attractivity of solutions on bounded and unbounded intervals of real line. the investigations of these and other similar problems form the scope for further research work in the theory of differential and integral inclusions under weaker upper semi-continuity conditions. some of the results in this direction will be reported elsewhere. references [1] agarwal, r.p., meehan, m. and o’regan, d., fixed point theory and applications, cambridge univ. press, 2001. 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[14] sadovskii, b.n., limit-compact and condensing operators, russian math. survey, 27 (1972), 85–155. cubo a mathematical journal vol.16, no¯ 02, (33–47). june 2014 voronovskaya type asymptotic expansions for multivariate quasi-interpolation neural network operators george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract here we study further the multivariate quasi-interpolation of sigmoidal and hyperbolic tangent types neural network operators of one hidden layer. we derive multivariate voronovskaya type asymptotic expansions for the error of approximation of these operators to the unit operator. resumen aqúı estudiamos extensiones de la cuasi-interpolación multivariada de operadores de redes neuronales de tipo sigmoidal y tangente hiperbólica de una capa oculta. obtenemos expansiones asintóticas del tipo voronovskaya para el error de aproximación de estos operadores para el operador unidad. keywords and phrases: multivariate neural network approximation, multivariate voronovskaya type asymptotic expansion. 2010 ams mathematics subject classification: 41a25, 41a36, 41a60. 34 george a. anastassiou cubo 16, 2 (2014) 1 background here we follow [5], [6]. we consider here the sigmoidal function of logarithmic type si (xi) = 1 1 + e−xi , xi ∈ r, i = 1, ..., n; x := (x1, ..., xn) ∈ r n, each has the properties lim xi→+∞ si (xi) = 1 and lim xi→−∞ si (xi) = 0, i = 1, ..., n. these functions play the role of activation functions in the hidden layer of neural networks. as in [7], we consider φi (xi) := 1 2 (si (xi + 1) − si (xi − 1)) , xi ∈ r, i = 1, ..., n. we notice the following properties: i) φi (xi) > 0, ∀ xi ∈ r, ii) ∑∞ ki=−∞ φi (xi − ki) = 1, ∀ xi ∈ r, iii) ∑∞ ki=−∞ φi (nxi − ki) = 1, ∀ xi ∈ r; n ∈ n, iv) ∫∞ −∞ φi (xi) dxi = 1, v) φi is a density function, vi) φi is even: φi (−xi) = φi (xi), xi ≥ 0, for i = 1, ..., n. we see that ([7]) φi (xi) = ( e2 − 1 2e2 ) 1 (1 + exi−1) (1 + e−xi−1) , i = 1, ..., n. vii) φi is decreasing on r+, and increasing on r−, i = 1, ..., n. let 0 < β < 1, n ∈ n. then as in [6] we get viii) ∞∑ ⎧ ⎪⎨ ⎪⎩ ki = −∞ : |nxi − ki| > n 1−β φi (nxi − ki) ≤ 3.1992e −n(1−β), i = 1, ..., n. cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 35 denote by ⌈·⌉ the ceiling of a number, and by ⌊·⌋ the integral part of a number. consider here x ∈ (∏n i=1 [ai, bi] ) ⊂ rn, n ∈ n such that ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., n; a := (a1, ..., an), b := (b1, ..., bn) . as in [6] we obtain ix) 0 < 1 ∑⌊nbi⌋ ki=⌈nai⌉ φi (nxi − ki) < 1 φi (1) = 5.250312578, ∀ xi ∈ [ai, bi] , i = 1, ..., n. x) as in [6], we see that lim n→∞ ⌊nbi⌋∑ ki=⌈nai⌉ φi (nxi − ki) ̸= 1, for at least some xi ∈ [ai, bi], i = 1, ..., n. we will use here φ (x1, ..., xn) := φ (x) := n∏ i=1 φi (xi) , x ∈ r n. (1) it has the properties: (i)’ φ (x) > 0, ∀ x ∈ rn, we see that ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ φ (x1 − k1, x2 − k2, ..., xn − kn) = ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ n∏ i=1 φi (xi − ki) = n∏ i=1 ( ∞∑ ki=−∞ φi (xi − ki) ) = 1. that is (ii)’ ∞∑ k=−∞ φ (x − k) := ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ φ (x1 − k1, ..., xn − kn) = 1, k := (k1, ..., kn), ∀ x ∈ r n. (iii)’ ∞∑ k=−∞ φ (nx − k) := ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ φ (nx1 − k1, ..., nxn − kn) = 1, ∀ x ∈ rn; n ∈ n. 36 george a. anastassiou cubo 16, 2 (2014) (iv)’ ∫ rn φ (x) dx = 1, that is φ is a multivariate density function. here ∥x∥ ∞ := max {|x1| , ..., |xn|}, x ∈ r n, also set ∞ := (∞, ..., ∞), −∞ := (−∞, ..., −∞) upon the multivariate context, and ⌈na⌉ : = (⌈na1⌉ , ..., ⌈nan⌉) , ⌊nb⌋ : = (⌊nb1⌋ , ..., ⌊nbn⌋) . for 0 < β < 1 and n ∈ n, fixed x ∈ rn, have that ⌊nb⌋∑ k=⌈na⌉ φ (nx − k) = ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ φ (nx − k) + ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ φ (nx − k) . in the last two sums the counting is over disjoint vector of k’s, because the condition ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ implies that there exists at least one ∣ ∣ kr n − xr ∣ ∣ > 1 nβ , r ∈ {1, ..., n}. it holds (v)’ ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ φ (nx − k) ≤ 3.1992e−n (1−β) , 0 < β < 1, n ∈ n, x ∈ (∏n i=1 [ai, bi] ) . furthermore it holds (vi)’ 0 < 1 ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) < (5.250312578) n , ∀ x ∈ (∏n i=1 [ai, bi] ) , n ∈ n. it is clear also that cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 37 (vii)’ ∞∑ ⎧ ⎪⎨ ⎪⎩ k = −∞ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ φ (nx − k) ≤ 3.1992e−n (1−β) , 0 < β < 1, n ∈ n, x ∈ rn. by (x) we obviously see that (viii)’ lim n→∞ ⌊nb⌋∑ k=⌈na⌉ φ (nx − k) ̸= 1 for at least some x ∈ (∏n i=1 [ai, bi] ) . let f ∈ c (∏n i=1 [ai, bi] ) and n ∈ n such that ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., n. we define the multivariate positive linear neural network operator (x := (x1, ..., xn) ∈ (∏n i=1 [ai, bi] ) ) gn (f, x1, ..., xn) := gn (f, x) := ∑⌊nb⌋ k=⌈na⌉ f ( k n ) φ (nx − k) ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) (2) := ∑⌊nb1⌋ k1=⌈na1⌉ ∑⌊nb2⌋ k2=⌈na2⌉ ... ∑⌊nbn⌋ kn=⌈nan⌉ f ( k1 n , ..., kn n ) (∏n i=1 φi (nxi − ki) ) ∏n i=1 (∑⌊nbi⌋ ki=⌈nai⌉ φi (nxi − ki) ) . for large enough n we always obtain ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., n. also ai ≤ ki n ≤ bi, iff ⌈nai⌉ ≤ ki ≤ ⌊nbi⌋, i = 1, ..., n. notice here that for large enough n ∈ n we get that e−n (1−β) < n−βj, j = 1, ..., m ∈ n, 0 < β < 1. thus be given fixed a, b > 0, for the linear combination ( an−βj + be−n (1−β) ) the (dominant) rate of convergence to zero is n−βj. the closer β is to 1 we get faster and better rate of convergence to zero. by acm (∏n i=1 [ai, bi] ) , m, n ∈ n, we denote the space of functions such that all partial derivatives of order (m − 1) are coordinatewise alsolutely continuous functions, also f ∈ cm−1 (∏n i=1 [ai, bi] ) . 38 george a. anastassiou cubo 16, 2 (2014) let f ∈ acm (∏n i=1 [ai, bi] ) , m, n ∈ n. here fα denotes a partial derivative of f, α := (α1, ..., αn), αi ∈ z +, i = 1, ..., n, and |α| := ∑n i=1 αi = l, where l = 0, 1, ..., m. we write also fα := ∂αf ∂xα and we say it is order l. we denote ∥fα∥ max ∞,m := max |α|=m {∥fα∥∞}, (3) where ∥·∥ ∞ is the supremum norm. we assume here that ∥fα∥ max ∞,m < ∞. next we follow [3], [4]. we consider here the hyperbolic tangent function tanh x, x ∈ r : tanh x := ex − e−x ex + e−x . it has the properties tanh 0 = 0, −1 < tanh x < 1, ∀ x ∈ r, and tanh (−x) = − tanh x. furthermore tanh x → 1 as x → ∞, and tanh x → −1, as x → −∞, and it is strictly increasing on r. this function plays the role of an activation function in the hidden layer of neural networks. we further consider ψ (x) := 1 4 (tanh (x + 1) − tanh (x − 1)) > 0, ∀ x ∈ r. we easily see that ψ (−x) = ψ (x), that is ψ is even on r. obviously ψ is differentiable, thus continuous. proposition 1. ([3]) ψ (x) for x ≥ 0 is strictly decreasing. obviously ψ (x) is strictly increasing for x ≤ 0. also it holds lim x→−∞ ψ (x) = 0 = lim x→∞ ψ (x) . infact ψ has the bell shape with horizontal asymptote the x-axis. so the maximum of ψ is zero, ψ (0) = 0.3809297. theorem 2. ([3]) we have that ∑∞ i=−∞ ψ (x − i) = 1, ∀ x ∈ r. thus ∞∑ i=−∞ ψ (nx − i) = 1, ∀ n ∈ n, ∀ x ∈ r. also it holds ∞∑ i=−∞ ψ (x + i) = 1, ∀x ∈ r. theorem 3. ([3]) it holds ∫∞ −∞ ψ (x) dx = 1. so ψ (x) is a density function on r. cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 39 theorem 4. ([3]) let 0 < α < 1 and n ∈ n. it holds ∞∑ ⎧ ⎪⎨ ⎪⎩ k = −∞ : |nx − k| ≥ n1−α ψ (nx − k) ≤ e4 · e−2n (1−α) . theorem 5. ([3]) let x ∈ [a, b] ⊂ r and n ∈ n so that ⌈na⌉ ≤ ⌊nb⌋. it holds 1 ∑⌊nb⌋ k=⌈na⌉ ψ (nx − k) < 1 ψ (1) = 4.1488766. also by [3] we get that lim n→∞ ⌊nb⌋∑ k=⌈na⌉ ψ (nx − k) ̸= 1, for at least some x ∈ [a, b]. in this article we will use θ (x1, ..., xn) := θ (x) := n∏ i=1 ψ (xi) , x = (x1, ..., xn) ∈ r n, n ∈ n. (4) it has the properties: (i)∗ θ (x) > 0, ∀ x ∈ rn, (ii)∗ ∞∑ k=−∞ θ (x − k) := ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ θ (x1 − k1, ..., xn − kn) = 1, where k := (k1, ..., kn), ∀ x ∈ r n. (iii)∗ ∞∑ k=−∞ θ (nx − k) := ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ θ (nx1 − k1, ..., nxn − kn) = 1, ∀ x ∈ rn; n ∈ n. (iv)∗ ∫ rn θ (x) dx = 1, that is θ is a multivariate density function. 40 george a. anastassiou cubo 16, 2 (2014) we obviously see that ⌊nb⌋∑ k=⌈na⌉ θ (nx − k) = ⌊nb⌋∑ k=⌈na⌉ n∏ i=1 ψ (nxi − ki) = ⌊nb1⌋∑ k1=⌈na1⌉ ... ⌊nbn⌋∑ kn=⌈nan⌉ n∏ i=1 ψ (nxi − ki) = n∏ i=1 ⎛ ⎝ ⌊nbi⌋∑ ki=⌈nai⌉ ψ (nxi − ki) ⎞ ⎠ . for 0 < β < 1 and n ∈ n, fixed x ∈ rn, we have that ⌊nb⌋∑ k=⌈na⌉ θ (nx − k) = ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ θ (nx − k) + ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ θ (nx − k) . in the last two sums the counting is over disjoint vector of k’s, because the condition ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ implies that there exists at least one ∣ ∣ kr n − xr ∣ ∣ > 1 nβ , r ∈ {1, ..., n}. il holds (v)∗ ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ θ (nx − k) ≤ e4 · e−2n (1−β) , 0 < β < 1, n ∈ n, x ∈ (∏n i=1 [ai, bi] ) . also it holds (vi)∗ 0 < 1 ∑⌊nb⌋ k=⌈na⌉ θ (nx − k) < 1 (ψ (1)) n = (4.1488766) n , ∀ x ∈ (∏n i=1 [ai, bi] ) , n ∈ n. it is clear that (vii)∗ ∞∑ ⎧ ⎪⎨ ⎪⎩ k = −∞ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ θ (nx − k) ≤ e4 · e−2n (1−β) , 0 < β < 1, n ∈ n, x ∈ rn. cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 41 also we get lim n→∞ ⌊nb⌋∑ k=⌈na⌉ θ (nx − k) ̸= 1, for at least some x ∈ (∏n i=1 [ai, bi] ) . let f ∈ c (∏n i=1 [ai, bi] ) and n ∈ n such that ⌈nai⌉ ≤ ⌊nbi⌋, i = 1, ..., n. we define the multivariate positive linear neural network operator (x := (x1, ..., xn) ∈ (∏n i=1 [ai, bi] ) ) fn (f, x1, ..., xn) := fn (f, x) := ∑⌊nb⌋ k=⌈na⌉ f ( k n ) θ (nx − k) ∑⌊nb⌋ k=⌈na⌉ θ (nx − k) (5) := ∑⌊nb1⌋ k1=⌈na1⌉ ∑⌊nb2⌋ k2=⌈na2⌉ ... ∑⌊nbn⌋ kn=⌈nan⌉ f ( k1 n , ..., kn n ) (∏n i=1 ψ (nxi − ki) ) ∏n i=1 (∑⌊nbi⌋ ki=⌈nai⌉ ψ (nxi − ki) ) . our considered neural networks here are of one hidden layer. in this article we find voronovskaya type asymptotic expansions for the above described neural networks quasi-interpolation normalized operators gn (f, x), fn (f, x), where x ∈ (∏n i=1 [ai, bi] ) is fixed but arbitrary. for other neural networks related work, see [2], [3], [4], [5], [6] and [7]. for neural networks in general, see [8], [9] and [10]. next we follow [1], pp. 284-286. about taylor formula -multivariate case and estimates let q be a compact convex subset of rn; n ≥ 2; z := (z1, ..., zn) , x0 := (x01, ..., x0n) ∈ q. let f : q → r be such that all partial derivatives of order (m − 1) are coordinatewise absolutely continuous functions, m ∈ n. also f ∈ cm−1 (q). that is f ∈ acm (q). each mth order partial derivative is denoted by fα := ∂αf ∂xα , where α := (α1, ..., αn), αi ∈ z +, i = 1, ..., n and |α| := ∑n i=1 αi = m. consider gz (t) := f (x0 + t (z − x0)), t ≥ 0. then g(j)z (t) = ⎡ ⎣ ( n∑ i=1 (zi − x0i) ∂ ∂xi )j f ⎤ ⎦(x01 + t (z1 − x01) , ..., x0n + t (zn − x0n)) , (6) for all j = 0, 1, 2, ..., m. example 6. let m = n = 2. then gz (t) = f (x01 + t (z1 − x01) , x02 + t (z2 − x02)) , t ∈ r, and g′z (t) = (z1 − x01) ∂f ∂x1 (x0 + t (z − x0)) + (z2 − x02) ∂f ∂x2 (x0 + t (z − x0)) . (7) 42 george a. anastassiou cubo 16, 2 (2014) setting (∗) = (x01 + t (z1 − x01) , x02 + t (z2 − x02)) = (x0 + t (z − x0)) , we get g′′z (t) = (z1 − x01) 2 ∂f 2 ∂x21 (∗) + (z1 − x01) (z2 − x02) ∂f2 ∂x2∂x1 (∗) + (z1 − x01) (z2 − x02) ∂f2 ∂x1∂x2 (∗) + (z2 − x02) 2 ∂f 2 ∂x22 (∗) . (8) similarly, we have the general case of m, n ∈ n for g (m) z (t) . we mention the following multivariate taylor theorem. theorem 7. under the above assumptions we have f (z1, ..., zn) = gz (1) = m−1∑ j=0 g (j) z (0) j! + rm (z, 0) , (9) where rm (z, 0) := ∫1 0 (∫t1 0 ... (∫tm−1 0 g(m)z (tm) dtm ) ... ) dt1, (10) or rm (z, 0) = 1 (m − 1) ! ∫1 0 (1 − θ) m−1 g(m)z (θ) dθ. (11) notice that gz (0) = f (x0) . we make remark 8. assume here that ∥fα∥ max ∞,q,m := max |α|=m ∥fα∥∞,q < ∞. then ∥ ∥ ∥ g(m)z ∥ ∥ ∥ ∞,[0,1] = ∥ ∥ ∥ ∥ ∥ [( n∑ i=1 (zi − x0i) ∂ ∂xi )m f ] (x0 + t (z − x0)) ∥ ∥ ∥ ∥ ∥ ∞,[0,1] ≤ (12) ( n∑ i=1 |zi − x0i| )m ∥fα∥ max ∞,q,m , that is ∥ ∥ ∥ g(m)z ∥ ∥ ∥ ∞,[0,1] ≤ ( ∥z − x0∥l1 )m ∥fα∥ max ∞,q,m < ∞. (13) hence we get by (11) that |rm (z, 0)| ≤ ∥ ∥ ∥ g (m) z ∥ ∥ ∥ ∞,[0,1] m! < ∞. (14) cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 43 and it holds |rm (z, 0)| ≤ ( ∥z − x0∥l1 )m m! ∥fα∥ max ∞,q,m , (15) ∀ z, x0 ∈ q. inequality (15) will be an important tool in proving our main results. 2 main results we present our first main result theorem 9. let 0 < β < 1, x ∈ ∏n i=1 [ai, bi], n ∈ n large enough, f ∈ ac m (∏n i=1 [ai, bi] ) , m, n ∈ n. assume further that ∥fα∥ max ∞,m < ∞. then gn (f, x) − f (x) = m−1∑ j=1 ⎛ ⎝ ∑ |α|=j ( fα (x) ∏n i=1 αi! ) gn ( n∏ i=1 (· − xi) αi , x ) ⎞ ⎠ + o ( 1 nβ(m−ε) ) , (16) where 0 < ε ≤ m. if m = 1, the sum in (16) collapses. the last (16) implies that nβ(m−ε) ⎡ ⎣gn (f, x) − f (x) − m−1∑ j=1 ⎛ ⎝ ∑ |α|=j ( fα (x) ∏n i=1 αi! ) gn ( n∏ i=1 (· − xi) αi , x ) ⎞ ⎠ ⎤ ⎦ (17) → 0, as n → ∞, 0 < ε ≤ m. when m = 1, or fα (x) = 0, for |α| = j, j = 1, ..., m − 1, then we derive that nβ(m−ε) [gn (f, x) − f (x)] → 0, as n → ∞, 0 < ε ≤ m. proof. consider gz (t) := f (x0 + t (z − x0)), t ≥ 0; x0, z ∈ ∏n i=1 [ai, bi]. then g(j)z (t) = ⎡ ⎣ ( n∑ i=1 (zi − x0i) ∂ ∂xi )j f ⎤ ⎦(x01 + t (z1 − x01) , ..., x0n + t (zn − x0n)) , (18) for all j = 0, 1, ..., m. by (9) we have the multivariate taylor’s formula f (z1, ..., zn) = gz (1) = m−1∑ j=0 g (j) z (0) j! + 1 (m − 1) ! ∫1 0 (1 − θ) m−1 g(m)z (θ) dθ. (19) 44 george a. anastassiou cubo 16, 2 (2014) notice gz (0) = f (x0). also for j = 0, 1, ..., m − 1, we have g(j)z (0) = ∑ α:=(α1,...,αn), αi∈z +, i=1,...,n, |α|:= ∑ n i=1 αi=j ( j! ∏n i=1 αi! )( n∏ i=1 (zi − x0i) αi ) fα (x0) . (20) furthermore g(m)z (θ) = ∑ α:=(α1,...,αn), αi∈z +, i=1,...,n, |α|:= ∑n i=1 αi=m ( m! ∏n i=1 αi! )( n∏ i=1 (zi − x0i) αi ) fα (x0 + θ (z − x0)) , (21) 0 ≤ θ ≤ 1. so we treat f ∈ acm (∏n i=1 [ai, bi] ) with ∥fα∥ max ∞,m < ∞. thus, by (19) we have for k n , x ∈ (∏n i=1 [ai, bi] ) that f ( k1 n , ..., kn n ) − f (x) = m−1∑ j=1 ∑ α:=(α1,...,αn), αi∈z +, i=1,...,n, |α|:= ∑n i=1 αi=j ( 1 ∏n i=1 αi! )( n∏ i=1 ( ki n − xi )αi ) fα (x) + r, (22) where r := m ∫1 0 (1 − θ) m−1 ∑ α:=(α1,...,αn), αi∈z +, i=1,...,n, |α|:= ∑n i=1 αi=m ( 1 ∏n i=1 αi! ) · ( n∏ i=1 ( ki n − xi )αi ) fα ( x + θ ( k n − x )) dθ. (23) by (15) we obtain |r| ≤ ( ∥ ∥x − k n ∥ ∥ l1 )m m! ∥fα∥ max ∞,m . (24) notice here that ∥ ∥ ∥ ∥ k n − x ∥ ∥ ∥ ∥ ∞ ≤ 1 nβ ⇔ ∣ ∣ ∣ ∣ ki n − xi ∣ ∣ ∣ ∣ ≤ 1 nβ , i = 1, ..., n. (25) so, if ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ we get that ∥ ∥x − k n ∥ ∥ l1 ≤ n nβ , and |r| ≤ nm nmβm! ∥fα∥ max ∞,m . (26) also we see that ∥ ∥ ∥ ∥ x − k n ∥ ∥ ∥ ∥ l1 = n∑ i=1 ∣ ∣ ∣ ∣ xi − ki n ∣ ∣ ∣ ∣ ≤ n∑ i=1 (bi − ai) = ∥b − a∥l1 , cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 45 therefore in general it holds |r| ≤ ( ∥b − a∥l1 )m m! ∥fα∥ max ∞,m . (27) call v (x) := ⌊nb⌋∑ k=⌈na⌉ φ (nx − k) . hence we have un (x) := ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) r v (x) = (28) ∑⌊nb⌋ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ : ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ φ (nx − k) r v (x) + ∑⌊nb⌋ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ : ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ φ (nx − k) r v (x) . consequently we obtain |un (x)| ≤ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∑⌊nb⌋ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ : ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ φ (nx − k) v (x) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ( nm nmβm! ∥fα∥ max ∞,m ) + 1 v (x) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⌊nb⌋∑ ⎧ ⎪⎨ ⎪⎩ k = ⌈na⌉ : ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ φ (nx − k) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ( ∥b − a∥l1 )m m! ∥fα∥ max ∞,m (by (v)’, (vi)’) ≤ nm nmβm! ∥fα∥ max ∞,m + (5.250312578) n (3.1992) e−n (1−β) ( ∥b − a∥l1 )m m! ∥fα∥ max ∞,m . (29) therefore we have found |un (x)| ≤ ∥fα∥ max ∞,m m! { nm nmβ + (5.250312578) n (3.1992) e−n (1−β) ( ∥b − a∥l1 )m } . (30) for large enough n ∈ n we get |un (x)| ≤ ( 2 ∥fα∥ max ∞,m n m m! )( 1 nmβ ) . (31) that is |un (x)| = o ( 1 nmβ ) , (32) 46 george a. anastassiou cubo 16, 2 (2014) and |un (x)| = o (1) . (33) and, letting 0 < ε ≤ m, we derive |un (x)| ( 1 nβ(m−ε) ) ≤ ( 2 ∥fα∥ max ∞,m n m m! ) 1 nβε → 0, (34) as n → ∞. i.e. |un (x)| = o ( 1 nβ(m−ε) ) . (35) by (22) we observe that ∑⌊nb⌋ k=⌈na⌉ f ( k n ) φ (nx − k) v (x) − f (x) = m−1∑ j=1 ⎛ ⎝ ∑ |α|=j ( fα (x) ∏n i=1 αi! ) ⎞ ⎠ (∑⌊nb⌋ k=⌈na⌉ φ (nx − k) (∏n i=1 ( ki n − xi )αi )) v (x) + (36) ∑⌊nb⌋ k=⌈na⌉ φ (nx − k) r v (x) . the last says gn (f, x) − f (x) − m−1∑ j=1 ⎛ ⎝ ∑ |α|=j ( fα (x) ∏n i=1 αi! ) gn ( n∏ i=1 (· − xi) αi , x ) ⎞ ⎠ = un (x) . (37) the proof of the theorem is complete. we present our second main result theorem 10. let 0 < β < 1, x ∈ ∏n i=1 [ai, bi], n ∈ n large enough, f ∈ ac m (∏n i=1 [ai, bi] ) , m, n ∈ n. assume further that ∥fα∥ max ∞,m < ∞. then fn (f, x) − f (x) = m−1∑ j=1 ⎛ ⎝ ∑ |α|=j ( fα (x) ∏n i=1 αi! ) fn ( n∏ i=1 (· − xi) αi , x ) ⎞ ⎠ + o ( 1 nβ(m−ε) ) , (38) where 0 < ε ≤ m. if m = 1, the sum in (38) collapses. the last (38) implies that nβ(m−ε) ⎡ ⎣fn (f, x) − f (x) − m−1∑ j=1 ⎛ ⎝ ∑ |α|=j ( fα (x) ∏n i=1 αi! ) fn ( n∏ i=1 (· − xi) αi , x ) ⎞ ⎠ ⎤ ⎦ (39) cubo 16, 2 (2014) voronovskaya type asymptotic expansions for multivariate . . . 47 → 0, as n → ∞, 0 < ε ≤ m. when m = 1, or fα (x) = 0, for |α| = j, j = 1, ..., m − 1, then we derive that nβ(m−ε) [fn (f, x) − f (x)] → 0, as n → ∞, 0 < ε ≤ m. proof. similar to theorem 9, using the properties of θ (x), see (4), (i)∗-(vii)∗ and (5). received: december 2011. revised: may 2012. references [1] g.a. anastassiou, advanced inequalities, world scientific publ. co., singapore, new jersey, 2011. [2] g.a. anastassiou, inteligent systems: approximation by artificial neural networks, intelligent systems reference library, vol. 19, springer, heidelberg, 2011. [3] g.a. anastassiou, univariate hyperbolic tangent neural network approximation, mathematics and computer modelling, 53(2011), 1111-1132. [4] g.a. anastassiou, multivariate hyperbolic tangent neural network approximation, computers and mathematics 61(2011), 809-821. [5] g.a. anastassiou, multivariate sigmoidal neural network approximation, neural networks 24(2011), 378-386. [6] g.a. anastassiou, univariate sigmoidal neural network approximation, submitted for publication, accepted, j. of computational analysis and applications, 2011. [7] z. chen and f. cao, the approximation operators with sigmoidal functions, computers and mathematics with applications, 58 (2009), 758-765. [8] s. haykin, neural networks: a comprehensive foundation (2 ed.), prentice hall, new york, 1998. [9] w. mcculloch and w. pitts, a logical calculus of the ideas immanent in nervous activity, bulletin of mathematical biophysics, 7 (1943), 115-133. [10] t.m. mitchell, machine learning, wcb-mcgraw-hill, new york, 1997. paper5.dvi cubo a mathematical journal vol.12, no¯ 03, (241–253). october 2010 on the weyl transform with symbol in the gel’fand-shilov space and its dual space yasuyuki oka department of mathematics, sophia university 7-1 kioicho, chiyoda-ku, tokyo 102-8554, japan email: yasuyu-o@hoffman.cc.sophia.ac.jp abstract in this paper, we claim two subjects. one is that the weyl transform with symbol in the gel’fand-shilov space s rr , r ≥ 1 2 , is a trace class operator. the other one is that the weyl transform with symbol in the generalized function (s rr ) ′, r ≥ 1 2 , is a continuous linear transformation from the gel’fand-shilov space s rr to (s r r ) ′. as r > 1, z. lozanovcrvenković and d. perišić have proved in [6] this result. our second claim includes their result. resumen en este artículo afirmamos dos asuntos. el primero es que la transformada de weyl con símbolo en el espacio de gel’fand-shilov s rr , r ≥ 1 2 , es un operador de clase trazo. el segundo asunto es que la transformación de weyl con símbolo en las funciones generalizadas (s rr ) ′, r ≥ 1 2 , es una transformación lineal continua del espacio gel’fand-shilov s rr to (s rr ) ′. como r > 1, z. lozanov-crvenković y d. perišić probaron en [6] este resultado. nuestro resultado incluye su resultado. key words and phrases: weyl transform, gel’fand-shilov space, fourier-wigner transform, trace class operator, schwartz’s kernel theorem. math. subj. class.: 46f05; 46f15; 81r15; 81s40. 242 yasuyuki oka cubo 12, 3 (2010) 1 introduction the subject of this article is to show the properties, as operators, of the weyl transform with the symbol in the gel’fand-shilov space s rr , r ≥ 1/2, and its dual space (s r r ) ′, r ≥ 1/2. the weyl transform was first considered by hermann weyl arising in quantum mechanics in [14] and the properties of the weyl transform as operators have been studied by many mathematicians. see for instance, [1], [6], [9], [11], [12], [13], [15] and others. these investigations are mainly to consider the correspondence between the functional space, in which the symbol belongs, and the operator class, in which the weyl transform belongs. there exist two remarkable results about these considerations: first, that the weyl transform with the symbol in schwartz class is a trace class operator in [13]; and secondly, that the weyl transform with the symbol in (s rr ) ′, r > 1, is a continuous and linear maps from s r r (r d ) to (s rr ) ′(rd ) in [6]. our discussion is principally aimed at slightly developing these two results. they depend on two areas of study: first, the correspondence between the weyl transform with the symbol in s rr , r ≥ 1/2, and the sequence space with some exponential decrease, and secondly, the study of the schwartz’s kernel theorem for (s rr ) ′, r ≥ 1/2. we consider these subjects in detail. the plan of the paper is as follows. in the next section we introduce some properties of the gel’fand-shilov space. in section 3 we treat the weyl transform with symbol in s rr . in section 4 we show the schwartz’s kernel theorem for (s rr ) ′ and the property of the weyl transform with symbol in generalized functions. through this article we always treat the index r ≥ 1/2. 2 the gel’fand-shilov space s rr and its dual (s r r ) ′ first of all, we give some notations. we use a multi-index α ∈ zd+, namely, α = (α1 ···αd ), where αi ∈ z and αi ≥ 0. so, for x ∈ rd , xα = x α1 1 ··· xαd d and ∂αx = ∂ α1 x1 ···∂αdxd , where ∂ α j x j = ( ∂ ∂x j )α j . definition 1 ([4]). let a, b ∈ (0,∞)d . for r = (r1,··· , r d ) and r i ≥ 0, 1 ≤ i ≤ d, s r,b r,a (rd ) = {ϕ ∈ c∞(rd ) | ∀δ ∈ (0,∞)d , ∀ρ ∈ (0,∞)d , ∃cδρ ≥ 0 s.t. |xk∂qx ϕ(x)| ≤ cδρ(a +δ)k(b +ρ)q kkr qqr , ∀k, q ∈ zd+}, where (a +δ)k = (a1 +δ1)k1 ···(ad +δd )kd , (b +ρ)q = (b1 +ρ1)q1 ···(bd +ρd ) qd . cubo 12, 3 (2010) on the weyl transform ... 243 the space s r,b r,a (rd ) is a fréchet space with the semi-norms ‖ϕ‖δρ = sup x,k,q |xk∂qx ϕ(x)| (a +δ)k(b +ρ)q kkr qqr , δi ,ρi = 1, 1 2 , 1 3 ··· . the space s rr (r d ) is given by the inductive limit s r r (r d ) = lim −→ a,b→∞ s r,b r,a (r d ). the gel’fand-shilov space is the subspace of the schwartz class s (rd ). let a ∈ (0,∞)d be a = r e a 1 r . for any a, b ∈ (0,∞)d , we define the space s r,br,a (rd ) by s r,b r,a (r d ) = {ϕ ∈ c∞(rd) | ∀δ,ρ ∈ (0,∞)d , ∃cδρ > 0 s.t. |∂ q x ϕ(x)| ≤ cδρ(b +ρ)q qqr e−aδ|x| 1 r , ∀k, q ∈ zd+}, where aδ = r e(a+δ) 1 r and ‖ϕ‖δρ = sup x,β |∂βx ϕ(x)| (b +ρ)βββr e−aδ|x| 1 r . the gel’fand-shilov spaces s rr (r d) enjoy the following properties [4]: proposition 1. let {ϕ j } be a sequence in s r r (r d ). then we obtain ϕ j −→ 0 as j −→ +∞ in s rr if and only if there are positive constants b and a such that sup x,β |∂βx ϕ j (x)| bβββr e−a|x| 1 r −→ 0 as j −→ +∞. proposition 2. (i) s rr ≡ {0} , 0 < r < 1 2 . (ii) for r1 < r2, s r1 r1 is included in s r2 r2 and s r1 r1 is dense in s r2 r2 . (iii) let ŝ rr be the image of the fourier transform of s r r . then ŝ r r = s rr . remark 1. as r = 1, the gel’fand-shilov space s 1 1 (rd ) is known to be isomorphism to the space of test functions of the fourier-hyperfunctions [7]. we define the hermite functions {hn (x)}n=0,1,2··· on r 1 by hn (x) = (2n n!)− 1 2 π − 1 4 (−1)n e x2 2 ( d dx )n e −x2 . 244 yasuyuki oka cubo 12, 3 (2010) it is known that the set {hn (x)}n=0,1,2,··· is a complete orthonormal system in l 2(r1). that is, for any f in l2(r1), f (x) = ∞ ∑ n=0 an hn (x) in l 2 (r 1 ), where an = ( f , hn) = ˆ r1 f (x)hn(x)dx. this expansion is called the hermite expansions and {an}n=0,1,2··· is called the hermite coefficients. for d-dimensions, the hermite functions on r d is defined by hα(x) = hα1 (x1) ⊗···⊗ hαd (xd ), α ∈ z d + , x ∈ r d . the set {hα(x)}α∈zd+ is also a complete orthonormal system in l2(rd ). proposition 3 ([17]). let φ ∈ s rr (r d ), r ≥ 1 2 . then there exist some constants c > 0 and l ∈ (0,∞)d such that φ = ∞ ∑ |α|=0 ( φ, hα ) hα and | ( φ, hα ) | ≤ c exp(−lα 1 2r ). conversely, if |aα| ≤ c exp(−lα 1 2r ) for some constants c > 0 and l ∈ (0,∞)d , then the series ∞ ∑ |α|=0 aαhα(x) converges to a function in s r r (r d ), where hα(x) is the hermite function. definition 2. we denote by (s rr ) ′(rd) the dual space of the gel’fand-shilov space s rr (r d). 3 the weyl transform with symbol in s rr as quantization from classical mechanics to quantum mechanics, h. weyl introduced the operator w (f) as follows: for any f ∈ s (r2d ), w (f)ϕ(ξ) = ï r2d f(x, y)[π(x, y)ϕ](ξ)dxd y, ϕ ∈ l2(rd ), (3.1) where [π(x, y)ϕ](ξ) = ei(x·ξ+ 1 2 x·y)ϕ(ξ+ y). we call this transform w (f) the weyl transform with symbol f. the weyl transform w (f) is also expressed by the following matrix element: for any ϕ, ψ ∈ l2(rd ), (w (f)ϕ,ψ) = ï r 2d f(x, y)(π(x, y)ϕ,ψ)dxd y = ï r2d f(x, y)v (ϕ,ψ)(x, y)dxd y, where v (ϕ,ψ)(x, y) is the fourier-wigner transform of ϕ and ψ defined by v (ϕ,ψ)(x, y) = (2π)− d 2 ˆ rd e ix·p ϕ( p + y 2 )ψ( p − y 2 )d p. cubo 12, 3 (2010) on the weyl transform ... 245 the fourier-wigner transform has the following property, see for example [5]. to be definite, we shall repeat here the proof. proposition 4. let ϕ,ψ ∈ s rr (r d ), r ≥ 1 2 . then v (ϕ,ψ) ∈ s rr (r 2d ) . p roo f . it follows from proposition 2 (iii) that a partial fourier transform of the first variables is a continuous map from s rr to s r r , so it suffices to show that if ϕ,ψ are in s r r (r d ), then ϕ( p + y 2 )ψ̄( p − y 2 ) is in s rr (r 2d). suppose ϕ,ψ ∈ s rr (r d ). since p α = α ∑ |k|=0 ( α k ) ( p + y 2 )k ( p − y 2 )α−k and y β = β ∑ |l|=0 ( β l ) ( p + y 2 )l (−1)|β−l| ( p − y 2 )β−l , we have that p α y β ∂ γ p∂ δ yϕ( p + y 2 )ψ̄( p − y 2 ) = α,β,γ,δ ∑ k,l,m,n ( α k ) ( β l ) ( γ m )( δ n ) (−1)|β−l| ( p + y 2 )k+l ( p − y 2 )α+β−k−l ×∂mp ∂ n yϕ( p + y 2 )∂ γ−m p ∂ δ−n y ψ̄( p − y 2 ). (3.2) set u = p + y 2 and v = p − y 2 , then (3.2) = α,β,γ,δ ∑ k,l,m,n ( α k ) ( β l ) ( γ m ) ( δ n ) (−1)|β+δ−l−n| ( 1 2 )|δ| u k+l v α+β−k−l ∂ m+n u ∂ γ−m+δ−n v ϕ(u)ψ̄(v). so we obtain that for any α, β γ, δ ∈ zd+, |pα yβ∂γp∂ δ yϕ( p + y 2 )ψ̄( p − y 2 )| ≤ α,β,γ,δ ∑ k,l,m,n ( α k ) ( β l ) ( γ m ) ( δ n ) |uk+l ∂m+nu ϕ(u)||v α+β−k−l ∂ γ−m+δ−n v ψ̄(v)| ≤ c1c2 α,β,γ,δ ∑ k,l,m,n ( α k )( β l ) ( γ m )( δ n ) a k+l 1 a α+β−k−l 2 b m+n 1 b γ−m+δ−n 2 × (k + l)(k+l)r (α+β− k − l)(α+β−k−l)r(m + n)(m+n)r (γ− m +δ− n)(γ−m+δ−n)r (3.3) for suitable constants a1, a2, b1, b2 ∈ (0,∞)d and c1, c2 > 0. thus we have that (3.3) ≤ c3 a α+β 3 b γ+δ 3 (α+β)(α+β)r(γ+δ)(γ+δ)r (3.4) for some constants a3, b3 ∈ (0,∞)d and c3 > 0. since (α+β)(α+β)r ≤ eαr eβrααrββr and (γ+δ)(γ+δ)r ≤ eγr eδrγγrδδr , 246 yasuyuki oka cubo 12, 3 (2010) if we put a4 = a3 er and b4 = b3 er , then we have (3.4) ≤ c3 a α+β 4 b γ+δ 4 α αr β βr γ γr δ δr . hence we obtain that there exist constants a4, b4 ∈ (0,∞)d and c3 > 0 such that |pα yβ∂γp∂ δ yϕ( p + y 2 )ψ̄( p − y 2 )| ≤ c3 a α+β 4 b γ+δ 4 α αr β βr γ γr δ δr for any α, β, γ and δ ∈ zd+. this completes the proof of proposition 4. ä a straightforward computation with (3.1) shows that if f(x, y) ∈ s (r2d ), then we have w (f)ϕ( p) = ˆ r d k ( p, p ′ )ϕ( p ′ )d p ′ , ϕ ∈ l2(rd), (3.5) where the kernel k ( p, p′) = f −1 1 f( p+p′ 2 , p′ − p). here f −1 1 f denotes the inverse fourier transform of f in the first variables. the weyl transform has the following fundamental properties, see for example [15]. proposition 5. (i) if the symbol f is in l1(r2d ), then the weyl transform w (f) is a bounded operator on l2(rd), (ii) let the symbol f be in l2(r2d). then the weyl transform w (f) is the hilbert schmidt operator on l2(rd ). conversely let φ be the hilbert-schmidt operator. then there exists f ∈ l2(rd ) such that φ = w (f). we obtain the following result concerning on the property of the weyl transform w (f) with the symbol f in s rr (r d ), r ≥ 1/2: theorem 1. let w (s rr (r 2d)) be the set of all the weyl transforms with the symbol in the gel’fand-shilov space s rr (r 2d). then w (s rr (r 2d)) = {r ∈ b(l2(rd )) |∃a, a′ ∈ (0,∞)d , ∃c > 0 such that |(rhα, hβ)| ≤ c e−a|α| 1 2r e−a ′|β| 1 2r , ∀α,β ∈ zd+}, where b(l2(rd )) is the set of all bounded operators on l2(rd ) and hα, hβ are the hermite functions. p roo f . let g = {r ∈ b(l2(rd )) |∃a, a′ ∈ (0,∞)d , ∃c > 0 such that |(rhα, hβ)| ≤ c e−a|α| 1 2r e−a ′|β| 1 2r , ∀α,β ∈ zd+}. by (3.5) and proposition 2 (iii), it is apparent that the symbol f ∈ cubo 12, 3 (2010) on the weyl transform ... 247 s r r (r 2d ) if and only if the kernel k ∈ s rr (r 2d ). by proposition 3 and fubini’s theorem, we have that |(w (f)hα, hβ)| ≤ ∣ ∣ ∣ ∣ ( ˆ r d k ( p, p ′ )hα( p ′ )d p ′ , hβ( p) ) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ï r2 d k ( p, p ′ )hα( p ′ )hβ( p)d p ′ d p ∣ ∣ ∣ ∣ = ∣ ∣(k , hα ⊗ hβ) ∣ ∣ ≤ c e−a|α| 1 2r e −a′|β| 1 2r for some constants a, a′ ∈ (0,∞)d and c > 0. therefore w (f) ∈ g . conversely, let r1 ∈ g . then ∞ ∑ |α|=0 ‖r1 hα‖2l2 (rd ) = ∞ ∑ |α|=0 (r1 hα, r1 hα) ≤ ∞ ∑ |α|=0 ∞ ∑ |β|=0 |(r1 hα, hβ)||(hβ, r1 hα)| = ∞ ∑ |α|=0 ∞ ∑ |β|=0 e −2a|α| 1 2r e −2a′|β| 1 2r < +∞. hence r1 is the hilbert-schmidt operator. therefore it follows from proposition 5 that there exists g ∈ l2(r2d ) such that w (g) = r1. then from (3.5) there exists c > 0 and a, a′ ∈ (0,∞)d such that |(r1 hα, hβ)| = |(w (g)hα, hβ)| = |(k , hα ⊗ hβ)| ≤ c e−a|α| 1 2r e −a′|β| 1 2r . by proposition 3, we obtain that k ∈ s rr (r 2d) and so is g. ä corollary 1. if f, g ∈ s rr (r 2d ), then there exists h ∈ s rr (r 2d ) such that w (h) = w (f)w (g) p roo f . let f, g ∈ s rr (r 2d ). then we have that |(w (f)w (g)hα, hβ)| = |(w (g)hα, w (f)∗hβ)| ≤ ∑ γ |(w (g)hα, hγ)||(hγ, w (f)∗hβ)| = ∑ γ |(w (g)hα, hγ)||(w (f)hγ, hβ)| ≤ c e−a|α| 1 2r e −b|β| 1 2r ∑ γ e −(a′+b′)|γ| 1 2r = c′ e−a|α| 1 2r e −b|β| 1 2r 248 yasuyuki oka cubo 12, 3 (2010) for suitable constants a, a′, b, b′ ∈ (0,∞)d and c, c′ > 0. hence we obtain that w (f)w (g) ∈ g . therefore it follows from theorem 1 that h is in s rr (r 2d ) such that w (h) = w (f)w (g). ä remark 2. it is known that w (f)w (g) = w (f ∗ 1 4 g), where (f ∗ 1 4 g)(x, y) = ï r2d f(x −ξ, y −η)g(ξ,η)e i 4 ×2( y·ξ−x·η) dξdη. so if f, g ∈ s rr (r 2d ), then (f ∗ 1 4 g) ∈ s rr (r 2d) from corollary 1. definition 3. we denote by s1 the family of all trace class operators defined as follows: for a ∈ b(l2(rd)), there exists an orthonormal basis {vk} of l2(rd ) such that ∑ k ‖avk‖l2 (rd ) < ∞. proposition 6. let a ∈ b(l2(rd)). if {v j } is an orthonormal basis of l2(rd ) then ∑ j ‖av j‖l2 (rd ) ≤ ∑ j,k | ( av j , vk ) |. p roo f . it is obvious as av j = 0, so it suffices to show as av j 6= 0. let w j be an unit vector for any index j. then we have ∑ j | ( a ∗ w j , v j ) | = ∑ j | ( w j , av j ) | = ∑ j ∣ ∣ ∣ ∣ ∣ ∑ k ( w j, vk )( vk, av j ) ∣ ∣ ∣ ∣ ∣ = ∑ j ∣ ∣ ∣ ∣ ∣ ∑ k ( a ∗ vk, v j ) ( w j, vk ) ∣ ∣ ∣ ∣ ∣ ≤ ∑ j,k | ( a ∗ vk, v j ) | = ∑ j,k | ( av j , vk ) |. (3.6) choose w j = av j ‖av j‖ . the inequality now follows. indeed from (3.6) we have ∑ j | ( w j , av j ) | = ∑ j ∣ ∣ ∣ ∣ ( av j ‖av j‖ , av j ) ∣ ∣ ∣ ∣ = ∑ j 1 ‖av j‖ | ( av j , av j ) |2 = ∑ j ‖av j‖ ≤ ∑ j,k | ( av j , vk ) |. ä we obtain the following result from theorem 1 and proposition 6: corollary 2. the weyl transform w (f) with symbol in s rr (r 2d) is of the trace class s1. remark 3. since the gel’fand-shilov classes are included in the schwartz class, the preceding corollary 2 can be seen also as a consequence of the results of a. voros [13], proving that the weyl transforms with symbol in the schwartz class are trace operators. cubo 12, 3 (2010) on the weyl transform ... 249 4 on the weyl transform with symbol in (s rr ) ′ we first show the schwartz’s kernel theorem for (s rr ) ′, r ≥ 1 2 , and give the property of the weyl transform with the symbol in (s rr ) ′, r ≥ 1 2 , as a corollary of the schwartz’s kernel theorem. s. -y. chung, d. kim and e. g. lee proved the schwartz kernel theorem for (s 1 1 )′ in [2] and z. lozanov-crvenković and d. perišić gave the schwartz kernel theorem for (s rr ) ′ as r > 1 in [6]. our result includes their results. we prove the following schwartz’s kernel theorem for (s rr ) ′, r ≥ 1 2 , along the idea in [2]: theorem 2. let k be a continuous and linear operator from s rr (r d2 x2 ) to ( s r r )′ (r d1 x1 ), r ≥ 1 2 . then there exists k in ( s r r )′ (r d1 x1 ×rd2x2 ), r ≥ 1 2 , such that 〈kψ,ϕ〉 = 〈k ,ϕ⊗ψ〉, where ϕ is in s rr (r d1 x1 ), r ≥ 1 2 , and ψ is in s rr (r d2 x2 ), r ≥ 1 2 . to prove the theorem 2, we begin from some preparations. we define the heat kernel e(x, t) by e(x, t) = ( 1 p 4πt )d e − |x| 2 4t , (x, t) ∈ rd × (0,∞). the heat kernel enjoys the following properties: · e(x, t) ∈ s (rdx ), · ˆ rd e(x, t)dx = 1, and · ( ∂ ∂t −∆ ) e(x, t) = 0 , in rd × (0,∞). moreover we obtain the following estimate on the heat kernel e(x, t): proposition 7 ([16]). for any α ∈ zd+, we have |∂αx e(x, t)| ≤ e(x, t)(α!) 1 2 (2t) −|α| (1 +|x|)α, x ∈ rd , 0 < t ≤ 1 2 . from this estimate, we immediately obtain the following properties: proposition 8. e(x, t) ∈ s rr (r d x ), r ≥ 1 2 . proposition 9. let e(x, t) is in s r,b r,a for any a, b > 0. then for every t > 0 and ε > 0, there is a constant c > 0 such that ‖e(x −·, t)‖δρ ≤ c exp[ε(|x| 1 r + (1/t)1/(2r−1))], x ∈ rd , 0 < t < t. ( r > 1 2 ) 250 yasuyuki oka cubo 12, 3 (2010) in the case where r = 1/2, we have the following inequality: ‖e(x −·, t)‖δρ ≤ cε,t eε|x| 2 , x ∈ rd , t > 0, ( r = 1 2 ) . moreover we need the several propositions, which are the result of c. dong and t. matsuzawa [3], to prove theorem 2 as follows: proposition 10 ([3]). let ϕ(x) ∈ s r,br,a (rd ), r ≥ 1/2. then u(x, t) ≡ ˆ rd e(x − y, t)ϕ( y)d y ∈ s r,br,a (r d ) , t > 0 and u(x, t) → ϕ(x) in ∈ s r,br,a (r d ) as t → 0. proposition 11 ([3]). if every c∞-function u(x, t) defined in rd+1+ = {(x, t) | x ∈ r d , t > 0} satisfies the conditions: ( ∂ ∂t −∆ ) u(x, t) = 0 , in rd+1+ , and for every t > 0 and ε > 0, there is a constant c > 0 such that |u(x, t)| ≤ c exp[ε(|x| 1 r + (1/t)1/(2r−1))], x ∈ rd , 0 < t < t. ( r > 1 2 ) in the case where r = 1/2, u(x, t) has the following inequality: |u(x, t)| ≤ cε,t eε|x| 2 , x ∈ rd , t > 0. ( r = 1 2 ) then u(x, t) can be expressed in the form u(x, t) = 〈u y , e(x − y, t)〉 with unique element u ∈ ( s r r )′ (r d ). p roo f o f t heorem 2. we show the proof of theorem 2 as r > 1 2 . since k is continuous, the bilinear form b on s r,b r,a (r d1 ) × s r,b ′ r,a′ (rd2 ), for any a, b ∈ (0,∞)d1 and a′, b′ ∈ (0,∞)d2 , b(ϕ,ψ) = 〈kψ,ϕ〉 , ϕ ∈ s r,br,a (rd1 ) , ψ ∈ s r,b′ r,a′ (rd2 ) is separately continuous. since s r,b r,a (r d1 ) and s r,b′ r,a′ (rd2 ) is fréchet space, b is continuous. hence we obtain that there exists a constant ca,a′,b,b′ > 0 such that |〈kψ,ϕ〉| ≤ ca,a′,b,b′ ‖ϕ‖δρ‖ψ‖δ′ρ′ (♯). set for (x1, x2) ∈ rd1 ×rd2 and t > 0, k t(x1, x2) = 〈ke(x2 −·, t), e(x1 −·, t)〉 . cubo 12, 3 (2010) on the weyl transform ... 251 now we show k t converges in ( s r r )′ (r d1 ×rd2 ) as t → 0. by (♯) and proposition 9, for any ε, ε′ > 0, there exists a constant cε,ε′ > 0 such that |k t(x1, x2)| ≤ cε,ε′ exp[ε(|x1| 1 r + (1/t)1/(2r−1))] exp[ε′(|x2| 1 r + (1/t)1/(2r−1))]. moreover we obtain ( ∂ ∂t −∆ ) k t(x1, x2) = 0 . therefore, by proposition 11, there exists k0 ∈ ( s r r )′ (r d1 ×rd2 ) such that k0 = lim t→0 k t in ( s r r )′ (r d1 ×rd2 ). for ϕ ∈ s rr (r d x1 ), ψ ∈ s rr (r d x2 ), 〈k t , ϕ⊗ψ〉 = ï r d1 +d2 k t(x1, x2)ϕ(x1)ψ(x2)dx1 dx2 = ï r d1 +d2 〈ke(x2 − y2, t)ψ(x2), e(x1 − y1, t)ϕ(x1)〉 dx1 dx2. since the riemann sum of an integral converges in s rr , we obtain 〈k t , ϕ⊗ψ〉 = 〈k ˆ r d2 e(x2 − y2, t)ψ(x2) , ˆ r d1 e(x1 − y1, t)ϕ(x1)〉. therefore, by proposition 10, we obtain 〈k0,ϕ⊗ψ〉 = 〈kψ,ϕ〉 , as t → 0. ä similarly, we can also show the proof of theorem 2 as r = 1 2 . remark 4. z. lozanov-crvenković and d. perišić also proved the schwartz kernel theorem for the spaces of tempered ultradistributions in [6] by means of the hermite expansions. we define the weyl transform with symbol t ∈ (s rr ) ′ by 〈w (t)ϕ,ψ〉 = 〈t, v (ϕ,ψ)〉, ϕ,ψ ∈ s rr (r d ), where v (ϕ,ψ̄) is the fourier-wigner transform of ϕ and ψ̄. it follows from proposition 4 that this definition is well defined. m. cappiello, t. gramchev and l. rodino also showed this subject in [1]. we obtain the following result from theorem 2. corollary 3. the map w from s (r2d) to the space of bounded operators on l2(rd ), defined by w (f)ϕ(ξ) = ï r2d f(x, y)[π(x, y)ϕ](ξ)dxd y , ϕ ∈ l2(rd), extends uniquely to a bijection from (s rr ) ′(r2d), r ≥ 1/2, to the space of continuous linear maps from s rr (r d ), r ≥ 1/2, to (s rr ) ′(rd), r ≥ 1/2. 252 yasuyuki oka cubo 12, 3 (2010) p roo f . let k be a continuous linear map from s rr (r d ) to (s rr ) ′(rd). by theorem 2, for any k, there exists k ∈ (s rr ) ′(r2d) such that 〈kϕ,ψ〉 = 〈k ,ϕ⊗ψ〉 , ϕ,ψ ∈ s rr (r d ). so we have 〈kϕ,ψ〉 = 〈k ,ϕ⊗ψ〉 = 〈f1sk , v (ϕ,ψ̄)〉, (4.1) where f1 is the fourier transform of the first variable and s is defined by sh(a, b) = h(a + b2 , a − b 2 ). set t = f1sk , (4.1) = 〈t, v (ϕ,ψ)〉 = 〈w (t)ϕ,ψ〉. since f1sk ∈ (s rr ) ′(r2d ), for any k, there exists t ∈ (s rr ) ′(r2d) such that k = w (t). ä remark 5. z. lozanov-crvenković and d. perišić gave the similar result for (s rr ) ′ as r > 1 in [6]. references [1] cappiello, m., gramchev, m.t. and rodino, l., gelfand-shilov spaces, pseudodifferential operators and localization operators, in modern trends in pseudodifferential operators, editors: toft, j., wong, m.w. and zhu, h., birkhäuser, 297–312. 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[9] pool, j.c.t., mathematical aspects of the weyl correspondence, j. math. phys. vol. 7, n.1, january (1966), 66–76. [10] simon, b., distributions and their hermite expansions, j. math. phys. vol. 12, n.1 (1971), 140–148. [11] simon, b., the weyl transform and l p functions on phase space, proc. amer. math. soc., 116 (1992), 1045–1047. [12] toft, j., continuity properties for modulation spaces with applications to pseudodifferential calculus, i, j. funct. anal., 207 (2), (2004), 399–429. [13] voros, a., an algebra of pseudodifferential operators and the asymptotics of quantum mechanics, j. funct. anal., 29 (1978), 104–132. [14] weyl, h., the theory of groups and quantum mechanics, dover, new york, 1950. [15] wong, m.w., weyl transforms, springer-verlag, new york, inc., 1998. [16] yoshino, k. and oka, y., asymptotic expansions of the solutions to the heat equations with hyperfunctions initial value, commun. korean math. soc., 23 (2008), n.4, 555–565. [17] zhang, g.-z., theory of distributions of s type and pansions, chinese math. acta., 4 (1963), 211–221. sawanoman2.dvi cubo a mathematical journal vol.12, no¯ 03, (187–202). october 2010 modulation spaces with aloc∞ -weights yoshihiro sawano department of mathematics, kyoto university, kyoto, 606-8502, japan, email: yosihiro@math.kyoto-u.ac.jp abstract in this paper we describe the function space m s,w p,q with w ∈ a loc ∞ together with some related results of weighted modulation spaces. resumen en este artículo describimos el espacio de la funciones m s,w p,q con w ∈ a loc ∞ junto con algunos resultados relacionados a espacios de modulación con peso. key words and phrases: modulation spaces, exponential weights. math. subj. class.: 41a17, 42b35. 188 yoshihiro sawano cubo 12, 3 (2010) 1 modulation spaces modulation spaces, which were initiated by feichtinger in 1983 (see [5]), were investigated for the purpose of measuring smoothness of functions and distributions in a way other than besov spaces. besov spaces as well as triebel-lizorkin spaces are very close to sobolev spaces and are used in partial differential equations. these spaces are defined by way of dilations. feichtinger took full advantage of the group structure of rn. recall that rn carries the structure of a lie group not with dilation but with addition. therefore, it seems natural that we consider the translation. the goal of the present paper is to combine the results in [17, 21]. the main results of [21] can be summarized as follows : quite a few of the results of usual modulation spaces m sp,q carries over to the a loc ∞ -weighted cases with 0 < p, q ≤ ∞. in the present paper we shall establish the following results on modulation spaces. to describe the result, we make a setup. assume that w : rn → (0,∞) is a measurable function with aloc∞ condition: there exists 1 < p < ∞ such that w satisfies the aloc p condition sup q:cube ( 1 |q| ˆ q w (x) dx ) ( 1 |q| ˆ q w (x)− 1 p−1 dx ) 1 p−1 < ∞. (1.1) suppose that the parameters p, q, s satisfy 0 < p < ∞, 0 < q < ∞, s ∈ r. (1.2) fix a window function ϕ ∈ c∞c so that it satisfies the non-degenerate condition: ˆ rn ϕ(x) dx 6= 0, supp(ϕ) ⊂ [−1, 1]n. (1.3) we write ϕm,x(z) = exp(2πim · z)ϕ(z − x) for m ∈ zn and x ∈ rn. we define ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ g > ( ∑ m∈zn 〈m〉q ( ˆ r n |〈f ,ϕm,x〉|pw (x) dx ) q p ) 1 q (1.4) for f ∈ c∞c , where we write 〈x〉 = √ 1 +|x|2. theorem 1. assume (1.1) and (1.2). then different choices of admissible ϕ satisfying (1.3) will yield equivalent norms. that is, if ϕ1,ϕ2 satisfy (1.3), then the norm equivalence ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ ϕ1 ≃ ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ ϕ2 (1.5) holds for f ∈ c∞c (r n). cubo 12, 3 (2010) modulation spaces with aloc∞ -weights 189 in view of (1.5), we shall write ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ instead of ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ g . as for this (new) modulation norm ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥, we have the following quantitiave information. lemma 1. there exist c > 0 and n ∈ n depending only on w and p, q, s such that |〈f ,ψ〉| ≤ c ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ ∑ |α|≤n sup x∈rn e n|x||∂αψ(x)| (1.6) holds for all ψ ∈ c∞c . denote by m s,wp,q the (abstract) completion of c ∞ c with ∥ ∥ ∥ f : m s,w p,q ∥ ∥ ∥ g . in view of (1.6), we see that m s,wp,q is a subset of d ′ satisfying |〈f ,ϕ〉| ≤ c ∑ |α|≤n sup x∈rn e n|x||∂αϕ(x)| (1.7) holds for all ϕ ∈ c∞c . in the present paper we shall prove the molecular decomposition suitable for m s,wp,q . definition 1 (molecule, atom). let s ∈ r. 1. suppose that k , n ∈ n are large enough and fixed. a ck -function τ : rn → c is said to be an (s; m, l)-molecule, if it satisfies |∂α(e−im·xτ(x))| ≤ 〈m〉−s e−n|x−l|, x ∈ rn for |α| ≤ k . 2. suppose that k , n ∈ n are large enough and fixed. a ck -function τ : rn → c is said to be an (s; m, l)-atom, if it satisfies |∂α(e−im·xτ(x))| ≤ 〈m〉−sχl+[−2,2]n , x ∈ rn for |α| ≤ k . 3. also set m s := {{ψs ml }m,l∈zn : each ψ s ml is an (s; m, l)-molecule} a s := {{as ml }m,l∈zn : each a s ml is an (s; m, l)-atom}. next, we introduce a sequence space m p,q to describe the condition of the coefficients of the molecular decomposition. 190 yoshihiro sawano cubo 12, 3 (2010) definition 2 (sequence space m p,q ). let 0 < p, q ≤ ∞. given a sequence λ = {λml }m,l∈zn , define ‖λ : mwp,q‖ >   ∑ m∈zn { ˆ rn ∣ ∣ ∣ ∣ ∣ ∑ l∈zn λmlχl+[0,1]n (x) ∣ ∣ ∣ ∣ ∣ p w (x) dx } q p   1 q . here a natural modification is made when p and/or q is infinite. mwp,q is the set of doubly indexed sequences λ = {λml }m,l∈zn for which the quasi-norm ‖λ : mwp,q‖ is finite. with these definitions in mind, we present a typical result in [21]. theorem 2. assume (1.1) and (1.2). 1. if λ = {λml }m,l∈zn ∈ m s,w p,q and {ψ s ml }m,l∈zn ∈ m s, then f := ∑ m,l∈zn λml ·ψsml (1.8) converges unconditionally in the topology of m s,w p,q . 2. there exists {as ml }m,l∈zn ∈ a s such that any f ∈ m s,w p,q admits the following decomposition: f = ∑ m,l∈zn λml · asml , (1.9) where λ = {λml }m,l∈zn satisfies ‖λ : ms,wp,q ‖ ≤ c ‖f : m s,w p,q ‖ (1.10) for some c > 0 independent of f . in the early 90’s, more and more applications were found out. for example, time-frequency analysis, which is a branch of signal analysis, deals with the translation and the modulation, so that modulation spaces come into play naturally. also, it is remarkable that modulation spaces are applied effectively to the pseudodifferential operators by sjöstrand, tachizawa and many researchers [12, 14, 15, 19, 22, 23, 24, 25]. modulation spaces are applicable to various partial differential equations. for example, baoxiang and chunyan used modulation spaces to investigate the kdv equation (see [3]). recently modulation spaces can be applied even to the modeling of wireless channels and the quantum mechanics [2]. now we describe the organization of this paper. in section 2 we describe other weighted modulation spaces and compare them with ours. section 3 is devoted to establishing theorem 1 as well as lemma 1. section 4 is intended as the proof of theorem 2. in section 5 we consider the weighted modulation space m s,wp∞ . finally in section 6 we present some examples. cubo 12, 3 (2010) modulation spaces with aloc∞ -weights 191 2 various weighted modulation spaces based on the standard notation of signal analysis, we adopt the following notations. ta f (x) := f (x − a), mb f (x) := e ib·x f (x), a, b ∈ rn, f ∈ s ′. fix ϕ ∈ c∞c be a positinve non-zero function. then define ‖f : m sp,q‖ > ( ˆ rn ( ˆ rn |〈f , m y txϕ〉|p dx ) q p 〈y〉s q d y ) 1 q for s ∈ r and 1 ≤ p, q ≤ ∞. denote by m sp,q the set of all tempered distributions f ∈ s ′ for which the norm is finite. an important observation is that the function space m sp,q does not depend on the specific choices of g. for more details we refer to [11, 18]. in general by weighted modulation norm we mean the following norm given by ‖f : mvp,q‖ > ( ˆ rn (ˆ rn |〈f , m y txϕ〉|p v(x, y) dx ) q p d y ) 1 q . note that m sp,q is recovered by setting v(x, y) = 〈y〉 s q. there are many important classes of weights. definition 3. 1. a weight v : r2n → [0,∞) is said to be a submultiplicative, if there exists a constant c > 0 such that v(x + y) ≤ c v(x) v( y) for all x, y ∈ r2n. 2. a weight v : r2n → [0,∞) is said to be subconvolutive, if v−1 ∈ l1(r2n) and v−1∗v−1 ≤ c v−1 for some constant c > 0. 3. a weight v : r2n → [0,∞) is said to satisfy the gelfand-raikov-shilov condition, if lim n→∞ v(n x) 1 n = 1 for all x 6= 0. 4. a weight v : r2n → [0,∞) is said to satisfy the beurling-domar condition, if ∞ ∑ j=1 log v(n x) n < ∞. 5. a weight v : r2n → [0,∞) is said to satisfy the logarithmic integral condition, if ˆ |x|≥1 log v(x) |x|n+1 dx < ∞. 192 yoshihiro sawano cubo 12, 3 (2010) example 1. 1. the function eα|x| with α ≥ 0 is a submultiplicative weight. similarly 〈x〉α with α ≥ 0 is a submultiplicative weight. 2. the function 〈x〉n+ε is a subconvolutive weight. we refer to [7] for more details of the submultiplicative, moderate and subconvolutive weights not only on rn but also on locally compact aberian groups. proposition 1. [13] the bourling-domar condition is stronger than the gelfand-raikovshilov condition. proof. this is just an easy consequence of the fact that the limit of a positive summable sequence is zero. in the present paper, we consider weights of the form v(x, y) = w (x)〈y〉s, where s ∈ r and w belongs to the class aloc∞ described just below. as the example w (x) = |x|α, α > −n shows, it can happen that v fails the submultiplicative condition or subconvolutive condition. another similar example shows that v does not necessarily satisfy the bouringdomar condition. before we go further, we recall the definition of alocp -weights. in the sequel by a “weight", we mean a non-negative measurable function w ∈ l1 l oc satisfying 0 < w < ∞ for a.e. and we define the local maximal operator mloc by m loc f (x) := sup x∈q q : cube,|q|≤1 1 |q| ˆ q |f ( y)| d y. let 1 ≤ p < ∞. then we define a loc p (w )=            ess. sup x∈rn mlocw (x) w (x) if p = 1 sup q : cube |q|≤1 (ˆ q w (x) dx |q| ) · (ˆ q w (x) 1 1−p dx |q| )p−1 if 1 < p < ∞. the quantity alocp (w ) is called the a loc p -norm of w . then it is easy to see that a loc p (w ) ≤ a loc q (w ), 1 ≤ q ≤ p < ∞. cubo 12, 3 (2010) modulation spaces with aloc∞ -weights 193 the class alocp of weights is the set of all weights w for which the norm a loc p (w ) is finite. we also define a loc ∞ := ⋃ 1≤p<∞ a loc p . we remark that |x|−n+ε ∈ aloc1 for all 0 < ε < n and that e α|x| ∈ aloc1 for all α ≥ 0. let w be a weight. then we define ‖f : lwp ‖ > (ˆ rn |f (x)|pw (x) dx ) 1 p , 1 ≤ p < ∞. here and below we assume that w ∈ aloc p with 1 ≤ p < ∞ for the sake of definiteness. 3 proof of theorem 1 now we prove theorem 1 and lemma 1. before we prove theorem 1, we first establish an auxiliary result (proposition 2) and then we prove theorem 1. proposition 2 will be strengthened after we prove lemma 1. 3.1 an auxiliary result on maximal operators we write pn (ψ) > ∑ α∈zn+,|α|≤n sup x∈rn e n|x||∂αψ(x)| for ψ ∈ c∞c . proposition 2. let k ∈ z, n > 0 and 0 < η ≤ 1. then we have sup ψ∈c∞c pn (ψ)≤1 |mkψ∗ f (x)| η ≤ c ∑ l∈z ˆ rn |mlϕ∗ f (x − y)|η 〈k − l〉nη enη|y| d y (3.11) for all f ∈ c∞c . proof. first let us consider the case when η = 1. note that ∑ l∈zn f ϕ(x + l)2 = (2π)− n 2 ∑ l∈zn f [ϕ∗ϕ](x + l) = (2π)− n 2 ∑ m∈zn ( ˆ rn ∑ l∈zn f [ϕ∗ϕ]( y + l) exp(−2πi y · m) d y ) exp(2πix · m) 194 yoshihiro sawano cubo 12, 3 (2010) > ∑ m∈zn ϕ∗ϕ(−2πm) exp(2πix · m) ≡ ϕ∗ϕ(0) from the poisson summation formula. consequently we obtain mkψ∗ f = cn ∑ l∈z mkψ∗ mlϕ∗ mlϕ∗ f . (3.12) now we shall estimate each summand. first of all, a repeated integration by parts yields that for all n > 0 there exists c = cn > 0 such that |mkψ∗ mlϕ( y)| ≤ c〈k − l〉−n e−n|y|. as a consequence we obtain |mkψ∗ mlϕ∗ mlϕ∗ f (x)| ≤ c〈k − l〉−n ˆ rn e −n|y||ml ϕ∗ f (x − y)| d y. inserting (3.12), we obtain the result when η = 1. namely we have proved |mkψ∗ f (x)| ≤ c ∑ l∈z 〈k − l〉−n ˆ rn e −n|y||ml ϕ∗ f (x − y)| d y (3.13) up to this point. of course, the constant c > 0 does depend on n. now we pass to the case when 0 < η < 1. we define mn,k f (x) := sup ψ∈c∞c , pn (ψ)≤1 y∈r, l∈z |mlψ∗ f (x − y)| 〈k − l〉n en|y| . then from (3.13) we deduce mn,k f (x) ≤ c sup y∈r l∈z ( 1 〈k − l〉n en|y| ∑ m∈z ˆ |mmϕ∗ f (x − y − z)| 〈m − l〉n en|z| d y ) ≤ c sup y∈r ( ∑ m∈z ˆ |mmϕ∗ f (x − y − z)| 〈m − k〉n en|y+z| d z ) ≤ c mn,k f (x)1−η ∑ m∈z ˆ |mmϕ∗ f (x − y)|η 〈m − k〉nη enη|y| d y. here we have used the peetre inequality 〈a + b〉 ≤ p 2〈a〉·〈b〉. as a result, we obtain |mkψ∗ f (x)|η ≤ mn,k f (x)η ≤ c ∑ m∈z ˆ |mmϕ∗ f (x − y)|η 〈m − k〉nη enη|y| d y, since mn,k f (x) < ∞. cubo 12, 3 (2010) modulation spaces with aloc∞ -weights 195 proposition 3. let w ∈ aloc p and f : rn → [0,∞) a measurable function. then we have { ˆ r n ( ˆ r n f(x − y)η d y ebη|y| ) p η w (x) dx } 1 p ≤ c ( ˆ r n f(x)pw (x) dx ) 1 p (3.14) for all p > pη and b ≫ 1. proof. by replacing p/η with p, we can assume that η = 1 and p > p. let ℓ ≥ 1. we denote χr = χ(−r,r)n rn . then define mloc≤ℓ f (x) > sup r≤ℓ χr ∗|f |(x). then there exists α > 0 such that ( ˆ r n m loc ≤ℓ f (x) p dx ) 1 p ≤ eαℓ ( ˆ r n |f (x)|p dx ) 1 p . (3.15) indeed, this inequality is true for ℓ = 1 by the definition of aloc p . since χr ∗χ1 ≥ χr+1 for r ≥ 1, we have m loc ≤k ≤ (m loc ≤1 ) k. as a consequece, we obtain (3.15). once we establish (3.15), (3.14) is an easy consequence of inequality ˆ r n f(x − y)e−b|y| d y ≤ ∞ ∑ j=1 ˆ (−2 j ,2 j )n f(x − y)e−2 j−1b d y ≤ 2n ∞ ∑ j=1 e −2 j−1b m loc ≤2 j f(x). the proof is therefore complete. 3.2 proof of theorem 1 let w ∈ aloc∞ throughout. then define ‖f m : l q (lwp )‖ > ( ∑ m∈zn ‖f m : lwp ‖ q ) 1 q for a family of measurable functions { f m}m∈zn . let 0 < p, q ≤ ∞ and s ∈ r. then the modulation norm (1.4) can be written as ‖f : m s,wp,q ‖ > ( ∑ m∈zn 〈m〉qs‖ mmϕ∗ f : lwp ‖ q ) 1 q . (3.16) we are now in the position of establishing theorem 1. 196 yoshihiro sawano cubo 12, 3 (2010) by proposition 2 we have |mkϕ2 ∗ f (x)|η ≤ c ∑ l∈z ˆ rn |mlϕ1 ∗ f (x − y)|η 〈k − l〉nη ebη|y| d y. if we invoke proposition 3, we obtain ‖mkϕ2 ∗ f ‖lwp ≤ c ∑ l∈z 1 〈k − l〉nη ‖mlϕ1 ∗ f ‖lwp if η < p/ p, n ≫ 1. hence it follows that ∑ k∈zn ( 〈k〉s‖mkϕ2 ∗ f ‖lwp )q ≤ c ∑ k∈zn ( ∑ l∈z 〈k〉s 〈k − l〉nη ‖ml ϕ1 ∗ f ‖lwp )q ≤ c ∑ l∈zn ( 〈l〉s‖mlϕ2 ∗ f ‖lwp )q , which implies ‖f : m s,wp,q ‖ϕ2 ≤ c‖f : m s,w p,q ‖ϕ1 . by symmetry theorem 1 was proved completely. 3.3 proof of lemma 1 instead of dealing with 〈f ,ψ〉 directly, we have only to deal with ψ∗ f (0), which is justified by the isomorphism ψ 7→ ψ(−·). proposition 3 and a normalization yield |ψ∗ f (0)|η ≤ c pn (ψ)η ∑ l∈z ˆ rn |ml ϕ∗ f ( y)|η 〈l〉nη enη|y| d y with 0 < η ≪ min( p, p, 1) 2 . ˆ rn |ml ϕ∗ f ( y)|η enη|y| d y > ˆ rn |ml ϕ∗ f ( y)|ηw ( y)η/p enη|y|w ( y)η/p d y ≤ (‖mlϕ∗ f ‖lwp ) η · ( ˆ rn ( w ( y)−η/p enη|y| )−p/( p−η) d y ) p−η η . since w− 1 p−1 ∈ aloc∞ , we see that w η/(p−η) ∈ aloc∞ . hence, if we choose s ≫ 1, then we obtain ˆ rn (e−nη|y|w ( y)−η/p)−p/( p−η) d y ≤ ∞ ∑ j=1 ˆ [−2 j ,2 j ] e −2 j−1 n p/( p−η) w ( y) η/(p−η) d y ≤ cs ∞ ∑ j=1 2 jn e−2 j−1 n p/( p−η) m≤2 j [χ1]( y) s w ( y) η/(p−η) d y cubo 12, 3 (2010) modulation spaces with aloc∞ -weights 197 < ∞. as a consequence, lemma 1 was proved. we define se as the set of all c ∞-functions f for which the norm pn (ψ) > ∑ α∈zn+,|α|≤n sup x∈rn exp(n|x|)|∂αψ(x)| is finite. s ′e is defined as the topological dual of se . we remark that s ′ e is a special case of gelfand-shilov spaces (see [16]). proposition 4. proposition 3 remains vaild for f ∈ s ′e . proof. keep to the same notation as proposition 3. the proof does not undergo any major change until the end of the proof of proposition 3. if mn,k f (x) were finite, then we would obtain |mkϕ∗ f (x)| η ≤ mn,k f (x)η ≤ c ∑ m∈z ˆ |mmγ∗ f (x − y)|η 〈m − k〉nη enη|y| d y. (3.17) however, this does not always work because mn,k f (x) can be infinite. we shall show that (3.17) still holds for all f ∈ s ′e (r) even when mn,k f (x) = ∞. for this purpose let us assume the most right-hand side (3.17) is finite. otherwise there is nothing to prove. assuming that the most right-hand side (3.17) is finite, we shall establish mn,k f (x) < ∞. since f ∈ s ′e (r), there exist n f > 0 such that mn,k f (x) < ∞ for all n ≥ n f . as a consequence (3.17) holds for such n and n. from this we deduce |mkϕ∗ f (x)| η ≤ c ∑ m∈z ˆ |mmγ∗ f (x − y)|η 〈m − k〉n f η en f η|y| d y. (3.18) the constant in (3.17) being dependent implicitly on n, c in (3.17) must be dependent on f . to emphasize this dependence, let us write this constant as c f . then we have |mkϕ∗ f (x)| η ≤ c f ∑ m∈z ˆ |mmγ∗ f (x − y)|η 〈m − k〉n f η en f η|y| d y ≤ c f ∑ m∈z 1 〈m − k〉nη ˆ |mmγ∗ f (x − y)|η enη|y| d y for all n with n ≤ n f . as a consequence for all n > 0, there exists c f such that |mkϕ∗ f (x)|η ≤ c f ∑ m∈z ˆ |mmγ∗ f (x − y)|η 〈m − k〉nη enη|y| d y. from the definition of the maximal operator mn,k f (x), we have mn,k f (x) ≤ c f sup y∈r ( ∑ m∈z ˆ |mmγ∗ f (x − y − z)|η 〈k − l〉nη〈m − l〉nη enη(|y|+|z|) d z ) 198 yoshihiro sawano cubo 12, 3 (2010) ≤ c f ∑ m∈z ˆ |mmγ∗ f (x − z)|η 〈k − m〉nη enη|z| d z < ∞. as a consequence (3.17) holds for all f ∈ s ′e (r). 4 proof of theorem 2 a fundamental technique in harmonic analysis is to represent functions or distributions as a linear combination of functions of an elementary form. we shall investigate the structure of weighted modulation spaces. we refer to [1, 4, 6, 8, 9, 10, 15, 20] for the definition of the molecules and atoms for different modulation spaces. now we prove theorem 2. 1. let n ∈ n be fixed. an integration by parts yields 〈m〉s ∣ ∣ ∣ ∣ ∣ ∑ l,m∈zn λml mkϕ∗ψsml(x) ∣ ∣ ∣ ∣ ∣ ≤ c ∑ l,m∈zn |λml| 〈k − m〉n exp(−n|x − l|) ≤ c ∞ ∑ j=1 ∑ l∈zn e−n j 〈k − m〉n m loc ≤ j ( ∑ m∈zn λmlχqm ) for some constant c depending only on n. as a result, we obtain the desired result by virtue of (3.15). 2. note that mm ∗ ϕ ∗ mmϕ ∗ ψ = cψ for all ψ ∈ se , since we have seen that ∑ m∈zn f ϕ(ξ + m)2 =: i 6= 0. we set aml (x) := 1 i ˆ l+[0,1]n mmϕ( y)mmϕ∗ f (x − y) d y. then we have f = ∑ l,m∈zn aml in s ′ e . since m−m aml (x) = 1 i ˆ l+[0,1]n mmϕ( y)〈f , exp(−im · ( y +∗))ϕ(x − y −∗)〉 d y, we have mm[∂ α(m−m aml )](x) = 1 i ˆ l+[0,1]n mmϕ( y)mm[∂ α ϕ]∗ f (x− y) d y. therefore, if we define λml > sup x∈l+[−2,2]n sup |α|≤m |∂α(m−m aml )(x)|, cubo 12, 3 (2010) modulation spaces with aloc∞ -weights 199 then, by proposition 3, we have ‖{λml }m,l∈zn : mwp,q‖ ≤ c‖f : m s,w p,q ‖. hence, it follows that f = ∑ m,l∈zn λml · aml λml is an atomic decomposition of f . this is the desired result. 5 weighted modulation space m s,w p,∞ a minor modification of the results above will yield a theory of the function space m s,w p,∞. we define the function space m s,wp,∞ as follows : definition 4. let 0 { ∑ l∈zn 〈m〉qs (ˆ rn |mmϕ∗ f (x)|pw (x) dx ) q p } 1 q for f ∈ s ′e . lemma 2. let 0 0, 0 < q < ∞, qε > n. then we have m s,w p,∞ ,→ m s−ε,w p,q . proof. this follows from a fundamental inequality ( ∑ m∈zn 〈m〉−qε|am|q ) 1 q ≤ sup m∈zn |am| ( ∑ m∈zn 〈m〉−qε ) 1 q which holds for all complex sequences {am}m∈zn . the atomic decomposition theorem can be formulated as follows: theorem 3. let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ r. assume that w ∈ aloc∞ . 1. the function space m s,w p,q does not depend on the choice of specific ϕ satisfying (1.3). 2. if λ = {λml }m,l∈zn ∈ m s,w p,q and {ψ s ml }m,l∈zn ∈ m s0 , then f := ∑ m,l∈zn λml ·ψ s ml (5.19) converges unconditionally in the topology of s ′e . 200 yoshihiro sawano cubo 12, 3 (2010) 3. there exists {as ml }m,l∈zn ∈ a s such that any f ∈ m s,w p,q admits the following decomposition: f = ∑ m,l∈zn λml · asml , (5.20) where λ = {λml }m,l∈zn satisfies ‖λ : ms,wp,q ‖ ≤ c ‖f : m s,w p,q ‖ (5.21) for some c > 0 independent of f . proof. almost all the proofs remains unchanged except for the convergence in (5.19). this will be established by lemma 2. 6 examples here we shall present some examples of weights. example 2. a weight wa(ξ) = exp(a|ξ|), a ∈ r belongs to the class of our admissible weights. it is interesting that m s,wa p,q is much larger than m s p,q = m s,w0 p,q for a < 0. example 3. if we define w (x) = (1 +|x|2) a 2 , then m s,w 2,2 is the weighted sobolev space. proposition 5. let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ r. if we define w (x) = (1 + |x|2) a 2 , then m s,w p,q ⊂ s ′. proof. in analogy with proposition 2, we can prove sup ψ∈c∞c qn (ψ)≤1 |mkψ∗ f (x)|η ≤ c ∑ l∈z ˆ rn |mlϕ∗ f (x − y)|η 〈k − l〉nη〈y〉nη d y (6.22) for all f ∈ c∞c , where qn (ψ) = ∑ |α|≤n sup x∈rn 〈x〉n|∂αψ(x)|. therefore, we can proceed as in the proof of lemma 1. references [1] balan, r., casazza, p.g., heil, c. and landau, z., density, overcompleteness, and localization of frames, ii. gabor systems, j. fourier anal. appl., 12 (3), 309–344, 2006. 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[25] tachizawa, k., the boundedness of pseudodifferential operators on modulation spaces, math. nachr., 168, 263–277, 1994. cubo a mathematical journal vol.15, no¯ 01, (49–75). march 2013 existence and stability in the α-norm for nonlinear neutral partial differential equations with finite delay taoufik chitioui, université de sfax, faculté des sciences de sfax, sfax, tunisie. chtioui.taoufik@yahoo.fr khalil ezzinbi, caddy ayyad university, université cadi ayyad, faculté des sciences semlalia, bp 2390, marrakech, morocco, ezzinbi@gmail.com amor rebey université de kairouan institut supérieur des mathématiques appliquées et de l’informatique, tunisie. ezzinbi@gmail.com abstract in this work, we study the existence, regularity and stability of solutions for some nonlinear class of partial neutral functional differential equations. we assume that the linear part generates a compact analytic semigroup on a banach space x, the delayed part is assumed to be continuous with respect to the fractional power of the generator. for illustration, some application is provided for some model with diffusion and nonlinearity in the gradient. resumen en este trabajo estudiamos la existencia, regularidad y estabilidad de soluciones para una clase de ecuaciones diferenciales parciales funcionales neutrales. asumimos que la parte lineal genera un semigrupo compacto anaĺıtico en un espacio de banach x, la parte retardada se asume continua respecto de la potencia fraccional del generador. como ejemplo, se muestra una aplicación para un modelo con difusión y no linealidad en el gradiente. keywords and phrases: neutal equation; analytic semigroup; fractional power; mild solution; strict solution. 2010 ams mathematics subject classification: 50 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) 1 introduction in this paper, we study the existence, regularity and stability of solutions in the α-norm for partial differential equations with finite delay. the following model provides an example of such a situation    ∂ ∂t [ v(t,x) − qv(t − r,x) + g( ∂ ∂x v(t − r,x)) ] = ∂2 ∂x2 [ v(t,x) − qv(t − r,x) +g( ∂ ∂x v(t − r,x)) ] + f ( v(t − r,x), ∂ ∂x [v(t,x) − qv(t − r,x)] ) for t ≥ 0 and x ∈ [0,π], v(t,0) − qv(t − r,0) = v(t,π) − qv(t − r,π) = 0 for t ≥ 0, v(θ,x) = v0(θ,x) for − r ≤ θ ≤ 0 and x ∈ [0,π], (1) where q, r are positive constants, the initial data v0 is given function and f, g are continuous functions. the previous system can be written as a neutral partial differential equation of the following form    d dt [ x(t) − g(t,xt) ] = −a [ x(t) − g(t,xt) ] + f(t,xt) for t ≥ 0, x0 = ϕ ∈ cα, (2) where −a generates an analytic semigroup (t(t))t>0 on a banach space x, cα := c([−r,0],d(a α)), r > 0, and 0 < α < 1, denotes the space of continuous functions from [−r,0] into d(aα), and the operator aα is the fractional α-power of a. this operator (aα,d(aα)) will be describe later. for x ∈ c([−r,b],d(aα)),b > 0, and t ∈ [0,b], xt denotes, as usual, the element of cα defined by xt(θ) = x(t + θ) for θ ∈ [−r,0].g and f are continuous functions from r+ × cα with values respectively in xα and x. this work was motivated by [4, 18]. in [4] the authors have developed a basic theory of partial neutral functional differential equations in fractional power spaces, they proved the existence and regularity of the solution of eq. (2) , but only in the case where g : cα → d(a α) is a bounded linear operator. they considered the following neutral partial differential equation    d dt d(xt) = −ad(xt) + f(xt) for t ≥ 0, x0 = ϕ ∈ cα, (3) where d is a bounded linear operator from cα into xα defined by dϕ = ϕ(0) −d0ϕ, for ϕ ∈ cα, where d0 is a bounded linear operator given by: d0ϕ = ∫0 −r dη(θ)ϕ(θ) for ϕ ∈ cα, and η : [−r,0] → l(xα) is of bounded variation and non-atomic at zero. that is var[−ε,0](η) → 0 as ε → 0. cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 51 which f is a globally lipschitz continuous mapping from cα into d(a α), and if x ∈ d(aα) and θ ∈ [−r,0] then η(θ)x ∈ d(aα) and aαη(θ)x = η(θ)aαx. it is well known, that if the phase space cα is the space of all continuous functions from [−r,0] into x (i.e. α = 0), equation (3) has been studied by several authors. for more details, we refer to the book of wu [29]. for example, wu and xia considered in [30] a system of partial neutral functional differential-difference equations defined on the unit circle s1, which is a model for a continuous circular array of resistively coupled transmission lines with mixed initial boundary conditions. they obtained equations of the form ∂ ∂t [x(.,t) − qx(.,t − r)] = k ∂2 ∂ξ2 [x(.,t) − qx(.,t − r)] + f(xt) for t ≥ 0, where ξ ∈ s1, k a positive constant and 0 ≤ q < 1. the space of initial data was chosen to be c([−r,0];h1(s1)). motivated by this work, hale presented, in [19, 20], the basic theory of existence and uniqueness, and properties of the solution operator, as well as hopf bifurcation and conditions for the stability and instability of periodic orbits for a more general class of pnfde on the unit circle s1. for the sake of comparison, let us briefly restate the equations considered by hale in [19, 20]. if ϕ ∈ c([−r,0];h1(s1)), we write it as ϕ(ξ,θ) for ξ ∈ s1 and θ ∈ [−r,0]. for any function f̃ ∈ ck+1(c([−r,0]; r); r), k ≥ 1, we let f ∈ ck+1(c([−r,0];h1(s1));l2(s1)) be defined by f(ϕ)(ξ) = f̃(ϕ(ξ,.)), ξ ∈ s1. let d̃ ∈ l(c([−r,0]; r); r) be defined by    d̃ψ = ψ(0) − g̃(ψ), g̃(ψ) = ∫0 −r dη(θ)ψ(θ), where η is of bounded variation and non-atomic at 0. we define d ∈ l(c([−r,0];h1(s1));h1(s1)) as d(ϕ)(ξ) = d̃(ϕ(ξ,.)) for ξ ∈ s1. hale considered, in [19, 20], pnfde of the form ∂ ∂t dxt = k ∂2 ∂ξ2 dxt + f(xt) for t ≥ 0, (4) with c([−r,0];h1(s1)) as the space of initial data. he considered the laplace operator a0 = k ∂ 2 ∂ξ2 with domain h2(s1), which yields an operator generating an analytic semigroup. in [1, 2, 3], authors considered a natural generalization of the work of hale [19, 20]. we extended the study to the case when the linear part of pnfde is non-densely defined hille-yosida operator. in[27], travis and webb investigated the local existence of mild solutions and strong solutions of eq. (2) with respect to the α-norm, but in the particular case when g(., .) = 0. the existence of strong solutions is considered when f is locally hölder continuous in both of its variables, also in [26], they studied the existence and regularity of mild solution when f is lipschitz continuous with 52 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) the x-norm. here, we assume that g is a nonlinear function and is defined in a smaller space than cx, that is cα for some 0 < α < 1, the space of continuous function from [−r,0] into xα, which will be specified later. we prove the existence of the mild and strict solution. this paper is organized as follows. in section 2, we recall some preliminary results about analytic semigroups and fractional power associated to its generator and the definition of the measure of noncompactness. after that, we start to prove the existence and uniqueness of mild solutions in the α-norm for eq. (2). in section 3, we study the regularity of solution, we give sufficient conditions to get the existence of the strict solutions. in section 4, we state some properties of the solution operator associated to the autonomous case of eq. (2). also, we investigate the stability near an equilibrium. mainly, we prove that the equilibrium of the solution semigroup associated to the autonomous case is locally exponentially stable when its linearized solution semigroup around this equilibrium is exponentially stable. finally, to illustrate our results, we give in section 5 an application to a reaction diffusion equation. 2 existence of mild solutions let (x,‖.‖) be a banach space, and α be a constant such that 0 < α < 1 and −a be the infinitesimal generator of a bounded analytic semigroup of linear operator (t(t))t≥0 on x. we assume without loss of generality that 0 ∈ ρ(a). note that if the assumption 0 ∈ ρ(a) is not satisfied, one can substitute the operator a by the operator (a − σi) with σ large enough such that 0 ∈ ρ(a − σ). this allows us to define the fractional power aα for 0 < α < 1, as a closed linear invertible operator with domain d(aα) dense in x. the closeness of aα implies that d(aα), endowed with the graph norm of aα, |x| = ‖x‖ + ‖aαx‖, is a banach space. since aα is invertible, its graph norm |.| is equivalent to the norm |x|α = ‖a αx‖. thus, d(aα) equipped with the norm |.|α, is a banach space, which we denote by xα. the space cα := c([−r,0],xα), r > 0 denotes the space of continuous functions from [−r,0] into xα endowed with the uniform norm topology: ‖ϕ‖α := sup θ∈[−r,0] |ϕ(θ)|α for ϕ ∈ cα. also, the following properties are well known. theorem 2.1. [24] let 0 < α < 1. assume that the operator −a is the infinitesimal generator of an analytic semigroup (t(t))t≥0 on the banach space x satisfying 0 ∈ ρ(a). then we have i) t(t) : x −→ d(aα) for every t > 0, ii) t(t)aαx = aαt(t)x for every x ∈ d(aα) and t ≥ 0, iii) for every t > 0, aαt(t) is bounded on x and there exist mα > 0 and δ > 0 such that ‖aαt(t)‖ ≤ mαe −δtt−α ≤ mαt −α for t > 0, cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 53 iv) if 0 < α ≤ β < 1, then d(aβ) →֒ d(aα). v) there exists nα > 0 such that ‖(t(t) − i)a−α‖ ≤ nαt α for t > 0. vi) if t(t) is compact for each t > 0, then a−α is compact. now, we propose to find the existence of a mild solution for problem (2) using the sadovskii’s fixed point theorem. then, we obtain the uniqueness result of the solution by adding a hypothesis of lipschitz continuous on f. let e be a banach space. we introduce the kuratowskii measure of noncompactness χ(ω) of a set ω ⊂ e by χ(ω) = inf{ε > 0 : ω has a finite cover of diameter < ε}. lemma 2.1. [10] let e be a banach space and b, c ⊆ e be bounded set. then, the following properties are true : (1) b is relatively compact if and only if χ(b) = 0, (2) χ(b + c) ≤ χ(b) + χ(c), where b + c = {x + y : x ∈ b, y ∈ c}, (3) every lipschitz continuous function k from c to f satisfies: χ[k(ω)] ≤ lipkχ(ω), where lip k decides the smallest lipschitz constant of k. definition 2.2. [25] a mapping k from a set c in a banach space e is called a condensing operator if it is continuous and for every bounded noncompact set ω ⊆ c the inequality holds χ[k(ω)] < χ(ω). theorem 2.2. [25](sadovskii’s fixed point theorem). if a condensing mapping k maps a bounded convex closed set c in a banach space e into itself, then k has at least one fixed point in t. first of all, we study the existence of mild solutions, in order to do that, we assume the following assumptions. (h0) the operator −a is the infinitesimal generator of an analytic semigroup (t(t))t≥0 on the banach space x , moreover, we assume that 0 ∈ ρ(a). (h1) the semigroup (t(t))t≥0 is compact on x for t > 0. it means that t(t) is compact on x for t > 0. 54 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) (h2) g : [0,a] × cα → xα is continuous and for each a > 0 there exists 0 < lg < 1 such that |g(t,ϕ) − g(t,ψ)|α ≤ lg‖ϕ − ψ‖α for every t ∈ [0,a] and ϕ, ψ ∈ cα. (h3) the function f : [0,a] × cα → x satisfies the following conditions i) f : [0,a] × cα → x is continuous. ii) there exists a continuous nondecreasing function β : [0,a] → r+ such that ‖f(t,ϕ)‖ ≤ β(t)‖ϕ‖α for (t,ϕ) ∈ [0,a] × cα. definition 2.3. a continuous function x : [−r,a] −→ xα, for a > 0 is said to be a mild solution of eq. (2), if i) x(t) = t(t) [ ϕ(0) − g(0,ϕ) ] + g(t,xt) + ∫t 0 t(t − s)f(s,xs)ds for t ∈ [0,a], ii) x0 = ϕ. definition 2.4. a continuous function x : [−r,a] −→ xα is said to be a strict solution of eq. (2), if i) x(.) − g(.,x(.)) ∈ c 1([0,a],xα), ii) d dt (x(t) − g(t,xt)) = −a(x(t) − g(t,xt)) + f(t, xt) for t ∈ [0,a], iii) x0 = ϕ. now, we state our first result. theorem 2.3. assume that the hypothesis (h0)-(h3) hold. let ϕ ∈ cα. assume that the following condition holds lg + mα ∫a 0 β(s) (a − s)α ds < 1. (5) then eq. (2) has at least one mild solution on [0,a]. proof. let k > ‖ϕ‖α. we define the following set bk = {x ∈ c([0,a],xα) : x(0) = ϕ(0) and |x|∞ ≤ k}, where |x|∞ = sup t∈[0,a] |x(t)|α. for x ∈ bk, define the mapping x̃ : [−r,a] → xα by x̃(t) = { x(t) for t ∈ [0,a] ϕ(t) for t ∈ [−r,0]. cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 55 the function t 7→ x̃t is continuous from [0,a] to cα. now, define the operator k on bk by k(x)(t) = t(t)(ϕ(0) − g(0,ϕ)) + g(t, x̃t) + ∫t 0 t(t − s)f(s, x̃s)ds for t ∈ [0,a]. it is sufficient to show that k has a fixed point in bk. we first show that there is a positive number k > ‖ϕ‖α such that k(bk) ⊆ bk. if not, then for each k > ‖ϕ‖α, there exist xk ∈ bk and tk ∈ [0,a] such that |(kxk)(tk)|α > k. it follows that k < |(kxk)(tk)|α ≤ |t(tk)(ϕ(0) − g(0,ϕ))|α + |g(tk, x̃tk)|α + ∫tk 0 |t(tk − s)f(s, x̃s)|αds. let m = sup{‖t(t)‖ : t ∈ [0,a]}. then k < m|ϕ(0) − g(0,ϕ)|α + |g(tk, x̃tk) − g(tk,0)|α + |g(tk,0)|α + ∫tk 0 mα (tk − s) α β(s)‖x̃s‖αds. moreover ‖x̃s‖α ≤ k for all s ∈ [0,a] and x ∈ bk. then, we obtain k < m|ϕ(0) − g(0,ϕ)|α + |g(tk, x̃tk) − g(tk,0)|α + |g(tk,0)|α + ∫tk 0 kmα (tk − s) α β(s)ds. we shall show that the function g : t 7→ ∫t 0 β(s) (t − s)α ds is nondecreasing on [0,a]. let t,t′ ∈ [0,a] be such that t < t′. then we have g(t) = ∫t 0 β(t − s) sα ds ≤ ∫t 0 β(t′ − s) sα ds ≤ ∫t′ 0 β(t′ − s) sα ds = g(t′). therefore k ≤ m|ϕ(0) − g(0,ϕ)|α + lg‖x̃tk‖α + sup 0≤s≤a |g(s,0)|α + ∫a 0 kmα (a − s)α β(s)ds. dividing both sides by k and taking the lower limit as k → +∞, then we get that lg + mα ∫a 0 β(s) (a − s)α ds ≥ 1, which contradicts (5). consequently, there exists k ≥ 0 such k(bk) ⊆ bk. to prove that k has at least a fixed point on bk, we decompose k as follows k := k1 + k2, where k1(x)(t) = g(t, x̃t) for t ∈ [0,a]. and k2(x)(t) = t(t)(ϕ(0) − g(0,ϕ)) + ∫t 0 t(t − s)f(s, x̃s)ds for t ∈ [0,a]. 56 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) we claim that k1 is a strict contraction and k2 is compact. to see this, observe that for t ∈ [0,a] and x, y ∈ bk, we have by assumption (h2). |k1x(t) − k1y(t)|α = |g(t, x̃t) − g(t, ỹt)|α ≤ lg‖x̃t − ỹt‖α ≤ lg|x − y|∞ then k1 is a strict contraction. we will prove now the continuity of k2. let (x n)n ⊂ bk with xn → x in bk. then, the set λ = {(s, x̃ n s ),(s, x̃s) : s ∈ [0,a],n ≥ 1} is compact in [0,a] × cα. by hëıne’s theorem implies that f is uniformly continuous in λ and |k2(x n) − k2(x)|∞ = sup t∈[0,a] ∫t 0 aαt(t − s) ( f(s, x̃ns ) − f(s, x̃s) ) ds ≤ mα ∫a 0 ds sα sup s∈[0,a] ‖f(s, x̃ns ) − f(s, x̃s)‖ → 0 as n → +∞. and this yield the continuity of k2, then the continuity of k on bk. we next show that the operator k2 is compact. in order to apply ascoli theorem we have to show that the set {k2(x)(t) : x ∈ bk} is relatively compact for each t ∈]0,a]. let t ∈]0,a] be fixed, and γ > 0 be such that α < γ < 1. then ‖ (aγk2(x))(t) ‖ ≤ ‖ a γt(t)(ϕ(0) − g(0,ϕ)) ‖ + ‖ ∫t 0 aγt(t − s)f(s, x̃s)ds ‖ ≤ mγt −γ ‖ ϕ(0) − g(0,ϕ) ‖ +kmγ ∫t 0 (t − s)−γβ(s)ds < +∞. then for fixed t ∈]0,a], {(aγk2x)(t)} is bounded in x. appealing (h1) and (vi) of theorem 2.1, we deduce that a−γ : x → xα is compact, it follows that {k2(x)(t) : x ∈ bk} is relatively compact set in xα. next, we will show that {k2x : x ∈ bk} is an equicontinuous family of functions. for 0 ≤ t1 < t2 ≤ a, k2x(t2) − k2x(t1) = (t(t2) − t(t1))(ϕ(0) − g(0,ϕ)) + ∫t2 t1 t(t2 − s)f(s, x̃s)ds + ∫t1 0 (t(t2 − s) − t(t1 − s))f(s, x̃s)ds = (t(t2) − t(t1))(ϕ(0) − g(0,ϕ)) + ∫t2 t1 t(t2 − s)f(s, x̃s)ds +(t(t2 − t1) − i) ∫t1 0 t(t1 − s)f(s, x̃s)ds. cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 57 we obtain that |k2x(t2) − k2x(t1)|α ≤ ‖(t(t2) − t(t1))a α(ϕ(0) − g(0,ϕ))‖ + kmα‖β‖∞ ∫t2 t1 (t2 − s) −αds +‖(t(t2 − t1) − i) ∫t1 0 aαt(t1 − s)f(s, x̃s)ds‖ it’s clair to prove the first part tend to zero as |t2 − t1| → 0. since for t1 > 0 the set { ∫t1 0 aαt(t1 − s)f(s, x̃s)ds : x ∈ bk } is relatively compact in x, there is a compact set k̃ in x such that ∫t1 0 aαt(t1 − s)f(s, x̃s)ds ∈ k̃ for x ∈ bk. by banach-steinhaus’s theorem, we have ∥∥∥(t(t2 − t1) − i) ∫t1 0 aαt(t1 − s)f(s, x̃s)ds ∥∥∥ → 0 as t2 → t1, uniformly in x ∈ bk. using similar argument for 0 ≤ t2 < t1 ≤ a, we can conclude that {k2x(t), x ∈ bk} is an equicontinuous . using ascoli-arzla theorem, we deduce that k2 : bk → bk is compact, and k = k1 + k2 is a condensing operator. by the sadovskii’s fixed-point theorem 2.2, we conclude that k has at least one fixed point in bk, which is a mild solutions of eq. (2) on [0,a]. to prove result on uniqueness, we to assume that (h4) f : [0,a] × cα → x is continuous and lipschitzian with respect to the second variable. let lf > 0 be such that ‖f(t,ψ1) − f(t,ψ2)‖ 6 lf‖ψ1 − ψ2‖α (6) for every ψ1,ψ2 ∈ cα and t ∈ [0,a]. theorem 2.4. let ϕ ∈ cα. if the assumptions (h0), (h2) and (h4) are satisfied, then eq. (2) has a unique mild solution provided that lg + mαlf a1−α 1 − α < 1. (7) proof. consider the nonempty closed subset of c([0,a],xα) defined by ω(ϕ) := {x ∈ c([0,a],xα) : x(0) = ϕ(0)}. for x ∈ ω(ϕ), define the mapping x̃ : [−r,a] → xα by x̃(t) = { x(t) for t ∈ [0,a] ϕ(t) for t ∈ [−r,0]. 58 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) define the operator k : ω(ϕ) → ω(ϕ) by k(x)(t) = t(t)(ϕ(0) − g(0,ϕ)) + g(t, x̃t) + ∫t 0 t(t − s)f(s, x̃s)ds for t ∈ [0,a]. we shall show that it is a strict contraction. let x,y ∈ ω(ϕ) and t ∈ [0,a]. then |kx(t) − ky(t)|α 6 |g(t, x̃t) − g(t, ỹt)|α + ∫t 0 |t(t − s){f(s, x̃s) − f(s, ỹs)}|αds 6 lg‖x̃t − ỹt‖α + mα ∫t 0 ‖f(s, x̃s) − f(s, ỹs)‖(t − s) −αds 6 ( lg + mαlf a1−α 1 − α ) |x − y|∞ then |kx − ky|∞ 6 ( lg + mαlf a1−α 1 − α ) |x − y|∞. it follows that k is a strict contraction since lg + mαlf a1−α 1 − α < 1. by the contraction principle, we conclude that there exists a unique fixed point x for k in ω(ϕ), therefore eq. (2) has a unique mild solution on [−r,a]. the proof is completed. 3 existence of strict solutions for the regularity of the integral solutions, we suppose moreover the following assumptions: (h5) g and f are continuously differentiable and their partial derivatives are locally lipschitzian with respect to the second argument in the sense that; for any compact set k ⊂ [0,a] × cα, there exist positive constants l1, l2, l3 and l4 such that |d1g(t,ψ1) − d1g(t,ψ2)|α 6 l1‖ψ1 − ψ2‖α, ‖d2g(t,ψ1) − d2g(t,ψ2)‖l(cα,xα) 6 l2‖ψ1 − ψ2‖α, ‖d1f(t,ψ1) − d1f(t,ψ2)‖ 6 l3‖ψ1 − ψ2‖α, ‖d2f(t,ψ1) − d2f(t,ψ2)‖l(cα,x) 6 l4‖ψ1 − ψ2‖α, for (t,ψ1), (t,ψ2) ∈ k and t ∈ [0,a]. where d1 and d2 are the partial derivatives with respect to the first and second argument. theorem 3.1. assume that (h0),(h2), (h4), (h5) hold and condition (7) is true. let ϕ ∈ c1([−r,0],xα) be such that ϕ(0) − g(0,ϕ) ∈ d(a) and ϕ′(0) − d1g(0,ϕ) − d2g(0,ϕ)ϕ ′ = −a[ϕ(0) − g(0,ϕ)] + f(0,ϕ) then eq. (2) has a unique strict solution on [0,a]. cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 59 proof. let x be the mild solution of eq. (2). consider the equation    y(t) = t(t)[−a(ϕ(0) − g(0,ϕ)) + f(0,ϕ)] + d1g(t,xt) + d2g(t,xt)yt + ∫t 0 t(t − s)[d1f(s,xs) + d2f(s,xs)ys]ds for t ∈ [0,a], y0 = ϕ ′ ∈ cα. (8) we claim that eq. (8) has a unique solution on [0,a]. in fact, consider the operator p defined on λ := {x ∈ c([−r,a];xα) : x(t) = ϕ ′(t) for t ∈ [−r,0]} by py(t) =    t(t)[−a(ϕ(0) − g(0,ϕ)) + f(0,ϕ)] + d1g(t,xt) +d2g(t,xt)yt + ∫t 0 t(t − s)[d1f(s,xs) + d2f(s,xs)ys]ds for t ∈ [0,a], ϕ′(t) for t ∈ [−r,0]. let u,v ∈ λ. then for each t ∈ [0,a], we have |pu(t) − pv(t)|α 6 ‖d2g(t,xt)‖l(cα,xα) ‖ut − vt‖α +mα ∫t 0 ‖d2f(s,xs)‖l(cα,x)‖us − vs‖α ds (t − s)α 6 ( lg + mαlf a1−α 1 − α ) |u − v|∞. then p is a strict contraction. consequently, it has a unique mild solution y. define z : [−r,a] → xα by z(t) =    ϕ(0) + ∫t 0 y(s)ds for t ∈ [0,a] ϕ(t) for t ∈ [−r,0], we will show that z(t) = x(t) on [0,a]. for t ∈ [0,a], we have z(t) = ϕ(0) + ∫t 0 t(s)(−a)(ϕ(0) − g(0,ϕ))ds + ∫t 0 t(s)f(0,ϕ)ds + ∫t 0 d1g(s,xs) + d2g(s,xs)ysds + ∫t 0 ∫s 0 t(s − τ)[d1f(τ,xτ) + d2f(τ,xτ)yτ]dτds. moreover, we can see that zt = ϕ + ∫t 0 ysds for t ∈ [0,a]. (9) 60 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) then t 7→ zt and t 7→ ∫t 0 t(t − s)f(s,zs)ds are continuously differentiable on [0,a] and satisfy d dt ∫t 0 t(t − s)f(s,zs)ds = t(t)f(0,ϕ) + ∫t 0 t(t − s)[d1f(s,zs) + d2f(s,zs)ys]ds, (10) then (10) yields ∫t 0 t(s)f(0,ϕ)ds = ∫t 0 t(t − s)f(s,zs)ds − ∫t 0 ∫s 0 t(s − τ)[d1f(τ,zτ) + d2f(τ,zτ)yτ]dτds. on the other hand g(t,zt) = g(0,ϕ) + ∫t 0 d ds g(s,zs)ds = g(0,ϕ) + ∫t 0 d1g(s,zs) + d2g(s,zs)ysds. then z(t) = t(t)(ϕ(0) − g(0,ϕ)) + g(t,zt) − ∫t 0 d1g(s,zs) + d2g(s,zs)ysds + ∫t 0 t(t − s)f(s,zs)ds − ∫t 0 ∫s 0 t(s − τ)[d1f(τ,zτ) + d2f(τ,zτ)yτ]dτds + ∫t 0 (d1g(s,xs) + d2g(s,xs)ys)ds + ∫t 0 ∫s 0 t(s − τ)[d1f(τ,xτ) + d2f(τ,xτ)yτ]dτds. therefore |z(t) − x(t)|α ≤ |g(t,zt) − g(t,xt)|α + ∫t 0 |d1g(s,zs) − d1g(s,xs)|αds + ∫t 0 |d2g(s,zs)ys − d2g(s,xs)ys|αds + ∫t 0 |t(t − s)(f(s,zs) − f(s,xs))|αds + ∫t 0 ∫s 0 |t(s − τ)[d1f(τ,zτ) − d1f(τ,xτ)]|αdτds + ∫t 0 ∫s 0 |t(s − τ)[d2f(τ,zτ)yτ − d2f(τ,xτ)yτ]|αdτds. note that the sets {(s,zs) : s ∈ [0,a]} and {(s,xs) : s ∈ [0,a]} are compacts in [0,a] ×cα, since the mapping t → zt and t → xt are continuous on [0,a]. then, we deduce that ‖d1g(s,zs) − d1g(s,xs)‖α 6 l1‖zs − xs‖α, ‖d2g(s,zs) − d2g(s,xs)‖l(cα,xα) 6 l2‖zs − xs‖α, ‖d1f(s,zs) − d1f(s,xs)‖ 6 l3‖zs − xs‖α, ‖d2f(s,zs) − d2f(s,xs)‖l(cα,x) 6 l4‖zs − xs‖α, cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 61 for all s ∈ [0,a], x ∈ λ and z given in (9). let l = max{lf, l1, l2, l3,l4}. then |z(t) − x(t)|α ≤ lg + l ( t + ‖y‖∞t + mα 1 − α t1−α + mα (1 − α)(2 − α) t2−α + mα‖y‖∞ (1 − α)(2 − α) t2−α ) sup 0≤s≤t |z(s) − x(s)|α. we can choose t0 ∈ [0,a] such that lg + l ( t0 + ‖y‖∞t0 + mα 1 − α t1−α0 + mα (1 − α)(2 − α) t2−α0 + mα‖y‖∞ (1 − α)(2 − α) t2−α0 ) < 1. we deduce that x = z on [0,t0]. we claim that x(t) = z(t) for t ∈ [0,a]. we proceed by contradiction and assume that there exists t1 ∈ [0,a] such that x(t1) 6= z(t1). let t ∗ be the smallest number such that x(t) 6= z(t). then t∗ = inf{t ∈ [0,a] : |z(t) − x(t)|α > 0}. by continuity, one has x(t) = z(t) for t ∈ [0,t∗] and there exists ε > 0 such that |z(t) − x(t)|α > 0 for t ∈]t ∗,t∗ + ε[. it follows for t ∈ [t∗,t∗ + ε] that |z(t) − x(t)|α ≤ |g(t,zt) − g(t,xt)|α + ∫t t∗ |d1g(s,zs) − d1g(s,xs)|αds + ∫t t∗ |d2g(s,zs)ys − d2g(s,xs)ys|αds + ∫t t∗ |t(t − s)(f(s,zs) − f(s,xs))|αds + ∫t t∗ ∫s t∗ |t(s − τ)[d1f(τ,zτ) − d1f(τ,xτ)]|αdτds + ∫t t∗ ∫s t∗ |t(s − τ)[d2f(τ,zτ)yτ − d2f(τ,xτ)yτ]|αdτds. consequently, |z(t) − x(t)|α ≤ lg + l ( ε + ‖y‖∞ε + mα 1 − α ε1−α + mα (1 − α)(2 − α) ε2−α + mα‖y‖∞ (1 − α)(2 − α) ε2−α ) sup t∗≤s≤t∗+ε |z(s) − x(s)|α. if we choose ε such that lg + l ( ε + ‖y‖∞ε + mα 1 − α ε1−α + mα (1 − α)(2 − α) ε2−α + mα‖y‖∞ (1 − α)(2 − α) ε2−α ) < 1 then x(t) = z(t) for t ∈ [t∗,t∗ + ε] which gives a contradiction. consequently x(t) = z(t) for t ∈ [0,a] and t 7→ xt is continuously differentiable in [0,a] and t 7→ f(t,xt) ∈ c 1([0,a],x). to end the proof, we use the following lemma. 62 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) lemma 3.1. [24] let h : [0,a] → x be continuously differentiable and u satisfy u(t) = t(t)u0 + ∫t 0 t(t − s)h(s)ds for t ∈ [0,a]. if u0 ∈ d(a), then u is continuously differentiable on [0,a] and u′(t) = −au(t) + h(t) for t ∈ [0,a]. in our case, we have ϕ(0) − g(0,ϕ) ∈ d(a), t 7→ f(t,xt) is continuously differentiable on [0,a] and x(t) − g(t,xt) = t(t) [ ϕ(0) − g(0,ϕ) ] + ∫t 0 t(t − s)f(s,xs)ds for t ∈ [0,a]. by lemma 3.1, the mapping t 7→ x(t) − g(t,xt) is continuously differentiable on [0,a] and for t ∈ [0,a], d dt [ x(t) − g(t,xt) ] = −a [ x(t) − g(t,xt) ] + f(t,xt) for t ∈ [0,a]. these implies that x is a strict solution of eq. (2) on [0,a]. 4 the solution semigroup in the autonomous case and the linearized stability principle in this section, we suppose that f and g are autonomous. then eq. (2) becomes    d dt [ x(t) − g(xt) ] = −a [ x(t) − g(xt) ] + f(xt) for t ≥ 0, x0 = ϕ ∈ cα. (11) we can see that the mild solutions of eq. (11) satisfy the properties of a nonlinear strongly continuous semigroup on cα and we prove that this semigroup satisfies the translation property and a lipschitz property. for each t ≥ 0, define the nonlinear operator u(t) on cα by u(t)(ϕ) = xt(.,ϕ) where x(.,ϕ) is the unique mild solution of eq. (11) for the initial condition ϕ ∈ cα. one can prove the proposition. proposition 4.1. under the assumption as in the theorem (2.4), the family (u(t))t>0 is a nonlinear strongly continuous semigroup on cα. moreover (i) (u(t))t>0 satisfies the following translation property, for t > 0 and θ ∈ [−r,0], (u(t)(ϕ))(θ) = { (u(t + θ)(ϕ))(0), if t + θ > 0 ϕ(t + θ), if t + θ 6 0 cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 63 (ii) for all t > 0, there are two functions p, q ∈ l∞([0,t],r+) such that, for all ϕ1, ϕ2 ∈ cα, ‖u(t)(ϕ1) − u(t)(ϕ2)‖α ≤ p(t)e q(t)‖ϕ1 − ϕ2‖α, t ∈ [0,t]. (12) proof. proof of (ii). let x1 := x(.,ϕ1), x 2 := x(.,ϕ2), t > 0 and m > 1 such that sup{‖t(t)‖, t ∈ [0,t]} ≤ m. for t ∈ [0,t], we have ‖u(t)(ϕ1) − u(t)(ϕ2)‖α = ‖x 1 t − x 2 t‖α = sup −r≤θ≤0 |x1(t + θ) − x2(t + θ)|α ≤ (m + mlg)‖ϕ1 − ϕ2‖α + lg sup −r≤θ≤0 ‖x1t+θ − x 2 t+θ‖α + mlf ∫t 0 ‖x1s − x 2 s‖αds. letting t ∈ [0,r]. then, for θ ∈ [−r,0] such that t + θ ≥ 0, we have ‖x1t+θ − x 2 t+θ‖α = sup −r≤τ≤0 |x1(t + θ + τ) − x2(t + θ + τ)|α = sup −r+t+θ≤τ≤t+θ |x1(τ) − x2(τ)|α = max{‖ϕ1 − ϕ2‖α, sup 0≤τ≤t+θ |x1(τ) − x2(τ)|α} ≤ ‖ϕ1 − ϕ2‖α + ‖x 1 t − x 2 t‖α. then, ‖x1t − x 2 t‖α ≤ ( m + mlg + 1 1 − lg )‖ϕ1 − ϕ2‖α + mlf 1 − lg ∫t 0 ‖x1s − x 2 s‖αds. using gronwall’s lemma, we obtain ‖x1t − x 2 t‖α ≤ ( m + mlg + 1 1 − lg )e mlf 1−lg t ‖ϕ1 − ϕ2‖α. we can repeat the previous argument for t ∈ [r,2r], to see that for every t ∈ [r,2r], ‖u(t)(ϕ1) − u(t)(ϕ2)‖α ≤ ‖u(r)‖‖u(t − r)(ϕ1) − u(t − r)(ϕ2)‖α ≤ ( m + mlg + 1 1 − lg )2e mlf 1−lg t ‖ϕ1 − ϕ2‖α. for t ∈ [2r,3r] ‖u(t)(ϕ1) − u(t)(ϕ2)‖α ≤ ‖u(2r)‖‖u(t − 2r)(ϕ1) − u(t − 2r)(ϕ2)‖α ≤ ( m + mlg + 1 1 − lg )3e mlf 1−lg t ‖ϕ1 − ϕ2‖α. 64 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) inductively, for t ∈ [nr,(n + 1)r] with n ≥ 2, we obtain ‖u(t)(ϕ1) − u(t)(ϕ2)‖α ≤ ‖u(nr)‖‖u(t − nr)(ϕ1) − u(t − nr)(ϕ2)‖α ≤ ( m + mlg + 1 1 − lg )n+1e mlf 1−lg t ‖ϕ1 − ϕ2‖α. consequently, the estimate (12) is true. this ends the proof. in what follows, we study the stability of an equilibrium of the following autonomous equation:    d dt [ d(xt) − g(xt) ] = −a [ d(xt) − g(xt) ] + f(xt) for t ≥ 0, x0 = ϕ ∈ cα, (13) where f and g are lipschitz continuous on cα with constants respectively lf and lg and d : cα −→ xα is an operator defined by dϕ = ϕ(0) − d0ϕ with d0 a bounded linear operator from cα into xα such that lg + ‖d0‖ < 1. we are now interested by the stability of the equilibriums of equation (13). by equilibrium, we mean a constant mild solution x∗ of (13). without loss of generality, we can assume that x∗ = 0 and g(0) = f(0) = 0: we need the following assumption. (h6) f and g are fréchet-differentiable at 0 and g′(0) = 0. let l = f′(0). then, the linearized equation of eq. (13) around the equilibrium 0 is the following:    d dt dyt = −adyt + l(yt) for t ≥ 0, y0 = ϕ ∈ cα. (14) let (u(t))t>0 the nonlinear semigroup associated to eq. (13) and the linear semigroup (v(t))t>0 associated to the linear equation (14) in the same space cα. then, we have the following result. theorem 4.1. assume that the conditions (h0), (h2), (h4), (h5) and (h6) hold. then, for every t > 0 the derivative at zero of u(t) is v(t). the proof of this theorem is based on the following fundamental lemma lemma 4.2. let h : cα −→ xα be a continuous function such that there exists 0 < µ0 < 1 satisfying |h(ϕ1) − h(ϕ2)|α 6 µ0‖ϕ1 − ϕ2‖α let ϕ ∈ cα and h : [0,+∞[−→ xα be a continuous function. suppose that there exist continuous functions x,y : [−r,+∞[−→ xα such that { x(t) − y(t) = h(xt) − h(yt) + h(t), t ≥ 0, x0 = y0 = ϕ. cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 65 then, for each 0 < t ≤ r we have ‖xt − yt‖α 6 1 1 − µ0 sup 06s6t |h(s)|α, t ∈ [0,t]. proof. for t ≥ 0, we have ‖xt − yt‖α = sup −r≤θ≤0 |x(t + θ) − y(t + θ)|α = sup t−r≤s≤t |x(s) − y(s)|α = sup 0≤s≤t |x(s) − y(s)|α ≤ sup 0≤s≤t |h(xs) − h(ys)|α + sup 0≤s≤t |h(s)|α ≤ µ0 sup 0≤s≤t ‖xs − ys‖α + sup 0≤s≤t |h(s)|α = µ0‖xt − yt‖α + sup 0≤s≤t |h(s)|α then ‖xt − yt‖α 6 1 1 − µ0 sup 06s6t |h(s)|α proof. (of theorem 4.1) it suffices to show that for each ϕ ∈ cα, t ≥ 0 and ε > 0, there exists δ > 0 such that ‖u(t)ϕ − v(t)ϕ‖α 6 ε‖ϕ‖α, for ‖ϕ‖α 6 δ let t ≥ 0 be fixed and ϕ ∈ cα. we have (d − g)(u(t)ϕ) − d(v(t)ϕ) = ∫t 0 t(t − s)[f(u(s)ϕ) − f(v(s)ϕ)]ds − t(t)g(ϕ) + ∫t 0 t(t − s)[f(v(s)ϕ) − l(v(s)ϕ)]ds then, (d − g)(u(t)ϕ) − (d − g)(v(t)ϕ) = g(v(t)ϕ) − t(t)g(ϕ) + ∫t 0 t(t − s)[f(u(s)ϕ) − f(v(s)ϕ)]ds + ∫t 0 t(t − s)[f(v(s)ϕ) − l(v(s)ϕ)]ds let x,y : [−r,+∞[−→ xα and h : [0,+∞[−→ xα be defined by x(t) = { (u(t)ϕ)(0) if t ∈ [0,+∞[ ϕ(t) if t ∈ [−r,0] y(t) = { (v(t)ϕ)(0) if t ∈ [0,+∞[ ϕ(t) if t ∈ [−r,0] 66 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) and h(t) = g(v(t)ϕ) − t(t)g(ϕ) + ∫t 0 t(t − s)[f(u(s)ϕ) − f(v(s)ϕ)]ds + ∫t 0 t(t − s)[f(v(s)ϕ) − l(v(s)ϕ)]ds then, { (d − g)(xt) − (d − g)(yt) = h(t), t ≥ 0, x0 = y0 = ϕ. which is equivalent to { x(t) − y(t) = (d0 + g)(xt) − (d0 + g)(yt) + h(t), t ≥ 0, x0 = y0 = ϕ. using lemma (4.2), we obtain ‖xt − yt‖α 6 1 1 − (lg + ‖d0‖) sup 06s6t |h(s)|α, t ≥ 0. by virtue of the continuous differentiability of g and f at 0, we deduce that for ε > 0, there exists δ > 0 such that |g(v(t)ϕ) − t(t)g(ϕ)|α 6 ε‖ϕ‖α for ‖ϕ‖α 6 δ, and mα ∫t 0 |f(v(s)ϕ) − l(v(s)ϕ)| ds (t − s)α 6 ε‖ϕ‖α for ‖ϕ‖α 6 δ. then, for ‖ϕ‖α 6 δ, |h(t)|α ≤ 2ε‖ϕ‖α + mαlf ∫t 0 ‖u(s)ϕ − v(s)ϕ‖α ds (t − s)α , since for s ∈ [0,t] and t ∈ [0,r], ‖u(s)ϕ − v(s)ϕ‖α = sup −r≤θ≤0 |xs(θ) − ys(θ)|α = sup −r+s≤τ≤s |x(τ) − y(τ)|α = sup 0≤τ≤s |x(τ) − y(τ)|α ≤ sup 0≤τ≤t |x(τ) − y(τ)|α = ‖u(t)ϕ − v(t)ϕ‖α. then for t ∈ [0,r] fixed ‖u(t)ϕ − v(t)ϕ‖α ≤ 2ε‖ϕ‖α 1 − (lg + ‖d0‖) + mαlf 1 − (lg + ‖d0‖) ∫t 0 ‖u(s)ϕ − v(s)ϕ‖α (t − s)α ds cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 67 using gronwall’s lemma, we obtain ‖u(t)ϕ − v(t)ϕ‖α ≤ 2ε‖ϕ‖α 1 − (lg + ‖d0‖) exp( mαlft 1−α (1 − (lg + ‖d0‖))(1 − α) ) for ‖ϕ‖α 6 δ. we conclude that u(t) is differentiable at 0, for each t ∈ [0,t] and dϕu(t)(0) = v(t). now, suppose that t ∈ [t,2t] fixed. it follows that, for max {‖ϕ‖α,‖u(t − t)(ϕ)‖α} ≤ δ0, where δ0 > 0 is small enough ‖u(t)ϕ − v(t)ϕ‖α ≤ ‖u(t)u(t − t)(ϕ) − v(t)u(t − t)(ϕ)‖α +‖v(t)‖‖u(t − t)(ϕ) − v(t − t)(ϕ)‖α ≤ ε‖ϕ‖α. by steps, we conclude that u(t) is differentiable at 0, for each t ≥ 0 and dϕu(t)(0) = v(t). theorem 4.2. under the assumption as in the theorem (4.1), if the zero equilibrium of (v(t))t≥0 is exponentially stable, then the zero equilibrium of (u(t))t≥0 is locally exponentially stable, in the sense that there exist δ > 0, µ > 0 and k ≥ 1 such that ‖u(t)(ϕ)‖α ≤ ke −µt‖ϕ‖α for t ≥ 0 and ϕ ∈ cα with ‖ϕ‖α ≤ δ. moreover, if cα can be decomposed as cα = h1 ⊕ h2 where hi are v-invariant subspaces of cα, h1 is fnite-dimensional and with ω = lim h→∞ 1 h log‖v(h)/h2‖α, we have inf{|λ| : λ ∈ σ(v(t)/h1)} > e ωt, then, the zero equilibrium of (u(t))t≥0 is not stable, in the sense that there exist ε > 0 and a sequence (ϕn)n converging to 0 and a sequence (tn)n of positive reel numbers such that ‖u(tn)ϕn‖α > ε. the proof of this theorem is based on proposition 4.1 ,theorem 4.1 and the following result. theorem 4.3. (desch and schappacher [13]). let (v(t))t≥0 be a nonlinear strongly continuous semigroup on a subset ω of a banach space z. assume that x0 ∈ ω is an equilibrium of (v(t))t≥0 such that v(t) is fréchet-differentiable at x0 for each t ≥ 0, with w(t) the derivative at x0 of v(t), t ≥ 0. then, (w(t))t≥0 is a strongly continuous semigroup of bounded linear operators on z and, if the zero equilibrium of (w(t))t≥0 is exponentially stable, then the equilibrium x0 of (v(t))t≥0 is locally exponentially stable. moreover, if z can be decomposed as z = z1 ⊕ z2 where zi are w-invariant subspaces of z and z1 is fnite-dimensional and with ω = lim h→∞ 1 h log‖w(h)/z2‖, 68 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) we have inf{|λ| : λ ∈ σ(w(t)/z1)} > e ωt, then, the zero equilibrium x0 of (v(t))t≥0 is not stable, in the sense that there exist ε > 0 and a sequence (xn)n converging to x0 and a sequence (tn)n of positive reel numbers such that ‖v(tn)xn− x0‖ > ε. in the following, we will concentrate our study on the linear equation (14). let (av,d(av)) be the generator of the semigroup (v(t))t≥0 on cα. we have the result theorem 4.4. [4] assume that the conditions (h0), (h2), (h4), (h5) and (h6) hold. then, the operator (av,d(av)) is given by { d(av) = {ϕ ∈ cα, ϕ ′ ∈ cα, d(ϕ) ∈ d(a) and d(ϕ ′) = −ad(ϕ) + l(ϕ)}, avϕ = ϕ ′, ϕ ∈ d(av). let c be the space of continuous functions from [−r,0] into x provided with the uniform norm topology and let cd = {ϕ ∈ c : d(ϕ) = 0}. definition 4.3. [22] d is said to be stable if the zero solution of the difference equation { d(yt) = 0, t ≥ 0, y0 = ϕ ∈ cd, is exponentially stable. lemma 4.4. [4] if d is stable, then there exist positive constants a, b, c and d such that for any ε ∈]0,r] sufficiently small and any continuous function h from [0,+∞[ into x, the solution v of the equation d(vt) = h(t), t ≥ 0, satisfies the inequality ‖vt‖ ≤ e −a(t−ε) [ b‖v0‖ + c sup 0≤s≤ε |h(s)| ] + d sup max(ε,t−r)≤s≤t |h(s)|, t ≥ ε. (15) the estimate (15) is very interesting because, if |h(s)| is bounded on [0,+∞[, then the ultimate bound on vt as t → +∞ is determined by the bound on |h(s)| for s in the delay interval [t − r,t] as t → +∞. proposition 4.5. [20] let d(ϕ) = p∑ k=0 akϕ(−rk). then, d is stable iff p∑ k=0 |ak| < 1. in the sequel, we assume that cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 69 (h7) the operator d is stable. theorem 4.5. [4] assume that (h0), (h1), (h2), (h4), (h5) and (h7) hold. then the semigroup (u(t))t≥0 can be decomposed as follows u(t) = u1(t) + u2(t) for t ≥ 0, where u1(t) is an exponentially stable semigroup on cα and u2(t) is compact on cα for every t > 0. let (y,‖.‖) be a banach space. for a bounded linear operator b in y, we define ‖b‖ess := inf{c > 0 : χ(b(h)) ≤ cχ(h), for every bounded set h of y}, where χ(.) denotes the measure of noncompactness in y. the essential growth bound of (v(t))t≥0 in cα is given by ωess(v) := inf t>0 1 t log ‖v(t)‖ess. it follows from theorem 4.5, that ωess(v) < 0. let ω0(v) := inf t>0 1 t log ‖v(t)‖α be the growth bound of (v(t))t≥0 in cα. then, it is well known (see [14]) that ω0(v) = max{ωess(v),s ′(av)}, where s′(av) = sup{reλ : λ ∈ σ(av)\σess(av)} and σess(av) is the essential spectrum of av . consequently, the stability of (v(t))t≥0 is completely determined by s′(av). note that σ(av)\σess(av) contains a finite number of eigenvalues of av . we say that λ ∈ c is a characteristic value of equation (14) if there exists a nonzero x ∈ d(∆(λ))\{0} such that ∆(λ)x = 0, where ∆(λ) is defined by ∆(λ) := λd(eλ.i) + ad(eλ.i) − l(eλ.i), and the domain d(∆(λ)) is given by d(∆(λ)) := {x ∈ xα : d(e λ.x) ∈ d(a) and ad(eλ.x) − l(eλ.x) ∈ xα}. consequently, we deduce the following theorem. theorem 4.6. [4] assume that (h0), (h1), (h2), (h4), (h5),(h6) and (h7) hold. then, the following assertions hold 70 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) (i) λ is an eigenvalue of av iff λ is a characteristic value of equation (14). (ii) if s′(av) < 0, then (v(t))t≥0 is exponentially stable and consequently, the zero equilibrium of (u(t))t≥0 is locally exponentially stable. (iii) if s′(av) = 0, then there exists ϕ ∈ cα, ϕ 6= 0, such that ‖v(t)ϕ‖α = ‖ϕ‖α, for t ≥ 0. (iv) if s′(av) > 0, then there exists ϕ ∈ cα such that ‖v(t)ϕ‖α → +∞ as t → +∞ and consequently, the zero equilibrium of (u(t))t≥0 is instable. (v) assume that s′(av) ≤ 0 and let s0(av) := {λ ∈ pσ(av) : reλ = 0}. if each λ in s0(av) is a pole of order 1 of the resolvent operator of av, then (v(t))t≥0 is stable in the sense that there exists a positive constant m such that ‖v(t)‖α ≤ m, for all t ≥ 0. 5 example to apply our theoretical results, we consider the following model of partial differential equation with delay    ∂ ∂t [ v(t,x) − qv(t − r,x) + g( ∂ ∂x v(t − r,x)) ] = ∂2 ∂x2 [ v(t,x) − qv(t − r,x) +g( ∂ ∂x v(t − r,x)) ] + f ( v(t − r,x), ∂ ∂x [v(t,x) − qv(t − r,x)] ) for t ≥ 0 and x ∈ [0,π], v(t,0) − qv(t − r,0) = v(t,π) − qv(t − r,π) = 0 for t ≥ 0, v(θ,x) = v0(θ,x) for − r ≤ θ ≤ 0 and x ∈ [0,π], (16) where q, r are positive constants, u0 ∈ c([−r,0]× [0,π]; r) and f, g are lipschitz continuous functions. let x := l2([0,π]; r) equipped with the l2-norm ‖.‖2. consider the operator a : d(a) ⊂ x → x defined by ay = −y ′′ with domain d(a) = h2(0,π) ∩ h10(0,π). the spectrum σ(−a) of −a is equal to the point spectrum σp(−a) and is given by σ(−a) = σp(−a) = {−n 2 : n ≥ 1} and the associated eigenfunctions (en)n≥1 are given by en(s) = √ 2 π sin ns,s ∈ [0,π]. then ay = ∞∑ n=1 n2(y,en)en, y ∈ d(a). for each y ∈ d(a 1 2 ) := {y ∈ x : ∞∑ n=1 n(y,en)en ∈ x} the operator a 1 2 is given by a 1 2 y = ∞∑ n=1 n(y,en)en. it is well known that −a is the infinitesimal generator of an analytic semigroup (t(t))t≥0 on cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 71 x given by t(t)x = ∞∑ n=1 e−n 2 t(x,en)en, x ∈ x. it follows that (t(t))t≥0 is a compact semigroup on x and 0 ∈ ρ(a). this implies that the assumption (h0) and (h1) are satisfied. lemma 5.1. [27] if y ∈ d(a 1 2 ), then y is absolutely continuous, y ′ ∈ x and ‖y ′‖ = ‖a 1 2 y‖. let g : c1 2 → x be defined by g(ϕ)(x) = qϕ(−r)(x) − g( ∂ ∂x ϕ(−r)(x)) for ϕ ∈ c1 2 and x ∈ [0,π], and f : c1 2 → x be defined by f(ϕ)(x) = f ( ϕ(−r)(x), ∂ ∂x [ϕ(0)(x) − qϕ(−r)(x)] ) for ϕ ∈ c1 2 and x ∈ [0,π]. lemma 5.2. [4, 27] f and g are lipschitz continuous from c1 2 into x. let x(t) = v(t, .) for t ≥ 0 and ϕ(θ) = v0(θ,.) for θ ∈ [−r,0]. then, eq. (16) takes the following abstract form    d dt ( x(t) − g(t,xt) ) = −a ( x(t) − g(t,xt) ) + f(t,xt) for t ≥ 0, x0 = ϕ. (17) consequently, we have the existence and uniqueness of the mild solution of eq.(16). let v0 ∈ c1 2 such that (a) v0(0, .) − qv0(−r, .) + g( ∂ ∂x v0(−r, .)) ∈ h 2[0,π] ∩ h10[0,π] and ∂ ∂θ v0 ∈ c1 2 , (b) ∂ ∂θ v0(0,x) − q ∂ ∂θ v0(−r,x) + g ′( ∂ ∂x v0(−r,x)) ∂2 ∂x∂θ v0(−r,x) = −a [ v0(0,x) − qv0(−r,x) + g( ∂ ∂x v0(−r,x)) ] +f ( v0(−r,x), ∂ ∂x [v0(0,x) − qv0(−r,x)] ) for x ∈ [0,π] we deduce that all assumptions of theorem 3.1 are satisfied. hence every mild solution of eq. (16) is a strict solution. in the sequel, we assume that 0 < q < 1: this means that the operator d is stable. we also assume that f and g are continuously differentiable and f(0,0) = 0, g(0) = 0 and g′(0) = 0. which implies that zero is a solution of 16 and the linearized equation at zero of equation (16) has the following form    ∂ ∂t [ v(t,x) − qv(t − r,x) ] = ∂2 ∂x2 [ v(t,x) − qv(t − r,x) ] +av(t − r,x) + b ∂ ∂x [v(t,x) − qv(t − r,x)] for t ≥ 0 and x ∈ [0,π], v(t,0) − qv(t − r,0) = v(t,π) − qv(t − r,π) = 0 for t ≥ 0, v(θ,x) = v0(θ,x) for − r ≤ θ ≤ 0 and x ∈ [0,π], (18) 72 taoufik chitioui, khalil ezzinbi and amor rebey cubo 15, 1 (2013) we obtain a region of stability of equation (18) as a function of parameters a, b and q. lemma 5.3. [4] the spectrum σ(ã) of the operator ã = ∂2 ∂x2 +b ∂ ∂x is equal to the point spectrum σp(ã) and is given by {−n 2 − b 2 4 : n ≥ 1}. theorem 5.1. suppose that a < 0 and 1 + b2 4 + a q ≥ 0. then, for every r > 0, all characteristic values of eq. (18) have negative real parts. proof. suppose that a < 0. then, the characteristic values of eq. (18) are determined by the expression λ − ae−λr 1 − qe−λr = −n2 − b2 4 , n ≥ 1. (19) let kn = n 2 + b 2 4 , n ≥ 1. then, eq. (19) becomes eλr(λ + kn) = λq + knq + a. this implies that e2re(λ)r ( (re(λ) + kn) 2 + (im(λ))2 = q2 ( (re(λ) + kn + a q ) + (im(λ))2 ) . on the other hand, under the conditions a < 0 and 1 + b2 4 + a q ≥ 0, we have, for all n ≥ 1 and λ ∈ c, re(λ) + kn > re(λ) + kn + a q ≥ re(λ) + 1 + b2 4 + a q ≥ re(λ). then, if we assume that re(λ) ≥ 0, we obtain that e2re(λ)r < q2, which is a contradiction. then, re(λ) < 0. remark that the stability result is independent of the delay. finally, as an immediate consequence of the last theorem, we have the local stability of the zero equilibrium of equation (16). proposition 5.4. under the same assumptions as in theorem 5.1, zero equilibrium of equation (16) is locally exponentially stable. received: october 2012. revised: february 2013. cubo 15, 1 (2013) existence and stability in the α-norm for nonlinear neutral ... 73 references [1] m.adimy and k. ezzinbi, a class of linear partial neutral functional dierential equations with nondense domain, journal of differential equations, 147, (1998), 285-332. 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() cubo a mathematical journal vol.16, no¯ 03, (11–19). october 2014 a note on inextensible flows of curves on oriented surface onder gokmen yildiz department of mathematics, faculty of arts and sciences, bilecik seyh edebali university, bilecik/turkey ogokmen.yildiz@bilecik.edu.tr soley ersoy department of mathematics, faculty of arts and sciences, sakarya university, sakarya/turkey sersoy@sakarya.edu.tr melek masal department of mathematics teaching, faculty of education, sakarya university, sakarya/turkey mmasal@sakarya.edu.tr abstract in this paper, we investigate a general formulation for inextensible flows of curves on an oriented surface in r3. we obtain necessary and sufficient conditions as partial differential equations involving the geodesic curvature and the geodesic torsion for inextensible curve flow lying on an oriented surface. moreover, some special cases of inextensible curves on oriented surface are given. resumen en este art́ıculo investigamos una formulación general para flujos inextensibles de curvas sobre una superficie orientable en r3. obtenemos condiciones necesarias y suficientes para las ecuaciones diferenciales parciales que involucran la curva geodésica y la torsión geodésica para curvas inextensibles fluyendo sobre superficies orientadas. más aún, se entregan algunos casos especiales de curvas inextensibles sobre superficies orientadas. keywords and phrases: curvature flows, inextensible, oriented surface. 2010 ams mathematics subject classification: 53c44, 53a04, 53a05. 12 onder gokmen yildiz, soley ersoy & melek masal cubo 16, 3 (2014) 1 introduction the flow of the curve is said to be inextensible if its arclength preserved. curve design using splines is one of the most fundamental topic in cagd. inextensible flows of the curves have beautiful shapes preserving connection to their control polygon. on the other hand, physically inextensible curve and surface flows give rise to motion which no strain energy is induced. for example, the swinging motion of a cord of fixed length can be described by inextensible curve and surface flows. many authors have studied geometric flow problems and applications of inextensible curve flows, [1]–[10]. an evolution equation for inelastic planar curves was derived by [9] and also, the general formulation of inextensible flows of curves and developable surfaces in r3 was exposed by [10]. in this paper, we derive a general formulation for inextensible flows of curves according to darboux frame in r3. we give the necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equations involving the geodesic curvature and geodesic torsion. 2 preliminaries let s be an oriented surface in three-dimensional euclidean space e3 and α(s) be a curve lying on the surface s. suppose that the curve α(s) is spatial then there exists the frenet frame { −→ t , −→ n, −→ b } at each points of the curve where −→ t is unit tangent vector, −→ n is principal normal vector and −→ b is binormal vector, respectively. the frenet equation of the curve α(s) is given by −→ t′ = κ −→ n −→ n′ = −κ −→ t + τ −→ b −→ b′ = −τ −→ n where κ and τ are curvature and torsion of the curve α(s), respectively. since the curve α(s) lies on the surface s there exists another frame of the curve α(s) which is called darboux frame and denoted by { −→ t , −→ g , −→ n } . in this frame −→ t is the unit tangent of the curve, −→ n is the unit normal of the surface s and −→ g is a unit vector given by −→ g = −→ n × −→ t . since the unit tangent −→ t is common element of both frenet frame and darboux frame, the vectors −→ n, −→ b, −→ g and −→ n lie on the same plane. so that the relations between these frames can be given as follows     −→ t −→ g −→ n     =     1 0 0 0 cosϕ sinϕ 0 − sinϕ cosϕ         −→ t −→ n −→ b     where ϕ is the angle between the vectors −→ g and −→ n. the derivative formulae of the darboux frame cubo 16, 3 (2014) a note on inextensible flows of curves on oriented surface 13 is     . −→ t . −→ g . −→ n     =     0 kg kn − kg 0 τg −kn − τg 0         −→ t −→ g −→ n     where kg, kn and τg are called the geodesic curvature, the normal curvature and the geodesic torsions, respectively. here and in the following, we use ”dot” to denote the derivative with respect to the arc length parameter of a curve. the relations between the geodesic curvature, normal curvature, geodesic torsion and κ, τ are given as follows, [11] kg = κcosϕ, kn = −κsinϕ, τg = τ + dϕ ds . furthermore, the geodesic curvature kg and geodesic torsion τg of the curve α(s) can be calculated as follows, [11] kg = 〈 d −→ α ds , d 2−→ α ds2 × −→ n 〉 τg = 〈 d −→ α ds , −→ n × d −→ n ds 〉 . in the differential geometry of surfaces, for any curve α(s) lying on a surface s the following relationships are well-known, [11] iα(s) is a geodesic curve if and only if kg = 0, iiα(s) is an asymptotic line if and only if kn = 0, iiiα(s) is a principal line if and only if τg = 0. through each point on a surface there passes, in general, a geodesic in every direction. a geodesic is uniquely determined by an initial point and tangent at that point. all straight lines on a surface are geodesics. along all curved geodesics the principal normal coincides with the surface normal. along asymptotic lines osculating planes and tangent planes coincide, along geodesics they are normal. through a point of a non-developable surface there are two asymptotic lines which can be real or imaginary. 3 inextensible flows of curve lying on oriented surface throughout this paper, we suppose that α : [0,l] × [0,w) → m ⊂ e3 is a one parameter family of differentiable curves on orientable surface m in e3, where l is the arclength of the initial curve. let u be the curve parameterization variable, 0 ≤ u ≤ l. if the speed of curve α is denoted by v = ∥ ∥ ∥ ∂ −→ α ∂u ∥ ∥ ∥ then the arclength of α is 14 onder gokmen yildiz, soley ersoy & melek masal cubo 16, 3 (2014) s(u) = u∫ 0 ∥ ∥ ∥ ∥ ∂ −→ α ∂u ∥ ∥ ∥ ∥ du = u∫ 0 vdu. (3.1) the operator ∂ ∂s is given in terms of u by ∂ ∂s = 1 v ∂ ∂u . (3.2) thus, the arclength is ds = vdu. definition 3.1. let m be an orientable surface and α be a differentiable curve on m in e3. any flow of the curve α can be expressed with respect to darboux frame { −→ t , −→ g , −→ n } in the following form: ∂ −→ α ∂t = f1 −→ t + f2 −→ g + f3 −→ n. (3.3) here, f1, f2 and f3 are scalar speeds of the curve α. let the arclength variation be s(u,t) = u∫ 0 vdu. (3.4) in the euclidean space the requirement that a curve not to be subject to any elongation or compression can be expressed by the condition ∂ ∂t s(u,t) = u∫ 0 ∂v ∂t du = 0 , u ∈ [0,1]. (3.5) definition 3.2. a curve evolution α(u,t) and its flow ∂ −→ α ∂t on the oriented surface m in e3 are said to be inextensible if ∂ ∂t ∥ ∥ ∥ ∥ ∂ −→ α ∂u ∥ ∥ ∥ ∥ = 0. now, we research the necessary and sufficient condition of a flow to be inextensible. for this reason, we need to the following lemma. lemma 3.1. in e3, let m be an orientable surface and { −→ t , −→ g , −→ n } be a darboux frame of α on m. there exists following relation between the scalar speed functions f1, f2, f3 and the normal curvature kn, geodesic curvature kg of α the curve ∂v ∂t = ∂f1 ∂u − f2vkg − f3vkn. (3.6) cubo 16, 3 (2014) a note on inextensible flows of curves on oriented surface 15 proof. since ∂ ∂u and ∂ ∂t commute and v2 = 〈 ∂ −→ α ∂u , ∂ −→ α ∂u 〉 , we have 2v∂v ∂t = ∂ ∂t 〈 ∂ −→ α ∂u , ∂ −→ α ∂u 〉 = 2 〈 ∂ −→ α ∂u , ∂ ∂u ( f1 −→ t + f2 −→ g + f3 −→ n )〉 = 2v ( ∂f1 ∂u − f2vkg − f3vkn ) . this completes the proof. if we consider the conditions of being geodesic and asymptotic of a curve and lemma 3.1, we can give the following corollary. corollary 3.1. if a curve is a geodesic curve or an asymptotic curve, then there are the following equations ∂v ∂t = ∂f1 ∂u − f3vkn or ∂v ∂t = ∂f1 ∂u − f2vkg, respectively. theorem 3.1. let { −→ t , −→ g , −→ n } be the darboux frame of a curve α on m and ∂ −→ α ∂t = f1 −→ t +f2 −→ g + f3 −→ n be a differentiable flow of α in r3. then the flow is inextensible if and only if ∂f1 ∂s = f2kg + f3kn. (3.7) proof. suppose that the curve flow is inextensible. from the equations (3.4) and (3.6) for u ∈ [0,l] we see that ∂ ∂t s(u,t) = u∫ 0 ∂v ∂t du = u∫ 0 ( ∂f1 ∂u − f2vkg − f3vkn ) du = 0. (3.8) thus, it can be seen that ∂f1 ∂u = f2vkg + f3vkn. (3.9) considering the last equation and (3.2), we reach ∂f1 ∂s = f2kg + f3kn. conversely, by following a similar way as above, the proof is completed. from theorem 3.1, we have following corollary. 16 onder gokmen yildiz, soley ersoy & melek masal cubo 16, 3 (2014) corollary 3.2. ilet the curve α is a geodesic curve on m. then the curve flow is inextensible if and only if ∂f1 ∂s = f3kn. iilet the curve α is an asymptotic line on m. then the curve flow is inextensible if and only if ∂f1 ∂s = f2kg. now, we restrict ourselves to the arclength parameterized curves. that is, v = 1 and the local coordinate u corresponds to the curve arclength s. we require the following lemma. lemma 3.2. let m be an orientable surface in e3 and { −→ t , −→ g , −→ n } be a darboux frame of the curve α on m. then, the differentiations of { −→ t , −→ g , −→ n } with respect to t is ∂ −→ t ∂t = ( f1kg + ∂f2 ∂s − f3τg ) −→ g + ( f1kn + ∂f3 ∂s + f2τg ) −→ n ∂ −→ g ∂t = − ( f1kg + ∂f2 ∂s − f3τg ) −→ t + ψ −→ n ∂ −→ n ∂t = − ( f1kn + ∂f3 ∂s + f2τg ) −→ t − ψ −→ g where ψ = 〈 ∂ −→ g ∂t , −→ n 〉 . proof. since ∂ ∂t and ∂ ∂s are commutative, it seen that ∂ −→ t ∂t = ∂ ∂t ( ∂ −→ α ∂s ) = ∂ ∂s ( ∂ −→ α ∂t ) = ∂ ∂s ( f1 −→ t + f2 −→ g + f3 −→ n ) = ∂f1 ∂s −→ t + f1 ∂ −→ t ∂s + ∂f2 ∂s −→ g + f2 ∂ −→ g ∂s + ∂f3 ∂s −→ n + f3 ∂ −→ n ∂s . substituting the equation (3.7) into the last equation and using theorem 3.1, we have ∂ −→ t ∂t = ( f1kg + ∂f2 ∂s − f3τg ) −→ g + ( f1kn + ∂f3 ∂s + f2τg ) −→ n. now, let us differentiate the darboux frame with respect to t as follows; 0 = ∂ ∂t 〈 −→ t , −→ g 〉 = 〈 ∂ −→ t ∂t , −→ g 〉 + 〈 −→ t , ∂ −→ g ∂t 〉 = ( f1kg + ∂f2 ∂s − f3τg ) + 〈 −→ t , ∂ −→ g ∂t 〉 (3.10) 0 = ∂ ∂t 〈 −→ t , −→ n 〉 = 〈 ∂ −→ t ∂t , −→ n 〉 + 〈 −→ t , ∂ −→ n ∂t 〉 = ( f1kn + ∂f3 ∂s + f2τg ) + 〈 −→ t , ∂ −→ n ∂t 〉 (3.11) from (3.10) and (3.11), we have obtain ∂ −→ g ∂t = − ( f1kg + ∂f2 ∂s − f3τg ) −→ t + ψ −→ n cubo 16, 3 (2014) a note on inextensible flows of curves on oriented surface 17 and ∂ −→ n ∂t = − ( f1kn + ∂f3 ∂s + f2τg ) −→ t − ψ −→ g respectively, where ψ = 〈 ∂ −→ g ∂t , −→ n 〉 . if we take into consideration last lemma, we have following corollary. corollary 3.3. let m be an orientable surface in e3. iif the curve α is a geodesic curve, then ∂ −→ t ∂t = ( ∂f2 ∂s − f3τg ) −→ g + ( f1kn + ∂f3 ∂s + f2τg ) −→ n, ∂ −→ g ∂t = − ( ∂f2 ∂s − f3τg ) −→ t + ψ −→ n, ∂ −→ n ∂t = − ( f1kn + ∂f3 ∂s + f2τg ) −→ t − ψ −→ g , where ψ = 〈 ∂ −→ g ∂t , −→ n 〉 . iiif the curve α is an asymptotic line, then ∂ −→ t ∂t = ( f1kg + ∂f2 ∂s − f3τg ) −→ g + ( ∂f3 ∂s + f2τg ) −→ n, ∂ −→ g ∂t = − ( f1kg + ∂f2 ∂s − f3τg ) −→ t + ψ −→ n, ∂ −→ n ∂t = − ( ∂f3 ∂s + f2τg ) −→ t − ψ −→ g , where ψ = 〈 ∂ −→ g ∂t , −→ n 〉 . iiiif the curve is a curvature line, then ∂ −→ t ∂t = ( f1kg + ∂f2 ∂s ) −→ g + ( f1kn + ∂f3 ∂s ) −→ n, ∂ −→ g ∂t = − ( f1kg + ∂f2 ∂s ) −→ t + ψ −→ n, ∂ −→ n ∂t = − ( f1kn + ∂f3 ∂s ) −→ t − ψ −→ g , where ψ = 〈 ∂ −→ g ∂t , −→ n 〉 . theorem 3.2. suppose that the curve flow ∂ −→ α ∂t = f1 −→ t +f2 −→ g +f3 −→ n is inextensible on the orientable surface on m. in this case, the following partial differential equations are held: ∂kg ∂t = f2k 2 g + f3kgkn + f1 ∂kg ∂s + ∂ 2 f2 ∂s2 − 2∂f3 ∂s τg − f3 ∂τg ∂s − f1knτg − f2τ 2 g + ψkn, ∂kn ∂t = f2kgkn + f3k 2 n + f1 ∂kn ∂s + ∂ 2 f3 ∂s2 + 2∂f2 ∂s τg + f2 ∂τg ∂s + f1kgτg − f3τ 2 g − ψkg, ∂τg ∂t = f2kgτg − ∂f2 ∂s kn + ∂f3 ∂s kg + f3knτg + ∂ψ ∂s . proof. since ∂ ∂s ∂ −→ t ∂t = ∂ ∂t ∂ −→ t ∂s we get ∂ ∂s ∂ −→ t ∂t = ∂ ∂s [( f1kg + ∂f2 ∂s − f3τg ) −→ g + ( f1kn + ∂f3 ∂s + f2τg ) −→ n ] = ( ∂f1 ∂s kg + f1 ∂kg ∂s + ∂ 2 f2 ∂s2 − ∂f3 ∂s τg − f3 ∂τg ∂s ) −→ g + ( f1kg + ∂f2 ∂s − f3τg ) ∂ −→ g ∂s + ( ∂f1 ∂s kn + f1 ∂kn ∂s + ∂ 2 f3 ∂s2 + ∂f2 ∂s τg + f2 ∂τg ∂s ) −→ n + ( f1kn + ∂f3 ∂s + f2τg ) ∂ −→ n ∂s 18 onder gokmen yildiz, soley ersoy & melek masal cubo 16, 3 (2014) i.e., ∂ ∂s ∂ −→ t ∂t = ( ∂f1 ∂s kg + f1 ∂kg ∂s + ∂ 2 f2 ∂s2 − ∂f3 ∂s τg − f3 ∂τg ∂s ) −→ g + ( f1kg + ∂f2 ∂s − f3τg ) ( −kg −→ t + τg −→ n ) + ( ∂f1 ∂s kn + f1 ∂kn ∂s + ∂ 2 f3 ∂s2 + ∂f2 ∂s τg + f2 ∂τg ∂s ) −→ n + ( f1kn + ∂f3 ∂s + f2τg ) ( −kg −→ t − τg −→ g ) while ∂ ∂t ∂ −→ t ∂s = ∂ ∂t ( kg −→ g + kn −→ n ) = ∂kg ∂t −→ g + kg ∂ −→ g ∂t + ∂kn ∂t −→ n + kn ∂ −→ n ∂t . thus, from the both of above two equations, we reach ∂kg ∂t = f2k 2 g + f3kgkn + f1 ∂kg ∂s + ∂ 2 f2 ∂s2 − 2 ∂f3 ∂s τg − f3 ∂τg ∂s − f1knτg − f2τ 2 g + ψkn (3.12) and ∂kn ∂t = f2kgkn + f3k 2 n + f1 ∂kn ∂s + ∂ 2 f3 ∂s2 + 2 ∂f2 ∂s τg + f2 ∂τg ∂s + f1kgτg − f3τ 2 g − ψkg. (3.13) noting that ∂ ∂s ∂ −→ g ∂t = ∂ ∂t ∂ −→ g ∂s , it is seen that ∂ ∂s ∂ −→ g ∂t = ∂ ∂s [ − ( f1kg + ∂f2 ∂s − f3τg ) −→ t + ψ −→ n ] = − ( ∂f1 ∂s kg + f1 ∂kg ∂s + ∂ 2 f2 ∂s2 − ∂f3 ∂s τg − f3 ∂τg ∂s ) −→ t − ( f1kg + ∂f2 ∂s − f3τg ) ( kg −→ g + kn −→ n ) + ∂ψ ∂s n + ψ ( −kn −→ t − τg −→ g ) while ∂ ∂t ∂ −→ g ∂s = ∂ ∂t ( −kg −→ t + τg −→ n ) = − ∂kg ∂t −→ t − kg ∂ −→ t ∂t + ∂τg ∂t −→ n + τg ∂ −→ n ∂t . thus, we obtain ∂τg ∂t = f2kgτg − ∂f2 ∂s kn + ∂f3 ∂s kg + f3knτg + ∂ψ ∂s . (3.14) no other new formulas are obtained from the relation ∂ ∂s ∂ −→ n ∂t = ∂ ∂t ∂ −→ n ∂s . thus, we give the following corollary from last theorem. corollary 3.4. let m be an orientable surface in e3. iif the curve α is a geodesic curve on m, then we have ∂kn ∂t = f3k 2 n + f1 ∂kn ∂s + ∂2f3 ∂s2 + 2 ∂f2 ∂s τg + f2 ∂τg ∂s − f3τ 2 g and ∂τg ∂t = − ∂f2 ∂s kn + f3knτg + ∂ψ ∂s . iiif the curve α is an asymptotic line, we have ∂kg ∂t = f2k 2 g+f1 ∂kg ∂s + ∂2f2 ∂s2 − 2 ∂f3 ∂s τg − f3 ∂τg ∂s − f2τ 2 g cubo 16, 3 (2014) a note on inextensible flows of curves on oriented surface 19 and ∂τg ∂t = f2kgτg + ∂f3 ∂s kg + ∂ψ ∂s . iiiif the curve α is a curvature line, then we have ∂kg ∂t = f2k 2 g + f3kgkn + f1 ∂kg ∂s + ∂ 2 f2 ∂s2 + ψkn ∂kn ∂t = f2kgkn + f3k 2 n + f1 ∂kn ∂s + ∂ 2 f3 ∂s2 − ψkg. received: september 2013. accepted: may 2014. references [1] g. chirikjian, j. burdick, a modal approach to hyper-redundant manipulator kinematics, ieee trans. robot. autom. 10 (1994), 343–354. 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[9] d.y. kwon, f.c. park, evolution of inelastic plane curves, appl. math. lett., 12 (1999), 115–119. [10] d.y. kwon, f.c. park, d.p. chi, inextensible flows of curves and developable surfaces, appl. math. lett., 18 (2005), 1156–1162. [11] b. o’neill, elementary differential geometry, academic press inc. new york, 1966. introduction preliminaries inextensible flows of curve lying on oriented surface articulo 3.dvi cubo a mathematical journal vol.12, no¯ 02, (29–42). june 2010 the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel m. h. saleh, s. m. amer 1 and m. h. ahmed departement of mathematics, faculty of science, zagazig university, zagazig, egypt. email: amrsammer@hotmail.com abstract this paper concerned with applicability of the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel . the results are illustrated in hölder space. resumen este art́ıculo es concerniente a la aplicabilidad del método de mayorantes de kantorovich para ecuaciones integrales singulares no lineales con núcleo de hilbert. los resultados son aplicaciones en espacios de hölder. key words and phrases: nonlinear singular integral equations, kantorovich majorants method, hölder spaces. ams 2000-subject classification: 45f15, 45g10. 1. introduction there is a large literature on nonlinear singular integral equations with hilbert and cauchy kernel and related riemann boundary value problems for analytic functions,cf.the monograph by pogorzel1corresponding author 30 m. h. saleh, s. m. amer and m. h. ahmed cubo 12, 2 (2010) ski [16], guseinov a.i. and mukhtarov kh.sh. [9],kantorovich l.v.[11],muskhelishvill n.i.[14],and mikhlin s.g.and prossdorf s.[13].the method of singular integral equations on closed contour has been intensively investigated by many approximation methods, specially method of modified newtonkantorovich, reduction, collocation and mechanical quadratures, (see[1-6,10,12,15,17,19]). in this paper the method of kantorovich majorants[7,18,20], has been applied to the following class of nonlinear singular integral equations with hilbert kernel : ϕ(t) = λg(t, 1 2π ∫ 2π 0 g(t,σ,ϕ(σ)) cot σ − t 2 dσ), (1.1) where λ is a numerical parameter, where v(t) = sg(t,σ,ϕ(σ)) = 1 2π ∫ 2π 0 g(t,σ,ϕ(σ)) cot σ − t 2 dσ, then equation (1.1) takes the form: ϕ(t) = λg(t,v(t)). now, we consider the equation: b(ϕ) = 0, (1.2) where (bϕ)(t) = ϕ(t) − λg(t,v(t)). (1.3) 2. formulation of the problem let f : s̄(ϕ 0 ,r) ⊂ x −→ y be a nonlinear operator defined on the closure of a ball s(ϕ 0 ,r) = {ϕ : ϕ ∈ x,‖ ϕ − ϕ 0 ‖< r}, in a banach space x into a banach space y. we give new conditions to ensure the convergence on newton-kantorovich approximations toward a solution of f(ϕ) = 0, under the hypothesis that f is frechet differentiable in s(ϕ 0 ,r), and that it’s derivative f̀ satisfies the local lipschitz condition : ‖ f̀(ϕ 1 ) − f̀(ϕ 2 ) ‖≤ k(r) ‖ ϕ 1 − ϕ 2 ‖,ϕ 1 ,ϕ 2 ∈ s̄(ϕ 0 ,r),o < r < r, (2.1) where k(r)is a non decreasing function on the interval [0,r] and k(r) = sup{‖ f̀(ϕ1 ) − f̀(ϕ2 ) ‖ ‖ ϕ 1 − ϕ 2 ‖ ϕ 1 ,ϕ 2 ∈ s̄(ϕ 0 ,r)ϕ 1 6= ϕ 2 }. (2.2) define a scalar function ψ : [0,r] → (0,∞) by ψ(r) = a + b ∫ r 0 w(t)dt − r, (2.3) cubo 12, 2 (2010) the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel 31 using the function w(r) = ∫ r 0 k(t)dt, (2.4) and a =‖ f̀(ϕ 0 )−1f(ϕ 0 ) ‖, b =‖ f̀(ϕ 0 )−1 ‖ . (2.5) theorem 2.1 [4,7] suppose that the equation (2.3)has a unique positive root r ∗ in [0,r] and ψ(r) ≤ 0. then the equation f(ϕ) = 0 has a unique solution ϕ ∗ in s(ϕ 0 ,r) and the newtonkantorovich approximations: ϕ n = ϕ n−1 − f̀(ϕ n−1 )−1f(ϕ n−1 ), n ∈ n, (2.6) are defined for all n ∈ n, belong to s(ϕ 0 ,r ∗ ) and converges to ϕ ∗ . moreover , the following estimate holds ‖ ϕ n+1 − ϕ n ‖≤ r n+1 − r n , ‖ ϕ ∗ − ϕ n ‖≤ r ∗ − r n , (2.7) where the sequence (r n )n∈n converges to r∗ , is defined by the recurrence formula r 0 = 0, r n+1 = r n − ψ(rn ) `ψ(r n ) , n ∈ n. (2.8) in the present paper , we investigate some sufficient conditions , which ensure that the class of nonlinear singular integral equations (1.1) verifies the hypotheses of theorem (2.1). 3. some auxiliary results definition 3.1[9] we denote by h δ , 0 < δ < 1,the hölder space of continuous functions , which satisfy the hölder condition with exponent δ with norm ‖ ϕ ‖ δ =‖ ϕ ‖ c +hδ(ϕ), (3.1) where ‖ ϕ ‖ c = max σ∈[0,2π] | ϕ(σ) |, and h δ(ϕ) = sup σ 1 6= σ 2 | ϕ(σ1) − ϕ(σ2) | | σ1 − σ2 |δ . lemma 3.1 [9] let the functions g(t,v(t)), g(t,σ,ϕ(σ)) and it’s partial derivatives up to second order, satisfy the following conditions | ∂ mg(t1,v(t1)) ∂vm − ∂ mg(t2,v(t2)) ∂vm |≤ cm(r){| t1 − t2 |δ + | v(t1 ) − v(t2 ) |}, (3.2) 32 m. h. saleh, s. m. amer and m. h. ahmed cubo 12, 2 (2010) and | ∂mg ϕ (t1,σ1,ϕ(σ1)) ∂ϕm − ∂mg ϕ (t2,σ2,ϕ(σ2)) ∂ϕm |≤ am(r){| t1 − t2 |δ + | σ1 − σ2 |δ + | ϕ(t1 ) − ϕ(t2 ) |} (3.3) where cm(r),am(r) are positive increasing functions m=0,1,2 and t i ,σ i ∈ [0, 2π], i = 1, 2. if ϕ(σ) ∈ h δ , then g(t,v(t)), g(t,σ,ϕ(σ)) ∈ h δ . lemma 3.2 if the functions g(t,v(t)) and g(t,σ,ϕ(σ)) satisfy the conditions of lemma(3.1), then the operator b(ϕ) has a frechet derivative at every fixed point in the space h δ and its derivative is given by b̀(ϕ)h = h(t) − λgv(t,v(t))sg ϕ (t,σ,ϕ(σ))h(σ). (3.4) moreover it satisfies lipschitz condition: ‖ b̀(ϕ 1 ) − b̀(ϕ 2 ) ‖≤ k(r) ‖ ϕ 1 − ϕ 2 ‖, (3.5) for all ϕ 1 ,ϕ 2 ∈ s(ϕ 0 ,r) and o < r < r. proof let ϕ(t)be any fixed point in the space 0,< δ < 1and h(t)be any arbitrary element in h δ , then we obtain : b(ϕ + h) − b(ϕ) = h(t) − λ[g(t,sg(t,σ,ϕ(σ) + h(σ))) − g(t,sg(t,σ,ϕ(σ)))] = b̀(ϕ)h + η(t,h), where 0 ≤ ξ ≤ 1 and η(t,h) = λ ∫ 1 0 (1 − ξ)[gv2 (t,sg(t,σ,ϕ(σ) + ξh(σ)))(sg ϕ (t,σ,ϕ(σ) + ξh(σ))h(σ))2 + gv(t,sg(t,σ,ϕ(σ) + ξh(σ)))sg ϕ2 (t,σ,ϕ(σ) + ξh(σ))h(σ)2]dξ. now , we shall prove that lim ‖h‖→0 ‖ η(t,h) ‖ ‖ h ‖ = 0. using the inequalities [9,13] ‖ ∫ b a y(s) s−x ds ‖≤ ρ 0 ‖ y ‖,where ρ0 is a positive constant ‖ uv ‖≤‖ u ‖‖ v ‖ for all u,v ∈ h δ        . (3.6) now; ‖ η(t,h) ‖ ≤ ‖ h(σ)2 ‖ ρ 0 [ ‖ gv2 (t,sg(t,σ,ϕ(σ))) ‖‖ (g ϕ (t,σ,ϕ(σ)))2 ‖ + ‖ gv(t,sg(t,σ,ϕ(σ))) ‖‖ g ϕ2 (t,σ,ϕ(σ)) ‖ ]. hence lim ‖h‖→0 ‖ η(t,h) ‖ ‖ h ‖ = 0, cubo 12, 2 (2010) the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel 33 which prove the differentiability of b(ϕ) in the sense of frechet and its derivative is given by (3.4). to prove the frechet derivative b̀(ϕ) satisfies lipschitz condition in the sphere s(ϕ 0 ,r) = {ϕ :‖ ϕ − ϕ 0 ‖< r}. we consider ‖ b̀(ϕ 1 )h − b̀(ϕ 2 )h ‖ = ‖ λgv(t,sg(t,σ,ϕ1(σ)))sg ϕ (t,σ,ϕ1(σ))h(σ) − λgv(t,sg(t,σ,ϕ2(σ)))sg ϕ (t,σ,ϕ2(σ))h(σ) ‖ ≤ | λ |‖ h ‖ [ ‖ gv(t,v1(t)) ‖‖ sg ϕ (t,σ,ϕ1(σ)) − sg ϕ (t,σ,ϕ2(σ)) ‖ + ‖ sg ϕ (t,σ,ϕ2(σ)) ‖‖ gv(t,v1(t)) − gv(t,v2(t)) ‖ ] ≤ ‖ h ‖ k(r) ‖ ϕ 1 − ϕ 2 ‖, where k(r) =| λ | ρ 0 [a1(r)d+ ‖ gϕ ‖ c1(r)a0(r)] , and d = max t | gv(t,sg(t,σ,ϕ(σ))) | then the lemma is proved. 4. solution of linear singular integral equation to find the operator b̀(ϕ 0 )−1, we investigate the solution of the equation h(t) − λgv(t,v(t)) 2π ∫ 2π 0 g ϕ (t,σ,ϕ(σ)) h(σ) cot σ − t 2 dσ = f(t). (4.1) for this aim we introduce the following theorem: theorem 4.1 if the functions g(t,v(t)) and g(t,σ,ϕ(σ)) satisfy the conditions of lemma(3.2), then the linear operator defined by (3.4) has a bounded inverse b̀(ϕ 0 )−1 for any fixed ϕ 0 ∈ h δ , (0 < δ < 1). proof let us transform the equation (4.1) by introducing new variables : s = eit,τ = eiσ,dτ = ieiσdσ, since 1 2 cot σ − t 2 dσ = ( 1 τ − s − 1 2τ )dτ, then equation (4.1) has the form h(s) − λxv(s,v(s)) πi ∫ γ ik ϕ (s,τ,ϕ(τ)) h(τ) ( 1 τ − s − 1 2τ )dτ = f(s), (4.2) where γ is a unit circle , gv(t,v(t)) = xv(s,v(s)) and g ϕ (t,σ,ϕ(σ)) = k ϕ (s,τ,ϕ(τ)). we introduce the sectionally holomorphic function of variable z as follows: h(z) = λxv(s,v(s)) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ)) τ − z h(τ)dτ − c, (4.3) 34 m. h. saleh, s. m. amer and m. h. ahmed cubo 12, 2 (2010) and h(∞) = −c = −λxv(s,v(s)) 4π ∫ γ ik ϕ (s,τ,ϕ(τ)) τ h(τ)dτ = −iλgv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ)) h(σ)dσ. according to sokhotoski formulae[9], we have h±(s) = ±iλxv(s,v(s)) 2 k ϕ (s,s,ϕ(s))h(s) + λxv(s,v(s)) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ)) τ − s h(τ)dτ − c. therefore h+(s) − h−(s) = iλxv(s,v(s))k ϕ (s,s,ϕ(s))h(s) h+(s) + h−(s) = λxv (s,v(s)) πi ∫ γ ik ϕ (s,τ,ϕ(τ )) τ −s h(τ)dτ − 2c        . (4.4) substituting from equation (4.4) into equation (4.2 )we have h(s) − f(s) + 2c = h+(s) + h−(s) + 2c. (4.5) hence we get h(s) = h+(s) + h−(s) + f(s), (4.6) therefore from (4.4) and (4.6) we have, h(s)[1 ± iλxv(s,v(s))k ϕ (s,s,ϕ(s))] = 2h±(s) + f(s), since1 ± iλxv(s,v(s))k ϕ (s,s,ϕ(s)) 6= 0, then the last conditions equivalent to the following h(s) = 2h+(s) 1+iλxv (s,v(s))k ϕ (s,s,ϕ(s)) + f (s) 1+iλxv (s,v(s))k ϕ (s,s,ϕ(s)) , h(s) = 2h−(s) 1−iλxv (s,v(s))k ϕ (s,s,ϕ(s)) + f (s) 1−iλxv (s,v(s))k ϕ (s,s,ϕ(s))          . (4.7) by equating the right hand side of equation (4.7) we get the riemann boundary value problem h +(s) = 1 + iλxv(s,v(s))k ϕ (s,s,ϕ(s)) 1 − iλxv(s,v(s))k ϕ (s,s,ϕ(s)) h −(s) + iλxv(s,v(s))k ϕ (s,s,ϕ(s)) 1 − iλxv(s,v(s))k ϕ (s,s,ϕ(s)) f(s). (4.8) it is well known that the index of equation (4.8) is zero[8],then 1 + iλxv(s,v(s))k ϕ (s,s,ϕ(s)) 1 − iλxv(s,v(s))k ϕ (s,s,ϕ(s)) = x+(s) x−(s) , where x(z) = 1 2π ∫ γ ln 1 + iλxv(s,v(s))ik ϕ (s,τ,ϕ(τ)) 1 − iλxv(s,v(s))k ϕ (s,τ,ϕ(τ)) dτ τ − z , cubo 12, 2 (2010) the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel 35 the problem (4.8)can be written in the form h+(s) x+(s) − h −(s) x−(s) = iλxv(s,v(s))k ϕ (s,s,ϕ(s))f(s) 1 − iλxv(s,v(s))k ϕ (s,s,ϕ(s))x+(s) . hence ,from [8], the boundary value problem (4.8) has the solution h(z) = x(z)[ λxv(s,v(s)) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ))f(τ) x+(τ)(1 − iλxv(s,v(s))k ϕ (s,τ,ϕ(τ))) dτ τ − s − c]. by sokhotski formulae h+(s) = iλxv(s,v(s))k ϕ (s,s,ϕ(s))f(s) 2(1 − iλxv(s,v(s))k ϕ (s,s,ϕ(s))) + λxv(s,v(s))x +(s) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ))f(τ) x+(τ)(1 − iλxv(s,v(s))k ϕ (s,τ,ϕ(τ))) dτ τ − s − cx+(s). (4.9) substituting from (4.9) into (4.7) we have, h(s) = f(s) u(s) + z(s)λxv(s,v(s)) u(s)πi ∫ γ ik ϕ (s,τ,ϕ(τ))f(τ) z(τ) dτ τ − s − 2cz(s) u(s) , (4.10) where u(s) = 1 + λ2x2v (s,v(s))k 2 ϕ (s,s,ϕ(s)), z(s) = √ u(s)eγ(s), and γ(s) = 1 2πi ∫ γ ln 1 + iλxv(s,v(s))ik ϕ (s,τ,ϕ(τ)) 1 − iλxv(s,v(s))k ϕ (s,τ,ϕ(τ)) dτ τ − s , since dτ τ − s = 1 2 cot σ − t 2 + i 2 dσ. hence z(eit) = z(s) = √ u(t)exp ( 1 4π ∫ 2π 0 ln 1 + iλgv(t,v(t))g ϕ (t,σ,ϕ(σ)) 1 − iλgv(t,v(t))g ϕ (t,σ,ϕ(σ)) dσ exp ( 1 4πi ∫ 2π 0 ln 1 + iλgv(t,v(t))g ϕ (t,σ,ϕ(σ)) 1 − iλgv(t,v(t))g ϕ (t,σ,ϕ(σ)) cot σ − t 2 dσ)). 36 m. h. saleh, s. m. amer and m. h. ahmed cubo 12, 2 (2010) now we determine the constant c as follows c = iλgv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ)) h(σ) dσ = = (1 + iz(t)λgv(t,v(t)) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))dσ)−1 [ iλgv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ))[ f(t) u(t) + z(t) 2πu(t) ∫ 2π 0 g ϕ (ξ,σ,ϕ(σ))f(ξ) z(ξ) cot ξ − σ 2 dξ + iz(t) 2πu(t) ∫ 2π 0 λgv (ξ,v(ξ))g ϕ (ξ,σ,ϕ(σ))f(ξ) z(ξ) dξ] dσ.} then h(t) = f(t) u(t) + λgv (t,v(t))z(t) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))f(σ) z(σ) cot σ − t 2 dσ + λgv(t,v(t))z(t) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))f(σ) z(σ) dσ − 2cz(t) u(t) = b̀(ϕ 0 )−1f(t). we shall prove that the operator b̀(ϕ 0 )−1 is bounded. it is easy to prove that v(t), γ(t) and z(t) ∈ h δ therefore by using inequality (3.6) we get ‖ b̀(ϕ 0 )−1 ‖δ ≤ ‖ 1 u ‖δ {1 + ρ 0 | λ |‖ z ‖δ‖ gv(t,v(t)) ‖δ‖ g ϕ (t,t,ϕ(t)) ‖δ‖ 1 z ‖δ + ρ 1 | λ |‖ z ‖δ‖ gv(t,v(t)) ‖δ +2c̃ ‖ z ‖δ}, (4.11) where ρ1 = 1 2π ∫ 2π 0 | g ϕ (t,σ,ϕ(σ)) z(σ) | dσ and c̃ = (1 + iz(t)λgv(t,v(t)) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))dσ)−1 [ iλgv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ))[ 1 u(t) + z(t) 2πu(t) ∫ 2π 0 g ϕ (ξ,σ,ϕ(σ)) z(ξ) cot ξ − σ 2 dξ + iz(t) 2πu(t) ∫ 2π 0 λgv(ξ,v(ξ))g ϕ (ξ,σ,ϕ(σ)) z(ξ) dξ]] dσ. we determine the norm of each term in right hand side of the above inequality. from definition (3.1) we have ‖ 1 u ‖ c =‖ 1 1 + λ2g2v(t,v(t))g 2 ϕ (t,t,ϕ(t)) ‖c≤ 1, cubo 12, 2 (2010) the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel 37 | 1 u(t1) − 1 u(t2) | ≤ | u(t1) − u(t2) |≤| λ2 || g2v(t1,v(t1))g2 ϕ (t1, t1,ϕ(t1)) − g2v(t2,v(t2))g2 ϕ (t2, t2,ϕ(t2)) | ≤ λ2[| gv(t1,v(t1))g ϕ (t1, t1,ϕ(t1)) − gv(t2,v(t2))g ϕ (t2, t2,ϕ(t2)) |] [ 2 ‖ gv(t,v(t)) ‖c‖ g ϕ (t,t,ϕ(t)) ‖c], since ‖ gv(t,v(t)) ‖c≤ c1(r) ‖ v ‖c + ‖ gv(t, 0) ‖c, similarly ‖ g ϕ (t,t,ϕ(t)) ‖c≤ a1(r) ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c, using conditions(3.2)and (3.3)we have | g ϕ (t1, t1,ϕ(t1)) − g ϕ (t2, t2,ϕ(t2)) |≤ a1(r)(2 + hδ(ϕ)) | t1 − t2 |δ, | gv(t1,v(t1)) − gv(t2,v(t2)) |≤ c1(r)(1 + hδ(v))) | t1 − t2 |δ . and | gv(t2,v(t2)) |≤| gv(t2, 0)) | +c1(r)(| v(t2) |), similarly | g ϕ (t1, t1,ϕ(t1)) |≤ a1(r) | ϕ | + | g ϕ (t1, t1, 0) | . hence | 1 u(t1) − 1 u(t2) |≤ λ2β. so ‖ 1 u ‖ δ ≤ r1, (4.12) where r1 = 1 + λ 2β and β = [(| g ϕ (t1, t1, 0) | +a1(r) | ϕ |)(c1(r)(1 + hδ(v)) | t1 − t2 |δ) + (| gv(t2, 0) | +c1(r) | v |)(a1(r)(2 + hδ(ϕ)) | t1 − t2 |δ)] [ (c1(r) ‖ v ‖c + ‖ gv(t, 0) ‖c)(a1(r) ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c)], to determine ‖ z ‖δ we get ‖ z ‖δ≤‖ √ u ‖δ (1+ ‖ γ ‖δ)e‖γ‖δ, (4.13) since ‖ u ‖ c ≤ √ 1 + λ2(c1 ‖ v ‖c + ‖ gv(t, 0) ‖c)2(a1 ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c)2. by 38 m. h. saleh, s. m. amer and m. h. ahmed cubo 12, 2 (2010) applying lagrange theorem: | √ u(t1) − √ u(t2) | = | 1 2 (1 + θ)−1/2 λ2 [g2v(t1,v(t1)) g 2 ϕ (t1, t1,ϕ(t1)) − g2v(t2,v(t2)) g2 ϕ (t2, t2,ϕ(t2))] | ≤ λ2β, where θ between λgv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) and λgv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2)). then ‖ √ u ‖δ≤ r2, (4.14) where r2 = √ 1 + (c1 ‖ v ‖c + ‖ gv(t, 0) ‖c)2(a1 ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c)2 + λ2β. also, we determine ‖ γ ‖δ, since γ(t) = 1 2π ∫ 2π 0 arctgλ gv(t,v(t)) g ϕ (t,σ,ϕ(σ)) cot σ − t 2 dσ + q, where q = 1 4π ∫ 2π 0 ln 1 + iλgv(t,v(t))g ϕ (t,σ,ϕ(σ)) 1 − iλgv(t,v(t))g ϕ (t,σ,ϕ(σ)) dσ, by using (3.6)we have ‖ γ ‖c≤ ρ 0 ‖ arctgλ gv(t,v(t)) g ϕ (t,t,ϕ(t)) ‖c + | q |≤ ρ 0 π 2 + | q |, | arctgλ gv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) − arctgλ gv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2)) | ≤ | λ 1 + θ21 [gv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) − gv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2))] | ≤ | λ | [(| g ϕ (t1, t1, 0) | +a1(r) | ϕ |)(c1(r)(1 + hδ(v)) | t1 − t2 |δ) + (| gv(t2, 0) | +c1(r) | v |)(a1(r)(2 + hδ(ϕ)) | t1 − t2 |δ)], where θ1 between λgv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) and λgv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2)). therefore ‖ γ ‖δ≤ r3, (4.15) where r3 = ρ 0 π 2 + | q | + | λ | [(| g ϕ (t1, t1, 0) | + a1(r) | ϕ | (c1(r)(1 + hδ(v)) | t1 − t2 |δ) + (| gv(t2, 0) | +c1(r) | v |)(a1(r)(2 + hδ(ϕ)) | t1 − t2 |δ)]. substituting from (4.14) and (4.15) into (4.13) we have ‖ z ‖δ≤ r2(1 + r3)er3. (4.16) cubo 12, 2 (2010) the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel 39 from (4.14) we can determine ‖ 1 z ‖δ, ‖ 1 z ‖δ≤ 1 ‖ √ u ‖δ (1+ ‖ γ ‖δ)e‖γ‖δ. but ‖ 1√ u ‖c≤‖ 1 √ 1 + λ2g2v(t2,v(t2)) g 2 ϕ (t2, t2,ϕ(t2)) ‖c≤ 1 and | 1√ u(t1) − 1√ u(t2) |≤| √ u(t1) − √ u(t2) |≤ λ2β then ‖ 1√ u ‖δ≤ r4, where r4 = (1 + λ 2 β). so that ‖ 1 z ‖δ≤ r4(1 + r3)er3. (4.17) then: ‖ b̀(ϕ 0 )−1 ‖≤ α0, where α0 = r1(1 + ρ | λ | r2(1 + r3)er3 )(‖ gv(t, 0) ‖c + c1(r)(1+ ‖ v ‖)(a1(r)(2+ ‖ ϕ ‖) + ‖ g ϕ (t,t, 0) ‖c)(r4(1 + r3)er3 ) + | ρ 1 || λ | r2(1 + r3)er3 )(‖ gv(t, 0) ‖c +c1(r)(1+ ‖ v ‖) + 2c̃r2(1 + r3)e r3, hence the theorem is proved. now ,we determine ‖ b̀(ϕ 0 )−1b(ϕ 0 ) ‖ as follows: ‖ b̀(ϕ 0 )−1b(ϕ 0 ) ‖≤ α0 ‖ b(ϕ0 ) ‖≤ µ0, (4.18) where µ0 = α0(‖ ϕ0 ‖ + | λ | c0(r)(1+ ‖ v ‖)+ ‖ g(t, 0) ‖c), since a =‖ b̀(ϕ 0 )−1b(ϕ 0 ) ‖, hence a ≤ b[‖ ϕ 0 ‖ + | λ | c0(r)(1+ ‖ v ‖)+ ‖ g(t, 0) ‖c], and b ≤ α0 therefore , the following theorem is valid. 40 m. h. saleh, s. m. amer and m. h. ahmed cubo 12, 2 (2010) theorem 4.2 suppose that the equation (2.3)has a unique positive root r ∗ in [0,r] and ψ(r) ≤ 0. then the equationb(ϕ) = 0has a unique solution ϕ ∗ in s(ϕ 0 ,r) and the newtonkantorovich approximations: ϕ n = ϕ n−1 − b̀(ϕ n−1 )−1b(ϕ n−1 ), n ∈ n, are defined for all n ∈ n, belong to s(ϕ 0 ,r ∗ ) and converges to ϕ ∗ . moreover , the following estimate holds ‖ ϕ n+1 − ϕ n ‖≤ r n+1 − r n , ‖ ϕ ∗ − ϕ n ‖≤ r ∗ − r n , where the sequence (r n )n∈n converges to r∗ , is defined by the recurrence formula r 0 = 0, r n+1 = r n − ψ(rn ) `ψ (r n ) , n ∈ n. we will illustrate the theorem 4.2 by the following example. consider the nonlinear function f(u) = 1 6 u3 + 1 6 u2 − 5 6 u + 1 3 , with derivative f̀(u) = 1 2 u 2 + 1 3 u − 5 6 , it’s clear that ‖ f̀(u1) − f̀(u2) ‖ ‖ u1 − u2 ‖ ≤ 1 6 [‖ 3(u1 + u2) ‖ +2] ≤ r + 1 3 , therefore we get k(r) = r + 1 3 . obviously, the scaler equation (2.3) takes the form ψ(r) = a + b 6 r3 + b 6 r2 − r. the equation ψ(r) = 0, (4.19) has a unique positive solution r∗ in[0,r] if and only if [ q 2 ]2 + [ p 3 ]3 > 0, where , p = − 1 3 − 6 b and q = 2 27 + 2 b + 6a b . hence , the function f(u) = 0 has a unique solution u∗ in s(0,r) and the assumptions of theorem (4.2) are verified. received: october 2008. revised: february 2009. cubo 12, 2 (2010) the method of kantorovich majorants to nonlinear singular integral equations with hilbert kernel 41 references [1] n. u. ahmed, semigroup theory with applications to systems and control, pitman research notes in mathematics series, 246. longman scientific & technical, harlow; john wiley & sons, new york, 1991. 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() cubo a mathematical journal vol.17, no¯ 02, (123–141). june 2015 calderón’s reproducing formula for q-bessel operator belgacem selmi faculté des sciences de bizerte, département de mathématiques, 7021 zarzouna, tunisie. belgacem.selmi@fsb.rnu.tn abstract in this paper a calderón-type reproducing formula for q-bessel convolution is established using the theory of q-bessel fourier transform [13, 17], obtained in quantum calculus. resumen en este trabajo se prueba una fórmula de tipo calderón para convolución q-bessel, usando la teoŕıa de q-bessel transformada de fourier [13, 17], obtenida en cálculo cuántico. keywords and phrases: q-calderon, q-calculus, q-bessel convolution, q-fourier bessel transform, q-measure. 2010 ams mathematics subject classification: 05a30, 33dxx, 44a15, 33d15. 124 belgacem selmi cubo 17, 2 (2015) 1 introduction calderón’s formula [1] involving convolutions related to the fourier transform is useful in obtaining reconstruction formula for wavelet transform, in decomposition of certain spaces and in characterization of besov spaces [6, 8, 10]. calderón’s reproducing formula was also established for bessel operator [4, 5]. this work is a continuation of a last work [9], and we establish formula for q-bessel convolution for both functions and measures witch generalize the above one. in the classical case this formula is expressed for a suitable function f as follows: f(x) = ∫ ∞ 0 (gt ∗ ht ∗ f)(x) dt t , (1) where g, h ∈ l2(r) and gt(x) = 1 t g(x t ), ht(x) = 1 t h(x t ), t > 0 satisfying ∫ ∞ 0 ĝ(xt)ĥ(xt) dx x = 1, for all t ∈ r \ {0}, where ĝ and ĥ is the usual fourier transform of g and h on r. if µ is a finite borel measure on the real line r, identity (1) has natural generalization as follow f(x) = ∫ ∞ 0 (f ∗ µt)(x) dt t , (2) where µt is the dilated measure of µ under some restriction on µ, the l p-norm of (2) has proved in [2]. a general form of (2) has been investigated in [3]. in this paper we study similar questions when in (1) and (2) the classical convolution ∗ is replaced by the q-bessel convolution ∗α,q on the half line generated by the q-bessel operator defined by ∆q,αf(x) = 1 x2α+1 dq [ x2α+1dqf ] (q−1x). (3) in this paper we prove that, for ϕ and ψ ∈ l1α,q(rq,+,dqσ(x)) satisfying ∫ ∞ 0 fα,q(ϕ)(ξ)fα,q(ψ)(ξ) dqξ ξ = 1 (4) we have f(x) = ∫ ∞ 0 (f ∗α,q ϕt ∗α,q ψt)(x) dqt t , f ∈ l1α,q(rq,+,dqσ(x)). (5) where dqσ(x) = (1 + q)−α γq2(α + 1) x2α+1dqx = bα,qx 2α+1dqx, ϕt(x) = 1 t2α+2 ϕ( x t ). in particular for ϕ ∈ l1α,q(rq,+,dqσ(x)) such that ∫ ∞ 0 [fα,q(ϕ)(ξ)] 2 dqξ ξ = 1, (6) cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 125 and for a suitable function f, put fε,δ(x) = ∫δ ε (f ∗α,q ϕt ∗α,q ϕt)(x) dqt t (7) then ‖fε,δ − f‖2,α,q −→ 0 as ε → 0 and δ → ∞. (8) in the case f ∈ l1α,q(rq,+,dqσ(x)) such that fα,qf ∈ l 1 α,q(rq,+,dqσ(x)) one has lim ε → 0 δ → ∞ fε,δ(x) = f(x), x ∈ r. (9) then we prove that for µ ∈ m ′ (rq,+), such that the q-integral cµ,α,q = ∫ ∞ 0 fα,q(µ)(λ) dqλ λ (10) is finite. then for all f ∈ l2α,q(rq,+,dqσ(x)), we have lim ε → 0 δ → ∞ fε,δ = cµ,α,qf. (11) where the limit is in l2α,q(rq,+,dqσ(x)). and if µ ∈ m ′ (rq,+) is such that the q-integral ∫ ∞ 0 |µ([0,y])| dqy y (12) is finite, for all f ∈ l2α,q(rq,+,dqσ(x)) lim ε → 0 δ → ∞ fε,δ = cµ,α,qf, in l 2 α,q(rq,+,dqσ(x)). (13) the outline of this paper is as follows: in section 2, basic properties of q-bessel transform on rq of functions and bounded measure and its underlying q-convolution structure are called and introduced here. in section 3, we give the first main result of the paper, the q-calderon’s reproducing formula for functions. section 4 is consecrate to establish the same result as in section 3 for finite measures. 2 preliminaries in this section we recall some basic result in harmonic analysis related to the q-bessel fourier transform. standard reference here is gasper & rahman [7]. 126 belgacem selmi cubo 17, 2 (2015) for a,q ∈ c the q-shifted factorial (a;q)k is defined as a product of k factors: (a;q)k = (1 − a)(1 − aq)...(1 − aq k−1), k ∈ n∗; (a;q)0 = 1. (14) if |q| < 1 this definition remains meaningful for k = +∞ as a convergent infinite product: (a;q)∞ = ∞∏ k=0 (1 − aqk). (15) we also write (a1, ...,ar;q)k for the product of r q-shifted factorials: (a1, · · · ,ar;q)k = (a1;q)k...(ar;q)k (k ∈ n or k = ∞). (16) a q-hypergeometric series is a power series (for the moment still formal) in one complex variable z with power series coefficients which depend, apart from q, on r complex upper parameters a1, ...,ar and s complex lower parameters b1, ...,bs as follows: rϕs(a1, · · · ,ar;b1, · · · ,bs;q,x) = ∞∑ k=0 (a1, · · · ,ar;q)k (b1, · · · ,bs;q)k(q;q)k [(−1)kq k(k−1) 2 ] 1+s−rxk (for r,s ∈ n). 2.1 q-exponential series eq(z) = 1ϕ0(0; −;q,z) = ∞∑ k=0 zk (q;q)k = 1 (z;q)∞ (| z |< 1) (17) eq(z) = 0ϕ0(−; −;q,−z) = ∞∑ k=0 q 1 2 k(k−1)zk (q;q)k = (−z;q)∞ (z ∈ c). (18) 2.2 q-derivative and q-integral the q-derivative of a function f given on a subset of r or c is defined by: dqf(x) = f(x) − f(qx) (1 − q)x (x 6= 0,q 6= 0), (19) where x and qx should be in the domain of f. by continuity we set (dqf)(0) = f ′ (0) provided f ′ (0) exists. the q-shift operators are (λqf)(x) = f(qx), (λ −1 q f)(x) = f(q −1x). (20) cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 127 for a ∈ r \ {0} and a function f given on (0,a] or [a,0), we define the q-integral by ∫a 0 f(x)dqx = (1 − q)a ∞∑ n=0 f(aqn)qn, (21) provided the infinite sum converges absolutely (for instance if f is bounded). if f(a) is given by the left-hand side of (21) then dqf = f. the right-hand side of (21) is an infinite riemann sum. for a q-integral over (0,∞) we define ∫ ∞ 0 f(x)dqx = (1 − q) +∞∑ −∞ f(qk)qk. (22) note that for n ∈ z and a ∈ rq, we have ∫ ∞ 0 f(qnx)dqx = 1 qn ∫ ∞ 0 f(x)dqx, ∫a 0 f(qnx)dqx = 1 qn ∫aqn 0 f(x)dqx. (23) the q-integration by parts is given for suitable functions f and g by: ∫b a f(x)dqg(x)dqx = [ f(x)g(x) ]b a − ∫b a dqf(x)g(x)dqx. (24) the q-logarithm logq is given by [19] logq x = ∫ dqx x = 1 − q logq logx. (25) for all a,b ∈ qz,a < b logq(b/a) = (1 − q) ∑ k:a≤qk≤b 1. (26) the improper integral is defined in the following way ∫ ∞/a 0 f(x)dqx = (1 − q) +∞∑ −∞ f ( qn a ) qn a . (27) we remark that for n ∈ z, we have ∫ ∞/qn 0 f(x)dqx = ∫ ∞ 0 f(x)dqx. (28) the following property holds for suitable function f ∫ ∞ 0 ∫x 0 f(x,y)dqydqx = ∫ ∞ 0 ∫ ∞ qy f(x,y)dqxdqy. (29) 128 belgacem selmi cubo 17, 2 (2015) 2.3 the q-gamma function the q-gamma function is defined by [7, 16] γq(z) = (q;q)∞ (qz;q)∞ (1 − q)1−z, 0 < q < 1;z 6= 0,−1,−2, ... (30) = ∫(1−q)−1 0 tz−1eq(−(1 − q)qt)dqt, (rez > 0) (31) moreover the q-duplication formula holds γq(2z)γq2( 1 2 ) = (1 + q)2z−1γ2q(z)γq2(z + 1 2 ). (32) 2.4 some q-functional spaces we begin by putting rq,+ = {+q k,k ∈ z}, r̃q,+ = {+q k,k ∈ z} ∪ {0} (33) and we denote by • l p α,q(rq,+), p ∈ [1,+∞[, ( resp. l ∞ α,q(rq,+) ) the space of functions f such that, ‖f‖p,α,q = ( ∫ ∞ 0 |f(x)|pdqσ(x)) 1 p < +∞. (34) (resp. ‖f‖∞,q = ess sup x∈rq,+ |f(x)| < +∞). (35) • sq,∗(rq) the q-analogue of schwartz space of even functions defined on rq such that d k q,xf(x) is continuous in 0 for all k ∈ n and nq,n,k(f) = sup x∈rq |(1 + x2)ndkq,xf(x)| < +∞. (36) • the q-analogue of the tempered distributions is introduced in [12] as follow: (i) a q-distribution t in rq is said to be tempered if there exists cq > 0 and k ∈ n such that: |〈t,f〉| ≤ cqnq,n,k(f); f ∈ sq,∗(rq). (37) (ii) a linear form t: sq,∗(rq) −→ c is said continuous if there exist cq > 0 and k ∈ n such that: |〈t,f〉| ≤ cqnq,n,k(f); f ∈ sq,∗(rq). (38) cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 129 • s ′ q,∗(rq) the space of even q-tempered distributions in rq. that is the topological dual of sq,∗(rq). • dq,∗(rq) the space of even functions infinitely q-differentiable on rq with compact support in rq. we equip this space with the topology of the uniform convergence of the functions and their q-derivatives. • cq,∗,0(rq) the space of even functions f defined on rq continuous on 0, infinitely q-differentiable and lim x→∞ f(x) = 0, ‖f‖cq,∗,0 = sup x∈rq |f(x)| < +∞. (39) • hq,∗(rq) the space of even functions f defined on rq continuous on 0 with compact support such that ‖f‖hq,∗ = sup x∈rq |f(x)| < +∞. (40) 2.5 q-bessel function the following properties of the normalized q-bessel function is given (see [13]) by jα(x;q 2 ) = γq2(α + 1) ∞∑ k=0 (−1)kqk(k−1) γq2(k + 1)γq2(α + k + 1) ( x 1 + q ) 2k. (41) this function is bounded and for every x ∈ rq and α > − 1 2 we have |jα(x;q 2 )| ≤ 1 (q;q2)2 ∞ , (42) ( 1 x dq ) jα(.;q 2)(x) = − (1 − q) (1 − q2α+2) jα+1(qx;q 2), (43) ( 1 x dq ) (x2αjα(x;q 2)) = 1 − q2α 1 − q x2(α−1)jα−1(x;q 2), (44) |dqjα(x;q 2)| ≤ (1 − q) (1 − q2α+2) x (q;q2)2 ∞ . (45) we remark that for λ ∈ c, the function jα(λx;q 2) is the unique solution of the q-differential system    ∆q,αu(x,q) = −λ 2u(x,q), u(0,q) = 1; dq,xu(x,q)|x=0 = 0, (46) 130 belgacem selmi cubo 17, 2 (2015) where ∆q,α is the q-bessel operator defined by ∆q,αf(x) = 1 x2α+1 dq [ x2α+1dqf ] (q−1x) (47) = q2α+1∆qf(x) + 1 − q2α+1 (1 − q)q−1x dqf(q −1x), (48) where ∆qf(x) = λ −1 q d 2 qf(x) = (d 2 qf)(q −1x), (49) and for k ∈ n and λ ∈ rq,+, ∆kq,xjα(λx;q 2) = (−1)kλ2kjα(λx;q 2). (50) 2.6 q-bessel translation operator tαq,x,x ∈ rq,+ is the q-generalized translation operator associated with the q-bessel transform is introduced in [13] and rectified in [17], where it is defined by the use of jackson’s q-integral and the q-shifted factorial as tαq,xf(y) = ∫+∞ 0 f(t)dα,q(x,y,t)t 2α+1dqt, α > −1 (51) with dα,q(x,y,z) = c 2 α,q ∫+∞ 0 jα(xt;q 2)jα(yt;q 2)jα(zt;q 2)t2α+1dqt where cα,q = 1 1 − q (q2α+2;q2)∞ (q2;q2)∞ . in particular the following product formula holds tαq,xjα(y,q 2) = jα(x,q 2)jα(y,q 2). it is shown in [18] that for f ∈ l1α,q(rq,+), t α q,xf ∈ l 1 α,q(rq,+) and ||tαq,xf||1,α,q = ||f||1,α,q. 2.7 the q-convolution and the q-bessel fourier transform the q-bessel fourier transform fα,q and the q-bessel convolution product are defined for suitable functions f,g as follows fα,q(f)(λ) = ∫ ∞ 0 f(x)jα(λx;q 2 )dqσ(x), cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 131 f ∗α,q g(x) = ∫+∞ 0 tαq,xf(y)g(y)dqσ(y). the q-bessel fourier transform fα,q is a modified version of the q-analogue of the hankel transform defined in [15]. it is shown in [13, 17, 14], that the q-bessel fourier transform fα,q satisfies the following properties: proposition 2.1. if f ∈ l1α,q(rq,+), then fα,q(f) ∈ cq,∗,0(rq,+) and ‖fα,q(f)‖cq,∗,0 ≤ bα,q||f||1,α,q. where bα,q = 1 (1 − q) (−q2;q2)∞(−q 2α+2;q2) (q2;q2)∞ . proposition 2.2. given two functions f, g ∈ l1α,q(rq,+), then f ∗α,q g ∈ l 1 α,q(rq,+), and fα,q(f ∗α,q g) = fα,q(f)fα,q(g). theorem 2.3. (inversion formula) 1. if f ∈ l1α,q(rq,+) such that fα,q(f) ∈ l 1 α,q(rq,+), then for all x ∈ rq,+, we have f(x) = ∫ ∞ 0 fα,q(f)(y)jα(xy;q 2 )dqσ(y). 2. fα,q(f) is an isomorphism of s∗,q(rq) and f 2 α,q(f) = id. • note that the inversion formula is valid for f ∈ l1α,q(rq,+) without the additional condition fα,q(f) ∈ l 1 α,q(rq,+). fα,q(f) can be extended to l 2 α,q(rq,+) and we have the following theorem: theorem 2.4. (q-plancherel theorem ) fα,q(f) is an isomorphism of l 2 α,q(rq,+), we have ||fα,q(f)||2,α,q = ||f||2,α,q, for f ∈ l 2 α,q(rq,+) and f−1α,q(f) = fα,q(f). proposition 2.5. (i) for f ∈ l p α,q(rq,+), p ∈ [1,∞[, g ∈ l 1 α,q(rq,+) we have f ∗α,q g ∈ l p α,q(rq,+) and ||f ∗αq g||p,α,q ≤ ||f||p,α,q||g||1,α,q. (ii) ∫ ∞ 0 fα,q(f)(ξ)g(ξ)dqσ(ξ) = ∫ ∞ 0 f(ξ)fα,q(g)(ξ)dqσ(ξ); f,g ∈ l 1 α,q(rq,+). 132 belgacem selmi cubo 17, 2 (2015) (iii) fα,q(t α q,xf)(ξ) = jα(ξx;q 2)fα,q(f)(ξ); f ∈ l 1 α,q(rq,+). specially, we choose q ∈ [0,q0] where q0 is the first zero of the function [17]: q 7→ 1φ1(0,q,q;q) under the condition log(1−q) log q ∈ z. definition 2.6. [11, 9] a bounded complex even measure µ on rq is a bounded linear functional µ on hq,∗(rq), i.e., for all f in hq,∗(rq), we have |µ(f)| ≤ c‖f‖hq,∗, (52) where c > 0 is a positive constant. denote the space of all such measure by m ′ (rq,+). note that µ ∈ m ′ (rq,+) can be identified with a function µ̃ on r̃q,+ such that µ̃ restricted to rq,+ is l 1 α,q(rq,+) : µ(f) = µ({0})f(0) + ∫ ∞ 0 µ̃(x)f(x)dq(x), (f ∈ hq,∗(rq)). for µ ∈ m ′ (rq,+) denote ‖µ‖ = |µ|(rq,+) where |µ| is the absolute value of µ. definition 2.7. the q-bessel fourier transform of a measure µ in m ′ (rq,+) is defined for all ϕ ∈ sq,∗(rq) by fα,qµ(λ) = bα,q ∫+∞ 0 jα(λx;q 2)dµ(x). (53) the q-bessel convolution product of a measure µ ∈ m ′ (rq,+) and a suitable function f on rq,+ is defined by µ ∗α,q f(x) = ∫ ∞ 0 tαq,xf(y)dµ(y). (54) proposition 2.8. (1) the q-bessel fourier transform fα,q of a measure µ in m ′ (rq,+) is the q-tempered distribution fα,qµ given by: 〈fα,qµ,ϕ〉 = 〈µ,fα,qϕ〉 = ∫+∞ 0 fα,qϕ(λ)dqµ(λ). (55) (2) for all x,λ ∈ rq,+ we have tαq,xfα,qµ(λ) = bα,q ∫+∞ 0 jα(xt;q 2)jα(λt;q 2)dqµ(t). (56) (3) for all µ ∈ m ′ (rq,+), fα,qµ is continuous on rq,+, and lim λ→∞ fα,qµ(λ) = µ({0}). (57) fα,q maps one to one m ′ (rq,+) into cb(rq,+), (the space of continuous and bounded functions on rq,+). cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 133 (4) if µ ∈ m ′ (rq,+) and f ∈ l p α,q(rq,+), p = 1,2 then µ ∗α,q f ∈ l p α,q(rq,+) and ‖µ ∗α,q f‖p,α,q ≤ ‖µ‖‖f‖p,α,q. (58) (5) for all µ ∈ m ′ (rq,+) and f ∈ l p α,q(rq,+), p = 1,2 we have fα,q(µ ∗α,q f) = fα,q(µ)fα,q(f). (59) definition 2.9. let µ ∈ m ′ (rq,+) and a > 0. we define the q-dilated measure µa of µ by ∫ ∞ 0 ϕ(x)dqµa(x) = ∫ ∞ 0 ϕ(ax)dqµ(x), ϕ ∈ hq,∗(rq). (60) proposition 2.10. (i) when µ = f(x)x2α+1dqx, with f ∈ l 1 α,q(rq,+), the measure µa, a > 0, is given by the function fa(x) = 1 a2α+2 f( x a ), x ≥ 0. (61) (ii) let µ ∈ m ′ (rq,+), then fα,q(µa)(λ) = fα,q(µ)(aλ), for all λ ≥ 0. (62) (iii) for µ ∈ m ′ (rq,+) and f ∈ l p α,q(rq,+),p = 1,2 we have lim a→0 µa ∗α,q f = µ(r̃q,+)f. (63) where the limit is in l p α,q(rq,+). (iv) let g ∈ l1α,q(rq,+) and f ∈ l p α,q(rq,+), 1 < p < ∞. then lim a→∞ f ∗α,q ga = 0 (64) where the limit is in l p α,q(rq,+). proof. statement of (i) and (ii) are obvious. a standard argument gives (iii). let us verify (iv). if f,g ∈ dq,∗(rq) then by (58) and (61) we have ‖f ∗α,q ga‖p,α,q ≤ ‖f‖1,α,q‖ga‖p,α,q = a −2(α+1)(p−1) p ‖f‖1,α,q‖g‖p,α,q → 0, as a → ∞. for arbitrary g ∈ l1α,q(rq,+) and f ∈ l p α,q(rq,+) the result follows by density. ✷ given a measure µ ∈ m ′ (rq,+). denote cµ,α,q = ∫ ∞ 0 fα,q(µ)(λ) dqλ λ . (65) 134 belgacem selmi cubo 17, 2 (2015) 3 q-calderón’s formula for functions in this section, we establish the q-calderón’s reproducing identity for functions using the properties of q-fourier bessel transform fα,q and q-bessel convolution ∗α,q. theorem 3.1. let ϕ and ψ ∈ l1α,q(rq,+) be such that following admissibility condition holds ∫ ∞ 0 fα,q(ϕ)(ξ)fα,q(ψ)(ξ) dqξ ξ = 1 (66) then for all f ∈ l1α,q(rq,+), the following calderón’s reproducing identity holds: f(x) = ∫ ∞ 0 (f ∗α,q ϕt ∗α,q ψt)(x) dqt t . (67) proof. taking q-bessel fourier transform of the right-hand side of (67), we get fα,q [∫ ∞ 0 (f ∗α,q ϕt ∗α,q ψt)(x) dqt t ] (ξ) = ∫ ∞ 0 fα,q(f)(ξ)fα,q(ϕt)(ξ)fα,q(ψt)(ξ) dqt t = fα,q(f)(ξ) ∫ ∞ 0 fα,q(ϕt)(ξ)fα,q(ψt)(ξ) dqt t = fα,q(f)(ξ) ∫ ∞ 0 fα,q(ϕ)(tξ)fα,q(ψ)(tξ) dqt t = fα,q(f)(ξ). now, by putting tξ = s, we get ∫ ∞ 0 fα,q(ϕ)(tξ)fα,q(ψ)(tξ) dqt t = ∫ ∞ 0 fα,q(ϕ)(s)fα,q(ψ)(s) dqs s = 1. hence, the result follows. ✷ the equality (67) can be interpreted in the following l2-sense. theorem 3.2. suppose ϕ ∈ l1α,q(rq,+) and satisfies ∫ ∞ 0 [fα,q(ϕ)(ξ)] 2 dqξ ξ = 1. (68) for f ∈ l1α,q(rq,+) ∩ l 2 α,q(rq,+), suppose that fε,δ(x) = ∫δ ε (f ∗α,q ϕt ∗α,q ϕt)(x) dqt t (69) then ‖fε,δ − f‖2,α,q −→ 0 as ε → 0 and δ → ∞. (70) cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 135 proof. taking q-bessel fourier transform of both sides of (69) and using fubini’s theorem, we get fα,q(f ε,δ )(ξ) = fα,q(f)(ξ) ∫δ ε [fα,q(ϕ)(tξ)] 2 dqt t by proposition 2.5, we have ‖ϕt ∗α,q ϕt ∗α,q f‖2,α,q ≤ ‖ϕt ∗α,q ϕt‖1,α,q‖f‖2,α,q ≤ ‖ϕt‖ 2 1,α,q‖f‖2,α,q. now using above inequality, minkowski’s inequality and relation (29), we get ‖fε,δ‖22,α,q = ∫ ∞ 0 | ∫δ ε (ϕt ∗α,q ϕt ∗α,q f)(x) dqt t |2dqσ(x) ≤ ∫δ ε ∫ ∞ 0 |(ϕt ∗α,q ϕt ∗α,q f)(x)| 2dqσ(x) dqt t ≤ ∫δ ε ‖ϕt ∗α,q ϕt ∗α,q f‖2,α,q dqt t ≤ ‖ϕt‖ 2 1,α,q‖f‖2,α,q ∫δ ε dqt t = ‖ϕt‖ 2 1,α,q‖f‖2,α,q logq( δ ε ). hence, by theorem 2.4, we get lim ε → 0 δ → ∞ ‖fε,δ − f‖22,α,q = lim ε → 0 δ → ∞ ‖fα,q(f ε,δ) − fα,q(f)‖ 2 2,α,q = lim ε → 0 δ → ∞ ∫ ∞ 0 |fα,q(f)(ξ) ( 1 − ∫δ ε [fα,q(ϕ)(tξ)] 2 dqt t ) |2dqσ(x) = 0. since |fα,q(f)(ξ) ( 1 − ∫δ ε [fα,q(ϕ)(tξ)] 2 dqt t ) | ≤ |fα,q(f)(ξ)|, therefore, by the dominated convergence theorem, the result follows. ✷ the reproducing identity (67) holds in the pointwise sense under different sets of nice conditions. theorem 3.3. suppose f, fα,qf ∈ l 1 α,q(rq,+). let ϕ ∈ l 1 α,q(rq,+) and satisfies ∫ ∞ 0 [fα,qϕ(tξ)] 2 dqt t = 1 (71) 136 belgacem selmi cubo 17, 2 (2015) then lim ε → 0 δ → ∞ fε,δ(x) = f(x), (72) where fε,δ is given by (69). proof. by proposition 2.5, we have ‖ϕt ∗α,q ϕt ∗α,q f‖1,α,q ≤ ‖ϕt‖ 2 1,α,q‖f‖1,α,q. now ‖fε,δ‖1,α,q = ∫ ∞ 0 | ∫δ ε (ϕt ∗α,q ϕt ∗α,q f)(x) dqt t |dqσ(x) ≤ ∫δ ε ∫ ∞ 0 |(ϕt ∗α,q ϕt ∗α,q f)(x)|dqσ(x) dqt t ≤ ∫δ ε ‖ϕt ∗α,q ϕt ∗α,q f‖1,α,q dqt t ≤ ‖ϕt‖ 2 1,α,q‖f‖1,α,q logq( δ ε ). therefore, fε,δ ∈ l1α,q(rq,+). also using fubini’s theorem and taking qbessel fourier transform of fε,δ, we get fα,qf ε,δ(ξ) = ∫ ∞ 0 jα(xξ;q 2) (∫δ ε (ϕt ∗α,q ϕt ∗α,q f)(x) dqt t ) dqσ(x) = ∫δ ε ∫ ∞ 0 jα(xξ;q 2 )(ϕt ∗α,q ϕt ∗α,q f)(x)dqσ(x) dqt t = ∫δ ε fα,qϕt(ξ)fα,qϕt(ξ)fα,qf(ξ) dqt t = fα,qf(ξ) ∫δ ε [fα,qϕ(tξ)] 2 dqt t . therefore by (71), |fα,qf ε,δ(ξ)| ≤ |fα,qf(ξ)|. it follows that fα,qf ε,δ ∈ l1α,q(rq,+). by inversion, we have f(x) − fε,δ(x) = ∫ ∞ 0 jα(xξ;q 2 ) [ fqf(ξ) − fα,qf ε,δ (ξ) ] dqσ(ξ). (73) putting gε,δ(x,ξ) = jα(xξ;q 2) [ fα,qf(ξ) − fα,qf ε,δ(ξ) ] (74) = jα(xξ;q 2)fqf(ξ) [ 1 − ∫δ ε [fα,qϕ(tξ)] 2 dqt t ] , cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 137 we get f(x) − fε,δ(x) = ∫ ∞ 0 jα(xξ;q 2) [ fα,qf(ξ) − fα,qf ε,δ(ξ) ] dqσ(ξ) = ∫ ∞ 0 gε,δ(x,ξ)dqσ(ξ). now using (71) and (74), we get lim ε → 0 δ → ∞ gε,δ(x,ξ) = 0. (75) since |gε,δ(x,ξ)| ≤ 1 (q;q2)2 ∞ |fα,qf(ξ)|, the dominated convergence theorem yields the result. ✷ 4 q-calderón’s formula for finite measures it is now possible to define analogues to (2) for the q-bessel convolution ∗α,q and investigate its convergence in the l2α,q(rq,+) q-norm. to this end we need some technical lemmas lemma 4.1. let µ ∈ m ′ (rq,+), for 0 < ε < δ < ∞ define gε,δ(x;q 2) = µ([x δ , x ε ]) x2α+2 , x > 0 (76) and kε,δ(λ;q 2) = ∫δ ε fα,q(µ)(qaλ) dqa a , λ ≥ 0. (77) then gε,δ ∈ l 1 α,q(rq,+) and fα,q(gε,δ)(λ;q 2 ) = kε,δ(λ;q 2 ) − µ({0}) logq( δ ε ), (78) where logq is given by (25). proof. we have by (25) and (29), | ∫ ∞ 0 gε,δ(x;q 2 )x2α+1dqx| ≤ ∫ ∞ 0 ( ∫ x ε x δ dq|µ|(y)) dqx x = ∫ ∞ 0 [ ∫ x ε 0 dq|µ|(y) − ∫ x δ 0 dq|µ|(y)] dqx x = ∫ ∞ 0 [ ∫ ∞ qεy dqx x − ∫ ∞ qδy dqx x ]dq|µ|(y) = ∫ ∞ 0 logq( ε δ )dq|µ|(y) = |µ|(r̃q,+) logq( ε δ ) < ∞. 138 belgacem selmi cubo 17, 2 (2015) using again relation (29) and q-fubini’s theorem we obtain fα,q(gε,δ)(λ) = ∫ ∞ 0 ∫ x ε x δ dqµ(y)jα(λx;q 2) dqx x = ∫ ∞ 0 ∫qδy qεy jα(λx;q 2) dqx x dqµ(y) = ∫ ∞ 0 ∫qδ qε jα(λxy;q 2 ) dqx x dqµ(y) = ∫qδ qε ∫ ∞ 0 jα(λxy;q 2)dqµ(y) dqx x = ∫qδ qε fα,qµ(λx) − µ({0}) dqx x = ∫δ ε fα,qµ(qλx) − µ({0}) dqx x = kε,δ(λ;q 2 ) − µ({0}) logq( δ ε ). ✷ lemma 4.2. let µ ∈ m ′ (rq,+), then for f ∈ l p α,q(rq,+),p = 1,2 and 0 < ε < δ < ∞, the function fε,δ(x;q2) = ∫δ ε f ∗α,q µa(x;q 2) dqa a (79) belongs to l p α,q(rq,+) and has the form fε,δ(x;q2) = f ∗α,q gqε,qδ(x;q 2) + µ({0})f(x) logq( δ ε ). (80) where gε,δ is given by (4.1). proof. applying q-fubini’s theorem we get fε,δ(x) = ∫δ ε ∫ ∞ 0 tαq,xf(ay)dqµ(y) dqa a = ∫ ∞ 0 ∫δ ε tαq,ayf(x) dqa a dqµ(y) = f(x)µ({0}) logq( δ ε ) + ∫ r̃q,+ ∫δy εy tαq,xf(a) dqa a dqµ(y) = f(x)µ({0}) logq( δ ε ) + ∫ r̃q,+ tαq,xf(a)( ∫q a ε q a δ dqa a )dqµ(y) = f(x)µ({0}) logq( δ ε ) + f ∗α,q gqε,qδ(x). cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 139 from this relation, inequality (58) and lemma 4.1 we deduce that fε,δ belongs to l p α,q(rq,+). ✷ lemma 4.3. let µ ∈ m ′ (rq,+), then for f ∈ l 2 α,q(rq,+), we have fα,q(f ε,δ)(λ;q2) = fq(f)(λ;q 2)kqε,qδ(λ;q 2), (81) where kε,δ, is the function defined in (67). proof. this follows from (59), (67) and (80). ✷ theorem 4.4. let µ ∈ m ′ (rq,+), be such that the q-integral cµ,α,q = ∫ ∞ 0 fα,q(µ)(λ) dqλ λ (82) be finite. then for all f ∈ l2α,q(rq,+), we have lim ε → 0 δ → ∞ ‖fε,δ − cµ,α,qf‖2,α,q = 0. (83) proof. by identity (81) and theorem 2.4 we have ‖fε,δ − cµ,α,qf‖ 2 2,α,q = ‖fα,q(f ε,δ ) − cµ,α,qfα,q(f)‖ 2 2,α,q = ‖fα,q(f)[kε,δ − cµ,α,q]‖ 2 2,α,q. or lim ε → 0 δ → ∞ kε,δ(λ) = cµ,α,q, for all λ > 0 the result follows from the dominate convergence theorem. ✷ lemma 4.5. let µ ∈ m ′ (rq,+), be such that the q-integral ∫ ∞ 0 |µ([0,y])| dqy y (84) be finite. then the q-integral cµ,α,q is finite and admits the representation cµ,α,q = ∫ ∞ 0 µ([0,y]) dqy y . (85) proof. from (76) we have gε,δ = µ([x δ , x ε ]) x2α+2 = gε − gδ, (86) 140 belgacem selmi cubo 17, 2 (2015) where g(y) = µ([0,y]) y2α+2 (87) and gε, gδ the dilated function of g. since g ∈ l 1 α,q(rq,+), we deduce from (62) and (78) that fα,qgε,δ(λ) = ∫δλ ελ fα,qµ(a) dqa a − µ({0}) logq( δ ε ) (88) = fα,qg(ελ) − fα,qg(δλ), for all λ > 0. or (84) implies necessarily µ({0}) = 0. hence when ε = 1 and δ → ∞, a combination of (88) and (57) gives fα,qg(λ) = ∫ ∞ λ fα,qµ(a) dqa a , for all λ > 0. (89) now the result follows from formula (84) by using the continuity of fα,q(µ). ✷ theorem 4.6. let µ ∈ m ′ (rq,+) such that ∫ ∞ 0 |µ([0,y])| dqy y (90) is finite and f ∈ l2α,q(rq,+). then lim ε → 0 δ → ∞ ‖fε,δ − cµ,α,qf‖2,α,q = 0. (91) proof. by (80) and (86) we have fε,δ = f ∗α,q gε − f ∗α,q gδ, (92) where g is as in (87). equation (91) is now a consequence of proposition 2.5. ✷ received: july 2014. accepted: may 2015. references [1] a. p. calderón, intermediate spaces and interpolation, the complex method,studia math, 24 (1964), 113-190. [2] b. rubin and e. shamir, calderón’s reproducing formula and singular integral operators on areal line,integral equations operators theory, 21 (1995), 77-92. [3] b. rubin, fractional integrals and potentials ,logman, harlow, (1996). cubo 17, 2 (2015) calderón’s reproducing formula for q-bessel operator 141 [4] m.a. mourou and k. trimã¨che, calderón’s reproducing formula associated with the bessel operator ,j. math.anal. appl., 219 (1998), 97-109. [5] p.s. pathak and g. pandey, calderón’s reproducing formula for hankel convolution,inter. j. math. and math. sci., vol 2006 issue 5, article id 24217. p.1-7. [6] j. l. ansorena and o. blasco, characterization of weighted besov spaces, math. nachr, 171 (1995), 5-17. [7] g. gasper and m. rahman, basic hypergeometric series, 2nd edn. cambridge university press, 2004. [8] h. q. bui, m. paluszynski and m. h. taibleson , a maximal function charaterization of weighted besov spaces-lipschitz and triebel-lizorkin spaces, studia. math, 119 (1996), 219-246. [9] a. nemri and b. selmi, calderón type formula in quantum calculus, indagationes mathematicae, vol 24, issues 3, (2013), 491-504. [10] a. nemri and b. selmi, some weighted besov spaces in quantum calculus, submitted. [11] nemri a, distribution of positive type in quantum calculus, j. non linear. math. phys., 4(2006)566583. [12] a. fitouhi and a. nemri, distribution and convolution product in quantum calculus, afri. diaspora j. math, 7(2008),nã¯â¿â 1 2 1, 39-57. [13] a. fitouhi, m. hamza and f. bouzeffour, the q-jα bessel function, j. approx. theory 115 (2002), 114-116. [14] m.haddad, hankel transform in quantum calculus and applications, fractional calculus and applied analysis, vol.9,issues 4 (2006), 371-384. [15] t. h. koornwinder and r. f. swarttouw, on q-analogues of the fourier and hankel transforms , trans. amer math. soc. 333, (1992), 445-461. [16] t. h. koornwinder, q-special functions, a tutorial arxiv:math/9403216v1. [17] a. fitouhi, l. dhaouadi and j. el kamel, inequalities in q-fourier analysis , j. inequal. pure appl. math, 171 (2006), 1-14. [18] a. fitouhi and l. dhaouadi, positivity of the generalized translation associated to the q-hankel transform, constr. approx, 34 (2011), 453-472. [19] v.g. kac and p. cheeung, quantum calculus, universitext, springer-verlag, new york, (2002). introduction preliminaries q-exponential series q-derivative and q-integral the q-gamma function some q-functional spaces q-bessel function q-bessel translation operator the q-convolution and the q-bessel fourier transform q-calderón's formula for functions q-calderón's formula for finite measures cubo a mathematical journal vol.11, no¯ 04, (49–57). september 2009 small singular values of an extracted matrix of a witten complex. d. le peutrec irmar, umr-cnrs 6625, université de rennes 1, campus de beaulieu, 35042 rennes cedex, france. email: dorian.lepeutrec@univ-rennes1.fr abstract it is shown how rather tricky induction processes, used for the accurate computation of exponentially small eigenvalues of witten laplacians, essentially amount to some gaussian elimination after the proper rewriting. resumen se muestra cómo un proceso de inducción bastante truculento se usa para el cálculo preciso de los valores propios pequeños de los laplacianos de witten utilizando esencialmente cantidades de eliminaciones gausianas después de una reescritura correcta. key words and phrases: induction process, witten laplacian, exponentially small eigenvalues, gaussian elimination. math. subj. class.: 81q10, 15a18, 34l15, 58j10, 58j37. 1 introduction and motivations the accurate computation of exponentially small eigenvalues of witten laplacians on 0-forms, or generators associated with reversible diffusion processes, relies on some rather tricky induction 50 d. le peutrec cubo 11, 4 (2009) process. in [2] [3], the induction scheme is modelled after the probabilistic picture of exit times. after [7] [8] [16] it appeared that this induction scheme could be extracted from its spectral analysis or probabilistic framework as a pure problem of finite dimensional linear algebra. the aim of this short text is to show that all these previous and rather involved inductions essentially amount, after a proper rewriting, to some gaussian elimination. we first recall that the witten laplacian writes ∆ (0) f,h = d (0),∗ f,h d (0) f,h = −h 2 ∆ + |∇f(x)|2 − h∆f(x) (1.1) on functions and more generally on differential forms with arbitrary degree, ∆f,h = ( d ∗ f,h + df,h )2 with df,h = e − f h (hd)e f h . (1.2) since it has a square structure, the eigenvalues of ∆ (0) f,h (resp. ∆f,h) are the squares of the singular values of d (0) f,h (resp. (d ∗ f,h +df,h)). remember that in the study of witten laplacians d denotes the exterior differential on a riemannian manifold, d∗ the codifferential, h > 0 is a small parameter considered in the limit h → 0 and f is a morse function. in the case when the manifold is rn with the euclidean metric, recall that the witten laplacian on functions (1.1) in l2(rn,dx) is unitary equivalent to the following operator −h(−2∇f(x).∇ + h∆) in l2(rn,e−2f/hdx). this last operator fits better with the probabilistic presentation ( [4] [17] [19]) and the simulated annealing framework ( [14]). the main purpose is the accurate computation of the smallest non zero eigenvalue of these operators among a finite collection of exponentially small eigenvalues, i.e. of order e− ck h as h → 0. the inverse of this eigenvalue can be interpreted as the longest lifetime of metastable states. the issue is the suitable control of errors (which are in absolute values larger than the final result) at every step of the induction process. a usual gramm-schmidt type orthonormalisation process as it is used in the semiclassical multiple wells problem ( [6] [9]) does not allow such a control. as this was pointed out in [7] [16], working with singular values rather than with eigenvalues of the square operator allows to use the fan inequalities ( [5] [18]) in their simplest form. these multiplicative inequalities propagate the control of the small relative errors on the singular values through the induction process. at the moment, this approach has been applied systematically only in the case of witten laplacian acting on functions. some cases with higher order witten laplacians can be considered. the only condition is the construction of global quasimodes, which is not completely elucidated for the moment except in the case of 0-forms. besides the simplification of previous proofs, this text aims at providing an abstract and general result to be referred to in the next future. cubo 11, 4 (2009) small singular values of an extracted matrix of a witten complex. 51 2 result let f (0) and f (1) be two complex hilbert spaces respectively of dimension m0 < +∞ and m1 < +∞ . let 〈 | 〉 denote the scalar product on f (0) or f (1) (without distinction), and let ‖ψ‖ and ‖a‖ = supψ 6=0 ‖aψ‖ ‖ψ‖ denote the norms of the vector ψ and of the linear application a associated with this scalar product. let moreover h0 and ε0 be two positive numbers. consider a linear application b(h) depending on h ∈ (0, h0]: b(h) : f (0) −→ f (1), and set a0(h) = b ∗ (h)b(h) ≥ 0 . let a1(h) = b(h)b ∗ (h) ≥ 0 , and note the intertwining relation: b(h)a0(h) = a1(h)b(h) . definition 1. for a number (resp. a linear operator) g(h), the notation g(h) = oε(e− α h ) means that, for all ε ∈ (0,ε0], there exists a constant cε > 0 such that: ∀h ∈ (0, h0] , |g(h)| ≤ cεe− α h (resp. ‖g(h)‖ ≤ cεe− α h ) . assumption 2.1. assume that there exist two bases (of f (0) and f (1) respectively) depending on (ε,h) ∈ (0,ε0] × (0, h0] and a positive number α independent of (ε,h) ∈ (0,ε0] × (0, h0] such that: ψ (0) k = ψ (0) k (ε,h) (k ∈ {1, . . . ,m0}) , 〈 ψ (0) k | ψ (0) k′ 〉 = δkk′ + oε(e− α h ) , ψ (1) j = ψ (1) j (ε,h) (j ∈ {1, . . . ,m1}) , 〈 ψ (1) j | ψ (1) j′ 〉 = δjj′ + oε(e− α h ) . assumption 2.2. assume furthermore that there exist an injective map j : {1, . . . ,m0} → {1, . . . ,m1} , a decreasing sequence (αk)k∈{1,...,m0} of real numbers, and a positive number d (independent of (ε,h) ∈ (0,ε0] × (0, h0]) such that: ∀ε ∈ (0,ε0], ∃cε > 1,∀k ∈ {1, . . . ,m0} , ∀h ∈ (0,h0], c−1ε e− αk+dε h ≤ ∣∣∣ 〈 ψ (1) j(k) | b(h)ψ(0)k 〉∣∣∣ ≤ cεe− αk−dε h ∀h ∈ (0,h0],∀j′ 6= j(k), ∣∣∣ 〈 ψ (1) j′ | b(h)ψ (0) k 〉∣∣∣ ≤ cεe− αk+α h . theorem 2.3. there exist positive numbers h′ 0 ≤ h0 and ε′0 ≤ ε0 such that, under assumptions 2.1 and 2.2, the eigenvalues 0 ≤ λ1(h) ≤ · · · ≤ λm0 (h) of a0(h) satisfy: 0 < λ1(h) < · · · < λm0 (h) , ∀k ∈ {1, . . . ,m0} , λk(h) = ∣∣∣ 〈 ψ (1) j(k) | b(h)ψ(0)k 〉∣∣∣ 2 (1 + oε(e− η h )) , where η > 0 is a real number independent of (ε, h) ∈ (0, ε′ 0 ] × (0, h′ 0 ]. 52 d. le peutrec cubo 11, 4 (2009) remark 2.4. more generally, vanishing eigenvalues can be included. it suffices to allow the value +∞ for the first values α1 = · · · = αℓ = +∞ and αm0 < · · · < αℓ+1 ∈ r , for some given ℓ ∈ {1, . . . ,m0}. in this last case, the eigenvalues of a0(h) satisfy: λ1 = · · · = λℓ = 0 and 0 < λℓ+1 < · · · < λm0 , while the above estimates hold for the non-zero eigenvalues. this theorem, or a modified form of this theorem according to remark 2.4, can be applied to simplify the final proof done in [7] for the case of the witten laplacian acting on 0-forms on a riemannian manifold without boundary or the one in [8] for some dirichlet realization in the case with a boundary. this final part of the analysis in [7] [8] has been reconsidered in [16], without giving all the possible simplifications. the reader can also find in [16] various illustrations in practical cases of this approach . once the quasimodes satisfying assumptions 2.1 and 2.2 are constructed, theorem 2.3 can be applied as soon as we work with a self-adjoint operator with a square structure. the application to witten laplacians on 0-forms with alternative boundary conditions is in progress. some examples of witten laplacians acting on p-forms for which quasimodes are constructed can be treated with this result and theorem 2.3 may be useful for a future generalization. while working with witten laplacians on 0-forms, the quasimodes ψ (0) k ’s are constructed globally after truncating e− f h , while the ψ (1) j ’s are introduced locally via a wkb approximation around saddle points of f, u (1) j(k) . note that the discussion in [7] [16] about sending u (1) j(1) to infinity when λ1 = 0 is replaced by consedering α1 = +∞ (according to remark 2.4) with an arbitrary additionnal ψ (1) j(1) . the application to some non self-adjoint fokker-planck operators with a distorted square strusture (see [1] [8] [11] [15]) seems more delicate (see remark 3.6). 3 proof let us begin by fixing the positive numbers ε′ 0 and η. we first choose ε′ 0 small enough such that: α ′ = α − dε′ 0 > 0 and α′′ = min k>k′ {αk′ − αk − 2dε′0} > 0 . then, we set: η = min{α, α′ ,α′′} = min{α′ ,α′′} . to prove theorem 2.3, it will be more convenient to work with matrices. let us give a definition and an easy application which will be very useful. cubo 11, 4 (2009) small singular values of an extracted matrix of a witten complex. 53 definition 2. a square matrix v (h) is said quasi-unitary if there exists an unitary matrix u such that: v (h) = u + oε(e−η/h). lemma 3.1. the product of quasi-unitary matrices is a quasi-unitary matrix. furthermore, to prove theorem 2.3, we need a particular case of fan inequalities that we recall here (we refer the reader to [18] for a proof). lemma 3.2. let b and c be respectively a compact and a bounded linear operator on a hilbert space h. the inequalities µn(bc) ≤ ‖c‖µn(b) µn(cb) ≤ ‖c‖µn(b) , where µn(b) is the n-th singular value of b, hold for all n ≤ dim h. we apply this lemma with h = h0 ⊥ ⊕ h1, while identifying b : h0 → h1 with j1bπ0 ∈ l(h), where π0 is the orthogonal projection h → h0 and j1 the embedding h1 → h. corollary 3.3. let h0, h1 be two hilbert spaces. let b be a compact linear operator from h0 to h1. assume that c ∈ l(h1) and d ∈ l(h0) are two invertible operators with: max { ‖c‖ , ∥∥c−1 ∥∥ , ‖d‖ , ∥∥d−1 ∥∥} ≤ 1 + ρ, for some ρ > −1. then the inequality (1 + ρ) −2 µn(b) ≤ µn(cbd) ≤ (1 + ρ)2µn(b) holds for all n ≤ min( dim h0, dim h1). remark 3.4. we will apply this corollary in the particular case when c and d depend on h ∈ (0, h ′ 0 ] and are quasi-unitary: c(h) = u + oε(e− η h ) and d(h) = v + oε(e− η h ) , where u and v are unitary matrices and ρ = oε(e− η h ). we obtain the equivalent relations: µn(cbd) = µn(b)(1 + oε(e− η h )) , µn(b) = µn(cbd)(1 + oε(e− η h )) . (3.1) from a0(h) = b ∗ (h)b(h), we deduce that the eigenvalues of a0(h) are the squares of the singular values of b(h): ∀k ∈ {1, . . . ,m0} , λk(h) = µ2m0+1−k(b(h)) ( µ1(b(h)) = ‖b(h)‖) . 54 d. le peutrec cubo 11, 4 (2009) in order to apply corollary 3.3, it will be easier to work with the singular values of b(h) than with the eigenvalues of a0(h). choose now two arbitrary orthonormal bases b(0) and b(1) (of f (0) and f (1) respectively). we make the identifications: b(h) = mat b(0), b(1) (b(h)) , b ∗ (h) = ( mat b(0), b(1) (b(h))) ∗ . let be b′(h) = (〈 ψ (1) j | b(h)ψ (0) k 〉) j,k = ( b ′ j k ) j,k . for i ∈ {1, . . . , ml} and l ∈ {0, 1}, we set cl = matb(l) ( ψ (l) 1 . . . ψ (l) ml ) , where ψ (l) i is written as a column vector in b(l). these change-of-coordinates matrices give b′(h) = c ∗ 1 b(h)c0 . remark 3.5. by assumption 2.1, the matrices c0 and c∗1 are quasi-unitary and assumption 2.2 implies, for h′ 0 small enough: ∀ 1 ≤ k′ < k ≤ m0 , b′j(k′) k′ = b′j(k) k.oε(e− η h ) , (3.2) ∀ 1 ≤ k ≤ m0 , ∀j 6= j(k) , b′j k = b′j(k) k.oε(e− η h ) . (3.3) we now simplify b′(h) by gaussian elimination in the following order: step 0: by permuting the rows, that is by left-multiplying with permutation matrices which are unitary, put the coefficients b′ j(k) k (for k in {1, . . . ,m0}) on the k-th row and k-th column. the new matrix has the form: b ′′ (h) =    b ′′ 1 1 = b ′ j(1) 1 b ′ j(2) 2 .oε(e− η h ) . . . b ′ j(m0) m0 .oε(e− η h ) ... b′′ 2 2 = b ′ j(2) 2 ... b ′ j(1) 1 .oε(e− η h ) ... . . . ... ... b′ j(2) 2 .oε(e− η h ) b ′′ m0 m0 = b ′ j(m0) m0 ... ... ...    . furthermore, the matrix b′′(h) satisfies the structure equations (3.2) and (3.3) with the injective map j : {1, . . . ,m0} → {1, . . . ,m1} replaced by the canonical injection i : {1, . . . ,m0} → {1, . . . ,m1}, i(k) = k. step 1: for j ∈ {1, . . . , m1} \ {m0}, replace the j-th row lj by lj − b′′j m0 b′′m0 m0 lm0 = lj − oε(e− η h ).lm0 . cubo 11, 4 (2009) small singular values of an extracted matrix of a witten complex. 55 step 2: then, for k ∈ {1, . . . , m0 − 1}, replace the k-th column ck by ck − b′′m0 k b′′m0 m0 cm0 = ck − oε(e− η h ).cm0 . due to the previous operations, only the m0-th row of the new matrix is changed by these operations. each operation of the two last steps preserves the structure of assumption 2.2, or more precisely the structure of remark 3.5 where we have replaced the injective map j by the canonical injection i. moreover, these operations correspond to left multiplications or right multiplications by quasi-unitary matrices. the new matrix only contains zeros on the m0-th row and m0-th column except for the (m0,m0)coefficient which is b′ j(m0) m0 = 〈 ψ (1) j(m0) | b(h)ψ(0)m0 〉 . when m0 ≥ 2, iterate the gaussian elimination, step 1 with the reference row m0 − ν and step 2 with the reference column m0 −ν, by taking successively ν = 1, . . . ,m0 − 2. at the end, we obtain a diagonal matrix d(h) ∈ mm0,m1 (c) such that: ∀k ∈ {1, . . . ,m0} , (d(h))k,k = 〈 ψ (1) j(k) | b(h)ψ(0)k 〉 (1 + oε(e−η/h)). moreover, by lemma 3.1, there exist two quasi-unitary matrices u(h) ∈ mm0 (c) and v (h) ∈ mm1 (c) satisfying d(h) = v (h)b ′ (h)u(h) = v (h)c ∗ 1 b(h)c0u(h) . using again lemma 3.1, v ′(h) = v (h)c∗ 1 and u′(h) = c0u(h) are quasi-unitary. from d(h) = v ′ (h)b(h)u ′ (h), we conclude using corollary 3.3 and (3.1). remark 3.6. a) the square self-adjoint structure a0(h) = b∗(h)b(h) is essential here to be able to conclude. even a small distortion, a0(h) = b∗(h)cb(h) with c = id +r with r = oε(e− η h ), in dimension 2, destroys the above arguments, due to ill-conditioning problem. in the decomposition a0(h) = b ∗ (h)b(h) + b ∗ (h)rb(h) = b ∗ (h)b(h) + b ∗ (h)oε(e− η h )b(h) , the remainder term b∗(h)oε(e− η h )b(h) cannot be put in general in the form b∗(h)b(h)oε(e− η h ): b ∗ (h)rb(h) = b ∗ (h)b(h) ( b(h) −1 rb(h) ) with ∥∥b(h)−1rb(h) ∥∥ ≤ ∥∥b(h)−1 ∥∥‖b(h)‖ ‖r‖ . for example, take η = 1 and b(h) = ( e − 4 h 0 0 e − 2 h ) and c(h) = ( 1 e − 1 h 0 1 ) . in this example the remainder factor equals b(h) −1 rb(h) = ( 0 e + 1 h 0 0 ) 56 d. le peutrec cubo 11, 4 (2009) with a norm of order e 1 h = e 2 h × e− ηh . b) a first attempt at the extension of this analysis to the non self-adjoint case related with kramersfokker-planck type operators, studied in [12] [13], led to the simple distortion a0(h) = b∗(h)cb(h) with c = id +oε(e− η h ). the previous remark shows that it cannot work without including some additional information about the intimate link between these non self-adjoint operators coming from kinetic theory and witten laplacians ( [1] [8] [11] [15]). acknowledgement: the author would like to thank f. hérau, t. jecko and f. nier for profitable discussions, and the anonymous referee for suggesting various improvements. received: august 2008. revised: september 2008. references [1] bismut, j. m. and lebeau, g., the hypoelliptic laplacian and ray-singer metrics, to appear in the princeton university press, 2008. [2] bovier, a. eckhoff, m. gayrard, v. and klein, m., metastability in reversible diffusion processes i: sharp asymptotics for capacities and exit times, jems vol.6 no. 4, pp. 399-424, 2004. [3] bovier, a. gayrard, v. and klein, m., metastability in reversible diffusion processes ii: precise asymptotics for small eigenvalues, jems vol.7 no. 4, pp. 66-99, 2004. [4] freidlin, m. i. and wentzell, a. d., random perturbations of dynamical systems, springer-verlag, new york, 1984. [5] gohgerg, i. and krejn, m., introduction à la théorie des opérateurs linéaires non autoadjoints dans un espace hilbertien, monographies universitaires de mathématiques, vol.39, dunod, 1971. [6] helffer, b., introduction to the semi-classical analysis for the schrödinger operator and applications. springer verlag. lecture notes in mathematics 1336, 1988. [7] helffer, b. klein, m. and nier, f., quantitative analysis of metastability in reversible diffusion processes via a witten complex approach, mémoire 105 société mathématique de france, 2006. [8] helffer, b. and nier, f., quantitative analysis of metastability in reversible diffusion processes via a witten complex approach: the case with boundary, matematica contemporanea vol.26 pp. 41-85, 2004. cubo 11, 4 (2009) small singular values of an extracted matrix of a witten complex. 57 [9] helffer, b. and sjöstrand, j.,, puits multiples en limite semi-classique ii -interaction moléculaire-symétries-perturbations. ann. inst. h. poincaré phys. théor. 42 (2), p. 127-212 (1985). [10] helffer, b. and sjöstrand, j., multiple wells in the semi-classical analysis iv. etude du complexe de witten, comm. partial differential equations vol.10 no. 3, pp. 245–340, 1985. [11] hérau, f. and nier, f., isotropic hypoellipticity and trend to the equilibrium for the fokker-planck equation with high degree potential, archive for rational mechanics and analysis 171 (2), p. 151-218, 2004. [12] hérau, f. sjöstrand, j. and stolk, c., semiclassical analysis for the kramers-fokkerplanck equation, comm. partial differential equations vol.30 no. 4-6, p. 689-760, 2005. [13] hérau, f. hitrik, m. and sjöstrand, j., tunnel effect for kramers-fokker-planck type operators, to appear in the ann. inst. henri poincaré, p.78, 2007. [14] holley, r. kusuoka, s. and stroock, d., asymptotics of the spectral gap with applications to the theory of simulated annealing, j. funct. anal. 83 (2), p.333-347, 1989. [15] lebeau, g., geometric fokker-planck equations, portugaliae mathematica. nova série 62, 2005. 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[19] stroock, d. w. and varadhan, s. r., multidimensionnal diffusion processes, springerverlag, berlin, 1979. articulo 4 cubo a mathematical journal vol.11, no¯ 03, (55–63). august 2009 periodic solutions of periodic difference equations by schauder’s theorem tetsuo furumochi1 department of mathematics, shimane university, 1060 nishikawatsucho, matsue 690-8504, japan email: furumochi@riko.shimane-u.ac.jp abstract in this paper, we discuss the existence problem of periodic solutions of the periodic difference equation x(n + 1) = f(n,x(n)), n ∈ z and the periodic difference equation with infinite delay x(n + 1) = f(n,xn), n ∈ z, where x and f are d-vectors, and z denotes the set of integers. we show the existence of periodic solutions by using schauder’s fixed point theorem, and illustrate an example. resumen en este artículo estudiamos el problema de existencia de soluciones periódicas para la ecuación en diferencia periódica x(n + 1) = f(n,x(n)), n ∈ z 1this paper is in final form and no version of it will be submitted for publication elsewhere, and partly supported in part by grant-in-aid for scientific research (c), no. 16540141, japan society for the promotion of science. 56 tetsuo furumochi cubo 11, 3 (2009) y la ecuación en diferencia periódica con retardo infinito x(n + 1) = f(n,xn), n ∈ z, donde x y f son d-vectores, y z denota el conjunto de los números enteros. mostramos la existencia de soluciones periódicas mediante el uso del teorema de punto fijo de schauder, exhibimos un ejemplo. key words and phrases: periodic solutions, difference equations, fixed point theorem. math. subj. class.: 39a10, 39a11, 39a20. 1 introduction the existence problem of periodic solutions of functional equations has been discussed in many books and papers. for example, see the books [1-3, 12, 15, 28, 30, 32] and papers [4-11, 13, 14, 16-27, 29, 33], and their references. in these books and papers, many kinds of functional equations have been studied. for example, volterra equations [2, 4-6, 13, 22, 23], ordinary and functional differential equations [1, 3, 10, 15-20, 27-29, 32, 33], integro-differential equations [7, 22], integral equations [8, 9, 21], and difference equations [11-14, 23-26, 30]. in this paper, we give some new existence results of periodic solutions for periodic difference equations by using schauder’s fixed point theorem and a convex liapunov function, and show that the existence problem of periodic solutions of a periodic difference equation with infinite delay can be reduced to the existence problem of periodic solutions of an auxiliary difference equation with finite delay. fixed point theorems are very useful tools in obtaining existence theorems for periodic solutions. since we use schauder’s second fixed point theorem later, first we state it for the sake of completeness. theorem 1 (schauder’s second theorem [31]). let (b,‖ · ‖) be a normed space, and let s be a nonempty convex subset of b. then every continuous mapping of s into a compact set c of s has a fixed point in c. 2 periodic difference equations let r+ = [0,∞), r = (−∞,∞), and let rd be the d-dimensional euclidean space. let f(n,x) : z × rd → rd be continuous in x for each fixed n ∈ z, and n-periodic in n for some n ∈ n with n > 1, where n denotes the set of positive integers. cubo 11, 3 (2009) periodic solutions of periodic difference ... 57 consider the periodic difference equation x(n + 1) = f(n,x(n)), n ∈ z. (1) for any n0 ∈ z and ξ ∈ rd, x(n) = x(n,n0,ξ) denotes the solution of eq.(1) with x(n0) = ξ. when we employ theorem 1 in order to prove the existence of a fixed point of a mapping, we need to define a suitable convex set in a banach space. in [27], grimmer introduced the concept of a convex liapunov function, and proved the existence of periodic solutions of functional differential equations by employing a fixed point theorem. moreover, in [20], the existence of periodic solutions of functional differential equations is proved by using a convex liapunov function and schauder’s fixed point theorem. here, first we state the definition of a convex liapunov function for the sake of completeness. definition. a function v (n,x) : z × rd → r+ is said to be a convex liapunov function if v (n,x) is continuous in x for each fixed n, and satifies the following conditions. (i) v (n,x) ≥ a(|x|) for a continuous function a(r) such that a(r) → ∞ as r → ∞, where | · | denotes the euclidean norm of rd. (ii) the set xρ := {x ∈ rd : v (n,x) ≤ ρ} is a convex set in rd for any n ∈ z and ρ > 0, provided that xρ is nonempty. now we have the following theorem. theorem 2. let v : z × rd → r+ be an n-periodic convex liapunov function. suppose that there exist an n0 ∈ z ∩ [0,n) and a constant ρ > max{v (n, 0) : 0 ≤ n ≤ n} such that for any ξ ∈ s := {x ∈ rd : v (n0,x) ≤ ρ}, we have v ( n0,x(n0 + n,n0,ξ) ) ≤ ρ. (2) then, eq.(1) has an n-periodic solution. proof. since v (n,x) is a convex liapunov function and ρ > max{v (n, 0) : 0 ≤ n ≤ n}, s is a nonempty compact convex subset of rd. let p be a mapping on s defined by p(ξ) := x(n0 + n,n0,ξ), ξ ∈ s. then (2) implies that p(s) is contained in s. moreover, the continuity of f(n,x) in x for each fixed n ∈ z implies that p : s → s is a continuous mapping. thus, by theorem 1, p has a fixed point ξ ∈ s, and x(n) = x(n,n0,ξ) is an n-periodic solution of eq.(1). now we show an example for theorem 2. 58 tetsuo furumochi cubo 11, 3 (2009) example. consider the 4-periodic difference equation        x (1) (n + 1) = αx (2) (n) + β cos nπ 2 , x (2) (n + 1) = γx (1) (n) + δ sin nπ 2 , where n ∈ z, and where α, β, γ, and δ are constants with √ 2 max(|α|, |γ|) < 1. let r be a positive constant with r ≥ √ 2 max(|β|, |δ|) 1 − √ 2 max(|α|, |γ|) , (3) and let v (n,x) := (x(1))2 + (x(2))2, where x := (x(1),x(2)). clearly, v is a 4-periodic convex liapunov function with a(r) = r2. the set s defined by s := {x ∈ r2 : |x| ≤ r} is a nonempty compact convex subset of r2 for the constant r > 0. for any ξ := (ξ(1),ξ(2)) ∈ s, let x (1) (n) = x (1) (n, 0,ξ), x (2) (n) = x (2) (n, 0,ξ), and let x(n) = ( x (1) (n),x (2) (n) ) . then, (3) implies − r√ 2 ≤ −|α|r − |β| ≤ αξ(2) − |β| ≤ x(1)(1) ≤ αξ(2) + |β| ≤ |α|r + |β| ≤ r√ 2 , − r√ 2 ≤ −|γ|r − |δ| ≤ γξ(1) − |δ| ≤ x(2)(1) ≤ γξ(1) + |δ| ≤ |γ|r + |δ| ≤ r√ 2 , which yields that |x(1)| ≤ r. thus we obtain x(1) ∈ s. by similar arguments, we have v ( 4,x(4, 0,ξ) ) ≤ r 2, and consequently x(4) ∈ s. thus, by theorem 2, this 4-periodic difference equation has a 4periodic solution x(n) with |x(n)| ≤ r for n ∈ z. 3 periodic difference equations with finite delay in this section, concerning the existence of periodic solutions of periodic difference equations with finite delay, we state some known results. for a fixed κ ∈ n, let b be the set of sequences φ : z ∩ [−κ, 0] → rd. for any φ ∈ b, define ‖φ‖ by ‖φ‖ := sup{|φ(k)| : k ∈ z ∩ [−κ, 0]}. for any α > 0, the set bα defined by bα := {φ ∈ b : ‖φ‖ ≤ α} is compact. for any sequence x(k) : z → rd and any fixed n ∈ z, the symbol xn denotes the restriction of x(k) on z ∩ [n − κ,n], that is, xn is an element of b defined by xn(k) := x(n + k), k ∈ z ∩ [−κ, 0]. cubo 11, 3 (2009) periodic solutions of periodic difference ... 59 consider the difference equation with finite delay x(n + 1) = f(n,xn), n ∈ z, (4) where f : z × b → rd is continuous in φ for each fixed n ∈ z, and n-periodic in n for some n ∈ n with n > 1. for any n0 ∈ z and any initial sequence φ ∈ b, there is a unique solution of eq.(4), denoted by x(n,n0,φ), such that it satisfies eq.(4) for n ∈ z ∩ [n0,∞) and x(n0 + k,n0,φ) = φ(k) for k ∈ z ∩ [−κ, 0]. in [26], concerning the existence of periodic solutions of eq.(4), the following theorem is proved by employing browder’s fixed point theorem. theorem 3 ([26]). if f(n,φ) in eq.(4) is n-periodic in n for some n ∈ n with n > 1, and if the solutions of eq.(4) are uniformly ultimately bounded for bound x, then eq.(4) has an n-periodic solution x(n) such that |x(n)| < x for n ∈ z. here the solutions of eq.(4) are said to be uniformly ultimately bounded for bound x, if there exists an x and if corresponding to any n0 ∈ z and α > 0, there exists a ν = ν(α) ∈ n such that φ ∈ bα implies that |x(n,n0,φ)| < x for n ∈ z ∩ [n0 + ν,∞). in theorem 3, uniform ultimate boundedness of solutions of eq.(4) is an important assumption. here we state a boundedness theorem due to shunian zhang without a proof. theorem 4 ([34]). suppose that there exists a liapunov function v : z × rd → r+, which satifies the following conditions; (i) a(|x|) ≤ v (n,x) ≤ b(|x|), where a, b : r+ → r+, a(r) and b(r) are continuous, increasing and a(r) → ∞ as r → ∞, (ii) ∆v(4)(n,x(n)) := v (n + 1,x(n + 1)) − v (n,x(n)) ≤ m − c(|x(n)|) whenever p ( v (n + 1,x(n + 1)) ) > v (k,x(k)) for k ∈ z ∩ [n − κ,n], where x(n) is a solution of eq.(4), m is a positive constant, c : r+ → r+ is continuous, increasing and c(r) → ∞ as r → ∞, and p : r+ → r+ is continuous, p(u) > u for u > 0, and κ ∈ n. then the solutions of eq.(4) are uniformly ultimately bounded for a bound x. 60 tetsuo furumochi cubo 11, 3 (2009) 4 periodic difference equations with infinite delay by combining liapunov’s method and theorems 3 and 4 in section 3, we can obtain a theorem which assures the existence of periodic solutions of periodic difference equations. but theorem 4 is applicable to difference equations with finite delay, and it seems to be open whether we can prove a theorem similar to theorem 4 for difference equations with infinite delay or not. in this section, we show that the existence problem of periodic solutions of periodic difference equations with infinite delay can be reduced to the existence problem of periodic solutions of auxiliary difference equations whose delay is equal to its period. let b be the set of bounded sequences φ : z− → rd, where z− denotes the set of nonpositive integers. for any φ ∈ b, define ‖φ‖ by ‖φ‖ := sup{|φ(k)| : k ∈ z−}. for any sequence x(k) : z → rd and any fixed n ∈ z, the symbol xn denotes the restriction of x(k) on z ∩ (−∞,n], that is, xn is an element of b defined by xn(k) = x(n + k), k ∈ z−. consider the periodic difference equation with infinite delay x(n + 1) = f(n,xn), n ∈ z, (5) where f(n,φ) : z × b → rd is continuous in φ for each fixed n ∈ z, and n-periodic in n for some n ∈ n with n > 1. for any n0 ∈ z and any initial sequence φ ∈ b, there is a unique solution of eq.(5), denoted by x(n,n0,φ), such that it satisfies eq.(5) for n ∈ z ∩ [n0,∞) and x(n0 + k,n0,φ) = φ(k) for k ∈ z−. corresponding to b, let bn be the set of sequences ψ : z ∩ [−n, 0] → rd. for any ψ ∈ bn , define a mapping ρ(ψ) : bn → bn by ρ(ψ)(k) := ψ(k) + n + k n ( ψ(−n) − ψ(0) ) , k ∈ z ∩ [−n, 0], and a mapping σ(ψ) : bn → b by σ(ψ)(k) :=        ψ(k), −n ≤ k ≤ 0, ρ(ψ)(k + jn), −(j + 1)n ≤ k < −jn, j ∈ n. then, we have the following lemma. lemma. the functional σ(ψ) is continuous. if ψ ∈ bn satisfies ψ(−n) = ψ(0), then σ(ψ)(k) is n-periodic on z−. cubo 11, 3 (2009) periodic solutions of periodic difference ... 61 proof. from the definition of σ(ψ), it is clear that the functional σ(ψ) is continuous. next, from the definition of ρ(ψ), if ψ(−n) = ψ(0), then we have ρ(ψ)(k) ≡ ψ(k) for k ∈ z ∩ [−n, 0], which together with the definition of σ(ψ), implies that σ(ψ)(k) is n-periodic on z−. for the functional f(n,φ) in eq.(4), define the functional g(n,ψ) : z × bn → rd by g(n,ψ) := f ( n,σ(ψ) ) , (n,ψ) ∈ z × bn. then, g(n,ψ) is continuous in ψ for each fixed n ∈ z, and n-periodic in n. corresponding to eq.(5), consider the auxiliary difference equation y(n + 1) = g(n,yn), n ∈ z, (6) where yn ∈ bn , that is, yn(k) = y(n + k), k ∈ z ∩ [−n, 0]. then, we have the following theorem. theorem 5. if eq.(5) has an n-periodic solution, then it is an n-periodic solution of eq.(6), and vice versa. proof. let x(n) be an n-periodic solution of eq.(5), and let yn ∈ bn be the restriction of x(k) on z ∩ [n − n,n]. then we have yn(−n) = yn(0), which together with lemma, implies σ(yn) = xn. thus, we obtain y(n + 1) = x(n + 1) = f(n,xn) = f ( n,σ(yn) ) = g(n,yn), which shows that y(n) is an n-periodic solution of eq.(6). the converse part can be proved similarly. received: january 28, 2008. revised: march 10, 2008. references [1] burton, t.a., stability and periodic solutions of ordinary and functional differential equations, dover, new york, 2005. 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[34] zhang, s., boundedness of finite delay systems, ann. of diff. eqs., 9(1993), 107–115. 09-furumochi () cubo a mathematical journal vol.13, no¯ 03, (49–56). october 2011 on strongly fβp-irresolute mappings ratnesh kumar saraf department of mathematics, government kamla nehru, girls college damoh (m.p.)-470661, india. and miguel caldas departamento de matemática aplicada, universidade federal fluminense rua mário santos braga s/n0,cep: 24020-140, niteroi-rj,brasil. email: gmamccs@vm.uff.br abstract in this paper, we introduce a new class of mappings called strongly fβp-irresolute mappings between fuzzy topological spaces. we obtain several characterizations of this class and study its properties and investigate the relationship with the known mappings. resumen en este trabajo presentamos una nueva clase de funciones llamadas funciones fuertemente fβp-irresolute entre espacios topológicos difusos. obtenemos varias caracterizaciones de esta clase, estudiamos sus propiedades e investigamos la relación con funciones conocidas. keywords: fuzzy topological spaces, fuzzy β-open sets, fuzzy β-preirresolute maps, strongly fuzzy β-preirresolute maps. 50 ratnesh kumar saraf and miguel caldas cubo 13, 3 (2011) mathematics subject classification: 54c10, 54d10. 1 introduction and preliminaries. the concept fuzzy has invaded almost all branches of mathematics with the introduction of fuzzy sets by zadeh [23] of 1965. the theory of fuzzy topological spaces was introduced and developed by chang [6] and since then various notions in classical topology have been extended to fuzzy topological spaces. recently professor el-naschie has been shown in [7] and [8] that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and ε∞ theory. thus our motivation in this paper is to define strongly fuzzy β-preirresolute (in short st-fβp-irresolute) mappings and investigate its properties. the new defined class of mapping is stronger that m-fuzzy β-continuous mappings and is a generalization of st-fαp-irresolute mappings. throughout this paper (x, τ), (y, σ) and (z, γ) (or simply x, y and z) represent non-empty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned. the fuzzy set a of x is called fuzzy α-open (fα-open) [5] (resp. fuzzy preopen (fp-open) [5], fuzzy β-open (fβ-open) [2]) if a ≤ int(cl(int(a)) (resp. a ≤ int(cl(a)), a ≤ cl(int(cl(a))), where cl(a) and int(a) denote the closure of a and the interior of a respectively. the fuzzy subset b of x is said to be fuzzy α-closed (fα-closed) (resp. fuzzy preclosed (fp-closed), fuzzy β-closed (fβ-closed)) if, its complement bc is fuzzy fα-open (resp. fp-open, fβ-open) in x. by fαo(x), fpo(x) and fβo(x) (resp. fαc(x), fpc(x), and fβc(x)) we denote the family of all fα-open, fp-open and fβ-open (resp. fα-closed, fp-closed and fβ-closed) sets of x. the intersection of all fuzzy β-closed sets containing a is called the β-closure of a and is denoted by βcl(a). the fuzzy β-interior [2] of a denoted by β-int(a), is defined by the union of all fuzzy β-open sets of x contained in a. a mapping f : x → y is said to be: (i) fuzzy completely weakly preirresolute [11] (resp. fαp-irresolute [5], m-fuzzy precontinuous [3], fβp-irresolute [17]) if, f−1(v) is fuzzy open (resp. fα-open), fp-open, fβ-open) in x for every fp-open set v of y. (ii) strongly m-fuzzy β-continuous [16] (resp. m-fuzzy β-continuous [15], st-fαp-irresolute [16]) if, f−1(v) is fuzzy open (resp. fβ-open, fα-open) in x for every fβ-open set v of y. (iii) fuzzy strongly continuous [12] if, f−1(v) is fuzzy clopen in x for every fuzzy subset v of y. a fuzzy point in x with support x ∈ x and value p (0 1. two fuzzy sets a and b are said to be quasi-coincident denoted by aqb, if there exists x ∈ x such that a(x) + b(x) > 1 [14] and by − q we denote ”is not q-coincident”. it is known [14] that a ≤ b if and only if aq(1 − b). cubo 13, 3 (2011) on strongly fβp-irresolute mappings 51 two non empty fuzzy subsets a and e are said to be fuzzy β-separated if there exist two fuzzy β-open subsets g and h such that a ≤ g, e ≤ h, a − qh and e − qg. a fuzzy subset which cannot be expressed as the union of two fuzzy β-separated subsets is said to be fuzzy β-connected sets. lemma 1.1. [22] let f : x → y be a mapping and xp be a fuzzy point of x. then: (1) f(xp)qb ⇒ xpqf−1(b), for every fuzzy set b of y. (2) xpqa ⇒ f(xp)qf(a), for every fuzzy set a of x. 2 st-fβp-irresolute mappings. definition 2.1. a mapping f : x → y is said to be strongly fuzzy βpreirresolute (briefly st-fβpirresolute) if, f−1(v) is fuzzy preopen in x for every fβ-open set v of y. from the definitions stated, we have the following diagram: a → b → c → d ↓ ↓ ↓ ↓ e → f → g → h where: a = st-mfβ-continuous; b = st-fαp-irresolute; c = stfβp-irresolute; d = mfβcontinuous; e = fuzzy completely weakly preirresolute; f= fαp-irrsesolute; g = mfp-continuous; h = fβp-irresolute. remark 2.1. however, converses of the above implications are not true in general, by [12, 16, 17] and the followings examples: (i) fαp-irrsesolute mapping does not imply fuzzy completely weakly preirresolute: let x = {a, b} and y = {x, y}. define fuzzy sets a(a) = 0.6, a(b) = 0.5; b(a) = 0, b(b) = 0.8; h(x) = 0.5, h(y) = 0.5; e(x) = 0.7, e(y) = 0.8. let τ = {0, a, 1}, γ = {0, b, 1}; σ = {0, h, 1} and υ = {0, e, 1}. the mapping f : (x, τ) → (y, σ) defined by f(a) = x , f(b) = y is fuzzy α-preirresolute but not fuzzy completely weakly preirresolute, because z(x) = 0.7, z(y) = 0.7 are fuzzy preopen in (y, σ) but f−1(z) is not fuzzy open in x. (ii) fuzzy completely weakly preirresolute mapping does not imply mfβ-continuous, see [[18], example 3.2]. (iii) mfβ-continuous mapping does not imply mfp-continuous, see [[19], example 3.1]. (iv) stfβp-irresolute mapping does not imply fαp-irrsesolute, see [[19], example 3.2]. 52 ratnesh kumar saraf and miguel caldas cubo 13, 3 (2011) (v) stfαp-irresolute mapping does not imply fuzzy completely weakly preirresolute, see [[16], example 3.1]. theorem 2.1. for a mapping f : x → y, the following are equivalent: (1) f is st-fβp-irresolute; (2) for every fuzzy point xt in x and every fβ-open set v of y containing f(xt), there exist a fp-open set u of x containing xt such that f(u) ≤ v; (3) for every fuzzy point xt in x and every fβ-open set v of y containing f(xt), there exist a fp-open set u of x containing xt such that xt ∈ u ≤ f −1(v); (4) for every fuzzy point xt in x, the inverse image of each β-neighbourhood of f(xt) is a preneighbourhood of xt; (5) for every fuzzy point xt in x and each β-neighbourhood e of f(xt), there exists an preneighbourhood a of xt such that f(a) ≤ e; (6) f−1(v) ≤ int(cl(f−1(v))) for every v ∈ fβo(y); (7) f−1(h) ∈ fpc(x) for every h ∈ fβc(y) ; (8) cl(int(f−1(e))) ≤ f−1(βcl(e)) for every fuzzy subset e of y; (9) f(cl(int(a))) ≤ βcl(f(a))) for every fuzzy subset a of x. proof. (1) ⇔ (2) ⇔ (3); (4) ⇒ (5): obvious (2) ⇒ (6): let v ∈ fβo(y) and xt ∈ f−1(v). by (2), there exists u ∈ fpo(x) containing xt such that f(u) ≤ v. thus we have xt ∈ u ≤ int(cl(u)) ≤ int(cl(f −1(v))) and hence f−1(v) ≤ int(cl(f−1(v))). (6) ⇒ (7): let h ∈ fβc(y). set v = y − h, then v ∈ fβo(y). by (6) we obtain f−1(v) ≤ int(cl(f−1(v))) and hence f−1(h) = x − f−1(y − h) = x − f−1(v) ∈ fpc(x). (7) ⇒ (8): let e be any fuzzy set of y. since βcl(e) ∈ fβc(y), then f−1(βcl(e)) ∈ fpc(x) and hence cl(int(f−1(βcl(e)))) ≤ f−1(βcl(e)). therefore we obtain cl(int(f−1(e))) ≤ f−1(βcl(e)). (8) ⇒ (9): let a be any fuzzy set of x. by (8), we have cl(int(a)) ≤ cl(int(f−1(f(a)))) ≤ f−1(βcl(f(a))) and hence f(cl(int(a))) ≤ βcl(f(a)). (9) ⇒ (1): let v ∈ fβo(y). since f−1(y − v) = x − f−1(v) is a fuzzy set of x and by (9), we obtain f(cl(int(f−1(y − v)))) ≤ βcl(f(f−1(y − v))) ≤ βcl(y − v) = y − βint(v) = y − v and hence x − int(cl(f−1(v))) = cl(int(x − f−1(v)))) = cl(int(f−1(y − v))) ≤ f−1(f(cl(int(f−1(y − v))))) ≤ f−1(y − v) = x − f−1(v). therefore, we have f−1(v) ≤ int(cl(f−1(v))) and hence f−1(v) ∈ fpo(x). thus, f is st-fβp-irresolute. (1) ⇒ (4): let xt be a fuzzy point in x and v be any β-neighbourhood of f(xt), then there exists g ∈ fβo(y) such that, f(xt) ∈ g ≤ v. now f −1(g) ∈ fpo(x) and xt ∈ f −1(g) ≤ f−1(v). thus f−1(v) is an preneighbourhood of xt in x. (5) ⇒ (2): let xt be a fuzzy point in x and v ∈ fβo(y) such that f(xt) ∈ v. then v is βneighbourhood of f(xt), so there is a preneighbourhood a of xt such that xt ∈ a, and f(a) ≤ v. hence there exists u ∈ fpo(x) such that xt ∈ u ≤ a, and so f(u) ≤ f(a) ≤ v. cubo 13, 3 (2011) on strongly fβp-irresolute mappings 53 theorem 2.2. for a function f : x → y, the following are equivalent: (1) f is st-fβp-irresolute; (2) for each fuzzy point xt of x and every e ∈ fβo(y) such that f(xt)qe, there exists a ∈ fpo(x) such that xtqa and f(a) ≤ e; (3) for every fuzzy point xt of x and every e ∈ fβo(y) such that f(xt)qe, there exists a ∈ fpo(x) such that xtqa and a ≤ f −1(e). proof. (1) ⇒ (2) let xt be a fuzzy point in x and e ∈ fβo(y) such that f(xt)qe. then f−1(e) ∈ fpo(x), and xtqf −1(e) by lemma 1.1. if we take a = f−1(e) then xtqa and f(a) = f(f−1(e)) ≤ e. (2) ⇒ (3) let xt be a fuzzy point in x and e ∈ fβo(y) such that f(xt)qe. then by (2), there exists a ∈ fpo(x) such that xtqa and f(a) ≤ e. hence we have xtqa and a ≤ f −1(f(a)) ≤ f−1(e). (3) ⇒ (1) let e ∈ fβo(y) and xt be a fuzzy point of x such that xt ∈ f−1(e). then f(xt) ∈ e. choose the fuzzy point xct (x) = 1 − xt(x). then f(x c t )qe. and so by (3), there exists a ∈ fpo(x) such that xct qa and f(a) ≤ e. now x c t qa implies x c t (x) + a(x) = 1 − xt(x) + a(x) > 1. it follows that xt ∈ a. thus xt ∈ a ≤ f −1(e). hence f−1(e) ∈ fpo(x). lemma 2.1. [1] let g : x → x × y be the graph of a mapping f : x → y. if a is a fuzzy set of x and b is a fuzzy of y, then g−1(a × b) = a ∩ f−1(b) theorem 2.3. a mapping f : x → y is st-fβp-irresolute if the graph mapping g : x → x × y, is st-fβp-irresolute. proof. let v be any fβ-open set of y, then by lemma 2.1, f−1(v) = 1x ∩ f −1(v) = g−1(1x × v). since v is fβ-open in y, 1x ×v is fβ-open in x×y. since g is st-fβp-irresolute g −1(1x ×v) ∈ fpo(x) and hence f−1(v) is fp-open in x and consequently f is st-fβp-irresolute. theorem 2.4. if f : x → y is st-fβp-irresolute and g : y → z is m-fuzzy β-continuous, then g ◦ f : x → z is st-fβp-irresolute. proof. straightforward. corollary 2.1. the composition of two st-fβp-irresolute mapping is st-fβp-irresolute. corollary 2.2. if f : x → y is fuzzy strongly continuous and g : y → z is st-fβp-irresolute, then g ◦ f : x → z is st-fβp-irresolute. proof. obvious. theorem 2.5. if f : x → y is m-fuzzy β-continuous and g : y → z is st-fβp-irresolute, then g ◦ f : x → z is st-fβp-irresolute. theorem 2.6. let {xi : i ∈ ω} be any family of fuzzy topological spaces. if f : x → ∏ xi is st-fβp-irresolute, then for each i ∈ ω, fi : x → xi is st-fβp-irresolute. 54 ratnesh kumar saraf and miguel caldas cubo 13, 3 (2011) proof. let pri be the projection of ∏ xi onto xi, we know that if a mapping is fuzzy continuous and fuzzy open, then it is m-fuzzy β-continuous [21]. so the mapping pri is m-fuzzy β-continuous. now for each i ∈ ω, fi = pri ◦ f : x → xi. it follows from theorem 2.1 that fi is st-fβp-irresolute since f is st-fβp-irresolute. 3 preservation of some fuzzy topological structure. in this section preservation of some fuzzy topological structure under the st-fβp-irresolute mapping are studied. let us recall the definition: a space x is said to be fuzzy β-compact [4] if for every fβ-open cover of x has a finite subcover, and x is fuzzy strongly compact [13] if for every fp-open cover of x has a finite subcover. theorem 3.1. every surjective st-fβp-irresolute image of a fuzzy strongly compact space is fuzzy β-compact. proof. let f : x → y be st-fβp-irresolute mapping of a fuzzy strongly compact space x onto a space y. let {gi : i ∈ ω} be any fβ-open cover of y. then {f −1(gi) : i ∈ ω} is a fp-open cover of x. since x is fuzzy strongly compact, there exist a finite subfamily {f−1(gij ) : j = 1, 2, ..., n} of {f−1(gi) : i ∈ ω} which covers x. it follows that {gij : j = 1, 2, ..., n} is a finite subfamily of {gi : i ∈ ω} which covers y. hence y is fβ-compact. theorem 3.2. let f : x → y be a st-fβp-irresolute mapping. if a is a fβ-connected subset of x, then f(a) is also fβ-connected in y. proof. suppose f(a) is not fβ-connected in y. then there exist fβ-separated subset g and h in y, such that f(a) = g ∪ h. since g and h are fβ-separated, there exist two fβ-open, subset u and v such that g ≤ u, h ≤ v, g − qv and h − qu. now f being st-fβp-irresolute so f−1(g) and f−1(h) are fp-open in x. thus fβ-open in x and a = f−1(f(a)) = f−1(g ∪ h) = f−1(g) ∪ f−1(h). it is easy to shows that f−1(g) and f−1(h) are fβ-separated in x. thus a is not fβ-connected in x. 4 conclusion. maps have always been of tremendous importance in all branches of mathematics and the whole science. on the other hand, topology plays a significant role in quantium physics, high energy physics and superstring theory [9, 10]. thus we have obtained a new class of mappings called strongly fβp-irresolute mappings between fuzzy topological spaces which are some generalized fuzzy continuity may have possible application in quantion physics, high energy physics and superstring theory. received: june 2009. revised: august 2010. cubo 13, 3 (2011) on strongly fβp-irresolute mappings 55 references [1] azad k.k., on fuzzy semicontinuity, fuzzy almost continuity and weakly continuity, j. math. anal. appl., 82 (1981), 14-32. 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[23] zadeh l.a., fuzzy sets, inform and control, 8(1965), 338-353. introduction and preliminaries. st-fp-irresolute mappings. preservation of some fuzzy topological structure. conclusion. raymonduniformestimates.dvi cubo a mathematical journal vol.12, no¯ 01, (67–81). march 2010 uniform spectral estimates for families of schrödinger operators with magnetic field of constant intensity and applications nicolas raymond laboratoire de mathématiques, université paris-sud 11, bâtiment 425, f-91405 email : nicolas.raymond@math.u-psud.fr abstract the aim of this paper is to establish uniform estimates of the bottom of the spectrum of the neumann realization of (i∇ + qa)2 on a bounded open set ω with smooth boundary when |∇ × a| = 1 and q → +∞. this problem was motivated by a question occurring in the theory of liquid crystals and appears also in superconductivity questions in large domains. resumen el objetivo de este artículo es establecer estimativas uniformes del espectro inferior de la realización de neumann de (i∇ + qa)2 sobre un conjunto ω abierto acotado con frontera suave cuando |∇ × a| = 1 y q → +∞. este problema fue motivado por una cuestión que ocurre en teoría de cristales líquidos y aparece también en cuestiones de supercontudividad en dominios grandes. key words and phrases: spectral theory, semiclassical analysis, neumann laplacian, magnetic field, liquid crystals. math. subj. class.: 35p, 35j10, 35q, 81q20, 82d30. 68 nicolas raymond cubo 12, 1 (2010) 1 introduction let ω ⊂ r3 be an open bounded set with c3 boundary and a ∈ c2(ω). we will study the quadratic form qa defined by: qa(u) = ∫ ω |(i∇ + qa)u|2 dx, ∀u ∈ h1(ω) and consider the associated selfadjoint operator, i.e the neumann realization of (i∇ + qa)2 on ω. we denote µω(q, a) or µ(q, a) the lowest eigenvalue of the previous operator. our purpose is to study the behavior of this eigenvalue as q tends to infinity and to control the uniformity of the estimates with respect to the magnetic field. we let a = {a ∈ c3(ω) : |b| = 1 where b = ∇ × a}. (1.1) the main estimates obtained in this paper are summarized in the two following theorems: theorem 1.1 (uniform lower bound). for all ǫ ∈]0, 1 2 [, there exists c = c(ω,ǫ) > 0 and q0 > 0, such that, for all q ≥ q0 and for all a ∈ a, µ(q, a) ≥ θ0q − c ( q1−2ǫ + (1 + |∇b|∞)q1/2+2ǫ ) . theorem 1.2 (uniform upper bound). for all δ ∈]0, 1/2[, there exists c = c(ω,δ) > 0 and q0 = q0(ω; δ) > 0, such that for all q ≥ q0 and all a ∈ a: µ(q, a) ≤ θ0q + c(q2δ + |b|2c1q2−4δ + |b|c1q1−δ + |b|c2q3/2−3δ + |b|2c2q2−6δ), where |b|2c1 = |b|∞ + |∇b|2∞ and |b|2c2 = |b|2c1 + |∇2b|2∞. remark 1.3. in those theorems, ǫ and δ are left undefined because they will play a role in the application to families of vector potentials where the semi-norms of b could become large. if the magnetic field is fixed and if we are not interested in uniformity, we take ǫ = 1 8 and δ = 1 3 to have the optimal estimates (relatively to the method), leading to the remainder o(q3/4) in the first case (in fact, one can hope a remainder o(q2/3)) and o(q2/3) in the second case (cf. [hm04]). let us also mention that, in dimension 2, the remainder is o(q1/2) (see [fh08]). � let us briefly recall the motivation of those estimates. cubo 12, 1 (2010) uniform spectral estimates for families ... 69 liquid crystals the first one occurs in the theory of liquid crystals. the asymptotic properties of the landaude gennes functional (cf. [bclp02, dg95, hp07, pan03, pan06]) lead to the analysis of the minimizers (or local minimizers) of the reduced functional: f(ψ, n) = ∫ ω |(i∇ + qn)ψ|2 + r|ψ|2 + g 2 |ψ|4dx, where r > 0, ψ ∈ h1(ω, c) and n ∈ c(τ), with c(τ) defined for τ > 0 by: c(τ) = {qnτ qt,q ∈ so3}, (1.2) where so3 denotes the set of the rotations in r 3 and nτ = (cos(τx3), sin(τx3), 0). (1.3) let us notice that, if n ∈ c(τ), ∇ × n + τn = 0 and consequently |∇ × n|∞ = τ. then, the analysis of the positivity of the hessian of the functional at ψ = 0 leads to a spectral problem and we are led to study the asymptotic properties of µ∗(q,τ) = inf n∈c(τ ) µ(q, n), (1.4) as qτ → +∞. in this context, x-b. pan has given estimates (cf. [pan06]) as: qτ → +∞ and τ → 0 and helffer and pan give some extensions in [hp07] including the case: qτ → +∞ and τ bounded. in this paper, we treat the case where: qτ → +∞ and τ → +∞. superconductivity the second one occurs in the theory of superconductivity in large domains. in [alm02, alm08], y. almog has analyzed properties of minimizers of ginzburg-landau’s functional when the size of ω tends to infinity; theorems 1.1 and 1.2 permit to treat another regime (for the linear problem); in fact, q will be allowed to tend to infinity. 70 nicolas raymond cubo 12, 1 (2010) organization of the paper the paper is organized as follows. first, we prove theorem 1.1 in section 2 and theorem 1.2 in section 3, then we will prove an agmon estimate in section 4 in order to study the localization of first eigenfunctions in the considered asymptotic regime. finally, section 5 will present the applications to the theory of liquid crystals and to the superconductivity in large domains. in each case we will show that the eigenfunctions become localized at the boundary. this corresponds to what is called surface smecticity in the first case and surface superconductivity in the second case (see [pan04]). 2 lower bound in this section, we give the proof of theorem 1.1. it is based on a localization technique through a partition of unity and the analysis of simplified models. 2.1 partition of unity for each r > 0, we consider a partition of unity (cf. [hm04]) with the property that there exists c = c(ω) > 0 such that: ∑ j |χrj |2 = 1 on ω ; (2.5) ∑ j |∇χrj|2 ≤ c r2 on ω. (2.6) each χrj is a c∞-cutoff function with support in the ball of center xj and radius r (denoted by bj ). we will choose r later for optimizing the error. we will use the ims formula (cf. [cfks86]): lemma 2.1. qa(u) = ∑ j qa(χju) − ∑ j ‖|∇χrj|u‖2, ∀u ∈ h1(ω). (2.7) so, in order to minimize qa(u), we will be reduced to the minimization of qa(v), with v supported in some bj , the price to pay being an error of order c r2 . 2.2 approximation by the constant magnetic field in a ball or a semi-ball we want to have estimates depending only on the magnetic field b = ∇ × a, that’s why we look for a canonical choice of a depending only on b; it is the aim of the following lemmas. let b a ball (or semi-ball) of center 0 and radius r > 0. cubo 12, 1 (2010) uniform spectral estimates for families ... 71 lemma 2.2. let f ∈ c2(b, r3). we assume the existence a constant c > 0 such that: |∇ × f| ≤ c|x|, for x ∈ b. then, there exists u ∈ c3(b) and α > 0 a constant such that: |f(x) − ∇u(x)| ≤ αc|x|2, for all x ∈ b. proof. the proof is similar to the one of poincaré’s theorem. let us define, for all x ∈ b: u(x) = ∫ 1 0 f(tx) · xdt. let us verify that u is suitable. as f ∈ c2(b, r3), we can extend f in a c2 function on r3, so by computing, we have, for x ∈ b: ∂iu(x) = fi(x) + 3∑ j=1,j 6=i ∫ 1 0 (∂ifj − ∂jfi) (tx)txjdt. � as a consequence, we immediately deduce the lemma which will give us uniformity in our further estimates : lemma 2.3. there exists c > 0 such that, for all a ∈ c2(b), there exists φ ∈ c3(b) verifying: |a(x) − alin(x) − ∇φ(x)| ≤ c|∇b|∞|x|2, for x ∈ b and where alin is defined by: a lin(x) = 1 2 b(0) ∧ x. 2.3 local estimates for the lower bound we now distinguish two cases: the balls inside ω and those which intersect the boundary. for the balls inside ω, we are reduced to the problem of dirichlet with constant magnetic field, and for the other balls, to the problem of neumann on an half plane with constant magnetic field. 72 nicolas raymond cubo 12, 1 (2010) 2.3.1 study inside ω let j such that bj does not intersect the boundary. we recall the inequality (problem of dirichlet with constant magnetic field), for all ψ ∈ c∞0 (ω): ∫ ω |(i∇ + qalin)ψ|2dx ≥ q ∫ ω |ψ|2dx. (2.8) using lemma 2.3 we make the change of gauge v 7→ e−iφv and with the classical inequality: |a + b|2 ≥ (1 − λ2)|a|2 − 1 λ2 |b|2, for λ > 0, we get: ∫ ω |(i∇ + q(a − ∇φ))(χjue−iφ)|2dx ≥ ( (1 − λ2) ∫ ω |(i∇ + qalin)(χjue−iφ)|2dx −c2q2 |∇b| 2 ∞ λ2 r4 ) ∫ ω |χju|2dx, where alin is defined in lemma 2.3. thus, we find with (2.8) applied to ψ = χjue −iφ: ∫ ω |(i∇ + qa)χju|2dx ≥ ( (1 − λ2)q − c2q2 |∇b| 2 ∞ λ2 r4 ) ∫ ω |χju|2dx. 2.3.2 study near the boundary we refer to [hm02, hm04], but we will control carefully the uniformity. we first recall some properties of the harmonic oscillator on an half axis (see [dh93, hm01]). harmonic oscillator on r+ for ξ ∈ r, we consider the neumann realization hn,ξ in l2(r+) associated with the operator − d 2 dt2 + (t + ξ)2, d(hn,ξ) = {u ∈ b2(r+) : u′(0) = 0}. (2.9) one knows that it has compact resolvent and its lowest eigenvalue is denoted µ(ξ); the associated l2-normalized and positive eigenstate is denoted uξ and is in the schwartz class. the function ξ 7→ µ(ξ) admits a unique minimum in ξ = ξ0 and we let: θ0 = µ(ξ0). we now introduce local coordinates near the boundary in order to compare with the harmonic oscillator on r+: local coordinates near the boundary let’s assume that 0 ∈ ∂ω. in a neighborhood v of 0, we take local coordinates (y1,y2) on ∂ω (via a c3 map φ). we denote n(φ(y1,y2)) the interior unit normal to the boundary at the point φ(y1,y2) and define local coordinates in v : φ(y1,y2,y3) = φ(y1,y2) + y3n(φ(y1,y2)). cubo 12, 1 (2010) uniform spectral estimates for families ... 73 more precisely, for a point x ∈ v , φ(y1,y2) is the projection of x on ∂ω ∩ v and y3 = d(x,∂ω). taking a convenient map φ, we can assume φ(0) = 0 and d0φ = id. let j such that bj ∩ ∂ω 6= ∅. we can assume that xj ∈ ∂ω and xj = 0 without loss of generality. after a change of variables, we have: ∫ ω |(i∇ + qa)χju|2dx = ∫ y3>0 |(i∇y + qã)χ̃ju|2(dφ)−1((dφ)−1)t| det(dφ)|dy, where the tilde denotes the functions in the new coordinates and ã = dyφ(a(φ(y))). there exists c > 0 (uniform in j) such that: ∫ y3>0 |(i∇y + qã)χ̃ju|2(dφ)−1((dφ)−1)t | det(dφ)|dy ≥ (1 − cr) ∫ y3>0 |(i∇y + qã)χ̃ju|2dy. we use again the approximation by the constant magnetic field (for semi-balls) on the support of χ̃j . more precisely, there exists α > 0 uniform in j, such that supp(χ̃j ) ⊂ b(xj,αr). then, we change our partition of unity: we replace the balls which intersect the boundary by φ(b(xj,αr)). there exists c > 0 such that for all j, there exists ã lin (defined in lemma 2.3) satisfying: ∫ y3>0 |(i∇y + qã)χ̃ju|2dy ≥ (1 − λ2) ∫ y3>0 |(i∇y + qã lin )χ̃ju|2dy −c 2q2 λ2 |∇b̃|2∞r4 ∫ ω |χ̃ju|2dy, where b̃ = ∇y × ã. in order to express b̃ as a function of b, we need the following lemma: lemma 2.4. with the previous notations, we have: b̃ = det(dφ)((dφ)−1)tb. proof. the result is standard. let us recall the proof for completness. let us introduce the 1-form ω: ω = a1dx1 + a2dx2 + a3dx3. in the new coordinates x = φ(y), we have, with the previous notations: ω = ã1dy1 + ã2dy2 + ã3dy3. 74 nicolas raymond cubo 12, 1 (2010) then, it remains to write: dω = (∇ × a)1dx2 ∧ dx3 + (∇ × a)2dx1 ∧ dx3 + (∇ × a)3dx1 ∧ dx2, and to express dxi as a function of (dyj ). the comatrix formula gives the conclusion. � we are reduced to the case of constant magnetic field (of intensity q) on r3+ = {y3 > 0} (see [hm02, hm04, lp00]) and we get: ∫ y3>0 |(i∇y + qã)χ̃ju|2dy ≥ θ0q ∫ y3>0 |χ̃ju|2dy. thus, we find: ∫ y3>0 |(i∇y + qã)χ̃ju|2dy ≥ ( (1 − λ2)qθ0 − c2q2 λ2 (1 + |∇b|2∞)r4 ) ∫ y3>0 |χ̃ju|2dy. 2.3.3 end of the proof we take, for 0 < ǫ < 1 2 , r = 1 q1/2−ǫ . we divide by q and we choose λ such that: λ2 = q λ2 (1 + |∇b|2∞)r4. then, the previous estimates lead to the existence of c > 0 and q0 > 0 depending only on ω such that for all q ≥ q0: qa(u) q ≥ ( qθ0 − c ( r + λ2 + 1 qr2 )) ∫ ω |u|2dx. we finally find, by the minimax principle: µ(q, a) q ≥ θ0 − c ( 1 q1/2−2ǫ + 1 q2ǫ + |∇b|∞ q1/2−2ǫ ) . 3 upper bound in this section, we give a proof of theorem 1.2. we refer to [hm04] and, in the next lines, we emphasize the crucial points where uniformity is concerned. in the case of the constant magnetic field on r3+, we know (cf. [hm02, lp00]) that the bottom of the spectrum is minimal when the magnetic field is tangent to the boundary. so, we will look for a quasimode localized near a point where the magnetic field is tangent. then, we will take as trial function some truncation of uξ0 . so, we fix x0 ∈ ∂ω such that: b(x0) · ν = 0. such a x0 exists; indeed, noticing that div(b) = 0, the stokes formula gives: ∫ ∂ω b · νdσ = ∫ ω div(b)dx = 0. cubo 12, 1 (2010) uniform spectral estimates for families ... 75 we take δ ∈]0, 1/2[ and we suppose that u is such that supp(u) ⊂ b(x0,αr), with r = 1 qδ . we assume that the support of u is small enough and after a change of coordinates, we can use the same arguments as in lemma 2.3 and take a gauge in which a satisfies: |a(y) − a0(y)| ≤ c|b|c2|y|3, where a0 = alin + r, with r = (r1,r2,r3) and rj homogeneous polynomial of order 2 and where ∇2b denotes the hessian matrix of b. we find: qa(u) ≤ (1 + cr)(qa0 (u) + c(|b|2c2r6q2‖u‖2 + qr3|b|c2‖u‖qa0 (u)1/2)). we let: u(y) = q1/4+δe−iξ0y2q 1/2 uξ0 (q 1/2y3)χ(4q δy3)χ(4q δ(y21 + y 2 2) 1/2). we have to compare: qa0 (u) and qalin (u). we get: qa0 (u) ≤ qalin (u) + cq2r4(1 + |∇b|2∞)‖u‖2 +2ℜ {∫ |y|≤αr,y3>0 (i∇ + qalin)u · (qa0 − qalin)udy } . we have to estimate the double product (cf. [hm04, section 6, p. 120]): ∣∣∣∣∣ℜ {∫ |y|≤αr,y3>0 (i∇ + qalin)u · (qa0 − qalin)udy }∣∣∣∣∣ ≤ c(1 + |∇b|∞)q 1−δ. moreover, using the exponential decrease of uξ0 , we have: q a lin (u) ≤ θ0q + cq2δ. 4 agmon’s estimates in order to estimate the asymptotic localization of the first eigenfunctions in the applications, we will need agmon’s estimates to have some exponential decrease inside ω; that is the aim of this section. let us introduce some notations (see [agm82, alm08, fh08]). for γ > 0 small enough, let ηγ be a smooth cutoff function such that: ηγ = { 1 if d(y) = d(y,∂ω) ≥ γ 0 if y /∈ ω , with |∇ηγ| ≤ c γ . 76 nicolas raymond cubo 12, 1 (2010) we let: ωγ = {y ∈ ω : d(y,∂ω) ≥ γ}. for α > 0, we let ξ(y) = ηγe αd(y). we finally denote µ0(q, a) the lowest eigenvalue of the dirichlet’s realization of (i∇ + qa)2 on ω. we have the following localization property: proposition 4.1. there exists c > 0 and γ0 > 0, depending only on ω such that for all 0 < ǫ ≤ 1 and α verifying: 0 < α < ( 1 1 + ǫ )1/2 (µ0 − µ)1/2, and for all 0 < γ ≤ γ0, if u is a normalized mode associated with the lowest eigenvalue µ = µ(q, a) of the neumann realization of (i∇ + qa)2, then: ‖ηγeαd(y)|u|‖h1(ω) ≤ c √ ǫγ ( µ0 + 1 µ0 − µ − (1 + ǫ)α2 )1/2 eαγ. proof. we consider the equation verified by u: (i∇ + qa)2u = µu. one multiplies by ξ2u and integrate by parts (using (i∇ + qa)u · ν = 0 on ∂ω) to get: |(i∇ + qa)(ξu)|22 = µ|ξu|22 + |(∇ξ)u|22. we have, for all ǫ > 0: |(∇ξ)u|2 ≤ (1 + 1 ǫ ) ∫ ω\ωγ |∇η|2e2αd(y)|u|2 + (1 + ǫ)α2 ∫ ω |ξu|2. we use that |u|2 = 1 to find: |ξu|2 ≤ c γ (1 + 1 ǫ ) e2αγ µ0 − µ − (1 + ǫ)α2 . moreover, the diamagnetic inequality gives: |∇|ξu||2 ≤ |(i∇ + qa)(ξu)|2. it follows that: ||ξ|u|||2h1(ω) ≤ c γ (1 + 1 ǫ )e2αγ µ0 + 1 µ0 − µ − (1 + ǫ)α2 . � 5 applications we now describe two applications of our main results. cubo 12, 1 (2010) uniform spectral estimates for families ... 77 5.1 application to an helical vector field in this section, we study µ∗(q,τ) defined in (1.4) as qτ → +∞ and τ → +∞. due to the definition (1.4), we will use the uniform analysis of µ(q, n) with n ∈ c(τ). 5.1.1 estimate of the first eigenvalue the main theorem in this section is the following: theorem 5.1. let c0 > 0 and 0 ≤ x < 12 . there exists c > 0 and q0 > 0 depending only on ω, c0 and x such that, if (q,τ) verifies qτ ≥ q0 and τ ≤ c0(qτ)x, (5.10) then: θ0 − c (qτ)1/4−x/2 ≤ µ ∗(q,τ) qτ ≤ θ0 + c (qτ)1/3−2x/3 . (5.11) remark 5.2. this statement was obtained for x = 0 in [hp07] and rough estimates where given in [bclp02] as τ q → 0. � proof. let us notice that: µ(qτ, qnτ q t τ ) = µ(q,qnτ q t). moreover, there exists c > 0 such that for all τ > 0 and n ∈ c(τ), if a = n τ and b = ∇× a, then: |b|∞ = 1, |∇b|∞ ≤ cτ, |∇2b|∞ ≤ cτ2. for the lower bound, we apply theorem 1.1 to the subfamily c(τ) of a and, using (5.10), we get: µ∗(q,τ) qτ ≥ θ0 − c ( 1 (qτ)1/2−2ǫ + 1 (qτ)2ǫ + c0 (qτ)x (qτ)1/2−2ǫ ) . we choose ǫ such that 1 2 − 2ǫ − x = 2ǫ, i.e: 2ǫ = 1 4 − x 2 . for the upper bound, we apply theorem 1.2 with a = nτ τ . then, we get: µ∗(q,τ) ≤ θ0qτ + c((qτ)2δ + (qτ)2−4δ+2x + (qτ)1−δ+x + (qτ)3/2−3δ+2x + (qτ)2−6δ+4x). we choose δ such that: 2δ = 1 − δ + x. thus, we take δ = 1+x 3 and the upper bound follows. � 78 nicolas raymond cubo 12, 1 (2010) 5.1.2 localization of the ground state near the boundary as τ → + ∞ we first state a proposition: proposition 5.3. for a ∈ a (cf. (1.1)), we denote µ0(q, a) the bottom of the spectrum of the dirichlet realization of (i∇ + qa)2. then, for all ǫ ∈]0, 1/2[, there exists c = c(ω,ǫ) > 0 and q0 = q0(ω,ǫ) s.t. if q ≥ q0, for all a ∈ a: µ0(q, a) q ≥ 1 − c ( 1 q2ǫ + |∇b|∞ q1/2−2ǫ ) . proof. we again use partition (2.5), formula (2.7) and the proof is the same as for theorem 1.1. � we deduce: corollary 5.4. let c0 > 0. for all x ∈ [0, 1/2[, there exists c = c(ω,x,c0) > 0, such that, for all n ∈ c(τ) and (q,τ) such that τ ≤ c0(qτ)x: µ0(qτ, n τ ) qτ ≥ 1 − c ( 1 qτ )1/4−x/2 . as an immediate consequence of proposition 4.1, we have the following theorem: theorem 5.5. for all x ∈ [0, 1/2[, there exists δ0 > 0, c > 0, c > 0 such that if (q,τ) verifies qτ ≥ δ0 and τ ≤ c0(qτ)x, then for all n ∈ c(τ) and u a l2-normalized solution of (i∇ + qn)2u = µ(q, n)u, in ω (i∇ + qn)u · ν = 0, on ∂ω we have: ‖η c√ qτ e((1−θ0) 1/2√qτ −r(qτ ))d(·,∂ω)|u|‖h1(ω) ≤ c, where r(qτ) = (qτ)3/8+x/4. proof. we apply proposition 4.1 with ǫ = (qτ)−1/4+x/2, α = (1 − θ0)1/2( √ qτ − λ), γ = 1√ qτ , λ = (qτ)1/4+x1/2, with x < x1 < 1 2 and we notice, by taking a truncated gaussian, that µ0(qτ, n τ ) ≤ cqτ. we finally take x1 = 1 2 ( 1 2 + x) and apply the estimate (5.11) of theorem 5.1. � cubo 12, 1 (2010) uniform spectral estimates for families ... 79 5.2 surface superconductivity in large domains 5.2.1 the problem let r > 0 and x0 ∈ ω. we denote ωr = {x0 + r(x − x0), x ∈ ω}. the aim of this part is to study the behaviour of µωr (q, a), where a ∈ a (cf.(1.1)) as q → +∞ and r → +∞. in his work [alm08], almog studies the regime q fixed and r → +∞ (see theorem 1.1 in [alm08]) and then makes q to tend to infinity (see section 3 of [alm08]). in this section, we give a theorem which treats another regime : q → +∞ and r with polynomial increase in q. we first observe the following scaling invariance: lemma 5.6. let r > 0. we have: µωr (q, a) = 1 r2 µω ( qr2, a(r·) r ) . 5.2.2 estimate of the lowest eigenvalue theorem 5.7. let c0 > 0 and y ≥ 0. there exists c>0, q0 > 0 and r0 > 0 depending only on ω, c0 and y such that, if (q,r) satisfies q ≥ q0, r ≥ r0 and r ≤ c0qy, then, for a ∈ a: θ0 − c (qr2) 1 4(1+2y) ≤ µωr (q, a) q ≤ θ0 − c (qr2) 1 3(1+2y) . proof. we use lemma 5.6. we let x = y 1 + 2y and notice that r ≤ c(y)(qr2)x with c(y) = c 1 1+2y 0 and a(r·) r ∈ a. then by the theorems 1.1 and 1.2, we have the wished conclusion by using the same arguments as for theorem 5.1. � 5.2.3 localization of the groundstate near the boundary in large domains in the case of large domains, we prove a quite analogous theorem with theorem 5.5: theorem 5.8. for all y ≥ 0, there exists δ0 > 0, δ1 > 0, c > 0, c > 0 such that if (q,r) verifies q ≥ δ0, r ≥ δ1 and r ≤ c0qy, then for all a ∈ a and u a l2-normalized solution of (i∇ + qa)2u = µωr (q, a)u, in ωr (i∇ + qa)u · ν = 0, on ∂ωr we have, ‖η c√ q e(1−θ0) 1/2( √ q−r(q,r))d(·,∂ωr)|u|‖h1(ωr) ≤ c, where r(q,r) = q 1/2− 1 8(1+2y) r − 1 4(1+2y) . 80 nicolas raymond cubo 12, 1 (2010) proof. after a rescaling, the proof is the same as the one of theorem 5.5. � received: october, 2008. revised: october, 2009. references [agm82] agmon. s., lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of n-body schrödinger operators, princeton university press, 1982. 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[pan06] pan, x-b.., landau-de gennes model of liquid crystals with small ginzburg-landau parameter, siam j. math. anal., 37:1616–1648, 2006. det-cubo.dvi cubo a mathematical journal vol.12, no¯ 03, (1–12). october 2010 partial fractions and q-binomial determinant identities wenchang chu dipartimento di matematica, università del salento, lecce-arnesano p. o. box 193, lecce 73100, italy email: chu.wenchang@unile.it chenying wang college of mathematics and physics, nanjing university of information science and technology nanjing 210044, p. r. china email: wang.chenying@163.com and wenlong zhang department of applied mathematics, dalian university of technology, dalian 116023, p. r. china email: wenlong.dlut@yahoo.com.cn abstract partial fraction decomposition method is applied to evaluate a general determinant of shifted factorial fractions, which contains several gaussian binomial determinant identities. 2 wenchang chu, chenying wang & wenlong zhang cubo 12, 3 (2010) resumen el método de descomposición en fracción parciales aplicado para evaluar un determinante general de fracciones factoriales trasladadas, la cual contiene varias identidades determinante binomial gaussiano. key words and phrases: the cauchy double alternant, partial fractions, q-binomial coefficients. math. subj. class.: 15a15, 11c20. binomial determinant evaluation plays an important role in combinatorial enumeration, particularly in plane partitions. this paper will establish a very general determinant identity through partial fraction decomposition method. it will be shown to be useful in q-binomial determinant evaluations with several interesting known and new formulae being exemplified. 1 partial fraction decomposition for two sequences {αk ,γk}k≥0, define the generalized shifted factorials by (x|α)0 = 1 and (x|α)n= n−1 ∏ k=0 (1 − xαk) with n ∈ n, (1a) ( y|γ)0 = 1 and ( y|γ)n = n−1 ∏ k=0 (1 − yγk) with n ∈ n. (1b) when αk = γk = q k for k ∈ n0, they will reduce to the usual shifted factorials (x; q)0 = 1 and (x; q)n = (1 − x)(1 − qx)···(1 − q n−1 x) with n ∈ n. (2) for the triangular matrix given by α=[αi j ]0≤i≤ j<∞, denote its j-th column by αj =(α0 j ,α1 j ,α2 j , ··· ,αj j ). then the main result may be stated as follows. theorem 1 (generalized cauchy determinant). let {xk} n k=0 be distinct complex numbers. then there holds the following determinant identity: det 0≤i, j≤n [ (xi|αj ) j (xi|γ) j+1 ] = ∏ 0≤i< j≤n(xi − x j )(αi j −γ j) ∏ n k=0 (xk|γ)n+1 . the very special case of this theorem with αi j = γi for i, j ∈ n0 results in the celebrated cauchy’s double alternant (cf. [6, 7]): det 0≤i, j≤n [ 1 1 − xiγ j ] = ∏ 0≤i< j≤n(xi − x j )(γi −γ j) ∏ 0≤i, j≤n(1 − xiγ j ) . (3) cubo 12, 3 (2010) partial fractions and q-binomial determinants 3 proof. expanding the rational function in partial fractions, we have (xi|αj ) j (xi|γ) j+1 = ∏ j−1 ι=0 (1 − xiαι j ) ∏ j k=0 (1 − xiγk) = j ∑ k=0 wk j 1 − xiγk where the connected coefficients are determined by the following limit relation wk j = lim xi → 1 γk (1 − xiγk) (xi|αj ) j (xi|γ) j+1 = ∏ j−1 ι=0 (αι j −γk) ∏ j ι=0: ι 6=k (γι −γk) . this leads us to the following determinant factorization det 0≤i, j≤n [ (xi|αj ) j (xi|γ) j+1 ] = det 0≤i,k≤n [ 1 1 − xiγk ] × det 0≤k, j≤n [ wk j ] . for the matrix [ wk j ] 0≤k, j≤n is upper triangular, its determinant is equal to the product of its diagonal entries: det 0≤k, j≤n [ wk j ] = n ∏ j=0 w j j = ∏ 0≤i< j≤n αi j −γ j γi −γ j . while the first determinant can be evaluated by cauchy’s double alternant (2). their combination yields the determinant identity stated in theorem 1. shifting the γ-parameters by γk → γk−1, we may state the determinant identity in theorem 1 in the following more convenient form. proposition 2 (determinant identity). let {xk} n k=0 be distinct complex numbers. then there holds the following determinant identity: det 0≤i, j≤n [ (xi|αj ) j (xi|γ) j ] = ∏ 0≤i< j≤n(xi − x j )(αi j −γ j−1) ∏n k=0 (xk|γ)n . letting αi j = p i yj and γk = q k further in proposition 2, we have the identity. corollary 3 (bibasic determinant evaluation formula). det 0≤i, j≤n [ (xi yj ; p) j (xi; q) j ] = q 2(n+13 ) ∏ 0≤i< j≤n (x j − xi ) n ∏ k=0 (q1−k yk; p)k (xk; q)n . from this corollary, we can derive numerous q-binomial determinant identities. 2 q-binomial determinant identities define the gaussian binomial coefficients by [ x n ] = (q1+x−n; q)n (q; q)n where n ∈ n0 and x ∈ c. applying corollary 3, we show now ten classes of q-binomial determinant identities. 4 wenchang chu, chenying wang & wenlong zhang cubo 12, 3 (2010) 2.1 expressing the q-binomial coefficient in terms of shifted factorials [ x i − j a ] = q −a j [ x i a ] (qa−x i ; q) j (q−x i ; q) j we derive the corresponding determinant formula det 0≤i, j≤n [ [ x i − j a ] ] = ∏ 0≤i< j≤n (q−x j − q−x i )(1 − q1+a+i− j ) (4a) × q2( n+1 3 )−a( n+1 2 ) (q; q)n+1n n ∏ k=0 [ x k a ][ n − 1 − x k n ]−1 . (4b) 2.2 rewriting the q-binomial coefficient in terms of shifted factorials [ a x i − j ] = (−1) j q−( j 2)+ j x i [ a x i ] (q−x i ; q) j (q1+a−x i ; q) j we get the corresponding determinant identity det 0≤i, j≤n [ q − j x i [ a x i − j ] ] = ∏ 0≤i< j≤n (q−x i − q−x j )(1 − q−a+i− j ) (5a) × q( n+1 3 )+(1+a)( n+1 2 ) (q; q)n+1n n ∏ k=0 [ a x k ][ a + n − x k n ]−1 . (5b) 2.3 reformulating the q-binomial coefficient in terms of shifted factorials [ a + x i − j x i − j ] = q −a j [ a + x i a ] (q−x i ; q) j (q−a−x i ; q) j we obtain the following determinant evaluation formula det 0≤i, j≤n [ [ a + x i − j x i − j ] ] = ∏ 0≤i< j≤n (q−x j − q−x i )(1 − q1+a+i− j ) (6a) × q2( n+1 3 )−2a( n+1 2 ) (q; q)n+1n n ∏ k=0 [ a + x k a ][ −1 − a + n − x k n ]−1 . (6b) 2.4 applying the q-binomial relation [ x i + j a ] = [ x i a ] (q1+x i ; q) j (q1−a+x i ; q) j cubo 12, 3 (2010) partial fractions and q-binomial determinants 5 we find the corresponding determinant formula det 0≤i, j≤n [ [ x i + j a ] ] = ∏ 0≤i< j≤n (qx j − qx i )(1 − q1+a+i− j ) (7a) × q2( n+1 3 )+(1−a)( n+1 2 ) (q; q)n+1n n ∏ k=0 [ x k a ][ x k − a + n n ]−1 . (7b) 2.5 observing the q-binomial relation [ a x i + j ] = (−1) j q(a−x i ) j−( j 2) [ a x i ] (q−a+x i ; q) j (q1+x i ; q) j we recover the determinant identity due to carlitz [4] (cf. chu [5] also) det 0≤i, j≤n [ q j x i [ a x i + j ] ] = ∏ 0≤i< j≤n (qx i − qx j )(1 − q−a+i− j ) (8a) × q( n+1 3 )+(1+a)( n+1 2 ) (q; q)n+1n n ∏ k=0 [ a x k ][ x k + n n ]−1 . (8b) 2.6 by invoking the q-binomial relation [ a + x i + j x i + j ] = [ a + x i a ] (q1+a+x i ; q) j (q1+x i ; q) j we recover another determinant identity due to carlitz [4] (see menon [9] also) det 0≤i, j≤n [ [ a + x i + j x i + j ] ] = ∏ 0≤i< j≤n (qx j − qx i )(1 − q1+a+i− j ) (9a) × q2( n+1 3 )+( n+1 2 ) (q; q)n+1n n ∏ k=0 [ a + x k a ][ x k + n n ]−1 (9b) which reduces, for q → 1, to the binomial determinant of ostrowski [10]. furthermore for δ = 0, 1, we can show the following determinant identity det 0≤i, j≤n [ c (δ) x i + j (q) ] = (2q)(1+n)(1+n+δ)+2 ∑n ι=0 xι q n(n+1)(1+2n+6δ)/6 (10a) × n ∏ k=0 (q; q2)1+k(q; q 2 )δ+x k (q2; q2)1+δ+x k+n ∏ 0≤i< j≤n (q2x i − q2x j ) (10b) where the q-catalan numbers due to andrews [2] has been slightly extended by c (δ) n (q) := (2q)1+δ+2n 1 − q2+2δ+2n [ δ+ 2n n ] 1 − q (−q; q)n (−q; q)δ+n . (11) 6 wenchang chu, chenying wang & wenlong zhang cubo 12, 3 (2010) when xk = k +ℓ, we get the following hankel determinant identity det 0≤i, j≤n [ c (δ) i+ j+ℓ (q) ] = (2q)(1+n)(1+δ+2n+2ℓ)qn(n+1)(4n+6ℓ+6δ−1)/6 (12a) × n ∏ k=0 (q; q)1+2k (q; q 2)δ+k+ℓ (q2; q2)1+δ+k+n+ℓ . (12b) letting δ = 0 and q → 1, we recover further the related results [1, 8, 11] on the classical catalan numbers cn = 1 n+1 (2n n ) : det 0≤i, j≤n [ ci+ j ] = 1, det 0≤i, j≤n [ ci+ j+1 ] = 1, det 0≤i, j≤n [ ci+ j+2 ] = n + 2. (13) 2.7 by means of the q-binomial relation [ x i + y j j ][ a + x i j ]−1 = q (y j −a) j (q−x i−y j ; q) j (q−a−x i ; q) j we get the following determinant identity det 0≤i, j≤n [ [ x i + y j j ][ a + x i j ]−1] = q 2(n+13 )− ∑n k=0 (2k a+n x k−kyk ) (q; q)n+1n ∏ n k=0 [ n−1−a−x k n ] (14a) × ∏ 0≤i< j≤n (qx i − qx j )(1 − q1+a−y j+i− j ). (14b) 2.8 in view of the q-binomial relation [ x i + y j + j y j ][ x i + y j y j ]−1 = (q1+x i+y j ; q) j (q1+x i ; q) j we obtain the corresponding determinant formula det 0≤i, j≤n [ [ x i + y j + j y j ][ x i + y j y j ]−1] = q2( n+1 3 )+( n+1 2 ) (q; q)n+1n n ∏ k=0 [ x k + n n ]−1 (15a) × ∏ 0≤i< j≤n (qx j − qx i )(1 − q1+y j +i− j ). (15b) 2.9 according to the q-binomial relation [ a + x i + y j j ][ x i + j j ]−1 = (q1+a+x i +y j− j ; q) j (q1+x i ; q) j cubo 12, 3 (2010) partial fractions and q-binomial determinants 7 we derive the corresponding determinant identity det 0≤i, j≤n [ [ a + x i + y j j ][ x i + j j ]−1] = q2( n+1 3 )+( n+1 2 ) (q; q)n+1n n ∏ k=0 [ x k + n n ]−1 (16a) × ∏ 0≤i< j≤n (qx j − qx i )(1 − q1+a+y j +i−2 j ). (16b) 2.10 similarly, the q-binomial relation [ x i + y j j ][ a + x i − j n − j ] = q (y j −a) j [ n j ][ a + x i n ] (q−x i −y j ; q) j (q−a−x i ; q) j leads us to the following binomial determinant evaluation formulae det 0≤i, j≤n [ [x i+y j j ][ a+x i − j n− j ] ] = ∏ 0≤i< j≤n (q−x j − q−x i )(1 − q1+a+i− j−y j ) (17a) × q ∑n k=0 (k−1−2a+yk)k (q; q)n+1n n ∏ k=0 [ n k ][ a+x k n ] [ −1−a+n−x k n ] , (17b) det 0≤i, j≤n [ [ x i+ j j ][a+x i +y j n− j ] ] = ∏ 0≤i< j≤n (q−x i − q−x j )(1 − q1+n−a−yn− j +i− j ) (18a) × q ∑ n k=0 (k−1+a−2n+yn−k)k (q; q)n+1n n ∏ k=0 [ n k ][ n+x k n ] [ −1−x k n ] ; (18b) where the last identity is derived from the first one under substitution j → n− j on the column index. 3 duplicate determinant identities performing the parameter replacements in proposition 2 xk → axk + c/xk , γk → dγk/(1 + acd 2 γ 2 k ), αi j → bαi j /(1 + ab 2 cα 2 i j ); 8 wenchang chu, chenying wang & wenlong zhang cubo 12, 3 (2010) and then applying factorizations xi − x j → (xi − x j )(a − c/xi x j ), αi j −γk → (bαi j − dγk)(1 − abcdαi j γk) (1 + ab2 cα2 i j )(1 + acd2γ2 k ) , 1 − xiγk → (1 − adγk xi )(1 − cdγk /xi ) 1 + acd2γ2 k , 1 − xkαi j → (1 − abxkαi j )(1 − bcαi j /xk) 1 + ab2 cα2 i j ; we find the following duplicate determinant identity. proposition 4. let {xk} n k=0 be distinct complex numbers. then there holds the following determinant identity: det 0≤i, j≤n [ (abxi|αj ) j(bc/xi |αj ) j (adxi|γ) j(cd/xi |γ) j ] = ∏ 0≤i< j≤n (bαi j − dγ j−1)(1 − abcdαi j γ j−1) × ∏ 0≤i< j≤n(xi − x j )(a − c/xi x j) ∏ n k=0 (adxk|γ)n(cd/xk|γ)n . this identity contains the following three determinant evaluations. corollary 5 (a = b = 1 and γk → 0 in proposition 4). det 0≤i, j≤n [ (xi|αj ) j (c/xi|αj ) j ] = ∏ 0≤i< j≤n { αi j (xi − x j )(1 − c/xi x j ) } . corollary 6 (d = 1 and αi j → 0 in proposition 4). det 0≤i, j≤n [ 1 (axi|γ) j (c/xi|γ) j ] = ∏ 0≤i< j≤n(x j − xi )(a − c/xi x j) ∏n k=0 (axk|γ)n(c/xk|γ)n n ∏ ℓ=1 γ ℓ ℓ−1. putting αi j = p i yj and γk = q k in proposition 4, we find the following determinant evaluation formula of factorial fractions with two different bases. corollary 7 (bibasic determinant identity). det 0≤i, j≤n [ (abxi yj ; p) j (bc yj /xi ; p) j (adxi ; q) j (cd/xi ; q) j ] = d( n+1 2 ) ∏ 0≤i< j≤n (x j − xi )(a − c/xi x j ) × q 2(n+13 ) n ∏ k=0 (q1−k b yk /d; p)k (q k−1 abcd yk ; p)k (adxk ; q)n (cd/xk ; q)n . cubo 12, 3 (2010) partial fractions and q-binomial determinants 9 when p = q and yk = 1, it reduces to the following determinant identity det 0≤i, j≤n [ (abxi ; q) j (bc/xi ; q) j (adxi ; q) j (cd/xi ; q) j ] = b( n+1 2 ) ∏ 0≤i< j≤n (xi − x j )(a − c/xi x j ) (19a) × q( n+1 3 ) n ∏ k=0 (d/b; q)k (q k−1abcd; q)k (adxk ; q)n(cd/xk ; q)n . (19b) the determinant evaluation formulae established in this section contain numerous q-binomial determinant identities as special cases, which will be illustrated by the following five examples. 3.1 expressing the q-binomial coefficients in terms of shifted factorials [ x i +a j ][ x i −b−c n− j ] [ x i +b j ][ x i −a−c n− j ] = q (a−b) j [ x i −b−c n ] [ x i −a−c n ] × (q1+x i−a−c−n; q) j (q −x i−a ; q) j (q1+x i−b−c−n; q) j (q−x i−b; q) j we establish from corollary 7 the determinant evaluation formula det 0≤i, j≤n [ [ x i +a j ][ x i −b−c n− j ] [ x i +b j ][ x i −a−c n− j ] q( j 2) ] = ∏ 0≤i< j≤n (qx i − qx j )(1 − qn−1+c−x i−x j ) (20a) × qb( n+1 2 ) (q; q)n+1n n ∏ k=0 [ n+a+b+c−k k ][ b−a k ][ x k−b−c n ] [ n k ][ x k+b n ][ n−1+b+c−x k n ][ x k−a−c n ] (20b) which contains, as special case, the following q-binomial determinant identity det 0≤i, j≤n [ [ λi+a j ][ λi−b n− j ] [ λi+b j ][ λi−a n− j ] q( j 2) ] = ∏ 0≤i< j≤n (qλi − qλ j )(1 − qn−1−λi−λ j) (21a) × qb( n+1 2 ) (q; q)n+1n n ∏ k=0 [ n+a+b−k k ][ b−a k ][ λk−b n ] [ n k ][ λk+b n ][ n−1+b−λk n ][ λk−a n ] . (21b) 3.2 rewriting the q-binomial coefficients in terms of shifted factorials [x i+y j j ][a−x i+y j j ] [ b+x i j ][ a+b−x i j ] = q 2 j(y j−b) (q−x i −y j ; q) j (q x i −a−y j ; q) j (q−x i −b; q) j (qx i −a−b; q) j we recover from corollary 7 the determinant identity due to joris van jeugt det 0≤i, j≤n   [x i+y j j ][a−x i +y j j ] [ b+x i j ][ a+b−x i j ] q( j 2)   = ∏ 0≤i< j≤n (qx i − qx j )(1 − qa−x i −x j ) (22a) × q ∑n k=0 kyk (q; q)n+1n n ∏ k=0 [1+a+b+yk−k k ][ b−yk k ] [ n k ][ b+x k n ][ a+b−x k n ] . (22b) 10 wenchang chu, chenying wang & wenlong zhang cubo 12, 3 (2010) this identity can further be specialized to the q-binomial determinant evaluation det 0≤i, j≤n   [ a+λi+ j j ][ a−λi+ j j ] [ b+λi j ][ b−λi j ] q( j 2)   = ∏ 0≤i< j≤n (q−λi − q−λ j )(1 − qλi+λ j ) (23a) × q ∑n k=0 k(a+k) (q; q)n+1n n ∏ k=0 [1+a+b k ][ b−a−k k ] [ n k ][ b+λk n ][ b−λk n ] . (23b) 3.3 reformulating the q-binomial coefficients in terms of shifted factorials [ x i +y j+ j x i −y j−a− j ] [ x i+y j x i−y j −a ][ x i +c+ j x i −a−c− j ] = q (c−y j ) j [ a+2c+2 j 2c−2y j ] [ a+2c a+2y j ][ c+x i a+2c ] × (q1+x i+y j ; q) j (q a−x i +y j ; q) j (q1+x i+c ; q) j (qa−x i +c; q) j we derive from corollary 7 the following determinant formula det 0≤i, j≤n [ [ x i +y j+ j x i −y j−a− j ] [ x i+y j x i−y j −a ][ x i +c+ j x i −a−c− j ] ] = ∏ 0≤i< j≤n (qx i − qx j )(1 − qa−1−x i −x j ) (24a) × q 2(n+23 )+( n+1 2 )(c−a)+n ∑n k=0 x k (q; q)n+1n [ a+2c+2n 2n ]n+1[2n n ]n+1 n ∏ k=0 [ a+2c+2k 2c−2yk ][ yk−c k ][ −k−a−c−yk k ] [ n k ][ a+2c a+2yk ][ x k+c+n a+2c+2n ] (24b) which reduces, for x i = bi and y j = 0, to the q-binomial determinant identity: det 0≤i, j≤n [ [ bi + j 2 j ][ bi + c + j 2c + 2 j ]−1] = ∏ 0≤i< j≤n (q−bi − q−b j )(1 − q1+bi+b j ) (25a) × q ∑n k=0 k(nb+c+k) (q; q)n+1n [2c+2n 2n ]n+1[2n n ]n+1 n ∏ k=0 [2c+2k 2c ][ −c k ][ −k−c k ] [ n k ][ bk+c+n 2c+2n ] . (25b) 3.4 according to corollary 6, the q-binomial relation q j x i [ x i + a − j x i + j ] = (−1) j q( j 2)−a j (q; q)x i +a (q; q)x i (q; q)a−2 j × 1 (q1+x i ; q) j (q−a−x i ; q) j yields the determinant evaluation formula det 0≤i, j≤n [ q j x i [ x i + a − j x i + j ] ] = ∏ 0≤i< j≤n (q−x i − q−x j )(1 − q1+a+x i+x j ) (26a) × n ∏ k=0 [ x k + a − n x k + n ] qn x k (q1+a−2n; q)2k . (26b) cubo 12, 3 (2010) partial fractions and q-binomial determinants 11 in particular for x i = c + bi, it becomes the q-binomial determinant identity det 0≤i, j≤n [ q bi j [ a + bi − j c + bi + j ] ] ∏ 0≤i< j≤n (q−bi − q−b j )(1 − q1+a+c+bi+b j) (27a) × q nb(n+12 ) n ∏ k=0 (q; q)a+bk−n (q; q)a−c−2k (q; q)c+bk+n (27b) which is the q-analogue of the determinant evaluated by amdeberhan and zeilberger [3, eq 2]. 3.5 in view of corollary 5, the q-binomial relation q − j x i [ x i + y j + j x i − y j + a − j ][ x i − y j + a x i + y j ] = (−1) j q(a−y j ) j−( j 2) (q; q)2y j −a+2 j(q; q)a−2y j × (q1+x i +y j ; q) j (q y j −x i−a ; q) j leads to the following determinant evaluation det 0≤i, j≤n [ q − j x i [ x i + y j + j x i − y j + a − j ][ x i − y j + a x i + y j ] ] (28a) = ∏ 0≤i< j≤n(q −x j − q−x i )(1 − q1+a+x i +x j ) ∏n k=0 (q; q)a−2yk (q; q)2yk−a+2k . (28b) in particular for x i = a + bi and y j = 0, the last identity gives det 0≤i, j≤n [ q −bi j [ a + bi + j c + bi − j ] ] = ∏ 0≤i< j≤n(q −b j − q−bi )(1 − q1+a+c+bi+b j) ∏n k=0 (q1+a+bk; q)c−a (q; q)a−c+2k (29) which results in the q-analogue of the binomial determinant identity due to amdeberhan and zeilberger [3, eq 1]. similarly, letting xi = q a+bi and αi j = q d j−i, we find from corollary 5 another determinant identity det 0≤i, j≤n [ [ a + bi + d j j ][ c − bi + d j j ] q( j 2) ] (30a) = n ∏ k=0 qk(c+d k) (q; q)2 k ∏ 0≤i< j≤n (q−bi − q−b j )(1 − qa−c+bi+b j ) (30b) which is the q-analogue of the result in amdeberhan and zeilberger [3, eq 14]. the list of examples can be endless. however, we are not going further to prolong it due to the space limitation. 12 wenchang chu, chenying wang & wenlong zhang cubo 12, 3 (2010) references [1] aigner, m., catalan-like numbers and determinants, j. combin. theory (ser. a) 87 (1999), 33–51. [2] andrews, g.e., catalan numbers, q-catalan numbers and hypergeometric series, j. combin. theory (ser. a) 44 (1987), 267–273. [3] amdeberhan, t. and zeilberger, d., determinants through the looking glass, adv. in appl. math. 27 (2001), 225–230. [4] carlitz, l., some determinants of q-binomial coefficients, j. reine angew. math. 226 (1967), 216–220. [5] chu, w., on the evaluation of some determinants with q-binomial coefficients, j. systems science & math. science 8:4 (1988), 361–366. [6] chu, w., generalizations of the cauchy determinant, publicationes mathematicae debrecen 58:3 (2001), 353–365. [7] chu, w., the cauchy double alternant and divided differences, electronic journal of linear algebra 15 (2006), 14–21. [8] cigler, j., operatormethoden für q-identitäten. vii: q-catalan-determinanten, österreich. akad. wiss. math.-natur. kl. sitzungsber. ii 208 (1999), 123–142. [9] menon, k.v., note on some determinants of q-binomial numbers, discrete math. 61:2-3 (1986), 337–341. [10] ostrowski, a.m., on some determinants with combinatorial numbers, j. reine angew. math. 216 (1964), 25–30. [11] radoux, c., nombres de catalan généralisés, bull. belg. math. soc. simon stevin 4:2 (1997), 289–292. arg516.dvi cubo a mathematical journal vol.12, no¯ 01, (149–159). march 2010 an improved convergence and complexity analysis for the interpolatory newton method ioannis k. argyros cameron university, department of mathematical sciences, lawton, ok 73505, usa email : iargyros@cameron.edu abstract we provide an improved compared to [5]–[7] local convergence analysis and complexity for the interpolatory newton method for solving equations in a banach space setting. the results are obtained using more precise error bounds than before [5]–[7] and the same hypotheses/computational cost. resumen nosotros entregamos aqúı un análisis de convergencia local y complejidad para el método de interpolación de newton para resolver ecuaciones en espacios de banach. los resultados mejoran los de [5]–[7] e son obtenidos usando mas precisas cotas de error y las mismas hipotesis y costo computacional. key words and phrases: newton’s method, local convergence, banach space, interpolatory newton method, complexity, radius of convergence. math. subj. class.: 65g99, 65h10, 65b05, 47h17, 49m15. 150 ioannis k. argyros cubo 12, 1 (2010) 1 introduction in this study we are concerned with the problem of approximating a simple solution α of the equation f (x) = 0, (1.1) where f is an operator defined on a convex subset d of a banach space x with values in a banach space y over the real or complex fields of dimension n , dim(x) = dim(y ) = n, 1 ≤ n ≤ +∞. we consider interpolatory iteration in for approximating x ∗ defined as follows: let xi be an approximation to α and let wi be the interpolatory polynomial of degree ≤ n − 1 such that w (j) i (xi) = f (j)(xi), j = 0, 1, . . . , n − 1 (n ≥ 2). (1.2) the next approximation x∗i+1 is a zero of wi. for n = 2 we obtain newton’s method: x∗i+1 = xi − f ′(xi) −1f (xi) (i ≥ 0). (1.3) we approximate xi+1 by applying a number of newton iterations to wi(x) = 0. let {xi} be the interpolatory newton iteration inn given by: z0 = xi zj+1 = zj − w ′ i(zj ) −1wi(zj ), j = 0, 1, . . . , k − 1 (1.4) xi+1 = zk, k = [log 2n]. a local convergence analysis and the corresponding complexity of method (1.4) was studied in the elegant paper by traub and wozniakowski [7]. relevant works can be found in [1]–[7], and the references there. here we are motivated by paper [7] and optimization considerations. in particular using more precise estimates on the distances ‖xi − α‖ (i ≥ 0) we show that under the same hypotheses and computational cost as in [5]–[7], we can provide a larger convergence radius, sharper error bounds on the distances and consequently a finer complexity for method (1.4). numerical examples are introduced which compare favorably with results to the corresponding ones in [5]–[7]. 2 local convergence analysis of method (1.4) let γ ≥ 0. we introduce the closed ball u = u (α, γ) = {x ∈ x | ‖x−α‖ ≤ γ}, and the parameters aj = aj (γ) = sup x∈u ∥ ∥ ∥ ∥ f ′(α)−1 f (j)(x) j! ∥ ∥ ∥ ∥ , (j ≥ 2) (2.1) cubo 12, 1 (2010) an improved convergence and complexity analysis ... 151 provided that f (j) exists. moreover we introduce the parameter a by a = a(γ) = sup x∈u ‖f ′(α)−1[f ′(x) − f ′(α)]‖ 2‖x − α‖ . (2.2) the foundation of our approach and what makes it more precise than the corresponding one in [7] is the fact that we use (2.2) instead of (2.1) (for j = 2) to obtain upper bounds on the crucial quantity ‖w′j (x) −1f ′(α)‖. indeed, on the one hand note that a ≤ a2 (2.3) holds in general and a2 a can be arbitrarily large [1], [2]. on the other hand see (2.28), (2.46), and remark 2.4. let us set a = a a2 , a2 6= 0. (2.4) note that a ∈ [0, 1]. we showed in [3] the following improvement of theorem 2.1 in [6] and theorem 2.1 in [5] respectively: theorem 2.1. if f is twice differentiable in u , (2.2) holds and a2γ ≤ 1 2(1 + a) , (2.5) xi ∈ u, (2.6) then the next approximation x∗i+1 generated by newton method (1.3) is well defined, and satisfies for all i ≥ 0: ‖x∗i+1 − α‖ ≤ a2 1 − 2aa2 ‖xi − α‖ 2 ≤ 1 2 ‖xi − α‖ (2.7) and x∗i+1 − α = 1 2 f ′(α)−1f ′(α)(xi − α) 2 + o(‖xi − α‖ 2). (2.8) theorem 2.2. if f is n-times differentiable, n ≥ 3 in u , (2.2) holds, and nanγ n−1 1 − aa2γ < ( 2 3 )n−1 (2.9) xi ∈ u, then the polynomial wi has a unique zero in u ∗ = u ∗ ( α, γ 2 ) and defining x∗i+1 as the zero of wi in u ∗ the following estimates hold for all i ≥ 0 ‖x∗i+1 − α‖ ≤ an(1 + ‖x ∗ i+1 − α‖/‖xi − α‖) n 1 − aa2‖x ∗ i+1 − α‖ ‖xi − α‖ n ≤ 1 2 ‖xi − α‖, (2.10) 152 ioannis k. argyros cubo 12, 1 (2010) and x∗i+1 − α = (−1)n n! f ′(α)−1f (n)(α)(xi − α) n + o(‖xi − α‖ n). (2.11) we can show the main local convergence theorem for method (1.4): theorem 2.3. if f is n-times differentiable, n ≥ 3 in u , (2.2) holds, and 0 ≤ ã2γ ≤ 1 3 + 2a (2.12) where, ã2 = a2 + n(n−1) 2 an(2γ) n−2 1 − aa2γ − nan ( 3 2 )n−1 γn−1 (2.13) x0 ∈ u, (2.14) then sequence {xi} (i ≥ 0) generated by interpolary-newton iteration inn is well defined, remains in u for all i ≥ 0, converges to α so that the following estimates hold for all i ≥ 0: ei+1 = ‖xi+1 − α‖ ≤ { 1 2 + 3 2 ( 1 2 )k } ei, (2.15) ei+1 ≤ ci,ne n i (2.16) where, ci,n = ( 1 + e∗i+1 ei ) [ an 1 − aa2e ∗ i+1 + (ã2(1 + hi)) 2k−1 ] (( 1 + e∗i+1 ei ) ei )2k−n , (2.17) for e∗i+1 = ‖x ∗ i+1 − α‖, hi = o(ei), 0 ≤ hi ≤ 3 + 2a 2 , k = [log 2n], (2.18) lim i→∞ ci,n = an + δã n−1 2 where δ = 0 if 2k > n and δ = 1, if 2k = n, (2.19) xi+1 − α = fn(xi − α) n + bi,k + o(‖xi − α‖ n), (2.20) where bi,1 = f2(xi − α) 2, (2.21) bi,j+1 = f2b 2 i,j , j = 1, 2, . . . , k − 1, (2.22) and fj = (−1)j j! f ′(α)−1f (j)(α) for j = 2 and n. (2.23) cubo 12, 1 (2010) an improved convergence and complexity analysis ... 153 the proof is similar to theorem 3.1 in [7], but there are differences where we use (2.2) instead of (2.1) (for i = 2). proof. we shall first show using induction on j ≥ 0 that w′j (zj) is invertible and zj ∈ u . set f (j)(x) − w (j) i (x) = r (j) n (x; xi), x ∈ u, j = 0, 1, 2, (2.24) where, ‖f ′(α)−1r(j)n (x; xi)‖ ≤ j! ( n j ) an‖x − xi‖ n−1. (2.25) we can write w′i(x) = f ′(x) − r′n(x; xi) = f ′(α)[i + f ′(α)−1{f ′(x) − f ′(α)} − f ′(α)−1r′n(x; xi)] (2.26) and in view of (2.2), (2.12) and (2.24) for x ∈ u we get in turn ‖f ′(α)−1[w′j (x) − f ′(α)]‖ ≤ 2aa2‖x − α‖ + nan‖x − xi‖ n−1 (2.27) ≤ 2aa2γ + nan(2γ) n−1 ≤ 2 3 + 2a < 1. (2.28) it follows from (2.28) and the banach lemma on invertible operators [4] that w′i(x) is invertible for all x ∈ u , and ‖w′i(x) −1f ′(α)‖ ≤ 1 1 − 2aa2‖x − α‖ − nan‖x − xi‖n−1 . (2.29) since the denominator in (2.13) is positive we get nanγ n−1 1 − aa2γ < ( 2 3 )n−1 (2.30) and from theorem 2.2 wi has a unique zero x ∗ i+1 in u ∗ and (2.10) holds. using (2.24) and (2.29) we get for x ∈ u ∥ ∥ ∥ ∥ w′j (x ∗ i+1) −1 w ′′ i (x) 2 ∥ ∥ ∥ ∥ ≤ ‖w′i(x ∗ i+1) −1f ′(α)‖ ∥ ∥ ∥ ∥ f ′(α)−1 w′′i (x) 2 ∥ ∥ ∥ ∥ ≤ a2 + n(n−1) 2 an‖x − xi‖ n−2 1 − 2aa2‖x ∗ i+1 − α‖ − nan‖x ∗ i+1 − xi‖ n−1 ≤ a2 + n(n−1) 2 an(2γ) n−2 1 − aa2γ − nan ( 3 2 γ )n−1 = ã2. (2.31) it follows from theorem 3.1 and (2.12) that for z1 = xi − f ′(xi) −1f (xi) ‖z1 − α‖ ≤ 1 2 ‖xi − α‖. (2.32) 154 ioannis k. argyros cubo 12, 1 (2010) since x∗i+1 ∈ u ∗, ‖z1 − x ∗ i+1‖ ≤ γ, we shall show zj+1 ∈ dj = { x : ‖x − x∗i+1‖ ≤ 1 2 ‖zj − x ∗ i+1‖ } ∩ u. (2.33) set wi(x) = wi(zj ) + w ′ i(zj )(x − zj) + r2(x; zj ), (2.34) where, r2(x; y) = ∫ 1 0 w′′i (y + t(x − y))(x − y) 2(1 − t)dt. (2.35) note that zj+1 is the solution of equation x = h(x) = x∗i+1 + w ′(xi+1) −1 { r2(x; zj ) − r2(x; x ∗ i+1) } . (2.36) we shall show h is contractive on dj . it follows from (2.12), (2.31) and (2.36): ‖h(x) − x∗i+1‖ ≤ ã2(‖x − zj‖ 2 + ‖x − x∗i+1‖ 2) ≤ 2 + 3a 2 ã2‖zj − x ∗ i+1‖ ≤ 1 2 ‖zj − x ∗ i+1‖. (2.37) moreover we have ‖h(x) − α‖ ≤ ‖x∗i+1 − α‖ + ‖h(x) − x ∗ i+1‖ ≤ ( 1 2 + 1 2 ) γ = γ. (2.38) it follows by the contraction mapping principle [4], (2.37) and (2.38) that zj+1 is the unique zero of h in dj . it follows that xi+1 = zk ∈ u , and ‖xi+1 − α‖ ≤ ‖xi+1 − x ∗ i+1‖ + ‖x ∗ i+1 − α‖ ≤ ( 1 2 )k ‖z0 − x ∗ i+1‖ + 1 2 ‖xi − α‖ ≤ [ 3 2 ( 1 2 )k + 1 2 ] ‖xi − α‖ ≤ 7 8 ‖xi − α‖, (2.39) which shows xi ∈ u and (2.15) hold true. set ej = ‖zj − x ∗ i+1‖ and x = zj+1 in (2.36). then we get ej+1 ≤ ã2 ( 1 + ej+1 ej )2 1 − ã2ej+1 e2j ≤ ã2(1 + hi)ej , (2.40) where hi = o(ej ) and 0 ≤ hi ≤ 2+3a 2 compare to (2.7). in view of ej = o(ej ) we can set cubo 12, 1 (2010) an improved convergence and complexity analysis ... 155 hi = o(ei). it follows from (2.10) and (2.40) ei+1 ≤ ‖xi+1 − x ∗ i+1‖ + ‖x ∗ i+1 − α‖ = ek + ‖x ∗ i+1 − α‖ ≤ [ ã2(1 + hi) ]2k−1 ‖xi − x ∗ i+1‖ 2k + an 1 − aa2e ∗ i+1 ( 1 + e∗i+1 ei )n eni ≤ ( 1 + e∗i+1 ei )n ( an 1 − aa2e ∗ i+1 + [ ã2(1 + hi) ]2k−1 [( 1 + e∗j+1 ei ) ei ]2k−n ) eni = ci,ne n i . (2.41) in view of e ∗ i+1 ei and hi tending to zero we get lim i→∞ ci,n = an + δã n−1 2 , (2.42) where δ = 0 if 2k > n and δ = 1 otherwise. hence, (2.16) holds. furthermore, we have zj+1 − x ∗ i+1 = w ′ i(x ∗ i+1) −1 w ′′ i (x ∗ i+1) 2 (zj − x ∗ i+1) 2 + o(ẽ3j ) = f ′(α)−1 f ′′(α) 2 (zj − x ∗ i+1) 2 + o(e∗i+1ẽ 2 j + ẽ 3 j ) = f2(zj − x ∗ i+1) 2 + o(ẽ2j ). (2.43) therefore, we get zk − x ∗ i+1 = f2 ( f2 · · · · (f2(xi − x ∗ i+1) 2)2 · · · )2 + o(e2ki ) = f2 ( f2 · · · · (f2(xi − α) 2)2 · · · )2 + o(e2ki ). (2.44) in view of (2.21), (2.22), and (2.44) we have zk − x ∗ i+1 = bi,k + o(e 2k i ). (2.45) in view of (2.11) and (2.45) we deduce xi+1 − α = zk − x ∗ i+1 + x ∗ i+1 − α = bi,k + fn(xi − α) n + o(eni ), (2.46) which shows (2.20). that completes the proof of the theorem. remark 2.4. the less precise estimate (using (2.1) for j = 2 instead of sharper (2.2) that is actually needed) ‖f ′(α)−1[w′j (x) − f ′(α)]‖ ≤ 2a2‖x − α‖ + nan‖x − xi‖ n−1 (2.47) 156 ioannis k. argyros cubo 12, 1 (2010) was used in [7] instead of (2.28), together with 0 ≤ ã2γ ≤ 1 5 (2.48) instead of weaker (2.12). if a = a2 our results theorem 2.1, theorem 2.2 and theorem 2.3 reduce to the corresponding theorem 2.1 in [6], theorem 2.1 in [5] and theorem 3.1 in [7] respectively. otherwise our results constitute improvements with advantages already stated in the introduction. we now give conditions under which inn enjoys a “type of global convergence”. let f (x) = ∞ ∑ i=1 1 ι! f (i)(xi − α) i (2.49) be analytic in d = u 0(α, r), and ‖f ′(α)−1f (i)(α)‖ ι! ≤ ki−1 (2.50) for i ≥ 2 and r ≥ 1 k . as in [7], one way to find k is to use cauchy’s formula ‖f ′(α)−1f (i)(α)‖ ι! ≤ m ri , (2.51) where, m = sup x∈d ‖f ′(α)−1f (x)‖. (2.52) let k = max [ 1 r , m r2 ] . then m r ≤ kr ≤ (kr)i−1 (2.53) and m ri ≤ ki−1. (2.54) we can show: theorem 2.5. if (2.2) and (2.50) hold then the interpolary newton method (1.4) converges provided that x0 ∈ u (α, γn), where γn = xn k (2.55) and xn, 0 < xn < x∞, satisfies the equation (3 + 2a) [ x (1 − x)3 + n(n − 1) 4(1 − x)2 ( 2x 1 − x )n−1 ] = 1 − ax (1 − x)3 − n (1 − x)2 [ 3x 2(1 − x) ]n−1 (2.56) cubo 12, 1 (2010) an improved convergence and complexity analysis ... 157 and xn → x∞, where x∞ ≥ .12 (2.57) is the positive solution of equation x (1 − x)3 = 1 4 + 2a . (2.58) proof. in view of (2.50) we have for f (x) = x 1 − kx , (2.59) that ‖f ′(α)−1f (i)(x)‖ ≤ f (i)(‖x − α‖). (2.60) using f (i)(x) = i!ki−1 (1 − kx)i+1 (i ≥ 2), (2.61) we get ai(γ) ≤ ki−1 (1 − kγ)i+1 (i ≥ 2). (2.62) it follows from (2.13) and (2.62) that ã2γ ≤ [ kγ (1−kγ)3 + n(n−1) 4(1−kγ)2 ( 2kγ 1−kγ )n−1 ] 1 − akγ (1−kγ)3 − n (1−kγ)2 ( 3kγ 2(1−kγ) )n−1 = 1 3 + 2a . (2.63) letting kγ = x we see that x satisfies equation (2.56). it is simple calculus to show that x = x(n) is an increasing function of n and x∞ = lim n→∞ x(n) satisfies equaiton (2.58). remark 2.6. if a = a2 (i.e. a = 1) our theorem 2.5 reduces to theorem 3.2 in [7]. otherwise it is an improvement, since the limit of sequence x(n) in [7] is .12 which is smaller than ours implying by (2.55) that we provide a larger radius of convergence. in particular if r is related to 1 k , say r = c1 k , then γn = xn k = xn c1r ≤ x∞ c1r . (2.64) the rest of the results introduced in [7] are improved. in particular with the notation introduced in [7] we have for i: ei = gie n i−1, gi ≤ g, g = g(n) =              a2 1 − 2aa2γ , n = 2 (1 + q)n [ an 1 − aa2 γ 2 + ( 7 2 ã2 )2k−1 [(1 + q)γ]2 k −n, n > 2 (2.65) 158 ioannis k. argyros cubo 12, 1 (2010) where ã2 is given by (2.13), q = 1 2 + 3 2 ( 1 2 )k , and k = [log 2n]. ii: if the total number of arithmetic operations necessary to solve a system of n linear equations is o(nβ ), β ≤ 3, then d(inn) =            o ( n β[log 2n] + n 2 ( n + n − 2 n − 2 ) ([log 2n] − 1 ) for n ≥ 2, (3 + 2a)[log 2n] + o(1), for n = 1. (2.66) remark 2.7. if a = a2 our results reduce to the ones in [7]. otherwise they constitute an improvement. we complete this study with an example to show that strict inequality can hold in (2.3): example 2.8. let x = y = r, x∗ = 0 and define function f on u = u (0, 1) by f (x) = ex − 1. (2.67) using (2.1), (2.2), (2.4) and (2.66) we obtain a = e − 1 2 < e 2 = a2 (2.68) and a = .632120588. (2.69) it follows from (2.5) that our radius of convergence is given by γa = .112699836. (2.70) the corresponding radius γt w given in theorem 2.1 in [6] or [7] is: γt w = 1 4a2 = .09196986. (2.71) that is γt w < γa. (2.72) received: october, 2008. revised: january, 2009. references [1] argyros, i.k., a unifying local-semilocal convergence analysis and applications for twopoint newton-like methods in banach space, j. math. anal. applic., 298 (2004), 374–397. cubo 12, 1 (2010) an improved convergence and complexity analysis ... 159 [2] argyros, i.k., approximate solution of operator equations with applications, world scientific publ. comp., river edge, new jersey, 2005. [3] argyros, i.k., an improved convergence and complexity analysis of newton’s method for solving equations, (to appear). [4] kantorovich, l.v. and akilov, g.p., functional analysis in normed spaces, moscow, 1959. [5] ortega, j.m. and rheinboldt, w.c., iterative solution of nonlinear equations in several variables, academic press, new york, 1970. [6] traub, j.f. and wozniakowski, h., strict lower and upper bounds on iterative computational complexity . in analytic computational complexity, j.f. traube, ed., academic press, new york, 1976, pp. 15–34. [7] traub, j.f. and wozniakowski, h., convergence and complexity of newton iteration for operator equations, j. assoc. comput. machinery, 26, no. 2 (1979), 250–258. () cubo a mathematical journal vol.13, no¯ 03, (91–115). october 2011 uncertainty principle for the riemann-liouville operator hleili khaled faculty of applied mathematics,département de mathématiques et d’informatique, institut national des sciences appliquées et de thechnologie, centre urbain nord bp 676 1080 tunis cedex, tunisia, email: khaled.hleili@gmail.com omri slim département de mathématiques appliquées, institut préparatoire aux études d’ingénieurs, campus universitaire mrezka 8000 nabeul, tunisia. email: slim.omri@issig.rnu.tn and lakhdar t. rachdi département de mathématiques, faculté des sciences de tunis, 2092 el manar ii, tunisia. email: lakhdartannech.rachdi@fst.rnu.tn abstract a beurling-hörmander theorem’s is proved for the fourier transform connected with the riemann-liouville operator. nextly, gelfand-shilov and cowling-price type theorems are established. resumen se demuestra el teorema de beurling-hörmander por la transformada de fourier conectada con el operador de riemann-liouville. además, se establecen teoremas tipo de gelfand-shilov y cowling-price. 92 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) keywords: beurling-hörmander theorem, gelfand-shilov theorem, cowlingprice theorem, fourier transform, riemann-liouville operator. mathematics subject classification: 43a32; 42b10. 1 introduction the uncertainty principles play an important role in harmonic analysis and have been studied by many authors, and from many points of view [12, 15]. these principles state that a function f and its fourier transform f̂ cannot be simultaneously sharply localized. theorems of hardy, morgan, gelfand-shilov, or cowlong-price,... are established for several fourier transforms [8, 14, 19, 20, 21], the most recent being the well known beurling-hörmander theorem’s which has been proved by hörmander [16], who took an idea of beurling [4]. this theorem states that if f is an integrable function on r with respect to the lebesgue measure, and if ∫∫ r2 |f(x)||f̂(y)|e|xy| dxdy < +∞, then f = 0 almost everywhere. later, bonami, demange and jaming [5] have generalized the above theorem and have established a strong multidimensional version of this uncertainty principle [15], by showing the following result if f is a square integrable function on rn with respect to the lebesgue measure, then ∫ rn ∫ rn |f(x)||f̂(y)| (1 + |x| + |y|)d e |〈x/y〉| dxdy < +∞, if and only if f may be written as f(x) = p(x)e−〈ax/x〉, where a is a real positive definite symmetric matrix and p is a polynomial with degree(p) < d − n 2 . in particular for d 6 n, f is identically zero. the beurling-hörmander uncertainty principle in its weak and strong forms has been studied by many authors, and for various fourier transforms. in particular, bouattour and trimèche [6] have showed this theorem for the hypergroup of chébli-trimèche, kamoun and trimèche [17] have proved an analogue of the beurling-hörmander theorem for some singular partial differential operators, trimèche [22] has showed this uncertainty principle for the dunkl transform, we cite also yakubovich [26], who has established the same result for the kontorovich-lebedev transform. the beurling-hörmander uncertainty principle implies many other known quantitative uncertainty principles as those of gelfand-shilov [13], cowling-price [8], morgan [3, 19] or also the one of hardy [14]. in [2], the third author with the others have considered the singular partial differential opercubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 93 ators defined by    ∆1 = ∂ ∂x , ∆2 = ∂2 ∂r2 + 2α + 1 r ∂ ∂r − ∂2 ∂x2 ; (r,x) ∈]0, +∞[×r ; α ≥ 0, and they associated to ∆1 and ∆2 the following integral transform, called the riemann-liouville operator which is defined on c∗(r 2) ( the space of continuous functions on r2, even with respect to he first variable ) by rα(f)(r,x) =    α π ∫ 1 −1 ∫ 1 −1 f(rs √ 1 − t2,x + rt)(1 − t2)α− 1 2 (1 − s2)α−1 dtds, if α > 0, 1 π ∫ 1 −1 f(r √ 1 − t2,x + rt) dt√ (1 − t2) ; if α = 0. the fourier transform connected with the operator rα is defined by fα(f)(µ,λ) = ∫ +∞ 0 ∫ r f(r,x)ϕµ,λ(r,x)dνα(r,x), where ϕµ,λ(r,x) = rα ( cos(µ.)e−iλ. ) (r,x). dνα is the measure defined on [0, +∞[×r by, dνα(r,x) = r2α+1 2αγ(α + 1) √ 2π dr ⊗ dx. many harmonic analysis results are established for the fourier transform fα (inversion formula, plancherel’s formula, paley-winer and plancherel’s theorems...). the aim of this work is to establish the beurling-hörmander theorem for the fourier transform fα and to deduce the analogues of the gelfand-shilov and the cowling-price theorems for this transform. more precisely, in the second section, we give some basic harmonic analysis results related to the fourier transform fα. the third section is devoted to establish the main result of this paper, that is the the beurling-hörmander theorem 94 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) . let f be a square integrable function on [0, +∞[×r with respect to the measure dνα. let d be a real number, d > 0. if ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. then i) for d 6 2, f = 0. ii) for d > 2, there exist a positive constant a and a polynomial p on r2 even with respect to the first variable, such that f(r,x) = p(r,x)e−a(r 2 +x 2 ) , with degree(p) < d 2 − 1, where γ+ = [0, +∞[×r ∪ { (it,x) | (t,x) ∈ [0, +∞[×r , t 6 |x| } . θ is the function defined on the set γ+ by θ(µ,λ) = ( √ µ2 + λ2,λ). dγ̃α the measure defined on the set γ+ by ∫ ∫ γ+ g(µ,λ) dγ̃α(µ,λ) = 1 π ( ∫ +∞ 0 ∫ r g(µ,λ)(µ2 + λ2)− 1 2 µdµdλ + ∫ r ∫ |λ| 0 g(iµ,λ)(λ2 − µ2)− 1 2 µdµdλ ) . the last section of this paper contains the following results that are respectively the gelfand-shilov and the cowling-price theorems for fα . let p,q be two conjugate exponents, p,q ∈]1, +∞[. let d,ξ,η be non negative real numbers such that ξη > 1. let f be a measurable function on r2, even with respect to the first variable, such that f ∈ l2(dνα). if ∫ +∞ 0 ∫ r |f(r,x)|e ξp |(r,x)|p p (1 + |(r,x)|)d dνα(r,x) < +∞, and ∫∫ γ+ |fα(f)(µ,λ)|e ηq|θ(µ,λ)|q q (1 + |θ(µ,λ)|)d dγ̃α(µ,λ) < +∞, then i) for d 6 1, f = 0. 2i) for d > 1, we have cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 95 a) f = 0 for ξη > 1. b) f = 0 for ξη = 1, and p 6= 2. c) f(r,x) = p(r,x)e−a(r 2 +x 2 ), for ξη = 1, and p = q = 2, where a > 0, and p is a polynomial on r2 even with respect to the first variable, with degree(p) < d − 1. . let ξ,η,ω1,ω2 be non negative real numbers such that ξη > 1 4 . let p,q be two exponents, p,q ∈ [1, +∞], and let f be a measurable function on r2, even with respect to the first variable such that f ∈ l2(dνα). if ∥∥∥ eξ|(.,.)| 2 (1 + |(., .)|)ω1 f ∥∥∥ p,να < +∞, and ∥∥∥ eη|θ(.,.)| 2 (1 + |θ(., .)|)ω2 fα(f) ∥∥∥ q,γ̃α < +∞, then i) for ξη > 1 4 , f = 0. ii) for ξη = 1 4 , there exist a positive constant a and a polynomial p on r2, even with respect to the first variable, such that f(r,x) = p(r,x)e−a(r 2 +x 2 ) . 2 the fourier transform associated with the riemann-liouville operator it’s well known [2] that for all (µ,λ) ∈ c2, the system    ∆1u(r,x) = −iλu(r,x), ∆2u(r,x) = −µ 2 u(r,x), u(0,0) = 1 , ∂u ∂r (0,x) = 0 , ∀x ∈ r, admits a unique solution ϕµ,λ, given by ∀(r,x) ∈ r2; ϕµ,λ(r,x) = jα(r √ µ2 + λ2)e−iλx, where jα(z) = 2αγ(α + 1) zα jα(z) = γ(α + 1) +∞∑ n=0 (−1)n n!γ(α + n + 1) ( z 2 )2n, z ∈ c, (2.1) 96 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) and jα is the bessel function of the first kind and index α [9, 10, 18, 25]. the modified bessel function jα has the following integral representation [18, 25], for all z ∈ c, we have jα(z) =    2γ(α + 1) √ πγ(α + 1 2 ) ∫ 1 0 (1 − t2)α− 1 2 cos(zt)dt, if α > − 1 2 ; cos(z), if α = − 1 2 . (2.2) from the relation (2.2), we deduce that for all z ∈ c, we have ∣∣jα(z) ∣∣ 6 e|im(z)|. (2.3) from the properties of the modified bessel function jα, we deduce that the eigenfunction ϕµ,λ satisfies the following properties sup (r,x)∈r2 |ϕµ,λ(r,x)| = 1, (2.4) if and only if (µ,λ) belongs to the set γ = r2 ∪ { (it,x) | (t,x) ∈ r2 , |t| ≤ |x| } . the eigenfunction ϕµ,λ has the following mehler integral representation ϕµ,λ(r,x) =    α π ∫ 1 −1 ∫ 1 −1 cos(µrs √ 1 − t2)e−iλ(x+rt)(1 − t2)α− 1 2 (1 − s2)α−1 dtds; if α > 0, 1 π ∫ 1 −1 cos(rµ √ 1 − t2)e−iλ(x+rt) dt√ 1 − t2 ; if α = 0. this integral representation allows to define the so-called riemann-liouville operator associated with ∆1,∆2 by rα(f)(r,x) =    α π ∫ 1 −1 ∫ 1 −1 f(rs √ 1 − t2,x + rt)(1 − t2)α− 1 2 (1 − s2)α−1 dtds; if α > 0, 1 π ∫ 1 −1 f(r √ 1 − t2,x + rt) dt√ (1 − t2) ; if α = 0. where f is a continuous function on r2, even with respect to the first variable. the transform rα generalizes the ”mean operator” defined by r0(f)(r,x) = 1 2π ∫ 2π 0 f(r sin θ,x + r cos θ) dθ. in the following, we denote by cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 97 dmn+1 the measure defined on [0, +∞[×rn by, dmn+1(r,x) = √ 2 π 1 (2π) n 2 dr ⊗ dx. lp(dmn+1) the space of measurable functions f on [0, +∞[×rn, such that ‖f‖p,mn+1 = ( ∫ +∞ 0 ∫ rn |f(r,x)|p dmn+1(r,x) ) 1 p < +∞, if p ∈ [1, +∞[, ‖f‖∞ ,mn+1 = ess sup(r,x)∈[0,+∞ [×rn |f(r,x)| < +∞, if p = +∞. dνα the measure defined on [0, +∞[×r, by dνα(r,x) = r2α+1 2αγ(α + 1) √ 2π dr ⊗ dx. lp(dνα) the space of measurable functions f on [0, +∞[×r such that ‖f‖p,να < +∞. γ+ = [0, +∞[×r ∪ { (it,x) | (t,x) ∈ [0, +∞[×r , t 6 |x| } . bγ+ the σ-algebra defined on γ+ by bγ+ = {θ −1(b) , b ∈ b([0, +∞[×r)}, where θ is the bijective function defined on the set γ+ by θ(µ,λ) = ( √ µ2 + λ2,λ). dγα the measure defined on bγ+ by ∀ a ∈ bγ+ ; γα(a) = να(θ(a)). lp(dγα) the space of measurable functions f on γ+, such that ‖f‖p,γα < +∞. dγ̃α the measure defined on bγ+ by dγ̃α(µ,λ) = 2α+ 1 2 γ(α + 1) √ π(µ2 + λ2)α+ 1 2 dγα(µ,λ). s∗(r 2) the shwartz’s space formed by the infinitely differentiable functions on r2, rapidly decreasing together with all their derivatives, and even with respect to the first variable. then we have the following properties. 98 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) proposition 2.1. i) for all non negative measurable function g on γ+, we have ∫ ∫ γ+ g(µ,λ) dγα(µ,λ) = 1 2αγ(α + 1) √ 2π ( ∫ +∞ 0 ∫ r g(µ,λ)(µ2 + λ2)αµdµdλ + ∫ r ∫ |λ| 0 g(iµ,λ)(λ2 − µ2)αµdµdλ ) . ii) for all measurable function f on [0, +∞[×r, the function foθ is measurable on γ+. furthermore if f is non negative or integrable function on [0, +∞[×r with respect to the measure dνα, then we have ∫ ∫ γ+ (f ◦ θ)(µ,λ) dγα(µ,λ) = ∫ +∞ 0 ∫ r f(r,x) dνα(r,x). iii) for all non negative measurable function f, respectively integrable on [0, +∞[×r with respect to the measure dm2, we have ∫ ∫ γ+ (f ◦ θ)(µ,λ) dγ̃α(µ,λ) = ∫ +∞ 0 ∫ r f(r,x) dm2(r,x). (2.5) in the following we shall define the fourier transform fα associated with the operator rα, and we shall give some properties that we use in the sequel. definition 2.1. the fourier transform fα associated with the riemann-liouville operator rα is defined on l1(dνα) by ∀(µ,λ) ∈ γ ; fα(f)(µ,λ) = ∫ +∞ 0 ∫ r f(r,x)ϕµ,λ(r,x) dνα(r,x). then, for all (µ,λ) ∈ γ, fα(f)(µ,λ) = f̃α(f) ◦ θ(µ,λ), (2.6) where for all (µ,λ) ∈ [0, +∞[×r, f̃α(f)(µ,λ) = ∫ +∞ 0 ∫ r f(r,x)jα(rµ)e −iλx dνα(r,x). (2.7) moreover, the relation (2.4) implies that the fourier transform fα is a bounded linear operator from l1(dνα) into l ∞ (dγα), and that for all f ∈ l1(dνα), we have ‖fα(f)‖∞ ,γα 6 ‖f‖1,να. (2.8) theorem 2.1 (inversion formula). let f ∈ l1(dνα) such that fα(f) ∈ l1(dγα), then for almost every (r,x) ∈ [0, +∞[×r, we have f(r,x) = ∫ ∫ γ+ fα(f)(µ,λ)ϕµ,λ(r,x) dγα(µ,λ) = ∫ +∞ 0 ∫ r f̃α(f)(µ,λ)jα(rµ)e iλx dνα(µ,λ). cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 99 lemma 2.2. let rα be the mapping defined for all non negative measurable function g on [0, +∞[×r by rα(g)(r,x) = 2γ(α + 1) √ πγ(α + 1 2 ) ∫ 1 0 (1 − s2)α− 1 2 g(rs,x) ds = 2γ(α + 1)r−2α √ πγ(α + 1 2 ) ∫ r 0 (r2 − s2)α− 1 2 f(s,x) ds, r > 0. (2.9) then for all non negative measurable functions f,g on [0, +∞[×r, we have ∫ +∞ 0 ∫ r f(r,x)rα(g)(r,x) dνα(r,x) = ∫ +∞ 0 ∫ r wα(f)(r,x)g(r,x) dm2(r,x), (2.10) where wα is the classical weyl transform defined for all non negative measurable function on [0, +∞[×r by wα(f)(r,x) = 1 2α+ 1 2 γ(α + 1 2 ) ∫ +∞ r (t2 − r2)α− 1 2 f(t,x)2tdt. (2.11) proposition 2.2. for all f ∈ l1(dνα), the function wα(f) belongs to l1(dm2), and we have ‖wα(f)‖1,m2 6 ‖f‖1,να. (2.12) moreover, for all (µ,λ) ∈ [0, +∞[×r, we have f̃α(f)(µ,λ) = (λ2 ◦ wα)(f)(µ,λ), (2.13) where λ2 is the usual fourier transform defined on l 1(dm2) by λ2(g)(µ,λ) = ∫ +∞ 0 ∫ r g(r,x) cos(rµ)e−iλx dm2(r,x). remark 2.1. it’s well known [23, 24] that the transforms f̃α and λ2 are topological isomorphisms from s∗(r 2) onto itself. then by the relation (2.13), we deduce that the classical weyl transform wα is also a topological isomorphism from s∗(r 2) onto itself. proposition 2.3. for all f ∈ s∗(r2), we have w −1 α (f) = (−1) 1+[α+ 1 2 ] w[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )1+[α+ 1 2 ] (f) ) , (2.14) where ( ∂ ∂t2 ) (f)(t,x) = 1 t ∂f ∂t (t,x). proof. for σ ∈ r, σ > 0, let us define the so-called fractional transform hσ, defined on s∗(r2) by hσ(f)(r,x) = 1 2σγ(σ) ∫ +∞ r (t2 − r2)σ−1f(t,x)2tdt = wσ− 1 2 (f)(r,x). 100 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) from the remark 2.1, it follows that for all real number σ > 0, the mapping hσ is a topological isomorphism from s∗(r 2) onto itself. moreover, we have the following properties for all σ,δ ∈ r; σ,δ > 0 and for every f ∈ s∗(r2), we have ( hσ ◦ hδ ) (f) = hσ+δ(f). for all σ ∈ r, σ > 0, and for every integer k, we have hσ(f) = (−1) k hσ+k (( ∂ ∂t2 )k (f) ) . (2.15) where ∂ ∂t2 is the linear continuous operator defined on s∗(r 2) by ∂ ∂t2 (f)(t,x) = 1 t ∂f ∂t (t,x). the relation (2.15) allows us to extend the mapping hσ on r, by setting hσ(f)(r,x) = (−1) k hσ+k (( ∂ ∂t2 )k (f) ) , where k is any integer such that σ + k > 0, σ ∈ r. the extension hσ, σ ∈ r satisfies ( hσ ◦ hδ ) (f) = hσ+δ(f), σ,δ ∈ r, f ∈ s∗(r2), and h0(f) = f, for all f ∈ s∗(r2). in particular, for all σ ∈ r, the transform hσ is a topological isomorphism from s∗(r2) onto itself, and the isomorphism inverse is given by h −1 σ = h−σ. thus, for all real number σ, we have h −1 σ (f) = (−1) 1+[σ] h1+[σ]−σ (( ∂ ∂t2 )1+[σ] (f) ) . in particular w −1 α (f) = h −1 α+ 1 2 (f) = (−1)1+[α+ 1 2 ] h[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )1+[α+ 1 2 ] (f) ) . 3 the beurling-hörmander theorem for the riemann-liouville operator in this section, we shall establish the main result of this paper, that is the beurling-hörmander theorem for the fourier transform fα. we recall firstly the following result that has been established by bonami, demange and jaming [5]. cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 101 theorem 3.1. let f be a measurable function on r × rn, even with respect to the first variable such that f ∈ l2(dmn+1), and let d be a real number, d ≥ 0. if ∫ +∞ 0 ∫ rn ∫ +∞ 0 ∫ rn |f(r,x)||λn+1(f)(s,y)| (1 + |(r,x)| + |(s,y)|)d e |(r,x)||(s,y)| dmn+1(r,x) dmn+1(s,y) < +∞, then there exist a positive constant a and a polynomial p on r × rn, even with respect to the first variable, such that f(r,x) = p(r,x)e−a(r 2 +|x| 2 ) , with degree(p) < d − (n + 1) 2 . in the following, we will establish some intermediary results that we use nextly. lemma 3.2. let f ∈ l2(dνα) such that ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞, (3.1) then the function f belongs to the space l1(dνα). proof. from the hypothesis, and the relations (2.5) and (2.6), we have ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) = ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |f(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. we assume of course that f 6= 0. then, there exists (µ0,λ0) ∈ [0, +∞[×r, such that (µ0,λ0) 6= (0,0), f̃α(f)(µ0,λ0) 6= 0, and |f̃α(f)(µ0,λ0)| ∫ +∞ 0 ∫ r |f(r,x)| e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞, hence ∫ +∞ 0 ∫ r |f(r,x)| e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. let h be the function defined on [0, +∞[ by h(s) = es|(µ0,λ0)| (1 + s + |(µ0,λ0)|) d , then the function h admits a minimum attained at s0 =    d |(µ0,λ0)| − 1 − |(µ0,λ0)|, if d |(µ0,λ0)| > 1 + |(µ0,λ0)|; 0, if d |(µ0,λ0)| 6 1 + |(µ0,λ0)|. 102 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) consequently, ∫ +∞ 0 ∫ r |f(r,x)| dνα(r,x) 6 1 h(s0) ∫ +∞ 0 ∫ r |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. lemma 3.3. let f ∈ l2(dνα) such that ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. then, there exists a > 0 such that the function f̃α(f) is analytic on the set ba = { (µ,λ) ∈ c2 | ∣∣im(µ) ∣∣ < a , ∣∣im(λ) ∣∣ < a } . proof. from the proof of the lemma 3.2, there exists (µ0,λ0) 6= (0,0), such that ∫ +∞ 0 ∫ r |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. let a > 0, such that 0 < 2a < |(µ0,λ0)|. then we have ∫ +∞ 0 ∫ r |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) = ∫ +∞ 0 ∫ r |f(r,x)|e2a|(r,x)| e|(r,x)|(|(µ0,λ0)|−2a) (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. let g be the function defined on [0, +∞[ by g(s) = es(|(µ0,λ0)|−2a) (1 + s + |(µ0,λ0)|) d , then g admits a minimum attained at s0 =    d |(µ0,λ0)| − 2a − 1 − |(µ0,λ0)|, if d |(µ0,λ0)| − 2a > 1 + |(µ0,λ0)|; 0, if d |(µ0,λ0)| − 2a 6 1 + |(µ0,λ0)|. consequently, ∫ +∞ 0 ∫ r |f(r,x)|e2a|(r,x)| dνα(r,x) 6 1 g(s0) ∫ +∞ 0 ∫ r |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. (3.2) cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 103 on the other hand, from the relation (2.1) we deduce that for all (r,x) ∈ [0, +∞[×r, the function (µ,λ) 7−→ jα(rµ)e−ixλ is analytic on c2 [7], even with respect to the first variable, and by the relation (2.3) we have ∣∣jα(rµ)e−iλx ∣∣ 6 e|(r,x)|(|im(µ)|+|im(λ)|). (3.3) from the relations (2.7), (3.2), and (3.3), it follows that the function f̃α(f) is analytic on ba, even with respect to the first variable. corollary 3.1. let f ∈ l2(dνα); f 6= 0; and let d be a real number, d > 0. if ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. then for all real number a; a > 0, we have να ({ (r,x) ∈ r2 | f̃α(f)(r,x) 6= 0 and |(r,x)| > a }) > 0. proof. from lemma 3.2, the function f belongs to l1(dνα), and consequently the function f̃α(f) is continuous on r2, even with respect to the first variable. then for all a > 0, the set { (r,x) ∈ r2 | f̃α(f)(r,x) 6= 0 and |(r,x)| > a } , is on open subset of r2. assume that να ({ (r,x) ∈ r2 | f̃α(f)(r,x) 6= 0 and |(r,x)| > a }) = 0, then for all (r,x) ∈ r2; |(r,x)| > a, we have f̃α(f)(r,x) = 0. applying lemma 3.3 and analytic continuation, we deduce that f̃α(f) vanishes on r 2, and by theorem 2.1, it follows that f = 0. lemma 3.4. let f ∈ l2(dνα) and let d be a real number d > 0. if ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞, then the function wα(f), belongs to l 2(dm2), where wα is the mapping defined by the relation (2.11). proof. from the hypothesis and the relations (2.5) and (2.6), we have ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) = ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |f(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. 104 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) by the same way as inequality (3.2) of the lemma 3.3, there exists b ∈ r, b > 0, such that ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)|e b|(µ,λ)| dm2(µ,λ) < +∞. (3.4) consequently, the function f̃α(f) lies in l 1(dνα) and by theorem 2.1, we get f(r,x) = ∫ +∞ 0 ∫ r f̃α(f)(µ,λ)jα(rµ)e iλx dνα(µ,λ); a.e. in particular the function f is bounded and ‖f‖∞ ,να 6 ‖f̃α(f)‖1,να. (3.5) now, we have |wα(f)(r,x)| 6 1 2α+ 1 2 γ(α + 1 2 ) ∫ +∞ r (t2 − r2)α− 1 2 |f(t,x)|2tdt = r2α+1 2α+ 1 2 γ(α + 1 2 ) ∫ +∞ 1 (u2 − 1)α− 1 2 |f(ru,x)|2udu. using minkowski’s inequality for integrals [11], we get ( ∫ +∞ 0 ∫ r |wα(f)(r,x)| 2 dm2(r,x) ) 1 2 6 1 2α+ 1 2 γ(α + 1 2 ) ( ∫ +∞ 0 ∫ r ( ∫ +∞ 1 r 2α+1(u2 − 1)α− 1 2 |f(ru,x)|2udu )2 dm2(r,x) ) 1 2 6 1 2α+ 1 2 γ(α + 1 2 ) ∫ +∞ 1 (u2 − 1)α− 1 2 ( ∫ +∞ 0 ∫ r r 4α+2 |f(ru,x)|2 dm2(r,x) ) 1 2 2udu = γ(α + 1) 1 2 2 α 2 − 3 4 π 1 4 γ(α + 1 2 ) ( ∫ +∞ 1 (u2 − 1)α− 1 2 u −2α− 1 2 du )( ∫ +∞ 0 ∫ r |f(t,x)|2t2α+1dνα(t,x) ) 1 2 = γ(α + 1) 1 2 2 α 2 − 7 4 π 1 4 γ(α + 1 2 ) ( ∫ 1 0 (1 − s)α− 1 2 s 9 4 ds )( ∫ +∞ 0 ∫ r |f(t,x)|2t2α+1dνα(t,x) ) 1 2 = cα ( ∫ +∞ 0 ∫ r |f(t,x)|2t2α+1dνα(t,x) ) 1 2 and by the relations (3.2) and (3.5), we get ( ∫ +∞ 0 ∫ r |wα(f)(r,x)| 2 dm2(r,x) ) 1 2 6 mα‖f‖ 1 2 ∞ ,να ( ∫ +∞ 0 ∫ r |f(t,x)|e2a|(t,x)|dνα(t,x) ) 1 2 < +∞. remark 3.1. let f be a function satisfying the hypothesis (3.1), then from the relations (3.2) and (3.4), we can prove that the function f belongs to the schwartz’s space s∗(r 2). since the weyl transform wα is an isomorphism from s∗(r 2) onto itself, then the function wα(f) belongs to s∗(r 2), in particular wα(f) ∈ l2(dm2). cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 105 remark 3.2. let σ be a positive real number such that σ + σ2 > d > 0. then, the function t 7−→ eσt (1 + t + σ)d , is increasing on [0, +∞[. theorem 3.5. let f ∈ l2(dνα), and let d be a real number, d > 0. if ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞, then ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) < +∞. proof. from the hypothesis and the relations (2.5) and (2.6), we have ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |f(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. (3.6) i) if d = 0, then by fubini’s theorem we have ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |wα(f)(r,x)||f̃α(f)(µ,λ)|e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ r |wα(f)(r,x)|e |(r,x)||(µ,λ)| dm2(r,x) ) dm2(µ,λ) 6 ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ r wα(|f|)(r,x)e |(r,x)||(µ,λ)| dm2(r,x) ) dm2(µ,λ). (3.7) using the relation (2.10), we deduce that ∫ +∞ 0 ∫ r wα(|f|)(r,x)e |(r,x)||(µ,λ)| dm2(r,x) = ∫ +∞ 0 ∫ r |f(r,x)|rα(e |(.,.)||(µ,λ)|)(r,x) dνα(r,x), (3.8) but for all (r,x) ∈ [0, +∞[×r rα(e |(.,.)||(µ,λ)|)(r,x) 6 e|(r,x)||(µ,λ)|. (3.9) combining the relations (3.6), (3.7), (3.8), and (3.9), we get ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |wα(f)(r,x)||f̃α(f)(µ,λ)|e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ r |f(r,x)|e|(r,x)||(µ,λ)| dνα(r,x) ) dm2(µ,λ) < +∞. 106 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) ii) if d > 0, let bd = { (u,v) ∈ [0, +∞[×r | |(u,v)| 6 d } . . by fubini’s theorem, we have ∫∫ bc d ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)||wα(f)(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ bc d |f̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ r wα(|f|)(r,x) e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d × dm2(r,x) ) dm2(µ,λ), and by the relation (2.10), we get ∫∫ bc d ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)||wα(f)(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ bc d |f̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ r |f(r,x)|rα ( e|(.,.)||(µ,λ)| (1 + |(., .)| + |(µ,λ)|)d ) (r,x) × dνα(r,x) ) dm2(µ,λ). (3.10) however, by the relation (2.9) and remark 3.2, we have for all (µ,λ) ∈ bcd rα ( e|(.,.)||(µ,λ)| (1 + |(., .)| + |(µ,λ)|)d ) (r,x) 6 e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d . (3.11) combining the relations (3.10) and (3.11), we obtain ∫∫ bc d ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)||wα(f)(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ bc d |f̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ r |f(r,x)| e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d ) dνα(r,x) ) dm2(µ,λ) 6 ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |f̃α(f)(µ,λ)||f(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. . ∫∫ bd ∫∫ bc d |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ bd |f̃α(f)(µ,λ)| ( ∫∫ bc d wα(|f|)(r,x) e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d dm2(r,x) ) dm2(µ,λ). cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 107 but for (µ,λ) ∈ bd, ∫∫ bc d wα(|f|)(r,x) e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d dm2(r,x) = ∫ +∞ 0 ∫ r |f(r,x)|rα ( e|(.,.)||(µ,λ)| (1 + |(., .)| + |(µ,λ)|)d 1bc d ) (r,x) dνα(r,x) 6 ∫∫ bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x). hence, ∫∫ bd ∫∫ bc d |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ( ∫∫ bd |f̃α(f)(µ,λ)|dm2(µ,λ) )( ∫∫ bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x) ) . in virtue of the relation (2.8), we have ∫∫ bd ∫∫ bc d |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ‖f‖1,ναm2(bd) ( ∫∫ bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x) ) . (3.12) on he other hand, from corollary 3.1 and the relation (3.6), there exists (µ0,λ0) ∈ [0, +∞[×r, |(µ0,λ0)| > d, f̃α(f)(µ0,λ0) 6= 0, and ∫∫ bc d |f(r,x)| e|(µ0,λ0)||(r,x)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞, (3.13) so, by remark 3.2, ∫∫ bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x) 6 ∫∫ bc d |f(r,x)| e|(µ0,λ0)||(r,x)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. (3.14) the relations (3.12), (3.13), and (3.14) imply that ∫∫ bd ∫∫ bc d |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) < +∞. 108 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) finally . ∫∫ bd ∫∫ bd |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 e d 2 ( ∫∫ bd |f̃α(f)(µ,λ)|dm2(µ,λ) )( ∫∫ bd |wα(f)(r,x)| dm2(r,x) ) 6 e d 2 m2(bd)‖fα(f)‖∞ ,γα ‖wα(f)‖1,m2, and therefore by the relations (2.8) and (2.12), we deduce that ∫∫ bd ∫∫ bd |wα(f)(r,x)||f̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 e d 2 m2(bd)‖f‖21,να < +∞, and the proof of theorem 3.5 is complete. theorem 3.6 (beurling-hörmander for rα). let f ∈ l2(dνα), and let d be a real number, d > 0. if ∫ ∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. then i) for d 6 2, f = 0. ii) for d > 2, there exist a positive constant a and a polynomial p, even with respect to the first variable, such that f(r,x) = p(r,x)e−a(r 2 +x 2 ) , with degree(p) < d 2 − 1. proof. let f ∈ l2(dνα), satisfying the hypothesis. from proposition 2.2, lemma 3.2, and lemma 3.4, we deduce that the function wα(f) belongs to the space l1(dm2) ∩ l2(dm2) and that f̃α(f) = λ2 ◦ wα(f). thus from theorem 3.5, we get ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r ∣∣wα(f)(r,x) ∣∣∣∣λ2 ( wα(f) ) (µ,λ) ∣∣e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d dm2(r,x)dm2(µ,λ) < +∞. applying theorem 3.1, when f is replaced by wα(f), we deduce that if d 6 2, wα(f) = 0, and by remark 2.1, f = 0. cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 109 if d > 2, then there exist a > 0 and a polynomial q even with respect to the first variable such that wα(f)(r,x) = q(r,x)e −a(r 2 +x 2 ) = ∑ 2p+q6m ap,qr 2p x q e −a(r 2 +x 2 ) . in particular, the function wα(f) belongs to the space s∗(r 2). from remark 2.1, the function f belongs to s∗(r 2) and from the relation (2.14), we get f(r,x) = h−α− 1 2 ( q(t,y)e−a(t 2 +y 2 ) ) (r,x) = (−1)[α+ 1 2 ]+1 h[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )[α+ 1 2 ]+1( p(t,y)e−a(t 2 +y 2 ) )) (r,x) = ∑ 2p+q6m ap,q(−1) [α+ 1 2 ]+1 h[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )[α+ 1 2 ]+1( t 2p y q e −a(t 2 +y 2 ) )) (r,x). (3.15) however, for all k ∈ n, ( ∂ ∂t2 )k( t 2p y q e −a(t 2 +y 2 ) ) = ( min(p,k)∑ j=0 c j k 2jp! (p − j)! (−2a)k−jt2(p−j) ) y q e −a(t 2 +y 2 ) , (3.16) and for all σ ∈ r, σ > 0, hσ ( t 2p y q e −a(t 2 +y 2 ) ) (r,x) = 1 2σγ(σ) ( p∑ j=0 c j p γ(σ + p − j) aσ+p−j r 2j ) x q e −a(r 2 +x 2 ) . (3.17) combining the relations (3.15), (3.16) and (3.17), we deduce that f(r,x) = p(r,x)e−a(r 2 +x 2 ) . where p is a polynomial, even with respect to the first variable and degree(p) = degree(q). 4 applications of beurling-hörmander theorem in this section, we shall deduce from the precedent beurling-hörmander theorem two most important uncertainty principles for the fourier transform fα, that are the gelfand-shilov and the cowling-price theorems. lemma 4.1. let p be a polynomial on r2, p 6= 0, with degree(p) = m. then there exist two positive constants a and c such that ∀t > a, p(t) = ∫ 2π 0 ∣∣p(t cos(θ),t sin(θ) ∣∣dθ > ctm. 110 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) proof. let p be a polynomial on r2, p 6= 0 and with degree(p) = m. we have p(t) = ∫ 2π 0 ∣∣ m∑ j=0 aj(θ)t j ∣∣dθ, where the functions aj, 0 6 j 6 m, are continuous on [0,2π]. it’s clear that the function p is continuous on [0, +∞[, and by dominate convergence theorem’s, we have p(t) ∼ cmt m (t −→ +∞), (4.1) where cm = ∫ 2π 0 |am(θ)|dθ > 0. now the relation (4.1) involves that there exists a > 0 such that ∀t > a, p(t) > cm 2 t m . theorem 4.2 (gelfand-shilov for rα). let p,q be two conjugate exponents, p,q ∈]1, +∞[. let ξ,η be non negative real numbers such that ξη > 1. let f be a measurable function on r2, even with respect to the first variable, such that f ∈ l2(dνα). if ∫ +∞ 0 ∫ r |f(r,x)|e ξp |(r,x)|p p (1 + |(r,x)|)d dνα(r,x) < +∞, and ∫∫ γ+ |fα(f)(µ,λ)|e ηq |θ(µ,λ)|q q (1 + |θ(µ,λ)|)d dγ̃α(µ,λ) < +∞ ; d > 0. then i) for d 6 1, f = 0. ii) for d > 1, we have a) f = 0 for ξη > 1. b) f = 0 for ξη = 1, and p 6= 2. c) f(r,x) = p(r,x)e−a(r 2 +x 2 ) for ξη = 1 and p = q = 2, where a > 0 and p is a polynomial on r2 even with respect to the first variable, with degree(p) < d − 1. proof. let f be a function satisfying the hypothesis. since ξη > 1, and by a convexity argument, cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 111 we have ∫∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)2d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) 6 ∫∫ γ+ ∫ +∞ 0 ∫ r |f(r,x)||fα(f)(µ,λ)| (1 + |(r,x)|)d(1 + |θ(µ,λ)|)d e ξη|(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) 6 ( ∫∫ γ+ |fα(f)(µ,λ)| (1 + |θ(µ,λ)|)d e ηq|θ(µ,λ)|q q dγ̃α(µ,λ) ) × ( ∫ +∞ 0 ∫ r |f(r,x)| (1 + |(r,x)|)d e ξp|(r,x)|p p dνα(r,x) ) < +∞. (4.2) then from the beurling-hörmander theorem, we deduce that i) for d 6 1, f = 0. ii) for d > 1, there exist a positive constant a, and a polynomial p on r2, even with respect to the first variable such that f(r,x) = p(r,x)e−a(r 2 +x 2 ) , (4.3) with degree(p) < d − 1, and by a standard calculus, we obtain f̃α(f)(µ,λ) = q(µ,λ)e − 1 4a (µ 2 +λ 2 ) , (4.4) where q is a polynomial on r2, even with respect to the first variable, with degree(p) = degree(q). on the other hand, from the relations (2.5), (2.6), (4.2), (4.3) and (4.4), we get ∫ +∞ 0 ∫ r ∫ +∞ 0 ∫ r |p(r,x)||q(µ,λ)| (1 + |(r,x)|)d(1 + |(µ,λ)|)d e ξη|(r,x)||(µ,λ)|−a(r 2 +x 2 ) × e− 1 4a (µ 2 +λ 2 ) dνα(r,x)dµdλ < +∞, so ∫ +∞ 0 ∫ +∞ 0 ϕ(t) (1 + t)d ψ(ρ) (1 + ρ)d e ξηtρ e −at 2 e − 1 4a ρ 2 t 2α+2 ρdtdρ < +∞, (4.5) where ϕ(t) = ∫ 2π 0 ∣∣p(t cos(θ),t sin(θ)) ∣∣∣∣ cos(θ) ∣∣2α+1dθ, and ψ(ρ) = ∫ 2π 0 ∣∣q(ρ cos(θ),ρ sin(θ)) ∣∣dθ. . suppose that ξη > 1. if f 6= 0, then each of the polynomials p and q is not identically zero, let m = degree(p) = degree(q). from lemma 4.1, there exist two positive constants a and c such that ∀t > a, ϕ(t) > ctm, 112 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) and ∀ρ > a, ψ(ρ) > cρm. then, the inequality (4.5) leads to ∫ +∞ a ∫ +∞ a eξηtρ (1 + t)d(1 + ρ)d e −at 2 e − 1 4a ρ 2 dtdρ < +∞. (4.6) let ε > 0, such that ξη − ε = σ > 1. the relation (4.6) implies that ∫ +∞ a ∫ +∞ a eεtρ (1 + t)d(1 + ρ)d e σtρ e −at 2 e − 1 4a ρ 2 dtdρ < +∞. (4.7) however, for all t > a > d ε and ρ > a, we have eερt (1 + t)d(1 + ρ)d > eεa 2 (1 + a)2d , and by the relation (4.7) it follows that ∫ +∞ a ∫ +∞ a e σtρ e −at 2 e − 1 4a ρ 2 dtdρ < +∞. (4.8) let f(t) = ∫ +∞ a e σρt− 1 4a ρ 2 dρ, then f can be written f(t) = eaσ 2 t 2 ( ∫ +∞ a e − 1 4a ρ 2 dρ + 2aσe− a2 4a ∫ t 0 e aσs−aσ 2 s 2 ds ) , in particular f(t) > eaσ 2 t 2 ∫ +∞ a e − 1 4a ρ 2 dρ. thus ∫ +∞ a ∫ +∞ a e σtρ e −at 2 e − 1 4a ρ 2 dtdρ = ∫ +∞ a e −at 2 f(t)dt > ∫ +∞ a e − 1 4a ρ 2 dρ ∫ +∞ a e a(σ 2 −1)t 2 dt = +∞, because σ > 1. this contradics the relation (4.8) and shows that f = 0. . suppose that ξη = 1 and p 6= 2. in this case we have p > 2 or q > 2. suppose that q > 2, then from the second hypothesis and the relation (4.4), we have ∫ +∞ 0 ψ(ρ)e− ρ2 4a e ηqρq q (1 + ρ)d ρdρ < +∞. (4.9) if f 6= 0, then the polynomial q is not identically zero, and by lemma 4.1 and the relation (4.9), it follows that ∫ +∞ 0 e− ρ2 4a e ηqρq q (1 + ρ)d dρ < +∞, cubo 13, 3 (2011) uncertainty principle for the riemann-liouville operator 113 which is impossible because q > 2. the proof of theorem 4.2 is complete. theorem 4.3 (cowling-price for rα). let ξ,η,ω1,ω2 be non negative real numbers such that ξη > 1 4 . let p,q be two exponents, p,q ∈ [1, +∞], and let f be a measurable function on r2, even with respect to the first variable such that f ∈ l2(dνα). if ∥∥∥ eξ|(.,.)| 2 (1 + |(., .)|)ω1 f ∥∥∥ p,να < +∞, (4.10) and ∥∥∥ eη|θ(.,.)| 2 (1 + |θ(., .)|)ω2 fα(f) ∥∥∥ q,γ̃α < +∞, (4.11) then i) for ξη > 1 4 , f = 0. ii) for ξη = 1 4 , there exist a positive constant a and a polynomial p on r2, even with respect to the first variable, such that f(r,x) = p(r,x)e−a(r 2 +x 2 ) . proof. let p′ and q′ be the conjugate exponents of p respectively q. let us pick d1,d2 ∈ r, such that d1 > 2α + 3 and d2 > 2. finally, let d be a positive real number such that d > max ( ω1 + d1 p′ ,ω2 + d2 q′ ,1 ) . from hölder’s inequality and the relations (4.10) and (4.11), we deduce that ∫ +∞ 0 ∫ r |f(r,x)|eξ|(r,x)| 2 (1 + |(r,x)|) ω1+ d1 p ′ dνα(r,x) < +∞, and ∫ ∫ γ+ |fα(f)(µ,λ)|e η|θ(µ,λ)| 2 (1 + |θ(µ,λ)|) ω2+ d2 q ′ dγ̃α(µ,λ) < +∞. consequently we have ∫ +∞ 0 ∫ r |f(r,x)|eξ|(r,x)| 2 (1 + |(r,x)|)d dνα(r,x) < +∞, and ∫ ∫ γ+ |fα(f)(µ,λ)|e η|θ(µ,λ)| 2 (1 + |θ(µ,λ)|)d dγ̃α(µ,λ) < +∞. then, the desired result follows from theorem 4.2. remark 4.1. the hardy’s theorem is a special case of theorem 4.3 when p = q = +∞. received: july 2010. revised: august 2010. 114 hleili khaled, omri slim and lakhdar t. rachdi cubo 13, 3 (2011) references [1] g. andrews, r. askey and r. roy, special functions, cambridge university press, new-york 1999. 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[26] s.b. yakubovich, uncertainty principles for the kontorovich-lebedev transform, math. modell. anal., 13 no. 2 (2008), pp 289–302. introduction the fourier transform associated with the riemann-liouville operator the beurling-hörmander theorem for the riemann-liouville operator applications of beurling-hörmander theorem cubo a mathematical journal vol.19, no¯ 01, (39–51). march 2017 maximal functions and properties of the weighted composition operators acting on the korenblum, α-bloch and α-zygmund spaces gabriel m. antón marval1, rené e. castillo2, julio c. ramos-fernández3 1 area de matemáticas, universidad nacional experimental de guayana, puerto ordaz 8050, estado boĺıvar, venezuela. 2 departamento de matemáticas, universidad nacional de colombia, ap360354 bogotá, colombia. 3 departamento de matemáticas, universidad de oriente, cumaná 6101, estado sucre, venezuela. gabman@gmail.com, recastillo@unal.edu.co, jcramos@udo.edu.ve abstract using certain maximal analytic functions, we obtain new characterizations of the continuity and compactness of the weighted composition operators when acts between korenblum spaces, α-bloch spaces and when acts from certain weighted banach spaces of analytic functions with a logarithmic weight into α-bloch spaces. as consequence of our results, we obtain a new characterization of the continuity and compactness of composition operators acting between α-zygmund spaces. resumen usando ciertas funciones anaĺıticas maximales, obtenemos nuevas caracterizaciones de la continuidad y compacidad de operadores de composición con pesos cuando actúan entre espacios de korenblum, espacios α-bloch y cuando actúan desde ciertos espacios de banach de funciones anaĺıticas con un peso logaŕıtmico en espacios α-bloch. como consecuencia de nuestros resultados, obtenemos una nueva caracterización de la continuidad y la compacidad de operadores de composición actuando entre espacios α-zygmund. keywords and phrases: weighted banach spaces of analytic functions, bloch space, weighted composition operators. 2010 ams mathematics subject classification: 30d45, 47b33. 40 g. antón marval, r. castillo, j. ramos-fernández cubo 19, 1 (2017) 1 introduction over recent years, there has been a growing interest in the study of the properties of the weighted composition operators when acts between banach spaces of analytic functions on d, the open unit disk of the complex plane c. for fixed holomorphic functions u : d → c and φ : d → d, we can define the linear operator wu,φ : h(d) → h(d), where h(d) denotes the space of all holomorphic functions, by wu,φ(f) := u · (f ◦ φ). which is known as the weighted composition operator with symbols u and φ. clearly, if u ≡ 1 we have w1,φ(f) = f ◦φ = cφ(f), the composition operator cφ, and if φ(z) = id(z) = z for all z ∈ d, we obtain wu,id(f) = u · f = mu(f), the multiplication operator mu. furthermore, we can see that wu,φ is 1-1 on h(d) unless that u ≡ 0 or φ is a constant function. montes-rodŕıguez in [16] and contreras and hernández-dı́az in [9] characterized the continuity, the compactness and calculated the essential norm of wu,φ : h ∞ v → h ∞ w in terms of a quotient involving the associated weight of v, under the requirement that the weights (a weight function is a bounded, continuous and positive function defined on d) v and w are radial (v(|z|) = v(z) for all z ∈ d), non-increasing (i.e. v(r1) ≥ v(r2) for all 0 < r1 < r2 < 1) and typical (that is, lim|z|→1− v(z) = 0). here, h ∞ v is the weighted banach spaces of analytic functions or growth spaces which consist of functions f ∈ h(d) such that ∥f∥h∞ v = sup z∈d v(z)|f(z)| < ∞. (1.1) the associated weight of v, denoted by ṽ, is defined by ṽ(z) = ( sup ∥f∥h∞ v ≤1 {|f(z)|} ) −1 , z ∈ d. (1.2) the weighted banach spaces of analytic functions or growth spaces h∞ v are natural generalizations of h∞, the space of all bounded analytic functions on the open unit disk d. it is known that h∞v is a banach space with the norm defined in (1.1). initially, interest in the weighted banach spaces of analytic functions were oriented to studying the growth conditions of analytic functions and the duality of these spaces, being present in various areas such as complex analysis, fourier analysis, spectral theory and partial differential equations. some examples may be found in [3]. the associated weight ṽ of v was introduced by anderson and duncan in [1] and studied by bierstedt, bonet and taskinen in [2]. the associated weight is a very important tool in the study of the weighted banach spaces of analytic functions, it is known that the space h∞ !v is isometrically equal to h∞ v , that is ∥f∥ h∞ !v = ∥f∥ h∞ v for all f ∈ h∞ v and v is dominated by its associated weight, that is, ṽ ≥ v > 0 on d. we said that a weight v is essential if ṽ ∼ v, that is, if there is a constant cv > 0 such that v(z) ≤ ṽ(z) ≤ cvv(z) for all z ∈ d. we refer to the interested reader the works [2] and [4] for more properties of the associated weight and the space h∞v . cubo 19, 1 (2017) maximal functions and properties of the weighted composition . . . 41 the korenblum’s spaces (also known as bergman spaces) h∞ α are particular cases of the growth spaces, they are obtained when the weights are the functions vα(z) = ( 1 − |z|2 )α with z ∈ d, where the parameter α is positive and fixed. clearly the weight vα is radial, typical, non-increasing and essential. we refer to [12] for more details about the korenblum’s spaces. the bloch-type space bv are related with the growth spaces, consist of all analytic function f on d such that f′ ∈ h∞ v . bv is a banach space with the norm ∥f∥bv = |f(0)| + ∥f ′∥h∞ v = |f(0)| + ∥f∥ !bv, where, ∥f∥ !bv = sup z∈d v(z)|f′(z)|. when v(z) = 1 − |z|2, the space bv becomes the classical bloch space and it is denoted by b, we refer to [23] for more details about bloch’s space. when v(z) = ( 1 − |z|2 )α , where α > 0 is fixed, we get the α-bloch spaces which is denoted by bα (see [24]). the study of the properties of composition operators on bloch-type spaces began with the celebrated work of madigan and matheson in [14], they characterized continuity and compactness for composition operators on the classical bloch space b. their results have been extended by a big numerous of authors (see [18] and the lot of references therein). however, the properties of the weighted composition operators acting on bloch-type space is still in development and there is not much references about this subject, it is remarkable the work of ohno, stroethoff and zhao [17], where they characterized the continuity and the compactness of the weighted composition operators acting between α-bloch spaces in terms of the continuity and compactness of certain weighted composition operators acting between certain growth spaces. in [22], tjani shows that the composition operator cφ : b → b is compact if and only if ∥cφ (αλ)∥b → 0 as |λ| → 1, where αλ(z) = (λ − z) / ( 1 − λz ) is the mobius transformation of the unit disk. a similar result for weighted composition operators from hardy spaces into logarithmic bloch spaces was obtained by colonna and li in [8]. also, giménez, malavé and ramos-fernández in [11] extended the tjani’s result to composition operators cφ : b → b µ , where the weight µ was taken to be a non-vanishing, complex valued holomorphic function satisfying a reasonable geometric condition on the euclidean disk d(1, 1). more recently, by changing the functions αλ, used by tjani, to certain maximal analytic functions σa with a ∈ d, ramos-fernández et. al. [15, 19, 6, 5] show that tjani’s result can be extended to the case of composition operators acting on α-bloch and weighted bloch spaces. the main objective of this note is to show that the technical developed by ramos-fernández et. al. [15, 19, 6, 5], using certain maximal analytic functions, can be used to obtain new characterizations of the continuity and compactness of the weighted composition operators acting on korenblum and α-bloch spaces. in fact, in section 2, we use maximal functions in certain log-growth space and in korenblum spaces to give a new characterization of the continuity and compactness of 42 g. antón marval, r. castillo, j. ramos-fernández cubo 19, 1 (2017) weighted composition operators on these spaces. as consequence of our results in section 2, in section 3, we give similar results for wu,φ acting between α-bloch spaces and in section 4, we give new characterizations of the continuity and compactness of the composition operators acting between α-zygmund spaces. 2 the case of weighted composition operators acting on certain growth spaces we consider first the case of weighted composition operators wu,φ from h ∞ vlog to h∞ β . our goal is to find maximal functions ga ∈ h ∞ vlog with a ∈ d in the sense that |ga(a)| ≥ k sup ∥f∥ h∞ v log ≤1 {|f(a)|} (2.1) for some constant k > 0. here vlog(z) = ( log ( e 1 − |z|2 )) −1 is a weight defined on d which is radial, typical and non-increasing. the weight vlog is related with the bloch space b, in fact it is easy to see that b is continuously contained into h∞vlog . a calculation tell us that we have to consider the functions ga defined by ga(z) = 1 − |a|2 2 (1 − az) (1 − log (1 − az)) , (2.2) with z ∈ d, where log(w) = log |w| + iarg(w) denotes the logarithm whose imaginary part lies in the interval (−π,π], which is holomorphic in the open euclidean disk with center at 1 and radius 1. then, following the ideas in [15] and [5], we have obtained the following result: theorem 2.1. let β > 0 be fixed, φ : d → d and u : d → c holomorphic functions. then (1) the weighted composition operator wu,φ is continuous from h ∞ vlog to h∞ β if and only if sup a∈d ∥wu,φ (ga)∥h∞ β < ∞. (2.3) (2) the operator wu,φ : h ∞ vlog → h∞ β is compact if and only if lim |a|→1− ∥wu,φ (ga)∥h∞ β = 0. (2.4) proof. suppose first that wu,φ : h ∞ vlog → h∞ β is continuous. observe that for each a ∈ d, we cubo 19, 1 (2017) maximal functions and properties of the weighted composition . . . 43 have ∥ga∥h∞ v log = sup z∈d 1 log ( e 1−|z|2 ) · 1 − |a|2 2 |1 − az| |1 − log (1 − az)| ≤ sup z∈d |1 − log |1 − az|| + |arg (1 − az)| log ( e 1−|z|2 ) ≤ sup z∈d 1 − log (1 − |z|) + π log ( e 1−|z|2 ) ≤ sup z∈d ⎛ ⎝ log ( e 1−|z|2 ) log ( e 1−|z|2 ) + log(1 + |z|) log ( e 1−|z|2 ) ⎞ ⎠ + π ≤ 1 + log(2) + π. then, we can see that there exists a constant k > 0 such that ∥wu,φ (ga)∥h∞ β ≤ ∥wu,φ∥ ∥ga∥hv log ≤ k ∥wu,φ∥ and (2.3) follows. conversely, if (2.3) is true, then for each s ∈ d we have (1 − |s|2)β vlog (φ(s)) |u(s)| = 2(1 − |s|2)β ∣ ∣gφ(s) (φ(s)) ∣ ∣ |u(s)| ≤ 2 sup a∈d ∥wu,φ (ga)∥h∞ β and the item (1) follows from the continuity’s theorem of montes-rodŕıguez [16] (see also contreras and hernández-dı́az [9]) since the weight vlog is essential. now we are going to show the item (2). in virtue of tjani’s lemma in [21] and since ga is a bounded sequence in h∞ vlog which converges to zero uniformly on compact subsets of d, we conclude that if the operator wu,φ : h ∞ vlog → h∞ β is compact then the relation (2.4) holds. conversely, if the relation (2.4) is true, then for every ε > 0 we can find r1 ∈ (1/2, 1) such that ∥wu,φ (ga)∥h∞ β < ε whenever r1 < |a| < 1. hence, if z ∈ d satisfies |φ(z)| > r1, then we can write (1 − |z|2)β vlog (φ(z)) |u(z)| = 2(1 − |z|2)β ∣ ∣gφ(z) (φ(z)) ∣ ∣ |u(z)| ≤ sup w∈d (1 − |w|2)β ∣ ∣gφ(z) (φ(w)) ∣ ∣ |u(w)| = ∥ ∥wu,φ ( gφ(z) )∥ ∥ h∞ β < ε. this shows that lim |φ(z)|→1− (1 − |z|2)β vlog (φ(z)) |u(z)| = 0 and wu,φ : h ∞ vlog → h∞ β is compact in virtue of a result due to montes-rodŕıguez [16] (see also contreras and hernández-dı́az [9]) and the fact that vlog is an essential weight. ! next, we are going to consider the case of the weighted composition operator wu,φ acting from h∞ α to h∞ β , where α and β are positive parameters fixed. we need to find maximal functions σ (α) a ∈ h ∞ α in the sense of the relation (2.1), changing, of course, the space h ∞ vlog by h∞α . 44 g. antón marval, r. castillo, j. ramos-fernández cubo 19, 1 (2017) in this case, for a ∈ d, we consider functions σ (α) a defined by σ(α) a (z) = 1 − |a| (1 − az)α+1 , (2.5) where z ∈ d. then, the argument in the proof of theorem 2.1 allow us to show the following result: theorem 2.2. let α,β > 0 be fixed and u : d → c, φ : d → d holomorphic functions. (1) the weighted composition operators wu,φ is bounded from h ∞ α to h ∞ β if and only if sup a∈d ∥ ∥ ∥wu,φ ( σ(α)a )∥ ∥ ∥ h∞ β < ∞. (2.6) (2) the weighted composition operators wu,φ : h ∞ α → h ∞ β is compact if and only if lim |a|→1− ∥ ∥ ∥wu,φ ( σ(α) a )∥ ∥ ∥ h∞ β = 0. (2.7) proof. the result follows arguing as in the proof of theorem 2.1. indeed, if (2.6) is true, then for any s ∈ d, we have (1 − |s|2)β ( 1 − |φ(s)| 2 )α |u(s)| = (1 − |s| 2)β (1 + |φ(s)|) ∣ ∣ ∣σ (α) φ(s) (φ(s)) ∣ ∣ ∣ |u(s)| ≤ 2 sup a∈d ∥ ∥ ∥wu,φ ( σ(α) a )∥ ∥ ∥ h∞ β and the continuity of wu,φ : h ∞ α → h∞ β follows from the continuity’s theorem of montes-rodŕıguez [16] (see also contreras and hernández-dı́az [9]). conversely, if wu,φ : h ∞ α → h∞ β is continuous, then we have sup a∈d ∥ ∥ ∥wu,φ ( σ(α) a )∥ ∥ ∥ h∞ β ≤ ∥wu,φ∥ sup a∈d ∥ ∥ ∥σ(α)a ∥ ∥ ∥ h∞ α = ∥wu,φ∥ sup a∈d sup z∈d ( 1 − |z|2 )α 1 − |a| |1 − az| α+1 ≤ 2α ∥wu,φ∥ . this shows the equivalence in the item 1. now, since σ (α) a converges to zero uniformly on compact subsets of d as |a| → 1 − and∥ ∥ ∥σ (α) a ∥ ∥ ∥ h∞ α ≤ 2α for all a ∈ d, tjani’s lemma in [21] implies that if wu,φ : h ∞ α → h∞ β is compact, then lim |a|→1− ∥ ∥ ∥wu,φ ( σ(α) a )∥ ∥ ∥ h∞ β = 0. cubo 19, 1 (2017) maximal functions and properties of the weighted composition . . . 45 conversely, if the relation (2.7) is true, then for every ε > 0 we can find r1 ∈ (1/2, 1) such that∥ ∥ ∥wu,φ ( σ (α) a )∥ ∥ ∥ h∞ β < ε whenever r1 < |a| < 1. hence, if z ∈ d satisfies |φ(z)| > r1, then we can write (1 − |z|2)β ( 1 − |φ(z)| 2 )α |u(z)| = 2(1 − |z| 2)β ∣ ∣ ∣σ (α) φ(z) (φ(z)) ∣ ∣ ∣ |u(z)| ≤ 2 sup w∈d (1 − |w|2)β ∣ ∣ ∣σ (α) φ(z) (φ(w)) ∣ ∣ ∣ |u(w)| = ∥ ∥ ∥wu,φ ( σ (α) φ(z) )∥ ∥ ∥ h∞ β < ε. this shows that lim |φ(z)|→1− (1 − |z|2)β ( 1 − |φ(z)| 2 )α |u(z)| = 0 and wu,φ : h ∞ α → h ∞ β is compact in virtue of a result due to montes-rodŕıguez [16] (see also contreras and hernández-dı́az [9]). ! as a consequence of our results, we obtain a recent result about the continuity and compactness of composition operators acting between α-bloch spaces due to malavé and ramos-fernández [15]. for a ∈ d we set λ(α) a (z) = αa ∫ z 0 σ(α) a (s)ds = (1 − |a|) [ 1 (1 − az)α − 1 ] . (2.8) then we have the following result: corolary 1. let α,β > 0 be fixed and φ : d → d an holomorphic function. then (1) the operator cφ from b α to bβ is bounded if and only if sup a∈d ∥ ∥ ∥cφ ( λ(α) a )∥ ∥ ∥ bβ < ∞. (2) the operator cφ : b α → bβ is compact if and only if lim |a|→1− ∥ ∥ ∥cφ ( λ(α) a )∥ ∥ ∥ bβ = 0. proof. this result follows from the fact that cφ : b α → bβ is continuous (resp. compact) if and only if wφ′,φ : h ∞ α → h∞ β is continuous (resp. compact). ! 3 application. a new criterion for the continuity and compactness of weighted composition operators acting between α-bloch spaces now, we are going to use the functions σ (α) a , λ (α) a and ga defined in (2.5), (2.8) and (2.2) to characterize the continuity and compactness of the weighted composition operator wu,φ acting between 46 g. antón marval, r. castillo, j. ramos-fernández cubo 19, 1 (2017) α-bloch spaces. according to the continuity’s results in [17, theorem 2.1] and [16, theorem 2.1] (see also [9, proposition 3.1]) we arrive to the following result: theorem 3.1. let α,β > 0 fixed, φ : d → d and u : d → c holomorphic functions. then (1) if 0 < α < 1, then the operator wu,φ from b α into bβ is continuous if and only if u ∈ bβ and sup a∈d ∥ ∥ ∥iucφ ( λ(α) a )∥ ∥ ∥ bβ < ∞. (2) if α = 1, the operator wu,φ from b into b β is continuous if and only if sup a∈d ∥jucφ (ga)∥bβ < ∞ and sup a∈d ∥ ∥ ∥iucφ ( λ(1) a )∥ ∥ ∥ bβ < ∞. (3) if α > 1, then the operator wu,φ from b α into bβ is continuous if and only if sup a∈d ∥ ∥ ∥jucφ ( σ(α−1) a )∥ ∥ ∥ bβ < ∞ and sup a∈d ∥ ∥ ∥iucφ ( λ(α) a )∥ ∥ ∥ bβ < ∞. where for u ∈ h(d) fixed and f ∈ h(d), we define the operators iu(f) and ju(f) by iu(f)(z) = ∫ z 0 f′(w)u(w)dw and ju(f)(z) = ∫ z 0 f(w)u′(w)dw, with z ∈ d. example 1. if u ≡ 1, then ju is the null operator and we obtain that cφ : b α → bβ is continuous if and only if sup a∈d ∥ ∥ ∥cφ ( λ(α)a )∥ ∥ ∥ bβ < ∞. again, we have extended a result in [15]. the same argument in the paragraph before theorem 3.1 allow us to show a result about the compactness of the weighted composition operators wu,φ acting between α-bloch spaces. in fact, by [17, theorem 3.1], [16, theorem 2.1] (or [9, corollary 4.3]) and the item 2 in our theorem 2.1 and theorem 2.2, imply the following result: theorem 3.2. let α,β > 0 be fixed, φ : d → d and u : d → c holomorphic functions. then (1) if 0 < α < 1, then wu,φ : b α → bβ is compact if and only if u ∈ bβ and lim |a|→1− ∥ ∥ ∥iucφ ( λ(α) a )∥ ∥ ∥ bβ = 0. (2) if α = 1, then wu,ϕ : b → b β is compact if and only if lim |a|→1− ∥jucφ (ga)∥bβ = 0 and lim |a|→1−i ∥ ∥ ∥iucφ ( λ(1) a )∥ ∥ ∥ bβ = 0. cubo 19, 1 (2017) maximal functions and properties of the weighted composition . . . 47 (3) if α > 1, then wu,φ : b α → bβ is compact if and only if lim |a|→1− ∥ ∥ ∥jucφ ( σ(α−1) a )∥ ∥ ∥ bβ = 0, and lim |a|→1− ∥ ∥ ∥iucφ ( λ(α) a )∥ ∥ ∥ bβ = 0. example 2. if u ≡ 1, then we have that the composition operator cφ : b α → bβ is compact if and only if lim |a|→1− ∥ ∥ ∥cφ ( λ(α)a )∥ ∥ ∥ bβ = 0. this extend a result in [15]. 4 application. continuity and compactness of the composition operators acting between α-zygmund spaces for α > 0 fixed, the α-zygmund space, denoted by zα, consist of all holomorphic functions f on d such that ∥f∥ !zα := sup z∈d ( 1 − |z|2 )α |f′′(z)| < ∞. clearly, f ∈ zα if and only if f ′ ∈ bα. also, it is easy to see that zα is a banach space with the norm ∥f∥zα = |f(0)| + |f ′(0)| + ∥f∥ !zα = |f(0)| + ∥f ′∥bα . we will use our results in the above sections to characterize the continuity and the compactness of the composition operator cφ acting between α-zygmund spaces in terms of the composition of certain special functions in zα. the key of our result repose in the following lemma: lemma 4.1. let α,β > 0 be fixed and φ : d → d an holomorphic function. then (1) the operator cφ from zα to zβ is bounded if and only if wφ′,φ : b α → bβ is bounded. (2) the operator cφ : zα → zβ is compact if and only if wφ′,φ : b α → bβ is compact. proof. suppose first that the operator cφ : zα → zβ is bounded. there exists a constant l > 0 such that ∥cφ (g)∥zβ ≤ l ∥g∥zα for all g ∈ zα. thus, for any f ∈ b α , we have that g(z) = ∫z 0 f(s)ds, belongs to zα and therefore, ∥wφ′,φ (f)∥bβ = ∥φ ′ · f ◦ φ∥bβ = ∥φ ′ · g′ ◦ φ∥bβ = ∥ ∥(g ◦ φ) ′ ∥ ∥ bβ ≤ |(g ◦ φ) (0)| + ∥ ∥(g ◦ φ) ′ ∥ ∥ bβ = ∥cφ(g)∥zβ ≤ l ∥g∥zα = l ∥f∥bα since g(0) = 0. this shows that wφ′,φ : b α → bβ is bounded. 48 g. antón marval, r. castillo, j. ramos-fernández cubo 19, 1 (2017) conversely, if wφ′,φ : b α → bβ is bounded, then there exists a constant l > 0 such that ∥wφ′,φ (g)∥bβ ≤ l ∥g∥bα for all g ∈ b α . thus, for any f ∈ zα, we have that g = f ′ ∈ bα and hence ∥ ∥(f ◦ φ) ′ ∥ ∥ bβ = ∥wφ′,φ (g)∥bβ ≤ l ∥f ′∥bα . furthermore, since f ∈ h(d), we can write |f (φ(0))| ≤ |f(0)| + |f′(0)| + ∫ φ(0) 0 ∫ s 0 |f′′(w)| |dw| |ds|. therefore, multiplying and dividing by ( 1 − |w|2 )α inside the integral symbol, we can find a constant k > 0, which can depend on φ(0), such that |f (φ(0))| ≤ k ∥f∥zα. we conclude then that ∥cφ (f)∥zβ = |f (φ(0))| + ∥ ∥(f ◦ φ) ′ ∥ ∥ bβ ≤ (k + l) ∥f∥zα and the operator cφ : zα → zβ is bounded. this shows the item (1). now we are going to show the item (2). suppose first that wφ′,φ : b α → bβ is a compact operator and let {gn} be a bounded sequence in zα, then the sequence {g ′ n} is a bounded sequence in bα. hence, by passing to a subsequence, we can suppose that {wφ′,φ (g ′ n)} converges in b β to a function h ∈ bβ. also, {gn (φ(0))} is a bounded sequence in c and, by passing to a subsequence, we can suppose that {gn (φ(0))} goes to z0 ∈ c as n → ∞. then, we can define the function g(z) = z0 + ∫z 0 h(s)ds, which belongs to zβ and we have that ∥cφ (gn) − g∥zβ = |gn (φ(0)) − g(0)| + ∥wφ ′,φ (g ′ n ) − h∥ bβ → 0 as n → ∞. that is, cφ : zα → zβ is a compact operator. conversely, if cφ : zα → zβ is a compact operator and {fn} is a bounded sequence in b α , then the sequence {gn} defined by gn(z) = ∫ z 0 fn(s)ds is bounded in zα and hence, by passing to a subsequence, we can suppose that {cφ (gn)} converges in zβ to a function h ∈ zβ. therefore, we can write ∥wφ′,φ (fn) − h ′∥bβ = ∥wφ′,φ (g ′ n) − h ′∥bβ = ∥ ∥(cφ (gn) − h) ′ ∥ ∥ bβ ≤ ∥cφ (gn) − h∥zβ → 0 as n → ∞ and the result follows since h′ ∈ bβ. ! as consequence of above lemma, [17, theorem 2.1], [16, theorem 2.1] (or [9, proposition 3.1]) and our theorem 2.1, theorem 2.2 and theorem 3.1, we have: cubo 19, 1 (2017) maximal functions and properties of the weighted composition . . . 49 corolary 2. let α,β > 0 be fixed and φ : d → d an holomorphic function. then, for each case, the following propositions are equivalents: (1) case 0 < α < 1. (a) the operator cφ from zα to zβ is continuous, (b) φ ∈ zβ and w(φ′)2,φ : h ∞ α → h ∞ β is continuous, (c) φ ∈ zβ and sup w∈d (1 − |w|2)β (1 − |φ(w)| 2 )α |φ′(w)| 2 < ∞. (d) φ ∈ zβ and sup a∈d ∥ ∥ ∥w(φ′)2,φ ( σ(α)a )∥ ∥ ∥ h∞ β < ∞. (e) φ ∈ zβ sup a∈d ∥ ∥ ∥iφ′cφ ( λ(α) a )∥ ∥ ∥ bβ < ∞. (2) case α = 1. (a) the operator cφ from zα to zβ is continuous, (b) wφ′′,φ : h ∞ vlog → h∞ β and w(φ′)2,φ : h ∞ 1 → h∞ β are continuous, (c) sup w∈d (1 − |w|2)β vlog (φ(w)) |φ′′(w)| < ∞, and sup w∈d (1 − |w|2)β 1 − |φ(w)| 2 |φ′(w)| 2 < ∞. (d) sup a∈d ∥wφ′′,φ (ga)∥h∞ β < ∞ and sup a∈d ∥ ∥ ∥w(φ′)2,φ ( σ(1) a )∥ ∥ ∥ h∞ β < ∞ (e) sup a∈d ∥jφ′cφ (ga)∥bβ < ∞ and sup a∈d ∥ ∥ ∥iφ′cφ ( λ(1) a )∥ ∥ ∥ bβ < ∞. (3) case α > 1. (a) the operator cφ from zα to zβ is continuous, (b) wφ′′,φ : h ∞ α−1 → h∞ β and w(φ′)2,φ : h ∞ α → h∞ β are continuous, (c) sup w∈d (1 − |w|2)β (1 − |φ(w)| 2 )α−1 |φ′′(w)| < ∞, and sup w∈d (1 − |w|2)β (1 − |φ(w)| 2 )α |φ′(w)| 2 < ∞. (d) sup a∈d ∥ ∥ ∥wφ′′,φ ( σ(α−1) a )∥ ∥ ∥ h∞ β < ∞ and sup a∈d ∥ ∥ ∥w(φ′)2,φ ( σ(α) a )∥ ∥ ∥ h∞ β < ∞ (e) sup a∈d ∥ ∥ ∥jφ′cφ ( σ(α−1) a )∥ ∥ ∥ bβ < ∞ and sup a∈d ∥ ∥ ∥iφ′cφ ( λ(α) a )∥ ∥ ∥ bβ < ∞. similar results can be found for the compactness of the operator cφ from zα to zβ. also, it is possible to write a characterizations using the zygmund’s norm and the fact that ∥f∥zα = |f(0)| + ∥f′∥bα. here, we can use the operators ĩuf(z) = ∫z 0 iu(f(s))ds and j̃uf(z) = ∫z 0 ju(f(s))ds. 50 g. antón marval, r. castillo, j. ramos-fernández cubo 19, 1 (2017) acknowledgement. the authors would like to thank the referees for their valuable comments which helped to improve the manuscript. references [1] j. m. anderson and j. duncan, duals of banach spaces of entire functions. glasgow math. j. 32 (1990), no. 2, 215–220. [2] k. d. bierstedt, j. bonet and j. taskinen, associated weights and spaces of holomorphic functions. studia math. 127 (1998), 137–168. 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() cubo a mathematical journal vol.17, no¯ 03, (43–51). october 2015 (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity in bitopological spaces carlos carpintero & ennis rosas department of mathematics, universidad de oriente, nucleo de sucre cumana, venezuela. facultad de ciencias basicas, universidad del atlantico, barranquilla, colombia. carpintero.carlos@gmail.com, ennisrafael@gmail.com sabir hussain department of mathematics, college of science, qassim university, p.o.box 6644, buraydah 51482, saudi arabia. sabiriub@yahoo.com, sh.hussain@qu.edu.sa abstract the aim of this paper is to introduce and characterize the notions of (i, j)-ω-semiopen sets as a generalization of (i, j)-semiopen sets in bitopological spaces. we also define and discuss the properties of (i, j)-ω-semicontinuous functions. resumen el objetivo de este art́ıculo es introducir y caracterizar las nociones de conjuntos (i, j)ω-semiabiertos como una generalización de conjuntos (i, j)-semiabiertos en espacios bitopológicos. también definimos y discutimos las propiedades de funciones (i, j)-ωsemicontinuas. keywords and phrases: bitopological spaces, (i, j)-ω-semiopen sets, (i, j)-ω-semiclosed sets. 2010 ams mathematics subject classification: 54a05,54c05,54c08. 44 carlos carpintero, sabir hussain & ennis rosas cubo 17, 3 (2015) 1 introduction and preliminaries the concept of a bitopological space was introduced by kelly [3]. on the other hand, s. bose [1], introduced the concept of (i, j)-semiopen sets in bitopological spaces. recently, as generalization of closed sets, the notion of ω-closed sets was introduced and studied by hdeib [2]. a point x ∈ x is called a condensation point of a, if for each u ∈ τ with x ∈ u, the set u ∩ a is uncountable. a is said to be ω-closed [2], if it contains all of its condensation points. the complement of a ω-closed set is said to be ω-open. it is well known that a subset w of a space (x, τ) is ω-open if and only if for each x ∈ w, there exists u ∈ τ such that x ∈ u and u\w is countable. in this paper, we introduce the concept of (i, j)-ω-semiopen sets as a generalization of (i, j)-semiopen sets in bitopological spaces. we also define and discuss the properties of (i, j)-ω-semicontinuous functions. for a subset a of x, the closure of a and the interior of a are denoted by cl(a) and int(a), respectively. a subset a of a bitopological space (x, τ1, τ2) is said to be (i, j)-semi open, if a ⊆ τi-cl(τj-int(a)), where i 6= j, i, j = 1, 2. the complement of a (i, j)-semiopen set is said to be a (i, j)-semiclosed. the (i, j)-semiclosure of a, denoted by (i, j)-scl(a) is defined by the intersection of all (i, j)-semiclosed sets containing a. the (i, j)-semi interior of a, denoted by (i, j)-sint(a) is defined by the union of all (i, j)-semiopen sets contained in a. the family of all (i, j)-semiopen (respectively (i, j)-semiclosed) subsets of a space (x, τ1, τ2) is denoted by (i, j) − so(x), (respectively (i, j) − sc(x)). a function f : (x, τ1, τ2) 7→ (y, σ1, σ2) is said to be (i, j)-semi continuous, if the inverse image of every σi-open set in (y, σ1, σ2) is (i, j)-semi open in (x, τ1, τ2), where i 6= j, i, j = 1, 2. a σi-open set u in (y, σ1, σ2) means u ∈ σi. 2 (i, j)-ω-semiopen sets a set x equipped with two topologies is called a bitopological space. throughout this paper, spaces (x, τ1, τ2) (or simply x) always means a bitopological spaces on which no separation axioms are assumed unless explicitly stated. definition 2.1. let x be a bitopological space and a ⊆ x. then a is said to be (i, j)-ω-semiopen, if for each x ∈ a there exists a (i, j)-semiopen ux containing x such that ux − a is a countable set. the complement of a (i, j)-ω-semiopen set is a (i, j)-ω-semiclosed set. the family of all (i, j)-ω-semiopen (respectively (i, j)-ω-semiclosed) subsets of a space (x, τ1, τ2) is denoted by (i, j) − ω − so(x), (respectively (i, j) − ω − sc(x)). also the family of all (i, j) − ωsemiopen sets of (x, τ1, τ2) containing x is denoted by (i, j) − ω − so(x, x). note that every (i, j)-semiopen set is a (i, j)-ω-semiopen. the following example shows that the converse is not true in general. example 2.2. let x = {a, b, c}, τ1 = {∅, {a, b}, x}, τ2 = {∅, {b, c}, x}. then {a, c} is a (i, j)-ωsemiopen but not (i, j)-semiopen. cubo 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 45 theorem 2.3. let x be a bitopological space and a ⊆ x. then a is said to be (i, j)-ω-semiopen if and only if for every x ∈ a, there exists a (i, j)-semiopen set ux containing x and a countable subset c such that ux − c ⊆ a. proof. let a be a (i, j)-ω-semiopen set and x ∈ a, then there exists a (i, j)-semiopen subset ux containing x such that ux − a is countable. let c = ux − a = ux ∩ (x − a). then ux − c ⊆ a. conversely, let x ∈ a. then there exists a (i, j)-ω-semiopen subset ux containing x and a countable subset c such that ux − c ⊆ a. thus ux − a ⊆ c and ux − a is countable. theorem 2.4. let x be a bitopological space and c ⊆ x. if c is a (i, j)-ω-semiclosed set, then c ⊆ k ∪ b, for some (i, j)-ω-semiclosed subset k and a countable subset b. proof. if c is a (i, j)-ω-semiclosed set, then x−c is a (i, j)-ω-semiopen set and hence by theorem 2.3, for every x ∈ x − c, there exists a (i, j)-semiopen set u containing x and a countable set b such that u − b ⊆ x − c. thus c ⊆ x − (u − b) = x − (u ∩ (x − b)) = (x − u) ∪ b, let k = x − u. then k is a (i, j)-semiclosed set such that c ⊆ k ∪ b. theorem 2.5. the union of any family of (i, j) − ω-semiopen sets is (i, j)-ω-semiopen set. proof. if {aα : α ∈ i} is a collection of (i, j)-ω-semiopen subsets of x, then for every x ∈ ⋃ α∈i aα, x ∈ aα, for some α ∈ i. hence, there exists a (i, j)-ω-semiopen subset u containing x, such that u − aα is countable. now as u − ( ⋃ α∈i aα) ⊆ u − aα, and thus u − ( ⋃ α∈i aα) is countable. therefore ⋃ α∈i aα is a (i, j)-ω-semiopen set. definition 2.6. the union of all (i, j)-ω-semiopen sets contained in a ⊆ x is called the (i, j)-ωsemi-interior of a and is denoted by (i, j) − ω-sint(a). the intersection of all (i, j)-ω-semiclosed sets of x containing a is called the (i, j)-ω-semiclosure of a and is denoted by (i, j)-ω − scl(a). remark 2.7. the (i, j)-ω-scl(a) is a (i, j)-ω-semiclosed set and the (i, j)-ω-sint(a) is a (i, j)ω-semiopen set. theorem 2.8. let x be a bitopological space and a, b ⊆ x. then the following properties hold: (1) (i, j)-ω-sint((i, j)-ω-sint(a)) = (i, j)-ω-sint(a). (2) if a ⊂ b, then (i, j)-ω-sint(a) ⊂ (i, j)-ω-sint(b). (3) (i, j)-ω-sint(a ∩ b) ⊂ (i, j)-ω-sint(a) ∩ (i, j) − ω − sint(b). (4) (i, j)-ω-sint(a) ∪ (i, j)-ω-sint(b) ⊂ (i, j)-ω-sint(a ∪ b). (5) (i, j)-ω-sint(a) is the largest (i, j)-ω-semiopen subset of x contained in a. (6) a is (i, j)-ω-semiopen if and only if a = (i, j)-ω-sint(a). (7) (i, j)-ω-scl((i, j)-ω-scl(a)) = (i, j)-ω-scl(a). 46 carlos carpintero, sabir hussain & ennis rosas cubo 17, 3 (2015) (8) if a ⊂ b, then (i, j)-ω-scl(a) ⊂ (i, j)-ω-scl(b). (9) (i, j)-ω-scl(a) ∪ (i, j)-ω-scl(b) ⊂ (i, j)-ω-scl(a ∪ b). (10) (i, j)-ω-scl(a ∩ b) ⊂ (i, j)-ω-scl(a) ∩ (i, j)-ω-scl(b). proof. (1), (2), (6), (7) and (8) follow directly from the definition of (i, j)-ω-semiopen and (i, j)ω-semiclosed sets. (3), (4) and (5) follow from (2). (9) and (10) follow by applying (8). example 2.9. let x be the real line, τ1 = {∅, re, q c} and τ2 = {∅, re, q, q c}. take a = (0, 1), b = (1, 2), then (i, j)-ω-scl(a ∩ b) ⊂ (i, j)-ω-scl(a) ∩ (i, j)-ω-scl(b). example 2.10. let x be the real line, τ1 = {∅, re, q} and τ2 = {∅, re, q}. the collection of (i, j) − so(x) is {∅, re, q}. take a = q, b = {π}. then (i, j)-ω-scl(a) = q, (i, j)-ω-scl(b) = {π} and (i, j)-ω-scl(a) ∪ (i, j)-ω-scl(b) ⊂ (i, j)-ω-scl(a ∪ b). theorem 2.11. let x be a bitopological space. suppose a ⊆ x and x ∈ x. then x ∈ (i, j)-ωscl(a) if and only if u ∩ a 6= ∅ for every u ∈ (i, j)-ω-so(x, x). proof. suppose that x ∈ (i, j)-ω-scl(a) and we show that u ∩ a 6= ∅, for all u ∈ (i, j)-ωso(x, x). suppose on the contrary that there exists u ∈ (i, j)-ω-so(x, x) such that u ∩ a = ∅, then a ⊆ x − u and x − u is a (i, j)-ω-semiclosed set. this follows that (i, j)-ω-scl(a) ⊆ (i, j)ω-scl(x − u) = x − u. since x ∈ (i, j)-ω-scl(a), we have x ∈ x − u and hence x /∈ u. which contradicts the fact that x ∈ u. therefore, u ∩ a 6= ∅. conversely, suppose that u ∩ a 6= ∅ for every u ∈ (i, j)-ω-so(x, x). we shall prove that x ∈ (i, j)-ω-scl(a). suppose on the contrary that x /∈ (i, j)-ω-scl(a). let u = x − (i, j)-ω-scl(a), then u ∈ (i, j)-ω-so(x, x) and u ∩ a = (x − ((i, j)-ω-scl(a))) ∩ a ⊆ (x − a) ∩ a = ∅. this is a contradiction to the fact that u ∩ a 6= ∅. hence x ∈ (i, j)-ω-scl(a). theorem 2.12. let x be a bitopological space and a ⊂ x. then the following properties hold: (1) (i, j)-ω-scl(x\a) = x\(i, j)-ω-sint(a); (2) (i, j)-ω-sint(x\a) = x\(i, j)-ω-scl(a). proof. (1). let x ∈ x\(i, j)-ω-scl(a). then there exists v ∈ (i, j)-ω-so(x, x) such that v ∩ a = ∅ and hence we obtain x ∈ (i, j)-ω-sint(x\a). this shows that x\(i, j)-ω-scl(a) ⊂ (i, j)-ωsint(x\a). now consider x ∈ (i, j)-ω-sint(x\a). since (i, j)-ω-sint(x\a) ∩ a = ∅, we obtain x /∈ (i, j)-ω-scl(a). therefore, we have, (i, j)-ω-scl(x\a) = x\(i, j)-ω-sint(a). (2). let x ∈ x\(i, j)-ω-sint(x−a). since (i, j)-ω-sint(x\a)∩a = ∅, we have x /∈ (i, j)-ω-scl(a) implies x ∈ x\(i, j)-ω-scl(a). now consider x ∈ x\(i, j)-ω-scl(a), then there exists v ∈ (i, j)-ωso(x, x) such that v ∩ a = ∅, hence we obtain that (i, j)-ω-sint(x\a) = x\(i, j)-ω-scl(a). definition 2.13. let x be a bitopological space and b ⊆ x. then b is a (i, j)-ω-semineighbourhood of a point x ∈ x if there exists a (i, j)-ω-semiopen set w such that x ∈ w ⊂ b. cubo 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 47 theorem 2.14. let x be a bitopological space and b ⊆ x. b is a (i, j)-ω-semiopen set if and only if it is a (i, j)-ω-semineighbourhood of each of its points. proof. let b be a (i, j)-ω-semiopen set of x. then by definition b is a (i, j)-ω-semineighbourhood of each of its points. conversely, suppose that b is a (i, j)-ω-semineighbourhood of each of its points. then for each x ∈ b, there exists sx ∈ (i, j)-ω-so(x, x) such that sx ⊂ b. then b = ⋃ {sx : x ∈ b}. since each sx is a (i, j)-ω-semiopen and arbitrary union of (i, j)-ω-semiopen sets is (i, j)-ω-semiopen, b is a (i, j)-ω-semiopen in x. theorem 2.15. if each nonempty (i, j)-ω-semiopen set of a bitopological space x is uncountable, then (i, j)-scl(a) = (i, j)-ω-scl(a), for each subset a ∈ τ1 ∩ τ2. proof. clearly (i, j)-ω-scl(a) ⊆ (i, j)-scl(a). on the other hand, let x ∈ (i, j)-scl(a) and b be a (i, j)-ω-semiopen set containing x. using theorem 2.3, there exists a (i, j)-semiopen set v containing x and a countable set c such that v − c ⊆ b. follows (v − c) ∩ a ⊆ b ∩ a and so (v ∩ a) − c ⊆ b ∩ a. now x ∈ v, x ∈ (i, j)-scl(a) such that v ∩ a 6= ∅ where v ∩ a is a (i, j)-ω-semiopen set, since v is a (i, j)-semiopen set and a ∈ τ1 ∩ τ2. using the hypothesis, each nonempty (i, j)-ω-semiopen set of x is uncountable and so is (v ∩a)\c. thus b∩a is uncountable. therefore, b ∩ a 6= ∅ implies that x ∈ (i, j)-ω-scl(a). theorem 2.16. let x be a bitopological space. if every (i, j)-ω-semiopen subset of x is τi-open in x. then (x, (i, j)-ω-so(x)) is a topological space. proof. 1. ∅, x belong to (i, j)-ω-so(x) 2. let u, v ∈ (i, j)-ω-so(x) and x ∈ u ∩ v. then there exists (i, j)-semi open sets g, h in x containing x such that g\u and h\v are countable. since (g ∩ h)\(u ∩ v) = (g ∩ h) ∩ ((x\u) ∪ (x\v)) ⊆ (g ∩ (x\u)) ∪ (h ∩ (x\v)) implies that (g ∩ h)\(u ∩ v) is a countable set and by hypothesis, the intersection of two (i, j)-semi open set is (i, j)-semi open. hence u ∩ v ∈ (i, j)-ωso(x)). 3. the union follows directly. 3 (i, j)-ω-semicontinuous functions definition 3.1. a function f : (x, τ1, τ2) → (y, σ1, σ2) is said to be a (i, j)-ω-semicontinuous, if the inverse image of every σi-open set of y is (i, j)-ω-semiopen in (x, τ1, τ2), where i 6= j, i, j=1, 2. definition 3.2. a function f : (x, τ1, τ2) → (y, σ1, σ2) is said to be a (i, j)-semicontinuous, if the inverse image of every σi-open set of y is (i, j)-semiopen in (x, τ1, τ2), where i 6= j, i, j=1, 2. theorem 3.3. every (i, j)-semicontinuous function is (i, j)-ω-semicontinuous. 48 carlos carpintero, sabir hussain & ennis rosas cubo 17, 3 (2015) proof. the proof follows from the the fact that every (i, j)-semiopen set is (i, j)-ω-semiopen. however, the converse may be false. example 3.4. let x = {a, b, c}, τ1 = {∅, {a}, {b}, {a, b}, x}, τ2 = {∅, {a}, x}, σ1 = {∅, {a, b}, x}, σ2 = {∅, {a, c}, x}. then the identity function f : (x, τ1, τ2) → (x, σ1, σ2) is (i, j)-ω-semicontinuous but not (i, j)-semicontinuous. theorem 3.5. for a function f : (x, τ1, τ2) → (y, σ1, σ2), the following statements are equivalent: (1) f is (i, j)-ω-semicontinuous; (2) for each point x ∈ x and each σi-open set f in y such that f(x) ∈ f, there is a (i, j)-ωsemiopen set a in x such that x ∈ a, and f(a) ⊂ f; (3) the inverse image of each σi-closed set in y is a (i, j)-ω-semiclosed in x; (4) for a ⊆ x, f((i, j)-ω-scl(a)) ⊂ σi-cl(f(a)); (5) for b ⊆ y, (i, j)-ω-scl(f−1(b)) ⊂ f−1(σi-cl(b)); (6) for c ⊆ y, f−1(σi-int(c)) ⊂ (i, j)-ω-sint(f −1(c)). proof. (1)⇒(2): let x ∈ x and f be a σi-open set of y containing f(x). by (1), f −1(f) is (i, j)ω-semiopen in x. let a = f−1(f). then x ∈ a and f(a) ⊂ f. (2)⇒(1): let f be σi-open in y and let x ∈ f −1(f). then f(x) ∈ f. by (2), there is a (i, j)-ωsemiopen set ux in x such that x ∈ ux and f(ux) ⊂ f implies x ∈ ux ⊂ f −1(f). hence f−1(f) is a (i, j)-ω-semiopen in x. (1)⇔(3): this follows from the fact that for any subset b of y, f−1(y\b) = x\f−1(b). (3)⇒(4): let a ⊆ x. since a ⊂ f−1(f(a)), we have a ⊂ f−1(σi-cl(f(a))). by hypothesis f −1(σicl(f(a))) is a (i, j)-ω-semiclosed set in x and hence (i, j)-ω-scl(a)) ⊂ f−1(σi-cl(f(a))). follows f((i, j)-ω-scl(a))) ⊂ f(f−1(σi-cl(f(a))) ⊆ σi-cl(f(a)). (4)⇒(3): let f be any σi-closed subset of y. then f((i, j)-ω-scl(f −1(f)) ⊂ σi-cl(f(f −1(f))) ⊂ σicl(f) = f. therefore, the (i, j)-ω-scl(f−1(f)) ⊂ f−1(f). consequently, f−1(f) is a (i, j)-ωsemiclosed set in x. (4)⇒(5): let b ⊆ y. now, f((i, j)-ω-scl(f−1(b))) ⊂ σi-cl(f(f −1(b))) ⊂ σi-cl(b). consequently, (i, j)-ω-scl(f−1(b)) ⊂ f−1(σi-cl(b)). (5)⇒(4): let b = f(a) where a ⊆ x. then, (i, j)-ω-scl(a) ⊂ (i, j)-ω-scl(f−1(b)) ⊂ f−1(σicl(b)) = f−1(σi-cl(f(a))), and hence f((i, j)-ω-scl(a)) ⊂ σi-cl(f(a)). (1)⇒(6): let b ⊆ y. clearly, f−1(σi-int(b)) is a (i, j)-ω-semiopen and we have f −1(σi-int(b)) ⊂ (i, j)-ω-sint(f−1σi-int(b)) ⊂ (i, j)-ω-sint(f −1b). (6)⇒(1): let b be a σi-open set in y. then σi-int(b) = b and f −1(b) ⊂ f−1(σi-int(b)) ⊂ (i, j)ω-sint(f−1(b)). hence, we have f−1(b) = (i, j)-ω-sint(f−1(b)). this implies that f−1(b) is a (i, j)-ω-semiopen in x. cubo 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 49 definition 3.6. the graph g(f) of f : (x, τ1, τ2) → (y, σ1, σ2) is said to be (i, j)-ω-semiclosed in x × y, if for each (x, y) ∈ (x× y) \ g(f), there exists u ∈ (i, j)-ω-so(x, x), i, j = {1, 2} with i 6= j and a σi-open set v of y containing y such that (u × v) ∩ g(f) = ∅. lemma 3.7. the graph g(f) of f : (x, τ1, τ2) → (y, σ1, σ2) is (i, j)-ω-semiclosed in x × y if and only if for each (x, y) ∈ (x × y) \ g(f), there exists u ∈ (i, j)-ω-so(x, x), i, j = {1, 2} with i 6= j and a σi-open set v of y containing y such that f(u) ∩ v = ∅. proof. the proof is an immediate consequence of definition 3.6. theorem 3.8. if a function f : (x, τ1, τ2) → (y, σ1, σ2) is a (i, j)-ω-semicontinuous function and (y, σi) is t1 i = {1, 2}, then g(f) is (i, j)-ω-semiclosed. proof. let (x, y) ∈ (x × y) \ g(f). then y 6= f(x). since (y, σi) is t1, there exist a σi-open set v and w of y such that f(x) ∈ v and y /∈ w and v ∩ w = ∅. since f is (i, j)-ω-semicontinuous, there exists u ∈ (i, j)-ω-so(x, x) such that f(u) ⊂ v. therefore, f(u) ∩w = ∅. therefore, by lemma 3.7, g(f) is (i, j)-ω-semiclosed. definition 3.9. a bitopological space x is said to be a (i, j)-ω-semi-t2 space, if for each pair of distinct points x, y ∈ x, there exist u, v ∈ (i, j)-ω-so(x) containing x and y, respectively, such that u ∩ v = ∅. theorem 3.10. if f : (x, τ1, τ2) → (y, σ1, σ2) is a (i, j)-ω-semicontinuous injective function and (y, σi) is a t2 space, then (x, τ1, τ2) is a ω-semi-t2 space. proof. the proof follows from the definition. theorem 3.11. if f : (x, τ1, τ2) → (y, σ1, σ2) is an injective (i, j)-ω-semicontinuous function with a (i, j)-ω-semiclosed graph, then x is a (i, j)-ω-semi-t2 space. proof. let x1 and x2 be any pair of distinct points of x. then f(x1) 6= f(x2), so (x1, f(x2)) ∈ (x × y)\g(f). since the graph g(f) is (i, j)-ω-semiclosed, there exist a (i, j)-ω-semiopen set u containing x1 and v ∈ σi containing f(x2) such that f(u)∩v = ∅. since f is (i, j)-ω-semicontinuous, f−1(v) is a (i, j)-ω-semiopen set containing x2 such that u∩f −1(v) = ∅. hence x is (i, j)-ω-semit2. definition 3.12. a collection {uα : α ∈ i} of (i, j)-semiopen sets in a bitopological space x is called a (i, j)-semiopen cover of a subset a of x, if a ⊆ ⋃ α∈i uα. definition 3.13. a bitopological space x is said to be (i, j)-semi lindeloff, if every (i, j)-semi open cover of x has a countable subcover. a subset a of bitopological space x is said to be (i, j)-semi lindeloff relative to x, if every cover of a by (i, j)-semiopen sets of x has a countable subcover. theorem 3.14. if x is a bitopological space such that every (i, j)-semiopen subset is (i, j)-semi lindeloff relative to x. then every subset is (i, j)-semi lindeloff relative to x 50 carlos carpintero, sabir hussain & ennis rosas cubo 17, 3 (2015) theorem 3.15. for a bitopological space x. the following properties are equivalent: (1) x is (i, j)-semi lindeloff. (2) every countable cover of x by (i, j)-semiopen sets has a countable subcover. proof. (2)⇒(1): since every (i, j)-semiopen set is (i, j)-ω-semiopen, the proof follows. (1)⇒(2): let {uα : α ∈ i} be any cover of x by (i, j)-ω-semiopen sets of x. for each x ∈ x, there exists an αx ∈ i such that x ∈ uαx. since uαx is a (i, j)-ω-semiopen, then there exists a (i, j)-semiopen set vαx such that x ∈ vαx and vαx −uαx is countable. the family {vα : α ∈ i} is a (i, j)-semiopen cover of x and x is (i, j)-semi lindeloff. therefore there exists a countable subcover αxi with i ∈ n such that x = ⋃ i∈n vαx i . since x = ⋃ i∈n [(vαx i − uαx i ) ∪ uαx i ] = ⋃ i∈n [(vαx i − uαx i ) ⋃ i∈n uαx i ]. since vαx i − uαx i is a countable set, for each α(xi), there exists a countable subset iα(xi) of i such that vαx i − uαx i ⊆ ⋃ iα(x i ) uα and therefore x ⊆ ⋃ i∈n ( ⋃ α∈iα(x i ) uα) ∪ ( ⋃ i∈n uα(xi)). definition 3.16. a bitopological space x is called pairwise lindeloff if each pairwise open cover of x has a countable subcover. theorem 3.17. let f : (x, τ1, τ2) → (y, σ1, σ2) be a (i, j)-ω-semicontinuous function. if x is (i, j)-semi lindeloff, then y is pairwise lindeloff. proof. let {uα : α ∈ i} be any cover of y by σi-open sets. then {f −1(uα) : α ∈ i} is a (i, j)-ωsemiopen cover of x. since x is (i, j)-semi lindeloff, there exists a countable subset i0 of i such that x = ⋃ α∈i0 uα. therefore, y is a pairwise lindeloff. definition 3.18. a function f : (x, τ1, τ2) → (y, σ1, σ2) is said to be: 1 (i, j)-ω-semiopen if f(u) is a (i, j)-ω-semiopen set in y for every τi-open set u of x. 2 (i, j)-ω-semiclosed if f(u) is a (i, j)-ω-semiclosed set in y for every τi-closed set u of x. theorem 3.19. for a function f : (x, τ1, τ2) → (y, σ1, σ2), the following properties are equivalent: (1) f is a (i, j)-ω-semiopen. (2) f(τi − int(u)) ⊆ (i, j)-ω-scl(f(u)), for each subset u of x. (3) τi − int(f −1(v)) ⊆ f−1((i, j)-ω-sint(v), for each subset v of y. proof. (1)⇒(2): let u be any subset of x. then τi − int(u) is a τi-open set of x. then f(τi − int(u)) is a (i, j)-ω-semiopen set of y. since f(τi − int(u)) ⊆ f(u), f(τi − int(u)) = (i, j)ω-sint(f(τi − int(u))) ⊆ (i, j)-ω-sint(f(u)). (2)⇒(3):let v be any subset of y. then f(τi − int(f −1(v))) ⊆ (i, j)-ω-sint(f(f−1(v))). hence τi − int(f −1(v)) ⊆ f−1((i, j)-ω-sint(v)). (3)⇒(1): let u be any τi-open set of x. then τi − int(u) = u. now, v = τi − int(v) ⊆ τi − cubo 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 51 int(f−1(f(v)) ⊆ f−1((i, j)-ω-sint(f(v)))). which implies that f(v) ⊆ f(f−1((i, j)-ω-sint(f(v)))) ⊆ (i, j)-ω-sint(f(v)). hence f(v) is a (i, j)-ω-semiopen set of y. thus f is (i, j)-ω-semiopen. theorem 3.20. let f : (x, τ1, τ2) → (y, σ1, σ2) be a function, then f is a (i, j)-ω-semiclosed function if and only if for each subset v of x, the (i, j)-ω-scl(f(v)) ⊆ f(τi − cl(v))). proof. let f be a (i, j)-ω-semiclosed function and v be any subset of x. then f(v) ⊆ f(τi −cl(v)) and f(τi − cl(v)) is a (i, j)-ω-semiclosed set of y. hence (i, j)-ω-scl(f(v)) ⊆ (i, j)-ω-scl(f(τi − cl(v))) = f(τi − cl(v))). conversely, let v be a τi-closed set of x. then f(v) ⊆ (i, j)-ωscl(f(v)) ⊆ f(τi − cl(v))) = f(v). hence f(v) is a (i, j)-ω-semiclosed set of y. therefore, f is a (i, j)-ω-semiclosed function definition 3.21. a bitopological space x is said to be (i, j)-ω-semiconnected, if x cannot be expressed as the union of two nonempty disjoint (i, j)-ω-semiopen sets. definition 3.22. a bitopological space x is said to be pairwise connected [5], if it cannot be expressed as the union of two nonempty disjoint sets u and v such that u is τi-open and v is τj-open, where i, j = {1, 2} and i 6= j. theorem 3.23. a (i, j)-ω-semicontinuous image of a (i, j)-ω-semiconnected space is pairwise connected. proof. the proof is clear. received: march 2015. accepted: may 2015. references [1] s. bose, semi-open sets, semi-continuity and semi-open mappings in bitopological spaces, bull. calcutta math. soc., 73(1981), 237-246. [2] h. z. hdeib, ω-closed mappings, revista colombiana mat., 16(1982), 65-78. [3] j. c. kelly, bitopological spaces, proc. london math. soc., 13, pp. 71-89, (1963). [4] h. maki, r. chandrasekhara rao and a. nagoor gani, on generalizing semi-open sets and preopen sets, pure appl. math. math. sci, 49 (1999), pp 17-29. [5] w. j. pervin, connectedness in bitopological spaces, ind. math., 29 (1967), 369-372. introduction and preliminaries (i, j)–semiopen sets (i,j)–semicontinuous functions cubo a mathematical journal vol.11, no¯ 02, (7–13). may 2009 n-person games with crossing externalities miklos n. szilagyi department of electrical & computer engineering, university of arizona, tucson, az 85721-0104 email: szilagyi@ece.arizona.edu abstract we report computer simulation experiments based on our agent-based simulation tool to model uniform n -person games with crossing payoff functions for the case when the agents are greedy simpletons who imitate the action of that of their neighbors who received the highest payoff for its previous action. the payoff (reward/penalty) functions are given as two straight lines: one for the cooperators and another for the defectors. the payoff curves are functions of the ratio of cooperators to the total number of agents. even if the payoff functions are linear, four free parameters determine them. in this investigation only crossing payoff functions are considered. we have investigated the behavior of the agents systematically. the results show that the solutions are non-trivial and in some cases quite irregular. they show drastic changes in case of the leader game in the narrow parameter range of 1.72 ≤ p ≤ 1.75. this behavior is similar to that observed by [3] for the n -person chicken game. irregular solutions were also found for the reversed stag hunt game. 8 miklos n. szilagyi cubo 11, 2 (2009) resumen nosotros reportamos experimentos de simulación de ordenadores basados en nuestra herramienta de simulación agent-based para el modelo de juegos n -personas uniforme con funciones “crossing payoff” para el caso cuando los agentes son “greedy simpletons” los cuales imitan la acción de que sus vecinos quienes recibieron alto pago por sus acciones previas. las funciones de pago (recompenza/castigo) son dadas como dos lineas rectas: una para los cooperadores y otra para opositores. las curvas de pago son funciones del radio de cooperadores para el número total de agentes. incluso si las funciones de pago son lineales, cuatro parametros libres determinan ellas. en esta investigación son consideradas solamente funciones “crossing payoff”. investigamos sistematicamente el comportamiento de los agentes. los resultados muestran que las soluciones son no-triviales y en algunos casos muy irregulares. ellas muestran cambios drásticos en el caso de juegos lider en el estrecho rango de parametros 1.72 ≤ p ≤ 1.75. este comportamento es similar al observado por [3] para juegos de n -persons chicken. soluciones irregulares fueran encontradas para el juego inverso stag-hunt. key words and phrases: agent-based simulation, cooperation, n -person games. math. subj. class.: 91a06, 91a25. 1 introduction externality is an economic term. it refers to actions that are external to a firm’s interests but within the interests of someone else. in n -person game theory externalities refer to the environment of the participating agents. they are the rewards and penalties they receive for their actions. in binary choice games the externalities are the payoff functions that determine these rewards and penalties [1]. there are an infinite variety of n -person games. in a previous paper [2] we listed the 13 most important characteristics that determine a game. in this short review we restrict ourselves to the investigation of uniform games with linear payoff functions only. the agents are considered to be greedy simpletons who imitate the action of that of their neighbors who received the highest payoff for its previous action. we assume that the individual agents may cooperate with each other for the collective interest or may defect, i.e., pursue their selfish interests. their decisions to cooperate or defect will accumulate over time to produce a result that will determine the success or failure of the given artificial society. cubo 11, 2 (2009) n -person games with crossing externalities 9 figure 1: payoff (reward/penalty) functions for defectors (d) and cooperators (c). the horizontal axis (x) represents the ratio of the number of cooperators to the total number of agents in the neighborhood; the vertical axis is the payoff provided by the environment. in this case, d(x) = x and c(x) = 0.6 + 0.14x. the broken lines represent the fact that the values c(0) and d(1) are not possible by definition. the payoffs for both choices are similar to those shown in figure 1. the horizontal axis represents the number of cooperators related to the total number of agents. we assume that the payoffs are linear functions of this ratio x. point p corresponds to the punishment when all agents defect, point r is the reward when all agents cooperate, t is the temptation to defect when everybody else cooperates, and s is the sucker’s payoff for cooperating when everyone else defects. c(0) and d(1) are impossible by definition, but we will follow the generally accepted notation by extending both lines for the full range of 0 ≤ x ≤ 1 and denoting c(0) = s and d(1) = t that makes it simpler to define the payoff functions. we connect by straight lines point s with point r (cooperators’ payoff function c) and point p with point t (defectors’ payoff function d). thus the payoff to each agent depends on its choice and on the distribution of other players among cooperators and defectors. the payoff function c(x) is the same for all cooperators and d(x) is the same for all defectors (uniform game). the two lines cross each other at the point xcross. there are 12 different orderings of the values of p, r, s, and t that lead to crossing payoff lines. each of them represents a different type of game. for the payoff functions shown in figure 1, for example, we have t > r > s > p which is the case of the chicken game. the chicken game is the only n -person game with crossing payoff functions that has been investigated in the literature [3]. it has the following properties: 1. both payoff functions increase with the increasing number of cooperators. 10 miklos n. szilagyi cubo 11, 2 (2009) 2. in the region of low cooperation the cooperators have a higher reward than the defectors. 3. when the cooperation rate is high, there is a higher payoff for defecting behavior than for cooperating behavior. 4. as a consequence, the slope of the d function is greater than that of the c function and the two payoff functions intersect. 5. all agents receive a lower payoff if all defect than if all cooperate. in this short paper we look at other n -person games with crossing payoff functions. for the greedy simpletons only the relative payoffs count; therefore, a thorough investigation is possible. a more detailed treatment of this subject can be found in [4]. 2 simulation we used our agent-based model developed for simulated social and economic experiments with a large number of decision-makers operating in a stochastic environment [5]. the simulation environment is a two-dimensional array of the participating agents. its size is limited only by the computer’s virtual memory. the behavior of a few million interacting agents can easily be observed on the computer’s screen. there are two actions available to each agent, and each agent must choose between them (cooperation or defection). the cooperators and the defectors are initially distributed randomly over the array. in the iterative game the aggregate cooperation proportion changes in time, i.e., over subsequent iterations. at each iteration, every agent chooses an action according to the rewards to its neighbors. the software tool draws the array of agents in a window on the computer’s screen, with each agent in the array colored according to its most recent action. the updating occurs simultaneously for all agents. after a certain number of iterations the proportion of cooperators stabilizes to either a constant value or oscillates around such a value. the experimenter can view and record the evolution of the society of agents as it changes in time. the outcome of the game strongly depends on the ”personalities” of the agents, i.e., on the type of their responses to the environment. the software tool allows for a number of different personalities and their arbitrary combinations. in this work we assume that all agents are greedy simpletons. throughout this work the total number of agents is 500 · 500 = 250, 000, the initial ratio of cooperators is 50%, and the neighborhood of each agent is one layer deep, i.e., each agent has exactly eight neighbors except those that are situated at the borders of the array. our free parameters are the four constants p, r, s, and t . they are chosen in such a way that the cubo 11, 2 (2009) n -person games with crossing externalities 11 two lines cross each other. we have performed a systematic investigation of games for hundreds of values of these parameters. the global ratio x(t) of the total number of cooperators in the entire array as a function of time (iterations) was observed for all parameter values. (note that x is different from x that refers to an agent’s immediate neighbors only.) the final ratio of cooperators xfinal around which x(t) oscillates represents the solution of the game. if at x < xcross the cooperators get a higher payoff than the defectors and at x > xcross the defectors get a higher payoff than the cooperators, then the intersection point xcross of the two payoff functions is a nash equilibrium for rational players. indeed, under these conditions at x < xcross the number of cooperators increases and this number decreases when x > xcross. this is, however, not true for the greedy simpletons because x refers to the immediate neighbors only while the final ratio of cooperators xfinal represents the ratio of the total number of cooperators in the entire array. let us first find out how the value of xfinal changes as a function of xcross. to investigate this dependence, we fix the values of the payoff lines’ slopes at r−s = −1 and t −p = 3 and move the two lines so that xcross changes but the payoff value ycross = 2.2 corresponding to xcross remains constant. we expect a monotonically increasing xfinal(xcross) dependence. the result is shown in figure 2. figure 2: the solution of the game xfinal as a function of xcross the values of (r − s) = −1, (t − p ) = 3, and ycross = 2.2 are fixed. for xcross < 0.2 the solution is total defection and it is overwhelming cooperation for xcross > 0.85. the transition in between these two limiting values, however, is quite interesting. as wee see, in the region 0.75 < xcross < 0.80 the value of xfinal even decreases as xcross grows. it is interesting to note the transitions between different games as the value of xcross changes. for 12 miklos n. szilagyi cubo 11, 2 (2009) 0 ≤ xcross0.2 we have the battle of the sexes game, for 0.3 ≤ xcross0.7 it is the benevolent chicken game, and for 0.8 ≤ xcross ≤ 1.0 we have the leader game. xcross = 0.25 and xcross = 0.75 are borderline cases in between two games. the transitions between games are relatively smooth. figure 3: rotating the d(x) function around the intersection point of the two lines xcross = 0.7, ycross = 2.2. c(x) = 2.9 − x. to investigate the role of the relative angle between the two payoff lines, we fix the c(x) function as c(x) = 2.9 − x, the intersection point of the two lines at xcross = 0.7, ycross = 2.2, and rotate the d(x) function around this point (figure 3). under these conditions for −10.0 ≤ p ≤ 0.5 we have the benevolent chicken game, for 0.6 ≤ p ≤ 1.8 it is the leader game, for 2.0 ≤ p ≤ 2.1 we have the reversed benevolent chicken game, and for 2.3 ≤ p ≤ 2.9 it is the reversed stag hunt game. p = 0.56, p = 1.9, and p = 2.2 are borderline cases in between two games. the transitions between games are quite smooth. we present the result of the simulation xfinal as a function of p in figure 4. for the region −10.0 ≤ p ≤ −0.6 the result is nearly constant around the value of xfinal = 0.82. therefore, we only show the result for the region −1.0 ≤ p ≤ 2.9. several remarkable features can be immediately noticed. first, neither the benevolent chicken nor the reversed benevolent chicken games are sensitive to the rotation of the d(x) function. however, the leader and the reversed stag hunt games behave quite strangely. the behavior of the reversed stag hunt game is quite remarkable. at p = 2.275, xfinal suddenly jumps from 0.66 to 0.80, then rises to 0.83, then at p = 2.394 it has a dip down to 0.76, rises again to 0.81, then at p = 2.4 jumps down to 0.65, rises again and finally at p = 0.525 suddenly cubo 11, 2 (2009) n -person games with crossing externalities 13 changes from 0.63 to 0.23, and at p = 2.65 reaches its final value of 0.11. the character of the x(t) function also drastically changes at these points. the x(t) function for p = 2.24 wildly fluctuates in between 0.40 and 0.91 so that xfinal = 0.65. for p = 2.41, xfinal = 0.65 again, but there are practically no fluctuations. the graphics outputs are also quite different. figure 4: the solution of the game xfinal as a function of p . the d(x) function is rotated around the fixed intersection point. the leader game behaves even more strangely. there are several wild fluctuations in between xfinal = 0.64 and xfinal = 0.90 in the narrow region of 1.72 ≤ p ≤ 1.75. we found similar behavior in the n -person chicken game [3]. received: may 3, 2008. revised: june 14, 2008. references [1] schelling, t.c., hockey helmets, concealed weapons, and daylight savings, journal of conflict resolution, 17(3)(1973), 381–428. [2] szilagyi, m.n., n -person prisoners’ dilemmas, cubo mathemática educacional, 5(3)(2003), 469–499. [3] szilagyi, m.n., agent-based simulation of the n -person chicken game, advances in dynamical games (s. jorgensen, m. quincampoix, and t.l. vincent, eds.) 9(2006), 695–703. [4] szilagyi, m.n. and somogyi, i., agent-based simulation of n-person games with crossing payoff functions, complex systems, 17(2008), 427–439. [5] szilagyi, m.n. and szilagyi, z.c., a tool for simulated social experiments, simulation, 74(1)(2000), 4–10. n02-crossing cmm.dvi cubo a mathematical journal vol.12, no¯ 01, (181–193). march 2010 on n (k)-contact metric manifolds a.a. shaikh department of mathematics, university of burdwan, golapbag, burdwan-713104, west bengal, india email : aask2003@yahoo.co.in and c.s. bagewadi department of mathematics, kuvempu university, jana sahyadri, shankaraghatta-577 451, karnataka, india email : prof bagewadi@yahoo.com abstract the object of the present paper is to study a type of contact metric manifolds, called n (k)contact metric manifolds admitting a non-null concircular and torse forming vector field. among others it is shown that such a manifold is either locally isometric to the riemannian product en+1(0) × sn(4) or a sasakian manifold. also it is shown that such a contact metric manifold can be expressed as a warped product i×ep ∗ m , where ( ∗ m , ∗ g) is a 2n-dimensional manifold. resumen el objetivo del presente art́ıculo es estudiar un tipo de variedades métricas de contacto, llamadas n (k)-variedades métricas de contacto admitiendo un campo de vectores concircular y forma torse. es demostrado también que tales variedades son o localmente isométricas a productos riemannianos en+1(0)×sn(4) o una variedade sasakian. es demostrado que tales variedades métricas de contacto pueden ser expresadas como un producto deformado i×ep ∗ m , donde ( ∗ m , ∗ g) es una variedad 2n-dimensional. 182 a.a. shaikh and c.s. bagewadi cubo 12, 1 (2010) key words and phrases: contact metric manifold, k-nullity distribution, n (k)-contact metric manifold, concircular vector field, torse forming vector field, η-einstein, sasakian manifold, warped product. math. subj. class.: 53c05, 53c15, 53c25. 1 introduction a contact manifold is a smooth (2n + 1)-dimensional manifold m 2n+1 equipped with a global 1-form η such that η ∧ (dη)n 6= 0 everywhere. given a contact form η, there exists a unique vector field ξ, called the characteristic vector field of η, satisfying η(ξ)=1 and dη(x, ξ)=0 for any vector field x on m 2n+1. a riemannian metric g is said to be associated metric if there exists a tensor field φ of type (1, 1) such that η(x) = g(x, ξ), dη(x, y ) = g(x, φy ) and φ2x = −x + η(x)ξ (1.1) for all vector fields x, y on m 2n+1. then the structure (φ, ξ, η, g) on m 2n+1 is called a contact metric structure and the manifold m 2n+1 equipped with such a structure is said to be a contact metric manifold [2]. it can be easily seen that in a contact metric manifold, the following relations hold : φξ = 0, η ◦ φ = 0, g(φx, φy ) = g(x, y ) − η(x)η(y ) (1.2) for any vector field x, y on m 2n+1. given a contact metric manifold m 2n+1(φ, ξ, η, g) we define a (1, 1) tensor field h by h = 1 2 £ξφ, where £ denotes the operator of lie differentiation.then h is symmetric and satisfies hξ = 0, hφ = −φh, t r.h = t r.φh = 0. (1.3) if ∇ denotes the riemannian connection of g, then we have the following relation ∇x ξ = −φx − φhx. (1.4) a contact metric manifold m 2n+1(φ, ξ, η, g) for which ξ is a killing vector field is called a k-contact manifold. a contact metric manifold is sasakian if and only if r(x, y )ξ = η(y )x − η(x)y, (1.5) where r is the riemannian curvature tensor of type (1, 3). in 1988, s. tanno [7] introduced the notion of k-nullity distribution of a contact metric manifold as a distribution such that the characteristic vector field ξ of the contact metric manifold belongs to the distribution. the contact metric manifold with ξ belonging to the k-nullity distribution is called n (k)-contact metric manifold and such a manifold is also studied by various cubo 12, 1 (2010) on n (k)-contact metric manifolds 183 authors. generalizing this notion in 1995, blair, koufogiorgos and papantoniou [4] introduced the notion of a contact metric manifold with ξ belonging to the (k, µ)-nullity distribution, where k and µ are real constants. in particular, if µ = 0, then the notion of (k, µ)-nullity distribution reduces to the notion of k-nullity distribution. the present paper deals with a study of n (k)-contact metric manifolds. the paper is organised as follows. section 2 is concerned with the discussion of n (k)-contact metric manifolds. in section 3, we obtain a necessary and sufficient condition for a n (k)-contact metric manifold to be an η− einstein manifold. section 4 is devoted to the study of n (k)-contact metric manifolds admitting a non-null concircular vector field and it is proved that such a manifold is either locally isometric to the riemannian product en+1(0) × sn(4) or a sasakian manifold. the last section deals with a study of n (k)-contact metric manifolds admitting a non-null torse forming vector field and it is shown that such a torse forming vector field reduces to a unit proper concircular vector field. hence a n (k)-contact metric manifold admits a proper concircular vector field, namely, the characteristic vector field ξ, and it is proved that a n (k)-contact metric manifold is a subprojective manifold in the sense of kagan [1]. finally it is shown that a n (k)-contact metric manifold can be expressed as a warped product i×ep ∗ m , where ( ∗ m , ∗ g) is a 2n-dimensional manifold. 2 n (k)-contact metric manifolds let us consider a contact metric manifold m 2n+1(φ, ξ, η, g). the k-nullity distribution [7] of a riemainnian manifold (m, g) for a real number k is a distribution n (k) : p → np(k) = {z ∈ tpm : r(x, y )z = k[g(y, z)x − g(x, z)y ]} for any x, y ∈ tpm. hence if the characteristic vector field ξ of a contact metric manifold belongs to the k-nullity distribution, then we have r(x, y )ξ = k[η(y )x − η(x)y ]. (2.1) thus a contact metric manifold m 2n+1(φ, ξ, η, g) satisfying the relation (2.1) is called a n (k)contact metric manifold. from (1.5) and (2.1) it follows that a n (k)-contact metric manifold is a sasakian manifold if and only if k = 1. also in a n (k)-contact metric manifold, k is always a constant such that k ≤ 1 [7]. the (k, µ)-nullity distribution of a contact metric manifold m 2n+1(φ, ξ, η, g) is a distribution [4] n (k, µ) : p → np(k, µ) = [ z ∈ tpm : r(x, y )z = k{g(y, z)x − g(x, z)y } +µ{g(y, z)hx − g(x, z)hy } ] 184 a.a. shaikh and c.s. bagewadi cubo 12, 1 (2010) for any x, y∈ tpm , where k, µ are real constants. hence if the characteristic vector field ξ belongs to the (k, µ)-nullity distribution, then we have r(x, y )ξ = k{η(y )x − η(x)y } + µ{η(y )hx − η(x)hy }. (2.2) a contact metric manifold m 2n+1(φ, ξ, η, g) satisfying the relation (2.2) is called a n (k, µ)-contact metric manifold or simply a (k, µ)-contact metric manifold. in particular, if µ = 0, then the relation (2.2) reduces to (2.1) and hence a n (k)-contact metric manifold is a n (k, 0)-contact metric manifold. let m 2n+1(φ, ξ, η, g) be a n (k)-contact metric manifold. then the following relations hold ([5], [7]): qφ − φq = 4(n − 1)hφ, (2.3) h2 = (k − 1)φ2, k ≤ 1, (2.4) qξ = 2nkξ, (2.5) r(ξ, x)y = k[g(x, y )ξ − η(y )x], (2.6) where q is the ricci operator, i.e., g(qx, y ) = s(x, y ), s being the ricci tensor of type (0, 2). in view of (1.1)-(1.2), it follows from (2.3)– (2.6) that in a n (k)-contact metric manifold, the following relations hold: t r.h2 = 2n(1 − k), (2.7) s(x, φy ) + s(φx, y ) = 2(2n − 2)g(φx, hy ), (2.8) s(φx, φy ) = s(x, y ) − 2nkη(x)η(y ) − 2(2n − 2)g(hx, y ), (2.9) qφ + φq = 2φq + 2(2n − 2)hφ, (2.10) η(r(x, y )z) = k[g(y, z)η(x) − g(x, z)η(y )], (2.11) s(φx, ξ) = 0 (2.12) for any vector field x, y on m 2n+1. also in a n (k)-contact metric manifold the scalar curvature r is given by ([4], [5]) r = 2n(2n − 2 + k). (2.13) we now state a result as a lemma which will be used later on. lemma 2.1. [3] let m 2n+1(φ, ξ, η, g) be a contact metric manifold with r(x, y )ξ=0 for all vector fields x, y. then the manifold is locally isometric to the riemannian product en+1(0) × sn(4). cubo 12, 1 (2010) on n (k)-contact metric manifolds 185 3 η-einstein n (k)-contact metric manifolds definition 3.1. a n (k)-contact metric manifold m 2n+1 is said to be η-einstein if its ricci tensor s of type (0, 2) is of the form s = ag + bη ⊗ η, (3.1) where a, b are smooth functions on m 2n+1. from (3.1) it follows that (i) r = (2n + 1)a + b, (ii) 2nk = a + b, (3.2) which yields by virtue of (2.13) that a = 2n − 2 and b = 2n(k − 1) + 2. obviously a and b are constants as k is a constant. hence by virtue of (3.1) we can state the following: proposition 3.1 in an η-einstein n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1), the ricci tensor is of the form s = (2n − 2)g + {2n(k − 1) + 2}η ⊗ η. (3.3) let m 2n+1(φ, ξ, η, g)(n > 1) be a n (k)-contact metric manifold. now we have (r(x, y ) · s)(u, v ) = −s(r(x, y )u, v ) − s(u, r(x, y )v ), which implies that (r(x, ξ) · s)(u, v ) = −s(r(x, ξ)u, v ) − s(u, r(x, ξ)v ). (3.4) first we suppose that a n (k)-contact metric manifold is an η-einstein manifold. then we have s(x, y ) = ag(x, y ) + bη(x)η(y ), (3.5) where a and b are given by a = 2n − 2 and b = 2n(k − 1) + 2. using (3.5), (2.5) and (2.6) in (3.4) we obtain (r(x, ξ) · s)(u, v ) = k[(2nk − a)g(x, u )η(v ) + g(x, v )η(u ) (3.6) −2bη(x)η(u )η(v )]. from (3.2)(ii) it follows that b = 2nk − a. (3.7) in view of (3.7), (3.6) reduces to (r(x, ξ) · s)(u, v ) = kb[g(x, u )η(v ) + g(x, v )η(u ) (3.8) −2η(x)η(u )η(v )]. 186 a.a. shaikh and c.s. bagewadi cubo 12, 1 (2010) putting v = ξ in (3.8) we obtain (r(x, ξ) · s)(u, ξ) = k{2n(k − 1) + 2}[g(x, u ) − η(x)η(u )]. (3.9) hence we can state the following: theorem 3.1. if a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g) (n > 1) is η-einstein, then the relation (3.9) holds. next, we suppose that in a n (k)-contact metric manifold m 2n+1(n > 1) the relation (3.9) holds. then using (2.5) and (2.6) in (3.4) we get (r(x, ξ) · s)(u, ξ) = k[2nkg(x, u ) − s(x, u )]. (3.10) by virtue of (3.9) and (3.10) we obtain k[s(x, u ) − (2n − 2)g(x, u ) − {2n(k − 1) + 2}η(x)η(u )] = 0. this implies either k = 0, or, s(x, u ) = (2n − 2)g(x, u ) + {2n(k − 1) + 2}η(x)η(u ). (3.11) if k = 0, then from (2.1) we have r(x, y )ξ = 0 for all x, y. hence by lemma 2.1, it follows that the manifold is locally isometric to the riemannian product en+1(0) × sn(4). again (3.11) implies that the manifold is η-einstein. hence we can state the following: theorem 3.2. if in a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) the relation (3.9) holds, then either the manifold is locally isometric to the riemannian product en+1(0) × sn(4) or the manifold is η-einstein. combining theorem 3.1 and theorem 3.2 we can state the following: theorem 3.3. a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1)(k 6= 0) is an η-einstein manifold if and only if the relation (3.9) holds. 4 n (k)-contact metric manifolds admitting a non-null concircular vector field definition 4.1. a vector field v on a riemannian manifold is said to be concircular vector field [6] if it satisfies an equation of the form ∇x v = ρx for all x, (4.1) where ρ is a scalar. cubo 12, 1 (2010) on n (k)-contact metric manifolds 187 we suppose that a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) admits a non-null concircular vector field. then we have (4.1). differentiating (4.1) covariantly we get ∇y ∇x v = ρ∇y x + dρ(y )x. (4.2) from (4.2) it follows that (since the torsion tensor t (x, y ) = ∇x y − ∇y x − [x, y ] = 0) ∇y ∇x v − ∇x∇y v − ∇[x,y ]v = dρ(x)y − dρ(y )x. (4.3) hence by ricci identity we obtain from (4.3) r(x, y )v = dρ(x)y − dρ(y )x, (4.4) which implies that r̃(x, y, v, z) = dρ(x)g(y, z) − dρ(y )g(x, z), (4.5) where r̃(x, y, v, z) = g(r(x, y )v, z). replacing z by ξ in (4.5) we get η(r(x, y )v ) = dρ(x)η(y ) − dρ(y )η(x). (4.6) again from (2.11) we have η(r(x, y )v ) = k[g(y, v )η(x) − g(x, v )η(y )]. (4.7) by virtue of (4.6) and (4.7) we have dρ(x)η(y ) − dρ(y )η(x) = k[g(y, v )η(x) − g(x, v )η(y )]. (4.8) putting x = φx and y = ξ in (4.8), and then using (1.2) we get dρ(φx) = −kg(φx, v ). (4.9) substituting x by φx in (4.9), we obtain by virtue of (1.1) that dρ(x) − dρ(ξ)η(x) = k[g(x, v ) − η(x)η(v )]. (4.10) now we have g(x, v ) 6= 0 for all x. for, if g(x, v ) = 0 for all x, then g(v, v ) = 0 which means that v is a null vector field, contradicts to our assumption. hence multiplying both sides of (4.10) by g(x, v ) we have dρ(x)g(x, v ) − dρ(ξ)g(x, v )η(x) = kg(x, v )[g(x, v ) − η(x)η(v )]. (4.11) also from (4.5) we get for z = v (since r̃(x, y, v, v ) = 0) dρ(x)g(y, v ) = dρ(y )g(x, v ). (4.12) putting y = ξ in (4.12) and then using (1.1) we obtain dρ(x)η(v ) = dρ(ξ)g(x, v ). (4.13) 188 a.a. shaikh and c.s. bagewadi cubo 12, 1 (2010) since η(x) 6= 0 for all x, multiplying both sides of (4.13) by η(x), we have dρ(x)η(x)η(v ) = dρ(ξ)η(x)g(x, v ). (4.14) by virtue of (4.11) and (4.14) we get [dρ(x) − kg(x, v )][g(x, v ) − η(x)η(v )] = 0. (4.15) hence it follows from (4.15) that either dρ(x) = kg(x, v ) for all x (4.16) or, g(x, v ) − η(x)η(v ) = 0 for all x. (4.17) first we consider the case of (4.16). by virtue of (4.16) we obtain from (4.5) that r̃(x, y, v, z) = k[−g(y, v )g(x, z) + g(x, v )g(y, z)]. (4.18) let {ei : i = 1, 2,...., 2n+1} be an orthonormal basis of the tangent space at any point of the manifold. then putting x = z = ei in (4.18) and taking summation over i, 1 ≤ i ≤ 2n + 1, we get s(y, v ) = −2nkg(y, v ). (4.19) now (∇z s)(y, v ) = ∇z s(y, v ) − s(∇z y, v ) − s(y, ∇z v ). (4.20) using (4.1) and (4.19) in (4.20) we obtain (∇z s)(y, v ) = ρ[−2nkg(y, z) + s(y, z)]. (4.21) setting y = z = ei in (4.21) and then taking summation over 1 ≤ i ≤ 2n + 1, we get 1 2 dr(v ) = ρ[−2nk(2n + 1) + r], (4.22) where r denotes the scalar curvature of the manifold. since in a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) k is a constant, by virtue of (2.13) it follows that r is constant and hence (4.22) yields (since r 6= 2nk(2n + 1)) ρ = 0, which implies by virtue of (4.4) that r(x, y )v = 0 for all x and y . this yields s(y, v ) = 0, which implies by virtue of (4.19) that k = 0. if k = 0 then from (2.1) we have r(x, y )ξ = 0 for all x and y and hence by lemma 2.1, it follows that the manifold is locally isometric to the riemannian product en+1(0) × sn(4). next we consider the case (4.17). differentiating (4.17) covariantly along z, we get (∇z η)(x)η(v ) + (∇z η)(v )η(x) = (∇z g)(x, v ) = 0. (4.23) cubo 12, 1 (2010) on n (k)-contact metric manifolds 189 now we have (∇x η)(y ) = ∇x η(y ) − η(∇x y ) = ∇x g(y, ξ) − g(∇x y, ξ). = (∇x g)(y, ξ) + g(y, ∇x ξ). that is, (∇x η)(y ) = g(y, ∇x ξ). (4.24) by virtue of (4.24) we get from (4.23) that η(v )g(x, ∇z ξ) + η(x)g(v, ∇z ξ) = 0. (4.25) in view of (1.4), (4.25) yields [g(x, φz) + g(x, φhz)]η(v ) + [g(v, φz) + g(v, φhz)]η(x) = 0. (4.26) putting x = ξ in (4.26) we get g(x, φz) + g(v, φhz) = 0. (4.27) substituting z by φz in (4.27), we obtain by virtue of (1.1), hφ = −φh and hξ = 0 that −g(v, z) + η(v )η(z) + g(v, hz) = 0. (4.28) using (4.17) in (4.28) we get g(v, hz) = 0 for all z. since h is symmetric, the above relation implies that g(hv, z) = 0 for all z, which gives us hv = 0. but since v is non-null, by our assumption, we must have h = 0 and hence from (2.4) it follows that k = 1. therefore the manifold is sasakian. hence summing up all the cases we can state the following: theorem 4.1. if a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) admits a non-null concircular vector field, then either the manifold is locally isometric to the riemannian product en+1(0) × sn(4) or the manifold is sasakian. 5 n (k)-contact metric manifolds admitting a non-null torse forming vector field definition 5.1. a vector field v on a riemannian manifold is said to be torse forming vector field ([6], [8]) if the 1-form ω(x) = g(x, v ) satisfies the equation of the form (∇x ω)y = ρg(x, y ) + π(x)ω(y ), (5.1) where ρ is a non-vanishing scalar and π is a non-zero 1-form given by π(x) = g(x, p ). 190 a.a. shaikh and c.s. bagewadi cubo 12, 1 (2010) if the 1-form π is closed, then the vector field v is called a proper concircular vector field. in particular if the the 1-form π is zero, then the vector field v reduces to a concircular vector field. let us consider a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) admitting a unit torse forming vector field u corresponding to the non-null torse forming vector field v . hence if t (x) = g(x, u ), then we have t (x) = ω(x) √ ω(x) . (5.2) by virtue of (5.2), it follows from (5.1) that (∇x t )(y ) = βg(x, y ) + π(x)t (y ), (5.3) where β = α √ ω(v ) is a non-zero scalar. since u is a unit vector field, substituting y by u in (5.3) yields π(x) = −βt (x) and hence (5.3) reduces to the following (∇x t )(y ) = β[g(x, y ) + t (x)t (y )]. (5.4) the relation (5.4) implies that the 1-form t is closed. differentiating (5.4) covariantly we obtain by virtue of ricci identity that −t (r(x, y )z) = (xβ)[g(y, z) + t (y )t (z)] − (y β)[g(x, z) + t (x)t (z)] (5.5) +β2[g(y, z)t (x) + g(x, z)t (y )]. setting z = ξ in (5.5) and then using (2.1) we get (xβ)[η(y ) + t (y )η(u )] − (y β)[η(x) + t (x)η(u )] (5.6) +(k + β2)[g(y, z)t (x) + g(x, z)t (y )] = 0. putting x = u in (5.6) we obtain [k + β2 + (u β)][η(y ) − η(u )t (y )] = 0, which implies that either [k + β2 + (u β)] = 0 (5.7) or, η(y ) − η(u )t (y ) = 0. (5.8) we first consider the case of (5.7). from (5.5) it follows that s(y, u ) = [2nβ2 + (u β)]t (y ) − (2n − 1)(y β), (5.9) which yields for y = ξ that (ξβ) = (u β)η(u ). (5.10) cubo 12, 1 (2010) on n (k)-contact metric manifolds 191 again, setting y = ξ in (5.6) we obtain by virtue of (5.10) that [1 − (η(u ))2][(xβ) − (k + β2)t (x)] = 0. (5.11) in this case η(y ) − η(u )t (y ) 6= 0 for all y and hence 1 − (η(u ))2 6= 0. consequently, (5.11) gives us (xβ) = (k + β2)t (x). (5.12) again, from π(x) = −βt (x) it follows that y π(x) = −[(y β)t (x) + β(y t (x))]. (5.13) in view of (5.13) we obtain dπ(x, y ) = −βdt (x, y ). since t is closed, π is also closed and hence the vector field v is a proper concircular vector field in this case. next, we consider the case of (5.8). the relation (5.8) implies that (η(u ))2 = 1 and hence η(u ) = ±1. consequently (5.8) reduces to η(y ) = ±t (y ). (5.14) differentiating (5.14) covariantly along x, we obtain by virtue of (5.14) that (∇x η)(y ) = ±β[g(x, y ) − η(x)η(y )], (5.15) which yields by virtue of (1.4) that g(x + hx, φy ) = ±β[g(x, y ) − η(x)η(y )]. (5.16) replacing y by φy in (5.16) and then using (1.2) we get −g(x, y ) − g(hx, y ) + η(x)η(y ) = ±βg(x, φy ). (5.17) again setting x = hx in (5.17) we obtain by virtue of (1.1) and (2.4) that −g(hx, y ) + (k − 1)[g(x, y ) − η(x)η(y )] = ±βg(hx, φy ). (5.18) putting x = y = ei in (5.18) and then taking summation over 1 ≤ i ≤ 2n + 1 we get by virtue of (1.3) that k = 1 (5.19) and hence the manifold is sasakian. 192 a.a. shaikh and c.s. bagewadi cubo 12, 1 (2010) let us now suppose that the manifold is non-sasakian. then k < 1 [4]. hence from (5.17) and (5.18) it follows that (k − β2)[g(x, y ) − η(x)η(y )] = ∓2βg(x, φy ) (5.20) which yields by contraction k = ±β2. since β 6= 0, it follows that (xβ) = 0 for any x and hence β is constant. consequently we obtain π(x) = −βt (x) where β is constant, it follows that the 1-form π is also closed and hence the vector field v is a proper concircular vector field. considering all the cases we can state the following: theorem 5.1. in a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1), a non-null torse forming vector field is a proper concircular vector field. from (1.4) and (5.4) it follows that in a n (k)-contact metric manifold the characteristic vector field ξ is a unit torse forming vector field and hence by virtue of theorem 5.1, we can state the following: theorem 5.2. a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) admits a proper concircular vector field. again, it is known that if a riemannian manifold admits a proper concircular vector field, then the manifold is a subprojective manifold in the sense of kagan ([1]). since a n (k)-contact metric manifold admits a concircular vector field, namely, the vector field ξ, in view of the known result we can state the following: theorem 5.3. a n (k)contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) is a subprojective manifold in the sense of kagan. by virtue of theorem 5.2 and theorem 4.1 we can state the following: theorem 5.4. a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) is either locally isometric to the riemannian product en+1(0) × sn(4) or a sasakian manifold. k. yano [8] proved that if a riemannian manifold m 2n+1 admits a concircular vector field, it is necessary and sufficient that there exists a coordinate system with respect to which the fundamental quadratic differential form may be written as ds 2 = (dx1)2 + ep ∗ gλµ dx λ dx µ , (5.21) where ∗ gλµ= ∗ gλµ (x ν ) are the function of xν only (λ, µ, ν = 2, 3, ...... , 2n) and p = p(x1) 6= constant, is a function of x1 only. since a n (k)-contact metric manifold admits a proper concircular vector field, namely, the characteristic vector field ξ, by virtue of the above it follows that there exists a coordinate system with respect to which the fundamental quadratic differential form can be written as (5.21). consequently the manifold can be expressed as a warped product i×ep ∗ m , where ( ∗ m , ∗ g) is a 2n-dimensional manifold. hence we can state the following: cubo 12, 1 (2010) on n (k)-contact metric manifolds 193 theorem 5.5. a n (k)-contact metric manifold m 2n+1(φ, ξ, η, g)(n > 1) can be expressed as a warped product i×ep ∗ m , where ( ∗ m , ∗ g) is a 2n-dimensional manifold. received: july, 2008. revised: january, 2009. references [1] adati, t., on subprojective spaces iii, tohoku math. j., 3(1951), 343–358. [2] blair, d.e., contact manifolds in riemannian geometry, lecture notes in math., 509, springer-verlag, 1976. [3] blair, d.e., two remarks on contact metric structure , tohoku math. j., 29(1977), 319–324. [4] blair, d.e., koufogiorgos, t. and papantoniou, b.j., contact metric manifolds satisfying a nullity condition, israel j. of math., 19(1995), 189–214. [5] shaikh, a.a. and baishya, k.k., on (k, µ)-contact metric manifolds, j. diff. geom. and dyn. sys., 11(1906), 253–261. [6] schouten, j.a., ricci calculas (second edition), springer-verlag, 1954, 322. [7] tanno, s., ricci curvatures of contact riemannian manifolds, tohoku math. j., 40(1988), 441–448. [8] yano, k., on the torse forming direction in riemannian spaces, proc. imp. acad. tokyo, 20(1940), 340–345. [9] yano, k., concircular geometry i-iv, proc. imp. acad. tokyo, 16(1940), 195200, 354–350. cubo a mathematical journal vol.14, no¯ 02, (15–41). june 2012 weak and entropy solutions for a class of nonlinear inhomogeneous neumann boundary value problem with variable exponent stanislas ouaro laboratoire d’analyse mathématique des equations (lame), ufr. sciences exactes et appliquées, université de ouagadougou, 03 bp 7021 ouaga 03, ouagadougou, burkina faso email: souaro@univ-ouaga.bf, ouaro@yahoo.fr abstract we study the existence and uniqueness of weak and entropy solutions for the nonlinear inhomogeneous neumann boundary value problem involving the p(x)-laplace of the form − div a(x,∇u) + |u|p(x)−2 u = f in ω, a(x,∇u).η = ϕ on ∂ω, where ω is a smooth bounded open domain in rn, n ≥ 3, p ∈ c(ω) and p(x) > 1 for x ∈ ω. we prove the existence and uniqueness of a weak solution for data ϕ ∈ l(p−) ′ (∂ω) and f ∈ l(p−) ′ (ω), the existence and uniqueness of an entropy solution for l1−data f and ϕ independent of u and the existence of weak solutions for f dependent on u and ϕ ∈ l(p−) ′ (ω). resumen estudiamos la existencia y unicidad de soluciones y entroṕıa débil para el problema no lineal inhomogéneos de neumann con valores de frontera que involucra el p(x)laplace de la forma − div a(x,∇u) + |u|p(x)−2 u = f en omega, a(x,∇u).η = ϕ sobre ∂ω, donde omega es en un dominio abierto suave y acotado en rn, n ≥ 3, p ∈ c(ω) y p(x) > 1 para x ∈ ω. probamos la existencia y unicidad de una solución débil para ϕ ∈ l(p−) ′ (∂ω) and f ∈ l(p−) ′ (ω), la existencia y unicidad de una solución de entroṕıa para l1−data f y ϕ independiente de u y la existencia de soluciones débiles para f dependiente sobre u y ϕ ∈ l(p−) ′ (ω). keywords and phrases: generalized lebesgue and sobolev spaces; weak solution; entropy solution; p(x)-laplace operator. 2010 ams mathematics subject classification: 35j20, 35j25, 35d30, 35b38, 35j60. 16 stanislas ouaro cubo 14, 2 (2012) 1 introduction the purpose of this paper is to study the existence and uniqueness of weak and entropy solutions to the following nonlinear inhomogeneous neumann problem involving the p(x)-laplace    −div a(x,∇u) + |u|p(x)−2 u = f in ω, a(x,∇u).η = ϕ on ∂ω, (1.1) where ω ⊂ rn (n ≥ 3) is a bounded open domain with smooth boundary and η is the unit outward normal on ∂ω. the study of various mathematical problems with variable exponent has recieved considerable attention in recent years (see [4,7,8-15,17,19,24-27,29,30,33,34]). these problems concern applications (see [21,22,31,32,35]) and raise many difficult mathematical problems. the operator −div a(x,∇u) is called p(x)-laplace, which becomes p-laplace when p(x) ≡ p (a constant). it possesses more complicated nonlinearities than the p-laplace. for related results involving the p-laplace, see [2,3]. in [2], the authors studied the problem    −div a(x,∇u) + γ(u) ∋ φ in ω, a(x,∇u).η + β(u) ∋ ψ on ∂ω, (1.2) where η is the unit outward normal on ∂ω, ψ ∈ l1(∂ω) and φ ∈ l1(ω). the nonlinearities γ and β are maximal monotone graphs in r2 such that 0 ∈ γ(0) and 0 ∈ β(0). they proved under a range condition the existence and uniqueness of weak and entropy solutions to the problem (1.2). following these ideas, ouaro and soma [24] proved the existence and uniqueness of weak and entropy solutions for a class of homogeneous nonlinear neumann boundary value problem of the form    −div a(x,∇u) + |u|p(x)−2 u = f in ω, ∂u ∂ν = 0 on ∂ω, (1.3) where ω ⊂ rn (n ≥ 3) is a bounded open domain with smooth boundary and ∂u ∂ν is the outer unit normal derivative on ∂ω. in this paper, our aim is to prove the existence and uniqueness of weak and entropy solutions to the nonlinear neumann boundary value problem (1.1) in order to generalize the results in [24]. the paper is presented as follows. in section 2, we introduce some fundamental preliminary results that we use in this work. the existence and the uniqueness of weak solution for (1.1) is proved in section 3 when the data f and ϕ belongs to l(p−) ′ . in section 4, we prove some existence results of weak solution to the problem (1.1) for an f assumed to depend on u and for a boundary datum ϕ ∈ l(p−) ′ (∂ω). finally, in section 5, we prove the existence and the uniqueness of an entropy solution of (1.1) when the data f and ϕ belongs to l1. cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 17 2 assumptions and preliminaries in this work, we study the problem (1.1) for a variable exponent p(.) which is continuous, more precisely, we assume that    p(.) : ω → r is a continuous function such that 1 < p− ≤ p+ < +∞, (2.1) where p− := ess inf x∈ω p(x). we denote p+ := ess sup x∈ω p(x). for the vector fields a(., .), we assume that a(x,ξ) : ω × rn → rn is carathéodory and is the continuous derivative with respect to ξ of the mapping a : ω × rn → r, a = a(x,ξ), i.e. a(x,ξ) = ∇ξa(x,ξ) such that: • the following equality holds true a(x,0) = 0, (2.2) for almost every x ∈ ω. • there exists a positive constant c1 such that |a(x,ξ)| ≤ c1(j(x) + |ξ| p(x)−1 ) (2.3) for almost every x ∈ ω and for every ξ ∈ rn where j is a nonnegative function in lp ′ (.)(ω), with 1/p(x) + 1/p′(x) = 1. • there exists a positive constant c2 such that for almost every x ∈ ω and for every ξ,η ∈ r n with ξ 6= η, (a(x,ξ) − a(x,η)).(ξ − η) > 0. (2.4) • the following inequalities hold true |ξ| p(x) ≤ a(x,ξ).ξ ≤ p(x)a(x,ξ) (2.5) for almost every x ∈ ω and for every ξ ∈ rn. remark 2.1. since for almost every x ∈ ω, a(x,.) is a gradient and is monotone then the primitive a(x,.) of a(x,.) is necessarily convex. as the exponent p(.) appearing in (2.3) and (2.5) depends on the variable x, we must work with lebesgue and sobolev spaces with variable exponents. 18 stanislas ouaro cubo 14, 2 (2012) we define the lebesgue space with variable exponent lp(.)(ω) as the set of all measurable function u : ω → r for which the convex modular ρp(.)(u) := ∫ ω |u| p(x) dx is finite. if the exponent is bounded, i.e., if p+ < ∞, then the expression |u|p(.) := inf { λ > 0 : ρp(.)(u/λ) ≤ 1 } defines a norm in lp(.)(ω), called the luxembourg norm. the space (lp(.)(ω), |.|p(.)) is a separable banach space. moreover, if 1 < p− ≤ p+ < +∞, then lp(.)(ω) is uniformly convex, hence reflexive, and its dual space is isomorphic to lp ′ (.)(ω), where 1 p(x) + 1 p′(x) = 1. finally, we have the hölder type inequality: ∣ ∣ ∣ ∣ ∫ ω uvdx ∣ ∣ ∣ ∣ ≤ ( 1 p− + 1 p′− ) |u|p(.) |v|p′(.) , (2.6) for all u ∈ lp(.)(ω) and v ∈ lp ′ (.)(ω). now, let w1,p(.)(ω) := { u ∈ lp(.)(ω) : |∇u| ∈ lp(.)(ω) } , which is a banach space equipped with the following norm ‖u‖1,p(.) := |u|p(.) + |∇u|p(.) . the space ( w1,p(.)(ω),‖u‖1,p(.) ) is a separable and reflexive banach space; more details can be found in [17]. an important role in manipulating the generalized lebesgue and sobolev spaces is played by the modular ρp(.) of the space l p(.)(ω). we have the following result (cf. [15]): lemma 2.2. if un,u ∈ l p(.)(ω) and p+ < +∞, then the following properties hold: (i) |u|p(.) > 1 ⇒ |u| p− p(.) ≤ ρp(.)(u) ≤ |u| p+ p(.) ; (ii) |u|p(.) < 1 ⇒ |u| p+ p(.) ≤ ρp(.)(u) ≤ |u| p− p(.) ; (iii) |u|p(.) < 1 (respectively = 1;> 1) ⇔ ρp(.)(u) < 1 (respectively = 1;> 1); (iv) |un|p(.) → 0 (respectively → +∞) ⇔ ρp(.)(un) → 0 (respectively → +∞); (v) ρp(.) ( u/ |u|p(.) ) = 1. for a measurable function u : ω −→ r, we introduce the following notation: ρ1,p(.)(u) := ∫ ω |u| p(x) dx + ∫ ω |∇u| p(x) dx. we have the following lemma (cf. [33]): lemma 2.3. if u ∈ w1,p(.)(ω), then the following properties hold true: (i) ‖u‖1,p(.) < 1(respectively = 1;> 1) ⇔ ρ1,p(.)(u) < 1(respectively = 1;> 1); (ii) ‖u‖1,p(.) < 1 ⇔ ‖u‖ p+ 1,p(.) ≤ ρ1,p(.)(u) ≤ ‖u‖ p− 1,p(.) ; (iii) ‖u‖1,p(.) > 1 ⇔ ‖u‖ p− 1,p(.) ≤ ρ1,p(.)(u) ≤ ‖u‖ p+ 1,p(.) . cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 19 put p∂(x) := (p(x))∂ :=    (n−1)p(x) n−p(x) , if p(x) < n ∞, if p(x) ≥ n. we have the following useful result (cf. [13,34]). proposition 2.4. let p ∈ c(ω) and p− > 1. if q ∈ c(∂ω) satisfies the condition 1 ≤ q(x) < p∂(x), ∀x ∈ ∂ω, then, there is a compact embedding w1,p(x)(ω) →֒ lq(x)(∂ω). in particular, there is a compact embedding w1,p(x)(ω) →֒ lp(x)(∂ω). let us introduce the following notation: given two bounded measurable functions p(.),q(.) : ω → r, we write q(.) ≪ p(.) if ess inf x∈ω (p(x) − q(x)) > 0. 3 weak solution in this section, we study the existence and uniqueness of a weak solution of (1.1) where the data ϕ ∈ l(p−) ′ (∂ω) and f ∈ l(p−) ′ (ω). the definition of weak solution is the following: definition 3.1. a weak solution of (1.1) is a measurable function u : ω −→ r such that u ∈ w1,p(.)(ω), and ∫ ω a(x,∇u).∇vdx + ∫ ω |u|p(x)−2 uvdx − ∫ ∂ω ϕvdσ = ∫ ω fvdx, ∀ v ∈ w1,p(.)(ω), (3.1) where dσ is the surface measure on ∂ω. let e denote the generalized sobolev space w1,p(.)(ω). if we denote the functional j : e → r by j(u) = ∫ ω a(x,∇u)dx + ∫ ω 1 p(x) |u|p(x)dx − ∫ ∂ω ϕudσ − ∫ ω fudx, then 〈j′(u),v〉 = ∫ ω a(x,∇u).∇vdx + ∫ ω |u|p(x)−2 uvdx − ∫ ∂ω ϕvdσ − ∫ ω fvdx, for all u,v ∈ e. therefore, the weak solution of (1.1) corresponds to the critical point of the functional j. the main result of this section is the following: theorem 3.2. assume that (2.1)-(2.5) hold. then there exists a unique weak solution of (1.1). proof. * existence. with the techniques that became standard by now, it is not difficult to 20 stanislas ouaro cubo 14, 2 (2012) verify that j is well-defined on e, is of class c1(e,r) and is weakly lower semi-continuous (see for example [6,19,24,25,26,28]). to end the proof of the existence part, we just have to prove that j is bounded from below and coercive. using (2.5) and since e is continuously embedded in lp−(ω), we have j(u) = ∫ ω a(x,∇u)dx + ∫ ω 1 p(x) |u|p(x)dx − ∫ ∂ω ϕudσ − ∫ ω fudx ≥ ∫ ω 1 p(x) |∇u|p(x)dx + ∫ ω 1 p(x) |u|p(x)dx − ‖ϕ‖ (p−)′,∂ω ‖u‖p−,∂ω − ‖f‖(p−)′,ω ‖u‖p−,ω ≥ 1 p+ ρ1,p(.)(u) − c‖ϕ‖(p−)′,∂ω ‖u‖1,p(.) − c‖u‖1,p(.), where ‖u‖p−,ω = (∫ ω |u|p−dx ) 1 p− and ‖u‖p−,∂ω = (∫ ∂ω |u|p−dσ ) 1 p− . as ϕ ∈ l(p−) ′ (∂ω), then ‖ϕ‖ (p−)′,∂ω < +∞. also, for the coercivity of j, we will work with u such that ‖u‖1,p(.) > 1. then, by lemma 2.3 we obtain that j(u) ≥ 1 p+ ‖u‖ p− 1,p(.) − c3‖u‖1,p(.). as p− > 1, then j is coercive. if ‖u‖1,p(.) < 1, we have that j(u) ≥ 1 p+ ‖u‖ p+ 1,p(.) − c3‖u‖1,p(.) ≥ −c3 > −∞. therefore, j is bounded from below. since the functional j is proper, lower semi-continuous and coercive, then it has a minimizer which is a weak solution of (1.1). ∗ uniqueness. let u1 and u2 be two weak solutions of (1.1). with u1 as weak solution, we take v = u1 − u2 in (3.1) to get ∫ ω a(x,∇u1).∇(u1−u2)dx+ ∫ ω |u1| p(x)−2 u1(u1−u2)dx− ∫ ∂ω ϕ(u1−u2)dσ = ∫ ω f(x)(u1−u2)dx. (3.2) similarly, with u2 as weak solution, we take ϕ = u2 − u1 to obtain ∫ ω a(x,∇u2).∇(u2−u1)dx+ ∫ ω |u2| p(x)−2 u2(u2−u1)dx− ∫ ∂ω ϕ(u2−u1)dσ = ∫ ω f(x)(u2−u1)dx. (3.3) after adding (3.2) and (3.3), we obtain ∫ ω (a(x,∇u1) − a(x,∇u2)) .(∇u1 −∇u2)+ ∫ ω ( |u1| p(x) u1 − |u2| p(x) u2 ) (u1 −u2)dx = 0. (3.4) cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 21 using (2.4), we deduce from (3.4) that ∫ ω ( |u1(x)| p(x) u1(x) − |u2(x)| p(x) u2(x) ) (u1(x) − u2(x))dx = 0. (3.5) since p− > 1, the following relation is true for any ξ,η ∈ r, ξ 6= η (cf. [14]) ( |ξ|p(x)−2ξ − |η|p(x)−2η ) (ξ − η) > 0. (3.6) therefore, from (3.5), we get ( |u1(x)| p(x) u1(x) − |u2(x)| p(x) u2(x) ) (u1(x) − u2(x)) = 0, a.e. x ∈ ω. (3.7) now, we use (3.6) to get u1(x) = u2(x) a.e. x ∈ ω. (3.8) and uniqueness is true � 4 weak solutions for a right-hand side dependent on u in this section, we show the existence result of weak solution to some general problem. more precisely, we prove that there exists at least one weak solution to the problem    − div a(x,∇u) + |u|p(x)−2 u = f(x,u) in ω, a(x,∇u).η = ϕ on ∂ω, (4.1) where ϕ ∈ l(p−) ′ (∂ω). we study (4.1) under the assumptions (2.1)-(2.5) and the following additional assumptions on f. f(x,t) : ω × r −→ r is carathéodory and there exists two positive constants c4, c5 such that |f(x,t)| ≤ c4 + c5|t| β(x)−1, (4.2) for every t ∈ r and for almost every x ∈ ω with 0 ≤ β(.) ≪ p(.). let f(x,t) = ∫t 0 f(x,s)ds. as mentioned before, we look for distributional solution of (4.1) in the following sense: definition 4.1. a weak solution of (4.1) is a measurable function u : ω −→ r such that u ∈ w1,p(.)(ω) and for all v ∈ w1,p(.)(ω) ∫ ω a(x,∇u).∇vdx + ∫ ω |u|p(x)−2 uvdx − ∫ ∂ω ϕvdσ = ∫ ω f(x,u)vdx. (4.3) 22 stanislas ouaro cubo 14, 2 (2012) we have the following existence result: theorem 4.2. assume that (2.1)-(2.5) and (4.2) hold. then, the problem (4.1) admits at least one weak solution. proof. let g(u) = ∫ ω f(x,u)dx, for all u ∈ e. the functional g is of class c1(e,r) with the derivative given by 〈 g′(u),v 〉 = ∫ ω f(x,u)vdx, ∀u,v ∈ e. consequently, j(u) = ∫ ω a(x,∇u)dx + ∫ ω 1 p(x) |u|p(x)dx − ∫ ∂ω ϕudσ − ∫ ω f(x,u)dx, u ∈ e is such that j is of class c1(e,r) and is lower semi-continuous. we then have to prove that j is bounded from below and coercive in order to complete the proof. from (4.2), we have |f(x,t)| ≤ c ( 1 + |t|β(x) ) and then j(u) ≥ 1 p+ ∫ ω |∇u|p(x)dx + 1 p+ ∫ ω |u|p(x)dx − ∫ ∂ω ϕudσ − c ∫ ω |u|β(x)dx − cmeas(ω). let m > 1 be a fixed real number (to be chosen later) and ǫ := ess inf x∈ω (p(x) − β(x)). we have j(u) ≥ 1 2p+ ρ1,p(.)(u) + ∫ {|u|≤m} ( 1 2p+ |u|p(x) − c|u|β(x) ) dx + ∫ {|u|>m} ( 1 2p+ |u|p(x) − c|u|β(x) ) dx − cmeas(ω) − c′′‖u‖1,p(.) ≥ 1 2p+ ρ1,p(.)(u) + ∫ {|u|>m} ( 1 2p+ |u|p(x) − c|u|β(x) ) dx − c′′‖u‖1,p(.) − (m β+ + 1)cmeas(ω) ≥ 1 2p+ ρ1,p(.)(u) + ∫ {|u|>m} |u|β(x) ( 1 2p+ |u|p(x)−β(x) − c) ) dx − c′′‖u‖1,p(.) − (m β+ + 1)cmeas(ω) ≥ 1 2p+ ρ1,p(.)(u) + ( 1 2p+ mǫ − c ) ∫ {|u|>m} |u|β(x)dx − c′′‖u‖1,p(.) − (m β+ + 1)cmeas(ω) ≥ 1 p+ ‖u‖ p− 1,p(.) − c′′‖u‖1,p(.) − (m β+ + 1)cmeas(ω), for all m > max((2p+c) 1 ǫ ,1) and all u ∈ e with ‖u‖1,p(.) > 1. since 1 < p− it follows that j(u) −→ +∞ as ‖u‖e −→ +∞. consequently, j is bounded from below and coercive. the proof is then complete. assume now that f+(x,t) = ∫t 0 f+(x,s)ds is such that there exists c6 > 0, c7 > 0 such that |f+(x,t)| ≤ c6 + c7|t| β(x)−1, (4.4) where 0 ≤ β(.) ≪ p(.). then we have the following result: cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 23 theorem 4.3 under assumptions (2.1)-(2.5) and (4.4), the problem (4.1) admits at least one weak solution. proof. as f = f+ − f−, let f−(x,t) = ∫t 0 f−(x,s)ds. then j(u) = ∫ ω a(x,∇u)dx + ∫ ω 1 p(x) |u|p(x)dx + ∫ ω f−(x,u)dx − ∫ ω f+(x,u)dx − ∫ ∂ω ϕudσ ≥ ∫ ω a(x,∇u)dx + ∫ ω 1 p(x) |u|p(x)dx − ∫ ω f+(x,u)dx − ∫ ∂ω ϕvdσ. therefore, similarly as in the proof of theorem 4.2, the result of theorem 4.3 follows immediately. 5 entropy solutions in this section, we study the existence of entropy solution for the problem (1.1) when the data f ∈ l1(ω) and ϕ ∈ l1(∂ω). we first recall some notations. for any k > 0, we define the truncation function tk by tk(s) := max{−k,min{k,s}}. let ω be a bounded open subset of rn of class c1 and 1 ≤ p(.) < +∞. it is well known( see [20] or [23]) that if u ∈ w1,p(.)(ω), it is possible to define the trace of u on ∂ω. more precisely, there is a bounded operator τ from w1,p(.)(ω) into lp(.)(∂ω) such that τ(u) = u|∂ω whenever u ∈ c(ω). set t 1,p(.)(ω) = { u : ω −→ r, measurable such that tk(u) ∈ w 1,p(.)(ω), for any k > 0 } . in [1], the authors have proved the following proposition 5.1 let u ∈ t 1,p(.)(ω). then there exists a unique measurable function v : ω −→ rn such that ∇tk(u) = vχ{|u| 0. the function v is denoted by ∇u. moreover if u ∈ w1,p(.)(ω) then v ∈ ( lp(.)(ω) )n and v = ∇u in the usual sense. it is easy to see that, in general, it is not possible to define the trace of an element of t 1,p(.)(ω). in demension one it is enough to consider the function u(x) = 1 x for x ∈]0,1[. therefore, we are going to define following [2,3], the trace for the elements of a subset t 1,p(.) tr (ω) of t 1,p(.)(ω). t 1,p(.) tr (ω) will be the set of functions u ∈ t 1,p(.)(ω) such that there exists a sequence (un)n ⊂ w1,p(.)(ω) satisfying the following conditions: (c1) un → u a.e in ω. (c2) ∇tk(un) → ∇tk(u) in l1(ω) for any k > 0. (c3) there exists a measurable function v on ∂ω, such that un → v a.e in ∂ω. the function v is the trace of u in the generalized sense introduced in [2,3]. in the sequel the trace of u ∈ t 1,p(.) tr (ω) on ∂ω will be denoted by tr(u). if u ∈ w 1,p(.)(ω),tr(u) coincides with τ(u) in 24 stanislas ouaro cubo 14, 2 (2012) the usual sense. moreover, for u ∈ t 1,p(.) tr (ω) and for every k > 0, τ(tk(u)) = tk (tr(u)) and if ϕ ∈ w1,p(.)(ω) ∩ l∞(ω) then (u − ϕ) ∈ t 1,p(.) tr (ω) and tr(u − ϕ) = tr(u) − tr(ϕ). we can now introduce the notion of entropy solution of (1.1). definition 5.2. a measurable function u is an entropy solution to problem (1.1) if u ∈ t 1,p(.) tr (ω), |u|p(x)−2 u ∈ l1(ω) and for every k > 0, ∫ ω a(x,∇u).∇tk(u−v)dx+ ∫ ω |u|p(x)−2 utk(u−v)dx ≤ ∫ ∂ω ϕtk(u−v)dσ+ ∫ ω f(x)tk(u−v)dx (5.1) for all v ∈ w1,p(.)(ω) ∩ l∞(ω). our main result in this section is the following: theorem 5.3. assume (2.1)-(2.5), f ∈ l1(ω) and ϕ ∈ l1(∂ω). then, there exists a unique entropy solution u to problem (1.1). the following propositions are useful for the proof of theorem 5.3. proposition 5.4. assume (2.1)-(2.5), f ∈ l1(ω) and ϕ ∈ l1(∂ω). let u be an entropy solution of (1.1). if there exists a positive constant m such that ∫ {|u|>k} kq(x)dx ≤ m (5.2) then ∫ {|∇u|α(.)>k} kq(x)dx ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) + m, for all k > 0, where α(.) = p(.)/(q(.) + 1). proof. taking v = 0 in the entropy inequality (5.1) and using (2.5), we get ∫ ω |∇tk(u)| p(x)dx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) for all k > 0. therefore, defining ψ := 1 k tk(u), we have for all k > 0, ∫ ω kp(x)−1|∇ψ|p(x)dx = 1 k ∫ ω |∇tk(u)| p(x)dx ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω). from the above inequality, from the definition of α(.) and (5.2), we get ∫ {|∇u|α(.)>k} kq(x)dx ≤ ∫ {|∇u|α(.)>k}∩{|u|≤k} kq(x)dx + ∫ {|u|>k} kq(x)dx ≤ ∫ {|u|≤k} kq(x) ( |∇u|α(x) k ) p(x) α(x) dx + m ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) + m, for all k > 0. proposition 5.5. assume (2.1)-(2.5), f ∈ l1(ω) and ϕ ∈ l1(∂ω). let u be an entropy solution of (1.1), then ∫ ω |∇tk(u)| p(x)dx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) for all k > 0 (5.3) cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 25 and ∥ ∥ ∥ |u|p(x)−2 u ∥ ∥ ∥ 1 = ∥ ∥ ∥ |u|p(x)−1 ∥ ∥ ∥ 1 ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω). (5.4) proof. the inequality (5.3) is already obtained in the proof of proposition 5.2. let’s prove (5.4). taking ϕ = 0 in (5.1), we get for all k > 0, ∫ ω |u|p(x)−2 utk(u)dx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) , then ∫ {|u|>k} |u|p(x)−2 utk(u)dx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) . from the inequality above, we obtain k ∫ {u>k} |u|p(x)−2 udx − k ∫ {u<−k} |u|p(x)−2 udx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) , which imply ∫ {u>k} |u|p(x)−2 udx − ∫ {u<−k} |u|p(x)−2 udx ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω). the last inequality means ∫ {|u|>k} |u|p(x)−1dx ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) for all k > 0. (5.5) we use fatou’s lemma in (5.5) by letting k goes to 0 to obtain (5.4). proposition 5.6. assume that (2.1)-(2.5) hold, f ∈ l1(ω) and ϕ ∈ l1(∂ω). let u be an entropy solution of (1.1), then ∫ {|u|≤k} |∇tk(u)| p−dx ≤ c(k + 1) for all k > 0. (5.6) proof. note that ∫ {|u|≤k} |∇tk(u)| p−dx = ∫ {|u|≤k,|∇u|>1} |∇tk(u)| p−dx + ∫ {|u|≤k,|∇u|≤1} |∇tk(u)| p−dx ≤ ∫ {|u|≤k,|∇u|>1} |∇tk(u)| p−dx + meas(ω) ≤ ∫ {|u|≤k} |∇tk(u)| p(x)dx + meas(ω). since ∫ {|u|≤k} |∇tk(u)| p(x)dx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) , we obtain ∫ {|u|≤k} |∇tk(u)| p−dx ≤ k ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) + meas(ω) for all k > 0. 26 stanislas ouaro cubo 14, 2 (2012) proposition 5.7. assume that (2.1)-(2.5) hold, f ∈ l1(ω) and ϕ ∈ l1(∂ω). let u be an entropy solution of (1.1). then meas{|u| > h} ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) hp−−1 for all h ≥ 1, (5.7) and meas{|∇u| > h} ≤ ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) hp−−1 for all h ≥ 1. (5.8) proof. ∫ ω |u|p(x)−1dx = ∫ {|u|≤h} |u|p(x)−1dx + ∫ {|u|>h} |u|p(x)−1dx ≥ ∫ {|u|>h} |u|p(x)−1dx ≥ ∫ {|u|>h} hp(x)−1dx ≥ hp−−1meas{|u| > h} since h ≥ 1. then, by (5.4) we deduce (5.7). we next prove (5.8). for k,λ ≥ 0, set φ(k,λ) = meas{|∇u|p− > λ, |u| > k}. we have φ(k,0) ≤ meas{|u| > k}. for k ≥ 1, we obtain by (5.7) φ(k,0) ≤ ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) k1−p−. using the fact that the function λ 7−→ φ(k,λ) is nonincreasing, we get for k > 0 and λ > 0, that φ(0,λ) = 1 λ ∫λ 0 φ(0,λ)ds ≤ 1 λ ∫λ 0 φ(0,s)ds ≤ 1 λ ∫λ 0 [ φ(0,s) + (φ(k,0) − φ(k,s)) ] ds ≤ φ(k,0) + 1 λ ∫λ 0 (φ(0,s) − φ(k,s))ds. now, let us observe that φ(0,s) − φ(k,s) = meas{|u| ≤ k, |∇u|p− > s}. then, thanks to (5.6), we get ∫+∞ 0 (φ(0,s) − φ(k,s))ds = ∫ {|u|≤k} |∇u|p−dx ≤ c(k + 1), cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 27 where c = max ( meas(ω),‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) . it follows that φ(0,λ) ≤ c(k + 1) λ + ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) k1−p−, for all k ≥ 1,λ > 0. in particular, we have φ(0,λ) ≤ c(k + 1) λ + ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) k1−p−, for all k ≥ 1,λ ≥ 1. we now set fλ(k) = c(k + 1) λ + ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) k1−p−, for all k ≥ 1, where λ ≥ 1 is a fixed real number. the minimization of fλ in k gives φ(0,λ) ≤ ( ‖f‖l1(ω) + ‖ϕ‖l1(∂ω) ) λ−(1/(p−) ′ ), (5.9) for all λ ≥ 1. setting λ = hp− in (5.9) gives (5.8). proof of theorem 5.3. ∗ uniqueness of entropy solution. let h > 0 and u1,u2 be two entropy solutions of (1.1). we write the entropy inequality (5.1) corresponding to the solution u1, with th(u2) as a test function, and to the solution u2, with th(u1) as a test function. upon addition, we get    ∫ {|u1−th(u2)|≤k} a(x,∇u1).∇(u1 − th(u2))dx + ∫ {|u2−th(u1)|≤k} a(x,∇u2).∇(u2 − th(u1))dx + ∫ ω |u1| p(x)−2 u1tk(u1 − th(u2))dx + ∫ ω |u2| p(x)−2 u2tk(u2 − th(u1))dx ≤ ∫ ∂ω ϕ ( tk(u1 − th(u2)) + tk(u2 − th(u1)) ) dσ + ∫ ω f ( tk(u1 − th(u2)) + tk(u2 − th(u1)) ) dx. (5.10) define now e1 := {|u1 − u2| ≤ k, |u2| ≤ h}, e2 := e1 ∩ {|u1| ≤ h}, and e3 := e1 ∩ {|u1| > h}. we start with the first integral in (5.10). by (2.5), we have 28 stanislas ouaro cubo 14, 2 (2012)    ∫ {|u1−th(u2)|≤k} a(x,∇u1).∇(u1 − th(u2))dx = ∫ {|u1−th(u2)|≤k}∩{|u2|≤h} a(x,∇u1).∇(u1 − th(u2))dx + ∫ {|u1−th(u2)|≤k}∩{|u2|>h} a(x,∇u1).∇(u1 − th(u2))dx = ∫ {|u1−th(u2)|≤k}∩{|u2|≤h} a(x,∇u1).∇(u1 − u2)dx + ∫ {|u1−hsign(u2)|≤k}∩{|u2|>h} a(x,∇u1).∇u1dx ≥ ∫ {|u1−th(u2)|≤k}∩{|u2|≤h} a(x,∇u1).∇(u1 − u2)dx = ∫ e1 a(x,∇u1).∇(u1 − u2)dx = ∫ e2 a(x,∇u1).∇(u1 − u2)dx + ∫ e3 a(x,∇u1).∇(u1 − u2)dx = ∫ e2 a(x,∇u1).∇(u1 − u2)dx + ∫ e3 a(x,∇u1).∇u1dx − ∫ e3 a(x,∇u1).∇u2dx ≥ ∫ e2 a(x,∇u1).∇(u1 − u2)dx − ∫ e3 a(x,∇u1).∇u2dx. (5.11) using (2.3) and (2.6), we estimate the last integral in (5.11) as follows:    ∣ ∣ ∣ ∣ ∫ e3 a(x,∇u1).∇u2dx ∣ ∣ ∣ ∣ ≤ c1 ∫ e3 ( j(x) + |∇u1| p(x)−1 ) |∇u2|dx ≤ c1 ( |j|p′(.) + ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) |∇u2|p(.),{h−k<|u1|≤h}, (5.12) where ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} = ∥ ∥ ∥ |∇u1| p(x)−1 ∥ ∥ ∥ lp ′(.)({h<|u1|≤h+k}) . the quantity c1 ( |j|p′(.) + ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) can be written as follows c1 ( |j|p′(.) + ∣ ∣ ∣ |∇th+k(u1)| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) < +∞, since th+k(u1) ∈ w 1,p(.)(ω) and j ∈ lp ′ (.)(ω). we deduce by proposition 5.7 that c1 ( |j|p′(.) + ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) |∇u2|p(.),{h−k<|u1|≤h} converges to 0 as h → +∞. therefore, from (5.11) and (5.12), we obtain ∫ {|u1−th(u2)|≤k} a(x,∇u1).∇(u1 − th(u2))dx ≥ ih + ∫ e2 a(x,∇u1).∇(u1 − u2)dx, (5.13) where ih converges to zero as h → +∞. we may adopt the same procedure to treat the second term in (5.10) to obtain ∫ {|u2−th(u1)|≤k} a(x,∇u2).∇(u2 − th(u1))dx ≥ jh − ∫ e2 a(x,∇u2).∇(u1 − u2)dx, (5.14) cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 29 where jh converges to zero as h → +∞. now set for all h,k > 0 kh = ∫ ω |u1| p(x)−2 u1tk(u1 − th(u2))dx + ∫ ω |u2| p(x)−2 u2tk(u2 − th(u1))dx. we have |u1| p(x)−2 u1tk(u1 − th(u2)) −→ |u1| p(x)−2 u1tk(u1 − u2) a.e in ω as h → +∞, and ∣ ∣ ∣ |u1| p(x)−2 u1tk(u1 − th(u2)) ∣ ∣ ∣ ≤ k|u1| p(x)−1 ∈ l1(ω). then by lebesgue theorem, we deduce that lim h→+∞ ∫ ω |u1| p(x)−2 u1tk(u1 − th(u2))dx = ∫ ω |u1| p(x)−2 u1tk(u1 − u2)dx. (5.15) similarly, we have lim h→+∞ ∫ ω |u2| p(x)−2 u2tk(u2 − th(u1))dx = ∫ ω |u2| p(x)−2 u2tk(u2 − u1)dx. (5.16) using (5.15) and (5.16), we get lim h→+∞ kh = ∫ ω ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) tk(u1 − u2)dx. (5.17) we next examine the right-hand side of (5.10). for all k > 0, f ( tk(u1 −th(u2))+tk(u2 −th(u1)) ) −→ f ( tk(u1 −u2)+tk(u2 −u1) ) = 0 a.e in ω as h → +∞, ϕ ( tk(u1−th(u2))+tk(u2−th(u1)) ) −→ ϕ ( tk(u1−u2)+tk(u2−u1) ) = 0 a.e in ∂ω as h → +∞, and ∣ ∣ ∣ f(x) ( tk(u1 − th(u2)) + tk(u2 − th(u1)) ) ∣ ∣ ∣ ≤ 2k|f| ∈ l1(ω), ∣ ∣ ∣ ϕ ( tk(u1 − th(u2)) + tk(u2 − th(u1)) ) ∣ ∣ ∣ ≤ 2k|ϕ| ∈ l1(∂ω). lebesgue theorem allows us to write lim h→+∞ [∫ ∂ω ϕ ( tk(u1 − th(u2)) + tk(u2 − th(u1)) ) dσ + ∫ ω f ( tk(u1 − th(u2)) + tk(u2 − th(u1)) ) dx ] = 0. (5.18) using (5.13), (5.14), (5.17) and (5.18), we get    ∫ {|u1−u2|≤k} ( a(x,∇u1) − a(x,∇u2) ) . ( ∇u1 − ∇u2 ) dx + ∫ ω ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) tk(u1 − u2)dx ≤ 0. (5.19) 30 stanislas ouaro cubo 14, 2 (2012) therefore ∫ ω ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) tk(u1 − u2)dx = 0. (5.20) for x fixed in ω, s 7−→ |s|p(x)−2 s is nondecreasing and vanishes at 0. then, ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) tk(u1 − u2) ≥ 0, ∀x ∈ ω and ∀k > 0. now, using inequality above and (5.20), for all k ∈ r+ there exist ωk ⊂ ω with meas(ωk) = 0 such that for all x ∈ ω\ωk, ( |u1(x)| p(x)−2 u1(x) − |u2(x)| p(x)−2 u2(x) ) tk(u1(x) − u2(x)) = 0. therefore, ( |u1(x)| p(x)−2 u1(x) − |u2(x)| p(x)−2 u2(x) ) (u1(x) − u2(x)) = 0, for all x ∈ ω\ ⋃ k∈n∗ ωk. (5.21) now, using (5.21) and (3.6), we get u1 = u2 a.e. in ω. ∗ existence of entropy solution. let fn = tn(f) and ϕn = tn(ϕ); then (fn)n and (ϕn)n are in l(p−) ′ (ω) and l(p−) ′ (∂ω) respectively and are strongly converging to f in l1(ω) and to ϕ in l1(∂ω) respectively. moreover ‖fn‖l1(ω) ≤ ‖f‖l1(ω) and ‖ϕn‖l1(∂ω) ≤ ‖ϕ‖l1(∂ω), for all n ∈ n. next, we consider the problem    −div a(x,∇un) + |un| p(x)−2 un = fn in ω, a(x,∇un).η = ϕn on ∂ω. (5.22) it follows from theorem 3.2 that there exists a unique un ∈ w 1,p(.)(ω) such that ∫ ω a(x,∇un).∇vdx + ∫ ω |un| p(x)−2 unvdx = ∫ ∂ω ϕnvdσ + ∫ ω fnvdx (5.23) for all v ∈ w1,p(.)(ω). our aim is to prove that these approximated solutions un tend, as n goes to infinity, to a measurable function u which is an entropy solution to the limit problem (1.1). to start with, we prove the following lemma: lemma 5.8. for any k > 0, ‖tk(un)‖1,p(.) ≤ 1 + c where c = c(k,ϕ,f,p−,p+,meas(ω)) is a positive constant. proof. by taking v = tk(un) in (5.23), we get ∫ ω a(x,∇un).∇tk(un) + ∫ ω |un| p(x)−2 untk(un)dx = ∫ ∂ω ϕntk(un)dσ + ∫ ω fntk(un)dx. cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 31 since all the terms in the left-hand side of equality above are nonnegative and ∫ ∂ω ϕntk(un)dσ + ∫ ω fntk(un)dx ≤ k ( ‖ϕn‖l1(∂ω) + ‖fn‖l1(ω) ) ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) ; by using (2.5) we obtain ∫ ω |∇tk(un)| p(x)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) (5.24) and ∫ ω |un| p(x)−2 untk(un)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . (5.25) the inequality (5.25) is equivalent to ∫ {|un|≤k} |tk(un)| p(x)dx + ∫ {|un|>k} |un| p(x)−2 untk(un)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . therefore, ∫ {|un|≤k} |tk(un)| p(x)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . (5.26) furthermore ∫ {|un|>k} |tk(un)| p(x)dx = ∫ {|un|>k} kp(x)dx ≤    kp+meas(ω) if k ≥ 1, meas(ω) if k < 1. this allows us to write ∫ {|un|>k} |tk(un)| p(x)dx ≤ (1 + kp+)meas(ω). (5.27) relations (5.26) and (5.27) give ∫ ω |tk(un)| p(x)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + (1 + kp+)meas(ω). (5.28) hence, adding (5.24) and (5.28), it yields ρ1,p(.)(tk(un)) ≤ 2k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + (1 + kp+)meas(ω) = c(k,ϕ,f,p+,meas(ω)). (5.29) if ‖tk(un)‖1,p(.) ≥ 1, we have ‖tk(un)‖ p− 1,p(.) ≤ ρ1,p(.)(tk(un)) ≤ c(k,ϕ,f,p+,meas(ω)), which is equivalent to ‖tk(un)‖1,p(.) ≤ ( c(k,ϕ,f,p+,meas(ω)) ) 1 p− = c(k,ϕ,f,p−,p+,meas(ω)). 32 stanislas ouaro cubo 14, 2 (2012) the above inequality gives ‖tk(un)‖1,p(.) ≤ 1 + c(k,ϕ,f,p−,p+,meas(ω)). then, the proof of lemma 5.8. is complete. from lemma 5.8. we deduce that for any k > 0, the sequence (tk(un)) is uniformly bounded in w1,p(.)(ω) and so in w1,p−(ω). then, up to a subsequence we can assume that for any k > 0, tk(un) converges weakly to σk in w 1,p−(ω), and so tk(un) converges strongly to σk in l p−(ω). we next prove the following proposition: proposition 5.9. assume that (2.1)-(2.5) hold and un ∈ w 1,p(.)(ω) is the weak solution of (5.22). then the sequence (un)n is cauchy in measure. in particular, there exists a measurable function u and a subsequence still denoted (un)n such that un −→ u in measure. proof. let s > 0 and define e1 := {|un| > k}, e2 := {|um| > k} and e3 := {|tk(un) − tk(um)| > s} where k > 0 is to be fixed. we note that {|un − um| > s} ⊂ e1 ∪ e2 ∪ e3, and hence meas{|un − um| > s} ≤ meas(e1) + meas(e2) + meas(e3). (5.30) let ǫ > 0. using proposition 5.7, we choose k = k(ǫ) such that meas(e1) ≤ ǫ/3 and meas(e2) ≤ ǫ/3. (5.31) since tk(un) converges strongly in l p−(ω), then it is a cauchy sequence in lp−(ω). thus meas(e3) ≤ 1 sp− ∫ ω |tk(un) − tk(um)| p−dx ≤ ǫ 3 , (5.32) for all n,m ≥ n0(s,ǫ). finally, from (5.30), (5.31) and (5.32), we obtain meas{|un − um| > s} ≤ ǫ for all n,m ≥ n0(s,ǫ). (5.33) relations (5.33) mean that the sequence (un)n is cauchy sequence in measure and the proof of proposition 5.9. is complete. note that as un −→ u in measure, up to a subsequence, we can assume that un −→ u a.e. in ω. in the sequel, we need the following two technical lemmas. lemma 5.10. ( cf.[30, lemma 5.4] ) let (vn)n be a sequence of measurable functions in ω. if vn converges in measure to v and is uniformly bounded in l p(.)(ω) for some 1 ≪ p(.) ∈ l∞(ω), then vn −→ v strongly in l1(ω). cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 33 the second technical lemma is a well known result in measure theory (cf. [16] ). lemma 5.11. let (x,m,µ) be a measure space such that µ(x) < +∞. consider a measurable function γ : x −→ [0,+∞] such that µ({x ∈ x : γ(x) = 0}) = 0. then, for every ǫ > 0, there exists δ > 0, such that µ(a) < ǫ, for all a ∈ m with ∫ a γdµ < δ. we now set to prove that the function u in the proposition 5.9 is an entropy solution of (1.1). let v ∈ w1,p(.)(ω) ∩ l∞(ω). for any k > 0, choose tk(un − v) as a test function in (5.23). we get ∫ ω a(x,∇un).∇tk(un − v)dx + ∫ ω |un| p(x)−2 untk(un − v)dx = ∫ ∂ω ϕn(x)tk(un − v)dσ + ∫ ω fn(x)tk(un − v)dx. (5.34) we have the following proposition: proposition 5.12. assume that (2.1)-(2.5) hold and un ∈ w 1,p(.)(ω) be the weak solution of (5.22). then (i) ∇un converges in measure to the weak gradient of u; (ii) for all k > 0, ∇tk(un) converges to ∇tk(u) in (l 1(ω))n. (iii) for all t > 0, a(x,∇tt(un)) converges to a(x,∇tt(u)) in ( l1(ω) )n strongly and in ( lp ′ (.)(ω) )n weakly. (iv) un converges to some function v a.e. on ∂ω. proof. (i) we claim that the sequence (∇un)n is cauchy in measure. indeed, let s > 0, and consider e1 := {|∇un| > h} ∪ {|∇um| > h}, e2 := {|un − um| > k} and e3 := {|∇un| ≤ h, |∇um| ≤ h, |un − um| ≤ k, |∇un − ∇um| > s}, where h and k will be chosen later. note that {|∇un − ∇um| > s} ⊂ e1 ∪ e2 ∪ e3. (5.35) let ǫ > 0. by proposition 5.7 (relation (5.8)), we may choose h = h(ǫ) large enough such that meas(e1) ≤ ǫ/3, (5.36) 34 stanislas ouaro cubo 14, 2 (2012) for all n,m ≥ 0. on the other hand, by proposition 5.9 meas(e2) ≤ ǫ/3, (5.37) for all n,m ≥ n0(k,ǫ). moreover, since a(x,ξ) is continuous with respect to ξ for a.e every x ∈ ω, by assumption (2.5) there exists a real valued function γ : ω −→ [0,+∞] such that meas({x ∈ ω : γ(x) = 0}) = 0 and (a(x,ξ) − a(x,ξ′)).(ξ − ξ′) ≥ γ(x), (5.38) for all ξ,ξ′ ∈ rn such that |ξ| ≤ h, |ξ′| ≤ h, |ξ − ξ′| ≥ s, for a.e x ∈ ω. let δ = δ(ǫ) be given by lemma 5.11., replacing ǫ and a by ǫ/3 and e3 respectively. as un is a weak solution of (5.22), using tk(un − um) as a test function, we get ∫ ω a(x,∇un).∇tk(un − um)dx + ∫ ω |un| p(x)−2 untk(un − um)dx = ∫ ∂ω ϕntk(un − um)dσ + ∫ ω fntk(un − um)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . similarly for um, we have ∫ ω a(x,∇um).∇tk(um − un)dx + ∫ ω |um| p(x)−2 umtk(um − un)dx = ∫ ∂ω ϕmtk(um − un)dσ + ∫ ω fmtk(um − un)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . after adding the last two inequalities, it yields    ∫ {|un−um|≤k} (a(x,∇un) − a(x,∇um)).(∇un − ∇um)dx + ∫ ω ( |un| p(x)−2 un − |um| p(x)−2 um ) tk(un − um)dx ≤ 2k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . since the second term of the above inequality is nonnegative, we obtain by using (5.38) ∫ e3 γ(x)dx ≤ ∫ e3 (a(x,∇un) − a(x,∇um)).(∇un − ∇um)dx ≤ 2k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) < δ, where k = δ/4 ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . from lemma 5.11, it follows that meas(e3) ≤ ǫ/3. (5.39) cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 35 thus using (5.35), (5.36), (5.37) and (5.39), we get meas({|∇un − ∇um| > s}) ≤ ǫ, for all n,m ≥ n0(s,ǫ) (5.40) and then the claim is proved. consequently, (∇un)n converges in measure to some measurable function v. in order to end the proof of (i), we need the following lemma: lemma 5.13. (a) for a.e. t ∈ r, ∇tt(un) converges in measure to vχ{|u| 0, {∣ ∣χ{|un| δ } ⊂ {∣ ∣χ{|un| δ } ≤ meas {|u| = t} + meas {un < t < u} + meas {u < t < un} + meas {un < −t < u} + meas {u < −t < un} . (5.41) note that meas {|u| = t} ≤ meas {t − h h} , for all h > 0. due to proposition 5.9, we have for all fixed h > 0, meas {|u − un| > h} → 0 as n → +∞. since meas {t 0, one can find n such that for all n > n, meas {un < t < u} < ǫ/2 + ǫ/2 = ǫ by choosing h and then n. each of the other terms in the right-hand side of (5.41) can be treated in the same way as for meas {un < t < u}. thus, meas {∣ ∣χ{|un| δ } → 0 as n → +∞. finally, since ∇tt(un) = ∇unχ{|un| 0, k > 0 and consider e4 = {|∇un − ∇um| > s, |un| ≤ k, |um| ≤ k} , e5 = {|∇um| > s, |un| > k, |um| ≤ k} , e6 = {|∇un| > s, |um| > k, |un| ≤ k} and e7 = {0 > s, |um| > k, |un| > k} . note that {|∇tk(un) − ∇tk(um)| > s} ⊂ e4 ∪ e5 ∪ e6 ∪ e7. (5.42) let ǫ > 0. by proposition 5.7, we may choose k(ǫ) such that meas(e5) ≤ ǫ 4 ,meas(e6) ≤ ǫ 4 and meas(e7) ≤ ǫ 4 . (5.43) therefore, using (5.40), (5.42) and (5.43), we get meas({|∇tk(un) − ∇tk(um)| > s}) ≤ ǫ, for all n,m ≥ n1(s,ǫ). (5.44) consequently, ∇tk(un) converges in measure to ∇tk(u). then, using lemmas 5.8 and 5.10, (ii) follows. (iii) by lemmas 5.10 and 5.13, we have that for all t > 0, a(x,∇tt(un)) converges to a(x,∇tt(u)) in ( l1(ω) )n strongly and a(x,∇tt(un)) converges to χt ∈ (l p′(.)(ω))n in (lp ′ (.)(ω))n weakly. since each of the convergences implies the weak l1-convergence, χt can be identified with a(x,∇tt(u)); thus, a(x,∇tt(u)) ∈ (l p′(.)(ω))n. the proof of (iii) is then complete. (iv) as un is a weak solution of (5.22), using tk(un) as a test function, we get ∫ ω |tk(un)| p(x) dx ≤ ∫ ω |un| p(x)−2 untk(un)dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) . (5.45) cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 37 we deduce from (5.24) and (5.45) that ∫ ω |tk(un)| p− dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + meas(ω), (5.46) and ∫ ω |∇tk(un)| p− dx ≤ k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + meas(ω). (5.47) furthermore, tk(un) converges weakly to tk(u) in w 1,p−(ω) and since for every 1 ≤ p ≤ +∞, τ : w1,p(ω) → lp(∂ω),u 7→ τ(u) = u|∂ω is compact, we deduce that tk(un) converges strongly to tk(u) in l p−(∂ω) and so, up to a subsequence, we can assume that tk(un) converges to tk(u), a.e. on ∂ω. in other words, there exists c ⊂ ∂ω such that tk(un) converges to tk(u) on ∂ω\c with µ(c) = 0 where µ is the area measure on ∂ω. now, we use hölder inequality, (5.46) and (5.47) to get ∫ ω |tk(un)|dx ≤ (meas(ω)) 1 (p−) ′ ( k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + meas(ω) ) 1 p− , (5.48) and ∫ ω |∇tk(un)|dx ≤ (meas(ω)) 1 (p−) ′ ( k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + meas(ω) ) 1 p− . (5.49) by using fatou’s lemma in (5.48) and (5.49) we get as n goes to +∞, ∫ ω |tk(u)|dx ≤ (meas(ω)) 1 (p−) ′ ( k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + meas(ω) ) 1 p− , (5.50) and ∫ ω |∇tk(u)|dx ≤ (meas(ω)) 1 (p−) ′ ( k ( ‖ϕ‖l1(∂ω) + ‖f‖l1(ω) ) + meas(ω) ) 1 p− . (5.51) for every k > 0, let ak := {x ∈ ∂ω : |tk(u(x))| < k} and c ′ = ∂ω\ ⋃ k>0 ak. we have µ(c′) = 1 k ∫ c′ |tk(u)|dx ≤ 1 k ∫ ∂ω |tk(u)|dx ≤ c1 k ‖tk(u)‖w1,1(ω) ≤ c1 k ‖tk(u)‖l1(ω) + c1 k ‖∇tk(u)‖l1(ω) . according to (5.50) and (5.51), we deduce by letting k → +∞ that µ(c′) = 0. let us define in ∂ω the function v by v(x) := tk(u(x)) if x ∈ ak. 38 stanislas ouaro cubo 14, 2 (2012) we take x ∈ ∂ω\ (c ∪ c′); then there exists k > 0 such that x ∈ ak and we have un(x) − v(x) = (un(x) − tk(un(x))) + (tk(un(x)) − tk(u(x))) . since x ∈ ak, we have |tk(u(x))| < k and so |tk(un(x))| < k, from which we deduce that |un(x)| < k. therefore un(x) − v(x) = (tk(un(x)) − tk(u(x))) → 0, as n → +∞. this means that un converges to v a.e. on ∂ω. the proof of the proposition 5.12 is then complete. we are now able to pass to the limit in the identity (5.34). for the right-hand side, the convergence is obvious since fn converges strongly to f in l 1(ω), ϕn converges strongly to ϕ in l 1(∂ω) and tk(un − v) converges weakly-∗ to tk(u − v) in l ∞(ω) and a.e in ω and to tk(u − v) in l ∞(∂ω) and a.e in ∂ω. for the second term of (5.34), we have ∫ ω |un| p(x)−2 untk(un − v)dx = ∫ ω ( |un| p(x)−2 un − |v| p(x)−2 v ) tk(un − v)dx + ∫ ω |v|p(x)−2 vtk(un − v)dx. the quantity ( |un| p(x)−2 un − |v| p(x)−2 v ) tk(un − v) is nonnegative and since for all x ∈ ω, s 7−→ |s|p(x)−2 s is continuous, we get ( |un| p(x)−2 un − |v| p(x)−2 v ) tk(un − v) −→ ( |u|p(x)−2 u − |v|p(x)−2 v ) tk(u − v)dx a.e in ω. then, it follows by fatou’s lemma that lim inf n→+∞ ∫ ω ( |un| p(x)−2 un − |v| p(x)−2 v ) tk(un−v)dx ≥ ∫ ω ( |u|p(x)−2 u − |v|p(x)−2 v ) tk(u−v)dx. let us show that |v|p(x)−2 v ∈ l1(ω). we have ∫ ω ∣ ∣ ∣ |v|p(x)−2 v ∣ ∣ ∣ dx = ∫ ω |v|p(x)−1dx ≤ ∫ ω ( ‖v‖∞ )p(x)−1 dx. if ‖v‖∞ ≤ 1, then ∫ ω ∣ ∣ ∣ |v|p(x)−2 v ∣ ∣ ∣ dx ≤ meas(ω) < +∞. if ‖v‖∞ > 1, then ∫ ω ∣ ∣ ∣ |v|p(x)−2 v ∣ ∣ ∣ dx ≤ ∫ ω ( ‖v‖∞ )p+−1 dx = ( ‖v‖∞ )p+−1 meas(ω) < +∞. cubo 14, 2 (2012) weak and entropy solutions for a class of nonlinear ... 39 hence |v|p(x)−2 v ∈ l1(ω). since tk(un −v) converges weakly-∗ to tk(u−v) in l ∞(ω) and |v|p(x)−2 v ∈ l1(ω), it follows that lim n→+∞ ∫ ω |v|p(x)−2 vtk(un − v)dx = ∫ ω |v|p(x)−2 vtk(u − v)dx. next, we write the first term in (5.34) in the following form ∫ {|un−v|≤k} a(x,∇un).∇undx − ∫ {|un−v|≤k} a(x,∇un).∇vdx. (5.52) set l = k + ‖v‖∞, the second integral in (5.52) equals to ∫ {|un−v|≤k} a(x,∇tl(un)).∇vdx. since a(x,∇tl(un)) is uniformly bounded in ( lp ′ (.)(ω) )n (by (2.3) and (5.24) ), by proposition 5.12−(iii), it converges weakly to a(x,∇tl(u)) in ( lp ′ (.)(ω) )n . therefore lim n→+∞ ∫ {|un−v|≤k} a(x,∇tl(un)).∇vdx = ∫ {|u−v|≤k} a(x,∇tl(u)).∇vdx. moreover a(x,∇un).∇un is nonnegative and converges a.e in ω to a(x,∇u).∇u. thanks to fatou’s lemma, we obtain lim inf n→+∞ ∫ {|un−v|≤k} a(x,∇un).∇undx ≥ ∫ {|u−v|≤k} a(x,∇u).∇udx. gathering results, we obtain ∫ ω a(x,∇u).∇tk(u − v)dx + ∫ ω |u|p(x)−2 utk(u − v)dx ≤ ∫ ∂ω ϕtk(u − v)dσ + ∫ ω ftk(u − v)dx. we conclude that u is an entropy solution of (1.1). acknowledgment the author want to express is deepest thanks to the editor and anonymous referee’s for comments on the paper. received: january 2011. revised: september 2011. references [1] a. alvino, l. boccardo, v. ferone, l. orsina & g. trombetti; existence results for non-linear elliptic equations with degenerate coercivity, ann. mat. pura appl. 182 (2003), 53-79. 40 stanislas ouaro cubo 14, 2 (2012) [2] f. andreu, n. igbida, j.m. mazón & j. toledo; l1 existence and uniqueness results for quasilinear elliptic equations with nonlinear boundary conditions, ann. i.h. poincaré an., 24 (2007), 61-89. 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[35] v. zhikov; on passing to the limit in nonlinear variational problem. math. sb. 183 (1992), 47-84. introduction assumptions and preliminaries weak solution weak solutions for a right-hand side dependent on u entropy solutions cubo a mathematical journal vol.11, no¯ 04, (1–13). september 2009 a new kupka type continuity, λ-compactness and multifunctions m. caldas departamento de matemática aplicada, universidade federal fluminense, rua mário santos braga, s/n, 24020-140, niterói, rio de janeiro, brasil. email: gmamccs@vm.uff.br e. hatir eǧitim fakültesi, selçuk üniversitesi, matematik bölümü, 42090 konya, turkey. email: hatir10@yahoo.com s. jafari department of economics, copenhagen university, oester farimagsgade 5, bygning 26, 1353 copenhagen k, denmark. email: jafari@stofanet.dk and t. noiri 2949-1 shiokita-cho, hinagu, yatsushiro-shi, kumamoto-ken, 869-5142, japan. email: t.noiri@nifty.com 2 m. caldas, e. hatir, s. jafari and t. noiri cubo 11, 4 (2009) abstract in this paper, we introduce a new kupka type function called λ-kupka and we investigate some of its properties. also we obtain several characterizations and some basic properties concerning upper (lower) λ-continuous multifunctions. resumen en este artículo introducimos un nueva función del tipo kupka llamada λ-kupka e investigamos algunas de sus propiedades. además, obtenemos diversas caracterizaciones y algunas propiedades básicas referentes a multifunciones continuas superiores e inferiores λcontinuas. key words and phrases: topological spaces, λ-sets, λ-open sets, λ-closed sets, λ-kupka continuity, multifunction. math. subj. class.: 54b05, 54c08; secondary: 54d05. 1 introduction maki [6] introduced the notion of λ-sets in topological spaces. a λ-set is a set a which is equal to its kernel(= saturated set), i.e. to the intersection of all open supersets of a. arenas et al. [1] introduced and investigated the notion of λ-closed sets and λ-open sets by involving λ-sets and closed sets. kupka [5] introduced firm continuity in order to study compactness. in the same spirit, we introduce and investigate the notion of λ-kupka continuity to study λ-compactness. kupka inspired by a number of characterizations of uc spaces (called also atsuji spaces) [7] to characterize compact spaces. in doing this, he asked the question that what kind of continuity should replace uniform to be sufficiently strong to characterize compact spaces. he answered to this question by introducing a new type of continuity between topological spaces called firm continuity. he obtained several characterizations of compact spaces. this enabled them to introduce a new type function called of firm contra-λ continuous and we use it to study and obtain characterizations of strong λ-closed spaces. lastly we obtain several characterizations concerning upper (lower)λ-continuous multifuntions. throughout this paper we adopt the notations and terminology of [6] and [1] and the following conventions: (x,τ), (y,σ) and (z,ν) (or simply x, y and z) will always denote topological spaces on which no separation axioms are assumed unless explicitly stated. we denote the interior and the closure of a set a by int(a) and cl(a), respectively. a subset a of a space (x,τ) is called λ-closed [1] if a = l∩d, where l is a λ-set and d is a closed set. the complement of a λ-closed set is called λ-open. we denote the collection of all λ-open sets by λo(x,τ) . we set λo(x,x) = {u | x ∈ u ∈ λo(x,τ)}. a point x in a topological space (x,τ) cubo 11, 4 (2009) a new kupka type continuity, λ-compactness and multifunctions 3 is called a λ-cluster point of a [3] if a ∩ u 6= ∅ for every λ-open set u of x containing x. the set of all λ-cluster points of a is called the λ-closure of a and is denoted by λcl(a) ( [1, 3]). definition 1. let b be a subset of a space (x,τ). b is a λ-set (resp. v -set) [6] if b = bλ (resp. b = b v ), where : b λ = ⋂ {u | u ⊃ b, u ∈ τ} and bv = ⋃ {f | b ⊃ f, fc ∈ τ} 2 λ-compactness and λ-kupka continuity definition 2. a space (x,τ) is said to be λ-compact [2] (also called λo-compact [4]) if every cover of x by λ-open sets has a finite subcover. it should be noted that in this paper we use the notation λ-compact instead of λo-compact. in [4], ganster et al. give some proper examples of λo-compact spaces and establish their relationships with some other strong compactness notions. for the convenience of the interested reader we mention an example of λo-compact spaces from [4]: let τ1 be the cofinite topology on x and τ2 be the point generated topology on x with respect to a point p ∈ x. let τ = τ1 ∩τ2. then (x,τ) is hereditarily compact and λo-compact. theorem 2.1. a topological space (x,τ) is λ-compact if and only if for every family {ai | i ∈ i} of λ-closed sets in x satisfying ⋂ i∈i {ai} = ∅, there is a finite subfamily ai1, ...,ain with ⋂ 1≤k≤n {ai} = ∅. proof. straightforward. recall that a function f : (x,τ) → (y,σ) is said to be λ-irresolute [2] if f−1(v ) is λ-open in (x,τ) for every λ-open set v of (y,σ). theorem 2.2. if f : (x,τ) → (y,σ) is a λ-irresolute surjection and (x,τ) is a λ-compact space, then (y,σ) is λ-compact. proof. let {vi | i ∈ i} be any cover of y by λ-open sets of (y,σ). since f is λ-irresolute {f−1(vi) | i ∈ i} is a cover of x by λ-open sets of (x,τ). by λ-compactness of (x,τ), there exists a finite subset i0 of i such that x = ⋃ i∈i0 {f−1(vi)}. since f is surjective, we obtain y = f(x) = ⋃ i∈i0 {vi}. this shows that (y,σ) is λ-compact. definition 3. a function f : x → y , where x and y are topological spaces, is said to have property ϕ [5] if for every open cover ∇ of y there exists a finite cover (the members of which need not be necessarily open) {a1,a2, ...,an} of x such that for each i ∈ {1, 2, ...,n}, there exists a set ui ∈ ∇ such that f(ai) ⊂ ui. 4 m. caldas, e. hatir, s. jafari and t. noiri cubo 11, 4 (2009) definition 4. a function f : x → y is said to be firmly continuous [5] if for every open cover ∇ of y there exists a finite open cover ξ of x such that for every u ∈ ξ there exists a set g ∈ ∇ such that f(u) ⊂ g. definition 5. a function f : x → y , where x and y are topological spaces, is said to have property ψ if for every λ-open cover ∇ of y there exists a finite cover (the members of which need not be necessarily λ-open) {a1,a2, ...,an} of x such that for each i ∈ {1, 2, ...,n}, there exists a set ui ∈ ∇ such that f(ai) ⊂ ui. lemma 2.3. a topological space x is λ-compact if and only if for every topological space y and every λ-irresolute function f : x → y , f has the property ψ. proof. suppose that the topological space x is λ-compact and the function f : x → y is λ-irresolute. let ξ be a λ-open cover of y . the set f(x) is λ-compact relative to y . this means that there exists a finite subfamily {u1,u2, ...,un} of ξ which cover f(x). then the sets a1 = f −1 (u1),a2 = f −1 (u2), ...,an = f −1 (un) form a cover of x such that f(ai) ⊂ ui for each i ∈ {1, 2, ...,n}. conversely, assume that x is a topological space such that for every topological space y and every λ-irresolute function f : x → y , f has property ψ. it follows that the identity function idx : x → x has also property ψ. therefore for every λ-open cover ξ of x, there exists a finite cover a1,a2, ...,an of x such that for each i ∈ {1, 2, ...,n} there exists a set ui ∈ ξ such that ai = idx(ai) ⊂ ui. then {u1,u2, ...,un} is a subcover of ξ. since ξ was an arbitrary λ-open cover of x, the space x is λ-compact. definition 6. a function f : x → y is said to be λ-kupka continuous if for every λ-open cover ∇ of y there exists a finite λ-open cover ξ of x such that for every u ∈ ξ, there exists a set g ∈ ∇ such that f(u) ⊂ g. remark 2.4. it should be noted that if the topological space x is λ-compact and y is an arbitrary topological space, then every λ-irresolute function f : x → y is λ-kupka continuous. lemma 2.5. let x, y , z and w be topological spaces. let g : x → y and h : z → w be λ-irresolute functions and let f : y → z be λ-kupka continuous. then the functions f ◦g : x → z and h ◦ f : y → w are λ-kupka continuous. lemma 2.6. let x and y be topological spaces. suppose that f : x → y is a λ-irresolute function which has the property ψ. then f is λ-kupka continuous. theorem 2.7. the following statements are equivalent for a topological space (x,τ): (1) x is λ-compact; (2) the identity function idx : x → x is λ-kupka continuous; (3) every λ-irresolute function from x to x is λ-kupka continuous; (4) every λ-irresolute function from x to a topological space y is λ-kupka continuous; (5) every λ-irresolute function from x to a topological space y has the property ψ; (6) for each topological space y and each λ-irresolute function f : y → x, f is λ-kupka continuous. cubo 11, 4 (2009) a new kupka type continuity, λ-compactness and multifunctions 5 proof. (1)⇒ (2): suppose that x is λ-compact. the identity function idx : x → x is λ-irresolute and by remark 2.4 idx is λ-kupka continuous. (2)⇒ (3): let f : x → x be any λ-irresolute function. by (2) the identity function idx : x → x is λ-kupka continuous and hence by lemma 2.5 f = idx ◦ f : x → x is λ-kupka continuous. (3)⇒ (4): let f : x → y be any λ-irresolute function. the identity idx : x → x is λirresolute and by (3) idx is λ-kupka continuous. it follows from lemma 2.5 that f = f ◦ idx : x → y is λ-kupka continuous. (4) ⇒ (5): this is obvious. (5) ⇒ (1): this follows immediately from lemma 2.3. (6) ⇒ (2): let idx : x → x be the identity function. then idx is λ-irresolute and by (6) idx is λ-kupka continuous. (1) ⇒ (6): let ∇ be a λ-open cover of x. by hypothesis, the space x is λ-compact. then there is a finite subcover {u1,u2, ...,un} of ∇. assume that ai = f−1(ui) for i ∈ i, where i = {1, 2, ...,n}. it follows that f(ai) ⊂ ui for every i ∈ i. this shows that f is λ-kupka continuous. remark 2.8. observe that if f : x → y is λ-irresolute, then for each point x in the space x and each λ-open set v of y containing f(x), there exists a λ-open set u containing x such that f(u) is contained in v . 3 λ-compactness and multifunctions let λ be a directed set. now we introduce the following notions which will be used in this paper. a net ξ = {xα | α ∈ λ} λ-accumulates at a point x ∈ x if the net is frequently in every u ∈ λo(x,x), i.e. for each u ∈ λo(x,x) and for each α0 ∈ λ, there is some α ≥ α0 such that xα ∈ u. the net ξ λ-converges to a point x of x if it is eventually in every u ∈ λo(x,x). we say that a filterbase θ = {fα | α ∈ γ} λ-accumulates at a point x ∈ x if x ∈ ⋂ α∈γ λcl(fα). given a set s with s ⊂ x, a λ-cover of s is a family of λ-open subsets uα of x for each α ∈ i such that s ⊂ ⋃ α∈i uα. a filterbase θ = {fα | α ∈ γ} λ-converges to a point x in x if for each u ∈ λo(x,x), there exists an fα in θ such that fα ⊂ u. recall that a multifunction (also called multivalued function ) f on a set x into a set y , denoted by f : x → y , is a relation on x into y , i.e. f ⊂ x × y . let f : x → y be a multifunction. the upper and lower inverse of a set v of y are denoted by f +(v ) and f−(v ): f + (v ) = {x ∈ x | f(x) ⊂ v } and f−(v ) = {x ∈ x | f(x) ∩ v 6= ∅} we begin with the following notions: definition 7. a point x in a space x is said to be a λ-complete accumulation point of a subset 6 m. caldas, e. hatir, s. jafari and t. noiri cubo 11, 4 (2009) s of x if card(s ∩ u) = card(s) for each u ∈ λo(x,x), where card(s) denotes the cardinality of s. definition 8. in a topological space x, a point x is called a λ-adherent point of a filterbase θ on x if it lies in the λ-closure of all sets of θ. theorem 3.1. a space x is λ-compact if and only if each infinite subset of x has a λ-complete accumulation point. proof. let the space x be λ-compact and s an infinite subset of x. let k be the set of points x in x which are not λ-complete accumulation points of s. now it is obvious that for each point x in k, we are able to find u(x) ∈ λo(x,x) such that card(s ∩ u(x)) 6= card(s). if k is the whole space x, then θ = {u(x) | x ∈ x} is a λ-cover of x. by the hypothesis x is λ-compact, so there exists a finite subcover ψ = {u(xi)}, where i = 1, 2, ...,n such that s ⊂ ⋃ {u(xi) ∩s | i = 1, 2, ...,n}. then card(s) = max{card(u(xi)∩s) | i = 1, 2, ...,n} which does not agree with what we assumed. this implies that s has a λ-complete accumulation point. now assume that x is not λ-compact and that every infinite subset s ⊂ x has a λ-complete accumulation point in x. it follows that there exists a λ-cover ξ with no finite subcover. set δ = min{card(φ) | φ ⊂ ξ, where φ is a λ-cover of x}. fix ψ ⊂ ξ for which card(ψ) = δ and ⋃ {u | u ∈ ψ} = x. let n denote the set of natural numbers. then by hypothesis δ ≥ card(n). by well-ordering of ψ by some minimal well-ordering " ∼ ", suppose that u is any member of ψ. by minimal well-ordering " ∼" we have card({v | v ∈ ψ,v ∼ u}) < card({v | v ∈ ψ}). since ψ can not have any subcover with cardinality less than δ, then for each u ∈ ψ we have x 6= ⋃ {v | v ∈ ψ,v ∼ u}. for each u ∈ ψ, choose a point x(u) ∈ x \ ⋃ {v ∪{x(v )} | v ∈ ψ,v ∼ u}. we are always able to do this if not one can choose a cover of smaller cardinality from ψ. if h = {x(u) | u ∈ ψ}, then to finish the proof we will show that h has no λ-complete accumulation point in x. suppose that z is a point of the space x. since ψ is a λ-cover of x then z is a point of some set w in ψ. by the fact that u ∼ w , we have x(u) ∈ w . it follows that t = {u | u ∈ ψ and x(u) ∈ w} ⊂ {v | v ∈ ψ,v ∼ w}. but card(t) < δ. therefore card(h ∩ w) < δ. but card(h) = δ ≥ card(n) since for two distinct points u and w in ψ, we have x(u) 6= x(w). this means that h has no λ-complete accumulation point in x which contradicts our assumptions. therefore x is λ-compact. theorem 3.2. for a space x the following statements are equivalent: (1) x is λ-compact; (2) every net in x with a well-ordered directed set as its domain λ-accumulates to some point of x. proof. (1) ⇒ (2): suppose that (x,τ) is λ-compact and ξ = {xα | α ∈ λ} a net with a well-ordered directed set λ as domain. assume that ξ has no λ-adherent point in x. then for each point x in x, there exist v (x) ∈ λo(x,x) and an α(x) ∈ λ such that v (x) ∩ {xα | α ≥ α(x)} = ∅. this implies that {xα | α ≥ α(x)} is a subset of x \v (x). then the collection c = {v (x) | x ∈ x} is a λ-cover of x. by hypothesis of the theorem, x is λ-compact and so c has a finite subfamily {v (xi)}, where i = 1, 2, ...,n such that x = ⋃ {v (xi)}. suppose that the corresponding elements cubo 11, 4 (2009) a new kupka type continuity, λ-compactness and multifunctions 7 of λ be {α(xi)}, where i = 1, 2, ...,n. since λ is well-ordered and {α(xi)}, where i = 1, 2, ...,n is finite, the largest element of {α(xi)} exists. suppose it is {α(xl)}. then for γ ≥ {α(xl)}, we have {xδ | δ ≥ γ} ⊂ ⋂n i=1(x \ v (xi)) = x \ ⋃n i=1 v (xi) = ∅ which is impossible. this shows that ξ has at least one λ-adherent point in x. (2) ⇒ (1): now it is enough to prove that each infinite subset has a λ-complete accumulation point by utilizing theorem 3.1. suppose that s ⊂ x is an infinite subset of x. according to zorn’s lemma, the infinite set s can be well-ordered. this means that we can assume s to be a net with a domain which is a well-ordered index set. it follows that s has a λ-adherent point z. therefore z is a λ-complete accumulation point of s. this shows that x is λ-compact. theorem 3.3. a space x is λ-compact if and only if each filterbase in x has at least one λadherent point. proof. suppose that x is λ-compact and θ = {fα | α ∈ γ} a filterbase in it. since all finite intersections of fα’s are non-empty, it follows that all finite intersection of λcl(fα)’s are also non-empty. now it follows from theorem 2.1 that ⋂ α∈γ λcl(fα) is non-empty. this means that θ has at least one λ-adherent point. now suppose θ is any family of λ-closed sets. let each finite intersection be non-empty. the sets fα with their finite intersection establish a filterbase θ. therefore θ λ-accumulates to some point z in x. it follows that z ∈ ⋂ α∈γ fα. now we have by theorem 2.1 that x is λ-compact. theorem 3.4. a space x is λ-compact if and only if each filterbase on x with at most one λ-adherent point is λ-convergent. proof. suppose that x is λ-compact, x a point of x and θ is a filter base on x. the λ-adherence of θ is a subset of {x}. then the λ-adherence of θ is equal to {x} by theorem 3.3. assume that there exists v ∈ λo(x,x) such that for all f ∈ θ, f ∩ (x \ v ) is non-empty. then ψ = {f \ v | f ∈ θ} is a filterbase on x. it follows that the λ-adherence of ψ is non-empty. however ⋂ f∈θ λcl(f \ v ) ⊂ ( ⋂ f∈θ λcl(f)) ∩ (x \ v ) = {x} ∩ (x \ v ) = ∅. but this is a contradiction. hence for each v ∈ λo(x,x), there exists an f ∈ θ with f ⊂ v . this shows that θ λ-converges to x. to prove the converse, it suffices to show that each filterbase in x has at least one λ-accumulation point. assume that θ is a filterbase on x with no λ-adherent point. by hypothesis, θ λ-converges to some point z in x. suppose fα is an arbitrary element of θ. then for each v ∈ λo(x,z), there exists fβ ∈ θ such that fβ ⊂ v . since θ is a filterbase, there exists a γ such that fγ ⊂ fα ∩ fβ ⊂ fα ∩ v , where fγ non-empty. this means that fα ∩ v is non-empty for every v ∈ λo(x,z) and correspondingly for each α, z is a point of λcl(fα). it follows that z ∈ ⋂ α λcl(fα). therefore z is a λ-adherent point of θ which is a contradiction. this shows that x is λ-compact. now, we further investigate properties of λ-compactness by 1-lower and 1-upper λ-continuous functions. we begin with the following notions and in what follows r denotes the set of real 8 m. caldas, e. hatir, s. jafari and t. noiri cubo 11, 4 (2009) numbers. definition 9. a function f : x → r is said to be 1-lower (resp. 1-upper) λ-continuous at the point y in x if for each λ > 0, there exists a λ-open set u(y) such that f(x) > f(y) \ λ(resp. f(x) > f(y)+λ) for every point x in u(y). the function f is 1-lower (resp. 1-upper) λ-continuous in x if it has these properties for every point x of x. theorem 3.5. a function f : x → r is 1-lower λ-continuous if and only if for each η ∈ r, the set of all x such that f(x) ≤ η is λ-closed. proof. it is obvious that the family of sets τ = {(η,∞) | η ∈ r} ∪ r establishes a topology on r. then the function f is 1-lower λ-continuous if and only if f : x → (r,τ) is λ-continuous. the interval (−∞,η] is closed in (r,τ). it follows that f−1((−∞,η]) is λ-closed. therefore the set of all x such that f(x) ≤ η is equal to f−1((−∞,η]) and thus is λ-closed. corollary 3.6. a subset s of x is λ-compact if and only if the characteristic function xs is 1-lower λ-continuous. theorem 3.7. a function f : x → r is 1-upper λ-continuous if and only if for each η ∈ r, the set of all x such that f(x) ≥ η is λ-closed. corollary 3.8. a subset s of x is λ-compact if and only if the characteristic function xs is 1-upper λ-continuous. theorem 3.9. if the function f(x) = supi∈ifi(x) exists, where fi are 1-lower λ-continuous functions from x into r, then f(x) is 1-lower λ-continuous. proof. suppose that η ∈ r. let f(x) < η and therefore for every i ∈ i, fi(x) < η. it is obvious that {x ∈ x | f(x) ≤ η} = ⋂ i∈i{x ∈ x | fi(x) ≤ η}. since each fi is 1-lower λcontinuous, then each set of the form {x ∈ x | fi(x) ≤ η} is λ-closed in x by theorem 3.5. since an arbitrary intersection of λ-closed sets is λ-closed, then f(x) is 1-lower λ-continuous. theorem 3.10. if the function g(x) = infi∈ifi(x) exists, where fi are 1-upper λ-continuous functions from x into r, then g(x) is 1-upper λ-continuous. theorem 3.11. let f : x → r be a 1-lower λ-continuous function, where x is λ-compact. then f assumes the value m = infx∈xf(x). proof. suppose η > m. since f is 1-lower λ-continuous, then the set k(η) = {x ∈ x | f(x) ≤ η} is a non-empty λ-closed set in x by infimum property. hence the family {k(η) | η > m} is a collection of non-empty λ-closed sets with finite intersection property in x. by theorem 2.1 this family has non-empty intersection. suppose z ∈ ⋂ η>m k(η). therefore f(z) = m as we wished to prove. theorem 3.12. let f : x → r be a 1-upper λ-continuous function, where x is a λ-compact space. then f attains the value m = supx∈xf(x). cubo 11, 4 (2009) a new kupka type continuity, λ-compactness and multifunctions 9 proof. it is similar to the proof of theorem 3.11. it should be noted that if a function f at the same time satisfies conditions of theorem 3.11 and theorem 3.12, then f is bounded and attains its bound. here, we give some characterizations of λ-compact spaces by using lower (resp. upper) λcontinuous multifunctions. definition 10. a multifunction f : x → y is said to be lower (resp. upper) λ-continuous if x \ f−(s) (resp. f−(s)) is λ-closed in x for each open (resp. closed) set s in y . lemma 3.13. for a multifunction f : x → y , the following statements are equivalent: (1) f is lower λ-continuous; (2) if x ∈ f−(u) for a point x in x and an open set u ⊂ y , then v ⊂ f−(u) for some v ∈ λo(x); (3) if x /∈ f +(d) for a point x in x and a closed set d ⊂ y , then f +(d) ⊂ k for some λ-closed set k with x /∈ k; (4) f−(u) ∈ λo(x) for each open set u ⊂ y . lemma 3.14. for a multifunction f : x → y , the following statements are equivalent: (1) f is upper λ-continuous; (2) if x ∈ f +(v ) for a point x in x and an open set v ⊂ y , then f(u) ⊂ v for some u ∈ λo(x); (3) if x /∈ f−(d) for a point x in x and a closed set d ⊂ y , then f−(d) ⊂ k for some λ-closed set k with x /∈ k; (4) f +(u) ∈ λo(x) for each open set u ⊂ y . recall that a relation, denoted by ≤, on a set x is said to be a partial order for x if it satisfies the following properties: (i) x ≤ x holds for every x ∈ x (reflexitivity), (ii) if x ≤ y and y ≤ x, then x = y (antisymmetry), (iii)if x ≤ y and y ≤ z, then x ≤ z (transivity). a set equipped with an order relation is called a partially ordered set (or poset). theorem 3.15. the following two statements are equivalent for a space x: (1) x is λ-compact. (2) every lower λ-continuous multifunction from x into the closed sets of a space assumes a minimal value with respect to set inclusion relation. proof. (1) ⇒ (2): suppose that f is a lower λ-continuous multifunction from x into the closed subsets of a space y . we denote the poset of all closed subsets of y with the set inclusion relation "⊆" by λ. now we show that f : x → λ is a lower λ-continuous function. we will show that n = f−({s ⊂ y | s ∈ λ and s ⊆ c}) is λ-closed in x for each closed set c of y . let z /∈ n, then f(z) 6= s for every closed set s of y . it is obvious that z ∈ f−(y \c), where y \c is open in y . by lemma 3.13 (2), we have w ⊂ f−(y \ c) for some w ∈ λo(z). hence f(w) ∩ (y \ c) 6= ∅ for each w in w . so for each w in w , f(w) \ c 6= ∅. consequently, f(w) \ s 6= ∅ for every closed 10 m. caldas, e. hatir, s. jafari and t. noiri cubo 11, 4 (2009) subset s of y for which s ⊆ c. we consider that w ∩ n = ∅. this means that n is λ-closed. it is clear to observe that f assumes a minimal value. (2) ⇒ (1): suppose that x is not λ-compact. it follows that we have a net {xi | i ∈ λ}, where λ is a well-ordered set with no λ-accumulation point by [5, theorem 3.2]. we give λ the order topology. let mj = λcl({xi | i ≥ j}) for every j in λ. we establish a multifunction f : x → λ where f(x) = {i ∈ λ | i ≥ jx}, jx is the first element of all those j’s for which x /∈ mj. since λ has the order topology, f(x) is closed. by the fact that {jx | x ∈ x} has no greatest element in λ, then f does not assume any minimal value with respect to set inclusion. we now show that f − (u) ∈ λo(x) for every open set u in λ. if u = λ, then f−(u) = x which is λ-open. suppose that u ⊂ λ and z ∈ f−(u). it follows that f(z) ∩ u 6= ∅. suppose j ∈ f(z) ∩ u. this means that j ∈ u and j ∈ f(z) = {i ∈ λ | i ≥ jx}. therefore mj ≥ mjx . since z /∈ mjx , then z /∈ mj. there exists w ∈ so(z) such that w ∩ {xi | i ∈ λ} = ∅. this means that w ∩ mj = ∅. let w ∈ w . since w ∩ mj = ∅, it follows that w /∈ mj and since jw is the first element for which w /∈ mj, then jw ≤ j. therefore j ∈ {i ∈ λ | i ≥ jw} = f(w). by the fact that j ∈ u, then j ∈ f(w) ∩ u. it follows that f(w) ∩ u 6= ∅ and therefore w ∈ f−(u). so we have w ⊂ f−(u) and thus z ∈ w ⊂ f−(u). therefore f−(u) is λ-open. this shows that f is lower λ-continuous which contradicts the hypothesis of the theorem. so the space x is λ-compact. theorem 3.16. the following two statements are equivalent for a space x: (1) x is λ-compact. (2) every upper λ-continuous multifunction from x into the subsets of a t1-space attains a maximal value with respect to set inclusion relation. proof. its proof is similar to that of theorem 3.15. the following result concerns the existence of a fixed point for multifunctions on λ-compact spaces. theorem 3.17. suppose that f : x → y is a multifunction from a λ-compact domain x into itself. let f(s) be λ-closed for s being a λ-closed set in x. if f(x) 6= ∅ for every point x ∈ x, then there exists a nonempty, λ-closed set c of x such that f(c) = c. proof. let λ = {s ⊂ x | s 6= ∅,s ∈ λc(x) and f(s) ⊂ s}. it is evident that x belongs to λ. therefore λ 6= ∅ and also it is partially ordered by set inclusion. suppose that {sγ} is a chain in λ. then f(sγ) ⊂ sγ for each γ. by the fact that the domain is λ-compact, s = ⋂ γ sγ 6= ∅ and also s ∈ λc(x). moreover, f(s) ⊂ f(sγ) ⊂ sγ for each γ. it follows that f(s) ⊂ sγ. hence s ∈ λ and s = inf{sγ}. it follows from zorn’s lemma that λ has a minimal element c. therefore c ∈ λc(x) and f(c) ⊂ c. since c is the minimal element of λ, we have f(c) = c. cubo 11, 4 (2009) a new kupka type continuity, λ-compactness and multifunctions 11 4 some properties of (m,n)-λ-compact spaces we begin with the following notions which will be used in the sequel. definition 11. a space (x,τ) is said to be (m,n)-λ-compact if from every λ-open covering {ui | i ∈ i} of x whose cardinality i, denoted by card i, is at most n one can select a subcovering {uij | j ∈ j} of x whose card j is at most m. definition 12. a subset a of the space (x,τ) is said to be (m,n)-λ-compact relative to x if form every cover {ui | i ∈ i} of a by λ-open sets of x whose card is at most n, one can select a subcover {uij | j ∈ j} of a whose card j is at most m. definition 13. a space (x,τ) is said to be completely (m,n)-λ-compact if every subset of x is (m,n)-λ-compact relative to x. remark 4.1. observe that a (1,n)-λ-compact space is a n-λ-compact space and (1,∞)-λ-compact space is the usual λ-compact space. a (1,ω)-λ-compactness is λ-compactness in the fréchet sense and a (ω,∞)-λ-compact space is a λ-lindelöf space. definition 14. a family {ui | i ∈ i} of subsets of a set x is said to have the m-intersection property if every subfamily of cardinality at most m has a non-void intersection. theorem 4.2. a space (x,τ) is (m,n)-λ-compact if and only if every family {pi} of λ-closed sets pi ⊆ x having the m-intersection property also has the n-intersection property. proof. the proof is a consequence of the following equivalent statements: (1) x is (m,n)-λ-compact; (2) if {ui | i ∈ i} is a λ-open cover of x such that card i ≤ n, then there is a subcover {uij } of x such that card j ≤ m; (3) if {ui | i ∈ i} is a family of λ-open sets such that x − (∪iuij ) 6= ∅ whenever card j ≤ m, then x − (∪iuij ) 6= ∅ whenever card j ≤ n; (4) if {pi | i ∈ i} is a family of λ-closed sets having the m-intersection property then {pi} has also the n-intersection property. theorem 4.3. if a space x is (m,n)-λ-compact and if y is a λ-closed subset of x, then y is (m,n)-λ-compact relative to x. proof. suppose that {ui | i ∈ i} is a cover of y by λ-open sets of x such that card i ≤ n. by adding uj = x −y , we obtain a λ-open cover of x with cardinality at most n. by eliminating uj, we have a subcover of {ui} whose cardinality is at most m. theorem 4.4. if x is a space such that every λ-open subset of x is (m,n)-λ-compact relative to x, then x is completely (m,n)-λ-compact. proof. let y ⊂ x and {ui | i ∈ i} be a cover of y by λ-open sets of x such that card i ≤ n. then the family {ui | i ∈ i} is a cover of ∪iui by λ-open sets of x. then, there is a 12 m. caldas, e. hatir, s. jafari and t. noiri cubo 11, 4 (2009) subfamily {uij | j ∈ j} of card j ≤ m which covers ∪iui. this subfamily also covers the set y and so y is (m,n)-λ-compact relative to x. theorem 4.5. let x be a space and {yk | k ∈ k} be a family of subsets. if every yk is (m,n)λ-compact relative to x for some m ≥ card k, then ∪{yk | k ∈ k} is (m,n)-λ-compact relative to x. proof. if {ui | i ∈ i} is a cover of y = ∪kyk by λ-open sets of x, then it is a cover of yk by λ-open sets of x for every k ∈ k. if card i ≤ n, then {ui} contains a subfamily {uij | jk ∈ jk} for which card jk ≤ m and is a covering of yk. the union of these families is a λ-open subfamily of {ui} which covers y and its cardinality is at most m. definition 15. a point x ∈ x is called an m-λ-accumulation point of a set s in x if for every λ-open set ux containing x, we have card (ux ∩ s) > m. observe that if m = 0, 1 or ω, then the relation card (ux ∩ s) > m means that ux ∩ s 6= ∅, not finite or not countable. theorem 4.6. let x be a space and s a subset of x of cardinality greater than m (i.e. s ⊂ x and card s > m). if x is (m,n)-λ-compact for some n > m, then s has a λ-accumulation point in x. if x is (m,∞)-λ-compact, then s has an m-λ-accumulation point in x. proof. assume that s ⊂ x of cardinality at most n which has no λ-accumulation points in x. then, for each x ∈ x, there is a λ-open set ux such that at most one point of s belongs to ux. suppose u is the union of all sets ux which contain no points of s. let us denote the union of all sets ux which contain the point s ∈ s. then u and us are λ-open sets. therefore {u,us} is a λ-open cover of x of cardinality at most n. if x is (m,n)-λ-compact, then this cover contains a subcover of cardinality at most n. if x is (m,n)-λ-compact, then this cover contains a subcover of cardinality at most m. but this subcover must contain every us since s ∈ s is covered only by us. thus card s ≤ m. if the cardinality of a set s is greater than m, then s has at least one λ-accumulation point in x. the two other cases can be proved by the same token with a little modification. received: april 2008. revised: june 2008. references [1] arenas, f. g. dontchev, j. and ganster, m., on λ-sets and dual of generalized continuity, questions answers gen. topology, 15(1997), 3-13. [2] caldas, m. jafari, s. and navalagi, g., more on λ-closed sets in topological spaces, rev. colomb. mat. 4(2007)2, 355-369. cubo 11, 4 (2009) a new kupka type continuity, λ-compactness and multifunctions 13 [3] caldas, m. and jafari, s., on some low separation axioms via λ-open and λ-closure operator, rend. circ. mat. palermo (2), 54(2005), 195-208. [4] ganster, m. jafari, s. and steiner, m., on some very strong compactness conditions, (submitted). [5] kupka, i., a strong type of continuity natural for compact spaces, tatra mt. math. publ., 14(1998), 17-27. [6] maki, h., generalized λ-sets and the associated closure operator, the special issue in commemoration of prof. kazusada ikeda’ retirement, 1. oct. 1986, 139-146. [7] waterhouse, w., on uc spaces, amer. math. monthly, 72(1965), 634-635. articulo 1 cubo a mathematical journal vol.11, no¯ 05, (51–56). december 2009 k-theory of an algebra of pseudodifferential operators on a noncompact manifold patrícia hess and severino t. melo instituto de matemática e estatística, universidade de são paulo, rua do matão 1010, 05508-090 são paulo, brazil emails: phess@ime.usp.br, toscano@ime.usp.br abstract let a denote the c*-algebra of bounded operators on l 2(r × s1) generated by: all multiplications a(m ) by functions a ∈ c∞(s1), all multiplications b(m ) by functions b ∈ c([−∞, +∞]), all multiplications by 2π-periodic continuous functions, λ = (1 − ∆r×s1 ) −1/2 , where ∆r×s1 is the laplacian operator on l 2(r × s1), and ∂tλ, ∂xλ, for t ∈ r and x ∈ s 1 . we compute the k-theory of a and of its quotient by the ideal of compact operators. resumen denotemos a la c*-algebra de operadores acotados sobre l2(r × s1) gerados por todas las multiplicaciones a(m ) por funciones a ∈ c∞(s1), todas las multiplicaciones b(m ) por funciones b ∈ c([−∞, +∞]), todas las multiplicaciones por funciones continuas 2π-periódicas, λ = (1 − ∆r×s1 ) −1/2 , donde ∆r×s1 es el operador de laplace sobre l 2(r × s1), y ∂tλ, ∂xλ, para t ∈ r y x ∈ s 1 . nosotros calculamos la k-teoria de a y su cuociente por el ideal de los operadores compactos. key words and phrases: k-theory, pseudodifferential operators. math. subj. class.: 46l80, 47g30. 52 p. hess and s.t. melo cubo 11, 5 (2009) 1 introduction let b denote an n-dimensiomal compact riemannian manifold. write ω = r×b1 and let ∆ω denote its laplacian. define a as the c*-algebra of bounded operators on l2(ω) generated by: (i) all a(mx), operators of multiplication by a ∈ c ∞ (b), [a(mx)f ](t, x) = a(x)f (t, x) ; (ii) all b(mt), operators of multiplication by b ∈ c([−∞, +∞]), [b(mt)f ](t, x) = b(t)f (t, x) ; (iii) every multiplication by 2π-periodic continuous functions; (iv) λ = (1 − ∆ω) −1/2 ; (v) 1 i ∂ ∂t λ, t ∈ r; (vi) lλ, where l ia a first order differential operator on b. a contains a class of classical pseudodifferential operators, including all a = lλn , where l is a differential operator of order n with coefficients approaching periodic functins at infinity. a : l2(ω) → l2(ω) is fredholm if and only if so is l : hn (ω) → l2(ω) and they have the same index. the structure of a is given in [4], so we will mention some of those results. the principal symbol extends to a ∗-homomorphism σ : a → cb(s ∗ ω), where cb(s ∗ ω) denotes the algebra of continuous funtions on the co-sphere bundle of ω. let e denote the kernel of σ and kω := k(l 2 (ω)) the ideal of compact operators on l2(ω). theorem 1. there is a ∗-isomorphism ψ : e kω −→ c(s1 ×{−1, +1},kz×b). (1) in fact, e is the commutator ideal of a. composing the canonical projection e → e/kω with ψ, we can extend the map e → c(s1 × {−1, +1},kz×b) and obtain a ∗-homomorphism γ : a→ c(s1 ×{−1, +1},l(l2(z × b))). from now on, we will just work in the case that the manifold b is the circle s1. therefore, the class of generators lλ can be replaced by the operator 1 i ∂ ∂β λ, β ∈ s1, since s1 has trivial tangent bundle. definition 2. let d be a c*-algebra and i its commutator ideal. we call symbol space of d the compact space m such that we have d/i ∼= c(m ) by the gelfand map. with this definition, we observe that the map σ is the composition a → a/e → c(ma), where ma is the symbol space of a. given a ∈a, we say that the symbol of a is the continuous function σa ∈ c(ma) associated to the class a in a/e by the gelfand isomorphism. cubo 11, 5 (2009) k-theory of an algebra of pseudodifferential operators ... 53 theorem 3. the symbol space of a is the set ma = {(t, x, (τ, ξ), e iθ ); t ∈ [−∞, +∞], x ∈ s1, (τ, ξ) ∈ r2 : τ 2 + ξ2 = 1, θ ∈ r e θ = t se |t| < ∞}, and the symbols of the generators (or classes of them) of a are given below as functions on (t, x, (τ, ξ), eiθ ) in the same order that they appear in the definition: a(x), b(t), eijθ, 0, τ, ξ. (2) for each ϕ ∈ r, let uϕ be the operator on l 2 (s1) given by uϕf (z) = z −ϕf (z), z ∈ s1, and let yϕ := fduϕf −1 d , where fd : l 2 (s1) → ℓ2(z) is the discrete fourier transform. so, yϕ is an unitary operator such that, for all k ∈ z, (yku)j = uj+k, and yϕyω = yϕ+ω. consider the following functions defined on r and taking values in the algebra of bounded operators on l2(s1): b4(τ ) = λ̃(τ ) , b5(τ ) = −τ λ̃(τ ) , b6(τ ) = 1 i ∂ ∂β λ̃(τ ), τ ∈ r, β ∈ s1, (3) where λ̃(τ ) = (1 + τ 2 − ∆s1 ) −1/2. proposition 4. for each generator a of a, γa(e 2πiϕ,±1) is given respectively by: a(mx) , b(±∞) , y−j , yϕbi(ϕ − mj)y−ϕ , i = 4, 5, 6, where bi(ϕ−mj ) ∈l(ℓ 2 (z, l2(s1))), i = 4, 5, 6, are the operators of multiplicaion by the sequences bi(ϕ − j) ∈ l 2 (s 1 ), and bi’s are given in (3). in this note, we announce results about the calculus of the k-theory of a. their proofs will appear in a forthcoming paper. 2 c*-subalgebras of a now, we define two c*-subalgebras of a and mention some facts about their structure. with the results mentioned here, we can compute their k-theory in the next section. let a† the c*-algebra of bounded operators on l2(ω) generated by the class of operators of multiplication by a ∈ c∞(s1), λ, 1 i ∂tλ and 1 i ∂β λ, and let a ⋄ denote the c*-algebra generated by the same operators of a† and operators of multiplication by 2π-periodic continuous functions. based on cordes, [1], and melo, [4], we can prove the following result. theorem 5. let e† and e⋄ denote the commutator ideals of a† and a⋄, respectively. then e† ∼= c0(r,k(l 2 (s 1 ))) and e⋄ ∼= c(s1,k(l2(z × s1))). 54 p. hess and s.t. melo cubo 11, 5 (2009) the ∗-isomorphism e⋄ ∼= c(s1,k(l2(z × s1))) can be extended to a ∗-homomorphism γ′ : a⋄ → c(s1,l(l2(z × s1))) given by γ′a(z) = γa(z,±1) for every a ∈a ⋄ ⊂a. knowing these isomorphisms, we are able to describe the symbol spaces of both c*-subalgebras of a. theorem 6. the symbol space of a† is homeomorphic to s1 × s1 and the symbol space of a⋄ is homeomorphic to s1 × s1 × s1. moreover, the symbol of an operator in a† or a⋄ coincides with the symbol of same operator regarded as an element of a. we can see a† as a c*-subalgebra of l(l2(r × s1)) as well. so, if we conjugate a† with f ⊗ is1 , where f : l 2 (r) → l2(r) is the fourier transform and is1 is the identity on l 2 (s 1 ), we obtain an algebra that can be viewed as a subalgebra of cb(r,l(l 2 (s 1 ))). let b† be this algebra. the idea is to define an action, and then, to show that that a⋄ has a crossed product structure. consider α the following translation-by-one automorphism: [α(b)](τ ) = b(τ − 1), τ ∈ r, l ∈b†. let b⋄ the algebra a⋄ conjugated with the fourier transform. analogously to [5, theorem 8], we obtain the next result. theorem 7. let α be as described above. we have: b⋄ ∼= b † ⋊αz where b† ⋊α z is the envelopping c*-algebra([2], 2.7.7) of the banach algebra with involution ℓ1(z,b†) of all summable z-sequences in b†, equipped with the product (a · b)(n) = ∑ k∈z akα k (bn−k), n ∈ z, a = (ak)k∈z, b = (bk)k∈z, and involution a∗(n) = αn(a∗ −n). 3 k-theory in this section, we compute the k-groups of the algebras that we were defined in this work. the main idea is to compute the connecting maps in the k-theory six-term exact sequence associated to the c*-algebra short exact sequence induced by the principal symbol 0 −→ e = kerσ i −→ a π −→ a e ∼= imσ −→ 0, (4) where i is the inclusion and π is the canonical projection. but before we begin the computation involving the largest algebra, we start with a† and a⋄using the same idea as above. cubo 11, 5 (2009) k-theory of an algebra of pseudodifferential operators ... 55 studying the k-theory six-term exact sequence induced by the sequence 0 −→ e† i −→ a† π −→ a† e† −→ 0, we just obtain partial results about the k-theory of a†, because we do not have a explicit description of the range of the exponential map. the same thing happens with the exponential map in the sequence involving a⋄. to compute the index map δ⋄1 : k1(a ⋄/e⋄) → k0(e ⋄ ), it was necessary to use the fedosov index formula [3]. theorem 8. δ⋄1 is surjective. from theorem 7, we have b⋄ ∼= b†⋊αz. then the k-groups of b † and b⋄ fit in the pimsnervoiculescu exact sequence, [6, theorem 2.4]: k0(b † ) id∗−α −1 ∗ −→ k0(b † ) ϕ∗◦i∗ −→ k0(b ⋄ ) ↑ ↓ k1(b ⋄ ) ϕ∗◦i∗ ←− k1(b † ) id∗−α −1 ∗ ←− k1(b † ) (5) now, we have enough information to conclude that k0(a † ) ∼= z , k1(a † ) ∼= z 2 , ki(a ⋄ ) ∼= z 3 , i = 0, 1. the next step is to work with a. in the begining of this section the sequence induced by the symbol was mentioned. we will rewrite it, quotiented by the ideal of compact operators kω : 0 −→ e kω i −→ a kω π −→ a e −→ 0, (6) where i is the inclusion and π is the canonical projection. by the isomorphism (1), we know that e/kω can be viewed as two copies of c(s 1,k(l2(z × s 1 ))). then, ki ( e kω ) ∼= z ⊕ z, i = 0, 1. we know the range of σ by theorem 3. with the help of [5, proposition 3], we can state the next result. proposition 9. k0(a/e) ∼= k1(a/e) ∼= z 6. from sequence (6), we have: z 2 ∼= k0(e/kω) i∗ −→ k0(a/kω) π∗ −→ k0(a/e) ∼= z 6 δ1 ↑ ↓ δ0 z 6 ∼= k1(a/e) π∗ ←− k1(a/kω) i∗ ←− k1(e/kω) ∼= z 2 (7) 56 p. hess and s.t. melo cubo 11, 5 (2009) the image of δ1 is determined the same way as the image of δ ⋄ 1 . theorem 10. δ1 is surjective. corollary 11. k0(a/kω) has no torsion. theorem 12. given the short exact sequence 0 −→kω i −→ a π −→ a kω −→ 0, where i is the inclusion and π is the canonical projection, the following statements hold: (i) δ1 : k1( a kω ) → z is surjective (there exists a matrix of operators in a with fredholm index equal to one); (ii) k0(a) ∼= k0(a/kω) ∼= z 5 ; (iii) k1(a) ∼= z 4 and k1(a/kω) ∼= z 5. received: december, 2008. revised: april, 2009. references [1] cordes, h.o., on the two-fold symbol chain of a c*-algebra of singular integral operators on a polycylinder, revista matemática iberoamericana, 2(1 y 2):215–234, 1986. [2] dixmier, j., c*-algebras, north holland, new york, 1977. [3] fedosov, b.v., index of an elliptic system on a manifold, funct. anal. and its application, 4(4):312–320, 1970. [4] melo, s.t., a comparison algebra on a cylinder with semi-periodic multiplications, pacific j. math., 142(2):281–304, 1990. [5] melo, s.t. and silva, c.c., k-theory of pseudodifferential operators with semi-periodic symbols, k-theory, 37:235–248, 2006. [6] pimsner, m. and voiculescu, d., exact sequences for k-groups and ext-groups of certain cross-product c*-algebras, j. operator theory, 4(1):93–118, 1980. b4-fonte c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2604 .dvi cubo a mathematical journal vol.13, no¯ 01, (45–60). march 2011 on the solution of generalized equations and variational inequalities ioannis k. argyros cameron university, department of mathematics sciences, universidad nacional autonoma de mexico, lawton, ok 73505, usa. email: iargyros@cameron.edu and säıd hilout poitiers university, laboratoire de mathématiques et applications, bd. pierre et marie curie, téléport 2, b.p. 30179, 86962 futuroscope chasseneuil cedex, france. email: said.hilout@math.univ-poitiers.fr abstract uko and argyros provided in [18] a kantorovich–type theorem on the existence and uniqueness of the solution of a generalized equation of the form f (u)+g(u) ∋ 0, where f is a fréchet–differentiable function, and g is a maximal monotone operator defined on a hilbert space. the sufficient convergence conditions are weaker than the corresponding ones given in the literature for the kantorovich theorem on a hilbert space. however, the convergence was shown to be only linear. in this study, we show under the same conditions, the quadratic instead of the linear convergenve of the generalized newton iteration involved. 46 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) resumen uko y argyros estudian en [18] un teorema tipo-kantorovich en el existencia y unicidad de la solución de una ecuación generalizada de la forma f (u) + g(u) ∋ 0, donde f es una función fréchet–diferenciable, y g es un operador monotono máximo definido en un espacio de hilbert. las condiciones de convergencia suficientes son más débiles que los correspondientemente dadas en la literatura para el teorema de kantorovich en un espacio de hilbert. sin embargo, la convergencia ha demostrado ser sólo lineal. en este estudio, mostramos en las mismas condiciones, la ecuación cuadrática en lugar de la lineal convergente de la iteración generalizada de newton involucradas. keywords: generalized equation, variational inequality, nonlinear complementarity problem, nonlinear operator equation, kantorovich theorem, generalized newton’s method, center–lipschitz condition. ams subject classification: 65k10, 65j99, 49m15, 49j53, 47j20, 47h04, 90c30, 90c33. 1 introduction let h be a hilbert space, let c be a let h be a hilbert space, let c be a closed convex subset of h with non–empty interior d, let f : c 7−→ h be a continuous function that is fréchet–differentiable on d, and let g be a non–empty maximal monotone operator defined on h × h, fixed all through this paper. then there exists α ≥ 0 (the monotonicity modulus of g) such that: [x1, y1] ∈ g and [x2, y2] ∈ g =⇒ (y2 − y1, x2 − x1) ≥ α ‖x1 − x2‖2. (1.1) it is well known (cf. [10]) that g is closed in the sense that [xm, ym] ∈ g, lim m→∞ xm = x and lim m→∞ ym = y =⇒ [x, y] ∈ g. (1.2) in the sequel, we will regard the statements [x, y] ∈ g, g(x) ∋ y, −y + g(x) ∋ 0, and y ∈ g(x) as synonymous. given any u0 ∈ d and r > 0, u (u0, r) will designate the closed ball {x ∈ h : ‖x − u0‖ ≤ r}, and u (u0, r) will designate the corresponding open ball. we are interested in the solvability of the generalized equation: f (u) + g(u) ∋ 0. (1.3) uko and argyros provided in [18] a weak kantorovich theorem that generalizes the kantorovich theorem for generalized equations, which weakened work by uko [16] on the hilbert space cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 47 version of the classical kantorovich theorem [6, 9] for the solvability of nonlinear operator equations. the kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of equations arising in various fields. an important extension of the kantorovich theorem was obtained recently by argyros [1], [2], who used a combination of lipschitz and center–lipschitz conditions in place of the lipschitz conditions used by kantorovich. in the present paper, we will formulate and prove an extension of the kantorovich theorem for the generalized equation (1.3). the depth and scope of this theorem is such that when we specialize it to nonlinear operator equations we get results that are weaker than the kantorovich theorem. our approach will be iterative, and the solution of problem (1.3) will be obtained as the limit of the solutions of the generalized newton subproblems (gnm): f ′(um) um+1 + g(um+1) ∋ f ′(um) um − f (um), m = 0, 1, . . . . (1.4) a well known example (cf. [11]) of a maximal monotone operator is obtained on setting g(x) = ∂φ(x) ≡ {v ∈ h : φ(x) − φ(y) ≤ (v, x − y) ∀y ∈ h} where, φ : h 7−→ (−∞, ∞] is a proper lower semicontinuous convex function. in this case problem (1.3) becomes the variational inequality: f (u) + ∂φ(u) ∋ 0. (1.5) such problems were introduced in the early sixties by stampacchia [15] and have found important applications in the physical and engineering sciences and in many other fields [1]–[21]. the generalized newton iterates (1.4) are obtained as solutions of mildly nonlinear generalized equations of the form: a z + g(z) ∋ b. (1.6) in this study, we recover the desired quadratic convergence of the (gnm), not attained in [18] (under the same hypotheses and computational cost). 2 semilocal convergence analysis of (gnm) uko and argyros showed in [18], and with a different approach in [2, case 3, p. 387] the following result on majorizing sequences for the (gnm): lemma 2.1. let η, α, b, m and m0 be nonnegative constants. 48 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) let: c0 > −α, η = b c0 + α , and 0 ≤ m0 ≤ m. suppose: (4 m0 + m + √ m 2 + 8 m m0) ≤ 4 (c0 + α). (2.1) the inequality in (2.1) is strict if m0 = 0. then, scalar sequence {sk} (k ≥ 0) given by: s0 = 0, s1 = η, sk+1 = sk + m (sk − sk−1)2 2 (1 − m0 sk) (2.2) is well defined, nondecreasing, bounded above by s⋆⋆, and converges to its unique least upper bound s⋆ ∈ [0, s⋆⋆], where s⋆⋆ = 2 η 2 − θ , (2.3) θ = 4 m m + √ m 2 + 8 m0 m (m0 6= 0). (2.4) moreover, the following estimates hold for all k ≥ 0: m0 s ⋆ ≤ 1, (2.5) 0 ≤ sk+1 − sk ≤ ( θ 2 )k η, (2.6) 0 ≤ s⋆ − sk ≤ ( θ 2 )k s⋆⋆. (2.7) in the next result, we show that under the same sufficient convergence condition (2.1) for (gnm), we can improve upon the linear error estimates (2.6), and (2.7), and show instead the quadratic convergence of the majorizing sequence {sn}. it is convenient for us to set: l0 = m0 c0 + α , and l = m c0 + α . cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 49 lemma 2.2. assume there exist constants l0 ≥ 0, l ≥ 0, and η ≥ 0, such that: q0 = l η ≤ 1 2 , (2.8) where, l = 1 8 ( l + 4 l0 + √ l2 + 8 l0 l ) . (2.9) the inequality in (2.8) is strict if l0 = 0. then, sequence {tk} (k ≥ 0) given by t0 = 0, t1 = η, tk+1 = tk + l1 (tk − tk−1)2 2 (1 − l0 tk) (k ≥ 1), (2.10) is well defined, nondecreasing, bounded above by t⋆⋆, and converges to its unique least upper bound t⋆ ∈ [0, t⋆⋆], where l1 = { l0 if k = 1 l if k > 1 , t⋆⋆ = 2 η 2 − δ , (2.11) 1 ≤ δ = 4 l l + √ l2 + 8 l0 l < 2 for l0 6= 0. (2.12) moreover the following estimates hold: l0 t ⋆ < 1, (2.13) 0 ≤ tk+1 − tk ≤ δ 2 (tk − tk−1) ≤ · · · ≤ ( δ 2 )k η, (k ≥ 1), (2.14) tk+1 − tk ≤ ( δ 2 )k (2 q0) 2 k −1 η, (k ≥ 0), (2.15) 0 ≤ t⋆ − tk ≤ ( δ 2 )k (2 q0) 2 k −1 η 1 − (2 q0)2k , (2 q0 < 1), (k ≥ 0). (2.16) proof. if l0 = 0, then (2.13) holds trivially. in this case, for l > 0, an induction argument shows that tk+1 − tk = 2 l (2 q0) 2 k (k ≥ 0), and therefore tk+1 = t1 + (t2 − t1) + · · · + (tk+1 − tk) = 2 l k ∑ m=0 (2 q0) m, 50 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) and t⋆ = lim k→∞ tk = 2 l ∞ ∑ k=0 (2 q0) 2 k . clearly, this series converges, since k ≤ 2k, 2 q0 < 1, and is bounded above by the number 2 l ∞ ∑ k=0 (2 q0) k = 4 l (2 − l η) . if l = 0, then in view of (2.10), 0 ≤ l0 ≤ l, we deduce: l0 = 0, and t⋆ = tk = η (k ≥ 1). in the rest of the proof, we assume that l0 > 0. the result until estimate (2.14) follows from lemma 1 in [2] (see also [1], [3]). note that in particular newton–kantorovich–type convergence condition (2.8) is given in [2, page 387, case 3 for δ given by (2.12). the factor η is missing from the left hand side of the inequality three lines before the end of page 387]. in order for us to show (2.15) we need the estimate: 1 − ( δ 2 )k+1 1 − δ 2 η ≤ 1 l0 ( 1 − ( δ 2 )k−1 l 4 l ) (k ≥ 1). (2.17) for k = 1, (2.17) becomes ( 1 + δ 2 ) η ≤ 4 l − l 4 l l0 or ( 1 + 2 l l + √ l2 + 8 l0 l ) η ≤ 4 l0 − l + √ l2 + 8 l0 l l0 (4 l0 + l + √ l2 + 8 l0 l) in view of (2.8), it suffices to show: l0 (4 l0 + l + √ l2 + 8 l0 l) (3 l + √ l2 + 8 l0 l) (l + √ l2 + 8 l0 l) (4 l0 − l + √ l2 + 8 l0 l) ≤ 2 l, which is true as equality. let us now assume estimate (2.17) is true for all integers smaller or equal to k. we must show (2.17) holds for k being k + 1: cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 51 1 − ( δ 2 )k+2 1 − δ 2 η ≤ 1 l0 ( 1 − ( δ 2 )k l 4 l ) (k ≥ 1). or ( 1 + δ 2 + ( δ 2 )2 + · · · + ( δ 2 )k+1) η ≤ 1 l0 ( 1 − ( δ 2 )k l 4 l ) . (2.18) by the induction hypothesis to show (2.18), it suffices 1 l0 ( 1 − ( δ 2 )k−1 l 4 l ) + ( δ 2 )k+1 η ≤ 1 l0 ( 1 − ( δ 2 )k l 4 l ) or ( δ 2 )k+1 η ≤ 1 l0 (( δ 2 )k−1 − ( δ 2 )k) l 4 l or δ2 η ≤ l (2 − δ) 2 l l0 . in view of (2.8) it suffices to show 2 l l0 δ 2 l (2 − δ) ≤ 2 l, which holds as equality by the choice of δ given by (2.12). that completes the induction for estimates (2.17). we shall show (2.15) using induction on k ≥ 0: estimate (2.15) is true for k = 0 by (2.8), (2.10), and (2.12). in order for us to show estimate (2.15) for k = 1, since t2 − t1 = l (t1 − t0)2 2 (1 − l0 t1) , it suffices: l η2 2 (1 − l0 η) ≤ δ l η2 or l 1 − l0 η ≤ 16 l l l + √ l2 + 8 l0 l (η 6= 0) or η ≤ 1 l0 ( 1 − l + √ l2 + 8 l0 l 16 l ) (l0 6= 0, l 6= 0). but by (2.8) η ≤ 4 l + 4 l0 + √ l2 + 8 l0 l . 52 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) it then suffices to show 4 l + 4 l0 + √ l2 + 8 l0 l ≤ 1 l0 ( 1 − l + √ l2 + 8 l0 l 16 l ) or l + √ l2 + 8 l0 l 16 l ≤ 1 − 4 l0 l + 4 l0 + √ l2 + 8 l0 l or l + √ l2 + 8 l0 l 16 l ≤ l + √ l2 + 8 l0 l l + 4 l0 + √ l2 + 8 l0 l or l ≥ 0, which is true by (2.9). let us assume (2.18) holds for all integers smaller or equal to k. we shall show (2.18) holds for k replaced by k + 1. using (2.10), and the induction hypothesis, we have in turn tk+2 − tk+1 = l 2 (1 − l0 tk+1) (tk+1 − tk)2 ≤ l 2 (1 − l0 tk+1) (( δ 2 )k (2 q0) 2 k −1 η )2 ≤ l 2 (1 − l0 tk+1) (( δ 2 )k−1 (2 q0) −1 η ) (( δ 2 )k+1 (2 q0) 2 k+1 −1 η ) ≤ ( δ 2 )k+1 (2 q0) 2 k+1 −1 η, since, l 2 (1 − l0 tk+1) (( δ 2 )k−1 (2 q0) −1 η ) ≤ 1, (k ≥ 1). (2.19) indeed, we can show instead of (2.19): tk+1 ≤ 1 l0 ( 1 − ( δ 2 )k−1 l 4 l ) , which is true, since by (2.14), and the induction hypothesis: cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 53 tk+1 ≤ tk + δ 2 (tk − tk−1) ≤ t1 + δ 2 (t1 − t0) + · · · + δ 2 (tk − tk−1) ≤ η + ( δ 2 ) η + · · · + ( δ 2 )k η = 1 − ( δ 2 )k+1 1 − δ 2 η ≤ 1 l0 ( 1 − ( δ 2 )k−1 l 4 l ) . that completes the induction for estimate (2.15). using estimate (2.18) for j ≥ k, we obtain in turn for 2 q0 < 1: tj+1 − tk = (tj+1 − tj ) + (tj − tj−1) + · · · + (tk+1 − tk) ≤ (( δ 2 )j (2 q0) 2 j −1 + ( δ 2 )j−1 (2 q0) 2 j−1 −1 + · · · + ( δ 2 )k (2 q0) 2 k −1 ) η ≤ ( 1 + (2 q0) 2 k + ( (2 q0) 2 k )2 + · · · ) ( δ 2 )k (2 q0) 2 k −1 η = ( δ 2 )k (2 q0) 2 k −1 η 1 − (2 q0)2k . (2.20) estimate (2.16) follows from (2.20) by letting j −→ ∞. that completes the proof of lemma 2.2. ♦ we need the result [18], [3],[10], [11]. lemma 2.3. let g be a maximal monotone operator satisfying condition (1.1), and let a be a bounded linear operator mapping h into h. if there exists c ∈ r, such that c > −α, and (a x, x) ≥ c ‖x‖2, ∀x ∈ h. (2.21) then, for any b ∈ h, the problem a z + g(z) ∋ b has a unique solution z ∈ h. we now use lemma 2.2 instead of lemma 2.1 to improve, in particular the error estimates, in the weak kantorovich–type existence theorem for problem (1.3) given in [18]. 54 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) theorem 2.1. let g be a maximal monotone operator satisfying condition (1.1), and suppose that there exist u0 ∈ d, and v0 ∈ h, such that v0 ∈ g(u0), (2.22) (f ′(u0) x, x) ≥ c0 ‖x‖2, ∀x ∈ h, (2.23) ‖f ′(x) − f ′(y)‖ ≤ m ‖x − y‖ ∀x, y ∈ d, (2.24) ‖f ′(x) − f ′(u0)‖ ≤ m0 ‖x − u0‖ ∀x ∈ d, (2.25) ‖f (u0) + v0‖ ≤ b, (2.26) where b ≥ 0, m ≥ m0 ≥ 0 and c0 > −α. let η = b c0 + α , and suppose condition (2.8), and u (u0, t ⋆) ⊆ d hold, where the t⋆ is the limit of the sequence {tm} defined in lemma 2.2. then equation (1.3) has a unique solution u in u (u0, r), where r = 2 η 1 + √ c0 + α − 2 m0 η , (2.27) and the newton iterations generated from (gnm) converge to u, and satisfy the estimates: ‖um − um−1‖ ≤ tm − tm−1, (2.28) ‖um − u0‖ ≤ tm, (2.29) ‖u − um‖ ≤ t⋆ − tm. (2.30) moreover, if there exists t ≥ 0, such that: m0 (r + t ) < 2 (c0 + α), (2.31) then, the solution is unique in u (u0, t ) ∩ d. this solution is also unique in the sets u (u0, t⋆), and u (u0, r) ∪ (d ∩ u (u0, r)), where r = 2 (c0 + α) m0 − r . furthemore, estimates (2.14)–(2.16) hold. proof. it follows from condition (2.23), and lemma 2.2, that the first iterate u1 is defined uniquely in (1.4). using (1.4), (2.22), and the monotonicity condition (1.1), we obtain α ‖u1 − u0‖2 + (v0 + f (u0) + f ′(u0) (u1 − u0), u1 − u0) ≤ 0. cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 55 rewriting this in the form α ‖u1 − u0‖2 + (f ′(u0) (u1 − u0), u1 − u0) ≤ (−f (u0) − v0, u1 − u0) and making use of (2.23) and (2.26), we see that: ‖u1 − u0‖ ≤ b c0 + α = η = t1 − t0. (2.32) if m0 = 0, then r = η ≤ t⋆, and r = ∞. in this case f ′(x) = f ′(u0), for all x ∈ d, and f (x) = f (u0) + ∫ 1 0 f ′(s x + (1 − s) u0) (x − u0) ds = f (u0) + f ′(u0) (x − u0), ∀x ∈ d. therefore, the unique solution of equation (1.3) in d is u = u1. since ‖ u1 − u0 ‖≤ η ≤ t⋆, the conclusion of the theorem holds in this case. if η = 0, then r = t⋆ = 0, r = 2 (c0 + α) m0 , and the unique solution of equation (1.3) in u (u0, r) ∪ (d ∩ u (u0, r)) is u = u0. in the rest of the proof, we assume that m0 > 0, and η > 0. in this case, it follows from (2.8) that: 2 m0 ≤ 4 m0 + m + √ m 2 + 8 m m0 4 ≤ c0 + α η . therefore, r and r are well defined. we prove by induction that the um are well defined and conditions (2.28) and (2.29) hold for m = 0, 1, . . . . it follows from (2.32), that the induction hypothesis is true if m = 1. we assume that m ≥ 1, and that the induction hypothesis holds for m. then it follows from (2.25) and (2.29), that ‖f ′(um) − f ′(u0)‖ ≤ m0‖um − u0‖ ≤ m0 tm. therefore (f ′(u0) x − f ′(um) x, x) ≤ ‖f ′(u0) − f ′(um)‖ ‖x‖2 ≤ m0 tm ‖x‖2 for all x ∈ h, which implies, because of (2.23), that: (f ′(um) x, x) ≥ (c0 − m0 tm) ‖x‖2, ∀x ∈ h. (2.33) 56 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) on the other hand, it follows from (1.4), that c0 − m0 tm > −α. therefore we conclude from lemma 2.2, that the iterate um+1 is defined uniquely in (1.4). it follows from (1.4) that g(um) ∋ vm ≡ −f (um−1) − f ′(um−1) (um − um−1) (2.34) and by using conditions (2.24) and (2.28), we see that ‖f (um) + vm‖ = ‖f (um) − f (um−1) − f ′(um−1) (um − um−1)‖ = ‖ ∫ 1 0 [f ′((1 − s) um−1 + s um) − f ′(um−1)] (um − um−1) ds‖ ≤ m 2 ‖um − um−1‖2 ≤ m 2 (tm − tm−1)2. (2.35) on the other hand, it follows from (1.1), (2.34), and (1.4) that: α‖um+1 − um‖2 + (vm + f (um) + f ′(um) (um+1 − um), um+1 − um) ≤ 0. rewriting this in the form α ‖um+1 − um‖2 + (f ′(um) (um+1 − um), um+1 − um) ≤ (f (um) + vm, um − um+1) and making use of (2.33) and (2.35), we see that ‖um+1 − um‖ ≤ m (tm − tm−1)2 2 (c0 + α − m0 tm) = tm+1 − tm. hence ‖um+1 − u0‖ ≤ ‖um+1 − um‖ + ‖um − u0‖ ≤ tm+1 − tm + tm = tm+1. it follows that (2.28) and (2.29) also hold when m is replaced with m + 1 and hence, by induction, that they hold for all positive integral values of m. this implies that ‖um+q − um‖ ≤ m+q ∑ k=m+1 ‖uk − uk−1‖ ≤ m+q ∑ k=m+1 (tk − tk−1) = tm+q − tm. since {tm} is a cauchy sequence, it follows that {um} is also a cauchy sequence converging to some u ∈ u (u0, t⋆). on letting q tend to infinity we see that (2.30) holds. since lim m→∞ vm = −f (u) and lim m→∞ um = u, it follows from (2.34), and property (1.2) that u solves problem (1.3). cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 57 to prove the uniqueness assertions, let v be any solution of problem (1.3). then since −f (v) ∈ g(v), it follows from (1.2), (2.34), and (2.22) that: α ‖v − u0‖2 ≤ −(v0 + f (v), v − u0) = −(f (v) − f (u0) − f ′(u0) (v − u0), v − u0) − (f ′(u0) (v − u0), v − u0) − (f (u0) + v0, v − u0). therefore, by (2.23), and the center–lipschitz condition (2.25), we get: (c0 + α) ‖v − u0‖ ≤ ‖f (v) − f (u0) − f ′(u0) (v − u0)‖ + ‖f (u0) + v0‖ ≤ ‖ ∫ 1 0 [f ′(s v + (1 − s) u0) − f ′(u0)] (v − u0) ds‖ + b ≤ m0 ‖v − u0‖ ∫ 1 0 (s ‖v − u0‖ + (1 − s) ‖v − u0‖) ds + b = m0 ‖v − u0‖2 2 + b. by solving this quadratic inequality, we see that either ‖v − u0‖ ≤ r or ‖v − u0‖ ≥ r. in particular case v = u, condition (2.13) implies that the condition ‖u − u0‖ ≥ r cannot hold since ‖u − u0‖ ≤ t⋆ ≤ c0 + α m0 < r. it follows thet ‖u − u0‖ ≥ r, and hence u ∈ u (u0, r). if condition (2.31) holds, and v is a solution of problem (1.3) in d ∩ u (u0, r). then, since −f (u) ∈ g(u), and −f (v) ∈ g(v), it follows from (1.2) that: α ‖u − v‖2 ≤ (f (v) − f (u), u − v) = −(f (u) − f (v) − f ′(u0)(u − v), u − v) − (f ′(u0) (u − v), u − v). if we now apply (2.23), and the center–lipschitz condition (2.25), we see that (c0 + α) ‖u − v‖ ≤ ‖f (u) − f (v) − f ′(u0) (u − v)‖ ≤ ‖ ∫ 1 0 [f ′(s u + (1 − s) v) − f ′(u0)](u − v) ds‖ ≤ m0 ‖u − v‖ ∫ 1 0 (s ‖u − u0‖ + (1 − s) ‖v − u0‖) ds = m0 ‖u − v‖ ‖u − u0‖ + ‖v − u0‖ 2 ≤ m0 ‖u − v‖ r + t 2 . 58 ioannis k. argyros and säıd hilout cubo 13, 1 (2011) therefore, it follows from (2.31) that u = v. this proves the uniqueness of the solution in u (u0, t ) ∩ d. if we set t = t⋆, then condition (2.31) reduces to the condition m0 t⋆ < c0 + α + √ c0 + α − 2 m0 η, which is true by (2.8). this shows that the solution of problem (1.3) is unique in u (u0, t ⋆) ∩ d. let r0 = 0, r1 = η, rm+1 = rm + m0 (rm − rm−1)2 2 (c0 + α − m0 rm) , then, the argument employed in the proof of lemma 2.2 shows that rm converges monotonically to a non–negative number r, such that m0 r ≤ c0 + α. since 2 (c0 + α) rm+1 − 2 m0 rm+1 rm = 2 (c0 + α) rm − 2 m0 rm rm−1 = · · · = 2 (c0 + α) r1 − 2 m0 r1 r0 = 2 η (c0 + α). by letting m −→ ∞, we obtain that r satisfies the equation −2 (c0 + α) r + m0 r2 + 2 η (c0 + α) = 0. therefore, r is given by the expression (2.27). on the other hand, it follows from (2.10), and an easy induction argument that the inequality rm ≤ tm holds for all m. this shows that r ≤ t⋆, and we conclude that the solution of problem (1.3) is unique in u (u0, r) ∪ (d ∩ u (u0, r)). that completes the proof of theorem 2.1. ♦ we complete this study with a numerical example. example 1. let x = y = r, d = ( 8 9 , 10 9 ), u0 = 1, and define function f on d by f (x) = 6 x3 − 1. set g = 0, f (x) = f ′(u0) −1 f (x), then we have f ′(u0) = i, α = 0, c0 = 1, m0 = 1.9, m = 2, η = .258646. the classical newton–kantorovich condition [1]–[3], [5], [6], [9]: m η ≤ 1 2 (2.36) is violated, since m η = .517292 > .5. cubo 13, 1 (2011) on the solution of generalized equations and variational inequalities 59 however, condition (2.1) becomes 3.999999528 < 4. condition (2.36) implies (2.1), but not necessarily vice verca unless if m = m0. note also that: m0 ≤ m holds in general, and m m0 can be arbitrarily large [2], [3]. we also have: θ = δ = 1.017145084, θ 2 = .508572542, s⋆⋆ = .526315727, q0 = .499999941, 2 q0 = .999999882. by comparing (2.6) with (2.15), we see that is an improvment by (2 q0) 2 k −1 at each k–th–step. several special cases, and other applications can also be found in [18] (see also [1]–[17], [19], [20]). received: october 2009. revised: november 2009. references [1] argyros, i.k.,on the newton-kantorovich hypothesis for solving equations, j. comput. appl. math., 11 (2004), 103–110. 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[21] wang, z., shen, z.,kantorovich theorem for variational inequalities, applied mathematics and mechanics (english edition), 25 (2004), 1291–1297. cubo a mathematical journal vol.11, no¯ 03, (101–113). august 2009 fixed point theory and nonlinear periodic systems ronald grimmer southern illinois university at carbondale, carbondale, il 62901, usa email: rgrimmer@math.siu.edu and min he kent state university, mathematical science, trumbull campus, warren, oh 44483, usa email: mhe@kent.edu abstract this work is concerned with a nonlinear periodic system, which depends on parameters. we investigate continuity with respect to parameters of the periodic solution of the system. applying a fixed point theorem and the results regarding parameters for c0semigroups, we obtained some convenient conditions for determining both existence of a unique periodic solution and continuity in parameters of the periodic solution. the results are applied to a nonlinear wave equation with forced and damped boundary conditions. resumen este trabajo tiene que ver con un sistema no lineal periódico que depende de parametros. investigamos continuidad con respecto a los parametros de la solución periódica 102 ronald grimmer and min he cubo 11, 3 (2009) del sistema. aplicando un teorema de punto fijo y resultados considerando parametros para c0-semigrupos, obtenemos algunas condiciones convenientes para determinar existencia de una única solución periódica y también continuidad en terminos de los parametros para la solución periódica. los resultados son aplicados para una ecuación de onda no lineal con condiciones de frontera de tipo forzado y damped. key words and phrases: c0-semigroup, periodic system, parameter, continuity. math. subj. class.: 47d62, 45k05, 35l20. 1 introduction boundary value conditions in partial differential equations lead to problems involving parameters. if these equations are reformulated as abstract cauchy problems, the equations will depend on these parameters in a way that is reflected by the domain of the operators involved. for such parameter dependent equations it is natural to want continuity and differentiability with respect to parameters of solutions of the equations. recent studies [3, 4, 6, 7, and references therein] have established fundamental theory about continuity and differentiability with respect to parameters. in this work, we are particularly interested in a class of nonlinear periodic systems that depend on parameters and study continuity in parameters of its solutions. in our subsequent paper, we will discuss differentiability with respect to parameters of the solutions. consider a nonlinear wave equation with forced and damped boundary conditions utt = uxx + ηf (ut) for t ≥ 0, u(x, 0) = u0(x), ut(x, 0) = u1(x) for x ∈ [0, 1], µut(0, t) − γux(0, t) = f1(t), δut(1, t) + γux(1, t) = f2(t), µ, γ, δ > 0, (1.1) where f1(t) and f2(t) are both ρ-periodic and continuously differentiable. f satisfies a uniform lipschitz condition let v = ut and w = ux, then equation (1.1) is written as vt = wx + ηf (v), wt = vx, µv(0, t) − γw(0, t) = f1(t), δv(1, t) + γw(1, t) = f2(t), µ, γ, δ > 0. (1.2) if the change of variables v̄ = v − xδ−1f2, w̄ = w + (1 − x)γ−1f1 cubo 11, 3 (2009) fixed point theory and nonlinear periodic systems 103 is made, equation (1.2) has the form: v̄t = w̄x + ηf (v̄ + xδ −1 f2(t)) + γ −1 f1(t) − xδ−1f ′2(t), w̄t = v̄x + δ −1 f2(t) + (1 − x)γ−1f ′1(t), µv̄(0, t) − γw̄(0, t) = 0, δv̄(1, t) + γw̄(1, t) = 0, µ, γ, δ > 0. (1.3) further, the associated abstract equation of (1.3) is given by dz(t) dt = a(ε)z(t) + f̄ (t, z(t), ε), z(0) = z0 (1.4) on x = l2[0, 1] × l2[0, 1], where a(ε) = [ 0 ∂x ∂x 0 ] , ε = (µ, γ, δ, η) ∈ r4 + , z = [ v̄ w̄ ] , d(a(ε)) = {[ v̄ w̄ ] ∈ 2 ∏ i=1 h 1 [0, 1] ∣ ∣ ∣ ∣ ∣ µv̄(0) = γw̄(0), δv̄(1) = −γw̄(1), µ, γ, δ > 0 } , f̄ (t, z, ε)x = [ ηf (v̄ + xδ −1 f2(t)) + γ −1 f1(t) − xδ−1f ′2(t) δ −1 f2(t) + (1 − x)γ−1f ′1(t) ] . for convenience, write v̄ as v,w̄ as w, f̄ as f1, and α = µ γ , β = γ δ , we then have dz(t) dt = a(ε)z(t) + f1(t, z(t), ε), z(0) = z0 (1.5) on x = l2[0, 1] × l2[0, 1], where a(ε) = [ 0 ∂x ∂x 0 ] , ε = (α, β, δ, η) ∈ r4 + , z = [ v w ] , d(a(ε)) = {[ v w ] ∈ 2 ∏ i=1 h 1 [0, 1] ∣ ∣ ∣ ∣ ∣ αv(0) = w(0), v(1) = −βw(1), α, β > 0 } , f1(t, z, ε)x = [ ηf (v + xδ −1 f2(t)) + γ −1 f1(t) − xδ−1f ′2(t) δ −1 f2(t) + (1 − x)γ−1f ′1(t) ] . by examining equation (1.5), we see that a) f1 is a nonlinear and ρ-periodic function, b) ε is a multi-parameter, c) a(ε) is linear and densely defined, and d) d(a(ε)) is dependent on the parameter ε. 104 ronald grimmer and min he cubo 11, 3 (2009) our work is motivated by this example to consider the abstract nonlinear periodic equation dz(t) dt = a(ε)z(t) + f (t, z(t), ε), z(0) = z0 (1.6) on a banach space x with norm ‖ · ‖, where a(ε) is linear and densely defined, ε ∈ p is a parameter (where p is an open subset of a finite-dimensional normed linear space p with norm | · |), f (t + ρ, z, ε) = f (t, z, ε) for some ρ > 0, and f is continuous in (t, z, ε) ∈ r × x × p . note that the above example illustrates a fact that the occurrence of parameters in boundary conditions causes the domain of the operator a(ε) to depend on the parameters. this phenomenon is not considered in any papers known to the authors. the goal of this work is to determine conditions concerning a) existence and uniqueness of the periodic (weak) solution of equation (1.6) and b) continuity with respect to parameter ε of the solution of (1.6). based on semigroup theory, when a(ε) generates a c0-semigroup t (t, ε), the weak solution of (1.6) will have the form z(t, ε) = t (t, ε)z0 + ∫ t 0 t (t − s, ε)f (s, z(s, ε), ε) ds. it is clear that the continuity in parameter ε of semigroup t (t, ε) will play a key role in attaining our goals. to this end we first discuss the parameter properties of c0-semigroup t (t, ε) and present a method which can be especially useful for dealing with operator a(ε) that has a domain dependent on ε in section 2. we then apply a fixed point theorem to show both existence of a unique periodic solution and continuity in ε of the periodic solution of (1.6) in section 3. in the last section, we shall illustrate the results and technique in an application to a nonlinear wave equation with variable boundary conditions. 2 continuity in parameters of c0-semigroups as noted in section 1, the semigroup theory indicates that the continuity in parameters of c0semigroup t (t, ε) generated by a(ε) in (1.6) can easily derive the analogous result for the solution of equation (1.6). hence, we will focus on discussing continuity property of c0-semigroup t (t, ε). in this section, we will first state a continuity result for the case that operators have constant domains (the domain is independent of parameters). then we will present a method that is especially useful in handling the case that operators have variable domains (the domain is dependent of parameters). in the sequel we use “a is a hille-yosida operator" to mean that there are constants m ≥ 1 and ω ∈ r such that λ > ω implies λ ∈ ρ(a) (the resolvent set of a) and ‖(λi − a)−n‖ ≤ m (λ − ω)n for λ > ω, n ∈ n. cubo 11, 3 (2009) fixed point theory and nonlinear periodic systems 105 in the special case when an operator has a constant domain, one has the result which states that the strong continuity in parameters of the operator implies the continuity in parameters of its semigroup. theorem 2.1. assume that (2.1) d(a(ε)) = d for all ε ∈ p . (2.2) there are constants m ≥ 1 and ω ∈ r such that ||(λi − a(ε))−n|| ≤ m (λ − ω)n for λ > ω, n ∈ n, and all ε ∈ p. (2.3) for each x ∈ d, a(ε)x is continuous in ε ∈ p . then (λi−a(ε))−1x is continuous in ε for each x ∈ x. further, the c0-semigroup t (t, ε) generated by a(ε) is strongly continuous in ε ∈ p , and the continuity is uniform on bounded t-intervals. in particular, for any ε ∈ p, h ∈ p with ε + h ∈ p , and any t0 ∈ [0, ∞), sup 0≤t≤t0 ‖t (t, ε + h)x − t (t, ε)x‖ = ◦(1) as |h| → 0, for each x ∈ x. proof. see [8] for the proof of continuity of (λi − a(ε))−1x. the proof of continuity of c0semigroup t (t, ε) is similar to the proof of the c0-semigroup approximation theorem in [1, theorem 3.17], and it is omitted. � the next theorem gives a convenient condition to determine the continuity in parameters of c0-semigroup when the domain of the operator is dependent on parameters. we remark that the following assumption is naturally possessed by many hyperbolic and parabolic types of boundary value problems (see an example in section 4). assumption q. let ε0 ∈ p be given. then for each ε ∈ p there exists bounded operators q1(ε), q2(ε) : x → x with bounded inverses q−11 (ε) and q −1 2 (ε), such that a(ε) = q1(ε)a(ε0)q2(ε). note that if a(ε1) = q1(ε1)a(ε0)q2(ε1), then a(ε) = q1(ε)a(ε0)q2(ε) = q1(ε)q −1 1 (ε1)q1(ε1)a(ε0)q2(ε1)q −1 2 (ε1)q2(ε) = q̃1(ε)a(ε1)q̃2(ε). thus, having such a relationship for some ε0 implies a similar relationship at any other ε1 ∈ p . without loss of generality then, we may just consider the continuity of the semigroup t (t, ε) at ε = ε0 ∈ p . consider an auxiliary operator ã(ε) = q2(ε)a(ε)q −1 2 (ε) = q2(ε)q1(ε)a(ε0). 106 ronald grimmer and min he cubo 11, 3 (2009) lemma 2.2. assume that assumption q and (2.2) are satisfied and suppose that (2.4) qi(ε)x and q −1 2 (ε)x are continuous in ε for x ∈ x, i = 1, 2. then the c0-semigroup t̃ (t, ε) generated by ã(ε) is strongly continuous at ε0, and the continuity is uniform on bounded t-intervals. in particular, for any h ∈ p with ε0 + h ∈ p , and any t0 ∈ [0, ∞), sup 0≤t≤t0 ‖t̃ (t, ε0 + h)x − t̃ (t, ε0)x‖ = ◦(1) as |h| → 0, for each x ∈ x. proof. note that ã(ε) = q2(ε)q1(ε)a(ε0). clearly, d(ã(ε)) = d(a(ε0)) for all ε ∈ p . thus, ã(ε) has a fixed domain. also (2.4) implies that ã(ε) is continuous in ε, that is, ã(ε) satisfies (2.3). since (λi − ã(ε))n = q2(ε)(λi − a(ε))nq−12 (ε), thus (λi − ã(ε))−n = q2(ε)(λi − a(ε))−nq−12 (ε). because q2(ε) and q −1 2 (ε) are bounded, (2.2) implies that ã(ε) is a hille-yosida operator for all ε ∈ p . now applying theorem 2.1, we have that t̃ (t, ε) is continuous in ε. � theorem 2.3. assume that assumption q, (2.2) and (2.4) are satisfied, then the c0-semigroup t (t, ε) generated by a(ε) is strongly continuous at ε0, and the continuity is uniform on bounded t-intervals. in particular, for any h ∈ p with ε0 + h ∈ p , and any t0 ∈ [0, ∞), sup 0≤t≤t0 ‖t (t, ε0 + h)x − t (t, ε0)x‖ = ◦(1) as |h| → 0, for each x ∈ x. proof. in fact, ã(ε) = q2(ε)a(ε)q −1 2 (ε). by the definition of c0-semigroup [2], it is clear that t (t, ε) = q −1 2 (ε)t̃ (t, ε)q2(ε). further, for x ∈ x and ε0 + h ∈ bδ(ε0) = {ε | |ε − ε0| ≤ δ}, ‖t (t, ε0 + h)x − t (t, ε0)x‖ = ‖q−1 2 (ε0 + h)t̃ (t, ε0 + h)q2(ε0 + h)x − q−12 (ε0)t̃ (t, ε0)q2(ε0)x‖ ≤ ‖q−1 2 (ε0 + h)t̃ (t, ε0 + h)‖ ‖q2(ε0 + h)x − q2(ε0)x‖ + ‖q−1 2 (ε0 + h)‖ ‖(t̃ (t, ε0 + h) − t̃ (t, ε0))q2(ε0)x‖ + ‖(q−1 2 (ε0 + h) − q−12 (ε0))t̃ (t, ε0)q2(ε0)x‖ . note that ‖q−1 2 (ε0 +h)t̃ (t, ε0 +h)‖, ‖q−12 (ε0 +h)‖ are uniformly bounded. moreover, from lemma 2.2 and (2.4), t̃ (t, ε), q2(ε), and q −1 2 (ε) are continuous in ε. thus, we have t (t, ε) is continuous in ε. � cubo 11, 3 (2009) fixed point theory and nonlinear periodic systems 107 3 continuity in parameters of periodic solutions of (1.6) the goal of this section is to obtain existence and continuity in parameters of a unique periodic solution z(t, ε) of (1.6). we first study a special case of (1.6) dz(t) dt = a(ε)z(t) + f (t, ε), z(0) = z0 (3.1) on a banach space x, where a(ε) is linear and densely defined, ε ∈ p is a parameter, f (t + ρ, ε) = f (t, ε) for some ρ > 0, and f (t, ε) is continuous in (t, ε) ∈ r × p . we aim for obtaining the existence and continuity in ε of the periodic solution of (3.1) and then apply this result together with a fixed point theorem to show that (1.6) has a unique periodic solution, which is continuous in parameter ε. for convenient reference, we state the fixed point theorem that will be needed in the later proofs. theorem 3.1.[5, p7] if f is a closed subset of a banach space x , g is a subset of a banach space y, ty : f → f, y ∈ g is a uniform contraction on f and tyx is continuous in y for each fixed x in f, then the unique fixed point g(y) of ty, y in g, is continuous in y. as discussed in section 2, several convenient conditions have been obtained for determining continuity in parameter ε of c0-semigroup t (t, ε). thereby, in this section, we will just assume that t (t, ε) is continuous in parameter ε. theorem 3.2. assume that (3.2) t (t, ε)z is continuous in ε for each z ∈ x, and ‖t (t, ε)‖ ≤ m (t0) for some m (t0) > 0 and all ε ∈ p, t ∈ [0, t0], and (3.3) ‖t (n ρ, ε)‖ ≤ k < 1 for some integer n with n ρ < t0 and all ε ∈ p . then there exists a unique ρ-periodic solution of (3.1), say z(t, ε), which is continuous in ε for ε ∈ p . proof. first, we know that the weak solution of (3.1) can be expressed as z(t, ε) = t (t, ε)z0 + ∫ t 0 t (t − s, ε)f (s, ε) ds. to show that there is a unique ρ-periodic solution which is continuous in ε, it suffices to show that the operator k(ε) has a unique fixed point where k(ε)z = t (ρ, ε)z + ∫ ρ 0 t (ρ − s, ε)f (s, ε) ds. 108 ronald grimmer and min he cubo 11, 3 (2009) consider the n th-iterate, kn (ε). we first note that kn (ε) is a uniform contraction on x. in fact, note that k n (ε)z = t (n ρ, ε)z + ∫ n ρ 0 t (n ρ − s, ε)f (s, ε) ds and then, for all ε ∈ p and z1, z2 ∈ x, ‖kn (ε)z1 − kn (ε)z2‖ = ‖t (n ρ, ε)(z1 − z2)‖ ≤ ‖t (n ρ, ε)‖ · ‖z1 − z2‖ ≤ k‖z1 − z2‖ (since ‖t (n ρ, ε)‖ ≤ k < 1). in addition, kn (ε) is continuous in ε for each fixed z ∈ x. to see this, let ε be an arbitrary point in p . it is clear, from (3.2) and continuity of f (t, ε), that t (n ρ − s, ε)f (s, ε) is continuous at ε = ε0. furthermore, ∫ n ρ 0 t (n ρ − s, ε)f (s, ε) ds is continuous at ε = ε0 (3.4) in fact, for each fixed s ∈ [0, n ρ] and h ∈ p with ε0 + h ∈ p , ‖t (n ρ − s, ε0 + h)f (s, ε0 + h) − t (n ρ − s, ε0)f (s, ε0)‖ ≤ ‖t (n ρ − s, ε0 + h)‖ · ‖f (s, ε0 + h) − f (s, ε0)‖ +‖(t (n ρ − s, ε0 + h) − t (n ρ − s, ε0))f (s, ε0)‖ → 0 as |h| → 0, (by (3.2) and continuity of f (t, ε)). also, by continuity of f (t, ε), there is b(ε0, δ(ε0)) (δ(ε0) > 0) such that ‖f (s, ε)‖ ≤ n (ε0) for some n (ε0) > 0 and (s, ε) ∈ [0, ρ] × b(ε0, δ(ε0)). thus ‖t (ρ − s, ε0 + h)f (s, ε0 + h) − t (ρ − s, ε0)f (s, ε0)‖ ≤ 4m (t0) · n (ε0) and (3.4) follows from the dominated convergence theorem. above all, kn (ε)z is continuous at ε = ε0. since ε0 is arbitrarily chosen, k n (ε)z is continuous in ε ∈ p . taking f = x, y = ε, and ty = kn (ε), we have that theorem 3.1 implies that there exists a unique fixed point z0(ε) of k n (ε) which is continuous in ε. thus, z0(ε) is the unique fixed point of k(ε) and it is continuous in ε. finally, using the same argument as above, we have that the unique ρ-periodic weak solution of (3.10) z(t, ε) = t (t, ε)z0(ε) + ∫ t 0 t (t − s, ε)f (s, ε) ds. is continuous in ε. � now we discuss equation (1.6). cubo 11, 3 (2009) fixed point theory and nonlinear periodic systems 109 lemma 3.3. assume that (3.2) and (3.3) are satisfied. then (i − t (n ρ, ε))−1z is continuous in ε for each z ∈ x. proof. first note that from (3.3), it is easy to show that (i − t (ρ, ε))−1 exists. also, ‖(i − t (n ρ, ε))−1‖ ≤ 1 1 − k . = h. next consider the operator defined on x: j(ε)z = t (n ρ, ε)z + y where y is a given point in x. then we have ‖j(ε)z1 − j(ε)z2‖ ≤ ‖t (n ρ, ε)‖ · ‖z1 − z2‖ ≤ k‖z1 − z2‖. thus, j(ε) is a uniform contraction. also it is obvious that j(ε)z is continuous in ε by (3.2). from theorem 3.1 it follows that there is a unique fixed point of j(ε), say z(ε), such that z(ε) = t (n ρ, ε)z(ε) + y and z(ε) is continuous in ε. furthermore, z(ε) = (i − t (n ρ, ε))−1y and (i − t (n ρ, ε))−1y is continuous in ε. � let p c[r, ρ] = {g ∈ c(r, x) ∣ ∣ g(t + ρ) = g(t), t ∈ r} together with the sup norm, ‖ · ‖∞. consider the equation z ′ (t) = a(ε)z(t) + f (t, g(t), ε) z(0) = z0 (3.5) on a banach space (x, ‖ · ‖), where f (t + ρ, g, ε) = f (t, g, ε) for some ρ > 0 and g ∈ p c[r, ρ], and f (t, g, ε) is continuous in (t, g, ε) ∈ r × p c[r, ρ] × p . lemma 3.4. assume that (3.2) and (3.3) are satisfied. then there exists a unique ρ-periodic solution of (3.5), say z(t, g, ε), which is continuous with respect to ε for ε ∈ p . also z(0, g, ε) = (i − t (n ρ, ε))−1 ∫ n ρ 0 t (ρ − s, ε)f (s, g(s), ε) ds. (3.6) proof. let f2(t, ε) = f (t, g(t), ε). then f2(t + ρ, ε) = f2(t, ε). also it is obvious that f2(t, ε) is continuous in (t, ε) because f (t, z, ε) is continuous in (t, z, ε). therefore, by theorem 3.2, there is a unique ρ-periodic weak solution z(t, ε, g) of (3.5) which is continuous in ε. in particular, z(0, ε, g) is continuous in ε. moreover, using the same argument as that in the proof of theorem 3.2 we see that z(0, ε, g) = t (n ρ, ε)z(0, ε, g) + ∫ n ρ 0 t (n ρ − s, ε)f (s, g(s), ε) ds. thus, (3.6) holds. � 110 ronald grimmer and min he cubo 11, 3 (2009) define j1(ε) : p c[r, ρ] → p c[r, ρ] by j1(ε)g(t) = t (t, ε)z(0, ε, g) + ∫ t 0 t (t − s, ε)f (s, g(s), ε) ds. lemma 3.5. assume that (3.2)and (3.3) are satisfied . in addition, assume that ‖f (t, z1, ε) − f (t, z2, ε)‖ ≤ l(ε1)‖z1 − z2‖, (3.7) where l(ε1) is continuous in ε1 ∈ p in a neighborhood of ε = 0 and l(0) = 0. then for small |ε1|, the operator j1(ε) has a unique fixed point g(·, ε) ∈ p c[r, ρ] which is continuous in ε. proof. first note that it is clear that (p c[r, ρ], ‖ · ‖∞) is a banach space. next since l(0) = 0, then, by continuity of l(ε1), there is δ0 such that |ε1| < δ0 implies l(ε 1 ) ≤ 1 4 min{ 1 hm 2(t0)t0 , 1 m (t0)t0 }. now for ε ∈ p with |ε1| < δ0, ‖t (t, ε)‖ · ‖z(0, g1, ε) − z(0, g2, ε)‖ = ‖t (t, ε)‖ · ‖(i − t (n ρ, ε))−1 ∫ n ρ 0 t (n ρ − s, ε)[f (s, g1, ε) − f (s, g2, ε)] ds‖ ≤ m (t0)‖(i − t (n ρ, ε))−1‖ ∫ n ρ 0 ‖t (n ρ − s, ε)‖ · ‖f (s, g1, ε) − f (s, g2, ε)‖ ds ≤ m (t0) · h · m (t0) ∫ n ρ 0 ‖f (s, g1, ε) − f (s, g2, ε)‖ ds ≤ h · m 2(t0) · t0 · l(ε1)‖g1 − g2‖ ≤ 1 4 ‖g1 − g2‖, and ∫ t 0 ‖t (t − s, ε)‖ · ‖f (s, g1(s), ε) − f (s, g2(s), ε)‖ ds ≤ m (t0)l(ε1) ∫ t 0 ‖g1(s) − g2(s)‖ ds ≤ m (t0) · t0l(ε1)‖g1 − g2‖ ≤ 1 4 ‖g1 − g2‖. hence ‖j1(ε)g1 − j1(ε)g2‖ ≤ ‖t (t, ε)‖ · ‖z(0, g1, ε) − z(0, g2, ε)‖ + ∫ t 0 ‖t (t − s, ε)‖ · ‖f (s, g1(s), ε) − f (s, g2(s), ε)‖ ds ≤ 1 4 ‖g1 − g2‖ + 1 4 ‖g1 − g2‖ ≤ 1 2 ‖g1 − g2‖. cubo 11, 3 (2009) fixed point theory and nonlinear periodic systems 111 therefore j1(ε) is a uniform contraction. furthermore, j1(ε)g is continuous in ε for fixed g. therefore from theorem 3.1, it follows that j1(ε) has a unique fixed point, say g(·, ε) ∈ p c[r, ρ], which is continuous in ε. � theorem 3.6 assume that (3.2), (3.3) and (3.7) are satisfied. then for small |ε1|, there exists a unique ρ-periodic weak solution of (1.6), say z(t, ε), which is continuous in ε for ε ∈ p . proof. this is an immediate result from lemma 3.4 and lemma 3.5. 4 application to a nonlinear wave equation consider the nonlinear wave equation mentioned in section 1 utt = uxx + ηf (ut), for t ≥ 0, u(x, 0) = u0(x), ut(x, 0) = u1(x) for x ∈ [0, 1], µut(0, t) − γux(0, t) = f1(t), δut(1, t) + γux(1, t) = f2(t), µ, γ, δ > 0, (4.1) where f1(t) and f2(t) are both ρ-periodic and continuously differentiable. f satisfies a uniform lipschitz condition. as shown in section 1, the associated abstract equation of (4.1) is given by: dz(t) dt = a(ε)z(t) + f1(t, z(t), ε), z(0) = z0 (4.2) on x = l2[0, 1] × l2[0, 1], where a(ε) = [ 0 ∂x ∂x 0 ] , ε = (α, β, δ, η) ∈ r4 + , z = [ v w ] , d(a(ε)) = {[ v w ] ∈ 2 ∏ i=1 h 1 [0, 1] ∣ ∣ ∣ ∣ ∣ αv(0) = w(0), v(1) = −βw(1), α, β > 0 } . f1(t, z, ε)x = [ ηf (v + xδ −1 f2(t)) + γ −1 f1(t) − xδ−1f ′2(t) δ −1 f2(t) + (1 − x)γ−1f ′1(t) ] now make the change of variables z̃ = [ ṽ w̃ ] = u [ v w ] where u = 1 2 [ 1 1 1 −1 ] 112 ronald grimmer and min he cubo 11, 3 (2009) then, a1(ε) ≡ u a(ε)u −1 = [ ∂x 0 0 −∂x ] . where d(a1(ε)) = { [ ṽ w̃ ] ∈ 2 ∏ i=1 h 1 [0, 1] ∣ ∣ ∣ ∣ ∣ (1 − α)ṽ(0) = (1 + α)w̃(0), (1 + β)ṽ(1) = −(1 − β)w̃(1). } define α̃ = 1 − α 1 + α , β̃ = 1 − β 1 + β α̃, β̃ ∈ (−1, 1]. a1(ε) has the domain d(a1(ε)) = { [ ṽ w̃ ] ∈ 2 ∏ i=1 h 1 [0, 1] ∣ ∣ ∣ ∣ ∣ α̃ṽ(0) = w̃(0), ṽ(1) = −β̃w̃(1), α̃, β̃ ∈ (−1, 1] } . if |α̃β̃| < 1, then a1(ε) generates a c0-semigroup, t1(t, ε), on x = l2[0, 1]×l2[0, 1], endowed with the norm ‖(f, g)‖ ≡ ‖f‖2 + ‖g‖2. it can easily be shown using the method of characteristics that this semigroup satisfies ‖t1(t, ε)‖ = 1 for 0 ≤ t < 1, ‖t1(t, ε)‖ = max{|α̃|, |β̃|} ≤ 1 for 1 ≤ t < 2, and ‖t1(t, ε)‖ ≤ |α̃β̃| < 1 for t ≥ 2. thus the semigroup is eventually contracting. it follows from t (t, ε) = u t1(t, ε)u −1 that t (t, ε) must have the same properties with respect to the norm |‖z‖| ≡ ‖u z‖ on x = l2[0, 1] × l2[0, 1] when α̃, β̃ > 0. now consider the operator a1(ε). take ε0 = (0, 0). for ε = (α̃,β̃) with α̃,β̃ ∈ (−1, 1] a1(ε) = 1 1 + α̃β̃ q(ε)a1(0)q(ε) where q(ε) = [ 1 β̃ −α̃ 1 ] . obviously, q(ε) is continuous in ε and is bounded. so is q−1(ε). let q2(ε) = 1 1+α̃β̃ q(ε) and q1(ε) = q(ε). it is clear that the hypotheses of theorem 2.3 are satisfied for any ε0 = (α̃0, β̃0) in r + × r+. that is, we have strong continuity in parameter ε of semigroup t1(t, ε) and thus strong continuity of semigroup t (t, ε). above all, the associated semigroup is eventually contracting and is continuous in ε. thus, it follows from theorem 3.6 that when |η| is small there is a unique ρ-periodic weak solution of (4.2) and thus also a unique ρ-periodic weak solution of (4.1) and it is l2 continuous in ε = (α, δ, γ, η). received: june 1, 2008. revised: july 25, 2008. cubo 11, 3 (2009) fixed point theory and nonlinear periodic systems 113 references [1] davies, e.d., one-parameter semigroups, academic press, london, 1980. [2] goldstein, j., semigroups of linear operators and applications, oxford university press, new york, 1985. [3] grimmer, r. and he, m., differentiability with respect to parameters of semigroups, semigroup forum, 59, (1999), 317–333. [4] grimmer, r. and he, m., integrodifferential equations with parameter dependent operators, differential and integral equations, 15, (2002), 33–45. [5] hale, j.k., ordinary differential equations, wiley-interscience, new york, 1969. [6] he, m., a perturbation theorem and its application, annals of differential equations, 15, (1999), 352–361. [7] he, m., on continuity in parameters of integrated semigroups, dynamical systems and differential equations, supplement volume 2003, (2003), 403–412. [8] he, m., a class of integrodifferential equations with memory, semigroup forum, 73, (2006), 427–443. 12-minhe_periodicsystem () cubo a mathematical journal vol.13, no¯ 03, (153–184). october 2011 sum and difference compositions in discrete fractional calculus michael holm the university of nebraska-lincoln, abstract we introduce fractional sum and difference operators, study their behavior and develop a complete theory governing their compositions. this theory is then applied to solve a general, fractional initial value problem. resumen introducimos operadores de suma y diferencia fraccionaria, estudiamos su comportamiento y desarrollamos una teora completa que rige sus composiciones. aplicamos esta teoŕıa para resolver un problema fraccionado de valor inicial. keywords: discrete fractional calculus, fractional sum, fractional difference, composition rule, fractional initial value problem. mathematics subject classification: 34 1 introduction gottfried leibniz and guilliaume l’hôpital are believed to have first sparked curiosity into the idea of fractional calculus during a 1695 overseas correspondence on the possible meaning for 1a special thank you to dr. allan peterson for his input throughout my work on this paper. 154 michael holm cubo 13, 3 (2011) a one-half derivative. by the late 19th-century, combined efforts by several mathematicians–most notably liouville, grünwald, letnikov and riemann–led to a fairly solid understanding of fractional calculus in the continuous setting. since then, fractional calculus has been used as a tool to study and model a variety of applied problems. indeed, podlubney [9] outlines fractional calculus applications in the fields of viscoelasticity, capacitor theory, electrical circuits, electroanalytical chemistry, neurology, diffusion, control theory and statistics. significantly less is known, however, about discrete fractional calculus. to the author’s knowledge, significant work to develop this area did not appear until the mid-1950’s, with the majority of interest shown within the past thirty years. diaz and osler [4] published a 1974 paper introducing a fractional difference defined as an infinite series, a generalization of the binomial formula for the nth-order difference ∆nf. although this definition agrees with the one presented in this paper for whole-order differences, it differs elsewhere. the fractional difference given in this paper is based on the one first given by gray and zhang [5] in 1988. they developed a special case for one composition rule given in this paper as well as versions of a power rule and leibniz’ formula. however, gray and zhang worked exclusively with the nabla operator, and so their results still differ from the few corresponding results found in this paper. a recent interest in discrete fractional calculus has been shown by atici and eloe, who discussed in [1] properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutivity of fractional sums. one year later, they presented in [2] more rules for composing fractional sums and differences, but they left many important cases unresolved or untouched. in addition, [1] and [2] pay little attention to function domains or to lower limits of summation and differentiation, two details vital for a correct and careful treatment of the power rule and the sum and difference composition rules—their neglect leads to domain confusion and, worse, to false or ambiguous claims. the goal of this paper is to develop and present a complete and precise theory for composing fractional sums and differences. careful attention is given to detail as several side matters are addressed along the way, including correcting and broadening the power rule stated incorrectly in [1], [2] and [3]. we conclude this paper in section 4 by using the tools developed in sections 1-3 to solve a general fractional initial value problem. 1.1 motivation we consider throughout this paper real-valued functions defined on a shift of the natural numbers: f : na → r, where na := n0 + {a} = {a, a + 1, a + 2, ...} (a ∈ r fixed). cubo 13, 3 (2011) sum and difference compositions in . . . 155 just as y(t) = t∫ a (t − s)n−1 (n − 1) ! f(s)ds, t ∈ [a, ∞) is the unique solution to the initial-value problem { y(n)(t) = f(t), t ∈ [a, ∞) y(i)(a) = 0, i = 0, 1, ..., n − 1 , so the discrete function y(t) = t−n∑ s=a (t − s − 1) n−1 (n − 1)! f(s), t ∈ na is the unique solution to the initial-value problem { ∆ny(t) = f(t), t ∈ na ∆iy(a) = 0, i = 0, 1, ..., n − 1 . in this latter case, it is easy to see that the solution y must satisfy y(a) = y(a + 1) = · · · = y(a + n − 1) = 0 and that y is found by simplifying n iterated sums of f, each taken from τ = a to τ = t (see [6]). we denote this solution y with the symbol ∆−na f, and we call y = ∆ −n a f the n th-order sum of f. 1.2 definitions the above discussion motivates the following definition for an arbitrary, real-order sum. definition 1.1. let f : na → r and ν > 0 be given. then the νth-order fractional sum of f (”the νth-sum of f”) is given by (∆−νa f)(t) := 1 γ (ν) t−ν∑ s=a (t − σ(s))ν−1f(s), for t ∈ na+ν. (1.1) also, we define the trivial sum ∆−0a f(t) := f(t), for t ∈ na. remark 1.1. • the name fractional sum is a misnomer, strictly speaking. early mathematicians working in the area had in mind rational order operators, but both in the general theory and in this paper, we allow sums of arbitrary real order—the symbols ∆−5a f, ∆ − 7 3 a f, ∆ − √ 2 a f and ∆−πa f all represent legitimate fractional sums. • for ease of notation, we will throughout this paper use the symbol ∆−νa f(t) in place of (∆−νa f) (t), where the t in ∆ −ν a f(t) represents an input for the function ∆ −ν a f, and not for the function f. 156 michael holm cubo 13, 3 (2011) • the fractional sum ∆−νa f(t) is a definite integral and depends on its lower limit of summation a. in fact, it makes sense to write ∆−νa f only if we know a priori that f is defined on na. the lower limit of summation, therefore, provides us with an important tool for keeping correct track of function domains throughout our work–omitting the subscript leads to domain confusion and general ambiguity. • the σ-function is used in (1.1) because of its tie to the more general theory of time scales, where for a discrete time scale such as na, σ(s) denotes the next point in the time scale after s. in this case, σ(s) = s + 1, for all s ∈ na. • the term (t − σ(s))ν−1 appearing in (1.1) is a use of the generalized falling function. the generalized falling function is given by t ν := γ (t + 1) γ (t + 1 − ν) , for any t, ν ∈ r for which the right-hand side is well-defined. hence, (t − σ(s))ν−1 = γ (t − s) γ (t − s − ν + 1) . we will also use the identities ν ν = γ (ν + 1), ∆tν = νtν−1 and tν+1 = (t − ν)tν, which hold whenever both sides are well-defined (see theorem 3.1 for a proof of the second identity). with a definition for the νth-fractional sum in hand, we give the following traditional definition for the νth-fractional difference [7]. definition 1.2. let f : na → r and ν ≥ 0 be given, and let n ∈ n be chosen such that n − 1 < ν ≤ n. then the νth-order fractional difference of f (“the νth-difference of f”) is given by (∆νaf) (t) = ∆ ν af(t) := ∆ n ∆ −(n−ν) a f(t), for t ∈ na+n−ν. (1.2) remark 1.2. • in this traditional definition, fractional-order differences are defined as the next higher whole-order difference acting on a small-order sum. later (theorem 2.1), we will derive an equivalent form for ∆νaf whose use is essential in many applications. • definition 1.2 agrees with the definition for standard whole-order differences: for any ν = n ∈ n0, ∆ v af(t) = ∆ n ∆ −(n−ν) a f(t) = ∆ n ∆ −0 a f(t) = ∆ n f(t), for t ∈ na. • when using (1.2), we often wish to apply the following well-known binomial formula for the whole-order difference ∆n: ∆ n f(t) = n∑ i=0 (−1)i ( n i ) f(t + n − i). cubo 13, 3 (2011) sum and difference compositions in . . . 157 • it is important to note that, whereas whole-order differences do not depend on any starting point or lower limit a, fractional differences do. to demonstrate, we know that the second difference of a function f : na → r at any point t ∈ na is given by ∆ 2 f(t) = f(t + 2) − 2f(t + 1) + f(t), an expression that in no way depends on a, but only on the values f takes on the set {t, t + 1, t + 2}. however, the fractional difference ∆ 1.5 a f(t) = ∆ 2 ∆ −0.5 a f(t) = ∆ −0.5 a f(t + 2) − 2∆ −0.5 a f(t + 1) + ∆ −0.5 a f(t) does depend on the lower limit of summation a. this dependence frustrates our traditional notion of a difference. however, the dependence of ∆νaf on a does vanish as ν → n − (see theorem 2.2). for this reason, we write ∆nf for whole-order differences and ∆νaf for general fractional differences. • in the above setting, it is important to think of ν > 0 as being situated between two natural numbers. for such a ν with n − 1 < ν ≤ n (n ∈ n0), we call the fractional difference equation ∆νa+ν−ny(t) = f(t) a ν th-order equation, and we identify it with and compare it to the whole-order difference equation ∆ny(t) = f(t). 1.3 domains when working with fractional sums and differences, it is crucial to understand their domains. we first look at domains for sums and then at domains for differences. consider the first-order definite sum of a function f at a point t ∈ na, for which we sum up the values of f(τ) from τ = a to τ = t − 1. as we know, this definite sum ∆−1a f(t) represents the ‘area’ under the graph of f from a to t, where the height of the function on the interval [t, t + 1] is given by the value f(t). in light of this, one may choose to consider the value ∆−1a f(a), thinking of this as the trivial area under f from t = a to t = a, which is zero. using definition 1.1 for ∆−1a f(a), we find that ∆ −1 a f(a) = 1 γ (1) t−1∑ s=a (t − σ(s))1−1f(s) ∣ ∣ ∣ ∣ ∣ t=a = a−1∑ s=a f(s). therefore, if we insist on considering ∆−1a f(a) as a legitimate value, then we must hold the convention that ∑a−1 s=a f(s) = 0. likewise, if we recognize the n values ∆−na f(a), ∆ −n a f(a + 1), · · ·, ∆ −n a f(a + n − 1) as having legitimate mathematical meaning, we must have ∆ −n a f(a) = ∆ −n a f(a + 1) = · · · = ∆ −n a f(a + n − 1) = 0, 158 michael holm cubo 13, 3 (2011) which leads us to adopt the convention that 1 γ (n) a+i−n∑ s=a (a + i − σ(s))n−1f(s) = 0, for i ∈ {0, 1, ..., n − 1} . this idea extends further to fractional sums as follows: for any ν > 0 with n − 1 < ν ≤ n, ∆ −ν a f(a + ν − n) = ∆ −ν a f(a + ν − n + 1) = · · · = ∆ −ν a f(a + ν − 1) = 0. in light of this, it is convenient and sensible to ignore these zeros and to define d { ∆ −ν a f } := na+ν, as given in definition 1.1. it can be shown that the first nontrivial value ∆−νa f takes on is the value ∆ −ν a f(a + ν) = f(a). however, we will at certain times recall and use the fact that the above discussed zeros exist before this point in the domain. we next use fractional sum domains to determine fractional difference domains. whereas whole-order differences are domain preserving operators (i.e. d ( ∆nf ) = d (f) , for all n ∈ n0), we find that a fractional difference operator shifts the domain of its argument. using definition 1.2, we find that for f : na → r and ν > 0 with n − 1 < ν ≤ n, d {∆νaf} = d { ∆ n ∆ −(n−ν) a f } = d { ∆ −(n−ν) a f } = na+n−ν. note that whereas the domain shift for a fractional sum is a large shift by ν to the right, the domain shift for a fractional difference is a relatively small shift by n − ν to the right. we next focus on the domains of two-operator sum and difference compositions. consider, for example, the composition ∆ −ν a+µ∆ −µ a f(t) = 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1 ( 1 γ (µ) s−µ∑ r=a (s − σ(r))µ−1f(r) ) . notice that the lower limit of the outer operator ∆−νa+µ must match the domain of the inner function ∆ −µ a f(t), which is na+µ. hence, the domain of the entire composition is na+µ+ν. the summary below shows the domains of all four possible sum and difference compositions. summary 1.1. let f : na → r and ν, µ > 0 be given, with n − 1 < ν ≤ n and m − 1 < µ ≤ m. then • d {∆−νa f} = na+ν • d {∆ ν af} = na+n−ν • d { ∆−νa+µ∆ −µ a f } = na+µ+ν • d { ∆νa+µ∆ −µ a f } = na+µ+n−ν • d { ∆−ν a+m−µ ∆ µ af } = na+m−µ+ν • d { ∆νa+m−µ∆ µ af } = na+m−µ+n−ν cubo 13, 3 (2011) sum and difference compositions in . . . 159 2 unifying the fractional sum and difference we show here that fractional sums and differences can be unified by a common definition, for which an appropriate version of leibniz’ rule is useful. let g : na+ν × na → r be given. then ∆ ( t−ν∑ s=a g(t, s) ) = t−ν∑ s=a ∆tg(t, s) + g(t + 1, t + 1 − ν), for t ∈ na+ν. (2.1) leibniz’ rule (2.1) is used in proving the following theorem, which gives a different but equivalent way of defining a fractional difference, one which mirrors the definition for the fractional sum. moreover, theorem 2.1 will allow us to substantially extend results from previous papers– most notably the power and composition rules. theorem 2.1. let f : na→ r and ν > 0 be given, with n − 1 < ν ≤ n. then the following two definitions for the fractional difference ∆νaf : na+n−ν → r are equivalent: ∆ ν af(t) := ∆ n ∆ −(n−ν) a f(t), (2.2) ∆ ν af(t) :=    1 γ (−ν) t+ν∑ s=a (t − σ(s))−ν−1f(s), n − 1 < ν < n ∆nf(t), ν = n (2.3) proof. let f and ν be given as in the statement of the theorem. we assume that (2.2) is the correct definition for the fractional difference and show that (2.3) is equivalent, for t ∈ na+n−ν. if ν = n, then definitions (2.2) and (2.3) are clearly equivalent, since ∆ ν af(t) = ∆ n ∆ −(n−ν) a f(t) = ∆ n ∆ −0 a f(t) = ∆ n f(t). if n − 1 < ν < n, then direct application of (2.2) yields ∆ ν af(t) = ∆n∆−(n−ν)a f(t) = ∆n   1 γ (n − ν) t−(n−ν)∑ s=a (t − σ(s))n−ν−1f(s)   = ∆n−1 γ (n − ν) · ∆   t−(n−ν)∑ s=a (t − σ(s))n−ν−1f(s)   (apply (2.1)), = ∆n−1 γ (n − ν) [ t−(n−ν)∑ s=a ( (n − ν − 1)(t − σ(s))n−ν−2f(s) ) 160 michael holm cubo 13, 3 (2011) + (t + 1 − σ(t + 1 − (n − ν)))n−ν−1f(t + 1 − (n − ν)) ] = ∆n−1 [ t−(n−ν)∑ s=a (t − σ(s))n−ν−2 γ (n − ν − 1) f(s) + f(t + 1 − (n − ν)) ] = ∆n−1   1 γ (n − ν − 1) t+1−(n−ν)∑ s=a (t − σ (s))n−ν−2f(s)   = ∆n−1   1 γ (n − ν − 1) t−(n−ν−1)∑ s=a (t − σ (s))n−ν−2f(s)   . repeating these steps n − 2 more times, we find that ∆ ν af(t) = ∆ n−1   1 γ (n − ν − 1) t−(n−ν−1)∑ s=a (t − σ (s))n−ν−2f(s)   = ∆n−2   1 γ (n − ν − 2) t−(n−ν−2)∑ s=a (t − σ (s))n−ν−3f(s)   = · · · · · · · · · · · · · · · · · · · = ∆n−n   1 γ (n − ν − n) t−(n−ν−n)∑ s=a (t − σ (s)) n−ν−(n+1) f(s)   = 1 γ (−ν) t+ν∑ s=a (t − σ (s))−ν−1f(s). note that since n − 1 < ν < n in the above work, the term 1 γ (n−ν−k) exists for each k = 1, 2, ..., n . furthermore, for each point t ∈ na+n−ν (say t = a + n − ν + m, for some m ∈ n0), (t − σ(s))n−ν−k−1 = γ (t − s) γ (t − s − n + ν + k + 1) = γ (a + n − ν + m − s) γ (a + m + k + 1 − s) exists and is well-defined for each k ∈ {1, 2, ..., n} and s ∈ {a, a + 1, ..., t − (n − ν − k)} = {a, a + 1, ..., a + m + k} . note, finally, that although definition (2.3) appears to be valid for all t ∈ na−ν, it only defines the νth-fractional difference on na+n−ν. in definition (2.3), one may wonder about the continuity of ∆νaf with respect to ν. we certainly desire for every function f : na → r, for example, that ∆1.99a f be very close to ∆ 2f. however, the term 1 γ (−ν) in (2.3) blows up as ν → n−! the following theorem addresses this matter. cubo 13, 3 (2011) sum and difference compositions in . . . 161 theorem 2.2. let f : na → r be given. then the fractional difference ∆νaf is continuous with respect to ν ≥ 0. more specifically, for each ν > 0, let tν,m := a + ⌈ν⌉ − ν + m be a fixed but arbitrary point in d {∆νaf} . then for each m ∈ n0, ν 7→ ∆νaf(tν,m) is continuous on [0, ∞). proof. let f : na → r be given, and fix n ∈ n and m ∈ n0. it is enough to show that ∆ ν af(a + n − ν + m) is continuous with respect to ν on (n − 1, n) (2.4) ∆ ν af(a + n − ν + m) → ∆ n f(a + m) as ν → n− (2.5) ∆ ν af(a + n − ν + m) → ∆ n−1 f(a + m + 1) as ν → (n − 1)+ (2.6) note that for any fixed ν > 0 with n − 1 < ν < n, ∆ ν af(a + n − ν + m) = 1 γ (−ν) t+ν∑ s=a (t − σ(s))−ν−1f(s) ∣ ∣ ∣ ∣ ∣ t=a+n−ν+m = 1 γ (−ν) a+n+m∑ s=a (a + n − ν + m − σ(s))−ν−1f(s) = 1 γ (−ν) a+n+m∑ s=a γ (a + n − ν + m − s) γ (a + n + m − s + 1) f(s) = a+n+m∑ s=a 1 γ (a + n + m − s + 1) γ (a + n − ν + m − s) γ (−ν) f(s) = a+n+m−1∑ s=a ( (a + n − ν + m − s − 1) · · · (−ν) (a + n + m − s)! f(s) ) + f(a + n + m) = n+m∑ i=1 ( (i − 1 − ν) · · · (−ν + 1) (−ν) i! f(a + n + m − i) ) + f(a + n + m). the above line demonstrates (2.4), the continuity of ν → ∆νaf(a + n − ν + m) on (n − 1, n). next, we take ν → n− to show (2.5): lim ν→ n− ∆ ν af(a + n − ν + m) = lim ν→ n− [ n+m∑ i=1 ( (i − 1 − ν) · · · (−ν) i! f(a + n + m − i) ) + f(a + n + m) ] = n+m∑ i=1 ( (i − 1 − n) · · · (−n) i! f(a + n + m − i) ) + f(a + n + m) = n∑ i=1 ( (i − 1 − n) · · · (−n) i! f(a + n + m − i) ) + f(a + n + m) (2.7) 162 michael holm cubo 13, 3 (2011) since the argument is zero for i = n + 1, ..., n + m, = n∑ i=1 ( (−1)i (n) · · · (n − i + 1) i! f(a + n + m − i) ) + f(a + n + m) = n∑ i=1 ( (−1)i ( n i ) f(a + n + m − i) ) + f(a + n + m) = n∑ i=0 (−1)i ( n i ) f(a + n + m − i) = n∑ i=0 (−1)i ( n i ) f((a + m) + n − i) = ∆nf(a + m). finally, we take ν → (n − 1)+ to show (2.6): lim ν→ (n−1)+ ∆ ν af(a + n − ν + m) = lim ν→ (n−1)+ [ n+m∑ i=1 ( (i − 1 − ν) · · · (−ν) i! f(a + n + m − i) ) + f(a + n + m) ] = n+m∑ i=1 ( (i − n) · · · (−n + 1) i! f(a + n + m − i) ) + f(a + n + m) = n−1∑ i=1 ( (−1)i (n − 1) · · · (n − i) i! f(a + n + m − i) ) + f(a + n + m) = n−1∑ i=1 ( (−1)i ( n − 1 i ) f(a + n + m − i) ) + f(a + n + m) = n−1∑ i=0 ( (−1)i ( n − 1 i ) f(a + m + 1 + (n − 1) − i) ) = ∆n−1f(a + m + 1). remark 2.1. • the above statement (2.7) shows explicitly why a fractional difference’s dependence on its lower limit a vanishes as the order of differentiation approaches a whole number. • theorem 2.2 implies that for any f : na → r and m ∈ n0, the sequence ∆ 1.9 a f(a + m + 0.1), ∆ 1.99 a f(a + m + 0.01), ∆ 1.999 a f(a + m + 0.001), ... cubo 13, 3 (2011) sum and difference compositions in . . . 163 approaches the value ∆2f(a + m). this notion of ”order continuity” adds a beauty to fractional calculus absent from standard whole-order calculus. theorem 2.2 shows that definition (2.3) in theorem 2.1 is a good and reasonable definition for the fractional difference. hence, we may now unify the fractional sum and difference into a single definition: definition 2.1. let f : na → r and ν > 0 be given. then (i) the νth-fractional sum of f is given by ∆ −ν a f(t) := 1 γ (ν) t−ν∑ s=a (t − σ(s))ν−1f(s), t ∈ na+ν. (ii) the νth-fractional difference of f is given by ∆ ν af(t) :=    1 γ (−ν) t+ν∑ s=a (t − σ(s))−ν−1f(s), ν 6∈ n ∆nf(t), ν = n ∈ n , t ∈ na+n−ν to demonstrate the importance of definition 2.1, we offer the following theorem generalizing the binomial representation of whole-order differences to fractional sums and differences. theorem 2.3. let f : na → r and ν > 0 be given, with n − 1 < ν ≤ n. for each t ∈ na+n−ν, ∆ ν af(t) = ν+t−a∑ k=0 (−1)k ( ν k ) f(t + ν − k). (2.8) for each t ∈ na+ν, ∆ −ν a f(t) = −ν+t−a∑ k=0 (−1)k ( −ν k ) f(t − ν − k) (2.9) = −ν+t−a∑ k=0 ( ν + k − 1 k ) f(t − ν − k). (2.10) proof. let f, ν and n be given as in the statement of the theorem, and let t ∈ na+n−ν be given 164 michael holm cubo 13, 3 (2011) by t = a + n − ν + m, for some m ∈ n0. then ∆ ν af(t) = 1 γ (−ν) t+ν∑ s=a (t − σ(s))−ν−1f(s) = t+ν∑ s=a γ (t − s) γ (t − s + ν + 1)γ (−ν) f(s) = a+n+m∑ s=a γ (a + n − ν + m − s) γ (a + n + m − s + 1)γ (−ν) f(s) = n+m∑ s=0 γ (n + m − s − ν) γ (n + m − s + 1)γ (−ν) f(s + a) = f(a + n + m) + n+m−1∑ s=0 (n + m − 1 − s − ν) · · · (−ν) γ (n + m − s + 1) f(s + a) = f(a + n + m) + n+m−1∑ s=0 (−1)n+m−s (ν) · · · (ν − (n + m − s) + 1) γ (n + m − s + 1) f(s + a) = n+m∑ s=0 (−1)n+m−s ( ν n + m − s ) f(s + a) = n+m∑ k=0 (−1)k ( ν k ) f(a + n + m − k) = n+m∑ k=0 (−1)k ( ν k ) f((a + n − ν + m) + ν − k) = ν+t−a∑ k=0 (−1)k ( ν k ) f(t + ν − k), proving (2.8) note that when ν = n, this reduces to the traditional binomial formula: ∆ n f(t) = n∑ k=0 (−1)k ( n k ) f(t + n − k), for t ∈ na. a very similar argument proves (2.9). moreover, when ν = n, we interpret ( −n k ) = γ (−n + 1) k!γ (−n − k + 1) = (−n) · · · (−n − k + 1) k! = (−1)k ( n + k − 1 k ) . in any case, we may write ( −ν k ) = (−1)k ( ν + k − 1 k ) , whose substitution into (2.9) yields (2.10). although (2.10) is probably more useful, (2.9) more closely resembles the traditional binomial formula. cubo 13, 3 (2011) sum and difference compositions in . . . 165 3 fractional sum and difference composition rules we turn now to the main focus of this paper. having set the stage by defining and developing many properties of fractional sums and differences, we have the necessary tools to study the following four compositions:    ∆−νa+µ∆ −µ a f(t) = 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1 s−µ∑ r=a (s−σ(r)) µ−1 γ (µ) f(r) ∆νa+µ∆ −µ a f(t) = 1 γ (−ν) t+ν∑ s=a+µ (t − σ(s))−ν−1 s−µ∑ r=a (s−σ(r)) µ−1 γ (µ) f(r) ∆−ν a+m−µ ∆ µ af(t) = 1 γ (ν) t−ν∑ s=a+m−µ (t − σ(s))ν−1 s+µ∑ r=a (s−σ(r)) −µ−1 γ (−µ) f(r) ∆νa+m−µ∆ µ af(t) = 1 γ (−ν) t+ν∑ s=a+m−µ (t − σ(s))−ν−1 s+µ∑ r=a (s−σ(r)) −µ−1 γ (−µ) f(r) whose domains are as given in summary 1.1. definition 2.1 is the tool allowing us to write the above compositions in this more uniform and often more useful way. it will be helpful to keep the above representations and their domains in mind as we develop a rule governing each composition. to work effectively with these compositions, however, we first need a general and precise power rule for summation and differentiation. much of the following proof may be found in [1]—however, not having kept track of domains or lower limits, the power rule given in [1] is incorrect as stated. the precise power rule presented in lemma 3.1 below corrects and extends this previous version. lemma 3.1. [power rule for summation and differentiation] let a ∈ r and µ > 0 be given. then, ∆(t − a)µ = µ(t − a)µ−1, (3.1) for any t for which both sides are well-defined. furthermore, for ν > 0 and µ ∈ r\ (−n) , ∆ −ν a+µ(t − a) µ = µ−ν (t − a) µ+ν , for t ∈ na+µ+ν, (3.2) and ∆ ν a+µ(t − a) µ = µν (t − a) µ−ν , for t ∈ na+µ+n−ν. (3.3) proof. it is easy to show (3.1) using the definition of the delta difference and properties of the gamma function. for (3.2) and (3.3), first notice that (t − a)µ, (t − a)µ+ν and (t − a)µ−ν are each well-defined and positive on their respective domains na+µ, na+µ+ν and na+µ+n−ν. to prove (3.2), we suppose µ ∈ r\ (−n) and consider the two cases ν = 1 and ν ∈ (0, 1)∪(1, ∞) 166 michael holm cubo 13, 3 (2011) separately. for ν = 1, we see from direct calculation that ∆ −1 a−µ(t − a) µ = ∆−1a−µ∆ ( (t − a) µ+1 µ + 1 ) , using (3.1) = t−1∑ s=a+µ ( (s + 1 − a) µ+1 µ + 1 − (s − a) µ+1 µ + 1 ) = (t − a) µ+1 µ + 1 − µµ+1 µ + 1 = µ−1 (t − a) µ+1 . for ν > 0 with ν 6= 1, define g1(t) := ∆ −ν a+µ(t − a) µ = 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1(s − a)µ and g2(t) := µ −ν (t − a) µ+ν , each for t ∈ na+µ+ν. we will show that both g1 and g2 solve the first-order initial value problem { (t − a − (µ + ν) + 1)∆g(t) = (µ + ν)g(t), for t ∈ na+µ+ν g(a + µ + ν) = γ (µ + 1) . (3.4) since g1(a + µ + ν) = 1 γ (ν) a+µ∑ s=a+µ (a + µ + ν − σ(s))ν−1(s − a)µ = 1 γ (ν) (ν − 1)ν−1µµ = γ (µ + 1), and g2(a + µ + ν) = µ −ν (µ + ν) µ+ν = γ (µ + 1) γ (µ + 1 + ν) γ (µ + ν + 1) = γ (µ + 1), both g1 and g2 satisfy the initial condition in (3.4). cubo 13, 3 (2011) sum and difference compositions in . . . 167 we next show that g1 satisfies the difference equation in (3.4). for t ∈ na+µ+ν, ∆g1(t) = ∆ [ 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1(s − a)µ ] (apply (2.1)) = 1 γ (ν) t−ν∑ s=a+µ (ν − 1)(t − σ(s))ν−2(s − a)µ + (t + 1 − (t + 2 − ν))ν−1 γ (ν) (t + 1 − ν − a)µ = ν − 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−2(s − a)µ + (t + 1 − ν − a)µ. also, we may manipulate g1 directly: g1(t) = 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1(s − a)µ = 1 γ (ν) t−ν∑ s=a+µ (t − σ(s) − (ν − 2))(t − σ(s))ν−2(s − a)µ = 1 γ (ν) t−ν∑ s=a+µ [(t − a − (µ + ν) + 1) − (s − a − µ)] (t − σ(s))ν−2(s − a)µ = t − a − (µ + ν) + 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−2(s − a)µ − 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−2(s − a)µ+1 = h(t) − k(t), where    h(t) := t−a−(µ+ν)+1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−2(s − a)µ k(t) := 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−2(s − a)µ+1 . integrating k by parts, k(t) = 1 γ (ν)   (t−ν+1)−1∑ s=a+µ (s − a)µ+1∆ ( − (t − s)ν−1 ν − 1 )   = 1 γ (ν) [ ( (s − a)µ+1 ( − (t − s)ν−1 ν − 1 )) ∣ ∣ ∣ ∣ ∣ s=t−ν+1 s=a+µ 168 michael holm cubo 13, 3 (2011) − (t−ν+1)−1∑ s=a+µ ( − (t − σ(s))ν−1 ν − 1 (µ + 1)(s − a)µ ) ] = 1 γ (ν) [ − γ (ν) ν − 1 (t − ν + 1 − a)µ+1 + µ + 1 ν − 1 t−ν∑ s=a+µ (t − σ(s))ν−1(s − a)µ ] = 1 ν − 1 [ µ + 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1(s − a)µ − (t − ν + 1 − a)µ+1 ] . it follows from the above work that • (t − a − (µ + ν) + 1)∆g1(t) = (ν − 1)h(t) + (t + 1 − ν − a) µ+1 • (µ + 1)g1(t) − (ν − 1)k(t) = (t + 1 − ν − a) µ+1. hence, (t − a − (µ + ν) + 1)∆g1(t) = (ν − 1)h(t) + (µ + 1)g1(t) − (ν − 1)k(t) = (ν − 1)g1(t) + (µ + 1)g1(t) = (µ + ν)g1(t). finally, g2 also satisfies the difference equation in (3.4): (t − a − (µ + ν) + 1)∆g2(t) = (t − a − (µ + ν) + 1) [ µ −ν(µ + ν) (t − a) µ+ν−1 ] , by (3.1) = (µ + ν)µ−ν [ (t − a − (µ + ν − 1)) (t − a) µ+ν−1 ] = (µ + ν)µ−ν(t − a)µ+ν = (µ + ν)g2(t). by the uniqueness of solutions to initial value problems for first order difference operators, we conclude that g1 ≡ g2 on na+µ+ν. we employ (3.1) and (3.2) in showing (3.3) as follows: for t ∈ na+µ+n−ν, ∆ ν a+µ(t − a) µ = ∆n [ ∆ −(n−ν) a+µ (t − a) µ ] = ∆n [ γ (µ + 1) γ (µ + 1 + n − ν) (t − a)µ+n−ν ] , by (3.2) = γ (µ + 1) γ (µ + 1 + n − ν) ( (µ + n − ν) · · · (µ + 1 − ν) ) (t − a)µ−ν, by (3.1) cubo 13, 3 (2011) sum and difference compositions in . . . 169 = γ (µ + 1) γ (µ + 1 + n − ν) γ (µ + n − ν + 1) γ (µ + 1 − ν) (t − a)µ−ν = µν(t − a)µ−ν. in the special case ν ∈ {µ + 1, µ + 2, ...}, we have µ + 1 − ν ∈ (−n0), and so µ ν = γ (µ + 1) γ (µ + 1 − ν) from (3.3) is not well-defined. in this case, we interpret the right hand side of (3.3) as zero, which is as we desire. 3.1 composing a sum with a sum the rule for composing two fractional sums depends on an appropriate application of power rule (3.2) presented in lemma 3.1 (see [1]). theorem 3.1. let f : na → r be given and suppose ν, µ > 0. then ∆ −ν a+µ∆ −µ a f(t) = ∆ −ν−µ a f(t) = ∆ −µ a+ν∆ −ν a f(t), for t ∈ na+µ+ν. proof. suppose f : na → r and ν, µ > 0. then for t ∈ na+µ+ν, ∆ −ν a+µ∆ −µ a f(t) = 1 γ (ν) t−ν∑ s=a+µ (t − σ(s))ν−1 ( 1 γ (µ) s−µ∑ r=a (s − σ(r))µ−1f(r) ) = 1 γ (ν)γ (µ) t−ν∑ s=a+µ s−µ∑ r=a (t − σ(s))ν−1(s − σ(r))µ−1f(r) = 1 γ (ν)γ (µ) t−(ν+µ)∑ r=a t−ν∑ s=r+µ (t − σ(s))ν−1(s − σ(r))µ−1f(r) . letting x = s − σ(r), we continue: = 1 γ (ν)γ (µ) t−(ν+µ)∑ r=a   t−ν−r−1∑ x=µ−1 (t − x − r − 2)ν−1xµ−1  f(r) = 1 γ (µ) t−(ν+µ)∑ r=a   1 γ (ν) (t−r−1)−ν∑ x=µ−1 ((t − r − 1) − σ(x))ν−1xµ−1  f(r) = 1 γ (µ) t−(ν+µ)∑ r=a ( [ ∆ −ν µ−1(t µ−1) ] ∣ ∣ ∣ ∣ ∣ t−r−1 f(r) ) = 1 γ (µ) t−(ν+µ)∑ r=a γ (µ) γ (µ + ν) (t − r − 1)µ−1+νf(r), using (3.2) 170 michael holm cubo 13, 3 (2011) = 1 γ (ν + µ) t−(ν+µ)∑ r=a (t − σ(r)) (ν+µ)−1 f(r) = ∆−(ν+µ)a f(t) = ∆−ν−µa f(t). since ν and µ are arbitrary, we conclude more generally ∆ −ν a+µ∆ −µ a f(t) = ∆ −ν−µ a f(t) = ∆ −µ a+ν∆ −ν a f(t), for t ∈ na+ν+µ. remark 3.1. in applying (3.2) above, we are allowed to write ∆ −ν µ−1 [ τ µ−1 ] = γ (µ) γ (µ + ν) τ µ−1+ν , for τ ∈ nµ−1+ν. since we are working with t ∈ na+µ+ν and r ∈ {a, ..., t − µ − ν} , it is indeed appropriate to evaluate these terms at t − r − 1 ∈ nµ+ν−1. 3.2 composing a difference with a sum before studying the more general composition ∆νa+µ ◦ ∆ −µ a , we first consider the special case when ν ∈ n0. note that atici and eloe [2] show (3.5) below for the case µ > k. lemma 3.2. let f : na → r be given. for any k ∈ n0 and µ > 0 with m − 1 < µ ≤ m, we have ∆ k ∆ −µ a f(t) = ∆ k−µ a f(t), for t ∈ na+µ (3.5) ∆ k ∆ µ af(t) = ∆ k+µ a f(t), for t ∈ na+m−µ (3.6) proof. let f, µ, m and k be given as in the statement of the lemma. case 3.1. (µ = m) observe that for t ∈ na+1, ∆∆ −1 a f(t) = ∆ [ t−1∑ s=a f(s) ] = t∑ s=a f(s) − t−1∑ s=a f(s) = f(t). likewise, for any k ∈ n, ∆ k ∆ −k a f(t) = ∆ k−1 [ ∆∆ −1 a+k−1 ( ∆ −(k−1) a f(t) )] = ∆k−1∆−(k−1)a f(t) = · · · · · · · = f(t), for t ∈ na+k. cubo 13, 3 (2011) sum and difference compositions in . . . 171 so, for any t ∈ na+m,    ∆k∆−ma f(t) = ∆ k−m [ ∆m∆−ma f(t) ] = ∆k−mf(t), if k ≥ m ∆k∆−ma f(t) = ∆ k∆−k a+m−k [ ∆ −(m−k) a f(t) ] = ∆k−ma f(t), if k < m . for (3.6), it is clear already that whole order difference operators commute. case 3.2. (m − 1 < µ < m) we first show that ∆∆ µ af(t) = ∆ 1+µ a f(t), for t ∈ na+m−µ. given t ∈ na+m−µ and using definition 2.1 for ∆ µ af, we find ∆∆ µ af(t) = ∆ [ 1 γ (−µ) t+µ∑ s=a (t − σ(s))−µ−1f(s) ] (apply (2.1)) = 1 γ (−µ) t+µ∑ s=a (−µ − 1)(t − σ(s))−µ−2f(s) + f(t + µ + 1) = 1 γ (−µ − 1) t+µ∑ s=a (t − σ(s))−µ−2f(s) + f(t + µ + 1) = 1 γ (−µ − 1) t+µ+1∑ s=a (t − σ(s))−µ−2f(s) = 1 γ (−µ − 1) t−(−µ−1)∑ s=a (t − σ(s)) (−µ−1)−1 f(s) = ∆−(−µ−1)a f(t) = ∆1+µa f(t). therefore, for any k ∈ n, ∆ k ∆ µ af(t) = ∆ k−1 [∆∆µaf(t)] = ∆k−1∆1+µa f(t) = · · · · · · ·· = ∆k+µa f(t), for t ∈ na+m−µ, proving (3.6) the results presented in section 2 allow us to employ identical work as shown above to prove (3.5), since we may replace each µ above with a negative µ. we now have all the tools in hand to write down a rule for composing fractional differences with fractional sums. 172 michael holm cubo 13, 3 (2011) theorem 3.2. let f : na → r be given, and suppose ν, µ > 0 with n − 1 < ν ≤ n. then ∆ ν a+µ∆ −µ a f(t) = ∆ ν−µ a f(t), for t ∈ na+µ+n−ν. proof. given f, ν, n and µ as in the statement of the theorem and t ∈ na+µ+n−ν, ∆ ν a+µ∆ −µ a f(t) = ∆ n ∆ −(n−ν) a+µ ∆ −µ a f(t) = ∆n∆−(n−ν+µ)a f(t), using theorem 3.1, = ∆n−(n−ν+µ)a f(t), using lemma 3.2, = ∆ν−µa f(t). remark 3.2. one may wonder if the correct domain has been chosen in theorem 3.2. to check this, we write out    d { ∆νa+µ∆ −µ a f } = na+µ+n−ν d { ∆ ν−µ a f(t) } = na+µ−ν, if ν < µ d { ∆ ν−µ a f(t) } = na+⌈ν−µ⌉−(ν−µ), if ν ≥ µ , and notice that in both cases, d { ∆νa+µ∆ −µ a f } ⊆ d { ∆ ν−µ a f(t) } . we reason that the composition rule holds on na+µ+n−ν, the intersection of the domains of the left and right hand sides of the equation. 3.3 composing a sum with a difference for the remaining two composition rules—those whose inner operation is differentiation—we may not simply add the two operators’ orders. this comes as no surprise, however, in light of the fundamental theorem of calculus: t−1∑ s=a (∆f(s)) = f(t) − f(a). theorem 3.3. let f : na → r be given, and suppose k ∈ n0 and ν, µ > 0 with m − 1 < µ ≤ m. then, ∆ −ν a ∆ k f(t) = ∆k−νa f(t) − k−1∑ j=0 ∆jf(a) γ (ν − k + j + 1) (t − a)ν−k+j, (3.7) for t ∈ na+ν, and ∆ −ν a+m−µ∆ µ af(t) = ∆ µ−ν a f(t) − m−1∑ j=0 ∆ j−(m−µ) a f(a + m − µ) γ (ν − m + j + 1) (t − a − m + µ)ν−m+j, (3.8) for t ∈ na+m−µ+ν cubo 13, 3 (2011) sum and difference compositions in . . . 173 proof. although (3.7) is merely a special case of (3.8), it is significant in its own right, and its proof (found in [2]) provides a stepping stone to (3.8). (3.7) let k ∈ n0 be given, and suppose ν > 0 with ν 6∈ {1, 2, ..., k − 1} . then summing by parts with t ∈ na+ν, ∆ −ν a ∆ k f(t) = 1 γ (ν) t−ν∑ s=a (t − σ(s))ν−1 [ ∆ k f(s) ] = 1 γ (ν) (t−ν+1)−1∑ s=a (t − σ(s))ν−1∆ ( ∆ k−1 f(s) ) , = 1 γ (ν) [ (t − s)ν−1∆k−1f(s) ∣ ∣ ∣ ∣ ∣ s=t−ν+1 s=a − t−ν∑ s=a ( − (ν − 1)(t − σ(s))ν−2∆k−1f(s) ) ] = ∆k−1f(t − ν + 1) − (t − a)ν−1 γ (ν) ∆ k−1 f(a) + 1 γ (ν − 1) t−ν∑ s=a ( (t − σ(s))ν−2∆k−1f(s) ) = 1 γ (ν − 1) t−ν+1∑ s=a ( (t − σ(s))ν−2∆k−1f(s) ) − ∆k−1f(a) γ (ν) (t − a)ν−1 = ∆−(ν−1)a [ ∆ k−1 f(t) ] − ∆k−1f(a) γ (ν) (t − a)ν−1 = ∆1−νa ∆ k−1 f(t) − ∆k−1f(a) γ (ν) (t − a)ν−1. continuing summation by parts (k − 1)-more times yields ∆ −ν a ∆ k f(t) = ∆1−νa ∆ k−1 f(t) − ∆k−1f(a) γ (ν) (t − a)ν−1 = ∆2−νa ∆ k−2 f(t) − ∆k−1f(a) γ (ν) (t − a)ν−1 − ∆k−2f(a) γ (ν − 1) (t − a)ν−2 = · · · · · · · · · · · · · · · · · · · · · · ·· = ∆k−νa f(t) − k∑ i=1 ∆k−if(a) γ (ν − i + 1) (t − a)ν−i = ∆k−νa f(t) − k−1∑ j=0 ∆jf(a) γ (ν − k + j + 1) (t − a)ν−k+j, for t ∈ na+ν. 174 michael holm cubo 13, 3 (2011) note that our assumption ν 6∈ {1, 2, ..., k − 1} implies that ν − k + j + 1 6∈ (−n0), and so the above expression is well-defined. next, suppose that ν ∈ {1, 2, ..., k − 1}. then k − ν ∈ n, and so for t ∈ na+ν, ∆ −ν a ∆ k f(t) = ∆k−ν∆ −(k−ν) a+ν ∆ −ν a ∆ k f(t), using theorem 3.2 = ∆k−ν [ ∆ −k a ∆ k f(t) ] , using theorem 3.1 = ∆k−ν  f(t) − k−1∑ j=0 ∆jf(a) γ (j + 1) (t − a)j   , applying the previous case = ∆k−νf(t) − k−1∑ j=0 ∆jf(a) γ (j + 1) [ ∆ k−ν(t − a)j ] = ∆k−νf(t) − k−1∑ j=k−ν ∆jf(a) γ (j + 1) γ (j + 1) γ (j + 1 − k + ν) (t − a)j−k+ν, = ∆k−νf(t) − k−1∑ j=0 ∆jf(a) γ (j + 1 − k + ν) (t − a)j−k+ν, since 1 γ (ν+1−k) = · · · = 1 γ (−1) = 1 γ (0) = 0, by convention. the above two cases prove (3.7). (3.8) suppose now that ν, µ > 0 with m − 1 < µ ≤ m. defining g(t) := ∆ −(m−µ) a f(t) and b := a + m − µ (the first point in the domain of g), we have for t ∈ na+m−µ+ν, ∆ −ν a+m−µ∆ µ af(t) = ∆−ν a+m−µ∆ m (g(t)) , and applying (3.7), = ∆m−ν a+m−µg(t) − m−1∑ j=0 ∆jg(b) γ (ν − m + j + 1) (t − b)ν−m+j = ∆m−ν a+m−µ∆ −(m−µ) a f(t) − m−1∑ j=0 ∆j∆ −(m−µ) a f(b) γ (ν − m + j + 1) (t − b)ν−m+j = ∆µ−νa f(t) − m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (ν − m + j + 1) (t − a − m + µ)ν−m+j, where in this last step, we applied theorem 3.2 twice. remark 3.3. • theorem 3.2 allows us to write (3.8) in the equivalent form ∆ −ν a+m−µ∆ µ af(t) = ∆ µ a+ν∆ −ν a f(t) − m−1∑ j=0 ∆ j−(m−µ) a f(a + m − µ) γ (ν − m + j + 1) (t − a − m + µ)ν−m+j, cubo 13, 3 (2011) sum and difference compositions in . . . 175 for t ∈ na+m−µ+ν. • when 0 < µ ≤ 1, the term ∆ j−(m−µ) a f(a + m − µ) in (3.8) simplifies nicely to f(a). more generally for any m − 1 < µ ≤ m, we have that for j ∈ {0, ..., m − 1} , ∆ j−(m−µ) a f(a + m − µ) = 1 γ (m − µ − j) t+j−(m−µ)∑ s=a (t − σ(s))m−µ−j−1f(s) ∣ ∣ ∣ ∣ ∣ t=a+m−µ = 1 γ (m − µ − j) a+j∑ s=a (a + m − µ − σ(s))m−µ−j−1f(s) = 1 γ (m − µ − j) j∑ k=0 (m − µ − σ(k))m−µ−j−1f(k + a) = j∑ k=0 ( m − µ − σ(k) j − k ) f(k + a). 3.4 composing a difference with a difference we conclude this section with a rule for composing two fractional differences. one quickly observes the similarity between the composition rule (3.9) below and the rule for ∆−ν a+m−µ ◦ ∆ µ a given in (3.8). the special case µ ∈ n0 in (3.9) below was developed by atici and eloe in [2]—we extend their result here to the more general case µ > 0. theorem 3.4. let f : na → r be given and suppose ν, µ > 0 with n − 1 < ν ≤ n and m − 1 < µ ≤ m. then for t ∈ na+m−µ+n−ν, ∆ ν a+m−µ∆ µ af(t) = ∆ ν+µ a f(t) − m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (−ν − m + j + 1) (t − a − m + µ)−ν−m +j, (3.9) where the terms in the summation vanish for ν ∈ n0, by our convention for γ . proof. let f, ν and µ be given as in the statement of the theorem. recall that lemma 3.2 proves (3.9) in the case when ν = n. 176 michael holm cubo 13, 3 (2011) if n − 1 < ν < n, then for t ∈ na+m−µ+n−ν, we have ∆ ν a+m−µ∆ µ af(t) = ∆n [ ∆ −(n−ν) a+m−µ ∆ µ af(t) ] = ∆n [ ∆ −n+ν+µ a f(t)− m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (n − ν − m + j + 1) (t − a − m + µ)n−ν−m+j ] , by (3.8) = ∆ν+µa f(t)− m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (n − ν − m + j + 1) ∆ n [ (t − a − m + µ)n−ν−m+j ] = ∆ν+µa f(t) − m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (−ν − m + j + 1) (t − a − m + µ)−ν−m+j. by the same token as (3.9), ∆ µ a+n−ν ∆ ν af(t) = ∆ µ+ν a f(t) − n−1∑ j=0 ∆ j−n+ν a f(a + n − ν) γ (−µ − n + j + 1) (t − a − n + ν)−µ−n+j, where the terms in the summation vanish for µ ∈ n0. combining this with (3.9), we may write two further rules for composing fractional differences: corollary 3.1. let f : na → r and ν, µ > 0 be given, with n − 1 < ν ≤ n and m − 1 < µ ≤ m. then, ∆ ν a+m−µ∆ µ af(t) = ∆ µ a+n−ν ∆ ν af(t) + n−1∑ j=0 ∆ j−n+ν a f(a + n − ν) γ (−µ − n + j + 1) (t − a − n + ν)−µ−n+j − m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (−ν − m + j + 1) (t − a − m + µ)−ν−m+j, for t ∈ na+m−µ+n−ν, and ∆ ν a+n−ν∆ ν af(t) = ∆ 2ν a f(t) − n−1∑ j=0 ∆ j−n+ν a f(a + n − ν) γ (−ν − n + j + 1) (t − a − n + ν)−ν−n+j, for t ∈ na+2(n−ν). cubo 13, 3 (2011) sum and difference compositions in . . . 177 4 application and examples to explicitly solve a nonhomogeneous, νth-order fractional initial value problem, we need many of the tools developed thus far in sections 2 and 3. specifically, we apply the general power rule (3.3) in lemma 3.1 and the two composition rules found in theorems 3.2 and 3.3. theorem 4.1. let f : na → r and ν > 0 be given, with n − 1 < ν ≤ n. consider the νth-order fractional difference equation ∆ ν a+ν−ny(t) = f(t), t ∈ na, (4.1) and the corresponding fractional initial value problem { ∆νa+ν−ny(t) = f(t), t ∈ na ∆iy(a + ν − n) = ai, i ∈ {0, 1, ..., n − 1} ; ai ∈ r . (4.2) the general solution to (4.1) is given by y(t) = n−1∑ i=0 αi (t − a) i+ν−n + ∆−νa f(t), for αi ∈ r, t ∈ na+ν−n, and the unique solution to (4.2) is y(t) = n−1∑ i=0   i∑ p=0 i−p∑ k=0 (−1)k i! (i − k)n−ν ( i p )( i − p k ) ap  (t − a)i+ν−n + ∆−νa f(t), for t ∈ na+ν−n. proof. let f and ν be given as in the statement of the theorem. for arbitrary but fixed αi ∈ r, define y : na+ν−n → r by y(t) := n−1∑ i=0 αi (t − a) i+ν−n + ∆−νa f(t). here, we extend the usual domain of the fractional sum ∆−νa f to the larger set na+ν−n to include the n zeros of ∆−νa f, as discussed in section 1.3. i.e., ∆ −ν a f(a + ν − n) = · · · = ∆ −ν a f(a + ν − 1) = 0. to show that y is the general solution of (4.1), we must show that any function of y’s form is a solution to (4.1) and that every solution to (4.1) is of y’s form. beginning with the former, 178 michael holm cubo 13, 3 (2011) observe that for t ∈ na, ∆ ν a+ν−ny(t) = ∆νa+ν−n [ n−1∑ i=0 αi (t − a) i+ν−n + ∆−νa f(t) ] = n−1∑ i=0 αi∆ ν a+ν−n (t − a) i+ν−n + ∆νa+ν−n∆ −ν a f(t). at this point, we would like to apply the power rule (3.3) in the summation and theorem 3.2 on the second term, but neither may be applied directly due to the mismatching lower limit on the operator ∆νa+ν−n. however, this problem is quickly remedied by throwing away the zero terms involved: ∆ ν a+ν−n∆ −ν a f(t) = 1 γ (−ν) t+ν∑ s=a+ν−n (t − σ(s))−ν−1∆−νa f(s) = 1 γ (−ν) t+ν∑ s=a+ν (t − σ(s))−ν−1∆−νa f(s) = ∆νa+ν∆ −ν a f(t), and ∆ ν a+ν−n (t − a) i+ν−n = 1 γ (−ν) t+ν∑ s=a+ν−n (t − σ(s))−ν−1(s − a)i+ν−n = 1 γ (−ν) t+ν∑ s=a+ν−n+i (t − σ(s))−ν−1(s − a)i+ν−n = ∆νa+i+ν−n (t − a) i+ν−n . therefore, ∆ ν a+ν−ny(t) = n−1∑ i=0 αi∆ ν a+i+ν−n (t − a) i+ν−n + ∆νa+ν∆ −ν a f(t) = f(t), applying theorems 3.1 and 3.2. next, we show that every solution of (4.1) has y’s form. suppose that z : na+ν−n → r is a solution to (4.1). then we may apply theorem 3.3 to solve (4.1) for z: ∆ ν a+ν−nz(t) = f(t), for t ∈ na ⇒ ∆−νa ∆ ν a+ν−nz(t) = ∆ −ν a f(t), for t ∈ na+ν−n ⇒ ∆0a+ν−nz(t) − n−1∑ i=0 ∆ i−(n−ν) a+ν−n z(a) γ (ν − n + i + 1) (t − a)ν−n+i = ∆−νa f(t) cubo 13, 3 (2011) sum and difference compositions in . . . 179 ⇒ z(t) = n−1∑ i=0 ( ∆i+ν−n a+ν−n y(a) γ (i + ν − n + 1) ) (t − a)i+ν−n + ∆−νa f(t), (4.3) for t ∈ na+ν−n. as before, we have extended the domain of ∆ −ν a f to na+ν−n. since z has the same form as y, we have shown that y(t) = n−1∑ i=0 αi (t − a) i+ν−n + ∆−νa f(t), t ∈ na+ν−n is the general solution of (4.1). the next task is to find the particular αi ∈ r which make y a solution to (4.2). from (4.3), it is clear that these constants have the form αi = ∆i+ν−n a+ν−n y(a) γ (i + ν − n + 1) , i ∈ {0, ..., n − 1} , but we must write each αi in terms of the initial conditions {ap} n−1 p=0 . in other words, we must find a way to write each ∆i+ν−n a+ν−n y(a) in terms of {∆py(a + ν − n)} n−1 p=0 . to accomplish this, we need • the binomial representation of a fractional difference from theorem 2.3: ∆ ν ag(t) = t−a+ν∑ k=0 (−1)k ( ν k ) g(t + ν − k), for t ∈ na+n−ν. • the following well-known formula (see [6]): g(t + m) = m∑ k=0 ( m k ) ∆ k g(t), for m ∈ n0. (4.4) applying the above two facts directly yields ∆ i+ν−n a+ν−ny(a) = i∑ k=0 (−1)k ( i + ν − n k ) y(a + i + ν − n − k) = i∑ k=0 (−1)k ( i + ν − n k ) y((a + ν − n) + (i − k)) = i∑ k=0 (−1)k ( i + ν − n k )i−k∑ p=0 ( i − k p ) ∆ p y(a + ν − n) = i∑ k=0 i−k∑ p=0 (−1)k ( i + ν − n k )( i − k p ) ap 180 michael holm cubo 13, 3 (2011) = i∑ p=0 i−p∑ k=0 (−1)k ( i + ν − n k )( i − k p ) ap = i∑ p=0 i−p∑ k=0 (−1)k γ (i + ν − n + 1) k!γ (i + ν − n − k + 1) (i − k)! p!(i − k − p)! ap = γ (i + ν − n + 1) · i∑ p=0 i−p∑ k=0 (−1)k i! γ (i − k + 1) γ (i − k + 1 + ν − n) i! p!(i − p)! (i − p)! k!(i − p − k)! ap = γ (i + ν − n + 1) i∑ p=0 i−p∑ k=0 (−1)k i! (i − k)n−ν ( i p )( i − p k ) ap. it follows that αi = i∑ p=0 ( i−p∑ k=0 (−1)k i! (i − k)n−ν ( i p )( i − p k ) ) ap, for i ∈ {0, ..., n − 1} . moreover, this solution is unique since the initial value problem (4.2) was solved directly, with no restricting assumptions or information lost. when actually solving an initial value problem of the form (4.2), the constants αi may seem cumbersome to calculate, especially by hand. however, we may write out more convenient and user-friendly expressions for the first several αi for use in solving lower order problems. to do this, we define g : n0 → r by g(t) := t n−ν . then the first five αi (provided they are required) are given by: α0 = g (0) a0, α1 = g (1) a1 − (g (0) − g (1)) a0, α2 = g (2) 2 a2 − (g (1) − g (2)) a1 + g (0) − 2g (1) + g (2) 2 a0, α3 = g (3) 6 a3 − g (2) − g (3) 2 a2 + g (1) − 2g (2) + g (3) 2 a1 − g (0) − 3g (1) + 3g (2) − g(3) 6 a0, α4 = g (4) 24 a4 − g (3) − g (4) 6 a3 + g (2) − 2g (3) + g (4) 4 a2 − g (1) − 3g (2) + 3g (3) − g (4) 6 a1 + g (0) − 4g (1) + 6g (2) − 4g (3) + g (4) 24 a0. cubo 13, 3 (2011) sum and difference compositions in . . . 181 note that if ν = n in (4.2), we are studying the well-known whole-order initial value problem { ∆ny(t) = f(t), t ∈ na ∆iy(a) = ai, i ∈ {0, 1, ..., n − 1} ; ai ∈ r . in this case, the solution given in theorem 4.1 simplifies considerably: y(t) = n−1∑ i=0   i∑ p=0 ( i−p∑ k=0 (−1)k i! ( i p )( i − p k ) ) ap  (t − a)i + ∆−na f(t) = n−1∑ i=0   i∑ p=0 ap i! ( i p )i−p∑ k=0 (−1)k ( i − p k )  (t − a)i + ∆−na f(t) = n−1∑ i=0 ai i! (t − a)i + ∆−na f(t), for t ∈ na, (4.5) since i−p∑ k=0 (−1)k ( i − p k ) = { 0, i − p > 0 1, i − p = 0 . one may prefer to write everything in (4.5) in terms of y: y(t) = n−1∑ i=0 ∆iy(a) i! (t − a)i + ∆−na ∆ n y(t), which produces a version of taylor’s theorem for functions y : na → r . in particular, since ∆ −n a ∆ n y(t) = 1 γ (n) t−n∑ s=a (t − σ(s))n−1∆ny(s) → 0 pointwise as n → ∞, we may write y(t) = ∞∑ i=0 ∆iy(a) i! (t − a)i, for t ∈ na. (4.6) however, (4.6) turns out to be just another version of formula (4.4) given in the proof of theorem 4.1. to see this, let t ∈ na be given by t = a + m, for some m ∈ n0. then (4.6) becomes y(t) = y(a + m) = ∞∑ i=0 ∆iy(a) i! m i = m∑ i=0 ( m i ) ∆ i y(a). we close with two examples demonstrating ideas presented in this paper. example 4.1. consider the following 2.7th-order initial value problem { ∆2.7−0.3y(t) = t 2, t ∈ n0 y(−0.3) = 2, ∆y(−0.3) = 3, ∆2y(−0.3) = 5 . (4.7) 182 michael holm cubo 13, 3 (2011) note that (4.7) is a specific instance of (4.2) from theorem 4.1, with a = 0, ν = 2.7, n = 3, f(t) = t2, a0 = 2, a1 = 3, a2 = 5 therefore, the solution to (4.7) is given by y(t) = n−1∑ i=0 αi(t − a) i+ν−n + ∆−νa f(t), for t ∈ na+ν−n = 2∑ i=0 αit i−0.3 + ∆−2.7 0 t 2 , for t ∈ n−0.3, where, α0 = 0 0.3 a0 α1 = 1 0.3 a1 − ( 0 0.3 − 10.3 ) a0, α2 = 20.3 2 a2 − ( 1 0.3 − 20.3 ) a1 + 00.3 − 2 · 10.3 + 20.3 2 a0, =⇒ α0 ≈ 1.541, α1 ≈ 3.962, α2 ≈ 3.684. our only remaining task is to calculate ∆ −2.7 0 t 2 = 1 γ (2.7) t−2.7∑ s=0 (t − σ(s))1.7s2 = 1 γ (2.7) t−2.7∑ s=2 (t − σ(s))1.7s2 = ∆−2.7 2 t 2 = γ (3) γ (5.7) t 4.7 ≈ 0.0276t 4.7 . therefore, the unique solution to (4.7) is y(t) ≈ 1.541t−0.3 + 3.962t0.7 + 3.684t1.7 + 0.0276t4.7, for t ∈ n−0.3. example 4.2. consider the composed difference operator ∆νa+m−µ ◦ ∆ µ a, where ν and µ are two positive non-integers who sum to an integer. by how much does the fractional composition operator ∆νa+m−µ ◦ ∆ µ a differ from the whole-order operator ∆ ν+µ? let f : na → r be given and choose m, n, p ∈ n so that n − 1 < ν < n, m − 1 < µ < m and ν + µ = p. then n + m = p + 1 and d { ∆ ν a+m−µ∆ µ af } = na+m−µ+n−ν = na+(n+m)−(ν+µ) = na+1 ⊆ d { ∆ p f } . cubo 13, 3 (2011) sum and difference compositions in . . . 183 applying composition rule (3.9) from theorem 3.4, we have for t ∈ na+1, ∆ ν a+m−µ∆ µ af(t) = ∆paf(t) − m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (−ν − m + j + 1) (t − a − m + µ)−ν−m+j. observe that lim ν→ n− µ→ (m−1)+ (t − a − m + µ)−ν−m+j = (t − a − 1) j−p−1 ∈ (0, ∞) and lim ν→ n− 1 γ (−ν − m + j + 1) = 0. it follows that lim ν→ n− µ→ (m−1)+ ( ∆ p af(t) − ∆ ν a+m−µ∆ µ af(t) ) = 0, as expected (compare this result to (3.6) in lemma 3.2). we are also interested in how these two operators compare for large t: lim t→ ∞ ( ∆ p af(t) − ∆ ν a+m−µ∆ µ af(t) ) = lim t→ ∞   m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (−ν − m + j + 1) (t − a − m + µ)−ν−m+j   = m−1∑ j=0 ∆ j−m+µ a f(a + m − µ) γ (−ν − m + j + 1) lim t→ ∞ γ (t − a − m + µ + 1) γ (t − a + p + 1 − j) = 0, by applying the squeeze theorem, since t − a − m + µ + 1, t − a + p + 1 − j ≥ 2 for t ∈ na+2 implies that 0 < γ (t − a − m + µ + 1) γ (t − a + p + 1 − j) ≤ γ (t − a + 1) γ (t − a + 2) = 1 t − a + 1 t→ ∞ → 0. we learn here that the discrepancy between a pth-order difference and two composed fractionalorder differences who sum to p depends explicitly on how far t is away from the first point in their common domain, a + 1. furthermore, we see this discrepancy vanish as t grows large. the following table shows the first seven of these discrepancies in the specific case f(t) = et, a = 0 and ν = µ = 1 2 : t 1 2 3 4 5 6 7 ∆ 1 2 1 2 ∆ 1 2 0 et − ∆et 1 8 1 16 5 128 7 256 21 1,024 33 2,048 429 32 ,768 received: august 2010. revised: september 2010. 184 michael holm cubo 13, 3 (2011) references [1] fm atici and pw eloe, a transform method in discrete fractional calculus, international journal of difference equations, 2 2007 165–176. [2] fm atici and pw eloe, initial value problems in discrete fractional calculus, proc. amer. math. soc., 137 2009 981–989. [3] fm atici and pw eloe, discrete fractional calculus with the nabla operator, electronic journal of qualitative theory of differential equations, spec. ed. i 2009 1–12.b kuttner, on differences of fractional order, proc. london math. soc., 7 1957 453–466. [4] jb diaz and tj olser, differences of fractional order, mathematics of computation, 28 1974. [5] henry l gray and n zhang, on a new definition of the fractional difference, mathematics of computation, 50 1988 513–529. [6] w kelley and a peterson, difference equations: an introduction with application, second edition, academic press, new york, new york, 2000. [7] ks miller and b ross, fractional difference calculus, proceedings of the international symposium on univalent functions, fractional calculus and their applications, nihon university, koriyama, japan, 1988 139–152. [8] k oldham and j spanier, the fractional calculus: theory and applications of differentiation and integration to arbitrary order, dover publications, inc., mineola, new york, 2002. [9] i podlubny, fractional differential equations, academic press, new york, new york, 1999. introduction motivation definitions domains unifying the fractional sum and difference fractional sum and difference composition rules composing a sum with a sum composing a difference with a sum composing a sum with a difference composing a difference with a difference application and examples articulo 9.dvi cubo a mathematical journal vol.12, no¯ 02, (127–143). june 2010 examples of a complex hyperpolar action without singular orbit naoyuki koike department of mathematics, faculty of science, tokyo university of science, 1-3 kagurazaka shinjuku-ku, tokyo 162-8601, japan email: koike@ma.kagu.tus.ac.jp abstract the notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as a counterpart to the hyperpolar action on a symmetric space of compact type. as examples of a complex hyperpolar action, we have hermann type actions, which admit a totally geodesic singular orbit (or a fixed point) except for one example. all principal orbits of hermann type actions are curvature-adapted and proper complex equifocal. in this paper, we give some examples of a complex hyperpolar action without singular orbit as solvable group free actions and find complex hyperpolar actions all of whose orbits are non-curvature-adapted or non-proper complex equifocal among the examples. also, we show that some of the examples possess the only minimal orbit. resumen la noción de una acción hiperpolar compleja sobre un espacio simétrico de tipo no compacto fue recientemente introducida como el análogo de la acción hiperpolar sobre un espacio simétrico de tipo compacto. como ejemplos de una acción hiperpolar complejas, nosotros tenemos acciones de tipo hermann, las cuales admiten una orbita (o un punto fijo) singular totalmente geodesica excepto para un ejemplo. todas las orbitas principales de acciones de tipo hermann son curvatura-adaptadas y unifocales complejas propias. en 128 naoyuki koike cubo 12, 2 (2010) este art́ıculo, nosotros damos algunos ejemplos de una acción hiperpolar compleja sin orbitas singulares como grupo soluble de acciones libres y encontramos acciones complejas hiperpolares cuyas orbitas son no curvatura-adaptadas o no propias unifocales complejas. también, mostramos que algunos de los ejemplos poseen solamente orbitas minimales. key words and phrases: symmetric space, complex hyperpolar action, complex equifocal submanifold. ams (mos) subj. class.: 53c35; 53c40 1 introduction in symmetric spaces, the notion of an equifocal submanifold was introduced in [30]. this notion is defined as a compact submanifold with globally flat and abelian normal bundle such that the focal radius functions for each parallel normal vector field are constant. however, this conditions of the equifocality is rather weak in the case where the symmetric spaces are of non-compact type and the submanifold is non-compact. so we [13, 14] have recently introduced the notion of a complex equifocal submanifold in a symmetric space g/k of non-compact type. this notion is defined by imposing the constancy of the complex focal radius functions instead of focal radius functions. here we note that the complex focal radii are the quantities indicating the positions of the focal points of the extrinsic complexification of the submanifold, where the submanifold needs to be assumed to be complete and of class cω (i.e., real analytic). on the other hand, heintze-liu-olmos [10] has recently defined the notion of an isoparametric submanifold with flat section in a general riemannian manifold as a submanifold such that the normal holonomy group is trivial, its sufficiently close parallel submanifolds are of constant mean curvature with respect to the radial direction and that the image of the normal space at each point by the normal exponential map is flat and totally geodesic. we [14] showed the following fact: all isoparametric submanifolds with flat section in a symmetric space g/k of non-compact type are complex equifocal and that conversely, all curvature-adapted and complex equifocal submanifolds are isoparametric ones with flat section. here the curvature-adaptedness means that, for each normal vector v of the submanifold, the jacobi operator r(·, v)v preserves the tangent space of the submanifold invariantly and the restriction of r(·, v)v to the tangent space commutes with the shape operator av, where r is the curvature tensor of g/k. furthermore, as a subclass of the class of complex equifocal submanifolds, we [15] defined that of the proper complex equifocal submanifolds in g/k as a complex equifocal submanifold whose lifted submanifold to h0([0, 1], g) (g := lie g) through some pseudo-riemannian submersion of h0([0, 1], g) onto g/k is proper complex isoparametric in the sense of [13], where we note that h0([0, 1], g) is a pseudo-hilbert space consisting of certain kind of paths in the lie algebra g of g. let g/k be a symmetric space of non-compact type and h be a closed subgroup of g which admits an embedded complete flat submanifold meeting all h-orbits orthogonally. then the h-action on g/k is called a complex hyperpolar action. this action was named thus because this action has not necessarily a singular orbit (which should be called a pole of this action) but the complexified action cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 129 has a singular orbit. note that all cohomogeneity one actions are complex hyperpolar. we [14] showed that principal orbits of a complex hyperpolar actions are isoparametric submanifolds with flat section and hence they are complex equifocal. conversely we [17] have recently showed that all homogeneous complex equifocal submanifolds occurs as principal orbits of complex hyperpolar actions. let h′ be a symmetric subgroup of g (i.e., there exists an involution σ of g with (fix σ)0 ⊂ h′ ⊂ fix σ), where fix σ is the fixed point group of σ and (fix σ)0 is the identity component of fix σ. then the h′-action on g/k is called a hermann type action. a hermann type action admits a totally geodesic orbit or a fixed point. except for one example, the totally geodsic orbit is singular (see theorem e of [17]). we [15] showed that principal orbits of a hermann type action are proper complex equifocal and curvature-adapted. we [17] have recently showed that all complex hyperpolar actions of cohomogeneity greater than one on g/k admitting a totally geodesic orbit and all complex hyperpolar actions of cohomogeneity one on g/k admitting reflective orbit are orbit equivalent to hermann type actions (see theorems b, c and remark 1.1 in [17]). let g/k be a symmetric space of non-compact type, g = f + p (f := lie k) be the cartan decomposition associated with (g, k), a be the maximal abelian subspace of p, ã be the cartan subalgebra of g containing a and g = f + a + n be the iwasawa’s decomposition. let a, ã and n be the connected lie subgroups of g having a, ã and n as their lie algebras, respectively. let π : g → g/k be the natural projection. the symmetric space g/k is identified with the solvable group an with a left-invariant metric through π|an . in this paper, we first prove the following fact for a complex hyperpolar action without singular orbit. theorem a. any complex hyperpolar action on g/k(= an ) without singular orbit is orbit equivalent to the free action of some solvable group contained in ãn . next we give some examples of a complex hyperpolar action without singular orbit as the free actions of solvable groups contained in an (see examples 1 and 2 of section 3), which contain examples of cohomogeneity one actions without singular orbit constructed by j. berndt and h. tamaru [3] as special cases (see also [1]). among these examples, we find complex hyperpolar actions all of whose orbits are non-proper complex equifocal or non-curvature-adapted. as its result, we have the following facts. theorem b. (i) for any symmetric space g/k of non-compact type and any positive integer r with r ≤ rank(g/k), there exists a complex hyperpolar action without singular orbit such that the cohomogeneity is equal to r and that any of the orbits is not proper complex equifocal. (ii) let g/k be one of su (p, q)/s(u (p) × u (q)) (p < q), sp(p, q)/sp(p) × sp(q) (p < q), so∗(2n)/u (n) (n : odd), e−146 /spin(10) · u (1) or f −20 4 /spin(9). then, for any positive integer r with r ≤ rank(g/k), there exists a complex hyperpolar action without singular orbit such that the cohomogeneity is equal to r and that any of the orbits is not curvature-adapted. also, among those examples, we find complex hyperpolar actions possessing the only minimal orbit. as its result, we have the following fact. 130 naoyuki koike cubo 12, 2 (2010) theorem c. for any irreducible symmetric space g/k of non-compact type and any positive integer r ≤ [ 1 2 (rank(g/k) + 1)], there exists a complex hyperpolar action without singular orbit such that the cohomogeneity is equal to r and that the only orbit is minimal. 2 complex equifocal submanifolds in this section, we recall the notions of a complex equifocal submanifold and a proper complex equifocal submanifold. we first recall the notion of a complex equifocal submanifold. let m be an immersed submanifold with abelian normal bundle in a symmetric space n = g/k of non-compact type. denote by a the shape tensor of m . let v ∈ t ⊥x m and x ∈ txm (x = gk). denote by γv the geodesic in n with γ̇v(0) = v. the strongly m -jacobi field y along γv with y (0) = x (hence y ′(0) = −avx) is given by y (s) = (pγv|[0,s] ◦ (d co sv − sdsisv ◦ av))(x), where y ′(0) = ∇̃vy, pγv|[0,s] is the parallel translation along γv|[0,s] and dcosv (resp. dsisv) is given by dcosv = g∗ ◦ cos( √ −1ad(sg−1∗ v)) ◦ g−1∗( resp. dsisv = g∗ ◦ sin( √ −1ad(sg−1∗ v))√ −1ad(sg−1∗ v) ◦ g−1∗ ) . here ad is the adjoint representation of the lie algebra g of g. all focal radii of m along γv are obtained as real numbers s0 with ker(d co s0v −s0dsis0v ◦av) 6= {0}. so, we call a complex number z0 with ker(dcoz0v−z0d si z0v ◦acv) 6= {0} a complex focal radius of m along γv and call dim ker(dcoz0v−z0d si z0v ◦acv) the multiplicity of the complex focal radius z0, where a c v is the complexification of av and d co z0v (resp. dsiz0v) is a c-linear transformation of (txn ) c defined by dcoz0v = g c ∗ ◦ cos( √ −1adc(z0g−1∗ v)) ◦ (gc∗)−1( resp. dsisv = g c ∗ ◦ sin( √ −1adc(z0g−1∗ v))√ −1adc(z0g−1∗ v) ◦ (gc∗)−1 ) , where gc∗ (resp. ad c) is the complexification of g∗ (resp. ad). here we note that, in the case where m is of class cω, complex focal radii along γv indicate the positions of focal points of the extrinsic complexification m c(→֒ gc/kc) of m along the complexified geodesic γcι∗v, where g c/kc is the anti-kaehlerian symmetric space associated with g/k and ι is the natural immersion of g/k into gc/kc. see section 4 of [14] about the definitions of gc/kc, m c(→֒ gc/kc) and γcι∗v. also, for a complex focal radius z0 of m along γv, we call z0v (∈ (t ⊥x m )c) a complex focal normal vector of m at x. furthermore, assume that m has globally flat normal bundle, that is, the normal holonomy group of m is trivial. let ṽ be a parallel unit normal vector field of m . assume that the number (which may be 0 and ∞) of distinct complex focal radii along γṽx is independent of the choice of x ∈ m . furthermore assume that the number is not equal to 0. let {ri,x | i = 1, 2, · · ·} be the set of all complex focal radii along γṽx , where |ri,x| < |ri+1,x| or ”|ri,x| = |ri+1,x| & re ri,x > re ri+1,x” or ”|ri,x| = |ri+1,x| & re ri,x = re ri+1,x & im ri,x = −im ri+1,x < 0”. let ri (i = 1, 2, · · · ) be complex valued functions on m defined by assigning ri,x to each x ∈ m . we call these functions ri (i = 1, 2, · · · ) complex focal radius functions for ṽ. we call riṽ a complex focal normal vector field for ṽ. if, for each parallel unit normal vector field ṽ of m , the number of distinct complex focal radii cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 131 along γṽx is independent of the choice of x ∈ m , each complex focal radius function for ṽ is constant on m and it has constant multiplicity, then we call m a complex equifocal submanifold. let n = g/k be a symmetric space of non-compact type and π be the natural projection of g onto g/k. let (g, θ) be the orthogonal symmetric lie algebra of g/k, f = {x ∈ g | θ(x) = x} and p = {x ∈ g | θ(x) = −x}, which is identified with the tangent space tek n . let 〈 , 〉 be the ad(g)-invariant non-degenerate symmetric bilinear form of g inducing the riemannian metric of n . note that 〈 , 〉|f×f (resp. 〈 , 〉|p×p) is negative (resp. positive) definite. denote by the same symbol 〈 , 〉 the bi-invariant pseudo-riemannian metric of g induced from 〈 , 〉 and the riemannian metric of n . set g+ := p, g− := f and 〈 , 〉g± := −π∗g−〈 , 〉 + π ∗ g+ 〈 , 〉, where πg− (resp. πg+ ) is the projection of g onto g− (resp. g+). let h 0([0, 1], g) be the space of all l2-integrable paths u : [0, 1] → g (with respect to 〈 , 〉g± ). let h0([0, 1], g−) (resp. h0([0, 1], g+)) be the space of all l2-integrable paths u : [0, 1] → g− (resp. u : [0, 1] → g+) with respect to −〈 , 〉|g−×g− (resp. 〈 , 〉|g+×g+ ). it is clear that h0([0, 1], g) = h0([0, 1], g−) ⊕ h0([0, 1], g+). define a non-degenerate symmetric bilinear form 〈 , 〉0 of h0([0, 1], g) by 〈u, v〉0 := ∫ 1 0 〈u(t), v(t)〉dt. it is easy to show that the decomposition h0([0, 1], g) = h0([0, 1], g−)⊕h0([0, 1], g+) is an orthogonal time-space decomposition with respect to 〈 , 〉0. for simplicity, set h0± := h0([0, 1], g±) and 〈 , 〉0,h0 ± := −π∗ h0 − 〈 , 〉0+π∗h0 + 〈 , 〉0, where πh0 − (resp. πh0 + ) is the projection of h0([0, 1], g) onto h0− (resp. h 0 +). it is clear that 〈u, v〉0,h0 ± = ∫ 1 0 〈u(t), v(t)〉g± dt (u, v ∈ h0([0, 1], g)). hence (h0([0, 1], g), 〈 , 〉0,h0 ± ) is a hilbert space, that is, (h0([0, 1], g), 〈 , 〉0) is a pseudo-hilbert space. let h1([0, 1], g) be the hilbert lie group of all absolutely continuous paths g : [0, 1] → g such that the weak derivative g′ of g is squared integrable (with respect to 〈 , 〉g± ), that is, g−1∗ g′ ∈ h0([0, 1], g). define a map φ : h0([0, 1], g) → g by φ(u) = gu(1) (u ∈ h0([0, 1], g)), where gu is the element of h1([0, 1], g) satisfying gu(0) = e and g−1u∗ g ′ u = u. we call this map the parallel transport map (from 0 to 1). this submersion φ is a pseudo-riemannian submersion of (h0([0, 1], g), 〈 , 〉0) onto (g, 〈 , 〉). let π : g → g/k be the natural projection. it follows from theorem a of [13] (resp. theorem 1 of [14]) that, in the case where m is curvature adapted (resp. of class cω ), m is complex equifocal if and only if each component of (π ◦ φ)−1(m ) is complex isoparametric. see [13] about the definition of a complex isoparametric submanifold in a pseudo-hilbert space. in particular, if each component of (π ◦ φ)−1(m ) are proper complex isoparametric, that is, for each normal vector v, there exists a pseudo-orthonormal base of the complexified tangent sapce consisting of the eigenvectors of the complexified shape operator for v, then we call m a proper complex equifocal submanifold. next we recall the notion of an infinite dimensional anti-kaehlerian isoparametric submanifold. let m be an anti-kaehlerian fredholm submanifold in an infinite dimensional anti-kaehlerian space v and a be the shape tensor of m . see [14] about the definitions of an infinite dimensional antikaehlerian space and anti-kaehlerian fredholm submanifold in the space. denote by the same symbol j the complex structures of m and v . fix a unit normal vector v of m . if there exists x(6= 0) ∈ t m with avx = ax +bjx, then we call the complex number a+b √ −1 a j-eigenvalue of av (or a complex principal curvature of direction v) and call x a j-eigenvector for a + b √ −1. also, we call the space of all j-eigenvectors for a + b √ −1 a j-eigenspace for a + b √ −1. the j-eigenspaces are orthogonal to one another and each j-eigenspace is j-invariant. we call the set of all j-eigenvalues of av the j-spectrum of av and denote it by specj av. the set specj av \ {0} is described as follows: specj av \ {0} = {λi | i = 1, 2, · · · } 132 naoyuki koike cubo 12, 2 (2010) ( |λi| > |λi+1| or ”|λi| = |λi+1| & re λi > re λi+1” or ”|λi| = |λi+1| & re λi = re λi+1 & im λi = −im λi+1 > 0” ) . also, the j-eigenspace for each j-eigenvalue of av other than 0 is of finite dimension. we call the j-eigenvalue λi the i-th complex principal curvature of direction v. assume that m has globally flat normal bundle. fix a parallel normal vector field ṽ of m . assume that the number (which may be ∞) of distinct complex principal curvatures of direction ṽx is independent of the choice of x ∈ m . then we can define functions λ̃i (i = 1, 2, · · · ) on m by assigning the i-th complex principal curvature of direction ṽx to each x ∈ m . we call this function λ̃i the i-th complex principal curvature function of direction ṽ. if m satisfies the following condition (aki), then we call m an anti-kaehlerian isoparametric submanifold: (aki) for each parallel normal vector field ṽ, the number of distinct complex principal curvatures of direction ṽx is independent of the choice of x ∈ m , each complex principal curvature function of direction ṽ is constant on m and it has constant multiplicity. let {ei}∞i=1 be an orthonormal system of txm . if {ei}∞i=1 ∪ {jei}∞i=1 is an orthonormal base of txm , then we call {ei}∞i=1 a j-orthonormal base. if there exists a j-orthonormal base consisting of j-eigenvectors of av, then av is said to be diagonalized with respect to the j-orthonormal base. if m is anti-kaehlerian isoparametric and, for each v ∈ t ⊥m , the shape operator av is diagonalized with respect to a j-orthonormal base, then we call m a proper anti-kaehlerian isoparametric submanifold. for arbitrary two unit normal vector v1 and v2 of a proper anti-kaehlerian isoparametric submanifold, the shape operators av1 and av2 are simultaneously diagonalized with respect to a j-orthonormal base. let m be a proper anti-kaehlerian isoparametric submanifold in an infinite dimensional antikaehlerian space v . let {ei | i ∈ i} be the family of distributions on m such that, for each x ∈ m , {ei(x) | i ∈ i} is the set of all common j-eigenspaces of av’s (v ∈ t ⊥x m ). the relation txm = ⊕ i∈i ei holds. let λi (i ∈ i) be the section of (t ⊥m )∗ ⊗ c such that av = reλi(v)id + imλi(v)j on ei(π(v)) for each v ∈ t ⊥m , where π is the bundle projection of t ⊥m . we call λi (i ∈ i) complex principal curvatures of m and call distributions ei (i ∈ i) complex curvature distributions of m . in the case where m is a real analytic submanifold in a symmetric space g/k of non-compact type, it is shown that m is complex equifocal if and only if (πc ◦ φc)−1(m c) is anti-kaehlerian isoparametric, where πc is the natural projection of gc onto gc/kc and φc is the parallel transport map for gc (which is defined in similar to the above φ). also, it is shown that m is proper complex equifocal if and only if (πc ◦ φc)−1(m c) is proper anti-kaehlerian isoparametric. 3 proof of theorems a and b in this section, we first prove theorem a. proof of theorem a. let h be a complex hyperpolar action on g/k(= an ) without singular orbit, h = lr (l : semi-simple, r : solvable) be the levi decomposition of h and l = klalnl (kl : compact, al : abelian, nl : nilpotent) be the iwasawa decomposition of l. since kl is compact, it has a fixed point p0 by the cartan’s fixed point theorem. suppose that kl · p 6⊂ alnlr · p for some cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 133 p ∈ g/k. then we have dim h · p0 < dim h · p, which implies that h · p0 is a singular orbit. this contradicts the fact that the h-action has no singular orbit. hence it follows that kl · p ⊂ alnlr · p for any p ∈ g/k. therefore we can show that the alnlr-action has the same orbits as the h-action. the group alnlr is decomposed into the product of some compact subgroup t ′ and some solvable normal subgroup s′ admitting a maximal compact normal subgroup s′k contained in the center of s′ such that s′/s′k is simply connected (see theorem 6 of [19]). since t ′ is compact, it is shown by the same argument as above that the s′-action has the same orbit as the alnlr-action (hence the h-action). take any p ∈ g/k and any g ∈ s′ with g 6= e. since s′ acts on g/k effectively, there exists p1 ∈ g/k with g(p1) 6= p1. the section σp1 through p1 is mapped into the section σg(p1) through g(p1) by g. since the s ′-action has no singular orbit, we have σp1 ∩ σg(p1) = ∅. let q be the intersection of h·p with σp1 . then g(q) is the intersection of h·p with σg(p1). hence we have g(q) 6= q. therefore s′ acts on each h-orbit effectively. since the isotropy group s′p of s ′ at any p ∈ g/k is compact, it is contained in a conjugate of s′k (see theorem 4 of [19]). hence s ′ p is contained in the center of s′. therefore, since the s′p-action has a fixed point p and it is effective, it is trivial. thus the s′-action is free. let s′ := lie s′ (the lie algebra of s′), s̃′ be a maximal solvable subalgebra of g containing s′ and s̃′ be the connected subgroup of g with lie s̃′ = s̃′. since g is a real semi-simple lie algebra and s̃′ is a maximal solvable subalgebra of g, s̃′ contains a cartan subalgebra ã′ of g. let t′ (resp. a′) be the toroidal part (resp. the vector part) of ã′. there exists a cartan decomposition g = f′ + p′ of g with t′ ⊂ f′ and a′ ⊂ p′. let g = g′0 + ∑ λ∈△′ g′λ be the root space decomposition with respect to a′ (i.e., g′0 is the centralizer of a ′ in g and g′λ = {x ∈ g | ad(a)(x) = λ(a)x for all a ∈ a′} and △′ = {λ ∈ (a′)∗ \ {0} | g′λ 6= {0}}). let n′ := ∑ λ∈△′ + g′λ, where △′+ is the positive root system with respect to some lexicographic ordering of a′. the algebra ã′ + n′ is a maximal solvable subalgebra of g. according to a result of [21], we may assume that s̃′ = ã′ + n′ by retaking ã′ if necessary. by imitating the proof of lemma 5.1 of [3], it is shown that a′ is a maximal abelian subspace of p′ because the s′-action has flat section. there exists g ∈ g satisfying ad(g)(f′) = f, ad(g)(p′) = p, ad(g)(a′) = a and ad(g)(ã′) = ã, where ad is the adjoint representation of g, a and ã are as in introduction. let s := ad(g)(s′) and s be the connected subgroup of g with lie s = s. since the s-action is conjugate to the s′-action and s ⊂ ãn , we obtain the statement of theorem a. q.e.d. let a be a maximal abelian subspace of p. fix a lexicographic ordering of a. let g = g0 + ∑ λ∈△ gλ, p = a + ∑ λ∈△+ pλ and f = f0 + ∑ λ∈△+ fλ be the root space decompositions of g, p and f with respect to a, where we note that gλ = {x ∈ g | ad(a)x = λ(a)x for all a ∈ a} (λ ∈ △), pλ = {x ∈ p | ad(a)2x = λ(a)2x for all a ∈ a} (λ ∈ △+), fλ = {x ∈ f | ad(a)2x = λ(a)2x for all a ∈ a} (λ ∈ △+ ∪ {0}). also, let g = f+a+n be the iwasawa decomposition of g and g = kan be the corresponding iwasawa decomposition of g, where we note that n = ∑ λ∈△+ gλ. now we shall give examples of a solvable group contained in an whose action on g/k(= an ) is complex hyperpolar. denote by π the natural projection of g onto g/k. since g/k is of non-compact type, π gives a diffeomorphism of an onto g/k. denote by 〈 , 〉 the left-invariant metric of an induced from that of g/k by π|an . also, 134 naoyuki koike cubo 12, 2 (2010) denote by 〈 , 〉g the bi-invariant metric of g inducing that of g/k. note that 〈 , 〉 6= ι∗〈 , 〉g, where ι is the inclusion map of an into g. let l be a r-dimensional subspace of a + n and set s := (a + n)⊖l , where (a + n) ⊖ l denotes the orthogonal complement of l in a + n with respect to 〈 , 〉e, where e is the identity element of g. if s is a subalgebra of a + n and lp := prp(l ) (prp : the orthogonal projection of g onto p) is abelian, then the s-action (s := expg(s)) is a complex hyperpolar action without singular orbit. we shall give examples of such a subalgebra s of a + n and investigate the structure of the s-orbit. example 1. let b be a r(≥ 1)-dimensional subspace of a and sb := (a + n) ⊖ b. it is clear that bp(= b) is abelian and that sb is a subalgebra of a + n. hence the sb-action (sb := expg(sb)) on g/k is a complex hyperpolar action without singular orbit. example 2. let {λ1, · · · , λk} be a subset of a simple root system π of △ such that hλ1 , · · · , hλk are mutually orthogonal, b be a subspace of a ⊖ span{hλ1 , · · · , hλk } (where b may be {0}) and li (i = 1, · · · , k) be a one-dimensional subspace of rhλi + gλi with li 6= rhλi , where hλi is the element of a defined by 〈hλi , ·〉λi(·) and rhλi is the subspace of a spanned by hλi . set l := b + k∑ i=1 li. then, it follows from lemma 3.1 (see the below) that lp is abelian and that sb,l1,··· ,lk := (a + n) ⊖ l is a subalgebra of a + n. hence the sb,l1,··· ,lk -action (sb,l1,··· ,lk := expg(sb,l1,··· ,lk )) on g/k is a complex hyperpolar action without singular orbit. lemma 3.1. let l and sb,l1,··· ,lk be as in example 2. then lp is abelian and sb,l1,··· ,lk is a subalgebra of a + n. proof. let h ∈ b and xi ∈ li (i = 1, · · · , k). since λi(h) = 0 and (xi)p ∈ rhλi ⊕ pλi , we have [h, (xi)p] = 0. fix i, j ∈ {1, · · · , k} (i 6= j). since λi and λj are simple roots and 〈hλi , hλj 〉 = 0, we have [(xi)p, (xj )p] = 0. thus lp is abelian. let v, w ∈ sb,l1,··· ,lk . since sb,l1,··· ,lk = (a ⊖ (b + k∑ i=1 rhλi )) ⊕ ( ∑ λ∈△+\{λ1,··· ,λk} gλ) ⊕ ( k∑ i=1 ((rhλi + gλi ) ⊖ li)), v and w are described as v = v0 + ∑ λ∈△+\{λ1,··· ,λk} vλ + k∑ i=1 vi and w = w0 + ∑ λ∈△+\{λ1,··· ,λk} wλ + k∑ i=1 wi, respectively, where v0, w0 ∈ a ⊖ (b + k∑ i=1 rhλi ), vλ, wλ ∈ gλ and vi, wi ∈ (rhλi + gλi ) ⊖ li. easily we have [v, w ] ≡ ∑ λ,µ∈△+\{λ1,··· ,λk} [vλ, wµ] + ∑ λ∈△+\{λ1,··· ,λk} k∑ i=1 ([vλ, wi] + [vi, wλ]) + k∑ i=1 k∑ j=1 [vi, wj ] (mod sb,l1,···lk ). since λ1, · · · , λk are simple roots, [vλ, wµ], [vλ, wi], [vi, wλ] and [vi, wj ] (λ, µ ∈ △+\{λ1, · · · , λk}, 1 ≤ i, j ≤ k) belong to sb,l1,··· ,lk . therefore we have [v, w ] ∈ sb,l1,··· ,lk . thus sb,l1,···lk is a subalgebra of a+n. q.e.d. cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 135 for the orbit sb,l1,··· ,lk · e, we have the following facts. lemma 3.2. let sb,l1,··· ,lk be as in example 2, ξ0 ∈ b, ξiti := 1 cosh(|λi|ti) ξi − 1 |λi| tanh(|λi|ti)hλi be a unit vector of li (i = 1, · · · , k), where ξi is a unit vector of gλi . denote by a the shape tensor of the orbit sb,l1,··· ,lk · e (⊂ an ). then, for aξ0 and aξiti , the following statements (i) ∼ (vii) hold: (i) for x ∈ a ⊖ (b + k∑ i=1 rhλi ), we have aξ0 x = aξiti x = 0 (i = 1, · · · , k). (ii) for x ∈ ker(ad(ξi)|gλi ) ⊖ rξ i, we have aξ0 x = 0 and aξiti x = −|λi| tanh(|λi|ti)x. (iii) assume that 2λi ∈ △+. for x ∈ g2λi , we have aξ0 ([θξi, x]) = 0 and aξiti x = −2|λi| tanh(|λi|ti)x − 1 2 cosh(|λi|ti) [θξi, x], aξiti ([θξi, x]) = − |λi| 2 cosh(|λi|ti) x − |λi| tanh(|λi|ti)[θξi, x], where θ is the cartan involution of g with fix θ = f. (iv) for x ∈ (rξi + rhλi ) ⊖ li, we have aξ0 x = 0 and aξiti x = −|λi| tanh(|λi|ti)x. (v) for x ∈ (gλj ⊖ rξj ) + ((rξj + rhλj ) ⊖ lj ) + g2λj (j 6= i), we have aξ0 x = aξiti x = 0. (vi) for x ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have aξ0 x = µ(ξ0)x. (vii) let ki := exp ( π√ 2|λi| (ξi + θξi) ) , where exp is the exponential map of g. then ad(ki) ◦ aξiti = −aξiti ◦ ad(ki) holds over n ⊖ k∑ i=1 (gλi + g2λi ), where ad is the adjoint representation of g. proof. let pr1a+n (resp. pr 2 a+n) be the projection of g onto a + n with respect to the decomposition g = f + (a + n) (resp. g = (f0 + ∑ λ∈△+ pλ) + (a + n)), prf (resp. prp) be the projection of g onto f (resp. p) with respect to the decomposition g = f + p and prf0 be the projection of g onto f0 with respect to the decomposition g = f0 + (a + ∑ λ∈△ gλ). then we have (3.1) prp ◦ pr1a+n = prp and prf ◦ pr2a+n = prf − prf0 . let h ∈ a, n1, n2 ∈ n and e ∈ gλ (λ ∈ △+). denote by ad(h)∗ (resp. ad(e)∗) the adjoint operator of ad(h) (resp. ad(e)) : a + n → a + n with respect to 〈 , 〉e. easily we can show (3.2) ad(h)∗ = ad(h). for simplicity, we denote prf(·) (resp. prp(·)) by (·)f (resp. (·)p). from (3.1) and the skewsymmetricness of ad(·) with respect to 〈 , 〉ge , we have 〈ad(e)n1, n2〉e = 〈ad(ef)((n1)p) + ad(ep)(((n1)f), (n2)p〉ge = −〈(n1)p, ad(ef)((n2)p)〉ge − 〈(n1)f, ad(ep)((n2)p)〉ge = −〈(n1)p, (pr1a+n(ad(ef)n2))p〉ge −〈(n1)f, (pr2a+n(ad(ep)n2))f + prf0 (ad(ep)n2)〉 g e = −〈n1, pr1a+n(ad(ef)n2)〉e + 〈n1, pr2a+n(ad(ep)n2)〉e 136 naoyuki koike cubo 12, 2 (2010) and hence prn(ad(e) ∗n2) = prn(−pr1a+n(ad(ef)n2) + pr2a+n(ad(ep)n2)), where prn is the projection of a + n onto n. also, we have 〈ad(e)h, n2〉e = −λ(h)〈e, n2〉e = −〈h, 〈e, n2〉ehλ〉e and hence pra(ad(e) ∗n2) = −〈e, n2〉ehλ, where pra is the projection of a + n onto a. also, we can show ad(e)∗h = 0. therefore, we have (3.3) ad(e)∗ =    0 on a −〈e, ·〉e ⊗ hλ − prn ◦ pr1a+n ◦ ad(ef) +prn ◦ pr2a+n ◦ ad(ep) on n on the other hand, according to the koszul’s formula, we have 〈aξx, y 〉e = 1 2 (〈[x, y ], ξ〉e − 〈[y, ξ], x〉e + 〈[ξ, x], y 〉e) = 1 2 〈(ad(ξ) + ad(ξ)∗)x, y 〉e for any x, y ∈ te(sb,l1,··· ,lk · e)sb,l1,··· ,lk and any ξ ∈ t ⊥e (sb,l1,··· ,lk · e) = b + k∑ i=1 li. that is, we have (3.4) aξ = 1 2 prt ◦ (ad(ξ) + ad(ξ)∗), where prt is the orthogonal projection of a + n onto sb,l1,··· ,lk . from (3.2) and (3.4), we have aξ0 x =    0 (x ∈ sb,l1,··· ,lk ⊖ ∑ µ∈△+\{λ1,··· ,λk} gλ) µ(ξ0)x (x ∈ gµ), where µ ∈ △+ \ {λ1, · · · , λk}. from (3.3) and (3.4), we have aξiti x = 0 (x ∈ a ⊖ (b + k∑ i=1 rhλi )). set gkλj := ker(ad(ξ j )|gλj ) and g i λj := im(ad(θξj )|g2λj ) (j = 1, · · · , k). then we have gλj = g k λj ⊕ giλj . by simple calculations, it is shown that this decomposition is orthogonal with respect to 〈 , 〉e. if x ∈ gkλj ⊖ rξ j , then it follows from (3.2), (3.3), (3.4), λi, λj ∈ π and 〈hλi , hλj 〉 = 0 (when i 6= j) that aξiti x = { −|λi| tanh(|λi|ti)x (i = j) 0 (i 6= j). if x ∈ g2λj , then it follows from (3.2), (3.3), (3.4), λi, λj ∈ π and 〈hλi , hλj 〉 = 0 (when i 6= j) that aξiti x =    −2|λi| tanh(|λi|ti)x − 1 2 cosh(|λi|ti) [θξi, x] (i = j) 0 (i 6= j). also, we have aξiti ([θξi, x]) = − |λi| 2 cosh(|λi|ti) x − |λi| tanh(|λi|ti)[θξi, x]. cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 137 let x := tanh(|λj|tj )ξj + 1|λj | cosh(|λj |tj ) hλj , which is a unit vector of (rξ j + rhλj ) ⊖ lj . from (3.2), (3.3), (3.4), λi, λj ∈ π and 〈hλi , hλj 〉 = 0 (when i 6= j), we have aξiti x = − 1 2 |λi| tanh(|λi|ti)x + 1 2 cosh(|λi|ti) prt (ad(ξ i)∗x) − 1 2|λi| tanh(|λi|ti)prt (ad(hλ)∗x) = { −|λi| tanh(|λi|ti)x (i = j) 0 (i 6= j). this completes the proof of (i) ∼ (vi). finally we shall show the statement (vii). let x ∈ n ⊖ k∑ i=1 (gλi + g2λi ) and ki be as in the statement (vii). from (3.2), (3.3), (3.4), λj ∈ π (j = 1, · · · , k) and 〈hλi , hλj 〉 = 0 (when i 6= j), we have aξiti x = 1 cosh(|λi|ti) [ξip, x] − 1 |λi| tanh(|λi|ti)[hλi , x]. by operating ad(ki) to both sides of this relation, we have ad(ki)(aξiti x) = −aξiti (ad(ki)x), where we use ad(ki)(ξ i p) = −ξip and ad(ki)(hλi ) = −hλi . thus the statement (vii) is shown. q.e.d. also, we have the following fact. lemma 3.3. let sb,l1,··· ,lk be as in example 2 and l̄i be the orthogonal projection of li onto gλi . set sb,̄l1,··· ,̄lk := (a + n) ⊖ (b + k∑ i=1 l̄i) and sb,̄l1,··· ,̄lk := expg(sb,̄l1,··· ,̄lk ). then the sb,̄l1,··· ,̄lk -action is conjugate to the sb,l1,··· ,lk -action. proof. denote by ∇ the levi-civita connection of the left-invariant metric of an . let h be a vector of b, ξi be a unit vector of l̄i (i = 1, · · · , k) and γξi be the geodesic in an with γ̇ξi (0) = ξi. let ti be a real number with 1 cosh(|λi|ti) ξi − tanh(|λi|ti)hλi ∈ li (i = 1, · · · , k). denote by the same symbols h, ξi and hλi the left-invariant vector fields arising from h, ξ i and hλi , respectively. by using the relation (5.4) of section 5 of [20] (arising the koszul formula for the left-invariant vector fields), we can show ∇ξ1 ξ1 = |λ1|hλ1 , ∇ξ1 hλ1 = −|λ1|ξ1 ∇ξ1 ξi = ∇ξ1 h = ∇hλ1 ξ 1 = ∇hλ1 ξ i = ∇hλ1 hλ1 = ∇hλ1 h = 0, where i = 2, · · · , k. from ∇ξ1 ξ1 = |λ1|hλ1 , ∇ξ1 hλ1 = −|λ1|ξ1, ∇hλ1 ξ 1 = ∇hλ1 hλ1 = 0, it follows that exp r{ξ1, hλ1} is a totally geodesic subgroup of an . hence γ̇ξ1 (t) is expressed as γ̇ξ1 (t) = a(t)(hλ1 )γξ1 (t) + b(t)(ξ 1)γ ξ1 (t). furthermore, we have ∇γ̇ξ1 γ̇ξ1 = (a ′ + |λ1|b2)hλ1 + (b′ −|λ1|ab)ξ1 = 0, that is, a′ = −|λ1|b2 and b′ = |λ1|ab. by solving this differential equation under the initial conditions a(0) = 0 and b(0) = 1, we have a(t) = − tanh(|λ1|t) and b(t) = 1cosh(|λ1|t) . hence we obtain 138 naoyuki koike cubo 12, 2 (2010) γ̇ξ1 (t) = 1 cosh(|λ1|t) (ξ1)γ ξ1 (t) − tanh(|λ1|t)(hλ1 )γξ1 (t). from ∇ξ1 ξ i = ∇ξ1 h = ∇hλ1 ξ i = ∇hλ1 h = 0 (i = 2, · · · , k), it follows that ξi (i = 2, · · · , k) and h are parallel along γξ1 (with respect to ∇). denote by pγ ξ1 |[0,t] the parallel translation along γξ1|[0,t] (with respect to ∇) and lγξ1 (t) the left translation by γξ1 (t). from the above facts, we have t ⊥γ ξ1 (t1) (sb,̄l1,··· ,̄lk ) = pγξ1 |[0,t1] (b + k∑ i=1 l̄i) = (lγ ξ1 (t1))∗(b + k∑ i=2 l̄i + l1) = (lγ ξ1 (t1))∗(t ⊥ e sb,l1,̄l2,··· ,̄lk ), which implies γξ1 (t1) −1sb,̄l1,··· ,̄lk γξ1 (t1) = sb,l1,̄l2,··· ,̄lk . by repeating the same discussion, we obtain (γξ1 (t1) · · · γξk (tk))−1sb,̄l1,··· ,̄lk (γξ1 (t1) · · · γξk (tk)) = sb,l1,··· ,lk . thus the sb,̄l1,··· ,̄lk -action is conjugate to the sb,l1,··· ,lk -action. q.e.d. for parallel submanifolds of a proper complex equifocal submanifold and a curvature-adapted complex equifocal submanifold, we have the following facts. lemma 3.4. (i) all parallel submanifolds of a proper complex equifocal submanifold are proper complex equifocal. (ii) all parallel submanifolds of a curvature-adapted complex equifocal submanifold are curvatureadapted and complex equifocal. proof. first we shall show the statement (i). let m be a proper complex equifocal submanifold in a symmetric space g/k of non-compact type and ṽ be the parallel normal vector field of m which is not a focal normal vector field. denote by η ev the end-point map for ṽ and mev := ηev(m ), which is a parallel submanifold of m . the vector field ṽ is regarded as a parallel normal vector field of the complexification m c along m . let ṽl be the horizontal lift of ṽ to h0([0, 1], gc) by the antikaehlerian submersion πc ◦ φc : h0([0, 1], gc) → gc/kc, which is a parallel normal vector field of m̃ c(:= (πc ◦ φc)−1(m c)). set m̃ c evl := ηevl (m̃ c), where η evl is the end-point map for ṽ l. note that m̃ c evl = (π c ◦ φc)−1((m ev) c). denote by ã and ãev l the shape tensors of m̃ c and m̃ c evl , respectively. let {λi | i ∈ i} be the set of all complex principal curvatures of m̃ c and ei be the complex curvature distribution for λi. then, according to lemma 3.2 of [16], we have (3.5) ãev l w |(ei)u = (λi)u(w) 1 − (λi)u(ṽlu ) id (i ∈ i, u ∈ m̃ c evl ), where we note that tη evl (u)m̃ c evl = tum̃ c(= ⊕ i∈i (ei)u). this implies that m̃ c evl is proper antikaehlerian isoparametric, that is, m ev is proper complex equifocal. thus the statement (i) is shown. next we shall show the statement (ii). let m be a curvature-adapted complex equifocal submanifold in g/k and ṽ be the parallel normal vector field of m . set m ev := ηev(m ). denote by a and a ev the shape tensors of m and m ev, repsectively. let w ∈ t ⊥x m . without loss of generality, we may assume that x = ek. let a be a maximal abelian subspace of p := tek (g/k) containing t ⊥ ek m and p = a + ∑ α∈△+ pα be the root space decomposition with respect to a. let x ∈ ker(av − λ id) ∩ ker(aw − µ id) ∩ pα cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 139 (λ ∈ spec av, µ ∈ spec aw, α ∈ △+). let w̃ be the parallel tangent vector field on the (flat) section σ of m through ek with w̃ek = w. since m ev is regarded as a partial tube over m , it follows from (ii) of corollary 3.2 in [15] that (3.6) (aev) ewη ev (ek) ((η ev )∗x) = 1 α(v) − λ tanh α(v) {−α(v)α(w) tanh α(v) +λ ( 1 − tanh α(v) α(v) ) α(w) + µ tanh α(v)}(η ev )∗x. let z be the element of p with expg(z)k = ηev(ek). for simplicity, set g := expg(z). since g∗ : p → tη ev(ek)(g/k) is the parallel translation along the normal geodesic γz (⇔ def γz (t) := expg(tz)k), it follows from (3.1) of [15] that (η ev)∗x = g∗(d co v (x) − dsiv (avx)) = ( cosh α(v) − λ sinh α(v) α(v) ) g∗x ∈ g∗pα. also, we have g−1∗ (t ⊥ η ev (ek) m ev) = t ⊥ ek m ⊂ a. hence we have r((ηev)∗x, w̃η ev (ek))w̃ηev (ek) = −α(w)2(η ev)∗x, which together with (3.6) implies [(aev) ewη ev (ek) , r(·, w̃η ev (ek))w̃ηev (ek)]((ηev)∗x) = 0. therefore, it follows from the arbitrariness of x that [(aev) ewη ev (ek) , r(·, w̃η ev (ek))w̃ηev (ek)] vanishes over (η ev)∗(ker(av − λ id) ∩ ker(aw − µ id) ∩ pα). since m is curvature-adapted, we have ⊕ λ∈spec av ⊕ µ∈spec aw ⊕ α∈△+ (η ev)∗(ker(av − λ id) ∩ ker(aw − µ id) ∩ pα) = tη ev(ek)mev. hence we have [(aev) ewη ev (ek) , r(·, w̃η ev (ek))w̃ηev (ek)] = 0. therefore, it follows from the arbitrariness of w that m ev is curvature-adapted. it is clear that mev is complex equifocal. thus the statement (ii) is shown. q.e.d. for the sb-action and the sb,l1,··· ,lk -action, we have the following facts. proposition 3.5. (i) all orbits of the sb-action are curvature-adapted but they are not proper complex equifocal. (ii) let λ1, · · · , λk (∈ △+) be as in example 2. if the root system △ of g/k is non-reduced and 2λi0 ∈ △+ for some i0 ∈ {1, · · · , k}, then all orbits of the sb,l1,··· ,lk -action are not curvature-adapted. also, if b 6= {0}, then they are not proper complex equifocal. proof. first we shall show the statement (i). the group sb acts isometrically on (an, 〈 , 〉). denote by a the shape tensor of the orbit sb · e in an . since 〈 , 〉 is left-invariant, it follows from the koszul formula that 〈avx, y 〉 = 〈ad(v)x, y 〉 for any v ∈ l = t ⊥e (sb · e) and x, y ∈ s = te(sb · e). hence we have av|a⊖l = 0 and av|gλ = λ(v)id (λ ∈ △+), where v ∈ t ⊥e (sb · e) = l (⊂ p). therefore, the orbit sb · e is curvature-adapted but it is not proper complex equifocal by (ii) of theorem 1 of [14]. hence so are all orbits of the sb-action by lemma 3.3. 140 naoyuki koike cubo 12, 2 (2010) next we shall show the statement (ii). assume that the root system △ of g/k is non-reduced. denote by a the shape tensor of the orbit sb,l1,··· ,lk ·e (⊂ an ). also, let ξ0 ∈ b and ξiti : 1 cosh(|λi|ti) ξi − 1 |λi| tanh(|λi|ti)hλi (ξi ∈ gλi ) be a unit (tangent) vector of li. then, according to lemma 3.2, we see that (3.7) aξ0|sb,l1,···lk ∩(a+ p k i=1 gλi ) = 0, aξ0|gµ = µ(ξ0)id (µ ∈ △+ \ k ∪ i=1 {λi}), aξiti |a⊖(b+pk j=1 rhλj ) = 0, aξiti |ker(ad(ξi)|gλi )⊖rξi = −|λi| tanh(|λi|ti)id aξiti |(rξi+rhλi )⊖li = −|λi| tanh(|λi|ti)id and that, in case of 2λi ∈ △+, aξiti |im(ad(θξi)|g2λi )+g2λi has two eigenvalues µ+i := − 3 2 |λi| tanh(|λi|ti) + 1 2 |λi| √ 2 − tanh2(|λi|ti) and µ−i := − 3 2 |λi| tanh(|λi|ti) − 1 2 |λi| √ 2 − tanh2(|λi|ti) with the same multiplicity. note that gλi = ker(ad(ξ i)|gλi ) ⊕ im(ad(θξ i)|g2λi ). the eigenspace for µ+i (resp. µ − i ) is spanned by z+ ξi,y := [θξi, y ] + |λi| ( sinh(|λi|ti) − √ sinh2(|λi|ti) + 2 ) y ′s (y ∈ g2λi ) (resp. z− ξi,y := [θξi, y ] + |λi| ( sinh(|λi|ti) + √ sinh2(|λi|ti) + 2 ) y ′s (y ∈ g2λi ))). denote by r the curvature tensor of 〈 , 〉. also, denote by xf (resp. xp) the f-component (resp. the p-component) of x ∈ g. then we have (3.8) ( r(z± ξi,y , ξiti )ξ i ti ) p = −a[[(z± ξi,y )p, (ξ i ti )p], (ξ i ti )p] = a(−[[z± ξi,y , ξiti ], ξ i ti ]p + [[(z ± ξi,y )f, (ξ i ti )f], (ξ i ti )p] +[[(z± ξi,y )f, (ξ i ti )p], (ξ i ti )f] + [[(z ± ξi,y )p, (ξ i ti )f], (ξ i ti )f]) for some non-zero constant a, where we note that a = 1 if the metric of g/k is induced from the restriction of the killing form of g to p. also we have (3.9) [[(z± ξi,y )p, (ξ i ti )f], (ξ i ti )f] = 0, (3.10) [[(z± ξi,y )f, (ξ i ti )f], (ξ i ti )p] = − tanh(|λi|ti) |λi| cosh(|λi|ti) [[[θξi, y ]f, ξ i f ], hλi ] and (3.11) [[(z± ξi,y )f, (ξ i ti )p], (ξ i ti )f] = |λi| tanh(|λi|ti) cosh(|λi|ti) [[θξi, y ]p, ξ i f ]. let η (resp. η̄) be the element of a + n with ηf = [[θξ i, y ]f, ξ i f ] (resp. η̄p = [[θξ i, y ]p, ξ i f ]). then it follows from (3.8) ∼ (3.11) that (r(z± ξi,y , ξiti )ξ i ti )p = −a[[z±ξi,y , ξ i ti ], ξiti ]p + a|λi| tanh(|λi|ti) cosh(|λi|ti) (2ηp + η̄p), cubo 12, 2 (2010) examples of a complex hyperpolar action without singular orbit 141 that is, (3.12) r(z± ξi,y , ξiti )ξ i ti = −a[[z± ξi,y , ξiti ], ξ i ti ] + a|λi| tanh(|λi|ti) cosh(|λi|ti) (2η + η̄). we have [ξi, θξi] = bhλi for some non-zero constant b. by simple calculation, we have (3.13) [[z± ξi,y , ξiti ], ξ i ti ] = 2|λi| tanh2(|λi|ti) ( − 3b|λi| 2 sinh(|λi|ti) + sinh(|λi|ti) ∓ √ sinh2(|λi|ti) + 2 ) y + tanh2(|λi|ti)[θξi, y ]. from (3.12) and (3.13), it follows that r(z± ξi,y , ξiti )ξ i ti belongs to im ad(θξi) ⊕ g2λi . hence r(·, ξiti )ξ i ti preserves im ad(θξi) ⊕ g2λi invariantly. it is clear that so is also aξiti . from (3.12) and (3.13), we have [r(·, ξiti )ξ i ti , aξiti ]|im ad(θξi)⊕g2λi 6= 0, under a suitable choice of ti. therefore, sb,l1,··· ,lk · e is not curvature-adapted under suitable choices of l1, · · · , lk. then, so are all orbits of the sb,l1,··· ,lk -action by lemma 3.4. furthermore, it follows from lemma 3.3 that all orbits of the sb,l1,··· ,lk -action are not curvature-adapted under arbitrary choices of l1, · · · , lk. also, it follows from the second relation of (3.7) that sb,l1,··· ,lk · e (hence all orbits of the sb,l1,··· ,lk -action) is not proper complex equifocal in case of b 6= {0}. q.e.d. from this proposition, we obtain the statements of theorem b. also, we have the following fact. proposition 3.6. if b = {0}, then the sb,l1,··· ,lk -action possesses the only minimal orbit. proof. according to lemma 3.3, the sb,l1,··· ,lk -action is conjugate to sb,̄l1,··· ,̄lk -action, where l̄i is the orthogonal projection of li onto gλi . hence they are orbit equivalent to each other. hence we suffice to show that the statement of this proposition holds for the sb,̄l1,··· ,̄lk action. let ξ i be a unit vector of l̄i. take p ∈ an . we can express as p = γξ1 (t1) · · · γξk (tk) for some t1, · · · , tk ∈ r, where γξi is the geodesic with γ̇ξi (0) = ξ i. set l̂i := r{ 1cosh(|λi|ti) ξ i − 1 |λi| tanh(|λi|ti)hλi } (i = 1, · · · , k). for simplicity, set ξiti := 1 cosh(|λi|ti) ξi − 1 |λi| tanh(|λi|ti)hλi . according to the proof of lemma 3.3, we have (γξ1 (t1) · · · γξk (tk))−1sb,̄l1,··· ,̄lk (γξ1 (t1) · · · γξk (tk)) = sb,̂l1,··· ,̂lk . hence the orbit sb,̄l1,··· ,̄lk · p is congruent to the orbit sb,̂l1,··· ,̂lk · e. denote by a the shape tensor of s b,̂l1,··· ,̂lk · e. according to lemma 3.2, we have tr aξiti = −|λi| tanh(|λi|ti) × (dim gλı + 2dim g2λi ) (i = 1, · · · , k). hence the orbit s b,̂l1,··· ,̂lk · e is minimal if and only if t1 = · · · = tk = 0, where we note that t ⊥e (sb,̂l1,··· ,̂lk · e) = r{ξ 1 t1 , · · · , ξktk } because of b = {0}. that is, the orbit sb,̄l1,··· ,̄lk · p is minimal if and only if p = e. thus the orbit sb,̄l1,··· ,̄lk -action posseses the only minimal orbit sb,̄l1,··· ,̄lk · e. this completes the proof. q.e.d. 142 naoyuki koike cubo 12, 2 (2010) from this proposition, we obtain the statement of theorem c. at the end of this paper, we propose the following question. question. is any complex hyperpolar action without singular orbit on a symmetric space of noncompact type orbit equivalent to either the sb-action (b ⊂ a) as in example 1 or the sb,l1,··· ,lk -action (li : a one dimensional subspace of gλi (i = 1, · · · , k), b ⊂ a ⊖ span{hλi | i = 1, · · · , k}) as in example 2 ? received: september 2008. revised: april 2009. references [1] j. berndt, homogeneous hypersurfaces in hyperbolic spaces, math. z. 229 (1998) 589-600. 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[31] b. wu, isoparametric submanifolds of hyperbolic spaces trans. amer. math. soc. 331 (1992) 609–626. cubo_reliability_versija5.dvi cubo a mathematical journal vol.12, no¯ 01, (7–13). march 2010 analysis of the component-based reliability in computer networks saulius minkevičius vilnius university, faculty of mathematics and informatics naugarduko 24, 03225 vilnius, lithuania email : stst@ktl.mii.lt abstract performance in terms of reliability of computer networks motivates this paper. limit theorems on the extreme queue length and extreme virtual waiting time in open queueing networks in heavy traffic are derived and applied to a reliability model for computer networks where we relate the time of failure of a computer network to the system parameters. resumen desempeño en términos de fiabilidad de redes de computador notiva este art́ıculo. teoremas ĺımite sobre la duración extrema de cola y el tiempo de espera virtual extremo en redes de cola abierta en trafico pesado sao derivados y aplicados a un modelo de fiabilidad para redes de computador donde relacionamos el tiempo de falha de una red de computador al sistema de parámetros. key words and phrases: performance evaluation, reliability theory, queueing theory, mathematical models of technical systems, open queueing network, heavy traffic, the probability limit theorem, extreme values, queue length of customers, virtual waiting time of a customer. math. subj. class.: 60k25, 60g70, 60f17. 8 saulius minkevičius cubo 12, 1 (2010) 1 introduction probabilistic models and queueing networks have long been used to study the performance and reliability of computer systems [1, 2] and to analyse the performance and reliability of computer networks and of distributed information systems [3, 4]. in this paper, we will first briefly review the works related to using the queueing theory of computer system reliability, and then present some new results on the estimation of the time of failure of a computer network. in one of the first papers of this kind [6], the reliability of execution of programs in a distributed computing system is considered, showing that a program, which runs on multiple processing elements that have to communicate with other processing elements for remote data files, can be executed successfully despite that certain system components may be unreliable. in order to analyse the performance of multimedia service systems which have unreliable resources and to estimate their capacity requirements, a capacity planning model using an open queueing network is presented in [9], and in [5] a novel model for a reliable system composed of n unreliable systems, which can hinder or enhance one anothers reliability, is discussed. in [10], the management policy of an m /g/1 queue with a single removable and non-reliable server is discussed and analytic results are explored, using an efficient matlab program to calculate the optimal threshold of the management policy and to evaluate system performance. in [11], the authors consider a single machine subject to breakdown and employ a fluid queue model with repair. in [12], the behaviour of a heterogeneous finite-source system with a single server is considered and applications in the field of telecommunications and reliability theory are treated. in this paper, first we present probability limit theorems on the extreme queue length of customers and the extreme virtual waiting time of a customer in heavy traffic for open queueing networks. we consider open networks with the “first come, first served” service discipline at each station and general distributions of interarrival and service times. the basic components of the queueing network are arrival processes, service processes, and routing processes among the different queues. the queueing network that we study has k single server stations, each of which has an associated infinite capacity waiting room. every station has an arrival stream from outside the network, and the arrival streams are assumed to be mutually independent renewal processes. customers are served in the order of arrival and after service they are randomly routed to either another station in the network, or out of the network entirely. service times and routing decisions form mutually independent sequences of independent identically distributed random variables. 2 the mathematical model let us consider the mutually independent sequences of independent identically distributed random variables { z (j) n , n ≥ 1 } , { s (j) n , n ≥ 1 } and { φ (j) n , n ≥ 1 } for j = 1, 2, . . . , k; defined on the probability space. the random variables z (j) n and s (j) n are strictly positive, and φ (j) n have supcubo 12, 1 (2010) analysis of the component-based reliability ... 9 port in {0, 1, 2, . . . , k}. we define µj = ( m [ s (j) n ])−1 > 0, σj = d ( s (j) n ) > 0 and λj = ( m [ z (j) n ])−1 > 0, aj = d ( z (j) n ) > 0, j = 1, 2, ..., k; with all of these terms assumed finite. denote pij = p ( φ (i) n = j ) > 0, j = 1, 2, . . . , k. in the context of the queueing network being considered, the random variables z (j) n represent an interarrival time from outside the network at the station j, while s (j) n is the nth service time at the station j, and φ (j) n is a routing indicator for the nth customer served at the station j. if φ (i) n = j (which occurs with probability pij ), then the nth customer served at the station i is routed to the station j. when φ (i) n = 0, the associated customer leaves the network. the matrix p is called a routing matrix. we observe that this system is quite general, encompassing tandem systems, acyclic networks of gi/g/1 queues, networks of gi/g/1 queues with feedback and open queueing networks. let us define qj(t) as the queue length of customers at the jth station of the queueing network at time t, β̂j = λj + k ∑ i=1 µi · pij − µj > 0, σ̂2j = (λj ) 3·dz(j)n + k ∑ i=1 (µi) 3 ·ds(i)n ·(pij )2 +(µj )3·ds(j)n > 0, j = 1, 2, . . . , k and t > 0. also, let us define wj (t) as the virtual waiting time of a customer at the jth station of a queueing network at time t (one must wait until a customer arrives at the jth station of the queueing network to be served at time t), β̃j = λj + k ∑ i=1 µi · pij µj − 1, σ̃2j = k ∑ i=1 p2ij · µi · ( σj + ( µi µj )2 · σi ) + λj · ( σj + ( λj µj )2 · aj ) , j = 1, 2, · · · , k and t > 0. we will also assume that the following “overload conditions” are fulfilled: λj + k ∑ i=1 µi · pij > µj , j = 1, 2, . . . , k. (1) note that these conditions guarantee that the length of all the queues will grow indefinitely with probability one. the results of the present paper are based on the following theorems: theorem 1. if conditions (1) are fulfilled, then lim n→∞ p    sup 0≤s≤t qj (ns) − β̂j · n · t σ̂j · √ n < x    = ∫ x −∞ exp ( − y 2 2t ) dy, 0 ≤ t ≤ 1 and j = 1, 2, . . . , k. and 10 saulius minkevičius cubo 12, 1 (2010) theorem 2. if conditions (1) are fulfilled, then lim n→∞ p    sup 0≤s≤t vj (ns) − β̃j · n · t σ̃j · √ n < x    = ∫ x −∞ exp ( − y 2 2t ) dy, 0 ≤ t ≤ 1 and j = 1, 2, . . . , k. proof. these theorems are proved in [7], and the proof is therefore omitted here so as not to lengthen this short paper. 3 the reliability of a computer network now we present a technical example from the computer network practice. assume that queues arrive at a computer vj at the rate λj per hour during business hours, j = 1, 2, . . . ,p+r. these queues are served at the rate µj per hour in the computer vj , j = 1, 2, . . . ,p+r. after service in the computer vj , with probability pj (usually pj ≥ 0.9), they leave the network and with probability pji, i 6= j, 1 ≤ i ≤p+r (usually 0 < pji ≤ 0.1) arrive at the computer vi, i = 1, 2, . . . ,p+r. also, we assume the computer vi fails when the virtual waiting time of queues is more than kj , j = 1, 2, . . . , p, and the computer vj fails when the queue length of queues is more than γi, i = 1, 2, . . . , r. in this section, we prove the following theorem on the probability that a computer network fails due to overload. theorem 3. if t ≥ max ( max 1≤j≤p kj β̂j , max 1≤i≤r γi β̃i ) and conditions (1) are fulfilled, the computer network becomes unreliable (all computers fail). proof. at first, using theorem 1 and theorem 2, we get that for x > 0 lim n→∞ p    sup 0≤s≤t qj (ns) − β̂j · n · t σ̂j · √ n < x    = ∫ x −∞ exp ( − y 2 2t ) dy, j = 1, 2, . . . , p. (2) and lim n→∞ p    sup 0≤s≤t vi(ns) − β̃i · n · t σ̃i · √ n < x    = ∫ x −∞ exp ( − y 2 2t ) dy, i = 1, 2, . . . , r. (3) cubo 12, 1 (2010) analysis of the component-based reliability ... 11 let us investigate a computer network which consists of the elements (computers) αj that are indicators of stations xj , j = 1, 2, . . . , p and elements (computers) νi that are indicators of stations yi, i = 1, 2, . . . , r denote xj = { 1, if the element αj is reliable 0, if the element αj is not reliable, j = 1, 2, . . . , p and yi = { 1, if the element νi is reliable 0, if the element νi is not reliable, i = 1, 2, . . . , r. note that {xj = 1} = { sup 0≤s≤t qj(ns) < kj}, j = 1, 2, . . . , p and {yi = 1} = { sup 0≤s≤t vi(ns) < γi}, i = 1, 2, . . . , r. denote the structural function of the system of elements, connected by scheme 1 from p + r (see, for example, [8]), as follows: φ(x1, x2, . . . , xp, y1, y2, . . . , yr, t) = { 1, ∑p j=1 xj + ∑r i=1 yi ≥ 1 0, ∑p j=1 xj + ∑r i=1 yi < 1. assume y = ∑p j=2 xi + ∑r i=1 yi. estimate the reliability function of the system (computer network) using the formula of conditional probability h(x1, x2, . . . , xp, y1, y2, . . . , yr, t) = eφ(x1, x2, . . . , xp, y1, y2, . . . , yr, t) = p (φ(x1, x2, . . . , xp, y1, y2, . . . , yr, t) = 1) = p ( p ∑ j=1 xj + r ∑ i=1 yi ≥ 1) = p (x1 + y ≥ 1) = p (x1 + y ≥ 1|y = 1) · p (y = 1) + p (x1 + y ≥ 1|y = 0) · p (y = 0) = p (x1 ≥ 0) · p (y = 1) + p (x1 ≥ 1) · p (y = 0) ≤ p (y = 1) + p (x1 ≥ 1) = p (y = 1) + p (x1 = 1) ≤ p (y ≥ 1) + p (x1 = 1) = p ( p ∑ j=2 xj + r ∑ i=1 yi ≥ 1) + p (x1 = 1) ≤ · · · ≤ p ∑ j=1 p (xj = 1) + r ∑ i=1 p (yi ≥ 1) ≤ · · · ≤ p ∑ j=1 p (xj = 1) + r ∑ i=1 p (yi = 1) = p ∑ j=1 p ( sup 0≤s≤t qj (ns) ≤ kj ) + r ∑ i=1 p ( sup 0≤s≤t vi(ns) ≤ γi). thus, 0 ≤ h(x1, x2, . . . , xp, y1, y2, . . . , yr, t) ≤ p ∑ j=1 p ( sup 0≤s≤t qj (ns) ≤ kj ) + r ∑ i=1 p ( sup 0≤s≤t vi(ns) ≤ γi). (4) 12 saulius minkevičius cubo 12, 1 (2010) applying theorem 1, we obtain that 0 ≤ lim n→∞ p ( sup 0≤s≤t qj (ns) < kj ) = lim n→∞ p    sup 0≤s≤t qj(ns) − β̂j · n · t σ̂j · √ n < kj − β̂j · n · t σ̂j · √ n    = ∫ −∞ −∞ exp ( − y 2 2t ) dy = 0. (5) j = 1, 2, . . . , p. from (5) it follows that for kj < ∞, lim n→∞ p ( sup 0≤s≤t qj(ns) < kj ) = 0, j = 1, 2, . . . , p. (6) similarly as in (5) (6) we prove that for γi < ∞ lim n→∞ p ( sup 0≤s≤t vi(ns) < γi ) = 0, i = 1, 2, . . . , r. (7) consequently, lim n→∞ h(x1, x2, . . . , xp, y1, y2, . . . , yr, t) = 0 (see (5)-(7)), which completes the proof. finally, we provide an expression for h(x1, x2, . . . , xp, y1, y2, . . . , yr, t), t > 0 by proving the following theorem. theorem 4. h(x1, x2, . . . , xp, y1, y2, . . . , yr, t) is equal to exp(− ∑p j=1 p ( sup 0≤s≤t qj(ns) < kj ) − ∑r i=1 p ( sup 0≤s≤t vi(ns) < γi)). proof. let λj , j = 1, 2, . . . , p and qi, i = 1, 2, . . . , r be the traffic intensities related to each server. then the probability of stopping this system is equal to e− ∑p j=1 λj − ∑ r i=1 qj (see, for example, [13]). but λj = m xj = p (xj = 1) = p ( sup 0≤s≤t qj (ns) < kj ), j = 1, 2, . . . , p. (8) and qi = m yi = p (yi = 1) = p ( sup 0≤s≤t vi(ns) < γi), j = 1, 2, . . . , .r. (9) applying (8) and (9), we obtain that h(x1, x2, . . . , xp, y1, y2, . . . , yr, t) is equal to e − p ∑ j=1 λj − r ∑ i=1 qj = e − ∑p j=1 p ( sup 0≤s≤t qj (ns) 0 if dt > 0 for all t ∈ b(x). a natural question is to study the brill-nother theory (for all multidegrees, not just for the total degree of the sheaf or line bundle) of a general element of m[τ ]. see [7] for the case of binary curves we conclude by giving examples of stable curves x such that ωx contains no spanned line bundle (section 4). we don’t touch a very important topic: the limit linear series introduced by d. eisenbud and j. harris for nodal curves of compact type, i.e. such that ‖x‖ has no loop and no multiple edge ([11]). see [18] for the positive characteristic case and [13] for the case of nodal curves with two smooth irreducible components. a quick glance at g. farkas’s survey on the geometry of mg and at the references quoted theirin shows the importance of this theory. however, a quick look at the examples in [11] (resp. [13]) and at example 2 (resp. example 3) here shows that the brill-noether theory coming from the limit linear series has nothing in common with the one considered here in section 1 or the one studied by l. caporaso in [7]. theorems 1 and 1 and section 1 are contained in [2] (in which also degree 2 sheaves are considered). section 2 is contained in [3]. 1 curves with gonality 1 remark 1. let x be a reduced and quasi-projective curve, p ∈ x, and f a sheaf on x with pure rank 1 and depth 1. the germ fp of f at p is a torsion free ox,p -module with rank 1. hence there exists an inclusion j : fp →֒ m with m a free ox,p -module with rank 1. the minimal integer dimk(m/fp ) for all such pairs (j, m ) is an important invariant of the germ fp . call ℓ(f, p ) this integer. we have ℓ(f, p ) ≥ 0 and ℓ(f, p ) = 0 if and only if fp is a free ox,p -module. this invariant may be computed on the formal completion of ox,p . let mx,p be the maximal ideal of the local ring ox,p . notice that mx,p is a free ox,p -module if and only if p ∈ xreg. hence if p ∈ sing(x) and fp ∼= mx,p , then ℓ(f, p ) = 1. now assume that x is projective. 106 e. ballico cubo 12, 1 (2010) fix a finite set s ⊆ sing(x) and let f : c → x be the partial normalization of x in which we normalize only the points of s. the torsion of f ∗(f ) is supported on the finite set f −1(s). set g := f ∗(f )/tors(f ∗(f )). g is a coherent sheaf on c with depth 1 and pure rank 1. since x and c are projective, the integers deg(f ) and deg(g) are well-defined and satisfy the riemmann-roch formulas χ(f ) = deg(f ) + χ(ox ), χ(g) = deg(g) + χ(oc ) even if x or c are not connected. we have deg(g) = deg(f ) − ∑ p ∈s ℓ(f, p ) (1) we need this formula only when each point of s is an ordinary node of x. in this case we may decompose f into ♯(s) partial normalizations of a single node. hence for nodes it is sufficient to prove it when ♯(s) = 1, say s = {p }. in this case (1) is obviously true if fp is free. if f = ip , then (1) holds, because ℓ(ip , p ) = 1 and g is the ideal sheaf of the two points f −1(p ). for an arbitrary fp use the next result. remark 2. take the set-up of the first part of remark 1. assume that p is either an ordinary node or an ordinary cusp of x. assume p ∈ sing(f ). by the classification of torsion free modules on simple curves singularities ([15], or, for nodes, [21], pp. 163–166) the germ of f at each p is formally equivalent to the maximal ideal mx,p of the local ring ox,p . hence remark 1 gives ℓ(f, p ) = 1. remarks 1 and 2 immediately give the following result. corollary 1. let x be a reduced projective curve and f a coherent sheaf on x with depth 1 and pure rank 1. fix s ⊆ sing(f ). assume that each point of s is an ordinary node or an ordinary cusp of x. let h : d → x (resp. f : c → x) be the partial normalization of x in which we normalize only the points of s (resp. the points of s and the singular points of x at which f is locally free). set l := f ∗(f )/tors(f ∗(f )) and r := h∗(f )/tors(h∗(f )). then deg(l) = deg(r) = deg(f ) − ♯(s). lemma 1. let x, y reduced, projective curves and f : y → x a finite surjective morphism. let a be a coherent sheaf on y . then deg(f∗(a)) = deg(a) + χ(ox ) − χ(oy ). proof. obviously, h0(x, f∗(a)) = h 0(y, a). since f is finite, r1f∗(a) = 0. hence the leray spectral sequence of f gives h1(x, f∗(a)) = h 1(y, a). thus χ(a) = χ(f∗(a)). since χ(a) = deg(a) + χ(oy ) and χ(f∗(a)) = deg(f∗(a)) + χ(ox ), we are done. remark 3. for any x (even not connected) and any l ∈ pic(x) we have ∑ t ∈b(x) deg(l|t ) = deg(l) (2) notice that (2) is true for non-locally free l if we only assume that l is locally free at each point of x lying on at least two irreducible components. cubo 12, 1 (2010) brill-noether theories for rank 1 sheaves ... 107 proof of theorem 1. fix any l ∈ d1(x). since deg(l|c) ≥ 0 for all c ∈ b(x), there is tl ∈ b(x) such that deg(l|tl) = 1, while the morphism hl : x → p r, r := h0(x, l) − 1, contracts to points all other components. let y1, . . . , ys be the closures in x of the connected components of x\tl. since l|tl is spanned, tl ∼= p 1, and hl|tl is bijective. this implies that hl(yi) = hl(yi ∩ t ) for all i. the second part of the statement of theorem 1 shows how to construct from any t ∈ s(x) a morphism hlt : x → p 1 such that the spanned line bundle h∗lt (op1 (1)) has degree 1. obviously, the last (resp. first) construction is the inverse of the first (resp. last) one. to check the last assertion take l constructed in a similar way by taking the degree d veronese embedding t →֒ pd of t ∼= p1, instead of the isomorphism φ. lemma 2. let t be an integral projective curve. there is no spanned rank 1 torsion-free sheaf f on t such that sing(f ) 6= ∅ and deg(f ) = 1. proof. assume the existence of such a sheaf f . since sing(f ) 6= ∅, t is singular. hence pa(t ) ≥ 1 = deg(f ). since f 6= ox and f is spanned, h 0(x, f ) ≥ 2. the contradiction comes from clifford’s inequality ([12], theorem a at p. 532). proof of theorem 2. assume the existence of f ∈ s1(x)\d1(x) and set c := ♯(sing(f )) and b := ♯(sing(f ) ∩ sing(x)′′). by assumption c > 0. if b = 0, then lemma 2 and the last sentence of remark 3 gives a contradiction. hence we may assume b > 0. let h : d → x be the partial normalization of x in which we normalize only the points of sing(f ) ∩ sing(x)′′. set r := h∗(f )/tors(h∗(f )). corollary 1 gives deg(r) = 1−b ≤ 0. r is a spanned sheaf with pure rank 1 and depth 1. hence b = 1 and r ∼= od. hence c = b and f has a unique singular point. since f has no torsion, the natural map h∗ : h0(x, f ) → h0(d, r) is injective. hence h0(x, r) ≥ 2. thus the only singular point, p , of sing(f ) is a disconnecting node of x. hence d = xp and h = fp . since xp has two connected components, we get h 0(x, f ) = 2. hence to prove all the assertions of theorem 2 it is sufficient to check that a := fp ∗(oxp ) ∈ s1(x)\d1(x). lemma 1 gives deg(a) = 1. obviously, a is not locally free at p . we have h0(x, a) = h0(xp , oxp ) = 2. let a′ be the subsheaf of a spanned by h0(x, a). if a′ = a, then we are done. assume a′ 6= a. hence deg(a′) ≤ deg(a) − 1 ≤ 0. since x is connected, we get h0(x, a′) ≤ 1. since h0(x, a′) = h0(x, a) = 2, we get a contradiction. example 1. fix integers g > q > 0. let x be a genus g stable curve with 2 irreducible components x1 and x2 such that pa(x1) = q and pa(x2) = g − q. since pa(x) = 1, we have ♯(x1 ∩ x2) = 1. the only point p of x1 ∩ x2 is a disconnecting node of x. it is easy to check that the degree 1 spanned sheaf associated by theorem 2 to this disconnecting node satisfies the basic inequality (5) (see section 3) if and only if g = 2q. 108 e. ballico cubo 12, 1 (2010) 2 the deformation theory of maps to p1 fix a positive multidegree d for τ . set δ(d) : ∑ t ∈b(x) dt (the total degree of d). for any positive multidegree d = {dt } let g 1(τ, d) denote the set of all pairs (x, f ) such that x ∈ m[τ ] and f : x → p1 is a morphism with multidegree d, i.e. such that deg(f ∗(op1 (1)) = dt for every t ∈ b(x). in this section we prove the following result. theorem 3. fix a topological type τ for nodal and connected projective curves and a positive multidegree d for τ . either g1(τ, d) = ∅ or g1(τ, d) is smooth and of pure dimension 2δ(d) + 2g − 2 − s, where x is any element of m[τ ], g := pa(x), and s := ♯(sing(x)). remark 4. let x be a connected projective curve with only ordinary nodes as singularities. let θx denote the dual of the cotangent sheaf ω 1 x . since θx is the dual of a generically rank 1 coherent sheaf, θx has no torsion. it is easy to check that θx = (ω 1 x /tors(ω 1 x )) ∗. fix p ∈ sing(x). it is well-known that the connected component of tors(ω1x ) supported by p has length 1 and that ω1x /tors(ω 1 x ) is not locally free at p (see [10], formula (4.1), or [16], p. 33). since x is gorenstein, every depth 1 sheaf on x is reflexive. thus ω1x /tors(ω 1 x ) ∼= θ∗x . hence sing(θx ) = sing(x). here we present an alternative proof. we claim that the germ at p of the sheaf ω1x /tors(ω 1 x ) is isomorphic to a colength 1 module f of the trivial ox,p -module of ωx,p . it would be sufficient to prove the claim, because no such f is locally free by the classification of torsion free modules over an ordinary node ([21], huitième partie, propositions 2 and 3). the claim is just part (2) of [5], lemma 6.1.2, in which the following notation is used: λ is the colength which we want compute, µ is the milnor number of x at p (µ = 1 for an ordinary node), δ is the genus of the singularity (δ = 1 for an ordinary node) κ (the cuspidal number) is equal to the multiplicity minus the number of branches by part (1) of [5], lemma 6.1.2 (hence κ = 0 for an ordinary node); the formula says that µ ≥ λ ≥ δ + κ, i.e. 1 ≥ λ ≥ 1 + 0. lemma 3. let x be a connected projective curve with only ordinary nodes as singularities. then deg(θx ) = 2 · χ(ox ) + ♯(sing(x)). proof. since x is gorenstein, every torsion free coherent sheaf f on x is reflexive and deg(f ∗) = − deg(f ). since θx ∼= (ω 1 x /tors(ω 1 x )) ∗, it is sufficient to prove that deg((ω1x /tors(ω 1 x )) = −2 · χ(ox ) − ♯(sing(x)). since deg(ωx ) = −2·χ(ox ), it is sufficient to prove the existence of an inclusion j : ω 1 x /tors(ω 1 x ) →֒ ωx whose cokernel is a torsion sheaf with length ♯(sing(x)). this assertion is the last part of remark 4, i.e. [5], lemma 6.1.2. let x be a nodal and connected projective curve, m a smooth and projective variety and f : x → m be a morphism whose restriction to each irreducible component of x is non-constant. the latter condition implies tx/m = 0, where tx/m (also denoted with tx/f /y or tf ) is the subsheaf of θx defined in [21], p. 387. our assumption on f is called “non-degenerate ” in [21], cubo 12, 1 (2010) brill-noether theories for rank 1 sheaves ... 109 definition 3.4.5. let n ′f denote the cokernel of the natural map θx → θm ; the same sheaf is denoted with nf in [21], definition 3.4.5. since tx/m = 0, we have an exact sequence of coherent sheaves on x ([21], p. 162). 0 → θx f̃ → f ∗(θm ) → n ′ f → 0 (3) we are interested in the functor of locally trivial deformations of the map f with m fixed, mainly when m = p1. since f is non-degenerate, the vector space h0(x, n ′f ) is the tangent space to the functor of locally trivial deformations of the map f with m fixed, while the vector space h1(x, n ′f ) is an obstruction space for the same functor ([21], lemma 3.4.7 (iii) and theorem 3.4.8). since x is nodal, saying “locally trivial ” means that we only look at nearby pairs (x′, f ′) with x′ a nodal curve with ‖x′‖ = ‖x‖, i.e. in which no node is smoothed, i.e. with the topological type of x. proposition 1. let x be a nodal and connected projective curve. let f : x → p1 be a morphism such that f|t is not constant for every irreducible component t of x. set g := pa(x), s := ♯(sing(x)) and d := deg(f ). the the functor of locally trivial deformations of f is smooth at f and of dimension 2d + 2g − 2 − s. proof. both θx and f ∗(θp1) are sheaves with depth 1 and pure rank 1. hence the injectivity of the map f̃ in the exact sequence (3) gives that n ′f is supported by finitely many points of x. hence h1(x, n ′f ) = 0 and h 0(x, n ′f ) = deg(f ∗(θp1 )) − deg(θx ) = 2d + 2g − 2 − s (lemma 3). proof of theorem 3. the theorem is just a restatement of proposition 1. let x be a nodal and connected projective curve. let gon1(x) denote the minimal integer d such that there is a degree d morphism x → p1. obviously, gon1(x) ≥ ∑ t ∈b(x) gon1(t ). thus gon1(x) ≥ ♯(b(x)) and the inequality is strict if at least one of the components of x is not isomorphic to p1. there are topological types τ such that h0(x, f ∗(op1 (1)) ≥ 3 for any nodal curve x with topological type τ and f computing gon1(x). a stupid example is given by the topological type of a reducible conic. a more interesting example is given by the graph curves x of genus g ≥ 4 ([4]) for which gon1(x) ≥ ♯(b(x)) = 3g − 3 and hence h 0(x, f ∗(op1 (1)) ≥ g − 1 (riemann-roch). let τ be a topological type for nodal and connected projective curves. let gon1,−(τ ) (resp. gon1,+(τ )) denote the minimal (resp. maximal) integer d such that there is x ∈ τ such that gon1(x) = d. question 1. compute gon1,−(τ ) and gon1,+(τ ) for every topological type τ . 3 caporaso’s compactification: the basic inequality fix an integer g ≥ 2. in [6] l. caporaso constructed a compactification over mg of the set of all line bundles with fixed degree on smooth genus g curves. recall that for every stable genus g curve x the canonical sheaf ωx is an ample line bundle. r. pandharipande proved that caporaso’s 110 e. ballico cubo 12, 1 (2010) compactification is equivalent to the moduli scheme of equivalence classes of all slope-semistable (with respect to the polarization ωx ) coherent sheaves with depth 1, pure rank 1 and degree d ([19], theorem 10.3.1; see [21] for the construction of this moduli space). this is a very nice result, because it shows that, at least if we take the canonical polarization, the brill-noether theory of elements of caporaso’s compactification or of semistable sheaves are the same. we will use the set-up of [6], because it has a very important feature: if x is reducible, then we may refine the degree, prescribing (at least for line bundles) the multidegree, i.e. the degree of the restriction to each irreducible components. easy examples show that for certain multidegrees a brill-noether locus may be empty, while for other multidegrees with the same total degree the corresponding moduli space is non-empty ([6], proposition 12). this is not an exceptional situation: it happens very frequently. we recall the definition of semibalanced or balanced line bundle and of basic inequality ([17], definition 1.1). a semistable curve x is called quasistable if any two exceptional irreducible components of x (i.e. smooth rational component intersecting the other components at two points) are disjoint. x is quasistable if and only if either x is stable or its stable reduction u : x → y have the following property: there is s ⊆ sing(y ) such that u|u−1(x\s) : u−1(x\s) → x\s is an isomorphism and u−1(p ) ∼= p1 for all p ∈ s. if x is quasistable, but not stable, then (u, y, s) are uniquely determined by x, while x is uniquely determined by the pair (y, s). let x be a quasistable curve of genus g ≥ 2. for any proper subcurve a of x set ka := ♯(a ∩ x\a) and wa := deg(ωx|a) = −2 · χ(oa) + ka. fix l ∈ pic(x). l is said to be semibalanced if the following inequality (called the basic inequality) deg(l) · wz /(2g − 2) − kz /2 ≤ deg(l|z) ≤ deg(l) · wz /(2g − 2) + kz /2 (4) holds for every proper connected subcurve z of x. l is said to be balanced if it is balanced and deg(l|e) = 1 for every exceptional component e of x. let x be a stable curve. fix s ⊆ sing(x). let us : x[s] → x be the partial normalization of x in which we normalize exactly the point of s. thus x[s] is nodal ♯(sing(x[s]) = ♯(sing(x)) − ♯(s) and χ(o[s]) = ♯(s) + χ(ox ). x[s] may be disconnected. let vs : xs → x denote the stable reduction of the quasistable model of the pair (x, s), i.e. xs is quasistable with ♯(s) exceptional components, each of them mapped to a different point of s and vs|v −1 s (x\s) : v −1 s (x\s) → x\s is an isomorphism. xs is connected and pa(xs ) = pa(x). x[s] is a subcurve of xs and us = vs|x[s]. remark 5. fix a pure rank 1 torsion free sheaf f on x. write uf , vf , x[f ] and xf instead of xs , vf , x[s] and xs if s := sing(f ). set f̃ := u ∗ f (f )/tors(u ∗ f (f )). since f̃ has no torsion and it is localy free at each singular point of x[f ], f̃ is a line bundle. fix p ∈ sing(f ). the classification of all pure rank 1 torsion free sheaves on a nodal singularity gives that the germ of f at p is isomorphic to the maximal ideal of the local ring ox,p ([21], huitième partie, propositions 2 and 3). hence deg(f̃ ) = deg(f ) − ♯(s). there is a unique line bundle f on xs such that f |x[f ] = f̃ and deg(f |e) = 1 for every exceptional curve e of xf . we have deg(f ) = deg(f̃ )+♯(s) = deg(f ). since f has no torsion, the pull-back map f ∗ induces an inclusion f ∗ : h0(x, f ) → h0(x[s], f̃ ). hence h0(x[s], f̃ ) ≥ h 0(x, f ). now assume that f is spanned by a linear subspace v ⊆ h0(x, f ). cubo 12, 1 (2010) brill-noether theories for rank 1 sheaves ... 111 since the tensor product is a right exact functor, f ∗(v ) spans f̃ . recall that xf is obtained from x[f ] adding ♯(s) disjoint exceptional curves. since deg(f |e) = 1 for every exceptional curve e of xf , ♯(s) mayer-vietoris exact sequences give h 0(xs , f ) = h 0(x[s], f̃ ) and that f is spanned if and only if f̃ is spanned (see lemmas 4 and 5)). lemma 4. fix a nodal curve x and s ⊆ sing(x). fix l ∈ pic(x[s]) and let m be the only line bundle on xs such that m|x[s] ∼= l and deg(m|e) for every exceptional curve e of vs. we have h0(x[s], l) = h 0(xs , m ). if l is spanned, then h 0(xs , m ) ≤ h 0(x[s], l) + ♯(s). m is spanned if and only if l is spanned and for any exceptional curve e of vs there is f ∈ h 0(x[s], l) vanishing at one of the point of e ∩ x[s], but not vanishing at the other point of e ∩ x[s]. proof. by induction on s we reduce to the case ♯(s) = 1. this inductive procedure is the reason for not requiring the stability of x (if ♯(s) ≥ 2), since this assumption would be lost in the inductive step. set {p } := s. let e be the only new exceptional curve of x[s] and p1, p2 the points of e ∪ x[s]. we have a mayer-vietoris exact sequence on xs : 0 → m → m|x[s] ⊕ m|e → m|{p1, p2} → 0 (5) since the restriction map h0(e, m|e) → h0({p1, p2}, m|{p1, p2}) is bijective, (5) gives that the restriction map ρ : h0(xs , m ) → h 0(x[s], l) is bijective. hence h 0(x[s], l) = h 0(xs , m ). the bijectivity of ρ gives that l is spanned if and only if m is spanned at each point of x[s]. since e ∼= p1 and deg(m|e) = 1, m is spanned at every point of e if and only if the restriction map η : h0(xs , m ) → h 0(e, m|e) is surjective. the cohomology exact sequence of (5) gives that η is surjective if and only if the restriction map β : h0(x[s], l) → l|{p1, p2} is surjective. since l is spanned, the surjectivity of β is equivalent to require the existence of f ∈ h0(x[s], l) vanishing at p1, but not at p2. lemma 5. fix a nodal curve x and a sheaf f on x with depth 1 and pure rank 1. assume that f is spanned. then f̃ and f are spanned and h0(xf , f ) = h 0(x[f ], f̃ ). proof. we know that f̃ is spanned. apply lemma 4 to s := sing(f ) and get equality h0(xf , f ) = h0(x[f ], f̃ ). to check the spannedness of f we need to check the last condition of lemma 4. fix p ∈ sing(f ) and let e be the corresponding exceptional curve. set {p1, p2} := e ∩ x[f ]. since the germ of f at p is isomorphic to the maximal ideal of ox,p and ox,p is an ordinary node, the fiber f |{p } of f at p is a 2-dimensional vector space. since f is spanned at p and the natural map u∗f : h 0(x, f ) → h0(x[f ], l) is injective, we get h 0(x[f ], i{p1,p2} ⊗ l) ≤ h 0(x[f ], l). hence the last part of lemma 4 gives the spannedness of f . summary of the section: we hope to have convinced the reader that in many cases we may separately study the basic inequality and the geometric properties (spannedness, very ampleness and so on) of a potential element of a brill-noether theory. 112 e. ballico cubo 12, 1 (2010) 4 stable curves with ωx without spanned locally free subsheaves let x be a stable curve of genus g. the dualizing sheaf ωx is spanned if and only if x has no disconnecting node, i.e. there is no p ∈ sing(x) such that x\{p } is not connected ([8], theorem d, or [1], part (a) of theorem 1.2, or [9], part (b) of theorem 3.3). since x is nodal, x\{p } has two connected comnponents if p is a disconnecting node of x. here we give two examples of genus g stable curves x such that there is no locally free spanned subsheaf l of ωx with h 0(x, l) ≥ 2 (i.e. l 6= ox ). since ωx is locally free, in any such example ωx is not spanned, i.e. x has a disconnecting node. the first example works for any genus g ≥ 2. for the second example we need to assume g large, say g ≥ 9. in the second example we have no injective map ox →֒ ωx . example 2. let x be a chain of g curves of genus 1, i.e. assume x = ∪ g i=1ti, pa(ti) = 1 for all i, ti ∩ tj 6= ∅ if and only if |i − j| ≤ 1 and ♯(ti ∩ ti+1) = 1 for all i ∈ {1, . . . , g − 1}. notice that x has g − 1 disconnecting nodes. the proofs of [8], theorem d, or of of [1], theorem 1.2, or an easy exercise left to the reader show that the subsheaf f of ωx spanned by h 0(x, ωx ) has the property that ωx /f = ⊕p ∈sing(x)′′ kp , where kp denote the skyscraper sheaf supported by p and such that h0(x, kp ) = 1. hence deg(f ) = 2g − 2 − ♯(sing(x) ′′) = g − 1 and f is not locally free at any point of sing(x)′′. every spanned subsheaf of ωx is contained in f . if l is a locally free subsheaf of f , then the torsion sheaf f/l must have every point of sing(x)′′ in its support. thus deg(l) ≤ deg(f ) − (g − 1) ≤ 0. if l is also spanned, we get l ∼= ox . example 3. let x be a genus g stable curve such that there is an irreducible component t ∼= p1. set k := ♯(x ∩ x\t ). notice that ωx|t has degree k − 2. assume that at least k − 1 of the points of x ∩ x\t are disconnecting node of x. let f be the subsheaf of ωx spanned by h0(x, ωx ) and let l be any locally free subsheaf of f . since ωx is not spanned at any disconnecting node, the inclusion map j : l →֒ ωx drops rank at each disconnecting node of x. hence deg(l|t ) ≤ deg(ωx|t )− (k − 1) ≤ −1. thus l is not spanned and any section of l vanishes identically on t . in particular l 6= ox . question 2. is it possible to describe all genus g stable curves x such that there is no locally free spanned subsheaf l of ωx with h 0(x, l) ≥ 2 and/or the ones for which there is no injective map ox →֒ ωx ? is it true that any example with the latter property has a smooth rational component t such that every section of ωx vanishes 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[21] seshadri, c., fibrés vectoriels sur les courbes algébriques, astérisque, 96, 1982. cubo a mathematical journal vol.11, no¯ 02, (117–126). may 2009 network oligopolies with multiple markets lászló kapolyi system consulting zrt., 1126. budapest, béla király út 30/c, hungary email: system@system.hu abstract cournot oligopolies with multiple markets are examined when transportation costs from each manufacturer to all markets are also included into the profit functions. under general and realistic conditions the equilibrium always exists, and based on the kuhn-tucker conditions a computer method is introduced to compute it. the dynamic extension of the model is also introduced with gradient adjustment, and the asymptotic stability of the equilibrium is examined. in the case of independent markets and linear price and cost functions the solution algorithm is simplified and the global asymptotic stability of the equilibrium is proved. resumen son examinados oligopolios cournot con mercados multiples cuando los costos de transporte de todas las manofacturas de todos los mercados son incluidos en la función de utilidad. bajo condiciones generales y realistas siempre existe equilibrio, y basados en condiciones de kuhntucker un método computacional es introducido para calcularlo. la extensión dinámica del modelo es también presentada con adaptación gradiente, es examinada la estabilidad asintótica. en el caso de mercados independientes y funciones de precio y costo lineales el algoritmo de solución es simplificado y es provada la estabilidad asintótica global del equilibrio. key words and phrases: oligopoly, multiple market, equilibrium, stability. math. subj. class.: 91a06, 91a80. 118 lászló kapolyi cubo 11, 2 (2009) 1 introduction the classical oligopoly theory considers a homogeneous market and several firms offering a certains product to the market. this simple model has been examined and extended by many authors in the last few decades. models with product differentiation, multi-product oligopolies, labor managed firms, rent-seeking games were analysed to mention only a few. a comprehensive summary of these models and their properties is given in okuguchi (1976) and in okuguchi and szidarovszky (1999). emition control, waste management, technology spill-over, secondary market competition were also included into oligopoly models, however very few attempts were made to examine the effect of multiple markets. they were mostly treated as multiproduct oligopolies, where the same good sold in different markets was considered as different products. if we introduce transportation costs into the models, their structure becomes different because of the additional additive cost terms. in this paper we make the first attempt to analyze the properties of oligopoly models including transportation costs to multiple markets. the paper is organized as follows. after the mathematical model is formulated, conditions are given for the existence of the noncooperative nash equilibrium based on the theory of concave games. then a computer method is introduced to find the equilibrium. in applying this method we have to find either feasible solutions for a system of usually nonlinear equalities and inequalities, or optimal solutions of a nonlinear programming problem. these methods can be significantly simplified in the cases of independent markets and linear cost and price functions. a similar algorithm will be then proposed to find the completely cooperative solution. a gradient adjustment process is then formulated as a dynamic extension of the model, and its asymptotical stability will be examined. in the case of independent markets and linear cost and price functions we will show that the equilibrium is always globally asymptotically stable. 2 mathematical model consider a network of n firms that produce the same product or offer the same service to m different markets. let xki be the quantity offered by firm k to market i, then the supply on market i is given as si = ∑n k=1 xki and the output of firm k is qk = ∑m i=1 xki. the production cost ck(qk) of firm k depends on qk, and the price on each market i depends on the supplies on all markets, since consumers might have choices where they want to purchase the goods. so the price on market i is given as pi(s1, . . . , sm ). assume also that there is a transportation cost (including possible duties) for each firm for supplying its product on the different markets: tk1(xk1) + · · · + tkm (xkm ). based on the above notation, the profit of firm k equals πk = m ∑ i=1 xkipi(s1, . . . , sm ) − ck(qk) − m ∑ i=1 tki(xki). (1) it is also assumed that each xki value is nonnegative and bounded from above: xki ∈ [0, lki] . in this way an n -person game is defined is which the n firms are the players, the strategy of each player is a vector xk = (xk1, . . . , xkm ) ∈ [0, lk1] × · · · × [0, lkm ] = sk, and the payoff function of player k is its cubo 11, 2 (2009) network oligopolies with multiple markets 119 profit, πk. the solution of this game is the noncooperative nash equilibrium, which is a set of strategy vectors x ∗ k (1 ≤ k ≤ n ) such that for all players k, 1. x∗k ∈ sk; 2. πk(x ∗ 1, . . . , x ∗ k−1, xk, x ∗ k+1, . . . , x ∗ n ) ≤ πk(x ∗ 1, . . . , x ∗ n ) (2) with all xk ∈ sk. assume that all functions pi, ck and tki are twice continuously differentiable. notice that ∂πk ∂xki = pi + xki ∂pi ∂si + ∑ j 6=i xkj ∂pj ∂si − c′k − t ′ ki = pi + m ∑ l=1 xkl ∂pl ∂si − c′k − t ′ ki, (3) furthermore ∂ 2 πk ∂x 2 ki = 2 ∂pi ∂si + xki ∂ 2 pi ∂s 2 i + ∑ j 6=i xkj ∂ 2 pj ∂s 2 i − c′′k − t ′′ ki (4) and ∂ 2 πk ∂xki∂xkj = ∂pi ∂sj + ∂pj ∂si + m ∑ l=1 xkl ∂ 2 pl ∂si∂sj − c′′k . (5) for the sake of convenient notation let j p be the jacobian of p = (p1, . . . , pm ) and hl the hessian of pl. then the hessian matrix of πk with respect to vector xk can be conveniently written as j p + j t p + m ∑ l=1 xklhl − c ′′ k · e − dk, (6) where e is the m × m matrix with all unity elements, and dk = diag (t ′′ k1, . . . , t ′′ km ) . now we make the following assumptions: (a) −p is monotonic in the sense that ( s (1) − s(2) )t ( p ( s (1) ) − p ( s (2) )) ≤ 0 (7) for all s(1) = ( s (1) 1 , . . . , s (1) m ) and s(2) = ( s (2) 1 , . . . , s (2) m ) ; (b) pl is concave for all l; 120 lászló kapolyi cubo 11, 2 (2009) (c) t ′′ki ≥ 0 for all k and i; (d) c′′k ≥ 0. notice that (a) implies that j p + j t p is negative semidefinite (see for example ortega and rheinboldt, 1970). condition (b) implies that all matrices hl are negative semidefinite, and (c) implies that dk is positive semidefinite. consequently the hessian of πk with respect to xk is negative semidefinite, so πk is concave in xk. therefore the nikaido–isoda theorem (forgo et al., 1999) implies that there is at least one nash equilibrium. here we used the fact that e is positive semidefinite with eigenvalues 0 and m (see for example okuguchi and szidarovszky, 1999). 3 computation of the equilibrium the equilibrium output of any firm k is the optimal solution of the problem maximize m ∑ i=1 xkipi(s1, . . . , sm ) − ck(qk) − m ∑ i=1 tki(xki) (8) s.to 0 ≤ xki ≤ lki for all i. the kuhn–tucker conditions are sufficient and necessary because of the concavity of πk. they can be written as follows. there exist constants uk1, . . . , ukm , vk1, . . . , vkm such that uki ≥ 0, vki ≥ 0 (all i), (9) 0 ≤ xki ≤ lki (all i), (10) ( ∂πk ∂xk1 , . . . , ∂πk ∂xkm ) + (uk1, vk1, . . . , ukm , vkm )                1 −1 1 −1 ... 1 −1                = (0, . . . , 0), (11) ukixki = 0 (all i), (12) vki (lki − xki) = 0 (all i). (13) here we gave only the nonzero elements of the matrix and used the fact that the constraints of problem (8) have the form xki ≥ 0 and lki − xki ≥ 0. the (xk1, . . . , xkm ) (k = 1, . . . , n ) part of any feasible solution of this equality-inequality system provides nash-equilibrium. cubo 11, 2 (2009) network oligopolies with multiple markets 121 we can also rewrite this feasibility problem as a single objective optimization problem: minimize n ∑ k=1 m ∑ i=1 (ukixki + vki(lki − xki)) subject to uki ≥ 0, vki ≥ 0, 0 ≤ xki ≤ lki (all k and i) (14) ∂πk ∂xki + uki − vki = 0 (all k and i), which can be solved by routine methods. the dimension of this problem can be reduced by introducing new variables αki = uki − vki. (15) from the last constraint we have αki = − ∂πk ∂xki (16) and the nonnegativity of uki and vki implies that vki = uki − αki ≥ −αki, that is, αki + vki ≥ 0 for all k and i. substituting (15) and (16) into the objective function we have a simplified model: minimize n ∑ k=1 m ∑ i=1 ( − ∂πk ∂xki xki + vkilki ) subject to vki ≥ 0 vki ≥ ∂πki ∂xki (all k and i) (17) 0 ≤ xki ≤ lki. in the objective function lki > 0, so the objective is minimal if and only if vki is minimal. from the first two constraints of (17) we see that the minimal value of vki is the larger of 0 and ∂πk ∂xki . hence a more simple version of the optimization model is the following: minimize n ∑ k=1 m ∑ i=1 ( − ∂πk ∂xki xki + lki · max { 0; ∂πk ∂xki }) (18) subject to 0 ≤ xki ≤ lki. 4 the case of independent markets in this section we assume that the market price pi depends on only the supply si on market i, so pi = pi(si). in this special case πk = m ∑ i=1 xkipi(si) − ck(qk) − m ∑ i=1 tki(xki) (19) 122 lászló kapolyi cubo 11, 2 (2009) and therefore ∂πk ∂xki = pi + xkip ′ i − c ′ k − t ′ ki, (20) furthermore ∂ 2 πk ∂x 2 ki = 2p ′ i + xkip ′′ i − c ′′ k − t ′′ ki (21) and with j 6= i, ∂ 2 πk ∂xki∂xkj = −c′′k . (22) therefore the hessian of πk has the more special diagonal form        2p ′ 1 − t ′′ k1 + xk1p ′′ 1 2p ′ 2 − t ′′ k2 + xk2p ′′ 2 . . . 2p ′ m − t ′′ km + xkm p ′′ m        − c′′k · e. (23) recall that e is positive semidefinite, so this matrix is negative semidefinite if for all i, (a’) p′i < 0; (b’) p′′i ≤ 0 in addition to assumptions (c) and (d). the nikaido–isoda theorem implies that under conditions (a’), (b’), (c) and (d) there is at least one nash-equilibrium. in this special case the optimization problem (17) reduces to the following: minimize n ∑ k=1 m ∑ i=1 (xki (−pi − xkip ′ i + c ′ k + t ′ ki) + vkilki) subject to vki ≥ 0 (24) vki ≥ pi + xkip ′ i − c ′ k − t ′ ki 0 ≤ xki ≤ lki. assume next that all function pi, ck and tki are linear pi(si) = ai − bisi, ck(qk) = ak + bkqk and tki = αki + βkixki. then p ′ i = −bi, c ′ k = bk, t ′ ki = βki, so we have a simple optimization problem: minimize n ∑ k=1 m ∑ i=1 (xki (−ai + bisi + xkibi + bk + βki) + vkilki) subject to vki ≥ 0 (25) vki − ai + bisi + xkibi + bk + βki ≥ 0 0 ≤ xki ≤ lki. cubo 11, 2 (2009) network oligopolies with multiple markets 123 in this problem all constraints are linear (notice that si = ∑n k=1 xki), and the objective function can be rewritten as follows: n ∑ k=1 m ∑ i=1 vkilki + n ∑ k=1 m ∑ i=1 xki(−ai + bisi + xkibi + bk + βki) = n ∑ k=1 m ∑ i=1    vkilki + xki (βki + bk − ai) + bixki  2xki + ∑ l 6=k xli      (26) which is a quadratic function of its unknows. for the actual solution of this problem standard software is available. 5 cooperative solutions the most commonly applied cooperative solutions (such as the shapley value, see for example, forgó et al., 1999) maximizes the total profit of all firms, and then this maximal profit is distributed among the firms in a well defined, concept-dependent way. the total profit equals π = n ∑ k=1 πk = m ∑ j=1 sj pj (s1, . . . , sm ) − n ∑ k=1 ck(qk) − n ∑ k=1 m ∑ j=1 tkj (xkj ). (27) the maximal solution of this function can be obtained in the following way. notice first that ∂π ∂xki = pi + si ∂pi ∂si + ∑ j 6=i sj ∂pj ∂si − c′k − t ′ ki (28) and the constraints can be rewritten as xki ≥ 0 lki − xki ≥ 0, so the kuhn–tucker conditions guarantee the existence of nonnegative constants uki and vki such that uki ≥ 0, vki ≥ 0 (all k and i) 0 ≤ xki ≤ lki ( ∂π ∂x11 , . . . , ∂π ∂x1m , . . . , ∂π ∂xn 1 , . . . , ∂π ∂xn m , ) + (u11, v11, . . . , un m , vn m )                1 −1 1 −1 . . . 1 −1                = (0, . . . , 0) (29) 124 lászló kapolyi cubo 11, 2 (2009) ukixki = 0 vki(lki − xki) = 0. from the third condition we see that for all k and i, ∂π ∂xki = uki − vki. (30) so the set of feasible solutions of this system of equalities and inequalities contains the optimal solutions. if π is concave as an n m -variable function, then the kuhn-tucker conditions are sufficient and necessary, so every feasible solution of system (29) is optimal for problem (27). another way is the direct optimization of the objective function π subject to the constraints 0 ≤ xki ≤ lki for all k and i. 6 dynamic extensions assuming continuous time scales – as usual is the theory of dynamic economics – it is assumed that the firms adjust their production levels proportionally to their marginal profits. this concept can be mathematically modeled as ˙xki = kki ∂πk ∂xki = kki  pi + m ∑ j=1 xkj ∂pj ∂si − c′k − t ′ ki   (31) for all k and i, where kki > 0 is a constant known as the speed of adjustment of firm k. the motivation of this construct is the following. if ∂πk ∂xki > 0, then the profit of firm k increases by the increasing value of xki. if ∂πk ∂xki < 0, then the profit increases by decreasing value of xki. if this partial derivative is zero, then the current production level is at a stationary point, so the firm does not want changes. notice first that the interior equilibria are the steady states of this system. using linearization, the local asymptotic stability of the equilibrium is guaranteed if all eigenvalues of the jacobian of the right hand side functions are in the left half of the complex plane. let gki denote the right hand side of equation (k, i). simple differentiation shows that ∂gki ∂xki = kki  2 · ∂pi ∂si + xki ∂ 2 pi ∂s 2 i + ∑ j 6=i xkj ∂ 2 pj ∂s 2 i − c′′k − t ′′ ki   (32) ∂gki ∂xkj = kki ( ∂pi ∂sj + ∂pj ∂si + m ∑ l=1 xkl ∂ 2 pl ∂si∂sj − c′′k ) (33) and if l 6= k, then ∂gki ∂xlj = kki ( ∂pi ∂sj + m ∑ r=1 xkr ∂ 2 pr ∂si∂sj ) (34) for all j. based on these derivatives, the jacobian can be represented in a block form (j kl) where j kl is an m × m matrix: j kl =    kk ( j p + j t p + ∑ j xkj hj − c ′′ k · e − dk ) if l = k kk ( j p + ∑ j xkj hj ) if l 6= k (35) cubo 11, 2 (2009) network oligopolies with multiple markets 125 with kk = diag(kk1, kk2, . . . , kkm ) and the notations of (6). the equilibrium is locally asymptotically stable if all eigenvalues of the jacobian have negative real parts. the structure of the jacobian is complicated, so we will only illustrate this condition in the linear case, when pi(si) = ai − bisi, ck(qk) = ak + bkqk and tki = αki + βkixki (as in constructing the optimization problem (25)). then with the notation p = diag(−b1, −b2, . . . , −bm ), the jacobian is        2k1p k1p . . . k1p k2p 2k2p . . . k2p ... ... ... kn p kn p . . . 2kn p        where each block is diagonal. by interchanging the rows and columns, this matrix can be rearranged as a blockdiagonal matrix diag(a1, a2, . . . , am ) with n × n blocks, ai =        −2k1ibi −k1ibi . . . −k1ibi −k2ibi −2k2ibi . . . −k2ibi ... ... ... −kn ibi −kn ibi . . . −2kn ibi        so the equilibrium is locally asymptotically stable if the eigenvalues of each block have negative real parts. this is guaranteed if the eigenvalues of matrix        2k1i k1i . . . k1i k2i 2k2i . . . k2i ... ... ... kn i kn i . . . −2kn i        have positive real parts for all i. this matrix can be rewritten as d + k · 1t with di = diag(k1i, k2i, . . . , kn i), ki = (k1i, k2i, . . . , kn i) t , and 1t = (1, 1, . . . , 1). the characteristic polynomial has the form ϕi(λ) = det ( d + k · 1t − λi ) = det (d − λi) det ( i + (d − λi)−1k · 1t ) = n ∏ l=1 (kli − λ) [ 1 + n ∑ l=1 kli kli − λ ] . the roots of the first factor are all positive. let h(λ) denote the bracketed factor. clearly lim λ→±∞ h(λ) = 1, lim λ→kli±0 h(λ) = ∓∞, h ′ (λ) = n ∑ l=1 kli (kli − λ)2 > 0. 126 lászló kapolyi cubo 11, 2 (2009) so h(λ) locally strictly increasing. therefore there is one root between each consecutive pair of poles (which are the positive kki numbers) and an additional root after the largest pole. we found this way n real positive roots. equating the bracketed term with zero leads to a polynomial equation of degree n , so we found all roots, they are real and positive. so the equilibrium is always locally asymptotically stable. notice that in the linear case system (31) is also linear, so local stability implies global asymptotical stability, therefore independently of the initial conditions the trajectories of the system converge to the nash-equilibrium. 7 conclusions in this paper cournot oligopolies were examined with multiple markets, when additional transportation costs were included into the profit functions of the firms. after the mathematical model was formulated sufficient conditions were presented for the existence of the nash equilibrium, and computer method were presented for determining the equilibrium based on the kuhn–tucker conditions. this method was simplified in the case of independent markets, and was further modified to find the completely cooperative solution. a gradient adjustment dynamic model was finally introduced and its asymptotical stability examined. in the case of independent markets and linear price and cost functions the equilibrium is always globally asymptitocally stable with respect to gradient adjustments. received: april 02, 2008. revised: may 05, 2008. references [1] forgó, f., szép, j. and szidarovszky, f., introduction to the theory of games. kluwer academic publishers, dordrecht/london, 1999. [2] okuguchi, k., expectations and stability in oligopoly models. springer-verlag, berlin/new-york, 1976. [3] okuguchi, k. and szidarovszky, f., the theory of oligopoly with multi-product firms. springer verlag, berlin/new york, 1999. [4] ortega, j. and rheinboldt, w., iterative solution of nonlinear equations in several variables. academic press, new york, 1970. [5] puu, t., attractors, bifurcations, and chaos: nonlinear phenomena in economics (2nd ed.). springerverlag, berlin/new york, 2003. [6] vives, x., oligopoly pricing. mit press, cambridge, mass, 1999. n08-network cubo a mathematical journal vol.11, no¯ 04, (87–107). september 2009 pseudo-differential operators with smooth symbols on modulation spaces joachim toft department of mathematics and systems engineering, växjö university, sweden email: joachim.toft@vxu.se abstract let m p,q (ω0) be the modulation space with parameters p,q and weight function ω0. if ∂ α a/ω ∈ l∞ for all α, then we prove that the pseudo-differential operator at(x,d) is continuous from m p,q (ω0ω) to m p,q (ω0) . more generally, if b is a translation invariant bf-space, then we prove that at(x,d) is continuous from m(ω0ω)(b) to m(ω0)(b). we use these results to establish identifications between such spaces with different weights. resumen sea m p,q (ω0) el espacio de modulación con parámetros p,q y función de peso ω0. si ∂ α a/ω ∈ l∞ para todo α, entonces probamos que el operador pseudo-diferencial at(x,d) es continuo de m p,q (ω0ω) a m p,q (ω0) . en general, si b es una translación invariante en el espacio-bf, entonces probamos que at(x,d) es continuo de m(ω0ω)(b) en m(ω0)(b). usamos estos resultados para establecer las identificaciones entre dichos espacios con diferentes pesos. key words and phrases: pseudo-differential operators, modulation spaces, coorbit spaces, bfspaces, sobolev spaces, besov spaces. math. subj. class.: 35s05, 47b37, 47g30, 42b35. 88 joachim toft cubo 11, 4 (2009) 1 introduction in this paper we establish continuity properties for certain pseudo-differential operators with smooth symbols when acting on general class of modulation spaces. these modulation spaces involve the usual modulation spaces, as well as certain type of weighted spaces related to wiener amalgam spaces. furthermore, we establish bijectivity properties for multiplication operators and fourier multipliers, and use these properties to establish identification properties between modulation spaces with different weights. in particular we cover theorem 2.1 in [30], where tachizawa considers pseudo-differential operators with symbols in s(ω)(r 2d ), the set of all smooth functions a on r2d such that (∂αa)/ω ∈ l ∞ (r2d). here ω is an appropriate weight function on r2d, which takes the form of ω(x,ξ) = 〈x〉t〈ξ〉s (1) in [30], where s,t ∈ r and 〈x〉 = (1 + |x|2)1/2. (we use the usual notation for function and distribution spaces, see e. g. [22].) in this context, tachizawa extends calderon-vaillancourt’s theorem, and proves that if ω0 is appropriate, and p,q ∈ [1,∞], then the corresponding pseudodifferential operators are continuous from the modulation space m p,q (ω0ω) to m p,q (ω0) . (cf. section 2 for the definition of modulation spaces and pseudo-differential operators.) tachizawa’s result were thereafter generalized in theorem 3.2 in the report [38], where the conditions on the weight ω are relaxed in the sense that it is only assumed that ω should be v-moderate for some polynomial v. a similar and interesting result comparing to [30, 38], concerns [32, theorem 5.3], where teofanov discuss similar properties in context of ultra-modulation spaces. in this approach, the condition on v here above is relaxed in the sense that v is permitted to grow subexponentially, instead of polynomially. this in turn implies that symbols to the pseudo-differential operators might grow subexponentially. however, the classes of pseudo-differential operators in [32] do not contain those in [30] or [38], because the symbols in [32] have to fulfill certain conditions of gelfandshilov type, which is not the case in [30, 38]. other related results are theorem 3 in [25], and theorem 3, corollary 2 and remark 3 in [26], where pilipović and teofanov consider mapping properties for pseudo-differential operators with symbols in ultra-modulation space and which fulfill certain ellipticity conditions. in section 3 we generalize [38, theorem 3.2], and prove continuity for such pseudo-differential operators on a broad class of modulation spaces, which contains the modulation spaces in [38], and their fourier transforms. these modulation spaces are in turn special cases of so called coorbit spaces (see [11, 12] for the definition of coorbit spaces, and [9] for an updated definition of modulation spaces). (see theorem 3.2 and theorem 3.2′.) furthermore we establish bijectivity properties for pseudo-differential operators, if they, in addition, are appropriate multiplication operators or fourier multipliers. (see corollary 3.6.) thereafter we give links on how these results can be used to establish identification properties between modulation spaces with different weights. (see remark 3.7 and theorems 3.9.) here we also present some immediate consequences cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 89 in modulation space theory and for spaces related to wiener amalgam spaces. (see corollary 3.6′, theorem 3.9′ and theorem 3.9′′.) the (classical) modulation spaces mp,q, p,q ∈ [1,∞], as introduced by feichtinger in [6], consist of all tempered distributions whose short-time fourier transforms (stft) have finite mixed l p,q norm. it follows that the parameters p and q to some extent quantify the degrees of asymptotic decay and singularity of the distributions in mp,q. the theory of modulation spaces was developed further and generalized in [11–13, 16], where feichtinger and gröchenig established the theory of coorbit spaces. in particular, the modulation space m p,q (ω) , where ω denotes a weight function on phase (or time-frequency shift) space, appears as the set of tempered (ultra-) distributions whose stft belong to the weighted and mixed lebesgue space l p,q (ω) . a major idea behind the design of these spaces was to find useful banach spaces, which are defined in a way similar to besov spaces, in the sense of replacing the dyadic decomposition on the fourier transform side, characteristic to besov spaces, with a uniform decomposition. from the construction of these spaces, it turns out that modulation spaces and besov spaces in some sense are rather similar, and sharp embeddings between these spaces can be found in [1, 29, 35, 37]. (see also [15, 23] for other embeddings.) during the last 15 years many results have been proved which confirm the usefulness of the modulation spaces in time-frequency analysis, where they occur naturally. for example, in [13, 17, 21], it is shown that all modulation spaces admit reconstructible sequence space representations using gabor frames. parallel to this development, modulation spaces have been incorporated into the calculus of pseudo-differential operators, in the sense of (i) the study of continuity of pseudo-differential operators with smooth symbols acting on modulation spaces, and (ii) the use of modulation spaces as symbol classes. tachizawa pioneered topic (i) in [30]. (see at the above.) in [28], sjöstrand founded topic (ii) and introduced the modulation space m∞,1, which contains non-smooth functions, as a symbol class. he proved that the symbol class m∞,1 corresponds to an algebra of operators which are bounded on l2. gröchenig and heil thereafter proved in [17, 18] that each operator with symbol in m∞,1 is continuous on all modulation spaces mp,q, p,q ∈ [1,∞]. this extends sjöstrand’s result since m 2,2 = l 2. some generalizations to operators with symbols in general unweighted modulation spaces were obtained in [19, 35], and in [36, 38, 39] some further extensions involving weighted modulation spaces are presented. modulation spaces in pseudodifferential calculus is currently an active field of research (see e. g. [18–20, 25, 31, 32]). 90 joachim toft cubo 11, 4 (2009) 2 preliminaries in this section we discuss basic properties for modulation spaces and other related spaces. the proofs are in many cases omitted since they can be found in [4–6, 11–13, 17, 33–36]. we start by recalling some properties of the involved weight functions. the positive function ω ∈ l∞loc(rd) is called v-moderate for some appropriate function v ∈ l∞loc(rd), if there is a constant c > 0 such that ω(x1 + x2) ≤ cω(x1)v(x2), x1,x2 ∈ rd. (2) if v can be chosen as polynomial, then ω is called polynomially moderate. the function v is called submultiplicative, if (2) holds for ω = v. as in [36] we let p(rd) be the set of all polynomially moderate functions on rd. we also let p0(r d ) be the set of all smooth ω ∈ p(rd) such that (∂αω)/ω is bounded for every α. note that if ω ∈ p(rd), then ω(x) + ω(x)−1 ≤ p(x), for some polynomial p on rd. in most of the applications, it is no restriction to assume that the weight functions belong to p0, which is a consequence of the following lemma. (see also [36].) lemma 2.1. assume that ω ∈ p(rd). then there is a function ω0 ∈ p(rd) and a constant c > 0 such that c−1ω ≤ ω0 ≤ cω. proof. the assertion follows by letting ω0 = ω ∗ ϕ for some 0 ≤ ϕ ∈ s (rd) \ 0. the duality between a topological vector space and its dual is denoted by 〈 · , · 〉. for admissible a and b in s ′(rd), we set (a,b) = 〈a,b〉, and it is obvious that ( · , · ) on l2 is the usual scalar product. next let v1 and v2 be vector spaces such that v1 ⊕ v2 = rd and v2 = v ⊥1 , and assume that v0 ∈ s ′(v1) and v ∈ s ′(rd) are such that v(x1,x2) = (v0 ⊗ 1)(x1,x2), when xj ∈ vj for j = 1, 2. then v(x1,x2) is identified with v0(x1), and we set v(x1,x2) = v(x1). in order to discuss modulation spaces, we recall the definition of short-time fourier transform. assume that χ ∈ s ′(rd) \ 0 and let τxχ(y) = χ(y − x) when x,y ∈ rd. then the short-time fourier transform vχf of f ∈ s ′(rd) with respect to the window function χ is the distribution in s ′(r2d), defined by the formula (vχf)(x,ξ) = f (f · τxχ)(ξ). here f denotes the fourier transform on s ′(rd), which takes the form ff(ξ) = f̂(ξ) = (2π) −d/2 ∫ f(y)e −i〈y,ξ〉 dy when f ∈ s (rd). we note that vχf is well-defined (as an element in s ′), since it is the partial fourier transform of the tempered distribution (x,y) 7→ f(y)χ(y − x) with respect to the y-variable. cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 91 (cf. [14].) if f,χ ∈ s (rd), then vχf is given by the formula (vχf)(x,ξ) = (2π) −d/2 ∫ f(y)χ(y − x)e−i〈y,ξ〉 dy. assume that χ ∈ s (rd) \ 0, p,q ∈ [1,∞] and ω ∈ p(r2d) are fixed. then the modulation space mp,q (ω) (rd) consists of all f ∈ s ′(rd) such that ‖f‖m p,q (ω) ≡ ( ∫ ( ∫ |vχf(x,ξ)ω(x,ξ)|p dx )q/p dξ )1/q < ∞ (3) (with the obvious modifications when p = ∞ and/or q = ∞). furthermore, the space w p,q (ω) (rd) consists of all f ∈ s ′(rd) such that ‖f‖w p,q (ω) ≡ ( ∫ ( ∫ |vχf(x,ξ)ω(x,ξ)|q dξ )p/q dx )1/p < ∞. (4) note that the latter space is related to certain types of wiener amalgam spaces. (cf. definition 4 in [13].) we recall that w p,q (ω) = fm q,p (ω0) when ω0(x,ξ) = ω(−ξ,x) ∈ p(r2d). in fact, let χ̌(x) = χ(−x) as usual. then parseval’s formula and a change of the order of integration shows that |f −1(f̂ τξχ̂)(x)| = |f (f τxχ̌)(ξ)|. (5) hence, by applying the l q,p (ω) norm, the assertion follows. the convention of indicating weight functions with parenthesis is used also in other situations. for example, if ω ∈ p(rd), then lp (ω) (rd) is the set of all measurable functions f on rd such that fω ∈ lp(rd), i. e. such that ‖f‖lp (ω) ≡ ‖fω‖lp is finite. the following proposition is a consequence of well-known facts in [6, 17]. here and in what follows, we let p′ denotes the conjugate exponent of p, i. e. 1/p + 1/p′ = 1. proposition 2.2. assume that p,q,pj,qj ∈ [1,∞] for j = 1, 2, ω,ω1,ω2,v ∈ p(r2d) are such that ω is v-moderate, χ ∈ m1 (v) (rd) \ 0, and let f ∈ s ′(rd). then the following is true: 1. f ∈ mp,q (ω) (rd) if and only if (3) holds, i. e. mp,q (ω) (rd) is independent of the choice of χ. moreover, mp,q (ω) is a banach space under the norm in (3), and different choices of χ give rise to equivalent norms; 2. f ∈ w p,q (ω) (rd) if and only if (4) holds, i. e. w p,q (ω) (rd) is independent of the choice of χ. moreover, w p,q (ω) is a banach space under the norm in (4), and different choices of χ give rise to equivalent norms; 3. if p1 ≤ p2, q1 ≤ q2 and ω2 ≤ cω1 for some constant c, then s (r d ) ⊆mp1,q1 (ω1) (r d ) ⊆mp2,q2 (ω2) (r d ) ⊆ s ′(rd), s (r d ) ⊆w p1,q1 (ω1) (r d ) ⊆w p2,q2 (ω2) (r d ) ⊆ s ′(rd). 92 joachim toft cubo 11, 4 (2009) proposition 2.2 permits us to be rather vague about to the choice of χ ∈ m1 (v) \ 0 in (3) and (4). for example, if c > 0 is a constant and ω is a subset of s ′, then ‖a‖m p,q (ω) ≤ c for every a ∈ ω, means that the inequality holds for some choice of χ ∈ m1 (v) \ 0 and every a ∈ ω. evidently, for any other choice of χ ∈ m1 (v) \ 0, a similar inequality is true although c may have to be replaced by a larger constant, if necessary. next we discuss weight functions which are common in the applications. for any s,t ∈ r, set σt(x) = 〈x〉t, σs,t(x,ξ)〈ξ〉s〈x〉t, (6) when x,ξ ∈ rd. then it follows that σt ∈ p0(rd) and σs,t ∈ p0(r2d) for every s,t ∈ r, and σt is σ|t|-moderate and σs,t is σ|s|,|t|-moderate. obviously, σs(x,ξ) = (1 + |x|2 + |ξ|2)s/2, and σs,t = σt ⊗ σs. moreover, if ω ∈ p(rd), then ω is σt-moderate provided t is chosen large enough. for conveniency we use the notations lps , m p,q s and m p,q s,t instead of l p (σs) , m p,q (σs) and m p,q (σs,t) respectively. remark 2.3. assume that p,q,q1,q2 ∈ [1,∞] and ω ∈ p(r2d). then the following properties for modulation spaces hold: 1. if q1 ≤ min(p,p′) and q2 ≥ max(p,p′) and ω(x,ξ) = ω(x), then m p,q1 (ω) ⊆ lp (ω) ⊆ mp,q2 (ω) , w p,q1 (ω) ⊆ lp (ω) ⊆ w p,q2 (ω) ; 2. s0 0 = ⋂ s∈r m ∞,1 s,0 ; 3. if p ≥ q, then mp,q (ω) ⊆ w p,q (ω) . if instead q ≥ p, then w p,q (ω) ⊆ mp,q (ω) ; 4. m1,∞(rd) and w 1,∞(rd) are convolution algebras such that if m(rd) is the set of all measures on rd with bounded mass, then m ⊆ w 1,∞ ⊆ m1,∞; 5. if ω is a subset of p(r2d) such that for any polynomial p on r2d, there is an element ω ∈ ω such that p/ω is bounded, then s (r d ) = ⋂ ω∈ω m p,q (ω) (r d ), s ′ (r d ) = ⋃ ω∈ω m p,q (1/ω) (r d ); 6. if s,t ∈ r are such that t ≥ 0, then m 2 s,0 = h 2 s , m 2 0,s = l 2 s, and m 2 t = l 2 t ∩ h2t . (see e. g. [4–6, 10–13, 17, 35, 36].) cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 93 we refer to [6, 11–13, 17, 27] for more facts about modulation spaces and w p,q (ω) -spaces. as anounced in the introduction we consider in section 3 mapping properties for pseudodifferential operators when acting on certain types of coorbit spaces, which are defined by imposing certain types of translation invariant solid bf-space norms on the short-time fourier transforms. (cf. [6, 8, 11, 12].) this familly of coorbit spaces contains the modulation and wiener amalgam spaces. in the following we recall the definition of these spaces. definition 2.4. assume that b is a banach space of complex-valued measurable functions on rd and v ∈ p(rd). then b is called a translation invariant bf-space on rd (with respect to v), if there is a constant c such that the following conditions are fulfilled: 1. s (rd) ⊆ b ⊆ s ′(rd) (continuous embeddings); 2. if x ∈ rd and f ∈ b, then τxf ∈ b, and ‖τxf‖b ≤ cv(x)‖f‖b; (7) 3. if f,g ∈ l1loc(rd) satisfy g ∈ b and |f| ≤ |g|, then f ∈ b and ‖f‖b ≤ c‖g‖b. here the condition (3) in definition 2.4 means that a translation invariant bf-space is a solid bf-space in the sense of (a.3) in [8]. it follows from this condition that if f ∈ b and h ∈ l∞, then f · h ∈ b, and ‖f · h‖b ≤ c‖f‖b‖h‖l∞. (8) example 2.5. assume that p,q ∈ [1,∞], and let lp,q 1 (r2d) be the set of all f ∈ l1loc(r2d) such that ‖f‖lp,q1 ≡ ( ∫ ( ∫ |f(x,ξ)|p dx )q/p dξ )1/q if finite. also let l p,q 2 (r2d) be the set of all f ∈ l1loc(r2d) such that ‖f‖lp,q2 ≡ ( ∫ ( ∫ |f(x,ξ)|q dξ )p/q dx )1/p is finite. then it follows that l p,q 1 and l p,q 2 are translation invariant bf-spaces with respect to v = 1. more generally, assume that ω,v ∈ p(r2d) are such that ω is v-moderate, and let lp,q j,(ω) (r2d), for j = 1, 2, be the set of all f ∈ l1loc(r2d) such that ‖f‖lp,qj,(ω) ≡ ‖f ω‖lp,qj is finite. then l p,q j,(ω) is a translation invariant bf-space with respect to v. remark 2.6. the conclusion in the latter part of example 2.5 is also a consequence of the first part in that example and the following observation. assume that ω0,v,v0 ∈ p(rd) are such that ω is v-moderate, and assume that b is a translation invariant bf-space on rd with respect to v0. also let b0 be the banach space which consists of all f ∈ l1loc(rd) such that ‖f‖b0 ≡ ‖f ω‖b is finite. then b0 is a translation invariant bf-space with respect to v0v. 94 joachim toft cubo 11, 4 (2009) for translation invariant bf-spaces we make the following observation. proposition 2.7. assume that v ∈ p(rd), and that b is a translation invariant bf-space with respect to v. then the convolution mapping (ϕ,f) 7→ ϕ ∗ f from c∞ 0 (rd) × b to s ′ extends uniquely to a continuous mapping from l1 (v) (rd) × b to b, and ‖ϕ ∗ f‖b ≤ c‖ϕ‖l1 (v) ‖f‖b, for some constant c which is independent of ϕ ∈ l1 (v) and f ∈ b. proposition 2.9 is a consequence of the results in [6, 8]. in order to be more self-contained we give here a short motivation. proof. first assume that ϕ ∈ c∞ 0 and that f ∈ b. then minkowski’s inequality and (8) give ‖ϕ ∗ f‖b = ∥∥∥ ∫ f( · − y)ϕ(y) dy ∥∥∥ b ≤ ∫ ‖f( · − y)ϕ(y)‖b dy = ∫ ‖f( · − y)‖b|ϕ(y)|dy ≤ c ∫ ‖f‖b v(y)|ϕ(y)|dy = c‖f‖b‖ϕ‖l1 (v) , which proves the result in this case. for general ϕ ∈ l1 (v) , the result follows from the fact that c0 is dense in l1 (v) . next we consider the general type of modulation spaces which we are especially interested in. definition 2.8. assume that b is a translation invariant bf-space on r2d, ω ∈ p(r2d), and that χ ∈ s (rd) \ 0. then the modulation space m(ω) = m(ω)(b) consists of all f ∈ s ′(rd) such that ‖f‖m(ω) = ‖f‖m(ω)(b) ≡ ‖vχf ω‖b is finite. assume that ω ∈ p(r2d) is fix, and consider the familly of distribution spaces which consists of all spaces of the form m(ω)(b) such that b is a translation invariant bf-space on r 2d. then it follows by remark 2.6 that this familly is invariant under ω. consequently we do not increase the set of possible spaces in definition 2.8 by permitting ω that are not identically 1. from this observation it seems to be superfluous to include the weight ω in definition 2.8. however, it will be convenient for us to permit such ω dependency when investigating mapping properties for pseudo-differential operators in section 3, when acting on spaces of the form m(ω)(b). obviously, we have m p,q (ω) (r d ) = m(ω)(b1) and w p,q (ω) (r d ) = m(ω)(b2) cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 95 when b1 = l p,q 1 (r2d) and b2 = l p,q 2 (r2d) (cf. example 2.5). it follows that many properties which are valid for the modulation spaces also hold for the spaces of the form m(ω)(b). for example we have the following proposition, which shows that the definition of m(ω)(b) is independent of the choice of χ. we omit the proof since it can be found in e. g. [8,11,12]. it also follows by similar arguments as in the proof of proposition 11.3.2 in [17]. proposition 2.9. assume that b is a translation invariant bf-space with respect to v0 ∈ p(r2d) for j = 1, 2. also assume that ω,v ∈ p(r2d) are such that ω is v-moderate, m(ω)(b) is the same as in definition 2.8, and let χ ∈ m1 (v0v) (rd) \ 0 and f ∈ s ′(rd). then f ∈ m(ω)(b) if and only if vχf ω ∈ b, and different choices of χ gives rise to equivalent norms in m(ω)(b). next we recall some facts in chapter xviii in [22] concerning pseudo-differential operators. assume that t ∈ r is fixed and that a ∈ s (r2d). then the pseudo-differential operator at(x,d) is the continuous operator on s (rd), defined by the formula (at(x,d)f)(x) = (opt(a)f)(x) = (2π) −d ∫ ∫ a((1 − t)x + ty,ξ)f(y)ei〈x−y,ξ〉 dydξ. (9) the definition of at(x,d) extends to any a ∈ s ′(r2d), and then at(x,d) is continuous from s (rd) to s ′(rd). moreover, for every fixed t ∈ r, it follows that there is a one to one correspondance between such operators, and pseudo-differential operators of the form at(x,d). (see e. g. [22].) if t = 1/2, then at(x,d) is equal to the weyl operator a w (x,d) for a. if instead t = 0, then the standard (kohn-nirenberg) representation a(x,d) is obtained. consequently, for every a ∈ s ′(r2d) and s,t ∈ r, there is a unique b ∈ s ′(r2d) such that as(x,d) = bt(x,d). by straight-forward applications of fourier’s inversion formula, it follows that as(x,d) = bt(x,d) ⇐⇒ b(x,ξ) = ei(t−s)〈dx,dξ〉a(x,ξ). (10) (cf. [22].) in the next section we discuss continuity for pseudo-differential operators with symbols in s(ω)(r 2d ), the set of all smooth functions a on r2d such that ∂αa/ω ∈ l∞(r2d). here ω ∈ p(r2d). if ω = 1, then we use the notation s0 0 (r2d) instead of s(ω)(r 2d ). 3 continuity for pseudo-differential operators with symbols in s(ω) in this section we discuss continuity for operators in op(s(ω0)) when acting on modulation spaces. in the first part we prove in theorem 3.2 below that if ω,ω0 ∈ p, t ∈ r and a ∈ s(ω), then at(x,d) is continuous from m(ω0ω)(b) to m(ω0)(b). in particular, theorem 2.1 in [30] as well as theorem 2.2 in [36] are covered. 96 joachim toft cubo 11, 4 (2009) in the second part we present some applications and prove that certain properties which are valid for sobolev spaces carry over to modulation spaces. we start by giving some remarks on s(ω)(r 2d ) when ω ∈ p(r2d). by straight-forward computations it follows that s(ω)(r 2d ) agrees with s(ω,g) when g(x,ξ)(y,η) = |y|2 + |η|2 is the standard euclidean metric on r2d. (see section 18.4–18.6 in [22].) since the metric g is constant it follows that it is trivially slowly varying and σ-temperate, where σ denotes the standard symplectic form on r2d. moreover, from the fact that ω is σt-moderate when t is large enough, it follows by straight-forward computations that ω is σ,g-temperate. the following lemma is therefore a consequence of theorem 18.5.10 in [22]. lemma 3.1. assume that ω ∈ p(r2d), s,t ∈ r, and that a,b ∈ s ′(r2d) are such that as(x,d) = bt(x,d). then a ∈ s(ω)(r2d) ⇐⇒ b ∈ s(ω)(r2d). we have now the following result. theorem 3.2. assume that t ∈ r, ω,ω0 ∈ p(r2d), a ∈ s(ω)(r2d), t ∈ r, and that b is a translation invariant bf-space on r2d. then at(x,d) is continuous from m(ω0ω)(b) to m(ω0)(b). we need some preparations for the proof, and start by recalling minkowski’s inequality in a somewhat general form. assume that dµ is a positive measure, and that f ∈ l1(dµ; b) for some banach space b. then minkowski’s inequality asserts that ∥∥∥ ∫ f(x) dµ(x) ∥∥∥ b ≤ ∫ ‖f(x)‖b dµ(x). we also need some lemmas. lemma 3.3. assume that ω ∈ p(r2d), a ∈ s(ω)(r2d), f ∈ s (rd), χ ∈ s (rd), χ2 = σsχ and 0 ≤ s ∈ r. if φ(x,ξ,z,ζ) = a(x + z,ξ + ζ) ω(x,ξ)〈z〉s〈ζ〉s (11) and h(x,ξ,y) = ∫ ∫ φ(x,ξ,z,ζ)χ2(z)〈ζ〉sei〈y−x−z,ζ〉 dzdζ, then vχ(a(·,d)f)(x,ξ) = (2π)−d(f,ei〈 · ,ξ〉h(x,ξ, · ))ω(x,ξ). (12) proof. for simplicity we assume that a is real-valued. by straight-forward computations we get vχ(a(·,d)f)(x,ξ) = (a(·,d)f,τxχei〈·,ξ〉) = (f,a(·,d)∗(τxχei〈·,ξ〉)) = (2π) −d (f,e i〈 · ,ξ〉 h̃(x,ξ, · ))ω(x,ξ), (13) cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 97 where h̃(x,ξ,y) = (2π) d e −i〈y,ξ〉 (a(·,d)∗(τxχei〈·,ξ〉))(y)/ω(x,ξ) = ∫ ∫ a(z,ζ) ω(x,ξ) χ(z − x)ei〈y−z,ζ−ξ〉 dzdζ = ∫ ∫ φ(x,ξ,z − x,ζ − ξ)χ2(z − x)〈ζ − ξ〉sei〈y−z,ζ−ξ〉 dzdζ. if z − x and ζ − ξ are taken as new variables of integrations, it follows that the right-hand side is equal to h(x,y,ξ). this proves the assertion. lemma 3.4. let s ≥ 0 be an even integer, φ and h be the same as in lemma 3.3, and set φβ (x,ξ,z,ζ) = ∂ β z φ(x,ξ,z,ζ), χ2,γ = ∂ γ χ2. (14) also let ψβ (x,ξ,y, · ) be the inverse partial fourier transform of φβ (x,ξ,y,η) with respect to the η variable, and let hβ,γ (x,ξ,y) = ∫ ψβ (x,ξ,y − z − x,z)χ2,γ (y − z − x) dz. (15) then there are constants cβ,γ which depend on β, γ, s and d only such that h(x,ξ,y) = ∑ |β+γ|≤s cβ,γhβ,γ (x,ξ,y). proof. by integrating by parts we get h(x,ξ,y) = ∫ ∫ φ(x,ξ,z,ζ)χ2(z)〈ζ〉s/2ei〈y−x−z,ζ〉 dzdζ = ∫ ∫ φ(x,ξ,z,ζ)χ2(z)(1 − ∆z)s/2(ei〈y−x−z,ζ〉) dzdζ = ∑ |β+γ|≤n cβ,γh̃β,γ (x,ξ,y), where h̃β,γ (x,ξ,y) = (2π) −d/2 ∫ ∫ φβ (x,ξ,z,ζ)χ2,γ (z)e i〈y−x−z,ζ〉 dzdζ. if we take y−x−z and ζ as new variables of integrations, and perform the integration with respect to the ζ variable, it follows that h̃β,γ = hβ,γ , which gives the result. for the next lemma we observe that if f ∈ s ′(rd) is fixed, then there are positive constants s0, n and c0 such that |vχ0f(x,ξ)| ≤ c0〈x,ξ〉n, where χ0 = σ−s and s ≥ s0. (16) 98 joachim toft cubo 11, 4 (2009) lemma 3.5. assume that ω ∈ p(r2d), a ∈ s(ω)(r2d), ϕ ∈ s (rd), and f ∈ s ′(rd). if s ≥ 0 is large enough and χ0 = σ−s, then there is a constant c such that |vϕ(at( · ,d)f)(x,ξ)| ≤ c(f(x, · ) ∗ χ0)(ξ), (17) where f(x,ξ) = |vχ0f(x,ξ)ω(x,ξ)|. (18) proof. it is no restriction to assume that a is real-valued, and by lemmas 2.1 and 3.1 it follows that we may assume that t = 0 and that ω ∈ p0. furthermore, by lemma 3.3, lemma 3.4 and (13), the result follows if we prove the following: 1. the right-hand side of (12) is well-defined for the fixed f ∈ s ′(rd) when s is chosen large enough, and that (12) holds also in this case; 2. for each multi-indices β and γ, there is a constant c such that iβ,γ (x,ξ) ≡ |(f,ei〈 · ,ξ〉hβ,γ (x,ξ, · ))ω(x,ξ)| ≤ c(f(x, · ) ∗ σ−s)(ξ). (9) let c0, s0 and n be chosen such that (16) is fulfilled, let n1 be an even and large integer, and let φβ be as in (14). the assertion (1) follows if we prove that for each multi-indices α and β, there is a constant cα,β = cn1,α,β such that |(∂αφβ )(x,ξ,z,ζ)| ≤ cα,β〈z〉−n1〈ζ〉−n1. (19) in order to prove (19) we choose m ≥ 0 and s ≥ m + n1 such that ω ∈ p0 is σm -moderate, and assume first that α = β = 0. then (11) and the facts that a ∈ s(ω) give |φ(x,ξ,z,ζ)| = |a(x + z,ξ + ζ)| ω(x,ξ)〈z〉s〈ζ〉s ≤ c1 |a(x + z,ξ + ζ)|〈z,ζ〉m ω(x + z,ξ + ζ)〈z〉s〈ζ〉s ≤ c2〈z〉 −n1〈ζ〉−n1. for general α and β, (19) follows from these computations in combination with leibnitz rule, using the facts that (∂γa)/ω ∈ l∞ and (∂γω)/ω ∈ l∞ for each multi-index γ. this gives (1). assume that n2 ≥ 0 is arbitrary. then it follows by choosing n1 in (19) large enough, that for some constant c it holds |∂αψβ (x,ξ,y − z,z)| ≤ c〈y〉−n2〈z〉−n2 (20) for every multi-index α such that |α| ≤ n2. if n3 is a fixed integer, then it follows from (15) and (20) that hβ,γ (x,ξ,y) = σ−n3 (y − x)ϕβ,γ (x,ξ,y − x), (21) where ϕβ,γ satisfies |∂αϕβ,γ (x,ξ,y)| ≤ c〈y〉−n3, |α| ≤ n3, cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 99 for some constant c, provided n2 was chosen large enough. hence, for any fixed s ≥ 0, it follows by choosing n3 large enough that |f (ϕβ,γ (x,ξ, · ))(η)| ≤ c〈η〉−s, (22) for some constant c. by choosing s > d, it follows from (21), (22) and straight-forward computations that iβ,γ (x,ξ) = |(f,ei〈·,ξ〉χ0(· − x)ϕβ,γ (x,ξ, · − x))| = |f ((f τxχ0)ϕβ,γ (x,ξ, · − x))(ξ)| ≤ (2π)−d/2 ∫ |f (f τxχ0)(ξ − η)||f (ϕβ,γ (x,ξ, · − x))(η)|dη, ≤ c ∫ |vχ0f(x,ξ − η)|χ0(η) dη, (23) where c = (2π) −d/2 ∫ sup x,ξ | ( f (ϕβ,γ (x,ξ, · − x))(η)| ) dη = (2π) −d/2 ∫ sup x,ξ | ( f (ϕβ,γ (x,ξ, · ))(η)| ) dη ≤ c1 ∫ 〈η〉−s dη < ∞. this gives (17), and the proof is complete. proof of theorem 3.2. we use the same notations as in lemma 3.5, and set g = |vχ(at( · ,d)f)|. since ω0 ∈ p, it follows that ω0(x,ξ) ≤ cω0(x,ξ − η)〈η〉s0 , for some constants c and s0. by lemma 3.5 we get g(x,ξ)ω0(x,ξ) ≤ c1 ∫ f(x,ξ − η)〈η〉−sω0(x,ξ) dη ≤ c2 ∫ f(x,ξ − η)ω0(x,ξ − η)〈η〉s0−s dη, = c2 ∫ fη,ω0 (x,ξ)〈η〉s0−s dη, for some constants c1 and c2, where fη,ω0 (x,ξ) = f(x,ξ − η)ω0(x,ξ − η). 100 joachim toft cubo 11, 4 (2009) now choose s1,s2 ∈ r in such way that s1 = s−s0 and b is a translation invariant bf-space with respect to σs2 , and let ω1 = ω0ω. then it follows for some constant c and minkowki’s inequality that ‖at(x,d)f‖m(ω0)(b) = ‖gω0‖b ≤ c1 ∫ ‖fη,ω0‖b〈η〉−s1 dη ≤ c2 ∫ ‖f ω0‖b〈η〉s2−s1 dη = c3‖f‖m(ω0ω)(b), where c3 = c2‖σs2−s1‖l1. since s can be chosen arbitrary large, it follows that s1 can be chosen larger than s2 + d, which implies that c3 < ∞. this gives the result. next we show that [36, theorem 2.2] is essentially a consequence of theorem 3.2. corollary 3.6. assume that t ∈ r, ω ∈ p0(r2d), ω0 ∈ p(r2d) are such that ω(x,ξ) = ω(x) or ω(x,ξ) = ω(ξ), and that b is a translation invariant bf-space on r2d. then ωt(x,d) is a homeomorphism from m(ω0ω)(b) to m(ω0)(b). proof. since it follows from the assumptions that ω ∈ s(ω), theorem 3.2 shows that ωt(x,d) is continuous from m(ω0ω)(b) to m p,q (ω) (b). on the other hand, since ω(x,ξ) = ω(x) or ω(x,ξ) = ω(ξ), it follows that the inverse of ωt(x,d) on s ′ (rd) is equal to (1/ω)t(x,d). hence theorem 3.2 together with the obvious fact that 1/ω ∈ p0 give ‖f‖m(ω0ω)(b) = ‖(1/ω)t(x,d)(ωt(x,d)f)‖m(ω0ω)(b) ≤ c‖ωt(x,d)f‖m(ω0)(b) for some constant c. this proves that ωt(x,d) is a bijective map from m(ω0ω)(b) to m(ω0)(b), and the result follows. remark 3.7. we remark that an immediate consequence of corollary 3.6 is that if b is a translation invariant bf-space on r2d, ω(x,ξ) = ω1(x)ω2(ξ) where ωj ∈ p0(rd) for j = 1, 2, and ω0 ∈ p(r2d), then m(ω0ω)(b) = {f ∈ s ′(rd) ; ω1(x)ω2(d)f ∈ m(ω0)(b) } = {f ∈ s ′(rd) ; ω2(d)(ω1f) ∈ m(ω0)(b) }. in particular, if s,t ∈ r, b = lp,q 1 or b = l p,q 2 , and ω(x,ξ) = σs,t(x,ξ) = 〈x〉t〈ξ〉s, then m p,q (σs,tω0) (r d ) = {f ∈ s ′(rd) ; 〈x〉t〈d〉sf ∈ mp,q (ω0) (r d ) } = {f ∈ s ′(rd) ; 〈d〉s(〈 · 〉tf) ∈ mp,q (ω0) (r d ) } cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 101 and w p,q (σs,tω0) (r d ) = {f ∈ s ′(rd) ; 〈x〉t〈d〉sf ∈ w p,q (ω0) (r d ) } = {f ∈ s ′(rd) ; 〈d〉s(〈 · 〉tf) ∈ w p,q (ω0) (r d ) }. remark 3.8. for certain ω it is possible to use remark 2.12 in [36] to prove that the continuity assertions in theorem 3.2 also holds when the symbols for the pseudo-differential operators belong to m ∞,1 (ω) (r2d). note that σs,t(x,d) here above, appears frequently in harmonic analysis and in the pseudodifferential calculus. for example, if p ∈ [1,∞], then recall that f ∈ s ′(rd) belongs to the sobolev space hps (r d ) if and only if ‖f‖hps ≡ ‖σs(d)f‖lp is finite. it is well-known that if s = n is a positive integer and 1 < p < ∞, then hps agrees with {f ∈ lp ; ∂αf ∈ lp when |α| ≤ n }. (see [2].) in the following theorems we prove that similar properties in a somewhat extended form also hold for general spaces of the form m(ω)(b). theorem 3.9. assume that n1,n2 ≥ 0 are integers, ω ∈ p(r2d), b is a translation invariant bf-spaces on r2d, and assume that f ∈ s ′(rd). then the following conditions are equivalent: 1. f ∈ m(σn1,n2 ω)(b); 2. xβ∂αf ∈ m(ω)(b) for all multi-indices α and β such that |α| ≤ n1 and |β| ≤ n2; 3. ∂α(xβf) ∈ m(ω)(b) for all multi-indices α and β such that |α| ≤ n1 and |β| ≤ n2; 4. f,xn2j f, ∂ n1 k f, x n2 j ∂ n1 k f ∈ m(ω)(b) for all 1 ≤ j,k ≤ d; 5. f,xn2j f, ∂ n1 k f, ∂ n1 k (x n2 j f) ∈ m(ω)(b) for all 1 ≤ j,k ≤ d. proof. we only prove the equivalences (1) ⇐⇒ (2) ⇐⇒ (4). the equivalences (1) ⇐⇒ (3) ⇐⇒ (5) follow by similar arguments and are left for the reader. 102 joachim toft cubo 11, 4 (2009) let m0 be the set of all f ∈ m(ω)(b) such that xβ∂αf ∈ m(ω)(b) when |α| ≤ n1 and |β| ≤ n2, and let m̃0 be the set of all f ∈ m(ω)(b) such that x n2 j f,∂ n1 k f,x n2 j ∂ n1 k f ∈ m(ω)(b) for j,k = 1, . . . ,d. we shall prove that m0 = m̃0 = m(σn1,n2 ω)(b). obviously, m0 ⊆ m̃0. since the symbol ξα of the operator dα belongs to s(σn1,n2 ) when |α| ≤ n, it follows from theorem 3.2 that the embedding m(σn1,n2 ω)(b) ⊆ m0 holds. the result therefore follows if we prove that m̃0 ⊆ m(σn1,n2 ω)(b). in order to prove this, assume first that n1 = n, n2 = 0, f ∈ m̃0, and choose open sets ω0 = {ξ ∈ rd ; |ξ| < 2 }, and ωj = {ξ ∈ rd ; 1 < |ξ| < d|ξj| }. then ⋃d j=0 ωj = r d, and there are non-negative functions ϕ0, . . . ,ϕd in s 0 0 such that supp ϕj ⊆ ωj and ∑d j=0 ϕj = 1. in particular, f = ∑d j=0 fj when fj = ϕj (d)f. the result follows if we prove that fj ∈ m(σn,0ω)(b) for every j. now set ψ0(ξ) = σn (ξ)ϕ0(ξ) and ψj (ξ) = ξ −n j σn (ξ)ϕj (ξ) when j = 1, . . . ,d. then ψj ∈ s00 for every j. hence theorem 3.2 gives ‖fj‖m(σn,0ω)(b) ≤ c1‖σn (d)fj‖m(ω)(b) = c1‖ψj (d)∂nj f‖m(ω)(b) ≤ c2‖∂ n j f‖m(ω)(b) < ∞ and ‖f0‖m(σn,0ω)(b) ≤ c1‖σn (d)f0‖m(ω)(b) = c1‖ψ0(d)f‖m(ω)(b) ≤ c2‖f‖m(ω)(b) < ∞ for some constants c1 and c2. this proves that ‖f‖m(σn,0ω)(b) ≤ c ( ‖f‖m(ω)(b) + d∑ j=1 ‖∂nj f‖m(ω)(b) ) , (24) and the result follows in this case. if we instead split up f into ∑ ϕjf, then similar arguments show that ‖f‖m(σ0,n ω)(b) ≤ c ( ‖f‖m(ω)(b) + d∑ k=1 ‖xnk f‖m(ω)(b) ) , (25) and the result follows in the case n1 = 0 and n2 = n from this estimate. the general case now follows if we combine (24) with (25). the proof is complete. cubo 11, 4 (2009) pseudo-differential operators with smooth symbols ... 103 we finish the section by stating the previous results in the special cases of modulation spaces and corresponding wiener amalgam related spaces. in fact, by letting b = l p,q 1 or b = l p,q 2 , the following results are immediate consequences of the previous ones. theorem 3.2′. assume that ω,ω0 ∈ p(r2d), a ∈ s(ω)(r2d), t ∈ r, and that p,q ∈ [1,∞]. then at(x,d) is continuous from m p,q (ω0ω) (rd) to mp,q (ω0) (rd), and from w p,q (ω0ω) (rd) to w p,q (ω0) (rd). we note that if t = 0 and ω0 = σs1,s2 where s1,s2 ∈ r, then theorem 3.2′ agrees with theorem 1.1 in [30]. corollary 3.6′. assume that ω ∈ p0(r2d), ω0 ∈ p(r2d) are such that ω(x,ξ) = ω(x) or ω(x,ξ) = ω(ξ), and that p,q ∈ [1,∞]. then ωt(x,d) is a homeomorphism from mp,q(ω0ω)(r d ) to m p,q (ω0) (rd), and from w p,q (ω0ω) (rd) to w p,q (ω0) (rd). theorem 3.9′. assume that n1,n2 ≥ 0 are integers, ω ∈ p(r2d), p,q ∈ [1,∞], and that f ∈ s ′(rd). then the following conditions are equivalent: 1. f ∈ mp,q (σn1,n2 ω) (rd); 2. xβ∂αf ∈ mp,q (ω) (rd) for all multi-indices α and β such that |α| ≤ n1 and |β| ≤ n2; 3. ∂α(xβf) ∈ mp,q (ω) (rd) for all multi-indices α and β such that |α| ≤ n1 and |β| ≤ n2; 4. f,xn2j f, ∂ n1 k f, x n2 j ∂ n1 k f ∈ m p,q (ω) (rd) for all 1 ≤ j,k ≤ d; 5. f,xn2j f, ∂ n1 k f, ∂ n1 k (x n2 j f) ∈ m p,q (ω) (rd) for all 1 ≤ j,k ≤ d. theorem 3.9′′. assume that n1,n2 ≥ 0 are integers, ω ∈ p(r2d), p,q ∈ [1,∞], and that f ∈ s ′(rd). then the following conditions are equivalent: 1. f ∈ w p,q (σn1,n2 ω) (rd); 2. xβ∂αf ∈ w p,q (ω) (rd) for all multi-indices α and β such that |α| ≤ n1 and |β| ≤ n2; 3. ∂α(xβf) ∈ w p,q (ω) (rd) for all multi-indices α and β such that |α| ≤ n1 and |β| ≤ n2; 4. f,xn2j f, ∂ n1 k f, x n2 j ∂ n1 k f ∈ w p,q (ω) (rd) for all 1 ≤ j,k ≤ d; 5. f,xn2j f, ∂ n1 k f, ∂ n1 k (x n2 j f) ∈ w p,q (ω) (rd) for all 1 ≤ j,k ≤ d. the following result was presented in [39, remark 1.3]. since the facts here do not seems to be well-known, we give some explicit motivations. 104 joachim toft cubo 11, 4 (2009) corollary 3.10. assume that p,q ∈ [1,∞] and ω ∈ p(r2d) is such that ω(x,ξ) = ω(x). then the following is true: 1. mp,q (ω) (rd) →֒ c(rd) if and only if q = 1; 2. w p,q (ω) (rd) →֒ c(rd) if and only if q = 1. proof. by corollary 3.6′ it follows that we may assume that ω = 1. if f ∈ w∞,1, then it follows that f (fϕ) ∈ l1 for every ϕ ∈ s , which implies that fϕ is a continuous function. since ϕ ∈ s is arbitrary chosen, it follows that f is continuous. this gives m p,1 ⊆ w p,1 ⊆ w∞,1 ⊆ c, (26) which proves one part of the assertion. next assume that q > 1, and let f be the characteristic function of the cube [0, 1]d. then f /∈ c, and it follows by straight-forward computations that f ∈ w 1,q ⊆ m1,q. since mp,q and w p,q increases with the parameters p and q, it follows that m p,q * c, and w p,q * c, when q > 1. (27) hence (26) and (27) give the result. remark 3.11. by using techniques of ultra-distributions, pilipović and teofanov prove in [24– 26, 31, 32] parallel results comparing to theorem 3.2′. here they consider generalized modulation spaces, where less growth restrictions are assumed on the weight function ω. it is for example not necessary that ω should be bounded by polynomials. received: may 2008. revised: september 2008. references [1] baoxiang, w. and chunyan, h., frequency-uniform decomposition method for the generalized bo, kdv and nls equations, j. differential equations, 239 (2007), 213–250. [2] bergh, j. and löfström, j., interpolation spaces, an introduction, springer-verlag, berlin heidelberg newyork, 1976. 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[35] toft, j., continuity properties for modulation spaces with applications to pseudo-differential calculus, i, j. funct. anal. (2), 207 (2004), 399–429. [36] toft, j., continuity properties for modulation spaces with applications to pseudo-differential calculus, ii, ann. global anal. geom., 26 (2004), 73–106. [37] toft, j., convolution and embeddings for weighted modulation spaces in: p. boggiatto, r. ashino, m. w. wong (eds) advances in pseudo-differential operators, operator theory: advances and applications 155, birkhäuser verlag, basel 2004, pp. 165–186. [38] toft, j., continuity and schatten-von neumann properties for pseudo-differential operators and toeplitz operators on modulation spaces, the erwin schrödinger international institute for mathematical physics, preprint esi 1732 (2005). [39] toft, j., continuity and schatten properties for pseudo-differential operators on modulation spaces in: j. toft, m. w. wong, h. zhu (eds) modern trends in pseudo-differential operators, operator theory: advances and applications 172, birkhäuser verlag, basel, 2007, pp. 173–206. articulo 7 cubo a mathematical journal vol.10, n o ¯ 04, (109–117). december 2008 positive solutions for elliptic boundary value problems with a harnack-like property toufik moussaoui department of mathematics, e.n.s., p.o. box 92, 16050 kouba, algiers, algeria email: moussaoui@ens-kouba.dz and radu precup department of applied mathematics, babeş–bolyai university, 400084 cluj, romania email: r.precup@math.ubbcluj.ro abstract the aim of this paper is to present some existence results of positive solutions for elliptic equations and systems on bounded domains of r n (n ≥ 1). the main tool is krasnosel’skii’s compression-expansion fixed point theorem. resumen el objetivo de este art́ıculo es presentar algunos resultados de existencia de soluciones positivas para ecuaciones elipticas y sistemas sobre dominios acotados de r n (n ≥ 1). la principal herramienta es el teorema de punto fijo compresión-expansión de krasnosel’skii. 110 toufik moussaoui and radu precup cubo 10, 4 (2008) key words and phrases: positive solution, elliptic boundary value problem, elliptic systems, harnack-like inequality, krasnosel’skii’s compression-expansion fixed point theorem. math. subj. class.: 47h10, 35j65. 1 introduction in this paper, we are concerned with the existence of positive solutions for the elliptic boundary value problem { −∆u = λ f (x, u) , in ω, u = 0, on ∂ω, (1.1) and for the elliptic system        −∆u = α g (x, u, v) , in ω, −∆v = β h (x, u, v) , in ω, u = v = 0, on ∂ω. (1.2) here ω is a bounded regular domain of r n (n ≥ 1), f : ω×r+ −→ r+ and g, h : ω×r 2 + −→ r+ are continuous functions, and λ, α and β are real parameters. by a positive solution of problem (1.1) we mean a function u ∈ c1 ( ω, r ) which satisfies (1.1) (with ∆u in the sense of distributions), and with u (x) > 0 for all x ∈ ω. a positive solution to problem (1.2) is a vector-valued function (u, v) ∈ c1 ( ω, r2 ) satisfying (1.2), with u, v ≥ 0 and u + v > 0 in ω. the main assumption will be a global weak harnack inequality for nonnegative superharmonic functions. by a superharmonic function in a domain ω ⊂ rn we mean a function u ∈ c1(ω, r) with ∆u ≤ 0 in the sense of distributions, i.e., ∫ ω ∇u · ∇v ≥ 0 for every v ∈ c∞ 0 (ω, r) satisfying v(x) ≥ 0 on ω. we shall assume that the following global weak harnack inequality holds:            there exists a compact set k ⊂ ω and a number η > 0 such that u(x) ≥ η‖u‖0 for all x ∈ k and every nonnegative superharmonic function u ∈ c1(ω, r) with u = 0 on ∂ω. (1.3) here by ‖u‖ 0 we denote the sup norm in c ( ω, r ) , i.e., ‖u‖0 = sup x∈ω |u(x)|. the connection between such type of inequalities and krasnosel’skii’s compression-expansion theorem when applied to boundary value problems was first explained in [4]. also in [4] (see also [1]), several comments on weak harnack type inequalities can be found. cubo 10, 4 (2008) positive solutions for elliptic boundary value ... 111 by a cone in a banach space e we mean a closed convex subset c of e such that c 6= {0} , λc ⊂ c for all λ ∈ r+, and c ∩ (−c) = {0} . our main tool in proving the existence of positive solutions to problems (1.1) and (1.2) is krasnosel’skii’s compression-expansion theorem [3], [2]: theorem 1. let e be a banach space, c ⊂ e a cone in e, and assume that t : c −→ c is a completely continuous map such that for some numbers r and r with 0 < r < r, one of the following conditions is satisfied: (i) ‖t u‖ ≤ ‖u‖ for ‖u‖ = r and ‖t u‖ ≥ ‖u‖ for ‖u‖ = r, (ii) ‖t u‖ ≥ ‖u‖ for ‖u‖ = r and ‖t u‖ ≤ ‖u‖ for ‖u‖ = r. then t has a fixed point with r ≤ ‖u‖ ≤ r. 2 existence results for problem 1.1 in this section, e is the banach space c0(ω, r) = {u ∈ c(ω, r) : u = 0 on ∂ω} endowed with norm ‖.‖ 0 , and c is the cone c = {u ∈ c0(ω, r+) : u(x) ≥ η‖u‖0 for all x ∈ k}. (2.1) in order to state our results we introduce the notation f0 = lim sup y→0+ max x∈ω f (x, y) y and f ∞ = lim inf y→∞ min x∈k f (x, y) y f 0 = lim inf y→0+ min x∈k f (x, y) y and f∞ = lim sup y→∞ max x∈ω f (x, y) y . also, for a function h : ω → r, by h| k we mean the function h| k (x) = h (x) if x ∈ k and h| k (x) = 0 if x ∈ ω \ k. for example, if 1 is the constant function 1 on ω, then 1| k (x) = 1 if x ∈ k and 1| k (x) = 0 for x ∈ ω \ k. theorem 2. suppose (1.3) holds. then for each λ satisfying 1 f ∞ η ‖(−∆)−1 1| k ‖ 0 < λ < 1 f0‖(−∆)−11‖0 (2.2) there exists at least one positive solution of problem (1.1). proof. let λ be as in (2.2) and let ǫ > 0 be such that 1 (f ∞ − ǫ)η ‖(−∆)−1 1| k ‖ 0 ≤ λ ≤ 1 (f0 + ǫ)‖(−∆)−11‖0 . (2.3) 112 toufik moussaoui and radu precup cubo 10, 4 (2008) we know that u is a solution of problem (1.1) if and only if u = λ (−∆)−1f u where f : c(ω, r) −→ c(ω, r), f u(x) = f (x, u(x)) . hence, a solution to problem (1.1) is a fixed point of the operator t : c −→ c0(ω, r) given by t u = λ (−∆)−1f u. we shall prove that the hypotheses of theorem 1 are satisfied. we have that the operator t satisfies { −∆(t u) = λ f (x, u) , in ω, t u = 0, on ∂ω. then by the global weak harnack inequality (1.3), one has t (c) ⊂ c. moreover, t is completely continuous by the arzela-ascoli theorem. furthermore, by the definition of f0, there exists an r > 0 such that f (x, u) ≤ (f0 + ǫ)u for 0 < u ≤ r and x ∈ ω. (2.4) let u ∈ c with ‖u‖0 = r. then using (2.4), the monotonicity of operator (−∆) −1 and of norm ‖.‖ 0 , and (2.3), we obtain ‖t u‖ 0 = λ ∥ ∥(−∆)−1f u ∥ ∥ 0 ≤ λ(f0 + ǫ)‖u‖0 ∥ ∥(−∆)−11 ∥ ∥ 0 ≤ ‖u‖0. hence ‖t u‖0 ≤ ‖u‖0 for ‖u‖0 = r. (2.5) by the definition of f ∞ , there is r > r such that f (x, u) ≥ (f ∞ − ǫ)u for u ≥ ηr and x ∈ k. then, if u ∈ c with ‖u‖0 = r, we have ‖t u‖0 = λ ∥ ∥ (−∆)−1f u ∥ ∥ 0 ≥ λ ∥ ∥ (−∆)−1 (f u)| k ∥ ∥ 0 ≥ λ(f ∞ − ǫ)η‖u‖0 ∥ ∥(−∆)−1 1| k ∥ ∥ 0 ≥ ‖u‖0. hence ‖t u‖0 ≥ ‖u‖0 for ‖u‖0 = r. (2.6) inequalities (2.5) and (2.6) show that the expansion condition (i) in theorem 1 is satisfied. now theorem 1 guarantees the existence of a fixed point u of t with r ≤ ‖u‖ 0 ≤ r. cubo 10, 4 (2008) positive solutions for elliptic boundary value ... 113 similarly, we have the following result: theorem 3. suppose (1.3) holds. then for each λ satisfying 1 f 0 η ‖(−∆)−1 1| k ‖ 0 < λ < 1 f∞‖(−∆)−11‖0 (2.7) there exists at least one positive solution of problem (1.1). proof. let λ be as in (2.7) and let ǫ > 0 be such that 1 (f 0 − ǫ)η ‖(−∆)−1 1| k ‖ 0 ≤ λ ≤ 1 (f∞ + ǫ)‖(−∆)−11‖0 . (2.8) by the definition of f 0 , there exists an r > 0 such that f (x, u) ≥ (f 0 − ǫ)u for 0 < u ≤ r and x ∈ k. if u ∈ c and ‖u‖0 = r, then ‖t u‖0 = λ ∥ ∥ (−∆)−1f u ∥ ∥ 0 ≥ λ ∥ ∥ (−∆)−1 (f u)| k ∥ ∥ 0 ≥ λ(f 0 − ǫ)η‖u‖0 ∥ ∥(−∆)−1 1| k ∥ ∥ 0 ≥ ‖u‖0. hence ‖t u‖0 ≥ ‖u‖0 for ‖u‖0 = r. (2.9) by the definition of f∞, there is r0 > 0 such that f (x, u) ≤ (f∞ + ǫ)u for u ≥ r0 and x ∈ ω. let m be such that f (x, u) ≤ m for all u ∈ [0, r0] and x ∈ ω, and let r be such that r > r and m ≤ (f∞ + ǫ) r. if u ∈ c with ‖u‖0 = r, then 0 ≤ u (x) ≤ (f∞ + ǫ) r for all x ∈ ω. consequently, also using (2.8), we obtain ‖t u‖ 0 = λ ∥ ∥(−∆)−1f u ∥ ∥ 0 ≤ λ(f∞ + ǫ)r ∥ ∥(−∆)−11 ∥ ∥ 0 ≤ r = ‖u‖0. hence ‖t u‖0 ≤ ‖u‖0 for ‖u‖0 = r. (2.10) inequalities (2.9) and (2.10) show that the compression condition (ii) in theorem 1 is satisfied. now theorem 1 guarantees the existence of a fixed point u of t with r ≤ ‖u‖ 0 ≤ r. 114 toufik moussaoui and radu precup cubo 10, 4 (2008) 3 existence results for problem 1.2 in this section, we are concerned with the existence of positive solutions to the dirichlet problem (1.2) for elliptic systems. here e will be the banach space c0(ω, r 2 ) := c0(ω, r) × c0(ω, r) endowed with the norm ‖(., .)‖0 given by ‖(u, v)‖0 = ‖u‖0 + ‖v‖0 and the cone in e will be c × c, where c is given by (2.1). in order to state our results in this section we introduce the notation g0 = lim sup y+z→0+ max x∈ω g (x, y, z) y + z and g ∞ = lim inf y+z→∞ min x∈k g (x, y, z) y + z g 0 = lim inf y+z→0+ min x∈k g (x, y, z) y + z and g∞ = lim sup y+z→∞ max x∈ω g (x, y, z) y + z . the limits h0, h0, h∞ and h∞ are defined similarly. theorem 4. suppose (1.3) holds. in addition assume that there are numbers p, q > 0 with 1 p + 1 q = 1 such that 1 g ∞ η ‖(−∆)−1 1| k ‖ 0 < α < 1 p g0‖(−∆)−11‖0 (3.1) and 1 h ∞ η ‖(−∆)−1 1| k ‖ 0 < β < 1 q h0‖(−∆)−11‖0 . (3.2) then there exists at least one positive solution (u, v) of problem (1.2). proof. let α, β be as in (3.1), (3.2) and let ǫ > 0 be such that 1 (g ∞ − ǫ)η ‖(−∆)−1 1| k ‖ 0 ≤ α ≤ 1 p (g0 + ǫ)‖(−∆)−11‖0 and 1 (h ∞ − ǫ)η ‖(−∆)−1 1| k ‖ 0 ≤ β ≤ 1 q (h0 + ǫ)‖(−∆)−11‖0 . it is easily seen that a vector-valued function (u, v) is a solution of problem (1.2) if and only if u = α (−∆)−1g (u, v) v = β (−∆) −1 h (u, v) where g, h : c(ω, r2) −→ c(ω, r), g(u, v)(x) = g (x, u(x), v(x)) , h (u, v) (x) = h (x, u (x) , v (x)) . cubo 10, 4 (2008) positive solutions for elliptic boundary value ... 115 hence, (u, v) is a positive solution of (1.2) if it is a fixed point of the operator t : c × c −→ c0(ω, r 2 ), t = (t1, t2) where t1 (u, v) = α (−∆) −1g (u, v) , t2 (u, v) = β (−∆) −1h (u, v) . we shall prove that the hypotheses of theorem 1 are satisfied. clearly the operator t = (t1, t2) satisfies        −∆(t1u) = α g (x, u, v) , in ω, −∆(t2v) = β h (x, u, v) , in ω, t1u = t2v = 0, on ∂ω. then by the global weak harnack inequality (1.3), we have t (c × c) ⊂ c × c. moreover, t is completely continuous by the arzela-ascoli theorem. by the definitions of g0 and h0, there exists an r > 0 with g (x, u, v) ≤ (g0 + ǫ)(u + v) for u, v ≥ 0, 0 < u + v ≤ r and x ∈ ω and h (x, u, v) ≤ (h0 + ǫ)(u + v) for u, v ≥ 0, 0 < u + v ≤ r and x ∈ ω. let (u, v) ∈ c × c with ‖(u, v)‖ 0 = r. we have ‖t1 (u, v)‖0 = α ∥ ∥(−∆)−1g (u, v) ∥ ∥ 0 ≤ α(g0 + ǫ)‖u + v‖0 ∥ ∥(−∆)−11 ∥ ∥ 0 ≤ 1 p ‖u + v‖0 ≤ 1 p (‖u‖0 + ‖v‖0) = 1 p ‖(u, v)‖0. then ‖t1 (u, v) ‖0 ≤ 1 p ‖(u, v)‖0. similarly, we have ‖t2 (u, v)‖0 = β ∥ ∥(−∆)−1h (u, v) ∥ ∥ 0 ≤ β(h0 + ǫ)‖u + v‖0 ∥ ∥(−∆)−11 ∥ ∥ 0 ≤ 1 q ‖u + v‖0 ≤ 1 q (‖u‖0 + ‖v‖0) = 1 q ‖(u, v)‖0. 116 toufik moussaoui and radu precup cubo 10, 4 (2008) thus ‖t2 (u, v) ‖0 ≤ 1 q ‖(u, v)‖0. combining the above two inequalities, we obtain ‖t (u, v) ‖0 = ‖t1 (u, v) ‖0 + ‖t2 (u, v) ‖0 ≤ ( 1 p + 1 q )‖(u, v)‖0 = ‖(u, v)‖0. next by the definitions of g ∞ and h ∞ , there is r > 0 such that g (x, u, v) ≥ (g ∞ − ǫ)(u + v) for u, v ≥ 0, u + v ≥ ηr and x ∈ k and h (x, u, v) ≥ (h ∞ − ǫ)(u + v) for u, v ≥ 0, u + v ≥ ηr and x ∈ k. let (u, v) ∈ c ×c with ‖(u, v)‖ 0 = r. then for each x ∈ k, u (x) ≥ η ‖u‖ 0 and v (x) ≥ η ‖v‖ 0 . hence (u + v) (x) ≥ η (‖u‖ 0 + ‖v‖ 0 ) , that is (u + v) (x) ≥ ηr for all x ∈ k. consequently, g (u, v) (x) ≥ (g ∞ − ǫ) (u + v) (x) for all x ∈ k. furthermore ‖t1 (u, v) ‖0 = α ∥ ∥(−∆)−1g (u, v) ∥ ∥ 0 ≥ α ∥ ∥(−∆)−1 g (u, v)| k ∥ ∥ 0 ≥ α(g ∞ − ǫ) ∥ ∥(−∆)−1 (u + v)| k ∥ ∥ 0 ≥ α(g ∞ − ǫ) ∥ ∥(−∆)−1 u| k ∥ ∥ 0 ≥ α(g ∞ − ǫ)η‖u‖0 ∥ ∥(−∆)−1 1| k ∥ ∥ 0 ≥ ‖u‖0. similarly, we have ‖t2 (u, v) ‖0 ≥ ‖v‖0. the above two inequalities give ‖t (u, v) ‖0 ≥ ‖(u, v)‖0. thus condition (i) in theorem 1 is satisfied. now theorem 1 guarantees the existence of a fixed point (u, v) of t with r ≤ ‖(u, v)‖ 0 ≤ r. in a similar way, one can prove: theorem 5. suppose (1.3) holds. in addition assume that there are numbers p, q > 0 with 1 p + 1 q = 1 such that 1 g 0 η ‖(−∆)−1 1| k ‖ 0 < α < 1 p g∞‖(−∆)−11‖0 and 1 h 0 η ‖(−∆)−1 1| k ‖ 0 < β < 1 q h∞‖(−∆)−11‖0 . then there exists at least one positive solution (u, v) of problem (1.2). received: april 2008. revised: april 2008. cubo 10, 4 (2008) positive solutions for elliptic boundary value ... 117 references [1] d. gilbarg and n. trudinger, elliptic partial differential equations of second order, springer, berlin, 1983. [2] a. granas and j. dugundji, fixed point theory, springer, new york, 2003. [3] m.a. krasnoselskii, topological methods in the theory of nonlinear integral equations, cambridge university press, new york, 1964. [4] r. precup, positive solutions of semi-linear elliptic problems via krasnoselskii type theorems in cones and harnack’s inequality, mathematical analysis and applications, aip conf. proc., 835, amer. inst. phys., melville, ny, 2006, 125–132. n9-moussaoui-precup2 cubo a mathematical journal vol.14, no¯ 02, (153–173). june 2012 local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle yavar kian centre de physique théorique cnrs-luminy, case 907, 13288 marseille, france. email: yavar.kian@cpt.univ-mrs.fr abstract consider the mixed problem with dirichelet condition associated to the wave equation ∂2tu − divx(a(t,x)∇xu) = 0, where the scalar metric a(t,x) is t-periodic in t and uniformly equal to 1 outside a compact set in x, on a t-periodic domain. let u(t,0) be the associated propagator. assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the floquet operator u(t,0) and we establish sufficient conditions for local energy decay. resumen considere el problema mixto con condiciones de dirichlet asociadas a la ecuación de onda ∂2tu − divx(a(t,x)∇xu) = 0, donde la métrica escalar a(t;x) es t-periódica en t y uniformemente igual a 1 fuera de un conjunto compacto en x, sobre un dominio t-periódico. sea u(t,0) el propagador asociado. asumiendo que las perturbaciones son non-trapping, probamos la continuación meromorfa de la resolvente de corte del operador de floquet u(t,0) y establecemos condiciones suficientes para la decadencia local de enerǵıa. keywords and phrases: time-dependent perturbation, moving obstacle, local energy decay, wave equation. 2010 ams mathematics subject classification: 35b40, 35l15 . 154 yavar kian cubo 14, 2 (2012) 1 introduction let ω be an open domain in r1+n, n ≥ 3 with c∞ boundary ∂ω. introduce the sets ω(t) = {x ∈ rn : (t,x) ∈ ω}, o(t) = rn \ ω(t), t ∈ r. we assume that there exists ρ1 > 0 such that for all t ∈ r o(t) ⊂ {x : |x| ≤ ρ1}. (1.1) moreover there exists t > 0 such that o(t + t) = o(t), t ∈ r. (1.2) for each (t,x) ∈ ∂ω, let ν(t,x) = (νt(t,x),νx(t,x)) be the exterior unit normal vector to ∂ω at (t,x) ∈ ∂ω pointing into ω. then, we assume that there exists 0 < c < 1 such that |νt| < c|νx|. (1.3) consider the following mixed problem    utt − divx(a(t,x)∇xu) = 0, (t,x) ∈ ω, u|∂ω = 0 (u,ut)(s,x) = (f1(x),f2(x)) = f(x), x ∈ ω(s), (1.4) where the perturbation a(t,x) ∈ c∞(rn+1) is a scalar function which satisfies the conditions: (i) c ≥ a(t,x) ≥ c > 0, (t,x) ∈ rn+1, (ii) there exists ρ > ρ1 such that a(t,x) = 1 for |x| ≥ ρ, (iii) there exists t > 0 such that a(t + t,x) = a(t,x), (t,x) ∈ rn+1. (1.5) throughout this paper we assume n ≥ 3. consider the set h(t) which is the closure of the space c∞0 (ω(t)) × c ∞ 0 (ω(t)) with respect to the norm ‖f‖ h(t) =    ∫ ω(t) ( |∇xf1| 2 + |f2| 2 ) dx    1 2 , f = (f1,f2) ∈ c ∞ 0 (ω(t)) × c ∞ 0 (ω(t)). let us introduce some general properties of solutions of (1.4). we show, in section 1, that for f ∈ h(s) there exists a unique solution of (1.4) and we introduce the propagator u(t,s) : h(s) ∋ (f1,f2) = f 7→ u(t,s)f = (u,ut)(t,x) ∈ h(t) (1.6) with u the solution of (1.4). moreover, we prove that u(t,s) is a bounded operator satisfying the following estimate ‖u(t,s)‖l(h(s),h(t)) ≤ be a|t−s|. (1.7) cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 155 the goal of this paper is to establish sufficient conditions for a local energy decay taking the form ‖χu(t,s)χ‖l(h(s),h(t)) ≤ cχp(t − s), t ≥ s, (1.8) with p(t) ∈ l1(r+) and χ ∈ c∞0 (|x| ≤ ρ + 1). we study problem (1.4) under a non-trapping condition. more precisely, let u(t,s,x,x0) be the kernel of the propagator u(t,s) and consider the following (h1) for all r > 0, there exists t1(r) > 0 such that u(t,s,x,x0) ∈ c ∞ ({(t,s,x,x0) : |x| ≤ r, |x0| ≤ r, |t − s| ≥ t1(r)}) . from [15], we know that singularities propagate along null-bicharacteristics (with consideration of their reflections from ∂ω). thus, one can show that condition (h1) is equivalent to the requirement that all null-bicharacteristics of (1.4) with consideration of reflections from ∂ω go out to infinity as |t − s| → +∞. let us recall that the non-trapping condition (h1) is necessary for (1.8) since for some trapping perturbations we may have solutions with exponentially increasing energy (see [7] for ω = r1+n and [22] for a(t,x) = 1). on the other hand, even for non-trapping periodic perturbations some parametric resonances could lead to solutions with exponentially growing energy (see [6] for time-periodic potentials). to exclude the existence of such solutions we must consider a second assumption. many authors have investigated the local energy decay of wave equations. the main hypothesis is that the perturbations are non-trapping. for a(t,x) = a0(x) independent of time and fixed obstacles, the meromorphic continuation and estimates of the cut-off resolvent χ ( −divx(a0(x)∇x.) − λ 2 )−1 χ, where χ ∈ c∞0 (r n) and λ ∈ c, are the main arguments for estimate (1.8) (see [24], [25], [27] and [28]). from these results, by considering the connection between the fourier transform in time of the solutions and the stationary operator −divx(a0(x)∇x.) − λ 2, one can deduce (1.8). for time dependent metric a(t,x) or moving obstacle, since the domain or the hamiltonian −divx(a(t,x)∇x.) are time-dependent, we cannot apply these arguments. however, the analysis of the floquet operator u(t,0) makes it possible to obtain (1.8) with t-periodic perturbations and moving obstacle. in [8] the authors have extended the lax-phillips theory to problem (1.4) with a(t,x) = 1 and they have established a local energy decay (1.8). by using the compactness of the local evolution operator, deduced from a propagation of singularities, and the rage theorem of georgiev and petkov (see [9]), bachelot and petkov have shown in [1] that in the case of odd dimensions, the decay of the local energy associated to the wave equation with time periodic potential is exponential for initial data with compact support included in a subspace of finite codimension. petkov has extended this result to even dimensions (see [21]), by using the meromorphic continuation of the cut-off resolvent of the floquet operator associated to this problem. let us introduce the cut-off resolvent, associated to the floquet operator u(t,0), defined by rψ1,ψ2(θ) = ψ1(u(t,0) − e −iθ ) −1ψ2 : h(0) → h(0), ψ1,ψ2 ∈ c ∞ 0 (r n ). 156 yavar kian cubo 14, 2 (2012) according to (1.7), rψ1,ψ2(θ) is a family of bounded operators analytic with respect to θ on {θ ∈ c : i(θ) > at}. applying some arguments of [26], in section 2, we show the meromorphic continuation of rψ1,ψ2(θ) to c for n odd and to {θ ∈ c : θ /∈ 2πz+ir −} for n even. let us recall that the meromorphic continuation of rψ1,ψ2(θ) is closely related to the asymptotic expansion of χu(t,0)χ, χ ∈ c∞0 (r n), as t → +∞ (see section 2 and the main theorem in [26]). consequently, it seems natural to consider the meromorphic continuations of rψ1,ψ2(θ) that imply (1.8). consider the following assumption. (h2) there exist φ1,φ2 ∈ c ∞ 0 (r n), satisfying φ1(x) = φ2(x) = 1 for |x| ≤ ρ + t + 2, such that the operator rφ1,φ2(θ) admits an analytic continuation from {θ ∈ c : im(θ) ≥ a > 0} to {θ ∈ c : im(θ) ≥ 0}, for n ≥ 3, odd, and to {θ ∈ c : im(θ) > 0} for n ≥ 4, even. moreover, for n even, rφ1,φ2(θ) admits a continuous continuation from {θ ∈ c : im(θ) > 0} to {θ ∈ c : im(θ) ≥ 0,θ 6= 2kπ,k ∈ z} and we have lim sup λ→0 im(λ)>0 ‖rφ1,φ2(λ)‖ < ∞. assuming (h1) and (h2) fulfilled, we obtain the following. theorem 1. assume (1.1), (1.2), (1.3), (1.5), (h1) and (h2) fulfilled. then, estimate (1.8) is fulfilled with    p(t) = e−δt for n ≥ 3 odd, p(t) = 1 (t + 1) ln2(t + e) for n ≥ 4 even. (1.9) let us remark that, assuming (h1) fulfilled, (h2) is a necessary and sufficient condition for estimate (1.8) with p(t) satisfying (1.9). moreover, if (h2) is not fulfilled, even the uniform estimate in time of the local energy ‖χu(t,0)χ‖l(h(0),h(t)) may not hold. for example, if rφ1,φ2(θ) has a pole θ0 ∈ c with i(θ0) > 0, one can establish the estimate ‖χu(t,0)χ‖l(h(0),h(t)) ≥ ce i(θ 0 ) t t and deduce existence of a solution with compactly supported initial data and exponentially growing local energy. it has been established in [6] that these phenomenon can occur even with a nontrapping condition. the goal of (h2) is to avoid existence of such solutions. remark 1. let the metric (aij(t,x))1≤i,j≤n be such that for all i, j = 1 · · ·n we have (i) there exists ρ > 0 such that aij(t,x) = δij, for |x| ≥ ρ, with δij = 0 for i 6= j and δii = 1, (ii) there exists t > 0 such that aij(t + t,x) = aij(t,x), ∀(t,x) ∈ r n+1, (iii)aij(t,x) = aji(t,x),∀(t,x) ∈ r n+1, (iv) there exist c > c > 0 such that c|ξ|2 ≥ n∑ i,j=1 aij(t,x)ξiξj ≥ c|ξ| 2, ∀(t,x) ∈ r1+n, ξ ∈ rn. cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 157 if we replace a(t,x) in (1.4) we get the following mixed problem    utt − n∑ i,j=1 ∂ ∂xi ( aij(t,x) ∂ ∂xj u ) = 0, (t,x) ∈ ω, u|∂ω = 0, (u,ut)(s,x) = (f1(x),f2(x)) = f(x), x ∈ ω(s). (1.10) all the results of this paper remain valid for the mixed problem (1.10) and their proofs follow from the same arguments. notice that the estimate ‖ψ1u(nt,0)ψ2‖l(h(0)) ≤ cψ1,ψ2 (n + 1) ln2(n + e) , n ∈ n, (1.11) implies (1.8). on the other hand, if (1.11) is valid, the assumption (h2) for n even is fulfilled. indeed, for large a >> 1 and im(θ) ≥ at we have rψ1,ψ2(θ) = −e iθ ∞∑ n=0 ψ1u(nt,0)ψ2e inθ and applying (1.11), we conclude that rψ1,ψ2(θ) admits an analytic continuation from {θ ∈ c : im(θ) ≥ a > 0} to {θ ∈ c : im(θ) > 0}. moreover, rψ1,ψ2(θ) is bounded for θ ∈ r. in section 4, we give some examples of metrics a(t,x) and moving obstacle o(t) such that (1.11) is fulfilled. 2 general properties the purpose of this section is to establish some general properties of solutions of problem (1.4). we will study the global well posedness of (1.4) and we will prove estimate (1.7). we start by fixing the notion of solutions of (1.4). definition 2.1. a distribution u(t,x) ∈ d′(ω) is called a solution of (1.4) if the following conditions hold: (i) (u(t, .),ut(t, .)) ∈ h(t) for each t ∈ r; extended inside o(t) by setting u(t,x) = 0, the functions t 7−→ ∇xu(t, .), t 7−→ ut(t, .) are continuous with values in l2(rn), 158 yavar kian cubo 14, 2 (2012) (ii) (u(s, .),ut(s, .)) = (f1,f2) = f (iii) ∂2tu − divx(a(t,x)∇xu) = 0 in ω in the sense of distributions. in the next result we obtain the existence and uniqueness of solutions of (1.4). theorem 2. assume (1.1), (1.2), (1.3) and (1.5) fulfilled. then, for each f ∈ h(s) there exists a unique solution u(t, .) of (1.4) with the property that for each t > 0 sup |t−s|≤d |s|≤2d ‖(u(t, .),ut(t, .))‖h(t) ≤ cd ‖f‖h(s) (2.1) proof. first we treat the existence and uniqueness of the solution for small |t − s|. given z ∈ ω(s), consider the cone cz,s = {(t,x) ∈ r 1+n : |x − z| ≤ |t − s|}. for |t − s| small enough and for z outside a small neighborhood of ∂ω(s) we obtain cz,s ⊂ ω. consequently, for (t,x) ∈ cz,s the solution u(t,x) of the mixed problem coincides with the solution of the cauchy problem { utt − divx(a(t,x)∇xu) = 0, (t,x) ∈ r × r n, (u,ut)(s,x) = (f1(x),f2(x)) = f(x), x ∈ r n, (2.2) with f extended by 0 for x ∈ o(s). thus, for |t − s| ≤ ǫ and ǫ sufficiently small, we will determine u(t,x) in some small neighborhood of ∂ω∩ {|t − s| ≤ ǫ}. given (s,z) with z ∈ ∂ω(s), we establish the existence and uniqueness of u(t,x) in some space-time neighborhood of (s,z). covering the compact set {s} × ∂ω(s) by a finite number of such neighborhoods and using the local uniqueness result for the points where these neighborhoods overlap, we deduce the existence and uniqueness for small |t − s|. introduce in a neighborhood of (s,z), z ∈ ∂ω(s), local coordinates (t,y), y′ = (y1, . . . ,yn−1), so that (s,z) is transformed into (0,0), while the boundary ∂ω is given by yn = g(t,y′) with g a c∞ function such that ∇y′g(0,0) = 0. since ν(t,y′,g(t,y′)) = 1 √ 1 + |gt(t,y′)| 2 + |∇y′g(t,y′)| 2 (−gt(t,y ′ ),−∇y′g(t,y ′ ),1), statement (1.3) implies that |gt(t,y ′)| < c(|∇y′g(t,y ′)| + 1) . thus, we have |gt(0,0)| < c. if we choose a sufficiently small neighborhood of (0,0) we can assume that |gt(t,y ′)| < c. changing variables xj = yj, j = 1, . . . ,n − 1, xn = yn − g(t,y ′) we transform ∂2t − divx(a(t,x)∇x·) cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 159 into the operator p(t,x,∂t,∂x) with principal symbol σ(p(t,x,∂t,∂x)) = − τ 2 + 2gtτξn − 2ξnb(t,x)ξ ′ · ∇x′g + b(t,x) |ξ ′| 2 + ( b(t,x) |∇x′g| 2 − g2t + b(t,x) ) ξ2n, where b(t,x) = a(t,y). here (τ,ξ′,ξn) are the variable dual to (t,x ′,xn). statement (1.3) and property (1.5) imply that b(t,x) |∇x′g| 2 − g2t + b(t,x) > 0. (2.3) consider the problem    p(t,x,∂t,∂x)u = 0 in rt × r n−1 x′ × r + xn , u(t,x′,0) = 0 in rt × r n−1 x′ , (u(0,x),ut(0,x)) = f(x). (2.4) we suitably extend the coefficients of p(t,x,∂t,∂x) to r 1+n preserving the strict hyperbolicity of p(t,x,∂t,∂x) with respect to t. for the mixed problem (2.4) we can apply the results of miyatake [18] and hörmander [10], chapter xxiv. notice that the inequality (2.3) guarantees that the boundary xn = 0 is timelike in the sense of hörmander [10]. the result of miyatake [18] says that if ∇xf1, f2 ∈ l 2 loc ( r n−1 x′ × r+xn ) with f1 = f2 for xn = 0, then for |t| ≤ δ there exists a unique solution u(t,x) ∈ h1 loc ( r n−1 x′ × r + xn ) of (2.4) satisfying the estimate ∑ j+|β| ∥ ∥ ∥ ∂ j t∂ β xu(t,x) ∥ ∥ ∥ l2 loc ( r n−1 x ′ ×r+xn ) ≤ cδ ∑ j+|β| ∥ ∥ ∥ ∂ j t∂ β xu(0,x) ∥ ∥ ∥ l2 loc ( r n−1 x ′ ×r+xn ) with a constant cδ depending on δ. notice that (1.5) implies that the boundary xn = 0 is noncharacteristic for p(t,x,∂t,∂x). so u(t,x) ∈ c ∞ ( r + xn;d ′(rn) ) (see theorem b.2.9 in hörmander [10]) and the trace u|xn=0 is meaningful. the same argument shows that ∇xu(t, .) and ut(t, .) depend continuously on t. thus we obtain the existence and uniqueness of the solution of (1.4) in ω∩ {|t − s| ≤ ǫ}. we can determine ǫ > 0 uniformly with respect to s, provided |s| ≤ 2d. making a construction by steps of length ǫ, we cover the interval |t − s| ≤ d and the proof is complete. following theorem 2, we can introduce the propagator u(t,s) defined by (1.6). combining the results of theorem 2 and the periodicity of o(t) and a(t,x), we deduce the following. proposition 1. assume (1.1), (1.2), (1.3) and (1.5) fulfilled. then, we have u(t + t,s + t) = u(t,s), (2.5) ‖u(t,s)‖l(h(s),h(t)) ≤ be a|t−s|. (2.6) proof. the proof of (2.5) is trivial. let us show estimate (2.6). applying (2.1), we obtain sup |s|,|t|≤t ‖u(t,s)‖l(h(s),h(t)) = c < ∞. 160 yavar kian cubo 14, 2 (2012) let t,s ∈ r and let 0 ≤ t′,s′ < t be such that t = lt + t′ and s = kt + s′ with k,l ∈ z. then, applying (2.5), we obtain u(t,s) = u(t′,0)u((k − l)t,0)u(s′,0) = u(t′,0)u(t,0)k−lu(s′,0). it follows that ‖u(t,s)‖l(h(s),h(t)) ≤ c 2(1 + c)|k−l| ≤ c2eln(1+c)|k−l| ≤ c2eln(1+c)|t−s| and we obtain (2.6) with a = ln(1 + c). notice that, combing the arguments used in the proof of theorem 2 with estimate (2.6), we can show that the duhamel’s principal holds. let p1 and p2 be the projectors of c 2 defined by p1(h) = h1, p2(h) = h2, h = (h1,h2) ∈ c 2 and let p1,p2 ∈ l(c,c2) be defined by p1(h) = (h,0), p2(h) = (0,h), h ∈ c. denote by v(t,s) : l2(ω(s)) → ḣ1(ω(t)) the operator defined by v(t,s) = p1u(t,s)p 2. notice that for h ∈ l2(ω(s)), w = v(t,s)h is the solution of    ∂2t(w) − divx(a(t,x)∇xw) = 0, w|∂ω = 0, (w,∂tw)|t=s = (0,h). let g(t,x) be a function defined on ω such that, for a1 > a (with a the constant of (2.6)), e−a1tg(t,x) ∈ l2(ω) and g(t,x) = 0 for |x| ≥ b with b ≥ ρ + 1. then there exists a unique solution v of    ∂2t(v) − divx(a(t,x)∇xv) = g(t,x), v|∂ω = 0, (v,∂tv)|t=s = (0,0). moreover, this solution can be written in the following way v(t, .) = ∫t s v(t,τ)g(τ, .)dτ. (2.7) cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 161 3 the meromorphic continuation of the cut-off resolvent rψ1,ψ2(θ) the goal of this section is to prove the meromorphic continuation of rψ1,ψ2(θ), assuming (h1) fulfilled. the main result of this section is the following. theorem 3. assume (h1), (1.1), (1.2), (1.3) and (1.5) fulfilled. let ψ1, ψ2 ∈ c ∞ 0 (r n). then, rψ1,ψ2(θ) admits a meromorphic continuation from {θ ∈ c : i(θ) > at} to c for n ≥ 3 odd and to c′ = {θ ∈ c : θ /∈ 2πz + ir−} for n ≥ 4 even. moreover, for n ≥ 4 even, there exists ǫ0 > 0 such that, for |θ| ≤ ǫ0, we have rψ1,ψ2(θ) = ∑ k≥−m ∑ j≥−mk rkjθ k(logθ)−j. (3.1) here rk,j ∈ l(h(0)) and, for k < 0 or j > 0, rk,j is a finite rank operator. to prove theorem 4, we will use some results of [26] and [13]. for this purpose, we introduce some tools and definitions of [26]. let γ ∈ c∞(r) be such that γ(t) = 1 for t ≥ −2t 3 − t 10 and γ(t) = 0 for t ≤ −2t 3 − 2t 10 . set v1(t,s) = γ(t − s)v(t,s). we recall that the fourier-bloch-gelfand transform f is defined by f(φ)(t,θ) = +∞∑ k=−∞ ( φ(t + kt, ·)eikθ ) , φ ∈ c∞0 (r × r n). applying (2.6), for i(θ) > at, with a > 0 the constant of (2.6), we can define f(χ1v1(t,s)χ2)(t,θ) = +∞∑ k=−∞ ( χ1v1(t + kt,s)χ2e ikθ ) , χ1, χ2 ∈ c ∞ 0 (r n ) and f′(χ1v1(t,s)χ2)(t,θ) = e itθ t f(χ1v1(t,s)χ2)(t,θ), χ1, χ2 ∈ c ∞ 0 (r n ). we will use the following definition of meromorphic continuation of a family of bounded operators. definition 1. let h1 and h2 be hilbert spaces. a family of bounded operators q(t,s,θ) : h1 → h2 is said to be meromorphic with respect to θ in a domain d ⊂ c, if q(t,s,θ) is meromorphically dependent on θ for θ ∈ d and for any pole θ = θ0 the coefficients of the negative powers of θ−θ0 in the appropriate laurent extension are finite-rank operators. denote c′ = {z ∈ c : z 6= 2kπ − iµ, k ∈ z, µ ≥ 0} and consider the following meromorphic continuation. 162 yavar kian cubo 14, 2 (2012) definition 2. we say that the family of operators q(t,s,θ), which are c∞ with respect to t and s, for t ∈ r and 0 ≤ s ≤ 2t 3 , and t-periodic with respect to t, has the property (s′) if: 1) for odd n the operators q(t,s,θ), θ ∈ c, and its derivatives with respect to t form a finitely-meromorphic family; 2) for even n the operators q(t,s,θ) and its derivatives with respect to t form a finitelymeromorphic family for θ ∈ c′ . moreover, in a neighborhood of θ = 0 in c′, q(t,s,θ) has the form q(t,s,θ) = θ−m ∑ j≥0 ( θ rt,s(logθ) )j pj,t,s(logθ) + c(t,s,θ), (3.2) where c(t,s,θ) is analytic with respect to θ, rt,s is a polynomial, the pj,t,s are polynomials of order at most lj and log is the logarithm defined on c \ ir −. moreover, c(t,s,θ) and the coefficients of the polynomials rt,s and pj,t,s are c ∞ and t-periodic with respect to t and c∞ with respect to s for 0 ≤ s ≤ 2t 3 . remark 2. notice that if q(t,s,θ) satisfies (s′) then ∂tq(t,s,θ) satisfies also (s ′). in [26] vainberg proposed a general approach to problems with time-periodic perturbations including potentials, moving obstacles and high order operators, provided that the perturbations are non-trapping. one of the main results of [26] is the following. theorem 4. (theorem 10, [26]) assume that the mixed problem (1.4) is well posed, the duhamel’s principal holds and let (2.6) and (h1) be fulfilled. then, for all b ≥ ρ+1, there exists t2(b) > t1(b) and an operator r(t,s) : l2(ω(s)) → ḣ1(ω(t)) such that the following conditions are fulfilled: (i) r(t + t,s + t) = r(t,s), (ii) r(t,s) is bounded, (iii) for all χ1, χ2 ∈ c ∞ 0 (|x| ≤ b), f ′(χ1r(t,s)χ2)(t,θ) admits a meromorphic continuation to the lower half plane satisfying property (s′) and χ1r(t,s)χ2 = χ1v(t,s)χ2 for t − s ≥ t2(b). in [26] vainberg established the result of theorem 4 for s = 0. in [13] it has been proven that this result can be generalized to 0 ≤ s ≤ 2t 3 . combining these results with the properties established in section 1, we obtain a meromorphic continuation of the fourier-bloch-gelfand transform of the solutions of (1.4) with initial data (0,g) and their derivatives of order 1 with respect to t. lemma 3.1. assume (h1), (1.1), (1.2), (1.3) and (1.5) fulfilled. then, for all ψ1, ψ2 ∈ c ∞ 0 (r n) and all 0 ≤ s ≤ 2t 3 , f′(ψ1v1(t,s)ψ2)(t,θ) and f ′(ψ1∂tv1(t,s)ψ2)(t,θ) admit a meromorphic continuation with respect to θ, continuous with respect to s ∈ [ 0, 2t 3 ] , from {θ ∈ c : i(θ) > at} to c for n ≥ 3 odd and to c′ = {θ ∈ c : θ /∈ 2πz + ir−} for n ≥ 4 even. cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 163 moreover, for n ≥ 4 even, there exists ǫ0 > 0 such that, for |θ| ≤ ǫ0, we have f′(ψ1v1(t,s)ψ2)(t,θ) = ∑ k≥−m ∑ j≥−mk qkj(s)θ k(logθ)−j. (3.3) f′(ψ1∂tv1(t,s)ψ2)(t,θ) = ∑ k≥−m ∑ j≥−mk skj(s)θ k (logθ)−j. (3.4) here qkj(s), skj(s) ∈ l(h(s),h(0)) and are continuous with respect to s for 0 ≤ s ≤ 2t 3 . proof. according to section 1, the mixed problem (1.4) is well posed, the duhamel’s principal holds, and (2.6), (2.7) are fulfilled. thus, we can apply the results of theorem 4. choose b ≥ ρ + 1 such that suppψ1∪suppψ2 ⊂ {x : |x| ≤ b}. take hb ∈ c ∞(r) such that hb(t) = 1 for t ≥ t2(b) + 6t 5 and hb(t) = 0 for t ≤ t2(b) + t. then, for all 0 ≤ s ≤ 2t 3 , statement (iii) of theorem 4 implies hb(t)ψ1v1(t,s)ψ2 = hb(t)ψ1r(t,s)ψ2. thus, f′(hb(t)ψ1v1(t,s)ψ2)(t,θ) admits a meromorphic continuation satisfying property (s ′). from now on, we assume that t2(b) = k0t with k0 ∈ n. for i(θ) > at we have f′(ψ1v1(t,s)ψ2)(t,θ) = f ′(hb(t)ψ1v1(t,s)ψ2)(t,θ) + f ′[(1 − hb(t))ψ1v1(t,s)ψ2](t,θ). (3.5) since 1 − hb(t) = 0 for t ≥ t2(b) + 6t 5 = (k0 + 1)t + t 5 , for i(θ) > at, we get f′[(1 − hb(t))ψ1v1(t,s)ψ2](t,θ) = e iθ [ k0+1∑ k=1 (ψ1v(kt,s)ψ2e ikθ ) + γ(−s)ψ1v(0,s)ψ2 ] . thus, f′[(1−hb(t))ψ1v1(t,s)ψ2](t,θ) admits an analytic continuation to c. combining the meromorphic continuation of f′(hb(t)ψ1v1(t,s)ψ2)(t,θ), the analytic continuation of f′[(1−hb(t))ψ1v1(t,s)ψ2](t,θ) and representation (3.5), we obtain the meromorphic continuation of f′(ψ1v1(t,s)ψ2)(t,θ). it remains to prove the meromorphic continuation of f ′(ψ1∂tv1(t,s)ψ2)(t,θ). notice that ∂tv(t,s) = p2u(t,s)p 2 and, for i(θ) > at, f′(ψ1v1(t,s))(t,θ) is well defined. for i(θ) > at, we have ∂t [f ′ (hb(t)ψ1v1(t,s)ψ2)(t,θ)] = iθ t f′(hb(t)ψ1v1(t,s)ψ2)(t,θ) + f ′ (h′b(t)ψ1v1(t,s)ψ2)(t,θ) + f′(hb(t)ψ1∂tv1(t,s))(t,θ) thus, for i(θ) > at, we get f′(hb(t)ψ1∂tv1(t,s)ψ2)(t,θ) =∂t [f ′(hb(t)ψ1v1(t,s)ψ2)(t,θ)] − f ′(h′b(t)ψ1v1(t,s)ψ2)(t,θ) − iθ t f′(hb(t)ψ1v1(t,s)ψ2)(t,θ) (3.6) 164 yavar kian cubo 14, 2 (2012) since f′(hb(t)ψ1v1(t,s)ψ2)(t,θ) admits a meromorphic continuation satisfying property (s ′) ∂t [f ′(hb(t)ψ1v1(t,s)ψ2)(t,θ)] and iθ t f′(hb(t)ψ1v1(t,s)ψ2)(t,θ) admit also a meromorphic continuation satisfying property (s′). moreover, since h′b(t) = 0 for t ≥ t2(b) + 6t 5 , f′(h′b(t)ψ1v1(t,s)ψ2)(t,θ) admits an analytic continuation with respect to θ. it follows that f′(hb(t)ψ1∂tv1(t,s)ψ2)(t,θ) admits a meromorphic continuation satisfying property (s′). we conclude by repeating the arguments used for proving the meromorphic continuation of f′(ψ1v1(t,s)ψ2)(t,θ). consider the operator defined by u(t,s) = p1u(t,s)p 1. for all h ∈ ḣ1(ω(s)), w = u(t,s)h is the solution of    ∂2tw − divx(a(t,x)∇xw) = 0, w|∂ω = 0, (w,wt)|t=s = (h,0). let γ1 ∈ c ∞(r) be such that γ1(t) = 1 for t ≥ − t 20 and γ1(t) = 0 for t ≤ − t 10 . set u1(t,s) = γ1(t − s)u(t,s). applying (2.6), for i(θ) > at and ψ1 ψ2 ∈ c ∞ 0 (r n) we can define f′(ψ1u1(t,s)ψ2)(t,θ) and f′(ψ1∂tu1(t,s)ψ2)(t,θ). from the results of lemma 1 we obtain the following meromorphic continuation of f′(ψ1u1(t,s)ψ2)(t,θ) and f ′(ψ1∂tu1(t,s)ψ2)(t,θ). lemma 3.2. assume (h1), (1.1), (1.2), (1.3) and (1.5) fulfilled. then, for all ψ1, ψ2 ∈ c ∞ 0 (r n), f′(ψ1u1(t,s)ψ2)(t,θ) and f ′(ψ1∂tu1(t,s)ψ2)(t,θ) admit a meromorphic continuation with respect to θ, continuous with respect to s ∈ [ 0, 2t 3 ] , from {θ ∈ c : i(θ) > at} to c for n ≥ 3 odd and to c′ for n ≥ 4 even. moreover, for n ≥ 4 even, there exists ǫ0 > 0 such that, for |θ| ≤ ǫ0, we have f′(ψ1u1(t,0)ψ2)(t,θ) = ∑ k≥−m ∑ j≥−mk mkjθ k (logθ)−j. (3.7) f′(ψ1∂tu1(t,0)ψ2)(t,θ) = ∑ k≥−m ∑ j≥−mk nkjθ k(logθ)−j. (3.8) here mkj, nkj ∈ l(h(0)) and, for k < 0 or j > 0, mkj, nkj are a finite rank operator. cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 165 proof. let α ∈ c∞(r) be such that α(t) = 0 for t ≤ t 2 and α(t) = 1 for t ≥ 2t 3 . for all h ∈ ḣ1(ω(0)), z = α(t)u(t,0)h is the solution of    ∂2tz − divx(a(t,x)∇xz) = [∂ 2 t,α](t)u(t,0)h, z|∂ω = 0, (z,∂tz)|t=0 = (0,0). (3.9) we deduce from the cauchy problem (3.9) the following representation u(t,0) = α(t)u(t,0) = ∫t 0 v(t,s)[∂2t,α](s)u(s,0)ds, t ≥ t. (3.10) since [∂2t,α](t) = 0 for t > 2t 3 , the formula (3.10) becomes u(t,0) = ∫ 2t 3 0 v(t,s)[∂2t,α](s)u(s,0)ds, t ≥ t. let r > 0 be such that suppψ1 ∪ suppψ2 ⊂ {x : |x| ≤ r}. choose b = r + ρ + t + 1 and take χ ∈ c∞0 (|x| ≤ b) such that χ(x) = 1 for |x| ≤ r + ρ + t. the finite speed of propagation implies ψ1u(t,0)ψ2 = ∫ 2t 3 0 ψ1v(t,s)χ[∂ 2 t,α](s)u(s,0)ψ2ds, t ≥ t. (3.11) thus, for i(θ) > at, we obtain f′(ψ1u1(t,0)ψ2)(t,θ) =f ′ [∫ 2t 3 0 ψ1v1(t,s)χ[∂ 2 t,α](s)u(s,0)ψ2ds ] (t,θ) − ∫ 2t 3 0 ψ1v1(0,s)χ[∂ 2 t,α](s)u(s,0)ψ2ds + e iθψ1ψ2 and it follows f′ [ψ1u1(t,0)ψ2] (t,θ) = ∫ 2t 3 0 f′ [ψ1v1(t,s)χ] (t,θ)[∂ 2 t,α](s)u(s,0)ψ2ds − ∫ 2t 3 0 ψ1v1(0,s)χ[∂ 2 t,α](s)u(s,0)ψ2ds + e iθψ1ψ2. combining this representation with the meromorphic continuation of f′(ψ1v1(t,s)χ)(t,θ) established in lemma 1, we prove the meromorphic continuation of f′(ψ1u(t,0)ψ2)(t,θ) as well as (3.7). 166 yavar kian cubo 14, 2 (2012) it remains to prove the meromorphic continuation of f′(ψ1∂tu(t,0)ψ2)(t,θ). let β ∈ c ∞ 0 (r n). the formula (3.11) implies that, for t ≥ t, we have ∂tu(t,0)β = ∫ 2t 3 0 ∂tv(t,s)[∂ 2 t,α](s)u(s,0)βds. by density, this leads to ψ1∂tu(t,0)ψ2 = ∫ 2t 3 0 ψ1∂tv(t,s)χ[∂ 2 t,α](s)u(s,0)ψ2ds, t ≥ t and, for i(θ) > at, we get f′(ψ1∂tu1(t,0)ψ2)(t,θ) = ∫ 2t 3 0 f′(ψ1∂tv1(t,s)χ)(t,θ)[∂ 2 t,α](s)u(s,0)ψ2ds − ∫ 2t 3 0 ψ1v1(0,s)χ[∂ 2 t,α](s)u(s,0)ψ2ds. we conclude by combining this representation with the results of lemma 1. proof of theorem 4. by definition, we can write γ1(t)ψ1u(t,0)ψ2 = ( γ1(t)ψ1u(t,0)ψ2 γ1(t)ψ1v(t,0)ψ2 γ1(t)ψ1∂tu(t,0)ψ2 γ1(t)ψ1∂tv(t,0)ψ2 ) . moreover, for i(θ) > at, we have f′ [γ1(t)ψ1u(t,0)ψ2] (t,θ) = e iθ ∞∑ k=0 ( ψ1u(t + kt,0)ψ2e ikθ ) = −e−iθrψ1,ψ2(θ) (3.12) and we obtain rψ1,ψ2(θ) = −e iθf′ [γ1(t)ψ1u(t,0)ψ2] (t,θ) = − ( eiθf′(ψ1u1(t,0)ψ2)(t,θ) e iθf′(ψ1v1(t,0)ψ2)(t,θ) eiθf′(ψ1∂tu1(t,0)ψ2)(t,θ) e iθf′(ψ1∂tv1(t,0)ψ2)(t,θ) ) . thus, combining the results of lemma 1 and lemma 2, we prove theorem 4. � 4 local energy decay the goal of this section is to prove theorem 1, assuming (h1) and (h2) fulfilled. for this purpose, we show how assumption (h2) alter the meromorphic continuation of rψ1,ψ2(θ) established in section 2. then, by integrating on a suitable contour, we prove the local energy decay. we treat separately the case of odd and even dimensions. we start with n odd. cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 167 lemma 4.1. assume n ≥ 3 odd, (1.1), (1.2), (1.3), (1.5), (h1) and (h2) fulfilled. then, for all ψ1, ψ2 ∈ c ∞ 0 (|x| ≤ ρ + 1), we get ‖ψ1u(t,s)ψ2‖l(h(s),h(t)) ≤ ce −δ(t−s), t ≥ s. (4.1) proof. notice that, for i(θ) > at, f′ [γ1(t)φ1u(t,0)φ2] (t,θ) is t-periodic with respect to t and 2π-periodic with respect to θ(see [26] theorem ). applying (3.12), we get f′ [γ1(t)φ1u(t,0)φ2] (dt,θ) = f ′ [γ1(t)φ1u(t,0)φ2] (t,θ) = −e −iθrφ1,φ2(θ). (4.2) moreover, from [26] we have the following inversion formula (see lemma 1 of [26]) φ1u(dt,0)φ2 = 1 2π ∫ [i(a+1)t−π,i(a+1)t+π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ. (4.3) we will show (4.1), by combining these statements with assumption (h2). first, assumption (h2) and (4.2) imply that f′ [γ1(t)φ1u(t,0)φ2] (dt,θ) has no poles on {θ : i(θ ≥ 0)}. it follows that there exists δ > 0 such that f′ [γ1(t)φ1u(t,0)φ2] (dt,θ) has no poles on {θ : i(θ) ≥ −δt, −π ≤ re(θ) ≤ π}. consider the contour c1 defined by c1 = [i(a+1)t+π,i(a+1)t−π]∪[i(a+1)t−π,−iδt−π]∪[−iδt−π,−iδt+π]∪[−iδt+π,i(a+1)t+π]. the cauchy formula implies ∫ c1 e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ = 0. also, since f′ [γ1(t)φ1u(t,0)φ2] (t,θ) is 2π-periodic with respect to θ we have ∫ [i(a+1)t−π,−iδt−π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ = − ∫ [−iδt+π,i(a+1)t+π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ and we obtain ∫ [i(a+1)t−π,i(a+1)t+π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ = ∫ [−iδt−π,−iδt+π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ. (4.4) it is obvious that ∫ [−iδt−π,−iδt+π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ = e −δ(dt) ∫ [−π,π] e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ − iδt)dθ and combining this with (4.4) and the inversion formula (4.3), we get ‖φ1u(dt,0)φ2‖l(h(0)) ≤ ce −δ(dt). (4.5) now let ψ1, ψ2 ∈ c ∞ 0 (|x| ≤ ρ + 1), and let t, s ∈ r be such that t ≥ s. we write t = t ′ + mt and s = s′ + kt with 0 ≤ t′,s′ < t and m, k ∈ n. the finite speed of propagation implies ψ1u(t,s)ψ2 = ψ1u(t ′,0)φ1u((m − k)t,0)φ2u(0,s ′ )ψ2. 168 yavar kian cubo 14, 2 (2012) then, applying (4.5) and theorem 2, we obtain ‖ψ1u(t,s)ψ2‖l(h(s),h(t)) ≤ c‖φ1u((m − k)t,0)φ2‖l(h(0)) ≤ c ′e−δ((m−k)t) ≤ c′e−δ(t−s). lemma 4.2. assume n ≥ 4 even, (1.1), (1.2), (1.3), (1.5), (h1) and (h2) fulfilled. then, for all ψ1, ψ2 ∈ c ∞ 0 (|x| ≤ ρ + 1), we get ‖ψ1u(t,s)ψ2‖l(h(s),h(t)) ≤ cp(t − s), t ≥ s (4.6) with p(t) = 1 (t + 1) ln 2 (t + e) . proof. repeating the arguments used in the proof of lemma 3, we obtain that f′ [γ1(t)φ1u(t,0)φ2] (t,θ) has no poles on {θ ∈ c′ : im(θ) ≥ 0}. moreover, representation (3.1) implies that there exists ǫ0 > 0 such that for θ ∈ c ′ with |θ| ≤ ǫ0 we have f′ [γ1(t)φ1u(t,0)φ2] (t,θ) = ∑ k≥−m ∑ j≥−mk rkjθ k(logθ)−j (4.7) and assumption (h2) implies that in this representation we have rkj = 0 for k < 0 or k = 0 and j < 0. it follows that, for θ ∈ c′ with |θ| ≤ ǫ0, we obtain the following representation f′ [γ1(t)φ1u(t,0)φ2] (t,θ) = a(θ) + bθ m0 log(θ)−µ + o θ→0 ( θm0 log(θ)−µ ) (4.8) with a(θ) analytic with respect to θ for |θ| ≤ ǫ0 , b a finite-dimensional operator, m0 ≥ 0 and µ ≥ 1. since f′ [γ1(t)φ1u(t,0)φ2] (t,θ) has no poles on {θ ∈ c ′ : im(θ) ≥ 0}, there exists 0 < δ ≤ ǫ0 t and 0 < ν < ǫ0 sufficiently small such that f ′ [γ1(t)φ1u(t,0)φ2] (t,θ) has no poles on {θ ∈ c : im(θ) ≥ −δt, −π ≤ re(θ) ≤ −ν, ν ≤ re(θ) ≤ π}. consider the contour σ = γ1 ∪ ω ∪ γ2 where γ1 = [−iδt − π,−iδt − ν], γ2 = [−iδ + ν,−iδ + π]. the contour ω of c, is a curve connecting −iδt − ν and −iδt + ν symmetric with respect to the axis re(θ) = 0. the part of ω lying in {θ : im(θ) ≥ 0} is a half-circle with radius ν, ω∩{θ : re(θ) < 0, im(θ) ≤ 0} = [−ν−iδt,−ν] and ω∩{θ : re(θ) > 0, im(θ) ≤ 0} = [ν,ν−iδt]. thus, ω is included in the region where we have no poles of f′ [γ1(t)φ1u(t,0)φ2] (t,θ). consider the closed contour c2 = [i(a + 1)t + π,i(a + 1)t − π] ∪ [i(a + 1)t − π,−iδt − π] ∪ σ ∪ [−iδt + π,i(a + 1)t + π]. an application of the cauchy formula yields ∫ c2 e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ = 0. cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 169 applying the same arguments as those used in the proof of lemma 3, we obtain ∫ [i(a+1)t−π,i(a+1)t+π] f [γ1(t)φ1u(t,0)φ2] (dt,θ)dθ = ∫ σ e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ and the inversion formula (4.3) implies φ1u(dt,0)φ2 = 1 2π ∫ σ e−idθf′ [γ1(t)φ1u(t,0)φ2] (t,θ)dθ, d ∈ n. (4.9) combining this representation with (4.8) and applying some arguments used in lemma 2 and lemma 3 of [13], we obtain (4.6). combining the results of lemma 3 and lemma 4, we prove theorem 1. 5 examples of metrics a(t,x) and obstacles o(t) in this section we will apply some properties of solutions of the wave equations with non-trapping metrics independent of t and fixed obstacle to construct time periodic metrics and moving obstacles such that conditions (h1) and (h2) are fulfilled. for this purpose, we assume that (h1) is fulfilled for the metrics a(t,x) and obstacle o(t) that we consider and we will establish examples for (h2). in order to prove (h2), we will modify the size t of the period of a(t,x). this choice is justified by the properties of u(t,s). let t1 > 0 and let ((at(t,x),ot(t)))t≥t1 be a family of couples of functions and obstacles such that the following conditions are fulfilled: (h3i) at(t,x) and ot(t) are t-periodic with respect to t and at(t,x) satisfies (1.5), (h3ii) for all t ≥ t1, if (a(t,x),o(t)) = (at(t,x),ot(t)) then conditions (1.1), (1.2), (1.3) and (h1) are fulfilled, (h3iii) there exist a function a1(x) and an obstacle o independent of t such that for (a(t,x),o(t)) = (a1(x),o) condition (h1) is fulfilled and, for all t1 ≤ t ≤ t, we have at(t,x) = a1(x) and ot(t) = o. let h be the closure of the space c∞0 (r n \ o) × c∞0 (r n \ o) with respect to the norm ‖f‖h =    ∫ rn\o ( |∇xf1| 2 + |f2| 2 ) dx    1 2 , f = (f1,f2) ∈ c ∞ 0 (r n \ o) × c∞0 (r n \ o). 170 yavar kian cubo 14, 2 (2012) consider the following cauchy problem    vtt − divx(a1(x)∇xv) = 0, t ∈ r, x ∈ r n \ o v|∂o = 0, t ∈ r, (v,vt)(0) = f, (5.1) and the associate propagator v(t) : h ∋ f 7−→ (v,vt)(t) ∈ h. let u be solution of (2.2). for t1 ≤ t ≤ t we have ∂2tu − divx(a1(x)∇xu) = ∂ 2 tu − divx(at(t,x)∇xu) = 0. it follows that for (a(t,x),o(t)) = (at(t,x),ot(t)) we get u(t,s) = v(t − s), t1 ≤ s < t ≤ t (5.2) and h(t) = h, t1 ≤ t ≤ t. (5.3) the asymptotic expansion of χv(t)χ as t → +∞ has been studied by many authors (see [25], [24] and [28]). it has been proven that, for non-trapping metrics and for n ≥ 3, the local energy decreases. to prove (h2), we will apply the following result. theorem 5. assume n ≥ 3. let φ ∈ c∞0 (r n). then, we have ‖φv(t)φ‖l(h) ≤ cφp(t) (5.4) with { p(t) =e−δt for n odd, p(t) = 〈t〉 1−n for n even. estimate (5.4) has been established by vainberg in [24], [25] but also by vodev in [27] and [28]. for n ≥ 4 even we will use the following identity. lemma 5.1. let ψ ∈ c∞0 (|x| ≤ ρ+ 1+t1) be such that ψ = 1, for |x| ≤ ρ+ 1 2 + t1. then, we have u(t1,0) − v(t1) = ψ(u(t1,0) − v(t1)) = (u(t1,0) − v(t1))ψ. (5.5) proof. first, notice that (5.3) implies h(0) = h. now, choose g ∈ h(0) = h and let w be the function defined by (w,wt)(t) = u(t,0)(1−ψ)g. the finite speed of propagation implies that, for 0 ≤ t ≤ t1 and |x| ≤ ρ + 1 2 , we get w(t,x) = 0. moreover, we have divx(a1(x)∇x) = ∆x = divx(a(t,x)∇x), for |x| > ρ. (5.6) thus, w is solution on 0 ≤ t ≤ t1 of the problem    wtt − divx(a1(x)∇xw) = 0, t ∈ r, x ∈ r n \ o w|∂o = 0, t ∈ r, (w,wt)(0) = (1 − ψ)g cubo 14, 2 (2012) local energy decay for the wave equation with a time-periodic ... 171 and it follows that (u(t1,0) − v(t1))(1 − ψ) = 0. (5.7) now, let u and v be the functions defined by (u,ut)(t) = u(t,0)g and (v,vt)(t) = v(t)g with g ∈ h. applying (5.6), we can easily show that on (1 − ψ)u is the solution of { ∂2t((1 − ψ)u)) − ∆x((1 − ψ)u)) = [∆x,ψ]u, (((1 − ψ)u),((1 − ψ)u)t)(0) = (1 − ψ)g, and (1 − ψ)v is the solution of { ∂2t(((1 − ψ)v)) − ∆x((1 − ψ)v)) = [∆x,ψ]v, (((1 − ψ)v),((1 − ψ)v)t)(0) = (1 − ψ)g. we have (1 − ψ)(u(t1,0) − v(t1)) = 0. (5.8) combining (5.7) and (5.8), we get (5.5). combining the arguments used in the proofs of lemma 7, 8 and 9 and theorem 14 of [13] with the identity (5.5), we obtain the following. theorem 6. assume n ≥ 3 and let ((at(t,x),ot(t)))t≥t1 satisfy (h3i), (h3ii), (h3iii). then, for t large enough and for (a(t,x),o(t)) = (at(t,x),ot(t)), assumption (h2) is fulfilled. received: november 2011. revised: november 2011. references [1] a. bachelot and v. petkov, existence des opérateurs d’ondes pour les systèmes hyperboliques avec un potentiel périodique en temps, ann. inst. h. poincaré (physique théorique), 47 (1987), 383-428. [2] c. o. bloom and n. d. kazarinoff, energy decays locally even if total energy grows algebraically with time, j. differential equation, 16 (1974), 352-372. [3] j-f. bony and v. petkov, resonances for non trapping time-periodic perturbation, j. phys. a: math. gen., 37 (2004), 9439-9449. 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[28] g. vodev, local energy decay of solutions to the wave equation for non-trapping metrics, ark. math, 42 (2004), no 2, 379-397. introduction general properties the meromorphic continuation of the cut-off resolvent r1,2() local energy decay examples of metrics a(t,x) and obstacles o(t) cubo a mathematical journal vol.10, n o ¯ 04, (73–83). december 2008 an intersection theorem and its applications mircea balaj department of mathematics, university of oradea, romania email: mbalaj@uoradea.ro and donal o’regan department of mathematics, national university of ireland, galway, ireland email: donal.oregan@nuigalway.ie abstract in this paper we obtain a very general intersection theorem for the values of a map. from this we derive existence theorems for two types of vectorial equilibrium problems, an analytic alternative and a minimax inequality involving three real functions. resumen en este art́ıculo obtenemos un teorema general de intersección para los valores de una aplicación. a través de este resultado deducimos teoremas de existencia para dos tipos de problemas de equilibrio vectoriales, una alternativa anaĺıtica y una desigualdad minimax envolviendo tres funciones reales. key words and phrases: the better admissible class, fixed point, quasiconvex map, equilibrium problem. math. subj. class.: 54c60, 49j35, 91b50. 74 mircea balaj and donal o’regan cubo 10, 4 (2008) 1. introduction and preliminaries a multimap (or simply a map) t : x ⊸ y is a function from a set x into the power set 2y of y , that is a function with the values t (x) ⊆ y for x ∈ x. to a map t : x ⊸ y we associate two other maps t c : x ⊸ y and t − : y ⊸ x defined by t c(x) = y \ t (x), and respectively t −(y) = {x ∈ x : y ∈ t (x)} the values of t − are called the fibers of t . let t : x ⊸ y be a map. as usual the set {(x, y) ∈ x × y : y ∈ t (x)} is called the graph of t . for a ⊆ x, and b ⊆ y let t (a) = ⋃ x∈a t (x) and t −(b) = {x ∈ x : t (x) ∩ b 6= ∅}. for topological spaces x and y a map t : x ⊸ y is said to be: upper semicontinuous (u.s.c.) if for any closed set f ⊆ y the set t −(f ) is closed in x; lower semicontinuous (l.s.c.) if for any open set u ⊆ y the set t −(u ) is open in x; compact if t (x) is contained in a compact subset of y ; closed if its graph is closed in x × y . the following lemma collects known facts about u.s.c. or l.s.c. maps (see for example [7] for assertion (i), [16] for assertion (ii) and [9] for assertion (iii)). lemma 1 let x and y be topological spaces and t : x ⊸ y be a map. (i) if t has compact values, then it is u.s.c. if and only if for each x ∈ x, any net {xt} converging to x and any net {yt} with yt ∈ t (xt) for all index t, there exists a subnet {yt′} of {yt} and y ∈ t (x) such that {yt′} converges to y. (ii) t is l.s.c. in x ∈ x if and only if for any y ∈ t (x) and any net {xt} converging to x, there exists a net {yt} converging to y, with yt ∈ t (xt) for each t. (iii) if y is compact and t is closed, then t is u.s.c.. if x is a subset of a topological vector space we denote by cox and x the convex hull and the closure of x respectively. let y be a convex set in a topological vector space and x be a topological space. the better admissible class b of mappings from y into x (see [15]) is defined as follows: t ∈ b(y, x) ⇔ t : y ⊸ x is a mapping such that for any nonempty finite subset a of y and any continuous mapping p : t (co a) → co a the composition p ◦ t|co a : co a ⊸ co a has a fixed point. the class b(y, x) includes many important classes of mappings, such as u kc (y, x) in [14], kkm (y, x) in [3] and a(y, x) in [2], as proper subclasses. definition 1. let x be a convex set in a vector space and y a vector space. a mapping t : x ⊸ y is called: (i) quasiconvex, if for every convex subset c of y , t −(c) is a convex set; (ii) convex, if for each x1, x2 ∈ x and λ ∈ (0, 1), λt (x1) + (1 − λ)t (x2) ⊆ t (λx1 + (1 − λ)x2); cubo 10, 4 (2008) an intersection theorem and its applications 75 (iii) concave, if for each x1, x2 ∈ x and λ ∈ (0, 1), t (λx1 + (1 − λ)x2) ⊆ λt (x1) + (1 − λ)t (x2). lemma 2 if a map t : x ⊸ y is convex then it is quasiconvex. proof. let c be a convex subset of y , x1, x2 ∈ t − (c) and λ ∈ (0, 1). if y1 ∈ t (x1) ∩ c, y2 ∈ t (x2) ∩ c, then λy1 + (1 − λ)y2 ∈ (λt (x1) + (1 − λ)t (x2)) ∩ c ⊆ t (λx1 + (1 − λ)x2) ∩ c, hence λx1 + (1 − λ)x2 ∈ t − (c). let us describe in short the contents on the next sections. we obtain first a very general intersection theorem involving three maps, one of them from the class b. two types of applications of this result will be given in the last two sections. the first one, offers existence theorems for the following types of vectorial equilibrium problems: let x be a topological space, y be a convex set in a topological vector space, z be a topological vector space and v be nonempty set. let f : y × z ⊸ v , c : z ⊸ v and p : x ⊸ z. (i) find x0 ∈ x such that f (y, z) ⊆ c(z) for each y ∈ y and z ∈ p (x0); and respectively, (ii) find x0 ∈ x such that f (y, z) ∩ c(z) 6= ∅ for each y ∈ y and z ∈ p (x0). finally, we obtain an analytic alternative and a minimax inequality involving three real functions. from now all (topological) vector spaces will be assumed real and all topological (vector) spaces will be assumed hausdorff. 2. an intersection theorem theorem 1. let x be a topological space, y be a convex set in a topological vector space and z be a nonempty set. let p : x ⊸ z, q : y ⊸ z two maps satisfying the following conditions: (i) for each y ∈ y , {x ∈ x : p (x) ⊆ q(y)} is closed; (ii) p has convex values and qc is quasiconvex; (iii) there exists a compact mapping t ∈ b(y, x) such that for each y ∈ y , p (t (y)) ⊆ q(y). then there exists x0 ∈ x such that p (x0) ⊆ ⋂ y∈y q(y). 76 mircea balaj and donal o’regan cubo 10, 4 (2008) proof. let s : y ⊸ x be the map defined by s(y) = {x ∈ x : p (x) * q(y)}. suppose that the conclusion of theorem is false. then x = ⋃ y∈y s(y). let x0 = t (y ). since x0 is compact there exists a finite set a = {y1, y2, . . . , yn} ⊆ y such that x0 = ⋃n i=1 (s(yi) ∩ x0). let {α1, α2, . . . , αn} be a partition of unity on x0 subordinated to the cover {s(yi) ∩ x0 : 0 ≤ i ≤ n}. recall that this means that        αi : x0 → [0, 1] is continuous, for each i ∈ {1, 2, . . . , n}; αi(x) > 0 ⇒ x ∈ s(yi); ∑n i=1 αi(x) = 1 for each x ∈ x0. define f : t (co a) → co a by f (x) = n ∑ i=1 αi(x)yi for all x ∈ t (co a). since f is continuous and t ∈ b(y, x), f ◦ t|a : coa ⊸ coa has a fixed point. hence there exists ỹ ∈ coa such that ỹ ∈ f (t (ỹ)). then, for some x̃ ∈ t (ỹ) we have ỹ = f (x̃). let i = {i ∈ {1, . . . , n} : αi(x̃) > 0}. then ỹ = f (x̃) ∈ co{yi : i ∈ i}. for each i ∈ i, x̃ ∈ s(yi), hence p (x̃) ∩ qc(yi) 6= ∅. by (ii) it follows that p (x̃) ∩ q c (ỹ) 6= ∅, or equivalently, p (x̃) * q(y). since x̃ ∈ t (ỹ), we get p (t (y)) * q(y), which contradicts (iii). proposition 2. if z is topological space, then condition (i) in theorem 1 is fulfilled in any of the following cases: (i1) p has open fibers; (i2) p is l.s.c. and q has closed values; proof. if p has open values then for each y ∈ y the set {x ∈ x : p (x) * q(y)} = ⋃ z∈qc(y) p −(z) is open, hence {x ∈ x : p (x) ⊆ q(y)} = x \ {x ∈ x : p (x) * q(y)} is closed. by the definition of lower semicontinuity it follows that if (i2) holds then each set {x ∈ x : p (x) ⊆ q(y)} is closed. 3. equilibrium theorems in [5], [6], [10-13], for a suitable choice of the sets y, z and v and of the maps f : y × z ⊸ v and c : z ⊸ v the authors study, all or part of the following problems: cubo 10, 4 (2008) an intersection theorem and its applications 77 (i) find z0 ∈ z such that f (y, z0) ⊆ c(z0) for all y ∈ y ; (ii) find z0 ∈ z such that f (y, z0) ∩ c(z0) 6= ∅ for all y ∈ y ; (iii) find z0 ∈ z such that f (y, z0) * c(z0) for all y ∈ y ; (iv) find z0 ∈ z such that f (y, z0) ∩ c(z0) = ∅ for all y ∈ y . each existence result concerning problem (i) (respectively, (ii)), yields an existence theorem for problem (iv) (respectively, (iii)), if we take into account the following equivalences: f (y, z) ⊆ c(z) ⇔ f (y, z) ∩ cc(z) = ∅ and f (y, z) ∩ c(z) 6= ∅ ⇔ f (y, z) * cc(z). for this reason we can fix our attention on problems (i) and (ii), only. in this section we study equilibrium problems more general than (i) and (ii): let x be a topological space, y be a convex set in a topological vector space, z be a topological vector space and v be a nonempty set. let f : y × z ⊸ v , c : z ⊸ v and p : x ⊸ z. (v) find x0 ∈ x such that f (y, z) ⊆ c(z) for each y ∈ y and z ∈ p (x0); and respectively, (vi) find x0 ∈ x such that f (y, z) ∩ c(z) 6= ∅ for each y ∈ y and z ∈ p (x0). of course, when x = z and p (z) = {z} for all z ∈ z, problem (v) (respectively (vi)), reduces to problem (i) (respectively (ii)). theorem 3. suppose that the maps f , c and p satisfy the following conditions: (i) one of the following two requirements is fulfilled: (i1) p has open fibers; (i2) p is l.s.c., c is closed map and for each y ∈ y , f (y, ·) is l.s.c. (ii) f and cc are convex maps, p has convex values; (iii) there exists a compact mapping t ∈ b(y, x) such that f (y, z) ⊆ c(z), for each y ∈ y and z ∈ p (t (y)). then there exists x0 ∈ x such that f (y, z) ⊆ c(z) for each y ∈ y and z ∈ p (x0). proof. let q : y ⊸ z be the map defined by q(y) = {z ∈ z : f (x, z) ⊆ c(z)}. we prove that if (i2) holds, then q has closed values. let y ∈ y and {zt}t∈∆ be a net in q(y) converging to z ∈ z. if v ∈ f (y, z), since f (y, ·) is l.s.c., there exists a net {vt}t∈∆ converging to v such that vt ∈ f (y, zt), for all t ∈ ∆. since zt ∈ q(y), vt ∈ f (y, zt) ⊆ c(zt). the map c is closed, hence v ∈ c(z). thus, f (y, z) ⊆ c(z), hence z ∈ q(y). by proposition 2, in both cases (i1) and (i2), condition (i) in theorem 1 is satisfied. 78 mircea balaj and donal o’regan cubo 10, 4 (2008) we show next that the map qc is convex. let y1, y2 ∈ y , λ ∈ (0.1) and z ∈ λq c (y1) + (1 − λ)qc(y2). there exist z1, z2 ∈ z such that z = λz1 + (1 − λ)z2 and v1, v2 ∈ v such that vi ∈ f (yi, zi) ∩ c c (zi), for i = 1, 2. since the maps f and c c are convex, λv1 + (1 − λ)v2 ∈ λf (y1, z1) + (1 − λ)f (y2, z2) ⊆ f (λy1 + (1 − λ)y2, λz1 + (1 − λ)z2), and similarly, λv1 + (1 − λ)v2 ∈ c c (λz1 + (1 − λ)z2). thus, λv1 + (1 − λ)v2 ∈ f (λy1 + (1 − λ)y2, z) ∩ c c (z), hence z ∈ q(λy1 + (1 − λ)y2). hence qc is convex and by lemma 2, it is quasiconvex. it is clear that condition (iii) is equivalent to the requirement similarly denoted in theorem 1, hence all requirements of this theorem are fulfilled. consequently, there exists x0 ∈ x such that p (x0) ⊆ ⋂ y∈y q(y), that is, f (y, z) ⊆ c(z), for each y ∈ y and z ∈ p (x0). theorem 4. suppose that the maps f , c and p satisfy the following conditions: (i) one of the following two requirements is fulfilled: (i1) p has open fibers; (i2) p is l.s.c., c is u.s.c. with compact values and for each y ∈ y , f (y, ·) is closed. (ii) f is concave map, cc is convex map and p has convex values; (iii) there exists a compact mapping t ∈ b(y, x) such that f (y, z) ∩ c(z) 6= ∅, for each y ∈ y and z ∈ p (t (y)). then there exists x0 ∈ x such that f (y, z) ∩ c(z) 6= ∅ for each y ∈ y and z ∈ p (x0). proof. the proof is similar to that of theorem 3. let q : y ⊸ z be the map defined by q(y) = {z ∈ z : f (x, z) ∩ c(z) 6= ∅}. we show first that if (i2) holds, then q has closed values. let y ∈ y and {zt}t∈∆ be a net in q(y) converging to z ∈ z. then, for each t ∈ ∆, there exists vt ∈ f (yt, zt) ∩ c(zt). since c is u.s.c. with compact values, by lemma 1 (i), there exist a subnet {vt′} of {vt} and v ∈ c(z) such that vt′ → v. since f (y, ·) is closed, v ∈ f (y, z). therefore f (y, z) ∩ c(z) 6= ∅, hence z ∈ q(y). let y1, y2 ∈ y , λ ∈ (0.1) and z ∈ λq c (y1) + (1 − λ)q c (y2). there exist z1, z2 ∈ z such that z = λz1 + (1 − λ)z2 and f (y1, z1) ⊆ c c (z1), f (y2, z2) ⊆ c c (z2). by (ii) we infer that f (λy1 + (1 − λ)y2, λz1 + (1 − λ)z2) ⊆ λf (y1, z1) + (1 − λ)f (y2, z2) ⊆ λc c (z1) + (1 − λ)c (z2) ⊆ cc(λz1 + (1 − λ)z2). it follows that z ∈ qc(λy1 + (1 − λ)y2), hence the map q c is convex. the maps p and q satisfy all the requirements of theorem 1 and the desired conclusion follows from this theorem. cubo 10, 4 (2008) an intersection theorem and its applications 79 4. analytic alternative, minimax inequality definition 2. (see [1]). let x and y be convex sets in two vector spaces. we say that a function q : y × z → r is (y, z)-quasiconvex if for any finite subset {(y1, z1), . . . , (yn, zn)} of y × z, and each y ∈ co {y1, . . . , yn} there exists z ∈ co {z1, . . . zn} such that q(y, z) ≤ max1≤i≤n q(yi, zi). it is clear that any function q : y × z → r quasiconvex on y × z is (y, z)-quasiconvex but example 2 in [1] shows that the converse is not true. definition 3. let x and z be topological spaces. a function p : x×z → r is said to be marginally upper semicontinuous in x (see [8]) if for every open subset u of z the function x → infz∈u p(x, z) is upper semicontinuous on x. any function upper semicontinuous in x is marginally upper semicontinuous in x but the example given in [8], p.249 shows that the converse is not true. theorem 5. let x be topological space, y and z be convex sets in topological vector spaces, p : x × z → r, q : y × z → r, t : x × y → r be functions and α, β, λ be real numbers. suppose that the following conditions are satisfied: (i) one of the following requirements is fulfilled: (i1) for each z ∈ z the set {x ∈ x : p(x, z) < α} is open; (i2) p is marginally upper semicontinuous in x and for each y ∈ y the set {z ∈ z : q(y, z) ≥ β} is closed; (ii) for each x ∈ x the set {z ∈ z : p(x, z) < α} is convex; (iii) q is (y, z)-quasiconvex; (iv) for x ∈ x, y ∈ y and z ∈ z the following implication holds: p(x, z) < α and q(y, z) < β ⇒ t(x, y) < λ; (v) the map t : y ⊸ x defined by t (y) = {x ∈ x : t(x, y) ≥ λ} is compact and belongs to the class b(y, x). then at least one of the following assertions holds: (a) there exists x0 ∈ x such that p(x0, z) ≥ α, for all z ∈ z. (b) there exists z0 ∈ z such that q(y, z0) ≥ β, for all y ∈ y . proof. define the maps p : x ⊸ z, q : y ⊸ z, t : x ⊸ y by p (x) = {z ∈ z : p(x, z) < α}, q(y) = {z ∈ z : q(y, z) ≥ β}, and 80 mircea balaj and donal o’regan cubo 10, 4 (2008) t (y) = {x ∈ x : t(x, y) ≥ λ}. if (i1) holds, then p has open fibers, if (i2) holds, then q has closed values and we claim that p is l.s.c. indeed, since p is marginally upper semicontinuous in x, for each open u ⊆ z the set {x ∈ x : p (x) ∩ u 6= ∅} = {x ∈ x : infz∈u p(x, z) < α} is open. hence, according to proposition 2, condition (i) in theorem 1 holds. let c be a convex subset of z, y1, y2 ∈ q c (c) and y ∈ co{y1, y2}. then there exist z1, z2 ∈ c such that q(y1, z1) < β, q(y2, z2) < β. since q is (y, z)-quasiconvex, there exists z ∈ co{z1, z2} ⊆ c such that q(y, z) ≤ max{q(y1, z1), q(y2, z2)] < β. thus y ∈ qc(c), hence q is quasiconvex. we prove that for each y ∈ y , p (t (y)) ⊆ q(y). suppose that for some y ∈ y there exists x ∈ t (y) and z ∈ p (x) \ q(y). by x ∈ t (y), we get t(x, y) ≥ λ. on the other hand, since z ∈ p (x) \ q(y), we have p(x, z) < α, q(y, z) < β and, by (iv), we get t(x, y) < λ; a contradiction. therefore the maps p, q, t satisfy all the requirement of theorem 1. according to this theorem there exists x0 ∈ x such that p (x0) ⊆ ⋂ y∈y q(y). suppose that both assertions in the conclusion of theorem are false. this means that: (a’) p (x) 6= ∅, for all x ∈ x; (b’) for each z ∈ z there exists y ∈ y such that z /∈ q(y). the following contradiction completes the proof: ∅ 6= p (x0) ⊆ ⋂ y∈y q(y) = ∅. theorem 6. let x be a topological compact space, y and z be two convex sets in topological vector spaces and p : x × z → r, q : y × z → r, t : x × y → r functions. suppose that the following conditions are fulfilled: (i) one of the following requirements is fulfilled: (i1) p is u.s.c. in x; (i2) p is marginally upper semicontinuous in x and q is u.s.c. in z; (ii) p is quasiconvex in z; (iii) q is (y, z)-quasiconvex; (iv) for x ∈ x, y ∈ y and z ∈ z the following implication holds: t(x, y) ≤ p(x, z) + q(y, z); (v) for each λ < infy∈y supx∈x t(x, y) the map t : y ⊸ x, defined by t (y) = {x ∈ x : t(x, y) ≥ λ} belongs to the class b(y, x). cubo 10, 4 (2008) an intersection theorem and its applications 81 then, infy∈y supx∈x t(x, y) ≤ supx∈x infz∈z p(x, z) + supz∈zinfy∈y q(y, z), with the convention ∞ + (−∞) = ∞. proof. we may suppose that inf y∈y sup x∈x t(x, y) > −∞, supx∈x infz∈z p(x, z) < ∞, supz∈zinfy∈y q(y, z) < ∞. by way of contradiction suppose that inf y∈y sup x∈x t(x, y) > supx∈x infz∈z p(x, z) + supz∈zinfy∈y q(y, z) and choose α, β, λ ∈ r such that supx∈xinfz∈z p(x, z) < α, supz∈zinfy∈y q(y, z) < β, λ < infy∈y supx∈x t(x, y), and α + β < λ. we prove that condition (iv) in theorem 5 is fulfilled. let x ∈ x, y ∈ y and z ∈ z such that p(x, z) < α and q(y, z) < β. since α + β < λ, by condition (iv) in the theorem that must be proved, we get t(x, y) ≤ p(x, z) + q(y, z) < α + β < λ. it is easy to see that all the requirements of theorem 5 are fulfilled. we prove that none of assertions (a), (b) of the conclusion of theorem 5 can take place. if (a) happens, then α ≤ infz∈z p(x0, z) ≤ supx∈x infz∈z p(x, z); a contradiction. if (b) happens, then β ≤ infy∈y q(y, z0) ≤ supz∈z infy∈y q(y, z); a contradiction. corollary 7. let x, y and z be convex subsets of three topological vector spaces, x being compact and p : x × z → r, q : y × z → r, t : x × y → r three functions satisfying conditions (i), (ii), (iii), (iv) of theorem 6 and (v’) t is upper semicontinuous on x × y and for each y ∈ y, t(., y) is quasiconcave on x. then, infy∈y supx∈x t(x, y) ≤ supx∈xinfz∈z p(x, z) + supz∈z infy∈y q(y, z), with the convention ∞ + (−∞) = ∞. proof. it suffices to prove that condition (v) in theorem 6 is fulfilled. obviously for each λ < infy∈y supx∈x t(x, y) the map t defined in condition (v) of theorem 6 has nonempty values. moreover, by (v’) the values of t are convex. since t is upper semicontinuous on x × y the map t is closed. since x is compact, by lemma 1, t is upper semicontinuous with compact values. consequently t is a kakutani map. since, k(y, x) ⊂ b(y, x), it follows that condition (v) from theorem 6 is satisfied. 82 mircea balaj and donal o’regan cubo 10, 4 (2008) the results obtained in this section generalize theorems 19, 20 and corollary 21 in [1], where the corresponding map t , in each result, had the kkm property. obviously, the condition t ∈ b(y, x) is a weaker one. received: february 2008. revised: march 2008. references [1] m. balaj, coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities, nonlinear anal., 68 (2008), 3962–3971. 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[16] n.x. tan and p.n. tinh, on the existence of equilibrium points of vector functions, numerical functional analysis optimiz., 19 (1998), 141–156. n6-bo_cubo cubo a mathematical journal vol.11, no¯ 05, (99–115). december 2009 on tikhonov functionals penalized by bregman distances i.r. bleyer and a. leitão department of mathematics, federal university of st. catarina, p.o. box 476, 88040-900, florianópolis, brazil emails: ismaelbleyer@gmail.com, acgleitao@gmail.com abstract we investigate tikhonov regularization methods for linear and nonlinear ill-posed problems in banach spaces, where the penalty term is described by bregman distances. we prove convergence and stability results. moreover, using appropriate source conditions, we are able to derive rates of convergence in terms of bregman distances. we also analyze an iterated tikhonov method for nonlinear problems, where the penalization is given by an appropriate convex functional. resumen investigamos métodos de regularización de tikhonov para problemas no lineales mal-puestos en espacios de banach, donde el término de penalización es descrito por distancias de bregman. probamos resultados de convergencia y estabilidad. además, usando condiciones apropriadas, somos capazes de obtener tasas de convergencia en términos de las distancias de bergman. también analizamos un método de iterados de tikhnov para problemas no lineales donde la penalización es dada por un funcional convexo apropriado. key words and phrases: tikhonov functionals, bregman distances, total variation regularization. math. subj. class.: 65j20, 47j06, 47j25. 100 i.r. bleyer and a. leitão cubo 11, 5 (2009) 1 introduction in this paper we study non-quadratic regularization methods for solving ill-posed operator equations of the form f(u) = y , (1) where f : d(f) ⊂ u → h is an operator between infinite dimensional banach spaces. both linear and nonlinear problems are considered. tikhonov method is widely used to approximate solutions of inverse problems modeled by operator equations in hilbert spaces [11, 5]. in this article we investigate a tikhonov methods, which consist of the minimization of functionals of the type jδα (u) = 1 2 ‖f(u) − yδ‖2 + αh (u) , (2) where α ∈ r+ is called regularization parameter, h (·) is a proper convex functional, and the noisy data yδ satisfy ‖y − yδ‖ < δ . (3) the method presented above represents a generalization of the classical tikhonov regularization. therefore, the following questions arise: • for α > 0, does the solution (2) exist? does the solution depends continuously on the data yδ? • is the method convergent? (i.e., if the data y is exact and α → 0, do the minimizers of (2) converge to a solution of (1)?) • is the method stable in the sense that: if α = α(δ) is chosen appropriately, do the minimizers of (2) converge to a solution of (1) as δ → 0? • what is the rate of convergence? how should the parameter α = α(δ) be chosen in order to get optimal convergence rates? the first point above is answered in [6]. throughout this article we assume the following assumptions. assumption 1.1. (a1) given the banach spaces u and h one associates the topologies τu and τh , respectively, which are weaker than the norm topologies; (a2) the topological duals of u and h are denoted by u∗ and h ∗, respectively; (a3) the norm ‖·‖ is sequentially lower semi-continuous with respect to τh , i.e., for uk → u with respect to the τu topology, h (u) ≤ lim infk h (uk); cubo 11, 5 (2009) on tikhonov functionals penalized ... 101 (a4) d(f) has non-empty interior with respect to the norm topology and is τu -closed. moreover, d(f) ∩ dom h 6= ∅; (a5) f : d(f) ⊆u → h is continuous from (u ,τu) to (h ,τh ); (a6) the functional h : [0, +∞] → h is proper, convex, bounded from below and τu lower semicontinuous; (a7) for every m > 0 , α > 0, the sets mα (m) = { u ∈ u | jδα (u) ≤ m } are τu compact, i.e. every sequence (uk) in mα (m) has a subsequence, which is convergent in u with respect to the τu topology. the goal of this paper is to answer the last three questions posed above. we obtain convergence rates and error estimates with respect to the generalized bregman distances, originally introduced in [3]. even though this tool does not satisfy symmetry requirement nor the triangular inequality, it is the main ingredient to this work. this paper is organized as follow: in section 2 we consider the linear case and give quantitative estimates for the minimizers of (2), for exact and for noisy data. in section 3 contains similar results as the section 2 for nonlinear problems. in section 4 we briefly discuss a iterative method for the nonlinear case, the main results contains convergence analysis. 2 convergence analysis for linear problems in this section we consider only the linear case. equation (1) will be denoted by fu = y, and the operator is defined from a banach space to a hilbert space. the main results of this section were proposed originally in [4, 8]. 2.1 rates of convergence for source condition of type i error estimates for the solution error can be obtained only under additional smoothness assumption on the data, the so called source conditions. at a first moment we assume that y ∈ r(f) and let u be an h-minimizing solution by definition a.2. we assume that there exist at least one element ξ in ∂h (u) which belongs to the range of adjoint of the operator f . note that r(f∗) ⊆ u∗ and ∂h (u) ⊆ u∗. summarizing, we have ξ ∈ r(f∗) ∩ ∂h (u) 6= ∅ , (4) where u is such that fu = y . (5) 102 i.r. bleyer and a. leitão cubo 11, 5 (2009) we can rewrite the source condition (4) as following: there exists an element ω ∈ h such that ξ = f∗ω. note that under this assumption we can define the dual pairing for (ψ,u) ∈ u∗ ×u , where ψ ∈ r(f∗) as 〈ψ , u〉 = 〈f∗ν , u〉 := 〈ν , fu〉 h , for some ν ∈h . theorem 2.1 (stability). let (3) hold and let u be an h-minimizing solution of (1) such that the source condition (4) and (5) are satisfied. then, for each minimizer uδα of (2) the estimate df ∗ω h ( uδα,u ) ≤ 1 2α (α‖ω‖ + δ) 2 (6) holds for α > 0. in particular, if α ∼ δ, then df ∗ω h ( uδα,u ) = o (δ). proof. we note that ∥∥fu − yδ ∥∥2 ≤ δ2, by (5) and (3). since uδα is a minimizer of the regularized problem (2), we have 1 2 ∥∥fuδα − y δ ∥∥2 + αh ( uδα ) ≤ δ2 2 + αh (u) . let df ∗ω h ( uδα,u ) the bregman distance between uδα and u, so the above inequality becomes 1 2 ∥∥fuδα − y δ ∥∥2 + α ( df ∗ω h ( uδα,u ) + 〈 f∗ω , uδα − u 〉) ≤ δ2 2 . hence, using (3) and cauchy-schwarz inequality we can derive the estimate 1 2 ∥∥fuδα − y δ ∥∥2 + 〈 αω , fuδα − y δ 〉 h + αdf ∗ω h ( uδα,u ) ≤ δ2 2 + α‖ω‖δ . using the the equality ‖a + b‖ 2 = ‖a‖ 2 + 2 〈a , b〉 + ‖b‖ 2 , it is easy to see that 1 2 ∥∥fuδα − y δ + αω ∥∥2 + αdf ∗ω h ( uδα,u ) ≤ α2 2 ‖ω‖ 2 + αδ ‖ω‖ + δ2 2 , which yields (6) for α > 0. theorem 2.2 (convergence). if u is an h-minimizing solution of (1) such that the source condition (4) and (5) are satisfied, then for each minimizer uα of (2) with exact data, the estimate df ∗ω h (uα,u) ≤ α 2 ‖ω‖ 2 holds true. proof. the proof is analogous to the proof of theorem 2.1, taking δ = 0. cubo 11, 5 (2009) on tikhonov functionals penalized ... 103 2.2 rates of convergence for source condition of type ii in this section we use another source condition, which is stronger than the one used in previous subsection. this condition corresponds the existence of some element ξ ∈ ∂h (u) ⊂u∗ in the range of the operator f∗f , i.e. ξ ∈ r(f∗f) ∩ ∂h (u) 6= ∅ , (7) where u is such that f∗fu = f∗y . (8) note that in (8) we do not require y ∈ r(f). moreover, the definition a.2 is given in context of least-squares solution. the condition (7) is equivalent to the existence of ω ∈u\ {0} such that ξ = f∗fω, where f∗ is the adjoint operator of f and f∗f : u → u∗. theorem 2.3 (stability). let (3) hold and let u be an h-minimizing solution of (1) such that the source condition (7) as well as (8) are satisfied. then the following inequalities hold for any α > 0: df ∗f ω h ( uδα,u ) ≤ df ∗f ω h (u − αω,u) + δ2 α + δ α √ δ2 + 2αdf ∗f ω h (u − αω,u), (9) ∥∥fuδα − fu ∥∥ ≤ α‖fω‖ + δ + √ δ2 + 2αdf ∗f ω h (u − αω,u) . (10) proof. since uδα is a minimizer of (2), it follows from algebraic manipulation and from the definition of bregman distance that 0 ≥ 1 2 [∥∥fuδα − y δ ∥∥2 − ∥∥fu − yδ ∥∥2 ] + αh ( uδα ) − αh (u) = 1 2 [∥∥fuδα ∥∥2 − ‖fu‖2 ] − 〈 f ( uδα − u ) , yδ 〉 h − αdf ∗f ω h (u,u) + α 〈 fω , f ( uδα − u )〉 h + αdf ∗f ω h ( uδα,u ) . (11) notice that ∥∥fuδα ∥∥2 − ‖fu‖2 = ∥∥f ( uδα − u + αω )∥∥2 − ‖f (u − u + αω)‖2 + 2 〈 fuδα − fu , fu − αfω 〉 h . moreover, by (8), we have 〈 f ( uδα − u ) , yδ − fu 〉 h = 〈 f ( uδα − u ) , yδ − y 〉 h . therefore, it follows from (11) that 1 2 ∥∥f ( uδα − u + αω )∥∥2 + αdf ∗f ω h ( uδα,u ) ≤ 〈 f ( uδα − u ) , yδ − y 〉 h + αdf ∗f ω h (u,u) + 1 2 ‖f (u − u + αω)‖ 2 for every u ∈ u , α ≥ 0 and δ ≥ 0. 104 i.r. bleyer and a. leitão cubo 11, 5 (2009) replacing u by u − αω in the last inequality, using (3), relations 〈a , b〉 ≤ | 〈a , b〉 | ≤ ‖a‖ ‖b‖, and defining γ = ∥∥f ( uδα − u + αω )∥∥ we obtain 1 2 γ2 + αdf ∗f ω h ( uδα,u ) ≤ δγ + αdf ∗f ω h (u − αω,u) . we estimate separately each term on the left hand side by right hand side. one of the estimates is an inequality in the form of a polynomial of the second degree for γ, which gives us the inequality γ ≤ δ + √ δ2 + 2αdf ∗f ω h (u − αω,u) . this inequality together with the other estimate, gives us (9). now, (10) follows from the fact that∥∥f ( uδα − u )∥∥ ≤ γ + α‖fω‖. theorem 2.4 (convergence). let α ≥ 0 be given. if u is a h-minimizing solution of (1) satisfying the source condition (7) as well as (8), then the following inequalities hold true: df ∗f ω h (uα,u) ≤ d f ∗f ω h (u − αω,u) , ‖fuα − fu‖ ≤ α‖fω‖ + √ 2αdf ∗f ω h (u − αω,u) . proof. the proof is analogous to the proof of theorem 2.3, taking δ = 0. notice that here α can be taken equal to zero. corollary 2.5. let the assumptions of the theorem 2.3 hold true. further, assume that h is twice differentiable in a neighborhood u of u and there exists a number m > 0 such that for any v ∈ u and u ∈ u the inequality 〈h′′(u)v , v〉 ≤ m ‖v‖ 2 (12) hold true. then, for the parameter choice α ∼ δ 2 3 we have d ξ h ( uδα,u ) = o ( δ 4 3 ) . moreover, for exact data we have d ξ h (uα,u) = o ( α2 ) . proof. using taylor’s expansion at the point u we obtain h (u) = h (u) + 〈h′(u) , u − u〉 + 1 2 〈h′′(µ)(u − u) , u − u〉 for some µ ∈ [u,u]. let u = u − αω in the above equality. for sufficiently small α, it follows from assumption (12) and the definition of the bregman distance, with ξ = h′(u), that d ξ h (u − αω,u) = 1 2 〈h′′(µ)(−αω) , − αω〉 ≤ α2 m 2 ‖ω‖ 2 u . note that d ξ h (u − αω,u) = o ( α2 ) , so the desired rates of convergence follow from theorems 2.3 and 2.4. cubo 11, 5 (2009) on tikhonov functionals penalized ... 105 3 convergence analysis for nonlinear problems this section points out the convergence analysis for the nonlinear problems. we need to assume a nonlinear condition. in contrast with other classical conditions, the following analysis covers the case when both u and h are banach spaces. assumption 3.1. assume that an h-minimizing solution u of (1) exists and that the operator f : d(f) ⊆ u → h is gâteaux differentiable. moreover, assume that there exists ρ > 0 such that, for every u ∈ d(f) ∩ bρ (u) ‖f (u) − f (u) − f ′ (u) (u − u)‖ ≤ cd ξ h (u,u) , c > 0 (13) and ξ ∈ ∂h (u). this assumption was proposed originally in [9]. 3.1 rates of convergence for source condition of type i for nonlinear operators we cannot define an adjoint operator. therefore the assumptions are done with respect to the linearization of the operator f . in comparison with the source condition (4) introduced on previous section, we assume that ξ ∈ r(f ′ (u) ∗ ) ∩ ∂h (u) 6= ∅ (14) where u solves f (u) = y . (15) the derivative of operator f is defined between the banach space u and l (u ,h ), the space of the linear transformations from u to h . when we apply the derivative at u ∈ u we have a linear operator f ′ (u) : u → h and so we can define its adjoint, f ′ (u) ∗ : h ∗ → u∗. the source condition (14) is stated as follows: there exists an element ω ∈ h ∗ such that ξ = f ′ (u) ∗ ω ∈ ∂h (u) . (16) theorem 3.2 (stability). let the assumptions 1.1, 3.1 and relation (3) hold true. moreover, assume that there exists ω ∈ h ∗ such that (16) is satisfied and c‖ω‖ h ∗ < 1. then, the following estimates hold: ∥∥f ( uδα ) − f (u) ∥∥ ≤ 2α‖ω‖ h ∗ + 2 ( α2 ‖ω‖ 2 h ∗ + δ2 ) 1 2 , d f ′(u)∗ω h ( uδα,u ) ≤ 2 1 − c‖ω‖ h ∗ [ δ2 2α + α‖ω‖ 2 h ∗ + ‖ω‖ h ∗ ( α2 ‖ω‖ 2 h ∗ + δ2 ) 1 2 ] . in particular, if α ∼ δ, then ∥∥f ( uδα ) − f (u) ∥∥ = o (δ) and df ′(u)∗ω h ( uδα,u ) = o (δ). 106 i.r. bleyer and a. leitão cubo 11, 5 (2009) proof. since uδα is the minimizer of (2), it follows from the definition of the bregman distance that 1 2 ∥∥f ( uδα ) − yδ ∥∥2 ≤ 1 2 δ2 − α ( d f ′(u)∗ω h ( uδα,u ) + 〈 f ′ (u) ∗ ω , uδα − u 〉) . by using (3) and (15) we obtain 1 2 ∥∥f ( uδα ) − f (u) ∥∥2 ≤ ∥∥f ( uδα ) − yδ ∥∥2 + δ2 . now, using the last two inequalities above, the definition of bregman distance, the nonlinearity condition and the assumption ( c‖ω‖ h ∗ − 1 ) < 0, we obtain 1 4 ∥∥f ( uδα ) − f (u) ∥∥2 ≤ 1 2 (∥∥f ( uδα ) − yδ ∥∥2 + δ2 ) ≤ δ2 − αd f ′(u)∗ω h ( uδα,u ) + α 〈 ω , − f ′ (u) ( uδα − u )〉 ≤ δ2 − αd f ′(u)∗ω h ( uδα,u ) + α ‖ω‖ h ∗ ∥∥f ( uδα ) − f (u) ∥∥ +α‖ω‖ h ∗ ∥∥f ( uδα ) − f (u) − f ′ (u) ( uδα − u )∥∥ = δ2 + α ( c‖ω‖ h ∗ − 1 ) d f ′(u)∗ω h ( uδα,u ) + α ‖ω‖ h ∗ ∥∥f ( uδα ) − f (u) ∥∥ (17) ≤ δ2 + α‖ω‖ h ∗ ∥∥f ( uδα ) − f (u) ∥∥ (18) from (18) we obtain an inequality in the form of a polynomial of second degree for the variable γ = ∥∥f ( uδα ) − f (u) ∥∥. this gives us the first estimate stated by the theorem. for the second estimate we use (17) and the previous estimate for γ. theorem 3.3 (convergence). let the assumptions 1.1 and 3.1 hold true. moreover, assume the existence of ω ∈ h ∗ such that (16) is satisfied and c‖ω‖ h ∗ < 1. then, the following estimates hold: ‖f (uα) − f (u)‖ ≤ 4α‖ω‖h ∗ , d f ′(u)∗ω h (uα,u) ≤ 4α‖ω‖ 2 h ∗ 1 − c‖ω‖ h ∗ . proof. the proof is analogous to the proof of theorem 3.2, taking δ = 0. 3.2 rates of convergence for source condition of type ii in this subsection we consider a source condition similar to the one in (7), namely we assume the existence of ξ ∈ r(f ′ (u) ∗ f ′ (u)) ∩ ∂h (u) 6= ∅ . the assumption above is equivalent the existence of an element ω ∈ u with ξ = f ′ (u) ∗ f ′ (u) ω ∈ ∂h (u) . (19) cubo 11, 5 (2009) on tikhonov functionals penalized ... 107 theorem 3.4 (stability). let the assumptions 1.1, 3.1 hold as well as estimate (3). moreover, let h be a hilbert space and assume the existence of an h-minimizing solution u of (1) in the interior of d(f). assume also the existence of ω ∈ u such that (19) is satisfied and c‖f ′ (u) ω‖ < 1. then, for α sufficiently small the following estimates hold: ‖f ( uδα ) − f (u) ‖ ≤ α ‖f ′ (u) ω‖ + g(α,δ) , d ξ h ( uδα,u ) ≤ αs + (cs)2/2 + δg(α,δ) + cs (δ + α‖f ′ (u) ω‖) α (1 − c‖f ′ (u) ω‖) , (20) where g(α,δ) = δ + √ (δ + cs) 2 + 2αs (1 + c‖f ′ (u) ω‖) and s = d ξ h (u − αω,u). proof. since uδα is the minimizer of (2), it follows that 0 ≥ 1 2 ∥∥f ( uδα ) − yδ ∥∥2 − 1 2 ∥∥f (u) − yδ ∥∥2 + α ( h ( uδα ) − h (u) ) = 1 2 ∥∥f ( uδα )∥∥2 − 1 2 ‖f (u)‖ 2 + 〈 f (u) − f ( uδα ) , yδ 〉 h + α ( h ( uδα ) − h (u) ) = φ ( uδα ) − φ (u) . (21) where φ (u) = 1 2 ‖f (u) − q‖ 2 + αd ξ h (u,u) − 〈 f (u) , yδ − q 〉 h + α〈ξ , u〉, q = f (u) − αf ′ (u) ω and ξ is given by source condition (19). from (21) we have φ ( uδα ) ≤ φ (u). by the definition of φ (·), taking u = u − αω and setting v = f ( uδα ) − f (u) + αf ′ (u) ω we obtain 1 2 ‖v‖ 2 + αd ξ h ( uδα,u ) ≤ αs + t1 + t2 + t3 , (22) where s is given in the theorem, and t1 = 1 2 ‖f (u − αω) − f (u) + αf ′ (u) ω‖ 2 , t2 = ∣∣〈f ( uδα ) − f (u − αω) , yδ − y 〉 h ∣∣ , t3 = α 〈 f ′ (u) ω , f ( uδα ) − f (u − αω) − f ′ (u) ( uδα − (u − αω) )〉 h . the next step is to estimate the constants tj , j = 1, 2, 3 above. we use the nonlinear condition (13), cauchy-schwarz, and some algebraic manipulation to obtain t1 ≤ c2s2 2 , t2 ≤ ∣∣〈v , yδ − y 〉 h ∣∣ + ∣∣〈f (u − αω) − f (u) + αf ′ (u) ω − , yδ − y 〉 h ∣∣ ≤ ‖v‖ ∥∥yδ − y ∥∥ + cdξh (u − αω,u) ∥∥yδ − y ∥∥ ≤ δ ‖v‖ + δcs, 108 i.r. bleyer and a. leitão cubo 11, 5 (2009) and t3 = α 〈 f ′ (u) ω , f ( uδα ) − f (u) − f ′ (u) ( uδα − u )〉 h +α〈f ′ (u) ω , − (f (u − αω) − f (u) + αf ′ (u) ω)〉 h ≤ α ‖f ′ (u) ω‖ ∥∥f ( uδα ) − f (u) − f ′ (u) ( uδα − u )∥∥ +α‖f ′ (u) ω‖ ‖f (u − αω) − f (u) + αf ′ (u) ω‖ ≤ α ‖f ′ (u) ω‖cd ξ h ( uδα,u ) + α‖f ′ (u) ω‖cd ξ h (u − αω,u) = αc‖f ′ (u) ω‖d ξ h ( uδα,u ) + αcs‖f ′ (u) ω‖ . using these estimates in (22), we obtain ‖v‖ 2 + 2αd ξ h ( uδα,u ) [1 − c‖f ′ (u) ω‖] ≤ 2δ ‖v‖ + 2αs + (cs)2 +2δcs + 2αcs‖f ′ (u) ω‖ . analogously as in the proof of theorem 2.3, each term on the left hand side of the last inequality is estimated separately by the right hand side. this allows the derivation of an inequality described by a polynomial of second degree. from this inequality, the theorem follows. theorem 3.5 (convergence). let assumptions 1.1, 3.1 hold and assume h to be a hilbert space. moreover, assume the existence of an h-minimizing solution u of (1) in the interior of d(f), and also the existence of ω ∈ u such that (19) is satisfied, and c‖f ′ (u) ω‖ < 1. then, for α sufficiently small the following estimates hold: ‖f (uα) − f (u)‖ ≤ α‖f ′ (u) ω‖ + √ (cs) 2 + 2αs (1 + c‖f ′ (u) ω‖) , d ξ h (uα,u) ≤ αs + (cs)2/2 + αcs‖f ′ (u) ω‖ h α ( 1 − c‖f ′ (u) ω‖ h ) , (23) where s = d ξ h (u − αω,u). proof. the proof is analogous to the proof of theorem 3.4, taking δ = 0. corollary 3.6. let assumptions of the theorem 3.4 hold true. moreover, assume that h is twice differentiable in a neighborhood u of u, and that there exists a number m > 0 such that for all u ∈ u and for all v ∈ u , the inequality 〈h′′(u)v , v〉 ≤ m ‖v‖ 2 holds. then, for the choice of parameter α ∼ δ 2 3 we have d ξ h ( uδα,u ) = o ( δ 4 3 ) , while for exact data we obtain d ξ h ( uδα,u ) = o ( α2 ) . proof. the proof is similar to the proof of corollary 2.5 and is based on theorems 3.4 and 3.5. 4 an iterated tikhonov method for nonlinear problems on this section we investigate an iterative method based on bregman distances for nonlinear problems. we consider the operator f : u → h defined between a banach space and a hilbert cubo 11, 5 (2009) on tikhonov functionals penalized ... 109 space, fréchet differentiable with closed and convex domain d(f). the operator equation (1) is ill-posed in the sense of hadamard, the solution does not need to be unique, so we define s (y) = {u ∈ d(f) | f (u) = y} . the method was originally proposed by osher in [7], who generalized the ideas of the method rof [10] (see [1] for further details). the analyzed method generalizes the iterated tikhonov method and is given by uk+1 ∈ argmin { 1 2 ∥∥f (u) − yδ ∥∥2 + αkdξkh (u,uk) } , (24) where the subgradient required is updated by the rule ξk+1 = ξk − 1 αk f ′ (uk+1) ∗ ( f (uk+1) − y δ ) . (25) algorithm 1 generalized tikhonov with bregman distance require: u0 ∈ d(f) ∩ dom h, ξ0 ∈ ∂h (u0) 1: k = 0 2: αk > 0 3: repeat 4: uk+1 ∈ argmin { 1 2 ∥∥f (u) − yδ ∥∥2 + αkdξkh (u,uk) } 5: ξk+1 = ξk − 1 αk f ′ (uk+1) ∗ ( f (uk+1) − y δ ) 6: k = k + 1 7: αk > 0 8: until convergence remark 4.1. it is easy to see that the definition (25) is equivalent to ξk+1 = ξ0 − k∑ j=0 1 αj f ′ (uj+1) ∗ ( f (uj+1) − y δ ) . (26) we obtain monotonicity of residuals directly from the above definitions. lemma 4.2. the iterates defined by algorithm 1 satisfy the estimate ∥∥yδ − f (uk+1) ∥∥ ≤ ∥∥yδ − f (uk) ∥∥ . proof. defining jδα (u) = 1 2 ∥∥f (u) − yδ ∥∥2 + αkdξkh (u,uk), the lemma follows the fact that uk+1 is a minimizer of (24), i.e., j δ α (uk+1) ≤ j δ α (uk). under a nonlinearity condition on f we prove a monotonicity result for the bregman distance, i.e., d ξk+1 h (u,uk+1) ≤ d ξk h (u,uk). 110 i.r. bleyer and a. leitão cubo 11, 5 (2009) lemma 4.3. let yδ ∈h be the given data. if for some uk and ξk, the iterate uk+1 in (24) satisfies ∥∥yδ − f (uk+1) − f ′ (uk+1) (u − uk+1) ∥∥ ≤ c ∥∥yδ − f (uk+1) ∥∥ , for some 0 < c < 1, then d ξk+1 h (u,uk+1) − d ξk h (u,uk) + d ξk h (uk+1,uk) ≤ − 1 − c αk ∥∥yδ − f (uk+1) ∥∥2 . (27) proof. this result follows from the equality (see [1] for details) d ξk+1 h (u,uk+1) − d ξk h (u,uk) + d ξk h (uk+1,uk) = 〈ξk+1 − ξk , uk+1 − u〉 . using (25) on the right hand side, summing ±(f (uk+1) −y δ ) on the second term (inside the inner product), using cauchy-schwarz and the assumptions, we conclude that estimate (27) holds. the subsequent results are obtained assuming that the nonlinear operator f is such that d(f) ⊆ l2 (ω) and ω ⊂ rn is a bounded lipschitz domain, and assuming that the regularization convex functional is given by h (u) = 1 2 ‖u‖ 2 l2(ω) + |u|bv (ω) . (28) lemma 4.4. if h (·) is the convex functional defined by (28), then 1 2 ‖v − u‖ 2 l2(ω) ≤ d ξ h (v,u) for every u,v ∈ d(f) and ξ ∈ ∂h (u). proof. this proof is straightforward, once we establish some auxiliary properties concerning calculus of subgradients. for a complete proof we refer the reader to [2]. assumption 4.5. let f : d(f) ⊂ l2 (ω) → h be a weakly sequentially closed nonlinear operator, f ′ (·) be locally bounded. moreover, suppose that the nonlinearity condition ‖f (v) − f (u) − f ′ (u) (v − u)‖ ≤ η ‖u − v‖l2(ω) ‖f (u) − f (v)‖ (29) is satisfied for every u, v ∈ bρ (u) ∩ d(f), where η,ρ > 0 and bρ (u) denotes the open ball around u of radius ρ in l2 (ω) and u ∈ s (y) ∩ dom h. remark 4.6. we can rewrite the left side of the inequality given in (29) as ‖f ′ (u) (v − u)‖ ≤ ( 1 + η ‖u − v‖l2(ω) ) ‖f (u) − f (v)‖ . the next result gives the mean result about the sequence of iterates from algorithm 1 is well-defined. cubo 11, 5 (2009) on tikhonov functionals penalized ... 111 proposition 4.7. let assumption 4.5 hold, k ∈ n and uk,ξk be a pair of iterates according to algorithm 1. then, there exists a minimizer uk+1 for (24) and ξk+1 given by (25) satisfies ξk+1 ∈ ∂h (uk+1). proof. if there exists an u such that jδαk (u) is finite, then there is a sequence (uj) ∈ d(f) ∩ bv (ω) such that limj j δ αk (uj ) → β, where β = inf { jδαk (u) | u ∈ d(f) } . in particular, d ξk h (uj,uk) ≤ m αk . by definition of the bregman distance, together with (28) and observing that 1 2 ‖uj‖ 2 l2(ω) − 〈ξk , uj〉 = 1 2 ‖uj − ξk‖ 2 l2(ω) − 1 2 ‖ξk‖ 2 l2(ω), we obtain |uj|bv (ω) ≤ m̃k, where m̃k ≥ 0 depends on the current iterates. thus, the existence of a minimizer follows from compactness arguments. it remains to prove that ξk+1 ∈ ∂h (uk+1). this result follows from the inequality φ2(v) ≥ φ2(uk+1) +〈−φ ′ 1(uk+1) , v − uk+1〉, where φ1(u) = 1 2 ∥∥yδ − f (u) ∥∥2 l2(ω) and φ2(u) = αkd ξk h (u,uk) (see [1, 2] for details). 4.1 main results the main results of this section give sufficient conditions to guarantee existence of a convergence subsequence in algorithm 1, (for both exact and noisy data). in particular, for noisy data, we introduce a stopping rule based on the discrepancy principle. for a complete proof we refer the reader to [2]. theorem 4.8 (convergence). let the assumption 4.5 hold, γ < min { 1 η , ρ 2 } for η, ρ as in (29), 0 < αk < ᾱ, h (u) < ∞. moreover, assume that the starting values u0, ξ0 ∈ l 2 (ω) satisfy d ξ0 h (u,u0) < γ2 8 for some u ∈ s (y). then, for exact data, the sequence (uk) has a subsequence converging to some u ∈ s (y) in the weak-∗ topology of bv (ω). moreover, if s (y) ∩ bρ (u) = {u}, then uk ∗ −⇀ u in bv (ω). proof. step 1: first we rewrite the assumption in the form 2 √ 2d ξ0 h (u,u0) < γ. assuming that the same condition holds for a pair of iterates uk, ξk we proof by induction that it also holds for the index k + 1. let uk+1 be the minimizer of jαk (·), so jαk (uk+1) ≤ jαk (u). thus we rewrite the inequality, then apply lemma 4.4 twice, and conclude that ‖uk+1 − u‖l2(ω) < γ. hence, assumption 4.5 is satisfied and the lemma 4.3 hold for all iterates. step 2: in this step we proof that ∑∞ i=0 1 αi ‖y − f (ui+1)‖ 2 < ∞. as in the previous step, it follows from lemma 4.3 that (27) holds for every k. adding up the first k terms and using the assumptions on the starting values, we conclude that d ξk h (u,uk) + k−1∑ i=0 d ξi h (ui+1,ui) + k−1∑ i=0 1 − ηγ αi ‖y − f (ui+1)‖ 2 ≤ γ2 8 . 112 i.r. bleyer and a. leitão cubo 11, 5 (2009) since all terms on the left hand side are positive, step 2 follows from the third term taking the limit as k tends to infinity. note that this series is convergent, by the convergence criterion for series follows f (uk) → y. step 3: we show the uniform limitation of the sequence (h (uk)). applying the bregman distance (it is always grater than zero) we have h (uk) ≤ h (u) − 〈ξk , u − uk〉. thus, by remark 4.1, ‖uk+1 − u‖l2(ω) < γ and the cauchy-schwarz inequality, we obtain h (uk) ≤ h (u) + γ ‖ξ0‖l2(ω) + k−1∑ i=0 1 αi ‖f (ui+1) − y‖ ‖f ′ (ui+1) (u − uk)‖ . in order to estimate the term inside the sum, note that for 0 ≤ i ≤ k−1, the estimate ‖f ′ (ui+1) (u − uk)‖ ≤ ‖f ′ (ui+1) (u − ui+1)‖ + ‖f ′ (ui+1) (uk − ui+1)‖ holds. now, using remark 4.6 twice, we find the bound (3 + 5ηγ) ‖f (ui+1) − y‖ for the previous estimate. substituting this estimate in the sum above and using step 2, the desired boundedness of the sequence (h (uk)) follows. step 4: we know that |h (uk)| = h (uk) ≤ n, for some n > 0 (see (28)). the remaining assertions of the theorem follow from standard compactness results (banach-alaoglu theorem). we use the closed graph theorem to ensure that the limit of the obtained sequence belongs to s (y). in the case of noisy data we use a generalized discrepancy principle as stopping rule. the stopping index is defined as the smallest integer k∗ satisfying ‖f (uk∗ ) − y δ‖ ≤ τδ (30) where τ > 1 still has to be chosen. theorem 4.9 (stability). let assumption 4.5 hold, γ < min { 1 η , ρ 2 } for η and ρ as in (29), 0 < α ≤ αk ≤ α, h (u) < ∞ and the starting values u0, ξ0 ∈ l 2 (ω) satisfy d ξ0 h (u,u0) < γ2 8 for an u ∈ s (y). moreover, let δm > 0 be a sequence such that δm → 0, and let the corresponding stopping indices k∗m be chosen according to (30) with τ > (1 + ηγ)/(1 − ηγ). then for every δm the stopping index is finite and the sequence ( uk∗ m ) has a subsequence converging to an u ∈ s (y) in the weak-∗ topology of bv (ω). moreover, if s (y) ∩ bρ (u) = {u}, then uk∗ m ∗ −⇀ u in bv (ω). proof. step 1: this step is analogous to step 1 in the proof of theorem 4.8. for each k such that k < k∗−1, we have ∥∥f (uk) − yδ ∥∥ > τδ. by induction one can prove that ‖uk+1 − u‖l2(ω) < γ, and that the nonlinear condition (27) holds. therefore, lemma 4.3 holds for c = 1 τ (1 + ηγ) + ηγ. step 2: we show that the stopping index k∗ is finite. analogous to step 2 in the proof of theorem 4.8, we sum up the first k∗ − 1 terms of (27), obtaining k∗−2∑ i=0 1 αi ∥∥yδ − f (ui+1) ∥∥2 < γ2 8 (1 − c) . (31) since for every k < k∗ − 1 the inequality ∥∥f (uk) − yδ ∥∥ > τδ holds, we use this inequality on the left hand side of the above estimate and conclude that k∗ < ( γ τδ )2 α 8 (1 − c) + 1 . cubo 11, 5 (2009) on tikhonov functionals penalized ... 113 step 3: in order to prove the convergence of the series in (31), notice that the right hand side of (31) does not depend on k∗. step 4: analogous to step 3 in the proof of theorem 4.8, we use the bregman distance, and remark 4.1 to conclude that h (uk∗ ) ≤ h (u) + |〈ξ0 , u − uk∗〉| + k∗−2∑ i=0 1 αi ∣∣〈f (ui+1) − yδ , f ′ (ui+1) (u − uk∗ ) 〉 h ∣∣ + 1 αk∗−1 ∣∣〈f (uk∗ ) − yδ , f ′ (uk∗ ) (u − uk∗ ) 〉 h ∣∣ . in the sequel we estimate the three terms on the right hand side of this inequality. for the first of them we have ‖u − uk∗‖l2(ω) < ρ. indeed, on step k ∗ − 1 we have uk∗ as minimizer of j δ αk (·), thus jδαk (uk∗ ) ≤ j δ αk (u). rearranging the terms and discarding some positive terms, it follows that d ξk∗−1 h (uk∗,uk∗−1) < δ2 2αk∗−1 + γ2 8 . finally, we apply lemma 4.4 with δ < δ = √ 3/4γ2α. to estimate the last two terms we use cauchy-schwarz, assumption 4.5, lemma 4.2 and 4.3, remark 4.6 together with steps 1, 2 and 3 above. summarizing, we obtain h (uk∗ ) < h (u) + ρ‖ξ0‖l2(ω) + (c + 3 + 4ηρ) (1 − c) γ2 8 + τδ2 (1 + ηρ) (1 + τ) α . step 5: this step is very similar to step 4 in the proof of theorem 4.8. we just need to show that f ( uk∗ m ) → y. this convergence follows from the estimate ∥∥f ( uk∗ m ) − y ∥∥ ≤ ∥∥f ( uk∗ m ) − yδm ∥∥ + ∥∥yδm − y ∥∥ ≤ (1 + τ) δm when δm goes to zero. a definitions definition a.1. given h (·) a convex functional, one can define the bregman distance with respect to h between the elements v,u ∈ dom h as dh (v,u) = { d ξ h (v,u) | ξ ∈ ∂h (u) } , where ∂h (u) denotes the subdifferential of h at u and d ξ h (v,u) = h (v) − h (u) − 〈ξ , v − u〉 . we remark that 〈 , 〉 denotes the standard dual pairing (duality product) with respect to u ∗ ×u . another important definition is the generalized solution, we introduce the notion of the hminimizing solution bellow. 114 i.r. bleyer and a. leitão cubo 11, 5 (2009) definition a.2. an element u ∈ dom h ∩ d(f) is called an h-minimizing solution of (1) if it minimizes the functional h among every possible solutions, that is, u = argmin{h (u) | f (u) = y} . whenever we need, we can choose the least-square solution instead the standard solution f (u) = y. acknowledgments the work of a.l. is supported by the brazilian national research council cnpq, grants 306020/2006– 8, 474593/2007–0, and by the alexander von humbolt foundation avh. received: december, 2008. revised: april, 2009. references [1] m. bachmayr, iterative total variation methods for nonlinear inverse problems, master’s thesis, johannes kepler universität, linz, january 2007. [2] i. r. bleyer, tikhonov functional and penalty with bregman distances (in portuguese), master’s thesis, federal university of santa catarina, florianópolis, december 2008. [3] l. bregman, the relaxation method for finding the common point of convex sets and its applications to the solution of problems in convex programming., ussr computational mathematics and mathematical physics, 7 (1967), pp. 200–217. [4] m. burger and s. osher, convergence rates of convex variational regularization, inverse problems, 20 (2004), pp. 1411–1421. [5] c. w. groetsch, the theory of tikhonov regularization for fredholm equation of the first kind, pitman, boston, 1984. [6] b. hofmann, b. kaltenbacher, c. pöschl, and o. scherzer, a convergence rates result for tikhonov regularization in banach spaces with non-smooth operators, inverse problems, 23 (2007), pp. 987–1010. [7] s. osher, m. burger, d. goldfarb, j. xu, and w. yin, an iterative regularization method for total variation-based image restoration, multiscale modeling & simulation, 4 (2005), pp. 460–489. cubo 11, 5 (2009) on tikhonov functionals penalized ... 115 [8] e. resmerita, regularization for ill-posed problems in banach spaces: convergence rates, inverse problems, 21 (2005), pp. 1303–1314. [9] e. resmerita and o. scherzer, error estimates for non-quadratic regularization and the relation to enhancement, inverse problems, 22 (2006), pp. 801–814. [10] l. i. rudin, s. osher, and e. fatemi, nonlinear total variation based noise removal algorithms, physica d, 60 (1992), pp. 259–268. [11] a. n. tikhonov, solution of incorrectly formulated problems and the regularization method, soviet math dokl, 4 (1963), pp. 1035–1038. english translation of dokl akad nauk sssr 151, 1963, 501-504. b7-article_cubo cubo a mathematical journal vol.11, no¯ 01, (1–20). march 2009 a generalization of wiman and valiron’s theory to the clifford analysis setting d. constales 1 department of mathematical analysis, ghent university, building s-22, galglaan 2, b-9000 ghent, belgium. email: dc@cage.ugent.be r. de almeida 2 departamento de matemática, universidade de trás-os-montes e alto douro, p-5000-911 vila real, portugal. email: ralmeida@utad.pt and r.s. kraußhar 3 department of mathematics, section of analysis, katholieke universiteit leuven, celestijnenlaan 200-b, b-3001 leuven (heverlee), belgium. email: soeren.krausshar@wis.kuleuven.be abstract the classical notions of growth orders, maximum term and the central index provide powerful tools to study the asymptotic growth behavior of complex-analytic functions. 1financial support from bof/goa 01ga0405 of ghent university gratefully acknowledged. 2partial supported by the r&d unit matemática e aplicações (uima) of the university of aveiro, through the portuguese foundation for science and technology (fct), co-financed by the european community fund feder, gratefully acknowledged. 3financial support from fwo project g.0335.08 gratefully acknowledged. 2 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) this leads to much insight into the structure of the solutions to many two dimensional partial differential equations that are related to boundary value problems from harmonic analysis in the plane. in this overview paper we show how the classical techniques and results from wiman and valiron can be extended to the clifford analysis setting in order to treat successfully analogous higher dimensional problems. resumen las nociones clásicas de orden de crecimiento, término máximo y de índice central proporcionan herramientas poderosas para estudiar el comportamiento de crecimiento asintótico de funciones complejas analíticas. esto nos revela la estructura de las soluciones de varias ecuaciones diferenciales parciales de dimensión dos que son relacionadas con problemas de valores en la frontera venidos de análisis armónico en el plano. mostramos como las técnicas clásicas y resultados de wiman y valiron pueden ser extendidas al contexto de análisis de clifford para tratar con éxito problemas análogos de dimensión grande. key words and phrases: monogenic functions, growth orders, growth type, maximum term, central index, valiron’s inequalities, asymptotic growth, partial differential equations. math. subj. class.: 30g35, 30d15. 1 introduction the study of the asymptotic growth behavior of holomorphic and meromorphic functions in one and several complex variables is one of the central topics in complex analysis. this line of investigation started with early works of e. lindelöf [22], a. pringsheim [24], a. wiman [26] and g. valiron [25] and had its major breakthrough in the 1920s by works of r. nevanlinna [23] and his school. their results turned out to be very useful in the study of complex partial differential equations, see e.g. [20, 21] and elsewhere. this provides a strong motivation to also develop analogous methods for other function classes and in higher dimensions. one natural higher dimensional generalization of complex analysis is clifford analysis. in this context one considers clifford algebra valued solutions of the generalized cauchy-riemann system df := ∂f ∂x0 + n∑ i=1 ∂f ∂xi ei = 0. (1) solutions to this system are often called monogenic or clifford holomorphic. many classical theorems from complex analysis, such as for instance the cauchy integral formula, the residue cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 3 theorem, laurent expansion theorems, etc. carry over to the higher dimensional context using this operator, see for instance [8, 5, 7]. nevertheless, as far as we know, questions concerning possible generalizations of wiman-valiron theory remained untouched for a long time. in [1] m.a. abul-ez and the first author introduced the notion of the growth order and the type for a particular subclass of entire clifford holomorphic functions. see also the follow-up papers [2, 3]. in our recent papers [8, 9, 11, 13] we developed the basics for a generalized wiman-valiron theory for general entire monogenic functions and for monogenic taylor series of finite convergence radius. we also managed to extend these techniques to the context of more general systems of partial differential equations, such as higher dimensional iterated cauchy-riemann systems [6, 7] and to polynomial cauchy-riemann systems equations with complex coefficients [10]. in this paper we give a concise overview over our results concerning the entire monogenic case. we show how the notions of growth orders, growth type, maximum term and the central index can be reasonably generalized to the clifford analysis context. we exhibit how these tools can be applied to get insight in the asymptotics of related function classes and in the structure of solutions to related higher dimensional partial differential equations. this line of investigation should be regarded as a starting point to develop analogous methods for larger classes of functions that are in kernels of elliptic differential operators. we hope to get more insight in the structure of the solutions to larger classes of higher dimensional partial differential equations. 2 preliminaries we begin by introducing the basic notions and concepts. for detailed information about clifford algebras and their function theory we refer for example to [8, 1] and [7]. 2.1 clifford algebras by {e1,e2, . . . ,en} we denote the canonical basis of the euclidean vector space r n. the attached real clifford algebra cl0n is the free algebra generated by r n modulo the relation x 2 = −‖x‖2e0, where x ∈ rn and e0 is the neutral element with respect to multiplication of the clifford algebra cl0n. in the clifford algebra cl0n the following multiplication rules hold eiej + ejei = −2δije0, i,j = 1, · · · ,n, where δij is the kronecker symbol. a basis for the clifford algebra cl0n is given by the set {ea : a ⊆ {1, · · · ,n}} with ea = el1el2 · · ·elr , where 1 ≤ l1 < · · · < lr ≤ n, e∅ = e0 = 1. each a ∈ cl0n can be written in the form a = ∑ a aaea with aa ∈ r. two examples of real clifford algebras are the complex number field c and the hamiltonian skew field h. 4 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) the conjugation anti-automorphism in the clifford algebra cl0n is defined by a = ∑ a aaea, where ea = elrelr−1 · · ·el1 and ej = −ej for j = 1, · · · ,n, e0 = e0 = 1. the linear subspace span r {1,e1, · · · ,en} = r ⊕ r n ⊂ cl0n is the so-called space of paravectors z = x0 + x1e1 + x2e2 + · · · + xnen which we simply identify with r n+1. the term x0 =: sc(z) is called the scalar part of the paravector z and x := x1e1 + · · · + xnen =: v ec(z) its vector part. a scalar product between two clifford numbers a,b ∈ cl0n is defined by 〈a,b〉 := sc(ab) and the clifford norm of an arbitrary a = ∑ a aaea is ‖a‖ = ( ∑ a |aa| 2 ) 1/2. any paravector z ∈ rn+1\{0} has an inverse element in rn+1 given by z−1 = z/‖z‖2. in order to present the calculations in a more compact form, the following notations will be used, where m = (m1, . . . ,mn) ∈ n n 0 is an n-dimensional multi-index: x m := x m1 1 · · ·xmn n , m! := m1! · · ·mn!, |m| := m1 + · · · + mn. by τ(i) we denote the multi-index (m1, . . . ,mn) with mj = δij for 1 ≤ j ≤ n. 2.2 clifford analysis one way to generalize complex function theory to higher dimensional hypercomplex spaces is offered by the riemann approach which considers clifford algebra valued functions defined in rn+1 that are annihilated by the generalized cauchy-riemann operator d := ∂ ∂x0 + n∑ i=1 ei ∂ ∂xi . (2) if u ⊂ rn+1 is an open set, then a real differentiable function f : u → cl0n is called left (right) monogenic or clifford holomorphic at a point z ∈ u if df(z) = 0 (or fd(z) = 0). functions that are left monogenic in the whole space are also called left entire. the notion of left (right) monogenicity in rn+1 provides indeed a powerful generalization of the concept of complex analyticity to clifford analysis. many classical theorems from complex analysis could be generalized to higher dimensions by this approach, we refer e.g. to [8]. one important tool is the generalized cauchy integral formula. let us denote by an+1 the n-dimensional surface “area” of the (n + 1)-dimensional unit ball, and by q0(z) = z ‖z‖n+1 the cauchy kernel function. then every function f that is left monogenic in a neighbourhood of the closure d of a domain d satisfies f(z) = 1 an+1 ∫ ∂d q0(z − w) dσ(w) f(w), (3) cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 5 where dσ(w) is the paravector-valued outer normal surface measure, i.e., dσ(w) = n∑ j=0 (−1)jejdw0 ∧ · · · ∧ d̂wj ∧ · · · ∧ dwn with d̂wj = dw0 ∧ · · · ∧ dwi−1 ∧ dwi+1 ∧ · · · ∧ dwn. it is important to mention that the set of left (right) monogenic functions forms only a clifford right (left) module for n > 1. in contrast to complex analysis, the ordinary powers of the hypercomplex variables are not null-solutions to the generalized cauchy-riemann system. in clifford analysis these are substituted by the fueter polynomials. these are defined by pm(z) = 1 |m|! ∑ π∈perm(m) zπ(m1) · · ·zπ(mn) where perm(m) is the set of all permutations of the sequence (m1, . . . ,mn) and zi := xi −x0ei for i = 1, . . . ,n and p0(z) := 1. in this paper we prefer to work with the slightly modified fueter polynomials vm(z) := m!pm(z) (4) which turns out to be more convenient in our calculations. these polynomials play the analogous role of the complex power functions in the taylor series representation of a monogenic function. more precisely, if f is a left monogenic function in a ball ‖z‖ < r, then for all ‖z‖ ≤ r with 0 < r < r f(z) = +∞∑ |m|=0 vm(z)am, where the elements am are clifford numbers which — as a consequence of cauchy’s integral formula (3) — are uniquely defined by am = 1 m!an+1 ∫ ‖z‖ 0 satisfying lim inf r→∞ m(r,f) r|s| = l < ∞, (9) then f(z) = |s|∑ |m|=0 vm(z)am. in order to characterize larger classes of monogenic functions by their asymptotic growth behavior it turned out to be convenient to introduce growth orders for monogenic functions [1, 3, 13]. for convenience we recall its definition. first we need, cf. e.g. [20]: cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 7 definition 1. let α ≥ 0. then the plus logarithm is defined by log + (α) := max{0, log(α)}. (10) in the same way as in the planar case (see [20]) one introduces in the clifford analysis setting (see also [2, 13]): definition 2. (order of growth) let f : rn+1 → cl0n be an entire function. then ρ(f) = ρ := lim sup r→∞ log + (log + m(r,f)) log(r) , 0 ≤ ρ ≤ ∞ (11) is called the order of growth of the function f. we further introduce λ(f) = λ : lim inf r→∞ log + (log + m(r,f)) log(r) , 0 ≤ λ ≤ ∞ (12) as the inferior order of growth of f. if ρ = λ, then we say that f is a function of regular growth. if ρ > λ then f has irregular growth. to get a finer classification of the growth behavior within the set of monogenic functions that have the same growth order, one further introduces the growth type of a monogenic function as follows, cf. [9]. definition 3. for an entire monogenic function f : rn+1 → cl0n of order ρ (0 < ρ < ∞) the (growth) type is defined by τ(f) = τ := lim sup r→∞ log + m(r,f) rρ . let us start with discussing some particular examples. in [13] we have proved that the following higher dimensional generalizations of the exponential function all have growth order equal to 1: (i) the monogenic plane wave exponential function from [1] defined for m ∈ rn \ {0} by f1(m,z) := (|m| + im)e −|m|x0ei, (ii) the monogenic generalization exponential function from [8] f2(z) = exp(x0,x1, . . . ,xn) = e x1+···+xn cos(x0 √ n) − ex1+···+xn 1 √ n (e1 + · · · + en) sin(x0 √ n) (iii) the quaternion-valued 3-fold periodic exponential function from [17] given by f3(z) := exp0(z) + e1exp1(z) + e2exp2(z) + e3exp3(z) 8 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) where exp0(z) = e x0 (cos( x1 √ 3 ) cos( x2 √ 3 ) cos( x3 √ 3 ) − sin( x1 √ 3 ) sin( x2 √ 3 ) sin( x3 √ 3 )) exp1(z) = e x0 √ 3 3 (sin( x1 √ 3 ) cos( x2 √ 3 ) cos( x3 √ 3 ) + cos( x1 √ 3 ) sin( x2 √ 3 ) sin( x3 √ 3 )) exp2(z) = e x0 √ 3 3 (cos( x1 √ 3 ) sin( x2 √ 3 ) cos( x3 √ 3 ) + sin( x1 √ 3 ) cos( x2 √ 3 ) sin( x3 √ 3 )) exp3(z) = e x0 √ 3 3 (sin( x1 √ 3 ) sin( x2 √ 3 ) cos( x3 √ 3 ) + cos( x1 √ 3 ) cos( x2 √ 3 ) sin( x3 √ 3 )). however, not all of these higher dimensional analogues of the exponential function turn out to be of the same type. for the first and the second example we can determine the value of m(r,f) exactly. we obtain that m(r,f1) = ‖|m| + im‖e |m|r, thus τ(f1) = |m|. for f2 we obtain that ‖f2(z)‖ = e x1+···+xn which implies that m(r,f2)e nr and therefore τ(f2) = n. for the third example we are able to establish a useful lower and upper bound estimate for the maximum modulus. by a direct calculation we obtain that √ 3 3 er ≤ m(r,f3) ≤ e r so that τ(f3) = 1. when |m| = 1, f1 and f3 thus share the same growth order and growth type. after having discussed some concrete examples, let us now turn to the more general framework. as a consequence of cauchy’s integral formula we can establish, cf. [13]: theorem 2. let f be a left entire function in rn+1. by fi we denote the function fi := ∂ ∂xi f and mi(r) := max‖z‖=r { ‖fi(z)‖ } where r > 0 and i ∈ {0, . . . ,n}. then ρ(f) = ρ′(f) and λ(f) = λ′(f), where ρ′(f) := lim sup r→∞ log + (log + (m′(r))) log(r) λ′(f) := lim inf r→∞ log + (log + (m′(r))) log(r) , for m′(r) := max i=0,1,...,n {mi(r)}. proof. we consider an arbitrary rectifiable curve from the origin to z. then f(z) = f(0) + 1∫ 0 n∑ i=0 xi fi(tz)dt. cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 9 for z ∈ rn+1 with ‖z‖ = r we get ‖f(z)‖ ≤ ‖f(0)‖ + r n∑ i=0 mi(r) ≤ ‖f(0)‖ + r(n + 1)m′(r). therefore m(r) ≤ ‖f(0)‖ + r(n + 1)m′(r). applying some properties of log+ we obtain that log + (m(r,f)) ≤ log+(‖f(0)‖) + log+(r(n + 1)) + log+(m′(r)) + log(2). this in turn leads to ρ(f) ≤ ρ′(f) and λ(f) ≤ λ′(f). to show the inequality in the other direction, we apply on fi cauchy’s integral formula: fi(z) = 1 an+1 ∫ ‖ζ−z‖=r−r qτ(i)(ζ − z)dσ(ζ)f(ζ). (13) applying the estimate (6) to (13) we hence obtain ‖fi(z)‖ ≤ 1 an+1 ∫ ‖ζ−z‖=r−r n (r − r)n+1 m(r)ds from which we then infer that mi(r) ≤ n (r − r) m(r,f). in particular, for m′(r) := max i=0,1,...,n {mi(r)} we have m′(r) ≤ n (r − r) m(r,f). (14) replacing r = 2r into (14) yields: m′(r) ≤ n r m(2r,f). thus, log + m′(r) ≤ log+ m(2r,f) + log+ ( n r ) . for what follows we may assume without loss of generality that r > n. hence, log + m′(r) ≤ log+ m(2r,f). furthermore, log + (log + m′(r)) log(r) ≤ log + (log + m(2r,f)) log(r) = log + (log + m(2r,f)) log(2r) log + (2r) log(r) 10 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) thus, we have log + (log + m′(r)) log(r) ≤ log + (log + m(2r,f)) log(2r) ( log 2 log(r) + 1 ) from which we can infer directly that ρ(f) ≥ ρ′(f) and λ(f) ≥ λ′(f). after having computed the growth order ρ(f) resp. λ(f) of a monogenic function f, we know the maximal value of the growth order of all partial derivatives. notice that cauchy’s integral formula was an important ingredient in the proof of this statement. to establish these types of results in a more general framework it is thus indeed important to work in classes of functions that are in the kernel of a differential operator that satisfy a cauchy type integral formula. see also our paper [7] where we treated more general solutions to higher dimensional iterated cauchy-riemann and dirac operators. the class of monogenic functions actually provides us with the canonical and easiest example of a function class which satisfies a cauchy type integral formula. 4 generalizations of some theorems from valiron to clifford analysis in this section we present some generalizations of some classical theorems from g. valiron to the clifford analysis setting. to this end we first define the maximum term and central index which are associated to the taylor series of a monogenic function. let us consider a left entire function f(z) = +∞∑ |l|=0 vl(z)al. let r > 0 be a fixed real. if f is transcendental, i.e. infinitely many al 6= 0, then lim |l|→∞ ‖al‖r |l| = 0. the following expression thus is well-defined: definition 4. (maximum term) let f : rn+1 → cl0n be a left entire function with the taylor series representation f(z) +∞∑ |l|=0 vl(z)al. furthermore, let r > 0 be a fixed real. then the associated maximum term is defined by µ(r) := µ(r,f) := max |l|≥0 {‖al‖r |l|}. (15) cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 11 we further introduce definition 5. (central indices) let f(z) = +∞∑ |l|=p vl(z)al be a left entire function. for r > 0 the index (or the indices) m with maximal length |m| with µ(r)‖am‖r |m| is (are) called central index (indices) which shall be denoted by ν(r) = ν(r,f) = m. by ν(0) we denote the indices l which satisfy |l| = p. the following theorem proved in [13], providing us with a direct generalization of valiron theorem, states a relation between the maximum term, central index and the maximum modulus. theorem 3. if f : rn+1 → cl0n is a left entire function, then for all 0 < r < r m(r) ≤ µ(r) [ |ν(r)|(1 + |ν(r)|)n−1 + ( r r − r ) n ] . (16) proof. the function f is assumed to be left entire. thus, it can be represented by f(z) = +∞∑ |l|=0 vl(z)al where infinitely many al 6= 0, since f is transcendental. from the maximum modulus theorem for monogenic functions we infer that for 0 < r < r: m(r) ≤ +∞∑ |l|=0 ‖al‖r |l| = |ν(r)|−1∑ |l|=0 ‖al‖r |l| + +∞∑ |l|=|ν(r)| ‖al‖r |l| ≤ |ν(r)|−1∑ |l|=0 µ(r) + +∞∑ |l|=|ν(r)| ‖al‖r |l|. (17) in view of |ν(r)|−1∑ |l|=0 1 = ∑ |l|=0 1 + ∑ |l|=1 1 + · · · + ∑ |l|=|ν(r)|−1 1 = 1 + ((n − 1) + 1)! (n − 1)!1! + · · · + [(n − 1) + (|ν(r) − 1)]! (n − 1)!(|ν(r)| − 1)! ≤ |ν(r)| [ [(n − 1) + |ν(r)| − 1]! (n − 1)!(|ν(r)| − 1)! ] where we use that for all n ≥ 1 the inequality (n − 1 + k)! (n − 1)!k! ≤ (n − 1 + (k + 1))! (n − 1)!(k + 1)! 12 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) holds, which itself can be verified by a straightforward induction over k. further, |ν(r)| [ [(n − 1) + |ν(r)| − 1]! (n − 1)!(|ν(r)| − 1)! ] = |ν(r)| [ (|ν(r)| + n − 2)(|ν(r)| + n − 3) · · · (|ν(r)| + 1)|ν(r)| (n − 1)! ] = |ν(r)| [ |ν(r)| + n − 2 n − 1 · |ν(r)| + n − 3 n − 2 · · · · · |ν(r)| + 1 2 · |ν(r)| 1 ] ≤ |ν(r)| [ (1 + |ν(r)| n − 1︸︷︷︸ ≤1+|ν(r)| )(1 + |ν(r)| n − 2︸︷︷︸ ≤1+|ν(r)| ) · · · · · (1 + |ν(r)| 1︸︷︷︸ =1+|ν(r)| ) ] ≤ |ν(r)| [ (1 + |ν(r)|)n−1 ] . inserting these results into (17) leads to m(r) ≤ µ(r)|ν(r)| [ (1 + |ν(r)|)n−1 ] + +∞∑ |l|=|ν(r)| ‖al‖r |l| ‖aν(r)‖r |ν(r)|r|l+ν(r)| ‖aν(r)‖r |ν(r)|r|l+ν(r)| = µ(r)|ν(r)| [ (1 + |ν(r)|)n−1 ] + µ(r) +∞∑ |l|=|ν(r)| ‖al‖r |l|r|ν(r)|r|l| ‖aν(r)‖r |ν(r)|r|l|r|ν(r)| ≤ µ(r)|ν(r)| [ (1 + |ν(r)|)n−1 ] + µ(r) +∞∑ |l|=|ν(r)| ( r r )|l|−|ν(r)| = µ(r) [ |ν(r)| [ (1 + |ν(r)|)n−1 ] + ( r r − r ) n ] . g. valiron has also proved that an entire complex-analytic function f of finite order shows the asymptotic behavior log(m(r,f)) ≈ log(m′(r)) where m′ is the maximum modulus of the derivative. the classical proof is based on the fact that one has the relation µ(r) ≤ m(r,f) for a complex-analytic function in one complex variable. in the framework of working with clifford algebra valued monogenic taylor series which are built with the fueter polynomials, we have a more complicated upper bound estimate of the form µ(r) ≤ n(n + 1) · · · (n + |ν(r)| − 1) ν(r)! m(r,f) for a central index ν(r). this is a consequence of the higher dimensional cauchy’s inequality. notice that this is a sharp upper bound, cf. [5]. adapting the classical methods based on cauchy’s inequality to the higher dimensional case provides us only with a weaker result in the clifford analysis setting. in [13] we proved that cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 13 proposition 1. for a left entire function f : rn+1 → cl0n of order ρ and inferior order λ set ρ1 := lim sup r→∞ log + log + µ(r) log(r) , ρ2 := lim sup r→∞ log + |ν(r)| log(r) , (18) and λ1 := lim inf r→∞ log + log + µ(r) log(r) , λ2 : lim inf r→∞ log + |ν(r)| log(r) . (19) then ρ ≤ ρ1 = ρ2 and λ ≤ λ1 = λ2. remark: in the two-dimensional complex case where we have µ(r) ≤ m(r) these methods allow one to establish the stronger result ρ = ρ1 = ρ2 and λ = λ1 = λ2, as shown for instance in [20, theorem 4.5]. with this proposition we may establish the following theorem. it provides us with a weaker analogy of valiron’s asymptotic result on the growth of the logarithm of the derivative of a given analytic function: theorem 4. if f : rn+1 → cl0n is left entire with ρ2(f) < ∞, then lim sup r→∞ log mi(r) log µ(r) ≤ 1 (20) where mi(r) := max ‖z‖=r {∥∥∥ ∂ ∂xi f(z) ∥∥∥ } for i = 1, . . . ,n. 5 the growth behavior and the taylor coefficients of a monogenic function in general it is difficult to determine the precise value of the maximum modulus. in many cases it is even complicated to just get a useful estimate on m(r,f) from below. in this section we present an explicit relation between the taylor coefficients and the growth order and the type of an entire monogenic function. this allows us to compute the growth type directly on the knowledge of the taylor coefficients without any knowledge on the maximum modulus of the function. notice that taylor series actually are a natural method to construct and to define entire monogenic functions. recall, that the product of two monogenic functions is not monogenic anymore in general. hence it is natural to construct entire monogenic functions in an additive way, for instance by its taylor series. the following two theorems provide us with higher dimensional generalizations in the clifford analysis setting of two theorems proved by lindelöf and pringsheim for complex analytic functions. in [8] we established 14 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) theorem 5. for an entire monogenic function f : rn+1 → cl0n, with taylor series representation f(z) = ∑ +∞ |m|=0 vm(z)am let π = lim sup |m|→+∞ |m| log |m| − log ‖ 1 c(n,m) am‖ . (21) then we have ρ(f) = π. remark: in the cases where ‖am‖ = 0 one puts lim sup |m|→+∞ |m| log |m| − log ‖ 1 c(n,m) am‖ := 0. the following theorem, proved in [9], also relates the growth type with the taylor coefficients of an entire monogenic function: theorem 6. let f : rn+1 → cl0n be an entire monogenic function with taylor series expansion f(z) = ∑ +∞ |m|=0 vm(z)am with order ρ (0 < ρ < +∞) and θ = lim sup |m|→+∞ |m| ( ‖am‖ ) ρ |m| . (22) then θ = τeρ, where τ is the type of f. in turn, theorem 6 allows us to construct immediately examples of entire monogenic taylor series of non-zero finite growth order ρ of any arbitrary real growth type 0 ≤ τ ≤ +∞. recalling from [9], we start with proposition 2. suppose that f : rn+1 → cl0n is an entire monogenic function. if ρ(f) = 0, then τ(f) = ∞ or f is a constant. proof. if ρ(f) = 0, then τ(f) = lim sup r→+∞ log + m(r,f). if τ(f) 6= ∞, then lim sup r→+∞ m(r,f) = eτ, which implies that ‖f(z)‖ ≤ eτ for all z ∈ rn+1. as a consequence of theorem 1, f must be a constant. example: consider p(z) to be an arbitrary left monogenic polynomial, i.e. there exist clifford numbers am ∈ cl0n and n ∈ n0 such that p(z) n∑ |m|=0 vm(z)am. from [13, theorem 3.1] we know that ‖p(z)‖ ≤ ( (n − 1 + n)! (n − 1)!n! + ε ) ‖an‖r n, (23) cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 15 where n is the index of length n for which ‖an‖ ≥ ‖am‖ for all |m| = n, with an arbitrarily small ε > 0 for r sufficiently large. thus, it follows with c(n) : ( (n−1+n)! (n−1)!n! + ε ) ‖an‖ that lim r→∞ log + (log + (m(r,p)) log(r) ≤ lim r→∞ log + (log + (c(n)rn) log(r) 0. thus, all monogenic polynomials satisfy ρ(p) = λ(p) = 0, like in the complex case. in view of proposition 2 the growth type τ equals +∞. more generally, we could establish, cf. [9]: proposition 3. let 0 < δ < +∞ and 0 < λ < +∞ be arbitrary real numbers. then f(z) = +∞∑ |m|=1 c(n, m)|m|− |m| δ vm ( ( λeδ nδ ) 1 δ z ) (24) is an entire monogenic function of growth order ρ = δ and growth type τ = λ. proof. by applying hadamard’s formula, one may directly conclude that the convergence radius of (24) is +∞. since the fueter polynomials vm are homogeneous polynomials of total degree |m|, f can directly be rewritten in the form f(z) = +∞∑ |m|=1 vm(z)am with am = c(n, m)|m| − |m| δ ( λeδ n δ ) |m| δ . according to theorem 5, the growth order of f therefore equals ρ(f) = lim sup |m|→+∞ |m| log |m| − log ‖ 1 c(n,m) am‖ = lim sup |m|→+∞ |m| log |m| − log ∣∣∣|m|− |m| δ ( λeδ n δ ) |m| δ ∣∣∣ = lim sup |m|→+∞ δ log |m| log |m| − log(λeδ nδ ) = δ. by theorem 6 we indeed furthermore obtain that τ(f) = 1 eδ lim sup |m|→+∞ c(n, m) δ |m| λeδ nδ = λ nδ lim sup m→+∞ max |m|=m c(n, m) δ m = λ nδ lim sup m→+∞ [ (n + m − 1)! (n − 1)!(m n )!n ] δ m = λ nδ lim sup m→+∞ [ 1 [ (n − 1)! ] δ m ( (n + m − 1)n+m−1+ 1 2 e−(n+m−1) ) δ m (( m n )m n + 1 2 e− m n )nδ m = λ nδ lim sup m→+∞ ( n + m − 1 m n ) δ λ. 16 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) by analogous calculations one can further show that proposition 4. let 0 < ρ < ∞. the functions g(z) = +∞∑ |m|=2 c(n, m) [ log |m| |m| ] |m| ρ vm(z) h(z) = +∞∑ |m|=2 c(n, m) [ 1 |m| log |m| ] |m| ρ vm(z) are entire monogenic functions of growth order ρ and τ(g) = +∞ and τ(h) = 0. 6 applications to partial differential equations in this section we show how the notions of the maximum term and the central indices can be applied to obtain some information on the structure of the solutions of certain class of higher dimensional partial differential equations. to proceed in this direction it turns out to be useful to first establish a relation between the asymptotic behavior of the maximum term of a monogenic function and that of their iterated radial derivatives. in [13] we established: theorem 7. let f : rn+1 → cl0n be a left entire function. then for all k ∈ n holds asymptotically 1 |ν(r)|k ∥∥∥[ek]f(z) − f(z) ∥∥∥ ≤ cµ(r)|ν(r)|− 1 2 +ε, r 6∈ f (25) where e := n∑ i=0 xi ∂ ∂xi is the euler operator on rn+1, c is a real positive constant and f is a set of finite logarithmical measure. as a direct consequence of theorem 7 one obtains proposition 5. let 0 < δ < 1 2 . we assume that ‖z‖ = r and that r be sufficiently large. suppose further that the relation ‖f(z)‖ > µ(r)|ν(r)|− 1 2 +δ (26) is satisfied for all those z that belong to a neighborhood vz0 of a point z0 in which we have ‖z0‖ = r and ‖f(z0)‖ = max ‖z‖=r {‖f(z)‖}. then for all k ∈ n holds asymptotically 1 |ν(r)|k [ek]f(z) − f(z) = o(1)f(z), r 6∈ f, (27) where f is again a set of finite logarithmical measure. cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 17 remark: this statement provides us with an analogy in the context of clifford analysis of the classical result [20, theorem 21.3] which states that entire complex-analytic functions that satisfy ‖f(z)‖ > m(r,f)[ν(r)]− 1 4 +δ have the asymptotic behavior f(m)(z) = ( ν(r) z ) m (1 + o(1))f(z). in the clifford analysis setting one thus obtains a similar asymptotic result when substituting the complex operator z d dz by the higher dimensional euler operator e. with these tools in hand we can study the structure of the solutions to some classes of partial differential equations. as a concrete example we present the following special case of an unpublished result from [12]: theorem 8. let f be an entire monogenic function of finite order ρ2 < ∞. let ‖z‖ = r and assume that r is sufficiently large. suppose further that the relation ‖f(z)‖ > µ(r)|ν(r)|− 1 2 +δ, r 6∈ f is satisfied for all those z that belong to a neighborhood vz0 of a point z0 in which we have ‖z0‖ = r and ‖f(z0)‖ = max ‖z‖=r {‖f(z)‖}. let mj[f] = aj k∏ i=0 (ei(f))ni, where aj are polynomials of degree j, and mj[f] has degree γmj = k∑ i=0 ni and weight γmj = k∑ i=0 ini. let q[f] = s∑ j=0 mj[f] be of degree γq and weight γq. if γqγm0 then the differential equation q[f] = 0 has no transcendental entire solutions. proof. if q[f] = 0, then m0[f] = − s∑ j=1 mj[f]. >from the definition of mj it follows that a0 [ k∏ i=0 (ei(f))ni ] m0 = − s∑ j=1 [ aj k∏ i=0 (ei(f))ni ] mj . applying proposition 5, we obtain that ‖a0‖|ν(r)| γm0 ‖f(z)‖γm0 ≤ s∑ j=1 ( ‖aj‖|ν(r)| γmj ‖f(z)‖ γmj ) . 18 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) since a0 is a non zero constant and aj are polynomials of degree j, taking the maximum over the norm, and applying (23) leads to |ν(r)|γm0 m(r,f)γm0 ≤ |ν(r)|γqm(r,f)γq−1 s∑ j=1 max ‖z‖r ‖aj‖ ‖a0‖ ≤ |ν(r)|γqm(r,f)γq−1rα. (28) therefore, in view of γq = γm0 one has m(r,f) ≤ |ν(r)|γq−γm0 rα. (29) for γq − γm0 < 0 it follows lim inf r→∞ m(r,f) rα ≤ lim inf r→∞ |ν(r)|γq−γm0 = 0 which implies that f is a polynomial, as a consequence of theorem 1. let us now consider the case where γq − γm0 > 0. since ρ2 < ∞, we have that |ν(r)| < r ρ2+ǫ for ǫ > 0. therefore, there exists a β > (γq − γm0 )(ρ2 + ǫ) such that lim inf r→∞ m(r,f) rβ+α ≤ lim inf r→∞ |ν(r)|γq−γm0 rβ ≤ lim inf r→∞ r(γq−γm0 )(ρ2+ǫ)−β = 0 which implies that f is a polynomial, as a consequence of theorem 1. concluding remarks: one can apply these techniques to obtain analogous statements for much more general classes of partial differential equations. in our recent paper [10] we were able to prove analogous statements for far more general systems involving polynomial expressions of the cauchy-riemann operator with arbitrary complex coefficients and radial differential operators. this paper gives a first impression in how one can extend the classical techniques from wiman and valiron from complex analysis to study much larger classes of higher dimensional elliptic operators and summarizes the first basic results. received: june 2008. revised: august 2008. references [1] m.a. abul-ez, d. constales, basic sets of polynomials in clifford analysis, complex variables 14 no. 1-4 (1990), 177–185. [2] m.a. abul-ez, d. constales, on the order of basic series representing clifford-valued functions, applied mathematics and computation 142 no. 2-3 (2003), 575–584. cubo 11, 1 (2009) a generalization of wiman and valiron’s theory to ... 19 [3] m.a. abul-ez, d. constales, on the convergence properties of basic series representing special monogenic functions, archiv der mathematik 81 no. 1 (2003), 62–71. [4] f. brackx, r. delanghe and f. sommen, clifford analysis. pitman 76, london, 1982. [5] d. constales and r.s. kraußhar, representation formulas for the general derivatives of the fundamental solution of the cauchy-riemann operator in clifford analysis and applications, zeitschrift für analysis und ihre anwendungen 21 no. 3 (2002), 579–597. [6] d. constales, r. de almeida and r.s. kraußhar, further results on the asymptotic growth of entire solutions of iterated dirac equations in rn, mathematical methods in the applied sciences 29 no. 5 (2006), 537–556. [7] d. constales, r. de almeida and r.s. kraußhar, on cauchy estimates and growth orders of entire solutions to iterated dirac and generalized cauchy-riemann equations, mathematical methods in the applied sciences 29 no. 14, (2006), 1663–1686. [8] d. constales, r. de almeida and r.s. kraußhar, on the relation between the growth and the taylor coefficients of entire solutions to the higher dimensional cauchy-riemann system in rn+1, journal of mathematical analysis and its applications 327 (2007), 763–775. [9] d. constales, r. de almeida and r.s. kraußhar, on the growth type of entire monogenic functions, archiv der mathematik 88 (2007), 153–163. [10] d. constales, r. de almeida and r.s. kraußhar, applications of the maximum term and the central index in the asymptotic growth analysis of entire solutions to higher dimensional polynomial cauchy-riemann equations, complex variables and elliptic equations 53 no. 3 (2008), 195–213. [11] d. constales, r. de almeida and r.s. kraußhar, basics of a generalized wimanvaliron theory for monogenic taylor series of finite convergence radius, to appear. [12] r. de almeida, n normal families and the growth behavior of polymonogenic functions, phd thesis, universidade de aveiro, portugal, 2006. [13] r. de almeida and r.s. kraußhar, on the asymptotic growth of monogenic functions, zeitschrift für analysis und ihre anwendungen 24 no. 4 (2005), 763–785. [14] r. delanghe, on regular points and liouville’s theorem for functions with values in a clifford algebra, simon stevin 44 (1970-71), 55–66. [15] r. delanghe, f. sommen and v. souček, clifford algebra and spinor valued functions, kluwer, dordrecht-boston-london, 1992. [16] r. fueter, functions of a hyper complex variable, lecture notes written and supplemented by e. bareiss, math. inst. univ. zürich, fall semester 1948/49. 20 d. constales, r. de almeida and r.s. kraußhar cubo 11, 1 (2009) [17] k. gürlebeck and a. hommel, on exponential functions in clifford analysis, to appear. [18] k. gürlebeck, k. habetha, and w. sprößig, funktionentheorie in der ebene und im raum, birkhäuser verlag, basel, 2006. [19] k. gürlebeck and w. sprößig, quaternionic and clifford calculus for physicists and engineers, john wiley & sons, chichester-new york, 1997. [20] g. jank and l. volkmann, meromorphe funktionen und differentialgleichungen, utb birkhäuser, basel 1985. [21] w. k. hayman, the local growth of power series: a survey of the wiman-valiron method, canadian mathematical bulletin 17 (1974), 317-358. [22] e. lindelöf, sur la détermination de la croissance des fonctions entìeres définies par un développement de taylor, darb. bull. 27 no.2 (1903), 213–226. [23] r. nevanlinna, zur theorie der meromorphen funktionen, acta matematica 46 (1925), 1–99. [24] a. pringsheim, elementare theorie der ganzen transzendenten funktionen von endlicher ordnung, mathematische annalen 58 (1904), 257–342. [25] g. valiron, lectures on the general theory of integral functions, chelsea, new york, 1949. [26] a. wiman, über den zusammenhang zwischen dem maximalbetrage einer analytischen funktion und dem größten gliede der zugehörigen taylorschen reihe, acta mathematica 37 (1914), 305–326. cubo 11-n01-2009 cubo a mathematical journal vol.11, no¯ 05, (1–22). december 2009 on some spectral problems and asymptotic limits occuring in the analysis of liquid crystals bernard helffer département de mathématiques, univ paris-sud and cnrs, 91405 orsay cedex, france email: bernard.helffer@math.u-psud.fr and xing-bin pan department of mathematics, east china normal university, shanghai 200062, p.r. china email : xbpan@math.ecnu.edu.cn abstract on the basis of de gennes’ theory of analogy between liquid crystals and superconductivity, the second author introduced the critical wave number qc3 of liquid crystals, which is an analog of the upper critical field hc3 for superconductors, and he predicted the existence of a surface smectic state, which was supposed to be an analog of the surface superconducting state. in this article we study this problem and our study relies on the landau-de gennes functional of liquid crystals in connection with a simpler functional called the reduced ginzburg-landau functional which appears to be relevant when some of the elastic constants are large. we discuss the behavior of the minimizers of these functionals. we describe briefly some results obtained by bauman-carme calderer-liu-phillips, and present more recent results on the reduced ginzburg-landau functional obtained by the authors. this paper is partially extracted of lectures given by the first author in recife and serrambi in august 2008. 2 bernard helffer and xing-bin pan cubo 11, 5 (2009) resumen sobre la base de teoria de gennes, la analogía entre cristales liquidos y superconductividad, el segundo autor introdujo el número de onda crítico qc3 de cristales liquidos, el cual es un análogo del campo crítico superior hc3 para superconductores, él predijo la existencia de una superficie “smetic state”, la qual fué supuesta ser un análogo de la superficie de estado de supercondutividad. en este artículo estudiamos este problema y nuestro estudio se basa en el funcional de landau-de gennes de cristales liquidos en conexión con un simple funcional llamado el funcional de ginzburg-landau reduzido que resulta ser relevante cuando algunas de las constantes elasticas son grandes. nosotros discutimos el comportamiento de los minimizadores de esos funcionales. describimos brevemente algunos resultados obtenidos por bauman-carme calderer-liu-phillips, y presentamos resultados mas recientes sobre el funcional de ginzburg-landau reduzido obtenidos por los autores. este artículo es parcialmente extraido de las conferencias dadas por el primier autor en recife y serrambi en agosto 2008. key words and phrases: liquid crystals, surface smectic state, landau-de gennes functional, reduced ginzburg-landau functional, critical wave number, critical elastic coefficients. math. subj. class.: 82d30, 82d55, 35j55, 35q55. 1 introduction in [p1], based on the de gennes analogy between liquid crystals and superconductivity [dg1, dgp], one of the authors (x. pan) introduced the critical wave number qc3 (which is an analog of the upper critical field hc3 for superconductors) and predicted the existence of a surface smectic state, which was supposed to be an analog of the surface superconducting state. the phase transition between the nematic state and smectic state of liquid crystals as the wave number varies around qc3 was studied in [p1]. bauman, calderer, liu and phillips [bclp] studied the phase transition of liquid crystals for a different set of parameters1. in this article (which is partially extracted of lectures given by one of the authors (b. helffer) in recife and serrambi in august 2008), we study this problem which relies on the landau-de gennes functional (modeling the properties of liquid crystals) in connection with a simpler functional called the reduced ginzburg-landau functional which appears to be relevant when some of the elastic constants are large. we discuss the behavior of the minimizers. we describe mainly some results obtained by bauman-carme caldererliu-phillips [bclp], pan [p1, p4, p5] and more recent results on the reduced ginzburg-landau functional obtained in [hp2, hp3]. we also add a few new results. all these results suggest that a liquid crystal with large ginzburg-landau parameter κ will be in the surface smectic state if the number qτ lies asymptotically between κ2 and κ2/θ0 with κ → ∞, where θ0 ∈ (0, 1) is the lowest eigenvalue of the schrödinger operator with a unit magnetic field in the half plane. 1in particular [p1] considers the case where k2 + k4 = 0 and in [bclp] it is assumed that c0 ≤ k2 + k4 ≤ c1 where c0 and c1 are fixed positive constants. cubo 11, 5 (2009) on some spectral problems and asymptotic ... 3 this is also a natural extension of what was done by fournais and helffer in superconductivity in continuation of previous results of lu-pan, bernoff-sternberg, helffer-morame and many others (see [fh4] and references therein). we will only present some of the results, emphasize on some points, recall some basic material and refer to the original papers or works in progress for more details and proofs. 2 some questions in the theory of liquid crystals and first answers 2.1 the model the model in liquid crystals can be described2 by the functional (ψ, n) 7→ ek[ψ, n] = ∫ ω { |∇qnψ| 2 − κ2|ψ|2 + κ2 2 |ψ|4 + k1 |div n| 2 + k2 |n · curl n + τ| 2 + k3 |n × curl n| 2 } dx, where: • ω ⊂ r3 is the region occupied by the liquid crystal, • ψ is a complex-valued function called the order parameter, • n is a real vector field of unit length called director field, • q is a real number called wave number, • ∇qn is the magnetic gradient: ∇qn = ∇ − iqn , • τ is a real number measuring the chiral pitch, • k = (k1,k2,k3) with k1 > 0, k2 > 0 and k3 > 0 is the triple of the elastic coefficients or frank coefficients, • κ > 0 depends on the material and on temperature and is called the ginzburg-landau parameter of the liquid crystal. this functional is called the landau-de gennes functional. we are interested in minimizing the functional over the pairs (ψ, n) ∈ h1(ω, c)×v (ω, s2), where h1(ω, c) is the standard sobolev space for complex-valued functions, v (ω, s2) consists of vector fields n such that div n ∈ l2(ω), curl n ∈ l2(ω, r3) and |n(x)|2 = 1 almost everywhere. we refer to [c], [bclp] and [p1, p4, p6] 2this is an already simplified model where boundary terms (see [bclp, p1]) have been eliminated. with the notations of these authors, we are considering as in [hp2] the case k2 + k4 = 0. 4 bernard helffer and xing-bin pan cubo 11, 5 (2009) for a more complete discussion of the mathematical issues and a discussion of the contents of the references to the physics literature [dg1, dg2, dgp]. we only mention here that the physical interpretation is that n is the molecular director field and that, if we write ψ(x) = ρ(x)eiφ(x) , where ρ(x) ≥ 0 and φ(x) is a real function, we recover the molecular mass density by δ(x) = ρ0(x) + ρ(x) cos φ(x) , where ρ0(x) is some given reference density. observing that we have the lower bound ek[ψ, n] ≥ − κ2 2 |ω|, (2.1) it is not too difficult to show that this functional admits minimizers. but the main questions are then: • what is the minimum of the energy ? • what is the nature of the minimizers ? of course the answer depends heavily on the various parameters and we will only be able to give answers in some asymptotic regimes. as in the theory of superconductivity, a special role will be played by some critical points of the functional, the pairs (0, n) , where n is a minimizer of the so called oseen-frank functional: n 7→ ekof [n] := ∫ ω { k1 |div n| 2 + k2 |n · curl n + τ| 2 + k3 |n × curl n| 2 } dx. these special solutions are called “nematic phases” and one is naturally asking if they are minimizers or local minimizers of the functional ek. in any case, the minimizers of ek satisfy some euler-lagrange equation. we do not write the complete system but note that the variation with respect to the order parameter leads to −∇2qnψ − κ 2ψ + κ2|ψ|2ψ = 0 in ω , (2.2) together with the boundary condition ν · ∇qnψ = 0 on ∂ω , (2.3) where ν denotes the normal to the boundary. using the maximum principle for u = |ψ|2 which can be seen as a solution of    −∆u + κ2u(1 − u) ≥ 0 in ω , ∂u ∂ν = 0 on ∂ω , one can show that ‖ψ‖l∞(ω) ≤ 1 . (2.4) cubo 11, 5 (2009) on some spectral problems and asymptotic ... 5 2.2 a universal upper bound for τ > 0, let us consider c(τ) the set of the s2-valued vector fields satisfying: curl n = −τn , div n = 0 . it has been shown in [bclp] that c(τ) consists of the vector fields nqτ such that, for some q ∈ so(3) , n q τ (x) ≡ qnτ(q tx) , x ∈ ω , (2.5) where nτ (y1,y2,y3) = (cos(τy3), sin(τy3), 0) , y ∈ r 3 . (2.6) this is also equivalent, as |n|2 = 1, to div n = 0 , n · curl n + τ = 0 , n × curl n = 0 . (2.7) so the last three terms in the functional ek vanish if and only if n ∈ c(τ) . as a consequence, if we denote by c(k1,k2,k3,κ,q,τ) = inf (ψ,n)∈h1(ω,c)×v (ω,s2) ek[ψ, n] , (2.8) the infimum of the energy over the natural maximal form domain of the functional, then c(k1,k2,k3,κ,q,τ) ≤ c(κ,q,τ) , (2.9) where c(κ,q,τ) = inf n∈c(τ) inf ψ∈h1(ω,c) gqn[ψ] , (2.10) and gqn[ψ] is the so called reduced ginzburg-landau functional which will be defined in the next subsection. 2.3 reduced ginzburg-landau functional given a vector field a, the reduced ginzburg-landau functional ga with magnetic potential a is defined on h1(ω, c) by ψ 7→ ga[ψ] = ∫ ω {|∇aψ| 2 − κ2|ψ|2 + κ2 2 |ψ|4} dx. (2.11) the standard ginzburg-landau functional with external vector field σhe (where he = curl f is a divergence free vector field on ω) (see [fh4] and references therein) takes the form egl[ψ, a] = gκσa[ψ] + κ 2σ2 ∫ r3 |curl a − he|2 dx (2.12) where 6 bernard helffer and xing-bin pan cubo 11, 5 (2009) • ω is a bounded and simply connected domain, • (ψ, a) ∈ h1(ω, c) × ḣ1 div,f(r 3, r3), • ḣ1 div,f(r 3, r3) = {a | div a = 0 , a − f ∈ ḣ1(r3, r3) } , • ḣ1(r3) denotes the homogeneous sobolev space, i.e. the closure of c∞0 (r 3 ) under the norm u 7→ ‖∇u‖l2(r3) , and ḣ 1 (r 3, r3) denotes the corresponding space of vector fields. so the oseen-frank energy in liquid crystals theory replaces the magnetic energy measuring the square of the l2 norm of curl a − he in r3 in ginzburg-landau theory. for convenience, we also write ga[ψ] as g[ψ, a]. so we have c(κ,q,τ) = inf n∈c(τ),ψ∈h1(ω,c) g[ψ,qn] ≤ 0 , (2.13) and if n ∈ c(τ) , then the following equality holds ek[ψ, n] = g[ψ,qn] . (2.14) 3 a limiting case: the case of large frank constants we have seen that in full generality (2.9) holds. conversely, it can be shown (see [bclp, p1, hp2]) that when the elastic parameters tend to +∞, the converse is asymptotically true. proposition 3.1. lim k1,k2,k3→+∞ c(k1,k2,k3,κ,q,τ) = c(κ,q,τ) . (3.1) so c(κ,q,τ) is a good approximation for the minimal value of ek for large kj’s. of course, a basic initial remark for the proof is that if (ψ, n) is a minimizer ek then we always have ekof [n] ≤ κ2|ω| 2 . (3.2) remark 3.2. in [bclp], the authors used instead the bound ekof [n] ≤ c(ω)q 2τ2 , (3.3) which is obtained from the universal upperbound c(k1,k2,k3,κ,q,τ) ≤ c(ω)q 2τ2 − κ2|ω| 2 . (3.4) this upper bound is obtained (see lemma 1 in [bclp]) by computing the energy of the pair (ψ, n) = (eiqx·nτ (x), nτ(x)), with nτ defined in (2.6). cubo 11, 5 (2009) on some spectral problems and asymptotic ... 7 this gives the two following controls ‖div n‖2l2(ω) ≤ κ2|ω| 2k1 , (3.5) and ‖curl n − τn‖2l2(ω) ≤ κ2|ω| 2 min{k2,k3} . (3.6) we also have ‖∇qnψ‖ 2 l2(ω) ≤ κ2|ω| 2 . (3.7) remark 3.3. 1. this limiting case where all the elastic coefficients tend to +∞ appears naturally in the transition from smectic-c to nematic phase (see [dg2]). 2. as observed by d. phillips in a conference in ryukoku university in japan, it would be also interesting to have the result with fixed k1 > 0, in the limit when k2 and k3 tend to +∞. a modification of the proof in [hp2] leads indeed (see lemma 3.4) to this result. for parameters in different regime (in particular c0 < k2 + k4 < c1), the same conclusion was obtained in [bclp]. 3. an interesting open problem is to control the rate of convergence in (3.1) (see [ray3]). proof of proposition 3.1 to illustrate the last point of the remark and to complete the proof of the proposition, we need the following result 3 lemma 3.4. let τ0 > 0 and c0 > 0. then for any ǫ > 0, there exists α > 0 such that if n ∈ v (ω, s 2 ), τ ∈ (0,τ0], and ‖curl n + τn‖l2(ω) ≤ α, ‖div n‖l2(ω) ≤ c0τ , then there exists q ∈ so(3) such that ‖n − nqτ ‖l4(ω) ≤ ǫ . (3.8) proof. we give the proof in the case of a fixed τ > 0. if it were not true, we will find ǫ0 > 0 and a bounded sequence nj ∈ v (ω, s 2 ) such that ‖div nj‖l2(ω) is bounded, lim j→+∞ ‖curl nj + τnj‖l2(ω) = 0 , 3the same conclusion was proved in [bclp] (lemma 4) in the case where k2 + k4 ≥ c0 > 0 and hence ‖∇n‖2 l2(ω) can be controlled by the energy. 8 bernard helffer and xing-bin pan cubo 11, 5 (2009) and inf q∈so(3) ‖nj − n q τ ‖l4(ω) ≥ ǫ0 . from the assumptions, it is clear that the sequence is bounded in v (ω, s2), and hence bounded in h1 loc (ω, r3) (see [p4], lemma 2.3), hence we can extract a subsequence, still denoted by nj, and find n∞ such that nj tends weakly to n∞ in h 1 loc (ω, r3). one can also show that |n∞| 2 = 1 a.e. in ω and that curl n∞ + τn∞ = 0. so n∞ belongs to c(τ) and there exists q ∈ so(3) such that n∞ = n q τ . now by compactness of the injection of h 1 loc (ω, r3) in l4 loc (ω, r3), we get that nj tends to n∞ in l 4 loc (ω, r3). let d be a compact subset of ω such that |ω \ d| < ǫ0/48. for large j we have ∫ d |nj − n∞| 4dx < ǫ40 3 . then ∫ ω |nj − n∞| 4 dx ≤ ∫ d |nj − n∞| 4 dx + 16|ω \ d| < 2ǫ40 3 , which leads to a contradiction. the control of the rate of convergence in proposition 3.1 should pass through a good knowledge of α(ǫ). the second step in the proof of proposition 3.1 consists in observing that, if (ψ, n) is a minimizer, we can, for any q ∈ so(3) , get the lower bound g[ψ,qn] = ‖∇qnψ‖ 2 l2(ω) − κ 2‖ψ‖2l2(ω) + κ2 2 ‖ψ‖4l4(ω) ≥ (1 − η)‖∇ qn q τ ψ‖2l2(ω) − κ 2‖ψ‖2l2(ω) + κ2 2 ‖ψ‖4l4(ω) − q2 η |ω|1/2‖n − nqτ ‖ 2 l4(ω) . (3.9) here we have used cauchy-schwarz inequality and (2.4). this we can rewrite, using (3.7), in the form g[ψ,qn] ≥ g[ψ,qnqτ ] − η 2 κ2|ω| − q2 η |ω|1/2‖n − nqτ ‖ 2 l4(ω) . (3.10) this gives g[ψ,qn] ≥ c(κ,q,τ) − η 2 κ2|ω| − q2 η |ω|1/2‖n − nqτ ‖ 2 l4(ω) . (3.11) putting together the estimates, we obtain ekof (n) ≤ η 2 κ2|ω| + q2 η |ω|1/2‖n − nqτ ‖ 2 l4(ω) , (3.12) cubo 11, 5 (2009) on some spectral problems and asymptotic ... 9 and c(κ,q,τ) ≥ c(k1,k2,k3,κ,q,τ) ≥ c(κ,q,τ) − η 2 κ2|ω| − q2 η |ω|1/2‖n − nqτ ‖ 2 l4(ω) . (3.13) these estimates are valid for any η > 0 and any q ∈ so(3) . the remainder on the right hand side can be chosen arbitrarily small by choosing first η small. then we can use (3.5) and (3.6) to have ‖curl n + τn‖l2(ω) and ‖div n‖l2(ω) small and using lemma 3.4 (and a good choice of q) to have ‖n − nqτ ‖l4(ω) small. 4 analysis of the reduced ginzburg-landau functional we now analyze the non-triviality of the minimizers realizing c(κ,q,τ). as for the ginzburglandau functional in superconductivity, this question is closely related to the analysis of the lowest eigenvalue λn1 (qn) of the neumann realization of the magnetic schrödinger operator −∇ 2 qn in ω that we met already when describing the euler-lagrange equation associated to the functional. namely λn1 = λ n 1 (qn) is the lowest eigenvalue of the following problem { −∇2qnφ = λ n 1 φ in ω , ν · ∇qnφ = 0 on ∂ω , (4.1) where ν is the unit outer normal of ∂ω. but the new point is that we will minimize λn1 (qn) over n ∈ c(τ). so we shall actually meet the quantity: µ∗(q,τ) = inf n∈c(τ) λn1 (qn) . (4.2) we preliminarily observe the lemma 4.1. if (ψ,qn) is a nontrivial minimizer of g, then g[ψ,qn] < 0 . the proof is simple. ψ is a solution of the (euler-lagrange) equation (2.2) with neumann condition (2.3). multiplying (2.2) by ψ and integrating over ω, we have, after an integration by parts and taking account of the boundary condition (2.3), ∫ ω {|∇qnψ| 2 − κ2(1 − |ψ|2)|ψ|2} dx = 0 , (4.3) and hence c(κ,q,τ) = gqn[ψ] = − κ2 2 ∫ ω |ψ|4 dx < 0 . (4.4) this gives: 1 2 (µ∗(q,τ) − κ 2 )‖ψ‖2l2(ω) ≤ 1 2 (λn1 (qn) − κ 2 )‖ψ‖2l2(ω) = c(κ,q,τ) . (4.5) 10 bernard helffer and xing-bin pan cubo 11, 5 (2009) hence, when c(κ,q,τ) < 0, we should have µ∗(q,τ) < κ 2. pushing forward, one has the main comparison statement (analogous to a statement in fournais-helffer [fh3] for surface superconductivity) in [hp2]: proposition 4.2. − κ2|ω| 2 [1 − κ−2µ∗(q,τ)] 2 ≤ c(κ,q,τ) (4.6) and c(κ,q,τ) ≤ − κ2 2 [1 − κ−2µ∗(q,τ)] 2 + sup n∈c(τ) sup φ∈sp(qn) ( ∫ ω |φ|2 dx)2∫ ω |φ|4 dx , (4.7) where sp(qn) is the eigenspace associated to λn1 (qn), and [ · ]+ denotes the positive part of the enclosed quantity. for the upper bound, we can minimize g[ψ,qn] over the pairs (ψ, n) with ψ = tψqn, where ψqn is an eigenfunction of −∇ 2 qn, t ∈ c, and n ∈ c(τ). for the lower bound, we have just to use the hölder inequality and (4.4) and (4.5). this completes the (sketch of the) proof that c(κ,q,τ) is strictly negative if and only if µ∗(κ,τ) < κ 2. 5 main questions as a consequence of proposition 4.2, we obtain that the transition from a nematic phase to a smectic phase is strongly related to the analysis of the solution of 1 − κ−2µ∗(q,τ) = 0 . (5.1) this is a pure spectral problem concerning a family indexed by n ∈ c(τ) of schrödinger operators with magnetic field −∇2qn. remark 5.1. in the analysis of (5.1), the monotonicity of q 7→ µ∗(q,τ) is an interesting open question (see fournais-helffer [fh3] for the phase transition from normal state to surface superconducting state of type ii superconductors). this will permit indeed to find a unique solution of (5.1) permitting a natural definition of the critical value qc3(κ,τ). we hope to answer this question in [hp3] in the case of a strictly convex domain. we have proved in [hp2] that if τ stays in a bounded interval, then qc3(κ,τ) and µ∗(q,τ) can be controlled in two regimes cubo 11, 5 (2009) on some spectral problems and asymptotic ... 11 • σ → +∞ , • σ → 0 , where σ = qτ , which, as it appears already in [bclp], is in some sense the leading parameter in the theory. we mention that if we examine the magnetic schrodinger operator −∇2qn, the parameter σ corresponds indeed to the intensity of the magnetic field corresponding to the magnetic potential qn, with n ∈ c(τ). this will be detailed in sections 7 and 8. 6 a simpler question a simpler question, which was first introduced in [p1], and partially solved in [p5] with the help of [p3, hm4], but can be completed under the additional assumption below by a careful control (see [hp2, hp3]) of the uniformity in the proof of [hm4], can be stated as follows: question 6.1. given a strictly convex open set ω ⊂ r3, find the direction h of the constant magnetic field giving asymptotically as σ → +∞ the lowest energy for the neumann realization in ω of the schrödinger operator with magnetic field σh. let us present shortly the answer to this question. we assume that assumption 6.2. at each point of ∂ω the curvature tensor has two strictly positive eigenvalues κ1(x) and κ2(x), so that 0 < κ1(x) ≤ κ2(x) . under this assumption, the set γh of boundary points where h is tangent to ∂ω, i.e. γh := {x ∈ ∂ω ∣∣ h · ν(x) = 0}, (6.1) is a regular submanifold of ∂ω: dt (h · ν)(x) 6= 0 , ∀x ∈ γh , (6.2) where dt denotes the tangential gradient along γh. the basic example where this assumption is satisfied is the ellipsoid. for any given h, let fh be the magnetic potential such that curl fh = h and div fh = 0 in ω, fh · ν(x) = 0 on ∂ω . we have the following two-term asymptotics of λn1 (σfh) of the neumann laplacian −∇ 2 σfh , (due to helffer-morame-pan [hm4, p3]). 12 bernard helffer and xing-bin pan cubo 11, 5 (2009) theorem 6.3. if ω and h are as above, then, as σ → +∞, λn1 (σfh) = θ0σ + γ̂hσ 2/3 + o(σ2/3−η) , (6.3) for some η > 0. moreover η is independent of h and the control of the remainder is uniform with respect to h. here θ0 ∈ (0, 1), δ0 ∈ (0, 1) and ν̂0 > 0 are spectral quantities (see in the appendix), and γ̂h is defined by γ̂h := inf x∈γh γ̃h(x) , (6.4) where γ̃h(x) := 2 −2/3ν̂0δ 1/3 0 |kn(x)| 2/3 ( 1 − (1 − δ0)|th(x) · h| 2 )1/3 . (6.5) here th(x) is the oriented, unit tangent vector to γh at the point x and kn(x) = |d t (h · ν)(x)| . note that the constant θ0 has been denoted by β0 in [lup1, lup2, lup3] and in [p1, p3, p4] etc. here is now the answer to the question 6.1. we have just to determine infh∈s2 γ̂h or equivalently inf h∈s2 inf x∈γh |kn(x)| 2/3 ( 1 − (1 − δ0)|th(x) · h| 2 )1/3 . so everything is reduced to the analysis of the map γh ∋ x 7→ kn(x) 2 ( 1 − (1 − δ0)|th(x) · h| 2 ) . as observed in the appendix of [hm3], where the comparison is done between the results of [p3] and [hm4], this last expression can be written in the form γh ∋ x 7→ κ1(x) 2 cos 2 φ(x) + κ2(x) 2 sin 2 φ(x) − (1 − δ0)(κ1(x) − κ2(x)) 2 sin 2 φ(x) cos2 φ(x) , where, for x ∈ ∂ω, φ(x) is defined by writing h = cos φ(x)u1(x) + sin φ(x)u2(x) , with (u1(x), u2(x)) being the orthonormal basis of the curvature tensor at x, associated to the eigenvalues κ1(x) and κ2(x). we easily observe that, is 0 < κ1 ≤ κ2 the function [0, 1] ∋ t 7→ κ21t + κ 2 2(1 − t) − (1 − δ0)(κ1 − κ2) 2t(1 − t) , admits a minimum at t = 1. hence, when minimizing over h and x ∈ γh, it is rather easy to show that the infimum is obtained by first choosing a point x0 of ∂ω where κ1(x) is minimum and then taking h = u1(x0). this leads to the following proposition which was conjectured and partially proved by x. pan [p5] and then completed in [hp2]. cubo 11, 5 (2009) on some spectral problems and asymptotic ... 13 proposition 6.4. under assumption 6.2, we have inf h∈s2 γ̂h = inf x∈∂ω (κ1(x)) 2/3 , (6.6) hence inf h∈s2 λn1 (σfh) = θ0σ + inf x∈∂ω (κ1(x)) 2/3σ2/3 + o(σ2/3−η) . (6.7) this answers question 6.1. 7 semi-classical case: qτ large when looking at the general problem, various questions occur. the magnetic field −qτn (corresponding when n ∈ c(τ) to the magnetic potential qn) is no more constant, so one should extend the analysis to this case. a first analysis [hm4, p3, hp2] (semi-classical in spirit) gives: theorem 7.1. as σ = qτ → +∞, µ∗(q,τ) = θ0 qτ + o((qτ) 2/3 ) , (7.1) where the remainder is controlled uniformly for 4 τ ∈ (0,τ0] . this is a consequence of λn1 (qn) = θ0 qτ + o((qτ) 2/3 ) , (7.2) with o((qτ)2/3) uniform with respect to n ∈ c(τ) and τ ∈ (0,τ0] . the reader could be astonished to have this uniformity. the first thing is to observe that, when ω has no holes, it is not the magnetic potential a = qn which is important in the analysis of the neumann groundstate energy of −∇2 a but the magnetic field which is −qτn. we observe that the magnetic field is of constant strength qτ and that its variation is controlled if τ ∈ (0,τ0]. the analysis of [hm4] which was devoted to the constant magnetic field case can go through (see [hp2]). this leads (assuming the monotonicity of µ∗ with respect to q), to obtain for the solution qc3(κ,τ) of (5.1) the expansion τ qc3(κ,τ) = κ2 θ0 + o(κ4/3) . (7.3) we refer to [hp2] and to the last section for a discussion of this critical wave number. we hope to give in [hp3] a two-term asymptotic of µ∗(q,τ) and consequently of qc3(κ,τ) for large κ (with τ ∈ (0,τ0]). 4this condition can be relaxed [ray2] at the price of a worse remainder. 14 bernard helffer and xing-bin pan cubo 11, 5 (2009) 8 perturbative case: qτ small a second analysis (perturbative in spirit) gives (see [hp2]) theorem 8.1. as σ = qτ → 0, µ∗(q,τ) = θ(τ)(qτ) 2 + o((qτ)4) , (8.1) where the remainder is controlled uniformly for τ ∈ (0,τ0], and τ 7→ θ(τ) is a continuous function on [0,τ0] such that θ(0) = inf h∈s2 1 |ω| ∫ ω |fh| 2 dx. (8.2) one can also give an asymptotic of c(κ,q,τ), see [hp2]. theorem 8.2. let ω be a bounded smooth domain in r3 and τ0 > 0. then, there exist positive constants c1(τ0), c2(τ0) and σ1(τ0), such that, for any q,τ,κ satisfying τ ∈ (0,τ0], σ = qτ ∈ (0,σ1], κ ≥ c1(τ0)σ , (8.3) we have ∣∣∣∣c(q,τ,κ) + κ2 2 |ω| − σ2|ω|θ(τ) ∣∣∣∣ ≤ c2(τ0)(1 + κ −2 )σ4 . (8.4) as a corollary and using (2.1), we immediately obtain under the same assumptions ∣∣∣∣ inf (ψ,n)∈h1(ω,c)×v (ω,s2) ek[ψ, n] + κ2 2 |ω| ∣∣∣∣ ≤ c(τ0)(qτ) 2 (1 + κ−2) . (8.5) this estimate is independent of the elastic parameters. remark 8.3. it would be good to have a second term in this last expansion. 9 coming back to the main functional we finish this survey by presenting the following results regarding the non-triviality of minimizers. similar but less accurate results have been obtained in [bclp]. for the corresponding results for superconductors see giorgi and phillips [giop] and lu and pan [lup1, lup2, lup3]. it is a consequence of (8.4) and of (2.9) that proposition 9.1. let ω be a bounded smooth domain in r3, τ0 > 0 and κ0 > 0. then, for any q,τ,κ satisfying τ ∈ (0,τ0], σ = qτ ∈ (0,σ1], κ ≥ c1(τ0)σ , κ ≥ κ0 , (9.1) the minimizers of ek are non trivial. cubo 11, 5 (2009) on some spectral problems and asymptotic ... 15 we note that the constants involved in the previous statement are independent of the elastic coefficients. the next proposition will show that, if σ/κ and the elastic constants are sufficiently large, the minimizers are the nematic phases. proposition 9.2. for any τ0 and any σ0, there exists a constant c > 0 such that, if τ ∈ (0,τ0] , qτ ≥ σ0 , qτ ≥ c(1 + q 2 )(1 + κ2) , then the functional ek has no minimizer with ψ 6= 0. proof. let (ψ, n) be a minimizer, with ψ not trivial. so ek[ψ, n] < 0 . we should keep in mind what we have obtained in (2.2). in particular, we deduce, like for getting (4.3), that ‖∇qnψ‖ 2 l2(ω) ≤ κ 2‖ψ‖2l2(ω) . (9.2) as in the proof of proposition 3.1, we can compare with some element in c(τ). for any q ∈ so(3) , we have ‖∇ qn q τ ψ‖2l2(ω) ≤ 2‖∇qnψ‖ 2 l2(ω) + 2q 2‖n − nqτ ‖ 2 l4(ω)‖ψ‖ 2 l4(ω) . (9.3) then, one can use (9.2) and the so called diamagnetic inequality5 and this, together with sobolev’s injection of h1(ω, c) in l4(ω, c), leads to ‖ψ‖2l4(ω) = ‖ |ψ| ‖ 2 l4(ω) ≤ c(ω)(‖∇|ψ|‖2l2(ω) + ‖ψ‖ 2 l2(ω)) ≤ c(ω)(‖∇qnψ‖ 2 l2(ω) + ‖ψ‖ 2 l2(ω)) . (9.4) hence we obtain, using the characterization of the groundstate energy, λn1 (qn q τ ) ≤ 2(κ 2 + c(ω)(1 + κ2)q2‖n − nqτ ‖ 2 l4(ω)) . (9.5) we now choose some q and observe that, for any σ0 > 0, there exists ĉ, such that, if qτ is larger than σ0, we have from the proof of theorem 7.1 (uniformly with respect to τ ∈ (0,τ0]) qτ ĉ ≤ λn1 (qn q τ ) . (9.6) this implies qτ ĉ ≤ 2(1 + q2‖n − nqτ ‖ 2 l4(ω))(1 + κ 2 ) . (9.7) the observation –and this is again independent of the elastic coefficients– is that qτ ĉ ≤ 2(1 + 4|ω|1/2q2)(1 + κ2) . (9.8) 5we recall that diamagnetic inequality says that, for any u ∈ h1 loc and any magnetic potential a in l2 loc we have |∇|u|| ≤ |∇au| , almost everywhere. 16 bernard helffer and xing-bin pan cubo 11, 5 (2009) the proof is finished by taking c = 4 max{1, 4|ω|1/2ĉ} . hence, this is only if we want to have a more precise information about the transition between nematic phase and smectic phase that we will have to use that some of the elastic constants are large. let us give a statement, in this direction. proposition 9.3. for any τ0 > 0, any k 0 1 > 0, any ǫ > 0, there exists κ0 and, for any q, c(ǫ,τ0,q) such that if 6 • κ ≥ κ0 , • τ ∈ (0,τ0] , • k1 ≥ k 0 1κ 2/τ, • minj=2,3 kj ≥ c(ǫ,τ0,q)κ 2 , • qτ ≥ (1 + ǫ)κ2/θ0 , then the functional has no minimizers with ψ 6= 0. proof. we have to improve the argument starting from (9.3) which we replace for any η > 0 by ‖∇ qn q τ ψ‖2l2(ω) ≤ (1 + η)‖∇qnψ‖ 2 l2(ω) + (1 + 1 η )q2‖n − nqτ ‖ 2 l4(ω)‖ψ‖ 2 l4(ω) . (9.9) this leads to replace (9.5) by λn1 (qn q τ ) ≤ (1 + η)κ 2 + c(ω,η)(1 + κ2)q2‖n − nqτ ‖ 2 l4(ω) , (9.10) with two free parameters η and q . we first choose η = ǫ/2. we can now use lemma 3.4 together with (3.5) and (3.6) in order to get c(ω,η)q2‖n − nqτ ‖ 2 l4(ω) ≤ ǫ 2 . this leads to our choice of q. this time we have to use the uniform asymptotic established in (7.2). remark 9.4. the results of this section have to be compared with similar, but less accurate, results of [bclp]. 6we can in the second item alternatively assume k1 ≥ k 0 1q 2τ . cubo 11, 5 (2009) on some spectral problems and asymptotic ... 17 application to the critical wave numbers proposition 9.3 admits a converse using what we know about c(κ,q,τ). this converse result is independent of the frank constants. more precisely, we can introduce (following [p1]): for κ > 0, τ > 0, we define q c3 (κ,τ) = inf{q > 0 : ψ = 0 is the minimizer of gqn for all n ∈ c(τ)} , qc3(κ,τ) = inf{q > 0 : ψ = 0 is the unique minimizer of gq′n for all q′ > q, n ∈ c(τ)}. (9.11) and q k c3(κ,τ) = inf{q > 0 : the (0, n) (n ∈ c(τ)) are minimizer of e k }, qk c3 (κ,τ) = inf{q > 0 : the (0, n) (n ∈ c(τ)) are the unique minimizers of ek for all q′ > q,}. (9.12) here we have explicated in the notation the dependence on k = (k1,k2,k3). it results of course of the universal estimate (2.9) that q k c3(κ,τ) ≤ qc3(κ,τ) and q k c3 (κ,τ) ≤ q c3 (κ,τ) , (9.13) for any elastic constants. proposition 9.3 implies that for large κ and sufficient large elastic constants the critical wave numbers satisfy τ lim kj →+∞ q k c3(κ,τ) ∼ κ2 θ0 and τ lim kj →+∞ qk c3 (κ,τ) ∼ κ2 θ0 . (9.14) a the de gennes family the family of operators h(ξ) h(ξ) = d2t + (t − ξ) 2 (a.1) on the half-line with neumann boundary condition at 0 appears initially in the analysis of the problem in the half plane r2+ := {x1 > 0} of the neumann realization of the magnetic schrodinger operator with constant magnetic field d2x1 + (dx2 − x1) 2. here we write dt = 1 i ∂ ∂t . the lowest eigenvalue of the operator h(ξ), ξ 7→ µ(ξ) admits a unique minimum at ξ0 > 0. for analyzing the variation of µ(ξ), it is useful to combine the following two formulas • the feynman-hellmann formula: µ′(ξ) = −2 ∫ +∞ 0 (t − ξ)uξ(t) 2 dt , 18 bernard helffer and xing-bin pan cubo 11, 5 (2009) where uξ is the normalized groundstate of h(ξ). • the bolley-dauge-helffer formula [dah]: µ′(ξ) = uξ(0) 2 (ξ2 − µ(ξ)) . this permits to show that µ(ξ) has a unique minimum, which is attained at ξ0 > 0. moreover lim ξ→+∞ µ(ξ) = 1 , lim ξ→−∞ µ(ξ) = +∞ . graph of µ(ξ) and comparison with dirichlet realization figure 1: de gennes model, computed by v. bonnaillie-noël two constants have played a role in the main text. the first one is θ0: 0 < θ0 = µ(ξ0) = inf ξ∈r µ(ξ) < 1 . (a.2) we have ξ20 = θ0 ∼ 0.59 . (a.3) the second one is: δ0 = 1 2 µ′′(ξ0) . (a.4) cubo 11, 5 (2009) on some spectral problems and asymptotic ... 19 b montgomery’s model we have also met the following family (depending on ρ) of quadratic oscillators: d2t + (t 2 − ρ)2 . (b.1) denoting by ν(ρ) the lowest eigenvalue of this operator, pan and kwek [pk] (see also [hel]) have shown that there exists a unique minimum of ν(ρ) leading to a new universal constant ν̂0 = inf ρ∈r ν(ρ) . (b.2) one has (feynman-hellmann formula) ρmin = 2 ∫ t2u2ρmin dt , where uρ denotes the normalized groundstate. numerical computations confirm that the minimum is attained for a positive value of ρ: ρmin ∼ 0.35 , and that this minimum is θmin ∼ 0.5698 . numerical computations also suggest that the minimum is non degenerate. this result is proved in [hel]. acknowledgements the first author would like to thank f. cardoso and a. sá barreto for their kind invitation at this conference and the opportunity to give a mini-course the week before, and he also thank his collaborators (particularly v. bonnaillie-noël, s. fournais), his student n. raymond (whose master thesis inspires partially this survey) and also p. bauman and d. phillips for useful discussions. the second author was partly supported by the national natural science foundation of china grant no. 10471125, 10871071, the science foundation of the ministry of education of china grant no. 20060269012, and the national basic research program of china grant no. 2006cb805902. received: november, 2008. revised: february, 2009. references [bclp] bauman, p., carme calderer, m., liu, c. and phillips, d., the phase transition between chiral nematic and smectic a∗ liquid crystals, arch. rational mech. anal., 165 (2002), 161–186. 20 bernard helffer and xing-bin pan cubo 11, 5 (2009) [bs] bernoff, a. and sternberg, p., onset of superconductivity in decreasing fields for general domains, j. math. phys., 39 (1998), 1272–1284. [c] calderer, m.c., studies of layering and chirality of smectic a∗ liquid crystals, mathematical and computer modelling, 34 (2001), 1273–1288. [dg1] de gennes, p.g., an analogy between superconductors and smectics a, solid state communications, 10 (1972), 753–756. [dg2] de gennes, p.g., some remarks on the polymorphism of smectics, molecular crystals and liquid crystals, 21 (1973), 49–76. [dgp] de gennes, p.g. and prost, j., the physics of liquid crystals, second edition, oxford science publications, oxford, 1993. [dah] dauge, m. and helffer, b., eigenvalues variation i, neumann problem for sturmliouville operators, j. differential equations, 104 (1993), 243–262. [dfs] del pino, m., felmer, p.l. and sternberg, p., boundary concentration for eigenvalue problems related to the onset of superconductivity, comm. math. phys., 210 (2000), 413–446. [fh1] fournais, s. and helffer, b., on the third critical field in ginzburg-landau theory, comm. math. phys., 266 (2006), 153–196. [fh2] fournais, s. and helffer, b., strong diamagnetism for general domains and applications, ann. inst. fourier, 57 (2007), 2389–2400. [fh3] fournais, s. and helffer, b., on the ginzburg-landau critical field in three dimensions, comm. pure appl. math., 62 (2009), 215–241. [fh4] fournais, s. and helffer, b., spectral methods in surface superconductivity, book in preparation. [giop] giorgi, t. and phillips, d., the breakdown of superconductivity due to strong fields for the ginzburg-landau model, siam j. math. anal., 30 (1999), 341–359. [hel] helffer, b., the montgomery model revisited, to appear in colloquium mathematicum. [hm1] helffer, b. and morame, a., magnetic bottles in connection with superconductivity, j. functional anal., 185 (2001), 604–680. [hm2] helffer, b. and morame, a., magnetic bottles for the neumann problem: the case of dimension 3, proceedings of the indian academy of sciences-mathematical sciences, 112 (2002), 710–84. [hm3] helffer, b. and morame, a., magnetic bottles for the neumann problem: curvature effects in the case of dimension 3 (general case) (expanded version), mp_arc 02–145 (2002). cubo 11, 5 (2009) on some spectral problems and asymptotic ... 21 [hm4] helffer, b. and morame, a., magnetic bottles for the neumann problem: curvature effects in the case of dimension 3 (general case), ann. sci. ecole norm. sup., 37 (2004), 105–170. [hp1] helffer, b. and pan, x.b., upper critical field and location of surface nucleation of superconductivity, ann. inst. henri poincaré, analyse non linéaire, 20 (2003), 145–181. [hp2] helffer, b. and pan, x.b., reduced landau-de gennes functional and surface smectic state of liquid crystals, j. functional anal., 255 (11) (2008), 3008–3069. [hp3] helffer, b. and pan, x.b., work in progress. [lip] lin, f.h. and pan, x.b., magnetic field-induced instabilities in liquid crystals, siam j. math. anal., 38 (2007), 1588–1612. [lup1] lu, k. and pan, x.b., gauge invariant eigenvalue problems in r2 and in r2+, trans. amer. math. soc., 352 (2000), 1247–1276. [lup2] lu, k. and pan, x.b., estimates of the upper critical field for the ginzburg-landau equations of superconductivity, physica d, 127 (1999), 73–104. [lup3] lu, k. and pan, x.b., surface nucleation of superconductivity in 3-dimension, j. diff. equations, 168 (2000), 386–452. [m] montgomery, r., hearing the zero locus of a magnetic field, comm. math. phys., 168 (1995), 651–675. [p1] pan, x.b., landau-de gennes model of liquid crystals and critical wave number, comm. math. phys., 239 (2003), 343–382. [p2] pan, x.b., superconductivity near critical temperature, j. math. phys., 44 (2003), 2639–2678. [p3] pan, x.b., surface superconductivity in 3-dimensions, trans. amer. math. soc., 356 (2004), 3899–3937. [p4] pan, x.b., landau-de gennes model of liquid crystals with small ginzburg-landau parameter, siam j. math. anal., 37 (2006), 1616–1648. [p5] pan, x.b., an eigenvalue variation problem of magnetic schrödinger operator in threedimensions, disc. contin. dyn. systems, ser. a, special issue for peter bates 60th birthday, 24 (2009), 933–978. [p6] pan, x.b., analogies between superconductors and liquid crystals: nucleation and critical fields, in: asymptotic analysis and singularities, advanced studies in pure mathematics, mathematical society of japan, tokyo, 47-2 (2007), 479–517. [p7] pan, x.b., critical fields of liquid crystals, in: moving interface problems and applications in fluid dynamics, khoo, b.c., li, z.l. and lin, p. eds., contemporary mathematics, 466 (2008), amer. math. soc., 121134. 22 bernard helffer and xing-bin pan cubo 11, 5 (2009) [pk] pan, x.b. and kwek, k.h., schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains. trans. amer. math. soc., 354 (2002), 4201–4227. [ray1] raymond, n., minimiseurs de la fonctionnelle de landau-de gennes–etude de la transition entre la phase nématique chirale et la phase smectique a∗ des cristaux liquides, master thesis, university paris sud, 2007. [ray2] raymond, n., uniform spectral estimates for families of schrödinger operators with magnetic field of constant intensity and applications, to appear in cubo mathematical journal (2009). [ray3] raymond, n., contribution to the asymptotic analysis of the landau-de gennes functional, submitted. b1-serrambicubo-09-02-01 cubo a mathematical journal vol.11, no¯ 03, (1–24). august 2009 uniformly continuous l1 solutions of volterra equations and global asymptotic stability leigh c. becker department of mathematics, christian brothers university, 650 e. parkway south, memphis, tn 38104-5581, email: lbecker@cbu.edu abstract the scalar linear volterra integro-differential equation x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds (e) is investigated, where a and b are continuous functions. liapunov functionals are constructed in order to obtain sufficient conditions so that solutions of (e) are absolutely riemann integrable on [0,∞) and have bounded derivatives. then some of these conditions are replaced with less stringent ones while others are eliminated altogether. under the new conditions, it is shown that one of the liapunov functionals is uniformly continuous which in turn implies that solutions of (e) are uniformly continuous. we then employ barbălat’s lemma to prove the zero solution of (e) is stable and that all solutions of (e) approach zero as t → ∞. examples illustrated with numerical solutions are provided. 2 leigh c. becker cubo 11, 3 (2009) resumen investigamos la siguiente ecuación linear integro-diferencial de volterra x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds (e) donde a y b son funciones continuas. funcionales de liapunov son construidos para obtener condiciones suficientes tal que las soluciones de (e) son absolutamente riemann integrables sobre el intervalo [0,∞) y tienen derivadas acotadas. algumas de estas condiciones son reemplazadas por otras menos rigurosas mientras que otras son eliminadas por completo. bajo las nuevas condiciones, se demuestra que uno de los funcionales de liapunov es uniformemente continuo, a su vez, esto implica que las soluciones de (e) son uniformemente continuas. usamos el lema de barbălat para provar que la solución nula de (e) es estable y que todas las soluciones de (e) se aproximam a cero cuando t → ∞. son presentados ejemplos con soluciones numéricas. key words and phrases: asymptotic stability, barbălat’s lemma, liapunov functionals, strongly positive definite functionals, uniformly continuous solutions, variation of parameters, volterra equations. math. subj. class.: 45j05, 45m10, 34k20, 45d05. 1 introduction we investigate the asymptotic behavior of solutions of the scalar linear homogeneous volterra integro-differential equation x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds (1.1) for t ≥ 0, where a and b are real-valued functions that are continuous on the respective domains [0,∞) and ω := {(t,s) : 0 ≤ s ≤ t < ∞}. in this setting, a solution of (1.1) satisfying a given initial condition exists on the entire interval [0,∞) and is unique (cf. section 2 for more details). employing liapunov functionals that were constructed for (1.1) by burton in [8, p. 122] and [9] and by becker in [5, p. 34], with some modifications, we obtain a number of conditions involving a and b so that the zero solution of (1.1) is stable and its other solutions approach zero as t → ∞. typically, one looks for conditions so that the derivative x′(t) is bounded on [0,∞). however, in this paper we suggest that in investigations of stability more emphasis ought to be placed on the uniform continuity of the solutions. the reason for this derives from the observation: every differentiable function with a bounded derivative is uniformly continuous—but not conversely as attested by the function f(t) = √ t 1 + t . (1.2) cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 3 even though its derivative f′ is unbounded on [0,∞), f is uniformly continuous on [0,∞). moreover, f(t) tends to zero as t → ∞. the thesis then in this paper is that conditions yielding uniformly continuous solutions will be less stringent than those yielding solutions with bounded derivatives. we use the following notation throughout this paper: • c[t0, t1] (resp. c[t0,∞)) will denote the set of all continuous real-valued functions on [t0, t1] (resp. [t0,∞)). • for ϕ ∈ c[0, t0], |ϕ|t0 := sup{|ϕ(t)|: 0 ≤ t ≤ t0}. • l1[0,∞) typically denotes the set of all real-valued functions f that are lebesgue measurable on [0,∞) and for which the lebesgue integral ∫ ∞ 0 |f| is finite. however, we use it to denote those functions in l1[0,∞) that are also continuous on [0,∞). for such a function, say g, the improper riemann integral ∫ ∞ 0 |g(t)|dt converges, i.e., limt→∞ ∫ t 0 |g(s)|ds exists and is finite. in short, by g ∈ l1[0,∞) we mean that g is continuous and absolutely riemann integrable on [0,∞). • l2[0,∞) will denote the set of all continuous real-valued functions that are square integrable on [0,∞). that is, h ∈ l2[0,∞) will mean that h is continuous on [0,∞) and h2 ∈ l1[0,∞). in section 2, we construct a liapunov functional and use it to find conditions that yield l 2 solutions of (1.1) with bounded derivatives. the idea of replacing those conditions with ones that yield uniformly continuous lp solutions is broached. in section 3, we modify the liapunov functional so as to lessen the stringency of the conditions in section 2 and to obtain l1 solutions of (1.1). furthermore, a couple of conditions are added so that the derivatives of solutions are bounded on [0,∞), which enables us to obtain a global asymptotic stability result. a well-known classical result of uniform asymptotic stability is obtained when a(t) is constant and positive and the kernel b is of convolution type, viz., that b depends only on the difference t − s. in section 4, we show that the conditions that were added to obtain the global asymptotic stability result in section 3 are actually unnecessary by arguing that the liapunov functional is uniformly continuous which in turn implies that solutions of (1.1) are uniformly continuous. the upshot is that “bounded derivative" conditions can be replaced with less restrictive “uniform continuity" conditions. one consequence of this is that the usual condition that a(t) be bounded below by a positive constant can be replaced with a(t) being merely nonnegative. 2 uniformly continuous l2 solutions in this section, we obtain conditions involving the continuous functions a and b so that all of the solutions of x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds (2.1) 4 leigh c. becker cubo 11, 3 (2009) belong to l2[0,∞). then we will add more conditions that will drive these l2 solutions to zero as t → ∞. but first let us define precisely what we mean by solutions of (2.1) and of the related nonhomogeneous equation x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds + f(t), (2.2) where f : [0,∞) → r is continuous. definition 2.1. a solution of (2.1) (resp. (2.2)) on [0,t ), where 0 < t ≤ ∞, with an initial value x0 ∈ r is a continuous function x: [0,t ) → r that satisfies (2.1) (resp. (2.2)) on (0,t ) such that x(0) = x0. it can be shown that a solution x(t) satisfying an initial condition x(0) = x0 exists on the entire interval [0,∞) and is unique (cf. [4, p. 5] or [7, pp. 23–27, p. 221]). for the sake of clarity, we will sometimes use the more explicit notation x(t, 0,x0) to denote such a solution. furthermore, for each t0 > 0 and each continuous initial function ϕ: [0, t0] → r, there is a unique continuous function x: [0,∞) → r that satisfies (2.2) on (t0,∞) such that x(t) ≡ ϕ(t) on [0, t0] (cf. [8, p. 179]). when the need for more clarity is warranted, we denote this solution by x(t,t0,ϕ). finally, we note that x(t) = 0 is a solution of (2.1) for 0 ≤ t < ∞, which is called its zero solution. the precise terminology that we use to describe the asymptotic behavior of solutions is given in definition 2.2 below and in definition 3.1 in section 3. definition 2.2. the zero solution of (2.1) is 1. stable if for every ǫ > 0 and every t0 ≥ 0, there exists a δ = δ(ǫ,t0) > 0 such that ϕ ∈ c[0, t0] with |ϕ|t0 < δ implies that |x(t,t0,ϕ)| < ǫ for all t ≥ t0. 2. globally asymptotically stable (asymptotically stable in the large) if it is stable and if every solution of (2.1) approaches zero as t → ∞. lemma 2.3. let a: [0,∞) → [0,∞) and b: ω → r be continuous functions. if a(t) − ∫ t 0 |b(t,s)|ds ≥ 0 (2.3) for all t ≥ 0 and if a(s) − ∫ t s |b(u,s)|du ≥ 0 (2.4) for all t ≥ s ≥ 0, then the zero solution of (2.1) is stable. suppose, in addition, that for some t1 ≥ 0 there is a constant k > 0 such that either a(t) − ∫ t 0 |b(t,s)|ds ≥ k (2.5) cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 5 for all t ≥ t1 or a(s) − ∫ t s |b(u,s)|du ≥ k (2.6) for all t ≥ s ≥ t1. then every solution x(t) of (2.1) belongs to l2[0,∞). proof. first we define the liapunov functional v : [0,∞) × c[0,∞) → [0,∞) by v (t,ψ(·)) := ψ2(t) + ∫ t 0 [ a(s) − ∫ t s |b(u,s)|du ] ψ 2 (s) ds. (2.7) it follows from (2.4) that v (t,ψ(·)) ≥ ψ2(t) for all t ≥ 0. for any t0 ≥ 0 and initial function ϕ ∈ c[0, t0], let x(t) = x(t,t0,ϕ) denote the unique solution of (2.1) on [0,∞) such that x(t) = ϕ(t) for 0 ≤ t ≤ t0. for brevity, let v (t) := v (t,x(·)), that is, the value of the functional v along the solution x(t) at t. taking the derivative of v with respect to t, we have v ′ (t) = 2x(t)x ′ (t) + a(t)x 2 (t) − ∫ t 0 |b(t,s)|x2(s) ds = 2x(t) [ −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds ] + a(t)x 2 (t) − ∫ t 0 |b(t,s)|x2(s) ds ≤ −a(t)x2(t) + ∫ t 0 |b(t,s)| · 2|x(t)||x(s)|ds − ∫ t 0 |b(t,s)|x2(s) ds ≤ −a(t)x2(t) + ∫ t 0 |b(t,s)| ( x 2 (t) + x 2 (s) ) ds − ∫ t 0 |b(t,s)|x2(s) ds. consequently, v ′ (t) ≤ − ( a(t) − ∫ t 0 |b(t,s)|ds ) x 2 (t) (2.8) for all t ≥ t0. thus, by (2.3), v ′(t) ≤ 0. this, together with v (t) ≥ x2(t), implies that x 2 (t) ≤ v (t) ≤ v (t0) (2.9) for all t ≥ t0. moreover, from v (t0) = ϕ 2 (t0) + ∫ t0 0 [ a(s) − ∫ t0 s |b(u,s)|du ] ϕ 2 (s) ds ≤ |ϕ|2t0m(t0), where m(t0) := 1 + ∫ t0 0 [ a(s) − ∫ t0 s |b(u,s)|du ] ds, this becomes |x(t)| ≤ |ϕ|t0 √ m(t0) (2.10) 6 leigh c. becker cubo 11, 3 (2009) for all t ≥ t0. this implies the zero solution is stable: for ǫ > 0, let δ = ǫ/ √ m(t0). then for ϕ ∈ c[0, t0] with |ϕ|t0 < δ, |x(t)| < δ √ m(t0) = ǫ (2.11) for all t ≥ t0. if (2.5) also holds, then (2.8) implies that v ′ (t) ≤ −kx2(t) for all t ≥ τ, where τ := max{t0, t1}. integrating, we obtain v (t) − v (τ) ≤ −k ∫ t τ x 2 (s) ds. consequently, x 2 (t) ≤ v (t) ≤ v (τ) − k ∫ t τ x 2 (s) ds (2.12) for all t ≥ τ. if, on the other hand (2.6) holds, then (2.7) and (2.9) imply x 2 (t) + k ∫ t t1 x 2 (s) ds ≤ v (t) ≤ v (t0) (2.13) for all t ≥ t1. either one, (2.12) or (2.13), implies that x2 ∈ l1[0,∞). we have just proved that under the conditions of lemma 2.3, the solution x(t,t0,ϕ) of (2.1) belongs to l2[0,∞). it then seems plausible that x2(t) → 0 as t → ∞. however, by itself convergence of an improper riemann integral of a function f on [0,∞) does not ensure that f approaches 0 as t → ∞ (cf. [13, p. 466]). but if f were also known to be uniformly continuous, then it would according to the next lemma attributed to barbălat [2]. barbălat’s lemma. if f : [0,∞) → r is both uniformly continuous and riemann integrable on [0,∞), then f(t) → 0 as t → ∞. a proof of this is given in [16, p. 866]. a proof for a nonnegative f can also be found in [12, p. 89]. note however that even if an l2 solution of (2.1) were also uniformly continuous, we could not use barbălat’s lemma to conclude anything since the uniform continuity of a function f does not imply the uniform continuity of f2, as exemplified by f(t) = t on [0,∞). nor does f ∈ l2[0,∞) imply that f ∈ l1[0,∞), as is the case with f(t) = (t + 1)−1 on [0,∞). nonetheless, an l2 function f that is uniformly continuous does approach zero. the following proof of this is adapted from the proof of barbălat’s lemma given in the aforementioned reference [16]. lemma 2.4. if f : [0,∞) → r is uniformly continuous on [0,∞) and if f2 is riemann integrable on [0,∞), then f(t) → 0 as t → ∞. cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 7 proof. suppose to the contrary that f(t) does not approach 0 as t → ∞. then an ǫ > 0 and a sequence {tn}∞n=1 exists on [0,∞) with tn → ∞ such that |f(tn)| ≥ ǫ for all n ≥ 1. since f is uniformly continuous, a δ > 0 exists for this ǫ with the property: for t ∈ [0,∞), |t−tn| ≤ δ implies that |f(t) − f(tn)| < ǫ 2 for all n ≥ 1. hence, |f(t)| ≥ |f(tn)| − |f(tn) − f(t)| > ǫ 2 for all t ∈ [tn, tn + δ] and n ≥ 1. this implies that ∫ tn+δ 0 f 2 (t) dt − ∫ tn 0 f 2 (t) dt = ∫ tn+δ tn f 2 (t) dt ≥ δǫ 2 4 > 0 for all n ≥ 1. however, this is a contradiction because the left-hand side converges to 0 as n → ∞ by the hypothesis that f2 is integrable on [0,∞). with lemma 2.4 at our disposal, we can find another condition to add to those of lemma 2.3 that will ensure that all solutions of (2.1) approach zero. theorem 2.5. let a: [0,∞) → [0,∞) and b: ω → r be continuous functions satisfying conditions (2.3) and (2.4) of lemma 2.3. if for some t1 ≥ 0 there are positive constants k and k such that either k + ∫ t 0 |b(t,s)|ds ≤ a(t) ≤ k (2.14) for all t ≥ t1 or k + ∫ t s |b(u,s)|du ≤ a(s) ≤ k (2.15) for all t ≥ s ≥ t1, then all solutions of (2.1) are uniformly continuous on [0,∞) and belong to l 2 [0,∞). furthermore, the zero solution is globally asymptotically stable. proof. we only need to show that all solutions of (2.1) tend to zero since stability has already been established in lemma 2.3. to this end, for any t0 ≥ 0 and ϕ ∈ c[0, t0], consider the corresponding solution x(t) = x(t,t0,ϕ). by (2.10), |x(t)| ≤ |ϕ|t0 √ m(t0) for all t ≥ t0. consequently, as a(t) ≤ k for t ≥ t1, |x′(t)| ≤ a(t)|x(t)| + ∫ t0 0 |b(t,s)||ϕ(s)|ds + ∫ t t0 |b(t,s)||x(s)|ds ≤ 2k|ϕ|t0 √ m(t0) + k|ϕ|t0 for t ≥ τ, where τ = max{t0, t1}. since x′(t) is bounded on [τ,∞), x(t) satisfies a lipschitz condition on [τ,∞). consequently, it is uniformly continuous on [τ,∞). this and the continuity of x(t) on [0,∞) imply x(t) is uniformly continuous on the entire interval [0,∞). by lemma 2.3, x 2 (t) ∈ l1[0,∞). therefore, by lemma 2.4, x(t) → 0 as t → ∞. 8 leigh c. becker cubo 11, 3 (2009) example 2.6. let k be a positive real number. all of the solutions of x ′ (t) = − ( k + 1 1 + t ) x(t) + ∫ t 0 cos s (1 + t)3 x(s) ds (2.16) are uniformly continuous on [0,∞) and belong to the set l2[0,∞). moreover, the zero solution is globally asymptotically stable. proof. since a(t) = k + 1/(1 + t) and b(t,s) = (cos s)/(1 + t)3, we have k + ∫ t 0 |b(t,s)|ds = k + ∫ t 0 | cos s| (1 + t)3 ds (2.17) ≤ k + t (1 + t)3 < k + 1 1 + t = a(t) for all t ≥ 0. thus, (2.14) is satisfied with k := k + 1. also, ∫ t s |b(u,s)|du ≤ ∫ t s 1 (1 + u)3 du < 1 2 (1 + s) −2 < 1 1 + s < k + 1 1 + s = a(s) for all (t,s) ∈ ω. therefore, all of the conditions of theorem 2.5 are satisfied. since the zero solution of (2.16) is globally asymptotically stable for each k > 0, no matter how small, it is plausible that the zero solution of x ′ (t) = − 1 1 + t x(t) + ∫ t 0 cos s (1 + t)3 x(s) ds (2.18) is also globally asymptotically stable. but then again it may not be in light of the scalar equation x ′ = −kx, which has a zero solution that is globally asymptotically stable when k > 0 but not for k = 0. however, the case for (2.18) is bolstered by the fact that the zero solution of x ′ = − x 1 + t (2.19) is globally asymptotically stable. additionally, the maple worksheet [3, cf. (6.1)] at the maple application center (www.maplesoft.com) shows the graphs of two numerical solutions of (2.18) approaching zero. in point of fact, the zero solution of (2.18) is globally asymptotically stable. we prove this next with the aid of the functional (2.20) below, which is the result of conflating, so to speak, the liapunov function v (t,x) = (1 + t)x2 used by yoshizawa for (2.19) in [18, p. 59] and the liapunov functional (2.7). example 2.7. the zero solution of (2.18) is globally asymptotically stable. proof. define the liapunov functional v (t,ψ(·)) := (1 + t)ψ2(t) + ∫ t 0 [ 1 1 + s − ∫ t s | cos s| (1 + u)2 du ] ψ 2 (s) ds. (2.20) cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 9 notice that v (t,ψ(·)) ≥ ψ2(t) for all t ≥ 0. for a given ϕ ∈ c[0, t0], let x(t) = x(t,t0,ϕ) denote the solution of (2.18) with x(t) = ϕ(t) on [0, t0]. for this particular solution, let v (t) := v (t,x(·)). taking the derivative of v with respect to t, we find for t ≥ t0 that v ′ (t) ≤ 2(1 + t)x(t) [ − 1 1 + t x(t) + ∫ t 0 cos s (1 + t)3 x(s) ds ] + x 2 (t) + 1 1 + t x 2 (t) − ∫ t 0 | cos s| (1 + t)2 x 2 (s) ds ≤ −x2(t) + 1 1 + t x 2 (t) + ∫ t 0 | cos s| (1 + t)2 2|x(t)||x(s)|ds − ∫ t 0 | cos s| (1 + t)2 x 2 (s) ds ≤ −x2(t) + 1 1 + t x 2 (t) + x 2 (t) ∫ t 0 | cos s| (1 + t)2 ds ≤ − ( 1 − 1 1 + t − t (1 + t)2 ) x 2 (t), which simplifies to v ′ (t) ≤ − ( t 1 + t )2 x 2 (t). (2.21) consequently, x 2 (t) ≤ v (t) ≤ v (t0) (2.22) for all t ≥ t0. this implies that the zero solution of (2.18) is stable by an argument much like the one from (2.9) to (2.11). it follows from (2.21) that v ′ (t) ≤ − 1 4 x 2 (t) (2.23) for all t ≥ τ, where τ = max{1, t0}. and so x 2 (t) ≤ v (t) ≤ v (τ) − 1 4 ∫ t τ x 2 (s) ds. therefore, x2(t) ∈ l1[0,∞). furthermore, it follows from (2.18) and (2.22) that x′(t) is bounded on [t0,∞). hence, by the uniform continuity argument in the proof of theorem 2.5, x(t) is uniformly continuous on [0,∞). therefore, x(t) → 0 as t → ∞. remark. t. a. burton, in a private note, points out that the liapunov functional (2.20) is strongly positive definite in the sense defined by lakshmikantham and leela in [15, p. 137]. it appears to be one of the few, if any, such nontrivial strongly positive definite functionals that have appeared in the literature to obtain an asymptotic stability result. 10 leigh c. becker cubo 11, 3 (2009) 3 uniformly continuous l1 solutions fulfillment of the conditions in lemma 2.3 in the previous section ensures that the solutions of x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds. (3.1) are in l2[0,∞). now we modify the liapunov functional (2.7) used to prove this lemma in order to obtain conditions that will ensure that all solutions are in l1[0,∞) and to replace the lower bounds for a(t) in (2.14) and (2.15) with less stringent ones. at the same time, we will also find conditions that imply either global asymptotic stability or the following types of stability: definition 3.1. the zero solution of (3.1) is 1. uniformly stable if for every ǫ > 0, there exists a δ = δ(ǫ) > 0 such that ϕ ∈ c[0, t0] with |ϕ|t0 < δ (any t0 ≥ 0) implies that |x(t,t0,ϕ)| < ǫ for all t ≥ t0. 2. uniformly asymptotically stable if it is uniformly stable and if there exists an η > 0 with the following property: for every ǫ > 0, there exists a t = t (ǫ) > 0 such that ϕ ∈ c[0, t0] with |ϕ|t0 < η (any t0 ≥ 0) implies that |x(t,t0,ϕ)| < ǫ for all t ≥ t0 + t . 3. uniformly asymptotically stable in the large if it is uniformly stable and if for every η > 0 and every ǫ > 0, there exists a t = t (η,ǫ) > 0 such that ϕ ∈ c[0, t0] with |ϕ|t0 < η (any t0 ≥ 0) implies that |x(t,t0,ϕ)| < ǫ for all t ≥ t0 + t . (in other words, x(t) ≡ 0 is uniformly asymptotically stable in the large if (2) is true for every η > 0.) the liapunov functional v (t,ψ(·)) that we employ in the next lemma to find conditions for the stability of the zero solution of (3.1) eliminates the need for condition (2.3). v (t,ψ(·)) was first derived by t. a. burton, with which he obtained asymptotic stability conditions like some of those in theorem 2.5 (cf. [8, pp. 122–123]). especially noteworthy in this regard is burton’s “rough algorithm” (to use his words) for deriving liapunov functionals that he describes on p. 121 in [8]. in (3.11) below, we tweak v (t,ψ(·)) a bit so that we can replace (2.6) with the less restrictive condition that a(t) ≥ k. lemma 3.2. let a: [0,∞) → [0,∞) and b: ω → r be continuous functions. if ∫ t s |b(u,s)|du ≤ a(s) (3.2) for all t ≥ s ≥ 0, then the zero solution of (3.1) is stable. furthermore, if for some t1 ≥ 0 there is a constant k > 0 such that a(t) ≥ k (3.3) for all t ≥ t1 and a constant λ ∈ (0, 1) such that ∫ t s |b(u,s)|du ≤ λa(s) (3.4) cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 11 for all t ≥ s ≥ t1, then every solution x(t) of (3.1) belongs to l1[0,∞). proof. first define the liapunov functional v : [0,∞) × c[0,∞) → [0,∞) by v (t,ψ(·)) := |ψ(t)| + ∫ t 0 [ a(s) − ∫ t s |b(u,s)|du ] |ψ(s)|ds. (3.5) by (3.2), v (t,ψ(·)) ≥ |ψ(t)| for all t ≥ 0. for any t0 ≥ 0 and ϕ ∈ c[0, t0], let x(t) = x(t,t0,ϕ) denote the solution of (3.1) on [0,∞) with x(t) = ϕ(t) for 0 ≤ t ≤ t0. then consider v (t) := v (t,x(·)) and its derivative. since x(t) is continuously differentiable on [t0,∞), |x(t)| has a right derivative dr|x(t)| given by dr|x(t)| = { x ′ (t) sgn x(t), if x(t) 6= 0 |x′(t)|, if x(t) = 0 (3.6) for all t ≥ t0 (cf. [14, p. 26]). thus, the right derivative of v for t ≥ t0 is drv (t) = dr|x(t)| + d dt ∫ t 0 [ a(s) − ∫ t s |b(u,s)|du ] |x(s)|ds ≤ −a(t)|x(t)| + ∫ t 0 |b(t,s)||x(s)|ds + a(t)|x(t)| − ∫ t 0 |b(t,s)||x(s)|ds and so drv (t) ≤ 0. (3.7) thus, |x(t)| ≤ v (t) ≤ v (t0) (3.8) for all t ≥ t0, where v (t0) = |ϕ(t0)| + ∫ t0 0 [ a(s) − ∫ t0 s |b(u,s)|du ] |ϕ(s)|ds ≤ m(t0)|ϕ|t0 and m(t0) := 1 + ∫ t0 0 [ a(s) − ∫ t0 s |b(u,s)|du ] ds. (3.9) for a given ǫ > 0, let δ = ǫ/m(t0). then for ϕ ∈ c[0, t0] with |ϕ|t0 < δ, we have |x(t)| ≤ v (t0) ≤ m(t0)|ϕ|t0 < δm(t0) = ǫ (3.10) for all t ≥ t0, which proves stability. now suppose (3.3) and (3.4) also hold. in that case, let γ := √ λ and vγ (t) := |x(t)| + ∫ t 0 [ γa(s) − 1 γ ∫ t s |b(u,s)|du ] |x(s)|ds. (3.11) by (3.4), vγ (t) ≥ |x(t)| (3.12) 12 leigh c. becker cubo 11, 3 (2009) for all t ≥ t1. and drvγ (t) ≤ −a(t)|x(t)| + ∫ t 0 |b(t,s)||x(s)|ds + γa(t)|x(t)| − 1 γ ∫ t 0 |b(t,s)||x(s)|ds (3.13) ≤ −(1 − γ)a(t)|x(t)| for all t ≥ τ, where τ := max{t0, t1}. then, because of (3.3), drv (t) ≤ −k(1 − γ)|x(t)|. (3.14) an integration (cf. [14, cor. 4.1, p. 27]) along with (3.12) yields |x(t)| ≤ vγ (t) ≤ vγ (τ) − k(1 − γ) ∫ t τ |x(s)|ds, (3.15) for all t ≥ τ. therefore, ∫ ∞ 0 |x(t)|dt converges. if in addition to the conditions of lemma 3.2, a(t) is bounded from above and condition (3.16) below is met, then all of the solutions of (3.1) tend to zero as t → ∞, as we will now prove. theorem 3.3. let a: [0,∞) → [0,∞) and b: ω → r be continuous functions such that ∫ t 0 |b(t,s)|ds ≤ a(t) (3.16) for all t ≥ 0 and ∫ t s |b(u,s)|du ≤ a(s), (3.17) for all t ≥ s ≥ 0. if for some t1 ≥ 0 there are positive constants k and k such that k ≤ a(t) ≤ k (3.18) for all t ≥ t1 and a constant λ ∈ (0, 1) such that ∫ t s |b(u,s)|du ≤ λa(s) (3.19) for all t ≥ s ≥ t1, then all solutions of (3.1) are uniformly continuous on [0,∞) and belong to l 1 [0,∞). moreover, the zero solution is globally asymptotically stable. proof. by lemma 3.2, the zero solution of (3.1) is stable. for any ϕ ∈ c[0, t0], consider the corresponding solution x(t) = x(t,t0,ϕ). by (3.8), |x(t)| ≤ v (t0) cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 13 for all t ≥ t0. this, together with (3.16) and (3.18), applied to (3.1) gives |x′(t)| ≤ a(t)|x(t)| + ∫ t0 0 |b(t,s)||ϕ(s)|ds + ∫ t t0 |b(t,s)||x(s)|ds ≤ kv (t0) + a(t0)|ϕ|t0 + v (t0)a(t) ≤ 2kv (t0) + a(t0)|ϕ|t0 for all t ≥ τ, where as before τ = max{t0, t1}. in short, x′(t) is bounded on [τ,∞). consequently, by the uniform continuity argument in the proof of theorem 2.5, x(t) is uniformly continuous on [0,∞). also, by lemma 3.2, x ∈ l1[0,∞). therefore, by barbălat’s lemma, x(t) → 0 as t → ∞. example 3.4. for any positive constants k and β, the zero solution of x ′ (t) = − ( k + 1 + β 1 + t ) x(t) + ∫ t 0 cos s (1 + t)2 x(s) ds, (3.20) is globally asymptotically stable. proof. condition (3.18) is satisfied since a(t) = k + 1+β 1+t is bounded by positive constants. condition (3.16) is also satisfied since ∫ t 0 |b(t,s)|ds = ∫ t 0 | cos s| (1 + t)2 ds ≤ t (1 + t)2 < 1 1 + t < a(t), for all t ≥ 0. furthermore, ∫ t s |b(u,s)|du ≤ ∫ t s 1 (1 + u)2 du < 1 1 + s < 1 1 + β ( k + 1 + β 1 + s ) = 1 1 + β a(s) for all t ≥ s ≥ 0. therefore, all the conditions of theorem 3.3 are satisfied. as the next example shows, the conditions of theorem 3.3 are easily met when a(t) is a positive constant and b is of convolution type (i.e., b depends only on the difference t−s). uniform asymptotic stability of the zero solution for this case was originally established by burton and mahfoud [10, p. 146]. a proof for a positive function b can also be found in burton’s monograph [7, pp. 55-57]. example 3.5. let a be a positive constant. if b: [0,∞) → r is continuous and ∫ ∞ 0 |b(t)|dt < a, (3.21) then the zero solution of x ′ (t) = −ax(t) + ∫ t 0 b(t − s)x(s) ds (3.22) is globally asymptotically stable. in point of fact, it is uniformly asymptotically stable in the large. 14 leigh c. becker cubo 11, 3 (2009) proof. define λ by λ := 1 a ∫ ∞ 0 |b(t)|dt. it follows from (3.21) that λ < 1 and ∫ t s |b(u − s)|du = ∫ t−s 0 |b(v)|dv ≤ λa for all t ≥ s ≥ 0. clearly then, conditions (3.16)–(3.19) are satisfied. therefore, by theorem 3.3, the zero solution of (3.22) is globally asymptotically stable. to show that the zero solution is also uniformly asymptotically stable in the large, let z(t) denote the principal solution, i.e., the solution of (3.22) with z(0) = 1. by lemma 3.2, z(t) ∈ l1[0,∞). (3.23) furthermore, (3.5) and (3.8) imply that |z(t)| ≤ 1 (3.24) for 0 ≤ t ≤ ∞. at this point we could invoke miller’s [17, pp. 493–498] classic result: for α ∈ r and b ∈ l1[0,∞), the zero solution of x ′ (t) = αx(t) + ∫ t 0 b(t − s)x(s) ds is uniformly asymptotically stable if and only if z(t) ∈ l1[0,∞). it would then follow from (3.21) and (3.23) that the zero solution of (3.22) is uniformly asymptotically stable. however, in order to show that it is in fact uniformly asymptotically stable in the large, we present a self-contained proof of that next. for t0 ≥ 0 and ϕ ∈ c[0, t0], let x(t) = x(t,t0,ϕ) denote the unique solution of (3.22) with x(t) = ϕ(t) for 0 ≤ t ≤ t0. hence, x ′ (t) = −ax(t) + ∫ t0 0 b(t − s)ϕ(s) ds + ∫ t t0 b(t − s)x(s) ds for t > t0. then, for x(t + t0) := x(t + t0, t0,ϕ), it follows from the chain rule that d dt x(t + t0) = −ax(t + t0) + ∫ t 0 b(t − s)x(s + t0) ds + f(t) (3.25) for t > 0, where f(t) := ∫ t0 0 b(t + u)ϕ(t0 − u) du. (3.26) in other words, x(t + t0) is the unique solution of y ′ (t) = −ay(t) + ∫ t 0 b(t − s)y(s) ds + f(t) (3.27) cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 15 with y(0) = x(t0) = ϕ(t0). by the variation of parameters formula (cf. [4, p. 14] or [7, p. 31, p. 223]), we have y(t) = z(t)ϕ(t0) + ∫ t 0 z(t − s)f(s) ds, or as y(t) = x(t + t0), x(t + t0) = z(t)ϕ(t0) + ∫ t 0 z(t − s) { ∫ t0 0 b(s + u)ϕ(t0 − u) du } ds. (3.28) using (3.28), we now show that the zero solution of (3.22) is uniformly stable. using (3.21) and (3.24), we obtain |x(t+t0)| ≤ |ϕ|t0 [ |z(t)| + ∫ t 0 |z(t − s)| { ∫ s+t0 s |b(v)|dv } ds ] ≤ |ϕ|t0 [ 1 + a ∫ ∞ 0 |z(t)|dt ] . (3.29) for any ǫ > 0, take δ := ǫ [ 1 + a ∫ ∞ 0 |z(t)|dt ] −1 . then from (3.29) it follows that |x(t + t0)| < ǫ for t ≥ 0, or that |x(t)| < ǫ for t ≥ t0, for any ϕ ∈ c[0, t0] with |ϕ|t0 < δ. now that uniform stability has been established, we must show that the rest of definition 3.1 (3) holds. choose any η > 0. for any t0 ≥ 0, choose any ϕ ∈ c[0, t0] with |ϕ|t0 < η. then it follows from (3.29) that |x(t + t0)| ≤ η [ |z(t)| + ∫ t 0 |z(t − s)|g(s) ds ] (3.30) where g(s) := ∫ s+t0 s |b(v)|dv. (3.31) since b ∈ l1[0,∞), g(s) → 0 as s → ∞. this and (3.23) imply that ∫ t 0 |z(t − s)|g(s) ds → 0 as t → ∞ (3.32) since the convolution of two functions approaches 0 as t → ∞ if one of them belongs to l1[0,∞) and the other one approaches 0 as t → ∞. by theorem 3.3, z(t) → 0 as t → ∞. (3.33) this and (3.32) imply that the right-hand side of (3.30) approaches 0 as t → ∞. consequently, for every ǫ > 0, there is a t = t (η,ǫ) > 0 such that |x(t + t0)| < ǫ for t ≥ t . in short, we have shown that for a given η > 0 and ǫ > 0, and any t0 ≥ 0, a t > 0 exists (independent of t0) such that |x(t,t0,ϕ)| < ǫ for all t ≥ t0 + t if ϕ ∈ c[0, t0] and |ϕ|t0 < η. 16 leigh c. becker cubo 11, 3 (2009) 4 a uniformly continuous liapunov functional we will establish in the next theorem that the conclusions reached in theorem 3.3 are in fact valid under the conditions of lemma 3.2. in other words, the conditions that were added to lemma 3.2 to obtain theorem 3.3 are superfluous. this result depends on the following two lemmas, the first of which is a consequence of the definition of uniform continuity and appears as an exercise in boas [6, p. 119]. lemma 4.1. let f ∈ c[0,∞). if limt→∞ f(t) exists and is finite, then f is uniformly continuous on [0,∞). consequently, we have: lemma 4.2. let f ∈ c[0,∞). if f is riemann integrable on [τ,∞) for some τ ≥ 0, then ∫ t 0 f(s) ds is uniformly continuous on [0,∞). theorem 4.3. let a: [0,∞) → [0,∞) and b: ω → r be continuous functions. if ∫ t s |b(u,s)|du ≤ a(s) (4.1) for all t ≥ s ≥ 0, then the zero solution of x ′ (t) = −a(t)x(t) + ∫ t 0 b(t,s)x(s) ds (4.2) is stable. furthermore, if for some t1 ≥ 0 there is a constant k > 0 such that a(t) ≥ k (4.3) for all t ≥ t1 and a constant λ ∈ (0, 1) such that ∫ t s |b(u,s)|du ≤ λa(s) (4.4) for all t ≥ s ≥ t1, then every solution x(t) of (4.2) belongs to l1[0,∞) and is uniformly continuous on [0,∞). moreover, the zero solution is globally asymptotically stable. proof. we have already established with lemma 3.2 that (4.1) implies the stability of the zero solution of (4.2). revisiting the proof of the lemma, let x(t) = x(t,t0,ϕ) be the solution corresponding to an initial function ϕ ∈ c[0, t0]. now consider v (t,ψ(·)) defined by (3.5). by (3.7), v (t) = v (t,x(·)) is decreasing on [t0,∞). consequently, as v (t) ≥ 0, limt→∞ v (t) exists and is finite. therefore, by lemma 4.1, v is uniformly continuous on [0,∞). now consider vγ (t) defined by (3.11). an integration of (3.13) yields vγ (t) − vγ (τ) ≤ −(1 − γ) ∫ t τ a(s)|x(s)|ds. cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 17 hence, ∫ t τ a(s)|x(s)|ds ≤ vγ (τ) 1 − γ for all t ≥ τ. and so a(t)|x(t)| is riemann integrable on [τ,∞). by lemma 4.2, ∫ t 0 a(s)|x(s)|ds is uniformly continuous on [0,∞). this suggests that the integral terms in v (t), namely w(t) := ∫ t 0 [ a(s) − ∫ t s |b(u,s)|du ] |x(s)|ds, may also be uniformly continuous on [0,∞). we now prove that this is the case. first define h: [0,∞) × [0,∞) → [0,∞) by h(t,s) := { a(s) − ∫ t s |b(u,s)|du if t ≥ s, a(s) if t < s. (4.5) for a fixed t∗ ∈ [0,∞), 0 ≤ h(t∗,s)|x(s)| ≤ a(s)|x(s)|. by a comparison test, ∫ ∞ 0 h(t ∗ ,s)|x(s)|ds ≤ ∫ ∞ 0 a(s)|x(s)|ds < ∞. consequently, the improper integral ∫ ∞ 0 h(t,s)|x(s)|ds defines a function, call it w(t), on the interval [0,∞). for t2 ≥ t1 ≥ 0, h(t2,s) ≤ h(t1,s). this implies w is decreasing on [0,∞). as w(t) ≥ 0, w(t) approaches a finite limit, say l, as t → ∞. this in turn implies that w(t) → l as t → ∞ because |w(t) − l| ≤ |w(t) − w(t)| + |w(t) − l| = ∣ ∣ ∣ ∣ ∫ t 0 h(t,s)|x(s)|ds − ∫ ∞ 0 h(t,s)|x(s)|ds ∣ ∣ ∣ ∣ + |w(t) − l| = ∣ ∣ ∣ ∣ − ∫ ∞ t h(t,s)|x(s)|ds ∣ ∣ ∣ ∣ + |w(t) − l| = ∫ ∞ t a(s)|x(s)|ds + |w(t) − l|. by lemma 4.1, w is uniformly continuous on [0,∞). we have established that v and w are uniformly continuous on [0,∞). hence, so is the difference v (t) − w(t) = |x(t)|. by lemma 3.2, |x| ∈ l1[0,∞). by barbălat’s lemma, |x(t)| → 0 as t → ∞. by lemma 4.1, x(t) is uniformly continuous on [0,∞). 18 leigh c. becker cubo 11, 3 (2009) example 4.4. every solution of x ′ (t) = −(t + 1)x(t) + ∫ t 0 2t (1 + t2 − s2)2 x(s) ds (4.6) belongs to l1[0,∞) and is uniformly continuous on [0,∞) and its zero solution is globally asymptotically stable. proof. as a(t) = t + 1, (4.3) is satisfied with k = 1 . as b(t,s) = 2t(1 + t2 − s2)−2, ∫ t s |b(u,s)|du = ∫ t s 2u (1 + u2 − s2)2 du = 1 − 1 1 + t2 − s2 . clearly then (4.1) is satisfied as ∫ t s |b(u,s)|du ≤ 1 + s = a(s) for all t ≥ s ≥ 0. also, for t ≥ s ≥ 2, ∫ t s |b(u,s)|du ≤ 1 − 1 1 + t2 − s2 < 1 + 1 s = 1 s a(s) ≤ 1 2 a(s). in other words, (4.4) is also satisfied with t1 = 2 and λ = 1/2. t 1 2 3 4 x t k3 k2 k1 0 1 2 3 figure 1: three numerical solutions of (4.6). the maple worksheet [3], which can be found at the maple application center website, uses the implicit trapezoidal rule and newton’s method for nonlinear systems to numerically approximate cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 19 solutions of scalar volterra integro-differential equations and draw their respective graphs. it was used to compute the three numerical solutions of (4.6) that are shown in fig. 1. a step size of h = 0.1 was used. one of the solutions satisfies the initial condition x(0) = 1. the other two solutions correspond to the initial functions ϕ(t) = 2 + t on the initial interval [0, 1] and ϕ(t) = −2 + sin(2t) on [0, 2]. finally, we add two integral conditions to theorem 4.3 so that we can drop (4.3), namely, the condition that a(t) be eventually bounded below by a positive constant. theorem 4.5. let a: [0,∞) → [0,∞) and b: ω → r be continuous functions satisfying conditions (4.1) and (4.4). if a constant l and a nonnegative function p ∈ l1[0,∞) exist such that ∫ t 0 e − ∫ t ξ a(u) du dξ ≤ l (4.7) for all t ≥ 0 and ∫ t s e − ∫ t ξ a(u) du|b(ξ,s)|dξ ≤ p(s) (4.8) for all t ≥ s ≥ 0, then the zero solution of (4.2) is globally asymptotically stable. proof. by lemma 3.2, the zero solution of (4.2) is stable because of (4.1). we will show that all of its solutions approach zero as t → ∞ by comparing them to the solutions of the equations y ′ (t) = − ( a(t) + 1 k ) y(t) + ∫ t 0 b(t,s)y(s) ds, (4.9) where k ∈ n (the set of natural numbers). note that for a given k ∈ n, (4.9) has a globally asymptotically stable zero solution on account of a(t)+1/k and b(t,s) satisfying all of the conditions of theorem 4.3. for any t0 ≥ 0 and ϕ ∈ c[0, t0], let x(t) be the solution of (4.2) with x(t) = ϕ(t) for 0 ≤ t ≤ t0. for k ∈ n, let yk(t) denote the solution of (4.9) with the same initial function—i.e., yk(t) = ϕ(t) for 0 ≤ t ≤ t0. now consider the difference x(t) − yk(t). for t ≥ t0, d dt [x(t) − yk(t)] = −a(t)[x(t) − yk(t)] + 1 k yk(t) + ∫ t 0 b(t,s)[x(s) − yk(s)] ds. multiplying this by µ(t) := exp ( ∫ t 0 a(v) dv ) and replacing a(t)µ(t) with µ′(t), we obtain d dt (µ(t)[x(t) − yk(t)]) = 1 k µ(t)yk(t) + µ(t) ∫ t 0 b(t,s)[x(s) − yk(s)] ds. then an integration from t0 to t yields x(t) − yk(t) = 1 k ∫ t t0 µ(ξ) µ(t) yk(ξ) dξ + ∫ t t0 µ(ξ) µ(t) ∫ ξ 0 b(ξ,s)[x(s) − yk(s)] dsdξ 20 leigh c. becker cubo 11, 3 (2009) for all t ≥ t0. as x(t) ≡ yk(t) on [0, t0], it follows that |x(t) − yk(t)| ≤ 1 k ∫ t 0 µ(ξ) µ(t) |yk(ξ)|dξ (4.10) + ∫ t 0 µ(ξ) µ(t) ∫ ξ 0 |b(ξ,s)| |x(s) − yk(s)|dsdξ for all t ≥ 0. by (3.9) and (3.10), |yk(t)| ≤ mk(t0)|ϕ|t0 (4.11) for all t ≥ t0, where mk(t0) := 1 + ∫ t0 0 [ a(s) + 1 k − ∫ t0 s |b(u,s)|du ] ds. this holds in fact for all t ≥ 0 as mk(t0) ≥ 1. moreover, the upper bound in (4.11) can be replaced by one that is independent of k; that is, |yk(t)| ≤ m1(t0)|ϕ|t0 for all t ≥ 0 and k ∈ n. this and (4.7) imply ∫ t 0 µ(ξ) µ(t) |yk(ξ)|dξ = ∫ t 0 e − ∫ t ξ a(u) du |yk(ξ)|dξ ≤ lm1(t0)|ϕ|t0 (4.12) for all t ≥ 0. as for the iterated integral in (4.10), by interchanging the order of integration and applying (4.8), we obtain ∫ t 0 µ(ξ) µ(t) ∫ ξ 0 |b(ξ,s)| |x(s) − yk(s)|dsdξ (4.13) = ∫ t 0 ( ∫ t s e − ∫ t ξ a(v) dv|b(ξ,s)|dξ ) |x(s) − yk(s)|ds ≤ ∫ t 0 p(s)|x(s) − yk(s)|ds. it follows then from (4.10), (4.12), and (4.13) that |x(t) − yk(t)| ≤ 1 k lm1(t0)|ϕ|t0 + ∫ t 0 p(s) |x(s) − yk(s)|ds for t ≥ 0. by gronwall’s inequality, |x(t) − yk(t)| ≤ 1 k lm1(t0)|ϕ|t0 e ∫ t 0 p(s) ds . consequently, |x(t)| ≤ |yk(t)| + 1 k lm1(t0)|ϕ|t0 e ∫ ∞ 0 p(s) ds < ∞ (4.14) for all t ≥ 0. this with yk(t) → 0 as t → ∞ for every k ∈ n implies that x(t) → 0 as t → ∞. cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 21 example 4.6. the zero solution of x ′ = −a(t)x (4.15) is globally asymptotically stable if a constant α > 0 exists such that ∫ t t0 a(u) du ≥ α(t − t0) (4.16) for t ≥ t0 ≥ 0. proof. condition (4.7) is satisfied with l = 1/α. the other three conditions in theorem 4.5 are trivially satisfied as b(t,s) in (4.2) is identically equal to zero. remark. in fact under condition (4.16), the zero solution of (4.15) is uniformly asymptotically stable in the large (cf. [11, p. 88]). example 4.7. the zero solution of x ′ (t) = −tx(t) + ∫ t 0 b(t,s) x(s) ds, (4.17) where b: ω → r is the function b(t,s) =      0, if 0 ≤ s < 1 3 (3t − 1)(3s − 1) (1 + t + s)5 , if s ≥ 1 3 , (4.18) is globally asymptotically stable. proof. since a(t) = t, ∫ t 0 e − ∫ t ξ a(u) du dξ = e −t2/2 ∫ t 0 e ξ2/2 dξ = √ 2d(t/ √ 2), (4.19) where d is dawson’s integral, namely, d(t) := e −t2 ∫ t 0 e ξ2 dξ. (4.20) it can be shown using an elementary argument that d is bounded on [0,∞), but that is a long established fact (cf. [1, p. 298]). from the information in [1] or through the use of a computer algebra system, we find that (4.19) has a absolute maximum value of 0.7651 . . . at t = 1.3069 . . . . consequently, (4.7) holds with l = 0.8. for t ≥ s ≥ 1/3, ∫ t s |b(u,s)|du = ∫ t s (3u − 1)(3s − 1) (1 + u + s)5 du ≤ (3s − 1) ∫ t s (3u − 1) (1 + u)5 du ≤ [ 3s − 1 (1 + s)4 ] s ≤ 0.2s. (4.21) 22 leigh c. becker cubo 11, 3 (2009) since b(t,s) = 0 for 0 ≤ s < 1/3, we conclude ∫ t s |b(u,s)|du ≤ λa(s) (4.22) for all t ≥ s ≥ 0, where λ = 0.2. thus, (4.1) and (4.4) hold for all t ≥ s ≥ 0. from (4.21) we see that ∫ t s e − ∫ t ξ a(u) du|b(ξ,s)|dξ ≤ ∫ t s |b(ξ,s)|dξ ≤ p(s), (4.23) where p(s) :=      0, if 0 ≤ s < 1 3 [ 3s − 1 (1 + s)4 ] s, if s ≥ 1 3 . (4.24) since p ∈ l1[0,∞), condition (4.8) holds. the graphs of four numerical solutions of (4.17) computed with the maple worksheet [3] are shown in fig. 2. one of the solutions satisfies the initial condition x(0) = 2. the others issue from the initial functions ϕ(t) = −5, ϕ(t) = −2 + sin(4t), and ϕ(t) = 3 + 2t on the initial interval [0, 1]. t 1 2 3 x t k4 k2 0 2 4 figure 2: four numerical solutions of (4.17). received: march 6, 2008. revised: july 11, 2008. cubo 11, 3 (2009) uniformly continuous l1 solutions of volterra equations ... 23 references [1] abramowitz, m. and stegun, i.a., handbook of mathematical functions with formulas, graphs, and mathematical tables, 2nd printing, national bureau of standards, applied mathematical series 55, 1964. [2] barbălat, i., systèms d’équations différentielle d’oscillations nonlinéaires, rev. roumaine math. pures appl., 4 (1959), 267–270. [3] becker, l.c., scalar volterra integro-differential equations, maple application center (www.maplesoft.com/applications), august 2007. [4] becker, l.c., principal matrix solutions and variation of parameters for a volterra integrodifferential equation and its adjoint, e. j. qualitative theory of diff. equ., no. 14 (2006), 1–22 (www.math.u-szeged.hu/ejqtde/2006/200614.html). [5] becker, l.c., stability considerations for volterra integro-differential equations, ph.d. dissertation, southern illinois university, carbondale, il, 1979. (a pdf version of the entire dissertation can be found at the web site: http://www.cbu.edu/~lbecker/research.htm.) [6] boas, jr., r.p., a primer of real functions, third edition, carus mathematical monographs, no. 13, maa, 1981. [7] burton, t.a., volterra integral and differential equations, second edition, mathematics in science and engineering, vol. 202, elsevier, amsterdam, 2005. [8] burton, t.a., stability and periodic solutions of ordinary and functional differential equations, dover publications, mineola, new york, 2005. [9] burton, t.a., stability theory for volterra equations, j. differential equations, 32 (1979), 101–118. [10] burton, t.a. and mahfoud, w.e., stability criteria for volterra equations, trans. amer. math. soc., 279 (1983), 143–174. [11] corduneanu, c., principles of differential and integral equations, chelsea publishing co., new york, 1977. [12] corduneanu, c., integral equations and stability of feedback systems, mathematics in science and engineering, vol. 104, academic press, new york, 1973. [13] fulks, w., advanced calculus, second edition, john wiley & sons, new york, 1969. [14] hartman, p., ordinary differential equations, reissue of 1982 second ed., in: classics in applied mathematics 38, siam, philadelphia, 2002. 24 leigh c. becker cubo 11, 3 (2009) [15] lakshmikantham, v. and leela, s., differential and integral inequalities, mathematics in science and engineering, vol. 55-i, academic press, new york, 1969. [16] logemann, h. and ryan, e.p., asymptotic behaviour of nonlinear systems, amer. math. monthly, 111 (2004), 864–889. [17] miller, r.k., asymptotic stability properties of linear volterra integrodifferential equations, j. differential equations, 10 (1971), 485–506. [18] yoshizawa, t., stability theory and the existence of periodic solutions and almost periodic solutions, applied math. sciences, vol. 14, springer-verlag, new york, 1975. 06-ucl1svegas cubo a mathematical journal vol.10, n o ¯ 03, (83–91). october 2008 many-ended complete minimal surfaces between two parallel planes in r 3 francisco brito∗ universidade federal de pernambuco – ufpe, ccen-departamento de matemática, 50.740.540, recife-pe, brazil email: brito@dmat.ufpe.br abstract we use some special convergent hadamard gap series to provide examples of complete minimal surfaces of many different conformal types between two parallel planes in three dimensional euclidean space. resumen nosotros usamos algunas series convergentes especiales de hadamard para dar ejemplo de superficies mı́nimas completas de varios diferentes tipos conforme entre dos planos paralelos en espacios euclideanos de dimensión tres. key words and phrases: complete minimal surfaces, hadamard gap series. math. subj. class.: 53a10. ∗i am grateful to profs. marcus v. m. wanderley and maria elisa oliveira for helpful hints and discussions. 84 francisco brito cubo 10, 3 (2008) 1 introduction in [7] f. xavier and l. p. m.jorge established the existence of complete non-planar minimal surfaces between two parallel planes in r3. their technique consisted of an artful use of runge’s theorem to prove the existence of holomorphic functions on the unit disc d with the right properties they needed. later their method was adapted by others to produce new surfaces as above with new features, like having cylindrical type as in [6] or being non-orientable as in [3]. another way for rendering complete minimal surfaces between two parallel planes in r3 was developed in [1]. this method consisted mainly in proving the existence of bounded holomorphic functions h in d, given by lacunary power series, and such that ∫ γ |h′(z)|2|dz| = ∞ for all divergent paths γ in the unit disc. in this paper we intend to show the flexibility of the second method by producing examples of complete minimal surfaces between two parallel planes in r3 of the following conformal types: 1. a disc with finitely many points removed. 2. any annulus, 0 < r < |z| < r. 3. any annulus as above with finitely many points removed. this work is organized as follows: in §2 we give some definitions and prove the lemmas that will be needed in the other sections. in the three remaining sections we describe the examples of the types above. remark 1.1 this work was written about fourteen years ago and circulated as a preprint for some time. meanwhile n.nadirashvili proved in [4] the existence of bounded complete minimal surfaces in r3. 2 some definitions and lemmas lacunary power series were defined in [1] with the restriction that they would have radius of convergence 1. this is just a mild technical point. here we use any positive real number r as radius of convergence and make the necessary changes for having an analogue of theorem 2 of [1]. definition 2.1 a convergent power series ∞ ∑ k=0 akz nk is lacunary if there exists a real number q > 1 such that nk+1 nk ≥ q for all k = 0, 1, . . . . lemma 2.2 let h(z) = ∞ ∑ k=0 akz nk be a lacunary power series of radius of convergence r > 0, and suppose that the following three conditions hold: cubo 10, 3 (2008) many-ended complete minimal surfaces ... 85 a) ∞ ∑ k=0 |ak|r nk converges. b) lim k→∞ r nk |ak|min { nk+1 nk , nk nk−1 } = ∞. c) ∞ ∑ k=0 r2nk |ak| 2 nk diverges. then h is bounded in dr = {z ∈ c; |z| < r}, and for all divergent paths γ in dr, one has that ∫ γ |h′(z)|2|dz| = ∞. proof. the change of variable z = rw together with (a) show that h is bounded. the same change of variable and theorem 2 of [1] finish the proof. lemma 2.3 if h satisfies the conditions of the above lemma in dr, h(z) = h(z −1) is a bounded holomorphic function in ar = {z ∈ c; |z| > r −1}, and for all divergent paths γ in ar tending to a point of |z| = r−1 one has that ∫ γ |h ′ (z)|2|dz| = ∞. proof. the change of variable z = w−1 and lemma 2.2 prove this assertion. now, given r,r ∈ r, 0 < r < r, let ωr,r be the annulus r < |z| < r. lemma 2.4 suppose that h1(z) = ∞ ∑ k=0 akz nk and h2(z) = ∞ ∑ k=0 bkz mk are lacunary power series that satisfy the conditions of lemma 2.2, and have radii of convergence r and r−1 respectively, with 0 < r < r. suppose further that h ′ 1 (z) and h ′ 2 (z−1) do not vanish in |z| = r and |z| = r respectively. then, for all divergent paths γ in ωr,r one has that ∫ γ |h ′ 1 (z)|2|(h2(z −1))′|2|dz| = ∞. proof. a divergent path in ωr,r either approach |z| = r or |z| = r. suppose that γ is a divergent path that approaches |z| = r. since h ′ 1 is holomorphic in a neighborhood of that circle and does not vanish at any point of it, it follows that there is a perhaps smaller neighborhood u of |z| = r having compact closure u , and such that inf z∈γ∩u {|h ′ 1 (z)|2} = a > 0. consequently, if γ̃ denotes the portion of γ inside u one has that ∫ γ |h ′ 1 (z)|2|(h2(z −1)) ′ |2|dz| ≥ a2 ∫ γ̃ |(h2(z −1)) ′ |2|dz| = ∞ by lemma 2.3. the rest of the proof follows in a similar way. in the next sections we will use mainly the weierstrass representation for minimal surfaces in r 3 (see [5]) and the three lemmas above. 86 francisco brito cubo 10, 3 (2008) 3 minimal immersions of a disc with finitely many points removed let ω = dr − {a1, a2, . . . , an}, where dr is the open disc of radius r centered at the origin and a1,a2,. . . ,an are distinct points of dr. theorem 3.1 there exist complete minimal immersions m of ω between two parallel planes of r 3. furthermore the ends of m corresponding to the points a1,a2,. . . ,an are all planar and have index one. proof. to avoid notational inconveniences we will prove separately the cases n = 1 and n > 1. in the first case we take ω = dr − {a}, for some a ∈ dr. using the weierstrass representation, set f (z) = (z − a)−2 and g(z) = (z − a)2h′(z) where h is any function as in lemma 2.2. clearly the data above defines a minimal immersion m of ω in r3. moreover, m is also complete for the metric is given by λ(z)|dz| = 1 2 {|z − a|−2 + |z − a|2|h′(z)|2}|dz|, and because x3(z) = re(h(z)), it follows from the properties of h that the third coordinate of m is bounded. the end corresponding to a is of course planar because f g is holomorphic at that point and have index one because f has a pole of order two at a and f g2 vanishes at that same point. for information on the behavior of ends of complete minimal surfaces see [2]. now we consider the case n > 1. in the weierstrass representation for m set f (z) = f (z)−1 exp    n ∑ j=1 aj fj (z)    and g(z) = h′(z)f (z), where h is a function as in lemma 1.1, f (z) = n ∏ j=1 (z − aj ) 2 the functions fj satisfy (z−aj) 2f ′ j (z) = f (z), and the aj are constants to be chosen so that ∫ σ f (z)dz = 0 for all closed curves σ in ω. since f g and f g2 have holomorphic extensions to all of |z| < r, it follows that this will be enough to exclude the possibility of real periods appearing in the weierstrass representation of m. an easy computation shows that the choice aj = f ′′ j (aj )(f ′ j (aj )) −2, j = 1, . . . , n solves the problem. cubo 10, 3 (2008) many-ended complete minimal surfaces ... 87 observe that the metric λ(z)|dz| on m is given by 2λ(z)|dz| = {|f (z)|−1 + |f (z)||h ′ (z)|2} ∣ ∣ ∣ ∣ ∣ exp { n ∑ k=1 akfk(z) } ∣ ∣ ∣ ∣ ∣ |dz|, and since there is a positive real c such that ∣ ∣ ∣ ∣ ∣ exp { n ∑ k=1 akfk(z) } ∣ ∣ ∣ ∣ ∣ ≥ c, z ∈ ω, it follows from the properties of h and that f has poles of order 2 at the aj that m is complete. also, x3(z) = re ∫ h ′ (z) exp{ n ∑ j=1 aj fj (z)}dz is bounded in ω. this can be seen in the following way: exp { n ∑ k=1 akfk(z) } and its derivatives as well as h are all bounded holomorphic functions in ω. as a matter of fact they are bounded in dr, so, by integration by parts, it follows that ∫ h ′ (z) exp{ n ∑ j=1 aj fj (z)}dz is bounded in dr, so x3 is also bounded. by an argument similar to the one done in the case n = 1 we conclude that the ends corresponding to the points aj are all planar and have index one. 4 complete minimal annuli between two parallel planes in r 3 all the examples of complete minimal annuli in [6] have the conformal type of an annulus of the form r−1 < |z| < r. here we give examples of all possible annuli 0 < r < |z| < r < ∞. theorem 4.1 given any annulus ωr,r there is a complete minimal immersion of it in r 3 with one coordinate bounded. proof. take any two functions h1 and h2 as in lemma 1.3, say h1(z) = ∞ ∑ k=0 akz nk and h2(z) = ∞ ∑ l=0 alz ml , with radii of convergence r and r−1 respectively, and such that all the nk and ml are simultaneously either even or odd. then in the weierstrass representation we set in dr,r, f (z) = 1 and g(z) = h ′ 1 (z)h ′ 2 (z), where h2(z) = h2(z −1). because f is constant, and g is an even function where defined, it follows that for all closed curves γ in ωr,r one has that ∫ γ f dz = ∫ γ f gdz = ∫ γ f g2dz = 0. 88 francisco brito cubo 10, 3 (2008) thus, the minimal surface so obtained is in fact well defined. furthermore, since the metric λ(z)|dz| is given by 2λ(z)|dz| = ( 1 + |h ′ 1 (z)|2|h ′ 2 (z)|2 ) |dz| it follows from lemma 1.3 that it is complete. it remains to prove only that one of the coordinates of that immersion is bounded. since x3(z) = re ∫ gdz = re ∫ h ′ 1 (z)h ′ 2 (z)dz, it is enough to prove that ∫ gdz is a bounded holomorphic function in ωr,r. consider ρ ∈ r such that r < ρ < r and define the sets a1 = ωr,r ∩ {z ∈ c such that |z| ≤ ρ} and a2 = ωr,r ∩ {z ∈ c such that |z| ≥ ρ}. we then observe that h ′ 1 (z), its derivatives and h2 are bounded in a1 and the same happens to h ′ 2 , its derivatives and h1 in a2. so, by integration by parts we can conclude that ∫ gdz is bounded in both a1 and a2, hence in ωr,r. 5 the case of a annulus with finitely many points removed let a = {a1, . . . , an} be a set of distinct points of ωr,r such that a ⋂ (−a) = ∅, and set ω = ωr,r − a. theorem 5.1 there is a complete minimal immersion of ω between two parallel planes of r3. furthermore, the ends corresponding to a1, . . . , an are planar. proof. first suppose n = 1 and take ω = ωr,r − {a} with a ∈ ωr,r. consider holomorphic functions h1 and h2 as in theorem 3.1, and define the surface m using the weierstrass representation by taking f (z) = (z − a)−2 and g(z) = (z − a)2h ′ 1 (z)h ′ 2 (z), keeping the notation of theorem 3.1. then f is holomorphic in a neighborhood of |z| ≤ r, and has residue zero at a, so ∫ σ f dz = 0, for all closed curves σ inside ω. because f g = h ′ 1 h ′ 2 is an even holomorphic function in ωr,r it follows that ∫ σ f gdz = 0 for all closed curves σ in ω too. now we must make one more choice in order to have ∫ σ f g2dz = 0 for all closed curves σ in ω. the idea is to start with h1 and h2 having no low powers in their power series expansions. since f (z)g2(z) = (z − a)2(h ′ 1 (z))2(h ′ 2 (z))2 is holomorphic in ω, by expanding (z − a)2, it is clear that the only term that may cause problems is −2az(h ′ 1 (z))2(h ′ 2 (z))2 because the other two terms are even functions. hence if the functions h1 and h2 are chosen to satisfy theorem 3.1 and have the cubo 10, 3 (2008) many-ended complete minimal surfaces ... 89 form h1(z) = ∞ ∑ k=1 akz 1+2 2 k and h2(z) = ∞ ∑ k=1 bkz 1+2 2 k there is no term in z−1 in the laurent expansion of z(h ′ 1 (z))2(h ′ 2 (z))2. hence, for all closed curves σ in ω, ∫ σ f g2dz = 0 as wanted. besides, as in theorem 2.1, the end corresponding to the point a is planar and have index one. in order to study the case n > 1 define the following holomorphic functions in ω: f (z) = n ∏ j=1 (z − aj ) 2, fk(z) = (z − ak) −2f (z) and g ′ k (z) = zfk(z)fk(−z) for k = 1, . . . , n, and finally h(z) = n ∑ j=1 aj gj (z), where the constants aj are to be determined so that, if the immersion m of ω is defined in terms of the weierstrass representation by setting f (z) = (f (z))−1 exp h(z) and g(z) = f (z) exp { − 1 2 h(z) } h ′ 1 (z)h ′ 2 (z), then f has residue zero at all the points aj . as before, the functions h1 and h2 are chosen satisfying the conditions of lemma 1.3 and the exponents are chosen so that ∫ |z|=ρ f (z)g(z)2dz = 0, for r < ρ < r. it must be pointed out that once these constants aj are determined the rest is done quite easily as follows: first we observe that h is an even holomorphic function in ωr,r, and the same happens to h ′ 1 and h ′ 2 , thus f g is an even holomorphic function in ωr,r and so ∫ σ f gdz = 0 for all closed curves σ in ωr,r as wanted. furthermore, 2λ(z)|dz| = |f (z)|−1| exp h(z)| + |f (z)||h ′ 1 (z)|2|h ′ 2 (z)|2, hence, repeating the reasoning in the proof of theorem 2.1 we conclude that λ(z)|dz| is complete and x3 is bounded. now we determine the constants aj . first, we observe that all the poles of f have order two and that for j = 1, . . . , n, (z − aj ) 2f (z) = exp h(z) fj (z) , thus d dz {(z − aj ) 2f (z)} = exp h(z) f 2 j (z) [ h ′ (z)fj (z) − f ′ j (z) ] = exp h(z) f 2 j (z) [ fj (z) { n ∑ k=1 akzfk(z)fk(−z) } − f ′ j (z) ] . 90 francisco brito cubo 10, 3 (2008) so, the residue of f at aj is zero if and only if fj (aj ) { n ∑ k=1 akaj fk(aj )fk(−aj ) } − f ′ j (aj ) = 0. since fk(aj ) = 0 for k 6= j, fj (aj ), fj (−aj ) 6= 0 and aj 6= −ak, for 1 ≤ j, k ≤ n, it follows that aj f 2 j (aj )fj (−aj )aj − f ′ j (aj ) = 0, for each j, 1 ≤ j ≤ n, hence aj = f ′ j (aj ) aj f 2 j (aj )fj (−aj ) , for j = 1, . . . , n. to finish the proof it is enough to show that we can choose h1 and h2 in such a way that ∫ |z|=ρ f (z)g(z)2dz = 0, for r < ρ < r. since f (z)g2(z) = f (z)[h ′ 1 (z)h ′ 2 (z)]2, and f has degree 2n, if we define h1(z) = ∞ ∑ k=1 akz 1+2n+2 2 k and h2(z) = ∞ ∑ k=1 bkz 1+2n+2 2 k we are done. also the observations about the ends in the other cases are valid here without change. the assumption that (−a) ∩ a = ∅ is not really needed. it is just a technical difficulty that can be easily overcome as follows: corollary 5.2 if a is any finite subset of ωr,r there is a complete minimal immersion of ω = ωr,r − a between two parallel planes of r 3. furthermore, the ends corresponding to the points of a are planar. proof. induction on the number of elements of a shows that there exists a transformation π : ω −→ ω ′ , where π(z) = z2 p for some positive integer p and ω ′ satisfies the condition of theorem 5.1. it is clear that (π , ω) is an unramified covering of ω ′ , and by theorem 5.1, there is a complete minimal immersion x of ω ′ between two parallel planes of r3. so x◦π is also a complete minimal immersion of ω in r3 with the same properties as before. received: february 2008. revised: june 2008. cubo 10, 3 (2008) many-ended complete minimal surfaces ... 91 references [1] f.f. de brito, power series with hadamard gaps and hyperbolic complete minimal surfaces, duke math. journal 68 (1992), 297–300. [2] l.p.m. jorge and w. meeks, the topology of complete minimal surfaces opf finite gaussian curvature, topology 22 (1983), 203–221. [3] f. lopez, a non orientable complete minimal surface in r3 between two parallel planes, proceedings of the amer. math. soc 103 (1988), 913–917. [4] n. nadirashvili, hadamard and calabi-yau’s conjectures on negatively curved and minimal surfaces, invent. math. 126 (1996), 457–465. [5] r. osserman, a survey of minimal surfaces, van nostrand, new york, 1969. [6] h. rosenberg and e. toubiana, a cylindrical type minimal surface in a slab in r3, bull. sc. math. 111 (1987), 241–245. [7] f. xavier and l.p.m. jorge, a complete minimal surface in r3 between two parallel planes, ann. of math. 112 (1980), 203–206. n07 cubo a mathematical journal vol.11, no¯ 01, (145–162). march 2009 on a new notion of holomorphy and its applications wolfgang sproessig freiberg university of mining and technology, faculty of mathematics and informatics, agricola-strasse 1, 09596 freiberg, germany. email: sproessig@math.tu-freiberg.de and le thu hoai hanoi university of technology, faculty for applied mathematics and informatics, dai co viet street 1, 10000 hanoi, vietnam. abstract this paper devotes a new general notion of holomorphy which works in the continous and discrete cases. with the help of methods of a general operator theory the so called l-holomorphy is introduced. realizations of this calculus follow. new versions of taylorand taylor–gontcharov formulae are deduced. the results are applied for the solution of higher order systems of differential equations. resumen este artículo es dedicado a una nueva noción de holomorfía la cual funciona en los casos continuo y discreto. con la ayuda de métodos de la teoría general de operadores la llamada l-holomorfia es presentada. realizaciones de este cálculo siguen. nuevas versiones de fórmulas de taylor-y taylor-gontcharov son deducidas. los resultados son aplicados para la solución de sistemas de orden superior de ecuaciones diferenciales. key words and phrases: generalized holomorphic functions, taylor-gontcharov formulae, plemelj projections, higher order boundary value problems. 146 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) 1 introduction. the aim of this article is to introduce a very general notion of holomorphy by the help of three general operators in banach spaces which have to satisfy some conditions. this introduction is oriented at the theory of right invertible operators. we refer to the well-known book of v.s. ryabenskij [11] (1987), w. schempp and f.j. delvos [2] (1990) and the article by m. tasche [16] (1981). the advantage of our approach is the fact that holomorphy can be considered in the continuous and discrete case within one calculus. we continue the line of action we have followed in books [6],[7],[5]. in the second part we present a large number of realisations. here we use above all results of the common research with k. gürlebeck confer again in [6], [7] and [4]. finally, some classes of boundary value problems of higher order will be considered. in that connection new formulae of taylorand taylor-gontcharov type are obtained. all our considerations take place in the scale of sobolev and besov spaces as well as its discrete analogue. 2 a general holomorphy let x,y,z be banach spaces. we introduce the bounded linear operators t,tr and p with the following properties (i) t : x → im t ⊂ y is injective. (ii) tr : y → z is a generalized trace operator . (iii) the operator p : imtr ∩ y → y satisfies the property ptrpu = pu. furthermore, we assume (i) im trt ⊂ kerp , (ii) im t ∩ ker tr = {0}. remark 1. we also have imt ∩ imp = {0}. indeed, let u ∈ imt ∩ imp = {0} then u = pw = tv and u = pw = ptrpw = ptrtv = 0. theorem 1. (mean value formula) set imt ⊕ imp =: y1 ⊂ y. there is a unique linear operator l with d(l) = y1 and l : d(l) → x, such that u = ptru + tlu. proof. let u ∈ d(l). then u permits the representation u = pv + tw, cubo 11, 1 (2009) on a new notion of holomorphy and its applications 147 with v ∈ imtr ∩ y, and w ∈ x. applying ptr from the left it follows ptru = ptrp v + ptrt w = p v. in this way the first item of the desired formula is obtained. in order also to obtain the second item we have to use the injectivity of the operator t . on the linear set im t there exists a linear operator l̃ with l̃tw = w. the operator l̃ can be extended to an linear operator l on y1 setting lz := l̃z1, where z = z1 + z2 with z1 ∈ im t and z2 ∈ im p . the additivity follows from l(z + z′) = l(z1 + z2 + z ′ 1 + z′ 2 ) = l̃(z1 + z ′ 1 ) = l̃z1 + l̃z ′ 1 = lz + lz′. the monogeneity with a real constant λ is also fulfilled. indeed, we have l(λz) = l̃(λz1) = λl̃z1 = λlz. now we obtain easily lu = lptru + lt w = w and our decomposition formula is completely proved. the uniqueness follows from tlu − tl1u = 0 leads to lu = l1u, where l1 is another linear operator which has to fulfil the decomposition formula. # corollary 1. the following relations between the operators l,p and t are valid: (i) the operator l is the left-inverse to the operator t, i.e. lt = i. (ii) set r := tl then r is a projection onto y1 with imr = imt. (iii) it holds kerl = imptr. proof. the relation (i) follows by the definition of l. indeed, let v ∈ x, then ltv = l̃tv = v. (ii) obviously, tl fulfils the idempotential property and so we have r2 = r. it is immediately clear that im r ⊂ im t. conversely, let v ∈ im t then v = tw and rv = rtw = tltw = tw = v, 148 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) i.e. im t ⊂ im r. to prove the relation (iii) we have to argue as follows: let u ∈ ker l, then u = ptru + tlu = ptru ∈ im ptr. on the other hand it follows from u ∈ im ptr that u = ptrv with v ∈ y and u = ptrv = ptrptrv + tlu = ptrv + tlu, which leads to tlu = 0 and because of the injectivity of the operator t : x → im t we conclude lu = 0, i.e. u ∈ ker l. # definition 1. elements u ∈ ker l∩y are called l-holomorphic. the operator l is called algebraic derivative.the operator ptr is called the initial projection and the operator t is denoted as general teodorescu transform. from the point of view of a general operator theory t is also called algebraical integral. corollary 2. set pr := trp : imtr ∩ y → z and qr := i − pr. the following properties are valid: (i) the operators pr,qr are idempotent, i.e. we have p 2 r = pr and q 2 r = qr and furthermore qrpr = prqr = 0. (ii) an element ξ ∈ z is the generalized trace of an element u from kerl if and only if prξ = ξ. (iii) we have qrξ = trtlu. proof. (i). it is sufficient to show p 2 r ξ = trptrpξ = trpξ = prξ, with ξ ∈ z. in order to prove (ii) let ξ = tru ∈ z and u ∈ ker l. then we have u = ptru + tlu = ptru = pξ. it now follows ξ = tru = trp ξ = prξ. conversely, let us assume ξ = prξ, then tru = ξ = prξ = trpξ = trptru. on the other hand theorem 1 yields tru = trptru + trtlu. hence trtlu = 0. because of imt ∩kertr = {0} follows tlu = 0 and such lu = 0, i.e. u ∈ ker l. for (iii) we have tru = trptru + trtlu. therfore, it holds trtlu = tru − trptru = ξ − trp ξ = ξ − prξ = qrξ. # denotation the operators pr,qr are called general plemelj projections. cubo 11, 1 (2009) on a new notion of holomorphy and its applications 149 remark 2. the condition imtl ∩ kertr = {0} can be seen as a very general formulation of a maximum principle. 3 types of l–holomorphy 3.1 l-holomorphy in r1 a trivial example is given by consideration of all functions u ∈ c1[0, 1] with l := d dt , t := t∫ 0 ·dτ , p := i and tr : c1[0, 1] → r1 with tru = u(0). then we get the well-known mean-value theorem: u(t) = u(0) + t∫ 0 u̇(τ)dτ = ptru + tlu. this is just the main-theorem of differential-integral calculus. the class of all l-holomorphic functions consist of all real constants. also a slightly modification of the trace operator and the generalized teodorescu transfrom does not change the triviality of the class of l-holomorphic functions. indeed, let u ∈ c1[0, 1], take l := d dt , p := i and tru := 1 2 [u(0) + u(1)], then (tu)(t) := t∫ 0 u(τ)dτ − 1 2 1∫ 0 u(τ)dτ . because of imptr = ker l we have again the space of all constants for the class of l-holomorphic functions. by using the so-called riemann-liouville integral of order α (cf. [14],[9]) we obtain a more interesting example. for this reason let u ∈ c[0, 1], 0 < α < 1. we consider the absolut continuous function (iα a+ u)(t) := 1 γ(α) t∫ 0 1 (t − τ)1−α u(τ)dτ , which has almost everywhere a derivative in l1[0, 1]. take now (lu)(t) := 1 γ(1 − α) d dt (i 1−α a+ u)(t) , (tu)(t) := (iα a+ u)(t) 150 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) and with n = [α] + 1 (ptru)(t) := n−1∑ k=0 (t − a)α−k−1 γ(α − k) dn−k−1 dtn−k−1 i n−α a+ u(t). (iα a+ u)(t) is called riemann-liouville fractional integral and (dα a+ u)(t) is denoted by riemann– liouville fractional derivative. the main-value theorem holds again. 3.2 notions of holomorphy in the complex plane the original notion of the holomorphy forms in natural way a class of l-holomorphic function. we have only to set l := ∂z . in more detailed we have the following: let g ⊂ c be a bounded domain with sufficient smooth boundary curve then the mean-value formula is written as 1 2πi ∫ γ u(t) t − z dt − 1 2πi ∫ g 1 t − z (∂u)(t)dξ dη = { u(z) , z ∈ g 0 , z ∈ c \ g . we have only to identify l := ∂ = 1 2 (∂ξ + i∂η) (tξ + iη) , t := − 1 2πi ∫ g 1 t − z · dξ dη , p := 1 2πi ∫ γ 1 t − z · dγt . the trace operator tr is defined as non-tangential limit from inner points tending to the boundary γ. remark 3. it is quite curious that the initial projection acts on the boundary. it seems that ”initial values” are ”smudged” over the surface. another example in the complex plane can be given by l := ∂ , (t ·)(z) = − 1 2πi ∫ g [ 1 t − z − 1 t + z ] · dξ dη and (p ·)(z) = − 1 2πi ∫ γ [ 1 t − z − 1 t + z ] · dγt . the trace operator is definded as before. this model goes back to j. ryan (cf. [8]). cubo 11, 1 (2009) on a new notion of holomorphy and its applications 151 3.3 l-holomorphy models generated by matrices a further model for l-holomorphy is given by: let {ei} n i=1 be a family of orthogonal matrices of order n with entries 0, 1,−1 as well as the property e∗ i ej + e ∗ j ei = 0 (i 6= j) furthermore, set e(a) = n∑ i=1 eiai , a = (a1, ...,an) t and e∗(a) = n∑ i=1 e∗ i ai and ∇ = (∂1, ...,∂n) t . take l := d(∇) , t := 1 σn ∫ g d ∗ (y−x) |y−x|n · dy and p := −1 σn ∫ γ d ∗ (y−x) |y−x|n · dγy then it holds (pu)(x) + tl(∇)u(x) = { u(x) , x ∈ g 0 , x ∈ rk \ g . here σn denotes the area of the n-dimensional unit sphere. (cf. [13],[15]). 3.4 dzuraev’s model also dzuraev’s model from 1982 [3] is worthy of being mentioned: let u := (u1,u2), z = x2 + ix3 and ∂ ∂z := 1 2 ( ∂ ∂x2 + i ∂ ∂x3 ) , y = y2 + iy3. further, let ∂x = ( ∂ ∂x1 2 ∂ ∂z −2 ∂ ∂z ∂ ∂x1 ) , e(y − x) = −1 |y − x|3 ( y1 − x1 −(y − z) y − z y1 − x1 ) and n(y) = ( n1 n2 − in3 −(n2 + in3) n1 ) . then take l := ∂x , t := 1 σ3 ∫ g e(y − x) · dy and p := 1 σ3 ∫ γ e(y − x)n(y) · dγy . the trace operator tr means in both cases the non-tangential limit to the boundary γ from inside of g. 4 quaternionic holomorphic functions real quaternions: the algebra of real quaternions h is defined by the basis elements e0 = 1 , e1,e2,e3, 152 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) which obey the arithmetic rules: e2 0 = 1 , e1e2 = −e2e1 = e3 , e2e3 = −e3e2 = e1 , e3e1 = −e1e3 = e2 . each quaternion a ∈ h permits the representation a = 3∑ k=0 akek (ak ∈ r ; k = 0, 1, 2, 3) . addition and multiplication in h turn it into a non-commutative number field. the main-involution in h is called quaternionic conjugation and defined by e0 = e0 , ek = −ek (k = 1, 2, 3) . which can be extended onto h by r-linearity. therefore we have a = a0 − 3∑ k=1 akek = a0 − a. note that aa = aa = 3∑ k=1 a2 k =: |a|2 h . if a ∈ h \ {0} then the quaternion a−1 := a |a|2 is the inverse to a. for a,b ∈ h we have abba. complex quaternions: the set of complex quaternions, which we also need, is denoted by h(c) and consist of all elements of the form a = 3∑ k=0 akek (ak ∈ c ; k = 0, 1, 2, 3) . by definition we state: iek = eki, k = 0, 1, 2, 3. here i denotes the usual imaginary unit in c. elements of h(c) can also be represented in the form a = a1 + ia2 (ak ∈ h; k = 1, 2). notice that the quaternionic conjugation acts only on the quaternionic units and not on the pure complex number i. cubo 11, 1 (2009) on a new notion of holomorphy and its applications 153 let x = wk p (g),y = wk+1 p (g),z = w k−(1/p)+1 p (γ); k = 0, 1, 2, ...; 1 < p < ∞. further, let l := d = 3∑ i=1 ∂iei (dirac operator (mass zero)), (tu)(x) := − 1 σ3 ∫ g e(x − y)u(y)dy (teodorescu transform), (pu)(x) := (fγu)(x) = 1 σ3 ∫ γ e(x − y)n(y)u(y)dγy (cauchy − fueter operator), (tru)(ξ) := n.t. − lim z→ξ∈γ z∈g u(z), with e(x) = d 1|x| and n = ∑ 3 i=1 eini the outward pointing unit vector of the normal. the class of l-holomorphic functions are just the solutions of the mosil–teodorescu system. we now consider so called dirac operators with mass. we will use the same spaces as above. then the general operators l,t and p are given by l := d + iα (dirac operator with mass), (tu)(x) := − 1 σ3 ∫ g eiα(x − y)u(y)dy (teodorescu type transform), (pu)(x) := 1 σ3 ∫ γ eiα(x − y)n(y)u(y)dγy (cauchy − fueter − typeoperator), (tru)(ξ) := n.t. − lim z→ξ∈γ z∈g u(z). for the description of the kernel function of this new teodorescu transform we have to use besselfunctions of third kind so called macdonald functions. we have eiα(x) := − ( iα 2π )(3/2) [ |x|−1/2k3/2(iα|x|)ω − k1/2(iα|x|) ] , where ω ∈ s2 and kµ(t) denotes. 5 discrete quaternionic holomorphic functions one advandage of our notion of l-holomorphy is its applicability also on lattices. we will present a calculus which was obtained by k. guerlebeck in 1988 [4] (cf. also [6]). for this reason we have to represent the domain on the lattice and to define what are inner and outer points relatively to the ”discrete boundary” and to say what the discrete boundary means. this boundary has to 154 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) approximate the original domain. it is necessary to disdinguish between a right and a left parts of the boundary. the approximating discrete domain is here always an axes–parallel polyeder with side faces, edges and corner points. more exactly holds r 3 h := {(ih,jh,kh) : i,j,k integer, h > 0}, gh := g ∩ r 3 h , γh := {x ∈ gh : dist(x, cogh) ≤ √ 3h}. let v ± i,h x the translation of x by ±h in xi-direction, then γh,ℓ(r) := {x ∈ γh : ∃i : v ± i,h x /∈ gh} (left(right) side planes), γh,ℓ(r);i := {x ∈ γh : v ± i,h x /∈ gh}, γh,ℓ(r);i,j := γh,ℓ(r);i ∩ γh,ℓ(r);j (left(right) edges), γh,ℓ(r);i,j,k := γh,ℓ(r);i,j ∩ γh,ℓ(r);k (left(right) corners). let be x = w 1 2,h (gh), y = l2,h(gh), z = w 1 2 2,h (gh). then (lu)(x) := (d ± h u)(x) = 3∑ i=1 ei[u(v ± i,h x) − u(x)] 1 h (discr. dirac operator), (tu)(x) := (t ± h u)(x) (discrete teodorescu transform) =   ∑ intgh∪γh,ℓ(r) + ∑ left(right) corners − ∑ left(right) edges  e± h (x − y)u(y)h3, where e± h are the discrete fundamental solutions of d± h . the discrete cauchy–fueter operator is introduced as follows (pu)(x) := (f ± h u)(x) = 3∑ i=1  − ∑ si + ∑ sij − ∑ sijk  e± h (x − v ∓ i,h y)n(y)u(y)h2 + 3∑ i=1 ∑ y∈γh,ℓ(r);m,j,k m 6=j 6=k h±(x − y)eiu(y)h 2, where si = γh,ℓ;i ∪ γh,r;i, sij := γh,ℓ;j − v + i,h γh,ℓ, sijk := γh,ℓ;j,k − v + i,h γh,ℓ;i,k. the corresponding mean value formulae are given as follows u(x) = (f ± h u)(x) + t ± h d ± h u(x) much more complicated is to find a suitable discrete fundamental solution, which is given by eh(x) as solution of a suitable difference equation −∆heh(x) = − 3∑ i=1 d − i,h d + i,h eh(x) = δh(x) = { h−3,x = 0 0,x ∈ r3 h \ {0} cubo 11, 1 (2009) on a new notion of holomorphy and its applications 155 expressed by using the fourier-transform we have eh(x) = 1 √ 2π 3 rhf ( 1 d2 ) . the function d is defined as follows d2 = 4 h2 ( sin 2 hξ1 2 + sin 2 hξ2 2 + sin 2 hξ3 2 ) and rhu is the restriction of the continuous function u onto the lattice r 3 h . we have |eh| ≤ c|x| m with a certain m > 0 depending on the properties of the difference operator e ± h (x) := d ∓ j,h eh(x). 6 l-holomorphy on the sphere meanwhile is also existing the notion of holomorphy on the sphere. a good reference is doctoral thesis of p. van lancker [17] the following operators has to be used γs + α α ∈ c \ n ∪ (−n). lα : = ω(γs + α) (günter’s gradient), tα : = − ∫ ω eα(ω,ξ) · ds(ω) (teodorescu transform), pc,α : = − ∫ −c eα(ω,ξ)n(ω) · dc(ω) (cauchy-fueter type operator). a corresponding borel-pompeiu formula is given by pc,αu + tαdαu = { u in ω 0 in s \ ω . we will consider the fundamental solution of günter’s gradient. let α ∈ c \ n ∪ {−2 − n}. then eα(ω,ξ) = π σ3 sin πα kα(−ξ,ω)ω, where σ3 is the surface area of the unit sphere. further, we define kα(−ξ,ω)ω = c 3/2 α (ω · ξ) + ξωc 3/2 α−1(ω · ξ), with the so-called gegenbauer polynomials cµ α (t). using kummer’s function 2f1(a,b; c; z) we get the representation c3/2 α (z) = γ(α + 3) γ(α + 1) 1 4 2f1(−α,α + 3; 2; 1 − z z ) z ∈ c \ {−∞, 1}. 156 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) kummer’s function is for |z| < 1 defined by 2f1(a,b; c; z) := ∞∑ k=0 (ak)(ak) (c)k zk k! , (a)k = γ(α + k) γ(α) . solutions of dαu = 0 in ω are called inner spherical holomorphic functions of order α in ω. we have dαeα(ω,ξ) = δ(ξ − ω) . a good reference for this topic is [1]. further we introduce a singular integral operator of bitzadse’s type (sc,αu)(ξ) : = 2 lim ε→0 ∫ c\bε(ξ) eα(ω,ξ)n(ω)u(ω)ds(ω) = 2v.p. ∫ c eα(ω,ξ)n(ω)u(ω)ds(ω). one can prove the algebraical identity s2 c,α = i. let ω+ := ω , ω− := coω. applying the general trace operator as non-tangential limit on the sphere towards the boundary c we get plemeljsokkotzkij-type formulae. n.t. − lim t→ξ t∈ω± (fc,αu)(t) 1 2 [±i + sc,α]u(ξ) =: { pc,αu(ξ), t ∈ ω + −qc,αu(ξ), t ∈ ω − . the operators qc,α := 1 2 [i − sc,α], pc,α : 1 2 [i + sc,α] are called plemelj projections. the space l2(γ) is now decomposed into the hardy spaces l2(c) = hs α (ω + ) ⊕ hsα(ω−) ↑ ↑ pc,α qc,α (cf. [12]). 7 taylor type formula using ideas of the theory of right invertible operators (cf. d. przeworska-rolewicz, [10]) one has with ym = d(l m ) ⊂ y (m is a natural number) the operators lj : ym → xm−j, p : zm−j → ym−j, ptr : ym−j → ym−j, tj : xm−j → ym (0 ≤ j ≤ m − 1). here we have ym ⊆ . . . ⊆ y2 ⊆ y1 and l 0 = t 0 = i. cubo 11, 1 (2009) on a new notion of holomorphy and its applications 157 proposition 1. the following properties are fulfiled (i) the operators tjptrlj (0 ≤ j ≤ m − 1) are projections on ym. (ii) the projections tjptrlj (0 ≤ j ≤ m−1) are complementary on ym, i.e. (t jptrlj)(tkptrlk) = (tkptrlk)(tjptrlj) = 0 for all 0 ≤ j,k ≤ m − 1 and k 6= j. proof. (i) indeed, using the assumption ptrp = p and corollary 1 we obtain (tjptrlj)(tjptrlj) = tjptrljtjptrlj = tjptrptrlj = tjptrlj, i.e. tjptrlj are projections on ym. to prove property (ii) we also use corollary 1. it is immediately clear that ljtj = i from lt = i. because of ptrt = 0 and ljtj = i follows for j < k: (tjptrlj)(tkptrlk) = tjptrljtkptrlk = tjptrtk−jptrlk = 0, i.e. (tjptrlj)(tkptrlk) = 0 (0 ≤ j < k ≤ m). taking into account relation in the corollary from above, the commutative property is obtained. indeed, from property lptr = 0 we have (tkptrlk)(tjptrlj) = tkptrlktjptrlj = tkptrlk−jptrlj = 0, i.e. (tkptrlk)(tjptrlj) = 0 (0 ≤ j < k ≤ m). hence all tjptrlj(0 ≤ j ≤ m) are complementary on ym. # then the next corollary is clear. corollary 3. the operator pm := m−1∑ j=0 tjptrlj = t 0ptrl0 ⊕ t 1ptrl1 ⊕ . . . ⊕ tm−1ptrlm−1 is a projection on ym−1. corollary 4. the operators pm,t m and lm have the following relations (i) the operator tm is the right-inverse to the operator lm, i.e. lmtm = i. (ii) the operators lm,pm satisfy the property l mpm = 0. 158 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) (iii) it holds pmt m = 0. proof. the relation (i) is simple to be obtained from corollary 1. to prove (ii), one use assumption lptr = 0 and ljtj = i for 0 ≤ j ≤ m − 1 as mentioned above then lmpm = m−1∑ j=0 lmtjptrlj = m−1∑ j=0 lm−jptrlj = 0. the same for relation (iii) with assumption ptrt = 0: pmt m = pm := m−1∑ j=0 tjptrljtm = pm := m−1∑ j=0 tjptrtm−j = 0. theorem 2. (the taylor type formula) let l be a right invertible operator that defined from an injection t and an initial operator p. then for m = 1, 2, ... the following identity holds on ym u = m−1∑ j=0 tjptrlju + tmlmu. proof. we have ker tm = {0} by assumption t is an injection and im tm ⊂ ym = d(l m ). corollary 3 shows that pm is a projection and pmt m = 0. furthermore, it is simple to show that im tm ∩ im pm = {0}. indeed, let u ∈ im t m ∩ im pm then u = pmv = t mw, (v ∈ ym−1,w ∈ x). since pmt m = 0 we get u = pmv = pmpmv = pmt mw = 0. let b be the (unique) right inverse to tm then (from the mean value formula) u = pmu + t mbu with d(b) := imtm ⊕ impm. now we will show that lm also satisfies above formula. by applying the mean value formula for lju we get lju = ptrlju + tlj+1u (0 ≤ j ≤ m − 1) rewrite in more detail and acting operators tj (0 ≤ j ≤ m − 1) to both sides we have t 0l0u = t 0ptrl0u + tlu, tlu = tptrlu + t 2l2u, · · · tm−1lm−1u = tm−1ptrlm−1u + tmlmu. cubo 11, 1 (2009) on a new notion of holomorphy and its applications 159 sum up all equabilities we obtain u = t 0l0u = t 0ptrl0u + tptrlu + . . . + tm−1ptrlm−1u + tmlmu = pmu + t mlmu. then the property of uniqueness of right inverse operator leads to b = lm. this completes the proof of our theorem. example 15. (realisation in r1) we continue the first example in section 3.1.for all functions u ∈ c1[0, 1], recall that l := d dt , t := t∫ 0 ·dτ , p := i and tr : c1[0, 1] → r1 with tru = u(0). then we have tjptr(lju)(t) = (lju)(0) tj j! and (tmu)(t) = t∫ 0 (t − τ)m−1 (m − 1)! u(τ)dτ . hence the theorem 2 yields the classical taylor’s formula u(t) = m−1∑ j=0 (lju)(0) tj j! + t∫ 0 (t − τ)m−1 (m − 1)! (lmu)(τ)dτ . example 16. (taylor formula for fractional operators) in [9] j.d. munkhammar gave taylor’s formula based on fractional caculus. let u(t) ∈ c1([a,b]) then the riemann-liouville fractional integral of order α is (tu)(t) := iα a+ u(t) = 1 γ(α) t∫ a u(s) (t − s)1−α ds , and the riemann–liouville fractional derivative of order α as follow (lu)(t) := dα a+ u(t) = 1 γ(1 − α) d dt t∫ a u(s) (t − s)α ds 160 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) where α ∈]0, 1[ and γ is a well known gamma function. hence dα a+ iα a+ = i. let α > 0, m ∈ z+ and u(t) ∈ c[α]+m+1([a,b]), the taylor formula is u(t) = m−1∑ k=−m d α+k a+ u(t0) γ(α + k + 1) (t − t0) α+k + i α+m a+ d α+m a+ u(t) for all a ≤ t0 < t ≤ b. 8 taylor-gontcharov’s formula for high order genaralized dirac operators corollary 5. (the taylor-gontcharov’s formula) a generalization of the taylor formula leads to u = m−1∑ j=0 t0t1...tjpjlj...l1l0u + t1...tmlm...l1u with l0 = t0 = i. example 17. (realisation on a lattice) let gh be the lattice of the bounded domain g and ∆h = d + h d − h be the discretized laplace operator. we consider the following problem ∆hu = f on gh, trγpγhu = g0 on γh, trγhd − h u = g1 on γh. γh is the "‘numerical"’ boundary of g for a meshwidth h. the unique solution is then given by u = f − h g0 + t − h f + h (trγht − h f + h ) −1t− h d − h g1 + t − h qht + h f with bergman projection ph = f + h (trγht − h f + h ) −1trγht − h the operators in taylor-gontcharov’ s formula are chosen as follows l1 := d − h , l2 := d + h , p1 := f − h , p2 := f + h , t1 := t − h , t2 := t + h received: april 2008. revised: august 2008. cubo 11, 1 (2009) on a new notion of holomorphy and its applications 161 references [1] delanghe, r., sommen, f., soucek, v., clifford algebra and spinor valued functions, kluwer, dordrecht. (1992). [2] delvos, f.j. and schempp w., boolean methods in interpolation and approximation, longman higher education division, wiley & sons inc. new york. (1990). [3] dzuraev, a.d., on the moisil-teodorescu system. in: begehr, h. jeffrey, a. (eds) partial differential equations with complex analysis. pitman res. notes math. ser. 262: 186–203. (1982). [4] gürlebeck k., grundlagen einer diskreten räumlich verallgemeinerten funktionentheorie und ihrer anwendungen, habilitation, tu chemnitz. (1988). [5] guerlebeck k., habetha k. and sproessig w., holomorphic functions in the plane and n-dimensional space, birkhauser, basel. (2008). [6] gürlebeck k. and sprößig w., quaternionic analysis and elliptic boundary value problems, birkhäuser, basel. (1990). [7] k. gürlebeck and sprössig w., quaternionic and clifford calculus for physicists and engineers , john wiley, chichester. (1997). [8] gürlebeck, k., kähler, u., ryan, j., sprößig, w., clifford analysis over unbounded domains, advances in applied mathematics 19, (1997), 216 239. [9] munkhammar, j.d. (2004) fractional calculus and the taylor series, project report, department of mathematics, uppsala university, uppsala. [10] d. przeworska-rolewicz., algebraic theory of right invertible operators, study mathematica, t. xlviii. (1973). [11] ryabenskij v.s., the method of difference potentials for some problems of continuum mechanics. moscow, nauka(russian). (1987). [12] ryan, j., plemelj projection operators over domain manifolds, mathematische nachrichten 223, (2001), 89-102. [13] saak, e.m., on the theory of multidimensional elliptic systems of first order, sov.math. dokl. vol. 18,no. 3. (1975). [14] samko, s.g., kilbas, a.a., marichev, o.l., fractional integral and derivatives: theory and applications, gordon and breach, amsterdam, (1993). [15] sprössig, w., analoga zu funktionentheoretischen sätzen im rn, beiträge zur analysis 12, (1978), 113–126. 162 wolfgang sproessig and le thu hoai cubo 11, 1 (2009) [16] tasche m., eine einheitliche herleitung verschiedener interpolationsformeln mittels der taylorschen formel der operatorenrechnung, zamm 61, (1981), 379–393. [17] van lancker, p., clifford analysis on the unit sphere, thesis university of ghent. (1997). articulo 6.dvi cubo a mathematical journal vol.12, no¯ 02, (77–96). june 2010 generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes s. albeverio institut für angewandte mathematik, universität bonn, wegelerstr. 6, d-53115 bonn, germany sfb 611,hcm, bonn, bibos, bielefeld bonn, cerfim, locarno and usi (switzerland) email: albeverio@uni-bonn.de and ya. belopolskaya st.petersburg state university for architecture and civil engineering, 2-ja krasnoarmejskaja 4, 190005, st.petersburg, russia email: yana@yb1569.spb.edu abstract we reduce the construction of a weak solution of the cauchy problem for the navier-stokes system to the construction of a stochastic problem solution. under suitable conditions we solve the stochastic problem and prove that simultaneously we obtain a weak (generalized) solution to the cauchy problem for the navier-stokes system. resumen nosotros reducimos la construcción de una solución débil de un problema de cauchy para el sistema de navier-stokes para la construcción de la resolución de un problema estocástico. bajo condiciones convenientes resolvimos el problema estocástico y probamos que simultáneamente obtenemos una solución débil (generalizada) para el problema de cauchy del sistema de navier-stokes. 78 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) key words and phrases: stochastic flows, diffusion processes, nonlinear parabolic equations, cauchy problem. ams subj. class.: 60h10, 60j60 , 35g05, 35k45 1 introduction the main purpose of this article is to construct both strong and weak solutions (in certain functional classes) of the cauchy problem for the navier-stokes (n-s) system ∂u ∂t + (u,∇)u = ν∆u − ∇p, u(0,x) = u0(x), x ∈ r3 (1.1) div u = 0. (1.2) here u(t,x) ∈ r3,x ∈ r3, t ∈ [0,∞) is the velocity of the fluid at the position x at time t and ν > 0 is the viscosity coefficient. p(t,x) is a scalar field called the pressure which appears in the equation to enforce the incompressibility condition (1.2). there exists a number of papers [1] – [4] and others where the system (1.1), (1.2) was treated from the probabilistic point of view on the base of stochastic models. in particular in our previous paper [1] the system (1.1), (1.2) was reduced to a probabilistic problem presented in the form of the following system of equations dξ(τ) = −u(t − τ,ξ(τ))dτ + σdw(τ), (1.3) u(t,x) = e0,x[u0(ξ(t)) + ∫ t 0 ∇p(t − τ,ξ(τ))dτ] (1.4) p(t,x) = 2e[ ∫ ∞ 0 γ(t,x + b(t))dt] = 2e[ ∫ ∞ 0 tr[∇u]2(t,x + b(t))dt]. (1.5) here σ = √ 2ν, w(t) and b(t) are independent standard wiener processes valued in r3, tr[∇u]2 = ∑3 i,k=1 ∇iuk∇kui. it was shown in [1] that if the initial value u0 is a c3function the functions u(t,x),p(t,x) given by (1.4), (1.5) are c2+α solutions of (1.1), (1.2) for 0 < α < 1. in the present paper we consider an alternative probabilistic system which allows to construct a weak (distributional) solution to (1.1), (1.2). the approach developed here is based on the theory of stochastic flows due to kunita [5], [6] and the results due to belopolskaya and dalecky [7], [8]. the article is composed as follows. in section 2 we give some preliminary information concerning different analytical approaches to the notion of a solution of the navier-stokes system. here we recall some common ways to exclude the pressure and to obtain a closed equation for the velocity, introduce necessary functional spaces and state various notions of solutions to (1.1), (1.2). in section 3 we state our approach and prove main results. in the last section we compare our approach and results with the euler-lagrange approach to incompressible fluids developed by constantin and iyer [9],[10]. 2 preliminaries within a classical approach to the n-s system one excludes the pressure from (1.1),(1.2) and investigates the resulting nonlinear pseudo-differential equation. to this end first one can derive formally cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 79 from (1.1),(1.2) the relation −∆p(t,x) = γ(t,x), (2.1) where γ(t,x) = 3 ∑ k,j=1 ∇kuj∇juk = tr[∇u]2 = ∇ · ∇ · u ⊗ u. (2.2) then given an r3-valued vector field u(t) over r3 the operator p is defined by pu(t) = u(t) − ∇∆−1∇ · u(t). (2.3) here and below u · v denotes the inner product in r3 of the vectors u and v. the map p called the leray projection is a projection of the space l2(r3) ≡ l2(r3)3 of square integrable vector fields to the space of divergence free vector fields. since the formal solution of the poisson equation (2.1) is given by p = ∆−1γ = ∆−1∇ · ∇ · u ⊗ u (2.4) one can present ∇p in the form ∇p = ∇∆−1∇ · ∇ · u ⊗ u keeping in mind that divu = 0. substituting this expression for ∇p into (1.1) one obtains the following cauchy problem ∂u ∂t = ν∆u − p∇ · (u ⊗ u), u(0) = u0. (2.5) when (2.5) is solved then the pressure is reconstructed from the poisson equation (2.1). the leray projection p is used to solve the n-s system both in numerous analytical papers (see, e.g., [11] for references) and in papers where the n-s system is studied from the probabilistic point of view [2],[3], [10]. in this paper we avoid the direct application of the leray projection and construct the solution of (1.1), (1.2) via stochastic processes associated with (1.1) and (2.1). to give a rigorous definition of a solution for the n-s system we have to specify the required functional spaces. let d = d(r3) = c∞c denote the space of all infinitely differentiable real valued functions on r3 with compact support equipped with the schwartz topology and let d′ be its topological dual. let 〈φ,ψ〉 = ∫ r3 φ(x)ψ(x)dx denote the natural coupling between φ ∈ d and ψ ∈ d′. if it will not lead to misunderstandings we will use the same notation for vector fields u and v as well, that is 〈h,u〉 = ∫ r3 3 ∑ k=1 hk(x)uk(x)dx. let d((0,t )×r3) = (d′((0,t )×r3))3 denote the space of r3-valued vector fields h with components hk ∈ d and d′ denote the space dual to d(r3). the leray weak solution of the n-s system on [0,t ] × r3 is a vector field u(t,x) in (d′((0,t ) × r3))3 such that u is locally square integrable on (0,t ) × r3, satisfies div u = 0 and there is a distribution p ∈ d′((0,t ) × r3) such that ∂u ∂t = ν∆u − ∇ · (u ⊗ u) − ∇p, lim t→0 u(t) = u0 (2.6) 80 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) holds in the sense of distributions. the kato mild solution is a solution u to the following integral equation u(t) = et∆u0 − ∫ t 0 e(t−s)∆p∇ · (u ⊗ u)(s)ds. (2.7) note that instead of looking for u(t,x) and p(t,x) one may look for their fourier images û(λ) = (2π)− 3 2 ∫ r3 e−iλ·xu(x)dx. the leray and kato approaches stated in original terms and in terms of the fourier transformation of the navier-stokes system were developed in a number of papers (see, e.g., references in the book by lemarie-rieusset [11]). below we will need as well the following functional spaces: the space c(rd,rn) of bounded continuous functions mapping rd to rn, the space c(r3,r1) = c(r3) of bounded continuous real functions f with the norm ‖f‖∞ = supx∈r3|f(x)|; the space c(r3) of bounded continuous vector functions with the norm ‖u‖∞ = supx∈r3‖u(x)‖, where ‖ · ‖ is the norm in r3; the space c0(r 3) of continuous vector functions with compact supports; the banach space lq(r3) of integrable functions f with norm ‖f‖q = ( ∫ r3 ‖f(x)‖qdx) 1 q ; the space ck(r3) of k-times differentiable functions with the norm ‖g‖ck = ∑ |β|≤k ‖dβg‖∞; the space ck,α(r3) (for a natural number k) of vector fields whose k-th derivatives are hölder continuous with exponent α, 0 < α ≤ 1 with norm ‖g‖ck,α = ‖g‖ck + [g]k+α and [g]k+α = ∑ |β|=k sup x,y∈r3 |dβg(x) − dβg(y)| |x − y|α . let z denote the set of all integers, and suppose that k ∈ z is positive and 1 < q < ∞. denote by wk,q = wk,q(r3) the set of all real functions h defined on r3 such that h and all its distributional derivatives ∇αh of order |α| = ∑ αj ≤ k belong to lq(r3). it is a banach space with norm ‖h‖k,p = ( ∑ |α|≤k ∫ r3 |dαh(x)|qdx) 1 q . (2.8) denote by w k,q 0 the subspace of functions from w k,q = wk,q(r3) with compact supports. finally we will need some spaces of locally integrable functions. let g ⊂ r3 be a bounded domain, p be a positive integer and f : g → r1 be a lebesgue measurable function. the set of functions {f : ∫ k |f(x)|pdx < ∞ for all compact subsets k ⊂ g} is denoted by lp loc and called a space of plocally integrable functions. although l p loc (g) are not normed spaces they are readily topologized. namely a sequence {un} converges to u in lploc(g) if {un} → u in lp(k) for each open k having compact closure in g and ‖u‖p,loc = ( ∫ k ‖u(x)‖pdx) 1 p < ∞. in a natural way one can define the spaces wk,q and l p loc (g) of vector fields with components in wk,p and in l p loc (g). cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 81 3 a probabilistic approach to the navier-stokes system let us come back to the navier-stokes system written in the form ∂u ∂t + (u,∇)u = σ 2 2 ∆u − ∇p, u(0,x) = u0(x), x ∈ r3 (3.1) −∆p = γ (3.2) with γ defined by (1.3). our main purpose in this section is to construct a diffusion process that allows us to obtain a a weak solution to (3.1), (3.2) via its probabilistic representation. to be more precise we intend to reduce the system (3.1), (3.2) to a certain system of stochastic equations and to construct its solution. then we have to verify that in this way we have constructed a weak solution of (3.1), (3.2). as above let w(t),b(t) be standard r3-valued independent wiener processes defined on a probability space (ω,f,p). given a bounded measurable function f(x) and a stochastic process ξ(t) we denote es,xf(ξ(t)) ≡ ef(ξs,x(t)) the conditional expectation under the condition ξ(s) = x. given a function g(t,x) ∈ r3, a smooth (in x) function q(t,x) ∈ r1, t ∈ (0,∞),x ∈ r3 and a constant σ we consider stochastic processes ξg(t) and λ(t) satisfying the stochastic equations dξgy(t) = g(t,ξ g y(t))dt − σdw(t), ξgy(0) = y ∈ r3, λ(t) = u0 − ∫ t 0 ∇q(τ,φg0,τ )dτ, (3.3) where φ g 0,t denotes the stochastic map in r 3 generated by the process ξg(t), φ g 0,t(y) = ξ g y(t). the map φ g 0,t : r 3 → r3 is called a stochastic flow. the processes ξg(t) and λ(t) are auxiliary ones. the main role in our considerations is played by the stochastic flow ψt,0 which is an inverse flow to φ0,t, ψt,0(φ0,t(y)) = y. to construct the flow ψt,0 we need the process ŵ(θ) = w(t − θ) − w(t) which is proved to be the standard wiener process. here we use the results of the kunita theory of stochastic flows [5],[6] and extend them to the case of stochastic processes associated with nonlinear pdes. actually we consider the closed system dψt,θ(x) = −u(θ,ψt,θ(x))dθ + σdŵ(θ), ψt,t(x) = x, (3.4) u(t,x) = e[u0(ψt,0(x)) − ∫ t 0 ∇p(τ,ψt,τ (x))dτ], (3.5) −2∇p(t,x) = e[ ∫ ∞ 0 1 τ γ(t,x + b(τ))b(τ)dτ], (3.6) where γ is given by (1.3) and look for a solution u(t,x),p(t,x),ψt,θ(x) of this system under some assumptions on the initial data u0 to be specified below. to construct the solution of (3.4)– (3.6) we consider its differential prolongation. namely, we consider the following formal relation dη x(θ) = −∇u(θ,ψt,θ(x))ηx(θ)dθ, ηx(t) = i, (3.7) 82 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) where i is the identity matrix acting in r3, and one of two formal relations for ∇u(t,x) ∇u(t,x) = e [ ∇u0(ψt,0(x))ηx(t) − ∫ t 0 ∇2p(τ,ψt,τ (x))ηx(τ)dτ ] . (3.8) or ∇u(t,x) = e [ ∇u0(ψt,0(x))ηx(t) − ∫ t 0 ∇p(τ,ψt,τ (x)) σ(t − τ) ∫ t τ ηx(θ)dŵ(θ)dτ ] . (3.9) note that to derive the second term in the right hand side of (3.9) we need a specific integration by parts formula called the bismut-elworthy-li formula [12]. since the system (3.3)– (3.8) is a closed system with respect to (ψt,0(x),η x(t),u(t,x),p(t,x),∇u(t,x)), we aim to prove the existence and uniqueness theorem for its solution. at the end we check that the functions (u(t,x),p(t,x)) given by (3.4)– (3.5) satisfy (3.1), (3.2). to construct the solution of (3.4)– (3.8) we consider a system of successive approximations and prove their convergence. set u 1(t,x) = u0(x), ψ 0 t,0(x) = x, p 1(t,x) = 0 (3.10) and consider stochastic processes ψkt,θ(x), vector fields u k(t,x) and scalar functions pk(t,x) given by the following relations dψkt,θ = −uk(θ,ψkt,θ)dθ + σdŵ(θ), ψkt,t = x, (3.11) uk+1(t,x) = e[u0(ψ k t,0(x)) − ∫ t 0 ∇pk+1(τ,ψkt,τ (x))dτ], (3.12) −2pk+1(t,x) = ∫ ∞ 0 e[γk(t,x + b(τ))]dτ, (3.13) where γk(t,x) = tr[∇uk]2(t,x)]. (3.14) finally, we consider η x,k t,θ , ∇uk+1(t,x) and ∇pk+1(t,x) defined respectively by dη x,k t,θ = −∇uk(θ,ψkt,θ)η x,k t,θ dθ, η x,k t,t = i, (3.15) and ∇uk+1(t,x) = e[∇u0(ψk+1t,0 (x))η x,k t,0 − ∫ t 0 ∇2pk+1(τ,ψkt,τ(x))η x,k t,τ dτ], (3.16) −2∇pk+1(t,x) = ∫ ∞ 0 1 τ e[γk(t,x + b(τ))b(τ)]dτ. (3.17) note that for k = 1 we have dψ 1 t,θ = −u0(θ,ψ1t,θ)dθ + σdŵ(θ), ψ1t,t = x, cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 83 that is we can solve the stochastic equation (3.4) independently on (3.5)-(3.6). then given the process ψ1t,0(x) and keeping in mind the properties of the function p 1 that satisfies the poisson equation −∆p1(t,x) = γ0(t,x), (3.18) we compute u1(t,x) from (3.12). next we compute ∇u1(t,x), ∇p1(t,x) from (3.16), (3.17) and proceed to k = 2. to prove the convergence of the successive approximations obtained in this way we need to derive some apriori estimates. let g(t,x) ∈ r3 be a given bounded lipschitz continuous function on [0,∞)×r3. set g(t,ψ(t,x)) = g(t) ◦ ψ(t)(x) for any functions ψ(t,x) ∈ r3). consider the stochastic equation dψ g t,θ = −g(θ) ◦ ψg t,θ dθ + σdŵ(θ), ψ g t,t(x) = x (3.19) and define the vector fields ug(t,x) and ∇pg(t,x) by ug(t,x) = e[u0(ψ g t,0(x)) − ∫ t 0 ∇pg(τ,ψgt,τ(x))dτ], (3.20) −2pg(t,x) = ∫ ∞ 0 e[γg(t,x + b(τ))]dτ, (3.21) where γg(t,x) = tr[∇g]2(t,x). (3.22) we derive formally from (3.21) by the integration by parts formula (bismut – elworthy – li formula [12]) that −2∇pg(t,x) = ∫ ∞ 0 e[ 1 τ γg(t,x + b(τ))b(τ)]dτ. (3.23) below we will describe the conditions on γ which justify (3.23). condition c 3.1 let g(t,x) ∈ r3 be a divergent free vector field depending on time and defined on [0,t ] × r3 for a certain constant t > 0. we assume that g(t) belongs to c1,α(r3), 0 < α ≤ 1 for a fixed t ∈ [0,t ] and satisfies the following estimates: 1. ‖g(t)‖q,loc ≤ ng(t) for some q to be specified below, ‖g(t)‖∞ ≤ kg(t) and ‖g(t,x) − g(t,y)‖ ≤ lg(t)‖x − y‖, ‖∇g(t,x) − ∇g(t,y)‖ ≤ l1g(t)‖x − y‖. 2. ‖∇g(t)‖∞ ≤ k1g(t), ‖∇g(t)‖r,loc ≤ n1g (t). here kg(t),lg(t), ng(t) and k 1 g(t),l 1 g(t),n 1 g (t) are positive continuous functions defined on an interval [0,t ] with t > 0, r = m and r = q for 1 < q < 3 2 < 3 < m < ∞. set ψg(τ) = ψ g t,τ(x) and consider the stochastic equation ψg(τ) = x − ∫ t τ g(τ1,ψ g(τ1))dτ1 + ∫ t τ σdŵ(τ1), (3.24) 84 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) with 0 ≤ τ ≤ t < t . when we are interested in the particular dependence of the process ψg(τ) on the parameters t,x we write ψg(τ) = ψ g t,x(τ) or ψ g(τ) = ψ g t,τ(x). lemma 3.1 assume that c 3.1 holds. then there exists a unique solution ψg(τ) of (3.24) and the following estimates e‖ψg(τ)‖2 ≤ 3[‖x‖2 + σ2(t − τ) + (t − τ) ∫ t τ [k2g(τ1)]dτ1], (3.25) e‖ψgt,x(τ) − ψ g t,y(τ)‖ ≤ ‖x − y‖e r t τ lg(θ)dθ, (3.26) e‖ψg(τ) − ψg1 (τ)‖ ≤ ∫ t τ ‖g(τ1) − g1(τ1)‖∞dτ1e r t τ lg(θ)dθ (3.27) hold. proof. the proof of the estimates of this lemma is standard and based on estimates of classical and stochastic integrals. we only show the proof of (3.26). in view of c 3.1 we have e‖ψgt,x(τ) − ψ g t,y(τ)‖ ≤ ‖x − y‖ + ∫ t τ lg(τ1)‖ψgt,x(τ1) − ψ g t,y(τ1)‖dτ1 where 0 ≤ τ ≤ t ≤ t with some constant t to be chosen later. finally, by gronwall’s lemma, we get e‖ψgt,x(τ) − ψ g t,y(τ)‖ ≤ ‖x − y‖e r t τ lg(θ)dθ. 2 along with (3.19)-(3.22) we need the equations for the mean square derivative ηg(t) = ∇ψgt,0(x) of the diffusion process ψ g t,0(x) that satisfies (3.19), and the gradient v(t,x) = ∇ug(t,x) of the function ug(t,x) given by (3.20). lemma 3.2 assume that c 3.1 holds. then the process ηx,g(τ) = ∇ψgt,τ(x) satisfies the stochastic equation dηx,g(τ) = −∇g(τ,ψx,gt,τ (x))ηx,g(τ)dτ, ηx,g(t) = i. (3.28) the process ηx,g(τ) possesses the following properties. the determinant det ηg(τ) is equal to 1, i. e. det ηg(τ) = jt,τ = 1 and e‖ηx,g(τ)‖ ≤ e r t τ k 1 g (θ)dθ (3.29) e‖ηx,g(τ) − ηy,g(τ)‖ ≤ c‖x − y‖ (3.30) with some positive constant c depending on t,τ and g. in addition the following integration by part formula is valid ∫ r3 f(ψ g t,x(τ))dx = ∫ r3 f(x)dx, f ∈ l1(r3). (3.31) proof. under c 3.1 the first statement immediately follows from general results of the stochastic differential equation theory. by direct computation one can check that jt,τ satisfies the linear equation djt,τ = −div g(ψgt,τ)jt,τdτ, jt,t = i cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 85 and since div g = 0 we get the second statement. besides ψ g t,τ is a c 1 stochastic diffeomorphism (see [5]) and hence the integration by part formula (3.31) holds. finally (3.29) is deduced from the inequality e‖ηx,g(τ)‖ ≤ 1 + ∫ t τ k1g(θ)e‖ηx,g(θ)‖dθ by the gronwall lemma. one can easily check that for the solution ηx,g(t) of (3.28) we have e‖ηx,g(τ) − ηy,g(τ)‖ ≤ ∫ t τ e‖∇g(θ,ψg t,θ (x)) − ∇g(θ,ψg t,θ (y))‖dθe r t τ k 1 g (θ)dθ ≤ e ∫ t τ l1g(θ)e‖ψ g t,θ (x)) − ψg t,θ (y))‖dθ and by (3.26) we derive (3.30). 2 let us state conditions on the initial data u0 of the n-s system. we say that c 3.2 holds when i) for some 0 < α ≤ 1 the initial vector field u0 ∈ c1+α0 (r3) satisfies the estimates ‖u0‖∞ ≤ k0, ‖∇u0‖∞ ≤ k10, ‖u0‖r,loc ≤ m0, ‖∇u0‖r,loc ≤ m10 with some positive constants k0,k 1 0,m0,m 1 0 and r. ii) u0 and ∇u0 are lipschitz-continuous with positive lipschitz constants l0 and l10 respectively. keeping in mind conditions c 3.1 and c 3.2 we derive estimates for ug(t) defined by (3.20) on a certain time interval [0,t ] and its gradient ∇ug(t,x). lemma 3.3 assume that g(t,x) satisfies c 3.1 and u0 satisfies c 3.2 with r = q and r = m for 1 < q < 3 2 < 3 < m < ∞. then the vector field ug(t,x) given by (3.20) satisfies the estimate ‖ug(t)‖∞ ≤ k0 + ∫ t 0 cqmk 1 g(τ)[‖∇g(τ)‖q,loc + ‖∇g(τ)‖m,loc]dτ. (3.32) under the conditions of this lemma the proof of (3.32) can be easily obtained by a direct computation from (3.20) using the estimates of the newton potential given in lemma 3.4 below. lemma 3.4 let g ⊂ r3 be a bounded domain and γg ∈ lq(g) ∩ lm(g) for some 1 ≤ q < 3 2 < 3 < m < ∞ and −∆p(t,x) = γg(t,x), x ∈ g. then ‖∇pg‖∞ ≤ cqm(‖γg‖q,loc + ‖γg‖m,loc) and ‖∇i∇jpg‖∞ ≤ c(‖γg‖q,loc + [γg]α). 3. let γg ∈ lr(g) for 1 < r < ∞.then pg ∈ w 2,r(r3) and the calderonzygmund inequality holds ‖∇2pg‖r,loc ≤ c1‖γg‖r,loc. 86 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) the proof of these estimates for a solution of the poisson equation can be found in the book by gilbarg and trudinger ([13] th 9.9). the probabilistic proof of some of these estimates can be found in [4]. 2 lemma 3.5 assume that the conditions of lemma 3.3 hold and ug(t,x) is given by (3.20). then the function ∇ug(t,x) admits a representation of the form ∇ug(t,x) = e[∇u0(ψgt,0(x))η x,g(t) − ∫ t 0 ∇2pg(τ,ψgt,τ (x))ηx,g(τ)dτ] (3.33) and the estimate ‖∇ug(t)‖∞ ≤ e r t 0 k 1 g (θ)dθ k 1 0 + ∫ t 0 e r t τ k 1 g (θ)dθ k 1 g(τ)[‖∇g(τ)‖q,loc + ‖∇g(τ)‖m,loc]dτ (3.34) holds for 1 < q < 3 2 < 3 < m < ∞, 0 ≤ t ≤ t . proof. the formal differentiation of (3.20) in x justified by c 3.1, c 3.2 and the results of lemma 3.4 yields (3.33). to verify the estimate (3.34) we use the above estimates for the process ηx,g(t) and the estimates of the newton potential derivative from lemma 3.4. hence we obtain ‖∇ug(t)‖∞ ≤ k10 ∫ t 0 k1g(θ)dθ+ (3.35) ∫ t 0 cqme r t τ k 1 g (θ)dθ[‖tr[∇g(τ)]2‖m,loc + ‖tr[∇g(τ)]2‖q,loc]dτ] that immediately leads to (3.34). 2 now we have to derive the estimate for the function ‖∇u(t)‖r,loc. lemma 3.6 assume that the conditions of lemma 3.3 hold. then for 1 < r < ∞ the gradient of the function ug(t,x) given by (3.20) satisfies the estimate ‖∇ug(t)‖r,loc ≤ e2 r t 0 k 1 g (θ)dθ [ ‖∇u0‖r,loc + c ∫ t 0 ‖∇g(τ)‖r,locdτ ] , (3.36) where 0 ≤ t ≤ t and c depends on r and t . proof. let us derive the lpestimate for ∇ug(t,x) given by (3.33). to derive the estimate for ‖∇ug(t)‖rr,loc = ∫ k ‖∇ug(t,x)‖rdx (where k is an arbitrary compact in g) we apply first the triangle inequality to obtain ‖∇ug(t)‖r,loc ≤ α1 + α2, where α1 = ( ∫ k e[‖∇u0(ψgt,0(x))ηx,g(t)‖r]dx ) 1 r , α2 = ( ∫ k ∫ t 0 e‖∇2pg(τ,ψgt,τ(x)))ηx,g(τ)‖rdτdx ) 1 r . to estimate α1 we apply the hölder inequality and take into account the inequality (3.29) for the process ηx,g(τ). besides we recall that ψt,τ(x) preserves the volume. as a result we have α1 ≤ ( ∫ k (e[‖∇u0(ψgt,0(x))‖2]e[‖ηx,g(t)‖2]) r 2 dx ) ) 1 r ≤ ‖∇u0‖r,loce r t 0 k 1 g (θ)dθ. cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 87 to derive the estimate for α2 we deduce from the calderon-zygmund inequality (see lemma 3.4) and the estimate of ηx,g(t) that αr2 ≤ cr ∫ t 0 e r τ 0 k 1 g (θ)dθk1g(τ) ∫ k ‖∇g(τ,x)‖rdxdτ. combining the above estimates for α1 and α2 we obtain the required estimate ‖∇ug(t)‖r,loc ≤ e r t 0 k 1 g (θ)dθ [‖∇u0‖r,loc+ cr ∫ t 0 e r τ 0 k 1 g (θ)dθ k 1 g(τ)‖∇g(τ)‖r,locdτ ] . finally we get ‖∇ug(t)‖r,loc ≤ e2 r t 0 k 1 g (θ)dθ [ ‖∇u0‖r,loc + c ∫ t 0 ‖∇g(τ)‖r,locdτ ] , where c depends on r and t. 2 theorem 3.7 assume that conditions c 3.1 and c 3.2 hold. then there exists an interval ∆1 = [0,t1] and functions α(t), β(t), κ(t) bounded for t ∈ ∆1, such that, if for all t ∈ ∆1, ‖g(t)‖∞ ≤ κ(t) and ‖∇g(t)‖∞ ≤ α(t), ‖∇g(t)‖r,loc ≤ βr(t) then the function ‖∇ug(t,x)‖ (where ug(t,x) is given by (3.20)) satisfies the estimates ‖ug(t)‖∞ ≤ κ(t), ‖∇ug(t)‖2∞ ≤ α(t), ‖∇ug(t)‖2r,loc ≤ βr(t) (3.37) for r = q and r = m and 1 < m < 3 2 < 3 < q < ∞. proof. analyzing the above estimates (3.35), (3.36) for the functions ug(t,x) and ∇ug(t,x) we note that to prove the required estimates it is enough to construct the solutions of the following integral equations α(s) = e r t s α(θ)dθk10 + cqm ∫ t s e r τ s α(θ)dθα(τ)[nq(τ) + nm(τ)]dτ, (3.38) nr(s) = e r t s α(θ)dθ‖∇u0‖r + cr ∫ t s e r τ s α(θ)dθ nr(τ)α(τ)dτ (3.39) for r = q and r = m and cqm = max(cq,cm) and β(s) = e r t s α(θ)dθ β0 + cqm ∫ t s e r τ s α(θ)dθ α(τ)β(τ)dτ, (3.40) where β(τ) = nq(τ) + nm(τ), and ‖∇u0‖q,loc + ‖∇u0‖m,loc = nq(0) + nm(0) = β0. to construct the solution of the above system of integral equations (3.40)-(3.42) we consider the system of odes dα ds = −α2(s) − cqmα(s)β(s), α(t) = k10, (3.41) dβ ds = −α(s)β(s) − cqmα(s)β(s), β(t) = β0. (3.42) 88 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) by classical results of the ode theory we know that there exists an interval [0,t1] depending on k 1 0,n 1 0 and c,cqm such that the system (3.41), (3.42) has a bounded solution defined on this interval. to prove the convergence for k → ∞ of functions uk(t,x),∇uk(t,x) we need one more auxiliary estimate. actually, we have proved that uk(t) is lipschitz-continuous with the lipschitz constant independent of k. it remains to prove that ∇uk(t) has the same property. lemma 3.8 assume that c 3.1 and c 3.2 hold. then the function ∇ug(t,x) defined in lemma 3.5 admits a representation of the form ∇ug(t,x) = e[∇u0(ψgt,0(x))ηx,g(t)− ∫ t 0 1 σ(t − τ) ∇pg(τ,ψgt,τ (x)) ∫ t τ ηx,g(θ)dŵ(θ)dτ] (3.43) and satisfies the estimate ‖∇ug(t,x) − ∇ug(t,y)‖ ≤ ng1 (t)‖x − y‖ if t ∈ [0,t1] for any x,y ∈ g where g is a compact in r3 and the positive function ng1 (t) depending on the parameters in conditions c 3.1 and c 3.2 is bounded over the interval [0,t1] defined in theorem 3.7. proof. to derive (3.43) we compute directly the gradient of the first term in (3.20) and apply the bismut-elworthy -li formula [12] to compute the gradient of the second term in this relation. next we use the representation (3.43) to deduce the lipschitz estimate for the gradient of the function u(t,x) . as a result we have ‖∇ug(t,x) − ∇ug(t,y)‖ ≤ κ1 + κ2 + κ3 + κ4, where κ1(t) = e[‖∇u0(ψgt,0(x)) − ∇u0(ψ g t,0(y))‖‖η x,g(t)‖], κ2(t) = e[‖∇u0(ψgt,0(y))‖‖ηx,g(0) − ηy,g(0)‖], κ3(t) = ∫ t 0 e [ ‖∇pg(τ,ψgt,τ (x)) − ∇pg(τ,ψ g t,τ (y))‖ σ(t − τ) ‖ ∫ t τ ηx,g(θ)dŵ(θ)‖ ] dτ, κ4(t) = ∫ t 0 1 σ(t − τ) e [ ‖∇pg(τ,ψgt,τ(y))‖ ∫ t τ [ηx,g(θ) − ηy,g(θ)]dŵ(θ)‖ ] dτ. one can easily check using the estimates stated in lemmas 3.3 – 3.5 that under the conditions c 3.1, c 3.2 κ1(t) ≤ l10e‖ψ g t,0(x) − ψ g t,0(y)‖e r t 0 k 1 g (θ)dθ ≤ ‖x − y‖l10e r t τ [lg(θ)+k 1 g (θ)]dθ = θ1‖x − y‖ and κ2(t) ≤ k10e‖ηx,g(t) − ηy,g(t)‖ ≤ ‖x − y‖ ∫ t τ k1g(θ)e r t θ k 1 g (θ1)dθ1dθ = θ2‖x − y‖. cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 89 to derive the estimates for κ3 and κ4 we apply the inequalities ‖∇i∇jpg‖∞ ≤ c(‖γg‖q,loc + [γg]1,g), ‖∇i∇jpg‖r,loc ≤ ‖γg‖r,loc from lemma 3.4. this yields κ3(t) ≤ ∫ t 0 e [ ‖∇2pg(τ)‖∞ ‖ψgt,0(x) − ψ g t,0(y)‖ σ √ t − τ ‖ηx,g(τ)‖ ] dτ ≤ ∫ t 0 (e‖ψgt,τ (x) − ψt,τ (y)‖2) 1 2 σ √ t − τ (‖γg(τ)‖q,loc + [γg(τ)]1,g)e r t τ k 1 g (θ)dθ dτ ≤ θ1‖x − y‖ ∫ t 0 ‖γg(τ)‖q,loc + [γg(τ)]1,g σ √ t − τ e r t τ k 1 g (θ)dθdτ ≤ θ3θ1‖x − y‖σ−1 ( sup0≤τ≤t[β(τ)] √ t + ∫ t 0 s(τ)√ t − τ dτ ) and κ4(t) ≤ ∫ t 0 cqm(‖γg(τ)‖q,loc + ‖γg(τ)‖m,loc) σ √ t − τ (e‖ηx,g(τ) − ηy,g(τ)‖2) 1 2 dτ ≤ 2sup0≤τ≤tβ(τ)‖x − y‖ ∫ t 0 θ2cqm σ √ t − τ dτ = 2θ4‖x − y‖ θ2cqm σ √ t. here θ3 = e r t 0 k 1 g (τ)dτ, s(τ) = [γg(τ)]1,g and β(τ) is defined in theorem 3.7. finally, combining the above estimates for κi, i = 1, 2, 3, 4, we obtain s(t) ≤ θ5 + θ6 ∫ t 0 s(τ)√ t − τ dτ and applying the gronwall lemma we derive the estimate s(t) ≤ θ5eθ6 √ t where θi, i = 5, 6 depend on the parameters in conditions c 3.1 and c 3.2, σ and t1 for 0 ≤ t ≤ t1, where t1 is defined in theorem 3.7. 2 the estimates of theorem 3.7 and lemma 3.8 allow to prove the uniform convergence on compacts of the successive approximations (3.10)-(3.14) for the solutions of the system (3.4) – (3.6) in c([0,t1], c1,α(k)) ∩ c([0,t1],lm(k) ∩ lq(k)) for 1 < q < 32 < 3 < m < ∞ and arbitrary compact k in g. in particular, they justify the possibility to differentiate the system (3.10)-(3.14) in x for each k and to consider the following equations dη k,x t,θ = −∇uk(θ,ψkt,θ)η x,k t,θ dθ, η x,k t,t = i, (3.44) where i is the identity matrix acting in r3 and ∇uk+1(t,x) = e[∇u0(ψk+1t,0 (x))η x,k t,0 − ∫ t 0 1 σ(t − τ) ∇pk+1(τ,ψkt,τ (x)) ∫ t τ η x,k t,θ dŵ(θ)dτ], (3.45) −2∇pk+1(t,x) = ∫ ∞ 0 1 τ e[γk(t,x + b(τ))b(τ)]dτ, (3.46) where γk = tr[∇uk]2. 90 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) now we can prove the following assertion. theorem 3.9 assume that c 3.2 holds. then if k → ∞ the functions uk(t),∇uk(t,x) determined by (3.11) and (3.45) uniformly converge on compacts to limiting functions u(t), ∇u(t) satisfying (3.4) and (3.8)and u(t) ∈ c([0,t1], c1,α), ∇u(t) ∈ c([0,t1], c0,α), 0 < α ≤ 1 for all t ∈ [0,t1]. here [0,t1] is the interval where the solution (α(t),β(t)) of the system (3.41), (3.42) is bounded. in addition the estimates ‖∇u(t)‖∞ ≤ α(t) , ‖∇u(t)‖q,loc ≤ β(t) hold for 1 < q < 32, t ∈ [0,t1]. proof. by theorem 3.7 we know that the mapping φ(t,x,g) = e [ u0(ψ g t,0(x)) − ∫ t 0 ∇pg(τ,ψgt,τ(x))dτ ] acts in the space c1,α ∩ lq,loc ∩ lm,loc (for a fixed t ∈ [0.t1]) with 1 < q < 32 < 3 < m < ∞. consider the successive approximations (3.10) –(3.14) and (3.44) – (3.46), set sk+1(t,x) = ‖uk+1(t,x) − uk(t,x)‖, n k+1(t,x) = ‖∇uk+1(t,x) − ∇uk(t,x)‖ and l k(t) = ‖sk(t)‖∞, mkr(t) = ‖sk(t)‖r,loc, ρk(t) = ‖nk(t)‖∞, ζkr (t) = ‖nk(t)‖r,loc. then we obtain n k+1(t,x) ≤ l10(e[‖ψkt,0(x) − ψk−1t,0 (x)‖‖η x,k t,0 ‖]+ e[‖ψkt,0(x)‖‖η x,k t,0 − η x,k−1 t,0 ‖]) + ∫ t 0 1 σ(t − τ) e[‖∇pk+1(τ,ψkt,τ (x))− ∇pk(τ,ψk−1t,τ (x))‖‖ ∫ t τ η x,k t,θ dŵ(θ)‖]dτ+ ∫ t 0 1 σ(t − τ) e [ ‖∇pk(τ,ψkt,τ (x))‖ ∫ t τ [η x,k t,θ − ηx,k−1 t,θ ]dŵ(θ)‖ ] dτ. (3.47) recall that by lemmas 3.2, 3.3 we know that sup x e‖ψkt,0(x) − ψk−1t,0 (x)‖ ≤ ∫ t 0 [ ‖uk(τ) − uk−1(τ)‖∞]dτe r t 0 α(τ)dτ, sup x e‖ηx,kt,0 − η x,k−1 t,0 ‖ ≤ ∫ t 0 ‖∇uk(τ) − ∇uk−1(τ)‖∞dτe r t 0 α(τ)dτ + sup x ∫ t 0 e‖∇uk−1(τ,ψkt,τ (x)) − ∇uk−1(τ,ψk−1t,τ (x))‖dτe r t 0 α(τ)dτ and applying the estimates from theorem 3.7 we get ρ k+1(t) ≤ e r t 0 α(τ)dτ[l10 ∫ t 0 sup x e‖uk(τ,ψkt,τ(x)) − uk−1(τ,ψk−1t,τ (x))‖dτ + ∫ t 0 ρk(τ)dτ + sup x ∫ t 0 e‖∇uk−1(τ,ψkt,τ(x)) − ∇uk−1(τ,ψk−1t,τ (x))‖dτ]+ cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 91 ∫ t 0 c[‖∇uk(τ)∇uk(τ)‖q + ‖∇uk(τ)∇uk(τ)‖m](e‖ηk(τ) − ηk−1(τ)‖2∞) 1 2 σ √ t − τ dτ+ ∫ t 0 1 σ √ t − τ sup x e‖∇pk+1(τ,ψkt,τ(x)) − ∇pk(τ,ψk−1t,τ (x))‖2) 1 2 e r t τ α(θ)dθdτ. to derive the estimate for the last term of the above inequality we recall (see lemma 3.1 and lemma 3.4) that for 1 < q < 3 2 the estimate ‖∇pk(t,x) − ∇pk(t,y)‖ ≤ ‖∇2pk(t)‖∞‖x − y‖ ≤ c[‖γk(t)‖q,loc + [γk(t)]1,g]‖x − y‖ holds and as a result we obtain e‖∇pk(τ,ψkt,τ (x)) − ∇pk(τ,ψk−1t,τ (x))‖ ≤ c[β(τ) + s(τ)]e‖ψkt,τ (x) − ψk−1t,τ (x)‖. in addition ‖∇pk+1(t) − ∇pk(t)‖∞ ≤ cqm[ ‖γk+1(t) − γk(t)‖q,loc+ ‖γk+1(t) − γk(t)‖m,loc] ≤ cqmα(t)[ ‖∇uk+1(t) − ∇uk(t)‖q,loc+ ‖∇uk(t) − ∇uk−1(t)‖q,loc+ ‖∇uk+1(t) − ∇uk(t)‖m,loc + ‖∇uk(t) − ∇uk−1(t)‖m,loc]. it follows from (3.47) that nk+1(t,x) ≤ c(t)[ ∫ t 0 e‖∇uk(τ,ψkt,τ(x)) − ∇uk−1(τ,ψk−1t,τ (x))‖dτ+ ∫ t 0 nk(τ,x)dτ] + ∫ t 0 1 σ √ t − τ c1[‖∇uk(τ)∇uk(τ)‖q+ ‖∇uk−1(τ)∇uk−1(τ)‖m]r(e‖ηx,k(τ) − ηx,k−1(τ)‖2) 1 2 dτ + ∫ t 0 1 σ √ t − τ e r t τ α(θ)dθ(e‖∇pk+1(τ,ψkt,τ(x)) − ∇pk(τ,ψk−1t,τ (x))‖2) 1 2 dτ. by the hölder inequality we derive that for any positive f(τ) ∈ lr,loc and 1m1 + 1 r = 1 and m1 < 2 we have for any compact g ⊂ r3 ∫ k [ ∫ t 0 1 σ √ t − τ f(τ,x)dτ]rdx ≤ 1 σr t r(2−m1) 2m1 ∫ t 0 ∫ k f r(τ,x)dxdτ. (3.48) then from (3.47) and (3.48) we have for r > 2 ζk+1r (t) ≤ c2(t)[ ∫ t 0 ∫ k [e‖uk(τ,ψkt,τ(x)) − uk−1(τ,ψk−1t,τ (x))‖rdxdτ]+ ∫ t 0 ζkr (τ)dτ + ∫ t 0 ∫ k ‖∇uk−1(τ,ψkt,τ (x)) − ∇uk−1(τ,ψk−1t,τ (x))‖rdxdτ] 92 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) + ∫ t 0 1 σ √ t − τ c[[‖∇uk(τ)∇uk−1(τ)‖q + ‖∇uk(τ)∇uk−1(τ)‖m]r ∫ k (e‖ηx,k(τ) − ηx,k−1(τ)‖2) r2 dx]dτ + ∫ t 0 1 σ √ t − τ e r t τ α(θ)dθ ∫ k (e‖∇pk+1(τ,ψkt,τ (x)) − ∇pk(τ,ψk−1t,τ (x))‖2) r 2 dxdτ. for the function mkr(t) = ‖uk(t) − uk−1(t)‖r,loc using the apriori estimates proved in lemmas 3.2 – 3.8 and theorem 3.9 we obtain m k+1 r (t) ≤ c(t)[( ∫ t 0 ∫ k e‖uk(τ,ψkt,τ(x)) − uk−1(τ,ψk−1t,τ (x))‖rdxdτ) 1 r +( 1 σ t 2−m1 2m1 ∫ t 0 ∫ k e‖∇uk+1(τ,ψkt,τ (x))∇uk(τ,ψkt,τ (x))− ∇uk−1(τ,ψkt,τ(x))∇uk−1(τ,ψk−1t,τ (x))‖rdxdτ) 1 r ] ≤ c1(t) [ ( ∫ t 0 mkr(τ)dτ ) 1 r + ( ∫ t 0 ∫ k α(τ)e‖ψkt,τ (x) − ψk−1t,τ (x)‖rdxdτ ) 1 r + 1 σ t 1 m1 − 1 2 ( ∫ t 0 [ρk+1(τ) + ρk(τ)]ζkr (τ)dτ ) 1 r ] . since uk and ∇uk are proved to be uniformly bounded on [0,t1] and ‖∇u1(t, ·) − ∇u0(·)‖r,loc ≤ const < ∞, ‖u1(t, ·) − u0(·)‖r,loc ≤ const < ∞, both for r = m and r = q we obtain that there exists a bounded on [0,t1] positive function c2(t) such that the function κk(t) = ρk(t) + ζkm(t) + m k r satisfies the estimate κk(t) ≤ [c2(t)] k k! and hence limk→∞ κ k(t) = 0. since all summands defining κk(t) are positive we deduce that all of them converges to 0 as k → ∞. as a results we deduce that for each t ∈ [0,t1) the family uk(t, ·) converges uniformly on compacts and the limiting function u(t, ·) ∈ c1,α ∩ lm,loc. in addition, we can check that the limiting function ∇u(t,x) is lipschitz continuous in x. in fact, by lemma 3.8 and theorem 3.9 for each t ∈ [0,t1], we have for any x,y ∈ g ‖∇uk(t,x) − ∇uk(t,y)‖ ≤ s(t)‖x − y‖, where s(t) and t1 were defined above in lemma 3.8 and the estimate is uniform in k. to prove the uniqueness of the solution of (3.4)-(3.6) constructed above we assume first that there exist two solutions u1(t,x), u2(t,x) to (3.4)-(3.6) possessing the same initial data u1(0,x) = u2(0,x) = u0(x). cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 93 computations similar to those used to prove the convergence of the family (uk(t),∇uk(t)) allow to check that both [∇u1(t) − ∇u2(t)]∞ = 0 and ‖∇u1(t) − ∇u2(t)‖m,loc = 0. finally, we know that a stochastic equation with lipschitz coefficients has a unique solution of the cauchy problem. this implies the uniqueness of the solution to (3.4)-(3.6). 2 summarizing the above results we see that the following statement is valid. theorem 3.10 assume that c 3.2 holds. then there exists a unique solution ψt,x(s),u(t,x),p(t,x) of the system (3.4)-(3.6) for all t from the interval [0,t1], with t1 given by theorem 3.7 and x ∈ k for any compact k ⊂ g. in addition the process ψt,x(s) is the markov process in r3 and u ∈ c([0,t1], c 1,α(k)) ∩ c([0,t1], lq,loc ∩ lm,loc) for 1 < q < 32 < 3 < m < ∞. proof. first we note that as soon as we know that u(t,x) is locally lipschitz continuous by classical sde theory we know that the silution ψt,0(x) of the equation (3.4) is the markov process in r3. all other assertions of the theorem are already proved above. to fulfill our program we have to check that the functions u(t,x),p(t,x) that satisfy (3.5) and (3.6) define a weak solution of (1.5),(1.2). let us come back to the kunita theory of stochastic flows [5], [6] and recall that given a distribution u0 ∈ d′ and a stochastic flow ψut,0 one can define a stochastic flow u0 ◦ ψut,0 as another distribution satisfying 〈u0 ◦ ψut,0,h〉 = 〈u0,h ◦ φu0,tj0,t〉. here φu0,t is the inverse flow to ψut,0. since any locally integrable function is a distribution, given u0 and the solution ψt,0,u(t),p(t) of (3.4)-(3.6) constructed above we consider a process λ(t) ∈ d′ of the form λ(t) = u0 − ∫ t 0 ∇pu(τ) ◦ φu0,τdτ. next we consider the process λ(t) ◦ ψut,0 = u0 ◦ ψut,0 − ∫ t 0 ∇pu(τ) ◦ ψut,τdτ and verify that a weak solution u(t) of (3.1) admits the representation u(t) = e[λ(t)◦ψut,0] and satisfies (3.2). by the generalized ito formula we derive λ(t) ◦ ψut,0 = u0 + ∫ t 0 σ2 2 ∆[u(θ) ◦ ψuθ,0]dθ+ (3.49) ∫ t 0 ∇[u(θ) ◦ ψuθ,0]σdw(θ) − ∫ t 0 ∇[u(θ) ◦ ψuθ,0]u(θ)dθ − ∫ t 0 ∇pu(θ)dθ, where (3.49) is considered in a weak sense. hence for lu = −(u,∇)u + σ 2 2 ∆u and the test function h ∈ d we have e [ ∫ r3 ∫ t 0 l(u(τ) ◦ ψuτ,s(x))dτ · h(x)dx ] = (3.50) e [ ∫ t 0 〈u(τ) ◦ ψuτ,0,l∗h〉dτ ] = ∫ t 0 l〈e[u(τ ◦ ψuτ,0)],h〉dτ. 94 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) as a result we deduce from (3.49) and (3.50) u(t) = e[λ(t) ◦ ψut,0] = u0 + ∫ t 0 le[u(τ) ◦ ψuτ,0]dτ − ∫ t 0 ∇pu(τ)dτ. differentiating each term with respect to t we check that the function u(t) = e[λ(t) ◦ ψut,0] solves the cauchy problem (1.1). as soon as the function p(t) was constructed as a solution of the poisson equation (3.2) we can verify that (1.2) holds as well. to summarize the obtained results we state the following: theorem 3.11. assume that c 3.2 holds. then the functions u(t), p(t) that solve (3.5),(3.6) satisfy (3.1)-(3.2) in a weak sense for t ∈ [0,t1] where t1 is defined in theorem 3.9. remark 3.13. we have proved that under condition c 3.2 the system (3.4)-(3.6) gives rise to a weak solution of (3.1)-(3.2). moreover, when the initial data are smoother, say u0 ∈ c2,α, α ∈ [0, 1] similar considerations can be applied to verify that the pair u(t,x),p(t,x) given by (3.5)-(3.6) stands for a classical c2-smooth solution of (3.1), (3.2). 4 lagrangian and stochastic approach to the n-s system the probabilistic approach developed in the previous section is in a sense an analogue of the lagrangian approach developed for the euler system which coincides with (1.1), (1.2) when σ = 0. the classical lagrangian path starting at y is governed by the newton equation ∂2φ̃0,t(y) ∂t2 = f φ̃ (t,y). (4.1) the force f in (4.1) has the form f φ̃ (t,y) = −∇p(t, φ̃0,t(y)) = −[(∇φ̃0,t(y))∗]−1∇[p(t, φ̃0,t(y))] (4.2) and the incompressibility condition yields det(∇φ̃0,t(y)) = 1. one can deduce from (4.1) that ∂ ∂t [ ∂φ̃k0,t(y) ∂t ∂φ̃k0,t(y) ∂yi ] = − ∂q(t, φ̃0,t(y)) ∂yi , (4.3) where q(t,y) = p(t,y) − 1 2 ‖ ∂φ̃0,t(y) ∂t ‖2 (4.4) summation over repeated indices is assumed. integrating (4.3) in time we get ∂φ̃k0,t(y) ∂t ∂φ̃k0,t(y) ∂yi = u0(y) − ∂n(t, φ̃0,t(y)) ∂yi , (4.5) where n(t,y) = ∫ t 0 q(τ,y)dτ (4.6) and u0(y) = ∂φ̃0,t(y) ∂t |t=0 is the initial velocity. cubo 12, 2 (2010) generalized solutions of the cauchy problem for the navier-stokes system and diffusion processes 95 consider the inverse diffeomorphism ψ̃t,0 = [φ̃0,t] −1, come back to (4.4), multiply it by [∇ψ̃t,0] and put y = ψ̃t,0(x). as a result we obtain by the chain rule the relation u i(t,x) = (u j 0(ψ̃t,0(x))∇xiψ̃ j t,0(x) − ∫ t 0 ∇xiq(τ,ψ̃t,τ (x))dτ. (4.7) hence the euler equations are equivalent to the system consisting of (4.7) and the relation ∆n(t,x) = ∂ ∂xi {uk0(ψ̃t,0(x)) ∂ψ̃ k t,0(x) ∂xi }, where n is given by (4.6). finally due to divu = 0 one can rewrite the equation of state (4.7) in the form u(t) = p{u0(ψ̃t,0)∇ψ̃t,0} = p{[∇ψ̃t,0]∗u0(ψ̃t,0)}, (4.8) where p = i − ∇∆−1∇ is the leray projector. the euler pressure is determined up to additive constants by p(t,x) = ∂n(t,x) ∂t + (u(t,x),∇)n(t,x) + 1 2 ‖u(t,x)‖2. when σ 6= 0 one can develop an analogue of the lagrange approach as follows. let us choose φ0,t to be generated by the stochastic equation dφ0,θ = u(θ,φ0,θ)dθ + σdw(θ), φ0,0(y) = y, (4.9) next set ψθ,0 = [φ0,θ] −1, (4.10) and finally obtain the closed system by choosing u(t) = ep[(∇ψt,0)(u0 ◦ ψt,0)]. (4.11) the system (4.9) – (4.11) was studied by constantin and iyer [9], [10]. in [14] the existence and uniqueness of the solution to (4.9) – (4.11) was proved by the successive approximation technique. the main result due to constantin and iyer reads as follows: theorem 4.1 let k ≥ 1 and u0 ∈ ck+1,α be divergence free. then there exists a time interval [0,t ] with t = t (k,α,l,‖u0‖k+1,α) but independent of viscosity σ and a pair φ0,t(x),u(t,x) such that u ∈ c([0,t ],ck+1,α) and (u,φ) satisfy (4.9)-(4.11). further there exists u = u(k,α,l,‖u0‖k+1,α) such that ‖u(t)‖k+1,α ≤ u for t ∈ [0,t ] and u satisfies the n-s system. comparing the system (3.4) – (3.6) and the system (4.9) – (4.11) we can check that the process ψt,0 given by (4.10) has the same distribution as the solution of (3.4). at the other hand the representations for the velocity u and the pressure p in the above systems are different. in the system (3.4) – (3.6) we avoid using the leray projection and use instead the probabilistic representation of the poisson equation. this allows us to construct both strong (classical) and weak (distributional) solutions of the cauchy problem for the n-s system. at the very end we remark that the approach developed in the previous section does not allow to construct a solution to the euler system as a limit of the solution to (1.1), (1.2) when σ goes to 0, since the appriori estimates in lemma 3.5 and lemma 3.8 cease to be valid. acknowledgement. the authors gratefully acknowledge the financial support of dfg grant 436 rus 113/823. received: february 2009. revised: march 2009. 96 s. albeverio and ya. belopolskaya cubo 12, 2 (2010) references [1] s. albeverio and ya. belopolskaya, probabilistic approach to hydrodynamic equations. in the book probabilistic methods in hydrodynamics. world scientific (2003) 1-21. [2] y. le jan and a. sznitman, stochastic cascades and 3-dimensional navier stokes equations, prob. theory relat. fields, 109 (1997) 343-366. [3] m. ossiander, a probabilistic representation of solution of the incompressible navier-stokes equations in r3. prob. theory relat. fields, 133 2 (2005) 267-298. [4] b. busnello, f. flandoli and m. romito, a probabilistic representation for the vorticity of a 3d viscous fluid and for general systems of parabolic equations, proc. edinburgh math. soc., 48 2 (2005) 295-336. [5] h. kunita, stochastic flows acting on schwartz distributions. j. theor. pobab. 7 2 (1994) 247-278. [6] h. kunita, generalized solutions of stochastic partial differential equations. j. theor. pobab.7 2 (1994) 279-308. [7] ya. belopolskaya and yu. dalecky, investigation of the cauchy problem for systems of quasilinear equations via markov processes . izv vuz matematika. n 12 (1978) 6-17. [8] ya. belopolskaya and yu. dalecky, stochastic equations and differential geometry. kluwer (1990). [9] p. constantin, an eulerian-lagrangian approach to the navier-stokes equations. commun. math. phys. bf 216 (2001) 663-686. [10] p. constantin and g. iyer, a stochastic lagrangian representation of the 3-dimensional incompressible navier-stokes equations. commun. pure and appl. math. 61 3 (2007) 330 345. [11] p.g. lemarie-rieusset, recent developments in the navier-stokes problem, chapman&hall, crc (2002). [12] k. d. elworthy and xue-mei li, differentiation of heat semigroups and applications, j. funct. anal. 125, no.1, (1994) 252-286. [13] d. gilbarg and n.s.trudinger, elliptical partial differential equations, second eddition. springer-verlag berlin heidelberg new york tokyo (1983). [14] g. a. iyer, stochastic lagrangian formulation of the incompressible navier-stokes and related transport equations. phd dissertation dept. math. univ. chicago (2006). articulo 1.dvi cubo a mathematical journal vol.12, no¯ 02, (1–17). june 2010 existence results for semilinear differential evolution equations with impulses and delay nadjet abada département de mathématiques, université mentouri constantine, email: n65abada@yahoo.fr mouffak benchohra 1 laboratoire de mathématiques, université de sidi bel abbès, bp 89, 22000 sidi bel abbès, algérie. email: benchohra@univ-sba.dz and hadda hammouche département de mathematiques, université kasdi merbah ouargla, email: h.hammouche@yahoo.fr abstract in this paper, we establish sufficient conditions for the existence of mild and extremal solutions for some densely defined impulsive functional differential equations in separable banach spaces with local and nonlocal conditions. we shall rely for the existence of mild solutions on a fixed point theorem due to burton and kirk for the sum of completely continuous and contraction operators, and for the existence of extremal solutions on dhage’s fixed point theorem. 1corresponding author 2 n. abada , m. benchohra and h. hammouche cubo 12, 2 (2010) resumen en este art́ıculo establecemos condiciones suficientes para la existencia de soluciones suaves y extremas para algunas ecuaciones diferenciales funcionales impulsivas densamente definidas en espacios de banach separables con condiciones locales y no locales. para la existencia de soluciones suaves usaremos un teorema de punto fijo debido a burton y kirk para la suma de un operador completamente continuo y otro contractivo; para la existencia de soluciones extremas usaremos el teorema de punto fijo de dhage. key words and phrases: densely defined operator, impulsive functional differential equations, fixed point, semigroups, mild solutions, extremal mild solutions, nonlocal condition, separable banach space. ams (mos) subj. class.: 34a37, 34k30, 34k35, 34k45. 1 introduction in this paper, we establish sufficient conditions for the existence of mild and extremal mild solutions of first order impulsive functional equations in a separable banach space (e. |.|) of the form: y′(t) − ay(t) = f(t,yt), a.e. t ∈ j = [0,b] , t 6= tk, k = 1, . . . ,m (1) ∆y|t=tk = ik(y(t − k )), k = 1, . . . ,m (2) y(t) = φ(t), t ∈ [−r, 0] , (3) where f : j × d → e is a given function, d = {ψ : [−r, 0] → e,ψ is continuous every where except for a finite number of points s at which ψ (s−) ,ψ (s+) exist and ψ (s−) = ψ (s)}, φ ∈ d, 0 < r < ∞, 0 = t0 < t1 < ... < tm < tm+1 = b, ik ∈ c (e,e) , k = 1, 2, . . . ,m, a : d(a) ⊂ e → e is the infinitesimal generator of a c0-semigroup t (t), t ≥ 0, and e a real separable banach space with norm |.|. we denote by yt the element of d defined by yt(θ) = y(t + θ), θ ∈ [−r, 0]. here yt(·) represents the history of the state from t − r, up to the present time t. functional differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. a good guide to the literature for functional differential equations is the books by hale [18] and hale and verduyn lunel [19] and kolmanovskii and myshkis [23] and the references therein. impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. there has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see for instance the monoghraphs by bainov and simeonov [5], benchohra et al [6], lakshmikantham et al [24], and samoilenko and perestyuk [29]. in the case where the impulses are absent (i.e ik = 0,k = 1, . . . ,m) and f is a single or multivalued map and a is a densely defined linear operator generating a c0-semigroup cubo 12, 2 (2010) impulsive semilinear functional differential equations 3 of bounded linear operators the problem (1)–(3) has been investigated on compact intervals in, for instance, the monographs by ahmed [1], hu and papageorgiou [21], kamenskii et al [22] and wu [30] and the papers of benchohra and ntouyas [7, 8, 9]. during the last decades problems of the form (1)–(3) have received much attention. some existence results were given in the monograph ahmed [2] and benchohra et al [6] and the papers by ahmed [3, 4], cardinali and rubbioni [15], liu [25] and rogovchenko [27, 28], and the references therein. the plan of this paper is as follows: in section 2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following sections. in section 3, we prove existence of mild solutions for problem (1)–(3). our approach will be based for the existence of mild solutions, on a fixed point theorem of burton and kirk [10] for the sum of a contraction map and a completely continuous map. in section 4, we shall prove the existence of extremal solutions of the problem (1)–(3), and our approach here is based on the concept of upper and lower solutions combined with a fixed point theorem on ordered banach spaces established recently by dhage [16]. in section 5 we study the impulsive functional differential equations with non-local initial conditions, of the form y′(t) − ay(t) = f(t,yt), a.e. t ∈ j = [0,b] , t 6= tk, k = 1, . . . ,m (4) ∆y|t=tk = ik(y(t − k )), k = 1, . . . ,m (5) y(t) + ht (y) = φ(t), t ∈ [−r, 0] , (6) where ht : pc([−r,b],e) → e is given function (see section 2 for the definition of pc([−r,b],e)). the non-local condition can be applied in physics with better effect than the classical initial condition y (0) = y0. for example, ht (y) may be given by ht (y) = p∑ i=1 ciy(ti + t), t ∈ [−r, 0] where ci, i = 1, ...,p, are given constants and 0 < t1 < ... < tp ≤ b. at time t = 0, we have h0 (y) = p∑ i=1 ciy (ti) non-local conditions were initiated by byszewski [11] (see also [12, 13, 14]) in which we refer for motivation and other references. finally, section 6 is devoted to an example illustrating the abstract theory considered in the previous sections. 2 preliminaries in this section, we introduce notations and definitions, preliminaries facts which are used throughout this paper. for ψ ∈ d the norm of ψ is defined by ‖ψ‖d = sup{|ψ(θ)| : θ ∈ [−r, 0]}, and b(e) denotes the banach space of bounded linear operators from e into e, with norm ‖n‖b(e) = sup{|n(y)| : |y| = 1}. 4 n. abada , m. benchohra and h. hammouche cubo 12, 2 (2010) l1(j,e) denotes the banach space of measurable functions y : j −→ e which are bochner integrable normed by ‖y‖l1 = ∫ b 0 |y(t)|dt. it is well known that the operator a generates a (c0) semigroup if a satisfies: (i) d(a) = e. (ii) the hille-yosida condition, that is, there exists m ≥ 0 and ω ∈ r such that (ω,∞) ⊂ ρ(a) and sup{(λ − ω)n|(λi − a)−n| : λ, n ∈ n} ≤ m, where ρ(a) is the resolvent set of a and i is the identity operator in e. for more details on semigroup theory we refer the interested reader to the books of ahmed [1] and pazy [26]. in order to define a mild solution of problems (1)–(3) and (4)–(6), we shall consider the space pc ([−r,b] ,e) ={y : [−r,b] → e : y (t) is continuous everywhere except for some tk at which y ( t − k ) and y ( t + k ) , k = 1, 2, . . . ,m exist and y ( t − k ) = y (tk)}. obviously, for any t ∈ [0,b] and y ∈ pc ([−r,b] ,e), we have yt ∈ d and pc ([−r,b] ,e) is a banach space with the norm ‖y‖ = sup {|y (t)| : t ∈ [−r,b]} . definition 2.1. a map f : j × d → e is said to be carathéodory if (i) the function t 7−→ f(t,x) is measurable for each x ∈ d; (ii) the function x 7−→ f(t,x) is continuous for almost all t ∈ j. 3 existence of mild solutions in this section, we give our main existence result for problem (1)–(3). before stating and proving this result, we give the definition of the mild solution. definition 3.1. a function y ∈ pc ([−r,b] ,e) is said to be a mild solution of problem (1)–(3) if y(t) = φ(t), t ∈ [−r, 0], and y is a solution of impulsive integral equation y(t) = t (t)φ(0) + ∫ t 0 t (t − s)f(t,ys)ds + ∑ 0 0 in the banach space e. let m = sup{‖t (t)‖b(e) : t ∈ j}; (h2) there exist constants dk > 0, k = 1, ...,m with m m∑ k=1 dk < 1 such that |ik (y) − ik (x)| ≤ dk |y − x| for each y,x ∈ e (h3) the function f : j × d → e is carathéodory; (h4) there exists a function p ∈ l1(j, r+) and a continuous nondecreasing function ψ : [0,∞) → (0,∞) such that |f(t,x)| ≤ p(t)ψ(‖x‖d), a.e. t ∈ j, for all x ∈ d with ∫ ∞ do ds ψ (s) > d1 ‖p‖l1 , where d0 = m(‖φ‖ + m∑ k=1 |ik(0)|) 1 − m m∑ k=1 dk , d1 = m 1 − m m∑ k=1 dk . theorem 3.2. assume that (h1)-(h4) hold. then the problem (1)–(3) has at least one mild solution on [−r,b]. proof. transform the problem (1)-(3) into a fixed point problem. consider the two operators: a,b : pc ([−r,b] ,e) → pc ([−r,b] ,e) defined by a(y) (t) :=    0, if t ∈ [−r, 0]; ∑ 0 0 there exists a positive constant l such that for each y ∈ bq = {y ∈ pc([−r,b],e) : ‖y‖ ≤ q} we have ‖b (y)‖ ≤ l. so choose y ∈ bq, then we have for each t ∈ j, |b(y)(t)| = ∣∣∣∣t (t)φ(0) + ∫ t 0 t (t − s)f(s,ys)ds ∣∣∣∣ ≤ m|φ(0)| + mψ(q) ∫ b 0 p(s) ds; then we have ‖b(y)‖ ≤ m‖φ‖ + mψ(q)‖p‖l1 := l. step 3: b maps bounded sets into equicontinuous sets of pc([−r,b],e). we consider bq as in step 2 and let τ1,τ2 ∈ j\ {t1, ..., tm} , τ1 < τ2.thus if ǫ > 0 and ǫ ≤ τ1 < τ2 we have cubo 12, 2 (2010) impulsive semilinear functional differential equations 7 |b(y)(τ2) − b(y)(τ1)| ≤ |t (τ2)φ(0) − t (τ1)φ(0)| +ψ(q) ∫ τ1−ǫ 0 ‖t (τ2 − s) − t (τ1 − s)‖b(e)p(s)ds +ψ(q) ∫ τ1 τ1−ǫ ‖t (τ2 − s) − t (τ1 − s)‖b(e)p(s)ds +ψ(q) ∫ τ2 τ1 ‖t (τ2 − s)‖b(e)p(s)ds. as τ1 → τ2 and ǫ become sufficiently small, the right-hand side of the above inequality tends to zero, since t (t) is a strongly continuous operator and the compactness of t (t) for t > 0 implies the continuity in the uniform operator topology ([1, 26]). this proves the equicontinuity for the case where t 6= ti,k = 1, 2, . . . ,m + 1. it remains to examine the equicontinuity at t = ti. first we prove equicontinuity at t = t−i . fix δ1 > 0 such that {tk : k 6= i} ∩ [ti − δ1, ti + δ1] = ∅. for 0 < h < δ1 we have |b(y)(ti − h) − b(y)(ti)| ≤ | (t (ti − h) − t (ti)) φ(0)| + ∫ ti−h 0 | (t (ti − h − s) − t (ti − s)) f(s,ys)|ds +ψ(q)m ∫ ti ti−h p(s)ds; which tends to zero as h → 0. define b̂0(y)(t) = b(y)(t), t ∈ [0, t1] and b̂i(y)(t) = { b(y)(t), if t ∈ (ti, ti+1] b(y)(t+i ), if t = ti. next we prove equicontinuity at t = t+i . fix δ2 > 0 such that {tk : k 6= i} ∩ [ti − δ2, ti + δ2] = ∅. for 0 < h < δ2 we have |b̂(y)(ti + h) − b̂(y)(ti)| ≤ | (t (ti + h) − t (ti)) φ(0)| + ∫ ti 0 | (t (ti + h − s) − t (ti − s)) f(s,ys)|ds +ψ(q)m ∫ ti+h ti p(s)ds. 8 n. abada , m. benchohra and h. hammouche cubo 12, 2 (2010) the right-hand side tends to zero as h → 0. the equicontnuity for the cases τ1 < τ2 ≤ 0 and τ1 ≤ 0 ≤ τ2 follows from the uniform continuity of φ on the interval [−r, 0]. as consequence of steps 1 to 3 together with arzelá-ascoli theorem it suffices to show that b maps b into a precompact set in e. let 0 < t < b be fixed and let ǫ be a real number satisfying 0 < ǫ < t. for y ∈ bq we define bǫ(y)(t) = t (t)φ(0) + t (ǫ) ∫ t−ǫ 0 t (t − s − ǫ)f(s,ys)ds. since t (t) is a compact operator, the set yǫ(t) = {bǫ(y)(t) : y ∈ bq} is precompact in e for every ǫ, 0 < ǫ < t. moreover, for every y ∈ bq we have |b(y)(t) − bǫ(y)(t)| ≤ ψ(q) ∫ t t−ǫ ‖t (t − s)‖b(e)p(s)ds ≤ ψ(q)m ∫ t t−ǫ p(s)ds. therefore, there are precompact sets arbitrarily close to the set yǫ(t) = {bǫ(y)(t) : y ∈ bq}. hence the set y (t) = {b(y)(t) : y ∈ bq} is precompact in e. hence the operator b : pc ([−r,b] ,e) → pc ([−r,b] ,e) is completely continuous. step 4: a is a contraction let x,y ∈ pc([−r,b],e). then for t ∈ j |a(y)(t) − a(x)(t)| = ∣∣∣∣∣ ∑ 0 0 such that ‖x‖ ≤ n‖y‖, whenever x ≤ y. we equip the space x = c(j,e) with the order relation ≤ induced by a regular cone c in e, that is for all y,y ∈ x : y ≤ y if and only if y(t) −y(t) ∈ c, ∀t ∈ j. in what follows will assume that the cone c is normal. cones and their properties are detailed in [17, 20]. let a,b ∈ x be such that a ≤ b. then, by an order interval [a,b] we mean a set of points in x given by [a,b] = {x ∈ x | a ≤ x ≤ b}. definition 4.2. let x be an ordered banach space. a mapping t : x → x is called isotone increasing if t (x) ≤ t (y) for any x,y ∈ x with x < y. similarly, t is called isotone decreasing if t (x) ≥ t (y) whenever x < y. definition 4.3. we say that x ∈ x is the least fixed point of g in x if x = gx and x ≤ y whenever y ∈ x and y = gy. the greatest fixed point of g in x is defined similarly by reversing the inequality. if both least and greatest fixed point of g in x exist, we call them extremal fixed point of g in x. very recently dhage has proved the following. theorem 4.1. [16]. let [a,b] be an order interval in a banach space x and let b1,b2 : [a,b] → x be two functions satisfying: (a) b1 is a contraction, (b) b2 is completely continuous, (c) b1 and b2 are strictly monotone increasing, and (d) b1(x) + b2(x) ∈ [a,b], ∀x ∈ [a,b]. further if the cone c in x is normal, then the equation x = b1(x) + b2(x) has a least fixed point x∗ and a greatest fixed point x ∗ ∈ [a,b]. moreover x∗ = lim n→∞ xn and x ∗ = lim n→∞ yn, where {xn} and {yn} are the sequences in [a,b] defined by xn+1 = b1(xn) + b2(xn), x0 = a and yn+1 = b1(yn) + b2(yn), y0 = b. we need the following definitions in the sequel. definition 4.4. we say that a function v ∈ pc([−r,b],e) is a lower mild solution of problem (1)–(3) if v(t) = φ(t), t ∈ [−r, 0], and v(t) ≤ t (t)φ(0) + ∫ t 0 t (t − s)f (s,vs) ds + ∑ 0 0 such that |ht(u)| ≤ α, u ∈ pc([−r,b],e) and for each k > 0 the set {φ(0) − h0(y), y ∈ pc([−r,b],e),‖y‖ ≤ k} is precompact in e (a2) there exists a function p ∈ l1(j, r+) and a continuous nondecreasing function ψ : [0,∞) → (0,∞) such that |f(t,x)| ≤ p(t)ψ(‖x‖d), a.e. t ∈ j, for all x ∈ d with ∫ ∞ d̃0 ds ψ (s) > d1 ‖p‖l1 , and d̃0 = m[‖φ‖d + α + m∑ k=1 |ik(0)|] 1 − m m∑ k=1 dk . then the problem 4)–(6) has at least one mild solution on [−r,b]. proof. transform the problem (4)–(6) into a fixed point problem. consider the two operators : b1 : pc ([−r,b] ,e) → pc ([−r,b] ,e) defined by b1(y)(t) =    φ(t) − ht(y), if t ∈ [−r, 0]; t (t) (φ(0) − h0(y)) + ∫ t 0 t (t − s)f (s,ys) ds, if t ∈ j, 14 n. abada , m. benchohra and h. hammouche cubo 12, 2 (2010) and a1(y)(t) =    0, if t ∈ [−r, 0]; ∑ 0 0, bk > 0, k = 1, . . . ,m, φ ∈ d = {ψ : [−r, 0] × [0,π] → ir; ψ is continuous everywhere except for a countable number of points at which ψ(s−),ψ(s+) exist with ψ(s−) = ψ(s)}, 0 = t0 < t1 < t2 < ... < tm < tm+1 = b, z(t + k ) = lim (h,x)→(0+,x) z(tk + h,x),z(t − k ) = lim (h,x)→(0−,x) z(tk + h,x) and q : [0,b] × ir → ir is a given function. let y(t)(x) = z(t,x), t ∈ j, x ∈ [0,π], ik(y(t − k ))(x) = bkz(t − k ,x), x ∈ [0,π], k = 1, . . . ,m f(t,φ)(x) = q(t,φ(θ,x)), θ ∈ [−r, 0], x ∈ [0,π], φ(θ)(x) = φ(θ,x), θ ∈ [−r, 0], x ∈ [0,π]. take e = l2[0,π] and define a : d(a) ⊂ e → e by aw = w′′ with domain d(a) = {w ∈ e,w,w′ are absolutely continuous, w′′ ∈ e,w(0) = w(π) = 0}. then aw = ∞∑ n=1 n2(w,wn)wn, w ∈ d(a) where ( , ) is the inner product in l2 and wn(s) = √ 2 π sinns, n = 1, 2, . . . is the orthogonal set of eigenvectors in a. it is well known (see [26]) that a is the infinitesimal generator of an analytic semigroup t (t), t ∈ [0,b] in e and is given by t (t)w = ∞∑ n=1 exp(−n2t)(w,wn)wn, w ∈ e. cubo 12, 2 (2010) impulsive semilinear functional differential equations 15 since the analytic semigroup t (t) is compact, there exists a constant m ≥ 1 such that ‖t (t)‖b(e) ≤ m. also assume that there exists an integrable function σ : [0,b] → ir+ such that |q(t,w(t − r,x))| ≤ σ(t)ω(|w|) where ω : [0,∞) → (0,∞) is continuous and nondecreasing with ∫ ∞ 1 ds s + ω(s) = +∞. assume that there exists a function l̃ ∈ l1([0,b], ir+) such that |q(t,w) − q(t,w)| ≤ l̃(t)|w − w|, t ∈ [0,b], w,w ∈ ir. we can show that problem (1)-(3) is an abstract formulation of problem (7)-(10). since all the conditions of theorem 3.2 are satisfied, the problem (7)-(10) has a solution z on [−r,b] × [0,π]. acknowledgement. this work was completed when the second author was visiting the ictp in trieste as a regular associate. it is a pleasure for him to express gratitude for its financial support and the warm hospitality. received: december 2008. revised: january 2009. references [1] n. u. ahmed, semigroup theory with applications to systems and control, pitman research notes in mathematics series, 246. longman scientific & technical, harlow; john wiley & sons, new york, 1991. 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[30] j. wu, theory and applications of partial functional differential equations, applied mathematical sciences 119, springer-verlag, new york, 1996. paper3.dvi cubo a mathematical journal vol.12, no¯ 03, (203–212). october 2010 analytic continuation and applications of eigenvalues of daubechies’ localization operator kunio yoshino department of natural sciences, faculty of knowledge engineering, tokyo city university, tokyo 158-8557, japan email: yoshinok@tcu.ac.jp abstract in this paper we introduce generating functions of eigenvalues of daubechies’ localization operator, study their analytic properties and give analytic continuation of these eigenvalues. making use of generating functions, we establish a reconstruction formula of symbol functions of daubechies’ localization operator with rotational invariant symbols. resumen introducimos funciones generadas por los autovalores del operador de localización de daubechies, estudiamos sus propiedades analíticas y damos continuación analítica de los autovalores. haciendo uso de las funciones generadas, establecemos la fórmula de reconstrucción de funciones símbolo del operador de localización de daubechies con símbolos rotacional invariante. key words and phrases: hermite functions, daubechies (localization) operator, borel transform, asymptotic expansion. math. subj. class.: 33, 44, 46f. 204 kunio yoshino cubo 12, 3 (2010) 1 introduction the daubechies (localization) operator was introduced by ingrid daubechies in [4], where she mainly treated localization operators with rotational invariant symbols. in particular, she expressed eigenvalues as mellin transforms of symbol functions. she also proved that hermite functions are eigenfunctions of localization operators with rotational invariant symbols. so far, the theory of localization operators has been studied by several researchers in various fields ([2], [5], [7], [10], [11], [12]). in this paper we will study analytic properties of generating functions of eigenvalues of daubechies’ localization operators. we will also give an analytic continuation of eigenvalues of daubechies’ localization operator. making use of generating functions, we will establish the reconstruction formula of symbol functions of daubechies’ localization operators with rotational invariant symbol. for the simplicity, we will confine ourselves to the 1-dimensional case in this paper. in section 2 we will introduce daubechies’ localization operator. in section 3 we will give the analytic continuation of eigenvalues of daubechies’ localization operator. in section 4 we will define the generating function of eigenvalues. in final section 5 we will establish the reconstruction formulas for rotational invariant symbol function. 2 daubechies’ localization operator according to [4], we define the localization operator pf as follows. definition 1 ([4]). daubechies’ localization operator is pf ( f )(x) = (2π)−1 ˆ ˆ r2 f( p, q)φp,q (x) < φp,q , f > d pd q, (1) where f( p, q) ∈ l1(r2), f (x) ∈ l2(r), φp,q(x) = π−1/4 ei px e−(x−q) 2/2, and < φp,q , f > denotes the inner product ˆ r φp,q(x) f (x) dx. the function f( p, q) is called the symbol function of the operator pf . daubechies obtained the following results. cubo 12, 3 (2010) eigenvalues of daubechies’ localization operator 205 proposition 1 ([4]). suppose that f( p, q) ∈ l1(r2). then (i) if f( p, q) ≥ 0, then pf is a positive operator. (ii) pf is bounded operator. that is, ||pf ( f )||l2 ≤ (2π) −1/2||f ||l2 ||f||l1 , ( f ∈ l 2(r)). (iii) pf is a trace class operator. proposition 2 ([4]). suppose f( p, q) = f̃(r2), where r2 = p2 + q2. then (i) the hermite functions hm (x) are eigenfunctions of the operator pf : pf (hm )(x) = λm hm (x), m ∈ n. (ii) secondly, λm = 1 m! ˆ ∞ 0 e −s s m f̃(2s)ds, m ∈ n, where the hermite functions hm (x) are defined by hm (x) = (−1)m(2m m! p π)−1/2 exp(x2/2) dm dxm exp(−x2), m ∈ n. for details on hermite functions, we refer the reader to [6, 7, 10]. in what follows we assume that (i) f( p, q) ∈ l1(r2). (ii) f( p, q) is rotational invariant, that is, f( p, q) = f̃(r2), where r2 = p2 + q2. 3 analytic continuation of eigenvalues of daubechies operator in this section we consider the analytic continuation of the eigenvalues {λm } ∞ m=0. by proposition 2 we have λm = 1 m! ˆ ∞ 0 e −s s m f̃(2s)ds. 206 kunio yoshino cubo 12, 3 (2010) we put λ(z) = 1 γ(z + 1) ˆ ∞ 0 e −s s z f̃(2s)ds, z ∈ c, re(z) > 0. where γ(z) is euler’s gamma function. then we have the following proposition. proposition 3. λ(z) have following properties: (i) λ(z) is holomorphic in the right half plane re(z) > 0. (ii) there exists a positive constant c such that |λ(z)| ≤ c p |z| e π 2 |y|, z = x + i y ∈ c, x > 0. (iii) λ(z) interpolates the eigenvalues {λm } ∞ m=0, that is, λ(m) = λm, m ∈ n. (iv) there exists a positive constant c such that |λm| ≤ c p |m| , m ∈ n. proof. the proof is as follows. (i) we can prove the holomorphicity of λ(z) by morea’s theorem and lebesgue’s dominated convergence theorem. (iii) is obvious. (iv) follows from (ii) and (iii). so we will prove statement (ii). by stirling’s formula, γ(z + 1) ∼ zz e−z p 2πz, re(z) > 0 cubo 12, 3 (2010) eigenvalues of daubechies’ localization operator 207 and e−s sx ≤ e−x xx for s ≥ 0, we have |λ(z)| = ∣ ∣ ∣ ∣ 1 γ(z + 1) ˆ ∞ 0 e −s s z f̃(2s) ds ∣ ∣ ∣ ∣ ≤ 1 |γ(z + 1)| ˆ ∞ 0 e −s|sz||f̃ (2s)| ds ≤ c |zz e−z| p 2π|z| ˆ ∞ 0 e −s s x|f̃(2s)| ds ≤ c e yarg(z) xx e−x p 2π|z| ˆ ∞ 0 e −x x x|f̃(2s)| ds ≤ c p 2π|z| e π 2 |y| ˆ ∞ 0 |f̃ (2s)| ds ≤ c′ p |z| e π 2 |y|. remark 1. the function λ(z) is the unique analytic continuation of eigenvalues {λm } ∞ m=0 because of (ii) in proproposition 3 and carlson’s theorem [3]. 4 generating functions of eigenvalues of daubechies operator in this section we introduce two generating functions λ(w) and g(t) of the eigenvalues {λm } ∞ m=0. we begin with λ(w). put λ(w) = ∞ ∑ m=0 λm w m, (|w| < 1). due to (iv) in proposition 3, the right-hand side is a convergent series if |w| < 1. we will show some analytic properties of λ(w). proposition 4. suppose that {λm} ∞ m=0 are eigenvalues of pf . then (i) the function λ(w) is given by the integral λ(w) = ˆ ∞ 0 e −s(1−w) f̃(2s)ds, re(w) < 1. (ii) λ(w) is holomorphic in the left-half plane {w ∈ c : re(w) < 1} and is bounded in its closure {w ∈ c : re(w) ≤ 1}. (iii) λ(iv) ∈ c0(r) for v ∈ r, that is, λ(iv) ∈ c(r) and lim|v|→∞ λ(iv) = 0. 208 kunio yoshino cubo 12, 3 (2010) proof. we prove the three parts. (i) by (ii) in proposition 2, we have λ(w) = ∞ ∑ m=0 λm w m = ∞ ∑ m=0 wm m! ˆ ∞ 0 e −s s m f̃(2s) ds = ˆ ∞ 0 e −s f̃(2s) ∞ ∑ m=0 (ws)m m! ds = ˆ ∞ 0 e −s(1−w) f̃(2s) ds. (ii) for re(w) ≤ 1, we have |λ(w)| ≤ ˆ ∞ 0 |e−s(1−w)||f̃(2s)| ds ≤ ˆ ∞ 0 |f̃ (2s)| ds = ||f̃||l1 . (iii) λ(iv) is the fourier transform of the l1 function e−s f̃(2s), for s ≥ 0. hence it belongs to c0(r n) by the riemann–lebesgue theorem [9]. proposition 5. suppose that f( p, q) is positive. if lim sup m→∞ λm 1/m = 1, then w = 1 is a singular point of λ(w). proof. since f( p, q) is positive, then pf is a positive operator by proposition 1. therefore, all the eigenvalues of pf are nonnegative. by the cauchy–hadamard formula, the radius of convergence of the power series ∞ ∑ m=0 λm w m is 1. by vivanti’s theorem, w = 1 is a singular point of λ(w). proposition 6. suppose the support of f̃(2s) is contained in [0, a]. then there exists a positive constant c such that (i) |λm| ≤ c am m! , m ∈ n. (ii) λ(w) is an entire function of exponential type. proof. we prove the two parts of the proposition. cubo 12, 3 (2010) eigenvalues of daubechies’ localization operator 209 (i) since the support of f̃(2s) is contained in the closed interval [0, a], by (ii) in proposition 2, we have λm = 1 m! ˆ a 0 e −s f̃(2s)sm ds ≤ am m! ˆ a 0 |f̃(2s)| ds. therefore, the inequality |λm| ≤ c am m! is valid. (ii) since |λ(w)| ≤ ˆ a 0 |f̃(2s)|e−s(1−u) ds ≤ ea|u| ˆ a 0 |f̃(2s)|ds, we have |λ(w)| ≤ c ea|u|, w = u + iv ∈ c. now we consider following formal power series: ∞ ∑ m=0 m!λm t −m−1. in general, the series on right-hand side is divergent. but this formal power series is an asymptotic expansion of the hilbert transform of f̃ (2s)e−s. namely, if we put g(t) = ˆ ∞ 0 f̃(2s)e−s t − s ds, t ∈ c\[0,∞], then g(t) has following properties. proposition 7. for the function g(t) we have (i) g(t) is laplace transform of λ(w). (ii) ∞ ∑ m=0 m!λm t −m−1 is an asymptotic expansion of g(t). proof. 210 kunio yoshino cubo 12, 3 (2010) (i) by (ii) in proposition 4, λ(w) is bounded in left-half plane. so we can consider the laplace transform of λ(w) along the negative real axis: ˆ −∞ 0 λ(w)e−tw dw = ˆ −∞ 0 { ˆ ∞ 0 e −s(1−w) f̃(2s) ds } e −tw dw = ˆ ∞ 0 f̃(2s) e−s { ˆ −∞ 0 e w(s−t) dw } ds = ˆ ∞ 0 f̃(2s)e−s t − s ds = g(t), for re t < 0. (ii) secondly, g(t) = ˆ ∞ 0 f̃ (2s)e−s t − s ds = 1 t ˆ ∞ 0 f̃(2s)e−s 1 − st−1 ds = 1 t ˆ ∞ 0 f̃ (2s)e−s { n ∑ m=0 (st−1 )m + (st−1 )n+1 1 − st−1 } ds = n ∑ m=0 m!λm t −m−1 + 1 tn+1 ˆ ∞ 0 f̃(2s)e−s sn+1 t − s ds. hence if |t| ≥ r and 0 < δ ≤ arg(t) ≤ 2π−δ, then we have |g(t) − n ∑ m=0 m!λm t −m−1| ≤ (n + 1)! r sin δ λn+1 |t|n+1 ≤ c n! p n + 1 r sin δ|t|n+1 . proposition 8. suppose that support of f̃(2s) is contained in [0, a]. then (i) g(t) is holomorphic in c\[0, a]. (ii) ∞ ∑ m=0 m!λm t −m−1 converges in |t| > a. proof. (i) from the assumption on the support of f̃(2s), we have g(t) = ˆ a 0 f̃(2s)e−s t − s ds. so g(t) is holomorphic in c\[0, a]. cubo 12, 3 (2010) eigenvalues of daubechies’ localization operator 211 (i) by (i) in proposition 6, |λm| ≤ c am m! , m ∈ n. hence ∞ ∑ m=0 m!λm t −m−1 converges if |t| > a. remark 2. the function λ(w) is the borel transform of g(t). for details on the borel and hilbert transforms, we refer the reader to [1, 8, 9]. 5 reconstruction of symbol functions in this section we establish our main results. theorem 1. the function f̃(2s) = (2π)−1 esf(λ(iv))(s), is valid in distribution sense, where f(λ(iv)) = ˆ ∞ −∞ e −isv) λ(iv)dv is fourier transform of λ(iv). proof. by (i) in proposition 4, we have λ(iv) = ˆ ∞ 0 e −s(1−iv) f̃(2s) ds = ˆ ∞ 0 e isv e −s f̃(2s) ds. this means that λ(iv) is the inverse fourier transform of e−s f̃(2s). since f̃(2s) is an l1function, then e−s f̃(2s) is a tempered distribution. hence, as tempered distribution, we have f̃(2s) = esf(λ(iv))(s). theorem 2. the function f̃(2s) is given by the formula f̃ (2s) = es lim t→0 −1 2πi (g(s + it) − g(s − it)) . 212 kunio yoshino cubo 12, 3 (2010) proof. it is well known that the boundary value lim t→0 −1 2πi [g(s + it) − g(s − it)] is the inverse map of the hilbert transform [8]. since g(t) is hilbert transform of e−s f̃(2s), we have lim t→0 −1 2πi (g(s + it) − g(s − it)) = e−s f̃(2s). references [1] anderson, m., topics in complex analysis, springer-verlag, new york, berlin, heidelberg, 1996. [2] ashino, r., boggiatto, p. and wong, m.w., advanced in pseudo-differential operators, birkhäuser-verlag, basel, berlin, boston, 2000. [3] boas, r.p., entire functions, academic press, new york, 1954. [4] daubechies, i., a time frequency localization operator; a geometric phase space approach, ieee. trans. inform. theory, vol.34, pp. 605–612, 1988. [5] daubechies, i., ten lectures on wavelets, rutgers university and at & t bell laboratories, 1992. [6] folland, g.b., harmonic analysis in phase space, princeton univ. press, 1989. [7] gröhenig, k., foundations of time-frequency analysis, birkhäuser-verlag, basel, berlin, boston, 2000. [8] morimoto, m., an introduction to sato’s hyperfunction, translation of math. monograph, vol. 129, amer. math. soc. providence, rhode island, 1993. [9] rudin, w., real and complex analysis, mcgraw-hill book company, new york, 1987. [10] wong, m.w., weyl transforms, springer-verlag, new york, 1998. [11] wong, m.w., localization operators on the weyl-heisenberg group, geometry, analysis and applications, proceedings of the international conference (editor:p.s.pathak), pp. 303–314, 2001. [12] wong, m.w., wavelet transforms and localization operator, birkhäuser-verlag, basel, berlin, boston, 2002. cha1.dvi cubo a mathematical journal vol.12, no¯ 03, (153–165). october 2010 existence of periodic solutions for a class of second-order neutral differential equations with multiple deviating arguments1 chengjun guo school of applied mathematics, guangdong, university of technology 510006, p.r.china email: guochj817@163.com donal o’regan department of mathematics, national university of ireland, galway, ireland email: donal.oregan@nuigalway.ie and ravi p. agarwal department of mathematical sciences, florida institute of technology, melbourne, florida 32901, usa email: agarwal@fit.edu abstract using kranoselskii fixed point theorem and mawhin’s continuation theorem we establish the existence of periodic solutions for a second order neutral differential equation with multiple deviating arguments. 1this project is supported by grant 10871213 from nnsf of china, by grant 06021578 from nsf of guangdong. 154 chengjun guo, donal o’regan & ravi p. agarwal cubo 12, 3 (2010) resumen usando el teorema del punto fijo de kranoselskii y el teorema de continuación de mawhin establecemos la existencia de soluciones periódicas de una ecuación diferencial neutral de segundo orden con argumento de desviación multiple. key words and phrases: periodic solution, multiple deviating arguments, neutral differential equation, kranoselskii fixed point theorem, mawhin’s continuation theorem. math. subj. class.: 34k15; 34c25. 1 introduction in this paper, we discuss the second-order neutral differential equation with multiple deviating arguments of the form x ′′ (t) + cx ′′ (t −τ) + a(t)x(t) + g(t, x(t −τ1 (t)), x(t −τ2 (t))··· , x(t −τn (t))) = p(t), (1.1) where |c| < 1, τ is a constant, τi (t)(i = 1, 2,··· , n), a(t) and p(t) are real continuous functions defined on r with positive period t and g(t, x1 , x2,··· , xn) ∈ c(r × r × r × ··· × r, r) and is t−periodic in t. periodic solutions for differential equations were studied in [2-4, 6-10, 12, 15] and we note that most of the results in the literatue concern delay problems. there are only a few papers[1, 5, 11, 13, 14] which discuss neutral problems. for the sake of completeness, we first state kranoselskii fixed point theorem and mawhin’s continuation theorem [3]. theorem a (kranoselskii). suppose that ω is a banach space and x is a bounded, convex and closed subset of ω. let u, s : x →ω satisfy the following conditions: (1) u x + s y ∈ x for any x, y ∈ x ; (2) u is a contraction mapping; (3) s is completely continuous. then u + s has a fixed point in x . let x and y be two banach space and l : d oml ⊂ x −→ y is a linear mapping and n : x −→ y is a continuous mapping. the mapping l will be called a fredholm mapping of index zero if d imk erl = cod imi ml < +∞, and i ml is closed in y. if l is a fredholm mapping of index zero, there exist continuous projectors p : x −→ x and q : y −→ y such that cubo 12, 3 (2010) existence of periodic solutions for a class ... 155 i mp = k erl and i ml = k erq = i m(i − q). it follows that l|d oml∩k erp : (i − p)x −→ i ml has an inverse which will be denoted by kp . if ω is an open and bounded subset of x, the mapping n will be called l−compact on ω if q n(ω) is bounded and kp (i − q)n(ω) is compact. since i mq is isomorphic to k erl, there exists an isomorphism j : i mq −→ k erl. theorem b (mawhin’s continuation theorem[3]). let l be a fredholm mapping of index zero, and let n be l−compact on ω. suppose (1) for each λ ∈ (0, 1) and x ∈ ∂ω, lx 6= λn x and (2) for each x ∈ ∂ω∩ k er(l), q n x 6= 0 and d e g(q n,ω∩ k er(l), 0) 6= 0. then the equation lx = n x has at least one solution in ω∩ d(l). 2 main results now we make the following assumption on a(t): (h1) ( π t )2 > m = max t∈[0,t] a(t) ≥ a(t) ≥ m = mint∈[0,t] a(t) > 0. our main results are the following theorems. theorem 2.1 suppose (h1) holds and also assume there exists a constant k1 > 0 such that (h2) ‖g‖0 ≤ m − 3|c|m −‖p‖0 , where ‖g‖0 = max {t∈[0,t],|x1|≤k1 ,··· ,|xn|≤k1}|g(t, x1 , x2,··· , xn)| and ‖p‖0 = max t∈[0,t]|p(t)|. then eq.(1.1) possesses a nontrivial t−periodic solution. theorem 2.2 suppose (h1) holds and also assume (h3) |g(t, x1 , x2,··· , xn)| ≤ γ ∑ n i=1 |xi|. then eq.(1.1) has at least one t−periodic solution as 0 < γ < 1 n [(1 −|c|)m −|c|m]. in order to prove the main theorems we need some preliminaries. set x := {x|x ∈ c2(r, r), x(t + t) = x(t),∀t ∈ r} and x(0)(t) = x(t) and define the norm on x as follows ||x|| = max t∈[0,t]|x(t)|+ max t∈[0,t]|x ′ (t)|+ max t∈[0,t]|x ′′ (t)|. 156 chengjun guo, donal o’regan & ravi p. agarwal cubo 12, 3 (2010) remark 2.3 if x ∈ x , then it follows that x(i)(0) = x(i)(t)(i = 0, 1, 2). in order to prove our main results, we need the following lemma [10]. lemma 2.4 ([10]). suppose that m is a positive number and satisfies 0 < m < ( π t )2. then for any function ϕ defined in [0, t], the following equation    x ′′ (t) + m x(t) = ϕ(t), x(0) = x(t), x ′ (0) = x ′ (t) has a unique solution x(t) = ´ t 0 g(t, s)ϕ(s)ds, where g(t, s)    w(t − s), (k − 1)t ≤ s ≤ t ≤ kt w(t + t − s), (k − 1)t ≤ t ≤ s ≤ kt(k ∈ n), w(t) = cos α(t− t 2 ) 2αsin αt 2 and α = p m. here max t∈[0,t] ´ t 0 |g(t, s)|ds = 1 m . proof of theorem 2.1: for ∀x ∈ x , define the operators u : x −→ x and s : x −→ x respectively by (u x)(t) = −cx(t −τ) (2.1) and (sx)(t) = cx(t −τ) + ´ t 0 g(t, s)[−cx ′′ (s −τ)(m − a(s))x(s) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds. (2.2) it is clear that a fixed point of u + s is a t−periodic solution of eq.(1.1). we are going to demonstrate that u and s satisfy the conditions of theorem a. let x, y ∈ x and |x| ≤ k1,|y| ≤ k1(here k1 is as in the statement of theorem 2.1). now we prove that |u x + s y| ≤ k1 holds. first, we have the following equality: ´ t 0 g(t, s)x ′′ (s −τ)ds = m ´ t 0 g(t, s)x(s −τ)ds. (2.3) cubo 12, 3 (2010) existence of periodic solutions for a class ... 157 in fact, we have from lemma 2.4 ´ t 0 g(t, s)x ′′ (s −τ)ds = ´ t 0 cosα(t−s− t 2 ) 2αsin tα 2 d[x ′ (s −τ)] + ´ t t cos α(t−s+ t 2 ) 2αsin tα 2 d[x ′ (s −τ)] = cos α(t−s− t 2 ) 2αsin tα 2 x ′ (s −τ)|t 0 −α ´ t 0 sinα(t−s− t 2 ) 2αsin tα 2 d[x(s −τ)] + cos α(t−s+ t 2 ) 2αsin tα 2 x ′ (s −τ)|tt −α ´ t t sin α(t−s+ t 2 ) 2αsin tα 2 d[x(s −τ)] = −α[ sin α(t−s− t 2 ) 2αsin tα 2 x(s −τ)|t 0 + sin α(t−s+ t 2 ) 2αsin tα 2 x(s −τ)|tt ] +α2[ ´ t 0 cos α(t−s− t 2 ) 2αsin tα 2 x(s −τ)ds + ´ t t cos α(t−s+ t 2 ) 2αsin tα 2 x(s −τ)ds] = m ´ t 0 g(t, s)x(s −τ)ds, (2.4) so (2.3) holds. from (h1), (h2) and (2.1)-(2.3), we have |(u y)(t) + (sx)(t)| ≤ |(u y)(t)|+|(sx)(t)| ≤ 2|c|k1 +| ´ t 0 g(t, s)(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds|+|c|k1 ≤ 2|c|k1 + m−mm k1 + ‖g‖0 m +|c|m|| ´ t 0 g(t, s)x(s −τ)ds| ≤ 3|c|k1 + m−mm k1 + ‖g‖0+‖p‖0 m ≤ k1, x, y ∈ x , (2.5) where ‖g‖0 and ‖p‖0 are given in (h2). set k2 = ρ0[(m−m)k1+|c|k3+‖g‖0+‖p‖0] 1−2|c| , (2.6) where ρ0 = t 2 sin tα 2 , k3 = mk1+‖g‖0+‖p‖0 1−|c| (2.7) and g = {x ∈ x : |x(t)| ≤ k1,|x ′ (t)| ≤ k2,|x ′′ (t)| ≤ k3}. it is clear that g is a bounded, convex and closed subset of x . (1) for ∀x, y ∈ g, we will show that | d d t [(u y)(t) + (sx)(t)]| ≤ k2 (2.8) 158 chengjun guo, donal o’regan & ravi p. agarwal cubo 12, 3 (2010) and | d 2[(u y)(t)+(s x)(t)] d t2 | ≤ k3. (2.9) from (2.1) we have d d t [(u x)(t)] = −cx ′ (t −τ) (2.10) and d2[(u x)(t)] d t2 = −cx ′′ (t −τ). (2.11) also from lemma 2.4 and (2.2) we have d d t [(sx)(t)] = ´ t 0 g t(t, s)[(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds + cx ′′ (t −τ), (2.12) where g t(t, s)    w̃(t − s), (k − 1)t ≤ s ≤ t ≤ kt w̃(t + t − s), (k − 1)t ≤ t ≤ s ≤ kt(k ∈ n) and w̃(t) = sin α(t− t 2 ) 2 sin αt 2 , since d d t [(sx)(t)] = { ´ t 0 g t(t, s)[(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds + cx ′′ (t −τ)} ′ = { ´ t 0 cos α(t−s− t 2 ) 2αsin tα 2 [(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds + cx ′′ (t −τ)} ′ +{ ´ s t cos α(t−s+ t 2 ) 2αsin tα 2 [(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds + cx ′′ (t −τ)} ′ = α{ ´ t 0 cos α(t−s− t 2 ) 2αsin tα 2 [(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds + cx ′′ (t −τ)} +α{ ´ s t cosα(t−s+ t 2 ) 2αsin tα 2 [(m − a(s))x(s) − cx ′′ (s −τ) + p(s) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]ds + cx ′′ (t −τ)}. cubo 12, 3 (2010) existence of periodic solutions for a class ... 159 note ´ t 0 |g t(t, s|ds ≤ t 2 sin αt 2 = ρ0 and d2[(s x)(t)] d t2 = p(t) − a(t)x(t) − g(t, x(t −τ1 (t)), x(t −τ2 (t))··· , x(t −τn (t))). (2.13) from (2.6),(2.7) and (2.10)-(2.13), we have | d d t [(u y)(t) + (sx)(t)]| ≤ | d d t [(u y)(t)]|+| d d t [(sx)(t)]| ≤ 2|c|k2 +ρ0[(m − m)k1 +|c|k3 +‖g‖0 +‖p‖0] ≤ k2 (2.14) and | d 2[(u y)(t)+(s x)(t)] d t2 | = |(m − a(t))x(t) − c y ′′ (t −τ) + p(t) −g(t, x(t −τ1 (t)), x(t −τ2 (t))··· , x(t −τn (t)))| ≤ (m − m)k1 +|c|k3 +‖g‖0 +‖p‖0 ≤ k3. (2.15) from (2.5), (2.14) and (2.15), we have u x + s y ∈ g for ∀x, y ∈ g. (2) u is a contraction mapping. let x, y ∈ g and we from (2.1) that ‖u x −u y‖ = max t∈[0,t]|cx(t −τ) − c y(t −τ)|+ max t∈[0,t]|cx ′ (t −τ) − c y ′ (t −τ)| +max t∈[0,t]|cx ′′ (t −τ) − c y ′′ (t −τ)| = |c|[max t∈[0,t]|x(t −τ) − y(t −τ)|+ max t∈[0,t]|x ′ (t −τ) − y ′ (t −τ)| +max t∈[0,t]|x ′′ (t −τ) − y ′′ (t −τ)|] = |c|‖x − y‖. since |c| < 1, u is a contraction mapping. (3) s is completely continuous. we can obtain the continuity of s from the continuity of a(t), p(t) and g(t, x(t−τ1 (t)), x(t− τ2(t))··· , x(t − τn(t))) for t ∈ [0, t], x ∈ g. in fact, suppose that xk ∈ g and ‖xk − s‖ → 0 as 160 chengjun guo, donal o’regan & ravi p. agarwal cubo 12, 3 (2010) k → +∞. since g is closed convex subset of x , we have x ∈ g. then |sxk − sx| = c[xk(t −τ) − x(t −τ)] + c[xk (t −τ) − x(t −τ)] + ´ t 0 g(t, s){(m − a(s))(xk (s) − x(s)) − c[x ′′ k (s −τ) − x ′′ (s −τ)] −[ g(s, xk (s −τ1(s)), xk (s −τ2(s))··· , xk (s −τn (s))) −g(s, x(s −τ1 (s)), x(s −τ2 (s))··· , x(s −τn (s)))]}ds. (2.16) using the lebesgue dominated convergence theorem, we have from (2.12), (2.13) and (2.16) that limk→+∞ ‖sxk − sx‖ = 0. then s is continuous. next, we prove that sx is relatively compact. it suffices to show that the family of functions {sx : x ∈ g} is uniformly bounded and equicontinuous on [0, t]. from (2.2), (2.12) and(2.13), it is easy to see that {sx : x ∈ g} is uniformly bounded and equicontinuity. since s is continuous and is relatively compact, s is completely continuous. by theorem a (kranoselskii fixed point theorem), we have a fixed point x of u + s. that means that x is a t−periodic solution of eq.(1.1). in order to prove theorem 2.2, we need some preliminaries. set z := {x|x ∈ c1(r, r), x(t + t) = x(t),∀t ∈ r} and x(0)(t) = x(t) and define the norm on z as follows ||x|| = max {max t∈[0,t]|x(t)|, max t∈[0,t]|x ′ (t)|}, and set y := { y|y ∈ c(r, r), y(t + t) = y(t),∀t ∈ r}. we define the norm on y as follow ||y||0 = max t∈[0,t]|y(t)|. thus both (z,||·||) and (y ,||·||0) are banach spaces. remark 2.5 if x ∈ z, then it follows that x(i)(0) = x(i)(t)(i = 0, 1). define the operators l : z −→ y and n : z −→ y respectively by lx(t) = x ′′ (t), t ∈ r, (2.17) cubo 12, 3 (2010) existence of periodic solutions for a class ... 161 and n x(t) = −cx ′′ (t −τ) − a(t)x(t) + p(t) −g(t, x(t −τ1 (t)), x(t −τ2 (t)),··· , x(t −τn (t))), t ∈ r. (2.18) clearly, k erl = {x ∈ z : x(t) = c ∈ r} (2.19) and i ml = { y ∈ y : ´ t 0 y(t)dt = 0} (2.20) is closed in y . thus l is a fredholm mapping of index zero. let us define p : z → z and q : y → y /i m(l) respectively by p x(t) = 1 t ´ t 0 x(t)dt = x(0), t ∈ r, (2.21) for x = x(t) ∈ x and q y(t) = 1 t ´ t 0 y(t)dt, t ∈ r (2.22) for y = y(t) ∈ y . it is easy to see that i mp = k erl and i ml = k erq = i m(i − q). it follows that l|d oml∩k erp : (i − p)z −→ i ml has an inverse which will be denoted by kp . let ω be an open and bounded subset of z, we can easily see that q n(ω) is bounded and kp (i − q)n(ω) is compact. thus the mapping n is l−compact on ω. that is, we have the following result. lemma 2.6. let l, n, p and q be defined by (2.17), (2.18), (2.21) and (2.22) respectively. then l is a fredholm mapping of index zero and n is l−compact on ω, where ω is any open and bounded subset of z. in order to prove theorem 2.2, we need the following lemma [12]. lemma 2.7 ([12 and remark 2.5]). let x(t) ∈ c(n)(r, r) ∩ ct . then ||x(i)||0 ≤ 12 ´ t 0 |x(i+1)(s)|ds, i = 1, 2,··· , n − 1, where n ≥ 2 and ct := {x|x ∈ c(r, r), x(t + t) = x(t),∀t ∈ r}. now, we consider the following auxiliary equation x ′′ (t) +cλx ′′ (t −τ) + a(t)λx(t) = λp(t) −λg(t, x(t −τ1 (t)), x(t −τ2 (t)),··· , x(t −τn (t))), (2.23) where 0 < λ < 1. lemma 2.8. suppose that conditions of theorem 2.2 are satisfied. if x(t) is a t−periodic 162 chengjun guo, donal o’regan & ravi p. agarwal cubo 12, 3 (2010) solution of eq.(2.23), then there are positive constants d i (i = 0, 1), which are independent of λ, such that ||x(i)||0 ≤ d i , t ∈ [0, t], i = 0, 1. (2.24) proof: suppose that x(t) is a t−periodic solution of (2.23). we have from (h3) and (2.23) that |x ′′ (t)| ≤ max t∈[0,t]|c||x ′′ (t)|+ m||x||0 +||p||0 +γn||x||0 . (2.25) from (2.25), we have max t∈[0,t]|x ′′ (t)| ≤ 1 1−|c| [(m +γn)||x||0 +||p||0 ]. (2.26) on the other hand, from lemma 2.4 and (2.23), we get x(t) = ´ t 0 g̃(t, s)λ[(m − a(s))x(s) + p(s) − cx ′′ (s −τ) −g(s, x(s −τ1 (s)), x(s −τ2 (s)),··· , x(s −τn (s))]ds, (2.27) where g̃(t, s)    w̃(t − s), (k − 1)t ≤ s ≤ t ≤ kt w̃(t + t − s), (k − 1)t ≤ t ≤ s ≤ kt(k ∈ n), (2.28) w̃(t) = cosα1(t− t2 ) 2α1 sin α1 t 2 , (2.29) α1 = p λm and max t∈[0,t] ´ t 0 |g̃(t, s)|ds = 1 λm . (2.30) from (h3), (2.27) and (2.30), we have ‖x‖0 = max t∈[0,t]| ´ t 0 g̃(t, s)λ[(m − a(s))x(s) + p(s) − cx ′′ (s −τ) −g(s, x(s −τ1 (s)), x(s −τ2 (s)),··· , x(s −τn (s))]ds| ≤ 1 m [(m − m)‖x‖0 +‖p‖0 +|c|max t∈[0,t]|x ′′ (t)|+γn‖x‖0 ]. (2.31) from (2.31), we have ‖x‖0 ≤ |c|max t∈[0,t]|x ′′ (t)|+‖p‖0 m−γn . (2.32) thus combining (2.26) and (2.32), we see that max t∈[0,t]|x ′′ (t)| ≤ m+m m(1−|c|)−m|c|−γn = ξ (2.33) and ‖x‖0 ≤ |c|ξ+‖p‖0 m−γn = d0. (2.34) cubo 12, 3 (2010) existence of periodic solutions for a class ... 163 finally from lemma 2.4, (2.33) and (2.34), we get ||x ′ ||0 ≤ d1. (2.35) the proof of lemma 2.8 is complete. proof of theorem 2.2: suppose that x(t) is a t-periodic solution of eq.(2.23). by lemma 2.8, there exist positive constants d i(i = 0, 1) which are independent of λ such that (2.24) is true. consider any positive constant d > max 0≤i≤1{d i } +‖p‖0 . set ω := {x ∈ z : ||x|| < d}. we know that l is a fredholm mapping of index zero and n is l-compact on ω(see [3]). recall k er(l) = {x ∈ z : x(t) = c ∈ r} and the norm on z is ||x|| = max {max t∈[0,t]|x(t)|, max t∈[0,t]|x ′ (t)|}. then we have x = d or x = −d for x ∈ ∂ω∩ k er(l). (2.36) from (h3) and (2.36), we have(if d is chosen large enough) a(t)d + g(t, d , d,··· , d) −‖p‖0 > 0 for t ∈ [0, t] (2.37) and x ′ (t) = 0 and x ′′ (t) = 0, for t ∈ [0, t]. (2.38) finally from (2.18), (2.22), (2.37) and (2.38), we have (q n x) = 1 t ´ t 0 [−cx ′′ (t −τ) − a(t)x(t) + p(t)]dt −g(t, x(t −τ1 (t)), x(t −τ2 (t)),··· , x(t −τn (t)))]dt 6= 0, ∀x ∈ ∂ω∩ k er(l). then, for any x ∈ k erl ∩∂ω and η ∈ [0, 1], we have xh(x,η) = −ηx2 − x t (1 −η) ´ t 0 [cx ′′ (t −τ) + a(t)x(t) − p(t) +g(t, x(t −τ1 (t)), x(t −τ2 (t)),··· , x(t −τn (t)))dt]dt 6= 0. 164 chengjun guo, donal o’regan & ravi p. agarwal cubo 12, 3 (2010) thus d e g{q n, ω∩ k er(l), 0} = d e g{− 1 t ´ t 0 [cx ′′ (t −τ) + a(t)x(t) − p(t) +g(t, x(t −τ1 (t)), x(t −τ2 (t)),··· , x(t −τn (t)))]dt,ω∩ k er(l), 0} = d e g{−x,ω∩ k er(l), 0} 6= 0. from lemma 2.8 for any x ∈ ∂ω∩ d om(l) and λ ∈ (0, 1) we have lx 6= λn x. by theorem b (mawhin’s continuation theorem), the equation lx = n x has at least a solution in d om(l)∩ω, so there exists a t-periodic solution of eq.(1.1). the proof is complete. references [1] chen, y.s., the existence of periodic solutions for a class of neutral differential difference equations, bull. austral. math. soc., 33 (1992), 508–516. [2] chen, y.s., the existence of periodic solutions of the equation x′(t) = −f (x(t), x(t − r)), j. math. anal. appl., 163 (1992), 227–237. [3] gaines, r.e. and mawhin, j.l., coincidence degree and nonlinear differential equation, lecture notes in math., vol.568, springer-verlag, 1977. [4] guo, z.m. and yu, j.s., multiplicity results for periodic solutions to delay differential difference equations via critical point theory, j. diff. eqns., 218 (2005), 15–35. [5] guo, c.j. and guo, z.m., existence of multiple periodic solutions for a class of threeorder neutral differential equations, acta. math. sinica, 52(4) (2009), 737–751. [6] guo, c.j. and guo, z.m., existence of multiple periodic solutions for a class of secondorder delay differential equations, nonlinear anal-b: real world applications, 10(5) (2009), 3825–3972. [7] hale, j.k., theory of functional differential equations, springer-verlag, 1977. 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() cubo a mathematical journal vol.13, no¯ 02, (37–57). june 2011 remarks on the generation of semigroups of nonlinear operators on p-fréchet spaces, 0 < p < 1 sorin g. gal university of oradea, department of mathematics, str. universitatii 1, 410087 oradea, romania email: galso@uoradea.ro abstract in this paper we study the convergence properties of the crandall-liggett sequence jn t/n (a)(x) = ( i − t n a )−n (x), n ∈ n, for a a nonlinear operator on some important non-locally convex f-spaces (called p-fréchet spaces with 0 < p < 1) and the generation of the corresponding strongly continuous one-parameter nonlinear semigroups. resumen en este trabajo se estudian las propiedades de convergencia de la secuencia de crandallliggett jn t/n (a)(x) = ( i − t n a )−n (x), n ∈ n para a un operador lineal en algunos importantes f-espacios no-localmente convexos (llamado p-fréchet espacios con 0 < p < 1) y la generación de los correspondientes semigrupos fuertemente continuos no lineales con un parámetro. keywords and phrases:: p-fréchet space, 0 < p < 1, cauchy problem, affine semigroup, nonlinear semigroup, crandall-liggett type theorem. mathematics subject classification: 47h06, 47h20. 38 sorin g. gal cubo 13, 2 (2011) 1. introduction it is well known that an f-space (x, +, ·, || · ||) is a linear space (over the field k = r or k = c) such that ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ x, ||x|| = 0 if and only if x = 0, ||λx|| ≤ ||x||, for all scalars λ with |λ| ≤ 1, x ∈ x, and with respect to the metric d(x, y) = ||x − y||, x is a complete metric space (see e.g. [4, p. 52] or [7]). in addition, if there exists 0 < p < 1 with ||λx|| = |λ|p||x||, for all λ ∈ k, x ∈ x, then || · || will be called a p-norm and x will be called p-fréchet space. (this is only a slight abuse of terminology. note that in e.g. [1] these spaces are called p-banach spaces). it is known that the f-spaces are not necessarily locally convex spaces. three classical examples of p-fréchet spaces, non-locally convex, are the hardy space hp with 0 < p < 1 that consists in the class of all analytic functions f : d → c, d = {z ∈ c; |z| < 1} with the property ||f|| = 1 2π sup{ ∫ 2π 0 |f(reit)|pdt; r ∈ [0, 1)} < +∞, the sequence space lp = {x = (xn)n; ||x|| = ∞∑ n=1 |xn| p < ∞} for 0 < p < 1, and the lp[0, 1] space, 0 < p < 1, given by lp = lp[0, 1] = {f : [0, 1] → r; ||f|| = ∫ 1 0 |f(t)|pdt < ∞.} some important characteristics of the f-spaces are given by the following remarks. remarks. 1) three of the basic results in functional analysis hold in f-spaces too : the principle of uniform boundedness (see e.g. [4, p. 52]), the open mapping theorem and the closed graph theorem (see e.g. [7, p. 9-10]). 2) the hahn-banach theorem fails in non-locally convex f-spaces. more exactly, if in an fspace the hahn-banach theorem holds, then that space is a necessarily locally convex space (see e.g. [6, chapter 4]). the beginning of a theory of semigroups of linear operators on p-fréchet spaces, 0 < p < 1, was developed in the very recent paper [5]. one of the main result in [5] is the chernoff-type formula eta(x) = ĺımn→ ∞ ( i − t n a )−n (x), for a a bounded linear operator on a p-fréchet space with 0 < p < 1. the aim of the present paper is to look for similar results, that is for convergence properties of the sequence jn t/n (a)(x) = ( i − t n a )−n (x), n ∈ n, in the case when a is a nonlinear operator on a p-fréchet space with 0 < p < 1. a very careful examination of the proofs in [3] shows us that because of the property ||λx|| = |λ|p||x|| with 0 < p < 1, the estimate for ||jn t/n (a)(x) − jm t/m (a)(x)|| does not converges to zero as m, n → ∞ and in fact the sequence jn t/n (a)(x), n ∈ n, is not, in general, a convergent one. cubo 13, 2 (2011) remarks on the generation of semigroups . . . 39 however, by using techniques in functional analysis, we will be able to prove that the sequence (jn t/n (a)(x))n∈n contains some convergent subsequences in the spaces l p and hp with 0 < p < 1, while this kind of result seems to fail in the space lp[0, 1], 0 < p < 1. moreover, in the simplest nonlinear case when a is an affine operator, we prove that the sequence (jn t/n (a)(x))n∈n is still convergent and some results in the case of banach spaces in [6] will be extended to p-fréchet spaces (0 < p < 1) too. the plan of the paper goes as follows. in section 2 we study the case when a is an affine operator on an arbitrary p-fréchet space, 0 < p < 1, section 3 deals with the case when a is a nonlinear lipschitz operator on lp, 0 < p < 1, while the sections 4 and 5 deal with the similar problem in the spaces hp and lp[0, 1], respectively, with 0 < p < 1. 2. affine semigroups as we will see, the affine case is closely connected to the linear case. first we need a result in operator theory on p-fréchet spaces (well-known in the case of classical banach spaces). lemma 2.1 let a, b : x → x be bounded linear operators on the p-fréchet space (x, || · ||), 0 < p < 1. if a is bijection and |||a−1b||| < 1 then a + b is bounded linear bijection on x. proof. since a is a bijection, as a consequence of the open mapping theorem it follows that a−1 is a bounded linear operator (see e.g. [1, theorem 14, p. 20 and corollary 2, p. 23]). next we reason as in the case of banach spaces. let y ∈ x be arbitrary fixed and define ty(x) = a −1(y) − (a−1b)(x). then the equation (a + b)(x) = y is equivalent to the equation ty(x) = x. but ||ty(x1) − ty(x2)|| ≤ |||a −1b||| · ||x1 − x2||, which shows that ty is a contraction in the complete metric space x (with respect to the metric d(x1, x2) = ||x1 − x2||). therefore it has a unique fixed point x, which shows that a + b is bijective and the lemma is proved. the first result on affine semigroups is the following. theorem 2.2 let (x, || · ||) be a p-fréchet space, 0 0 sufficiently small. then b is invertible and if we define jλ(a)(x) = (i − λa) −1(x), (here i defines the identity operator) then t (t)(x) = ĺım n→ +∞ jnt/n(a)(x) = e tb(x) + b−1[etb(x0)] − b −1(x0), is a strongly continuous semigroup of nonlinear (affine) operators on x, (where according to [4], etb(x) = ĺımn→ +∞ j n t/n (b)(x) is a strongly continuous semigroup of linear operators on x). 40 sorin g. gal cubo 13, 2 (2011) proof. by easy calculation we can write jλ(a)(x) = jλ(b)(x + λx0) = (i − λb) −1(x + λx0), and in general jnλ (a)(x) = (i − λb) −n(x) + λ [ n∑ k=1 (i − λb)−k(x0) ] . but it is easy to show that for any operator g we have the identity (i − g)(i + g + g2 + ... + gn−1) = i − gn. replacing g by jλ(b), by lemma 2.1 it follows that i − jλ(b) is invertible and we immediately obtain [ n∑ k=1 (i − λb)−k(x0) ] = λjλ(b)[i − jλ(b)] −1[i − jnλ (b)] −n(x0). but jλ(b)[i − jλ(b)] −1 = (i − λb)−1[i − jλ(b)] −1 = {[i − (i − λb)−1][i − λb]}−1 = {−λb}−1 = − 1 λ b−1, which implies that [ n∑ k=1 (i − λb)−k(x0) ] = −b−1[(i − λb)−n](x0). taking λ = t n , passing to limit with n → +∞ and taking into account the important remark after the theorem 2.11 in [4] which says that etb(x) = ĺım n→ +∞ (i − t n b)−n, we arrive at ĺım n→ +∞ jnt/n(a)(x) = e tb + b−1[etb(x0)] − b −1(x0). also, simple calculations show that if we denote t (t)(x) = etb(x) + b−1[etb(x0)] − b −1(x0), then t (0) = i, {t (t), t ≥ 0} has the semigroup property, t (·)(x) is continuous as function of t, a(x) = ĺımhց 0 t (h)(x)−x h , for all x ∈ x and t ′(t)(x) = b[t (t)(x)] + x0, which proves the theorem. remarks. 1) according to theorem 2.2, t (t)(u0) is the unique solution of the cauchy problem u′(t) = b[u(t)] + x0, u(0) = u0. (the uniqueness of the solution follows from lemma 2.12 in [5] concerning the uniqueness of the solution for the inhomogeneous cauchy problem in p-fréchet spaces, 0 < p < 1.) cubo 13, 2 (2011) remarks on the generation of semigroups . . . 41 2) let us give a simple example satisfying theorem 2.2. let (x, || · ||) be a p-fréchet space, 0 < p < 1, and define a : x → x by a(x) = b(x) + x0, where b(x) = −x for all x ∈ x and x0 ∈ x is fixed. b obviously is strictly dissipative and a obviously is nonlinear, strictly dissipative, with ||(i − λa)−1||lip = 1 1 + λ < 1, for all λ > 0. we see that b−1 = b, etb(x) = xe−t and t (t)(x) = (x − x0)e −t + x0 and in this case t (t)(u0) is the unique solution to the nonlinear cauchy problem du dt = −u(t) + x0, u(0) = u0. 3) from the proof of theorem 2.2, it easily follows the following. corollary 2.3 in the case when (x, ||·||) is a banach space (i.e. a p-fréchet space with p = 1), the statement of theorem 2.2 still remains true. in what follows, let us consider some concepts introduced in [6] for banach spaces. they remain unchanged for the case of p-fréchet spaces too. definition 2.4 by an affine semigroup (s(t) : t ≥ 0) on a p-fréchet space x, 0 < p < 1, we mean a family of continuous affine transformations on x with the properties : (i) s(0)=i, s(t+s)=s(t)[s(s)], for all t, s ≥ 0 ; (ii) for each x ∈ x, t → s(t)(x) is a continuous function from [0, +∞) into x. (iii) any family (s(t) : t ≥ 0) of affine transformations on x can be written in the form s(t)(x) = t (t)(x) + z(t), for all t ≥ 0, x ∈ x, where t (t)(x) = s(t)(x) − s(t)(0) is its linear part and z(t) = s(t)(0) is its translation part (z : [0, +∞) → x). (iv) let us denote by x = x × r. it is a p-fréchet space, endowed with the p-norm ||(x, r)|| = máx{||x||, |r|p}. if (s(t) : t ≥ 0) is a family of affine transformations on x of the form s(t)(x) = t (t)(x)+z(t), for all t ≥ 0, x ∈ x, where t (t)(x) is its linear part and z(t) is its translation part, the augmented family associated with (s(t) : t ≥ 0), is a family (t (t); t ≥ 0) of linear transformations on x, defined by t (t)[x, r] = [t (t)(x) + rz(t), r]. having introduced these concepts, propositions 1.1 and 1.2 proved in [6] for banach spaces, hold (with the same proofs) for p-fréchet spaces too, summarized as follows. theorem 2.5 (i) let (s(t) : t ≥ 0) be a family of affine transformations on the p-fréchet space x, 0 < p < 1, with its linear part (t (t) : t ≥ 0) and its translation part z(t); t ≥ 0). then (s(t) : t ≥ 0) is an affine semigroup on x if and only if (t (t) : t ≥ 0) is a linear semigroup on x and z(·) is a continuous map from [0, +∞) into x satisfying z(t + s) = t (t)[z(s)] + z(t), s, t ≥ 0. 42 sorin g. gal cubo 13, 2 (2011) (ii) let (s(t) : t ≥ 0) be a family of affine transformations on the p-fréchet space x, 0 < p < 1, and let (t (t) : t ≥ 0) be the augmented family on x, associated with s(·). then (s(t) : t ≥ 0) is an affine semigroup on x, if and only if (t : t ≥ 0) is a linear semigroup on x. remark. while proposition 2.1 in [6] remains valid in the case of p-fréchet spaces too, 0 < p < 1, the other results in [6, section 2] (i.e. corollary 2.2, proposition 2.3, proposition 2.4 and corollary 2.5) seem to be not valid. the reason is that they use the fundamental theorem of calculus in banach spaces, which, as it was pointed out in [5], does not hold in p-fréchet spaces, 0 < p < 1. it would be of interest to see what other results for affine semigroups on banach spaces in [6], would remain valid for p-fréchet spaces too, 0 < p < 1. 3. nonlinear semigroups on lp, 0 < p < 1 before to starting the study in the concrete lp-case, 0 < p < 1, let us briefly recall the problem and make a useful remark, valid in any p-fréchet space, 0 < p < 1. for (x, || · ||x) a p-fréchet space, 0 < p ≤ 1 (the case p = 1 means that x is a banach space), let a : x → x be a nonlinear operator and let us consider the abstract cauchy problem d dt u(t) = a[u(t)], t ≥ 0, u(0) = x, where the solution is u : r+ → x and x ∈ x is fixed. the nonlinear operator a is considered a lipschitz mapping, that is ||a(x) − a(y)||x ≤ |||a|||lip||x − y||x, for every x, y ∈ x, where |||a|||lip = sup{||a(x)−a(y)||x/||x−y||x; x, y ∈ x, x 6= y} < +∞. if we replace this differential equation by the difference equation 1 ε [uε(t) − uε(t − ε)] = a[uε(t)], t ≥ 0, with initial condition uε(s) = x, −ε ≤ s ≤ 0, then we easily get by recurrence that uε(t) = ( i − t n a )−n (x), for ε = t n . remark. without loss of generality, we may suppose a(0) = 0. indeed, if we suppose that a(0) 6= 0, then denoting b(u) = a(u) − a(0) we get b(0) = 0 and if v(t) is solution of the abstract cauchy problem d dt v(t) = b[v(t)], v(0) = u0, then u(t) = v(t)+ta(0) is a solution of the above (in a) mentioned problem. moreover, if for a fixed ω ∈ r, the operator a − ωi is dissipative, then b − ωi also is dissipative. indeed, from b − ωi = cubo 13, 2 (2011) remarks on the generation of semigroups . . . 43 (a − ωi) − a(0), since a − ωi is injective and surjective, it easily follows that b − ωi is injection and surjection and , in addition, from the relationship (b − ωi)−1(y) = (a − ωi)−1(y + a(0)), we get ||(b − ωi)−1||lip = ||(a − ωi) −1||lip ≤ 1. in what follows, we denote the p-norm in lp by || · ||p. the first main result of this section is the following. theorem 3.1 let a : lp → lp, 0 < p < 1, be nonlinear and lipschitz, such that a(0) = 0 and there exists ω ∈ r with a − ωi dissipative. then the sequence in lp defined by jn t/n (x) = (i − t n a)−n(x), n ∈ n, t ≥ 0, x ∈ lp, contains a subsequence j nk t/nk (x), k ∈ n (the same subsequence for all x ∈ lp and all t ∈ r+ = the set of all rational numbers ≥ 0), convergent to an element of lp in the weak topology of lp. proof. by a(0) = 0 we get (i − t n a)−n(0) = 0, for all n ∈ n. the dissipative property implies ||[i − t(a − ωi)]−1||lip ≤ 1, for all t ≥ 0, which is equivalent to ||(i − t 1+tω a)−1||lip ≤ |1 + tω| p, for all t ≥ 0 with 1 + tω 6= 0. for λ = t 1+tω we get ||(i − λa)−1||lip ≤ (1 − λω) −p, in particular for all λ > 0 with λω < 1. (note that for n sufficiently great, depending on t and ω, we have t n ω < 1.) therefore, ||(i − t n a)−1||lip ≤ (1 − t n ω)−p and by mathematical induction ||(i − t n a)−n||lip ≤ (1 − t n ω)−np. but it is known that the sequence (1+ s n )n converges (for n → +∞) to es, and for any s ∈ r it is monotonically increasing, for all n ≥ [|s|]+1 (see e.g. [10, p. 263]), which implies that (1− t n ω)−np converges to etωp, monotonically decreasing, for all n ≥ [|tw|] + 1. therefore, the greatest value of (1 − t n ω)−np is for n = [|tω|] + 1, which means that there exists m = m(t, p, ω) > 0 (depending only on t, p and ω) such that ||(i − t n a)−n||lip ≤ m, for all n ∈ n. we obtain ||jnt/n(x) − j n t/n(y)||p ≤ (1 − t n ω)−np||x − y||p ≤ m||x − y||p, (3.1) for all x, y ∈ lp. taking y = 0 we have jn t/n (0) = 0 and denoting jn t/n (x) = (gn,r(t)(x))r ∈ l p, we obtain ||jn t/n (x)||p ≤ m||x||p, i.e. +∞∑ r=1 |gn,r(t)(x)| p ≤ m||x||p < +∞, (3.2) for all n ∈ n. 44 sorin g. gal cubo 13, 2 (2011) now, since lp, 0 < p < 1, has a schauder basis (see e.g. [7, p. 20]), it follows that it is separable, denote by y a countable dense subset of lp. also, denote by r+, the set of all positive nonnegative rational numbers and define gn : n×r+ ×y → k (where k = r or c), by gn(r, t, y) = gn,r(t)(y). since e := n ×r+ × y is countable and by (2) the sequence (gn)n is pointwise bounded on e, by the cantor’s diagonal process (see e.g. [11, p. 156-157]), there exists a subsequence gnk , k ∈ n, pointwise convergent on e. denote gr(t)(y) = ĺımk→ +∞ gnk,r(t)(y), for all (r, t, y) ∈ e. we will show that in fact there exists the limit ( in r ), ĺımk→ ∞ gnk,r(t)(x), for all r ∈ n, t ∈ r+ and x ∈ l p. for this purpose, we will show that (gnk,r(t)(x))k is a cauchy sequence in r (i.e. it is convergent). for this purpose, let x ∈ lp and y ∈ y. we have |gnk,r(t)(x) − gns,r(t)(x)| ≤ |gnk,r(t)(x) − gnk,r(t)(y)|+ |gnk,r(t)(y) − gns,r(t)(y)| + |gns,r(t)(y) − gns,r(t)(x)|. taking into account (1) too, we immediately obtain |gnk,r(t)(x) − gns,r(t)(x)| ≤ 2m 1/p · ||x − y||1/pp + |gnk,r(t)(y) − gns,r(t)(y)|. now, since y is dense in lp, for x ∈ lp and ε > 0, let y ∈ y such that 2m1/p||x − y|| 1/p p < ε 2 , which implies |gnk,r(t)(x) − gns,r(t)(x)| < ε 2 + |gnk,r(t)(y) − gns,r(t)(y)|. but for this y, the sequence (gnk,r(t)(y))k is convergent, i.e. it is a cauchy sequence, which implies that there exists l0 such that for all k, s ≥ l0 we have |gnk,r(t)(y) − gns,r(t)(y)| < ε 2 . this leads to |gnk,r(t)(x) − gns,r(t)(x)| < ε, for all k, s ≥ l0, i.e. (gnk,r(t)(x))k is a cauchy sequence in r. therefore, we can write gr(t)(x) = ĺım k→ ∞ gnk,r(t)(x), for all t ∈ r+ and x ∈ l p. by (2) it follows m∑ r=1 |gnk,r(t)(x)| p ≤ m||x||p < +∞, for all k, m ∈ n. passing here to limit with k → ∞, we get m∑ r=1 |gr(t)(x)| p ≤ m||x||p < +∞, cubo 13, 2 (2011) remarks on the generation of semigroups . . . 45 for all m ∈ n, which obviously implies ∞∑ r=1 |gr(t)(x)| p ≤ m||x||p < +∞, i.e. g(t)(x) := (gr(t)(x))r belongs to l p. now, we will show that for any x∗ ∈ (l1)∗, i.e. of the form (see e.g. [8, pp. 36-37]) x∗(z) =∑∞ j=1 ujzj, for all z ∈ l 1, where (uj)j ∈ m, with m denoting the space of all bounded sequences, we have x∗(j nk t/nk (x)) → x∗(g(t)(x)), when k → ∞, for any fixed t ∈ r+, x ∈ lp. note that x∗(g(t)(x)) has sense for g(t)(x) ∈ lp, because lp ⊂ l1. it is obvious that each functional of the form x∗i (x) = xi, for all x = (xi)i ∈ l 1, is linear and continuous on l1, since |x∗i (x)| = |xi| ≤ ∑∞ j=1 |xj| = ||x||l1 and for k → ∞, x∗i [j nk t/nk (x)] = gnk,i(t)(x) → gi(t)(x) = x∗i (g(t)(x)), for all i ∈ n. then, obviously that for any y∗ ∈ span{x∗1, ..., x ∗ i , ..., } =: y ∗ we also have y∗[j nk t/nk (x)] → y∗[g(t)(x)], for k → ∞. we show that y∗ is dense in (l1)∗ in the weak topology on (l1)∗. indeed, let x∗ ∈ (l1)∗ be arbitrary, x∗(u) = ∑∞ i=1 αiui, for all u = (uj)j ∈ l 1, where α = (αj)j ∈ m. since z ∗ n(u) =∑n j=1 αjuj = ∑n j=1 αjx ∗ j (u), it follows z ∗ n ∈ y ∗ and we get |x∗(u) − z∗n(u)| ≤ +∞∑ j=n+1 |αjuj| ≤ ||α||m +∞∑ i=n+1 |ui| ≤ m0 +∞∑ i=n+1 |ui| → 0, for n → ∞. this implies that z∗n → x∗ in the weak topology (i.e. the density of y∗ in (l1)∗ in the weak topology) and that for any ε > 0 and any u1, u2 ∈ l 1, x∗ ∈ (l1)∗, there exists y∗ ∈ y∗, such that |x∗(uj) − y ∗(uj)| < ε, j = 1, 2. for u1 = j nk t/nk (x) and u2 = g(t)(x), we get |x∗[j nk t/nk (x)] − x∗[g(t)(x)]| ≤ |x∗[j nk t/nk (x)] − y∗[j nk t/nk (x)]|+ |y∗[j nk t/nk (x)] − y∗[g(t)(x)]| + |y∗[g(t)(x)] − x∗[g(t)(x)]| < 2ε + |y∗[j nk t/nk (x)] − y∗[g(t)(x)]| < 3ε, for all k > k0, with k0 depending on ε, t and x. this shows that for any x∗ ∈ (l1)∗, if k → ∞ then we have x∗[jnk t/nk (x)] → x∗[g(t)(x)], for any fixed t ∈ r+ and x ∈ l p. finally, since according to [7], p. 27, l1 is the so-called banach envelope of lp and (l1)∗ = (lp)∗ (with the same dual norms too), the theorem is proved. remarks. 1) we may repeat the reasonings in the proof of theorem 3.1 for the sequence (jn t/n (x), n ∈ n, n 6= nk), where nk is the subsequence in theorem 3.1, so that by mathematically 46 sorin g. gal cubo 13, 2 (2011) induction we easily obtain that the sequence (jn t/n (x))n∈n has at most a countable set of limit points in the weak topology of lp, denote that set by t ∗(t)(x), where t ∈ r+, x ∈ l p. for any fixed t ∈ r+, an element a ∈ t ∗(t) is in fact a mapping a : lp → lp. 2) from the proof of theorem 3.1, we easily can derive that in addition, the functions gn,r(t) : lp → r are lipschitz functions, i.e. |gn,r(t)(x) − gn,r(t)(y)| ≤ m1/p||x − y||1/pp , which implies that the family (gn,r(t))n,r∈n is equicontinuous. also, for any x ∈ l p, t ∈ r+, the sequence (gn,r(t)(x))n,r∈n is bounded . unfortunately we cannot apply the classical arzela-ascoli theorem in lp, because lp is not locally compact. however, we may impose some additional properties to the nonlinear operator a, which could imply better convergence results in theorem 3.1, as follows. consider on lp the so called lexicographic order, i.e. for x = (xj)j, y = (yj)j ∈ l p, we write x ≤ y if and only if xj ≤ yj, for all j ∈ n and x < y if and only if x ≤ y and there is a j with xj < yj. the following simple result holds. lemma 3.2 suppose that a : lp → lp is a dissipative nonlinear operator, a(0)=0, a is convex and non-increasing with respect to the above order, i.e. a[αx + (1 − α)y] ≤ αa(x) + (1 − α)a(y), for all x, y ∈ lp, α ∈ [0, 1] and x < y implies a(x) ≥ a(y). we have : (i) i − λa is concave and non-decreasing, for any λ > 0 ; (ii) b := (i − λa)−1 is convex and non-decreasing, for any λ > 0 ; (iii) bn is convex and non-decreasing, for any λ > 0. the proof is an easy exercise and it is left to the reader. remark. lemma 3.2 says that if a is convex and non-increasing, then so is jn t/n (x), which obviously implies that the functions gn,r(t) : l p → r in the proof of theorem 3.1 are convex and non-decreasing. corollary 3.3 denote by t (t)(x) ∈ lp the weak limit in lp of the sequence (j nk t/nk (x))k, for all t ∈ r+ and x ∈ l p, where (nk)k is the subsequence in theorem 3.1. we have (i) t (0) = i ; (ii) ||t (t)(x) − t (t)(y)||l1 ≤ e tω||x − y|| 1/p p , for all t ∈ r+, x, y ∈ l p ⊂ l1 ; (iii) for any t, s ∈ r+ and a ∈ t ∗(t + s), there exist b ∈ t ∗(t) and c ∈ t ∗(s) such that a(x) = b[c(x)], for all x ∈ lp. proof. (i) it is obvious by the definition of j nk t/nk (x) ; (ii) first, passing to limit with k → +∞ in the following inequality in the proof of theorem cubo 13, 2 (2011) remarks on the generation of semigroups . . . 47 3.1 ||j nk t/nk (x) − j nk t/nk (y)||p ≤ (1 − t nk ω)−nkp||x − y||p, we easily get limk→ ∞ ||j nk t/nk (x) − j nk t/nk (y)||p ≤ limk→ ∞ (1 − t nk ω)−nkp||x − y||p = e tωp ||x − y||p. let x∗ ∈ (lp)∗ be with |||x∗|||(lp)∗ ≤ 1. according to [7, p. 27], it can be extended to a x∗ ∈ (l1)∗, preserving its norm, i.e. |||x∗|||(l1)∗ ≤ 1. we have |x∗[t (t)(x)] − x∗[t (t)(y)]| = |x∗[t (t)(x) − t (t)(y)]| ≤ |x∗[t (t)(x)] − x∗[j nk t/nk (x)]| + |x∗[j nk t/nk (x)] − x∗[j nk t/nk (y)]|+ |x∗[j nk t/nk (y)] − x∗[t (t)(y)]| := ak + |x ∗[j nk t/nk (x)] − x∗[j nk t/nk (y)]| + bk, where limk→ ∞ ak = limk→ ∞ bk = 0 by the definitions of t (t)(x) and t (t)(y). on the other hand, |x∗[j nk t/nk (x)] − x∗[j nk t/nk (y)]| ≤ |||x∗|||(l1)∗ ||j nk t/nk (x) − j nk t/nk (y)||l1 ≤ ||j nk t/nk (x) − j nk t/nk (y)||l1 ≤ ||j nk t/nk (x) − j nk t/nk (y)||1/pp . passing in the above two inequalities to limit with k → ∞, we get |x∗[t (t)(x)] − x∗[t (t)(y)]| ≤ etω||x − y||1/pp , for all x∗ ∈ (l1)∗, with |||x∗|||(l1)∗ ≤ 1. passing here to supremum with such x ∗, by a classical result in functional analysis for normed spaces, it follows sup|||x∗||| (l1)∗ ≤1|x ∗[t (t)(x)] − x∗[t (t)(y)]| = ||t (t)(x) − t (t)(y)||l1 ≤ e tω ||x − y||1/pp . (iii) let q ∈ n, t ∈ r+, x ∈ l p be fixed. for any x∗ ∈ (lp)∗, we have limk→ +∞ x ∗[j nk qt/nk (x)] = x∗([t (qt)](x)), which immediately implies limk→ +∞ x ∗(j qnk qt/qnk (x)) = limk→ +∞ x ∗(j qnk t/nk (x)) = ĺım k→ +∞ x∗([t (t)]q(x)). applying the same reasonings as in the proof of theorem 3.1, there exists a subsequence of (qnk)k, let us denote it by (qk)k, such that ĺım k→ +∞ x∗[j qk qt/qk (x)] = x∗([t (t)]q(x)), 48 sorin g. gal cubo 13, 2 (2011) which shows that for all t ∈ r+, if a ∈ t ∗(qt), then there exists b ∈ [t ∗(t)]q with a = b. then, for l, k, r, s ∈ n, we easily get that for any a ∈ t ∗( l k + r s ) = t ∗ ( ls+rk ks ) , there exists d ∈ [t ∗( 1 ks )]ls+kr with a = d. on the other hand, denoting ks = t, we have j n(ls+kr) t/n (x) = (i − t n a)−n(ls+kr)(x) = (i − t n a)−nls[(i − t n a)−nkr(x)], so for the above d, there exists a subsequence (nj)j with d = ĺımj→ +∞ x ∗(j nj(ls+kr) t/nj (x)) and there exist b ∈ [t ∗(t)]ls and c ∈ [t ∗(t)]kr such that d(x) = b[c(x)]. the corollary is proved. the next result shows that for some particular nonlinear operators, the whole sequence (jn t/n (x))n is convergent in the l p space. theorem 3.4 let a : lp → lp, 0 0 such that |fk(α) − fk(β)| ≤ m|α − β|, for all k ∈ n, α, β ∈ r. then, for any t ≥ 0 and x ∈ lp, the sequence jn t/n (x) = (i− t n a)−n(x), n ∈ n is strongly convergent to a limit in lp. proof. first by definition it easily follows that a is a lipschitz operator with respect to the || · ||p-norm in l p. then, we can write jn t/n (x) = (gn,k(t)(xk))k, where gn,k(t) : r → r are given by gn,k(t)(u) = (i − t n fk) −n(u). by the hypothesis, it follows that each sequence (gn,k(t)(u))k is convergent in the banach space r, denote gk(t)(u) = ĺımn→ +∞ gn,k(t)(u). we know that lp has the basis {e1, e2, ..., en, ..., }, where ei = (δin)n∈n. due to the particular form of jn t/n (x) = (gn,k(t)(xk))k, we have gn,k(t)(0) = 0 for all k ∈ n and it is obvious that if x ∈ span{e1, ..., ei, ...} =: y, then j n t/n (x) becomes a sequence with only a finite number of non-zero elements. this means that for such x, jn t/n (x) is convergent in lp. also, obviously y is dense in lp. let x ∈ lp and ε > 0 be arbitrary. there exists y ∈ y such that ||x − y||p < ε. we get ||jnt/n(x) − j m t/m(x)||p ≤ ||j n t/n(x) − j n t/n(y)||p + ||j n t/n(y) − j m t/m(y)||p+ ||jmt/m(y) − j m t/m(x)||p ≤ 2m||x − y||p + ||j n t/n(y) − j m t/m(y)||p < 2mε + ||j n t/n(y) − j m t/m(y)||p, where ||jn t/n ||lip ≤ m = 1 (see the proof of theorem 3.1, where we take ω = 0). since (jn t/n (y))n is convergent in l p, it is a cauchy sequence and therefore given δ > 0, there is a n0 such that ||j n t/n (y) − jm t/m (y)||p < δ, for all m, n > n0. together with the above inequality this implies that (jn t/n (x))n is a cauchy sequence in the complete metric space l p, i.e. it is convergent in lp. the theorem is proved. as an application of theorem 3.1, we obtain the following corollary 3.5 let a : lp → lp, 0 < p < 1, be nonlinear, lipschitz, such that a(0) = 0, there exists ω ∈ r with a − ωi dissipative and a is weakly continuous (that is for any x∗ ∈ (lp)∗, if limn→ ∞ x ∗(an) = x ∗(a), then limn→ ∞ x ∗[a(an)] = x ∗[a(a)] ). cubo 13, 2 (2011) remarks on the generation of semigroups . . . 49 for x ∈ lp and t ∈ r+, let us consider as in the statement and proof of theorem 3.1, the sequence in lp, uk(x)(t) = j nk t/nk (x) = (gk,r(x)(t))r∈n, convergent (as k → ∞) in the weak topology of lp, to u(x)(t) = (gr(x)(t))r∈n. let us suppose that for all r ∈ n, x ∈ lp, the real functions gr(x)(t) are left differentiable with respect to t ∈ r+, that is there exists (finite) [gr(x)] ′ −(t) = limh→ 0,h∈r+ gr(x)(t) − gr(x)(t − h) h , t ∈ r+, and that for all k, r ∈ n, x ∈ lp, the real functions gk,r(x)(t) are differentiable (in the classical sense) with respect to t ∈ [0, σ), satisfying in addition the relation limtkր t[gk,r(x)] ′(tk) = [gr(x)] ′ −(t), for all t ∈ [0, σ) ∩ r+ and all tk ∈ [0, σ) with tk ր t. here, for s < 0 we take by convention gr(x)(s) = gr(x)(0), gk,r(x)(s) = gk,r(x)(0), which gives sense to [gr(x)] ′ −(0) = 0 and [gk,r(x)] ′(s) = 0, s ≤ 0. then, v(t) = u(x)(t) is a solution of the cauchy problem v′−(t) = a[v(t)], t ∈ [0, σ) ∩ r+, v(0) = x, where v′−(t) is defined componentwise as above and v(s)=v(0), for s < 0. proof. let x∗ ∈ (lp)∗ be arbitrary. according to [7, p. 27], it can be extended to a x∗ ∈ (l1)∗, preserving its norm. by the considerations from the beginning of this section, it follows that uk(x)(t) satisfies the difference equation uk(x)(t) − uk(x)(t − t/nk) t nk = a[uk(x)(t)], t ≥ 0. this obviously implies x∗ [ uk(x)(t) − uk(x)(t − t/nk) t nk − a(uk(x)(t)) ] = 0, t ≥ 0. but by theorem 3.1 we have limk→ ∞ x ∗[uk(x)(t)] = x ∗[u(x)(t)], for all t ∈ r+, x ∈ l p. taking into account the weak continuity of a, first we obtain limk→ ∞ x ∗(a[uk(x)(t)]) = x ∗(a[u(x)(t)]). next we will show that limk→ ∞ x ∗ [ uk(x)(t) − uk(x)(t − t/nk) t nk ] = x∗([u(x)]′−(t)) (3), for all t ∈ r+, x ∈ l p. 50 sorin g. gal cubo 13, 2 (2011) for this purpose, we reason as in the proof of theorem 3.1, that is first we prove (3) for any x∗r ∈ (l p)∗, r ∈ n of the form x∗r(x) = xr, for all x = (x1, ..., xr, ...) ∈ l p. this one reduces to limk→ ∞ gk,r(x)(t) − gk,r(x)(t − t/nk) t nk = [gr(x)] ′ −(t), for all t ∈ r+, x ∈ l p, r ∈ n. by the mean value theorem, there exists ξt,k ∈ (t−t/nk, t) such that gk,r(x)(t)−gk,r(x)(t−t/nk) t nk = [gk,r(x)] ′(ξt,k), which by the hypothesis immediately implies that at the limit with k → ∞ we obtain (3). also, it is clear that (3) holds for any y∗ ∈ span{x∗1, ..., x ∗ r, ...} = y ∗. reasoning now exactly as at the end of proof in theorem 3.1 (since y∗ is dense in (l1)∗ in the weak topology on (l1)∗), we easily get that (3) is satisfied for all x∗ ∈ (l1)∗. in conclusion, we get x∗[(u(x))′−(t) − a(u(x)(t))] = 0, for all t ∈ r+ and all x ∗ ∈ (l1)∗. passing here to supremum with |||x∗|||(l1)∗ ≤ 1 and taking into account a classical result in functional analysis (since l1 is a normed space), we obtain ||[u(x)]′−(t) − a(u(x)(t))||l1 = 0, t ∈ r+, x ∈ l p, which implies [u(x)]′−(t) = a(u(x)(t)), for all t ∈ r+, x ∈ l p. also, obviously u(x)(0) = x, which proves the corollary. a consequence of theorem 3.4 is the following corollary 3.6 for x = (x1, ..., xk, ...) ∈ l p and 0 < p < 1, let us consider as in the statement and proof of theorem 3.4, the operator a, the sequence un(t) := j n t/n (x) = (gn,k(t)(xk))k∈n ∈ l p, where gn,k(t) : r → r are given by gn,k(t)(u) = (i − tn fk) −n(u) and u(t) = (gk(t)(xk))k∈n ∈ l p with limn→ ∞ ||un(t) − u(t)||p = 0, for all t ≥ 0. if, in addition, un(t), u(t) are differentiable with respect to t ∈ [0, σ] such that limn→ ∞ ∞∑ k=1 ||g′n,k(xk) − g ′ k(xk)|| p = 0, where ||g′n,k(xk) − g ′ k(xk)|| := supt∈[0,σ]|g ′ n,k(t)(xk) − g ′ k(t)(xk)|, then v(t) = u(t) represents the unique solution of the nonlinear cauchy problem d dt v(t) = a[v(t)], t ∈ [0, σ], v(0) = x. cubo 13, 2 (2011) remarks on the generation of semigroups . . . 51 proof. by the considerations from the beginning of this section, it follows that un(t) = j n t/n (x) satisfies the difference equation un(t) − un(t − t/n) t n = a[un(t)], t ≥ 0. passing here to limit (with n → ∞) in the || · ||p-norm in lp, since a is lipschitz in lp (see the proof of theorem 3.4 ), it follows that limn→ ∞ a(un(t)) = a(u(t)), for all t ∈ [0, σ]. for the left-hand side, we have ∥ ∥ ∥ ∥ u′(t) − un(t) − un(t − t/n) t n ∥ ∥ ∥ ∥ p ≤ ∥ ∥ ∥ ∥ u′(t) − u(t) − u(t − t/n) t n ∥ ∥ ∥ ∥ p + ∥ ∥ ∥ ∥ u(t) − u(t − t/n) t n − un(t) − un(t − t/n) t n ∥ ∥ ∥ ∥ p , where limn→ ∞ ||u ′(t) − u(t)−u(t−t/n) t n ||p = 0 by the definition of derivative, while by the mean value theorem we obtain ∥ ∥ ∥ ∥ u(t) − u(t − t/n) t n − un(t) − un(t − t/n) t n ∥ ∥ ∥ ∥ p = ∞∑ k=1 ∣ ∣ ∣ ∣ [gk(t)(xk) − gn,k(t)(xk)] − [gk(t − t/n)(xk) − gn,k(t − t/n)(xk)] t/n ∣ ∣ ∣ ∣ p = ∞∑ k=1 |g′k(ξt,k,n)(xk) − g ′ n,k(ξt,k,n)(xk)| p ≤ ∞∑ k=1 ||g′k(xk) − g ′ n,k(xk)|| p, which by the hypothesis implies limn→ ∞ ∥ ∥ ∥ ∥ u(t) − u(t − t/n) t n − un(t) − un(t − t/n) t n ∥ ∥ ∥ ∥ p = 0 and proves the corollary. example. a simple example satisfying the conditions (and the conclusions) in corollary 3.6 is given as follows. define the non-linear strictly decreasing continuous function f : r → r, by f(x) = −x if x < 0, f(x) = −2x if x ≥ 0 and a : lp → lp by a(x) = (f(x1), ..., f(xk), ...), for all x = (x1, ..., xk, ...) ∈ l p. it is easy to check that |f(α) − f(β)| ≤ 2|α − β|, for all α, β ∈ r, which implies that a is lipschitz nonlinear operator. also, it is easy to check that for all λ > 0, the operator i − λa is invertible, with x = (x1, ..., xk, ...), (i − λa) −1(x) = (g(x1), ..., g(xk), ...), g(xk) = xk 1+λ if xk < 0, g(xk) = xk 1+2λ if xk ≥ 0, and ||(i − λa)−1||lip ≤ ( 1 1 + λ )p ≤ 1, 52 sorin g. gal cubo 13, 2 (2011) which shows that a is dissipative. simple calculation shows that un(t) = (gn(t)(x1), ..., gn(t)(xk), ...), where gn(t)(xk) = xk (1+(t/n))n if xk < 0, gn(xk) = xk (1+2(t/n))n if xk ≥ 0, u(t) = (g(t)(x1), ..., g(t)(xk), ...), where g(t)(xk) = xke −t if xk < 0, g(t)(xk) = xke −2t if xk ≥ 0. it is easy to prove that all the conditions in corollary 3.6 are satisfied with σ = 1, which shows that u(t) defined as above is the unique solution of the nonlinear cauchy problem d dt v(t) = a[v(t)], t ∈ [0, 1], v(0) = x. 4. nonlinear semigroups on hp, 0 < p < 1 in this section we consider the hp space, 0 < p < 1, where we denote its p-norm by || · ||p. the main result is the following. theorem 4.1 let a : (hp, || · ||p) → (hp, || · ||p), 0 < p < 1, be nonlinear and lipschitz, such that a(0) = 0 and there exists ω ∈ r with a − ωi dissipative. then the sequence in hp defined by jn t/n (x) = (i − t n a)−n(x), n ∈ n, t ≥ 0, x ∈ hp, contains a subsequence j nk t/nk (x), k ∈ n (the same subsequence for all x ∈ hp and all t ∈ r+ = the set of all rational numbers ≥ 0), uniformly convergent on compacts in d. proof. by a(0) = 0 we get (i − t n a)−n(0) = 0, for all n ∈ n. the dissipative property implies ||i − t(a − ωi)−1||lip ≤ 1, for all t ≥ 0, which is equivalent to ||(i − t 1+tω a)−1||lip ≤ |1 + tω| p, for all t ≥ 0 with 1 + tω 6= 0. for λ = t 1+tω we get ||(i − λa)−1||lip ≤ (1 − λω) −p, in particular for all λ > 0 with λω < 1. (note that for n sufficiently great, depending on t and ω, we have t n ω < 1.) therefore, ||(i − t n a)−1||lip ≤ (1 − t n ω)−p and by mathematical induction ||(i − t n a)−n||lip ≤ (1 − t n ω)−np. but it is known that the sequence (1+ s n )n converges (for n → +∞) to es, and for any s ∈ r is monotonically increasing, for all n ≥ [|s|] + 1 (see e.g. [10, p. 263]), which implies that (1 − t n ω)−np converges to etωp, monotonically decreasing, for all n ≥ [|tw|] + 1. therefore, the greatest value of (1 − t n ω)−np is for n = [|tω|] + 1, which means that there exists m = m(t, p, ω) > 0 (depending only on t, p and ω) such that ||(i − t n a)−n||lip ≤ m, for all n ∈ n. cubo 13, 2 (2011) remarks on the generation of semigroups . . . 53 we obtain ||jnt/n(x) − j n t/n(y)||p ≤ (1 − t n ω)−np||x − y||p ≤ m||x − y||p, (4) for all x, y ∈ hp. taking y = 0 we have jn t/n (0) = 0 and we obtain ||jnt/n(x)||p ≤ m(t, p, ω)||x||p < +∞, (5) for all n ∈ n. since hp, 0 < p < 1, has a schauder basis (see e.g. [9]), it follows that it is separable, denote by y a countable dense subset of hp. also, denote by r+, the set of all nonnegative rational numbers and define e = r+ × y. obviously e is a countable set, let us denote it by e = {e1, ..., ej, ..., } with the distinct elements two by twos, ej = (rj, yj) and for each e = (t, y) ∈ e, denote sn(e) = j n t/n (y) ∈ hp. obviously, sn(e) are analytic functions in d, for all n ∈ n and all e ∈ e. according to e.g. [7, p. 35, (3.4)], the point evaluations ϕz(x) = x(z), z ∈ d, are linear and bounded functionals on hp, 0 < p < 1 and the following inequality holds |x(reiθ)| ≤ 21/p||x||p(1 − r) −1/p, for all x ∈ hp and z = reiθ. together with (5), this implies that for all z = reiθ, |z| ≤ r0 < 1, i.e. 0 < r ≤ r0, e = (t, y) ∈ e, we obtain |sn(e)(z)| = |ϕz[sn(e)]| ≤ 2 1/p ||sn(e)||p 1 (1 − r)1/p ≤ 21/p 1 (1 − r0) 1/p m(t, p, ω)||y||p. in other words, for any fixed e = (t, y) ∈ e, the sequence of analytic functions (sn(e))n∈n, is uniformly bounded on each compact subset of d, which by the classical montel’s theorem implies that it contains a subsequence uniformly convergent on compact subsets of d. for e1 ∈ e, there exists a subsequence of (sn(e1))n∈n, denoted by (s1,n(e1))n∈n, which is uniformly convergent on compact subsets of d. for e2 ∈ e, reasoning analogously, the sequence (s1,n(e2))n∈n contains in turn, a subsequence denoted by (s2,n(e2))n∈n, which is uniformly convergent on compact subsets of d. in general, for em ∈ e, there exists a subsequence of the previous one, (sm,n(em))n∈n, uniformly convergent on compact subsets of d. continuing this process gives rise to the infinite array of analytic functions in d, s1,1, s1,2, s1,3, ..., s2,1, s2,2, s2,3, ..., s1,1, s1,2, s3,3, ..., 54 sorin g. gal cubo 13, 2 (2011) ......................................... s1,m, s2,m, s3,m, ..., and so on, such that the first row means that (s1,n(e1))n∈n uniformly converges on compact subsets of d, the second row means that (s2,n(ej))n∈n is uniformly convergent on compact subsets of d for j = 1, 2, the third row means that (s3,n(ej))n∈n is uniformly convergent on compact subsets of d, for j = 1, 2, 3, and so on. as a consequence, we can consider the diagonal sequence (sn,n)n∈n, which has the property that (sn,n(ej))n∈n is uniformly convergent on compact subsets of d, for all j ∈ n, that is (sn,n(e))n∈n is uniformly convergent on compact subsets of d, for all e ∈ e. now, let us denote a = r+ × h p. we will show that in fact (sn,n(e))n∈n is uniformly convergent on compact subsets of d, for all e ∈ a. indeed, let e = (r, x) ∈ a and since y is dense in hp, let yk ∈ y, k ∈ n, satisfying ||x − yk||p → 0, when k → ∞. denoting ak = (r, yk) ∈ e, by (4) we have ||sn,n(e) − sn,n(ak)||p ≤ m(r, p, ω)||x − yk||p, for all k ∈ n. it is enough to show that (sn,n(e)(z))n∈n is a cauchy sequence with respect to the uniform norm (denoted by || · ||) in each compact disk, dr in d. indeed, this is immediate by the inequalities ||sn,n(e) − sm,m(e)|| ≤ ||sn,n(e) − sn,n(ak)|| + ||sn,n(ak) − sm,m(ak)|| + ||sm,m(ak) − sm,m(e)|| ≤ 2m(r, p, ω)||x − yk||p + ||sn,n(ak) − sm,m(ak)|| and by the above properties. the theorem is proved. remarks. 1) by relation (3.4) in [7, p. 35], it is evident that if limn→ ∞ ||xn − x||p = 0, then (xn)n is uniformly convergent on compact subsets of d. in general, the converse is not valid. as a consequence, if we denote by x the uniform limit of (xn)n on compact subsets of d, then x is an analytic function in d, but in general it does not belong to hp, 0 < p < 1. 2) we can repeat the reasonings in the proof of theorem 4.1 for the sequence (jn t/n (x), n ∈ n, n 6= nk), where nk is the subsequence in theorem 4.1, so that by mathematically induction we easily obtain that the sequence (jn t/n (x))n∈n has at most a countable set of limit points in the locally convex topology of uniform convergence on compact subsets in d. if we denote that set by t ∗(t)(x), where t ∈ r+, x ∈ h p, then for any fixed t ∈ r+, an element a ∈ t ∗(t) is in fact a mapping a : hp → hol(d), where hol(d) denotes the spaces of all holomorphic (analytic) functions in d. corollary 4.2. let a : (hp, || · ||p) → (hp, || · ||p), 0 < p < 1, be nonlinear and lipschitz, such that a(0) = 0, there exists ω ∈ r with a−ωi dissipative and a : (hp, t ) → (hp, t ) is continuous, where t represents the locally convex topology of uniform convergence on compact subsets in d. cubo 13, 2 (2011) remarks on the generation of semigroups . . . 55 for x ∈ hp and t ∈ r+, let us consider as in the statement of theorem 4.1, the sequence in hp, uk(x)(t) = j nk t/nk (x), k ∈ n, uniformly convergent on compacts in d (as k → ∞) to u(x)(t). let us suppose that for x ∈ hp and all z ∈ d, the complex valued function [u(x)(t)](z) is left derivable with respect to the real variable t ∈ r+, that is there exists (finite) ∂[u(x)(t)(z)]− ∂t = limh→ 0,h∈r+ [u(x)(t)](z) − [u(x)(t − h)](z) h , t ∈ r+, and also suppose that limk→ ∞ [uk(x)(t)](z) − [uk(x)(t − t/nk)](z) t nk = ∂[u(x)(t)(z)]− ∂t , for all t ∈ r+ ∩ [0, σ), x ∈ h p, z ∈ d. here, for s < 0 we take by convention [u(x)(s)](z) = [u(x)(0)](z), [uk(x)(s)](z) = [uk(x)(0)](z), for all z ∈ d, which gives sense to ∂[u(x)(0)(z)]− ∂t , for all z ∈ d. then, v(t) = u(x)(t) is a solution (analytic in d but not necessarily in hp) of the cauchy problem ∂[v(t)(z)]− ∂t = a[v(t)](z), t ∈ [0, σ) ∩ r+, z ∈ d v(0)(z) = x(z), z ∈ d, where ∂[v(t)(z)]− ∂t is defined as above and v(s)=v(0), for s < 0. proof. by the considerations from the beginning of the section 3, it follows that uk(x)(t) satisfies the difference equation uk(x)(t) − uk(x)(t − t/nk) t nk = a[uk(x)(t)], t ≥ 0. but by theorem 4.1 and by the continuity assumption on a, we have limk→ ∞ a[uk(x)(t)](z) = a[u(x)(t)](z), for all z ∈ d. therefore, passing to limit with k → ∞ in the above difference equation, by the hypothesis we immediately obtain ∂[u(x)(t)(z)]− ∂t = a[u(x)(t)](z), t ∈ [0, σ) ∩ r+, z ∈ d, z ∈ d. also, obviously u(x)(0) = x, which proves the corollary. 5. nonlinear semigroups on lp[0, 1], 0 < p < 1 in this section we consider the lp[0, 1] space, 0 < p < 1, where we denote its p-norm by || · ||p. the main result is the following. 56 sorin g. gal cubo 13, 2 (2011) theorem 5.1. let a : (lp[0, 1], || · ||p) → (lp[0, 1], || · ||p), 0 < p < 1, be nonlinear and lipschitz, such that a(0) = 0 and there exists ω ∈ r with a − ωi dissipative. then, for any fixed t ≥ 0, x ∈ lp[0, 1], the sequence in lp[0, 1] defined by jn t/n (x) = (i − t n a)−n(x), n ∈ n, contains a subsequence ak(t, x) := j nk t/nk (x), k ∈ n, such that limk→ ∞ 1 k1/p k∑ i=1 ai(t, x)(s) = 0, a.e. s ∈ [0, 1]. proof. reasoning exactly as in the proof of theorem 4.1, relations (4)-(5), we get that ||jnt/n(x)||p ≤ m(t, p, ω)||x||p, where || · ||p is the p-norm in l p[0, 1]. in other words, for any fixed t ≥ 0 and x ∈ lp[0, 1], the sequence (jn t/n (x))n is bounded in the p-norm of l p[0, 1], 0 < p < 1. according to [2], this implies that for any t ≥ 0 and x ∈ lp[0, 1], there exists a subsequence ai(t, x) := j ni t/ni (x), i ∈ n, such that limn→ ∞ 1 n1/p n∑ i=1 ai(t, x)(s) = 0, a.e. s ∈ [0, 1]. remark. unfortunately, a sequence (ai(t, x))i∈n, satisfying the relation proved by theorem 5.1, can satisfy (in the sense that does not produce a contradiction) limi→ ∞ ai(t, x, ω)(s) = ∞, a.e. s ∈ [0, 1], which is the worst possible divergence result. if to this fact we add that the dual space of lp[0, 1], 0 < p < 1, is {0}, then it seems that in this space, in general we cannot derive any result on the convergence of some subsequences of (jn t/n (x))n. acknowledgement. the author thanks the referee for the very useful remarks. received: september 2009. revised: december 2009. referencias [1] a. bayoumi, foundations of complex analysis in non-locally convex spaces, north-holland mathematics studies, elsevier, vol. 193, amsterdam, 2003. [2] s.d. chatterji, a general strong law, inventiones math., 9(1970), 235-245. [3] m.g. crandall and t.m. liggett, generation of semigroups of non-linear transformations on general banach spaces, amer. j. math., 93(1971), 265-298. [4] n. dunford and j.t. schwartz, linear operators, part i, interscience, new york, 1964. cubo 13, 2 (2011) remarks on the generation of semigroups . . . 57 [5] s.g. gal and j.a. goldstein, semigroups of linear operators on p-fréchet spaces, 0 < p < 1, acta math. hungar., 114(1-2)(2007), 13-36. [6] j.a. goldstein, s. oharu and a. vogt, affine semigroups on banach spaces, hiroshima math. j., 18(1988), no. 2, 433-450. [7] n.j. kalton, n.t. peck and j.w. roberts, an f-space sampler, london mathematical society lecture notes series, vol. 89, cambridge university press, 1984. [8] i. muntean, course and problems in functional analysis, vol. ii (romanian), faculty of mathematics, ”babes-bolyaiüniversity press, cluj, 1988. [9] p. oswald, on schauder bases in hardy spaces, proc. roy. soc. edinburg, sect. a, 93(1982/83), no. 3-4, 259-263. [10] t. popoviciu, course of mathematical analysis, part 2, functions. limits. (romanian), ”babes-bolyaiüniversity, faculty of mathematics-mechanics, cluj, 1972. [11] w. rudin, principles of mathematical analysis, third edition, mcgraw-hill inc., 1976. introduction affine semigroups nonlinear semigroups on lp, 0 0, p = const > 0, j: rn → r is a c1 nonnegative function. in addition, j is symmetric (j(z) = j(−z)), and ∫ rn j(z)dz = 1. the initial data u0 ∈ c 0(ω), 0 ≤ u0(x) < m , x ∈ ω. here, (0, t ) is the maximal time interval on which the solution u exists. the time t may be finite or infinite. when t is infinite, then we say that the solution u exists globally. when t is finite, then the solution u develops a singularity in a finite time, namely, lim t→t ‖u(·, t)‖∞ = m, where ‖u(·, t)‖∞ = supx∈ω |u(x, t)|. in this last case, we say that the solution u quenches in a finite time, and the time t is called the quenching time of the solution u. recently, nonlocal diffusion has been the subject of investigation of many authors (see, [1]-[7], [10]-[12], [14]-[18], [20], [24], [27], and the references cited therein). nonlocal evolution equations of the form ut = ∫ rn j(x − y)(u(y, t) − u(x, t))dy, and variations of it, have been used by several authors to model diffusion processes (see, [3], [4], [10], [17], [18]). the solution u(x, t) can be interpreted as the density of a single population at the point x, at the time t, and j(x − y) as the probability distribution of jumping from location y to location x. then, the convolution (j ∗ u)(x, t) = ∫ rn j(x − y)u(y, t)dy is the rate at which individuals are arriving to position x from all other places, and −u(x, t) = − ∫ rn j(x − y)u(x, t)dy is the rate at which they are leaving location x to travel to any other site (see, [17]). let us notice that the reaction term (m − u)−p in the equation (1) can be rewritten as follows (m − u(x, t)) −p = ∫ rn j(x − y) (m − u(x, t)) −p dy. therefore, in view of the above equality, the reaction term (m − u)−p can be interpreted as a force that decreases the rate at which individuals are leaving location x to travel to any other site. due to the presence of the term (m − u)−p, we shall see later that the phenomenon of quenching occurs cubo 12, 1 (2010) quenching for discretizations ... 25 for the density u(x, t). on the other hand, the integral in (1) is taken over ω. thus, there is no individuals that enter or leave the domain ω. it is the reason why in the title of the paper, we have added neumann boundary condition. in the current paper, we are interested in the numerical study of the phenomenon of quenching using a discrete form of (1)-(2). let us notice that, setting v = m − u, the problem (1)-(2) is equivalent to vt(x, t) = ∫ ω j(x − y)(v(y, t) − v(x, t))dy − v−p in ω × (0, t ), (3) v(x, 0) = v0(x) > 0 in ω, (4) where v0(x) = m − u0(x). consequently, the solution u of (1)-(2) quenches at the time t if and only if the solution v of (3)-(4) quenches at the time t , that is, lim t→t vmin(t) = 0, where vmin(t) = minω v(x, t). thus, by convenience, we shall often utilize the problem (3)-(4) instead of (1)-(2). we start by the construction of an explicit adaptive scheme as follows. let i be a positive integer, and let h = 2/i. define the grid xi = −1 + ih, 0 ≤ i ≤ i, and approximate the solution v of (3)-(4) by the solution u (n) h = (u (n) k )k∈γ of the following discrete equations δtu (n) k = ∑ l∈γ∗ hn j(xk − xl) ( u (n) l − u (n) k ) − ( u (n) k )−p , k ∈ γ, n ≥ 0, (5) u (0) k = ϕk , k ∈ γ, (6) where γ = {(j1, · · · , jn ); 0 ≤ j1, · · · , jn ≤ i} , γ∗ = {(j1, · · · , jn ); 0 ≤ j1, · · · , jn ≤ i − 1} , xk = (xk1 , · · · , xkn ) , xl = (xl1 , · · · , xln ) , and δtu (n) k = u (n+1) k − u (n) k ∆tn . in order to permit the discrete solution to reproduce the properties of the continuous one when the time t approaches the quenching time t , we need to adapt the size of the time step so that we take ∆tn = min { h2, τ ( u (n) hmin )p+1} , where u (n) hmin = mink∈γ u (n) k , τ ∈ (0, 1). let us notice that the restriction on the time step ensures the positivity of the discrete solution. to facilitate our discussion, we need to define the notion of numerical quenching time. 26 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) definition 1.1. we say that the discrete solution u (n) h of (5)-(6) quenches in a finite time if limn→∞ u (n) hmin = 0, and the series ∑∞ n=0 ∆tn converges. the quantity ∑∞ n=0 ∆tn is called the numerical quenching time of the discrete solution u (n) h . in the present paper, under some conditions, we show that the discrete solution quenches in a finite time and estimate its numerical quenching time. we also show that the numerical quenching time converges to the real one when the mesh size goes to zero. a similar result has been obtained by nabongo and boni in [25] within the framework of the phenomenon of quenching for local parabolic problems. one may also consult the papers of the same authors in [22] and [23] for numerical studies of the phenomenon of quenching where semidiscretizations in space have been utilized. the remainder of the paper is organized as follows. in the next section, we reveal certain properties of the continuous problem. in the third section, we exhibit some features of the discrete scheme. in the fourth section, under some assumptions, we demonstrate that the discrete solution quenches in a finite time, and estimate its numerical quenching time. in the fifth section, the convergence of the numerical quenching time is analyzed, and finally, in the last section, we show some numerical experiments to illustrate our analysis. 2 local existence in this section, we shall establish the existence and uniqueness of solutions of (1)-(2) in ω × (0, t ) for all small t . some results about quenching are also given. let t0 > 0 be fixed, and define the function space yt0 = {u; u ∈ c([0, t0], c(ω))} equipped with the norm defined by ‖u‖yt0 = max0≤t≤t0 ‖u(·, t)‖∞ for u ∈ yt0 . it is easy to see that yt0 is a banach space. introduce the set xt0 = { u; u ∈ yt0 , ‖u‖yt0 ≤ b0 } , where b0 = ‖u0‖∞+m 2 . we observe that xt0 is a nonempty bounded closed convex subset of yt0 . define the map r as follows r : xt0 → xt0 , r(v)(x, t) = u0(x) + ∫ t 0 ∫ ω j(x − y)(v(y, s) − v(x, s))dyds + ∫ t 0 (m − v(x, s))−pds. theorem 2.1. assume that u0 ∈ c(ω). then r maps xt0 into xt0 , and r is strictly contractive if t0 is appropriately small relative to ‖u0‖∞. proof. due to the fact that ∫ ω j(x − y)dy ≤ ∫ rn j(x − y)dy = 1, a straightforward computation reveals that |r(v)(x, t) − u0(x)| ≤ 2‖v‖yt0 t + ( m − ‖v‖yt0 )−p t, which implies that ‖r(v)‖yt0 ≤ ‖u0‖∞ + 2b0t0 + (m − b0) −pt0. if t0 ≤ b0 − ‖u0‖∞ 2b0 + (m − b0)−p , (7) cubo 12, 1 (2010) quenching for discretizations ... 27 then ‖r(v)‖yt0 ≤ b0. therefore, if (7) holds, then r maps xt0 into xt0 . now, we are going to prove that the map r is strictly contractive. letting v, z ∈ xt0 and setting α = v − z, we discover that |(r(v) − r(z))(x, t)| ≤ ∣∣∣∣ ∫ t 0 ∫ ω j(x − y)(α(y, s) − α(x, s))dyds ∣∣∣∣ + ∣∣∣∣ ∫ t 0 ((m − v(x, s))−p − (m − z(x, s))−p)ds ∣∣∣∣ . use taylor’s expansion to obtain |(r(v) − r(z))(x, t)| ≤ 2‖α‖yt0 t + t‖v − z‖yt0 p ( m − ‖β‖yt0 )−p−1 , where β is a function which is localized between v and z. we deduce that ‖r(v) − r(z)‖yt0 ≤ 2‖α‖yt0 t0 + t0‖v − z‖yt0 p ( m − ‖β‖yt0 )−p−1 , which implies that ‖r(v) − r(z)‖yt0 ≤ (2t0 + t0p(m − b0) −p−1)‖v − z‖yt0 . if t0 ≤ 1 4 + 2p(m − b0)−p−1 , (8) then ‖r(v) − r(z)‖yt0 ≤ 1 2 ‖v − z‖yt0 . hence, we see that r(v) is a strict contraction in yt0 , and the proof is complete. it follows from the contraction mapping principle that for appropriately chosen t0, r has a unique fixed point u ∈ yt0 which is a solution of (1)-(2). if ‖u‖yt0 < m , then taking as initial data u(·, t0) ∈ c(ω) and arguing as before, it is possible to extend the solution up to some interval [0, t1) for certain t1 > t0. hence, we conclude that if the maximal time interval of existence of the solution, (0, t ), is finite then the solution quenches in a finite time in l∞(ω) norm, namely, limt→t ‖u(·, t)‖∞ = m . remark 2.1. let us notice that we can define the map r in the space yt0 = c 1,2(ω × [0, t0]). besides, if u0 ∈ c 1(ω), then arguing as in the proof of theorem 2.1, it is not hard to see that theorem 2.1 remains valid. consequently, if u0 ∈ c 1(ω), then the solution u of (1)-(2) belongs to c1,2(ω × [0, t )) when t is finite. the following lemma is a version of the maximum principle for nonlocal problems. lemma 2.1. let a ∈ c0(ω × [0, t )), and let u ∈ c0,1(ω × [0, t )) satisfy the following inequalities ut − ∫ ω j(x − y)(u(y, t) − u(x, t))dy + a(x, t)u(x, t) ≥ 0 in ω × (0, t ), (9) u(x, 0) ≥ 0 in ω. (10) then, we have u(x, t) ≥ 0 in ω × (0, t ). 28 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) proof. let t0 be any positive quantity satisfying t0 < t . since a(x, t) is bounded in ω × [0, t0], then there exists λ such that a(x, t) − λ > 0 in ω × [0, t0]. define z(x, t) = e λtu(x, t) and let m = min x∈ω,t∈[0,t0] z(x, t). due to the fact that z is continuous in ω × [0, t0], then it achieves its minimum in ω × [0, t0]. consequently, there exists (x0, t0) ∈ ω × [0, t0] such that m = z(x0, t0). we get z(x0, t0) ≤ z(x0, t) for t ≤ t0 and z(x0, t0) ≤ z(y, t0) for y ∈ ω. this implies that zt(x0, t0) ≤ 0, ∫ ω j(x0 − y)(z(y, t0) − z(x0, t0))dy ≥ 0. (11) with the aid of the first inequality of the lemma, it is not hard to see that zt(x0, t0) − ∫ ω j(x0 − y)(z(y, t0) − z(x0, t0))dy + (a(x0, t0) − λ)z(x0, t0) ≥ 0. we deduce from (11) that (a(x0, t0) − λ)z(x0, t0) ≥ 0. since a(x0, t0) − λ > 0, we get z(x0, t0) ≥ 0. this implies that u(x, t) ≥ 0 in ω × [0, t0], and the proof is complete. an immediate consequence of the above lemma is that the solution u of (1)-(2) is nonnegative in ω × (0, t ) because the initial data u0(x) is nonnegative in ω. now, let us give a result about quenching which says that the solution u of (1)-(2) always quenches in a finite time. this assertion is stated in the theorem below. theorem 2.2. the solution u of (1)-(2) quenches in a finite time, and its quenching time t satisfies the following estimate t ≤ (m − a)p+1 p + 1 , where a = 1 |ω| ∫ ω u0(x)dx. proof. since (0, t ) is the maximal time interval of existence of the solution u, our aim is to show that t is finite and satisfies the above inequality. due to the fact that the initial data u0(x) is nonnegative in ω, we know from lemma 2.1 that the solution u(x, t) of (1)-(2) is nonnegative in ω × (0, t ). integrating both sides of (1) over (0, t), we find that u(x, t) − u0(x) = ∫ t 0 ∫ ω j(x − y)(u(y, s) − u(x, s))dyds + ∫ t 0 (m − u(x, s))−pds for t ∈ (0, t ). (12) integrate again in the x variable and apply fubini’s theorem to obtain ∫ ω u(x, t)dx − ∫ ω u0(x)dx = ∫ t 0 (∫ ω (m − u(x, s))−pdx ) ds for t ∈ (0, t ). (13) set w(t) = 1 |ω| ∫ ω u(x, t)dx for t ∈ [0, t ). cubo 12, 1 (2010) quenching for discretizations ... 29 taking the derivative of w in t and using (13), we arrive at w′(t) = ∫ ω 1 |ω| (m − u(x, t))−pdx for t ∈ (0, t ). it follows from jensen’s inequality that w′(t) ≥ (m − w(t))−p for t ∈ (0, t ), or equivalently (m − w)pdw ≥ dt for t ∈ (0, t ). (14) integrate the above inequality over (0, t ) to obtain t ≤ (m − w(0))p+1 p + 1 . since the quantity on the right hand side of the above inequality is finite, we deduce that u quenches in a finite time at the time t which obeys the above inequality. use the fact that w(0) = a to complete the rest of the proof. 3 properties of the semidiscrete scheme in this section, we give some results about the discrete maximum principle of nonlocal problems for our subsequent use. the lemma below is a discrete version of the maximum principle for nonlocal parabolic problems. lemma 3.1. for n ≥ 0, let u (n) h , a (n) h be two vectors such that δtu (n) k ≥ ∑ l∈γ∗ hn j(xk − xl) ( u (n) l − u (n) k ) + a (n) k u (n) k , k ∈ γ, n ≥ 0, u (0) k ≥ 0, k ∈ γ. then, we have u (n) k ≥ 0, k ∈ γ, n > 0 when ∆tn ≤ 1 2n ‖j‖∞+‖a (n) h ‖∞ , where ‖a (n) h ‖∞ = supk∈γ |a (n) k |. proof. if u (n) h ≥ 0, then a straightforward computation reveals that u (n+1) k ≥ u (n) k ( 1 − 2n ‖j‖∞∆tn − ‖a (n) h ‖∞∆tn ) , k ∈ γ, n ≥ 0. (15) to obtain the above inequalities, we have used the fact that ∑ l∈γ∗ hn j(xk − xl) ≥ 0, k ∈ γ, and ∑ l∈γ∗ hn j(xk − xl) ≤ ‖j‖∞ ( i−1∑ l=0 h )n = 2n ‖j‖∞, k ∈ γ. 30 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) making use of (15) and an argument of recursion, we easily check that u (n+1) h ≥ 0, n ≥ 0. this finishes the proof. an immediate consequence of the above result is the following comparison lemma. its proof is straightforward. lemma 3.2. for n ≥ 0, let u (n) h , v (n) h and a (n) h be three vectors such that δtu (n) k − ∑ l∈γ∗ hn j(xk − xl) ( u (n) l − u (n) k ) + a (n) k u (n) k ≥ δtv (n) k − ∑ l∈γ∗ hn j(xk − xl) ( v (n) l − v (n) k ) + a (n) k v (n) k , k ∈ γ, n ≥ 0, u (0) k ≥ v (0) k , k ∈ γ. then, we have u (n) k ≥ v (n) k , k ∈ γ, n > 0 when ∆tn ≤ 1 2n ‖j‖∞+‖a (n) h ‖∞ . remark 3.1. for n ≥ 0, introduce the vector z (n) h defined as follows z (n) k = ‖ϕh‖∞ − u (n) k , k ∈ γ, where u (n) h is the solution of (5)-(6). a straightforward computation reveals that δtz (n) k ≥ ∑ l∈γ∗ hn j(xk − xl) ( z (n) l − z (n) k ) , k ∈ γ, n ≥ 0, z (0) k ≥ 0, k ∈ γ. it follows from lemma 3.1 that ‖ϕh‖∞ ≥ u (n) k , k ∈ γ, n > 0 when h is small enough. 4 the numerical quenching time in this section, under some assumptions, we show that the discrete solution quenches in a finite time and estimate its numerical quenching time. our result concerning the numerical quenching time is stated in the following theorem. theorem 4.1. assume that the initial data at (6) satisfies 2n ‖j‖∞‖ϕh‖ p+1 ∞ < 1. then, the discrete solution u (n) h of (5)-(6) quenches in a finite time, and its quenching time t ∆th obeys the following estimate t ∆th ≤ τ ϕ p+1 hmin 1 − (1 − τ ′)p+1 , where τ ′ = a min{h2ϕ −p−1 hmin , τ} and a = 1 − 2n ‖j‖∞‖ϕh‖ p+1 ∞ . cubo 12, 1 (2010) quenching for discretizations ... 31 proof. we know from remark 3.1 that ‖u (n) h ‖∞ ≤ ‖ϕh‖∞. since ∑ l∈γ∗ hn j(xk − xl) ≤ 2 n ‖j‖∞, k ∈ γ, exploiting (5), we see that δtu (n) k ≤ 2n ‖j‖∞‖ϕh‖∞ − ( u (n) k )−p , k ∈ γ, n ≥ 0, or equivalently δtu (n) k ≤ − ( u (n) k )−p ( 1 − 2n ‖j‖∞‖ϕh‖∞ ( u (n) k )p) , k ∈ γ, n ≥ 0. use the fact that ‖u (n) h ‖∞ ≤ ‖ϕh‖∞, n ≥ 0 to arrive at δtu (n) k ≤ − ( u (n) k )−p ( 1 − 2n ‖j‖∞‖ϕh‖ p+1 ∞ ) , k ∈ γ, n ≥ 0. these estimates may be rewritten as follows u (n+1) k ≤ u (n) k − a∆tn ( u (n) k )−p , k ∈ γ, n ≥ 0. (16) let k0 ∈ γ be such that u (n) k0 = u (n) hmin . replacing k by k0 in (16), we note that u (n+1) k0 ≤ u (n) hmin − a∆tn ( u (n) hmin )−p , n ≥ 0, which implies that u (n+1) hmin ≤ u (n) hmin − a∆tn ( u (n) hmin )−p , n ≥ 0, (17) because u (n+1) k0 ≥ u (n+1) hmin . we observe that a∆tn(u (n) hmin )−p−1 = a min { h2(u (n) hmin )−p−1, τ } . (18) exploiting (17), we see that u (n+1) hmin ≤ u (n) hmin , n ≥ 0, and by induction, we note that u (n) hmin ≤ u (0) hmin = ϕhmin. in view of (18), we discover that a∆tn ( u (n) hmin )−p−1 ≥ a min { h2ϕ −p−1 hmin , τ } = τ ′. (19) therefore, employing (17), we get u (n+1) hmin ≤ u (n) hmin (1 − τ ′) , n ≥ 0. (20) using an argument of recursion, we find that u (n) hmin ≤ u (0) hmin (1 − τ ′) n = ϕhmin (1 − τ ′) n , n ≥ 0. (21) 32 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) this implies that u (n) hmin goes to zero as n approaches infinity. now, let us estimate the discrete quenching time. the restriction on the time step and (21) lead us to ∞∑ n=0 ∆tn ≤ τ ϕ p+1 hmin ∞∑ n=0 ( (1 − τ ′) p+1 )n . use the fact that the series on the right hand side of the above inequality converges towards 1 1−(1−τ ′)p+1 to complete the rest of the proof. remark 4.1. due to (20), an argument of recursion reveals that u (n) hmin ≤ u (q) hmin (1 − τ ′) n−q , n ≥ q. in view of the above estimates, the restriction on the time step allows us to write ∞∑ n=q ∆tn ≤ τ ( u (q) hmin )p+1 ∞∑ n=q ( (1 − τ ′) p+1 )n−q . since the series on the right hand side of the above inequality converges towards 1 1−(1−τ ′)p+1 , we infer that ∑∞ n=q ∆tn ≤ τ ( u (q) hmin ) p+1 1−(1−τ ′)p+1 , or equivalently t ∆th − tq ≤ τ ( u (q) hmin )p+1 1 − (1 − τ ′) p+1 . apply taylor’s expansion to obtain (1−τ ′)p+1 = 1−(p+1)τ ′+o(τ ′). this implies that τ 1−(1−τ ′)p+1 = τ τ ′((p+1)+o(1)) . due to the fact that τ ′ = a min{h2ϕ −p−1 hmin , τ}, if we choose τ = h2, then we note that τ ′ τ = a min{ϕ −p−1 hmin , 1}, which implies that τ τ ′ = o(1) with the choice τ = h2. in the sequel, we pick τ = h2. under the assumption of the above theorem, we have seen that the discrete solution quenches in a finite time, and an estimation of its numerical quenching time has been given. in the theorem below, we derive an upper bound of the numerical quenching time taking into account the assumption of the earlier theorem. theorem 4.2. assume that bh2 < 1, where b = 1+2n ‖j‖∞‖ϕh‖ p+1 ∞ . then, under the hypothesis of theorem 4.1, the discrete solution u (n) h of (5)-(6) quenches in a finite time, and its quenching time t ∆th satisfies the following estimate t ∆th ≥ h2 min{1, ϕ p+1 hmin } 1 − (1 − bh2)p+1 . proof. we know from theorem 4.1 that the discrete solution quenches in a finite time. thus, our purpose is to establish the above estimate. employing (5), we note that δtu (n) k ≥ −2n ‖j‖∞‖ϕh‖∞ − ( u (n) k )−p , k ∈ γ, n ≥ 0. (22) cubo 12, 1 (2010) quenching for discretizations ... 33 to obtain the above inequalities, we have used the fact that j is nonnegative, ‖u (n) h ‖∞ ≤ ‖ϕ‖∞ and ∑ l∈γ∗ hn j(xk − xl) ≤ 2 n ‖j‖∞, k ∈ γ. since 0 ≤ u (n) k ≤ ‖ϕh‖∞, k ∈ γ, the inequalities (22) become δtu (n) k ≥ −b ( u (n) k )−p , k ∈ γ, n ≥ 0. (23) taking into account the restriction on the time step, it is not hard to see that ∆tn ( u (n) k )−p−1 ≤ ∆tn ( u (n) hmin )−p−1 ≤ h2, k ∈ γ, n ≥ 0. (24) in view of (23) and (24), we infer that u (n+1) k ≥ u (n) k ( 1 − bh2 ) , k ∈ γ, n ≥ 0, which implies that u (n+1) hmin ≥ u (n) hmin ( 1 − bh2 ) , n ≥ 0. (25) by induction, we realize that u (n) hmin ≥ u (0) hmin ( 1 − bh2 )n = ϕhmin ( 1 − bh2 )n , n ≥ 0. (26) now, let us estimate the numerical quenching time. the restriction on the time step and (26) reveal that ∞∑ n=0 ∆tn ≥ ∞∑ n=0 min { h2, h2ϕ p+1 hmin (( 1 − bh2 )p+1)n} , (27) which implies that ∞∑ n=0 ∆tn ≥ min { h2, h2ϕ p+1 hmin } ∞∑ n=0 (( 1 − bh2 )p+1)n . use the fact that the series on the right hand side of the above inequality converges towards 1 1−(1−bh2)p+1 to complete the rest of the proof. remark 4.2. apply taylor’s expansion to obtain ( 1 − bh2 )p+1 = 1 − b(p + 1)h2 + o(h2), which implies that h 2 1−(1−bh2)p+1 = 1 b(p+1)+o(1) . 5 convergence of the numerical quenching time in this section, we denote by vh(t) = (v(xk , t))k∈γ . under some hypotheses, we prove that the discrete solution quenches in a finite time, and its numerical quenching time converges to the real one when the mesh size goes to zero. we need the following lemma. 34 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) lemma 5.1. let f ∈ c1(ω). then, we have ∫ ω f (x)dx = ∑ j∈γ∗ hn f (xj ) + o(h) the proof of the above lemma is based on the fact that, if g ∈ c1([−1, 1]), then ∫ 1 −1 g(x)dx = ∑i−1 j=0 hg(xj ) + o(h). making use of the above result, and exploiting fubini’s theorem, one easily proves the above lemma. in order to obtain the result concerning the convergence of the numerical quenching time, we firstly prove that the discrete solution approaches the real one in any interval ω×[0, t −τ ] with τ ∈ (0, t ). this result is stated in the following theorem. theorem 5.1. assume that the problem (3)-(4) admits a solution v ∈ c1,2(ω × [0, t − τ ]) such that mint∈[0,t −τ ] vmin(t) = α > 0 with τ ∈ (0, t ). suppose that the initial data at (6) satisfies ‖ϕh − vh(0)‖∞ = o(1) as h → 0. (28) then, the problem (5)-(6) admits a unique solution u (n) h for h small enough, n ≤ r, and the following relation holds sup 0≤n≤r ‖u (n) h − vh(tn)‖∞ = o(‖ϕh − vh(0)‖∞ + h) as h → 0, where r is a positive integer such that ∑r−1 j=0 ∆tj ≤ t − τ , and tn = ∑n−1 j=0 ∆tj . proof. the problem (5)-(6) admits for each n ≥ 0, a unique solution u (n) h . let d ≤ r be the greatest integer such that ‖u (n) h − vh(tn)‖∞ < α 2 for n < d. (29) making use of (28), we note that d ≥ 1 for h small enough. an application of the triangle inequality renders u (n) hmin ≥ vhmin(tn) − ‖u (n) h − vh(tn)‖∞ ≥ α − α 2 = α 2 for n < d. (30) exploit taylor’s expansion and use lemma 5.1 to obtain δtv(xk , tn) = vt(xk , tn) + ∆tn 2 vtt(xk , t̃n), k ∈ γ, n < d, ∫ ω j(xk − y)(v(y, tn) − v(xk , tn))dy = ∑ l∈γ∗ hn j(xk − xl)(v(xl, tn) − v(xk , tn)) + o(h), k ∈ γ, n < d, cubo 12, 1 (2010) quenching for discretizations ... 35 which implies that δtv(xk , tn) = ∑ l∈γ∗ hn j(xk − xl)(v(xl, tn) − v(xk , tn)) −(v(xk , tn)) −p + ∆tn 2 vtt(xk , t̃n) + o(h), k ∈ γ, n < d. introduce the error e (n) h defined as follows e (n) k = u (n) k − v(xk , tn), k ∈ γ, n < d. invoking the mean value theorem, it is easy to see that δte (n) k = ∑ l∈γ∗ hn j(xk − xl) ( e (n) l − e (n) k ) + p ( ξ (n) k )−p−1 e (n) k − ∆tn 2 vtt(xk , t̃n) + o(h), k ∈ γ, n < d, where ξ (n) k is an intermediate value between v(xk , tn) and u (n) k . we infer that there exists a positive constant q such that δte (n) k ≤ ∑ l∈γ∗ hn j(xk − xl) ( e (n) l − e (n) k ) + p ( ξ (n) k )−p−1 e (n) k +qh, k ∈ γ, n < d, (31) because v ∈ c1,2, j ∈ c1(rn ), and ∆tn = o(h 2). introduce the vector z (n) h defined as follows z (n) k = e(l+1)tn (‖ϕh − vh(0)‖∞ + qh), k ∈ γ, n < d, where l = p ( α 2 )−p−1 . a straightforward computation reveals that δtz (n) k ≥ ∑ l∈γ∗ hn j(xk − xl) ( z (n) l − z (n) k ) + p ( ξ (n) k )−p−1 +qh, k ∈ γ, n < d, z (0) k ≥ e (0) k , k ∈ γ. we deduce from lemma 3.2 that z (n) k ≥ e (n) k , k ∈ γ, n < d. 36 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) in the same way, we also show that z (n) k ≥ −e (n) k , k ∈ γ, n < d, which implies that ‖e (n) h ‖∞ ≤ ‖z (n) h ‖∞, n < d, or equivalently ‖u (n) h − vh(tn)‖∞ ≤ e (l+1)tn (‖ϕh − vh(0)‖∞ + qh), n < d. (32) now, let us reveal that d = r. to prove this result, we argue by contradiction. assume that d < r. replacing n by d in (32), and using (29), we discover that α 2 ≤ ‖u (d) h − vh(td)‖∞ ≤ e (l+1)t (‖ϕh − vh(0)‖∞ + qh). since the term on the right hand side of the second inequality goes to zero as h tends to zero, we deduce that α 2 ≤ 0, which is impossible. consequently, d = r, and the proof is complete. now, we are in a position to prove the main result of this section. theorem 5.2. assume that the problem (3)-(4) has a solution v which quenches in a finite time t such that v ∈ c1,2(ω × [0, t )). suppose that the initial data at (6) satisfies the condition (28). then, under the hypotheses of theorem 4.1, the solution u (n) h of (5)-(6) quenches in a finite time, and its numerical quenching time t ∆th obeys the following relation lim h→0 t ∆th = t. proof. let 0 < ε < t /2. in view of remark 4.1, we know that τ 1−(1−τ ′ )p+1 is bounded. thus, there exists a positive constant ρ such that τ ρp+1 1 − (1 − τ ′)p+1 ≤ ε 2 . (33) since u quenches at the time t , there exists a time t0 ∈ (t − ε/2, t ) such that 0 < vhmin(t) < ρ 2 for t ∈ [t0, t ). let q be a positive integer such that tq = ∑q−1 n=0 ∆tn ∈ [t0, t ). invoking theorem 5.1, we know that the problem (5)-(6) admits a unique solution u (n) h such that ‖u (q) h − vh(tq)‖∞ ≤ ρ 2 . an application of the triangle inequality gives u (q) hmin ≤ vhmin(tq) + ‖u (q) h − vh(tq)‖∞, which implies that u (q) hmin ≤ ρ 2 + ρ 2 = ρ. taking into account theorem 4.1, we know that the discrete solution u (n) h quenches in a finite time t ∆th . it follows from remark 4.1 and (33) that |t ∆th − t | ≤ |t ∆t h − tq| + |tq − t | ≤ ε 2 + ε 2 = ε. this finishes the proof. cubo 12, 1 (2010) quenching for discretizations ... 37 6 numerical results in this section, we give some computational experiments to illustrate the theory given in the previous section. we consider the problem (3)-(4) in the case where n = 1, j(x) = { 15 16 (1 − x2)2 if |x| ≤ 1, 0 if |x| > 1, v0(x) = 4 + cos(πx). we consider the explicit scheme defined in (5)-(6). we also use the following implicit scheme u (n+1) i − u (n) i ∆tn = i−1∑ l=0 hj(xi − xl) ( u (n+1) l − u (n+1) i ) −(u (n) i ) −p−1u (n+1) i , 0 ≤ i ≤ i, u (0) i = ϕi, 0 ≤ i ≤ i. in both cases, we take ϕi = 4 + cos(πxi). as in the case of the explicit scheme, here, we also choose ∆tn = h 2 ( u (n) hmin )p+1 . let us again remark that for the above implicit scheme, existence and positivity of the discrete solution are also guaranteed using standard methods (see, for instance [9]). in the following tables, in rows, we present the numerical quenching times, the numbers of iterations, the cpu times and the orders of the approximations corresponding to meshes of 16, 32, 64, 128. we take for the numerical quenching time tn = ∑n−1 j=0 ∆tj which is computed at the first time when ∆tn = |tn+1 − tn| ≤ 10 −16. the order (s) of the method is computed from s = log((t4h − t2h)/(t2h − th)) log(2) . numerical experiments for p = 1 table 1: numerical quenching times, numbers of iterations, cpu times (seconds) and orders of the approximations obtained with the explicit euler method i tn n cpu time s 16 4.843163 1272 8 32 4.553751 4849 40 64 4.507777 18651 308 2.65 128 4.501537 59860 3008 2.89 38 théodore k. boni and diabaté nabongo cubo 12, 1 (2010) table 2: numerical quenching times, numbers of iterations, cpu times (seconds) and orders of the approximations obtained with the implicit euler method i tn n cpu time s 16 4.903291 1071 5.2 32 4.563995 4097 43 64 4.509770 15674 358 2.64 128 4.501537 59860 9360 2.71 remark 6.1. we observe from tables 1-2 that the numerical quenching time of the discrete solution is approximately equal to 4.5. in what follows, we also give some plots to illustrate our analysis. in figures 1-2, we can appreciate that the discrete solution quenches in a finite time. 0 200 400 600 800 1000 1200 1400 0 5 10 15 20 0 1 2 3 4 5 ni u (i ,n ) figure 1: evolution of the discrete solution (explicit scheme). 0 200 400 600 800 1000 1200 0 5 10 15 20 0 1 2 3 4 5 ni u (i ,n ) figure 2: evolution of the discrete solution (implicit scheme). received: october, 2008. revised: october, 2009. references [1] andren, f., mazon, j.m., rossi, j.d. and toledo, j., the neumann problem for nonlocal nonlinear diffusion equations, j. evol. equat., 8 (2008), 189–215. 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[28] walter, w., differential-und integral-ungleucungen, springer, berlin., (1964). c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2607 .dvi cubo a mathematical journal vol.13, no¯ 01, (103–123). march 2011 engineering design under imprecise probabilities: computational complexity vladik kreinovich department of computer science, university of texas at el paso, 500 w. university, el paso, texas 79968, usa. email: vladik@utep.edu abstract in engineering design problems, we want to make sure that a certain quantity c of the designed system lies within given bounds – or at least that the probability of this quantity to be outside these bounds does not exceed a given threshold. we may have several such requirements – thus the requirement can be formulated as bounds [f c(x), f c(x)] on the cumulative distribution function fc(x) of the quantity c; such bounds are known as a p-box. the value of the desired quantity c depends on the design parameters a and the parameters b characterizing the environment: c = f (a, b). to achieve the design goal, we need to find the design parameters a for which the distribution fc(x) for c = f (a, b) is within the given bounds for all possible values of the environmental variables b. the problem of computing such a is called backcalculation. for b, we also have ranges with different probabilities – i.e., also a p-box. thus, we have backcalculation problem for p-boxes. for p-boxes, there exist efficient algorithms for finding a design a that satisfies the given constraints. the next natural question is to find a design that satisfies additional 104 vladik kreinovich cubo 13, 1 (2011) constraints: on the cost, on the efficiency, etc. in this paper, we prove that that in general, the problem of finding such a design is computationally difficult (np-hard). we show that this problem is np-hard already in the simplest possible linearized case, when the dependence c = f (a, b) is linear. we also provide an example when an efficient algorithm is possible. resumen en los problemas de diseño de ingenieŕıa, en los que quiere asegurarse de que una determinada cantidad c del sistema se encuentra dentro de los ĺımites dados – o por lo menos que la probabilidad de esa cantidad fuera de estos ĺımites no superen un determinado umbral. es posible que haya varios requisitos – la exigencia puede formularse como ĺımites [f c(x), f c(x)] en la función de distribución acumulada fc(x) de la cantidad de c, esos ĺımites son conocidos como p-caja. el valor de la cantidad deseada c depende de los parámetros de diseño a y los parámetros b caracterizar el medio ambiente: c = f (a, b). para lograr el objetivo de diseño, tenemos que encontrar los parámetros de diseño a para que la distribución de fc(x) para c = f (a, b) esté dentro de los ĺımites dados por todos los valores posibles de las variables ambientales b. el problema de la informática se llama a retrocálculo. por b, también tienen rangos con diferentes probabilidades, es decir, también una p-box. por lo tanto, tenemos un problema de retrocálculo para p-cajas. para p-cajas, existen algoritmos eficientes para encontrar un diseño a que satisface las restricciones dadas. la pregunta lógica es encontrar un diseño que satisfaga restricciones adicionales: el coste, la eficiencia, etc. en este trabajo, demostramos, en general, el problema de encontrar un diseño que es computacionalmente dif́ıcil (nphard). se demuestra que este problema es np-hard ya en el caso lineal más simple posible, cuando la dependencia c = f (a, b) es lineal. también ofrecemos un ejemplo, cuando un algoritmo eficiente es posible. keywords: engineering design, imprecise probability, computational complexity, p-boxes, nphard ams subject classification: 65k10, 65j99, 49m15, 49j53, 47j20, 47h04, 90c30, 90c33. 1 engineering design problems and the notion of backcalculation: deterministic case one of the main objective of engineering design is to guarantee that the value of a certain quantity (or several quantities) c is within a given range [c, c]. for example, when we design a car engine, cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 105 we must make sure: • that its power is at least as much as needed for the loaded car to climb the steepest mountain roads, • that the concentration of undesirable substances in the exhaust does not exceed the required threshold, etc. the value of the quantity c usually depends on the parameters a describing the design and on the parameters b of the environment: c = f (a, b). for example, the concentration of a substance in a car exhaust depends: • on the parameter(s) that describe the design of the car exhaust filters, and • on the concentration of the chemicals in the original fuel. we need to select a design a in such a way that c = f (a, b) ∈ [c, c] for all possible values of the environmental parameter(s) b. in this paper, we consider the simplest case when: • the design of each system is characterized by a single parameter a, and • the environment is also characterized by a single parameter b. we will show that already in this simple case, the design problem is, in general, computationally difficult (np-hard). we also show that in some cases, a feasible algorithm is possible. (some of our results first appeared in a conference paper [5].) to be able to find a design that satisfies the given constraint on c for all possible values of the environmental parameter b, we need to know which values of b are possible, i.e., we need to know the range [b, b] of possible values of b. thus, we arrive at the following problem: • we know the desired range [c, c]; • we know the dependence c = f (a, b); • we know the range [b, b] of possible values of b; • we want to describe the set of all values of a for which f (a, b) ∈ [c, c] for all b ∈ [b, b]. this design-related problem is sometimes called a backcalculation problem, to emphasize its difference from the forward calculation problem, when • we are given a design a and • we want to estimate the value of the desired characteristic c = f (a, b). 106 vladik kreinovich cubo 13, 1 (2011) 2 linearized problem in many engineering situations, the intervals of possible values of a and b are reasonably narrow: a ≈ ã and b ≈ b̃ for some ã and b̃. in such situations, we can expand the dependence c = f (a, b) in taylor series and ignore terms which are quadratic and higher order in terms of ∆a def = a − ã and ∆b def = b − b̃. as a result, we get a simple linear dependence c ≈ c0 + ka · a + kb · b. (2.1) we can simplify this expression even further if we take into account that the numerical value of each of the quantities a and b depends on the choice of the starting point and on the choice of a measuring unit. if we change the starting point and the measuring unit, then the new numerical value can be obtained from the original one by an appropriate linear transformation. for example, if we know the temperature tc in celsius, then we can compute the temperature tf in the fahrenheit scale as tf = 32 + 1.8 · tc . we can use this possibility to simplify the above expression for c. specifically, we can change the starting points and the measuring units in such a way that: • the new numerical value for a is described by the linear expression c0 + ka · a, and • the new numerical value for b is described by the linear expression kb · b. in these new scales, the dependence of c on a and b takes the simplest form c = a + b. we will show that the design problem becomes computationally difficult (np-hard) already for this simplest case. 3 from guaranteed bounds to p-boxes ideally, it is desirable to provide a 100% guarantee that the quantity c never exceeds the threshold c. in practice, however, too many unpredictable factors affect the performance of a system and thus, such a guarantee is not realistically possible. what we can realistically guarantee is that the probability of exceeding c is small enough. in other words, we set some threshold εc > 0 and we require that prob(c ≤ c) ≥ 1 − εc. in addition to this requirement, we can also require that the excess of c over c be not too large. this can be done, e.g., by requiring that for some value c1 > c, the probability prob(c ≤ c1) is bounded from below by the value 1−ε1 for some smaller ε1 < ε. we can several such requirements for different values ci and εi. similarly, instead of the idealized exact inequality c ≥ c, in practice, we can only require that prob(c ≥ c) ≤ δ for some small probability δ > 0. cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 107 from the mathematical viewpoint, all such constraints are lower or upper bounds on the values of the cumulative distribution function fc(x) def = prob(c ≤ x). by combining the bounds corresponding to all the constraints, we can thus conclude that the cdf fc(x) must satisfy, for every x, the inequalities f c(x) ≤ fc(x) ≤ f c(x), (3.1) where f c(x) is the largest of all the lower bounds on fc(x) and f c(x) is the smallest of all the upper bounds on fc(x). in other words, for every x, the corresponding value fc(x) must belong to the interval [f c(x), f c(x)]. this x-dependent interval is known as a probability box, or a p-box, for short; see, e.g., [2]. similarly, for the environmental parameter b, we rarely know guaranteed bounds b and b. at best, we know that for a given bound b, the probability of exceeding this bound is small, i.e., that prob(b ≤ b) ≥ 1 − εb for some small εb. so here too, instead of a single bound, in effect, we have a p-box [f b(x), f b(x)]. 4 towards formulating the design (backcalculation) problem for p-boxes in the deterministic approach to design, we assume that we can manufacture an object with the exact value a of the corresponding parameter – or at least the value which is guaranteed to be within the given bounds [a, a]. in manufacturing, however, it is not practically possible to always guarantee that the value a is within the given interval. at best, we can guarantee that, e.g., the probability of a ≤ a is greater than or equal to 1 − εa for some small value εa. in other words, the design restriction on a can also be formulated in terms of p-boxes. thus, we arrive at the following problem. 5 backcalculation problem for p-boxes we are given: • the desired p-box [f c(x), f c(x)] for c; • the dependence c = f (a, b); and • the p-box [f b(x), f b(x)] describing b. 108 vladik kreinovich cubo 13, 1 (2011) our objective is to find a p-box [f a(x), f a(x)] for which: • for every probability distribution fa(x) ∈ [f a(x), f a(x)], • for every probability distribution fb(x) ∈ [f b(x), f b(x)], and • for all possible correlations between a and b, the distribution of c = f (a, b) is within the given p-box [f c(x), f c(x)]. 6 reminder: forward calculation for p-boxes in order to analyze the backcalculation problem for p-boxes, let us first describe how the corresponding forward calculation problem is solved. let us assume that we know: • the p-box [f a(x), f a(x)] for a; and • the p-box [f b(x), f b(x)] describing b. the objective of forward calculation is to find the range [f c(x), f c(x)] of possible values of fc(x) for c = f (a, b) for all possible distributions fa(x) ∈ [f a(x), f a(x)] and fb(x) ∈ [f b(x), f b(x)] and all possible correlations between a and b. it turns out that these calculations are best done in terms not of the original cdfs and p-boxes, but rather in terms of their inverses – quantile functions. for a cdf fa(x), quantiles a0, . . . , an are described as values for which fa(ai) = i n . since the cdf is monotonic, the quantiles are also monotonic: if i < j, then ai ≤ aj . when instead of the exact cdf, we only know a p-box [f a(x), f a(x)], then instead of the exact quantiles ai we only know interval bounds [ai, ai] for these quantiles: • the lower bounds ai are quantiles of the function f c(x), and • the upper bounds ai are quantiles of the function f c(x). these bounds are also monotonic: if i < j, then ai ≤ aj and ai ≤ aj . for the function c = f (a, b) = a + b, forward calculation can be easily described in terms of the quantile bounds: once we know the bounds [ai, ai] and [bi, bi] corresponding to a and b, we can compute the quantile bounds for c as follows: ci = max j (aj + bi−j ); (6.1) ci = min j (aj−i + bn−j ). (6.2) these formulas were first described in [8]. cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 109 7 formulation of the problem in terms of quantile bounds, the backcalculation problem takes the following form: • we know the quantile intervals [bi, bi] corresponding to the environmental variable b; • we are given the intervals [̃ci, c̃i] that should contain the quantiles for c = a + b; • we must find the bounds ai and ai for which, for the values ci and ci are determined by the formulas (6.1) and (6.2), we have [ci, ci] ⊆ [̃ci, c̃i]. 8 in effect, we have two different problems: finding ai and finding ai an important consequence of the formulas (6.1) and (6.2) is that: • the lower bounds ci for c are determined only by the lower bounds ai and bi for a and b, and • the upper bounds ci for c are determined only by the upper bounds ai and bi for a and b. thus, in effect, we can formulate the problem of finding the values ai and the problem of finding the values ai as two separate yet similar problems. without losing generality, in the following text, we will only consider the following problem of finding ai: • we know the values bi; • we are given the values c̃i; • we must find the values a0 ≤ . . . ≤ an for which c̃i ≤ max j (aj + bi−j ). (8.1) 9 a simple example before we start describing how to solve the general problem, let us first describe a simple example. in the general p-box case, to describe the uncertainty of a variable x, we use n + 1 quantile intervals [xi, xi] for i = 0, 1, . . . , n. p-box uncertainty is a generalization of the case of interval uncertainty, in which the uncertainty in each variable x is characterized by a single interval [x, x]. this case corresponds to n = 0. 110 vladik kreinovich cubo 13, 1 (2011) in this case, the inequality for a0 takes the form c̃0 ≤ b0 + a0 or, equivalently, c̃0 − b0 ≤ a0 – the same form as for interval uncertainty. we are looking for the simplest possible example which is different from interval uncertainty. after a single interval, the next simplest example is when we use two intervals, i.e., when n = 1. in this case, we need to find two values a0 ≤ a1 that satisfy the following inequalities: c̃0 ≤ b0 + a0; (9.1) c̃1 ≤ max(b0 + a1, b1 + a0). (9.2) depending on which term in the max expression is the largest, we have two cases: b0 + a1 ≤ b1 + a0 and b0 + a1 ≥ b1 + a0. in the first case, the following inequalities must be satisfied: c̃0 ≤ b0 + a0; b0 + a1 ≤ b1 + a0; (9.3) c̃1 ≤ b1 + a0. the first of these three inequalities from (9.3) is equivalent to c̃0 − b0 ≤ a0. (9.4) similarly, the third inequality from (9.3) is equivalent to c̃1 − b1 ≤ a0. (9.5) thus, these two inequalities are equivalent to max(̃c0 − b0, c̃1 − b1) ≤ a0. (9.6) the second inequality from (9.3) is equivalent to a1 − a0 ≤ b1 − b0. (9.7) thus, in the first case, the values a0 and a1 must satisfy the following two inequalities: a0 ≥ max(̃c0 − b0, c̃1 − b1); 0 ≤ a1 − a0 ≤ b1 − b0. (9.8) graphically, the values that satisfy these inequalities fill the following area: cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 111 a0 6 a1 � � � � � � � � � � � � � � � � v0 v0 v1 where v0 def = max(̃c0 − b0, c̃1 − b1) and v1 def = v0 + b1 − b0. in the second case, the following inequalities must be satisfied: c̃0 ≤ b0 + a0; b0 + a1 ≥ b1 + a0; (9.9) c̃1 ≤ b0 + a1. by moving the unknowns to one side and all the other terms to the other side, we conclude that we must satisfy the following inequalities: a0 ≥ c̃0 − b0; a1 − a0 ≥ b1 − b0; (9.10) a1 ≥ c̃1 − c̃0. (since b1 − b0 ≥ 0, the second inequality automatically implies that a1 ≥ a0.) graphically, we have the following representation: 112 vladik kreinovich cubo 13, 1 (2011) a0 6 a1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� w0 w1 w2 where w0 def = c̃0 − b0, w1 def = w0 + (b1 − b0), and w2 def = c̃1 − c̃0. overall, the set of all possible design values is the union of these two sets. let us illustrate this union on a simple numerical example when b0 = c̃0 = 0, b1 = 1, and c̃1 = 2. in this case, the first case corresponds to the following inequalities a0 ≥ 1; 0 ≤ a1 − a0 ≤ 1. (9.11) and the second case leads to the following inequalities: a0 ≥ 0; a1 − a0 ≥ 1; (9.12) a1 ≥ 2. thus, the union of the two corresponding sets has the following form: a0 6 a1 � � � � � � � � � � � � � � �� 1 1 2 cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 113 10 a designed system usually consists of several subsystems a designed system usually consists of several subsystems. so, instead of selecting a single p-box for a single design parameter a, we need to design p-boxes corresponding to all these subsystems. let s denote the number of these subsystems, and let a (s) i , b (s) i , and c̃ (s) i denote quantile bounds corresponding to the s-th subsystem, s = 1, 2, . . . , s. thus, we arrive at the following problem: • we know the values b (s) i ; • we are given the values c̃ (s) i ; • we must find, for each s = 1, . . . , s, the values a (s) 0 ≤ . . . ≤ a (s) n for which c̃ (s) i ≤ max j (a (s) j + b (s) i−j ). (10.1) 11 there exist effective algorithms for backcalculation for p-boxes, there are efficient algorithms for solving the backcalculation problem; see, e.g., [1, 3, 4, 7]. 12 need for additional cost constraints in general, as we can see, the backcalculation problem has many possible solutions. some design solutions require less efforts, some require more efforts. it is therefore desirable not just to find a solution, but rather to find a solution which satisfies given constraints on the manufacturing efforts such as cost, energy expenses, etc. the values a (s) i are the lower bounds on the design parameters. the smaller the lower bounds, the easier it is to maintain them. thus, the cost of maintaining a lower bound increases with the value a (s) i . in this paper, we show that the problem is np-hard even for the simplest case when the corresponding effort is simply proportional to the value a (s) i , and thus, the overall effort of maintaining all the characteristics a (s) i is equal to the weighted linear combination e = s∑ s=1 n∑ i=0 w (s) i · a (s) i . (12.1) 114 vladik kreinovich cubo 13, 1 (2011) the corresponding constraint is that this effort should not exceed a given value e. as we have mentioned, we may have several (c) constraints corresponding to different type of effort – cost, energy consumption, etc. thus, in general, the constrained backcalculation problem takes the following form. 13 formulation of the problem in precise mathematical terms we are given: • positive integers n, s, and c; • the values b (s) i corresponding to different s = 1, . . . , s and i = 0, . . . , n; • the values c̃ (s) i corresponding to different s = 1, . . . , s and i = 0, . . . , n; • the values ec corresponding to different c = 1, . . . , c, and • the values w (s) c,i corresponding to different s = 1, . . . , s, c = 1, . . . , c, and i = 0, . . . , n. we must find for each s = 1, . . . , s, the values a (s) 0 ≤ . . . ≤ a (s) n for which the following two sets of inequalities are satisfied: c̃ (s) i ≤ max j (a (s) j + b (s) i−j ); (13.1) s∑ s=1 n∑ i=0 w (s) c,i · a (s) i ≤ ec. (13.2) 14 our main result our main result is that the above problem is np-hard. 15 proof main idea. formally, np-hard means that an arbitrary problem from a certain class np can be reduced to this problem; see, e.g., [6]. thus, to prove that a problem is np-hard, it is sufficient to prove that a known np-hard problem can be reduced to it. indeed, cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 115 • by definition of np-hardness, every problem with the class np can be reduced to the known np-hard problem; • since this known problem can be reduced to our problem, • we can therefore conclude that every problem form the class np can be reduced to our problem; • in other words, we can conclude that our problem is np-hard. in our proof, as such a known np-hard problem, we take the knapsack problem; see, e.g., [6]. in this problem, we are given a set of s objects, for each of which we know its volume vs > 0 and its price ps > 0. we also know the total volume v of a knapsack and the threshold price p . within the restriction on the volume, we must select some of the s objects in such a way that the total price of all the selected objects is at least p . to describe this problem in precise terms, for each object i, we define a new variable xs such that xs = 1 if the s-th object is taken and xs = 0 if the s-th object is not taken. in terms of these new variables, the overall volume of all the selected objects is equal to s∑ s=1 vs · xs, and the overall price of all the selected objects is equal to s∑ s=1 ps · xs. thus, the knapsack problem takes the following form: find the values xs ∈ {0, 1} for which s∑ s=1 vs · xs ≤ v (15.1) and s∑ s=1 ps · xs ≥ p. (15.2) we will prove that this problem can be reduced to the above backcalculation problem, i.e., that for each instance v1, . . . , vs , p1, . . . , ps , v, p of the knapsack problem there is an instance of the backcalculation problem whose solution can effectively lead to the solution of the original knapsack problem. towards reduction: selection of p-boxes. in our reduction, we will use the same pair of p-boxes b, c for all s subsystems, the only difference will be in the weights. let us denote the common value of b (s) i for all s = 1, . . . , s by bi and the common value of c̃ (s) i by ci. in these notations, the constrains on the unknowns a (s) i take a simplified form ci ≤ max j (a (s) j + bi−j ). (15.3) in our reduction, we take n = 1. for n = 1, for every s, the inequalities (15.3) lead to the following constraints on the corresponding two unknown a (s) 0 ≤ a (s) 1 : c0 ≤ a (s) 0 + b0; (15.4) 116 vladik kreinovich cubo 13, 1 (2011) c1 ≤ max(a (s) 0 + b1, a (s) 1 + b0). (15.5) specifically, we take b0 = c0 = 0, b1 = 1, and c1 = 2. for these values, the above inequalities take the following form: a (s) 0 ≥ 0; (15.6) 2 ≤ max(a (s) 0 + 1, a (s) 1 ). (15.7) the largest of the two values is greater than or equal to 2 if and only if (at least) one of these two values is greater than or equal to 2. thus, the second constraint (15.7) means that: • either 2 ≤ a (s) 0 + 1 and thus a (s) 0 ≥ 1 • or a (s) 1 ≥ 2. analysis of the selected p-boxes. let us show that out of all possible solutions a (s) 0 ≤ a (s) 1 satisfying these two inequalities, only two solutions satisfy the additional constraint a (s) 0 + a (s) 1 ≤ 2: • a (s) 0 = a (s) 1 = 1 and • a (s) 0 = 0 and a (s) 1 = 2. indeed, we know that for each solution, we have: • either a (s) 0 ≥ 1 • or a (s) 1 ≥ 2. in the first case a (s) 0 ≥ 1, we have a (s) 0 + a (s) 1 = 2a (s) 0 + (a (s) 1 − a (s) 0 ). (15.8) the first term in the right-hand side is ≥ 2, the second term is always non-negative – since a (s) 1 ≥ a (s). thus, the only possibility for the right-hand side sum to be ≤ 2 is when the value 2a (s) 0 is exactly 2, and the difference a (s) 1 − a (s) 0 is exactly 0. in this case, we have a (s) 0 = a (s) 1 = 1. in the second case a (s) 1 ≥ 2, due to a (s) 0 ≥ 0 (condition (15.6)), the only possibility for the sum a (s) 0 + a (s) 1 to be ≤ 2 is when a (s) 0 = 0 and a (s) 1 = 2. reduction and the final part of the proof. we will reduce the given instance of the knapsack problem to the following system with 3 constraints: • in the first constraint, we take w (s) 1,i = 1 for all s and i, and we take e1 = 2 · s. • in the second constraint, we take w (s) 2,0 = vs, w (s) 2,1 = 0, and e2 = v . cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 117 • in the third constraint, we take w (s) 2,0 = 0, w (s) 2,1 = ps, and e3 = p − 2 · s∑ s=1 ps. (15.9) let us first analyze the first constraint s∑ s=1 (a (s) 0 + a (s) 1 ) ≤ 2 · s. (15.10) as we have shown in the previous section, each simple sum a (s) 0 + a (s) 1 is at least 2, and it is only equal to 2 when either a(s) = 0 or a (s) 0 = 1. thus, the only possibility for the sum s∑ s=1 (a (s) 0 + a (s) 1 ) of s such simple sums to not exceed 2s is when each of these sums is equal to exactly 2, i.e., when for every s, • either either a(s) = 0 or a (s) 0 = 1, and • a (s) 1 = 2 − a (s) 0 . for these values a (s) i , the second constraint takes the form s∑ s=1 vs · a (s) 0 ≤ v, (15.11) and the third constraint takes the form s∑ s=1 ps · a (s) 1 ≤ 2 · s∑ s=1 ps − p. (15.12) substituting the expression a (s) 1 = 2 − a (s) 0 into this inequality, we get s∑ s=1 ps · (2 − a (s) 0 ) ≤ 2 · s∑ s=1 ps − p, (15.13) i.e., equivalently, 2 s∑ s=1 ps − s∑ s=1 ps · a (s) 0 ≤ 2 s∑ s=1 ps − p, (15.14) which, in its turn, is equivalent to s∑ s=1 ps · a (s) 0 ≥ p, (15.15) thus, for every solution to the constrained backcalculation problem, the values xs def = a (s) 0 form a solution to the knapsack problem: each of them is equal to 0 or 1, and they satisfy the corresponding inequalities (15.11) and (15.15). 118 vladik kreinovich cubo 13, 1 (2011) vice versa, one can easily check that if the values xs form a solution to the knapsack problem, then the corresponding values a (s) 0 = xs and a (s) 1 = 2 − xs form a solution to the constrained backcalculation problem. the reduction is proven, so the backcalculation problem is indeed, in general, np-hard. 16 situation in which a feasible algorithm is possible let us describe an example of an optimization problem in which a feasible algorithm is possible. optimal backcalculation: towards the formulation of the problem in precise terms. in general, there are many combinations of the values ai that satisfy the desired constraints. which combination should we choose? in the design case, a natural requirement is to select the values ai which will be the easiest to implement. to describe this idea in precise terms, we must analyze how easy is it to implement different values of ai. in general, the value ai is the value for which f (ai) = i n , i.e., we which we must guarantee that prob(a ≤ ai) = f (ai) ≤ f (ai) = i n . (16.1) in particular: • once we select the value a0, we must guarantee that the probability prob(a ≤ a0) of the actual value a being below a0 is 0: prob(a ≤ a0) = 0; • once we select a1, we must guarantee that the probability prob(a ≤ a1) of the actual value a being below a1 does not exceed 1/n: prob(a ≤ a1) ≤ 1/n; • once we select a2, we must guarantee that the probability prob(a ≤ a2) of the actual value a being below a2 does not exceed 2/n: prob(a ≤ a2) ≤ 2/n; • etc. when we motivated the need to take into account partial information about the probabilities, we have mentioned that the most difficult task is to guarantee that the actual a never gets below a cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 119 threshold. the corresponding restriction is related to the value a0: we must guarantee that a ≥ a0. the smaller this value a0, the weaker this constraint and thus, the easiest to satisfy. thus, it is reasonable to select the smallest possible value a0. once this value is selected and the corresponding bound is guaranteed, we must guarantee that the inequalities a ≥ ai be guaranteed with the probabilities ≥ 1 − i/n. the closer this probability to 1, the more stringent is the corresponding requirement, and thus, the more difficult the corresponding task. so, after we have selected a0, the most difficult of the remaining tasks is to select the value a1, the value for which the guaranteed probability of violating the restriction a ≥ a1 is the smallest (probability = 1/n). thus, to make the design as easy to implement as possible, we should make this restriction the least difficult to implement – i.e., we should select the value a1 as small as possible. once we have fixed a0 and a1, the most difficult of the remaining tasks is to to guarantee that a ≥ a2 with probability ≥ 1 − (2/n). thus, we must select the corresponding threshold a2 to be as small as possible. as a result, we arrive at the following formulation of the backcalculation problem. formulation of the optimal backcalculation problem in precise mathematical terms. out of all the tuples a0 ≤ . . . ≤ an that satisfy the inequalities (8.1), • we first select all the tuples for which the value a0 is the smallest possible; • out of the selected tuples, we select all the tuples for which the value a1 is the smallest possible; • then, out of the newly selected tuples, we select those for which the value a2 is the smallest possible; • etc. in mathematical terms, we can say that a tuple a = (a0, . . . , an) is better than a tuple a ′ = (a′0, . . . , a ′ n) if one of the following conditions hold: • either a0 < a ′ 0; • or a0 = a ′ 0 and a1 < a ′ 1; • or a0 = a ′ 0, a1 = a ′ 1, and a2 < a ′ 2; • . . . • or a0 = a ′ 0, . . . , ai−1 = a ′ i−1, and ai < a ′ i; • . . . 120 vladik kreinovich cubo 13, 1 (2011) • or a0 = a ′ 0, . . . , an−1 = a ′ n−1, and an < a ′ n. in computer science, this relation is known as a lexicographic (alphabetic) order, since this is exactly how words are placed in a dictionary or in a lexicon: a word w = ℓ0ℓ1 . . . consisting of the letters ℓ0, ℓ1, . . . , is placed before a word w ′ = ℓ′0ℓ ′ 1 . . . consisting of the letters ℓ ′ 0, ℓ ′ 1, . . . if one of the following conditions hold: • either the letter ℓ0 precedes the letter ℓ ′ 0 (e.g., ∅ apple goes before ∅ zebra); • or ℓ0 = ℓ ′ 0 and the next letter ℓ1 precedes the corresponding letter ℓ ′ 1 (e.g., ∅ abuse goes before ∅ alpha); • . . . • or ℓ0 = ℓ ′ 0, . . . , ℓi−1 = ℓ ′ i−1, and the next letter ℓi precedes the corresponding letter ℓ ′ i; • . . . in these terms, we must select the tuple which is the smallest in the lexicographic order. towards a solution to the optimal backcalculation problem. to find the optimal solution, let us start with the value a0. for i = 0, the condition (8.1) becomes a0 + b0 ≤ c̃0, i.e., equivalently, a0 ≥ c̃0 −b0. the smallest real number that satisfies this inequality is the value a0 = c̃0 −b0. thus, we should take a0 = c̃0 − b0. (16.2) let us now assume that we have already selected the values a0, . . . , ai−1, and that we are now selecting the value ai. the i-th condition (8.1) has the form c̃i ≤ max j (bi−j + aj ) = max [ max j≤i−1 (bi−j + aj ), b0 + ai ] . (16.3) if c̃i ≤ max j≤i−1 (bi−j + aj ), (16.4) then the condition (16.3) is already satisfied. in this case, the only restriction on ai is that ai ≥ ai−1; thus, the smallest possible value of ai is ai = ai−1. if the inequality (16.4) is not satisfied, then to satisfy (16.3), we must satisfy the inequality c̃i ≤ b0 + ai, i.e., equivalently, ai ≥ c̃i − b0. thus, the smallest possible value here is ai = c̃i − b0. so, we arrive at the following algorithm: cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 121 algorithm for optimal backcalculation. ([1, 3, 4, 7]) we know: • the interval quantiles [bi, bi] for the environmental variable b; and • the interval quantiles [̃ci, c̃i] for c. we want to find the lexicographically optimal interval quantiles [ai, ai] for a for which c satisfies the given constraints. in this algorithm, we compute the values a0, a1, . . . one by one. • first, we compute a0 = c̃0 − b0. • once the values a0, . . . , ai−1 are computed, we check whether c̃i ≤ max j≤i−1 (bi−j + aj ). (16.5) if this inequality is satisfied, we take ai = ai−1; otherwise, we take ai = c̃i − b0. comment. in the derivation of the algorithm, we, in fact, proved that the result a of applying this algorithm always satisfies the inequalities (8.1): indeed, we have selected each value ai in such a way that the i-th inequality (8.1) is satisfied. we have also shown that no tuple satisfying (8.1) can be (lexicographically) better than the result a of using this algorithm – and thus, this result a is indeed optimal. algorithm illustrated on the above simple example. in the above example when b0 = c̃0 = 0, b1 = 1, and c̃1 = 2, we do the following: • first, we compute a0 = c̃0 − b0 = 0 − 0 = 0. • next, we check whether c̃1 ≤ max j≤0 (bi−j + aj ) = b1 + a0. (16.6) here, we have 2 = c̃1 6≤ b1 + a0 = 1 + 0 = 1, so we take a1 = c̃1 − b0 = 2 − 0 = 2. one can easily check that the resulting tuple (a0, a1) = (0, 2) is indeed lexicographically optimal: • it has the smallest possible value of a0 = 0, and • out of all the tuples with this value of a0, it has the smallest possible value of a1. 122 vladik kreinovich cubo 13, 1 (2011) a0 6 a1 � � � � � � � � � � � � � � �� 1 1 2 t acknowledgments this work was supported in part by nsf grant hrd-0734825 and by grant 1 t36 gm078000-01 from the national institutes of health. received: june 2009. revised: november 2009. references [1] s. ferson, using approximate deconvolution to estimate cleanup targets in probabilistic risk analyses, in: p. t. kostecki, e. j. calabrese, and m. bonazountas (eds.), hydrocarbon contaminated soils, amherst scientific publishers, amherst, massachusetts, 1995, pp. 245– 254. [2] s. ferson. ramas risk calc 4.0. crc press, boca raton, florida, 2002. [3] s. ferson, v. kreinovich, and w. t. tucker, untangling equations involving uncertainty: deconvolutions, updates, and backcalculations, proceedings of the nsf workshop on reliable engineering computing, savannah, georgia, september 15–17, 2004. [4] s. ferson and t. f. long, deconvolution can reduce uncertainty in risk analyses, in: m. newman and c. strojan (eds.), risk assessment: measurement and logic, ann arbor press, ann arebor, michigan, 1997. [5] v. kreinovich, expert knowledge is needed for design under uncertainty: for p-boxes, backcalculation is, in general, np-hard, proceedings of the 2009 annual conference of the north american fuzzy information processing society nafips’09, cincinnati, ohio, june 14–17, 2009. [6] c. h. papadimitriou, computational complexity, addison wesley, 1993. cubo 13, 1 (2011) engineering design under imprecise probabilities: computational complexity 123 [7] w. t. tucker and s. ferson, setting cleanup targets in a probabilistic assessment, in: s. mishra (ed.), groundwater quality modeling and management under uncertainty, american society of civil engineers, reston, virginia, 2003. [8] r. c. williamson and t. downs, probabilistic arithmetic i: numerical methods for calculating convolutions and dependency bounds, international journal of approximate reasoning, 1990, vol. 4, pp. 89–158. cubo a mathematical journal vol.11, n o ¯ 02, (55–84). may 2009 cournot models: dynamics, uncertainty and learning ferenc szidarovszky systems & industrial engineering department, the university of arizona, tucson, arizona, 85721-0020, usa email: szidar@sie.arizona.edu and vernon l. smith and steven rassenti interdisciplinary center for economic science, george mason university, 3330 washington boulevard arlington, va 22201 emails: vsmith2@gmu.edu, srassent@gmu.edu abstract this chapter gives an overview of the recent developments in the theory of dynamic oligopolies including some new results. we will discuss the cournot classical model and its extensions to product differentiation, multiproduct models, price adjusting oligopolies, labor managed and rent seeking games. the dynamic process based on these models will be analyzed. from the theoretical point of view we will investigate models with and without full information, with partial cooperation among the firms, and under the assumption that the information about the production levels of the rivals has time delay. we will also introduce and discuss special learning procedures based on repeated price information. we will also briefly discuss investigations based on laboratory experiments, in which more realistic cases can be examined. resumen damos una descripción general de los recientes desarrollos de la teoria de dinámica de oligopolios incluyendo algunos resultados novos. discutimos el modelo clásico de cournot y sus extensiones 56 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) a la diferenciación producto, modelos multiproducto, ajuste de precios de oligopolios, labores dirigidas y juegos de rent seeking. la dinámica de procesos basados en estos modelos será analizada. desde el punto de vista teórico investigaremos modelos con y sin información completa, con cooperación parcial entre las firmas, y bajo la suposición que la información al respecto de los niveles de producción de los rivales tiene un tiempo de retrazo. introduzimos y discutimos procesos especiales de aprendizaje basados en información de precios repetidos. también brevemente discutimos investigaciones sobre experimentos de laboratórios, en los cuales casos mas realisticos son examinados. key words and phrases: n-person game, oligopoly, dynamic systems, stability learning. math. subj. class.: 91a20, 91a80. 1 introduction since the pioneering work of [1] oligopoly models are the most frequently discussed topics in the literature of mathematical economics. they describe the interaction of manufacturers and service suppliers through some market demand structure. most of the authors consider this problem as a game in which the supplied quantities (or selected prices) are the strategies and the payoff functions are defined as the profits of the firms. in a competitive environment the nash equilibrium is the solution of the game, in which none of the firms can increase its profit by changing its production level alone. in an n -person oligopoly there is no guarantee in general for the existence of such equilibrium, and even if equilibrium exists, there is the possibility of the existence of multiple equilibria ([5]). several versions of the cournot model has been developed in the literature including oligopolies without and with product differentiation, multiproduct oligopolies, labor managed and rent seeking models as well as oligopsonies, in which the firms also compete on the factor market. a comprehensive summary of the different model variants is presented in [6]. if the state of an oligopoly is a nash equilibrium at a certain time, then no firm has the interest to move away from the equilibrium, therefore the state will remain at the equilibrium for all future times. however in a disequibilium state at least one of the firms is able to increase its profit by changing strategy, so the state of the system will change. if the new state is an equilibrium, then it will remain the state of the system for all future times. otherwise another change of the state will occur, and so, a dynamic process will develop. the dynamics of this process depend on the desires of the firms, and also on the accuracy and timing of the available information. during this process the firms are able to monitor repeated information on the demand structure and the actions of the competitors, which rises the possibility of some learning mechanisms. the research on dynamic behavior of the firms can be done in two fundamentally different ways. the mathematical models are always based on certain assumptions on the objectives of the firms, on the types and analytical properties of the functions involved, and on the information structure. those assumptions very often differ from the economic reality. in more realistic cases it is very often impossible to obtain nice mathematical results, so simulation is used. the applied simulation methodology still depends on the basic assumptions of the model, it simple generates experimental results in the absence of analytical tools. cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 57 another methodology is based on actual laboratory experiments, when real people make the decisions under computer generated environment, and by monitoring their repeated actions we are able to gain a detailed understanding of their priorities, information usage, and decision making mechanism. in this paper we will give a brief overviews of the most important problems arising in examining dynamic oligopolies, and in addition we will offer some new results on this topic. this chapter is developed as follows. static oligopoly models will be first introduced in session 2 and the existence and uniqueness of the equilibrium will be discussed briefly. dynamic models with full information on the demand structure will be then examined, and session 4 will be devoted to partial cooperation among the firms. session 5 will deal with the effect of uncertainty in the information on the demand structure. information delays will be considered in session 6 when we will discuss how the stability of the equilibrium is lost through hopf bifurcation giving the possibility of the birth of limit cycles. some special learning schemes will be introduced in session 7. the fundamentals of experimental economics will be outlined in session 8. finally, some conclusions will be drawn. 2 the cournot model and its extensions consider an economy of n firms that produce the same product or offer the same service to a homogeneous market. let k = 1, 2, . . . , n denote the firms and xk the produced or offered quantity of firm k. the total output of the industry is q = ∑n k=1 xk, and we assume that the market price f is a function of q. if ck(xk) is the cost of firm k, then its profit is given as ϕk(x1, . . . , xn ) = xkf (q) − ck(xk). (2.1) let lk denote the capacity limit of firm k. then an n -person noncooperative game is defined, in which the firms are the players, interval [0, lk] is the set of strategies of player k, and ϕk is its payoff function. this model is known as the classical cournot model, or the single-product quantity adjusting oligopoly without product differentiation. in the literature of oligopoly theory it is usually assumed that functions f and ck (k = 1, 2, . . . , n ) are twice continuously differentiable and for all xk ∈ [0, lk] and q ∈ [0, ∑n k=1 lk], (a) f ′ (q) < 0; (b) xkf ′′ (q) + f ′ (q) < 0; (c) f ′ (q) − c′′k (xk) < 0. notice that the payoff ϕk of player k does not depend explicitly on the outputs of each individual competitor, it depends on only the output qk = ∑ l 6=k xl of the rest of the industry. for each feasible value of qk we can easily determine the best strategy choice of firm k, which is called its best reply. under assumptions (a)-(c), function ϕk is strictly concave in xk with any fixed value of qk, and simple differentiation shows that 58 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) arg max xk {xkf (xk + qk) − ck(xk)} =            0, if f (qk) − c ′ k(0) < 0; lk, if f (qk + lk) + lkf ′ (qk + lk) −c′k(lk) > 0; zk, otherwise. (2.2) where zk is the unique solution of the strictly monotonic equation f (qk + xk) + xkf ′ (qk + xk) − c ′ k(xk) = 0 (2.3) in [0, lk]. since the value of zk depends on qk, we may write zk = rk(qk). in our future analysis we will need the derivative of the best response function. in the neighborhood of an interior optimum, rk is differentiable as the consequence of the implicit function theorem. by differentiating equation (2.3) implicitly with respect to qk, we have f ′ · (1 + r′k) + r ′ k · f ′ + xkf ′′ · (1 + r′k) − c ′′ k · r ′ k = 0 implying that r ′ k(qk) = − f ′ (q) + xkf ′′ (q) 2f ′(q) + xkf ′′(q) − c′′ k (xk) . assumptions (b) and (c) imply that −1 < r′k(qk) < 0. clearly a strategy vector x ∗ = (x ∗ 1, . . . , x ∗ n ) is a nash-equilibrium if and only if for all k, (i) x ∗ k ∈ [0, lk]; (ii) x ∗ k = rk ( ∑ l 6=k x ∗ l ) . it is well known (see for example, [5]) that under conditions (a)-(c) there is a unique nash equilibrium. assume next that the firms produce different but related goods. let x1, . . . , xn denote again the produced quantities and let fk(x1, . . . , xn ) denote the price of the product of firm k. then the profit of this firm is given as ϕk(x1, . . . , xn ) = xkfk(x1, . . . , xn ) − ck(xk). (2.4) this model is known as a single product oligopoly with product differentiation. in bertrand (or price adjusting) oligopolies we consider a single product oligopoly with product differentiation in which each firm selects its price. if p1, . . . , pn are the selected prices and dk(p1, . . . , pn ) is the demand function of the product of firm k, then the profit of this firm can be obtained again by equation (2.4), where for all k, fk = pk and cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 59 xk = dk(p1, . . . , pn ), so the profit functions now depend on only the price selections. multiproduct oligopolies are obtained by assuming that the firms produce m different items. let x (m) k denote the output of firm k of product m, then the firm’s production can be characterized by its production vector xk = (x (1) k , . . . , x (m) k ). the vector q = ∑n k=1 xk shows the total production vectors of the industry. if ck(xk) is the production cost of firm k, then its profit is given as ϕk(x1, . . . , xn ) = x t k f (q) − ck(xk). (2.5) here f (q) = ( f1(q), . . . , fm (q) ) with fm(q) (1 ≤ m ≤ m ) being the price function of product m. consider again the classical cournot model (2.1) and let lk(xk) denote the labor force needed by firm k to produce output xk. then the profit per labor of this firm can be given as ψk(x1, . . . , xn ) = ϕk(x1, . . . , xn ) lk(xk) = xkfk(q) − ck(xk) lk(xk) . (2.6) if the firms maximize profits per labor instead of their total profits, then the oligopoly is said to be labor managed. let n denote the number of agents involved in rent seeking activity. let xk be the expenditure of agent k and fk(xk) its production function for lotteries, then the probability that agent k will win the rent is pk = fk(xk) ∑n l=1 fl(xl) . if the rent is normalized to 1, then the expected net rent of agent k is given as πk = fk(xk) ∑n l=1 fl(xl) − xk. (2.7) this model is known as a rent-seeking game. by introducing the new variables yk = fk(xk) and ck = f −1 k it is clear that the payoff function (2.7) has the new form πk = yk ∑n l=1 yl − ck(yk). (2.8) notice that function form (2.8) reduces to (2.1) by selecting f (q) = 1/q, so rent-seeking games are usually considered as special oligopolies. the equilibria of the above extensions can be defined in the same way as it has been presented for the classical cournot model. existence and uniqueness results are represented in [6]. 60 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) 3 dynamic models with full information in this session we consider only the classical cournot model, and assume that the firms know the true price function and the simultaneous output values of all competitors. in this case for firm k, the best output choice is rk(qk) as it was shown in session 2.2. if xk is the current output of the firm and rk(qk) is its desired output, then under continuous time scales the firm will change its output in the direction toward the desired output, since it cannot ”jump” with the output value instantaneously. therefore the output change of the firms can be described by the system of ordinary differential equations ẋk(t) = kk (rk(qk(t)) − xk(t)) (k = 1, 2, . . . , n ) (3.1) where kk > 0 is a constant being called the speed of adjustment of firm k. here we assume that the firms know their best response functions, and instantaneous information is available about the market price. if p (t) denotes the price at time period t, then p (t) = f (xk(t) + qk(t)) from which firm k is able to calculate the output if the rest of the industry: qk(t) = f −1 (p (t)) − xk(t). in this way the firms have all necessary information to proceed with output adjustments (3.1). an output vector x ∗ = (x ∗ 1, . . . , x ∗ n ) is a steady state of system (3.1) if and only if for all k, x ∗ k = rk   ∑ l 6=k x ∗ l   , that is, when x ∗ is a nash equilibrium. example 1 assume linear price and cost functions: f (q) = b − aq, ck(xk) = akxk + bk (k = 1, 2, . . . , n ). in this case equation (2.3) can be written as b − a(qk + xk) + xk(−a) − ak = 0 implying that xk = − qk 2 + b − ak 2a , so the best response of firm k is cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 61 rk(qk) = − qk 2 + b − ak 2a by assuming interior optimum. therefore the dynamic system (3.1) can be rewritten as follows: ẋk = kk  − 1 2 ∑ l 6=k xl − xk + b − ak 2a   (3.2) for k = 1, 2, . . . , n . ∇ let now x ∗ = (x ∗ 1, . . . , x ∗ n ) denote an interior equilibrium of the classical cournot model. we can examine the local asymptotic stability of this equilibrium with respect to the dynamic process (3.1). the jacobian j c of the system at the equilibrium has the special structure        −k1 k1r1 · · · k1r1 k2r2 −k2 · · · k2r2 . . . . . . . . . . . . kn rn kn rn · · · −kn        (3.3) where rk = r ′ k(q ∗ k) with q ∗ k = ∑ l 6=k x ∗ l . we have seen earlier that −1 < rk < 0. the characteristic polynomial of j c can be written as ϕ(λ) = det(d + ab t − λi) with d = diag(−k1(1 + r1), . . . , −kn (1 + rn )), a = (k1r1, . . . , kn rn ) t , and b t = (1, . . . , 1). in obtaining a closed form representation of ϕ(λ) we can use the well-known fact that with any n -element vectors a and b, det(i + ab t ) = 1 + a t b. therefore ϕ(λ) = det(d − λi) · det(i + (d − λi)−1abt ) = n ∏ k=1 (−kk(1 + rk) − λ) · [ 1 + n ∑ k=1 kkrk −kk(1 + rk) − λ ] . (3.4) the main result of this session can be formulated as theorem 1 all eigenvalues of j c have negative real parts implying the local asymptotic stability of the equilibrium. proof the roots of function (3.4) are λ = −kk(1 + rk), which are all negative, and the roots of the bracketed factor. this equation clearly can be rewritten as 62 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) s ∑ i=1 αi βi − λ + 1 = 0 (3.5) where αi < 0, βi < 0 and the βi values are different. this equation is equivalent to a polynomial equation of degree s, so there are s real or complex roots. let g(λ) denote the left hand side, then lim λ→±∞ g(λ) = 1, lim λ→±βi g(λ) = ±∞, g ′ (λ) = s ∑ i=1 αi (βi − λ)2 < 0. the graph of function g is shown in figure 1. clearly there is a root before β1 and one root between βi and βi+1 for i = 1, 2, . . . , s − 1. since βi < 0 for all i, we found s real negative roots. hence all eigenvalues of j c are real and negative, which completes the proof. 2 1 2 1s s g 1 figure 1: the graph of function g. assume next that the time scales are discrete. if xk(t) denote the production level of firm k at time period t, then its best choice with given xl(t) (l 6= k) values is rk(qk(t)), where qk(t) = ∑ l 6=k xl(t). in many industries the firms are unable to make large changes in their production levels during a single time period, therefore they select levels in the direction toward their best choices. this dynamism can be conveniently modelled as xk(t + 1) = αkxk(t) + (1 − αk)rk(qk(t)) (3.6) for k = 1, 2, . . . , n , where 0 ≤ αk < 1 is a given constant for all k. in order to examine the asymptotic behavior of the equilibrium notice first that the jacobian of this system at the equilibrium can be given as follows: cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 63 j d =        α1 (1 − α1)r1 · · · (1 − α1)r1 (1 − α2)r2 α2 · · · (1 − α2)r2 . . . . . . . . . . . . (1 − αn )rn (1 − αn )rn · · · αn        . (3.7) we can rewrite this matrix similarly to the continuous case: j d = d + ab t with d = diag ((α1 − 1)r1 + α1, . . . , (αn − 1)rn + αn ) , a = ((1 − α1)r1, . . . , (1 − αn )rn ) t , and b t = (1, . . . , 1). therefore the characteristic polynomial of j d is given as ϕ(λ) = det(d + ab t − λi) = det(d − λi) det(i + (d − λi)−1abt ) = n ∏ k=1 ((αk − 1)rk + αk − λ) · [ 1 + n ∑ k=1 (1 − αk)rk (αk − 1)rk + αk − λ ] . (3.8) in this case theorem 1 can be modified as follows: theorem 2 all eigenvalues of j d are inside the unit circle if and only if n ∑ k=1 (1 − αk)rk (αk − 1)rk + αk + 1 > −1. (3.9) in this case the equilibrium is locally asymptotically stable. if the left hand side is smaller than −1, then the equilibrium is unstable. proof the eigenvalues are λ = (αk − 1)rk + αk, which are all positive, and the roots of the bracketed factor. first we show that the roots (αk − 1)rk + αk are inside the unit circle. since they are positive it is sufficient to show that (αk − 1)rk + αk < 1, which is obviously true, since it can be rewritten as (αk − 1)(−rk − 1) > 0, where both factors are negative. the other roots are the solutions of equation (3.5) where αi < 0 and 0 < βi < 1 for all i. the graph of function g is the same as it is shown in figure 1 with the only difference 64 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) that all βi values are between 0 and 1. all roots between the pairs βi and βi+1 are inside the unit circle, and the smallest root is also inside the unit circle if and only if g(−1) > 0. it is easy to see that this inequality is equivalent to condition (3.9). 2 if all firms select the best response, then αk = 0 for all k, and in this case condition (3.9) can be rewritten as n ∑ k=1 rk 1 − rk > −1, (3.10) which certainly holds if all values rk are sufficiently close to zero. in the case of symmetric oligopolies the rk best responses are identical, so rk ≡ r. in this further special case relation (3.10) simplifies to the following: n r 1 − r > −1 which can be rewritten as r > −1 n − 1 . (3.11) example 2 consider again the linear case given in the previous example. since r′k(qk) = − 1 2 for all k, relation (3.11) holds only for n = 2, so the equilibrium is asymptotically stable for only duopolies with αk = 0 (k = 1, 2, . . . , n ). in this case relation(3.9) simplifies as n ∑ k=1 αk − 1 αk + 3 > −1 (3.12) which certainly holds if the αk values are sufficiently close to 1. ∇ in this session we have focused on the classical cournot model, however its extensions and all model variants can be examined similarly to this case. 4 models with partial cooperation a usual way how the firms might increase their profits is to seek certain cooperation with the rivals. a common way to model such ”partial” cooperation can be given as follows. let ϕk denote the profit of firm k (1 ≤ k ≤ n ), and let γkl denote a nonnegative constant for all k and l, which shows the cooperation level of cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 65 firm k toward firm l. then firm k maximizes ϕk + ∑ l 6=k γklϕl that is, it takes a certain portion of the profits of the competitors into account in addition to its own profit. thus the payoff function of firm k becomes ψk(x1, . . . , xn ) = (xkf (q) − ck(xk)) + ∑ l 6=k γkl(xlf (q) − cl(xl)) (4.1) in the case of the classical cournot model. for the sake of mathematical convenience assume that γkl ≡ γk for all k and l, that is, each firm has identical cooperation levels toward its rivals. in this special case ψk(x1, . . . , xn ) = (xk + γkqk)f (q) − ck(xk) − γk ∑ l 6=k cl(xl). (4.2) with given value of qk, the best response of firm k can be obtained by simple differentiation. assuming interior optimum, then at the optimum f (xk + qk) + (xk + γkqk)f ′ (xk + qk) − c ′ k(xk) = 0. (4.3) the derivative of the left hand side with respect to xk is the following: 2f ′ (q) + (xk + γkqk)f ′′ (q) − c′′k (xk). assume that conditions (a) and (c) introduced in session 2.2 are satisfied with a modified version of condition (b): (b’) (xk + γkqk)f ′′ (q) + f ′ (q) < 0 for all xk ∈ [0, lk], qk ∈ [0, ∑ l 6=k ll] and q = xk + qk. then ψk is strictly concave in xk, so the payoff maximizing xk value is unique. if it is interior, then it can be obtained as the unique solution of the monotonic equation (4.3). let rk(qk) denote the solution as before. by differentiating equation (4.3) implicitly with respect to qk it is easy to see that r ′ k(qk) = − (1 + γk)f ′ (q) + (xk + γkqk)f ′′ (q) 2f ′(q) + (xk + γkqk)f ′′(q) − c′′ k (xk) . (4.4) clearly both the numerator and denominator are negative, so r ′ k(qk) is always negative. in addition, r ′ k(qk) > −1 if the following stronger version of condition (c) is satisfied: (c’) (1 − γk)f ′ (q) − c′′k (xk) < 0 for all 0 ≤ xk ≤ lk and 0 ≤ q ≤ ∑n l=1 ll. under conditions (a), (b’) and (c’) all results of the previous session remain true for this case with the only difference that r ′ k(qk) is now given by equation (4.4). 66 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) 5 models with uncertain price function in this session we will examine again the classical cournot model, however the methodology and the results to be discussed here can be extended to other model variants. assume now that the firms do not know the true price function f , but they have certain estimates of it. let fk denote the estimated price function by firm k. then this firm believes that its profit is ϕk(x1, . . . , xn ) = xkf k(q) − ck(xk). (5.1) assume that conditions (a)-(c) are satisfied with f being replaced by f k. then with all fixed qk, firm k has a believed profit maximizing output rk(qk), which is usually different than the ”true” best response of the firm. consider first continuous time scales and assume that similarly to the full information case, each firm adjusts its production level into the direction toward its believed best reply. then we have a modified version of system (3.1): ẋk(t) = kk ( rk ( qk(t) ) − xk(t) ) (k = 1, 2, . . . , n ) (5.2) where qk(t) is the estimate of qk(t) by firm k. notice that the true price function is not known by the firm, so it cannot compute the true value of qk. instead the following method is used. the true market price, which the firm observes is f (xk + qk). on the other hand, firm k believes that it equals f k(xk + qk), so in fact, firm k solves the equation f k(xk + qk) = f (xk + qk) to get its estimate qk = (f −1 k ◦ f )(xk + qk) − xk. (5.3) for the sake of simplicity introduce the function gk = f −1 k ◦ f . then system (5.2) can be rewritten as ẋk(t) = kk ( rk (gk (xk(t) + qk(t)) − xk(t)) − xk(t) ) (5.4) for k = 1, 2, . . . , n . notice that the steady state of this system (if exists) is usually different than the nash equilibrium, since rk usually differs from the true best response function rk. we can refer to this steady state as the ”believed” equilibrium, which will be denoted as x ∗ = (x ∗ 1, . . . , x ∗ n ). the jacobian j c of the system has a similar form to matrix (3.3):        k1 (r1(g1 − 1) − 1) k1r1g1 · · · k1r1g1 k2r2g2 k2 (r2(g2 − 1) − 1) · · · k2r2g2 . . . . . . . . . . . . kn rn gn kn rn gn · · · kn (rn (gn − 1) − 1)        (5.5) cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 67 with gk = g ′ k(x ∗ k + q ∗ k). in the full information case, fk ≡ f for all k, therefore gk is the identity function with gk = 1. in this case j c reduces to matrix (3.3). otherwise fk is only an estimate if f , and if this estimate is sufficiently good, then gk is close to the identity function with gk ≈ 1. the location of the eigenvalues of j c can be similarly examined as it was shown in the full information case. observe that j c = d + ab t where in this case d = diag (−k1(1 + r1), . . . , −kn (1 + rn )), a = (k1r1g1, . . . , kn rn gn ) t and b t = (1, . . . , 1). the characteristic polynomial of this matrix is also similar to (3.4): n ∏ k=1 (−kk(1 + rk) − λ) · [ 1 + n ∑ k=1 kkrkgk −kk(1 + rk) − λ ] . (5.6) since f is strictly decreasing and we may assume that all estimates fk are also strictly decreasing (otherwise the firms’ estimates are irrealistic), gk is strictly increasing with nonnegative derivative gk. therefore the proof of theorem 1 can be used without any changes to show that the ”believed” equilibrium is locally asymptotically stable in this case as well. assume next that the time scales are discrete. in this case the discrete dynamic system (3.6) is modified as xk(t + 1) = αkxk(t) + (1 − αk)rk (gk (xk(t) + qk(t)) − xk(t)) (5.7) for k = 1, 2, . . . , n , where we used equation (5.3) again. the steady state of this system (if exists) is usually different than the nash equilibrium, so we might refer to the steady state as a ”believed” equilibrium similarly to the continuous case. the jacobian j c of system (5.7) has the special structure        α1 + (1 − α1)r1(g1 − 1) (1 − α1)r1g1 · · · (1 − α1)r1g1 (1 − α2)r2g2 α2 + (1 − α2)r2(g2 − 1) · · · (1 − α2)r2g2 . . . . . . . . . . . . (1 − αn )rn gn (1 − αn )rn gn · · · αn + (1 − αn )rn (gn − 1)        (5.8) where rk and gk are as before. in the full information case gk = 1, so j c reduces to matrix (3.7). otherwise fk is only an estimate of f , and if this estimate is sufficiently good, then gk is close to the identity function with gk ≈ 1. this matrix can also be rewritten as j d = d+ab t with d = diag ((α1 − 1)r1 + α1, . . . , (αn − 1)rn + αn ), a = ((1 − α1)r1g1, . . . , (1 − αn )rn gn ) t and b t = (1, . . . , 1). the characteristic polynomial of j d has a similar form to the previously discussed cases: n ∏ k=1 ((αk − 1)rk + αk − λ) · [ 1 + n ∑ k=1 (1 − αk)rkgk (αk − 1)rk + αk − λ ] . (5.9) 68 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) it is easy to see that theorem 2 remains valid in this case with the only difference that condition (3.9) has to be modified as n ∑ k=1 (1 − αk)rkgk (αk − 1)rk + αk + 1 > −1. (5.10) 6 the effect of information delay in this session we will examine how delayed information affects the asymptotic behavior of the equilibrium. for the sake of simplicity we will consider only the classical cournot model. assume that at time period t firm k obtains a delayed information on the production level of the rest of the industry, qk(sk), where t− sk is the delay. if firm k uses this latest information to form its best response, then the dynamic system becomes ẋk(t) = kk (rk (qk(sk)) − xk(t)) (k = 1, 2, . . . , n ). (6.1) if the delay is known and it is denoted by dk(t), then sk = t − dk(t), and so equation (6.1) becomes a difference-differential equation. however the delay is uncertain in real economies, therefore a convenient modelling way is offered by considering it as a random variable and replacing the random right hand sides of equation (6.1) by their expected values. in this way a volterra-type integro-differential equation is obtained: ẋk(t) = kk (∫ t 0 w(t − s, tk, mk)rk (qk(s)) ds − xk(t) ) . (6.2) the weighting function w is defined as w(t − s, t, m) = { 1 t e − t−s t , if m = 0; 1 m! ( m t )m+1 (t − s)me− m(t−s) t , if m ≥ 1, (6.3) where t > 0 is a real and m ≥ 0 is an integer parameter. this weighting function has the following properties: (a) ∫ ∞ 0 w(s, t, m)ds = 1; (b) if m = 0, then weights are exponentially decreasing with the largest weight given to the most current data. if m ≥ 1, then the most current data has zero weight, the weight is increasing to a maximal value at t − s = t , and decreases thereafter. (c) with increasing value of m, the weighting function becomes more peaked around t − s = t , as m → ∞ the weighting function converges to the dirac delta function centered at t − s = t . (d) if t → 0, then the weighting function converges to the dirac delta function centered at zero. theorem 3 system (6.2) is equivalent to a system of ordinary differential equations by introducing additional unknown functions. cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 69 proof assume first that m = 0 and let p be any function of time. introduce the new function p0(t) = ∫ t 0 1 t e − t−s t p (s)ds. by simple differentiation, ṗ0(t) = 1 t (p (t) − p0(t)) . assume next that m ≥ 1, and for all l = 0, 1, . . . , m introduce the functions pl(t) = ∫ t 0 1 l! ( m t )l+1 (t − s)le− m(t−s) t p (s)ds. then by differentiation, ṗl(t) = m t (pl−1(t) − pl(t)) and ṗ0(t) = m t (p (t) − p0(t)) . by selecting pk(s) = rk (qk(s)), the integro-differential equation system is equivalent to the following system of ordinary differential equations: ẋk(t) = kk (pkm(t) − xk(t)) (1 ≤ k ≤ n ) ṗkl(t) = qk tk (pk,l−1(t) − pkl(t)) (1 ≤ k ≤ n, 1 ≤ l ≤ mk) ṗk0(t) = qk tk (pk(t) − pk0(t)) (1 ≤ k ≤ n ) with qk = { 1, if mk = 0; mk, if mk ≥ 1. (6.4) 2 linearizing equation (6.2) we have for all k, ẋkδ(t) = kk  rk ∫ t 0 w(t − s, tk, mk) · ∑ l 6=k xlδ(s)ds − xkδ (t)   (6.5) where rk = r ′ k(q ∗ k), and xkδ (t) is the deviation of xk(t) from its equilibrium level. as it is usual in the theory of ordinary differential equations we look for the solution as xkδ (t) = vke λt . by substituting this form into (6.5) and letting t → ∞ we have (λ + kk)vk − kkrk ∫ t 0 w(s, tk, mk)e −λs ds ∑ l 6=k vl = 0 70 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) and by using simple integration and the definition of the gamma function this equation simplifies as (λ + kk)vk − kkrk ( 1 + λtk qk )−(mk+1) ∑ l 6=k vl = 0. (6.6) introduce function ak(λ) = (λ + kk) ( 1 + λtk qk )mk+1 to see that equations (6.6) are equivalent to a determinantal equation det        a1(λ) −k1r1 · · · −k1r1 −k2r2 a2(λ) · · · −k2r2 . . . . . . . . . . . . −kn rn −kn rn · · · an (λ)        = 0. (6.7) notice that by introducing d = diag (a1(λ) + k1r1, . . . , an (λ) + kn rn ) , a = (−k1r1, . . . , −kn rn ) t and b t = (1, . . . , 1) this equation can be rewritten as det(d + ab t ) = det(d) · det(i + d−1abt ) = n ∏ k=1 (ak(λ) + kkrk) · [ 1 − n ∑ k=1 kkrk ak(λ) + kkrk ] = 0. first we prove that all roots of equation ak(λ) + kkrk = 0 (6.8) have negative real parts. clearly λ 6= 0, and assume that with some root λ, reλ ≥ 0. then |λ + kk| > kk and ∣ ∣ ∣ ∣ 1 + λtk qk ∣ ∣ ∣ ∣ > 1 implying that kk > |kkrk| = |ak(λ)| > kk which is an obvious contradiction. as the asymptotic behavior of the equilibrium is concerned we have to examine the locations of the solutions of equation cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 71 1 − n ∑ k=1 kkrk ak(λ) + kkrk = 0. (6.9) notice first that it is equivalent to a polynomial equation, so there are finitely many roots. in the general case computer methods are needed to locate the roots, however in special cases analytic results can be obtained. assume now that the firms are identical and the equilibrium is symmetric. then k1 = . . . = kn = k, t1 = . . . = tn = t , m1 = . . . = mn = m, q1 = . . . = qn = q, and r1 = . . . = rn = r showing that equation (6.9) reduces to the following: (λ + k) ( 1 + λt q )m+1 + (1 − n )kr = 0. (6.10) consider first the case of t = 0, which corresponds to models without information delay. in this case (6.10) has a unique root λ < 0, so the equilibrium is locally asymptotically stable. note that this symmetric case is a special case of theorem 1. assume next that t > 0 and m = 0. then (6.10) becomes quadratic: λ 2 t + λ(1 + kt ) + k (1 + (1 − n )r) = 0. since all coefficients are positive, all roots have negative real parts implying the local asymptotic stability of the equilibrium (see for example, [8]). consider next the case of t > 0 and m = 1. in this case (6.10) is a cubic equation: λ 3 t 2 + λ 2 (2t + t 2 k) + λ(1 + 2kt ) + k (1 + (1 − n )r) = 0. (6.11) all coefficients are positive and the routh-hurwitz criterion implies that all roots have negative real parts if and only if (2t + t 2 k)(1 + 2kt ) > t 2 k (1 + (1 − n )r) , (6.12) which can be rewritten as 2t 2 k 2 + t k (4 + r(n − 1)) + 2 > 0. (6.13) the discriminant of the left hand side is r(n − 1) [8 + r(n − 1)] , where r(n − 1) < 0. therefore we have the following cases. case 1. if 8 + r(n − 1) > 0, then the left hand side of (6.13) has no real root, so it always holds. therefore the equilibrium is locally asymptotically stable. 72 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) case 2. if 8 + r(n − 1) = 0, then there is a unique root t k = −4 − r(n − 1) 4 = − (8 + r(n − 1)) + 4 4 = 1 and if t k 6= 1, then the equilibrium is locally asymptotically stable. case 3. if 8 + r(n − 1) < 0, then there are two real roots (t k) ∗ 1,2 = −4 − r(n − 1) ± √ (4 + r(n − 1)) 2 − 16 4t 2 . notice that −4 − r(n − 1) = −(8 + r(n − 1)) + 4 > 0, so both roots are positive. hence the equilibrium is locally asymptotically stable if t k < (t k) ∗ 1 or t k > (t k) ∗ 2, where (t k) ∗ 1 < (t k) ∗ 2. the equilibrium is unstable if (t k) ∗ 1 < t k < (t k) ∗ 2. figure 2 shows the stability region of the equilibrium. 1 8 n 0-1 r kt 1 figure 2: stability region in the (r, kt ) space from the above analysis we can draw the following interesting conclusions. if n < 9, then −8/(n − 1) < −1, so case 1. occurs regardless of the value of r, so the equilibrium is always locally asymptotically stable. that is, we need at least 9 firms to have instability. if n = 9, then case 1. occurs for r > −1 and case 2. occurs for r = −1. assume next that n ≥ 10. if r > − 8 n−1 , then the equilibrium is always locally asymptotically stable, if r = − 8 n−1 , then it occurs if kt 6= 1 (that is, always except a particular value of kt ), and if r < − 8 n−1 , then kt has to be sufficiently small or sufficiently large to guarantee local asymptotical stability. it is also interesting to note that the stability conditions depend on only the product kt and not on the individual values of these parameters. it shows a certain compensation between the speed of adjustment and average information delay. cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 73 we also know from the above analysis that by crossing the critical values (t k) ∗ 1 and (t k) ∗ 2 with the value of t k, stability is lost or gained. it is interesting to examine what happens at these critical values. we will show that hopf-bifurcation occurs (see for example, [4]) implying the possibility of limit cycles around the equilibrium. at the critical values inequality (6.13) as well as (6.12) become equality implying that equation (6.11) can be rewritten as 0 = λ 3 t 2 + λ 2 (2t + t 2 k) + λ t 2 k (1 + (1 − n )r) 2t + t 2k + k(1 + (1 − n )r) = ( λt 2 + (2t + t 2 k) ) ( λ 2 + k(1 + (1 − n )r) 2t + t 2k ) showing that one eigenvalue is negative, λ1 = − 2+t k t , and the other two are pure complex. differentiating equation (6.11) implicitly with respect to t we have 3λ 2 λ̇t 2 + λ 3 2t + 2λλ̇(2t + t 2 k) + λ 2 (2 + 2t k) + λ̇(1 + 2t k) + λ2k = 0 implying that λ̇ = −2t λ3 − λ2(2 + 2t k) − 2kλ 3λ2t 2 + 2λ(2t + t 2k) + (1 + 2t k) . (6.14) by letting α 2 = k(1 + (1 − n )r) 2t + t 2k clearly λ̇ = 2t α 3 i + α 2 (2 + 2t k) − 2kαi −3α2t 2 + 2αi(2t + t 2k) + (1 + 2t k) with real part reλ̇ = −2α4t 2(2 + 2t k) + 4α2t (t α2 − k)(2 + t k) (−2α2t 2)2 + 4α2(2t + t 2k)2 where we used the simple fact that 1 + 2t k = t 2 α 2 . the numerator is 4α 2 t (α 2 t − t k2 − 2k). the firs factor is positive, and the second factor can be rewritten as 74 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) k(1 + (1 − n )r) 2 + t k − t k2 − 2k = k 2 + t k · ( (1 + (1 − n )r) − (2 + t k)2 ) = k 2 + t k ( 1 + 2(t k + 1) 2 t k − (2 + t k)2 ) = 1 (2 + t k)t ( −(t k)3 − 2(t k)2 + (t k) + 2 ) = 1 − (t k)2 t 6= 0 (6.15) where we used the equality form of inequality (6.13). since reλ̇ 6= 0, there is the possibility of a limit cycle around the equilibrium. the discrete version of this model and its asymptotical properties can be discussed similarly. 7 learning in oligopoly models for the sake of simplicity we will discuss the classical cournot model again and assume linear cost and price functions. therefore assume that the cost function of firm k is ck(xk) = αkxk + βk, and the true price function is f (q) = b − aq, where q = ∑n k=1 xk as before. assume that the firms have only limited knowledge on the price function, and during the dynamic process they repeatedly update their beliefs of the price function giving rise of a learning process. in this session we will examine three cases. case 1. assume that the firms know the value of q, where the price becomes zero. in this case firm k believes that the price function is fk(q) = εk ( b a − q ) , but does not know that εk = a is the true value. case 2. if the firms know only the slope of the price function, then firm k believes that the price function is fk(q) = εk − aq, but does not know that εk = b is the true value. case 3. if the firms know the price at q = 0 but they are uncertain about the slope, then firm k believes that the price function is fk(q) = b − εkq, but does not know the true value εk = a. as we will see, the dynamic learning processes will be significantly different in the above cases. starting with case 1. we examine the game first from the viewpoint of firm k. it believes that the profit of each firm (including itself) is ϕ (k) l (x1, . . . , xn ) = xlεk ( b a − q ) − (αlxl + βl) (l = 1, 2, . . . n ). (7.1) by assuming interior optima, the believed best response of firm l is xl = b a − αl εk − q. (7.2) by adding these equations q = n b a − 1 εk n ∑ l=1 αl − n q cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 75 implying that firm k believes that at the equilibrium the total production of the industry is q (k) = 1 n + 1 ( n b a − ∑n l=1 αl εk ) and the corresponding equilibrium price is fk(q (k) ) = εk ( b a − q(k) ) = 1 n + 1 ( bεk a + n ∑ l=1 αl ) . (7.3) firm k also produces the corresponding believed equilibrium level xk = b a − αk εk − q(k) = b (n + 1)a − αk εk + ∑n l=1 αl εk(n + 1) . (7.4) therefore in reality, the total production level of the industry becomes q = n ∑ k=1 xk = n b (n + 1)a − n ∑ k=1 αk εk + ( n ∑ l=1 αl )( n ∑ k=1 1 εk ) 1 n + 1 (7.5) with the corresponding equilibrium price p = b − aq = b n + 1 + a n ∑ k=1 αk εk − a n + 1 ( n ∑ l=1 αl )( n ∑ k=1 1 εk ) . (7.6) the actual price is usually different than the believed prices by the firms. for firm k, the discrepancy between the actual and believed price is d (k) = p − fk(q (k) ) = b n + 1 ( 1 − εk a ) + a n ∑ k=1 αk εk − 1 n + 1 ( n ∑ l=1 αl )[ a n ∑ k=1 1 εk + 1 ] . (7.7) based on this price discrepancy firm k thinks as follows. if d (k) = 0, then the believed price equals the actual price, so the believed price is considered correct. if d (k) > 0, then the believed price is too low, so firm k wants to increase its estimate on the price function by increasing εk. if d (k) < 0, then its price estimate was too high, so the firm wants to decrease it by decreasing the value of εk. by assuming continuous time scales this adjustment process can be modelled as ε̇k = kkd (k) (k = 1, 2, . . . , n ), (7.8) and in the discrete case as εk(t + 1) = εk(t) + kkd (k) (k = 1, 2, . . . , n ). (7.9) here kk > 0 is the speed of adjustment of firm k. 76 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) first we prove that systems (7.8) and (7.9) have only one steady state εk = a (k = 1, 2, . . . , n ) which is the full information case. notice first that d (k) = 0 for all k may occur only if the εk values are identical. let ε denote this common value, then 0 = b n + 1 ( 1 − ε a ) + a ε n ∑ k=1 αk − 1 n + 1 ( n ∑ k=1 αk ) [ an ε + 1 ] = b n + 1 ( 1 − ε a ) + ( n ∑ k=1 αk ) ( a ε − 1 ) 1 n + 1 . if ε > a, then both terms are negative; if ε < a, then both are positive, and if ε = a, then they are zero. hence the only steady state is εk = a for all k. in order to analyze the asymptotical stability of systems (7.8) and (7.9) notice first that ∂d (k) ∂εk = − b (n + 1)a + a ε 2 k ( −αk + 1 n + 1 n ∑ l=1 αl ) and for l 6= k, ∂d (k) ∂εl = a ε 2 l ( −αl + 1 n + 1 n ∑ k=1 αk ) . for the sake of simple notation introduce the variable γk = 1 n + 1 n ∑ l=1 αl − αk. (7.10) the jacobian of the continuous system (7.8) has the form j c =         k1 ( − b (n +1)a + aγ1 ε2 1 ) k1aγ1 ε2 1 · · · k1aγ1 ε2 1 k2aγ2 ε2 2 k2 ( − b (n +1)a + aγ2 ε2 2 ) · · · k2aγ2 ε2 2 . . . . . . . . . . . . kn aγn ε2 n kn aγn ε2 n · · · kn ( − b (n +1)a + aγn ε2 n )         (7.11) = d + ab t with d = diag ( −k1b (n + 1)a , . . . , −kn b (n + 1)a ) , a = ( k1aγ1 ε 2 1 , . . . , kn aγn ε 2 n )t and b t = (1, . . . , 1). therefore the characteristic polynomial of this matrix is the following: cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 77 ϕ(λ) = n ∏ k=1 ( −kkb (n + 1)a − λ )  1 + n ∑ k=1 kkaγk ε2 k −kkb (n +1)a − λ   . (7.12) the eigenvalues are λ = −kkb (n +1)a < 0 and the solutions of equation n ∑ k=1 kkaγk ε2 k − kkb (n +1)a − λ + 1 = 0 which has the same form as equation (3.5) by assuming that γk ≤ 0 for all k. notice that this condition holds if the marginal costs αk are close to each other. then by repeating the proof of theorem 1 we can show that all eigenvalues have negative real parts implying the local asymptotical stability of the equilibrium. the jacobian of the discrete system (7.9) is similarly j d = i + j c , which has the same structure as j c , but the identity matrix has to be added to the diagonal matrix d. therefore its characteristic polynomial has the form n ∏ k=1 ( 1 − kkb (n + 1)a − λ )  1 + n ∑ k=1 kkaγk ε2 k 1 − kkb (n +1)a − λ   . (7.13) by repeating the proof of theorem 2 we can show that all eigenvalue of j d are inside the unit circle at the equilibrium εk = a if γk ≤ 0 and kkb (n +1)a < 2 for all k, furthermore n ∑ k=1 kkγk(n + 1) 2(n + 1)a − kkb > −1. (7.14) in case 2. we assume that firm k believes that the price function is fk(q) = εk − aq, and the firms learn about the value of εk. then firm k believes that the profit of any firm l (including itself) is ϕ (k) l (x1, . . . xn ) = xl(εk − aq) − (αlxl + βl) (7.15) so the believed best response of firm l is xl = εk − αl a − q, (7.16) the total output of the industry is believed to be q (k) = n εk − ∑n l=1 αl (n + 1)a (7.17) with price fk(q (k) ) = εk + ∑n l=1 αl n + 1 . (7.18) 78 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) based on this belief firm k produces the amount xk = εk − αk a − q(k) = εk − (n + 1)αk + ∑n l=1 αl (n + 1)a . (7.19) in reality however each firm thinks in the same way but believes in its own εl value in the price function, so the actual total production level of the industry becomes q = n ∑ k=1 xk = 1 (n + 1)a ( n ∑ k=1 εk − n ∑ l=1 αl ) (7.20) with actual equilibrium price p = b − aq = b − 1 n + 1 ( n ∑ k=1 εk − n ∑ l=1 αl ) . (7.21) based on the discrepancy d (k) = p − fk(q (k) ) = 1 n + 1 ( (n + 1)b − n ∑ l=1 εl − εk ) (7.22) the dynamic process become similar to (7.8) and (7.9) with the only difference that in this case d (k) is given in equation (7.22). similarly to the previous case we can prove that there is a unique steady state εk = b for all k, which corresponds to the full information case. clearly d (k) = 0 for all k, if the εk values are identical. let ε denote this common value, then (n + 1)b − n ε − ε = 0 implying that ε = b. notice that systems (7.8) and (7.9) are both linear in this case, so local asymptotical stability implies global asymptotical stability. the coefficient matrix in the continuous case is j c = 1 n + 1        −2k1 −k1 · · · −k1 −k2 −2k2 · · · −k2 . . . . . . . . . . . . −kn −kn · · · −2kn        (7.23) and in the discrete case j d = i + j c . similarly to theorems 1 and 2 we can easily prove that the continuous system is always asymptotically stable and the discrete system is asymptotically stable if and only if for all k, kk < 2(n + 1), and n ∑ k=1 kk 2(n + 1) − kk < 1. cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 79 we turn our attention next to case 3, when the firms learn about the slope of the price function. in this case we assume the firm k believes that the price function is fk(q) = b − εkq, the profit of firm l (l = 1, 2, . . . , n ) is believed by firm k to be ϕ (k) l (x1, . . . , xn ) = xl(b − εkq) − (αlxl + βl), (7.24) so the best believed output choice is xl = b − αl εk − q, and the believed total production of the industry is q (k) = n b − ∑n l=1 αl (n + 1)εk . (7.25) the believed equilibrium price, fk(q (k) ) = b + ∑n l=1 αl n + 1 , (7.26) is the same for all firms. firm k also believes that its equilibrium output is xk = b − (n + 1)αk + ∑n l=1 αl (n + 1)εk . (7.27) therefore in reality the total production of the industry becomes q = n ∑ k=1 xk = 1 n + 1 (( b + n ∑ l=1 αl ) n ∑ k=1 1 εk − (n + 1) n ∑ k=1 αk εk ) (7.28) with actual equilibrium price p = b − aq = b − a n + 1 (( b + n ∑ l=1 αl ) n ∑ k=1 1 εk − (n + 1) n ∑ k=1 αk εk ) . (7.29) based on the discrepancy between the actual and believed price d (k) = 1 n + 1 ( n b − a ( b + n ∑ l=1 αl ) n ∑ k=1 1 εk + a(n + 1) n ∑ k=1 αk εk − n ∑ l=1 αl ) (7.30) the dynamic process is similar to the previous cases (7.8) and (7.9). note that d (k) is the same for all firms, and therefore a set of εk (k = 1, 2, . . . , n ) values is a steady state of the system if and only if b ( n − a n ∑ k=1 l εk ) + (n + 1)a n ∑ k=1 αk εk − ( n ∑ l=1 αl )( a n ∑ k=1 1 εk + 1 ) = 0. 80 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) since this is a single equation for n unknowns with a feasible solution εk = a, there are infinitely many steady states. that is, there is the possibility that all firms believe in wrong price functions but none of them notices it since believed and actual prices still coincide. in this case no learning is possible in this way. in the above discussed cases we always assumed that the firms use instantaneous information about prices, however in reality price informations are always delayed. the effect of information lag in the learning process can be similarly examined to the cases being demonstrated in session 2.6. 8 laboratory experiments in november 2002 the nobel committee awarded vernon smith the prize in economics for a body of work spanning a half-century that demonstrated that controlled laboratory experiments could be used to study economic behavior, exactly as experiments in the hard sciences study physical phenomena. smith summarized his methodological breakthrough in a highly referenced 1982 paper, ”microeconomic systems as an experimental science.” today, economic experiments are widely undertaken to explore three main avenues of research: to inform economic theory, to test-bed newly designed institutions under stressful environmental conditions, and to understand how brain activity leads to economic behavior. as economic theorists we should be very concerned with rescuing our theories from the doldrums of mathematical curiosity by subjecting them to the rigors of laboratory scrutiny. this chapter has so far presented a review and some new developments in oligopoly theory that are enmeshed with an abundant theoretical literature in this area, but have we not discussed how to assess whether those results relate to what people really do. we can greatly increase the value of our theories to the society that invests in them by becoming concerned with how people really organize themselves to form, sustain, and adapt rules of order in order to generate beneficial outcomes for themselves. each laboratory that conducts experiments with human subjects approaches its research with different auxiliary hypotheses but uses the common analytic framework that has three main components: the environment, the institution, and the behavior of the human subjects. the environment includes the subjects’ preferences or incentives for achieving various allocations in the exchange system, subjects’ productive capabilities and system technical constraints on achieving those allocations, and knowledge about the initial conditions and allocation in the exchange system in which subjects will participate. the experimenter controls the environment using induced values, a mapping of outcomes to different monetary earnings, and carefully worded instructions. for example, in a simple oligopoly environment the experimenter may privately inform each experimental subject i through computerized instructions that he will be playing the role of a producer of a fictitious good, and that he will be able to, in any given period during the upcoming experiment, produce up to ui units of the good at a cost of $ci per unit, and that if he can sell those units he will earn a cash profit, paid to him by the experimenter, which will be the difference between the market price, p, he sells each unit for, and his cost of production. the institution consists of a set of rules that completely specify (1) what messages subjects are allowed to send and when they can send them, (2) how these messages are translated into reallocations of environmental conditions, and (3) what feedback the subject receives about the messages that were sent and the reallocations they produced. the experimenter typically controls for the institution with a computer network that instantiates these rules. for example, in one simple oligopoly environment a ’producer’ may only be able to make only one offer of a given total quantity that he is willing to sell at a particular per unit price in cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 81 a given period, and the institution may gather those offers, order them from lowest to highest price, and sell as many of them as possible at a uniform price to buyers whose bids to buy have been ordered from highest to lowest. the institution in this simple example may decide not to reveal publicly what offers and bids are submitted or what was the volume traded in the period. the behavior is what the experimenter can then observe in the form of messages that are actually sent by subjects as they participate in the experiment. by controlling the environment, e, and the institution, i, and creating experimental ’treatments’ which can selectively alter either of them, the experimenter can estimate the behavioral response function bi(e, i) for each subject, i. institutions where there is repeated interaction among subjects precipitate learning and the behavioral response functions themselves become functions of time, b t i(e, i), that depend upon the sequence of information delivered to the subject by the institution and the perceived reallocations in the environment. the experimental economics laboratory typically uses economic theory to predict the form of the behavioral functions bi(·, ·), and thus predicts the environmental reallocations that will be produced as the subjects interact. it becomes possible to compare predicted outcomes to actual outcomes and predicted behavioral functions to estimated behavioral functions to ask how well does the theory perform in the laboratory? consider the following example of a two-player game that has been extensively studied in the laboratory. in this case the messages are very simple. subject one must move first and decide whether to stop the game immediately, in which case he receives a payment of $10 and subject two also receives a payment of $10, or pass the decision on to subject two. if subject one chooses to pass, subject two now must choose between an allocation which pays subject one $0 and himself $40, or an allocation which pays subject one $ 15 and himself $25. using game theory, nash equilibrium predicts that when this game is played once by anonymous traders who have complete information, subject 2 would always choose the (0, 40), and that subject 1, realizing what subject 2 would do, will always to stop the game immediately for a ($10, $10) allocation. in fact, when this experiment is run typically only 50% of the subject 1’s elect to stop, and, conditional on passing to subject 2, 75% of the subject 2’s choose the allocation ($15, $25). thus in the typical population the expected payoff of subject 1 if he passes is .75 × 15 = 11.25, greater than 10. this simple experiment demonstrates the failure of a theory that does not fully account for the evolved tendency of human subjects to divine opportunity through understanding the history and intentions of those with whom they interact. a greater irony reveals itself in this scenario when we relax the nash assumptions by repeating the game and giving subjects incomplete information in the form of their only their own payoffs: then the nash prediction is robust! fouraker and siegel [3] conducted the first extensive experimental study of basic oligopoly theory more than 40 years ago. they hypothesized that although the original quantity adjusting cournot model does not directly discuss the information conditions of the agents, those conditions as well as the number of agents might have important consequences during repeated economic interactions. they were right: in cournot quantity adjusting experiments where subjects either knew (complete info.) or didn’t know (incomplete info.) their rivals payoff function, the distribution of outcomes suffered more variability under complete information, while the cournot prediction was more robust with less information. there are many experiments in various other environments that verify that when provided information subjects attempt to use it in not always an entirely predictable manner. further analysis provided by the fouraker/siegel data showed that both rivalistic (the tendency to increase your output when you observe others are producing more than you) and cooperative (the tendency to decrease your output when you observe others are producing more than you) signaling behaviors are more prevalent when information is complete. this result was most prevalent in 82 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) duopolies and transferred strongly to a bertrand price-setting environment. since the original oligopoly experiments, there has been an exponential growth of published oligopoly theory that deals with a multitude of mathematical nuances and assumptions, but comparatively few experimental tests that reign in the relevance of all those theories to human behavior. an interesting experimental study executed by [2] points out an environmental condition, low profitability under competition, that may confound the predictions of extant theory in a price setting scenario. they create a simple single product multi-period environment where 5 independent producers have identical cost structures such that they can each produce 100 units at cost c per unit, 100 units at cost c + d per unit, and 100 additional units at cost c + d + e per unit. the demand function is linear and represented by more than a hundreds robot buyers who simply reveal their value for consuming the product during the experiment. the demand function intersects the aggregate supply function at 1200 units, where the marginal cost of production is c + d + e. at the competitive price the sellers’ share of surplus is an order of magnitude less than the buyers’. each period during the experiment the oligopolists must post a single price at which they are willing to sell their product and the infinitesimal buyers queue up electronically and buy in order of lowest price available. buyers are rationed uniformly amongst producers tied in price. a competitive nash equilibrium exists that has each producer offering his 300 units at price c + d + e and earning 100d + 200e, since a higher price would exclude the producer completely and a lower price would produce a loss on his higher cost units. but this is not what the average group of oligopolists does in this environment. they engage in an offering strategy that creates price cycles that tend to rise rapidly and fall slowly. the highest price producer (perhaps 2 or 3) is always excluded in a given period, but the remaining active sellers are rewarded with supra-competitive profits. these subjects, under much more complicated environmental and institutional conditions than in the simple two person game discussed above, and without direct communication, create a continuous tension between competitive (gradual undercutting) and cooperative (raise prices) behavior that serves them well in the long run. our experimental philosophy has been shaped largely by our interpretation of friedrich von hayek’s view that economics is the study of the co-emergent order of the brain and our social institutions. this philosophical framework leads us to look for ecologically rational explanations of both individual and institutional behavior. in particular we assume that individuals are adapted to specific functional needs that arose over evolutionary time, but that are now expressed by our modern brains through interactions with our current institutions and environments. however, as our brains learn to cope with their modern scenario they encounter cognitive opportunity costs that we attempt to overcome by the development of institutions that can perform the requisite computations and reallocations cheaply and efficiently. economic experiments are crucial in testing the theory used to prescribe institutional innovation and in test-bedding the prescription before its implementation. 9 conclusions in this paper dynamic oligopoly models were examined. after introducing the classical cournot model and its extensions, the stability of single product oligopoly was discussed with the assumption that full and instantaneous information was available to all firms concerning the market price. the equilibrium is always locally asymptotically stable with continuous time scales, however in order to preserve stability in the discrete case we have to assume that either the derivatives of the best response functions of all firms are cubo 11, 2 (2009) cournot models: dynamics, uncertainty and learning 83 sufficiently small, or the firms do not change much their output levels. then we assumed that there was partial cooperation among the firms when each firm took a certain portion of the profits of the rivals into account in its payoff function. under realistic conditions we could obtain the same stability results as in the previous case. the firms are usually uncertain in the market as they use only an estimate of the price function in their decisions on the best choices on production levels. we could show that similar stability conditions hold for this case as well, however the production level might converge to a steady state which differs from the nash equilibrium. we also investigated the effect of information delay in market price, and showed that stability can be lost. we have derived conditions for stability and instability of the equilibrium and showed that at the critical value of the bifurcation parameter hopf bifurcation occurs giving the possibility of the birth of limit cycles. three particular learning processes were then introduced and examined when the firms had only limited information on the linear price function. it was interesting to see that the number of steady states (that is, the possibility of learning) and the asymptotic properties of the learning process were different for different types of uncertainty. all theoretical results discussed in these sessions were based on certain specific assumptions on cost and market demand structure as well as on particular assumptions on the behavior of the decision makers. such special conditions are not always satisfied in real economies, and in addition, the decision making managers do not think and decide always as we expect them to do. the most appropriate methodology in examining human decision making and economic processes without special conditions is based on laboratory experiments with realistic environment. experimental economy is this very important procedure in which the actual decisions of the participants are repeated, observed and analyzed and hence we are able to gain the right insight into the minds of the decision making humans. received: march 14, 2008. revised: may 15, 2008. references [1] cournot, a., recherches sur les principes mathèmatiques de la thèorie de richesses, hachette, paris (english translation (1960): researches into the mathematical principles of the theory of wealth. kelley, new york.), 1838. [2] durham, y., mccabe, k., olson, m., rassenti, s., and smith, v., oligopoly competition in fixed cost environments, international journal of industrial organization, fall, 2003. [3] fouraker, l.e., and siegel, s., bargaining behavior, mcgraw-hill book company/new york, 1963. [4] guckenheimer, j. and holmes, p., nonlinear oscillations, dynamic systems and bifurcations of vector fields, springer-verlag, berlin/heidelberg/new york, 1983. [5] okuguchi, k., expectations and stability in oligopoly models, springer-verlag, berlin/heidelberg/new york, 1976. [6] okuguchi, k. and szidarovszky, f., the theory of oligopoly with multi-product firms, springerverlag, berlin/heidelberg/new york (2nd edition), 1999. 84 ferenc szidarovszky, vernon l. smith and steven rassenti cubo 11, 2 (2009) [7] smith, v.l., microeconomic systems as an experimental science, american economic review, december, 1982. [8] szidarovszky, f. and bahill, t.a., linear system theory, crc press, boca raton/london (2nd edition), 1998. n05-cournot cubo a mathematical journal vol.11, no¯ 04, (59–71). september 2009 estimates for solutions to nonlinear degenerate elliptic equations fabrizio cuccu, dipartimento di matematica e informatica, università di cagliari, via ospedale 72, 09124 cagliari, italy. email: fcuccu@unica.it petar popivanov institute of mathematics and informatics, bulgarian academy of sciences, sofia 1113, bulgaria. email: popivano@math.bas.bg and giovanni porru dipartimento di matematica e informatica, università di cagliari, via ospedale 72, 09124 cagliari, italy. email: porru@unica.it abstract we find estimates of the l∞ norm of solutions to special nonlinear degenerate elliptic partial differential equations in terms of norms of the data. we also discuss a special isoperimetric inequality involved in the definition of the ellipticity of the above equations. resumen encontramos estimaciones de la norma l∞ de las soluciones de ecuaciones diferenciales parciales elípticas degeneradas nolineales en términos de la norma de los datos. además, discutimos una desigualdad isoperimétrica especial involucrada en la definición de la elipticidad de las ecuaciones anteriormente descritas. 60 fabrizio cuccu, petar popivanov and giovanni porru cubo 11, 4 (2009) key words and phrases: degenerate elliptic equations; isoperimetric inequality. math. subj. class.: 35b45; 35j70. 1 introduction. we are interested in the estimation of the l∞ norm of generalized solutions of a special class of nonlinear second order degenerate elliptic partial differential equations in divergent form. the sobolev type spaces w 1,p 0 (ψ,d) to whom our solutions belong are defined below. our main tools in the proposed investigation are a variant fubini’s theorem (see for example [7]) and a generalization of the famous isoperimetric inequality (see [2]). the condition (3) (see below) is inspired by the main assumption of [8]. unfortunately, in [8] there are no geometrical conditions leading to the fulfillment of such an assumption, and there are not nontrivial examples satisfying it. we propose here the proof of (3) in the two dimensional case and for functions φ(x) = λ|x|β, ψ(x) = λ|x|γ with 0 < λ ≤ λ and γ + 1 > β ≥ γ ≥ 0. thus, the complete study of the generalized isoperimetric inequality (3) is an open and, we think, a difficult problem. our main result is the a priori estimate of the l∞ norm of the solution by means of appropriate norms of ψ and fψ 1 s−1 , s > s0, and relies heavily on the parameter α of condition (3). here s0 is a special number which depends on p and α and it is sharp. in fact, as simple examples show, our equation (1) even in the linear case can possess unbounded solutions for s = s0. our results are very precise when φ(x) and ψ(x) are constants, as in this case the classical isoperimetric inequality holds. we propose a special example when the corresponding a priori estimate is sharp. the paper is organized as follows: main results, special cases illustrating the main theorem, proof of (3) in dimension two and for special radially symmetric functions φ and ψ. 2 main results let d be a bounded smooth domain in rn and let 1 < p < n. consider the following non linear second order degenerate elliptic equation in divergent form − ( (aijuxiuxj ) p−2 2 aijuxi ) xj = f(x), x ∈ d. (1) the summation convention from 1 to n over repeated indices is in effect. the matrix [aij] = [aij(x,u,∇u)] is assumed to be symmetric and elliptic in the sense 0 ≤ φ(x)|ξ|p ≤ ( aijξiξj )p 2 ≤ ψ(x)|ξ|p ∀ξ ∈ rn, (2) with φ 6≡ 0 and ψ ∈ l1(d). the functions φ and ψ are subject to the following condition: there exist two constants c > 0 and α ∈ (p−1 p , 1) such that, for each borel set e ⊂ d with smooth cubo 11, 4 (2009) estimates for solutions to nonlinear degenerate elliptic equations 61 boundary ∂e we have ∫ ∂e φ(x)dσ ≥ c (∫ e ψ(x)dx )α . (3) note that (1) is the euler equation of the functional ∫ d [( aijuxiuxj )p 2 − p u f ] dx. we consider solutions u ∈ w 1,p 0 (ψ,d), where w 1,p 0 (ψ,d) is the completion of c1 0 (d) with respect to the norm ‖u‖ = (∫ d ( |u|p + ψ(x)|∇u|p ) dx )1 p . we shall use the following formula ( [7] pag. 37). if g(x) ≥ 0 is measurable in the sense of borel in an open set ω and if u ∈ c0,1(ω) then ∫ ω g(x)|∇u|dx = ∫ ∞ 0 dτ ∫ fτ g(x)dσ, (4) where fτ = {x ∈ ω : |u(x)| = τ}, and dσ stands for the (n − 1)-hausdorff measure. the equality (4) has been extended to functions u ∈ w 1,1loc (d) (see for example [6], theorem 1.1). a detailed theory of sobolev spaces including w 1,1 loc (d) can be found in [7]. if u ∈ w 1,p 0 (ψ,d) is a solution of equation (1) we shall denote dt = {x ∈ d : u(x) > t}. by putting ω = dt with t > 0, equality (4) yields ∫ dt g(x)|∇u|dx = ∫ sup u t dτ ∫ fτ g(x)dσ, (5) with sup u = sup x∈d u(x), fτ = {x ∈ d : u(x) = τ}. we note that ∂dt ⊂ ft. lemma 2.1. if condition (3) holds and if φ(x) ≤ h(x) ≤ ψ(x) then for almost all t > 0 we have ∫ ft h(x)|∇u|p−1dσ ≥ cp (v (t)) αp ( −v ′(t) )p−1 , where v (t) = ∫ dt ψ(x)dx. 62 fabrizio cuccu, petar popivanov and giovanni porru cubo 11, 4 (2009) proof. by using the hölder inequality we find (∫ ft h(x)dσ )p = (∫ ft ( h(x)|∇u|p−1 )1 p ( h(x) |∇u| )p−1 p dσ )p ≤ ∫ ft h(x)|∇u|p−1dσ (∫ ft h(x) |∇u| dσ )p−1 . it follows that ∫ ft h(x)|∇u|p−1dσ ≥ (∫ ft φ(x)dσ )p (∫ ft ψ(x) |∇u| dσ )p−1 . (6) equality (5) with g(x) = ψ(x) |∇u| yields v (t) = ∫ sup u t dτ ∫ fτ ψ(x) |∇u|dσ. hence, for almost all t > 0 we have v ′ (t) = − ∫ ft ψ(x) |∇u|dσ. by using the latter equation and condition (3), from inequality (6) we get the statement of the lemma. theorem 2.2. assume conditions (2) and (3). let s > 1 p(1−α) , and let fψ−1+ 1 s ∈ ls(d). if u ∈ w 1,p 0 (ψ,d) is a solution of equation (1) then we have ‖u‖l∞(d) ≤ 1 c p p−1 s(p − 1) ps(1 − α) − 1‖fψ 1 s −1‖ 1 p−1 ls(d) (∫ d ψ(x)dx )ps(1−α)−1 s(p−1) . proof. putting h(x) = ( aijuxiuxj )p 2 |∇u|p , condition (2) implies φ(x) ≤ h(x) ≤ ψ(x). according to the definition of weak solution of (1) we have that for every v ∈ w 1,p 0 (ψ,d) ∫ d ( aijuxiuxj )p−2 2 aijuxivxj dx = ∫ d f(x)v(x)dx. since u ∈ w 1,p 0 (ψ,d), for t > 0 we can take v = (u(x) − t)+. we find ∫ dt ( aijuxiuxj )p 2 dx = ∫ dt f(x)(u(x) − t)dx, where dt = {x ∈ d : u(x) > t}. cubo 11, 4 (2009) estimates for solutions to nonlinear degenerate elliptic equations 63 recalling our definition of h(x), by the latter equation we have ∫ dt h(x)|∇u|pdx = ∫ dt f(x)dx ∫ u t dτ = ∫ sup u t dτ ∫ dτ f(x)dx. (7) as it concerns the left hand side, we apply (5) with g(x) = h(x)|∇u|p−1. we find ∫ dt h(x)|∇u|pdx = ∫ sup u t dτ ∫ fτ h(x)|∇u|p−1dσ. the latter equation and (7) yield ∫ sup u t dτ ∫ fτ h(x)|∇u|p−1dσ = ∫ sup u t dτ ∫ dτ f(x)dx. after a differentiation, for almost all t we get ∫ ft h(x)|∇u|p−1dσ = ∫ dt f(x)dx ≤ ∫ dt |f(x)|dx. (8) applying lemma 2.1 we find c p (v (t)) αp ( −v ′(t) )p−1 ≤ ∫ dt |f(x)|ψ 1−s s ψ s−1 s dx ≤ ‖fψ 1s −1‖ls(d) (∫ dt ψ(x)dx )s−1 s . rearranging we have 1 ≤ 1 c p p−1 ‖fψ 1s −1‖ 1 p−1 ls(d) (v (t)) s(1−pα)−1 s(p−1) (−v ′(t)). finally, integrating over (0, sup u) we get sup u ≤ 1 c p p−1 s(p − 1) ps(1 − α) − 1‖fψ 1 s −1‖ 1 p−1 ls(d) (v (0)) ps(1−α)−1 s(p−1) . being v (0) ≤ ∫ d ψ(x)dx we find sup u ≤ 1 c p p−1 s(p − 1) ps(1 − α) − 1‖fψ 1 s −1‖ 1 p−1 ls(d) (∫ d ψ(x)dx )ps(1−α)−1 s(p−1) . (9) if u is a solution of equation (1) then −u is a solution the same equation with −f in place of f. therefore, the estimate (9) also holds for −u. the theorem follows. 3 special cases illustrating theorem 2.2 we shall begin this section with the case when φ(x) = λ and ψ(x) = λ. then condition (3) holds with α = 1 − 1 n , c = λ λ n−1 n nω 1 n n , 64 fabrizio cuccu, petar popivanov and giovanni porru cubo 11, 4 (2009) where ωn is the measure of the unit ball in r n. for 1 n p , theorem 2.2 yields ‖u‖l∞(d) ≤ λ ( nλω 1 n n ) p p−1 ns(p − 1) ps − n ‖f‖ 1 p−1 ls(d) |d| ps−n n s(p−1) . (10) however, in this special case we can prove an inequality sharper than (10). indeed, since λ ≤ h(x), by the inequality (8) we find λ ∫ ft |∇u|p−1dσ ≤ ‖f‖ls(d)|dt| s−1 s . (11) instead of (6) we use the inequality ∫ ft |∇u|p−1dσ ≥ (∫ ft dσ )p (∫ ft 1 |∇u| dσ )p−1 . putting µ(t) = |dt| we know that µ ′ (t) = − ∫ ft 1 |∇u|dσ. using this equality and the familiar isoperimetric inequality (see, for example, [2]) ∫ ft dσ ≥ nω 1 n n (µ(t)) n−1 n we find ∫ ft |∇u|p−1dσ ≥ n p ω p n n (µ(t)) p(n−1) n (−µ′(t))p−1 . hence, by (11) we get λn p ω p n n (µ(t)) p(n−1) n (−µ′(t))p−1 ≤ ‖f‖l s(d)(µ(t)) s−1 s , and 1 ≤ µ −1+ 1 p−1 ( p n − 1 s ) (t) ( λnpω p n n ) 1 p−1 (−µ′(t))‖f‖ 1 p−1 ls(d) . (12) integration over (0, sup u) yields sup u ≤ 1 (nλ) 1 p−1 ω p n(p−1) n s(p − 1) ps − n ‖f‖ 1 p−1 ls(d) |d| ps−n n s(p−1) . the same estimate can be found for −u. therefore, ‖u‖l∞(d) ≤ 1 (nλ) 1 p−1 ω p n(p−1) n s(p − 1) ps − n ‖f‖ 1 p−1 ls(d) |d| ps−n n s(p−1) . the latter inequality improves (10) by the factor λ/λ. cubo 11, 4 (2009) estimates for solutions to nonlinear degenerate elliptic equations 65 in case f is bounded in d then we can take s → ∞ and we find ‖u‖l∞(d) ≤ 1 (nλ) 1 p−1 ω p n(p−1) n p − 1 p (sup x∈d |f(x)|) 1 p−1 |d| p n(p−1) . (13) in the case λ = λ, f is a constant and d is a ball, the inequality (13) is sharp. indeed, if r is the radius of the ball and f = a > 0, with u(x) = u(r) for |x| = r, the equation reads as λ ( r n−1|u′|p−1 )′ = r n−1 a. integrating we find −u′ = r 1 p−1 (nλ) 1 p−1 a 1 p−1 , u(r) = (p − 1)a 1 p−1 p(nλ) 1 p−1 [ r p p−1 − r p p−1 ] , and ‖u‖l∞(d) = u(0) = (p − 1)a 1 p−1 p(nλ) 1 p−1 r p p−1 . (14) equation (14) yields (13) with the equality sign. we shall consider now the case when aij = |x|βδij, β ≥ 0, δij being the kronecker symbol. in this case condition (2) holds with φ(x) = ψ(x) = |x| βp2 . if we look for condition (3) when e are balls centered in the origin we find α = 1 − 1 n+β . we think that this value of α is correct for all borel sets e, but we can prove this fact in case of n = 2 only (see the next section). let us show that the conclusion of theorem 2.2 is generally false if fψ 1 s −1 ∈ ls(d) with s = n+β p and p ≥ 2. we have ( (aijuxiuxj ) p−2 2 aijuxi ) xj = ( (|x|β|∇u|2) p−2 2 |x|βuxi ) xi . if u(x) is a radial function and u(x) = v(r) for |x| = r, we have uxi = v′ xir and ( (|x|β|∇u|2) p−2 2 |x|βuxi ) xi = r 1−n ( r n−1+ βp 2 |v′|p−2v′ )′ . consider problem (1) when ω is a ball b centered in the origin and f = f(r) is a radial function. then the solution v is radial and satisfies the equation −r1−n ( r n−1+ βp 2 |v′|p−2v′ )′ = f(r). (15) let b be the ball with radius 1/e. consider the unbounded function v(r) = log(− log r). 66 fabrizio cuccu, petar popivanov and giovanni porru cubo 11, 4 (2009) we find v ′ = 1 r log r and −r1−n ( r n−1+ βp 2 |v′|p−2v′ )′ = r 1−n ( r n+ βp 2 −p (− log r)1−p )′ = r βp 2 −p (− log r)1−p [ n + βp 2 − p + (1 − p) 1 log r ] . hence, our function v satisfies equation (15) with f(r) = c(r)r βp 2 −p (− log r)1−p, where c(r) is a bounded function for r < 1 e . if s = n+β p we have ( f(r)ψ ( 1 s −1) )s = c̃(r)r βp 2 −n−β (− log r)(1−p) n+β p . if β > 0 it is easy to see that f(r)ψ( 1 s −1) ∈ ls(b) for s ≤ n+β p and p ≥ 2. the same computations with β = 0 show that when f ∈ ls(d) with s = n p and p > n n−1 we may have unbounded solutions of equation (1). 4 proof of the isoperimetric inequality in dimension two and for radially symmetric functions φ and ψ we shall deal with φ(x) = λ|x|β, ψ(x) = λ|x|γ, 0 < λ ≤ λ, γ + 1 > β ≥ γ ≥ 0 in this section. we use polar coordinates (ρ,θ). if e is a given set, we define a new set e′ according to the following rule. for every ρ > 0, if fρ = {x ∈ r2 : |x| = ρ} we replace e ∩ fρ with the arc with radius ρ, having the same 1-dimensional measure as e ∩ fρ, centered in (ρ, 0). the sets e′ are situated symmetrically with respect to θ = 0. we have ∫ e |x|γdx = ∫ e′ |x|γdx. indeed, if we integrate from ρ to ρ + dρ we find ργldρ, where l is the 1-dimensional measure of e ∩ fρ (which is equal to the 1-dimensional measure of e′ ∩ fρ). moreover, we have ∫ ∂e |x|βds ≥ ∫ ∂e′ |x|βds′. indeed, if β = 0 this is the classical isoperimetric inequality (see, for example, [5]). if β > 0 we can apply this inequality to the region of e enclosed between ρ and ρ + dρ. the boundary integral of this elementary part of e is ρβ(ds + 2l) (l has the same meaning as before). similarly, the cubo 11, 4 (2009) estimates for solutions to nonlinear degenerate elliptic equations 67 boundary integral of the part of e′ enclosed between ρ and ρ + dρ is ρβ(ds′ + 2l). hence, ds ≥ ds′ for each value of ρ. the inequality follows. therefore, in order to prove condition (3) we can confine ourselves to sets e having the representation e = {r ≤ ρ ≤ r, −h(ρ) ≤ θ ≤ h(ρ)} with r ≥ 0. of course, 0 ≤ h(ρ) ≤ π. consider first the case h(ρ) = π for 0 ≤ ρ ≤ r, that is the disc centered in the origin and of radius r. we find ∫ ∂e λ|x|βds = 2πλrβ+1, ∫ e λ|x|γdx = 2πλ γ + 2 r γ+2 . therefore, with α = β + 1 γ + 2 (20) we have ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α = 2πλ ( 2πλ γ+2 )α . let h(ρ) < π for 0 ≤ ρ ≤ r. define the new set dτ = {0 ≤ ρ ≤ r, −τ ≤ θ ≤ τ}, with τ such that ∫ e |x|γdx = ∫ dτ |x|γdx. this value of τ ∈ (0,π) exists because dπ is the disc with radius r and d0 is the segment (0,r), e ⊂ dπ but e 6= dπ and f(τ) = ∫ dτ |x|γdx is a continuous monotonically increasing function for 0 < τ < π. let us show that ∫ ∂e |x|βds ≥ 2 ∫ l |x|βds′ = 2 β + 1 r β+1 , (21) where l is the segment θ = τ, 0 ≤ ρ ≤ r. indeed, if ds is the length of the part of the arc ∂e between ρ and ρ + dρ, and if ds′ is the length of the part of the segment l between ρ and ρ + dρ, we have ds ≥ 2ds′. we notice that the segment l is situated at θ = τ, whereas ∂e has one part situated at θ ≥ 0, and the symmetric part situated at θ ≤ 0. easy computations yield ∫ dτ |x|γdx = 2τ γ + 2 r γ+2 . 68 fabrizio cuccu, petar popivanov and giovanni porru cubo 11, 4 (2009) therefore, with α as in (20) we have ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ 2λ β+1( 2τλ γ+2 )α > 2λ β+1( 2πλ γ+2 )α . now we consider r > 0. if h(r) = π for r ≤ ρ ≤ r we can replace e by the ball h(r) = π for 0 ≤ ρ ≤ r. the integral of λ|x|γ over the ball is greater than the integral over e, whereas, the integral of λ|x|β over the boundary of the ball is smaller than the integral over ∂e. hence, in this situation we find ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ 2πλ ( 2πλ γ+2 )α . let h(r) < π for r ≤ ρ ≤ r. define the set gτ = {r ≤ ρ ≤ r, −τ ≤ θ ≤ τ}, with τ such that ∫ e |x|γdx = ∫ gτ |x|γdx. we have 0 < τ < π. arguing as in the proof of (21), now we find ∫ ∂e |x|βds ≥ 2 ∫ l |x|βds′ = 2 β + 1 (r β+1 − rβ+1), (22) where l is the segment θ = τ, r ≤ ρ ≤ r. let us show that we also have ∫ ∂e |x|βds ≥ 2 ∫ γ |x|βds′ = 2τ rβ+1, (23) where γ is the arc ρ = r, 0 < θ < τ. indeed, if ds is the length of the part of the arc ∂e between θ and θ + dθ, θ > 0, and if ds′ is the length of the part of the arc γ between θ and θ + dθ, we have ds ≥ ds′. recall that ∂e is symmetric with respect to θ = 0. if we add (22) and (23) we get ∫ ∂e |x|βds ≥ 1 β + 1 (r β+1 − rβ+1) + τ rβ+1. on the other side, direct computation yields ∫ e |x|γdx = ∫ gτ |x|γdx = 2τ ∫ r r ρ γ+1 dρ = 2τ γ + 2 (r γ+2 − rγ+2). therefore, with α as in (20) we find ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ λ β+1 (r β+1 − rβ+1) + τ λ rβ+1 ( 2τλ γ+2 )α (rγ+2 − rγ+2)α . cubo 11, 4 (2009) estimates for solutions to nonlinear degenerate elliptic equations 69 if τ ≥ 1 β+1 we get λ β+1 (r β+1 − rβ+1) + τ λ rβ+1 ( 2τλ γ+2 )α (rγ+2 − rγ+2)α ≥ λ β+1 r β+1 ( 2πλ γ+2 )α (rγ+2 − rγ+2)α ≥ λ β+1( 2πλ γ+2 )α . to discuss the case τ < 1 β+1 we consider the function f(r,t) = r β+1 − rβ+1 + t rβ+1 ( t(rγ+2 − rγ+2) )α for 0 < t < 1 and 0 < r < r. it is easy to see that ∂f ∂t = 0 for t = α 1 − α r β+1 − rβ+1 rβ+1 . with t ∈ (0, 1), the function f(r,t) is positive, strictly decreasing for t < t and strictly increasing for t > t. then two cases are considered. a) 0 ≤ r ≤ rα 1 β+1 . then we have t ≥ 1. therefore, recalling that α is given by equation (20) we find f(r,t) ≥ f(r, 1) = r β+1 ( rγ+2 − rγ+2 )α ≥ 1, ∀t ∈ (0, 1). b) rα 1 β+1 < r < r. (i) from γ + 1 > β ≥ γ ≥ 0 we find 1 2 ≤ γ + 1 γ + 2 ≤ α = β + 1 γ + 2 < 1, which implies 2α − 1 ≥ 0 and α(γ + 1) − β(1 − α) > 0. (ii) since α rβ+1 < 1 rβ+1 we have f(r,t) ≥ f(r,t) = 1 1 − α r β+1 − rβ+1 ( α 1−α rβ+1−rβ+1 rβ+1 (rγ+2 − rγ+2) )α ≥ 1 1 − α r β+1 − rβ+1 ( 1 1−α rβ+1−rβ+1 rβ+1 (rγ+2 − rγ+2) )α = (1 − α)α−1 (r β+1 − rβ+1)1−αrα(β+1) (rγ+2 − rγ+2)α . (iii) as we know from the theory of homogeneous functions, r β+1 − rβ+1 ≥ cβ(r − r)(r + r)β 70 fabrizio cuccu, petar popivanov and giovanni porru cubo 11, 4 (2009) and r γ+2 − rγ+2 ≤ cγ(r − r)(r + r)γ+1, for suitable positive constants cβ and cγ. this implies (r β+1 − rβ+1)1−αrα(β+1) (rγ+2 − rγ+2)α ≥ c 1−α β cαγ r α(β+1) (r − r)2α−1(r + r)α(γ+1)−β(1−α) ≥ c 1−α β cαγ r α(β+1) r2α−1(2r)α(γ+1)−β(1−α) = c 1−α β cαγ 2 α(γ+1)−β(1−α) . therefore, for this kind of sets e there is cβ,γ > 0 such that ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ cβ,γ λ λα . let h(ρ) = π for 0 ≤ ρ ≤ r and h(ρ) < π for r < ρ ≤ r. if there is a set gτ = {r ≤ ρ ≤ r, −τ ≤ θ ≤ τ}, with τ < π such that ∫ e |x|γdx = 2 ∫ gτ |x|γdx, then we can argue as in the previous case and find ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ cβ,γ λ (2λ)α . otherwise we must have ∫ e |x|γdx ≥ 2 ∫ gπ |x|γdx. since ∫ br |x|γdx ≥ ∫ e |x|γdx and gπ = br \ br, this implies that ∫ br |x|γdx ≥ 2 ∫ br |x|γdx − 2 ∫ br |x|γdx, from which we get ( r r )γ+2 ≥ 1 2 . (24) on the other side, since br ⊂ e ⊂ br we have ∫ e |x|γdx ≤ ∫ br |x|γdx, ∫ ∂e λ|x|βds ≥ ∫ ∂br λ|x|βds. hence, ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ ∫ ∂br λ|x|βds (∫ br λ|x|γdx )α = 2πλ β+1( 2πλ γ+2 )α ( r r )β+1 . (25) cubo 11, 4 (2009) estimates for solutions to nonlinear degenerate elliptic equations 71 estimates (24) and (25) yield ∫ ∂e λ|x|βds (∫ e λ|x|γdx )α ≥ 2πλ β+1( 4πλ γ+2 )α . the case e has the representation h(ρ) ≤ π for 0 ≤ ρ ≤ r, can be reduced to one of the previous cases. indeed, let b = sup{ρ : h(ρ) = π}. if b = r we replace e with the ball ẽ with radius r. if b < r we replace e with the set ẽ having the representation h̃(ρ) = π for 0 ≤ ρ ≤ b, and h̃(ρ) = h(ρ) for b < ρ ≤ r. the integral of λ|x|γ over ẽ is greater than the corresponding integral over e, whereas, the integral of λ|x|β over ∂ẽ is smaller than the corresponding integral over ∂e. the case e has the representation h(ρ) ≤ π for r ≤ ρ ≤ r can be treated similarly. if b is as before and b = r we replace e with the ball ẽ with radius r. if b < r we replace e with the set ẽ having the representation h̃(ρ) = π for 0 ≤ ρ ≤ b, and h̃(ρ) = h(ρ) for b < ρ ≤ r. this way we have completed the proof of (3) in our special case. received: may 2008. revised: september 2008. references [1] agmon, s. douglis, a. and nirenberg, l., estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. i, commun. pure appl. math., 12 (1959), 623–727. [2] evans, l. c. and gariepy, r. f., measure theory and fine properties of functions, crc press 1992. [3] federer, h., curvature measures transaction ams, vol. 93 (1959), 418–451. [4] gilbarg, d. and trudinger, n. s., elliptic partial differential equations of second order, springer verlag, berlin, 1977. [5] kawohl, b., rearrangements and convexity of level sets in pde, lectures notes in mathematics, 1150, berlin, 1985. [6] malý, j. swanson, d. and ziemer, w. p., the coarea formula for sobolev mappings, arxiv:math.ca/0112008 v1, 1 dec. 2001, 1–16. [7] mazya, v., sobolev spaces, springer, berlin, 1985. [8] novruzov, a. a., on the maximum principle of elliptic equations of the second order with non-negative characteristic form, trans. acad. sci. azerb. ser. phys.-tech. math. sci., 20 (2000), 91–96. articulo 5 cubo a mathematical journal vol.10, n o ¯ 03, (1–12). october 2008 proximal-resolvent methods for mixed variational inequalities muhammad aslam noor and khalida inayat noor mathematics department, comsats institute of information technology, islamabad, pakistan email: noormaslam@hotmail.com email: khalidanoor@hotmail.com abstract it is well-known that the mixed variational inequalities are equivalent to the fixed point problem. we use this alternative equivalent formulation to suggest and analyze some new proximal resolvent methods for solving mixed variational inequalities. we also study the convergence of these new methods under some mild conditions. these new iterative methods include the projection, extragradient and proximal methods as special cases. results obtained in this paper represent a refinement and improvement of the previously known results. resumen es bien conocido que las desigualdades variacionales mescladas son equivalentes a problemas de punto fijo. nosotros usamos esta formulación alternativa equivalente para sugerir y analizar nuevos métodos resolventes proximales para resolver desigualdes variacionales mesclasdas. también estudiamos la convergencia de estos nuevos métodos bajo algunas condiciones blandas. estos nuevos métodos iterativos incluyen como casos especiales la prejección, métodos extragradiente y proximales. los resultados en este 2 muhammad aslam noor and khalida inayat noor cubo 10, 3 (2008) art́ıculo representam un refinamiento y perfeccionamiento de resultados previamente conocidos. key words and phrases: variational inequalities, resolvent method, fixed point, proximal methods, convergence. math. subj. class.: 49j40, 90c30. 1. introduction variational inequalities, which were introduced and considered by stampacchia [26] in 1964, have had a great impact and influence in the development of almost all branches of pure and applied sciences. it has been shown that the variational inequalities provide a simple, unified, natural, novel and general framework to study a wide class of problems arising in various branches of pure and applied sciences. the ideas and techniques of variational inequalities are being used in a variety of diverse fields and proved to innovative and productive, see [1-26] and the references therein. in recent years, variational inequalities have been extended and generalized in several directions. a useful and important generalization of variational inequalities is called the mixed variational inequality or variational inequality of the second kind containing the nonlinear term. due to the presence of the nonlinear term, the projection method and its variant forms including the wienerhopf equations can not be extended for solving the mixed variational inequality. to overcome these drawbacks, some iterative methods have been developed and investigated for solving mixed variational inequalities using the technique of auxiliary principle technique, the origin of which can be traced back to lions and stampacchia [10] and glowinski, lions and tremolieres [7]. this technique has been used by several researchers to develop implicit and explicit methods for solving the mixed variational inequalities and the equilibrium problems, see [14-23] and the references therein. we would like to mention that, if the nonlinear term in the mixed variational inequalities is a proper, convex and lower-semicontinuous, then it has been shown [14] that the mixed variational inequalities are equivalent to the fixed point problem. this alternative equivalent formulation has been used to suggest and analyze several iterative methods for solving the mixed variational inequalities. the convergence of these resolvent iterative methods requires that the underlying operator is strong monotone and lipschitz continuous. secondly it is very difficult to evaluate the resolvent of the operator. these facts have motived to modify the resolvent iterative method. noor [16-20] used the technique of updating the solution to suggest and analyze some modified extraresolvent type method. the extraresolvent method overcomes this difficulty by using the technique of updating the solution, which modified the resolvent method by performing additional step and resolvent at each step according to double resolvent formula. it is worth mentioning that the convergence of the extraresolvent method requires that the solution exists and the operator to be monotone and lipschitz continuous. when the operator is not lipschitz continuous or when the lipschitz continuous is not known, the extraresolvent method and its variant forms require an armijo-like line search procedure to compute the step size with a new resolvent needed for cubo 10, 3 (2008) proximal-resolvent methods for mixed ... 3 each trial, which leads to expensive computation. to overcome these draw backs, many authors have suggested and proposed some modified methods for solving mixed variational inequalities. we also note that if the nonlinear term involving the mixed variational inequalities is an indicator function of a convex set in the hilbert space, then the mixed variational inequalities are equivalent to the classical variational inequalities. he at el. [9] and noor [19] have considered a class of modified proximal-extragradient methods for solving the classical variational inequalities, which uses a better step-size rule (inexactness criteria) and includes the proximal and the extragradient methods as a special cases. they have shown the convergence of this approximate proximal method requires either monotonicity or pseudomonotonicity. it has been shown [9] that these proximalextragradient methods are numerically efficient and robust. it is worth mentioning that there are no such methods for solving the mixed variational inequalities. inspired and motivated by the research going in this dynamic field, we suggest some new proximal-resolvent methods for solving the mixed variational inequalities. we show that the convergence of our methods requires only the pseudomonotonicity, which is a weaker condition than monotonicity. results obtained in this paper include the results of he et al [9] and noor [19] as special cases and improve the convergence criteria of methods of he et al [9]. our results can also be viewed as a significant extension and generalization of the previously known methods for solving the mixed variational inequalities and related optimization problems. 2. formulation let k be a nonempty closed and convex set in a real hilbert space h, whose inner product and norm are denoted by 〈·, ·〉 and ‖.‖ respectively. let t : h −→ h be a nonlinear operator and s be a nonexpansive operator. let pk be the projection of h onto the convex set k. let ϕ : h −→ r ∪ {∞} be a continuous function. it is well known that the subdifferential ∂ϕ(.) of a proper, convex and lower-semicontinuous function ϕ is a maximal monotone operator. we consider the problem of finding u ∈ h such that 〈t u, v − u〉 + ϕ(v) − ϕ(u) ≥ 0, ∀v ∈ h, (1) which is known as the mixed variational inequality introduced or variational inequality of the second type, see glowinski, lions and tremolieres [7] and lions and stampacchia [10].. we note that, if the function ϕ in the mixed variational inequality is a proper, convex and lower-semicontinuous, then problem (1) is equivalent to finding u ∈ h such that 0 ∈ t u + ∂ϕ(u), (2) which is known as the problem of finding a zero of sum of two (or more ) monotone operators. here ∂ϕ is the subdifferential of the function ϕ. it is well known that a large class of problems arising in industry, ecology, finance, economics, transportation, network analysis and optimization 4 muhammad aslam noor and khalida inayat noor cubo 10, 3 (2008) can be formulated and studied in the framework of (1) and (2), see [3-6, 15-24] and the references therein. if ϕ is an indicator function of a closed convex set k in h, that is, ϕ(u) = ik (v) = { 0, if v ∈ k; +∞, otherwise. then the mixed variational inequalities (1) are equivalent to finding u ∈ k such that 〈t u, v − u〉 ≥ 0, ∀v ∈ k, (3) which is known as the classical variational inequality, introduced and studied by stampacchia [26] in 1964. for the numerical methods, formulations and applications of the mixed variational inequalities, readers are advised to see [1-25] and the references therein. we now recall some well known concepts and results. definition 2.1[3]. for any maximal operator t, the resolvent operator associated with t, for any ρ > 0, is defined as jt (u) = (i + ρt ) −1(u), ∀u ∈ h. it is well known that an operator t is maximal monotone if and only if its resolvent operator jt is defined everywhere. it is single-valued and nonexpansive. that is, ‖jt u − jt v‖ ≤ ‖u − v‖, ∀u, v ∈ h. if ϕ(.) is a proper, convex and lower-semicontinuous function, then its subdifferential ∂ϕ(.) is a maximal monotone operator. in this case, we can define the resolvent operator jϕ(u) = (i + ρt ) −1(u), ∀u ∈ h associated with the subdifferential ∂ϕ(.). the resolvent operator jϕ has the following useful characterization, see[3,20]. lemma 2.1. for a given z ∈ h, u ∈ h satisfies the inequality 〈u − z, v − u〉 + ρϕ(v) − ρϕ(u) ≥ 0, ∀v ∈ h (4) if and only if u = jϕ(z, where jϕ = (i + ρ∂ϕ) −1 is the resolvent operator. it is well-known that the resolvent operator jϕ is a nonexpansive operator, that is, ‖jϕ(u) − jϕ(v)‖ ≤ ‖u − v‖, ∀u, v ∈ h. lemma 2.1 plays a very important and significant role in the analysis of the mixed variational inequalities. if the proper, convex and semi-lowercontinuous function ϕ is an indicator function cubo 10, 3 (2008) proximal-resolvent methods for mixed ... 5 of a closed convex set k in h, then jϕ ≡ pk , is the projection operator from h onto the closed convex set k. in this case, lemma 2.1 reduces to the following well known result, which is known as the projection lemma. lemma 2.2 . let k be a closed convex set k in h. then, for a given z ∈ h, u ∈ k satisfies the inequality 〈u − z, v − u〉 ≥ 0, ∀v ∈ k, if and only if u = pk z, where pk is the projection of h onto the closed convex set k. it is also known that the projection operator pk is nonexpansive. for the applications of lemma 2.2, see [1-25]. definition 2.2. ∀u, v ∈ h, the operator t : h −→ h with respect the function ϕ is said to be pseudomonotone, if 〈t u, v − u〉 + ϕ(v) − ϕ(u) ≥ 0 implies 〈t v, v − u〉 + ϕ(v) − ϕ(u) ≥ 0. note that monotonicity implies pseudomonotonicity but the converse is not true [5]. 3. main results in this section, we use the projection technique to suggest some iterative methods for solving the variational inequalities. for this purpose, we need the following result, which can be proved by invoking lemma 2.1. lemma 3.1. the function u ∈ h is a solution of the mixed variational inequality (1) if and only if u ∈ h satisfies the relation u = jϕ[u − ρt u], (5) where ρ > 0 is a constant and jϕu = (i + ρ∂ϕ) −1(u) is the resolvent operator. lemma 3.1 implies that problems (1) and (5) are equivalent. this alternative formulation is very important from the numerical analysis point of view and has played a significant part in suggesting several numerical methods for solving variational inequalities and complementarity problems, see [1-7,10-20]. we now define the projection residue vector by the relation r(u) = u − jϕ[u − ρt u] = u − y, y = jϕ[u − ρt u]. invoking lemma 3.1, one can easily show that u ∈ h is a solution of (1) if and only if u ∈ h is a zero of the equation r(u) = 0. 6 muhammad aslam noor and khalida inayat noor cubo 10, 3 (2008) for a positive constant α, we consider u = u − αr(u) = u − α{u − jϕ[u − ρt u]}, which is another fixed point problem. we use alternative fixed point formulation to suggest and analyze the following iterative method for solving the mixed variational inequality (1). algorithm 3.1. for a given u0 ∈ h, compute the approximate solution un+1 by the iterative scheme un+1 = jϕ[un − γnr(un+1)] = jϕ[un − γn{un − jϕ[un − ρt un+1]}], n = 0, 1, 2, . . . , or equivalently yn = jϕ[un − ρt un+1] un+1 = jϕ[un − γn{un − yn}], n = 0, 1, 2, . . . which can be considered as a proximal point method and appears to be a new one. if ϕ is the indicator function of a closed convex set k, then jϕ ≡ pk , the projection of h onto k. consequently, algorithm 3.1 collapse to the following algorithm for solving classical variational inequalities (3). algorithm 3.2. for a given u0 ∈ h, compute the approximate solution un+1 by the iterative scheme un+1 = pk [un − γnr(un+1)] = pk [un − γn{un − pk [un − ρt un+1]}], n = 0, 1, 2, . . . , or equivalently yn = pk [un − ρt un+1] un+1 = pk [un − γn{un − yn}], n = 0, 1, 2, . . . which can be considered as a proximal-extragradient method. note that for γn = 1, algorithm 3.1 reduces to: algorithm 3.3. for a given u0 ∈ h, compute the approximate solution un+1 by the iterative scheme un+1 = jϕ[un − ρt un+1], n = 0, 1, 2 . . . which is known as the proximal method and convergence requires only pseudomonotonicity, see noor [20]. in recent years, proximal methods have been considered and studied extensively. several conditions have been studied which are easy to implement, see [9, 17-20]. cubo 10, 3 (2008) proximal-resolvent methods for mixed ... 7 we now use the technique of updating the solution to rewrite the fixed-point formulation (5) as: y = jϕ[u − ρt u] (6) u = jϕ[y − ρt y], which can be written as u = jϕ[jϕ[u − ρt u] − ρt jϕ[u − ρt u]], which is another fixed point formulation of the mixed variational inequalities (1). here we use this equivalent alternative formulation to suggest the following method for solving mixed variational inequalities (1). algorithm 3.4. for a given u0 ∈ h, find the approximate solution un+1 by the iterative schemes: yn = jϕ[un − ρt un+1] un+1 = jϕ[yn − ρt yn], n = 0, 1, 2, . . . algorithm 3.5. for a given u0 ∈ h, find the approximate solution un+1 by the iterative schemes: un+1 = jϕ[jϕ[un − ρt un+1] − ρt jϕ[un − ρt un+1]], n = 0, 1, 2, . . . algorithms 3.4 and algorithm 3.5 are called the two-step or predictor-corrector implicit iterative resolvent methods for solving the mixed variational inequalities (1) and appear to be new ones. if ϕ is the indicator function of a closed convex set k, then algorithm 3.5 is equivalent to the following implicit projection iterative method for solving the classical variational inequalities (3), which are mainly due to noor [16-18]. algorithm 3.6. for a given u0 ∈ h, find the approximate solution un+1 by the iterative schemes: un+1 = pk [pk [un − ρt un+1] − ρt pk [un − ρt un+1]], n = 0, 1, 2, . . . now we look at algorithm 3.4 from a different angle. consider y defined by (6) as an approximate solution of the mixed variational inequality (1) and define w = jϕ[u − γ(u − y)] z = u − ρt w. we use this formulation to suggest the following iterative method algorithm 3.7. for a given u0 ∈ h, calculate the approximate solution un+1 by the iterative schemes; yn = jϕ[un − ρt un+1] wn = jϕ[un − γ(un − yn)] un+1 := zn = un − ρt wn, n = 0, 1, 2, . . . 8 muhammad aslam noor and khalida inayat noor cubo 10, 3 (2008) which is called the modified extraresolvent method and appears to be a new one. note that for γ = 1, algorithm 3.7 reduces to algorithm 3.8. for a given u0 ∈ h, compute the approximate solution un+1 by the iterative scheme yn = jϕ[un − ρt un+1] un+1 = un − ρt yn, n = 0, 1, 2, . . . which is called the extraresolvent method for solving the mixed variational inequality (1). for a positive constant α, consider u = u − α(u − z). here the positive constant α can be viewed as a step length along the direction −(u − z). we use this fixed-point formulation to suggest the following iterative method. algorithm 3.9. for a given u0 ∈ h, compute the following iterative schemes: yn = jϕ[un − ρnt un+1] wn = jϕ[un − γn(un − yn)] zn = jϕ[un − ρnt wn] (7) un+1 = un − α(un − zn), n = 0, 1, 2, . . . (8) α = ‖zn − wn‖ 2 + ‖un − zn‖ 2 − △(wn) 2‖un − zn‖2 (9) where △(wn) ≤ ν(‖zn − wn‖ 2 + ‖un − zn‖ 2), ν < 1 = ν{2〈wn − zn, wn − un + ρnt wn + ρnϕ ′(wn)〉 − ‖wn − zn‖ 2}. (10) here △(wn) is known as the inexactness criteria which can be viewed as stepsize and ϕ ′(.) is the differential of the convex function ϕ. for α = 1 and zn = wn, algorithm 3.9 is exactly algorithm 3.8. if y = w, then algorithm 3.9 reduces to: algorithm 3.10. for a given u∈h, compute the approximate solution un+1 by the iterative schemes yn = jϕ[un − ρnt un+1] wn = jϕ[un − γ(un − yn)] un+1 := zn = un − α(un − wn), n = 0, 1, 2, . . . α = ‖un − yn‖ 2 + ‖un − wn‖ 2 − △(yn) 2‖un − wn‖2 cubo 10, 3 (2008) proximal-resolvent methods for mixed ... 9 which is an approximate extraresolvent method for solving (1). remark 3.1. algorithms 3.5-3.10 are called the approximate proximal extraresolvent methods, which are new ones. we would like to point out that if the nonlinear term ϕ in the mixed variational inequality (1) is an indicator function of a closed convex set k, then the resolvent jϕ = pk is the projection operator of h onto the closed convex set k. consequently, algorithms 3.1-3.10 reduce to algorithms for variational inequalities (3) which appear to be new ones for the variational inequalities (3). in a similar way, one can obtain several new and known algorithms as special cases of algorithm 3.9. this shows that algorithm 3.9 is more flexible and unifies several recently proposed (implicit or explicit ) algorithms for solving the mixed variational inequalities. we now study the convergence analysis of algorithm 3.9. the analysis is in the spirit of he, yang and yuan [9] and noor [19]. to convey the idea and for the sake of completeness, we include the details. theorem 3.1. let the operator t be pseudomonotone. if u ∈ k be a solution of the mixed variational inequality (1) and un+1 be the approximate solution obtained from algorithm 3.9, then ‖un+1(α) − u‖ 2 ≤ ‖un − u‖ 2 − (1 − ν)2 4 {‖un − wn‖ 2 + ‖un − zn‖ 2}. (11) proof. let u ∈ k be a solution of (1). then 〈t u, v − u〉 + ϕ(v) − ϕ(u) ≥ 0, ∀v ∈ k, implies that 〈t v, v − u〉 + ϕ(v) − ϕ(u) ≥ 0, (12) since t is pseudomonotone. taking v = wn in (12), we have 〈t wn, wn − u〉 + ϕ(wn) − ϕ(u) ≥ 0, which can be written as 〈t wn, zn − u〉 ≥ 〈t wn, zn − wn〉 + ϕ(u) − ϕ(wn). (13) taking z = [un − ρnt wn], u = zn and v = u in (4), we have 〈un − ρnt wn − zn, un − u〉 + ρnϕ(u) − ρnϕ(zn) ≥ 0, from which we have 〈un − zn, un − u〉 ≥ 〈un − u, ρnt wn〉 + ρnϕ(zn) − ρnϕ(u). (14) from (13) and (14), we have 〈un − zn, zn − wn〉 ≥ 〈ρnt wn, zn − wn〉 + ρn(ϕ(zn) − ϕ(wn) ≥ ρn〈t wn + ϕ ′(wn), zn − wn〉. (15) 10 muhammad aslam noor and khalida inayat noor cubo 10, 3 (2008) consider ‖un − u‖ 2 − ‖un+1(α) − u‖ 2 = ‖un − u‖ 2 − ‖un − α(un − zn) − u‖ 2 ≥ ‖un − u‖ 2 − ‖un − u − α(un − zn)‖ 2 = 2α〈un − u, un − zn〉 − α 2‖un − zn‖ 2 = 2α‖un − zn‖ 2 + 2α〈zn − u, un − zn〉 − α 2‖un − zn‖ 2 . (16) combining (10), (15) and (16), we obtain ‖un − u‖ 2 − ‖un+1(α) − u‖ 2 ≥ α{‖zn − wn‖ 2 + ‖un − zn‖ 2 − △(wn)} − α 2‖un − zn‖ 2 , (17) which is a quadratic in α and has a maximum at α ∗ = ‖zn − wn‖ 2 + ‖un − zn‖ 2 − △(wn) 2‖un − wn‖2 . (18) from (10), (17) and (18), we have the required result (11). 2 theorem 3.2. let h be a finite dimensional subspace. if u ∈ k be a solution of (1) and un+1 be the approximate solution obtained from algorithm 3.9, then limn−→∞(un) = u. proof. let u ∈ h be a solution of (1). from (11), it follows that the sequence {‖u − un‖} is nonincreasing and consequently {un} is bounded. furthermore, we have ∞ ∑ n=1 (1 − ν)2 4 {‖zn − wn‖ 2 + ‖un − zn‖ 2} ≤ ‖u0 − u‖ 2, which implies that lim n−→∞ ‖zn − wn‖ = 0 (19) lim n−→∞ ‖un − zn‖ = 0. (20) thus we see that the sequences {wn} and {zn} are also bounded. also from (19) and (20), we have ‖r(wn)‖ = ‖wn − jϕ[wn − ρt wn]‖ = ‖wn − zn + zn − jϕ[wn − ρt wn]‖ ≤ ‖wn − zn‖ + ‖jϕ[un − ρt wn] − jϕ[wn − ρt wn]‖ ≤ ‖wn − zn‖ + ‖un − wn‖ = 0. thus lim n−→∞ r(wn) = 0. (21) let û be a cluster point of { wn} and the subsequence {wni} converges to û. since r(u) is a continuous function of u, it follows that lim n−→∞ r(wni ) = r(û) = 0, cubo 10, 3 (2008) proximal-resolvent methods for mixed ... 11 which shows that û is a solution of the mixed variational inequality (1). from (19) and (20), we know that limn−→∞(yni ) = û = limn−→∞(zni ). hence from (11), we have ‖un+1 − û‖ 2 ≤ ‖un − û‖ 2 , ∀n ≥ 0, which shows that the sequence {un} converges to û, the required result. 2 conclusion. in this paper, we have suggested and analyzed some new proximal extraresolvent methods for pseudomonotone mixed variational inequalities and related optimization problems. the convergence of the new methods require only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. it has been shown [9] that a special case of algorithm 3.9 is numerically efficient and robust in solving traffic equilibrium problems. the results obtained are encouraging. the comparison of these new methods with the other methods is an interesting open problem for further research. acknowledgement. i wish to express my deepest gratitude to prof. dr. claudio cuevas, editor-in-chief, cubo journal, for this invitation. i am also grateful to dr. s. m. junaid zaidi, rector, ciit, for the excellent research facilities. received: january 2008. revised: april 2008. references [1] a. bnouhachem, a self-adaptive method for mixed quasi variational inequality, j. math. anal. appl., 309 (2005), 136–150. 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[26] g. stampacchia, formes bilineaires coercivities sur les ensembles coercivities sur les ensembles convexes, c. r. acad. sci. paris, 258 (1964), 4413–4416. n01 mostaf.dvi cubo a mathematical journal vol.12, no¯ 01, (83–93). march 2010 a fixed point theorem of reich in g-metric spaces zead mustafa and hamed obiedat the hashemite university, department of mathematics, p.o. box 150459, zarqa 13115, jordan emails: zmagablh@hu.edu.jo, hobiedat@hu.edu.jo abstract in this paper we prove some fixed point results for mapping satisfying sufficient contractive conditions on a complete g-metric space, also we showed that if the g-metric space (x, g) is symmetric, then the existence and uniqueness of these fixed point results follows from reich theorems in usual metric space (x, dg), where (x, dg) the metric induced by the g-metric space (x, g). resumen en este artículo nosotros provamos algunos resultados de punto fijo para aplicaciones satisfaciendo condiciones suficientes de contractividad sobre un espacio g-métrico completo, también mostramos que si el espacio g-métrico (x, g) es simétrico, entonces la existencia y unicidad de estos resultados de punto fijo siguen de teoremas de reich en espacios métricos usuales (x, dg), donde (x, dg) es la métrica inducida por el espacio g-métrico (x, g). key words and phrases: metric space, generalized metric space, d-metric space, 2-metric space. math. subj. class.: 47h10, 46b20. 84 zead mustafa and hamed obiedat cubo 12, 1 (2010) 1 introduction the study of fixed points of a functions satisfying certain contractive conditions has been at the center of vigorous research activity, because it has a wide range of applications in different areas such as, variational, linear inequalities, optimization and parameterize estimation problems. in 2005, z. mustafa and b. sims introduced a new class of generalized metric spaces (see [2], [3]), which are called g-metric spaces as generalization of metric space (x, d), to develop and to introduce a new fixed point theory for a variety of mappings in this new setting, also to extend known metric space theorems to a more general setting. the g-metric space is as follows. definition 1. let x be a nonempty set, and let g : x × x × x → r+, be a function satisfying the following properties: (g1) g(x, y, z) = 0 if x = y = z; (g2) 0 < g(x, x, y) ; for all x, y ∈ x, with x 6= y; (g3) g(x, x, y) ≤ g(x, y, z), for all x, y, z ∈ x, with z 6= y; (g4) g(x, y, z) = g(x, z, y) = g(y, z, x) = . . ., (symmetry in all three variables); and (g5) g(x, y, z) ≤ g(x, a, a) + g(a, y, z), for all x, y, z, a ∈ x, (rectangle inequality ). then the function g is called a generalized metric, or, more specifically a g-metric on x, and the pair (x, g) is called a g-metric space. example 1. ([2]) let (x, d) be a usual metric space, and define gs and gm on x × x × x to r + by gs(x, y, z) = d(x, y) + d(y, z) + d(x, z), and gm(x, y, z) = max{d(x, y), d(y, z), d(x, z)} for all x, y, z ∈ x. then (x, gs) and (x, gm) are g-metric spaces. definition 2. ([3]) let (x, g) be a g-metric space, and let (xn) be a sequence of points of x. a point x ∈ x is said to be the limit of the sequence (xn) if limn,m→∞ g(x, xn, xm) = 0, and one say that the sequence (xn) is g-convergent to x. thus, that if xn −→ 0 in a g-metric space (x, g), then for any ǫ > 0, there exists n ∈ n such that g(x, xn, xm) < ǫ, for all n, m ≥ n , (we mean by n the natural numbers). proposition 1. ([3]) let (x, g) be g-metric space. then the following are equivalent. (1) (xn) is g-convergent to x. cubo 12, 1 (2010) a fixed point theorem of reich in g-metric spaces 85 (3) g(xn, xn, x) → 0, as n → ∞. (4) g(xn, x, x) → 0, as n → ∞. (5) g(xm, xn, x) → 0, as m, n → ∞. definition 3. ([3]) let (x, g) be a g-metric space, a sequence (xn) is called g-cauchy if given ǫ > 0, there is n ∈ n such that g(xn, xm, xl) < ǫ, for all n, m, l ≥ n . that is g(xn, xm, xl) −→ 0 as n, m, l −→ ∞. proposition 2. ([3]) in a g-metric space, (x, g), the following are equivalent. 1. the sequence (xn) is g-cauchy. 2. for every ǫ > 0, there exists n ∈ n such that g(xn, xm, xm) < ǫ, for all n, m ≥ n . definition 4. ([3]) let (x, g) and (x′, g′) be two g-metric spaces, and let f : (x, g) → (x′, g′) be a function, then f is said to be g-continuous at a point a ∈ x if and only if, given ǫ > 0, there exists δ > 0 such that x, y ∈ x; and g(a, x, y) < δ implies g′(f (a), f (x), f (y)) < ǫ. a function f is g-continuous at x if and only if it is g-continuous at all a ∈ x. proposition 3. ([3]) let (x, g), and (x ′ , g ′ ) be two g-metric spaces. then a function f : x −→ x ′ is g-continuous at a point x ∈ x if and only if it is g-sequentially continuous at x; that is, whenever (xn) is g-convergent to x we have (f (xn)) is g-convergent to f (x). definition 5. ([3]) a g-metric space (x, g) is called symmetric g-metric space if g(x, y, y) = g(y, x, x) for all x, y ∈ x. it is clear that, any g-metric space where g derives from an underlying metric via gs or gm in example 1 is symmetric. the following example presents the simplest instance of a nonsymmetric g-metric and so also one which does not arise from any metric in the above ways. example 2. ([3]) let x = {a, b}, and let, g(a, a, a) = g(b, b, b) = 0, g(a, a, b) = 1, g(a, b, b) = 2 and extend g to x × x × x by symmetry in the variables. then it is easily verified that g is a g-metric, but g(a, b, b) 6= g(a, a, b). proposition 4. ([3]) let (x, g) be a g-metric space, then the function g(x, y, z) is jointly continuous in all three of its variables. proposition 5. ([3]) every g-metric space (x, g) induces a metric space (x, dg) defined by dg(x, y) = g(x, y, y) + g(y, x, x), ∀x, y ∈ x. 86 zead mustafa and hamed obiedat cubo 12, 1 (2010) note that if (x, g) is symmetric, then dg(x, y) = 2g(x, y, y), ∀x, y ∈ x. (1.1) however, if (x, g) is not symmetric then it holds by the g-metric properties that 3 2 g(x, y, y) ≤ dg(x, y) ≤ 3g(x, y, y), ∀x, y ∈ x. (1.2) definition 6. ([3]) a g-metric space (x, g) is said to be g-complete ( or complete g-metric ) if every g-cauchy sequence in (x, g) is g-convergent in (x, g). proposition 6. ([3])a g-metric space (x, g) is g-complete if and only if (x, dg) is a complete metric space. theorem 1.1 (reich,[4]). let (x, d) be a complete metric space, and t be a function mapping x into it self, satisfy the following condition, d(t (x), t (y)) ≤ ad(x, t (x)) + bd(y, t (y)) + cd(x, y), ∀x, y ∈ x. (1.3) where a, b, c are nonnegative numbers satisfying a + b + c < 1. then, t has a unique fixed point (i.e., there exists u ∈ x; t u = u). 2 main results in this section, we will present several fixed point results on a complete g-metric space. theorem 2.1. let (x, g) be a complete g-metric space, and let t : x −→ x be a mapping satisfies the following condition g(t (x), t (y), t (z)) ≤ k{g(x, t (x), t (x)) + g(y, t (y), t (y)) + g(z, t (z), t (z))} (2.1) for all x, y, z ∈ x, where k ∈ [0, 1/3). then t has a unique fixed point (say u), and t is gcontinuous at u. proof. suppose that t satisfies condition (2.1), then for all x, y ∈ x, we have g(t x, t y, t y) ≤ k[g(x, t x, t x) + 2g(y, t y, t y)], and (2.2) g(t y, t x, t x) ≤ k[g(y, t y, t y) + 2g(x, t x, t x)]. (2.3) suppose that (x, g) is symmetric. then from the definition of metric (x, dg) and (1.1), we have dg(t x, t y) ≤ kdg(x, t x) + 2kdg(y, t y), ∀x, y ∈ x. (2.4) cubo 12, 1 (2010) a fixed point theorem of reich in g-metric spaces 87 in this line, since 0 < k + 2k < 1 , then the metric condition (2.4) will be a special case of the reich condition (1.3), so the existence and uniqueness of the fixed point follows from theorem (1.1). however, if (x, g) is not symmetric then we can conclude that dg(t x, t y) = g(t x, t y, t y) + g(t y, t x, t x) ≤ 3kg(x, t x, t x) + 3kg(y, t y, t y), ∀x, y ∈ x. so, by the definition of the metric (x, dg) and (1.2), we get dg(t x, t y) ≤ 2kdg(x, t x) + 2kdg(y, t y), ∀x, y ∈ x, and, the metric condition gives no information about this map since 0 < 2k + 2k need not be less than 1. but the existence of a fixed point can be proved using properties of a g-metric. let x0 ∈ x, be an arbitrary point, and define the sequence (xn) by xn = t n(x0), then the condition (2.1) implies that g(xn, xn+1, xn+1) ≤ kg(xn−1, xn, xn) + 2kg(xn, xn+1, xn+1), hence g(xn, xn+1, xn+1) ≤ k 1 − 2k g(xn−1, xn, xn). let q = k 1−2k , then 0 < q < 1 since 0 ≤ k < 1/3. so, g(xn, xn+1, xn+1) ≤ q g(xn−1, xn, xn). continuing in the same argument, we will find g(xn, xn+1, xn+1) ≤ q n g(x0, x1, x1). (2.5) moreover, for all n, m ∈ n; n < m we have by repeated use the rectangle inequality and using equation (2.5) that g(xn, xm, xm) ≤ g(xn, xn+1, xn+1) + g(xn+1, xn+2, xn+2) + g(xn+2, xn+3, xn+3) + ... + g(xm−1, xm, xm) ≤ (qn + qn+1 + ... + qm−1)g(x0, x1, x1) ≤ q n 1−q g(x0, x1, x1), and so, lim g(xn, xm, xm) = 0, as n, m −→ ∞. thus (xn) is g-cauchy sequence, then by completeness of (x, g), there exists u ∈ x such that (xn) is g-convergent to u. assume on the contrary that t (u) 6= u. then g(xn+1, t (u), t (u)) ≤ k {g(xn, xn+1, xn+1) + 2g(u, t (u), t (u))}. taking the limit as n −→ ∞, and using the fact that the function g is continuous on its variable, this leads to g(u, t (u), t (u)) ≤ 2k g(u, t (u), t (u)). this contradiction implies that u = t (u). to prove uniqueness, suppose that u and v are two fixed points for t , then g(u, v, v) ≤ k g(u, t (u), t (u)) + 2kg(v, t (v), t (v)) = 0, which implies that u = v. 88 zead mustafa and hamed obiedat cubo 12, 1 (2010) to show that t is g-continuous at u, let (yn) ⊆ x be a sequence converges to u in (x, g), then we can deduce that g(u, t (yn), t (yn)) ≤ k {g(u, t (u), t (u)) + 2g(yn, t (yn), t (yn))}. (2.6) moreover, from g-metric axioms we have, g(yn, t (yn), t (yn)) ≤ g(yn, u, u) + g(u, t (yn), t (yn)), so, equation (2.6) implies that g(u, t (yn), t (yn)) ≤ 2k 1−2k g(yn, u, u). taking the limit as n → ∞, from which we see that g(yn, t (yn), t (yn)) → 0 and so, by proposition 3, t (yn) → u = t u, therefor t is g-continuous at u. this completes the prove of theorem (2.1). corollary 1. let (x, g) be a complete g-metric spaces, and let t : x −→ x be a mapping satisfying the following condition for some m ∈ n g(t m(x), t m(y), t m(z)) ≤ k { g(x, t m(x), t m(x)) + g(y, t m(y), t m(y))+ g(z, t m(z), t m(z)) } (2.7) for all x, y, z ∈ x, where k ∈ [0, 1/3). then t has unique fixed point (say u), and t m is gcontinuous at u. proof. from previous theorem we see that t m has a unique fixed point (say u), that is, t m(u) = u, and t m(u) is g-continuous at u. but, t (u) = t (t m(u)) = t m+1(u) = t m(t (u)), so t (u) is another fixed point for t m and by uniqueness t u = u. theorem 2.2. let (x, g) be a complete g-metric space, and let t : x −→ x, be a mapping satisfying the following condition g(t (x), t (y), t (z)) ≤ αg(x, y, z) + β { g(y, t (y), t (y)) + g(z, t (z), t (z))+ g(x, t (x), t (x)) } (2.8) for all x, y, z ∈ x, where 0 ≤ α + 3β < 1. then t has unique fixed point (say u), and t is g-continuous at u. proof. suppose that t satisfies condition (2.8). then for all x, y ∈ x g(t x, t y, t y) ≤ αg(x, y, y) + β[g(x, t x, t x) + 2g(y, t y, t y)], and (2.9) g(t y, t x, t x) ≤ αg(y, x, x) + β[g(y, t y, t y) + 2g(x, t x, t x)]. (2.10) suppose that (x, g) is symmetric. then from the definition of metric (x, dg) and (1.1) we get dg(t x, t y) ≤ αdg(x, y) + βdg(x, t x) + 2βdg(y, t y), ∀x, y ∈ x. (2.11) cubo 12, 1 (2010) a fixed point theorem of reich in g-metric spaces 89 since 0 < α + 3β < 1, then the metric condition (2.11) becomes the same as reich condition (1.3), so the existence and uniqueness of the fixed point follows from theorem (1.1). however, if (x, g) is not symmetric, then we conclude that dg(t x, t y) = g(t x, t y, t y) + g(t y, t x, t x) ≤ α[g(x, y, y) + g(y, x, x)] + 3βg(x, t x, t x) + 3βg(y, t y, t y), ∀x, y ∈ x. so, by the definition of the metric (x, dg) and (1.2) we get dg(t x, t y) ≤ αdg(x, y) + 2βdg(x, t x) + 2βdg(y, t y), ∀x, y ∈ x. the metric condition gives no information about this map since 0 < α + 2β + 2β need not be less 1, but this can be proved by g-metric. let x0 ∈ x, be an arbitrary point, and define the sequence (xn) by xn = t n(x0), then by (2.8) we can verify that g(xn, xn+1, xn+1) ≤ α g(xn−1, xn, xn) + β{g(xn−1, xn, xn) + 2g(xn, xn+1, xn+1)} then (1 − 2 β)g(xn, xn+1, xn+1) ≤ (α + β)g(xn−1, xn, xn) therefore, g(xn, xn+1, xn+1) ≤ α + β 1 − 2β g(xn−1, xn, xn). let q = α+β 1−2β , then 0 ≤ q < 1 since 0 ≤ α + 3 β < 1. so, g(xn, xn+1, xn+1) ≤ qg(xn−1, xn, xn). continuing in the same argument, we will find g(xn, xn+1, xn+1) ≤ q ng(x0, x1, x1). for all n, m ∈ n; n < m, we have by repeated use the rectangle inequality that g(xn, xm, xm) ≤ g(xn, xn+1, xn+1) + g(xn+1, xn+2, xn+2) + g(xn+2, xn+3, xn+3) + ... + g(xm−1, xm, xm) ≤ (qn + qn+1 + ... + qm−1)g(x0, x1, x1) ≤ q n 1−q g(x0, x1, x1). then, lim g(xn, xm, xm) = 0, as n, m −→ ∞, thus (xn) is g-cauchy sequence. due to the completeness of (x, g), there exists u ∈ x such that (xn) is g-convergent to u in (x, g). suppose that t (u) 6= u. then g(xn, t (u), t (u)) ≤ αg(xn−1, u, u) + β {g(xn−1, xn, xn) + 2g(u, t (u), t (u))}, and by taking the limit as n −→ ∞, and using the fact that the function g is continuous, we get that g(u, t (u), t (u)) ≤ 2 β g(u, t (u), t (u)). this contradiction implies that u = t (u). to prove uniqueness, suppose that u and v are two fixed points for t . then g(u, v, v) ≤ αg(u, v, v) + β {g(u, t (u), t (u)) + 2βg(v, t (v), t (v))} = 0 + αg(u, v, v), which is implies that u = v, since 0 < α < 1. 90 zead mustafa and hamed obiedat cubo 12, 1 (2010) to show that t is g-continuous at u, let (yn) ⊆ x be a sequence converges to u in (x, g), then we deduce that g(u, t (yn), t (yn)) ≤ αg(u, yn, yn) + β{g(u, t (u), t (u)) + 2g(yn, t (yn), t (yn))} = αg(u, yn, yn) + 2βg(yn, t (yn), t (yn)). (2.12) but, by g-metric axioms we have g(yn, t (yn), t (yn)) ≤ g(yn, u, u) + g(u, t (yn), t (yn)), so equation (2.12) implies that g(u, t (yn), t (yn)) ≤ α 1−2β g(u, yn, yn) + 2β 1−2β g(yn, u, u). taking the limit as n → ∞, from which we see that g(yn, t (yn), t (yn)) → 0, and so by proposition 3, t (yn) → u = t u. so, t is g-continuous at u. this completes the proof of theorem (2.2). corollary 2. let (x, g) be a complete g-metric spaces, and let t : x −→ x be a mapping satisfying, the following condition for some m ∈ n g(t m(x), t m(y), t m(z)) ≤ αg(x, y, z) + β        g(x, t m(x), t m(x))+ g(y, t m(y), t m(y))+ g(z, t m(z), t m(z))        (2.13) for all x, y, z ∈ x,where 0 ≤ α + 3β < 1. then t has unique fixed point (say u), and t m is g-continuous at u. proof. we use the same argument in corollary 1. theorem 2.3. let (x, g) be complete g-metric space, and let t : x −→ x be a mapping satisfying the condition g(t (x), t (y), t (z)) ≤ αg(x, y, z) + β max { g(x, t (x), t (x)), g(y, t (y), t (y)), g(z, t (z), t (z)) } (2.14) for all x, y, z ∈ x, where 0 ≤ α + β < 1. then t has unique fixed point (say u), and t is g-continuous at u. proof. suppose that t satisfies condition (2.14). then for all x, y ∈ x g(t x, t y, t y) ≤ αg(x, y, y) + β max{g(x, t x, t x), g(y, t y, t y)}, and (2.15) g(t y, t x, t x) ≤ αg(y, x, x) + β max{g(y, t y, t y), (x, t x, t x)}. (2.16) suppose that (x, g) is symmetric. then from the definition of metric (x, dg) and (1.1) we get. dg(t x, t y) ≤ αdg(x, y) + β max{dg(x, t x), dg(y, t y)}, ∀x, y ∈ x. (2.17) cubo 12, 1 (2010) a fixed point theorem of reich in g-metric spaces 91 in this line since 0 ≤ α + β < 1, then the metric condition (2.17) will be a special case of the reich condition (1.3). therefor the existence and uniqueness of the fixed point follows from theorem (1.1). however, if (x, g) is not symmetric then dg(x, y) = g(t x, t y, t y) + g(t y, t x, t x) ≤ α[g(x, y, y) + g(y, x, x)] + 2β max{g(x, t x, t x), g(y, t y, t y)}. so, by definition of the metric (x, dg) and (1.2), we will have dg(t x, t y) ≤ αdg(x, y) + 2β max{ 2 3 dg(x, t x), 2 3 dg(y, t y)}, ∀x, y ∈ x. the metric condition gives no information about this map since α + 4β 3 need not be less than 1. but the existence of a fixed point can be proved using properties of a g-metric. let x0 ∈ x, be arbitrary point, and define the sequence (xn) by xn = t n(x0), then by (2.14) we get. g(xn, xn+1, xn+1) ≤ α g(xn−1, xn, xn) + β max {g(xn−1, xn, xn), g(xn, xn+1, xn+1)}. (2.18) we see that there are two cases : (1) suppose max {g(xn−1, xn, xn), g(xn, xn+1, xn+1)} = g(xn−1, xn, xn). then in this case, equation (2.18) implies that g(xn, xn+1, xn+1) ≤ (α + β)g(xn−1, xn, xn). (2) suppose max {g(xn−1, xn, xn), g(xn, xn+1, xn+1)} = g(xn, xn+1, xn+1). then in this case, equation (2.18) implies that g(xn, xn+1, xn+1) ≤ α g(xn−1, xn, xn) + βg(xn, xn+1, xn+1), therefore g(xn, xn+1, xn+1) ≤ α 1 − β g(xn−1, xn, xn), but in this case we have g(xn−1, xn, xn) ≤ g(xn, xn+1, xn+1), hence g(xn−1, xn, xn) ≤ g(xn, xn+1, xn+1) ≤ α 1 − β g(xn−1, xn, xn) which is a contradiction since α 1−β < 1. then, it must be the case (1) is true, which says that g(xn, xn+1, xn+1) ≤ (α + β)g(xn−1, xn, xn). let q = α + β, then 0 ≤ q < 1 since 0 ≤ α + β < 1, therefor g(xn, xn+1, xn+1) ≤ q g(xn−1, xn, xn). 92 zead mustafa and hamed obiedat cubo 12, 1 (2010) continuing in the same argument, we will find g(xn, xn+1, xn+1) ≤ q n g(x0, x1, x1). for all n, m ∈ n; n < m we have by repeated use the rectangle inequality that g(xn, xm, xm) ≤ g(xn, xn+1, xn+1) + g(xn+1, xn+2, xn+2) + g(xn+2, xn+3, xn+3) + ... + g(xm−1, xm, xm) ≤ (qn + qn+1 + ... + qm−1)g(x0, x1, x1) ≤ q n 1−q g(x0, x1, x1). this proved that, lim g(xn, xm, xm) = 0, as n, m −→ ∞, thus (xn) is g-cauchy sequence. due to the completeness of (x, g), there exists u ∈ x such that (xn) is g-convergent to u in (x, g). suppose that t (u) 6= u. then by condition (2.14), we have g(xn, t (u), t (u)) ≤ αg(xn−1, u, u) + β max {g(xn−1, xn, xn), g(u, t (u), t (u))}. taking the limit as n −→ ∞, and using the fact that the function g is continuous, we get g(u, t (u), t (u)) ≤ β g(u, t (u), t (u)), this contradiction implies that u = t (u). to prove uniqueness, suppose that u and v are two fixed points for t . then by (2.14) we have g(u, v, v) ≤ αg(u, v, v) + β max {g(u, t (u), t (u)), g(v, t (v), t (v))} = α g(u, v, v), since α < 1 this implies that u = v. to show that t is g-continuous at u, let (yn) ⊆ x be a sequence converging to u in (x, g), then g(u, t (yn), t (yn)) ≤ αg(u, yn, yn) + β max {g(u, t (u), t (u)), g(yn, t (yn), t (yn))}, hence g(u, t (yn), t (yn)) ≤ αg(u, yn, yn) + βg(yn, t (yn), t (yn)) (2.19) but, by g-metric axioms we have g(yn, t (yn), t (yn)) ≤ g(yn, u, u) + g(u, t (yn), t (yn)). thus equation (2.19) implies that, g(u, t (yn), t (yn)) ≤ α 1−β g(u, yn, yn) + β 1−β g(yn, u, u). taking the limit as n → ∞, from which we see that g(u, t (yn), t (yn)) → 0 and so, by proposition (3), we have t (yn) → u = t u which implies that t is g-continuous at u. this completes the proof of theorem (2.3). corollary 3. let (x, g) be a complete g-metric spaces, and let t : x −→ x be a mapping satisfying the following condition for some m ∈ n g(t m(x), t m(y), t m(z)) ≤ αg(x, y, z) + β max        g(x, t m(x), t m(x)), g(y, t m(y), t m(y)), g(z, t m(z), t m(z))        (2.20) for all x, y, z ∈ x, where 0 ≤ α + β < 1. then t has unique fixed point (say u), and t m is g-continuous at u. cubo 12, 1 (2010) a fixed point theorem of reich in g-metric spaces 93 proof. we use the same argument in corollary 1. received: september, 2008. revised: october, 2009. references [1] mustafa, z. and sims, b., some remarks concerninig d–metric spaces, proceedings of the internatinal conferences on fixed point theorey and applications, valencia (spain), july (2003). 189–198. [2] mustafa, z., a new structure for generalized metric spaces – with applications to fixed point theory, phd thesis, the university of newcastle, australia, 2005. [3] mustafa, z. and sims, b., a new approach to generalized metric spaces, journal of nonlinear and convex analysis, volume 7, no. 2 (2006). 289–297. [4] reich, s., some remarks concerning contraction mappings, canad. math. bull. 14, (1971), 121–124. mr 49 ♯ 1501. cubo a mathematical journal vol.10, n o ¯ 02, (61–74). july 2008 c(n)-almost automorphic solutions of some nonautonomous differential equations khalil ezzinbi université cadi ayyad, faculté des sciences semlalia, département de mathématiques, bp. 2390, marrakech, morocco email: ezzinbi@ucam.ac.ma valerie nelson department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore, md 21251, usa email: valerie.nelson@morgan.edu and gaston n’guérékata department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore, md 21251, usa email: gaston.n’guerekata@morgan.edu abstract this paper is concerned with the study of properties of c(n)-almost automorphic functions and their uniform spectra. we apply the obtained results to prove massera type theorems for the nonautonomous differential equation in c k : x′(t) = a(t)x(t)+f (t), t ∈ r and a(t) is τ periodic and the equation x′(t) = ax(t) + f (t), t ∈ r where the operator a generates a quasi-compact semigroup in a banach space, and in both cases f is c(n)-almost automorphic. 62 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) resumen en este art́ıculo estudiamos las propriedades de funciones c(n)-casi automoficas. aplicamos los resultados obtenidos para provar teoremas de tipo massera para la ecuación diferencial no autonoma en c k : x′(t) = a(t)x(t) + f (t), t ∈ r, a(t) es τ -periódica y para la ecuación x′(t) = ax(t) + f (t), t ∈ r donde el operador a genera un semigrupo casi compacto en un espacio de banach, en ambos casos f es una función c(n)-casi automorfica. key words and phrases: evolution equation, mild solution, almost automorphy, uniform spectrum. math. subj. class.: 47d06, 34g10, 45m05 1 introduction let us consider in c k equations of the form dx dt = a(t)x + f (t), (1.1) where a(t) is a (generally unbounded) linear operator which is τ -periodic, and f is a c(n)-almost automorphic) function on r. we will prove a massera type result for the above differential equation and present conditions under which every bounded solution of this equation is c(n+1)-almost automorphic. the concept of c(n)-almost automorphic functions was introduced by ezzinbi, fatajou and n’guérékata in [9] as a generalization of c(n)-almost periodicity (see for instance [1, 2, 3, 5, 13]). in their work [9] , the authors study the existence of c(n)-almost automorphic solutions, (n ≥ 1), for the following partial neutral functional differential equation d dt dut = adut + l(ut) + f (t) for t ∈ r (1.2) where a is a linear operator on a banach space x satisfying the following well-known hille-yosida condition (h 0 ) there exist m̄ ≥ 1 and ω ∈ r such that (ω, +∞) ⊂ ρ(a) and |r(λ, a)n| ≤ m̄ (λ − ω)n for n ∈ n and λ > ω, where ρ(a) is the resolvent set of a and r(λ, a) = (λi − a)−1 for λ ∈ ρ(a). d : c → x is a bounded linear operator, where c = c([−r, 0] ; x) is the space of continuous functions from [−r, 0] cubo 10, 2 (2008) c(n)-almost automorphic solutions ... 63 to x endowed with the uniform norm topology. for the well posedness of equation (1.2), we assume that d has the following form dϕ = ϕ(0) − ∫ 0 −r [dη(θ)] ϕ(θ) for ϕ ∈ c, for a mapping η : [−r, 0] → l(x) of bounded variation and non atomic at zero, which means that there exists a continuous nondecreasing function δ : [0, r] → [0, +∞) such that δ(0) = 0 and ∣∣∣∣ ∫ 0 −s [dη(θ)] ϕ(θ) ∣∣∣∣ ≤ δ(s) sup −r≤θ≤0 |ϕ(θ)| for ϕ ∈ c and s ∈ [0, r] , where l(x) denotes the space of bounded linear operators from x to x. for every t ≥ σ, the history function ut ∈ c is defined by ut(θ) = u(t + θ) for θ ∈ [−r, 0] . l is a bounded linear operator from c to x and f is a continuous function from r to x. another important problem studied in [9] is the following massera type result. consider the differential equations dx dt = dx(t) + e(t), (1.3) where d is a constant d × d matrix and e :→ rd is c(n)-almost automorphic function. then if equ. (1.3) has a bounded solution on r + , it has a c(n+1)-almost automorphic solution. moreover every bounded solution on r is c(n+1)-almost automorphic. in the present paper we continue the study of elementary properties of c(n)-almost automorphic functions and apply them to investigate the c(n)-almost automorphic functions solutions to the non autonomous periodic equation (1.1). the work is organized as follows. in section 2, we review the concept of c(n)-almost periodic functions and present further properties of c(n)-almost automorphic functions with values in a hilbert space. in section 3, we discuss some results related to the uniform spectrum of c(n)almost automorphic functions. our main results (theorem 4.2 and 4.11)are presented in section 4. 2 c(n)-almost periodic and c(n)-almost automorphic functions we recall some properties about c(n)-almost periodic and c(n)-almost automorphic functions. let bc(r, x) be the space of all bounded and continuous functions from r to x, equipped with the uniform norm topology. let h ∈ bc(r, x) and τ ∈ r, we define the function hτ by hτ (s) = h(τ + s) for s ∈ r. 64 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) let cn(r, x) be the space of all continuous function which have a continuous n-th derivative on r and cn b (r, x) be the subspace of cn(r, x) of functions satisfying sup t∈r n∑ i=0 ‖h(i)(t)‖ < ∞, h(i) denotes the i-the derivative of h. then cn b (r, x) is a banach space provided with the following norm ‖h‖n = sup t∈r n∑ i=0 ‖h(i)(t)‖. definition 2.1. a bounded continuous function h : r → x is said to be almost periodic if {hτ : τ ∈ r} is relatively compact in bc(r, x). definition 2.2. a continuous function θ : r×x → x is said to be almost periodic in t uniformly in x if for any compact k in x and for every sequence of real numbers (s′ n )n there exists a subsequence (sn)n such that lim n→∞ θ(t + sn, x) exists uniformly in (t, x) ∈ r×k. definition 2.3. [3] let ε > 0 and h ∈ cn b (r, x). a number τ ∈ r is said to be a ‖ · ‖n − ε almost period of the function f if ‖hτ − h‖n < ε. the set of all ‖ · ‖n − ε almost period of the function h is denoted by e (n) (ε, f ). definition 2.4. [3] a function h ∈ cn b (r, x) is said to be a almost periodic function if for every ε > 0, the set e(n)(ε, h) is relatively dense in r. definition 2.5. ap (n)(x) is the space of the cn-almost periodic functions. since it is well known that for any almost periodic functions h1 and h2 and ε > 0, there exists a relatively dense set of their common ε almost period. consequently, we get the following result. proposition 2.6. h ∈ ap (n)(x) if and only if h(i) ∈ ap (x) for i = 0, 1, 2, ..., n. since ap (x) equipped with uniform norm topology is a banach space, then we get the following result. proposition 2.7. ap (n)(x) provided with the norm ‖ · ‖n is a banach space. example. the following example of a cn-almost periodic function has been given in [5]. let g(t) = sin(αt) + sin(βt), where α β /∈ q. then the function h(t) = eg(t) is cn-almost periodic for any n ≥ 1. in [5], one can find example of function which is cn-almost periodic but not cn+1-almost periodic. cubo 10, 2 (2008) c(n)-almost automorphic solutions ... 65 definition 2.8. [18] a continuous function h : r → x is said to be almost automorphic if for every sequence of real numbers (s′ n )n there exists a subsequence (sn)n such that k(t) = lim n→∞ h(t + sn) exists for all t in r and lim n→∞ k(t − sn) = h(t) for all t in r. remark. by the pointwise convergence, the function k is just measurable and not necessarily continuous. if the convergence in both limits is uniform, then h is almost periodic. the concept of almost automorphy is then larger than the one of the almost periodicity. if h is almost automorphic, then its range is relatively compact, thus bounded in norm. let p(t) = 2 + cost + cos √ 2t and h : r → r such that h = sin 1 p . then h is almost automorphic, but h is not uniformly continuous on r, it follows that h is not almost periodic. definition 2.9. [18] a continuous function h : r → x is said to be compact almost automorphic if for every sequence of real numbers (s′ n )n, there exists a subsequence (sn)n such that lim m→∞ lim n→∞ h(t + sn − sm) = h(t) uniformly on any compact set in r. theorem 2.10. [18] if we equip aac(x), the space of compact almost automorphic x-valued functions, with the sup norm, then aac(x) is a banach space. theorem 2.11. [18] if we equip aa(x), the space of almost automorphic x-valued functions, with the sup norm, then aa(x) turns out to be a banach space. definition 2.12. a continuous function θ : r×x → x is said to be almost automorphic in t with respect to x if for every sequence of real numbers (s′ n )n, there exists a subsequence (sn)n such that lim m→∞ lim n→∞ θ(t + sn − sm, x) = θ(t, x) for t ∈ r and x ∈ x. now we recall the concept of cn-almost automorphic functions recently introduced in [9] as a generalization of the one of cn-almost periodic functions. definition 2.13. a continuous function h : r → x is said to be cnalmost automorphic for n ≥ 1 if for i = 0, 1, ..., n, the i-th derivative h(i) of h is almost automorphic. we will denote by aa(n)(x) the space of all cn-almost automorphic x-valued functions. definition 2.14. ([9]) a continuous function h : r → x is said to be cn-compact almost automorphic if for i = 0, 1, ..., n, the i-th derivative h(i) of h is compact almost automorphic. we denote by aa (n) c (x) the space of all c n -compact almost automorphic x-valued functions. since aa(x) and aac(x) are banach spaces, then we get also the following result. 66 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) proposition 2.15. ([9]) aa(n)(x) and aa (n) c (x) provided with the norm |.|n are banach spaces. the following superposition result is easy to prove. proposition 2.16. let f ∈ aa(n)(x) and a ∈ b(x). then af ∈ aa(n)(x). proposition 2.17. let λ ∈ aa(n)(r, k) and f ∈ a(n)(x) where x is a banach space over the field k. then (λf )(t) := λ(t)f (t) is in aa(n)(x). we also have the following results theorem 2.18. let x be a hilbert space and f ∈ aa(n)(x). then the function f (t) = ∫ t 0 f (s)ds ∈ aa(n+1)(x) iff rf is bounded in x. proof. we have just to prove the only if part. it comes by induction. the case n = 0 is known ([18] theorem 2.4.6). assume now that f is in aa(n)(x), and that the theorem is true for n − 1; then f ∈ aa(n)(x). but we have f ′ = f and so f ′ ∈ aa(n)(x), from which we conclude that f ∈ aa(n+1)(x). theorem 2.19. let ν ∈ aa(n)(r, ls(x, y )) and f ∈ aa (n) (r, x). then νf ∈ aa(n)(r, y ) for two banach spaces x and y . proof. it suffices to observe that ν(i)f (n−i) : r → y is almost automorphic, for each i = 0, 1, ...n. 3 uniform spectrum of a function in bc(r, x) let us consider the following simple ordinary differential equation in a complex banach space x x ′ (t) − λx = f (t), (3.1) where f ∈ bc(x). if reλ 6= 0, the homogeneous equation associated with this has an exponential dichotomy; so, (3.1) has a unique bounded solution which we denote by xf,λ(·). moreover, from the theory of ordinary differential equations, it follows that for every fixed ξ ∈ r, xf,λ(ξ) := { ∫ ξ −∞ eλ(ξ−t)f (t)dt (if reλ < 0) − ∫ +∞ ξ eλ(ξ−t)f (t)dt (if reλ > 0). (3.2) = { ∫ 0 −∞ e−ληf (ξ + η)dη (if reλ < 0) − ∫ +∞ 0 e−ληf (ξ + η)dη (if reλ > 0). (3.3) as is well known, the differentiation operator d is a closed operator on bc(r, x). the above argument shows that ρ(d) ⊃ c\ir and xf,λ = (d − λ) −1f for every λ ∈ c\ir and f ∈ bc(r, x). cubo 10, 2 (2008) c(n)-almost automorphic solutions ... 67 hence, for every λ ∈ c with reλ 6= 0 and f ∈ bc(r, x) the function [(λ − d)−1f ](t) = ŝ(t)f (λ) ∈ bc(r, x). moreover, (λ − d)−1f is analytic on c\ir. definition 3.1. let f be in bc(r, x). then, i) α ∈ r is said to be uniformly regular with respect to f if there exists a neighborhood u of iα in c such that the function (λ − d)−1f , as a complex function of λ with reλ 6= 0, has an analytic continuation into u. ii) the set of ξ ∈ r such that ξ is not uniformly regular with respect to f ∈ bc(r, x) is called uniform spectrum of f and is denoted by spu(f ). observe that, if f ∈ bu c(r, x), then α ∈ r is uniformly regular if and only if it is regular with respect to f (cf. [15]). we now list some properties of uniform spectra of functions in bc(r, x). proposition 3.2. let g, f, fn ∈ bc(r, x) such that fn → f as n → +∞ and let λ ⊂ r be a closed subset satisfying spu(fn) ⊂ λ for all n ∈ n. then the following assertions hold: i) spu(f ) = spu(f (h + ·)); ii) spu(αf (·)) ⊂ spu(f ), α ∈ c; iii) sp(f ) ⊂ spu(f ); iv) spu(bf (·)) ⊂ spu(f ), b ∈ l(x); v) spu(f + g) ⊂ spu(f ) ∪ spu(g); vi) spu(f ) ⊂ λ. we also recall the following important result (see [15] for the proof). proposition 3.3. let f ∈ bc(r, x). then spu(f ) = spc(f ), where spc(f ) denotes the carleman spectrum of f . from the above properties, the following is obtained: proposition 3.4. ([3]) let f ∈ c (n) b (x). then spu(f (i) ) ⊂ spu(f (i−1) ), f or every i = 1, 2, ..., n. now we can state and prove. 68 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) lemma 3.5. let f ∈ aa(n)(x) and φ ∈ l1(r) whose fourier transform has compact support supp(φ) . then the function g := φ ∗ f ∈ aa(n)(x); moreover spu(g) ⊂ spu(f ) ∩ supp(φ). proof. let’s assume n = 0. and let (s′ n ) be an arbitrary sequence of real numbers. since f ∈ aa(x), there exists a subsequence (sn) such that h(t − s) := lim n→∞ f (t − s + sn) is well-defined for each t, s ∈ r, and lim n→∞ h(t − s − sn) = f (t − s) each t, s ∈ r. note that ‖f (t − s + sn)φ(s)‖ ≤ ‖f‖∞‖φ(s)‖. and since φ ∈ l 1 (r), we may deduce by the lebesgue’ dominated convergence theorem that lim n→∞ g(t + sn) = ∫ r lim n→∞ f (t − s + sn)φ(s)ds = ∫ r h(t − s)φ(s)ds = (h ∗ φ)(t) for each t ∈ r. similarly we can prove that lim n→∞ (h ∗ φ)(t − sn) = (φ ∗ f )(t) for each t ∈ r. thus φ ⋆ f ∈ aa(x). now we know that g is cn with derivatives: g(k) = φ ∗ f (k) (if k ≤ n). so, for each k ≤ n, g(k) ∈ aa(x), and the lemma follows. 4 applications to differential equations consider in a (complex) banach space x the linear equation x ′ (t) = ax(t) + f (t), t ∈ r, (4.1) where a : d(a) ⊂ x → x is a linear operator, and f ∈ c(r, x). we first generalize [9] theorem 3.20 as follows. lemma 4.1. suppose f ∈ aa(n)(x) and a ∈ l(x). then every bounded solution of eq.(4.2) is in aa(n+1)(x). proof. it suffices to observe that since a is bounded, then x(n+1)(t) = ax(n)(t) + f (n)(t). cubo 10, 2 (2008) c(n)-almost automorphic solutions ... 69 we have the following massera type result. theorem 4.2. let f ∈ aa(n)(ck). if eq. (4.1) has a bounded solution on r+, then it has a aa(n+1)(ck) solution. moreover every bounded solution of the differential equation x′(t) = a(t)x(t) + f (t), t ∈ r, (4.2) where a(t) : r → mk(c) is τ -periodic, is in aa (n+1) (c k ). proof. the proof is similar to theorem 3.1 [14]. first let us note that by floquet’s theory and without loss of generality we may assume that a(t) = a is independent of t. next we will show that the problem can be reduced to the one-dimensional case. in fact, if a is independent of t, by a change of variable if necessary, we may assume that a is of jordan normal form. in this direction we can go further with assumption that a has only one jordan box. that is, we have to prove the theorem for equations of the form   x′1(t) x′2(t) . . . x′ k (t)   =   λ 1 0 . . . 0 0 λ 1 . . . 0 . . . . . . . . . . . . 0 0 0 . . . λ     x1(t) x2(t) . . . xk(t)   +   f1(t) f2(t) . . . fk(t)   , t ∈ r. now if x is a bounded solution of the above system on r+, then by theorem 3.14 [9], it has an almost almost automorphic solution on r. since f ∈ aa(n)(c), then by lemma 4.1 above, we deduce that x ∈ aa(n+1)(c). the following is easy to establish. corollary 4.3. consider the differential equation x′(t) = ax(t) + f (t), t ∈ r (4.3) where f ∈ aa(n)(rk), and a ∈ b(rk) such that the real part of each of its eigenvalues is negative. then eq.(4.3) has a unique solution in x ∈ aa(n+1)(rk). we also have the following result. theorem 4.4. let a ∈ b(rk) and suppose that eq.(4.3) has a unique aa(1)(rk) solution for each f ∈ aak. then the map t : aa(rk) → aa(1)(rk), f → x is linear and continuous, that is there exists c > 0 such that ‖x‖1 ≤ c‖f‖0 where ‖ · ‖0 denotes the usual sup norm in aa(r k ) proof. linearity of t is obvious. let us prove its continuity. 70 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) first, let us consider the map s : aa(1)(rk) → aa(rk) given by sx(t) = f (t). that is, x is the unique aa(1)(rk) solution to acp. s is defined as (sx)(t) = x′(t) − ax(t) = f (t), thus sx = f so st f = f . also t sx = t f = x. we deduce that s = t −1. on another hand we have ‖sx‖0 ≤ ‖x ′ ‖0 + k‖x‖0 ≤ k1(‖x ′ ‖0 + ‖x‖0) where k1=max (1, k). thus we have ‖sx‖0 ≤ k1‖x‖1. that means s is continuous. and since s is injective, then s−1 = t is continuous ([16] 1.6.6 corollary page 44) this ends our proof. now we investigate the existence of c(n) almost automorphic solutions for the following equation x′(t) = ax(t) + f (t) for t ∈ r (4.4) where a is the infinitesimal generator of a strongly continuous semigroup (t (t)) t≥0 in a banach space x. definition 4.5. we say that a functon is a mild solution of equation (4.4) if for any σ and t ≥ σ, we have x(t) = t (t − σ)x(σ) + ∫ t σ t (t − s)f (s)ds. for simplicity, mild solution will be called solution in the sequel. we need to recall some preliminary results on quasi compact semigroups.we first introduce the kuratowski measure of noncompactness α(.) of bounded sets k in a banach space x by α(k) = inf {ε > 0 : k has a finite cover of balls of diameter < ε} . for a bounded linear operator b on x, |b| α is defined by |b| α = inf {ε > 0 : α(b(k)) ≤ εα(k) for any bounded set k of x} . the essential growth bound ωess (t ) of the semigroup (t (t))t≥0 is defined by ωess (t ) = lim t→+∞ 1 t log |t (t)| α , = inf t>0 1 t log |t (t)| α . cubo 10, 2 (2008) c(n)-almost automorphic solutions ... 71 definition 4.6. the essential spectrum σess(a) of a is the set of λ ∈ σ(a) : the spectrum of a, such that one of the following conditions holds: (i) im(λi − a) is not closed, (ii) the generalized eigenspace mλ(a) = ⋃ k≥1 ker(λi − a)k is of infinite dimension, (iii) λ is a limit point of σ(a) \ {λ}. the essential radius of any bounded operator t in y is defined by ress(t ) = sup{|λ| : λ ∈ σess(t )}. definition 4.7. we say that the semigroup (t (t))t≥0 is quasi compact if ωess (t ) < 0. theorem 4.8. the semigroup (t (t))t≥0 is quasi compact if for some t0 > 0, we have ress(t (t0)) < 1. lemma 4.9. if the semigroup (t (t))t≥0 is quasi compact. then, σ+(a) = {λ ∈ σ(a) : re(λ) ≥ 0} is a finite set of the eigenvalues of a which are not in the essential spectrum. theorem 4.10. [9] assume that the semigroup (t (t))t≥0 is quasi compact. then x is decomposed as follows x = s ⊕ v, where x is t −invariant and there are positive constants α and n such that |t (t) x| ≤ n e−αt |x| for t ≥ 0 and x ∈ s. (4.5) moreover v is a finite dimensional space and the restriction of t to v becomes a group. let p − and p + denote respectively the projection operators respectively of x into s and v . theorem 4.11. assume that the semigroup (t (t))t≥0 is quasi compact and the input function f is c(n)-almost automorphic. if equation (4.4) has a bounded solution on r+, then it has a c(n)-almost automorphic solution. moreover every bounded solution of equation (4.4) on r is a c(n)-almost automorphic solution. proof of theorem. let b be a matrix be such that t (t) = etb in v . 72 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) let u be a bounded solution of equation (4.4) on r+. the function z(t) = p +u(t) is a bounded solution on r + of the following ordinary differential equation z′(t) = bz(t) + p +f (t) for t ≥ 0. (4.6) moreover, the function t → p +f (t) is c(n)-almost automorphic from r to rd. by theorem 4.2 we get that the reduced system (4.6) has a c(n)-almost automorphic solution z̃ and the function v defined by v(t) = z̃ (t) + ∫ t −∞ t (t − s) p −f (s) ds for t ∈ r, is a bounded solution of equation (4.4) on r. we claim that v is c(n)-almost automorphic. in fact, let y be defined by y(t) = ∫ t −∞ t (t − s) p −f (s) ds for t ∈ r. then y ∈ c (n) b (r, x). clearly y is a. a. by [19]. also we have y′(t) = p −f (t) + y(t). so y′ is a. a. in general y(i) = p −f (i−1)(t) + y(i−1)(t), i = 1, 2, ..., n, which implies that y is c(n) almost automorphic. let u be a bounded solution on r, then u is given by the following formula u (t) = z (t) + ∫ t −∞ t (t − s) p −f (s) ds for t ∈ r, where z(t) = p +u(t) for t ∈ r is a solution of the reduced system (4.6), which is c(n)-almost automorphic by theorem 4.2 and arguing as above, one can prove that the function t → ∫ t −∞ t (t − s) p − f (s) ds for t ∈ r, is also c(n)-almost automorphic. received: december 2007. revised: february 2008. references [1] m. adamczak, c(n)-almost periodic functions, comment. math. prace mat. 37 (1997), 1–12. 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[6] c. corduneanu, almost periodic solutions for some class of functional differential equations, funct. diff. equ., vol. 14, no. 2-3-4 (2007), 223–229. [7] t. diagana, g. n’guérékata and nguyen van minh, almost automorphic solutions of evolution equations, proc. amer. math. soc. 132 (2004), 3289–3298. [8] k. ezzinbi, s. fatajou and g.m. n’guérékata, massera type theorem for the existence of c(n)-almost periodic solutions for partial functional differential equations with infinite delay, nonlinear analysis, theory, methods and applications, to appear. [9] k. ezzinbi, s. fatajou and g.m. n’guérékata, c(n)-almost automorphic solutions for partial neutral functional differential equations, applicable analysis, vol. 86, issue 9 (2007), 1127–1146. [10] y. hino, and s. murakami, almost automorphic solutions for abstract functional differential equations, j. math. anal. appl. 286 (2003), 741–752. [11] y. hino, t. naito, n.v. minh, and j.s. shin, almost periodic solutions of differential equations in banach spaces, taylor & francis, london new york, 2002. [12] b.m. levitan and v.v. zhikov, almost periodic functions and differential equations, moscow univ. publ. house 1978. english translation by cambridge university press 1982. [13] j. liang, l. maniar, g. n’guérékata and ti-jun xiao, existence and uniqueness of c(n)-almost periodic solution to some ordinary differential equations, nonlinear analysis, 66 (2007), 1899–1910. [14] j. liu, g. n’guérékata and nguyen van minh, a massera type theorem for almost automorphic solutions of differential equations, j. math. anal. appl. 299 (2004), no. 2, 587– 599. [15] j. liu, g. n’guérékata, nguyen van minh and vu quoc phong, bounded solutions of parabolic equations in continuous function spaces, funkciolaj ekvacioj. [16] r.e. megginson, an introduction to banach space theory, graduate texts in mathematics, 183 springer, new york-berlin-milan-london. 74 khalil ezzinbi, valerie nelson and gaston n’guérékata cubo 10, 2 (2008) [17] g.m. n’guérékata, almost automorphic functions and applications to abstract evolution equations, contemporary math. 252 (1999), 71–76. [18] g.m. n’guérékata, almost automorphic and almost periodic functions in abstract spaces, kluwer, amsterdam, 2001. [19] g.m. n’guérékata, existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, semigroup forum, vol. 69 (2004), no. 1, 80–89. [20] g.m. n’guérékata, topics in almost automorphy, springer, new york, 2005. n5 scubo.dvi cubo a mathematical journal vol.12, no¯ 03, (49–69). october 2010 on the group of strong symplectic homeomorphisms augustin banyaga department of mathematics, the pennsylvania state university, university park, pa 16802 email: banyaga@math.psu.edu abstract we generalize the “hamiltonian topology” on hamiltonian isotopies to an intrinsic “symplectic topology” on the space of symplectic isotopies. we use it to define the group ss ym peo (m,ω) of strong symplectic homeomorphisms, which generalizes the group hameo(m,ω) of hamiltonian homeomorphisms introduced by oh and müller. the group ss ym peo(m,ω) is arcwise connected, is contained in the identity component of s ym peo(m,ω); it contains hameo(m,ω) as a normal subgroup and coincides with it when m is simply connected. finally its commutator subgroup [ss ym peo(m,ω), ss ym peo(m,ω)] is contained in hameo(m,ω). resumen generalizamos la “topología hamiltoniano” sobre isotopias hamiltonianas para una “topología simpléctica” intrinseca en el espacio de isotopias simplécticas. nosotros usamos esto para definir el grupo ss ym peo(m,ω) de homeomorfismos simplécticos fuertes, el qual generaliza el grupo hameo(m,ω) de homeomorfismos hamiltonianos introducido por oh y müller. el grupo ss ym peo(m,ω) es conexo por arcos, es contenido en la componente identidad de s ym peo(h,ω); este contiene hameo(m,ω) como un subgrupo normal y coincide con este cuando m es simplemente conexa. finalmente su subgrupo conmutador [ss ym peo(m,ω), ss ym peo(m,ω)] es contenido en hameo(m,ω). 50 augustin banyaga cubo 12, 3 (2010) key words and phrases: hamiltonian homeomorphisms, hamiltonian topology, symplectic topology, stromg symplectic homeomorphisms, c0 symplectic topology. math. subj. class.: msc2000:53d05; 53d35. 1 introduction no natural metric on the group s ym p(m,ω) of symplectic diffeomorphisms of a symplectic manifold (m,ω) is known. in this paper we construct a “hofer-like” metric, depending on several ingredients. however, we prove that all these metrics are equivalent and hence define a natural metric topology on s ym p(m,ω) ( theorem 1’). we use this natural topology on s ym p(m,ω) to define a new group of symplectic homeomorphisms, herein called the group of strong symplectic homeomorphisms (theorem 2). this group may carry a calabi invariant. the eliashberg-gromov symplectic rigidity theorem says that the group s ym p(m,ω) of symplectomorphisms of a closed symplectic manifold (m,ω) is c0 closed in the group diff ∞(m) of c∞ diffeomorphisms of m [7],[9]. this means that the “symplectic” nature of a sequence of symplectomorphisms survives topological limits. also lalonde-mcduff-polterovich have shown in [11] that for a symplectomorphism, being “hamiltonian” is topological in nature. these phenomenons attest that there is a c0 symplectic topology underlying the symplectic geometry of a closed symplectic manifold (m,ω). according to oh-müller ([13]), the automorphism group of the c0 symplectic topology is the closure of the group s ym p(m,ω) in the group h omeo(m) of homeomorphisms of m endowed with the c0 topology. that group, denoted s ym peo(m,ω) has been called the group of symplectic homeomorphisms: s ym peo(m,ω) =: s ym p(m,ω). the c0 topology on h omeo(m) coincides with the metric topology coming from the metric d( g, h) = max(su px∈m d0( g(x), h(x)), su px∈m d0( g −1 (x), h −1 (x)) where d0 is a distance on m induced by some riemannian metric [3]. on the space p h omeo(m) of continuous paths γ : [0, 1] → h omeo(m), one has the distance d(γ,µ) = su pt∈[0,1] d(γ(t),µ(t)). consider the space p h am(m) of all isotopies φh = [t 7→ φ t h ] where φt h is the family of hamiltonian diffeomorphisms obtained by integration of the family of vector fields x h for a cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 51 smooth family h(x, t) of real functions on m, i.e. d dt φ t h (x) = x h (φ t h (x)) and φ0 h = id. recall that x h is uniquely defined by the equation i(x h )ω = dh where i(.) is the interior product. the set of time one maps of all hamiltonian isotopies {φt h } form a group, denoted h am(m,ω) and called the group of hamiltonian diffeomorphisms. definition: the hamiltonian topology [13] on p h am(m) is the metric topology defined by the distance dham(φh ,φh′ ) = ||h − h ′ ||+ d(φh ,φh′ ) where ||h − h ′ || = ˆ 1 0 osc(h − h ′ )dt. and the oscillation of a function u is osc(u) = maxx∈m u(x) − minx∈m u(x). let h ameo(m,ω) denote the space of all homeomorphisms h such that there exists a continuous path λ ∈ p h omeo(m) such that λ(0) = id, λ(1) = h and there exists a cauchy sequence (for the dham distance) of hamiltonian isotopies φhn , which c 0 converges to λ ( in the d metric). the following is the first important theorem in the c0 symplectic topology [13]: theorem (oh-müller): the set h ameo(m,ω) is a topological group. it is a normal subgroup of the identity component s ym peo0 (m,ω) in s ym peo(m,ω). if h 1(m,r) 6= 0, then h ameo(m,ω) is strictly contained in s ym peo0 (m,ω). remark: it is still unknown in general if the inclusion h ameo(m,ω) ⊂ s ym peo0 (m,ω) is strict. the group h ameo(m,ω) is the topological analogue of the group h am(m,ω) of hamiltonian diffeomorphisms. 52 augustin banyaga cubo 12, 3 (2010) the goal of this paper is to construct a subgroup of s ym peo0 (m,ω), denoted ss ym peo(m,ω) and nicknamed the group of strong symplectic homeomorphisms, containing h ameo(m,ω), that is: h ameo(m,ω) ⊂ ss ym peo(m,ω) ⊂ s ym peo0 (m,ω). like h ameo(m,ω), the group ss ym peo(m,ω) is defined using a blend of the c0 topology and the hofer topology on the space i so(m,ω) of symplectic isotopies of (m,ω). we believe that ss ym peo(m,ω) is “more right” than the group s ym peo(m,ω) for the c0 symplectic topology. in particular the flux homomorphism seems to exist on ss ym peo(m,ω). this will be the object of a future paper. the results of this paper have been announced in [1]. the c0 counter part of the c∞ contact topology is been worked out in [5], [6]. 2 the symplectic topology on i so(m,ω) let i so(m,ω) denote the space of symplectic isotopies of a closed symplectic manifold (m,ω). recall that a symplectic isotopy is a smooth map φ : m × [0, 1] → m such that for all t ∈ [0, 1], φt : m → m, x 7→ φ(x, t) is a symplectic diffeomorphism and φ0 = id. the “lie algebra” of s ym p(m,ω) is the space s ym p(m,ω) of symplectic vector fields, i.e the set of vector fields x such that i x ω is a closed form. let φt be a symplectic isotopy, then φ̇t(x) = dφt dt (φ −1 t (x)) is a smooth family of symplectic vector fields. by the theorem of existence and uniqueness of solutions of ode’s, φ ∈ i so(m,ω) 7→ φ̇t is a 1-1 correspondence between i so(m,ω) and the space c∞([0, 1], s ym p(m,ω)) of smooth families of symplectic vector fields. hence any distance on c∞([0, 1], s ym p(m,ω)) gives rise to a distance on i so(m,ω). an intrinsic topology on the space of symplectic vector fields. we define a norm ||.|| on s ym p(m,ω) as follows: first we fix a riemannian metric g (which may be the one we used to define d0 above, or any other riemannian metric), and a basis b = {h1, .., hk } of harmonic 1-forms. for hodge theory, we refer to [14]. cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 53 recall that the space harm1 (m, g) of harmonic 1-forms is a finite dimensional vector space over r and its dimension is the first betti number of m. on harm1 (m, g), we put the following “euclidean” norm: for h ∈ harm1 (m, g) , h = ∑ λi hi , define: |h |b := ∑ |λi|. this norm is equivalent to any other norm since harm1 (m, g) is a finite dimensional vector space. here we choose this one for convenience in the calculations and estimates to come later. given x ∈ s ym p(m,ω), we consider the hodge decomposition of i x ω [14] : there is a unique harmonic 1-form h x and a unique function u x such that i x ω = h x + du x recall that the function u x is given by the following formula: u x = δg(i(x )ω), where δ is the codifferential and g is the green operator (see [14]). this defines a decompsition of x ∈ s ym p(m,ω) as : x = #h x + x u x ,where #h x is defined by the equation i(#h x )ω = h x and x u x is the hamiltonian vector field with u x as hamiltonnian. we now define a norm ||.|| on the the vector space s ym p(m,ω) by: ||x|| = |h x |b + osc(u x ). (1) it is easy to see that this is a norm. let us just verify that ||x|| = 0 implies that x = 0. indeed |h x |b = 0 implies that i x ω = du x , and osc(u x ) = 0 implies that u x is a constant, therefore du x = 0. remark: this norm is not invariant by s ym p(m,ω). hence it does not define a finsler metric on s ym p(m,ω). the norm ||.|| defined above depends of course on the riemannian metric g and the basis b of harmonic 1-forms. however, we have the following: theorem 1: all the norms ||.|| defined by equation (1) using different riemannian metrics and different basis of harmonic 1-forms are equivalent. hence the topology on the space s ym p(m,ω) of symplectic vector fields defined by the norm (1) is intrinsic : it is independent of the choice of the riemannian metric g and of the basis b of harmonic 1-forms. 54 augustin banyaga cubo 12, 3 (2010) for each symplectic isotopy φ = (φt), consider the hodge decomposition of i(φ̇t )ω i(φ̇t)ω = h φ t + du φ t where h φt is a harmonic 1-form. we define the length l(φ) of the isotopy φ = (φt) by: l(φ) = ˆ 1 0 (|h φ t |+ osc(u φ t ))dt = ˆ 1 0 ||φ̇t||dt. one also writes ˆ 1 0 ||φ̇t||dt = |||φ̇t|||. in the expressions above, we have written |h φt | for |h φ t |b , where b is a fixed basis of harm1 (m, g), for a fixed riemannian metric g. we define the distance d0(φ,ψ) between two symplectic isotopies φ = (φt) and ψ = (ψt) by: d0(φ,ψ) = |||φ̇t −ψ̇t||| := ˆ 1 0 (|h φ t − h ψ t |+ osc(u φt − u ψt ))dt. denote by φ−1 = (φ−1t ) and by ψ −1 = (ψ−1t ) the inverse isotopies. remarks: 1. the distance d0(φ,ψ) 6= l(ψ −1 φ) unless ψ and φ are hamiltonian isotopies ( see proposition 1). 2. l(φ) 6= l(φ−1) unless φ is hamiltonian. indeed, h φ −1 t = −h φ t but u φ t is very differerent from uφ −1 t . the formula of the difference u φ t − u φ −1 t follows from propositions 3, 4 and 5. in view of the remarks above, we define a more “symmetrical” distance d by: d(φ,ψ) = (d0(φ,ψ) + d0(φ −1 ,ψ −1 ))/2 following [13], we define the symplectic distance on i so(m,ω) by: ds ym p (φ,ψ) = d(φ,ψ) + d(φ,ψ). definition: the symplectic topology on i so(m,ω) is the metric topology defined by the distance ds ym p. theorem 1’: the symplectic topology on i so(m,ω) is canonical: it is independent of all choices involved in its definition. cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 55 we may also define another distance d∞ on i so(m,ω): d ∞ 0 (φ,ψ) = su pt∈[0,1] (|h φ t − h ψ t |) + su pt∈[0,1] osc(u φt − u ψt )) d ∞ (φ,ψ) = (d ∞ 0 (φ,ψ) + d ∞ 0 (φ −1 ,ψ −1 ))/2 and d ∞ s ym p (φ,ψ) = d(φ,ψ) + d ∞ (φ,ψ) proposition 1: let φ = (φt),ψ = (ψt) be two hamiltonian isotopies and σt = (ψt) −1φt then |||σ̇t||| = |||φ̇t −ψ̇t||| = ˆ 1 0 osc(u φ t − u ψt )dt proof: this follows immediately from the equation σ̇t = (ψt −1 )∗(φ̇t −ψ̇t), which is a consequence of proposition 4 stated in section 4. corollary: the distance ds ym reduces to the hamiltonian distance dham when φ and ψ are hamiltonian isotopies. the symplectic topology reduces to the “hamiltonian topology” of [13] on paths in h am(m,ω). a hofer-like metric on s ym p(m,ω)0 for any φ ∈ s ym p(m,ω), define: e0(φ) = in f (l(φ)) where the infimum is taken over all symplectic isotopies φ from φ to the identity. the following result was proved in [2]. theorem: the map e : s ym p(m,ω)0 → r∪ {∞} : e(φ) =: (e0(φ) + e0(φ −1 ))/2 is a metric on the identity component s ym p(m,ω)0 in the group s ym p(m,ω), i.e. it satisfies (i) e(φ) ≥ 0 and e(φ) = 0 iff φ is the identity. (ii) e(φ) = e((φ)−1) (iii) e(φ.ψ) ≤ (eφ) + e(ψ). the restriction to h am(m,ω) is bounded from above by the hofer norm. 56 augustin banyaga cubo 12, 3 (2010) recall that the hofer norm [10] of a hamiltonian diffeomorphism φ is ||φ||h = in f (l(φh )) where the infimum is taken over all hamiltonian isotopies from φ to the identity. the hofer-like metric above depends on the choice of a riemannian metric g and a basis b of harmonic 1-forms. hence it is not “natural”. however, by theorem 1, all the metrics constructed that way are equivalent; so they define a natural topology on s ym p(m,ω)0 . 3 strong symplectic homeomorphisms definition: a homeomorphism h is said to be a strong symplectic homeomorphism if there exists a continuous path λ : [0, 1] → h omeo(m) such that λ(0) = id; λ(1) = h and a sequence φ n = (φn t ) of symplectic isotopies, which converges to λ in the c0 topology (induced by the norm d) and such that φn is cauchy for the metric ds ym p . we will denote by ss ym peo(m,ω) the set of all strong symplectic homeomorphisms. this set is well defined independently of any riemannian metric or any basis of harmonic 1-forms. clearly, if m is simply connected, the set ss ym peo(m,ω) coincides with the group h ameo(m,ω). we denote by ss ym peo(m,ω)∞ the set defined like in ss ym peo(m,ω) but replacing the norm ds ym p by the norm d ∞ s ym p. let p h omeo(m) be the set of continuous paths γ : [0, 1] → h omeo(m) such that γ(0) = id, and let p ∞(h arm1 (m) be the space of smooth paths of harmonic 1-forms. we have the following maps: a1 : i so(m,ω) → p h omeo(m),φ 7→ φ(t) a2 : i so(m,ω) → p ∞(h arm1 (m),φ 7→ h φt a3 : i so(m,ω) → c ∞(m × [0, 1],r),φ 7→ uφ let q be the image of the mapping a = a1 × a2 × a3 and q the closure of q inside i (m,ω) =: p h omeo(m) × p ∞(h arm1 (m) × c∞(m × [0, 1],r), with the symplectic topology, which is the c0 topology on the first factor and the metric topology from d on the second and third factor. then ss ym peo(m,ω) is just the image of the evaluation map of the path at t= 1 of the image of the projection of q on the first factor. this defines a surjective map: a : q → ss ym peo(m,ω) cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 57 the symplectic topology on ss ym peo(m,ω) is the quotient topology induced by a. our main results are : theorem 2: the set q is a topological group. theorem 3: let (m,ω) be a closed symplectic manifold. then ss ym peo(m,ω) is an arcwise connected topological group (with the sympectic topology), containing h ameo(m,ω) as a normal subgroup, and contained in the path component of the identity s ym peo0 (m,ω) of s ym peo(m,ω). if m is simply connected, ss ym peo(m,ω) = h ameo(m,ω). finally, the commutator subgroup [ss ym peo(m,ω), ss ym peo(m,ω)] of ss ym peo(m,ω) is contained in h ameo(m,ω). conjectures: 1. let (m,ω) be a closed symplectic manifold, then [ss ym peo(m,ω), ss ym peo(m,ω)] = h ameo(m,ω). 2. the inclusion ss ym peo(m,ω) ⊂ s ym peo0 (m,ω) is strict. 3. the results in theorem 3 hold for ss ym peo(m,ω)∞ . conjecture 3 is supported by a result of muller asserting that h ameo(m,ω) coincides with h ameo(m,ω)∞ which is defined by replacing the l(1,∞) hofer norm by the l∞ norm [12]. measure preserving homeomorphisms on a symplectic 2n dimensional manifold (m,ω), we consider the measure µω defined by the liouville volume ωn. let h omeo µω 0 (m) be the identity component in the group of homeomorphisms preserving µω. we have: s ym peo0 (m,ω) ⊂ h omeo µω 0 (m). oh and müller [13] have observed that h ameo(m,ω) is a sub-group of the kernel of fathi’s mass-flow homomorphism [8]. this is a homomorphism θ : h omeo µω 0 (m) → h1(m,r)/γ, where γ is some sub-group of h1(m,r). fathi proved that if the dimension of m is bigger than 2, then k erθ is a simple group. this leaves open the following question [13]: is h omeo µω 0 (s2) = s ym peo0 (s 2,ω) a simple group? but s ym peo0 (s 2,ω) contains h ameo(s2 ,ω) as a normal subgroup. the question is to decide if the inclusion h ameo(s 2 ,ω) ⊂ s ym peo0 (s 2 ,ω) 58 augustin banyaga cubo 12, 3 (2010) is strict. since ss ym peo(s2 ,ω) = h ameo(s2 ,ω), our conjecture 2 implies that h omeo µω 0 (s2) = s ym peo0 (s 2,ω) is not a simple group, a conjecture of [13]. questions 1. is ss ym peo(m,ω) a normal subgroup of s ym peo0 (m,ω)? 2. is [s ym peo0 (m,ω), s ym peo0 (m,ω)] contained in h ameo(m,ω)? 4 proofs of the results 4.1 proof of theorem 1 if b and b′ are two basis of harm1 (m, g), then elementary linear algebra shows that |.|b and |.|b′ are equivalent. this implies that the corresponding norms on s ym p(m,ω) are also equivalent. let us now start our construction with a riemannian metric g and a basis b = (h1, ..hk ) of harm1 (m, g). we saw that for any x ∈ s ym p(m,ω), i x ω = h x + du x and we wrote h x = ∑ λi hi . let g′ be another riemannian metric. the g′-hodge decomposition of i x ω is: i x ω = h ′ x + du ′ x where h ′ x is g′-harmonic. consider the g′-hodge decompositions of the members hi of the basis b i.e. hi = h ′ i + dvi where h′ i is g′ harmonic. b′ = (h′ 1 , ..h′ k ) is a basis of harm1 (m, g′). indeed suppose that ∑ r i h ′ i = 0 . the 1-form ∑ r i hi = d( ∑ r i vi) is g-harmonic and exact : ∑ r i hi = d( ∑ r i vi). but an exact harmonic form must be identically zero. therefore all r i are zero since {hi } form a basis. hence {h′ i } are linearly independent. the 1-form h ′′ x =: ∑ λi h ′ i is a g′harmonic form representing the cohomology class of i x ω. by uniqueness, h ′ x = h ′′ x . hence |h ′ x |b′ = ∑ |λi| = |h x |b cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 59 furthermore h ′ x = ∑ λi (hi − dvi ) = h x + dv where v = − ∑ λi vi. hence i x ω = h ′ x + du ′ x = h x + d(v + u ′ x ) by uniqueness in the g-hodge decomposition of i x ω, u x = v + u ′ x . denote by ||x||g′ , resp. ||x||g , the norm of x using the riemannian metric g ′ and the basis b ′, resp. using the riemannian metric g and the basis b. then: ||x||g′ = |h ′ x |b′ + osc(u ′ x ) = |h ′ x |b′ + osc(u x − v) ≤ |h ′ x |b′ + osc(u x ) + osc(−v) = |h x |b + osc(u x ) + osc(v) = ||x||g + osc(v). let c = 2maxi|vi|, since v = − ∑ λi vi, we get the following inequality: osc(v) ≤ 2max(|v|) ≤ c|h x |b = c|h ′ x |b′ therefore ||x||g′ ≤ ||x||g + osc(v) ≤ ||x||g + c|h x |b ≤ ||x||g + c(|h x |b + osc(u x )) = (c + 1)||x||g similarly, ||x||g =|h x |b + osc(u x ) = |h x |b + osc(u ′ x + v) ≤ |h x |b + osc(u ′ x ) + osc(v) =|h ′ x |b′ + osc(u ′ x ) + osc(v) = ||x||g′ + osc(v) ≤ ||x||g′ + c|h ′ x |b′ ≤ ||x||g′ + c(|h ′ x |b′ + osc(u ′ x ) = (c + 1)||x||g′ hence the metrics ||x||g and ||x||g are equivalent for the purpose of the proof of the main theorem, we fix a riemannian metric g and a basis b = (h1, .., hk ) of harm 1 (m, g). the norm of a harmonic 1-form h will be simply denoted |h | and the norm of a symplectic vector field x will be simply denoted ||x||. 4.2 proof of theorem 3 we prove first that the set ss ym peo(m,ω)subsets ym peo(m,ω) is closed under composition and inverse maps. let hi ∈ ss ym peo(m,ω) i = 1, 2 and let λi be continuous paths in h omeo(m) with λi (0) = id, λi (1) = hi and let φ n i be ds ym p cauchy sequences of symplectic isotopies converging c0 to λi . then φ n 1 .(φn 2 )−1 converges c0 to the path λ1(t)(λ2(t)) −1. here φn 1 .(φn 2 )−1(t) = φn 1 (t).(φn 2 (t))−1. 60 augustin banyaga cubo 12, 3 (2010) by definition of the distance ds ym p, φ n is a ds ym p cauchy sequence if and only if both φ n and (φn)−1 are d0 cauchy and dcauchy sequences. main lemma: if φn = (φn t ) and ψn t = (ψn t ) are ds ym p cauchy sequences in i so(m), so is ρn t = φn t ψn t . the proof of the main lemma is very delicate; it will take most of the remaining part of this paper. the estimates are much more involved than in the hamiltonian case, due to the fact that the decomposition of a symplectic isotopy into a hamiltonian one and a harmonic one does not behave nicely with respect to the product of isotopies. it will be enough to prove that ρnt is a d0 cauchy sequence. indeed since (φ n)−1 and (ψn)−1 are d0 cauchy by assumption, the main lemma applied to their product implies that their product is also d0 cauchy. hence (ψn)−1(φn)−1 = (φnψn)−1 = (ρn t )−1 is a d0 cauchy sequence. this will conclude the proof that ss ym peo(m,ω) is a group. we leave the details to the reader. we will use the following estimate: proposition 2: there exists a constant e such that for any x ∈ s ym p(m,ω), and h ∈ harm1 (m, g) |h (x )| =: su px∈m |h (x)(x (x))| ≤ e||x||.|h | proof: let (h1, .., hr ) be the chosen basis for harmonic 1-forms and let e = maxi e i and e i = su pv (su px∈m |hi (x)(v (x))| where v runs over all symplectic vector fields v such that ||v || = 1. without loss of generality, we may suppose x 6= 0 and set v = x /||x||. let h = ∑ λi hi . then h (x ) = ||x|| ∑ λi hi (v ). hence |h (x )| ≤ ||x|| ∑ |λi|su px (|hi (x)(v )(x)|) ≤ ||x|| ∑ |λi|e = e||x||.|h |. we will also need the following standard facts: proposition 3: let φ be a diffeomorphism, x a vector field and θ a differential form on a smooth manifold m. then (φ −1 ) ∗ [i x φ ∗ θ] = iφ∗ x θ proposition 4: if φt,ψt are any isotopies, and if we denote by ρt = φtψt, and by φ t = (φ)−1t then ρ̇t = φ̇t + (φt)∗ψ̇t cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 61 and φ̇ t = −((φ) −1 t )∗(φ̇t) proposition 5: let θt be a smooth family of closed 1-forms and φt an isotopy, then φ ∗ t θt −θt = dvt where vt = ˆ t 0 (θt(φ̇s) ◦φs)ds proof of the main lemma: if φt,ψt are symplectic isotopies, and if ρt = φtψt, propositions 3, 4 and 5 give: i(ρ̇t)ω = h φ t + h ψ t + dk (φ,ψ) (i) where k = k (φ,ψ) = uφt + (u ψ t ) ◦ (φt) −1 + vt(φ,ψ), and vt(φ,ψ) = ˆ t 0 (h ψ t (φ̇s ) ◦φ −1 s )ds. (i i) let now φn t ,ψn t be cauchy sequences of symplectic isotopies, and consider the sequence ρnt = φ n t ψ n t . we have: |||ρ̇ n t −ρ̇ m t ||| = ˆ 1 0 |h φ n t − h φ m t + h ψ n t − h ψ m t |+ osc(k (φ n ,ψ n ) − k (φ m ,ψ m ))dt ≤ ˆ 1 0 |h φ n t − h φ m t )|dt + ˆ 1 0 |h ψ n t − h ψ m t )|dt + ˆ 1 0 osc(u φ n t − u φ m t )dt + ˆ 1 0 osc(u ψ n t ) ◦ (φ n t ) −1 − u ψ m t ◦ (φ m t ) −1 )dt + ˆ 1 0 osc(vt (φ n ,ψ n ) − vt(φ m ,ψ m )dt = |||φ̇n t −φ̇ m t|||+ ˆ 1 0 |h ψ n t − h ψ m t )|dt + a + b where a = ˆ 1 0 osc(u ψ n t ) ◦ (φ n t ) −1 − u ψ m t ◦ (φ m t ) −1 )dt and b = ˆ 1 0 osc(vt (φ n ,ψ n ) − vt(φ m ,ψ m )dt. (i i i) 62 augustin banyaga cubo 12, 3 (2010) we have: a ≤ ˆ 1 0 osc(u ψ n t ) ◦ (φ n t ) −1 − u ψ m t ◦ (φ n t ) −1 )dt + ˆ 1 0 osc(u ψ m t ) ◦ (φ n t ) −1 − (u ψ m t ) ◦ (φ m t ) −1 )dt = ˆ 1 0 osc(u ψ n t − u ψ m t )dt + c where c = ˆ 1 0 osc(u ψ m t ◦ (φ n t ) −1 − u ψ m t ◦ (φ m t ) −1 )dt. hence |||ρ̇ n t −ρ̇ m t ||| ≤ |||φ̇ n t −φ̇ m t ||| + ˆ 1 0 |h ψ n t − h ψ m t )|dt + ˆ t 0 osc(u ψ n t − u ψ m t )dt + b + c = |||φ̇ n t −φ̇ m t |||+|||ψ̇ n t −ψ̇ m t |||+ b + c we now show that c → 0 when m, n → ∞. sub-lemma 1 (reparametrization lemma [13]): ∀ǫ ≥ 0,∃m0 such that c = ˆ 1 0 osc(u ψ m t ◦ (φ n t ) −1 − u ψ m t ◦ (φ m t ) −1 )dt =: ||u ψ m t ◦ (φ n t ) −1 − u ψ m t ◦ (φ m t ) −1 )|| ≤ ǫ if m ≥ m0 and n large enough remark: this is the “reparametrization lemma” of oh-müller [13] (lemma 3.21. (2)). for the convenience of the reader and further references, we include their proof. proof: for short, we write um for u ψ m t and µ n t for (φn t )−1. first, there exists m0 large such that ||um −um0 || ≤ ǫ/3 for m ≥ m0, since (um ) is a cauchy sequence for the distance d(un , um ) = ´ 1 0 osc(un − um )dt. therefore ||um ◦µ n t − um ◦µ m t ))|| ≤ ||um ◦µ n t − um0 ◦µ n t ))||+||um0 ◦µ n t − um0 ◦µ m t ))||+||um0 ◦µ m t − um ◦µ m t ))|| = ||um − um0 ||+||um0 ◦µ n t − um0 ◦µ m t ))||+||um0 − um|| ≤ (2/3)ǫ+||um0 ◦µ n t − um0 ◦µ m t ))|. by uniform continuity of um0 , there exists a positive δ such that if d(µ m t ,µn t ) ≤ δ, then max osc((um0 ◦µ n t − um0 ◦µ m t )) ≤ ǫ/3. hence ||um0 ◦µ n t − um0 ◦µ m t ))|| ≤ ǫ/3 for n, m large. recall that µn t is a dcauchy sequence. cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 63 to show that ρ̇n t is a cauchy sequence, the only thing which is left is to show that b → 0 when n, m → ∞. let us denote vt(φ n,ψn) by vn t , h ψ n t by h t n or h n and (φ n t )−1 by µn t . for a function on m, we consider the norm |f | = su px∈m |f (x)| we have: |v n t − v m t | = | ˆ t 0 (h t n(µ̇ n s ) ◦µ n s − h t m(µ̇ m s ) ◦µ m s )ds| ≤ ˆ 1 0 |((h t n − h t m)(µ̇ n s )) ◦µ n s |ds + ˆ 1 0 |h t m(µ̇ n s −µ̇ m s )) ◦µ m s |ds + ˆ 1 0 |h t m (µ̇ n s ) ◦µ n s − h t m(µ̇ n s ) ◦µ m s |ds the last integral can be estimated as follows: ˆ 1 0 |h t m(µ̇ n s ) ◦µ n s − h t m (µ̇ n s ) ◦µ m s |ds ≤ ˆ 1 0 |h t m(µ̇ n s ) ◦µ n s − h t m(µ̇ n0 s ) ◦µ n s |ds (1) + ˆ 1 0 |h t m(µ̇ n0 s ) ◦µ n s − h t m(µ̇ n0 s ) ◦µ m s |ds (2) + ˆ 1 0 |h t m(µ̇ n0 s ) ◦µ m s − h t m(µ̇ n s ) ◦µ m s |ds (3) for some integer n0. proposition 2 gives e|h m|d0((φ n)−1, (φn0 )−1) ≤ 2e|h m|d((φ n), (φn0 )−1) as an upper bound for (1) and (3). it also gives the following estimates: ˆ 1 0 |((h t n − h t m)(µ̇ n s )) ◦µ n s |ds ≤ e|h t n − h t m| ˆ 1 0 ||µ̇ n s )||ds = e.|h t n − h t m|.l((φ n ) −1 ) 64 augustin banyaga cubo 12, 3 (2010) and ˆ 1 0 |(h t m(µ̇ n s −µ̇ m s )) ◦µ m s |ds ≤ e.|h t m| ˆ 1 0 ||(µ̇ n s −µ̇ m s )||ds = e|h t m|d0((φ n ) −1 , (φ m )) −1 ) ≤ 2e|h t m|d(φ n ,φ m ). therefore, we get the following estimate: |v n t − v m t | ≤ e.|h t n − h t m|l(φ n ) −1 ) + e|h t m|2(d(φ n ,φ m ) + 2d(φ n ,φ n0 )) + g where g = ˆ 1 0 |h t m(µ̇ n0 s ) ◦µ n s − h t m(µ̇ n0 s ) ◦µ m s |ds since osc(vn t − vm t ) ≤ 2|vn t − vm t |, we see that ˆ 1 0 osc(v n t − v m t )dt ≤ 2e(l(φ n ) −1 ) ˆ 1 0 |h t n − h t m|dt +e2(d(φ m ,φ n ) + 2ed(φ n ,φ n0 ) ˆ 1 0 |h t m|dt) + ˆ 1 0 gdt we need the following facts: sub-lemma 2 (reparametrization lemma): ∀ǫ ≥ 0,∃n0 such that l = ˆ 1 0 gdt = ˆ 1 0 ( ˆ 1 0 |h t m(µ̇ n0 s ) ◦µ n s − h t m(µ̇ n0 s ) ◦µ m s |ds ) dt ≤ ǫ for n ≥ n0 and m sufficiently large. proposition 6: l((φn))−1 and ´ 1 0 |h t m|dt are bounded for every n, m. we finish first the estimate for ´ 1 0 osc(vnt − v m t )dt using sub-lemma 2 and proposition 6. putting together all the information we gathered, we see that: ˆ 1 0 osc(v n t − v m t )dt ≤ 2e(l(φ n ) −1 ) ˆ 1 0 |h t n − h t m|dt +e(2d(φ m ,φ n )) + 2ed(φ n ,φ n0 )( ˆ 1 0 |h t m|dt) + l ≤ 2el((φ n ) −1 )d(φ n ,φ m ) + e(2d(φ m ,φ n ) + 2ed(φ n ,φ n0 ) ˆ 1 0 |h t m|dt + l therefore: ˆ 1 0 osc(v n t − v m t )dt → 0 cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 65 when n, m → ∞, and n0 is chosen sufficiently large now let n0 → ∞ as well.. this finishes the proof of the main lemma. proof of sub-lemma 2: g = ˆ 1 0 |h t m(µ̇ n0 s ) ◦µ n s − h t m(µ̇ n0 s ) ◦µ m s |ds ≤ ˆ 1 0 |h t m(µ̇ n0 s ) ◦µ n s − h t m0 (µ̇ n0 s ) ◦µ n s |ds + ˆ 1 0 |h t m0 (µ̇ n0 s ) ◦µ n s − h t m0 (µ̇ n0 s ) ◦µ m s |ds + ˆ 1 0 |h t m0 (µ̇ n0 s ) ◦µ m s − h t m(µ̇ n0 s ) ◦µ m s |ds for some m0. exactly like in the proof of sub-lemma 1 g(t, n, m) ≤ 2e|h t m − h t m0 |.(l(ψ n0 ) −1 ) + f where f = ˆ 1 0 |h t m0 (µ̇ n0 s ) ◦µ n s − h t m0 (µ̇ n0 s ) ◦µ m s |ds by uniform continuity of h tm0 (µ̇ n0 s ), f → 0 when n, m → ∞ since µ n t is cauchy. by similar arguments as in the sub-lemma 1, g → 0 and hence l → 0 when m, n → ∞ and m0 → ∞. we have just proved that the subset ss ym p(m,ω) of s ymeo(m,ω) is closed under composition and inversion. this concludes the proof that ss ym peo(m,ω) is a group. the fact that it is arcwise connected in the ambiant topology of h omeo(m) is obvious from the definition. h ameo(m,ω) is a normal subgroup of ss ym peo(m,ω) since it is normal in s ym peo(m,ω) [13]. let h, g ∈ ss ym peo(m,ω) and let φn,ψn be symplectic isotopies which form cauchy sequences and c0 converge to h, g. by the main lemma the sequence φn.ψn.(φn)−1(φn)−1 is a cauchy sequence. it obviously converges c0 to the commutator h gh−1 g−1 ∈ ss ym peo(m,ω). it is a standard fact that φn.ψn.(φn)−1(ψn)−1 is a hamiltonian isotopy. indeed let φt and ψt be symplectic isotopies, and let σt = φtψtφ −1 t ψ −1 t , then σ̇t = x t + yt + zt +ut 66 augustin banyaga cubo 12, 3 (2010) with x t = φ̇t , yt = (φt)∗ψ̇t, zt = −(φtψtφ −1 t )∗φ̇t, and ut = −(σt)∗ψ̇t. by proposition 5, i(x t+zt)ω and i(yt+ut)ω are exact 1-forms. hence σt is a hamiltonnian isotopy. by proposition 1, the metric d coincides with the one for hamiltonian isotopies. hence φ n.ψn.(φn)−1(ψn)−1 is a cauchy sequence for dham. therefore: [ss ym peo(m,ω), ss ym peo(m,ω)] ⊂ h ameo(m,ω)]. this concludes the proof of theorem 3 proof of theorem 2: we now prove that ss ym peo(m,ω), with the symplectic topology, is a topological group. in fact, we prove that q ( see section 3) is a topological group. recall that an element of q is a couple (γ, v = (h , u) where γ ∈ p h omeo(m), h ∈ l(1,∞)([0, 1], harm1 (m,ω) , u ∈ l(0,1)([0, 1]xm,r), and there exists a ds ym p cauchy sequence of symplectic isotopies φn(t) such that φn(1) → γ , in the c 0 topology and limn→∞ (h n, un ) = (h , u). here we wrote h n for h φn and un for u φn n . the product and the inverse in q are given by: (γ, (h , u)).(γ ′ , (h ′ , u ′ )) = (γγ ′ , (h + h ′ , u + u ′ ◦γ+ v)) (γ, (h , u)) −1 = (γ −1 , (−h ,−(u ◦γ+ w)) where v is the limit of the cauchy sequence vn(t) given by formula (ii): vn(t) = ˆ 1 0 (h ′ n(σ̇n(s)) ◦σn (s))ds, with σn(s) = (φ ′ n(s)) −1. and w the limit of a similar sequence in which σn is replaced by φn . part i. let us first show that the inversion is continuous: let (γk, (h k, uk )) be a sequence converging to (γ, (h , u)), for each k, there is a cauchy sequence φkn of symplectic isotopies such that φkn → γk as n → ∞ in the c 0 topology, h kn → h k, u k n → uk . we need only to show that wk → w, that is (*) l imn,k→∞ ˆ 1 0 |h k n (φ̇ k n(s)) ◦φ k n (s) − h n (φ̇n(s)) ◦φn (s)|ds = 0. we have the following inequalities : ||φ̇ k n −φ̇n|| ≤ ||φ̇ k n − v k ||+||v k − v ||+||v −φ̇n|| and each term in the right hand of this inequality → 0 as n, k → ∞. similarly, |h k n − h n| ≤ |h k n − h k |+|h k − h |+|h − h n| cubo 12, 3 (2010) on the group of strong symplectic homeomorphisms 67 and each term in the right hand of this inequality → 0 as n, k → ∞. formula (*) follows from these inequalities and the techniques developped in this paper (including the reparametrisation lemma). we leave the details to the reader. part ii. now we prove that the composition is continuous: let (γk, v k = (h k, uk )) and (γ′k, v ′k = (h ′k, u′k )) converging to (γ, (h , u)) and (γ′, (h ′, u′)). by part i, if σ̇kn → u k and σ̇′ k n → u ′k,then by part i, u k → u. here we denoted by σkn, and σ′ k n respectively (φ k n) −1, (φ′kn ) −1. we only need to prove: 1) uk ◦γk → u ◦γ 2) vk → v. the proof of (1) goes along the lines explained in this paper ( including the reparamareization lemma ) and the details are left tothe reader. the proof of (2) follows from part i and uses the inequalities: ||σ̇ k n −ρ̇n|| ≤ ||σ̇ k n −u k ||+||u k −u||+||u −ρ̇n|| each of the three parts of the second member of the inequality → 0as n, k → ∞. the details are left tothe reader. this concludes the proof of theorem 2. appendix: for the convenience of the reader, we give here the proofs of propositons 3, 4, and 5. proof of proposition 3: let θ be a p-form, x a vector field and φ a diffeomorphism. for any x ∈ m and any vector fields y1, ..yp−1, we have: (φ−1)∗[i x φ ∗θ](x)(y1, ..., yp−1 ) = (i x φ ∗θ)(φ−1(x))(d xφ −1(y1(x), ...(d xφ −1(yp−1(x)) = (φ∗θ)(φ−1(x))(xφ−1(x), d xφ −1(y1(x)), ...(d xφ −1(yp−1(x)) = θ(φ(φ−1(x))(dφ−1(x)φ(xφ−1(x)), dφ−1(x)φd xφ −1(y1(x)), ...dφ−1 (x)φd xφ −1(yp−1(x) = θ(x)((φ∗ x )x, y1(x), ..yp−1(x)) = (i(φ∗ x )θ)(x)(y1, .., yp−1 ) since dφ−1(x)φd xφ −1 = d x(φφ −1) = id. therefore (φ−1)∗[i x φ ∗θ] = i(φ∗ x ))θ proof of proposition 4: this is just the chain rule. see [10] page 145. 68 augustin banyaga cubo 12, 3 (2010) proof of proposition 5: for a fixed t, we have d ds φ ∗ s θt = φ ∗ s (lφ̇s θt), where l x is the lie derivative in the direction x . since θ is closed, we have: d ds φ ∗ s θt = φ ∗ s (d iφ̇s θt) = d(φ ∗ s (θt(φ̇s)) = d(θt(φ̇s) ◦φs). hence for every u φ ∗ uθt −θt = ˆ u 0 d ds φ ∗ s θt ds = d( ˆ u 0 (θt(φ̇s) ◦φs)ds) now set u = t. acknowledgement i would like to thank claudio cuevas for soliciting this paper for cubo. i am also very grateful to the referee for an extensive list of good remarks, questions, and suggestions which drastically improved the final form of this paper. in particular i owe to him/her the idea to finish the proof of theorem 1. references [1] banyaga, a., on the group of symplectic homeomorphisms, c.r. acad. sci. paris ser., 1 346(2008), 867–872. [2] banyaga, a., a hofer-like metric on the group of symplectic diffeomorphisms, contemp. math., vol. 512 (2010), 1–23. [3] banyaga, a., sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, comment. math. helv., 53(1978), pp.174–227. [4] banyaga, a., the structure of classical diffeomorphisms groups, mathematics and its applications vol 400. kluwer academic publisher’s group, dordrecht, the netherlands, 1997. [5] banyaga, a. and spaeth, p., the group of contact homeomorphisms, preprint, 2008. [6] banyaga, a. and spaeth, p., the c0 contact topology and the group of contact homeomorphisms, arxiv 0812.2461. 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[13] oh, y-g. and müller, s, the group of hamiltonian homeomorphisms and c0symplectic topology, j. symp. geometry, 5(2007), 167–225. [14] warner, f., foundations of differentiable manifolds and lie groups, scott, foresman and company, 1971. cubo a mathematical journal vol.10, n o ¯ 03, (43–55). october 2008 the modulo two homotopy groups of the l2-localization of the ravenel spectrum ippei ichigi and katsumi shimomura department of mathematics, faculty of science, kochi university, kochi, 780-8520, japan email: 95sm004@math.kochi-u.ac.jp email: katsumi@math.kochi-u.ac.jp abstract the ravenel spectra t (m) for non-negative integers m interpolate between the sphere spectrum and the brown-peterson spectrum. let l2 denote the bousfield-ravenel localization functor with respect to v−1 2 bp . in this paper, we determine the homotopy groups π∗(l2t (m) : z/2) = [m2, l2t (m)]∗ for m > 1, where m2 denotes the modulo two moore spectrum. resumen el espectro de ravenel t (m) para enteros no negativos m interpola entre el espectro esferico y el espectro de brown-peterson. denotemos por l2 el funtor de localización de bousfield-ravenel con respecto a v−1 2 bp . en este art́ıculo, determinamos el grupo de homotopia π∗(l2t (m) : z/2) = [m2, l2t (m)]∗ para m > 1, donde m2 denota el espectro de moor modulo dos. key words and phrases: homotopy groups, bousfield-ravenel localization, ravenel spectrum. math. subj. class.: 55q99, 55q51, 20j06. 44 ippei ichigi and katsumi shimomura cubo 10, 3 (2008) 1 introduction let s(2) denote the stable homotopy category of 2-local spectra, and bp ∈ s(2) denote the brownpeterson ring spectrum. then, bp∗ = π∗(bp ) = z(2)[v1, v2, . . . ] and bp∗(bp ) = π∗(bp ∧bp ) = bp∗[t1, t2, . . . ], which form a hopf algebroid. the adams-novikov spectral sequence for computing the homotopy groups π∗(x) of a spectrum x has the e2-term e ∗ 2 (x) = ext∗bp∗(bp )(bp∗, bp∗(x)). let l2 : s(2) → s(2) be the bousfield-ravenel localization functor with respect to v −1 2 bp . then, the e2-term e ∗ 2 (l2s 0) for the sphere spectrum s0 is determined in [12], but the homotopy groups π∗(l2s 0) stay undetermined. the ravenel spectrum t (m) for m > 0 is a ring spectrum characterized by bp∗(t (m)) = bp∗[t1, t2, . . . tm] ⊂ bp∗(bp ) as a bp∗(bp )-comodule. the spectrum t (m) interpolates between the sphere spectrum and the brown-peterson spectrum, and so the homotopy groups π∗(l2t (m)) seem accessible if m is sufficiently large. indeed, π∗(l2t (∞)) = π∗(l2bp ) is determined by ravenel [8]. let mk denote the mod k moore spectrum defined by the cofiber sequence s0 w2 s0 wi mk wj s1. (1.1) for m = 1, t (1) ∧ m2 is the mahowald spectrum x〈1〉 and the homotopy groups of l2x〈1〉 are determined in [11]. but even the homotopy groups of l2t (1) ∧ m4 are too complicated to be determined completely (cf. [2], [3]). consider a spectrum t (m)/(va 1 ) defined as a cofiber of the self-map va 1 : σ2at (m) → t (m) defined by the generator v1 ∈ π2(t (m)). we use the notation: vm(0) = t (m) ∧ m2 and vm(1)a = t (m)/(v a 1 ) ∧ m2, (1.2) and abbreviate vm(1)1 to vm(1). in this paper, we consider the case where m > 1, and determine π∗(l2vm(1)) and π∗(l2vm(0)). the adams-novikov e2-term e ∗ 2 (l2vm(1)) for m > 1 is determined by ravenel [10] as follows: e∗ 2 (l2vm(1)) = km(2)∗ ⊗ ∧(h1,0, h1,1, h2,0, h2,1) (1.3) for generators hi,j ∈ e 1,2 m+i+j+1 −2 j+1 2 (l2vm(1)) and km(2)∗ = v −1 2 z/2[v2, v3, . . . , vm+2]. we show that vm(1) is a t (m)-module spectrum with m2-action, and then that all additive generators of the e2-term are permanent cycles and the extension problem of the spectral sequence is trivial. theorem 1.4. π∗(l2vm(1)) = km(2)∗ ⊗ ∧(h1,0, h1,1, h2,0, h2,1) as a z/2-module. let α : σ8m2 → m2 denote the adams map such that bp∗(α) = v 4 1 , and ka 2 denote a cofiber of αa. then, we show that vm(1)4a = t (m) ∧ k a 2 in lemma 2.4 and denote the telescope of vm(1)4 α → · · · α → vm(1)4a α → vm(1)4a+4 α → · · · by vm(1)∞. by the v1-bockstein spectral sequence, we determine the adams-novikov e2-term e ∗ 2 (l2vm(1)∞), whose structure is given in [4] without cubo 10, 3 (2008) the modulo two homotopy groups ... 45 proof. here we give a proof of it. consider the integers en and an defined by en = 8n − 1 7 and an =          1 n = 0 3ek+1 − 1 n = 3k + 1 6ek+1 n = 3k + 2 12ek+1 n = 3k + 3. (1.5) we introduce modules em(2)∗ = v −1 2 z(2)[v1, v2, . . . , vm+2], q(k) = em−1(2)∗/(2, v ak 1 )[xk+1]〈xk/v ak 1 〉, where xn ∈ em(2)∗ is an element defined in (4.1) such that xn ≡ v 2 n m+2 modulo (2, v1), and xn/v an 1 ∈ e0 2 (l2vm(1)∞) by proposition 4.3. we also introduce homology classes ζ and ζn of e1 2 (vm(0)), which correspond to elements vm+2h1,1 and v 2 l ek m+2ζl ∈ e 1 2 (l2vm(1)) for n = 3k + l with l ∈ {1, 2, 3}, respectively, where ζl corresponds to h1,0 if l = 1, and h2,l−2 if l = 2, 3. proposition 1.6. (cf. [4]) the e2-term of adams-novikov spectral sequence for computing π∗(l2vm(1)∞) is isomorphic to the direct sum of q(0) ⊗ ∧(h1,0, h2,0, h2,1) and the tensor product of ∧(ζ) and em−1(2)∗/(2, v ∞ 1 ) ⊕ ⊕ k>0 q(k) ⊗ ∧(ζk+1, ζk+2) as a z/2[v1]-module. by noticing that xn ∈ e 0 2 (l2vm(1)an ) survives to π∗(l2vm(1)an ) in lemma 5.1, we see that all additive generators of proposition 1.6 are permanent cycles. theorem 1.7. the homotopy groups π∗(l2vm(1)∞) are isomorphic to the adams-novikov e2term given in proposition 1.6. consider the cofiber sequence vm(0) wη v−11 vm(0) wp vm(1)∞ w σvm(0) (1.8) for the localization map η. here, we introduce algebras km(1)∗ = z/2[v1, v2, . . . , vm+1] and km(1)∗ = v −1 1 km(1)∗. ravenel showed the following proposition 1.9. (cf. [10]) the homotopy groups π∗(v −1 1 vm(0)) are isomorphic to km(1)∗ ⊗ ∧(h1,0). 46 ippei ichigi and katsumi shimomura cubo 10, 3 (2008) there is a relation between h1,0 and ζ, which is shown in section four: lemma 1.10. the induced homomorphism p∗ from p in (1.8) assigns h1,0/v j 1 ∈ e1 2 (v−1 1 vm(0)) to ζ/v j−2 1 ∈ e1 2 (l2vm(1)∞). observing the correspondence in the adams-novikov e2-terms, we obtain corollary 1.11. the homotopy groups π∗(l2vm(0)) are isomorphic to the direct sum of σ −1q(0)⊗ ∧(h1,0, h2,0, h2,1) and the tensor product of ∧(ζ) and km(1)∗ ⊕ σ −1km(1)∗/(2, v ∞ 1 , v∞ 2 ) ⊕ ⊕ k>0 σ−1q(k) ⊗ ∧(ζk+1, ζk+2) as a z/2[v1]-module. in the next section, we observe about an action of the moore spectrum m2 on vm(1)t and a ring structure of vm(1)4t, in order to study the adams-novikov differential and the extension problem of the spectral sequence in the following sections. we prove theorem 1.4 in section three. section four is devoted to show proposition 1.6. we end by proving theorem 1.7 in the last section. 2 the spectrum t (m) ∧ ktk we work in the stable homotopy category of spectra localized at the prime two. let bp denote the brown-peterson spectrum. then, we have the adams-novikov spectral sequence e s,t 2 (x) = ext s,t γ (a, bp∗(x)) =⇒ π∗(x). here (a, γ) is the associated hopf algebroid such that (a, γ) = (bp∗, bp∗(bp )) = (z(2)[v1, v2, . . . ], bp∗[t1, t2, . . . ]) for the hazewinkel generators vk ∈ bp2k+1−2 and the generators tk ∈ bp2k+1−2(bp ). let mk and k t k for k = 2, 4 and t > 0 denote spectra defined by the cofiber sequences s0 w2 s0 wi mk wj s1 and σ8tmk wαt mk witk ktk wjtk σ8t+1mk. here α denotes the adams map such that bp∗(α) = v 4 1 . note that m4 and k t 4 are ring spectra (cf. [5]). the ravenel spectrum t (m) is characterized by bp∗(t (m)) = a[t1, . . . , tm] ⊂ γ as γ-comodules, and is a ring spectrum, whose multiplication and unit map we denote by µ and ι, respectively. throughout the paper, we fix a positive integer m. let (a, γm) = (a, γ/(t1, t2, . . . , tm)) be the hopf algebroid associated with (a, γ), and consider a spectrum x such that bp∗(x) = m ⊗a a[t1, . . . , tm] for a γ-comodule m . then, we have an isomorphism e∗ 2 (x) = ext∗ γm (a, m ) (2.1) cubo 10, 3 (2008) the modulo two homotopy groups ... 47 by the change of rings theorem (cf. [10]). by observing the reduced cobar complex for the ext group, we have lemma 2.2. the e2-term has the vanishing line of the slope 1/(qm − 1) if m is (−1)-connected. hereafter, we put qm = 2 m+2 − 2 (2.3) which is the degree of u1 = vm+1 and s1 = tm+1. this shows π2(t (m)) = bp2 = z(2){v1} if m > 0. let t (m)/(va 1 ) for an integer a > 0 denote the cofiber of ṽa 1 : σ8at (m) → t (m), where ṽ1 : σ 8t (m) → t (m) is the composite ṽ1 : σ 8 t (m) = s8 ∧ t (m) t (m) ∧ t (m)wv1∧t (m) wµ t (m). lemma 2.4. for k = 2, 4 and a > 0, t (m)/(v4a 1 ) ∧ mk = t (m) ∧ k a k . in particular, t (m) ∧ ka 2 ∧ m4 = t (m)/(v 4a 1 ) ∧ m2 ∧ m4 = t (m) ∧ m2 ∧ k a 4 . proof. since π8(t (m)∧mk) = bp8/(k) = z/k{v 4 1 , v1v2} by lemma 2.2, we see that v 4 1 ∧mk = ι∧ αi ∈ π8(t (m)∧mk). indeed, both of these elements are assigned to v 4 1 ∈ bp8(t (m)∧mi) under the homomorphism induced from the unit map of bp . it extends to v4 1 ∧mk = ι∧α : mk → t (m)∧mk, since [mk, t (m) ∧ mk]8 = π8(t (m) ∧ mk). indeed, π9(t (m) ∧ mk) = bp9/(k) = 0. we further extend it to a self-map a = ṽ4 1 ∧ mk = t (m) ∧ α : t (m) ∧ mk → t (m) ∧ mk by the ring structure of t (m). now the cofiber of aa is t (m)/(v4a 1 ) ∧ mk = t (m) ∧ k a k . � this lemma implies vm(1)4a = t (m) ∧ k a 2 (2.5) for the spectrum vm(1)4a in (1.2). lemma 2.6. let f denote one of the spectra mk and k a k for k = 2, 4 and a > 0. then, there is a pairing νf : f ∧ f → t (m) ∧ f such that νf ◦ (f ∧ if ) = ι ∧ f : f → t (m) ∧ f for m > 0. here if : s 0 → f denotes the inclusion to the bottom cell. proof. the pairing for f = m4 or k a 4 is the composite (ι∧f ∧f )(t (m)∧µf ) for the multiplication µf of the ring spectrum of f (see [5]). for f = m2, we see that π0(t (m)∧m2) = bp0/(2) = z/2 and π1(t (m)∧m2) = bp1/(2) = 0 by lemma 2.2, and so [m2, t (m) ∧ m2]0 = z/2. 48 ippei ichigi and katsumi shimomura cubo 10, 3 (2008) note that m2 ∧ m4 = m2 ∨ σm2. then, by lemma 2.4, t (m) ∧ m2 ∧ k a 4 = t (m)/(v4a 1 ) ∧ m2 ∧ m4 = t (m)/(v 4a 1 ) ∧ (m2 ∨ σm2) = t (m)/(v4a 1 ) ∧ m2 ∨ σt (m)/(v 4a 1 ) ∧ m2 = t (m) ∧ k a 2 ∨ σt (m) ∧ ka 2 . we also see that t (m) ∧ ka 2 ∧ ka 4 = t (m)/(v4a 1 ) ∧ ka 2 ∧ m4 = t (m)/(v 4a 1 ) ∧ (ka 2 ∨ σka 2 ), and so t (m) ∧ ka 2 ∧ ka 4 ∧ m2 = t (m) ∧ k a 2 ∧ ka 2 ∨ σt (m) ∧ ka 2 ∧ ka 2 . then, t (m) ∧ m2∧k a 4 ∧ka 4 ∧m2 = t (m)∧k a 2 ∧ka 4 ∧m2 ∨ σt (m)∧k a 2 ∧ka 4 ∧m2 = t (m)∧ka 2 ∧ka 2 ∨ σt (m)∧ka 2 ∧ka 2 ∨ σt (m)∧ka 2 ∧ka 2 ∧m2. let µk : k a 4 ∧ka 4 → ka 4 denote the multiplication of the ring spectrum ka 4 , and ν̃ be the composite t (m) ∧ m2 ∧ m2 t (m) ∧ t (m) ∧ m2wt (m)∧νm2 wµ∧m2 t (m) ∧ m2. then the desired pairing is a composite ka 2 ∧ ka 2 wι∧k∧k t (m) ∧ ka 2 ∧ ka 2 winc∧ka2 t (m) ∧ m2 ∧ ka4 ∧ ka4 ∧ m2 wswitch t (m)∧m2∧m2∧k a 4 ∧ka 4 wν̃ t (m) ∧ m2∧ka4 ∧ka4 wt (m)∧m2∧µk t (m) ∧ m2∧ka4 wprj t (m)∧ka2 . � corollary 2.7. the spectra vm(0) and vm(1)4a for a > 0 are ring spectra. we say that a spectrum x has m2-action, if there is a pairing ϕx : x ∧ m2 → x such that ϕx (x ∧i) = idx . here i : s 0 → m2 is the inclusion of (1.1) and idx : x → x denotes the identity map. lemma 2.8. vm(1)t has m2-action. proof. since t (m) is an associative ring spectrum, t (m)/(vt 1 ) is a t (m)-module spectrum. the action ϕvm(1)t is defined by the composite vm(1)t∧m2 = t (m)/(v t 1 )∧m2∧m2 wt (m)/(vt1)∧νm2 t (m)/(vt 1 )∧t (m)∧m2 w t (m)/(vt1)∧m2 = vm(1)t. � since vm(1)t is a t (m)-module spectrum, it implies the following corollary 2.9. vm(1)t is a vm(0)-module spectrum. 3 the homotopy groups of l2vm(1) note that if bp∗(x) is (2, v1)-nil, then bp∗(l2x) = v −1 2 bp∗(x), since l2 is smashing (cf. [8], [9]). therefore, the adams-novikov e2-term e ∗ 2 (l2vm(1)t) is ext ∗ γ (a, v−1 2 bp∗/(2, v t 1 )[t1, . . . , tm]), which is isomorphic to e∗ 2 (l2vm(1)t) = ext ∗ γm (a, v−1 2 bp∗/(2, v t 1 )) cubo 10, 3 (2008) the modulo two homotopy groups ... 49 by (2.1). consider a spectrum em(2) = v −1 2 bp 〈m + 2〉 for the johnson-wilson spectrum bp 〈m + 2〉. then we obtain a hopf algebroid (em(2)∗, σm(2)) = (v −1 2 z(2)[v1, v2, . . . , vm+2], em(2)∗ ⊗a γm ⊗a em(2)∗). since v −1 2 bp∗/j w1⊗ηr em(2)∗/j ⊗a γm for an invariant regular ideal j = (2b, va 1 ) is a faithfully flat extension, we have an isomorphism ext∗ γm (a, bp∗/j) ∼= ext ∗ σm(2) (em(2)∗, em(2)∗/j) by a theorem of hopkins’ (cf. [1, th. 3.3]). note that m + 2 is the smallest number n, for which v −1 2 bp∗/j w1⊗ηr v−12 bp 〈n〉∗/j ⊗a γm is a faithfully flat extension. we use the abbreviation h∗m = ext∗ σm(2) (em(2)∗, m ) (3.1) for a σm(2)-comodule m . we compute the ext group h ∗m by the reduced cobar complex ˜ω∗ σm(2) m (cf. [10]). since the differentials of the cobar complex are defined by the right unit ηr : em(2)∗ → σm(2) and the diagonal ∆ : σm(2) → σm(2) ⊗em(2)∗ σm(2), we write down here some formulas on them based on the hazewinkel and the quillen formulas: vn = 2ℓn − ∑n−1 k=1 ℓkv 2 k n−k ∈ q ⊗ a = q[ℓ1, ℓ2, . . . ], ηr(ℓn) = ∑n k=0 ℓkt 2 k n−k ∈ q ⊗ γ = q ⊗ a[t1, t2, . . . ] and ∑ i+j=n ℓi∆(t 2 i j ) = ∑ i+j+k=n ℓit 2 i j ⊗ t 2 i+j k ∈ q ⊗ γ ⊗a γ. (3.2) hereafter, we put v2 = 1 and use the following notation: ui = vm+i and si = tm+i. since the structure maps preserve degrees, we may recover v2’s from its degrees. then, we obtain the following two lemmas immediately from (3.2) by a routine computation: lemma 3.3. the right unit ηr : a → γm/(2) acts as follows: ηr(vn) = vn for n ≤ m + 1, ηr(u2) = u2 + v1s 2 1 + v2 m+1 1 s1, ηr(u3) ≡ u3 + s 4 1 + s1 + v1r1 mod (2, v 2 m+2 1 ), ηr(u4) ≡ u4 + s 4 2 + s2 + v3s 8 1 + v2 m+1 3 s1 mod (2, v1) for r1 = s 2 2 + v1u2s 2 1 . this yields the relations in σm(2): s4 1 + s1 ≡ v1r1 mod (2, v 2 m+2 1 ) and s4 2 + s2 + v3s 8 1 + v2 m+1 3 s1 ≡ 0 mod (2, v1). (3.4) 50 ippei ichigi and katsumi shimomura cubo 10, 3 (2008) lemma 3.5. the diagonal ∆ behaves on the generators si as follows: ∆(s1) = s1 ⊗ 1 + 1 ⊗ s1, ∆(s2) = s2 ⊗ 1 + 1 ⊗ s2 + v1s1 ⊗ s1, ∆(s3) ≡ s3 ⊗ 1 + 1 ⊗ s3 + v2s 2 1 ⊗ s2 1 mod (2, v 1 ). lemma 3.6. let z denote an element defined by r4 1 +r1 +v 2 3 s4 1 +v2 m+2 3 s2 1 = v1z. then the cochains r1, z ∈ ˜ω 1 σm(2) em(2)∗/(2) are cocycles. besides, z ≡ u2s 2 1 modulo (v2 1 ). proof. since v1 ∈ ˜ω 0 σm(2) em(2)∗/(2) and s1 ∈ ˜ω 1 σm(2) em(2)∗/(2) are both cocycles, so is r1 by the relation v1r1 = s 4 1 + s1 ∈ σm(2) in (3.4). furthermore, v3 ∈ ˜ω 0 σm(2) em(2)∗/(2) is a cocycle. it follows similarly from its definition that z is a cocycle. by the definition of r1, r4 1 + r1 ≡ s 8 2 + s2 2 + v1u2s 2 1 ≡ v1u2s 2 1 + v2 3 s16 1 + v2 m+2 3 s2 1 modulo (2, v2 1 ) by (3.4). � we now work as [6]. lemma 3.7. ut 2 ∈ e0 2 (vm(1)) and u t 2 h2,0 ∈ e 1 2 (vm(1)) for each t > 0 are permanent cycles. proof. for t = 1, the lemma is seen by lemma 2.2. consider the cofiber sequence σ2vm(0) v1 → vm(0) i1 → vm(1) j1 → σ3vm(0). put d(u t 2 ) = v1k ′ t ∈ ˜ω1 σm(2) em(2)∗/(2) by virtue of lemma 3.3, and let kt ∈ e 1 2 (vm(0)) be the homology class of the cocycle k ′ t. then, k1 = h1,1, v1kt = 0 and kt+1 = 〈k1, v1, kt〉. indeed, 〈k1, v1, kt〉 is the class of k ′ 1 ηr(u t 2 ) + u2k ′ t = d(u t+1 2 )/v1 = k ′ t+1. besides, δ(ut 2 ) = kt for the connecting homomorphism associated to the cofiber sequence. let ξ1 ∈ πqm−1(vm(0)) denote the homotopy element detected by k1. then, v1ξ1 = ξ1v1 = 0. suppose now that ut 2 ∈ e0 2 (vm(1)) is a permanent cycle. then, kt is a permanent cycle that detects the element ξt = j1u t 2 by the geometric boundary theorem. since v1ξt = 0, the toda bracket {ξ1, v1, ξk} is defined, which is detected by the massey product 〈k1, v1, kt〉. note here that the toda bracket is defined since vm(0) is a ring spectrum. it follows that kt+1 is a permanent cycle and detects a homotopy element, which we denote by ξt+1. since the massey product 〈v1, k1, v1〉 is zero in the e2-term e 0,qm+4 2 (vm(0)), we see that {v1, ξ1, v1} = 0 by lemma 2.2. now we compute v1{ξ1, v1, ξk} = {v1, ξ1, v1}ξk = 0, and ξt+1 is pulled back to u t+1 2 under the map j1. turn to ut 2 h2,0. in this case a similar argument works. for the connecting homomorphism δ, δ(ut 2 h2,0) = 〈h 2 1,0, v1, kt〉, which detects a homotopy element {η 2 0 , v1, ξt}, where η0 denotes an element detected by h1,0. applying v1 shows {v1, η 2 0 , v1}ξt = 0. indeed, {v1, η 2 0 , v1} is detected by e s,2qm+4+s 2 (vm(0)) for s > 2. � lemma 3.8. the elements h1,0, h1,1 ∈ e 1 2 (vm(0)) and h2,1 ∈ e 1 2 (l2vm(0)) are permanent cycles. proof. h1,0, h1,1 are seen immediately by lemma 2.2. cubo 10, 3 (2008) the modulo two homotopy groups ... 51 the cobar module ˜ω 4,4qm+6 γm bp∗/(2) is generated by v 3 1 s ⊗4 1 and v2s ⊗4 1 by degree reason. the first generator cobounds v2 1 s2 ⊗ s1 ⊗ s1, and we obtain e 4,4qm+6 2 (vm(0)) = z/2{v2h 4 1,0}. put d3(h2,1) = av2h 4 1,0 ∈ e 4,4qm+6 2 (vm(0)) for a ∈ z/2. let w be an element fit in d(s3) = v2s 2 1 ⊗ s2 1 + v1w by virtue of lemma 3.5. then, d(w) = 0 in the cobar complex ˜ω 3 σm(2) em(2)∗/(2), and we see that s⊗4 1 cobounds s2 3 ⊗ s1 ⊗ s1 + v1w 2 ⊗ s2 + (r1 ⊗ s1 + s1 ⊗ r1 + v1r1 ⊗ r1) ⊗ s2 (in which we set v2 = 1). it follows that d3(h2,1) = av2h 4 1,0 = 0 ∈ e 4 2 (l2vm(0)) as desired. indeed, v2h 4 1,0 = v1gh 2 1,0 = 0, since v2h 2 1,0 = v1g for an element g and v1h 2 1,0 = 0 by d(s2) = v1s1 ⊗ s1. � proof of theorem 1.4. every element x ∈ es 2 (l2vm(1)) is decomposed as x = x ′x′′ for x′ ∈ z/2[u2]⊗∧(h2,0) and x ′′ ∈ km−1(2)∗⊗∧(h1,0, h1,1, h2,1). note that km−1(2)∗⊗∧(h1,0, h1,1, h2,1) ⊂ e∗ 2 (l2vm(0)). since x ′ (resp. x′′) is a permanent cycle of the adams-novikov spectral sequence for computing π∗(l2vm(1)) (resp. π∗(l2vm(0))) by lemma 3.7 (resp. 3.8), we obtain that the element x is a permanent cycle from corollary 2.9. we see that the extension problem is trivial by lemma 2.8. indeed, z/2 = π0(m2) acts on π∗(l2vm(1)). � 4 the elements xn we introduce the integer bn for n ≥ 0 by bn =      an − 8 n ≡ 1 (3) an − 3 n ≡ 2 (3) 0 n ≡ 0 (3), and the elements xn ∈ em(2)∗ defined by xn = x 2 n−1 + v bn 1 yn−1, where yn =          0 n ≤ 0 or n ≡ 2 (3) x0 n = 1 x2 + v 2 1 v4 3 x2 1 + v4 1 v2 m+3 3 x1 n = 3 xn−2yn−3 n ≡ 0, 1 (3) and n ≥ 4. (4.1) we also consider cocycles zn ∈ σm(2): zn =      s2 n+1 1 n = 0, 1 r2 n−1 1 n = 2, 3 xn−3zn−3 n > 3. (4.2) proposition 4.3. for the differential d : ω0 σm(2) em(2)∗/(2) → ω 1 σm(2) em(2)∗/(2) of the cobar complex, d(xn) = v an 1 zn. 52 ippei ichigi and katsumi shimomura cubo 10, 3 (2008) proof. for n = 0 and 1, it is immediate from lemma 3.3, and the cases for n = 2 and 3 follow from the computation d(x2) = d(u 4 2 + v3 1 u2) = v 4 1 s8 1 + v4 1 s2 1 = v6 1 r2 1 by (3.4). for n = 4, d(x4) ≡ d(x 4 2 + v18 1 x2 + v 20 1 v4 3 x2 1 + v22 1 v2 m+3 3 x1) ≡ v24 1 r8 1 + v24 1 r2 1 + v24 1 v4 3 s8 1 + v24 1 v2 m+3 3 s4 1 ≡ v26 1 z2 ≡ v26 1 x1z1 mod (2, v 28 1 ) by the definition of z. suppose inductively that d(x3k+1) = v a3k+1 1 x3k−2z3k−2 mod (2, v a3k+1+2 1 ) for k > 0. d(x2 3k+1 ) ≡ v 2a3k+1 1 x2 3k−2 z2 3k−2 mod (2, v 2a3k+1+4 1 ) d(v a3k+2−3 1 y3k+1) ≡ d(v a3k+2−3 1 x3k−1y3k−2) ≡ v a3k+2−3 1 x3k−1(v1z 2 3k−2 + v3 1 z3k−1) mod (2, v a3k+2−3+a3k−1 1 ) and the sum shows d(x3k+2) ≡ v a3k+2 1 x3k−1z3k−1 mod (2, v a3k+2+2 1 ). similarly, d(x4 3k+2 ) ≡ v 4a3k+2 1 x4 3k−1 z4 3k−1 mod (2, v 4a3k+2+8 1 ) d(v a3k+4−8 1 y3k+3) ≡ d(v a3k+4−8 1 x3k+1y3k) ≡ v a3k+4−8 1 x3k+1(v 6 1 z2 3k + v8 1 z3k+1) mod (2, v a3k+4−8+a3k+1 1 ) and we have d(x3k+4) = v a3k+4 1 x3k+1z3k+1 mod (2, v a3k+4+2 1 ), which completes the induction. � proof of lemma 1.10. it suffices to show that h1,0/v j 1 ∈ e1 2 (l2vm(1)∞) equals ζ/v j−2 1 . the element h1,0/v j 1 is represented by s1/v j 1 . we make a computation in the cobar complex d(u2 2 /v j+2 1 ) = s4 1 /v j 1 = s1/v j 1 + r1/v j−1 1 d(v2 3 u2 2 /v j+1 1 ) = v2 3 s4 1 /v j−1 1 d(v2 m+2 3 u2/v j 1 ) = v2 m+2 3 s2 1 /v j−1 1 d(x2 2 /v j+11 1 ) = r4 1 /v j−1 1 by lemma 3.3 and proposition 4.3. the sum yields the homologous relation s1/v j 1 ∼ z/v j−2 1 by lemma 3.6, and so h1,0/v j 1 = ζ/v j−2 1 in e1 2 (l2vm(1)∞). � proof of proposition 1.6. we consider the v1-bockstein spectral sequence given by the short exact sequence 0 → em(2)∗(vm(1)) ϕ → em(2)∗(vm(1)∞) v1 → em(2)∗(vm(1)∞) → 0 for ϕ given by ϕ(x) = x/v1. let b ∗ denote the z/2[v1]-module of the proposition. then, it is easy to see that bs contains the image of ϕ∗ : e s 2 (l2vm(1)) → e s 2 (l2vm(1)∞) and that proposition 4.3 defines a homomorphism f : bs → es 2 (l2vm(1)∞). we also consider the composite ∂ = δ ◦ f : bs → es+1 2 (l2vm(1)), where δ : e s 2 (l2vm(1)∞) → e s+1 2 (l2vm(1)) denotes the connecting homomorphism associated to the short exact sequence. by [7, remark 3.11], it suffices to show the sequence 0 w coker ∂ wϕ∗ b∗ wv1 b∗ w∂ im ∂ w 0 (4.4) cubo 10, 3 (2008) the modulo two homotopy groups ... 53 is exact. we decompose e∗ 2 (l2vm(1)) into the direct sum of mc = km−1(2)∗[u 2 2 ]{u2}⊗∧(h10, h20, h21), mi = km−1(2)∗[u 2 2 ]{h11} ⊗ ∧(h10, h20, h21) and n ⊗ ∧(ζ) = km−1(2)∗[u 2 2 ] ⊗ ∧(h10, h20, h21, ζ). we notice that for non-negative integers n and r with r < 8, there exist uniquely non-negative integers t and q such that n = 8qt + req. by this fact, we decompose summands of n as follows: km−1(2)∗[u 2 2] = km−1(2)∗ ⊕ ⊕ k≥1 xkkm−1(2)∗[xk+1] a , km−1(2)∗[u 2 2]h10 = ⊕ q≥0 ( ( x3q+2km−1(2)∗[x3q+3] a ⊕ x3q+3km−1(2)∗[x3q+4] b ) ζ3q+4 ⊕ km−1(2)∗[x3q+2]ζ3q+1 a ) , km−1(2)∗[u 2 2]h20 = ⊕ q≥0 ( x3q+3km−1(2)∗[x3q+4]ζ3q+5 c ⊕ ( x3q+1km−1(2)∗[x3q+2] d ⊕ km−1(2)∗[x3q+3] a ) ζ3q+2 ) , km−1(2)∗[u 2 2]h21 = ⊕ q≥0 ( x3q+1km−1(2)∗[x3q+2] e ⊕ x3q+2km−1(2)∗[x3q+3] f ⊕ km−1(2)∗[x3q+4] a ) ζ3q+3, km−1(2)∗[u 2 2]h10h20 = ⊕ q≥0 ( km−1(2)∗[x3q+3]ζ3q+4ζ3q+2 a ⊕ x3q+3km−1(2)∗[x3q+4]ζ3q+4ζ3q+5 b ⊕km−1(2)∗[x3q+2]ζ3q+1ζ3q+2 d ) , km−1(2)∗[u 2 2]h20h21 = ⊕ q≥0 ( km−1(2)∗[x3q+4]ζ3q+3ζ3q+5 c ⊕ ( x3q+1km−1(2)∗[x3q+2] b ⊕km−1(2)∗[x3q+3] f ) ζ3q+2ζ3q+3 ) , km−1(2)∗[u 2 2]h10h21 = ⊕ q≥0 ( ( km−1(2)∗[x3q+3]x3q+2 b ⊕km−1(2)∗[x3q+4] b ) ζ3q+4ζ3q+3⊕km−1(2)∗[x3q+2]ζ3q+1ζ3q+3 e ) , km−1(2)∗[u 2 2]h10h20h21 = ⊕ k≥1 km−1(2)∗[xk+1]ζkζk+1ζk+2 b . here, m x and m ′ x for modules m and m ′ mean that m and m ′ are isomorphic under a bockstein differential dr for some r so that dr(m ) = m ′, which is seen by proposition 4.3. let nc (resp. ni ) be the direct sum of single (resp. double) underlined submodules of n , and put ˜m = q(0) ⊗ ∧(h1,0, h2,0, h2,1), ˜n = ⊕ k>0 q(k) ⊗ ∧(ζk+1, ζk+2). then we have the three exact sequences 0 → mc ϕ∗ → ˜m v1 → ˜m → mi → 0, 0 → nc ϕ∗ → ˜n v1 → ˜n → ni → 0 and 0 → km−1(2)∗ → em−1(2)∗/(2, v ∞ 1 ) → em−1(2)∗/(2, v ∞ 1 ) → 0, the direct sum of which yields the sequence (4.4). � 5 the adams-novikov e∞-term for π∗(l2t (m) ∧ m2) we first show that all elements of the adams-novikov e2-term for π∗(l2vm(1)∞) are permanent cycles. take an element x/vt 1 ∈ e0 2 (l2vm(1)∞). then x ∈ e 0 2 (l2vm(1)t). thus, if x = y 2/vt 1 for 54 ippei ichigi and katsumi shimomura cubo 10, 3 (2008) some y ∈ e0 2 (l2vm(1)4t), then x is a permanent cycle. so it is sufficient to show that d3(xn) = 0 ∈ e3 2 (l2vm(1)an ) for each n ≥ 0. we consider the integer εn = { 2 n 6≡ 0 (3) 0 n ≡ 0 (3) so that vm(1)an+εn is a ring spectrum by corollary 2.7. lemma 5.1. d3(xn) = 0 ∈ e 3 2 (l2vm(1)an ) for n ≥ 0. proof. for n = 0, it is shown in lemma 3.7. suppose that d3(xn) = ξ ∈ e 3 2 (l2vm(1)an ) for n > 0. send this to e 3 2 (l2vm(1)an−1 ), and we see that ξ = d3(xn) = d3(x 2 n−1) ∈ e 3 2 (l2vm(1)an−1 ). then, the map v εn−1 1 : e3 2 (l2vm(1)an−1 ) → e3 2 (l2vm(1)an−1+εn−1 ) assigns v εn−1 1 ξ to v 2εn−1 1 ξ = d3((v εn−1 1 xn−1) 2), which is zero, since v εn−1 1 xn−1 ∈ e 0 2 (l2vm(1)an−1+εn−1 ) and vm(1)an−1+εn−1 is a ring spectrum. it follows that ξ = v an−1−εn−1 1 ξ′ for some ξ′ ∈ e3 2 (l2vm(1)an−an−1+εn−1 ). note that this works even if n = 1, though vm(1) is not a ring spectrum. consider the commutative diagram vm(1) vm(1) ∗ vm(1) vm(1)an−a+1 vm(1)an+1 vm(1)a vm(1)an−a+1 vm(1)an−a vm(1)an vm(1)a vm(1)an−a, uvan−a1 uvan1 w u w u van−a1 u wv a 1 u wiv wjv u pwva1 wi′v wj′v (a = an−1 − εn−1) in which rows and columns are cofiber sequences. let 〈x〉 ∈ π∗(x) denote a homotopy element detected by x ∈ e∗ 2 (x). noticing that xn ∈ e 0 2 (l2vm(1)an−1−εn−1 ) is a permanent cycle, we see that jv∗(〈xn〉) = 〈v an−an−1+εn−1 1 ζn〉 and j ′ v∗ (〈xn〉) = 〈ξ ′〉, and so p∗(〈v an−an−1+εn−1 1 ζn〉) = 〈ξ ′〉. since 〈ζn〉 ∈ π∗(l2vm(1)) by theorem 1.4, we obtain 〈ξ ′〉 = 0, and 〈xn〉 is in the image under the map i ′ v∗ . it follows that there is a permanent cycle x′n ∈ e0 2 (l2vm(1)an ), whose leading term is xn, such that iv∗(〈x ′ n〉) = 〈xn〉 ∈ π∗(l2vm(1)an−1−εn−1 ). the lemma now follows by replacing xn by x ′ n. � received: february 2008. revised: april 2008. cubo 10, 3 (2008) the modulo two homotopy groups ... 55 references [1] m. hovey and h. sadofsky, invertible spectra in the e(n)-local stable homotopy category, j. london math. soc., 60 (1999), 284–302. 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[12] k. shimomura and x. wang, the adams-novikov e2-term for π∗(l2s 0) at the prime 2, math. z., 241 (2002), 271–311. n04 articulo 15.dvi cubo a mathematical journal vol.12, no¯ 02, (235–259). june 2010 on semisubmedian functions and weak plurisubharmonicity chia-chi tung1 dept. of mathematics and statistics, minnesota state university, mankato, mankato, mn 56001, usa email: chia.tung@mnsu.edu abstract in this note subharmonic and plurisubharmonic functions on a complex space are studied intrinsically. for applications subharmonicity is characterized more effectually in terms of properties that need be verified only locally off a thin analytic subset; these include the submean-value inequalities, the spherical (respectively, solid) monotonicity, near as well as weak subharmonicity. several results of gunning [9, k and l] are extendable via regularity to complex spaces. in particular, plurisubharmonicity amounts (on a normal space) essentially to regularized weak plurisubharmonicity, and similarly for subharmonicity (on a general space). a generalized hartogs’ lemma and constancy criteria for certain matrix-valued mappings are given. resumen en esta nota son estudiadas intŕınsicamente las funciones subarmonicas y plurisubarmonicas sobre un espacio complejo. para aplicaciones, subarmonicidad es caracterizada mas eficientemente en términos de propiedades que necesitan ser verificadas solamente localmente en un subconjunto anaĺıtico delgado; estas aplicaciones incluyen la desigualdad del valor-submedio, la monotonicidad esférica (respectivamente, sólida), bien como subarmonicidad debil. varios resultados de gunning [9, k and l] son extendibles v́ıa regularidad a espacios complejos. en particular, plurisubarmonicidad (sobre un espacio normal) 1supports by the ”globale methoden in der komplexen geometrie” grant of the german research society dfg and the faculty improvement grant of minnesota state university, mankato, are gratefully acknowledged. 236 chia-chi tung cubo 12, 2 (2010) importa esencialmente para plurisubarmonicidad débil regularizada y similarmente para subarmoniciada (sobre un espacio general). son dados un lema de hartogs generalizado y un criterio de constancia para ciertas aplicaciones matriz-valuada. key words and phrases: subharmonicity, seminear subharmonicity, jensen function, weak subharmonicity, weak plurisubharmonicity 2000 math. subj. class.: primary: 31c05; secondary: 31c10, 32c15 1 introduction it is well-known that subharmonic functions are widely used in potential theory and partial differential equations, and their complex analogue, the plurisubharmonic functions, have played a significant role in the development of higher dimensional complex analysis. in this work such functions are studied on a general complex space from an intrinsic viewpoint, the main purpose here being to show that they admit intrinsic distributional representations despite the presence of singularities. in what follows let y denote a (reduced) complex space of pure dimension m > 0. this means that y is a hausdorff space which admits a countable basis of open sets and an open covering {uj} together with homeomorphisms αj : uj → vj where vj is a pure m-dimensional analytic subset in some open subset of cnj such that each mapping αj ◦ α −1 k : αk(uj ∩ vk) → αj (uj ∩ vk) is biholomorphic. a riemann covering of an m-dimensional complex space y at a point a ∈ y is a holomorphic map of a neighborhood of a into cm with discrete fibers. a mappping φ : y → [−∞,∞) is called subharmonic (φ ∈ sh (y )) if: (i) φ is upper semicontinuous y (φ ∈ cusc(y )); (ii) for any compact set k ⊂ y and any function h : k → r which is continuous in k and (locally) semiharmonic ([24]) in int (k), and h(z) ≥ φ(z) for every z ∈ ∂k, it follows that h(a) ≥ φ(a) for every a ∈ k ([17, p. 1][10, p. 16]). subharmonicity is both a global and a local property (lemma 3.2), the link between the two being rested with a maximum principle for almost everywhere solidly submedian functions (proposition 3.1). in fact, via local riemann coverings characterizations of subharmonicity can be given, as in the euclidean case, in terms of the local property of being solidly, respectively, spherically, submedian. with the continuity assumption subharmonicity is also equivalent, by virtue of a maximum principle for seminearly subharmonic functions (proposition 3.3), to near subharmonicity, spherical (respectively, solid) monotonicity. for applications it gains in usefulness if such a property needs only be verified locally off a thin analytic subset (theorems 3.1 and 3.3). especially, in the definition of subharmonicity, it suffices to require the condition (ii) to hold for closed pseudoballs k lying in y off some thin analytic subset, instead of arbitrary compact subsets of y (by lemma 3.1 and theorem 3.1; cf. [1, p. 36]). a locally integrable (or an upper semicontinuous) function ψ : y → [−∞,∞) is said to be regularized at a ∈ y (respectively, regularized in y ) if νp ∆ (a) ψ(a) = lim ε→0 〈ψ ⌋∆〉a,ε (1.1) for some standard domain ∆ at a (respectively, the said condition holds for each a ∈ y ). an upper semicontinuous function ψ : y → [−∞,∞) is said to be strictly regularized in y, if ψ satisfies not only the condition (1.1) for any standard domain ∆ at every point of y, but also the condition cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 237 νp ∆ (a) ψ(a) = lim ε→0 [ψ⌋∆]a,ε, (1.2) for each pseudoball ∆ at every point a of y. here νp(a) denotes the multiplicity of a light holomorphic map p at a ([21, p. 22]; for a topological treatment see radò and reichelderfer [18]), and 〈ψ ⌋∆〉a,ε (respectively, [ψ⌋∆]a,ε) the solid (respectively, spherical) mean-value function of ψ ((2.3)(2.4)). every plurisubharmonic function is strictly regularized (propositions 3.2 and 4.2). a mapping φ : y → [−∞,∞) is plurisubharmonic (φ ∈ psh (y ) in the sense of lelong and oka) iff φ ∈ cusc(y ) and the pull-back of φ by any holomorphic map from the unit disc ∆ ⊂ c into y is subharmonic in ∆. this definition turns out, owing to a notable result of fornæss and narasimhan ([5, p. 64]), to be equivalent to one in an apparently stronger sense ([15, p. 356]): a mapping φ : y → [−∞,∞) is plurisubharmonic if φ has a plurisubharmonic extension into the ambient space of a local embedding of y at every point. consequently plurisubharmonicity of a mapping is a local property. in contrast to ”subharmonicity”, such a mapping is necessarily radially submedian ((4.2)) and, if not identically equal to −∞ on any open subset, a jensen function (proposition 4.2). on a normal complex space the (possibly weaker) property of being semiradially submedian suffices to characterize plurisubharmonicity (theorem 4.1). a locally integrable function with nonnegative local distributional laplacian is called weakly subharmonic. with c2-differentiability such a function can be regarded as possessing (off a thin analytic subset) nonnegative local spherical mean-radial derivative. by a similar token the requirement of semidefinite positivity of the distributional levi form of locally integrable functions gives rise to a class pshw(y ) of weakly plurisubharmonic functions. denote by sh {6≡−∞}(y ) the set of all subharmonic functions in y not identically equal to −∞ on any open subset of y. it turns out that a weakly subharmonic function φ belongs to sh {6≡−∞}(y ) if and only if φ is regularized in y (theorem 3.4). owing to the strict regularity of plurisubharmonic functions, several results of gunning [9, k and l] are extendable to complex spaces. some of these are indicated in §4 and §5. especially, given φ ∈ pshw(y ), there exists a unique locally integrable, regularized function ψ : y → [−∞,∞) such that ψ = φ almost everywhere and ψ ∈ sm rad(y ) ∩ psh {6≡−∞}(yreg); if further y is normal, then a function φ : y → [−∞,∞) is weakly plurisubharmonic and regularized in y if and only if φ ∈ psh {6≡−∞}(y ) (theorem 4.2 and corollary 4.1). as applications, a generalized hartogs’ lemma, conditions for the constancy of certain matrix-valued mappings (including extensions of two results of bochner and montgomery [2, p. 155]), and the maximum modulus principle for weakly real-analytic, subharmonic mappings, are given. also, some lemmas and examples of recurring use on subharmonicity and plurisubharmonicity are gathered in an appendix. the author is indebted to the referee for suggestions which led to the improvement of this paper. 2 preliminaries denote by ‖z‖ the euclidean norm of z = (z1, · · · ,zm) ∈ c m, where each component zj = xj +iyj. let the space cm be oriented so that the eulidean kähler form υm := ((i/2π) ∂∂̄ ‖z‖2)m is positive. in what follows let x,y denote (reduced) complex spaces of pure dimension m > 0, and let p : y → cm be a holomorphic map. set a′ := p(a), p[a] := p−a′, and ra := ‖p [a]‖ for each a ∈ y. if u ⊆ y is an open set, a ∈ u and r > 0, set u[a](r) := {z ∈ u | ra(z) < r}, and u[a][r] := {z ∈ u | ra(z) ≤ r}. 238 chia-chi tung cubo 12, 2 (2010) denote by b[a′](r) the open ball in c m with center a′ and radius r, omitting the subscript if a′ = 0. let dυ (respectively, dσr) be the euclidean volume element of c m (respectively, the sphere s(r) = ∂b(r)) and | b(r) | (respectively, | s |) the volume of b(r) (respectively, | s(1) |). a complex space x together with a holomorphic map p : x → ω, where ω is an open, connected subset of cm, is called a semi-riemann domain (of dimension m > 0) if there exists a thin analytic set σ in ω such that σp := p −1(σ) is thin in x, and the restriction p : x0 := x \ σp → ω0 := ω \ σ has discrete fibers; the map p = (p1, · · · ,pm) : x → ω is called a riemann semicovering. if σ = ∅, then (x,p) is called a riemann domain and the map p a riemann covering. if p : x → ω is, in addition, a local homeomorphism, then (x,p) is said to be unramified. every proper modification of an affine algebraic variety is a semi-riemann domain (as such it is parabolic in the sense of [22, pp. 73–74]). riemann domains, in particular, analytic coverings of cm, play a fundamental role in complex analysis; in fact, every pure m-dimensional complex space is locally an analytic covering of a domain in cm (hence a riemann domain). let (x,p) be a semi-riemann domain; denote by x∗ the largest open subset of x on which p is locally biholomorphic. for each open subset d ⊆ x, set d0 := d ∩ x0 and d∗ := d ∩ x∗. suppose that p : ũ → ω, is a holomorphic map defined on an open neighborhood ũ of a point a ∈ y such that: i) p−1(a′) ∩ ũ = {a}; ii) for a sufficiently small ball u′ := b[a′](r) in c m, ua := p−1(u′) ∩ ũ = p−1(u′) ∩ ũ is connected and the mapping p⌋ua : ua → u ′ is an analytic covering (biholomorphic if a ∈ d∗); iii) every branch v ka , k = 1, · · · ,sa, of ua contains a; and iv) sa = deg (p⌋ua) = ν y p (a) (2.1) ([21, proposition 1.3]). such an ua is called, for convenience, a pseudoball (of radius r) at a ([24, p. 557]). a pseudopolydisc ∆[a](r) over a polydisc (of polyradius r) in c m is similarly defined. an open neighborhood ∆ of a point a in a complex space y is called a standard domain at a if and only if there exists a riemann covering p : ∆ → ω exhibiting ∆ as either a pseudoball (of radius r) or a pseudopolydisc (of polyradius r) at a. if p : y → ω is a riemann semicovering and a ∈ y 0, define d(a) (respectively, dpd(a)) := the supremum of r > 0 (respectively, r = (r1, · · · ,rm) > (0, · · · , 0)) such that a pseudoball (respectively, polydisc) ∆[a](r) exists. the notions of ck-differential forms, the exterior differentiation d, the operators ∂, ∂̄ and dc := (1/4πi)(∂ − ∂̄), are well-defined on a complex space y despite the presence of singularities ([23, chapter 4]). if g ⊆ y is open subset, denote by dg the (maximal) boundary manifold of greg in yreg, the manifold of simple points of y, oriented to the exterior of greg ([23, p. 218]). if p : g → c m is a holomorphic map and a ∈ g, the form υp : = dd c r2a = i 2π ∂∂̄ r2a (2.2) is nonnegative ([23, §4]) and independent of a. the poincaré form σa := 2 dcra r2m−1a ∧ υm−1p is d-closed. also set dυ̃ := p∗dυ, dσ[a],r := (p [a])∗dσr, where dυ (respectively, dσr ) denotes the euclidean volume element of cm (respectively, s(r)). if p : y → cm is a riemann semicovering and φ ∈ cusc(u), where u = u[a](r0) ⊂ y, the associated spherical mean-value function is defined by cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 239 [φ⌋u]a,r := ∫ du[a](r) φσa, 0 < r < r0, (2.3) ([24, (3.2)]). similarly, if φ ∈ cusc(∆), where ∆ = ∆[a](r0) is a standard domain in y, the associated solid mean-value function is defined by 〈φ⌋∆〉a,r := 1 vol (∆(a′,r)) ∫ ∆a(r) φ(z) dṽ(z), 0 < r < r0, (2.4) where vol (∆(a′,r)) denotes the euclidean volume of the induced domain ∆(a′,r) := p(∆a(r)). let j dg : dg → y denote the inclusion mapping and dσ = dσ dg the (lebesgue) surface measure on y ∗ ∩ dg induced by the local patches p u := p : u → b[a′](r) on an unramified neighborhood u of a point a ∈ dg. 3 subharmonicity an upper semicontinuous map φ : y → [−∞,∞) is called (locally) semispherically submedian in y (φ ∈ sm sm.sph(y )), if there exists a thin analytic subset a of y such that: (i) at every z ∈ y \a, there is a pseudoball ∆ of radius r0 for which the inequality νp ∆ (z) φ(z) ≤ [φ⌋∆]z,r, 0 < r < r0, (3.1) holds; (ii) if φ admits an exceptional peak point, that is, a point z∗ ∈ a with φ(z∗) = supy φ, then φ(z∗) ≤ lim infn→∞ φ(an) for all sequences {an} in ∆\a converging to z∗. similarly, the set sm sm.sol(y ) of (locally) semisolidly submedian maps is defined by requiring (in place of (3.1)) the inequality νp ∆ (z) φ(z) ≤ 〈φ⌋∆〉z,r, 0 < r < r0, (3.2) to hold for some pseudoball ∆ = ∆z(r0). if in the preceding the set a can be taken to be the empty set, then φ is called spherically submedian (respectively, solidly submedian) in y. an upper semicontinuous map φ : y → [−∞,∞) is said to be weakly jensen if there exists a thin analytic subset a of y such that at each z ∈ y0(φ) := y \a, the inequality (3.2) holds with respect to some standard domain ∆, where, in the case of a pseudoball (respectively, pseudopolydisc), r0 denotes the radius (respectively, the polyradius) of ∆. (in view of proposition 4.1 and theorem 3.1) the following extension of [24, proposition 3.2] generalizes the maximum principle for subharmonic (and plurisubharmonic) functions: proposition 3.1. (maximum principle for weakly jensen functions) let d be a domain in y and φ : d → [−∞,∞) a weakly jensen function with finite supremum m. assume that (i) the peak set p := {z ∈ d |φ(z) = m} is nonempty; (ii) if no peak point of φ lies in d0 := d0(φ), then every sequence {an} in d0 converging to a point z0 ∈ p satisfies the inequality φ(z0) ≤ lim inf n→∞ φ(an). then φ = constant. 240 chia-chi tung cubo 12, 2 (2010) proof. let f := {z ∈ d |φ(z) < m}. assume at first that φ admits a peak point in d0. choose a standard domain ∆ ⊆ d0 at such a point a for which the inequality (3.2) holds. then the proof of [24, proposition 3.2] carries over here and yields the desired conclusion. now consider the case where φ has no peak point in d0. let z0 be any peak point of φ, and {an} a sequence in d0 converging to z0. choose a sequence of standard domains ∆[an](rn) (relative to a riemann covering pn ) with norm ‖rn‖ → 0 as n → ∞ such that the solid submean-value property (3.2) holds. suppose that f ∩b[z0](rk) 6= ∅ for every standard domain b[z0](rk). then φ(z) < m for all z in a neighborhood of each point in f ∩ b[z0](rk). since ∆[an](rn) ∩ b[z0](rk) 6= ∅ for sufficiently large n and an ∈ f, it follows from [23, proposition 5.2.2] and [24, (2.4)] that vol (∆(a′n,rn)) νpn (an) φ(an) ≤ ∫ ∆an (rn) φ(z) dṽ(z) < ∫ ∆an (rn) m dṽ(z) = m vol (∆(a′n,rn)) deg(pn ⌋∆). this inequality and the relation [24, (2.5)] imply that φ(z0) ≤ lim inf n→∞ φ(an) < m, hence a contradiction. therefore f ∩ b[z0](rk) = ∅ for some b[z0](rk). thus the set p is open and nonvoid. then by the connectedness of d one must have f = ∅. consequently φ(z) ≡ m in d. in the following, if v k is a branch of a standard domain ∆ at point a ∈ y, and φ ∈ l1(∆), denote by φ̃k a locally integrable function on ∆ ′ such that p∗φ̃k = φ on v k. since each branch v k contains a, the function φ̃k can be chosen to be continuous in ∆ ′, provided so is φ in ∆. lemma 3.1. let (x,p) be a riemann domain. assume that φ : x → [−∞,∞) is upper semicontinuous, and for each pseudoball u ⊂ x and every continuous function h : u → r with h ≥ φ⌋∂u such that h is semiharmonic in u, one has h ≥ φ⌋u. then νp(a) φ(a) ≤ [φ]a,r for all a ∈ x and all r ∈ (0,d(a)), (in particular, φ is spherically submedian). proof. let u be a pseudoball at a ∈ x, and r ∈ (0,d(a)). there exists, for each branch v k, k = 1, · · · , l, of u , a decreasing sequence of continuous functions φ̃nk on u ′ a′ [r] converging pointwise to φ̃k. the poisson integral hn := pa′,r(φ̃ n k ) is continuous and harmonic in u ′ a′ (r), and hn = φ̃ n k ≥ φ̃k on ∂u′a′ (r). by the subharmonicity of φ̃k and the formula [25, (4.16)], one has φ̃k(z ′) ≤ hn(z ′) = ∫ ∂u′[a′](r) pa′,r(z ′,ζ′) φ̃nk (ζ ′) dσ[a′],r, for each z′ ∈ u′a′ (r). hence by the monotone convergence theorem, φ̃k(z ′) ≤ lim n→∞ hn(z ′) = ∫ ∂u′[a′](r) pa′,r(z ′,ζ′) φ̃k(ζ ′) dσ[a′],r. in particular, by the definition (2.3), cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 241 φ̃k(a ′) ≤ [φ̃k]a′,r = ∫ ∂u′[a′](r) 1 r | s | r2 ‖a′ − ζ′‖2m φ̃k(ζ ′) dσ[a′],r. consequently φ satisfies the spherical submean-value inequality (3.1) for all r ∈ (0,d(a)). observe that for each ψ ∈ l1(b[a′][r0]), the fubini type formula holds: ∫ b[a′](r) ψ dυ[a′] = ∫ r 0 ( ∫ s[a′](t) ψ dσ[a′],t ) dt, 0 < r < r0. (3.3) lemma 3.2. let y, be a pure m-dimensional complex space and φ : y → [−∞,∞). then φ ∈ sh (y ) if and only if φ is locally subharmonic in y. proof. using the formula (3.3), the following preliminary assertion can be proved (in the same way as in [24, lemma 3.1]): for every pseudoball ∆ = ∆[a](r0) ⊂ y and for each φ ∈ l 1(∆), the inequality ”[φ⌋∆]a,r ≥ (constant) c for all r ∈ (0,r0)” implies that ”〈φ⌋∆〉a,r ≥ c for all r ∈ (0,r0)”. suppose that φ is locally subharmonic in y. let k ⊂ y be a compact subset and h : k → r a continuous function which is semiharmonic in int (k) and h(z) ≥ φ(z) for all z ∈ ∂k. it suffices to show that the difference ψ := φ − h satisfies the maximum principle on each connected component g of int (k). suppose now that supg ψ is attained at a point of g. by assumption, at each point a ∈ g there is an open neighborhood q such that φ ∈ sh (q). then by lemma 3.1 and the preliminary assertion, φ is solidly submedian in a neighborhood u of a ∈ q. thus by virtue of the gauss mean-value formula for semiharmonic functions one has νp(a) ψ(a) ≤ 〈ψ ⌋u〉a,r for all r ∈ (0,r0). therefore, by the maximum principle for solidly submedian functions (proposition 3.1), ψ = (constant) c in g, so that for every sequence {an} ⊂ g converging to a point a ∈ ∂g, one must have 0 ≥ (φ − h)(a) ≥ lim sup n→∞ (φ − h)(an) = c. consequently h(z) ≥ φ(z) for all z ∈ k, as desired. theorem 3.1. (1) sm sm.sph(y ) = sm sm.sol(y ) = sh (y ). (2) if φ ∈ sh (y ), then φ is solidly submedian (relative to any pseudoball) at every point of y. proof. the preliminary assertion in lemma 3.2 shows that sm sm.sph(y ) is a subset of sh sm.sol(y ). as in the proof of lemma 3.2, the inclusion ” sm sm.sol(y ) ⊆ sh (y )” is a consequence of proposition 3.1. finally the assertion (2) follows from lemma 3.1 and the preliminary assertion in lemma 3.2. remark 3.1. the second assertion of the above theorem gives a generalization of lemma 15.2 of [7] to complex spaces. also, it will be seen later (theorem 3.4) that if φ is only weakly subharmonic, then the solid submean-value inequality holds almost everywhere in y. corollary 3.1. if φ : y → c is semiharmonic, then there exists ψ ∈ c0(y ) ∩ sh (y ) such that ψ = φ almost everywhere in y. 242 chia-chi tung cubo 12, 2 (2010) proof. since φ is semiharmonic in any given pseudoball u ⊂ y, there exists, by [24, theorems 4.2], a continuous function ψ u in u such that ψ u = φ almost everywhere and ψ u has the solid mean-value property. hence theorem 3.1 implies that ψ u is subharmonic in u. the conclusion follows then from lemma 3.2. an upper semicontinuous function φ : y → [−∞,∞) is said to be spherically (respectively, solidly) monotone at z ∈ y if there exists a pseudoball u at z of radius r0 such that [φ⌋u]z,r ≤ [φ⌋u]z,s, if 0 < r ≤ s < r0, (3.4) (respectively, 〈φ⌋u〉z,r ≤ 〈φ⌋u〉z,s, if 0 < r ≤ s < r0). (3.5) let p u : u → ω be an unramified riemann covering at a point a ∈ yreg. the radial derivative of φ ∈ c1(u) at z ∈ dua(r)\p −1(a′) (for sufficiently small r > 0) is defined by j∗a,r(d cφ ∧ υm−1p u ) (z) = r2m−1 2 (rp u ,a φ)(z) σa, (3.6) where ja,r : dua(r) → x, denotes the inclusion (for more detail see [24, p. 567]). the spherical mean radial derivative [rp u ,z(φ)]z,r := ∫ du[z](r) rp u ,z(φ) σz, z ∈ u, exists for sufficiently small r > 0. it follows from the identity (3.6) (and [24, p. 571]) that d dρ [φ⌋u]z,ρ ] ρ=r = [rp u ,z(φ)]z,r, ∀z ∈ u, (3.7) for sufficiently small r > 0. for a real-valued φ ∈ c2(yreg) the spherical monotonicity at a point a ∈ yreg amounts to the condition that φ possesses (locally) nonnegative spherical mean radial derivative at a, that is, there is an unramified riemann covering p u at a such that [rp u ,a(φ)]a,r ≥ 0 for sufficiently small r > 0. let △p u denote the pull-back (under p u ) of the euclidean laplace operator on cm. by differentiating under the integral sign and using the divergence theorem, it is easy to show that d dρ [φ⌋b]a,ρ ] ρ=r = 1 | s |r2m−1 ∫ b[a](r) △pu φdυ[a],r. consequently one has, for each z ∈ u, [rp u ,z(φ)]z,r = r 2m 〈△p u φ〉z,r, (3.8) for sufficiently small r > 0. cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 243 lemma 3.3. if p u : u → ω is an unramified riemann covering and φ ∈ c2(u) is real-valued and solidly submedian, then △p u φ ≥ 0. proof. suppose that △p u φ < 0 in u[a](r0) for some a ∈ u and r0 > 0. by the relations (3.7)-(3.8) one has d dρ [φ⌋u]a,ρ ∣ ∣ ρ=r < 0, ∀r ∈ (0,r0). hence it follows from the relation (1.2) that νp u (a) φ(a) > 1 | s |ρ2m−1 ∫ dx[a](ρ) φdσ[a],ρ, ∀ρ ∈ (0,r0). (3.9) thus, multiplying (3.9) by| s |ρ2m−1 and integratingover (0,r), 0 < r < r0, yields |b(r)|νp u (a) φ(a) > ∫ x[a](r) φdυ̃, which implies that νp u (a) φ(a) > 〈φ⌊u〉a,r, a contradiction. therefore the lemma is proved. an upper semicontinuous map φ : y → [−∞,∞) is called nearly subharmonic at z ∈ y if there exists a pseudoball u at z of radius r0 such that 〈φ⌋u〉z,r ≤ [φ⌋u]z,r ∀r ∈ (0,r0); (3.10) φ is said to be seminearly subharmonic in y (φ ∈ sm sm.near(y ) if there exists at each a ∈ y an open neighborhood q and a thin analytic subset aq of q such that φ is nearly subharmonic at every point of q\aq. the class msm.sph(y ) (respectively, msm.sol(y )) of functions semispherically monotone (respectively, semisolidly monotone) in y is similarly defined. note that by lemma 3.1 and theorem 3.1-(2), if φ ∈ sh (y ), then φ is spherically monotone and nearly subharmonic relative to every pseudoball at any point of y. lemma 3.4. if φ is spherically monotone at a ∈ y (with u and r0 as in (3.4) for z = a), and φ is not identically equal to −∞ in u, then the function ”r 7→ [φ⌋u]a,r” is continuous on (0,r0). proof. since ∫ du[a](r) φdσ[a],r = sa ∑ k=1 ∫ dv k [a] (r) φdσ[a],r = sa ∑ k=1 ∫ s[a′](r) φ̂k dσ[a′],r, ∀r ∈ (0,r0), it suffices to prove that the function ”r 7→ [φ⌋v k]a,r” is continuous on (0,r0). let ψ denote the induced function of φ⌋v k on u′. there exists a monotonically decreasing sequence {ψ(µ)} of c∞ subharmonic functions (with increasing domains) converging pointwise on u′ to ψ. let ε > 0 and t ∈ (0,r0) be given. suppose that there exists a decreasing sequence {tn} converging to t such that [ψ ⌋u′]a′,t + ε ≤ [ψ ⌋u ′]a′,tn for each n. then, for a sufficiently large n, [ψ ⌋u′]a′,t + ε ≤ [ψ (µ) ⌋u′]a′,tn → n→∞ [ψ(µ)⌋u′]a′,t, 244 chia-chi tung cubo 12, 2 (2010) where the last limit relation follows from the stokes theorem. applying the monotone convergence theorem to the sequence {ψ(n ) − ψ(µ)} for a sufficiently large n, one obtains [ψ ⌋u′]a′,t + ε ≤ lim µ→∞ [ψ(µ)⌋u′]a′,t = [ψ ⌋u ′]a′,t, a contradiction. the conclusion now follows from the spherical monotonicity of φ. if u is a pseudoball at a0 ∈ y of radius r0, and φ ∈ l 1(u), applying the formula (3.3) to the restriction of φ to each branch of u, one obtains, for every r ∈ (0,r0), the relation |b(r)| 〈φ⌋u〉a0,r = ∫ r 0 [φ⌋u]a0,t |s|t 2m−1 dt. (3.11) theorem 3.2. sh (y ) ⊆ msph(y ) ⊆ msol(y ) = sm near(y ). proof. it will be first shown that, for a given φ ∈ cusc(y ), ”subharmonicity” implies ”spherical monotonicity”. let u be a pseudoball at a ∈ y of radius r0 such that φ is subharmonic in u. let wε and the smooth approximations φ̃k,ε of φ̃k (extended to be 0 outside wε), k = 1, · · · , l, be defined as in the proof of theorem 4.2 of [24]. it will be shown that each φ̃k,ε is solidly submedian in wε. let mb(r) := 1 |b(r)| χb(r) be the measure defined by a uniform unit mass distribution over the ball b(r). take z′ ∈ u′. there exists rz′ > 0 such that 〈φ̃k⌋u ′〉z′,r = 1 |b(r)| ∫ b[z′](r) φ̃k(y) dυ(y) = (m b(r) ∗ φ̃k) (z ′), 0 < r < rz′. thus for every z ∈ v j\{a}, φ̃k,ε(z ′) = (hε ∗ φ̃k) (z ′) ≤ hε ∗ (mb(r) ∗ φ̃k) (z ′), ∀r ∈ (0,rz′ ). hence (following an argument of [14, p. 205]) the inequality φ̃k,ε(z ′) ≤ (m b(r) ∗ φ̃k,ε) (z ′) = 〈φ̃k,ε⌋u ′〉z′,r holds at each z′ ∈ wε\{a ′} for sufficiently small r > 0. thus by lemma 3.3, △p u φ̃k,ε is nonnegative in wε\{a ′}, and therefore by the formulas (3.7)-(3.8), the function ”r 7→ [φ̃k,ε]z′,r” is increasing on (0,r0 − ε). moreover, for a fixed r ∈ (0,r0), [φ̃k⌋u ′]z′,r = lim ε→0 [φ̃k,ε⌋wε]z′,r, ∀z ′ ∈ u′. it follows from this that the function ”r 7→ [φ⌋u]z,r” is increasing on (0,r0) for every z ∈ u. thus φ is spherically monotone at every point of y. by approximating the last integral in the formula (3.11) by its riemann sums, it follows easily that the ”spherical monotonicity” of φ at a point z ∈ y implies its ”solid monotonicity” at the same point. to prove the assertion that φ is solidly monotone at a point z ∈ y if and only if it is nearly subharmonic at the same point, it may be assumed without loss of generality that φ is not identically cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 245 equal to −∞ in u. then by lemma 3.4, differentiation of the formula (3.11) yields for each z ∈ u and r ∈ (0,r0), r 2m d dρ 〈φ⌋u〉z,ρ ] ρ=r = [φ⌋u]z,r − 1 |b(r)| ∫ r 0 [φ⌋u]z,t t 2m−1 dt. therefore the desired equivalence follows. to prove that ”spherical monotonicity” at a point implies ”near subharmonicity” at the same point, suppose that (with q and u as above) the inequality (3.4) holds for each z ∈ q. then the relation (3.11) implies that |b(r)| 〈φ⌋u〉z,r ≤ [φ⌋u]z,r | s |r2m 2m , ∀r ∈ (0,r0), so that the inequality (3.13) holds, thus proving the desired claim. an upper semicontinuous function ψ : y → [−∞,∞) is called: (1) strongly regularized in y, if ψ satisfies both the conditions (1.1) and (1.2) for all pseudoballs ∆ at every point of y ; (2) a jensen function (ψ ∈ j (y )), if at each z ∈ y the inequality (3.2) holds for ψ with respect to every standard domain ∆ = ∆z (r0), where, in the case of a pseudoball (respectively, pseudopolydisc), r0 = dpd(a) (respectively, r0 = d(a)). observe that the expression [24, (3.9)] (where the proof, with slight modifications, remains valid in the case of a pseudopolydisc) shows that every continuous function ψ : y → r is strictly regularized in y. this property shall be extended to jensen (hence plurisubharmonic) functions. if ψ : y → [−∞,∞), define ψ(∗)(a) := lim sup z→a ψ(z), ∀a ∈ y. since every subharmonic function is weakly jensen, it follows from the maximum principle (proposition 3.1) that ψ = ψ(∗) for all ψ ∈ sh (y ) (compare [9, theorem j2(c)]). proposition 3.2. (1) if ψ ∈ sh (y ), then ψ is strongly regularized in y. in particular, if ψ ∈ sh (y ) is not identically zero, then the set {z ∈ y |ψ(z) 6= 0} is of positive measure at some point of y. (2) if ψ ∈ j (y ), then ψ is strictly regularized in y. proof. consider first the case ψ ∈ sh (y ). for any pseudoball ∆ at a point a ∈ y and a fixed small positive r, lim sup ε→0 〈ψ⌋∆〉a,ε ≤ lim sup ε→0 〈 sup z∈∆a(r)\{a} ψ(z)〉a,ε = lim ε→0 〈1⌋∆〉a,ε sup z∈∆a(r)\{a} ψ(z) = νp ∆ (a) sup z∈∆a(r)\{a} ψ(z). here (on the above right-hand side) the relation (3.9) of [24] is invoked. thus lim sup ε→0 〈ψ⌋∆〉a,ε ≤ νp ∆ (a) ψ(a). (3.12) 246 chia-chi tung cubo 12, 2 (2010) it follows then from the solid submean-value property of ψ that the limit relation (1.1) holds. the relation (1.2) can be similarly proved. if ψ ∈ j (y ), then the inequality (3.12) remains valid for any standard domain ∆ at each point a ∈ y, hence the limit relation (1.1) follows from the solid submean-value property (3.2) a maximum principle for continuous nearly subharmonic functions in cm was proved in [6, lemma 3.1, p. 5]. this result is generalized and strengthened below for later applications: proposition 3.3. (maximum principle for seminearly subharmonic functions) assume φ is a realvalued, continuous, seminearly subharmonic function in a domain d ⊆ y. if φ is bounded above and admits a peak point in d, then φ = constant in d. proof. let p := {z ∈ d |φ(z) = m} be the peak set of φ. there exists at every a ∈ d an open neighborhood q and a thin analytic subset aq of q such that φ is nearly subharmonic at every point of q\aq. let d̂ := ∪(q\aq). consider first the case c0 ∩ p 6= ∅ for some component c0 of d̂. let c0 be any such component and u a pseudoball of radius r0 at a point a0 ∈ c0 ∩ p such that the inequality 〈φ⌋u〉z,r ≤ [φ⌋u]z,r ∀r ∈ (0,r0), (3.13) holds for each z ∈ u. observe that the mapping r 7→ [φ⌋u]a0,r is continuous in r. suppose there exists t1 ∈ (0,r0) such that [φ⌋u]a0,t1 < m − 1 n (for n > 1 |m| ). following an idea of [6, p. 5 (last paragraph)], it will be shown that this assumption leads to a contradiction to the inequality (3.13). let ρ∗ be the infimum of all t ∈ (0, t1) such that the spherical means [φ⌋u]a0,t are bounded above by [φ⌋u]a0,t1. then ρ ∗ > 0, for, otherwise, there is a sequence {tµ} converging to 0 with [φ⌋u]a0,tµ ≤ [φ⌋u]a0,t1 for each µ, such that, by virtue of [24, proposition 3.1], deg (p u ) φ(a0) ≤ [φ⌋u]a0,t1 < m − 1 n . (3.14) hence a contradiction results. the fubini type formula (3.3) implies that |b(ρ∗)| 〈φ⌋u〉a0,ρ∗ = l ∑ j=1 ∫ v j [a0] (ρ∗) φdυ[a0] = ∫ ρ∗ 0 ( l ∑ j=1 ∫ s[a′ 0 ](t) φ̃j dσ[a′0],t ) dt = ∫ ρ∗ 0 [φ⌋u]a0,t |s|t 2m−1 dt > [φ⌋u]a0,t1 |s|(ρ∗)2m 2m . this leads to a contradiction 〈φ⌋u〉a0,ρ∗ > [φ⌋u]a0,ρ∗. therefore one must have [φ⌋u]a0,r ≥ m for every r ∈ (0,r0). then the formula (3.11) yields that 〈φ⌋u〉a0,r ≥ m for such r. since φ ≤ m, this implies that φ = m in a neighborhood of a0. thus c0 ⊆ p. consequently, by the continuity of φ, φ = m also in d. cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 247 now suppose that c0 ∩ p = ∅ for all components c0 of d̂. let z∗ be any point of p. then z∗ ∈ aq for every q (chosen as above). there exists a sequence {an} in q\(aq ∪ p) (for a fixed q) converging to z∗. choose a sequence of pseudoballs u[an](rn) such that φ is nearly submedian at each point of un = u[an](rn). then (as shown in the equation (3.14)), deg (p u ) φ(an) ≤ [φ⌋un]an,tn < m − 1 n , (where tn depends on an). thus by the continuity of φ, one has φ(z∗) < m, a contradiction! hence the preceding argument showing that φ = m in a neighborhood of a0 implies the same for an. consequently the sequence {an} lies in p, again a contradiction! therefore c0 ∩ p 6= ∅ for some component c0 of d̂, and thereby the proposition is proved. theorem 3.3. for a continuous function φ : y → r the following conditions are equivalent: (i) φ ∈ msm.sph(y ); (ii) φ ∈ msm.sol(y ); (iii) φ ∈ sm sm.near(y ); (iv) φ ∈ sh (y ). proof. in view of theorem 3.2, it suffices to prove that every element φ ∈ sm sm.near(y ) is subharmonic in y. let k ⊂ y be a compact set and h : k → r a continuous function such that h is semiharmonic in int (k) and h(z) ≥ φ(z) for every z ∈ ∂k. suppose that φ is seminearly subharmonic in y. then at each point a in a connected component g of int (k), there is an open neighborhood q and a thin analytic subset aq of q such that φ is nearly subharmonic at each point of q\aq. thus given z ∈ q\aq, there is a pseudoball u = uz(r0) in which the inequality (3.13) holds, and, consequently, 〈φ − h⌋u〉z,r ≤ [φ − h⌋u]z,r, ∀r ∈ (0,r0). thus by proposition 3.3, the function φ − h satisfies the maximum principle on g. it follows that φ − h ≤ 0 in int (k). this proves the subharmonicity of φ. a locally lipschitz function is semiharmonic if it has almost everywhere vanishing local radial derivative (this will be shown in a later work). with c2-differentiability, ”subharmonicity” betokens similarly the nonnegativity (off a thin analytic subset) of the local spherical mean radial derivative: proposition 3.4. (1) let φ ∈ c0(y ) ∩ c2(yreg) be real-valued. assume that for some thin analytic subset a of y there exists, at each a ∈ y \a, an unramified riemann covering p u : u → ω satisfying one of the following: (a) φ has (locally) nonnegative spherical mean radial derivative at each point of u; (b) △p u φ ≥ 0 in u. then φ is subharmonic in y. (2) if φ ∈ sh(y ) ∩ c2(yreg), then both the conditions (a) and (b) are valid for any unramified riemann covering (on an open set in yreg). proof. let p u : u → ω be an unramified riemann covering at a ∈ yreg. clearly for any φ ∈ c 2(u) the relation (3.8) implies that the conditions (a) and (b) are equivalent; moreover, by virtue of the formula (3.7) each of these implies that φ is spherically monotone in u. thus the first assertion follows from theorem 3.3. since each subharmonic function is solidly submedian, the second assertion is a consequence of the preceding and lemma 3.3. definition 3.1. a locally integrable function φ : y → [−∞,∞) is called weakly subharmonic in y (φ ∈ shw(y )) if there exists at each a ∈ y a standard domain ∆ = ∆a(r) such that for all nonnegative u ∈ c20 (∆), 248 chia-chi tung cubo 12, 2 (2010) ∫ ∆a(r) φddcu ∧ υm−1p u ≥ 0, ∀r ∈ (0,r). (3.15) observe that, if p : d → ω, is a riemann semicovering on an open set d ⊆ y and if φ ∈ shw(d), then the inequality (3.15) holds with d in place of ∆a(r) for all nonnegative u ∈ c 2 0 (d). theorem 3.4. (1) there exists a bijection between shw(y ) and sh {6≡−∞}(y ) taking each φ ∈ shw(y ) to a unique element φ̂ ∈ sh {6≡−∞}(y ) such that φ̂ = φ almost everywhere in y. (2) a function φ : y → [−∞,∞) is weakly subharmonic and regularized in y if and only if φ ∈ sh {6≡−∞}(y ). proof. suppose that φ ∈ shw(y ). let a ∈ y and φ̃k,ε be defined as in the proof of theorem 3.2 (for u = ∆ as in the above definition). the weak subharmonicity of φ implies that the induced function φ̃k ∈ shw(∆ ′), hence the function φ̃k,ε is weakly subharmonic in wε for every ε > 0 (by the proof of the assertion ”(1) implies (2)” in [24, theorem 4.2, p. 564]). suppose that the form ddcφ̃k,ε ∧ υ m−1 [a′] is negative at some z′ ∈ wε. then dd cφ̃k,ε ∧ υ m−1 [a′] < 0 in a neighborhood q ⊆ wε of z′, hence for all u ∈ c20 (q) with u ≥ 0 and u = 1 in a neighborhood of z ′, one has ∫ ∆′ φ̃k,ε dd cu ∧ υm−1 [a′] = ∫ ∆′ uddcφ̃k,ε ∧ υ m−1 [a′] < 0, a contradiction. thus △id φ̃k,ε ≥ 0 in wε, so that (by the proof of theorems 3.2) the function φ̃k,ε is spherically monotone, and for every z′ ∈ wε\{a ′}, φ̃k,ε(z ′) ≤ 〈φ̃k,ε〉z′,r (3.16) for sufficiently small r > 0. it follows from the argument of [13, p. 20] that the function ε 7→ φ̃k,ε(z ′) is nondecreasing for fixed z′ ∈ wε. define ψ ∆ (z) := min k lim ε→0 {φ̃k,ε(z ′) |v k ∋ z}, ∀z ∈ ∆. (3.17) then ψ ∆ is upper semicontinuous in ∆. moreover ψ ∆ (z) = φ(z) for almost every z ∈ ∆. also, the formula (3.17) implies that for each z ∈ ∆, ψ ∆ (z) ≤ min k {φ̃k,ε(z ′) |v k ∋ z} ≤ min k 1 |b(r)| ∫ b[z′](r) φ̃k,ε(y) dv(y), for sufficiently small positive ε and r, where the last inequality follows from (3.16). letting ε → 0 this inequality implies that νp(z) ψ∆ (z) ≤ 〈ψ∆〉z,r, cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 249 for sufficiently small r > 0. therefore by theorem 3.1, ψ ∆ ∈ sh {6≡−∞}(∆). it follows from proposition 3.2 that setting ψ := ψ ∆ on each ∆ (as above) defines a subharmonic function ψ : y → [−∞,∞) which coincids with φ almost everywhere in y. conversely, given φ̂ ∈ sh {6≡−∞}(y ), it will first be shown that φ̂ is locally integrable in y. for this purpose a preliminary observation is needed: if uz(r) ⊂ y, 0 < r < d(z), is a pseudoball at z and φ̂(z) > −∞, then φ̂ is integrable over uz(r). this is an immediate consequence of the local solid submean-value inequality for φ̂. let i be the set of all points of y where φ̂ is locally integrable. clearly i is an open set and since locally φ̂ 6≡ −∞, it follows from the preliminary observation that i is nonempty. on the other hand, if b ∈ y \i, the preliminary observation implies that φ̂ ≡ −∞ almost everywhere in some pseudoball ub(r); for, otherwise, φ̂(z) > −∞ for some point z ∈ ub(r) for each (sufficiently small) r > 0; hence there is a sequence {zn} ⊂ y converging to b such that φ̂(zn) > −∞ for every n. if 0 < r < d(b)/2 and n is sufficiently large, one has ‖zn − b‖ < r and φ̂ is locally integrable over uzn (r); but since b ∈ uzn (r), it follows that b ∈ i, a contradiction. thus ub(r) ⊂ y \i, so that y \i is open. applied to each component yµ of y, this implies that yµ\i = ∅, hence φ̂ is locally integrable in y. it remains to show that φ̂ satisfies the condition (3.15). let ∆ be a standard domain at a ∈ y and denote by φ̃k the function on ∆′ induced by φ̂⌋v k. then φ̃k is subharmonic, hence solidly submedian, in ∆′. it follows as in the proof of theorem 3.2 that the smooth approximations φ̃kε of φ̃k are solidly submedian in wε. consequently by lemma 3.3, △id φ̃ k ε ≥ 0 in w ′ ε (for a given open set ∆′′ ⋐ ∆′). then by the l1-convergence of φ̃kε to φ̃k, for each nonnegative g ∈ c 2 0 (∆ ′), ∫ ∆′ φ̃k ddcg ∧ υm−1 [a′] = lim ε→0 ∫ ∆′ g ddcφ̃kε ∧ υ m−1 [a′] ≥ 0. (3.18) let u ∈ c20 (∆) with u ≥ 0. since the function ũ k induced by u⌋v k has compact support in ∆′, there exists a sequence fn ∈ c 2 0 (∆ ′) such that fn → △id ũ k in l1(∆′). by solving the dirichlet problem ddc (w υm−1 [a′] ) = 1 4m fn υ m [a′] in ∆ ′, w = 0 on ∂dn, where dn is an open neighborhood of spt(fn) with smooth boundary, one obtains a sequence {wn} in c20 (∆ ′) such that △id wn tends to △id ũ k in l1(∆′). without loss of generality it may be assumed that each wn is nonnegative. then by the inequality (3.18), ∫ v k φ̂ddcu ∧ υm−1p = ∫ ∆′\{a′} φ̃k ddcũk ∧ υm−1 [a′] ≥ 0. the proof of the assertion (1) is thereby completed. the second assertion follows from the preceding proof and the regularity of a subharmonic function. 4 plurisubharmonicity let π̃ : cm\{0} → pm−1(c) denote the natural projection. the induced map π : s2m−1 → pm−1(c) defines a fiber bundle with fiber π−1([a]) = la := {ζa |ζ ∈ c, ‖ζ‖ = 1} over [a] ∈ p m−1(c), where 250 chia-chi tung cubo 12, 2 (2010) ‖a‖ = 1. let j(r) : s 2m−1 → cm be the dilation: j(r)(z) = rz. the fubini-study form ω̈ on p m−1(c) pulls back under π̃ to the projective form ω := ddc log ‖z‖2 on cm\{0}. thus π∗(ω̈m−1) = j∗(r)(π̃ ∗(ω̈m−1)) = j∗(r)(ω m−1). (4.1) an upper semicontinuous function φ : y → [−∞,∞) is said to be (locally) radially submedian (φ ∈ sm rad(y )) if there exists at every point a ∈ y a pseudoball u of radius r0 such that for all a ∈ s2m−1, νp u (a) φ(a) ≤ 1 2π 2π ∫ 0 ∑ k φ̃k(a ′ + reiθa) dθ, 0 < r < r0, (4.2) (where the sum ranges over all branches of u) for some r0 = r0(a, a) ≤ r0. an upper semicontinuous function φ : y → [−∞,∞) is said to be (locally) semiradially submedian (φ ∈ sm sm.rad(y )) if φ belongs, locally at every a ∈ y, to sm rad(q\aq) for some thin analytic subset aq in an open neighborhood q of a. a peak point of φ⌋q belonging to aq is called a local exceptional peak point. proposition 4.1. psh (y ) ⊆ sm rad(y ) ⊆ sh (y ). proof. let u be a pseudoball at a ∈ y. the first inclusion relation follows from the fact that given φ ∈ psh (y ), each induced function φ̃k is subharmonic at a ′ 0 when restricted to the complex line la = {z ∈ c m |z = a′0 + ζa, ζ ∈ c}, for each a ∈ s 2m−1. to prove the second inclusion relation, suppose that φ ∈ sm rad(y ). the expression (4.1) allows one to evaluate the spherical means of each φ̃k : u ′ → r by integrating along fibers of the bundle map π (as in [20, p. 202]). the detail is given below for completeness. without loss of generality assume that a′0 = 0. write z = e iθ a, a ∈ s2m−1, 0 ≤ θ ≤ 2π. let ia : la →֒ c m, ja : la →֒ la, ja : la →֒ s 2m−1 be the inclusion mappings. then j ∗ a j ∗ (r)(d c ln ‖z‖2) = j∗ a i ∗ a (dc ln ‖z‖2) = j∗ a (dc ln ‖ζ‖2) = 1 2π dθ. therefore for small r > 0 one has νp(a0) φ(a0) = ∫ [a]∈pm−1(c) νp(a0) φ(a0) ω̈ m−1 ≤ ∫ [a]∈pm−1(c) [ ∫ 2π 0 l ∑ j=1 φ̃j (a ′ 0 + r e iθ a) dθ 2π ] ω̈m−1 = l ∑ j=1 ∫ s2m−1 φ̃j (a ′ 0 + rz) j ∗ r (σ0) = l ∑ j=1 ∫ dv j [a0] (r) φσa0 = [φ⌋u]a0,r thus φ is spherically submedan in y therefore, by theorem 3.1, φ ∈ sh (y ). theorem 4.1. (1) if φ ∈ sm sm.rad(y ) and is continuous at every local exceptional peak point (if any), then φ is subharmonic. (2) if y is a normal space, then sm sm.rad(y ) = psh (y ). cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 251 proof. by proposition 4.1 and theorem 3.1, a function φ ∈ sm sm.rad(y ) that is continuous at every local exceptional peak point is semisolidly submedian, hence subharmonic, in y. to prove the second assertion, note that in the definition of the set sm sm.rad(y ), the local analytic set aq at a point z can be chosen (without loss of generality) so that q\aq ⊆ qreg. the desired conclusion follows then from the plurisubharmonicity test in cm ([26, p. 73]) and the extension theorem of grauert and remmert for plurisubharmonic functions on a normal complex space ([8, satz 3, p. 181]). a function φ : y → [0,∞) is called logarithmically subharmonic (respectively, logarithmically plurisubharmonic), if log φ ∈ sh (y ) (respectively, log φ ∈ psh (y ); for c2-functions in cm, another definition of logarithmic plurisubharmonicity is given in [3, p. 200]). if f = (f 1 , · · · ,f n ) : y → cn and c = (c1, · · · ,cn ), where each cj is an arbitrary positive constant, set ⌊f⌋c := ‖f1‖ c1 + · · · + ‖fn ‖ cn . if f is a holomorphic map, then the function ⌊f⌋c is logarithmically plurisubharmonic (see example 6.3 of the appendix); furthermore, log ⌊f⌋c is a jensen function, provided f has thin zero set. this is in fact a special case of the next proposition (which generalizes the classical jensen’s inequality): proposition 4.2. psh (y ){6≡−∞} ⊆ j (y ). proof. let φ ∈ psh {6≡−∞}(y ). by theorems 4.1 and 3.4, φ is subharmonic and locally integrable in y. the submean-value inequality relative to pseudoballs follows from theorem 3.1. suppose now that u = ∆a(r) ⋐ y is a pseudopolydisc of polyradius r. each induced function φ̃k is plurisubharmonic in p(u)\{a′}, hence also in p(u) (by [8, satz 3, p. 181]), so that it is subharmonic in each variable zj, 1 ≤ j ≤ m. repeated integration yields the inequality φ(a) ≤ vol(∆(a′,r))−1 ∫ ∆(a′,r) φ̃k(z ′) dv(z′). from this and the identity (2.1) the the submean-value inequality (3.2) follows. definition 4.1. a locally integrable function ψ : y → [−∞,∞) is said to be weakly plurisubharmonic in y (ψ ∈ pshw(y )) if at each point a ∈ y there exists a standard domain ∆ such that the levi form of ψ, l(ψ,λ), is positive semidefinite in ∆, that is, for some thin analytic subset a of ∆ (with ∆\a ⊆ ∆∗) and for all nonnegative u ∈ c20 (∆), 〈l(ψ,λ),u〉∆ := ∫ ∆\a ψ ∑ µ,ν ∂2u ∂pµ ∂p̄ν λµλ̄ν dṽ ≥ 0, ∀λ ∈ c m. (4.3) similarly, a locally integrable function ψ : y → [−∞,∞) is said to be weakly pluriharmonic in y (ψ ∈ phw(y )) if at each a ∈ y there exists a standard domain ∆ in which the above levi form l(ψ,λ) ≡ 0. theorem 4.2. (compare [13, theorem 3, p. 21] and [9, theorem k15]) (1) psh {6≡−∞}(y ) ⊆ pshw(y ). (2) assume that φ ∈ pshw(y ). then there exists a unique locally integrable, regularized function ψ : y → [−∞,∞) such that ψ = φ almost everywhere and ψ ∈ sm rad(y ) ∩ psh {6≡−∞}(yreg). 252 chia-chi tung cubo 12, 2 (2010) proof. let φ ∈ psh {6≡−∞}(y ). by theorems 4.1 and 3.4, φ is locally integrable in y. consider first the special case of a c2-functions φ : y → [−∞,∞). suppose that y is nonsingular. for any standard domain ∆ ⊂ y and α ∈ c2(∆), denote by α̃ the function on ∆′ induced by α and set l(α,λ)(z) := ∑ µ,ν ∂2α̃ ∂zµ ∂z̄ν (z′) λµλ̄ν, z ′ ∈ ∆′, λ ∈ cm. then φ ∈ psh (u) if and only if l(φ,λ)(z) ≥ 0 for every z ∈ ∆ and every λ ∈ cm; but that is clearly equivalent to the requirement that there exists a thin analytic subset a of ∆ such that for all nonnegative u ∈ c20 (∆), ∫ ∆\a ul(φ,λ) υmp ≥ 0. by [9, lemma k14], the latter condition is in turn equivalent to the requirement (4.3) (with d = ∆). if y is singular, then by using local desingularizations of y, it is easily seen that a c2-function φ : y → r belongs to psh (y ) precisely when it satisfies the condition (4.3) of weak plurisubharmonicity. thus the c2 case of the assertion (1) is established. for the general case, observe that each induced function φ̃k : ∆ ′ → r is plurisubharmonic in u′ := ∆′, hence so are the smooth approximations φ̃k,ε in wε. let u ∈ c 2 0 (∆) and denote by ũk the function induced by u⌋v k in ∆′. since the support of ũk lies in wε whenever ε is sufficiently small, and for such values ε it follows from the first part of the proof that ∫ ∆′ φ̃k,ε l(ũk,λ) dv(z) ≥ 0, λ ∈ c m. as ε → 0, the function φ̃k,ε converges to φ̃k in l 1-norm on the support of ũk. consequently φ satisfies the condition (4.3). thus the assertion (1) is established in the general case. now suppose that φ ∈ pshw(y ). let ∆ be a pseudoball at a ∈ y relative to which the condition (4.3) holds. define ψ∆ : ∆ → [−∞,∞) by νp ∆ (z) ψ∆(z) := lim r→0 ∑ {〈φ⌋v k〉z,r |v k ∋ z}, ∀z ∈ ∆. then ψ∆ is regularized in ∆ and ψ∆ = φ almost everywhere; moreover, the levi form l(ψ∆,λ) ≥ 0 in ∆. consider the smooth approximations ψ̃k,ε in wε of the induced function ψ̃k of ψk := ψ∆⌋v k. the second half of the proof of [9, theorem k15] shows that ψ̃k is plurisubharmonic in ∆ ′\{a′}, hence also in ∆′ (by grauert and remmert’s extension theorem [8, satz 3, p. 181]). it follows from proposition 3.2 that the function ψ given by ψ := ψ∆ on each ∆ is well-defined. also, ψ is locally integrable, radially submedian and regularized in y. since ψ∆ is plurisubharmonic in each open set ∆\{a}, so is the function ψ in yreg. remark 4.1. the above theorem and its proof (of the first assertion) imply that, if d ⊆ y is a normal, open subset and φ ∈ psh {6≡−∞}(d), then for any riemann semicovering p : d → ω, the inequality (4.3) remains valid: 〈l(φ,λ),u〉d ≥ 0 for all nonnegative u ∈ c 2 0 (d) and every λ ∈ c m. cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 253 corollary 4.1. let y be a normal space. a function φ : y → [−∞,∞) is weakly plurisubharmonic and regularized in y if and only if φ ∈ psh {6≡−∞}(y ). proof. suppose that φ ∈ pshw(y ) is regularized in y. then by theorem 4.2, φ is plurisubharmonic in yreg. the grauert and remmert’s extension theorem ([8, satz 3, p. 181]) asserts that φ⌋yreg admits a plurisubharmonic extension ψ to y, which, by subharmonicity, must coincide with φ (proposition 3.2). conversely, every element of psh {6≡−∞}(y ) is weakly plurisubharmonic and regularized in y by theorem 4.2-(1) and proposition 3.2. remark 4.2. (compare [9, corollary k18]) let y be a normal space. a function φ : y → [−∞,∞) is pluriharmonic if and only if φ is weakly pluriharmonic and regularized in y. 5 applications as a consequence of theorem 3.1, subharmonicity and plurisubharmonicity are closed under a basic sequential limiting process (lemma 6.1). by allowing to decrease the values of a subharmonic (respectively, plurisubharmonic) function on a null set, the failure of the preservation of subharmonicity (respectively, plurisubharmonicity) under more general limit operations can be remedied. a mapping φ : y → [−∞,∞) is called presubharmonic (φ ∈ sh[y ]) (respectively, preplurisubharmonic, (φ ∈ psh[y ])) if φ admits a subharmonic (respectively, plurisubharmonic) majorant φ̂ : y → [−∞,∞) such that φ = φ̂ almost everywhere. lemma 5.1. (compare [9, theorem l8]) a mapping φ : y → [−∞,∞) is presubharmonic (respectively, preplurisubharmonic) if and only if the upper envelop φ[∗](z) := max (φ(z),φ(∗)(z)), ∀z ∈ y, (5.1) is subharmonic (respectively, plurisubharmonic), with φ[∗] = φ almost everywhere, in y ; hence φ[∗] is the least such majorant of φ. proof. the sufficiency part of the lemma is trivial. suppose that φ admits a subharmonic majorant φ̂ which agrees with φ almost everywhere. then the same argument as in the proof of [9, theorem l8] shows that φ(z) ≤ φ[∗](z) ≤ φ̂[∗](z) = φ̂(z) for all z ∈ y and hence φ(z) = φ[∗](z) almost everywhere. on the other hand, if φ[∗](a) < φ̂(a) at some a ∈ y, then by the definition (5.1), there are constants ε > 0, δ > 0, such that supz∈∆a(ε) φ(z) = φ̂(a) − ε. but since φ̂ is subharmonic and φ̂ = φ almost everywhere in y, it follows from the regularity of φ̂ (proposition 3.2) that, for some standard domain ∆ at a, νp ∆ (a) φ̂(a) = lim r→0 〈φ⌋∆〉a,r ≤ νp ∆ (a) (φ̂(a) − ε), a contradiction, thus proving the desired claim. the remaining case of preplurisubharmonicity is entirely similar. thanks to corollary 4.1 and lemma 5.1, the preservation of preplurisubharmonicity on a complex space can be assured in reference to: (1) the pointwise limit of a monotonically decreasing sequence in 254 chia-chi tung cubo 12, 2 (2010) psh[y ]; (2) the supremum and lim sup of a sequence in psh[y ] that is locally uniformly bounded from above (as in [9, theorems l9]); and, for a normal space y, (3) the supremum of a family {φt}t∈t ⊂ psh[y ] that is locally uniformly bounded from above (as in [9, theorems l10]), and (4) the operation of forming ψt0 := lim supt→t0 φt, for {φt}t∈t ⊂ psh[y ] that is locally uniformly bounded from above (where t0 ∈ t and t is an open (or closed) subset of r ∪{∞} or c m). similar assertions hold on a general space for presubharmonic functions. an application of such preservation properties is given by the following generalization of the hartogs’ lemma: corollary 5.1. (hartogs’ lemma for presubharmonic functions) assume that {φ t } t>0 is a family in sh[y ] that is locally uniformly bounded from above. then (1) the function g := lim supt→∞ φt is presubharmonic in y ; (2) if for some g ∈ c0(y ), g ≤ g in y, then for every compact set k ⊂ y and every positive ε, there is a positive number n such that supk φt < g + ε for all t ≥ n. proof. let ft := sups≥t φs. then g = limt→∞ ft. by the preceding remark, both ft and g are presubharmonic in y, (alternatively, this can be proved using theorem 3.1 and lemma 6.2). since {ft} is a decreasing sequence, so is also the sequence {f [∗] t }, hence the limit function limt→∞ f [∗] t is subharmonic. it follows from lemma 5.1 that g[∗] is equal to limt→∞ f [∗] t outside a set of measure zero, and hence by proposition 3.2, everywhere in y. the second assertion follows by showing that the nested family et := {z ∈ k |f [∗] t (z) − g(z) ≥ ε}, for t > 0, has an empty intersection (by the same argument as in [12, p. 93]). in a similar vein it is of interest to see if the theorem of lelong-norguet-bremermann [13, p. 54] on the representation of plurisubharmonic functions remains valid for a p -convex domain d in a general complex space, where p = psh (d), that is, whether each ψ ∈ psh (d) admits an representation ψ = ( lim sup k→∞ 1 k log ||fk|| )[∗] , for some sequence of holomorphic functions fk in d. let p : y → ω be a riemann semicovering and g ⋐ y a weak stokes domain ([24, p. 568]). the (real) energy of a function τ ∈ c1(g) is defined by e g (τ) := ∫ g dτ ∧ dcτ̄ ∧ υm−1p ([25, § 3]). if τ ∈ c1,1(g) ([24, p. 562]) is real-valued, then its energy is given by e g (τ) = ∫ dg τ dcτ ∧ υm−1p − ∫ g τ ddcτ ∧ υm−1p . (5.2) suppose now that p : y → ω is an unramified riemann covering. by the expression [24, (5.7)] for the normal derivatives one has ∫ dg τ,dcτ ∧ υm−1p = (−1) m(m−1) 2 1 2‖s‖ ∫ dg τ ∂ντ dσdg. (5.3) cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 255 also, since the energy e g (τ) is never negative [24, (3.7)], the expression (5.2) (applied to the pseudoannular region g = ua(r)\ua(s), where 0 < s < r < r0) and the identity (5.6) of [24] imply that the spherical mean [τ rp,a(τ)]a,r of a subharmonic c 2-function τ : y → [0,∞) is nonnegative and increasing in r (for small r > 0). if φ = (φ 1 , · · · ,φ n ) : y → cn, set τρ := ‖φ‖ 2ρ for each positive constant ρ. a mapping φ : y → cn is called subharmonic, respectively, semiharmonic, if so are the real and imaginary parts of each component of φ (regarded as an r-valued function); φ is called weakly real-analytic if each component φ j is continuous in y and real-analytic in yreg. kellog ([11, theorem iv, p. 213]) showed that a real-valued harmonic function on a closed, regular plane region is nonconstant unless it has vanishing normal derivative at every boundary point. the relations (5.2) and (5.3) lead to a generalization of this result to subharmonic mappings: proposition 5.1. (1) assume that g ⋐ y is a stokes domain and φ : g → cn, with components φj ∈ c 1,1(g) ∩ c2(g), is subharmonic in g. if for some constant ρ ≥ 1 2 , ∫ dg τρ d cτρ ∧ υ m−1 p = 0, (5.4) then φ = constant in g. (2) assume that y is a connected complex space and φ : y → cn a weakly real-analytic subharmonic map. then φ is a constant if and only if there exist a point a ∈ yreg and ρ ≥ 1 2 such that the function [τρ rp,a(τρ)]a,r is monotonically decreasing in r (for small r > 0) relative to an unramified riemann covering p at a. proof. by example 6.1 of the appendix and lemma 3.2, the function τρ = ‖φ‖ 2ρ is subharmonic in g for all ρ ≥ 1 2 . (1) observe that ddc‖φj‖ 2 = φ̄jdd cφj + φjdd cφ̄j + dφj ∧ d cφ̄j − d cφj ∧ dφ̄j (5.5) almost everywhere in g. writing φj = uj + ivj (in terms of its real and imaginary parts), one has ‖φ‖2 = ∑n j=1(‖uj‖ 2 + ‖vj‖ 2). thus the mapping φ̃ := (|u1|, |v1|, · · · , |un |, |vn |) : g → r 2n is continuous in g and subharmonic in g with ‖φ̃‖2 = ‖φ‖2. therefore it can be assumed without loss of generality that each φj is real-valued. then the expression (5.5) implies that ddcτ1 ∧ υ m−1 p ≥ 2 n ∑ j=1 (duj ∧ d cuj + dvj ∧ d cvj ) ∧ υ m−1 p ≥ 0 (5.6) almost everywhere in g. to prove the assertion (1), observe that by the condition (5.4) and proposition 3.4, the energy (5.2) (for τρ) is nonpositive. since the integral eg (τρ) is always nonnegative, the norm of the gradient, ‖ ▽ τρ‖, must vanish locally almost everywhere in g. consequently τρ, hence also τ1, is constant in g. hence the relation (5.6) implies that φ = constant in g. for the assertion (2), assume that relative to an unramified riemann covering p at a point a ∈ yreg, the function [τρ rp,a(τρ)]a,r is decreasing in r for some ρ ≥ 1 2 . then by the expression (5.2), the energy e d (τρ) 256 chia-chi tung cubo 12, 2 (2010) for the pseudoannular region g := ua(r)\ua(s), 0 < s < r < r0, is nonpositive. it follows that ‖ ▽ τρ‖ 2 = 0 in g. therefore τρ is constant in every component of g\z, where z is the zero set of f = ‖φ‖2. since each component φj of φ is real analytic in y off a thin analytic subset, the same is true for the function f; hence (choosing a to be a point where f is real analytic and r0 sufficiently small) the closure of every component of g\z must contain a ([16, lemma 1, p. 96]). this implies that τρ is constant in g; in particular, so is τ1. it follows as above from the relation (5.6) that φ = constant in g. hence by the identity theorem for real-analytic functions, φ = constant in y . corollary 5.2. let y be a connected complex space and φ = (φ 1 , · · · ,φ n ) : y → cn a weakly real-analytic and subharmonic map. if ‖φ‖ attains a local maximum at some point of y, then φ = constant. proof. observe that the function τ 1 = ‖φ‖2 : y → r is solidly submedian in y. assume that ‖φ‖ attains a local maximum at a point z0 ∈ y. hence proposition 3.1 (applied to a riemann covering p u of a small open, connected neighborhood u of z0) implies that τ1 is constant in u. thus τ1 has vanishing mean spherical radial derivative at some point of u, and by proposition 5.1-(2), φ = constant in u. since each φj is real analytic in yreg, φj = constant in yreg by the identity theorem for real-analytic functions, whence the conclusion follows. by means of the solid submean-value property of subharmonic functions on a general space, two results of bochner and montgomery on matrix-valued mappings [2, theorems 10 and 11, p. 155] can be extended: proposition 5.2. assume that y is a connected complex space. (1) if φ = (φ 1 , · · · ,φ kn ) : y → rkn is continuous, subharmonic such that the image of each map g j := (φ (j−1)n +1 , · · · ,φ jn ) : y → rn, 1 ≤ j ≤ k, is contained in the unit sphere in rn, then φ = constant. (2) every continuous mapping φ : y → u(n) (a unitary group) with semiharmonic components is a constant. 6 appendix lemma 6.1. if {φn} is a monotonically decreasing sequence of subharmonic (respectively, plurisubharmonic) functions in a complex space y, then the limit function lim φn is subharmonic (respectively, plurisubharmonic) in y. lemma 6.2. assume that (i) y is locally irreducible; (ii) ψ : y → [−∞,∞) is locally integrable and locally bounded from above; and (iii) ψ is solidly submedian with respect to every pseudoball in y. then ψ is presubharmonic in y. proof. since ψ is locally bounded from above, it can be proved in the same way as in [9, lemma l6-(c)] that ψ[∗] is upper semicontinuous in y. let ∆ be a pseudoball at at point z ∈ y. then either ψ[∗](z) = ψ(z), hence the solid submean-value inequality holds at z, or there exists a sequence {ζn} in ∆\{z} tendng to z such that ψ [∗](z) = limn→∞ ψ(ζn); in the latter case one has ψ[∗](z) ≤ lim inf n→∞ 〈ψ ⌋∆〉ζn ,r ≤ lim inf n→∞ 〈ψ[∗] ⌋∆〉ζn,r = 〈ψ [∗] ⌋∆〉z,r, cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 257 for sufficiently small r > 0. thus ψ[∗] is subharmonic in y by theorem 3.1. also, ψ[∗](z) ≤ lim inf n→∞ 〈ψ ⌋∆〉ζn,r = 〈ψ ⌋∆〉z,r. it follows then from the regularity of ψ[∗] that lim r→0 〈ψ[∗] ⌋∆〉z,r = lim r→0 〈ψ ⌋∆〉z,r. consequently ψ[∗] = ψ almost everywhere in y. lemma 6.3. (compare [17, 3.13]) let g : i → r be an increasing function on a finite (or infinite) interval i ⊆ [−∞,−∞) such that g⌋i ∩ r is convex. assume that φ ∈ sh (y ) (respectively, φ ∈ psh (y )) with φ(y ) ⊂ i. then g(φ) ∈ sh (y ) (respectively, g(φ) ∈ psh (y )). proof. suppose that φ ∈ sh (y ). let ∆ be a pseudoball at a point a ∈ y of radius r0. by theorem 3.1, one has φ(a) ≤ 〈φ⌋∆〉a,r for every r ∈ (0,r0). note that, since g(φ) = g +(φ) − g−(φ), the function g(φ) is locally integrable on y. using the supporting line argument of [19, p. 115], it can be shown that the jensen’s inequality for convex functions holds for the pair (g,φ): g(〈φ⌋∆〉a,r) ≤ 〈g ◦ φ⌋∆〉a,r, ∀r ∈ (0,r0). therefore, by theorem 3.1, the composition g ◦ φ is subharmonic in y. the corresponding assertion for φ ∈ psh (y ) follows then from the definition of plurisubharmonicity. example 6.1. assume that φ = (φ 1 , · · · ,φ n ) : y → cn is a subharmonic map with components φ j ∈ c2(y \a) for some thin analytic subset a of y. then for all constants c, ρ with c > 1, cρ ≥ 1, the function (⌊φ⌋ĉ)ρ ∈ sh (y ) (where ĉ = (c, · · · ,c)). proof. it is a consequence of [17, 3.23, p. 19], proposition 3.4 and lemma 3.2 that the function g := (⌊φ⌋ĉ)1/c is subharmonic in y, hence so is gcρ = (⌊φ⌋ĉ)ρ by lemma 6.3, provided cρ ≥ 1. lemma 6.4. (compare [17, 2.13]) assume that φ : y → [0,∞) is upper semicontinuous and φ 6≡ 0. then φ is logarithmically subharmonic in y if and only if φeh ∈ sh(d) for every subdomain d ⋐ y and every continuous function h : d → r which is semiharmonic in d. in particular, if φ is logarithmically subharmonic in y, then φ ∈ sh (y ). the converse is false. proof. by virtue of proposition 3.1 and lemmas 3.2 and 6.3, the desired conclusion follows from the same argument as in [17, 2.13]. example 6.2. if φk : y → [0,∞) is upper semicontinuous and logarithmically subharmonic for 1 ≤ k ≤ l then so are ∑l k=1 φk and max1≤k≤l φk. example 6.3. (compare [4, p. 119]) let fj = (fj1, · · · ,fjn ) : y → c n be a holomorphic map for each 1 ≤ j ≤ s. then for all constants ρ > 0 and c(j) = (cj1, · · · ,cjn ) with cjk > 0, the functions (⌊fj⌋ c(j) )ρ and max 1≤j≤s ⌊fj⌋ c(j) are logarithmically plurisubharmonic in y. 258 chia-chi tung cubo 12, 2 (2010) remark 6.1. if φ : y → [0,∞) is logarithmically plurisubharmonic, it can be shown as in lemma 6.4 that, for every subdomain d ⋐ y and every continuous function h : d → r which is semiharmonic in d, one has φeh ∈ psh (d) (in particular, φ ∈ psh (y )). in the converse direction, using theorem 4.1 the following lemma can be proved (as in [12, theorem 2.6.1]): lemma 6.5. (compare also [3, proposition vi.4.9] and, respectively, [17, 3.12]) let (x,p) be a normal semi-riemann domain. assume that φ : x → [0,∞) is upper semicontinuous and locally φ 6≡ 0. then φ is logarithmically plurisubharmonic (respectively, logarithmically subharmonic) in x if and only if φ |e p ckpk | ∈ psh (x) (respectively, φ |e p ckpk | ∈ sh (x)) for all constants (c1, · · · ,cm) ∈ c m. received: january 2009. revised: may 2009. references [1] h. behnke and h. grauert, analysis in non-compact complex spaces,analytic functions (h. behnke and h. grauert,ed.), princeton univ. press, princeton, new jersey, 1960. [2] s. bochner and w. martin, several complex variables, princeton university press, princeton, new jersey, 1948. [3] d’angelo, john p., inequalities from complex analysis, the carus math. mono. number 28,math. assoc. amer., washington dc, 2002. [4] demailly, jean-pierre, courants positifs et théorie de l’intersection, gaz. des math. no. 53 (1992), 131-159. [5] j. fornæss and r. narasimhan, the levi problem on complex spaces with singularities, math. ann. 248 (1980), 47-72. [6] p. freitas and j. p. matos, characterization of harmonic and subharmonicfunctions via meanvalue properties, arxiv:math/010607801v1 (2001), 1-10. [7] d. gilbarg and n. s. trudinger, elliptic partial differential equations of second order, springer-verlag, berlin-heidelberg-new york, 1977. [8] h. grauert and r. remmert, plurisubharmonischen funktionen in komplexen räume, math. zeit. 65 (1956), 175-194. [9] r. c. gunning, introduction to holomorphic functions of several variables, vol. i: function theory, wadsworth, belmont, california, 1990. [10] l. hörmander, an introduction to complex analysis in several variables, d. van nostrand, princeton, new jersey, 1966. [11] w. kellog, foundations of potential theory, dover, new york, 1953. [12] p. lelong, fonctionnelles analytiques et fonctions entièrs (n variables), l’univ. de montréal, 1968. cubo 12, 2 (2010) on semisubmedian functions and weak plurisubharmonicity 259 [13] p. lelong, plurisubharmonic functions and positive differential forms, gordon and breach, new york, london, paris, 1969. [14] e. h. lieb and m. loss, analysis,graduate studies in math. vol. 14, amer.math. soc., 1996. [15] r. narasimhan, the levi problem for complex spaces, math. ann. 142 (1961), 355-362. [16] r. narasimhan, introduction to the theory of analytic spaces, lecture notes in math. 25, springer, berlin-heidelberg-new york, 1966. [17] t. radó, subharmonic functions, ergebnisse der mathematik und ihrer grenzgebiete, springer, berlin, 1937. [18] t. radó and p. reichelderfer, continuous transformations in analysis, springer-verlag, berlin-heidelberg, 1955. [19] h. l. royden, real analysis, third edition, macmillan, new york, london, 1988. [20] b. v. shabat, introduction to complex analysis part ii, amer. math. soc., 1992. [21] w. stoll, the multiplicity of aholomorphic map, invent. math. 2,(1966), 15-58. [22] w. stoll, value distribution on parabolicspaces, lecture notes in math. 600, springer,berlinheidelberg-new york, 1977. [23] c. tung, the first main theoremof value distribution on complex spaces, memoiredell’accademia nazionale dei lincei, serie viii,vol. xv, (1979), sez.1, 91-263. [24] c. tung, semi-harmonicity, integral means and euler type vector fields, advances in applied clifford algebras, 17 (2007), 555-573. [25] c. tung, integral products, bochner-martinelli transforms and applications, taiwanese journal of mathematics, 13 no. 5 (2009), 1583-1608. [26] v. vladmirov, methods of the theory of functions of many complex variables, the mit press, cambridge and london, 1966. argyros-hilout-625.dvi cubo a mathematical journal vol.12, no¯ 01, (161–174). march 2010 convergence conditions for the secant method ioannis k. argyros department of mathematics sciences, lawton, ok 73505, usa email : iargyros@cameron.edu and saïd hilout poitiers university, laboratoire de mathématiques et applications, 86962 futuroscope chasseneuil cedex, france email : said.hilout@math.univ–poitiers.fr abstract we provide new sufficient convergence conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a banach space. our new idea uses recurrent functions, lipschitz–type and center–lipschitz–type instead of just lipschitz–type conditions on the divided difference of the operator involved. it turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. numerical examples are also provided in this study. resumen son dadas nuevas condiciones suficientes para la convergencia del método de la secante para una solución localmente única de una ecuación no lineal en un espacio de banach. estas ideas nuevas usan funciones recurrentes, tipo-lipschitz y tipo centro-lipschitz sobre la diferencia dividida de los operadores envolvidos. resulta que esta manera las cotas de errores son mas 162 ioannis k. argyros and saïd hilout cubo 12, 1 (2010) precisas que las anteriores y bajo nuestras hipótesis de convergencia nosotros podemos cubrir casos donde las condiciones previas eran violadas. ejemplos numéricos son dados en este estudio. key words and phrases: secant method, banach space, majorizing sequence, divided difference, fréchet–derivative. math. subj. class.: 65h10, 65b05, 65g99, 65n30, 47h17, 49m15. 1 introduction in this study we are concerned with the problem of approximating a locally unique solution x⋆ of equation f (x) = 0, (1.1) where f is a fréchet–differentiable operator defined on a convex subset d of a banach space x with values in a banach space y. a large number of problems in applied mathematics and also in engineering are solved by finding the solutions of certain equations. for example, dynamic systems are mathematically modeled by difference or differential equations, and their solutions usually represent the states of the systems. for the sake of simplicity, assume that a time–invariant system is driven by the equation ẋ = t (x), for some suitable operator t , where x is the state. then the equilibrium states are determined by solving equation (1.1). similar equations are used in the case of discrete systems. the unknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). except in special cases, the most commonly used solution methods are iterative–when starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. iteration methods are also applied for solving optimization problems. in such cases, the iteration sequences converge to an optimal solution of the problem at hand. since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework. we consider the secant method in the form xn+1 = xn − δf (xn−1, xn)−1 f (xn) (n ≥ 0), (x−1, x0 ∈ d) (1.2) where δf (x, y) ∈ l(x , y) (x, y ∈ d) is a consistent approximation of the fréchet–derivative of f [5], [13]. bosarge and falb [7], dennis [9], potra [16], argyros [1]–[5], gutiérrez [10] and others [11], [15], [18], have provided sufficient convergence conditions for the secant method based on lipschitz–type conditions on δf . cubo 12, 1 (2010) convergence conditions for the secant method 163 the sufficient convergence condition for the secant method used in most references is given ℓ c + 2 √ ℓ η ≤ 1, (1.3) where, ℓ, c, η are non–negative parameters to be precised later. this hypothesis is easily violated. indeed, let ℓ = 1, η = .18, and c = .185. then, (1.3) does not holds, since ℓ c + 2 √ ℓ η = 1.033528137. hence, there is not guarantee that an equation using the information (ℓ, c, η) has a solution that can be found using secant method (1.2). in this study we are motived by optimization considerations, and the above observation. here, using recurrent functions, lipschitz–type and center–lipschitz–type conditions, we provide a semilocal convergence analysis for (1.2). it turns out that our error bounds are more precise and our convergence conditions hold in cases where the corresponding hypotheses mentioned in earlier references mentioned above are violated. newton’s method is also examined as a special case. numerical examples are also provided in this study. 2 semilocal convergence analysis of the secant method we need the following result on majorizing sequences for the secant method (1.2). lemma 2.1. let ℓ0 > 0, ℓ > 0, c > 0, and η ∈ [0, c] be given parameters. assume: ℓ0 (c + η) < 1. (2.4) set δ0 = ℓ (c + η) c (1 − ℓ0 (c + η)) . (2.5) moreover, assume: there exists δ ∈ [ max{ℓ, δ0}, 1 c ) , such that estimate ℓ q + ℓ0 ( 2 q1/ √ 5 1 − q(p−1)/ √ 5 + q2 + q ) + ℓ − δ ≤ 0 (2.6) is satisfied for q = δ c, and p = 1 + √ 5 2 . then, scalar sequence {tn} (n ≥ −1) given by t−1 = 0, t0 = c, t1 = c + η, tn+2 = tn+1 + ℓ (tn+1 − tn−1) (tn+1 − tn) 1 − ℓ0 (tn+1 − t0 + tn) (2.7) 164 ioannis k. argyros and saïd hilout cubo 12, 1 (2010) is non–decreasing, bounded above by t⋆⋆ = c q ∞ ∑ n=1 qun , (2.8) and converges to some t⋆ ∈ [0, t⋆⋆], where {un} is the fibonacci’s sequence: u−1 = u0 = 1, un+1 = un + un−1, n ≥ 1. moreover, the following a priori estimates hold: 0 ≤ t⋆ − tn ≤ c (q1/ √ 5)p n q ( 1 − qpn (p−1)/ √ 5 ) (n ≥ 0). (2.9) proof. we shall show using induction on k: ℓ (tk+1 − tk−1) 1 − ℓ0 (tk+1 − t0 + tk) ≤ δ (tk − tk−1), (2.10) 0 ≤ tk+1 − tk ≤ tk − tk−1, (2.11) and tk+1 − tk ≤ quk−1 c. (2.12) note that if (2.10) holds, then by (2.7), we have: tk+2 − tk+1 = ℓ (tk+1 − tk−1) (tk+1 − tk) 1 − ℓ0 (tk+1 − t0 + tk) ≤ δ (tk − tk−1) (tk+1 − tk). (2.13) estimate (2.10) can also be written as: tk+1 − tk ≤ αk (tk − tk−1), (2.14) where, αk = 1 ℓ ( δ (1 − ℓ0 (tk+1 − t0 + tk)) − ℓ ) . (2.15) for k = 0, (2.14) becomes η ≤ 1 ℓ ( δ (1 − ℓ0 (c + η)) − ℓ ) c, or δ ≥ δ0, which is true by the choice of δ. estimates (2.11) and (2.12) also hold for k = 0 by the initial conditions. cubo 12, 1 (2010) convergence conditions for the secant method 165 assume that (2.10) (i.e., (2.13)), (2.11), and (2.12) hold for all k ≤ n. by the induction hypotheses we have 0 ≤ tk+2 − tk+1 ≤ δ quk−1−1 c (tk+1 − tk) = quk−1 (tk+1 − tk) < tk+1 − tk. (2.16) that is (2.11) holds for n = k + 1. on the other hand: 0 ≤ tk+2 − tk+1 ≤ quk−1 (tk+1 − tk) ≤ quk−1 quk c = quk+1 c, (2.17) which shows (2.12) for n = k + 1. we also have the estimate 0 ≤ tk+2 − t0 ≤ (t1 − t0) + (t2 − t1) + · · · + (tk+2 − tk+1) ≤ c q (qu0 + qu1 + · · · + quk ) < t⋆. (2.18) clearly, we have: uk = 1√ 5 [( 1 + √ 5 2 )k+1 − ( 1 − √ 5 2 )k+1] ≥ 1√ 5 ( 1 + √ 5 2 )k = pk√ 5 , (2.19) so, for any k ≥ 0, m ≥ 0, we get in turn: 0 ≤ tk+m − tk ≤ (tk − tk+1) + (tk+1 − tk+2) + · · · + (tk+m − tk+m−1) ≤ c q (quk + quk+1 + · · · + quk+m−1 ) ≤ c q (qp k/ √ 5 + qp k+1/ √ 5 + · · · + qp k+m−1/ √ 5). (2.20) using bernoulli’s inequality, we obtain: tk+m − tk ≤ c q qp k/ √ 5 (1 + q(p k+1−pk )/ √ 5 + q(p k+2−pk )/ √ 5 · · · + q(p k+m−1−pk )/ √ 5) = c q qp k/ √ 5 (1 + qp k (p−1)/ √ 5 + qp k (p2−1)/ √ 5 · · · + qp k (pm−1−1)/ √ 5) ≤ c q qp k/ √ 5 (1 + qp k (p−1)/ √ 5 + qp k (1+2 (p−1)−1)/ √ 5 · · · + qp k (pm−1−1)/ √ 5) = c q qp k/ √ 5 ( 1 + qp k (p−1)/ √ 5 + ( qp k (p−1)/ √ 5 )2 · · · + ( qp k (p−1)/ √ 5 )m−1) = c q qp k/ √ 5 1 − qp k (p−1) m/ √ 5 1 − qpk (p−1)/ √ 5. (2.21) in particular, from (2.18) we have: 166 ioannis k. argyros and saïd hilout cubo 12, 1 (2010) tk+m ≤ c q (qu0 + · · · + quk+m−1 ) + c (2.22) in order for us to show (2.10), it suffices: δ (tk − tk−1) ≤ αk+1 (2.23) or δ (tk − tk−1) ≤ 1 ℓ ( δ (1 − ℓ0 (tk+2 − t0 + tk+1)) − ℓ ) or ℓ δ quk−1−1 + δ ℓ0 (tk+2 + tk+1 − t0) ≤ δ − ℓ or ℓ δ quk−1−1 c + δ ℓ0 ( c q (qu0 + qu1 + · · · + quk+1 ) + c+ c q (qu0 + qu1 + · · · + quk ) + c − c ) ≤ δ − ℓ or ℓ δ quk−1−1 c + ℓ0 ( 2 (qu0 + qu1 + · · · + quk ) + quk+1 + δ c ) ≤ δ − ℓ or ℓ quk−1 + ℓ0 ( 2 (qu0 + qu1 + · · · + quk ) + quk+1 + q ) ≤ δ − ℓ (2.24) or ℓ q + ℓ0 ( 2 q1/ √ 5 1 − q(p−1)/ √ 5 + q2 + q ) + ℓ − δ ≤ 0, which is true by the choice of δ and q given by (2.6). the induction for (2.10)–(2.12) is now complete. it follows from (2.21) that scalar sequence {tn} is cauchy in the complete space r, and as such it converges to some t⋆ ∈ [0, t⋆⋆]. by letting m −→ 0 in (2.21), we obtain (2.9). that completes the proof of lemma 2.1. ♦ we shall also provide another result where condition (2.6) is dropped from the hypotheses of lemma 2.1: lemma 2.2. let ℓ0 > 0, ℓ > 0, c > 0, and η ∈ (0, c] be given parameters. assume: ℓ0 (c + η) < 1; cubo 12, 1 (2010) convergence conditions for the secant method 167 there exists δ ∈ ( max{ℓ + 2 ℓ0, δ0, δ1}, min{ 1 c , δ∞} ) , (2.25) and, v1 ≥ δ1, (2.26) where, δ1 = q1 c , is such that q1 is the unique zero in (0, 1) of equation q(t) = ℓ0 t 4 + ℓ0 t 2 + ℓ t − ℓ = 0, δ∞ = q∞ c , is such that q∞ is the unique positive zero of equation f∞(t) = ℓ0 ( 2 t1/ √ 5 1 − t(p−1)/ √ 5 + t ) + ℓ − δ = 0, and v1 is the unique positive zero of equation f0(t) = ℓ0 t 2 + (3 ℓ0 + ℓ) t + 2 ℓ0 + ℓ − δ = 0; and ℓ0 t 2 uk + ℓ0 t uk + ℓ tuk−2 − ℓ ≥ 0 (k ≥ 1) for t ≥ δ1. (2.27) then, the conclusions of lemma 2.1 hold true. proof. we follow the proof of lemma 2.1 until estimate (2.23). then, let us define functions fk, gk by fk(t) = ℓ t uk−1 + ℓ0 ( 2 (1 + tu1 + · · · + tuk ) + tuk+1 + t ) + ℓ − δ, (2.28) and gk(t) = gk(t) t uk−1 , (2.29) where, gk(t) = ℓ0 t 2 uk + ℓ0 t uk + ℓ tuk−2 − ℓ. (2.30) 168 ioannis k. argyros and saïd hilout cubo 12, 1 (2010) we need to find the relationship between two consecutive functions fk: fk+1(t) = ℓ t uk + ℓ0 ( 2 (t0 + tu1 + · · · + tuk+1 ) + tuk+2 + t ) + ℓ − δ = ℓ tuk−1 − ℓ tuk−1 + ℓ tuk + ℓ0 ( 2 (t0 + tu1 + · · · + tuk )+ tuk+1 + tuk+1 + tuk+2 + t ) + ℓ − δ = fk(t) + ℓ (t uk − tuk−1 ) + ℓ0 (tuk+1 + tuk+2 ) = fk(t) + gk(t). (2.31) using hypothesis 2 ℓ0 + ℓ−δ < 0, and (2.28), we get: f0(0) = 2 ℓ0 + ℓ−δ < 0, fk(0) = δ −ℓ < 0 (k ≥ 1). moreover for sufficiently large t, we also have fk(t) > 0 for all k ≥ 0. it then follows from the intermediate value theorem that: for each k ≥ 0, there exists vk ≥ 0, with fk(vk) = 0. each vk is the unique positive zero of fk, since f ′ k(t) > 0 (k ≥ 0). that is the graph of function fk crosses the positive axis only once. by the definition of fk and vk, there exists a polynomial pk−1 of degree k −1, with pk−1(s) > 0 (s > 0), such that: fk(t) = (t − vk) pk−1(t). to show (2.14), we must have fk(t) ≤ 0. that is q = t ≤ vk (k ≥ 0). (2.32) in view of (2.28) and (2.31), we get fk+1(vk) = fk(vk) + gk(vk) = gk(vk) = qk(vk) v uk−1 k ≥ 0 for vk ≥ δ1 (by (2.27)). therefore, if vk ≥ δ1, then vk+1 ≤ vk, and lim k−→∞ vk = q∞ exists. note that v1 ≥ δ1 by (2.23). then, it follows v2 ≤ v1. assume vm ≥ δ1 (m ≤ n), then vm+1 ≤ vm. we must also show vm+1 ≥ δ1. but this is true, since vm+1 ≥ δ∞ ≥ δ1 by (2.23). then, estimate (2.32) certainly holds if δ ≤ δ∞, which is true by the choice of δ. that completes the proof of lemma 2.2. ♦ we shall study the secant method (1.2) for triplets (f, x−1, x0) belonging to the class c(ℓ, ℓ0, η, c, δ) defined as follows: definition 2.3. let ℓ, ℓ0, η, c, δ be non–negative parameters satisfying the hypotheses of lemma 2.2. we say that a triplet (f, x−1, x0) belongs to the class c(ℓ, ℓ0, η, c, δ) if: cubo 12, 1 (2010) convergence conditions for the secant method 169 (c1) f is a nonlinear operator defined on a convex subset d of a banach space x with values in a banach space y; (c2) x−1 and x0 are two points belonging to the interior d0 of d and satisfying the inequality ‖ x0 − x−1 ‖≤ c; (c3) f is fréchet–differentiable on d0, and there exists an operator δf : d0 ×d0 → l(x , y) such that: the linear operator a = δf (x−1, x0) is invertible, its inverse a −1 is bounded and: ‖ a−1 f (x0) ‖ ≤ η; ‖ a [δf (x, y) − f ′(z)] ‖ ≤ ℓ (‖ x − z ‖ + ‖ y − z ‖); ‖ a [δf (x, y) − f ′(x0)] ‖ ≤ ℓ0 (‖ x − x0 ‖ + ‖ y − x0 ‖) for all x, y, z ∈ d. (c4) the set dc = {x ∈ d; f is continuous at x} contains the closed ball u (x0, t⋆) = {x ∈ x |‖ x − x0 ‖≤ t⋆} where t⋆ is given in lemma 2.1. we present the following semilocal convergence theorem for secant method (1.2). theorem 2.4. if (f, x−1, x0) ∈ c(ℓ, ℓ0, η, c, δ), then sequence {xn} (n ≥ −1) generated by secant method (1.2) is well defined, remains in u (x0, t ⋆) for all n ≥ 0 and converges to a unique solution x⋆ ∈ u (x0, t⋆) of equation f (x) = 0. moreover the following estimates hold for all n ≥ 0 ‖ xn+2 − xn+1 ‖≤ tn+2 − tn+1, (2.33) and ‖ xn − x⋆ ‖≤ t⋆ − tn, (2.34) where the sequence {tn} (n ≥ 0) given by (2.7). furthermore, if u ( x0, 1 2 (c + 1 ℓ0 ) ) ⊆ d, t⋆⋆ < 1 2 (c + 1 ℓ0 ) (2.35) the solution x⋆ is unique in u (x0, t ⋆). finally, if t⋆⋆ < 1 ℓ0 − r, u (x0, r) ⊆ d, where t⋆⋆ is given by (2.8), then the solution x⋆ is unique in u (x0, r). 170 ioannis k. argyros and saïd hilout cubo 12, 1 (2010) proof. we first show operator l = δf (xk, xk+1) is invertible for xk, xk+1 ∈ u (x0, t⋆). it follows from (2.7), (2.8), (c2) and (c3) that: ‖ i − a−1 l ‖=‖ a−1 (l − a) ‖ ≤ ‖ a−1(l − f ′(x0)) ‖ + ‖ a−1(f ′(x0) − a) ‖ ≤ ℓ0 (‖ xk − x0 ‖ + ‖ xk+1 − x0 ‖ + ‖ x0 − x−1 ‖) ≤ ℓ0 (tk − t0 + tk+1 − t0 + c) ≤ ℓ0 (t⋆ − t0 + t⋆ − t0 + c) ≤ ℓ0 ( 2 [ η 1 − δ + c ] − c ) ≤ 1 (2.36) since δ ≤ δ∞. according to the banach lemma on invertible operators [5], [13], and (2.36), l is invertible and ‖ l−1 a ‖≤ ( 1 − ℓ0 (‖ xk − x0 ‖ + ‖ xk+1 − x0 ‖ +c) )−1 . (2.37) the second condition in (c3) implies the lipschitz condition for f ′ ‖ a−1 (f ′(u) − f ′(v)) ‖≤ 2 ℓ ‖ u − v ‖, u, v ∈ d0. (2.38) by the identity, f (x) − f (y) = ∫ 1 0 f ′(y + t(x − y)) dt (x − y) (2.39) we get ‖ a−10 [f (x) − f (y) − f ′(u)(x − y)] ‖≤ ℓ (‖ x − u ‖ + ‖ y − u ‖) ‖ x − y ‖ (2.40) and ‖ a−10 [f (x) − f (y) − δf (u, v) (x − y)] ‖≤ ℓ (‖ x − v ‖ + ‖ y − v ‖ + ‖ u − v ‖) ‖ x − y ‖ (2.41) for all x, y, u, v ∈ d0. by a continuity argument (2.38)–(2.41) remain valid if x and/or y belong to dc. we first show (2.33). if (2.33) holds for all n ≤ k and if {xn} (n ≥ 0) is well defined for n = 0, 1, 2, · · · , k then ‖ x0 − xn ‖≤ tn − t0 < t⋆ − t0, n ≤ k. (2.42) that is (1.2) is well defined for n = k + 1. for n = −1, and n = 0, (2.33) reduces to ‖ x−1 − x0 ‖≤ c, and ‖ x0 − x1 ‖≤ η. suppose (2.33) holds for n = −1, 0, 1, · · · , k (k ≥ 0). using (2.37), (2.41) and f (xk+1) = f (xk+1) − f (xk) − δf (xk−1, xk) (xk+1 − xk) (2.43) cubo 12, 1 (2010) convergence conditions for the secant method 171 we obtain in turn ‖ xk+2 − xk+1 ‖ = ‖ δf (xk, xk+1)−1 f (xk+1) ‖ ≤ ‖ δf (xk, xk+1)−1 a ‖ ‖ a−1 f (xk+1) ‖ ≤ ℓ (‖ xk+1 − xk ‖ + ‖ xk − xk−1 ‖) 1 − ℓ0 [‖ xk+1 − x0 ‖ + ‖ xk − x0 ‖ +c] ‖ xk+1 − xk ‖ ≤ ℓ (tk+1 − tk + tk − tk−1) 1 − ℓ0 [tk+1 − t0 + tk − t0 + t0 − t−1] (tk+1 − tk) = tk+2 − tk+1. (2.44) the induction for (2.33) is now complete. it follows from (2.33) and lemma 2.2 that sequence {xn} (n ≥ −1) is cauchy in a banach space x , and as such it converges to some x⋆ ∈ u (x0, t⋆) (since u (x0, t ⋆) is a closed set). by letting k → ∞ in (2.44), we obtain f (x⋆) = 0. estimate (2.34) follows from (2.33) by using standard majoration techniques [1], [5], [13]. we shall first show uniqueness in u (x0, t ⋆). let y⋆ ∈ u (x0, t⋆) be a solution of equation (1.1). set m = ∫ 1 0 f ′(y⋆ + t (y⋆ − x⋆)) dt. it then by (c3): ‖ a−1 (a − m) ‖ ≤ ℓ0 (‖ y⋆ − x0 ‖ + ‖ x⋆ − x0 ‖ + ‖ x0 − x−1 ‖) ≤ ℓ0 ( 2 (t⋆ − t0) + t0 ) ≤ ℓ0 (2 t⋆⋆ − c) < 1, (2.45) since δ ≤ δ∞. it follows from (2.35), and the banach lemma on invertible operators that m−1 exists on u (x0, t ⋆). using the identity: f (x⋆) − f (y⋆) = m (x⋆ − y⋆), (2.46) we deduce x⋆ = y⋆. finally, we shall show uniqueness in u (x0, r). as in (2.45), we arrive at ‖ a−1 (a − m) ‖< ℓ0 (t⋆⋆ + r) ≤ 1. hence, again we conclude x⋆ = y⋆. that completes the proof of theorem 2.4. ♦ 172 ioannis k. argyros and saïd hilout cubo 12, 1 (2010) remark 2.5. returning back to the example given in the introduction, say ℓ0 = .1, we obtain δ0 = 2.047714554, 1 c = 5.405. condition (2.4) is true, since ℓ0 (c + η) = .0365 < 1. choose δ = 2.5 ∈ ( δ, 1 c ) to obtain q = .4625. then condition (2.6) becomes .423156212 < q. that is our results can apply, whereas the ones using (1.3) cannot. remark 2.6. let us define the majoring sequence {wn} used in [4], [5] (under condition (1.3)): w−1 = 0, w0 = c, w1 = c + η, wn+2 = wn+1 + ℓ (wn+1 − wn−1) (wn+1 − tn) 1 − ℓ (wn+1 − w0 + wn) . (2.47) note that in general ℓ0 ≤ ℓ (2.48) holds , and ℓ ℓ0 can be arbitrarily large [3], [5]. in the case ℓ0 = ℓ, then tn = wn (n ≥ −1). otherwise tn+1 − tn ≤ wn+1 − wn, (2.49) 0 ≤ t⋆ − tn ≤ w⋆ − wn, w⋆ = lim n−→∞ wn. (2.50) note also that strict inequality holds in (2.49) for n ≥ 1, if ℓ0 < ℓ. the proof of (2.49), (2.50) can be found in [5]. note that the only difference in the proofs is that the conditions of lemma 2.2 are used here, instead of the ones in [4]. however this makes no difference between in the proofs. we complete this study with an example example to show that ℓ0 < ℓ. example 2.7. let x = y = c[0, 1] be the space of real–valued continuous functions defined on the interval [0, 1] with norm ‖ x ‖= max 0≤s≤1 |x(s)|. let θ ∈ [0, 1] be a given parameter. consider the "cubic" integral equation u(s) = u3(s) + λ u(s) ∫ 1 0 q(s, t) u(t) dt + y(s) − θ. (2.51) here the kernel q(s, t) is a continuous function of two variables defined on [0, 1] × [0, 1]; the parameter λ is a real number called the "albedo" for scattering; y(s) is a given continuous function cubo 12, 1 (2010) convergence conditions for the secant method 173 defined on [0, 1] and x(s) is the unknown function sought in c[0, 1]. equations of the form (2.51) arise of gasses [5], [8]. for simplicity, we choose u0(s) = y(s) = 1, and q(s, t) = s s + t , for all s ∈ [0, 1], and t ∈ [0, 1], with s + t 6= 0. if we let d = u (u0, 1 − θ), and define the operator f on d by f (x)(s) = x3(s) + λ x(s) ∫ 1 0 q(s, t) x(t) dt + y(s) − θ, (2.52) for all s ∈ [0, 1], then every zero of f satisfies equation (2.51). we define the divided difference δ f (x, y) by δ f (x, y) = ∫ 1 0 f ′(y + t (x − y)) dt. we have the estimate max 0≤s≤1 | ∫ s s + t dt| = ln 2. therfore, if we set b =‖ f ′(u0)−1 ‖, then, we obtain: η = b (|λ| ln 2 + 1 − θ), ℓ = b (|λ| ln 2 + 3 (2 − θ)) and ℓ0 = 1 2 b (2 |λ| ln 2 + 3 (3 − θ)). note that ℓ0 < ℓ for all θ ∈ [0, 1]. received: october, 2008. revised: january, 2009. references [1] argyros, i.k., the theory and application of abstract polynomial equations, st.lucie/crc/ lewis publ. mathematics series, 1998, boca raton, florida, u.s.a. 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[9] dennis, j.e., toward a unified convergence theory for newton–like methods, in nonlinear functional analysis and applications (l.b. rall, ed.), academic press, new york, (1971), 425–472. [10] gutiérrez, j.m., a new semilocal convergence theorem for newton’s method, j. comput. appl. math., 79 (1997), 131–145. [11] hernández, m.a., rubio, m.j. and ezquerro, j.a., secant–like methods for solving nonlinear integral equations of the hammerstein type, j. comput. appl. math., 115 (2000), 245–254. [12] huang, z., a note of kantorovich theorem for newton iteration, j. comput. appl. math., 47 (1993), 211–217. [13] kantorovich and l.v., akilov, g.p., functional analysis, pergamon press, oxford, 1982. [14] laasonen, p., ein überquadratisch konvergenter iterativer algorithmus, ann. acad. sci. fenn. ser i, 450 (1969), 1–10. [15] ortega, j.m. and rheinboldt, w.c., iterative solution of nonlinear equations in several variables, academic press, new york, 1970. [16] potra, f.a., sharp error bounds for a class of newton-like methods, libertas mathematica, 5 (1985), 71–84. [17] schmidt, j.w., untere fehlerschranken fur regula-falsi verhafren, period. hungar., 9 (1978), 241–247. [18] yamamoto, t., a convergence theorem for newton-like methods in banach spaces, numer. math., 51 (1987), 545–557. [19] wolfe, m.a., extended iterative methods for the solution of operator equations, numer. math., 31 (1978), 153–174. cubo a mathematical journal vol.11, no¯ 03, (115–124). august 2009 bounded solutions and periodic solutions of almost linear volterra equations muhammad n. islam and youssef n. raffoul department of mathematics, university of dayton, dayton, oh 45469-2316, usa emails: muhammad.islam@notes.udayton.edu, youssef.raffoul@notes.udayton.edu abstract this article addresses boundedness and periodicity of solutions of certain volterra type equations. these equations are studied under a set of assumptions on the functions involved in the equations. the equations will be called almost linear when these assumptions hold. resumen este artículo es concerniente a acotomiento y periocidad de ciertas ecuaciones de tipo volterra. estas ecuaciones son estudiadas bajo un conjunto de condiciones sobre las funciones envolvidas en las ecuaciones. las ecuaciones serán llamadas casi lineales cuando estas condiciones sean válidas. key words and phrases: volterra integral equation, integrodifferential equation, resolvent, krasnoselsii’s fixed point theorem, bounded solution, periodic solution. math. subj. class.: 45d05,45j05. 116 muhammad n. islam and youssef n. raffoul cubo 11, 3 (2009) 1 introduction. consider the following scalar equations: x(t) = a(t) + ∫ t 0 c(t,s)g(x(s))ds, t ≥ 0, (1.1) and x ′ (t) = a(t)h(x(t)) + ∫ t −∞ c(t,s)g(x(s))ds + p(t), t ∈ (−∞,∞). (1.2) we assume that the functions h and g are continuous and that there exist positive constants h, h∗, g, g∗ such that |h(x) − hx| ≤ h∗, (1.3) and |g(x) − gx| ≤ g∗. (1.4) equations (1.1) and (1.2) will be called almost linear if (1.3) and (1.4) hold. in [2] burton introduced this concept of almost linear equations and studied certain important properties of the resolvent kernel of a linear volterra equation. throughout this paper we assume a(t) in (1.1) is continuous for t ≥ 0, and a(t),p(t) in (1.2) are continuous for −∞ < t < ∞. also, we assume that c(t,s) in (1.1) is continuous for 0 ≤ s ≤ t < ∞, and c(t,s) in (1.2) is continuous for −∞ < s ≤ t < ∞. in section 2 we obtain the boundedness of solutions of (1.1) using the respective resolvent kernels. in section 3 we study (1.2) and show the existence of a periodic solution by employing krasnoselskii’s fixed point theorem. the literature on the resolvent is massive. however, for many interesting results on resolvents of volterra integral and integrodifferential equations we refer to [1], [3], [4], [6–8], [10–16], [18] and [19]. burton [8] contains a large number of existing studies on the resolvents of volterra integral equations which also includes many recent works related to the resolvent. on krasnoselskii’s fixed point theorem and it’s application in integral equations we refer the reader to [5], [9] and [17]. 2 on solutions of (1.1). we rewrite (1.1), x(t) = a(t) + ∫ t 0 c(t,s)[g(x(s)) − gx(s)]ds + ∫ t 0 c(t,s)gx(s)ds. (2.1) let a(t) = a(t) + ∫ t 0 c(t,s)[g(x(s)) − gx(s)]ds (2.2) cubo 11, 3 (2009) almost linear volterra equations 117 and b(t,s) = gc(t,s). then (2.1) becomes x(t) = a(t) + ∫ t 0 b(t,s)x(s)ds. (2.3) let r(t,s) be the resolvent kernel associated with (2.3). then r(t,s) exists and satisfies r(t,s) = −b(t,s) + ∫ t s r(t,u)b(u,s)du. (2.4) then any solution x(t) of (2.3) satisfies x(t) = a(t) − ∫ t 0 r(t,s)a(s)ds. (2.5) theorem 2.1 assume a(t) is bounded for t ≥ 0. also assume sup t≥0 ∫ t 0 |c(t,s)|ds < ∞, (2.6) and sup t≥0 ∫ t 0 |r(t,s)|ds < ∞. (2.7) then any solution x(t) of (1.1) is bounded. proof. from (2.2),using (1.4) and (2.6) we obtain |a(t)| ≤ |a(t)| + g∗ ∫ t 0 |c(t,s)|ds < ∞. therefore from (2.5) and (2.7), we get |x(t)| ≤ |a(t)| + ∫ t 0 |r(t,s)||a(s)|ds < ∞. this concludes the proof of theorem 2.1. assume a′(t) and ct(t,s) both exist and are continuous. now differentiating (1.1), one gets x ′ (t) = a ′ (t) + c(t,t)g(x(t)) + ∫ t 0 ct(t,s)g(x(s))ds (2.8) = a ′ (t) + c(t,t)[g(x(t)) − gx(t)] + ∫ t 0 ct(t,s)[g(x(s)) − gx(s)]ds +c(t,t)gx(t) + ∫ t 0 ct(t,s)gx(s)ds. 118 muhammad n. islam and youssef n. raffoul cubo 11, 3 (2009) let f(t) = a ′ (t) + c(t,t)[g(x(t)) − gx(t)] + ∫ t 0 ct(t,s)[g(x(s)) − gx(s)]ds. (2.9) then (2.8) becomes x ′ (t) = gc(t,t)x(t) + ∫ t 0 gct(t,s)x(s)ds + f(t). (2.10) let b(t,s) = gct(t,s), a(t) = gc(t,t). then (2.10) becomes x ′ (t) = a(t)x(t) + ∫ t 0 b(t,s)x(s)ds + f(t), x(0) = a(0). (2.11) let z(t,s) be the resolvent kernel associated with (2.11). then z(t,s) exists and satisfies zs(t,s) = −z(t,s)a(s) − ∫ t s z(t,u)b(u,s)du, z(t,t) = 1. (2.12) then from the variation of parameters formula, any solution x(t) of (2.11) has the form x(t) = z(t, 0)a(0) + ∫ t 0 z(t,s)f(s)ds. (2.13) theorem 2.2 assume a′(t) is bounded. also assume sup t≥0 ∫ t 0 |ct(t,s)|ds < ∞, (2.14) and sup t≥0 ∫ t 0 |z(t,s)|ds < ∞. (2.15) in addition, we assume that |c(t,t)| and |z(t, 0)| are bounded. then any solution x(t) of (2.11) is bounded. proof. applying (1.4) and (2.14) in (2.9), we get |f(t)| ≤ |a′(t)| + |c(t,t)|g∗ + ∫ t 0 |ct(t,s)|g∗ds < ∞. therefore from (2.13) one obtains |x(t)| ≤ |z(t, 0)||a(0)| + ∫ t 0 |z(t,s)||f(s)|ds < ∞. cubo 11, 3 (2009) almost linear volterra equations 119 this concludes the proof of theorem 2.2. properties in (2.7) and (2.15) are known as integrability properties of resolvent. conditions to ensure (2.7) can be found in [11], [14] and [18], and conditions to ensure (2.15) can be found in [10], [12], [13] and [19]. 3 periodic solutions of (1.2) in this section we investigate the existence of a periodic solution of (1.2) using krasnoselskii’s fixed point theorem. we start with a statement of krasnoselskii’s fixed point theorem. theorem krasnoselskii [17]. let k be a closed convex non-empty subset of a banach space m. suppose that a and b map k into m such that (i) x,y ∈ k, implies ax + by ∈ k, (ii) a is continuous and ak is contained in a compact subset of m, (iii) b is a contraction mapping. then there exists z ∈ k with z = az + bz. in this section we assume that sup −∞ 0 such that a(t + t ) = a(t), p(t + t ) = p(t), c(t + t,s + t ) = c(t,s). (3.4) 120 muhammad n. islam and youssef n. raffoul cubo 11, 3 (2009) we assume that ∫ t 0 a(t)dt 6= 0. (3.5) let m be the complete metric space of continuous t -periodic functions φ : (−∞,∞) → (−∞,∞) with the supremum metric. then, for any positive constant m the set pt = {f ∈ m : ||f|| ≤ m}, (3.6) is a closed convex subset of m. let k(t) = p(t) + ∫ t −∞ c(t,s) [ g(x(s)) − gx(s) ] ds + ∫ t −∞ c(t,s)gx(s)ds. then we may write (3.3) as x ′ (t) − ha(t)x(t) = −ha(t)x(t) + a(t)h(x(t)) + k(t). (3.7) assume (3.4) and (3.5) hold. multiply both sides of (3.7) with e−h ∫ t 0 a(s)ds and then integrate both sides from t − t to t, to obtain x(t)[e −h ∫ t t−t a(s)ds − 1]e−h ∫ t−t 0 a(s)ds = ∫ t t−t [ − ha(u)x(u) + a(u)h(x(u)) + k(u) ] e −h ∫ u 0 a(s)ds du. now, multiplying both sides by eh ∫ t−t 0 a(s)ds , we get x(t)[e −h ∫ t t−t a(s)ds − 1] = ∫ t t−t [ − ha(u)x(u) + a(u)h(x(u)) + k(u) ] e −h ∫ u t−t a(s)ds du. due to the periodicity of a(t) we note that e−h ∫ t t−t a(s)ds = e −h ∫ t 0 a(s)ds . substituting k by the expression given earlier and then dividing by e −h ∫ t t−t a(s)ds − 1, we arrive at x(t) = 1 e −h ∫ t 0 a(s)ds − 1 { ∫ t t−t a(u)[h(x(u)) − hx(u)]e−h ∫ u t−t a(s)ds du + ∫ t t−t ∫ u −∞ c(u,s)[g(x(s)) − gx(s)] ds e−h ∫ u t−t a(s)ds du + ∫ t t−t ∫ u −∞ c(u,s)gx(s) ds e −h ∫ u t−t a(s)ds du + ∫ t t−t p(u)e −h ∫ u t−t a(s)ds du } . (3.8) define mappings a and b from pt into m as follows. cubo 11, 3 (2009) almost linear volterra equations 121 for φ ∈ pt , (aφ)(t) = 1 e −h ∫ t 0 a(s)ds − 1 { ∫ t t−t a(u)[h(φ(u)) − hφ(u)]e−h ∫ u t−t a(s)ds du + ∫ t t−t ∫ u −∞ c(u,s)[g(φ(s)) − gφ(s)] ds e−h ∫ u t−t a(s)ds du } and for ψ ∈ pt , (bψ)(t) = 1 e −h ∫ t 0 a(s)ds − 1 { ∫ t t−t ∫ u −∞ c(u,s)gψ(s) ds e −h ∫ u t−t a(s)ds du + ∫ t t−t p(u)e −h ∫ u t−t a(s)ds du } . it can easily be verified that both (aφ)(t) and (bψ)(t) are t -periodic and continuous in t. assume sup −∞ b}, we have γϕ = m • b ∪ m b = (a ∪ m b) ∪ b. since γϕ is connected, either a ∪ m b ∩ b 6= ∅ or (a ∪ m b ) ∩ b 6= ∅. because of formula (1) and the form of m b, we can find y0 ∈ ϕ(b) such that y0 ∈ (m b ∩ b). therefore, since ϕ(b) is connected, formula (1) implies that m b ∩ a = m b ∩ a = ∅. thus, using formula (1) again, we obtain (m b ∪ b) ∩ a = (m b ∪ b) ∩ a = ∅, but it is a contradiction, because γϕ = m b ∪ b ∪ a is connected. we can show in an analogous way that the set {(x, y) ∈ m • b : x ≥ a}, i.e. γϕ↾i := {(x, y) ∈ γϕ : a ≤ x ≤ b}, is connected, too. since any interval j ⊂ l can be expressed as the union of closed intervals of l that have a point in common, the graph γϕ↾j can be expressed, in view of the above conclusions, as the union of connected sets of l 2 that have a point in common. this is sufficient (see e.g. [mu, p. 150]) in order the graph γϕ↾j to be connected which completes the proof. the map ϕ is determined by a gδ-relation in l 2 if its graph γϕ is a gδ-subset of l 2 , i.e. if γϕ = ⋂ m∈n gm, where all gm ⊂ l 2 are open. map ϕ is determined by a connectivity gδ-relation if it has both the above properties. in [ass] resp. [afp], we have shown that m -maps resp. n -maps in r are determined by connectivity gδ-relations in r 2 . 3. statements on linear continua the following fixed point theorem is intuitively obvious (for l = r, cf. lemma 2.4 in [ass]). theorem 1. let i = [a, b] ⊂ l be a closed interval of a linear continuum l and ϕ : i ⊸ l be a multivalued mapping with a connected graph. assume that either i ⊂ ϕ(i) or ϕ(i) ⊂ i. then ϕ has a fixed point in i. cubo 10, 4 (2008) simple fixed point theorems on linear continua 33 proof. denote by γϕ ⊂ i × l ⊂ l 2 the graph of ϕ and define the sets p, p1 and p2 as p := {(x, x) ∈ l2}, p1 := {(x, y) ∈ l 2 : x < y}, p2 := {(x, y) ∈ l 2 : y < x}. obviously, p1 and p2 are nonempty disjoint open sets in l 2 and l 2 = p ∪ p1 ∪ p2. assume that fix ϕ := {x ∈ i : x ∈ ϕ(x)} = ∅, i.e. p ∩ γϕ = ∅. • if i ⊂ ϕ(i), then there exist points c, d ∈ [a, b] such that a ∈ ϕ(c) and b ∈ ϕ(d). moreover, a < c (otherwise, a ∈ ϕ(a) and a is a fixed point) and d < b (otherwise, b ∈ ϕ(b) and b is a fixed point). then d < b ⇒ (d, b) ∈ p1 ∩ γϕ ⇒ p1 ∩ γϕ 6= ∅ and a < c ⇒ (c, a) ∈ p2 ∩ γϕ ⇒ p2 ∩ γϕ 6= ∅. from the above arguments, we have γϕ ⊂ p1 ∪ p2, where γϕ is connected and p1 ∪ p2 is disconnected which is a contradiction. • if ϕ(i) ⊂ i, then a < p, for all p ∈ ϕ(a) (otherwise, a ∈ ϕ(a) and a is a fixed point) and q < b, for all q ∈ ϕ(b) (otherwise, b ∈ ϕ(b) and b is a fixed point). then a < p ⇒ (a, p) ∈ p1 ∩ γϕ ⇒ p1 ∩ γϕ 6= ∅ and q < b ⇒ (b, q) ∈ p2 ∩ γϕ ⇒ p2 ∩ γϕ 6= ∅. from the above arguments, we have γϕ ⊂ p1 ∪ p2, where γϕ is connected and p1 ∪ p2 is disconnected which is again a contradiction. the following slight generalization of a one-dimensional version of the brouwer theorem is well-known, because it can be easily deduced from the evident intermediate value property. corollary 1. if a single-valued map f : i → i, where i ⊂ l is a closed interval of a linear continuum l, has a connected graph, then f has a fixed point in i. example 4. the function f : [−1, 1] → [−1, 1] defined by f (x) := { sin 1 x , for x ∈ [−1, 1] \ {0}, 1, for x = 0. is not continuous, but has a connected graph. it admits, in fact, infinitely many fixed points in [−1, 1] (see fig. 4). 34 jan andres, karel pastor and pavla šnyrychová cubo 10, 4 (2008) -1 0.5-0.5 1 0.5 0 1 -0.5 0-1 figure 4: function f from example 4. lemma 2. let ϕ : l ⊸ l have a connected graph γϕ. if ϕ has an n-orbit, for some n ∈ n, then it also has a fixed point. proof. let {x1, x2, . . . , xn} be an n-orbit of ϕ. assume that fix ϕ := {x ∈ l : x ∈ ϕ(x)} = ∅. denote a := min{xi : i = 1, . . . n} and b := max{xi : i = 1, . . . n}. there exist k, l ∈ {1, 2, . . . n} such that xk ∈ ϕ(a) and xl ∈ ϕ(b). then xk > a and xl < b (otherwise, ϕ has a fixed point xk or xl). hence, (a, xk) ∈ p1 := {(x, y) ∈ γϕ : x < y} and (b, xl) ∈ p2 := {(x, y) ∈ γϕ : y < x}. since the sets p1 and p2 are nonempty disjoint open sets in γϕ and γϕ = p1 ∪ p2 (we suppose fix ϕ = ∅), we obtain a contradiction with the connectedness of γϕ. the class of maps with a connected graph is rather large. in particular, it trivially contains maps determined by connectivity relations, and since in r upper and lower semicontinuous maps with closed connected values are determined by connectivity gδ-relations (see [ass] or [afp]), they also have connected graph. since the maps satisfying the assumptions of theorem 1 possess a fixed point, so obviously do their iterates whose graph is not necessarily connected like e.g. the a map ϕ : [0, 1] ⊸ [ 1 4 , 3 4 ], where ϕ(x) :=        { 1 4 , 3 4 }, for x ∈ [0, 1) , [ 1 4 , 3 4 ] , for x = 1, cubo 10, 4 (2008) simple fixed point theorems on linear continua 35 because ϕ2 : [0, 1] ⊸ [ 1 4 , 3 4 ], where ϕ2(x) = { 1 4 , 3 4 }, for x ∈ [0, 1]. on the other hand, if ϕ has still connected values (i.e. if it is determined by a connectivity relation; cf. lemma 1), then all the iterates ϕn, n ∈ n, of ϕ have the same property. indeed. it directly follows from the definition of a connectivity relation that, for any (possibly degenerate) interval i ⊂ l, ϕ(i) is connected, i.e. an interval. thus, ϕ2(i) = ϕ(ϕ(i)) must be also connected, i.e. ϕ2 is determined by a connectivity relation and, in particular, it has a connected graph. by induction, we get that it holds for all the iterates, as claimed. moreover, there exist maps with a disconnected graph or disconnected values whose some iterate determines a connectivity relation like the mapping ϕ : [0, 1] ⊸ [ 1 4 , 3 4 ], where ϕ(x) :=                                      3 4 , for x ∈ [ 0, 1 2 ) \ { 1 4 }, [ 1 4 , 3 4 ] , for x = 1 4 , { 1 4 , 3 4 }, for x = 1 2 , [ 1 4 , 3 4 ] , for x = 3 4 , 1 4 , for x ∈ ( 1 2 , 1 ] \ { 3 4 }, because ϕ2 : [0, 1] ⊸ [ 1 4 , 3 4 ], ϕ2(x) = [ 1 4 , 3 4 ] (see fig. 5). figure 5: maps ϕ and ϕ2. if, in theorem 1, ϕ = ξn, for some n ∈ n, then a natural question therefore arises whether or not mapping ξ itself admits a fixed point. as a partial answer, we can give the two following corollaries. corollary 2. let ϕ : l ⊸ l be a multivalued mapping with a connected graph. assume that, for some n ∈ n, the n-th iterate ϕn of ϕ has also a connected graph and that there exists a closed 36 jan andres, karel pastor and pavla šnyrychová cubo 10, 4 (2008) interval i ⊂ l of a linear continuum l such that either i ⊂ ϕn(i) or ϕn(i) ⊂ i. then ϕ has a fixed point. proof. according to theorem 1, ϕn has a fixed point in i. if it is not at the same time a fixed point of ϕ, then a nontrivial k-orbit of ϕ occurs, for some k|n. by means of lemma 2, ϕ must have a fixed point. corollary 3. let ϕ : l ⊸ l be a multivalued mapping with a connected graph and connected values (i.e. let ϕ determine a connectivity relation; cf. lemma 1). assume that, for the n-th iterate ϕn, n ∈ n, of ϕ there exists a closed interval i ⊂ l of a linear continuum l such that either i ⊂ ϕn(i) or ϕn(i) ⊂ i. then ϕ has a fixed point. proof. since ϕn has, by the above arguments, a connected graph, an application of corollary 2 completes the proof. the following example demonstrates that the graph connectedness in corollaries 2 and 3 cannot be avoided. example 5. the mapping ϕ : [0, 1] ⊸ [0, 1] with closed connected values (observe that ϕ([0, 1]) = [0, 1], see fig. 6), where ϕ(x) :=            [ 1 2 , 1 ] , for x = 0, −x + 1, for x ∈ (0, 1 2 ) ∪ ( 1 2 , 1), 0, for x = 1 2 , [ 0, 1 2 ] , for x = 1, has the second iterate ϕ2 : [0, 1] ⊸ [0, 1], where ϕ2(x) =        [ 0, 1 2 ] , for x = 0, x, for x ∈ (0, 1 2 ) ∪ ( 1 2 , 1), [ 1 2 , 1 ] , for x ∈ { 1 2 , 1}, which is an m -mapping (see fig. 6), but despite the fact that the set of fixed points of ϕ2 is the whole interval [0, 1], ϕ itself is fixed point free. cubo 10, 4 (2008) simple fixed point theorems on linear continua 37 figure 6: maps ϕ and ϕ2 from example 5. as a simple example of an application of corollary 3, let us consider a continuous (singlevalued) function f : (0, ∞) → (0, ∞), where f (x) := 1 x , whose second iterate f 2 : (0, ∞) → (0, ∞) is f 2(x) = x. one can readily check that, for i = [ 1 4 , 1 2 ], we have f ([ 1 4 , 1 2 ]) = [2, 4], i.e. f (i) 6⊂ i and i 6⊂ f (i), but f 2([ 1 4 , 1 2 ]) = [ 1 4 , 1 2 ]. thus, according to corollary 3, f has a fixed point. observe that the only fixed point of f , x = 1 6∈ [ 1 4 , 1 2 ]. on the other hand, e.g. for the interval [ 1 2 , 2], we already have f ([ 1 2 , 2]) = [ 1 2 , 2], and it is sufficient to apply theorem 1, according to which f has a fixed point in [ 1 2 , 2]. for m -maps, theorem 1 can be improved in the form of the following lemma. lemma 3 (cf. [ap], lemma 2.2). let ϕ : l ⊸ l be an m -map. assume that ik ⊂ l, k = 0, 1, . . . , n − 1, are closed intervals such that ik+1 ⊂ ϕ(ik), for k = 0, 1, . . . , n − 1, and in = i0, which we write as i0 → i1 → · · · → in = i0. then the n-th iterate ϕ n of ϕ (i.e. the n-fold composition of ϕ with itself) has a fixed point x0 (i.e. x0 ∈ ϕ n (x0)) with xk+1 ∈ ϕ(xk), xn = x0, where xk ∈ ik, for k = 0, 1, . . . , n − 1. we will finally show how lemma 3 can be employed for restricting the problem of coexistence of periodic orbits from noncompact linearly ordered spaces to closed intervals. theorem 2. let an m -mapping ϕ : l ⊸ l have an n-orbit {x1, . . . , xn}, and let a := min{x1, . . . , xn}, b := max{x1, . . . , xn}. then there exist a closed inteval i, [a, b] ⊂ i ⊂ l, and an m -mapping ϕ̂ : i ⊸ i such that ϕ̂(x) = ϕ(x) for every x ∈ (a, b) and, for every k ∈ n, the existence of a k-orbit of ϕ̂ implies the existence of a k-orbit of ϕ. proof. if ϕ([a, b]) = [a, b], it suffices to put ϕ̂ = ϕ and i = [a, b]. on the contrary, let [d, c] := ϕ([a, b]). now, the proof splits into the following cases. i. ϕ((b, c]) 6⊂ [d, c]. setting s := inf{x ∈ (b, c] : ϕ(x) 6⊂ [d, c]}, then due to the upper semicontinuity of ϕ just one of the following possibilities occurs: 38 jan andres, karel pastor and pavla šnyrychová cubo 10, 4 (2008) 1. c ∈ ϕ(s) if ϕ([d, a]) ⊂ [d, c], we put i = [d, c] and define ϕ̂(x) =        ϕ(x), for every x ∈ [d, s), ϕ(x) ∩ [d, c], for x = s, c, for every x ∈ (s, c]. the points forming a k-orbit, k ∈ n \ {1}, of ϕ̂ form the same orbit of ϕ, because the point c can only form a 1-orbit of ϕ̂ in addition to ϕ. on the other hand, the existence of 1-orbit of ϕ on [a, b] follows from theorem 1. if ϕ([d, a]) 6⊂ [d, c], then we consider t := sup{x ∈ [d, a) : ϕ(x) 6⊂ [d, c]}. if c ∈ ϕ(t), we put i = [d, c] and define ϕ̂(x) =        c, for every x ∈ [d, t) ∪ (s, c] ϕ(x) ∩ [d, c], for x = t, s, ϕ(x), for every x ∈ (t, s). if c 6∈ ϕ(t), then d ∈ ϕ(t). indeed, supposing d 6∈ ϕ(t), either (if t < a) the upper semicontinuity of ϕ leads to a contradiction with the definition of t or (if t = a) we obtain a contradiction with the fact that ϕ(a) is a connected interval and ϕ(a) ∩ [a, b] 6= ∅. we put i = [d, c] and define ϕ̂(x) =            d, for every x ∈ [d, t), ϕ(x) ∩ [d, c], for x = t, s, ϕ(x), for every x ∈ (t, s), c, for every x ∈ (s, c]. the points forming a k-orbit, k ∈ n \ {1}, of ϕ̂ form the same orbit of ϕ, because the points c or d can only form a 1-orbit of ϕ̂ in addition to ϕ. again, theorem 1 implies the existence of a fixed point on [a, b]. 2. c 6∈ ϕ(s) and d ∈ ϕ(s). setting e := min{y ∈ ϕ(x) : x ∈ [b, c]} and r := min{x ∈ [b, c] : e ∈ ϕ(x)}, there are the following possibilities depending on function values of ϕ on [e, a]: a) ϕ([e, a]) ⊂ [e, c]. if ϕ((s, c]) ≤ c (i.e., y ≤ c, for every y ∈ ϕ(x), where x ∈ (s, c]), it suffices to put i = [e, c] and ϕ̂ = ϕ. otherwise, setting q := inf{x ∈ (s, c] : ϕ(x) > c}, we put i = [e, c] and define ϕ̂(x) =        ϕ(x), for every x ∈ [e, q), ϕ(x) ∩ [e, c], for x = q, c, for every x ∈ (q, c]. cubo 10, 4 (2008) simple fixed point theorems on linear continua 39 b) ϕ([e, a]) 6⊂ [e, c]. we consider u := sup{x ∈ [e, a) : ϕ(x) 6⊂ [e, c]}, and put i = [e, c]. the definition of ϕ̂ depends on the relation of e and ϕ(u), and on the relation of ϕ((s, r)) and c. if e ∈ ϕ(u) and ϕ((s, r)) ≤ c, then we define ϕ̂(x) =        e, for every x ∈ [e, u) ∪ (r, c], ϕ(x) ∩ [e, c], for x = u, r, ϕ(x), for every x ∈ (u, r). if e ∈ ϕ(u) and m := inf{x ∈ (s, r) : ∃y ∈ ϕ(x), y > c} ∈ (s, r), then we define ϕ̂(x) =            e, for every x ∈ [e, u), ϕ(x) ∩ [e, c], for x = u, m, ϕ(x), for every x ∈ (u, m), c, for every x ∈ (m, c], and the same arguments as those at the end of part i., 1. conclude this case. if e 6∈ ϕ(u), then c ∈ ϕ(u). if, moreover, ϕ((s, r]) ≤ c, we define ϕ̂(x) =            c, for every x ∈ [e, u), ϕ(x) ∩ [e, c], for x = u, r, ϕ(x), for every x ∈ (u, r), e, for every x ∈ (r, c]. the points forming a k-orbit of mapping ϕ̂, for k ∈ n\{1, 2}, form the same k-orbit of ϕ, because the points c, e can only form a 2-orbit {e, c} of ϕ̂ in addition to ϕ. the existence of a 2-orbit of ϕ is also guaranteed, because it holds [e, a] → [b, c] → [e, a]. finally, if e 6∈ ϕ(u), c ∈ ϕ(u) and m ∈ (s, r), we define ϕ̂(x) =        c, for every x ∈ [e, u) ∪ (m, c], ϕ(x) ∩ [e, c], for x = u, m, ϕ(x), for every x ∈ (u, m). ii. ϕ((b, c]) ⊂ [d, c]. we will discuss functional values of ϕ on [d, a]. 1. ϕ([d, a]) ⊂ [d, c]. it suffices to put i = [d, c] and ϕ̂ = ϕ. 2. ϕ([d, a]) 6⊂ [d, c]. we consider v := sup{x ∈ [d, a) : ϕ(x) 6⊂ [d, c]}. 40 jan andres, karel pastor and pavla šnyrychová cubo 10, 4 (2008) if d ∈ ϕ(v), we put i = [d, c] and define ϕ̂(x) =        d, for every x ∈ [d, v), ϕ(x) ∩ [d, c], for x = v, ϕ(x), for every x ∈ (v, c]. if d 6∈ ϕ(v), we set f := max{y ∈ ϕ(x) : x ∈ [d, a]} and p := max{x ∈ [d, a] : f ∈ ϕ(x)}. there are two possibilities w.r.t. functional values of ϕ on (c, f ]: a) ϕ((c, f ]) ⊂ [d, f ]. if ϕ([d, v)) ≥ d, it suffices to put i = [d, f ] and ϕ̂ = ϕ. otherwise, setting n := sup{x ∈ [d, v) : ϕ(x) < d}, we put i = [d, f ] and define ϕ̂(x) =        d, for every x ∈ [d, n), ϕ(x) ∩ [d, f ], for x = n, ϕ(x), for every x ∈ (n, f ]. b) ϕ((c, f ]) 6⊂ [d, f ]. we consider w := inf{x ∈ (c, f ] : ϕ(x) 6⊂ [d, f ]}, and put i = [d, f ]. the definition of ϕ̂ depends on the relation of f and ϕ(w) and on the relation of ϕ((p, v)) and d. if f ∈ ϕ(w) and ϕ((p, v)) ≥ d, then we define ϕ̂(x) =        f, for every x ∈ [d, p) ∪ (w, f ], ϕ(x) ∩ [d, f ], for x = p, w, ϕ(x), for every x ∈ (p, w). if f ∈ ϕ(w) and n := sup{x ∈ (p, v) : ∃y ∈ ϕ(x), y < d} ∈ (p, v), then we define ϕ̂(x) =            d, for every x ∈ [d, n), ϕ(x) ∩ [d, f ], for x = n, w, ϕ(x), for every x ∈ (n, w), f, for every x ∈ (w, f ]. we can use the same ideas as in the previous cases to conclude this situation. if f 6∈ ϕ(w), then d ∈ ϕ(w). if, moreover, ϕ((p, v)) ≥ d, we define ϕ̂(x) =            f, for every x ∈ [d, p), ϕ(x) ∩ [d, f ], for x = p, w, ϕ(x), for every x ∈ (p, w), d, for every x ∈ (w, f ]. cubo 10, 4 (2008) simple fixed point theorems on linear continua 41 and the analogous arguments as before conclude this case, jointly with the fact that by lemma 3 ϕ has a 2-orbit, because [d, a] −→ [b, f ] −→ [d, a]. finally, if f 6∈ ϕ(w), d ∈ ϕ(w) and n ∈ (p, v), we define ϕ̂(x) =        d, for every x ∈ [d, n) ∪ (w, f ], ϕ(x) ∩ [d, f ], for x = n, w, ϕ(x), for every x ∈ (n, w). 4. concluding remarks if lemma 3 could be generalized for maps determined by connectivity gδ-relations on linear continua, then a sharkovskii-type theorem might be formulated for these maps by means of the appropriately modified statements like theorem 2. on the real line, this was already done in [ass]. the sharkovskii-type theorems establish an order relationship among the periods that the mapping can possess by means of a new (sharkovskii’s) ordering of positive integers. we already pointed out that this sharkovskii phenomenon is in principle one-dimensional. nevertheless, there exists a two-dimensional analogy in the sense that an order relationship can be replaced by forcing relations on braid types (see [ha], [m1], [m2]). more precisely, one braid type is larger than the second if whenever a homeomorphism has a periodic orbit of the first type, then it also has a periodic orbit of the second type. the theory of braid types on surface dynamics was developed by several authors, but the standard reference for us is here the paper [bo] by p. boyland. higher than two-dimensional analogies of sharkovskii’s theorem require special structure of maps. for triangular maps, it was achieved (in a single-valued case) by p. kloeden [kl] and further extended (in a multivalued case) in [afp], [ap], [aps]. since we were able to do it in [aps] on a cartesian product of linear continua l1 × · · ·× ln , where ln was only a closed interval, a natural question arises whether theorem 2 can be extended to triangular m -maps on l1 ×· · ·× ln , where ln is not necessarily a closed interval. a combination of theorem 1 and corollary 1 in [a3] leads directly to the following random fixed point theorem (for definitions and more details, see [a3]). theorem 3. let φ : ω × l ⊸ l be a random operator, where ω is a complete measurable space and l is a complete separable metric linear continuum. assume that, for each ω ∈ ω, there exists a closed interval iω ⊂ l such that φ(ω, ·) : iω ⊸ l has a connected graph and either iω ⊂ φ(ω, iω) or iω ⊃ φ(ω, iω). then φ has a random fixed point, i.e. a measurable function x : ω → l such that x(ω) ∈ φ(ω, x(ω)), for a.a. ω ∈ ω. 42 jan andres, karel pastor and pavla šnyrychová cubo 10, 4 (2008) for more sophisticated fixed point theorems, where closed subsets are covered by their images or just intersect their images, see e.g. [a1] and the references therein. received: january 2008. revised: february 2008. references [a1] j. andres, some standard fixed–point theorems revisited, atti sem. mat. fis. univ. modena 49 (2001), 455–471. [a2] j. andres, period three implications for expansive maps in rn , j. difference eqns appl. 10, 1 (2004), 17–28. [a3] j. andres, randomization of sharkovskii–type theorems, proc. amer. math. soc. 136 (2008), 1385–1395. [ap] j. andres and k. pastor, on a multivalued version of the sharkovskii theorem and its application to differential inclusions, iii, topol. meth. nonlin. anal., 22 (2003), 369– 386. [afp] j. andres, t. fürst and k. pastor, full analogy of sharkovsky’s theorem for lower semicontinuous maps, j. math. anal. appl. 340 (2008), 1132–1144. [ag] j. andres and l. górniewicz, topological fixed point principles for boundary value problems, kluwer, dordrecht, 2003. [aps] j. andres, k. pastor and p. šnyrychová, a multivalued version of sharkovskii’s theorem holds with at most two exceptions, j. fixed point theory appl. 2 (2007), 153–170. [as] d. alcaraz and m. sanchiz, a note on šarkovskii’s theorem in connected linearly ordered spaces, int. j. bifurc. chaos 13, 7 (2003), 1665–1671. [ass] j. andres, p. šnyrychová and p. szuca, sharkovskii’s theorem for connectivity gδ–relations, int. j. bifurc. chaos, 16, 8 (2006), 2377–2393. [bo] p. boyland, topological methods in surface dynamics, topol. appl. 58 (1994), 224– 298. [b1] r.f. brown, fixed point theory, in “history of topology (chapter 10)” (ed. by i.m. james), elsevier, amsterdam, 1999, pp. 271–299. [b2] r.f. brown, fixed points of n–valued multimaps of the circle, bull. polish acad. sci. math. 54 (2006), 153–162. [b3] r.f. brown, the lefschetz number of an n–valued multimaps, jp jour. fixed point theory appl. 2 (2007), 53–60. cubo 10, 4 (2008) simple fixed point theorems on linear continua 43 [dz] z. dzedzej, fixed point index theory for a class of nonacyclic multivalued maps, dissertationes math. 235 (1985), 1–58. [g1] l. górniewicz, present state of the brouwer fixed point theorem for multivalued mappings, ann. sci. math. québec 22, 2 (1998), 169–179. [g2] l. górniewicz, topological fixed point theory of multivalued mappings (2nd edition). springer, berlin, 2006. [ha] m. handel, the forcing partial order on three times punctured disk, ergod. th. dynam. sys. 17, (1997), 593–610. [ka] j. kampen, on fixed points of maps and iterated maps and applications, nonlin. anal. 42, (2000), 509–532. [kl] p.e. kloeden, on sharkovsky’s cycle coexisting ordering, bull. austral. math. soc. 20 (1979), 171–177. [mi] d. miklaszewski, the role of various kinds of continuity in the fixed point theory of set–valued mappings, lecture notes in nonlin. anal., vol. 7, j. schauder center for nonlinear studies, n. copernicus univ., toruń, 2005. [m1] t. matsuoka, braids of periodic points and a 2–dimensional analogue of sharkovskii’s ordering, in: “dynamical systems and nonlinear oscillations” (g. ikegami, ed.) world sci. press, singapore, 1986, pp. 58–72. [m2] t. matsuoka, periodic points and braid theory, in: “handbook of topological fixed point theory” (ed. by r.f. brown, m. furi, l. górniewicz and b. jiang), springer, berlin, 2005, pp. 171–216. [mu] j. r. munkres, topology. a first course, prentice–hall, new jersey, 1975. [s1] h. schirmer, an index and a nielsen number for n–valued multifunctions, fund. math. 124 (1984), 207–219. [s2] h. schirmer, a topologist’s view of sharkovsky’s theorem, houston j. math. 11, 3 (1985), 385–395. [sh] a.n. sharkovskii, coexistence of cycles of a continuous map of a line into itself, ukrain. math. j. 16 (1964), 61–71 (in russian); int. j. bifurc. chaos 5 (1995), 1263– 1273 (english translation). [sk] r. skiba, fixed points of multivalued weighted maps, lecture notes in nonlin. anal., vol. 9, j. schauder center for nonlinear studies, n. copernicus univ., toruń, 2007. [sn] p. šnyrychová, periodic points for maps in rn, acta univ. palacki. olomuc., fac. rer. nat., mathematica 42 (2003), 87–104. n3-cl5 articulo 4.dvi cubo a mathematical journal vol.12, no¯ 02, (43–52). june 2010 on subsets of ideal topological spaces v. renukadevi department of mathematics, anja college, sivakasi-626 124, tamil nadu, india. email: renu siva2003@yahoo.com abstract we define some new collection of sets in ideal topological spaces and characterize them in terms of sets already defined. also, we give a decomposition theorem for α − i−open sets and open sets. at the end, we discuss the property of some collection of subsets in ⋆−extremally disconnected spaces. resumen definimos una nueva colección de conjuntos en espacios topológicos ideales y caracterizamos estos en términos de conjuntos ya definidos. también damos un teorema de descomposición para α−i− abiertos y conjuntos abiertos. finalmente discutimos la probabilidad de algunas colecciones de subconjuntos en espacios disconexos ⋆− extremos. key words and phrases: ⋆−extremally disconnected spaces, t−i−set, α−i−open set, pre−i−open set, semi−i−open set, semi⋆ − i−open, semipre⋆ − i−open, ci−set, bi−set, b1i−set, b2i−set, b3i−set, δ −i−open, ri−open, i−locally closed set, weakly i−locally closed set, air−set, di−set. 2000 ams subject classification: primary: 54 a 05, 54 a 10 44 v. renukadevi cubo 12, 2 (2010) 1 introduction by a space, we always mean a topological space (x, τ ) with no separation properties assumed. if a ⊂ x, cl(a) and int(a) will, respectively, denote the closure and interior of a in (x, τ ). an ideal i on a topological space (x, τ ) is a nonempty collection of subsets of x which satisfies (i) a ∈ i and b ⊂ a implies b ∈ i and (ii) a ∈ i and b ∈ i implies a ∪ b ∈ i. given a topological space (x, τ ) with an ideal i on x and if ℘(x) is the set of all subsets of x, a set operator (.)⋆ : ℘(x) → ℘(x), called a local f unction [14] of a with respect to τ and i, is defined as follows: for a ⊂ x, a⋆(i, τ )={x ∈ x | u ∩ a /∈ i for every u ∈ τ (x)} where τ (x) = {u ∈ τ | x ∈ u}. we will make use of the basic facts concerning the local function [11, theorem 2.3] without mentioning it explicitly. a kuratowski closure operator cl⋆() for a topology τ ∗(i, τ ), called the ⋆ − topology, finer than τ is defined by cl⋆(a) = a ∪ a⋆(i, τ ) [16]. when there is no chance for confusion, we will simply write a⋆ for a⋆(i, τ ) and τ ⋆ or τ ⋆(i) for τ ⋆(i, τ ). int⋆(a) will denote the interior of a in (x, τ ⋆). if i is an ideal on x, then (x, τ, i) is called an ideal space. a subset a of an ideal space (x, τ, i) is τ ⋆ − closed or ⋆ − closed [11](resp.⋆ − perf ect[10] ) if a⋆ ⊂ a(resp.a = a⋆). a subset a of an ideal space (x, τ, i) is said to be a t − i − set[8] if int(a) = int(cl⋆(a)). a subset a of an ideal space (x, τ, i) is said to be δ − i − open[2](resp. α − i − open [8],pre − i − open [6],semi − i − open [8], strong β − i − open[9]) if int(cl⋆(a)) ⊂ cl⋆(int(a))(resp. a ⊂ int(cl⋆(int(a))), a ⊂ int(cl⋆(a)), a ⊂ cl⋆(int(a), a ⊂ cl⋆(int(cl⋆(a))). we will denote the family of all δ − i−open (resp. α − i−open, pre−i−open, semi−i−open, strong β−i−open) sets by δio(x)(resp.αio(x), p io(x), sio(x), sβio(x)). the largest prei−open set contained in a is called the pre−i −interior of a and is denoted by piint(a). for any subset a of an ideal space (x, τ, i), piint(a) = a ∩ int(cl⋆(a)) [15, lemma 1.5]. 2 subsets of ideal topological spaces let (x, τ, i) be an ideal space. a subset a of x is said to be a semi⋆ − i−open set [7] if a ⊂ cl(int⋆(a)). a subset a of x is said to be a semi⋆ − i−closed set [7] if its complement is a semi⋆ − i−open set. clearly, a is semi⋆ − i−closed if and only if int(cl⋆(a)) ⊂ a if and only if int(cl⋆(a)) = int(a) and so semi⋆ − i−closed sets are nothing but t − i−sets. a is said to be a semipre⋆ − i−closed set if int(cl⋆(int(a))) ⊂ a. clearly, a is said to be a semipre⋆ − i−closed if and only if int(cl⋆(int(a))) = int(a) if and only if a is α⋆ − i−set [8]. clearly, x is both semi⋆ − i−closed and semipre⋆ − i−closed. the smallest semi⋆ − i−closed (resp. semipre⋆ − i−closed) set containing is called the semi⋆ − i − closure (resp.semipre⋆ − i − closure) of a and is denoted by sicl(a)(resp.spicl(a)). a subset a of an ideal space (x, τ, i) is said to be a bi−set[8] if a = u ∩ v where u is open and v is a t − i−set. the easy proof of the following theorem 2.1 is omitted which says that the arbitrary intersection of semi⋆ − i−closed (resp. semipre⋆ − i−closed) set is a semi⋆ − i−closed (resp. semipre⋆ − i−closed) set. theorem 2.1. let (x, τ, i) be an ideal space and a ⊂ x. if {aα | α ∈ △} is a family of semi ⋆ − i−closed (resp. semipre⋆ − i−closed) sets, then ∩aα is a semi ⋆ − i−closed (resp. semipre⋆ − i−closed) set. theorem 2.2. let (x, τ, i) be an ideal space and a ⊂ x. then the following hold. (a)sicl(a) = a ∪ int(cl⋆(a)). cubo 12, 2 (2010) on subsets of ideal topological spaces 45 (b)spicl(a) = a ∪ int(cl⋆(int(a))). proof. the proof follows from theorem 1.3 and theorem 3.1 of [5]. every semi⋆ − i−closed set is a semipre⋆ − i−closed set but not the converse as shown by the following example 2.3. theorem 2.4 below shows that the reverse direction is true if the set is semi−i−open. theorem 2.5 gives a characterization of t − i−sets. example 2.3. consider the ideal space (x, τ, i) where x = {a, b, c, d}, τ = {∅, {d}, {a, c}, {a, c, d}, x} and i = {∅, {c}, {d}, {c, d}}. if a = {a}, then int(cl⋆(int(a))) = int(cl⋆(∅)) = ∅ ⊂ a and so a is semipre⋆ − i−closed. since int(cl⋆(a)) = int(cl⋆({a})) = int({a, b, c}) = {a, c} * {a}, a is not semi⋆ − i−closed. theorem 2.4. let (x, τ, i) be an ideal space and a ⊂ x be semipre⋆ − i− closed. if a is semi−i−open, then a is semi⋆ − i−closed. proof. if a is semi−i−open, then a ⊂ cl⋆(int(a)) and so cl⋆(a) ⊂ cl⋆(int(a)). now int(cl⋆(a)) ⊂ int(cl⋆(int(a))) ⊂ a and so a is semi⋆ − i−closed. theorem 2.5. let (x, τ, i) be an ideal space and a ⊂ x. then the following are equivalent. (a) a is a t − i−set. (b) a is semi⋆ − i−closed. (c) a is a semipre⋆ − i−closed bi−set. proof. enough to prove that (c)⇒(a). suppose a is a semipre⋆−i−closed bi−set. then a = u ∩v where u is open and v is a t − i−set. now int(cl⋆(a)) = int(cl⋆(u ∩ v )) ⊂ int(cl⋆(u ) ∩ cl⋆(v )) = int(cl⋆(u )) ∩ int(cl⋆(v )) = int(cl⋆(u )) ∩ int(v ) = int(cl⋆(u ) ∩ int(v )) ⊂ int(cl⋆(u ∩ int(v )) = int(cl⋆(int(u ∩ v ))) = int(cl⋆(int(a))) ⊂ a and so int(cl⋆(a)) ⊂ int(a). but int(a) ⊂ int(cl⋆(a)) and so int(a) = int(cl⋆(a)) which implies that a is a t − i−set. the following example 2.6 shows that the union of two semi⋆ − iclosed (resp. semipre⋆ − i−closed) set is not a semi⋆ − iclosed (resp. semipre⋆ − i−closed) set. example 2.6. consider the ideal space(x, τ, i) of example 2.3. if a = {a, c} and b = {d}, then int(cl⋆(a)) = int(cl⋆({a, c})) = int({a, b, c}) = {a, c} = a and so a is semi⋆ − i−closed and hence semipre⋆ − i−closed. also, int(cl⋆(b)) = int(cl⋆({d})) = int({d}) = {d} = b. therefore, b is semi⋆ − i−closed and so semipre⋆ − i−closed. but int(cl⋆(int(a ∪ b))) = int(cl⋆(int({a, c, d}))) = int(cl⋆({a, c, d})) = int(x) = x * a ∪ b and so a ∪ b is not semipre⋆ − i−closed and hence a ∪ b is not semi⋆ − i−closed. a subset a of an ideal space (x, τ, i) is said to be a ci−set [8] if a = u ∩ v where u is open and v is a semipre⋆ − i−closed set. we will denote the family of all ci−set by ci(x). the following theorem 2.7 gives a characterization of bi−sets and ci−sets. theorem 2.7. let (x, τ, i) be an ideal space and a be a subset of x. then the following hold. (a) a is a bi−set if and only if there exists an open set u such that a = u ∩ sicl(a). (b) a is a ci−set if and only if there exists an open set u such that a = u ∩ spicl(a). proof. (a) suppose a is a bi−set. then a = u ∩ v where u is open and v is a t − i−set. since t − i−sets are semi⋆ − i−closed sets, sicl(v ) = v. now a = u ∩ a ⊂ u ∩ sicl(a) ⊂ u ∩ sicl(v ) = u ∩ v = a and so a = u ∩ sicl(a). conversely, suppose a = u ∩ sicl(a) for some 46 v. renukadevi cubo 12, 2 (2010) open set u. since sicl(a) is semi⋆ − i−closed, int(cl⋆(sicl(a))) ⊂ sicl(a). also, int(sicl(a)) ⊂ int(cl⋆(sicl(a))) ⊂ sicl(a) and so int(sicl(a)) = int(cl⋆(sicl(a))) which implies that sicl(a) is a t − i−set. therefore, a is a bi−set. (b) the proof is similar to that of (a). a subset a of an ideal space (x, τ, i) is said to be a a1i−set (resp. b1i−set [4](αim1−set [1]) ) if a = u ∩ v where u is open (resp.α − i−open ) and cl⋆(int(v )) = x. we will denote the family of all b1i−sets (resp. a1i−sets) by b1i(x) (resp.a1i(x)). clearly, a1i(x) ⊂ b1i(x). the following theorem 2.8 shows that b1i−sets and a1i−sets are nothing but α − i−open sets. theorem 2.8. let (x, τ, i) be an ideal space. then b1i(x) = αio(x) = a1i(x). proof. suppose a ∈ b1i(x). then a = u ∩ v where u is α − i−open and cl ⋆(int(v )) = x. since v ⊂ x = int(cl⋆(int(v ))), v ∈ αio(x). since αio(x) is a topology on x, a ∈ αio(x) and so b1i(x) ⊂ αio(x). suppose a ∈ αio(x). then a ⊂ int(cl⋆(int(a))) and so a = int(cl⋆(int(a)))∩(x−(int(cl⋆(int(a)))− a)) = int(cl⋆(int(a))) ∩ ((x − int(cl⋆(int(a)))) ∪ a). also, cl⋆(int((x − int(cl⋆(int(a)))) ∪ a)) ⊃ cl⋆(int(x − int(cl⋆(int(a)))) ∪ int(a)) = cl⋆(int(x − int(cl⋆(int(a))))) ∪ cl⋆(int(a)) ⊃ cl⋆(int(x − cl⋆(int(a))))∪ cl⋆(int(a)) ⊃ int(x−cl⋆(int(a)))∪cl⋆(int(a)) ⊃ int((x−cl⋆(int(a)))∪cl⋆(int(a))) = int(x) = x. therefore, a ∈ a1i(x) which implies that αio(x) ⊂ a1i(x). clearly, a1i(x) ⊂ b1i(x). this completes the proof. a subset a of an ideal space (x, τ, i) is said to be an ri−open set [17] if a = int(cl⋆(a)). we will denote the family of all ri−open sets by rio(x). in [17], it is established that rio(x) is a base for a topology τi and τs ⊂ τi ⊂ τ where τs is the semiregularization of τ. the following theorem 2.9 gives characterizations of pre−i−open sets in terms of ri−open sets. theorem 2.9. let (x, τ, i) be an ideal space and a ⊂ x. then the following are equivalent. (a) a is pre−i−open. (b) there exists an ri−open set g such that a ⊂ g and cl⋆(g) = cl⋆(a). (c)a = g ∩ d where g is ri−open and d is τ ⋆−dense. (d)a = g ∩ d where g is open and d is τ ⋆−dense. proof. (a)⇒(b). suppose a is pre−i−open. if g = int(cl⋆(a)), then a ⊂ g and int(cl⋆(g)) = int(cl⋆(int(cl⋆(a)))) = int(cl⋆(a)) = g which implies that g is an ri−open set containing a. also, cl⋆(a) ⊂ cl⋆(g) = cl⋆(int(cl⋆(a))) ⊂ cl⋆(a) which implies that cl⋆(a) = cl⋆(g). this proves (b). (b)⇒(c). suppose g is an ri−open set such that a ⊂ g and cl⋆(g) = cl⋆(a). if d = a ∪ (x − g), then a = g ∩ d and d is τ ⋆−dense. this proves (c). (c)⇒(d) is clear. (d)⇒(a) follows from lemma 4.3 of [3]. a subset a of an ideal space (x, τ, i) is said to be a a2i−set (resp. b2i−set [4](αim2−set [1]) ) if a = u ∩ v where u is open (resp.α − i−open ) and cl⋆(v ) = x. we will denote the family of all a2i−sets (resp. b2i−sets) by a2i(x) (resp.b2i(x)). clearly, a2i(x) ⊂ b2i(x). the following theorem 2.10 shows that a2i−sets and b2i−sets are nothing but pre−i−open sets. also, it shows that the converse of proposition 2.6 of [4] is true. theorem 2.10. let (x, τ, i) be an ideal space. then a2i(x) = p io(x) = b2i(x). cubo 12, 2 (2010) on subsets of ideal topological spaces 47 proof. by theorem 2.9(d), a2i(x) = p io(x). since a2i(x) ⊂ b2i(x), it is enough to prove that b2i(x) ⊂ a2i(x). suppose a ∈ b2i(x). then a = u ∩ v where u is α − i−open and cl⋆(v ) = x. now a ⊂ u ⊂ int(cl⋆(int(u ))) = int(cl⋆(int(u ∩ x))) = int(cl⋆(int(u ∩ cl⋆(v )))) ⊂ int(cl⋆(int(cl⋆(u ∩ v )))) = int(cl⋆(u ∩ v )) = int(cl⋆(a)) and so a ∈ p io(x). this completes the proof. clearly, a1i(x) ⊂ a2i(x). the following example 2.11 shows that an a2i−set need not be an a1i−set. example 2.11. consider the ideal space (x, τ, i) where x = {a, b, c}, τ = {∅, {c}, x} and i = {∅, {c}}. if a = {a, c}, then a is an a2i−set. but cl ⋆(int(a)) = int(a) ∪ (int(a))⋆ = {c} 6= x. hence a is not an a1i−set. a subset a of an ideal space (x, τ, i) is said to be an αin5−set [1] if a = u ∩ v where u is α −i−open and v is ⋆−closed. we will denote the family of all αin5−sets of an ideal space (x, τ, i) by αin5(x). a subset a of an ideal space (x, τ, i) is said to be an i−locally closed [6] (resp. weakly i−locally closed [13]) set if a = u ∩ v where u is open and v is a ⋆−perfect (resp. ⋆−closed) set. by theorem 2.9 of [15], a is weakly i−locally closed if and only if a = u ∩ cl⋆(a) for some open set u. the family of all weakly i−locally closed sets is denoted by w ilc(x). clearly, every weakly i−locally closed set is an αin5−set but not the converse as shown by the following example 2.12. theorem 2.13 below gives a characterization of αin5−sets. example 2.12. [4, example 2.2]consider the ideal space (x, τ, i) where x = {a, b, c}, τ = {∅, {a}, {a, c}, x} and i = {∅, {b}, {c}, {b, c}}. if a = {a, b}, then int(cl⋆(int(a))) = int(cl⋆(int({a, b}))) = int(cl⋆({a})) = int({a, b, c}) = x ⊃ a and so a is α − i−open and hence an αin5−set. but there is no open set u such that a = u ∩ cl ⋆(a) where cl⋆(a) = x. hence a is not a weakly i−locally closed set. theorem 2.13. let (x, τ, i) be an ideal space and a ⊂ x. then a is αin5−set if and only if a = u ∩ cl⋆(a) for some u ∈ αio(x). proof. if a is an αin5−set, then a = u ∩ v where u is α − i−open and v is ⋆−closed. since a ⊂ v, cl⋆(a) ⊂ cl⋆(v ) = v and so u ∩ cl⋆(a) ⊂ u ∩ v = a ⊂ u ∩ cl⋆(a) which implies that a = u ∩ cl⋆(a). conversely, suppose a = u ∩ cl⋆(a) for some u ∈ αio(x). since cl⋆(a) is ⋆−closed, a is an αin5−set. a subset a of an ideal space (x, τ, i) is said to be an ir−closed set [1] if a = cl⋆(int(a)). a subset a of an ideal space (x, τ, i) is said to be an αai−set [4](αin2−set [1]) (resp.air−set [1]) if a = u ∩v where u is an α−i−open (resp. open) set and v is an ir−closed set. air−sets are called as ai−sets in [4]. we will denote the family of all αai −sets (resp.air−sets) by αai(x)(resp.air(x)). clearly, every air−set is an αai −set but the converse is not true [4, example 2.2]. theorem 2.14 below shows that αai −sets are nothing but semi−i−open sets which shows that the reverse direction of proposition 2.4 of [4] is true and each such set is both a strong β − i−open set and an αin5−set. theorem 2.14. let (x, τ, i) be an ideal space. then αai(x) = sβio(x) ∩ αin5(x) = sio(x). proof. suppose a ∈ αai(x). then a = u ∩v where u ∈ αio(x) and v is an ir−closed set. now a = u ∩ v ⊂ int(cl⋆(int(u ))) ∩ cl⋆(int(v )) ⊂ cl⋆(int(cl⋆(int(u ))) ∩ int(v )) = cl⋆(int(cl⋆(int(u )) ∩ int(v ))) ⊂ cl⋆(int(cl⋆(int(u )∩int(v )))) = cl⋆(int(cl⋆(int(u∩v )))) = cl⋆(int(u∩v )) = cl⋆(int(a)) ⊂ 48 v. renukadevi cubo 12, 2 (2010) cl⋆(int(cl⋆(a))) and so a ∈ sβio(x). since v is ⋆−closed, a ∈ αin5(x) and so αai(x) ⊂ sβio(x) ∩ αin5(x). conversely, suppose a ∈ sβio(x) ∩ αin5(x). a ∈ sβio(x) implies that a ⊂ cl⋆(int(cl⋆(a))) and a ∈ αin5(x) implies that a = u ∩ cl ⋆(a) where u ∈ αio(x). since a ⊂ u, a ⊂ u ∩ cl⋆(int(cl⋆(a))) ⊂ u ∩ cl⋆(a) = a and so a = u ∩ cl⋆(int(cl⋆(a))). since cl⋆(int(cl⋆(a))) is ir−closed, a ∈ αai(x) and so sβio(x) ∩ αin5(x) ⊂ αai(x). therefore, αai(x) = sβio(x) ∩ αin5(x). if a ∈ sio(x), then a ∈ sβio(x) by proposition 1(d) of [9]. moreover, if v = a ∪ (x − cl⋆(int(a))), then a = v ∩cl⋆(int(a)). also, int(cl⋆(int(v ))) = int(cl⋆(int(a∪(x −cl⋆(int(a)))))) ⊃ int(cl⋆(int(a)∪int(x−cl⋆(int(a))))) = int(cl⋆(int(a)) ∪ cl⋆(int(x−cl⋆(int(a))))) ⊃ int(cl⋆(int(a)) ∪ int(x − cl⋆(int(a)))) ⊃ int(int(cl⋆(int(a)) ∪ (x − cl⋆(int(a))))) = int(x) = x ⊃ v and so v is α − i−open. therefore, a ∈ αin5(x) and hence sio(x) ⊂ sβio(x) ∩ αin5(x). conversely, suppose a ∈ sβio(x) ∩ αin5(x). a ∈ αin5(x) implies that a = u ∩ v where u is α − i−open and v is ⋆−closed. since a ∈ sβio(x), a ⊂ cl⋆(int(cl⋆(a))) = cl⋆(int(cl⋆(u ∩ v ))) ⊂ cl⋆(int(cl⋆(int(cl⋆(int(u ))) ∩ v ))) ⊂ cl⋆(int(cl⋆(int(cl⋆(int(u )))) ∩ v )) = cl⋆(int(cl⋆(int(u )) ∩ v )) = cl⋆(int(cl⋆(int(u ))) ∩ int(v )) ⊂ cl⋆(int(cl⋆(int(u ) ∩ int(v )))) = cl⋆(int(cl⋆(int(u ∩ v )))) = cl⋆(int(u ∩ v )) = cl⋆(int(a)). therefore, a ∈ sio(x) which implies that sβio(x) ∩ αin5(x) ⊂ sio(x). hence sβio(x) ∩ αin5(x) = sio(x). corollary 2.15. let (x, τ, i) be an ideal space and a ⊂ x. then the following are equivalent. (a) a is α − i−open. (b) a is pre−i−open and semi−i−open [4, proposition 1.1]. (c) a is a b2i−set and αai−set[4, theorem 2.3]. proof. (a) and (b) are equivalent by proposition 1.1 of [4]. (b) and (c) are equivalent by theorem 2.10 and theorem 2.14. corollary 2.16. let (x, τ, i) be an ideal space and a ⊂ x. then the following are equivalent. (a) a is open. (b) a is α − i−open and air − set. (c) a is pre−i−open and air−set. (d) a is α − i−open and weakly i−locally closed. (e) a is α − i−open and bi−set. (f) a is α − i−open and ci−set. proof. (a) and (b) are equivalent by theorem 2.1 of [4]. that (b) implies (c) is clear. (c) and (d) are equivalent by proposition 2.2 of [4]. (d) implies (e) and (e) implies (f) are clear. (f) implies (a) follows from proposition 3.3 of [8]. a subset a of an ideal space (x, τ, i) is said to be a a3i−set (resp.b3i−set [4](αin1−set [1]) ) if a = u ∩ v where u is open (resp.α − i−open ) and cl⋆(int(v )) ⊂ v. we will denote the family of all a3i−sets (resp. b3i−sets) by a3i(x) (resp.b3i(x)). clearly, a3i(x) ⊂ b3i(x). the following example 2.17 shows that the reverse direction is not true. example 2.18 below shows that a2i−sets and a3i−sets are independent concepts. theorem 2.19 below gives a characterization of air−sets in terms of a3i−sets. cubo 12, 2 (2010) on subsets of ideal topological spaces 49 example 2.17. consider the ideal space (x, τ, i) of example 2.12. if a = {a, b}, then int(cl⋆(int(a))) = int(cl⋆(int({a, b}))) = int(cl⋆({a})) = int(x) = x ⊃ a and so a is an α − i−open set and hence a is a b3i−set. since cl ⋆(int(a)) * a and x is the only open set containing a, a is not an a3i−set. example 2.18. (a) consider the ideal space (x, τ, i) of example 2.12. if a = {a, b}, then a is not an a3i−set. since cl ⋆(a) = a ∪ a⋆ = {a, b} ∪ x = x, and so a is an a2i−set. (b) consider the ideal space (x, τ, i) where x = {a, b, c, d}, τ = {∅, {d}, {a, c}, {a, c, d}, x} and i = {∅, {c}, {d}, {c, d} }. if a = {a, b, c}, then cl⋆(int(a)) = int(a) ∪ (int(a))⋆ = {a, c} ∪ {a, b, c} = {a, b, c} = a and so a is an a3i−set. since cl ⋆(a) = a ∪ a⋆ = {a, b, c} 6= x, a is not an a2i−set. theorem 2.19. let (x, τ, i) be an ideal space . then air(x) = sio(x) ∩ a3i(x). proof. suppose a ∈ air(x). clearly, a ∈ a3i(x). by theorem 3.3 of [1], a ∈ sio(x). therefore, air(x) ⊂ sio(x)∩a3i (x). conversely, suppose a ∈ sio(x)∩a3i (x). a ∈ a3i(x) implies that a = u ∩ v where u is open and cl⋆(int(v )) ⊂ v. a ∈ sio(x) implies that a ⊂ cl⋆(int(a)) and so a = a ∩ cl⋆(int(a)) = (u ∩ v ) ∩ cl⋆(int(u ∩ v )) ⊂ u ∩ cl⋆(int(u ∩ v )) = u ∩ cl⋆(u ∩ int(v )) ⊂ u ∩ cl⋆(u ) ∩ cl⋆(int(v )) ⊂ u ∩ v = a and so a = u ∩ cl⋆(int(u ∩ v )) = u ∩ cl⋆(int(a)). since cl⋆(int(a)) is ir−closed, a ∈ air(x). therefore, air(x) = sio(x) ∩ a3i(x). corollary 2.20. let (x, τ, i) be an ideal space and a ⊂ x. then the following are equivalent. (a)a ∈ air(x). (b)a ∈ sio(x) ∩ a3i(x). (c) a ∈ αai(x) ∩ a3i(x). (d)a ∈ sβio(x) ∩ αin5(x) ∩ a3i(x). (e)a ∈ sβio(x) ∩ w ilc(x). proof. (a), (b), (c) and (d) are equivalent by theorem 2.14 and theorem 2.19. (a) and (e) are equivalent by theorem 2.10 of [15]. by remark 3.3 of [8], every bi−set is a ci−set but the reverse direction is not true. the following theorem 2.22 gives characterizations of bi−sets in terms of ci−sets. a subset a of an ideal space (x, τ, i) is said to be an αbi−set (αin3−set [1]) if a = u ∩v where u ∈ αio(x) and v is a t−i−set. a subset a of an ideal space (x, τ, i) is said to be an αci−set [4](αin4−set [1]) if a = u ∩ v where u ∈ αio(x) and v is a α⋆ − i−set. clearly every αbi−set is an αci−set [1, proposition 3.2(c)] but not the converse [1, example 3.4]. we will denote the family of all αbi−sets (resp.αci−sets) in (x, τ, i) by αbi(x) (resp.αci(x)). we define di(x) = {a ⊂ x | int(a) = piint(a)} and if a ∈ di, then a is called a di−set. the following lemma 2.21 characterizes αbi−sets and αci−sets, the proof, which is similar to the proof of theorem 2.7, is omitted. corollary 2.23 follows from theorem 2.22. lemma 2.21. let (x, τ, i) be an ideal space and a be a subset of x. then the following hold. (a) a is a αbi−set if and only if there exists an α − i−open set u such that a = u ∩ sicl(a). (b) a is an αci−set if and only if there exists an α − i−open set u such that a = u ∩ spicl(a). theorem 2.22. let (x, τ, i) be an ideal space and a ⊂ x. then the following are equivalent. (a) a is a di−set and a ci−set. (b) a is a δ − i−open set and a ci−set. 50 v. renukadevi cubo 12, 2 (2010) (c) a is a bi−set. (d) a is an αbi−set and a ci−set. proof. (a)⇒(b). suppose a ∈ di(x) ∩ ci(x). if a ∈ di(x), then int(a) = piint(a). now int(cl⋆(a)) = cl⋆(a) ∩ int(cl⋆(a)) ⊂ cl⋆(a ∩ int(cl⋆(a))) = cl⋆ (piint(a)) = cl⋆(int(a)) and so a is a δ − i−open set. this proves (b). (b)⇒(c). suppose a is a δ − i−open set and a ci−set. then, by theorem 2.4 of [12], int(cl ⋆(a)) = int(cl⋆(int(a))) and so a∪int(cl⋆(a)) = a∪int(cl⋆(int(a))) which implies that sicl(a) = spicl(a). if a is a ci−set, then theorem 2.7, a = u ∩ spicl(a) for some open set u and so a = u ∩ sicl(a) for some open set u which implies that a is a bi−set. (c)⇒(a). clearly, every bi−set is a ci−set. if a is a bi−set, then a = u ∩ v where u is open and int(cl⋆(v )) = int(v ). now piint(a) = a ∩ int(cl⋆(a)) = a ∩ int(cl⋆(u ∩ v )) ⊂ a ∩ int(cl⋆(u ) ∩ cl⋆(v )) = a ∩ int(cl⋆(u )) ∩ int(cl⋆(v )) = (u ∩ v ) ∩ int(cl⋆(u )) ∩ int(v ) = u ∩ int(v ) = int(u ∩ v ) = int(a). but always, int(a) ⊂ piint(a) and so int(a) = piint(a) which implies that a is a di−set. this proves (a). (c)⇒(d) is clear. (d)⇒(c). if a is an αbi−set, then a = u ∩ v where u is α − i−open and int(cl ⋆(v )) = int(v ). now a ⊂ u implies that a ⊂ int(cl⋆(int(u ))) and so int(cl⋆(a)) ⊂ int(cl⋆(int(cl⋆(int(u ))))) = int(cl⋆(int(u ))) ⊂ int(cl⋆(u )). again,a ⊂ v implies that int(cl⋆(a)) ⊂ int(cl⋆(v )) = int(v ). therefore, int(cl⋆(a)) ⊂ int(cl⋆(u )) ∩ int(v ) ⊂ cl⋆(int(u ) ∩ int(v )) ⊂ cl⋆(int(u ∩ v )) = cl⋆(int(a)) and so int(cl⋆(a)) = int(cl⋆(int(a))) which implies that a ∪ int(cl⋆(a)) = a ∪ int(cl⋆(int(a))). hence sicl(a) = spicl(a). since a is a ci−set, by theorem 2.7, a = g ∩ spicl(a) for some open set g and so a = g ∩ sicl(a). therefore, a is a bi−set. corollary 2.23. let (x, τ, i) be an ideal space. then the following hold. (a) every bi−set is a di−set. (b) every bi−set is a αbi−set. (c) every di−set is a δ − i−open set (proof follows from (a)⇒(b) of theorem 2.22). the following theorem 2.24 characterizes αbi−open sets in terms of δ − i−open sets and αci−open sets. example 2.25 below shows that δ − i−openness and αci−openness are independent concepts. theorem 2.24. let (x, τ, i) be an ideal space. then αbi(x) = δio(x) ∩ αci(x). proof. clearly, αbi(x) ⊂ αci(x). if a ∈ αbi(x), then a = u ∩ v where u is α − i−open and v is a t − i−set. a ⊂ u implies that int(cl⋆(a)) ⊂ int(cl⋆(u )) ⊂ int(cl⋆(int(cl⋆(int(u ))))) ⊂ int(cl⋆(int(u ))) ⊂ cl⋆(int(u )). also, a ⊂ v implies that int(cl⋆(a)) ⊂ int(cl⋆(v )) = int(v ) and so int(cl⋆(a)) ⊂ cl⋆(int(u )) ∩ int(v ) ⊂ cl⋆(int(u ) ∩ int(v )) = cl⋆(int(u ∩ v )) = cl⋆(int(a). therefore, a ∈ δio(x). hence αbi(x) ⊂ δio(x) ∩ αci(x). conversely, suppose a ∈ δio(x) ∩ αci(x). a ∈ δio(x) implies that int(cl⋆(a)) = int(cl⋆(int(a))) and so sicl(a) = spicl(a). a ∈ αci(x) implies that a = u ∩ spicl(a) for some α − i−open set u by lemma 2.21 and so a = u ∩ sicl(a) for some α −i−open set u which implies that a ∈ αbi(x). therefore, δio(x) ∩ αci(x) ⊂ αbi(x). this completes the proof. example 2.25. (a) let x = {a, b, c, d}, τ = {∅, {d}, {a, b}, {a, b, d}, x} and i = {∅, {c} }. if a = {a, c}, then int(cl⋆(int(a))) = int(cl⋆(int({a, c}))) = int(cl⋆(∅)) = ∅ = int(a). therefore, a is cubo 12, 2 (2010) on subsets of ideal topological spaces 51 an α⋆ − i−set and hence an αci−set. but int(cl ⋆(a)) = int({a, b, c}) = {a, b} and cl⋆(int(a)) = cl⋆(∅) = ∅ and so a is not a δ − i−set. (b) let x = {a, b, c, d}, τ = {∅, {a}, {c}, {a, c}, x} and i = {∅, {a} }. if a = {a, b, c}, then a is neither open nor an α⋆ − i−set and so a is not an αci−set. but int(cl ⋆(a)) = int({a, b, c, d}) = x and cl⋆(int(a)) = cl⋆({a, c}) = x and so a is a δ − i−set. an ideal space (x, τ, i) is said to be ⋆−extremally disconnected [7] if the τ ⋆−closure (⋆−closure) of every open set is open. clearly, b3i(x) ⊂ αci(x). by example 3.6 of [1] the reverse direction is not true. the following theorem 2.26 shows that for ⋆−extremally disconnected spaces, the two collection of sets coincide. example 2.27 below shows that αci (x) = b3i(x) does not imply that the space is ⋆−extremally disconnected. theorem 2.26. let (x, τ, i) be a ⋆−extremally disconnected ideal space. then b3i(x) = αci(x). proof. enough to prove that αci(x) ⊂ b3i(x). suppose a ∈ αci(x). then a = u ∩ v where u is α − i−open and int(cl⋆(int(v ))) = int(v ). since (x, τ, i) is ⋆−extremally disconnected, cl⋆(int(v )) is open and so int(v ) = int(cl⋆(int(v ))) = cl⋆(int(v )). therefore, a ∈ b3i(x). this completes the proof. example 2.27. consider the ideal space (x, τ, i) where x = {a, b, c}, τ = {∅, {b}, {c}, {b, c}, x} and i = {∅, {a}}. if a = {b}, a is open and cl⋆(a) = {b} ∪ {a, b} = {a, b}, which is not open. hence (x, τ, i) is not ⋆−extremally disconnected but ℘(x) = αci(x) = b3i(x). received: november 2008. revised: february 2009. references [1] a. açikgöz and ş.yüksel, some new sets and decompositions of ai−r−continuity, α − i−continuity, continuity via idealization, acta math. hungar., 114(1 2)(2007), 79 89. [2] a. açikgöz, t. noiri and ş.yüksel, on δ −i−open sets and decomposition of α−continuity, acta math. hungar., 102(2004), 349 357. [3] a. açikgöz, ş.yüksel and noiri, α − i−preirresolute functions and β − i−preirresolute functions, bull. malays. math. sci. soc., (2)28(1)(2005), 1 8. [4] g. aslim, a. caksu guler and t. noiri, on decompositions of continuity and some weaker forms of continuity via idealization, acta math. hungar., 109(3)(2005), 183 190. [5] á. császár, on the γ−interior and γ−closure of a set, acta math. hungar., 80(1998), 89 93. [6] j. dontchev, on pre−i−open sets and a decomposition of i−continuity, banyan math.j., 2(1996). [7] e. ekici and t. noiri, ⋆−extremally disconnected ideal topological spaces, acta math. hungar., to appear. [8] e. hatir and t. noiri, on decomposition of continuity via idealization, acta math. hungar., 96(4)(2002), 341 349. 52 v. renukadevi cubo 12, 2 (2010) [9] e. hatir, a. keskin and t. noiri, on a new decomposition of continuity via idealization, jp jour. geometry and topology, 3(2003), 53 64. [10] e. hayashi, topologies defined by local properties, math. ann., 156(1964), 205 215. [11] d. jankovic and t.r. hamlett, new topologies from old via ideals, amer. math. monthly, 97 no.4 (1990), 295 310. [12] v. jeyanthi and d. sivaraj, a note on δ − i−sets, acta math. hungar., 114(1 2)(2007), 165 169. [13] a. keskin, ş. yüksel and t. noiri, decompositions of i−continuity and continuity, commun. fac. sci. univ. ank. serials a1, 53(2004), 67 75. [14] k. kuratowski, topology, vol. i, academic press, new york, 1966. [15] v. renukadevi, note on ir−closed and air−sets, acta math. hungar., to appear. [16] r. vaidyanathaswamy, the localization theory in set topology, proc. indian acad. sci. math. sci., 20 (1945), 51 61. [17] s. yüksel, a. açikgöz and t. noiri, on δ − i−continuous functions, turk. j. math., 29(2005), 39 51. articulo 13.dvi cubo a mathematical journal vol.12, no¯ 02, (199–215). june 2010 fixed point theory for compact absorbing contractions in extension type spaces donal o’regan department of mathematics, national university of ireland, galway, ireland email: donal.oregan@nuigalway.ie abstract several new fixed point results for self maps in extension type spaces are presented in this paper. in particular we discuss compact absorbing contractions. resumen son presentados en este art́ıculo varios resultados nuevos de punto fijo para autoaplicaciones en espacios de tipo extensión. en particular discutimos contracciones compactas absorbentes. key words and phrases: extension spaces, fixed point theory, compact absorbing contractions. ams (mos) subj. class.: 47h10 1 introduction in sections 2, 3 and 4 we present new results on fixed point theory in extension type spaces. section 2 discusses compact self-maps on nes, anes and sanes spaces whereas section 3 discusses compact absorbing contractions. in section 4 we provide an alternative approach using projective 200 donal o’regan cubo 12, 2 (2010) limits. these results improve those in the literature; see [1-3, 5, 8-11, 14-15] and the references therein. our results were motivated in part from ideas in [1, 2, 9, 12, 15]. for the remainder of this section we present some definitions and known results which will be needed throughout this paper. suppose x and y are topological spaces. given a class x of maps, x(x,y ) denotes the set of maps f : x → 2y (nonempty subsets of y ) belonging to x , and xc the set of finite compositions of maps in x . we let f(x) = {z : fixf 6= ∅ for all f ∈x(z,z)} where fixf denotes the set of fixed points of f . the class a of maps is defined by the following properties: (i). a contains the class c of single valued continuous functions; (ii). each f ∈ac is upper semicontinuous and closed valued; and (iii). bn ∈f(ac) for all n ∈{1, 2, ....}; here b n = {x ∈ rn : ‖x‖≤ 1}. remark 1.1. the class a is essentially due to ben-el-mechaiekh and deguire [6]. a includes the class of maps u of park (u is the class of maps defined by (i), (iii) and (iv). each f ∈uc is upper semicontinuous and compact valued). thus if each f ∈ ac is compact valued the class a and u coincide and this is what occurs in section 2 since our maps will be compact. the following result can be found in [6, proposition 2.2] (see also [9 pp. 286] for a special case). theorem 1.1. the hilbert cube i∞ (subset of l2 consisting of points (x1,x2, ...) with |xi| ≤ 1 2i for all i) and the tychonoff cube t (cartesian product of copies of the unit interval) are in f(ac). we next consider the class uκc (x,y ) (respectively a κ c (x,y )) of maps f : x → 2 y such that for each f and each nonempty compact subset k of x there exists a map g ∈uc(k,y ) (respectively g ∈ac(k,y )) such that g(x) ⊆ f(x) for all x ∈ k. theorem 1.2. i∞ and t are in f(aκc ) (respectively f(u κ c )). proof: let f ∈ aκc (i ∞,i∞) and we must show fixf 6= ∅. now by definition there exists g ∈ ac(i ∞,i∞) with g(x) ⊆ f(x) for all x ∈ i∞, so theorem 1.1 guarantees that there exists x ∈ i∞ with x ∈ gx. in particular x ∈ f x so fixf 6= ∅. thus i∞ ∈f(aκc ). 2 notice [14] that uκc is closed under compositions. the class u κ c include (see [3]) the kakutani maps, the acyclic maps, the o’neill maps, the approximable maps and the maps admissible with respect to gorniewicz. for a subset k of a topological space x, we denote by covx (k) the set of all coverings of k by open sets of x (usually we write cov (k) = covx (k)). given a map f : x → 2 x and α ∈ cov (x), a point x ∈ x is said to be an α–fixed point of f if there exists a member u ∈ α such that x ∈ u and f(x) ∩u 6= ∅. given two maps single valued f, g : x → y and α ∈ cov (y ), f and g are said to be α–close if for any x ∈ x there exists ux ∈ α containing both f(x) and g(x). cubo 12, 2 (2010) fixed point theorems 201 we say f and g are α-homotopic if there is a homotopy hh : x → y (0 ≤ t ≤ 1) joining f and g such that for each x ∈ x the values ht(x) belong to a common ux ∈ α for all t ∈ [0, 1]. the following results can be found in [4, lemma 1.2 and 4.7]. theorem 1.3. let x be a regular topological space and f : x → 2x an upper semicontinuous map with closed values. suppose there exists a cofinal family of coverings θ ⊆ covx (f(x)) such that f has an α–fixed point for every α ∈ θ. then f has a fixed point. remark 1.2. from theorem 1.3 in proving the existence of fixed points in uniform spaces for upper semicontinuous compact maps with closed values it suffices [5 pp. 298] to prove the existence of approximate fixed points (since open covers of a compact set a admit refinements of the form {u[x] : x ∈ a} where u is a member of the uniformity [13 pp. 199] so such refinements form a cofinal family of open covers). note also uniform spaces are regular (in fact completely regular) [7 pp. 431] (see also [7 pp. 434]). note in theorem 1.3 if f is compact valued then the assumption that x is regular can be removed. for convenience in this paper we will apply theorem 1.3 only when the space is uniform. let x, y and γ be hausdorff topological spaces. a continuous single valued map p : γ → x is called a vietoris map (written p : γ ⇒ x) if the following two conditions are satisfied: (i). for each x ∈ x, the set p−1(x) is acyclic (ii). p is a proper map i.e. for every compact a ⊆ x we have that p−1(a) is compact. let d(x,y ) be the set of all pairs x p ⇐ γ q → y where p is a vietoris map and q is continuous. we will denote every such diagram by (p,q). given two diagrams (p,q) and (p′,q′), where x p ′ ⇐ γ′ q ′ → y , we write (p,q) ∼ (p′,q′) if there are maps f : γ → γ′ and g : γ′ → γ such that q′ ◦f = q, p′◦f = p, q◦g = q′ and p◦g = p′. the equivalence class of a diagram (p,q) ∈ d(x,y ) with respect to ∼ is denoted by φ = {x p ⇐ γ q → y} : x → y or φ = [(p,q)] and is called a morphism from x to y . we let m(x,y ) be the set of all such morphisms. for any φ ∈ m(x,y ) a set φ(x) = q p−1 (x) where φ = [(p,q)] is called an image of x under a morphism φ. consider vector spaces over a field k. let e be a vector space and f : e → e an endomorphism. now let n(f) = {x ∈ e : f(n)(x) = 0 for some n} where f(n) is the nth iterate of f, and let ẽ = e\n(f). since f(n(f)) ⊆ n(f) we have the induced endomorphism f̃ : ẽ → ẽ. we call f admissible if dimẽ < ∞; for such f we define the generalized trace tr(f) of f by putting tr(f) = tr(f̃) where tr stands for the ordinary trace. let f = {fq} : e → e be an endomorphism of degree zero of a graded vector space e = {eq}. we call f a leray endomorphism if (i). all fq are admissible and (ii). almost all ẽq are trivial. for such f we define the generalized lefschetz number λ(f) by λ(f) = ∑ q (−1)q tr (fq). 202 donal o’regan cubo 12, 2 (2010) a linear map f : e → e of a vector space e into itself is called weakly nilpotent provided for every x ∈ e there exists nx such that f nx (x) = 0. assume that e = {eq} is a graded vector space and f = {fq} : e → e is an endomorphism. we say that f is weakly nilpotent iff fq is weakly nilpotent for every q. it is well known [9, pp 53] that any weakly nilpotent endomorphism f : e → e is a leray endomorphism and λ(f) = 0. let h be the c̆ech homology functor with compact carriers and coefficients in the field of rational numbers k from the category of hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. thus h(x) = {hq(x)} is a graded vector space, hq(x) being the q–dimensional c̆ech homology group with compact carriers of x. for a continuous map f : x → x, h(f) is the induced linear map f⋆ = {f⋆ q} where f⋆ q : hq(x) → hq(x). with c̆ech homology functor extended to a category of morphisms (see [10 pp. 364]) we have the following well known result (note the homology functor h extends over this category i.e. for a morphism φ = {x p ⇐ γ q → y} : x → y we define the induced map h (φ) = φ⋆ : h(x) → h(y ) by putting φ⋆ = q⋆ ◦p −1 ⋆ ). recall the following result [8 pp. 227]. theorem 1.4. if φ : x → y and ψ : y → z are two morphisms (here x, y and z are hausdorff topological spaces) then (ψ ◦ φ)⋆ = ψ⋆ ◦ φ⋆. two morphisms φ, ψ ∈ m(x,y ) are homotopic (written φ ∼ ψ) provided there is a morphism χ ∈ m(x × [0, 1],y ) such that χ(x, 0) = φ(x), χ(x, 1) = ψ(x) for every x ∈ x (i.e. φ = χ ◦ i0 and ψ = χ ◦ i1, where i0, i1 : x → x × [0, 1] are defined by i0(x) = (x, 0), i1(x) = (x, 1)). recall the following result [9, pp. 231]: if φ ∼ ψ then φ⋆ = ψ⋆. let φ : x → y be a multivalued map (note for each x ∈ x we assume φ(x) is a nonempty subset of y ). a pair (p,q) of single valued continuous maps of the form x p ← γ q → y is called a selected pair of φ (written (p,q) ⊂ φ) if the following two conditions hold: (i). p is a vietoris map and (ii). q (p−1(x)) ⊂ φ(x) for any x ∈ x. definition 1.1. a upper semicontinuous map φ : x → y is said to be strongly admissible [9, 10] (and we write φ ∈ ads(x,y )) provided there exists a selected pair (p,q) of φ with φ(x) = q (p−1(x)) for x ∈ x. cubo 12, 2 (2010) fixed point theorems 203 definition 1.2. a map φ ∈ ads(x,x) is said to be a lefschetz map if for each selected pair (p,q) ⊂ φ with φ(x) = q (p−1(x)) for x ∈ x the linear map q⋆ p −1 ⋆ : h(x) → h(x) (the existence of p−1⋆ follows from the vietoris theorem) is a leray endomorphism. when we talk about φ ∈ ads it is assumed that we are also considering a specified selected pair (p,q) of φ with φ(x) = q (p−1(x)). remark 1.3. in fact since we specify the pair (p,q) of φ it is enough to say φ is a lefschetz map if φ⋆ = q⋆ p −1 ⋆ : h(x) → h(x) is a leray endomorphism. however for the examples of φ, x known in the literature [9] the more restrictive condition in definition 1.2 works. we note [9, pp 227] that φ⋆ does not depend on the choice of diagram from [(p,q)], so in fact we could specify the morphism. if φ : x → x is a lefschetz map as described above then we define the lefschetz number (see [9, 10]) λ (φ) (or λx (φ)) by λ (φ) = λ(q⋆ p −1 ⋆ ). if we do not wish to specify the selected pair (p,q) of φ then we would consider the lefschetz set λ (φ) = {λ(q⋆ p −1 ⋆ ) : φ = q (p −1)}. definition 1.3. a hausdorff topological space x is said to be a lefschetz space provided every compact φ ∈ ads(x,x) is a lefschetz map and λ(φ) 6= 0 implies φ has a fixed point. definition 1.4. a upper semicontinuous map φ : x → y with closed values is said to be admissible (and we write φ ∈ ad(x,y )) provided there exists a selected pair (p,q) of φ. definition 1.5. a map φ ∈ ad(x,x) is said to be a lefschetz map if for each selected pair (p,q) ⊂ φ the linear map q⋆ p −1 ⋆ : h(x) → h(x) (the existence of p −1 ⋆ follows from the vietoris theorem) is a leray endomorphism. if φ : x → x is a lefschetz map, we define the lefschetz set λ (φ) (or λx (φ)) by λ (φ) = { λ(q⋆ p −1 ⋆ ) : (p,q) ⊂ φ } . definition 1.6. a hausdorff topological space x is said to be a lefschetz space provided every compact φ ∈ ad(x,x) is a lefschetz map and λ(φ) 6= {0} implies φ has a fixed point. recall the following result [8]. theorem 1.5. every open subset of the tychonoff cube is a lefschetz space. the following concepts will be needed in section 4. let (x,d) be a metric space and s a nonempty subset of x. for x ∈ x let d(x,s) = infy∈s d(x,y). also diams = sup{d(x,y) : x,y ∈ s}. we let b(x,r) denote the open ball in x centered at x of radius r and by b(s,r) we denote ∪x∈s b(x,r). for two nonempty subsets s1 and s2 of x we define the generalized hausdorff distance h to be h(s1,s2) = inf{ǫ > 0 : s1 ⊆ b(s2,ǫ), s2 ⊆ b(s1,ǫ)}. 204 donal o’regan cubo 12, 2 (2010) now suppose g : s → 2x . then g is said to be hemicompact if each sequence {xn}n∈n in s has a convergent subsequence whenever d(xn,g (xn)) → 0 as n →∞. now let i be a directed set with order ≤ and let {eα}α∈i be a family of locally convex spaces. for each α ∈ i, β ∈ i for which α ≤ β let πα,β : eβ → eα be a continuous map. then the set { x = (xα) ∈ ∏ α∈i eα : xα = πα,β (xβ ) ∀α, β ∈ i, α ≤ β } is a closed subset of ∏ α∈i eα and is called the projective limit of {eα}α∈i and is denoted by lim← eα (or lim←{eα,πα,β} or the generalized intersection [1, 2] ∩α∈i eα.) 2 preliminary fixed point theory the fixed point theory presented in this section can partly be found in [9, 14]. however for the convenience of the reader we present the following elementary approach. by a space we mean a hausdorff topological space. let x and y be spaces. a space y is an neighborhood extension space for q (written y ∈ nes(q)) if ∀x ∈ q, ∀k ⊆ x closed in x, and for any continuous function f0 : k → y , there exists a continuous extension f : u → y of f0 over a neighbourhood u of k in x. let x ∈ nes(compact) and f ∈ uκc (x,x) a compact map. now let k = f(x). we know [12] that k can be embedded as a closed subset k⋆ of t ; let s : k → k⋆ be a homeomorphism. also let i : k →֒ x be an inclusion. let u be an open neighbourhood of k⋆ in t and hu : u → x be a continuous extension of is−1 : k⋆ → x on u (guaranteed since x ∈ nes(compact)). let ju : k ⋆ →֒ u be the natural embedding so hu ju = is −1. finally let g = ju sf hu . notice g ∈uκc (u,u). we now assume (2.1) g ∈uκc (u,u) has a fixed point. then there exists x ∈ u with x ∈ gx. let y = hu (x), so y ∈ hu ju sf (y) i.e. y = hu ju s (q) for some q ∈ f (y). since hu ju (z) = is −1(z) for z ∈ k⋆, we have hu ju s (q) = i (q), so y ∈ f(y). theorem 2.1. let x ∈ nes(compact) and f ∈ uκc (x,x) a compact map. also assume (2.1) holds with k, k⋆, u, s, i, ju and hu as described above. then f has a fixed point. we discuss theorem 2.1 for the class ad(x,x). let x ∈ nes(compact) and f ∈ ad(x,x) a compact map. also let k, k⋆, u, s, i, ju and hu as described above. let (p,q) be a selected pair for f . now since f hu ∈ ad(u,x) then [9, section 40] guarantees that there exists a selected pair (p′,q′) of f hu with (q ′)⋆ (p ′)−1⋆ = q⋆ p −1 ⋆ (hu )⋆. notice (q′)⋆ (p ′)−1⋆ (ju )⋆ s⋆ = q⋆ p −1 ⋆ (hu )⋆ (ju )⋆ s⋆ = q⋆ p −1 ⋆ since hu ju s = is −1 s. next note g = ju sf hu ∈ ad(u,u) has a selected pair (p ′,ju sq ′) (since ju sq ′ (p′)−1(x) ⊆ ju sf hu (x) = g(x) for x ∈ u) and from theorem 1.5 we know u is a lefschetz space so (ju sq ′)⋆ (p ′)−1⋆ is a leray endomorphism. notice (ju )⋆ s⋆ (q ′)⋆ (p ′)−1⋆ = (ju sq ′)⋆ (p ′)−1⋆ cubo 12, 2 (2010) fixed point theorems 205 and from above (q′)⋆ (p ′)−1⋆ (ju )⋆ s⋆ = q⋆ p −1 ⋆ so [8, page 314, see (1.3)] (here e ′ = u′, e′′ = u′′, u = (q′)⋆ (p ′)−1⋆ , v = (ju )⋆ s⋆, f ′ = (ju sq ′)⋆ (p ′)−1⋆ and f ′′ = q⋆ p −1 ⋆ ) guarantees that q⋆ p −1 ⋆ is a leray endomorphism and λ (q⋆ p −1 ⋆ ) = λ ((ju sq ′)⋆ (p ′)−1⋆ ). thus λ (f) is well defined. next suppose λ (f) 6= {0}. then there exists a selected pair (p,q) as described above with λ (q⋆ p −1 ⋆ ) 6= 0. let p ′ and q′ be as described above with λ ((ju sq ′)⋆ (p ′)−1⋆ ) = λ (q⋆ p −1 ⋆ ) 6= 0. now since u is a lefschetz space there exists x ∈ u with x ∈ ju sq ′ (p′)−1(x) i.e. x ∈ g(x) so (2.1) is satisfied. combining with theorem 2.1 we have the following result. theorem 2.2. let x ∈ nes(compact) and f ∈ ad(x,x) a compact map. then λ (f) is well defined and if λ (f) 6= {0} then f has a fixed point. remark 2.1. theorem 2.2 says that nes(compact) spaces are lefschetz spaces (for the class ad). remark 2.2. essentially the same reasoning as in theorem 2.2 establishes: let x ∈ nes(compact) and f ∈ ads(x,x) a compact map. then λ (f) is well defined and if λ (f) 6= 0 then f has a fixed point i.e. nes(compact) spaces are lefschetz spaces (for the class ads). a space y is a approximate neighborhood extension space for q (written y ∈ anes(q)) if ∀α ∈ cov (y ), ∀x ∈ q, ∀k ⊆ x closed in x, and any continuous function f0 : k → y , there exists a neighborhood uα of k in x and a continuous function fα : uα → y such that fα|k and f0 are α-close. let x ∈ anes(compact) be a uniform space and f ∈uκc (x,x) a compact upper semicontinuous map with closed values. also let α ∈ covx (k) where k = f(x). to show f has a fixed point it suffices (theorem 1.3 with remark 1.2) to show f has an α–fixed point. let α′ = α∪{x\k} and let k⋆, s and i be as above. since x ∈ anes(compact) there exists an open neighborhood uα of k⋆ in t and fα : uα → x a continuous function such that fα|k⋆ and s −1 are α′–close and as a result fα juα s : k → x and i : k → x are α–close; here juα : k ⋆ →֒ uα is the natural imbedding. finally let gα = juα sf fα. notice gα ∈u κ c (uα,uα) is a compact upper semicontinuous map with closed values. we now assume (2.2) gα ∈u κ c (uα,uα) has a fixed point for each α ∈ covx (f(x)). we still have α ∈ covx (k) fixed and we let x be a fixed point of gα. now let y = fα(x) so y ∈ fα juα sf(y) i.e. y = fα juα s(q) for some q ∈ f(y). now since fα juα s and i are α–close there exists u ∈ α with fα juα s(q) ∈ u and i(q) ∈ u i.e. q ∈ u and y = fα juα s(q) ∈ u. thus q ∈ u and y ∈ u so y ∈ u and f(y) ∩u 6= ∅ since q ∈ f (y). theorem 2.3. let x ∈ anes(compact) be a uniform space and f ∈ uκc (x,x) a compact upper semicontinuous map with closed values. also assume (2.2) holds with k, k⋆, uα, s, juα , i and fα as described above. then f has a fixed point. we discuss theorem 2.3 for the class ad(x,x). first however we need the following definition. a space y is a strongly approximate neighborhood extension space for q (written y ∈ sanes(q)) if ∀α ∈ cov (y ), ∀x ∈ q, ∀k ⊆ x closed in x, and any continuous function f0 : k → y , there 206 donal o’regan cubo 12, 2 (2010) exists a neighborhood uα of k in x and a continuous function fα : uα → y such that fα|k and f0 are α close and α-homotopic. let x ∈ sanes(compact) be a uniform space and f ∈ ad(x,x) a compact map. also let k, k⋆, uα, s, juα , i and fα as described above. let (p,q) be a selected pair for f . now since f fα ∈ ad(uα,x) then [9, section 40] guarantees that there exists a selected pair (p ′ α,q ′ α) of f fα with (q′α)⋆ (p ′ α) −1 ⋆ = q⋆ p −1 ⋆ (fα)⋆. as a result (q′α)⋆ (p ′ α) −1 ⋆ (juα )⋆ s⋆ = q⋆ p −1 ⋆ (fα)⋆ (juα )⋆ s⋆ = q⋆ p −1 ⋆ since fα juα s is α homotopic to i (note fα|k⋆ and s −1 are α′-homotopic by definition). next note gα = juα sf fα ∈ ad(uα,uα) has a selected pair (p ′ α,juα sq ′ α) and from theorem 1.5 we have that (juα sq ′ α)⋆ (p ′ α) −1 ⋆ is a leray endomorphism. now since (juα )⋆ s⋆ (q ′ α)⋆ (p ′ α) −1 ⋆ = (juα sq ′ α)⋆ (p ′ α) −1 ⋆ and from above (q′α)⋆ (p ′ α) −1 ⋆ (juα )⋆ s⋆ = q⋆ p −1 ⋆ then [8, page 314, see (1.3)] guarantees that q⋆ p −1 ⋆ is a leray endomorphism and we have λ (q⋆ p −1 ⋆ ) = λ ((juα sq ′ α)⋆ (p ′ α) −1 ⋆ ). thus λ (f) is well defined. next suppose λ (f) 6= {0}. then there exists a selected pair (p,q) as described above with λ (q⋆ p −1 ⋆ ) 6= 0. let p ′ α and q ′ α be as described above with λ ((juα sq ′ α)⋆ (p ′ α) −1 ⋆ ) = λ (q⋆ p −1 ⋆ ) 6= 0. now since uα is a lefschetz space there exists x ∈ uα with x ∈ juα sq ′ α (p ′ α) −1(x) i.e. x ∈ gα(x) so (2.2) is satisfied. combining with theorem 2.3 we have the following result. theorem 2.4. let x ∈ sanes(compact) be a uniform space and f ∈ ad(x,x) a compact map. then λ (f) is well defined and if λ (f) 6= {0} then f has a fixed point. remark 2.3. theorem 2.4 says that sanes(compact) uniform spaces are lefschetz spaces (for the class ad). remark 2.4. essentially the same reasoning as in theorem 2.4 establishes: let x ∈ sanes(compact) be a uniform space and f ∈ ads(x,x) a compact map. then λ (f) is well defined and if λ (f) 6= 0 then f has a fixed point i.e. sanes(compact) uniform spaces are lefschetz spaces (for the class ads). one could in fact generalize theorem 2.2 and theorem 2.4 by using some results in [1]. let x be a subset of a hausdorff topological space and let x be a uniform space. then x is said to be schauder admissible if for every compact subset k of x and every open covering α ∈ covx (k) there exists a continuous function πα : k → e such that (i). πα and i : k →֒ x are α–close; (ii). πα(k) is contained in a subset c of x with c a lefschetz space; and (iii). πα and i : k →֒ x are homotopic. remark 2.5. for example we could take c ∈ nes(compact) or c ∈ sanes(compact) in (ii) above (for both ad and ads maps). the following result can be found in [1]. cubo 12, 2 (2010) fixed point theorems 207 theorem 2.5. let x be a subset of a hausdorff topological space and let x be a uniform space. also suppose x is schauder admissible. if f ∈ ad(x,x) is a compact map then λ (f) is well defined and if λ (f) 6= {0} then f has a fixed point (i.e. schauder admissible uniform spaces are lefschetz spaces (for the class ad)). let x be a hausdorff topological space and let α ∈ cov(x). x is said to be schauder admissible α-dominated if there exists a schauder admissible space xα and two continuous functions rα : xα → x, sα : x → xα such that rα sα : x → x and i : x → x are α–close and also that rα sα ∼ idx . x is said to be almost schauder admissible dominated if x is schauder admissible α-dominated for every α ∈ cov(x). the following result can be found in [1]. theorem 2.6. let x be a subset of a hausdorff topological space and let x be a uniform space. also suppose x is almost schauder admissible dominated. if f ∈ ad(x,x) is a compact map then λ (f) is well defined and if λ (f) 6= {0} then f has a fixed point (i.e. almost schauder admissible dominated uniform spaces are lefschetz spaces (for the class ad)). remark 2.6. a similar result holds if f ∈ ad(x,x) is replaced by f ∈ ads(x,x) in theorem 2.5 and 2.6. 3 asymptotic fixed point theory let x be a hausdorff topological space. a map f ∈ ad(x,x) is said to be a compact absorbing contraction (written f ∈ cac(x,x) or f ∈ cac(x)) if there exists y ⊆ x such that (i). f(y ) ⊆ y ; (ii). f |y ∈ ad(y,y ) (automatically satisfied) is a compact map with y a lefschetz space; (iii). for every compact k ⊆ x there is an integer n = n(k) such that f n(k) ⊆ y . remark 3.1. examples of lefschetz spaces y can be found in section 2. for example y could be nes(compact) or a sanes(compact) uniform space. remark 3.2. if y = u is an open subset of x then (iii) could be changed to (iii)’. for every x ∈ x there exists an integer n = n(x) such that f n(x)(x) ⊆ y = u. to see this we show (iii)’ implies (iii). for each x ∈ x there exists n(x) such that f n(x)(x) ⊆ y = u so by upper semicontinuity there exists an open neighborhood ux of x in x such that f n(x)(y) ⊆ y = u for y ∈ ux. let k be a compact subset of x. then there exists an open covering {ux1, ....,uxn} of k. let n = max{n(x1), ...,n(xn)} and so for x ∈ k we have f n(x) ⊆ u = y , so (iii) is true. remark 3.3. for a discussion on compact absorbing contractions see [9, section 42] and [12, section 15.5]. 208 donal o’regan cubo 12, 2 (2010) theorem 3.1. let x be a hausdorff topological space and f ∈ cac(x,x). then λ (f) is well defined and if λ (f) 6= {0} then f has a fixed point. proof: let y be as described above. let (p,q) be a selected pair for f so in particular q p−1(y ) ⊆ f(y ). consider f |y and let q ′, p′ : p−1(y ) → y be given by p′(u) = p(u) and q′(u) = q(u). notice (p′,q′) is a selected pair for f |y . now since y is a lefschetz space then q ′ ⋆ (p ′)−1⋆ is a leray endomorphism. now [9, proposition 42.2, pp 208] guarantees (see (iii)) that the homeomorphism q′′⋆ (p ′′)−1⋆ : h(x,y ) → h(x,y ) is weakly nilpotent (here p′′, q′′ : (γ,p−1(y )) → (x,y ) are given by p′′(u) = p(u) and q′′(u) = q(u)). then [9, pp 53] guarantees that q′′⋆ (p ′′)−1⋆ is a leray endomorphism and λ (q ′′ ⋆ (p ′′)−1⋆ ) = 0. also [9, property 11.5, pp 52] guarantees that q⋆ p −1 ⋆ is a leray endomorphism (with λ (q⋆ p −1 ⋆ ) = λ (q′⋆ (p ′)−1⋆ )) so λ (f) is well defined. next suppose λ (f) 6= {0}. then there exists a selected pair (p,q) of f with λ (q⋆ p −1 ⋆ ) 6= 0. let (p′,q′) be as described above with λ (q⋆ p −1 ⋆ ) = λ (q ′ ⋆ (p ′)−1⋆ ). then λ (q ′ ⋆ (p ′)−1⋆ ) 6= 0 so since y is a lefschetz space then there exists x ∈ y with x ∈ f |y (x) i.e. x ∈ f x. 2 remark 3.4. a map f ∈ ads(x,x) is said to be a compact absorbing contraction (written f ∈ cacs(x,x) or f ∈ cacs(x)) if there exists y ⊆ x such that (i). f(y ) ⊆ y ; (ii). f |y ∈ ads(y,y ) (automatically satisfied) is a compact map with y a lefschetz space; (iii). for every compact k ⊆ x there is an integer n = n(k) such that f n(k) ⊆ y . essentially the same reasoning as in theorem 3.1 establishes the following: let x be a hausdorff topological space and f ∈ cacs(x,x). then λ (f) is well defined and if λ (f) 6= 0 then f has a fixed point. 4 fixed point theory in fréchet spaces we now present another approach based on projective limits. let e = (e,{| · |n}n∈n ) be a fréchet space with the topology generated by a family of seminorms {|· |n : n ∈ n}; here n = {1, 2, ....}. we assume that the family of seminorms satisfies (4.1) |x|1 ≤ |x|2 ≤ |x|3 ≤ ....... for every x ∈ e. a subset x of e is bounded if for every n ∈ n there exists rn > 0 such that |x|n ≤ rn for all x ∈ x. for r > 0 and x ∈ e we denote b(x,r) = {y ∈ e : |x−y|n ≤ r∀n ∈ n}. to e we associate a sequence of banach spaces {(en, | · |n)} described as follows. for every n ∈ n we consider the equivalence relation ∼n defined by (4.2) x ∼n y iff |x−y|n = 0. we denote by en = (e /∼n, | · |n) the quotient space, and by (en, | · |n) the completion of e n with respect to | · |n (the norm on e n induced by | · |n and its extension to en are still denoted cubo 12, 2 (2010) fixed point theorems 209 by | · |n). this construction defines a continuous map µn : e → en. now since (4.1) is satisfied the seminorm | · |n induces a seminorm on em for every m ≥ n (again this seminorm is denoted by | · |n). also (4.2) defines an equivalence relation on em from which we obtain a continuous map µn,m : em → en since em /∼n can be regarded as a subset of en. now µn,m µm,k = µn,k if n ≤ m ≤ k and µn = µn,m µm if n ≤ m. we now assume the following condition holds: (4.3) { for each n ∈ n, there exists a banach space (en, | · |n) and an isomorphism (between normed spaces) jn : en → en. remark 4.1. (i). for convenience the norm on en is denoted by | · |n. (ii). in many applications en = e n for each n ∈ n. (iii). note if x ∈ en (or e n) then x ∈ e. however if x ∈ en then x is not necessaily in e and in fact en is easier to use in applications (even though en is isomorphic to en). for example if e = c[0,∞), then en consists of the class of functions in e which coincide on the interval [0,n] and en = c[0,n]. finally we assume (4.4) { e1 ⊇ e2 ⊇ ........ and for each n ∈ n, |jn µn,n+1 j −1 n+1 x|n ≤ |x|n+1 ∀ x ∈ en+1 (here we use the notation from [1, 2] i.e. decreasing in the generalized sense). let lim← en (or ∩∞1 en where ∩ ∞ 1 is the generalized intersection [1, 2]) denote the projective limit of {en}n∈n (note πn,m = jn µn,m j −1 m : em → en for m ≥ n) and note lim← en ∼= e, so for convenience we write e = lim← en. for each x ⊆ e and each n ∈ n we set xn = jn µn(x), and we let xn, intxn and ∂xn denote respectively the closure, the interior and the boundary of xn with respect to | · |n in en. for r > 0 and x ∈ en we denote bn(x,r) = {y ∈ en : |x−y|n ≤ r}. let m ⊆ e and consider the map f : m → 2e . assume for each n ∈ n and x ∈ m that jn µn f (x) is closed. let n ∈ n and mn = jn µn(m). since we first consider volterra type operators we assume (note this assumption is only needed in theorems 4.1 and 4.2) (4.5) if x, y ∈ e with |x−y|n = 0 then hn(f x,f y) = 0; here hn denotes the appropriate generalized hausdorff distance (alternatively we could assume ∀n ∈ n,∀x, y ∈ m if jn µn x = jn µn y then jn µn f x = jn µn f y and of course here we do not need to assume that jn µn f (x) is closed for each n ∈ n and x ∈ m). now (4.5) guarantees that we can define (a well defined) fn on mn as follows: for y ∈ mn there exists a x ∈ m with y = jn µn(x) and we let fn y = jn µn f x (we could of course call it f y since it is clear in the situation we use it); note fn : mn → c(en) and note if there exists a z ∈ m with y = jn µn(z) then jn µn f x = jn µn f z from (4.5) (here c(en) 210 donal o’regan cubo 12, 2 (2010) denotes the family of nonempty closed subsets of en). in this paper we assume fn will be defined on mn i.e. we assume the fn described above admits an extension (again we call it fn) fn : mn → 2 en (we will assume certain properties on the extension). now we present some lefschetz type theorems in fréchet spaces. our first two results are motivated by fredholm type operators. theorem 4.1. let e and en be as described above, c ⊆ e and f : c → 2 e. also assume for each n ∈ n and x ∈ c that jn µn f (x) is closed and also for each n ∈ n that fn : cn → 2 en as described above is a closed map with x /∈ fn(x) in en for x ∈ ∂ cn. suppose the following conditions are satisfied: (4.6) { for each n ∈ n, fn ∈ cac(cn,cn) and fn : cn → 2 en is hemicompact, (4.7) for each n ∈ n, λcn (fn) 6= {0} and (4.8) { for each n ∈{2, 3, ....} if y ∈ cn solves y ∈ fn y in en then jk µk,n j −1 n (y) ∈ ck for k ∈{1, ...,n− 1}. then f has a fixed point in e. proof: for each n ∈ n there exists yn ∈ cn with yn ∈ fn yn in en. lets look at {yn}n∈n . notice y1 ∈ c1 and j1 µ1,k j −1 k (yk) ∈ c1 for k ∈ n\{1} from (4.8). note j1 µ1,n j −1 n (yn) ∈ f1 (j1 µ1,n j −1 n (yn)) in e1; to see note for n ∈ n fixed there exists a x ∈ e with yn = jn µn (x) so jn µn (x) ∈ fn (yn) = jn µn f(x) on en so on e1 we have j1 µ1,n j −1 n (yn) = j1 µ1,n j −1 n jn µn (x) ∈ j1 µ1,n j −1 n jn µn f(x) = j1 µ1,n µn f(x) = j1 µ1 f(x) = f1(j1 µ1 (x)) = f1(j1 µ1,n j −1 n jn µn (x)) = f1 (j1 µ1,n j −1 n (yn)). now (4.6) guarantees that there exists is a subsequence n⋆1 of n and a z1 ∈ c1 with j1 µ1,n j −1 n (yn) → z1 in e1 as n →∞ in n ⋆ 1 and z1 ∈ f1 z1 since f1 is a closed map. note z1 ∈ c1 since x /∈ f1(x) in e1 for x ∈ ∂ c1. let n1 = n ⋆ 1 \{1}. now j2 µ2,n j −1 n (yn) ∈ c2 for n ∈ n1 together with (4.6) guarantees that there exists a subsequence n⋆2 of n1 and a z2 ∈ c2 with j2 µ2,n j −1 n (yn) → z2 in e2 as n → ∞ in n ⋆ 2 and z2 ∈ f2 z2. also z2 ∈ c2. note from (4.4) and the uniqueness of limits that j1 µ1,2 j −1 2 z2 = z1 in e1 since n ⋆ 2 ⊆ n1 (note j1 µ1,n j −1 n (yn) = j1 µ1,2 j −1 2 j2 µ2,n j −1 n (yn) for n ∈ n⋆2 ). let n2 = n ⋆ 2 \{2}. proceed inductively to obtain subsequences of integers n⋆1 ⊇ n ⋆ 2 ⊇ ......, n ⋆ k ⊆{k,k + 1, ....} and zk ∈ ck with jk µk,n j −1 n (yn) → zk in ek as n → ∞ in n ⋆ k and zk ∈ fk zk. also zk ∈ ck. note jk µk,k+1 j −1 k+1 zk+1 = zk in ek for k ∈{1, 2, ...}. also let nk = n ⋆ k \{k}. fix k ∈ n. now zk ∈ fk zk in ek. note as well that zk = jk µk,k+1 j −1 k+1 zk+1 = jk µk,k+1 j −1 k+1 jk+1 µk+1,k+2 j −1 k+2 zk+2 = jk µk,k+2 j −1 k+2 zk+2 = ..... = jk µk,m j −1 m zm = πk,m zm cubo 12, 2 (2010) fixed point theorems 211 for every m ≥ k. we can do this for each k ∈ n. as a result y = (zk) ∈ lim←en = e and also note zk ∈ ck for each k ∈ n. thus for each k ∈ n we have jk µk (y) = zk ∈ fk zk = jk µk f y in ek so y ∈ f y in e. 2 remark 4.2. of course one could remove x /∈ fn(x) in en for x ∈ ∂ cn for each n ∈ n if c is a closed subset of e. the proof follows as in theorem 4.1 except in this case zk ∈ ck (but not necessarily in ck). also from y = (zk) ∈ lim←en = e and πk,m (ym) → zk in ek as m → ∞ we can conclude that y ∈ c = c (note q ∈ c iff for every k ∈ n there exists (xk,m) ∈ c, xk,m = πk,n (xn,m) for n ≥ k with xk,m → jk µk (q) in ek as m → ∞). thus zk = jk µk (y) ∈ ck and so jk µk (y) ∈ jk µk f (y) in ek as before. note in fact we can remove the assumption that c is a closed subset of e if we assume f : y → 2e with c ⊆ y and cn ⊆ yn for each n ∈ n. remark 4.3. if we replace fn : cn → 2 en is hemicompact in (4.6) with fn : cn → 2 en is hemicompact then one can remove x /∈ fn(x) in en for x ∈ ∂ cn and fn : cn → 2 en is a closed map for each n ∈ n in the statement of theorem 4.1 since if for each n ∈ n, fn : cn → 2 en is hemicompact then we automatically have that zk ∈ ck. essentially the same reasoning as in theorem 4.1 (with remark 4.2) yields the following result. theorem 4.2. let e and en be as described above, c ⊆ e and f : c → 2 e. also assume c is a closed subset of e, for each n ∈ n and x ∈ c that jn µn f (x) is closed and also for each n ∈ n that fn : cn → 2 en is as described above. suppose the following conditions are satisfied: (4.9) for each n ∈ n, fn ∈ cac(cn,cn) is hemicompact, (4.10) for each n ∈ n, λ cn (fn) 6= {0} and (4.11) { for each n ∈{2, 3, ....} if y ∈ cn solves y ∈ fn y in en then jk µk,n j −1 n (y) ∈ ck for k ∈{1, ...,n− 1}. then f has a fixed point in e. remark 4.4. note we can remove the assumption in theorem 4.2 that c is a closed subset of e if we assume f : y → 2e with c ⊆ y and cn ⊆ yn for each n ∈ n. our result two results are motivated by urysohn type operators. in this case the map fn will be related to f by the closure property (4.16). theorem 4.3. let e and en be as described above, c ⊆ e and f : c → 2 e. also for each n ∈ n assume there exists fn : cn → 2 en and suppose the following conditions are satisfied: (4.12) for each n ∈ n, fn ∈ cac(cn,cn) (4.13) for each n ∈ n, λcn (fn) 6= {0} 212 donal o’regan cubo 12, 2 (2010) (4.14) { for each n ∈{2, 3, ....} if y ∈ cn solves y ∈ fn y in en then jk µk,n j −1 n (y) ∈ ck for k ∈{1, ...,n− 1} (4.15)                      for any sequence {yn}n∈n with yn ∈ cn and yn ∈ fn yn in en for n ∈ n and for every k ∈ n there exists a subsequence nk ⊆{k + 1,k + 2, .....}, nk ⊆ nk−1 for k ∈{1, 2, ....}, n0 = n, and a zk ∈ ck with jk µk,n j −1 n (yn) → zk in ek as n →∞ in nk and (4.16)                  if there exists a w ∈ c and a sequence {yn}n∈n with yn ∈ cn and yn ∈ fn yn in en such that for every k ∈ n there exists a subsequence s ⊆{k + 1,k + 2, .....} of n with jk µk,n j −1 n (yn) → w in ek as n →∞ in s, then w ∈ f w in e. then f has a fixed point in e. remark 4.5. notice to check (4.15) we need to show that for each k ∈ n the sequence {jk µk,n j −1 n (yn)}n∈nk−1 ⊆ ck is sequentially compact. proof: fix n ∈ n. now there exists yn ∈ cn with yn ∈ fn yn in en. lets look at {yn}n∈n . notice y1 ∈ c1 and j1 µ1,k j −1 k (yk) ∈ c1 for k ∈ {2, 3, ...}. now (4.15) with k = 1 guarantees that there exists a subsequence n1 ⊆{2, 3, ....} and a z1 ∈ c1 with j1 µ1,n j −1 n (yn) → z1 in e1 as n →∞ in n1. look at {yn}n∈n1. now j2 µ2,n j −1 n (yn) ∈ c2 for k ∈ n1. now (4.15) with k = 2 guarantees that there exists a subsequence n2 ⊆ {3, 4, ...} of n1 and a z2 ∈ c2 with j2 µ2,n j −1 n (yn) → z2 in e2 as n → ∞ in n2. note from (4.4) and the uniqueness of limits that j1 µ1,2 j −1 2 z2 = z1 in e1 since n2 ⊆ n1 (note j1 µ1,n j −1 n (yn) = j1 µ1,2 j −1 2 j2 µ2,n j −1 n (yn) for n ∈ n2). proceed inductively to obtain subsequences of integers n1 ⊇ n2 ⊇ ......, nk ⊆{k + 1,k + 2, ....} and zk ∈ ck with jk µk,n j −1 n (yn) → zk in ek as n →∞ in nk. note jk µk,k+1 j −1 k+1 zk+1 = zk in ek for k ∈{1, 2, ...}. fix k ∈ n. note zk = jk µk,k+1 j −1 k+1 zk+1 = jk µk,k+1 j −1 k+1 jk+1 µk+1,k+2 j −1 k+2 zk+2 = jk µk,k+2 j −1 k+2 zk+2 = ..... = jk µk,m j −1 m zm = πk,m zm for every m ≥ k. we can do this for each k ∈ n. as a result y = (zk) ∈ lim←en = e and also note y ∈ c since zk ∈ ck for each k ∈ n. also since yn ∈ fn yn in en for n ∈ nk and jk µk,n j −1 n (yn) → zk = y in ek as n →∞ in nk we have from (4.16) that y ∈ f y in e. 2 cubo 12, 2 (2010) fixed point theorems 213 remark 4.6. from the proof we see that condition (4.14) can be removed from the statement of theorem 4.3. we include it only to explain condition (4.15) (see remark 4.5). remark 4.7. suppose in theorem 4.3 we have (4.15)⋆                      for any sequence {yn}n∈n with yn ∈ cn and yn ∈ fn yn in en for n ∈ n and for every k ∈ n there exists a subsequence nk ⊆{k + 1,k + 2, .....}, nk ⊆ nk−1 for k ∈{1, 2, ....}, n0 = n, and a zk ∈ ck with jk µk,n j −1 n (yn) → zk in ek as n →∞ in nk instead of (4.15) and f : c → 2e is replaced by f : y → 2e with c ⊆ y and cn ⊆ yn for each n ∈ n and suppose (4.16) is true with w ∈ c replaced by w ∈ y . then the result in theorem 4.3 is again true. the proof follows the reasoning in theorem 4.3 except in this case zk ∈ ck (but not necessarily in ck) and y ∈ y . in fact we could replace cn ⊆ yn above with cn a subset of the closure of yn in en if y is a closed subset of e (so in this case we can take y = c if c is a closed subset of e). to see this note zk ∈ ck, y = (zk) ∈ lim←en = e and πk,m (ym) → zk in ek as m →∞ and we can conclude that y ∈ y = y . in fact in this remark we could replace (in fact we can remove it as mentioned in remark 4.6) (4.14) with (4.14)⋆ { for each n ∈{2, 3, ....} if y ∈ cn solves y ∈ fn y in en then jk µk,n j −1 n (y) ∈ ck for k ∈{1, ...,n− 1} and the result above is again true. essentially the same reasoning as in theorem 4.3 (with remark 4.7) yields the following result. theorem 4.4. let e and en be as described above, c ⊆ e and f : c → 2 e. also assume c is a closed subset of e and for each n ∈ n that fn : cn → 2 en and suppose the following conditions are satisfied: (4.17) { for each n ∈{2, 3, ....} if y ∈ cn solves y ∈ fn y in en then jk µk,n j −1 n (y) ∈ ck for k ∈{1, ...,n− 1} (4.18) for each n ∈ n, fn ∈ cac(cn,cn) (4.19) for each n ∈ n, λ cn (fn) 6= {0} 214 donal o’regan cubo 12, 2 (2010) (4.20)                      for any sequence {yn}n∈n with yn ∈ cn and yn ∈ fn yn in en for n ∈ n and for every k ∈ n there exists a subsequence nk ⊆{k + 1,k + 2, .....}, nk ⊆ nk−1 for k ∈{1, 2, ....}, n0 = n, and a zk ∈ ck with jk µk,n j −1 n (yn) → zk in ek as n →∞ in nk and (4.21)                  if there exists a w ∈ c and a sequence {yn}n∈n with yn ∈ cn and yn ∈ fn yn in en such that for every k ∈ n there exists a subsequence s ⊆{k + 1,k + 2, .....} of n with jk µk,n j −1 n (yn) → w in ek as n →∞ in s, then w ∈ f w in e. then f has a fixed point in e. remark 4.8. condition (4.17) can be removed from the statement of theorem 4.4. remark 4.9. note we can remove the assumption in theorem 4.4 that c is a closed subset of e if we assume f : y → 2e with c ⊆ y and cn ⊆ yn (or cn a subset of the closure of yn in en if y is a closed subset of e) for each n ∈ n with of course w ∈ c replaced by w ∈ y in (4.21). received: march 2009. revised: may 2009. references [1] r.p. agarwal and d.o’regan, a lefschetz fixed point theorem for admissible maps in fréchet spaces, dynamic systems and applications, 16(2007), 1–12. [2] r.p. agarwal and d.o’regan, fixed point theory for compact absorbing contractive admissible type maps, applicable analysis, 87(2008), 497–508. [3] r.p. agarwal, d.o’regan and s. park, fixed point theory for multimaps in extension type spaces, jour. korean math. soc., 39(2002), 579–591. [4] h. ben-el-mechaiekh, the coincidence problem for compositions of set valued maps, bull. austral. math. soc., 41(1990), 421–434. [5] h. ben-el-mechaiekh, spaces and maps approximation and fixed points, jour. computational and appl. mathematics, 113(2000), 283–308. [6] h. ben-el-mechaiekh and p. deguire, general fixed point theorems for non–convex set valued maps, c.r. acad. sci. paris, 312(1991), 433–438. [7] r. engelking, general topology, heldermann verlag, berlin, 1989. cubo 12, 2 (2010) fixed point theorems 215 [8] g. fournier and l. gorniewicz, the lefschetz fixed point theorem for multi-valued maps of non-metrizable spaces, fundamenta mathematicae, 92(1976), 213–222. [9] l. gorniewicz, topological fixed point theory of multivalued mappings, kluwer acad. publishers, dordrecht, 1999. [10] l. gorniewicz and a. granas, some general theorems in coincidence theory, j. math. pures et appl., 60(1981), 361–373. [11] a. granas, fixed point theorems for approximative anr’s, bull. acad. polon. sc., 16(1968), 15–19. [12] a. granas and j. dugundji, fixed point theory, springer, new york, 2003. [13] j.l. kelley, general topology, d. van nostrand reinhold co., new york, 1955. [14] d. o’regan, fixed point theory on extension type spaces and essential maps on topological spaces, fixed point theory and applications, 2004(2004), 13-20. [15] d. o’regan, asymptotic lefschetz fixed point theory for anes(compact) maps, rendiconti del circolo matematico di palermo, 58(2009), 87-94. mil1final.dvi cubo a mathematical journal vol.12, no¯ 01, (15–21). march 2010 on some problems of james miller b. bhowmik, s. ponnusamy department of mathematics, indian institute of technology madras, chennai-600 036, india emails: ditya@iitm.ac.in, samy@iitm.ac.in and k.-j. wirths institut für analysis, tu braunschweig, 38106 braunschweig, germany email : kjwirths@tu-bs.de abstract we consider the class of meromorphic univalent functions having a simple pole at p ∈ (0, 1) and that map the unit disc onto the exterior of a domain which is starlike with respect to a point w0 6= 0, ∞. we denote this class of functions by σ ∗(p, w0). in this paper, we find the exact region of variability for the second taylor coefficient for functions in σ∗(p, w0). in view of this result we rectify some results of james miller. resumen consideramos la clase de funciones univalentes meromoforficos teniendo un polo simple en p ∈ (0, 1) y la aplicación del disco unitario sobre el exterior de un dominio el cual es estrellado con respecto al punto w0 6= 0, ∞. denotamos esta clase de funciones por σ ∗(p, w0). en este art́ıculo encontramos la región exacta de variabilidad del segundo coeficiente de taylor para funciones in σ∗(p, w0). en vista de estos resultados nosotros rectificamos algunos resultados de james miller. 16 b. bhowmik, s. ponnusamy and k.-j. wirths cubo 12, 1 (2010) key words and phrases: starlike, meromorphic, and schwarz’ functions, taylor coefficient. math. subj. class.: 30c45. 1 introduction let d := {z : |z| < 1} be the unit disc in the complex plane c. let σ∗ denote the class of functions g(z) = 1 z + d0 + d1z + d2z 2 + · · · which are univalent and analytic in d except for the simple pole at z = 0 and map d onto a domain whose complement is starlike with respect to the origin. functions in this class is referred to as the meromorphic starlike functions in d. this class has been studied by clunie [4] and later an extended version by pommerenke [10], and many others. another related class of our interest is the class s(p) of univalent meromorphic functions f in d with a simple pole at z = p, p ∈ (0, 1), and with the normalization f (z) = z + ∑∞ n=2 an(f )z n for |z| < p. if f ∈ s(p) maps d onto a domain whose complement with respect to c is convex, then we call f a concave function with pole at p and the class of these functions is denoted by co(p). in a recent paper, avkhadiev and wirths [2] established the region of variability for an(f ), n ≥ 2, f ∈ co(p) and as a consequence two conjectures of livingston [7] in 1994 and avkhadiev, pommerenke and wirths [1] were settled. in this paper, we consider the class σ∗(p, w0) of meromorphically starlike functions f such that c \ f (d) is a starlike set with respect to a finite point w0 6= 0 and have the standard normalization f (0) = 0 = f ′(0) − 1. we now recall the following analytic characterization for functions in σ∗(p, w0). theorem a. f ∈ σ∗(p, w0) if and only if there is a probability measure µ(ζ) on ∂d = {ζ : |ζ| = 1} so that f (z) = w0 + pw0 (z − p)(1 − zp) exp ( ∫ ∂d 2 log(1 − ζz)dµ(ζ) ) where w0 = − 1 p + 1/p − 2 ∫ |ζ|=1 ζdµ(ζ) . the necessary part of theorem a has been proved by miller [9] while the sufficiency part has been established by yuh lin [6, theorem 1]. in [8, 9], miller discussed a numbers of properties of the class σ∗(p, w0). see also [3, 6, 11] for some other basic results such as bounds for |f (z) − w0|. we may state an equivalent formulation of theorem a (see also [11]). a function f is said to be in σ∗(p, w0) if and only if there exists an analytic function p (z) in d with p (0) = 1 and re p (z) > 0, z ∈ d, (1.1) cubo 12, 1 (2010) on some problems of james miller 17 where p (z) = −zf ′(z) f (z) − w0 − p z − p + pz 1 − pz . (1.2) we may write p (z) in the following power series form p (z) = 1 + b1z + b2z 2 + · · · . also, each f ∈ σ∗(p, w0) has the taylor expansion f (z) = z + ∞ ∑ n=2 an(f )z n, |z| < p. (1.3) to recall the next result, we need to introduce a notation. let p(b1) denote the class of analytic functions p (z) satisfying p (0) = 1, p ′(0) = b1 and re p (z) > 0 in d. in 1972, miller [8] obtained estimations for the second taylor coefficient a2(f ). indeed, he showed that theorem b. if f (z) ∈ σ∗(p, w0), then the second coefficient is given by a2(f ) = 1 2 w0 ( b2 − p 2 − 1 p2 − 1 w02 ) where b2 is the second coefficient of a function in p(b1), i.e. the region of variability for a2(f ) is contained in the disc ∣ ∣ ∣ ∣ a2(f ) + 1 2 w0 ( p2 + 1 p2 + 1 w02 ) ∣ ∣ ∣ ∣ ≤ |w0|. (1.4) further there is a p0, 0.39 < p0 < 0.61, such that if p < p0, then re a2(f ) > 0. in 1980, miller [9, equation (9)] also proved a sharp estimate regarding the second taylor coefficient. in fact, he showed that ∣ ∣ ∣ ∣ a2(f ) − 1 + p2 p − w0 ∣ ∣ ∣ ∣ ≤ |w0|, f ∈ σ ∗(p, w0). (1.5) the aim of this paper is to find the region of variability for the second coefficient a2(f ) of functions in σ∗(p, w0) for any fixed pair (p, w0). also we find the exact region of variability for a2(f ) for fixed p, and as a consequence of this we show that re a2(f ) > 0 for all values of p ∈ (0, 1) which miller did not seem to expect as we see in the last part of theorem b. 2 region of variability of second taylor coefficients for functions in σ∗(p, w0) theorem 2.1. let f ∈ σ∗(p, w0) having the expansion (1.3). then for a fixed pair (p, w0), the exact region of variability of the second taylor coefficient a2(f ) is the disc determined by the 18 b. bhowmik, s. ponnusamy and k.-j. wirths cubo 12, 1 (2010) inequality ∣ ∣ ∣ ∣ ∣ a2(f ) − ( p + 1 p + w0 ) + 1 4 w0 ( p + 1 p + 1 w0 )2 ∣ ∣ ∣ ∣ ∣ ≤ |w0| ( 1 − 1 4 ∣ ∣ ∣ ∣ p + 1 p + 1 w0 ∣ ∣ ∣ ∣ 2 ) . (2.2) proof. the proof uses the representation formula (1.1), i.e. f ∈ σ∗(p, w0) if and only if re p (z) > 0 in d with p (0) = 1, where p is given by (1.2). since it is convenient to work with the class of schwarz functions, we can write each such p as p (z) = 1 + ω(z) 1 − ω(z) , z ∈ d, (2.3) where ω : d → d is holomorphic with ω(0) = 0 so that ω(z) has the form ω(z) = c1z + c2z 2 + · · · . (2.4) using (1.2) and the power series representations of p (z) and f (z), it is easy to compute      b1 = p + 1 p + 1 w0 , and b2 = p 2 + 1 p2 + 1 w02 + 2a2(f ) w0 . (2.5) now eliminating w0 from (2.5), we get b2 = p 2 + 1 p2 + [ b1 − ( p + 1 p )]2 + 2a2(f ) [ b1 − ( p + 1 p )] . (2.6) using the power series representations of p (z) and ω(z), it follows by comparing the coefficients of z and z2 on both sides that b1 = 2c1 and b2 = 2(c 2 1 + c2). inserting the above two relations in (2.6), we get 2(c2 1 + c2) = p 2 + 1 p2 + [ 2c1 − ( p + 1 p )]2 + 2a2(f ) [ 2c1 − ( p + 1 p )] . now solving the above equation for a2(f ), we get a2(f ) = 1 p + p ( c2 1 − c2 + p 2 − 2c1p 1 + p2 − 2c1p ) . (2.7) now, since w0 and p are fixed, we have c1 fixed. hence using the well known estimate |c2| ≤ 1−|c1| 2 for unimodular bounded function ω(z), the last equation results the following estimate ∣ ∣ ∣ ∣ a2(f ) − 1 p − p ( c2 1 + p2 − 2c1p 1 + p2 − 2c1p ) ∣ ∣ ∣ ∣ ≤ p(1 − |c1| 2) |1 + p2 − 2c1p| . cubo 12, 1 (2010) on some problems of james miller 19 now, as b1 = 2c1, substituting c1 = 1 2 (p + 1/p + 1/w0) in the above equation we get the required estimate as given in (2.2). a point on the boundary of the disc described by (2.2) is attained for the unique extremal functions given by (1.2) and (2.3), where ω(z) = z(c1 + cz) 1 + c1cz , |c| = 1. the points in the interior of the disc described in (2.2) are attained for the same functions, but with |c| < 1. remark. comparison of theorem b and theorem 2.8 below, shows that the exact region of variability of a2(f ) found by miller is for the case c1 = 0 only. a little computation reveals that both variability regions are the same for c1 = 0, i.e., ∣ ∣ ∣ ∣ a2(f ) − 1 + p2 + p4 p(1 + p2) ∣ ∣ ∣ ∣ ≤ p 1 + p2 . this also shows that (1.5) gives the precise region of variability only for the case c1 = 0. in all other cases, the boundaries of the discs described by (1.4) and (1.5) have only one point in common with the disc described by (2.2) because, in the both cases, on the boundaries of the discs described by (1.4) and (1.5), we need |b2| = 2. now, as b2 = 2(c2 + c1 2), this means that |c2 + c1 2| = 1. according to the coefficients bounds for unimodular bounded function, this is only possible for a unique c2 if c1 6= 0. in the following theorem, we describe the exact region of variability of the second taylor coefficient of f ∈ σ∗(p, w0), where only p is fixed. theorem 2.8. let f ∈ σ∗(p, w0) having the expansion (1.3) and let p be fixed. then the exact set of variability of the second taylor coefficient a2(f ) is given by |a2(f ) − 1/p| ≤ p. (2.9) proof. we may rewrite (2.7) as a2(f ) = 1 p + p m, (2.10) where m = c2 1 − c2 + p 2 − 2c1p 1 + p2 − 2c1p . we wish to prove that |m| ≤ 1. since ω′(0) = c1, we have |c1| ≤ 1. now we fix c1 ∈ d. then c 2 1 − c2 varies in the closed disc ∆(c1) := {z : |z − c 2 1 | ≤ 1 − |c1| 2}. 20 b. bhowmik, s. ponnusamy and k.-j. wirths cubo 12, 1 (2010) the map t (ζ) = ζ + p2 − 2c1p 1 + p2 − 2c1p maps the disc ∆(c1) onto the disc with center c2 1 + p2 − 2c1p 1 + p2 − 2c1p and radius 1 − |c1| 2 |1 + p2 − 2c1p| . therefore, in order to prove |m| ≤ 1, it suffices to show that ∣ ∣ ∣ ∣ c2 1 + p2 − 2c1p 1 + p2 − 2c1p ∣ ∣ ∣ ∣ + 1 − |c1| 2 |1 + p2 − 2c1p| ≤ 1. this is equivalent to |c1 − p| 2 + 1 − |c1| 2 = re (1 + p2 − 2c1p) ≤ |1 + p 2 − 2c1p|. we see that equality is attained in the above inequality if and only if c1 is real. now for real c1, we have t (∆(c1)) = d if and only if c1 = p or w0 = −p 1 − p2 . hence the extremal functions for the inequality (2.9) are given by (1.2) with p (z) as in (2.3) with ω(z) = z(p + cz) 1 + pcz , |c| = 1, and the points in the interior of the disc described by (2.9) are attained for the same functions, but with |c| < 1. we observe that for real c1 we can obtain m = 1 only for c2 = c 2 1 − 1. this results in other starlike centers, but the extremal function is always the same, since a2(f ) = p + 1/p is attained in the class s(p) only for f (z) = z/((1 − zp)(1 − z/p)), see for instance [5]. remark. this result ensures us that re a2(f ) > 0 for all p ∈ (0, 1). in the article [8, theorem 1], miller hoped for a possibility that for p > .61, the real part of a2(f ) may be negative. but in view of our theorem we conclude that his hope was in vain. remark. in [9], miller has obtained an estimate for the real part of the third coefficient a3(f ) for all p. however, in geometric function theory, the classical question of finding the exact region of variability for an(f ), n ≥ 3, f ∈ σ ∗(p, w0), remains an open problem. received: march, 2008. revised: september, 2009. cubo 12, 1 (2010) on some problems of james miller 21 references [1] avkhadiev, f.g., pommerenke, c. and wirths, k.-j., on the coefficients of concave univalent functions, math. nachr., 271(2004), 3–9. [2] avkhadiev, f.g. and wirths, k.-j., a proof of the livingston conjecture, forum math., 19(2007), 149–158. [3] bhowmik, b. and ponnusamy, s., coefficient inequalities for concave and meromorphically starlike univalent functions, ann. polon. math., 93(2008), 177–186. [4] clunie, j., on meromorphic schlicht functions, j. london math. soc., 34(1959), 215–216. [5] jenkins, j.a., on a conjecture of goodman concerning meromorphic univalent functions, michigan math. j., 9(1962), 25–27. [6] yuh lin, c., on the representation formulas for the functions in the class σ∗(p, w0), proc. amer. math. soc., 103(1988), 517–520. [7] livingston, a.e., convex meromorphic mappings, ann. polon math., 59(3)(1994), 275– 291. [8] miller, j., starlike meromorphic functions, proc. amer. math. soc., 31(1972), 446–452. [9] miller, j., convex and starlike meromorphic functions, proc. amer. math. soc., 80(1980), 607–613. [10] pommerenke, c., on meromorphic starlike functions, pacific j. math., 13(1963), 221–235. [11] yulin, z. and owa, s., some remarks on a class of meromorphic starlike functions, indian j. pure appl. math., 21(9)(1990), 833–840. cubo a mathematical journal vol.10, n o ¯ 02, (135–144). july 2008 iterative regularization methods for a discrete inverse problem in mri a. leitão universidade federal de santa catarina, departamento de matemática, p.o. box 476, 88040-900, florianópolis – brasil email: aleitao@mtm.ufsc.br and j.p. zubelli impa instituto nacional de matemática pura e aplicada, estrada dona castorina 110, 22460-320, rio de janeiro – brasil email: zubelli@impa.br abstract we propose and investigate efficient numerical methods for inverse problems related to magnetic resonance imaging (mri). our goal is to extend the recent convergence results for the landweber-kaczmarz method obtained in [7], in order to derive a convergent iterative regularization method for an inverse problem in mri. resumen nosotros investigamos y proponemos métodos numéricos eficientes para problemas inversos relacionados con resonancia magnética de imagen (mri). nuestro objetivo es extender resultados recientes de convergencia para el método de landweber-kaczmarz 136 a. leitão and j.p. zubelli cubo 10, 2 (2008) obtenido en [7], a fin de obtener un método de regularización iterativo convergente para un problema inverso en mri. key words and phrases: magnetic resonance imaging, mri, tomography, medical imaging, inverse problems. math. subj. class.: 94a08, 47a52, 65d18 1 introduction magnetic resonance imaging, also known as mr–imaging or simply mri, is a noninvasive technique used in medical imaging to visualize body structures and functions, providing detailed images in arbitrary planes. unlike x-ray tomography it does not use ionizing radiation, but uses a powerful magnetic field to align the magnetization of hydrogen atoms in the body. radio waves are used to systematically alter the alignment of this magnetization, causing the hydrogen atoms to produce a rotating magnetic field detectable by the scanner. more specifically, when a subject is in the scanner, the hydrogen nuclei (i.e., protons, found in abundance in the human body as water) align with the strong magnetic field. a radio wave at the correct frequency for the protons to absorb energy pushes some of the protons out of alignment. the protons then snap back to alignment, producing a detectable rotating magnetic field as they do so. since protons in different areas of the body (e.g., fat and muscle) realign at different speeds, the different structures of the body can be revealed. the image to be identified in mri corresponds to a complex valued function p : [0, 1]×[0, 1]→c and the image acquisition process is performed by so-called receivers. due to the physical nature of the acquisition process, the information gained by the receivers does not correspond to the unknown image, but instead, to p multiplied by receiver dependent sensitivity kernels. in real life applications, the sensitivity kernels are not precisely known and have to be identified together with p. this corresponds to a version of the blind deconvolution problem that has been investigated by many authors. see for example [2, 12, 14] our main goal in this article is to investigate efficient iterative methods of kaczmarz type for the identification problem related to mri. we extend the convergence results for the loping landweber-kaczmarz method in [7] and derive a convergent iterative regularization method for this inverse problem. this article is outlined as follows. in section 2 the description of a discrete mathematical model for magnetic resonance imaging is presented. in section 3 we derive the corresponding inverse problem for mri. in section 4 we investigate efficient iterative regularization methods for this inverse problem. using a particular hypothesis on the sensitivity kernels, we are able to derive convergence and stability results for the proposed iterative methods. cubo 10, 2 (2008) iterative regularization methods ... 137 2 the direct problem in what follows we present a discrete model for mri. in our approach, we follow the notation introduced in [1]. the image to be identified is considered to be a discrete function p : {1, . . . , phor} × {1, . . . , pver} =: b → c , where phor, pver ∈ n0 are known. therefore, the number of degrees of freedom related to this parameter is pnum := phor × pver (typical values are phor = pver = 256; pnum = 65536). as mentioned above, the image acquisition process is performed by several receivers, denoted here by rj , j = 0, . . . , rnum − 1, where rnum ∈ n0 is given (typically one faces the situation where rnum << pnum). due to the physical nature of the acquisition process, the information gained by the receivers does not correspond to the unknown image, but instead, to p multiplied by receiver dependent sensitivity kernels sj = s(rj ) : b → c , j = 0, . . . , rnum − 1 . in real life applications, the sensitivity kernels sj are not precisely known and have to be identified together with p. this fact justifies the following ansatz: (a1) the sensitivity kernels sj can be written as linear combination of the given basis functions bn : b → c, for n = 1, . . . , bnum, and bnum ∈ n0. in other words, we assume the existence of coefficients bj,n ∈ c such that sj (m) = bnum∑ n=1 bj,n bn(m) , m ∈ b , j = 0, . . . , rnum − 1 . (2.1) in the sequel we make use the abbreviated notations bj := (bj,n) bnum n=1 and (bj ) := (bj ) rnum−1 j=0 . notice that the coefficient vectors bj = b(rj ) are receiver dependent. the measured data for the inverse problem is given in a subset of the fourier space of the image p, i.e. there exists a known subset m ⊂ b (consisting of pproj elements) such that the receiver dependent measurement mj = m(rj ) satisfies mj := p[f(p × sj )] , j = 0, . . . , rnum − 1 . where f is the discrete fourier transform (dft) operator defined by f : {f | f : b → c} → {f̂ | f̂ : b → c} f 7→ (f(f ))(m) := pnum−1∑ n=0 f (n) exp ( − 2πi pnum nm ) , 138 a. leitão and j.p. zubelli cubo 10, 2 (2008) and p is the operator defined by p : { f | f : b → c } → { g | g : m → c } =: y f 7→ (p[f ])(m) := f (m) , m ∈ m . notice that, due to ansatz (a1) and the linearity of f and p, the measured data mj ∈ y can be written in the form mj = bnum∑ n=1 bj,n p[f(p × bn)] , j = 0, . . . , rnum − 1 . (2.2) remark 2.1. the numerical evaluation of the dft requires naively o(p2num) arithmetical operations. however, in practice the dft must be replaced by the fast fourier transform (fft), which can be computed by the cooley-tukey algorithm1 and requires only o(pnum log(pnum)) operations. 3 the inverse problem next we use the discrete model discussed in the previous section as a starting point to formulate an inverse problem for mri. the unknown parameters to be identified are the discrete image function p and the sensitivity kernels sj . due to the ansatz (a1), the parameter space x consists of pairs of the form (p, (bj )), i.e. x := { (p, (bj )) ; p ∈ c pnum , (bj ) ∈ (c bnum ) rnum } . it is immediate to observe that x can be identified with c(pnum+bnum×rnum), while y can be identified with c pproj . the parameter to output operators fi : x → y are defined by fi : (p, (bj )) 7→ bnum∑ n=1 bi,n p[f(p × bn)] , i = 0, . . . , rnum − 1 . (3.1) due to the experimental nature of the data acquisition process, we shall assume that the data mi in (2.2) is not exactly known. instead, we have only approximate measured data m δ i ∈ y satisfying ‖m δ i − mi‖ ≤ δi , (3.2) with δi > 0 (noise level). therefore, the inverse problem for mri can be written in the form of the following system of nonlinear equations fi(p, (bj )) = m δ i , i = 0, . . . , rnum − 1 . (3.3) it is worth noticing that the nonlinear operators fi’s are continuously fréchet differentiable, and the derivatives are locally lipschitz continuous. 1the fft algorithm was published independently by j.w. cooley and j.w. tukey in 1965. however, this algorithm was already known to c.f. gauss around 1805. cubo 10, 2 (2008) iterative regularization methods ... 139 4 iterative regularization in this section we analyze efficient iterative methods for obtaining stable solutions of the inverse problem in (3.3). 4.1 an image identification problem our first goal is to consider a simplified version of problem (3.3). we assume that the sensitivity kernels sj are known, and we have to deal with the problem of determining only the image p. this assumption can be justified by the fact that, in practice, one has very good approximations for the sensitivity kernels, while the image p is completely unknown. in this particular case, the inverse problem reduces to f̃i(p) = m δ i , i = 0, . . . , rnum − 1 , (4.1) where f̃i(p) = fi(p, (bj )), the coefficients (bj ) being known. this is a much simpler problem, since f̃i : x̃ → y are linear and bounded operators, defined at x̃ := {f | f : b → c}. we follow the approaches in [7, 5] and derive two iterative regularization methods of kaczmarz type for problem (4.1). both iterations can be written in the form p δ k+1 = p δ k − ωkαksk , (4.2) where sk := f̃[k](p δ k ) ∗ (f̃[k](p δ k ) − m δ i ) , (4.3) ωk := { 1 ‖f̃[k](p δ k ) − mδ i ‖ > τ δ[k] 0 otherwise . (4.4) here τ > 2 is an appropriately chosen constant, [k] := (k mod rnum) ∈ {0, . . . , rnum − 1} (a group of rnum subsequent steps, starting at some multiple k of rnum, is called a cycle), p δ 0 = p0 ∈ x̃ is an initial guess, possibly incorporating some a priori knowledge about the exact image, and αk ≥ 0 are relaxation parameters. distinct choices for the relaxation parameters αk lead to the definition of the two iterative methods: 1) if αk is defined by αk := { ‖sk‖ 2/‖f̃[k](p δ k )sk‖ 2 ωk = 1 1 ωk = 0 , (4.5) the iteration (4.2) corresponds to the loping steepest-descent kaczmarz method (lsdk) [5]. 2) if αk ≡ 1, the iteration (4.2) corresponds to the loping landweber-kaczmarz method (llk) [7]. 140 a. leitão and j.p. zubelli cubo 10, 2 (2008) the iterations should be terminated when, for the first time, all pk are equal within a cycle. that is, we stop the iteration at the index kδ ∗ , which is the smallest multiple of rnum such that p k δ ∗ = p k δ ∗ +1 = · · · = pkδ ∗ +rnum−1 . (4.6) convergence analysis of the lsdk method from (3.1) follows that the operators f̃i are linear and bounded. therefore, there exist m > 0 such that ‖f̃i‖ ≤ m , i = 0, . . . , rnum − 1 . (4.7) since the operators f̃i are linear, the local tangential cone condition is trivially satisfied (see (4.16) below). thus, the constant τ in (4.4) can be chosen such that τ > 2. moreover, we assume the existence of p ∗ ∈ bρ/2(p0) such that f̃i(p ∗ ) = mi , i = 0, . . . , rnum − 1 , (4.8) where ρ > 0 and (mi) rnum−1 i=0 ∈ y rnum corresponds to exact data satisfying (3.2). in the sequel we summarize several properties of the lsdk iteration. for a complete proof of the results we refer the reader to [5, section 2]. lemma 4.1. let the coefficients αk be defined as in (4.5), the assumption (4.8) be satisfied for some p∗ ∈ x̃, and the stopping index kδ ∗ be defined as in (4.6). then, the following assertions hold: 1) there exists a constant α > 0 such that αk > α, for k = 0, . . . , k δ ∗ . 2) let δi > 0 be defined as in (3.2). then the stopping index k δ ∗ defined in (4.6) is finite. 3) pδ k ∈ bρ/2(p0) for all k ≤ k δ ∗ . 4) the following monotony property is satisfied: ‖p δ k+1 − p ∗ ‖ 2 ≤ ‖p δ k − p ∗ ‖ 2 , k = 0, 1, . . . , kδ ∗ , (4.9) ‖p δ k+1 − p ∗ ‖ 2 = ‖p δ k − p ∗ ‖ 2 , k > kδ ∗ . (4.10) moreover, in the case of noisy data (i.e. δi > 0) we have ‖f̃i(p δ k δ ∗ ) − m δ i ‖ ≤ τ δi , i = 0, . . . , rnum − 1 . (4.11) next we prove that the lsdk method is a convergent regularization method in the sense of [3]. theorem 4.2 (convergence). let αk be defined as in (4.5), the assumption (4.8) be satisfied for some p∗ ∈ x̃, and the data be exact, i.e. mδ i = mi in (3.2). then, the sequence p δ k defined in (4.2) converges to a solution of (4.1) as k → ∞. cubo 10, 2 (2008) iterative regularization methods ... 141 proof. notice that, since the data is exact, we have ωk = 1 for all k > 0. the proof follows from [5, theorem 3.5]. theorem 4.3 (stability). let the coefficients αk be defined as in (4.5), and the assumption (4.8) be satisfied for some p∗ ∈ x̃. moreover, let the sequence {(δ1,m, . . . , δrnum,m)}m∈n (or simply {δm}m∈n) be such that limm→∞(maxi δi,m) = 0, and let m δm i be a corresponding sequence of noisy data satisfying (3.2) (i.e. ‖mδm i − mi‖ ≤ δi,m, i = 0, . . . , rnum − 1, m ∈ n). for each m ∈ n, let km ∗ be the stopping index defined in (4.6). then, the lsdk iterates pδm k m ∗ converge to a solution of (4.1) as m → ∞. proof. the proof follows from [5, theorem 3.6]. convergence analysis of the llk method the convergence analysis results for the llk iteration are analog to the ones presented in theorems 4.2 and 4.3 for the lsdk method. in the sequel we summarize the main results that we could extend to the llk iteration. theorem 4.4 (convergence analysis). let αk ≡ 1, the assumption (4.8) be satisfied for some p∗ ∈ x̃, the operators f̃i satisfy (4.7) with m = 1, and the stopping index k δ ∗ be defined as in (4.6). then, the following assertions hold: 1) let δi > 0 in (3.2). then the stopping index k δ ∗ defined in (4.6) is finite. 2) pδ k ∈ bρ/2(p0) for all k ≤ k δ ∗ . 3) the monotony property in (4.9), (4.10) is satisfied. moreover, in the case of noisy data, (4.11) holds true. 4) for exact data, i.e. δi = 0 in (3.2), the sequence p δ k defined in (4.2) converges to a solution of (4.1) as k → ∞. 5) let the sequence {δm}m∈n, the corresponding sequence of noisy data m δm i , and the stopping indexes km ∗ be defined as in theorem 4.3. then, the llk iterates pδm k m ∗ converge to a solution of (4.1) as m → ∞. proof. the proof follows from corresponding results for the llk iteration for systems of nonlinear equations in [7]. notice that the assumption m = 1 in theorem 4.4 is nonrestrictive. indeed, since the operators f̃i are linear, it is enough to scale the equations in (4.1) with appropriate multiplicative constants. 142 a. leitão and j.p. zubelli cubo 10, 2 (2008) 4.2 identification of image and sensitivity our next goal is to consider the problem of determining both the image p as well as the sensitivity kernels sj in (3.3). the llk and lsdk iterations can be extended to the nonlinear system in a straightforward way (p δ k+1, (bj ) δ k+1) = (p δ k , (bj ) δ k ) − ωkαksk , (4.12) where sk := f ′ [k](p δ k , (bj ) δ k ) ∗ (f[k](p δ k , (bj ) δ k ) − m δ i ) , (4.13) ωk := { 1 ‖f[k](p δ k , (bj ) δ k ) − mδ i ‖ > τ δ[k] 0 otherwise . (4.14) in the llk iteration we choose αk ≡ 1, and in the lsdk iteration we choose αk := { ‖sk‖ 2/‖f ′[k](p δ k , (bj ) δ k )sk‖ 2 ωk = 1 1 ωk = 0 . (4.15) in order to extend the convergence results in [7, 5] for these iterations, we basically have to prove two facts: 1) assumption (14) in [7]. 2) the local tangential cone condition [7, eq. (15)], i.e. the existence of (p0, (bj )0) ∈ x and η < 1/2 such that ‖fi(p, (bj )) − fi(p̄, (b̄j )) − f ′ i (p, (bj ))[(p, (bj )) − (p̄, (b̄j ))]‖y ≤ η‖fi(p, (bj )) − fi(p̄, (b̄j ))‖y , (4.16) for all (p, (bj )), (p̄, (b̄j )) ∈ bρ(p0, (bj )0), and all i = 1, . . . , rnum. the first one represents no problem. indeed, the fréchet derivatives of the operators fi are locally lipschitz continuous. thus, for any (p0, (bj )0) ∈ x and any ρ > 0 we have ‖f ′ i (p, (bj ))‖ ≤ m = mρ,p0,(bj )0 for all (p, (bj )) in the ball bρ(p0, (bj )0) ⊂ x. the local tangential cone condition however, does not hold. indeed, the operators fi are second order polynomials of the variables bj,n and p. therefore, it is enough to verify whether the real function f (x, y) = xy satisfies |f (x, y) − f (x̄, ȳ) − f ′(x, y)((x − x̄, y − ȳ))| ≤ η|f (x, y) − f (x̄, ȳ)| , in some vicinity of a point (x0, y0) ∈ r 2 containing a local minimizer of f . this, however, is not the case. cubo 10, 2 (2008) iterative regularization methods ... 143 therefore, the techniques used to prove convergence of the llk and lsdk iterations in [7, 5] cannot be extended to the nonlinear system (3.3). it is worth noticing that the local tangential cone condition is a standard assumption in the convergence analysis of adjoint type methods (landweber, steepest descent, levenberg-marquardt, asymptotical regularization) for nonlinear inverse problems [3, 4, 6, 9, 10, 11, 13]. thus, none of the classical convergence proofs for these iterative methods can be extended to system (3.3) in a straightforward way. motivated by the promising numerical results and efficient performance of the llk and lsdk iterations for problems known not to satisfy the local tangential cone condition (see [7, 8, 5]), we intend to use iteration (4.12) for computing approximate solutions of system (3.3). this numerical investigation will be performed in a forthcoming article. 5 conclusions we presented the description of a discrete mathematical model for magnetic resonance imaging and derived the corresponding inverse problem for mri. we investigate efficient iterative regularization methods for this inverse problem. an iterative method of kaczmarz type for obtaining approximate solutions for the inverse problem is proposed. using a particular assumption on the sensitivity kernels, we are able to prove convergence and stability results for the proposed iterative methods. the convergence analysis presented in this article extends the results for the loping landweberkaczmarz method in [7]. moreover, we prove that our method is a convergent iterative regularization method in the sense of [3]. acknowledgments the work of a.l. was supported by the brazilian national research council cnpq, grants 306020/20068 and 474593/2007-0. j.p.z. was supported by cnpq under grants 302161/2003-1 and 474085/20031. this work was developed during the permanence of the authors in the special semester on quantitative biology analyzed by mathematical methods, october 1st, 2007 -january 27th, 2008, organized by ricam, austrian academy of sciences. received: march 2008. revised: may 2008. 144 a. leitão and j.p. zubelli cubo 10, 2 (2008) references [1] f. bauer and s. kannengiesser, an alternative approach to the image reconstruction for parallel data acquisition in mri, mathematical methods in the applied sciences, 30 (2007), 1437–1451 [2] m. bertero and p. boccacci, introduction to inverse problems in imaging, bristol: iop, institute of physics publishing, 1998. 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[14] j. p. zubelli, r. marabini, c. sorzano, and g. herman, three-dimensional reconstruction by chanine’s method from electron microscopic projections corrupted by instrumental aberrations, inverse probl. 19, 4 (2003), 933–949. n9 cubo a mathematical journal vol.11, no¯ 01, (73–100). march 2009 on mapping properties of monogenic functions k. gürlebeck and j. morais institute of mathematics / physics, bauhaus-university, weimar, germany. email: guerlebe@fossi.uni-weimar.de email: jmorais@mat.ua.pt abstract main goal of this paper is to study the description of monogenic functions by their geometric mapping properties. at first monogenic functions are studied as general quasi-conformal mappings. moreover, dilatations and distortions of these mappings are estimated in terms of the hypercomplex derivative. then pointwise estimates from below and from above are given by using a generalized bohr’s theorem and a borelcarathéodory theorem for monogenic functions. finally it will be shown that monogenic functions can be defined as mappings which map infinitesimal balls to special ellipsoids. resumen el principal objetivo de este artículo es estudiar la descripción de funciones monogénicas a través de las propiedades geométricas de sus mapeos. primero son estudiadas funciones monogénicas como aplicaciones casi-conformes generales. además, dilataciones y distorciones de estas aplicaciones son estimadas en términos de la derivada hipercompleja. entonces estimativas puntuales por abajo y por arriba son dadas usando un teorema de bohr generalizado y un teorema de borel-carathéodory para funciones monogénicas. finalmente es demostrado que funciones monogénicas pueden ser definidas como mapeos que aplican bolas infinitesimales en elipsoides especiales. key words and phrases: monogenic functions, quasi-conformal mappings, geometric mapping properties. 74 k. gürlebeck and j. morais cubo 11, 1 (2009) math. subj. class.: 30g35. 1 introduction quaternionic analysis provides us with a spatial analogue of the one-dimensional complex function theory in the plane. generalizing ideas from the complex case to the higher dimensional real euclidean space, quaternionic analysis is applied to construct solutions of some classes of partial differential equations for vector-valued functions in three or four dimensions (see, e.g., [17, 18, 31, 32]), involving the dirac operator or a generalized cauchy-riemann operator, respectively. we are mainly interested in the study of spatial generalizations of holomorphic or antiholomorphic functions. the crucial fact of such functions (transformations) is that they describe essentially mappings of the unit disk onto or into the unit disk transforming partial differential equations to other differential equations and moreover, they embrace cauchy’s inequalities, maximum modulus principle, bohr’s theorem, schwarz lemma, hadamard theorem and others. since they provide with the best description of the pointwise behavior from a given function, at first one has to ask whether or not those results can be generalized to ”holomorphic” (resp. anti-holomorphic) functions in higher dimensions (sections 3 and 4). in addition, to deal with these results it is also necessary to study the mapping properties of such functions more detailed. it is already known, as the ideas of [7] and [5] show, that these functions preserve geometrical properties like length, distance and angles, while mapping domains to the ball. therefore they are applied as well for the transformation of differential equations. our main task is to characterize such mappings which map technically relevant domains to mathematically simple domains and to find out if such class of functions leave the differential equations and/or some geometrical properties invariant. at this point it is of interest to know that in contrast to the situation in the plane, the set of conformal mappings is restricted only to the set of möbius transformations. but the theory of generalized holomorphic functions (by historical reasons they are also called monogenic functions, cf. [8]) does not cover the set of möbius tranformations in rn+1, and since the möbius transformations are not monogenic, one can only expect that monogenic functions represent certain quasi-conformal mappings. on the other hand, the class of all quasi-conformal mappings is much bigger than the class of monogenic functions. the question arises if monogenic functions correspond to a special subclass of quasi-conformal mappings. in the paper [28], the concept of monogenic-conformal mappings realized by functions in r n+1 and with values in the clifford algebra cl0,n was already considered. together with the geometric interpretation of the hypercomplex derivative (see [17]), dilatations and distortions of these mappings can be estimated. compared with the related work, the advantage of our approach lies in the possibility to study the description of monogenic functions by their geometric mapping properties. the local mapping properties of a monogenic function or of a real analytic function are mainly determined by the behaviour of the linear part of their taylor expansions. according to this, in [23] the geometric behaviour of the linear part of a monogenic function was considered. cubo 11, 1 (2009) on mapping properties of monogenic functions 75 as a consequence, it is shown that monogenic functions can be defined as mappings which map infinitesimal balls to special ellipsoids and vice versa. here we extend this result considering the all series expansion of a function. the text is based on recent publications of the authors [19, 20, 21, 22, 23] and contains new material as well. 2 preliminaries let h := {a = a0 + a1e1 + a2e2 + a3e3, ai ∈ r, i = 0, 1, 2, 3} be the algebra of the real quaternions, where the imaginary units ei (i = 1, 2, 3) are subject to the multiplication rules e2 1 = e2 2 = e2 3 = −1, e1e2 = e3 = −e2e1, e2e3 = e1 = −e3e2, e3e1 = e2 = −e1e3. through this paper we shall denote by sc(a) := a0 the scalar part of a and by vec(a) := a1e1 + a2e2 + a3e3 its vector part. analogously to the complex case, the (quaternion-)conjugate element of a is the quaternion a := sc(a) − vec(a) = a0 − a1e1 − a2e2 − a3e3. also, we shall use the euclidean norm |a|2 = aa = ( a2 0 + a2 1 + a2 2 + a2 3 )1/2 . the real vector space r3 is to be embedded in the subset a := spanr{1, e1, e2} of h via the identification of each element x = (x0,x) = (x0,x1,x2) ∈ r 3 with the paravector (also called reduced quaternion) x := x0 + x = x0 + x1e1 + x2e2 ∈ a. as a consequence, no distinction will be made between x as a point in r3 or its correspondent reduced quaternion. also, we emphasize that a is only a real vector space but not a subalgebra of h. for more details on the real algebra of quaternions we refer e.g. to [8], [31], [18], [15]. let now ω be an open subset of r3 with piecewise smooth boundary. a quaternion-valued function or, briefly, an h-valued function is a mapping f : ω −→ h such that f(x) = 3∑ i=0 fi(x)ei, where e0 = 1 and the coordinate-functions fi (i = 0, 1, 2, 3) are real-valued in ω. properties such as continuity, differentiability or integrability are ascribed coordinate-wisely. for continuously real-differentiable functions f : ω −→ h, the operator d = ∂x0 + e1∂x1 + e2∂x2 (1) 76 k. gürlebeck and j. morais cubo 11, 1 (2009) is called generalized cauchy-riemann operator. the conjugate generalized cauchy-riemann operator is defined by d = ∂x0 − e1∂x1 − e2∂x2. (2) a function f : ω −→ h is called left (resp. right) monogenic in ω if df = 0 in ω (resp.,fd = 0 in ω). remark 1. in general, left (resp. right) monogenic functions are not right (resp. left) monogenic. from now on, we refer only to left monogenic functions. for simplicity, we will call them monogenic. however, all results achieved to left monogenic functions can easily be adapted to right monogenic functions. the generalized cauchy-riemann operator (1) and its conjugate (2) factorize the laplace operator in r3. in fact, it holds ∆3f = ddf = ddf, which implies that any monogenic function is also a harmonic function. analogously as in the complex one-dimensional case 1 2 d̄ defines a derivative of monogenic functions. this was shown in [16], where 1 2 d̄f was called hypercomplex derivative of f. a monogenic function f : ω −→ h with an identically vanishing hypercomplex derivative (i.e. a function from ker d ∩ ker d̄) is called hyperholomorphic constant (see again [16]). it is immediately clear that such function depends only on x1 and x2. additionally, we introduce the following notations: br := br(0) will denote the ball of radius r in r3 centered at the origin, sr = ∂br its boundary and dσr (resp. dvr) the lebesgue measure on sr (resp. br). for simplicity, in the case r = 1 we omit r in the notations. we will also denote by l2(sr; x; r) (resp. l2(br; x; r)) the r-linear hilbert space of square integrable functions on sr (resp. br) with values in x (x = r or a). in the case x = r we abbreviate l2(sr; r; r) (resp. l2(br; r; r)) briefly by l2(sr) (resp. l2(br)). also, the real-valued inner product in l2(sr; a; r) (resp. l2(br; a; r)) is given by 〈f,g〉 l2(sr;a;r) = ∫ sr sc(fg)dσr, (3) respectively, 〈f,g〉 l2(br;a;r) = ∫ br sc(fg)dvr, (4) for any f,g ∈ l2(sr; a; r) (resp. l2(br; a; r)). each homogeneous harmonic polynomial pn of degree n can be written in spherical coordinates as pn(x) = r npn(ω), ω ∈ sr, (5) cubo 11, 1 (2009) on mapping properties of monogenic functions 77 its restriction, pn(ω), to the boundary of the ball br is called spherical harmonic 1 of degree n. from (5), it is clear that a homogeneous polynomial is determined by its restriction to sr. denoting by hn(sr) the space of real-valued spherical harmonics of degree n on sr, it is well-known (see [2] and [29]) that dim hn(sr) = 2n + 1. it is also known (see [2] and [29]) if n 6= m, the spaces hn(sr) and hm(sr) are orthogonal in l2(sr). let us denote the homogeneous monogenic polynomials of degree n by hn. in an analogous way to the spherical harmonics, the restriction of hn to the boundary of the ball br is called spherical monogenic2 of degree n. now let m+(ω; a; n) be the space of a-valued homogeneous monogenic polynomials of degree n in ω ⊂ r3. in [27], it is shown that the space m+(ω; a; n) has dimension 2n + 3. later, this result was generalized for arbitrary higher dimensions by r. delanghe in [12]. consider, for each n ∈ n0, a basis {h ν n : ν1, ..., dim m+(ω; a; n)} of m+(ω; a; n). since the coordinates of hν n are harmonic, for arbitrary n,k = 0, 1, ..., we have ‖hν n ‖2 l2(br;a;r) = r2n+3 2n + 3 ‖hν n ‖2 l2(s;a;r). (6) 3 homogeneous monogenic polynomials in [9] and [10], a special r-linear complete orthonormal system of a-valued homogeneous monogenic polynomials defined in the unit ball of r3 is explicitly constructed. the main idea of this construction is based on the already referred factorization of the laplace operator. the authors took a system of real-valued homogeneous harmonic polynomials and applied the d operator in order to obtain a system of a-valued homogeneous monogenic polynomials. for an easier description, we introduce spherical coordinates x0 = r cos θ, x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, where 0 < r < ∞, 0 < θ ≤ π, 0 < ϕ ≤ 2π. each point x = (x0,x1,x2) ∈ r 3 \ {0} admits a unique representation x = rω, where r|x| and |ω| = 1. therefore, ωi = xi r for i = 0, 1, 2. as described, the homogeneous monogenic polynomials (solid spherical monogenics) {x0,† n , xm,† n , y m,† n : m = 1, ...,n + 1}, (7) 1this restriction is also called surface spherical harmonic by some authors (see [8]). 2such restriction is also called surface inner spherical monogenic (see [8]). 78 k. gürlebeck and j. morais cubo 11, 1 (2009) formed by the extensions to the ball of {x0 n ,xm n ,y m n : m = 1, ...,n + 1} are obtained by applying the operator 1 2 d to the system of homogeneous harmonic polynomials u 0,† n+1 ,u m,† n+1 ,v m,† n+1 , m = 1, . . . ,n + 1, with the notations u 0,† n+1 := rn+1u0 n+1 = rn+1pn+1(cos θ) u m,† n+1 := rn+1um n+1 = rn+1pm n+1 (cos θ) cos mϕ v m,† n+1 := rn+1v m n+1 = rn+1pm n+1 (cos θ) sin mϕ, m = 1, ...,n + 1. hereby pn+1 stands for the legendre polynomial of degree n + 1 and the functions p m n+1 are the associated legendre functions3. the set {u0 n+1 ,um n+1 ,v m n+1 : m = 1, . . . ,n + 1} denotes the standard orthogonal basis of spherical harmonics of degree n + 1 in r3 (considered, e.g., in [34]) with respect to the inner product 〈f,g〉 l2(s) = ∫ s fg dσ. moreover, their norms are given by ‖ u0 n+1 ‖l2(s) = 2 √ π 2n + 3 ‖ um n+1 ‖l2(s) = ‖ v m n+1 ‖l2(s) = √ 2π 2n + 3 (n + 1 + m)! (n + 1 − m)! . we begin by considering the following norm estimates already obtained in [9] which will be used later on. proposition 1. for a given fixed n ∈ n0, the spherical monogenics x 0 n , xm n and y m n (m = 1, . . . ,n + 1) are orthogonal to each other with respect to the inner product (3) and their norms are given by ‖ x0 n ‖l2(s;a;r) = √ π(n + 1) ‖ xm n ‖l2(s;a;r) = ‖ y m n ‖l2(s;a;r) = √ π 2 (n + 1) (n + 1 + m)! (n + 1 − m)! . remark 2. a similar result can be obtained for the homogeneous monogenic polynomials (7) if one takes into account relation (6). 3these functions were introduced in 1877 by ferrers. for that reason, some authors (c.f. [34]) call them ferrers functions. cubo 11, 1 (2009) on mapping properties of monogenic functions 79 in the second and third sections we will look more closely to the pointwise behavior of a given function. for that reason in what follows we present pointwise estimates of our basis polynomials (7), already obtained by the authors in [21]. proposition 2. let n ∈ n0. for the homogeneous monogenic polynomials (7) the following estimates hold: |x0,† n (x)| ≤ 1 2 √ π (n + 1)(2r)n ‖ x0 n ‖l2(s;a;r) |xm,† n (x)| ≤ 1 2 √ π (n + 1)(2r)n ‖ xm n ‖l2(s;a;r) |y m,† n (x)| ≤ 1 2 √ π (n + 1)(2r)n ‖ y m n ‖l2(s;a;r), with m = 1, . . . ,n + 1. an interesting point to note here is that the real part of these polynomials are again related with the set {u0 n ,um n ,v m n : m = 1, . . . ,n}. these relations are given in the next theorem: theorem 1. given a fixed n ∈ n0, we have the following relations: sc(x0 n ) = (n + 1) 2 u0 n (θ,ϕ) sc(xm n ) = (n + m + 1) 2 um n (θ,ϕ) sc(y m n ) = (n + m + 1) 2 v m n (θ,ϕ), for m = 1, . . . ,n. proof. we just prove the relation for the spherical harmonics sc(xl n ) (l = 0, . . . ,n). the proof for sc(y m n ) (m = 1, . . . ,n) is similar. taking results from [9], the real part of the spherical monogenics xl n (l = 0, ...,n) is given by sc(xl n ) = al,n(θ) cos(lϕ) with al,n(θ) = 1 2 ( sin 2 θ d dt [ p l n+1 (t) ] t=cos θ + (n + 1) cos θp l n+1 (cos θ) ) . it is well known that the legendre polynomials, together with the associated legendre functions, satisfy in particular the recurrence formula (1 − t2)(p l n+1 (t))′ = (n + l + 1)p l n (t) − (n + 1)tp l n+1 (t), (8) 80 k. gürlebeck and j. morais cubo 11, 1 (2009) for l = 0, . . . ,n+ 1. now, making the change of variable cos θ = t in al,n(θ) and using the previous recurrence formula, it follows immediately that al,n(arccos t) = 1 2 [ (1 − t2)(p l n+1 (t))′ + (n + 1)tp l n+1 (t) ] = (n + l + 1) 2 p l n (t). making again a change of variable t = cos θ our statement is proved. as a consequence, it turns out the result: theorem 2. for a fixed n ∈ n0, the spherical harmonics { sc(x0 n ), sc(xm n ), sc(y m n ) : m = 1, . . . ,n } are orthogonal in l2(s). with such relations together with the norms of the spherical harmonics, we are ready to establish, as well, the l2-norms of sc(x 0 n ), sc(xm n ) and sc(y m n ) (m = 1, ...,n). proposition 3. for a fixed n ∈ n0, the norms of the spherical harmonics sc(x 0 n ), sc(xm n ) and sc(y m n ) are given by ‖sc(x0 n )‖l2(s) = (n + 1) √ π 2n + 1 and ‖sc(xm n )‖l2(s) = ‖sc(y m n )‖l2(s)(n + 1 + m) √ π 2 1 (2n + 1) (n + m)! (n − m)! , for m = 1, . . . ,n. remark 3. using a different measure, we have established theorem 2 and the previous proposition already in [21]. for future use we need also the next results: proposition 4. given a fixed n ∈ n0, the spherical harmonics { sc(xn+1 n ei), sc(y n+1 n ei) : i = 1, 2 } are orthogonal to each other with respect to the inner product (3) and their norms are given by ‖sc(xn+1 n ei)‖l2(s) = ‖sc(y n+1 n ei)‖l2(s) = 1 2 √ π(n + 1)(2n + 2)!. cubo 11, 1 (2009) on mapping properties of monogenic functions 81 the case i = 1 was already studied by the authors in [20]. for i = 2 the proof is similar. remark 4. the orthogonality is ensured if one takes into account the following representations ([9], proposition 3.4.3) xn+1 n = −cn+1,n cos(nϕ)e1 + c n+1,n sin(nϕ)e2 (9) y n+1 n = −cn+1,n sin(nϕ)e1 − c n+1,n cos(nϕ)e2. since some of our further results are not only restricted to the unit ball, from now on we represent by x0,†,∗r n ,xm,†,∗r n ,y m,†,∗r n (m = 1, ...,n+1) the normalized basis functions x0,† n ,xm,† n ,y m,† n in l2(br; a; r). based on these functions, in [9] and [10] the following orthonormal basis is constructed, therein restricted to the unit ball. theorem 3. for each n, the set of 2n + 3 homogeneous monogenic polynomials { x0,†,∗r n ,xm,†,∗r n ,y m,†,∗r n , m1, ...,n + 1 } (10) forms an orthonormal basis in m+(br; a; n) with respect to the inner product (4). remark 5. the estimates stated in proposition 2 are still valid for this new system of polynomials (10). in particular, taking into account relation (6) it follows: |x0,†,∗r n (x)| = √ 2n + 3 r2n+3 |x0,† n (x)| ‖x 0,† n ‖l2(s;a;r) and moreover, from proposition 2 |x0,† n (x)| = ∣∣∣rnx0,†n ( x r )∣∣∣ ≤ rn 1 2 √ π (n + 1)2n ∣∣∣ x r ∣∣∣ n ‖ x0 n ‖l2(s;a;r) for 0 < |x| = r < r. theorem 3 makes it possible to define the fourier expansion of a square integrable a-valued monogenic function in l2(br). moreover, each monogenic function can be decomposed in an orthogonal sum of a monogenic "main part" (g) of the function and a hyperholomorphic constant (h). more precisely, it holds: lemma 1. a monogenic l2-function f : ω ⊂ r 3 −→ a can be decomposed into f = f(0) + g + h, (11) where the functions g and h have fourier series g(x) = ∞∑ n=1 ( x0,†,∗r n (x)α0 n + n∑ m=1 [ xm,†,∗r n (x)αm n + y m,†,∗r n (x)βm n ] ) h(x) = ∞∑ n=1 [ xn+1,†,∗r n (x)αn+1 n + y n+1,†,∗r n (x)βn+1 n ] . 82 k. gürlebeck and j. morais cubo 11, 1 (2009) the associated fourier coefficients α0 n ,αm n ,βm n (m = 1, ...,n + 1) are real-valued. 4 bohr’s theorem for monogenic functions during the last years the standard bohr’s phenomena attracted a lot of attention. in 1914, h. bohr discovered that there exists a radius r ∈ (0, 1) such that if a power series of a holomorphic function converges in the unit disk and its sum has a modulus less than 1, then for |z| < r the sum of the absolute values of its terms is again less than 1. the significance of the theorem is that such radius does not depend on the function.4 to be more precise, the classical bohr’s theorem says that: theorem 4. [6] let f be a bounded analytic function in the open unit disk, with taylor expansion f(z) = ∞∑ n=0 anz n convergent in the unit disk and with modulus less than 1. then ∞∑ n=0 |an|r n < 1 for 0 ≤ r < 1 3 . this result, known as bohr’s inequality, is true for 0 < r < 1 3 and the constant 1 3 cannot be improved, that is, the inequality fails for any r ≥ 1 3 . originally, this theorem was proved for 0 ≤ r < 1 6 , but soon improved to the sharp result by m. riesz, i. schur, and n. wiener independently. in bohr’s paper [6] his own proof was published as well as a proof by wiener based on function theory methods. later, s. sidon gave a different proof [35], which was subsequently rediscovered by m. tomić [36]. recently, multi-dimensional analogues and other generalizations of bohr’s theorem are treated by several mathematicians such as aizenberg [1], beneteau, dahlner and khavinson [3], boas and khavinson [4], dineen and timoney [14], paulsen, popescu and singh [30], and many others. in several of these papers, the proof of bohr’s inequality or of bohr-type inequalities, respectively, in the theory of holomorphic functions of one or n variables is based on the orthogonality of the powers of the complex variable(s). to use similar ideas in the quaternionic case, it seems to be natural to work with the fourier expansion of monogenic functions. it is not so simple as in the complex case to switch between the taylor expansion and the fourier series of a function. the reason for this is that the taylor expansion with respect to the fueter variables does not give us orthogonal summands. it should be also remarked that in some papers (see, e.g., [30]) the idea is to work with the fourier coefficients of the boundary values of a holomorphic function. we prefer here to consider (analogously to the original formulation of bohr’s theorem) only monogenic functions in the ball. it follows directly from the supposed boundedness of the monogenic functions that they are also square integrable in the ball and therefore we can work with fourier series there. the existence of integrable boundary values needs additional assumptions. 4for the physicists, the notion "bohr radius" is associated to niels bohr, the founder of the quantum theory and winner of the nobel prize in physics in 1922. cubo 11, 1 (2009) on mapping properties of monogenic functions 83 in the remainder of this section, we collect generalizations and different modifications of this theorem (see [19, 21]) and we show that the result can be extended to the all class of monogenic functions with |f(x)| < 1 in b. in [19], the first version of a quaternionic bohr type theorem was obtained, therein restricted to the case of functions with f(0) = 0. theorem 5. [19] let f be a square integrable a-valued monogenic function with f(0) = 0 and |f(x)| < 1 in b and let ∞∑ n=1 [ x0,†,∗ n α0 n + n+1∑ m=1 ( xm,†,∗ n αm n + y m,†,∗ n βm n ) ] be its fourier expansion. then ∞∑ n=1 ∣∣∣x0,†,∗n α0n + n+1∑ m=1 ( xm,†,∗ n αm n + y m,†,∗ n βm n )∣∣∣ < 1 holds in the ball {x : |x| < 0.047}. this result is adapted very well to the complex situation. the absolute value is taken from all summands of the same degree n. in the complex case this is also a first important step. all the considered functions with f(0) = 0 are orthogonal to the constants. this is used later on to estimate all fourier coefficients of a general holomorphic function with |f(z)| ≤ 1 by the first fourier coefficient (see, e.g., [30]). however, it is important to remark that in the quaternionic context, the set of ”constants” is much bigger. hereby constants are also monogenic functions which have an identically vanishing hypercomplex derivative. then it is immediately clear that the constant function and all monogenic functions which depend only on x1 and x2 are the so called hyperholomorphic constants. moreover, if we, as in this paper, consider only a-valued functions then a non-trivial hyperholomorphic constant must have values in span r {e1, e2}. with these observations it seems to be natural to study at first the class of functions which are orthogonal to the non-trivial hyperholomorphic constants in l2(b; a; r) with |f(x)| < 1 in b (see [21]). this approach is also supported by the fact that in lemma 1 it is shown that each monogenic function can be decomposed in an orthogonal sum of a monogenic ”main part” of the function and a hyperholomorphic constant. remind that an orthonormal basis of the subspace of hyperholomorphic constants is given by the set {xn+1,† n ,y n+1,† n }∞ n=0 . the fourier representation in the hypothesis of the next theorem describes the general form of these main parts. the non-trivial hyperholomorphic constants in the decomposition do not influence the real part of the function at the origin because their image lies in span r {e1, e2}. theorem 6. [21] let f be an a-valued monogenic function such that f(x) − f(0) is orthogonal to the hyperholomorphic constants with respect to the inner product (4) with |f(x)| < 1 in b and 84 k. gürlebeck and j. morais cubo 11, 1 (2009) let ∞∑ n=0 [ x0,†,∗ n α0 n + n∑ m=1 ( xm,†,∗ n αm n + y m,†,∗ n βm n ) ] be its fourier expansion. then ∞∑ n=0 [ |x0,†,∗ n ||α0 n | + n∑ m=1 ( |xm,†,∗ n ||αm n | + |y m,†,∗ n ||βm n | ) ] < 1 holds in the ball of radius r, with 0 ≤ r < 0.004. proof. we give only the main ideas of the proof. for more details see [21]. at first it is important to note that in the previous series the sum which contains the variable m runs now only from 1 to n. this fact expresses the supposed orthogonality to the hyperholomorphic constants xn+1,† n and y n+1,† n . since the basis polynomials are homogeneous, the value of f at the origin is f(0) = ∞∑ n=0 ( x0,†,∗ n (0)α0 n + n∑ m=1 [ xm,†,∗ n (0)αm n + y m,†,∗ n (0)βm n ] ) = 1 2 √ 3 π α0 0 ∈ r. without loss of generality we assume that f(0) is positive (otherwise we work with −f). since the associated fourier coefficients are real-valued, the real part of f is given by sc(f) = ∞∑ n=0 { sc(x0,†,∗ n )α0 n + n∑ m=1 [ sc(xm,†,∗ n )αm n + sc(y m,†,∗ n )βm n ] } . basically, the main idea of the proof is to find relations between the general fourier coefficients and the coefficient of the zeroth term, i.e., α0 0 . multiplying both sides of the equation sc(1 − f) = 1 − sc(f) (12) by the solid spherical harmonics {sc(x 0,†,∗ k ), sc(x p,†,∗ k ), sc(y p,†,∗ k ) : p = 1, ...,k}, integrating over the ball and applying the modulus we get the following relations: |α0 k | ≤ max b |x 0,†,∗ k | 2 √ π 3 ‖sc(x 0,†,∗ k )‖2 l2(b) ( 2 √ π 3 − α0 0 ) |α p k | ≤ max b |x p,†,∗ k | 2 √ π 3 ‖sc(x p,†,∗ k )‖2 l2(b) ( 2 √ π 3 − α0 0 ) |β p k | ≤ max b |y p,†,∗ k | 2 √ π 3 ‖sc(y p,†,∗ k )‖2 l2(b) ( 2 √ π 3 − α0 0 ) , p = 1, ...,k. cubo 11, 1 (2009) on mapping properties of monogenic functions 85 with some calculations, applying propositions 1-3 we arrive at 1 2 √ 3 π α0 0 + ∞∑ n=1 [ |x0,†,∗ n ||α0 n | + n∑ m=1 ( |xm,†,∗ n ||αm n | + |y m,†,∗ n ||βm n | ) ] ≤ 1 2 √ 3 π α0 0 + 1 √ 3π ( 2 √ π 3 − α0 0 ) ∞∑ n=1 (4r)n(n + 1)4(2n + 3). the principal significance is that 1 2 √ 3 π α0 0 + ∞∑ n=1 [ |x0,†,∗ n ||α0 n | + n∑ m=1 ( |xm,†,∗ n ||αm n | + |y m,†,∗ n ||βm n | ) ] < 1 if 2 3 ∞∑ n=1 (4r)n(n + 1)4(2n + 3) < 1. we see that the last series converges for r < 1 4 , and therefore, the inequality is satisfied for 0 ≤ r < 0.004. as we have seen, the set {xn+1,† n ,y n+1,† n } belonging to h play a special role. in order to extend the previous result, next we present some important properties of this function and/or of its coordinates. for simplicity we restrict ourselves to the unit ball. lemma 2. the hyperholomorphic constant h can be written as h = h1e1 + h2e2, where its coordinates have fourier series h1(x) = ∞∑ n=1 ( [xn+1,†,∗ n (x)]1α n+1 n + [y n+1,†,∗ n (x)]1β n+1 n ) h2(x) = ∞∑ n=1 ( [xn+1,†,∗ n (x)]2α n+1 n + [y n+1,†,∗ n (x)]2β n+1 n ) . moreover, the following properties hold: proposition 5. the harmonic functions h1 and h2 are orthogonal with respect to the inner product (3). proof. because relation (6) we just need to ensure the orthogonality in l2(s). for technical reasons, we rewrite the function h as follows h(x) = ∞∑ n=1 √ 2n + 3 rn ( xn+1,∗ n (θ,ϕ)αn+1 n + y n+1,∗ n (θ,ϕ)βn+1 n ) . 86 k. gürlebeck and j. morais cubo 11, 1 (2009) by definition of the real-valued inner product in l2(s), using representation (9) and the previous expression we have < h1,h2 >l2(s)= ∞∑ n=1 ∞∑ n′=1 √ 2n + 3 rn ‖xn+1n ‖l2(s;a;r) √ 2n′ + 3 rn ′ ‖xn ′+1 n′ ‖l2(s;a;r) ∫ s an(θ,ϕ)bn′ (θ,ϕ)dσ where an(θ,ϕ) = −c n+1,n (θ) ( cos(nϕ)αn+1 n + sin(nϕ)βn+1 n ) = an(θ)an(ϕ) bn′ (θ,ϕ) = c n ′ +1,n ′ (θ) ( sin(n′ϕ)αn ′ +1 n′ − cos(n′ϕ)βn ′ +1 n′ ) = bn′ (θ)bn′ (ϕ). if the degrees of the summands in the series of h1 and h2 are different (n 6= n ′ ) then the orthogonality is ensured. therefore it is only necessary to prove that the previous integral vanishes for n = n′. and since ∫ s an(θ,ϕ)bn(θ,ϕ)dσ = ∫ π 0 an(θ)bn(θ) sin θdθ ∫ 2π 0 an(ϕ)bn(ϕ)dϕ, it is enough to prove that the second integral on the right-hand side is zero. moreover, ∫ 2π 0 an(ϕ)bn(ϕ)dϕ = [ (αn+1 n ) 2 − (βn+1 n ) 2 ]∫ 2π 0 cos(nϕ) sin(nϕ)dϕ + αn+1 n βn+1 n (∫ 2π 0 sin 2 (nϕ)dϕ − ∫ 2π 0 cos 2 (nϕ)dϕ ) = 0. proposition 6. the harmonic functions h1 and h2 verify the following relation: sup b |h1(x)| = sup b |h2(x)|. the proof follows immediately from representation (9). we are thus led to the following generalization of theorem 6. theorem 7. let f be an a-valued monogenic function with |f(x)| < 1 in b and let ∞∑ n=0 [ x0,†,∗ n α0 n + n+1∑ m=1 ( xm,†,∗ n αm n + y m,†,∗ n βm n ) ] be its fourier expansion. then ∞∑ n=0 [ |x0,†,∗ n ||α0 n | + n+1∑ m=1 ( |xm,†,∗ n ||αm n | + |y m,†,∗ n ||βm n | ) ] < 1 holds in the ball of radius r, with 0 ≤ r < 0.004. cubo 11, 1 (2009) on mapping properties of monogenic functions 87 proof. considering f written as in (11) (lemma 1) f(x) = f(0) + g(x) + h(x), with f(0) = g(0) + h(0). the study of the function g was already considered in theorem 6. we showed that |g(x)| ≤ ∞∑ n=1 [ |x0,†,∗ n ||α0 n | + n∑ m=1 ( |xm,†,∗ n ||αm n | + |y m,†,∗ n ||βm n | ) ] ≤ 1 √ 3π ( 2 √ π 3 − |α0 0 | ) ∞∑ n=1 (4r)n(n + 1)4(2n + 3). we consider now the function h written as fourier series h(x) = ∞∑ n=1 ( xn+1,†,∗ n (x)αn+1 n + y n+1,†,∗ n (x)βn+1 n ) . the proof for the function h follows the same idea as the previous one. according to its fourier expansion, in this case we should find relations with α1 0 and/or β1 0 . however, it is important to note that the function h has no real part, and therefore, it is not possible to apply straight the previous idea. because h lies in span r {e1, e2}, the interesting point is that multiplying h at right by e1 (resp., by e2) the real part is different of zero, standing then for −h1 (resp., −h2). moreover, since there are two different fourier coefficients in the zeroth term, it is natural to ask: which coefficient should be compared, α1 0 or β1 0 ? multiplying the function h at right by e1 we obtain h̃(0) + h̃(x) := he1(0) + he1(x) = (x 1,†,∗ 0 e1)α 1 0 + (y 1,†,∗ 0 e1)β 1 0 + ∞∑ n=1 [ (xn+1,†,∗ n e1)(x)α n+1 n + (y n+1,†,∗ n e1)(x)β n+1 n ] where (x 1,†,∗ 0 e1)α 1 0 + (y 1,†,∗ 0 e1)β 1 0 = 1 2 √ π α1 0 + 1 2 √ π β1 0 e3. for this case, taking into account the previous assumption, it is obviously that we should consider for zeroth term the coefficient α1 0 . in a similar way, considering a new function ˜̃ h := he2, β1 0 should be surely considered. having disposed of this preliminary step let us return to the proof. it follows easily multiplying h at right by e1 sc(h̃) = sc(he1) = ∞∑ n=1 [ sc(xn+1,†,∗ n e1)α n+1 n + sc(y n+1,†,∗ n e1)β n+1 n ] . 88 k. gürlebeck and j. morais cubo 11, 1 (2009) taking into account proposition 6, for 0 < δ2 ≤ 1 we multiply both sides of the equation sc ( δ2 2 − h̃ ) = δ2 2 − sc(h̃) by the orthonormal solid spherical harmonics sc(x k+1,†,∗ k e1) ( resp. sc(y k+1,†,∗ k e1)), integrating over the ball and taking the modulus it follows |αk+1 k | ≤ max b |x k+1,†,∗ k | 2 √ π 3 ‖sc(x k+1,†,∗ k )‖2 l2(b) (√ π 3 δ2 − |α 1 0 | ) |βk+1 k | ≤ max b |x k+1,†,∗ k | 2 √ π 3 ‖sc(x k+1,†,∗ k )‖2 l2(b) (√ π 3 δ2 − |α 1 0 | ) now with some calculations, using propositions 1-3 and applying the maximum modulus principle, the previous inequalities can be rewritten as follows |αk+1 k | ≤ 2 √ 3 2 k √ 2k + 3(k + 1) (√ π 3 δ2 − |α 1 0 | ) |βk+1 k | ≤ 2 √ 3 2 k √ 2k + 3(k + 1) (√ π 3 δ2 − |α 1 0 | ) . in a similar way, from the study of the function ˜̃ h we obtain |αk+1 k | ≤ 2 √ 3 2 k √ 2k + 3(k + 1) (√ π 3 δ2 − |β 1 0 | ) |βk+1 k | ≤ 2 √ 3 2 k √ 2k + 3(k + 1) (√ π 3 δ2 − |β 1 0 | ) . using the previous inequalities we end with |h(x)| ≤ ∞∑ n=1 ( |xn+1,†,∗ n ||αn+1 n | + |y n+1,†,∗ n ||βn+1 n | ) ≤ 1 √ 3π ( 2 √ π 3 δ2 − |α 1 0 | − |β1 0 | ) ∞∑ n=1 (4r)n(n + 1)2(2n + 3). finally, we obtain |f(x)| ≤ 1 2 √ 3 π |α0 0 − α1 0 e1 − β 1 0 e2| + 1 √ 3π ( 2 √ π 3 (1 + δ2) − |α 0 0 | − |α1 0 | − |β1 0 | ) ∞∑ n=1 (4r)n(n + 1)4(2n + 3), and since |f(x)| < 1 it is clear that 1 √ 3π ( 2 √ π 3 (1 + δ2) − |α 0 0 | − |α1 0 | − |β1 0 | 2 √ π 3 − |α0 0 | − |α1 0 | − |β1 0 | ) ∞∑ n=1 (4r)n(n + 1)4(2n + 3) < 1. cubo 11, 1 (2009) on mapping properties of monogenic functions 89 of crucial importance is the fact that the coefficient 2 √ π 3 − |α0 0 | − |α1 0 | − |β1 0 | 2 √ π 3 (1 + δ2) − |α 0 0 | − |α1 0 | − |β1 0 | is bounded from above by 1. a simple calculation shows that the last series converges for r < 1 4 , and therefore, the inequality is satisfied for 0 ≤ r < 0.004. this shows that such a radius exists in the three-dimensional euclidean ball. it has to be studied how the estimate for the bohr radius can be improved. 5 real part theorems for monogenic functions in the remainder of this section, we refer to the results as "real part theorems" in honor to the first assertion of such a kind, the classical (improved) hadamard’s real part theorem (1892). his work on functions of a complex variable was one of the first to examine the general theory of analytic functions. since then, the acceptance of his work is worldwide. looking back to all of these years one can say that time has shown that his topic has a wide range of applications. some important indicators for such a development is that based on it, many recent results with strong applications are still coming out. moreover, they provide with the best description of the pointwise behavior of analytic functions from a given space. a lot of results and extended list of references concerning these and other fundamental inequalities, as well as their applications, can be found in the book by kresin and maz’ya (see [33]). in the complex case, hadamard’s real part theorem contains only the modulus of the function in the left-hand side of an inequality and bounds the growth of a function by the growth of its real part. more precisely, for r < r the inequality |f(z) − f(0)| ≤ cr r − r sup |ξ|≤r |re (f(ξ) − f(0)) |, (13) holds for analytic functions on a closed disk of radius r centered at the origin. such an inequality appeared first in 1892 (see [24]) with c = 4. later, borel and carathéodory found the sharp constant c = 2. as a matter of this fact, a more general estimate for |f(z)| with f(0) 6= 0 was noticed by carathéodory (see landau [25, 26]) |f(z)| ≤ 2r r − r sup |ξ|≤r |ref(ξ)| + r + r r − r |f(0)|. (14) having in mind the type of inequalities we want to prove, we will restrict ourselves to the three-dimensional case. the frequent use of quaternionic analysis in the study of three-dimensional 90 k. gürlebeck and j. morais cubo 11, 1 (2009) problems motivate us to consider functions defined in r3 and with values in the reduced quaternions. it is also known that already in the four-dimensional case (quaternion-valued functions) there are a lot of non-trivial monogenic functions with vanishing scalar part. for such functions we cannot get the result as desired. in [20, 22] it is shown that it is possible to generalize borel-carathéodory and hadamard’s real part theorems to monogenic functions, therein restricted to the unit ball in the euclidean space r 3. in this section we generalize these results for an arbitrary ball of radius r, analogously to the complex case. remark 6. several proofs of bohr’s inequality are based on estimating all fourier coefficients by the first one. we will observe that the proof of both theorems follows from this idea. here we consider relations between the fourier coefficients of the function and the fourier coefficients of its real part. the referred relations come with the next lemma: lemma 3. let f be a square integrable a-valued monogenic function and n ∈ n0. then the fourier coefficients of f are given by α0 n = ‖x0,† n ‖l2(br;a;r) ‖sc(x 0,† n )‖ 2 l2(br) ∫ br sc(f)sc(x0,† n )dvr αm n = ‖xm,† n ‖l2(br;a;r) ‖sc(x m,† n )‖ 2 l2(br) ∫ br sc(f)sc(xm,† n )dvr βm n = ‖y m,† n ‖l2(br;a;r) ‖sc(y m,† n )‖ 2 l2(br) ∫ br sc(f)sc(y m,† n )dvr, m = 1, ...,n αn+1 n = ‖xn+1,† n ‖l2(br;a;r) ‖sc(x n+1,† n e1)‖ 2 l2(br) ∫ br sc(he1)sc(x n+1,† n e1)dvr βn+1 n = ‖y n+1,† n ‖l2(br;a;r) ‖sc(y n+1,† n e1)‖ 2 l2(br) ∫ br sc(he1)sc(y n+1,† n e1)dvr. originally the fourier coefficients are defined by the inner product of the function f and elements of the space m+(r3; a,n). now as we can see these coefficients, up to a factor, are also associated with the scalar part of f. proof. we give only some ideas of the proof. for more details see [20]. according to lemma 1, f cubo 11, 1 (2009) on mapping properties of monogenic functions 91 can be written as fourier series respecting the decomposition (11) f(x) = f(0) + ∞∑ n=1 ( x0,†,∗r n (x)α0 n + n∑ m=1 [ xm,†,∗r n (x)αm n + y m,†,∗r n (x)βm n ] ) ︸︷︷︸ =g + ∞∑ n=1 [ xn+1,†,∗r n (x)αn+1 n + y n+1,†,∗r n (x)βn+1 n ] ︸︷︷︸ =h . we will present the proof only for the coefficients α0 n of g. the remaining coefficients αm n and βm n (m = 1, ...,n) are obtained in a similar way. as described, we aim to compare each fourier coefficient α0 n with sc(f). we have seen several times before that sc(f) = ∞∑ n=0 ( sc(x0,†,∗r n )α0 n + n∑ m=1 [ sc(xm,†,∗r n )αm n + sc(y m,†,∗r n )βm n ] ) . multiplying both sides of the previous expression by the solid spherical harmonics {sc(x 0,†,∗r k ), sc(x p,†,∗r k ), sc( p = 1, ...,k} (k ≥ 1) and integrating over the ball we get the desired relations. for the study of the coefficients αn+1 n and βn+1 n we multiply the equation sc(he1) = ∞∑ n=0 [ sc(xn+1,†,∗r n e1)α n+1 n + sc(y n+1,†,∗r n e1)β n+1 n ] by the orthogonal solid spherical harmonics sc(x k+1,†,∗r k e1) (resp. sc(y k+1,†,∗r k e1)) (k ≥ 1) and integrating over the ball carries our results. lemma 4. let f be a square integrable a-valued monogenic function. for each n ∈ n0, the fourier coefficients satisfy the inequalities |α0 n | ≤ 2 √ π 3 √ r3 ‖x0,† n ‖l2(br;a;r) ‖sc(x 0,† n )‖l2(br) sup |ξ|≤r |sc (f(ξ)) | |αm n | ≤ 2 √ π 3 √ r3 ‖xm,† n ‖l2(br;a;r) ‖sc(x m,† n )‖l2(br) sup |ξ|≤r |sc (f(ξ)) | |βm n | ≤ 2 √ π 3 √ r3 ‖y m,† n ‖l2(br;a;r) ‖sc(y m,† n )‖l2(br) sup |ξ|≤r |sc (f(ξ)) |, m = 1, ...,n |αn+1 n | ≤ 2 √ π 3 √ r3 ‖xn+1,† n ‖l2(br;a;r) ‖sc(x n+1,† n e1)‖l2(br) sup |ξ|≤r |sc (he1(ξ)) | |βn+1 n | ≤ 2 √ π 3 √ r3 ‖y n+1,† n ‖l2(br;a;r) ‖sc(y n+1,† n e1)‖l2(br) sup |ξ|≤r |sc (he1(ξ)) |. 92 k. gürlebeck and j. morais cubo 11, 1 (2009) the previous inequalities are basic results to prove the following theorem: theorem 8 (real-part theorem). let f be a square integrable a-valued monogenic function in br. then, for 0 ≤ r < r 2 we have the inequality |f|r ≤ |f(0)| + √ 2 3 8r (r − 2r)3 ( a1(r,r) sup |ξ|≤r |sc (f(ξ)) | + a2(r,r) sup |ξ|≤r |sc (he1(ξ)) | ) , where |f|r = max |x|=r |f(x)| and a1(r,r) = 8r2(2r − r) r − 2r + 6r2 a2(r,r) = 4r 2 − 6rr + 3r2. proof. considering f written as in (11) and taking into account the maximum modulus principle we have |f|r ≤ |f(0)| + |g|r + |h|r. let us start with the study of the function g. using the previous lemma it follows that |g|r ≤ 2 √ π 3 √ r3 sup |ξ|≤r |sc (f(ξ)) | ∞∑ n=1 [ |x0,†,∗r n | ‖x0 n ‖l2(br;a;r) ‖sc(x0 n )‖l2(br) + n∑ m=1 ( |xm,†,∗r n | ‖xm,† n ‖l2(br;a;r) ‖sc(x m,† n )‖l2(br) + |y m,†,∗r n | ‖y m,† n ‖l2(br;a;r) ‖sc(y m,† n )‖l2(br) )] . applying proposition 2 and taking into account remark 5 it follows |g|r ≤ 2 √ 2 3 sup |ξ|≤r |sc (f(ξ)) | ∞∑ n=1 ( 2r r ) n (n + 1)2(n + 2). in the same way, we can study the function h. |h|r = |h̃|r ≤ 2 √ 2 3 sup |ξ|≤r |sc (he1(ξ)) | ∞∑ n=1 ( 2r r ) n (n + 1)(n + 2). finally, we obtain |f|r ≤ |f(0)| + 2 √ 2 3 sup |ξ|≤r |sc (f(ξ)) | ∞∑ n=1 ( 2r r ) n (n + 1)2(n + 2) + 2 √ 2 3 sup |ξ|≤r |sc (he1(ξ)) | ∞∑ n=1 ( 2r r ) n (n + 1)(n + 2). cubo 11, 1 (2009) on mapping properties of monogenic functions 93 now, note that the last series are convergent for 0 ≤ r < r 2 . the previous theorem states that a monogenic l2-function f : ω ⊂ r 3 −→ a is bounded by a combination of its real part and one of its other components. this result is a more general estimate but, in fact, it is not a complete analogy to the complex case. therein, an analytic function is only bounded by its real part. however, restricting ourselves to the class of functions which are orthogonal to the subspace of the non-trivial hyperholomorphic constants in l2(br; a; r), we get a stronger result: corollary 1. let f̃ be a square integrable a-valued monogenic function in br orthogonal to the non-trivial hyperholomorphic constants with respect to the inner product (3). then, for 0 ≤ r < r 2 we have the following inequality: |f̃|r ≤ |f̃(0)| + 8ra(r,r) (r − 2r)4 sup |ξ|≤r |sc(f̃(ξ))| where a(r,r) = 8r2(2r − r) + 6r2(r − 2r). remark 7. replacing f̃(x) by f(x) − f(0) in the resulting relation from the previous corollary, we arrive at |f(x) − f(0)|r ≤ 8ra(r,r) (r − 2r)4 sup |ξ|≤r |sc (f(ξ) − f(0)) | with a(r,r) as in the previous theorem, which is a refinement of hadamard’s real part theorem. we observe that in the constants of our estimates, the factor 1/(r − 2r) occurs and not r −r as it could be expected from the complex case. as we will see in the next section, to explain this is because monogenic functions do not map balls to balls in the small but balls to ellipsoids. 6 first applications as in the case of holomorphic functions in the complex plane we have to ask if also the growth of the derivative (here of the hypercomplex derivative) can be bounded by the growth of the function. if this is possible then, consequently, the behaviour of the derivative can be estimated by the scalar part of the monogenic function. the following theorem gives a first result. theorem 9. let f be a square integrable a-valued monogenic function in br. then, for 0 ≤ r < r 2 we have the following inequality: ∣∣∣∣( 1 2 d)f(x) ∣∣∣∣ r ≤ 8 √ 3 r2(2r2 + 7rr + 2r2) (r − 2r)5 sup |ξ|≤r |sc (f(ξ)) |. 94 k. gürlebeck and j. morais cubo 11, 1 (2009) proof. we consider f written as in (11) (lemma 1). since the referred series is convergent in l2, it converges uniformly to f in each compact subset of br. also the series of all partial derivatives converges uniformly to the corresponding partial derivatives of f in compact subsets of br. applying the hypercomplex derivative 1 2 d term by term to the series, it follows formally ( 1 2 d)f = ∞∑ n=1 [ ( 1 2 d)x0,†,∗r n α0 n + n∑ m=1 ( ( 1 2 d)xm,†,∗r n αm n + ( 1 2 d)y m,†,∗r n βm n ) + ( 1 2 d)xn+1,†,∗r n αn+1 n + ( 1 2 d)y n+1,†,∗r n βn+1 n ] the proof follows from the idea applied in theorem 8 with the estimates of the fourier coefficients from lemma 4 and taking into account the following equalities for the homogeneous monogenic polynomials (7) and their derivatives: ( 1 2 d)xl,† n = (n + l + 1)x l,† n−1 ( 1 2 d)y m,† n = (n + m + 1)y m,† n−1, for l = 0, ...,n and m = 1, ...,n. 7 m-conformal mappings the concept of monogenic-conformal mappings described by paravector-valued real differentiable functions in ω ⊂ rn+1 and with values in the clifford algebra cl0,n (in the cauchy-riemann sense) was introduced by malonek in [28]. let z∗ ∈ s be a fixed point and {sm} a regular sequence of subdomains which is shrinking to z∗ if m tends to infinity and whereby z∗ belongs to all sm. in [28] it is shown that a function f realizes locally in the neighborhood of a fixed point z = z∗ a left m-conformal mapping if and only if f is left monogenic and its left derivative is different from zero. this result is described by the limit of a ”quotient” of a 2-form (surface area) and a 3-form (volume). however, the geometric properties of such a result are not directly visible. here we show that the description of monogenic functions can be now formulated easily by accessible geometric mapping properties. for simplicity in what follows we focus our attention on the case of the dirac operator. let r 0,3 be the real vector space r3 endowed with a quadratic form of signature (0, 3) and let (ε0,ε1,ε2) be an associated orthonormal basis for r0,3. then r0,3 generates the clifford algebra r0,3 which is a real linear associative algebra of dimension 23 and with identity 1. the multiplication in r0,3 is given according to the multiplication rules ε2 i = −1, i = 0, 1, 2 εiεj + εjεi = 0, i 6= j, 0 ≤ i,j ≤ 2. cubo 11, 1 (2009) on mapping properties of monogenic functions 95 we introduce the dirac operator ∂ = ε0∂x0 + ε1∂x1 + ε2∂x2. (15) we are mainly interested in the case of vector-valued functions f : ω ⊂ r3 −→ r0,3 defined as f(x) = f0(x)ε0 + f1(x)ε1 + f2(x)ε2, (16) where its coordinates fi (i = 0, 1, 2) are real-valued functions defined in ω. continuously realdifferentiable functions f : ω −→ r0,3 which satisfy ∂f = 0 ⇐⇒    ∂x0f0 + ∂x1f1 + ∂x2f2 = 0 ∂x0f1 − ∂x1f0 = 0 ∂x0f2 − ∂x2f0 = 0 ∂x1f2 − ∂x2f1 = 0 , (17) are said to be (left) monogenic in ω. moreover, as ∂2 = −∆, where ∆ is the laplace operator in r 3, (left) monogenic functions in ω are also harmonic in ω. let f : ω ⊂ r3 −→ r0,3 be an arbitrary real-differentiable function. clearly then, f(x) := f(0) + f(x) + r(x) being f its linear part and r stands for the rest (degree n ≥ 2). under the above hypotheses, we claim that f is a general linear function given as in (16), where the coordinates fi (i = 0, 1, 2) are given by f0(x) = a0x0 + a1x1 + a2x2 f1(x) = b0x0 + b1x1 + b2x2 (18) f2(x) = c0x0 + c1x1 + c2x2. let us denote by e the ellipsoid generated by the quadratic form e := {(x0,x1,x2) : x2 0 α2 + x2 1 β2 + x2 2 γ2 = 1}, (19) where α, β and γ are the lengths of the semi-axes. the next theorem characterizes the local mapping properties of the linear part f of an arbitrary function f . theorem 10. let f be a linear function. then, the function f is monogenic if and only if it maps a ball to an ellipsoid centered at the origin with the property that the reciprocal of the length of one semi-axis is equal to the sum of the reciprocals of the lengths of the other two semi-axes. 96 k. gürlebeck and j. morais cubo 11, 1 (2009) proof. for simplicity we just prove the sufficient condition. the necessary condition can be found in [23]. suppose that there exists a linear analytic function f which maps the unit ball b to an arbitrarily oriented ellipsoid ε with the referred property. we rotate this ellipsoid transforming it to an ellipsoid ε∗ such that the directions of its semi-axes y(0), y(1), y(2) coincide with the directions of the standard coordinate system (y0,y1,y2). such an ellipsoid is given by ε∗ : {(y0,y1,y2) : y2 0 α2 + y2 1 β2 + y2 2 γ2 ≤ 1}. let f̃ be the function whose image represents ε∗. we denote by d̃ the associated matrix to f̃ d̃ =   ã0 ã1 ã2 b̃0 b̃1 b̃2 c̃0 c̃1 c̃2   . we remind that ε∗ preserves the orientation, and therefore, it holds the property d̃y(i) = λy(i) (i = 0, 1, 2). it is easily seen that d̃ is a diagonal matrix and moreover, its elements satisfy the equation ã0 + b̃1 + c̃2 = 0. in this case the associated function is given by f̃(x) = −(b̃1 + c̃2)x0e0 + b̃1x1e1 + c̃2x2e2. it is easy to check that this function is monogenic (with respect to the dirac operator). now, as it was described before, we apply a rotation r to ε∗ in order to obtain ε. roughly speaking, for the rotation r =   r1,1 r1,2 r1,3 r2,1 r2,2 r2,3 r3,1 r3,2 r3,3   such that rtr = i = rrt , we have then that rd̃rt := a. note that the original function f is now represented by the symmetric matrix a, in fact ax = f(x) for x = (x0 x1 x2) t , being its coordinates given by f0(x) = (ã0r 2 1,1 + b̃1r 2 1,2 + c̃2r 2 1,3 )x0 + (ã0r1,1r2,1 + b̃1r1,2r2,2 + c̃2r1,3r2,3)x1 + (ã0r1,1r3,1 + b̃1r1,2r3,2 + c̃2r1,3r3,3)x2 f1(x) = (ã0r1,1r2,1 + b̃1r1,2r2,2 + c̃2r1,3r2,3)x0 + (ã0r 2 2,1 + b̃1r 2 2,2 + c̃2r 2 2,3 )x1 + (ã0r2,1r3,1 + b̃1r2,2r3,2 + c̃2r2,3r3,3)x2 f2(x) = (ã0r1,1r3,1 + b̃1r1,2r3,2 + c̃2r1,3r3,3)x0 + (ã0r2,1r3,1 + b̃1r2,2r3,2 + c̃2r2,3r3,3)x1 + (ã0r 2 3,1 + b̃1r 2 3,2 + c̃2r 2 3,3 )x2. it is easy to check, as desired, that the function f is monogenic. cubo 11, 1 (2009) on mapping properties of monogenic functions 97 remark 8. by a simple linear transformation of variables the previous theorem can be extended to a (small) ball with radius r. remark 9. the composition of a linear function with a translation allows to extend the result to an ellipsoid centered at an arbitrary point x̃. this composition preserves the monogenicity. next one has to show that theorem 10 can be generalized to arbitrary real-analytic functions which have the described local mapping properties. a monogenic function with non-vanishing linear part will map in the small balls to the special class of ellipsoids. non-vanishing linear part means that all directional first derivatives of the function are different from zero. equivalently this can be characterized by the jacobian determinant. for details and further relations to the hypercomplex derivative see the paper [11]. theorem 11. let f be a real-analytic function. then, the function f is monogenic if and only if it maps locally a ball to an ellipsoid with the property that the reciprocal of the length of one semi-axis is equal to the sum of the reciprocals of the lengths of the other two semi-axes. proof. if the function is monogenic then there is almost nothing to prove. the local mapping properties at a point x are determined by the linear part of the taylor expansion at x. theorem 10 leads to the stated result. if a real-analytic function has the supposed local mapping properties then we have to expand the function at x in a real taylor series. applying ideas from [15], chapter ii, paragraph 5.2.2 we can show after a longer calculation which we will skip here that the linear part of the taylor expansion satisfies at each point of the domain the dirac equation and so the function must be monogenic. monogenic functions as null solutions of the dirac operator can be mapped to monogenic (or anti-monogenic) functions in the sense of satisfied cauchy-riemann equations (for details see, e.g., [13]). this transformation is an isometry and so it becomes clear that the here discussed mapping properties of monogenic functions remain true under this transformation. the result of theorem 11 allows to describe the monogenic functions as a special class of quasi-conformal mappings. if we visualize quasi-conformal mappings in r3 by points, given by the lengths of the semi-axes of the associated ellipsoids, then the monogenic functions (with nonvanishing jacobian determinant) can be seen as a two-dimensional manifold in r3. acknowledgments: financial support from foundation for science and technology (fct) via the phd/grant sfrh/bd/19174/2004. 98 k. gürlebeck and j. morais cubo 11, 1 (2009) received: april 2008. revised: august 2008. references [1] l. aizenberg, multidimensional analogues of bohr’s theorem on power series. proc. amer. math. soc. 128, 1147-1155 (2000). 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[36] m. tomić, sur un théorème de h. bohr. math. scand. 11, 103-106. mr 31:316 (1962). cubo a mathematical journal vol.12, no¯ 03, (13–32). october 2010 a family of stationary solutions to the euler equations and generalized solutions juliana conceição precioso departamento de matemática, universidade estadual paulista, 15054-000, s.j.r. preto, sp, brazil email: precioso@ibilce.unesp.br abstract in this work, we present a interesting family of stationary solutions for the euler equations, which behaves in the same way that the approximated solutions presented in [6]. resumen en este trabajo, presentamos una familia interesante de soluciones estacionarias para las ecuaciones de euler, que se comportan de la misma manera que las soluciones aproximadas presentadas en [6]. key words and phrases: euler equations, incompressible flows, generalized solutions. math. subj. class.: 35d99. 14 juliana c. precioso cubo 12, 3 (2010) 1 introduction an ideal incompressible fluid moving inside d ⊂ rn is classically described by a velocity field u(t, x) and a pressure field p(t, x), subject to the classical euler equations: { ∂t u + (u ·∇)u +∇p = 0 ∇· u = 0, (1.1) with the boundary condition that u is tangent to ∂d. in classical continuous mechanics [1], the motion of an incompressible inviscid fluid in a compact domain d ⊂ rn can be seen as a geodesic on the group of all diffeomorphisms of d with unit jacobian determinant, g(d). this set is included in s(d) the semigroup of all borel maps h of d that satisfy ˆ d f (h(x))d x = ˆ d f (x)d x, ∀f ∈ c0(d). for more details see [1], [2] or [6]. we will denote v := { u : [0, t] × d −→ rn such that u ∈ c0(q), u(t,·) ∈ li p(d) uniformly in 0 ≤ t ≤ t, div u = 0, u(t,·) · n̂∣∣ ∂d = 0 } . note that the flow (t, x) 7→ g(t, x) describing the motion of fluid particles is defined by { ∂t g(t, x) = u(t, g(t, x)) g(0, x) = x. (1.2) by cauchy-lipschitz theorem, for each u ∈ v , there is a unique solution to (1.2) and for each time t the map g(t, x) = g(t,·) ∈ g(d). then, by elementary calculations, the euler equations can be replaced by the following equivalent equations: { ∂2tt g(t, x) +∇p(t, g(t, x)) = 0 det d x g(t, x) = 1. (1.3) lets call (1.3) by the “lagrangian formulation” of the euler equations. from a geometrical point of view, different from the natural pde point of view which consists in adressing the euler equations as an evolution equation with prescribed initial cubo 12, 3 (2010) a family of stationary solutions ... 15 velocity field, it is natural to solve the problem to minimize the action a( g) = 1 2 ˆ t 0 ˆ d |∂t g(t, x)|2 d xd t, among all trajectories on g(d) connecting g(0,·) = i d and g(t,·) = h. the corresponding system of pde’s is the lagrangian formulation of the euler equations (1.3). ebin and marsden showed local existence and uniqueness for this problem, namely, if h and i are sufficiently close in a sufficiently high order sobolev norm, then there is a unique geodesic connecting i d to h, see [9]. in the large, uniqueness can fail. however, in [12], a. i. shnirelman shows that existence of minimal geodesics may fail to a class of data. to solve the problem to find minimal geodesics in a generalized sense, in particular for data h in shnirelman class, was introduced suitable “young measure", (see [15] and [18]) of different ways as in [4], [6], [12] and [13]. in [4], was used a concept that takes into account the dynamics of the particles. to each path t ∈ [0, t] 7−→ z(t) ∈ d, one associates the probability that it is followed by some material particle. more precisely, was proposed a notion of the generalized flow, as been a probability measure on set ω = d[0,t] of all curves t ∈ [0, t] −→ z(t) ∈ d, namely, a borel probability measure µ, on product space ω = d[0,t], such that each projection µt for 0 ≤ t ≤ t is a lebesgue measure on d. the action in this context is express by ˆ ω ˆ t 0 1 2 |z′ (t)|2 dµt(z)d t. brenier showed the existence of generalized solutions and, later in [5], the existence and uniqueness of the pressure gradient linked to them through a suitable poisson equation, but did not obtain for them a complete set of equations beyond the classical euler equations. however, in [6] it was possible. the problem to find minimal geodesics was reformulated in terms of a pair of measures associated to the field u, solution of the euler equations, in the following way: given a smooth trajectory t ∈ [0, t] 7→ g(t, x) ∈ g(d), we define the measures (respectively nonnegative and vector-valued) c(t, x, a) = δ(x − g(t, a)), m(t, x, a) = ∂t g(t, a)δ(x − g(t, a)), (1.4) defined on q ′ = [0, t] × d × d. these measures satisfy ˆ d c(t, x, a)da = 1, (1.5) ∂t c +∇x · m = 0, (1.6) c(0, x, a) = δ(x − a); c(t, x, a) = δ(x − h(a)). (1.7) 16 juliana c. precioso cubo 12, 3 (2010) moreover, the measure m is absolutely continuous with respect to c, with a density v ∈ l2(q ′ , d c), so that m = cv, and the action is given by a( g) = 1 2 ˆ 1 0 ˆ d |v(t, x, a)|2 c(t, x, a)d xda, or equivalently, a( g) = k [c, m], where k [c, m] := sup x {〈c, f〉+〈m,φ〉}, (1.8) where (c, m) is of the form (1.4) and x = { (f,φ) ∈ c0(q′ ) × ( c0(q ′ ) )n ; f(t, x, a) + 1 2 |φ(t, x, a)|2 ≤ 0 } . then, brenier defined the relaxed problem, as the problem to look for pairs of measures (c, m) that minimize k [c, m] and are admissible in the sense of (1.5), (1.6) and (1.7), but do not necessarily satisfy (1.4). also was showed that, for d = [0, 1]n and each data h ∈ s(d), the relaxed problem always has solutions (c, m) and that there is a unique locally bounded measure ∇x p in the interior of q = [0, t] × d, depending only h, such that ∂t(cv) +∇x(cv ⊗ v) + c∇x p = 0, holds in the sense of distributions on the interior of q ′ , where c is a extension of c for which the product c(t, x, a)∇x p(t, x) is well-defined. moreover, was showed that for any h ∈ s([0, 1]3) of the form h(x1, x2, x3) = (h(x1, x2), x3), that , for any ε > 0, there is a uε ∈ v such that k (uε) + 1 2ε ||guε (t,·) − h||2l2(d) ≤ iε(h) +ε, where iε(h) = infu∈v { k (u) + 1 2ε ||gu(t,·) − h||2l2(d) } and k(uε) = 1 2 ˆ t 0 ˆ d |uε(t, x)|2 d xd t = 1 2 ˆ t 0 ˆ d |∂t gε(t, x)|2 d xd t = a( gε). in addition, the measures (cε, mε) associated with uε, through (1.4), converge, as ε → 0 to the generalized solutions of the relaxed problem. moreover, the fields uε satisfy ∇x · uε = 0, ∂t uε + (uε ·∇) uε → −∇p, cubo 12, 3 (2010) a family of stationary solutions ... 17 weakly, as ε tends to zero. as observed in [6], with each solution (c, m) we may associated a measure-valued solution µ, in the sense of diperna and majda, by setting ˆ q×rd f (t, x,ξ)dµ(t, x,ξ) = ˆ q′ f (t, x, v(t, x, a))d c(t, x, a), for any continuous function f ∈ q ×rd with at most quadratic growth as ξ → ∞. for more details, see [6] and [7]. in [4], brenier shows explicit examples of non trivial generalized solutions. a typical example is when d is the unit disk in 2d and h(x) = −x. we know that the problem of the minimal action has two trivial solutions g+(t, x) = eiπt x and g−(t, x) = e−iπt x with the same pressure field p(x) = π 2|x|2 2 . we have another (generalized) solution (c, m) to the same problem which is given by ˆ q′ f (t, x, a)c(t, x, a)d td xda = ˆ [0,1]×d ˆ 1 0 f (t, g(t, a,θ), a)dθd tda, ˆ q′ f (t, x, a)m(t, x, a)d td xda = ˆ [0,1]×d ˆ 1 0 ∂tg(t, a,θ) f (t, g(t, a,θ), a)dθd tda, for all continuous function f , where g(t,θ, a) = a cos(πt) + ( 1 −|a|2 ) 1 2 e2iπθ sin(πt) ∈ d. note that each particle initially located at a ∈ d splits up along a circle of radius ( 1 −|a|2 ) 1 2 sin(πt), with center a cos(πt), that moves across the unit disk and shrinks down to the point −a when t = 1. in general, is very difficult to obtain explicit examples of non trivial generalized solutions and the explicit examples constructed by brenier, are based on the model presented in [4], which takes in account a concept purely lagrangian of young measures, the so-called generalized flows. beyond supplying an application of the model developed in [6], the results of this paper give an interesting information for the limit of a sequence of the stationary solutions, showing that they are associated with measures that satisfy the euler equations in a specified weak sense. for another point of view, the results of the paper give example of as a sequence of highly oscillatory solutions still can have a limit that is solution in some sense. namely, we exhibited a family uε (which behavior as the“approximated solutions" argued above) such that when ε → 0, the velocity field gets more and more oscillatory, but the measures (cε, mε) associated to the field uε converges to a solution (c, m) of the equations ˆ d c(t, x, a)da = 1, ∂t c +∇x · m = 0, 18 juliana c. precioso cubo 12, 3 (2010) and ∂t(cv) +∇x(cv ⊗ v) + c∇x p = 0. equivalent vector fields to those ones treaties in this work have been studied as application of high order essentially no oscillatory (eno) schemes for smooth solutions of navierstokes and euler equations, (see [14]), in problems involving the taylor-green vortex, (see [8], [3] and [10]) and to explore a discrete singular convolution algorithm (dsc) for solving certain mechanics problems, (see [16] and [17]). 2 a family of (stationary) solutions in this work we consider the following family of stationary solutions to the euler equations: un(x, y) = ( −cos(x) sin (n y) , 1 n sin(x) cos (n y) , 0 ) , pn(x, y) = − 1 4 ( cos(2x) + 1 n2 cos (2n y) ) . note that, |d pn(x, y)| ≤ c, and when n goes to infinity the pressure field strongly converges to p(x, y) = − 1 4 cos(2x). then, is easy verify that (un ·∇) un +∇p → 0, when n → ∞. for a moment, let us observe the behavior of family un. for n = 1 we have, u1(x, y) = (−cos(x) sin( y), sin(x) cos( y), 0) and { ẋ = −cos(x) sin( y) ẏ = sin(x) cos( y). (2.1) then, we have (ẋ, ẏ) = (0, 0) ⇔ (x, y) = ( (2k + 1)π 2 , (2l + 1)π 2 ) or (kπ, lπ), where k, l ∈ z, (see figure 1). for n = 2 we have, (ẋ, ẏ) = (0, 0) ⇔ (x, y) = ( (2k + 1)π 2 , (2l + 1)π 4 ) or ( kπ, lπ 2 ) , where k, l ∈ z, (see figure 2). thus, for n we have, (ẋ, ẏ) = (0, 0) ⇔ (x, y) = ( (2k + 1)π 2 , (2l + 1) π 2n ) or ( kπ, lπ n ) , where k, l ∈ z. note that, when n → ∞ the velocity field gets more and more oscillatory. in the next section we will show that the measures (cn, mn) defined by cn(t, x, a) = δ ( x − gun (t, a) ) , mn(t, x, a) = un(t, x)δ(x − gun (t, a)) converges to the solution (c, m) of the equations:ˆ d c(t, x, a)da = 1, (2.2) cubo 12, 3 (2010) a family of stationary solutions ... 19 –2 –1 0 1 2 y –2 –1 1 2 x figure 1: phase portrait of the velocity field u1(x, y) = (−cos(x) sin( y), sin(x) cos( y), 0) –2 –1 0 1 2 y –2 –1 1 2 x figure 2: phase portrait of the velocity field u2(x, y) = ( −cos(x) sin(2 y), 1 2 sin(x) cos(2 y), 0 ) ∂t c +∇x · m = 0 (2.3) ∂t(cv) +∇x · (cv ⊗ v) + c∇x p = 0, (2.4) by the consistency theorem in [6], or its generalization for variable density in [11], we know that if un is a solution of the euler equations, then the pair of the measures (cn, mn) defined as below satisfy the equations (2.2), (2.3), (2.4). 20 juliana c. precioso cubo 12, 3 (2010) 3 the limite (c, m) in this section we build explicitly the limite (c, m). for this, in first we rewrite the field un(x, y) = ( −cos(x) sin(n y), 1 n sin(x) cos(n y), 0 ) as un(x, y) = ( u11(x, n y), 1 n u21(x, n y), 0 ) , where u11(x, n y) = −cos(x) sin(n y) and u21(x, n y) = sin(x) cos(n y). of here in ahead, we will omit third coordinate of the fields u ′ ns. we also observe that the field has period 2π. now, we define { x = x(t,γ,δ) y = y(t,γ,δ) the solution of    d x d t = u11(x, y) = cos(x) sin( y) d y d t = u21(x, y) = sin(x)cos( y) x(0,γ,δ) = γ y(0,γ,δ) = δ. (3.1) let be 0 ≤ i ≤ n − 1, 0 ≤ α1 ≤ 2π, and 2πi n ≤ α2 ≤ 2π n (i + 1), where i, n ∈ n. this is: • for n = 1, i = 0 and we have 0 ≤ α2 ≤ 2π. • for n = 2, 0 ≤ i ≤ 1 and we have { 0 ≤ α2 ≤ π, if i = 0 π ≤ α2 ≤ 2π, if i = 1. • for n = k, 0 ≤ i ≤ k − 1 and we have    0 ≤ α2 ≤ 2π k , if i = 0 2π k ≤ α2 ≤ 4π k , if i = 1 . . . 2π(k − 1) k ≤ α2 ≤ 2π, if i = (k − 1). cubo 12, 3 (2010) a family of stationary solutions ... 21 α 2 α 2 α α1 1 0 0 2 π π2 2 π2 π n=1 π n=2 α 2 α 1 0 π2 2 π π n=k k 2π k 2π (k-1) figure 3: then, i counts the cells (in the vertical line) from 0 to 2π for each n, as we can observe in figure 3. now, we define    xin(t,α1,α2) := x ( t,α1, n ( α2 − 2πi n )) yin(t,α1,α2) := 1 n y ( t,α1, n ( α2 − 2πi n )) + 2πi n . (3.2) note that, by the definition above • for n = 1, we have i = 0, thus { x01(t,α1,α2) := x(t,α1,α2), y01 (t,α1,α2) := y(t,α1,α2). • for n = 2, we have 0 ≤ i ≤ 1, thus    x02(t,α1,α2) := x(t,α1, 2α2) y02 (t,α1,α2) := 1 2 y(t,α1, 2α2) , if i = 0 and    x12(t,α1,α2) := x(t,α1, 2(α2 −π)) y12 (t,α1,α2) := 1 2 y(t,α1, 2(α2 −π)) +π , if i = 1. • for n = k, we have 0 ≤ i ≤ k − 1, thus    x0k(t,α1,α2) := x(t,α1, kα2) y0k(t,α1,α2) := 1 k y(t,α1, kα2) , if i = 0, 22 juliana c. precioso cubo 12, 3 (2010) ...    xk−1k (t,α1,α2) := x ( t,α1, k ( α2 − 2(k − 1)π k )) yk−12 (t,α1,α2) := 1 k y ( t,α1, k ( α2 2(k − 1)π k )) + 2(k − 1)π k , if i = k − 1. then, we conclude that 0 ≤ xin ≤ 2π, 2πi n ≤ yin ≤ 2π(i + 1) n and    xin(0,α1,α2) = x ( 0,α1, n ( α2 − 2πi n )) = α1 yin(0,α1,α2) = 1 n y ( 0,α1, n ( α2 − 2πi n )) + 2πi n = α2. moreover, by (3.2) we have    d xin d t (t,α1,α2) = d x d t ( t,α1, n ( α2 − 2πi n )) d yin d t (t,α1,α2) = 1 n d y d t ( t,α1, n ( α2 − 2πi n )) (3.3) therefore, by (3.1) and (3.3) d xin d t = u11 ( x ( t,α1, n ( α2 − 2πi n )) , y ( t,α1, n ( α2 − 2πi n ))) = u11 ( xin(t,α1,α2), n y i n(t,α1,α2) − 2πi ) = u11 ( xin(t,α1,α2), n y i n(t,α1,α2) ) = u1n ( xin(t,α1,α2), y i n(t,α1,α2) ) , and d yin d t = 1 n u21 ( x ( t,α1, n ( α2 − 2πi n )) , y ( t,α1, n ( α2 − 2πi n ))) = 1 n u21 ( xin(t,α1,α2), n y i n(t,α1,α2) − 2πi ) = 1 n u21 ( xin(t,α1,α2), n y i n(t,α1,α2) ) = u2n ( xin(t,α1,α2), y i n(t,α1,α2) ) . now defining { xn(t,α1,α2) := xin(t,α1,α2) yn(t,α1,α2) := yin(t,α1,α2) if 2πi n ≤ α2 ≤ 2π(i + 1) n , cubo 12, 3 (2010) a family of stationary solutions ... 23 we conclude that    d xn d t (t,α1,α2) = u1n (xn(t,α1,α2), yn(t,α1,α2)) d yn d t (t,α1,α2) = u2n (xn(t,α1,α2), yn(t,α1,α2)) xn(0,α1,α2) = α1 yn(0,α1,α2) = α2. (3.4) in the remain of the work, for simplicity, we will use the following notation: d i = (0, 2π)i = (0, 2π)×···×(0, 2π), i times, where i = 1,··· , 4. now, we are ready to show the following result: theorem 3.1. consider (xn, yn) solution of (3.4). let { cn(t, x, y,α1,α2) = δ((x, y) − (xn(t,α1,α2), yn(t,α1,α2))) mn(t, x, y,α1,α2) = un(x, y)δ((x, y) − (xn(t,α1,α2), yn(t,α1,α2))) . then, 〈ϕ, cn〉 → 1 2π ˆ d3 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dβ2 dγ and 〈φ, mn〉 → 1 2π ˆ d3 ˆ t 0 φ 1(x(t,α1,β2),γ,α1,γ, t)u 1 1(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 dγ, whenever n → ∞, for any ϕ ∈ c∞0 (d4 × (0, t)) and φ ∈ ( c∞0 (d4 × (0, t)) )2 . proof. let ϕ ∈ c∞0 (d4 × (0, t)) and cn(t, x, y,α1,α2) = δ((x, y) − (xn, yn)(t,α1,α2)) then, we have 〈ϕ, cn〉 = ˆ d2 ˆ t 0 ϕ(xn(t,α1,α2), yn(t,α1,α2),α1,α2, t)d tdα1 dα2 = n−1∑ i=0 ˆ 2π n (i+1) 2π n i ˆ 2π 0 ˆ t 0 ϕ ( xin(t,α1,α2), y i n (t,α1,α2) ,α1,α2, t ) d tdα1 dα2 = n−1∑ i=0 ˆ 2π n (i+1) 2π n i ˆ 2π 0 ˆ t 0 ϕ ( x ( t,α1, n ( α2 − 2πi n )) , 1 n y ( t,α1, n ( α2 − 2πi n )) + 2πi n ,α1,α2, t ) d tdα1 dα2 24 juliana c. precioso cubo 12, 3 (2010) now, make β2 = n ( α2 − 2πi n ) = nα2 − 2πi, namely, α2 = β2 n + 2πi n . then, we have 〈ϕ, cn〉 = n−1∑ i=0 ˆ d2 ˆ t 0 ϕ ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n , α1, β2 n + 2πi n , t ) d tdα1 dβ2 n = an + bn, where an = 1 2π n−1∑ i=0 (ˆ d2 ˆ t 0 ϕ ( x(t,α1,β2), 2πi n ,α1, 2πi n , t ) d tdα1 dβ2 ) 2π n (3.5) and bn = n−1∑ i=0 1 n ˆ d2 ˆ t 0 [ ϕ ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) −ϕ ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )] d tdα1 dβ2. note that, by mean value theorem, we have ∣∣∣∣ϕ ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) − −ϕ ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )∣∣∣∣ = ∣∣∣∣ ∂ϕ ∂y 1 n y(t,α1,α2) + ∂ϕ ∂β2 β2 n ∣∣∣∣ ≤ ≤ (∣∣∣∣ ∣∣∣∣ ∂ϕ ∂y ∣∣∣∣ ∣∣∣∣ l∞((0,2π)4×(0,t)) |y(t,α1,α2)| n + ∣∣∣∣ ∣∣∣∣ ∂ϕ ∂β2 ∣∣∣∣ ∣∣∣∣ l∞((0,2π)4×(0,t)) |β2| n ) ≤ ≤ 2π n (∣∣∣∣ ∣∣∣∣ ∂ϕ ∂y ∣∣∣∣ ∣∣∣∣ l∞ + ∣∣∣∣ ∣∣∣∣ ∂ϕ ∂β2 ∣∣∣∣ ∣∣∣∣ l∞ ) ≤ 2π n ∥ dϕ ∥l∞ . then, we obtain |bn| ≤ n−1∑ i=0 1 n ˆ d2 ˆ t 0 ∣∣∣∣ [ ϕ ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) −ϕ ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )]∣∣∣∣ d tdα1 dβ2 ≤ n−1∑ i=0 1 n 2π n ∥ dϕ ∥l∞ ˆ 2π 0 ˆ 2π 0 ˆ t 0 d tdα1 dβ2 = 2π n ∥ dϕ ∥l∞ 4π2 t − 2π n2 ∥ dϕ ∥l∞ 4π2 t ≤ 2π n ∥ dϕ ∥l∞ 4π2 t. cubo 12, 3 (2010) a family of stationary solutions ... 25 therefore, bn → 0, when n → ∞. now, define the function ψ by ψ(γ) := ˆ d2 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dβ2 thus, we rewrite (3.5) as an = 1 2π n−1∑ i=0 ψ ( 2πi n ) 2π n = 1 2π n−1∑ i=0 ψ ( 2πi n ) ( 2π(i + 1) n − 2πi n ) . by this form, if γi = 2πi n then, { γ0,γ1,··· ,γn } is a partition of the (0, 2π) and an = 1 2π n−1∑ i=0 ψ(γi ) ( γi+1 −γi ) is a riemann sum. therefore, an → 1 2π ˆ 2π 0 ψ(γ)dγ, when n → ∞. then, we conclude that 〈ϕ, cn〉 → 1 2π ˆ d3 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dβ2 dγ, when n → ∞, for any ϕ ∈ c∞0 (d4 × (0, t)) . now, consider φ ∈ ( c∞0 (d4 × (0, t)) )2 and let mn = un(x, y)δ((x, y) − (xn(t,α1,α2), yn(t,α1,α2))) = ( u11(x, n y), 1 n u21(x, n y) ) δ((x, y) − (xn(t,α1,α2), yn(t,α1,α2))) . then, we obtain 〈φ, mn〉 = ˆ d2 ˆ t 0 φ(xn(t,α1,α2), yn(t,α1,α2),α1,α2, t) [ u11(xn(t,α1,α2), n yn(t,α1,α2)), 1 n u21(xn(t,α1,α2), n yn(t,α1,α2)) ] d tdα1 dα2 = n−1∑ i=0 ˆ 2π n (i+1) 2π n i ˆ 2π 0 ˆ t 0 φ(xin(t,α1,α2), y i n(t,α1,α2),α1,α2, t) [ u11(x i n(t,α1,α2), n y i n(t,α1,α2)), 1 n u21(x i n(t,α1,α2), n yin(t,α1,α2)) ] d tdα1 dα2, 26 juliana c. precioso cubo 12, 3 (2010) and therefore, making β2 = n ( α2 − 2πi n ) we have 〈φ, mn〉 = n−1∑ i=0 ˆ d2 ˆ t 0 1 n φ 1 ( x(t,α1,β2), 1 n y(t,α1,α2) + 2πi n ,α1, β2 n + 2πi n , t ) u11(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 + + n−1∑ i=0 ˆ d2 ˆ t 0 1 n2 φ 2 ( x(t,α1,β2), 1 n y(t,α1,α2) + 2πi n ,α1, 2πi n , t ) u21(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 = a1n + b1n + a2n + b2n, where, a1n = 1 2π n−1∑ i=0 (ˆ d2 ˆ t 0 1 n φ 1 ( x(t,α1,β2), 2πi n ,α1, 2πi n , t ) u11(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 ) 2π, a2n = 1 2πn n−1∑ i=0 (ˆ d2 ˆ t 0 1 n φ 2 ( x(t,α1,β2), 2πi n ,α1, 2πi n , t ) u21(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 ) 2π, b1n = n−1∑ i=0 ˆ d2 ˆ t 0 1 n [ φ 1 ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n , α1, β2 n + 2πi n , t ) −φ1 ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )] u11(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2, b2n = n−1∑ i=0 ˆ d2 ˆ t 0 1 n2 [ φ 2 ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) −φ2 ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )] u21(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2, as we seen before, we conclude that ∣∣∣∣φ i ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) − −φi ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )∣∣∣∣ ≤ 2π n ∥ dφ ∥l∞ , i = 1, 2. cubo 12, 3 (2010) a family of stationary solutions ... 27 thus, we have the estimate |b1n| ≤ n−1∑ i=0 1 n ˆ d2 ˆ t 0 ∣∣∣∣φ 1 ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) −φ1 ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )∣∣∣∣|u 1 1(x(t,α1,β2), y(t,α1,β2))|d tdα1 dβ2, ≤ 2π n ∥ dφ ∥l∞ 4π2 tc. therefore, b1n → 0, when n → ∞. now, we go to study the term b2n. |b2n| ≤ n−1∑ i=0 1 n2 ˆ d2 ˆ t 0 ∣∣∣∣φ 2 ( x(t,α1,β2), 1 n y(t,α1,β2) + 2πi n ,α1, β2 n + 2πi n , t ) −φ2 ( x(t,α1,β2), 2πi n ,α1, 2πi n , t )∣∣∣∣|u 2 1(x(t,α1,β2), y(t,α1,β2))|d tdα1 dβ2 ≤ 2π n2 ∥ dφ ∥l∞ 4π2 tc, and, therefore, also b2n → 0, when n → ∞. defining the function ψ by ψ i (γ) := ˆ d2 ˆ t 0 φ i (x(t,α1,β2),γ,α1,γ, t)u i 1(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2, i = 1, 2, we obtain, a1n = 1 2π n−1∑ i=0 ψ 1 ( 2πi n ) 2π n = 1 2π n−1∑ i=0 ψ 1 ( 2πi n ) ( 2π(i + 1) n − 2πi n ) . by this form, if γi = 2πi n then, { γ0,γ1,··· ,γn } is a partition of the (0, 2π) and a1n = 1 2π n−1∑ i=0 ψ 1(γi ) ( γi+1 −γi ) is a riemann sum. therefore, a1n → 1 2π ˆ 2π 0 ψ 1(γ)dγ, when n → ∞. for the last term, we have that a2n = 1 2πn n−1∑ i=0 ψ 2 ( 2πi n ) 2π n 28 juliana c. precioso cubo 12, 3 (2010) = 1 2πn n−1∑ i=0 ψ 2 ( 2πi n ) ( 2π(i + 1) n − 2πi n ) , and therefore, a2n → 0, when n → ∞. then, we conclude that 〈φ, mn〉 → 1 2π ˆ d3 ˆ t 0 φ 1(x(t,α1,β2),γ,α1,γ, t)u 1 1(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 dγ, when n → ∞, for any φ ∈ ( c∞0 (d4 × (0, t)) )2 . by the last theorem we can conclude that 〈ϕ, cn〉 → 1 2π ˆ d3 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dβ2 dγ and 〈φ, mn〉 → 1 2π ˆ d3 ˆ t 0 φ 1(x(t,α1,β2),γ,α1,γ, t)u 1 1(x(t,α1,β2), y(t,α1,β2))d tdα1 dβ2 dγ, whenever n → ∞, for any ϕ ∈ c∞0 (d4 × (0, t)) and φ ∈ ( c∞0 (d4 × (0, t)) )2 . now, note that 2π〈ϕ, cn〉 → ˆ d3 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dβ2 dγ is equivalent to ˆ 2π 0 〈ϕ, cn〉 dβ2 → ˆ d3 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dγdβ2 and, therefore, ˆ 2π 0 [ 〈ϕ, cn〉− ˆ d2 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dγ ] dβ2 → 0. then, we conclude that 〈ϕ, cn〉 → ˆ d2 ˆ t 0 ϕ(x(t,α1,β2),γ,α1,γ, t)d tdα1 dγ. cubo 12, 3 (2010) a family of stationary solutions ... 29 of completely analogous way, we also conclude that 〈φ, mn〉 → ˆ d2 ˆ t 0 φ 1(x(t,α1,β2),γ,α1,γ, t)u 1 1 ( x(t,α1,β2), y(t,α1,β2) ) d tdα1 dγ. thus, the limite (c, m) is given by { c(x, y,α1,α2, t) = δ ( (x,α2) − (x(t,α1,β2), y) ) m(x, y,α1,α2, t) = δ ( (x,α2) − (x(t,α1,β2), y) ) ( u11(x, y(t,α1,β2)), 0 ) . (3.6) 4 solution to the relaxed euler equations in this section we will conclude our work showing that the pair (c, m), build in the before section, satisfy the relaxed euler equations. theorem 4.1. the pair of measures (c,m) defined in (3.6) satisfy the following equations ˆ d2 c(t, x, y,α1,α2)dα1 dα2 = 1, ∂t c +∇· m = 0, ∂t(cv) +∇· (cv ⊗ v) + c∇p = 0. in the sense of distributions. proof. note that 〈1, c〉 = ˆ d2 ˆ t 0 d tdα1 dα2 = ˆ d2 ˆ t 0 d td xd y. then, we obtain ˆ d2 ˆ t 0 (ˆ d2 c(t, x, y,α1,α2)dα1 dα2 − 1 ) d td xd y = 0 and, therefore, ˆ d2 c(t, x, y,α1,α2)dα1 dα2 = 1. now, we will show that pair (c, m) satisfy the equation ∂t c + ∇ · m = 0. consider ϕ ∈ c∞0 (d4 × (0, t)), thus 〈ϕ(x, y,α1,α2, t),∂t c(t, x, y,α1,α2) +∇(x, y) · m(t, x, y,α1,α2)〉 = = − ˆ d2 ˆ t 0 ∂tϕ(x(t,α1,β2), y,α1, y, t)d tdα1 d y − ˆ d2 ˆ t 0 ∂xϕ(x(t,α1,β2), y,α1, y, t)u 1 1 ( x(t,α1,β2), y(t,α1,β2) ) d tdα1 d y 30 juliana c. precioso cubo 12, 3 (2010) = − ˆ d2 ˆ t 0 ∂t ( ϕ(x(t,α1,β2), y,α1, y, t) ) d tdα1 d y = − ˆ d2 ( ϕ(α1, y,α1, y, t) −ϕ(β2, y,α1, y, 0) ) dα1 d y = 0. finally, we will show that the pair (c, m) also satisfy the equation ∂t(cv)+∇·(cv⊗v)+c∇p = 0. first note that [ d iv(x, y)(u ⊗ u) ]i = ui d iv(x, y) u+u·∇(x, y) ui = u·∇(x, y) ui , because d iv(x, y) u = 0. let ϕ ∈ ( c∞0 (d4 × (0, t)) )2 , then 〈ϕ1(x, y,α1,α2, t),∂t ( c(t, x, y,α1,α2)v1(x, y,α1,α2, t)〉+ +〈ϕ1(x, y,α1,α2, t),∇(x, y) · (c(t, x, y,α1,α2)v(x, y,α1,α2, t) ⊗ v(x, y,α1,α2, t))1〉 +〈ϕ1(x, y,α1,α2, t), c(x, y,α1,α2, t)∂x p(x, y)〉 = = − ˆ d2 ˆ t 0 u11(x(t,α1,β2), y(t,α1,β2))∂tϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y − ˆ d2 ˆ t 0 [ u11(x(t,α1,β2), y(t,α1,β2)) ]2 ∂xϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y + ˆ d2 ˆ t 0 ∂x p(x(t,α1,β2), y)ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y = − ˆ d2 ˆ t 0 u11(x(t,α1,β2), y(t,α1,β2))∂t [ ϕ 1(x(t,α1,β2), y,α1, y, t) ] d tdα1 d y + ˆ d2 ˆ t 0 [ u11(x(t,α1,β2), y(t,α1,β2)) ]2 ∂xϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y − ˆ d2 ˆ t 0 [ u11(x(t,α1,β2), y(t,α1,β2)) ]2 ∂xϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y + ˆ d2 ˆ t 0 ∂x p(x(t,α1,β2), y)ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y = ˆ d2 ˆ t 0 ∂t [ u11(x(t,α1,β2), y(t,α1,β2)) ] ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y + ˆ d2 ˆ t 0 ∂x p(x(t,α1,β2), y)ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y = ˆ d2 ˆ t 0 [ ∂x u 1 1(x(t,α1,β2), y(t,α1,β2))∂t x(t,α1,β2) +∂y u11(x(t,α1,β2), y(t,α1,β2))∂t y(t,α1,β2) ] ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y + ˆ d2 ˆ t 0 ∂x p(x(t,α1,β2), y)ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y = ˆ d2 ˆ t 0 ∇(x, y) u11(x(t,α1,β2), y(t,α1,β2)) cubo 12, 3 (2010) a family of stationary solutions ... 31 u11(x(t,α1,β2), y(t,α1,β2))ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y + ˆ d2 ˆ t 0 ∂x p(x(t,α1,β2), y)ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y. since that p(x) = − 1 4 cos(2x) we have that ∂x p = 1 2 sin(2x) and then, ∂x p(x(t,α1,β2), y) = ∂x p(x(t,α1,β2), y(x(t,α1,β2)). therefore, we conclude that 〈ϕ1(x, y,α1,α2, t),∂t ( c(t, x, y,α1,α2)v1(x, y,α1,α2, t)〉+ +〈ϕ1(x, y,α1,α2, t),∇(x, y) · (c(t, x, y,α1,α2)v(x, y,α1,α2, t) ⊗ v(x, y,α1,α2, t))1〉+ +〈ϕ1(x, y,α1,α2, t), c(x, y,α1,α2, t)∂x p(x, y)〉 = = ˆ d2 ˆ t 0 [ ∇(x, y) u11(x(t,α1,β2), y(t,α1,β2))u11(x(t,α1,β2), y(t,α1,β2)) +∂x p(x(t,α1,β2), y(t,α1,β2)) ] ϕ 1(x(t,α1,β2), y,α1, y, t)d tdα1 d y. now, note that u11(x, n y) = −cos(x) sin(n y) and u21(x, n y) = sin(x) cos(n y), namely, u11(x, z) = −cos(x) sin(z) and u21(x, z) = sin(x) cos(z), where z = n y. then, ∇u11 · u1 = −sin(x) cos(x) = − 1 2 sin(2x), and, therefore, ∇u11 · u1 +∂x p = − 1 2 sin(2x) + 1 2 sin(2x) = 0. then, ∇(x, y) u11 ( x(t,α1,β2), y(t,α1,β2) ) · u1(x(t,α1,β2), y(t,α1,β2))+ +∂x p(x(t,α1,β2), y(t,α1,β2)) = 0 and we can conclude that the pair of measures (c, m = cv) satisfy ∂t(cv) +∇· (cv ⊗ v) + c∇p = 0. acknowledgement. the author would like to thank lucas c. f. ferreira, helena j. nussenzveig lopes and milton c. lopes filho for their useful comments and weber f. pereira for many fruitful discussions. the author gratefully acknowledge fapesp thematic project #2007/51490-7. 32 juliana c. precioso cubo 12, 3 (2010) references [1] arnold, v.i., sur la géométrie différentielle des groupes de lie de dimension infine et ses applications à l’hidrodynamique, ann. inst. fourier, 16, 319–361, 1966. 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[18] young, l.c., lectures on the calculus of variations and optimal control theory, chelsea, new york, 1980. cubo a mathematical journal vol.11, no¯ 03, (65–77). august 2009 a localized heat source undergoing periodic motion: analysis of blow-up and a numerical solution c.m. kirk department of mathematics, california polytechnic state university, san luis obispo, ca 93407 email: ckirk@calpoly.edu abstract a localized heat source moves with simple periodic motion along a one-dimensional reactive-diffusive medium. blow-up will occur regardless of the amplitude or frequency of motion. numerical results suggest that blow-up is delayed by increasing the amplitude or by increasing the frequency of motion. a brief survey is presented of the literature concerning numerical studies of nonlinear volterra integral equations with weakly singular kernels that exhibit blow-up solutions. resumen una fuente de calor localizada se mueve con un movimiento periódico simple a lo largo de un medio reactivo-difuso unidimensional. “blow-up” ocurrirá considerando la amplitud o frecuencia del movimiento. resultados numéricos sugieren que el “blow-up” es retardado por aumento de amplitud o frecuencia de movimiento. un breve informe es presentado de la literatura al respecto de estudios numéricos de ecuaciones integrales de volterra no lineales con nucleo débilmente singular que exhiben soluciones “blow-up”. 66 colleen m. kirk cubo 11, 3 (2009) key words and phrases: moving heat source, blow-up, integral equations, numerical simulation. math. subj. class.: 45d05, 45m99, 35k60, 65-04, 65r20. introduction a number of studies have addressed blow-up in a reactive-diffusive medium due to a localized heat source. generally this problem is modeled by a parabolic partial differential equation (pde) with a nonlinear source term. the highly localized nature of the heat source can be represented by a dirac delta function. see [30] and [31] for surveys of this literature. the general model often takes the form: x q tt (x, t) − txx(x, t) = δ(x − x0)f [t (x, t)] for 0 < t and x ∈ d (1) t (x, 0) = t0(x) f or x ∈ d with appropriate boundary conditions on d. usually t0 is non-negative and continuous. in most studies, q = 0. [10] and [12] address degenerate problems with q 6= 0. variations of (1) include systems of equations ([20], [26], [23]), higher dimensions ([11], [19]), and problems that include non-local features ([27], [12]). the model can also include motion of sources ([25], [18], [19], [20]) and sources of varying size and shape [19]. the delta function δ(x − x0) reflects the intense localization. some studies have allowed for less intense, perhaps more realistic, localization [19]. one question of interest is whether or not the solution undergoes blow-up in finite time. in this article we will examine a localized heat source that moves with simple periodic motion (x0(t) = a cos(ωt)) along a one-dimensional reactive-diffusive medium. blow-up will occur regardless of the amplitude and frequency of motion. this particular motion is suggested by [18] which addresses various types of motion in one-dimension. this problem is initially modeled as a nonlinear parabolic pde. the analysis is carried out by converting to the corresponding volterra integral equation (vie). we develop a numerical method to model the solution. the results of the numerical study suggest that blow-up is delayed either by increasing the amplitude or by increasing the frequency of motion. intuitively this makes sense since increasing the amplitude allows the heat source to oscillate over a wider spatial domain, allowing the heat more time to dissipate as the source moves into cooler surroundings. increasing the frequency causes the source to move more quickly, generally allowing the heat less opportunity to accumulate. furthermore the numerical results agree with the analytical results that can be obtained for this specific kind of motion. numerical modeling of nonlinear vies with blow-up solutions can be a difficult problem. little research has been done in this area. specifically, rigorous numerical analysis of such schemes is essentially non-existent. a brief survey of the literature regarding numerical studies of nonlinear vies with weakly singular kernels that exhibit blow-up solutions is presented in the last section of this paper. cubo 11, 3 (2009) a localized heat source undergoing periodic ... 67 theory consider the following nonlinear pde: tt (x, t) − txx(x, t) = δ(x − x0)f [t (x, t)] for 0 < t and x ∈ (−∞, ∞) t (x, 0) = t0(x) for x ∈ (−∞, ∞) t (x, t) → 0 as |x| → ∞ this problem models the heating of a reactive-diffusive material by a highly-localized source. the delta function reflects the strong localization of the heat source. t0 is non-negative, continuous, and approaches 0 as |x| → ∞. here we present the conversion from the pde to the corresponding integral equation. first apply the appropriate one-dimensional free space green’s function. t (x, t) = t ∫ 0 ∞ ∫ −∞ g(x, t|ξ, s)δ(ξ − x0)f [t (x0, s)]dξds + ∞ ∫ −∞ g(x, t|ξ, 0)t0(ξ)dξ, −∞ < x < ∞, t ≥ 0 let x = x0 and utilize the delta function property: t (x0, t) = t ∫ 0 g(x0, t|x0, s)f [t (x0, s)]ds + ∞ ∫ −∞ g(x0, t|ξ, 0)t0(ξ)dξ, −∞ < x < ∞, t ≥ 0 define u(t) ≡ t (x0, t) and h(t) ≡ ∞ ∫ −∞ g(x0, t|ξ, 0)t0(ξ)dξ and k(t, s) ≡ g(x0, t|x0, s) = h(t−s) 2 √ π(t−s) exp [ −(x0(t)−x0(s)) 2 4(t−s) ] so that: u(t) = h(t) + t ∫ 0 k(t, s)f (u(s))ds (2) of interest is whether or not (2) has a blow-up solution. a solution u(t) is a blow-up solution if u(t) → ∞ as t → ̂t < ∞. the following theorem is used to prove the existence of a unique, continuous solution to (2) up to some lower bound time t∗. various versions of this theorem appear in [32], [24], [18] and [19]. theorem 1. (existence theorem): equation (2) has a unique, continuous solution for 0 ≤ t < t∗, where t∗ < ∞ can be determined as the smallest root of i(t∗) = λ ≡ max 0≤m<∞ [ m f (m+h) ] and t∗ = ∞ if i(t) < λ for all t ≥ 0. 68 colleen m. kirk cubo 11, 3 (2009) the proof of this theorem involves the creation of a mapping from the space of continuous functions u(t) that satisfy: 0 ≤ u(t) ≤ m < ∞, 0 ≤ t ≤ t1 where m is the smallest root of m f (m+h̄) = 1 f ′(m) .the mapping is defined as the integral operator k where: k[u(t)] = h(t) + t ∫ 0 k(t, s)f [u(s)]ds it can be shown that k is a contraction mapping (with the sup norm) from the space into itself if certain conditions are satisfied. then by the contraction mapping theorem, a unique continuous solution exists. those required conditions lead to the lower bound on the blow-up time. the following theorem provides an upper bound on the blow-up time. theorem 2. (non-existence theorem): let k(t, s) be a non-increasing function in t. whenever there exists a t∗∗ < ∞ such that i(t∗∗) = κ ≡ ∫ ∞ h dz f (z) , it follows that (2) can not have a continuous solution for t ≥ t∗∗. a contradiction argument is used to prove non-existence of the solution. this method exploits the non-increasing nature of the typical kernel. the theorem must be modified if the kernel does not have this property. various versions of this theorem appear in [32], [24], [18] and [19]. the bounds on the blow-up time can then be summarized: if u(t) → ∞ as t → ̂t < ∞, then t∗ < ̂t < t∗∗. now consider a moving source: x0 = x0(t) in one-dimension along an infinite rod. assume x0(t) is continuously differentiable. this is a reasonable assumption since x0(t) is a position function. then: u(t) = h(t) + t ∫ 0 1 2 √ π(t−s) exp ( − (x0(t)−x0(s)) 2 4(t−s) ) f [u(s)]ds specifically we consider the special case of simple periodic motion as suggested by [18]: x0(t) = a cos(ωt), a > 0, ω > 0. let f (z) = ez and h(t) = 0 so that u(t) = t ∫ 0 1 2 √ π(t − s) exp ( − (a cos(ωt) − a cos(ωs)) 2 4(t − s) ) e u(s) ds (3) for this motion, blow-up will always occur. to see this, apply a theorem from [18]: cubo 11, 3 (2009) a localized heat source undergoing periodic ... 69 theorem 3. let x0(t) be bounded and lipschitz continuous, with constants x ∗∗ > 0, v ∗∗ > 0 such that |x0(t) − x0(t′)| ≤ v∗∗|t − t′|, 0 ≤ t < ∞, 0 ≤ t′ < ∞. then a continuous solution of (3) can not exist for t ≥ t∗∗ = πκ2 exp(x∗∗v∗∗). note, of course, that in our example |x0(t)| ≤ a, |x0(t) − x0 (t′) | ≤ aω |t − t′| , 0 ≤ t < ∞, 0 ≤ t′ < ∞ hence theorem 3 guarantees that blow-up will necessarily occur. the upper bound is obtained directly from theorem 3: t∗∗ = πκ2ea 2ω . recall: κ ≡ ∞ ∫ h dz f (z) , which for this example becomes: κ = 1. so then t∗∗ = πea 2ω. now consider the lower bound on the blow-up time. a comparison kernel can be introduced in order to modify theorem 1 for problems such as this one for which i(t) is difficult to obtain. (note that a comparison kernel was also needed to prove theorem 3 since the kernel in this problem is not necessarily non-increasing.) apply the following theorem from [18]: theorem 4. let k(t, s) ≤ k(t, s), 0 ≤ s < t < ∞, where k(t, s) is continuous for 0 ≤ s < t and integrable as s → t. then theorem 1 holds with i(t) replaced by i(t) where i(t) ≡ ∫ t 0 k(t, s)ds, 0 ≤ t < ∞ the appropriate comparison kernel in this case is k(t, s) = 1 2 √ π(t−s) since h = 0 and f (z) = ez, we have λ = 1 e . the lower bound on the blow-up time is obtained from i(t∗) = 1 e . so t∗ = π e2 . together the bounds on the blow-up time are: π e2 < ̂t < πe a2ω numerical method the numerical computation of the solution to problem (3) is based on a standard product quadrature approach. discretize the interval of integration (0, t) into subintervals: (ti, ti+1), i = 1...n. ui = u(ti) is the approximation of u(t) at t = ti. the nonlinear portion of the integrand is represented by the canonical lagrange functions. we have exp(u(s)) ≈ s−ti+1 ti−ti+1 e u(ti) + s−ti ti+1−ti e u(ti+1) for s ∈ (ti, ti+1). the typical integration rule for un+1 can then be written: 70 colleen m. kirk cubo 11, 3 (2009) un+1 = 1 2 √ π n ∑ i=1 ti+1 ∫ ti 1√ tn+1 − s exp { −a2[cos(ωtn+1) − cos(ωs)]2 4(tn+1 − s) } exp(u(s))ds = 1 2 √ π n ∑ i=1 e u(ti) ti − ti+1 ti+1 ∫ ti s − ti+1√ tn+1 − s exp { −a2[cos(ωtn+1) − cos(ωs)]2 4(tn+1 − s) } ds + 1 2 √ π n ∑ i=1 e u(ti+1) ti+1 − ti ti+1 ∫ ti s − ti√ tn+1 − s exp { −a2[cos(ωtn+1) − cos(ωs)]2 4(tn+1 − s) } ds (4) for the approximation of the solution at time tn+1. to address the singular behavior when i = n, apply the mean value theorem to the term cos(ωt) − cos(ωs) to obtain: exp { −a2[cos(ωt)−cos(ωs)]2 4(t−s) } = exp { −a2ω2[sin(ω˜t)(t−s)]2 4(t−s) } = exp { −a2ω2[sin(ω˜t)]2 4 (t − s) } where s < ˜t < t. then choose a small value 0 < ∆mn << (tn, tn+1). consider just the integral from the last term in equation (4) to illustrate this step for i = n : tn+1 ∫ tn s−tn √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds = tn+1−∆mn ∫ tn s−tn √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds + tn+1 ∫ tn+1−∆mn s−tn √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds evaluate analytically the integral for (tn+1 − ∆mn, tn+1) : tn+1 ∫ tn+1−∆mn s−tn √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds = tn+1 ∫ tn+1−∆mn s−tn √ tn+1−s exp { −a2ω2[sin(ω˜t)]2 4 (tn+1 − s) } ds = 1 r5/2 [ √ ∆mnr 3/2 exp(−r∆mn) + r √ π erf( √ r √ ∆mn) { r(tn+1 − tn) − 12 } ] where r ≡ a 2ω2 sin2 ˜t 4 and tn+1 − ∆mn < ˜t < tn+1. then the governing equation becomes: cubo 11, 3 (2009) a localized heat source undergoing periodic ... 71 un+1 = 1 2 √ π n ∑ i=1 [ eu(ti ) ti−ti+1 ti+1 ∫ ti s−ti+1 √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds] + 1 2 √ π n ∑ i=1 [ e u(ti+1) ti+1−ti ti+1 ∫ ti s−ti √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds] = 1 2 √ π n−1 ∑ i=1 [ eu(ti ) ti−ti+1 ti+1 ∫ ti s−ti+1 √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds] + 1 2 √ π eu(tn ) tn−tn+1 tn+1 ∫ tn s−tn+1 √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds + 1 2 √ π n−1 ∑ i=1 [ e u(ti+1) ti+1−ti ti+1 ∫ ti s−ti √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds] + 1 2 √ π e u(tn+1) tn+1−ti tn+1 ∫ tn s−tn √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds un+1 = 1 2 √ π n−1 ∑ i=1 [ eu(ti ) ti−ti+1 ti+1 ∫ ti s−ti+1 √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds] + 1 2 √ π eu(tn ) tn−tn+1 tn+1−∆mn ∫ tn s−tn+1 √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds + 1 2 √ π eu(tn ) tn−tn+1 1 r5/2 [ √ ∆mnr 3/2 exp(−r∆mn) + − r2 √ π erf( √ r √ ∆mn)] + 1 2 √ π n−1 ∑ i=1 [ e u(ti+1) ti+1−ti ti+1 ∫ ti s−ti √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds] + 1 2 √ π eu(tn+1) tn+1−tn tn+1−∆mn ∫ tn s−tn √ tn+1−s exp { −a2[cos(ωtn+1)−cos(ωs)] 2 4(tn+1−s) } ds + 1 2 √ π eu(tn+1) tn+1−tn 1 r5/2 [ √ ∆mnr 3/2 exp(−r∆mn) + r √ π erf( √ r √ ∆mn) { r(tn+1 − tn) − 12 } ] where r ≡ a 2ω2 sin2 ˜t 4 and tn+1 − ∆mn < ˜t < tn+1. newton’s method is then used to solve the implicit nonlinear equations that arise. the results of the numerical calculations follow here. for these calculations, n = 100 and ∆mn = 1 100 ( t n ) = t 10,000 where t is the stopping time. the numerical code is run in matlab. the intervals (ti, ti+1) are chosen to be uniform. first we compare various amplitude values. in plot 1, we keep frequency fixed at ω = 1 and plot results for a = 1, a = 10 and a = 20. note that the blow-up time seems to increase from about ̂t ≈ 1.2 to ̂t ≈ 9.5 to ̂t ≈ 16 as the amplitude increases. intuitively this makes sense since increasing the amplitude allows the heat source to oscillate over a wider spatial domain, allowing the heat more time and space to dissipate as the source moves into cooler surroundings. note the oscillatory behavior for a = 10 and a = 20. this behavior is expected since the oscillating source moves more quickly near the center of its path of motion and moves more slowly near the extremes of its path. hence one would expect the medium (where the source is located) to heat up less readily when the source is near the center of the path and to heat 72 colleen m. kirk cubo 11, 3 (2009) up more readily when the source is near the extremes of the path. figure 1: 0 5 10 15 0 1 2 3 4 5 t u (t ) u(t): a = 1, w = 1 u(t): a = 10, w = 1 u(t): a = 20, w = 1 we do not see this oscillatory behavior when a = 1. this is because blow-up occurs very quickly, before the source changes direction even once. see plot 2 for more detailed pre-blow-up behavior for a = 1. figure 2: 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 t u (t ) u(t): a = 1, w = 1 now vary the frequency ω while keeping the amplitude a = 10 fixed. in plot 3, we show results for ω = 0.1, ω = 0.5 and ω = 1. as frequency increases, the blow-up time seems to increase from about ̂t ≈ 1.1 to ̂t ≈ 6.2 to ̂t ≈ 9.5. this behavior is consistent with intuition. increasing the frequency causes the source to move more quickly, generally allowing the heat less opportunity to accumulate. cubo 11, 3 (2009) a localized heat source undergoing periodic ... 73 figure 3: 0 2 4 6 8 10 0 1 2 3 4 5 t u (t ) u(t): a = 10, w = 0.1 u(t): a = 10, w = 0.5 u(t): a = 10, w = 1.0 if the frequency is small enough, the source may move so slowly that the heat accumulates and blow-up occurs before the source changes direction even once. indeed this seems to be the case with ω = 0.1, where blow-up occurs well before t = π 0.1 and no oscillatory behavior is seen. see plot 4. when ω = 1, the source completes an oscillation before blow-up occurs. figure 4: 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 t u (t ) u(t): a = 10, w = 0.1 furthermore these numerical results agree with the analytical results that can be obtained for this specific kind of motion. for example, consider the case ω = 1, a = 1, with ̂t ≈ 1.2 in plot 2. the numeric blow-up time lies within the analytical bounds: 0.425 < ̂t < 8.539. these bounds, however, are fairly crude. also note that the numerical solution remains bounded: u(t) < m = 1 for t < 0.425, as required by another analytical result we obtained in the theory section of this paper. 74 colleen m. kirk cubo 11, 3 (2009) numerical approaches to nonlinear vies with blow-up solutions we have not yet carried out convergence or error analysis of our numerical scheme. in fact, the numerical modeling of nonlinear volterra integral equations with blow-up solutions can be a difficult problem. little research has been done in this area. specifically, rigorous numerical analysis of such schemes is essentially non-existent. here we present a brief review of the literature regarding numerical studies of nonlinear vies of the second kind with weakly singular kernels that exhibit blow-up solutions. there is a solid body of research on the numerical analysis of nonlinear vies of the second kind with weakly singular kernels that do not exhibit blow-up solutions. see [4], [5], [6], [7], [1], [16]. these papers provide very good summaries of the existing literature. they also describe in detail some of the important techniques used to address these problems. one successful approach involves collocation solutions, with approximations in the space of piecewise polynomials. appropriate quadrature formulas are usually used to approximate the integrals involved. generally large nonlinear systems must then be solved. special issues arise with the introduction of weakly singular kernels. for example, the order of convergence is reduced near the singularity when polynomial splines and uniform meshes are used. these issues can often be successfully addressed using graded meshes which are finer near the region of the singularity; by carrying out a variable transformation; or by using other basis functions which more closely reflect the nature of the solution near the singularity. see [5], [9], [8], [28], [17]. the convergence, superconvergence, and stability properties of these collocation solutions are then studied. in [5], h. brunner includes extensive details of these methods. he also lists many references for these methods, as well as for other methods used to address these problems. research on numerical methods for nonlinear vies of the second kind with weakly singular kernels that do exhibit blow-up in finite time is still at the early stages. the development of computational approaches and rigorous analysis of these approaches is limited. see [2] and [3] and [5]. despite this, researchers have attempted to compute solutions numerically to various interesting applied problems. without rigorous analysis of errors and convergence, however, these authors often rely upon available analytical information to help verify their numerical results. it can be useful to employ knowledge of the asymptotic behavior of the solution near the time of blow-up. if the solution is known to exhibit certain asymptotic behavior near blow-up, then the numerical solution can be tracked until it reflects this behavior, suggesting the onset of blow-up. for example, in [21], d. g. lasseigne and w. e. olmstead analyze the ignition of a combustible solid due to excessive reactant consumption. an integral equation is obtained that models the perturbation of temperature in the reaction zone. before carrying out the numerical analysis, an asymptotic analysis is performed on the governing integral equation. then the behavior of the evolving numerical solution is compared to the expected asymptotic solution form. eventually, the numerical solution matches the known behavior pattern. this provides extra assurance that thermal runaway is indeed occurring. l. r. ritter, w. e. olmstead and v. a. volpert follow a cubo 11, 3 (2009) a localized heat source undergoing periodic ... 75 similar approach in [29] to model numerically the initiation of a polymerization wave. their goal is to understand how various parameters in a frontal polymerization process determine whether the onset of a wave front will be initiated or inhibited. in [10] and [12], c. y. chan and h. y. tian employ computational methods to estimate blow-up time. in [10] they examine a degenerate parabolic problem with a nonlinear source in a one-dimensional material of finite length. they obtain analytical expressions for the bounds on the blow-up time as functions of the length of the domain. then they use an iterative approach to approximate the solution over the discretized spatial domain at the estimated blow-up time. this estimated blow-up time is then refined, shifted upward or downward, depending on whether the growth of the iterated solution lies within or exceeds certain tolerances. they provide numerical results indicating the blow-up time is a decreasing function of the length of the domain, as expected. these same authors apply a similar technique in [12] to a problem with a nonlinear source of local and nonlocal features. again they use an iterative process to refine the analytical bounds. note that in none of these studies is a rigorous numerical analysis carried out. the authors instead acquire confidence in their numerical solutions partly by making judicious use of known analytical results. clearly some understanding of the analytical solution is important in choosing the best numerical approach and in acquiring confidence in the numerical results. h. brunner proposes in [5] another possible approach to these problems using collocation methods. he suggests the possibility of obtaining two different collocation solutions vh and uh generated from different sets of collocation points such that the corresponding iterated numerical solutions satisfy: vith ≤ u(t) ≤ uith for a chosen mesh ih. furthermore, if the vie of interest stems from an originating pde, one might instead examine the original pde for blow-up behavior. the research into these corresponding pdes is more advanced than that of analogous vies. see [2] and [3] and [5] for surveys of these studies. received: july 28, 2008. revised: 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[34] stuart, a.m. and floater, m,s., on the computation of blow-up, europ. j. appl. math., 1(1990), 47–71. 10-cmkirklhsupm cubo a mathematical journal vol.11, no¯ 02, (127–138). may 2009 an inter-group conflict and its relation to oligopoly theory tamar kugler department of management and organizations, eller college of management, university of arizona, tucson, arizona, 85721-0108, email: tkugler@eller.arizona.edu and ferenc szidarovszky systems & industrial engineering department, the university of arizona, tucson, arizona, 85721-0020, usa email: szidar@sie.arizona.edu abstract a game theoretical model of inter-group conflicts is revisited. in this model members of each group contribute to secure a public good which becomes then available to all members regardless if they contributed or not, and the groups compete for an exogenous prize simultaneously. we first show that the best response of each group member is mathematically equivalent to that in oligopolies with isoelastic price and linear cost functions. then a complete equilibrium analysis is given showing that, except in a very special case, there is a unique equilibrium. and finally, a dynamic extension of the game is introduced and analysed, where the players are able to increase their contributions at any time during a given time period. 128 tamar kugler and ferenc szidarovszky cubo 11, 2 (2009) resumen un modelo de juego teórico de conflictos de intergrupos es revisado. en este modelo los miembros de cada grupo contribuyen para asegurar un bien público, el cual queda disponible para todos los miembros sin importar si éstos contribuyen o no, y los grupos compiten por un premio simultaneamente. mostramos que la mejor respuesta de todo grupo es matemáticamente equivalente a oligopolios con funciones de precio isoelásticas y costo lineal. un completo análisis de equilibrio es dado, demostrando que, salvo en casos muy especiales, existe un único equilibrio. finalmente, es presentada y analizada una extensión dinámica del juego, donde los jugadores son capaces de aumentar sus contribuiciones en cualquer momento durante un determinado período. key words and phrases: oligopoly, n-person games, intergroup conflict. math. subj. class.: 91a07, 91a40. 1 introduction in this paper we will investigate an inter-group conflict: a game involving groups of players in which conflict arises in both the group and the individual levels simultaneously. in the classical approach the groups were considered as the players, the payoff of each group was computed as the sum of the payoffs of its members. however, if group members have selfish interest that does not always coincide with group interest, then this “group equilibrium” approach is irrealistic. therefore it is more appropriate to model how the satisfaction of group objectives affects individual members and to include these consequences into the payoffs of the members. in this case a multiplayer game can be defined in which the players are the members of all groups and therefore we have to consider conflict only among the players. in this study we will follow this approach. the game we will examine in this paper has been introduced and studied by [4] and further analysed in [1]. assume that the members of n groups (n ≧ 2) contribute to a public good and simultaneously the groups compete to win an exogenous prize s. this external prize can be thought of as a mechanism to increase contribution by creating a between group competition. let n(i) denote the number of members of group i. each member k of each group i receives an initial endowment of yki. the decision of each member is to decide on the contributed amount xki. then this member will keep yki − xki for herself. the overall contribution of group i is xi = ∑n(i) k=1 xki, which serves two purposes. first, it generates a public good for the group. let gi denote the maximal public good that can be generated if all members contribute their entire endowment. otherwise the generated public good is proportional to the contributed amount: gixi/yi, where yi = ∑n(i) k=1 yki is the total endowment of group i. second, the group contribution probabilistically determines the group’s success in winning the exogenously determined prize. it is assumed that only one group can win the prize and higher group contribution implies higher winning probability. therefore the payoff of member cubo 11, 2 (2009) an inter-group conflict and its relation to oligopoly theory 129 k of group i has the following form: ϕki = s · xi x · xki xi + gi xi yi + (yki − xki), (1) where x = ∑n i=1 xi is the total contribution of all members of all groups. the first term gives the expected share of the external prize assuming that in the case of winning it, the members’ shares are proportional to their contributions. the second term is the public good generated by the group which is assumed to become available equally to all group members, and the third term is that part of the initial endowment which is not contributed. in the first term xi cancels, and it is not defined for x = 0. in this case when no contributions are made, no prize is awarded and no public good is generated. since no game is played, the payoffs of all members of all groups are zero, or alternatively, we can assume that they can keep their endowments. in this way a ∑n i=1 n(i) – player game is defined, in which the members are the players, the strategy set and payoff function of member k of group i is [0, yki] and ϕki, respectively. in this paper we will examine the existence and uniqueness of the nash equilibrium of this game. the paper is developed as follows. first we will demonstrate a strong analogy between this game and oligopolies with isoelastic price functions. they have mathematically identical best response functions with the only difference that the parameters corresponding to marginal costs are not restricted to positive values. then the existence and the uniqueness of the equilibrium will be proved. a simple dynamic extension of the game will be introduced next. the last section will conclude the paper. 2 relation to oligopoly games the game presented above has been introduced to model conflicts between noncooperative groups. despite the different interpretation, its structure has a close resemblence to a special oligopoly game. in this section we will demonstrate the relationship between these seemingly different lines of research. consider a market of n firms producing identical product. let xk denote the output of firm k with capacity limit lk. assume that the cost of firm k depends on its own output level, ck(xk), but the unit price depends on the total production of all firms. assuming hyperbolic price, a∑n l=1 xl , and linear cost functions, αk + βkxk, the profit of firm k can be given as ϕk = axk ∑n l=1 xl − (αk + βkxk). (2) an n -person noncooperative game is defined above, where the firms are the players, the strategy set of firm k is the closed interval [0, lk], and its payoff function is ϕk. this family of games, known as oligopoly games, is one of the most frequently discussed topics in mathematical economics [2], [3]. 130 tamar kugler and ferenc szidarovszky cubo 11, 2 (2009) we will next show that games (1) and (2) are very similar, their equilibrium problems are equivalent. consider first game (1) and member k of group i, and assume that the contribution of the other participants are known to her. if we introduce notation qki = x − xki, which is the total contributions of all others, then clearly, ϕki = sxki xki + qki + (yki − xki) + gixki yi + gi ∑ l 6=k xli yi . (3) notice that the last term and yki do not depend on xki, so the best response of this player depends on only qki, and it is the maximizer of function sxki xki + qki − ( 1 − gi yi ) xki (4) on interval [0, yki]. consider next game (2), and for firm k introduce the notation qk = ∑ l 6=k xl. then maximizing ϕk is equivalent to the maximization of function axk xk + qk − βkxk. (5) obviously functions (4) and (5) are equivalent with a and βk being replaced by s and 1 − gi yi , respectively. consequently the best responses of the players are the same and therefore the equilibria of the two games are also equivalent to each other. notice that in oligopoly theory the marginal cost βk has to be always positive, while 1 − gi yi can be also zero or negative. therefore the existence results known from oligopoly theory cannot be directly applied. 3 best responses and equilibria our public good contribution game is based on relations (1) and (4) which is mathematically equivalent to a generalized oligopoly game, where marginal costs are not restricted to negative values. as we will see later, the dynamic extension of the public good contribution game is fundamentally different than that of oligopolies. for the sake of simple notation we will use function (5), which will be denoted by fk. by simple differentiation, ∂fk ∂xk = aqk (xk + qk) 2 − βk (6) and ∂ 2 fk ∂x 2 k = − 2aqk (xk + qk) 3 < 0, (7) cubo 11, 2 (2009) an inter-group conflict and its relation to oligopoly theory 131 so fk is strictly concave in xk, so the best response of firm k is unique. assume first that qk = 0. then fk = { 0 if xk = 0 a − βkxk if xk > 0, so we have three cases. if βk > 0, then firm k’s interest is to produce as small as possible positive amount, so no best response exists. this is also the case in game (1), when the players can keep their endowments if no contributions are made by i. if βk = 0, then the best response is the entire interval (0, lk], and if βk < 0, then the best response is the maximum feasible amount lk. assume next that qk > 0. the concavity of fk implies that the best response of player k is given as rk(qk) =      0 if ∂fk ∂xk |xk=0≦ 0 lk if ∂fk ∂xk |xk=lk ≧ 0 z ∗ k otherwise (8) where z∗k is the unique solution of equation ∂fk ∂zk = aqk (zk + qk) 2 − βk = 0 (9) in interval (0, lk). notice that in the case when βk ≦ 0, the second case of (8) occurs, so rk(qk) = lk. notice that in the case of qk = 0 we had a similar case, lk was always a best response but in the case of βk = 0 there were infinitely may other best responses in interval (0, lk). otherwise with notation z ∗ k = √ aqk βk − qk (10) we have rk(qk) =      0 if z∗k ≦ 0 lk if z ∗ k ≧ lk z ∗ k otherwise. (11) this function is illustrated in figure 1. in order to reduce the equilibrium problem to a singlevariable equation we have to rewrite the best response definitions in terms of the total production level q = ∑n k=1 xk of all players. we assume next again that qk > 0. the first case of (11) occurs when qk = q, that is, if √ aq βk − q ≦ 0. this is the case as q = 0 or q ≧ a βk . the second case of (11) occurs when qk = q − lk, or √ a(q − lk) βk − (q − lk) ≧ lk, 132 tamar kugler and ferenc szidarovszky cubo 11, 2 (2009) qk rk(qk) lk a 4βk a βk ∑ l 6=k ll figure 1: graph of best response rk(qk) which can be rewritten as q(1 − q βk a ) ≧ lk. (12) the third case occurs when xk = √ a(q − xk) βk − (q − xk) or xk = q ( 1 − q βk a ) . (13) in summary, the best response function of player k is equivalent with the following: r̄k(q) =      0 if q = 0 or q ≧ a βk lk if gk(q) ≧ lk gk(q) otherwise (14) where gk(q) = q ( 1 − q βk a ) (15) for all k. in analysing the case of qk = 0 we have seen that in that case zero cannot occur as the best response, so at any equilibrium q 6= 0. function (14) is illustrated in figure 2. in examining the existence and uniqueness of the equilibrium we have to consider several possibilities. cubo 11, 2 (2009) an inter-group conflict and its relation to oligopoly theory 133 gk(q) q r̄k(q) lk a βk ∑n l=1 ll figure 2: graph of function r̄k(q) notice first that q = 0, when all xk = 0, cannot be equilibrium. consider next the case when only one xk > 0 and all other xl = 0 (l 6= k). then qk = 0, and ql > 0 for l 6= k, so βk ≦ 0 necessarily. if βk = 0, then any xk ∈ (0, lk] is possible and if βk < 0, then xk = lk. for all other players βl > 0 and xk = q ≧ a βl . assume next that xk > 0 for at least two players. then qk > 0 for all players. consider next equation h(q) = n ∑ k=1 r̄k(q) − q = 0. (16) in the left hand side r̄k(q) is either the truncated parabola (14) for βk > 0 or the horizontal line lk for βk ≦ 0 (it is also the largest best response if qk = 0). we will now prove that equation (16) has a unique solution. observe first that for a truncated parabola r̄′k(0) = 1. since n ≧ 2, there are either at least two r̄k functions being truncated parabolas, or for at least one k, r̄k(q) ≡ lk. in all cases the right hand side limit of h(q) at zero is always positive. at q = ∑n l=1 ll, the value of h(q) is nonpositive, since r̄k(q) cannot be greater than lk. since h(q) is continuous, there is at least one solution q > 0. assume next that there are two solutions q(1) and q(2) (q(1) < q(2)). we will first prove that there is a q ∗ ∈ [q(1), q(2)] such that h(q∗) ≦ 0 and h′(q∗+) ≧ 0 where h′(q∗+) denotes the right hand side derivative. assume that q(1) does not satisfy these conditions. then h′(q(1)+) < 0 and there is a small ε > 0 such that h′(q(1) + ε) < 0 implying that point (q(1) + ε, h(q(1) + ε)) is under the horizontal axis. however h(q(2)) = 0, therefore the graph of h(q) between q(1) + ε and q(2) cannot be always nonincreasing. since function h can have only finitely many breakpoints, there 134 tamar kugler and ferenc szidarovszky cubo 11, 2 (2009) has to be a q∗ such that h(q∗) ≦ 0 and h′(q∗+) ≧ 0. in a small neighborhood on the right hand side of q∗, h(q) = (c + lq − aq2) − q (17) where the first term is the sum of l parabolic and some constant segments. if h(q∗) ≦ 0, then c + (l − 1)q∗ − aq∗2 ≦ 0 or a ≧ c q∗2 + l − 1 q∗ . (18) similarly, h′(q∗+) ≧ 0 implies that l − 2aq∗ − 1 ≧ 0, so a ≦ l − 1 2q∗ . (19) relations (18) and (19) are contradictory except in the following special cases. if l = 0, then there is no player with truncated parabola as her best response, all have lk as their best responses with zero derivatives, so h′(q∗) = −1 and hence h′(q∗) ≧ 0 is impossible. if l = 1 and c = 0, then for one player, r̄k(q ∗ ) is truncated parabola and for all other players r̄l(q ∗ ) = 0. since all truncated parabolas are under the 45 degree line, there is no larger solution of equation (16) then q ∗. we have therefore contradiction in all cases. the unique solution of equation (16) gives the unique equilibrium if for at least two players, xk > 0. if at the solution only one xk is positive, then there is the possibility of infinitely many equilibria. this is the case, when for one player βk = 0, all other βl > 0, and max l 6=k a βl < lk. (20) then x̄l = 0 for l 6= k and x̄k ∈ [ maxl 6=k a βl ; lk ] are all equilibria. otherwise for all k, x̄k = r̄k(q̄), where q̄ is the solution of equation (16). this is the case in game (1) if for a group gi = yi and this group has only one member. 4 dynamic extensions consider discrete time scales, t = 0, 1, 2, . . . , and assume that at each time period the players are able to increase their contribution levels, but they cannot decrease the already pledged amounts. at t = 0 each player’s initial contribution is zero. at each time period each player checks if cubo 11, 2 (2009) an inter-group conflict and its relation to oligopoly theory 135 x1 x2 l1 l2 a β1 a β2 figure 3: the set of steady states of the dynamical process additional contribution increases her payoff or not by computing her marginal profit: ∂ϕk ∂xk = aqk (xk + qk) 2 − βk. (21) if βk < 0, then this derivative is always positive, so player k will always increase her contribution until it reaches the maximum lk level. if βk = 0, then we have a simple situation, since in the case of qk > 0 the player’s interest is to increase contribution and if qk = 0, then firm k’s interest is to keep the current level, so until qk remains zero, firm k will keep her zero initial contribution. if βk > 0, then the player stops contributing if aqk (xk + qk) 2 − βk ≦ 0 or xk = lk. (22) since the contributions of each player form a monotonic and bounded sequence, the process always converges regardless of the amounts of the contribution increases. hence the steady states of the dynamical process can be characterized as follows: if βk < 0, then xk = lk; if βk = 0, then xk { = 0 if qk = 0 = lk if qk 6= 0; 136 tamar kugler and ferenc szidarovszky cubo 11, 2 (2009) x1 x2 l1 l2 figure 4: the case of a single steady state and if βk > 0, then xk { ≧ √ aqk βk − qk if √ aqk βk − qk < lk = lk otherwise. (23) consider the two-person case, when q1 = x2 and q2 = x1 and assume the general case shown in figure 1 with both β1 and β2 being positive. figure 3 shows the set of all steady states. we also assume sufficiently large l1 and l2 values. if the l1 and l2 values are relatively small, we might have the point (l1, l2) as the only steady state, as it is shown in figure 4. any dynamic process starts at the origin and proceeds along a sequence of horizontal and vertical segments. the reached steady state depends on the amounts of the increments. 5 an example by assuming that βk > 0 for all k, we will first determine the interior equilibrium (when 0 < xk < lk for all players). then for all k, r̄k(q) = q(1 − q βk a ). by adding this relation for all k, a single-variable equation can be obtained for q: q = q n ∑ k=1 ( 1 − q βk a ) . (24) cubo 11, 2 (2009) an inter-group conflict and its relation to oligopoly theory 137 since q = 0 cannot be an equilibrium, we have n − q a n ∑ k=1 βk = 1, so q̄ = (n − 1)a ∑n k=1 βk (25) and therefore x̄k = (n − 1)a ∑n k=1 βk ( 1 − βk(n − 1) ∑n l=1 βl ) . (26) in the special case of n = 2, q̄ = a β1 + β2 (27) and x̄k = a β1 + β2 ( 1 − βk β1 + β2 ) = aβl (β1 + β2) 2 with l 6= k. (28) this is always positive, and is below lk if a lk < (β1 + β2) 2 βl . (29) notice that figure 3 shows such a case, when the unique equilibrium is the intercept of the two curves. 6 conclusions we could show that the multilevel inter-group conflict where groups can compete for an external price while the members are contributing for a common public good is mathematically equivalent to generalized oligopolies with hyperbolic price and linear cost functions. to the best of our knowledge, this similarity has not been noted before in the literature of mathematical economics. however, the well-known results of oligopoly theory cannot be applied without additional considerations, since the parameter which replaces marginal costs can have zero and negative values, which have no economic sense in oligopoly models. except for a very special case the nash-equilibrium is always unique, and can be obtained by solving a single-variable algebraic equation. the dynamic extension of the game might have infinitely many steady states, and since the contribution sequences are monotonic and bounded, they always converge. this dynamic model is 138 tamar kugler and ferenc szidarovszky cubo 11, 2 (2009) fundamentally different than dynamic oligopolies ([3]). a simple formula has been derived for the interior equilibrium of the game under the additional condition that βk > 0 or gi < yi. received: april 21, 2008. revised: may 12, 2008. references [1] kugler, t. and szidarovszky, f., modeling inter-group competitions: a game theoretical analysis of between and within group conflicts, submitted for publication, 2008. [2] okuguchi, k., expectations and stability in oligopoly models, springer-verlag, berlin/heidelberg/new york, 1976. [3] okuguchi, k. and szidarovszky, f., the theory of oligopoly with multi-product firms, 2nd edn, springer-verlag, berlin/heidelberg/new york, 1999. [4] rapoport, a. and amaldoss, w., social dilemmas embedded in between-group competitions: effects of contest and distribution rules, in m. foddy, m. smithson, s. schneider and m. hogg, eds ’resolving social dilemmas’, psychology press, philadelphia. n09-inter-group_conflicts_v1 yminstanton.dvi cubo a mathematical journal vol.12, no¯ 03, (99–120). october 2010 self-dual and anti-self-dual solutions of discrete yang-mills equations on a double complex volodymyr sushch koszalin university of technology, sniadeckich 2, 75-453 koszalin, poland email: volodymyr.sushch@tu.koszalin.pl abstract we study a discrete model of the su (2) yang-mills equations on a combinatorial analog of r 4. self-dual and anti-self-dual solutions of discrete yang-mills equations are constructed. to obtain these solutions we use both techniques of a double complex and the quaternionic approach. interesting analogies between instanton, anti-instanton solutions of discrete and continual self-dual, anti-self-dual equations are also discussed. resumen estudiamos el modelo discreto de las ecuaciones de yang-mills su (2) sobre un análogo combinatório de r4. soluciones auto-dual y anti-auto-dual para las ecuaciones discretas de yang-mills son construidas. para obtener estas soluciones usamos las técnicas de doble complejo y abordage cuaternionico. interesantes analogías entre soluciones instantones y anti-instantones de ecuaciones discretas y continuas auto-dual y anti-auto-dual son discutidas. key words and phrases: yang-mills equations, self-dual and anti-self-dual equations, instantons and anti-instantons, difference equations. math. subj. class.: 81t13, 39a12. 100 volodymyr sushch cubo 12, 3 (2010) 1 introduction it is well known that the self-dual and anti-self-dual connections are the absolute minima of the lagrangian for a 4-dimensional non-abelian gauge theory. the first self-dual solution the one instanton to the su(2) yang-mills equations on r4 was obtained by belavin et al [3]. later other more general multi-instanton solutions were described in [5, 11]. since then numerous extensions have been made. classical references are the books by atiyah [1], freed and uhlenbeck [8]. in this paper we study a discrete analog of the su(2) yang-mills equations on a combinatorial analog of r4. the ideas presented here are strongly influenced by book of dezin [6]. we develop discrete models of some objects in differential geometry, including the hodge star operator, the differential and the covariant exterior differential operator, in such a way that they preserve the geometric structure of their continual analogs. we continue the investigations which were originated in [7, 19, 20, 21]. the purpose of this paper is to construct the self-dual and anti-self-dual solutions of discrete su(2) yang-mills equations which imitate the corresponding solutions of continual theory. the geometrical discretisation techniques used here extend those introduced in [6] and [19]. a combinatorial model of r4 based on the use of the double complex construction is taken from [21]. there are many other approaches to the discretisation of yang-mills theories. numerous papers have been written on this subject. see, for example, [2, 4, 9, 10, 12, 13, 15, 18, 16] and the references therein. most of them are based on the lattice discretisation scheme. however, in the case of the lattice formulation there are difficulties in keeping geometrical properties of an origin gauge theory. an alternative geometrical discretisation scheme of a field theory can be found in [17]. the paper is organized as follows. in section 2 we review some basic facts of the su(2) yang-mills theory on r4. we begin by recalling the connection between the lie group su(2) and the space of quaternions. finally, we write down the basic instanton and anti-instanton solutions in quaternionic form. the notations here are compiled from [1] and [14]. section 3 contains a brief summary of definitions and properties due to the double complex construction. we repeat here the relevant material from [21]. this article is also the main reference for this section. in particular, we introduce discrete matrix-valued forms (analog of differential forms) and define analogs of the main continual operations on them. in section 4 using the quaternionic approach we present the discrete yang-mills equations. we write out components of the discrete curvature 2-form in quaternionic form. the discrete self-dual and anti-self-dual equations are described. we try to be as close to concubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 101 tinual su(2) yang-mills theory as possible. hence we discuss conditions when the discrete curvature will be su(2)-valued. finally, section 5 is devoted to self-dual and anti-self-dual solutions of the discrete yangmills equations. we construct these solutions as discrete quaternionic 1-forms and discuss some analogies with continual instanton and anti-instanton solutions. 2 quaternions and su (2)-connection in this section we briefly recall some well known settings of the smooth yang-mills theory in euclidean 4-dimensional space (see, for example, [14]). we begin with a brief review of some preliminaries about quaternions. the quaternions are formed from real numbers by adjoining three symbols i, j, k and an arbitrary quaternion x can be written as x = x1 + x2i + x3j + x4k, (2.1) where x1, x2, x3, x4 ∈ r. the symbols i, j, k satisfy the following identities i 2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. (2.2) it is clear that the space of quaternions is isomorphic to r4. by analogy with the complex numbers x1 is called the real part of x and x2i + x3j + x4k is called the imaginary part. in further we will write im x = x2i + x3j + x4k. the conjugate quaternion of x is defined by x̄ = x1 − x2i − x3j − x4k. then the norm |x| of a quaternion can be introduced as follows |x| 2 = xx̄ = x 2 1 + x 2 2 + x 2 3 + x 2 4. (2.3) if x 6= 0, then it has a unique inverse x−1 given by x −1 = x̄/|x| 2 . (2.4) the algebra of quaternions can be represented as a sub-algebra of the 2 × 2 complex matrices m(2,c). we identify the quaternion (2.1) with a matrix f (x) ∈ m(2,c) by setting f (x) = ( x1 + x2 i x3 + x4 i −x3 + x4 i x1 − x2 i ) . (2.5) 102 volodymyr sushch cubo 12, 3 (2010) here i is the imaginary unit. it is well known that the unit quaternions, i.e., they have norm |x| = 1, form a group and this group is isomorphic to su(2). the following 2 × 2 complex matrices i = ( i 0 0 −i ) , j = ( 0 1 −1 0 ) , k = ( 0 i i 0 ) (2.6) realize a representation of the lie algebra su(2) of the group su(2). note that multiplying by −i these tree matrices we obtain the standard pauli matrices. matrices (2.6) correspond to the units i, j, k given by (2.2). thus the lie algebra su(2) can be viewed as the pure imaginary quaternions with basis i, j, k. let now a be an su(2)-connection. this means that a is an su(2)-valued 1-form and we can write a = ∑ µ aµ(x)dx µ , (2.7) where aµ(x) ∈ su(2) and x = (x1, ..., x4 ) is a point of r 4. the connection a is also called a gauge potential. define a gauge transformation by a function g(x) taking value in su(2). then the gauge potential a must transform like a → g −1 a g + g −1 d g. (2.8) let us define the curvature 2-form f by f = d a + a ∧ a, (2.9) where ∧ denotes the exterior multiplication. consider also the covariant exterior differential operator d a given by d a ω = dω+ a ∧ω+ (−1) p+1 ω∧ a, (2.10) where ω is a su(2)-valued p-form. the yang-mills action s can be expressed in terms of the 2-forms f and ∗f as s = −tr ˆ r4 f ∧∗f, (2.11) where ∗ is the hodge star operator. in r4 the operator ∗2 is either an involution or antiinvolution, i.e., ∗2 = ±1. the yang-mills lagrangian l = −tr(f ∧ ∗f) is invariant under the gauge transformation (2.8). by the physical requirement it is clear that the action s should be finite. hence the curvature f should be square integrable. this means that f → 0 as |x| → ∞. cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 103 consequently, we must describe the boundary condition at infinity for the connection a. by virtue of gauge freedom (2.8) we have a ∼ g −1 d g as |x| → ∞, (2.12) where ∼ implies asymptotic behaviour. here and subsequently we do not specify the rate of decay. written in terms of the covariant exterior differential operator d a the euler-lagrange equations for the extrema of (2.11) have the form d a f = 0, d a ∗ f = 0. (2.13) these equations are the yang-mills equations. the first equation of (2.13) is known also as the bianchi identity. in 4-dimensional yang-mills theories the following equations f = ∗f, f = −∗ f (2.14) are called self-dual and anti-self-dual respectively. these equations are first-order non-linear equations for the potential a which imply the second-order yang-mills equations (2.13). solutions of (2.14) – the self-dual and anti-self-dual connections – are called also instantons and anti-instantons [8]. it is known that the self-dual and anti-self-dual connections are the absolute minima of the action s. the connection 1-form a can be defined also as taking values in the space of pure imaginary quaternions. to express a in quaternion form we consider the quaternion differential dx = dx1 + dx2i + dx3j + dx4k and the conjugate quaternion of dx dx̄ = dx1 − dx2i − dx3j − dx4 k. let f (x) be a function of the quaternion variable x with quaternion values. then we can write a as a = im( f (x)dx), (2.15) where f (x) = f1(x) + f2(x)i + f3(x)j + f4(x)k. using the rules of multiplication (2.2) we have a1(x) = f2(x)i + f3(x)j + f4(x)k, a2(x) = f1(x)i + f4(x)j − f3(x)k, 104 volodymyr sushch cubo 12, 3 (2010) a3(x) = −f4(x)i + f1(x)j + f2(x)k, a4(x) = f3(x)i − f2(x)j + f1(x)k. using (2.15) we can rewrite (2.9) as follows f = im(d f (x) ∧ dx + f (x)dx ∧ f (x)dx). (2.16) note that calculation of the imaginary part of f (x)dx and computing its curvature commute. let us take the following expression for f (x): f (x) = x̄ 1 +|x|2 . (2.17) then the connection 1-form a is defined by a = im { x̄dx 1 +|x|2 } . (2.18) the explicit components aµ can be written as a1(x) = −x2i − x3j − x4k 1 +|x|2 , a2(x) = x1i − x4j + x3k 1 +|x|2 , a3(x) = x4i + x1j − x2k 1 +|x|2 , a4(x) = −x3i + x2j + x1k 1 +|x|2 . (2.19) putting (2.17) in (2.16) we get the pure imaginary expression f = dx̄ ∧ dx (1 +|x|2)2 . (2.20) it is easy to show that the 2-form dx̄ ∧ dx is anti-self-dual. hence f is anti-self-dual too and the connection (2.18) describes an anti-instanton . see for details [1]. similarly, if we take a = im { xdx̄ 1 +|x|2 } , (2.21) then we obtain the self-dual 2-form f = dx ∧ dx (1 +|x|2)2 . (2.22) thus the curvature is self-dual and (2.21) describes an instanton . 3 double complex we will need the double complex construction described in [21]. in with section for the convenience of the reader we repeat the relevant material from [21] without proofs, thus making our presentation self-contained. cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 105 let the tensor product c(4) = c ⊗ c ⊗ c ⊗ c of an 1-dimensional complex c be a combinatorial model of euclidean space r4 (see for details also [6]). the 1-dimensional complex c is defined in the following way. let c0 denotes the real linear space of 0-dimensional chains generated by basis elements x j (points), j ∈ z. it is convenient to introduce the shift operators τ,σ in the set of indices by τ j = j + 1, σ j = j − 1. (3.1) we denote the open interval (x j , xτ j ) by e j . we’ll regards the set {e j } as a set of basis elements of the real linear space c1 of 1-dimensional chains. then the 1-dimensional complex (combinatorial real line) is the direct sum of the introduced spaces c = c0 ⊕c1. the boundary operator ∂ on the basis elements of c is given by ∂x j = 0, ∂e j = xτ j − x j . (3.2) the definition is extended to arbitrary chains by linearity. multiplying the basis elements x j , e j in various ways we obtain basis elements of c(4). let s ( p) k , where k = (k1, k2 , k3, k4) and ki ∈ z, be an arbitrary basis element of c(4). then a p-dimensional chain is given by c p = ∑ k ∑ p c k ( p) s ( p) k , c k ( p) ∈ r. (3.3) we suppose that the superscript ( p) contains the whole requisite information about the quantity and places of 1-dimensional elements e j in s ( p) k . for example, the 1-dimensional basis elements ei k of c(4) can be written as e 1 k = e k1 ⊗ xk2 ⊗ xk3 ⊗ xk4 , e 2 k = xk1 ⊗ e k2 ⊗ xk3 ⊗ xk4 , e 3 k = xk1 ⊗ xk2 ⊗ e k3 ⊗ xk4 , e 4 k = xk1 ⊗ xk2 ⊗ xk3 ⊗ e k4 (3.4) and for the 2-dimensional basis elements ε i j k we have ε 12 k = e k1 ⊗ e k2 ⊗ xk3 ⊗ e k4 , ε 23 k = xk1 ⊗ e k2 ⊗ e k3 ⊗ xk4 , ε 13 k = e k1 ⊗ xk2 ⊗ e k3 ⊗ xk4 , ε 24 k = xk1 ⊗ e k2 ⊗ xk3 ⊗ e k4 , ε 14 k = e k1 ⊗ xk2 ⊗ xk3 ⊗ e k4 , ε 34 k = xk1 ⊗ xk2 ⊗ e k3 ⊗ e k4 . (3.5) using (3.2) we define the boundary operator ∂ on chains of c(4) in the following way: if c p , cq are chains of the indicated dimension, belonging to the complexes being multiplied, then ∂(c p ⊗ cq ) = ∂c p ⊗ cq + (−1) p c p ⊗∂cq . (3.6) for example, for the basis element ε24 k we have ∂ε 24 k = ∂(xk1 ⊗ e k2 ) ⊗ xk3 ⊗ e k4 − xk1 ⊗ e k2 ⊗∂(xk3 ⊗ e k4 ) 106 volodymyr sushch cubo 12, 3 (2010) = ∂xk1 ⊗ e k2 ⊗ xk3 ⊗ e k4 + xk1 ⊗∂e k2 ⊗ xk3 ⊗ e k4 − xk1 ⊗ e k2 ⊗∂xk3 ⊗ e k4 − xk1 ⊗ e k2 ⊗ xk3 ⊗∂e k4 = xk1 ⊗ xτk2 ⊗ xk3 ⊗ e k4 − xk1 ⊗ xk2 ⊗ xk3 ⊗ e k4 − xk1 ⊗ xk2 ⊗ xk3 ⊗ xτk4 + xk1 ⊗ xk2 ⊗ xk3 ⊗ xk4 . for convenience we also introduce the shift operators τi and σi which act in the set of indices k = (k1, k2, k3 , k4), ki ∈ z, as τi k = (k1, ...τki , ...k4 ), σi k = (k1, ...σki , ...k4 ), (3.7) where τ and σ are given by (3.1). let us introduce the construction of a double complex. together with the complex c(4) we consider its double, namely the complex c̃(4) of exactly the same structure. define the one-to-one correspondence ∗ : c(4) → c̃(4), ∗ : c̃(4) → c(4) (3.8) in the following way. let s ( p) k be an arbitrary p-dimensional basis element of c(4), i.e., the product s ( p) k = sk1 ⊗ sk2 ⊗ sk3 ⊗ sk4 contains exactly p of 1-dimensional elements e ki and 4 − p of 0-dimensional elements xki , p = 0, 1, 2, 3, 4, ki ∈ z. then ∗ : s ( p) k → ±s̃ (4−p) k , ∗ : s̃ (4−p) k → ±s ( p) k , (3.9) where s̃ (4−p) k = ∗sk1 ⊗∗sk2 ⊗∗sk3 ⊗∗sk4 and ∗ski = ẽ ki if ski = xki and ∗ski = x̃ki if ski = e ki . in the first of mapping (3.9) we take "+" if the permutation (( p), (4 − p)) of (1, 2, 3, 4) is even and "−" if the permutation (( p), (4 − p)) is odd. recall that in symbol ( p) the number of basis element is contained. for example, for the 2-dimensional basis element ε13 k = e k1 ⊗ xk2 ⊗ e k3 ⊗ xk4 we have ∗ε 13 k = −ε̃24 k since the permutation (1, 3, 2, 4) is odd. the mapping ∗ : s̃ (4−p) k → ±s ( p) k is defined by analogy. proposition 3.1. let cr ∈ c(4) be an r-dimensional chain (3.3). then we have ∗∗ cr = (−1) r(4−r) cr . (3.10) proof. see [21]. now we consider a dual object of the complex c(4). let k (4) be a cochain complex with gl(2,c)-valued coefficients, where gl(2,c) is the lie algebra of the group gl(2,c). recall that gl(2,c) consists of all complex 2 × 2 matrices m(2,c) with bracket operation [·,·]. cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 107 we suppose that the complex k (4), which is a conjugate of c(4), has a similar structure: k (4) = k ⊗ k ⊗ k ⊗ k , where k is a conjugate of the 1-dimensional complex c. basis elements of k can be written as x j , e j . then an arbitrary basis element of k (4) is given by sk ( p) = sk1 ⊗ sk2 ⊗ sk3 ⊗ sk4 , where sk j is either xk j or ek j . for example, we denote the 1-, 2dimensional basis elements of k (4) by ek i , εk i j respectively, cf. (3.4), (3.5). for a p-dimensional cochain ϕ ∈ k (4) we have ϕ = ∑ k ∑ p ϕ ( p) k s k ( p) , (3.11) where ϕ ( p) k ∈ gl(2,c). we will call cochains forms, emphasizing their relationship with the corresponding continual objects, differential forms. we define the pairing operation < · , · > for arbitrary basis elements εk ∈ c(4), s k ∈ k (4) by the rule < εk, as k >= { 0, εk 6= sk a, εk = sk , a ∈ gl(2,c). (3.12) here for simplicity the superscript ( p) is omitted. the operation (3.12) is linearly extended to cochains. the operation ∂ (3.6) induces the dual operation d c on k (4) in the following way: < ∂εk, as k >=< εk, ad c s k > . (3.13) for example, if ϕ = ∑ k ϕk x k, where xk = xk1 ⊗ xk2 ⊗ xk3 ⊗ xk4 , is a 0-form, then d c ϕ = ∑ k 4 ∑ i=1 (∆iϕk)e k i , (3.14) where ∆iϕk = ϕτi k −ϕk and e k i is the 1-dimensional basis elements of k (4). the coboundary operator d c is an analog of the exterior differentiation operator. now we describe a cochain product on the forms of k (4). see [6] for details. we denote this product by ∪. in terms of the homology theory this is the so-called whitney product. first we introduce the ∪-product on the chains of the 1-dimensional complex k . for the basis elements of k the ∪-product is defined as follows x j ∪ x j = x j , e j ∪ x τ j = e j , x j ∪ e j = e j , j ∈ z, supposing the product to be zero in all other case. to arbitrary forms the ∪-product be extended linearly. let us introduce an r-dimensional complex k (r), r = 1, 2, 3, in an obvious notation. let sk ( p) be an arbitrary p-dimensional basis element of k (r). it is convenient to write the basis element of k (r + 1) in the form sk ( p) ⊗ s j , where sk ( p) is a basis element of k (r) 108 volodymyr sushch cubo 12, 3 (2010) and s j is either e j or x j , j ∈ z. then, supposing that the ∪-product in k (r) has been defined, we introduce it for basis elements of k (r + 1) by the rule (s k ( p) ⊗ s j ) ∪ (s k (q) ⊗ s µ ) = q( j, q)(s k ( p) ∪ s k (q) ) ⊗ (s j ∪ s µ ), (3.15) where the signum function q( j, q) is equal to −1 if the dimension of both elements s j , sk (q) is odd and to +1 otherwise. the extension of the ∪-product to arbitrary forms of k (r + 1) is linear. note that the coefficients of forms multiply as matrices. proposition 3.2. let ϕ and ψ be arbitrary forms of k (4). then d c (ϕ∪ψ) = d c ϕ∪ψ+ (−1) p ϕ∪ d c ψ, (3.16) where p is the dimension of a form ϕ. the proof of proposition 3.2 is totally analogous to one in [6, p. 147] for the case of discrete forms with real coefficients. the complex of the cochains k̃ (4) over the double complex c̃(4) with the operator d c defined in it by (3.13) has the same structure as k (4). the operation (3.8) induces the respective mapping ∗ : k (4) → k̃ (4), ∗ : k̃ (4) → k (4) by the rule: < c̃, ∗ϕ >=< ∗c̃, ϕ >, < c, ∗ψ̃ >=< ∗c, ψ̃ >, (3.17) where c ∈ c(4), c̃ ∈ c̃(4), ϕ ∈ k (4), ψ̃ ∈ k̃ (4). hence for the basic elements of k (4) or k̃ (4) we have relations (3.9). it is obviously that proposition 3.1 is true for any r-dimensional cochain cr ∈ k (4). so we have ∗∗ϕ = (−1) r(4−r) ϕ for any discrete r-form ϕ on k (4) and note that the same relation holds for the hodge star operator. thus this operator is a combinatorial analog of the hodge star operator. let us introduce the following operation ι̃ : k (4) → k̃ (4), ι̃ : k̃ (4) → k (4) by setting ι̃s k ( p) = s̃ k ( p) , ι̃s̃ k ( p) = s k ( p) , (3.18) where sk ( p) and s̃k ( p) are basis elements of k (4) and k̃ (4). hence for a p-form ϕ ∈ k (4) we have ι̃ϕ = ϕ̃. recall that the coefficients of ϕ̃ ∈ k̃ (4) and ϕ ∈ k (4) are the same. cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 109 proposition 3.3. the following hold ι̃ 2 = i d, ι̃∗ = ∗ι̃, ι̃d c = d c ι̃, (3.19) ι̃(ϕ∪ψ) = ι̃ϕ∪ ι̃ψ, where ϕ, ψ ∈ k (4). proposition 3.4. let h be a discrete 0-form. then for an arbitrary p-form ϕ ∈ k (4) we have ι̃∗ (h ∪ϕ) = h ∪ ι̃ ∗ϕ. (3.20) proof. see [21]. note that the definition of inner product in the double complex and a discrete analog of the yang-mills actions (2.11) can be found in [21]. 4 quaternions and discrete forms let us consider a discrete 0-form with coefficients belonging to m(2,c). we put f = ∑ k f k x k , (4.1) where xk = xk1⊗xk2⊗xk3 ⊗xk4 is the 0-dimensional basis element of k (4), k = (k1, k2, k3, k4 ), ki ∈ z. suppose that the matrices f k ∈ m(2,c) look like (2.5), i. e. f k = ( f 1 k + f 2 k i f 3 k + f 4 k i −f 3 k + f 4 k i f 1 k − f 2 k i ) , (4.2) where f s k ∈ r, s = 1, 2, 3, 4. then f k in quaternionic form can be expressed as f k = f 1 k + f 2 k i + f 3 k j + f 4 k k. (4.3) hence the form (4.1) can be considered as a discrete form with quaternionic coefficients. we will call it simply the quaternionic form when no confusion can arise. in a proper way we define the quaternionic 0-form f̄ with coefficients f̄ k regarded as the conjugate quaternions of f k. let f −1 be the quaternionic form, where f −1 k is given by (2.4). then we have f ∪ f −1 = ∑ k f k f −1 k x k = ∑ k x k . (4.4) proposition 4.1. let f be a discrete 0-form and f 6= 0. then we have d c f ∪ f −1 = −f ∪ d c f −1 . (4.5) 110 volodymyr sushch cubo 12, 3 (2010) proof. by definition (3.14) and according to (4.4), we have d c ( f ∪ f −1) = 0. using proposition 3.2 we immediately obtain (4.5). let us denote by e the following quaternionic 1-form e = ∑ k e k = ∑ k (e k 1 + e k 2i + e k 3j + e k 4 k), (4.6) where ek i is the 1-dimensional basis elements of k (4). let a ∈ k (4) be a discrete 1-form. we define the discrete su(2)-connection a to be a = ∑ k 4 ∑ i=1 a i k e k i , (4.7) where ai k ∈ su(2) and k = (k1, k2, k3, k4 ), ki ∈ z. using (4.3) and (4.6) we write (4.7) in quaternionic form as a = im( f ∪ e) = im ( ∑ k f k e k ) . (4.8) then the ai k are given by a 1 k = f 2 k i + f 3 k j + f 4 k k, a 2 k = f 1 k i + f 4 k j − f 3 k k, a 3 k = −f 4 k i + f 1 k j + f 2 k k, a 4 k = f 3 k i − f 2 k j + f 1 k k. (4.9) define the quaternionic 0-form x by x = ∑ k κx k , κ = k1 + k2i + k3j + k4k, (4.10) where ki ∈ z. it is easy to check that d c x = e. (4.11) therefore we can rewrite (4.8) as a = im( f ∪ d c x). (4.12) let g be a quaternionic 0-form (4.1) with the components of unit norm, i.e., |gk| = 1 for any k. it means that the corresponding discrete form is su(2)-valued. we now define a gauge transformation for the discrete potential a which is analogous to (2.8). this is a → g −1 ∪ a ∪ g + g −1 ∪ d c g, (4.13) where a is given by (4.8) or (4.12). note that the gauge transformed discrete form a is su(2)-valued too. it is not so obviously as in the continual case but follows immediately from the definition of ∪-multiplication and formula (3.16). more generally, if we assume that the gauge transformation g is an arbitrary quaternionic 0-form, then we take the imaginary part cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 111 of g−1 ∪ a ∪ g+ g−1 ∪ d c g in (4.13). for a deeper discussion of gauge invariant discrete models of the yang-mills theory we refer the reader to [19, 21]. an arbitrary discrete 2-form f ∈ k (4) can be written as follows f = ∑ k ∑ i< j f i j k ε k i j , (4.14) where f i j k ∈ gl(2,c), εk i j is the 2-dimensional basis element of k (4) and 1 ≤ i, j ≤ 4, k = (k1, k2, k3, k4 ), ki ∈ z. let f is given by f = d c a + a ∪ a. (4.15) combining (4.7) and (4.15) and using (3.12), (3.13) and (3.15), we obtain f i j k = ∆i a j k −∆j a i k + a i k a j τi k − a j k a i τj k , (4.16) where ∆i a j k = a j τi k − a j k and τi k is given by (3.7). let us define a discrete analog of the exterior covariant differentiation operator (2.10) as follows d c a ω = d c ω+ a ∪ω+ (−1) p+1 ω∪ a, (4.17) where ω is an arbitrary p-form of k (4) looking like (3.11). then a discrete analog of equations (2.13) can be written as d c a f = 0, d c a ∗ ι̃f = 0, (4.18) where ι̃ is given by (3.18). it is easy to check that the combinatorial bianchi identity: d c f + a ∪ f − f ∪ a = 0 (4.19) holds for the discrete curvature form (4.15) (cf. (2.13)). remark 4.2. in the continual case the curvature form f (2.9) takes values in the algebra su(2) for any su(2)-valued connection form a. unfortunately, this is not true in the discrete case because, generally speaking, the components ai k a j τi k − a j k ai τj k of the form a∪ a (see (4.16)) do not belong to su(2). to define an su(2)-valued discrete analog of the curvature 2-form we use the quaternionic form of a (4.8) and put in (4.15). then the discrete curvature form f is given by f = im{d c f ∪ e + ( f ∪ e) ∪ ( f ∪ e)}. (4.20) it should be noted that in the discrete case calculation of the imaginary part of f ∪ e and computing its curvature do not commute. 112 volodymyr sushch cubo 12, 3 (2010) proposition 4.3. if a = im(x−1 ∪ d c x), where x is given by (4.10), then f = 0. proof. using (4.5) and putting f = x−1 in (4.20) we get f = im(d c (x −1 ∪ d c x) + (x −1 ∪ d c x) ∪ (x −1 ∪ d c x) = im(d c x −1 ∪ d c x − d c x −1 ∪ x ∪ x −1 ∪ d c x). according to (4.4) the form x ∪ x−1 has unit components. hence d c x −1 ∪ x ∪ x −1 ∪ d c x = d c x −1 ∪ d c x. we now write down the components of (4.14) using quaternions. putting (4.9) in (4.16) we find that f 12 k = (∆1 f 1 k −∆2 f 2 k − f 3 k f 3 τ1 k − f 4 k f 4 τ1 k − f 3 k f 3 τ2 k − f 4 k f 4 τ2 k )i + (∆1 f 4 k −∆2 f 3 k + f 2 k f 3 τ1 k + f 4 k f 1 τ1 k + f 1 k f 4 τ2 k + f 3 k f 2 τ2 k )j + (−∆1 f 3 k −∆2 f 4 k + f 2 k f 4 τ1 k − f 3 k f 1 τ1 k − f 1 k f 3 τ2 k + f 4 k f 2 τ2 k )k − f 2 k f 1 τ1 k − f 3 k f 4 τ1 k + f 4 k f 3 τ1 k + f 1 k f 2 τ2 k + f 4 k f 3 τ2 k − f 3 k f 4 τ2 k , f 13 k = (−∆1 f 4 k −∆3 f 2 k + f 3 k f 2 τ1 k − f 4 k f 1 τ1 k − f 1 k f 4 τ3 k + f 2 k f 3 τ3 k )i + (∆1 f 1 k −∆3 f 3 k − f 2 k f 2 τ1 k − f 4 k f 4 τ1 k − f 4 k f 4 τ3 k − f 2 k f 2 τ3 k )j + (∆1 f 2 k −∆3 f 4 k + f 2 k f 1 τ1 k + f 3 k f 4 τ1 k + f 4 k f 3 τ3 k + f 1 k f 2 τ3 k )k + f 2 k f 4 τ1 k − f 3 k f 1 τ1 k − f 4 k f 2 τ1 k − f 4 k f 2 τ3 k + f 1 k f 3 τ3 k + f 2 k f 4 τ3 k , f 14 k = (∆1 f 3 k −∆4 f 2 k + f 3 k f 1 τ1 k + f 4 k f 2 τ1 k + f 2 k f 4 τ4 k + f 1 k f 3 τ4 k )i + (−∆1 f 2 k −∆4 f 3 k − f 2 k f 1 τ1 k + f 4 k f 3 τ1 k + f 3 k f 4 τ4 k − f 1 k f 2 τ4 k )j + (∆1 f 1 k −∆4 f 4 k − f 2 k f 2 τ1 k − f 3 k f 3 τ1 k − f 3 k f 3 τ4 k − f 2 k f 2 τ4 k )k − f 2 k f 3 τ1 k + f 3 k f 2 τ1 k − f 4 k f 1 τ1 k + f 3 k f 2 τ4 k − f 2 k f 3 τ4 k + f 1 k f 4 τ4 k , f 23 k = (−∆2 f 4 k −∆3 f 1 k + f 4 k f 2 τ2 k + f 3 k f 1 τ2 k + f 1 k f 3 τ3 k + f 2 k f 4 τ3 k )i + (∆2 f 1 k −∆3 f 4 k − f 1 k f 2 τ2 k + f 3 k f 4 τ2 k + f 4 k f 3 τ3 k − f 2 k f 1 τ3 k )j + (∆2 f 2 k +∆3 f 3 k + f 1 k f 1 τ2 k + f 4 k f 4 τ2 k + f 4 k f 4 τ3 k + f 1 k f 1 τ3 k )k + f 1 k f 4 τ2 k − f 4 k f 1 τ2 k + f 3 k f 2 τ2 k − f 4 k f 1 τ3 k + f 1 k f 4 τ3 k − f 2 k f 3 τ3 k , f 24 k = (∆2 f 3 k −∆4 f 1 k + f 4 k f 1 τ2 k − f 3 k f 2 τ2 k − f 2 k f 3 τ4 k + f 1 k f 4 τ4 k )i cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 113 + (−∆2 f 2 k −∆4 f 4 k − f 1 k f 1 τ2 k − f 3 k f 3 τ2 k − f 3 k f 3 τ4 k − f 1 k f 1 τ4 k )j + (∆2 f 1 k +∆4 f 3 k − f 1 k f 2 τ2 k − f 4 k f 3 τ2 k − f 3 k f 4 τ4 k − f 2 k f 1 τ4 k )k − f 1 k f 3 τ2 k + f 4 k f 2 τ2 k + f 3 k f 1 τ2 k + f 3 k f 1 τ4 k − f 2 k f 4 τ4 k − f 1 k f 3 τ4 k , f 34 k = (∆3 f 3 k +∆4 f 4 k + f 1 k f 1 τ3 k + f 2 k f 2 τ3 k + f 2 k f 2 τ4 k + f 1 k f 1 τ4 k )i + (−∆3 f 2 k −∆4 f 1 k + f 4 k f 1 τ3 k + f 2 k f 3 τ3 k + f 3 k f 2 τ4 k + f 1 k f 4 τ4 k )j + (∆3 f 1 k −∆4 f 2 k + f 4 k f 2 τ3 k − f 1 k f 3 τ3 k − f 3 k f 1 τ4 k + f 2 k f 4 τ4 k )k + f 4 k f 3 τ3 k + f 1 k f 2 τ3 k − f 2 k f 1 τ3 k − f 3 k f 4 τ4 k − f 2 k f 1 τ4 k + f 1 k f 2 τ4 k . to obtain the components of (4.20) we must take the imaginary part of these equations. proposition 4.4. the discrete curvature 2-form f (4.15) is su(2)-valued if and only if −f 2 k f 1 τ1 k − f 3 k f 4 τ1 k + f 4 k f 3 τ1 k + f 1 k f 2 τ2 k + f 4 k f 3 τ2 k − f 3 k f 4 τ2 k = 0, f 2 k f 4 τ1 k − f 3 k f 1 τ1 k − f 4 k f 2 τ1 k − f 4 k f 2 τ3 k + f 1 k f 3 τ3 k + f 2 k f 4 τ3 k = 0, −f 2 k f 3 τ1 k + f 3 k f 2 τ1 k − f 4 k f 1 τ1 k + f 3 k f 2 τ4 k − f 2 k f 3 τ4 k + f 1 k f 4 τ4 k = 0, f 1 k f 4 τ2 k − f 4 k f 1 τ2 k + f 3 k f 2 τ2 k − f 4 k f 1 τ3 k + f 1 k f 4 τ3 k − f 2 k f 3 τ3 k = 0, −f 1 k f 3 τ2 k + f 4 k f 2 τ2 k + f 3 k f 1 τ2 k + f 3 k f 1 τ4 k − f 2 k f 4 τ4 k − f 1 k f 3 τ4 k = 0, f 4 k f 3 τ3 k + f 1 k f 2 τ3 k − f 2 k f 1 τ3 k − f 3 k f 4 τ4 k − f 2 k f 1 τ4 k + f 1 k f 2 τ4 k = 0. proof. from the above it follows immediately. proposition 4.5. let e is given by (4.6). then the 2-form e ∪ ē is self-dual, i.e., e ∪ ē = ∗ι̃(e ∪ ē), (4.21) and ē ∪ e is anti-self-dual, i.e., ē ∪ e = −∗ ι̃(ē ∪ e). (4.22) proof. denote e i = ∑ k e k i , εi j = ∑ k ε k i j . recall that ek i and εk i j are the 1-dimensional and 2-dimensional basic elements of k (4) (see also (3.4) and (3.5)). from this by (3.15) we obtain e i ∪ e j = εi j and e j ∪ e i = −εi j for all i < j. then we have e ∪ ē = (e1 + e2i + e3j + e4k) ∪ (e1 − e2i − e3j − e4k) = −2{(e1 ∪ e2 + e3 ∪ e4)i + (e1 ∪ e3 − e2 ∪ e4)j + (e1 ∪ e4 + e2 ∪ e3)k} = −2{(ε12 +ε34)i + (ε13 −ε24)j + (ε14 +ε23)k}. 114 volodymyr sushch cubo 12, 3 (2010) using (3.17) and (3.19) we get ∗ι̃(e ∪ ē) = −2ι̃{(ε̃34 + ε̃12)i + (−ε̃24 + ε̃13)j + (ε̃23 + ε̃14)k} = e ∪ ē. in the same way we obtain (4.22). corollary 4.6. for any quaternionic 0-form f the form f ∪ e ∪ ē is self-dual and f ∪ ē ∪ e is anti-self-dual. proof. this follows immediately from (3.20). discrete self-dual and anti-self-dual equations (discrete analogs of equations (2.13)) are defined by f = ι̃∗ f, f = −ι̃∗ f, (4.23) where f is the discrete curvature form (4.4). using (4.5), by the definitions of ι̃ and ∗, the first equation (self-dual) of (4.23) can be rewritten as follows f 12 k = f 34 k , f 13 k = −f 24 k , f 14 k = f 23 k . (4.24) by analogue with the continual case solutions of (4.23) (or (4.24)) are called instantons and anti-instantons respectively. 5 discrete instanton and anti-instanton in further analogy with the continual case consider the discrete su(2)-connection a. let a be the quaternionic 1-form (4.8), where the components of f are given by f k = κ̄ 1 +|κ|2 , (5.1) where κ = k1 + k2i + k3j + k4k, ki ∈ z. putting the last in (4.9) we obtain a 1 k = −k2i − k3j − k4k 1 +|κ|2 , a 2 k = k1i − k4j + k3k 1 +|κ|2 , a 3 k = k4i + k1j − k2k 1 +|κ|2 , a 4 k = −k3i + k2j + k1k 1 +|κ|2 . (5.2) it is convenient to denote mi = 1 (1 +|κ|2)(1 +|τiκ| 2) , i = 1, 2, 3, 4. (5.3) recall that the shift operator τi is given by (3.7). substituting (5.2) in (4.16) and using (5.3) we find that f 12 k = {m1(1 + k 2 2 − k 2 1 − k1) + m2(1 + k 2 1 − k 2 2 − k2)}i cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 115 + {m1(k4 k1 + k2 k3) − m2(k3 k2 + k4 k1)}j + {m1(k2 k4 − k1 k3) + m2(k1 k3 − k2 k4)}k + m1(k1 k2 + k2) − m2(k1 k2 + k1), f 13 k = {m1(k2 k3 − k1 k4) + m3(k1 k4 − k2 k3)}i + {m1(1 + k 2 3 − k 2 1 − k1) + m3(1 + k 2 1 − k 2 3 − k3)}j + {m1(k1 k2 + k3 k4) − m3(k3 k4 + k1 k2)}k + m1(k1 k3 + k3) − m3(k1 k3 + k1), f 14 k = {m1(k1 k3 + k2 k4) − m4(k2 k4 + k1 k3)}i + {m1(k3 k4 − k1 k2) + m4(k1 k2 − k3 k4)}j + {m1(1 + k 2 4 − k 2 1 − k1) + m4(1 + k 2 1 − k 2 4 − k4)}k + m1(k1 k4 + k4) − m4(k1 k4 + k1), f 23 k = {−m2(k2 k4 + k1 k3) + m3(k1 k3 + k2 k4)}i + {m2(k3 k4 − k1 k2) + m3(k1 k2 − k3 k4)}j − {m2(1 + k 2 3 − k 2 2 − k2) + m3(1 + k 2 2 − k 2 3 − k3)}k + m2(k2 k3 + k3) − m3(k2 k3 + k2), f 24 k = {m2(k2 k3 − k4 k1) + m4(k1 k4 − k2 k3)}i + {m2(1 + k 2 4 − k 2 2 − k2) + m4(1 + k 2 2 − k 2 4 − k4)}j − {m2(k1 k2 + k3 k4) − m4(k3 k4 + k1 k2)}k + m2(k2 k4 + k4) − m4(k2 k4 + k2), f 34 k = −{m3(1 + k 2 4 − k 2 3 − k3) + m4(1 + k 2 3 − k 2 4 − k4)}i + {m3(−k2 k3 − k1 k4) + m4(k1 k4 + k2 k3)}j + {m3(k2 k4 − k1 k3) + m4(k1 k3 − k2 k4)}k + m3(k3 k4 + k4) − m4(k3 k4 + k3). proposition 5.1. the 2-form f with components f i j k above is su(2)-valued if and only if k1 = k2 = k3 = k4. (5.4) proof. from proposition 4.4 f is su(2)-valued if and only if mi (ki k j + k j ) − m j (ki k j + ki ) = 0 for any ki ∈ z, i, j = 1, 2, 3, 4 and i < j. it follows immediately (5.4). 116 volodymyr sushch cubo 12, 3 (2010) thus, the su(2)-valued discrete curvature 2-form f can be written in the quaternionic form as follows f = ∑ k, ki=µ mµ(2 − 2µ){(ε k 12 −ε k 34)i + (ε k 13 +ε k 24)j + (ε k 14 −ε k 23)k}. (5.5) from (5.2) here we have mµ = 1 2(1+4µ2)(1+µ+2µ2) . since ki = µ, in (5.5) we can write ε µ i j instead of εk i j . if we consider the 0-form ω = ∑ µ mµ(1 −µ)x µ , µ ∈ z (5.6) and use the following relation (see the proof of proposition 4.5) ē ∪ e = 2{(ε12 −ε34)i + (ε13 +ε24)j + (ε14 −ε23)k}, then f can be written as f = ω∪ ē ∪ e. (5.7) in view of corollary 4.6 f is anti-self-dual, i.e., f = −ι̃∗ f. thus under condition (5.4) a with components (5.1) describes an anti-instanton. in the same manner we can see that the following quaternionic 1-form a = im( f ∪ ē), (5.8) where f has the components f k = κ 1 +|κ|2 , (5.9) leads to an instanton solution of (4.24). indeed, substituting (5.8) and (5.9) in (4.16) we now obtain f 12 k = {−m1(1 + k 2 2 − k 2 1 − k1) − m2(1 + k 2 1 − k 2 2 − k2)}i + {m1(k4 k1 − k2 k3) + m2(k3 k2 − k4 k1)}j + {m1(−k2 k4 − k1 k3) + m2(k1 k3 + k2 k4)}k + m1(k1 k2 + k2) − m2(k1 k2 + k1), f 13 k = {m1(−k2 k3 − k1 k4) + m3(k1 k4 + k2 k3)}i − {m1(1 + k 2 3 − k 2 1 − k1) + m3(1 + k 2 1 − k 2 3 − k3)}j + {m1(k1 k2 − k3 k4) + m3(k3 k4 − k1 k2)}k cubo 12, 3 (2010) self-dual and anti-self-dual solutions ... 117 + m1(k1 k3 + k3) − m3(k1 k3 + k1), f 14 k = {m1(k1 k3 − k2 k4) + m4(k2 k4 − k1 k3)}i + {m1(−k3 k4 − k1 k2) + m4(k1 k2 + k3 k4)}j − {m1(1 + k 2 4 − k 2 1 − k1) + m4(1 + k 2 1 − k 2 4 − k4)}k + m1(k1 k4 + k4) − m4(k1 k4 + k1), f 23 k = {m2(−k2 k4 + k1 k3) + m3(−k1 k3 + k2 k4)}i + {m2(k3 k4 + k1 k2) − m3(k1 k2 + k3 k4)}j − {m2(1 + k 2 3 − k 2 2 − k2) + m3(1 + k 2 2 − k 2 3 − k3)}k + m2(k2 k3 + k3) − m3(k2 k3 + k2), f 24 k = {m2(k2 k3 + k4 k1) − m4(k1 k4 + k2 k3)}i + {m2(1 + k 2 4 − k 2 2 − k2) + m4(1 + k 2 2 − k 2 4 − k4)}j + {m2(k1 k2 − k3 k4) + m4(k3 k4 − k1 k2)}k + m2(k2 k4 + k4) − m4(k2 k4 + k2), f 34 k = −{m3(1 + k 2 4 − k 2 3 − k3) + m4(1 + k 2 3 − k 2 4 − k4)}i + {m3(−k2 k3 + k1 k4) + m4(−k1 k4 + k2 k3)}j + {m3(k2 k4 + k1 k3) − m4(k1 k3 + k2 k4)}k + m3(k3 k4 + k4) − m4(k3 k4 + k3). again, under condition (5.4) we can write f as f = ∑ µ mµ(2µ− 2){(ε µ 12 +ε µ 34 )i + (ε µ 13 −ε µ 24 )j + (ε µ 14 +ε µ 23 )k}, where µ ∈ z. therefore f = ω∪ e ∪ ē, (5.10) where ω is given by (5.6). thus the discrete curvature form (5.10) is self-dual and we can say that (5.8) describes an instanton. now to complete the analogy with the continual case we describe more precisely how the anti-instanton given by (5.1) behaves as |κ| → ∞. it is clear that f k is asymptotically κ̄ |κ|2 = κ−1. then a ∼ im(x −1 ∪ d c x) as |κ| → ∞. (5.11) here x is given by (4.10). by virtue of proposition 4.3 the discrete curvature f = 0 at infinity. 118 volodymyr sushch cubo 12, 3 (2010) proposition 5.2. the anti-instanton (5.1) has the same form at ∞ as it has near 0. proof. introduce the quaternionic 0-form y = ∑ k yk x k , where yk = 1 κ and remind κ = k1 + k2i + k3j + k4k. clearly, y = x −1. we first compute x ∪ f ∪ e ∪ x−1, where f is given by (5.1). to do this, take (4.10), (4.11) and use the ∪-product definition. we have x ∪ f ∪ e = ( ∑ k κx k ) ∪ ( ∑ k κ̄ 1 +|κ|2 x k ) ∪ e = ( ∑ k |κ|2 1 +|κ|2 x k ) ∪ e = e − ( ∑ k 1 1 +|κ|2 x k ) ∪ e = d c x − ( ∑ k 1 1 +|κ|2 x k ) ∪ d c x. from this by (4.5) we get x ∪ f ∪ e ∪ x −1 = −x ∪ d c x −1 + ( ∑ k κ 1 +|κ|2 x k ) ∪ d c x −1 . 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[21] sushch, v., a gauge-invariant discrete analog of the yang-mills equations on a double complex, cubo a math. journal, 8, (2006), no. 3, 61–78. cubo a mathematical journal vol.11, no¯ 05, (57–70). december 2009 dispersive estimates for the schrödinger equation with potentials of critical regularity fernando cardoso, claudio cuevas universidade federal de pernambuco, departamento de matemática, cep. 50540-740 recife-pe, brazil. emails: fernando@dmat.ufpe.br, cch@dmat.ufpe.br and georgi vodev département de mathématiques, umr 6629 du cnrs, université de nantes, 2 rue de la houssinière, bp 92208, fr-44322 nantes cedex 03, france email : vodev@math.univ-nantes.fr abstract we prove l 1 → l ∞ dispersive estimates with a logarithmic loss of derivatives for the schrödinger group e it(−∆+v ) for a class of real-valued potentials v ∈ c (n−3)/2(rn), v (x) = o(〈x〉−δ), where n = 4, 5, δ > 3 if n = 4 and δ > 5 if n = 5. resumen probamos l 1 → l ∞ estimativas dispersivas con una perdida logaritmica de derivadas para el grupo de schrödinger eit(−∆+v ) para una clase de potenciales a valores reales v ∈ c(n−3)/2(rn), v (x) = o(〈x〉−δ), donde n = 4, 5, δ > 3 si n = 4 y δ > 5 si n = 5. key words and phrases: schrodinger equation, dispersive estimates. math. subj. class.: 35l15, 35b40, 47f05. 58 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) 1 introduction and statement of results the present paper is a continuation of our previous one [2] where l1 → l∞ dispersive estimates without loss of derivatives for the schrödinger group eit(−∆+v ) have been proved for potentials v ∈ ck(rn), v (x) = o(〈x〉−δ ), where n = 4, 5, k > (n − 3)/2, δ > 3 if n = 4 and δ > 5 if n = 5. to be more precise, denote by g0 and g the self-adjoint realizations of the operators −∆ and −∆ + v on l2(rn), n ≥ 4, where v ∈ l∞(rn) is a real-valued potential satisfying |v (x)| ≤ c〈x〉−δ, ∀x ∈ rn, (1.1) with constants c > 0, δ > (n + 2)/2. in fact, we are interested in finding the biggest possible class of real-valued potentials for which the perturbed schrödinger group satisfies the following analogue of the well-known dispersive estimate for the free one: ∥∥eitgpac ∥∥ l1→l∞ ≤ c|t|−n/2, t 6= 0, (1.2) where pac denotes the spectral projection onto the absolutely continuous spectrum of g. there have been many works studying this problem. in general, the proof of (1.2) goes in studying separately and in a different maner three regions of frequencies (1) low ones belonging to an interval [0,ε], 0 < ε ≪ 1, (assuming additionally that zero is neither an eigenvalue nor a resonance), (2) intermediate ones in [ε,ε−1], and (3) high frequencies in [ε−1, +∞). it became clear that in dimensions one, two and three no regularity of the potential is needed to prove (1.2). in higher dimensions the same conclusion remains true as far as the frequencies from the first two regions are concerned, but it is no longer true at high frequencies. in fact, from purely mathematical point of view the problem of proving (1.2) turns out to be interesting and difficult at high frequencies, only. that is why, in the present paper we will be only interested in the following high frequency analogue of (1.2): ∥∥eitgχa(g) ∥∥ l1→l∞ ≤ c|t|−n/2, t 6= 0, (1.3) where χa ∈ c ∞ ((−∞, +∞)), χa(λ) = 0 for λ ≤ a, χa(λ) = 1 for λ ≥ a + 1, a ≫ 1. note that when n = 1 the estimate (1.3) is proved by goldberg and shlag [4] for potentials v ∈ l1, while in dimension n = 2 it is proved by moulin [7] for potentials satisfying sup y∈r2 ∫ r2 |v (x)| |x − y|1/2 dx < +∞. when n = 3 rodnianski and shlag [10] proved (1.3) for small potentials belonging to a subclass of kato class with kato norm satisfying sup y∈r3 ∫ r3 |v (x)| |x − y| dx < 4π, while for large potentials goldberg [3] proved (1.3) for v ∈ l3/2−ǫ ∩ l3/2+ǫ, 0 < ǫ ≪ 1. the situation, however, changes drastically when n ≥ 4. indeed, in this case goldberg and visan [5] cubo 11, 5 (2009) dispersive estimates for the schrödinger equation ... 59 showed the existence of potentials v ∈ ck0 (r n ), ∀k < (n − 3)/2, for which (1.3) fails to hold. on the other hand, journé, soffer and sogge [6] proved (1.3) for potentials satisfying (1.1) with δ > n as well as the following regularity condition v̂ ∈ l1. (1.4) note that (1.3) has been recently proved by moulin and vodev [8] for potentials satisfying (1.1) with δ > n − 1 and (1.4). without any regularity conditions on the potential vodev [12] proved dispersive estimates with a loss of (n − 3)/2 derivatives. more precisely, it was shown in [12] that under the condition (1.1) only, we have the estimates ∥∥eitgχa(g)f ∥∥ l∞ ≤ c|t|−n/2 ∥∥∥〈g〉(n−3)/4 f ∥∥∥ l1 , (1.5) ∥∥eitgχa(g)f ∥∥ l∞ ≤ c|t|−n/2 ∥∥∥〈x〉n/2+ǫ f ∥∥∥ l2 , (1.6) for every 0 < ǫ ≪ 1. so, the natural question which arises when n ≥ 4 is that one of finding the smallest possible regularity of the potential in order to have (1.3). in other words, is it possible to replace the condition (1.4) by another one requiring less regularity on the potential? in view of the counterexample of [5] mentioned above, when n ≥ 4 it is quite natural to make the following conjecture 1. the dispersive estimate (1.3) holds true for all potentials v ∈ c (n−3)/2 0 (r n ). to our best knowledge, this is still an open problem. the following weaker statement, however, is more likely to be valid. conjecture 2. the dispersive estimate (1.3) holds true for all potentials v ∈ ck0 (r n ), where k > (n − 3)/2. indeed, when n = 4 or n = 5 conjecture 2 follows from the recent results of [2]. however, it is still open when n ≥ 6. in fact, in [2] more general potentials are treated not necessarily compactly supported. to describe this in more detials, introduce the spaces ckδ (r n ) and vkδ (r n ) of all functions v ∈ ck(rn) satisfying ‖v ‖ ck δ := sup x∈rn ∑ 0≤|α|≤k0 〈x〉δ |∂αx v (x)| +ν sup x∈rn ∑ |β|=k0 〈x〉δ sup x′∈rn:|x−x′|≤1 ∣∣∂βx v (x) − ∂βx v (x′) ∣∣ |x − x′|ν < +∞, ‖v ‖ vk δ := sup x∈rn ∑ 0≤|α|≤k0 〈x〉δ+|α| |∂αx v (x)| 60 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) +ν sup x∈rn ∑ |β|=k0 〈x〉δ+k0+ν sup x′∈rn:|x−x′|≤1 ∣∣∂βx v (x) − ∂βx v (x′) ∣∣ |x − x′|ν < +∞, where k0 ≥ 0 is an integer and ν = k − k0 satisfies 0 ≤ ν < 1. in [2] we have proved the following theorem 1.1. let n = 4 or n = 5 and let v ∈ ckδ (r n ) with k > (n − 3)/2, δ > 3 if n = 4 and δ > 5 if n = 5. then, the dispersive estimate (1.3) holds true. it is not clear, however, if this still holds with k = (n− 3)/2. using the results of [2] we prove in the present paper the following theorem 1.2. let n = 4 or n = 5 and let v ∈ c (n−3)/2 δ (r n ), δ > 3 if n = 4 and δ > 5 if n = 5. then, we have the dispersive estimate ∥∥eitgχa(g)f ∥∥ l∞ ≤ cǫ|t| −n/2 ∥∥∥ ( log ( 2 + g2 ))2+ǫ f ∥∥∥ l1 , (1.7) for every 0 < ǫ ≪ 1. moreover, for every 2 ≤ p < +∞ we have the optimal dispersive estimate ∥∥eitgχa(g) ∥∥ lp ′ →lp ≤ c|t|−n(1/2−1/p), (1.8) where 1/p + 1/p′ = 1. note that it is not clear if (1.7) and (1.8) hold true when n ≥ 6. to prove the dispersive estimates above one needs to bound the quantity a(t,h) = |t|n/2 ∥∥eitgψ(h2g) ∥∥ l1→l∞ uniformly in both t and h, where ψ ∈ c∞0 ((0, +∞)) and 0 < h ≪ 1 is a semi-classical parameter. it was shown in [12] that, under the assumption (1.1) only, we have the bound a(t,h) ≤ ch−(n−3)/2 (1.9) in all dimensions n ≥ 4, where c > 0 is a constant independent of t and h. on the other hand, if we suppose (1.1) fulfilled with δ > n − 1 as well as (1.4), then we have the optimal bound (see [8]) a(t,h) ≤ c. (1.10) note that (1.10) still holds under the assumptions of theorem 1.1 (see [2]). to prove the estimates (1.7) and (1.8) we show in the present paper that, under the assumptions of theorem 1.2, we have the bound a(t,h) ≤ c log 1 h . (1.11) it is an open problem, however, to show that (1.11) still holds for potentials v ∈ c (n−3)/2 0 (r n ) when n ≥ 6. indeed, the strategy of proving (1.11) proposed in [1] leads to the study of a finite number (∼ n/2) of operators, tj (t,h), t > 0, j = 0, 1, ..., with explicit kernels defined as follows tj (t,h) = i ∫ t 0 ei(t−τ )g0ψ1(h 2g0)v tj−1(τ,h)dτ, t0(t,h) = e itg0ψ(h2g0), cubo 11, 5 (2009) dispersive estimates for the schrödinger equation ... 61 where ψ1 ∈ c ∞ 0 ((0, +∞)), ψ1 = 1 on supp ψ. roughly speaking, one needs to show that if v ∈ c (n−3)/2 0 (r n ), then each tj satisfies the bound ‖tj (t,h)‖l1→l∞ ≤ cjt −n/2 log 1 h , j ≥ 1. (1.12) in the present paper we prove (1.12) with j = 1 in all dimensions n ≥ 4 (actually for a larger class of potentials see section 3). however, this is hard to show for j ≥ 2. in fact, under the assumption (1.1) only, we have the bounds (see [1]) ‖tj (t,h)‖l1→l∞ ≤ cjt −n/2hj−n/2, j ≥ 1. (1.13) on the other hand, without any regularity assumption on v , the kernel of tj behaves like cjt −n/2h−j(n−3)/2. these observations show that if one wants to prove (1.12) (for j ≥ 2) it suffices to do it for j < n/2 only, and secondly one should better avoid using the kernels of tj for this purpose, unless one menages to show that some regularity on the potential improves the behaviour in h of the kernels (which is far from being clear when j ≥ 2). note that (1.11) follows from (1.12) with j = 1 and the following theorem proved in [2]. theorem 1.3. if n = 4 we suppose v ∈ cεδ (r 4 ) with ε > 0, δ > 3, while if n = 5 we suppose v ∈ c1δ (r 5 ) with δ > 5. then, there exist constants c,ε0 > 0 so that we have the estimate ∥∥∥∥∥∥ eitgψ(h2g) − 1∑ j=0 tj (t,h) ∥∥∥∥∥∥ l1→l∞ ≤ chε0t−n/2. (1.14) note again that it is a difficult open problem to show that (1.14) holds true for potentials v ∈ c (n−3)/2 0 (r n ) when n ≥ 6. however, (1.14) holds in all dimensions n ≥ 4 for potentials satisfying (1.1) with δ > n− 1 as well as (1.4) (see appendix b of [8]). it is shown in [12], [1] that, under the assumption (1.1) only, we have the bound (1.14) with h−(n−4)/2 in place of hε0 in the right-hand side. 2 proof of theorem 1.2 in this section we will show that theorem 1.2 follows from the estimate (1.11). to this end we will use the identity χa(σ) = ∫ 1 0 ψ(θσ) dθ θ , where ψ(σ) = σχ′a(σ) ∈ c ∞ 0 ((0, +∞)). so we can write χa(σ) ( log ( 2 + σ2 ))−2−ǫ = ∫ 1 0 ψθ(θσ) dθ θ (log (4θ−2)) 2+ǫ , 62 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) where ψθ(σ) = ψ(σ) ( 1 − log(θ2/2 + σ2/4) log(θ2/4) )−2−ǫ belongs to c∞0 ((0, +∞)) uniformly in θ and having a support independent of θ. therefore, we have eitgχa(g) ( log ( 2 + g2 ))−2−ǫ = ∫ 1 0 eitgψθ(θg) dθ θ (log (4θ−2)) 2+ǫ . hence, by (1.11) we get ∥∥∥eitgχa(g) ( log ( 2 + g2 ))−2−ǫ∥∥∥ l1→l∞ ≤ ∫ 1 0 ∥∥eitgψθ(θg) ∥∥ l1→l∞ dθ θ (log (4θ−2)) 2+ǫ ≤ c|t|−n/2 ∫ 1 0 dθ θ (log (2θ−1)) 1+ǫ ≤ c|t|−n/2 ∫ 1 0 −d log ( 2θ−1 ) (log (2θ−1)) 1+ǫ ≤ c|t|−n/2, which proves (1.7). to prove (1.8) we will use the following estimate proved in [12] (see theorem 3.1) ∥∥eitgψ(θg) − eitg0ψ(θg0) ∥∥ l2→l2 ≤ ch. (2.1) on the other hand, by (1.11) we have ∥∥eitgψ(θg) − eitg0ψ(θg0) ∥∥ l1→l∞ ≤ cǫh −ǫ|t|−n/2, (2.2) for every 0 < ǫ ≪ 1. an interpolation between (2.1) and (2.2) leads to the estimate ∥∥eitgψ(θg) − eitg0ψ(θg0) ∥∥ lp ′ →lp ≤ cǫh 1−(1+ǫ)(1−2/p)|t|−n(1/2−1/p), (2.3) for every 2 ≤ p ≤ +∞, where 1/p + 1/p′ = 1. let 2 ≤ p < +∞. by (2.3) we get ∥∥eitgχa(g) − eitg0χa(g0) ∥∥ lp ′ →lp ≤ ∫ 1 0 ∥∥eitgψ(θg) − eitg0ψ(θg0) ∥∥ lp ′ →lp dθ θ ≤ c|t|−n(1/2−1/p) ∫ 1 0 θ−1+1/p−ǫ(1/2−1/p)dθ ≤ c|t|−n(1/2−1/p) ∫ 1 0 θ−1+1/(2p)dθ ≤ c|t|−n(1/2−1/p), (2.4) provided ǫ > 0 is taken small enough. clearly, (1.8) follows from (2.4). 3 study of the operator t1 in all dimensions n ≥ 4 let γ = t/2 if 0 < t ≤ 2, γ = 1 if t ≥ 2, and decompose the operator t1 as follows t1 = (∫ γ 0 + ∫ t t−γ ) ... + ∫ t−γ γ ... := t (1) 1 + t (2) 1 . in this section we will prove the following cubo 11, 5 (2009) dispersive estimates for the schrödinger equation ... 63 proposition 3.1. let n ≥ 4 and let v ∈ v (n−3)/2−k δ (r n ), where 0 ≤ k < (n− 3)/2 and δ > 2 + k. then, we have the estimate ∥∥∥t (1)1 (t,h) ∥∥∥ l1→l∞ ≤ ct−n/2h−k log 1 h . (3.1) moreover, if v ∈ vmδ (r n ) with an integer 0 ≤ m < (n − 3)/2, and if k ≥ 0 is such that n − 1 − 2m − δ < k < (n − 3)/2 − m, then we have ∥∥∥t (2)1 (t,h) ∥∥∥ l1→l∞ ≤ ct−n/2h−k, t ≥ 2. (3.2) remark. it is proved in [12] that if v satisfies (1.1) with δ > (n + 1)/2, then we have ‖t1(t,h)‖l1→l∞ ≤ ct −n/2h−(n−3)/2. note also that (3.2) with m = k = 0 is proved in [8] (see appendix b), and this is enough for the proof of theorem 1.2. proof. we will make use of the fact that the kernel of the operator eitg0ψ(h2g0) is of the form kh(|x − y|, t), where kh(σ,t) = σ−2ν (2π)ν+1 ∫ ∞ 0 eitλ 2 ψ(h2λ2)jν (σλ)λdλ = h −nk1(σ/h,t/h 2 ), (3.3) where jν (z) = z νjν (z), jν (z) = (h + ν (z) + h − ν (z)) /2 is the bessel function of order ν = (n− 2)/2. thus, the kernel of t1 is of the form t (x,y,t,h) = ∫ t 0 ∫ rn k̃h(|x − ξ|, t − τ)kh(|y − ξ|,τ)v (ξ)dξdτ, where k̃h denotes the kernel of the operator e itg0ψ1(h 2g0). it is shown in [12] (see proposition 2.1) that the function kh satisfies the bound |kh(σ,t)| ≤ c(hσ) s−(n−1)/2t−s−1/2, ∀t,σ > 0, 0 < h ≤ 1, 0 ≤ s ≤ (n − 1)/2. (3.4) set ah(σ,t) = t n/2eiσ 2/4tkh(σ,t) = a1(σ/h,t/h 2 ). (3.5) clearly, (3.4) can be rewritten as |ah(σ,t)| ≤ c ( t hσ )s , ∀t,σ > 0, 0 < h ≤ 1, 0 ≤ s ≤ (n − 1)/2. (3.6) denote a′h = dah/dt. we need the following lemma 3.2. for every t,σ > 0, 0 < h ≤ 1, 0 ≤ s ≤ (n − 1)/2 and every integer k ≥ 0 such that k + s ≤ n/2, we have the bound ∣∣∂kσah(σ,t) ∣∣ ≤ c ( 1 σ )k ( t hσ )s . (3.7) 64 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) moreover, if k + s ≤ (n − 2)/2, we have ∣∣∂kσa ′ h(σ,t) ∣∣ ≤ ct−1 ( 1 σ )k ( t hσ )s . (3.8) proof. in view of (3.5), it suffices to prove (3.7) and (3.8) with h = 1. consider first the case 0 < σ ≤ 1. we will use that jν (z) = z 2νgν(z) with a function gν(z) analytic at z = 0. hence, given any integer k ≥ 0 we have ∂kσk1(σ,t) = ∫ ∞ 0 eitλ 2 ψk(λ)g (k) ν (σλ)dλ, where ψk ∈ c ∞ 0 ((0, +∞)) and g (k) ν (z) = d kgν (z)/dz k. let t ≥ 1. then, in the same way as in the proof of proposition 2.1 of [12] (see (2.10)) we have ∣∣∂kσk1(σ,t) ∣∣ ≤ ck,mt−m−1/2, (3.9) for every integer m ≥ 0, and hence for all real m ≥ 0. using (3.9) we get ∣∣∂kσa1(σ,t) ∣∣ ≤ ctn/2 k∑ j=0 ∣∣∣∂jσ ( eiσ 2/4t )∣∣∣ ∣∣∂k−jσ k1(σ,t) ∣∣ ≤ ct(n−1)/2−m k∑ j=0 ∣∣∣∂jσ ( eiσ 2/4t )∣∣∣ ≤ ct(n−1)/2−m, which proves (3.7) (with h = 1) in this case. let 0 < t ≤ 1. then we have ∣∣∂kσk1(σ,t) ∣∣ ≤ ck. (3.10) using (3.10) we get ∣∣∂kσa1(σ,t) ∣∣ ≤ ctn/2 k∑ j=0 ∣∣∣∂jσ ( eiσ 2/4t )∣∣∣ ∣∣∂k−jσ k1(σ,t) ∣∣ ≤ ctn/2 k∑ j=0 ∣∣∣∂jσ ( eiσ 2/4t )∣∣∣ ≤ ctn/2−k ≤ cts. consider now the case σ ≥ 1. we will use that jν (z) = e izb+ν (z) + e −izb−ν (z) with functions b ± ν satisfying ∣∣∂jzb ± ν (z) ∣∣ ≤ cjz(n−3)/2−j, ∀z ≥ z0, (3.11) for every integer j ≥ 0 and every z0 > with a constant cj > 0 depending on j and z0. we can write k1 = k + 1 + k − 1 with k ± 1 defined by replacing in the definition of k1 the function jν (z) by e±izb±ν (z). then the functions a±1 (σ,t) = t n/2eiσ 2/4tk±1 (σ,t) cubo 11, 5 (2009) dispersive estimates for the schrödinger equation ... 65 can be written in the form a±1 (σ,t) = t n/2 ∫ ∞ 0 eit(λ±σ/2t) 2 b̃±ν (σλ)ϕ(λ)dλ, where ϕ(λ) = (2π)−ν−1λ1+2νψ(λ2), b̃±ν (z) = z −2νb±ν (z). hence ∂kσa ± 1 (σ,t) = t n/2 ∫ ∞ 0 k∑ j=0 cj∂ j σ ( eit(λ±σ/2t) 2 ) ∂k−jσ b̃ ± ν (σλ)ϕ(λ)dλ. using the identity ∂jσ ( eit(λ±σ/2t) 2 ) = (∓2t)−j∂ j λ ( eit(λ±σ/2t) 2 ) and integrating by parts, we get ∂kσa ± 1 (σ,t) = k∑ j=0 tn/2−jeiσ 2/4t ∫ ∞ 0 eitλ 2 ±iσλϕ̃(λ)b±ν,j (λ,σ)dλ, where ϕ̃ ∈ c∞0 ((0, +∞)), ϕ̃ = 1 on supp ϕ, and b±ν,j (λ,σ) = cj (±2) −j ∂ j λ ( ∂k−jσ b̃ ± ν (σλ)ϕ(λ) ) . it is easy to deduce from (3.11) that the functions b±ν,j satisfy the bounds ∣∣∂ℓλb ± ν,j (λ,σ) ∣∣ ≤ cℓ,jσ−(n−1)/2−k+j, (3.12) for all integers ℓ,j ≥ 0. using (3.12), in the same way as in the proof of proposition 2.1 of [12] (see (2.13)), we get ∣∣∣∣ ∫ ∞ 0 eitλ 2 ±iσλϕ̃(λ)b±ν,j (λ,σ)dλ ∣∣∣∣ ≤ cm,jt −m−1/2σ−(n−1)/2−k+j+m, (3.13) for all integers m, and hence for all real m. by (3.13) with m = (n − 1)/2 − s − j we obtain ∣∣∂kσa1(σ,t) ∣∣ ≤ cσ−k ( t σ )s , which is the desired result in this case. to prove (3.8) with h = 1, observe that a′1(σ,t) = t n/2eiσ 2/4t ( k′1(σ,t) + n 2t k1(σ,t) − iσ2 4t2 k1(σ,t) ) . (3.14) on the other hand, integrating by parts twice with respect to the variable λ2 and using that the function gν (z) = z −2νjν (z) = z −νjν (z) satisfies the equation g′′ν (z) + (n − 1)z −1g′ν (z) + gν (z) = 0, 66 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) we get k′1(σ,t) + n 2t k1(σ,t) − iσ2 4t2 k1(σ,t) = t −1 ( k (0) 1 (σ,t) + k (1) 1 (σ,t) ) , (3.15) where k (j) 1 (σ,t) = (σ t )j ∫ ∞ 0 eitλ 2 ψ(j)(λ)g(j)ν (σλ)dλ, j = 0, 1, where ψ(j) ∈ c∞0 ((0, +∞)), g (0) ν (z) = gν (z), g (1) ν (z) = dgν (z)/dz. by (3.14) and (3.15), a′1(σ,t) = t −1 ( a (0) 1 (σ,t) + a (1) 1 (σ,t) ) , (3.16) where a (j) 1 (σ,t) = t n/2eiσ 2/4tk (j) 1 (σ,t), j = 0, 1. now, in the same way as above one can see that the functions a (j) 1 satisfy (3.7) with h = 1, provided k + s ≤ (n − 2)/2. 2 the kernel of the operator t1 is of the form t (x,y,t,h) = ∫ t 0 ∫ rn e−iϕ(t − τ)−n/2τ−n/2ãh(|x − ξ|, t − τ)ah(|y − ξ|,τ)v (ξ)dξdτ, where ϕ = |x − ξ|2 4(t − τ) + |y − ξ|2 4τ , and ãh is defined by replacing in the definition of ah the function kh by k̃h. observe that by lemma 3.2 we have the bounds ∣∣∂αξ ah(|x − ξ|, t) ∣∣ ≤ c(t/h)k+ǫ|x − ξ|−|α|−k−ǫ, (3.17) ∣∣∂αξ a ′ h(|x − ξ|, t) ∣∣ ≤ ct−1(t/h)k+ǫ|x − ξ|−|α|−k−ǫ, (3.18) for every 0 ≤ ǫ ≪ 1, 0 ≤ k < (n − 3)/2, and all multi-indices α such that |α| ≤ (n − 2)/2 − k − ǫ. define the functions f(1) and f(2) by replacing ∫ t 0 in the definition of the function t by ∫ γ 0 and ∫ t/2 γ , respectively. let φ ∈ c∞0 (r), φ(λ) = 1 for |λ| ≤ 1/2, φ(λ) = 0 for |λ| ≥ 1, and write 1 = ∞∑ q=0 φq(λ), where φ0 = φ, φq(λ) = φ̃(2 −qλ), q ≥ 1, with a function φ̃ ∈ c∞0 (r), φ̃(λ) = 0 for |λ| ≤ 1/2 and |λ| ≥ 1. we can write f(1) = ∞∑ p=1 ∞∑ q=0 f(1)p,q , f (2) = ∞∑ q=0 f(2)q , where f(1)p,q = ∫ γ 0 ∫ rn e−iϕ(t − τ)−n/2τ−n/2φp (γ τ ) φq(|ξ|)ãh(|x − ξ|, t − τ)ah(|y − ξ|,τ)v (ξ)dξdτ cubo 11, 5 (2009) dispersive estimates for the schrödinger equation ... 67 = t1−n ∫ ∞ t/γ ∫ rn e−iϕ ( µ µ − 1 )n/2 µn/2−2φp (γµ/t) ãh(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)vq(ξ)dξdµ, f(2)q = ∫ t/2 1 ∫ rn e−iϕ(t − τ)−n/2τ−n/2φq(|ξ|)ãh(|x − ξ|, t − τ)ah(|y − ξ|,τ)v (ξ)dξdτ = t1−n ∫ t 2 ∫ rn e−iϕ ( µ µ − 1 )n/2 µn/2−2ãh(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)vq(ξ)dξdµ, where we have made a change of variables µ = t/τ and set vq(ξ) = φq(|ξ|)v (ξ). clearly, it suffices to prove the following proposition 3.3. under the assumptions of proposition 3.1, there exist constants c,ε′ > 0 such that we have the bounds ∣∣∣f(1)p,q ∣∣∣ ≤ c2−εp−ε ′qt−n/2h−k−ε, (3.19) for every 0 < ε ≪ 1, and ∣∣∣f(2)q ∣∣∣ ≤ c2−ε ′qt−n/2h−k, t ≥ 2. (3.20) indeed, by (3.19) we have ∣∣∣f(1) ∣∣∣ ≤ ct−n/2h−k(εhε)−1 = c′t−n/2h−k log 1 h , if we take ε so that h−ε = 2, while (3.20) yields ∣∣∣f(2) ∣∣∣ ≤ ct−n/2h−k, t ≥ 2. proof. let ρ ∈ c∞0 (r n ) be a real-valued function such that ∫ ρ(x)dx = 1, and set ρθ(x) = θ−nρ(x/θ), 0 < θ ≤ 1, vq,θ = ρθ ∗ vq. let k0 ≥ 0 be an integer such that (n − 3)/2 − k = k0 + ν with 0 ≤ ν < 1. since v ∈ vk0+νδ (r n ), we have ∣∣∂αξ vq(ξ) ∣∣ ≤ c2−q(δ+|α|), 0 ≤ |α| ≤ k0, (3.21) ∣∣∂αξ vq(ξ) − ∂ α ξ vq(ξ ′ ) ∣∣ ≤ c2−q(δ+k0+ν)|ξ − ξ′|ν, |ξ − ξ′| ≤ 1, |α| = k0. (3.22) it is easy to see that these bounds imply ∣∣∂αξ vq,θ(ξ) ∣∣ ≤ c2−q(δ+|α|), 0 ≤ |α| ≤ k0, (3.23) ∣∣∂αξ vq,θ (ξ) ∣∣ ≤ cθ−1+ν 2−q(δ+k0+ν), |α| = k0 + 1, (3.24) ∣∣∂αξ vq(ξ) − ∂ α ξ vq,θ(ξ) ∣∣ ≤ cθ2−q(δ+|α|+1), 0 ≤ |α| ≤ k0 − 1, (3.25) ∣∣∂αξ vq(ξ) − ∂ α ξ vq,θ(ξ) ∣∣ ≤ cθν 2−q(δ+|α|+ν), |α| = k0. (3.26) integrating by parts with respect to the variable ξ as in section 4 of [12] (see the proof of (4.15)) we obtain the following 68 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) lemma 3.4. let 0 ≤ m < (n − 1)/2 be an integer and let w(µ, ·) ∈ cm0 (r n ). then we have the estimate ∣∣∣∣∣ t1−n ∫ ∞ t/γ ∫ rn e−iϕ ( µ µ − 1 )n/2 µn/2−2φp (γµ/t) w(µ,ξ)dξdµ ∣∣∣∣∣ ≤ ct−n/2 (2p/γ) (n−3)/2−m ∑ 0≤|α|≤m ∫ rn |ξ − y| −2m+|α|−1 ∣∣∂αξ w(∞,ξ) ∣∣ dξ +ct−n/2 (2p/γ) (n−3)/2−m ∑ 0≤|α|≤m ∫ rn |y − ξ|−1 ∣∣ξ − y − γt−1(x − y) ∣∣−2m+|α| ∣∣∂αξ w(t/γ,ξ) ∣∣ dξ +ct−n/2−1 (2p/γ) (n−3)/2−m−1 ∑ 0≤|α|≤m ∫ 2pt/γ 2p−1t/γ ∫ rn ( |y − ξ|−1 ∣∣ξ − y − µ−1(x − y) ∣∣−2m+|α| + ∣∣ξ − y − µ−1(x − y) ∣∣−2m+|α|−1 ) ∣∣∂αξ w(µ,ξ) ∣∣ dξdµ +ct−n/2 (2p/γ) (n−3)/2−m ∑ 0≤|α|≤m ∫ 2pt/γ 2p−1t/γ ∫ rn |y − ξ|−1 ∣∣ξ − y − µ−1(x − y) ∣∣−2m+|α| × ∣∣∂µ∂αξ w(µ,ξ) ∣∣ dξdµ. (3.27) we would like to apply this lemma with a function w of the form w(µ,ξ) = ãh(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)q(ξ) where q ∈ cm0 (r n ) is independent of the variable µ. in view of (3.17) and (3.18) we have ∣∣∂αξ w(µ,ξ) ∣∣ ≤ c ∑ |α1|+|α2|+|α3|=|α| ∣∣∣∂α1ξ ãh(|x − ξ|, t(µ − 1)/µ) ∣∣∣ ∣∣∣∂α2ξ ah(|y − ξ|, t/µ) ∣∣∣ ∣∣∣∂α3ξ q(ξ) ∣∣∣ ≤ ch−k−ε(t/µ)k+ε ∑ |α1|+|α2|+|α3|=|α| |x − ξ|−|α1||y − ξ|−|α2|−k−ε ∣∣∣∂α3ξ q(ξ) ∣∣∣ , (3.28) ∣∣∂µ∂αξ w(µ,ξ) ∣∣ ≤ tµ−2 ∣∣∂αξ (ã ′ h(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)q(ξ)) ∣∣ +tµ−2 ∣∣∂αξ (ãh(|x − ξ|, t(µ − 1)/µ)a ′ h(|y − ξ|, t/µ)q(ξ)) ∣∣ ≤ ctµ−2 ∑ |α1|+|α2|+|α3|=|α| ∣∣∣∂α1ξ ã ′ h(|x − ξ|, t(µ − 1)/µ) ∣∣∣ ∣∣∣∂α2ξ ah(|y − ξ|, t/µ) ∣∣∣ ∣∣∣∂α3ξ q(ξ) ∣∣∣ +ctµ−2 ∑ |α1|+|α2|+|α3|=|α| ∣∣∣∂α1ξ ãh(|x − ξ|, t(µ − 1)/µ) ∣∣∣ ∣∣∣∂α2ξ a ′ h(|y − ξ|, t/µ) ∣∣∣ ∣∣∣∂α3ξ q(ξ) ∣∣∣ ≤ cµ−1h−k−ε(t/µ)k+ε ∑ |α1|+|α2|+|α3|=|α| |x − ξ|−|α1||y − ξ|−|α2|−k−ε ∣∣∣∂α3ξ q(ξ) ∣∣∣ . (3.29) by (3.27), (3.28) and (3.29), one can easily get the estimate ∣∣∣∣∣ t1−n ∫ ∞ t/γ ∫ rn e−iϕ ( µ µ − 1 )n/2 µn/2−2φp (γµ/t) w(µ,ξ)dξdµ ∣∣∣∣∣ cubo 11, 5 (2009) dispersive estimates for the schrödinger equation ... 69 ≤ ct−n/2h−k−ε ( γ2−p )m+k+ε−(n−3)/2 m(m,q), (3.30) where m(m,q) = ∑ 0≤|α|≤m sup rn 〈ξ〉n+2ε ′ −2m−1−k+|α| ∣∣∂αξ q(ξ) ∣∣ , for every 0 < ε′ ≪ 1 and 0 < ε ≤ ε′. consider first the case k0 < (n − 3)/2. then, we are going to apply (3.30) with m = k0, q = vq − vq,θ, and m = k0 + 1, q = vq,θ, respectively. choose ε ′ such that δ ≥ k + 2 + 4ε′. in view of (3.25) and (3.26), we have m(k0,vq − vq,θ) ≤ cθ ν (2 q ) ν−ε′ , (3.31) while by (3.23) and (3.24), we have m(k0 + 1,vq,θ) ≤ cθ ν−1 (2 q ) ν−1−ε′ . (3.32) combining (3.30), (3.31) and (3.32) we conclude ∣∣∣f(1)p,q ∣∣∣ ≤ ct−n/2h−k−ε ( γ2−p )ε 2 −ε′q (( θ2p+q/γ )ν + ( θ2p+q/γ )ν−1) . (3.33) taking θ = γ2−p−q we deduce (3.19) from (3.33). let now k0 = (n − 3)/2. this implies that n is odd and ν = 0. then we apply (3.30) with m = k0 and q = vq. in view of (3.21) we have m(k0,vq) ≤ c2 −ε′q, (3.34) with some constant 0 < ε′ ≪ 1, so in this case (3.19) follows from (3.30) and (3.34). proceeding as in the proof of (3.30) we obtain in the same way the estimate ∣∣∣∣∣ t1−n ∫ t 2 ∫ rn e−iϕ ( µ µ − 1 )n/2 µn/2−2w(µ,ξ)dξdµ ∣∣∣∣∣ ≤ ch−k ( t−n/2 + tm+k−n+3/2 ) m(m,q) +ch−ktm+k−n+3/2m(m,q) ∫ t 2 µ(n−3)/2−m−k−1dµ ≤ c′h−kt−n/2m(m,q), (3.35) where we have used that m + k < (n − 3)/2. on the other hand, since v ∈ vmδ (r n ) with δ > n − 1 − 2m − k, it is easy to check that we have the bound m(m,vq) ≤ c2 −ε′q, (3.36) with some constant 0 < ε′ ≪ 1. now, combining (3.35) and (3.36) leads to (3.20). 2 70 fernando cardoso, claudio cuevas and georgi vodev cubo 11, 5 (2009) acknowledgements this work was carried out while g.v. was visiting the universidade federal de pernambuco, recife, in august september 2008 with the support of the agreement brazil-france in mathematics proc. 69.0014/01-5. the first two authors have also been partially supported by the cnpq-brazil. received: november, 2008. revised: april, 2009. references [1] f. cardoso and g. vodev, semi-classical dispersive estimates for the wave and schrödinger equations with a potential in dimensions n ≥ 4, cubo, 10 (2008), 1-14. [2] f. cardoso, c. cuevas and g. vodev, dispersive estimates for the schrödinger equation in dimensions four and five, asymptot. anal. 62 (2009), 125-145. [3] m. goldberg, dispersive bounds for the three dimensional schrödinger equation with almost critical potentials, gafa 16 (2006), 517-536. [4] m. goldberg and w. schlag, dispersive estimates for schrödinger operators in dimensions one and three, commun. math. phys. 251 (2004), 157-178. [5] m. goldberg and m. visan, a counterexample to dispersive estimates for schrödinger operators in higher dimensions, commun. math. phys. 266 (2006), 211-238. [6] j.-l. journé, a. soffer and c. sogge, decay estimates for schrödinger operators, commun. pure appl. math. 44 (1991), 573-604. [7] s. moulin, high frequency dispersive estimates in dimension two, ann. h. poincaré 10 (2009), 415-428. [8] s. moulin and g. vodev, low-frequency dispersive estimates for the schrödinger group in higher dimensions, asymptot. anal. 55 (2007), 49-71. [9] w. schlag, dispersive estimates for schrödinger operators in two dimensions, commun. math. phys. 257 (2005), 87-117. [10] i. rodnianski and w. schlag, time decay for solutions of schrödinger equations with rough and time-dependent potentials, invent. math. 155 (2004), 451-513. [11] g. vodev, dispersive estimates of solutions to the schrödinger equation, ann. h. poincaré 6 (2005), 1179-1196. [12] g. vodev, dispersive estimates of solutions to the schrödinger equation in dimensions n ≥ 4, asymptot. anal. 49 (2006), 61-86. b5-ccv2 cubo a mathematical journal vol.10, n o ¯ 03, (145–159). october 2008 asymptotic constancy and stability in nonautonomous stochastic differential equations john a.d. appleby∗ school of mathematical sciences, dublin city university, dublin 9, ireland email: john.appleby@dcu.ie url: http://webpages.dcu.ie/~applebyj james p. gleeson department of applied mathematics, university college cork, cork, ireland email: j.gleeson@ucc.ie and alexandra rodkina department of mathematics and computer science, the university of the west indies, mona, kingston 7, jamaica email: alexandra.rodkina@uwimona.edu.jm abstract this paper considers the asymptotic behaviour of a scalar non-autonomous stochastic differential equation which has zero drift, and whose diffusion term is a product of a function of time and space dependent function, and which has zero as a unique ∗the first author is partially supported by an albert college fellowship awarded by dublin city university’s research advisory panel. the third author is supported by the mona research fellowship programm awarded by the university of the west indies, jamaica and by london mathematical society 146 john a.d. appleby et al. cubo 10, 3 (2008) equilibrium solution. we classify the pathwise limiting behaviour of solutions; solution either tends to a non-trivial, non-equilibrium and random limit, or the solution hits zero in finite time. in the first case, the exact rate of decay can always be computed. these results can be inferred from the square integrability of the time dependent factor, and the asymptotic behaviour of the corresponding autonomous stochastic equation, where the time dependent multiplier is unity. resumen este art́ıculo considera el comportamiento asintótico de una ecuación diferencial estocastica escalar no-autónoma la cual tiene cero desviación y cujo término de difusión es un producto de una función de equilibrio. nosotros clasificamos el comportamiento limite por caminos de las soluciones; la solución atiende a un no equilibrio y ĺımite randon no trivial, o la solución encuentra cero en tiempo finito. en el primer caso, las tasas de decaimiento siempre pueden ser calculadas. estos resultados pueden ser inferidos de la integrabilidad al cuadrado del factor dependiente del tiempo, y el comportamiento asintótico de la correspondiente ecuación estocatica autónoma, donde el multiplicador dependiente del tiempo es la unidad. key words and phrases: brownian motion, almost sure asymptotic stability, asymptotic constancy, stochastic differential equation, nonautonomous, feller’s test, explosions. math. subj. class.: 60h10, 93e15. 1 introduction this note considers the asymptotic behaviour of solutions of the “separable” stochastic differential equation dx(t) = σ(t)g(x(t)) db(t). (1.1) a solution of this equation with initial condition ξ is denoted by x(·, ξ). it is presumed that zero is a point equilibrium, so x(t, 0) = 0 is a solution of (1.1). a standard deterministic change of time scale reduces this equation to an autonomous equation dx̃(t) = g(x̃(t)) db̃(t), (1.2) from which it can be shown that the condition that σ ∈ l2([0, ∞); r) largely determines whether the solution tends to the equilibrium or to a non–trivial and non–equilibrium limit. another feature which is examined is the relationship between the process x̃ hitting zero in a finite amount of time, or tending to zero as t → ∞ (in the case when x̃ remains strictly positive) and the corresponding properties of x. as will be seen, a complete picture of the dynamics of (1.1) can be deduced in terms of conditions on g and σ. cubo 10, 3 (2008) asymptotic constancy and stability in sdes 147 even though the time–change technique employed is well–known, some novel features appear in the analysis. first, we are unaware of an extensive literature concerning the pathwise convergence of solutions of stochastic differential equations to non–equilibrium limits. second, we determine here sharp upper and lower estimates in terms of the rate at which the noise intensity fades on the almost sure rate of convergence of the solution to this non–equilibrium limit, in the case when σ ∈ l2([0, ∞); r). this requires a delicate use of the law of the iterated logarithm, partly correcting an error on the asymptotic behaviour of a tail martingale established in [1] and used in [2]. finally, in the case where σ 6∈ l2([0, ∞); r), the results here, taken in conjunction with work in [3, 4] would enable exact almost sure rates of convergence to zero of solutions of (1.1) to be established. 2 existence of solutions in this paper we deal with highly nonlinear stochastic differential equations, sde, whose solutions can hit zero at finite time, due to the non-lipshitz behavior of the diffusion coefficients. moreover, for non-autonomous equations, it is convenient for the completeness of our exposition, to state carefully and to prove an existence result. this result is a corollary of well-known existence result and martingale time changing theorem. let (ω, f, (f(t))t≥0, p) be a complete probability space with filtration (f(t))t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while f(0) contains all p-null sets). let (b(t))t≥0 be a scalar standard brownian motion, defined on the probability space (ω, f, (f(t))t≥0, p). since we will consider equations with deterministic initial conditions, it is enough to work with the natural filtration of b: that is f(t) ≡ fb(t) where fb(t) = σ(b̃(s) : 0 ≤ s ≤ t). suppose that the function σ obeys σ ∈ c([0, ∞); r). (2.1) we define the local martingale m = {m (t), 0 ≤ t < ∞, fb(t)} by m (t) = ∫ t 0 σ(s) db(s), t ≥ 0, (2.2) with square variation 〈m〉 given by 〈m〉(t) = ∫ t 0 σ2(s) ds. let t ∗ be given by t ∗ = ∫ ∞ 0 σ2(s) ds, (2.3) where we define t ∗ = ∞ if σ 6∈ l2([0, ∞); r). 148 john a.d. appleby et al. cubo 10, 3 (2008) define also, for each 0 ≤ s ≤ t ∗, the fb—stopping time t (s) = inf{t ≥ 0 : 〈m〉(t) ≥ s}. (2.4) by the martingale time change theorem, there exists a standard brownian motion {b̃(t); 0 ≤ t ≤ t ∗; g(t)}, such that b̃(s) = m (t (s)), g(s) = fb(t (s)), 0 ≤ s ≤ t ∗, and moreover m (t) = b̃(〈m〉(t)), fb(t) = g(〈m〉(t)), t ≥ 0. in what follows we presume that g : r → r obeys g(0) = 0. (2.5) since we do not want the equation to have any other equilibria in (0, ∞) we ask that the non– degeneracy condition g(x) > 0, for all x 6= 0, (2.6) also be satisfied. we are now in a position to state an existence result for the solution of the autonomous stochastic differential equation. proposition 2.1. suppose that g obeys (2.5) and (2.6) and that there exists a strictly increasing function (2.7a) q : [0, ∞) → [0, ∞) with q(0) = 0 such that ∫ ǫ 0 1 q2(u) du = ∞, for all ǫ > 0, and that g and q both obey |g(x) − g(y)| ≤ q(|x − y|), x, y ∈ r. (2.7b) then there exists a unique strong non-exploding solution x̃ of dx̃(t) = g(x̃(t)) db̃(t), t ≥ 0, x(0) = ξ > 0, (2.8) on the complete probability space (ω, f, (g(t))t≥0, p). this result is a corollary of the result of yamada and watanabe (see e.g. [7], proposition 2.13, page 291, and [7], theorem 5.4., page 332). the main concern of this paper is the asymptotic behaviour of solutions of the nonautonomous equations dx(t) = σ(t)g(x(t)) db(t), t ≥ 0, x(0) = ξ > 0. (2.9) before conducting this asymptotic analysis however, we must verify that this equation has a welldefined solution. this is accomplished by the following result. cubo 10, 3 (2008) asymptotic constancy and stability in sdes 149 proposition 2.2. suppose that g and q obey (2.5), (2.6), (2.7a) and (2.7b) and σ obeys (2.1). then there exists a unique strong non-exploding solution x of (2.9) on the complete probability space (ω, f, (fb(t))t≥0, p). proof. by proposition 2.1, x̃ is the unique strong solution of (2.8). consider x̃(t) for t ∈ [0, t ∗), where t ∗ is defined as in (2.3), and define x(t) = x̃ ( ∫ t 0 σ2(s)ds ) , t ≥ 0. (2.10) then, as x̃(s) is g(s)-measurable, x̃(t) is g ( ∫ t 0 σ2(s)ds ) -measurable. but g (〈m〉(t)) = fb(t), so x(t) is fb(t)-measurable. now, by [7], proposition 3.4.8, we get for t ≥ 0 ∫ 〈m〉(t) 0 g(x̃(u))db̃(u) = ∫ t 0 g(x(s))dm (s) = ∫ t 0 σ(s)g(x(s))db(s), and, therefore, as 〈m〉(t) = ∫ t 0 σ2(s)ds, we get x(t) = x̃ ( ∫ t 0 σ 2(s)ds ) = x̃ (〈m〉(t)) = x̃(0) + ∫ 〈m〉(t) 0 g(x̃(u))db̃(u) = ξ + ∫ t 0 σ(s)g(x(s))db(s). therefore x defined by (2.10) is a strong solution of (2.9) on (ω, f, (fb(t))t≥0, p). to show uniqueness, suppose that there is another solution of (2.9), y . then ỹ defined by ỹ (s) = y (t (s)) obeys (2.8). hence, as (2.8) has a unique solution, we have ỹ = x̃, and therefore it follows that y = x. this completes the proof. in this paper, we choose to write explicitly the dependence of solutions on their initial conditions, which are always assumed to be deterministic. thus, the value at time t ≥ 0 of the process y with initial condition y (0) = ξ is denoted by y (t, ξ). 3 main result in this section, we state and discuss the main results of the paper concerning the asymptotic behaviour of non–autonomous equation. at the end of the section we present an example of a non– autonomous linear equation which can be analysed without the use of the theorems established here, but whose behaviour illustrates the results proven. as seen in the proof of proposition 2.2 the non–autonomous equation (2.9) is equivalent to (2.8), under a deterministic time change. however the subject of proposition 2.2 is the existence 150 john a.d. appleby et al. cubo 10, 3 (2008) for non–autonomous equation. the following proposition by contrust, focusses on the relation between the solutions of the two equations. proposition 3.1. let ξ > 0 be deterministic. suppose that g obeys (2.5), (2.6). let σ obey (2.1). suppose that x(·, ξ) is the unique strong solution of (2.9) with x(0, ξ) = ξ. let t ∗ be given by (2.3), and t be defined by (2.4). (i) if σ ∈ l2([0, ∞); r), then there exists a standard brownian motion b̃ = {b̃(t); 0 ≤ t ≤ t ∗; g(t)} where g(t) = fb(t (t)) and b̃(t) = b(t (t)) such that the process x̃ = {x̃(t); 0 ≤ t ≤ t ∗; g(t)} defined by x̃(t) = x(t (t)) obeys (2.8). (ii) if σ 6∈ l2([0, ∞); r), then there exists a standard brownian motion b̃ = {b̃(t); 0 ≤ t < ∞; g(t)} where g(t) = fb (t (t)) and b̃(t) = b(t (t)) such that the process x̃ = {x̃(t); 0 ≤ t < ∞; g(t)} defined by x̃(t) = x(t (t)) obeys (2.8). before stating the first result on asymptotic behaviour we present some notation and an important auxiliary result. suppose that x̃(·, ξ) is the solution of (2.8), where ξ > 0. define s̃0(ξ) = inf{t ≥ 0 : x̃(t, ξ) = 0}. (3.1) let us suppose that for δ > 0 we may define the function v : (0, ∞) → (0, ∞) by v(x) = 2 ∫ δ x ∫ δ y dz g2(z) dy, x > 0. (3.2) the following result is due to feller (see e.g. [7], theorem 5.5.29, page 348). proposition 3.2. let ξ > 0 be deterministic, and x̃(·, ξ) be a strong solution of (2.8). if s̃0(ξ) is as defined in (3.1), then lim t→s̃0(ξ) x̃(t, ξ) = 0, sup 0≤t 0 be deterministic. suppose that g obeys (2.5), (2.6). let σ obey (2.1). suppose that x(·, ξ) is the unique strong solution of (2.9) with x(0, ξ) = ξ. let v be defined as in (3.2), and suppose that lim x→0+ v(x) = ∞. then we have the following case distinction: (a) if σ ∈ l2((0, ∞); r), then there exists an almost surely positive and fb(∞)–measurable random variable l = l(ξ, ω) such that lim t→∞ x(t, ξ) = l(ξ) > 0, a.s. (3.4) (b) if σ 6∈ l2((0, ∞); r), then lim t→∞ x(t, ξ) = 0, a.s. and s0(ξ) defined by (3.3) obeys s0(ξ) = ∞, a.s. the proof of this result and proofs of subsequent results in the this section, are postponed to the final section of the paper. when σ ∈ l2([0, ∞); r) the rate at which convergence to l(ξ) occurs can be determined exactly. theorem 3.4. let ξ > 0 be deterministic. suppose that g obeys (2.5), (2.6), and let g ∈ c1((0, ∞); (0, ∞)). let σ obey (2.1) and σ ∈ l2([0, ∞); r). suppose that x(·, ξ) is the unique strong solution of (2.9) with x(0, ξ) = ξ. let v be defined as in (3.2), and suppose that lim x→0+ v(x) = ∞. let l(ξ) be the almost surely positive and fb(∞)–measurable random variable defined by (3.4). then: (i) if σ obeys ∫ ∞ t σ 2(s) ds > 0, for all t ≥ 0, (3.5) then lim sup t→∞ x(t, ξ) − l(ξ) √ 2 ∫ ∞ t σ2(s) ds log log (∫ ∞ t σ2(s) ds )−1 = g(l(ξ)), a.s., (3.6a) lim inf t→∞ x(t, ξ) − l(ξ) √ 2 ∫ ∞ t σ2(s) ds log log (∫ ∞ t σ2(s) ds )−1 = −g(l(ξ)), a.s. (3.6b) 152 john a.d. appleby et al. cubo 10, 3 (2008) (ii) if σ does not obey (3.5) i.e., if there exists τ ≥ 0 such that ∫ ∞ t σ2(s) ds = 0 for all t ≥ τ , then x(t, ξ) = x(τ, ξ) = l(ξ), for all t ≥ τ , a.s. of course, in case (b) in theorem 3.3, the rate of convergence cannot be so easily computed. however pathwise rates of decay to zero for nonlinear autonomous stochastic differential equations have been found in [3, 4], and could readily be applied here. finally, the distribution of the random limit l is in principle well–understood, by using the forward kolmogorov equation for the process x̃. theorem 3.5. let ξ > 0 be deterministic. suppose that g obeys (2.5), (2.6). let σ obey (2.1) and σ ∈ l2([0, ∞); r). let t ∗ be given by (2.3). suppose that x(·, ξ) is the unique strong solution of (2.9) with x(0, ξ) = ξ. let v be defined as in (3.2), and suppose that lim x→0+ v(x) = ∞, and let l(ξ) be the almost surely positive and fb(∞)–measurable random variable defined by (3.4). then p[l(ξ) ≤ x] = ∫ x 0 γ(t ∗; y) dy, x ≥ 0, where ∂γ ∂t (t; y) = 1 2 ∂2 ∂y2 ( g2(y)γ(t; y) ) , (t, y) ∈ [0, t ∗] × (0, ∞), and γ(0; y) = δξ(y), y ∈ r, where δξ is the δ-function. the result holds because l(ξ) = limt→t ∗+ x̃(t, ξ) = x̃(t ∗, ξ). moreover, as x̃ is a diffusion process with known infinitesimal generator and deterministic initial condition ξ, we can deduce its distribution function from the forward kolmogorov equation, and therefore, the distribution of l(ξ) is also known. it remains merely to classify the behaviour in the case when limx→0 v(x) < ∞. theorem 3.6. let ξ > 0 be deterministic. suppose that g obeys (2.5), (2.6). let σ obey (2.1). suppose that x(·, ξ) is the unique strong solution of (2.9) with x(0, ξ) = ξ. let v be defined as in (3.2), and suppose that lim x→0+ v(x) < ∞. then we have the following case distinction: (a) if σ ∈ l2((0, ∞); r), then there exists an almost surely positive and fb(∞)–measurable random variable l = l(ξ, ω) such that lim t→∞ x(t, ξ) = l(ξ) > 0, a.s. on {s̃0(ξ) ≥ t ∗}, cubo 10, 3 (2008) asymptotic constancy and stability in sdes 153 and lim t→s0(ξ) − x(t, ξ) = 0, a.s. on {s̃0(ξ) < t ∗}, where s̃0(ξ) defined by (3.1) and s0(ξ) is defined by (3.3). (b) if σ 6∈ l2((0, ∞); r), then lim t→s0(ξ) − x(t, ξ) = 0, a.s., where s0(ξ) defined by (3.3) obeys s0(ξ) < +∞, a.s. in case (a) the rate of convergence to the non–trivial random limit is the same as given in theorem 3.4, but only a.s. on the event {s̃0(ξ) > t ∗}. the probability of the event {s̃0(ξ) < t ∗} can be computed for the process x̃ obeying (2.8), by considering the limit p[s̃0(ξ) < t ∗] = lim a→0+ p[s̃a(ξ) < t ∗] where for ξ > a > 0 we define s̃a(ξ) = inf{t ≥ 0 : x̃(t, ξ) = a}. it is possible to compute the moment generating function of s̃a(ξ), λ 7→ e[e −λs̃a(ξ)] for λ ≥ 0, by solving an appropriate sturm– liouville problem, from which the probability p[s̃a(ξ) < t ∗] can in principle be determined by inverse transform methods. the interested reader can refer to [6, chapter 4.11] for further details on computation of the moment generating function. 3.1 an example a simple example of a process which can be analysed completely without appealing to these results (but which is consistent with them) is the unique strong solution of x(t) = ξ + ∫ t 0 σ(s)x(s) db(s), t ≥ 0, where ξ > 0 and σ ∈ c([0, ∞); r). this equation has explicit solution x(t, ξ) = ξ exp ( ∫ t 0 σ(s) db(s) − 1 2 ∫ t 0 σ2(s) ds ) , t ≥ 0. here we identify g(x) = x, x ≥ 0, and have limx→0+ v(x) = ∞. hence theorems 3.3, 3.4 and 3.5 can be applied to this stochastic differential equation. in the case when σ ∈ l2([0, ∞); r), the martingale convergence theorem (cf., e.g., [8, proposition iv.1.26]) ensures that limt→∞ ∫ t 0 σ(s) db(s) exists and is almost surely finite. therefore, there is an almost surely positive and almost surely finite fb(∞)—measurable random variable l(ξ) given by l(ξ) = ξ exp ( ∫ ∞ 0 σ(s) db(s) − 1 2 ∫ ∞ 0 σ2(s) ds ) 154 john a.d. appleby et al. cubo 10, 3 (2008) such that lim t→∞ x(t, ξ) = l(ξ) > 0, a.s. this chimes with part (a) of theorem 3.3. in the case when σ 6∈ l2([0, ∞); r), we have that lim t→∞ ∫ t 0 σ2(s) ds = +∞, lim t→∞ ∫ t 0 σ(s) db(s) ∫ t 0 σ2(s) ds = 0, a.s. the latter fact resulting from the strong law of large numbers for martingales (cf., e.g., [8, exercise v.i.16]). therefore lim t→∞ 1 ∫ t 0 σ2(s) ds log x(t) = − 1 2 , a.s. hence limt→∞ x(t, ξ) = 0, a.s., which agrees with part (b) of theorem 3.3. moreover, in the case when σ ∈ l2([0, ∞); r), l(ξ) is lognormally distributed; this is obvious by observation of the formula for l(ξ), but can also be confirmed by solving the partial differential equation for the transition density γ in theorem 3.5. in the case when σ ∈ l2([0, ∞); r), the rate of convergence in theorem 3.4 can be obtained, if it is shown that lim sup t→∞ ∫ ∞ t σ(s) db(s) √ 2 ∫ ∞ t σ2(s) ds log log (∫ ∞ t σ2(s) ds )−1 = 1, a.s., (3.7a) lim inf t→∞ ∫ ∞ t σ(s) db(s) √ 2 ∫ ∞ t σ2(s) ds log log (∫ ∞ t σ2(s) ds )−1 = −1, a.s. (3.7b) this can be established as follows: define t ∗ = ∫ ∞ 0 σ2(s) ds, and let m be the local martingale defined in (2.2), and with square variation 〈m〉. then, by the martingale time change theorem there exists a brownian motion b̃ such that m (t) = b̃(〈m〉(t)), 0 ≤ t < ∞. thus lim sup t→∞ ∫ ∞ t σ(s) db(s) √ 2 ∫ ∞ t σ2(s) ds log log (∫ ∞ t σ2(s) ds )−1 = lim sup t→∞ m (∞) − m (t) √ 2(〈m〉(∞) − 〈m〉(t)) log log (〈m〉(∞) − 〈m〉(t)) −1 = lim sup t→∞ b̃(〈m〉(∞)) − b̃(〈m〉(t)) √ 2(〈m〉(∞) − 〈m〉(t)) log log (〈m〉(∞) − 〈m〉(t)) −1 = lim sup s→t ∗− b̃(t ∗) − b̃(s) √ 2(t ∗ − s) log log(t ∗ − s)−1 = lim sup u→0+ b̃(t ∗) − b̃(t ∗ − u) √ 2u log log(u)−1 = 1, a.s., cubo 10, 3 (2008) asymptotic constancy and stability in sdes 155 since b̄ defined by b̄(t) = b̃(t ∗) − b̃(t ∗ − t), 0 ≤ t ≤ t ∗, is also a standard brownian motion, and therefore subject to the law of the iterated logarithm. (3.7) was stated in [2], but a proof was not supplied there. a variant of this result is proven in [1] but this proof contains an error as it is incorrectly stated there that ∫ ∞ t x(s) db(s) = ∫ 1/t 0 1 s x(1/s) dw (s) for a process x ∈ l2([0, ∞); r) a.s., where w is the standard brownian motion given by w (t) = tb(1/t) for t > 0 and w (0) = 0. 4 proofs 4.1 proof of theorem 3.3 by proposition 3.2, it follows that limx→0+ v(x) = ∞ implies s̃0(ξ) = +∞, a.s. in case (a), when σ ∈ l2([0, ∞); r), t (s) → ∞ as s → t ∗−. hence, as x̃(s) = x(t (s)) for s ∈ [0, t ∗], lim t→∞ x(t) = lim s→t ∗− x(t (s)) = lim s→t ∗− x̃(s) = x̃(t ∗) > 0, a.s., because t ∗ < +∞ = s̃0(ξ), a.s. in case (b), when σ 6∈ l2([0, ∞); r), t (s) → ∞ as s → ∞. hence, as x̃(s) = x(t (s)) for s ∈ [0, ∞), we have s0(ξ) = t (s̃0(ξ)) = +∞, a.s., and lim t→∞ x(t) = lim s→∞ x(t (s)) = lim s→∞ x̃(s) = lim s→s̃0(ξ) x̃(s) = 0, a.s., because s̃0(ξ) = ∞, a.s. 4.2 proof of theorem 3.4 the proof of part (ii) is straightforward, because for t ≥ τ we have x(t) = x(τ ) + ∫ t τ σ(s)g(x(s)) db(s) = x(τ ), as ∫ ∞ τ σ2(s) ds = 0 and the continuity of σ imply that σ(t) = 0 for all t ∈ [τ, ∞). to prove part (i) we proceed as follows. because σ ∈ l2([0, ∞); r), and limx→0+ v(x) = ∞, by proposition 3.1 and proposition 3.2, the process x̃ = {x(t); 0 ≤ t ≤ t ∗; g(t)} defined by x̃(t) = x(t (t)) is strictly positive a.s. and obeys x̃(t) = ξ + ∫ t 0 g(x̃(s)) db̃(s), 0 ≤ t ≤ t ∗, 156 john a.d. appleby et al. cubo 10, 3 (2008) where t ∗ and t are defined by (2.3) and (2.4). since g in c1((0, ∞); (0, ∞)) we may define the function h ∈ c2((0, ∞); r) by h(x) = ∫ x 1 1 g(u) du, x ∈ r. then the process ỹ = {ỹ (t); 0 ≤ t ≤ t ∗; g(t)} defined by ỹ (t) = h(x̃(t)) is well–defined. since h ∈ c2((0, ∞); r) and x̃(t) > 0 for all t ∈ [0, t ∗] a.s. ỹ is an itô–process, which by itô’s rule, has decomposition for 0 ≤ t ≤ t ∗ given by ỹ (t) = h(x̃(t)) = h(ξ) + b̃(t) − 1 2 ∫ t 0 g′(x̃(s)) ds. since x̃ is almost surely positive and continuous, g ∈ c1((0, ∞); (0, ∞)), it follows that lim t→t ∗− 1 t ∗ − t ∫ t ∗ t g′(x̃(s)) ds = g′(x̃(t ∗)), a.s. therefore lim t→t ∗− 1 √ 2(t ∗ − t) log log(1/(t ∗ − t)) ∫ t ∗ t g′(x̃(s)) ds = 0, a.s. then for t ∈ [0, t ∗] h(x̃(t ∗)) − h(x̃(t)) = b̃(t ∗) − b̃(t) − 1 2 ∫ t ∗ t g′(x̃(s)) ds, and lim sup t→t ∗− h(x̃(t ∗)) − h(x̃(t)) √ 2(t ∗ − t) log log(1/(t ∗ − t)) = lim sup t→t ∗− b̃(t ∗) − b̃(t) √ 2(t ∗ − t) log log(1/(t ∗ − t)) = lim sup s→0+ b̃(t ∗) − b̃(t ∗ − s) √ 2s log log(1/s) . now, because b̄ = {b̄(t) : 0 ≤ t ≤ t ∗; fb̄(t)} defined by b̄(t) = b̃(t ∗) − b̃(t ∗ − t) is a standard brownian motion, by the law of the iterated logarithm for brownian motion we have lim sup s→0∗+ b̃(t ∗) − b̃(t ∗ − s) √ 2s log log(1/s) = lim sup s→0∗+ b̄(s) √ 2s log log(1/s) = 1, a.s. hence lim sup t→t ∗− h(x̃(t ∗)) − h(x̃(t)) √ 2(t ∗ − t) log log(1/(t ∗ − t)) = 1, a.s. since h is in c1((0, ∞); r) and x̃ has continuous sample paths, we have lim t→t ∗− h(x̃(t ∗)) − h(x̃(t)) x̃(t ∗) − x̃(t) = h′(x̃(t ∗)) = 1 g(x̃(t ∗)) , a.s. cubo 10, 3 (2008) asymptotic constancy and stability in sdes 157 hence lim sup t→t ∗− x̃(t ∗) − x̃(t) √ 2(t ∗ − t) log log(1/(t ∗ − t)) = lim sup t→t ∗− x̃(t ∗) − x̃(t) h(x̃(t ∗)) − h(x̃(t)) h(x̃(t ∗)) − h(x̃(t)) √ 2(t ∗ − t) log log(1/(t ∗ − t)) = g(x̃(t ∗)), a.s. therefore, as t (s) → ∞ as s ↑ t ∗, and 〈m〉(t (s)) = s for s ∈ [0, t ∗], we have lim sup t→∞ l(ξ) − x(t) √ 2 ∫ ∞ t σ2(s) ds log log(1/ ∫ ∞ t σ2(s) ds) = lim sup t→∞ x(∞) − x(t) √ 2(〈m〉(∞) − 〈m〉(t)) log log(1/〈m〉(∞) − 〈m〉(t)) = lim sup s↑t ∗ x(t (t ∗)) − x(t (s)) √ 2(〈m〉(t (t ∗)) − 〈m〉(t (s))) log log(1/〈m〉(t (t ∗)) − 〈m〉(t (s))) = lim sup s↑t ∗ x̃(t ∗) − x̃(s) √ 2(t ∗ − s) log log(1/(t ∗ − s)) = g(x̃(t ∗)) = g(l(ξ)), a.s. an analogous argument gives lim inf t→t ∗− x̃(t ∗) − x̃(t) √ 2(t ∗ − t) log log(1/(t ∗ − t)) = −g(x̃(t ∗)), a.s., from which we can infer that lim inf t→∞ l(ξ) − x(t) √ 2 ∫ ∞ t σ2(s) ds log log(1/ ∫ ∞ t σ2(s) ds) = −g(l(ξ)), a.s., as required. 4.3 proof of theorem 3.6 by proposition 3.2, it follows that limx→0+ v(x) < +∞ implies s̃0(ξ) < +∞, a.s. in case (b) when σ 6∈ l2([0, ∞); r), t (s) → ∞ as s → ∞. hence, as x̃(s) = x(t (s)) for s ∈ [0, ∞), we have s0(ξ) = t (s̃0(ξ)) < +∞, a.s., and lim s→s0(ξ) − x(s) = lim t→s̃0(ξ) − x(t (t)) = lim t→s̃0(ξ) − x̃(t) = 0, a.s., because s̃0(ξ) < +∞, a.s. 158 john a.d. appleby et al. cubo 10, 3 (2008) in case (a) when σ ∈ l2([0, ∞); r), t (s) → ∞ as s → t ∗−. define the event a = {ω : s̃0(ξ) ≥ t ∗}. then because x̃(s) = x(t (s)) for s ∈ [0, t ∗], we have s0(ξ) = t (s̃0(ξ)), and so a = {ω : s̃0(ξ) ≥ t ∗} = {ω : s0(ξ) = +∞}. clearly, as s̃0(ξ) ≥ t ∗ on a, we have lim t→t ∗− x̃(t) = x̃(t ∗) > 0, a.s. on a. thus, as x̃(s) = x(t (s)) for s ∈ [0, t ∗], lim t→∞ x(t) = lim s→t ∗− x(t (s)) = lim s→t ∗− x̃(s) = x̃(t ∗) > 0, a.s. on a. hence { lim t→∞ x(t, ξ) = l(ξ) > 0, s0(ξ) = +∞ } = {s̃0(ξ) ≥ t ∗}, a.s. on the other hand, consider the event ā, where ā = {ω : s̃0(ξ) < t ∗} = {ω : s0(ξ) < +∞}, by virtue of the fact that s0(ξ) = t (s̃0(ξ)) < t (t ∗) = +∞. then lim t→s̃0(ξ) − x̃(t) = 0, a.s. on ā. thus, as x̃(s) = x(t (s)) for s ∈ [0, t ∗], lim t→s0(ξ) − x(t) = lim s→s̃0(ξ) − x(t (s)) = lim s→s̃0(ξ) − x̃(s) = 0, a.s. on ā. hence { lim t→s0(ξ) − x(t, ξ) = 0, s0(ξ) < +∞ } = {s̃0(ξ) < t ∗}, a.s. received: july 2008. revised: august 2008. references [1] j.a.d. appleby almost sure subexponential decay rates of scalar itô-volterra equations, electron. j. qual. theory differ. equ., proc. 7th coll. qtde, paper no.1, 1–32, 2004. [2] j.a.d. appleby and d. mackey, almost sure polynomial asymptotic stability of scalar stochastic differential equations with damped stochastic perturbations, electron. j. qual. theory differ. equ., proc. 7th coll. qtde, paper no.2, 1–33, 2004. cubo 10, 3 (2008) asymptotic constancy and stability in sdes 159 [3] j.a.d. appleby, x. mao and a. rodkina, pathwise superexponential decay rates of solutions of autonomous stochastic differential equations, stochastics, 77(3), 241–270, 2005. [4] j.a.d. appleby, a. rodkina and h. schurz, pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equation, disc. con. dynam. sys. ser. b., 6(4), 667–696, 2006. [5] n. ikeda and s. watanabe, stochastic differential equations and diffusion processes, north holland, new york, 2nd edition, 1981. [6] k. itô and h.p. mckean, diffusion processes and their sample paths, springer, berlin, 2nd edition, 1974. [7] i. karatzas and s.e. shreve, brownian motion and stochastic calculus, volume 113 of graduate texts in mathematics, springer, new york, 1991. [8] d. revuz and m. yor, continuous martingales and brownian motion. third edition, springer, new york, 1999. n13 cubo a mathematical journal vol.11, no¯ 04, (109–125). september 2009 a characterization of the product hardy space h1 luiz antonio pereira gomes departamento de matemática, universidade estadual de maringá, av. colombo 5790, 87020-000 maringa pr, brazil email: lapgomes@uem.br and eduardo brandani da silva departamento de matemática, universidade estadual de maringá, av. colombo 5790, 87020-000 maringa pr, brazil email: ebsilva@wnet.com.br abstract a characterization of the product space h1 such as the two parameters space h 1,2 0 is obtained, where h 1,2 0 is a particular case of spaces h p,q s , which are generalizations of spaces studied by j. peetre and h. triebel. resumen se obtiene una caracterización del espacio producto h1 como el espacio a dos parámetros h 1,2 0 , donde h 1,2 0 es un caso particular de los espacios h p,q s , los cuales son generalizaciones de los espacios estudiados por j. peetre y h. triebel. key words and phrases: product spaces, singular integral operators, vector-valued operators. math. subj. class.: 42b30, 46e40, 47b38. 110 gomes and da silva cubo 11, 4 (2009) 1 introduction recent advances in the theory of product hardy and bmo spaces (see [10], [11] and [18] for instance) have called the attention of many authors, which have achieve results about old and new problems of this rich area. one of these problems concern to the characterizations of hardy spaces. in a fundamental work within the theory of hardy spaces hp over product of semi planes (or with two parameters), r. f. gundy and e. m. stein [13] proved that two parameters space h1 may be characterized by double and partial hilbert transforms, using area integrals and maximal functions, with equivalent norms. after this initial work, several authors obtained other characterizations of the two parameters space h1. for more details see, for instance, [3], [4], [6], [7], [14], [19], [20] and [21]. in this work a characterization of the hardy space h1 over product of semi-planes such as the two parameters space h 1,2 0 is obtained. this space is a particular case of the two parameters spaces h p,q s (when s = (0, 0), p = (1, 1) and q = (2, 2)). the h p,q s spaces are generalizations of the one parameter spaces hp,qs , studied by j. peetre and h. triebel. for the one parameter case, a characterization of space h1, such as that obtained in this work, was initially obtained by j. peetre [15,16]. later, h. triebel obtained in [24] another proof by different arguments and after him, a new proof was achieved by j. l. rubio de francia, f. j. ruiz and j. l. torrea [17]. one of ingredients for the proof of the h1 characterization obtained in this work, consists of the theorems about singular integral vector operators contained in [12]. 2 spaces h1(ir × ir) and hp,q s (ir × ir) in the product case 2.1 notations. the notations and basic results used through this work are introduced here. the letter c always denotes a constant which may assume different values in a sequence of inequalities. s(ir2) denotes the class of rapidly decreasing functions (at infinity). let e be a banach space. s′(ir2,e) is the class of all continuous linear maps t defined over s(ir2) with values in e (that is, if φj → φ in s(ir2) then t(φj) → t(φ)). if e is a banach space in relation to the norm ||.||e and p = (p1,p2) with 0 < p1,p2 ≤ ∞, l p (ir 2 ,e) is the space of all functions f defined over ir2 with values in e, such that ‖f(x)‖e is lebesgue measurable, and ‖f‖lp (ir2,e) = (∫ ir (∫ ir ‖f(x)‖p1e dx1 )p2/p1 dx2 )1/p2 < ∞ with usual modifications when some of pi are equal to ∞. we observe that if pi = p, for i = 1, 2, then lp (ir2,e) = lp(ir2,e). cubo 11, 4 (2009) a characterization of the product hardy space h1 111 to avoid any confusion, we write lp (e) and ‖.‖lp (e) instead of lp (ir2,e) and ‖.‖lp (ir2,e), and when e = ic, the field of complex numbers, lp and ‖.‖lp are posited. given a banach space e and q = (q1,q2) with 0 < q1,q2 ≤ ∞, the (multi-)sequence spaces ℓ q (zz 2 ,e) (ℓq(e) to avoid confusion) are defined in a analogous way. if e is a banach space, the fourier transform of a function f ∈ l1(ir2,e) is defined by ff(x) = f̂(x) = ∫ ∫ ir2 e −2πix·y f(y) dy , where x · y = x1.y1 + x2.y2. the following notation is used, � = {(0, 0), (1, 0), (0, 1), (1, 1)}. 2.2 definition. let e be a hilbert space and f ∈ l1(ir2,e). their hilbert transforms hkf, k ∈ �, are the elements of s′(ir2,e) defined by : (1) f(h10f) = −i sgx f(f)(x,y), (2) f(h01f) = −i sgy f(f)(x,y), (3) f(h11f) = (−i sgx)(−i sgy)f(f)(x,y), (4) (h00f) = f. spaces h1(ir×ir,e) and bmo(ir×ir,e) are defined, which generalize the product spaces h 1 (ir × ir) and bmo(ir × ir) for the vectorial case. 2.3 definition. let e be a hilbert space. h1(ir × ir,e) is the vector space of function f in l1(ir2,e) such that their hilbert transforms, hkf, k ∈ � \ {(0, 0)}, belong to l1(ir2,e). we equipped space h1(ir × ir,e) with the norm: ‖f‖h1(ir×ir,e) = ∑ k∈� ‖hkf‖l1(ir2,e) , where h00f = f. 2.4 definition. let e be a hilbert space. a function g from ir2 to e belongs to bmo(ir× ir,e), if it may be represented as g = ∑ k∈� hkgk , (1) 112 gomes and da silva cubo 11, 4 (2009) where h00g00 = g00 and ∑ k∈� ||gk||l∞(ir2,e) < ∞. we equipped the space bmo(ir×ir,e) with the norm: ||g||bmo(ir×ir,e) = inf{ ∑ k∈� ||gk||l∞(ir2,e)} , where the infimum takes over all representations of g in the form (1). chang-fefferman proved in [6] that for real value functions, the product space bmo(ir×ir) is the dual of the product space h1(ir×ir). this result is valid also for spaces bmo(ir×ir,e), where e is a hilbert space; therefore, the product space bmo(ir×ir,e) is the dual of the product space h1(ir × ir,e). results on the action of singular vector integral operators with product kernel over the product spaces h1(ir × ir,e) and bmo(ir × ir,e) are given by the two following theorems. proofs are provided in gomes-silva [12]. 2.5 theorem. let e, f and g be banach spaces and k1 and k2 kernels in l 2 loc(ir 2 ,l(e,f)) and l2loc(ir 2 ,l(f,g)), respectively, satisfying ∫ |x−y′|>γ|y−y′| ‖kj(x,y) − kj(x,y′)‖lj dx ≤ c · γ−δ , j = 1, 2, (1) for every γ ≥ 2 and some δ > 0, where l1 = l(e,f) and l2 = l(f,g). let t1 and t2 be bounded linear operators from l2(ir,e) to l2(ir,f) and from l2(ir,f) to l2(ir,g), respectively, satisfying t1f(x) = ∫ ir k1(x,u) f(u) du , (2) for every f ∈ l2c(ir,e), and t2f(y) = ∫ ir k2(y,v) f(v) dv , (3) for every f ∈ l2c(ir,f). let t be a linear operator from l2c(ir2,e) to m(ir2,g) satisfying tf(x,y) = ∫ ∫ ir2 k2(y,v) k1(x,u) f(u,v) du dv , (4) for every f ∈ l2c(ir2,e) and (x,y) /∈ sup f. suppose that t has a bounded extension from l 2 (ir 2 ,e) to l2(ir2,g). then, t has a bounded extension from h1(ir × ir,e) to l1(ir2,g); that is, there exists a constant c > 0, such that ‖tf‖l1(ir2,g) ≤ c ‖f‖h1(ir×ir,e) , cubo 11, 4 (2009) a characterization of the product hardy space h1 113 for all f ∈ h1(ir × ir,e). 2.6 theorem. let e be a banach space, f and g hilbert spaces and k1 and k2 kernels in l 1 loc(ir 2 ,l(e,f)) and l1loc(ir 2 ,l(f,g)), respectively, satisfying ∫ |x′−y|>γ|x−x′| ‖kj(x,y) − kj(x′,y)‖lj dx ≤ c · γ−δ , j = 1, 2, (1) for every γ ≥ 2 and some δ > 0, where l1 = l(e,f) and l2 = l(f,g). let t1 and t2 be bounded linear operators from l2(ir,e) to l2(ir,f) and from l2(ir,f) to l2(ir,g), respectively, satisfying t1f(x) = ∫ ir k1(x,u) f(u) du , (2) for every f ∈ l∞c (ir,e), and t2f(y) = ∫ ir k2(y,v) f(v) dv , (3) for every f ∈ l∞c (ir,f). let t be a linear operator from l∞c (ir2,e) to m(ir2,g) satisfying tf(x,y) = ∫ ∫ ir2 k2(y,v) k1(x,u) f(u,v) du dv , (4) for every f ∈ l∞c (ir2,e) and (x,y) /∈ sup f. suppose that t has a bounded extension from l 2 (ir 2 ,e) to l2(ir2,g). then, t is a bounded linear operator from l∞c (ir 2 ,e) to bmo(ir × ir,g); that is, there exists a constant c > 0, such that ‖tf‖bmo(ir×ir,g) ≤ c ‖f‖l∞(ir2,e) , for all f ∈ l∞c (ir2,e). 2.7 lemma. there exists ϕ ∈ s(ir), such that (1) sup fϕ = {t ∈ ir : 2−1 ≤ |t| ≤ 2} ; (2) |fϕ(t)| > 0 se 2−1 < |t| < 2 ; (3) ∑∞ i=−∞ fϕ(2−it) = 1 se t 6= 0 . for the proof see berg-löfströn [2] 114 gomes and da silva cubo 11, 4 (2009) 2.8 system of test functions. let ϕ be given as in the lemma 2.7 and for each i ∈ zz let ϕi be the function given by ϕi(t) = 2 i ϕ(2 i t). the family (ϕi)i∈zz is called a system of test functions over ir. since fϕi(t) = fϕ(2−it) for each i ∈ zz, and from 2.7(1), 2.7(2) and 2.7(3), it follows that (1) sup fϕi = {t ∈ ir : 2i−1 ≤ |t| ≤ 2i+1} ; i ∈ zz ; (2) |fϕi(t)| > 0 se 2i−1 < |t| < 2i+1 ; (3) ∑∞ i=−∞ fϕi(t) = 1 se t 6= 0 . 2.9 definition. let s = (s1,s2), p = (p1,p2) and q = (q1,q2), such that sn ∈ ir, 0 < pn < ∞ and 0 < qn ≤ ∞, n = 1, 2. let (ϕi)i∈zz and (ψj)j∈zz be systems of test functions as in 2.8. then, h p,q s (ir × ir) = h p,q s (ir × ir,ϕ,ψ) is the vector space of all functions f in lp (ir2) ∩ s′(ir2) with real values, satisfying (2s1i+s2jϕiψj ∗ f)ij ∈ lp (ℓq). spaces h p,q s (ir×ir) are equipped with the following quasi-norm (it is a norm if min (p1,p2,q1,q2) ≥ 1) : ‖f‖ϕ,ψ h p,q s = ‖(2s1i+s2jϕiψj ∗ f)ij‖lp (ℓq) . (1) to avoid any confusion, we simply denote ‖f‖ϕ,ψ h p,q s by ‖f‖ h p,q s . when s = (s,s), p = (p,p) and q = (q,q), then ‖f‖ϕ,ψ h p,q s = ‖(2s(i+j)ϕiψj ∗ f)ij‖lp(ℓq ) and the space h p,q s (ir × ir) is simply denoted by hp,qs (ir × ir). the next result shows that the quasi-norm 2.9(1) is independent of the systems of test functions (ϕi)i∈zz and (ψj)j∈zz. 2.10 theorem. let (αi)i∈zz, (βj)j∈zz, (ϕk)k∈zz and (ψl)l∈zz be systems of test functions as in 2.8. let s, p and q, as in definition 2.9. then the quasi-norms ‖.‖α,β h p,q s and ‖.‖ϕ,ψ h p,q s are equivalents, that is, there are positive constants c1 and c2, such that c1 · ‖f‖α,β h p,q s ≤ ‖f‖ϕ,ψ h p,q s ≤ c2 · ‖f‖α,β h p,q s . (1) for the proof see schmeisser-triebel [22]. cubo 11, 4 (2009) a characterization of the product hardy space h1 115 2.11 remark. from the proof of theorem 2.10 it follows that condition 2.8(3) of the systems (αk)k and (βl)l is unnecessary, that is, ∞∑ k=−∞ fαk(t) = ∞∑ l=−∞ fβl(t) = 1 , t 6= 0 , to obtain inequalities of the type ‖f‖α,β h p,q s ≤ c · ‖f‖ϕ,ψ h p,q s . from systems (αk)k and (βl)l another kind of condition may be demanded, such as, ∞∑ k=−∞ [fαk(t)]2 = ∞∑ l=−∞ [fβl(t)]2 = 1 , t 6= 0 . this will be considered in the next section. 3 the characterization h1(ir × ir) = h1,2 0 (ir × ir) 3.1 lemma. let ϕ ∈ s(ir) such that ϕ̂(0) = 0 and |ϕ̂(t)| > 0 if 2−1 < |t| < 2. defining ϕj(x) = 2 j ϕ(2 j x), j ∈ zz, one has (1) ∑ j∈zz |ϕ̂j(t)|2 ≤ c ; (2) ∑ j∈zz |ϕj(x)|2 ≤ c · |x|−2; (3) ( ∑ j∈zz |ϕj(x − y) − ϕj(x)|2) 1 2 ≤ c · |y| |x|2 , if |x| > 2|y|. for the proof see torrea [23]. 3.2 theorem. let ϕ and ψ be as in the lemma 3.1. then ‖(ϕiψj ∗ f)ij‖l1(ir2,ℓ2) ≤ c · ‖f‖h1(ir×ir) (1) for all f ∈ h1(ir × ir). proof. let us consider the linear operator defined on l2c(ir 2 ) by tf = (ϕiψj ∗ f)ij ∈ m(ir2,ℓ2(zz2)). 116 gomes and da silva cubo 11, 4 (2009) the operator t is well defined: if f ∈ l2c(ir2), then, by plancherel’s theorem and by 3.1(1), ∫ ∫ ir2 ∑ j∈zz ∑ i∈zz |ϕiψj ∗ f(x,y)|2dxdy = = ∑ j∈zz ∑ i∈zz ∫ ∫ ir2 |ϕiψj ∗ f(x,y)|2dxdy = ∑ j∈zz ∑ i∈zz ∫ ∫ ir2 |ϕ̂i(s)ψ̂j(t)f̂(s,t)|2dsdt = ∫ ∫ ir2 ( ∑ i∈zz |ϕ̂i(s)|2)( ∑ j∈zz |ψ̂j(t)|2)|f̂(s,t)|2dsdt ≤ c · ∫ ∫ ir2 |f̂(s,t)|2dsdt = c · ‖f‖l2(ir2) , (2) thus, it follows ∑ j∈zz ∑ i∈zz |ϕiψj ∗ f(x,y)|2 < ∞ for almost all (x,y); that is, tf(x,y) ∈ ℓ2(zz2). to show that tf is a measurable function, it is enough to verify that the map (x,y) −→ tf(x,y).α is measurable for all α ∈ ℓ2(zz2), since ℓ2(zz2) is separable. if α = (αij)ij tf(x,y).α = ∑ j∈zz ∑ i∈zz (ϕiψj ∗ f(x,y))αij = ∑ j∈zz ∑ i∈zz αijϕiψj ∗ f(x,y) , which is measurable. the inequality 3.2(2) shows that t is a bounded operator from l2(ir2) to l2(ir2,ℓ2(zz2)). for each n ∈ in, let us consider the operators tn, tn 1 and tn 2 defined in the following way: t n is defined on l2c(ir 2 ) by t n f = (ϕiψj ∗ f; −n ≤ i,j ≤ n) ∈ m(ir2,ℓ2(zz2)) ; t n 1 is defined on l2(ir) by t n 1 f = (ϕi ∗ f; −n ≤ i ≤ n) ∈ m(ir,ℓ2(zz)) ; cubo 11, 4 (2009) a characterization of the product hardy space h1 117 t n 2 is defined on l2c(ir,ℓ 2 (zz)) by t n 2 g = (ψj ∗ gi; −n ≤ i,j ≤ n) ∈ m(ir,ℓ2(zz2)) . our next step it will be to show that these operators satisfy the hypothesis of theorem 2.5. analogously for operator t , it is easy to verify that for each n ∈ in, tn is well defined and tnf is a measurable function. moreover, from 3.2(2) it follows that operators tn are all bounded from l 2 (ir 2 ) to l2(ir2,ℓ2(zz2)), with ‖tn‖ bounded by a constant regardless of n. the operators tn 1 are bounded from l2(ir) to l2(ir,ℓ2(zz)) with ‖tn 1 ‖ bounded by a constant regardless n, since by the plancherel’s theorem and 3.1(1), ∫ ir n∑ i=−n |ϕi ∗ f(x)|2dx = n∑ i=−n ∫ ir |ϕ̂i(s)|2|f̂(s)|2ds ≤ c · ∫ ir |f̂(s)|2ds = c · ‖f‖l2(ir). now, for each n ∈ in, the kernel kn 1 defined by k n 1 (x) : λ ∈ ic −→ kn 1 (x).λ = (ϕi(x)λ; −n ≤ i ≤ n) ∈ ℓ2(zz) is well defined, belongs to l2loc(ir,l(ic,ℓ 2 (zz))), verifies the condition 2.5(1) with l1 = l(ic,ℓ 2 (zz)) and for all f ∈ l2c(ir), t n 1 f(x) = ∫ ir k n 1 (x − y)f(y)dy. (3) indeed, kn 1 is well defined: if ϕ ∈ s(ir), then ‖kn 1 (x).λ‖ℓ2(zz) = ( n∑ i=−n |ϕi(x)|2) 1 2 |λ| ≤ c(n)|λ| (4) for all λ ∈ ic and all x ∈ ir. on the other hand, since l(ic,ℓ2(zz)) is isometric in ℓ2(zz), and the map x ∈ ir −→ n∑ i=−n αiϕi(x) is measurable for all α = (αi)i ∈ ℓ2(zz), it follows that kn1 is measurable. now, if a ⊂ ir is a compact set, then by 3.2(4) 118 gomes and da silva cubo 11, 4 (2009) ∫ a ‖kn 1 (x)‖2l1dx ≤ c(n)|a| < ∞ , where l1 = l(ic,ℓ 2 (zz)). this shows that kn 1 belongs to l2loc(ir,l1). to prove 3.2(3), since the map u ∈ ir −→ (ϕi(u)f(u); −n ≤ i ≤ n) ∈ ℓ2(zz) is integrable, we have t n 1 f(x) = (ϕi ∗ f(x); −n ≤ i ≤ n) = ( ∫ ir ϕi(x − u)f(u)du; −n ≤ i ≤ n) = ∫ ir (ϕi(x − u)f(u); −n ≤ i ≤ n)du = ∫ ir k n 1 (x − u)f(u)du. finally, if |x − u′| > γ|y − u′|, with γ ≥ 2, then by 3.1(3) we obtain ‖kn 1 (x − u) − kn 1 (x − u′)‖l1 = ( n∑ i=−n |ϕi(x − u) − ϕi(x − u′)|2) 1 2 = ( n∑ i=−n |ϕi(x − u′ − (u − u′)) − ϕi(x − u′)|2) 1 2 ≤ c · |u − u ′| |x − u|2 , where c is a constant regardless of n. therefore, the kernel kn 1 verifies the condition 2.5(1) with l1 = l(ic,ℓ 2 (zz)) and constant c regardless of n. the boundness of the operators tn 2 from l2(ir,ℓ2(zz)) to l2(ir,ℓ2(zz2)), with ‖tn 2 ‖ bounded by a constant regardless of n, follows from 3.1(1) using an analogous reasoning which was done for tn 1 . now, let us verify that for each n ∈ in, there exists a kernel kn 2 in l2loc(ir,l(ℓ 2 (zz),ℓ 2 (zz 2 ))), satisfying 2.5(1) with l2 = l(ℓ 2 (zz),ℓ 2 (zz 2 )), such that t n 2 g(y) = ∫ ir k n 2 (y − v)g(v)dv, (5) for all g ∈ l2c(ir,ℓ2(zz)). indeed, let kn2 be defined by k n 2 (y) : α = (αi)i ∈ ℓ2(zz) −→ kn2 (y).α = (ψj(y).αi; −n ≤ i,j ≤ n) ∈ ℓ2(zz2). cubo 11, 4 (2009) a characterization of the product hardy space h1 119 this function is well defined: if ψ ∈ s(ir), it follows that ‖kn 2 (y).α‖ℓ2(zz2) = ( n∑ j=−n n∑ i=−n |ψj(y)αi|2) 1 2 (6) ≤ ( n∑ j=−n |ψj(y)|2) 1 2 ( ∑ i∈zz |αi|2) 1 2 ≤ c(n).‖α‖ℓ2(zz) for all α = (αi)i ∈ ℓ2(zz) and all y ∈ ir. the measurability of kn2 follows from kn2 = ∑n i,j=−n k n 2,ij, where each kn 2,ij is defined by k n 2,ij(y).α = (....., 0,ψj(y)αi, 0, .....) and it is measurable, since each ψj is measurable. the fact that ‖kn2 (y)‖2l2 is locally integrable, where l2 = l(ℓ 2 (zz),ℓ 2 (zz 2 )), follows from 3.2(6). the verification of 3.2(5) is analougous to 3.2(3). as in case of the kernel kn 1 , it follows from 3.1(3) that ‖kn 2 (y − v) − kn 2 (y − v′)‖l2 ≤ c · |v − v′| |y − v′|2 where c is a constant regardless of n. then, get kn 2 satisfies 2.5(1) with constant regardless of n. to complete the proof that operators tn, tn 1 and tn 2 satisfy the hypothesis of theorem 2.5, we observe that the map (u,v) ∈ ir2 −→ (ϕi(u)ψj(v)f(u,v); −n ≤ i,j ≤ n) ∈ ℓ2(zz2) is integrable when f ∈ l2c(ir2); then we have t n f(x,y) = ∫ ∫ ir2 k n 2 (y − v)kn 1 (x − u)f(u,v)dudv for all f ∈ l2c(ir2). therefore, by theorem 2.5, ‖tnf‖l1(ir2,ℓ2(zz2)) ≤ c · ‖f‖h1(ir×ir) (7) for all n and all f ∈ h1(ir × ir), where c is a constant regardless of n. finally, applying the theorem of monotone convergence in 3.2(7), 3.2(1) is obtained as requested. as a consequence of theorem 3.2, the following result gives us a part of the characterization h 1 (ir × ir) = h1,2 0 (ir × ir). 120 gomes and da silva cubo 11, 4 (2009) 3.3 corollary. the space h1(ir × ir) is continuously embedded in h1,2 0 (ir × ir), that is, there is a positive constant c, such that ‖f‖ h 1,2 0 (ir×ir) ≤ c · ‖f‖h1(ir×ir) for all f ∈ h1(ir × ir). proof. it is enough to observe the test functions used to define the space h 1,2 0 (ir×ir) satisfies the hypothesis of the theorem 3.2. the next theorem will be fundamental to prove the contrary immersion in the corollary 3.3; that is, the space h 1,2 0 (ir × ir) is continuously embedded in the space h1(ir × ir). 3.4 theorem. let ϕ and ψ be given as in the lemma 3.1. then ‖(ϕiψj ∗ f)ij‖bmo(ir×ir,ℓ2) ≤ c · ‖f‖l∞(ir2) , for all f ∈ l∞c (ir2), where bmo(ir×ir,ℓ2) is the topological dual of the space h1(ir×ir,ℓ2). proof. it is enough to follow the proof of theorem 3.2, using in this case theorem 2.6. 3.5 theorem. let o be the space of the real functions f ∈ s(ir2) with real values, such that (1) f̂ ∈ c∞c (ir2), (2) sup f̂ ∩ [(ir × {0}) ∪ ({0} × ir)] = ∅. then o is a dense subspace of h1,2 0 (ir × ir). proof. it is enough to adapt the arguments used by h. sato in [19] to obtain a dense subspace of h1(ir × ir). 3.6 theorem. a function f in l1(ir2) belongs to h1(ir × ir) if, and only if f belongs to h 1,2 0 (ir × ir). moreover, there is a constant c > 0, such that c −1 .‖f‖h1(ir×ir) ≤ ‖f‖h1,20 (ir×ir) ≤ c · ‖f‖h1(ir×ir). (1) cubo 11, 4 (2009) a characterization of the product hardy space h1 121 proof. it is enough to prove the first inequality in 3.6(1), since the second was proved in corollary 3.3. let f ∈ o and g ∈ l∞c (ir2) such that ‖g‖l∞(ir2) ≤ 1. let α = (αi)i∈zz e β = (βj)j∈zz systems of test functions as given in 2.8, but with the condition 2.8(3) replaced by∑∞ i=−∞[α̂i(s)] 2 = 1 , s 6= 0 and ∑∞ j=−∞[β̂j(t)] 2 = 1, t 6= 0 (see remark 2.11). thus, using the polarization formula and plancherel’s theorem, ∫ ∫ ir2 f(x,y)g(x,y)dxdy = (2) = 1 4 [ ∫ ∫ ir2 |f + g|2dxdy − ∫ ∫ ir2 |f − g|2dxdy] = 1 4 [ ∫ ∫ ir2 ∞∑ i=−∞ [α̂i(s)] 2 ∞∑ j=−∞ [β̂j(t)] 2|f(f + g)|2dsdt − − ∫ ∫ ir2 ∞∑ i=−∞ [α̂i(s)] 2 ∞∑ j=−∞ [β̂j(t)] 2|f(f − g)|2dsdt] = 1 4 [ ∞∑ j=−∞ ∞∑ i=−∞ ∫ ∫ ir2 |f(αiβj ∗ (f + g))|2dsdt − − ∞∑ j=−∞ ∞∑ i=−∞ ∫ ∫ ir2 |f(αiβj ∗ (f − g))|2dsdt] = ∫ ∫ ir2 1 4 [ ∞∑ j=−∞ ∞∑ i=−∞ (|αiβj ∗ (f + g)|2 − |αiβj ∗ (f − g)|2)]dxdy = ∫ ∫ ir2 ∞∑ j=−∞ ∞∑ i=−∞ (αiβj ∗ f)(αiβj ∗ g)dxdy. now, by theorem 3.4, ‖(αiβj ∗ g)ij‖bmo(ir×ir,ℓ2) ≤ c · ‖g‖l∞(ir2) , (3) for all g ∈ l∞c (ir2). this shows that (αiβj ∗ g)ij ∈ bmo(ir × ir,ℓ2(zz2)). on the other hand, if we denote by h the hilbert transform in one variable and with the convention h0ϕ = ϕ and h1ϕ = hϕ, then for each k = (l,m) ∈ �, we have hk(αiβj ∗ f)(x,y) = (hlαi hmβj ∗ f)(x,y) . (4) it is enough to prove to k = (1, 0), since the another cases are similar. indeed, by definition 2.2, f[h10(αiβj ∗ f)](s,t) = −i sgs f(αiβj ∗ f)(s,t) = −i sgs α̂i(s)β̂j(t)f̂(s,t) = f(hαi)(s) β̂j(t) f̂(s,t) = f[hαi.βj ∗ f](s,t) , 122 gomes and da silva cubo 11, 4 (2009) and 3.6(4) is obtained to k = (1, 0). moreover, the sequences (hαi)i∈zz and (hβj)j∈zz are systems of test functions satisfying 2.8(1) and 2.8(2), since f[hαi](s) = −i sgs α̂i(s) and f[hβj](t) = −i sgt β̂j(t). thus, taking into account the remark 2.11, theorem 2.10 is applied to obtain ‖(αiβj ∗ f)ij‖h1(ir×ir,ℓ2) = ∑ k∈� ‖(hk(αiβj ∗ f))ij‖l1(ir2,ℓ2) (5) = ∑ (l,m)∈� ‖(hlαi hmβj ∗ f)ij‖l1(ir2,ℓ2) ≤ c · ‖f‖ h 1,2 0 (ir×ir) . this shows that (αiβj ∗ f)ij ∈ h1(ir × ir,ℓ2(zz2)). using 3.6(2), 3.6(3) and 3.6(5) and using the fact that bmo(ir × ir,ℓ2(zz2)) is the dual of h1(ir × ir,ℓ2(zz2)), | ∫ ∫ ir2 f.g dxdy| ≤ c · ‖(αiβj ∗ f)ij‖h1(ir×ir,ℓ2) ‖(αiβj ∗ g)ij‖bmo(ir×ir,ℓ2) ≤ c · ‖f‖ h 1,2 0 (ir×ir) . taking the supremum over all functions g in l∞(ir2), such that ‖g‖l∞(ir2) ≤ 1, ‖f‖l1(ir2) ≤ c · ‖f‖h1,20 (ir×ir) (6) for all f ∈ o. if f belongs to o, then hkf belongs too, for each k = (l,m) ∈ �. therefore, 3.6(6) and theorem 2.10 implies ‖f‖h1(ir×ir) = ∑ k∈� ‖hkf‖l1(ir2) (7) ≤ c · ∑ k∈� ‖hkf‖h1,20 (ir×ir) = c · ∑ k∈� ‖(ϕiψj ∗ hkf)ij‖l1(ir2,ℓ2) = c · ∑ (l,m)∈� ‖(hlϕi hmψj ∗ f)ij‖l1(ir2,ℓ2) ≤ c · ‖(ϕiψj ∗ f)ij‖l1(ir2,ℓ2) = c · ‖f‖ h 1,2 0 (ir×ir) , for all f ∈ o. finally, we may prove the inequality 3.6(7) is true for all f ∈ h1,2 0 (ir×ir). let f ∈ h1,2 0 (ir×ir). by theorem 3.5, o is dense in h1,2 0 (ir×ir); then there exists a sequence (fn)n of elements of o such that fn → f in the norm of h1,20 (ir × ir), from which it follows (fn)n is a cubo 11, 4 (2009) a characterization of the product hardy space h1 123 cauchy sequence in h 1,2 0 (ir × ir). from 3.6(7) (fn)n is a cauchy sequence in h1(ir × ir), from which results there exists an element g ∈ h1(ir×ir) such that fn → g in the norm of h1(ir×ir), since h1(ir × ir) is a complete space. by corollary 3.3, h1(ir × ir) is continuously embedded in h 1,2 0 (ir × ir); then fn → g in the norm of h1,20 (ir × ir) and hence g = f. thus, for all ε > 0, there is n ∈ in, such that ‖fn −f‖h1(ir×ir) < ε and ‖fn −f‖h1,20 (ir×ir) < ε. therefore, by 3.6(7), ‖f‖h1(ir×ir) ≤ ‖f − fn‖h1(ir×ir) + ‖fn‖h1(ir×ir) < ε + c · ‖fn − f‖h1,20 (ir×ir) + c · ‖f‖h1,20 (ir×ir) < (c + 1)ε + c · ‖f‖ h 1,2 0 (ir×ir) , for all ε > 0 and f ∈ h1,2 0 (ir × ir). this implies that 3.6(7) is true for all f ∈ h1,2 0 (ir × ir). proof is complete. acknowledgment. the authors would like to thank professor dicesar lass fernandez of campinas state university for many valuable discussions and insights in this research. received: april 2008. revised: october 2008. references [1] a. benedek and r. panzone, the spaces lp with mixed norm, duke math. j. 28 (1961), 301–324. 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[25] h. triebel, theory of function spaces, birkhauser verlag, basel, boston and stuttgart, 1983. articulo 8 c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2605 .dvi cubo a mathematical journal vol.13, no¯ 01, (61–71). march 2011 q− fractional inequalities george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a., email: ganastss@gmail.com abstract here we give q−fractional poincaré type, sobolev type and hilbert-pachpatte type integral inequalities, involving q−fractional derivatives of functions. we give also their generalized versions. resumen estudiamos el tipo q−fraccional poincaré, el tipo sobolev y el tipo integral de inecuaciones de hilbert-pachpatte, involucrando a q−fraccional derivados de funciones. damos también las versiones generalizadas. keywords: q−fractional derivative, q−fractional integral, q−fractional poincaré inequality, q− fractional sobolev inequality, q−fractional hilbert-pachpatte inequality. ams subject classification: 26a24, 26a33, 26a39, 26d10, 26d15, 33d05, 33d60, 81p99. 1 introduction here we follow [4] in all of this section, see also [3]. 62 george a. anastassiou cubo 13, 1 (2011) let q ∈ (0, 1), we define [α] q := 1 − qα 1 − q , (α ∈ r) . (1) the q−analog of the pochhammer symbol (q−shifted factorial) is defined by: (a; q)0 = 1, (a; q)k = k−1 ∏ i=0 ( 1 − aqi ) (k ∈ n ∪ {∞}) . the expansion to reals is (a; q) α = (a; q)∞ (aqα; q)∞ (α ∈ r) ; (2) also define the q−analog (a − b) (α) = aα ( b a ; q ) ∞ ( qα b a ; q ) ∞ , a, b ∈ r, a 6= 0. notice that (a − b) (α) = aα ( b a ; q ) α . the q−gamma function is defined by γq (x) = (q; q)∞ (qx; q)∞ (1 − q) 1−x , (x ∈ r − {0, −1, −2, ...}) . (3) clearly γq (x + 1) = [x]q γq (x) . (4) the q−derivative of a function f (x) is defined by (dqf ) (x) = f (x) − f (qx) x − qx , (x 6= 0) , (5) (dqf ) (0) = lim x→0 (dqf ) (x) , (6) and the q−derivatives of higher order: d 0 q f = f, d n q f = dq ( d n−1 q f ) , n = 1, 2, 3, ... (7) the q−integral is defined by (iq,0f ) (x) = ∫ x 0 f (t) dqt = x (1 − q) ∞ ∑ k=0 f ( xq k ) q k, (0 < q < 1) , (8) and (iq,af ) (x) = ∫ x a f (t) dqt = ∫ x 0 f (t) dqt − ∫ a 0 f (t) dqt. (9) cubo 13, 1 (2011) q− fractional inequalities 63 by [2], we see that: if f (x) ≥ 0, then it is not necessarily true that ∫ b a f (x) dqx ≥ 0. in the case of a = xqn, then (9) becomes ∫ x xqn f (t) dqt = x (1 − q) n−1 ∑ k=0 f ( xq k ) q k , (10) see also [2]. double q−integration is defined the usual iterative way. also we define i 0 q,af = f, i n q,af = iq,a ( i n−1 q,a f ) , n = 1, 2, 3, ... (11) the following are valid: (dqiq,af ) (x) = f (x) , (12) (iq,adqf ) (x) = f (x) − f (a) . (13) denote [n] q ! = [1] q [2] q ... [n] q , n ∈ n; [0] q ! = 1, [ n k ] q = [n] q ! [k] q ! [n − k] q ! . in the next we work on (0, b), b > 0, and let a ∈ (0, b). also the required q−derivatives and q−integrals do exist. definition 1. the fractional q−integral is ( i α q,af ) (x) = xα−1 γq (α) ∫ x a ( q t x ; q ) α−1 f (t) dqt (14) = 1 γq (α) ∫ x a (x − qt) (α−1) f (t) dqt, ( a < x, α ∈ r+ ) . the usual fractional integral (see also [1]) is the limit case of (14) as q ↑ 1, since lim q↑1 x α−1 ( q t x ; q ) α−1 = (x − t) α−1 . (15) clearly ( i α q,af ) (a) = 0. (16) we mention 64 george a. anastassiou cubo 13, 1 (2011) theorem 2. let α, β ∈ r+. the q−fractional integration has the semigroup property ( i β q,ai α q,af ) (x) = ( i α+β q,a f ) (x) , (a < x) . (17) corollary 3. for α ≥ n (n ∈ n) it holds ( d n q i α q,af ) (x) = ( i α−n q,a f ) (x) , (a < x) . (18) we mention the fractional q−derivative of caputo type: definition 4. the fractional q−derivative of caputo type is ( ∗d α q,af ) (x) =    ( i−αq,a f ) (x) , α ≤ 0; ( i ⌈α⌉−α q,a d ⌈α⌉ q f (x) ) , α > 0, (19) where ⌈.⌉ denotes the ceiling of the number. next we mention the highlight of this introductory section. again all here come from [4]. so the following is the fractional q−taylor formula of caputo type. theorem 5. let α ∈ r+ − n, a < x. then ( i α q,a ∗d α q,af ) (x) = f (x) − ⌈α⌉−1 ∑ k=0 ( dkq f ) (a) [k] q ! x k ( a x ; q ) k . (20) also we give theorem 6. let α ∈ r+ − n, β ∈ r+, α > β > 0, a < x. then ( i β q,a ∗d α q,af ) (x) = ( ∗d α−β q,a f ) (x) − (21) ⌈α⌉−1 ∑ k=⌈α−β⌉ ( dkq f ) (a) γq (k − α + β + 1) x k−α+β ( a x ; q ) k−α+β . 2 main results we need the following q−hölder’s inequality. proposition 7. let x > 0, 0 < q < 1; p1, q1 > 1 such that 1 p1 + 1 q1 = 1; n ∈ n. then ∫ x xqn |f (t)| |g (t)| dqt ≤ ( ∫ x xqn |f (t)| p1 dqt ) 1 p 1 ( ∫ x xqn |g (t)| q1 dqt ) 1 q 1 . (22) proof. by the discrete hölder’s inequality we have ∫ x xqn |f (t)| |g (t)| dqt = x (1 − q) n−1 ∑ k=0 ∣ ∣f ( xq k )∣ ∣ ∣ ∣g ( xq k )∣ ∣q k = cubo 13, 1 (2011) q− fractional inequalities 65 x (1 − q) n−1 ∑ k=0 ( ∣ ∣f ( xq k )∣ ∣ ( q k ) 1 p 1 )( ∣ ∣g ( xq k )∣ ∣ ( q k ) 1 q 1 ) ≤ ( x (1 − q) n−1 ∑ k=0 ∣ ∣f ( xq k )∣ ∣ p1 q k ) 1 p 1 ( x (1 − q) n−1 ∑ k=0 ∣ ∣g ( xq k )∣ ∣ q1 q k ) 1 q 1 = ( ∫ x xqn |f (t)| p1 dqt ) 1 p 1 ( ∫ x xqn |g (t)| q1 dqt ) 1 q 1 . we present a q−fractional poincaré type inequality. theorem 8. let x > 0, 0 < w ≤ x, 0 < q < 1; α > 0, p1, q1 > 1 such that 1 p1 + 1 q1 = 1; n ∈ n. set ∆ (w) := f (w) − ⌈α⌉−1 ∑ k=0 ( dkq f ) (wqn) [k] q ! w k (qn; q) k . then ∫ x 0 |∆ (w)| q1 wq1(α−1) dqw ≤ 1 (γq (α)) q1 · ( ∫ x 0 ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt )q1 dqw ) 1 p 1 · ( ∫ x 0 ( ∫ w wqn ∣ ∣ ∗d α q,wqn f (t) ∣ ∣ q1 dqt )q1 dqw ) 1 q 1 . (23) proof. by q−fractional taylor’s formula (20) we get ∆ (w) = ( i α q,wqn ∗d α q,wqn f ) (w) = wα−1 γq (α) ∫ w wqn ( q t w ; q ) α−1 ( ∗d α q,wqn f ) (t) dqt. (24) here by (14) and (19), we see that ( ∗d α q,wqn f ) (t) = t⌈α⌉−α−1 γq (⌈α⌉ − α) ∫ w wqn ( q s t ; q ) ⌈α⌉−α−1 d ⌈α⌉ q f (s) dq (s) , (25) all wqn ≤ t ≤ w. here we observe trivially that ∣ ∣ ∣ ∣ ∫ x xqn f (t) dqt ∣ ∣ ∣ ∣ ≤ ∫ x xqn |f (t)| dqt. (26) furthermore we see that ( q t w ; q ) α−1 = ( q t w ; q ) ∞ ( qα t w ; q ) ∞ = ∏∞ i=0 ( 1 − q t w qi ) ∏∞ i=0 ( 1 − qα t w qi ) = ∏∞ i=0 ( 1 − t w qi+1 ) ∏∞ i=0 ( 1 − t w qi+α ) > 0. (27) 66 george a. anastassiou cubo 13, 1 (2011) hence by (22) we obtain |∆ (w)| ≤ wα−1 γq (α) ∫ w wqn ( q t w ; q ) α−1 ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣dqt ≤ wα−1 γq (α) ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) 1 p 1 · ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt ) 1 q 1 . (28) consequently we derive |∆ (w)| wα−1 ≤ 1 γq (α) ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) 1 p 1 · (29) ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt ) 1 q 1 , and |∆ (w)| q1 wq1(α−1) ≤ 1 (γq (α)) q1 ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) q 1 p 1 · (30) ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt ) . applying q−hölder’s inequality (which is also valid on [0, x]) on (30), we see that ∫ x 0 |∆ (w)| q1 wq1(α−1) dqw ≤ 1 (γq (α)) q1 · ∫ x 0   ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) q 1 p 1 · ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt )  dqw ≤ 1 (γq (α)) q1 · ( ∫ x 0 ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt )q1 dqw ) 1 p 1 · ( ∫ x 0 ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt )q1 dqw ) 1 q 1 , (31) proving the claim. next we give a q−fractional sobolev type inequality. theorem 9. here all terms and assumptions as in theorem 8. additionaly let r1, r2 > 1 : 1 r1 + 1 r2 = 1. then ( ∫ x 0 ( |∆ (w)| wα−1 )r1 dqw ) 1 r 1 ≤ 1 γq (α) · cubo 13, 1 (2011) q− fractional inequalities 67    ∫ x 0 ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) r 2 1 p 1 dqw    1 r2 1 · ( ∫ x 0 ( ∫ w wqn ∣ ∣ ∗d α q,wqn f (t) ∣ ∣ q1 dqt ) r 1 r 2 q 1 dqw ) 1 r 1 r 2 . (32) proof. as in the proof of theorem 8 we get (29), so that ( |∆ (w)| wα−1 )r1 ≤ 1 (γq (α)) r1 ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) r 1 p 1 · (33) ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt ) r 1 q 1 . hence ∫ x 0 ( |∆ (w)| wα−1 )r1 dqw ≤ 1 (γq (α)) r1 · (34) ∫ x 0   ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) r 1 p 1 · ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt ) r 1 q 1  dqw (by q−hölder’s inequality on [0, x]) ≤ 1 (γq (α)) r1    ∫ x 0 ( ∫ w wqn ( ( q t w ; q ) α−1 )p1 dqt ) r 2 1 p 1 dqw    1 r 1 · (35) ( ∫ x 0 ( ∫ w wqn ∣ ∣ ( ∗d α q,wqn f ) (t) ∣ ∣ q1 dqt ) r 1 r 2 q 1 dqw ) 1 r 2 , proving the claim. it follows a q−fractional hilbert-pachpatte type inequality. theorem 10. let for i = 1, 2 that xi > 0, 0 < wi ≤ xi, 0 < q < 1; α > 0, p1, q1 > 1 such that 1 p1 + 1 q1 = 1; n ∈ n. call ∆i (wi) = fi (wi) − ⌈α⌉−1 ∑ k=0 ( dkq fi ) (wiq n) [k] q ! w k i (q n; q) k , f (w1) = ∫ w1 w1q n ( q t1 w1 ; q )p1 α−1 dqt1, (36) 68 george a. anastassiou cubo 13, 1 (2011) and g (w2) = ∫ w2 w2q n ( q t2 w2 ; q )q1 α−1 dqt2. then ∫ x1 0 ∫ x2 0 |∆1 (w1)| |∆2 (w2)| (w1w2) α−1 ( f (w1) p1 + g(w2) q1 )dqw1dqw2 ≤ (37) x 1 p 1 1 x 1 q 1 2 (γq (α)) 2 ( ∫ x1 0 ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) dqw1 ) 1 q 1 · ( ∫ x2 0 ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) dqw2 ) 1 p 1 . proof. we notice by (20) that ∆i (wi) = w α−1 i γq (α) ∫ wi wiq n ( q ti wi ; q ) α−1 ( ∗d α q,wiq n fi ) (ti) dqti, (38) for i = 1, 2. therefore we derive |∆1 (w1)| ≤ w α−1 1 γq (α) ∫ w1 w1q n ( q t1 w1 ; q ) α−1 ∣ ∣ ( ∗d α q,w1q n f1 ) (t1) ∣ ∣dqt1 ≤ w α−1 1 γq (α) ( ∫ w1 w1q n ( q t1 w1 ; q )p1 α−1 dqt1 ) 1 p 1 · ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) 1 q 1 . (39) similarly we obtain |∆2 (w2)| ≤ w α−1 2 γq (α) ∫ w2 w2q n ( q t2 w2 ; q ) α−1 ∣ ∣ ( ∗d α q,w2q n f2 ) (t2) ∣ ∣dqt2 ≤ w α−1 2 γq (α) ( ∫ w2 w2q n ( ( q t2 w2 ; q ) α−1 )q1 dqt2 ) 1 q 1 · ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) 1 p 1 . (40) consequently we get |∆1 (w1)| |∆2 (w2)| ≤ (w1w2) α−1 (γq (α)) 2 (f (w1)) 1 p 1 (g (w2)) 1 q 1 · ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) 1 q 1 · ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) 1 p 1 (41) (by young’s inequality) ≤ (w1w2) α−1 (γq (α)) 2 ( f (w1) p1 + g (w2) q1 ) · cubo 13, 1 (2011) q− fractional inequalities 69 ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) 1 q 1 · ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) 1 p 1 . (42) therefore ∫ x1 0 ∫ x2 0 |∆1 (w1)| |∆2 (w2)| (w1w2) α−1 ( f (w1) p1 + g(w2) q1 )dqw1dqw2 ≤ 1 (γq (α)) 2 ( ∫ x1 0 ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) 1 q 1 dqw1 ) · ( ∫ x2 0 ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) 1 p 1 dqw2 ) ≤ (43) x 1 p 1 1 x 1 q 1 2 (γq (α)) 2 ( ∫ x1 0 ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) dqw1 ) 1 q 1 · ( ∫ x2 0 ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) dqw2 ) 1 p 1 , (44) proving the claim. we continue with a generalized q−fractional poincaré type inequality. theorem 11. let x > 0, 0 < w ≤ x, 0 < q < 1; α > β > 0, p1, q1 > 1 : 1 p1 + 1 q1 = 1; n ∈ n. set k (w) = ( ∗d α−β q,wqn f ) (w) − ⌈α⌉−1 ∑ k=⌈α−β⌉ ( dkq f ) (wqn) γq (k − α + β + 1) w k−α+β (qn; q) k−α+β . then ∫ x 0 ( |k (w)| wβ−1 )q1 dqw ≤ 1 (γq (β)) q1 · (45) ( ∫ x 0 ( ∫ w wqn ( ( q t w ; q ) β−1 )p1 dqt )q1 dqw ) 1 p 1 · ( ∫ x 0 ( ∫ w wqn ∣ ∣ ∗d α q,wqn f (t) ∣ ∣ q1 dqt )q1 dqw ) 1 q 1 . proof. by (21) we get k (w) = i β q,wqn ( ∗d α q,wqn f ) (w) = wβ−1 γq (β) ∫ w wqn ( q t w ; q ) β−1 ( ∗d α q,wqn f ) (t) dqt. (46) rest of proof goes as in the proof of theorem 8. next comes a generalized q−fractional sobolev’s type inequality. 70 george a. anastassiou cubo 13, 1 (2011) theorem 12. here all terms and assumptions as in theorem 11. additionaly let r1, r2 > 1 : 1 r1 + 1 r2 = 1. then ( ∫ x 0 ( |k (w)| wβ−1 )r1 dqw ) 1 r 1 ≤ 1 γq (β) · (47)    ∫ x 0 ( ∫ w wqn ( ( q t w ; q ) β−1 )p1 dqt ) r 2 1 p 1 dqw    1 r2 1 · ( ∫ x 0 ( ∫ w wqn ∣ ∣ ∗d α q,wqn f (t) ∣ ∣ q1 dqt ) r 1 r 2 q 1 dqw ) 1 r 1 r 2 . proof. as in the theorem 9, using (46). we finish with a generalized q−fractional hilbert-pachpatte type inequality. theorem 13. let for i = 1, 2 that xi > 0, 0 < wi ≤ xi, 0 < q < 1; α > β > 0, p1, q1 > 1 : 1 p1 + 1 q1 = 1; n ∈ n. call ki (wi) = ( ∗d α−β q,wiq n fi ) (wi) − ⌈α⌉−1 ∑ k=⌈α−β⌉ ( dkq fi ) (wiq n) γq (k − α + β + 1) w k−α+β i (q n; q) k−α+β , f ∗ (w1) = ∫ w1 w1q n ( q t1 w1 ; q )p1 β−1 dqt1, (48) g ∗ (w2) = ∫ w2 w2q n ( q t2 w2 ; q )q1 β−1 dqt2. then ∫ x1 0 ∫ x2 0 |k1 (w1)| |k2 (w2)| (w1w2) β−1 ( f ∗(w1) p1 + g∗(w2) q1 )dqw1dqw2 ≤ x 1 p 1 1 x 1 q 1 2 (γq (β)) 2 · (49) ( ∫ x1 0 ( ∫ w1 w1q n ∣ ∣ ∗d α q,w1q n f1 ∣ ∣ q1 (t1) dqt1 ) dqw1 ) 1 q 1 · ( ∫ x2 0 ( ∫ w2 w2q n ∣ ∣ ∗d α q,w2q n f2 ∣ ∣ p1 (t2) dqt2 ) dqw2 ) 1 p 1 . proof. similar to the proof of theorem 10, using (21). received: october 2009. revised: november 2009. cubo 13, 1 (2011) q− fractional inequalities 71 references [1] george anastassiou, fractional differentiation inequalities, springer, n. york, heidelberg, 2009. [2] h. gauchman, integral inequalities in q−calculus, computers and mathematics with applications, 47 (2004), 281-300. [3] p. rajkovic, s. marinkovic, m. stankovic, fractional integrals and derivatives in q−calculus, applicable analysis and discrete mathematics, 1 (2007), 311-323. [4] m. stankovic, p. rajkovic, s. marinkovic, on q−fractional derivatives of riemannliouville and caputo type, arxiv: 0909.0387 v1[math.ca] 2 sept. 2009. cubo a mathematical journal vol.11, no¯ 05, (23–38). december 2009 resonances and ssf singularities for magnetic schrödinger operators jean-françois bony, vincent bruneau université bordeaux i, institut de mathématiques de bordeaux, umr cnrs 5251, 351, cours de la libération, 33405 talence, france emails: bony@math.u-bordeaux1.fr, vbruneau@math.u-bordeaux1.fr philippe briet centre de physique théorique, cnrs-luminy, case 907, 13288 marseille, france email : briet@cpt.univ-mrs.fr and georgi raikov departamento de matemáticas, facultad de matemáticas, pontificia universidad católica de chile, vicuña mackenna 4860, santiago de chile email : graikov@mat.puc.cl abstract the aim of this note is to review recent articles on the spectral properties of magnetic schrödinger operators. we consider h0, a 3d schrödinger operator with constant magnetic field, and h̃0, a perturbation of h0 by an electric potential which depends only on the variable along the magnetic field. let h (resp. h̃) be a short range perturbation of h0 (resp. of h̃0). in the case of (h, h0), we study the local singularities of the krein spectral shift function (ssf) and the distribution of the resonances of h near the landau levels which play the role of spectral thresholds. in the case of (h̃, h̃0), we study similar problems near the eigenvalues of h̃0 of infinite multiplicity. 24 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) resumen el objetivo de esta nota es reseñar artículos recientes sobre las propiedades espectrales de operadores de schrödinger magnéticos. consideramos h0, el operador tridimensional de schrödinger con campo magnético constante, y h̃0, perturbación de h0 por un potencial eléctrico que depende sólo de la variable a lo largo del campo magnético. sea h, respectivamente h̃, perturbación de corto alcance de h0, respectivamente h̃0. en el caso del par (h, h0), estudiamos las singularidades locales de la función de corrimiento espectral (ssf) de krein y la distribución de las resonancias de h cerca de los niveles de landau que tienen el papel de umbrales espectrales. en el caso del par (h̃, h̃0), investigamos problemas similares cerca de los valores propios de multiplicidad infinita de h̃0. key words and phrases: magnetic schrödinger operators, resonances, spectral shift function. math. subj. class.: 35p25, 35j10, 47f05, 81q10. 1 introduction let h0 := (−i∇ − a) 2 − b, be the (shifted) 3d schrödinger operator with constant magnetic field b = (0, 0,b), b > 0, selfadjoint in l2(r3), and essentially self-adjoint on c∞0 (r 3 ). here a = ( − bx2 2 , bx1 2 , 0 ) is a magnetic potential generating the magnetic field b = curl a. it is well-known that the spectrum of h0 is absolutely continuous and coincides with the interval [0,∞), i.e. σ(h0) = σac(h0) = [0,∞). moreover, the landau levels 2bq, q ∈ z+ := {0, 1, 2, . . .}, play the role of thresholds in σ(h0) (see [10, 15, 3] and section 3 below). for x = (x1,x2,x3) ∈ r 3 we write x = (x⊥,x3) where x⊥ := (x1,x2) ∈ r 2 are the variables on the plane perpendicular to the magnetic field b. further, let v : r3 → r be the electric potential. throughout the article, we assume that v ∈ l∞(r3) ∩ c(r3), and v does not vanish identically. moreover, we will suppose that v satisfies one of the following decay assumptions: |v (x)| = o(〈x⊥〉 −m⊥〈x3〉 −m3 ), m⊥ > 2, m3 > 1, x = (x⊥,x3) ∈ r 3, (1.1) |v (x)| = o(〈x〉−m0 ), m0 > 3, x ∈ r 3, (1.2) |v (x)| = o(〈x⊥〉 −m⊥ exp (−n|x3|)), m⊥ > 2, ∀n > 0, x = (x⊥,x3) ∈ r 3. (1.3) note that each of the conditions (1.2) and (1.3) implies (1.1). on the domain of h0 define the perturbed operator h := h0 + v. cubo 11, 5 (2009) resonances and ssf singularities ... 25 we introduce the krein spectral shift function (ssf) ξ(·; h,h0) for the operator pair (h,h0) (see section 2 for its definition), and study its behavior near the landau levels. in theorem 4.1 we show that at least in the case of v of constant sign, the ssf ξ(·; h,h0) has a singularity at each landau levels, i.e. it either blows up to +∞ if v ≥ 0, or blows down to −∞ if v ≤ 0, as the energy approaches the landau level in an appropriate manner. the singularities of ξ(·; h,h0) are described in the terms of the spectral characteristics of compact operators of berezin-toeplitz type, studied in section 3. one of the possible explanation of the singularities of the ssf ξ(·; h,h0) is the accumulation of resonances of h to the landau levels. in section 5 we define the resonances of h as the poles of an appropriate meromorphic extension of its resolvent, and in theorem 5.1 we obtain upper and lower estimates of the number of the resonances in a vicinity of a given landau level. the lower bound confirms our conjecture that the singularities of the ssf at the landau levels and the accumulation of the resonances of h to these levels, are intimately related. further, in section 6 we introduce a modified operator pair (h̃,h̃0). here h̃0 := h0 + v0 where v0 = v0(x3) decays fast enough at infinity, while h̃ := h̃0 + v where v satisfies (1.1). the remarkable property of the operator h̃0 is that, under appropriate assumption on v0, it has infinitely many eigenvalues of infinite multiplicity, most of which are embedded in the continuous spectrum. we study the asymptotic behavior of the ssf ξ(·; h̃,h̃0) near these eigenvalues of h̃0 of infinite multiplicity. finally, in section 7, under the assumption that the perturbation v is axisymmetric, we investigate the asymptotic behavior as κ → 0 of the unitary group associated with h̃0 + κv , and relate it to certain dynamic resonances for the operator h̃. 2 the spectral shift function assume that v satisfies (1.1). then the resolvent difference (h − i)−1 − (h0 − i) −1 is a traceclass operator, and there exists a unique ξ = ξ(·; h,h0) ∈ l 1 (r; (1 + e2)−1de) such that the lifshits-krein trace formula tr (f(h) − f(h0)) = ∫ r ξ(e; h,h0)f ′ (e)de holds for each f ∈ c∞0 (r), and ξ(e; h,h0) = 0 for each e ∈ (−∞, inf σ(h)) (see the original works [17, 14] or [24, chapter 8]). the function ξ(·; h,h0) is called the spectral shift function (ssf) for the operator pair (h,h0). for almost every e ∈ σac(h) = [0,∞) the ssf ξ(e; h,h0) coincides with the scattering phase for the operator pair (h,h0) according to the birman-krein formula det s(e; h,h0) = e −2πiξ(e;h,h0) where s(e; h,h0) is the scattering matrix for the operator pair (h,h0) (see the original work [4] or [24, chapter 8]). on the other hand, for almost every e < 0 the value −ξ(e; h,h0) coincides with the number of the eigenvalues of h less than e, counted with their multiplicity. 26 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) a priori, the ssf ξ(·; h,h0) is defined as an element of l 1 (r; (1 + e2)−1de). the following proposition provides more precise information on ξ(·; h,h0): proposition 2.1. [7, proposition 2.5] (i) the ssf ξ(·; h,h0) is bounded on every compact subset of r \ 2bz+. (ii) the ssf ξ(·; h,h0) is continuous on r \ (2bz+ ∪ σpp(h)) where σpp(h) is the set of the eigenvalues of h. 3 auxiliary toeplitz operators we have h0 = h0,⊥ ⊗ i‖ + i⊥ ⊗ h0,‖ where i⊥ and i‖ are the identities in l 2 (r 2 x⊥ ) and l2(rx3 ) respectively, h0,⊥ := ( −i ∂ ∂x1 + bx2 2 )2 + ( −i ∂ ∂x2 − bx1 2 )2 − b is the (shifted) landau hamiltonian, self-adjoint in l2(r2x⊥ ), and h0,‖ := − d2 dx23 is the 1d free hamiltonian, self-adjoint in l2(rx3 ). note that h0,⊥ = a ∗a where a∗ := −2ieb|z| 2/4 ∂ ∂z e−b|z| 2/4, z = x1 + ix2, is the creation operator, and a := −2ie−b|z| 2/4 ∂ ∂z̄ eb|z| 2/4, z̄ = x1 − ix2, is the annihilation operator. moreover, [a,a∗] = 2b. therefore, σ(h0,⊥) = ∪ ∞ q=0{2bq}. furthermore, ker h0,⊥ = ker a = { f ∈ l2(r2)|f = ge−b|z| 2/4, ∂g ∂z̄ = 0 } is the classical fock-segal-bargmann space (see e.g. [11]), and ker (h0,⊥ − 2bq) = (a ∗ ) q ker h0,⊥, q ≥ 1. cubo 11, 5 (2009) resonances and ssf singularities ... 27 evidently, dim ker (h0,⊥ − 2bq) = ∞ for each q ∈ z+. let u : r2 → r. fix q ∈ z+. introduce the toeplitz operator pqupq : pql 2 (r 2 ) → pql 2 (r 2 ) where pq is the orthogonal projection onto ker (h0,⊥ − 2bq). obviously, if u ∈ l ∞ (r 2 ), then the operator pqupq is bounded. denote by sr the schatten-von neumann class of order r ∈ [1,∞). then u ∈ lr(r2) implies pqupq ∈ sr, r ∈ [1,∞) (see [20, lemma 5.1] or [9, lemma 3.1]). moreover, if u is sufficiently regular, then p0up0 is unitarily equivalent to a ψdo with anti-wick symbol ω(y,η) := u(b−1/2η,b−1/2y), (y,η) ∈ t ∗r, (see [20]), while pqupq with q ≥ 1 is unitarily equivalent to p0 ( q∑ s=0 q! (2b)s(s!)2(q − s)! ∆ su ) p0 (see [7, lemma 9.2]). for further references, in the following three theorems we describe the eigenvalue asymptotics for the toeplitz operator pqupq, q ∈ z+, under the assumptions that u admits a power-like decay, exponential decay, or is compactly supported, respectively. theorem 3.1. [20, theorem 2.6] let 0 ≤ u ∈ c1(r2), and u(x⊥) = u0(x⊥/|x⊥|)|x⊥| −α (1 + o(1)), |∇u(x⊥)| = o(|x⊥| −α−1 ), as |x⊥| → ∞, with α > 0, and 0 < u0 ∈ c(s 1 ). fix q ∈ z+. then tr 1(s,∞)(pqupq) = ψα(s)(1 + o(1)), s ↓ 0, (3.1) where ψα(s) := s −2/α b 4π ∫ s1 u0(t) 2/αdt. (3.2) theorem 3.2. [22, theorem 2.1, proposition 4.1] let 0 ≤ u ∈ l∞(r2) and ln u(x⊥) = −µ|x⊥| 2β (1 + o(1)), |x⊥| → ∞, with β ∈ (0,∞), µ ∈ (0,∞). fix q ∈ z+. then tr 1(s,∞)(pqupq) = ϕβ (s)(1 + o(1)), s ↓ 0, (3.3) 28 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) where ϕβ (s) :=    b 2µ1/β | ln s|1/β if 0 < β < 1, 1 ln (1+2µ/b) | ln s| if β = 1, β β−1 (ln | ln s|)−1| ln s| if 1 < β < ∞, s ∈ (0,e−1). (3.4) theorem 3.3. [22, theorem 2.2, proposition 4.1] let 0 ≤ u ∈ l∞(r2). assume that supp u is compact, and u ≥ c > 0 on an open subset of r2. fix q ∈ z+. then tr 1(s,∞)(pqupq) = ϕ∞(s)(1 + o(1)), s ↓ 0, (3.5) where ϕ∞(s) := (ln | ln s|) −1| ln s|, s ∈ (0,e−1). (3.6) remark: relations (3.1) and (3.3) with β < 1 are semiclassical in the sense that they are equivalent to tr 1(s,∞)(pqupq) = b 2π ∣∣{x⊥ ∈ r2|u(x⊥) > s }∣∣(1 + o(1)), s ↓ 0. the asymptotic order in relation (3.3) with β = 1 which corresponds to gaussian decay of v , is semiclassical, but the coefficient is not. finally, even the asymptotic order in (3.3) with β > 1, and (3.5) is not semiclassical. moreover the main terms of these asymptotics do not depend on the landau levels. 4 singularities of the ssf at the landau levels let v satisfy (1.1). for x⊥ ∈ r 2, λ ≥ 0, set w(x⊥) := ∫ r |v (x⊥,x3)|dx3, (4.1) wλ = wλ(x⊥) := ( w11 w12 w21 w22 ) , where w11 := ∫ r |v (x⊥,x3)| cos 2 ( √ λx3)dx3, w12 = w21 := ∫ r |v (x⊥,x3)| cos ( √ λx3) sin ( √ λx3)dx3, w22 := ∫ r |v (x⊥,x3)| sin 2 ( √ λx3)dx3. we have rank pqwpq = ∞, rank pqwλpq = ∞, λ ≥ 0. cubo 11, 5 (2009) resonances and ssf singularities ... 29 theorem 4.1. [9, theorems 3.1, 3.2] let v satisfy (1.2), and v ≥ 0 or v ≤ 0. fix q ∈ z+. then we have ξ(2bq − λ; h,h0) = o(1), λ ↓ 0, if v ≥ 0, and for each ε ∈ (0, 1) we have −tr 1((1−ε)2 √ λ,∞) (pqwpq) + o(1) ≤ ξ(2bq − λ; h,h0) ≤ −tr 1((1+ε)2 √ λ,∞) (pqwpq) + o(1), λ ↓ 0, if v ≤ 0. moreover, for each ε ∈ (0, 1) we have ± 1 π tr arctan ( ((1 ± ε)2 √ λ)−1pqwλpq ) + o(1) ≤ ξ(2bq + λ; h,h0) ≤ ± 1 π tr arctan ( ((1 ∓ ε)2 √ λ)−1pqwλpq ) + o(1), λ ↓ 0, if ±v ≥ 0. remark: the proof of theorem 4.1 is based on a representation of the ssf obtained by a. pushnitski in [19]. if we assume that w admits a power-like or exponential decay at infinity, or has a compact support, we can combine the results of theorems 3.1 – 3.3 for u = w , with theorem 4.1 and obtain explicitly the main asymptotic term of ξ(2bq + λ; h,h0) as λ ↓ 0 or λ ↑ 0: corollary 4.1. [9, corollaries 3.1 – 3.2], [21, corollary 2.1] let (1.2) hold with m0 > 3. (i) assume that the hypotheses of theorem 3.1 hold with u = w and α > 2. then we have ξ(2bq − λ; h,h0) = − b 2π ∣∣∣ { x⊥ ∈ r 2|w(x⊥) > 2 √ λ }∣∣∣ (1 + o(1)) = −ψα(2 √ λ) (1 + o(1)), λ ↓ 0, (4.2) if v ≤ 0, and ξ(2bq + λ; h,h0) = ± b 2π2 ∫ r2 arctan ((2 √ λ)−1w(x⊥))dx⊥ (1 + o(1)) = ± 1 2 cos (π/α) ψα(2 √ λ) (1 + o(1)), λ ↓ 0, if ±v ≥ 0, the function ψα being defined in (3.2). (ii) assume that the hypotheses of theorem 3.2 hold with u = w. then we have ξ(2bq − λ; h,h0) = −ϕβ (2 √ λ) (1 + o(1)), λ ↓ 0, β ∈ (0,∞), 30 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) if v ≤ 0, the functions ϕβ being defined in (3.4). if, in addition, v satisfies (1.1) for some m⊥ > 2 and m3 > 2, we have ξ(2bq + λ; h,h0) = ± 1 2 ϕβ (2 √ λ) (1 + o(1)), λ ↓ 0, β ∈ (0,∞), if ±v ≥ 0. (iii) assume that the hypotheses of theorem 3.3 hold with u = w. then we have ξ(2bq − λ; h,h0) = −ϕ∞(2 √ λ) (1 + o(1)), λ ↓ 0, if v ≤ 0, the function ϕ∞ being defined in (3.6). if, in addition, v satisfies (1.1) for some m⊥ > 2 and m3 > 2, we have ξ(2bq + λ; h,h0) = ± 1 2 ϕ∞(2 √ λ) (1 + o(1)), λ ↓ 0, if ±v ≥ 0. as a corollary we obtain the following result which could be regarded as generalized levinson formulae: corollary 4.2. let v satisfy (1.2), and v ≤ 0. fix q ∈ z+. then lim λ↓0 ξ(2bq + λ; h,h0) ξ(2bq − λ; h,h0) = 1 2 cos π α if w admits a power-like decay with decay rate α > 2 (i.e. if u = w satisfies the hypotheses of theorem 3.1), or lim λ↓0 ξ(2bq + λ; h,h0) ξ(2bq − λ; h,h0) = 1 2 if w decays exponentially or has a compact support (i.e. if u = w satisfies the hypotheses of theorems 3.2 – 3.3). remark: the classical levinson formula relates the number of the negative eigenvalues of −∆ + v and lime↓0 ξ(e; −∆ + v,−∆) (see the original work [16] or the survey article [23]). 5 resonances near the landau levels one of the possible explanations of the singularities of the ssf described in theorem 4.1, is the accumulation of resonances of h at the landau levels. in this section we define the resonances of h as an appropriate extension of the resolvent (h − z)−1 defined a priori for z ∈ c+ := {ζ ∈ c | im ζ > 0}, following the exposition of [5]. in theorem 5.1 below we establish upper and lower estimates on the number of resonances of h in a ring centered at a given landau level; the lower cubo 11, 5 (2009) resonances and ssf singularities ... 31 bounds imply accumulation of the resonances to the landau level. for z ∈ c+ we have (h0 − z) −1 = ∞∑ q=0 pq ⊗ ( h0,‖ + 2bq − z )−1 . for each q ∈ z+ and n > 0 the operator-valued function z 7→ ( h0,‖ + 2bq − z )−1 ∈ l(e−n〈x3〉l2(r),en〈x3〉l2(r)), admits a holomorphic extension from c \ [2bq,∞) to the 2-sheeted covering pq : {ζ ∈ c \ {0}, im ζ > −n} ∋ k 7→ k 2 + 2bq ∈ c \ {2bq}. this extension however depends on q ∈ z+. let π1(c \ 2bz+) be the fundamental group of c \ 2bz+, and g be the subgroup of π1(c \ 2bz+) generated by { a21,a2a1a −1 2 a −1 1 | a1,a2 ∈ π1(c \ 2bz+) } . we define pg : m 7→ c \ 2bz+ as the connected infinite-sheeted covering such that p∗g(π1(m)) = g. fix a base point in m. let f be the connected component of p−1g (c \ [0,∞)) containing this base point. by definition, the functions m ∋ z 7→ √ z − 2bq have a positive imaginary part on f. set f+ := f ∩ p −1 g (c+). for λ0 ∈ c and ε > 0 put d(λ0,ε) := {λ ∈ c| |λ − λ0| < ε}, d(λ0,ε) ∗ := {λ ∈ c| 0 < |λ − λ0| < ε}. let d∗q ⊂ m be the connected component of p −1 g (d(2bq, 2b) ∗ ) that intersects f+. there exists an analytic bijection zq: d(0, √ 2b)∗ ∋ k 7→ zq(k) ∈ d ∗ q, such that pg(zq(k)) = 2bq + k 2 and z−1q (d ∗ q ∩ f+) is the first quadrant of d(0, √ 2b)∗. for n > 0 set mn := { z ∈ m | im √ z − 2bq > −n,∀q ∈ z+ } . evidently, ∪n >0mn = m. proposition 5.1. [5, proposition 1] (i) for each n > 0 the operator-valued function z 7→ (h0 − z) −1 ∈ l ( e−n〈x3〉l2(r3); en〈x3〉l2(r3) ) has a holomorphic extension from the open upper half-plane to mn . (ii) suppose that v satisfies (1.3) with m⊥ > 0. then for each n > 0 the operator-valued function z 7→ (h − z)−1 ∈ l ( e−n〈x3〉l2(r3); en〈x3〉l2(r3) ) , has a meromorphic extension from the open upper half plane to mn . moreover, the poles and the range of the residues of this extension do not depend on n. 32 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) we define the resonances of h as the poles of the meromorphic extension of the resolvent (h − z)−1. the multiplicity of a resonance z0 is defined as rank 1 2iπ ∫ γ (h − z)−1dz, where γ is an appropriate circle centered at z0. in what follows, resq(h0 + κv ) denotes the set of the resonances of h0 + κv in d ∗ q . theorem 5.1. [5, theorem 2] assume that v satisfies (1.3), and v ≥ 0 or v ≤ 0. fix q ∈ z+. then for any δ > 0 there exist κ0,r0 > 0 such that: (i) for any 0 < r < r0 and 0 ≤ κ ≤ κ0, we have #{z = zq(k) ∈ resq(h0 + κv ) |r < |k| < 2r} = o(tr 1(r,8r)(κpqwpq)). (ii) for any 0 ≤ κ ≤ κ0, h0 + κv has no resonances in {z = zq(k) | 0 < |k| < r0, ∓im k ≤ 1 δ |re k|} if ±v ≥ 0. (iii) if w satisfies ln w(x⊥) ≤ −c〈x⊥〉 2, then for any 0 ≤ κ ≤ κ0, h0 + κv has an infinite number of resonances in {z = zq(k) | 0 < |k| < r0, ∓im k > 1 δ |re k|} if ±v ≥ 0. more precisely, there exists a decreasing sequence (rℓ)ℓ∈n of positive numbers, rℓ ↓ 0 such that, #{z = zq(k) ∈ resq(h0 + κv ) | rℓ+1 < |k| < rℓ, ∓ im k > 1 δ |re k|} ≥ tr 1(2rℓ+1,2rℓ)(κpqwpq). the setting is summarized by the following figure: im k re k v ≥ 0 r0 re k im k v ≤ 0 r0 figure 1: resonances near a landau level for v of definite sign. they are essentially concentrated near the semi-axis k = ∓i[0,∞) for ±v ≥ 0. the physical sheet is in white. cubo 11, 5 (2009) resonances and ssf singularities ... 33 6 ssf for a pair of modified magnetic schrödinger operators in this section we replace the unperturbed operator h0 by h̃0 = h0 + v0 where v0 is an electric potential which depends only on the variable x3, and decays fast enough as |x3| → ∞. under appropriate assumptions on v0, the operator h̃0 has infinitely many eigenvalues of infinite multiplicity, most of which are embedded in the continuous spectrum. we introduce the perturbed operator h̃ := h̃0 + v where v satisfies (1.1), and investigate the asymptotic behavior of the ssf ξ(·; h̃,h̃0) near the eigenvalues of h̃0 of infinite multiplicity. let v0 ∈ c 1 (r; r) satisfy the assumption |v0(x)| = o(〈x〉 −δ0 ), x ∈ r, δ0 > 1. (6.1) set h‖ := h0,‖ + v0 = − d2 dx23 + v0(x3). we have σess(h‖) = σac(h‖) = [0,∞), σpp(h‖) ∩ (0,∞) = ∅, σsc(h‖) = ∅. for simplicity, assume that inf σ(h‖) > −2b. then h̃0 := h0,⊥ ⊗ i‖ + i⊥ ⊗ h‖ = h0 + i⊥ ⊗ v0 is the hamiltonian of a non-relativistic spinless quantum particle subject to a classical electromagnetic field (e, b) with b = (0, 0,b) and e = −(0, 0,v′0). let λ ∈ σdisc(h‖). then 2bq + λ with q ∈ z+ is an eigenvalue of h̃0 of infinite multiplicity; if q ≥ 1, this eigenvalue is embedded in σac(h̃0). assume now that v : r3 → r satisfies (1.1) with m⊥ > 2 and m3 > 1, and set h̃ := h̃0 + v. then (h̃ − i)−1 − (h̃0 − i) −1 is trace-class, and the ssf ξ(·; h̃,h̃0) is well defined. set z := {e = 2bq + µ | q ∈ z+, µ ∈ σdisc(h‖) or µ = 0}. similarly to proposition 2.1, we can show that the ssf ξ(·; h̃,h̃0) is bounded on every compact subset of r \ z, and is continuous on r \ (z ∪ σpp(h̃)). let λ ∈ σdisc(h‖), and ψ be the eigenfunction satisfying h‖ψ = λψ, ‖ψ‖l2(r) = 1, ψ = ψ̄. by analogy with (4.1) set w̃(x⊥) := ∫ r |v (x⊥,x3)|ψ(x3) 2dx3. 34 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) theorem 6.1. [2, theorem 6.1] let v0 satisfy (6.1), and v satisfy (1.1). assume inf σ(h‖) > −2b. let λ ∈ σdisc(h‖). fix q ∈ z+. then for each ε ∈ (0, 1) we have tr 1((1+ε)λ,∞)(pqw̃pq) + o(1) ≤ ξ(2bq + λ + λ; h̃,h̃0) ≤ tr 1((1−ε)λ,∞)(pqw̃pq) + o(1), ξ(2bq + λ − λ; h̃,h̃0) = o(1), as λ ↓ 0, if v ≥ 0, and −tr 1((1−ε)λ,∞)(pqw̃pq) + o(1) ≤ ξ(2bq + λ − λ; h̃,h̃0) ≤ −tr 1((1+ε)λ,∞)(pqw̃pq) + o(1), ξ(2bq + λ + λ; h̃,h̃0) = o(1), as λ ↓ 0, if v ≤ 0. similarly to corollary 4.2, we can combine the result of theorem 6.1 with those of theorems 3.1 – 3.3, and obtain explicit asymptotic formulae describing the singularity of ξ(·; h̃,h̃0) at 2bq+λ under explicit assumptions about the decay of w̃ at infinity. we omit here this obvious corollary, and refer the reader to [2, corollary 6.1]. 7 dynamical resonances for axisymmetric perturbations v in this section we assume that the perturbation v is axisymmetric, and investigate the asymptotics as κ → 0 of the unitary group e−it(h̃0+κv ), t ≥ 0. for m ∈ z introduce the operator h (m) 0,⊥ := − 1 ̺ d d̺ ̺ d d̺ + ( m ̺ − b̺ 2 )2 − b, self-adjoint in l2(r+; ̺d̺). we have σ(h (m) 0,⊥ ) = ∪ ∞ q=m− {2bq}, m− := max{0,−m}, dim ker(h (m) 0,⊥ − 2bq) = 1, ∀q ≥ m−. for m ∈ z, q ∈ z, q ≥ m−, let ϕq,m satisfy h (m) 0,⊥ ϕq,m = 2bqϕq,m, ‖ϕq,m‖l2(r+;̺d̺) = 1. let the multiplier by v0 be h0,‖-compact. set h (m) 0 := h (m) 0,⊥ ⊗ i‖ + i⊥ ⊗ h0,‖, h̃ (m) 0 := h (m) 0,⊥ ⊗ i‖ + i⊥ ⊗ h‖ = h (m) 0 + i⊥ ⊗ v0, where i⊥ is the identity operator in l 2 (r+; ̺d̺); σ(h (m) 0 ) = σess(h̃ (m) 0 ) = [2m−b,∞). cubo 11, 5 (2009) resonances and ssf singularities ... 35 let (̺,φ,x3) be the cylindrical coordinates in r 3. the operator h (m) 0 (resp. h̃ (m) 0 ), m ∈ z, is unitarily equivalent to the restriction of h0 (resp., of h̃0) onto ker (l − m) with l := −i ∂ ∂φ . hence, the operator h0 (resp., h̃0) is unitarily equivalent to the orthogonal sum ⊕m∈zh (m) 0 (resp., ⊕m∈zh̃ (m) 0 ). let the multiplier by v : r3 → r be h0-bounded with zero relative bound. suppose that v is axisymmetric i.e. ∂v ∂φ = 0. let κ ∈ r. on d(h̃0) introduce the operator h̃κ := h̃0 + κv self-adjoint in l2(r3), and on d(h̃ (m) 0 ) introduce the operator h̃(m) κ := h̃ (m) 0 + κv, m ∈ z, self-adjoint in l2(r+ × r; ̺d̺dx3). let λ ∈ σdisc(h‖). for z ∈ c+, m ∈ z, q ∈ z+, q ≥ m−, set fq,m(z) := 〈(h̃ (m) 0 − z) −1 (i − pq,m)v φq,m,v φq,m〉 where 〈·, ·〉 is the scalar product in l2(r+ × r; ̺d̺dx3), φq,m(̺,x3) = ϕq,m(̺)ψ(x3) so that h̃ (m) 0 φq,m = (2bq + λ)φq,m, and pq,m := |φq,m〉〈φq,m|. we will say that the fermi golden rule fq,m,λ is valid if the limit fq,m(2bq + λ) = lim δ↓0 fq,m(2bq + λ + iδ) exists and is finite, and im fq,m(2bq + λ) > 0. for j ∈ z+ set vj (x3) := x j 3 dj v0 dx j 3 , vj = x j 3 ∂j v ∂x j 3 . we will say that the condition cν , ν ∈ z+, holds true if the multipliers by vj , j = 0, 1, are h0,‖compact, the multipliers by vj , j ≤ ν, are h0,‖-bounded, the multiplier by v is h0-bounded with zero relative bound, and the multipliers by vj , j = 1, . . . ,ν, are h0-bounded. theorem 7.1. [2, theorem 4.1] let v be axisymmetric. fix n ∈ z+, and assume that the condition cν holds with ν ≥ n + 5. suppose that inf σ(h‖) > −2b. let λ ∈ σdisc(h‖). fix m ∈ z, q ∈ z+, q > m−, and suppose that the fermi golden rule fq,m,λ is valid. then there exists a function g ∈ c∞0 (r; r) such that g = 1 near 2bq + λ, and 〈e−ih̃ (m) κ tg(h̃(m) κ )φq,m, φq,m〉 = a(κ)e −iλq,m (κ)t + b(κ, t), t ≥ 0, (7.1) where λq,m(κ) = 2bq + λ + κ〈v φq,m, φq,m〉 − κ 2fq,m(2bq + λ) + oq,m,v (κ 2 ), κ → 0. (7.2) moreover, a and b satisfy the estimates |a(κ) − 1| = o(κ2), 36 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) |b(κ, t)| = o(κ2| ln |κ||(1 + t)−n), |b(κ, t)| = o(κ2(1 + t)−n+1), as κ → 0 uniformly with respect to t ≥ 0. remarks: (i) the numbers λq,m appearing in (7.1) can be interpreted as resonances for the operator h̃m, and hence of h̃. another definition of the resonances of h̃ which is in the spirit of [1] and [12] and includes x3-analyticity assumptions concerning v0 and v , can be found in [2, section 3] (see also [13] and [6] where no axial symmetry of v is assumed). (ii) note that the numbers 〈v φq,m, φq,m〉 appearing in (7.2) are eigenvalues of the toeplitz operator pqw̃pq appearing in theorem 6.1. (iii) the proof of theorem 7.1 is based on appropriate mourre estimates (see [18]), and the approach developed in [8]. acknowledgements. g. raikov thanks the organizers of the second symposium on scattering and spectral theory, pernambuco, brazil, august 2008, for having invited him to present there the results of this paper. j.-f. bony and v. bruneau were partially supported by the french anr grant no. jc0546063. v. bruneau and g. raikov were partially supported by the chilean science foundation fondecyt under grants 1050716 and 7060245 and by the cooperation program cnrsconicyt rqwus41561. g. raikov was partially supported by núcleo científico icm p07-027-f “mathematical theory of quantum and classical magnetic systems". received: december, 2008. revised: march, 2009. references [1] aguilar, j. and combes, j.m., a class of analytic perturbations for one-body schrödinger hamiltonians, comm. math. phys., 22 (1971), 269–279. [2] astaburuaga, m.a., briet, ph., bruneau, v., fernández, c. and g. raikov, dynamical resonances and ssf singularities for a magnetic schrödinger operator, serdica math. j., 34 (2008), 179–218. [3] avron, j., herbst, i. and simon, b., schrödinger operators with magnetic fields. i. general interactions, duke math. j., 45 (1978), 847–883. [4] birman, m.š. and kreı̆n, m.g., on the theory of wave operators and scattering operators, dokl. akad. nauk sssr, 144 (1962), 475–478 (russian); english translation in soviet math. doklady, 3 (1962). cubo 11, 5 (2009) resonances and ssf singularities ... 37 [5] bony, j.-f., bruneau, v. and raikov, g.d., resonances and spectral shift function near landau levels, ann. inst. fourier, 57 (2007), 629–671. [6] bruneau, v., khochman, a. and raikov, g., perturbation of a magnetic schrödinger operator near an embedded infinite-multiplicity eigenvalue, to appear in: spectral and scattering theory for quantum magnetic systems, contemporary mathematics, 500 (2009), 47–62. [7] bruneau, v., pushnitski, a. and raikov, g.d., spectral shift function in strong magnetic fields, algebra i analiz, 16 (2004), 207–238; see also st. petersburg math. j., 16 (2005), 181–209. [8] cattaneo, l., graf, g.m. and hunziker, w., a general resonance theory based on mourre’s inequality, ann. henri poincaré, 7 (2006), 583–601. [9] fernández, c. and raikov, g.d., on the singularities of the magnetic spectral shift function at the landau levels, ann. henri poincaré, 5 (2004), 381–403. [10] fock, v, bemerkung zur quantelung des harmonischen oszillators im magnetfeld, z. physik, 47 (1928), 446–448. [11] hall, b.c., holomorphic methods in analysis and mathematical physics, in: first summer school in analysis and mathematical physics, cuernavaca morelos, 1998, 1–59, contemp. math., 260, ams, providence, ri, 2000. [12] hunziker, w., distortion analyticity and molecular resonance curves, ann. inst. h. poincaré phys. théor., 45 (1986), 339–358. [13] khochman, a., resonances and spectral shift function for a magnetic schrödinger operator, j. math. phys. 50 (2009), no. 4, 043507, 16 pp. [14] krein, m.g., on the trace formula in perturbation theory, mat. sb., 33 (1953), 597–626 (russian). [15] landau, l., diamagnetismus der metalle, z. physik, 64 (1930), 629–637. [16] levinson, n., on the uniqueness of the potential in a schrödinger equation for a given asymptotic phase, danske vid. selsk. mat.-fys. medd., 25 (1949), no. 9, 1–29. [17] lifshits, i.m., on a problem in perturbation theory, uspekhi mat. nauk, 7 (1952), 171–180 (russian). [18] mourre, e., absence of singular continuous spectrum for certain self-adjoint operators, comm. math. phys., 78 (1981), 391–408. [19] pushnitskĭı, a., a representation for the spectral shift function in the case of perturbations of fixed sign, algebra i analiz, 9 (1997), 197–213 (russian); english translation in st. petersburg math. j., 9 (1998), 1181–1194. 38 j.-f. bony, ph. briet, v. bruneau and g. raikov cubo 11, 5 (2009) [20] raikov, g.d., eigenvalue asymptotics for the schrödinger operator with homogeneous magnetic potential and decreasing electric potential. i. behaviour near the essential spectrum tips, comm. pde, 15 (1990), 407–434; errata: comm. pde, 18 (1993), 1977–1979. [21] raikov, g.d., spectral shift function for schrödinger operators in constant magnetic fields, cubo 7 (2005), 171–199. [22] raikov, g.d. and warzel, s., quasi-classical versus non-classical spectral asymptotics for magnetic schrödinger operators with decreasing electric potentials, rev. math. phys., 14 (2002), 1051–1072. [23] robert, d., semiclassical asymptotics for the spectral shift function, in: differential operators and spectral theory, ams translations ser. 2, 189, 187–203, ams, providence, ri, 1999. [24] yafaev, d.r., mathematical scattering theory. general theory. translations of mathematical monographs, 105 ams, providence, ri, 1992. b2-bbbr9 cubo a mathematical journal vol.11, no¯ 02, (1–6). may 2009 some theoretical issues concerning hamming coding erika griechisch university of pécs, college of natural sciences, institute of mathematics and informatics, ifjúság útja 6, pécs, 7624, hungary email: griechisch.erika@gmail.com abstract the security of telecommunication largely depends on effective and safe coding. national security as well as the safety of the entire society also depends on how information is exchanged between government agencies. the security of information can also be guaranteed by a safe and effective coding system. a hamming code is a linear error-correcting code which can detect and correct single-bit errors. it can also detect, but not correct up to two simultaneous bit errors. for each integer m > 1 there is a code with the parameters {2m − 1, 2m − m − 1, 3}. the factorization of abelian groups and the complete factor problem of 2-groups are closely related to the error-correcting hamming codes. in this paper we will deal with the rédei property of 2-groups. resumen la seruridad en telecomunicaciones depende ampliamente de efectivos e seguros códigos. la seguridad nacional bien como la seguridad de la sociedad entera también depende de como la información es intercambiada entre agencias de govierno. la seguridad de información también puede ser garantizada por efectivos y seguros códigos. un código hamming es un código linear error-corrección el cual puede detectar y corregir errores single-bit. este puede también detectar, pero no corrigir dos errores bit simultaneos. para todo entero m > 1 hay un código con los parametros {2m − 1, 2m − m − 1, 3}. la factorización de grupos abelianos y el problema de factor completo de 2-grupos son relativamente proximos de los códigos hamming error-corrector. este artículo trabaja con la propiedad de rédei de 2-grupos. 2 erika griechisch cubo 11, 2 (2009) key words and phrases: factorization of abelian groups, full-rank tiling, rédei property, dancing link, exhaustive search. math. subj. class.: 20k01, 05b45, 52c22, 68r05. 1 introduction let g be a finite abelian group, with identity element e. let a1, . . . , an be given subsets of g. then a1 · · · an = {a1 · · · an | ai ∈ ai} is a factorization of g, if g = a1 · · · an and each g ∈ g can be uniquely represented in the form a1 · · · an. a subset a of g is normalized if e ∈ a. the factorization is called normalized if each factor is normalized,. let 〈a〉 denote the smallest subgroup of g that contains a. it is called the span of a in g. if g is a direct product of cyclic groups of order t1, . . . , tn, then the type of g is (t1, . . . , tn). a group of type (p, . . . , p), where p is prime, is called an elementary-p-group, and the group of type (t1, . . . , tn), where each ti is a power of p is called a p-group. in this short paper we will restrict our attention to p-groups. definition 1. g has the rédei property if from each normalized factorization g = ab it follows that either 〈a〉 6= g or 〈b〉 6= g. in the special case, when g = {e}, g has the rédei property by definition. the reason is the following. {e} has only one factorization, namely {e}{e}. in this case 〈a〉 = 〈b〉 = g. in 1970 l. rédei conjectured if g = ab is a normalized factorization of g and g is of type (p, p, p), then either 〈a〉 6= g or 〈b〉 6= g. this was published as problem 5 in [3]. the following facts are known about the rédei property. let p be a prime and let fp be a family of p-groups whose types are depicted in table 1 or a subgroup of such a group. szabó proved in [4], that if g is a p-group with the rédei property, then g is a member of the fp family. p = 2 (2 α , 2 β , 2, 2) α ≥ 3, β ≥ 2 (2 α , 2, 2, 2, 2, 2) α ≥ 3 (2 2 , 2 2 , 2, 2, 2, 2, 2, 2, 2) p = 3 (3 α , 3 β , 3) α ≥ 2, β ≥ 2 (3 α , 3, 3, 3) α ≥ 2 (3, 3, 3, 3, 3) p ≥ 5 (pα, pβ , p) α ≥ 1, β ≥ 1 table 1: the fp family we will show, that a group of type (4, 4, 2, 2) does not have the rédei property. as a consequence of this fact is that the earlier list will change. this is the main result of this paper. cubo 11, 2 (2009) some theoretical issues concerning hamming coding 3 2 mathematical results lemma 1. let g be a group of type (4, 4, 2, 2). then g has a full-rank factorization. proof. let x1, x2, y1, y2 be a basis of g, where |x1| = |x2| = 4, |y1| = |y2| = 2. set a = {e, x1, x2, x1x2y1, x1x2y1, x1x 2 2y1y2, x 2 1x2y1y2, x2y1y2, x 2 1x 2 2y1y2} and let b = {e, y2, x1x 2 2y2, x1x 3 2y1, x 2 1x2, x 2 1x 3 2y2, x 3 1x2y1y2, x 3 1x 2 2}. it can be easily verified, that the product ab is direct. for convenience we exhibited the elements a and b in table 2 using only their exponents. a b 0000 0000 1000 0001 0100 1201 1110 1310 1101 2100 1211 2301 2111 3111 2211 3200 table 2: factors a and b table 3 summaries the elements of product ab. clearly 〈a〉 = 〈b〉 = g. furthermore e, x1 ∈ a + x1 ∈ 〈a〉. e, x2 ∈ a + x2 ∈ 〈a〉. x1, x2 ∈ 〈a〉 + x1x2y1 ∈ a + y1 ∈ 〈a〉. x1x2y2 ∈ a, x1, x2 ∈ 〈a〉 + y2 ∈ 〈a〉. thus x1, x2, y1, y2 ∈ 〈a〉 and so 〈a〉 = g. e, y2 ∈ b + y2 ∈ 〈b〉. x1x 2 2y2, y2 ∈ b + x1x 2 2 ∈ 〈b〉. x 2 1x2 ∈ b, x1x 2 2 ∈ 〈b〉 + x 3 1x 3 2 ∈ 〈b〉. x 3 1x 2 2 ∈ b, x2 ∈ 〈b〉 + x2 ∈ 〈b〉. x1x 2 2, x2 ∈ 〈b〉 + x1 ∈ 〈b〉. x1x 3 2y1 ∈ b, x1, x2, y2 ∈ 〈b〉 + y1 ∈ 〈b〉. 4 erika griechisch cubo 11, 2 (2009) 0000 0001 1201 1310 2100 2301 3111 3200 0000 0000 0001 1201 1310 2100 2301 3111 3200 1000 1000 1001 2201 2310 3100 3301 0111 0200 0100 0100 0101 1301 1010 2200 2001 3211 3300 1110 1110 1111 2311 20000 3210 3011 0201 0310 1101 1101 1100 2300 2011 3201 3000 0210 0301 1211 1211 1210 2010 2101 3311 3110 0300 0011 2111 2111 2110 3310 3001 0211 0010 1200 1311 2211 2211 2210 3010 3101 0311 0110 1300 1011 table 3: the product a and b similarly, x1, x2, y1, y2 ∈ 〈b〉 and therefore 〈b〉 = g. 2 notice that the construction in lemma 1 was accomplised by an exhaustive computer search using d.e. knuth [1] dancing links algorithm. theorem 1 let f ′2 be a family of 2-groups whose types are given in table 4 or a subgroup of such a group. if a 2-group g has the rédei property, then g is a member of the f ′2 family. proof. let g be a group of type (2 α(1) , . . . , 2 α(r) , 2 β(1) , . . . , 2 β(s) , 2 γ(1) , . . . , 2 γ(t) ), where α(1) ≥ · · · ≥ α(r) ≥ 3, β(1) = · · · = β(s) = 2, γ(1) = · · · = γ(t) = 1. (2 α , 2 β , 2) α, β ≥ 2 (2 α , 2, 2, 2, 2, 2) α ≥ 3 (2 2 , 2, 2, 2, 2, 2, 2, 2, 2) table 4: the f ′2 family suppose that g has the rédei property. it is sufficient to shown that g is a member of f ′2 family. if r + s ≥ 3, then g has a subgroup h of type (4, 4, 4). now by [5], h has a full-rank factorization and so by theorem 1 in [4] g also has a full-rank factorization. for the remaining part of the the proof we may assume that 0 ≤ r + s ≤ 2. cubo 11, 2 (2009) some theoretical issues concerning hamming coding 5 we distinguish between the following cases listed in table 5. case r s t 1 0 0 ≤ 9 2 0 1 ≤ 8 3 0 2 ≤ 1 4 1 0 ≤ 4 5 1 1 ≤ 1 6 2 0 ≤ 1 table 5: cases case 1 if r = 0, s = 0 and t ≥ 10, then g has a subgroup h of the type (2, ..., 2). by [2], h admits a full-rank factorization and so does g as well. thus t ≤ 9 as required case 2 if r = 0, s = 1 and t ≥ 9, then g has a subgroup of the type (4, 9 ︷︸︸︷ 2, . . . , 2), then g has a subgroup h of type ( 10 ︷︸︸︷ 2, . . . , 2). by [2], h has a full-rank factorization so does g as well. thus t ≤ 8 as required. case 3 if r = 0, s = 2 and t ≥ 2, then g has a subgroup of the type (4, 4, 2, 2) which has full-rank factorization by lemma 1. so g has a full-rank factorization also. thus t ≤ 1 as required. case 4 if r = 1, s = 0, t ≥ 5, then g has a subgroup of the type (8, 2, 2, 2, 2, 2) and this subgroup has full-rank factorization by [4]. so g also has a full-rank factorization. thus t ≤ 4 as required. case 5 if r = 1, s = 1, t ≥ 2, then g has a subgroup of the type (4, 4, 2, 2) which has full-rank factorization by lemma 1. so g has a full-rank factorization also. thus t ≤ 1 as required. case 6 if r = 2, s = 0, t ≥ 2, then g has a subgroup of the type (8, 8, 2, 2) which has a subgroup of type (4, 4, 2, 2). thus g has a full-rank factorization by lemma 1. therefore t ≤ 1 is required. thus the proof is completed. 2 acknowledgement the help of péter császár providing the implementation of the exact cover algorithm, is highly appreciated. received: march 21, 2008. revised: april 29, 2008. 6 erika griechisch cubo 11, 2 (2009) references [1] knuth, d.e., dancing links, in millennial perspectives in computer science, j. davies, b. roscoe, and j. woodcock, eds., palgrave macmillan, basingstoke, 2000, pp. 187–214. [2] östergard, p.r.j. and vardy, a., resolving the existence of full-rank tilings of binary hamming spaces, siam journal of discrete mathematics, vol. 18, no. 2 (2004), pp. 382–387. [3] rédei, l., lückenhafte polynome über endlichen körpern, birkhäuser verlag, basel 1970, (english translation: lacunary polynomials over finite fields, north-holland, amsterdam, 1973). [4] szabó, s., factoring finite abelian groups by subsets with maximal span, siam journal of discrete mathematics, vol. 20, no. 4 (2006), pp. 920–931. [5] szabó, s., topics in factorization of abelian groups, birkhäuser, 2004. n01-factor articulo 2.dvi cubo a mathematical journal vol.12, no¯ 02, (19–27). june 2010 differences of weighted composition operators between weighted banach spaces of holomorphic functions and weighted bloch type spaces elke wolf mathematical institute, university of paderborn, d-33095 paderborn, germany. email: lichte@math.uni-paderborn.de abstract we consider analytic self-maps φ1, φ2 of the open unit disk as well as analytic maps ψ1,ψ2. these maps induce differences of weighted composition operators acting between weighted banach spaces of holomorphic functions and weighted bloch type spaces. in this article we give necessary and sufficient conditions for such a difference to be bounded resp. compact. resumen nosotros consideramos auto aplicaciones φ1, φ2 del disco unitario abierto bien como aplicaciones anaĺıticas ψ1,ψ2. estas aplicaciones inducen diferencias de composición de operadores con peso actuando entre espacios de banach pesados de funciones holomorfas y espacios de tipo bloch con peso. en este art́ıculo damos condiciones necesarias y suficientes para que tal diferencia sea acotada, respectivamente, compacta. key words and phrases: weighted composition operators, weighted bloch type spaces, weighted banach spaces of holomorphic functions. math. subj. class. 2000: 47b33, 47b38. 20 elke wolf cubo 12, 2 (2010) 1 introduction for analytic self-maps φ1,φ2 of d and analytic maps ψ1,ψ2 the corresponding weighted composition operators ψicφi are defined by ψicφif = ψif ◦ φi, i = 1, 2. composition operators and weighted composition operators acting on various spaces of analytic functions have recently been of much interest, see for example [14], [8], [12], [2], [4], [13]. differences of them have been studied e.g. in [3], [9], [16], [17], [18]. let v and w be strictly positive, continuous and bounded functions (weights) on d and h(d) be the set of all analytic functions on d. in this article we are interested in differences ψ1cφ1 −ψ2cφ2 acting between weighted banach spaces of holomorphic functions h∞v := {f ∈ h(d); ‖f‖v := sup z∈d v(z)|f(z)| < ∞} and the weighted bloch type spaces bw of functions f ∈ h(d) satisfying ‖f‖bw := supz∈d v(z)|f ′(z)| < ∞. our aim is to give necessary and sufficient conditions for a difference ψ1cφ1 − ψ2cφ2 : h ∞ v → bw to be bounded resp. compact in terms of the involved weights and the analytic maps φ1,φ2,ψ1,ψ2. 2 notation and auxiliary results an introduction to the concept of composition operators can be found in the monographs [5] and [15]. in this article we are especially interested in radial weights (i.e. weights with v(z) = v(|z|) for every z ∈ d) which satisfy additionally the lusky condition (l1) (due to lusky [10]) (l1) inf k∈n v(1 − 2−k−1) v(1 − 2−k) > 0. when dealing with differences of weighted composition operators we need some geometric data. recall that for any z ∈ d, ϕz is the möbius transformation which interchanges the origin and z, namely ϕz(w) = z−w 1−zw ,w ∈ d. the pseudohyperbolic distance ρ(z,w) for z,w ∈ d is defined by ρ(z,w) = |ϕz(w)|. moreover, we have ϕ ′ z(w) = |z|2−1 (1−zw)2 for z,w ∈ d. let us recall some auxiliary results. the next lemma is taken from [3], see also [6]. lemma 1. (bonet-lindström-wolf [3]) let v be a radial weight satisfying the lusky condition (l1) and let f ∈ h∞v . then there exists a constant cv > 0 (depending on the weight v) such that |f(z) − f(p)| ≤ cv‖f‖v max { 1 v(z) , 1 v(p) } ρ(z,p) for all z,p ∈ d. theorem 2. (harutyunyan-lusky, [7] theorem 2.1) let v and w be radial weights which are continuously differentiable with respect to |z| with lim|z|→1 v(z) = lim|z|→1 w(z) = 0 and such that h ∞ w is isomorphic to l∞. if lim supr→1 ( − w ′(r) v(r) ) < ∞, then d : h∞v → h ∞ w ,f → f ′ is bounded. cubo 12, 2 (2010) differences of weighted composition operators ... 21 for conditions when h∞w is isomorphic to l∞ we refer the reader to [11] and [7]. by [7] we know that the following weights have the desired properties: w(z) = (1 − |z|)α,α > 0,w(z) = e − 1 1−|z| , z ∈ d. for the study of the compactness of the difference ψ1cφ1 − ψ2cφ2 we need the following result. proposition 3. (cowen-maccluer, [5] proposition 3.11) let x and y be h∞v or bw. then ψ1cφ1 − ψ2cφ2 : x → y is compact if and only if for every bounded sequence (fn)n∈n in x such that fn → 0 uniformly on the compact subsets of d, then (ψ1cφ1 − ψ2cφ2 )fn → 0 in y . 3 main result in the sequel we consider weights v of the following type: let ν be a holomorphic function on d, non-vanishing and strictly positive on [0, 1[. moreover we assume that ν is decreasing on [0, 1[ and satisfies limr→1 ν(r) = 0. then we define the corresponding weight v by v(z) := ν(|z| 2) for every z ∈ d. furthermore, we suppose the boundedness of the function ν′ on d. next, we give some illustrating examples of weights of this type: (i) consider ν(z) = (1−z)α, α ≥ 1. then the corresponding weight is the so-called standard weight v(z) = (1 − |z|2)α. (ii) selecting ν(z) = e − 1 (1−z)α , α ≥ 1, we obtain the weight v(z) = e − 1 (1−|z|2)α . (iii) choose ν(z) = sin(1 − z) and the corresponding weight is given by v(z) = sin(1 − |z|2). fix a point p ∈ d and an analytic self-map φ of d. we introduce a function vφ(p)(z) := ν(φ(p)z) for every z ∈ d. since ν is holomorphic on d, the function vφ(p) is also holomorphic on d. furthermore, vφ(p)(φ(p)) = ν(|φ(p)|2) = v(φ(p)) and v′ φ(p) (z) = φ(p)ν′(φ(p)z) for every z ∈ d, i.e. v′ φ(p) (φ(p)) = φ(p)ν′(|φ(p)|2). we start with considering boundedness of operators ψ1cφ1 − ψ2cφ2 : h ∞ v → bw and give first a necessary condition in terms of the involved weights and then a sufficient condition. proposition 4. let w be a weight and v be a weight as described in the beginning of this section. let ψ1,ψ2 ∈ h(d) and φ1,φ2 be analytic self-maps of d. if ψ1cφ1 − ψ2cφ2 : h ∞ v → bw is bounded, then the following conditions are satisfied (a) supz∈d w(z) ∣ ∣ ∣ ∣ ψ ′ 1(z) v(φ1(z)) 1 2 ϕ2 φ2(z) (φ1(z)) + 2 ψ1(z) v(φ1(z)) 1 2 ϕφ2(z)(φ1(z))ϕ ′ φ2(z) (φ1(z)) ∣ ∣ ∣ ∣ < ∞, (b) supz∈d w(z) ∣ ∣ ∣ ∣ ψ ′ 2(z) v(φ2(z)) 1 2 ϕ2 φ1(z) (φ2(z)) + 2 ψ2(z) v(φ2(z)) 1 2 ϕφ1(z)(φ2(z))ϕ ′ φ1(z) (φ2(z)) ∣ ∣ ∣ ∣ < ∞, (c) supz∈d ∣ ∣ ∣ ψ1(z)w(z)φ1(z)ν ′(|φ1(z)| 2) v(φ1(z)) ∣ ∣ ∣ ρ(φ1(z),φ2(z)) < ∞, (d) supz∈d ∣ ∣ ∣ ψ2(z)w(z)φ2(z)ν ′(|φ2(z)| 2) v(φ2(z)) ∣ ∣ ∣ ρ(φ1(z),φ2(z)) < ∞, 22 elke wolf cubo 12, 2 (2010) proof (a) fix a point p ∈ d and put fφ1(p)(z) := ( 2 vφ1(p)(z) − vφ1(p)(φ1(p)) vφ1(p)(z) 2 ) 1 2 and gφ1(p)(z) := fφ1(p)(z)ϕ 2 φ2(p) (z) for every z ∈ d. next, we get ‖gφ1(p)‖v ≤ sup z∈d ∣ ∣ ∣ ∣ v(z)2 2 vφ1(p)(z) − v(z)2 vφ1(p)(φ1(p)) vφ1(p)(z) 2 ∣ ∣ ∣ ∣ 1 2 ≤ (3m) 1 2 where m = supz∈d v(z) and therefore the constant does not depend on the choice of p. thus, gφ1(p) ∈ h∞v and g ′ φ1(p) (z) = f′ φ1(p) (z)ϕ2 φ2(p) (z)+2fφ1(p)(z)ϕ ′ φ2(p) (z)ϕφ2(p)(z) for every z ∈ d, where f ′ φ1(p) (z) = ( − v ′ φ1(p) (z) vφ1(p)(z) 2 + v ′ φ1(p) (z)vφ1(p)(φ1(p)) vφ1(p)(z) 3 )( 2 vφ1(p)(z) − vφ1(p)(φ1(p)) vφ1(p)(z) 2 )− 1 2 and ϕ′ φ2(p) (z) = |φ2(p)| 2−1 (1−φ2(p)z)2 for every z ∈ d. we get fφ1(p)(φ1(p)) = 1 v(φ1(p)) 1 2 and f′ φ1(p) (φ1(p)) = 0 and hence gφ1(p)(φ1(p)) = ϕ 2 φ2(p) (φ1(p)) v(φ1(p)) 1 2 as well as g′ φ1(p) (φ1(p)) = 2 ϕ ′ φ2(p) (φ1(p))ϕφ2(p)(φ1(p)) v(φ1(p)) 1 2 . now, w(p) ∣ ∣ ∣ ∣ ∣ ψ′1(p)ϕ 2 φ2(p) (φ1(p)) v(φ1(p))) 1 2 + 2 ψ1(p)ϕφ2(p)(φ1(p))ϕ ′ φ2(p) (φ1(p)) v(φ1(p))) 1 2 ∣ ∣ ∣ ∣ ∣ = w(p) ∣ ∣ ∣ ψ ′ 1(p)gφ1(p)(φ1(p)) + ψ1(p)φ ′ 1(p)g ′ φ1(p) (φ1(p)) − ψ ′ 2(p)gφ1(p)(φ2(p)) − ψ2(p)φ ′ 2(p)g ′ φ1(p) (φ2(p)) ∣ ∣ ∣ ≤ ‖ψ1cφ1 − ψ2cφ2‖‖gφ1(p)‖v < ∞. thus, (a) follows, and we can show (b) analogously. for the proof of condition (c) we fix a point p ∈ d and put fφ1(p)(z) := vφ1(p)(φ1(p)) vφ1(p)(z) − ( vφ1(p)(φ1(p)) vφ1(p)(z) ) 1 2 = v(φ1(p)) vφ1(p)(z) − ( v(φ1(p)) vφ1(p)(z) ) 1 2 and gφ1(p)(z) := fφ1(p)(z)ϕ 2 φ2(p) (z) for every z ∈ d. hence ‖gφ1(p)‖v ≤ 2m and we get g′φ1(p)(z) = f ′ φ1(p) (z)ϕ2φ2(p)(z) + 2fφ1(p)(z)ϕφ2(p)(z)ϕ ′ φ2(p) (z) for every z ∈ d, where f′φ1(p)(z) = − vφ1(p)(φ1(p))v ′ φ1(p) (z) vφ1(p)(z) 2 + 1 2 vφ1(p)(φ1(p)) 1 2 v′ φ1(p) (z) vφ1(p)(z) 3 2 thus, we obtain fφ1(p)(φ1(p)) = 0 and f ′ φ1(p) (φ1(p)) = − 1 2 φ1(p)ν ′(|φ1(p)| 2) v(φ1(p)) . hence gφ1(p)(φ1(p)) = 0 and g′ φ1(p) (φ1(p)) = − 1 2 φ1(p)ν ′(|φ1(p)| 2)ϕ2 φ2(p) (φ1(p)) v(φ1(p)) . finally, 1 2 w(p) ∣ ∣ ∣ ∣ ∣ φ1(p)ν ′(|φ1(p)| 2)ϕ2 φ2(p) (φ1(p)) vφ2(p)(φ1(p)) ∣ ∣ ∣ ∣ ∣ = w(p) ∣ ∣ ∣ ψ′1(p)gφ1(p)(φ1(p)) + ψ1(p)φ ′ 1(p)g ′ φ1(p) (φ1(p)) − ψ ′ 2(p)gφ1(p)(φ2(p)) − ψ2(p)φ ′ 2(p)g ′ φ1(p) (φ2(p)) ∣ ∣ ∣ ≤ ‖ψ1cφ1 − ψ2cφ2‖‖gφ1(p)‖v < ∞. cubo 12, 2 (2010) differences of weighted composition operators ... 23 the claim follows. we can show (d) analogously. proposition 5. let v and w be weights. if (a) there is a weight u such that the operator d : h∞v → h ∞ u ,f → f ′ is bounded and additionally supz∈d max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ρ(φ1(z),φ2(z)) < ∞ as well as supz∈d max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } |φ′1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| < ∞, (b) supz∈d max { 1 v(φ1(z)) , 1 v(φ2(z)) } w(z)|ψ′1(z) − ψ ′ 2(z)| < ∞, (c) supz∈d max { 1 v(φ1(z)) , 1 v(φ2(z)) } w(z) max{|ψ′1(z)|, |ψ ′ 2(z)|}ρ(φ1(z),φ2(z)) < ∞ then ψ1cφ1 − ψ2cφ2 : h ∞ v → bw is bounded. proof let f ∈ h∞v . using lemma 1 we obtain sup z∈d w(z)|((ψ1cφ1 − ψ2cφ2 )f) ′(z)| ≤ sup z∈d w(z)|ψ′1(z) − ψ ′ 2(z)||f(φ1(z))| + sup z∈d w(z)|ψ′2(z)||f(φ1(z)) − f(φ2(z))| + sup z∈d w(z)|f′(φ1(z))||φ ′ 1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| + sup z∈d w(z)|φ′2(z)ψ2(z)||f ′(φ1(z)) − f ′(φ2(z))| ≤ sup z∈d w(z)|ψ′1(z) − ψ ′ 2(z)| max { 1 v(φ1(z)) , 1 v(φ2(z)) } ‖f‖v + sup z∈d max{|ψ′1(z)|, |ψ ′ 2(z)|} max { 1 v(φ1(z)) , 1 v(φ2(z)) } ρ(φ1(z),φ2(z))‖f‖v + sup z∈d w(z) u(φ1(z)) ‖f′‖u|φ ′ 1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| + sup z∈d max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ρ(φ1(z),φ2(z))‖f ′‖u ≤ sup z∈d w(z)|ψ′1(z) − ψ ′ 2(z)| max { 1 v(φ1(z)) , 1 v(φ2(z)) } ‖f‖v + sup z∈d max{|ψ′1(z)|, |ψ ′ 2(z)|} max { 1 v(φ1(z)) , 1 v(φ2(z)) } ρ(φ1(z),φ2(z))‖f‖v + sup z∈d max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ‖d‖‖f‖v|φ ′ 1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| + sup z∈d max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ρ(φ1(z),φ2(z))‖d‖‖f‖v and the claim follows. next, we turn our attention to compactness of ψ1cφ1 − ψ2cφ2 : h ∞ v → bw. proposition 6. let w be a weight and v be a weight as described in the beginning of this section. let ψ1,ψ2 ∈ h(d) and φ1,φ2 be analytic self-maps of d. if ψ1cφ1 − ψ2cφ2 : h ∞ v → bw is bounded, then the following conditions are satisfied 24 elke wolf cubo 12, 2 (2010) (a) lim sup|φ1(z)|→1 w(z) ∣ ∣ ∣ ∣ ψ ′ 1(z) v(φ1(z)) 1 2 ϕ2 φ2(z) (φ1(z)) + 2 ψ1(z) v(φ1(z)) 1 2 ϕφ2(z)(φ1(z))ϕ ′ φ2(z) (φ1(z)) ∣ ∣ ∣ ∣ = 0, (b) lim sup|φ2(z)|→1 w(z) ∣ ∣ ∣ ∣ ψ ′ 2(z) v(φ2(z)) 1 2 ϕ2 φ1(z) (φ2(z)) + 2 ψ2(z) v(φ2(z)) 1 2 ϕφ1(z)(φ2(z))ϕ ′ φ1(z) (φ2(z)) ∣ ∣ ∣ ∣ = 0, (c) lim sup|φ1(z)|→1 ∣ ∣ ∣ ψ1(z)w(z)φ1(z)ν ′(|φ1(z)| 2) v(φ1(z)) ∣ ∣ ∣ ρ(φ1(z),φ2(z)) = 0, (d) lim sup|φ2(z)|→1 ∣ ∣ ∣ ψ2(z)w(z)φ2(z)ν ′(|φ2(z)| 2) v(φ2(z)) ∣ ∣ ∣ ρ(φ1(z),φ2(z)) = 0, proof (a) consider a sequence (zn)n ⊂ d such that |φ1(zn)| → 1 if n → ∞. we set fφ1(zn)(z) := vφ1(zn)(φ1(zn)) 1 6 ( 3 2 1 vφ1(zn)(z) 2 − vφ1(zn)(φ1(zn)) vφ1(zn)(z) 3 ) 1 3 and gφ1(zn)(z) := fφ1(zn)(z)ϕ 2 φ2(zn) (z) for every z ∈ d. thus ‖gφ1(zn)‖v ≤ supz∈d vφ1(zn)(φ1(zn)) 1 6 ∣ ∣ ∣ 3 2 v(z)3 vφ1(zn)(z) 2 − v(z)3vφ1(zn)(φ1(zn)) vφ1(zn)(z) 3 ∣ ∣ ∣ 1 3 ≤ m 1 6 ( 5 2 m ) 1 3 for every n ∈ n, where m := supz∈d v(z). thus, (gφ1(zn))n∈n is a bounded sequence in h ∞ v which converges to zero uniformly on the compact subsets of d. moreover, g′φ1(zn)(z) = f ′ φ1(zn) (z)ϕ2φ2(zn)(z) + 2fφ1(zn)(z)ϕφ2(zn)(z)ϕ ′ φ2(zn) (z) for every z ∈ d, where f ′ φ1(zn) (z) = vφ1(zn)(φ1(zn)) 1 6 ( 3 2 1 vφ1(zn)(z) 2 − vφ1(zn)(φ1(zn)) vφ1(zn)(z) 3 )− 2 3 · · ( − v′ φ1(zn) (z) vφ1(zn)(z) 3 + vφ1(zn)(φ1(zn)) vφ1(zn)(z) 4 v′φ1(zn)(z) ) for every n ∈ n. by proposition 3, the fact that ψ1cφ1 − ψ2cφ2 is compact yields ‖(ψ1cφ1 − ψ2cφ2 )gφ1(zn)‖bw → 0 if n → ∞. finally, ‖(ψ1cφ1 − ψ2cφ2 )gφ1(zn)‖bw ≥ w(zn) ∣ ∣ ∣ ∣ ∣ ψ′1(zn) v(φ1(zn)) 1 2 ϕ2φ2(zn)(φ1(zn)) + 2 ψ1(zn)ϕφ2(zn)(φ1(zn))ϕ ′ φ2(zn) (φ1(zn)) v(φ1(zn)) 1 2 ∣ ∣ ∣ ∣ ∣ thus, (a) follows and we can show (b) analogously. consider now fφ1(zn)(z) := vφ1(zn)(φ1(zn)) vφ1(zn)(z) − ( vφ1(zn)(φ1(zn)) vφ1(zn)(z) ) 1 2 = v(φ1(zn)) vφ1(zn)(z) − ( v(φ1(zn)) vφ1(zn)(z) ) 1 2 and gφ1(zn)(z) := fφ1(zn)(z)ϕ 2 φ2(zn) (z) for every z ∈ d. cubo 12, 2 (2010) differences of weighted composition operators ... 25 then ‖gφ1(zn)‖v ≤ supz∈d v(z) ∣ ∣ ∣ ∣ vφ1(zn)(φ1(zn)) vφ1(zn)(z) − ( vφ1(zn)(φ1(zn)) vφ1(zn)(z) ) 1 2 ∣ ∣ ∣ ∣ ≤ 2m for every n ∈ n. thus (gφ1(zn))n is a bounded sequence in h ∞ v and gφ1(zn) → 0 uniformly on every compact subset of d. moreover gφ1(zn)(φ1(zn)) = 0 and g ′ φ1(zn) (φ1(zn)) = − 1 2 v ′ φ1(zn) (φ1(zn)) v(φ1(zn)) ϕ2 φ2(zn) (φ1(zn)). since ψ1cφ1 − ψ2cφ2 is compact, by proposition 3 we have ‖(ψ1cφ1 − ψ2cφ2 )gn‖bw → 0 if n → ∞. thus, ‖(ψ1cφ1 − ψ2cφ2 )gφ1(zn)‖bw = sup z∈d w(z)|((ψ1cφ1 − ψ2cφ2 )gφ1(zn)) ′(z)| ≥ 1 2 w(zn)|ψ1(zn)φ ′ 1(zn)φ1(zn)|ρ(φ1(zn),φ2(zn)) 2 |ν ′(|φ1(zn)| 2)| v(φ1(zn)) . finally, lim sup|φ1(z)|→1 w(z)|ψ(z)||φ ′ 1(z)||φ1(z)| |ν′(|φ1(z)| 2)| v(φ1(z)) = 0, and (c) holds. (d) follows analogously. proposition 7. let v and w be weights. if (a) there is a weight u such that the operator d : h∞v → h ∞ u ,f → f ′ is bounded and additionally lim supmax{|φ1(z)|,|φ2(z)|}→1 max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ρ(φ1(z),φ2(z)) = 0 as well as lim supmax{|φ1(z)|,|φ2(z)|}→1 max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } |φ′1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| = 0, (b) lim supmax{|φ1(z)|,φ2(z)|}→1 max { 1 v(φ1(z)) , 1 v(φ2(z)) } w(z)|ψ′1(z) − ψ ′ 2(z)| = 0, (c) lim supmax{|φ1(z)|,|φ2(z)|}→1 max { 1 v(φ1(z)) , 1 v(φ2(z)) } w(z) max{|ψ′1(z)|, |ψ ′ 2(z)|}ρ(φ1(z),φ2(z)) = 0 then ψ1cφ1 − ψ2cφ2 : h ∞ v → bw is compact. proof let (fn)n∈n be a sequence in h ∞ v with ‖fn‖v ≤ 1 and fn → 0 uniformly on compact subsets of d. by the assumption, for any ε > 0 there is 0 < δ < 1 such that δ < max{|φ1(z)|, |φ2(z)|} < 1 implies max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ρ(φ1(z),φ2(z)) < ε max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } |φ′1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| < ε max { 1 v(φ1(z)) , 1 v(φ2(z)) } w(z)|ψ′1(z) − ψ ′ 2(z)| < ε max { 1 v(φ1(z)) , 1 v(φ2(z)) } w(z) max{|ψ′1(z)|, |ψ ′ 2(z)|}ρ(φ1(z),φ2(z)) < ε 26 elke wolf cubo 12, 2 (2010) then applying lemma 1 sup z∈d w(z)|((ψ1cφ1 − ψ2cφ2 )fn) ′(z)| ≤ sup z∈d w(z)|ψ′1(z) − ψ ′ 2(z)||fn(φ1(z))| + sup z∈d w(z)|ψ′2(z)||fn(φ1(z)) − fn(φ2(z))| + sup z∈d w(z)|f′n(φ1(z))||φ ′ 1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| + sup z∈d w(z)|φ′2(z)ψ2(z)||f ′ n(φ1(z)) − f ′ n(φ2(z))| ≤ sup {z; max{|φ1(z)|,|φ2(z)|}>δ} w(z)|ψ′1(z) − ψ ′ 2(z)| max { 1 v(φ1(z)) , 1 v(φ2(z)) } ‖fn‖v + sup {z; max{|φ1(z)|,|φ2(z)|}>δ} max{|ψ′1(z)|, |ψ ′ 2(z)|} max { 1 v(φ1(z)) , 1 v(φ2(z)) } ρ(φ1(z),φ2(z))‖fn‖v + sup {z; max{|φ1(z)|,|φ2(z)|}>δ} w(z) u(φ1(z)) ‖f′‖u|φ ′ 1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| + sup {z; max{|φ1(z)|,|φ2(z)|}>δ} max { w(z) u(φ1(z)) , w(z) u(φ2(z)) } ρ(φ1(z),φ2(z))‖f ′ n‖u + sup {z; max{|φ1(z)|,φ2(z)|}≤δ} w(z)|ψ′1(z) − ψ ′ 2(z)||fn(φ1(z))| + sup {z; max{|φ1(z)|,φ2(z)|}≤δ} w(z)|ψ′2(z)||fn(φ1(z))| + sup {z; max{|φ1(z)|,φ2(z)|}≤δ} w(z)|ψ′2(z)||fn(φ2(z))| + sup {z; max{|φ1(z)|,φ2(z)|}≤δ} w(z)|f′n(φ1(z))||φ ′ 1(z)ψ1(z) − φ ′ 2(z)ψ2(z)| + sup {z; max{|φ1(z)|,φ2(z)|}≤δ} w(z)|φ′2(z)ψ2(z)||f ′ n(φ1(z))| + sup {z; max{|φ1(z)|,φ2(z)|}≤δ} w(z)|φ′2(z)ψ2(z)||f ′ n(φ2(z))|. the claim follows. received: october 2008. revised: february 2009. references [1] k.d. bierstedt, j. bonet and j. taskinen, associated weights and spaces of holomorphic functions, studia math. 127 (1998), no. 2, 137-168. 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[18] e. wolf, differences of weighted composition operators, to appear in collectanea math. cubo a mathematical journal vol.10, n o ¯ 01, (19–32). march 2008 on the index of clifford algebras of quadratic forms syouji yano department of mathematics, graduate school of science osaka university, toyonaka, osaka, 560-0043 – japan email: yano@gaia.math.wani.osaka-u.ac.jp abstract in this paper, we determine the index of the clifford algebras of 6-dimensional quadratic forms over a field whose characteristic is unequal to 2. in the case that the characteristic is equal to 2, we compute the clifford algebras of the scharlau’s transfer of 4-dimensional quadratic forms with trivial arf invariant, and then investigate how the index of the clifford algebra of q depends on orthogonal decompositions of q when q is a low dimensional quadratic form. resumen en este art́ıculo determinamos el ı́ndice de la algebra de clifford de formas quadraticas 6-dimensionales cuja caracteŕıstica es distinta de dos. en el caso de caracteristica dos cálculamos la algebra de clifford de la traslación de scharlau de formas quadraticas 4-dimensionales con art invariante trivial e se investiga como el indice de la algebra de clifford de q depende de la descomposición ortogonal de q quando q es una forma quadrática de dimensión baja. 20 syouji yano cubo 10, 1 (2008) key words and phrases: quadratic forms, clifford algebras, index. math. subj. class.: 15a66. 1 introduction in his book [5], knus classified the clifford algebras c(q), the even clifford algebras c0(q) and the discriminant algebras z(q) of low dimensional quadratic forms q over a field f . in the case of dimension 6, knus showed the following classification table 1 [5, appendix a]. table 1 q : dimf q = 6 z(q) c0(q) c(q) ν = 0 f × f d4 × d4 m2(d4) ν = 0,νl = 0 l l ⊗ d4 ? ν = 0,νl = 1 l m2(l ⊗ d2) ? ν = 0,νl = 3 l m4(l) ? ν = 1 f × f m2(d2) × m2(d2) m4(d2) ν = 1,νq = 1 l m2(l ⊗ d2) m2(d4) ν = 1,νq = 2 l m2(l ⊗ d2) m4(d2) ν = 2, 1 ∈ q(h(f 2)⊥) l m4(l) m8(f) ν = 2, 1 /∈ q(h(f 2)⊥) l m4(l) m4(d2) ν = 3 f × f m4(f) × m4(f) m8(f) here q is the 8-dimensional quadratic form defined by nz ⊥ −q, where nz denotes the reduced norm form on z(q), and l is a separable quadratic extension over f . in the first column of the table 1, ν,νq and νl denote the witt index of q,q and ql, respectively. in the third and fourth columns of the table 1, dn denotes a central division f-algebra of dimension n 2 . in this paper we study the question marks of the table 1. for the 8-dimensional quadratic form q = nz ⊥ −q, it is known that c(q) ≃ m2(c(q)) and indc(q) = indc(q). hence indc(q) is determined by q. the solutions of second and third question marks of the table 1 are given as in the table 2 by considering how the form q is decomposed into the orthogonal sum of subform of 2 or 4 dimensions. in the case of ch(f) 6= 2, izhboldin and karpenko [8, theorem 16.10] proved that an 8-dimensional quadratic form φ has the trivial arf invariant and satisfies indc(φ) ≤ 4 if and only if φ is isometric either to (1) an orthogonal sum of two quadratic forms which are each similar to 2-fold pfister forms or (2) a scharlau’s transfer of a 4-dimensional quadratic form which is similar to a 2-fold pfister form over a quadratic extension of f . the solution of cubo 10, 1 (2008) on the index of clifford algebras of quadratic forms 21 first question mark of the table 1 is given as in the table 3 by applying this result to the form q. in the case of ch(f) = 2, we will prove that the if part of izhboldin and karpenko’s theorem also holds. whether the only if part of izhboldin and karpenko’s theorem holds or not for ch(f) = 2 is not known, but we will give some sufficient conditions for q to decomposed into an orthogonal sum of 2-fold pfister forms. we summarize in the following tables 2, 3 and 4 all the results we proved in this paper on indc(q) of a 6-dimensional anisotropic quadratic form q with non-trivial arf invariant. the table 2 gives a classification of c(q) in the case that νl ≥ 1 and the characteristic of f is arbitrary. the table 3 (resp. the table 4) gives a classification of c(q) in the case that νl = 0 and ch(f) 6= 2 (resp. ch(f) = 2). any positive condition of q such that indc(q) = 8 is not known. we use the following notations. gpr(f) : a set of similar forms of r-fold pfister forms over f . gp2(f)n := {⊥ n i=1 πi | πi ∈ gp2(f)} (gp2(f)1 = gp2(f).) e : a set of separable quadratic extensions of f . s ∗ e/f (gp2(e)) : image of gp2(e) by scharlau’s transfer s ∗ e/f . s = ∪e∈es ∗ e/f gp2(e) ∪ gp2(f)2. table 2 ch(f) ≥ 0,q : dimf q = 6,ν = 0,z(q) = l c0(q) c(q) νl = 1,νq = 0,q is of type e7 m2(l ⊗ d2) m4(d2) νl = 1,νq = 0,q is not of type e7 (q ∈ gp2(f)2) m2(l ⊗ d2) m2(d4) νl = 1,νq = 1 m2(l ⊗ d2) m2(d4) νl = 1,νq = 2 m2(l ⊗ d2) m4(d2) νl = 3,νq = 0,q is of type e7 m4(l) m4(d2) νl = 3,νq = 0,q is not of type e7 (q ∈ gp3(f)) m4(l) m8(f) νl = 3,νq = 2 m4(l) m4(d2) table 3 ch(f) 6= 2,q : dimf q = 6,ν = 0,z(q) = l c0(q) c(q) νl = 0,νq = 0,q /∈ s l ⊗ d4 d8 νl = 0,νq = 0,q ∈ s l ⊗ d4 m2(d4) νl = 0,νq = 1 l ⊗ d4 m2(d4) 22 syouji yano cubo 10, 1 (2008) table 4 ch(f) = 2,q : dimf q = 6,ν = 0,z(q) = l c0(q) c(q) νl = 0,νq = 0,q /∈ s l ⊗ d4 ? ∈ {d8,m2(d4)} νl = 0,νq = 0,q ∈ s l ⊗ d4 m2(d4) νl = 0,νq = 1 l ⊗ d4 m2(d4) by these results, we can make the following table 5 on the 8-dimensional quadratic forms with trivial arf invariant if ch(f) 6= 2. table 5 ch(f) 6= 2,q : dimf q = 8,z(q) = f × f c0(q) c(q) ν = 0,q 6∈ s d8 × d8 m2(d8) ν = 0,q ∈ s, q does not have a norm splitting m2(d4) × m2(d4) m4(d4) ν = 0,q is of type e7 m4(d2) × m4(d2) m8(d2) ν = 0,q ∈ gp3(f) m8(f) × m8(f) m16(f) ν = 1 m2(d4) × m2(d4) m4(d4) ν = 2 m4(d2) × m4(d2) m8(d2) ν = 4 m8(f) × m8(f) m16(f) as an application of tables 2, 3 and 4, we will show a minkowski-hasse type theorem in theorem 4.5. 2 notation and definition in this section we recall the basic notations on the quadratic forms. let f be a field of arbitrary characteristic. a quadratic space (v,q) over f is a pair of a finite dimensional f-vector space v and a quadratic form q : v −→ f such that q satisfies: 1. q(λv) = λ 2 q(v) for λ ∈ f,v ∈ v ; 2. bq : v × v −→ f defined by bq(v,w) = q(v + w) − q(v) − q(w) is an f-bilinear form. a quadratic form q is called regular if bq is nonsingular. we assume that all the quadratic forms are regular throughout this paper. a morphism of quadratic spaces φ : (v,q) −→ (v ′,q′) is an f-linear map φ : v −→ v ′ such that q(x) = q ′ (φ(x)) for all x ∈ v . if φ is an f-isomorphism, then it is called isometry. a quadratic form which represents 0 for some nonzero element in v is called isotropic, otherwise it is called anisotropic. a 2-dimensional isotropic quadratic space defined by cubo 10, 1 (2008) on the index of clifford algebras of quadratic forms 23 qh(x) = x1x2 for x = (x1,x2) ∈ f 2 is called hyperbolic space and denoted by h(f) = (f 2 ,qh). a quadratic form q is decomposed to an orthogonal sum of n-hyperbolic forms and an anisotropic form q0, i.e.,q ≃ q n h ⊥ q0. then n is uniquely determined by q and is called the witt index of q and denoted by ν(q). if ch(f) 6= 2, then n-dimensional quadratic form is isometric to a diagonal form q(x) = ∑n i=1 aix 2 i , (x = (x1, · · · ,xn) ∈ f n ) the q is denoted by < a1, · · · ,an >. in characteristic 2, the dimension of a regular quadratic form is always even and the diagonal quadratic forms are not regular. we can decompose 2m-dimensional quadratic form into q(x) = ∑m i=1(aix 2 2i−1 + x2i−1x2i + bix 2 2i). this q is denoted by [a1,b1] ⊥ · · · ⊥ [am,bm]. in general the signed discriminant δ(q) of 2m-dimensional quadratic form q is defined to be δ(q) = (−1)m det bq as an element of f • /f •2 . if ch(f) = 2, then the signed discriminant of q is trivial. in this case, for a quadratic form q = [a1,b1] ⊥ · · · ⊥ [am,bm], we define the classical arf invariant α(q) of q by α(q) = a1b1 +· · ·+ambm as an element of f/℘(f), where ℘(f) = {x + x2,x ∈ f}. we have δ(q ⊥ q′) = δ(q)δ(q′) and α(q ⊥ q′) = α(q) + α(q′). a form ≪ a1, · · · ,an ≫=< 1,a1 > ⊗ · · · ⊗ < 1,an > if ch(f) 6= 2, and a form [[b,a1, · · · ,an−1 ≫= [1,b]⊗ ≪ a1, · · · ,an−1 ≫ if ch(f) = 2 are called an n-fold pfister form. we denote by gpn(f) the set of all similar forms to n-fold pfister forms let f ⊂ e be a field extension. we can extend a quadratic form q : v −→ f to a form qe : e ⊗ v −→ e by putting qe ( ∑ i λi ⊗ vi ) = ∑ i λ 2 iq(vi) + ∑ i σ(g), hence m = m ∩ mi for some i; in particular if m is a maximal subgroup of g, then m = mi ∈ m. so m contains all the c complements of n which are maximal; since the union of these complements does not cover n , we need at least c + 1 subgroups in m. (2) let lg(n ) be the smallest index of a proper subgroup of g supplementing n . by lemma 3.2 in [14] a minimal cover m of g contains at least lg(n ) subgroups which supplement n . on the other hand, if all the subgroups in m are supplements of n , then by [8, lemma 1] we have lg(n ) ≤ σ(g) − 1. in any case we conclude σ(g) ≥ lg(n ) + 1 ≥ l(s) r + 1. corollary 6. let n be a minimal normal subgroup of a group g. if σ(g) < σ(g/n ), then 1. if n is abelian, complemented and non-central, then |n| + 1 ≤ σ(g); 2. if n = sr where s is a non-abelian simple group, then 5r + 1 ≤ σ(g). proposition 7. let n be a non-solvable normal subgroup of a finite group g. then σ(g) ≤ |n|−1. proof. consider the centralizers in g of the nontrivial elements of n : if there exists an element g ∈ g which does not belong to ⋂ 16=n∈n cg(n) then the subgroup 〈g〉 acts fixed point freely on n . by the classification of finite simple groups (see e.g. [15]), it follows that n is solvable, a contradiction. hence σ(g) ≤ |n| − 1. corollary 8. if n is a non-abelian minimal normal subgroup of g and δg(n ) > 1, then σ(g) = σ(g/n ). proof. assume by contradiction that σ(g) < σ(g/n ). since δg(n ) > 1, there exists a maximal subgroup m of g, such that g/mg is a primitive group of type iii and m/mg is a common complement of the two minimal normal subgroups of the socle h/mg × n mg/mg of g/mg. in particular m is a non-normal complement of n and it has |n| conjugates, hence |n| + 1 ≤ σ(g) by lemma 5. this contradicts proposition 7. corollary 9. let p a large prime not of the form (qk − 1)/(q − 1) where q is a prime power and k an integer; then σ(alt(5) ≀ alt(p)) < σ(alt(p)). proof. by proposition 7, σ(alt(5) ≀ alt(p)) < | alt(5)|p. on the other hand, by theorem [12, 4.4], σ(alt(p)) ≥ (p − 2)! > 60p for a large enough prime not of the form (qk − 1)/(q − 1). cubo 10, 3 (2008) on the structure of primitive n-sum groups 199 proposition 10. let g be a finite group. if v is a complemented normal abelian subgroup of g and v ∩ z(g) = 1, then σ(g) < 2|v |. in particular, if v is a minimal normal subgroup, then σ(g) ≤ 1 + q + · · · + qn where q = | endg(v )| and |v | = q n. proof. let h be a complement of v in g; we shall prove that g is covered by the family of subgroups a = {hv | v ∈ v } ∪ {ch (v)v | 1 6= v ∈ v }. let g = hw ∈ g, where h ∈ h, w ∈ v . if h /∈ ch (v) for every v ∈ v \ {1}, then cv (h) = 1 and the cardinality of the set {h v | v ∈ v } is |v : cv (h)| = |v |. therefore {h v | v ∈ v } = {hv | v ∈ v } and g = hw ∈ hv for some v ∈ v . thus σ(g) ≤ |a| ≤ |v | + (|v | − 1) < 2|v |. in particular, if v is h-irreducible, then endg(v ) = endh (v ) = f is a finite field. we may identify h with a subgroup of gl(n, q), where |f| = q and dimf v = n. in this case g is covered by a = {h v | v ∈ v } ∪ {ch (w )v | w ≤ v, dimf w = 1}, so σ(g) ≤ qn + (1 + · · · + qn−1). corollary 11. let h be a finite group, v an h-module, g = v ⋊ h the semidirect product of v by h and assume that cv (h) = 0. then 1. if h1(h, v ) 6= 0, then σ(g) = σ(h); 2. if σ(h) ≥ 2|v |, then h1(h, v ) = 0. proof. assume by contradiction that σ(g) < σ(h). by lemma 5, c + 1 ≤ σ(g) where c is the number of complements of v in g. if h1(h, v ) 6= 0, then there are at least two conjugacy classes of complements for v in g and, since cv (h) = 0, any conjugacy class consists of |v | complements, hence c ≥ 2|v | and σ(g) > 2|v | against proposition 10. corollary 12. let v the fully deleted module for alt(n) over f2 and let g be the semidirect product of v by alt(n). 1. if n = p is a large odd prime not of the form (qk − 1)/(q − 1) where q is a prime power and k an integer, then σ(g) < σ(alt(n)). 2. if n is even, then σ(g) = σ(alt(n)) proof. 1) since |v | = 2p−1 (see e.g. [11, prop. 5.3.5]), proposition 10 gives that σ(g) < 2|v | < 2p. on the other hand, by theorem [12, 4.4], σ(alt(p)) ≥ (p − 2)! > 2p for a large enough prime not of the form (qk − 1)/(q − 1). 2) this follows from corollary 11 and the fact that h1(alt(n), v ) 6= 0 whenever n is even (see e.g. [2, p. 74]). corollary 13. let v 6= w be non-frattini non-central abelian minimal normal subgroups of g. then 1. if δg(v ) > 1, then σ(g) = σ(g/v ); 200 eloisa detomi and andrea lucchini cubo 10, 3 (2008) 2. σ(g) = min{σ(g/v ), σ(g/w )}. proof. 1) by a result in [3], the number c of complements of v in g is c = | der(g/v , v )| = | endg/v (v )| δg(v )−1| der(g/cg(v ), v )| hence c ≥ 2|v | whenever δg(v ) > 1. if σ(g) < σ(g/v ), then by lemma 5 and proposition 10, 2|v | < c + 1 ≤ σ(g) < 2|v |, a contradiction. 2) if v and w are g-equivalent, then by (1) σ(g) = σ(g/v ) = σ(g/w ). so assume that v and w are not g-equivalent and, by contradiction, that σ(g) < min{σ(g/v ), σ(g/w )}. a complement of v in g has at least |v | conjugates and it is a maximal subgroup of g, so we can find at least |v | complements of v . in the same way there are at least |w | distinct complements of |w | in g. moreover, since v and w are not g-equivalent, v and w cannot have a common complement. arguing as in lemma 5 we see that all the complements of v and w have to appear in a minimal cover of g. therefore σ(g) ≥ |v | + |w | ≥ min{2|v |, 2|w |}, against proposition 10. 3 the structure of σ-primitive groups we collect some known properties of σ-primitive groups and some consequences of the previous section. corollary 14. let g be a non-abelian σ-primitive group. then: 1. z(g) = 1; 2. the frattini subgroup of g is trivial; 3. if n is a minimal normal subgroup of g, then δg(n ) = 1; 4. there is at most one abelian minimal normal subgroup of g; 5. the socle soc(g) = g1 × · · · × gn is a direct product of non-g-equivalent minimal normal subgroups and at most one of them is abelian. 6. g is a subdirect product of the monolithic primitive groups xi = g/rg(gi) associated to the minimal normal subgroups gi, 1 ≤ i ≤ n. proof. part (1) is theorem 4 in [8]. if φ(g) is the frattini subgroup of g and h is a proper subgroup of g, then also h φ(g) is a proper subgroup of g. hence we can assume that φ(g) is contained in every subgroup of a minimal cover of g so that σ(g) = σ(g/ φ(g)) and therefore (2) holds. parts (3) and (4) follows from corollaries 8 and 13. then (3) and (4) implies (5). to prove (6) we consider the intersection r = ⋂n i=1 rg(gi). if r 6= 1, then r contains a minimal normal cubo 10, 3 (2008) on the structure of primitive n-sum groups 201 subgroup n of g. by (2) and (5), n is non-frattini and g-equivalent to gi for some 1 ≤ i ≤ n. hence by proposition 4 (5), rg(gi)n 6= rg(gi), in contradiction with n ≤ r ≤ rg(gi). definition 15. let x be a primitive monolithic group and let n be its socle. for any non-empty union ω = ⋃ i ωin of cosets of n in x with the property that 〈ω〉 = x, define σω(x) to be the minimum number of supplements of n in g needed to cover ω. then we define σ ∗(x) = min { σω(x) | ω = ⋃ i ωin, 〈ω〉 = x } . proposition 16. let g be a non-abelian σ-primitive group, g1, . . . , gn the minimal normal subgroups, and x1, . . . xn the monolithic primitive groups associated to gi, i = 1, . . . n. then σ(g) ≥ ∑n i=1 σ∗(xi). proof. let m be a set of σ = σ(g) maximal subgroups whose union is g. define m¬gi = {m ∈ m | m 6≥ gi}; note that • m¬gi 6= ∅ for each 1 ≤ i ≤ n; otherwise every maximal subgroup of m would contain gi and the set {m/gi | m ∈ m} would cover g/gi with σ(g) < σ(g/gi) subgroups. • m¬gi ∩m¬gj = ∅ for i 6= j; indeed if there exists m ∈ m¬gi ∩m¬gj , then gimg/mg and gj mg/mg are minimal normal subgroups of the primitive group g/mg, hence δg(gi) ≥ 2, contrary to corollary 14. therefore m contains the disjoint union of the non-empty sets m¬gi , 1 ≤ i ≤ n, and we are reduced to prove that |m¬gi| ≥ σ ∗(xi), for every i. let us fix an index i and let π : g 7→ x be the projection of g over x = xi. we set n = soc x ∼= gi, mi = {m ∈ m | m ≥ gi} = m\m¬gi and ω = { π(g) | g ∈ g \ ⋃ m∈mi m } . by minimality of the cover m, g 6= ⋃ m∈mi m hence ω 6= ∅. moreover, as gi ≤ m ∈ mi and π(gi) = soc x = n , we get that for every x ∈ ω the coset xn is contained in ω. if 〈ω〉 = h 6= x, then g is covered by the set mi ∪ {π −1(h)} and this actually is a minimal cover of g, since |mi| + 1 ≤ σ. but then, as π −1(h) ≥ gi, we would have σ(g/gi) ≤ |mi| + 1 = σ(g), a contradiction. hence 〈ω〉 = x. now we shall prove that |m¬gi| ≥ σω(x) ≥ σ ∗(x). by [9, proposition 11] the kernel r = rg(gi) of the projection πi of g over x has the property that if h is a proper subgroup of g such that hgi = g then hr 6= g. therefore every maximal subgroup m ∈ m¬gi contains r, m = π−1(π(m )) and π(m ) is a a maximal subgroup of x supplementing n. clearly, as ⋃ m∈m¬g i m covers g \ ⋃ m∈mi m , we have that ⋃ m∈m¬g i π(m ) covers ω. therefore |{π(m ) | m ∈ m¬gi}| = |m¬gi| ≥ σω(x) ≥ σ ∗(x). 202 eloisa detomi and andrea lucchini cubo 10, 3 (2008) remark 17. for every primitive monolithic group xi, σ ∗(xi) ≤ lxi (soc(xi)), where lxi (soc(xi)) is the smallest index of a proper subgroup of xi supplementing soc(xi). indeed, if a supplement of ni = soc(xi) in xi has non trivial intersection with a coset gni, then |gni ∩ m| = |ni ∩ m| = |gni|/|g : m|, and therefore we need at least lxi (soc(xi)) supplements to cover gni. so in particular the previous proposition implies that σ(g) ≥ ∑n i=1 lxi (ni). lemma 18. let n be a normal subgroup of a group x. if a set of subgroups covers a coset yn of n in x, then it also covers every coset yαn with α prime to |y|. proof. let s be an integer such that sα ≡ 1 mod |y|. as s is prime to |y|, by a celebrated result of dirichlet, there exists infinitely many primes in the arithmetic progression {s + |y|n | n ∈ n}; we choose a prime p > |x| in {s + |y|n | n ∈ n}. clearly, yp = ys. as p is prime to |x|, there exists an integer r such that pr ≡ 1 mod |x|. hence, if yn ⊆ ∪i∈i mi, for every g ∈ y αn we have that gp ∈ (yα)pn = (yα)sn = yn ⊆ ∪i∈i mi and therefore also g = (g p)r belongs to ∪i∈i mi. corollary 19. let g be a non-abelian σ-primitive group, n a minimal normal subgroup and x the monolithic primitive groups associated to n . then: 1. if x = n , then g = n ; 2. if |x/n| is a prime, then g = x. proof. note that if x = n , then there is only one coset of n in x hence ω = n , σ∗(n ) = σn (n ) = σ(n ). by proposition 16, σ ∗(n ) = σ(n ) ≤ σ(g). as n = x is a homomorphic image of g, we get g = n . now let |x/n| be a prime. let ω be a non-empty union of cosets of n in x with the property that 〈ω〉 = x; then ω contains a coset yn which is a generator for x/n . by lemma 18 we have that if ⋃ i mi covers ω, then ⋃ i mi covers every coset of n with the exception, at most, of the subgroup n itself. hence, σ(x) ≤ σω(x) + 1 that is σ ∗(x) ≥ σ(x) − 1. by proposition 16, σ(g) ≥ ∑n i=1 σ∗(xi). moreover, by remark 16, σ ∗(xi) ≥ 2. therefore, as σ(g) ≤ σ(x), there is no room for another minimal normal subgroup in g. corollary 20. if n = alt(n), n 6= 6, is a normal subgroup of g, then either σ(g) = σ(g/n ) or g ∈ {sym(n), alt(n)}. proof. it is sufficient to consider a σ-primitive image of g and then apply corollary 19. actually, the corollary holds also for n = 6, thanks to the following proposition. proposition 21. let g be a σ-primitive group and let o∞(g) be the smallest normal subgroup of g such that g/ o∞(g) is solvable. if g is non solvable, then g/ o∞(g) is a cyclic group. cubo 10, 3 (2008) on the structure of primitive n-sum groups 203 proof. by corollary 14, g is a subdirect product of the monolithic primitive groups xi associated to the minimal normal subgroups gi, 1 ≤ i ≤ n; call ni = soc(xi) ∼= gi. let m be a set of σ = σ(g) maximal subgroups whose union is g and define m¬gi = {m ∈ m | m 6≥ gi}. let mi be the minimal index of a supplement of ni in xi: by remark 17, σ(g) ≥ ∑n i=1 mi. let r = o∞(g) and assume by contradiction that g/r is not cyclic. then, by tomkinson’s result [14], σ(g/r) = q + 1 where q is the order of the smallest chief factor a = h/k of g/r having more than a complement in g/r. as g is not solvable, then σ(g) < σ(g/r) = q + 1. since g is the subdirect product of the xi’s, without loss of generality we can assume that a is a chief factor of x = x1. if n = soc(x) is an elementary abelian p-group, then, by corollary 6 and corollary 14 (1), |n| + 1 ≤ σ(g) < q + 1. therefore |n| < q and a is a chief factor, say u/v , of an irreducible linear group x/n ≤ gl(n ) acting on n . by clifford theorem, u is a completely reducible linear group hence op(u ) = 1. then, by theorem 3 in [4], |u/u ′| < |n| < q, against |a| = |u/v | = q. assume now that n = s is a simple non-abelian group. then a is isomorphic to a chief factor of a subgroup of out(s) hence q = |a| ≤ | out(s)| < m1 (see e.g. lemma 2.7 [4]). but σ(g) ≥ ∑n i=1 mi ≥ m1 > q, against σ(g) < q + 1. we are left with the case n = sr where s is a simple non-abelian group. then x/n is isomorphic to a subgroup of out(s) ≀ sym(r). if a is isomorphic to a chief factor of a transitive subgroup of sym(r), then theorem 2 in [4] gives that q = |a| ≤ 2r < (n1) r ≤ m1, where n1 is the minimal index of a subgroup of s. but this contradicts m1 ≤ σ(g) ≤ q. therefore a has to be a chief factor of a subgroup of out(s)r. then q = |a| ≤ | out(s)|r ≤ nr 1 ≤ m1 gives the final contradiction. lemma 22. let g be a non-solvable transitive permutation group of degree n. then either σ(g) ≤ 4n or every non-abelian composition factor of g is isomorphic to an alternating group of odd degree. proof. let g be a non-solvable transitive permutation group of degree n. we can embed g into a wreath product of its primitive components, let say g ≤ k1 ≀ k2 ≀ · · · ≀ kt where ki is a primitive permutation group of degree ni and n1n2 · · · nt = n (see for example [7]). let kj be a non-solvable component and assume that kj is not an alternating or symmetric group of odd degree; then g has an homomorphic image g which is embedded in a wreath product k ≀ h where k = kj is a permutation group of degree a = nj and h has degree b with ab ≤ n. if k does not contain alt(a) then |k| ≤ 4a [13] and g has a non-solvable normal subgroup of order at most 4ab. by proposition 7 this implies that σ(g) ≤ σ(g) ≤ 4ab ≤ 4n. so assume that k contains alt(a) where a is even. we identify g with its image in k ≀ h: g is a transitive group of degree ab, with a system of imprimitivity b with blocks of size a and k is the permutation group induced on a block by its stabilizer. let m1 be the set of subgroups g ∩ m where m is a maximal intransitive subgroup of sym(ab) and let m2 be the set of subgroups g∩(m ≀h) where m ∼= sym(a/2)≀sym(2) is a maximal imprimitive subgroup of sym(a); if t ∈ m2 and b ∈ b, then the permutation group induced on b by the stabilizer tb is isomorphic to the imprimitive proper subgroup sym(a/2) ≀ sym(2) of k, 204 eloisa detomi and andrea lucchini cubo 10, 3 (2008) hence t is a proper subgroup of g. now let x ∈ g: if x is not a cycle of length ab then there exists t ∈ m1 containing x; otherwise there exists t ∈ m2 containing x. hence the set m1 ∪m2 covers g with ab/2 ∑ i=1 ( ab i ) + 1 2 ( a a/2 ) ≤ 2ab ≤ 2n proper subgroups. therefore σ(g) ≤ 2n. proposition 23. let g be a σ-primitive group with a non-abelian minimal normal subgroup n . if g/n cg(n ) is not cyclic, then all the non-abelian composition factors of g/n cg(n ) are alternating groups of odd degree. proof. let n = sr, where s is a non-abelian simple group. by corollary 6, 5r + 1 ≤ σ(g). denote by x the monolithic primitive group associated to the g-group n ; then x is a subgroup of aut(s) ≀ sym(r). let k be the image of x in sym(r). if k is solvable, then, by schreier conjecture, x/ soc(x) ∼= g/n cg(n ) is solvable. by proposition 21 it follows that g/n cg(n ) is cyclic. thus, if g/n cg(n ) is not cyclic, then k is non-solvable. since 5 r + 1 ≤ σ(g) ≤ σ(k), the previous lemma implies that every non-abelian composition factor of k is an alternating group of odd degree. then, by schreier conjecture, the same holds for g/n cg(n ). theorem 24. let g be a σ-primitive group with no abelian minimal normal subgroups. then either g is a primitive monolithic group and g/ soc(g) is cyclic, or g/ soc(g) is non-solvable and all the non-abelian composition factors of g/ soc(g) are alternating groups of odd degree. proof. by corollary 14, g is a subdirect product of the monolithic primitive groups xi associated to the minimal normal subgroups gi, 1 ≤ i ≤ n. by proposition 23 and proposition 21, for every i, g/gicg(gi) ∼= xi/ soc(xi) is either cyclic or non-solvable and all of its non-abelian composition factors are alternating groups of odd degree. therefore either g/ soc(g) is solvable (hence cyclic by proposition 21) or non-solvable and all of its non-abelian composition factors are alternating groups of odd degree. we are left to prove that if g/ soc(g) is cyclic then n = 1. assume by contradiction that n ≥ 2. let ui be the number of distinct prime divisors of the order of the cyclic groups xi/ soc(xi) and assume that u1 ≤ · · · ≤ un. step 1. let mi be the minimal index of a supplement of soc(xi) in xi; then mi ≥ ui if soc(xi) = s is a simple group, then xi/s is isomorphic to a subgroup of out(s), and thus ui ≤ 2 ui ≤ | out(s)| ≤ mi (see e.g. lemma 2.7 [4]). if soc(xi) = s r where r 6= 1, then xi/ soc(xi) is isomorphic to a subgroup y of out(s) ≀ sym(r). let k be the intersection of y with the base subgroup (out(s))r of the wreath product cubo 10, 3 (2008) on the structure of primitive n-sum groups 205 out(s) ≀ sym(r) and let a be the number of distinct prime divisors of |k|; since |k| divides | out(s)|r, we get that 2a ≤ | out(s)| ≤ ni where ni is the minimal index of a subgroup of s. now b = ui − a is smaller or equal than the number of distinct prime divisors of the order of y /k which is isomorphic to a non trivial subgroup of sym(r), hence 1 ≤ b < r and thus ui = a + b ≤ (2 a)b ≤ (2a)r ≤ (ni) r ≤ mi whenever a > 0. if a = 0, then xi/ soc(xi) is isomorphic to a subgroup of sym(r) and thus ui < r ≤ (ni) r ≤ mi. this proves the first step. let π be the projection of g over x = x1 and call n = soc x. note that there exist precisely u1 maximal subgroups of the cyclic group x/n ; let h1, . . . , hu1 be the maximal subgroups of g such that their images in x/n give all the maximal subgroups of x/n . let m be a set of σ = σ(g) maximal subgroups whose union is g and define a to be the set of maximal subgroups of m containing g1, b = m \ a and ω = { π1(g) | g ∈ g \ ⋃ m∈a m } . step 2. assume that ω contains a coset yn such that 〈yn〉 = x/n . by lemma 18, if ω is covered by σω(x) maximal subgroups, then the same subgroups cover every coset yαn with α prime to |y|. all the other elements of x are covered by the u1 maximal subgroups π(h1), . . . , π(hu1 ), since the images of these elements are not generators of x/n . then σ(x) ≤ σω(x)+u1. on the other hand, by proposition 16, σω(x)+ ∑ i6=1 σ∗(xi) ≤ σ(g) < σ(x), hence ∑ i6=1 σ∗(xi) < u1. remark 17 and step 1 give that ∑ i6=1 ui ≤ ∑ i6=1 mi < u1, and this contradicts the minimality of u1. step 3. assume that ω does not contain a coset yn such that 〈yn〉 = x/n . then ω is covered by the images in x of the subgroups h1, . . . , hu1 and thus, by definition of ω, g is covered by the subgroups in a and h1, . . . , hu1 . it follows that |b|+|a| = σ(g) ≤ u1 +|a|, hence, by step 1, |b| ≤ u1 ≤ m1, against lemma 3.2 in [14]. this final contradiction implies that g has to be a primitive monolithic group and proves the proposition. 4 there is no group for which σ(g) = 11 in this section we will show that σ(g) can never be equal to 11. the first trivial observation is that σ(g) 6= 11 whenever g is solvable, since in this case by tomkinson’s result σ(g) = q + 1, for a prime power q. assume by contradiction that there exists a primitive 11-sum group g. by corollary 14, soc(g) is the direct product of n non g-equivalent minimal normal subgroups g1, . . . , gn, where at most one of them is abelian. 206 eloisa detomi and andrea lucchini cubo 10, 3 (2008) lemma 25. suppose that g is a primitive 11-sum group. then g has no abelian minimal normal subgroups. proof. assume by contradiction that g1 is abelian. by corollary 14, g1 is a complemented noncentral factor of g, hence, by corollary 6, |g1| + 1 ≤ σ(g) = 11. moreover, by proposition 10, 11 = σ(g) < 2|g1|. hence |g1| can only be 7, 2 3 or 32. actually, if |g1| = 7, then the bound in proposition 10 gives σ(g) ≤ 1 + 7, against σ(g) = 11. note that, by proposition 16, σ(g) = 11 ≥ ∑n i=1 σ∗(xi) where xi are the monolithic groups associated to the gi’s; since g1 is the only abelian subgroup and σ ∗(xi) ≥ 5 if gi is non-abelian, then g1 is the unique minimal normal subgroup of g and g ≤ g1 ⋊ aut(g1). if |g1| = 9, then g ≤ f 2 3 ⋊ gl(2, 3); hence g is solvable, a contradiction. thus |g1| = 8 and g = f 3 2 ⋊ gl(3, 2), since every proper subgroup of gl(3, 2) is solvable. let m = {m1, · · · , m11} be a set of 11 maximal subgroups covering g. in [6] it is proved that σ(gl(3, 2)) = 15 and, in particular, that one needs at least 7 subgroups to cover the seven point stabilizers of gl(3, 2). it follows that all the 8 complement of g1 in g occur in m, let say they are m1, . . . , m8. as in the proof of proposition 10, for every point stabilizer g ∈ gl(3, 2) there exists an element vg ∈ g1 such that gvg does not belong to any complement of g1 in g. hence the remaining subgroups m9, m10, m11 of m have to cover all the elements gvv where g is a point stabilizer. since m9, m10 and m11 contain g1, this would imply that we can cover the seven point stabilizers of gl(3, 2) with only three subgroups, a contradiction. theorem 26. there is no group g with σ(g) = 11. proof. suppose that g is a primitive 11-sum group and let g1, . . . , gn be its minimal normal subgroups. by the previous lemma every gi is non-abelian. if gi = alt(5) for some i, then, by corollary 20, g = alt(5) or sym(5). otherwise, σ∗(xi) ≥ lxi (gi) > 5 for every i and proposition 16 implies that there is at most one minimal normal subgroup in g. by the same argument, if g1 = s r, where s is a simple non-abelian group, since lx1 (g1) ≥ 5 r and, by lemma 5, 5r +1 ≤ σ(g) = 11, we have that g1 = s and lx1 (g1)+1 ≤ 11. therefore g is an almost-simple group with socle s and lg(s) ≤ 10, in particular s ∈ {alt(n) | 5 ≤ n ≤ 10} ∪ {sym(n) | 5 ≤ n ≤ 10} ∪ {psl(2, q) | 7 ≤ q ≤ 8}. thanks to the works of maroti [12] and bryce et al. [6], we can exclude most of these cases: indeed σ(alt(n)) ≥ 2n−2 if n 6= 7, 9, σ(alt(5)) = 10, σ(alt(9)) ≥ 80, σ(sym(n) = 2n−1 if n is odd and n 6= 9, σ(sym(9)) ≥ 172, σ(psl(2, 7)) = 15, σ(pgl(2, 7)) = 29, σ(psl(2, 8)) = 36. moreover, σ(aut(alt(6))) ≤ σ(c2 × c2) = 3 and σ(sym(6)) = 13 (see e.g. [1]). the remaining cases are g = alt(7), sym(8), sym(10), m10, pgl(2, 9) and aut(psl(2, 8)). • g 6= alt(7). assume by contradiction σ(alt(7)) = 11. there are seven maximal subgroups of alt(7) isomorphic to alt(6); since σ(alt(6)) = 16 > 11, each of them has to appear in a minimal cover of g. moreover, there are two conjugacy classes with 15 maximal subgroups isomorphic to cubo 10, 3 (2008) on the structure of primitive n-sum groups 207 psl(3, 2) and since σ(psl(3, 2)) = σ(psl(2, 7)) = 15 > 11 we have that σ(alt(7)) is at least 7 + 15 + 15. • g 6= sym(8). if σ(sym(8)) ≤ 11 then, since σ(sym(7)) = 26 and σ(alt(8)) ≥ 26, arguing as in the previous case we get that a minimal cover m of sym(8) contains the 8 point stabilizers and alt(8). let g1 = (1, 2, 3, 4, 5, 6, 7, 8), g2 = (1, 2, 3, 7, 4, 5, 6, 8) and g3 = (1, 2, 3, 5, 4, 6, 7, 8); any couple of them generate sym(8) so that we need at least 3 more subgroups in m, and thus σ(sym(8)) > 11. • g 6= sym(10). if σ(sym(10)) ≤ 11, then, as σ(sym(9)) = 28 and σ(alt(10)) ≥ 28, a minimal cover m of sym(10) contains 10 point stabilizers and alt(10). but these subgroups do not cover the 10-cycles. thus σ(sym(10)) > 11. • g 6= m10. in m10 there are 180 elements of order 8. the only maximal subgroups containing elements of order 8 are the sylow 2-subgroups and each of them contains 4 of these elements; thus we need at least 180/4 = 45 subgroups to cover the elements of order 8. • g 6= pgl(2, 9). in pgl(2, 9) there are 144 elements of order 10. the only maximal subgroups containing elements of order 10 are the normalizers of the sylow 5-subgroups and each of them contains 4 of these elements; thus we need at least 144/4 = 36 subgroups to cover the elements of order 10. • g 6= aut(psl(2, 8)). in aut(psl(2, 8)) \ psl(2, 8) there are 336 elements of order 9. the only maximal subgroups containing elements of this kind are the normalizers of the sylow 3-subgroups; each of them contains 12 of these elements thus we need at least 336/12 = 28 subgroups to cover aut(psl(2, 8)). 5 direct products proposition 27. let g = h1 × h2 be the direct product of two subgroups. let ni be the smallest normal subgroup of hi such that hi/ni is a direct product of simple groups. if h1/n1 and h2/n2 have at most one non-abelian simple group s in common and the multiplicity of s in h1/n1 is at most one, then either σ(g) = min{σ(h1), σ(h2)}, or the cyclic group cp is an epimorphic image of both h1 and h2 and σ(g) = p + 1. proof. let g be a counterexample with minimal order. we first prove that g is a σ-primitive group. as φ(g) = φ(h1) × φ(h2), we have φ(g) = 1. let n be a minimal normal subgroup of g and assume by contradiction that σ(g) = σ(g/n ). if n ≤ h1, then, by minimality of |g|, we have that either σ(g/n ) = σ(h1/n × h2) = min{σ(h1/n ), σ(h2)} ≥ min{σ(h1), σ(h2)} ≥ σ(g), and so σ(g) = min{σ(h1), σ(h2)}, or cp is a common factor of h1/n n1 and h2/n2, and σ(g/n ) = p+1; in this case σ(g) = σ(g/n ) = p + 1. now assume that n is not contained in h1 or h2. then n is a central minimal normal subgroup of g, n = cp ∼= n1n/n1 ∼= n2n/n2 and g has a factor group isomorphic to cp × cp; therefore σ(g) ≤ p + 1. on the other hand, n = n h2 ∩ h1 ∼= n is 208 eloisa detomi and andrea lucchini cubo 10, 3 (2008) a central minimal normal subgroup of g contained in h1; by the previous case, σ(g) < σ(g/n ). since δg(n ) ≥ 2, n has at least |n| = p complements; hence, by lemma 5, σ(g) ≥ p + 1 and therefore σ(g) = p + 1. thus a counterexample g with minimal order is a σ-primitive group. if g is solvable, then either g ∼= c2p and σ(g) = p+1 or g is monolithic: the second possibility cannot occur as g is the direct product of two non trivial normal subgroups. so from now on we may assume that g is non solvable, and in particular, by proposition 21, that h1/n1 and h2/n2 have no common abelian factor. now observe that if m is a maximal subgroup of g and m does not contain h1 and h2, then g/mg is a primitive group with nontrivial normal subgroups h1mg/mg and h2mg/mg. if h1mg/mg = h2mg/mg, then g/mg = h1mg/mg = h2mg/mg is a central factor of g/mg and h1/n1 and h2/n2 have a common abelian factor, a contradiction. thus h1mg/mg 6= h2mg/mg, and since g/mg = h1mg/mg × h2mg/mg is a primitive group, h1mg/mg and h2mg/mg are isomorphic simple groups. therefore, if h1/n1 and h2/n2 have no simple groups in common, then every maximal subgroup m of g contains either h1 or h2, and we obtain the result arguing as in lemma 4 of [8]. so, we assume that h1/n1 and h2/n2 have precisely one non-abelian simple group s in common and the multiplicity of s in h1/n1 is one: let ki ≥ ni be the normal subgroups of hi such that h1/k1 = s and h2/k2 = s n, being n the multiplicity of s in h2/n2 , and set k = k1 × k2. let m be a minimal cover of g given by σ(g) maximal subgroups of g. we set: m1 = {l ∈ m | l ≥ h1} = {h1 × m | m a maximal subgroup of h2}, m2 = {l ∈ m | l ≥ h2} = {m × h2 | m a maximal subgroup of h1}, m3 = m \ (m1 ∪ m2). then we define the two sets ω1 = h1 \ ⋃ m×h2∈m2 m, ω2 = h2 \ ⋃ h1×m∈m1 m, and their images under the projection πki of hi over hi/ki ωi = {πki (w) | w ∈ ωi}. as h1/k1 is not cyclic, we can cover ω1 with |ω1| subgroups. hence we can cover h1 = { ⋃ m×h2∈m2 m} ∪ ω1 with the images of the maximal subgroups in m2 plus |ω1| maximal subgroups, and thus σ(h1) ≤ |m2| + |ω1|. on the other hand, |m2| + |m3| ≤ σ(g) < σ(h1), and we obtain that |ω1| > |m3|. now observe that the elements of the set ω1 × ω2 can not belong to any of the subgroup of m1 or m2, thus the set ω1 × ω2 has to be covered by the subgroups of m3. if m ∈ m3, cubo 10, 3 (2008) on the structure of primitive n-sum groups 209 then g/mg is a primitive group and g/mg = h1mg/mg × h2mg/mg = s × s; in particular m ≥ k and m/k is a maximal subgroup of diagonal type of g/k. this means that there exists an automorphism α of s and an index i ∈ {1, . . . n}, such that the set (m/k) ∩ (ω1 × ω2) is given by elements of the type (x, y1, y2, . . . , yn) where x ∈ ω1, (y1, y2, . . . , yn) ∈ ω2 and yi = x α. for every y ∈ s we denote by sy the number of vectors (y1, y2, . . . , yn) such that (y1, y2, . . . , yn) ∈ ω2 and yi = y: note that ∑ y∈s sy = |ω2| = |ω1 × ω2|/|ω1|. on the other hand |(m/k) ∩ (ω1 × ω2)| ≤ ∑ y∈s sy = |ω1 × ω2|/|ω1| < |ω1 × ω2|/|m3|, since |ω1| > |m3|. this implies that we can not cover ω1 × ω2 with the |m3| subgroups of m3, a contradiction. theorem 28. let g = h1 × h2 be the direct product of two subgroups. if no alternating group alt(n) with n odd is a homomorphic image of both h1 and h2, then either σ(g) = min{σ(h1), σ(h2)} or σ(g) = p + 1 and s = cp is a homomorphic image of both h1 and h2. proof. let g be a counterexample with minimal order. let ni be the minimal normal subgroup of hi such that hi/ni is a direct product of simple groups. as in the proof of proposition 27, it is easy to see that g is a σ-primitive group, h1/n1 and h2/n2 have at least one simple group s in common and s is non-abelian. by corollary 14, g has at most one abelian minimal normal subgroup, so we can assume that every minimal normal subgroup of h1 is non-abelian. let k be a normal subgroup of g with g/k ∼= s. note that δg(g/k) ≥ 2, indeed δg(g/k) coincides with the multiplicity of s in g/(n1 ×n2). hence, by corollary 14 (3), no minimal normal subgroup of g is g-equivalent to g/k. this implies in particular that s is an epimorphic image of h1/ soc(h1), and consequently s is an homomorphic image of x/n where x is a monolithic primitive group associated to a minimal normal subgroup n of h1. by the remark above n is nonabelian, so n = t r with t a non-abelian simple group. since x is a subgroup of aut(t ) ≀ sym(r) and s is non-abelian, s is an homomorphic image of a transitive group y of degree r. then y satisfies the assumption of lemma 22 and, since s is not an alternating group of odd degree, we get σ(y ) ≤ 4r. since, by corollary 6, 5r + 1 ≤ σ(g) ≤ σ(y ), we get a contradiction. received: july 2008. revised: august 2008. 210 eloisa detomi and andrea lucchini cubo 10, 3 (2008) references [1] a. abdollahi, f. ashraf and s.m. shaker, the symmetric group of degree six can be covered by 13 and no fewer proper subgroups, bull. malays. math. sci. soc., 30(2) (2007), no. 1, 57–58. 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[15] y.m. wang, finite groups admitting a fixed-point-free automorphism group, northeast. math. j., 9(4) (1993), 516–520. n16 c:/documents and settings/profesor/escritorio/cubo/cubo/cubo/cubo 2011/cubo2011-13-01/vol13 n\2721/art n\2608 .dvi cubo a mathematical journal vol.13, no¯ 01, (125–136). march 2011 multiple objective programming involving differentiable (hp, r)-invex functions xiaoling liu, dehui yuan, shengyun yang, guoming lai department of mathematics and inform., tech.,hanshan normal university, chaozhou, guangdong, 521041, china. email: ydhliu@gmail.com, ydhlxl@sohu.com and chuanqing xu department of mathematics, beijing armed forced eng. institute, beijing 100072, china. abstract in this paper, we introduce new types of generalized convex functions which include locally (hp, r)-pre-invex functions and (hp, r)-invex functions. relationship between these two new classes of functions are established. we also present the conditions for optimality in differentiable mathematical programming problems where the functions considered are (hp, r)-invex functions introduced in this paper. resumen este trabajo, establece nuevos tipos de funciones convexas generalizadas que incluyen localmente funciones (hp, r) de pre-invex y funciones (hp, r)-invex. la relacin entre 126 xiaoling liu, dehui yuan and shengyun yang cubo 13, 1 (2011) estas dos nuevas clases de funciones estn establecidas. tambin se presentan las condiciones de optimalidad en diferenciables problemas de programacin matemtica, donde las funciones consideradas en este artculo son funciones (hp, r)-invex. keywords: differentiable mathematical programming. mathematics subject classification: 90b50. 1 introduction convexity plays a central role in many aspects of mathematical programming (see[22, 5, 8]) including analysis of stability [9, 16], sufficient optimality conditions and duality [12, 15, 11]. based on convexity assumptions, nonlinear programming problems can be solved efficiently. there have been many attempts to weaken the convexity assumptions in order to treat many practical problems. therefore, many concepts of generalized convex functions have been introduced and applied to mathematical programming problems in the literature [1, 2, 19, 27, 28, 25]. one of these concepts, invexity, was introduced by hanson in [11]. hanson has shown that invexity has a common property in mathematical programming with convexity that karush kuhn tucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions. ben-israel and mond [7] introduced the concept of pre-invex functions which is a special case of invexity. following [11] and [7], many authors have introduced concepts of generalized invexity and pre-invexity including strictly pseudoinvex functions and quasiinvex functions [13], prepseudoinvex and prequasiinvex functions [21] and r-pre-pseudoinvex functions [1]. the relationships between some of these generalized invex functions were studied in [20, 21]. recently, antczak [3] introduced new definitions of p-invex sets and (p, r)-invex functions which can be seen as generalization of invex functions. he also discussed nonlinear programming problems involving the (p, r)-invexity-type functions in [2, 4]. on the other hand, kaul et al. [14] introduced the classes of locally connected sets which generalizes the arcwise connected sets [6] and locally star-shaped sets [10]. based on the new class of sets, they [14, 15] introduced a new class of functions called locally connected functions. they [15]also defined the directional derivative (with respect to a vector function) of a real valued function, and also defined locally p -connected functions in terms of its right differential. motivated by [7, 14, 1, 2, 3, 4, 15, 18], yuan et al. introduced definition of a new class of sets, locally hp-invex sets, and definitions of classes of generalized convex functions called locally (hp, r, α)-pre-invex functions. and we give the concept of locally differentiable (hp, r)invex functions, discuss the relationship between (hp, r)-invexity and locally (hp, r, 1)-pre-invexity. based on the definition of locally differentiable (hp, r)-invexity, we have managed to deal with nonlinear programming problems under some assumptions. the rest of the paper is organized as follows: in section 2, we give some preliminary concepts cubo 13, 1 (2011) multiple objective programming involving differentiable (hp, r)-invex functions 127 regarding locally hp-convex sets, locally (hp, r, α)-preinvex function, and differentiable (hp, r)invex function, discuss the relationship between (hp, r)-invexity and locally (hp, r, 1)-pre-invexity. in section 3, we present the conditions for optimality in differentiable mathematical programming problems in which the functions considered belong to the classes of functions introduced in section. in section 4, we present the conditions of optimality for the following nonlinear differentiable fractional programming problems (mfp) with the same convexity assumption. 2 differentiable (hp, r)-invex functions let rn be the n-dimensional euclidean space, rn+ = {x ∈ r n|x > 0} and ṙn+ = {x ∈ r n|x > 0}. in this section, we give definitions of locally hp-invex set. definition 1. [3] let a1, a2 > 0, λ ∈ (0, 1) and r ∈ r. then the weighted r-mean of a1 and a2 is given by mr(a1, a2; λ) := { (λar1 + (1 − λ)a r 2) 1 r for r 6= 0, aλ1 a 1−λ 2 for r = 0. definition 2. [29] s ⊂ rn is a locally hp-invex set if and only if, for any x, u ∈ s, there exist a maximum positive number a(x, u) ≤ 1 and a vector function hp : s × s × [0, 1] → r n, such that hp(x, u; 0) = e u, hp(x, u; λ) ∈ ṙ n +, ln (hp(x, u; λ)) ∈ s, ∀ 0 < λ < a(x, u) f or p ∈ r. and hp(x, u; λ) is continuous on the interval (0, a(x, u)), where the logarithm and the exponentials appearing in the relation are understood to be taken componentwise. definition 3. [29] a function f : s → r defined on a locally hp-invex set s ⊂ r n is said to be locally (hp, r)-pre-invex on s if, for any x, u ∈ s, there exists a maximum positive number a(x, u) ≤ 1 such that f (ln (hp(x, u; λ))) ≤ ln ( mr(e f (x), ef (u); λα) ) , ∀ 0 < λ < a(x, u) f or p ∈ r where the logarithm and the exponentials appearing on the left-hand side of the inequality are understood to be taken componentwise. if u is fixed, then f is said to be locally (hp, r)-pre-invex at u. correspondingly, if the direction of above inequality is changed to the opposite one, then f is said to (hp, r)-pre-incave on s or at u. now, we introduce the classes of differentiable (hp, r)-invex functions. for convenience, we assume that s be a hp-invex set, hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ s, and f : s → r is differentiable on s. the symbol h′p(x, u; 0+) , (h′p1(x, u; 0+), · · · , h ′ pn(x, u; 0+)) t denotes the right derivative of hp at 0 with respect to variable λ for each given pair x, u ∈ s; ▽f (x) , (▽1f (x), . . . , ▽nf (x)) t denotes the differential of f at x, where ∂if (x) is partial differential of f with respect to the i-th componentwise; ▽f (u) e u denotes ( ▽1f (u) e u 1 , · · · , ▽nf (u) e un )t . 128 xiaoling liu, dehui yuan and shengyun yang cubo 13, 1 (2011) definition 4. let s be a hp-invex set, hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ s, and f : s → r is differentiable on s. if for all x ∈ s, one of the relations 1 r erf (x) > 1 r erf (u) [ 1 + r ▽f (u) eu t h′p(x, u; 0+) ] (>) f or r 6= 0, f (x) − f (u) > ▽f (u) e u t h′p(x, u; 0+) (>) f or r = 0, (2.1) holds, then f is said to be (hp, r)-invex (strictly (hp, r)-invex) at u ∈ s. if the inequalities (2.1) are satisfied at any point u ∈ s, then f is said to be (hp, r)-invex (strictly (hp, r)-invex) on s. remark 5. any function f satisfying (2.1) is called (hp, r)-invex (strictly (hp, r)-invex) on s. however, if hp(x, u; λ) = mp(e η(x,u)+u, eu; λ) and a(x, u) = 1 for all x, u ∈ s, we will say that f is (p, r)-invex (strictly (p, r)-invex) with respect to η on s; furthermore, f is r-invex (strictly (p, r)-invex) with respect to η on s in the case p = 0 and f is invex (strictly invex) with respect to η on s in the case p = 0 and r = 0. remark 6. in order to define an analogous class of (strict) (hp, r)-incave functions on the hp-invex set, the direction of the inequality (2.1) in the definition of these functions should be changed to the opposite one. theorem 7. let s ⊂ rn be a hp-invex set, hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ s, and f : s → r is differentiable on s. if f is (hp, r, α)-pre-invex ((hp, r, α)-pre-incave) on s and α = 1, then f is (hp, r)-invex ((hp, r)-incave) on s. proof. the theorem will be proved only in the case f is (hp, r, α)-pre-invex, and we just prove that the theorem is true when r > 0 (the proof in the case when r < 0 is analogous to the one when r > 0; only the directions of the inequalities should be changed to the opposite ones). firstly, we prove the theorem is true in the case when r > 0. since f : s → r is a (hp, r, α)pre-invex function and α = 1, by definition 3, we have erf (u) [ erf (ln(hp(x,u;λ)))−rf (u) − 1 ] λ−1 6 erf (x) − erf (u). by letting λ → 0, we get the inequality erf (x) − erf (u) > rerf (u) ▽f (u) eu t h′p(x, u; 0+) that is 1 r erf (x) > 1 r erf (u) [ 1 + r ▽f (u) eu t h′p(x, u; 0+) ] now, we prove the case r = 0. by definition 3, we have [f (ln (hp(x, u; λ))) − f (u)] λ −1 6 f (x) − f (u). by letting λ → 0, we get the inequality f (x) − f (u) > ▽f (u) eu t h′p(x, u; 0+) therefore, we get the desired result. � cubo 13, 1 (2011) multiple objective programming involving differentiable (hp, r)-invex functions 129 3 optimality for multiple objective programming in this section, we present the conditions for optimality in differentiable mathematical programming problems in which the functions considered belong to the classes of functions introduced earlier in this paper. consider the following form of optimization problem (vop) min f (x) g(x) 6 0, x ∈ s, where s ⊂ rn, f : s → rq, g : s → rm. let us denote by e the set of feasible solutions of (vop), i.e., the set of the form e := {x ∈ s|g(x) 6 0}. from now on, we assume that hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ s, fi : s → r(i = 1, · · · , q), gj : s → r(j = 1, · · · , m) are differentiable on s, f = (f1, · · · , fq), g = (g1, · · · , gm) and s is an hp-invex (nonempty) set. definition 8. x̄ ∈ e is said to be an efficient solution for problem (vop), if there exists no x ∈ e such that f (x) 6 f (x̄); x̄ ∈ e is said to be a weak efficient solution for problem (vop), if there exists no x ∈ e such that f (x) < f (x̄); theorem 9. let e be a hp-invex set with the respect to the same hp. assume that x̄ ∈ s is feasible for problem (vop), and there exists λ ∈ rq, µ ∈ rm such that ∑q i λi▽fi(x̄) + ∑m j=1 µj ▽gj (x̄) = 0, (3.1) ∑m j=1 µj gj (x̄) = 0, (3.2) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) > 0. (3.3) if fi(i = 1, · · · , q) are strictly (hp, r)-invex and gj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (vop). proof. here we prove only the cases when r > 0 or r = 0(the proof of the case when r < 0 is similar to the one when r > 0; the only changes arise from the form of inequalities defining the class of (hp, r)-invex functions). assume that x is an arbitrary feasible point for problem (vop). on the contrary to suppose that x̄ is not an efficient solution of problem (vop). thus, there exists x ∈ e such that f (x) 6 f (x̄). we first consider the case when r > 0. therefore, q ∑ i λi { erfi(x) − erfi(x̄) } 6 0. by hypothesis, f and gi(i = 1, · · · , m) are (hp, r)-invex at x̄ on s; therefore, for all x ∈ s, the 130 xiaoling liu, dehui yuan and shengyun yang cubo 13, 1 (2011) inequalities 1 r erfi(x) > 1 r erfi(x̄) [ 1 + r ▽fi(x̄) ex̄ t h′p(x, u; 0+) ] , i = 1, · · · , q, (3.4) 1 r ergj (x) > 1 r ergj (x̄) [ 1 + r ▽gj (x̄) ex̄ t h′p(x, x̄; 0+) ] , j = 1, · · · , m, (3.5) are true. denote i(x̄) , {j|µj > 0, j = 1, · · · , m}. by (3.2), we have gj (x̄) = 0 if j ∈ i(x̄) and µj = 0 if gj (x̄) 6= 0, thus gj (x) 6 gj (x̄) for j ∈ i(x̄). therefore, from (3.4) and (3.5), we have er(fi(x)−fi(x̄)) > 1 + r ▽fi(x̄) ex̄ t h′p(x, u; 0+) (3.6) r ▽gj (x̄) ex̄ t h′p(x, x̄; 0+) 6 0, j ∈ i(x̄). (3.7) by(3.6), (3.7) and (3.3), we deduce that ( ∑m i=1 λi▽fi(x̄) ex̄ + ∑ j∈i(x̄) µj ▽gj (x̄) ex̄ )t h′p(x, x̄; 0+) < 0. (3.8) notice that ∑m j=1 µj ▽gj (x̄) = ∑ j∈i(x̄) µj ▽gj (x̄), by (3.8), we have ( ∑m i=1 λi▽fi(x̄) + ∑m j=1 µj ▽gj (x̄) ex̄ )t h′p(x, x̄; 0+) < 0. this, together with (3.1), follows a contradiction 0 < 0. now, we prove the theorem is true in the case when r = 0. since f (x) 6 f (x̄), then ∑q i λi (fi(x) − fi(x̄)) 6 0. by definition 4, we have fi(x) − fi(x̄) > ▽fi(x̄) ex̄ t h′p(x, x̄; 0+), i = 1, . . . , q, gj (x) − gj (x̄) > ▽gj (x̄) ex̄ t h′p(x, x̄; 0+), j = 1, · · · , m, on the same line as the case when r > 0, we have the contradiction 0 < 0 again. therefore, x̄ is an efficient solution for problem (vop). � theorem 10. let e be a hp-invex set with the respect to the same hp. assume that x̄ ∈ s is feasible for problem (vop), and there exists λ ∈ rq, µ ∈ rm such that ∑q i λi▽fi(x̄) + ∑m j=1 µj ▽gj (x̄) = 0, (3.9) ∑m j=1 µj gj (x̄) = 0, (3.10) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) > 0. (3.11) if fi(i = 1, · · · , q), gj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (vop). cubo 13, 1 (2011) multiple objective programming involving differentiable (hp, r)-invex functions 131 proof. the theorem can be proved on simple lines as theorem 9. � the assumption on functions in theorem 9 ( or theorem 10) could also be given in another form. it is enough to assume that the lagrange function fi + ∑m j=1 µj gj (i = 1, . . . , m) are strictly (hp, r)-invex (or strictly (hp, r)-invex) . and so, the following two theorems are true. their proofs are on the same line as theorem 9, therefore we delete them here. theorem 11. assume that x̄ ∈ s is feasible for problem (vop), and there exists λ ∈ rq, µ ∈ rm satisfying (3.1), (3.2) and (3.3). if fi + ∑m j=1 µj gj (i = 1, . . . , m) are strictly (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (vop). theorem 12. assume that x̄ ∈ s is feasible for problem (vop), and there exists λ ∈ rq, µ ∈ rm satisfying (3.9), (3.10) and (3.11). if fi + ∑m j=1 µj gj (i = 1, . . . , m) are (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (vop). for weak efficient solution of problem (vop), we have the following theorems. theorem 13. assume that x̄ ∈ s is feasible for problem (vop), and there exists λ ∈ rq, µ ∈ rm satisfying (3.1), (3.2) and (3.3). if fi(i = 1, · · · , q), gj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then x̄ is a weak efficient solution of problem (vop). theorem 14. assume that x̄ ∈ s is feasible for problem (vop), and there exists λ ∈ rq, µ ∈ rm satisfying (3.1), (3.2) and (3.3). if fi + ∑m j=1 µj gj (i = 1, . . . , m) are (hp, r)-invex at x̄ on s, then x̄ is a weak efficient solution of problem (vop). 4 multiple objective fractional programming in this section, we present the conditions of optimality for the following nonlinear differentiable fractional programming problems (mfp). (mfp) min f (x) g(x) , ( f1(x) g1(x) , f2(x) g2(x) , · · · , fq(x) gq(x) )t s.t. h(x) = (h1(x), h2(x), · · · , hm(x)) 6 0 x ∈ s, where s ⊂ rn, f : s → rq, g : s → rq, h : s → rm, f = (f1, · · · , fq), g = (g1, · · · , gq), h = (h1, · · · , hm), are differentiable on s and s is an hp-invex (nonempty) set. moreover, for i = 1, . . . , q, gi(x) > 0 for all x ∈ s. let us denote by e the set of feasible solutions of (mfp), i.e., the set of the form e := {x ∈ s|h(x) 6 0}. we assume that hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ s. definition 15. x̄ ∈ e is said to be an efficient solution for problem (mfp), if there exists no x ∈ e such that f (x) g(x) 6 f (x̄) g(x̄) ; x̄ ∈ e is said to be a weak efficient solution for problem (mfp), if there exists no x ∈ e such that f (x) g(x) < f (x̄) g(x̄) ; 132 xiaoling liu, dehui yuan and shengyun yang cubo 13, 1 (2011) theorem 16. assume that x̄ ∈ s is feasible for problem (mfp), and there exists λ ∈ rq, u ∈ rq, µ ∈ rm such that ∑q i λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) = 0, (4.1) ∑m j=1 µj hj (x̄) = 0, (4.2) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) ≧ 0, (4.3) u = (u1, · · · , uq) ≧ 0, ui = fi(x̄)/gi(x̄), i = 1, . . . , q. (4.4) if fi −uigi(i = 1, · · · , q) are strictly (hp, r)-invex at x̄ on s, and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (mfp). proof. similar to theorem 9, here we prove only the cases when r > 0 or r = 0. assume that x is an arbitrary feasible point for problem (mfp). on the contrary to suppose that x̄ is not an efficient solution of problem (mfp). thus, there exists x ∈ e such that f (x) g(x) 6 f (x̄) g(x̄) = u. we first consider the case when r > 0. therefore, q ∑ i λi { er(fi(x)−uigi(x)) − 1 } 6 0. by the convexity of fi − uigi(i = 1, · · · , q) and hj (j = 1, · · · , m), we have er(fi(x)−uigi(x)) − 1 > r ( ▽fi(x̄) − ui▽gi(x̄) ex̄ )t h′p(x, u; 0+), i = 1, · · · , q, (4.5) 1 r erhj (x) > 1 r erhj (x̄) [ 1 + r ▽hj (x̄) ex̄ t h′p(x, x̄; 0+) ] , j = 1, · · · , m, (4.6) thus, by (4.2), (4.3), (4.5) and (4.6), we deduce that ( ∑q i=1 λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) ex̄ )t h′p(x, u; 0+) < 0 therefore, we get a contradiction 0 < 0. similarly, we can prove the theorem in the case when r = 0. � theorem 17. assume that x̄ ∈ s is feasible for problem (mfp), and there exists λ ∈ rq, u ∈ rq, µ ∈ rm satisfying (4.1), (4.2), (4.3) and (4.4). if one of the following holds: 1) fi(i = 1, · · · , q) are strictly (hp, r)-invex at x̄ on s, uigi(i = 1, · · · , q) are (hp, r)-incave at x̄ on s, and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s; 2) uigi(i = 1, · · · , q) are strictly (hp, r)-incave at x̄ on s, fi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s; then x̄ is an efficient solution of problem (mfp). cubo 13, 1 (2011) multiple objective programming involving differentiable (hp, r)-invex functions 133 proof. in the same way as theorem 16, here we prove only 1) for the cases when r > 0, 2) can be proved similar to 1). assume that x is an arbitrary feasible point for problem (mfp). on the contrary to suppose that x̄ is not an efficient solution of problem (mfp). thus, there exists x ∈ e such that f (x) g(x) 6 f (x̄) g(x̄) = u. therefore, q ∑ i λi { erfi(x) − eruigi(x) } 6 0. (4.7) 1) since fi(i = 1, · · · , q) are strictly (hp, r)-invex at x̄ on s, uigi(i = 1, · · · , q) are (hp, r)incave at x̄ on s, and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then er(fi(x)) − er(fi(x̄)) > r ( ▽fi(x̄) ex̄ )t h′p(x, u; 0+), i = 1, · · · , q, (4.8) er(uigi(x)) − er(uigi(x̄)) 6 r ( ui▽gi(x̄) ex̄ )t h′p(x, u; 0+), i = 1, · · · , q, (4.9) 1 r erhj (x) > 1 r erhj (x̄) [ 1 + r ▽hj (x̄) ex̄ t h′p(x, x̄; 0+) ] , j = 1, · · · , m, (4.10) notice that erfi(x̄)) = er(uigi(x̄)), by (4.2)-(4.4), (4.7)-(4.10), we have ( ∑q i=1 λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) ex̄ )t h′p(x, u; 0+) < 0. therefore, we get a contradiction 0 < 0. � theorem 18. assume that x̄ ∈ s is feasible for problem (mfp), and there exists λ ∈ rq, u ∈ rq, µ ∈ rm such that ∑q i λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) = 0, (4.11) ∑m j=1 µj hj (x̄) = 0, (4.12) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) ≧ 0, (4.13) u = (u1, · · · , uq) ≧ 0, ui = fi(x̄)/gi(x̄), i = 1, . . . , q. (4.14) if fi − uigi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (mfp). proof. the theorem can be proved on simple lines as theorem 16. theorem 19. assume that x̄ ∈ s is feasible for problem (mfp), and there exists λ ∈ rq, u ∈ rq, µ ∈ rm satisfying (4.11), (4.12), (4.13) and (4.14). if uigi(i = 1, · · · , q) are (hp, r)-incave at x̄ on s, fi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s, then x̄ is an efficient solution of problem (mfp). 134 xiaoling liu, dehui yuan and shengyun yang cubo 13, 1 (2011) proof. the theorem can be proved on simple lines as theorem 17. theorem 20. assume that x̄ ∈ s is feasible for problem (mfp), and there exists λ ∈ rq, u ∈ rq, µ ∈ rm such that ∑q i λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) = 0, ∑m j=1 µj hj (x̄) = 0, λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) ≧ 0, u = (u1, · · · , uq) ≧ 0, ui = fi(x̄)/gi(x̄), i = 1, . . . , q. if one of the following holds: 1) fi − uigi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (hp, r)-invex at x̄ on s; 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[29] yuan, d.h., liu x.l., yang s.y., nyamsuren d., and altannar c., optimality conditions and duality for nonlinear programming problems involving locally (hp, r, α)pre-invex functions and hp-invex sets,inrernational journal of pure and applied mathematics,41(4),561-576,2007. cubo a mathematical journal vol.10, n o ¯ 04, (1–13). december 2008 remarks on kkm maps and fixed point theorems in generalized convex spaces sehie park the national academy of sciences, republic of korea, department of mathematical sciences, seoul national university, seoul 151–747, korea email: shpark@math.snu.ac.kr abstract various types of φa-spaces (x, d; {φa}a∈〈d〉) are simply g-convex spaces. various types of generalized kkm maps on φa-spaces are simply kkm maps on g-convex spaces. therefore, our g-convex space theory can be applied to various types of φaspaces. as such examples, we obtain kkm type theorems and a very general fixed point theorem on φa-spaces. resumen varios tipos de φa-espacios (x, d; {φa}a∈〈d〉) son simplemente espacios g-convexos. varios tipos de aplicaciones kkm generalizadas sobre φa-espacios son aplicaciones simplemente kkm sobre espacios g-convexos. por lo tanto, nuestra teoria de espacios g-convexos puede ser aplicada a varios tipos de φa-espacios. como ejemplo obtenemos teoremas do tipo kkm y un teorema general de punto fijo sobre φa-espacios. key words and phrases: abstract convex space, generalized (g-) convex space, φa-space, l-spaces, f c-spaces, property (h); h-condition. math. subj. class.: 47h04, 47h10, 49j27, 49j35, 54h25, 91b50. 2 sehie park cubo 10, 4 (2008) 1 introduction the kkm theory, first called by the author, is the study on applications of equivalent formulations of the kkm principle due to knaster, kuratowski, and mazurkiewicz. the kkm principle provides the foundations for many of the modern essential results in diverse areas of mathematical sciences. since 1993, the author has initiated the study of the kkm theory on generalized convex spaces (or g-convex spaces) (x, d; γ) as a common generalization of various general convexities without linear structures due to other authors. we have established within such a frame the foundations of the kkm theory, as well as fixed point theorems and many other equilibrium results for multimaps. this direction of study has been followed by a number of other authors. in the last decade, some authors who introduced spaces of the form (x, {ϕa}) having a family {ϕa} of continuous functions defined on simplices claimed that such spaces generalize g-convex spaces without giving any justifications or proper examples. in fact, a number of modifications or imitations of the g-convex spaces have followed; for example, l-spaces due to ben-el-mechaiekh et al. [1], spaces having property (h) due to huang [10], f c-spaces due to ding [6,7], convexity structures satisfying the h-condition [22], and others. some authors also tried to generalize the kkm principle for their own settings. they introduced various types of generalized kkm maps; for example, generalized kkm maps on l-spaces [5,20], generalized r-kkm maps [2,8,9], and many others. in order to destroy such inadequate concepts and to upgrade the kkm theory, recently, we proposed new concepts of abstract convex spaces and the kkm spaces which are proper generalizations of g-convex spaces and adequate to establish the kkm theory; see [15-18]. moreover, we noticed that all spaces of the form (x, {ϕa}) can be unified to φa-spaces (x, d; {φa}a∈〈d〉) or spaces having a family {φa}a∈〈d〉 of singular simplices. in the present note, we show that various types of φa-spaces (x, d; {φa}a∈〈d〉) are simply g-convex spaces, and various types of generalized kkm maps on φa-spaces are simply kkm maps on g-convex spaces. therefore, our g-convex space theory can be applied to various types of φaspaces. as such examples, we obtain kkm type theorems and a very general fixed point theorem on φa-spaces. 2 abstract convex spaces in this section, we follow mainly [15,16]. let 〈d〉 denote the set of all nonempty finite subsets of a set d. definition. an abstract convex space (e, d; γ) consists of nonempty sets e, d, and a multimap γ : 〈d〉 ⊸ e with nonempty values. we may denote γa := γ(a) for a ∈ 〈d〉. cubo 10, 4 (2008) remarks on kkm maps and fixed point theorems ... 3 examples. in [15-17], we gave plenty of examples of abstract convex spaces. here we give only two classes of them as follows: 1. a convexity space (e, c) in the classical sense consists of a nonempty set e and a family c of subsets of e such that e itself is an element of c and c is closed under arbitrary intersection. for details, see [21], where the bibliography lists 283 papers. 2. a generalized convex space or a g-convex space (e, d; γ) consists of a topological space e, a nonempty set d, and a multimap γ : 〈d〉 ⊸ e such that for each a ∈ 〈d〉 with the cardinality |a| = n + 1, there exists a continuous function φa : ∆n → γ(a) such that j ∈ 〈a〉 implies φa(∆j ) ⊂ γ(j). here, ∆n is a standard n-simplex with vertices {ei} n i=0, and ∆j the face of ∆n corresponding to j ∈ 〈a〉; that is, if a = {a0, a1, . . . , an} and j = {ai0 , ai1 , . . . , aik } ⊂ a, then ∆j = co{ei0 , ei1 , . . . , eik }. for g-convex spaces; see [11-14,19] and references therein. from now on, in an abstract convex space (e, d; γ), e is assumed to be a topological space. definition. let (e, d; γ) be an abstract convex space and z a topological space. for a multimap f : e ⊸ z with nonempty values, if a multimap g : d ⊸ z satisfies f (γa) ⊂ g(a) := ⋃ y∈a g(y) for all a ∈ 〈d〉, then g is called a kkm map with respect to f . a kkm map g : d ⊸ e is a kkm map with respect to the identity map 1e . a multimap f : e ⊸ z is called a kc-map [resp., a ko-map] if, for any closed-valued [resp., open-valued] kkm map g : d ⊸ z with respect to f , the family {g(y)}y∈d has the finite intersection property. the following is the origin of the kkm theory; see [11,12]. the kkm principle. let d be the set of vertices of an n-simplex ∆n and g : d ⊸ ∆n be a kkm map (that is, co a ⊂ g(a) for each a ⊂ d) with closed [resp., open] values. then ⋂ z∈d g(z) 6= ∅. 3 φa-spaces recently, there have appeared authors in [2,3,6-10,20,22] and others who introduced spaces of the form (x, {ϕa}). some of them tried to rewrite some results on g-convex spaces by simply replacing γ(a) by ϕa(∆n) everywhere and claimed to obtain generalizations without giving any justifications or proper examples. 4 sehie park cubo 10, 4 (2008) motivated by this fact, we are concerned with a reformulation of the class of g-convex spaces as follows [17]: definition. a φa-space (x, d; {φa}a∈〈d〉) consists of a topological space x, a nonempty set d, and a family of continuous functions φa : ∆n → x (that is, singular n-simplices) for a ∈ 〈d〉 with the cardinality |a| = n + 1. any g-convex space is a φa-space. the converse also holds: theorem 1. a φa-space (x, d; {φa}a∈〈d〉) can be made into a g-convex space (x, d; γ). proof. this can be done in two ways. (1) for each a ∈ 〈d〉, by putting γa := x, we obtain a trivial g-convex space (x, d; γ). (2) let {γα}α be the family of maps γ α : 〈d〉 ⊸ x giving a g-convex space (x, d; γα). note that, by (1), this family is not empty. then, for each α and each a ∈ 〈d〉 with |a| = n + 1, we have φa(∆n) ⊂ γ α a and φa(∆j ) ⊂ γ α j for j ⊂ a. let γ := ⋂ α γ α , that is, γa := ⋂ α γ α a for each a ∈ 〈d〉. then φa(∆n) ⊂ γa and φa(∆j ) ⊂ γj for j ⊂ a. therefore, (x, d; γ) is a g-convex space. consequently, g-convex spaces and φa-spaces are essentially the same. definition. for a φa-space (x, d; {φa}a∈〈d〉), any map t : d ⊸ x satisfying φa(∆j ) ⊂ t (j) for each a ∈ 〈d〉 and j ∈ 〈a〉 is called a kkm map. theorem 2. (1) a kkm map g : d ⊸ x on a g-convex space (x, d; γ) is a kkm map on the corresponding φa-space (x, d; {φa}a∈〈d〉). (2) a kkm map t : d ⊸ x on a φa-space (x, d; {φa}) is a kkm map on a new g-convex space (x, d; γ). proof. (1) this is clear from the definition of a kkm map on a g-convex space. (2) define γ : 〈d〉 ⊸ x by γa := t (a) for each a ∈ 〈d〉. then (x, d; γ) becomes a g-convex space. in fact, for each a with |a| = n+1, we have a continuous function φa : ∆n → t (a) =: γ(a) cubo 10, 4 (2008) remarks on kkm maps and fixed point theorems ... 5 such that j ∈ 〈a〉 implies φa(∆j ) ⊂ t (j) =: γ(j). moreover, note that γa ⊂ t (a) for each a ∈ 〈d〉 and hence t : d ⊸ x is a kkm map on a g-convex space (x, d; γ). the following is a kkm theorem for φa-spaces. the proof is just a simple modification of the corresponding one in [12,13,19]: theorem 3. for a φa-space (x, d; {φa}a∈〈d〉), let g : d ⊸ x be a kkm map with closed [resp., open] values. then {g(z)}z∈d has the finite intersection property. (more precisely, for each n ∈ 〈d〉 with |n| = n + 1, we have φn (∆n) ∩ ⋂ z∈n g(z) 6= ∅). further, if (3) ⋂ z∈m g(z) is compact for some m ∈ 〈d〉, then we have ⋂ z∈d g(z) 6= ∅. proof. let n = {z0, z1, . . . , zn}. since g is a kkm map, for each vertex ei of ∆n, we have φn (ei) ∈ g(zi) for 0 ≤ i ≤ n. then ei 7→ φ −1 n g(zi) is a closed [resp., open] valued map such that ∆k = co{ei0, ei1 , . . . , eik } ⊂ ⋃k j=0 φ −1 n g(zij ) for each face ∆k of ∆n. therefore, by the original kkm principle, ∆n ⊃ ⋂n i=0 φ −1 n g(zi) 6= ∅ and hence φn (∆n) ∩ ( ⋂ z∈n g(z) ) 6= ∅. the second conclusion is clear. remarks. (1) we may assume that, for each a ∈ d and n ∈ 〈d〉, g(a) ∩ φn (∆n) is closed [resp., open] in φn (∆n). this is said by some authors that g has finitely closed [resp., open] values. however, by replacing the topology of x by its finitely generated extension, we can eliminate “finitely”; see [13]. (2) for x = ∆n, if d is the set of vertices of ∆n and γ = co, the convex hull, theorem 3 reduces to the original kkm principle and its open version; see [11,12]. (3) if d is a nonempty subset of a topological vector space x (not necessarily hausdorff), theorem 3 extends fan’s kkm lemma; see [11,12]. (4) note that any kkm theorem on spaces of the form (x, {ϕa}) can not generalize the original kkm principle or fan’s kkm lemma. 4 examples of φa-spaces in this section, we give some examples of spaces of the form (x, {ϕa}) given by other authors: (i) in 1998, ben-el-mechaiekh et al. [1] defined an l-space (e, γ), which is a particular form of our g-convex space (x, d; γ) for the case e = x = d. some authors incorrectly claimed that the class of l-spaces contains our class of g-convex spaces; for example, [4,5], which contain a 6 sehie park cubo 10, 4 (2008) number of particular results (with certain defects) of known ones. (ii) in 2003, the authors of [20] considered the l-space. (iii) [10] a topological space y is said to have property (h) if, for each n = {y0, . . . , yn} ∈ 〈y 〉, there exists a continuous mapping ϕn : ∆n → y . (iv) [6,7] (y, {ϕn }) is said to be a f c-space if y is a topological space and for each n = {y0, . . . , yn} ∈ 〈y 〉 where some elements in n may be same, there exists a continuous mapping ϕn : ∆n → y . this definition appears in a large number of papers of the same author and his followers. note that for each n , there should be infinitely many ϕn ’s. the author of [6,7] wrote in more than one dozen papers that: “it is easy to see that the class of f c-spaces includes the classes of convex sets in topological vector spaces, c-spaces (or h-spaces) [20], g-convex spaces, l-convex spaces [1], and many topological spaces with abstract convexity structure as true subclasses. hence, it is quite reasonable and valuable to study various nonlinear problems in f c-spaces.” there he failed to give any justification or any proper example of his space which is not g-convex. one wonders how could a pair (y, {ϕn }) generalize a triple (x, d; γ). (v) in [22], a pair (y, c) is introduced, where y is a topological space and c is a family of subsets of y such that (y, c) is similar to the convexity space in the classical sense. a pair (x, c) is said to have the selection property with respect to a topological space s if every multimap f : s ⊸ x admits a single-valued continuous selection whenever f is lower semicontinuous and nonempty closed convex valued. a pair (y, c) is said to satisfy h-condition if c has the following property: (h) for each finite subset {y0, . . . , yn} ⊂ y , there exists a continuous mapping f : ∆n → conv{y0, . . . , yn}, where ∆n is the standard n-simplex, such that f (∆j ) ⊂ conv{yj : j ∈ j} for each nonempty subset j ⊂ n = {0, 1, . . . , n}, where conv denotes the closed convex hull. for these definitions, we note the following remarks: (i) a pair (y, c) is a particular form of our abstract convex space (e, d; γ) with y = e = d and γa:=conv(a) = ⋂ {b ∈ c | a ⊂ b} for a ∈ 〈y 〉. then (y, c) becomes our abstract convex space (y ; γ). (ii) the selection property would be better to call the michael selection property. (iii) a pair (y, c) satisfying the h-condition is a particular form of our g-convex space (x, d; γ) with y = x = d such that γ is closed-valued. the following new result gives an example of φa-spaces: cubo 10, 4 (2008) remarks on kkm maps and fixed point theorems ... 7 theorem 4. if an abstract convex space (e, d; γ) has the michael selection property with respect to a simplex and if γ is closed-valued, then we have a φa-space (e, d; {φa}a∈〈d〉). corollary 4.1. [22, theorem 1] if a pair (y, c) has the selection property with respect to any simplex, then the pair satisfies the h-condition. in fact, just following the proof of [22, theorem 1], we can easily deduce the more general theorem 4. moreover, in [22], several results on the pairs (y, c) satisfying the h-condition are obtained. some of such results are particular forms of known results on g-convex spaces. 5 various kkm maps a number of authors tried to generalize the concept of kkm maps on particular forms of φa-spaces. in this section, we show that all of them are particular forms of our kkm maps. (i) in 2003 [20, definition 2], for an l-space (x, γ) and a topological space y , a correspondence g : y ⊸ x is called a generalized kkm-correspondence, if for all a = {y0, y1, . . . , yn} ∈ 〈y 〉, there exists a subset b = {x0, x1, . . . , xn} ∈ 〈x〉, such that for all j ⊆ {0, 1, . . . , n}, it is satisfied that φb (∆j ) ⊆ ⋃ j∈j g(yj ). note that a generalized kkm-correspondence becomes simply our kkm map on a φa-space (x, d; {φa}a∈〈d〉) by putting d := y and, for any a ∈ 〈d〉, by defining φa(∆|a|−1) := φb(∆|b|−1) for b ∈ 〈x〉 corresponding to a. (ii) in 2003 [2, definition 2.1], for a nonempty set x and a topological space y , t : x → 2y is said to be generalized relatively kkm (r-kkm) mapping if for any n = {x0, x1, . . . , xn} ∈ 〈x〉, there exists a continuous mapping φn : ∆n → y such that, for each ei0 , ei1 , · · · , eik , φn (∆k) ⊂ k ⋃ j=0 t xij , where ∆k is a standard k-simplex of ∆n with vertices ei0 , ei1 , · · · , eik . for a φa-space (y, x; {φn }n∈〈x〉), t : x → 2 y is simply a kkm map. (iii) let x be a nonempty set and y be a topological space with property (h). in 2005 [10], t : x → 2y is said to be a generalized r-kkm mapping if for each {x0, . . . , xn} ∈ 〈x〉, there 8 sehie park cubo 10, 4 (2008) exists n = {y0, . . . , yn} ∈ 〈y 〉 such that ϕn (∆k) ⊂ k ⋃ j=0 t xij , for all {i0, . . . , ik} ⊂ {0, . . . , n}. similarly to (ii), a generalized r-kkm map t : x → 2y is simply a kkm map for the φa-space (y, x; {φa}a∈〈x〉). the author of [8] claimed as follows: “the above class of generalized r-kkm mappings includes those classes of kkm mappings, h-kkm mappings, g-kkm mappings, generalized g-kkm mappings, generalized s-kkm mappings, glkkm mappings and gmkkm mappings defined in topological vector spaces, h-spaces, g-convex spaces, g-h-spaces, l-convex spaces and hyperconvex metric spaces, respectively, as true subclasses.” this is partially incorrect. in view of this claim and theorem 2, so many variants of kkm type theorems in [2-10,20,22] and a large number of other papers can be reduced to the ones in our g-convex space theory. we should recognize that, in the kkm theory on g-convex spaces, every argument is related to the finite intersection property of functional values of kkm maps having closed [resp., open] values, in other words, related to some n ∈ 〈d〉 in (x, d; γ). (iv) motivated by a large number of recent works on generalized kkm maps, we introduced the following definition in [19]: let (x, d, γ) be a g-convex space and i a nonempty set. a map f : i ⊸ x is called a generalized kkm map provided that for each n ∈ 〈i〉, there exists a function σ : n → d such that γσ(m) ⊂ f (m ) for each m ∈ 〈n〉. in [19], a unified account on results for such maps was given; for example, the kkm type theorem, characterizations of such maps, an equilibrium theorem implying minimax inequalities, variational inequalities, and so on. a little later than [19], similar results appeared in [4,5], which has trivial defects in certain aspects. 6 various kkm type theorems for particular forms of g-convex spaces, some authors obtained kkm type theorems or equivalents which can not be applicable even to the kkm principle for (∆n, v ; co) or to the ky fan lemma for (x ⊃ d; co), where x is a topological vector space. in this section, we give two kkm type theorems which improve corresponding ones in [2,20]: theorem 5. let x be a topological space, d a nonempty set, and g : d ⊸ x a map such that cubo 10, 4 (2008) remarks on kkm maps and fixed point theorems ... 9 (1) g is transfer closed-valued [that is, ⋂ z∈d g(z) = ⋂ z∈d g(z)]; (2) there exists a∗ ∈ y with g(a∗) compact. then, there exists a g-convex space (x, d; γ) such that g is a kkm map if and only if ⋂ z∈d g(z) 6= ∅. proof. (necessity) follows from theorem 3. (sufficiency) choose an x∗ ∈ ⋂ z∈d g(z) 6= ∅. define a map γ : 〈d〉 ⊸ x given by the constant function γ(a) = {x∗} for all a ∈ 〈d〉 with |a| = n + 1, and a function φa : ∆n → γ(a) by φa(λ) = x ∗ for all λ ∈ ∆n. then it is easy to verify that, with this g-convex space (x, d; γ), g is a kkm map. corollary 5.1. [20, theorem 1] let x and y be topological spaces and γ : y → x a transfer closed-valued correspondence on y such that there exists y∗ ∈ y with cl[γ(y∗)] compact. then, there exists an l-structure on x such that γ is a generalized kkm-correspondence if and only if ⋂ y∈y γ(y) 6= ∅. recall that several generalizations of [20, theorem 1] already appeared in [19]. theorem 6. for a φa-space (y, d; {φn }n∈〈d〉), let t : d → 2 y be a map such that t (z) is nonempty and closed for each z ∈ d. (i) if t is a kkm map, then for each n ∈ 〈d〉 with |n| = n + 1, φn (∆n) ∩ ⋂ x∈n t (x) 6= ∅. (ii) if the family {t (z) | z ∈ z} has finite intersection property, then t is a kkm map. proof. (i) apply theorem 3. (ii) just follow the sufficiency part of theorem 5. the following is the key result in [2] with almost a page proof: corollary 6.1. [2, theorem 3.1] let x be a nonempty set and y be a topological space. let t : x → 2y be a set-valued mapping such that t (x) is nonempty and compactly closed in y for each x ∈ x. (i) if t is a generalized r-kkm mapping, then for each n = {x0, x1, . . . , xn} ∈ 〈x〉, φn (∆n) ∩ ( ⋂ x∈n t x ) 6= ∅, where φn is the continuous mapping in touch with n in definition of a generalized r-kkm map10 sehie park cubo 10, 4 (2008) ping. (ii) if the family {t (x) | x ∈ x} has finite intersection property, then t is a generalized r-kkm mapping. proof. switch the topology of y to its compactly generated extension [13]. then we can eliminate ‘compactly’ and apply theorem 6. remark. in [2], its authors used the partition of unity subordinated to a cover of φn0 (∆n) which should be assumed hausdorff. they claim that, applying their theorem 3.1, they obtained new theorems which unify and extend many known results in recent literature. however, theirs are all disguised forms of known results and their practical applicability is doubtful. 7 fixed points of b-maps in this section, the well-known better admissible class b on g-convex spaces [14] can be introduced on φa-spaces. a φa-space (e, d; {φa}a∈〈d〉) is an abstract convex space (e, d; γ), where γn := φn (∆n) for each n ∈ 〈d〉 with |n| = n + 1, and hence there is a continuous function φn : ∆n → γn . this (e, d; γ) is not necessarily a g-convex space. definition. let (e, d; γ) be an abstract convex space, x a nonempty subset of e and y a topological space. we define the better admissible class b of maps from x into y as follows: f ∈ b(x, y ) ⇐⇒ f : x ⊸ y is a map such that, for any γn ⊂ x, where n ∈ 〈d〉 with the cardinality |n| = n+1, and for any continuous function p : f (γn ) → ∆n, there exists a continuous function φn : ∆n → γn such that the composition ∆n φn −→ γn f |γ n ⊸ f (γn ) p −→ ∆n has a fixed point. note that φn (∆n) is a compact subset of x. recall that for a φa-space (e, d; {φa}a∈〈d〉), by letting γn := φn (∆n), the above definition works. there are a large number of examples of b-maps; see [14] and references therein. we introduce particular types of subsets of abstract convex uniform spaces adequate to establish our fixed point theory. in fact, as in [14], we introduce the klee approximability of ranges of maps: definition. let (e, d; {φa}a∈〈d〉; u) be a uniform φa-space. a subset k of e is said to be klee approximable if, for each entourage u ∈ u, there exists a continuous function h : k → e satisfying cubo 10, 4 (2008) remarks on kkm maps and fixed point theorems ... 11 (1) (x, h(x)) ∈ u for all x ∈ k; (2) h(k) ⊂ φn (∆n) for some n ∈ 〈d〉 with |n| = n + 1; and (3) there exist a continuous function p : k → ∆n such that h = φn ◦ p. especially, for a subset x of e, k is said to be klee approximable into x whenever the range h(k) ⊂ φn (∆n) ⊂ x for some n ∈ 〈d〉 in condition (2). we have given a lot of examples of klee approximable subsets in [14]. now we have the following generalizations of the main result of [14]: theorem 7. let (e, d; {φa}a∈〈d〉; u) be a uniform φa-space, x ⊂ y subsets of e, and f : y ⊸ y a map such that f |x ∈ b(x, y ) and f (x) is klee approximable into x. then f has the almost fixed point property (that is, for any u ∈ u, there exist xu ∈ x such that f (xu ) ∩ u [xu ] 6= ∅). further if (e, u) is hausdorff, f is closed, and f (x) is compact in y , then f has a fixed point x0 ∈ y (that is, x0 ∈ f (x0)). proof. since k := f (x) is a klee approximable into x, for each symmetric entourage u ∈ u, there exists a continuous function h : k → x satisfying conditions (1) (3) of the definition of klee approximable subsets, and we have ∆n φn −→ γn f |γ n ⊸ k p −→ ∆n for some n ∈ 〈d〉 with |n| = n + 1 and γn := φn (∆n) ⊂ x. let p ′ := p|f (γn ). since f |x ∈ b(x, y ), the composition p ′ ◦ (f |γn ) ◦ φn : ∆n ⊸ ∆n has a fixed point au ∈ ∆n. let xu := φn (au ). then au ∈ (p ′ ◦ f ◦ φn )(au ) = (p ′ ◦ f )(xu ) and hence xu = φn (au ) ∈ (φn ◦ p ′ ◦ f )(xu ). since h = φn ◦ p by definition, we have xu = h(yu ) for some yu ∈ (f |γn )(xu ). therefore, for each entourage u ∈ u, there exist points xu ∈ x and yu ∈ f (xu ) such that (xu , yu ) = (h(yu ), yu ) ∈ u . so, for each u , there exist xu , yu ∈ x such that yu ∈ f (xu ) and yu ∈ u [xu ]. now suppose that f is closed and f (x) is compact. since f (x) is relatively compact, we may assume that the net yu in f (x) converges to some x0 ∈ f (x). since (xu , yu ) ∈ u for each u ∈ u, by the hausdorffness of e, the net xu also converges to x0. since the graph of f is closed in y × y and (xu , yu ) ∈ gr(f ), we have (x0, y0) ∈ gr(f ) and hence we have x0 ∈ f (x0). this completes our proof. 12 sehie park cubo 10, 4 (2008) note that, by choosing particular subclass of multimaps or particular types of φa-spaces, we can deduce a large number of known or new fixed point theorems from theorem 7. received: december 2007. revised: december 2007. references [1] h. ben-el-mechaiekh, s. chebbi, m. florenzano and j.-v. llinares, abstract convexity and fixed points, j. math. anal. appl. 222 (1998), 138–150. [2] l. deng and x. xia, generalized r-kkm theorems in topological space and their applications, j. math. anal. appl. 285 (2003), 679–690. [3] l. deng and m.g. yang, coincidence theorems with applications to minimax inequalities, section theorem, best approximation and multiobjective games in topological spaces, acta math. sinica, english ser. 22 (2006), 1809–1818. [4] x.p. ding, abstract convexity and generalizations of himmelberg type fixed-point theorems, comp. math. appl. 41 (2001), 497–504. [5] x.p. ding, generalized l-kkm type theorems in l-convex spaces with applications, comp. math. appl. 43 (2002), 1240–1256. [6] x.p. ding, maximal element theorems in product fc-spaces and generalized games, j. math. anal. appl. 305 (2005), 29–42. [7] x.p. ding, generalized kkm type theorems in f c-spaces with applications (i), j. glob. optim. 36 (2006), 581–596. [8] x.p. ding, new generalized r-kkm type theorems in general topological spaces and applications, acta math. sinica, english ser. 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[15] s. park, on generalizations of the kkm principle on abstract convex spaces, nonlinear anal. forum 11 (2006), 67–77. [16] s. park, elements of the kkm theory on abstract convex spaces, j. korean math. soc. 45(1) (2007), 1–27. [17] s. park, various subclasses of abstract convex spaces for the kkm theory, proc. nat. inst. math. sci. 2(4) (2007), 35–47. [18] s. park, equilibrium existence theorems in the kkm spaces, nonlinear analysis (2007), doi:10.1016/j.na.2007.10.058. [19] s. park and w. lee, a unified approach to generalized kkm maps in generalized convex spaces, j. nonlinear convex anal. 2 (2001), 157–166. [20] m.c. sánchez, j.-v. llinares and b. subiza, a kkm-result and an application for binary and non-binary choice function, econom. th. 21 (2003), 185–193. [21] b.p. sortan, introduction to axiomatic theory of convexity, kishyneff, 1984. [russian with english summary] [22] s.-w. xiang and h. yang, some properties of abstract convexity structures on topological spaces, nonlinear analysis 67 (2007), 803–808. n1-parkcubo cubo a mathematical journal vol.10, n o ¯ 04, (85–100). december 2008 a disc-cutting theorem and two-dimensional bifurcation of a reaction-diffusion system with inclusions martin väth* university of würzburg, math. institut am hubland, d-97074 würzburg, germany email: vaeth@mathematik.uni-wuerzburg.de abstract we provide a topological disc-cutting theorem which allows to prove that unilateral inclusions in a reaction-diffusion system of prey-predator type with a two-dimensional bifurcation parameter necessarily have a certain global branch of (global) bifurcation points. resumen presentamos un teorema “disc-cutting” topológico el cual permite probar que inclusiones unilaterales en un sistema de reación-difusión de tipo predador-presa con parametro de bifurcación 2-dimencional, necessariamente tiene una cierta rama global de puntos de bifucarción (global). *this paper was written in the framework of a heisenberg fellowship (az. va 206/1-2). financial support by the dfg is gratefully acknowledged. the author wants to thank e. vogt for valuable comments and suggestions. 86 martin väth cubo 10, 4 (2008) key words and phrases: global bifurcation, two-dimensional bifurcation, elliptic equation, inclusion, laplace operator. math. subj. class.: 35b32, 35j60, 35k47, 47h04, 47h11. 1 introduction although we provide in section 2 a general topological theorem about the existence of a global branch which is applicable to a large class of bifurcation problems with a parameter from a space of dimension at least 2, our main motivation for the result comes from the following particular problem. let ω ⊆ rn be a bounded domain with a lipschitz boundary, and let measurable (possibly empty) subsets ω0 ⊆ ω and γ0, γ ⊆ ∂ω be fixed with mes(γ0 ∩ γ) = 0. we consider the reactiondiffusion system ut = d1∆u + b11u + b12v + f1(d, x, u, v, ∇u, ∇v) = 0 on ω, vt ∈ d2∆v + b21u + b22v + f2(d, x, u, v, ∇u, ∇v) + { {0} on ω \ ω0, m0(d, x, u, v, ∇u, ∇v) on ω0, (1.1) with the boundary conditions                    u = v = 0 on γ0, ∂u ∂n = g1(d, x, u, v) on ∂ω \ γ0, ∂v ∂n = g2(d, x, u, v) on ∂ω \ (γ0 ∪ γ), ∂v ∂n ∈ g2(d, x, u, v) + m1(d, x, u, v) on γ. (1.2) here, d = (d1, d2) ∈ r 2 + is a bifurcation parameter, the nonlinearities fi and gi are small at (u, v) = 0, and mi are nonnegative interval functions specified later. the scalar parameters bij are assumed to satisfy b11 > 0, b12 < 0, b21 > 0, b22 < 0, b11 + b22 < 0, b11b22 − b12b21 > 0, (1.3) which means that system (1.1) is a special system of activator-inhibitor or prey-predator type such that in case d1 = d2 = 0 (i.e. without diffusion) the solution (0, 0) is stable. however, it is known (see e.g. [11] or [2, appendix] or [1]) that the stability of (1.1)/(1.2) with classical data m0 = m1 = 0 depends on d = (d1, d2). in fact, the domain ds of those d ∈ r 2 + where this system is exponentially stable is the right-hand side of the “envelope” of the sequence of hyperbolas cn := { (d1, d2) ∈ r 2 + : d2 = b12b21/κ 2 n d1 − b11/κn + b22 κn } . (1.4) cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 87 where κ1 ≤ κ2 ≤ · · · → ∞ denotes the sequence of eigenvalues of −∆, i.e. for which a (weak) nontrivial solution of the problem          −∆u = κnu on ω, u = 0 on γ0, ∂u ∂n = 0 on ∂ω \ γ0 (1.5) exists. d1 d2 c4 c3 c2 c1 ds figure 1: hyperbolas (1.4) determining ds using degree theory for multivalued maps, it was shown in [6] (for similar results and related systems, see also [2]–[5], [7]–[10]) that in the case of natural unilateral (possibly multivalued) functions m0 and/or m1 there is a destabilizing effect in the sense that even the stationary points admit a global bifurcation along certain paths in ds . in this paper, we will show by a purely topological argument that results of such a type actually imply the existence of certain global branches of bifurcation points (i.e. not only the bifurcation branch is global but actually even the set of bifurcation points itself). such phenomena can naturally arise only because our bifurcation parameter d is not only from a one-dimensional space. we point out that although we concentrate only on the system (1.1)/(1.2), the same results (and proofs) hold for all systems for which corresponding results along paths are available. in particular, this is the case when we consider instead of (1.2) the signorini type boundary conditions (see e.g. [9], [10])                    u = v = 0 on γ0, ∂u ∂n = 0 on ∂ω \ γ0, ∂v ∂n = 0 on ∂ω \ (γ0 ∪ γ), ∂v ∂n ≥ 0, v ≥ 0, ∂v ∂n · v = 0 on γ. 88 martin väth cubo 10, 4 (2008) 2 the disc-cutting theorem our main topological tool is based on a generalization of the whyburn lemma yielding global branches [12] and on the following result. theorem 2.1 (boundaries connect squaresides). let x be a topological space into which a square with (compact) “square-sides” a1, a2, a3, a4 and “square-interior” q is homeomorphically embedded. let v ⊆ x be open such that a1 ⊆ v and v ∩ a3 = ∅. then there is a connected subset c ⊆ q ∩ ∂v such that c ∩ ai 6= ∅ for i = 2, 4. the afore mentioned generalization of the whyburn lemma is the following (this version can be proved using only the [countable] axiom of dependent choices): theorem 2.2. let x be a regular space, a ⊆ x compact, and s ⊆ x. then for each open set u ⊇ a for which u ∩ s is compact and metrizable the following statements are equivalent: 1. for each open set ω ⊇ a with ω ⊆ u there is some x ∈ ∂ω in s. 2. there is a connected set c ⊆ (s ∩ u ) \ a such that c ∩ s intersects a and ∂u . for the proof of theorem 2.2 we refer to [12]. let us now use this result and some winding number theory to prove theorem 2.1. proof of theorem 2.1. we show first that it suffices to show the claim for the case x = r2 and q = (0, 1)×(0, 1) with a1 := [0, 1]×{0}, a2 := {1}×[0, 1], a3 := [0, 1]×{1}, and a4 := {0}×[0, 1]. to see that then the general case holds also, assume that we have a homeomorphism f of q onto a subset x0 of a general space x. note that x0 is the union of the sets ˜q := f (q) and ˜ai := f (ai). note that we do not assume that x is hausdorff, so x0 might not be closed, although it is compact. however, if v ⊆ x is open with ˜a1 ⊆ v and v ∩ ˜a3 = ∅, then v0 := f −1 (v ∩ x0) is open in q, and so there is some open v1 ⊆ r 2 with v1 ∩ q = v0. since v 0 ⊆ f −1 (v ∩ x0) is disjoint from the compact set f −1( ˜a3) = a3, and since r 2 is regular, we thus find a closed neighborhood of a3 which is disjoint from v0. eliminating this neighborhood from v1 if necessary, we may thus assume without loss of generality that v 1 ∩ a3 = ∅. by the special case of the theorem, we thus find a connected set c ⊆ q ∩ ∂v1 with c ∩ ai 6= ∅ for i = 2, 4. then ˜c := f (c) is connected, and its closure contains f (c) and thus intersects ˜ai = f (ai) for i = 2, 4. moreover, ˜c is contained in f (q ∩ ∂v1). note that, since q is open, v 1 ∩ q = (v1 ∩ q) ∩ q = v 0 ∩ q, and so ˜c ⊆ f (v 0 ∩ q) ⊆ ˜q ∩ v . moreover, ˜c is disjoint from v , since ˜c = f (c) is contained in x0 and disjoint from v ∩ x0 = f (v0) in view of c ∩ v0 = ∅. hence, we have indeed found a connected set ˜c ⊆ q ∩ v \ v = q ∩ ∂v whose closure intersects ˜ai for i = 2, 4. this proves the general case of the claim. we prove now the claim in the special case x := r2, q := (0, 1) × (0, 1) and ai as above. thus, let v ⊆ r2 be open with a1 ⊆ v and v ∩ a3 = ∅. without loss of generality, we may cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 89 assume in addition that v is contained in q0 := (−1, 2) × (−1, 1). indeed, otherwise, we could replace v by the intersection v0 := v ∩ {(x1, x2) ∈ (−1/2, 3/2) × (−1/2, 1) : x2 < 1 − dist(x1, [0, 1])} and note that a1 ⊆ v0, v 0 ∩ a3 = ∅, and q ∩ ∂v0 = q ∩ ∂v . put s := q ∩ ∂v and assume by contradiction that there is no connected subset c ⊆ s ∩ q with c ∩ ai 6= ∅ (i = 2, 4). applying theorem 2.2 in the space q with a := a2 and u := q \ a4, we find some open (in q) set ω ⊆ q with a2 ⊆ ω and ω ∩ a4 = ∅ such that the boundary b0 of ω with respect to q contains no point from ∂v . by the compactness of these sets, we thus find some closed neighborhood m ⊆ r2 of ∂v which is disjoint from b0. since ∂v is a compact subset of q0 and r 2 is regular, we may assume in addition that m ⊆ q0. covering the compact set v with sufficiently small open balls, we find an open set g ⊆ v ∪ m containing v such that the boundary of g consists of finitely many piecewise smooth closed curves. fix some a ∈ a1. since a ∈ v ⊆ g, the argument principle of complex analysis (or, in other words, the well-known connection between the degree of the identity function with the winding number of the boundary) implies that at least one of these curves must have nonzero winding number with respect to a. we think of such a closed curve as a continuous map γ : s1 → q0 (where s 1 denotes the unit circle). note that this curve lies completely in ∂g ⊆ m . in particular, γ(s1) is disjoint from b0 and thus contained in the union of the three sets ω2 := ω ∩ q, ω4 := (q \ ω) ∩ q = q \ ω, r := q0 \ (q ∪ {a}). since ω is open in q, it follows that ω2 is open in q and thus open in r 2 . analogously, also ω4 is open in r 2 . with the notation γ = (γ1, γ2), we define now a homotopy h : [0, 1] × s 1 → r2 \ {a} by h(t, s) :=      ((1 − t)γ1(s), γ2(s)) if s ∈ γ −1 (ω4), ((1 − t)γ1(s) + t, γ2(s)) if s ∈ γ −1 (ω2), (γ1(s), γ2(s)) otherwise. this map is indeed continuous by the glueing lemma, because γ can cross the boundary of ωi only at ai (i = 2, 4). we thus have shown that γ is homotopic (in r 2 \ {a}) to a curve which assumes only values in r. since r is obviously simply connected, γ is actually homotopic to a constant (in r 2 \ {a}). hence, the homotopy invariance of the winding number shows that γ actually has winding number 0 around a. this is the required contradiction. using theorems 2.1 and 2.2, we can now prove the following disc-cutting theorem: theorem 2.3 (disc-cutting). let x be a topological space into which a (compact) disc with “discinterior” q is homeomorphically embedded. let the “disc-boundary” be the union of four nonempty disjoint connected sets a1, a2, a3, a4, enumerated in order along the boundary. assume also that a2 and a4 both contain at least two points. 90 martin väth cubo 10, 4 (2008) let s ⊆ q be closed in q such that each compact smooth (via the embedding) injective path p in q ∪ a2 ∪ a4 with p ∩ ai 6= ∅ (i = 2, 4) contains some point from s. then there is a connected subset c ⊆ s such that c ∩ ai 6= ∅ for i = 1, 3. remark 2.1. one could also replace “smooth path” by “polygonal path” in the statement of theorem 2.3 with the obvious modification in the following proof. proof. we show first that it suffices to show the claim for the case x = r2 and the unit disc q. indeed, if f : q → x is a homeomorphism onto a subset of a general space x, let ˜q := f (q) and ˜ai := f (ai). let s ⊆ ˜q be as in the claim; in particular, s is closed in ˜q. then s0 := f −1 (s) is closed in q. the hypothesis on s means that each smooth path connecting a2 with a4 in q meets s0. the special case of the result thus implies that there is a connected subset c0 ⊆ s0 such that c0 ∩ ai 6= ∅ for i = 1, 3. then c := f (c0) ⊆ s is connected, and c ∩ ˜ai ⊇ f (c0) ∩ f (ai) 6= ∅. hence, the statement holds also in the general case. thus, to prove the theorem, we may assume without loss of generality that x = r2 and that q is the unit disc. assume by contradiction that a set c as in the claim does not exist. we apply theorem 2.2 in the space q with a := a1, u := q\a3, and s instead of s. observing that a ⊆ u , because a2 and a4 are nondegenerate, we find some open in q set ω ⊇ a1 with ω ∩ a3 = ∅ such that the boundary b0 of ω with respect to q contains no element of s. note that b0 is a closed subset of q and thus compact. note also that b0 is disjoint from s and from ai (i = 1, 3). we thus find an open neighborhood m ⊆ r2 of b0 which is disjoint from s ∪ a1 ∪ a3. moreover, if we let âi (i = 1, 3) be compact “intervals” of the circle boundary which contain the corresponding “intervals” ai (i = 1, 3) in their interior (with respect to the circle boundary) but still satisfy â1 ⊆ ω and â3 ⊆ q \ ω, and if we let âi (i = 2, 4) denote closure of the corresponding remaining intervals (contained in ai), we can apply theorem 2.1 with v := ω and the four “square-sides” âi. we thus find a connected set c ⊆ b0 such that there are points ai ∈ c ∩ âi for i = 2, 4, and so ai ∈ c ∩ ai (i = 2, 4). since c ⊆ b0 is connected, if follows that a2 and a4 belong to the same connected component of b0. since m ⊆ r 2 is an open neighborhood of b0, we may thus connect a2 and a4 by a smooth injective path in m . since m is disjoint from ai (i = 1, 3), we thus find a compact smooth injective path p in m ∩ (q ∪ a2 ∪ a4) with p ∩ ai 6= ∅ (i = 2, 4). since m ∩ s = ∅, this path cannot contain a point from s, contradicting the hypothesis. 3 the reaction-diffusion system with inclusions 3.1 detailed hypotheses we will consider the weak formulation of the stationary problem corresponding to (1.1)/(1.2), i.e. we will consider the weak formulation of d1∆u + b11u + b12v + f1(d, u, v, ∇u, ∇v) = 0 d2∆v + b21u + b22v + f2(d, u, v, ∇u, ∇v) ∈ −m0(d, u, v, ∇u, ∇v) in ω (3.1) cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 91 with boundary conditions            u = v = 0 on γ0, ∂u ∂n = g1(d, u, v) on ∂ω \ γ0, ∂v ∂n ∈ g2(d, u, v) + m1(d, x, u, v) on ∂ω \ γ0, (3.2) where we will assume that the (possibly multivalued) functions mi have the form m0(d, x, u, v, w, z) := [c0(d)m0(x, u, v, w, z), c0(d)m0(x, u, v, w, z)] and m1(d, x, u, v) := [c1(d)m1(x, u, v), c1(d)m1(x, u, v)], and where we assumed for the simplicity of notation that m 0 , m0, m1 and m1 vanish for x /∈ ω0 or x /∈ γ, respectively, where ω0 ⊆ ω and γ ⊆ ∂ω \ γ0 are measurable. in order to require nontrivial situations, we will assume that mesω0 > 0 or mesγ > 0 (or both). (3.3) for our considerations it will be crucial that mesγ0 > 0 (3.4) so that we can equip the space h of all functions from w 1,2(ω, r2) vanishing on γ0 with the scalar product 〈u, v 〉 := ∫ ω 〈∇u (x), ∇v (x)〉 dx, which under hypothesis (3.4) generates the inherited topology, see e.g. [13, theorem 4.8.1]. we assume (1.3), and by ds ⊆ r 2 + , we denote the (open) domain of stability mentioned in the introduction. note that all points of r 2 + ∩ ∂ds belong to some of the hyperbolas cn defined by (1.4). we will assume that all of the above functions are at least defined for d ∈ ds ∪ {d ∗} where the point d∗ ∈ cn ∩ ∂ds will be specified later on. for i = 0, 1, we fix exponents pi, qi, and q ∗ i according to the following restrictions. { pi ∈ [1, ∞), 1 ≤ q ∗ i < qi < ∞ arbitrary if n ≤ 2, p0 := n n−2 , p1 := n−1 n−2 , ∞ > q0 > q ∗ 0 := 2n n+2 , ∞ > q1 > q ∗ 1 := 2n−2 n if n > 2. moreover, we assume the following hypothesis. 1. ci, ci are continuous on ds ∪ {d ∗} and without zeros on ds . 2. for each d ∈ ds ∪ {d ∗} the following holds: the functions fi(d, · , u, v, w, z) and gi(d, · , u, v) are measurable, and fi(d, x, · , · , · , · ) and gi(d, x, · , · ) are continuous for almost all x. moreover, fi and gi satisfy the growth estimates |fi(d, x, u, v, w, z)| ≤ ad(x) + bd · ((|u| + |v|) p0 + ‖w‖ + ‖z‖) 2/q0 , 92 martin väth cubo 10, 4 (2008) and |gi(d, x, u, v, w, z)| ≤ ãd(x) +˜bd · ((|u| + |v|) p1 ) 2/q1 , where the quantities ‖ad‖lq0 (ω) , ‖ãd‖lq1 (∂ω\γ0) , bd, and ˜bd are locally bounded with respect to d. 3. for each d0 ∈ ds ∪ {d ∗} there are estimates of the form |fi(d, x, u, v, w, z) − fi(d0, x, u, v, w, z)| ≤ cd0 (d) ( ad0,d(x) + ((|u| + |v|) p0 + ‖w‖ + ‖z‖) 2/q ∗ 0 ) and |gi(d, x, u, v) − gi(d0, x, u, v)| ≤ c̃d0 (d) ( ãd0,d(x) + (|u| + |v|) 2p1/q ∗ 1 ) where ‖ad0,d‖l q ∗ 0 (ω) , ‖ãd0,d‖l q ∗ 1 (∂ω\γ0) ≤ 1 and cd0 (d), c̃d0 (d) → 0 as d → d0. 4. fi and gi become uniformly small at (u, v) = 0 in the sense that for each sufficiently small ball b in ds (and thus for each nonempty compact subset b ⊆ ds) the following holds: sup w,z∈rn sup d∈b |fi(d, x, u, v, w, z)| ≤ cb max { (|u| + |v|)2p0/q0 , |u| + |v| } lim (u,v,w,z)→0 sup d∈b fi(d, x, u, v, w, z) |u| + |v| + ‖w‖ + ‖z‖ = 0 sup w,z∈rn sup d∈b |gi(d, x, u, v)| ≤ cb max { (|u| + |v|)2p1/q1 , |u| + |v| } lim (u,v,w,z)→0 sup d∈b gi(d, x, u, v) |u| + |v| = 0 5. the functions m 0 ( · , u, v, w, z) and m0( · , u, v, w, z) are measurable, m0(x, · , · , · , · ) is lower semicontinuous, m0(x, · , · , · , · ) is upper semicontinuous, and the corresponding superposition operators m 0 (u, v, w, z)(x) := m 0 (x, u(x), v(x), w(x), z(x)) and m 0(u, v, w, z)(x) := m0(x, u(x), v(x), w(x), z(x)) send continuous (and thus measurable) functions into measurable functions. moreover, we require for some a0 ∈ lq0 (ω) and b0 < ∞ the growth estimates max {|m 0 (x, u, v, w, z)| , |m0(x, u, v, w, z)|} ≤ a0(x) + b0 · ((|u| + |v|) p0 + ‖w‖ + ‖z‖) 2/q0 . cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 93 6. the functions m 1 ( · , u, v) and m1( · , u, v) are measurable, m1(x, · , · ) is lower semicontinuous, m1(x, · , · ) is upper semicontinuous, and the corresponding superposition operators m 1(u, v)(x) := m1(x, u(x), v(x)) and m 1(u, v)(x) := m1(x, u(x), v(x)) send continuous (and thus measurable) functions into measurable functions. moreover, we require the following growth estimates for some a1 ∈ lq1 (γ) and b1 < ∞: max {|m 1 (x, u, v)| , |m1(d, x, u, v)|} ≤ a1(x) + b1 · (|u| + |v|) 2p1/q1 . 7. the following unilateral conditions hold: 0 = c 0 (d)m 0 (x, u, v, w, z) = c0(d)m0(x, u, v, w, z) if v > 0, 0 = c 0 (d)m 0 (x, u, 0, w, z) ≤ c0(d)m0(x, u, 0, w, z) 0 ≤ c0(d)m0(x, u, v, w, z) ≤ c0(d)m0(x, u, v, w, z) if v < 0, 0 = c 1 (d)m 1 (x, u, v) = c1(d)m1(x, u, v) if v > 0, 0 = c1(d)m1(x, u, 0) ≤ c1(d)m1(x, u, 0) 0 ≤ c1(d)m1(x, u, v) ≤ c1(d)m1(x, u, v) if v < 0. lim (u,v,w,z)→0 v<0 |m 0 (x, u, v, w, z)| v = −∞ for almost all x ∈ ω0, lim (u,v)→0 v<0 |m 1 (x, u, v)| v = −∞ for almost all x ∈ γ. 3.2 definition of weak solutions we consider the cone k := {u = (u1, u2) ∈ h : u2|ω0 ≥ 0 and u2|γ ≥ 0} and define operators a(d), g(d, · ), m (d, · ) : h → h by 〈a(d)u, v 〉 := ∫ ω 〈( d −1 1 b11 d −1 1 b12 d −1 2 b21 d −1 2 b22 ) u (x), v (x) 〉 dx, 〈g(d, u ), v 〉 := ∫ ω 〈( d −1 1 f1(d, u (x), ∇u (x)) d −1 2 f2(d, u (x), ∇u (x)) ) , v (x) 〉 dx + ∫ ∂ω\γ0 〈( g1(d, u (x)) g2(d, u (x)) ) , v (x) 〉 dx, 94 martin väth cubo 10, 4 (2008) and m (d, u ) := ⋂ v ∈k { z ∈ h : 〈z, v 〉 ∈ ∫ ω0 〈( 0 · d−1 1 d −1 2 m0(d, x, u (x), ∇u (x)) ) , v (x) 〉 dx + ∫ γ 〈( 0 m1(d, x, u (x)) ) , v (x) 〉 dx } := ⋂ v =(ṽ,v)∈k { z = ( 0 z ) ∈ h : ∫ ω0 d −1 2 c 0 (d)m 0 (x, u (x), ∇u (x))v(x) dx + ∫ γ c 1 (d)m 1 (x, u (x))v(x) dx ≤ 〈z, v 〉 ≤ ∫ ω0 d −1 2 c0(d)m0(x, u (x), ∇u (x))v(x) dx + ∫ γ c1(d)m1(x, u (x))v(x) dx } , respectively. we define weak solutions of problem (3.1)/(3.2) as solutions of the inclusion u − a(d)u − g(d, u ) ∈ m (d, u ). our hypotheses imply in particular (see e.g. [6]): proposition 3.1. f (d, u ) := a(d)u − g(d, u ) − m (d, u ) is an upper semicontinuous map with nonempty compact values. moreover, f is compact in the sense that if d0 ⊆ ds ∪ {d ∗} is compact and b ⊆ h is bounded then f (d0 × b) is precompact. 3.3 local and global bifurcation points note that (d, 0) ∈ ds × h is always a solution of (3.1)/(3.2). we call a pair (d, u ) ∈ ds × h a nontrivial solution if u = (u, v) 6= 0, and if (d, u, v) is a weak solution of (3.1)/(3.2). the local bifurcation points (in ds ) are the elements of the set blocal := {d ∈ ds : each neighborhood of (d, 0) ∈ ds × h contains a nontrivial solution} . we call a point d ∈ ds a global bifurcation point (with respect to a point d ∗ ∈ cn ∩ ∂ds) if there is a connected set c ⊆ ds × (h \ {0}) consisting only of nontrivial solutions such that (d, 0) ∈ c and such that c is a global branch in the sense that at least one of the following holds: 1. c is unbounded. 2. c reaches d∗, i.e. c contains some point (d∗, u ) which is a weak solution of (3.1)/(3.2). note that in the second case, we do not exclude u = 0, i.e. c might return to the trivial branch at the hyperbola point d∗ ∈ cn. we denote the set of global bifurcation points (with respect to d∗) by bglobal(d ∗ ). cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 95 proposition 3.2. each global bifurcation point is a local bifurcation point. moreover, blocal is closed in ds. in particular, bglobal(d∗) ∩ ds ⊆ blocal. in our considerations an important role will be played by the vertical asymptote of the rightmost hyperbola {(d1, d2) ∈ ds : d1 = b11/κ1} (3.5) and the corresponding part to the right of this asymptote, i.e. h := {(d1, d2) ∈ ds : d1 > b11/κ1} . (3.6) the following has been shown in [6]: proposition 3.3. h ∩ blocal = ∅. we also need another terminology. we say that a point d ∈ ∂ds is n-interior if d ∈ cn and if there is some eigenfunction e of −∆ for the eigenvalue κn, i.e. e = u is a weak solution of (1.5), such that, for some constant ε > 0, e ≥ ε > 0 almost everywhere on ω0 and e ≥ ε > 0 almost everywhere on γ. (3.7) recall in this connection that we require (3.3) for the case that γ is a smooth manifold with boundary and ω0 = ∅, we replace (3.7) by the milder requirement e(x) > 0 for almost all x ∈ γ. (3.8) we say that d ∈ ∂ds is (n, m)-interior if d ∈ cn ∩ cm and if there is a function e which is a linear combination of eigenfunctions to the eigenvalues κn and κm such that (3.7) or (3.8) holds, respectively. if d ∈ cn ∩ cm ∩ ∂ds and d is n-interior or m-interior then d is also (n, m)-interior. however, d might be (n, m)-interior without being n-interior or m-interior. using the main results from [6], we will prove now: lemma 3.1. let d ∈ ∂ds be n-interior or (n, m)-interior. then there is an open neighborhood u0 ⊆ r 2 of d such that u0 ∩ ds ∩ blocal = ∅. moreover, if the hypotheses are satisfied with d ∗ = d, then each continuous compact path γ in ds ∪ {d ∗} connecting d∗ = d with some point from (3.6) contains some point from bglobal(d ∗ ) ⊆ ds. lemma 3.1 would follow rather straightforwardly from the results of [6] if we would allow that the connected set c in the definition of global bifurcation points is contained in (ds ∪{d ∗}) × (h \ {0}). however, it might happen that c \({d∗}×h) fails to be connected. therefore, we need some additional arguments. we use the following result which is actually a consequence of theorem 2.2 (and can also be proved using only the [countable] axiom of dependent choices, see [12]): 96 martin väth cubo 10, 4 (2008) theorem 3.1. let x be a regular space, a ⊆ x compact, and s ⊆ x be closed. suppose that s is locally compact, metrizable and σ-compact. then for each open set u ⊇ a the following statements are equivalent: 1. there is a connected set c ⊆ s which intersects a and is either noncompact or intersects ∂u . 2. there is a connected set c ⊆ (s∩u )\a such that c ∩s intersects a and is either noncompact or intersects ∂u . proof of lemma 3.1. only the last claim is not immediately contained in some of the results from [6]. to see this last claim, we apply the main result from [6] first to show that there is a connected set c0 ⊆ (ds ∪ {d ∗}) × (h \ {0}) such that c0 intersects (γ ∩ ds ) × {0} and such that either c0 is unbounded or c0 intersects also {d ∗} × h. moreover, we will arrange it that, in the space x := r2 × h, c0 has the additional property that closures of bounded subsets of s := c0 are compact and consist only of (weak) solutions and satisfies c0 ∩ (r 2 × {0}) = (γ \ (u0 ∩ ds )) × {0} . (3.9) indeed, assume that γ = σ([a, b]) with some continuous σ : [a, b] → ds ∪ {d ∗} satisfying σ(a) = d∗ and σ(b) ∈ h. we extend σ to a continuous σ : [a, ∞) → ds ∪ {d ∗} with σ(s) ∈ h for all s ≥ b such that both components of σ(s) tend to ∞ as s → ∞. for all sufficiently small s0 ∈ (a, b) we have σ([a, s0]) ⊆ u0, and by the main result from [6], we find some connected set c1 ⊆ [a, ∞) × (h \ {0}) such that c0 := {(σ(s), u) : (s, u) ∈ c1} consists only of (nontrivial) weak solutions of (3.1)/(3.2) and such that c1 contains some point from [s0, b] × {0} and such that either c1 is unbounded or c1 contains some point from {a} × h or from ([a, s0) ∪ (b, ∞)) × {0} the set c0 has all required properties. indeed, since c0 consists only of nontrivial solutions, the closure of σ([a, ∞)) is contained in γ ∪ (u0 ∩ ds ) ∪ h, and no point of (u0 ∩ ds) ∪ h is a local bifurcation point, we obtain (3.9). the set c0 is connected, because it is the image of the connected set c1 under the continuous map t (s, u) := (σ(s), u). the set c0 contains t (c1) and thus intersects t ([s0, b] × {0}) ⊆ (γ ∩ ds ) × {0} and is either unbounded (by our choice of the extension of σ) or intersects {d∗} × h or (u0 ∪ h) × {0}. in the latter case, c0 actually intersects {d∗} × {0} by (3.9). to see these remaining properties, recall that with f from proposition 3.1 the weak solutions of (3.1)/(3.2) are the elements of {(d, u) ∈ (ds ∪ {d ∗}) × h : u ∈ f (d, u)} . cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 97 since c0 is contained in this set, f is upper semicontinuous, and the closure of σ([a, ∞)) is contained in ds ∪ {d ∗}, also s = c0 is contained in this set. the compactness of f described in proposition 3.1 and our choice of the extension of σ implies that closed bounded subsets of s are compact. hence, s = c0 has all required properties. in particular, s is locally compact and σ-compact. we apply theorem 3.1 with a := (γ \ u0) × {0} and u := x \ ({d ∗} × h). the connected set c0 witnesses that the first statement of theorem 3.1 is satisfied: note that this set indeed intersects a in view of (3.9), because c0 intersects (γ ∩ ds ) × {0}. hence, also the second statement of theorem 3.1 holds which means that there is a connected set c ⊆ (s ∩ u ) \ a such that the set c contains some point (d0, 0) with d0 ∈ γ \ u0 ⊆ γ ∩ ds and such that either c is noncompact (and thus unbounded) or intersects ∂u = {d∗} × h. thus, d0 ∈ bglobal(d ∗ ). 4 the main result theorem 4.1. let d0 := ds \ h where h is from (3.6). let d∗ ∈ ∂ds be m-interior or (n, m)-interior (n ≤ m) and such that the hypotheses described at the beginning of section 3 are satisfied with this d∗. then there is a connected set b ⊆ bglobal(d∗) ∩ d0 ⊆ blocal such that b intersects the d1-axis or some hyperbola ck “strictly under” d∗. more precisely, we have k ≥ n, and the case ck = cm is only possible if d ∗ is an intersection point of two different hyperbolas. in all cases, the intersection b ∩ ck does not contain d ∗ (i.e. is strictly under d∗). moreover, this branch b satisfies in addition the following: 1. if cn is the right-most hyperbola (i.e. if cn = c1) then b is unbounded. 2. otherwise (i.e. if cn 6= c1) the set b is unbounded, or b intersects some hyperbola ck “strictly over” d∗ (i.e. k ≤ n, and the case ck = cn is only possible if d ∗ is an intersection point of two different hyperbolas; b ∩ ck does not contain d ∗). moreover, for any k for which there is some k-interior point we have b ∩ck = ∅, and for any pair (k, ℓ) for which the intersection point is (k, ℓ)-interior point this intersection point is not contained in b. figure 2 illustrates qualitatively the four possibilities of branches b of bifurcation points if there is some n-interior point with cn 6= c1; one of these possibilities must (qualitatively) occur according to theorem 4.1. similarly, figure 3 illustrates the two possibilities of branches b if there is some 1-interior point. 98 martin väth cubo 10, 4 (2008) in particular, if there are n-interior points for every n, then the last statement of theorem 4.1 implies that only one possibility can occur: there must be a branch b which is unbounded and such that b intersects the d1-axis (possibly at (0, 0)). we point out that (contrary to what the figures might suggest) the theorem does not state that the branch b is pathwise connected, i.e. it might look “weird” (but it is connected in the topological sense). d1 d2 c4 c3 c2 c1 d0 l h figure 2: the four qualitative different possible branches b of bifurcation points if there is some 2-interior point (one of these must occur) d1 d2 c4 c3 c2 c1 d0 l h figure 3: the two qualitative different possible branches b of bifurcation points if there is some 1-interior point (one of these two must occur) proof. the last statement of theorem 4.1 is automatically satisfied by the first claim of lemma 3.1, since bglobal(d ∗) ⊆ blocal must be disjoint from any k-interior or (k, ℓ)-interior point. cubo 10, 4 (2008) a disc-cutting theorem and two-dimensional bifurcation ... 99 using this fact with d∗, we find some open neighborhood u0 ⊆ r 2 which is disjoint from bglobal(d ∗) ⊆ blocal. let l0 ⊆ h be some line which is parallel but strictly to the right of the line (3.5). let q, h0 ⊆ ds be that parts to the left and right of this line l0, respectively. lemma 3.1 implies blocal ∩ ds ⊆ q. using the one-point compactification x of q, we consider q as the disc-interior of some homeomorphically embedded disc, whose boundary corresponds to the union of the d1-axis, the line l0, the point ∞, and the “envelope” e = r 2 + ∩ ∂ds of all of the hyperbolas cn. let a2 be that part of the boundary which corresponds to l0 (without the two “boundary points” at ∞ and at the d1-axis), and let a4 correspond to u0 ∩ e. let a1 and a3 denote the ramining (compact) connected subsets of the boundary of the disc q. now we can apply the disc-cutting theorem with s = q ∩ bglobal(d∗). in fact, each continuous compact path in q connecting a2 with a4 must intersect s by lemma 3.1. hence, the disc-cutting theorem 2.3 implies the existence of a connected set b ⊆ s with b ∩ ai 6= ∅ for i = 1, 3. since b cannot intersect l0, the property b ∩ a3 means that either b is unbounded or that b intersects some point of some ck “strictly above” d ∗ . the property b ∩ a1 means that b intersect some point of some ck “strictly below” d ∗ or the d1-axis. received: march 2008. revised: march 2008. references [1] drábek, p. and kučera, m., reaction-diffusion systems: destabilizing effect of unilateral conditions, nonlinear anal., 12 (1988), no. 11, 1172–1192. [2] eisner, j., reaction-diffusion systems: destabilizing effect of conditions given by inclusions, math. bohem., 125 (2000), no. 4, 385–420. [3] , critical and bifurcation points of reaction-diffusion systems with conditions given by inclusions, nonlinear anal., 46 (2001), 69–90. [4] , reaction-diffusion systems: destabilizing effect of conditions given by inclusions ii, examples, math. bohem., 126 (2001), no. 1, 119–140. [5] eisner, j. and kučera, m., spatial patterning in reaction-diffusion systems with nonstandard boundary conditions, fields institute communications, 25 (2000), 239–256. [6] eisner, j., kučera, m. and väth, m., global bifurcation of a reaction-diffusion system with inclusions, (submitted). [7] kučera, m., bifurcation points of variational inequalities, czechoslovak math. j., 32 (1982), 208–226. 100 martin väth cubo 10, 4 (2008) [8] , a new method for obtaining eigenvalues of variational inequalities. multiple eigenvalues, czechoslovak math. j., 32 (1982), 197–207. [9] , influence of signorini boundary conditions on bifurcation in reaction-diffusion systems, proceedings of the 5th international isaac congress, catania 2005, 2007, (to appear). [10] kučera, m. and bosák, m., bifurcation for quasi-variational inequalities of reactiondiffusion type, saacm, 3 (1993), no. 2, 111–127. [11] turing, a.m., the chemical basis of morphogenesis, phil. trans. roy. soc. london, b (1952), 37–72. [12] väth, m., global solution branches and a topological implicit function theorem, ann. mat. pura appl., 186 (2007), no. 2, 199–227. [13] ziemer, w.p., weakly differentiable functions, springer, new york, berlin, heidelberg, 1989. n7-vaeth_disc cubo a mathematical journal vol.10, n o ¯ 03, (13–20). october 2008 poincaré type inequalities for linear differential operators george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152 u.s.a. email: ganastss@memphis.edu abstract various lp form poincaré type inequalities [1], forward and reverse, are given for a linear differential operator l, involving its related initial value problem solution y, ly, the associated green’s function h and initial conditions at point x0 ∈ r. resumen varias lp desigualdes de tipo poincaré [1], hacia adelante o atrás, son dadas para un operador diferencial linear l, envolviendo la solución y de un problema de valor inicial asociado, ly, la función green asociada h y las condiciones iniciales en un punto x0 ∈ r. key words and phrases: poincaré inequality, linear differential operator. math. subj. class.: 26d10. 14 george a. anastassiou cubo 10, 3 (2008) 1. background here we follow [2], pp. 145-154. let [a, b] ⊂ r, ai (x) , i = 0, 1, . . . , n − 1 (n ∈ n) , h (x) be continuous functions on [a, b] and let l = dn + an−1 (x) d n−1 + . . . + a0 (x) be a fixed linear differential operator on c n ([a, b]) . let y1 (x) , . . . , yn (x) be a set of linear independent solutions to ly = 0. here the associated green’s functions for l is h (x, t) := ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ y1 (t) . . . yn (t) y′ 1 (t) . . . y′n (t) . . . . . . . . . y (n−2) 1 (t) . . . y (n−2) n (t) y1 (x) . . . yn (x) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ y1 (t) . . . yn (t) y′ 1 (t) . . . y′n (t) . . . . . . . . . y (n−2) 1 (t) . . . y (n−2) n (t) y (n−1) 1 (t) . . . y (n−1) n (t) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ , (1) which is a continuous function on [a, b] 2 . consider a fixed x0 ∈ [a, b] , then y (x) = ∫ x x0 h (x, t) h (t) dt, ∀x ∈ [a, b] , (2) is the unique solution to the initial value problem ly = h; y(i) (x0) = 0, i = 0, 1, . . . , n − 1. (3) next we assume all of the above. 2. results we present the following poincaré type inequalities. theorem 1. let x0 1 : 1 p + 1 q = 1; ν > 0. cubo 10, 3 (2008) poincaré type inequalities ... 15 then 1) ‖y‖ lν (x0,b) ≤ ( ∫ b x0 ( ∫ x x0 |h (x, t)| p dt )ν/p dx )1/ν ‖ly‖ lq(x0,b) . (4) when ν = q we have 2) ‖y‖ lq (x0,b) ≤ ( ∫ b x0 ( ∫ x x0 |h (x, t)| p dt )q/p dx )1/q ‖ly‖ lq (x0,b) . (5) when ν = p = q = 2 we get 3) ‖y‖ l2(x0,b) ≤ ( ∫ b x0 ( ∫ x x0 h2 (x, t) dt ) dx )1/2 ‖ly‖ l2(x0,b) . (6) proof. from (2) we have |y (x)| ≤ ∫ x x0 |h (x, t)| |h (t)| dt ≤ ( ∫ x x0 |h (x, t)| p dt ) 1/p (∫ x x0 |h (t)| q dt ) 1/q ≤ ( ∫ x x0 |h (x, t)| p dt ) 1/p ( ∫ b x0 |h (t)| q dt ) 1/q . (7) that is |y (x)| ν ≤ ( ∫ x x0 |h (x, t)| p dt )ν/p ‖ly‖ ν lq (x0,b) , (8) therefore ∫ b x0 |y (x)| ν dx ≤ ( ∫ b x0 ( ∫ x x0 |h (x, t)| p dt )ν/p dx ) ‖ly‖ ν lq(x0,b) , (9) proving the claim. 2 we continue with theorem 2. let x0 > a and x ∈ [a, x0] , and p, q > 1 : 1 p + 1 q = 1; ν > 0. then 1) ‖y‖ lν (a,x0) ≤ ( ∫ x0 a ( ∫ x0 x |h (x, t)| p dt )ν/p dx ) 1/ν ‖ly‖ lq (a,x0) . (10) when ν = q we have 2) ‖y‖ lq(a,x0) ≤ ( ∫ x0 a ( ∫ x0 x |h (x, t)| p dt )q/p dx ) 1/q ‖ly‖ lq (a,x0) . (11) 16 george a. anastassiou cubo 10, 3 (2008) when ν = p = q = 2 we get 3) ‖y‖ l2(a,x0) ≤ ( ∫ x0 a ( ∫ x0 x h 2 (x, t) dt ) dx ) 1/2 ‖ly‖ l2(a,x0) . (12) proof. from (2) we have |y (x)| ≤ ∫ x0 x |h (x, t)| |h (t)| dt ≤ ( ∫ x0 x |h (x, t)| p dt )1/p (∫ x0 x |h (t)| q dt )1/q ≤ ( ∫ x0 x |h (x, t)| p dt ) 1/p (∫ x0 a |h (t)| q dt ) 1/q . (13) that is |y (x)| ν ≤ ( ∫ x0 x |h (x, t)| p dt )ν/p ‖ly‖ ν lq (a,x0) , (14) therefore ∫ x0 a |y (x)| ν dx ≤ ( ∫ x0 a ( ∫ x0 x |h (x, t)| p dt )ν/p dx ) ‖ly‖ ν lq (a,x0) , (15) proving the claim. 2 extreme cases follow proposition 3. here x0 < b, x ∈ [x0, b] , and p = 1, q = ∞. then 1) ‖y‖ lν (x0,b) ≤ ( ∫ b x0 ( ∫ x x0 |h (x, t)| dt )ν dx )1/ν ‖ly‖ l∞(x0,b) . (16) when ν = 1 we have 2) ‖y‖ l1(x0,b) ≤ ( ∫ b x0 ( ∫ x x0 |h (x, t)| dt ) dx ) ‖ly‖ l∞(x0,b) . (17) proof. from (2) we have |y (x)| ≤ ∫ x x0 |h (x, t)| |h (t)| dt ≤ ( ∫ x x0 |h (x, t)| dt ) ‖h‖ l∞(x0,b) . (18) cubo 10, 3 (2008) poincaré type inequalities ... 17 that is |y (x)| ν ≤ ( ∫ x x0 |h (x, t)| dt )ν ‖ly‖ ν l∞(x0,b) , (19) and ∫ b x0 |y (x)| ν dx ≤ ( ∫ b x0 ( ∫ x x0 |h (x, t)| dt )ν dx ) ‖ly‖ ν l∞(x0,b) , (20) proving the claim. 2 we continue with proposition 4. here x0 > a, x ∈ [a, x0] , and p = 1, q = ∞. then 1) ‖y‖ lν (a,x0) ≤ ( ∫ x0 a ( ∫ x0 x |h (x, t)| dt )ν dx )1/ν ‖ly‖ l∞(a,x0) . (21) when ν = 1 we get 2) ‖y‖ l1(a,x0) ≤ ( ∫ x0 a ( ∫ x0 x |h (x, t)| dt ) dx ) ‖ly‖ l∞(a,x0) . (22) proof. from (2) we have |y (x)| ≤ ∫ x0 x |h (x, t)| |h (t)| dt ≤ ( ∫ x0 x |h (x, t)| dt ) ‖h‖ l∞(a,x0) . (23) that is |y (x)| ν ≤ ( ∫ x0 x |h (x, t)| dt )ν ‖ly‖ ν l∞(a,x0) , (24) and ∫ x0 a |y (x)| ν dx ≤ ( ∫ x0 a ( ∫ x0 x |h (x, t)| dt )ν dx ) ‖ly‖ ν l∞(a,x0) , (25) proving the claim. 2 next we give reverse poincaré type inequalities. theorem 5. let x0 < b, x ∈ [x0, b] , and 0 0. assume h (x, t) ≥ 0 for x0 ≤ t ≤ x, and ly = h is of fixed sign and nowhere zero. then 1) ‖y‖ lν (x0,b) ≥ ( ∫ b x0 ( ∫ x x0 (h (x, t)) p dt )ν/p dx ) 1/ν ‖ly‖ lq (x0,b) . (26) 18 george a. anastassiou cubo 10, 3 (2008) when ν = p we get 2) ‖y‖ lp(x0,b) ≥ ( ∫ b x0 ( ∫ x x0 (h (x, t)) p dt ) dx )1/p ‖ly‖ lq(x0,b) . (27) when ν = 1 we obtain 3) ‖y‖ l1(x0,b) ≥ ( ∫ b x0 ( ∫ x x0 (h (x, t)) p dt )1/p dx ) ‖ly‖ lq(x0,b) . (28) proof. by (2) we have |y (x)| = ∫ x x0 h (x, t) |h (t)| dt, for all x0 ≤ x ≤ b. (29) from (29) by reverse hölder’s inequality we obtain |y (x)| ≥ ( ∫ x x0 (h (x, t)) p dt )1/p (∫ x x0 |h (t)| q dt )1/q ≥ ( ∫ x x0 (h (x, t)) p dt )1/p ( ∫ b x0 |h (t)| q dt ) 1/q , (30) for all x0 < x ≤ b. i.e. it holds |y (x)| ν ≥ ( ∫ x x0 (h (x, t)) p dt )ν/p ‖h‖ ν lq (x0,b) , (31) for all x0 ≤ x ≤ b, and ∫ b x0 |y (x)| ν dx ≥ ( ∫ b x0 ( ∫ x x0 (h (x, t)) p dt )ν/p dx ) ‖h‖ ν lq(x0,b) , (32) proving the claim. 2 we continue with theorem 6. let x0 > a, x ∈ [a, x0] , and 0 0. assume h (x, t) ≤ 0 for x ≤ t ≤ x0, and ly = h is of fixed sign and nowhere zero. then 1) ‖y‖ lν (a,x0) ≥ ( ∫ x0 a ( ∫ x0 x (−h (x, t)) p dt )ν/p dx ) 1/ν ‖ly‖ lq(a,x0) . (33) cubo 10, 3 (2008) poincaré type inequalities ... 19 when ν = p we get 2) ‖y‖ lp(a,x0) ≥ ( ∫ x0 a ( ∫ x0 x (−h (x, t)) p dt ) dx )1/p ‖ly‖ lq (a,x0) . (34) when ν = 1 we have 3) ‖y‖ l1(a,x0) ≥ ( ∫ x0 a ( ∫ x0 x (−h (x, t)) p dt )1/p dx ) ‖ly‖ lq(a,x0) . (35) proof. from (2) we have |y (x)| = ∣ ∣ ∣ ∣ ∫ x x0 h (x, t) h (t) dt ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ x0 x h (x, t) h (t) dt ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ x0 x (−h (x, t)) h (t) dt ∣ ∣ ∣ ∣ = ∫ x0 x (−h (x, t)) |h (t)| dt. (36) from (36) by reverse hölder’s inequality we obtain |y (x)| ≥ ( ∫ x0 x (−h (x, t)) p dt )1/p (∫ x0 x |h (t)| q dt )1/q ≥ ( ∫ x0 x (−h (x, t)) p dt )1/p (∫ x0 a |h (t)| q dt )1/q . (37) for all a ≤ x < x0. i.e. it holds |y (x)| ν ≥ ( ∫ x0 x (−h (x, t)) p dt )ν/p ‖ly‖ ν lq(a,x0) , (38) for all a ≤ x ≤ x0, and ∫ x0 a |y (x)| ν dx ≥ ( ∫ x0 a ( ∫ x0 x (−h (x, t)) p dt )ν/p dx ) ‖ly‖ ν lq (a,x0) , (39) proving the claim. 2 received: december 2007. revised: april 2008. 20 george a. anastassiou cubo 10, 3 (2008) references [1] g. acosta and r. g. durán, an optimal poincaré inequality in l1 for convex domains, proc. ams, 132 (1)2003, pp.195–202. [2] d. kreider, r. kuller, d. ostberg and f. perkins, an introduction to linear analysis, addison-wesley publishing company, inc., reading, mass., usa, 1966. n02 cubo a mathematical journal vol.11, no¯ 04, (15–28). september 2009 small data global existence and scattering for the mass-critical nonlinear schrödinger equation with power convolution in r3 george venkov department of differential equations, faculty of applied mathematics and informatics, technical university of sofia, 8 "kliment ohridski" str., 1756 sofia, bulgaria. email: gvenkov@tu-sofia.bg abstract the main purpose of the present paper is to consider the well-posedness of the l2critical nonlinear schrödinger equation of a hartree type i∂tψ + △ψ = (|x|−1 ∗ |ψ| 8 3 )ψ, (t,x) ∈ r+ × r3. more precisely, we shall establish the local existence of solutions for initial data ψ0 in l 2 (r 3 ), as well as the existence of global solutions for small initial data. moreover, we shall prove the existence of scattering operator. resumen el principal objetivo del artículo es considerar si la ecuación de schrödinger no lineal l 2crítica del tipo hartree i∂tψ + △ψ = (|x|−1 ∗ |ψ| 8 3 )ψ, (t,x) ∈ r+ × r3. está bien puesta o no. en efecto, estableceremos la existencia local de soluciones para datos iniciales ψ0 en l 2 (r 3 ), así como la existencia de soluciones globales para datos iniciales pequeños. más aún, probaremos la existencia del operador de scattering. key words and phrases: nonlinear schrödinger equation, power convolution, hartree equation, local and global existence. math. subj. class.: 35a05, 35q55. 16 george venkov cubo 11, 4 (2009) 1 introduction in this paper we consider the cauchy problem for the defocussing mass-critical nonlinear schrödinger equation of a hartree type i∂tψ + △ψ = (|x|−1 ∗ |ψ| 8 3 )ψ, (t,x) ∈ r+ × r3, (1.1) ψ(0,x) = ψ0(x), (1.2) where ∗ denotes the usual convolution operator in r3. here ψ = ψ(t,x) is a complex valued function and the initial value ψ0 : r 3 7→ c is given. equation (1.1) can be written in terms of the wave function ψ and the potential v as the schrödinger-poisson system of the form i∂tψ + △ψ = v ψ, (1.3) △v = −4π|ψ| 83 , (1.4) where the (−) sign in the poisson equation (1.4) corresponds to the repulsive type interaction. equation (1.1) is known as the schrödinger equation with nonlocal power nonlinearity of a hartree type. the main purpose of the present work is to obtain local and global existence, well-posedness and scattering of solutions to (1.1)–(1.2). when studying the problem of local and global existence of solution to the nonlinear schrödinger equation, one is primarily interested in the scaling symmetry of the equation under the transformation ψλ(t,x) = 1 λa ψ( t λ2 , x λ ), λ > 0, (1.5) with some constant a depending on the equation. on the other hand, after the above scaling the l p-norm has dimension, namely ‖ψλ‖lp(r3) = 1 λ a− 3 p ‖ψ‖lp(r3). (1.6) a simple calculation shows that for a = 3 2 the scaling transformation (1.5) leaves equation (1.1) unperturbed and, in addition, preserves the l2-norm. the above scaling symmetry of (1.1) is closely related to the so-called pseudoconformal symmetry, p[ψ](τ,y) = φ(τ,y) = 1 τ 3 2 e i y2 4τ ψ ( 1 τ , y τ ) . (1.7) this is a symmetry in the sense that, if ψ(t,x) is a solution to (1.1) on (t,x) ∈ [t1, t2] × r3, then φ(τ,y) is a solution to the same equation on (τ,y) ∈ [ 1 t2 , 1 t1 ] × r3. in general, the scaling and the pseudoconformal symmetries (1.5) and (1.7) relate (1.1) to a wide class of equations, referred to cubo 11, 4 (2009) small data global existence and scattering for the ... 17 as the mass-critical (l2-critical or pseudoconformal) nonlinear schrödinger equations. the name comes from the fact that the transforms (1.5) and (1.7) leave both the equation and the mass (the l 2-norm) invariant. mass is one of the basic structures used in physics and is defined by m(ψ(t)) = ∫ r3 |ψ(t,x)|2dx. (1.8) for (1.1), we shall prove (see corollary 2.4 bellow) that the mass is a conserved quantity, i.e. m(ψ(t)) = ‖ψ(t)‖2l2(r3) = ‖ψ0‖2l2(r3) = m(ψ0). (1.9) as in the papers [1,4,11,13,18–20], our results make use of mixed spaces of the type lq([0,t ],lr(r3)) for admissible q and r. thus, we make the following definition. definition 1. we say that the pair (q,r) of exponents is schrödinger–admissible if q and r satisfy 2 q = 3 ( 1 2 − 1 r ) , 2 ≤ q ≤ ∞. (1.10) in the frame of the mass-critical nls, equation (1.1) is similar to the schrödinger equation with local (pure power) nonlinearity, which in an arbitrary spatial dimension n ≥ 1 has the form i∂tψ + △ψ = |ψ| 4 n ψ, (t,x) ∈ r+ × rn. (1.11) the problems of global existence and well-posedness for solutions to (1.11) have been intensively studied, see for example [1–4, 18–21]. the local theory for (1.11) is due to cazenave and weissler, who in [4] constructed local-in-time solutions for arbitrary initial data in l2(rn) and also constructed global solutions for small initial data. their results can be summarized as follows: given ψ0 ∈ l2(rn), there exists a unique local solution ψ to (1.11) with ψ(0,x) = ψ0(x). the solution ψ has a conserved mass m(ψ(t)) = m(ψ0). moreover, if m(ψ0) is sufficiently small depending on n, then ψ is a global solution and ∫ r+ ∫ r3 |ψ(t,x)| 2(n+2) n dxdt ≤ m(ψ0). (1.12) the condition that ψ ∈ l2(n+2)/nt,x is natural for (1.11). this balanced space appears in the original strichartz inequality [17] and it is necessary in order to ensure local existence and uniqueness of solutions to (1.11). following the strategy developed for (1.11), we aim to establish the local well-posedness theory for (1.1) and to construct global solutions for sufficiently small l2-initial data. more precisely we shall use the following definition 2. a function ψ : [0,t∗) × r3 7→ c, 0 < t∗ ≤ ∞ is a l2(r3) solution to (1.1) if ψ ∈ c0([0,t ],l2(r3)) ∩ l14/3([0,t ],l14/5(r3)) for 0 < t < t∗, and we have the duhamel’s integral representation ψ(t) = u(t)ψ0 − i ∫ t 0 u(t − s)(|x|−1 ∗ |ψ(s)| 83 )ψ(s)ds, (1.13) 18 george venkov cubo 11, 4 (2009) for any t ∈ [0,t ]. here u(t) = eit∆ is the free schrödinger evolution group defined via the fourier transform f̂(ξ) = 1 (2π)3/2 ∫ r3 e −ix·ξ f(x)dx, (1.14) by êit∆f(ξ) = e −it|ξ|2 f(ξ). (1.15) we say that ψ is a global solution to (1.1) if t∗ = ∞. the first main result of the present paper is the following theorem 1.1. for every initial data ψ0 ∈ l2(r3) there exists a unique maximal solution ψ ∈ c 0 ([0,t ∗ ),l 2 (r 3 )) ∩ l143 ([0,t∗),l145 (r3)) of (1.1). furthermore: (i) ψ ∈ lq([0,t ],lr(r3)), for 0 < t < t∗ and every admissible pair (q,r); (ii) the mass is conserved, i.e. m(ψ(t)) = m(ψ0) for t ∈ [0,t∗); (iii) there exists a constant ε > 0 sufficiently small, such that if ‖ψ0‖l2(r3) < ε , then t∗ = ∞ and ψ ∈ lq(r+,lr(r3)) for every admissible pair (q,r); (iv) if t∗ < ∞, then ‖ψ‖lq ([0,t∗),lr (r3)) = ∞ for every r > 14/5; (v) ψ depends continuously on the initial data ψ0 ∈ l2(r3) in the space ψ ∈ c0([0,t∗),l2(r3)) ∩ l 14 3 ([0,t ∗ ),l 14 5 (r 3 )). there exists an extensive literature on the scattering theory for the schrödinger equation with convolution nonlinearity [8, 14, 15] and for the hartree equation [5–7, 9, 10], of which the existence of a wave operator is the question of crucial importance. let v(t) = u(t)ψ+ be a solution to the free schrödinger equation i∂tv + △v = 0, (1.16) with initial data ψ+ ∈ x (called the asymptotic state), where x = xψ0 is a suitable banach space, depending on the initial data. that question can be formulated as follows. does there exist a solution of (1.1)–(1.2), which behaves asymptotically as v when t → ∞ in a suitable sense, depending on the choice of the space x? if that is the case, then the map ω+ : x 7→ x is called the wave operator for positive times. in other words, a global strong x-solution ψ to the nonlinear equation (1.1) with an initial data ψ0 scatters in x to a solution v(t) = u(t)ψ+ if we have lim t→∞ ‖ψ(t) − u(t)ψ+‖x = 0, (1.17) or equivalently (by using the unitarity of u(t)) lim t→∞ ‖u(−t)ψ(t) − ψ+‖x = 0. (1.18) cubo 11, 4 (2009) small data global existence and scattering for the ... 19 suppose that for every asymptotic state ψ+ ∈ x, there exists a unique initial data ψ0 ∈ x, whose corresponding x-wellposed solution is global and scatters to v(t) as t → ∞. then, we can define the wave operator ω+ : x 7→ x in the sense of the space x by ω+ψ+ = ψ0. (1.19) the problem of the existence of ψ for given ψ+ is referred to as the problem of existence of the wave operator. when the wave operator ω+ is injective, we say that the cauchy problem (1.1)–(1.2) is asymptotically complete in x. the same problem can be constructed for negative times, but for definiteness, hereinafter, we shall restrict our attention only to positive time. a standard way to construct the wave operator ω+ consists in solving the cauchy problem for (1.1) with initial data ψ+ at t = ∞ in the form of the integral equation ψ(t) = u(t)ψ0 + i ∫ ∞ t u(t − s)(|x|−1 ∗ |ψ(s)| 83 )ψ(s)ds. (1.20) one usually solves (1.20) by a contraction method in a neighborhood of infinity in time (or in the time interval [t,∞) for t sufficiently large) and then continues that solution to all times. thus, the problem is an immediate consequence of the global well-posedness and uses the results of theorem 1.1. with our second main result, we shall construct scattering theory in l2(r3) with small initial data. in fact, we shall prove the following theorem 1.2. let ε > 0 be sufficiently small and consider the ball bε = {ψ ∈ l2(r3); ‖ψ‖l2 < ε}. let ψ ∈ c0([0,t∗),l2(r3)) ∩ l143 ([0,t∗),l145 (r3)) be the unique maximal solution of (1.1), given by part(iii) of theorem 1.1. then we have: (i) for any ψ± ∈ bε, there exists a unique ψ0 ∈ bε, such that lim t→±∞ ‖u(−t)ψ(t) − ψ±‖l2 = 0; (1.21) (ii) for any ψ0 ∈ bε, there exists unique ψ± ∈ bε, such that (1.21) is satisfied; (iii) the wave operators ω± : ψ± 7→ φ0 and the scattering operator s = ω−1+ ◦ ω− are homeomorphisms from bε onto itself and isometric in the l2(r3) norm. the paper is organized as follows. in section 2 we state some useful results and prove theorem 1.1. in section 3 we prove theorem 1.2 and give some generalization notes about the scattering problems for (1.1). we shall conclude this section by giving some of the notations, used in the paper. as usual, l r (r n ) = {ϕ ∈ s′; ‖ϕ‖lr < ∞}, where ‖ϕ‖lr = ( ∫ |ϕ(x)|rdx) 1r if 1 ≤ r < ∞ and ‖ϕ‖l∞ = ess.sup{|ϕ(x)|; x ∈ rn} if r = ∞. we use r′ for denoting the exponent dual to r and defined by 1/r + 1/r ′ = 1. given lebesgue exponents q, r and a function f(t,x) in lq(r,lr(r3)), we write ‖f‖lq (r,lr (r3)) = ( ∫ ‖f(t)‖q lr (r3) dt) 1 q . 20 george venkov cubo 11, 4 (2009) 2 the local existence result we shall start this section by collecting some preliminaries and useful results. as we shall see, the l 14 3 t l 14 5 x norm in space-time plays a fundamental role. this is better understood if we recall some of the estimates available for the corresponding linear problem. we begin by recalling the following properties of the free schrödinger evolution group u(t) = eit△ (see for instance [12, 13, 22]). lemma 2.1. let (q,r) be an admissible pair. then, for every ϕ ∈ l2(r3) the following estimate holds ‖u(t)ϕ‖lq (r,lr (r3)) ≤ c0‖ϕ‖l2(r3). (2.1) moreover, for every admissible pair (θ,ρ) and f ∈ lθ′ ([0,t ],lρ′ (r3)) we have ∥∥∥∥ ∫ · 0 u(· − s)f(s)ds ∥∥∥∥ lq ([0,t],lr (r3)) ≤ c‖f‖lθ′ ([0,t],lρ′ (r3)), (2.2) for 0 < t ≤ ∞. here the constants c0,c > 0 and depend only on the spatial exponents r and ρ. the classical strichartz estimates for the schrödinger equation [17] are one of the main tools in the study of local and global existence, time decay and scattering both for the linear and the nonlinear equation, due to the fact that they fit the assumptions required by the contraction argument (see for instance [5, 13, 17, 18, 22]). the strichartz type estimates for the inhomogeneous schrödinger equation i∂tv(t,x) + △v(t,x) = f(t,x), v(0,x) = f(x), (2.3) in r+ × r3 are given, up to the end-point, namely the pair (q,r) = (2, 6) by the following result due to keel and tao [13]. lemma 2.2. if (q,r) and (q̃, r̃) satisfy (1.10), then the solution to the cauchy problem (2.3) satisfies the estimate ‖v‖lq ([0,t∗),lr (r3)) + ‖v‖l∞([0,t∗),l2(r3)) ≤ c(‖f‖ lq̃ ′ ([0,t∗),lr̃ ′(r3)) + ‖f‖l2(r3)). (2.4) very important tool in our functional analysis background is the following lemma. lemma 2.3. (hardy-littlewood-sobolev inequality) for 0 < α < 3 consider the riesz potential iα(g)(x) = ∫ r3 g(y) |x − y|3−αdy. (2.5) then for any 1 < θ < r < ∞ and g ∈ lr(r3), we have ‖iα(g)‖lθ ≤ c‖g‖lr, (2.6) where 1 θ = 1 r − α 3 . cubo 11, 4 (2009) small data global existence and scattering for the ... 21 for the proof of lemma 2.3, see equation (31), chapter viii.4.2 in stein [16]. the first important result in the present study of (1.1) is the following conservation law. lemma 2.4. let ψ ∈ c0([0,t∗),l2(r3)) be a solution to (1.1) with initial data ψ(0) = ψ0 ∈ l 2 (r 3 ). then m(ψ(t)) = m(ψ0), (2.7) for any 0 ≤ t < t∗. proof. we multiply (1.1) by ψ to get iψ∂tψ + ψ△ψ = (|x|−1 ∗ |ψ| 8 3 )|ψ|2. (2.8) we conjugate (2.8) and subtract the result from the above expression to obtain i∂t|ψ|2 = (ψ△ψ − ψ△ψ) = ∇ · (ψ∇ψ − ψ∇ψ). (2.9) by integration over r3 we obtain the desired result d dt m(ψ(t)) = d dt ‖ψ(t)‖2l2(r3) = 0 (2.10) and the proof of the lemma is completed. we should note here that the specific power of 8/3 in the hartree-type nonlinearities of (1.1) does not allow the establishment of the energy conservation law (due mainly to the presence of the convolution). the latter would cause difficulties in proving, for example, global existence for solution of (1.1) at the h1-level. the arguments of theorem 1.1 rely primarily on the strichartz estimate (2.4) in lemma 2.2 and on the hölder inequality. let us denote by n(ψ) = (|x|−1 ∗ |ψ| 83 )ψ (2.11) the nonlinear term in the hartree equation (1.1). consider the form v (ψ1,ψ2)(t,x) = |x|−1 ∗ |ψ1(t,x)|α|ψ2(t,x)| 8 3 −α , 0 ≤ α ≤ 8 3 . (2.12) then n(ψ) = v (ψ,ψ)ψ and we shall need the following estimates. lemma 2.5. for any 3 < p < ∞ and r1,r2 satisfying 1p = α r1 + 8/3−α r2 − 2 3 , 0 ≤ α ≤ 8 3 we have ‖v (ψ1,ψ2)(t, ·)‖lp(r3) ≤ c‖ψ1(t, ·)‖αlr1 (r3)‖ψ2(t, ·)‖ 8 3 −α lr2 (r3) . (2.13) 22 george venkov cubo 11, 4 (2009) proof. it is sufficient to apply the hardy-littlewood-sobolev inequality lemma 2.3 and hölder inequality to get ‖| · |−1 ∗ |ψ1(t,x)|α|ψ2(t,x)| 8 3 −α‖lp ≤ c‖ψ1(t, ·)‖αlr1 ‖ψ2(t, ·)‖ 8 3 −α lr2 , provided 3 < p < ∞, 1 p = α r1 + 8/3 − α r2 − 2 3 . lemma 2.6. for any r̃ satisfying 2 ≤ r̃ ≤ 6 and for any r1,r2,r3, 2 ≤ rj ≤ 6,j = 1, 2, 3, satisfying 1 r̃′ = α r1 + 8/3 − α r2 + 1 r3 − 2 3 , (2.14) we have ‖v (ψ1,ψ2)(t, ·)ψ3(t, ·)‖lr̃′ (r3) ≤ c‖ψ1(t, ·)‖αlr1 (r3)‖ψ2(t, ·)‖ 8 3 −α lr2 (r3) ‖ψ3(t, ·)‖lr3 (r3). (2.15) proof. the hölder inequality implies ‖v (ψ1,ψ2)(t, ·)ψ3(t, ·)‖lr̃′ ≤ c‖v (ψ1,ψ2)(t, ·)‖lp ‖ψ3(t, ·)‖lr3 , where 1 r̃′ = 1 p + 1 r3 . applying further the estimate of lemma 2.5, we complete the proof of the lemma. the corresponding space time estimate follows easily from the above estimate. lemma 2.7. let 0 < t ≤ ∞ and consider the schrödinger-admissible pair (q,r) = (14/3, 14/5). then, we have the estimate ‖n(ψ)‖ l 14 11 ([0,t],l 14 9 (r3) ≤ c‖ψ‖ 11 3 l 14 3 ([0,t],l 14 5 (r3) . (2.16) proof. applying hölder inequality in time to (2.16) in lemma 2.6 we obtain ‖v (ψ1,ψ2)(t, ·)ψ3(t, ·)‖lq̃′ ([0,t],lr̃′ ) ≤ c‖ψ1(t, ·)‖αlq1 ([0,t],lr1 )‖ψ2(t, ·)‖ 8 3 −α lq2 ([0,t],lr2 ) ‖ψ3(t, ·)‖lq3 ([0,t],lr3 ), where 0 ≤ α ≤ 8/3 and 1 r̃′ = α r1 + 8/3 − α r2 + 1 r3 − 2 3 , (2.17) 1 q̃′ = α q1 + 8/3 − α q2 + 1 q3 . (2.18) cubo 11, 4 (2009) small data global existence and scattering for the ... 23 now we shall choose the couples (qj,rj) so that 1 qj = 3 2 ( 1 2 − 1 rj ) , j = 1, 2, 3 (2.19) and the relations (2.17), (2.18) are satisfied. indeed, the simplest choice is r = r̃ = r1 = r2 = r3, q = q̃ = q1 = q2 = q3. then (2.17) reads as 1 r′ = 11 3r − 2 3 , (2.20) while (2.18) becomes 1 q′ = 11 3q (2.21) which together give the couple (q,r) = (14/3, 14/5) and the proof is completed. lemma 2.8. let 0 < t ≤ ∞ and let (q,r) be a schrödinger-admissible pair. then there exists a constant c > 0, independent of t such that ∥∥∥∥ ∫ · 0 u(· − s)[n(ψ)(s) − n(χ)(s)]ds ∥∥∥∥ lq ([0,t],lr ) (2.22) ≤ c ( ‖ψ‖ 8 3 l 14 3 ([0,t],l 14 5 ) + ‖χ‖ 8 3 l 14 3 ([0,t],l 14 5 ) ) ‖ψ − χ‖ l 14 3 ([0,t],l 14 5 ) , for every ψ,χ ∈ l143 ([0,t ],l145 (r3)). proof. to prove the lemma, we shall use the estimates in lemma 2.1. then the estimate (2.22) follows directly from (2.15), (2.16), (2.2) and hölder inequality. proof of theorem 1.1 . the proof of (ii) follows from lemma 2.4. we shall prove the existence of solution to (1.1) by a fix point argument. let ψ0 ∈ l2(r3) with ‖ψ0‖l2(r3) < ε, where ε > 0 is sufficiently small. let r > 0 and consider the ball b2r(t) = {ψ ∈ c0([0,t ],l2)) ∩ l 14 3 ([0,t ],l 14 5 ); ‖ψ‖ l 14 3 ([0,t],l 14 5 ) ≤ 2r}, endowed with the metric d(ψ,χ) = ‖ψ − χ‖ l 14 3 ([0,t],l 14 5 ) . since the space l 14 3 ([0,t ],l 14 5 ) is reflexive, the ball b2r is weakly compact, implying b2r is a complete metric space. consider the map φ[ψ](t), defined by the right-hand side of the duhamel’s integral representation (1.13). then, for ψ ∈ b2r, using (2.1), (2.2) and (2.16), we can write ‖ψ‖ l 14 3 ([0,t],l 14 5 ) ≤ ‖u(·)ψ0‖ l 14 3 ([0,t],l 14 5 ) + c1‖ψ‖ 11 3 l 14 3 ([0,t],l 14 5 ) ≤ c0ε + c1‖ψ‖ 11 3 l 14 3 ([0,t],l 14 5 ) . (2.23) 24 george venkov cubo 11, 4 (2009) now, write r = c0ε and choose ε > 0 so small that there exists a positive number y, satisfying c1y 11 3 − y + r > 0, 0 < y ≤ 2r. for that purpose, it is sufficient to take c1(2r) 8 3 < 1 2 , or equivalently ε < 1 2c0(2c1) 3 8 , implying that φ[ψ] ∈ c0([0,t ],l2)) ∩ l143 ([0,t ],l145 ). on the other hand, from (2.22) it follows that ‖n(ψ) − n(χ)‖ l 14 11 ([0,t],l 14 9 ≤ c2 ( ‖ψ‖ 8 3 l 14 3 ([0,t],l 14 5 ) + ‖χ‖ 8 3 l 14 3 ([0,t],l 14 5 ) ) ‖ψ − χ‖ l 14 3 ([0,t],l 14 5 ) ≤ c22(2r) 8 3 ‖ψ − χ‖ l 14 3 ([0,t],l 14 5 ) , (2.24) for every ψ,χ ∈ b2r. if we choose r such that c22(2r) 8 3 ≤ 1 2 , or equivalently ε ≤ 1 2c0(4c2) 3 8 , we finally obtain that the map φ[ψ](t) is a strict contraction on the ball b2r. thus φ[ψ] has a fixed point ψ, which is the unique solution of (1.1) in b2r. so far, we have proved the statement of theorem 1.1, as well as the part (i). notice, that the strichartz estimate (2.4) implies a similar inequality, namely ‖ψ‖ l 14 3 ([0,t],l 14 5 ) ≤ c0‖ψ0‖l2 + c1‖ψ‖ 11 3 l 14 3 ([0,t],l 14 5 ) (2.25) where we can take both the constants in (2.23) and (2.25) to be the same. the second comment on the above proof is that from (2.23) and (2.25), the size of the ball b2r depends directly on the size of the norm ‖u(t)ψ0‖ l 14 3 ([0,t],l 14 5 (r3)) and it can be done small by taking either ‖ψ0‖l2(r3) small or the interval [0,t ] small. let us denote by t∗ the supremum of all t > 0 for which there exists a solution of (1.1) in c0([0,t ],l2(r3)) ∩ l143 ([0,t ],l145 (r3)). to prove (iii), observe that if ψ0 is sufficiently small, then (2.23) holds regardless of the value of t . thus we may accomplish the fixed point procedure in the ball b2r(∞), providing t∗ = ∞. further, we claim that if t∗ < ∞, then ‖ψ‖lq ([0,t∗),lr (r3)) = ∞ for every r > 14/5. indeed, on the contrary, let us assume that t∗ < ∞ and ‖ψ‖ l 14 3 ([0,t∗),l 14 5 ) < ∞. for any t ∈ [0,t∗) let cubo 11, 4 (2009) small data global existence and scattering for the ... 25 τ ∈ [0,t∗ − t). using duhamel’s formula (1.13), we can write ψ(t + τ) = u(t + τ)ψ0 − i ∫ t+τ 0 u(t + τ − s)n(ψ)(s)ds = u(τ)u(t)ψ0 − i ∫ t+τ 0 u(t + τ − s)n(ψ)(s)ds = u(τ)ψ(t) + iu(τ) ∫ t 0 u(t − s)n(ψ)(s)ds −i ∫ t+τ 0 u(t + τ − s)n(ψ)(s)ds = u(τ)ψ(t) − i ∫ t+τ t u(t + τ − s)n(ψ)(s)ds. (2.26) from (2.26) and the estimate (2.22) in lemma 2.8 we obtain ‖u(·)ψ(t)‖ l 14 3 ([0,t∗−t),l 14 5 ) ≤ c ( ‖ψ(t)‖ l 14 3 ([t,t∗),l 14 5 ) + ‖ψ‖ 11 3 l 14 3 ([t,t∗),l 14 5 ) ) . (2.27) observing now that ‖u(·)ψ‖ l 14 3 ([0,t],l 14 5 ) → 0 as t → 0 and taking t close enough to t∗, it follows that ‖u(·)ψ(t)‖ l 14 3 ([0,t∗−t),l 14 5 ) can be made small enough and the assumptions in (iii) are fulfilled. therefore, ψ can be extended after t∗, which contradicts the maximality. let (q,r) be a schrödinger-admissible pair with r ≥ 14 5 . then, from hölder inequality for t < t∗, we can write ‖ψ‖ l 14 3 ([0,t],l 14 5 ) ≤ ‖ψ‖1−α l∞([0,t],l2) ‖ψ‖αlq ([0,t],lr ), α ∈ (0, 1). (2.28) letting t → t∗, we obtain that ‖ψ‖lq ([0,t],lr ) = ∞, which proves the statement (iv). to prove (v), consider a sequence ψk 0 ∈ l2(r3), such that ψk 0 → ψ0 ∈ l2(r3) as k → ∞. thus, for k large enough, ‖ψk 0 ‖l2(r3) < ε. we can use the duhamel’s formula (1.13) to construct a sequence of solutions ψk ∈ l143 ([0,t ],l145 (r3)) to (1.1) with initial datum ψk 0 . applying the proof of (iii), we obtain that ψk → ψ in c0([0,t ],l2(r3)) ∩ l14/3([0,t ],l14/5(r3)) as k → ∞, and in fact in every lq([0,t ],lr(r3))) for (q,r) be an admissible pair. thus, the proof of the theorem is completed. 3 small data scattering theory in l2(r3) in this section we shall prove theorem 1.2. the arguments for proving the existence of the wave operator are standard and follows the exposition in [12,15]. we shall prove only the (+) case since the (−) case can be proved similarly. let ψ0 ∈ l2(r3) with ‖ψ0‖l2 < ε and ψ ∈ b2r, where the ball b2r is defined in the previous section. then, for t > t0, using duhamel’s integral formula (1.13) we have u(−t)ψ(t) = u(−t0)ψ(t0) − i ∫ t t0 u(−s)n(ψ)(s)ds. (3.1) 26 george venkov cubo 11, 4 (2009) therefore, using the estimate (2.16), we have ‖u(−t)ψ(t) − u(−t0)ψ(t0)‖l2 = ∥∥∥∥ ∫ t t0 u(−s)n(ψ)(s)ds ∥∥∥∥ l2 ≤ c ‖ψ‖ 11 3 l 14 3 ([t0,t],l 14 5 ) → 0, (3.2) as t0 → ∞. since u(−t0)ψ(t0) ∈ l2, then the proof of part (ii) is completed. to prove (i), assume that ψ+ ∈ l2(r3), ψ ∈ b2r and consider the map φ+[ψ](t) = u(t)ψ+ + i ∫ ∞ t u(t − s)n(ψ)(s)ds, t > t, (3.3) for some t = t(ψ+) large enough. then, using the same arguments as in the proof of part (iii) of theorem 1.1, we find that φ+ is a contraction on b2r and has a unique fixed point if ‖ψ+‖l2 < ε. using the global well-posedness result established in theorem 1.1 for small data, one can then extend this solution uniquely for any t ∈ [0,∞), and in particular ψ will take some value ψ0 = ψ(0) ∈ l2 at time t = 0. this gives existence of the wave operator ω+, defined by ψ0 = ω+φ+ = ψ+ + i ∫ ∞ 0 u(−s)n(ψ)(s)ds. (3.4) this proves part (i) of the theorem 1.2. to prove (iii), we shall use the following observations. since, the wave operators ω± are isometric in the space bε, it is clear that the scattering operator s : φ− 7→ φ+ is well defined as a map from bε onto itself and isometric in the l 2 norm, i.e. ‖sψ‖l2 = ‖ψ‖l2 . this completes the proof of the theorem 1.2. it is clear that the finiteness of the l 14/3 t l 14/5 x -norm is sufficient to yield global well-posedness and scattering results for small data in l2. as the l 2(n+2)/n t,x -norm for the schrödinger equation with pure power nonlinearity (1.11), this is the best possible choice of admissible indices for (1.1), which ensures the space-time integrability via the strichartz estimates and the contraction property of the nonlinear map. moreover, the above results does not depend on the repulsive character of the problem and can be proved in a similar way for the focussing analogue of (1.1). finally, we note that the global existence, well-posedness and scattering problems for (1.1) with an arbitrary initial data require spaces, strictly smaller than the l2 one. for example, in another work, following the approach of hayashi, naumkin and ozawa [9], we shall consider the above problems for data in the pseudoconformal space σs = {ψ0 ∈ hs; |x|sψ0 ∈ l2} for some 0 < s < 1, depending on the behavior of the leading term of the solution ψ for large time. received: march 2008. revised: july 2008. cubo 11, 4 (2009) small data global existence and scattering for the ... 27 references [1] bégout, p. and vargas, a., mass concentration phenomena for the l2-critical nonlinear schrödinger equation, trans. amer. math. soc. 359, (2007), 5257–5282. 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[22] yajima, k., existence of solutions for schrödinger evolution equations, comm. math. phys. 110, (1987), 415–426. articulo 2 cubo a mathematical journal vol.10, n o ¯ 04, (119–135). december 2008 fixed point results for set-valued and single-valued mappings in ordered spaces seppo heikkilä department of mathematical sciences, university of oulu, box 3000, fin-90014 university of oulu, finland email: sheikki@cc.oulu.fi abstract in this article we use a recursion principle and generalized iteration methods to prove existence and comparison results for fixed points of setand single-valued mappings in ordered spaces. resumen en este art́ıculo usamos el principio de recurrencia y métodos de iteración generalizados para provar resultados de exitencia y comparación para puntos fijos de aplicaciones conjunto (uni-)valoradas en espacios ordenados. key words and phrases: poset, set-valued mapping, fixed point, solution, maximal, minimal, sup-center, inf-center, recursion principle, generalized iteration methods, order compact, chain complete. math. subj. class.: 47h04, 47h07, 47h10. 120 seppo heikkilä cubo 10, 4 (2008) 1 introduction let p be a nonempty partially ordered set (poset). as an introductory result we show that a set-valued mapping f from p to the set 2p \∅ of nonempty subsets of p has minimal and maximal fixed points, that is, the set fix f = {x ∈ p | x ∈ f(x)} has minimal and maximal elements, if the following conditions hold. (c1) sup{c, y} ∈ p for some c ∈ p and for every y ∈ p . (c2) if x ≤ y in p , then for every z ∈ f(x) there exists a w ∈ f(y) such that z ≤ w, and for every w ∈ f(y) there exists a z ∈ f(x) such that z ≤ w. (c3) strictly monotone sequences of f[p ] = ⋃ {f(x) : x ∈ p } are finite. as for the proof, denote x0 = c, and choose y0 from f(x0). if y0 6≤ x0, then x0 < x1 := sup{c, y0}. apply then condition (c2) to choose y1 from f(x1) such that y0 ≤ y1. if y0 = y1, then stop. otherwise, y0 < y1, whence x1 = sup{c, y0} ≤ x2 := sup{c, y1}, and apply again condition (c2) to choose y2 from f(x2) such that y1 ≤ y2. continuing in a similar way, condition (c3) ensures that after a finite number of choices we get the situation, where yn−1 = yn ∈ f(xn). in view of the above construction we then have xn := sup{c, yn−1} = sup{c, yn}. denoting z0 := xn and w0 := yn then w0 ∈ f(z0) and w0 ≤ sup{c, w0} = z0. if w0 = z0, then z0 is a fixed point of f. otherwise, denoting z1 := w0, we have z1 < z0. in view of condition (c2) there exists a w1 ∈ f(z1) such that w1 ≤ w0. if equality holds, then z1 = w0 = w1 ∈ f(z1), so that z1 is a fixed point of f. otherwise, w1 < w0, denote z2 := w1, and choose by (c2) such a w2 ∈ f(z2) that w2 ≤ w1, and so on. condition (c3) implies that a finite number of steps yields the situation zm := wm−1 = wm ∈ f(zm). thus zm belongs to fix f. being a subset of f[p ], strictly monotone sequences of fix f are finite by condition (c3). this property implies in turn that fix f has minimal and maximal elements. the above described result will be generalized in section 3. for instance, we show that f has minimal and maximal fixed points when the above condition (c1) holds, condition (c2) is replaced by a stronger monotonicity condition, and (c3) is replaced by a chain completeness of the order closure of the range f[p ]. applications to single-valued mappings are also given. fixed points of a concrete mapping are approximated by using an algorithmic method developed from the above described reasoning. the obtained results are used in section 4 to derive fixed point results in ordered normed spaces and in ordered topological spaces. existence proofs require several consecutive applications of a recursion principle and generalized iteration methods introduced in [4, 6] and presented in section 2. cubo 10, 4 (2008) fixed point results for set-valued ... 121 2 recursions and iterations in posets given a nonempty set p , a relation x < y in p × p is called a partial ordering, if x < y implies y 6< x, and if x < y and y < z imply x < z. defining x ≤ y if and only if x < y or x = y, we say that p = (p, ≤) is a partially ordered set (poset). an element b of a poset p is called an upper bound of a subset a of p if x ≤ b for each x ∈ a. if b ∈ a, we say that b is the greatest element of a, and denote b = max a. a lower bound of a and the least element, min a, of a are defined similarly, replacing x ≤ b above by b ≤ x. if the set of all upper bounds of a has the least element, we call it a supremum of a and denote it by sup a. we say that y is a maximal element of a if y ∈ a, and if z ∈ a and y ≤ z imply that y = z. an infimum of a, inf a, and a minimal element of a are defined similarly. we say that a poset p is a lattice if inf{x, y} and sup{x, y} exist for all x, y ∈ p . w is called a chain if x ≤ y or y ≤ x for all x, y ∈ w . we say that w is well-ordered if nonempty subsets of w have least elements, and inversely well-ordered if nonempty subsets of w have greatest elements. in both cases w is a chain. a basis to our considerations is the following recursion principle (cf. [6], lemma 1.1.1). lemma 2.1. given a nonempty poset p , a subset d of 2p = {a : a ⊆ p } with ∅ ∈ d and a mapping f : d → p , there is a unique well-ordered chain c in p such that x ∈ c if and only if x = f (c xn else xn+1 = inf{c, gxn}. let g : p → p satisfy the hypotheses of theorem 3.2. the result corollary 3.1 can be applied to approximate the fixed points x∗ and x∗ of g introduced in theorem 3.2 in the following manner. assume that g, g : p → p satisfy the hypotheses given for g in corollary 3.1, and that g(x) ≤ g(x) ≤ g(x) for all x ∈ p. (3.7) since x∗ is increasing with respect to g, it follows from (3.7) that x∗ ≤ x∗ ≤ x∗, where x∗ and x∗ are obtained by algorithm (i) of corollary 3.1 with g replaced by g and g, respectively. since partial ordering is the only structure needed in the proofs, the above results can be applied to problems where only ordinal scales are available. on the other hand, these results have some practical value also in real analysis. we shall demonstrate this by an example where the above described method is applied to a system of the form xi = gi(x1, . . . , xm), i = 1, . . . , m, (3.8) where the functions gi are real valued functions of m real variables. example 3.3. in this example we approximate a solution x∗ = (x1, y1) of the system x = g1(x, y) := n1(x, y) 2 − |n1(x, y)| , y = g2(x, y) := n2(x, y) 3 − |n2(x, y)| , (3.9) where n1(x, y) = 11 12 x + 12 13 y + 1 234 and n2(x, y) = 15 16 x + 14 15 y − 7 345 , (3.10) by calculating such upper and lower estimates of (x1, y1) whose corresponding coordinates differ less than 10 −100 . the mapping g = (g1, g2), defined by (3.9), (3.10) maps the set p = {(x, y) ∈ r 2 : |x|+|y| ≤ 1 2 } into p , and is increasing on p . it follows from example 2.1 that c = (0, 0) is an order center of p , and that p is chain complete. thus the results of theorem 3.2 are valid. 130 seppo heikkilä cubo 10, 4 (2008) upper and lower estimates to the fixed point x∗ = (x1, y1) of g, and hence to a solution (x1, y1) of system (3.9), (3.10), can be obtained by applying the algorithm (i) given in corollary 3.1 to operators g and g, defined by { g(x, y) = (10−101ceil(10101g1(x, y)), 10 −101 ceil(10 101g2(x, y)), g(x, y) = (10−101floor(10101g1(x, y)), 10 −101 floor(10 101g2(x, y)), (3.11) where ceil(x) is the least integer ≥ x and floor(x) is the greatest integer ≤ x. the so defined operators g, g are increasing and map the set p = {(x, y) ∈ r2 : |x| + |y| ≤ 1 2 } into finite subsets of p , and (3.7) holds. we are going to show that the required upper and lower estimates are obtained by algorithm (i) of corollary 3.1 with g replaced by g and g, respectively. the following maple program is used in calculations of the upper estimate x∗ = (x1, y1). (n1, n2):=(11/12*x+12/13*y+1/234,15/16*x+14/15*y-7/345): (z,w):=(n1/(2-abs(n1)),n2/(3-abs(n2))):(g1,g2):=(ceil(10 101 z)/10 101 ,ceil(10 101 w)/10 101 ): (x0,y0):=(0,0);x:=x0:y:=y0:u:=g1:v:=g2:b[0]:=[x,y]: for k from 1 while abs(u-x)+abs(v-y)> 0 do: if u<=x and v<=y then (x,y):=(u,v) else (x,y):=(max(x,u),max(y,v)):fi: u:=g1:v:=g2:b[k]:=[x,y]:od:n:=k-1: x1:=x;y1=y; the above program yields the following results (n=1246).            x1 = −0.00775318684978081165491069304103701961947143138774717254950456999535 626408273278584836718225237250043, y1 : −0.01359961542461090148983671991312928002452425440128992737588059916178 38548683927620135569441397855721 in particular, (x1, y1) is the fixed point x∗ of g. replacing ’ceil’ by ’floor’ in the above program we obtain components of the fixed point x∗ = (x2, y2) of g (n:=1248).            x2 = −0.00775318684978081165491069304103701961947143138774717254950456999535 62640827327858483671822523725005, y2 = −0.01359961542461090148983671991312928002452425440128992737588059916178 385486839276201355694413978557215 the above calculated components of x∗ and x∗ are exact, and their differences are < 10−100. according to the above reasoning the exact fixed point x∗ of g belongs order interval [x∗, x∗]. in particular, both (x1, y1) and (x2, y2) approximate an exact solution (x1, y1) of system (3.9), (3.10) with the required precision. moreover, x1 ≤ x1 ≤ y1 and x2 ≤ y1 ≤ y2. remarks 3.1. the results of lemma 3.1 and its dual and corollary 3.1 could be combined to obtain upper and lower estimates also to fixed points of set-valued functions. cubo 10, 4 (2008) fixed point results for set-valued ... 131 4 special cases in this section we shall first present existence and comparison results for equations and inclusions in ordered normed spaces. next we formulate in ordered topological spaces some existence and comparison results derived in section 3. 4.1 equations and inclusions in ordered normed spaces definition 4.1. a closed subset e+ of a normed space e is called an order cone if e+ + e+ ⊆ e+, e+ ∩ (−e+) = {0} and ce+ ⊆ e+ for each c ≥ 0. the space e, equipped with an order relation ’≤’, defined by x ≤ y if and only if y − x ∈ e+ is called an ordered normed space. it is easy to see that the above defined order relation ≤ is a partial ordering in e. lemma 4.1. let c be a chain in an ordered normed space e, and assume that each monotone sequence of c has a weak limit in e. then c contains an increasing sequence which converges weakly to sup c and a decreasing sequence which converges weakly to inf c. this result holds also when weak convergence is replaced by strong convergence. proof. c has by [6], lemma 1.1.2 a well-ordered cofinal subchain w . since all increasing sequences of w have weak limits, there is by [2], lemma a.3.1 an increasing sequence (xn) in w which converges weakly to x = sup w = sup c. noticing that −c is a chain whose increasing sequences have weak limits, there exists an increasing sequence (xn) of −c which converges weakly to sup(−c) = − inf c. denoting yn = −xn, we obtain a decreasing sequence (yn) of c which converges weakly to inf c. in the case of strong convergence the conclusion follows from [6], proposition 1.1.5. in what follows, e is an ordered normed space having some of the following properties. (e0) bounded and monotone sequences of e have weak limits. (e1) x+ = sup{0, x} exists, and ‖x+‖ ≤ ‖x‖ for every x ∈ e. when c ∈ e and r ∈ [0, ∞), denote br(c) := {x ∈ e : ‖x − c‖ ≤ r}. recall (cf. e.g., [11]) that if a sequence (xn) of a normed space e converges weakly to x, then (xn) is bounded, i.e. supn ‖xn‖ < ∞, and ‖x‖ ≤ lim inf n→∞ ‖xn‖. (4.1) the next auxiliary result is needed in the sequel. 132 seppo heikkilä cubo 10, 4 (2008) lemma 4.2. let e be an ordered normed space with properties (e0) and (e1). if c ∈ e and r ∈ (0, ∞), then c is an order center of br(c), and for every chain c of br(c) both sup c and inf c exist and belong to br(c). proof. since sup{c, x} = (x − c)+ − c and inf{c, x} = c − (c − x)+, for all x ∈ e, (4.2) then (e1) and (4.2) imply that ‖ sup{c, x} − c‖ = ‖ inf{c, x} − c‖ = ‖(x − c)+‖ ≤ ‖x − c‖ ≤ r for every x ∈ br(c). thus both sup{c, x} and inf{c, x} belong to br(c). let c be a chain in br(c). since c is bounded there is by (e0) and lemma 4.1 an increasing sequence (xn) in c which converges weakly to x = sup c. since ‖xn − c‖ ≤ r for each n, it follows from (4.1) that ‖x − c‖ ≤ lim inf n→∞ ‖xn − c‖ ≤ r. thus x = sup c exists and belongs to br(c). similarly one can show that inf g[c] exists in e and belongs to br(c). applying theorem 3.2 and lemmas 4.1 and 4.2 we obtain the following fixed point results. theorem 4.1. given a subset p of e, assume that g : p → p is increasing, and that g[p ] ⊆ br(c) ⊆ p for some c ∈ e and r ∈ (0, ∞). then g has (a) minimal and maximal fixed points; (b) least and greatest fixed points x∗ and x ∗ in the order interval [x, x] of p , where x is the greatest solution of x = inf{c, g(x)}, and x is the least solution of x = sup{c, g(x)}. moreover, x∗, x∗, x and x are all increasing with respect to g. proof. let c be a chain in p . since g[c] is a chain in br(c), then both sup g[c] and inf g[c] exist in e and belongs to br(c) ⊆ p by lemma 4.2. because c is an order center of br(c) and ocl(g[p ]) ⊆ g[p ] ⊆ br(c) ⊆ p , then c is an order center of ocl(g[p ]) in p . the above proof shows that the hypotheses of theorem 3.2 are valid. in the set-valued case we have the following consequence of theorem 3.1. theorem 4.2. assume that p is a subset of e which contains br(c) for some c ∈ e and r ∈ (0, ∞). let f : p → 2p \ ∅ be an increasing mapping whose values are weakly compact in e, and whose range f[p ] is contained in br(c). then f has minimal and maximal fixed points. remarks 4.1. each of the following spaces has properties (e0) and (e1) (as for the proofs, see, e.g. [1, 2, 3, 5, 6, 7, 8, 10]): cubo 10, 4 (2008) fixed point results for set-valued ... 133 (a) a sobolev space w 1,p(ω) or w 1,p 0 (ω), 1 < p < ∞, ordered a.e. pointwise, where ω is a bounded domain in r n . (b) a finite-dimensional normed space ordered by a cone generated by a basis. (c) lp, 1 ≤ p ≤ ∞, normed by p-norm and ordered coordinatewise. (d) lp(ω), 1 ≤ p ≤ ∞, normed by p-norm and ordered a.e. pointwise, where ω is a σ-finite measure space. (e) a separable hilbert space whose order cone is generated by an orthonormal basis. (f) a weakly complete banach lattice or a umb-lattice (cf.[1]). (g) lp(ω, y ), 1 ≤ p ≤ ∞, normed by p-norm and ordered a.e. pointwise, where ω is a σ-finite measure space and y is any of the spaces (b)–(f). (h) newtonian spaces n 1,p(y ), 1 < p < ∞, ordered a.e. pointwise, where y is a metric measure space. thus the results of theorems 4.1–4.2 hold if e is any of the spaces listed in (a)–(h). 4.2 fixed point results in ordered topological spaces let p = (p, ≤) be an ordered topological space, i.e., for each a ∈ p the order intervals [a) = {x ∈ p : a ≤ x} and (a] = {x ∈ p : x ≤ a} are closed in the topology of p . in what follows, we assume that p has the following property: (c) each well-ordered chain c of p whose increasing sequences converge in p contains an increasing sequence which converges to sup c, and each inversely well-ordered chain c of p whose decreasing sequences converge in p contains a decreasing sequence which converges to inf c. corollary 4.1. the following ordered topological spaces have property (c). (a) ordered metric spaces. (b) order closed subsets of ordered normed spaces equipped with a norm topology. (c) order closed subsets of ordered normed spaces equipped with a weak topology. (d) ordered topological spaces which satisfy the second countability axiom. proof. (a) and (b) follow from the result of [6], proposition 1.1.5 and from its dual. (c) is a consequence of [2], appendix, lemma a.3.1 and its dual. (d) if p is an ordered topological spaces which satisfies the second countability axiom, then each chain of p is separable, whence p has property (c) by the result of [6], lemma 1.1.7 and its dual. the following result is a consequence of proposition 3.6. proposition 4.1. given an ordered topological space p with property (c), assume that g : p → p is an increasing function. 134 seppo heikkilä cubo 10, 4 (2008) (a) if g[p ] has an upper bound in p , and if g maps decreasing sequences of p to convergent sequences, then g has greatest and minimal fixed points. (b) if g[p ] has a lower bound in p , and if g maps increasing sequences of p to convergent sequences, then g has least and maximal fixed points. proof. (a) let d be an inversely well-ordered chain in p . since g is increasing, then g[d] is inversely well-ordered. every decreasing sequence of g[d] is of the form (g(xn)), where (xn) is a decreasing sequence in d. thus the hypotheses of (a) and property (c) imply that x∗ = inf g[d] exists and belongs to p . it then follows from proposition 3.6(b) that g has the greatest fixed point and a minimal fixed point. the conclusions of (b) is a similar consequence of proposition 3.6(a). the next fixed point result is a consequence of proposition 4.1 and lemma 4.1. corollary 4.2. let p be an order closed subset of an ordered normed space e whose (order) bounded and monotone sequences have weak or strong limits, and let g : p → p be increasing. (a) if g[p ] has an upper bound in p , and if g maps decreasing sequences of p to (order) bounded sequences, then g has the greatest and a minimal fixed point. (b) if g[p ] has a lower bound in p , and g maps increasing sequences of p to (order) bounded sequences, then g has the least and a maximal fixed point. the next result is a consequence of theorem 3.2. theorem 4.3. given an ordered topological space p with property (c), assume that g : p → p is increasing and maps monotone sequences of p to convergent sequences. (a) if c is a sup-center of ocl(g[p ]) in p , then g has minimal and maximal fixed points. moreover, g has the greatest fixed point x∗ in (x], where x is the least solution of the equation x = sup{c, g(x)}. both x and x∗ are increasing with respect to g. (b) if c is an inf-center of ocl(g[p ]) in p , then g has minimal and maximal fixed points. moreover, g has the least fixed point x∗ in [x), where x is the greatest solution of the equation x = inf{c, g(x)}. both x and x∗ are increasing with respect to g. as a consequence of propositions 3.1 and 3.2 and theorem 3.1 we get the following result. proposition 4.2. let p be an ordered topological space with property (c), and let the values of f : p → 2p \ ∅ be compact. (a) assume that following hypothesis holds. (f+) if (xn) and (yn) are increasing and yn ∈ f(xn) for every n, then (yn) converges. if the set s+ = {x ∈ p : [x) ∩ f(x) 6= ∅} is nonempty, then f has a maximal fixed point. (b) assume that f satisfies the following hypothesis. cubo 10, 4 (2008) fixed point results for set-valued ... 135 (f−) if (xn) and (yn) are decreasing and yn ∈ f(xn) for every n, then (yn) converges. if the set s− = {x ∈ p : (x] ∩ f(x) 6= ∅} is nonempty, then f has a minimal fixed point. (b) assume that the hypotheses (f±) hold. if ocl(f[p ]) has a sup-center or an inf-center in p , then f has minimal and maximal fixed points. received: april 2008. revised: april 2008. references [1] g. birkhoff, lattice theory, amer. math. soc. publ. xxv, rhode island, 1940. [2] s. carl and s. heikkilä, nonlinear differential equations in ordered spaces, chapman & hall/crc, boca raton, 2000. [3] s. carl and s. heikkilä, elliptic problems with lack of compactness via a new fixed point theorem, j. differential equations 186 (2002), 122–140. [4] s. heikkilä, a method to solve discontinuous boundary value problems, nonlinear anal., 47: 2387–2394, 2001. [5] s. heikkilä, operator equations in ordered function spaces, in r.p. agarwal and d. o’regan (eds.), nonlinear analysis and applications: to v. lakshmikantham on his 80:th birthday, kluwer acad. publ., dordrecht, isbn 1-4020-1688-3 (2003), 595–616. [6] s. heikkilä and v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker, inc., new york, 1994. [7] juha kinnunen and olli martio, nonlinear potential theory on metric spaces, illinois j. math. 46, 3, 857–883, 2002. [8] j. lindenstraus and l. tzafriri, classical banach spaces ii, function spaces, springerverlag, berlin, 1979. [9] h.l. royden, real analysis, the macmillan company, london, 1968. [10] n. shanmugalingam, newtonian spaces: an extension of sobolev spaces to metric measure spaces, revista matemática iberoamericana, 16, 2, (2000), 243–279. [11] k. yoshida, functional analysis, springer-verlag, berlin, 1974. n10-fixor cubo a mathematical journal vol.10, n o ¯ 02, (75–82). july 2008 spectral rank for c∗-algebras takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, nishihara, okinawa 903-0213, japan email: sudo@math.u-ryukyu.ac.jp abstract we introduce a notion of dimension for c∗-algebras that we call spectral rank, based on spectrums of generators of c∗-algebras. we study some basic properties for this new rank and establish its fundamental theory. resumen introducimos la noción de dimensión para c∗-algebras que llamamos rango espectral; esta noción es basada en el espectro de los generadores de c∗-algebras. estudiamos algunas propriedades básicas para este nuevo rango y establecemos su teoria fundamental. key words and phrases: c*-algebra, rank, spectrum. math. subj. class.: 46l05 76 takahiro sudo cubo 10, 2 (2008) introduction there have been several attempts to introduce suitable ranks for c∗-algebras; the stable rank (and connected stable rank) of rieffel [4], the real rank of brown and pedersen [1], the completely positive (or decomposition) rank (or covering dimension) of winter [7], and the topological rank and (another) covering dimension of the author [5], [6], etc. for reference, see [2] or [3]. in this paper we introduce a yet another notion of dimension for c∗-algebras that we call spectral rank. this rank is based on spectrums of generators of c∗-algebras. we study some basic properties for this new rank that might become an interesting new invariant for c∗-algebras. in section 1, some basic properties of the spectral rank for c∗-algebras concerning their fundamental algebraic structures are discussed. in section 2, introduced is an approximate version of the spectral rank that we call approximate spectral rank. 1 spectral rank let a be a c∗-algebra. the spectrum of an element a of a is defined by sp(a) = {λ ∈ c | a − λ1 is not invertible in a + }, where a + = a when a is unital, and a + is the unitization of a by c of complex numbers when a is non-unital. note that the spectrum sp(a) is a non-empty closed subset of c and bounded by the norm ‖a‖. definition 1.1. let a be a c∗-algebra with (specific and initial) generators aj . define the spectral rank of a to be spr(a) = inf    ∑ j dim sp(aj ) | aj are generators of a    where dim(·) is the (covering) dimension for spaces. remark. this notion should not depend on generators but do depend in a sense, and certainly does not depend on their certain equivalences like adjoint unitary operations since sp(aj ) = sp(ad(u)aj ) where ad(u)aj = uaju ∗ for some unitary u, and some cancellative terms. in another view, we just look at generators of the algebraic part of a and ignore some unknown ones in the c∗-closure of the part in a, and always consider such a situation in what follows. proposition 1.2. let a be a c∗-algebra with unitary generators uj for j ∈ j a set. then spr(a) ≤ ∑ j dim sp(uj ) ≤ ∑ j 1 = |j|, where the second inequality is equality if a is universal. cubo 10, 2 (2008) spectral rank for c∗-algebras 77 proof. note that sp(uj ) is a closed subset of the torus t with dim t = 1. thus, dim sp(uj) ≤ 1. if a is universal, we have sp(uj ) = t. 2 proposition 1.3. let a be a c∗-algebra and b its quotient c∗-algebra, where generators of b are mapped from those of a by the quotient map. then spr(a) ≥ spr(b). proof. let π be the quotient map from a to b. let aj be generators of a. then π(aj ) are generators of b and note that sp(π(aj )) ⊂ sp(aj ). 2 example 1.4. let c(tn) be the c∗-algebra of all continuous functions on the n-torus tn. this is the universal c∗-algebra generated by mutually commuting n unitaries. hence, spr(c(tn)) = n. let t n θ be the noncommutative n-torus, which is defined to be the universal c∗-algebra generated by n unitaries uj such that ujui = e 2πiθij uiuj for 1 ≤ i, j ≤ n, where θ = (θij ) is a skew-adjoint n × n matrix over r. thus, spr(tn θ ) = n. let c∗(fn) be the full group c ∗ -algebra of the free group fn with n generators. this is the universal c∗-algebra generated by n unitaries with no relations. hence, spr(c∗(fn)) = n. proposition 1.5. let a be a c∗-algebra with isometry generators sj for j ∈ j a set. then spr(a) ≤ ∑ j dim sp(sj ) = ∑ j 2 = 2|j|, and the inequality spr(a) ≤ 2|j| holds in general, with {aj}j∈j generators of a. proof. note that sp(sj ) is the unit disk d with dim d = 2. thus, dim sp(sj ) = 2. in general, sp(aj ) is a closed subset of c, so that dim sp(aj ) ≤ 2. 2 example 1.6. let f be the toeplitz algebra, which is the universal c∗-algebra generated by an isometry. thus, spr(f) = 2. let c∗(nn) be the full semigroup c ∗ -algebra of the free semigroup nn with n generators, i.e., nn ∼= ∗ n n the n-fold free product of the semigroup n of natural numbers. this is the universal c∗-algebra generated by n isometries with no relations. hence, spr(c∗(nn)) = 2n. let on be the cuntz algebra (2 ≤ n < ∞), which is the universal c ∗ -algebra generated by n isometries sj such that ∑ n j=1 sj s ∗ j = 1. then spr(on) = 2n. let o∞ be the cuntz algebra generated by isometries sj (j ∈ n) such that ∑ n j=1 sj s ∗ j < 1. then spr(o∞) = ∞. proposition 1.7. let a be a c∗-algebra generated by compact operators. then spr(a) = 0. 78 takahiro sudo cubo 10, 2 (2008) proof. note that for a compact opearator t , we have dim sp(t ) = 0. 2 proposition 1.8. let a, b be c∗-algebras and a ⊕ b their direct sum. then spr(a ⊕ b) = spr(a) + spr(b). proof. suppose that aj are generators of a and bj are those of b. then aj ⊕ 0 and 0 ⊕ bj are generators of a ⊕ b. 2 example 1.9. for mn(c) the n × n matrix algebra over c, spr(mn(c)) = 0. let k be the c∗-algebra of all compact operators on a separable infinite-dimensional hilbert space. then spr(k) = 0. an af algebra that is an inductive limit of finite dimensional c∗-algebras (i.e., finite direct sums of some mn(c)) has spectral rank zero. proposition 1.10. let a be a c∗-algebra and i its c∗-subalgebra, where generators of i can be always taken from those of a. then spr(i) ≤ spr(a). in particular, we may take i as a closed two-sided ideal or hereditary c∗-subalgebra in this sense. proof. generators of i can be viewed as a part of those of a. 2 remark. the assumption for generators is necessary. indeed, there exist some c∗-algebras that are embeddable into af algebras, but they should have spectral rank non-zero, such as rotation c∗-algebras. this is an obstruction to our theory, but it seems that in this case the generators of those c∗-algebras are not to be visible in af. 2 proposition 1.11. let 0 → i → a → a/i → 0 be a short exact sequence of c∗-algebras, where generators of i are always taken from those of a. then spr(a) ≥ max{spr(i), spr(a/i)}. remark. it is likely but not true in general that spr(a) ≤ spr(i) + spr(a/i). for instance, the toeplitz algebra f = c∗(s) is decomposed into 0 → k → f → c(t) → 0, where k is generated by some elements like the finite rank projections 1 − sn(sn)∗ for n ∈ n and s is mapped to the generator of c(t). but spr(f) = 2, spr(k) = 0, and spr(c(t)) = 1. however, the generators of k are not a part of those of f. cubo 10, 2 (2008) spectral rank for c∗-algebras 79 proposition 1.12. let a be a c∗-algebra and a+ its unitization by c. then spr(a) = spr(a + ). proof. note that for the unit 1, sp(1) = {1} ⊂ c. thus, dim sp(1) = 0. 2 proposition 1.13. let 0 → i → a → b → 0 be an extension of c∗-algebras. then spr(a) ≤ max{spr(m (i)), spr(b)}. where m (i) is the multiplier algebra of i, and generators of a are viewed as part of those of the direct sum m (i) ⊕ b containing the pullback c∗-algebra associated with the extension. proof. it is well known that a is isomorphic to the pullback c∗-algebra in m (i) ⊕ b with the associated busby map from b to m (i)/i and the quotient map from m (i) to m (i)/i. 2 proposition 1.14. let a, b be c∗-algebras and a ⊗ b their c∗-tensor product with a c∗-norm. then max{spr(a), spr(b)} ≤ spr(a ⊗ b) ≤ spr(a) + spr(b). proof. the left inequality is clear since a, b are c∗-subalgebras of a ⊗ b preserving generators. assume first that a, b are unital. suppose that aj are generators of a and bj are those of b. then aj ⊗ 1 and 1 ⊗ bj are generators of a ⊗ b. if a or b are non-unital, then spr(a ⊗ b) ≤ spr(a + ⊗ b + ) ≤ spr(a + ) + spr(b + ) = spr(a) + spr(b) since a ⊗ b is a closed ideal of a+ ⊗ b+. 2 corollary 1.15. for mn(a) the n × n matrix algebra over a c ∗-algebra a, spr(mn(a)) = spr(a). furthermore, if b is a c∗-algebra with spr(b) = 0, then spr(a ⊗ b) = spr(a). proof. note that mn(a) ∼= a ⊗ mn(c). 2 proposition 1.16. let a be a c∗-algebra, g a finitely generated discrete group with n generators, and a ⋊α g a (full or reduced) c ∗-crossed product of a by an action α of g by automorphisms. then spr(a) ≤ spr(a ⋊α g) ≤ spr(a) + n. in particular, if g = z, then spr(a) ≤ spr(a ⋊α z) ≤ spr(a) + 1. 80 takahiro sudo cubo 10, 2 (2008) proof. the crossed product a ⋊α g is generated by a and the unitaries corresponding to the generators of g (on the universal, or a certain hilbert space). 2 corollary 1.17. let g be a finitely generated discrete group with n generators. let c∗(g) and c∗ r (g) be its full and reduced group c∗-algebras. then spr(c∗(g)) ≤ n, and spr(c∗ r (g)) ≤ n. proposition 1.18. let a be a c∗-algebra, n a finitely generated discrete semi-group with n generators, and a ⋊β n a (full or reduced) semi-group crossed product of a by an action β of n by endomorphisms. then spr(a) ≤ spr(a ⋊β n ) ≤ spr(a) + 2n. in particular, if n = n, then spr(a) ≤ spr(a ⋊β n) ≤ spr(a) + 2. proof. the crossed product a ⋊β n is generated by a and the isometries corresponding to the generators of n (on the universal, or a certain hilbert space). 2 corollary 1.19. let n be a finitely generated discrete semi-group with n generators. let c∗(n ) and c∗ r (n ) be its full and reduced semi-group c∗-algebras. then spr(c ∗ (n )) ≤ 2n, and spr(c∗ r (n )) ≤ 2n. proposition 1.20. let a be a continuous field c∗-algebra on a locally compact hausdorff space x with fibers the same c∗-algebra b. then spr(a) ≤ spr(c0(x)) + spr(b), where c0(x) is the c ∗-algebra of all continuous functions on x vanishing at infinity. if b is unital with spr(b) = 0, then spr(a) = spr(c0(x)). proof. assume first that b is unital. then a is assumed to be generated by generators of c0(x) and those of b. also, we obtain spr(c0(x)) ≤ spr(a). if b is non-unital, we can consider the unitization a + of a by adding the unit field on x taking the unit of the unitization b + of b. therefore, we obtain spr(a) = spr(a + ) ≤ spr(c0(x)) + spr(b + ) = spr(c0(x)) + spr(b). 2 cubo 10, 2 (2008) spectral rank for c∗-algebras 81 example 1.21. let t2 θ be the rational rotation c∗-algebra corresponding to a rational θ. this can be viewed as a continuous filed c∗-algebra on the 2-torus t2 with fibers the same mn(c) with n the period of θ. thus, spr(t 2 θ ) = spr(c(t2)) = 2. proposition 1.22. let a be a c∗-algebra and b a c∗-algebra deformed from a with generators and relations deformed from those of a. then spr(a) = spr(b). example 1.23. let tn θ be a noncommutative n-torus. this is deformed from c(tn), and spr(t n θ ) = spr(c(tn)) = n. proposition 1.24. let a, b be c∗-algebras and a∗b their c∗-free product with a (full or reduced) c∗-norm. then max{spr(a), spr(b)} ≤ spr(a ∗ b) ≤ spr(a) + spr(b). also, a ∗ b can be replaced with the unital c∗-free product a ∗c b. proof. the left inequality is clear since a, b are c∗-subalgebras of a ∗ b preserving generators. suppose that aj are generators of a and bj are those of b. then aj and bj are generators of a ∗ b. 2 2 approximate spectral rank definition 2.1. let a be a c∗-algebra. define the approximate spectral rank of a to be the minimum non-negative integer n = aspr(a) such that for any a ∈ a and ε > 0, there exists a c∗-subalgebra b of a with spr(b) ≤ n such that ‖a − b‖ ≤ ε for some b ∈ b. proposition 2.2. let a be an inductive limit of c∗-algebras an with spr(an) ≤ sn for some sn. then aspr(a) ≤ lim sn, where lim is the limit infimum. example 2.3. if a is an af-algebra, then aspr(a) = 0 = spr(a). let a be an at-algebra, i.e., an inductive limit of finite direct sums of matrix algebras over c(t). if a is an inductive limit of k direct sums of matrix algebras over c(t), then aspr(a) ≤ k. indeed, for such a k direct sum, spr(⊕ k j=1mnj (c(t))) = spr(⊕ k j=1(mnj (c) ⊗ c(t))) = k∑ j=1 spr(mnj (c) ⊗ c(t)) ≤ k∑ j=1 (spr(mnj (c)) + spr(c(t))) = k. 82 takahiro sudo cubo 10, 2 (2008) in particular, if t 2 θ is a simple noncommutative 2-torus, then it is an inductvie limit of 2 direct sums of matrix algebras over c(t). hence, aspr(t2 θ ) ≤ 2. proposition 2.4. let x be a locally compact hausdorff space with dim x finite. then aspr(c0(x)) ≤ dim x. proof. note that x can be viewed as a projective limit of the product spaces [0, 1]n, where dim x = n. thus, c0(x) is an inductive limit of c([0, 1] n ). since c([0, 1]n) ∼= ⊗nc([0, 1]), we obtain the conclusion. 2 remark. there exists a locally compact hausdorff space x with dim x = 1 but dim x+ = 0, where x+ is the one-point compactification of x. thus, aspr(c0(x)) = 1, but spr(c0(x)) = spr(c(x + )) = 0, where c0(x) + ∼= c(x+). also, aspr(c(x+)) = 0. remark. more fundamental properties for the approximate spectral rank could be obtained as the spectral rank in section 1, but their details would be considered somewhere else. received: february 2007. revised: april 2008. references [1] l.g. brown and g.k. pedersen, c∗-algebras of real rank zero, j. funct. anal. 99 (1991), 131–149. [2] j. dixmier, c∗-algebras, north-holland (1977). [3] g.k. pedersen, c∗-algebras and their automorphism groups, academic press (1979). [4] m.a. rieffel, dimension and stable rank in the k-theory of c∗-algebras, proc. london math. soc. 46 (1983), 301–333. [5] t. sudo, a topological rank for c∗-algebras, far east j. math. sci. 15 (1) (2004), 71–86. [6] t. sudo, a covering dimension for c∗-algebras, cubo a math. j. 8 (1) (2006), 87–96. [7] w. winter, covering dimension for nuclear c∗-algebras, j. funct. anal. 199 (2003), 535–556. n6 cubo a mathematical journal vol.10, n o ¯ 03, (57–64). october 2008 the extension of the formula by dupire shunsuke kaji department of mathematics, graduate school of science, osaka university, machikaneyamachou 1-1, toyonaka, osaka japan 560-0043, osaka japan 560-0043 email: kaji@math.sci.osaka-u.ac.jp abstract we provide the extension of dupire’s pde, as the partial integro-differential equations of market prices of call options with many maturities and strike prices for jump diffusion model. resumen nosotros damos la extensión de dupire pde, como las ecuaciones parciales integrodiferenciales de precios de mercado de opciones de llamada con muchos vencimientos y golpe de precios para modelos de difusión con saltos. key words and phrases: dupire, pde, jump-diffusion model. math. subj. class.: 35k15, 60h30. 58 shunsuke kaji cubo 10, 3 (2008) 1 introduction let (ω,f,p) be a probability space. on the space (ω,f,p) we set a standard brownian motion w = {wt}t∈[0,t ] from w0 = 0 and a poission random measure n(dtdx) on (0,t ]×r with intensity measure dtν(dx), where t ∈ (0,∞) and the measure ν on r satisfies ∫ r (1 + e2z) ∧ z2ν(dz) < ∞. (1) we consider a risk-neutral price process {sxt }t∈[0,t ] of a risk asset satisfing dsxt = σ(t,s x t )s x t dwt + (r − δ)s x t dt; sx 0 = x ∈ (0,∞), where r ≥ 0 denotes the interest rate and δ ≥ 0 the dividend rate. the function σ : [0,t ]×(0,∞) → [0,∞) has the lipschiz condition and is often called the volatility of the asset’s price. according to the well-known discussion of option pricing model, if for each t,k ∈ (0,∞) we have a unique solusion u(t,x,t,k) to the parabolic equation and boundary condition ∂u ∂t + 1 2 σ(t,x) 2 x2 ∂2u ∂x2 + (r − δ)x ∂u ∂x − ru = 0, (t,x) ∈ [0,t ) × (0,∞); u(t,x,t,k)|t=t = (x − k) +, x ∈ (0,∞), then a price of a call option with maturity t and strike price k is given by u(t,x,t,k)|t=0 = e −rt e[(sxt − k) +]. dupire[1] found that u(t,x,t,k) as a function of (t,k) satisfies the following dual equation to the last parabolic equation: ∂u ∂t = 1 2 σ(t,k) 2 k2 ∂2u ∂k2 − (r − δ)k ∂u ∂k − δu, (t,k) ∈ (t,∞) × (0,∞). but his approach is not enough mathematically. there are some works justifing rigorously his idea, for example, klebaner[4] etc. klebaner[4] gives the last equation by the meyer-tanaka formula. on the other hand, there are also works on option pricing model for jump-diffusion processes, for example, geometric lévy processes by fujiwara and miyahara[2]. recently, jourdain[3] provides the extension of dupire’s work for jump-diffusion processes by stochastic flow approch. now, we consider the following risk-neutral evolusion {xxt }t∈[0,t ] for the underlying risk asset’s prices: xxt = x + ∫ t 0 a(u,xxu )x x udwu + (r − δ) ∫ t 0 xxudu + ∫ (0,t]×r xxu−(e z − 1){n(dudz) − duν(dz)}, t ∈ [0,t ] cubo 10, 3 (2008) the extension of the formula by dupire 59 where a(t,x) : [0,t ]×(0,∞) → [0,∞) satisfies the lipschiz condition and has the second derivative with respect to x. then {xxt }t∈[0,t ] has the extended diffusion operator(see yoshida[5] p.408) (atf)(x) = 1 2 a(t,x) 2 x2f′′(x) + (r − δ)xf′(x) + ∫ r f(xez) − f(x) − (ez − 1)xf′(x)ν(dz). for each maturity t and strike price k we denote c(x,t,k) = e−rt e[(xxt − k) +] (2) by a call option price with an asset price x. in particular, in the case a(·, ·) ≡ a the last definition (2) is justified by fujiwara and miyahara[2]. if we moreover assume that a(·, ·) belongs to the class v = { f : [0,t ] × (0,∞) → r| sup (t,x)∈[0,t ]×(0,∞) 3 ∑ k=0 |xk ∂kf ∂xk (t,x)| < ∞ } , then jourdain[3] provides the following equation of (t,k): − ∂c ∂t + at c = 0, (t,k) ∈ (0,∞) × (0,∞), where (at f)(k) = 1 2 a(t,k) 2 k2f′′(k) − (r − δ)kf′(k) − δf(k) + ∫ r {f(ke−z) − f(k) − (e−z − 1)kf′(k)}ezν(dz). here notice that the assumption a(·, ·) ∈ v satisfies the lipschiz condition. in this note we provide the same result of the above without a(·, ·) ∈ v by using not only stochastic flow approch but also another one. 2 main result we fix x ∈ (0,∞) as follows. we have the following main theorem. theorem 2.1. c(x,t,k) as a function of (t,k) satisfies − ∂c ∂t + at c = 0, (t,k) ∈ (0,∞) × (0,∞) in weak sense; that is, ∫ ∞ 0 ∫ ∞ 0 c(x,t,k) { ∂ϕ ∂t (t,k) + a∗t ϕ(t,k) } dtdk = 0, ∀ϕ ∈ c∞ 0 ((0,∞) 2 ), where ∫ ∞ 0 ∫ ∞ 0 ψ(t,k)a∗t ϕ(t,k)dtdk = ∫ ∞ 0 ∫ ∞ 0 at ψ(t,k)ϕ(t,k)dtdk, ∀ϕ,∀ψ ∈ c∞ 0 ((0,∞) 2 ). 60 shunsuke kaji cubo 10, 3 (2008) 2.1 lemmas lemma 2.1. it follows that 0 ≤ c(x,t,k) ≤ e−δt x, (t,k) ∈ (0,∞) × (0,∞). (3) for every ϕ(·) ∈ c2 0 ((0,∞)) e−rt e[ϕ(xxt )] = ∫ ∞ 0 c(x,t,k)ϕ′′(k)dk, t ∈ (0,∞) (4) holds. remark 2.1. it follows from (3) that c(x,t,k) as a function of (t,k) is locally integrable on (0,∞) × (0,∞). thus the right-hand side of (4) is well-defined. proof: by (2) we have 0 ≤ ert c(x,t,k) ≤ e[xxt ]. moreover, since {e−(r−δ)txxt }t∈[0,t ] is a nonnegative local martingale with initial value x, the right-hand side of the last inequality is ≤ e(r−δ)t x. hence we get (3). finally, we compute from (2) that the right-hand side of (4) is = ∫ ∞ 0 e −rt e[(xxt − k) +]ϕ′′(k)dk = e−rt e [ ∫ ∞ 0 (xxt − k) + ϕ ′′(k)dk ] = e−rt e[ϕ(xxt )]. hence we get (4). before we moreover introduce lemmas, for every ϕ(·, ·) ∈ c∞ 0 ((0,∞) 2 ) we set a family {φh}h>0 of all functions φh(t,x) = 1 h {e[ϕ(t,xxt +h)] − e[ϕ(t,x x t )]}, (t,k) ∈ (0,∞) × (0,∞). lemma 2.2. limh↓0 ∫ ∞ 0 φh(t,x)dt = − ∫ ∞ 0 ∫ ∞ 0 e rt c(x,t,k) ∂3ϕ ∂t∂k2 (t,k)dtdk. proof: first, we set c̃(x,t,k) = ert c(x,t,k). by using (4), we have ∫ ∞ 0 φh(t,x)dt = ∫ ∞ 0 { ∫ ∞ 0 c̃(x,t + h,k) − c̃(x,t,k) h ∂2ϕ ∂k2 (t,k)dk } dt, cubo 10, 3 (2008) the extension of the formula by dupire 61 where h > 0. moreover we compute that the right-hand side of the last equality is = ∫ ∞ 0 { ∫ ∞ 0 c̃(x,t + h,k) − c̃(x,t,k) h ∂2ϕ ∂k2 (t,k)dt } dk = ∫ ∞ 0 { ∫ ∞ 0 c̃(x,t,k) 1 h ( ∂2ϕ ∂k2 (t − h,k) − ∂2ϕ ∂k2 (t,k) ) dt } dk, and so we have ∫ ∞ 0 φh(t,x)dt = ∫ ∞ 0 ∫ ∞ 0 c̃(x,t,k) 1 h ( ∂2ϕ ∂k2 (t − h,k) − ∂2ϕ ∂k2 (t,k) ) dtdk. then, by using the dominated convergence theorem, ϕ(·, ·) ∈ c∞ 0 ((0,∞) 2 ) and (3) imply that the right-hand side of the last equality converges to − ∫ ∞ 0 ∫ ∞ 0 c̃(x,t,k) ∂3ϕ ∂t∂k2 (t,k)dtdk as h ↓ 0. hence we get the desired result. we denote by the following operator depended on time t ∈ [0,∞) : (ãtf)(x) = 1 2 a(t,x) 2 x 2 f ′′(x) + { ∂ ∂x (a(t,x) 2 x 2) + (r − δ)x}f′(x) + { 1 2 ∂2 ∂x2 (a(t,x) 2 x 2) + (r − 2δ) } f(x) + ∫ r e 2z f(xez) − (2ez − 1)f(x) − (ez − 1)xf′(x)ν(dz). lemma 2.3. limh↓0 ∫ ∞ 0 φh(t,x)dt = ∫ ∞ 0 ∫ ∞ 0 e rt c(x,t,k) ( ãt ∂2ϕ ∂k2 (t,k) ) dtdk. proof: first, we divide a· into two parts as follows: (a·f)(x) = { 1 2 a(·,x) 2 x2f′′(x) + (r − δ)xf′(x) + ∫ |z|<1 f(xez) − f(x) − zxf′(x)ν(dz) − ∫ |z|<1 (ez − 1 − z)xf′(x)ν(dz) − ∫ |z|≥1 f(x) + (ez − 1)xf′(x)ν(dz) } + ∫ |z|≥1 f(xez)ν(dz) = (a0 · f)(x) + ∫ |z|≥1 f(xez )ν(dz). 62 shunsuke kaji cubo 10, 3 (2008) since ϕ(·, ·) ∈ c∞ 0 ((0,∞) 2 ) we can choose subintervals i1 = [α1,β1] and i2 = [α2,β2] of (0,∞) such that supp ϕ ⊂ i1 ×i2. we pick δ > 0 and set ĩ1 = {x|α1 ≤ x ≤ β1 + δ} and ĩ2 = {x|α2e −1 ≤ x ≤ β2e}. we denote by ‖ f ‖c(γ) = supx∈γ|f(x)|, where γ is a compact subset of (0,∞) 2 and f ∈ c(γ) = {f is a real-valued continuous function on γ}. then observe that a0 t ϕ(t, ·) belongs to c2((0,∞)), since ϕ(·, ·) ∈ c∞ 0 ((0,∞) 2 ) and a(t, ·) has the second derivative, and |a0uϕ(t,k)| ≤ 1 2 ‖ a2 ‖ c(ĩ1×i2) ‖ k2 ∂2ϕ ∂k2 ‖ c(i1×i2) 1i1×i2 (t,k) + |r − δ| ‖ k ∂ϕ ∂k ‖ c(i1×i2) 1i1×i2 (t,k) + ∫ |z|<1 z2 ν(dz) ‖ k2 ∂2ϕ ∂k2 + k ∂ϕ ∂k ‖ c(i1×ĩ2) 1 i1×ĩ2 (t,k) + ∫ |z|<1 ez − 1 − z ν(dz) ‖ k ∂ϕ ∂k ‖ c(i1×i2) 1i1×i2 (t,k) + ν(|z| ≥ 1) ‖ ϕ ‖ c(i1×i2) 1i1×i2 (t,k) + ∫ |z|≥1 |ez − 1| ν(dz) ‖ k ∂ϕ ∂k ‖ c(i1×i2) 1i1×i2 (t,k) ≤ { 1 2 ‖ a2 ‖ c(ĩ1×i2) ‖ k2 ∂2ϕ ∂k2 ‖ c(i1×i2) + (|r − δ| + ∫ |z|<1 ez − 1 − z ν(dz) + ∫ |z|≥1 |ez − 1| ν(dz))‖ k ∂ϕ ∂k ‖ c(i1×i2) + ∫ |z|<1 z2 ν(dz)‖ k2 ∂2ϕ ∂k2 + k ∂ϕ ∂k ‖ c(i1×ĩ2) + ν(|z| ≥ 1)‖ ϕ ‖ c(i1×i2) } × 1 i1×ĩ2 (t,k) = c1 × 1i1×ĩ2 (t,k), ∀u ∈ ĩ1, where c1 < ∞ holds since ϕ(·, ·) ∈ c ∞ 0 ((0,∞) 2 ), (1), and a(·, ·) is continuous. moreover, it is easy that we have ∣ ∣ ∣ ∣ ∣ ∫ |z|≥1 ϕ(t,kez)ν(dz) ∣ ∣ ∣ ∣ ∣ ≤ c21i1 (t ), where c2 is a positive constant not depending on t and k. therefore the inequality of the observation and the last inequality imply |auϕ(t,k)| ≤ (c1 + c2)1i1 (t ), ∀u ∈ ĩ1. here, fix t and by using appendix 3.2 it follows from ϕ(·, ·) ∈ c∞ 0 ((0,∞) 2 ) that φh(t,x) = 1 h ∫ t +h t e[(auϕ(t, ·))(x x u )]du = 1 h ∫ t +h t e[auϕ(t,x x u )]du, cubo 10, 3 (2008) the extension of the formula by dupire 63 for all 0 < h < δ. then the last two inequality and equality imply limh↓0φh(t,x) = e[at ϕ(t,x x t )]; |φh(t,x)| ≤ (c1 + c2)1i1 (t ), 0 < ∀h < δ. according to the dominated convergence theorem, the last two results imply limh↓0 ∫ ∞ 0 φh(t,x)dt = ∫ ∞ 0 e[at ϕ(t,x x t )]dt. (5) on the other hand, by using (4) we have from the above observation e−rt e[a0t ϕ(t,x x t )] = ∫ ∞ 0 c(x,t,k) ∂2 ∂k2 (a0t ϕ(t, ·))(k)dk moreover we have e−rt e[ ∫ |z|≥1 ϕ(t,xxt e z)ν(dz)] = ∫ |z|≥1 e−rt e[ϕ(t,xxt e z)]ν(dz) = ∫ |z|≥1 ∫ ∞ 0 c(x,t,k) ∂2 ∂k2 (ϕ(t,kez))dkν(dz) = ∫ ∞ 0 c(x,t,k) ∫ |z|≥1 e2z ∂2ϕ ∂k2 (t,kez)ν(dz)dk, where the second line of the last equality holds by (4). therefore the last two equalities imply e−rt e[at ϕ(t,x x t )] = ∫ ∞ 0 c(x,t,k){ ∂2 ∂k2 (a0t ϕ(t,k)) + ∫ |z|≥1 e 2z ∂ 2ϕ ∂k2 (t,kez)ν(dz)}dk, and so by computing the right-hand side of the last equality we have e−rt e[at ϕ(t,x x t )] = ∫ ∞ 0 c(x,t,k)(ãt ∂2ϕ ∂k2 (t,k))dk. hence (5) and the last equality imply the desired result. 2.2 proof of theorem 2.1 first, pick any ψ(t,k) ∈ c∞ 0 ((0,∞) 2 ). according to lemma 2.2 and 2.3, for all ϕ(t,k) ∈ c∞ 0 ((0,∞) 2 ) such that ert ∂ 2 ϕ ∂k2 = ψ, we have ∫ ∞ 0 ∫ ∞ 0 e rt c(x,t,k) { ∂3ϕ ∂t∂k2 (t,k) + ãt ∂2ϕ ∂k2 (t,k) } dtdk = 0, and so ∫ ∞ 0 ∫ ∞ 0 c(x,t,k) { ∂ψ ∂t (t,k) + ãt ψ(t,k) } dtdk = 0 64 shunsuke kaji cubo 10, 3 (2008) holds. on the other hand, we can compute the integral by parts ∫ ∞ 0 ∫ ∞ 0 ψ(t,k)ãt ϕ(t,k)dtdk = ∫ ∞ 0 ∫ ∞ 0 at ψ(t,k)ϕ(t,k)dtdk, ∀ϕ,∀ψ ∈ c∞ 0 ((0,∞) 2 ). hence the last two equalities imply the desired conclusion. 3 appendix appendix 3.1. let x ⊂ rd, where d is a positive integer, be a domain and ck(x), where k = 0, 1, 2, · · · ,∞, be a class of all real-valued functions on x which have continuous partial derivatives of order ≤ k if k < ∞; of order < ∞ if k = ∞. let ck 0 (x) be a class of all functions which belong to ck(x) and compact supports. appendix 3.2. (dynkin’s formula) for every f ∈ c2 0 ((0,∞)), e[f(xxt )] = f(x) + e [ ∫ t 0 (auf)(x x u )du ] , (t,x) ∈ [0,∞) × (0,∞) holds. acknowledgment: the author is grateful to j.sekine, assistant professor for useful advices. received: february 2008. revised: may 2008. references [1] dupire, b., pricing with a smile, risk, 7 (1994), pp.18–20. [2] fujiwara, t. and miyahara, y., the minimal entropy martingale measures for geometric lévy processes, finance stochast., 7 (2003), pp.509–531. [3] jourdain, b., stochastic flow approach to dupire’s formula, finance stochast., (2007). [4] klebaner, f., option price when the stock is a semimartingale, elect. comm. in probab., 7 (2002), pp.79–83. [5] yoshida, k., functional analysis, 4th ed. n05 cubo a mathematical journal vol.10, n o ¯ 02, (15–30). july 2008 dynamical inverse problem for the equation utt − ∆u − ∇ ln ρ · ∇u = 0 (the bc method) m.i. belishev saint-petersburg department of the steklov mathematical institute (pomi), 27 fontanka, st. petersburg 191023, russia, email: belishev@pdmi.ras.ru abstract a dynamical system of the form utt − ∆u − ∇ ln ρ · ∇u = 0, in r n + × (0, t ) u|t=0 = ut|t=0 = 0, in r n + uxn = f on ∂r n + × [0, t ], is considered, where r n + := { x = {x1, . . . , xn}| xn > 0 } ; ρ = ρ(x) is a smooth positive function (density) such that ρ, 1 ρ are bounded in r n +; f is a (neumann) boundary control of the class l2(∂r n + × [0, t ]); u = u f (x, t) is a solution (wave). with the system one associates a response operator rt : f 7→ uf |∂rn + ×[0,t ]. a dynamical inverse problem is to determine the density from the given response operator. fix an open subset σ ⊂ ∂rn+; let l2(σ×[0, t ]) be the subspace of controls supported on σ. a partial response operator rt σ acts in this subspace by the rule rt σ f = uf |σ×[0,t ]; let r2t σ be the operator corresponding to the same system considered on the doubled time interval [0, 2t ]. denote bt σ := { x ∈ rn+| {x 1, . . . , xn−1, 0} ∈ σ, 0 < xn < t } and assume ρ|σ to be known. we show that r 2t σ determines ρ|bt σ and propose an efficient 16 m.i. belishev cubo 10, 2 (2008) procedure recovering the density. the procedure is available for constructing numerical algorithms. the instrument for solving the problem is the boundary control method which is an approach to inverse problems based on their relations with control theory (belishev, 1986). our presentation is elementary and can serve as introduction to the bc method. resumen consideramos el sistema dinámico utt − ∆u − ∇ ln ρ · ∇u = 0, en r n + × (0, t ) u|t=0 = ut|t=0 = 0, en r n + uxn = f sobre ∂r n + × [0, t ], donde r n + := { x = {x1, . . . , xn}| xn > 0 } ; ρ = ρ(x) es una función positiva suave (densidad) tal que ρ, 1 ρ son limitada en r n +; f es un control en la frontera (neumann) de clase l2(∂r n + × [0, t ]); u = u f (x, t) es la solución (onda). con el sistema asociamos un operador respuesta rt : f 7→ uf |∂rn + ×[0,t ]. un problema dinámico inverso consiste en determinar la densidad desde el operador respuesta. fije un subconjunto abierto σ ⊂ ∂rn+; sea l2(σ × [0, t ]) el subespacio de los controles soportados em σ. un operador respuesta parcial rt σ actua en este subespacio mediante la regla rt σ f = uf |σ×[0,t ]; sea r 2t σ el operador correspondiente al mismo sistema considerado en el intervalo de tiempo [0, 2t ]. denote bt σ := {x ∈ rn+| {x 1, . . . , xn−1, 0} ∈ σ, 0 < xn < t } y suponga que ρ|σ es conocido. nosotros mostramos que r 2t σ determina ρ| b t σ y es propuesto un procedimento eficiente de recuperar la densidad. el procedimiento es encontrado por construción de algoritmos númericos. el instrumento de resolver el problema es el método de control en la frontera este es un abordage para problemas inversos basado en sus relaciones con teoria de control (belishev, 1986). nuestra presentación es elemental y puede servir como introducción al método bc. key words and phrases: dynamical inverse problem, response operator, determination of density, boundary control method. math. subj. class.: 35bxx, 35r30 cubo 10, 2 (2008) dynamical inverse problem for the equation ... 17 1 about the paper the problem under consideration comes from geophysics. we deal with a dynamical system of the form utt − ∆u − ∇ ln ρ · ∇u = 0 in r n + × (0, t ) (1) u|t=0 = ut|t=0 = 0 in r n + (2) uxn = f on γ × [0, t ], (3) where r n + := { x = {x1, . . . , xn}| xn > 0 } simulates the earth, γ := ∂rn+ is the earth surface, ρ = ρ(x) is a smooth function (density) satisfying 0 < ρ∗ ≤ ρ(·) ≤ ρ ∗ , f is a (neumann) boundary control of the class l2(γ×[0, t ]), u = u f (x, t) is a solution. the solution describes a wave initiated by the control and propagating into the earth from the surface. with the system one associates a response operator rt : f 7→ uf |∂rn + ×[0,t ]. from the physical viewpoint, f is a force applied at the surface, whereas uf |∂rn + ×[0,t ] is a displacement measured at the same surface. thus, r t is an ”input 7→ output” map representing the measurements, which the external observer implements at the surface. the dynamical inverse problem, which the paper deals with, is to determine the density from the given response operator. 2 results begin with certain of the notations. with a point x ∈ rn+ we associate a pair (γ, τ ) : γ := {x1, . . . , xn−1, 0} ∈ γ, τ := xn ≥ 0 of its semigeodesic coordinates and write x = x(γ, τ ). fix ξ > 0, let σ ⊂ γ be an open subset at the surface; the set bξ σ := {x ∈ rn+| x = x(γ, τ ), γ ∈ σ, τ ∈ (0, ξ)} is called a tube with a bottom σ and a top σξ = {x(γ, τ )| γ ∈ σ, τ = ξ}. also, introduce a subdomain ω ξ σ := {x ∈ rn+| dist (x, σ) < ξ} 1 (contoured with cdef on fig 1). the tube (shadowed on fig 1) is the part of the subdomain illuminated with rays emanating from σ in normal direction to the surface. figure 1: the tube bξ σ and the subdomain ωξ σ 1dist is the standard euclidean distance in rn + 18 m.i. belishev cubo 10, 2 (2008) let σ ⊂ γ and t > 0 be fixed. consider system (1)–(3) with the final moment t = 2t and introduce a partial response operator r2t σ acting in the (sub)space l2(σ × [0, 2t ]) of controls supported on σ by the rule r2t σ f := uf |σ×[0,2t ] 2 . as is well-known, the waves in system (1)–(3) propagate with the unit velocity. by this, the response operator depends on the density locally: r2t σ is determined by the behavior of ρ in the subdomain ωt σ only 3 . such a locality motivates the setup of the inverse problem: given operator r2t σ to recover ρ|ωt σ . however, since the substitution ρ → cρ with a constant c > 0 does not change system (1)–(2), the unique determination of density is impossible. avoiding such a nonuniqueness, it is natural to assume the boundary values of ρ to be known. the main result of the paper is theorem 1 let t > 0 be fixed, the operator r2t σ given, and the function ρ|σ known. then these data uniquely determine ρ|bt σ . the proof is constructive: we propose an efficient procedure recovering the density in the tube. moreover, the procedure is provided with an additional option that is visualization of waves: given f we recover uf | b t σ . 3 motivation and comments there are two reasons to deal namely with the version (1) of the general wave equation with variable coefficients. first, the interest is motivated by possible applications in geophysics (see, e.g., [6]). the second reason is the following. the instrument for solving the problem is the boundary control method (bcm), which is an approach to inverse problems based on their relations with control theory (belishev, 1986). in comparison with another versions (see [1] – [4]), the variant of the bcm available for equation (1) is the simplest one. as such, it has good chances for numerical realization. our presentation is elementary: along with the paper [3], this one can serve as an introduction to the bcm. the plan of the paper is as follows: • in section 4, the basic notions and objects (spaces, operators etc) are introduced • in section 5, we present the so-called amplitude formula (af) which solves the inverse problem: it recovers ρ|bt σ via r2t σ • sections 6–10 are devoted to the derivation of the af • in section 11, a certain additional option of the bcm is described: the af enables one to recover the solutions uf in the tube bt σ that is what we call a wave visualization • section 12 contains the concluding remarks; also, the extension of our results is shortly discussed. 2this operator represents the measurements implemented at the part of the earth surface 3in other words, r2tσ does not depend on ρ|rn + \ωt σ cubo 10, 2 (2008) dynamical inverse problem for the equation ... 19 simplifying the notations, we accept the convention: unless the otherwise is specified, the subset σ ⊂ γ is assumed fixed and we write r2t instead of r2t σ , ω ξ instead of ω ξ σ , bξ instead of bξ σ , etc. also, without loss of generality, we assume σ to be bounded and ∂σ smooth. 4 dynamical system with system (1)–(3) one associates (i) an outer space ft := {f ∈ l2, ρ0 (γ × [0, t ])| supp f ⊂ σ × [0, t ]} of controls acting from σ with the inner product (f, g) f t = ∫ σ×[0,t ] f (γ, t) g(γ, t) ρ0(γ)dγdt , where ρ0 := ρ|γ, dγ is the euclidean surface element on ∂r n +. in f t we single out a family of subspaces f t, ξ := { f ∈ ft | supp f ⊂ σ × [t − ξ, t ] } , ξ ∈ [0, t ] consisting of the delayed controls (t − ξ is the value of delay, ξ is the action time; ft, 0 = {0}, ft, t = ft ); (ii) an inner space of states (waves) ht := l2, ρ(ω t ) with the product (y, w) h t = ∫ ωt y(x) w(x) ρ(x)dx and the family of its subspaces 4 h ξ := { y ∈ ht | supp y ⊂ ωξ } , ξ ∈ [0, t ] . since the waves described by a hyperbolic system (1)–(3) propagate (from σ into rn+) with the speed 1, the inclusion supp uf (·, t) ⊂ ωt holds for all t ∈ [0, t ], whereas ωt is the subdomain filled with waves at the final moment t = t . correspondingly, we consider the waves as time depended elements of the space ht ; (iii) a control operator w t : ft → ft , w t f := uf (·, t ). this operator is continuous [7] and injective for any t > 0 [1]. by the above-mentioned hyperbolicity, for f ∈ ft, ξ one has supp uf (·, t ) ⊂ ωξ that yields the embedding w t ft, ξ ⊂ hξ; (iv) a response operator rt :ft → ft , rt f := uf |σ×[0,t ], which is also a continuous map 5 ; (v) a connecting operator ct : ft → ft , ct := (w t )∗w t . for f, g ∈ ft , one has ( uf (·, t ), ug(·, t ) ) h t = (w t f, w t g) h t = (ct f, g) f t , (4) 4recall that ωξ = {x ∈ rn + | dist (x, σ) < ξ} 5moreover, rt is a compact operator [7] 20 m.i. belishev cubo 10, 2 (2008) so that ct connects the metrics of the outer and inner spaces. by injectivity of w t , the operator ct is also injective. one of central points of the bcm is an explicit relation between the response and connecting operators. denote by st : ft → f2t the operator extending controls from σ × [0, t ] to σ × [0, 2t ] as odd (with respect to t = t ) functions of time; let j 2t : f2t → f2t be the integration: (j 2t f )(·, t) := ∫ t 0 f (·, s) ds; r2t : f2t → f2t the response operator of system (1)–(3) with the final moment t = 2t . lemma 1 the relation ct = 1 2 (st )∗j 2t r2t st (5) holds. proof choose f, g ∈ ft and denote f− := s t f . assume f , g to be such that uf− , ug are the classical solutions. blagoveschenskii’s function b(s, t) := ∫ r n + uf− (·, s) ug(·, t) ρdx s, t ∈ [0, 2t ] × [0, t ] is well defined and satisfies btt(s, t) − bss(s, t) = ∫ ω [ u f − (·, s) u g tt (·, t) − u f − ss (·, s) u g (·, t) ] ρdx = ∫ ω [ uf− (·, s) divρ∇ug(·, t) − divρ∇uf− (·, s) ug(·, t) ] dx = ∫ γ [ −uf− (·, s) u g xn (·, t) + u f − xn (·, s) ug(·, t) ] ρ0dγ = − ∫ γ [ (r2t f−)(·, s) g(·, t) − f−(·, s) (r t g)(·, t) ] ρ0dγ =: f (s, t) (in the second equality we have used equation (1) in the form ρutt = div(ρ∇u). finding b by the d’alembert formula (with regard to the initial conditions b(·, 0) = bt(·, 0) = 0), putting t = t , and taking into account the oddness of f−, we get b(t, t ) = 1 2 ∫ t 0 dt ∫ 2t −t t f (s, t) ds = − 1 2 ∫ t 0 dt ∫ 2t −t t ds ∫ γ (r 2t f−)(·, s) g(·, t) ρ0dγ := ∫ γ×[0,t ] 1 2 {∫ t 0 (r 2t f−)(·, s) ds − ∫ 2t −t 0 (r 2t f−)(·, s) ds } g(·, t) ρ0dγdt that can be easily transformed to b(t, t ) = ( 1 2 (st )∗j 2t r2t st f, g ) f t . (6) cubo 10, 2 (2008) dynamical inverse problem for the equation ... 21 on the other hand, we have b(t, t ) = ( u f − (·, t ), u g (·, t ) ) h t = ( u f (·, t ), u g (·, t ) ) h t = 〈see (4)〉 = (c t f, g) f t . (7) comparing (6) with (7) and taking into account the density (in ft ) of f, g used, we arrive at (5). � 5 amplitude formula here we present a relation that determines the density from the response operator. recall that x(γ, τ ) denotes the point in rn+ with the semigeodesic coordinates γ and τ , the subset σ ∋ γ is fixed, ρ0 = ρ|γ. fix ξ ∈ (0, t ); let f̌ ξ := {f ξ k } ∞ k=1 ⊂ f t, ξ be a linearly independent complete system 6 of delayed controls acting from σ and such that the corresponding solutions uf ξ k are classical. with the system we associate its gram matrix {gik} ∞ i,k=1 : gik := (c t f ξ i , f ξ k ) f t = ∫ σ×[t −ξ,t ] (ct f ξ i )(γ, t) f ξ k (γ, t) ρ0(γ)dγdt (8) and a sequence of numbers {βi} ∞ i=1 : βi := − (κ t , f ξ i ) f t = − ∫ σ×[t −ξ,t ] (t − t) f ξ i (γ, t) ρ0(γ)dγdt , (9) where κ t = κ t (γ, t) := t − t. for any integer n ≥ 1, a linear algebraic system n∑ k=1 gik α n k = βi, i = 1, . . . , n (10) is uniquely solvable (w.r.t. αn1 , . . . , α n n ) by injectivity of the operator ct . lemma 2 for a fixed (γ, ξ) ∈ σ × (0, t ), the relation ρ (x(γ, ξ)) = ρ0(γ) {[ ∂ ∂t lim n→∞ n∑ k=1 α n k ( c t f ξ k )] (γ, t) ∣∣∣∣ t=t −ξ+0 t=t −ξ−0 }2 (11) holds. relation (11) is a relevant version of the so-called amplitude formula (af), which is one of the main tools for solving inverse problems by the bcm: see [1], [2], [3]. the assertion of theorem 1 easily follows from the af. indeed, if ρ0 is known and r 2t is given, then one can find ct by (5) and, choosing a system f̌ ξ, determine the r.h.s. of (11). as (γ, ξ) runs over σ × (0, t ), the points x(γ, ξ) exhaust the tube bt . hence, ρ|bt is determined by r 2t . in sections 6–10 we derive the af. 6the completeness means that closft span f̌ ξ = ft, ξ 22 m.i. belishev cubo 10, 2 (2008) 6 propagation of jumps the well-known and typical for hyperbolic problems fact is that discontinuity of a control f in system (1)–(3) implies discontinuity of the corresponding wave uf , the latter discontinuity (singularity) propagating along the space-time rays and being supported on the characteristic surfaces. below we recall certain details. let θ0(t) := 1 2 [1 + sign t], t ∈ r be the heavyside function; the sequence {θk(·)} k=∞ k=−∞ : dθk dt = θk−1 is usually referred to as a smoothness scale 7 . choose a smooth control f ∈ ft and fix ξ ∈ (0, t ); by fξ(γ, t) := f (γ, t)θ0 (t − (t − ξ)) we denote its cut-off function on σ × [t − ξ, t ], so that fξ is an element of the subspace f t, ξ . in the generic case, the control fξ has a jump at the cross-section σ × {t = t − ξ} (see ab on fig 2), the value (amplitude) of the jump being equal to fξ( · , t)| t=t −ξ+0 t=t −ξ−0 = f ( · , t − ξ) − 0 = f ( · , t − ξ). t=0 t=t t=t-x figure 2: jumps in system (1)–(3) jumps of a neumann control induce jumps of a wave velocity. namely, the velocity u fξ t turns out to be discontinuous; its jump is supported on the characteristic surface {(x, t)| t = (t − ξ)+ dist (x, σ)}. this surface consists of the plane part abde and two conic parts bcd and aef (see fig 2; the arrows pick up the space-time rays, which the discontinuity propagates along). the jump on the conic parts is weaker than the one on the plane part and plays no role in the further considerations. the jump on the plane part is described as follows. in a (space-time) neighborhood of abde, the solution is sought in the form ufξ = ap + wp , ap(x, t) := p∑ k=1 ak(γ, τ ) θk (t − (t − ξ) − τ ) , (12) where x in the l.h.s. is x(γ, τ ); ap is an ansatz, which is a function of the class c p−1 loc ; ak are the socalled amplitude functions; wp ∈ c p loc is a smoother reminder. substituting such a representation in (1)–(3), one derives a recurrent system of ode’s 8 for the amplitude functions. as result, one 7θk with negative k’s are understood in the sense of distributions 8the so-called transport equations cubo 10, 2 (2008) dynamical inverse problem for the equation ... 23 arrives at the representation (for p = 1) of the form ufξ (x, t) = − ( ρ (x(γ, τ )) ρ0(γ, τ ) ) − 1 2 f (γ, t − ξ) θ1 (t − (t − ξ) − τ ) + w(x, t) θ2 (t − (t − ξ) − τ ) , (13) where x = x(γ, τ ) in the l.h.s. and w is a smooth function. for details of this technique see, e.g., [5]. at the final moment t = t , the wave ufξ (· , t ) is supported in the subdomain ωξ contoured with cdef on fig 2. the surface cdef coincides with the forward front of the wave. by (13), in a neighborhood of the plane part ed of the front, the representation u fξ (x, t ) = (w t fξ)(x) = − ( ρ (x(γ, τ )) ρ0(γ, τ ) ) − 1 2 f (γ, t − ξ) θ1 (ξ − τ ) + w(x) θ2(ξ − τ ) (14) holds with a smooth w. correspondingly, the velocity of the wave has a jump at ed: u fξ t (x(γ, τ ), t ) ∣∣∣∣ τ =ξ+0 τ =ξ−0 = 0 − u fξ t (x(γ, ξ − 0)) = 〈see (13)〉 = − ( ρ (x(γ, ξ)) ρ0(γ, ξ) ) − 1 2 f (γ, t − ξ) , γ ∈ σ . (15) so, up to the factor −( ρ ρ0 ) − 1 2 , the shape of the velocity jump reproduces the shape of the control jump. 7 dual system a dynamical system vtt − ∆v − ∇ ln ρ · ∇v = 0, in r n + × (0, t ) (16) v|t=0 = 0, vt|t=0 = y, in r n + (17) vxn = 0 on γ × [0, t ], (18) is said to be dual to system (1)–(3); its solution v = vy(x, t) describes a wave initiated by a velocity perturbation y and propagating (in the reversed time) into rn+. the term ’dual’ is motivated by the following relation between solutions of these systems. lemma 3 for any square summable f and y compactly supported in γ×[0, t ] and rn+ respectively, the equality ∫ r n + u f (· , t ) y ρdx = ∫ γ×[0,t ] f v y ρ0dγdt (19) is valid. 24 m.i. belishev cubo 10, 2 (2008) proof let f and y be compactly supported and such that the solutions uf and vy are classical. the relations 0 = ∫ r n + ×[0,t ] [ u f tt − ∆uf − ∇ρ · ∇uf ] vy ρdx dt = ∫ r n + ×[0,t ] [ ρu f tt − div ρ∇u f ] v y ρdx dt = ∫ r n + [ u f t vy − uf v y t ] ∣∣∣∣ t=t t=0 ρdx + ∫ t 0 dt ∫ γ [ u f xn vy − uf v y xn ] ρ0dγ + ∫ r n + ×[0,t ] u f [v y tt − ∆v y − ∇ρ · ∇v y ] ρdx dt = 〈 see (2), (17) 〉 = − ∫ r n + uf (·, t ) y ρdx + ∫ γ×[0,t ] f vy ρ0dγdt imply (19). by the density of the chosen f ’s and y’s in the corresponding l2spaces, the passage to the limit in the proper sense leads to the assertion of the lemma. � taking in (17) y ∈ ht , introduce an observation operator ot : ht → ft , ot y := vy|σ×[0,t ]. for f ∈ ft , relation (19) can be written in the form (w t f, y) h t = (f, ot y) f t , which yields ot = ( w t ) ∗ (20) and clarifies the duality of the systems. 8 jumps in dual system here we consider the dual system provided with the specific cauchy data (17): the velocity perturbation y is assumed to be discontinuous. recall the notations: ωξ = {x ∈ rn+| dist (x, σ) < ξ}, bξ = {x ∈ rn+| x = x(γ, τ ), γ ∈ σ, τ ∈ (0, ξ)}, ξ > 0. fix ξ ∈ (0, t ) and take a function y ∈ c∞(ωt ) ⊂ ht ; denote by yξ := { y in ωξ 0 in ωt \ ωξ its cut-off function onto the subdomain ω ξ . in the generic case, the function yξ has a jump at a surface ∂ωξ ∩ rn+ (see cdef on fig 3). cubo 10, 2 (2008) dynamical inverse problem for the equation ... 25 t=0 t=t t=t-x figure 3: jumps in the dual system (16)-(18) return to the dual system and replace y by yξ in (17). the well-known fact is that discontinuous data produce discontinuous waves. namely, the jump of the data at cdef implies the jump of the wave velocity, the latter one being supported on the characteristic surfaces. in particular, these surfaces contain the plane parts abde and ednm consisting of the space-time rays (see the arrows) emanating from the set σξ × {t = t } (see ed). the amplitude of the jumps at these parts can be found by standard geometrical optics technique, i.e., by the use of the relevant analog of representation (12) for the solution vyξ . omitting the details, the result is as follows. for a point (x0, t0) ∈ abde such that x0 = x(γ, τ ), γ ∈ σ, 0 < τ < ξ and t0 = t − (τ − ξ), one has v yξ t (x0, t) ∣∣∣∣ t=t0+0 t=t0−0 = 1 2 ( ρ (x(γ, ξ)) ρ (x(γ, τ )) ) 1 2 y (x(γ, ξ)) . (21) the meaning of this formula is quite transparent: the jump of data at the point x(γ, ξ) initiates the jump of velocity, which propagates (in the reversed time) along the ray {(x(γ, τ ), t) | t = t − (τ − ξ), τ ∈ [0, ξ]}, its amplitude being proportional to the jump of data up to the factor 1 2 ( ρ ρ ) 1 2 . 9 further, at the moment t = t − ξ the jump reaches the boundary γ and is reflected back into r n + (see the rays constituting the part ablk). as result, a trace of the velocity v yξ t on σ × [0, t ] turns out to be discontinuous, its jump being supported at the cross-section σ × {t = t − ξ} (see ab). the amplitude of this jump can be found from (21): the equality v yξ t (γ, t) ∣∣∣∣ t=t −ξ+0 t=t −ξ−0 = ( ρ (x(γ, ξ)) ρ (x(γ, 0)) ) 1 2 y (x(γ, ξ)) 9the part ednm also supports the jump moving into rn + in opposite direction. this jump plays no role in the further considerations. 26 m.i. belishev cubo 10, 2 (2008) holds 10 . by the definition of the observation operator, this equality can be written as ( ∂ ∂t ot yξ ) (γ, t) ∣∣∣∣ t=t −ξ+0 t=t −ξ−0 = ( ρ (x(γ, ξ)) ρ0(γ) ) 1 2 y (x(γ, ξ)) . (22) at last, putting y = 1 (so that 1ξ is an indicator of the subdomain ω ξ ), we obtain the important auxiliary relation ( ρ (x(γ, ξ)) ρ0(γ) ) 1 2 = ( ∂ ∂t ot 1ξ ) (γ, t) ∣∣∣∣ t=t −ξ+0 t=t −ξ−0 , (γ, ξ) ∈ σ × (0, t ) . (23) completing the derivation of (11) in the next two sections, we show how to express the r.h.s. of (23) through the inverse data (operator r2t ). 9 controllability. wave basis fix ξ ∈ (0, t ). in control theory, the set uξ := ran w t = {uf (·, t )| f ∈ ft, ξ} of waves produced by controls acting from σ (the action time is ξ) is said to be reachable (at the moment t = ξ). since the wave propagation speed is equal to 1, the waves constituting u ξ are supported in the metric neighborhood ω ξ of σ; as result, the embedding uξ ⊂ hξ holds. the property of system (1)–(3), which plays the key role in solving inverse problems, is that for any σ and ξ this embedding is dense: clos uξ = hξ. control theory interprets this fact as a local approximate boundary controllability 11 of the system. roughly speaking, it means that the reachable set is rich enough: any function supported in the subdomain ω ξ filled with waves, can be approximated (in l2metric) by a wave uf (·, t ) ∈ uξ with the properly chosen control f ∈ ft, ξ. the proof of this property relays on the fundamental holmgren–john–tataru uniqueness theorem (see [1] for detail). let f̌ ξ = {f ξ k }∞ k=1 ⊂ f t, ξ be a linearly independent complete system of controls acting from σ; denote by ǔξ = {u ξ k }∞ k=1 ⊂ h ξ, u ξ k := uf ξ k (·, t ) = w t f ξ k a system of the corresponding waves. by the controllability, the latter system is also complete: clos span ǔξ = hξ. with a slight abuse of terms, we call ǔξ a wave basis. denote by p ξ n the orthogonal projection in ht onto span {u ξ k }n k=1. since the waves form a complete system, one has s-limn→∞ p ξ n = p ξ, where p ξ projects in ht onto hξ, i.e., cuts off functions supported in ω t onto ω ξ . recall that 1ξ denotes the indicator of ω ξ . representing 1ξ = p ξ 1t = lim n→∞ p ξ n 1t , p ξ n 1t = n∑ k=1 αn k u ξ k , (24) 10this equality can be also derived from (14), (15) by the use of duality relation (19). the factor 1 2 is doubled owing to the contribution of the reflected rays 11this motivates the name ”boundary control method” cubo 10, 2 (2008) dynamical inverse problem for the equation ... 27 one can find the coefficients α ξ n as follows. multiplying the last equality in (24) by u ξ i , one gets n∑ k=1 gik α n k = βi, i = 1, . . . , n , (25) where gik := (u ξ i , u ξ k ) h t = (w t f ξ i , w t f ξ k ) h t = 〈see (4)〉 = (ct f ξ i , f ξ k ) f t = ∫ σ×[t −ξ,t ] (ct f ξ i )(γ, t) f ξ k (γ, t) ρ0(γ)dγdt (26) and βi := (p ξ n 1t , u ξ i ) h t = (1t , p ξ n u ξ i ) h t = (1t , u ξ i ) h t = ∫ ωt uf ξ i (·, t ) ρdx = 〈see (2)〉 = ∫ ωt ρdx ∫ t 0 (t − t) u f ξ i tt (·, t ) dt = ∫ t 0 dt (t − t) ∫ ωt divρ∇uf ξ i (·, t ) dx = − ∫ γ×[0,t ] (t − t) u f ξ i xn (·, t ) ρ0dγdt = 〈see (3)〉 = − ∫ σ×[t −ξ,t ] (t − t) f ξ i (γ, t) ρ0dγdt = − (κ t , f ξ i ) f t (27) with κ t (γ, t) := t − t. in the course of integration by parts, the integral over ∂ωt ∩ rn+ vanishes because the wave uf ξ i (·, t ) is supported into the smaller subdomain ωξ. also, note that the linear independence of the system f̌ ξ and the injectivity of the of the operator w t provide the linear independence of the system ǔξ in hξ. by the latter, the gram matrix g is invertible and, hence, system (25) is uniquely solvable w.r.t. αn1 , . . . , α n n . 10 completing the proof. applying ot to the both sides of (24), with regard to u ξ k = w t f ξ k , we have o t 1ξ = lim n→∞ n∑ k=1 α n k o t w t f ξ k = 〈see (20)〉 = lim n→∞ n∑ k=1 α n k c t f ξ k . (28) at last, substituting (28) into (23), we arrive at (11). the amplitude formula is derived and theorem 1 is proven. � the obtained results can be presented in the form of a procedure recovering the density. if the values of ρ0|σ are known and the response operator r 2t is given (as result of measurements on the part σ of the earth surface), the external observer can determine the density ρ in the tube bt as follows: step 1 find the operator ct by (5). fix ξ ∈ (0, t ) and choose a complete system of controls f̌ ξ ⊂ ft, ξ. find the gram matrix {gik} and the numbers βi by (26), (27). 28 m.i. belishev cubo 10, 2 (2008) step 2 solve system (25) and find αn1 , . . . , α n n for n = 1, 2, . . . . fixing a γ ∈ σ, determine ρ (x(γ, ξ)) by the af (11). step 3 varying (γ, ξ) ∈ σ × (0, t ), recover ρ| b t . 11 visualization of waves the amplitude formula enables one to recover not only the density but the waves themselves. choose a control f ∈ ft providing the wave uf to be a classical solution of (1)–(3). for a fixed (γ, ξ) ∈ σ × (0, t ), the representation u f (x(γ, ξ), t ) = ( ρ (x(γ, ξ)) ρ0(γ) ) − 1 2 {[ ∂ ∂t lim n→∞ n∑ k=1 η n k c t f ξ k ] (γ, t) ∣∣∣∣ t=t −ξ+0 t=t −ξ−0 } (29) holds, where the coefficients ηn1 , . . . , η n n satisfy the system n∑ k=1 gik η n k = θi, i = 1, . . . , n with θi := (c t f, f ξ i ) f t . indeed, putting y = uf (·, t ) in (22), we have ( ρ (x(γ, ξ)) ρ0(γ) ) 1 2 uf (x(γ, ξ), t ) = [ ∂ ∂t ot ( uf (·, t ) ) ξ ] (γ, t) ∣∣∣∣ t=t −ξ+0 t=t −ξ−0 . (30) using the wave basis by analogy with (24), we can represent ( uf (·, t ) ) ξ = p ξuf (·, t ) = lim n→∞ p ξ n uf (·, t ) , p ξ n uf (·, t ) = n∑ k=1 ηn k u ξ k (31) and find the coefficients ηn1 , . . . , η n n from the gram system n∑ k=1 gikη n k = θi , i = 1, . . . , n with the r.h.s. θi := ( uf (·, t ), u ξ i ) h t = 〈see (4)〉 = (ct f, f ξ i ) f t . then, substituting ot ( uf (·, t ) ) ξ = 〈see (31)〉 = lim n→∞ n∑ k=1 ηn k ot u ξ k = lim n→∞ n∑ k=1 ηn k ct f ξ k cubo 10, 2 (2008) dynamical inverse problem for the equation ... 29 in (30), we arrive at (29). thus, the external observer with the knowledge of the response operator r2t can make the waves visible in the tube bt under the part σ of the earth surface. this is what we call a visualization (see [1]–[3]). 12 comments • as regards the numerical realization of the procedure outlined in section 10, the most problematic point is step 2, which consists of solving system (25) for large n ’s. namely, since ct is a compact operator, the condition number of the matrix {gik} grows as n → ∞, so that (25) turns out to be an ill-posed system. the second problem is that the passage to the limit and the differentiation w.r.t. to t in (11) do not commute. actually, the difficulties of this type are unavoidable: they reflect the well-known strong ill-posedness of multidimensional inverse problems. however, certain affirmative results in numerical testing do exist and show that elaboration of workable bc-algorithms is not a hopeless endeavor [2], [6]. • for numerical realization, an important issue is the proper choice of the system f̌ ξ. for many reasons, the controls producing the waves with sharp forward front are preferable. in the problem with neumann boundary controls (2), it looks reasonable to simulate a complete system by the set {f ξ qp } m,n p=1, q=1: f ξ qp (γ, t) = δ′ ε ( t − [ (t − ξ) + p ξ m + 1 ]) fq(γ) , where δ′ ε (t) is a relevant regularization of the first derivative of the dirac function δ(t), {fq} ∞ q=1 is an orthonormal basis in l2(σ), m and n are large integers. also, note that in applications one deals as a rule with a certain prescribed set of ’standard’ controls. therefore, it is important to develop the algorithms with controls simulating the real sources. • the amplitude formulae (11) and (29) can be easily extended to the case of a curved boundary γ. namely, assume that σ ⊂ γ and t > 0 are such that the field of the normal rays, which form the tube bt , is regular. then, the only correction required is to replace ρ0 by ρ0 j , where j = j(γ, τ ) is the jacobian of the passage from the semideodesic coordinates in bt σ to the cartesian coordinates (see [3]). 13 acknowledgements i would like to thank prof. c. cuevas for kind invitation to write this paper for the journal. i’m grateful to s.v.belisheva and i.v.kubyshkin for assistance in computer graphics. the work is supported by the rfbr grants no. 08-01-00511 and the project nsh-8336.2006.1. received: november 2007. revised: january 2008. 30 m.i. belishev cubo 10, 2 (2008) references [1] m.i. belishev, boundary control in reconstruction of manifolds and metrics (the bc-method). invers problems., 13 (1997), no 5, r1–r45. [2] m.i. belishev, v.yu. gotlib, dynamical variant of the bc-method: theory and numerical testing. journal of inverse and ill-posed problems, 7, no 3: 221–240, 1999. [3] m.i. belishev, how to see waves under the earth surface (the bc-method for geophysicists). ill-posed and inverse problems, s.i.kabanikhin and v.g.romanov (eds). vsp, 55–72, 2002. [4] m.i. belishev, recent progress in the boundary control method. invers problems., 23 (2007), no 5, r1–r67. [5] m. ikawa. hyperbolic pdes and wave phenomena, translations of mathematical monographs, v. 189 ams; providence. rhode island, 1997. [6] s.i. kabanikhin, m.a. shishlenin, a.d. satybaev, direct methods of solving inverse hyperbolic problems. utrecht, the netherlands, vsp, 2004. [7] i. lasiecka, j-l. lions, r. triggiani, non homogeneous boundary value problems for second order hyperbolic operators. j. math. pures appl, v. 65 (1986), no 3, 142–192. n2 cubo a mathematical journal vol.10, n o ¯ 01, (1–10). march 2008 prime factorization of entire functions xinhou hua and rémi vaillancourt1 department of mathematics and statistics university of ottawa, ottawa, on, k1n 6n5, canada email: hua@mathstat.uottawa.ca, remi@uottawa.ca abstract let n be a prime number and let f (z) be a transcendental entire function. then it is proved that both [f (z) + cz] n and [f (z) + cz] −n are uniquely factorizable for any complex number c, except for a countable set in c. resumen sea n un número primo y f (z) una función entera transcendental. entonces ambos [f (z) + cz] n y [f (z) + cz] −n se factorizan de manera única para cualquier número complejo c, excepto para un conjunto numerable en c. key words and phrases: entire function, unique factorization. math. subj. class.: 30d05. 1this research was partially supported by the natural sciences and engineering research council of canada and the centre de recherches mathématiques of the université de montréal. 2 xinhou hua and rémi vaillancourt cubo 10, 1 (2008) 1 introduction the fundamental theorem of elementary number theory states that every integer n ≥ 2 can be expressed uniquely as the product of primes in the form n = p m1 1 · · · p mk k , for k ≥ 1, with distinct prime factors p1, . . . , pk and corresponding exponents m1 ≥ 1, . . . , mk ≥ 1 uniquely determined by n. for example, 2700 = 2 2 3 3 5 2 . in 1922, ritt ([14]) generalized this theorem to polynomials. to state his result, we introduce the following concepts. let f (z) be a nonconstant meromorphic function. a decomposition f (z) = f (g(z)) = f ◦ g(z) (1) will be called a factorization of f (z) with f (z) and g(z) being the left and right factors of f (z), respectively, where f (z) is meromorphic and g(z) is entire (g(z) may be meromorphic when f (z) is rational) (see [2], [4], [19]). a function f (z) is said to be prime (pseudo-prime) if f (z) is nonlinear and every factorization of the form (1) implies that either f (z) is fractional linear or g(z) is linear (either f (z) is rational or g(z) is a polynomial). example 1 ez + z is prime. this is stated by rosenbloom [15] and proved by gross [3]. example 2 (cos z)eaz+b + p(z) is prime, where a (6= 0) and b are constants, and p(z) is a nonconstant polynomial. this was conjectured by gross–yang [5] and proved by hua [7]. suppose that a function f (z) has two prime factorizations f (z) = f1 ◦ · · · ◦ fm(z) = g1 ◦ · · · ◦ gn(z), i.e., fi (i = 1, . . . , m) and gj (j = 1, . . . , n) are prime functions. if m = n and if there exist linear functions lj (j = 1, . . . , n − 1) such that f1(z) = g1 ◦ l −1 1 , f2(z) = l1 ◦ g2 ◦ l −1 2 , . . . , fn(z) = l −1 n ◦ gn(z), then the two factorizations are called equivalent. if any two prime factorizations of f (z) are equivalent, then f (z) is called uniquely factorizable. in particular, for an entire function cubo 10, 1 (2008) prime factorization of entire functions 3 f (z), if any two prime entire factorizations of f (z) are equivalent, then f (z) is called uniquely factorizable in the entire sense. ritt [14] proved the following result. proposition 1 let p(z) be a nonlinear polynomial. if p(z) has two prime factorizations p(z) = p1 ◦ · · · ◦ pm(z) = q1 ◦ · · · ◦ qn(z), where pi (i = 1, . . . , m) and qj (j = 1, . . . , n) are polynomials, then m = n. moreover, one factorization can be changed to another one by a sequence of applications of any of the following three ways: 1. replace pi and pi+1 by pi ◦ l and l −1 ◦ pi+1, respectively; 2. alternate pi and pi+1 when both are chebychev polynomials; 3. replace zk and zsh(zk) by zsh(z)k and zk, respectively, where h(z) is a polynomial, and s and k are natural numbers. example 3 z10 + 1 = (z5 + 1) ◦ z2 = (z2 + 1) ◦ z5. however, ritt’s result cannot be extended to rational functions. example 4 z3 ◦ z 2−4 z−1 ◦ z 2+2 z+1 = z(z−8)3 (z+1)3 ◦ z3. this example was given by michael zieve (see [1]). for transcendental functions, the diverse cases are very complex. for example, e z can have infinitely many nonlinear factors. example 5 for any integer n, e z = z 2 ◦ z3 ◦ · · · ◦ zn ◦ ez/n!. the following example shows that transcendental entire functions can have non-equivalent prime factorizations (see [10]). example 6 z 2 ◦ ( ze z2 ) = ( ze 2z ) ◦ z2. of course, there are functions which are uniquely factorizable. the following example is given by urabe [17]. 4 xinhou hua and rémi vaillancourt cubo 10, 1 (2008) example 7 for any two nonconstant polynomials p(z) and q(z), (z + e p(ez ) ) ◦ (z + q(ez)) is uniquely factorizable. the following result, proved by hua [6], shows that, for a given function, we can construct uncountably many uniquely factorizable functions. proposition 2 let f (z) be a transcendental entire function and n ≥ 3 be a prime number. then both f (zn) − czn and (zn − c)f (zn) are uniquely factorizable for any complex number c except for a countable set. in this paper, we prove the following two results. theorem 1 let f (z) be a transcendental entire function and n ≥ 3 be a prime number. then [f (z) − cz]n is uniquely factorizable for any complex number c except for a countable set. theorem 2 let f (z) be a transcendental entire function and n ≥ 3 be a prime number. then [f (z) − cz]−n is uniquely factorizable for any complex number c except for a countable set. 2 some lemmas the following lemmas will be used in the proof of the theorems. lemma 1 ([4]) suppose that p(z) is a nonconstant polynomial and g(z) is entire. then p(g(z)) is periodic if and only if g(z) is periodic. lemma 2 ([11]) let f (z) be a transcendental entire function. then for any complex number c except for a countable set, f (z) − cz is prime. remark. so far, there is no example with countably infinite exceptions. in [13], it is proved that there is at most one exception for f (z) = g(e z ), where g(z) is an entire function satisfying max|z|=r |g(z)| ≤ e kr for a positive constant k. in [8] and [18], some other functions f (z) are studied. cubo 10, 1 (2008) prime factorization of entire functions 5 lemma 3 ([12]) let f (z) be a transcendental entire function. we denote by ν(a, f ) the least order of almost all zeros of f (z) − a, where “almost all” means all with possibly finite exceptions. then ∑ a6=∞ ( 1 − 1 ν(a, f ) ) ≤ 1. lemma 4 ([16]) let f (z) and g(z) be prime entire functions. assume that both f (z) and f (z) = f (g(z)) are non-periodic. then f (z) is uniquely factorizable if and only if f (z) is uniquely factorizable in the entire sense. lemma 5 let f (z) be a nonconstant meromorphic function. then f (z)−cz is non-periodic for any complex number c with at most one exception. proof of lemma 5. suppose there exist two different numbers c and d such that f (z) − cz and f (z) − dz are periodic with period u and v, respectively. then f ′(z) is periodic and f ′ (z + u) = f ′ (z) = f ′ (z + v). let w be the period of f ′ (z). then there exist two nonzero integers m and k such that u = mw and v = kw. this implies that u = m k v. hence f (z) − cz = f (z + ku) − c(z + ku) = f (z + mv) − c(z + ku) = f (z + mv) − d(z + mv) + d(z + mv) − c(z + ku) = f (z) − dz + d(z + mv) − c(z + ku) = f (z) − cz + dmv − cku. therefore dmv = cku, and so, d = c, which is a contradiction. 2 the following lemma is a simple version of the so-called borel unicity theorem which can be found in [2] and [4]. lemma 6 let h0(z), . . . , hn(z) be rational functions and let g1(z), . . . , gn(z) be nonconstant entire functions such that n ∑ j=1 hj (z)e gj (z) = h0(z). then h0 = 0. lemma 7 let f (z) be a transcendental entire function. then f (z) − cz 6= p (z)ef1(z) for all c ∈ c with at most one exception, where p (z) is a polynomial and f1(z) is a nonconstant entire function. 6 xinhou hua and rémi vaillancourt cubo 10, 1 (2008) proof of lemma 7. suppose to the contrary that there exist two different constants c and d, two polynomials p1(z) and p2(z), and two nonconstant entire functions f1(z) and f2(z) such that f (z) − cz = p1(z)e f1(z) and f (z) − dz = p2(z)e f2(z). then cz − dz = p2(z)e f2(z) − p1(z)e f1(z). by lemma 6, cz − dz = 0; thus d = c which is a contradiction. 2 3 proof of theorem 1 let f (z) = [f (z) − cz]n = zn ◦ (f (z) − cz). obviously, z n is non-periodic. let z(f ) = {f (z) : f ′′(z) = 0}. then z(f ) is a countable set, and for any c 6∈ z(f ), f ′(z) − c has only simple zeros ([9, theorem f]). we combine z(f ) and all the exceptions (if any) in lemmas 1, 2, 5 and 7 to form an exceptional set e. then e is a countable set which may be empty. for any c ∈ c − e, we have the following properties: (p1) the function f (z) is non-periodic; (p2) the function f (z) − cz is prime; (p3) f ′ (z) − c has only simple zeros. (p4) f (z)−cz 6= p (z)ef1(z) for any polynomial p (z) and nonconstant entire function f1(z). next we assume c ∈ c − e. by lemma 4, we need only prove that f (z) is uniquely factorizable in the entire sense, which means, we just need to consider entire factors. assume that f (z) = g(z) ◦ h(z), (2) where g(z) and h(z) are nonconstant entire functions. we consider three cases. cubo 10, 1 (2008) prime factorization of entire functions 7 case 1. g(z) has at least two zeros, z1 and z2, of order m1 and m2, respectively, such that (n, m1) = (n, m2) = 1, that is, n and mi (i = 1, 2) have no common factors other than 1. then by (2) and the fact that n is prime, the order of any zero of h(z) − zi (i = 1, 2) should be a multiple of n. hence ν(zi, h) ≥ n ≥ 3 (i = 1, 2), which implies that ∑ a6=∞ ( 1 − 1 ν(a, f ) ) ≥ 1 − 1 3 + 1 − 1 3 > 1. this is a contradiction to lemma 3. case 2. g(z) has one zero, z0, of order m such that (n, m) = 1. then by (2) and the fact that n is prime, g(z) and h(z) can be written as g(z) = (z − z0) r g1(z) n , h(z) = z0 + h1(z) n , r = m (mod n), (3) where g1(z) and h1(z) are entire functions. obviously, 1 ≤ r < n. substituting (3) into (2) we have f (z) = h1(z) rn [g1(z0 + h1(z) n )] n , which implies that f (z) − cz = uh1(z) r g1(z0 + h1(z) n ) = [uz r g1(z0 + z n )] ◦ h1(z), (4) where u is an n-th root of unity. since f (z) − cz is prime, we have two subcases as follows. case 2.1. since the left factor uzrg1(z0 + z n ) is linear, then r = 1 and g1 is a constant. it follows from (3) that g(z) is linear. this is a trivial case. case 2.2. the right factor h1(z) is linear. let h1(z) = az + b (a, b ∈ c, a 6= 0). by (4), f (z) − cz = u(az + b)rg1[z0 + (az + b) n )]. (5) if g1(z) has a zero, then by differentiating (5) we see that f ′ (z) − c has a zero of order n − 1 ≥ 2, which is a multiple zero of f ′(z) − c. this contradicts (p3). therefore g1(z) has no zero. this implies that there exists a nonconstant entire function g2(z) such that g1(z) = e g2(z). by (5), f (z) − cz = u(az + b)reg2[z0+(az+b) n)] , which contradicts (p4). 8 xinhou hua and rémi vaillancourt cubo 10, 1 (2008) case 3. the order of any zero of g(z) is a multiple of n. then there exists an entire function g2(z) such that g(z) = g2(z) n . (6) it follows from (2) that [f (z) − cz]n = [g2 ◦ h(z)] n , and so, f (z) − cz = ug2(z) ◦ h(z) for an n-th root of unity, u. since f (z) − cz is prime, we have two subcases. case 3.1. the left factor ug2(z) is linear. it follows from (6) that g(z) = z n ◦ l(z) for a linear function l(z). therefore we get an equivalent factorization. case 3.2. the right factor h(z) is linear. this is a trivial case. the proof is complete. 2 4 proof of theorem 2 assume that [f (z) − cz]−n = g(z) ◦ h(z), where g(z) is a nonconstant meromorphic function and h(z) is a nonconstant entire function. then we have [f (z) − cz]n = 1 g(z) ◦ h(z). now, since the left-hand side is entire, the conclusion follows from lemma 4 and theorem 1. 2 5 open questions question 1 can n be 2 in theorems 1 and 2? question 2 what kind of rational functions are uniquely factorizable? question 3 is (z + ee z ) ◦ (z + ee z ) uniquely factorizable? received: november 2005. revised: january 2006. cubo 10, 1 (2008) prime factorization of entire functions 9 references [1] w. bergweiler, an example concerning factorization of rational functions (with correction), exposition. math., 11 (1993) 281–283. 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(edited by c. c. yang), marcel dekker, 1982. cubohua2911.pdf clanek&_nbsp_20.dvi cubo a mathematical journal vol.12, no¯ 01, (95–102). march 2010 an identity related to derivations of standard operator algebras and semisimple h∗-algebras1 irena kosi-ulbl faculty of mechanical engineering, university of maribor, smetanova 17, maribor, slovenia email : irena.kosi@uni-mb.si and joso vukman department of mathematics and computer science, faculty of natural sciences and mathematics, university of maribor, koroška 160, maribor, slovenia email : joso.vukman@uni-mb.si abstract in this paper we prove the following result. let x be a real or complex banach space, let l(x) be the algebra of all bounded linear operators on x, and let a(x) ⊂ l(x) be a standard operator algebra. suppose d : a(x) → l(x) is a linear mapping satisfying the relation d(an) = n ∑ j=1 a n−j d(a)aj−1 for all a ∈ a(x). in this case d is of the form d(a) = ab − ba, for all a ∈ a(x) and some b ∈ l(x), which means that d is a linear derivation. in particular, d is continuous. we apply this result, which generalizes a classical result of chernoff, to semisimple h∗−algebras. this research has been motivated by the work of herstein [4], chernoff [2] and molnár [5] and is a continuation of our recent work [8] and [9] .throughout, r will represent an associative ring. given an integer n ≥ 2, a ring r is said to be n−torsion free, if for x ∈ r, nx = 0 1this research has been supported by the research council of slovenia 96 irena kosi-ulbl and joso vukman cubo 12, 1 (2010) implies x = 0. recall that a ring r is prime if for a, b ∈ r, arb = (0) implies that either a = 0 or b = 0, and is semiprime in case ara = (0) implies a = 0. let a be an algebra over the real or complex field and let b be a subalgebra of a. a linear mapping d : b → a is called a linear derivation in case d(xy) = d(x)y + xd(y) holds for all pairs x, y ∈ r. in case we have a ring r an additive mapping d : r → r is called a derivation if d(xy) = d(x)y + xd(y) holds for all pairs x, y ∈ r and is called a jordan derivation in case d(x2) = d(x)x + xd(x) is fulfilled for all x ∈ r. a derivation d is inner in case there exists a ∈ r, such that d(x) = ax − xa holds for all x ∈ r. every derivation is a jordan derivation. the converse is in general not true. a classical result of herstein [4] asserts that any jordan derivation on a prime ring of characteristic different from two is a derivation. cusack [3] generalized herstein’s result to 2−torsion free semiprime rings. let us recall that a semisimple h∗−algebra is a semisimple banach ∗−algebra whose norm is a hilbert space norm such that (x, yz∗) = (xz, y) = (z, x∗y) is fulfilled for all x, y, z ∈ a (see [1]). let x be a real or complex banach space and let l(x) and f (x) denote the algebra of all bounded linear operators on x and the ideal of all finite rank operators in l(x), respectively. an algebra a(x) ⊂ l(x) is said to be standard in case f (x) ⊂ a(x). let us point out that any standard algebra is prime, which is a consequence of hahn-banach theorem. resumen en este artículo nosotros provamos el seguiente resultado. sea x un espacio de banach real o complejo, sea l(x) a algebra de todos los operadores linares acotados sobre x, y sea a(x) ⊂ l(x) una algebra de operadores estandar. suponga d : a(x) −→ l(x) una aplicación lineal verificando la relación d(an) = n ∑ j=1 a n−j d(a)aj−1 para todo a ∈ a(x). en este caso d es de la forma d(a) = ab − ba, para todo a ∈ a(x) y algún b ∈ l(x), lo que significa que d es una deriviación lineal. en particual, d es continua. nosotros aplicamos este resultado el cual generaliza un resultado clásico de chernoff, para h∗-algebras semisimple. este trabajo fué motivado por un trabajo de herstein [4], chernoff [2] y molnár [5] y este una continuación de nuestro reciente trabajo [8] y [9]. key words and phrases: prime ring, semiprime ring, banach space, standard operator algebra, h∗–algebra, derivation, jordan derivation. math. subj. class.: 46k15, 46h99, 13n15. let us start with the following result proved by chernoff [2] (see also [6] and [8]). theorem a. let x be a real or complex banach space and let a(x) be a standard operator algebra on x. let d : a(x) → l(x) be a linear derivation. in this case d is of the form d(a) = ab − ba, for all a ∈ a(x) and some b ∈ l(x). in particular, d is continuous. it is our aim in this paper to prove the following result which generalizes theorem a. theorem 1. let x be a real or complex banach space and let a(x) be a standard operator cubo 12, 1 (2010) an identity related to derivations of standard operator ... 97 algebra on x. suppose d : a(x) → l(x) is a linear mapping satisfying the relation d(an) = n ∑ j=1 an−j d(a)aj−1. for all a ∈ a(x). in this case d is of the form d(a) = ab − ba, for all a ∈ a(x) and some b ∈ l(x), which means that d is a linear derivation. in particular, d is continuous. proof. we have the relation d(an) = n ∑ j=1 an−j d(a)aj−1. (1) let a be from f (x) and let p ∈ f (x), be a projection with ap = p a = a. from the above relation one obtains d(p ) = p d(p ) + (n − 2) p d(p )p + d(p )p. (2) right multiplication of the relation (2) by p gives p d(p )p = 0. (3) putting a + p for a in the relation (1), we obtain n ∑ i=0 ( n i ) d ( a n−i p i ) = ( n−1 ∑ i=0 ( n − 1 i ) a n−1−i p i ) d(a + p )+ ( n−2 ∑ i=0 ( n − 2 i ) a n−2−i p i ) d(a + p )(a + p )+ ( n−3 ∑ i=0 ( n − 3 i ) a n−3−i p i ) d(a + p )(a + p )2 + · · · + (a + p )2d(a + p ) ( n−3 ∑ i=0 ( n − 3 i ) a n−3−i p i ) + (a + p )d(a + p ) ( n−2 ∑ i=0 ( n − 2 i ) a n−2−i p i ) + d(a + p ) ( n−1 ∑ i=0 ( n − 1 i ) a n−1−i p i ) . (4) using (1) and rearranging the equation (4) in sense of collecting together terms involving equal number of factors of p we obtain: 98 irena kosi-ulbl and joso vukman cubo 12, 1 (2010) n−1 ∑ i=1 fi (a, p ) = 0, where fi (a, p ) stands for the expression of terms involving i factors of p. replacing a by a + 2p, a + 3p, ..., a + (n − 1) p in turn in the equation (1), and expressing the resulting system of n − 1 homogeneous equations of variables fi (a, p ), i = 1, 2, ..., n − 1, we see that the coefficient matrix of the system is a van der monde matrix        1 1 · · · 1 2 22 · · · 2n−1 ... ... ... ... n − 1 (n − 1) 2 · · · (n − 1) n−1        . since the determinant of the matrix is different from zero, it follows that the system has only a trivial solution. in particular, fn−2 (a, p ) = n (n − 1) d ( a 2 ) − (n − 1) (n − 2) ( a 2 d(p ) + d(p )a2 ) − ((n − 2) (n − 3) + (n − 3) (n − 4) + · · · + 3 · 2 + 2 · 1) ( a2d(p )p + p d(p )a2 ) − 2 ((n − 2) + (n − 3) + (n − 4) + · · · + 3 + 2 + 1) (ad (a) p + p d (a) a) − 4 (1 · (n − 2) + 2 · (n − 3) + 3 · (n − 4) + · · · + (n − 3) · 2 + (n − 2) · 1) ad(p )a− 2 (n − 1) (ad (a) + d (a) a) = 0, and fn−1 (a, p ) = nd (a) − (p d (a) + d (a) p ) − (n − 1) (ad(p ) + d(p )a) − ((n − 2) + (n − 3) + (n − 4) + · · · + 2 + 1) (ad(p )p + p d(p )a) − (n − 2) p d (a) p = 0. the above equations reduce to n (n − 1) d ( a 2 ) = (n − 1) (n − 2) ( a 2 d(p ) + d(p )a2 ) + 1 3 (n − 3) (n − 2) (n − 1) ( a2d(p )p + p d(p )a2 ) + (n − 2) (n − 1) (ad (a) p + p d (a) a) + 4 (1 · (n − 2) + 2 · (n − 3) + 3 · (n − 4) + · · · + (n − 3) · 2 + (n − 2) · 1) ad(p )a+ 2 (n − 1) (ad (a) + d (a) a) , (5) cubo 12, 1 (2010) an identity related to derivations of standard operator ... 99 and 2nd (a) = 2 (p d (a) + d (a) p ) + 2 (n − 1) (ad(p ) + d(p )a) + (n − 2) (n − 1) (ad(p )p + p d(p )a) + 2 (n − 2) p d (a) p, (6) respectively. multiplying the relation (3) from both sides by a we obtain ad(p )a = 0, (7) which reduces the relation (5) to n (n − 1) d ( a2 ) = (n − 1) (n − 2) ( a2d(p ) + d(p )a2 ) + 1 3 (n − 3) (n − 2) (n − 1) ( a 2 d(p )p + p d(p )a2 ) + (n − 2) (n − 1) (ad (a) p + p d (a) a) + 2 (n − 1) (ad (a) + d (a) a) . (8) applying the relation (3) and the fact that ap = p a = a, we have p d(p )a = (p d(p )p )a = 0. similarly one obtains that ad(p )p = 0. the relations (8) and (6) can now be written as nd ( a2 ) = (n − 2) ( a2d(p ) + d(p )a2 ) + (n − 2) (ad (a) p + p d (a) a) + 2 (ad (a) + d (a) a) , (9) and nd (a) = p d (a) + d (a) p + (n − 1) (ad(p ) + d(p )a) + (n − 2) p d (a) p = 0, (10) respectively. right multiplication of the relation (10) by p gives d(a)p = d(p )a + p d(a)p. (11) similarly one obtains p d(a) = ad(p ) + p d(a)p. (12) multiplying the relation (11) from the right side and the relation (12) from the left side by a, we obtain d(a)a = d(p )a2 + p d(a)a, (13) and ad(a) = a2d(p ) + ad(a)p. (14) combining relations (9), (13) and (14) we obtain nd ( a2 ) = (n − 2) ( d(p )a2 + p d (a) a ) + (n − 2) ( a2d(p ) + ad (a) p ) + 2 (ad (a) + d (a) a) = (n − 2) (ad (a) + d (a) a) + 2 (ad (a) + d (a) a) . 100 irena kosi-ulbl and joso vukman cubo 12, 1 (2010) we have therefore d(a2) = d(a)a + ad(a) (15) for any a ∈ f (x). from the relation (10) one can conclude that d(a) ∈ f (x) for any a ∈ f (x). we have therefore a jordan derivation on f (x). since f (x) is prime it follows that d is a derivation by herstein’s theorem. applying theorem a one can conclude that d is of the form d(a) = ab − ba, (16) for all a ∈ a(x) and some b ∈ l(x). it remains to prove that the relation (16) holds on a(x) as well. let us introduce d1 : a(x) → l(x) by d1(a) = ab − ba and consider d0 = d − d1. the mapping d0 is, obviously, linear and satisfies the relation (1). besides, d0 vanishes on f (x). it is our aim to prove that d0 vanishes on a(x) as well. let a ∈ a(x), let p be an one-dimensional projection and s = a + p ap − (ap + p a). we have d0(s) = d0(a). and sp = p s = 0. we have d0(a n) = n ∑ j=1 an−j d0(a)a j−1 (17) for all a ∈ a(x). applying the above relation we obtain n ∑ j=1 sn−j d0(s)s j−1 = d0(s n) = d0(s n + p ) = d0((s + p ) n) = n ∑ j=1 (s + p )n−j d0(s + p )(s + p ) j−1 = n ∑ j=1 (s + p )n−j d0(a)(s + p ) j−1 = n ∑ j=1 (sn−j + p )d0(s)(s j−1 + p ) = n ∑ j=1 sn−j d0(a)s j−1+ n ∑ j=1 p d0(a)s j−1 + n ∑ j=1 s n−j d0(a)p + p d0(a)p. we have therefore n ∑ j=1 p d0(a)s j−1 + n ∑ j=1 sn−jd0(a)p + p d0(a)p = 0. (18) multiplying the above relation from both sides by p we obtain p d0(a)p = 0, (19) which reduces the relation (18) to n ∑ j=1 p d0(a)s j−1 + n ∑ j=1 s n−j d0(a)p = 0. (20) cubo 12, 1 (2010) an identity related to derivations of standard operator ... 101 right multiplication of the above relation by p gives n ∑ j=1 sn−jd0(a)p = 0. (21) let us prove that n−1 ∑ j=1 kj s n−1−jd0(a)p = 0 (22) holds where kj = 2 n−1−j − 2n−1, j = 1, 2, ..., n − 1. putting in the relation (21) 2a for a we obtain n ∑ j=1 2n−jsn−j d0(a)p = 0. multiplying the relation (21) by 2n−1 and subtracting the relation so obtained from the above relation we obtain the relation (22). since the relation (21) implies the relation (22) one can conclude by induction that d0(a)p = 0. since p is an arbitrary one-dimensional projection, it follows that d0(a) = 0, for any a ∈ a(x), which completes the proof of the theorem. let us point out that in case n = 3 theorem 1 reduces to theorem in [9]. theorem 2. let a be a semisimple h∗−algebra and let d : r → r be a linear mapping satisfying the relation d(xn) = n ∑ j=1 xn−j d(x)xj−1 for all x ∈ r. in this case d is a linear derivation. proof. the proof goes through using the same arguments as in the proof of theorem in [5] with the exception that one has to use theorem 1 instead of lemma in [5]. since in the formulation of the results presented in this paper we have used only algebraic concepts, it would be interesting to study the problem in a purely ring theoretical context. we conclude with the following conjecture. conjecture. let r be a semiprime ring with suitable torsion restrictions and let d : r → r be an additive mapping satisfying the relation d(xn) = n ∑ j=1 xn−j d(x)xj−1 for all x ∈ r. in this case d is a derivation. in case r has the identity element the conjecture above was proved in [8]. since semisimple h∗−algebras are semiprime, theorem 2 proves the conjecture above in a special case. received: june, 2008. revised: october, 2009. 102 irena kosi-ulbl and joso vukman cubo 12, 1 (2010) references [1] ambrose, w., structure theorems for a special class of banach algebras, trans. amer. math. soc., 57 (1945), 364–386. [2] chernoff, p.r., representations, automorphisms, and derivations of some operator algebras, j. funct. anal., 2 (1973), 275–289. [3] cusack, j., jordan derivations on rings, proc. amer. math. soc., 53 (1975), 321–324. [4] herstein, i.n., jordan derivations of prime rings, proc. amer. math. soc., 8 (1957), 1104– 1119. [5] molnár, l., on centralizers of an h∗−algebra, publ. math. debrecen, 46, 1–2 (1995), 89–95. [6] šemrl, p., ring derivations on standard operator algebras, j. funct. anal., 112 (1993), 318–324. [7] vukman, j., on automorphisms and derivations of operator algebras, glasnik mat., 19 (1984), 135–138. [8] vukman, j. and kosi-ulbl, i., a note on derivations in semiprime rings, internat. j. math. &math. sci., 20 (2005), 3347–3350. [9] vukman, j., on derivations of standard operator algebras and semisimple h∗−algebras, studia sci. math. hungar., 44 (2007), 57–63. cubo a mathematical journal vol.11, n o ¯ 02, (15–36). may 2009 optimal effort in heterogeneous agents population with global and local interactions arianna dal forno and ugo merlone statistics and applied mathematics department, university of torino, corso unione sovietica 218/bis, turin, i-10134, italy emails: dalforno@econ.unito.it, merlone@econ.unito.it abstract a game where agents interact in small teams is proposed; the interaction is examined when the population consists of different types of agent and a reward mechanism devised to increase competition is introduced. we prove that such a mechanism may expand the set of nash equilibria and, in particular, reduce the production level of some agents. finally, we extend our results to heterogeneous populations by means of agents based modeling. this way we can study the dynamics of adjustment of agents response and extend our results when considering local interaction and a egocentric knowledge of the population composition. resumen se propone un juego en el cual agentes interactuan en un pequeño grupo, la interacción es examinada cuando la población contituye diversos tipos de agentes y se introduce un mecanismo de recompenza para aumentar la competición. se demuestra que tal mecanismo puede expandir el conjunto de equilibrio de nash y, en particular, reduce el nivel 16 arianna dal forno and ugo merlone cubo 11, 2 (2009) de produción de algunos agentes. finalmente, extendemos nuestros resultados a poblaciones heteregeneas mediante un modelo de agente artificial. de esta manera es posible estudiar la dinámica de ajustamiento de respuesta de agentes y estender los resultados considerando interacción local y un conocimiento egocentrico de la composición de la población. key words and phrases: bounded rationality, mathematical organization theory, public goods, heterogeneous agents, agent based simulation. ams classification: 91a80,91b18,91a26,91d10. 1 introduction the problem of incentives and compensation is cardinal in modern economic literature; many papers address this problem considering moral hazard in agency relationships. other papers address the problem of optimal form of hierarchy in firms and give interesting results (for a survey, see e.g. [6] and [10]). the approach we use is different; we provide a model of firm which is very simple, and do not consider hierarchy explicitly. the model of firm we introduce may be interpreted as an interaction game [8] with agents playing bounded rationality strategies. this approach may fail to take into account some relevant phenomena but allows us to shed light on aspects that are usually neglected such as the interpretation of equilibria as corporate culture. we consider a population of 2n agents randomly paired in teams 1 ; each member supplies a non-observable individual effort in order to produce a good. members are rewarded according to their joint production, yet each agent bears its own private cost in providing effort. in a first analysis we limit our study to symmetric agents; when considering only rational agents this game has one single nash equilibrium. we are interested in outcomes which are less predictable than nash equilibria and, in particular, we consider a profile of strategies which is not only ideal from the firm’s prospective but also maximizes the workers’ welfare. to achieve this particular profile we introduce a class of agents which is able to commit to a coordinated effort even if they have incentive to shirk; the effort provided by these agents is the maximum feasible one and any other effort cannot be greater than this one. by contrast, we assume that some of our agents may not be fully aware of the set of alternatives from which they have to choose, or may have not the skills necessary to make whatever complicated calculations are needed to discover its optimal course of action, or, finally, do not clearly perceive the action-consequence relationship especially when they face uncertainty. in other words, we assume bounded rationality of some agents and consider their effort fixed to some level determined exogenously. 1it must be noted that teams we consider differ from the ones considered in [7] cubo 11, 2 (2009) optimal effort in heterogeneous ... 17 obviously, a population consisting of heterogeneous agents may affect behavior of rational agents and, consequently, the equilibria of the game. the results we provide are interesting since we find, for example, that the presence of low fixed effort individuals induces rational agents to work harder. the relative composition of the population is used to study the equilibria with heterogeneous agents and these equilibria are compared to the ones in homogeneous population. in [1] it is argued that a thorough understanding of internal incentives is critical to develop a viable theory of the firm. furthermore, in the literature individual vs group incentives plans have been compared. some of the drawbacks of individual incentives are well known in the literature (see for instance [12]). in this paper we do not approach these problems from the classical point of view of incentive and contracts literature (for a first introduction see [6]), rather we propose a simple ranking policy clearly showing some of these drawbacks. this way even a simple model, like the one we propose, encompasses some of the crucial points in the comparison between individual and group incentives plans. in particular, we rank agents according to their profit and, even if this should increase competition between agents, it just expands the equilibrium set of the game including only equilibria dominated by the nash equilibrium. furthermore, we show that this policy, in some cases, reverses the effect of fixed low effort individuals on rational agents in terms of optimal effort. performing simulation with artificial agents allows us to extend the theoretical results in two more directions. first, we can consider heterogeneous populations and observe the best reply dynamical adjustments when the population compositions vary. second, we can consider the effects of local interaction when the agents have no longer the knowledge about the global composition of the population, but they can observe only the composition of their neighborhood in order to make their decisions. the paper is structured in the following way: in section 2 we present the model of the game and some results about equilibria in the symmetric case; in section 3 we discuss how the proposed game may be seen as a model of a firm; in section 4 we introduce bounded rationality agents and some results about the case with the population partitioned in different classes of agents. section 5 defines the ranking policy and studies how this affects the equilibrium set and the performance of rational agents. section 6 provides a discussion of the results when the theoretical model is extended by artificial simulation. finally, in the last section we summarize our findings and provide possible interesting directions in further research. 2 the model a population of 2n agents is randomly partitioned, following a uniform distribution, in n couples of two players; the partition is not revealed to the agents. as a consequence, no agent knows who his/her mate is; this avoids some of the problems related to repeated games and simplify the analysis. in this one-shot game each team is supposed to produce a good; the final production is the only verifiable variable and depends on the vector of the non-observable efforts each agent 18 arianna dal forno and ugo merlone cubo 11, 2 (2009) supplies: this is a joint production model (for a survey, see e.g. [6]). agent i ∈ i = {1, 2, . . . , 2n} receives a monetary payoff for the final production of its team and supplies an effort ei which implies some cost to him/her; we assume ei ∈ e ⊆ r + , with e compact, convex and nonempty. for sake of simplicity we consider only the utility of the payoff each agent receives: if agent i is paired with agent j and the vector of their efforts is (ei, ej), their utility will be f (ei + ej); each agent’s disutility of effort is c (e). furthermore, we assume: • f : r+ → r • c : r+ → r • f, c ∈ c2 • f (0) = c (0) = 0 • f ′ > 0, c′ > 0 • f ′ (0) > c′ (0) • f ′′ < 0, c′′ > 0 when agents i and j are the members of the same team, their payoffs are respectively:    πi (ei, ej) = f (ei + ej) − c (ei) πj (ei, ej ) = f (ei + ej ) − c (ej ) (1) and they are not transferable. since the partition is random, agent i’s profit is a random variable πi depending also on the effort its randomly paired mate exerts. the expected value of its profits is e [πi (e,x)] = 1 2n − 1 n ∑ k=1 πi (ei, ek) , k 6= i where x is the random variable determining the partition and e = (ei, e−i). 2.1 equilibria of the game in some particular cases since for all players the set of strategies e is a compact, convex and nonempty subset of r + , and the payoff functions are continuous and concave w.r.t. the strategies, the existence of at least one equilibrium is guaranteed. we remark that the production function is symmetrical and agents have the same cost function. in this section we limit our study to the case in which the agents choose the same effort, i.e., we consider symmetric equilibria. therefore, the expected profit of each agent is actually (1). in cubo 11, 2 (2009) optimal effort in heterogeneous ... 19 addition, if we assume that players can commit to an effort, they decide the optimal effort e c solving the following problem: max e f (2e) − c (e) , e ∈ e ⊂ r+ where e is the set of feasible efforts. we call this profile of strategies 2 the coordination equilibrium ( e c , e c ) . obviously, this is not a nash equilibrium, in fact: proposition 1 the coordination equilibrium is pareto optimal, but even if the two team-mates decide to coordinate to this equilibrium, there is an incentive to shirk. furthermore, this profile of strategies maximizes the social welfare. proof: the payoff function is strictly concave so focs are sufficient to determine the interior coordination equilibrium. the coordination equilibrium is therefore characterized by: 2f ′ ( 2e c ) − c′ ( e c ) = 0. (2) let us consider the partial derivative of agent i ′ s profit with respect to its effort when the coordination equilibrium is played: ∂ ∂ei ( πi ( ei, ej = e c )) |ei=ec = f ′ ( 2e c ) − c′ ( e c ) < 2f ′ ( 2e c ) − c′ ( e c ) = 0. this partial derivative is negative: πi decreases with respect to ei. therefore, choosing a lower effort close enough to e c , player i has a larger profit. the second part is obvious since this profile of strategies is also the solution of max ei,ej 2f (ei + ej ) − c (ei) − c (ej ) , ei, ej ∈ e ⊂ r + . � to obtain this equilibrium we need something more, something that makes each player commit to provide effort e c ; for example, it could be achieved by signing an enforceable contract or assuming repetition of this situation (for a deep analysis refer to [5]). we assume the existence of such agents 3 and consider this profile of strategies for the following reasons. first of all, from the social point of view it would be desirable to have this situation; then, it is an important yardstick to which compare the other suboptimal equilibria; finally, in this situation the agents exert the maximal effort and this is one of the goals of the firm. if we do not assume the existence of such ideal agents, this game has a unique nash equilibrium ( e n , e n ) and it holds: 2each symmetric profile of strategies can, at least theoretically, be coordinate between players; nevertheless, we reserve this name to this particular one. 3with this assumption this profile of strategies constitutes an equilibrium in the sense that, these committed agents have no incentive to deviate. 20 arianna dal forno and ugo merlone cubo 11, 2 (2009) theorem 1 the effort exerted in the nash equilibrium is lower than the effort exerted in the coordination equilibrium, i.e., en < ec . proof: the nash equilibrium can be found solving      max ei f (ei + ej ) − c (ei) , max ej f (ei + ej ) − c (ej ) , ei, ej ∈ e and is characterized by the following foc: f ′ ( 2e n ) = c ′ ( e n ) . (3) by contradiction let e c ≤ en ; since f is concave and c is convex, it follows:    c ′ ( e c ) ≤ c′ ( e n ) , f ′ ( 2e c ) ≥ f ′ ( 2e n ) , by condition (2) on the coordination equilibrium and (3), this means    c ′ ( e c ) ≤ c′ ( e n ) c ′ ( e c ) = 2f ′ ( 2e c ) > f ′ ( 2e c ) ≥ f ′ ( 2e n ) = c ′ ( e n ) , clearly absurd. � obviously, this game is a public goods game [4]. 3 the game as a model of firms we propose this model to study the possible equilibria in a complex structure such as a firm; in fact, we can think of these equilibria as corporate culture (see e.g. [5]). corporate culture may be defined as the basic assumptions and beliefs that are shared by the members of a group or organization and that are used as norm. the organization’s problem is to identify a rule that allows relatively efficient transactions to take place and devise some way to communicate that rule to all current and potential trading partners. we do not expect that all the equilibria of the game we consider may represent efficient equilibria. the organization has an interest in preserving and promoting a good reputation to allow for future beneficial transactions; nevertheless, it is quite common to observe situations where there is cubo 11, 2 (2009) optimal effort in heterogeneous ... 21 no interest in reputation and members’ main interest is free riding. this situation is widespread in italian public sector offices and often it is claimed that a higher level of competition could avoid such misbehavior. the ranking procedure we introduce in section 5 may be compared to some individual merit compensation incentives used by organizations. although individual incentive systems often lead to improved performance, these programs may, at times, lead to employees competing with one another, with undesirable results. finally, it must be observed that the particular ranking policy we consider may be unrealistic since the principal usually cannot observe private costs of its agents; nevertheless there are always costs to take into account and our model can be easily modified to describe situations where the agents are provided with a fixed resource and have to allocate it efficiently. furthermore, very often there is a social mechanism which tends to reward people who obtain just a higher profit than the others’, without necessarily maximizing it. 4 some results in heterogeneous populations even if this model is very simple we assume that not all agents may: • be fully aware of the set of alternatives from which they have to choose • have the skill necessary to make whatever complicated calculations are needed to discover their optimal course of action • do clearly perceive the action-consequence relationship especially when facing uncertainty. for an analysis of some other motives that may conflict with the rational man paradigm the reader may refer to [11]. we are interested in considering simple “bounded rationality” strategies and how they affect the equilibria of the game with heterogeneous agents. in particular, we consider agents who stubbornly provide a fixed effort regardless of their results. the rationales of such a behavior may be different: for example, an agent, given the difficulties to find an optimal effort, may resolve to providing a fixed effort which it consider appropriate for the situation. consider then three classes of agents depending on their behaviors: • bounded rationality agents playing a fixed effort ē ∈ [ 0, e c ] • committed agents who play the optimal effort knowing the fact that all the agents of this class play the same coordinated effort • rational agents who play the optimal effort and do not commit to any coordinated effort. in the following, we assume that the type of agent is private information while the composition of the population is common knowledge. let e (ē) be the optimal effort an agent, either committed 22 arianna dal forno and ugo merlone cubo 11, 2 (2009) or rational, exerts when its team-mate is a bounded rationality agent playing effort ē. then e (ē) is the solution of the following problem: max e f (e + ē) − c (e) . proposition 2 consider a team consisting of a rational (committed) agent and a fixed effort agent. the higher is the fixed effort provided, the lower is the optimal effort of the rational (committed) agent: ē1 < ē2 =⇒ e (ē1) > e (ē2) . furthermore, a rational (committed) agent will always provide an effort lower than ec . proof: by contradiction let e (ē1) ≤ e (ē2); this means e (ē1) + ē1 < e (ē2) + ē2. since f ′ is decreasing: c ′ (e (ē1)) = f ′ (e (ē1) + ē1) > f ′ (e (ē2) + ē2) = c ′ (e (ē2)) absurd since e (ē1) ≤ e (ē2) =⇒ c ′ (e (ē1)) < c ′ (e (ē2)). for the second part we know by the first part of this proposition that e (ē) will be maximum when the fixed effort is null, so it is sufficient to prove that when ē = 0 then e (0) < e c ; e (0) is characterized by the foc f ′ (e (0)) = c ′ (e (0)). by contradiction assume e (0) ≥ ec , by monotonicity of f ′ and c ′ it follows:    2f ′ ( e c ) > f ′ ( e c ) ≥ f ′ (e (0)) c ′ ( e c ) ≤ c′ (e (0)) , clearly absurd. � this result states the fact that a rational agent will exert higher effort when paired with somebody exerting a lower effort. this is interesting in terms of free riding: it may appear that the rational agent’s behavior incentives free riding, but it must be recalled that rational agents are interested only in maximizing their profit. in section 5 we introduce an incentive to competition and the results will be different. in the following, we consider the equilibria resulting as the composition of population varies. 4.1 fixed effort agents vs committed agents let us consider the profit maximizing game; the population of 2n agents is partitioned in two subsets: 1. m fixed effort ē agents cubo 11, 2 (2009) optimal effort in heterogeneous ... 23 2. (2n − m) committed agents who coordinate on the best effort knowing the fact that there are m agents providing the fixed effort. a single committed agent does not know which kind of mate it will be paired with so it has to solve the following problem: max e 2n − m − 1 2n − 1 (f (2e) − c (e)) + m 2n − 1 (f (e + ē) − c (e)) . (4) let e ∗ be the optimal solution effort to problem (4), it holds: theorem 2 as the number of committed agents increases to 2n the optimal effort provided by the committed agents increases to the coordination equilibrium effort ec . as the effort ē provided by the fixed effort agents increases, the optimal effort provided by the committed agents decreases. proof: let us consider foc of problem (4): f (e ∗ , m, ē) = = 2 (2n − m − 1) f ′ (2e∗) + mf ′ (e∗ + ē) − (2n − 1) c′ (e∗) = 0. by implicit function theorem it is possible to write: ∂e ∗ ∂m = − ∂f/∂m ∂f/∂e∗ = = − −2f ′ (2e∗) + f ′ (e∗ + ē) 4 (2n − m − 1) f ′′ (2e∗) + mf ′′ (e∗ + ē) − (2n − 1) c′′ (e∗) . by assumptions on concavity/convexity of f and c the denominator is negative while, as it concerns the numerator, consider the following strictly concave functions g1 (e) = f (e + ē) − c (e) , g2 (e) = f (2e) − c (e) . since by proposition 2 we have e (ē) < e c , then ∀e such that e (ē) < e < ec it is:    g ′ 1 (e) < 0, g ′ 2 (e) > 0, because e (ē) is the optimal point of g1 (e) and e c is the optimal point of g2 (e). it follows: −g′2 (e ∗ ) + g ′ 1 (e ∗ ) = −2f ′ (2e∗) + f ′ (e∗ + ē) < 0. this way if e (ē) < e c it follows that ∂e ∗ /∂m < 0 and therefore the optimal effort to problem (4) is decreasing with respect to m. in the limit case, all agents are committed, and obviously, the 24 arianna dal forno and ugo merlone cubo 11, 2 (2009) optimal effort will be e c . finally, let us consider: ∂e ∗ ∂ē = − ∂f/∂ē ∂f/∂e∗ = = − mf ′′ (e ∗ + ē) 4 (2n − m − 1) f ′′ (2e∗) + mf ′′ (e∗ + ē) − (2n − 1) c′′ (e∗) < 0. and the second part of the theorem follows. � figure 1 summarizes the results stated in proposition 2 and theorem 2. figure 1: fixed vs committed as the number of committed players approaches the whole population, their effort approximates e c and, the higher is the effort provided by fixed effort agents, the lower is the optimal effort supplied by committed players. this result may be explained since when the population tends to consist of almost only committed agents, the probability for a single committed agent to be paired with a different kind of agent will be low. furthermore, when few committed agents face a population of fixed effort agents, they will expect a low probability of facing another committed agent. as it concerns the second part it may be interpreted analogously to proposition 2. 4.2 rational agents vs committed agents let us consider rational agents and recall that while their goal is to maximize their profit, they are not able to coordinate as committed agents do. it is obvious that if the population consists only of rational agents they will play the nash equilibrium. in this section we study the equilibria in a mixed population composed of both committed and rational agents. cubo 11, 2 (2009) optimal effort in heterogeneous ... 25 consider: 1. m rational agents playing the effort ei , i = 1, . . . , m 2. (2n − m) committed agents who play the best effort eco knowing the fact that there are m rational agents. a single committed agent does not know which kind of mate it will be paired with so it has to solve the following problem: max eco 2n − m − 1 2n − 1 (f (2eco) − c (eco)) + m 2n − 1 (f (eco + ej) − c (eco)) (5) j = 1, . . . , m while a rational agent i will solve: max ei 2n − m 2n − 1 (f (ei + eco) − c (ei)) + 1 2n − 1 ∑ j 6=i (f (ei + ej ) − c (ei)) . (6) proposition 3 in a mixed population composed of both committed and rational agents, the effort provided by the committed is not inferior to the effort provided by the rational agents. proof: let us consider foc of problem (5) and (6): { f (m, eco) = (2n − m − 1) [2f ′ (2eco) − c ′ (eco)] + m [f ′ (eco + ej ) − c ′ (eco)] = 0, g (m, ei) = (2n − m) [f ′ (ei + eco) − c ′ (ei)] + ∑ j 6=i [f ′ (ei + ej ) − c ′ (ei)] = 0. all rational agents solve the same problem, therefore we consider symmetrical equilibria: ei = er, i = 1, . . . , m { f (m, eco) = (2n − m − 1) [2f ′ (2eco) − c ′ (eco)] + m [f ′ (eco + er) − c ′ (eco)] = 0, g (m, er) = (2n − m) [f ′ (er + eco) − c ′ (er)] + (m − 1) [f ′ (2er) − c ′ (er)] = 0. (7) by contradiction let er > eco, it follows 2er > er + eco > 2eco. now consider f ′ (eco + er) − c ′ (eco): • if f ′ (eco + er) − c ′ (eco) ≥ 0, we have: er + eco > 2eco ⇒ f ′ (2eco) > f ′ (er + eco) and it follows 2f ′ (2eco) − c ′ (eco) > f ′ (2eco) − c ′ (eco) > 0 absurd since it contradicts the first foc in (7). 26 arianna dal forno and ugo merlone cubo 11, 2 (2009) • if f ′ (eco + er) − c ′ (eco) < 0, we have: er > eco ⇒ f ′ (er + eco) − c ′ (er) < f ′ (er + eco) − c ′ (eco) < 0 and it is 2er > er + eco ⇒ f ′ (2er) − c ′ (er) < f ′ (er + eco) − c ′ (er) . putting together we obtain: f ′ (2er) − c ′ (er) < f ′ (er + eco) − c ′ (er) < f ′ (er + eco) − c ′ (eco) < 0 absurd since it contradicts the second foc in (7). it follows er ≤ eco. � this may be easily interpreted since rational agents deviate and shirk in order to maximize their profit. theorem 3 as the number of committed agents increases to 2n, the optimal effort provided by the rational agents decreases to the best reply effort to the coordination equilibrium e ( e c ) and the optimal effort provided by the committed agents increases to the coordination equilibrium effort ec . proof: let us consider conditions (7), by implicit function theorem it is possible to write: ∂er ∂m = − ∂g/∂m ∂g/∂er = − −f ′ (er + eco) + f ′ (2er) (2n − m) f ′′ (er + eco) + 2 (m − 1) f ′′ (2eco) − (2n − 1) c′′ (er) . by assumptions on concavity/convexity of f and c the denominator is negative, and, as it concerns the numerator, since er < eco and f is a marginal decreasing function, it is positive. therefore, ∂er/∂m > 0. obviously, when a single rational agent faces only committed agents it will play the best reply effort to the coordination equilibrium e ( e c ) . as it concerns the second part of the statement, it is obvious that when a single committed agent faces only rational agents it will play the nash effort e n and, vice versa, if the population consists only of committed agents, they all will play the coordinated effort e c . furthermore, committed agent’s effort eco depends on m, the number of rational agents, both directly and indirectly, via the optimal effort of rational agents: eco = g (m, er (m)) . we consider the infinitesimal variation in eco as m increases: deco dm = ∂g ∂m + ∂g ∂er der dm . by theorem (2) ∂g/∂m is negative; we just proved that der/dm > 0, and by second part of theorem (2) also ∂g/∂er < 0. it follows the second part of the thesis. cubo 11, 2 (2009) optimal effort in heterogeneous ... 27 � figure 2 summarizes the results stated in proposition 3 and theorem 3. figure 2: rational vs committed when few rational agents face many committed agents, they expect that the latter will tend to exert an effort close to the committed one and, as a consequence, they will tend to play the best reply to the expected effort. vice versa, when few committed agents face many rational agents, their effort will be lower since the probability to be paired with agents maximizing their own profit will be higher. 4.3 fixed effort agents vs rational agents consider: 1. m fixed effort ē agents; 2. (2n − m) rational agents playing the effort ei , i = 1, . . . , 2n − m. a single rational agent i will solve: max ei m 2n − 1 (f (ei + ē) − c (ei)) + 1 2n − 1 ∑ j 6=i (f (ei + ej ) − c (ei)) . (8) theorem 4 assume ē > en (ē < en ), as the number of fixed effort agents increases to 2n, the optimal effort provided by the rational agents decreases (increases) to e (ē). 28 arianna dal forno and ugo merlone cubo 11, 2 (2009) proof: when a single rational agent faces only fixed effort agents it will play e (ē). by proposition (2) it is ē > e n ⇒ e (ē) < e ( e n ) = e n and, obviously, e (ē) ≤ er ≤ e n . let us consider foc of problem (8) when we consider symmetrical equilibria: ei = er, i = 1, . . . , 2n − m : h (m, er) = m [f ′ (er + ē) − c ′ (er)] + (2n − m − 1) [f ′ (2er) − c ′ (er)] = 0. (9) by implicit function theorem it is possible to write: ∂er ∂m = − ∂h/∂m ∂h/∂er = − f ′ (er + ē) − f ′ (2er) m [f ′′ (er + ē) − c′′ (er)] + (2n − m − 1) [2f ′′ (2er) − c′′ (er)] . and it follows ∂er/∂m < 0. similarly, the case in which ē < e n can be proved. � figure 3 summarizes the results stated in theorem 4. figure 3: rational vs fixed this may be explained taking into account the relative proportion of the population and the fact that the rational agents maximize their profit. 5 a ranking policy the firm may consider to introduce some mechanism to increase productivity, for example, rewarding employees who maximize their profit. nevertheless, such mechanisms must be devised carefully cubo 11, 2 (2009) optimal effort in heterogeneous ... 29 since a known problem of individual incentives is that they may lead employees competing with one another (see for instance [12]). in particular, we introduce a ranking policy and will show what may be some of the undesirable results for the agents. consider the following individual incentive plan: all agents are ranked according to their payoff and normalized in the sense that the agent with the higher payoff will get 1 and the one with the lower will get 0; should all the agents have the same payoff, every agent will get 1. formally: ri =              πi − min j∈i {πj} max j∈i {πj} − min j∈i {πj} if max j∈i {πj} 6= min j∈i {πj} 1 if max j∈i {πj} = min j∈i {πj} (10) this policy may recall the ”employee of the year” prize even if in a sketched way. obviously, it must be noted that usually firms give this kind of prize observing agents’ efforts and that in our model effort is not observable and costs are private. nevertheless, this ranking policy may be viewed as a social policy where the best individual is the one with higher personal profit, and this kind of pressure may not be ignored since here we do not assume any social norm. in this section we will assume that agents want to maximize rank and examine how this ranking policy can affect the equilibrium set. it is obvious that any profile of strategies where all players exert the same effort gives the maximum outcome. nevertheless, some may prefer to avoid tie results and this may give incentive to deviation. this can be explained in different ways: for example, if an agent is particularly competitive it may deviate to be the only winner, while, if some others expect such a deviation, they could anticipate it and behave consequently. in particular: theorem 5 consider the team game defined in section 2; if we introduce the ranking policy (10) the set of nash equilibria expands to the set (e∗) where e∗ = (e∗, e∗, . . . , e∗) ∈ r2n, e∗ ∈ [ 0, e n ] . proof: let all the players exert the same effort ê, then player i will exert an effort ei such that    f (ei + ê) − c (ei) ≥ f (ei + ê) − c (ê) , f (ei + ê) − c (ei) ≥ f (2ê) − c (ê) . the first condition guarantees that player i’s profit will be not lower than its mate’s j, while the second condition means that player i’s profit is not lower than the profits of the members of the other teams. it should be noted that the first condition is equivalent to ei ≤ ê. let us consider a profile of strategies e ∗ = (e ∗ , e ∗ , . . . , e ∗ ), where e ∗ ∈ [ 0, e n ] . if player i exerts effort ei = e ∗ his payoff is the same as the other players’ and its rank will be 1. furthermore, player i has no incentive to exert a lower effort since if we consider its profit, the partial derivative with respect to its effort is positive when ei = e ∗ : ∂ ∂ei (πi (ei, ej = e ∗ )) |ei=e∗ = f ′ (2e ∗ ) − c′ (e∗) > f ′ ( 2e n ) − c′ ( e n ) = 0. 30 arianna dal forno and ugo merlone cubo 11, 2 (2009) the second pure partial derivative is negative and this means that: ∂ ∂ei (πi (ei, e ∗ )) = f ′ (ei + e ∗ ) − c′ (ei) > 0 ∀ei ∈ [0, e ∗ ] so there is no incentive to deviate. finally, let us consider a profile of strategies e ∗ = (e ∗ , e ∗ , . . . , e ∗ ), where e ∗ > e n . player i has incentive to shirk since, considering its profit, the partial derivative with respect to its effort is positive when ei = e ∗ : ∂ ∂ei (πi (ei, ej = e ∗ )) |ei=e∗ = f ′ (2e ∗ ) − c′ (e∗) < f ′ ( 2e n ) − c′ ( e n ) = 0 � the set of nash equilibria is expanded by the ranking policy; this adds some further problems in predicting which equilibrium will be selected. one way to overcome this could be to assume that agents have a lexicographic utility and among different equilibria would prefer the one with higher profit. finally, even in very simple cases we show how this policy may not increase the productivity of agents. consider the situation in which a rational agent is employed in a firm where a corporate culture consisting in providing the same effort ē is established. without the incentive policy its optimal effort will be e (ē); this effort will be higher than ē if and only if ē < e n and, in particular, will be the same as the other agents’ if and only if ē = e n . now, introducing the incentive policy, it holds: theorem 6 consider 2n − 1 fixed effort ē agents, and a single rational agent, then the incentive policy never incentives the rational agent to provide an effort larger than ē. proof:the rational agent will solve: max e f (e + ē) − c (e) and take into account also its partner profit. let e ∗ := arg max [f (e + ē) − c (e)]; three cases are given: 1. e ∗ < ē: in this case the rational agent will provide this effort since [f (e + ē) − c (e)] < [f (2ē) − c (ē)] and will be the only one to get payoff 1; 2. e ∗ = ē: this case is trivial since all agents will get 1; 3. e ∗ > ē: in this case should the rational agent provide this effort its partner would free ride and would be the only one to get 1. cubo 11, 2 (2009) optimal effort in heterogeneous ... 31 clearly there is no incentive to provide efforts higher than ē. � this simple result is very important because it shows, in this simple model, some of the “undesirable results” [12]) of the individual incentive. in particular, if the fixed level ē is low, the effort of a new rational employee is bounded by this individual incentive plan. 6 simulation results in the following we propose a simulation approach in order to extend the theoretical results we derived in previous sections. the agent based simulations we performed were implemented on a customized version of the platform described in [2]. the platform has been modified simply including the behavioral classes we described in the theoretical analysis. in particular, we considered the same functional form which was used in human subject experiments in [3], that is, f (e) = 5 √ e and c (e) = e 2 ; when agents i and j are in the same team, their payoff is respectively    πi (ei, ej) = 5 √ ei + ej − e 2 i , πj (ei, ej ) = 5 √ ej + ei − e 2 j . (11) it is worth noting that, considering the one-shot game, the nash equilibrium is e n = 3 √ 25/32 ≃ .92100787466 while the coordination equilibrium is e c = 3 √ 25/2 ≃ 1.4620088691. we are interested in extending the results about optimal effort when considering a mixed population of 2n agents which is partitioned as follows: 1. rational agents 1, 2, . . . , m1 who maximize their individual profit; 2. m2 fixed effort ē agents; 3. (2n − m1 − m2) committed agents who coordinate on the best effort. for all the agents the population composition is common knowledge. as a consequence an equilibrium configuration can be found solving              max ei 2n−m1−m2 2n−1 5 √ ei + eco + 1 2n−1 ∑ j ≤ m1 j 6= i 5 √ ei + ej + m2 2n−1 5 √ ei + ē − e 2 i (i = 1, 2, . . . , m1) , max eco 2n−m1−m2−1 2n−1 5 √ 2eco + m1 2n−1 5 √ eco + er + m2 2n−1 5 √ eco + ē − e 2 co. as we assume that rational agents are symmetric an equilibrium configuration may be (e ∗ r , e ∗ co) where e ∗ r indicates the optimal effort for the m1 rational agents and e ∗ co indicates the optimal 32 arianna dal forno and ugo merlone cubo 11, 2 (2009) effort for the m2 committed agents. in the simulation, by the symmetry assumption, rational and committed agents decide their efforts dynamically as follows:      e t+1 r = arg max er 2n−m1−m2 2n−1 5 √ er + etco + m1−1 2n−1 5 √ er + etr + m2 2n−1 5 √ er + ē − e 2 r, e t+1 co = arg max eco 2n−m1−m2−1 2n−1 5 √ 2eco + m1 2n−1 5 √ eco + etr + m2 2n−1 5 √ eco + ē − e 2 co. (12) since in our case the best reply functions are given implicitly, in the simulation we use an iterative method in order to solve the maximization problem and therefore the best reply functions for each agent are numerically evaluated. in our simulations we compute the steady state for different population compositions. assuming 2n = 900 agents, we represent on the x−axis the number of fixed agents and on the y−axis the number of rational agents; it follows that point (x, y), where 0 ≤ x, y ≤ 900 and x + y ≤ 900, represents a population consisting of x fixed agents, y rational agents and 900 − x − y committed agents. for each population composition we compute the rational agent and committed agent steady effort assuming that all agents know the population composition. the respective efforts 4 are illustrated in figures 4 and 5. figure 4: rational (left) and committed (right) agent effort with global interaction and fixed effort level ē = 0.1 it can be observed that the results we obtain by simulation are consistent with all the theoretical we obtained in previous sections. in fact, when considering the restrictions for zero fixed agents in figure 4, we obtain respectively rational and committed agents’ efforts; combining opportunely these restrictions we obtain the results stated in proposition 3 and theorem 3, as depicted in figure 2. observe that the same results hold for figure 5 as we do not consider fixed effort agents. then, when considering the restrictions for zero rational agents in figures 4 (right) and 5 (right), we obtain respectively the committed agents’ efforts in a 0.1 and 1.2 fixed effort agents population; an opportune combination of these restrictions allows us to obtain the results stated in proposition 2 and theorem 2, as summarized in figure 1. 4recall that, given the population composition, efforts are defined only for x + y ≤ 900. cubo 11, 2 (2009) optimal effort in heterogeneous ... 33 finally, when considering the restrictions for zero committed agents in figures 4 (left) and 5 (left), i.e, y = 900 − x, x = 0, 100, 200, . . . , 900, we obtain respectively the rational agents’ efforts in a 0.1 and 1.2 fixed effort agents population. once again, with an appropriate combination of these restrictions we obtain the results stated in theorem 4 and summarized in figure 3. figure 5: rational (left) and committed (right) agent effort with global interaction and fixed effort level ē = 1.2 so far, in the simulations, we have assumed that the population composition was common knowledge and that agents could interact globally, as in a complete graph (see [13]). in the following, we analyze the consequence of a local interaction. we assume that agents can interact in a von neumann neighborhood, and do not know the population composition. this has important consequences in the dynamic process described in (12) as agents must have knowledge of their opponents’ best reply in order to make a decision. in particular, each agent assumes that all the agents in its neighborhood share its own information about neighborhood composition. we can say that the agents have egocentric thought about neighborhood in the sense of piaget [9]. figure 6: an example of egocentric neighborhood 34 arianna dal forno and ugo merlone cubo 11, 2 (2009) for example, in figure 6 assume that a rational agent interacts with two fixed effort agents and two committed agents. that is, its neighborhood consists of one rational, two committed and two fixed effort agents. this agent does not have the complete composition of its neighbors’ neighborhood; in our model we assume that the agents consider all their neighbors having their same neighborhood composition. that is, the local composition is assumed to be constant. figure 7: rational (left) and committed (right) agent effort with local interaction and fixed effort level ē = 0.1 figure 8: rational (left) and committed (right) agent effort with local interaction and fixed effort level ē = 1.2. in our simulations the agents are randomly located on a toroidal lattice; the results we obtain with this assumption and local interaction are presented in figures 7 and 8. comparing these results to those presented in figures 4 and 5, we can observe that the average efforts are quite close to those in global interaction. so, global interaction behavior is a good approximation of the average behavior we obtain when considering local interaction in von neumann neighborhood with egocentric agents. finally, it is interesting to observe that when agents are located according to their behavioral class instead of being randomly located, the results are quite similar to those obtained when considering homogeneous populations . cubo 11, 2 (2009) optimal effort in heterogeneous ... 35 7 conclusions and further research in this paper we described a game which tries to model some of the transactions taking place in a firm. we discussed some possible equilibria of the game in the simpler cases and observed how the game may be compared to a version of the prisoner’s dilemma. the results we found and the stability of equilibria are interpreted in terms of corporate culture. we considered also some “bounded rationality” agents and the impact their presence has on the ideal cases equilibria. then, we discussed a ranking incentive devised to increase competition between agents; we showed how this ranking policy affects the performance of the subjects and found the set of equilibria of the modified game. since the modified game has a continuum of equilibria we suggested one possible way to refine them. the results we found are interesting in terms of free riding and shirking and may help to shed light on how the different composition of employees may result in different equilibria. furthermore, we showed in the simple model of firm how an individual incentive plan may result in limiting the optimal effort of some agents. finally, by means of agent based simulation two more directions were investigated. first, we could consider heterogeneous populations with coexistence of all the kinds of behavior we studied in the theoretical analysis. this way, we could overcome the problem of agents having the best reply functions in implicit form by a numerical iterative method. the comparison of the results, when assuming that population composition was common knowledge across the agents, exhibited consistency with the theoretical results. second, we assumed that the agents could interact only in their von neumann neighborhood, without knowing the global population composition. since for practical reasons each agent must know the best reply of its opponents, we modeled a sort of egocentric thought about the neighbors’ neighborhood composition. that is, we assumed that each agent had an egocentric perspective in a sense close to piaget about the composition of its neighbors’ neighborhood. in this case the local interaction results were quite similar to the global ones, and therefore coherent with the theoretical analysis. in further research it would be interesting to analyze which equilibria may be selected in a dynamic setting of the game. it would be also interesting to consider also some more sophisticated models of bounded rationality. received: march 14, 2008. revised: may 9, 2008. 36 arianna dal forno and ugo merlone cubo 11, 2 (2009) references [1] g.p baker, m.c. jensen and k. murphy, compensation and incentives: practice vs. theory, j. of finance, xliii, no.3, (1988), 1–29. [2] a. dal forno and u. merlone, a multi-agent simulation platform for modeling perfectly rational and bounded-rational agents in organizations, j. of artif. societies and soc. sim., 5, no.2 (2002). [3] a. dal forno and u. merlone, from classroom experiments to computer code, j. of artif. societies and soc. sim., 7, no.3 (2004). [4] j.h. kagel and a.e. roth, the handbook of experimental economics, princeton university press, 1995, princeton, nj. [5] d.m. kreps, corporate culture and economic theory. in: perspectives on positive political economy, (eds.: j.e. alt and k.a. shepsle) cambridge university press, 1990, cambridge, uk, 91–143. [6] i. macho-stadler and d. pérez-castrillo, an introduction to the economics of information (incentive and contracts), oxford university press, 1997, oxford, uk. [7] j. marschack and r. radner, economic theory of teams, yale university press, 1972, new haven, conn. [8] s.e. morris, interaction games: a unified analysis of incomplete information, local interaction and random matching, 1997, santa fe institute working paper no. 97-08-072e. [9] j. piaget, the psychology of intelligence, routledge & kegan paul, 1950, oxon, uk. [10] r. radner, hierarchy: the economics of managing, j. of econ. liter., xxx, no.3, (1988), 1382–1415. [11] a. rubinstein, modeling bounded rationality, the mit press, 1998, cambridge, ma. [12] r.m. steers and j.s. black, organizational behavior (fifth edition), harpercollins college publishers, 1994, new york, ny. [13] s. wasserman and k. faust, social network analysis, cambridge university press, 1997, cambridge, uk. n03-dalfornomerlone cubo a mathematical journal vol.11, n o ¯ 02, (85–105). may 2009 dynamic oligopolies and intertemporal demand interaction carl chiarella school of finance and economics, university of technology sydney, p.o. box 123, broadway, nsw 2007, australia email: carl.chiarella@uts.edu.au and ferenc szidarovszky systems & industrial engineering department, the university of arizona, tucson, arizona, 85721-0020, usa email: szidar@sie.arizona.edu abstract dynamic oligopolies are examined with continuous time scales and under the assumption that the demand at each time period is affected by earlier demands and consumptions. after the mathematical model is introduced the local asymptotical stability of the equilibrium is examined, and then we will discuss how information delays alter the stability conditions. we will also investigate the occurrence of a hopf bifurcation giving the possibility of the birth of limit cycles. numerical examples will be shown to illustrate the theoretical results. resumen dinámica de oligopolios son examinados en escala de tiempo continuo y bajo la suposición que la demanda en todo tiempo periodico es afectada por la demanda e 86 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) consumo temprano. es presentado el modelo matemático y examinada la estabilidad asintótica local y entonces discutiremos como la información de retrazo altera las condiciones de estabilidad. también investigamos el acontecimiento de bifurcación de hofp dando la posibilidad de nacimiento de ciclos limites. ejemplos númericos son exhibidos para ilustrar los resultados teóricos. key words and phrases: noncooperative games, dynamic systems, time delay, stability. math. subj. class.: 91a20, 91a06. 1 introduction oligopoly models are the most frequently studied subjects in the literature of mathematical economics. the pioneering work of [3] is the basis of this field. his classical model has been extended by many authors, including models with product differentiation, multi-product, labor-managed oligopolies, rent-seeking games to mention a few. a comprehensive summary of single-product models is given in [6] and their multi-product extensions are presented and discussed in [7] including the existence and uniqueness of static equilibria and the asymptotic behavior of dynamic oligopolies. in both static and dynamic models the inverse demand function relates the demand and price of the same time period, however in many cases the demand for a good in one period will have an effect on the demand and price of the goods in later time periods. in the case of durable goods the market becomes saturated, and even in the case of non-durable goods the demand for and consumption of the goods in earlier periods will lead to taste or habit formation of consumers that will affect future demands. intertemporal demand interaction has been considered by many authors in analyzing international trade (see for example, [4] and [9]). more recently [8] have developed a two-stage oligopoly and examined the existence and uniqueness of the nash equilibrium. in this paper we will examine the continuous counterpart of the model of [8]. after the dynamic model is introduced, the asymptotic behavior of the equilibrium will be analyzed. we will show that under realistic conditions the equilibrium is always locally asymptotically stable. we will also show that this stability might be lost when the firms have only delayed information about the outputs of the competitors and about the demand interaction. 2 the mathematical model an n-firm single-product oligopoly is considered without product differentiation. let xk be the output of firm k and let ck(xk) be the cost of this firm. it is assumed that the effect of market saturation, taste and habit formation, etc. of earlier time periods are represented by a timecubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 87 dependent variable q, which is assumed to be driven by the dynamic rule q̇ = h ( n ∑ k=1 xk, q ) , (2.1) where h is a given bivariate function. it is also assumed that the price function depends on both the total production level of the industry and q. hence the profit of firm k can be formulated as πk = xkf (xk + sk, q) − ck(xk) (2.2) where sk = ∑ l 6=k xl. if lk denotes the capacity limit of firm k, then 0 ≤ xk ≤ lk and 0 ≤ sk ≤ ∑ l 6=k ll. the common domain of functions h and f is [0, ∑n k=1 lk] × r+, and the domain of the cost function ck is [0, lk]. assume that h is continuously differentiable, f and ck are twice continuously differentiable on their entire domains. with any fixed values of sk ∈ [0, ∑ l 6=k ll] and q ≥ 0, the best response of firm k is rk(sk, q) = arg max 0≤xk≤lk {xkf (xk + sk, q) − ck(xk)} , (2.3) which exists since πk is continuous in xk and the feasible set for xk is a compact set. as it is usual in oligopoly theory we make the following additional assumptions: (a) f ′ x < 0; (b) f ′ x + xkf ′′ xx ≤ 0; (c) f ′ x − c ′′ k < 0 for all k and feasible xk, sk and q. under these assumptions πk is strictly concave in xk, so the best response of each firm is unique and can be obtained as follows: rk(sk, q) =      0 if f (sk, q) − c ′ k(0) ≤ 0 lk if lkf ′ x(lk + sk, q) + f (lk + sk, q) − c ′ k(lk) ≥ 0 x ∗ k otherwise, (2.4) where x ∗ k is the unique solution of equation f (xk + sk, q) + xkf ′ x(xk + sk, q) − c ′ k(xk) = 0 (2.5) in interval (0, lk). 88 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) if we assume that at each time period each firm adjusts its output into the direction towards its best response, then the resulting dynamism becomes ẋk = kk · (rk(sk, q) − xk) (k = 1, 2, · · · , n), (2.6) where kk is the speed of adjustment of firm k, and if we add the dynamic equation (2.1) to these differential equations, an (n + 1)-dimensional dynamic system is obtained. clearly, x̄1, · · · , x̄n, q̄ is a steady state of this system if and only if h ( n ∑ k=1 x̄k, q̄ ) = 0 (2.7) and x̄k = rk   ∑ l 6=k x̄l, q̄   (2.8) for all k. the asymptotic behavior of the steady states will be examined in the next section. 3 stability analysis assume that x̄1, · · · , x̄n , q̄ is an interior steady state, that is, both f ( ∑ l 6=k x̄l, q̄) − c ′ k(0) and lkf ′ x(lk + ∑ l 6=k x̄l, q̄) + f (lk + ∑ l 6=k x̄l, q) − c ′ k(lk) are nonzero for all k. if x̄k = 0 or x̄k = lk, then clearly both ∂rk ∂sk and ∂rk ∂q are equal to zero. otherwise these derivatives can be obtained by implicitly differentiating equation (2.5) with respect to sk and q: rk = ∂rk ∂sk = − f ′ x + xkf ′′ xx 2f ′ x + xkf ′′ xx − c ′′ k (3.1) and r̄k = ∂rk ∂q = − f ′ q + xkf ′′ xq 2f ′ x + xkf ′′ xx − c ′′ k . (3.2) assumptions (b) and (c) imply that −1 < rk ≤ 0, (3.3) and if we assume that (d) f ′ q + xkf ′′ xq ≤ 0 cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 89 for all k and feasible xk, sk and q, then r̄k ≤ 0. (3.4) notice that relations (3.3) and (3.4) hold for all interior steady states. the jacobian of system given by the differential equations (2.1) and (2.6) has the special form j =        −k1 k1r1 · · · k1r1 k1r̄1 k2r2 −k2 · · · k2r2 k2r̄2 . . . . . . . . . . . . knrn knrn · · · −kn knr̄n h h · · · h h̄        where h = ∂h ∂x and h̄ = ∂h ∂q . the main result of this section is the following. theorem 1 assume that at the steady state, h̄ ≤ 0 and rkh̄ − r̄kh > 0 for all k. then the steady state is locally asymptotically stable. proof. we will prove that the eigenvalues of j have negative real parts at the steady state. the eigenvalue equation of j can be written as −kkuk + kkrk ∑ l 6=k ul + kkr̄kv = λuk (1 ≤ k ≤ n) (3.5) and h n ∑ k=1 uk + h̄v = λv. (3.6) let u = ∑n k=1 uk, then from (3.6), hu = (λ − h̄)v. assume first that h = 0. then the eigenvalues of j are h̄ and the eigenvalues of matrix      −k1 k1r1 · · · k1r1 k2r2 −k2 · · · k2r2 . . . . . . . . . knrn knrn · · · −kn      . notice that this matrix is the jacobian of continuous classical cournot dynamics and it is wellknown that its eigenvalues have negative real parts if kk > 0 and −1 < rk ≤ 0 for all k (see for example, [1]). in this case the assumptions of the theorem imply that h̄ < 0, so all eigenvalues of j have negative real parts. 90 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) assume next that h 6= 0. then from (3.6), u = λ − h̄ h v, (3.7) and from (3.5), uk = kkrku + kkr̄kv λ + kk(1 + rk) = kkrk(λ − h̄) + kkr̄kh (λ + kk(1 + rk))h v. (3.8) by adding these equations for all values of k and using (3.7) we have n ∑ k=1 kkrkλ + kk(r̄kh − rkh̄) λ + kk(1 + rk) = λ − h̄ , (3.9) since v 6= 0, otherwise (3.8) would imply that uk = 0 for all k. assume next that λ = a + ib is a root of equation (3.9) with a ≥ 0. let g(λ) denote the left hand side of this equation. then re g(λ) = re kk(rka + r̄kh − rkh̄) + ikkrkb a + kk(1 + rk) + ib = kk(rka + r̄kh − rkh̄)(a + kk(1 + rk)) + kkrkb 2 (a + kk(1 + rk)) 2 + b2 < 0 and re (λ − h̄) = a − h̄ ≥ 0, which is an obvious contradiction. 2 the conditions of the theorem are satisfied in the special model of [7, section 5.4] on oligopolies with saturated markets, where the dynamic rule of q is assumed to be linear, q̇ = n ∑ k=1 xk − c · q with some c > 0. in this case h = 1 and h̄ = −c, so h 6= 0 and rkh̄ − r̄kh = −crk − r̄k > 0 unless rk = r̄k = 0. 4 the effect of delayed information in this section, we assume that the conditions of theorem 1 hold, and the firms have only delayed information on their own outputs as well as on the output of the rest of the industry. it is also assumed that they have also delayed information on the value of parameter q. a similar situation occurs when the firms react to average past information rather than reacting to sudden market cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 91 changes. as in [2] we assume continuously distributed time lags, and will use weighting functions of the form w(t − s, t, m) = { 1 t e − t−s t if m = 0 1 m ( m t )m+1 (t − s)me− (t−s)m t if m ≥ 1 (4.1) where t > 0 is a real and m ≥ 0 is an integer parameter. the main properties and the applications of such weighting functions are discussed in detail in [2]. by replacing the delayed quantities by their expectations, equations (2.6) become a set of volterra-type integro-differential equations: ẋk(t) = kk ·  rk   ∫ t 0 w(t − s, tk, mk) ∑ l 6=k xl(s)ds, ∫ t 0 w(t − s, uk, pk)q(s)ds   − ∫ t 0 w(t − s, vk, lk)xk(s)ds ) (1 ≤ k ≤ n) (4.2) accompanied by equation (2.1). it is well-known that equations (4.2) are equivalent to a higher dimensional system of ordinary differential equations, so all tools known from the stability theory of ordinary differential equations can be used here. linearizing equation (4.2) around the steady state we have ẋkδ = kk ·  rk · ∫ t 0 w(t − s, tk, mk) ∑ l 6=k xlδ (s)ds +r̄k · ∫ t 0 w(t − s, uk, pk)qδ(s)ds − ∫ t 0 w(t − s, vk, lk)xkδ (s)ds ) (4.3) where xkδ and qδ are deviations of xk and q from their steady state levels. we seek the solution in the form xkδ (t) = uke λt and qδ = ve λt , then we substitute these into equation (4.3) and let t → ∞. the resulting equation will have the form − ( λ + kk ( 1 + λvk ck )−(lk+1) ) uk + kkrk ( 1 + λtk ak )−(mk+1) ∑ l 6=k ul + kkr̄k ( 1 + λuk bk )−(pk +1) v = 0 (4.4) where ak = { 1 if mk = 0 mk otherwise, bk = { 1 if pk = 0 pk otherwise, ck = { 1 if lk = 0 lk otherwise, 92 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) and we use the identity ∫ ∞ 0 w(s, t, m)e −λs ds = { (1 + λt ) −1 if m = 0 ( 1 + λt m )−(m+1) if m ≥ 1. linearizing equation (2.1) around the steady state we have q̇δ(t) = h · n ∑ k=1 xkδ (t) + h̄qδ(t) (4.5) and by substituting xkδ (t) = uke λt and qδ = ve λt into this equation we have h n ∑ k=1 uk + (h̄ − λ)v = 0. (4.6) for the sake of simplicity introduce the notation ak(λ) = λ + kk ( 1 + λvk ck )−(lk+1) , bk(λ) = kkrk ( 1 + λtk ak )−(mk+1) , and ck(λ) = kkr̄k ( 1 + λuk bk )−(pk +1) , then equation (4.4) can be rewritten as −ak(λ)uk + bk(λ) ∑ l 6=k ul + ck(λ)v = 0 (1 ≤ k ≤ n). (4.7) equations (4.7),(4.6) have nontrival solution if and only if det        −a1(λ) b1(λ) · · · b1(λ) c1(λ) b2(λ) −a2(λ) · · · b2(λ) c2(λ) . . . . . . . . . . . . bn(λ) bn(λ) · · · −an(λ) cn(λ) h h h h̄ − λ        = 0. (4.8) assume first that h = 0. then the conditions of theorem 1 imply that h̄ < 0. the eigenvalues are therefore λ = h̄, which is negative, and the roots of equation det      −a1(λ) b1(λ) · · · b1(λ) b2(λ) −a2(λ) · · · b2(λ) . . . . . . . . . bn(λ) bn(λ) · · · −an(λ)      = 0. (4.9) cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 93 by introducing d = diag ( − a1(λ) − b1(λ), · · · , −an(λ) − bn(λ) ) , 1 ⊤ = (1, · · · , 1) and b = ( b1(λ), . . . , bn(λ) )⊤ , this equation can be rewritten as det(d + b · 1⊤) = det(d) det(i + d−1b · 1⊤) = n ∏ k=1 ( − ak(λ) − bk(λ) ) · [ 1 − n ∑ k=1 bk(λ) ak(λ) + bk(λ) ] = 0 . (4.10) therefore in this case we have to examine the locations of the roots of equations ak(λ) + bk(λ) = 0 (1 ≦ k ≦ n) (4.11) and n ∑ k=1 bk(λ) ak(λ) + bk(λ) = 1 . (4.12) assume next that h 6= 0 and ak(λ) + bk(λ) 6= 0. by introducing the new variable u = ∑n k=1 uk, equation (4.7) can be rewritten as bk(λ)u = ( ak(λ) + bk(λ) ) uk − ck(λ)v . by combining this equation with (4.6) we have uk = hck(λ) + bk(λ)(λ − h̄) h ( ak(λ) + bk(λ) ) v , (4.13) where we assume that the denominator is nonzero. adding this equation for all values of k and using (4.6) again we see that n ∑ k=1 hck(λ) + bk(λ)(λ − h̄) ak(λ) + bk(λ) = λ − h̄ . (4.14) here we also used the fact that v 6= 0, since otherwise uk would be zero for all k from equation (4.13), and eigenvectors must be nonzero. notice first that in the absence of information lags tk = uk = vk = 0 for all k, and in this special case equation (4.14) reduces to (3.9) as it should. the analysis of the roots of equations (4.11), (4.12) and (4.14) in the general case requires the use of computer methods, however in the case of symmetric firms and special lag structures analytic results can be obtained. assume 94 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) symmetric firms with identical cost functions, same initial outputs, identical time lags and speeds of adjustment. then the firms also have identical trajectories, ak(λ) ≡ a(λ), bk(λ) ≡ b(λ), ck(λ) ≡ c(λ) and therefore uk ≡ u. because of this symmetry equations (4.7) and (4.6) become ( − a(λ) + (n − 1)b(λ) ) u + c(λ)v = 0 (4.15) and nhu + (h̄ − λ)v = 0 . (4.16) if h = 0, then from the conditions of theorem 1 we know that h̄ < 0. equation (4.16) implies that in this case either λ = h̄ or v = 0. in the first case this eigenvalue is negative, which cannot destroy stability. in the second case u 6= 0 and equation (4.15) implies that −a(λ) + (n − 1)b(λ) = 0 . (4.17) if h 6= 0, then u = − h̄ − λ nh v (4.18) and by substituting this relation into equation (4.15) we get a single equation for v: ( ( − a(λ) + (n − 1)b(λ) )( − h̄ − λ nh ) + c(λ) ) v = 0 . (4.19) notice that v 6= 0, otherwise from (4.18) u = 0 would follow and eigenvectors must be nonzero. therefore we have the following equation: nhc(λ) + (n − 1)b(λ)(λ − h̄) − a(λ)(λ − h̄) = 0 . notice first that in the case of h = 0 this equation reduces to (4.17) and λ = h̄. note next that this equation can be rewritten as the polynomial equation λ(λ − h̄) ( 1 + λt a )m+1( 1 + λu b )p+1( 1 + λv c )l+1 + (λ − h̄)k ( 1 + λt a )m+1( 1 + λu b )p+1 − nhkr̄ ( 1 + λt a )m+1( 1 + λv c )l+1 − (n − 1)(λ − h̄)kr ( 1 + λu b )p+1( 1 + λv c )l+1 = 0 . (4.20) cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 95 4.1 no information lag assume first that there is no information lag. then t = u = v = 0, and equation (4.20) specializes as λ(λ − h̄) + (λ − h̄)k − nhkr̄ − (n − 1)(λ − h̄)kr = 0 which is quadratic, λ 2 + λ ( − h̄ + k − (n − 1)kr ) + ( − h̄k − nhkr̄ + (n − 1)h̄kr ) = 0 . (4.21) under the conditions of theorem 1, all coefficients are positive. therefore the roots have negative real parts. we have already proved this fact in theorem 1 in the more general case. 4.2 time lag in q assume next that there is no time lag in the outputs but there is time lag in assessing the value of q. then t = v = 0, and if p = 0, then equation (4.20) becomes λ(λ − h̄)(1 + λu ) + (λ − h̄)k(1 + λu ) − nhkr̄ − (n − 1)(λ − h̄)kr(1 + λu ) = 0 or λ 3 u + λ 2 ( 1 + u (k − h̄ − nkr + kr) ) + λ ( − h̄ + k − nkr + kr + u (−kh̄ + nkrh̄ − krh̄) ) + ( − h̄k(r + 1) + kn(rh̄ − r̄h) ) = 0 . (4.22) under the conditions of theorem 1, all coefficients are positive. by applying the routh–hurwitz criterion all roots have negative real parts if and only if [ − h̄ + k(1 − u h̄) ( 1 − (n − 1)r ) ][ 1 − u h̄ + u k ( 1 − (n − 1)r ) ] > u ( − h̄k ( 1 − (n − 1)r ) − knhr̄ ) , (4.23) which is a quadratic inequality in u . by introducing the notation z = 1 − (n − 1)r > 0 it can be written as u 2 kh̄z(h̄ − kz) + u ( (h̄ − kz)2 + knhr̄ ) + (−h̄ + kz) > 0 , (4.24) where the leading coefficient is nonnegative and the constant term is positive. we have to consider next the following cases: case 1. if h ≦ 0, then (4.24) holds for all u > 0, since all coefficients are positive. 96 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) case 2. assume next that h > 0. if r̄ ≧ − (h̄ − kz)2 knh , then the linear coefficient is also nonnegative, so (4.24) holds for all u > 0. if r̄ < − (h̄ − kz)2 knh , then the linear coefficient of (4.24) is negative. we have now the following subcases. (a) assume first that h̄ = 0. then the conditions of theorem 1 imply that r̄ < 0 and h > 0. in this special case (4.24) becomes linear, so it holds if and only if u < −kz (kz)2 + knhr̄ . (4.25) the stability region in the (r̄, u ) space is illustrated in figure 1. figure 1: stability region in the (r̄, u ) space for h̄ = 0 (b) assume next that h̄ 6= 0, then h̄ < 0. the discriminant of (4.24) is zero, if r̄ = r̄ ∗ = −(kz − h̄)2 − 2(kz − h̄) √ −kh̄z knh . (4.26) in this case (4.24) has a real positive root u ∗ , and the equilibrium is locally asymptotically stable if u 6= u ∗. if r̄ > r̄ ∗ , then the discriminant is negative, (4.24) has no real roots, so it is satisfied for all u , that is, the equilibrium is locally asymptotically stable. if r̄ < r̄ ∗ , then the discriminant is positive, there are two real positive roots u ∗ 1 u ∗ 2 . the stability region in the (r̄, u ) space is shown in figure 2. cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 97 2 figure 2: stability region in the (r̄, u ) space for h̄ < 0 returning to case (a), assume that r̄ < − (kz)2 knh . starting from a very small value of u , increase its value gradually. until reaching the critical value u = −kz (kz)2 + knhr̄ , (4.27) the equilibrium is locally asymptotically stable. this stability is lost after crossing the critical value. we will next prove that at the critical value a hopf bifurcation occurs giving the possibility of the birth of limit cycles. notice first that since h̄ = 0, equation (4.22) has the special form λ 3 u + λ 2 (1 + u kz) + λkz + (−knhr̄) = 0 , (4.28) and (4.23) specializes as kz(1 + u kz) > −u knhr̄ . (4.29) at the critical value this inequality becomes equality, so at the critical value (4.28) can be rewritten as 0 = λ 3 u + λ 2 (1 + u kz) + λ −u knhr̄ (1 + u kz) − knhr̄ = ( λ 2 − knhr̄ 1 + u kz ) ( u λ + (1 + u kz) ) 98 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) so the roots are λ1 = − 1 + u kz u and λ23 = ±iα with α 2 = −knhr̄ 1 + u kz ( = kz u ) . so we have a negative real root and a pair of pure complex roots. select now u as the bifurcation parameter and consider the eigenvalues as functions of u . by implicitly differentiating equation (4.28) with respect to u , with the notation λ̇ = dλ du we have 3λ 2 λ̇u + λ 3 + 2λλ̇(1 + u kz) + λ 2 kz + λ̇kz = 0 implying that λ̇ = −λ3 − λ2kz 3λ2u + 2λ(1 + u kz) + kz . at the critical values λ = ±iα, so λ̇ = ±iα3 + α2kz −2α2u ± 2αi(1 + u kz) with real part reλ̇ = 2α 4 (4α4u 2) + 4α2(1 + u kz)2 > 0 (4.30) so all conditions of hopf bifurcation are satisfied. the other case of h̄ 6= 0 can be examined in a similar way. the details are omitted. we will however illustrate this case later in a numerical study. 4.3 time lag in sk assume next that the firms have instantaneous information about their own outputs and parameter q, but have only delayed information about the output of the rest of the industry. in this case u = v = 0, and t > 0. by assuming m = 0, equation (4.20) becomes λ(λ − h̄)(1 + λt ) + (λ − h̄)k(1 + λt ) − nhkr̄(1 + λt ) − (n − 1)(λ − h̄)kr = 0 , (4.31) which is again a cubic equation: λ 3 t + λ 2 (1 − h̄t + t k) + λ ( − h̄ + k − kh̄t − nhkt r̄ − (n − 1)kr ) + ( − kh̄ − nhkr̄ + (n − 1)krh̄ ) = 0 . cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 99 however, in contrast to the previous case there is no guarantee that the coefficients are all positive. under the conditions of theorem 1, the cubic and quadratic coefficients are positive, the linear coefficient is positive if nhkt r̄ < −h̄ + kz − kh̄t (4.32) and the constant term is positive if nhkr̄ < −kh̄z . (4.33) if these relations hold, then the routh-hurwitz stability criterion implies that the eigenvalues have negative real parts if and only if (1 − h̄t + t k)(−h̄ + kz − kh̄t − nhkt r̄) > t (−kh̄z − nhkr̄) (4.34) which can be written as a quadratic inequality of t : t 2 k(h̄ − k)(h̄ + nhr̄) + t ( (h̄ − k)2 − (n − 1)k2r ) + (kz − h̄) > 0 . (4.35) we have to consider now two cases: case 1. if h̄ = 0, then r̄ < 0 and h > 0, so all coefficients of (4.35) are positive, so it holds for all t > 0. notice that in this case (4.32) and (4.33) are also satisfied, so the equilibrium is locally asymptotically stable. case 2. if h̄ 6= 0, then h̄ < 0. the linear coefficient and the constant term of (4.35) are both positive, however the sign of the quadratic coefficient is indeterminate. therefore we have to consider two subcases: (a) assume first that h̄ + nhr̄ ≦ 0. then the quadratic coefficient is nonnegative, so (4.35) holds for all t > 0. in this case nhkt r̄ ≦ −h̄kt , so (4.32) also holds. assume first that h ≧ 0, then (4.33) is also satisfied, consequently the equilibrium is locally asymptotically stable. assume next that h < 0, then the conditions of theorem 1 imply that both r and h̄ are negative. therefore nhkr̄ ≦ −kh̄ < −kh̄ + kh̄(n − 1)r = −kh̄z , so (4.33) also holds, and the equilibrium is locally asymptotically stable again. (b) assume next that h̄ + nhr̄ > 0. in this case both h and r̄ must be negative. in this case r̄ < −h̄ nh , (4.36) the quadratic coefficient of (4.35) is negative, therefore (4.35) has two real roots, one is positive, t ∗ , and the other is negative. clearly, (4.35) holds if t < t ∗ . 100 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) relation (4.33) holds if r̄ > −h̄z nh , (4.37) and under conditions (4.36) and (4.37), relation (4.32) holds if t < kz − h̄ nhkr̄ + kh̄ = t ∗∗ . we will next prove that t ∗ < t ∗∗ , so this last condition is irrelevant for the local asymptotic stability of the equilibrium. let p(t ) denote the left hand side of (4.35), then t ∗ < t ∗∗ if p(t ∗∗ ) < 0. this inequality is the following: (kz − h̄)2(h̄ − k) k(nhr̄ + h̄) + (kz − h̄) [ (h̄ − k)2 − (n − 1)k2r ] k(nhr̄ + h̄) + (kz − h̄) < 0 which can be simplified as (kz − h̄)(h̄ − k) + (h̄ − k)2 − (n − 1)k2r + k(nhr̄ + h̄) < 0 . this relation is equivalent to the following: 0 > zh̄ + nhr̄ = h̄ − nh̄r + h̄r + nhr̄ = h̄(1 + r) − n(h̄r − hr̄) where both terms are negative, which completes the proof. figure 3 shows the stability region in the (r̄, t ) space. the occurrence of hopf bifurcation at the critical value t ∗ can be examined in the same way as shown before, the details are omitted, however a numerical study of his case will be presented in the next section. 5 numerical examples we will examine next a special case of the oligopoly model of [7] with saturated markets. it is assumed that q̇ = n ∑ k=1 xk − αq with some 0 < α < 1, and the market price and the cost functions are linear: f ( n ∑ k=1 xk, q ) = a − ( n ∑ k=1 xk + βq ) and ck(xk) = ckxk (k = 1, 2, . . . , n) . cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 101 figure 3: stability region in the (r̄, t ) space for h < 0 in this case the best response of firm k is rk(sk, q) = −sk − βq + a − ck 2 . we also assume that kk ≡ k and ck ≡ c. it is easy to see that h = 1, h̄ = −α, r = − 1 2 , r̄ = − β 2 and z = 1 − (n − 1)(− 1 2 ) = n + 1 2 . assume u > 0, t = v = 0 as in subsection 4.2. from case 2(b) we know that the equilibrium is locally asymptotically stable if r̄ > r̄ ∗ = − ( k(n+1) 2 + α )2 + 2 ( k(n+1) 2 + α )√ kα(n+1) 2 kn . if r̄ = r̄ ∗ , then it is locally asymptotically stable if u 6= u ∗ = √ kα(n+1) 2 kαz = √ 2 kα(n + 1) , and if r̄ < r̄ ∗ , then the equilibrium is locally asymptotically stable if u u ∗ 2 , where u ∗ 1 and u ∗ 2 are the positive roots of the left hand side of (4.24). we have selected the numerical values a = 25, c = 5, α = k = 1 100 and n = 3. in this case z = 2 and r̄ ∗ = − 3 + 2 √ 2 100 ≈ −0.0583 , 102 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) so if we select β > 3 + 2 √ 2 50 ≈ 0.1166 then we have two real roots of the left hand side of (4.24). so if β = 0.5, then it has the form u 2 6 1003 − u 66 1002 + 3 100 = 0 with the roots u ∗ 1,2 = 50(11 ± √ 119) , so the critical values are u ∗ 1 ≈ 4.564 and u ∗ 2 ≈ 1095.436 . in figure 4, figure 5 and figure 6 we have illustrated this phenomenon. figure 4 shows a shrinking cycle with u = 3. figure 5 shows the complete limit cycle with u = u ∗ 1 , and figure 6 illustrates an expanding cycle with u = 6. figure 4: shrinking cycle we will next illustrate a model with delay in the output of the rest of the industry as in subsection 4.3. assume now the dynamic equation q̇ = −γ n ∑ k=1 xk − αq cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 103 figure 5: complete cycle with positive γ and α. assume also that kk ≡ k, the price function and the cost functions are the same as in the previous case, so we have h = −γ, h̄ = −α, r = − 1 2 and r̄ = − β 2 , and so z = n+1 2 as before. we now assume t > 0, u = v = 0. conditions (4.36) and (4.37) are satisfied if − β 2 < α −nγ and − β 2 > α · n+1 2 −nγ , that is, 2α nγ < β < α(n + 1) nγ . in this case t ∗ is the positive solution of equation −t 2k(α + k)(nβγ − 2α) + t (2α2 + 4αk + k2(n + 1)) + ( 2α + k(n + 1) ) = 0 . we have selected the numerical values a = 25, ck = 5,α = k = γ = 1 10 and n = 3. the value of β has to be between 2 3 and 4 3 , so for the sake of simplicity we have chosen β = 1. then the quadratic equation (4.35) for t ∗ has the form −t 2 2 1000 + t · 10 100 + 6 10 = 0 104 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) figure 6: expanding cycle with positive root t ∗ = 5(5 + √ 37) ≈ 55.4138 . with this critical value a complete limit cycle is obtained, if t < t ∗ then the limit cycle changes to a shrinking one and if t > t ∗ it becomes an expanding cycle. the resulting figures are very similar to those presented in the previous case, so they are not shown here. 6 conclusions in this paper dynamic oligopolies were examined with continuous time scales and intertemporal demand interaction. the effect of earlier demands and consumptions were modelled by introducing an additional state variable, and in an n-firm oligopoly this concept resulted in an (n + 1)-dimensional continuous system. the local asymptotical stability of the equilibrium has been proved under general conditions. this stability however might be lost if the firms have only delayed information on the demand interaction, on their own outputs and also on the outputs of the competitors. by assuming continuously distributed time lags stability conditions were derived for important special cases and in cases when instability occurs the occurrence of hopf bifurcation was investigated giving the possibility of the birth of limit cycles. the theoretical results have been illustrated by computer studies. received: april 21, 2008. revised: may 09, 2008. cubo 11, 2 (2009) dynamic oligopolies and intertemporal demand interaction 105 references [1] bischi, g., chiarella, c., kopel, m. and szidarovszky, f., dynamic oligopolies: stability and bifurcations., springer-verlag, berlin/heidelberg/new york (in press), 2009. [2] chiarella, c. and szidarovszky, f., the birth of limit cycles in nonlinear oligopolies with continuously distributed information lags, in m. dror, p. l’ecyer, and f. szidarovszky, editors, modelling uncertainty, kluwer academic publishers, dordrecht, (2001), pp. 249–268. [3] cournot, a., recherches sur les principles mathématiques de la théorie de richessess, hachette, paris. (english translation (1960): researches into the mathematical principles of the theory of wealth, kelley, new york), 1838. [4] froot, k. and klemperer, p., exchange rate pass-through when market share matters, american economic review, 79(1989), 637–653. [5] guckenheimer, j. and holmes, p., nonlinear oscillations, dynamical systems and bifurcations of vector fields, springer-verlag, berlin/new york, 1983. [6] okuguchi, k., expectations and stability in oligopoly models, springer-verlag, berlin/heidelberg/new york, 1976. [7] okuguchi, k. and szidarovszky, f., the theory of oligopoly with multi-product firms. springer-verlag, berlin/heidelberg/new york, 2nd edition, 1999. [8] okuguchi, k. and szidarovszky, f., oligopoly with intertemporal demand interaction, journal of economic research, 8(2003), 51–61. [9] tivig, t., exchange rate pass-through in two-period duopoly, international journal of industrial organization, 14(1996), 631–645. n06-dymol_int_demand_v3 maskit.dvi cubo a mathematical journal vol.12, no¯ 01, (175–179). march 2010 a short note on m-symmetric hyperelliptic riemann surfaces ∗ rubén a. hidalgo departamento de matemáticas, universidad técnica federico santa maŕıa, valparáıso, chile email : ruben.hidalgo@usm.cl abstract we provide an argument, based on schottky groups, of a result due to b. maskit which states a necessary and sufficient condition for the double oriented cover of a planar compact klein surface of algebraic genus at least two to be a hyperelliptic riemann surface. resumen damos un argumento, basado en grupos de schottky, de un resultado debido a b. maskit el cual establece una condición necesária y suficiente para el cubrimiento duplo orientado de una superficie de klein compacta planar de genero algebrico al menos dos ser una superficie de riemann hipereliptica. key words and phrases: schottky groups, hyperelliptic riemann surfaces. math. subj. class.: 30f10, 30f40. ∗partially supported by projects fondecyt 1070271 and utfsm 12.09.02 176 rubén a. hidalgo cubo 12, 1 (2010) 1 preliminaries let us consider a collection of (g + 1) pairwise disjoint round circles on the riemann sphere, say c1,..., cg+1, bounding a common domain d of connectivity (g +1). if we denote by τj the reflection on the circle cj , then the group g = 〈τ1, ..., τg+1〉 is an extended kleinian group, isomorphic to the free product of (g + 1) copies of z2. we say that g is a planar extended schottky group of rank g. the region of discontinuity ω of a planar extended schottky group of rank g is connected (the complement of a cantor set for g ≥ 2) and s = ω/g is a planar compact klein surface of algebraic genus g, that is, holomorphically equivalent to the closure of d. quasiconformal deformation theory asserts that every planar compact klein surface of algebraic genus g is obtained in this way. let g = 〈τ1, ..., τg+1〉 a planar extended schottky group of rank g. let g + be its index two subgroup of orientation preserving transformations. it turns out that g+ is a (classical) schottky group of genus g, freely generated by the transformations aj = τg+1τj , for j = 1, ..., g. the closed riemann surface s+ = ω/g+ is the double oriented cover of the planar compact klein surface s = ω/g. any of the transformation in g − g+ induces an anticonformal involution τ : s+ → s+ (that is, a real structure on s+) so that s = s+/〈τ〉. it follows that the number of ovals of τ (its connected components of fixed points) is equal to (g + 1), in particular, (s+, τ ) is a m -symmetric riemann surface. let us denote by π : s+ → s the two-fold (branched) klein cover induced by τ and by p : ω → s+ the schottky covering of s+ induced by the schottky group g+. in [4] b. maskit proved the following result. theorem 1.1. let g be a planar extended schottky group of rank g ≥ 2, defined by tye circles c1,..., cg+1. then the riemann surface ω/g + is hyperelliptic if and only if there is a circle which is orthogonal to all cj , j = 1, ..., g + 1. the aim of this note is to provide a different proof of theorem 1.1 relaying more on the schottky groups spirit. we need to recall some extra definitions. let σ1,... σg+1 be pairwise disjoint simple loops on the riemann sphere, all of them bounding a common domain d of connectivity g + 1. assume that for each j = 1, ..., g + 1, there is a möbius transformation of order 2, say ej , so that ej permutes both topological discs discs bounded by σj (in particular, both fixed points of ej belong to σj ). the group k = 〈e1, ..., eg+1〉 is a kleinian group, isomorphic to a free product of g + 1 copies of z2, called a whittaker group of rank g [3]. if ω is the region of discontinuity of k, then ω is connected (the complement of a cantor set for g ≥ 2) and s = ω/k is an orbifold of signature (0, 2g + 2; 2, ..., 2), that is, the riemann sphere with exactly 2(g + 1) conical points, all of them of conical order 2. inside k there is exactly one index two torsion free subgroup, say k(2). it turns out that k(2) is a schottky group of rank g, called a hyperelliptic schottky group, which is freely generated by the transformations eg+1e1,..., eg+1eg. in this case, s (2) = ω/k(2) turns out to cubo 12, 1 (2010) a short note on m -symmetric hyperelliptic ... 177 be a hyperelliptic riemann surface, the hyperellitic involution (unique for g ≥ 2 [2]) is induced by any of the transformations in k − k(2). the projection of the fixed points of e1,..., eg+1 to s (2) provides the 2(g + 1) fixed points of the hyperelliptic involution. 2 the necessary part let us consider a planar extended schottky group g of rank g ≥ 2, say generated by the reflections τ1,..., τg+1 on a collection of (g+1) pairwise disjoint round circles on the riemann sphere, say c1,..., cg+1, bounding a common domain d of connectivity (g + 1). let g + be the index two orientation preserving schottky subgroup and let s+ = ω/g+, where ω is the region of discontinuity of g (the same as for g+). as before, we denote by τ : s+ → s+ the real structure induced on s+ by the action of g. let us denote by o1 = p (c1),...., og+1 = p (cg+1) the ovals of τ . let us assume s+ is a hyperelliptic riemann surface and let j : s+ → s+ be its hyperelliptic involution. as the hyperelliptic involution is unique [2], j and τ should commute, in particular, the collection of ovals of τ is invariant under j. the schottky group g+ is defined by the ovals o1,..., og+1, that is, by the normalizer (in the fundamental group) of them. it follows that the hyperelliptic involution lifts to a conformal automorphism ĵ : ω → ω under p : ω → s+, that is, jp = p ĵ. we have that ĵ2 ∈ g+. as j has fixed point, we may assume that ĵ also has fixed points, in particular, ĵ2 = i. it is known that ω is of class oad; that is, it admits no holomorphic function with finite dirichlet norm (see [1, pg 241]). it follows from this (see [1, pg 200]) that every conformal map from ω into the riemann sphere is a möbius transformation. in this way, ĵ is the restriction of a möbius transformation of order two. lemma 2.1. each oval has exactly two fixed points of j and each fixed point belongs to some oval. moreover, each oval is invariant under j. proof. let us denote by d1 and d2 the two connected components of s + − ∪ g+1 j=1oj . if one of the fixed points of j is not contained in ∪ g+1 j=1oj , then we should have that j(d1) = d1. but as d1 is planar (isomorphic to the closure of d) we will have that the restriction of j onto d1 coincides with a möbius transformation of order 2. it will follows then that j must have at most 4 fixed points on s, a contradiction. in particular, every fixed point of j is contained in some oval. also, if the oval ok contains a fixed point of j, then we should have that j(ok) = ok. in that case, we have that ok should have exactly two fixed points of j. as j contains exactly 2(g + 1) fixed points and we have exactly (g + 1) ovals, we have that: (i) each oval has exactly two fixed points of j and (ii) each oval is invariant unde! r j. by the previous lemma, for each k ∈ {1, ..., g + 1}, j(ok) = ok. it follows that we may choose liftings ĵ1,..., ĵg+1, of the hyperelliptic involution, each one of order 2 so that ĵk(ck) = ck and 178 rubén a. hidalgo cubo 12, 1 (2010) both fixed point of ĵk are contained in ck. let us consider the whittaker group ĝ = 〈ĵ1, ..., ĵg+1〉, and its hyperelliptic schottky group ĝ(2) = 〈ĵg+1ĵ1, ...., ĵg+1ĵg〉. we have, by the construction, that g+ = ĝ(2). lemma 2.2. ĵg+1ĵk = τg+1τk, k = 1, ..., g. proof. let us first observe that the circles ck, c ′ k = τg+1(ck) and ĵg+1(ck) are lifting of the oval ok. the circles c1, c ′ 1,..., cg, c ′ g (respectively, the circles c1, ĵg+1(c1),..., cg and ĵg+1(cg)) form a standard fundamental domain for g+. as ĵg+1(ck) must belong to the disc bounded by the circle cg+1 which does not contains the circle ck, we should have that ĵg+1(ck) should be one of the discs c ′ 1,...., c ′ g. but as mentioned, the only such disc which is a lifting of ok is exactly c ′ k . we have that ĵg+1(ck) = c ′ k . now, we have that the loxodromic transformations ĵg+1ĵk, τg+1τk ∈ g + send ck onto c ′ k and each maps the exterior of ck onto the interior of c ′ k . it follows that the transformation η = τkτg+1ĵg+1ĵk ∈ g + keeps invariant the circle ck and each of its bounded discs. as ck is contained on the region of discontinuity of g+, it follows that η cannot be loxodromic. as g+ only contains loxodromic transformations besides the identity, we should have η = i. let us recall that, if c is a circle on the riemann sphere and p and q are any two different points on it, then there is a unique orthogonal circle to it passing through these two given points. the previous fact together with the fact that a circle is uniquely determined by 3 points on it and the following lemma asserts the existence of a common orthogonal circle as desired to prove the necessary part of the theorem. lemma 2.3. let us consider two pairwise disjoint circles, say c1 and c2. let σj be the reflection of cj and tj be an elliptic transformation of order 2 preserving cj whose fixed points belong to cj . then σ2σ1 = t2t1 if and only if there is a circle c such that: (i) the fixed points of t1 and t2 belong to c and (ii) c is orthogonal to both c1 and c2. proof. we may normalize by a suitable möbius transformation in order to assume that c1 is the unit circle and c2 is the circle centered at the origin and a positive radius r > 1. in this case we cubo 12, 1 (2010) a short note on m -symmetric hyperelliptic ... 179 have that σ2σ1(z) = r 2z. the equality t2t1 = σ2σ1 then obligates to have that the fixed points of both t1 and t2 on a line through 0. 3 the sufficiency part let us assume we have (g + 1) circles, say c1,..., cg+1, each one of them orthogonal to a common circle c0. let us denote by τk the reflection on the circle ck, for k = 0, 1, .., g + 1. let us denote by ηk = τ0τk, for k = 1, ..., g + 1, which are elliptic transformations of order 2. let g be the planar extended schottky group generated by the reflections τ1,..., τg+1 and let ĝ be the whittaker group generated by the involutions η1,..., ηg+1. it easy to see that g + is the hyperelliptic subgroup of ĝ. it follows then that the uniformized surface by g+ is hyperelliptic, with hyperelliptic involution induced by η0. received: september 2008. revised: january 2009. references [1] ahlfors, l. and sario, l., riemann surfaces, princeton university press, princeton nj, 1960. [2] farkas, h. and kra, i., riemann surfaces, second edition. graduate texts in mathematics, 71, springer-verlag, new york, 1992. [3] keen, l., on hyperelliptic schottky groups, ann. acad. sci. fenn. series a.i. mathematica, 5, 1980. [4] maskit, b., remarks on m-symmetric riemann surfaces, contemporary math., 211 (1997), 433–445. cubo a mathematical journal vol.11, no¯ 05, (117–128). december 2009 structure of resolvents of elliptic cone differential operators: a brief survey juan b. gil penn state altoona, 3000 ivyside park, altoona, pa 16601, usa. email : jgil@psu.edu abstract the resolvent of an elliptic cone differential operator is surveyed under the aspect of its pseudodifferential structure and its asymptotic behavior as the spectral parameter tends to infinity. the exposition is descriptive and focuses on the case when the domain of the given operator is stationary. resumen se examina la resolvente de un operador diferencial de tipo cónico, elíptico, bajo el aspecto de su estructura pseudodiferencial y su comportamiento asintótico cuando el parámetro espectral tiende a infinito. la exposición es descriptiva y se enfoca en el caso cuando el dominio del operador dado es estacionario. key words and phrases: resolvents, trace asymptotics, manifolds with conical singularities. math. subj. class.: 58j35, 35p05, 47a10. 118 juan b. gil cubo 11, 5 (2009) 1 introduction the purpose of this paper is to give a brief descriptive account of joint work with thomas krainer and gerardo mendoza on resolvents of general cone differential operators whose symbols satisfy natural ellipticity conditions. cone operators arise particularly in the study of differential equations on a manifold with conical singularities – basic case of an incomplete riemannian manifold. the results presented here rely on the analytic and geometric approach developed in the series of papers [3]–[7]. there the reader can find details and further information, including complete proofs, examples, applications, as well as an extensive discussion on the existing literature in the subject. we start our survey by reviewing the necessary functional analytic framework. let h be a hilbert space and let d0 ⊂ h be a dense subspace. let a be a linear operator, initially defined as an unbounded operator a : d0 ⊂ h → h. we are interested in the closed extensions of a in h. in other words, we are looking for domains d ⊂ h with d0 ⊂ d to which a can be extended as a closed operator. there are two canonical such domains: dmin(a) = closure of d0 in h with respect to ‖ · ‖a, dmax(a) = {u ∈ h : au ∈ h}, where ‖u‖a = ‖u‖h + ‖au‖h. both domains are dense in h and the extension ad : d ⊂ h → h is closed if and only if d is a closed subspace of dmax(a) that contains dmin(a). thus, there is a one-to-one correspondence between the closed extensions of a and the closed subspaces of dmax(a)/dmin(a). if the operator a is fixed and there is no possible ambiguity, we will write dmin and dmax instead of dmin(a) and dmax(a). if ad is closed in h, so is ad − λ = ad − λi for every λ ∈ c. if ad − λ is invertible and (ad − λ) −1 is bounded in h, λ is said to be an element of res(ad), the resolvent set of ad. the family (ad −λ) −1 is called the resolvent of ad, and the set spec(ad) = c\ res(ad) is the spectrum of ad. a closed sector (or ray) λ ⊂ c is called a sector (or ray) of minimal growth for ad : d → h if there exists r > 0 such that ad − λ is invertible for every λ in λr = {λ ∈ λ : |λ| ≥ r}, and the resolvent satisfies either of the equivalent estimates ∥∥(ad − λ)−1 ∥∥ l (h) ≤ c/|λ|, ∥∥(ad − λ)−1 ∥∥ l (h,d) ≤ c, cubo 11, 5 (2009) structure of resolvents of elliptic cone ... 119 for some c > 0 and all λ ∈ λr. our research on elliptic cone operators has been guided by two basic goals: one is to find verifiable conditions on a and d for the resolvent of ad to exist and for a sector λ ⊂ c to be a sector of minimal growth for ad. this information is particularly relevant for nonselfadjoint operators. secondly, we are interested in describing the pseudodifferential structure and asymptotic properties of the resolvent as the spectral parameter λ tends to infinity. in this paper, we will discuss our progress and main difficulties around these goals. we finish this introduction by mentioning that the asymptotic information obtained for the resolvent can be directly applied, for instance, in the short-time asymptotic analysis of heat traces, and in the study of the meromorphic structure of zeta functions. this follows from the standard functional calculus, cf. [10], [15]. 2 cone operators let m be a smooth compact n-dimensional manifold with boundary y = ∂m . we fix a defining function x for y and choose a collar neighborhood [0, ε) × y of the boundary of m . let e be a smooth vector bundle over m . a cone differential operator of order m on sections of e is an element a = x−mp with p in diff m b (m ; e); the space of totally characteristic differential operators of order m, see [13]. thus a is a linear differential operator on c∞( ◦ m ; e) of order m, which near y , in local coordinates (x, y) ∈ (0, ε) × y , takes the form a = x−m ∑ k+|α|≤m akα(x, y)(xdx) kdαy (2.1) with coefficients akα smooth up to x = 0. these operators occur, for example, when introducing polar coordinates around a point or as laplace-beltrami operators corresponding to cone metrics, cf. [1]. every cone operator a ∈ x−m diffmb (m ; e) has a principal c-symbol cσσ(a) defined on the c-cotangent ct ∗m of m . over the interior of m , cσσ(a) is essentially the usual principal symbol of a. near the boundary y , cσσ(a) is of the form ∑ k+|α|=m akα(x, y)ξ kηα, see (2.1). the operator a is said to be c-elliptic if cσσ(a) is invertible on ct ∗m\0, and the family a − λ is c-elliptic with parameter λ ∈ λ ⊂ c if cσσ(a) − λ is invertible on ( ct ∗m × λ)\0. with a = x−mp one associates the (indicial) family p̂ (σ) = ∑ k+|α|≤m akα(0, y)σ kdαy , 120 juan b. gil cubo 11, 5 (2009) also called the conormal symbol of a. if a is c-elliptic, then p̂ (σ) is invertible for all σ ∈ c except a discrete set, specb(a), called the boundary spectrum of a. fix a positive b-density m on m and let l2b (m ; e) denote the l 2 space with respect to a hermitian form on e and the density m. for s ∈ n let hsb (m ; e) = {u ∈ l 2 b (m ; e) : p u ∈ l 2 b (m ; e) ∀p ∈ diff s b(m ; e)}. throughout this paper we will assume that a is a c-elliptic cone operator of order m > 0, and as reference hilbert space we choose, for instance, x−m/2l2b(m ; e). consider a as a densely defined unbounded operator a : c∞c ( ◦ m ; e) ⊂ x−m/2l2b (m ; e) → x −m/2l2b (m ; e). in [12] lesch showed that, in the situation at hand, dmax/dmin is finite dimensional and every closed extension of a, ad : d ⊂ x −m/2l2b (m ; e) → x −m/2l2b(m ; e), is fredholm. modulo dmin the elements of dmax are determined by their asymptotic behavior near the boundary of m . the structure of these asymptotics depends on the elements σ in the boundary spectrum of a with |ℑσ| < m/2. in [9] it was shown that dmin = dmax ∩ ( ⋂ ε>0 xm/2−εhmb (m ; e) ) , and dmin = x m/2hmb (m ; e) if and only if specb(a) ∩ {σ ∈ c : ℑσ = − m 2 } = ∅. moreover, there exists ε > 0 such that dmax →֒ x −m/2+εhmb (m ; e). the embedding (dmax, ‖·‖a) →֒ x −m/2l2b (m ; e) is therefore compact. thus, for every domain d with dmin ⊂ d ⊂ dmax and all λ ∈ c, ad − λ : d → x −m/2l2b(m ; e) is fredholm with ind(ad − λ) = ind ad. consequently, spec(ad) 6= c ⇒ ind ad = 0. conversely, if ind ad = 0, then spec(ad) is either discrete or all of c. remark 2.2. the complexity of the spectrum of a cone operator can already be observed in the simple case of the laplacian on the interval [0, 1], see [6]. in that case, the following situations are possible: cubo 11, 5 (2009) structure of resolvents of elliptic cone ... 121 • closed extensions with index zero whose spectrum is empty. • closed extensions with index zero whose spectrum is c. • a family of domains dβ with dβ → d0 (in a suitable sense) such that spec(∆dβ ) is discrete and independent of β, but spec(∆d0 ) = c. 3 the model operator and rays of minimal growth by means of a taylor expansion at x = 0, a cone operator a ∈ x−m diff mb (m ; e) induces a decomposition xma = p0 + xp̃1, where p̃1 ∈ diff m b (m ; e) and p0 is an operator with coefficients independent of x near y . we let y ∧ = [0, ∞) × y and consider p0 as an element of diff m b (y ∧ ; e). we call the operator x−mp0 ∈ x −m diff m b (y ∧ ; e) the model operator of a and denote it by a∧. if a is written as in (2.1) near the boundary, then a∧ = x −m ∑ k+|α|≤m akα(0, y)(xdx) kdαy . this operator acts on c∞c ( ◦ y ∧; e) and can be extended as a densely defined closed operator in x−m/2l2b (y ∧ ; e). the domains of the minimal and maximal closed extensions of a∧ are denoted by d∧,min and d∧,max, and like for a, the space d∧,max/d∧,min is finite dimensional. in fact, there is a natural isomorphism θ : dmax/dmin → d∧,max/d∧,min that allows passage from domains over m to domains over y ∧. with a domain d for a we associate a domain d∧ for a∧ defined via d∧/d∧,min = θ(d/dmin). (3.1) the model operator and its canonical domains d∧,min and d∧,max exhibit an important invariance property with respect to the natural r+-action on y ∧. this property is crucial for the characterization of domains and in the geometric study of resolvents of elliptic cone operators. for this reason, it has been incorporated in our systematic approach and is worth reviewing: let r+ ∋ ̺ 7→ κ̺ : x −m/2l2b (y ∧ ; e) → x−m/2l2b (y ∧ ; e) be the one-parameter group of isometries which on functions is defined by (κ̺f )(x, y) = ̺ m/2f (̺x, y). 122 juan b. gil cubo 11, 5 (2009) it is easily verified that a∧ satisfies κ̺a∧ = ̺ −ma∧κ̺, (3.2) thus the domains d∧,min and d∧,max are both κ-invariant. in particular, κ induces an action on d∧,max/d∧,min. a domain d for a cone operator a is said to be stationary if its associated domain d∧, see (3.1), is κ-invariant. the relation (3.2) implies a∧ − ̺ mλ = ̺mκ̺(a∧ − λ)κ −1 ̺ (3.3) for every ̺ > 0 and λ ∈ c. this property is called κ-homogeneity, see e.g. [14]. any intermediate space d∧ with d∧,min ⊂ d∧ ⊂ d∧,max gives rise to a closed extension a∧,d∧ : d∧ ⊂ x −m/2l2b (y ∧ ; e) → x−m/2l2b (y ∧ ; e). as opposed to a, even if the c-symbol of a∧ is invertible, not every such extension is fredholm. however, for certain values of λ ∈ c, a∧ − λ is better behaved: we define the background resolvent set of a∧ as bg-res(a∧) = {λ ∈ c : a∧,min − λ injective and a∧,max − λ surjective}. using the κ-homogeneity (3.3) one can prove that this set is a union of open sectors. moreover, if λ ∈ bg-res(a∧), then a∧,d∧ − λ : d∧ ⊂ x −m/2l2b (y ∧ ; e) → x−m/2l2b (y ∧ ; e) is fredholm with ind(a∧,d∧ − λ) = ind(a∧,min − λ) + dim d∧/d∧,min. the index is constant on connected components of bg-res(a∧). let λ be a sector in bg-res(a∧) and consider the grassmannian g = {d∧/d∧,min : ind(a∧,d∧ − λ) = 0 for λ ∈ λ} (3.4) of d-dimensional subspaces of d∧,max/d∧,min, where d = − ind(a∧,min − λ). one of the main reasons for considering the model operator in the context of spectral theory for cone operators is the following result: theorem 3.5. let a ∈ x−m diff mb (m ; e) be c-elliptic with parameter in λ. let d be a domain for a and let d∧ be its associated domain. if λ is a sector of minimal growth for a∧,d∧ , then it is a sector of minimal growth for ad. so, the question on the existence of rays of minimal growth for a cone operator ad is reduced to studying rays of minimal growth for the corresponding a∧,d∧ . the simplest case to study is when the domain d∧ is κ-invariant. cubo 11, 5 (2009) structure of resolvents of elliptic cone ... 123 proposition 3.6. suppose d∧ is κ-invariant. a sector λ is a sector of minimal growth for a∧,d∧ if and only if λ\{0} ⊂ bg-res(a∧) and a∧,d∧ − λ0 is invertible for some λ0 ∈ λ\{0}. if d∧ is not κ-invariant, it generates an orbit on the grassmannian g, see (3.4). in this case, we consider the attracting set of its κ-orbit as ̺ → 0: ω − (d∧) = {d ∈ g : ∃ ̺k → 0 such that d = lim k→∞ κ̺k (d∧/d∧,min)}. theorem 3.7. let λ0 ∈ bg-res(a∧). the ray γ through λ0 is a ray of minimal growth for a∧,d∧ iff a∧,d − λ0 is invertible for all d such that d/d∧,min ∈ ω − (d∧). remark 3.8. the above invertibility condition can be expressed in terms of the nonvanishing of a suitable finite determinant. the limiting set ω−(d∧) can be interpreted as the “principal object” associated with the domain of a. a nice and explicit application of the previous theorem to second order regular singular operators on a metric graph can be found in [6]. 4 structure of resolvents let λ be a closed sector in c and assume that a ∈ x−m diffmb (m ; e) is c-elliptic with parameter in λ. let ad be a closed extension of a in x −m/2l2b(m ; e) and let d∧ be the associated domain of d. by theorem 3.5 we know that if λ is a sector of minimal growth for a∧,d∧ , then it is a sector of minimal growth for ad. in particular, in such a sector the resolvent (ad − λ) −1 exists, thus admin − λ : dmin → x −m/2l2b is injective and admax − λ : dmax → x −m/2l2b is surjective. let kλ = ker(admax − λ) and rλ = rg(admin − λ). if λ ∈ res(ad), then dmax = kλ ⊕ d. (4.1) let bmin(λ) be the left-inverse of admin −λ with kernel r ⊥ λ and let bmax(λ) be the right-inverse of admax − λ with range k ⊥ λ . we then have (see [3, section 5]) (ad − λ) −1 = bmax(λ) + [ 1 − bmin(λ)(a − λ) ] πkλ,d bmax(λ), (4.2) where πkλ,d is the projection on kλ with kernel d according to (4.1). in fact, the projection can be replaced by πmaxπkλ,dπmax, where πmax is the projection onto the orthogonal complement of 124 juan b. gil cubo 11, 5 (2009) dmin in dmax. with similar computations one can also analyze the resolvent of the model operator a∧ on d∧, see [3, section 8]. if we are interested in the asymptotic properties of the resolvent, it is accustomed to use a suitable parameter-dependent pseudodifferential calculus to approximate the resolvent by means of a “good” parametrix. in [4] we showed: theorem 4.3. if λ is a sector of minimal growth for a∧,d∧ , then (ad − λ) −1 = b(λ) + gd(λ) for λ ∈ λ, where b(λ) is a parametrix of admin − λ with b(λ)(admin − λ) = 1 for λ sufficiently large, and gd(λ) is a smoothing operator of finite rank. in the proof of this theorem the first major step is the construction of the parametrix b(λ). an important aspect of our parametrix is that it is an actual left-inverse for λ sufficiently large. the family gd(λ) is then constructed as follows. under the given assumptions, there is an operator family k(λ) : cd → x−m/2l2b , with d = − ind admin , such that ( (a − λ) k(λ) ) : dmin ⊕ c d → x−m/2l2b is invertible for λ ∈ λr for some r > 0. its inverse can be written as ( (admin − λ) k(λ) )−1 = ( b(λ) t (λ) ) , where b(λ) is the parametrix of admin − λ, and t (λ) : x −m/2l2b → c d is a smooth family of operators with “nice” asymptotic properties. let e be any d-dimensional complement of dmin in d. if we split d = dmin ⊕ e and write ad − λ = ( (admin − λ) (a − λ)|e ) , then ( b(λ) t (λ) )( (admin − λ) (a − λ)|e ) = ( 1 b(λ)(a − λ)|e 0 t (λ)(a − λ)|e ) , so ad − λ is invertible if and only if t (λ)(a − λ) : e → c d is invertible. now, since t (λ)(a − λ) vanishes on dmin, it induces an operator on the quotient: f (λ) = [t (λ)(a − λ)] : dmax/dmin → c d, and ad − λ is invertible if and only if fd(λ) = f (λ)|d/dmin is invertible. on the other hand, 1 − b(λ)(a − λ) also vanishes on dmin, so it induces a map [ 1 − b(λ)(a − λ) ] : dmax/dmin → x −m/2l2b , cubo 11, 5 (2009) structure of resolvents of elliptic cone ... 125 and we end up with the decomposition (ad − λ) −1 = b(λ) + [ 1 − b(λ)(a − λ) ] fd(λ) −1t (λ). (4.4) this decomposition and the asymptotic properties of its components are crucial for the results presented in the next section. observe that both representations of the resolvent, (4.2) and (4.4), give a more refine picture of how the domain d affects it. in each case, the domain-dependent contribution is reduced to a family of linear operators acting on finite dimensional spaces. from these representations one can derive explicit krein-like formulas. 5 trace expansions under the assumptions of the previous section, if λ is a sector of minimal growth for ad, then for ℓ ∈ n sufficiently large, (ad −λ) −ℓ is an analytic family of trace class operators in x−m/2l2b (m ; e). in this section we give a complete asymptotic expansion of tr(ad − λ) −ℓ, as |λ| → ∞, in the case when the domain is stationary. theorem 5.1. suppose d is stationary. then, for any ϕ ∈ c∞(m ; end(e)) and ℓ ∈ n with mℓ > n, tr ( ϕ(ad − λ) −ℓ ) ∼ ∞∑ j=0 mj∑ k=0 αjkλ n−j m −ℓ log k λ as |λ| → ∞, with a suitable branch of the logarithm, with constants αjk ∈ c. the numbers mj vanish for j < n, and mn ≤ 1. in general, the αjk depend on ϕ, a, d, and ℓ, but the coefficients αjk for j < n and αn,1 do not depend on d. if both a and ϕ have coefficients independent of x near ∂m , then mj = 0 for all j > n. as mentioned in the introduction, this result has direct consequences in the asymptotic analysis of spectral functions defined by means of the resolvent. for some 0 < ε0 < π/2, let λ = {λ ∈ c : | arg λ| ≥ π 2 − ε0} be a sector of minimal growth for ad. then it is known that −ad generates an analytic semigroup in h given by e−tad = i 2π ∫ γ e−tλ(ad − λ) −1 dλ for t > 0, (5.2) where γ is a contour in λ such that for λ large, | arg λ| = π 2 − δ for some 0 < δ < ε0. if, in addition, the resolvent set of ad contains an open neighborhood v of the origin, then for z ∈ c with ℜz < 0, we define azd = i 2π ∫ γ λz (ad − λ) −1 dλ, (5.3) 126 juan b. gil cubo 11, 5 (2009) where γ is an infinite path in λ ∪ v that runs along a ray of minimal growth to a small circle centered at the origin and contained in v , then clockwise about the origin avoiding the negative real axis, and out of v along a ray of minimal growth. in both equations (5.2) and (5.3), the path γ is chosen to be positively oriented with respect to the spectrum of ad. now, theorem 5.1 together with (5.2) give the asymptotic expansion tr(ϕe−tad ) ∼ ∞∑ j=0 aj t j−n m + ∞∑ j=0 mj∑ k=0 ajkt j m log k t as t → 0+. moreover, if a is bounded from below on the minimal domain, then the ζ-function ζad (s) = tr(a −s d ) of any selfadjoint extension with stationary domain (e.g. the friedrichs extension) is holomorphic for ℜs > n/m and has a meromorphic extension to all of c with poles contained in the set { n−j m : j ∈ n0}. (5.4) this follows from theorem 5.1 together with (5.3), or via the formula a−s d = 1 γ(s) ∫ ∞ 0 ts−1e−tad dt, ℜs > 0, which implies ζad (s) = 1 γ(s) m(hd)(s), where m(hd) denotes the mellin transform of the function hd(t) = tr(e −tad ). if d is nonstationary, the analysis for the asymptotics passed the n-th term is considerably more involved. for instance, at the level of resolvents, these asymptotics may include rational functions in log λ and complex powers of λ. this case is discussed in [8]. with the results from [7], one gets the partial expansion tr ( ϕ(ad − λ) −ℓ ) ∼ n−1∑ j=0 αj,0λ n−j m −ℓ + αn,1λ −ℓ log λ + o(|λ|−ℓ) as |λ| → ∞, which implies tr(ϕe−tad ) ∼ n−1∑ j=0 aj t j−n m + an,1 log t + o(1) as t → 0 +. consequently, we get that ζad (s) extends meromorphically to ℜs > 0, but we do not know in general how this function behaves in all of c. the complexity of the nonstationary case has already been observed in simple situations. there are examples on the half-line (see [2]) where the ζ-function extends meromorphically with cubo 11, 5 (2009) structure of resolvents of elliptic cone ... 127 additional poles not contained in the set (5.4). moreover, for partial differential operators of laplace type (with coefficients independent of the radial variable x), the ζ-function may not admit a meromorphic extension to all of c due to the presence of logarithmic singularities, see e.g. [11]. acknowledgment the present exposition is based on a talk given at the “second symposium on scattering and spectral theory” in serrambi, pernambuco, brazil, august 2008. their financial support is greatly appreciated. received: february, 2009. revised: may, 2009. references [1] cheeger, j., on the spectral geometry of spaces with cone-like singularities, proc. nat. acad. sci., 76 (1979), 2103–2106. [2] falomir, h., pisani, p.a.g. and wipf, a., pole structure of the hamiltonian ζ-function for a singular potential, j. phys. a, 35 (2002), 5427–5444. [3] gil, j., krainer, t. and mendoza, g., geometry and spectra of closed extensions of elliptic cone operators, canad. j. math., 59 (2007), no. 4, 742–794. [4] , resolvents of elliptic cone operators, j. funct. anal., 241 (2006), no. 1, 1–55. [5] , on rays of minimal growth for elliptic cone operators, oper. theory adv. appl., 172 (2007), 33–50. [6] , a conic manifold perspective of elliptic operators on graphs, j. math. anal. appl., 340 (2008), 1296–1311. [7] , trace expansions for elliptic cone operators with stationary domains, preprint, 2008. [8] , dynamics on grassmannians and resolvents of cone operators, preprint, 2009. [9] gil, j. and mendoza, g., adjoints of elliptic cone operators, amer. j. math., 125 (2003), no. 2, 357–408. [10] gilkey, p., invariance theory, the heat equation, and the atiyah-singer index theorem, crc press, boca raton, ann arbor, 1996, second edition. [11] kirsten, k., loya, p. and park, j., the very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2/dr2 − 1/(4r2), j. math. phys., 47 (2006), no. 4, 043506, 27 pp. 128 juan b. gil cubo 11, 5 (2009) [12] lesch, m., operators of fuchs type, conical singularities, and asymptotic methods, teubnertexte zur math. vol 136, b.g. teubner, stuttgart, leipzig, 1997. [13] melrose, r., the atiyah-patodi-singer index theorem, research notes in mathematics, a k peters, ltd., wellesley, ma, 1993. [14] schulze, b.–w., pseudo-differential operators on manifolds with singularities, studies in mathematics and its applications, 24. north-holland publishing co., amsterdam, 1991. [15] seeley, r., complex powers of an elliptic operator, singular integrals, ams proc. symp. pure math. x, 1966, amer. math. soc., providence, 1967, pp. 288–307. b8-cuboproc cubo a mathematical journal vol.10, n o ¯ 03, (93–102). october 2008 dualities useful in bond percolation pedro ferreira de lima unversidade regional do cariri – urcac av. leão sampaio s/n, cep.63.010-970, crajubar-ce, brasil email: limapf@yahoo.com.br and andré toom universidade federal de pernambuco – ufpe, ccen-departamento de estat́ıstica, cep.50.740-540, recife-pe, brasil email: toom@de.ufpe.br abstract we state four facts about dual pairs of graphs drawn in a plane. these facts pertain respectively to finite non-oriented, infinite non-oriented, finite oriented, and infinite oriented graphs. we do not include proofs of the former two facts (although we have them), but show that these facts are “evident” in some naive sense. then we deduce the latter two facts from the former two ones. all of these facts can be used to obtain upper estimations for critical values of two-dimensional percolation models and we present three references and one example to illustrate this. resumen nosotros establecemos cuatro hechos acerca de pares duales de grafos en el plano. estos hechos se relacionan respectivamente con grafos finito no-orientado, infinito noorientado, finito orientado y infinito orientado. nosotros no incluimos demostraciones 94 pedro f. de lima and andré toom cubo 10, 3 (2008) de los dos anteriores hechos (aunque tenemos estas) pero demostramos que estos hechos son “evidentes” en algun sentido ingenuo. entonces deducimos los dos ultimos hechos desde los dos anteriores. todos estos hechos pueden ser usados para obtener estimativas por arriba para valores cŕıticos de modelos de infiltración en dimensión dos y presentamos tres referencias y un ejemplo para ilustrar esto. key words and phrases: percolation, critical values, oriented planar graphs, duality. math. subj. class.: 05c10, 60k35, 82b43, 94c15. 1 introduction we state four facts about dual pairs of graphs drawn in a plane. these facts pertain respectively to finite non-oriented, infinite non-oriented, finite oriented, and infinite oriented graphs. all of these facts can be used to obtain upper estimations for critical values of two-dimensional percolation models and we present three references and one example to illustrate this. we call the former two facts the “main lemma” and the latter two facts “theorem”. we do not prove the main lemma although we have a proof in [l.2002]; instead we show that it is “evident” in some naive sense. then we deduce our theorem from the main lemma. we consider graphs with a finite or countable sets v of vertices and e of edges. every edge connects two vertices, which are called its ends (and which may coincide). one and the same pair of vertices may be connected by several edges. a finite path is a finite sequence “ vertexedge-vertex-edge-. . . -edge-vertex” in which every edge connects the vertices between which it is placed in this sequence and in which some vertices and/or edges may coincide. an infinite path is an infinite sequence “vertex-edge-vertex-edge-. . . ” with the same properties.. a path is called self-avoiding if all the vertices in its sequence are different. a contour is a finite path in which the initial and final vertices coincide. a contour is self-avoiding if all its vertices are different except, of course, the first and last vertices, which coincide. we say that a graph g is connected if every two vertices of g are connected by a path in g. all the graphs considered in this paper are assumed to be connected unless stated otherwise. we assume that every vertex is an end of only a finite number of edges. therefore, if one of the sets v or e is finite, the other is finite also. if v and e are finite, we call g finite, otherwise g is infinite. we consider non-oriented and oriented percolation models on graphs depending on the way in which we attribute certain states to their edges. in the non-oriented model each edge of a graph can be open or closed, independently of all the other edges. in the oriented model we distinguish two directions of each edge, and every edge can be open or closed in each direction independently of the state of the other direction of the same edge and states of all the other edges. henceforth we shall write simply graphs instead of percolation models on graphs when it does not produce confusion. a path in a non-oriented graph is open if all its edges are open. a path in an oriented graph is open in a certain direction if all its edges are open in this direction. in a non-oriented cubo 10, 3 (2008) dualities useful in bond percolation 95 graph a contour is open if all its edges are open. in an oriented graph a contour is open in a certain direction if all its edges are open in this direction. now let us speak about drawing graphs in a plane ir2. a curve is a continuous function f : [0, 1] → ir2. the points f (0) and f (1) are called the ends of this curve. a curve is called self-avoiding if ∀x, y ∈ [0, 1] : x 6= y ⇒ f (x) 6= f (y). a curve is called polygonal if it is piecewise affine, that is there are 0 = t0 < t1 < . . . tn = 1 such that f (t) is an affine function of t in every segment [tk−1, tk], k = 1, . . . n. the points f (tk), 0 < k < n, are called corners and the sets {f (t) | tk−1 ≤ t ≤ tk}, k = 1, . . . , n are called pieces. henceforth we consider only polygonal curves. this approach is not unusual. we say that a graph g is drawn in a plane if the following conditions are satisfied:                                          1. each vertex v of g is represented by a point p (v) in the plane, so that different vertices are represented by different points. we denote the set of points representing all vertices of g by p (v ). 2. each edge e of g is represented by a polygonal curve fe, where: a) fe(0) and fe(1) represent the ends of this edge. b) the curve fe is self-avoiding, except when e is a loop, in which case fe(0) = fe(1). 3. if ei 6= ej are two differents edges, the corresponding curves have no common points, except common ends, which represent common ends of ei and ej when they have such ones. 4. each bounded subset of the plane intersects only a finite (or empty) set of curves representing edges. a closed curve is a polygonal function f from a circle s1 to ir2. a jordan curve is a closed self-avoiding curve, which means that the values of this function are different for different elements of s1. according to the well-known jordan theorem, any jordan curve separates the remaining part of the plane into two open regions, one bounded, the other unbounded, so that it intersects any curve whose ends belong to different regions. notice that every finite path of a graph drawn in a plane is represented by a curve and every contour is represented by a closed curve, which are self-avoiding if and only if the original path, respectively contour is self-avoiding. given a graph g drawn in a plane, we call p (g) the union of representations of its edges. since p (g) is closed, its complement is open. connected components of this complement are called faces of g. 96 pedro f. de lima and andré toom cubo 10, 3 (2008) we say that two connected graphs g and g′ drawn in the same plane are dual if they satisfy the following conditions. (here and elsewhere we may write simply vertices and edges, while in fact we mean their representations in the plane.)                            1. there is a 1-to-1 correspondence between the faces of g and vertices of g′, called duality, such that each face of g contains its dual vertex of g′. 2. there is a 1-to-1 correspondence between the vertices of g and faces of g′, called duality, such that each vertex of g is in the dual face of g′. 3. there is a 1-to-1 correspondence between the edges of g and edges of g′ called duality, such that representation of each edge of each graph crosses the representation of its dual edge of the dual graph only in one point, which is not a corner or end of any of them. representations of edges of g and g′, which are not dual, have no common points. this kind of duality is well-known and described in many textbooks. observe that this kind of duality is a symmetric relationship, that is, if g′ is dual of g, then g is dual of g′. of course, if a graph is finite, its dual is finite too. now from duality of graphs drawn in a plane we go to dualities of percolation models on these graphs. the rules (1 ) and (2 ) are the central point of our definitions. rule for a dual pair of non-oriented graphs (g, g′): every edge of graph g′ is open if and only if the dual edge of graph g is closed. } (1) given a non-oriented graph g, we say that:        a) vertices α and β are reachable from each other if there is an open finite self-avoiding path connecting them. b) vertex α and ∞ are reachable from each other if there is an open infinite self-avoiding path which starts at α. main lemma. let (g, g′) be a dual pair of non-oriented graphs, satisfying the rule (1 ). then:                  a) if g is finite: two vertices α and β are not reachable from each other in g if and only if there is an open self-avoiding contour in g′, whose representation in the plane leaves p (α) and p (β) in different regions. b) if g is infinite: a vertex α and ∞ are not reachable from each other in g if and only if there is an open self-avoiding contour in g′, whose representation leaves the point p (α) in the bounded region. the main lemma is “evident” in the same sense, in which all the basic topological facts are evident (jordan’s theorem, for example). we have a proof of it in [l.2002], but do not include it cubo 10, 3 (2008) dualities useful in bond percolation 97 here. let us notice that it is possible to turn the non-oriented percolation model into a graph by eliminating all closed edges. the resulting graph may be disconnected and the main lemma boils down to the following “evident” statements:                  a) two vertices of a finite graph drawn in a plane are not connected with a path in this graph if and only if there is a face containing a jordan curve separating the representations of these vertices. b) a vertex of an infinite graph drawn in a plane is not connected with infinity (i.e. there is no infinite self-avoiding path in this graph starting at this vertex) if and only if there is a face containing a jordan curve surrounding the representation of this vertex. rule for a dual pair of oriented graphs (g, g′): given dual edges e and e′, for each direction of e the corresponding direction of e′ is the direction from right to left when we go along e in the given direction. every edge of the graph g′ is open in a certain direction if and only if the dual edge of the graph g is closed in the corresponding direction.          (2) observe that in the case of oriented graphs the symmetry of duality becomes more complicated: given a direction of an edge e′ of g′, the corresponding direction of e is from left to right when we go along e′ in this direction. given an oriented graph g, we say that:          a) vertex β is reachable from vertex α if there is a a self-avoiding path connecting α and β, open in the direction from α to β. b) ∞ is reachable from a vertex α if there is an infinite self-avoiding path which begins at α and is open in the direction away from α. theorem. let (g, g′) be a dual pair of oriented graphs, satisfying the rule (2 ). then:                a) if g is finite: a vertex β is not reachable from another vertex α in g if and only if there is a self-avoiding contour in g′, open in such a direction that its representation in the plane leaves p (α) on the left side and p (β) on the right side. b) if g is infinite: ∞ is not reachable from a vertex α in g if and only if there is an open self-avoiding contour in g′, whose representation in the plane leaves the point p (α) in the finite area and surrounds it in the counter-clock direction. let us assume that the main lemma is proved and prove the theorem. proof in case a). let (g, g′) be a dual pair of oriented finite graphs. let us call a good path a self-avoiding path in the graph g connecting α and β, open in the direction from α for β. a good contour is a self-avoiding contour in the graph g′, which is open in such a direction that its representation is a closed curve that leaves the point p (α) on the left side and p (β) on the right side. 98 pedro f. de lima and andré toom cubo 10, 3 (2008) in one direction. let us suppose that there is a good path in g and there is a good contour c in g′ and obtain a contradiction. let us denote h∗ and c∗ the representations of h and c in the plane respectively. by our assumption, p (α) and p (β) are at different sides of c∗. from jordan theorem, c∗ and h∗ have at least one common point. let q∗ be the first point of intersection between c∗ and h∗ when we move along h∗ starting at p (α). so q∗ belongs to representations of two dual edges, e of g and e′ of g′. the edge e′ belongs to the contour c and is open in the direction of c. therefore the ends of e are on the opposite sides of c∗ and the direction of e′ in the direction of c corresponds to the direction of e from the left side to the right side of c. since p (α) is on the left side when we move along c in the counter-clock direction, the edge e of the path h is open in the direction from left to right of c. so both e and e′ are open in dual directions, which contradicts rule (2 ). in the other direction. let us assume that there is no good path in the graph g and prove that there is a good contour in the graph g′. let us classify vertices of g into three types as follows:        1) a vertex v of g is type 1 if there is an open path from α to v. 2) a vertex v is type 2 if there is a path from v to β without vertices type 1. 3) a vertex v is type 3 if it is neither type 1, nor type 2. notice that every vertex of g has exactly one type. given a dual pair (g, g′), every face of g′ is given the same type as the type of the corresponding vertex of g. let us introduce a dual pair (g, g ′ ) of non-oriented graphs, which have the same vertices and edges as g and g′, and their representations in the plane. let any edge of the graph g be open if and only if both ends of this edge are type 2 in g or both are not type 2 in g. after that, all the edges of g ′ are declared open or closed according to rule (1 ). since α is type 1 in g and β is type 2 in g, every path in g connecting α and β has a closed edge. therefore β is not reachable from α in g. hence, from the finite case of main lemma, there is an open self-avoiding contour c in the graph g ′ , whose representation in the plane separates p (α) and p (β). let us denote e′ 1 , e′ 2 , . . . , e′n the edges of that contour. all these edges are open, so all of their duals e1, e2, . . . , en, in g are closed. therefore every edge ei connects a vertex not type 2 with a vertex type 2 of g. let us denote these vertices u1, u2, . . . , un and v1, v2, . . . , vn respectively. let us prove that all the vertices u1, u2, . . . , un are type 1. suppose that a vertex uk is not type 1. we know that uk is connected with a vertex vk by an edge and that vk is type 2. there is a path from vk to β without vertices type 1, so there is a path from uk to β without vertices type 1, so uk is type 2. this is a contradiction. so every ui is type 1, therefore every ei is closed in g in the direction from ui to vi because otherwise vi would be type 1. therefore all e ′ i should be open in g ′ in such a direction that the faces ui are on the left side and the faces vi are on the right side of them. notice that no vertex type 2 can be inside the contour c. therefore ui are inside of our contour c and vi are outside of c because vi are type 2. thus c is a self-avoiding contour, which separates p (α) from p (β) and is open in such a direction that it leaves p (α) on the left side and p (β) on the right side. cubo 10, 3 (2008) dualities useful in bond percolation 99 proof in case b). let (g, g′) be a dual pair of infinite oriented graphs. a good path is a selfavoiding infinite path in g, which begins in α and is open in this direction. a good contour is a self-avoiding contour in the graph g′, which is open in that direction, whose representation in the plane goes around p (α) in the counter-clock direction. in one direction. let us suppose that there is a good path h in g and a good contour c in g′ and obtain a contradiction. let us denote h∗ and c∗ the representations of h and c in the plane respectively. due to the condition 4 of definition of graph drawn in a plane, there is only a finite set of vertices of g inside c∗. hence, since h is self-avoiding and infinite, it contains a vertex β, whose representation is in the exterior of c∗. so p (α) and p (β) are in different regions of ir2 \ c∗. hence, from jordan theorem, c∗ and h∗ have at least one common point. let q∗ be the first point of intersection of c∗ with h∗ when we go along h∗ starting at p (α). so q∗ belongs to representations of two dual edges, one of g and the other of g′. by the rule (2 ), these edges cannot be open at the same time in corresponding directions. but they are: the edge of h is open from inside to outside of the contour c∗ and the edge of c is open in the counter-clock direction. this makes a contradiction. in the other direction. let us assume that there is no good path and prove that there is a good contour. let us classify vertices of g into three types as follows (this classification is similar to that in the finite case, but not exactly the same):        1) a vertex v of g is type 1 if there is an open path from α to v. 2) a vertex v is type 2 if there is a path from v to ∞ without vertices type 1. 3) a vertex v is type 3 if it is neither type 1, nor type 2. notice that every vertex of g has exactly one type. given a dual pair (g, g′), every face of g′ is given the same type as the type of the corresponding vertex of g. as before, let us use a dual pair (g, g ′ ) of non-oriented graphs, which have the same vertices and edges and the same representations in the plane as g and g′. let any edge of the graph g be open if and only if the ends of that edge are both type 2 or both of another type in g. after that, let the edges of g ′ be open or closed according to the rule (1 ). since there is no good path in g, the set of vertices type 1 in g is finite. therefore every self-avoiding path in g beginning at α has a finite number of vertices type 1. so for each infinite self-avoinding path in g, starting at α, there is a last vertex ω type 1, all the subsequent vertices being type 2. according to the definition of g, the edge of g, which connects ω with its successor in this path, is closed. so ∞ is not reachable from α in g. therefore, due to the infinite case of main lemma, in the dual graph g ′ there is an open self-avoiding contour c, whose representation surrounds p (α). let us denote e′ 1 , e′ 2 , . . . , e′n the edges of that contour. since all these edges are open, all their duals e1, e2, . . . , en in g are closed. according to the definition of the graph g, each edge ei of g connects a vertex not type 2 with a vertex type 2 in g. let us denote these vertices u1, u2, . . . , un and v1, v2, . . . , vn respectively. like in the finite case, we can prove that each ui is type 1. so ei has to be closed 100 pedro f. de lima and andré toom cubo 10, 3 (2008) in g in the direction from ui to vi, because otherwise vi would have type 1. so all e ′ i should be open in g′ in such a direction that the faces ui are on the left side and the faces vi are on the right side of them. as before, no vertex of type 2 can be inside the contour c∗. therefore, every ui is inside of c ∗ and vi is outside of it because vi is type 2. so c is a self-avoiding contour in g ′, whose representation surrounds p (α), which is open in a direction, which leaves p (α) on the left side, that is in the counter-clock direction. � our proofs are over. let us present examples of use of our main lemma and theorem to obtain upper estimations of critical values in percolation. as usual, we denote il2 the non-oriented graph in which the set of vertices is zz 2 and two vertices are connected with an edge if the euclidean distance between them is 1. an example of application of an assertion similar to the item b) of main lemma to the case when g = il2 is on pp. 15-19 of grimmett’s book [g.1999]. also there is an example on pp. 6-13 of [t.2001]. an example of application of an assertion similar to the item a) of our theorem to a special class of cellular automata (a discrete analog of contact processes) is in [t.1968]. an example of application of the item b) of our theorem is in [t.2001] on pp. 16-20. it remains to present an example of use of item a) of the main lemma. let us consider a finite rectangular part of the graph il2 with the width w and height h. it is shown on the figure 1 with w = 8 and h = 6. α                                o x ↑y                                β figure 1. notice that we are using a rectangular coordinate system oxy with the origin at the left-lower corner of the picture. let us suppose that all the leftmost vertices with x = 0 are identified with α (say, one of them is α and the vertical edges connecting them are always open). all the rightmost vertices with x = w are identified with β (say, one of them is β and the vertical edges connecting them are always open). all the other edges are open with probability ǫ and closed with probability 1 − ǫ independently from each other. let us estimate the probability that α and β are reachable from each other. it is easy to prove that, if h ≤ const ·w and ǫ is small enough, say ǫ < 1/3, this probability tends to zero when w → ∞. to estimate the same probability for large ǫ is not so easy; we shall do it using the item a) of main lemma. according to it, α and β are not reachable from each other if and only if there is a contour in the dual picture, separating α from β. this contour must contain the vertice dual of the unbounded face of the original graph. cutting this cubo 10, 3 (2008) dualities useful in bond percolation 101 contour at this vertex, we obtain a path connecting the upper and lower sides of the dual graph, which is similar to the original one. let us denote δ = 1 − ǫ. thus the probability that α and β are not reachable from each other does not exceed w · ∞ ∑ k=h (3δ)k = w · (3δ)h 1 − 3δ . if w ≤ const · h and ǫ > 2/3, this quantity tends to zero when h → ∞. thus the probability that α and β are reachable from each other behaves differently for small vs. large values of ǫ when w ≍ h → ∞. now let us discuss some previous publications. whitney [w.1932, w.1933] proved some statements about dual graphs. his theorem 4 on page 77 of [w.1933] is similar to the finite case of our main lemma, but it gives only a criterion whether a graph is connected or not and is not concerned whether two particular vertices are connected. our main lemma is often believed to be “well-known”, but we cannot refer the reader to a source, where it is proved or even stated beyond a few special cases. we believe that this is not acceptable because mathematics is a general rigorous science and all important mathematical facts should be stated and proved in a general form. hammersley had some idea of duality when he wrote [h.1959], which provided the first upper estimation of a critical value in percolation. however, no general definition of duality was stated and no rule similar to our rule (1 ) was declared there. the most fundamental book on percolation is grimmett’s [g.1999]. on pages 16-18 and 283-287 he explains this kind of duality and its application to obtain an upper estimation of a critical value. however, he does all this only for the graph il2 and without a proof. instead of going into topological details, [g.1999] refers the reader to the page 386 of [k.1982], the first page of appendix “some results for planar graphs”, without specifying, which result of that appendix is to be used. however, [k.1982] deals mostly with periodic, therefore, infinite case and mostly with site percolation and matching pairs. finally, about duality of oriented graphs. according to our knowledge, our theorem and the very definition of duality of oriented graphs have never been mentioned in print except [t.1968, t.2001], in both cases without a proof, and [l.2002], unpublished. although our theorem allows to obtain upper estimations in oriented percolation models, these estimations can be obtained by other means also, albeit not so easily. let us present some examples. shnirman [s.1968] proved non-ergodicity of stavskaya process (a discrete-time contact process) without using duality. he considered the sequence of distributions for all natural t and proved by induction that all of them satisfy a certain infinite system of inequalities. his method was so complicated that it almost never was used again except [t.1972]. durrett [d.1984] obtained an upper estimation of critical value in a few oriented percolation models, as a corollary of his study of a certain contact process. liggett [l.1995] obtained upper estimations of critical values in some percolation models as by-products of his study of a certain growth model. for our example liggett proves that the critical value does not exceed 2/3. durrett’s and liggett’s numerical estimations 102 pedro f. de lima and andré toom cubo 10, 3 (2008) are better than that which we obtained in [t.2001], which was an educational text designed just to illustrate some ideas. however, the duality approach seems more general and can be amplified to obtain better estimations. acknowledgements. p. de lima did this work at the department of statistics of the federal university of pernambuco as a part of his master program supported by capes. a. toom’s research was supported by cnpq, grant 300991/98-3. received: february 2008. revised: june 2008. references [d.1984] r. durrett, oriented percolation in two dimensions, annals of probability, 12 (1984), n. 4, pp. 999–1040. [g.1999] j. grimmett, percolation, springer, 1999. [h.1959] j.m. hammersley, bornes supérieures de la probabilité critique das un processus de filtration, le calcul des probabilités et ses application, colloques internationaux du centre national de la recherche scientifique, lxxxvii (1959), paris, pp. 17–37. (in french.) [k.1982] h. kesten, percolation theory for mathematicians, birkhäuser, 1982. [l.1995] t.m. liggett, survival of discrete time growth models, with applications to oriented percolation, annals of applied probability, 5 (1995), pp. 613–636. [l.2002] p.f. de lima, teoremas de dualidade usados na percolação, master’s thesis, 2002. unpublished. deposited at the department of statistics of ufpe, brazil. (in portuguese.) [s.1968] m. shnirman, on the problem of ergodicity of a markov chain with infinite set of states, problemy kibernetiki, 20 (1968), pp. 115–124. (in russian.) [t.1968] a. toom, a family of uniform nets of formal neurons, soviet math. doklady, 9 (1968), pp. 1338–1341. [t.1972] a. toom, on invariant measures in non-ergodic random media, probabilistical methods of investigation, issue 41. ed. by a. kolmogorov. moscow university press, 1972, pp. 43–51. (in russian.) [t.2001] a. toom, contornos, conjuntos convexos e autômatas celulares, curso ministrado no 23o colóquio brasileiro de matemática. impa, rio de janeiro, rj, brazil, 2001. (in portugues.) [w.1932] h. whitney, non-separable and planar graphs, trans. amer. math. soc., 34 (1932) cambridge, usa, pp. 339–362. [w.1933] h. whitney, planar graphs, fund. math., 21 (1933), cambridge, usa, 1933, pp. 73–84. n08 cubo a mathematical journal vol.10, n o ¯ 03, (115–132). october 2008 limit cycles of liénard-type dynamical systems valery a. gaiko∗ department of mathematics, belarusian state university of informatics and radioelectronics, l. beda str. 6-4, minsk 220040 – belarus email: valery.gaiko@yahoo.com abstract in this paper, using geometric properties of the field rotation parameters, we present a solution of smale’s thirteenth problem on the maximum number of limit cycles for liénard’s polynomial system, generalize the obtained results for special classes of polynomial systems, and complete the global qualitative analysis of a piecewise linear dynamical system approximating a liénard-type polynomial system with an arbitrary number of finite singularities. resumen en este art́ıculo, usando propriedades geometricas del campo de rotación de parametros, nosotros presentamos una solución del problema trece de smale sobre el número máximo de ciclos ĺımite para el sistema polinomial de liénard, generalizamos los resultados obtenidos para clases especiales de sistemas polinomiales, y completamos el analisis cualitativo global de un sistema dinamico linear por pedazos aproximando un sistema polinomial de tipo liérnard con un número arbitrario finito de singularidades. ∗this work was supported by the netherlands organization for scientific research (nwo). 116 valery a. gaiko cubo 10, 3 (2008) key words and phrases: planar polynomial dynamical system, liénard’s polynomial system, generalized liénard’s cubic system, piecewise linear liénard-type dynamical system, hilbert’s sixteenth problem, smale’s thirteenth problem, field rotation parameter, bifurcation, limit cycle. math. subj. class.: 34c05, 34c07, 34c23, 37g05, 37g10, 37g15. 1 introduction we consider planar dynamical systems . x = pn(x,y), . y = qn(x,y), (1.1) where pn(x,y) and qn(x,y) are polynomials with real coefficients in the real variables x, y and not greater than n degree. first of all, we consider a special case of (1.1): a classical liénard’s polynomial system of the form . x = y, . y = −x + µ1 y + µ2 y 2 + µ3 y 3 + . . . + µ2k y 2k + µ2k+1 y 2k+1. (1.2) the main problem of qualitative theory of such systems is hilbert’s sixteenth problem on the maximum number and relative position of their limit cycles, i. e., closed isolated trajectories of (1.1). this problem was formulated as one of the fundamental problems for mathematicians of the xx century, however it has not been solved even in the simplest (quadratic, cubic, etc.) cases of the polynomial systems. in this paper, we suggest a new geometric approach to solving the problem in the case of liénard’s system (1.2). in this special case, it is considered as smale’s thirteenth problem becoming one of the main problems for mathematicians of the xxi century [16], [20]. in section 2 of this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we present a solution of smale’s thirteenth problem for liénard’s polynomial system (1.2). in section 3, by means of the same geometric approach, we generalize the obtained result and present a solution of hilbert’s sixteenth problem on the maximum number of limit cycles surrounding a singular point for an arbitrary polynomial system. in section 4, we consider generalized liénard’s cubic system with three finite singularities, for which the developed geometric approach can complete its global qualitative analysis: in particular, it easily solves the problem on the maximum number of limit cycles in their different distribution. in this section, we give also an alternative proof of the main theorem for the generalized liénard’s system applying the wintner–perko termination principle for multiple limit cycles. in section 5, by means of the same principle, we complete the global qualitative analysis of a piecewise linear dynamical system approximating a liénard-type polynomial system with an arbitrary number of finite singularities. cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 117 2 liénard’s polynomial system system (1.2) and more general liénard’s systems have been studied in numerous works (see, for example, [2], [16], [17], [19], [20]). it is easy to see that (1.2) has the only finite singularity: an anti-saddle at the origin. at infinity, system (1.2) for k > 1 has two singular points: a node at the “ends” of the y-axis and a saddle at the “ends” of the x-axis. for studying the infinite singularities, the methods applied in [2] for rayleigh’s and van der pol’s equations and also erugin’s two-isocline method developed in [10] can be used. following [10], we will study limit cycle bifurcations of (1.2) by means of a canonical system containing only the field rotation parameters of (1.2). it is valid the following theorem. theorem 2.1. liénard’s polynomial system (1.2) with limit cycles can be reduced to the canonical form . x = y ≡ p, . y = −x + µ1 y + y 2 + µ3 y 3 + . . . + y2k + µ2k+1 y 2k+1 ≡ q, (2.1) where µ1, µ3, . . . , µ2k+1 are field rotation parameters of (2.1). proof. vanish all odd parameters of (1.2), . x = y, . y = −x + µ2 y 2 + µ4 y 4 + . . . + µ2k y 2k, (2.2) and consider the corresponding equation dy dx = −x + µ2 y 2 + µ4 y 4 + . . . + µ2k y 2k y ≡ f(x,y). (2.3) since f(x,−y) = −f(x,y), the direction field of (2.3) (and the vector field of (2.2) as well) is symmetric with respect to the x-axis. it follows that for arbitrary values of the parameters µ2, µ4, . . . , µ2k system (2.2) has a center at the origin and cannot have a limit cycle surrounding this point. therefore, without loss of generality, all even parameters of system (1.2) can be supposed to be equal, for example, to one: µ2 = µ4 = . . . = µ2k = 1 (they could be also supposed to be equal to zero). to prove that the rest (odd) parameters rotate the vector field of (2.1), let us calculate the following determinants: ∆µ1 = pq ′ µ1 − qp ′µ1 = y 2 ≥ 0, ∆µ3 = pq ′ µ3 − qp ′µ3 = y 2 ≥ 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆µ2k+1 = pq ′ µ2k+1 − qp ′µ2k+1 = y 2 ≥ 0. by definition of a field rotation parameter [4], for increasing each of the parameters µ1, µ3, . . . , µ2k+1, under the fixed others, the vector field of system (2.1) is rotated in positive direction 118 valery a. gaiko cubo 10, 3 (2008) (counterclockwise) in the whole phase plane; and, conversely, for decreasing each of these parameters, the vector field of (2.1) is rotated in negative direction (clockwise). thus, for studying limit cycle bifurcations of (1.2), it is sufficient to consider canonical system (2.1) containing only its odd parameters, µ1, µ3, . . . , µ2k+1, which rotate the vector field of (2.1). the theorem is proved. 2 by means of canonical system (2.1), let us study global limit cycle bifurcations of (1.2) and prove the following theorem. theorem 2.2. liénard’s polynomial system (1.2) has at most k limit cycles. proof. according to theorem 2.1, for the study of limit cycle bifurcations of system (1.2), it is sufficient to consider canonical system (2.1) containing only the field rotation parameters of (1.2): µ1, µ3, . . . , µ2k+1. vanish all these parameters: . x = y, . y = −x + y2 + y4 + . . . + y2k. (2.4) system (2.4) is symmetric with respect to the x-axis and has a center at the origin. let us input successively the field rotation parameters into this system beginning with the parameters at the highest degrees of y and alternating with their signs. so, begin with the parameter µ2k+1 and let, for definiteness, µ2k+1 > 0 : . x = y, . y = −x + y2 + y4 + . . . + y2k + µ2k+1 y 2k+1. (2.5) in this case, the vector field of (2.5) is rotated in positive direction (counterclockwise) turning the origin into a nonrough unstable focus. fix µ2k+1 and input the parameter µ2k−1 < 0 into (2.5): . x = y, . y = −x + y2 + y4 + . . . + µ2k−1 y 2k−1 + y2k + µ2k+1 y 2k+1. (2.6) then the vector field of (2.6) is rotated in opposite direction (clockwise) and the focus immediately changes the character of its stability (since its degree of nonroughness decreases and the sign of the field rotation parameter at the lower degree of y changes) generating a stable limit cycle. under further decreasing µ2k−1, this limit cycle will expand infinitely, not disappearing at infinity (because of the parameter µ2k+1 at the higher degree of y). denote the limit cycle by γ1, the domain outside the cycle by d1, the domain inside the cycle by d2 and consider logical possibilities of the appearance of other (semi-stable) limit cycles from a “trajectory concentration” surrounding the origin. it is clear that, under decreasing the parameter µ2k−1, a semi-stable limit cycle cannot appear in the domain d2, since the focus spirals filling this domain will untwist and the distance between their coils will increase because of the vector field rotation. cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 119 by contradiction, we can also prove that a semi-stable limit cycle cannot appear in the domain d1. suppose it appears in this domain for some values of the parameters µ ∗ 2k+1 > 0 and µ∗ 2k−1 < 0. return to initial system (2.4) and change the inputting order for the field rotation parameters. input first the parameter µ2k−1 < 0 : . x = y, . y = −x + y2 + y4 + . . . + µ2k−1 y 2k−1 + y2k. (2.7) fix it under µ2k−1 = µ ∗ 2k−1 . the vector field of (2.7) is rotated clockwise and the origin turns into a nonrough stable focus. inputting the parameter µ2k+1 > 0 into (2.7), we get again system (2.6), the vector field of which is rotated counterclockwise. under this rotation, a stable limit cycle γ1 will immediately appear from infinity, more precisely, from a separatrix cycle of the poincaré circle form containing infinite singularities of the saddle and node types [2]. this cycle will contract, the outside spirals winding onto the cycle will untwist and the distance between their coils will increase under increasing µ2k+1 to the value µ ∗ 2k+1 . it follows that there are no values of µ∗ 2k−1 < 0 and µ∗ 2k+1 > 0, for which a semi-stable limit cycle could appear in the domain d1. this contradiction proves the uniqueness of a limit cycle surrounding the origin in system (2.6) for any values of the parameters µ2k−1 and µ2k+1 of different signs. obviously, if these parameters have the same sign, system (2.6) has no limit cycles surrounding the origin at all. let system (2.6) have the unique limit cycle γ1. fix the parameters µ2k+1 >0, µ2k−1 <0 and input the third parameter, µ2k−3 > 0, into this system: . x = y, . y = −x + y2 + . . . + µ2k−3 y 2k−3 + y2k−2 + . . . + µ2k+1 y 2k+1. (2.8) the vector field of (2.8) is rotated counterclockwise, the focus at the origin changes the character of its stability and the second (unstable) limit cycle, γ2, immediately appears from this point. under further increasing µ2k−3, the limit cycle γ2 will join with γ1 forming a semi-stable limit cycle, γ12, which will disappear in a “trajectory concentration” surrounding the origin. can another semistable limit cycle appear around the origin in addition to γ12? it is clear that such a limit cycle cannot appear either in the domain d1 bounded on the inside by the cycle γ1 or in the domain d3 bounded by the origin and γ2 because of the increasing distance between the spiral coils filling these domains under increasing the parameter µ2k−3. to prove impossibility of the appearance of a semi-stable limit cycle in the domain d2 bounded by the cycles γ1 and γ2 (before their joining), suppose the contrary, i. e., for some set of values of the parameters, µ∗ 2k+1 > 0, µ∗ 2k−1 < 0, and µ∗ 2k−3 > 0, such a semi-stable cycle exists. return to system (2.4) again and input first the parameters µ2k−3 > 0 and µ2k+1 > 0 : . x = y, . y = −x + y2 + . . . + µ2k−3 y 2k−3 + y2k−2 + y2k + µ2k+1 y 2k+1. (2.9) both parameters act in a similar way: they rotate the vector field of (2.9) counterclockwise turning the origin into a nonrough unstable focus. fix these parameters under µ2k−3 = µ ∗ 2k−3 , µ2k+1 = µ ∗ 2k+1 and input the parameter µ2k−1 < 0 into (2.9) getting again system (2.8). since, by our assumption, this system has two limit cycles 120 valery a. gaiko cubo 10, 3 (2008) for µ2k−1 > µ ∗ 2k−1 , there exists some value of the parameter, µ12 2k−1 (µ∗ 2k−1 < µ12 2k−1 < 0), for which a semi-stable limit cycle, γ12, appears in system (2.8) and then splits into a stable cycle, γ1, and an unstable cycle, γ2, under further decreasing µ2k−1. the formed domain d2 bounded by the limit cycles γ1, γ2 and filled by the spirals will enlarge since, on the properties of a field rotation parameter, the interior unstable limit cycle γ2 will contract and the exterior stable limit cycle γ1 will expand under decreasing µ2k−1. the distance between the spirals of the domain d2 will naturally increase, what will prevent the appearance of a semi-stable limit cycle in this domain for µ2k−1 < µ 12 2k−1 . thus, there are no such values of the parameters, µ∗ 2k+1 > 0, µ∗ 2k−1 < 0, and µ∗ 2k−3 > 0, for which system (2.8) would have an additional semi-stable limit cycle. obviously, there are no other values of the parameters µ2k+1, µ2k−1, and µ2k−3 for which system (2.8) would have more than two limit cycles surrounding the origin. therefore, two is the maximum number of limit cycles for system (2.8). this result agrees with [19], where it was proved for the first time that the maximum number of limit cycles for liénard’s system of the form . x = y, . y = −x + µ1 y + µ3 y 3 + µ5 y 5 (2.10) was equal to two. suppose that system (2.8) has two limit cycles, γ1 and γ2 (this is always possible if µ2k+1 ≫ −µ2k−1 ≫ µ2k−3 > 0), fix the parameters µ2k+1, µ2k−1, µ2k−3 and consider a more general system than (2.8) (and (2.10)) inputting the fourth parameter, µ2k−5 < 0, into (2.8): . x = y, . y = −x + y2 + . . . + µ2k−5 y 2k−5 + y2k−4 + . . . + µ2k+1 y 2k+1. (2.11) under decreasing µ2k−5, the vector field of (2.11) will be rotated clockwise and the focus at the origin will immediately change the character of its stability generating the third (stable) limit cycle, γ3. under further decreasing µ2k−5, γ3 will join with γ2 forming a semi-stable limit cycle, γ23, which will disappear in a “trajectory concentration” surrounding the origin; the cycle γ1 will expand infinitely tending to the poincaré circle at infinity. let system (2.11) have three limit cycles: γ1, γ2, γ3. could an additional semi-stable limit cycle appear under decreasing µ2k−5, after splitting of which system (2.11) would have five limit cycles around the origin? it is clear that such a limit cycle cannot appear either in the domain d2 bounded by the cycles γ1 and γ2 or in the domain d4 bounded by the origin and γ3 because of the increasing distance between the spiral coils filling these domains under decreasing µ2k−5. consider two other domains: d1 bounded on the inside by the cycle γ1 and d3 bounded by the cycles γ2 and γ3. as before, we will prove impossibility of the appearance of a semi-stable limit cycle in these domains by contradiction. suppose that for some set of values of the parameters µ∗ 2k+1 > 0, µ∗ 2k−1 < 0, µ∗ 2k−3 > 0, and µ∗ 2k−5 < 0, such a semi-stable cycle exists. return to system (2.4) again, input first the parameters µ2k−5 < 0, µ2k−1 < 0 and then the parameter µ2k+1 > 0 : . x = y, . y = −x+y2+. . .+µ2k−5y 2k−5 +. . .+µ2k−1y 2k−1 +y2k +µ2k+1y 2k+1 . (2.12) cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 121 fix the parameters µ2k−5, µ2k−1 under the values µ ∗ 2k−5 , µ∗ 2k−1 , respectively. under increasing µ2k+1, the node at infinity will change the character of its stability, the separatrix behavior of the infinite saddle will be also changed and a stable limit cycle, γ1, will immediately appear from the poincaré circle at infinity [2]. fix µ2k+1 under the value µ ∗ 2k+1 and input the parameter µ2k−3 > 0 into (2.12) getting system (2.11). since, by our assumption, (2.11) has three limit cycles for µ2k−3 < µ ∗ 2k−3 , there exists some value of the parameter µ23 2k−3 (0 < µ23 2k−3 < µ∗ 2k−3 ) for which a semi-stable limit cycle, γ23, appears in this system and then splits into an unstable cycle, γ2, and a stable cycle, γ3, under further increasing µ2k−3. the formed domain d3 bounded by the limit cycles γ2, γ3 and also the domain d1 bounded on the inside by the limit cycle γ1 will enlarge and the spirals filling these domains will untwist excluding a possibility of the appearance of a semi-stable limit cycle there. all other combinations of the parameters µ2k+1, µ2k−1, µ2k−3, and µ2k−5 are considered in a similar way. it follows that system (2.11) has at most three limit cycles. if we continue the procedure of successive inputting the odd parameters, µ2k−7, . . . , µ3, µ1, into system (2.4), it is possible first to obtain k limit cycles (µ2k+1 ≫ −µ2k−1 ≫ µ2k−3 ≫ −µ2k−5 ≫ µ2k−7 ≫ . . .) and then to conclude that canonical system (2.1) (i. e., liénard’s polynomial system (1.2) as well) has at most k limit cycles. the theorem is proved. 2 3 an arbitrary polynomial system let us consider an arbitrary polynomial system . x = pn(x,y,µ1, . . . ,µk), . y = qn(x,y,µ1, . . . ,µk), (3.1) where pn and qn are polynomials in the real variables x, y and not greater than n degree containing k field rotation parameters, µ1, . . . ,µk, and having an anti-saddle at the origin. generalizing the main result of the previous section, we will prove the following theorem. theorem 3.1. polynomial system (3.1) containing k field rotation parameters and having a singular point of the center type at the origin for the zero values of these parameters can have at most k − 1 limit cycles surrounding the origin. proof. vanish all parameters of (3.1) and suppose that the obtained system . x = pn(x,y, 0, . . . , 0), . y = qn(x,y, 0, . . . , 0) (3.2) has a singular point of the center type at the origin. let us input successively the field rotation parameters, µ1, . . . ,µk, into this system. suppose, for example, that µ1 > 0 and that the vector field of the system . x = pn(x,y,µ1, 0, . . . , 0), . y = qn(x,y,µ1, 0, . . . , 0) (3.3) 122 valery a. gaiko cubo 10, 3 (2008) is rotated counterclockwise turning the origin into a stable focus under increasing µ1. fix µ1 and input the parameter µ2 into (3.3) changing it so that the field of the system . x = pn(x,y,µ1,µ2, 0, . . . , 0), . y = qn(x,y,µ1,µ2, 0, . . . , 0) (3.4) would be rotated in opposite direction (clockwise). let be so for µ2 < 0. then, for some value of this parameter, a limit cycle will appear in system (3.4). there are three logical possibilities for such a bifurcation: 1) the limit cycle appears from the focus at the origin; 2) it can also appear from some separatrix cycle surrounding the origin; 3) the limit cycle appears from a so-called “trajectory concentration”. in the last case, the limit cycle is semi-stable and, under further decreasing µ2, it splits into two limit cycles (stable and unstable), one of which then disappears at (or tends to) the origin and the other disappears on (or tends to) some separatrix cycle surrounding this point. but since the stability character of both a singular point and a separatrix cycle is quite easily controlled [10], this logical possibility can be excluded. let us choose one of the two other possibilities: for example, the first one, the so-called andronov–hopf bifurcation. suppose that, for some value of µ2, the focus at the origin becomes non-rough, changes the character of its stability and generates a stable limit cycle, γ1. under further decreasing µ2, three new logical possibilities can arise: 1) the limit cycle γ1 disappears on some separatrix cycle surrounding the origin; 2) a separatrix cycle can be formed earlier than γ1 disappears on it, then it generates one more (unstable) limit cycle, γ2, which joins with γ1 forming a semi-stable limit cycle, γ12, disappearing in a “trajectory concentration” under further decreasing µ2; 3) in the domain d1 outside the cycle γ1 or in the domain d2 inside γ1, a semi-stable limit cycle appears from a “trajectory concentration” and then splits into two limit cycles (logically, the appearance of such semi-stable limit cycles can be repeated). let us consider the third case. it is clear that, under decreasing µ2, a semi-stable limit cycle cannot appear in the domain d2, since the focus spirals filling this domain will untwist and the distance between their coils will increase because of the vector field rotation. by contradiction, we can prove that a semi-stable limit cycle cannot appear in the domain d1. suppose it appears in this domain for some values of the parameters µ∗ 1 > 0 and µ∗ 2 < 0. return to initial system (3.2) and change the inputting order for the field rotation parameters. input first the parameter µ2 < 0 : . x = pn(x,y,µ2, 0, . . . , 0), . y = qn(x,y,µ2, 0, . . . , 0). (3.5) fix it under µ2 = µ ∗ 2 . the vector field of (3.5) is rotated clockwise and the origin turns into a unstable focus. inputting the parameter µ1 > 0 into (3.5), we get again system (3.4), the vector field of which is rotated counterclockwise. under this rotation, a stable limit cycle, γ1, will appear from some separatrix cycle. the limit cycle γ1 will contract, the outside spirals winding onto this cycle will untwist and the distance between their coils will increase under increasing µ1 to the value µ∗ 1 . it follows that there are no values of µ∗ 2 < 0 and µ∗ 1 > 0, for which a semi-stable limit cycle could appear in the domain d1. the second logical possibility can be excluded by controlling the stability character of the cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 123 separatrix cycle [10]. thus, only the first possibility is valid, i. e., system (3.4) has at most one limit cycle. let system (3.4) have the unique limit cycle γ1. fix the parameters µ1 > 0, µ2 < 0 and input the third parameter, µ3 > 0, into this system supposing that µ3 rotates its vector field counterclockwise: . x = pn(x,y,µ1,µ2,µ3, 0, . . . , 0), . y = qn(x,y,µ1,µ2,µ3, 0, . . . , 0). (3.6) here we can have two basic possibilities: 1) the limit cycle γ1 disappears at the origin; 2) the second (unstable) limit cycle, γ2, appears from the origin and, under further increasing the parameter µ3, the cycle γ2 joins with γ1 forming a semi-stable limit cycle, γ12, which disappears in a “trajectory concentration” surrounding the origin. besides, we can also suggest that: 3) in the domain d2 bounded by the origin and γ1, a semi-stable limit cycle, γ23, appears from a “trajectory concentration”, splits into an unstable cycle, γ2, and a stable cycle, γ3, and then the cycles γ1, γ2 disappear through a semi-stable limit cycle, γ12, and the cycle γ3 disappears through the andronov–hopf bifurcation; 4) a semi-stable limit cycle, γ34, appears in the domain d2 bounded by the cycles γ1, γ2 and, for some set of values of the parameters, µ ∗ 1 , µ∗ 2 , µ∗ 3 , system (3.6) has at least four limit cycles. let us consider the last, fourth, case. it is clear that a semi-stable limit cycle cannot appear either in the domain d1 bounded on the inside by the cycle γ1 or in the domain d3 bounded by the origin and γ2 because of the increasing distance between the spiral coils filling these domains under increasing the parameter µ3. to prove impossibility of the appearance of a semi-stable limit cycle in the domain d2, suppose the contrary, i. e., for some set of values of the parameters, µ ∗ 1 > 0, µ∗ 2 < 0, and µ∗ 3 > 0, such a semi-stable cycle exists. return to system (3.2) again and input first the parameters µ3 > 0, µ1 > 0 : . x = pn(x,y,µ1,µ3, 0, . . . , 0), . y = qn(x,y,µ1,µ3, 0, . . . , 0). (3.7) fix these parameters under µ3 = µ ∗ 3 , µ1 = µ ∗ 1 and input the parameter µ2 < 0 into (3.7) getting again system (3.6). since, by our assumption, this system has two limit cycles for µ2 > µ ∗ 2 , there exists some value of the parameter, µ12 2 (µ∗ 2 < µ12 2 < 0), for which a semi-stable limit cycle, γ12, appears in system (3.6) and then splits into a stable cycle, γ1, and an unstable cycle, γ2, under further decreasing µ2. the formed domain d2 bounded by the limit cycles γ1, γ2 and filled by the spirals will enlarge, since, on the properties of a field rotation parameter, the interior unstable limit cycle γ2 will contract and the exterior stable limit cycle γ1 will expand under decreasing µ2. the distance between the spirals of the domain d2 will naturally increase, what will prevent the appearance of a semi-stable limit cycle in this domain for µ2 < µ 12 2 . thus, there are no such values of the parameters, µ∗ 1 > 0, µ∗ 2 < 0, µ∗ 3 > 0, for which system (3.6) would have an additional semi-stable limit cycle. therefore, the fourth case cannot be realized. the third case is considered absolutely similarly. it follows from the first two cases that system (3.6) can have at most two limit cycles. 124 valery a. gaiko cubo 10, 3 (2008) suppose that system (3.6) has two limit cycles, γ1 and γ2, fix the parameters µ1 > 0, µ2 < 0, µ3 > 0 and input the fourth parameter, µ4 < 0, into this system supposing that µ4 rotates its vector field clockwise: . x = pn(x,y,µ1, . . . ,µ4, 0, . . . , 0), . y = qn(x,y,µ1, . . . ,µ4, 0, . . . , 0). (3.8) the most interesting logical possibility here is that when the third (stable) limit cycle, γ3, appears from the origin and then, under preservation of the cycles γ1 and γ2, in the domain d3 bounded on the inside by the cycle γ3 and on the outside by the cycle γ2, a semi-stable limit cycle, γ45, appears and then splits into a stable cycle, γ4, and an unstable cycle, γ5, i. e., when system (3.8) for some set of values of the parameters, µ∗ 1 , µ∗ 2 , µ∗ 3 , µ∗ 4 , has at least five limit cycles. logically, such a semi-stable limit cycle could also appear in the domain d1 bounded on the inside by the cycle γ1, since, under decreasing µ4, the spirals of the trajectories of (3.8) will twist and the distance between their coils will decrease. on the other hand, in the domain d2 bounded on the inside by the cycle γ2 and on the outside by the cycle γ1 and also in the domain d4 bounded by the origin and γ3, a semi-stable limit cycle cannot appear, since, under decreasing µ4, the spirals will untwist and the distance between their coils will increase. to prove impossibility of the appearance of a semi-stable limit cycle in the domains d3 and d1, suppose the contrary, i. e., for some set of values of the parameters, µ∗ 1 > 0, µ∗ 2 < 0, µ∗ 3 > 0, and µ∗ 4 < 0, such a semi-stable cycle exists. return to system (3.2) again, input first the parameters µ4 < 0, µ2 < 0 and then the parameter µ1 > 0 : . x = pn(x,y,µ1,µ2,µ4, 0, . . . , 0), . y = qn(x,y,µ1,µ2,µ4, 0, . . . , 0). (3.9) fix the parameters µ4, µ2 under the values µ ∗ 4 , µ∗ 2 , respectively. under increasing µ1, a separatrix cycle is formed around the origin generating a stable limit cycle, γ1. fix µ1 under the value µ ∗ 1 and input the parameter µ3 > 0 into (3.9) getting system (3.8). since, by our assumption, system (3.8) has three limit cycles for µ3 < µ ∗ 3 , there exists some value of the parameter µ23 3 (0 < µ23 3 < µ∗ 3 ) for which a semi-stable limit cycle, γ23, appears in this system and then splits into an unstable cycle, γ2, and a stable cycle, γ3, under further increasing µ3. the formed domain d3 bounded by the limit cycles γ2, γ3 and also the domain d1 bounded on the inside by the limit cycle γ1 will enlarge and the spirals filling these domains will untwist excluding a possibility of the appearance of a semi-stable limit cycle there. all other combinations of the parameters µ1, µ2, µ3, and µ4 are considered in a similar way. it follows that system (3.8) has at most three limit cycles. if we continue the procedure of successive inputting the field rotation parameters, µ5, µ6, . . . , µk, into system (3.2), it is possible to conclude that system (3.1) can have at most k − 1 limit cycles surrounding the origin. the theorem is proved. 2 cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 125 4 generalized liénard’s cubic system in [13], we considered generalized liénard’s cubic system of the form: . x = y, . y = −x + (λ − µ) y + (3/2) x2 + µxy − (1/2) x3 + αx2y . (4.1) this system has three finite singularities: a saddle (1, 0) and two antisaddles — (0, 0) and (2, 0). at infinity system (4.1) can have either the only nilpotent singular point of fourth order with two closed elliptic and four hyperbolic domains or two singular points: one of them is a hyperbolic saddle and the other is a triple nilpotent singular point with two elliptic and two hyperbolic domains. we studied global bifurcations of limit and separatrix cycles of (4.1), found possible distributions of its limit cycles and carried out a classification of its separatrix cycles. we proved also the following theorems. theorem 4.1. the foci of system (4.1) can be at most of second order. theorem 4.2. system (4.1) has at least three limit cycles. using the results obtained in [13] and applying the approach developed in this paper, we can easily prove a much stronger theorem. theorem 4.3. system (4.1) has at most three limit cycles with the following their distributions : ((1, 1), 1), ((1, 2), 0), ((2, 1), 0), ((1, 0), 2), ((0, 1), 2), where the first two numbers denote the numbers of limit cycles surrounding each of two anti-saddles and the third one denotes the number of limit cycles surrounding simultaneously all three finite singularities. theorem 4.3 agrees, for example, with the earlier results by iliev and perko [15], but it does not agree with a quite recent result by dumortier and li [5] published in the same journal. the authors of both papers use very similar methods: small perturbations of a hamiltonian system. in [15], the zeros of the melnikov functions are studied and, in particular, it is proved that at most two limit cycles can bifurcate from either the interior or exterior period annulus of the hamiltonian under small parameter perturbations giving a generalized liénard system. in [5], zeros of the abelian integrals are studied and it is “proved” that at most four limit cycles can bifurcate from the exterior period annulus. thus, dumortier and li “obtain” a configuration of four big limit cycles surrounding three finite singularities together with the fifth small limit cycle which surrounds one of the anti-saddles. the result by dumortier and li [5] also does not agree with the wintner–perko termination principle for multiple limit cycles [10], [18]. applying the method as developed in [3], [7]–[13], we can show that system (4.1) cannot have either a multiplicity-three limit cycle or more than three limit cycles in any configuration. that will be another proof of theorem 4.3 (the same approach can be applied to proving theorems 2.2 and 3.1 as well). but first let us formulate the wintner–perko termination principle [18] for the polynomial system . x = f (x, µ), (4.2µ) 126 valery a. gaiko cubo 10, 3 (2008) where x ∈ r2; µ ∈ rn; f ∈ r2 (f is a polynomial vector function). theorem 4.4 (wintner–perko termination principle). any one-parameter family of multiplicity-m limit cycles of relatively prime polynomial system (4.2µ) can be extended in a unique way to a maximal one-parameter family of multiplicity-m limit cycles of (4.2µ) which is either open or cyclic. if it is open, then it terminates either as the parameter or the limit cycles become unbounded; or, the family terminates either at a singular point of (4.2µ), which is typically a fine focus of multiplicity m, or on a (compound ) separatrix cycle of (4.2µ), which is also typically of multiplicity m. the proof of this principle for general polynomial system (4.2µ) with a vector parameter µ ∈ rn parallels the proof of the planar termination principle for the system . x = p(x,y,λ), . y = q(x,y,λ) (4.2λ) with a single parameter λ ∈ r (see [10], [18]), since there is no loss of generality in assuming that system (4.2µ) is parameterized by a single parameter λ; i. e., we can assume that there exists an analytic mapping µ(λ) of r into rn such that (4.2µ) can be written as (4.2 µ(λ)) or even (4.2λ) and then we can repeat everything, what had been done for system (4.2λ) in [18]. in particular, if λ is a field rotation parameter of (4.2λ), the following perko’s theorem on monotonic families of limit cycles is valid. theorem 4.5. if l0 is a nonsingular multiple limit cycle of (4.20), then l0 belongs to a oneparameter family of limit cycles of (4.2λ); furthermore: 1) if the multiplicity of l0 is odd, then the family either expands or contracts monotonically as λ increases through λ0; 2) if the multiplicity of l0 is even, then l0 bifurcates into a stable and an unstable limit cycle as λ varies from λ0 in one sense and l0 disappears as λ varies from λ0 in the opposite sense; i. e., there is a fold bifurcation at λ0. proof of theorem 4.3. the proof is carried out by contradiction. suppose that system (4.1) with three field rotation parameters, λ, µ, and α, has three limit cycles around, for example, the origin (the case when limit cycles surround another focus is considered in a similar way). then we get into some domain in the space of these parameters which is bounded by two fold bifurcation surfaces forming a cusp bifurcation surface of multiplicity-three limit cycles. the corresponding maximal one-parameter family of multiplicity-three limit cycles cannot be cyclic, otherwise there will be at least one point corresponding to the limit cycle of multiplicity four (or even higher) in the parameter space. extending the bifurcation curve of multiplicityfour limit cycles through this point and parameterizing the corresponding maximal one-parameter family of multiplicity-four limit cycles by a field rotation parameter, according to theorem 4.5, we cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 127 will obtain a monotonic curve which, by the wintner–perko termination principle (theorem 4.4), terminates either at the origin or on some separatrix cycle surrounding the origin. since we know absolutely precisely at least the cyclicity of the singular point (theorem 4.1) which is equal to two, we have got a contradiction with the termination principle stating that the multiplicity of limit cycles cannot be higher than the multiplicity (cyclicity) of the singular point in which they terminate. if the maximal one-parameter family of multiplicity-three limit cycles is not cyclic, on the same principle (theorem 4.4), this again contradicts to theorem 4.1 not admitting the multiplicity of limit cycles higher than two. moreover, it also follows from the termination principle that either the ordinary separatrix loop or the eight-loop cannot have the multiplicity (cyclicity) higher than two (in that way, it can be proved that the cyclicity of three other separatrix cycles [13] is at most two). therefore, according to the same principle, there are no more than two limit cycles in the exterior domain surrounding all three finite singularities of (4.1). thus, system (4.1) cannot have either a multiplicity-three limit cycle or more than three limit cycles in any configuration. the theorem is proved. 2 5 a piecewise linear dynamical system consider a liénard-type dynamical system . x = y − ϕ(x), . y = β − αx − y, α > 0, β > 0, (5.1) where ϕ(x) is a piecewise linear function containing k dropping sections and approximating an arbitrary polynomial of degree 2k + 1. the line β − αx − y = 0 and the curve y = ϕ(x) can be considered as the isoclines of zero and infinity, respectively, for the corresponding equation. such systems and equations may occur, for example, when tunnel diode circuits and some other problems are studied (see [1], [2], [6], [14]). suppose that the ascending sections of system (5.1) have an inclination k1 > 0 and the descending (dropping) sections have an inclination k2 < 0. then the phase plane of (5.1) can be divided onto 2k + 1 parts in every of which (5.1) is a linear system: the ascending sections are in k + 1 strip regions (i,iii,v, . . . , 2k + 1) and the descending sections are in other k such regions (ii,iv,v i, . . . , 2k). the parameters k1, k2, and also α can be considered as rotation parameters for the sewed vector field of (5.1) (see [2], [10]). system (5.1) can have an odd number of simple singular points: 1, 3, 5, . . . , 2k + 1. if (5.1) has the only singular point, this point will be always an antisaddle (center, focus or node). a focus (node) will be always stable in odd regions and unstable in even regions if k2 > 1. if system (5.1) has 2k + 1 singularities, then k of them are saddles (they are in even regions) and k + 1 others are antisaddles (foci or nodes) which are always stable (they are in odd regions). the pieces of the straight lines β = x2i−1α + y2i−1 and β = x2iα + y2i (i = 1, 2, . . . ,k), where (x2i−1,y2i−1) and 128 valery a. gaiko cubo 10, 3 (2008) (x2i,y2i) are the coordinates of the upper and lower corner points of the curve ϕ(x), respectively, form a discriminant curve separating the domains in the plane (α,β), where α ≤ k2, with different numbers of singular points. the points of the discriminant curve correspond to the sewed singularities of saddle-focus or saddle-node type (α < k2) and its corner points correspond to the unstable equilibrium segments (α = k2) which coincide with the dropping sections of the curve y = ϕ(x). in the case when k2 < 1, closed trajectories cannot exist and only bifurcations of singular points are possible in system (5.1). therefore, we will consider further only the case when k2 > 1 and (k1 − 1) 2 < 4k2 giving various bifurcations and, first of all, the bifurcations of limit cycles. studying all such bifurcations (local and global), we will give a proof of the following theorem. theorem 5.1. system (5.1) with k dropping sections and 2k + 1 singular points can have at most k + 2 limit cycles, k + 1 of which surround the foci one by one and the last, (k + 2)-th, limit cycle surrounds all of the singular points of (5.1). proof of theorem 5.1. to prove the theorem, we will study both local and global bifurcations of limit cycles. the limit cycle of system (5.1) will be called small if it belongs to at most two adjoining regions; the cycle will be called big if it belongs to at least three adjoining regions. 5.1 local bifurcations following [1], we will study first stability of the singular points on the line of sewing. suppose that the straight line β − αx − y = 0 passes through the corner point (x1,y1) of the curve y = ϕ(x) on the boundary of regions i, ii and that α > (k2 + 1) 2/4. then the region i (ii) will be filled by the pieces of trajectories of the stable (unstable) focus. introduce positive coordinates s0 (lower (x1,y1)) and s1 (upper (x1,y1)) on the line of sewing of regions i and ii; s2 (lower (x2,y2)) and s3 (upper (x2,y2)) on the line of sewing of regions ii and iii, etc. the maps s0 → s1 along the trajectories of region i and s1 → s0 along the trajectories of region ii are written as follows: s1 = s0e πσ1/ω1, s̄0 = s1e πσ2/ω2, (5.2) where σi, ωi (i = 1, 2) are the real and imaginary parts of the roots of the characteristic equation for a singular point of regions i, ii, respectively. the singular point (x1,y1) will be a sewed center (s̄0 = s0) iff σ1/ω1 + σ2/ω2 = 0, i. e., when α = α∗ ≡ (1 −k1/k2)/(k2 −k1 + 2). the sewed focus (x1,y1) will be stable (s̄0 < s0) when α > α ∗ and unstable (s̄0 > s0) when α < α ∗. consider the return map s0 → s̄0 along the trajectories of regions i and ii. for region i, we cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 129 will have s0 = δ0 sin ω1τ1 (ω1 cos ω1τ1 − σ1 sin ω1τ1 − ω1e −σ1τ1 ) ≡ δ0ζ(τ1), s1 = δ0 sin ω1τ1 (ω1 cos ω1τ1 + σ1 sin ω1τ1 − ω1e σ1τ1 ) ≡ δ0χ(τ1), (5.3) where δ0 is the distance from the boundary of regions i, ii to the singular point; ζ and χ are monotonic functions. the return map along the trajectories of region ii has a similar form. calculation of the first derivative for the return map gives ds̄0 ds0 = s0 s̄0 e2(σ1τ1+σ2τ2), (5.4) where τi (i = 1, 2) is motion time along the trajectories of regions i, ii, respectively; σi = (1+ki)/2 (i = 1, 2). studying the return map s0 → s̄0 by means of (5.4), we prove that at most one limit cycle can exist in regions i and ii (see also [1]). the same result can be obtained for regions iii and iv,. . . , 2k − 1 and 2k. consider now the map s̄0 = f(s0) sewed of two pieces: s̄0 = ξ(s0) along the trajectories in regions i, ii, . . . , 2k and s̄0 = ψ(s0) along the trajectories in all regions, i, ii, . . . , 2k, 2k + 1. the map s0 → s1 in region i is given by (5.3). the maps s1 → s3, s3 → s5, . . . , s2k−1 → s2k−2 (s2k−1 → s2k+1, s2k+1 → s2k, s2k → s2k−2), s2k−2 → s2k−4, . . . , s2 → s0 have similar forms. the derivatives for the functions ξ(s0), ψ(s0) are given by the following expressions, respectively: ds̄0 ds0 = s0 s̄0 e 2(σ1(τ1+τ + 3 +τ − 3 +...+τ2k−1)+σ2(τ + 2 +τ − 2 +...+τ + 2k−2 +τ − 2k−2 )) , (5.5) ds̄0 ds0 = s0 s̄0 e2(σ1(τ1+τ + 3 +τ − 3 +...+τ2k+1)+σ2(τ + 2 +τ − 2 +...+τ + 2k +τ − 2k )), (5.6) where τ1, τ2k−1, τ2k+1 are motion times in regions i, 2k − 1, 2k + 1 and τ + 2i (τ− 2i ), τ+ 2i+1 (τ− 2i+1 ), i = 1, 2, . . . ,k, are motion times in the upper (lower) parts of regions ii,iii, . . . , 2k, respectively. studying the return map s̄0 = f(s0) by means of (5.5) and (5.6), we prove that at most two limit cycles can be generated by the boundary of the domain filled by closed trajectories of (5.1) and that these two limit cycles can be only outside the boundary. suppose that a part of the straight line β − αx − y = 0 coincides with a dropping section of (5.1), for example, with the first one (α = k2). the dropping section of (5.1) will be an unstable equilibrium segment and regions i, ii (because of the condition (k1 − 1) 2 < 4k2) will be filled by trajectories of the stable foci. it is easy to obtain an explicit expression for the map of the half-line s0 into itself: s̄0 = s0 e 2πσ1/ω1 + δ(k2 − 1)(1 + e πσ1/ω1 ), (5.7) where δ is the width of regions ii. 130 valery a. gaiko cubo 10, 3 (2008) this map has the only stable fixed point, and we can show that two stable foci surrounded by unstable limit cycles (one by one) are generated from the ends of the equilibrium segment under the rotation of the line β − αx − y = 0 (see also [1]). the simplest type of separatrix cycles of (5.1) is a so-called eight-loop formed by two ordinary saddle loops. in the case of 2k + 1 simple singular points, a separatrix cycle can contain k + 1 saddle loops, the first and the last of which are ordinary loops with one rough saddle on each and the k − 1 others are separatrix digons with two rough saddles on each. such a separatrix cycle will be called nondegenerate. in the cases when the straight line β − αx − y = 0 passes through the corner points of the curve y = ϕ(x), we will have degenerate separatrix cycles of lips-type containing one or two sewed saddle-nodes. it is clear that the bifurcations of separatrix cycles do not depend on the parameter β (see [1]). the separatrix cycles can be formed or destroyed only under a variation of the parameter α. the character of their stability will be determined by the sign of the saddle quantities which are always positive in our case, when the saddles are inside or on the boundary of even regions ii, iv, . . . , 2k and k2 > 1 (the corresponding theorems are valid for the piecewise linear dynamical systems as well [2]). it follows that the separatrix cycles of (5.1) are always unstable (inside and outside) and, under a variation of α, a nondegenerate separatrix cycle can generate at most k + 1 small unstable limit cycles inside its loops (digons) or the only big unstable limit cycle outside it. 5.2 global bifurcations now we are able to consider the global bifurcations of limit cycles. suppose again that the zero isocline β−αx−y = 0 passes through the corner point (x1,y1) of the infinite isocline y = ϕ(x) and that α > α∗. in this case, the only singular point in the phase plane is a sewed stable focus and all trajectories of (5.1) tend to it when t → +∞. for decreasing α (k2 < α < α ∗), the sewed focus becomes unstable and a stable limit cycle is generated from the boundary curve of the domain filled by closed trajectories (immediately after passing the value α∗ by the parameter α). for α = k2, the first dropping section of (5.1) will coincide with a part of the straight line β − αx − y = 0 and an unstable equilibrium segment will appear inside the stable limit cycle. if we rotate the line β − αx − y = 0 around an interior point of the segment (changing both of the parameters, α and β), two unstable limit cycles surrounding stable foci (one by one) will be generated from the ends (x1,y1) and (x2,y2) of the equilibrium segment. under the further rotation of the line β − αx − y = 0, it will pass first through the next corner point, (x4,y4), and then, successively, through the points (x6,y6), . . . , (x2k,y2k). every time, the corner point becomes a sewed saddle-node generating an unstable limit cycle surrounding a stable focus. so, we will get a piecewise linear system with 2k + 1 singular points having at least k + 1 small unstable limit cycles surrounding the stable foci (one by one) inside a big stable limit cycle, k + 2, surrounding all of the singular points. under the further rotation of the zero isocline, all k + 1 small limit cycles simultaneously cubo 10, 3 (2008) limit cycles of liénard-type dynamical systems 131 disappear in a separatrix cycle consisting of k + 1 loops (digons), this separatrix cycle generates a big (unstable) limit cycle which combines with another big (stable) limit cycle of (5.1) forming a semi-stable (double) limit cycle which finally disappears in a so-called trajectory condensation. let us prove that system (5.1) cannot have more than k + 2 limit cycles. the proof is carried out by contradiction by means of the wintner–perko termination principle [2], [10], [18]. since a small limit cycle is always unique in the corresponding strip regions, suppose that system (5.1) with three field rotation parameters, k1, k2, and α, has three big limit cycles. then we get into some domain in the space of these parameters which is bounded by two fold bifurcation surfaces forming a cusp bifurcation surface of multiplicity-three limit cycles [10], [18]. the corresponding maximal one-parameter family of multiplicity-three limit cycles cannot be cyclic, otherwise there will be at least one point corresponding to the limit cycle of multiplicity four (or even higher) in the parameter space. extending the bifurcation curve of multiplicity-four limit cycles through this point and parameterizing the corresponding maximal one-parameter family of multiplicity-four limit cycles by a field rotation parameter, for example, by the parameter α, we will obtain a monotonic curve which, by the wintner–perko termination principle, terminates either at the boundary curve of the domain filled by closed trajectories of (5.1) or on some degenerate separatrix cycle of (5.1) [10], [18]. since we know at least the cyclicity of the boundary curve which is equal to two, we have got a contradiction with the termination principle stating that the multiplicity of limit cycles cannot be higher than the multiplicity (cyclicity) of the end bifurcation points in which they terminate [10], [18]. if the maximal one-parameter family of multiplicity-three limit cycles is not cyclic, using the same principle, this again contradicts with the cyclicity result for the boundary curve not admitting the multiplicity of limit cycles to be higher than two. moreover, it also follows from the termination principle that the degenerate separatrix cycles of (5.1) cannot have the multiplicity (cyclicity) higher than two. therefore, according to the same principle, there are no more than two big limit cycles in the exterior domain outside the boundary curve of (5.1). the same results can be obtained by means of the new geometric methods developed in [12]. the phase portraits and bifurcation diagrams for system (5.1) will be similar to that which were constructed in [1], [2]. thus, system (5.1) with 2k + 1 singular points cannot have more than k + 2 limit cycles, i. e., k + 2 is the maximum number of limit cycles of such system and the obtained distribution (k + 1 small limit cycles plus a big limit cycle) is the only possibility for their distribution. the theorem is proved. 2 received: may 2008. revised: june 2008. 132 valery a. gaiko cubo 10, 3 (2008) references [1] a.n. bautin, qualitative investigation of a piecewise linear system, j. appl. math. mech., 38 (1974), 691–700. [2] n.n. bautin and e.a. leontovich, methods and examples of the qualitative analysis of dynamical systems in a plane, nauka, moscow, 1990 (in russian). [3] f. botelho and v.a. gaiko, global analysis of planar neural networks, nonlinear anal., 64 (2006), 1002–1011. [4] g.f.d. duff, limit-cycles and rotated vector fields, ann. math., 67 (1953), 15–31. [5] f. dumortier and c. li, perturbation from an elliptic hamiltonian of degree four: (iv) figure eight-loop, j. diff. equat., 188 (2003), 512–554. [6] a.f. filippov, differential equations with discontinuous right-hand sides, kluwer, boston, 1988. [7] v.a. gaiko, qualitative theory of two-dimensional polynomial dynamical systems: problems, approaches, conjectures, nonlinear anal., 30 (1997), 1385–1394. [8] v.a. gaiko, hilbert’s sixteenth problem and global bifurcations of limit cycles, nonlinear anal., 47 (2001), 4455–4466. 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[16] yu. ilyashenko, centennial history of hilbert’s 16th problem, bul. amer. math. soc., 39 (2002), 301–354. [17] a. lins, w. de melo, and c.c. pugh, on liénard’s equation, in: lecture notes in mathematics, vol. 597, springer, berlin, 1977, pp. 335–357. [18] l. perko, differential equations and dynamical systems, springer, new york, 2002. [19] g.s. rychkov, the maximal number of limit cycles of the system . y =−x, . x=y− ∑ 2 i=0 aix 2i+1 is equal to two, diff. equat., 11 (1975), 301–302. [20] s. smale, mathematical problems for the next century, math. intellig., 20 (1998), 7–15. n10 l_fns1c.dvi cubo a mathematical journal vol.12, no¯ 03, (71–81). october 2010 l -random and fuzzy normed spaces and classical theory donal o’regan department of mathematics, national university of ireland, galway, ireland email: donal.oregan@nuigalway.ie and reza saadati department of mathematics and computer science, amirkabir university of technology, 424 hafez avenue, tehran 15914, iran email: rsaadati@eml.cc abstract in this paper we study l -random and l -fuzzy normed spaces and prove open mapping and closed graph theorems for these spaces. resumen en este artículo estudiamos espacios normados l -random and l -fuzzy. probamos el teorema de la aplicación abierta y el teorema del gráfico cerrado. key words and phrases: l -random normed space, l -fuzzy normed space, completeness, quotient space, open mapping and closed graph. math. subj. class.: please inform. 72 donal o’regan & reza saadati cubo 12, 3 (2010) 1 introduction and preliminaries in this paper we study l -random and l -fuzzy normed spaces and study completeness for these spaces. further we prove open mapping and closed graph theorems in this setting. the ideas here are motivated from the functional analysis literature. the plan in sections 1-3 is to present in detail the l -random normed space setting. in section 4 we see from the definition how easily the theory extends to the l -fuzzy normed space situation. let l = (l,≥l ) be a complete lattice, i.e. a partially ordered set in which every nonempty subset admits a supremum and infimum, and 0l = inf l, 1l = sup l. the space of lattice random distribution functions, denoted by ∆+ l , is defined as the set of all mappings f : r∪ {−∞,+∞} → l such that f is continuous and non-decreasing on r, f(0) = 0l , f(+∞) = 1l . now d+ l ⊆ ∆ + l is defined as d+ l = {f ∈ ∆+ l : l−f(+∞) = 1l }, where l − f (x) denotes the left limit of the function f at the point x. the space ∆+ l is partially ordered by the usual point-wise ordering of functions, i.e., f ≥ g if and only if f(t) ≥l g(t) for all t in r. the maximal element for ∆+ l in this order is the distribution function given by ε0(t) =    0l , if t ≤ 0, 1l , if t > 0. define the mapping t∧ from l 2 to l by: t∧(x, y) =      x, if y ≥l x, y, if x ≥l y. recall (see [4], [5]) that if {xn} is a given sequence in l, (t∧) n i=1 xi is defined recurrently by (t∧) 1 i=1 xi = x1 and (t∧) n i=1 xi = t∧((t∧) n−1 i=1 xi , xn) for n ≥ 2. a negation on l is any decreasing mapping n : l → l satisfying n (0l ) = 1l and n (1l ) = 0l . if n (n (x)) = x, for all x ∈ l, then n is called an involutive negation. in the following l is endowed with a (fixed) negation n . definition 1.1. a lattice random normed space (briefly, l -random normed space) is a triple (x , p , t ), where x is a vector space, t is a t–norm on the lattice l and p is a mapping from x × [0,∞) into d+ l such that the following conditions hold: (lrn1) p (x, t) = ε0(t) for all t > 0 if and only if x = 0; (lrn2) p (αx, t) = p ( x, t |α| ) for all x in x , α 6= 0 and t ≥ 0; (lrn3) p (x + y, t + s) ≥l t (p (x, t), p ( y, s)) for all x, y ∈ x and t, s ≥ 0. cubo 12, 3 (2010) l -random and fuzzy normed spaces ... 73 we note from (lpn2) that p (−x, t) = p (x, t) (x ∈ x , t ≥ 0). example 1.2. let l = [0, 1] × [0, 1] and operation ≤l defined by: l = {(a1, a2) : (a1, a2) ∈ [0, 1] × [0, 1] and a1 + a2 ≤ 1}, (a1, a2) ≤l (b1, b2) ⇐⇒ a1 ≤ b1, a2 ≥ b2, ∀a = (a1, a2), b = (b1, b2) ∈ l. then (l,≤l ) is a complete lattice (see [2]). in this complete lattice, we denote its units by 0l = (0, 1) and 1l = (1, 0). let (x ,‖ · ‖) be a normed space. let t (a, b) = (min{a1, b1}, max{a2, b2}) for all a = (a1, a2), b = (b1, b2 ) ∈ [0, 1] × [0, 1] and µ be a mapping defined by p (x, t) = ( t t +‖x‖ , ‖x‖ t +‖x‖ ) , ∀t ∈ r+. then (x , p , t ) is a l -random normed space. definition 1.3. let (x , p , t ) be a l -random normed space. (1) a sequence {xn} in x is said to be convergent to x in x if, for every t > 0 and ε ∈ l\{0l }, there exists a positive integer n such that p (xn − x, t) >l n (ε) whenever n ≥ n. (2) a sequence {xn} in x is called cauchy sequence if, for every t > 0 and ε ∈ l\{0l }, there exists a positive integer n such that p (xn − xm, t) >l n (ε) whenever n ≥ m ≥ n. (3) a l -random normed space (x , p , t ) is said to be complete if and only if every cauchy sequence in x is convergent to a point in x . theorem 1.4. if (x , p , t ) is a l -random normed space and {xn} is a sequence such that xn → x, then limn→∞ p (xn, t) = p (x, t). proof. the proof is the same as in [9]. let (x , p , t ) be a l -random normed space. for t > 0 we define the open ball b(x, r, t) with center x and radius r ∈ l \ {0l , 1l } as b(x, r, t) = { y ∈ x : p (x − y, t) >l n (r)}. henceforth we assume that t is a continuous t–norm on the lattice l such that for every µ ∈ l \ {0l , 1l }, there is a λ ∈ l \ {0l , 1l } such that t n−1(n (λ), ..., n (λ)) >l n (µ). lemma 1.5. let (x , p , t ) be a l -random normed space. let n be a continuous negator on l . define eλ,p : v → r + ∪ {0} by eλ,p (x) = inf{t > 0 : p (x, t) >l n (λ)} for each λ ∈ l \ {0l , 1l } and x ∈ v . then we have the following properties. 74 donal o’regan & reza saadati cubo 12, 3 (2010) (i) for any µ ∈ l \ {0l , 1l } there exists λ ∈ l \ {0l , 1l } such that eµ,p (x + y) ≤ eλ,p (x) + eλ,p ( y) for any x, y ∈ v . (ii) the sequence (xn)n∈n is convergent w.r.t. a l -random norm p if and only if eλ,p (xn − x) → 0. also the sequence (xn)n∈n is cauchy w.r.t. a l -random norm p if and only if it is cauchy w.r.t. eλ,p . proof. for (i), by the continuity of the t-norm t and the negator n , for every µ ∈ l \{0l , 1l } we can find a λ ∈ l \ {0l , 1l } such that t (n (λ), n (λ)) ≥l n (µ). by definition 1.1 we have p (x + y, eλ,p (x) + eλ,p ( y) + 2δ) ≥l t (p (x, eλ,m (x) +δ), p ( y, eλ,p ( y) +δ)) ≥l t (n (λ), n (λ)) ≥l n (µ), for every δ > 0, which implies that eµ,p (x + y) ≤ eλ,p (x) + eλ,p ( y) + 2δ. since δ > 0 was arbitrary, we have eµ,p (x + y) ≤ eλ,p (x) + eλ,p ( y). for (ii), we have p (xn − x,η) >l n (λ) ⇐⇒ eλ,p (xn − x) < η for every η > 0. 2 quotient spaces definition 2.1. let (v , p , t ) be a l -random normed space, w a linear manifold in v and let q : v −→ v /w be the natural map, q x = x + w . for t > 0, we define: p̄ (x + w , t) = sup{p (x + y, t) : y ∈ w }. cubo 12, 3 (2010) l -random and fuzzy normed spaces ... 75 theorem 2.2. let w be a closed subspace of a l -random normed space (v , p , t ). if x ∈ v and ǫ > 0, then there is an x′ in v such that x′ + w = x + w , eλ,p (x ′) < e ¯λ,p (x + w ) +ǫ. proof. by the properties of sup, there always exists y ∈ w such that eλ,p (x + y) < e ¯λ,p (x + w ) +ǫ. now it is enough to put x ′ = x + y. theorem 2.3. let w be a closed subspace of a l -random normed space (v , p , t ) and p̄ be given in the above definition. then: (1) p̄ is a l -random normed space, on v /w . (2) p̄ (q x, t) ≥l p (x, t). (3) if (v , p , t ) is a complete l -random normed space, then so is (v /w , p̄ , t ). proof. it is clear that p̄ (x + w , t) >l 0l . let p̄ (x + w , t) = 1l . by definition there is a sequence {xn} in w such that p (x+ xn, t) −→ 1l . thus, x+ xn −→ 0 or equivalently xn −→ (−x) and since w is closed, x ∈ w and x + w = w , the zero element of v /w . then we have p̄ ((x + w ) + ( y + w ), t) = p̄ ((x + y) + w , t) ≥l p ((x + m) + ( y + n), t) ≥l t (p (x + m, t1), p ( y + n, t2 )) for m, n ∈ w , x, y ∈ v and t1 + t2 = t. now if we take the sup, then we have p̄ ((x + w ) + ( y + w ), t) ≥l t (p̄ (x + w , t1), p̄ ( y + w , t2 )). therefore p̄ is a l -random norm on v /w . (2) by definition 2.1, we have p̄ (q x, t) = p̄ (x + w , t) = sup{p (x + y, t) : y ∈ w } ≥l p (x, t). note that, by lemma 1.5, eλ,p̄ (q x) = inf{t > 0 : p̄ (q x, t) >l n (λ)} ≤ inf{t > 0 : p (x, t) >l n (λ)} = eλ,p (x). (3) let {xn + w } be a cauchy sequence in v /w . then there exists n0 ∈ n such that for every n ≥ n0, eλ,p̄ ((xn + w ) − (xn+1 + w )) ≤ 2 −n. let y1 = 0. choose y2 ∈ w such that eλ,p (x1 − (x2 − y2), t) ≤ eλ,p̄ ((x1 − x2) + w ) + 1/2. however e ¯λ,p ((x1 − x2) + w ) ≤ 1/2 and so eλ,p (x1 − (x2 − y2)) ≤ 1/2 2. 76 donal o’regan & reza saadati cubo 12, 3 (2010) now suppose yn−1 has been chosen, so choose yn ∈ w such that eλ,p ((xn−1 + yn−1) − (xn + yn)) ≤ eλ,p̄ ((xn−1 − xn) + w ) + 2 −n+1. hence we have eλ,p ((xn−1 + yn−1) − (xn + yn)) ≤ 2 −n+2. however for every positive integer m > n and by lemma 1.5 for λ ∈ l there exists γ ∈ l, such that eλ,p ((xm + ym) − (xn + yn)) ≤ eγ,p ((xn+1 + yn+1) − (xn + yn)) + ···+ eγ,p ((xm + ym) − (xm−1 + ym−1)) ≤ m ∑ i=n 2−i. by lemma 1.5, {xn + yn} is a cauchy sequence in v . since v is complete, there is an x0 in v such that xn + yn −→ x0 in v . on the other hand, xn + w = q(xn + yn) −→ q(x0) = x0 + w . therefore, every cauchy sequence {xn + w } is convergent in v /w and so v /w is complete. thus (v /w , p̄ , t ) is a complete l -random normed space. theorem 2.4. let w be a closed subspace of a l -random normed space (v , p , t ). if two of the spaces v , w and v /w are complete, then so is the third one. proof. if v is a complete l -random normed space, then so are v /w and w . hence all that needs to be checked is that v is complete whenever both w and v /w are complete. suppose that w and v /w are complete l -random normed spaces and let {xn} be a cauchy sequence in v . since eλ,p̄ ((xn − xm) + w ) ≤ eλ,p (xn − xm) for each m, n ∈ n, the sequence {xn + w } is cauchy in v /w and so converges to y + w for some y ∈ w . thus there is a n0 ∈ n such that for every n ≥ n0, we have eλ,p̄ ((xn − y) + w ) < 2 −n. now by the last theorem there exist a sequence { yn} in v such that yn +w = (xn − y) +w , eλ,p ( yn) < eλ,p̄ ((xn − y) +w ) + 2 −n. thus we have limn eλ,p ( yn) ≤ 0 by lemma 1.5, p ( yn, t) → 1l for every t > 0, i.e. limn yn = 0. therefore, {xn − yn − y} is a cauchy sequence in w and thus is convergent to a point z ∈ w . this implies that {xn} converges to z + y and hence v is complete. 3 open mapping and closed graph theorems definition 3.1. a linear operator t : (v , p , t ) −→ (v ′, p ′, t ′) is said to be l -random bounded if there exist constants h ∈ r+ such that for every x ∈ v and for every t > 0, p ′(t x, t) ≥l p (x, t/h). (3.1) cubo 12, 3 (2010) l -random and fuzzy normed spaces ... 77 note that, by (3.1) we have eλ,p ′ (t x) = inf{t > 0 : p ′(t x, t) >l n (λ)} ≤ inf{t > 0 : p (x, t/h) >l n (λ)} = = h inf{t > 0 : p (x, t) >l n (λ)} = heλ,p (x). theorem 3.2. every linear operator t : (v , p , t ) −→ (v ′, p ′, t ′) is l -random bounded if and only if it is continuous. proof. by (3.1) every l -random bounded linear operator is continuous. now, we prove the converse. let the linear operator t be continuous but not l -random bounded. then, for each n in n there is a xn in v such that eλ,p ′ (t xn) ≥ neλ,p ( pn). if we let yn = xn neλ,p (xn ) then it is easy to see yn → 0 but t yn do not tend to 0. theorem 3.3. (open mapping theorem) if t is a l -random bounded linear operator from a complete l -random normed space (v , p , t ) onto a complete l -random normed space (v ′, p ′, t ) then t is an open mapping. proof. the theorem will be proved in several steps. step1: let e be a neighborhood of the 0 in v . we show 0 ∈ ( t(e) )o . let w be a balanced neighborhood of 0 such that w + w ⊂ e. since t(v ) = v ′ and w is absorbing, it follows that v ′ = ∪n t(nw ), so by theorem 3.17 in [6], there exists a n0 ∈ n such that t(n0w ) has nonempty interior. therefore, 0 ∈ ( t(w ) )o − ( t(w ) )o . on the other hand, ( t(w ) )o − ( t(w ) )o ⊂ t(w ) − t(w ) = t(w ) + t(w ) ⊂ t(e). thus the set t(e) includes the neighborhood ( t(w ) )o − ( t(w ) )o of 0. step 2: we show 0 ∈ (t(e))o. since 0 ∈ e and e is an open set, there exists 0l n. on the other hand, 0 ∈ t(b(0,ǫn, t ′ n )), where t ′ n = 1 2n t0, so by step 1, there exist 0l 0 such that b(0,σn, tn ) ⊂ t(b(0,ǫn, t ′ n )). since the set {b(0, r, 1/n)} is a countable local base at zero and t′n −→ 0 as n −→ ∞, so tn and σn can be chosen such that tn −→ 0 and σn −→ 0l as n → ∞. now we show b(0,σ1, t1) ⊂ (t(e)) o. suppose y0 ∈ b(0,σ1, t1 ). then y0 ∈ t(b(0,ǫ1, t ′ 1)) and so for 0l 0 the ball b( y0,σ2, t2) intersects t(b(0,ǫ1, t ′ 1)). therefore there exists x1 ∈ b(0,ǫ1, t ′ 1) such that t x1 ∈ b( y0,σ2, t2), i.e. p ′( y0 − t x1, t2 ) >l n (σ2) or equivalently y0 − t x1 ∈ b(0,σ2, t2) ⊂ t(b(0,ǫ1, t ′ 1)). by the similar argument there exist x2 in b(0,ǫ2, t ′ 2) such that p ′( y0 − (t x1 + t x2), t3) = p ′(( y0 − t x1) − t x2, t3) >l n (σ3). 78 donal o’regan & reza saadati cubo 12, 3 (2010) if this process is continued, it leads to a sequence {xn} such that xn ∈ b(0,ǫn, t ′ n ), p ′ ( y0 − ∑n−1 j=1 t x j, tn ) >l n (σn). now if n, m ∈ n and m > n, then p ( n ∑ j=1 x j − m ∑ j=n+1 x j , t ) = µ ( m ∑ j=n+1 x j , t ) ≥l t m−n(p (xn+1, tn+1), p (xm, tm )) where tn+1 + tn+2 + ··· + tm = t. put t ′ 0 = min {tn+1, tn+2,··· , tm }. since t ′ n −→ 0, there exists n0 ∈ n such that 0 < t ′ n ≤ t ′ 0 for n > n0. therefore, for m > n we have t m−n(p (xn+1, t ′ 0 ), p (xm, t ′ 0 )) ≥l t m−n(p (xn+1, t ′ n+1), p (xm, t ′ m )) ≥l t m−n(n (ǫn+1), n (ǫm)). hence, lim n−→∞ p ( m ∑ j=n+1 x j , t ) ≥l lim n−→∞ t m−n(n (ǫn+1), n (ǫm)) = 1l . that is, p ( ∑m j=n+1 x j , t ) −→ 1l , for all t > 0. thus the sequence { ∑n j=1 x j } is a cauchy sequence and consequently the series { ∑ ∞ j=1 x j } converges to some point x0 ∈ v , because v is a complete space. by fixing t > 0, there exists n0 ∈ n such that t > tn for n > n0, because tn −→ 0. thus p ′ ( y0 − t ( n−1 ∑ j=1 x j ) , t ) ≥l p ′ ( y0 − t ( n−1 ∑ j=1 x j ) , tn ) ≥l n (σn) and thus p ′ ( y0 − t ( ∑ n−1 j=1 x j ) , t ) −→ 1l . therefore, y0 = lim n t ( n−1 ∑ j=1 x j ) = t ( lim n n−1 ∑ j=1 x j ) = t x0. but, by proposition 1 of [7], p (x0, t0 ) = lim n p ( n ∑ j=1 x j , t0 ) ≥l t n(lim n (p (x1, t ′ 1), p (xn, t ′ n )) ≥l lim n t n−1(n (ǫ1), ..., n (ǫn)) >l n (α) hence x0 ∈ b(0,α, t0). step 3: let g be an open subset of v and x ∈ g. then we have t(g) = t x + t(−x + g) ⊃ t x + (t(−x + g))o. hence t(g) is open, because it includes a neighborhood of each of its point. cubo 12, 3 (2010) l -random and fuzzy normed spaces ... 79 corollary 3.4. every one-to-one l -random bounded linear operator from a complete l random normed space onto a complete l -random normed space has a l -random bounded inverse. definition 3.5. let t and t ′ be two continuous t-norms. then t ′ dominates t , denoted by t ′ ≫l t , if for all x1, x2, y1, y2 ∈ l , t [t ′(x1, x2), t ′( y1, y2)] ≤l t ′[t (x1, y1), t (x2, y2)]. theorem 3.6. (closed graph theorem) let t be a linear operator from the complete l -random normed space (v , p , t ) into the complete l -random normed space (v ′, p ′, t ). suppose for every sequence {xn} in v such that xn −→ x and t xn −→ y for some elements x ∈ v and y ∈ v ′ it follows that t x = y. then t is l -random bounded. proof. for any t > 0, x ∈ v and y ∈ v ′, define φ((x, y), t) = t ′(p (x, t), p ′( y, t)), where t ′ ≫l t . first we show that (v × v ′,φ, t ) is a complete l -random normed space. the properties of (lrn1) and (lrn2) are immediate from the definition. for the triangle inequality (lrn3), suppose that x, z ∈ v , y, u ∈ v ′ and t, s > 0, then t (φ((x, y), t),φ((z, u), s)) = t [t ′(p (x, t), p ′( y, t)), t ′(p (z, s), p ′(u, s))] ≤l t ′[t (p (x, t), p (z, s)), t (p ′( y, t), p ′(u, s))] ≤l t ′(p (x + z, t + s), p ′( y + u, t + s)) = φ((x + z, y + u), t + s). now if {(xn, yn)} is a cauchy sequence in v ×v ′, then for every ǫ ∈ l \ {0l } and t > 0 there exists n0 ∈ n such that φ((xn, yn) − (xm, ym ), t) >l n (ǫ) for m, n > n0. thus for m, n > n0, t ′(p (xn − xm, t), p ′( yn − ym, t)) = φ((xn − xm, yn − ym), t) = φ((xn, yn) − (xm, ym), t) >l n (ǫ). therefore {xn} and { yn} are cauchy sequences in v and v ′, respectively, and there exist x ∈ v and y ∈ v ′ such that xn −→ x and yn −→ y and consequently (xn, yn) −→ (x, y). hence (v × v ′,φ, t ) is a complete l -random normed space. the remainder of the proof is the same as the classical case. 4 l -fuzzy normed space we conclude the paper with the setting of l -fuzzy normed spaces. consider the l -fuzzy normed space (x , f , t ) in which f is a l -fuzzy set on x× ]0,+∞[ satisfying the following 80 donal o’regan & reza saadati cubo 12, 3 (2010) conditions for every x, y in x and t, s in (0,+∞): (a) 0l 0, ch denotes the set of φ ∈ c with ‖φ‖ < h. x′(t) denotes the right-hand derivative at t if it exists and is finite. it is supposed that f : r+ × c → rn, that f is continuous, and that f takes bounded sets into bounded sets. here, r+ = [0, ∞). then it is known [2, 6, 7, 15] that for each t0 ∈ r+ and each φ ∈ c there is at least one solution x(t0, φ) of (1) satisfying xt0 (t0, φ) = φ defined on an interval [t0, t0 + α) for some α > 0 and if there is an h1 < h with |x(t, t0, φ)| ≤ h1, then α = ∞. by means of liapunov’s second method, throughout this paper we work with wedges, denoted by wi : r+ → r+, which are continuous and strictly increasing. we also work with continuous cubo 11, 3 (2009) some general theorems on uniform boundedness ... 27 functionals v : r+ × c → r+ (called liapunov functionals) with v (t, 0) ≡ 0, whose derivative v ′ with respect to (1) is defined by v ′ (1) (t, φ) = lim δ→0+ sup[v (t + δ, xt+δ(t, φ)) − v (t, φ)]/δ. definition 1.1. solutions of (1) are uniformly bounded (u.b.) if for each b1 > 0 there exists b2 > 0 such that [t0 ≥ 0, ‖φ‖ ≤ b1, t ≥ t0] imply that |x(t, t0, φ)| < b2. solutions of (1) are uniformly ultimately bounded (u.u.b.) for bound b if for each b3 > 0 there exists t > 0 such that [t0 ≥ 0, ‖φ‖ ≤ b3, t ≥ t0 + t ] imply |x(t, t0, φ)| < b. because we are also going to state some stability results, it is necessary to tell the difference of conditions between stability and boundedness. when we discuss stability, we always assume, in addition to the above general assumptions: (i) f : r+ × c → rn, and f (t, 0) ≡ 0 so that x ≡ 0 is a solution of (1), and is called the zero solution. (ii) v : r+ × ch → r+, and v (t, 0) ≡ 0. (iii) wi(0) = 0 for each wedge wi(r). it is a common idea that stability theory can be generalized in a manner parallel to boundedness theory. but the fact is that the development of boundedness theory is much slower than that of stability theory. for instance, for the system of ordinary differential equations x ′ = f (t, x), x ∈ rn, (2) where f : r+ × d → rn continuous, and d ⊂ rn an open set with 0 ∈ d, two classical results may be stated as the following: theorem 1.1. let v : r+ × d → r+ be continuous and suppose (i) w1(|x|) ≤ v (t, x) ≤ w2(|x|), and (ii) v ′ (2) (t, x(t)) ≤ −w3(|x(t)|). then x ≡ 0 of (2) is uniformly asymptotically stable. theorem 1.2. let v : r+ × d → r+ be continuous and suppose (i) w1(|x|) ≤ v (t, x) ≤ w2(|x|), with w1(r) → ∞ as r → ∞, (ii) v ′ (2) (t, x(t)) ≤ −w3(|x(t)|) + m, m > 0,. and (iii) w3(u ) > m , for some u > 0. then the solutions of (2) are u.b. and u.u.b. 28 tingxiu wang cubo 11, 3 (2009) the parallel results of theorem 1.1 and theorem 1.2 for delay equations may be found in [15;p.190; p.202], and stated as the following. theorem 1.3. let v : r+ × ch → r+ be continuous with (i) w1(‖φ‖) ≤ v (t, φ) ≤ w2(‖φ‖), and (ii) v ′ (2) ≤ −w3(‖φ‖). then the zero solution of (1) is uniformly asymptotically stable. theorem 1.4. let v : r+ × c → r+ be continuous with (i) w1(|φ(0)|) ≤ v (t, φ) ≤ w2(|φ(0)|) + w3(‖φ‖), (ii) v ′ (1) (t, xt) ≤ 0, for |x(t)| large, (iii) w1(r) − w3(r) → ∞ as r → ∞. then solutions of (1) are u.b. theorem 1.3 is a direct parallel result of theorem 1.1 for delay equations. it has not proved to be useful. for applications, investigators gave the next theorem [15; p.192]. theorem 1.5. suppose that f (t, φ) is bounded for φ bounded. let v : r+×c → r+ be continuous with (i) w1(|φ(0)|) ≤ v (t, φ) ≤ w2(|φ(0)|) + w3(‖φ‖2), where ‖ · ‖2 denotes the l2-norm; (ii) v ′ (1) (t, xt) ≤ −w3(|x(t)|). then x ≡ 0 is uniformly asymptotically stable. in 1978, burton [1] eliminated the condition that f (t, φ) is bounded for φ bounded in theorem 1.5. since then, stability theory of this type has been developed very much. in 1989, burton and hatvani [4] gave the following quite general results. concepts of pim and ip used below will be defined in the next section. theorem 1.6. suppose that d, v : r+ × ch → r+ are continuous, η : r+ → r+ is pim, and the following conditions are satisfied. (i) w1(|x(t)|) ≤ v (t, xt) ≤ w2(|x(t)|) + w3( ∫ t t−h d(s, xs)ds); (ii) v ′ (1) (t, xt) ≤ −η(t)w4(d(t, xt)); (iii) d(t, φ) ≤ w5(‖φ‖); (iv) for some k ∈ (0, h) there is a wedge wk such that [t ∈ r+, u : [−2h, 0] → rn is continuous, |u(s)| < k for s ∈ [−2h, 0]] imply wk(inf{|u(r)| : −h ≤ r ≤ 0}) ≤ ∫ 0 −h d(t + s, us)ds. then x = 0 is uniformly asymptotically stable. cubo 11, 3 (2009) some general theorems on uniform boundedness ... 29 theorem 1.7. let v : r+ × ch → r+ be continuous with (i) w1(|φ(0)|) ≤ v (t, φ) ≤ w2(‖φ‖); and (ii) v ′ (1) (t, xt) ≤ −η1w3(|x′(t)|)−η2(t)w4(|x(t)|), where η1 > 0 is a constant, lims→∞ ∫ t∗+s t∗ η2(s)ds = ∞ uniformly with respect to t∗, and there are α > 0, r0 > 0 such that r > r0 implies w3(r) ≥ αr. then x=0 is uniformly asymptotically stable. in 1991, wang [10, 11] generalized and unified these two theorems and gave the following general and yet clean theorem. theorem 1.8. let d, v : r+ × ch → r+ with v continuous and d continuous along the solutions of (1). suppose that there are continuous functions η1, η2 : r+ → r+ and that the following conditions hold: (i) w1(|x(t)|) ≤ v (t, xt) ≤ w2(|x(t)| + ∫ t t−h d(s, xs)ds); (ii) v ′ (1) (t, xt) ≤ −γ1(t)w3(m(xt)) − γ2(t)w4(d(t, xt)); where γ1ǫ ip(s ) for some s > 0, γ2ǫ pim, and m(φ) = min−h≤s≤0 |φ(s)|; (iii) d(t, φ) ≤ w5(‖φ‖). then x = 0 is uniformly asymptotically stable. the research on stability of this type continues. in 1994, wang [12] improved theorem 1.8 with weaker decrescentness. but comparing with stability theory, boundedness theory develops much more slowly than stability theory does. for u.b. although theorem 1.4 had been proved before 1966, a parallel result like theorem 1.5 without the condition that f is bounded for φ bounded had not been given until 1986. in 1986, burton and zhang [5] showed theorem 1.9. let v : r+ × c → r+ be continuous with (i) w1(|φ(0)|) ≤ v (t, φ) ≤ w2(|φ(0)|) + w3( ∫ 0 −h w4(|φ(s)|)ds), (ii) v ′ (1) (t, xt) ≤ −w4(|x(t)|) + m, m > 0, (iii) w1(r), w4(r) → ∞, as r → ∞. then solutions of (1) is u.b. and u.u.b. in 1990, burton [3] showed theorem 1.10. let v : r+ × c → r+ continuous with (i) w1(|φ(0)|) ≤ v (t, φ) ≤ w2(|φ(0)|) + w3(‖φ‖2), (ii) v ′ (1) (t, xt) ≤ −w4(‖xt‖2) + m, m > 0, (iii) w1(r) → ∞, as r → ∞, w4(u/2) ≥ 12m for some u > 0. then solutions of (1) are u.b. and u.u.b. 30 tingxiu wang cubo 11, 3 (2009) in this paper, we are going to give some general theorems like theorem 1.8, generalize theorems like theorem 1.10, and investigate some examples. one application of uniform boundedness and ultimate uniform boundedness is to prove the existence of periodic solutions. [2] gives much discussion on periodic solutions. makay [8] discussed dissipativeness, which is weaker than u.u.b, and gave an interesting result on periodic solutions. based on this paper and [13], the author examines many common functional differential equations, and obtains not only u.b. and u.u.b., but also the existence of periodic solutions. for more examples or applications of this paper and [13], please see [14]. 2 preliminaries definition 2.1. a measurable function γ : r+ → r+ is said to be integrally positive with parameter α > 0 (ip(α)) if limt→∞ inf ∫ t+α t η(s)ds > 0. that is, η ∈ ip(α) implies that there exist t > 0, and γ > 0 such that for each t > t , ∫ t+α t η(s)ds ≥ γ. thus we also denote ip(α, γ) = ip(α). this definition is equivalent to the original one, which can be found in [4]. now we give a weaker definition than the last one. definition 2.2. a measurable function γ : r+ → r+ is said to be partially integrally positive with parameters α > 0, β > 0, and γ > 0 (pip(α, β, γ)) if there is a sequence {tn}∞1 with α ≤ tn+1 − tn ≤ β such that ∫ tn+α tn η(s)ds ≥ γ. clearly η ∈ ip(α, γ) implies η ∈ pip(α, β, γ) for any β ≥ α. lemma 2.1. let f : r+ → r+ be continuous and g(t) = ∫ t t−h f (s)ds. if g(t1) ≥ ε for some t1 ≥ 2h and ε > 0, then there is a closed interval [a, b] of length h containing t1 in which g(t) ≥ ε/2. the proof of this lemma, which was originally proved by t. krisztin, can be found in [9]. 3 main results definition 3.1. a functional d : r+ × c → r+ is said to be continuous along solutions of (1) if d(t, xt) is continuous on [t0, ∞) for each solution x(t, t0, φ) of (1) defined on [t0, ∞). denote m(φ) = min −h≤s≤0 |φ(s)| f or each φ ∈ c. theorem 3.1. let v : r+ × c → r+ be continuous and d : r+ × c → r+ be continuous along solutions of (1). let γ1 ∈ pip(α, β, γ1) and γ2 ∈ ip(h, γ2). denote c = max{β, h}. let w1, w2, w3, w4, w5 be wedges with w1(r) → ∞, as r → ∞. assume that (i) w1(|x(t)|) ≤ v (t, xt) ≤ w2(|x(t)| + ∫ t t−h d(u, xu)du); (ii) v (t, φ) ≤ w3(‖φ‖); cubo 11, 3 (2009) some general theorems on uniform boundedness ... 31 (iii) v ′ (1) (t, xt) ≤ −γ1(t)w4(m(xt)) − γ2(t)w5( ∫ t t−h d(u, xu)du) + m ; (iv) there is a ξ > 0 such that w4(ξ)γ1 > 10m c, and w5(ξ/2)γ2 > 10m c. then solutions of (1) are u.b. and u.u.b. proof. γ1 ∈ pip(α, β, γ1) implies that there is a sequence {tn}∞1 with α ≤ tn+1 − tn ≤ β such that ∫ tn+α tn γ1(u)du ≥ γ1. γ2 ∈ ip(h, γ2 ) implies that there is t1 > 0 such that for each t > t1, ∫ t+h t γ2(u)du ≥ γ2. first, we want to show u.b. that is, for each b1 > 0, there is a b2 > 0 such that [t0 ≥ 0, φ ∈ c, ‖φ‖ < b1 , and t ≥ t0 ] imply |x(t, t0, φ)| < b2. denote x(t) = x(t, t0, φ). let t2 = max{t1, t1}, u = w2(2ξ), and ∆ = max{2w3(b1) + (t2 + 5c)m, u}. let i0 = [t0, t0 + t2 + 5c], ik = [t0 + t2 + 5kc, t0 + t2 + 5(k + 1)c], k = 1, 2, 3, · · · . claim i. for each k = 0, 1, 2, · · · , there is a qk ∈ ik such that v (qk, xqk ) < ∆. we use mathematical induction to prove it. for k = 0, integrating (iii) from t0 to t ∈ i0, we have v (t, xt) ≤ v (t0, xt0 ) + (t2 + 5c)m ≤ w3(b1) + (t2 + 5c)m < ∆. (3) clearly, there is a q0 ∈ i0 such that v (q0, xq0 ) < ∆. in fact, q0 can be any number in i0. particularly, we take q0 = t0 + t2 + 5c. for k = n, assume there is a qn ∈ in such that v (qn, xqn ) < ∆. we want to show there is a qn+1 ∈ in+1 such that v (qn+1, xqn+1 ) < ∆. if this is false, then v (t, xt) ≥ ∆ on in+1. it is clear that there is a qnǫin such that v (qn, xqn ) = ∆, and v (t, xt) ≥ ∆ on [qn, t0 + t2 + 5(n + 1)c] ⋃ in+1. then w2 ( |x(t)| + ∫ t t−h d(u, xu)du ) ≥ v (t, xt) ≥ ∆ ≥ u (4) on [qn, t0 + t2 + 5(n + 1)c] ⋃ in+1. particularly, consider the interval in+1 = [t0 + t2 + 5(n + 1)c + c, t0 + t2 + 5(n + 1)c + 4c] ⊂ in+1. (4) implies that either there is some t∗ ∈ in+1 with ∫ t∗ t∗−h d(u, xu)du ≥ 1 2 w −1 2 (u ) = ξ 32 tingxiu wang cubo 11, 3 (2009) or |x(t)| ≥ 1 2 w −1 2 (u ) = ξ for each t ∈ in+1. case i. ∫ t∗ t∗−h d(u, xu)du ≥ ξ. by lemma 2.1, there are a and b with b − a = h and t∗ ∈ [a, b] such that for each t ∈ [a, b] ∫ t t−h d(u, xu)du ≥ 1 2 ξ. clearly [a, b] ⊂ in+1. integrating (iii) from qn to t0 + t2 + 5(n + 2)c, we have ∆ ≤ v (t0 + t2 + 5(n + 2)c, xt0+t2+5(n+2)c ) ≤ v (qn, xqn ) − ∫ t0+t2+5(n+2)c qn γ2(s)w5 ( ∫ s s−h d(u, xu)du ) + 10m c ≤ ∆ − w5( 1 2 ξ) ∫ a+h a γ2(s)ds + 10m c ≤ ∆ − w5( 1 2 ξ)γ2 + 10m c < ∆, a contradiction. case ii. |x(t)| ≥ ξ for each t ∈ in+1. note that in+1 contains three subintervals of length of c. therefore in+1 contains at least three members of {tn}, say s1, s2, and s3 with s1 < s2 < s3. then integrating (iii) from qn to t0 + t2 + 5(n + 2)c, we have ∆ ≤ v (t0 + t2 + 5(n + 2)c, xt0+t2+5(n+2)c ) ≤ v (qn, xqn ) − ∫ t0+t2+5(n+2)c qn γ1(u)w4(m(xu))du + 10m c ≤ ∆ − w4(ξ) ∫ s2+α s2 γ1(s)ds + 10m c ≤ ∆ − w4(ξ)γ1 + 10m c < ∆, a contradiction. so there is a qn+1 ∈ in+1 such that v (qn+1, xqn+1 ) < ∆. the mathematical induction is complete. therefore for each k = 0, 1, 2, · · · , there is a qk ∈ ik such that v (qk, xqk ) < ∆. now for each t ≥ t0, v (t, xt) ≤ ∆ if t ∈ i0 (see(3)), cubo 11, 3 (2009) some general theorems on uniform boundedness ... 33 or if t ∈ ik, k = 1, 2, 3, · · · , (iii) implies v (t, xt) ≤ v (qk, xqk ) + 5m c ≤ ∆ + 5m c if t ≥ qk; or v (t, xt) ≤ v (qk−1, xqk−1 ) + 10m c ≤ ∆ + 10m c if t < qk. that is w1(|x(t)|) ≤ v (t, xt) ≤ ∆ + 10m c for each t ≥ t0. take b2 = w −11 (∆ + 10m c). this proves u.b. next we are going to prove u.u.b. for bound b. b will be determined at the end of the proof. that is for each b3 > 0, there exists a t > 0 such that [t0 ≥ 0, ‖φ‖ < b3, t ≥ t0 + t ] imply that |x(t, t0, φ)| < b. the proof is similar to that of u.b. u and ξ are the same as before. let n = max {[ w3(b3) + t2m + 5m c w5(ξ/2)γ2 − 10m c ] , [ w3(b3) + t2m + 5m c w4(ξ)γ1 − 10m c ]} + 1, and t3 = 10n c + t2, where [x] denotes the greatest integer function. let j0 = [t0 + t2, t0 + t3 + 5c]. claim ii. there is a p0 ∈ j0 such that v (p0, xp0 ) < u . we show the claim by contradiction. assume v (t, xt) ≥ u for each t ∈ j0. then for each t ∈ j0, w2 ( |x(t)| + ∫ t t−h d(u, xu)du ) ≥ u (5) note that j0 can contain 2n + 1 subintervals of length 3c, say, j0i = [t0 + t2 + 5ic + c, t0 + t2 + 5ic + 4c], i = 0, 1, 2, · · · , 2n. on each j0i, (5) implies that either there is a u ∗ i ∈ j0i such that ∫ u∗i u∗ i −h d(u, xu)du ≥ ξ, or |x(t)| ≥ ξ on j0i. if the number of these {u∗i } is more than n + 1, by lemma 2.1, there are ai and bi with bi − ai = h and u∗i ∈ [ai, bi] such that for each t ∈ [ai, bi] ∫ t t−h d(u, xu)du ≥ 1 2 ξ. clearly [ai, bi] ⋂ [ai+1, bi+1] = ∅ and [ai, bi] ⊂ j0 for each such i. 34 tingxiu wang cubo 11, 3 (2009) integrating (iii) from t0 to t0 + t3 + 5c, we have 0 ≤ v (t0 + t3 + 5c, xt0+t3+5c)) ≤ v (t0, xt0 ) − ∫ t0+t3+5c t0 γ2(s)w5 ( ∫ s s−h d(u, xu)du ) ds + m (t3 + 5c) ≤ w3(‖xt0‖) − n ∑ i=1 ∫ bi ai γ2(s)w5 ( ∫ s s−h d(u, xu)du ) ds + m (t3 + 5c) ≤ w3(b3) − w5(ξ/2)γ2n + 10n m c + t2m + 5m c ≤ w3(b3) − n [w5(ξ/2)γ2 − 10m c] + t2m + 5m c < 0, by the choice of n a contradiction. this means that the number of {u∗i } is less than n + 1. suppose that there are more than n + 1 intervals of j0i on which |x(t)| ≥ ξ, say these intervals are j0i, i = 1, 2, 3, · · · , n + 1. clearly, each of these intervals contains at least three members of {tn}, say v1i, v2i, and v3i with v1i < v2i < v3i. then integrating (iii) from t0 to t0 + t3 + 5c, we have 0 ≤ v (t0 + t3 + 5c, xt0+t3+5c ) ≤ v (t0, xt0 ) − ∫ t0+t3+5c t0 γ1(s)w4(m(xs))ds + m (t3 + 5c) ≤ w3(‖xt0‖) − n ∑ i=1 ∫ v2i+α v2i γ1(s)w4(m(xs))ds + m (t3 + 5c) ≤ w3(b3) − w4(ξ)n γ1 + 10n m c + t2m + 5m c ≤ w3(b3) − n [w4(ξ)γ1 − 10m c] + t2m + 5m c < 0, by the choice of n a contradiction. therefore there must be a p0 ∈ j0 such that v (p0, xp0 ) < u . now define jk = [p0 + 5(k − 1)c, p0 + 5kc] for k = 1, 2, 3, · · · . claim iii. for each k = 1, 2, 3, · · · , there is a pk ∈ jk such that v (pk, xpk ) < u . we use mathematical induction, again. for k = 1, j1 = [p0, p0 + 5c] and by claim ii, we obviously can take p1 = p0 with v (p1, xp1 ) < u . assume that for k = n, there is a pn ∈ jn such that v (pn, xpn ) < u . we want to show for k = n + 1, there is a pn+1 ∈ jn+1 such that v (pn+1, xpn+1 ) < u . assume for the sake of contradiction that v (t, xt) ≥ u on jn+1. it is clear that there is a pn ∈ jn such that v (pn, xpn ) = u, and v (t, xt) ≥ u on [pn, p0 + 5nc] ⋃ jn+1. then w2 ( |x(t)| + ∫ t t−h d(u, xu)du ) ≥ v (t, xt) ≥ u (6) cubo 11, 3 (2009) some general theorems on uniform boundedness ... 35 on [pn, p0 + 5nc] ⋃ jn+1. particularly, consider the interval jn+1 = [p0 + 5nc + c, p0 + 5nc + 4c] ⊂ jn+1. (6) implies that either there is t∗ ∈ jn+1 with ∫ t∗ t∗−h d(u, xu)du ≥ ξ, or |x(t)| ≥ ξ for each t ∈ jn+1. case i. ∫ t∗ t∗−h d(u, xu)du ≥ ξ. by lemma 2.1, there are a and b with b − a = h and t∗ ∈ [a, b] such that for each t ∈ [a, b], ∫ t t−h d(u, xu)du ≥ ξ/2. clearly [a, b] ∈ jn+1. integrating (iii) from pn to p0 + 5(n + 1)c, we have u ≤ v (p0 + 5(n + 1)c, xp0+5(n+1)c ) ≤ v (pn, xpn ) − ∫ p0+5(n+1)c pn γ2(s)w5 ( ∫ s s−h d(u, xu)du ) ds + 10m c ≤ u − w5(ξ/2) ∫ a+h a γ2(s)ds + 10m c ≤ u − w5(ξ/2)γ2 + 10m c < u, a contradiction. case ii. |x(t)| ≥ ξ for each t ∈ jn+1. note that jn+1 contains three subintervals of length of c. therefore jn+1 contains at least three members of {tn}, say s1, s2, and s3 with s1 < s2 < s3. then integrating (iii) from pn to p0 + 5(n + 1)c, we have u ≤ v (p0 + 5(n + 1)c, xp0+5(n+1)c ) ≤ v (pn, xpn ) − ∫ p0+5(n+1)c pn γ1(s)w4(m(xu))du + 10m c ≤ u − w4(ξ) ∫ s2+α s2 γ1(s)ds + 10m c ≤ u − w4(ξ)γ1 + 10m c < u a contradiction. so there is a pn+1 ∈ jn+1 such that v (pn+1, xpn+1 ) 0 such that t ∈ jk. then (iii) implies w1(|x(t)|) ≤ v (t, xt) ≤ v (pk, xpk ) + 5m c ≤ u + 5m c if t ≥ pk; 36 tingxiu wang cubo 11, 3 (2009) or w1(|x(t)|) ≤ v (t, xt) ≤ v (pk−1, xpk−1 ) + 10m c ≤ u + 10m c if t < pk; the later case will not happen for k = 1 because of the choice of p1. take b = w −1 1 (u + 10m c) and t = t3 + 5c. then for each t ≥ t0 + t , |x(t)| < b. this proves u.u.b. corollary 3.1. let v : r+ × c → r+ be continuous with (i) w1(|x(t)|) ≤ v (t, xt) ≤ w2(|x(t)| + ∫ t t−h |x(s)|pds), where w1(r) → ∞, as r → ∞; and p > 0 is a constant; (ii) v ′ (1) (t, xt) ≤ −γ(t)w6( ∫ t t−h |x(s)|pds)+m , where γ ∈ ip(h, γ), and m > 0 is a constant; (iii) there is a ξ > 0 such that min{w6(ξ/2), w6(hξp)}γ > 20m h. then solutions of (1) are u.b. and u.u.b. proof. in theorem 3.1, take d(t, xt) = |x(t)|p. condition (ii) implies v ′ (1) (t, xt) ≤ − 1 2 γ(t)w6 ( ∫ t t−h |x(s)|pds ) − 1 2 γ(t)w6 ( ∫ t t−h |x(s)|pds ) + m ≤ − 1 2 γ(t)w6 (h(m(xt)) p ) − 1 2 γ(t)w6 ( ∫ t t−h |x(s)|pds ) + m. clearly, γ1(t) = 1 2 γ(t) ∈ pip(h, h, γ/2), γ2(t) = 12 γ(t) ∈ ip(h, γ/2). take w4(r) = w6(hr p ), and w5(r) = w6(r). the other conditions of theorem 3.1 can be verified easily. with a little stronger condition, we can state a cleaner corollary. corollary 3.2. let v : r+ × c → r+ be continuous with (i) w1(|x(t)|) ≤ v (t, xt) ≤ w2(|x(t)| + ∫ t t−h |x(s)|pds), where w1(r) → ∞, as r → ∞; and p > 0 is a constant; (ii) v ′ (1) (t, xt) ≤ −γ(t)w6( ∫ t t−h |x(s)|pds) + m , where γ ∈ ip(h, γ), m > 0 a constant, and w6(r) → ∞ as r → ∞. then solutions of (1) are u.b. and u.u.b. remark. in application, the inequality v (t, xt) ≤ w2(|x(t)|) + w7 ( ∫ t t−h d(u, xu)du ) (7) is more often seen. but condition (i) of theorem 3.1 looks cleaner and a little more convenient in proof. it can be shown that (7) and condition (i) are equivalent. proposition 3.1. (i) if w1 and w2 are two wedges on r+, then there are wedges w∗ and w ∗ such that w∗(s + t) ≤ w1(s) + w2(t) ≤ w ∗(s + t), f or s, t ∈ r+. cubo 11, 3 (2009) some general theorems on uniform boundedness ... 37 (ii) if w is a wedge, then there are wedges w1, w2, w3, and w4 such that w1(s) + w2(t) ≤ w (s + t) ≤ w3(s) + w4(t), f or each s, t ∈ r+. proposition 3.1(i) was proved in the both [9, proposition 5] and [10, lemma 2]. but proposition 3.1(i) only shows that (7) implies condition (ii) of theorem 3.1. to show that condition (ii) of theorem 3.1 implies (7), we need proposition 3.1(ii). proof of proposition 3.1(ii). obviously w (s + t) = 1 2 w (s + t) + 1 2 w (s + t) ≥ 1 2 w (s) + 1 2 w (t). so take w1(s) = 1 2 w (s), and w2(t) = 1 2 w (t). it is also clear that w (2s) + w (2t) ≥ w (s + t), since s + t ≤ max{2s, 2t}. now take w3(s) = w (2s), and w4(t) = w (2t). this proves proposition 3.1 (ii). 4 examples example 4a. consider the scalar equation x ′ (t) = −a(t)x(t) + ∫ t t−h b(s)x(s)ds + f (t, xt) (8) with a : r+ → r+ and b : [−h, ∞) → r continuous, and f (t, φ) : r+ × c → r continuous. theorem 4.1. suppose that the functions a and b of (8) satisfy: (a) there is a constant θ > 0 with 0 < θh < 1 such that |b(t)| − θa(t) ≤ 0; (b) ∫ t t−h a(s)ds ≤ b for some constant b > 0, and ∫ t t−h a(s)ds is pip(α, β, γ) for some constants α > 0, β > 0, and γ > 0; (c) |f (t, φ)| ≤ m for (t, φ) ∈ r+ × c, and m > 0 is a constant. then solutions of (8) are u.b. and u.u.b. proof. the conclusion follows theorem 3.1. find θ0 > 0 and δ > 0 such that θ0 = θ + δ and 0 < θ0h < 1. this can be done, for instance, by taking δ = 1−θh 2h . then for t ≥ 0, |b(t)| − θ0a(t) ≤ −δa(t). define v (t, xt) = |x(t)| + θ0 ∫ 0 −h ∫ t t+s a(u)|x(u)|duds 38 tingxiu wang cubo 11, 3 (2009) and d(t, xt) = a(t)|x(t)|. then we have |x(t)| ≤ v (t, xt) ≤ |x(t)| + θ0h ∫ t t−h a(u)|x(u)|du ≤ |x(t)| + θ0h ∫ t t−h d(s, xs)ds. therefore condition (i) of theorem 3.1 is satisfied. clearly, v (t, φ) ≤ ( 1 + θ0h ∫ t t−h a(s)ds ) ‖φ‖ ≤ (1 + θ0hb)‖φ‖ by condition (b). hence condition (ii) of theorem 3.1 is fulfilled. we also have v ′ (t, xt) ≤ −a(t)|x(t)| + ∫ t t−h |b(u)||x(u)|du + |f (t, xt)| + θ0ha(t)|x(t)| − θ0 ∫ t t−h a(u)|x(u)|du = (θ0h − 1)a(t)|x(t)| + ∫ t t−h [|b(u)| − θ0a(u)]|x(u)|du + m ≤ (θ0h − 1)a(t)|x(t)| − δ ∫ t t−h a(u)|x(u)|du + m (9) ≤ − 1 2 δ ∫ t t−h a(u)du m(xt) − 1 2 δ ∫ t t−h d(u, xu)du + m (10) this implies that condition (iii) of theorem 3.1 is satisfied. take w4(r) = r, and w5(r) = r. then w4(r) → ∞ and w5(r) → ∞ as r → ∞. therefore condition (iv) of theorem 3.1 is also fulfilled. now according to theorem 3.1, solutions of (8) are u.b. and u.u.b. remark: if we use (9), we need to assume a ∈ pip(α, β, γ) which is clearly stronger than ∫ t t−h a(s)ds ∈ pip(α, β, γ). so condition (iii) of theorem 3.1 is weaker than v ′ (1) (t, xt) ≤ −γ1(t)w4(|x(t)|) − γ2(t)w5 ( ∫ t t−h d(u, xu)du ) + m. theorem 4.2.consider equation (8) again. suppose that (a) there are constants k > 1, α > 0, β > 0, and γ > 0 such that −a(t) + kh|b(t)| := −γ(t) ≤ 0 and γ ∈ pip(α, β, γ); (b) ∫ t t−h |b(s)|ds ≤ b for each t ≥ 0 and some constant b ≥ 0; (c) |f (t, φ)| ≤ m for some constant m ≥ 0 and each (t, φ) ∈ r+ × c. cubo 11, 3 (2009) some general theorems on uniform boundedness ... 39 then solutions of (8) are u.b. and u.u.b. proof. the conclusion follows theorem 3.1. define d(t, φ) = |b(t)||φ(0)| and v (t, xt) = |x(t)| + k ∫ 0 −h ∫ t t+s |b(u)||x(u)|duds. then v ′ (t, xt) ≤ −a(t)|x(t)| + ∫ t t−h |b(s)||x(s)|ds + |f (t, x + t)| + k ∫ 0 −h |b(t)||x(t)|ds − k ∫ 0 −h |b(t + s)||x(t + s)|ds = (−a(t) + kh|b(t)|)|x(t)| + (1 − k) ∫ t t−h |b(s)||x(s)|ds + m = −γ(t)|x(t)| − (k − 1) ∫ t t−h d(s, xs)ds + m. all the other conditions of theorem 3.1 can be verified easily. therefore the solutions of (4a) are u.b. and u.u.b. received: january 27, 2008. revised: march 10, 2008. references [1] burton, t.a., uniform asymptotic stability in functional differential equations, proc. amer. math. soc., 68(1978), 195–199. 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[9] makay, g., on the asymptotic stability of the solutions of functional differential equations with infinite delay, j. differential equations, vol.108, no.1, 1994, p.139–151. [10] wang, t., stability in abstract functional differential equations, part i: general theorems, j. math. anal. appl., vol.186, no.2, sept. 1994, p.534–558. [11] wang, t., stability in abstract functional differential equations, part ii: applications, j. math. anal. appl., vol.186, no.3, sept. 1994, p.835–861. [12] wang, t., weakening condition w1(|φ(0)|) ≤ v (t, φ) ≤ w2(‖φ‖) for uniform asymptotic stability, nonlinear analysis, vol.23, no.2, 1994, p.251–264. [13] wang, t., uniform boundedness with the condition, v ′(t, xt) ≤ −γ(t)w4(m(xt)) −w5(d(t, xt)) + m , nonlinear analysis, no.2, pp.251–264, vol.23 (1994). [14] wang, t., periodic solutions and liapunov functionals, in the book, boundary value problems for functional differential equations, edited by j. henderson, world scientific, pp.289– 299, (1995). [15] yoshizawa, t., stability theory by liapunov’s second method, math. soc. japan, tokyo, 1966. 07-uni_boun cubo a mathematical journal vol.11, n o ¯ 02, (139–149). may 2009 a general purpose platform for data clustering analysis haiyan qiao and brandon edwards department of computer science and engineering california state university, san bernardino san bernardino, ca 92407, usa email: hqiao@csusb.edu abstract grouping objects into meaningful sets clustering is an important procedure in many fields of social sciences. yet, clustering analysis is a difficult problem because many factors come into play in devising a well tuned clustering technique for a given clustering problem. therefore, an easy-to-use clustering analysis tool is needed. in this paper, we have designed a general purpose clustering analysis platform, which integrates the different clustering algorithms and provides the clustering results in both textual and visual display. this platform will assist the users without any computing or programming background in the selection of appropriate clustering algorithm, improve the quality of clustering, and be extendable to the evaluation of clustering results. resumen agrupación de objetos en conjuntos meaningful-clustering es un importante procedimiento en muchos campos de las ciencias sociales. análisis de clustering es un problema dif́ıcil pues varios factores entran en juego en idear una bien afinada técnica de clustering para un determinado problema de clustering. por lo tanto, una análisis de 140 haiyan qiao and brandon edwards cubo 11, 2 (2009) clustering de fácil nanejo es necesário. en este art́ıculo, designamos una plataforma para propositos generales de análisis de clustering, la cual integra los diferentes algoritmos de clustering y prove los resultados de clustering mostrando a la vez textual y visual. esta plataforma asiste a los usuarios sin cualquer formación computacional o de programación en la seleción de algoritmos de clustering apropriados, perfeccionando la calidad del clustering y extendible para evaluación de resultados de clustering. key words and phrases: data clustering, clustering validation, clustering platform. math. subj. class.: 62h30, 91c20. 1 introduction with the increased advancement of computers and technologies for data collection and data storage, the task of finding patterns and discovering knowledge through data analysis techniques, e.g., data clustering, is getting exceedingly important. data clustering is to group data into clusters so that the objects within the same cluster are similar to each other, while the objects in different clusters are dissimilar to one another. for example, social network analysis, i.e., the identification and reorganization of community by certain types of interdependency among people, such as values, visions, ideas, financial exchange, friends, kinship, dislike, conflict, trade, web links, sexual relations, disease transmission, or airline routes [1], plays an important role on the study of social sciences. in the analysis of complex survey data, cluster analysis is usually the crucial step to discover patterns that will be used in the decision-makings such as the target audience for a product, the development of new products etc. data clustering analysis is also an essential technique for behavior studies and social-economic studies. although data clustering is an important procedure in many fields of the social sciences, clustering analysis is a difficult problem. many factors such as effective similarity measures, algorithms and initial parameters come into play in devising a well tuned clustering technique for a given clustering problem. moreover, it is well known that no clustering method can be universally applicable to all sorts of data structures in terms of data distribution, size, density, dimensionality etc. a large number of clustering algorithms have been developed for different contexts or purposes. the diversity, on one hand, provides us with many choices. on the other hand, the profusion of options causes confusion [2, 3]. how to choose an effective clustering algorithm and how to assess the quality of the clusters returned are critical for the success of data clustering. however, these fundamental questions are not well addressed in the research of data clustering. as the new developments in one discipline usually spread slowly in the relevant disciplines, the development and use of data clustering as a formal analysis procedure by anthropologists, psychologies, political scientists, and sociologists is not only promising but challenging as well. therefore, it is desirable to have a clustering tool that assists the researchers in different disciplines in clustering analysis. a few platforms for cluster analysis have been developed for some cubo 11, 2 (2009) a general purpose platform for data clustering analysis 141 special contexts or purposes; see [4, 5]. however, they are either limited in their scope or are far from being user-friendly. one such platform, named cluster 3.0 [4], allows a user to cluster gene expression data for use with bioinformatics. however, cluster 3.0 only offers k-means algorithm and hierarchical clustering method, and the results generated by the platform are not easy to interpret. another such platform, rapidminer [5], offers a variety of clustering. rapidminer, however, requires extensive knowledge and skills in programming, which are lacked for the general users. there is no platform so far that offers a simple and user-friendly environment for cluster analysis. in this paper, we will describe a generalized cluster analysis platform with easy-to-use interface. the platform provides the classic clustering algorithms with both textual and visual display of clustering results for any given data set. the platform serves the researchers in social sciences easy-to-use interface and the ability for future extension to include clustering evaluation criteria. the platform will assist the users in comparing clustering results of different clustering algorithms and making a wise decision of which cluster algorithm best serves their purpose. the rest of the paper is organized as follows. section 2 surveys the primary data clustering algorithms and clustering metrics. the general purpose clustering analysis platform is described and demonstrated in section 3. our conclusions and directions for future work are presented in section 4. 2 methodology to cluster a data set, one question we need to answer is which similarity measure is appropriate in a given situation. the most common similarity measurements within a set of data are distance measurements, such as euclidean distance, a special case (p = 2) of minkowski metric dp(xi, xj ) = ( n ∑ k=1 |xi,k − xj,k| p ) 1/p = ‖xi − xj‖p, where n is data space dimension. instead of direct use of minkowski distance, normalization of the continuous features is usually adopted in order to avoid the tendency of the largest-scaled feature to dominate the others [3]. there are other measures used for the clustering strings, such as cosine similarity, which is the most commonly used method for document clustering [2]. another important question for clustering analysis is clustering metrics. data clustering algorithms are classified into different categories based on different clustering metrics. a few categories of commonly used clustering algorithms are introduced below. 2.1 clustering analysis (a) k-means algorithm and its variants k-means algorithm is the most well-known centroid algorithm which assigns every data point 142 haiyan qiao and brandon edwards cubo 11, 2 (2009) to whichever cluster’s center is nearest. its basic procedure is to first select k points as initial group centroids, and to repeat the following steps until there is no change in the position of any cluster’s centroid. (1) assign each point in the data set to the cluster with the nearest centroid. (2) re-calculate the position of each cluster’s centroid. the clustering metric for k-means is to minimize total intra-cluster variance: v = k ∑ i=1 ∑ xj ∈si (xj − µi) 2 , where si, i = 1, 2, · · · k denote k different clusters and µi is the centroid or mean point of cluster si. a number of variations of this algorithm have been developed [6, 7, 8], such as fuzzy c-means [6], in which a data point, instead of belonging exclusively to a single cluster, belongs to all the clusters with a degree of membership and the sum of coefficients of being in clusters is one. compared with other clustering algorithms, the class of k-means algorithm and its variants has fast convergence rate and is relatively easy to implement. however, to apply the algorithm, the user must provide pre-specified value of k, i.e., the number of clusters, which is usually difficult to determine. in addition, this class of algorithms is susceptible to noise in the data set as each data point must belong to a cluster, an outlier can distort the shapes of clusters [2, 3]. (b) hierarchical methods the hierarchical methods group data points into a tree of clusters. the data points are clustered with either agglomerative (bottom up) or divisive (top-down) method and the clustering procedure is presented with the dendrogram [2, 3, 9]. the clustering metric in hierarchical methods is inter-group distance, which is defined as single-linkage, complete linkage, or average linkage. suppose d(r, s) denotes the distance between two clusters r and s. the single linkage is defined as the minimum distance of any pairwise points between two clusters, i.e., d(r, s) = min{d(i, j) : where point i is in cluster r and point j is in cluster s }. the complete linkage is defined as the maximum distance of any pairwise points between two clusters, i.e., d(r, s) = max{d(i, j) : where point i is in cluster r and point j is in cluster s }. the average linkage is defined as the average distance between all pairwise points between two clusters, i.e., d(r, s) = t rs/(n r ∗ n s), where t rs is the sum of all pairwise distances between cluster r and cluster s and n r and n s are respectively respectively the sizes of the clusters r and s. at each stage of agglomerative methods, the clusters r and s with minimum clustering metric are merged as a single cluster. the advantages of hierarchical methods are that a complete hierarchy of clusters is computed and visually illustrated. in addition, they do not need to specify the number of clusters in advance. cubo 11, 2 (2009) a general purpose platform for data clustering analysis 143 however, they cannot perform flexible adjustments once the splitting (divisive method) or merging (agglomative method) decisions are made during the clustering process. (c) density-based clustering algorithms in density-based clustering algorithms [10], clusters are interpreted as dense regions in the data space and are separated by regions of low object density (noise). these regions may have an arbitrary shape and the points inside a region may be arbitrarily distributed. for these algorithms, the clustering metric is density of data points in a region. the key idea is that for each data of a cluster the neighborhood of a given radius (eps) has to contain at least a minimum number of data points (minpts). the performance of density-based clustering thus relies on the parameters of eps and minpts. the density-based clustering algorithms have some advantages over k-means and hierarchical clustering methods. they are more robust in identifying clusters of arbitrary shapes and sizes and can separate from surrounding noise [2,3]. their disadvantage is that they might be sensitive to input parameters eps and minpts, which are difficult to determine in advance. (d) expectation-maximization (em) algorithm and its variants the em algorithm [11] is used in statistics to classify each point into the most likely probabilistic model and estimate the parameters of each model. em regards the data set as incomplete and divides each data point into two parts the observable features and the missing data. its basic procedure is first to initialize the distribution parameters and then repeat the following steps until the estimations of the distribution parameters are convergent. (1) expectation (e): computes an expectation of the likelihood by including the latent variables as if they were observed. (2) maximization (m): re-estimates the parameters by maximizing the expected likelihood found on the e step. the major disadvantages of em algorithms are the sensitivity to the selection of initial parameters, the possibility of convergence to a local optimum, and the slow convergence rate [11, 2, 3]. the variants of em have addressed these problems. 3 implementation of clustering platform to our knowledge, there is no platform so far that offers a simple and user-friendly environment for cluster analysis. we fill that gap by developing a general purpose cluster analysis platform, which implements some commonly used clustering algorithms and has an easy-to-use interface. the platform can be used for cluster analysis and validation of general data sets. 144 haiyan qiao and brandon edwards cubo 11, 2 (2009) 3.1 development of the clustering platform to implement the general purpose platform, we choose java programming language due to its good support of graphical user interfaces (gui) and platform independence. in the developing process, netbeans is used as the integrated development environment because of its visual tools that generate skeleton code. in addition, the gui builder of netbeans is used to support a sophisticated yet simplified swing application framework and beans binding. in the platform, the computation is conducted by matlab. matlab is designed for convenient numerical computations, especially matrix manipulation and includes many special functions developed for specific fields such as optimization, statistics etc. we choose matlab because it is a high performance language for computation, simulation, and visualization. in addition, there are a variety of matlab open source codes for clustering analysis. it saves us time not to write these algorithms from scratch. the platform brings together a variety of resources for performing cluster analysis using matlab [12]. the only problem to develop clustering algorithms in matlab is that matlab is not a free software. to make our platform accessible to users with no extra financial cost, we installed one license of matlab on our server [14] so users do not need to install matlab on their local machines. to configure matlab server and call matlab code from java interface, we first download the package from [13] and then do the following: 1. copy the exitform.fig, exitform.m, remotematlab.jar and startmatlabserver.m to our working directory. 2. copy xmlrpc libraries from apache and some commons libraries to our working directory. 3. to launch matlab server, type “startmatlabserver” in matlab command window. to be able to call matlab procedures located on the server from java program during the development of platform, we follow the steps below: 1. install netbeans. 2. import lib directory library into netbeans. 3. import remotematlab.jar into netbeans. 4. create instance of matlabcilent in java application. 3.2 running of the clustering platform to run the platform, a user does not need to install any software and only needs to download clusteringplatform.jar file from our website [14] and run it on his machine with internet access. after launching the file downloaded, the user will see a window as shown in figure 1. the top half of the window shows the data set to be clustered, with the listing of number of data points and the cubo 11, 2 (2009) a general purpose platform for data clustering analysis 145 number of attributes in the data set. the lower half of the window allows the user to choose which clustering algorithm to run on the current dataset and to decide the values of the parameters for the algorithm. for example, when the tab “k-means” is selected as shown in figure 1, the user needs to input the number of clusters k and number of iterations, and select similarity measure from the pull-down list. when the user clicks on the “run” button, k-means algorithm is called and runs on matlab server in the backend, and the results including output data file and visual display are sent back to the user and displayed in the platform on the user’s machine. to load the data set for clustering, the user can copy and paste the data set to the top window, or click on the menu “file” and choose “open” to open the right data file. the format of data set that the platform takes is similar to the relational database, i.e., each data entry is represented in one row, and each data entry consists of multiple attributes represented in columns separated by space or tab key; see the data set in figure 1. figure 1: clustering platform gui 146 haiyan qiao and brandon edwards cubo 11, 2 (2009) to illustrate the visual display of the clustering results, a few snapshots are taken. in figure 2, the hierarchical clustering result is displayed for a randomly picked data set. this dendrogram display gives the user the merge distance and the points contained within each cluster at the current level that the user is viewing. the user may pick at which level he would the dendrogram to be drawn. the default display for the dendrogram is the top level. in figure 3, em visual result is displayed. the chart displays the likelihood that each particular point belongs to a cluster. the darker the shade, the more likely it belongs to that cluster. figure 4 shows k-means visual results where each cluster center is depicted as a circle. figure 5 shows dbscan (density-based spatial clustering of applications with noise) visual results where the results display each cluster as a separate color. figure 2: hierarchical results cubo 11, 2 (2009) a general purpose platform for data clustering analysis 147 figure 3: em visual results figure 4: k-means visual results 4 conclusion and future work in this paper, a general purpose clustering analysis platform is introduced. the platform integrates a number of commonly-used clustering algorithms with friendly user interface and both textual and visual display of results. an advantage of the platform is that it is extendable, so it is easy to integrate more clustering algorithms in the future. another advantage is that it is easy to use, so we expect that it can satisfy the need of researchers in social sciences to cluster data without writing a single line of code. a drawback of the visual display of results in our platform is that it is not capable of visualizing clustering results effectively in high-dimensional data space. after a number of clustering algorithms have been implemented in this platform, it will be important to validate clustering results of different algorithms with different parameters. when assessing the output of a clustering algorithm, both the intraconnectivity and interconnectivity 148 haiyan qiao and brandon edwards cubo 11, 2 (2009) figure 5: dbscan visual results of clusters have to be taken into account to ensure that the clusters are compact and isolated. it is desirable to have a high intraconnectivity and a low degree of interconnectivity. however, there are no general standards for cluster validation in existing literature except in well-prescribed subdomains. our future work will address the problem of clustering evaluation and integrate the evaluation criteria into the platform to assist users in making better choice of clustering algorithms and improving clustering quality. received: june 4, 2008. revised: july 8, 2008. references [1] wikipedia, the free encyclopedia, “social network”, 25 jun. 2008. wikimedia foundation, inc., http://en.wikipedia.org/wiki/social network [2] r. xu, and d. wunsch ii, survey of clustering algorithms, ieee transactions on neural networks, 16(2005), no. 3, 645–678. [3] a. k. jain, m. n. murty and p. j. flyn, data clustering: a review, acm computing surveys, 31(1999), no. 3, 264–323. [4] http://bonsai.ims.u-tokyo.ac.jp/∼mdehoon/software/cluster/software.htm [5] http://www.rapid-i.com cubo 11, 2 (2009) a general purpose platform for data clustering analysis 149 [6] m. sato, y. sato and l. c. jain, fuzzy clustering models and applications, physica-verlag, 1997. [7] z. huang, extensions to the k-means algorithm for clustering large datasets with categorical values, data mining and knowledge discovery, 2(1998), 283–304. [8] g. hamerly and c. elkan, learning the k in k-means, technical report cs2002-0716, university of california, san diego, 2002. [9] n. jardine and r. sibson, the construction of hierarchic and non-hierarchic classifications, computer journal, 11(1968), 177–184. [10] m. ester, h. p. kriegel, j. sander, and x. xu, a density-based algorithm for discovering clusters in large spatial data sets with noise, proc. 2nd int. conf. on knowledge discovery and data mining, portland, or, 1996, 226–231. [11] g. mclachlan and t. krishnan, the em algorithm and extensions, new york: wiley, 1997. [12] http://www.dcorney.com/clusteringmatlab.html [13] http://plasmapowered.com/wiki/index.php/calling matlab from java [14] http://caplatform.ias.csusb.edu n10-haiyan_paper_v2 a mathematical journal vol. 7, no 3, (15 26). december 2005. some special classes of neutral functional differential equations constantin corduneanu the university of texas at arlington box 19408, uta, arlington tx 76019 concord@uta.edu abstract this paper is dedicated to the investigation of existence, mainly local, of solutions of two classes of neutral functional differential equations. a reduction method and fixed point methods are emphasized. resumen este art́ıculo está dedicado al estudio de existencia, principalmente local, de soluciones de dos clases de ecuaciones diferenciales funcionales neutrales. un método de reducción y de punto fijo son puestos con algún énfasis. key words and phrases: functional equation, neutral equation, existence of solution math. subj. class.: 34k40 1 introduction in several recent papers of the author [3], [4], [5], as well as in some joint papers with m. mahdavi [7], [8], certain types of neutral functional equations (including functional–differential ones) have been investigated in regard to the existence of so16 constantin corduneanu 7, 3(2005) lutions. such equation, in a rather general form, can be written as (u x)(t) = (v x)(t), t ∈ [0, t ], (1) or, in functional differential form, d dt (u x)(t) = (v x)(t), t ∈ [0, t ]. (2) in (1) and (2), u and v stand for some operators acting on convenient function spaces, whose elements are defined on [0, t ]. the case [0, t ), t ≤ ∞, has been also discussed, while the solution has been sought in various function spaces (usually, c, lp, 1 ≤ p < ∞). let us notice that equation (2), by integrating both sides on [0, t], t ≤ t , takes the form (1): (u x)(t) = c + ∫ t 0 (v x)(s)ds, t ∈ [0, t ]. (3) moreover, when v is a causal operator, the right hand side of (3) is also causal. the case of causal operators has been dealt with in the author’s recent book [6]. the basic method of investigating the existence of solutions to equations(1) and (2) consisted in reducing such equations to the simpler form x(t) = (w x)(t), t ∈ [0, t ], (4) and then applying existing results available for (4). in our book [2] we provided existence results for (4), based on the existing literature, while in [6] we have illustrated how this method works for neutral equations like (1) or (2). the aim of this paper is to investigate some classes of neutral equations encountered in the existing literature, by using mainly the above described method. in other words, to reduce such equations to the form (4), and then to apply known results. we are not necessarily intended to reobtain results already known in the literature, but to see what kind of results one obtains by means of the above described method. we shall particularly refer to the papers of t.a. burton [1] and loris faina [9], in which various classes of functional differntial equations, of neutral type, are investigated. 2 reduction of neutral equations to the form (4) let us start with the neutral equation x′(t) = f (t, x(t), x′(t)), t ∈ [0, t ], (5) investigated by loris faina [9] and other authors. this is the simplest form, in which x and f take scalar values, or values belonging to irn. the initial value condition attached to (5) will be x(0) = x0 ∈ irn, (6) 7, 3(2005) some special classes of neutral functional differential equations 17 if one deals with the vector case. formally, let us denote x′(t) = y(t), (7) which implies under rather general assumptions (see below) x(t) = x0 + ∫ t 0 y(s)ds. (8) the neutral equation (5) can be now written as y(t) = f ( t, x0 + ∫ t 0 y(s)ds, y(t) ) . (9) obviously, the right hand side of (9) engages only the values of y on the interval 0 ≤ s ≤ t (t ≤ t ). this means that equation (9) is an equation of the form (4), in which the right hand side is a causal operator (in y). there is, therefore, an equivalence between the initial value problem (5), (6), and the problem (8), (9). this equivalence will be further discussed when we precise the underlying function spaces. since equation (9) contains only the unknown y(t), we shall be able to investigate it in various function spaces (continuous or measurable functions), by using known results. in l. faina [9] there are more general neutral functional differential equations than (5). for instance, in (5) one assumes that f is a map from r × c(r) × l1(r) into irn, while the initial condition (6) is replaced by x(t) = ϕ(t), t ∈ (−∞, t0], t0 ∈ r fixed. in other terms, the infinite delay is dealt with. this case can be also covered by the scheme described above, though the procedure is more intricate. in t.a. burton [1], the following neutral functional differential equation is studied x′(t) = f (t, x(t), x′(t − h(t)) + g(t, x(t), x(t − h(t)), where 0 ≤ h(t) ≤ h0, h0 > 0 being fixed. to the above equation one attaches the typical initial condition for delay equations, namely x(t) = ϕ(t), t ∈ [−h0, 0]. we shall rewrite burton’s equation in the form x′(t) = f (t, x(t), x′(α(t))) + g(t, x(t), x(α(t))), (10) where α(t) is such that 0 ≤ α(t) ≤ t on some interval [0, t ]. the right hand side in (10) can be regarded as a causal operator in x. hence, (10) is also of the form (4). moreover, one can use the initial condition (6) for determining a (unique) solution to (10), (6). the literature on neutral functional equations is very rich, and a good amount of references can be found in our book [6]. in many more cases than those illustrated above, the method of reduction to equations of the form (1), with causal operator in the right hand side, can be successfully applied. in what follows, we shall dwell on the equations (5) and (9), trying to apply the reduction procedure in these cases, as well as other methods. 18 constantin corduneanu 7, 3(2005) 3 the equation (9) in the space c([0, t ], irn) we shall consider in this section the equation (9) in the space c([0, t ], irn). the assumption to be made are of such a nature that the right hand side of this equation represent a compact operator on c([0, t1], ir n), with t1 ≤ t. obviously, the arzelàascoli criterion of compactness in c([0, t ], irn) will be used. the compactness of the operator y(t) −→ f ( t, x0 + ∫ t 0 y(s)ds, y(t) ) , (11) on c([0, t ], irn) can be achieved under various sets of hypotheses. we shall describe such a set of hypeotheses, which will imply the existence of a local solution to the equation (9). such a solution will generate a continuously differentiable solution to the problem (5), (6). before we proceed with the statement of the hypotheses, it is instructive to look at a simple example for the equation (5). namely, we will choose f = 2x(t)x′(t) + 1. this leads to the integral x(t) = x2(t) + t + c, from which we derive x(t) = 1 2 ( 1 + √ 1 − 4c − 4t ) . choosing the initial value x0 = 1/2, we get c = 1/4, which means the solution is x(t) = 1 2 ( 1 + 2 √ −t ) . this shows that we have no solutions of (5) on any [0, t ], t > 0, for x0 = 1/2. therefore, a problem of the form (5), (6) may be deprived of (local) solutions, even though the right hand side of (5) is quite a usual function. we shall return now to the general problem (5), (6), and provide some conditions which assure the existence of local solutions. but before getting into details, we shall modify somewhat the equation (5), in order to encompass a larger category of situations. namely, let us rewrite (9) in the form y(t) = f ( t, x0 + ∫ t 0 y(s)ds; y ) , (9)′ and assume the right hand side in (9)′ is defined on the set [0, t ]×irn×c([0, t ], irn). we shall assume continuity of the map y −→ f, but further hypotheses will be formulated. compared to (9), the equation (9)′ involves now an operator in the right hand side, defined on the space c([0, t ], irn). the following hypotheses will be made, in view of obtaining the existence of solutions to the equation (9)′: h1 the map y −→ f ( t, x0 + ∫ t 0 y(s)ds; y ) (12) is continuous from [0, t ] × irn × c([0, t ], irn) into irn, and causal. 7, 3(2005) some special classes of neutral functional differential equations 19 h2 for each γ > 0, there exist two functions ω1(r) and ω2(r), continuous on [0, ∞), ω1(0) = ω2(0) = 0, and positive for r > 0, such that∣∣∣∣f ( t, x0 + ∫ t 0 y(s)ds; y ) − f ( u, x0 + ∫ u 0 y(s)ds; y )∣∣∣∣ ≤ ≤ ω1(|t − u|) + ω2(γ|t − u|), (13) for arbitrary t, u ∈ [0, t ], and all y ∈ c, with |y|c ≤ γ. let us notice the fact that choosing ω1(r) = αr, ω2(r) = βr, α, β > 0, the condition (13) becomes a lipschitz type continuity condition. h3 the map (12) takes bounded sets in c([0, t ], ir n), into bounded sets of irn. we can now prove the following (local) existence theorem for the equation (9)′. theorem 1. consider the functional equation (9)′ in the space c([0, t ], irn). assume that the map (12) satisfies the hypotheses h1, h2 and h3. then equation (9)′ has a local solution (i.e. defined on some interval [0, t1], t1 ≤ t ), provided f (0, x0; y) is independent of y. moreover, the equation x′(t) = f (t, x(t); x′), (5)′ under initial condition (6), has a local solution, which is continuously differentiable. remark 1. the localization is possible due to the causality of the operator (12) (hypothesis h1). remark 2. in case f does not depend on the last argument, the equation (5)′ becomes x′(t) = f (t, x(t)), while hypotheses h2 and h3 are automatically satisfied in case of continuity. the result of theorem 1 reduces to the classical peano’s existence theorem. the existence of the functions ω1(r) and ω2(r) is a simple consequence of the uniform continuity of f (t, x) on a set of the form [0, t ]×b, with b compact in irn. the imposition of hypothesis h2 is motivated by the fact that the last argument in f ( t, x0 + ∫ t 0 y(s)ds; y ) belongs to an infinite dimensional space, i.e. to c([0, t ], irn). proof of theorem 1. the equation (9)′ is, according to our hypotheses, a functional equation with causal operaor of the form (4). the hypotheses h1 and h2 assure the continuity of the map (12) from c([0, t ], irn) into itself. based on hypothesis h3, the map (12) from c([0, t ], irn) into c([0, t ], irn) is also compact. indeed, according to the hypothesis h3, the image of the ball |y|c ≤ γ is a bounded set in c([0, t ], irn). the inequality (13) in hypothesis h2 tells us that the image of the ball |x|c ≤ γ consists of equicontinuous functions on [0, t ], with values in irn. since γ > 0 is an arbitrary number, we conclude that the operator (12) is compact (takes bounded sets into relatively compact sets). hence theorem 3.1 in [6] applies directly, keeping also 20 constantin corduneanu 7, 3(2005) in mind that the operator (12) enjoys the property of fixed initial value. consequently, (9)′ has a local solution in some space c([0, t1], ir n), with t1 ≤ t. this result leads immediately to the existence of a local solution for the problem (5)′, 6. this ends the proof of theorem 1. remark 3. the case of measurable solutions to the equation (9)′, when the corresponding solutions to (5)′ will be absolutely continuous functions, can be treated in the same manner as in the continuous case. one has to use theorem 3.3 in [6], instead of theorem 3.1. we shall leave to the reader the task of formulating existence results. 4 existence of solutions to equation (10) if we denote again x′(t) = y(t), and take into account the initial condition (6), then equation (10) becomes y(t) = f ( t, x0 + ∫ t 0 y(s)ds, y(α(t)) ) + +g ( t, x0 + ∫ t 0 y(s)ds, y(α(t)) ) , (14) which is precisely of the form (4), with causal operator in the right hand side. this is due to the assumption on α(t), namely 0 ≤ α(t) ≤ t, t ∈ [0, t ]. we shall consider now a particular case of equation (14), as far as the function g is concerned. instead of the term f , we shall consider another operator-like term. more precisely, we shall deal with the functional equation y(t) + g(y(α(t))) = c + ∫ t 0 (w y)(s)ds, (15) under the following assumptions: 1) g : c([0, t ], irn) −→ c([0, t ], irn) is a contraction map on this space: |g(x) − g(y)|c ≤ λ|x − y|c , λ ∈ [0, 1); 2) w : c([0, t ], irn) −→ c([0, t ], irn) is a continuous causal operator, taking bounded sets of c([0, t ], irn) into bounded sets; 3) α : [0, t ] −→ [0, ∞) is continuous, and such that α(0) = 0 and 0 ≤ α(t) ≤ t for t ∈ [0, t ]. remark 4. the vector c ∈ irn is arbitrary, but it can be chosen in such a way to satisfy some kind of initial condition. for instance, if we assign to y the initial value y0, and assume (without loss of generality) that g(θ) = θ ∈ irn, then one obtains c = y0. 7, 3(2005) some special classes of neutral functional differential equations 21 in regard to the equation (15), the following existence result can be stated: theorem 2. consider the functional equation (15), under conditions 1), 2), 3) stated above. then, there exists a solution y = y(t), defined on some interval [0, t1] ⊂ [0, t ], for each c ∈ irn. this solution is such that y(t) + g(y(α(t))) is continuously differentiable. proof. the hypotheses accepted are of such a nature that allow the application of theorem 6.1 in [6], which yields the existence result. the idea of proof is based on the fact that the functional equation y(t)+g(y(α(t))) = f (t) is uniquely solvable in c([0, t ], irn). moreover, y(t) depends continuously of f (t), which allows to deal with (15) by contraction mapping principle, or by another fixed point methods. details of this approach can be found in our paper [5], where further existence results are obtained. 5 further considerations on equation (10) the idea of proof mentioned above can be adapted to other functional equations. for an illustration we will consider the equation (10), as well as the auxiliary equation x′(t) = g(t, x(t), x(α(t))) + f (t), (16) with f ∈ c([0, t ], irn). the attached initial condition is (6). the functional integral equation equivalent to (16), (6) is x(t) = x0 + ∫ t 0 f (s)ds + ∫ t 0 g(s, x(s), x(α(s)))ds. (17) let us assume the following conditions on the data in equation (17): 1) g : [0, t ] × irn × irn −→ irn is continuous, and satisfies the lipschitz condition |g(t, x, y) − g(t, x̄, ȳ)| ≤ l(|x − x̄| + |y − ȳ|), with l > 0; 2) f ∈ c([0, t ], irn); 3) α(t) is continuous on [0, t ], and α(0) = 0, 0 ≤ α(t) ≤ t. it is easy to see that the usual process of iteration leads to the following relationship: x(k+1)(t) − x(k)(t) = = ∫ t 0 [ g ( s, x(k)(s), x(k)(α(s)) ) − g ( s, x(k−1)(s), x(k−1)(α(s)) )] ds, k ≥ 1, 22 constantin corduneanu 7, 3(2005) with x(0)(t) = x0 + ∫ t 0 f (s)ds. we further derive on behalf of condition 1) |x(k−1)(t) − x(k)(t)| ≤ ≤ l ∫ t 0 [ |x(k)(s) − x(k−1)(s)| + |x(k)(α(s)) − x(k−1)(α(s))| ] ds, which leads to sup 0≤s≤t |x(k+1)(s) − x(k)(s)| ≤ 2l ∫ t 0 sup 0≤u≤s |x(k)(u) − x(k−1)(u)|ds, (18) if we keep in mind that 0 ≤ α(t) ≤ t. the inequality (18) can be processed in the usual manner, and one finds that lim x(k)(t) = x(t) as k −→ ∞, uniformly on [0, t ], with x(t) satisfying (16). the uniqueness can be also proven by the standard method, as well as the continuous dependence of the solution with respect to f ∈ c([0, t ], irn). the auxiliary result established above enables us to make some progress in regard to the equation (10). we rewrite it for the reader’s convenience, x′(t) = g(t, x(t), x(α(t))) + f (t, x(t), x′(α(t))), and regard it as a (nonlinear) perturbed equation associated to (16). of course, we preserve the initial condition (6). the following fixed point scheme can be attached to the equation (19): for each continuously differntiable u(t) on [0, t ], with values inirn, we shall attach the unique solution x(t) of the equation like (16) x′(t) = g(t, x(t), x(α(t))) + f (t, u(t), u′(α(t))) (19) with the initial condition (6). the existence and uniqueness of x(t), under the above scheme, is guaranteed under the conditions 1), 2) and 3) specified above. consequently, in the space c([0, t ], irn), or rather in the space c(1)([0, t ], irn), we have defined an opeator u −→ x, where u and x are related by the equation (19), with x satisfying also (6). let us denote by v the operator defined above, i.e. x(t) = (v u)(t), t ∈ [0, t ], (20) with u and x as described above. the operator v appears as a compound operator: first, u −→ f (t, u(t), u′(α(t))), and second f −→ x, with x the solution of (19), (6). as noticed earlier in this section, the second operator is continuous on c([0, t ], irn). the first operator involved, u −→ f (t, u(t), u′(α(t))) can be made continuous, under adequate hypotheses on the function f . it is obviously continuous from c(1)([0, t ], irn) into c([0, t ], irn) when f (t, u, v) is continuous on [0, t ] × irn × irn. 7, 3(2005) some special classes of neutral functional differential equations 23 instead of pursuing the above scheme, which can certainly lead to results of existence for (10), we shall attempt to apply the contraction mapping principle to the equation (10), but modifying somewhat the scheme presented above. namely, we will consider the scheme described by the following equation, attached to (10): x′(t) = g(t, x(t), x(α(t))) + f (t, x(t), u′(α(t))). (21) by means of (21) and (6), we shall define the operator on c(1)([0, t ], irn), say x(t) = (w u)(t), in the following manner. given u ∈ c(1)([0, t ], irn), the equation (21) can be solved in x under rather mild assumptions (as seen above, under lipschitz condition). the unique solution of (21), (6) will be denoted by x(t) = (w u)(t). from (2) we derive the following relationship between x = w u and y = w v, where u, v ∈ c(1)([0, t ], irn): x′(t) − y′(t) = g(t, x(t), x(α(t))) − g(t, y(t), y(α(t)))+ +f (t, x(t), u′(α(t))) − f (t, y(t), v′(α(t))). (22) assuming also a lipschitz condition on f (t, x, y), as we did already on g(t, x, y), we obtain |x′(t) = y′(t)| ≤ l(|x(t) − y(t)| + |x(α(t)) − y(α(t))|)+ +m|x(t) − y(t)| + m|u′(α(t)) − v′(α(t))|, for any t ∈ [0, t ], where l, m and m are positive numbers. the above inequality yields sup |x′(t) − y′(t(| ≤ (2l + m ) sup |x(t) − y(t)|+ +m sup |u′(α(t)) − v′(α(t))|, (23) with sup taken on [0, t ], or on any [0, t1], t1 ≤ t. but x(t) − y(t) = ∫ t 0 [x′(s) − y′(s)]ds, (24) because x(0) = y(0) = x0, according to (6). from (23) we derive sup |x(t) − y(t)| ≤ t sup |x′(s) − y′(s)|, (25) with sup taken on [0, t ]. taking into account (23), (24) and (25) we obtain sup |x′(t) − y′(t)| ≤ (2l + m )t sup |x′(t) − y′(t)|+ +m sup |u′(t) − v′(t)|. (26) since we want (26) to be a relation showing the fact that the operator w is a contraction on c(1)([0, t ], irn), we see from (26) that a first condition to be imposed is (2l + m )t < 1. (27) if we admit (27), then (26) allows us to write sup |x′(t) − y′(t)| ≤ m[1 − (2l + m )t ]−1 sup |u′(t) − v′(t)|, 24 constantin corduneanu 7, 3(2005) which really represents a contraction condition for w , as soon as λ = m[1 − (2l + m )t ]−1 < 1. (28) it is appropriate to notice the fact that the norm in c(1)([0, t ], irn) is (by our choice) |x0| + sup |x′(t)|. (29) accordingly, the norm for u − v should be |u0 − v0| + sup |u′(t) − v′(t)|, which is in advantage of the contraction inequality |w u − w v|c(1) ≤ λ|u − v|c(1) , (30) as derived from above. therefore, we can now state the following (global) existence result for the problem (10), (6): theorem 3. consider the problem (10), (6), and assume the following conditions are verified by the functions f and g: 1) f, g : [0, t ] × irn × irn −→ irn are continuous maps; 2) f and g satisfy the lipschitz type conditions |f (t, x, y) − f (t, x̄, ȳ)| ≤ l(|x − x̄| + |y − ȳ|), |g(t, x, y) − g(t, x̄, ȳ)| ≤ m|x − x̄| + m|y − ȳ|, with positive constants l, m and m; 3) the inequalities (27) and (28) are satisfied. then, there exists a unique solution x(t) ∈ c(1)([0, t ], irn), which can be approximated by the scheme described by the equation (21). the proof of theorem 3 has been carried out above, before its statement. remark 5. it is obvious from the inequalities (27) and (28) that severe restrictions must be imposed to the constants l, m, m and t . first, if l, m are fixed, it is obvious that the inequality (27) can be satisfied provided we choose t small enough: t < (2l + m )−1. this restriction suggests that we need to confine our investigation, possibly, to a smaller interval than the original interval [0, t ]. but this kind of restriction is in accordance with the fact we are looking for local solutions to our problem. second, once we choose t such that (27) takes place, there remains the inequality (28) to be satisfied. if the constants l, m and t are fixed, then the only way to satisfy (28) is to choose m small enough. in conclusion, local existence for the problem (10), (6) is always assured by choosing the constant m sufficiently small. 7, 3(2005) some special classes of neutral functional differential equations 25 other neutral equations can be investigated in regard to the existence of their solutions, using approaches described above. we suggest to the reader to try such procedures on equations of the form x′(t) = f (t, x(t), x(t − h)) + g(t, x(t), x′(t − h)), under an initial condition of the form x(t) = ϕ(t), t ∈ [−h, 0]. also, similar to the equation (15), is d dt [x(t) + g(x(t − h))] = (w x)(t), with initial datum x(t) = ϕ(t), t ∈ [−h, 0]. see our paper [4] for details. received: july 2003. revised: december 2003. references [1] t.a. burton, an existence theorem for neutral equations. nonlinear studies 5 (1998), 1-6. [2] c. corduneanu, integral equations and applications. cambridge univ. press, 1991. [3] c. corduneanu, neutral functional equations with abstract volterra operators. in ”advances in nonlinear dynamics”, gordon & breach, (1997), 229235. [4] c. corduneanu, neutral functional equations of volterra type. functional differential equations (israel), (1997), 265-270. [5] c. corduneanu, existence of solutions to neutral functional differential equations. j. diff. equations, 168 (2000), 93-101. [6] c. corduneanu, functional equations with causal operators, taylor & francis, 2002. [7] c. corduneanu, m. mahdavi, on neutral functional differential equations with causal operators. proc. third workshop of the int. inst. general system science, tianjin (china), 1998, 43-48. [8] c. corduneanu, m. mahdavi, on neutral functional differential equations with causal operators, ii. in ”integral methods in science and engineering”, chapman & hall crc press, research notes #418 (2000), 102-106. 26 constantin corduneanu 7, 3(2005) [9] loris faina, existence and continuous dependence for a class of neutral functional differential equations. annales polonici mathematici, lxiv.3 (1996), 215-226. cubo a mathematical journal vol.10, n o ¯ 03, (161–170). october 2008 existence and uniqueness of pseudo almost automorphic solutions to some classes of partial evolution equations j. blot, d. pennequin université paris 1 panthéon-sorbonne, laboratoire marin mersenne, centre p.m.f., 90 rue de tolbiac, 75647 paris cedex 13, france emails: blot@univ-paris1.fr, pennequi@univ-paris1.fr and gaston m. n’guérékata department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore, m.d. 21251 – usa email: gaston.n’guerekata@morgan.edu abstract we are concerned in this paper with the partial differential equation d dt [u(t)+f(t,u(t))] = au(t) t ∈ r, where a is a (generally unbounded) linear operator which generates a semigroup of bounded linear operators (t (t))t≥0. under appropriate sufficient conditions, we prove the existence and uniqueness of a pseudo almost automorphic mild solution to the equation. resumen nosotros consideramos en este art́ıculo la ecuación diferencial parcial d dt [u(t)+f(t,u(t))] = au(t) t ∈ r, donde a es un (generalmente no acotado) operador lineal que genera 162 j. blot et al. cubo 10, 3 (2008) un semigrupo de operadores lineales acotados (t (t))t≥0. bajo condiciones suficientes apropriadas, provamos la existencia y unicidad de una solución blanda pseudo casi automorfica para tal ecuación. key words and phrases: pseudo almost automorphic function, exponentially stable semigroup, partial differential equations. math. subj. class.: 34g10, 47a55. 1 introduction since the publication of the monograph [12], the study of almost automorphic function (a concept introduced by s. bochner in the literature in the mid sixties as a generalization of almost periodicity in the sense of bohr) has regained great interest. several extensions of the concept were introduced including asymptotic almost automorphy by n’guérékata ([10]), p-almost automorphy by diagana ([2]), and stepanov-like almost automorphy by n’guérékata and pankov ([14]). recently, j. liang et al. have suggested the notion of pseudo almost automorphic functions, i.e. functions that can be written uniquely as a sum of an almost automorphic function and an ergodic term, i.e. a function with vanishing mean (cf [6], and [7], [8]). this latter turns out to be more general than asymptotic almost automorphy. however it seems to be more complicated. there has been a considerable interest in the existence of (these various types of) almost automorphic solutions of evolution equations. semigroups theory and fixed point techniques have been frequently used for semilinear evolution equations. in [3] the authors studied the existence and uniqueness of an almost automorphic mild solution to the equation d dt [u(t) + f(t,u(t))] = au(t) + g(t,u(t)) t ∈ r, (1.1) where the functions f(t,u) and g(t,u) are almost automorphic in t, for each u. this latter motivated our recent paper [15], where we study the existence and uniqueness of a pseudo almost automorphic mild solution the semilinear evolution equations of the form du dt = au(t) + g(t,u(t)), t ∈ r, (1.2) where a is an unbounded sectorial operator with not necessarily dense domain in a banach space x and g : r × xα → x, where xα, α ∈ (0, 1), is any intermediate banach space between d(a) and x. in this paper, we study pseudo almost automorphic solutions to perturbations to equations: du dt = au(t) t ∈ r, (1.3) consisting of the class of abstract partial evolution equations of the form cubo 10, 3 (2008) existence and uniqueness of pseudo ... 163 d dt [u(t) + f(t,u(t))] = au(t) t ∈ r, (1.4) where a is the infinitesimal generator of an exponentially stable c0-semigroup acting on x, b,c are two densely defined closed linear operators on x, and f is continuous functions. under some appropriate assumptions, we establish the existence and uniqueness of an almost automorphic (mild) solution to eq. (1.4) using the banach fixed-point principle. we start this work by presenting some properties of pseuso almost automorphic functions in section 2 including an application to a volterra-like integral equation. our main result (theoreme 3.3) is presented in section 3. 2 preliminaries in this work, (x,‖·‖) will stand for a banach space. the collection of all bounded linear operators from x is denoted by b(x) — this is a banach space when it is equipped with its natural norm ‖a‖b(x) := sup x∈x,x 6=0 ‖ax‖x ‖x‖x . the fields of real and complex numbers, are respectively denoted by c and r. we let bc(r, x) denote the space of all x-valued bounded continuous functions r → x– it is a banach space when equipped with the sup norm ‖u‖∞ := sup t∈r ‖u(t)‖x for each u ∈ b(r, x). we will use the following well-known concepts in the sequel. definition 2.1. a continuous function f : r 7→ x is said to be almost automorphic if for every sequence of real numbers (s′n)n∈n there exists a subsequence (sn)n∈n ⊂ (s ′ n)n∈n such that g(t) := lim n7→∞ f(t + sn) is well-defined for each t ∈ r, and f(t) = lim n7→∞ g(t − sn) for each t ∈ r. similarly, definition 2.2. a continuous function f : r × x 7→ x is said to be almost automorphic in t ∈ r for each u ∈ x if every sequence of real numbers (σn)n∈n contains a subsequence (sn)n∈n such that g(t,u) := lim n7→∞ f(t + sn,u) is well defined for each t ∈ r and each u ∈ x and, f(t,u) = lim n7→∞ g(t − sn,u) exists for each t ∈ r and u ∈ x. 164 j. blot et al. cubo 10, 3 (2008) the following natural properties hold: if f,h : r 7→ x are almost automorphic functions and if λ ∈ r, then f + h, λf, and fλ are almost automorphic, where fλ(t) := f(t + λ). moreover, r(f) := {f(t), t ∈ r} is relatively compact. since the range of an almost automorphic function f is relatively compact on x, then it is bounded. almost automorphic functions constitute a banach space aa(x) when it is endowed with the sup norm. this naturally generalizes the concept of (bochner) almost periodic functions. definition 2.3. let x be a banach space. 1. a bounded continuous function with vanishing mean value can be defined as aa0(r, x) = { φ ∈ bc(r, x) : lim t →∞ 1 2t ∫ t −t ‖φ(σ)‖dσ = 0 } . 2. similarly we define aa0(r × x, x) to be the collection of all functions f : t 7→ f(t,x) ∈ bc(r × x, x) satisfying lim t →∞ 1 2t ∫ t −t ‖f(σ,x)‖dσ = 0 uniformly for x in any bounded subset of x. now we describe the sets paa(r, x) and paa(r × x, x) of pseudo almost automorphic functions: paa(r, x) = { f = g + φ ∈ bc(r, x), g ∈ aa(r, x) and φ ∈ aa0(r, x) } ; paa(r × x, x) = { f = g + φ ∈ bc(r × x, x), g ∈ aa(r × x, x) and φ ∈ aa0(r × x, x) } . in both cases above, g and φ are called respectively the principal and the ergodic terms of f. we have the following elementary properties of pseudo almost automorphic functions. theorem 2.4. ( [8] theorem 2.2). paa(r, x) is a banach space under the supremum norm. let now f,h : r → r and consider the convolution (f ⋆ h)(t) := ∫ r f(s)h(t − s)ds, t ∈ r, if the integral exists. remark 2.5. the operator j : paa(r, x) → paa(r, x) such that (jx)(t) := x(−t) is well-defined and linear. moreover it is an isometry and j2 = i. cubo 10, 3 (2008) existence and uniqueness of pseudo ... 165 remark 2.6. the operator ta defined by (tax)(t) := x(t + a) for a fixed a ∈ r leaves paa(r, x) invariant. let us now discuss conditions which do ensure the pseudo almost automorphy of the convolution function f ⋆ h of f with h where f is pseudo almost automorphic and h is a lebesgue mesurable function satisfying additional assumptions. let f : r → x and h : r → r; the convolution function (if it does exist) of f with h denoted f ⋆ h is defined by: (f ⋆ h)(t) := ∫ r f(σ)h(t − σ)dσ = ∫ r f(t − σ)h(σ)dσ = (h ⋆ f)(t), for all t ∈ r. hence, if f ⋆ h is well-defined, then f ⋆ h = h ⋆ f. let ϕ ∈ l1 and λ ∈ c. it is well-known that the operator aϕ,λ defined by aϕ,λu = λu + ϕ ⋆ u (2.1) acts continuously in bc(r, x) i.e., there exists k > 0 such that ‖aϕ,λu‖bc(r,x) ≤ k‖u‖bc(r,x),∀u ∈ bc(r, x) (2.2) moreover aϕ,λ leaves bc(r, x) invariant. now denote m := {pap(r, x),paa(r, x)} where pap(r,x) is the banach space of all pseudo almost periodic functions f : r → x. then we have. theorem 2.7. for ω ∈ m, aϕ,λ(ω) ⊂ ω. proof. . it is an immediate consequence of the remarks above. application: a volterra-like equation consider the equation x(t) = g(t) + ∫ +∞ −∞ a(t − σ)x(σ)dσ, t ∈ r, (2.3) where g : r → r is a continuous function and a ∈ l1(r). theorem 2.8. suppose g ∈ paa(r, x) and ‖a‖l1 < 1. then (2.3) above has a unique pseudo almost automorphic solution. proof. it is clear that the operator x ∈ paa(r, x) → ∫ +∞ −∞ a(t − σ)x(σ)dσ ∈ paa(r, x) 166 j. blot et al. cubo 10, 3 (2008) is well-defined. now consider γ : paa(r, x) → paa(r, x) such that (γx)(t) = g(t) + ∫ +∞ −∞ a(t − σ)x(σ)dσ, t ∈ r. we can easily show that ‖(γx) − (γy)‖ ≤ ‖a‖l1‖x − y‖. the conclusion is immediate by the principle of contraction. 3 main results this section is devoted to the proof of the main result of the paper, that is, the existence and uniqueness of an almost automorphic (mild) solution to eq. (1.4). for that we need to establish a few preliminary results. definition 3.1. a function u ∈ bc(r, x) is said to be a mild solution to eq. (1.4) if the function s → at (t − s)f(s,u(s)) is integrable on (−∞, t) for each t ∈ r and u(t) = −f(t,u(t)) − ∫ t −∞ at (t − s)f(s,u(s))ds for each t ∈ r. we now make the following assumptions. (h.1) the operator a is the infinitesimal generator of an exponentially stable semigroup (t (t))t≥0 such that there exist constants m > 0 and δ > 0 with ‖t (t)‖b(x) ≤ me −δt, ∀t ≥ 0. furthermore, the function σ → at (σ) defined from (0,∞) into b(x) is strongly (lebesgue) measurable and there exist a function γ : (0,∞) → [0,∞) such that sups≥s0 γ(s) < ∞ for any s0 > 0, and a constant ω > 0 with ρ := ∫ ∞ 0 e−ωsγ(s)ds < ∞ such that ‖at (s)‖b(x) ≤ e −ωs .γ(s), s > 0. (h.2) the function f : r × x 7→ x, (t,u) 7→ f(t,u) is jointly continuous and ‖f(t,u) − f(t,v)‖x ≤ k(t) .‖u − v‖, and for all t ∈ r, and ∀u,v ∈ x. here k ∈ l1(r, r+). (h.3) f = g + ψ ∈ paa(r × x, x), where g and ψ are the principal and the ergodic terms of f respectively and f(t,u) and g(t,u) are uniformly continuous on every bounded subset k ⊂ x uniformly in t ∈ r. cubo 10, 3 (2008) existence and uniqueness of pseudo ... 167 lemma 3.2. suppose that assumptions (h.1)-(h.2)-(h.3) hold. define the nonlinear operator λ1 by: for each ξ ∈ paa(x), (λ1ξ)(t) = ∫ t −∞ at (t − s)f(s,ξ(s))ds then λ1 maps paa(x) into itself. proof. set h defined by: h(.) = f(.,ξ(.)). since h ∈ paa(r, x) using [6, theorem 2.4] with assumption (h.3), we can write h = β + φ where β is the principal part and φ the ergodic term of h. using the same argument as in [11], we can prove that t 7→ ∫ t −∞ at (t − s)β(s)ds is in aa(x). now, set: ν(t) = − ∫ t −∞ at (t − s)φ(s)ds. we have: 1 2t ∫ t −t ‖ν(t)‖xdt ≤ 1 2t ∫ t −t ∫ t −∞ ‖a(t − s)φ(s)‖xdsdt ≤ 1 2t ∫ t −t ∫ t −∞ e−ω(t−s)γ(t − s)‖φ(s)‖xdsdt. let’s write: 1 2t ∫ t −t ∫ t −∞ e−ω(t−s)γ(t − s)‖φ(s)‖xdsdt = i1 + i2, where: i1 = 1 2t ∫ t −t ∫ −t −∞ e−ω(t−s)γ(t − s)‖φ(s)‖xdsdt and i2 = 1 2t ∫ t −t ∫ t −t e−ω(t−s)γ(t − s)‖φ(s)‖xdsdt. we prove know that i1 → 0 and i2 → 0 as t → ∞. indeed, for i1, let s0 > 0 and set m(s0) = sups≥s0 γ(s), and k = supt∈r ‖φ(t)‖x. we have: i1 ≤ k 1 2t ∫ t −t ∫ −t −∞ e−ω(t−s)γ(t − s)dsdt = k 2t ∫ ∫ d e−ω(t−s)γ(t − s)dsdt, where d = {(s,t) ∈ r2, |t| ≤ t, s ≤ −t}. we introduce also: d1 = {(s,t) ∈ d,t − s ≥ s0}, d2 = d \ d1. we have: ∫ ∫ d1 e−ω(t−s)γ(t − s)dsdt ≤ m(s0) ∫ ∫ d1 e−ω(t−s)dsdt ≤ m(s0) ∫ ∫ d e −ω(t−s) dsdt = m(s0)e −ωt ω ∫ t −t e −ωt dt ≤ 2t m(s0)e −ωt ω . 168 j. blot et al. cubo 10, 3 (2008) moreover, ∫ ∫ d2 e−ω(t−s)γ(t − s)dsdt ≤ ∫ −t −t −s0 ∫ s+s0 −t e−ω(t−s)γ(t − s)dtds ≤ ∫ −t −t −s0 ∫ s0 −t −s e−ωσγ(σ)dσds ≤ ∫ −t −t −s0 ∫ s0 0 e −ωσ γ(σ)dσds ≤ s0 ∫ s0 0 e−ωσγ(σ)dσ. so, for any t ≥ 1, we have: i1 ≤ k 2t ( 2t m(s0)e −ωt ω + s0 ∫ s0 0 e−ωσγ(σ)dσ ) ≤ k ( e−ωt m(s0) ω + s0 ∫ s0 0 e−ωσγ(σ)dσ ) . let ǫ > 0. we can find s0 > 0 such that ks0 ∫ s0 0 e−ωσγ(σ)dσ < ε/2. let us take such an s0. after, for t sufficiently large, ke−ωt m(s0) ω < ǫ/2, and so, for sufficiently large t , i1 ≤ ǫ. now, we consider i2. we have: i2 = 1 2t ∫ t −t ‖φ(s)‖xds ∫ t s e −ω(t−s) γ(t − s)dt ≤ 1 2t ∫ t −t ‖φ(s)‖xds ∫ t −s 0 e−ωσγ(σ)dσ ≤ ρ 1 2t ∫ t −t ‖φ(s)‖xds → 0 as t → ∞. now we are ready to state and prove the following. theorem 3.3. suppose that assumptions (h.1)-(h.2) hold. then eq. (1.4) has a unique pseudo almost automorphic (mild) solution if proof. define the nonlinear operator γ : aa(x) 7→ aa(x) by: γ(u) : t 7→ −f(t,u(t)) − ∫ t −∞ at (t − s)f(s,u(s))ds. we have: ‖γ(u)(t) − γ(v)(t)‖x ≤ ‖f(t,u(t)) − f(t,v(t))‖x + ∫ t −∞ ‖at (t − s)(f(s,u(s)) − f(s,v(s)))‖xds cubo 10, 3 (2008) existence and uniqueness of pseudo ... 169 ≤ k(t)‖u(t) − v(t)‖x + ∫ t −∞ e−ω(t−s)γ(t − s)k(s)‖u(x) − v(s)‖xds ≤ [ k(t) + ∫ t −∞ e−ω(t−s)γ(t − s)k(s)ds ] ‖u − v‖∞ ≤ (1 + ρ)‖k‖∞‖u − v‖∞. so, we obtain: ‖γ(u) − γ(v)‖∞ ≤ (1 + ρ)‖k‖∞‖u − v‖∞, and we can conclude using the banach’s fixed point principle. acknowledgements. this paper was written when the second author was visiting the ”laboratoire marin mersenne” of the university of paris 1 panthéon-sorbonne in june 2008. he is grateful to professors blot and pennequin for their invitation. received: june 2008. revised: august 2008. references [1] s. bochner, a new approach to almost periodicity, proc. nat. acad. sci. u.s.a., 48(1962), pp. 2039–2043. [2] t. diagana, existence of p-almost automorphic mild solutions to some abstract differential equations, intern. j. evol. equ., 1(1) (2005), 57–67. [3] t. diagana and g.m. n’guérékata, almost automorphic solutions to some classes of partial evolution equations, appl. math. lett., 20(2007), no. 4, 462–466. [4] t. diagana, g.m. n’guérékata and n.v. minh, almost automorphic solutions of evolution equations, proc. amer. math. soc., 132(2004), no. 11, pp. 3289–3298. [5] j.a. goldstein and g.m. n’guérékata, almost automorphic solutions of semilinear evolution equations, proc. amer. math. soc., 133(2005), no. 8, pp. 2401–2408. [6] j. liang, j. zhang and t.-j. xiao, composition of pseudo almost automorphic and asymptotically almost automorphic functions, j. math. anal. appl., j. math. anal. appl., 340(2008), no.2, 1493–1499. [7] j. liang, g.m. n’guérékata, t.-j. xiao and j. zhang, some properties of pseudo almost automorphic functions and applications to abstract differential equations, nonlinear analysis tma, (to appear). [8] t.-j. xiao, j. liang and j. zhang, pseudo almost automorphic functions to semilinear differential equations in banach spaces, semigroup forum, 76(2008), no.3, 518–524. 170 j. blot et al. cubo 10, 3 (2008) [9] a. lunardi, analytic semigroups and optimal regularity in parabolic problems, pnlde vol.16, birkhäauser verlag, basel, 1995. [10] g.m. n’guérékata, quelques remarques sur les fonctions asymptotiquement presque automorphes, ann. sci. math. québec, 7(2) (1983), 185–191. [11] g.m. n’guérékata, existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, semigroup forum, 69(2004), pp. 80–86. [12] g.m. n’guérékata, almost automorphic functions and almost periodic functions in abstract spaces, kluwer academic / plenum publishers, new york-london-moscow, 2001. [13] g.m. n’guérékata, topics in almost automorphy, springer-verlag, new york, 2005. [14] g.m. n’guérékata and a. pankov, stepanov-like almost automorphic functions and monotone evolution equations, nonlinear analysis, 68(9) (2008), 2658–2667. [15] g.m. n’guérékata and d. pennequin, pseudo almost automorphic solutions for hyperbolic semilinear evolution equations in intermediate banach spaces, dyn. contin. disc. impuls. syst., ser. a, proceedings of the 6th. conference on differential equation and dynamical systems, baltimore, md, usa, may 22–26, 2008, (to appear). [16] a. pazy, semigroups of linear operators and applications to partial differential equations, springer-verlag, 1983. [17] m. renardy and r.c. rogers, an introduction to partial differential equations, texts in appl. math. 13, springer-verlag, new york-berlinheidelberg-london-paris, 1992. n14 h4bott_20_01_2009.dvi cubo a mathematical journal vol.12, no¯ 01, (195–217). march 2010 projective squares in p2 and bott’s localization formula† jacqueline rojas∗ ufpb-ccen – departamento de matemática, cidade universitária, 58051-900, joão pessoa-pb – brasil email : jacq@mat.ufpb.br ramón mendoza ufpe-ccen – departamento de matemática, cidade universitária, 50740-540, recife-pe – brasil email : ramon@dmat.ufpe.br and eben da silva ufrpe/uast-ccen – departamento de matemática e f́ısica, fazenda saco, caixa postal 63, 56900-000, serra talhada-pe – brasil email : eben@uast.ufrpe.br abstract we give an explicit description of the hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree 4 in the projective plane that allows us to afford a natural embedding in a product of grassmann varieties. we also use this description to explain how to apply bott’s localization formula (introduced in 1967 in bott’s work [2]) to give an answer for an enumerative question as used by the first time by ellingsrud and strømme in [8] to compute the number of twisted cubics on a general calabi-yau threefold which is a complete intersection in some projective space and used later by kontsevich in [16] to count rational plane curves of degree d passing through 3d − 1 points in general position in the plane. †dedicated to israel vainsencher on the occasion of his 60th birthday. ∗partially supported by cnpq (edital casadinho n o 620108/2008-8) 196 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) resumen en este trabajo, damos una descripción expĺıcita del esquema de hilbert, que parametriza los subesquemas cerrados de dimensión cero y grado 4 del plano proyectivo, esto nos permite mapear este esquema en un producto de variedades de grassmann. usamos dicha construcción, para explicar como se utiliza la fórmula de localización de bott (introducida en 1967 por bott en [2]) para responder una pregunta de geometria enumerativa, tal como lo hicieron ellingsrud y strømme en [8], para calcular cuantas cúbicas torcidas existen en una variedad de calabiyau tri-dimensional, que es una intersección completa en algún espacio proyectivo, y que fue usada posteriormente por kontsevich en [16], para contar curvas planas racionales de grado d pasando por 3d − 1 puntos en posición general en el plano. key words and phrases: hilbert scheme, bott’s localization formula. math. subj. class.: 14c05, 14n05. 1 introduction enumerative geometry has been an active and attractive research subject in math for a long time. a typical problem in enumerative geometry asks for the number of geometric objects of a certain type that satisfy a given set of conditions. for example: 1. very easy: given two distinct points in the plane, how many lines go through all of them? (the answer a result from euclidean geometry is clearly one.) 2. easy: given 2n general lines in the plane, how many n –gons are there with its set of vertices meeting all of them? (easy combinatorial answer: {2n − 1}! = factorial of odd’s numbers between 1 and 2n − 1 (see section 4).) 3. medium: how many lines lie on a general cubic surface? (famous answer: 27.) or how many lines lie on a general quintic threefold? (answer: 2875. hermann schubert determined this number explicitly at page 72 in [20], see also the computation at page 281 of cox-katz’s book [4].) or in a more general way: how many lines lie on a general hypersurface of degree 2n − 3 in pn? (answer: see [11]) 4. hard: given 3d − 1 general points in the plane, how many plane rational curves of degree d pass through all of them? (answer: n (d). n (d) denotes the gromov-witten invariants, they have their origins in physics, in the topological sigma models introduced by witten in [22]. on the other hand, kontsevich in [16] found a formula that expresses n (d) in terms of n (e) for e < d, so a single initial datum is required for the recursion, namely, the case d = 1, which correspond to the fact that through two distinct points in the plane pass exactly one line. see kock-vainsencher [15] for an elementary introduction and chapter 9 in cox-katz’s book [4].) cubo 12, 1 (2010) projective squares in p2 ... 197 in the 19th century, geometers developed a powerful ”calculus” for solving enumerative problems. their method had no rigorous theoretical foundation, but it worked remarkably well. justifying their results was the subject of problem 15th on hilbert’s famous list. in the 20th century, enumerative geometry has been reconceptualized and made rigorous in terms of intersection theory on parameter spaces (see fulton [10], kleiman-laksov [14] and kleiman [13] for a survey). so, in order to give a correct answer to an enumerative question, the key issue in the study of parameter spaces is to find a compactification. for example, the kontsevich’s moduli space of stable maps is used in [16] to calculate n (d). in theorem 1.9 of nakajima’s book [19] is given an explicit description of the hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree n in a2. and second, what kind of techniques can be used on a given parameter space to solve an enumerative problem. usually, the answer to an enumerative problem is reduced to compute chern classes of some vector bundles. so, for example in [7] ellingsrud-göttsche study the chern and segre classes of tautological vector bundles on the hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree n over a smooth and projective surface over the complex numbers. the purpose of this article is to explain how to apply bott’s localization formula to give an answer to question 2 above when n = 4 using an explicit and elementary description of a parameter space for squares in the plane, that is, the hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree 4 in p2 2 notation and convention for any homogeneous ideal i in the ring c[x0, x1, x2], let id denote the homogeneous part of degree d , that is, i = ⊕∞ d=0id. and when we refer to the hilbert polynomial associated to the closed subscheme determined by the homogeneous ideal i we refer precisely to the hilbert polynomial associated to the c[x0, x1, x2]-module c[x0, x1, x2]/i (see pg. 51 in hartshorne’s book [12]). let i = {f ∈ c[x0, x1, x2] | for each i = 0, 1, 2, there is an ni such that f · x ni i ∈ i} be the saturation of the homogeneous ideal i in c[x0, x1, x2]. we say that i is saturated if i = i. let f denote the vector space of linear forms in the variables x0, x1, x2 and fd the vector space of homogeneous forms of degree d. let f1, ..., fs ∈ fd (f1, ..., fs ∈ c[x0, x1, x2]), we denote by [f1, ..., fs] (〈f1, ..., fs〉) the c-vector space generated by f1, ..., fs in fd (the ideal generated by f1, ..., fs in c[x0, x1, x2]). for each point p ∈ p2, let f p d denote the linear system of forms of degree d vanishing at the point p. let gn(fd) denote the grassmann variety parametrizing the n-dimensional vector subspaces of fd. set x = g2(f2) be the grassmannian of pencils of conics in p 2, with tautological sequence 0 −→a−→f2 −→f2 −→ 0 (2.1) 198 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) where a ⊂ f2 denote a subbundle of rank 2 with fiber over π ∈ g2(f2) given by the vector subspace π ⊂f2. 3 an explicit description of hilb4p2 3.1 hilbert scheme of points in p2 let hilbdp2 be the hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree d in p2. as we have a 1-1 correspondence between saturated homogeneous ideals of c[x0, x1, x2] and closed subschemes of p2 (see ex. 5.10 of chapter ii in hartshorne’s book [12]) then we set. hilbdp2 =    i ⊂ c[x0, x1, x2] | i is saturated homogeneous ideal in c[x0, x1, x2] such that the hilbert polynomial of the c[x0, x1, x2]-module c[x0, x1, x2]/i is equal to d.    . (3.1) it is known that hilbdp2 is nonsingular of dimension 2d (see [9]) and that for each positive integer d it embeds in the grassmann variety of codimension d subspaces of fd (see pg. 34 in [1], lecture 15 in [18]). having in mind (3.1) we are going to give an explicit description of all saturated homogeneous ideals in hilb4p2. for those who are interested in the scheme structure in more detail, we recommend the reading of [1] and [21]. naturally as suggested by bézout’s theorem we begin with a pair of conics in the plane. 3.2 quadruplets determined by conics for each π = [q1, q2] ∈ x we can associate the ideal iπ = 〈q1, q2〉⊂ c[x0, x1, x2] (iπ is a saturated ideal). the variety determined by iπ correspond to the intersection of two conics. thus, we have the following two possibilities:    if gcd(q1, q2) = 1 then according to bézout’s theorem the number of intersection points between q1 and q2 should be 2x2=4 points counted with multiplicities. if gcd(q1, q2) 6= 1 then we have that q1 = ℓℓ1, q2 = ℓℓ2 with ℓ, ℓ1 ∈ p ( f ) and ℓ2 ∈ p ( f / [ℓ1] ) . cubo 12, 1 (2010) projective squares in p2 ... 199 so in the general case we have the following pictures: . ........ ......... .......... ............. ................. ....................... ................................................... .................................................................................................................................................................................... ......... ........ ......... .......... ............. ................ . ....................... ......................... .......................... ............................ . ............................ .......................... ......................... ....................... . ................ ............. .......... ......... ........ . ...................... .................... ................... .............. .......................................... ......... . ........ .... ....... ...... . ........ ....... ..... ........ ........ ..... ........ ........ ....... . ......... ........ ..... ......... ......... .. ............ ....... . ............. ............ .......... ........... ......... .......... ............ ............. . .................... ..................... ....................... • • • •q1 q2 gcd(q1, q2) = 1 ℓ bb bb bb bb ℓ1 �� ℓ2 • gcd(q1, q2) 6= 1 thus it is natural to consider the following subvariety of x. let y = { [q1, q2] ∈ x | q1 = ℓℓ1, q2 = ℓℓ2 with ℓ, ℓ1 ∈ p ( f ) and ℓ2 ∈ p ( f / [ℓ1] )} . (3.2) let p be the intersection point of two lines ℓ1 and ℓ2, then y can be illustrated as follows: y =    . .................................................................................................................................................................................. ℓ •p    this figure suggest that we need a cubic form in order to obtain three points on the line ℓ. in the next section we are looking for that cubic form. 3.3 quadruplets generated in degree three now, we will describe the cubic homogeneous polynomial f ∈ c[x0, x1, x2], that we need to add to the ideal iπ = 〈ℓℓ1, ℓℓ2〉 in order to get a quadruplets of points in the plane. 3.1. lemma. let i = 〈ℓℓ1, ℓℓ2, f〉 ⊂ c[x0, x1, x2] be an ideal where ℓ, ℓ1 and ℓ2 are linear forms such that [ℓ1, ℓ2] ∈ g2(f) and f 6∈ 〈ℓℓ1, ℓℓ2〉 is a cubic homogeneous polynomial. then we have that 1. if f 6∈ 〈ℓ〉 and f ∈ 〈ℓ1, ℓ2〉 then i is saturated and the hilbert polynomial of the variety defined by i is 4. 2. if f 6∈ 〈ℓ〉 and f 6∈ 〈ℓ1, ℓ2〉 then i is saturated and the hilbert polynomial of the variety defined by i is 3. 3. if f ∈ 〈ℓ〉 then the saturation of i is i = 〈ℓ〉 and the hilbert polynomial of the variety defined by i is t + 1. proof. see [21]. 200 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) in the general case we have the following pictures: ℓ • • • bb bb bb bb ℓ1 �� ℓ2 • p q1 = ℓℓ1, q2 = ℓℓ2, f3 ∈ 〈ℓ1, ℓ2〉 \ 〈ℓ〉. ℓ q1 = ℓℓ1, q2 = ℓℓ2, f3 ∈ 〈ℓ〉. we conclude from lemma 3.1 that a good choice for a cubic form f such that the ideal 〈ℓℓ1, ℓℓ2, f〉 determines a quadruplets of points in the plane, will be to begin with f ∈ 〈ℓ1, ℓ2〉 \ 〈ℓℓ1, ℓℓ2〉. note that, the vector subspace of cubic forms ℓℓ1·f+ℓℓ2·f is equal to the 5-dimensional vector space ℓ·f p 2 of cubic forms that are multiple of the linear form ℓ times a conic passing through the point p ({p} = ℓ1∩ℓ2). on the other hand, the vector space of cubic forms passing through the point p, f p 3 , have dimension 9, so f ∈ 〈ℓ1, ℓ2〉\〈ℓℓ1, ℓℓ2〉 is varying in a 4-dimensional vector space. thus we have obtained a p3-bundle e1 over y (cf. (3.2)). in fact, we can consider e1 embedded in g2(f2) × g6(f3) as follows: e 1 ∋ ([ℓℓ1, ℓℓ2], f ) 7−→ ([ℓℓ1, ℓℓ2], ℓ ·f p 2 + [f ]) ∈ g2(f2) × g6(f3) (3.3) with f ∈ p(f p 3 /(ℓ ·f p 2 )). note that, to each point ([ℓℓ1, ℓℓ2], f ) ∈ e 1, we can associate the homogeneous ideal 〈ℓℓ1, ℓℓ2, f〉 in c[x0, x1, x2]. next, we will give a description of those points in e 1 whose associated ideal define a quadruplets in the plane. certainly, if f ∈ 〈ℓ〉 we do not obtain a quadruplet in the plane (cf. lemma 3.1). thus the problem now it is to know when a cubic form f ∈ 〈ℓ1, ℓ2〉\〈ℓℓ1, ℓℓ2〉 will be a multiple of the line ℓ. in fact, we have the following result. 3.2. lemma. let w = { [ℓ2, ℓℓ1] ∈ x | [ℓ, ℓ1] ∈ g2(f) } ⊂ y, which is illustrated as w =    . ............................................................................................................................................................ ℓ • p    where {p} = ℓ ∩ ℓ1. then we have that 1. if [ℓℓ1, ℓℓ2] ∈ y \ w then 〈ℓ1, ℓ2〉 ∩ 〈ℓ〉 = 〈ℓℓ1, ℓℓ2〉. therefore does not exist a cubic form f ∈ 〈ℓ1, ℓ2〉 \ 〈ℓℓ1, ℓℓ2〉 being a multiple of ℓ. 2. if [ℓ2, ℓℓ1] ∈ w then 〈ℓ, ℓ1〉 ∩ 〈ℓ〉 = 〈ℓ〉 and 〈ℓ〉3 = 〈ℓ 2, ℓℓ1〉3 ⊕ [ℓϕ] with ϕ(p) 6= 0, that is, ϕ ∈ f2 \ 〈ℓ, ℓ1〉2. thus the fiber of e 1 over [ℓ2, ℓℓ1] has exactly one point, does not define a quadruplet and all the others will do. in fact, the locus where f is a multiple of ℓ is given by the following section of e1|w cubo 12, 1 (2010) projective squares in p2 ... 201 e1|w ⊃ w 1 ∋ ([ℓ2, ℓℓ1], ℓϕ) 7−→ ([ℓ 2, ℓℓ1], ℓ ·f2) ∈ g2(f2) × g6(f3) ↑ ↑ w ∋ [ℓ2, ℓℓ1] (3.4) and the saturation of the ideal 〈ℓ2, ℓℓ1, ℓϕ〉 is equal to 〈ℓ〉. 3.4 quadruplets generated in degree four it follows from lemma 3.1 (2.) that it does not help to add any other new generator to the ideal 〈ℓℓ1, ℓℓ2, f〉 in order to get a quadruplets of points in p 2. and from (3.) and bézout’s theorem that, it is sufficient to choose a degree four homogeneous polynomial g ∈ c[x0, x1, x2] with g 6∈ 〈ℓ〉. thus we have obtained a p4-bundle e2 over w1 (cf. (3.4)). in fact, we can consider e2 embedded in g2(f2) × g6(f3) × g11(f4) as follows: e 2 ∋ ([ℓ2, ℓℓ1], ℓϕ, g̃) 7−→ ([ℓ 2, ℓℓ1], ℓ ·f2, ℓ ·f3 + [g]) ∈ g2(f2, ) × g6(f3) × g11(f4) (3.5) with ℓϕ ∈ p(fp3 /(ℓ ·f p 2 )) and g̃ ∈ p(f4/(ℓ ·f3)). in fact, we have that. 3.3. lemma. let i = 〈ℓ, g〉 ⊂ c[x0, x1, x2] be an ideal where ℓ is a linear form and g 6∈ 〈ℓ〉 is a quartic homogeneous polynomial. then we have that i is saturated and the hilbert polynomial of the variety defined by i is 4. proof. see [21]. 4 enumerative application now we are interested in giving an answer to the following enumerative question: how many squares are there with its set of vertices meeting eight general lines? ( • • • • ) more generally, how many n -gons are there with its set of vertices meeting 2n general lines? note that, each vertex in the n -gon is determined by the intersection of a pair of distinct lines. so, let ℓ1, ..., ℓ2n be 2n general given lines in p 2 and set pn = {n − gons having its vertices in exactly one pair of these distinct lines}. now, fix 2n + 2 general lines ℓ1, ..., ℓ2n +2 in p 2 and let pn +1,i = {(n + 1) − gons having one vertex over ℓ2n +2 and ℓi} for i = 1, ..., 2n + 1. note that: • pn +1,i ∩pn +1,j = ∅ for i 6= j; • pn +1,i are in bijection with pn for i = 1, ..., 2n + 1; 202 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) • pn +1 = ⋃2n +1 i=1 pn +1,i. thus, we have that #(pn +1) = ∑2n +1 i=1 #(pn +1,i) = (2n + 1) · #(pn ). using induction, we see that #(pn +1) = (2n + 1) ·(2n −1) · ... ·5 · 3 ·1 = {2n + 1}! = the factorial of odd’s numbers between 1 and 2n + 1. therefore, #(p4) = {7}! = 7 · 5 · 3 · 1 = 105. next we will use bott’s localization formula to find the answer to the enumerative problem( • • • • ) on an appropriate parameter space. the bott’s localization formula that we will apply express the integral of a homogeneous polynomial in the chern classes of a bundle on a smooth, compact variety with a c∗-action in terms of data given by the induced linear actions on the fiber of the bundle and the tangent bundle in the (isolated) fixed points of the action. in fact, bott’s residues formula said that. 4.1. theorem. let t be a torus and x be a smooth, complete variety with a t -action. let e1, . . . , es be t – equivariant vector bundles. then we have that. ∫ x p(e) ∩ [x] = ∑ f ⊂ xt (πf )∗ ( pt (e|f ) ∩ [f ]t ct df (nf x) ) (4.1) where • f is a (dimx − df )-dimensional component of x t ; • x t is the fixed point locus; • p(e) = p(e1, ..., es) is a homogeneous polynomial of degree dimx in the chern classes of the bundles e′j s. in fact, p(e) is a weighted homogeneous polynomial in the variables x i j = ci(ej ), where xij has degree i; • nf x denoted the normal bundle of f in x; • [f ]t is the t -equivariant fundamental class of f ; • ct df (nf x) denoted the top t -equivariant chern class of the normal bundle nf x; • pt (e|f ) = p(e1t , ..., est ), where eit denoted the quotient bundles associated to ei; • (πf )∗ denoted the proper pushforward of the morphism f if →֒ x πx −→ pt︸︷︷︸ πf . in spite of the possibly awe-inspiring appearance of (4.1) at first (in part because we do not explain what means each ingredient in the formula), we hope to convince the reader that it is rather simple to apply in practice. see [3], [5], [6] and the elementary exposition in [17] for details. see [8] and chapter 9 in [4] for applications. see also [11] for a computational improvement to bott’s application that have a close connection with cauchy’s residue formula. cubo 12, 1 (2010) projective squares in p2 ... 203 4.1 parameter space for squares let us consider the following two closed subvarieties of g2(f2)×g6(f3) and g2(f2)×g6(f3)× g11(f4) respectively. for each pencil of conics [q1, q2] ∈ g2(f2) , let c ∈ g6(f3) be the linear system defined as follows c =    q1 ·f + q2 ·f if gcd(q1, q2) = 1, ℓ ·f p 2 + [f ] if q1 = ℓℓ1, q2 = ℓℓ2 with ℓ ∈ p(f), [ℓ1, ℓ2] ∈ g2(f) and f ∈ p(f p 3 /(ℓ ·f p 2 )) where {p} = ℓ1 ∩ ℓ2. (4.2) let x 1 = { ([q1, q2], c) ∈ g2(f2) × g6(f3) | c is defined as in (4.2) } . (4.3) now for each ([q1, q2], c) ∈ x 1, let q ∈ g11(f4) be the linear system defined as follows: q =    q1 ·f2 + q2 ·f2 if gcd(q1, q2) = 1, ℓ ·f p 3 + f ·f if q1 = ℓℓ1, q2 = ℓℓ2, ℓ ∈ p(f), [ℓ1, ℓ2] ∈ g2(f) and f ∈ p(f p 3 /(ℓ ·f p 2 )) with {p} = ℓ1 ∩ ℓ2 such that f /∈ 〈ℓ〉, ℓ ·f3 + [g] if q1 = ℓℓ1, q2 = ℓℓ2, f = ℓϕ where ϕ ∈f2 \f p 2 and g̃ ∈ p(f4/(ℓ ·f3)). (4.4) let x 2 = { ([q1, q2], c, q) ∈ x 1 × g11(f4) | q is defined as in (4.4) } . (4.5) follows from (3.3), (4.2) and (4.3) that e1 is a subvariety of x1. in the same way follows from (3.5), (4.4) and (4.5) that e2 is a subvariety of x2. therefore, we have the following diagram for our parameter space x2. e2 →֒ x2 ւ ց w1 →֒ e1 →֒ x1 ↓ ↓ ↓ w →֒ y →֒ x (4.6) in fact, it is verified that x1 is the blowup of x along y with e1 being the exceptional divisor and also that x2 is the blowup of x1 along w1 with e2 being the exceptional divisor (see [1], [21]). on the other hand, for a ∈ c, ([x20, x0(x1 + ax2)], x0 · f2, x0 · f3 + [x 4 1]) are distinct points in x2, but its image in hilb4p2 is equal to the ideal 〈 x0, x 4 1 〉 . therefore x2 is not isomorphic to hilb4p2. nevertheless, can be verified that x2 is isomorphic to the the blowup of hilb4p2 along the 6-dimensional subvariety of aligned quadruplets (see [1]) ( 〈 x0, x 4 1 〉 is an aligned quadruplets). 204 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) 5 divisor of incidence to a line let ℓ be a line in p2 and dℓ be the hypersurface in x = g2(f2) defined by the condition ℓ ∩ q1 ∩ q2 6= ∅ for [q1, q2] ∈ x. dℓ =    ℓ • • • •    let d̃ℓ be the subvariety of ℓ × x defined by d̃ℓ = { (q, π) ∈ ℓ × x | q ∈ base locus of the pencil π } . note that: • d̃ℓ is a codimension two subvariety of ℓ × x. • the image of d̃ℓ under p2 : ℓ×x −→ x, the projection in the second coordinate, is equal to dℓ. 5.1 class of dℓ let a be the tautological subbundle of g2(f2) as in (2.1). let us consider the diagram of natural maps of vector bundles over ℓ × x, a→֒ f2 ց ↓ f2/f • 2 ∼= oℓ(2) here the fiber f2/f • 2 (q,π) is equal to f2/f2 q. note that the slant arrow vanishes at (q, π) ∈ ℓ × x if and only if (q, π) ∈ d̃ℓ. hence we have [d̃ℓ] = (c2(a ∨ ⊗oℓ(2))) ∩ [ℓ × x] = (c2(a) − 2h · c1(a)) ∩ [ℓ × x], where h = c1(oℓ(1)). pushing forward via p2 : ℓ × x −→ x, it follows that [dℓ] = −2c1(a) ∩ [x]. in fact, p2⋆(c2(p ⋆ 2a) ∩ [ℓ × x]) = c2(a) ∩ p2⋆[ℓ × x] = 0. cubo 12, 1 (2010) projective squares in p2 ... 205 c1(p ⋆ 1oℓ(1)) ∩ [ℓ × x] = c1(p ⋆ 1oℓ(1)) ∩ [p ⋆ 1(ℓ)], = p⋆1(c1(oℓ(1)) ∩ [ℓ]), = p⋆1([pt]), = [pt × x]. then p2⋆(c1(p ⋆ 2a) · c1(p ⋆ 1oℓ(1)) ∩ [ℓ × x]) = p2⋆(c1(p ⋆ 2a) ∩ [pt × x]), = c1(a) ∩ [x]. a local coordinate check shows that dℓ contains the blowup center y (see (3.2)) with multiplicity one. hence we find the formula for the class of the strict transform in x1, [d (1) ℓ ] = −2c1(a) ∩ [x 1] − [e1]. similarly, (omitting pullbacks) we get for the succeeding strict transform, [d (2) ℓ ] = −2c1(a) ∩ [x 2] − [e2,1] − [e2]. here we have omitted the pull-back in a and e2,1 denote the strict transform of e1. now a solution to the question ( • • • • ) in section 4 asks us to compute the degree of the self-intersection [d (2) ℓ ]8. thus from bott’s formula (cf. (4.1)) we have that. ∫ [x2] [d (2) ℓ ]8 = ∑ f ∫ [f ]t [2ct1 (af ) + c t 1 (o(e 2,1)f ) + c t 1 (o(e 2)f )] 8 ct df (nf x2) , (5.1) where df denotes the codimension of the component f in x. f is a component of x t the locus of fixed points for a suitable torus action, starting at x and following all the way up to x2. 6 fixed points at x2 let v be an n-dimensional complex vector space. then a general action of c∗ on v is diagonalized, so there is a basis {v1, ..., vn} of v such that t ·vi = λ(t)vi for all t ∈ c. in fact, λ is a character of the group c∗. so λ(t) = twi for some integer wi. we also have an induced action on gk(v ), the grassmann variety of k-planes in v, given by t · w = [t · w1, ..., t · wk] for any w = [w1, ..., wk] ∈ gk(v ). and the fixed points are given by: wi1,i2,...,ik = [vi1 , vi2 , ..., vik ] where (i1i2...ik) is a k-cicle in sn, so we have at all ( n k ) fixed points in gk(v ). 206 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) consider now the action of c∗ over fd given by t ◦ x i0 0 x i1 1 x i2 2 = t i0w0+i1w1+i2w2 xi00 x i1 1 x i2 2 with i0 + i1 + i2 = d and extend it by linearity. we also have an induced action on gn(fd), x 1 and x2 respectively. according to (4.6) the image of e2 and e1 in x are respectively w and y. and we also have that x2 \e2 ∼= x1 \w1 and x1 \e1 ∼= x\y. let e2,1 ⊂ x2 be the strict transform of e1, then we have that: fixed points in are in correspondence with fixed points in are in correspondence with fixed points in x2 \ (e2,1 ∪ e2) x1 \ e1 x \ y fixed points in are mapped on fixed points in are mapped on fixed points in e2,1 \ e2 e1 \ w1 y \ w e2 w1 w (6.1) so we will look for fixed points having in mind (6.1). 6.1 fixed points in x2 \ (e2,1 ∪ e2) if the weights (w0, w1, w2) are sufficiently general, we find the following 6 fixed points in x \ y: π1 = [x 2 0, x 2 1], π2 = [x 2 0, x1x2], π3 = [x 2 0, x 2 2], π4 = [x0x1, x 2 2], π5 = [x0x2, x 2 1], π6 = [x 2 1, x 2 2]. since this 6 fixed points lie off y then they lift (isomorphically) all the way up to x2. so their contribution can be obtained at once, down on x. of course the exceptional divisors give no contribution here. on the numerator of (5.1) we have for 2ct1 (aπi ) i = 1, ..., 6, fixed points in x \ y aπi decomposition of aπi into eigenspaces 2c t 1 (aπi ) π1 = [x 2 0, x 2 1] [x 2 0, x 2 1] t 2w0 + t2w1 2(2w0 + 2w1) π2 = [x 2 0, x1x2] [x 2 0, x1x2] t 2w0 + tw1+w2 2(2w0 + w1 + w2) π3 = [x 2 0, x 2 2] [x 2 0, x 2 2] t 2w0 + t2w2 2(2w0 + 2w2) π4 = [x0x1, x 2 2] [x0x1, x 2 2] t w0+w1 + t2w2 2(w0 + w1 + 2w2) π5 = [x0x2, x 2 1] [x0x2, x 2 1] t w0+w2 + t2w1 2(w0 + w2 + 2w1) π6 = [x 2 1, x 2 2] [x 2 1, x 2 2] t 2w1 + t2w2 2(2w1 + 2w2) on the denominator of (5.1) we get nπi x = tπi x 2 = tπi x = f2/aπi ⊗a ∨ πi . note that the eigen-decomposition of f2 is given by cubo 12, 1 (2010) projective squares in p2 ... 207 f2 = ∑ 0≤i≤j≤2 twi+wj = t2w0 + tw0+w1 + tw0+w2 + t2w1 + tw1+w2 + t2w2 . thus for π1 = [x 2 0, x 2 1] we have that tπ1 x = f2/aπ1⊗a ∨ π1 = f2/[x 2 0, x 2 1]⊗[x 2 0, x 2 1] ∨ = (tw0+w1 + tw0+w2 + tw1+w2 + t2w2 )(t−2w0 + t−2w1 ). next we give the eigen-decomposition of tπi x and c t 8 (tπi x), for i = 1, ..., 6: π1 = [x 2 0, x 2 1] ↔ (t w0+w1 + tw0+w2 + tw1+w2 + t2w2 )(t−2w0 + t−2w1 ), ↔ (w1 − w0)(w0 − w1)(w2 − w0)(w0 + w2 − 2w1)(w1 + w2 − 2w0)(w2 − w1)(2w2 − 2w0) (2w2 − 2w1), π2 = [x 2 0, x1x2] ↔ (t w0+w1 + tw0+w2 + t2w1 + t2w2 )(t−2w0 + t−(w1+w2)), ↔ (w1 − w0)(w0 − w2)(w2 − w0)(w0 − w1)(2w1 − 2w0)(w1 − w2)(2w2 − 2w0)(w2 − w1), π3 = [x 2 0, x 2 2] ↔ (t w0+w1 + tw0+w2 + t2w1 + tw1+w2 )(t−2w0 + t−2w2 ), ↔ (w2 − w0)(w0 − w2)(w1 − w0)(w0 + w1 − 2w2)(w1 + w2 − 2w0)(w1 − w2)(2w1 − 2w0) (2w1 − 2w2), π4 = [x0x1, x 2 2] ↔ (t 2w0 + tw0+w2 + t2w1 + tw1+w2 )(t−2w2 + t−(w0+w1)), ↔ (w1 − w2)(w2 − w0)(w0 − w2)(w2 − w1)(2w1 − 2w2)(w1 − w0)(2w0 − 2w2)(w0 − w1), π5 = [x0x2, x 2 1] ↔ (t 2w0 + tw0+w1 + tw1+w2 + t2w2 )(t−2w1 + t−(w0+w2)), ↔ (w2 − w1)(w1 − w0)(w0 − w1)(w1 − w2)(2w2 − 2w1)(w2 − w0)(2w0 − 2w1)(w0 − w2), π6 = [x 2 1, x 2 2] ↔ (t 2w0 + tw0+w1 + tw0+w2 + tw1+w2 )(t−2w1 + t−2w2 ), ↔ (w1 − w2)(w2 − w1)(w0 − w2)(w0 + w2 − 2w1)(w1 + w0 − 2w2)(w0 − w1)(2w0 − 2w2) (2w0 − 2w1). so the first six contributions to (5.1) are:    28(2w0+2w1) 8 (w1−w0)(w0−w1)(w2−w0)(w0+w2−2w1)(w1+w2−2w0)(w2−w1)(2w2−2w0)(2w2−2w1) + 28(2w0+w1+w2) 8 (w1−w0)(w0−w2)(w2−w0)(w0−w1)(2w1−2w0)(w1−w2)(2w2−2w0)(w2−w1) + 28(2w0+2w2) 8 (w2−w0)(w0−w2)(w1−w0)(w0+w1−2w2)(w1+w2−2w0)(w1−w2)(2w1−2w0)(2w1−2w2) + 28(w0+w1+2w2) 8 (w1−w2)(w2−w0)(w0−w2)(w2−w1)(2w1−2w2)(w1−w0)(2w0−2w2)(w0−w1) + 28(w0+w2+2w1) 8 (w2−w1)(w1−w0)(w0−w1)(w1−w2)(2w2−2w1)(w2−w0)(2w0−2w1)(w0−w2) + 28(2w1+2w2) 8 (w1−w2)(w2−w1)(w0−w2)(w0+w2−2w1)(w1+w0−2w2)(w0−w1)(2w0−2w2)(2w0−2w1) . 6.2 fixed points in e2,1 \ e2 since e2,1 \ e2 is isomorphic to e1 \ w1. then, we have to look for fixed points on y \ w (cf. (6.1)). we find after some computation the following 3 fixed points in y \ w. π7 = [x0x1, x0x2], π8 = [x1x0, x1x2], π9 = [x2x0, x2x1]. (6.2) 208 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) thus to determine the contributions to (5.1) in this case, we only have to calculate ct8 (ty1 x 1), ct1 (o(e 1)y1 ) for those fixed points y 1 ∈ e1 lying over πi and 2c t 1 (aπi ) for i = 7, 8, 9. according to (3.3) the fiber of e1 over [ℓℓ1, ℓℓ2] ∈ y \ w is given by e 1 [ℓℓ1,ℓℓ2] = p(〈ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2 〉) (6.3) where f indicates classes of f ∈ f p 3 modulo ℓ · f p 2 with {p} = ℓ1 ∩ ℓ2. note that, ℓ = x0, ℓ1 = x1, ℓ2 = x2 for π7 and so on. and can be verified that ([ℓℓ1, ℓℓ2], ℓ · f p + [f ]) ∈ e1 ⊂ x1 with f ∈ {ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2} are fixed points for the induced action of t = c ∗ on x1. thus we obtain 3 × 4 = 12 fixed points lying in e1 \ w1. in order to compute the contributions coming from this 12 fixed points to (5.1) we need to determine tangent and normal spaces. since the exact sequence of c∗–representations 0 → tπy → tπx → (nyx)π → 0 splits, we may write the following decomposition into eigen spaces for [ℓℓ1, ℓℓ2] ∈ y \ w, (nyx)[ℓℓ1,ℓℓ2] = t[ℓℓ1,ℓℓ2]x −t[ℓℓ1,ℓℓ2]y = t[ℓℓ1,ℓℓ2]g2(f2) −t[ℓℓ1,ℓℓ2]( ∼=y︷︸︸︷ p(f) × g2(f)) = (ℓℓ1 + ℓℓ2) ∨ ⊗ f2−(ℓℓ1+ℓℓ2)︷︸︸︷ (ℓ2 + ℓ21 + ℓ1ℓ2 + ℓ 2 2)− t[ℓℓ1,ℓℓ2]y︷︸︸︷(ℓ1 ℓ + ℓ2 ℓ + ℓ ℓ1 + ℓ ℓ2 ) = ℓ 2 2 ℓℓ1 + ℓ 2 1 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ . (6.4) note that from (6.3) and (6.4) we have the two descriptions, e 1 [ℓℓ1,ℓℓ2] = p((nyx)[ℓℓ1,ℓℓ2]) = p(〈ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2 〉) and (nyx)[ℓℓ1,ℓℓ2] = ℓ22 ℓℓ1 + ℓ21 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ . we can reconcile this two descriptions noting that to any normal vector ξ as in (6.4) we can associated a curve γt in x with tangent ξ at t = 0 such that γt ∈ x \ y for t 6= 0, so it lifts to a curve γ1t in x 1 whose tangent at t = 0 give a monomial in {ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2} associated to the normal direction corresponding to ξ as described in the following table: normal vector ξ curve with tangent ξ lifts to a curve in x1 ℓ 2 2 ℓℓ1 = (ℓℓ1) ∨⊗ℓ22 γt = [ℓℓ1 + tℓ 2 2, ℓℓ2] γ 1 t = (γt, ℓℓ2·f + (ℓℓ1 + tℓ 2 2)ℓ + (ℓℓ1 + tℓ 2 2)ℓ1 + tℓ 3 2) ℓ 2 1 ℓℓ2 = (ℓℓ2) ∨⊗ℓ21 γt = [ℓℓ1, ℓℓ2 + tℓ 2 1] γ 1 t = (γt, ℓℓ1·f + (ℓℓ2 + tℓ 2 1)ℓ + (ℓℓ2 + tℓ 2 1)ℓ2 + tℓ 3 1) ℓ1 ℓ = (ℓ)∨⊗ℓ1 γt = [ℓℓ1, ℓℓ2 + tℓ1ℓ2] γ 1 t = (γt, ℓℓ1·f + (ℓℓ2 + tℓ1ℓ2)ℓ + (ℓℓ2 + tℓ1ℓ2)ℓ2 + tℓ 2 1ℓ2) ℓ2 ℓ = (ℓ)∨⊗ℓ2 γt = [ℓℓ1, ℓℓ2 + tℓ 2 2] γ 1 t = (γt, ℓℓ1·f + (ℓℓ2 + tℓ 2 2)ℓ + (ℓℓ2 + tℓ 2 2)ℓ2 + tℓ1ℓ 2 2) (6.5) determination of ct8 (ty1 x 1), ct1 (o(e 1)y1 ) for those fixed points y 1 ∈ e1 lying over πi and 2ct1 (aπi ) for i = 7, 8, 9. cubo 12, 1 (2010) projective squares in p2 ... 209 on the other hand, for any π1 ∈ e1 lying over π ∈ y, we have that tπ1 x 1 = tπ1 e 1 + (ne1 x 1)π1 = tπ1 e 1 π + tπy + [π 1]. note that [π1] = oe1 (−1)π1 = o(e 1)π1 . (6.6) let y = [ℓℓ1, ℓℓ2] ∈ y\w and y 1 i = (y, fi)∈e 1 with fi ∈{ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2}. so for y 1 1 = (y, ℓ 3 1) ∈ e 1 we have that: ty11 x 1 = ty11 e 1 y + tyy + [y 1 1] = ty11 e 1 y︷︸︸︷ p([ ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2 ]) +ty( ∼= y︷︸︸︷ p(f) × g∈(f)) + [y 1 1] = ℓ2 ℓ1 + ℓ22 ℓ21 + ℓ32 ℓ31︸︷︷︸ t y1 1 e1y + ℓ1 ℓ + ℓ2 ℓ + ℓ ℓ1 + ℓ ℓ2︸︷︷︸ tyy + ℓ21 ℓℓ2︸︷︷︸ o e1 (−1) y1 1 . ay = ℓℓ1 + ℓℓ2 and o(e 1) (y,ℓ31) = oe1 (−1)(y,ℓ31) = ℓ31 f rom (6.5) ←→ ℓ 2 1 ℓℓ2 . we listed below, the eigen-decomposition of the tangent and first exceptional divisor at each fixed point y1i ∈ e 1, following the description above. fixed point type for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. tangent and first exceptional divisor y11 = ([x0x1, x0x2], x 3 1) ty11 x 1 = t(w2−w1) + t(2w2−2w1) + t(3w2−3w1) + t(w1−w0)+ t(w2−w0) + t(w0−w1) + t(w0−w2) + t(2w1−w0−w2), o(e1)y11 = t(2w1−w0−w2). y12 = ([x0x1, x0x2], x 2 1x2) ty12 x 1 = t(w1−w2) + t(w2−w1) + t(2w2−2w1) + t(w1−w0)+ t(w2−w0) + t(w0−w1) + t(w0−w2) + t(w1−w0), o(e1)y12 = t(w1−w0). y13 = ([x0x1, x0x2], x1x 2 2) ty13 x 1 = permute w1 and w2 in ty12 x 1, o(e1)y13 = t(w2−w0). y14 = ([x0x1, x0x2], x 3 2) ty14 x 1 = permute w1 and w2 in ty11 x 1, o(e1)y14 = t(2w2−w0−w1). thus the contribution to (5.1) at each y1i ∈ e 1 lying over y = [ℓℓ1, ℓℓ2] ∈ y \w is given by: 210 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) fixed point for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. contribution to the numerator in (5.1) contribution to the denominator in (5.1) y11 = (y, x 3 1 ) 2ct1 (ay) = 2(2w0 + w1 + w2)+ c t 1 (o(e 1) y1 1 ) = 2w1 − w0 − w2. c t 8 (ty1 1 x 1) = (w2 − w1)(2w2 − 2w1)(3w2 − 3w1)· (w1 − w0)(w2 − w0)(w0 − w1)· (w0 − w2)(2w1 − w0 − w2). y12 = (y, x 2 1x2) 2ct1 (ay) = 2(2w0 + w1 + w2)+ c t 1 (o(e 1) y1 2 ) = w1 − w0. c t 8 (ty1 2 x 1) = (w1 − w2)w2 − w1)(2w2 − 2w1)(w1 − w0)· (w2 − w0)(w0 − w1)(w0 − w2)(w1 − w0). y13 = (y, x1x 2 2) permute w1 and w2 in 2ct2 (ay) + c t 1 (o(e 1) y1 2 ). c t 8 (ty1 3 x 1) = permute w1 and w2 in c t 8 (ty1 2 x 1). y14 = (y, x 3 2) permute w1 and w2 in 2ct2 (ay) + c t 1 (o(e 1) y1 1 ). c t 8 (ty1 4 x 1) = permute w1 and w2 in c t 8 (ty1 1 x 1). in fact, if we make a cyclic permutation of x′is in the table above, we will obtain all the 12 contributions to (5.1) determined by the 3 fixed points πi ∈ y \w for i = 7, 8, 9 (cf. (6.2)). 6.3 fixed points in e2 according to (6.1) we have to look for fixed points in w. after some computation we find the following 6 fixed points in w: π10 = [x 2 0, x0x1], π11 = [x 2 0, x0x2], π12 = [x 2 1, x1x0], π13 = [x 2 1, x1x2], π14 = [x 2 2, x2x0], π15 = [x 2 2, x2x1]. (6.7) in order to determine the contributions to (5.1) in this case we have to calculate ct8 (tπ2 x 2), ct1 (o(e 2)π2 ) for those fixed points π 2 ∈ e2 lying over fixed points π1 ∈ w1 ⊂ e1, ct1 (o(e 1)π1 ) for those fixed points π1 ∈ e1 lying over πi and 2c t 1 (aπi ) for i = 10, ..., 15. we have from (3.3) that the fiber of e1 over [ℓ2, ℓℓ1] ∈ w is given by: e 1 [ℓ2,ℓℓ1] = p(〈ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 〉) (6.8) where f indicates classes of f ∈f p 3 modulo ℓ ·f p 2 with {p} = ℓ∩ℓ1. note that, ℓ = x0, ℓ1 = x1 for π10 and so on. and can be verified that ([ℓ 2, ℓℓ1], ℓ·f p +[f ]) ∈ e1 ⊂ x1 with f ∈{ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2} are fixed points for the induced action of t = c∗ on x1. thus we obtain: { 6 × 3 = 18 fixed points lying in e1 \ w1 if f ∈{ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2}, 6 × 1 = 6 fixed points lying in w1 if f = ℓℓ22. (6.9) in order to compute the contribution of this fixed points to (5.1) we need to determine tangent and normal spaces as we did in (6.4). cubo 12, 1 (2010) projective squares in p2 ... 211 we may write the following decomposition into eigen spaces for [ℓ2, ℓℓ1] ∈ w, (nyx)[ℓ2,ℓℓ1] = t[ℓ2,ℓℓ1]x −t[ℓ2,ℓℓ1]y = t[ℓ2,ℓℓ1]g2(f2) −t[ℓ2,ℓℓ1]( ∼=y︷︸︸︷ p(f) × g2(f)) = (ℓ2 + ℓℓ1) ∨ ⊗ f2−(ℓ 2+ℓℓ1)︷︸︸︷ (ℓℓ2 + ℓ 2 1 + ℓ1ℓ2 + ℓ 2 2)− t [ℓ2,ℓℓ1] y ︷︸︸︷(ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1 ) = ℓ 2 1 ℓ2 + ℓ1ℓ2 ℓ2 + ℓ 2 2 ℓ2 + ℓ 2 2 ℓℓ1 . (6.10) note that from (6.8) and (6.10) we have the two descriptions, e 1 [ℓ2,ℓℓ1] = p((nyx)[ℓ2,ℓℓ1]) = p(〈ℓℓ 2 2, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 〉) and (nyx)[ℓ2,ℓℓ1] = ℓ21 ℓ2 + ℓ1ℓ2 ℓ2 + ℓ22 ℓ2 + ℓ22 ℓℓ1 . again we can reconcile this two descriptions as we did in (6.5). in this case the correspondence is given by: ℓ 2 1 ℓ2 ℓ1ℓ2 ℓ2 ℓ 2 2 ℓ2 ℓ 2 2 ℓℓ1 l l l l ℓ31 ℓ 2 1ℓ2 ℓ1ℓ 2 2 ℓℓ 2 2 (6.11) now, let w1i = ([ℓ 2, ℓℓ1], fi) ∈ e 1 with fi ∈{ℓℓ 2 2, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2}. contributions to (5.1) coming from w1i for i = 2, 3, 4 note that the three points w1i for i = 2, 3, 4 lift (isomorphically) all the way up to x 2 since x2 \ e2 ∼= x1 \ w1. so their contribution can be obtained at once on x1. so for w12 = (w, ℓ 3 1) ∈ e 1 lying over w = [ℓ2, ℓℓ1] we have from (6.6) that tw12 x 1 = tw12 e 1 w + twy + [w 1 2] = tw12 e 1 w︷︸︸︷ p([ ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 ]) +tw( ∼= y︷︸︸︷ p(f) × g∈(f)) + [w 1 2] = ℓℓ22 ℓ31 + ℓ2 ℓ1 + ℓ22 ℓ21︸︷︷︸ t w1 2 e1w + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸︷︷︸ twy + ℓ21 ℓ2︸︷︷︸ o e1 (−1) w1 2 . (6.12) a[ℓ2,ℓℓ1] = ℓ 2 + ℓℓ1 and o(e 1) ([ℓ2,ℓℓ1],ℓ 3 1) = oe1 (−1)([ℓ2,ℓℓ1],ℓ31) = ℓ31 f rom (6.11) ←→ ℓ 2 1 ℓ2 . we listed below, the eigen-decomposition of the tangent and first exceptional divisor at each fixed point w1i ∈ e 1 for i = 2, 3, 4, following the description above. 212 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) fixed point for ℓ = x0 and ℓ1 = x1. tangent and first exceptional divisor w12 = ([x 2 0, x0x1], x 3 1) tw12 x 1 = t(w0+2w2−3w1) + t(w2−w1) + t(2w2−2w1) + t(w1−w0)+ t(w2−w0) + t(w2−w0) + t(w2−w1) + t(2w1−2w0), o(e1)w12 = t(2w1−2w0). w13 = ([x 2 0, x0x1], x 2 1x2) tw13 x 1 = t(w0+w2−2w1) + t(w1−w2) + t(w2−w1) + t(w1−w0)+ t(w2−w0) + t(w2−w0) + t(w2−w1) + t(w1+w2−2w0), o(e1)w13 = t(w1+w2−2w0). w14 = ([x 2 0, x0x1], x1x 2 2) tw14 x 1 = t(w0−w1) + t(2w1−2w2) + t(w1−w2) + t(w1−w0)+ t(w2−w0) + t(w2−w0) + t(w2−w1) + t(2w2−2w0), o(e1)w14 = t(2w2−2w0). thus the contribution to (5.1) at each w1i ∈ e 1 for i = 2, 3, 4 lying over w = [ℓ2, ℓℓ1] is given by: fixed point for ℓ = x0, ℓ1 = x1. contribution to the numerator in (5.1) 2c t 1 (aw)︸︷︷︸ 2(3w0+w1) + ct1 (o(e 1) w1 i ) contribution to the denominator in (5.1) c t 8 (tw1 i x 1) w12 = (w, x 3 1) 2(3w0 + w1) + (2w1 − 2w0) 4(w0 + 2w2 − 3w1)(w2 − w1) 3(w1 − w0) 2(w2 − w0) 2 w13 = (w, x 2 1x2) 2(3w0 + w1) + (w1 + w2 − 2w0) −(w0 + w2 − 2w1)(w2 − w1) 3(w1 − w0)(w2 − w0) 2 (w1 + w2 − 2w0) w14 = (w, x1x 2 2) 2(3w0 + w1) + (2w2 − 2w0) 4(w0 − w1) 2(w1 − w2) 3(w2 − w0) 3 making a cyclic permutation of x′is in the table above, we will obtain all the 18 contributions to (5.1) determined by the 6 fixed points πi ∈ w for i = 10, ..., 15 (cf. (6.7) and (6.9)). contributions to (5.1) coming from w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2) ∈ w 1 in order to determine the contributions to (5.1) in this case we have to calculate ct8 (tw2 x 2), ct1 (o(e 2)w2 ) for those fixed points w 2 ∈ e2 lying over the fixed point w11 ∈ w 1 ⊂ e1, ct1 (o(e 1)w11 ) and 2ct1 (aw). consider now the fiber of e2 over w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2). according to (3.5), it is just e 2 ([ℓ2,ℓℓ1],ℓ̃ℓ 2 2) = p(〈 ℓ̃41, ℓ̃ 3 1ℓ2, ℓ̃ 2 1ℓ 2 2, ℓ̃1ℓ 3 2, ℓ̃ 4 2 〉) (6.13) where g̃ indicates classes of g ∈f4 modulo ℓ·f3. and can be verified that ([ℓ 2, ℓℓ1], ℓ·f2, ℓ·f2+[g]) ∈ e2 ⊂ x2 with g ∈{ℓ41, ℓ 3 1ℓ2, ℓ 2 1ℓ 2 2, ℓ1ℓ 3 2, ℓ 4 2} are fixed points for the induced action of t = c ∗ on x2. thus we obtain 6 × 5 = 30 fixed points lying in e2. in order to compute the contribution of this 30 fixed points to (5.1) we need to determine tangent and normal spaces as we did in (6.4). cubo 12, 1 (2010) projective squares in p2 ... 213 we may write the following decomposition into eigen spaces for w11 = (w, ℓℓ 2 2) ∈ w 1 lying over w = [ℓ2, ℓℓ1], (nw1 x 1)w11 = tw11 x 1 −tw11 w 1 = tw11 e 1 w + twy + [w 1 1] −tw11 w 1 = tw11 e 1 w︷︸︸︷ p([ ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 ]) +tw( ∼= y︷︸︸︷ p(f) × g∈(f)) + [w 1 1] −tw11 w 1 = ℓ31 ℓℓ22 + ℓ21 ℓℓ2 + ℓ1 ℓ ︸︷︷︸ t w1 1 e1w + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸︷︷︸ twy + ℓ22 ℓℓ1︸︷︷︸ o e1 (−1) w1 1 − ( ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ1 ) ︸︷︷︸ t w1 1 w1 = ℓ 3 1 ℓℓ22 + ℓ 2 1 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ + ℓ 2 2 ℓℓ1 . (6.14) note that from (6.13) and (6.14) we have the two descriptions at w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2), e 2 w11 = p((nw1 x 1)w11 ) = p(〈 ℓ̃ 4 1, ℓ̃ 3 1ℓ2, ℓ̃ 2 1ℓ 2 2, ℓ̃1ℓ 3 2, ℓ̃ 4 2 〉) and (nw1 x 1)w11 = ℓ31 ℓℓ22 + ℓ21 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ + ℓ22 ℓℓ1 . we can reconcile this two descriptions as we did in (6.5) and (6.11). in this case the correspondence is given by: ℓ 3 1 ℓℓ22 ℓ 2 1 ℓℓ2 ℓ1 ℓ ℓ2 ℓ ℓ 2 2 ℓℓ1 l l l l l ℓ41 ℓ 3 1ℓ2 ℓ 2 1ℓ 2 2 ℓ1ℓ 3 2 ℓ 4 2 (6.15) now let w2i = (w 1 1, g̃i) ∈ e 2 with gi ∈ {ℓ 4 1, ℓ 3 1ℓ2, ℓ 2 1ℓ 2 2, ℓ1ℓ 3 2, ℓ 4 2}. for w 2 1 = (w 1 1, ℓ̃ 4 1) ∈ e 2 lying over w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2) we have from (6.6) changing 1 by 2 that: tw21 x 2 = tw21 e 2 w1 1 + tw11 w 1 + [w21] = ℓ31ℓ2 ℓ41 + ℓ22 ℓ21 + ℓ32 ℓ31 + ℓ42 ℓ41︸︷︷︸ t w2 1 e2 w1 1 + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸︷︷︸ t w1 1 w1 + ℓ31 ℓℓ22︸︷︷︸ o e2 (−1)w2 1 . o(e2)w21 = oe2 (−1)(w11,ℓ̃41) = ℓ41 ( f rom (6.15) ←→ ℓ 3 1 ℓℓ22 ). we listed below, the eigen-decomposition of the tangent and first exceptional divisor at each fixed point w2i ∈ e 2 for i = 1, ..., 5, following the description above. 214 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) fixed point for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. tangent and second exceptional divisor w21 = (w 1 1, x̃ 4 1) tw21 x 2 = t(w2−w1) + t(2w2−2w1) + t(3w2−3w1) + t(4w2−4w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(3w1−w0−2w2), o(e2)w21 = t(3w1−w0−2w2). w22 = (w 1 1, x̃ 3 1x2) tw22 x 2 = t(w1−w2) + t(w2−w1) + t(2w2−2w1) + t(3w2−3w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(2w1−w0−w2), o(e2)w22 = t(2w1−w0−w2). w23 = (w 1 1, x̃ 2 1x 2 2) tw23 x 2 = t(2w1−2w2) + t(w1−w2) + t(w2−w1) + t(2w2−2w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(w1−w0), o(e2)w23 = t(w1−w0). w24 = (w 1 1, x̃1x 3 2) tw24 x 2 = t(3w1−3w2) + t(2w1−2w2) + t(w1−w2) + t(w2−w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(w2−w0), o(e2)w24 = t(w2−w0). w25 = (w 1 1, x̃ 4 2) tw25 x 2 = t(4w1−4w2) + t(3w1−3w2) + t(2w1−2w2) + t(w1−w2)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(2w2−w0−w1), o(e2)w25 = t(2w2−w0−w1). following the description given in (6.12), we obtain the following eigen-decomposition for the tangent space of x1 at w11. tw11 x 1 = tw11 e 1 w + twy + [w 1 1] = tw11 e 1 w︷︸︸︷ p([ ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 ]) +tw( ∼= y︷︸︸︷ p(f) × g∈(f)) + [w 1 1] = ℓ31 ℓℓ22 + ℓ21 ℓℓ2 + ℓ1 ℓ ︸︷︷︸ t w1 1 e1w + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸︷︷︸ twy + ℓ22 ℓℓ1︸︷︷︸ o e1 (−1) w1 1 . (6.16) and o(e1) ([ℓ2,ℓℓ1],ℓℓ 2 2) = oe1 (−1)([ℓ2,ℓℓ1],ℓℓ22) = ℓℓ22 f rom (6.11) ←→ ℓ 2 2 ℓℓ1 , then ct1 (o(e 1)w11 ) = 2w2−w0−w1 doing ℓ = x0, ℓ1 = x1 and ℓ2 = x2. thus the contribution to (5.1) at each w2i ∈ e 2 for i = 1, ..., 5 lying over w11 is given by: cubo 12, 1 (2010) projective squares in p2 ... 215 fixed point for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. contribution to the numerator in (5.1) 2c t 1 (aw)︸︷︷︸ 2(3w0+w1) + c t 1 (o(e 1 ) w1 1 ) ︸︷︷︸ 2w2−w0−w1 + ct1 (o(e 2) w2 i ) contribution to the denominator in (5.1) c t 8 (tw2 i x 2) w21 = (w 1 1, x̃ 4 1) (5w0 + w1 + 2w2) + (3w1 − w0 − 2w2) 24(w2 − w1) 5(w1 − w0)(w2 − w0)(3w1 − w0 − 2w2) w22 = (w 1 1, x̃ 3 1x2) (5w0 + w1 + 2w2) + (2w1 − w0 − w2) −6(w2 − w1) 5(w1 − w0)(w2 − w0)(2w1 − w0 − w2) w23 = (w 1 1, x̃ 2 1x 2 2) (5w0 + w1 + 2w2) + (w1 − w0) 4(w2 − w1) 5(w1 − w0) 2(w2 − w0) w24 = (w 1 1, x̃1x 3 2) (5w0 + w1 + 2w2) + (w2 − w0) −6(w2 − w1) 5(w1 − w0)(w2 − w0) 2 w25 = (w 1 1, x̃ 4 2) (5w0 + w1 + 2w2) + (2w2 − w0 − w1) 24(w2 − w1) 5(w1 − w0)(w2 − w0)(2w2 − w0 − w1) making a cyclic permutation of x′is in the table above, we will obtain all the 30 contributions to (5.1) determined by the 6 fixed points w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2) ∈ w 1 (cf. (6.7) and (6.9)). so, there are altogether 66 fixed points as indicated bellow by the bold points. in fact, consider the diagram below, where we use ”•” to indicate the terminal fixed points and ”◦” to indicate the non-terminal ones. e2 ∋    • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ↓ w1 ∋ { ◦ ◦ ◦ ◦ ◦ ◦ e1 \ w1 ∋    • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •    ∈ e1 \ w1 ↓ ↓ ◦ ◦ ◦ ◦ ◦ ◦ ︸︷︷︸ w ◦ ◦ ◦ ︸︷︷︸ y\w • • • • • • ︸︷︷︸ x\y in the first line in the bottom we put the 15 = 6 (in x \ y)︸︷︷︸ terminal + 3 (in y \ w)︸︷︷︸ non-terminal + 6 (in w)︸︷︷︸ non-terminal fixed points in x. in the middle, we have 12 (respectively 18) terminal fixed points in e1 \ w1 that are mapped to the 3 (respectively 6) fixed points in y \ w (respectively w) by the the first blow-up map, and we also have 6 non-terminal fixed points in w1 that are mapped to the 6 fixed points in w (w1 is the second blow-up center and w1 ∼= w). 216 jacqueline rojas, ramón mendoza and eben da silva cubo 12, 1 (2010) at the top, we have 30 terminal fixed points in e2 that are mapped to the 6 fixed points in w1 by the the second blow-up map (this last 6 fixed points in w1 are mapped isomorphically to the 6 fixed points in w by the first blow-up map). finally using a maple script, we find one more time that there exist 105 squares whose vertices lie over 8 lines in general position in p2. acknowledgements the first and second author wish to express their gratitude to instituto nacional de matemática pura e aplicada (impa) and to departamento de matemática da universidade federal da paráıba campus 1, for providing the right environment for concluding this work. the first author was partially supported by conselho nacional de desenvolvimento cient́ıfico e tecnológico (cnpq). received: july, 2008. revised: january, 2009. references [1] avritzer, d. and vainsencher, i., hilb4p2, in proceedings of the conference at sitges, spain (1987), ed. s. xambó, springer-verlag lect. notes math. 1436 (1987), 30–59. 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[20] schubert, h., die n-dimensionale verallgemeinerung der anzahlen für die vielpunktig verührenden tangenten einer punktallgemeinen fläche m-ten grades, math. annalen 26 (1886), 52–73. [21] da silva, e.a., uma compactificação do espaço das quádruplas de pontos em p2, master dissertation, dm –ufpb, 2005. [22] witten, e., topological sigma models, comm. math. phys. 118, no. 3 (1988), 411–449. paper_bcdo_vaxjo.dvi cubo a mathematical journal vol.12, no¯ 03, (171–185). october 2010 generalized spectrograms and τ-wigner transforms boggiatto paolo, de donno giuseppe, oliaro alessandro department of mathematics, university of turin, via carlo alberto, 10, 10123 torino, italy email: paolo.boggiatto@unito.it email: giuseppe.dedonno@unito.it email: alessandro.oliaro@unito.it and bui kien cuong higher education department, hanoi pedagogical university 2, building g7-144 xuan thuy rd – hanoi, vietnam email: buikiencuong@yahoo.com abstract we consider in this paper wigner type representations w i gτ depending on a parameter τ ∈ [0, 1] as defined in [2]. we prove that the cohen class can be characterized in terms of the convolution of such w i gτ with a tempered distribution. we introduce furthermore a class of “quadratic representations” s pτ based on the τ-wigner, as an extension of the two window spectrogram (see [2]). we give basic properties of s pτ as subclasses of the general cohen class. 172 boggiatto paolo, et. al cubo 12, 3 (2010) resumen nosotros consideramos en este artículo representaciones de tipo wigner w i gτ dependiendo de um parámetro τ ∈ [0, 1] como definido en [2]. probamos que la clase cohen puede ser caracterizada en terminos de la convolución de tales w i gτ con una distribución temperada. introducimos también la clase de “representaciones cuadraticas” s pτ basado en el τ-wigner, como una extensión de dos ventanas espectrograma (ver [2]). nosotros damos propiedades básicas de s pτ como subclases de la clase cohen. key words and phrases: time-frequency representation, τ-wigner distribution, generalized spectrogram. math. subj. class.: 42b10, 47a07, 33c05. 1 introduction one of the basic problems in time-frequency analysis is the representation of the energy of a signal simultaneously with respect to time and frequency. considering for generality signals as square-integrable functions on rd , the classical mathematical tool used for this aim are sesquilinear maps q : l2(rd ) × l2(rd ) → l2(r2d ). for a given signal f , the function q( f , f )(x,ω), or for short q( f )(x,ω), plays a role corresponding to that of density of mass in classical mechanics or that of probability distribution in statistics. in contrast however to these situations, in the case of the energy of a signal the time-frequency distribution to be used is not unique. many proposals have been presented in the literature, each having advantages and drawbacks, see [5], [6], [7], [8], [9] for detailed presentations of these topics. this is due essentially to the presence of the heisenberg uncertainty principle which makes some of the natural requirements of a joint time-frequency distribution incompatible (see [11]). two of the most used time-frequency representations are the wigner distribution: w i g( f , f )(x,ω) = w i g( f )(x,ω) = ˆ rd e −2πitω f (x + t/2) f (x − t/2) dt (1.1) and the spectrogram s p g( f )(x,ω) = |vg( f )(x,ω)| 2 (1.2) where vg( f ) is the gabor transform (also known as short-time fourier transform) and is defined by vg( f )(x,ω) = ˆ rd e −2πitω f (t) g(x − t) dt (1.3) cubo 12, 3 (2010) generalized spectrograms ... 173 in dependence on the “window” g(x), which in the most generality can be supposed to be a tempered distribution. this paper is based on these two representations of which we present modifications depending on parameters. we shall analyze the properties of these new representations with respect to classical requirements such as reality of values, marginal distribution conditions, and their relations with the cohen class. this is a very general class of time-frequency representations, introduced by l. cohen, see [6], and widely studied since the 1970’s. it can be defined as the set of representations of the form c( f ) = σ∗ w i g( f ) (1.4) where, in our context, σ will be supposed to be a tempered distribution in s ′(r2d ) and will be called cohen kernel. the wide possibility of choice of the cohen kernel permits to cover most time-frequency representations. we recall next that some considerations concerning shifts of the ghost frequencies led in [2] to the introduction of the representations w i gτ( f , g)(x,ω) = ˆ rd e −2πitω f (x +τt) g(x − (1 −τ)t) dt (1.5) which are a parameterized version of the wigner representation in dependence on τ ∈ [0, 1]. it was also showed in [2] that these representations constitute the natural “quadratic form” counterparts to the τ-pseudo-differential operators which are extensions of the weyl calculus on rd ; classical references on this subject are shubin [14] and wong [15], see also [1] for generalizations concerning global hypo-ellipticity. in the present paper we analyze at first the role of (1.5) in the definition of the cohen class, showing that we can replace w i g( f ) in (1.4) by w i gτ( f ), for an arbitrary fixed τ ∈ [0, 1], getting equivalent definitions of the cohen class. in the second part of the paper, we propose a new form based on the two window spectrogram and the τ-wigner representation. the two window spectrogram was studied in [3]-[4] (called there generalized spectrogram) and is defined by s pφ,ψ( f , g)(x, w) = vφ f (x, w)vψ g(x, w). (1.6) using τ-wigner distribution, we generalize here definition (1.6) by replacing the classical wigner distribution with τ-wigner distributions. we obtain new representations that we shall call parameterized two window spectrograms and we study some of their basic properties such as positivity, support properties and boundedness in the l p context. we show that our definition is motivated by the fact that the parameterized two window spectrograms show in some basic cases reduced interference phenomena with respect to (1.6) without a loss in the quality of the time-frequency localization. finally we prove that among the variety of time174 boggiatto paolo, et. al cubo 12, 3 (2010) frequency representations they constitute a peculiarity as they do not belong to the cohen class 1. 2 τ-wigner representations and the cohen class in the definition (1.4) of the cohen class the wigner representation plays a special role and one natural question is if it can be replaced by another representation. in general this can be achieved under some additional conditions. more precisely suppose c0( f ) = σ0 ∗ w i g( f ) is a fixed representation in the cohen class; then, as long as ĉ0( f )/σ̂0 belongs to s ′(r2d) for every signal f ∈ s (rd ), we have w i g( f ) = f −1(ĉ0( f )/σ̂0). but even under this somewhat restrictive condition it does not necessarily happen that c0 → f −1(ĉ0( f )/σ̂0) is a convolution. actually only if this were the case we could write f −1(ĉ0( f )/σ̂0) = σ ′ ∗ c0( f ) for a suitable fixed σ′ ∈ s ′(r2d), and then for any generic representation in the cohen class c = σ∗ w i g, (with σ ∈ s ′(r2d )), we would obtain c( f ) = σ∗ w i g( f ) = (σ∗σ′) ∗ c0( f ). in this case, under the further condition that σ ∗ σ′ ∈ s ′(r2d ), we would have that every element in the cohen class could be expressed in terms of c0 instead of w i g. in view of these observations it is interesting, even if not surprising, that any w i gτ representation can replace the wigner representation in the expression of the cohen class. in order to prove this assertion we need the explicit expression of w i gτ as a member of the cohen class. we recall then from [2] the following result. proposition 1. the representation w i gτ( f ) belongs to the cohen class for every τ ∈ [0, 1], in particular w i gτ( f )(x,ω) = ( στ ∗ w i g( f ) ) (x,ω), (2.1) for every f ∈ s (rd ), where στ =    2d |2τ−1|d e2πi 2 2τ−1 xω for τ 6= 12 δ for τ = 12 (2.2) and δ is the dirac distribution. 1according to (1.4) we only consider signal independent kernels σ cubo 12, 3 (2010) generalized spectrograms ... 175 we have now the following proposition: proposition 2. let τ ∈ [0, 1] be fixed, then w i gτ can be used to express the entire cohen class, i.e. every representation c in the cohen class can be written in the form c( f ) = σ′ ∗ w i gτ( f ) for a suitable σ′ ∈ s ′(r2d). proof. let c( f ) = σ∗ w i g( f ) (2.3) with σ ∈ s ′(r2d), be the expression of c( f ) in the cohen class. from the previous proposition we have w i gτ( f ) = στ ∗ w i g( f ) and a straightforward computation yields: στ ∗σ1−τ = δ. we have therefore σ1−τ ∗ w i gτ( f ) = w i g( f ) and substituting in (2.3) we get formally: c( f ) = (σ∗σ1−τ) ∗ w i gτ( f ) this expression has actually a meaning if we show that σ∗σ1−τ is a well defined tempered distribution. as σ∗ σ1−τ = f −1(σ̂σ̂1−τ) and σ ∈ s ′(r2d ), this is equivalent to prove that σ̂1−τ is a multiplier of s ′(r2d). since ´ e2πi yρ d y dρ = 1 we have f σ1−τ(ξ, t) = e −πi(1−2τ)tξ (2.4) which is a c∞ function with derivatives with polynomial growth and therefore our assertion is proved. the thesis is then satisfied with σ′ = σ∗σ1−τ. we turn now our attention to the spectrograms with the aim of describing how the general context above applies to this specific case. as already pointed out in the introduction, the classical spectrogram, defined by s p g ( f )(x, w) = |vg f (x, w)| 2, (2.5) is a way to represent the energy of a signal f simultaneously with respect to time and frequency; vg f is the short-time fourier transform, or gabor transform, with window g, see 176 boggiatto paolo, et. al cubo 12, 3 (2010) for reference [13], [16], [10]. in [3], the two window spectrogram has been introduced and studied: it depends on two windows and it is defined by the skew-linear form s pφ,ψ( f , g)(x, w) = vφ( f )vψ( g)(x, w); (2.6) when φ = ψ, f = g, formula (2.6) becomes the classical spectrogram. the following relationship between wigner distribution and two window spectrogram holds (see [3]): s pφ,ψ( f , g)(x, w) = w i g(ψ̃,φ̃) ∗ w i g( f , g)(x, w), (2.7) where φ̃(s) := φ(−s) and ψ̃(s) := ψ(−s). relation (2.7), valid in suitable functional settings, for example when f , g,φ,ψ ∈ s (rd), gives us the expression of the two window spectrogram as an element of the cohen class, where σ in (1.4) is given now by w i g(ψ̃,φ̃). as proved in proposition 2, we can re-write s pφ,ψ( f , g) through the τ-wigner transform. in the special case of the two window spectrogram this can be made more explicit as showed by the following result. proposition 3. for every f , g,φ,ψ ∈ s (rd) and for every τ ∈ [0, 1], we have s pφ,ψ( f , g) = w i g1−τ(ψ̃,φ̃) ∗ w i gτ( f , g)(x, w). proof. since w i g1−τ(ψ̃,φ̃) = w i gτ(φ̃,ψ̃), (2.8) we have to prove that s pφ,ψ( f , g) = w i gτ(φ̃,ψ̃) ∗ w i gτ( f , g)(x, w). (2.9) let us observe that, by a simple change of variables, we can write w i gτ( f , g)(x − y, w −η) = ft→η ( e 2πiωt f (x − y −τt) g(x − y + (1 −τ)t) ) . since w i gτ(φ̃,ψ̃)( y,η) = ft→η ( φ̃( y +τt)ψ̃( y − (1 −τ)t) ) , by the standard properties of the fourier transform we get w i gτ(φ̃,ψ̃) ∗ w i gτ( f , g)(x, w) = ( φ̃( y +τt)ψ̃( y − (1 −τ)t), e2πiωt f (x − y −τt) g(x − y + (1 −τ)t) ) l2 (r2d y,t ) . finally, by the change of variables { y +τt = y y − (1 −τ)t = t cubo 12, 3 (2010) generalized spectrograms ... 177 in the l2-product, we have w i gτ(φ̃,ψ̃) ∗ w i gτ( f , g)(x, w) = ( φ̃(y )ψ̃(t), e2πiω(y −t) f (x − y ) g(x − t) ) l2(r2d y ,t ) . this shows that w i gτ(φ̃,ψ̃) ∗ w i gτ( f , g)(x, w) is independent of τ ∈ [0, 1], and so for every τ ∈ [0, 1], w i gτ(φ̃,ψ̃) ∗ w i gτ( f , g)(x, w) = w i g(φ̃,ψ̃) ∗ w i g( f , g)(x, w). from (2.8), (2.7) and this last identity, we get (2.9). 3 the parameterized two window spectrogram: definition and motivations so far we have been concerned with relationships between τ−wigner and spectrograms representations within the frame of the cohen class. in this section we want to consider relationships between these two types of representations under another point of view which will bring us to the definition of a further representation. we start with some preliminary remarks. it is well-known that the wigner transform can be expressed in function of the spectrogram by the following equality w i g( f , g)(x, w) = 2d e4πixwvg̃ f (2x, 2w), (3.1) and viceversa we have vg f (x, w) = 2 −d e −πixw w i g( f , g̃)( x 2 , w 2 ). (3.2) from (2.6) it is then clear that we can then rewrite the two window spectrogram as s pφ,ψ( f , g)(x, w) = 4 −d w i g( f ,φ̃)( x 2 , w 2 )w i g( g,ψ̃)( x 2 , w 2 ). (3.3) in view of this equality it is natural to introduce the following generalization of the spectrogram: definiton 4. let τ1,τ2 ∈ [0, 1] be two parameters, the parameterized two window spectrogram, denoted s p (τ1 ,τ2) φ,ψ ( f , g), is defined by s p (τ1 ,τ2 ) φ,ψ ( f , g)(x, w) = 4 −d w i gτ1 ( f ,φ̃)( x 2 , w 2 )w i gτ2 ( g,ψ̃)( x 2 , w 2 ), (3.4) where φ,ψ are window functions and f , g are signals in suitable functional or distributional spaces. 178 boggiatto paolo, et. al cubo 12, 3 (2010) remark 5. when τ1 = τ2 = 1/2, the parameterized two window spectrogram becomes the two window spectrogram s p (τ1 ,τ2 ) φ,ψ ( f , g)(x, w) = s pφ,ψ( f , g)(x, w). the introduction of this new family of parameterized representations is not due to pure search of mathematical generality. actually, as we describe next, the form s p (τ1 ,τ2) φ,ψ ( f , g) shows an interesting behavior for what concerns localization properties and reduction of interference disturbances in particular in the cases where frequencies occur in time intervals very close to one another. to this aim let us consider a signal f containing the frequency ω = 2 in the time interval [−4, 0] and the frequency ω = 3 in the time interval [0, 4]; we fix the window functions φ = χ[−10,10] and ψ = χ[− 110 , 1 10 ] , where χ[a,b] denotes the characteristic function of the interval [a, b] and we compare the pictures of the parameterized two window spectrograms s p (τ1 ,τ2 ) φ,ψ ( f , g) for different values of τ1 and τ2. the two window spectrogram s pφ,ψ( f , f ), corresponding to case τ1 = τ2 = 1 2 , is visualized in figure 1: figure 1: s p ( 12 , 1 2 ) φ,ψ ( f , f ) = s pφ,ψ( f , f ) as we can see, although the localization is good both in time and in frequency, the picture presents disturbing interference patterns. the explanation of this fact is the following. the gabor transform vφ f with a large window φ gives better information regarding frequencies, and the gabor transform vψ f with a narrow window ψ gives better information concerning time. when we consider the two window spectrogram s pφ,ψ( f , g) = vφ f vψ g we take a product of one gabor transform well localized in time and another one well localized in frequency, and so the reciprocal cut-off effect yields good localization both in time and cubo 12, 3 (2010) generalized spectrograms ... 179 frequency, see [4] for a detailed discussion on this subject. it could seem therefore that we have overcome the heisenberg uncertainty principle but of course it is not so. actually what is obtained in good localization, is “paid” terms of interference. more precisely, the fact that each gabor transform is well localized in one variable and, consequently, badly localized in the other, implies that the supports of the two gabor transforms also intersects in places where no frequency is present. this is what is observed in figure 1 and clearly represents a considerable drawback in the use of the classical two window spectrogram. let us consider now the parameterized two window spectrogram, with the same windows and signal as above. in picture 2 we have a representation of s p (0.3,0.3) φ,ψ ( f , f ) and s p (0.2,0.2) φ,ψ ( f , f ) (for simplicity we take here τ1 = τ2). s p (0.3,0.3) φ,ψ ( f , f ) s p (0.2,0.2) φ,ψ ( f , f ) figure 2: parameterized two window spectrogram for different values of τ1,τ2. as we observe from the pictures, although the windows φ and ψ are kept fixed, the interference between the two frequencies is considerably reduced when the parameter τ in s p (τ,τ) φ,ψ ( f , f ) becomes small, keeping on the other hand the good level of localization. incidentally we also remark that the improvement of frequency localization is only apparent as it is essentially the consequence of an effect of vertical contraction and horizontal dilation compensated in the picture by a relabeling of the axis. 4 properties of the parameterized two window spectrogram in this section we analyze some properties of the representation s p (τ1 ,τ2 ) φ,ψ ( f , g) with τ1,τ2 ∈ [0, 1]. more precisely we consider positivity, l p−boundedness and support property, we con180 boggiatto paolo, et. al cubo 12, 3 (2010) clude then our investigations by showing that the parameterized two window spectrogram does not belong to the cohen class. for what positivity is concerned we limit ourself to the following basic fact, we have s p (τ) φ ( f )(x, w) := s p (τ,τ) φ,φ ( f , f )(x, w) = 4 −d |w i gτ( f ,φ̃)(x, w)| 2 ≥ 0. and therefore the following property holds: proposition 6. for τ1 = τ2, f = g and φ = ψ the parameterized two window spectrogram is a positive time-frequency representation. we consider next the parameterized two window spectrogram in the context of the l p spaces. for this purpose we shall need the following proposition, which is proved in [2]. proposition 7. let us fix q and p satisfying q ≥ 2 and q′ ≤ p ≤ q, ( 1 q + 1 q′ = 1). then: i) for each τ ∈ (0, 1), w i gτ : l p′ (r) × l p(r) → lq(r2d ) is continuous, in particular: ‖w i gτ( g, f )‖lq ≤ 1 |1 −τ| d( 1 p − 1 q ) 1 |τ| d(1− 1 p − 1 q ) ‖g‖ l p ′ ‖f ‖l p . (4.1) ii) for τ = 0, w i g0( g, f )(x, w) = r( g, f )(x, w) and w i g0 : l q(r) × lq ′ (r) → lq(r2d) is continuous, in particular ‖w i g0( g, f )‖lq ≤ ‖g‖lq′ ‖f ‖lq . (4.2) iii) for τ = 1, w i g1( g, f )(x, w) = r( g, f )(x, w) and w i g1 : l q′ (r) × lq(r) → lq(r2d) is continuous, in particular ‖w i g1( g, f )‖lq ≤ ‖g‖lq ‖f ‖lq′ . (4.3) furthermore for p, q in the remaining cases the τ-wigner transform is not bounded as sesquilinear map: l p ′ (r) × l p(r) → lq(r2d). the l p behavior of the parameterized two window spectrogram is specified by the following proposition. theorem 8. let q ≥ 1, q j ≥ 2, p j ≥ 1, ( j = 1, 2) satisfy the following conditions: 1 q1 + 1 q2 = 1 q ; q ′ j ≤ p j ≤ q j , ( j = 1, 2), where 1 q j + 1 q′ j = 1. then i) the parameterized two window spectrogram s p(τ1 ,τ2 ) : l p ′ 1 × l p1 × l p ′ 2 × l p2 → lq is continuous (0 < τ1,τ2 < 1), in particular ‖s p (τ1 ,τ2) φ,ψ ( f , g)‖lq ≤ c‖f ‖ l p ′ 1 ‖φ‖l p1 ‖g‖ l p ′ 2 ‖ψ‖l p2 , (4.4) where c = c1c2 with c j = 1 |1−τ| d( 1p j − 1 q j ) 1 |τ| d(1− 1p j − 1 q j ) , j = 1, 2. cubo 12, 3 (2010) generalized spectrograms ... 181 ii) when τ1 = 1,τ2 = 0 then s p (1,0) : lq1 × lq ′ 1 × lq ′ 2 × lq2 → lq is continuous, in particular ‖s p (1,0) φ,ψ ( f , g)‖lq ≤ ‖f ‖lq1 ‖φ‖ l q ′ 1 ‖g‖ l q ′ 2 ‖ψ‖lq2 . (4.5) iii) when τ1 = 0,τ2 = 1 then s p (0,1) : lq ′ 1 × lq1 × lq2 × lq ′ 2 → lq is continuous, in particular ‖s p (0,1) φ,ψ ( f , g)‖lq ≤ ‖f ‖ l q ′ 1 ‖φ‖lq1 ‖g‖lq2 ‖ψ‖ l q ′ 2 . (4.6) iv) when τ1 = τ2 = 1 then s p (1,1) : lq1 × lq ′ 1 × lq2 × lq ′ 2 → lq is continuous, in particular ‖s p (1,1) φ,ψ ( f , g)‖lq ≤ ‖f ‖lq1 ‖φ‖ l q ′ 1 ‖g‖lq2 ‖ψ‖ l q ′ 2 . (4.7) v) when τ1 = τ2 = 0 then s p (0,0) : lq ′ 1 × lq1 × lq ′ 2 × lq2 → lq is continuous, in particular ‖s p (0,0) φ,ψ ( f , g)‖lq ≤ ‖f ‖ l q ′ 1 ‖φ‖lq1 ‖g‖ l q ′ 2 ‖ψ‖lq2 . (4.8) proof. it is an easy consequence of proposition 7 and the generalized hölder’s inequality ‖f g‖lq ≤ ‖f ‖lq1 ‖f ‖lq2 for 1 q1 + 1 q2 = 1 q , q1 ≥ q, we recall now some notations. we indicate with h(supp f ) the convex hull of supp f and with πx,πw the orthogonal projections on the first and the second factor in r d x ×r d w respectively. properties on the support of time-frequency representations is a widely studied subject because too large projections πx and πw of the support of a representation in comparison with the supports of the signal itself and its fourier transform respectively would indicate a “spreading” of the energy that is seen as disturbance in the applications, see for instance [12]. we have the following basic results. lemma 9. let w i gτ( f , g) be the τ-wigner representation defined by (1.5); then πx(suppw i gτ( f , g)) ⊂ h(supp f + supp g). (4.9) and πw(suppw i gτ( f , g)) ⊂ h(supp f̂ + supp ĝ). (4.10) proof. suppose that w i gτ( f , g)(x,ω) 6= 0, then there exists t ∈ r d such that f ( y1) 6= 0 and g( y2) 6= 0 with y1 = x +τt and y2 = x − (1 −τ)t. on the other hand x = λy1 +µy2 with λ = 1 −τ and µ = τ, i.e. x can be written as convex linear combination of y1 and y2. we have therefore 182 boggiatto paolo, et. al cubo 12, 3 (2010) that x belongs to the segment [ y1, y2] and (4.9) follows then immediately. to obtain (4.10) we just need to recall that w i gτ( f , g)(x, w) = w i gτ( f̂ , ĝ)(w,−x). and repeat the argument above with x replaced by ω. from (4.9), (4.10), and the equality supp( f g) = supp f ∩ supp g, we obtain the “support” property of the parameterized two window spectrogram. proposition 10. the support of the parameterized two window spectrogram satisfies the following properties: πx(supps p (τ1 ,τ2 ) φ,ψ ( f , g)) ⊂ h(supp f + suppφ̃) ∩ h(supp g + suppψ̃) (4.11) and πw(supps p (τ1 ,τ2 ) φ,ψ ( f , g)) ⊂ h(supp f̂ + supp ˆ̃φ) ∩ h(supp ĝ + supp ˆ̃ψ). (4.12) remark 11. the meaning of the proposition 10 becomes even more evident if we consider the case where f = g is a signal and we suppose that one window is well localized in time and the other one in frequency. assume for example that supp φ ⊂ bδ and supp ψ̂ ⊂ bδ, with bδ ball of radius δ > 0, then proposition 10 implies that supp s p (τ1 ,τ2 ) φ,ψ ( f , f ) ⊂ h(supp f + b δ) × h(supp f̂ + bδ), i.e. we have good localization both in time and in frequency, having reduced the spread of the energy to a ball of radius δ with respect to each variable. finally we prove that the parameterized two window spectrogram, in general, does not belong to the cohen class. let us consider for simplicity the case τ1 = τ2 := τ in definition 4, with τ 6= 12 ( actually for τ = 1 2 , the representation s p ( 12 , 1 2 ) φ,ψ ( f , g) belongs to the cohen class, since, as proved in [3], it coincides with s pφ,ψ( f , g) ). we denote for shortness s p τ φ,ψ( f , g) := s p (τ,τ) φ,ψ ( f , g); the following proposition holds. proposition 12. for τ 6= 12 there does not exist a tempered distribution σ = στ,φ,ψ ∈ s ′(r2d) such that s p τ φ,ψ = σ∗ w i g, (4.13) i.e. s pτ φ,ψ( f , g) = σ∗ w i g( f , g) for every f , g ∈ s (r d). cubo 12, 3 (2010) generalized spectrograms ... 183 proof. by definition 4 and simple changes of variables we have: s p τ φ,ψ( f , g) = 4 −d ˆ e −2πit ω2 f ( x 2 +τt ) φ̃ ( x 2 − (1 −τ)t ) dt ˆ e 2πit ω2 g ( x 2 +τt ) ψ̃ ( x 2 − (1 −τ)t ) dt = ˆ e −2πisω f (2τs)φ ( 2(1 −τ)s − x 2τ ) ds ˆ e −2πisω g(−2τs)ψ ( −2(1 −τ)s − x 2τ ) ds. by standard properties of the fourier transform we can write the inverse fourier transform of s pτ φ,ψ( f , g)(x,ω) in the following way: f −1 x→t ω→ξ ( s p τ φ,ψ( f , g)(x,ω) ) = = f −1 x→t [ f (2τξ)φ ( 2(1 −τ)ξ− x 2τ )] ∗ f −1 x→t [ g(−2τξ)ψ ( −2(1 −τ)ξ− x 2τ )] = (2τ)2d [ e 2πi(4τ(1−τ))tξ f (2τξ)φ̂(2τt) ] ∗ [ e −2πi(4τ(1−τ))tξ g(−2τξ) ψ̂(2τt) ] , where the convolution is performed in both the variables (t,ξ). finally, writing explicitly the convolution, we obtain f −1 x→t ω→ξ ( s p τ φ,ψ( f , g)(x,ω) ) = (2τ)2d e2πi(4τ(1−τ))tξ ˆ e −2πi(4τ(1−τ))tx f (2τ(ξ− x)) g(−2τx) dx ˆ e −2πi(4τ(1−τ))ξs φ̂(2τ(t − s))ψ̂(2τs) ds. (4.14) we observe that, by the definition of the wigner transform, f −1 x→t ω→ξ ( w i g( f , g) ) = f −1 x→t ω→ξ [ fs→ω ( f ( x + s 2 ) g ( x − s 2 ))] = ˆ e 2πixt f ( x + ξ 2 ) g ( x − ξ 2 ) dx. (4.15) now let us suppose that (4.13) holds for some tempered distribution σ; by taking the inverse fourier transform and using (4.14) and (4.15), the following should be verified for every f , g ∈ s (rd ): (2τ)2d e2πi(4τ(1−τ))tξ ˆ e −2πi(4τ(1−τ))tx f (2τ(ξ− x)) g(−2τx) dx ˆ e −2πi(4τ(1−τ))ξs φ̂(2τ(t − s))ψ̂(2τs) ds = σ̌(t,ξ) ˆ e 2πixt f ( x + ξ 2 ) g ( x − ξ 2 ) dx, (4.16) 184 boggiatto paolo, et. al cubo 12, 3 (2010) where σ̌(t,ξ) is the inverse fourier transform of σ. in particular, (4.16) should hold for f and g of the following type: f (s) = e−πλs 2 , g(s) = e−πµs 2 , for every λ,µ > 0. in this case we can compute explicitly the integrals involving f and g in (4.16) and we have: ˆ e −2πi(4τ(1−τ))tx e −πλ(2τξ−2τx)2 e −πµ(−2τx)2 dx = = e −4π λµ λ+µ τ2ξ2 ˆ e −2πi(4τ(1−τ))tx e −π ( 2(λ+µ)1/2τx− 2λτ (λ+µ)1/2 ξ )2 dx = (2τ √ λ+µ)−d e −4π λµ λ+µ τ2ξ2 e −2πi λ λ+µ 4τ(1−τ)tξ ˆ e −2πi 2(1−τ) (λ+µ)1/2 t y e −πy2 d y = (2τ √ λ+µ)−d e −2πi λ λ+µ 4τ(1−τ)tξ e −4π λµ λ+µ τ2ξ2 e −π 4(1−τ)2 λ+µ t2 . (4.17) similarly we obtain that ˆ e 2πixt f ( x + ξ 2 ) g ( x − ξ 2 ) dx = ( √ λ+µ)−d e −2πi λ λ+µ tξ e −π λµ λ+µ ξ2 e −π 1 λ+µ t2 . (4.18) now, replacing (4.17) and (4.18) in (4.16) we have for σ̌(t,ξ) the following expression σ̌(t,ξ) = (2τ)d e 2πi(4τ(1−τ))tξ−πitξ−2πi λ λ+µ (4τ(1−τ)−1)tξ e −π 4λµτ2−λµ λ+µ ξ2 e −π 4(1−τ)2−1 λ+µ t2 ˆ e −2πi(4τ(1−τ))ξs φ̂ ( 2τ(t − s) ) ψ̂(2τs) ds. (4.19) for τ 6= 12 we deduce then that σ̌(t,ξ) necessarily depends on the two parameters λ and µ, and this is impossible since σ in (4.13) is independent of f and g. remark 13. we also observe that in the case τ = 1/2 all terms in (4.19) involving the parameters λ and µ cancel, making σ independent of them, and confirming, as expected, that in this case the representation is in the cohen class. references [1] boggiatto, p., buzano, e. and rodino, l., hypoellipticity and spectral theory, akademie verlag, berlin, 1996. mathematical research, vol. 92. 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[16] wong, m.w., wavelet transform and localization operators, birkhäuser-verlag, basel, 2002. kdvbseg.dvi cubo a mathematical journal vol.12, no¯ 01, (41–58). march 2010 korteweg-de vries-burgers equation on a segment elena i. kaikina instituto de matemáticas, unam campus morelia, ap 61-3 (xangari), morelia cp 58089, michoacán, mexico email : ekaikina@matmor.unam.mx and leonardo guardado-zavala, hector f. ruiz-paredes, s. juarez zirate posgrado electrica, instituto tecnológico de morelia, cp 58120, morelia, michoacán, méxico emails: guardado@ps.itm.mx hruiz@sirio.tsemor.mx sjzirate@matmor.unam.mx abstract we study the following initial-boundary value problem for the korteweg-de vries-burgers equation on the interval (0, 1)    ut + uux − uxx + uxxx = 0, t > 0, x ∈ (0, 1) u(x, 0) = u0(x), x ∈ (0, 1) u(0, t) = u(1, t) = ux(1, t) = 0, t > 0. (0.1) we prove that if the initial data u0 ∈ l 2, then there exists a unique solution u ∈ c ( [0, ∞) ; l2 ) ∪ c ( (0, ∞) ; h1 ) of the initial-boundary value problem (0.1). we also obtain the large time asymptotic of solution uniformly with respect to x ∈ (0, 1) as t → ∞. 42 e.i. kaikina et. al. cubo 12, 1 (2010) resumen estudiamos el siguiente problema de valor inicial en la frontera para la ecuación de kortewegde vries-burgers en el intervalo (0, 1)    ut + uux − uxx + uxxx = 0, t > 0, x ∈ (0, 1) u(x, 0) = u0(x), x ∈ (0, 1) u(0, t) = u(1, t) = ux(1, t) = 0, t > 0. (0.1) provamos que si el dato inicial u0 ∈ l 2, entonces existe una única solución u ∈ c ( [0, ∞) ; l2 ) ∪ c ( (0, ∞) ; h1 ) del problema de valor inicial en la frontera (0.1). también obtenemos comportamiento asintótico de la solución con respecto a x ∈ (0, 1) cuando t → ∞. key words and phrases: dissipative nonlinear evolution equation, large time asymptotics, korteweg-de vries-burgers equation. math. subj. class.: 35q35. 1 introduction we study the global existence and large time asymptotic behavior of solutions to the initialboundary value problem for the korteweg-de vries-burgers equation in the interval (0, 1)    ut + uux − uxx + uxxx = 0, t > 0, x ∈ (0, 1) u(x, 0) = u0(x), x ∈ (0, 1) u(0, t) = u(1, t) = ux(1, t) = 0, t > 0. (1.1) the korteweg-de vries-burgers equation (1.1) is a simple universal model equation which appears as the first approximation in the description of the dispersive-dissipative nonlinear waves, and has many applications in various fields of physics, biology and engineering. in the case of the cauchy problem some estimates for the time decay rates of solutions to the korteweg-de vriesburgers type equations were found in papers [3], [4], [5] and the large time asymptotics of solutions was obtained in [6], [11], [12]. in the case of the boundary value problem on half-line the large time asymptotics of solutions were studied in papers [2], [7], [8],[9], [10]. as far as we know large time asymptotic behavior for solutions of the initial-boundary value problem for the korteweg-de vries-burgers equation (1.1) on the interval was not studied previously. in this paper we consider (1.1) in the case of the initial data belonging to l2. we note here that we do not assume the smallness condition on the data. in the case of large initial data it is more difficult than that of small data to obtain exact representation of large time asymptotics of solutions and there are a few results (see, e.g. [13]). another difficulty in the study of the boundary value problem for the korteweg-de vries-burgers equation (1.1) is that the linear operator −∂2x + ∂3x is not self-adjoint cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 43 and we can not apply the fourier method when we take the boundary value into account. to avoid this difficulty we apply the laplace transformation with respect to space variable to derive the green function of the resulting equation. for obtaining lp -estimates of the green function we use the method of papers [8] and [9]. to state the results of the present paper precisely we give some notations. let us denote h1 = { ϕ ∈ l2 (0, 1) ; ‖ϕ‖ h1 = ‖ϕ‖ l2 + ‖ϕx‖l2 < ∞ } . we introduce the function λ(x) ∈ l∞(0, 1) λ(x) = −△̃(−ξ0, 1 − x) △̃′(−ξ0, 1) , △̃(ξ, y) = 3∑ j=1 e−φj yφ′j (ξ) , △̃′(ξ, 1) = 3∑ j=1 e−φj y ( φ ′′ j (ξ) − ( φ ′ j (ξ) )2) , where △̃(ξ, y) = 3∑ j=1 e−φj yφ′j (ξ) , △̃′(ξ, y) = 3∑ j=1 e−φj y ( φ′′j (ξ) − ( φ′j (ξ) )2) . here φl(ξ) are the roots of the characteristic equation −p2 + p3 + ξ = 0, such that re φl(ξ) > 0, l = 1, 2, and re φ3(ξ) < 0, for all ξ ∈ d0 = { ξ ∈ c : re ξ ≥ 0, ξ /∈ [ 0, 4 27 ]} , and ξ0 ∈ c , reξ0 > 0 is the first root of the equation 3∑ j=1 e−φj φ ′ j (−ξ) = 0. (1.2) by the same letter c we denote different positive constants if it does not make confusion. we state the main result of this paper. theorem 1.1. suppose that the initial data u0 ∈ l2. then there exists a unique solution of (1.1) u ∈ c ( [0, ∞) ; l2 ) ∪ c ( (0, ∞) ; h1 ) such that the solution has the following asymptotics u(x, t) = aλ(x)e−ξ0t + o ( e−(ξ0+δ)t ) for t → ∞ uniformly with respect to x ∈ (0, 1) , where δ > 0 is a constant satisfying the condition such that |ξ0| + δ < |ξ1| , where ξ1 is the second root of (1.2), the constant a is defined by 44 e.i. kaikina et. al. cubo 12, 1 (2010) a = ∫ 1 0 u0(y)△̃(−ξ0, y)dy + ∫ ∞ 0 eξ0τ dτ ∫ 1 0 u(y, τ )uy(y, τ )△̃(−ξ0, y)dy. remark 1.1. by virtue of the numerical computations via program maple we have ξ0 ≈ 70 and ξ1 ≈ 200. we organize our paper as follows. in section 2 we solve the linear initial-boundary value problem corresponding to (1.1). in section 3 we prove the local existence of solutions to (1.1). section 4 is devoted to the proof of global existence of solutions to (1.1) for the case of small initial data. we prove theorem 1.1 in section 5 by using the time decay estimates of solutions obtained in section 4. 2 linear problem we consider the following linear initial-boundary value problem    ut − uxx + uxxx = f, t > 0, x ∈ (0, 1) , u(x, 0) = u0(x), x ∈ (0, 1) , u(0, t) = u(1, t) = ux(1, t) = 0, t > 0. (2.1) we define for x ∈ (0, 1) △̃(ξ, x) = 3∑ j=1 e−φj xφ′j (ξ) , (2.2) where φl(ξ) are the roots of the characteristic equation −p2 + p3 + ξ = 0, such that re φl(ξ) > 0, l = 1, 2, and re φ3(ξ) < 0, for all ξ ∈ d0 = { ξ ∈ c : re ξ ≥ 0, ξ /∈ [ 0, 4 27 ]} , denote by gf = θ(x) (∫ x 0 f (y)f1(x, y, t)dy + ∫ 1 x f (y)f2(x, y, t)dy ) , where θ(x) = { 1, x ∈ [0, 1] 0, x /∈ [0, 1] , f1(x, y, t) = − 1 2πi ∫ i∞ −i∞ eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ and f2(x, y, t) = − 1 2πi ∫ i∞ −i∞ dξeξt △̃(ξ, 1 − x)△̃(ξ, y) − △̃(ξ, y − x)△̃(ξ, 1) △̃(ξ, 1) . cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 45 proposition 2.1. let the initial data u0 ∈ l2 and f ∈ l2 then a solution u of (2.1) has the following representation u(x, t) = gu0 + ∫ t 0 g(t − τ )f (τ )dτ . proof. to derive an integral representation for solutions of the problem (2.1) we suppose that there exists a solution u(x, t) of problem (2.1), which is continued by zero outside of x < 0, x > 1 u(x, t) = 0 for all x /∈ [0, 1] , ∂jxu(0, t) = lim x→0+ ∂jxu(x, t), j = 0, 1, 2 ∂jxu(1, t) = lim x→1− ∂jxu(x, t), j = 0, 1, 2. also we suppose that f = 0. we denote operator p [φ(p, t)] = − 1 2πi ∫ i∞ −i∞ e(q−p) − 1 q − p φ(q, t)dq. we have for the laplace transform l {ku} = p  k(p)û + p2 2∑ j=1 ∂j−1x u(0, t) − e−p∂j−1x u(a, t) pj − p3 3∑ j=1 ∂j−1x u(0, t) − e−p∂j−1x u(a, t) pj   . since l {u} is analytic for all p ∈ c we have l {u} = û(p, t) = p [û(p, t)] . (2.3) applying the laplace transform with respect to x to problem (2.1) we obtain    p [ût + k(p)û(p, t) + b1(p, t) − e−pb2(p, t)] = 0, t > 0, x > 0, û(p, 0) = û0(p), u(0, t) = u(1, t) = ux(0, t) = 0, t > 0, (2.4) where b1(p, t) = p 2 2∑ j=1 ∂j−1x u(0, t) pj − p3 3∑ j=1 ∂j−1x u(0, t) pj , (2.5) b2(p, t) = p 2 2∑ j=1 ∂j−1x u(a, t) pj − p3 2∑ j=1 ∂j−1x u(a, t) pj . we rewrite (2.4) in the form ût + k(p)û(p, t) + b1(p, t) − e−pb2(p, t) = φ(p, t), (2.6) 46 e.i. kaikina et. al. cubo 12, 1 (2010) where some function φ(p, t) is analytic for all p ∈ c , |φ(p, t)| ≤ c(t) 1 + |e −p| |p| , |p| > 1 (2.7) and p [φ(p, t)] = 0. (2.8) now we prove that under this conditions φ(p, t) ≡ 0.to find φ(p, t) we introduce functions ω1(z, t) = 1 2πi ∫ i∞ −i∞ 1 q − z φ(q, t)dq, ω2(z, t) = e−z 2πi ∫ i∞ −i∞ eq q − z φ(q, t)dq. since φ(p, t) satisfies holder condition the functions ω1(z, ξ), ω2(z, ξ) are analytic in re z 6= 0. denote by ω+1,2(p, t) = lim z→p,re z<0 ω1,2(z, t), and ω−1,2(p, t) = lim z→p,re z>0 ω1,2(z, t). for re p = 0.since function φ(p, t) is analytic for all p ∈ c from estimate (2.7) we have ω−2 (p, ξ) = ω + 1 (p, ξ) = 0. another hand by sokhotsky-plemeli formula we get ω−2 (p, ξ) = v p e−p 2πi ∫ i∞ −i∞ 1 q − p φ(p, t)dq − 1 2 φ(p, t) ω+1 (p, ξ) = v p 1 2πi ∫ i∞ −i∞ 1 q − p φ(p, t)dq + 1 2 φ(p, t). and therefore for re p = 0 ω−2 (p, ξ) − ω + 1 (p, ξ) = p [φ(p, t)] − φ(p, t) = 0. thus for re p = 0 φ(p, t) = p [φ(p, t)] = 0 and therefore due to analycity φ(p, t) ≡ 0 for all p ∈ c . applying the laplace transformation to problem (2.6) with respect to time variable we write lt {û(p, t)} = ̂̂u(p, ξ) as ̂̂u(p, ξ) = 1 k(p) + ξ ( û0(p) − b̂1(p, ξ) + e−pb̂2(p, ξ) ) (2.9) cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 47 for p ∈ c, k(p) = −p2 + p3. here functions b̂1(p, ξ), b̂2(p, ξ) are the laplace transforms of b1(p, t), b2(p, t) with respect to time. in order to get the integral formula for solution , we need to know the functions b̂1(p, ξ), b̂2(p, ξ) . we will find its using the analytic condition of function ̂̂u for p ∈ c and re ξ > 0 ̂̂u(p, ξ) = p { ̂̂u(p, ξ) } . (2.10) via (2.9) we rewrite (2.10) in the form 1 k(p) + ξ ( û0(p) − b̂1(p, ξ) + e−pb̂2(p, ξ) ) (2.11) = 1 2πi ∫ i∞ −i∞ e(q−p) − 1 q − p 1 k(q) + ξ ( û0(q) − b̂1(q, ξ) + e−pb̂2(q, ξ) ) . let φl(ξ) are the roots of the characteristic equation −p2 + p3 + ξ = 0, such that re φl(ξ) > 0, l = 1, 2, and re φ3(ξ) < 0, for all ξ ∈ d0 = { ξ ∈ c : re ξ ≥ 0, ξ /∈ [ 0, 4 27 ]} . note that the functions φl(ξ) are analytic in the domain { ξ ∈ c : ξ /∈ ( −∞, 4 27 ]} . we represent p2 = ξ 1−p for |p| < 1 and p3 = −ξ 1− 1 p for |p| > 1, hence we get the asymptotics φ1(ξ) = { √ ξ + o(|ξ|), ξ → 0, im ξ > 0, 1 + o(|ξ|), ξ → 0, im ξ < 0, ei π 3 3 √ ξ + o (1) , |ξ| → ∞, (2.12) φ2(ξ) = { 1 + o(|ξ|), ξ → 0, im ξ > 0, √ ξ + o(|ξ|), ξ → 0, im ξ < 0, e−i π 3 3 √ ξ + o (1) , |ξ| → ∞, (2.13) and φ3(ξ) = { − √ ξ + o(|ξ|), |ξ| → 0, − 3 √ ξ + o (1) , |ξ| → ∞, (2.14) for all ξ ∈ c : ξ /∈ ( −∞, 4 27 ] (by √ ξ and 3 √ ξ we denote the main value of the analytic function, i.e.√ 1 = 3 √ 1 = 1.) by cauchy theorem we have for all p ∈ c 1 2πi ∫ i∞ −i∞ e(q−p) − 1 q − p 1 k(q) + ξ ( û0(q) − b̂1(q, ξ) + e−pb̂2(q, ξ) ) = 1 k(p) + ξ ( û0(p) − b̂1(p, ξ) + e−pb̂2(p, ξ) ) + e(φ3(ξ)−p) φ3 − p φ ′ 3(ξ) ( −û0(φ3) + b̂1(φ3, ξ) − e−φ3 b̂2(φ3, ξ) ) − 2∑ j=1 1 φj − p φ ′ j (ξ) ( û0(φj ) − b̂1(φj , ξ) + e−φj b̂2(φj , ξ) ) . 48 e.i. kaikina et. al. cubo 12, 1 (2010) using (2.11) we get    b̂2(φ3, ξ) = e φ 3 ( −û0(φ3) + b̂1(φ3, ξ) ) b̂1(φj , ξ) = û0(φj ) + e −φj b̂2(φj , ξ), j = 1, 2. (2.15) so we need to put in the initial-boundary value problem one boundary data in the point x = 0 and two boundary data in the point x = 1. let, for example, u(0, t) = ux(1, t) = u(1, t) = 0. thus from system (2.16) we get { −∂xxû(a, ξ) = eφ3(ξ)a (−û0(φ3) + ∂xû(0, ξ)(1 − φ3) − ∂xxû(0, ξ)) ∂xû(0, ξ)(1 − φj ) − ∂xxû(0, ξ) = û0(φj ) − e−φj a∂xxû(a, ξ), j = 1, 2, (2.16) which is equal to   e−φ1(ξ) 1 − φ1 −1 e−φ2(ξ) 1 − φ2 −1 e−φ3(ξ) 1 − φ3 −1     ∂xxû(1, ξ) ∂xû(0, ξ) ∂xxû(0, ξ)   =   û0(φ1) û0(φ2) û0(φ3)   . (2.17) denote the determinant of this system by △(φ1,φ2,φ3), then it has a form △(φ1,φ2,φ3) = ∣∣∣∣∣∣∣∣ 1 1 1 e−φ1(ξ) e−φ2(ξ) eφ3(ξ) φ1(ξ)e φ 1 (ξ) φ2(ξ)e φ 2 (ξ) φ3(ξ)e φ 3 (ξ) ∣∣∣∣∣∣∣∣ = e−φ1 (φ2 − φ3) + e−φ2 (φ3 − φ1) + e−φ3 (φ1 − φ2). in the domain ξ ∈ d0. since ∑3 j=1 φj = 1 and φ ′ 1(ξ) = − 1 (φ1 − φ2)(φ1 − φ3) ; φ ′ 2(ξ) = − 1 (φ2 − φ1)(φ2 − φ3) ; φ′3(ξ) = − 1 (φ3 − φ1)(φ3 − φ2) (2.18) we can rewrite △(φ1,φ2,φ3) as △(φ1,φ2,φ3) = v (ξ) 3∑ j=1 e−φj (ξ)φ′j (ξ). (2.19) where v (ξ) = (φ1 − φ2)(φ2 − φ3)(φ3 − φ1). since v (ξ) 6= 0 and re φl(ξ) > 0, l = 1, 2, re φ3(ξ) < 0 in domain ξ ∈ d0 we easily get for |ξ| ≫ 1, ξ ∈ d0 △(φ1,φ2,φ3) 6= 0. cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 49 by numeric computations we can check that △(φ1,φ2,φ3) 6= 0 for all |ξ| ≤ c, ξ ∈ d0 = { ξ ∈ c : re ξ ≥ 0, ξ /∈ [ 0, 4 27 ]} . therefore there exists a unique solution of the system (2.17)   ∂xxû(1, ξ) ∂xû(0, ξ) ∂xxû(0, ξ)   = ∫ 1 0 dyu0(y)   e−φ1(ξ) 1 − φ1 −1 e−φ2(ξ) 1 − φ2 −1 e−φ3(ξ) 1 − φ3 −1   −1   e−φ1y e−φ2y e−φ3y   . (2.20) since u(0, t) = ux(1, t) = u(1, t) = 0 by (2.9) and (2.5) we have ̂̂u(p, ξ) = 1 k(p) + ξ (û0(p) + (p − 1)ûx(0, ξ)) + ûxx(0, ξ) − e−paûxx(a, ξ)). (2.21) taking inverse laplace transform with respect to variable p we get û(x, ξ) = θ(x) ∫ 1 0 dyu0(y) 1 2πi ∫ +i∞ −i∞ dpepx 1 k(p) + ξ ×(e−py + (p − 1)ûx(0, ξ)) + ûxx(0, ξ) − e−pûxx(a, ξ)). by the method of residues we see that for ξ ∈ d0 û(x, ξ) = θ(x) ∫ x 0 dyu0(y)f̂1(x, y, ξ) + ∫ 1 x dyu0(y)f̂2(x, y, ξ), (2.22) where f1(x, y, ξ) = −φ′3eφ3(ξ)x ( e−yφ3(ξ) + (φ3 − 1)ûx(0, ξ) + ûxx(0, ξ) ) (2.23) − 2∑ j=1 φ ′ j (ξ)e −φj (1−x)ûxx(1, ξ) and f2(x, y, ξ) = −φ′3eφ3(ξ)x ((φ3 − 1)ûx(0, ξ) + ûxx(0, ξ)) (2.24) − 2∑ j=1 φ ′ j (ξ) ( e−φj (1−x)ûxx(1, ξ) − e−φj (ξ)(x−y) ) . denote △̃(ξ, y) = ∑3 j=1 e −φj yφ′j (ξ) . substituting (2.20) into (2.23) and (2.24) and using (2.18) we get f̂1(x, y, ξ) = − △̃(ξ, 1 − x)△̃(ξ, y) △̃(ξ, 1) , f̂2(x, y, ξ) = △̃(ξ, y − x)△̃(ξ, 1) − △̃(ξ, 1 − x)△̃(ξ, y) △̃(ξ, 1) . 50 e.i. kaikina et. al. cubo 12, 1 (2010) since φ′l(ξ) = o ( |ξ|− 1 2 ) , l = 1, 2, 3 for |ξ| < 1, ξ ∈ d0, we have △̃(ξ, 1 − x)△̃(ξ, y) △̃(ξ, 1) = (∑3 j=1 e −φj (ξ)(1−x)φ′j )(∑3 j=1 e −φj (ξ)yφ′j ) ∑3 j=1 e −φj (ξ)φ′j = o ( |ξ|− 1 2 ) (2.25) and △̃(ξ, y − x) = o ( |ξ|− 1 2 ) (2.26) for |ξ| < 1, ξ ∈ d0. due to the fact that re φl(ξ) > 0, l = 1, 2, re φ3(ξ) < 0 for |ξ| > 1, ξ ∈ d0, we obtain for |ξ| > 1 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y) = e−φ3(1−x+y) ( φ′3 )2 ( 1 + ∑2 i=1 o ( e(−φj +φ3)(1−x) φ ′ j φ′ 3 )) e−φ3 φ′3 ( 1 + ∑2 j=1 o ( e(−φj +φ3) φ′ j φ′ 3 )) ×  1 + 2∑ j=1 o ( e(−φj +φ3)y φ′j φ′3 )  = eφ3(x−y)φ ′ 3 ( 1 + 2∑ i=1 o ( e(−φj +φ3)y ) + 2∑ i=1 o ( e(−φj +φ3)(1−x) )) . (2.27) therefore taking the asymptotics (2.12)-(2.14) into account we find that f̂1(x, y, ξ) = o ( ξ− 2 3 e−c 3 √ |ξ|(x−y) ) (2.28) for ξ ∈ d0, |ξ| > 1, x > y. also from (2.27) we get f̂2(x, y, ξ) = △̃(ξ, y − x) − △̃(ξ, 1 − x)△̃(ξ, y) △̃(ξ, 1) = 2∑ j=1 ( φ′j e −φj (y−x) +o ( φ ′ 3e − re φj y+re φ3x ) + o ( φ ′ 3e − re φj (1−x)+re φ3(1−y) )) = o ( ξ− 2 3 e−c 3 √ |ξ|(y−x) ) (2.29) for ξ ∈ d0, |ξ| > 1, y > x. thus there exist the inverse laplace transforms for the functions f̂1(x, y, ξ) and f̂2(x, y, ξ). taking the inverse laplace transformation of (2.23) and (2.24) we obtain f1(x, y, t) = l−1 { f̂1(x, y, ξ) } = − 1 2πi ∫ γ0 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ and f2(x, y, t) = l−1 { f̂1(x, y, ξ) } = − 1 2πi ∫ γ0 dξeξt △̃(ξ, 1 − x)△̃(ξ, y) − △̃(ξ, y − x)△̃(ξ, 1) △̃(ξ, 1) , cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 51 where γ0 = ∂d0, i.e. γ0 = (−i∞, −i0) ∪ [ −i0, 4 27 − i0 ] ∪ [ 4 27 + i0, i0 ] ∪ (i0, i∞) . since functions f̂l(x, y, ξ) are symmetric with respect to φ1,φ2,φ3,using the relations φ1 (ξ) = φ2 ( ξ ) , φ2 (ξ) = φ1 ( ξ ) , φ3 (ξ) = φ3 ( ξ ) for all ξ ∈ d0 we can change the contour of integration γ0 to the imaginary axis (−i∞, i∞) to get f1(x, y, t) = − 1 2πi ∫ i∞ −i∞ eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ and f2(x, y, t) = − 1 2πi ∫ i∞ −i∞ dξeξt △̃(ξ, 1 − x)△̃(ξ, y) − △̃(ξ, y − x)△̃(ξ, 1) △̃(ξ, 1) . therefore taking inverse laplace transform of (2.22) with respect to ξ we obtain u(x, t) = gu0. thus by duhamel principle proposition is proved. lemma 2.2. we have the asymptotics for large time fj (x, y, t) = −e−ξ0tλ (x) △̃(−ξ0, y) + o ( e−(ξ0+δ)t ) (2.30) and estimates |∂nx fj (x, y, t)| ≤ ce−ξ0t {t} −α |x − y|2α−1−n (2.31) for x, y ∈ (0, 1) , x 6= y, t > 0, where α ∈ [ 0, n+1 2 ] , n = 0, 1, j = 1, 2. proof. we consider a curve in the complex left-half plane re ξ < 0 such that re φ1 (ξ) = 0, it is defined by the equation (iy) 2 − (iy)3 = ξ with y = im φ1 (ξ) . therefore there exists a contour c0 = { ξ ∈ c, re ξ < 0 : re ξ = o ( |ξ| 2 3 )} such that re φl(ξ) > 0, l = 1, 2, re φ3(ξ) < 0 for all ξ ∈ c0. we also consider a contour c1 = (−ξ0 − δ − i∞, −ξ0 − δ − i0) ∪ (−ξ0 − δ − i0, −i0) ∪ (i0, −ξ0 − δ + i0) ∪ (−ξ0 − δ + i0, −ξ0 − δ + i∞) . denote c0 ∩ c1 = {z1, z2} , im z1 > 0, im z2 < 0, re zl = −ξ0 − δ, l = 1, 2. 52 e.i. kaikina et. al. cubo 12, 1 (2010) we now define a contour c = c2 ∪ c3, where c2 = {ξ ∈ c1 : im z2 ≤ im ξ ≤ im z1} . c3 = {ξ ∈ c0 : im ξ > im z1 or im ξ < im z2} . note that the asymptotic formulas (2.12)-(2.14) are valid on the contour c. in view of them we have (2.25)-(2.29) for ξ ∈ c. therefore changing the contour of integration to c we obtain f1(x, y, τ ) = − 1 2πi ∫ ξ∈c2 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ − 1 2πi ∫ ξ∈c3 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ. (2.32) since △̃(x + i0, q) = △̃(x − i0, q) we get − 1 2πi ∫ −i0 −ξ 0 −δ−i0 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ − 1 2πi ∫ −ξ 0 −δ+i0 +i0 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ = −e−ξ0t △̃(−ξ0, 1 − x)△̃(−ξ0, y) △̃′(−ξ0, 1) . (2.33) also by (2.25) we have 1 2πi ∫ z1 −ξ 0 −δ+i0 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ + 1 2πi ∫ −ξ 0 −δ−i0 z2 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ = o ( e−(ξ0+δ)t ) . (2.34) taking into account (2.28) we get ∣∣∣∣∣ − 1 2πi ∫ ξ∈c3 eξt 1 △̃(ξ, 1) △̃(ξ, 1 − x)△̃(ξ, y)dξ ∣∣∣∣∣ < ce−(ξ0+δ)t ∫ ξ∈c3 e−ct|ξ| 2 3 +t(ξ 0 +δ)−c|x−y||ξ| 1 3 |ξ|− 2 3 dξ < ce−(ξ0+δ)tt−α |x − y|2α−1 (2.35) since c |ξ| 2 3 − (ξ0 + δ) ≥ 0 for ξ ∈ c3, where α ∈ [ 0, 1 2 ] . by (2.33)-(2.35) we have from (2.32) f1(x, y, t) = −e−ξ0t △̃ (−ξ0, 1 − x) △̃ (−ξ0, y) △̃′ (−ξ0, 1) + o ( e−(ξ0+δ)t ) cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 53 for x, y > 0, t ≥ 1, and moreover |f1(x, y, t)| ≤ ce−ξ0t ( 1 + {t}−α |x − y|2α−1 ) for all x, y ∈ (0, 1) , x 6= y, t > 0, where α ∈ [ 0, 1 2 ] . thus the result of the lemma is true for the case n = 0. consider the case n = 1. in view of the asymptotic formulas (2.12)-(2.14) we get ∂x△̃(ξ, 1 − x)△̃(ξ, y) △̃(ξ, 1) = (∑3 j=1 e −φj (ξ)(1−x) ( φ′j φj ))(∑3 j=1 e −φj (ξ)yφ′j ) ∑3 j=1 e −φj (ξ)φ′j = o(1) (2.36) and ∂x△̃(ξ, y − x) = o(1) (2.37) for |ξ| < 1, ξ ∈ c2 and in the same argument as in the proof of the estimate (2.25) we get ∂x△̃(ξ, 1 − x)△̃(ξ, y) △̃(ξ, 1) = e−φ3(y−x)φ3φ ′ 3 ( 1 + o ( e−c 3 √ |ξ|y ) + o ( e−c 3 √ |ξ|(1−x) )) (2.38) and ∂x△̃(ξ, y − x) = e−φ3(y−x)φ3φ ′ 3 ( 1 + o ( e−c 3 √ |ξ|y ) + o ( e−c 3 √ |ξ|(1−x) )) (2.39) for all |ξ| > 1, ξ ∈ c3. hence by the similar way to (2.33)-(2.35) we get |∂xf1(x, y, t)| ≤ e−ξ0t ∣∣∣∣∣ △̃(−ξ0, y)∂x△̃(−ξ0, 1 − x) △̃′(−ξ0, 1) ∣∣∣∣∣ + ce−ξ0t +ce−(ξ0+δ)t ∫ ξ∈c3 e−ct|ξ| 2 3 +t(ξ 0 +δ)−c|x−y||ξ| 1 3 |ξ|− 1 3 dξ ≤ e−ξ0t ( c + c {t}−α |x − y|2α−2 ) for all x, y ∈ (0, 1) , x 6= y, t > 0, where α ∈ [0, 1] .the function f2(x, y, t) is considered in the same way. lemma 2.2 is proved. now we prove the local existence for the linear problem (2.1). theorem 2.3. let the initial data u0 ∈ l2 and f ∈ l2. then for any t > 0 there exists a unique solution u ∈ c ( [0, t ] ; l2 ) ∪ c ( (0, t ] ; h1 ) of the linear initial-boundary value problem (2.1) such that sup t∈(0,t ] tα ‖∂nx u (t)‖l2 ≤ cλ provided that λ = ‖u0‖l2 + t 1−β sup t∈(0,t ] tβ ‖f (t)‖ l2 < ∞, where n = 0, 1, α ∈ ( n 2 , 1 ) , β ∈ (0, 1) . 54 e.i. kaikina et. al. cubo 12, 1 (2010) proof. ¿from proposition and estimates of lemma 2.2 we have ‖∂nx u (t)‖l2 ≤ t −α ∥∥∥∥ ∫ 1 0 u0(y) |x − y|2α−1−n dy ∥∥∥∥ l2 + ∥∥∥∥ ∫ t 0 τ −α (t − τ )−β dτ ∫ 1 0 (t − τ )β f (y, t − τ ) |x − y|2α−1−n dy ∥∥∥∥ l2 ≤ ct−α ( ‖u0(y)‖l2 + t 1−β sup t∈(0,t ] tβ ‖f (·, t)‖ l2 ) for n = 0, 1. so we have the estimate of the theorem. theorem 2.3 is proved. 3 local existence for the nonlinear problem in this section we prove the following result. theorem 3.1. suppose that the initial data u0(x) ∈ l2. then there exists a unique solution u(x, t) ∈ c ([0, t ] ; l∞) ∪ c ( (0, t ] ; h1 ) where t > 0 depends on ‖u0‖l2 . proof. we prove the local existence of solutions by the contraction mapping principle. let u(x, t) be a solution of the following linear problem    ut + n(w) − uxx + uxxx = 0, t > 0, x ∈ (0, 1) , u(x, 0) = u0(x), x ∈ (0, 1) , u(0, t) = u(1, t) = ux(1, t) = 0, t > 0, (3.1) where n(w) = wwx, w is taken from the closed ball h 1 ρ = { w ∈ c ( (0, t ] ; h1 ) ; sup t∈(0,t ] 1∑ n=0 tαn‖∂nx w‖l2 ≤ ρ } , where αn ∈ ( n 2 , 1 ) and satisfies the boundary condition w(0, t) = w(1, t) = wx(1, t) = 0. the initial-value problem (3.1) defines a mapping m by u = m(w) and we will show that m is the contraction mapping from h1ρ into itself for a sufficiently small t > 0. since w ∈ h1ρ, using the estimate ‖w‖ l∞ ≤ 2 ‖w‖ l2 ‖wx‖l2 , we have sup t∈(0,t ] tβ ‖n(w)(t)‖ l2 ≤ c sup t∈(0,t ] tβ ‖w(t)‖ 1 2 l2 ‖wx(t)‖ 3 2 l2 ≤ cρ2, where β = α0+3α1 2 < 1. via theorem 2.3 problem (3.1) has a unique solution u(x, t) ∈ c ( (0, t ] ; h1 ) , such that sup t∈(0,t ] tαn ‖∂nx u‖l2 ≤ cλ, cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 55 where λ = ‖u0‖l2 + t 1−β sup t∈(0,t ] tβ ‖n(w)(t)‖ l2 . therefore we obtain the estimate sup t∈(0,t ] 1∑ n=0 tαn‖∂nx u‖l2 ≤ c‖u0‖l2 + ct 1−βρ2 ≤ ρ (3.2) if t = (2cρ) − 1 1−β = ( 4c2 ‖u0‖l2 )− 1 1−β and c ‖u0‖l2 ≤ ρ 2 . thus the mapping m transforms the closed ball h1ρ with a center at the origin and a radius ρ into itself. analogously we can estimate the difference sup t∈(0,t ] 1∑ n=0 tαn ‖∂nx (u − ũ) ‖l2 ≤ 1 2 sup t∈(0,t ] 1∑ n=0 tαn ‖∂nx (w − w̃) ‖l2 for sufficiently small t > 0. therefore the mapping m is a contraction mapping in h1ρ and there exists a unique solution u(x, t) ∈ c ( (0, t ] ; h1 ) of the initial-value problem (1.1). theorem 3.1 is proved. remark 3.1. by virtue of estimate (3.2) we see that if the initial data u0 are small, i.e. the norm ‖u0‖l2 ≤ ε, where ε > 0 is sufficiently small, then there exists t ≥ 1, such that there exists a unique solution u, which is also small: supt∈(0,t ] ∑1 n=0 t αn ‖∂nx u‖l2 < cε. 4 large time asymptotics in this section we give sufficient conditions for global existence of solutions to the initial-boundary value problem (1.1) with small initial data ‖u0‖l2 < ε1, (4.1) where ε1 > 0 is sufficiently small. theorem 4.1. suppose that the initial data u0 ∈ l2 and satisfy (4.1). then there exists a unique solution u of (1.1) such that u ∈ c ( [0, ∞) ; l2 ) ∪ c ( (0, ∞) ; h1 ) . furthermore the solution has the following asymptotics u(x, t) = aλ(x)e−ξ0t + o ( e−(ξ0+δ)t ) (4.2) for t → ∞ uniformly with respect to x ∈ (0, 1) , where ξ0 > 0, δ > 0, the constant a, the function λ(x) ∈ l∞ are defined below in the proof. 56 e.i. kaikina et. al. cubo 12, 1 (2010) proof. according to theorem 3.1 and remark 3.1 we see that after some time t ≥ 1 the solution is small in the norm h1. therefore we can only consider the initial-boundary problem (1.1) for t ≥ 1 with small initial data u (x, 1) such that ‖u (·, 1)‖ h1 ≤ ε2, where ε2 > 0 is sufficiently small. let us prove that eξ0t ‖u(t)‖ h1 < ε (4.3) for all t ≥ 1, with some small ε > 0. by the contradiction we can find a maximal interval [1, t1] such that the estimate eξ0t ‖u(t)‖ h1 ≤ ε (4.4) is true for all t ∈ [1, t1] and estimate (4.3) is violated at time t = t1. from (4.4) and estimates (2.31) of lemma 2.2 we obtain for n = 0, 1 ∥∥∥u(n)x (·, t) ∥∥∥ l2 ≤ ce−ξ0t ‖u (·, 1)‖ l2 + ∫ t 1 dτ ∫ x 0 |uuy(y, τ )| ‖∂nx f1 (·, y, t − τ )‖l∞ dy + ∫ t 1 dτ ∫ 1 x |uuy(y, τ )| ‖∂nx f2 (·, y, t − τ )‖l∞ dy ≤ cε2e−ξ0t + ce−ξ0t ∫ t 1 eξ0τ ‖u(τ )‖ 1 2 l2 ‖ux(τ )‖ 3 2 l2 {t − τ}− 3 4 dτ ≤ cε2e−ξ0t + cε2e−ξ0t ∫ t 1 e−ξ0τ {t − τ}− 3 4 dτ ≤ c ( ε2 + ε 2 ) e−ξ0t for t ∈ [1, t1] . the contradiction obtained proves (4.3). now using the estimate (4.3) and lemma 2.2 we prove that the solution has asymptotics (4.2) for t → ∞ uniformly with respect to x > 0, where λ(x) = − △̃(−ξ0, 1 − x) △̃′(−ξ0, 1) , a = ∫ 1 0 u0(y)△̃(−ξ0, y)dy + ∫ ∞ 0 eξ0τ dτ ∫ 1 0 u(y, τ )uy(y, τ )△̃(−ξ0, y)dy. indeed, due to asymptotics (2.30) of lemma 2.2 we have u(x, t) = aλ(x)e−ξ0t + r(x, t), (4.5) where in view of (4.3) we have |r(x, t)| ≤ e−(ξ0+δ)t ‖u (·, 1)‖ h1 + c ∫ t 1 e−(ξ0+δ)(t−τ )dτ ∫ 1 0 |uuy| dy +e−ξ0t |λ (x)| ∫ ∞ t eξ0τ dτ ∫ 1 0 |uuy| ∣∣∣△̃(−ξ0, y) ∣∣∣dy = o ( e−(ξ0+δ)t ) for all t ≥ 1, where δ > 0. theorem 4.1 is proved. cubo 12, 1 (2010) korteweg-de vries-burgers equation ... 57 5 large initial data we consider the initial-boundary value problem (1.1) with any initial data ‖u0‖l2 ≤ c. multiplying equation (1.1) by u and integrating with respect to x ∈ (0, 1) we get d dt ‖u‖2 l2 + 2 ∫ 1 0 ( u2ux − uuxx + uuxxx ) dx = 0. we have ∫ 1 0 u2uxdx = 1 3 u3 ∣∣∣∣ 1 0 = 0, ∫ 1 0 uuxxdx = uux|10 − ∫ 1 0 u2xdx = − ∫ 1 0 u2xdx, ∫ 1 0 uuxxxdx = uuxx|10 − 1 2 u2x ∣∣1 0 = 1 2 u2x (0, t) in view of the boundary data, hence d dt ‖u‖2 l2 + 2 ∫ 1 0 u2xdx + u 2 x (0, t) = 0. integration with respect to t > 0 yields ‖u (t)‖ l2 + 2 ∫ t 0 ‖ux (τ )‖2l2 dτ ≤ ‖u0‖l2 for all t ∈ (0, ∞) . we see that the norm ‖u (t)‖ l2 ≤ ‖u0‖l2 for all t ≥ 0. by the standard continuation process via theorem 3.1 we obtain that there exists a unique global solution u ∈ c ( (0, ∞) ; h1 ) since the existence time t depends only on ‖u0‖l2 . moreover we see that for any ε > 0 there exists a time t > 0 such that ‖ux (t )‖2l2 < ε. by the inequality u2 (x, t ) = 2 ∫ x 0 uuydy ≤ 2 ‖u‖l2 ‖ux‖l2 we see that the norm ‖u (t )‖l∞ , and hence the norm ‖u (t )‖l2 , are also small by the estimate ‖u (t )‖ l2 ≤ ‖u (t )‖ l∞ . then we consider the initial-boundary value problem (1.1) for t ≥ t and apply theorem 4.1 whence the result of theorem 1.1 follows. theorem 1.1 is proved. acknowledgments. the work of one of the authors (e.i.k.) is partially supported by conacyt. received: september, 2008. revised: october, 2009. references [1] aikawa, h. and hayashi, n., holomorphic solutions of semilinear heat equations, complex variables, 16 (1991), pp. 115–125. 58 e.i. kaikina et. al. cubo 12, 1 (2010) [2] aikawa, h., hayashi, n. and saitoh, s., the bergman space on a sector and the heat equation, complex variables, 15 (1990), pp. 27–36. 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[12] naumkin, p.i. and shishmarev, i.a., asymptotic relationship as t → ∞ between solutions to some nonlinear equations, differential equations, 30 (1994), pp. 806–814. [13] shishmarev, i.a., tsutsumi, m. and kaikina, e.i., asymptotics in time for the nonlinear nonlocal schrödinger equations with a source, j. math. soc. japan, 51 (1999), pp. 463–484. cubo a mathematical journal vol.10, n o ¯ 01, (103–115). march 2008 continuous or discontinuous deformations of c∗-algebras takahiro sudo department of mathematical sciences faculty of science, university of the ryukyus nishihara, okinawa 903-0213, japan email: sudo@math.u-ryukyu.ac.jp abstract we study deformations of c ∗ -algebras that become continuous or discontinuous. resumen estudiamos deformación de c ∗ -algebras que son continuas o discontinuas. key words and phrases: c*-algebra, continuous field, crossed product. math. subj. class.: primary 46l05. 104 takahiro sudo cubo 10, 1 (2008) introduction continuous fields of c ∗ -algebras have been of interest in the theory of c ∗ -algebras (see dixmier [5, chapter 10]). in particular, continuous field c ∗ -algebras of continuous trace with hausdorff spectrums are well studied to classify them. in this case the continuous fields of c ∗ -algebras become locally trivial and they are built up by trivial continuous field c ∗ -algebras that are tensor products of the c ∗ -algebras of continuous functions on their base spaces with some fixed fibers. continuous deformations of c ∗ -algebras are in a particular case of continuous fields of c ∗ -algebras in the sense that their base spaces are the closed interval [0, 1] and the fibers on the half open interval (0, 1] are the same (cf. e-theory in blackadar [1]). it has been known that continuous deformations of c ∗ -algebras may have non-hausdorff spacetrums in general ([5, 10]). it is first obtained in [10] that there exists no continuous deformation from a c ∗ -algebra generated by isometries to a c ∗ -algebra generated by unitaries, in particular, no continuous deformation from cuntz and toeplitz algebras to the c ∗ -algebras of continuous functions on the tori. in this paper we investigate some interesting properties for continuous or discontinuous deformations of c ∗ -algebras beyond the result of [10], but using its ideas. we find it convenient to divide continuous deformations of c ∗ -algebras into two classes. one consists of degenerate continuous deformations of c ∗ -algebras and the other does of nondegenerate continuous deformations of c ∗ -algebras, that we define later. we find that it is easy to have degenerate continuous deformations of c ∗ -algebras, some of which are useful to provide some examples with non-hausdorff spectrums, and it is not easy to construct nondegenerate continuous deformations of c ∗ -algebras. indeed, we find that there exists no nondegenerate continuous deformations in some cases as given below. in section 1 we forcus on degenerate or nondegenerate continuous deformations of c ∗ algebras. in secion 2 we give some nondegenerate discontinuous deformations of c ∗ -algebras by considering crossed product c ∗ -algebras by the integer group z and the real group r and by semigroup crossed product c ∗ -algebras by the semigroup(s) of natural numbers, which would be of interest. refer to dixmier [5], pedersen [8] and murphy [7] for details of the c ∗ -algebra theory. 1 continuous deformations of c∗-algebras recall that a continuous deformation from a c ∗ -algebra a to another b means a continuous field c ∗ -algebra γ([0, 1], {at}t∈[0,1]) on the closed interval [0, 1] with fibers at given by a0 = b and at = a for 0 < t ≤ 1, where the continuous field c ∗ -algebra is defined and generated by giving continuous operator fields on [0, 1] such that their norm at fibers are cubo 10, 1 (2008) continuous or discontinuous deformations of c ∗ -algebras 105 continuous and the set of (or generated by) their evaluations at each point t ∈ [0, 1] is dense in at. refer to [5] for details of continuous fields of c ∗ -algebras. definition 1.1 we say that a continuous deformation from a c∗-algebra a to another b is degenerate if there exist continuous operator fields coming from some generators of a that are zero at 0 ∈ [0, 1]. we say that a continuous deformation from a c∗-algebra a to another b is nondegenerate if it is not degenerate, i.e., there exist no continuous operator fields coming from generators of a that are zero at 0 ∈ [0, 1]. proposition 1.2 let a, b be c∗-algebras. assume that we have the following splitting exact sequence: 0 → c0((0, 1], a) → e → b → 0, where c0((0, 1], a) is the c ∗-algebra of continuous a-valued functions on the half open interval (0, 1]. then the extension e is a continuous deformation from a to b. remark. a continuous deformation from a to b has the same decomposition as the extension e above, but its extension is not necessarily splitting. example 1.3 let a be a unital c∗-algebra. then we have the following natural splitting exact sequence: 0 → c0((0, 1], a) → e → c → 0, where the unit operator field f defined by f (t) = 1 ∈ a for (0, 1] and f (0) = 1 ∈ c is continuous in e. this continuous deformation is degenerate if a 6= c and nondegenerate if a = c. degenerate continuous deformations theorem 1.4 let a be a c∗-algebra. suppose that a has a non-trivial projection p, and let pap denote the c∗-subalgebra of a generated by the elements pap for a ∈ a. then there exists a continuous deformation from a to pap. also, if a is unital, then there exists a continuous deformation from a to pap ⊕ (1 − p)a(1 − p), where 1 − p can be replaced with a projection of a orthogonal to p. proof. we construct a continuous field c∗-algebra γ([0, 1], {at}t∈[0,1]) with fibers at given by at = a for 0 < t ≤ 1 and a0 = pap as follows. assume that constant continuous operator fields f on pap such as f (t) = f (s) ∈ pap for t, s ∈ [0, 1] are contained in γ([0, 1], {at}t∈[0,1]). and assume that other continuous operator fields of γ([0, 1], {at}t∈[0,1]) vanish at zero. more concretely, we can take the other way to prove the statement in the case that a is a unital c ∗ -algebra as follows. then any element a ∈ a can be viewed as the following matrix: a ( a11 a12 a21 a22 ) 106 takahiro sudo cubo 10, 1 (2008) for a11 = pap, a12 = pa(1 − p), a21 = (1 − p)ap, and a22 = (1 − p)a(1 − p). thus, we take the following matrix functions as continuous operator fields of γ([0, 1], {at}t∈[0,1]): a(t) ( a11(t) a12(t) a21(t) a22(t) ) with a(0) ( pap 0 0 0 ) for t ∈ [0, 1] such that a(1) = a. for the second assertion, we just replace a22(0) = 0 with a22(0) = (1 − p)a(1 − p). 2 example 1.5 there exists a continuous deformation from the matrix algebra mn(c) to mm(c) for n ≥ m ≥ 1 by theorem 1.4 since mm(c) ∼= pmn(c)p for p a rank m projection of mn(c). also, there exists a continuous deformation from the matrix algebra mn(c) to c k , where 1 ≤ k ≤ n by choosing k orthogonal rank 1 projections of mn(c). note that this continuous deformation has non hausdorff spectrum if k ≥ 2. there exists a continuous deformation from the c ∗ -algebra k of compact operators to mm(c) for any m ≥ 1 by theorem 1.4 since mm(c) ∼= p(k)p for p a rank m projection of k. also, there exists a continuous deformation from the c ∗ -algebra k to c k (k ≥ 1) and to c0(n) the c ∗ -algebra of sequences vanishing at infinity. let a be an af algebra, i.e., an inductive limit of finite dimensional c ∗ -algebras (or finite direct sums of matrix algebras over c). then, as shown in theorem 1.4 there exists a continuous deformation from a to its c ∗ -subalgebra mm(c) for some m ≥ 1. let a ⊕ b be the direct sum of c∗-algebras a, b. then there exists a continuous deformation from a ⊕ b to a. theorem 1.6 let a be a c∗-algebra and b a unital c∗-algebra. then there exists a continuous deformation from the c∗-tensor product a ⊗ b with a c∗-norm to a. proof. note that any c∗-tensor product a ⊗ b with a certain c∗-norm is generated by simple tensors a ⊗ b for a ∈ a and b ∈ b. we construct a continuous field c∗-algebra γ([0, 1], {at}t∈[0,1]) with fibers at given by at = a ⊗ b for t ∈ (0, 1] and a0 = a as follows. since b is unital, we assume that the constant operator fields on a ∼= a ⊗ c in a ⊗ b are continuous and other continuous operator fields vanish at zero. 2 example 1.7 let c(tn) be the c∗-algebra of continuous functions on the n-torus tn (n ≥ 0), where c(t0) = c. then there exists a continuous deformation from c(tn) to c(t m ) for n > m ≥ 0 since c(tn) ∼= c(tm) ⊗ c(tn−m). as for crossed product c ∗ -algebras by groups, cubo 10, 1 (2008) continuous or discontinuous deformations of c ∗ -algebras 107 theorem 1.8 let a be a unital c∗-algebra, γ a discrete group and a ⋊α γ the full crossed product c∗-algebra by an action α of γ on a. then there exists a continuous deformation from a ⋊α γ to either a or the full group c ∗-algebra c∗(γ) of γ. moreover, there exists a continuous deformation from the reduced crossed product c∗-algebra a ⋊α,r γ to either a or the reduced group c∗-algebra c∗r (γ) of γ. proof. note that the full crossed product c∗-algebra a ⋊α γ is generated by a and c ∗ (γ), and a and c ∗ (γ) are c ∗ -subalgebras of a ⋊α γ. we assume that the constant operator fields on a (or c ∗ (γ)) in a ⋊α γ are continuous and other continuous operator fields vanish at zero. also, we can replace a ⋊α γ with a ⋊α,r γ and c ∗ (γ) with c ∗ r (γ) respectively. 2 theorem 1.9 let a be a unital c∗-algebra, g a locally compact group and a ⋊α g the full crossed product c∗-algebra by an action α of g on a. then there exists a continuous deformation from a ⋊α g to the full group c ∗-algebra c∗(g) of g. moreover, there exists a continuous deformation from the reduced crossed product c∗-algebra a ⋊α,r g to the reduced group c∗-algebra c∗r (g) of g. proof. note that the full crossed product c∗-algebra a ⋊α g is generated by elements af for a ∈ a and f ∈ c∗(g), and c∗(g) is a c∗-subalgebra of a ⋊α g. we assume that the constant operator fields on c ∗ (g) in a ⋊α g are continuous and other continuous operator fields vanish at zero. also, we can replace a ⋊α g with a ⋊α,r g and c ∗ (g) with c ∗ r (g) respectively. 2 as for free products of c ∗ -algebras, theorem 1.10 let a, b be unital c∗-algebras. then there exists a continuous deformation from the (full or reduced) unital free product c∗-algebra a ∗c b (an amalgam over c) to a. proof. note that the (full or reduced) unital free product c∗-algebra a ∗c b is generated by a and b, where the unit of a is identified with that of b. we construct a continuous field c ∗ -algebra γ([0, 1], {at}t∈[0,1]) with fibers at given by at = a ∗c b for t ∈ (0, 1] and a0 = a by assuming the constant operator fields on a in a ∗c b are continuous and other continuous operator fields vanish at zero. 2 example 1.11 let c∗(f2) be the full group c ∗ -algebra of the free group f2 with two generators (see davidson [4]). then there exists a continuous deformation from c ∗ (f2) to c(t) since c ∗ (f2) ∼= c∗(z) ∗c c ∗ (z) and c ∗ (z) ∼= c(t) by the fourier transform. 108 takahiro sudo cubo 10, 1 (2008) nondegenerate continuous deformations example 1.12 let h3 be the real 3-dimensional heisenberg lie group and c ∗ (h3) its group c ∗ -algebra. since h3 is isomorphic to a semi-direct product r 2 ⋊ r, we have c ∗ (h3) ∼= c ∗ (r 2 ) ⋊ r ∼= c0(r 2 ) ⋊ r by the fourier transform. then it is known that c ∗ (h3) can be viewed as the continuous field c ∗ -algebra γ0(r, {at}t∈r) with fibers at = k for t 6= 0 and a0 = c0(r 2 ) since at ∼= c0(r) ⋊αt r ∼= k for t 6= 0 where the action α t of r on r is a shift and a0 ∼= c0(r) ⋊α0 r ∼= c0(r 2 ) since the action α 0 of r on r is trivial. therefore, the restriction of this continuous field c ∗ -algebra to [0, 1] gives a continuous deformation from k to c0(r 2 ). let h2n+1 be the real (2n + 1)-dimensional generalized heisenberg lie group and c ∗ (h2n+1) its group c ∗ -algebra. since h2n+1 is isomorphic to a semi-direct product r n+1 ⋊ r n , we have c ∗ (hn+1) ∼= c∗(rn+1) ⋊ rn ∼= c0(r n+1 ) ⋊ r n by the fourier transform. then it is known that c ∗ (h2n+1) can be viewed as the continuous field c ∗ -algebra γ0(r, {at}t∈r) with fibers at = k for t 6= 0 and a0 = c0(r 2n ) since at ∼= c0(r n ) ⋊αt r n ∼= k for t 6= 0 where the action αt of rn on rn is a shift and a0 ∼= c0(r n ) ⋊α0 r n ∼= c0(r 2n ) since the action α 0 of r n on r n is trivial. therefore, the restriction of this continuous field c ∗ -algebra to [0, 1] gives a continuous deformation from k to c0(r 2n ). more generally, proposition 1.13 let a be a c∗-algebra, g a locally compact group and a ⋊αt g the full crossed product c∗-algebras by actions αt of g on a for t ∈ [0, 1]. suppose that the actions {αt}t∈[0,1] are continuous in the sense that the maps from t ∈ [0, 1] to αt(a) for a ∈ a are continuous and that a ⋊αt g ∼= a ⋊αs g for t, s ∈ (0, 1] and α 0 is trivial. then there exists a continuous deformation from a ⋊α1 g to a ⊗ c ∗ (g). furthermore, similarly we can replace a ⋊αt g with their reduced crossed product c ∗-algebras and c∗(g) with its reduced group c ∗-algebra respectively. remark. even if g = r, the assumption a ⋊αt r ∼= a ⋊αs r for t, s ∈ (0, 1] are not true in general. for instance, let c(t 2 ) ⋊θ r be the crossed product c ∗ -algebra by the action θ of r on t 2 defined by θt(z, w) = (e 2πit z, e 2πiθt w) ∈ t2 where θ ∈ r, which is also called the foliation c ∗ -algebra of c(t 2 ) by r of connes [2]. then it is known that c(t 2 ) ⋊θ r ∼= k ⊗ (c(t) ⋊θ z), where c(t) ⋊θ z is the rotation algebra corresponding to θ. moreover, it is known that c(t) ⋊θ z ∼= c(t) ⋊θ′ z if and only if θ = θ ′ or θ = 1 − θ′ (mod 1). the proposition above gives a general procedure to construct nondegenerate continuous fields by crossed products c ∗ -algebras, but it is not easy to have continuous actions cubo 10, 1 (2008) continuous or discontinuous deformations of c ∗ -algebras 109 {αt}t∈[0,1] in the sense above and check the isomorphisms of their crossed product c ∗ algebras for t ∈ (0, 1]. as for tensor products of c ∗ -algebras, proposition 1.14 let a, b be c∗-algebras. suppose that the c∗-tensor product a ⊗ b with a c∗-norm is isomorphic to a. then there exists a continuous deformation from a⊗b to a. example 1.15 we have k ⊗ k ∼= k. a c∗-algebra a is stable if a ⊗ k ∼= a. let a be a simple separable nuclear c ∗ -algebra. then a ∼= a ⊗ o∞ if and only if a is purely infinite, where o∞ is the cuntz algebra generated by a sequence of othogonal isometries. a c ∗ -algebra a is simple, separable, unital and nuclear if and only if a⊗o2 ∼= o2, where o2 is the cuntz algebra generated by two orthogonal isometires with the sum of their range projections equal to the identity. see rørdam [9] for these significant results. 2 discontinuous deformations of c∗-algebras nondegenerate discontinuous deformations theorem 2.1 let a be a unital commutative c∗-algebra and a⋊α z the crossed product c ∗algebra of a by a non trivial action α of z. then there exists no nondegenerate continuous deformation from a ⋊α z to a. if a is nonunital and commutative, then there exists no nondegenerate continuous deformation from a ⋊α z to a + the unitization of a by c. proof. note that a is a c∗-subalgebra of a ⋊α z and a ⋊α z is generated by a and a unitary corresponding to the action α of z. let u be such a unitary. then we have the covariance relation: u au ∗ = α1(a) for a ∈ a. suppose that we had a continuous field c ∗ -algebra γ([0, 1], {at}t∈[0,1]) such that a0 = a and at = a ⋊α z for 0 < t ≤ 1. we may assume that (certain) constant continuous operator fields on a (or a + if a is non unital) are contained in γ([0, 1], {at}t∈[0,1]) (where the argument below is applicable to the case without constant continuous operator fields). also, we may assume that the operator field f defined by f (0) = u a unitary of a (or u a unitary of a + if a is nonunital) and f (t) = u for 0 < t ≤ 1 is also contained in it. then the operator field f bf ∗ for (certain) b ∈ a defined by f bf ∗ (t) = f (t)bf ∗ (t) = u bu ∗ = α1(b) and f bf ∗ (0) = ubu ∗ = uu ∗ b = b must be continuous. but this is impossible in general since b 6= α1(b) for some b ∈ a since α is non trivial so that (b − f bf ∗)(t) = b − α1(b) 6= 0 for t ∈ (0, 1] but (b − f bf ∗ )(0) = b − b = 0. 2 110 takahiro sudo cubo 10, 1 (2008) example 2.2 let c(t) be the c∗-algebra of continuous functions on the torus t and c(t) ⋊αθ z the crossed product c ∗ -algebra that is called a rotation algebra, where α θ is induced from the action of z on t by the multiplication e 2πiθt for t ∈ z (see weggeolsen [11]). by theorem 2.1, there exists no nondegenerate continuous deformation from c(t) ⋊αθ z to c(t). moreover, let c(t k ) ⋊αθ z be the crossed product c ∗ -algebra (which is one of noncommutative tori) by an action α θ by z on c(t k ), where θ = (θj ) k j=1 and α θ t (zj ) = (e 2πiθj tzj ) ∈ t k for t ∈ z. then there exists no nondegenerate continuous deformation from c(tk) ⋊αθ z to c(t k ). furthermore, theorem 2.3 let a be a unital simple c∗-algebra and a ⋊α z the crossed product c ∗algebra of a by a non trivial action α of z. then there exists no nondegenerate continuous deformation from a ⋊α z to a. if a is nonunital and simple, then there exists no nondegenerate continuous deformation from a ⋊α z to a + the unitization of a by c. proof. let u be a unitary corresponding to α. suppose that we had a continuous field c ∗ -algebra γ([0, 1], {at}t∈[0,1]) such that a0 = a and at = a ⋊α z for 0 < t ≤ 1. we may assume that the operator field f defined by f (0) = u a unitary of a (or u a unitary of a + if a is nonunital) and f (t) = u for 0 < t ≤ 1 is also contained in it. then the operator field f uf ∗ defined by f uf ∗ (t) = f (t)uf ∗ (t) = u uu ∗ = α1(u) and f uf ∗ (0) = uuu ∗ = u must be continuous. hence it follows that α1(u) = u since the operator field f uf ∗ − α1(u) is continuous and (f uf ∗ − α1(u))(t) = 0 for t ∈ (0, 1] so that (f uf ∗ − α1(u))(0) = 0. thus, u is fixed under α. therefore, the c ∗ -algebra c ∗ (u) generated by u is fixed under α. then a must have c ∗ (u) as a nontrivial quotient c ∗ -algebra, which contradicts to that a is simple. we use the similar argument for the case of a nonunital and simple. 2 example 2.4 let on be the cuntz algebra generated by n orthogonal isometries {sj} n j=1 such that ∑n j=1 sj s ∗ j = 1 (see cuntz [3] or the text books davidson [4] or wegge-olsen [11]). then by theorem 2.3 there exists no nondegenerate continuous deformation from on ⊗ k to mn∞ ⊗ k, where mn∞ is the uhf algebra. it is known that the c ∗ -tensor product on ⊗ k isomorphic to the crossed product c ∗ -algebra (mn∞ ⊗ k) ⋊α z (see rørdam [9]). moreover, theorem 2.5 let a be an either commutative or simple, unital c∗-algebra and a ⋊α γ the (reduced or full) crossed product c∗-algebra of a by a non trivial action α of γ a discrete cubo 10, 1 (2008) continuous or discontinuous deformations of c ∗ -algebras 111 group. then there exists no nondegenerate continuous deformation from a⋊α γ to a. if a is nonunital and either commutative or simple, then there exists no nondegenerate continuous deformation from a ⋊α γ to a + the unitization of a by c. proof. note that the (full or reduced) crossed product c∗-algebra a ⋊α γ is generated by a and the unitaries corresponding to generators of γ and a is a c ∗ -subalgebra of a ⋊α γ. let u be one of the unitaries. we apply the arguments given in the proofs of theorems 2.1 and 2.3 for the c ∗ -algebra generated by a and u . note that u may have torsion in the arguments. 2 as for crossed product c ∗ -algebras by continuous groups, theorem 2.6 let a be an either commutative or simple, unital (or non unital) c∗-algebra and a ⋊α r the crossed product c ∗-algebra of a by a non trivial action α of r. then there exists no nondegenerate continuous deformation from a ⋊α r to a. proof. note that the crossed product c∗-algebra a ⋊α r is generated by elements af for a ∈ a and f ∈ c∗(r). since c∗(r) ∼= c0(r) by the fourier transform, we identify elements of c ∗ (r) with those of c0(r). note that the unitization c0(r) + by c is isomorphic to c(t). now suppose that we had a continuous field c ∗ -algebra γ([0, 1], {at}t∈[0,1]) such that a0 = a and at = a ⋊α r for 0 < t ≤ 1. then we can have a extended continuous field c ∗ -algebra γ([0, 1], {bt}t∈[0,1]) such that b0 = a and bt the c ∗ -algebra generated by a and that c(t) for 0 < t ≤ 1 by assuming that the operator field from the unit of c(t) to the unit of a (or of a + if a nonunital) is continuous. suppose that a is commutative. since α is nontrivial, there exists b ∈ a such that u bu ∗ 6= b. indeed, if u bu ∗ = b for any b ∈ a, then a and c(t) commute. hence a and c0(r) commute. thus, a ⋊α r ∼= a ⊗ c∗(r) so that α must be trivial. therefore, we can adopt the argument given in the proof of theorem 2.1. suppose that a is simple. on the other hand, by the argument given in the proof of theorem 2.3, we have u uu ∗ = u, where the operator field from u to u ∈ a is continuous. thus, the c ∗ -algebra c ∗ (u) generated by u commutes with c(t) generated by u . hence c ∗ (u) commutes with c ∗ (r). then a has c ∗ (u) as a nontrivial quotient c ∗ -algebra, which is the contradiction. 2 remark. we can replace with r with t in the statement above. note that c∗(t) ∼= c0(z) by the fourier transform and c0(z) + ∼= c((z)+), where (z)+ is the one point compactification of z and it is identified with a closed subset of t. example 2.7 let c∗(h3) be the group c ∗ -algebra of the real 3-dimensional heisenberg lie group h3. then c ∗ (h3) ∼= c∗(r2) ⋊ r ∼= c0(r 2 ) ⋊ r since h3 ∼= r2 ⋊ r. hence there 112 takahiro sudo cubo 10, 1 (2008) exists no nondegenerate continuous deformation from c ∗ (h3) to c0(r 2 ) of c0(r 2 ) ⋊ r. furthermore, theorem 2.8 let a be an either commutative or simple, unital (or non unital) c∗-algebra and a⋊αr n the crossed product c∗-algebra of a by an action α of rn such that the restriction of α to any factor r of rn is non trivial. then there exists no nondegenerate continuous deformation from a ⋊α r n to a. proof. we use the same process as given in the proof of the theorem above. note that a ⋊α r n is generated by elements af for a ∈ a and f ∈ c∗(rn), and c∗(rn) ∼= c0(r n ) so that c0(r n ) + ∼= c((rn)+) ∼= c(sn), where (rn)+ is the one point compactification of r n and s n is the n-dimensional sphere. take a unitary u of c(s n ) that corresponds to a coordinate projection from s n (n ≥ 2) to t and gives a nontrivial action on a. 2 remark. we can replace with rn with tn in the statement above. note that c∗(tn) ∼= c0(z n ) by the fourier transform and c0(z n ) + ∼= c((zn)+), where (zn)+ is the one point compactification of z n and it is identified with a closed subset of t. example 2.9 let c∗(h2n+1) be the group c ∗ -algebra of the real (2n + 1)-dimensional heisenberg lie group h2n+1. then c ∗ (h2n+1) ∼= c∗(rn+1) ⋊ rn ∼= c0(r n+1 ) ⋊ r n since h2n+1 ∼= rn+1 ⋊ rn. hence there exists no nondegenerate continuous deformation from c ∗ (h2n+1) to c0(r n+1 ) of c0(r n+1 ) ⋊ r n . as for crossed product c ∗ -algebras by semigroups. theorem 2.10 let a be a unital c∗-algebra with no proper isometries and a ⋊α n the semigroup crossed product c∗-algebra of a by an action α of the additive semigroup n of natural numbers by proper isometries. then there exists no nondegenerate continuous deformation from a ⋊α n to a. if a is non unital and without proper isometries, then there exists no nondegenerate continuous deformation from a ⋊α n to the unitization a + by c. proof. suppose that we had a continuous field c∗-algebra γ([0, 1], {at}t∈[0,1]) with fibers at given by a0 = a and at = a ⋊α n for t ∈ (0, 1]. note that a ⋊α n is generated by a and a proper isometry. let s be such a isometry. then we have the covariance relation: sas ∗ = α1(a) for a ∈ a. since s ∗ s = 1 the unit of a (and a ⋊α n) (or 1 ∈ c of a + if a is non unital) the operator field f defined by f (t) = s ∗ s and f (0) = 1 in a is continuous. we may assume that the operator field g defined by g(t) = s for t ∈ (0, 1] and g(0) = a an element of a is continuous. then it follows that a ∗ a = 1. cubo 10, 1 (2008) continuous or discontinuous deformations of c ∗ -algebras 113 if a 6= 1, then the last equation is the contradiction since a has no proper isometries. if a = 1, then note that the operator field h defined by h(t) = ss ∗ for t ∈ (0, 1] and h(0) = 1 is continuous since the operator field g is so. hence, the operator field f − h is also continuous, which is impossible because f (t) − h(t) = 1 − ss∗ 6= 0 for t ∈ (0, 1] but f (0) − h(0) = 1 − 1 = 0. 2 example 2.11 it is known that on ∼= mn∞ ⋊α n (see [9]). since the uhf algebra mn∞ has no proper isometries, we obtain by the theorem above that there exists no nondegenerate continuous deformation from on to mn∞ . theorem 2.12 let a be a unital c∗-algebra with no proper isometries and a ⋊α n × the semigroup crossed product c∗-algebra of a by an action α of the multiplicative semigroup n × of natural numbers by proper isometries. then there exists no nondegenerate continuous deformation from a ⋊α n × to a. if a is non unital and without proper isometries, then there exists no nondegenerate continuous deformation from a ⋊α n × to the unitization a+ by c. proof. note that the semigroup crossed product c∗-algebra a ⋊α n × is generated by a and c ∗ (n × ), and c ∗ (n × ) is isomorphic to the infinite tensor product of c ∗ (n) over prime numbers since n × ∼= ⊕n over prime numbers, where c∗(n) is the c∗-algebra generated by a proper isometry, which is just the usual toeplitz algebra. thus, a ⋊α n corresponding to a and a certain proper isometry in c ∗ (n × ) is regarded as a c ∗ -subalgebra of a ⋊α n × . therefore, we can use the arguments as given in the proof of the theorem above. 2 example 2.13 following laca-raeburn [6], the hecke c∗-algebra of bost-connes is realized as the semigroup crossed product c ∗ -algebra c ∗ (q/z) ⋊α n × . thus, we obtain by the theorem above that there exists no nondegenerate continuous deformation from c ∗ (q/z) ⋊α n × to c ∗ (q/z). moreover, theorem 2.14 let a be a unital c∗-algebra with no proper isometries and a ⋊α n the (reduced or full) semigroup crossed product c∗-algebra of a by an action α of a discrete semigroup n by proper isometries. then there exists no nondegenerate continuous deformation from a ⋊α n to a. if a is non unital and without proper isometries, then there exists no nondegenerate continuous deformation from a ⋊α n to the unitization a + by c. proof. note that the (reduced or full) semigroup crossed product c∗-algebra a ⋊α n is generated by a and isometries corresponding to generators of n . let s be one of the 114 takahiro sudo cubo 10, 1 (2008) isometries. we apply the argument given in the proof of theorem 2.10 for the c ∗ -algebra generated by a and s. 2 as for free products of c ∗ -algebras, theorem 2.15 let a be a c∗-algebra that contains an either unitary or isometry generator. then there exists no nondegenerate continuous deformation from the (full or reduced) unital free product c∗-algebra a ∗c c(t) to c(t). proof. let u be a unitary generator of a and v the generating unitary of c(t). we assume that we had a nondegenerate continuous deformation from (full or reduced) free product c ∗ -algebra a ∗c c(t) to c(t). then we may assume that the constant operator field f by v is continuous and the operator field g from u to a certain unitary w of c(t) is also continuous. then (f g − gf )(t) = f (t)g(t) − g(t)f (t) = v u − u v 6= 0 for t ∈ (0, 1] but (f g − gf )(0) = f (0)g(0) − g(0)f (0) = v w − w v = 0 since c(t) is commutative, which leads to the contradiction. in the argument above we can replace u with a isometry generator s of a since we can assume that the operator field from s to a unitary of c(t) is continuous. 2 example 2.16 since c∗(f2) ∼= c(t) ∗c c(t), there exists no nondegenerate continuous deformation from the full group c ∗ -algebra c ∗ (f2) of f2 to c(t). similarly, theorem 2.17 let a be a c∗-algebra that contains an either unitary or isometry generator u , and a⋊α z be the crossed product c ∗-algebra by a non trivial action α of z on a. suppose that v u v ∗ 6= u , where v is the generaing unitary corresponding to α. then there exists no nondegenerate continuous deformation from a ⋊α z to c(t). proof. consider the operator field from v u v ∗ − u 6= 0 to v w v ∗ − w = v v ∗w − w = 0, where w is a certain unitary of c ∗ (z) ∼= c(t) (by the fourier transform). if we had a nondegenerate continuous deformation from a ⋊α z to c(t), this operator field should be continuous but it is impossible. 2 received: july 2007. revised: september 2007. cubo 10, 1 (2008) continuous or discontinuous deformations of c ∗ -algebras 115 references [1] b. blackadar, k-theory for operator algebras, second edition, cambridge, (1998). [2] a. connes, noncommutative geometry, academic press, (1990). [3] j. cuntz, k-theory for certain c∗-algebras, ann. of math., 113 (1981), 181–197. [4] k.r. davidson, c∗-algebras by example, fields institute monographs, ams. (1996). [5] j. dixmier, c∗-algebras, north-holland, (1962). [6] m. laca and i. raeburn, a semigroup crossed product arising in number theory, j. london math. soc., (2) 59 (1999), 330–344. [7] g.j. murphy, c∗-algebras and operator theory, academic press, (1990). [8] g.k. pedersen, c∗-algebras and their automorphism groups, academic press (1979). [9] m. rørdam and e. størmer, classification of nuclear c∗-algebras. entropy in operator algebras, ems 126 operator algebras and non-commutative geometry vii, springer, (2002). [10] t. sudo, k-theory of continuous deformations of c∗-algebras, acta math. sin. (engl. ser.) 23, no. 7 (2007) 1337–1340 (online 2006). [11] n.e. wegge-olsen, k-theory and c∗-algebras, oxford univ. press (1993). contf-calg2.pdf articulo 14.dvi cubo a mathematical journal vol.12, no¯ 02, (217–234). june 2010 fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols l. p. castro1 and a. s. silva2 department of mathematics, aveiro university, 3810-193 aveiro, portugal email: castro@ua.pt email: anabela.silva@ua.pt abstract we will present sufficient conditions for the fredholm property of wiener-hopf plus and minus hankel operators with different fourier matrix symbols in the c∗-algebra of semialmost periodic elements. in addition, under such conditions, we will derive a formula for the sum of the fredholm indices of these wiener-hopf plus hankel and wiener-hopf minus hankel operators. some examples are provided to illustrate the results of the paper. resumen presentaremos condiciones suficientes para garantizar la propiedad de fredholm de operadores de tipo wiener-hopf más y menos hankel con diferentes śımbolos de fourier matriciales en la c*-álgebra de elementos semi-casi periódicos. además, bajo tales condiciones, obtendremos una fórmula para la suma de los ı́ndices de fredholm de estos operadores wiener-hopf más hankel y wiener-hopf menos hankel. algunos ejemplos son dados para ilustrar los resultados del art́ıculo. key words and phrases: fredholm property, fredholm index, wiener-hopf operator, hankel operator, semi-almost periodic matrix-valued function 1corresponding author: castro@ua.pt 2sponsored by fundação para a ciência e a tecnologia (portugal) under grant number sfrh/bd/38698/2007. 218 l. p. castro and a. s. silva cubo 12, 2 (2010) math. subj. class.: 47b35, 47a05, 47a12, 47a20, 42a75. 1 introduction one of the objectives of the present paper is to obtain sufficient conditions for the fredholm property of matrix wiener-hopf plus and minus hankel operators of the form wφ1 ± hφ2 : [l 2 +(r)] n → [l2(r+)] n (n ∈ n) (1) with wφ1 and hφ2 being matrix wiener-hopf and hankel operators defined by wφ1 = r+f −1φ1 · f and hφ2 = r+f −1φ2 · fj , respectively. we denote by bn×n the banach algebra of all n × n matrices with entries in a banach algebra b, and bn will denote the banach space of all n dimensional vectors with entries in a banach space b. let l2(r) be the usual space of square-integrable lebesgue measurable functions on the real line r, and l2(r+) the corresponding one in the positive half-line r+ = (0, +∞). we are using the notation l2+(r) for the subspace of l 2(r) formed by all the functions supported on the closure of r+. in addition, r+ represents the operator of restriction from [l 2 +(r)] n into [l2(r+)] n , f denotes the fourier transformation, j is the reflection operator given by the rule jϕ(x) = ϕ̃(x) = ϕ(−x), x ∈ r, and (in general) φ1, φ2 ∈ [l ∞(r)]n×n are the so-called fourier matrix symbols. it is well-known that for such fourier matrix symbols (with lebesgue measurable and essentially bounded entries) the operators in (1) are bounded. we would like to point out that the operators presented in (1) have been central objects in several recent research programmes (cf. e.g. [1]–[8]). one of the reasons for such interest is related to the fact that eventual additional knowledge about regularity properties of (1) have direct consequences in different types of applications (see [9]–[12]). in the present work, the main purpose is to obtain conditions which will characterize the situation when wφ1 + hφ2 and wφ1 − hφ2 are at the same time fredholm operators, and to present a formula for the sum of their fredholm indices. all these will be done for matrices φ1 and φ2 in the class of semi-almost periodic elements (cf. definition 2.1). therefore, the present work deals with a more general situation than what was under consideration in [1, 7, 13], and some of the present results can be seen as a generalization of part of the results of the just mentioned works. however, the most general situation of considering the operators wφ1 + hφ2 and wφ1 − hφ2 independent of each other (with semi-almost periodic symbols) is not considered in the present paper and remains open. 2 preliminary results and notions the smallest closed subalgebra of l∞(r) that contains all the functions eλ (λ ∈ r), where eλ(x) = eiλx, x ∈ r, is denoted by ap and called the algebra of almost periodic functions: ap := algl∞(r){eλ : λ ∈ r}. in addition, we will also use the notation ap+ := algl∞(r){eλ : λ ≥ 0}, ap− := algl∞(r){eλ : λ ≤ 0} cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 219 for these two subclasses of ap (which are still closed subalgebras of l∞(r)). we will likewise make use of the wiener subclass of ap (denoted by ap w ) formed by all those elements from ap which allow a representation by an absolutely convergent series. therefore, ap w is precisely the (proper) subclass of all functions ϕ ∈ ap which can be written in an absolutely convergent series of the form: ϕ = ∑ j ϕj eλj , λj ∈ r , ∑ j |ϕj| < ∞ . we recall that all ap functions have a well-known mean value. the existence of such a number is provided in the next proposition. proposition 2.1. (cf., e.g., [14, proposition 2.22]) let a ⊂ (0, ∞) be an unbounded set and let {iα}α∈a = {(xα, yα)}α∈a be a family of intervals iα ⊂ r such that |iα| = yα − xα → ∞ as α → ∞. if ϕ ∈ ap , then the limit m (ϕ) := lim α→∞ 1 |iα| ∫ iα ϕ(x) dx exists, is finite, and is independent of the particular choice of the family {iα}. for any ϕ ∈ ap , the number that has just been introduced m (ϕ) is called the bohr mean value or simply the mean value of ϕ. in the matrix case the mean value is defined entry-wise. let ṙ := r ∪ {∞}. we will denote by c(ṙ) the set of all continuous functions ϕ on the real line for which the two limits ϕ(−∞) := lim x→−∞ ϕ(x), ϕ(+∞) := lim x→+∞ ϕ(x) exist and coincide. the common value of these two limits will be denoted by ϕ(∞). furthermore, c0(ṙ) will represent the collection of all ϕ ∈ c(ṙ) for which ϕ(∞) = 0. let c(r) := c(r) ∩ p c(ṙ), where c(r) is the usual set of continuous functions on the real line and p c(ṙ) is the set of all bounded piecewise continuous functions on ṙ. as mentioned above, we will deal with fourier symbols from the c∗-algebra of semi-almost periodic elements which is defined as follows. definition 2.1. the c∗-algebra sap of all semi-almost periodic functions on r is the smallest closed subalgebra of l∞(r) that contains ap and c(r): sap = algl∞(r){ap, c(r)}. in addition, it is possible to interpret the sap functions in a different form due to the following characterization of d. sarason [15]. theorem 2.1. let u ∈ c(r) be any function for which u(−∞) = 0 and u(+∞) = 1. if ϕ ∈ sap , then there is ϕl, ϕr ∈ ap and ϕ0 ∈ c0(ṙ) such that ϕ = (1 − u)ϕl + uϕr + ϕ0. (2) 220 l. p. castro and a. s. silva cubo 12, 2 (2010) the functions ϕl, ϕr are uniquely determined by ϕ, and independent of the particular choice of u. the maps ϕ 7→ ϕl, ϕ 7→ ϕr are c∗-algebra homomorphisms of sap onto ap. this theorem is also valid in the matrix case. let us now recall the so-called right and left ap factorizations. in such notions, we will use the notation gb for the group of all invertible elements of a banach algebra b. definition 2.2. a matrix function φ ∈ gap n×n is said to admit a right ap factorization if it can be represented in the form φ(x) = φ−(x)d(x)φ+(x) (3) for all x ∈ r, with φ− ∈ gap n×n − , φ+ ∈ gap n×n + , (4) and d is a diagonal matrix of the form d(x) = diag [ eiλ1x, . . . , eiλn x ] , λj ∈ r. the numbers λj are called the right ap indices of the factorization. a right ap factorization with d = in×n is referred to be a canonical right ap factorization. if in a right ap factorization besides condition (4) the factors φ± belong to ap w , then we say that φ admits a right apw factorization (it being clear in such a case that φ ∈ ap w ). it is said that a matrix function φ ∈ gap n×n admits a left ap factorization if instead of (3) we have φ(x) = φ+(x) d(x) φ−(x) for all x ∈ r, and φ± and d having the same property as above. note that from the last definition it follows that if an invertible almost periodic matrix function φ admits a right ap factorization, then φ̃ admits a left ap factorization, and also φ−1 admits a left ap factorization. the vector containing the right ap indices will be denoted by k(φ), i.e., in the above case k(φ) := (λ1, . . . , λn ). if we consider the case with equal right ap indices (k(φ) := (λ1, λ1, . . . , λ1)), then the matrix d(φ) := m (φ−)m (φ+) is independent of the particular choice of the right ap factorization. in this case, this matrix d(φ) is called the geometric mean of φ. in order to relate operators and to transfer certain operator properties between the related operators, we will also be using the known notion of equivalence after extension relation between bounded linear operators. cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 221 definition 2.3. consider two bounded linear operators t : x1 → x2 and s : y1 → y2, acting between banach spaces. we say that t is equivalent after extension to s if there are banach spaces z1 and z2 and invertible bounded linear operators e and f such that [ t 0 0 iz1 ] = e [ s 0 0 iz2 ] f, (5) where iz1 , iz2 represent the identity operators in z1 and z2, respectively. this relation between t and s will be denoted by t ∗ ∼ s. note that such operator relation between two operators t and s, if obtained, allows several consequences on the properties of these two operators. namely, t and s will have the same fredholm regularity properties (i.e., the properties that directly depend on the kernel and on the image of the operator). as we will realize in the next result, such kind of operator relation is valid for a diagonal operator constructed with our wiener-hopf plus and minus hankel operators and a corresponding pure wiener-hopf operator. lemma 2.1. let φ1, φ2 ∈ g[l ∞(r)]n×n . then dφ1,2 := [ wφ1 + hφ2 0 0 wφ1 − hφ2 ] : [l2+(r)] 2n → [l2(r+)] 2n (6) is equivalent after extension to the wiener-hopf operator wψ : [l 2 +(r)] 2n → [l2(r+)] 2n with fourier symbol ψ =   φ1 − φ2φ̃ −1 1 φ̃2 −φ2φ̃ −1 1 φ̃−11 φ̃2 φ̃ −1 1   . (7) we refer to [4, theorem 2.1] for a detailed proof of this lemma (where all the elements in the corresponding operator relation are given in explicit form and within the context of a so-called ∆relation after extension; see [16]). 3 the fredholm property in the present section we will work out characterizations for the fredholm property of wφ1 + hφ2 and wφ1 − hφ2 . we start with the general case (where no dependence between the sap matrices φ1 and φ2 is imposed), and in the last subsection (of the present section) we will consider a particular case where some relation between φ1 and φ2 will allow extra detailed descriptions. 3.1 general case we start by recalling a known fredholm characterization for wiener-hopf operators with sap matrix fourier symbols having lateral almost periodic representatives admitting right ap factorizations. 222 l. p. castro and a. s. silva cubo 12, 2 (2010) theorem 3.1. (cf. e.g., [14, theorem 10.11]) let φ ∈ sap n×n and assume that the almost periodic representatives φℓ and φr admit a right ap factorization. then the wiener-hopf operator wφ is fredholm if and only if: (i) φ ∈ gsap n×n ; (ii) the almost periodic representatives φℓ and φr admit canonical right ap factorizations (and therefore with k(φℓ) = k(φr) = (0, . . . , 0)); (iii) sp(d−1(φr)d(φℓ)) ∩ (−∞, 0] = ∅, where sp(d −1(φr)d(φℓ)) stands for the set of the eigenvalues of the matrix d−1(φr )d(φℓ) := [d(φr)] −1d(φℓ). the matrix version of sarason’s theorem (cf. theorem 2.1) applied to ψ in (7) says that if ψ ∈ gsap 2n×2n then this matrix function admits the following representation ψ = (1 − u)ψℓ + uψr + ψ0, (8) where ψℓ,r ∈ gap 2n×2n are defined for the particular ψ in (7) by ψℓ =   φ1ℓ − φ2ℓφ̃ −1 1r φ̃2r −φ2ℓφ̃ −1 1r φ̃−11r φ̃2r φ̃ −1 1r   (9) and ψr =   φ1r − φ2rφ̃ −1 1ℓ φ̃2ℓ −φ2rφ̃ −1 1ℓ φ̃−11ℓ φ̃2ℓ φ̃ −1 1ℓ   (10) (with φ1ℓ, φ1r and φ2ℓ, φ2r being the local representatives at ∓∞ of φ1 and φ2, respectively), u ∈ c(r), u(−∞) = 0, u(+∞) = 1, ψ0 ∈ [c0(ṙ)] 2n×2n . from (9) it follows that ψ̃−1 ℓ =   φ̃−11ℓ φ̃ −1 1ℓ φ̃2ℓ −φ2rφ̃ −1 1ℓ φ1r − φ2rφ̃ −1 1ℓ φ̃2ℓ   . (11) therefore, we obtain that ψr =   0 in in 0   ψ̃−1 ℓ   0 in in 0   . (12) these representations, and the above relation between the operator (6) and the pure wiener-hopf operator, lead to the following characterization in the case when ψℓ admits a right ap factorization. theorem 3.2. let ψ ∈ sap 2n×2n and assume that ψℓ admits a right ap factorization. in this case, the wiener-hopf plus and minus hankel operators wφ1 + hφ2 and wφ1 −hφ2 are both fredholm if and only if the following three conditions are satisfied: cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 223 (c1) ψ ∈ gsap 2n×2n ; (c2) ψℓ admits a canonical right ap factorization; (c3) sp[hd(ψℓ)] ∩ ir = ∅, where h =   0 in in 0  . proof. (i) let us assume that the wiener-hopf plus and minus hankel operators wφ1 + hφ2 and wφ1 − hφ2 are both fredholm operators. then, wψ is also fredholm due to the above presented equivalence after extension relation. therefore, using theorem 3.1 we obtain that ψ ∈ gsap 2n×2n , ψℓ and ψr admit canonical right ap factorizations and sp(d−1(ψr)d(ψℓ)) ∩ (−∞, 0] = ∅. (13) in particular, we realize that propositions (c1) and (c2) are already fulfilled. additionally, the canonical right ap factorization of ψℓ can be normalized into ψℓ = θ−λθ+, (14) where θ± have the same factorization properties as the original lateral factors of the canonical factorization but with m (θ±) = i, and where λ := d(ψℓ). let h =   0 in in 0   . (15) from (12) and (14) we derive that ψr = hψ̃ −1 ℓ h = hθ̃−1+ λ −1θ̃ −1 − h which shows that d(ψr) = hλ −1h (16) and therefore d−1(ψr) = hλh. (17) in this way, we conclude that sp[d−1(ψr)d(ψℓ)] = sp[hλhλ] = sp[(hλ)2]. thus, (13) turns out to be equivalent to sp[(hλ)2] ∩ (−∞, 0] = ∅ which leads to sp[hλ] ∩ ir = ∅ . therefore, the proposition (c3) is also satisfied. 224 l. p. castro and a. s. silva cubo 12, 2 (2010) (ii) let us now assume that (c1), (c2) and (c3) hold true. from condition (c1) we have ψ ∈ gsap 2n×2n . the left and right representatives of ψ are given by (9) and (10). due to the fact that ψℓ admits a canonical right ap factorization, it follows that ψ −1 ℓ admits a canonical left ap factorization and ψ̃−1 ℓ admits a canonical right ap factorization. therefore,   0 in in 0   ψ̃−1 ℓ   0 in in 0   = ψr (18) admits a canonical right ap factorization. these two canonical right ap factorizations and condition (c3) imply that sp(d−1(ψr)d(ψℓ)) ∩ (−∞, 0] = ∅. all these facts together with theorem 3.1 give us that wψ is a fredholm operator. using the equivalence after extension relation, we obtain that the wiener-hopf plus and minus hankel operators wφ1 + hφ2 and wφ1 − hφ2 are both fredholm operators. let us now think about the case of ψ ∈ sap w 2n×2n , where sap w denotes the algebra of all semi-almost periodic functions ϕ whose almost periodic representatives ϕℓ and ϕr (cf. (2)) belong to ap w . if ψ ∈ sap w 2n×2n , then in theorem 3.2 we can drop the assumption which states that ψℓ admits an ap factorization and also simplify the corresponding conditions (c1) and (c2): corollary 3.1. let ψ ∈ sap w 2n×2n . the wiener-hopf plus and minus hankel operators wφ1 + hφ2 and wφ1 − hφ2 are both fredholm if and only if the following three conditions are satisfied: (c1′) ψ ∈ gsap w 2n×2n ; (c2′) ψℓ admits a canonical right ap w factorization; (c3′) sp[hd(ψℓ)] ∩ ir = ∅, where h =   0 in in 0  . proof. the result is derived from theorem 3.2 and from the following known facts which apply to any φ ∈ gap w 2n×2n : (j) φ has a canonical right ap factorization if and only if φ has a canonical right ap w factorization; (jj) φ has a canonical right ap w factorization if and only if wφ is invertible. in fact, for our ψ ∈ sap w 2n×2n , note that if both operators wφ1 + hφ2 and wφ1 − hφ2 have the fredholm property, then by the above equivalence after extension relation we also have that the wiener-hopf operator wψ is a fredholm operator. therefore, wψℓ and wψr are invertible operators and from (jj) this is equivalent to ψℓ and ψr to admit canonical right ap w factorizations. thus, the assertion now follows from theorem 3.2 and proposition (j). 3.2 the case of φ1 = φ̃2 for some particular cases where φ1 and φ2 are dependent on each other, we can simplify the statement of theorem 3.2 by making use of consequent equivalence after extension operator relations. in the present subsection we will analyze the case of φ1 = φ̃2. cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 225 let φ2 ∈ gsap n×n and consider φ1 = φ̃2. in this case, the matrix ψ takes the form ψ =   0 −in φ−12 φ̃2 φ̃ −1 2   and the wiener-hopf operator wψ is equivalent after extension to the operator wφ−1 2 fφ2 . in fact, we have in this case: wψ = r+f −1   0 −in in φ̃ −1 2   fℓ0r+f−1   φ−12 φ̃2 0 0 in   f (where ℓ0 : [l 2(r+)] 2n → [l2+(r)] 2n denotes the zero extension operator). this together with the equivalence after extension relation between the operator (6) and wψ shows that dφ1,2 ∗ ∼ w φ −1 2 fφ2 (19) (due to the transitivity of the equivalence after extension relation). from theorem 2.1 we conclude that φ2 ∈ gsap n×n admits the following representation φ2 = (1 − u)φ2ℓ + uφ2r + φ20 (20) (with φ20 ∈ [c0(ṙ)] n×n ) and φ−12 φ̃2 = [(1 − u)φ2ℓ + uφ2r + φ20] −1[(1 − ũ)φ̃2ℓ + ũφ̃2r + φ̃20]. (21) therefore, from (21), we obtain that (φ−12 φ̃2)ℓ = φ −1 2ℓ φ̃2r, (φ −1 2 φ̃2)r = φ −1 2r φ̃2ℓ. (22) these representations and the above relation between wψ and wφ−1 2 fφ2 (when φ1 = φ̃2), allow us to construct the following result. theorem 3.3. let φ2 ∈ sap n×n and assume that φ−12ℓ φ̃2r admits a right ap factorization. in this case, the wiener-hopf plus and minus hankel operators w fφ2 + hφ2 and w fφ2 − hφ2 are both fredholm operators if and only if the following three conditions are satisfied: (l) φ2 ∈ gsap n×n ; (ll) φ−12ℓ φ̃2r admits a canonical right ap factorization; (lll) sp[d(φ−12ℓ φ̃2r)] ∩ ir = ∅. proof. (i) if w fφ2 ±hφ2 are both fredholm operators, then from a similar reasoning as in [5, proposition 2.6] it follows that φ2 ∈ g[l ∞(r)]n×n and therefore φ2 ∈ gsap n×n . the fredholm property of the wiener-hopf plus and minus hankel operators w fφ2 + hφ2 and w fφ2 − hφ2 implies that the operator wψ is fredholm and due to the transitivity of equivalence 226 l. p. castro and a. s. silva cubo 12, 2 (2010) after extension relations, it follows that the operator w φ −1 2 fφ2 has also the fredholm property (cf. (19)). employing theorem 3.1 we obtain that φ−12 φ̃2 ∈ gsap n×n , (φ−12 φ̃2)ℓ and (φ −1 2 φ̃2)r admit canonical right ap factorizations and sp[d−1((φ−12 φ̃2)r)d((φ −1 2 φ̃2)ℓ)] ∩ (−∞, 0] = ∅. (23) due to (22) we conclude that φ−12ℓ φ̃2r admits a canonical right ap factorization and we derive from (23) that sp[d−1(φ−12r φ̃2ℓ)d(φ −1 2ℓ φ̃2r)] ∩ (−∞, 0] = ∅. (24) a canonical right ap factorization of φ−12ℓ φ̃2r can be normalized into φ−12ℓ φ̃2r = θ−λθ+, (25) where θ± have the same factorization properties as the original lateral factors of the canonical factorization but with m (θ±) = i, and where λ := d(φ −1 2ℓ φ̃2r). thus, (25) allows φ−12r φ̃2ℓ = ( ˜ φ−12ℓ φ̃2r) −1 = θ̃−1+ λ −1θ̃−1 − which shows that d(φ−12r φ̃2ℓ) = λ −1 and therefore (24) turns out to be equivalent to sp[λ2] ∩ (−∞, 0] = ∅. from the eigenvalue definition, it therefore results in sp[λ] ∩ ir = ∅ which proves proposition (lll). (ii) let us now consider that (l)–(lll) hold true. the property (l) implies that φ−12 φ̃2 is also invertible in sap n×n . since φ−12ℓ φ̃2r = (φ −1 2 φ̃2)ℓ admits a canonical right ap factorization, then ( ˜ φ−12 φ̃2)ℓ = φ̃ −1 2ℓ φ2r admits a canonical left ap factorization and [( ˜ φ−12 φ̃2)ℓ] −1 = φ−12r φ̃2ℓ admits a canonical right ap factorization. these last two canonical right ap factorizations and condition (lll) imply that sp[d−1((φ−12 φ̃2)r)d((φ −1 2 φ̃2)ℓ)] ∩ (−∞, 0] = sp[d −1(φ−12r φ̃2ℓ)d(φ −1 2ℓ φ̃2r)] ∩ (−∞, 0] = ∅. all these facts together with theorem 3.1 show that w φ −1 2 fφ2 is a fredholm operator. using the equivalence after extension relations of (19) and lemma 2.1, we obtain that the wiener-hopf plus and minus hankel operators w fφ2 + hφ2 and w fφ2 − hφ2 have the fredholm property. cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 227 4 index formula in the present section we will be concentrated in obtaining a fredholm index formula for dφ1,2 , i.e., for the sum of wiener-hopf plus and minus hankel operators wφ1 ± hφ2 with fourier symbols φ1, φ2 ∈ gsap n×n such that ψℓ admits a right ap factorization. within this context, let us now assume that wφ1 + hφ2 and wφ1 − hφ2 are fredholm operators. 4.1 general situation let gsap0,0 denote the set of all functions ϕ ∈ gsap for which k(ϕℓ) = k(ϕr ) = 0. to define the cauchy index of ϕ ∈ gsap0,0 we need the lemma presented below. lemma 4.1. (see e.g. [14, lemma 3.12]) let a ⊂ (0, ∞) be an unbounded set and let {iα}α∈a = {(xα, yα)}α∈a be a family of intervals such that xα ≥ 0 and |iα| = yα − xα → ∞, as α → ∞. if ϕ ∈ gsap0,0 and arg ϕ is any continuous argument of ϕ, then the limit 1 2π lim α→∞ 1 |iα| ∫ iα ((argϕ)(x) − (argϕ)(−x))dx (26) exists, is finite, and is independent of the particular choices of {(xα, yα)}α∈a and arg ϕ. the limit (26) is denoted by indϕ and is usually called the cauchy index of ϕ. moreover, following [7, section 4.3] we can generalize the notion of cauchy index for sap presented in lemma 4.1 for functions with k(ϕℓ) + k(ϕr) = 0. the following theorem provides a formula for the fredholm index of matrix wiener-hopf operators with sap fourier symbols. theorem 4.1. (cf. e.g. [14, theorem 10.12]) let φ ∈ sap n×n . if the almost periodic representatives φℓ, φr admit right ap factorizations, and if wφ is a fredholm operator, then ind wφ = −ind[det φ] − n∑ k=1 ( 1 2 − { 1 2 − 1 2π arg ξk }) (27) where ξ1, . . . , ξn ∈ c\(−∞, 0] are the eigenvalues of the matrix d −1(φr)d(φℓ) and {·} stands for the fractional part of a real number. additionally, when choosing arg ξk in (−π, π), we have ind wφ = −ind [det φ] − 1 2π n∑ k=1 arg ξk. we will now be concerned with the question of finding a formula for the sum of the fredholm indices of wφ1 + hφ2 and wφ1 − hφ2 (i.e., ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ]). using the equivalence after extension relation presented in lemma 2.1, we conclude that ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] = ind wψ. 228 l. p. castro and a. s. silva cubo 12, 2 (2010) observing that wψ is a fredholm operator and using (27), we obtain indwψ = −ind[det ψ] − 2n∑ k=1 ( 1 2 − { 1 2 − 1 2π arg ηk }) (28) where ηk ∈ c\(−∞, 0] are the eigenvalues of the matrix of d −1(ψr)d(ψℓ) = (hd(ψℓ)) 2, with h =   0 in in 0   (cf. (16)–(17)). therefore, (28) can be rewritten as indwψ = −ind[det ψ] − 2n∑ n=1 ( 1 2 − { 1 2 − 1 π arg ζk }) (29) where ζk ∈ c\ir are the eigenvalues of the matrix hd(ψℓ). moreover, formula (28) is reduced to indwψ = −ind[det ψ] − 1 π 2n∑ k=1 β(ζk) (30) where β(ζk) := { arg(ζk) if ℜe ζk > 0 arg(−ζk) if ℜe ζk < 0 (31) when choosing the argument in (− π 2 , π 2 ). these conclusions are assembled in the following corollary. corollary 4.1. let ψ ∈ gsap 2n×2n and assume that ψℓ admits a right ap factorization. if wφ1 ± hφ2 are fredholm operators, then ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] = −ind[det ψ] − 2n∑ k=1 ( 1 2 − { 1 2 − 1 π arg ζk }) (32) where ζk ∈ c\ir are the eigenvalues of the matrix hd(ψℓ). moreover, making use of (31), formula (32) simplifies to the following one: ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] = −ind[det ψ] − 1 π 2n∑ k=1 β(ζk). (33) 4.2 the case of φ1 = φ̃2 for the particular case where φ1 = φ̃2 we can simplify formula (33) even further. in fact, when φ1 = φ̃2, employing the equivalence after extension relation (19), we deduce that ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] = −ind[det(φ −1 2 φ̃2)] − 1 π n∑ k=1 β(δk), (34) where δk ∈ c\ir are the eigenvalues of the matrix d(φ −1 2ℓ φ̃2r) and β(δk) = { arg(δk) if ℜe δk > 0 arg(−δk) if ℜe δk < 0 cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 229 with the argument in both cases being chosen in (− π 2 , π 2 ). in addition, let us now simplify the form of ind[det(φ−12 φ̃2)]. observing that the matrix φ −1 2ℓ φ̃2r has a canonical right ap factorization, it holds k(φ−12ℓ φ̃2r) = (0, . . . , 0) and consequently k(det(φ−12ℓ φ̃2r)) = 0. taking this into consideration, it follows that k((det φ−12 )ℓ) + k((det φ −1 2 )r) = k(det(φ −1 2ℓ )) + k(det(φ −1 2r )) = k(det(φ−12ℓ )) + k(det(φ2r) −1) = k(det(φ−12ℓ )) + k[( ˜det(φ2r)−1) −1] = k(det(φ−12ℓ )) + k( ˜det(φ2r)) = k(det(φ−12ℓ )) + k(det(φ̃2r)) = k(det(φ−12ℓ ) det(φ̃2r)) = k(det(φ−12ℓ φ̃2r)) = 0 (35) also because for any f ∈ gap we have k(f ) = k(f̃ −1) and [det φ]ℓ = det φℓ. applying a similar reasoning to φ̃2, we obtain k((det φ̃2)ℓ) + k((det φ̃2)r) = k(det(φ̃2ℓ)) + k(det(φ̃2r)) = k( ˜ det(φ̃2ℓ)−1) + k(det(φ̃2r)) = k(det(φ−12ℓ )) + k(det(φ̃2r)) = k(det(φ−12ℓ ) det(φ̃2r)) = k(det(φ−12ℓ φ̃2r)) = 0. (36) employing now (26), (35) and (36), the following computation holds true: ind[det(φ−12 φ̃2)] = ind[det φ −1 2 det φ̃2] = ind[det φ−12 ] + ind[det φ̃2] = ind[det φ2] −1 + ind[d̃et φ2] = ind[det φ2] −1 − ind[det φ2] = −ind[det φ2] − ind[det φ2] = −2 ind[det φ2]. thus, we have just concluded the following corollary. corollary 4.2. let φ1, φ2 ∈ gsap n×n such that φ1 = φ̃2 and assume that φ −1 2ℓ φ̃2r admits a right ap factorization. if wφ1 ± hφ2 are fredholm operators, then ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] = 2 ind[det φ2] − 1 π n∑ k=1 β(δk) (37) 230 l. p. castro and a. s. silva cubo 12, 2 (2010) where δk ∈ c\ir are the eigenvalues of the matrix d(φ −1 2ℓ φ̃2r) and β(δk) = { arg(δk) if ℜe δk > 0 arg(−δk) if ℜe δk < 0 (38) with the argument in both cases being chosen in (− π 2 , π 2 ). 5 examples in the present section we exemplify the above theory with two particular cases of corresponding fourier symbol matrices φ1 and φ2. 5.1 first example let φ1 = φ̃2, with φ2(x) = (1 − u(x))   eix 0 0 e−ix   + u(x)   e−ix 0 0 eix   +   0 − 1 x−i 1 x+i 0   (39) and where u is the real-valued function defined by u(x) = { 1 2 ex if x < 0 1 − 1 2 e−x if x ≥ 0. (40) from (39) and theorem 2.1, it becomes clear that φ2 ∈ sap 2×2. in addition, we will show that φ2 ∈ gsap 2×2. to this purpose, let us compute the determinant of φ2: det φ2(x) = det   (1 − u(x))eix + u(x)e−ix − 1 x−i 1 x+i (1 − u(x))e−ix + u(x)eix   = 1 + (2u(x) − 2u2(x))(cos(2x) − 1) + 1 x2+1 . recalling that u is a real-valued function given by (40), we obtain det φ2(x) =    1 + (ex − e2x)(cos(2x) − 1) + 1 x2+1 if x < 0 1 + (e−2x − e−x)(cos(2x) − 1) + 1 x2+1 if x ≥ 0 (41) let us first show that det φ2(x) 6= 0 for x ∈ (−∞, 0). in this domain ex − e2x belongs to (0, 1 4 ] and cos(2x) − 1 ∈ [−2, 0]. therefore, − 1 2 < (ex − e2x)(cos(2x) − 1) ≤ 0 and hence, 1 2 < 1 + (ex − e2x)(cos(2x) − 1) ≤ 1 cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 231 1.25 10 20 1.75 0−10 1.5 2.0 1.0 −20 figure 1: the range of det φ2 in the first example. (cf. figure 1). observing that 1 x2+1 ∈ (0, 1) (when x < 0), we conclude that for x < 0: det φ2 > 1 2 . (42) let us now consider x ∈ [0, +∞). in this case, we have e−2x − e−x ∈ [− 1 4 , 0]. this implies that 0 ≤ (e−2x − e−x)(cos(2x) − 1) < 1 2 . hence, 1 ≤ 1 + (e−2x − e−x)(cos(2x) − 1) < 3 2 . observing that 1 x2+1 ∈ (0, 1] (x ≥ 0) we conclude that for x ≥ 0: det φ2 > 1. (43) from (42) and (43), it follows that φ2 ∈ gsap 2×2. now, a direct computation yields that φ−12ℓ φ̃2r =   1 0 0 1   which obviously admits a canonical right ap factorization and d(φ−12ℓ φ̃2r) = i2×2. hence, sp[d(φ−12ℓ φ̃2r)] ∩ ir = {1} ∩ ir = ∅. 232 l. p. castro and a. s. silva cubo 12, 2 (2010) this allows us to conclude that the operators w fφ2 ± hφ2 have the fredholm property. thus, by using the above theory (cf. corollary 4.2) we are now in a position to compute the sum of their fredholm indices. for this case, we have ind[w fφ2 + hφ2 ] + ind[w fφ2 − hφ2 ] − 2 ind det(φ2) − 1 π 2∑ k=1 β(δk) where δk ∈ c\ir are the eigenvalues of the matrix d(φ −1 2ℓ φ̃2r) and β is given by (38). in addition, we have already seen that det φ2 is a real-valued positive function, and therefore its argument is zero. altogether, we have: ind[w fφ2 + hφ2 ] + ind[w fφ2 − hφ2 ] = 0 (since the eigenvalues of d(φ−12ℓ φ̃2r) are also real and positive, and therefore their arguments are also zero). 5.2 second example let φ1 = 1 + e −x 2 and φ2 = −(1 − u(x))e −ix + u(x)e−2ix, where u is the real-valued function defined by u(x) = 1 2 + 1 2 tanh(x). consequently, observing that ũ(x) = 1 − u(x) we have (cf. (7)) ψ =   1 + e−x 2 + ( u(x)e − ix 2 −(1−u(x))e ix 2 ) 2 1+e−x 2 (1−u(x))e−ix−u(x)e−2ix 1+e−x 2 −u(x)eix+(1−u(x))e2ix 1+e−x 2 1 1+e−x 2   . from theorem 2.1, it becomes clear that φ1 and φ2 ∈ sap and thus, the matrix ψ belongs to sap 2×2. since det ψ = 1, we conclude that ψ ∈ gsap 2×2. following (9), we obtain ψℓ =   1 + eix e−ix e2ix 1   . moreover, observing that ψℓ =   e−ix 1 1 0     e2ix 1 1 0   , we conclude that ψℓ admits a canonical right ap factorization and d(ψℓ) =   m (e−ix) m (1) m (1) 0     m (e2ix) m (1) m (1) 0   . since m (e−ix) = m (e2ix) = 0 and m (1) = 1, we obtain that d(ψℓ) = i2×2 and therefore, hd(ψℓ) =   0 1 1 0   , h =   0 1 1 0   . cubo 12, 2 (2010) fredholm property of matrix wiener-hopf plus and minus hankel operators with semi-almost periodic symbols 233 hence, sp[hd(ψℓ)] ∩ ir = {−1, 1} ∩ ir = ∅. these are sufficient conditions for these operators wφ1 ± hφ2 to have the fredholm property (cf. theorem 3.1). let us now calculate the sum of their fredholm indices. for this case, we have ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] − ind[det ψ] − 1 π 2∑ k=1 β(ζk) where ζk ∈ c\ir are the eigenvalues of the matrix hd(ψℓ) and β is given by (31). in addition, we have previously seen that det ψ = 1, therefore having a zero argument. altogether, we have ind[wφ1 + hφ2 ] + ind[wφ1 − hφ2 ] = 0. acknowledgement. this work was supported in part by unidade de investigação matemática e aplicações of universidade de aveiro through the portuguese science foundation (fct–fundação para a ciência e a tecnologia). received: march 2009. revised: may 2009. references [1] g. bogveradze, fredholm theory for wiener-hopf plus hankel operators, phd thesis, university of aveiro, aveiro, 2008. 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[8] a. p. nolasco and l. p. castro, a duduchava-saginashvili’s type theory for wiener-hopf plus hankel operators, j. math. anal. appl. 331 (2007), 329–341. [9] l. p. castro and d. kapanadze, exterior wedge diffraction problems with dirichlet, neumann and impedance boundary conditions, acta appl. math., 110 (2010), 289-311. 234 l. p. castro and a. s. silva cubo 12, 2 (2010) [10] l. p. castro, f.-o. speck and f. s. teixeira, explicit solution of a dirichlet-neumann wedge diffraction problem with a strip, j. integral equations appl. 15 (2003), 359–383. [11] l. p. castro, f.-o. speck and f. s. teixeira, on a class of wedge diffraction problems posted by erhard meister, oper. theory adv. appl. 147 (2004), 211–238. [12] e. meister, f.-o. speck and f. s. teixeira, wiener-hopf-hankel operators for some wedge diffraction problems with mixed boundary conditions, j. integral equations appl. 4 (1992), 229– 255. 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() cubo a mathematical journal vol.12, no¯ 01, (115–132). march 2010 entire functions in weighted l2 and zero modes of the pauli operator with non-signdefinite magnetic field grigori rozenblum department of mathematics, chalmers university of technology, and department of mathematics university of gothenburg, s-412 96 gothenburg, sweden email : grigori@math.chalmers.se and nikolay shirokov department of mathematics and mechanics, st. petersburg state university, russia email : nikolai.shirokov@gmail.com abstract for a real non-signdefinite function b(z), z ∈ c, we investigate the dimension of the space of entire analytical functions square integrable with weight e±2f , where the function f (z) = f (x1, x2) satisfies the poisson equation ∆f = b. the answer is known for the function b with constant sign. we discuss some classes of non-signdefinite positively homogeneous functions b, where both infinite and zero dimension may occur. in the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. the topic is closely related with the question of the dimension of the zero energy subspace (zero modes) for the pauli operator. 116 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) resumen para una función no signo definida b(z), z ∈ c, investigamos la dimensión del espacio de funciones analíticas enteras de cuadrado integrable con peso e±2f , donde la función f (z) = f (x1, x2) verifica la ecuación de poisson ∆f = b. la respuesta es conocida para la función b con signo constante. discutimos algunas clases de funciones b no signo definida e positivamente homogéneas, donde dimensión zero y infinita pueden ocurrir. en el caso anterior nosotros presentamos un método de construir funciones enteras con un comportamiento en infinito prescrito en diferentes direcciones. el tópico es estrechamente relacionado con la cuestión de la dimensión del subespacio de energía zero para el operador de pauli. key words and phrases: pauli operators, zero modes, entire functions. math. subj. class.: 30d15, 81q10, 47n50, 35q40. 1 introduction in 1979 y. aharonov and a. cacher in [1] discovered that the pauli operator in dimension 2 with a compactly supported bounded magnetic field b(x), x = (x1,x2), can possess zero modes, eigenfunctions with zero energy. the number of these zero modes (the dimension of the zero energy eigenspace) is finite and is determined by the total flux of the magnetic field. the zero modes problem has been investigated further on and the aharonov-casher formula was extended to rather singular and not compactly supported magnetic fields being signed measures with finite total variation ([2]). on the other hand, for sign-definite fields with infinite flux, the authors proved in [4] that the space of zero modes is infinite-dimensional, thus extending the aharonov-casher formula to this case. moreover, the infiniteness of zero modes was established in [4] for a class of magnetic fields with variable sign, such that in certain sense the part having one direction is infinitesimal with respect to the part with another direction, while both parts have infinite flux, as well for weakly perturbed constant magnetic fields. on the other hand, in [2] an example was constructed of a magnetic field consisting of tiny islands, sparsely placed in the plane, carrying positive magnetic field, on the background of annuli with negative field, such that both positive and negative parts of the field have infinite total flux, so that no zero modes exist. so, it was, generally, unclear, what is the situation with zero modes for the case when neither of sign-parts of the magnetic field prevails over the other one. after having been acquainted with [4], b.simon asked the first author (g.r.) about the number of zero modes for a very simple configuration of the field of this kind: some constant with one sign in one half-plane and a constant with different sign in another one. the answer (no zero modes at all) was found quite easily, but a more general question arose: how many zero modes are generated by the magnetic field which is constant in a sector in the plane and constant, with different sign, in the complement of this sector, or, more generally, by a non-signdefinite radial-homogeneous field. the present paper contains some results in this direction. for the sector case, it turns out that if one of the sectors is sufficiently small, the space of zero modes is infinite-dimensional. on the other hand, if the angles of the sectors are cubo 12, 1 (2010) zero modes 117 sufficiently close to π, zero modes are completely absent. somewhat similar situation takes place for fields with some other degree of homogeneity. starting from the paper [1], it became clear that the progress in the zero modes problem depends heavily on the properties of solutions of the poisson equation ∆f(x) = b(x) in terms of b(x). the zero modes are generated by entire analytical (or anti-analytical) functions u(z) of the variable z = x1 + ix2 such that u exp(±f) ∈ l2(r2). if b(x) = b > 0 is a nonzero constant the equation has a solution of the form f(x) = b 2 |x|2, and this fact obviously leads to the infiniteness of the dimension of the space of analytical functions with u exp(−f) ∈ l2(r2). however, generally, the boundedness of b does not guarantee by itself a quadratic estimate for f , moreover, it may happen that the poisson equation has no semi-bounded solutions, and such a straightforward reasoning about zero modes fails. we need a deeper analysis of entire functions, square integrable with weight exp(±2f), without the condition imposed, that f is subharmonic (the subharmonic case is investigated exhaustively in [3] and [5]). we start in section 2 by considering solutions of the poisson equations for a radial homogeneous right-hand side. in particular, for our ‘sector’ configuration of the field we construct a special solution of the poisson equation. this solution f is not semi-bounded, behaves at infinity as c|x|2 log |x| in all directions but four, with c depending on the direction and having variable sign. next, in sect.3, we construct entire analytical functions u such that u exp(−f) ∈ l2. such functions u must decay rather rapidly in directions where f is negative, and they may grow, but in a controllable way, in directions where f is positive. we present a method for constructing entire functions with such behavior. this construction can be put through provided the angle of the sector where b is negative is sufficiently small. so, it turns out that in this latter case there are infinitely many zero modes. we consider the case of a sector with an astute angle in sect.4. we show that in this case there are no zero modes at all, provided that the angle of the sector is sufficiently close to π. the reason for this is that, if an entire function is square integrable in a sector with sufficiently fast growing weight, it must be zero everywhere, disregarding its behavior outside the sector. this idea requires certain work for being implemented in our case, since for a sector type magnetic field the sets where the potential has fixed sign differ only slightly from the quarter-plane so that logarithmic effects must be taken into account. in the final section we briefly consider magnetic fields, radial homogeneous of some negative order. the main part of the results were obtained when the second author (n.sh.) was enjoying the hospitality of chalmers university of technology, being supported by a grant from the swedish royal academy of sciences (kva). 118 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) 2 general constructions 2.1 homogeneous solutions of the poisson equation we identify the real plane r2 with the complex plane c1, setting z = x1 +ix2; by dµ we denote the lebesgue measure on the plane. let b(x) be a real-valued function in r2 positively homogeneous of degree s. then, as it is well known, the solution of the poisson equation ∆f(x) = b(x) (2.1) can be looked for as a positively homogeneous function of degree s + 2. in fact, if the function b has the form b(ψ)rs in polar coordinates (r,ψ), we can look for f in the form ϕ(ψ)rs+2 and obtain an equation for ϕ ϕ(ψ)′′ + (s + 2)2ϕ(ψ) = b(ψ) (2.2) if s in not an integer, (2.2) has a unique solution, and thus f = ϕ(ψ)rs+2 is a solution for (2.1). however, if s is an integer, the equation (2.2) is solvable only for those b which are orthogonal to exp(i(s+ 2)ψ). if this orthogonality condition is not satisfied, the solution of (2.1) must contain a logarithmic factor, f(r,ψ) = a sin((s + 2)(ψ − ψ0))rs+2 log r + ϕ(ψ)rs+2, (2.3) with properly selected a and ψ0. such case will be referred to as the resonance one; if the orthogonality condition is met as well as for a noninteger s we have the non-resonance case. for 0 < α < π we denote by ω1 the sector ψ = arg z ∈ (0,α) and by ω2 the complementing sector in the plane. having two numbers, b1 < 0 < b2, we set b(x) ≡ b(x) = 2b1 for x ∈ ω1 and b(x) = 2b2 in ω2. by scaling, one can reduce the situation to the case b2 = 1 and we always suppose that it is already done. we are looking for a solution f(x) of the equation (2.1). since the homogeneity degree equals s = 0, the solution must contain a power-logarithmical term as in (2.3). it is convenient to write the solution of (2.1) in a somewhat different form. 2.2 a solution of the poisson equation for the sector configuration we are looking for an explicit formula for f . this function will be constructed step-wise. we start by elementary solutions separately in ω1 and ω2. these solutions do not fit together on the ray arg z = α. then some correction terms will be introduced. so, we start with φ(z) = b1x 2 2,z ∈ ω1; φ(z) = x22,z ∈ ω2. cubo 12, 1 (2010) zero modes 119 this function satisfies the equation (2.1) everywhere except the ray lα = {arg z = α} where it is discontinuous. to compensate this jump, as well as the jump of its derivative, we will use the branch of logarithm, continuous in the domain c(α) = c\lα. for the function ξ(z) = (ze −iα)2 2π log z the imaginary part has a jump on lα while the real part is continuous but has a discontinuous derivative. our first correction will make the whole solution continuous on lα. the jump of φ at the point z0 = r0e iα equals φ(z0)|ω1 − φ(z0)|ω2 = −c0r20 sin2 α, c0 = 1 − b1, therefore we set φ1(z) = φ(z) + c0 sin 2 α im (ξ(z)). (2.4) since the jump of the second summand in (2.4) at the point z0 = r0e iα equals c0r 2 0 sin 2 α, the function φ1 is continuous in c. we consider now the the derivatives of φ1 at lα. obviously, the derivative along the ray is continuous. the derivative across the ray has a jump, and we will compensate this jump by subtracting the real part of ξ(z) with a proper coefficient. we set f(z) = φ1(z) − c0 sin α cos α re (ξ(z)). since we added the real and imaginary parts of functions that are analytic in c(α), the poisson equation (2.1) will be satisfied by f in c(α). the function f and its derivatives are continuous everywhere, thus the distributional laplacian of f coincides with the classical laplacian, and therefore f is the solution we need. to get a better understanding of f , we represent it in a little bit different way: f(z) = φ(z) + c0 sin 2 α 2π re ( 1 i (zeiα)2 log z ) − (2.5) c0 sin α cos α 2π re ( (zeiα)2 log z ) = φ(z) − c0 sin α 2π re (( ze− iα 2 )2 log z ) . note that the behavior of f(z) for large |z| is determined by the second, power-logarithmic term in (2.5), except the directions where it vanishes, i.e., except the directions arg(z) = α 2 + kπ 4 , k = 0, 1, 2, 3. these half-lines divide c in four quarters, in two of those the function f grows as c|z|2 log |z|, with some positive c (depending on the direction), in the other two this functions tends to −∞, again like c|z|2 log |z| but with a negative c this time. in the next section we will construct entire analytical functions u(z) such that u exp(f) ∈ l2. 3 existence of zero modes the aim of this section is to establish the following fact concerning the sector configuration, as in subsection 2.2. theorem 3.1. suppose that the size α of the sector and b1 are sufficiently small. then the space of entire analytical functions u(z) satisfying u exp(f) ∈ l2 is infinite-dimensional. 120 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) 3.1 construction of a subharmonic function in this subsection we construct a subharmonic function of a special form, to be used further on in the construction of analytical functions with prescribed behavior at infinity. we fix some positive ǫ, to be determined later. consider two sectors θ◦j = {z : | arg z − πj| < ǫ}, j = 0, 1, and set θj = θ ◦ j ∩ {|z| > 1}. for some fixed σ, we cut each of the sectors into strips by straight lines im z = kσ; k = 0,±1,±2, . . . . starting from the boundary lying closest to the imaginary axis, we cut each such strip by lines parallel to the imaginary axis, into domains having area σ2. just a finite number of such domains are not polygons, a few of domains in each strip are triangles or trapezia, all the rest are unit squares. we will denote generically all these pieces of different form by q and the set of these domains by q; by qj we denote the set of pieces in θj. for each q ∈ q we select a point aq ∈ q in the following way. if q is a square we take the center of q as aq. otherwise we choose aq so that ∫ q (z − aq)dµ = 0. a simple geometrical consideration shows that the distance between such points is not less than σ/2. now we define vǫ(z) = re  σ2 ∑ q∈q ( log ( 1 − z aq ) + z aq + 1 2 z2 a2q )  . (3.1) it is clear that the series in (3.1) converges uniformly on compacts not containing the points aq and thus (3.1) defines a harmonic function in the plane, with these points removed. due to symmetry, we can express vǫ(z) via the sum only over the domains q belonging to q1, i.e., lying in the right half-plane, vǫ(z) = re  σ2 ∑ q∈q1 ( log ( 1 − z 2 a2q ) + z2 a2q )  . (3.2) the function vǫ(z) will be approximated by the real part of the integral wǫ(z) = ǫ∫ −ǫ dθ ∞∫ 1 ( log ( 1 − z 2 τ2 e−2iθ ) + z2 τ2 e−2iθ ) τdτ. (3.3) the behavior of wǫ(z) is studied in the appendix. let us estimate the difference vǫ(z) − re wǫ(z) for 2ǫ < | arg z| < π − 2ǫ, i.e. outside some sectorial neighborhood of θj (assuming ǫ < π/8). vǫ(z) − re wǫ(z) = (3.4) ∑ q∈q1 re   ∫∫ q ( log ( 1 − z 2 a2q ) + z2 a2q − log ( 1 − z 2 w2 ) − z 2 w2 ) dµ(w)   . to estimate a single term in (3.4), consider the function β(w) = log(1 − z 2 w2 ) + z 2 w2 , for 2ǫ < cubo 12, 1 (2010) zero modes 121 | arg z| < π − 2ǫ. we have ∫∫ q β(w)dµ(w) = β(aq) + β ′(aq) ∫∫ q (w − aq)dµ(w) + o   ∫∫ q |β′′(w)|dµ(w)   . since ∫∫ q (w − aq)dµ(w) = 0, we get the estimate |vǫ(z) − re wǫ(z)| ≤ c ∑ q∈q1 ∫ q |β′′(w)|dµ(w). (3.5) next, β′′(w) = 2z2 w2 3w2 − z2 (w2 − z2)2 + 6z2w−4, therefore, |β′′(w)| ≤ c|z|2|w|−4, so, finally, |vǫ(z) − re wǫ(z)| ≤ c|z|2 ∫∫ |w|≥1,| arg w|<ǫ |w|−4dµ(w) ≤ ǫ|z|2. (3.6) now we are going to estimate (3.4) in the sectors around x1-axis, | arg z − jπ| ≤ 2ǫ for j = 0 or j = 1. of course, vǫ has logarithmic singularities at all points z = ±aq and wǫ has not. we surround each point by a small disk, |z ± aq| ≤ σ4 and consider first this difference for z lying outside all these disks. we cut the angles into three parts, γ1 : |w| ≥ 2|z|; γ2 : |w| ≤ 1 2 |z|; γ3 : |w| ∈ ( 1 2 |z|, 2|z| ) . correspondingly, we denote by q1,j the set of those domains q ∈ q1 for which aq ∈ γj, j = 1, 2, 3. we suppress the z-dependence of these sets in notations. for q ∈ q1,1, w ∈ q we have 5|w| ≥ |z − w|, |z + w| ≥ |w|2 , therefore the quantity w 2(3w2 − z2)/(w2 − z2)2 is bounded and thus |β′′(w)| ≤ c|z2w−4|. (3.7) summing over q ∈ q1,1, we get ∑ q∈q1,1 ∫∫ q β′′(w)|dµ(w) ≤ c ∫∫ γ1 |z2w−4|dµ(w) ≤ c. (3.8) for q ∈ q1,2, w ∈ q, we note that 5|z| ≥ |z − w|, |z + w| ≥ |z|2 , therefore the quantity w2(3w2 − z2)/(w2 − z2)2 is bounded and we again arrive at (3.7). thus ∑ q∈q1,2 ∫∫ q |β′′(w)|dµ(w) ≤ c ∫∫ γ2 |z2w−4|dµ(w) ≤ cǫ|z2|. (3.9) 122 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) the region γ3 requires a harder work. in any q ∈ q1,3, we write re ( σ2 log ( 1 − z 2 a2q ) + z2 a2q ) − re ∫∫ q ( log ( 1 − z 2 w2 ) + z2 w2 ) dµ(w) = ∫∫ q re ( log(1 − z 2 a2q ) − log ( 1 − z 2 w2 )) dµ(w)+ (3.10) ∫∫ q re ( z2 a2q − z 2 w2 ) dµ(w). consider the second term in (3.10). we have z 2 a2 q − z 2 w2 = z2(z + w)w−2a−2q (z − w). the quantities |z|, |w|, |aq| are of the same order, while |z −w| ≤ 2σǫ−1/2. (of course, |z −w| ≤ σ √ 2 if q is a unit square, but if q is a triangle or a trapezium, only the bound by 2σǫ−1/2 is guaranteed.) therefore, the second term in (3.10) is majorized by cσǫ− 1 2 |z−1|, and since the quantity of domains in q1,3 is of order σ−2ǫ|z|2, we obtain the estimate ∑ q∈q1,3 ∫∫ q re ( z2 a2q − z 2 w2 ) dµ(w) ≤ cσ−1 √ ǫ|z|. (3.11) next we estimate the first term in (3.10). we transform the integrand as log ( 1 − z 2 a2q ) −log ( 1− z 2 w2 ) = log aq − z w − z +log aq + z w + z +2 log w aq . (3.12) in the second term in (3.12) we write ∣∣∣∣log aq + z w + z ∣∣∣∣ = ∣∣∣∣log ( 1 + aq − w z + w )∣∣∣∣ ≤ c√ ǫ|z| . (3.13) similarly, the third term in (3.12) is estimated as ∣∣∣∣log w aq ∣∣∣∣ = ∣∣∣∣log ( 1 + w − aq aq )∣∣∣∣ ≤ c√ ǫ|z| . (3.14) now we pass to the first term in (3.12). we split the sum into two: the sum over such q that |aq − z| ≤ 10σ/ √ ǫ and the sum over the remaining q. consider the first, finite sum (recall that |z − aq| > σ4 ): ∑ q∈q1,3,|aq−z|≤10σ/ √ ǫ ∫∫ q ∣∣∣∣log ( aq − z w − z )∣∣∣∣dµ(w) ≤ c| log σ|ǫ −1. (3.15) for the second sum we have |w − z| ≥ 5σ√ ǫ therefore for w ∈ q ∣∣∣∣log ( aq − z w − z )∣∣∣∣ = ∣∣∣∣log ( 1 + aq − w w − z )∣∣∣∣ ≤ c|w − z| −1, cubo 12, 1 (2010) zero modes 123 and thus ∑ q∈q1,3,|aq−z|≥ 10σ√ǫ ∫∫ q ∣∣∣∣log ∣∣∣∣ aq − z w − z ∣∣∣∣ ∣∣∣∣dµ(w) ≤ c ∫∫ |w|∈(|z|/2,2|z|),|w−z|≥ 5σ√ ǫ dµ(w) |w − z| ≤ cσ|z|. (3.16) summing the estimates (3.5), (3.6), (3.8), (3.9), (3.11), (3.13), (3.14), (3.15), (3.16), we obtain the following inequality. proposition 3.2. for a given ǫ and functions vǫ(z) and wǫ(z) defined as in (3.1) and (3.3), |vǫ(z) − re wǫ(z)| ≤ cǫ|z|2 + c′| log σ|ǫ−1 + cσ|z|, as |z| → ∞ and z avoids σ 4 -neighborhoods of the points aq. in particular, using the asymptotics (a.3) for wǫ, we have vǫ(z) = 1 2 |z|2 log |z| sin(2ǫ) cos(2ψ) + ǫo(|z|2) + o(log |σ|) + cσ|z|, (3.17) if z tends to infinity along the line z = |z|eiψ. it remains to estimate the difference in question for z in σ/4 neighborhood of the point aq0 for some q0 ∈ q. note that by our construction, there can be only one such point aq0 . here we can simply separate the term corresponding to q = q0 in the sum (3.4). for the sum of remaining terms the inequality we just obtained holds. this gives us the following estimate. proposition 3.3. for a given ǫ and functions ve(z) and wǫ(z) defined as in (3.1) and (3.3), ∣∣∣∣vǫ(z) − re wǫ(z) − σ 2 log ( 1 − z aq0 )∣∣∣∣ ≤ cǫ|z| 2 + c′| log σ|ǫ−1 + cσ|z|, (3.18) as |z| → ∞ and z lies in the σ-neighborhood of the point aq0 . the estimates (3.17), (3.18) lead to the following inequalities for the exponent of vǫ(z). proposition 3.4. for a positive κ, we have | exp(κvǫ(z))| ≤ exp(κ 1 2 |z|2 log |z| sin(2ǫ) cos(2ψ) + ǫo(κ|z|2) + o(log |σ|)), (3.19) as z tends to infinity along the line z = |z|eiψ. proof. if z tends to infinity along the line z = |z|eiψ avoiding the σ/4-neighborhoods of the points aq, (3.19) follows from (3.17). for z in these neighborhoods, we apply (3.18) and use that | exp(log ( 1 − z aq0 ) )| is bounded. 124 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) 3.2 construction of entire functions we return to our initial problem. we recall that b2 = 1 and that |b1| is sufficiently small (how small will be determined later). we set κ = c0 sin(α) π sin(2ǫ) ; c0 = 1 − b1. we chose σ = κ− 1 2 and define the function φ(z) as the weierstrass product: φ(z) = ∏ q∈q+ [( 1 − z2 a2q ) exp ( z2 a2q )] . we also denote φα(z) = φ ( ze−i α+π 2 ) . by our choice of σ, this function is related to the function vǫ considered in section 3.1: log |φα(z)|2 = σ−2vǫ ( ze−i α+π 2 ) = κvǫ ( ze−i α+π 2 ) . therefore, by (3.17), with z = reiψ log |φα(z)|2 ≤ 1 2 |z|2 log |z| sin(2ǫ)κ cos(2(ψ − α + π 2 )) +κ(ǫ + 1)o(|z|2) = −c sin α 2π |z|2 log |z| cos(2(ψ − α/2)) + sin α(ǫ + 1)/ sin(2ǫ)o(|z|2). (3.20) with ǫ chosen as α1/2 (thus σ ∼ α− 12 ), we have sin α(ǫ + 1)/ sin(2ǫ) ≤ cα1/2. for a function g(z), to be specified later, we consider the integral i(g) = ∫∫ c e2f(z)|φǫ(z)|2|g(z)|2dµ(z) = ∫∫ c e2f(z)+2 log |φǫ(z)||g(z)|2dµ(z). (3.21) from the estimates for f in (2.5) and for log φ in (3.20), we see that the terms with |z|2 log |z| in the exponent in (3.21) cancel and therefore i(g) ≤ c ∫∫ ω1 e|b1|x 2 2+c0α 1/2|z|2|g(z)|2dµ(z) +c ∫∫ ω2 e−x 2 2+c0α 1/2|z|2|g(z)|2dµ(z) = i1(g) + i2(g). (3.22) now we choose the function g(z). we take it in the form g(z) = exp(−1/4 ( ze−i α 2 )2 )p(z), cubo 12, 1 (2010) zero modes 125 where p(z) is an arbitrary polynomial. then (3.22) implies i1(g) ≤ c ∫∫ ω1 e|b1| sin 2 α|z|2+c0α1/2|z2|− 12 |z| 2 cos 2(ψ−α/2)dµ(z) ≤ ∫∫ ω1 e(|b1|α 2+c0α 1/2)|z2|− 1 2 |z2| cos(α)|p(z)|2dµ(z) < ∞, as soon as α, |b1|α are sufficiently small. further on, we split the integral i2(g) as i2(g)′ +i2(g)′′, so that i2(g) ′ involves integration over the region in ω2 where | arg z| < α1/5 or | arg z−π| < α1/10, and i2(g) ′′ involves the integration over the rest of ω2, i.e., the region where | arg z| > α1/5 and | arg z − π| > α1/10. this gives us i2(g) = i2(g) ′ + i2(g) ′′ ≤ ∫∫ ω2 e− 1 2 cos( 3 2 α1/5)|z|2+c0α1/2|z|2|p(z)|2dµ(z)+ ∫∫ ω2 e− sin 2 α1/5|z|2+c0α1/2|z|2|p(z)|2dµ(z). (3.23) both integrals in (3.23) converge, again, as soon as α is small enough. thus any entire analytical function u(z) of the form u(z) = φα(z) exp(−1/4 ( ze−i α 2 )2 )p(z) belongs to l2(c) with weight exp(2f(z)) and therefore the dimension of the corresponding subspace is infinite. 4 nonexistence of zero modes in this section we prove the following theorem about the non-existence of zero modes. theorem 4.1. suppose that the angle α is sufficiently close to π and |b1| < 12 . then the space of analytical functions u satisfying u exp(±f) ∈ l2 consists only of the zero function. 4.1 a half-plane we consider the case of α = π first. so, let us have b(x) = 2b1 < 0 in the half-plane c + = x2 > 0 and b(x) = 2b2 = 2 in the half-plane c − = x2 < 0. the potential, the solution of the equation ∆f = b can be taken in the form f(z) = b1x 2 2,x2 > 0; f(z) = x 2 2,x2 < 0. we will show that no nontrivial entire analytical function u(z) can belong to l2 with weight e f(z) or e−f(z). actually, a more general statement is correct. 126 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) proposition 4.2. let h(s),s ≥ 0 be a positive function, h(s) → ∞ as s → ∞. then the set of functions u(z), analytical in the half-plane x2 > 0 and continuous up to the boundary such that ∫∫ c+ ex2h(x2)|u(z)|2dλ(z) < ∞ (4.1) consists only of a zero function. it is clear that the absence of nontrivial entire functions in the case we started with follows from proposition 4.2 applied separately to half-planes c+ and c−. moreover, proposition 4.2 will be the destination point for other configurations of b to be considered: having obtained a lower estimate for the function f in some domain, we make a conformal mapping of this domain onto the upper half-plane, where the proposition can be applied. proof. it follows from the condition (4.1) that for any fixed y0 > 0, the function uy0 (x1) = u(x1 + iy0) belongs to l2(r 1) as a function of x1, moreover, u(x1 + ix2) tends to 0 as x2 → ∞, uniformly in x1 ∈ r1. thus u(x1 + ix2) is a bounded harmonic function in a half-plane cy0+ = x2 ≥ y0 with boundary values in l2(r 1), such function can be expressed by means of the fourier transform: u(x1 + ix2) = f−1ξ→x1e −(x2−y0)|ξ|2ûy0(|ξ|). therefore, ∫∫ x2>y0 ex2h(x2)|u(x1 + ix2)|2dµ(z) = ∫∫ x2>y0 ex2h(x2)−2(x2−y0)|ξ||ûy0 (ξ)|2dξdx2, and since h(s) → ∞ as s → ∞, the integral diverges unless uy0 ≡ 0 for any y0 > 0. the case of a sector configuration of b will be reduced to proposition 4.2 by means of a conformal mapping. the following subsection will be devoted to the proof that such special mapping is, in fact univalent. 4.2 univalentness property proposition 4.3. denote for r > 0 by c+r the upper half-plane c + with the disk |z| ≤ r removed. then for any a 6= 0 there exists a number ra > 1 such that the function ζ(ω) = ω(log ω − π 2 i)a is analytical and univalent in c + ra and maps it onto a set ωa ⊂ c. the boundaries of ωa are described by κ = − πa 2 log ρ + o( log log ρ log2 ρ ), ζ = ρeiκ, κ near 0; κ = π + πa 2 log ρ + o( log log ρ log2 ρ ), ζ = ρeiκκ near π. (4.2) cubo 12, 1 (2010) zero modes 127 in other words, the function ζ(ω) maps conformally the upper half-plane, with a disk cut away, onto a slightly, logarithmically, deformed half-plane, with a compact set cut away. proof. the fact that the function ξ is analytical in the domain c+ra and the asymptotic expressions (4.2) for the mapping of the boundaries follow directly from the definition of the function. what, actually, requires being checked is that the function is univalent for r = ra sufficiently large. we start with an intermediate mapping onto a strip. set z = z(ω) = log ω − π 2 i. (4.3) the mapping (4.3) transforms c+ onto the strip {z = x + iy : |y| < π 2 } and the domain cr onto the half-strip πς = {z = x + iy : |y| < π2 ,x > ς}, ς = log r. since ω = ie z, it is sufficient to check that the function ζ(z) = ezza is univalent in πς for ς large enough. we choose ς so that ς > 2π and moreover | arg(za)| < π 40 , | arg(1 + az−1)| < π 40 , z ∈ πς. (4.4) we show first that the function ζ(z) is univalent in any substrip d = {z = x + iy : x > ς, |y − yd| < 0.4π}, such that ς satisfies (4.4) and d ⊂ πς. let zd = ς + 1 + iyd, νd = arg(ζ ′(zd)) = yd + arg(z a d) + arg(1 + a/zd). (4.5) by (4.4), for any z = x + iy ∈ d, we have | arg ζ′(z) − arg ζ′(zd)| (4.6) ≤ |y − yd| + | arg za| + | arg(1 + a/z)| + | arg zad| + | arg(1 + a/zd)| ≤ 0.4π + 4 · π 40 = π 2 . now let z1,z2 be some points in d, z2−z1 |z2−z1| = e iχ. then we have ζ(z2) − ζ(z1) = ∫ [z1,z2] ζ′(t)dt = eiχ |z2−z1|∫ 0 ζ′(z1 + τe iχ)dτ = ei(χ+νd ) |z2−z1|∫ 0 e−iνdζ′(z1 + τe iχ)dτ, 128 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) and therefore, by (4.5), (4.6) |ζ(z1) − ζ(z2)| = ∣∣∣∣∣∣∣ |z2−z1|∫ 0 e−iνdζ′(z1 + τe iχ)dτ ∣∣∣∣∣∣∣ ≥ ∣∣∣∣∣∣∣ |z2−z1|∫ 0 re ( e−iνdζ′(z1 + τe iχ) ) dτ ∣∣∣∣∣∣∣ > 0. now we consider the whole half-strip πς. let us take arbitrary points z1,z2 ∈ πς. if | im (z1−z2)| < 0.8π then z1,z2 lie in some half-strip d of the type just considered and thus ζ(z1) = ζ(z2) is impossible. on the other hand, if im (z2 − z1) ≥ 0.8π, we have arg ζ(z2) − arg ζ(z1) = im (z2 − z1) + (arg(za2 ) − arg(za2 )) > 0.8π − 2π 40 > 0, and again ζ(z1) = ζ(z2) is impossible. 4.3 estimates for a large angle we consider the case when in the setting of section 2 the angle α is close to π. we are going to show that if θ = π − α is small enough then there are no nontrivial entire functions u(z) such that u exp(f) or u exp(−f) belong to l2(c). to do this, we consider two adjoining sectors ξ± with slightly curved boundaries, where the functions ±f are positive. by performing the conformal mapping of ξ± onto the upper half-plane, we arrive at the situation described in subsection 4.1. recall that in polar coordinates r,ψ the function f(z) has the form f(reiψ) = φ(z) − c0 sin θ 2π r2 log r cos(2(ψ − α/2)) + c0 sin θ 2π ψ sin(2(ψ − α 2 ))r2. recall that the polar angle ψ lies in [α,α + 2π) in this representation, c0 = 1 + |b1|. we suppose that α is close to π and consider two sectors in the complex plane, s = {z : ψ ∈ (α 2 + 7 4 π, α 2 + 9 4 π)} and t = {z : ψ ∈ (α 2 + 9 4 π,α + 2π) ∩ [α, α 2 + 3 4 π)} = t1 ∪ t2. in these sectors the log-quadratic part of f is negative, resp. positive. consider the sector s. if z = reiψ ∈ s, we have z ∈ ω1, φ(z) = b1r2 sin2 ψ, and therefore −f(z) = ( |b1| sin2 ψ − c0 sin θ 2π ψ sin(2(ψ − α 2 )) ) r2 (4.7) + c0 sin θ 2π cos(2(ψ − α 2 ))r2 log r. it follows from (4.7)that −f ≥ cr2 log r for sufficiently large r in s with arbitrarily small sectors near the boundary of s removed. to estimate f near the boundary of s, we chose θ so small that c0 sin θ 2π ( 9 4 π + α 2 ) ≤ 1 2 sin2( 9 4 π + α 2 )|b1|. (4.8) cubo 12, 1 (2010) zero modes 129 then, by (4.8), for z = reiψ, ψ close to 9 4 π + α 2 , we have ( |b1| sin2 ψ − c0 sin θ 2π ψ sin(2(ψ − α 2 )) ) r2 ≥ |b1| 2 r2 ≥ cr2. (4.9) for some positive constant c. for ψ close to 7 4 π+ α 2 we note that sin(2(ψ− α 2 )) is negative, therefore ( |b1| sin2 ψ − c0 sin θ 2π ψ sin(2(ψ − α 2 )) ) r2 ≥ cr2 (4.10) now, it follows from (4.9), (4.10) that for some constant a > 0, small enough, the quadratic term in (4.7) majorates the log-quadratic term in the domains s1(a) = {z = reiψ, |ψ−( 74π+ α 2 )| ≤ a log r } and s2(a) = {z = reiψ, |ψ − ( 94π + α 2 )| ≤ a log r }: ∣∣∣∣ c0 sin θ 2π cos(2(ψ − α 2 ))r2 log r ∣∣∣∣ ≤ c 2 r2, z ∈ s1(a) ∪ s2(a). therefore, in the domain s′ = s ∪ s1(a) ∪ s2(a), s′ slightly larger than the quarter-plane, we have −f(z) ≥ c 2 |z|2. now suppose that for some nontrivial analytical function u(z) we have ∫∫ s′ |u(z)|2e−2f(z)dµ(z) < ∞ (4.11) in order to obtain a contradiction, we make a conformal mapping of the domain s′ onto a domain covering the upper halve-plane. we will do it in two steps. let ζ = ρeiκ = (zei 7 4 π)2. under this mapping, the domain s′ is transformed conformally onto s̃ = {ζ : ρ > r20, −δ(ρ) < arg κ < π + δ(ρ), where δ(ρ) ∼ 2a log ρ . next we make a change of variables in the integral in (4.11) by setting v(ζ) = u( √ ζ)√ ζ . we obtain ∫∫ s̃ |v(ζ)|2e|ζ|dµ(ζ) < ∞. (4.12) as it follows from our construction, the domain s̃ is slightly, logarithmically, larger than the upper half-plane with a disk removed. now we map conformally this set onto the upper half-plane with a compact set removed. it is more convenient to do it by considering the inverse mapping. we set ζ = ζ(ω) = ω(log(ω − πi 2 ))a. if a is small enough and a is large enough, the image under this mapping of the upper half-plane with the disk |ω| < a removed, lies in s̃. by proposition 130 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) 4.3, the mapping ζ(ω) is univalent in this set, so the inverse, ω = ω(ζ) exists, maps the image of ζ(ω) onto the ca and its asymptotics as |ζ| → ∞ can be easily found. we change variables in the integral in (4.12) which gives ∫∫ c + a |v(ζ(ω))|2|ζ′(ω)|e c 4 |ω|| log |ω||adµ(ω) < ∞. (4.13) since |ζ′(ω)| behaves logarithmically at infinity, we can apply proposition 4.2 with h(x2) = (log(|x2| + 1))a − c. thus, (4.13) can hold only for v ≡ 0, or u ≡ 0. now we consider the sector t . from the expression for f(z) in (2.5) we obtain for small θ f(rei( α 2 + 3 4 π)) = ( 1 2 + o(θ) ) r2 ≥ 1 4 r2. (4.14) we consider now an auxiliary entire analytical function h(z) = − i 6 e−αz2, so that re h(z) = − 1 6 r2 sin 2(ψ − α 2 − π 2 ). then, by (4.14) and (4.9) for the function f̃(z) = f(z) − re h(z) we have f̃(z) ≥ 1 12 r2 for ψ = 9 4 π + α 2 , f̃(z) ≥ 1 24 r2 for ψ = 3 4 π + α 2 . so, the log-quadratic term in f̃ is positive in the sector t and the quadratic term in f̃ is positive on the boundaries of t . therefore, similar to the above consideration in the sector s, the function f̃ admits a quadratic lower estimate in a domain slightly larger than t : f̃(z) ≥ c|z|2,ψ ∈ [α 2 , 3 4 π + a log r ] ∪ [ 9 4 π + α 2 − a log r , α 2 + 2π]. if an entire analytical function u satisfies ∫∫ c e2f |u|2δµ(z) < ∞ then the function ũ = u exp(h) satisfies ∫∫ c e2f̃ |ũ|2δµ(z) < ∞. it remains to repeat the reasoning with the conformal mapping used for the sector s to show that the function ũ and therefore u must necessarily be zero. this concludes the proof of theorem 4.1. 5 the non-resonance case as it can be observed from the calculations above, the main trouble in the study of the ’sector’ configuration is created by power-logarithmic behavior of the function f . in the non-resonance case such terms are not present in f , therefore the analysis of zero modes is considerably easier. theorem 5.1. let b(z) = b(reiψ) be radial homogeneous of degree s ∈ (−2, 0] and β0 = (2π)−1 ∫ 2π 0 b(eiψ)dψ 6= 0. for s = 0,−1 we suppose that ∫ 2π 0 b(eiψ)ei(s+2)ψdψ = 0. then, if∫ 2π 0 |b(eiψ) − β0|2dψ is sufficiently small then the space of zero modes is infinite-dimensional. proof. suppose that β0 > 0 and set b̃(x) = b(x) − β0rs. then the solution f of the poisson equation ∆f = b can be represented as f = φ + f̃ , where φ(x) = β0(s + 2) −2|r|s+2 and cubo 12, 1 (2010) zero modes 131 f̃ = ϕ(ψ)rs+2 with ϕ(ψ) being a solution of ϕ′′(ψ) + (s + 2)2ϕ(ψ) = b̃(ψ). such solution exists (for s = 0 or s = −1 we use the orthogonality condition and require also that ϕ(ψ) is again orthogonal to ei(s+2)ψ ). moreover, the solution ϕ(ψ), by ellipticity, belongs to the sobolev space h2 on the circle s1 with estimate ||ϕ||h2(s1) ≤ c||b̃||l2(s1). thus, if the latter norm is small enough, then, by the embedding theorem, ||ϕ||c(s1) ia also small and can be made smaller than |β0|. in this case, it turns out that f(x) ≥ c|x|s+2 and therefore for any polynomial p(z) the integral ∫∫ c |p(z)|2e−2fdµ(z) converges. we explain here the role of the orthogonality condition in the cases s = 0 and s = −1. if it is violated, the solution of the poisson equation contains necessarily a log-power term, similar to the one considered in sections 3,4, and therefore will not be sign-definite. the same complication arises in the case s = −2. finally, we present a construction showing that in the nonresonance case the absence of zero modes can also occur. example 5.2. let s ∈ (−1, 0]. we construct the function f(z) = f(reip) = f(ψ)rs+2 in the following way. for some ǫ > 0, ǫ < 1 4 (1 + s), we consider two disjoint arcs i+,i− in s 1 having length π(s+ 2−ǫ)−1 < π 2 . we set f(ψ) = β+ > 0 on i+, f(ψ) = β− < 0 on i− with some constants β± and define f(ψ) in an arbitrary way on the complement of i±, to obtain a smooth function on the circle. denote by s± the sectors in c defined by the arcs i±. supposing that there exists an analytical function u(z) satisfying ∫∫ s+ e2f |u(z)|2dµ(z) < ∞, we make a conformal mapping z(ζ) of the upper half-plane c+ onto the sector s+, z = z0ζ (s+2−ǫ)−1 , |z0| = 1. then the integral transforms to (s + 2 − ǫ)−1 ∫∫ c+ e2f(z0ζ (s+2−ǫ)−1) |u(z0ζ(s+2−ǫ) −1 )|2|ζ|(s+2−ǫ) −1−1dµ(ζ). (5.1) for the exponent 2f(z0ζ (s+2−ǫ)−1 ) we have the lower estimate by |z| s+2 s+2−ǫ , and by proposition 4.2 the function u should be zero. in a similar way, the integral ∫∫ s− e−2f |u(z)|2dµ(z) cannot be finite unless u ≡ 0. a some integrals we present here the calculation of the integral in (3.3). we show first that v(z) = ∞∫ 1 ( t log(1 − z 2 t2 ) + z2 t2 ) dt = z2 − 1 2 log(1 − z2) − z 2 2 . (a.1) it is clear that v(0) = 0. we find the derivative of v(z). for |z| < 1 it is legal to differentiate under the integral sign, therefore v′(z) = −z ∞∫ 1 ( 1 t − z + 1 t + z − 2 t )dt = z log(1 − z2). 132 grigori rozenblum and nikolay shirokov cubo 12, 1 (2010) the derivative of the right-hand side in (a.1) gives the same expression. for |z| ≥ 1, im z 6= 0, we can continue analytically the expression in (a.1) separately to the upper and lower half-planes, and the corresponding branches of the logarithm should be used. next we consider the integral in (3.3). we represent it, using (a.1) as wǫ(z) = ǫ∫ −ǫ dϑ ∞∫ 1 ( log(1 − z 2 τ2 e−2iϑ) + z2 τ2 e−2iϑ ) τdτ = 1 2 ǫ∫ −ǫ (zeiθ)2 log(1 − (zeiθ)2)dθ − 1 2 ǫ∫ −ǫ (zeiϑ)2dϑ − 1 2 ǫ∫ −ǫ log(1 − (zeiϑ)2)dϑ. (a.2) we estimate the ϑ integral for small ǫ, obtaining wǫ(z) = 1 2 |z|2 log |z| sin(2ǫ) cos(2φ) + ǫo(|z|2) (a.3) as z tends to infinity along the line z = |z|eiφ. received: october, 2008. revised: october, 2008. references [1] aharonov, y. and casher, a., ground state of a spin1 2 charged particle in a twodimensional magnetic field, phys. rev. a (3), 19 (1979), no.6, 2461–2462. [2] erdös, l. and vugalter, v., pauli operator and aharonov-casher theorem for measure valued magnetic fields, comm. math. phys., 225 (2002), no. 2, 399–421. [3] hörmander, l., an introduction to complex analysis in several variables, north holland, princeton, amsterdam, 1973. [4] rozenblum, g. and shirokov, n., infiniteness of zero modes for the pauli operator with singular magnetic field, j. funct. anal., 233 (2006), no. 1, 135–172. [5] shigekawa, i., spectral properties of schrödinger operators with magnetic fields for a spin 1 2 particle, j. func. anal., 101 (1991), 255–285. introduction general constructions homogeneous solutions of the poisson equation a solution of the poisson equation for the sector configuration existence of zero modes construction of a subharmonic function construction of entire functions nonexistence of zero modes a half-plane univalentness property estimates for a large angle the non-resonance case some integrals cubo a mathematical journal vol.11, no¯ 04, (127–136). september 2009 on an inequality related to the radial growth of subharmonic functions juhani riihentaus department of physics and mathematics, university of joensuu, p.o. box 111, fi-80101 joensuu, finland. email: juhani.riihentaus@joensuu.fi abstract it is a classical result that every subharmonic function, defined and lp-integrable for some p, 0 < p < +∞, on the unit disk d of the complex plane c is for almost all θ of the form o((1 − |z|)−1/p), uniformly as z → eiθ in any stolz domain. recently pavlović gave a related integral inequality for absolute values of harmonic functions, also defined on the unit disk in the complex plane. we generalize pavlović’s result to so called quasi-nearly subharmonic functions defined on rather general domains in rn, n ≥ 2. resumen es un resultado clásico que toda función subarmónica definida y lp-integrable para algún p, 0 < p < +∞, sobre el disco unitario d del plano complejo c es para casi todo θ de la forma o((1 − |z|)−1/p), uniformemente cuando z → eiθ en cualquier dominio de stolz. recientemente, pavlović encontró una desigualdad integral relacionada para valores absolutas de funciones armónicas, también definidas en el disco unitario del plano complejo. generalizamos el resultado de pavlović a las así llamada funciones subarmónicas casi-cercanas definidas en dominios bastante generales en rn, n ≥ 2. key words and phrases: subharmonic function, quasi-nearly subharmonic function, accessible boundary point, approach region, integrability condition, radial order. 128 juhani riihentaus cubo 11, 4 (2009) math. subj. class.: 31b25, 31b05. 1 introduction 1.1 previous results. the following theorem is a special case of the original result of gehring [4, theorem 1, p. 77], and of hallenbeck [5, theorems 1 and 2, pp. 117-118], and of the later and more general results of stoll [23, theorems 1 and 2, pp. 301-302, 307]: theorem a: if u is a function harmonic in d such that i(u) := ∫ d | u(z) |p (1− | z |)β dm(z) < +∞, (1) where p > 0, β > −1, then lim r→1− | u(reiθ) |p (1 − r)β+1 = 0 (2) for almost all θ ∈ [0, 2π). observe that gehring, hallenbeck and stoll in fact considered subharmonic functions and that the limit in (2) was uniform in stolz approach regions (in stoll’s result in even more general regions). for a more general result, see [19, theorem, p. 31], [15, theorem, p. 233], [10, theorem 2, p. 73] and [18, theorem 3.4.1, pp. 198-199]. with the aid of [12, theorem a and theorem 1, pp. 433-434], pavlović showed that the convergence in (2) in theorem a is dominated. at the same time he pointed out that whole theorem a follows from his result: theorem b ([12, theorem 1, pp. 433-434]) if u is a function harmonic in d satisfying (1), where p > 0, β > −1, then j(u) := 2π∫ 0 sup 0 0 write dρ = {x ∈ d : δ(x) < ρ}. bn(x,r) is the euclidean ball in rn, with center x and radius r, and b(x) = bn(x, 1 3 δ(x)). we write bn = b(0, 1) and sn−1 = ∂bn. m is the lebesgue measure in rn, and νn = m(b n ). l1 loc (d) is the space of locally (lebesgue) integrable functions on d. the d-dimensional hausdorff (outer) measure in rn is denoted by hd, 0 ≤ d ≤ n. our constants c and k are always positive, mostly ≥ 1 and they may vary from line to line. (one exception: in the proof of theorem 2 we write k for ∂ω, just in order to follow our previous notation in [19].) on the other hand, c0 and r0 are fixed constants which are involved with the used (and thus fixed) admissible function ϕ (see 1.5 (5) below). similarly, if α > 0 is given, c1 = c1(c0,α), c2 = c2(c0,α) and c3 = c3(c0,α) are fixed constants, coming directly from [19, lemma 2.3, pp. 32-33] or [15, lemma 2.3, p. 234], and thus defined already there. 1.3 nearly subharmonic functions. we recall that an upper semicontinuous function u : d → [−∞, +∞) is subharmonic if for all bn(x,r) ⊂ d, u(x) ≤ 1 νn r n ∫ bn(x,r) u(y) dm(y). the function u ≡ −∞ is considered subharmonic. we say that a function u : d → [−∞, +∞) is nearly subharmonic, if u is lebesgue measurable, u + ∈ l1 loc (d), and for all bn(x,r) ⊂ d, u(x) ≤ 1 νn r n ∫ bn(x,r) u(y) dm(y). observe that in the standard definition of nearly subharmonic functions one uses the slightly stronger assumption that u ∈ l1 loc (d), see e.g. [6, p. 14]. however, our above, slightly more general definition seems to be more useful, see [21, proposition 2.1 (iii) and proposition 2.2 (vi), (vii), pp. 54-55]. 1.4 quasi-nearly subharmonic functions. a lebesgue measurable function u : d → [−∞, +∞) is k-quasi-nearly subharmonic, if u+ ∈ l1 loc (d) and if there is a constant k = k(n,u,d) ≥ 1 such that for all bn(x,r) ⊂ d, um (x) ≤ k νn r n ∫ bn(x,r) um (y) dm(y) (3) for all m ≥ 0, where um := sup{u,−m} + m. a function u : d → [−∞, +∞) is quasi-nearly subharmonic, if u is k-quasi-nearly subharmonic for some k ≥ 1. a lebesgue measurable function u : d → [−∞, +∞) is k-quasi-nearly subharmonic n.s. (in the narrow sense), if u+ ∈ l1 loc (d) and if there is a constant k = k(n,u,d) ≥ 1 such that for all 130 juhani riihentaus cubo 11, 4 (2009) bn(x,r) ⊂ d, u(x) ≤ k νn r n ∫ bn(x,r) u(y) dm(y). (4) a function u : d → [−∞, +∞) is quasi-nearly subharmonic n.s., if u is k-quasi-nearly subharmonic n.s. for some k ≥ 1. quasi-nearly subharmonic functions (perhaps with a different terminology), or, essentially, perhaps just functions satisfying a certain generalized mean value inequality, more or less of the form (3) or (4) above, have previously been considered or used at least in [3, 25, 8, 14, 24, 5, 11, 9, 23, 15, 10, 16, 17, 18, 13, 19, 20, 21, 7]. we recall here only that this function class includes, among others, subharmonic functions, and, more generally, quasisubharmonic and nearly subharmonic functions (for the definitions of these, see above and e.g. [6]), also functions satisfying certain natural growth conditions, especially certain eigenfunctions, polyharmonic functions, subsolutions of certain general elliptic equations. also, the class of harnack functions is included, thus, among others, nonnegative harmonic functions as well as nonnegative solutions of some elliptic equations. in particular, the partial differential equations associated with quasiregular mappings belong to this family of elliptic equations, see vuorinen [26]. observe that already domar [2] has pointed out the relevance of the class of (nonnegative) quasi-nearly subharmonic functions. to motivate the reader still further, we recall here the following, see e.g. [13, proposition 1, theorem a, theorem b, p. 91] and [21, proposition 2.1 and proposition 2.2, pp. 54-55]: (i) a k-quasi-nearly subharmonic function n.s. is k-quasi-nearly subharmonic, but not necessarily conversely. (ii) a nonnegative lebesgue measurable function is k-quasi-nearly subharmonic if and only if it is k-quasi-nearly subharmonic n.s. (iii) a lebesgue measurable function is 1-quasi-nearly subharmonic if and only if it is 1-quasinearly subharmonic n.s. and if and only if it is nearly subharmonic (in the sense defined above). (iv) if u : d → [0, +∞) is quasi-nearly subharmonic and p > 0, then up is quasi-nearly subharmonic. especially, if h : d → r is harmonic and p > 0, then | h |p is quasi-nearly subharmonic. (v) if u : d → [−∞, +∞) is quasi-nearly subharmonic n.s., then either u ≡ −∞ or u is finite almost everywhere in d, and u ∈ l1 loc (d). 1.5 admissible functions. a function ϕ : [0, +∞) → [0, +∞) is admissible, if it is strictly increasing, surjective and there are constants c0 = c0(ϕ) ≥ 1 and r0 > 0 such that ϕ(2t) ≤ c0 ϕ(t) and ϕ−1(2s) ≤ c0 ϕ−1(s) (5) cubo 11, 4 (2009) on an inequality related to the radial growth of subharmonic ... 131 for all s, t, 0 ≤ s, t ≤ r0. functions ϕ1(t) = t τ , τ > 0, or, more generally, nonnegative, increasing surjective functions ϕ2(t) which satisfy the ∆2-condition and for which the functions t 7→ ϕ2(t)t are increasing, are examples of admissible functions. further examples are ϕ3(t) = ct α [log(δ + t γ )] β , where c > 0, α > 0, δ ≥ 1, and β,γ ∈ r are such that α + βγ > 0. for more examples, see [15, 18]. let ϕ : [0, +∞) → [0, +∞) be an admissible function and let α > 0. one says that ζ ∈ ∂d is (ϕ,α)-accessible, shortly accessible, if γϕ(ζ,α) ∩ bn(ζ,ρ) 6= ∅ for all ρ > 0. here γϕ(ζ,α) = {x ∈ d : ϕ(|x − ζ|) < αδ(x) }, and it is called a (ϕ,α)-approach region, shortly an approach region, in d at ζ. choosing ϕ(t) = t (in the case of the unit disk d of the complex plane c) one gets the familiar stolz approach region. choosing ϕ(t) = tτ , τ ≥ 1, say, one gets more general approach regions, see [23]. 1.6 let 0 ≤ d ≤ n. a set e ⊂ rn is ahlfors-regular with dimension d if it is closed and there is a constant c4 > 0 so that c −1 4 r d ≤ hd(e ∩ bn(x,r)) ≤ c4rd for all x ∈ e and r > 0. the smallest constant c4 is called the regularity constant for e. simple examples of ahlfors-regular sets include d-planes and d-dimensional lipschitz graphs. also certain cantor sets and self-similar sets are ahlfors-regular. for more details, see [1, pp. 9-10]. 2 the results 2.1 first a partial generalization to pavlović’s result [12, theorem 1, pp. 433-434] or theorem b above. observe that though the constant c below in (6) does depend on k, it is, nevertheless, otherwise independent of the (k-)quasi-nearly subharmonic function u. theorem 1 let ω be a domain in rn, n ≥ 2, ω 6= rn, such that its boundary ∂ω is ahlforsregular with dimension d, 0 ≤ d ≤ n. let u : ω → [0, +∞) be a k-quasi-nearly subharmonic function. let ϕ : [0, +∞) → [0, +∞) be an admissible function, with constants r0 and c0. let α > 0 be arbitrary. let ρ0 := min{r0/21+α,r0/23αc0,ϕ(r0)/α}. let γ ∈ r be such that ∫ ω δ(x) γ u(x) dm(x) < +∞. then there is a constant c = c(n, ω,d,ϕ,α,γ,k) such that for all ρ ≤ ρ0, ∫ ∂ω sup x∈γϕ,ρ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) ≤ c ∫ ωρ′ δ(x) γ u(x) dm(x), 132 juhani riihentaus cubo 11, 4 (2009) where ρ′ = 4 3 ρ and γϕ,ρ(ζ,α) = {x ∈ γϕ(ζ,α) : δ(x) < ρ}. proof. proceeding as in [19, proof of theorem (with ψ = id), pp. 31-35] (cf. [15, proof of theorem, pp. 235-237]) and choosing k = ∂ω, one obtains ∫ ∂ω m ∂ω ρ (ζ) dh d (ζ) ≤ c ∫ ωρ′ δ(x) γ u(x) dm(x) where ρ′ = 4 3 ρ and m∂ωρ : ∂ω → [0, +∞], m ∂ω ρ (ζ) sup x∈γϕ,ρ(ζ,α) δ(x) n+γ u(x) [ϕ−1(δ(x))]d + hd(bn(x,c1c2 ϕ −1(δ(x))) ∩ ∂ω). here and below the constants c1 = c1(c0,α), c2 = c2(c0,α) and c3 = c3(c0,α) are, as pointed out above, directly from [19, proof of lemma 2.3, pp. 32-33] or [15, proof of lemma 2.3, pp. 234-235]. by this lemma one has, for each ζ ∈ ∂ω and for each x ∈ γϕ,ρ(ζ,α), bn(x,c1c2ϕ−1(δ(x))) ⊂ b n (ζ,c1c2c3ϕ −1 (δ(x))). since ∂ω is ahlfors-regular with dimension d, we have h d (b n (ζ,c1c2c3ϕ −1 (δ(x))) ∩ ∂ω) ≤ c4[c1c2c3ϕ−1(δ(x))]d where also c4 is a fixed constant. therefore m ∂ω ρ (ζ) sup x∈γϕ,ρ(ζ,α) δ(x) n+γ u(x) [ϕ−1(δ(x))]d + hd(bn(x,c1c2 ϕ −1(δ(x))) ∩ ∂ω) ≥ sup x∈γϕ,ρ(ζ,α) δ(x) n+γ u(x) [ϕ−1(δ(x))]d + hd(bn(ζ,c1c2c3 ϕ −1(δ(x))) ∩ ∂ω) ≥ sup x∈γϕ,ρ(ζ,α) δ(x) n+γ u(x) [ϕ−1(δ(x))]d + c4(c1c2c3) d[ϕ−1(δ(x))]d ≥ 1 1 + (c1c2c3) dc4 sup x∈γϕ,ρ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }. hence ∫ ∂ω sup x∈γϕ,ρ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) ≤ c ∫ ωρ′ δ(x) γ u(x) dm(x), concluding the proof. 2.2 theorem 1 seems to be useful in many situations. for example, with the aid of it one gets the following improvements to pavlović’s result [12, theorem 1, pp. 433-434] or theorem b above: theorem 2 let ω, d, u, ϕ, α, γ and ρ0 be as above in theorem 1. suppose moreover that h d (∂ω) < +∞. then there is a constant c = c(n, ω,d,ϕ,α,γ,k) such that ∫ ∂ω sup x∈γϕ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x)}dhd(ζ) ≤ c ∫ ω δ(x) γ u(x) dm(x). cubo 11, 4 (2009) on an inequality related to the radial growth of subharmonic ... 133 proof. by theorem 1 (we may clearly assume that ∫ ω δ(x) γ u(x) dm(x) < +∞), ∫ ∂ω sup x∈γϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) ≤ c ∫ ω ρ′ 0 δ(x) γ u(x) dm(x). write γ c ϕ,ρ0 (ζ,α) := {x ∈ γϕ(ζ,α) : δ(x) ≥ ρ0}. since sup x∈γϕ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) } ≤ sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γϕ−1(δ(x))]−du(x) } + sup x∈γϕ,ρ0 (ζ,α) {δ(x)n+γϕ−1(δ(x))]−du(x) }, we obtain: ∫ ∂ω sup x∈γϕ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) ≤ ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) + ∫ ∂ω sup x∈γϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) ≤ ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) + c ∫ ω ρ′ 0 δ(x) γ u(x) dm(x) ≤ ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) + c ∫ ω δ(x) γ u(x) dm(x). it remains to show that ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x)}dhd(ζ) ≤ c ∫ ω δ(x) γ u(x) dm(x) for some c = c(n, ω,d,ϕ,α,γ,k). for all x ∈ γcϕ,ρ0 (ζ,α) we have u(x) ≤ k νn( δ(x) 3 )n ∫ b(x) u(y) dm(y). 134 juhani riihentaus cubo 11, 4 (2009) using also the facts that 2 3 δ(x) ≤ δ(y) ≤ 4 3 δ(x) for all y ∈ b(x), one gets easily: ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x)}dhd(ζ) ≤ ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−d k νn( δ(x) 3 )n ∫ b(x) u(y) dm(y)}dhd(ζ) ≤ 3 n k νn ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {δ(x)γ [ϕ−1(δ(x))]−d ∫ b(x) u(y) dm(y)}dhd(ζ) ≤ ( 3 2 )|γ| 3 n k νn ∫ ∂ω sup x∈γcϕ,ρ0 (ζ,α) {[ϕ−1(δ(x))]−d ∫ b(x) δ(y) γ u(y) dm(y)}dhd(ζ) ≤ 3 |γ|+n k 2|γ|νn [ϕ −1 (ρ0)] −d h d (∂ω) ∫ ω δ(y) γ u(y) dm(y). thus ∫ ∂ω sup x∈γϕ(ζ,α) {δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dhd(ζ) ≤ c ∫ ω δ(x) γ u(x) dm(x), concluding the proof. corollary let u : bn → [0, +∞) be a subharmonic function and let p > 0, α > 1 and γ > −1 − max{ (n − 1)(1 − p), 0 }. then there is a constant c = c(n,γ,p,α) such that ∫ sn−1 sup x∈γid(ζ,α) {(1− | x |)γ+1u(x)p}dσ(ζ) ≤ c ∫ bn (1− | x |)γu(x)p dm(x). here id is the identity mapping of rn and σ is the spherical (lebesgue) measure in sn−1. remark observe that suzuki [24, theorem 2, pp. 272-273] has shown the following: if p > 0 and γ ≤ −1 − max{ (n − 1)(1 − p), 0 }, then the only nonnegative subharmonic function on a bounded domain d of rn with c2 boundary satisfying ∫ d δ(x) γ u(x) p dm(x) < +∞ (6) is the zero function. on the other hand, if p > 0 and γ > −1 − max{ (n − 1)(1 − p), 0 }, then there exist nonnegative non-zero subharmonic functions on d = bn satisfying (6). received: august 2008. revised: november 2008. references [1] g. david and s. semmes, analysis of and on uniformly rectifiable sets, math. surveys and monographs 38, amer. math. soc., providence, rhode island (1991). cubo 11, 4 (2009) on an inequality related to the radial growth of subharmonic ... 135 [2] y. domar, on the existence of a largest subharmonic minorant of a given function, ark. mat. 3, nr. 39 (1957), 429–440. 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[26] m. vuorinen, on the harnack constant and the boundary behavior of harnack functions, ann. acad. sci. fenn., ser. a i, math. 7 (1982), 259–277. articulo 9 a mathematical journal vol. 7, no 3, (87 94). december 2005. convergence rates in regularization for ill-posed variational inequalities nguyen buong 1 vietnamse academy of science and technology, institute of information technology 18, hoang quoc viet, q. cau giay, ha noi, vietnam nbuong@ioit.ncst.ac.vn abstract in this paper the convergence rates for ill-posed inverse-strongly monotone variational inequalities in banach spaces are obtained on the base of choosing the regularization parameter by the generalized discrepancy principle. resumen en este art́ıculo se obtienen tasas de convergencia para desigualdades variacionales en problemas inversos mal puestos fuertemente monótonos en espacios de banach, sobre la base de la elección del parámetro de regularización por medio del principio de discrepancia generalizada. key words and phrases: monotone operators, hemi-continuous, strictly convex banach space, frechet differentiable and tikhonov regularization. math. subj. class.: 47h17; cr: g1.8. 1the author would like to express his thanks to the referees for their valuable remarks. this work was supported by the national fundamental research program in natural sciences. 88 nguyen buong 7, 3(2005) 1 introduction. let x be a real reflexive banach space having the e-property and x∗, the dual space of x, be strictly convex. for the sake of simplicity, the norms of x and x∗ will be denoted by the symbol ‖.‖. we write 〈x∗, x〉 instead of x∗(x) for x∗ ∈ x∗ and x ∈ x. let a be a hemi-continuous and monotone operator from x into x∗, and k be a closed convex subset of x. for a given f ∈ x∗, consider the variational inequality: find an element x0 ∈ k such that 〈a(x0) − f, x − x0〉 ≥ 0, ∀x ∈ k. (1.1) variational inequalities and their approximations have been extensively studied in the last two decates. existence and approximations of solutions of variational inequalities for various classes of operators in hilbert and banach spaces have been considered in [1]-[5], [7], [8], [10], [11] and [13]. we mention, in particular, the paper [3], [11], where the operator method or iterative method of regularization are considered. further, in [7] the convergence rates of the operator method of regularization is investigated under the inverse-strongly monotone a in hilbert space when the parameter of regularization α is chosen a priory. in the banach space x, the operator method of regularization is the following variational inequality 〈ah(xτα) + αu (x τ α − x 0) − fδ, x − xτα〉 ≥ 0, x τ α ∈ k, ∀x ∈ k, (1.2) where ah are also monotone operators from x into x∗ and approximate a in the sense ‖ah(x) − a(x)‖ ≤ hg(‖x‖) (1.3) with a nonegative continuous and bounded (image of bounded set is bounded) function g(t), u is the normalized duality mapping of x, i.e., u is the mapping from x onto x∗ satisfying the condition (see [14]) 〈u (x), x〉 = ‖x‖2, ‖u (x)‖ = ‖x‖, fδ are the approximations of f : ‖fδ − f‖ ≤ δ, τ = (h, δ), and x0 is some element in x playing the role of a criterion selection. by the choice of x0, we can influence which solution we want to approximate. in [11], it is showed the existence and uniqueness of the solution xτα for every α > 0 and for arbitrary ah, fδ. and, the regularized solution xτα converges to x0 ∈ s0, the set of solutions of (1.1) which is assumed to be nonempty, with ‖x0 − x0‖ = min x∈s0 ‖x − x0‖, if (h+δ)/α, α → 0. moreover, for each fixed τ = (δ, h) the papameter of regularization α can be chosen by the discrepancy principle ρ(α) = (k − 1)(δ + h)p + δp + g(‖xτα‖)h p, 0 1, 7, 3(2005) convergence rates in regularization for ill-posed variational inequalities 89 where ρ(α) = α‖xτα − x0‖, under the conditions: x0 ∈ int k and ‖ah(x0) − fδ‖ > (k − 1)(δ + h)p + δp + g(‖x0‖)hp for 0 < δ < δ < 1, 0 < h < h < 1. the case x0 ∈ ∂k also is considered when xτα ∈ int k. in this paper, under the condition x0 ∈ k\s0 without the restriction xτα ∈ int k we shall show that the parameter of regularization α = α(δ, h) can be chosen by the generalized discrepancy principle ρ(α) = (δ + h)pα−q, p, q > 0, (1.4) for arbitrary monotone operator a, and on the base of the result we can estimate the convergence rates when a is an inverse-strongly monotone operator, i.e., a possesses the property 〈a(x) − a(y), x − y〉 ≥ 1 β ‖a(x) − a(y)‖2, ∀x, y ∈ x, (1.5) where β is some positive constant. in facts, variational inequalities with inversestrongly monotone operator belong to a class of nonlinear ill-posed problems (see [7]). note that the generalized discrepancy principle for parameter choice is presented first in [6] for the ill-posed operator equation a(x) = f (1.6) when a is a linear and bounded operator in hilbert space. recently, it is considered and applied in estimating convergence rates of the regularized solution for equation (1.6) involving an m-accretive (in general nonlinear) operator (see [9]). later, the symbols ⇀ and → denote weak convergence and convergence in norm, respectively, and the notation a ∼ b is meant that a = o(b) and b = o(a). 2. main result to obtain the result on the convergence rate for {xτ α(δ,h) } as in [6] we need the following lemmas. lemma 1. for each p, q, δ, h > 0, there exists at least a value α such that (1.4) holds. proof. it follows from [11] that ρ(α) is a continuous and nondecreasing function on [α0, +∞), α0 > 0. moreover, ρ(α) > 0 ∀ α 6= 0. indeed, if α1 6= 0 with ρ(ατ1 ) = 0, then xτα1 = x 0 and from (1.2) it follows 〈ah(x0) − fδ, x − x0〉 ≥ 0, ∀x ∈ k. after passing δ and h to zero in this inequality we see x0 ∈ s0. this contradicts the assumption x0 ∈ k\s0. therefore, αqρ(α) → +∞, as α → +∞. on the other hand, since 0 ≤ ρ(α) = α‖xτα − x 0‖ ≤ δ + hg(‖x0‖‖) + 2α‖x0 − x0‖ 90 nguyen buong 7, 3(2005) (see also [11]), we have αqρ(α) → 0, as α → +0. hence, there exists a value α such that (1.4) holds. lemma 2. limδ,h→0 α(δ, h) = 0. proof. let δn, hn → 0, and αn = α(δn, hn) → ∞ as n → ∞. from (1.3), 〈ahn (x τn αn ) + αnu (x τn αn − x0) − fδn , x − x τn αn 〉 ≥ 0, ∀x ∈ k, (2.1) the monotone property of ahn and x 0 ∈ k it follows ‖xτnαn − x 0‖ ≤ ‖ahn (x 0) − fδn‖/αn → 0, as n → ∞. therefore, xτnαn → x 0, as n → ∞. on the other hand, by using the monotone property of ahn and the property of u we can write (2.1) in the form 〈ahn (x) − fδn , x − x τn αn 〉 ≥ −αn〈u (xτnαn − x 0), x − xτnαn〉 ≥ −αn‖xτnαn − x 0‖‖x − xτnαn‖ ≥ −ρ(αn)‖x − xτnαn‖ ≥ −(δn + hn)pα−qn ‖x − x τn αn ‖ → 0, as n → ∞. it means that 〈a(x0) − f, x − x0〉 ≥ 0, ∀x ∈ k, i.e., x0 is a solution of (1.1). it contradicts x0 /∈ s0. thus, α(δ, h) remains bounded as δ, h → 0. let δn, hn → 0 as n → ∞, and meantime αn → c > 0. since α1+qn ‖xτnαn − x 0‖ = (δn + hn)p, we have ‖xτnαn − x 0‖ → 0, as n → ∞. again, x0 ∈ s0. hence, limδ,h→0 α(δ, h) = 0. lemma 3. if 0 < p < q, then limδ,h→0(δ + h)/α(δ, h) = 0. proof. it is easy to see that[ δ + h α(δ, h) ]p [(δ + h)pα(δ, h)−q]α(δ, h)q−p = ρ(α(δ, h))α(δ, h)q−p = α(δ, h)‖xτα(δ,h) − x 0‖α(δ, h)q−p ≤ [ δ + hg(‖x0‖) + 2α(δ, h)‖x0 − x0‖ ] α(δ, h)q−p → 0 as δ, h → 0. therefore, lim δ,h→0 [ δ + h α(δ, h) ]p = 0. the lemma is proved. 7, 3(2005) convergence rates in regularization for ill-posed variational inequalities 91 lemma 4. let 0 0 such that, for sufficiently small δ, h > 0, the relation c1 ≤ (δ + h)pα(δ, h)−1−q ≤ c2 holds. proof. from (δ + h)pα(δ, h)−1−q = α(δ, h)−1ρ(α(δ, h)) = ‖xτα(δ,h) − x 0‖ ≤ δ α(δ, h) + h α(δ, h) g(‖x0‖) + 2‖x0 − x0‖ and lemma 3, it implies the existence of a positive constant c2 in the lemma. on the other hand, as x is reflexive and {xτ α(δ,h) } is bounded, there exists a subsequence of the sequence {xτ α(δ,h) } that converges weakly to some element x̃0 in k such that ‖x̃0 − x0‖ ≤ lim inf ‖xτα(δ,h) − x 0‖. we can conclude that x̃0 6= x0. indeed, if x̃0 = x0, then from the monotone hemicontinuous property of ah and (1.2) it follows 〈ah(x) + αu (x − x0) − fδ, x − xτα〉 ≥ 0, ∀x ∈ k. after passing δ and h in the last inequality to zero we obtain 〈a(x) − f, x − x̃0〉 ≥ 0, ∀x ∈ k which is equivalent to (1.1). it is meant that x̃0 ∈ s0. it contradicts x0 /∈ s0. therefore, there exists a constant c1 in the lemma. to estimate the convergence rates for {xτ α(δ,h) } we assume that 〈u (x) − u (y), x − y〉 ≥ mu‖x − y‖s, mu > 0, s ≥ 2, ∀x, y ∈ x. (2.2) it is well-known that when x ≡ h, the hilbert space, mu = 1, s = 2, and when x = lp or wp, mu = p − 1, s = 2 for the case 1 2 we have to use the duality mapping u s satisfying the condition 〈u s(x), x〉 = ‖x‖s, ‖u s(x)‖ = ‖x‖s−1, s ≥ 2 instead of u . then, mu s = 22−p/p and s = p in (2.2) (see [12]). theorem 1. assume that the following conditions hold: (i) a is an inverse-strongly-monotone operator in x with ‖a(x) − a(x0) − a′(x0)(x − x0)‖ ≤ τ̃‖a(x) − a(x0)‖, ∀x ∈ x, where τ̃ is some positive constant; 92 nguyen buong 7, 3(2005) (ii) there exists an element z ∈ x such that a′(x0)∗z = u (x0 − x0); (iii) the parameter α is chosen by (1.4) with p < q. then, we have ‖xτα(δ,h) − x0‖ = o((δ + h) θ), θ = p (1 + q)(2s − 1) . proof. from (1.1) (1.3) it follows 〈a(xτα(δ,h)) − a(x0), x τ α(δ,h) − x0〉 + α(δ, h) × 〈u (xτα(δ,h) − x 0) − u (x0 − x0), xτα(δ,h) − x0〉 ≤ (δ + hg(‖xα(δ,h)‖))‖xα(δ,h) − x0‖ +α(δ, h)〈u (x0 − x0), x0 − xτα(δ,h)〉. (2.3) thus, by using (1.5) and the monotone property of u we obtain ‖a(xτα(δ,h)) − a(x0)‖ ≤ o( √ δ + h + α(δ, h))‖xτα(δ,h) − x0‖ 1/2. on the other hand, from (2.2), (2.3) and the monotone property of a which is followed from (1.5) we have mu‖xτα(δ,h) − x0‖ s ≤ 〈u (xτα(δ,h) − x 0) − u (x0 − x0), xτα(δ,h) − x0〉 ≤ δ + c̃0h α(δ, h) ‖xτα(δ,h) − x 0)‖ + 〈z, a′(x0)(x0 − xτα(δ,h))〉 where c̃0 is some positive constant, and∣∣〈z, a′(x0)(x0−xτα(δ,h))〉∣∣ ≤ ‖z‖(τ̃ + 1)‖a(xτα(δ,h)) − a(x0)‖ ≤ ‖z‖(τ̃ + 1)o( √ δ + h + α(δ, h))‖xτα(δ,h)) − x0‖ 1/2. now, from lemma 4 it implies that α(δ, h) ≤ c−1/(1+q)1 (δ + h) p/(1+q). and δ + h α(δ, h) ≤ c2(δ + h)1−pα(δ, h)q ≤ c2c −q/(1+q) 1 (δ + h) 1−p(δ + h)pq/(1+q) ≤ c2c −q/(1+q) 1 (δ + h) 1−p/(1+q). in final, we have mu‖xτα(δ,h)−x0‖ s−1/2 ≤ max{1, c̃0}c2c −q/(1+q) 1 (δ + h) 1−p/(1+q) × ‖xτα(δ,h) − x0‖ 1/2 + o( √ δ + h + α(δ, h)) ≤ o((δ + h)1−p/(1+q)‖xτα(δ,h) − x0‖ 1/2 + o((δ + h)p/2(1+q)). 7, 3(2005) convergence rates in regularization for ill-posed variational inequalities 93 using the implication a, b, c ≥ 0, s > t, as ≤ bat + c =⇒ as = o(bs/(s−t) + c) we obtain ‖xτα(δ,h) − x0)‖ = o((δ + h) θ). received: july 2004. revised: august 2004. references [1] alber, ya.i. : on solution of the equations and variational inequalities with maximal monotone mappings, soviet math. dokl. 247 (1979), 1292-1297. 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[8] liskovets, o.a.: regularization for the problems involving monotone operators in the discrete approximation of the spaces and operators, zh. vychisl. mat. i mat. fiz. 27 (1987), 3-15. [9] nguyen buong: generalized discrepancy principle and ill-posed equation involving accretive operators, nonlinear funct. analysis & appl. 9 (2004), 73-78. [10] noor, m.a. and rassias, t.m.: projection methods for monotone variational inequalities, j. of math. anal. and appl. 237 (1999), 405-412. 94 nguyen buong 7, 3(2005) [11] ryazantseva, i.p.: on solving variational inequalities with monotone operators by method of regularization , zh. vychisl. mat. i mat. fiz. 23 (1983), 479-483. [12] ryazantseva, i.p.: on one algorithm for nonlinear monotone equations with unknown estimate errors in the data, zh. vychisl. mat. i mat. fiz. 29 (1989), 1572-1576. [13] ryazantseva, i.p.: continuous method of regularization of the first order for monotone variational inequalities in banach space, diff. equations, belorussian, 39 (2003), 113-117. [14] vainberg, m.m.: variational method and method of monotone operators, moscow, mir, 1972. articulo 5.dvi cubo a mathematical journal vol.12, no¯ 02, (53–75). june 2010 generalized quadrangles and subconstituent algebra 1 fernando levstein famaf-ciem,unc, universidad nacional de córdoba medina allende y haya de la torre. cp 5000 córdoba, argentina email: levstein@famaf.unc.edu.ar and carolina maldonado departamento de matemática, centro de ciências exatas e da natureza, universidade federal de pernambuco av. prof. luiz freire, s/n cidade universitária recife, brasil email: cmaldona@famaf.unc.edu.ar abstract the point graph of a generalized quadrangle gq(s, t) is a strongly regular graph γ = srg(ν, κ, λ, µ) whose parameters depend on s and t. by a detailed analysis of the adjacency matrix we compute the terwilliger algebra of this kind of graphs (and denoted it by t ). we find that there are only two non-isomorphic terwilliger algebras for all the generalized quadrangles. the two classes correspond to wether s2 = t or not. we decompose the algebra into direct sum of simple ideals. considering the action t × cx −→ cx we find the decomposition into irreducible t -submodules of cx (where x is the set of vertices of the γ). resumen el grafo de puntos de un cuadrángulo generalizado gq(s, t) es un grafo fuertemente regular γ = srg(ν, κ, λ, µ) cuyos parámetros dependen de s y t. mediante un análisis detallado de 1this work supported by facepe, ccen ufpe and ciem-famaf unc, conicet. 54 f. levstein and c. maldonado cubo 12, 2 (2010) la matriz de adyacencia, calculamos el álgebra de terwilliger (t -álgebra) de esta familia de grafos. encontramos que para todos los cuadrángulos generalizados, existen solo dos tipos no isomorfos de t -álgebras asociadas. dichas clases dependen de si s2 = t o no. descomponemos el álgebra en suma directa de ideales simples. considerando la acción t × cx −→ cx encontramos la descomposición de cx en t -submódulos irreducibles. (x es el conjunto de vértices de γ) key words and phrases: strongly regular graphs, generalized quadrangles, terwilliger algebra. ams (mos) subj. class.: 05e30 1 introduction the subconstituent algebra was first introduced by p. terwilliger in his paper [13]. it was defined on a class of combinatorial objects known as association schemes (see also [2, 3]). it is a noncommutative, finite dimensional, semisimple c algebra. we will denote it by t . it has been studied for many examples such as p and qpolynomial association schemes [6], distance-regular graph that supports a spin model [7], group association schemes [4, 5], strongly regular graphs [17]. in [8] it was given an explicit description of the t -algebra of the hypercube and more generally in [10] of a hamming scheme h(d; q). the case of the johnson schemes it was analyzed in [9]. in this paper we focus on the t -algebra of a special family of strongly regular graphs, which are examples of association schemes: generalized quadrangles gq(s, t) . they are indeed a subfamily of partial geometries defined in [1]. a strongly regular graph is associated to them, so we can study the t -algebra of such a family. we show that there are only two non-isomorphic t -algebras for all the generalized quadrangles. the two classes correspond to whether s2 = t or not. we obtain the dimension of t in both cases. this is in agreement with the result expected from [17] that gives dimensions of the t -algebra attached to a strongly regular graph. the particular class of gq(s, s2) has a combinatorial characterization given by j.a. thas in [16]. with a detailed analysis of the adjacency matrix, we obtain restriction on the parameters (s, t) (also given in 1.2.2 of [12]). the paper is organized as follows: in section 2 we give the basic definitions and comment on some known basic results of algebraic combinatorics. in section 3 we analyze the blocks of the matrices in t and we give a basis of t in proposition 3.21. in section 4 we find the simple ideals of t (propositions 4.3, 4.4) and in theorem 4.5 we decompose t into direct sum of simple ideals. finally in section 5 we give the irreducible t -submodules of the action t × cx −→ cx (where x is the set of vertices of the γ). cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 55 2 definitions 2.1 strongly regular graphs definition 2.1. (see [11]) a strongly regular graph γ = srg(ν, κ, λ, µ) is a graph with ν vertices that is regular of degree κ and that has the following properties: • for any two adjacent vertices x, y there are exactly λ vertices adjacent to x and to y • for any two nonadjacent vertices x, y there are exactly µ vertices adjacent to x and to y 2.2 generalized quadrangles definition 2.2. (see [1] , [12]) a generalized quadrangle gq(s, t) is a system of points and lines with an incidence relation satisfying the axioms (1) − (4) below. we will use standard geometric language. a point incident with a line is said to lie on the line and the line is said to pass through the point. if two lines are incident with the same point, we say that they intersect. axioms 1. for any two distinct points there is at most one line passing through them; 2. there are exactly r = t + 1 lines passing for each point; 3. there are exactly k = s + 1 points lying on each line; 4. if a point p does not lie on the line l, then there is exactly one line passing through p and intersecting l if two points lie on a common line, we say that they are collinear and we write x ∼ y. the point graph of a generalized quadrangle is the graph with the points of the quadrangle as vertices, and edges {x, y} such that x ∼ y. it is well known by [1, 12] that the point graph of a gq(k − 1, r − 1) is a (possibly trivial) γ = srg(ν, κ, λ, µ) with: ν = k (1 + (k − 1)(r − 1)) , κ = r(k − 1), λ = k − 2, µ = r (1) 2.3 bose-mesner algebra let γ = srg(ν, κ, λ, µ) be a strongly regular graph, x be the set of vertices and ∂ : x × x → {0, 1, 2} be the path-length distance for γ. let m atx (c) denote the c-algebra of matrices with complex entries, where the rows and columns are indexed by x. 56 f. levstein and c. maldonado cubo 12, 2 (2010) definition 2.3. the adjacency matrix of γ of is the following (0, 1)-matrix in m atx (c): (a)xy = { 1 if ∂(x, y) = 1 0 otherwise proposition 2.4. (see [11]) let γ = srg(ν, κ, λ, µ) be a strongly regular graph, a the adjacency matrix of γ and i, j ∈ m atx (c) the identity and the full ones matrix respectively. then aj = κj (2) a2 + (µ − λ)a + (µ − κ)i = µj (3) proof. by definitions 2.1 and 2.3; a is a symmetric matrix with κ 1’s on each row and column. this proves equation (2). to prove (3) we observe that defining (a2)xy = { 1 if ∂(x, y) = 2 0 otherwise , axioms of definition 2.1 imply that i + a + a2 = j (a2 6= j − i) otherwise γ would be a complete graph). computing: (a2)xy = σz∈x axzazy = |{z : ∂(x, z) = 1 and ∂(z, y) = 1}| =       κ if x = y λ if ∂(x, y) = 1 µ if ∂(x, y) = 2 therefore a2 = κi + λa + µa2 = κi + λa + µ(j − i − a) which implies the (3). definition 2.5. (see [2], [3] ) the bose-mesner algebra of a strongly regular graph γ is the 3-dimensional algebra of matrices in m atx (c) which are linear combinations of i, j and a. we denoted it by a. that this is indeed an algebra is a consequence of equations (2) and (3) in proposition 2.4. the following facts are well known in algebraic combinatorics (see [2, 3]). cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 57 the algebra a consists of symmetric commuting matrices and identifying c x = {f : x → c} we can consider for all m ∈ a the action: m × cx → cx . since {i, j, a} consists of symmetric commuting matrices , they are diagonalyzed simultaneously by a unitary matrix. that is, we have a decomposition of cx into common eingenspaces of i, j, a. the number of eigenspaces is 2 + 1 since any strongly regular graph has diameter= 2 (diameter:= the greatest distance in the graph). therefore, let γ be a strongly regular graph, c x = v0 ⊕ v1 ⊕ v2 be such a decomposition and let ei, i = 0, 1, 2 be the orthogonal projections ei : c x → vi expressed in matrix form with respect to the canonical basis {ei} i = 1...|x|. then, e0 = 1 |x|j (j the matrix of all 1,s) e0 + e1 + e2 = i eiej = δij ei the ei are called the primitive idempotents of γ. 2.4 dual bose-mesner algebra definition 2.6. (see [13]) the ith dual idempotent with respect to the vertex x denoted by e∗i := e ∗ i (x) is the diagonal matrix in m atx (c) defined by (e∗i )yy = { 1 if ∂(x, y) = i 0 if ∂(x, y) 6= i lemma 2.7. the matrices {e∗i }2i=0 satisfy the following equations: e∗0 + e ∗ 1 + e ∗ 2 = i (4) e∗i t = e∗i (5) e∗i e ∗ j = δij e ∗ i (6) proof. its follows straightforward from definition above. definition 2.8. let γ be a strongly regular graph. for x ∈ x, the dual bose-mesner algebra of γ with respect to x, is the 3-dimensional algebra of matrices in m atx (c) which are linear combinations of {e∗i }2i=0. we denoted it by a∗ := a∗(x). 58 f. levstein and c. maldonado cubo 12, 2 (2010) that this is indeed an algebra is a consequence of equations (4),(5) and (6) in the previous lemma. 2.5 terwilliger algebra definition 2.9. (see [13]) let γ be a strongly regular graph and x be its set of vertices. the subconstituent or terwilliger algebra of γ with respect to the vertex x ∈ x is the algebra generated by the bose-mesner algebra a := a(x) and the dual bose-mesner algebra a∗ := a∗(x). we denote this algebra by t := t (x). remark 2.10. t is closed under the conjugate-transpose map, so it is semi-simple. 3 t -algebra of gq(k − 1, r − 1). in this section we consider a connected strongly regular graph γ = srg(ν, κ, λ, µ) coming from a generalized quadrangle gq(k − 1, r − 1). we fix x0 ∈ x and we analyze the associated t (x0)-algebra . in the following we analyze the structure of the matrices belonging to t in a more detailed way . lemma 3.1. for all t ∈ t , t is generated by a, e∗0 , e∗1 , e∗2 proof. by definition t is generated by the algebras a = 〈{i, j, a}〉 and a∗ = 〈{e∗0 , e∗1 , e∗2 }〉. that is t consist on sum and products of matrices in {i, j, a, e∗0 , e∗1 , e∗2 }. equation (3) shows that j can be obtained as a linear combination of a2, a, i and equation (4) shows that the identity is the sum of {e∗i }2i=0. remark 3.2. it is well known that for the point graph of a generalized quadrangle the isomorphism class of t (x) is independent on the vertex x, since the group of automorphism of the graph γ acts transitively on x preserving the distance. then any automorphism g : x → x x → y, induces an isomorphism tg : t (x) → t (y). m x → m y where m yuv := m x g−1ug−1v , for m x ∈ t (x), m y ∈ t (y); u, v ∈ x and then t (x) ≃ t (y) in view of lemma 3.1 we consider the products e∗i ae ∗ j i, j = 0, 1, 2 where a is the adjacency matrix and e∗i the dual idempotents of definitions 2.3 and 2.6 respectively. cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 59 3.1 block analysis we will use an order of the set of vertices x that allows us to analyze the matrices in t (x0) in a convenient way. let x0 be a fixed vertex of x. take ω0 = {x0}, ωi = {y ∈ x | ∂(x0, y) = i} we consider the matrices in m atx (c) indexed by the blocks ωi × ωj . just to give examples, we have: e∗0 =       x0 ω1 ω2 x0 1 0 0 ω1 0 0 0 ω2 0 0 0       e∗1 =       x0 ω1 ω2 x0 0 0 0 ω1 0 i 0 ω2 0 0 0       e∗1 ae ∗ 2 =       x0 ω1 ω2 x0 0 0 0 ω1 0 0 a|ω1×ω2 ω2 0 0 0       we will denote p := a|ω1×ω1 q := a|ω1×ω2 s := a|ω2×ω2 and iik := i|ωi×ωk , jik := j|ωi×ωk , that is the submatrix of i or j of size ωi × ωk. then a|x0×ω1 = j01 = (1, ..., 1) and since a is symmetric we have a|ω2×ω1 = q t , a|ω1×x0 = j t 01 = (1, ..., 1) t . then a looks like: 60 f. levstein and c. maldonado cubo 12, 2 (2010) a =                x0 ω1 ω2 x0 0 1...1 0 1 ω1 ... p q 1 ω2 0 q t s                the following lemma gives some descriptions of blocks of a. lemma 3.3. let γ = srg(ν, κ, λ, µ) be a srg associated to a generalized quadrangle gq(k − 1, r − 1) (that is the parameters (ν, κ, λ, µ) satisfy equations in (1)). let jkl, p, q, s be defined as above. then 1. a|x0×ω1 = j01 2. j10 = j t 01 3. |ω1| = r(k − 1); |ω2| = (r − 1)(k − 1)2 4. p is a block of size |ω1| × |ω1| with (k − 2) 1′s on each row and column, 5. q is a block of size |ω1| × |ω2| with (r − 1)(k − 1) 1′s on each row and r 1′s on each column and 6. s has size |ω2| × |ω2| with r(k − 2) 1′s on each row and column. proof. • (1) holds since by definition of a, the block indexed by x0 × ω1 is the set of neighbors of x0. • (2) holds since a is symmetric. • (3) holds since |ω1| = κ (the degree of γ) and 1 + |ω1| + |ω2| = ν (the number of vertices of γ). parameters κ, ν are given in equations (1). • assertion (4) holds since for a fixed x ∈ ω1 there are λ = k − 2 neighbors of x in ω1. • on the same way for a fixed x ∈ ω2 there are µ = r neighbors of x in ω1 which implies that q has r 1’s on each column. the number of 1’s on each row of q is |ω1| − (k − 2) − 1. • the number of 1’s on each row and column of s is |ω1| − r. cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 61 remark 3.4. we have already discussed that in order to describe t we should analyze the products among the matrices in {e∗i ae∗j }i,j=0,1,2. that is essentially the products among the blocks j01, j10, p, q, q t and s. in the following subsections we analyze the structure of each block ωi × ωj and finally we give a basis for each one. 3.2 ω1 × ω1-block we start giving expressions for some products belonging to the ω1 × ω1-block: {p n, qqt, p j11, j11p, j10j01}. we describe the powers of p . lemma 3.5. p satisfies p 2 = (k − 3)p + (k − 2)i11 proof. the ω1 × ω1-block has size r(k − 1) × r(k − 1) and p has (k − 2) 1’s on each row and column. it is indexed by the vertices in ω1. it has a one in the (xi, xj ) entry if and only if the common neighbors xi, xj of x0 form an edge of the graph γ. as the equation for p does not depend on the order of the vertices of ω1 we will consider a special ordering in which p has a simple form. we label the vertices in the following way: ω0 = {x0} and l1, l2...lr the r lines passing through the point x0. we call x1,1, x1,2, ...x1,k−1 the (k − 1) points lying on the l1 \ {x0}; x2,1, x2,2, ...x2,k−1 the points lying on the l2 \ {x0} and so on. all the points lying on the same line are collinear points. then any two of them form an edge on the point graph of the generalized quadrangle. if we order the vertices of the ω1 × ω1-block with the order of the lines, that is l1 l2 . . . lr ︷︸︸︷ x1,1, x1,2, ...x1,k−1; ︷︸︸︷ x2,1, x2,2, ...x2,k−1; . . . ; ︷︸︸︷ xr,1, xr,2, ...xr,k−1 p has the form: p =                              j-i 0 ... ... ... 0 0 j-i ... ... ... 0 ... ... ... ... ... ... ... ... ... ... ... ... 0 ... ... ... j-i 0 0 ... ... ... 0 j-i                              and is not difficult to see that p 2 = (k − 3)p + (k − 2)i11, which implies the lemma. 62 f. levstein and c. maldonado cubo 12, 2 (2010) corollary 3.6. the matrices p, i11 and j11 are independent and p 2 depends on p and i11. proof. p, i11 and j11 are independent, otherwise the relation among them should be p = j11−i11. but this would imply that the graph is not connected. since we omit these cases we have the conclusion. lemma 3.7. using the same ordering as above for ω1 and any order for ω2 we have qqt = (r − 1)(k − 2)i11 − (r − 1)p + (r − 1)j11 j10j01 = r(k − 1)j11 p j11 = (k − 2)j11 proof. equating the ω1 × ω1-block of (3) we have j10j01 + p 2 + qqt + (µ − λ)p + (µ − κ)i11 = µj11. replacing the parameters λ, µ, κ by equation (1) and p 2 as in the previous lemma, we get j11 + (k − 3)p + (k − 2)i11 + qqt + (r − k + 2)p − r(k − 2)i11 = rj11. which implies the expression for qqt. the other equations are easy to check. proposition 3.8. the products p n, qqt, j10j01 y p j11 can be expressed as a linear combinations of p, i11, j11 and they are linearly independent. proof. it follows directly from lemmas 3.5 and 3.7. 3.3 ω1 × ω2block now we give expressions for the products p q, qs, j11q, qj22, j12s lemma 3.9. using the same ordering for ω1 as in the lemma 3.5 the following equation holds: p q = j12 − q proof. the ω1 × ω2 -block has size r(k − 1) × (r − 1)(k − 1)2. from lemma 3.3, we now that q has (r − 1)(k − 1) 1’s on each row and r 1’s on each column. by hypothesis, the rows of q are indexed by the vertices of the lines l1, l2, ...lr. the columns are indexed by the set ω2 (the vertices which are not neighbors of x0). let (xij , y) be an entry of the product p q where y ∈ ω2 and xij is the ith vertex of the line lj. then (p q)(xij ,y) = r∑ m=1 k−1∑ n=1 p(xij ,xmn)q(xmn,y). since p vanishes on the vertices lying on different lines (p(xij ,xkl) = 0 for i 6= k), (p q)(xij ,y) = k−1 ∑ n=1 p(xij ,xin)q(xin,y). cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 63 each vertex of ω2 has exactly one neighbor on the line li (fourth axiom of definition 2.2). therefore for y ∈ ω2 there exist a unique xiny ∈ li such that q(xij , y) = { 1 if j = ny 0 if j 6= ny then (p q)(xij ,y) = ∑k−1 n=1 p(xij ,xin)q(xin,y) = p(xij ,xiny ) = (j − i)(xij ,xiny ) = { 0 if j = ny 1 if j 6= ny = (j − q)(xij ,xiny ), which proves the lemma. lemma 3.10. q and s satisfy: qs = (r − 1)j12 + (k − 1 − r)q, j11q = rj12, qj22 = (r − 1)(k − 1)j12, j12s = r(k − 2)j12 proof. the ω1 × ω2-block of identity (3) for a gives p q + qs + (r − k + 2)q = rj12. replacing p q by the result of the lemma 3.9 we have the first equation. for the other equations, we use that q has (r − 1)(k − 1) 1′s on each row and r 1′s on each column, and s has r(k − 2) 1′s on each row and column. proposition 3.11. the products p nq, snq, j11q, qj22, j12s can be expressed as linear combinations of q and j12. proof. using lemmas 3.5 and 3.9 we can prove inductively that p nq is a linear combination of q and j12. on the same way lemma 3.10 proves inductively the assertion for s nq . the other equations were also proved in lemma 3.10. 3.4 ω2 × ω2-block in the following, we give an expression for sn, qtq and j22s. lemma 3.12. qtq = −s2 + r(k − 2)i22 + (k − 2 − r)s + rj22, sj22 = r(k − 2)j22 (7) proof. the ω2 × ω2-block of identity (3) for a gives the first equation. the matrix s has r(k − 2) 1′s on each row and column thus we get the second equation. 64 f. levstein and c. maldonado cubo 12, 2 (2010) proposition 3.13. s3 = ((k − 1 − r) + (k − 2 − r)) s2 + (r(k − 2) − (k − 1 − r)(k − 2 − r)) s − ((k − 1 − r) + r(k − 2)) i22 + (r(r − 1)(k − 2)) j22. equivalently if we denote λ1 = k − r − 1, λ2 = k − 2, λ3 = −r, then s satisfies the equation (s − λ1i22) (s − λ2i22) (s − λ3i22) = r(r − 1)j22 (8) proof. postmultiplying qtq given in (3.12) by s we have q t qs = −s3 + r(k − 2)s + (k − 2 − r)s2 + r2(k − 2)j22 replacing qs by the expression given in the lemma 3.10 q t ((k − 1 − r)q + (r − 1)j22) = −s3 + r(k − 2)s + (k − 2 − r)s2 + r2(k − 2)j22, s 3 = −(k − 1 − r)qtq − r(r − 1)j22 + r(k − 2)s + (k − 2 − r)s2 + r2(k − 2)j22. replacing qtq by 3.12 we have the first equation, that is equivalent to s3 − ((k − 1 − r) + (k − 2 − r)) s2 − (r(k − 2) − (k − 1 − r)(k − 2 − r)) s + ((k − 1 − r) + r(k − 2)) i22 = (r(r − 1)(k − 2)) j22 at this moment we can not tell whether s2, s, i22 and j22 are independent or not. in what follows we are going to show that s2 depends on s i22 and j22 if and only if the parameters of the generalized quadrangle satisfy (k − 1)2 = r − 1. corollary 3.14. denoting λ0 = r(k − 2), λ1 = k − r − 1, λ2 = k − 2, λ3 = −r s satisfies the equation (s − λ0i22) (s − λ1i22) (s − λ2i22) (s − λ3i22) = 0 proof. by lemma the ω2 ×ω2-block has size (r−1)(k−1)2 ×(r−1)(k−1)2. s has r(k−2) 1’s on each row and on each column. so we have sj22 = r(k − 2)j22. thus, if we multiply (8) by s − r(k − 2)i22 we have the corollary. cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 65 this corollary implies that s has at most four different eigenvalues. we know that r(k − 2) is an eigenvalue associated to the one dimensional eigenspace generated by (1, 1, ..., 1). then by perronfrobenious theorem it has multiplicity one. let di = dim vλi , where vλi is the eigenspace corresponding to λi. we have the following linear system of equations on d0 and the unknowns: {di}3i=1 tri = ∑3 i=0 di = (r − 1)(k − 1)2, trs = ∑3 i=0 λidi = 0 and trs2 = ∑3 i=0 λi 2 di = r(k − 2)(r − 1)(k − 1)2, then tri = ∑3 i=1 di = (r − 1)(k − 1)2 − 1, trs = ∑3 i=1 λidi = −r(k − 2) and trs2 = ∑3 i=1 λi 2 di = r(k − 2)(r − 1)(k − 1)2 − (r(k − 2))2 , with set of solutions d1 = r(k − 2) d2 = r(k−1)2(r−2) (k+r−2) and d3 = (k−2)(r−1)((k−1)2−(r−1)) (k+r−2) . as the dimensions are non negative integers we have (k−1)2 ≥ (r−1), which is known as the inequality of d.g. higman.(page 3 of [12]) in general k + r − 2 must divide both (k − 2)(r − 1)((k − 1)2 − (r − 1)) and r(k − 1)2(r − 2) if the parameters correspond to a generalized quadrangle. dimensions {di}3i=1 are always positive integers unless (k − 1)2 = r − 1, in which case d3 = 0 and λ3 is not an eigenvalue. thus we have the following: proposition 3.15. s has λ3 = −r as eigenvalue if and only if the parameters r and k satisfy (k − 1)2 > r − 1. proof. it follows by the comments above. corollary 3.16. the matrices s, i22, j22 are linearly independent. s 2 depends on such matrices if and only if (k − 1)2 = r − 1 proof. we have seen in proposition 3.13 that the vector space generated by {sn}n≥0 has dimension 3 or 4. this depends on the minimal polynomial of s and we have shown it has 3 different eigenvalues if and only if (k − 1)2 = r − 1. proposition 3.17. the products {qtq, j22s, {sn}n≥0} can be expressed as a linear combinations of s, i22 and j22 , if and only if the parameters r, k of the generalized quadrangle satisfy (k − 1)2 = r − 1. otherwise s2, s, i22 and j22 span these products. 66 f. levstein and c. maldonado cubo 12, 2 (2010) proof. follows directly from lemma 3.10 and corollary 3.16 . theorem 3.18. the following spanning set are basis for the corresponding blocks. {x0} × ωi = 〈j0i〉 i = 0, 1, 2 ω1 × ω1 = 〈{i11, j11, p }〉 ω1 × ω2 = 〈{j12, q}〉 ω2 × ω2 = 〈{i22, j22, s}〉 ⇔ (k − 1)2 = r − 1 = 〈{ i22, j22, s, s 2 }〉 ⇔ (k − 1)2 6= r − 1 proof. it follows straightforward from propositions 3.8, 3.11 and 3.17. 3.5 basis for t as a vector space the previous block-analysis allows to give a basis (as a vector space) of the t -algebra attached to a gq(k − 1, r − 1). actually we have analyzed the blocks of arbitrary matrices in t . to be rigorous we should embed each block in m atx (c) . to do this we propose the following definition 3.19. let b an arbitrary block indexed by the vertices in {ωi × ωj} i, j = 0, ...2. we identify the block b with a matrix ι(b) in m atx (c) in the following way: ι(b)xy = { bxy if (x, y) ωi × ωj 0 otherwise example 3.20. let b be a block-matrix indexed by ω2 × ω1. then ι(b) =       x0 ω1 ω2 x0 0 0 0 ω1 0 0 0 ω2 0 b 0       proposition 3.21. if the parameters of gq(k − 1, r − 1) satisfy (k − 1)2 6= r − 1 then t = 〈{ {ι(jij )}2i,j=0, {ι(ijj )}2j=1, ι(p ), ι(q), ι(qt), ι(s), ι(s2) }〉 otherwise t = 〈{ {ι(jij )}2i,j=0, {ι(ijj )}2j=1, ι(p ), ι(q), ι(qt), ι(s) }〉 . therefore dim(t ) = 16 or dim(t ) = 15 respectively. proof. by theorem 3.18 ,the matrices {ι(jmn)}2m,n=0, {ι(imm)}2m=1, ι(p ), ι(q), ι(qt), ι(s) cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 67 and eventually ι(s2) (when (k − 1)2 6= r − 1) give a basis (as a vector space) of a subalgebra of t . this subalgebra contains the adjacency matrix a and the dual idempotents {e∗i } since a = ι(j00) + ι(j10) + ι(j10) t + ι(p ) + ι(q) + ι(qt) + ι(s) e∗m = ι(imm). therefore it coincides with t . 4 simple ideals of t in this section we decompose t as a direct sum of orthogonal simple ideals. we will guide us by the expression given by proposition 3.21. there is one ideal present in every t -algebra: the ideal m linearly generated by {ι(jmn)}2m,n=0. definition 4.1. for m, n = 0, 1, 2 let mmn ∈ m atx (c) be: mmn = 1√ |ωm||ωn| ι(jmn) proposition 4.2. the vector subspace m = 〈 {mmn}2m,n=0 〉 is a simple ideal of t and m ≃ end(c3). proof. it not difficult to prove that mmnmpq = δnpmmq m, n, p, q = 0, 1, 2 which implies the proposition. using standard techniques we compute the following basis for the second ideal. let us denote n11 = 1 k−1 ι ((k − 2)i11 − p ) , n12 = 1 (k−1) √ (k−1)(r−1) ι ((k − 1)q − j12) , n21 = n t 12, n22 = 1 (k−1)2(r−1) ι ((k − 1)qtq − rj22) we have the following proposition 4.3. the vector subspace n = 〈 {nmn}2m,n=1 〉 is a simple ideal of t orthogonal to the ideal m and n ≃ end(c2). proof. it not difficult to prove that nmnnpq = δnpnmq m, n, p, q = 1, 2 m n = 0 ∀ m ∈ m, n ∈ n , which implies the proposition. 68 f. levstein and c. maldonado cubo 12, 2 (2010) now we give the expressions for the remaining one-dimensional ideals of t . one can easily prove the following: proposition 4.4. the matrices p11 = 1 k−1 ι ( p + i11 − 1r j11 ) r22 = 1 (r−1)(k−2+r) ι ( s2 − (k − 1 − 2r)s − r(k − 1 − r)i22 − rj22 ) s22 = 1 (k−1)(k−2+r) ι ( s 2 − (2k − r − 3)s + (k − 1 − r)(k − 2)i22 − (k−2)(r−1)(k−1) j22 ) are idempotents and orthogonal to the ideals m and n . moreover, if (k − 1)2 = (r − 1) r22 = 1 r−1 ι ( s − (k − 1 − r)i22 − 1k−1 j22 ) , s22 = 0 if not, r22 y s22 are linearly independent and orthogonal. then p = 〈p11〉, r = 〈r22〉, s = 〈s22〉 are ideals of t , orthogonal among them and orthogonal to m and to n . we get directly the following: theorem 4.5. let m, n , p, r, s ⊆ t be the simple ideals described above. then, the t -algebra of a gq(k−1, r−1) has the following decomposition as a direct sum of orthogonal simple ideals: t = m ⊕ n ⊕ p ⊕ r ⊕ s ≃ end(c3) ⊕ end(c2) ⊕ end(c1) ⊕ end(c1) ⊕ end(c1) ⇐⇒ (k − 1)2 6= r − 1 t = m ⊕ n ⊕ p ⊕ r ≃ end(c3) ⊕ end(c2) ⊕ end(c1) ⊕ end(c1) ⇐⇒ (k − 1)2 = r − 1 proof. it follows straightforward from propositions 4.2, 4.3 and 4.4. 5 decomposition of cx into irreducible t -submodules in this section we consider the action of the t -algebra t × cx −→ cx (x is the set of vertices of the generalized quadrangle). we have that t cx ⊆ cx and since i ∈ t it holds t cx = cx . in the following we give a decomposition of cx into irreducible left t -submodules. cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 69 5.1 isotypic left t -submodules let t = m ⊕ n ⊕ p ⊕ r ⊕ s be the decomposition of theorem 4.5. we can associate to each simple ideal a left t −submodule in the following way: {simple ideals of t } : → { left t -submodules of cx } z → zcx they are indeed left t -submodules since by the orthogonality of the simple ideals we have t zcx ⊆ zcx for any simple ideal z ∈ {m, n , p, r, s} . we call them isotypic t -submodules. then the decomposition of cx is : c x = mcx ⊕ n cx ⊕ pcx ⊕ rcx ⊕ scx (9) scx = 0 ⇐⇒ (k − 1)2 = r − 1 (10) 5.2 irreducible left t -submodules in this section we decompose each of the left isotypic t -submodules into irreducible left t submodules. to give the needed definitions we use as a guide the simple ideal n = {n11, n12, n21, n22} associated to the left isotypic t -submodule n cx . the matrices of the basis satisfy nij nkl = δjk nil i, j, k, l = 1, 2 (11) in particular, {nii}i=1,2 are idempotents and they have a (not unique) decomposition as a sum of rk(nii) projectors of rank one.(here rk(a) denote rank of a.) that is, there exist {n (j)11 } rk(n11) j=1 , {n (l) 22 } rk(n22) l=1 one-rank projectors such that (12) n11 = rk(n11)∑ j=1 n (j) 11 , n22 = rk(n22)∑ l=1 n (l) 22 which satisfy (13) n (j) ii n (k) ii = δjk n (j) ii for i = 1, 2 (14) 70 f. levstein and c. maldonado cubo 12, 2 (2010) remark 5.1. by equation (11) we have for example, n21 = n21n11 then n21 = rk(n11)∑ j=1 n21 n (j) 11 the remark carries out to define the following subspaces of n cx definition 5.2. for i = 1, . . . , rk(n11) w n (i) 11 := { n (i) 11 v + (n21n (i) 11 )w v, w ∈ cx ; } then we have: proposition 5.3. for i = 1, . . . , rg(n11); wn (i)11 is an irreducible left t -submodule of dimension 2 and w n (i) 11 ≃ w n (j) 11 . proof. equation (9) and the fact that mutually different ideals are orthogonal implies that w n (i) 11 ⊆ n cx and that w n (i) 11 is a left t -submodule. w n (i) 11 is two dimensional since n (i) 11 is a one-rank projector ∀ i = 1, . . . , rk(n11). therefore given {ej}|x|j=1 the canonical basis of c x , the subspace 〈 { n (i) 11 ej }|x| j=1 〉 has dimension one as well has 〈 { n21n (i) 11 ej }|x| j=1 〉 , which implies that w n (i) 11 has dimension two. it is irreducible since if we consider a one dimensional subspace, it should be of the form {( αn (i) 11 + βn21n (i) 11 ) v; v ∈ cx } but the following actions of t would imply n11 (αn (i) 11 + βn21n (i) 11 )c x ⊆ αn (i)11 cx ⇒ β = 0 n22 (αn (i) 11 + βn21n (i) 11 ) c x ⊆ βn21n (i)11 ⇒ α = 0 (which is a contradiction since it was a one dimensional subspace.) it is easy to check that w n (i) 11 ≃ w n (j) 11 considering the isomorphism: σn : ( n (i) 11 + n21n (i) 11 ) c x −→ ( n (j) 11 + n21n (j) 11 ) c x n (i) 11 v + n21n (i) 11 w −→ n (j) 11 v + n21n (j) 11 w which preserve the action of t . proposition 5.4. n cx = rk(n11)⊕ j=1 w n (j) 11 cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 71 proof. we have that ∑rk(n11) j=1 wn (j) 11 ⊆ n cx . conversely, n11 c x ⊆ ∑rk(n11) j=1 wn (j)11 since by equation (13) n11 c x = ( ∑rk(n11) j=1 n (j) 11 ) c x ⊆ ∑rk(n11) j=1 wn (j)11 . also n21c x ⊆ ∑rk(n11) j=1 wn (j)11 , since n21 c x = n21n11 c x = n21 ( ∑rk(n11) j=1 n (i) 11 ) c x = ( ∑rk(n11) j=1 n21n (i) 11 ) c x = ∑rk(n11) j=1 ( n21n (i) 11 ) c x but we also have n12c x ⊆ ∑rk(n11) j=1 wn (j)11 , since by equation (9) n12c x = n12n cx by equation (11) = n11n cx = n11c x analogously n22c x = n22n cx = n21n cx = n21c x . which implies ∑ j w n (j) 11 ⊇ n cx and therefore the equality holds. we will prove that it is a direct sum by comparing dim   rk(n11)∑ j=1 w n (j) 11   with dimn cx . we have rk(n11) 2-dimensional subspaces. by equation (14) and by definition of wn (j)11 given in 5.2; it follows that rk(n11)∑ j=1 dimw n (j) 11 = 2 rk(n11) = 2 tr(n11) = 2 r(k − 2). 72 f. levstein and c. maldonado cubo 12, 2 (2010) on the other hand, we obtain the dimension of n cx , computing the rank of the projection n : cx → n cx n = n11 + n22 which has the form =     0 0 0 0 ((k−2)i11−p ) k−1 0 0 0 ((k−1)qt q−rj22) (k−1)2(r−1)     . it is easy to check that rk(n ) = tr(n ) = tr ( ((k−2)i11−p ) k−1 ) + tr ( ((k−1)qtq−rj22) (k−1)2(r−1) ) by lemmas 3.3 and 3.5 = k−2 k−1 |ω1| + (k−1)r−r(k−1)2(r−1) |ω2| = k−2 k−1 r(k − 1) + (k − 2)r = 2r(k − 2) analogously we can decompose the other isotypic left t -submodules. considering the matrices mij , p11, r22, s22 we define (the same way as for wn (i)11 ), definition 5.5. wm00 := { m00u + m10m00v + m20m00w u, v, w ∈ cx } w p (i) 11 := { p (i) 11 u u ∈ cx , i = 1, . . . , rk(p11) } w r (i) 22 := { r (i) 22 u u ∈ cx , i = 1, . . . , rk(r22) } w s (i) 22 := { s (i) 22 u u ∈ cx , i = 1, . . . , rk(s22) } then we have the following theorem 5.6. c x = wm00 ⊕ r(k−2) j=1 wn (j)11 ⊕r−1 j=1 wp (j)11 ⊕dr j=1 wr(j)22 ⊕ds j=1 ws(j)22 and mcx = wm00 where wm00 is an irreducible left t -module of dimension 3 pcx = ⊕r−1j=1 wp (j)11 where wp j11 are irreducible left t -modules of dimension 1 rcx = ⊕drj=1 wr(j)22 where wr(j)22 are irreducible left t -modules of dimension 1 scx = ⊕dsj=1 ws(j)22 where ws(j)22 are irreducible left t -modules of dimension 1 cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 73 where dr = r(r−2)(k−1)2 (k−2+r) , ds = (r−1)(k−2)((k−1)2−(r−1)) (k−2+r) and rk(m00) = 1, rk(p11) = (r − 1), rk(r22) = dr , rk(s22) = ds proof. the proof is analogous to the one given for the decomposition of n cx . the number of irreducible left t -submodules that appear on each decomposition depends on the rank of the projections to corresponding isotypic leftt -submodule: m : cx → mcx m = m00 + m11 + m22 =      1 0 0 0 j11√ |ω1||ω1| 0 0 0 j22√ |ω2||ω2|      . p : cx → pcx p := p11 r : cx → rcx r := r22 s : cx → scx s := s22 from the definition of such matrices, and computing its trace, we get the corresponding ranks. corollary 5.7. k (1 + (k − 1)(r − 1)) = 3 + 2r(k − 2) + r − 1 + r(r−2)(k−1) 2 (k−2+r) + (r−1)(k−2)((k−1)2−(r−1)) (k−2+r) proof. one can get the equation by computing the dimensions of the decomposition given in theorem 5.6. remark 5.8. in subsections 5.1 , 5.2 we can exchange ”left” by ”right” considering the action of the t -algebra c x × t −→ cx that gives c x t = cx acknowledgement. this work was finished during a visiting research position at the department of mathematics of ufpe, recife-brasil (september-november 2008) supported by facepe and ciem-conicet. we would like to thank professors manoel lemos and fernando souza for the invitation. 74 f. levstein and c. maldonado cubo 12, 2 (2010) some topics of this paper were developed in the regular seminar of the research group of estruturas discretas of such a department. we also wish to thank paul terwilliger for introducing us into the subject. received: november 2008. revised: february 2009. references [1] r.c. bose, strongly regular graphs, partial geometries and partially balanced designs. pac. j. math. 13, 389-419 (1963) [2] a. e. brouwer, a. m. cohen and a. neumaier, distance-regular graphs. ergebnisse der mathematik und ihrer grenzgebiete. 3. folge, 18. berlin etc.: springer-verlag. xvii, 495 p. 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[10] f. levstein, c. maldonado and d. penazzi, the terwilliger algebra of a hamming scheme h(d, q). eur. j. comb. 27, no. 1, 1-10 (2006). [11] j. h. van lint and r. wilson, a course in combinatorics, cambridge u.p.(1992) [12] s. e. payne and j. a. thas, finite generalized quadrangles, pitman advanced publishing (1984). [13] paul terwilliger, the subconstituent algebra of an association scheme. i. j. algebr. comb. 1, no.4, 363-388 (1992). [14] paul terwilliger, the subconstituent algebra of an association scheme. ii. j. algebr. comb. 2, no.1, 73-103 (1993). cubo 12, 2 (2010) generalized quadrangles and subconstituent algebra 75 [15] paul terwilliger, the subconstituent algebra of an association scheme. iii. j. algebr. comb. 2, no.2, 177-210 (1993). [16] j.a. thas, combinatorial characterizations of generalized quadrangles with parameters s = q and t = q2. geom. dedicata 7, 223-232 (1978). [17] m. tomiyama and n. yamazaki, the subconstituent algebra of a strongly regular graph. kyushu j. math. 48, no.2, 323-334 (1994). cubo a mathematical journal vol.11, no¯ 04, (73–86). september 2009 circulant matrices, gauss sums and mutually unbiased bases, i. the prime number case monique combescure institut de physique nucléaire de lyon (ipnl), 4 rue enrico fermi f-69622 villeurbanne cedex, france. email: mcombe@ipnl.in2p3.fr abstract in this paper, we consider the problem of mutually unbiased bases in prime dimension d. it is known to provide exactly d + 1 mutually unbiased bases. we revisit this problem using a class of circulant d × d matrices. the constructive proof of a set of d + 1 mutually unbiased bases follows, together with a set of properties of gauss sums, and of bi-unimodular sequences. resumen en este artículo consideramos el problema de bases insesgadas mutuamente en dimensión prima d. se sabe cómo obtener exactamente d + 1 bases insesgadas mutuamente. revisamos el problema usando una clase de matrices circulantes d×d. la demostración constructiva obtiene un conjunto de d + 1 bases insesgadas mutuamente junto con un conjunto de propiedades de sumas gausianas y de sucesiones biúnimodulares. key words and phrases: mutually unbiased bases, circulant matrices, gauss sums. math. subj. class.: 81p68, 15a30, 11t23 1 introduction mutually unbiased bases (mub’s) are a set {b0, ...,bn} of orthonormal bases of cd such that the scalar product in cd of any vector in bj with any vector in bk, ∀j 6= k is of modulus d−1/2. starting 74 monique combescure cubo 11, 4 (2009) from the natural base b0 consiting of vectors v1 = (1, 0, ...0), v2 = (0, 1, 0, ..., 0), ...,vd = (0, 0, ..., 1), it is known that this problem reduces to find n unitary hadamard matrices pj such that p ∗ j pk is also a unitary hadamard matrix ∀j 6= k. (a unitary matrix is hadamard if all its entries are of modulus d−1/2). the problem has been solved in prime power dimension d = pn, p,n ∈ n,p prime, and yields exactly d+1 mub’s which is the maximum number of mub’s ( [2] and references herein contained). if d is factorizable in pm1 1 p m2 2 ... with pi 6= pj prime numbers, it is known that one has at least n = min p mi i [5]. in this paper we show that in prime dimension d = p, the discrete fourier transform f together with a suitable circulant matrix c allow to construct a set of d + 1 mub’s. in addition this construction allows to obtain, as a by-product, a set of properties of gauss sums of the following form : ∣∣∣∣∣ d−1∑ k=0 exp ( 2iπ d [ lk(k + 1) 2 + jk ])∣∣∣∣∣ = √ d, ∀j ∈ fd, l ∈ fd coprime with d, d ≥ 3 (1.1) (fd is the field of residues modulo d). a direct proof of this property is given in [12]. similar results on generalized gauss sums appear in [1]. the definition of f and of circulant matrices is given below. the natural role played by circulant matrices in this context is a new result. the circulant unitary matrices are known to be in one-to-one correspondence with the bi-unimodular sequences c = (c1,c2, ...,cd) [4], namely sequences such that |cj| = |(fc)j| = 1, where f is the discrete fourier transform. not surprisingly gauss sums appear naturally in this context, since suitable gauss sequences are examples of bi-unimodular sequences. at the end of this paper, we consider the case of non-prime dimension, and show that gauss sums properties can be deduced in the odd and in the even dimension cases. in a forthcoming work we shall consider the case of prime power dimensions and show that the theory of block-circulant matrices with circulant blocks solve the mub problem in that case. a d × d matrix is hadamard if all its entries have equal moduli [7], and h ∗ h = d 1l definition 1. a d×d matrix h is hadamard if |hj,k| is constant ∀j,k = 1, ...,d, and h∗h = d 1l. we call h a “unitary hadamard matrix” if |hj,k| = d−1/2, ∀j,k = 1, ...,d, and d∑ l=1 h ∗ j,lhl,k = δj,k definition 2. a d × d matrix c is called circulant [6], and denoted circ(c0,c1, ...,cd−1), if all its cubo 11, 4 (2009) circulant matrices, gauss sums and mutually unbiased bases ... 75 rows and columns are successive circular permutations of the first. it is of the form c =    c0 cd−1 . . c1 c1 c0 . . c2 . . . . . cd−1 cd−2 . . c0    proposition 1.1. (i) the set c of all d × d circulant matrices is a commutative algebra. (ii) c is a subset of normal matrices (ii) let v = circ(0, 0, ..., 1). clearly v d = 1l. then c is circular if and only if it commutes with v , and one has for any sequence c = (c1, ...,cd) ∈ cd, c = circ(c1, ...,cd) : c = pc(v ) = c01l + cd−1v + ... + c1v d−1 where pc is the polynomial pc(x) = d∑ k=0 ckx −k proof : see [6] for the product of two circulant matrices c, c′ (that therefore commute with v ), one has v cc ′ = cv c ′ = cc ′ v which establishes that cc′ is indeed circulant. moreover it is well-known that there is a close link between the circulant matrices and the discrete fourier transform. namely the latter diagonalizes all the circulant matrices. the discrete fourier transform is defined by the following d × d unitary matrix f with matrix elements : fj,k = d −1/2 exp ( 2iπjk d ) , j,k = 0, 1, ...,d − 1 (1.2) proposition 1.2. (i) the circulant matrix v = circ(0, 0, ..., 1) is such that f ∗ v f = u ≡ diag(1,ω,ω2, ...,ωd−1) where ω = exp ( 2iπ d ) (1.3) (ii) let c = circ(c0,c1, ...,cd−1) be a circulant d × d matrix. then f ∗ cf = √ d diag(ĉ0, ĉ−1, ..., ĉ−(d−1)) where ĉl = 1√ d d−1∑ k=0 ckω kl (1.4) 76 monique combescure cubo 11, 4 (2009) proof : (i) it is enough to check that vk, the k-th column vector of f which has components (vk)j = ω jk √ d , j,k = 0, 1, ...,d − 1 is eigenvector of v with eigenvalue ωk, which is immediate since (v vk)j = ω k(j+1) √ d = ω k (vk)j (ii) one has (proposition 1.3(ii)) c = d−1∑ k=0 ckv −k thus f ∗ cf = d−1∑ k=0 ck(f ∗ v f) −k = d−1∑ k=0 cku −k = diag(d0, ...,dd−1) with dl = d−1∑ k=0 ckω −lk = √ d ĉ−l lemma 1.3. for any k ∈ n, we denote by [k] the rest of the division of k by d. given any sequence c = (c0, ...,cd−1) ∈ cd its autocorrelation function obeys d−1∑ k=0 c̄kc[j+k] = d−1∑ l=0 |ĉl|2ω−jl where the fourier transform ĉ of c has been defined in (1.4). proof : for any j = 0, ...,d − 1, one has d−1∑ l=0 |ĉl|2ω−jl = 1 d d−1∑ l=0 ω −jl d−1∑ k=0 c̄kω −lk d−1∑ k′=0 ck′ω k′l = d−1∑ k,k′=0 c̄kck′ 1 d d−1∑ l=0 ω l(k′−k−j) = d−1∑ k=0 c[j+k]c̄k since d −1 d−1∑ l=0 ω l(k′−j−k) = δk′,[j+k] it is known ( [4], [13]) that circulant unitary hadamard matrices are in one-to-one correspondance with bi-unimodular sequences c = (c0,c1, ...,cd−1). definition 3. a sequence c = (c0,c1, ...,cd−1) is called bi-unimodular if one has |cj| = |ĉj| = 1, ∀j = 1, ...,d, where ĉj is defined by (1.4). cubo 11, 4 (2009) circulant matrices, gauss sums and mutually unbiased bases ... 77 proposition 1.4. let (c0, ...,cd−1) be a bi-unimodular sequence. then the circulant matrix c = d −1/2 circ(c0,c1, ...,cd−1) is an unitary hadamard matrix. proof : this is a standard “if and only if” statement. one uses lemma 1.5 : d−1∑ k=0 c̄kc[j+k] d−1∑ l=0 |ĉl|2ω−jl but since |ĉl| = 1, ∀l = 0, ...,d − 1, the rhs is simply dδj,0, and therefore d−1∑ k=0 c̄kc[k+j] = d δj,0 which proves the unitarity of the circulant hadamard matrix c. in all that follows we call indifferently f or p0 the discrete fourier transform. in [8], the authors introduce for any dimension d being the power of a prime number a set of operators called “rotation operators” which can be viewed as “circulant matrices” (this property is however not put forward explicitely by the authors). in this paper, restricting ourselves to the prime number case, we show that these operators can be used to define a set of d + 1 mutually unbiased bases in dimension d. mutually unbiased bases are extensively studied in the framework of quantum information theory. they are defined as follows : definition 4. a set {b1,b2, ...,bm} of orthonormal bases of cd is called mub if for any vector b (k) j ∈ bk and any b (k′) j′ ∈ bk′ one has |b(k)j · b (k′) j′ | = d −1/2 , ∀k 6= k′ = 1, ...,m, ∀j,j′ = 1, ...,d where the dot represents the hermitian scalar product in cd. remark 1.5. it is trivial to show that if the orthonormal bases bk are the column vectors of an unitary matrix ak, then the property that must satisfy the ak’s in order that {1l,b1, ...,bm} are mub’s is that all ak, k = 1, ...,m and a∗kak′, k 6= k′ = 1, ...,m are unitary hadamard matrices. namely if uj, vk are column vectors for unitary matrices a, a′ respectively, then u j · vk = (a∗a′)j,k thus unitary hadamard matrices play a major role in the mub problem. it is known that the maximum number of mub’s in any dimension d is d + 1, and that this number is attained if d = pm, p being a prime number. in this paper, restricting ourselves to 78 monique combescure cubo 11, 4 (2009) m = 1, we revisit the proof of this property, using circulant matrices introduced by [8]. we then show that it implies beautiful properties of gauss sums, namely the following ( [12]) : proposition 1.6. let d ≥ 3 be an odd number. then ∀k = 1, ...,d−1 coprime with d the sequences g (k) := ( exp ( iπkj(j + 1) d )) j=0,...,d−1 (1.5) are bi-unimodular. thus (1.1) holds true. this property will appear as a subproduct of our study of mub’s for d a prime number ≥ 3 via the circulant matrices method. as stressed above, the link between circulant matrices and bi-unimodular sequences is well established. what is new here is the fact that the mub problem via a circulant matrix method allows to recover the bi-unimodularity of gauss sequences. the crucial role played by the gauss sequence is due to the crucial role played by the discrete fourier transform (or in other therms the fourier-vandermonde matrices) in the mub problem for prime numbers. let us introduce it now explicitely. it is known since schwinger ( [11]) that a simple toolbox of unitary d × d matrices sometimes refered to as “generalized pauli matrices” u,v allows to find mub’s. u, v generate the discrete weyl-heisenberg group [14]. denote by ω the primitive root of unity (1.3). the matrix u is simply u = diag(1,ω,ω 2 , ...,ω d−1 ) which generalizes the pauli matrix σz to dimensions higher than two. the matrix v generalizes σx : v = circ(0, 0, ..., 1) =    0 1 0 . . 0 0 0 1 . . 0 . . . . . . 0 0 0 . . 1 1 0 0 . . 0    then one has the following result : theorem 1.7. (i) the u, v matrices obey the ω-commutation rule : v u = ωuv (ii) the discrete fourier transform matrix p0 = f defined by (1.2), namely p0 = 1√ d    1 1 1 . . 1 1 ω ω 2 . . ω d−1 1 ω 2 ω 4 . . ω 2(d−1) . . . . . . 1 ω d−1 ω 2(d−1) . . ω (d−1)(d−1)    cubo 11, 4 (2009) circulant matrices, gauss sums and mutually unbiased bases ... 79 diagonalizes v , namely v = p0up ∗ 0 = p ∗ 0 u ∗ p0 (iii) one has p 4 0 = 1l (ii) is simply proposition 1.4(i). for the proof of (iii) reminiscent to the properties of continuous fourier transform, it is enough to check that p 2 0 = w =    1 0 0 0 . . 0 0 0 0 0 . . 1 . . . . . . . . . . . . . . 0 0 1 0 . . 0 0 1 0 0 . . 0    thus w = 1l in dimension d = 2, and w 2 = 1l, ∀d ≥ 3. in [5] we have shown that for d odd one can add to the general toolbox of unitary schwinger matrices u, v a diagonal matrix d of the form d = diag(1,ω,ω 3 , ...,ω k(k+1)/2 , ..., 1) so that the mub problem for odd prime dimension reduces to properties of u, v, d, and that certain properties of quadratic gauss sums follow as a by-product. 2 the d=2 case we have u = σz and v = σx, σz, σx being the usual pauli matrices. since uv = σy, finding mub’s in dimension d = 2 amounts to diagonalize σx, σy. one has : σx = p0σzp ∗ 0 σy = p1σzp ∗ 1 with p0 = 1√ 2 ( 1 1 1 −1 ) , p1 = 1√ 2 ( 1 i i 1 ) p1 is circulant. proposition 2.1. the set {1l,p0,p1} defines three mub’s in dimension 2. since the matrices p0, p1 are trivially unitary hadamard matrices, it is enough to check that p ∗ 0 p1 is itself a unitary hadamard matrix, which holds true since p ∗ 0 p1 = e iπ/4 √ 2 ( 1 1 −i −i ) 80 monique combescure cubo 11, 4 (2009) 3 the prime dimension d ≥ 3 d being prime, we denote by fd the galois field of integers mod d. let us recall the definition of the “rotation operator” of [8], which, as already stressed in nothing but a circulant matrix in the odd prime dimension d. definition 5. define r as an unitary operator commuting with v and diagonalizing v u. proposition 3.1. (i) r is a circulant matrix. (ii) rk is also circulant ∀k ∈ z. this follows from proposition 1.3. therefore we are led to consider a subclass of circulant matrices that are unitary. they must satisfy : ∀k = 0, ...,d − 1 |ck| = d−1/2 and ∀k = 1, ...,d − 1 ∑d−1 j=0 c̄jc[d−k+j] = 0 (orthogonality condition). now it remains to show that such a matrix r exists. in [5] we have constructed a unitary matrix p1 that diagonalizes v u. it is defined as p1 = d −1 p0 for any d odd integer (not necessarily prime). we have established the following result : proposition 3.2. (i) for any odd integer d, the matrix p∗ 0 p1 is a unitary hadamard matrix. (ii) for d ≥ 3 odd integer, let pk := d−kp0. then p∗0 pk is a unitary hadamard matrix for all k coprime with d. (iii) the matrix d defined above is such that |trdk| = √ d, ∀k ∈ fd coprime with d for the simple proof of this result see [5]. (iii) is a simple consequence of (i) and(ii). namely the eigenvector of v u belonging to the eigenvalue 1 has components vj = d −1/2 ω − j(j+1) 2 since p∗ 0 p1 is unitary hadamard matrix, the element of p ∗ 0 p1 of the first row and first column is simply d −1 d−1∑ j=0 ω − j(j+1) 2 cubo 11, 4 (2009) circulant matrices, gauss sums and mutually unbiased bases ... 81 and since its modulus must be d−1/2 we obtain (iii) for k = 1. the proof for any k coprime with d can be obtained similarly using (ii). the “problem” is that p1 is not circulant. however the circulant matrix r is obtained from p1 by multiplying the kth column vector of p1 by a phase which is ω − k(k−1) 2 this operation preserves the fact that it is unitary and that it diagonalizes v u. we thus have : v u = p1up ∗ 1 = rur ∗ (3.1) proposition 3.3. let r be the matrix : r = d −1/2 circ (1,ω −1 ,ω −3 , ...,ω −k(k+1)/2 , ..., 1) it is a unitary hadamard matrix. proof : by construction it is a unitary hadamard matrix, since p1 is. to prove the fact that it is circulant, it is enough to know that (p1)j,k = d −1/2 ω jk− j(j+1) 2 thus rjk = d −1/2 ω jk− j(j+1) 2 − k(k−1) 2 = d −1/2 ω − (j−k)(j−k+1) 2 (3.2) since jk − j(j + 1) 2 − k(k − 1) 2 = −(j − k)(j − k + 1) 2 thus all column vectors are obtained from the first by the circularity property. furthermore the rk have the property that they diagonalize v ku, ∀k = 1, ...,d − 1: theorem 3.4. (i) r = αp0dp∗0 where α := d −1/2 ∑d−1 k=0 ω −k(k+1)/2 is a complex number of modulus 1. (ii) rd = αd1l where 1l denotes the unity d × d matrix. (iii) r k u(r ∗ ) k = v k u, ∀k = 0, ...,d the proof is extremely simple : (i) we have shown that cj = ω −j(j+1) 2 is a bi-unimodular sequence, thus α is a complex number of modulus one. moreover from proposition 1.4 (ii), the unitary matrix r = d−1/2circ(cj) is such that r̂ = p ∗ 0 rp0 = diag(ĉ−k) 82 monique combescure cubo 11, 4 (2009) but ĉ−k = 1√ d d−1∑ j=0 ω −jk− j(j+1) 2 αω k(k+1) 2 since d−1∑ j=0 ω − j(j+1) 2 = d−1∑ j=0 ω − (j+k)(j+k+1) 2 = d−1∑ j=0 ω −jk− j(j+1) 2 − k(k+1) 2 therefore r̂ = αdiag(ω k(k+1) 2 ) = αd (ii) is simply a consequence of (i) since dd = 1l. (iii) is obtained by recurrence. namely it is true for k = 1 by (3.1). for k ≥ 2 one has : r k u(r ∗ ) k = rv k−1 ur ∗ = v k−1 rur ∗ since r commutes with v . but v k−1 rur ∗ = v k−1 (v u) = v k u there is a direct link between the matrices pk that diagonalize v u k and the rk that diagonalize v k u : theorem 3.5. α being the complex number of modulus one defined above, we have for any k = 0, ...,d − 1 pk = α k p ∗ 0 r −k p 2 0 proof : in [5] we have proven that pk = d −k p0 diagonalizes v u k. but one has d −k = d d−k = α k−d p ∗ 0 r d−k p0 so the result follows immediately. corollary 3.6. for any k = 1, ...,d − 1, rk is a unitary hadamard circulant matrix when d ≥ 3 is prime. proof : rk is circulant and unitary since r is. therefore we have only to check that it is hadamard. we have : r −k = α −k p 2 0 p ∗ 0 pkp 2 0 but we recall that p 2 0 equals the permutation matrix w . thus all matrix elements of rk equal, up to a phase, some matrix elements of p∗ 0 pk. but we have established in [5] that the matrix p ∗ 0 pk is unitary hadamard ∀k = 1, ...,d − 1, thus all its matrix elements are equal in modulus to d−1/2. this completes the proof. now we show how this property of the matrix r reflects itself in gauss sums properties. cubo 11, 4 (2009) circulant matrices, gauss sums and mutually unbiased bases ... 83 proposition 3.7. (i) let d be an odd prime. then for any k = 1, 2, ...,d − 1 rk is an unitary hadamard matrix if and only if one has ∣∣∣∣∣∣ d−1∑ j=0 exp ( iπ d [kj 2 + j(k + 2m)] ) ∣∣∣∣∣∣ = √ d, ∀m = −d + 1, ...,d − 1 (ii) under the same conditions rk is a unitary hadamard matrix if and only if ∣∣∣∣∣∣ k−1∑ j=0 exp ( iπ k dj 2 + (k + 2m)j ) ∣∣∣∣∣∣ = √ k proof : since rk = αkp0d k p ∗ 0 , the matrix elements of rk are (r k )m,l = α k d d−1∑ j=0 ω j(m−l)+k j(j+1) 2 since rk is circulant, it is unitary hadamard matrix if and only if the matrix elements of the first column (l=0) are of modulus d−1/2, thus if and only if ∣∣∣∣∣∣ d−1∑ j=0 exp ( iπ d [kj 2 + j(k + 2m)] ) ∣∣∣∣∣∣ = √ d which proves (i). (ii) using the reciprocity theorem for gauss sums ( [3]), we have for all integers a,b with ac 6= 0 and ac + d even that the quantity s(a,b,d) := d−1∑ j=0 exp ( πi d (aj 2 + bj) ) obeys s(a,b,d) = ∣∣∣∣ d a ∣∣∣∣ 1/2 exp ( πi 4 [sgn(ad) − b2/ad] ) s(−d,−b,a) thus |s(a,b,d)| = √ d if and only if |s(−d,−b,a)| = √ a. applying if to a = k = 1, ...,d − 1 coprime with d and b = 2m + k, and taking the complex conjugate yields the result. namely ad + b = dk + k + 2m is even for all k = 0, ...,d − 1 since d is odd. corollary 3.8. let d ≥ 3 be a prime number. then for any k = 1, 2, ...,d − 1 the sequences g (k) := ( ω k j(j+1) 2 ) j=0,...,d−1 are bi-unimodular. remark 3.9. this property is known, but has an extension in the non prime odd dimensions. see next section. 84 monique combescure cubo 11, 4 (2009) it is known that the diagonalization of v uk, k = 0, ...,d− 1 provides a set of d + 1 mub’s for a prime p. here we show that the same is true with the diagonalization of v ku, k = 0, ...,d − 1. theorem 3.10. let d ≥ 2 be a prime dimension. then the orthonormal bases defined by the unitary matrices 1l,p0,r,r2, ...,rd−1 provide a set of d + 1 mub’s. proof : for d = 2 this has been already proven in section 2. for d ≥ 3 (thus odd, since it is prime), it is enough to check that : (i) p0,r k , k = 1, ...,d − 1 are unitary hadamard matrices, together with (ii) p∗ 0 r k , k = 1, ...,d − 1 and (rk′ )∗rk, 1 ≤ k′ < k ≤ d − 1. since (i) has been already established, it remains to show (ii). since rk = αkp0d k p ∗ 0 we have p ∗ 0 r k = α k d k p ∗ 0 which is trivially an unitary hadamard matrix (since p0 is, α is of modulus 1 and d is diagonal and unitary). for (r∗)k ′ r k it is trivial since (r ∗ ) k′ r k = r k−k′ which is unitary hadamard for any k 6= k′, k,k′ = 1, ...,d − 1. 4 the case of arbitrary odd dimension we have shown in [5] that for any odd dimension d ≥ 3 the matrices p∗ 0 pk is an unitary hadamard matrix provided k is co-prime with d. this can be transfered to a similar property for the matrix r−k, and therefore to the bi-unimodularity of the sequence g(k) defined in (1.5). proposition 4.1. let d ≥ 3 be an odd integer, and k be any number coprime with d. then (i) rk is an unitary hadamard matrix. (ii) the sequence g(k) is bi-unimodular. (iii) both properies are equivalent. this implies proposition 1.10. theorem 4.2. let d be odd and k > 2 be the smallest divisor of d. then the orthonormal bases defined by the unitary matrices { 1l,f,r,r 2 , ...,r k−1 } provide a set of k + 1 mub’s in dimension d. 5 the case of arbitrary even dimension let d be even and ω = exp ( 2iπ d ) cubo 11, 4 (2009) circulant matrices, gauss sums and mutually unbiased bases ... 85 we denote ω1/2 = eiπ/d. one defines the discrete fourier transform f as usually. the theory of circulant matrices is also pertinent for even dimensions d ≥ 4. namely in that case the matrix d ′ = diag(1,ω −1/2 , ...,ω −k2/2 , ...,ω −1/2 ) has been shown ( [5]) to be such that the unitary hadamard matrix p1 = d ′ f diagonalizes v u. but the circulant matrix r obtained by multiplying the k-th column vector of p1 by ω −k2/2 also diagonalizes v u : proposition 5.1. the circulant matrix whose matrix elements are rj,k = 1√ d ω −(j−k)2/2 diagonalizes v u and is such that f∗r is an unitary hadamard matrix. proof : (p1)j,k = 1√ d ω − j2 2 +jk thus rj,k = 1√ d ω − (j−k)2 2 rj,k only depends on j − k thus is circulant (and unitary). therefore it is diagonalized by f , namely there exists an unitary diagonal matrix d′′ such that f ∗ rf = d ′′ again this implies that f∗r is an unitary hadamard matrix. corollary 5.2. (i) the orthonormal bases defined by the unitary matrices {1l,f,r} provide a set of 3 mub’s in arbitrary even dimension d. (ii) one has the following property of quadratic gauss sums for d even : ∣∣∣∣∣ d−1∑ k=0 exp ( ik 2 π d )∣∣∣∣∣ = √ d remark 5.3. r2 is circulant, unitary, but not hadamard. thus it does not help to find more than 3 mub’s in even dimensions. in dimensions d = 2n, another method is necessary to prove that there exists d + 1 mub’s. acknowledgments : it is a pleasure to thank b. helffer for his interest in this work, and to bahman saffari for interesting discussions and for providing to me reference [12]. received: august 2008. revised: september 2008. 86 monique combescure cubo 11, 4 (2009) references [1] albouy, o. and kibler, m., su2 nonstandard bases : the case of mutually unbiased bases, symmetry, integrability and geometry : methods and applications, (2007) [2] bandyopadhyay, s. boykin, p. o. roychowdhury, v. and vatan, f., a new proof of the existence of mutually unbiased bases, algorithmica, 34, 512-528, (2002) [3] berndt, b. c. evans, r. j. and williams, k. s., gauss and jacobi sums, canadian mathematical society series of monographs and advanced texts, vol 21, wiley, (1998) [4] bjrck, g. and saffari, b., new classes of finite unimodular sequences with unimodular fourier transforms. circulant hadamard matrices with complex entries, c. r. acad. sci. paris, 320 serie 1, (1995), 319-324 [5] combescure, m., the mutually unbiased bases revisited, contemporary mathematics, (2007), to appear [6] davis, p. j., circulant matrices, wiley, (1979) [7] hadamard, j., rsolution d’une question relative aux dterminants, bull. sci. math. 17, 2460246 (1893) [8] klimov, a. b. muoz, c. and romero, j. l., geometrical approach to the discrete wigner function, arxiv:quant-ph/0605113, (2006) [9] klimov, a. b. sanchez-soto, l. l. and de guise, h., multicomplementary operators via finite fourier transform, journal of physics a 38, 2747–2760 (2005) [10] planat, m. and rosu, h., mutually unbiased phase states, phase uncertainties, and gauss sums, eur phys. j. d 36, 133-139, (2005) [11] schwinger, j., unitary operator bases, proc nat. acad. sci. u.s.a. 46, 560 (1960) [12] saffari, b., quadratic gauss sums, to appear [13] turyn, r., sequences with small correlation, in error correcting codes, h. b. mann ed., wiley (1968), 195-228 [14] weyl, h., gruppentheorie and quantenmechanik, hirzel, leipzig, (1928) articulo 6 cubo a mathematical journal vol.10, n o ¯ 01, (33–41). march 2008 on some what fuzzy faintly semicontinuous functions g. palanichetty and g. balasubramanian ramanujan institute for advanced studies in mathematics university of madras, department of mathematics i.r.t. polytechnic college, krishnagiri – 635 108 tamilnadu chepauk, chennai – 600 005,india email: gpalanichetty@yahoo.co.in abstract in this paper the concept of some what fuzzy faintly semicontinuous functions, some what fuzzy faintly semiopen functions,weakly some what fuzzy faintly semiopen functions are introduced. some characterizations and interesting properties of these functions are discussed. resumen el concepto de funciones fuzzy débil semicontinuas, funciones fuzzy débil semiabiertas son introducidas. caracterizaciones e propriedades de este tipo de funciones son discutidas. key words and phrases: some what fuzzy faintly semicontinuous functions, some what fuzzy faintly semiopen functions, fuzzy semidense set, fuzzy semiseparable space, fuzzy dsspace. math. subj. class.: 54a40. 34 g. palanichetty and g. balasubramanian cubo 10, 1 (2008) 1 introduction the concept of fuzzy sets was introduced by zadeh [8]. based on the concept of fuzzy sets, c.l.chang [4] introduced and developed the concept of fuzzy topological spaces. since then various important notions in the classical topology such as continuous functions [4] have been extended to fuzzy topological spaces. the concept of some what continuous functions was introduced and studied by karl r. gentry and hughes b.hoyle [5] in general topological spaces and this concept was extended to fuzzy topological spaces in [6].. in [2] the concept of fuzzy faintly α-continuous functions was introduced and studied. also some characterizations of fuzzy faintly continuous functions are given. the purpose of this paper is to introduce and study the concept of some what fuzzy faintly semicontinuous functions and it is infact a generalization of fuzzy faintly continuous functions introduced and studied in [2]. section 2 deals with preliminaries. section 3 deals with some what fuzzy faintly semicontinuous functions and some of its characterizations. section 4 deals with some what fuzzy faintly semiopen functions and some of its basic properties and characterizations. 2 preliminaries in this paper by (x, t ) we mean fuzzy topological space in the sense of [4]. let λ be a fuzzy set. we define closure of λ = clλ = ∧{µ/µ ≥ λ, µ is fuzzy closed } and interior of λ = intλ = ∨{σ/σ ≤ λ, σ is fuzzy open}.in [7] the semiinterior and semiclosure of a fuzzy set λ are defined as s-int(λ) = ∨ {µ/µ ≤ λ, µ is fuzzy semiopen} and s-cl(λ) = ∧{µ/µ ≥ λ, µ is fuzzy semiclosed} and also shown that s-cl(1 − λ) = 1−s-int(λ) and s-int(1 − λ) = 1−scl(λ) . a fuzzy set λ in (x, t ) is called proper if λ 6= 0 and λ 6= 1.a fuzzy point xp in x is a fuzzy set in x defined by xp (y) = { p, p ∈ (0, 1] for y = x, y ∈ x; 0, for y 6= x, y ∈ x. x and p are respectively called the support and the value of xp. a fuzzy point xp is said to be quasi-coincident [1] with α, denoted by xpqα , if and only if p > α ′ (x) or p + α(x) > 1 where α ′ denotes the complement of α.. a fuzzy subset λ in a fuzzy topological space x is said to be a q− neighbourhood for a fuzzy point xp if and only if there exists a fuzzy open subset β such that xpqβ ≤ λ. a fuzzy set λ is called fuzzy θ-open [1] if and only if intθ (λ) = λ where intθ (λ) = ∨{xp ∈ x/ for some open qneighborhood β of xp, clβ ≤ λ} and clθ (λ) = ∧ {µ/µ > λ, µ is θ closed}. λ is called θ-closed if clθ(λ) = λ. a fuzzy set λ cubo 10, 1 (2008) on some what fuzzy faintly semicontinuous functions 35 in a fuzzy topological space (x, t ) is called dense (θdense, semidense) if there exists no fuzzy closed set (θclosed set, semiclosed set ) µ such that λ < µ < 1. a fuzzy set λ in (x, t ) is called fuzzy semiopen [3] if for some fuzzy open set v, v ≤ λ ≤ cl (v) , equivalently λ is called fuzzy semiopen if λ ≤ cl int(λ) .if λ and µ are any two fuzzy sets in x and y respectively, we define λ × µ : x × y → i as follows: (λ × µ) (x, y) = min (λ (x) , µ (y)) . a fuzzy topological space x is product related to a fuzzy topological space y if for any fuzzy set v in x and ξ in y whenever λ ′ (= 1 − λ) � v and µ′ (1 − µ) � ξ imply λ′ × 1 ∨ 1 × µ′ ≥ v × ξ, where λ is a fuzzy open set in x and µ is a fuzzy open set in y,there exists a fuzzy open set λ1 and a fuzzy open set µ1in y such that λ ′ 1 ≥ v or µ ′ 1 ≥ ξ and λ ′ 1 × 1 ∨ 1 × µ′1 = λ ′ × 1 ∨ 1 × µ′. if (x, t ) and (y, s) are any two fuzzy topological spaces, we define a product fuzzy topology t × s on x × y to be that fuzzy topology for which b = {λ × µ/λ ∈ t, µ ∈ s} forms a base. 3 some what fuzzy faintly semicontinuous functions in [2] the concept of fuzzy faintly continuous function is given as follows : definition 3.1 let f : (x, t1) → (y, t2) be a function from the fuzzy topological space (x, t1) to the fuzzy topological space (y, t2). f is called fuzzy faintly continuous if f −1 (λ) is fuzzy open for every fuzzy θ− open set λ in y . in [7] the concept of somewhat fuzzy semicontinuous function is introduced as follows. definition 3.2 let f : (x, t ) → (y, s) be a function from the fuzzy topological space (x, t ) to the fuzzy topological space (y, s). f is called somewhat fuzzy semicontinuous if λ ∈ s and f −1 (λ) 6= 0 implies there exists a fuzzy semiopen set µ of x such that µ ≤ f −1(λ). we are now ready to make the following: definition 3.3 let (x, t ) and (y, s) be any two fuzzy topological spaces. a function f : (x, t ) → (y, s) is called some what fuzzy faintly semicontinuous if for every fuzzy θopen set λ in y such that f −1(λ) 6= 0, there exists a fuzzy semiopen set 0 6= µ in (x, t ) such that µ ≤ f −1(λ). clearly every fuzzy faintly continuous function [2] is some what fuzzy faintly semicontinuous but the converse is not true as the following example shows. 36 g. palanichetty and g. balasubramanian cubo 10, 1 (2008) example 3.4 consider example 2.3 [2]. let x = y = i = [0, 1]. let µ1, µ2, µ3 be fuzzy sets on i defined as follows: µ1 (x) = { 0, 0 ≤ x ≤ 1 2 2x − 1, 1 2 ≤ x ≤ 1 µ2 (x) =    1, 0 ≤ x ≤ 1 4 −4x + 2, 1 4 ≤ x ≤ 1 2 0, 1 2 ≤ x ≤ 1 and µ3 (x) = { x, 0 ≤ x ≤ 1 4 ; 1, 1 4 ≤ x ≤ 1. clearly t = {0, µ2, 1}, s = {0, µ1, µ2, µ1 ∨ µ2, 1} are two fuzzy topologies on i. let f : (i, t ) → (i, s) be defined as follows f (x) = x, for each x in i. now we can verify f −1 (µ3) = µ3 and f −1 (1) = 1 are fuzzy semiopen subsets of (i, t ) and also µ3 and 1 are the only fuzzy θ-open subsets of (i, s). it is easy to see that µ3 and 1 are fuzzy semiopen sets in (i, t ) . hence f is somewhat fuzzy faintly semicontinuous; but since f −1(µ3) = µ3 6∈ t1, f is not fuzzy faintly continuous. proposition 3.5 let (x, t ) and (y, s) be any two fuzzy topological spaces. let f : (x, t ) → (y, s) be a function.then the following are equivalent. (a) f is some what fuzzy faintly semicontinuous. (b) if λ is a fuzzy θclosed set of y such that f −1(λ) 6= 1, then there exists a proper fuzzy semiclosed set µ of x such that µ ≥ f −1(λ). (c) if λ is a fuzzy semidense set, then f (λ) is a fuzzy θ dense set in y . proof. (a)⇒ (b) suppose f is some what fuzzy faintly semicontinuous and λ is any fuzzy θ-closed set in y such that f −1 (λ) 6= 1. therefore clearly 1 − λ is fuzzy θ open and f −1 (1−λ) = 1−f −1(λ) 6= 0. but by (a) there exists a fuzzy semiopen set µ∗ in (x, t ) such that µ ∗ 6= 0 and µ∗ ≤ f −1(1 − λ). therefore 1 − µ∗ ≥ 1 − f −1 (1 − λ) = 1 − [ 1 − f −1 (λ) ] = f −1 (λ) . put 1 − µ∗ = µ. clearly µ is proper fuzzy semiclosed set and µ ≥ f −1(λ). this shows (a) ⇒ (b). (b)⇒ (c) let λ be a fuzzy semidense set in x and suppose f (λ) is not fuzzy θ dense in y . then there exists a fuzzy θclosed set say µ ∗ such that f (λ) < µ ∗ < 1. (a) now µ ∗ < 1 ⇒ f −1(µ∗) 6= 1. then by (b) there exists a proper fuzzy semiclosed set σ in (x, t ) such that σ ≥ f −1(µ∗). but by (a), f −1(µ∗) > f −1[f (λ)] ≥ λ that is σ > λ. this cubo 10, 1 (2008) on some what fuzzy faintly semicontinuous functions 37 implies there exists a proper fuzzy semiclosed set σ such that σ > λ which is a contradiction, since λ is fuzzy semidense set. this proves (b) ⇒ (c). (c) ⇒ (a) let λ be any fuzzy θ-open set in (y, s) and suppose f −1(λ) 6= 0 and hence λ 6= 0. we want to show that f is some what fuzzy faintly semicontinuous. that is we want to show that there exists a fuzzy semiopen set µ in (x, t ) such that 0 6= µ ≤ f −1(λ).that is we want to show that s-int[f −1 (λ)] 6= 0.suppose s-int[f −1(λ)] = 0.then s-cl [ 1 − f −1(λ) ] = 1 − s-int[f −1(λ)] = 1 − 0 = 1. this means 1−f −1(λ) is fuzzy semidense in x. now by (c), f [1−f −1(λ)] is fuzzy θ-dense in y. that is clθf [1 − f −1 (λ)] = 1; but f [1 − f −1(λ)] = f [f −1(1 − λ)] ≤ 1 − λ < 1 (since λ 6= 0). since 1−λ is fuzzy θ-closed and f [1−f −1(λ)] ≤ 1−λ,clθ f [1−f −1 (λ)] ≤ 1−λ.that is 1≤1−λ ⇒ λ≤0 ⇒ λ = 0, which is a contradiction to the fact that λ 6= 0. therefore we must have s-int[f −1 (λ)] 6= 0. this proves that f is some what fuzzy faintly semicontinuous. 2 proposition 3.6 let (x, t ) and (y, s) be fuzzy topological spaces and f : (x, t ) → (y, s) be some what fuzzy faintly semicontinuous. let a be a subset of x such that χa ∧ µ 6= 0 for some 0 6= µ in (x, t ) . let t /a be the induced fuzzy topology on a. then f /a : (a, t /a) → (y, s) is some what fuzzy faintly semicontinuous. proof. suppose λ is a fuzzy θ open set in (y, s) such that f −1(λ) 6= 0. since f is some what fuzzy faintly semicontinuous, there exists a fuzzy semiopen set µ in (x, t ) such that µ 6= 0 and µ ≤ f −1(λ). but µ/a ∈ t /a and µ/a 6= 0 (since χa ∧ µ 6= 0 for all µ ∈ t ). now (f /a) −1 (λ) (x) = λ [(f /a) (x)] = λf (x) > µ (x) = µ/a (x) for x ∈ a. also µ is a fuzzy semiopen set in (x, t ) implies µ/a is fuzzy semiopen in (a, t /a) and µ/a < (f /a) −1 (λ). this proves that f /a is some what fuzzy faintly semicontinuous. 2 definition 3.7 let x be a set, t and t ′ be any two fuzzy topologies for x. we say that t ′ is weakly semiequivalent to t if λ 6= 0 is a fuzzy semiopen set in (x, t ) , then there is a fuzzy semiopen set µ in (x, t ′) such that µ 6= 0 and µ ≤ λ. proposition 3.8 let (x, t ) and (y, s) be any two fuzzy topological spaces and suppose that f : (x, t ) → (y, s) is some what fuzzy faintly semicontinuous. let t ′ be a fuzzy topology weakly semiequivalent to t . then f : (x, t ′) → (y, s) is some what fuzzy faintly semicontinuous. proof. let λ be a fuzzy θopen set in s such that f −1 (λ) 6= 0. since f : (x, t ) → (y, s) is some what fuzzy faintly semicontinuous , there is a fuzzy semiopen set µ in (x, t ) and µ 6= 0 38 g. palanichetty and g. balasubramanian cubo 10, 1 (2008) such that µ ≤ f −1(λ). but by the hypothesis t ′ is weakly semiequivalent to t. therefore there exists a fuzzy semiopen set µ ∗ such that µ ∗ in (x, t ′ ) and µ ∗ 6= 0 and µ∗ < µ. but µ < f −1 (λ) implies µ ∗ ≤ f −1(λ). this means f : (x, t ′) → (y, s) is some what fuzzy faintly semicontinuous. 2 definition 3.9 let y be a set. let s and s′ be any two fuzzy topologies in y. we say that s ′ is weakly θ-equivalent to s if λ 6= 0 is a fuzzy θ-open set in (y, s′) , then there exists a fuzzy θ-open set µ in (y, s) such that µ ≤ λ. proposition 3.10 let (x, t ) and (y, s) be any two fuzzy topological spaces and suppose f : (x, t ) → (y, s) is some what fuzzy faintly semicontinuous. let s′ be a fuzzy topology weakly θ-equivalent to s. then f : (x, t ) → (y, s′) is some what fuzzy faintly semicontinuous. proof. let λ be a fuzzy θ-open set in (y, s′) such that f −1 (λ) 6= 0. since s′ is weakly θ-equivalent to s, there exists a fuzzy θ-open set λ ∗ in (y, s) such that 0 6= λ∗ ≤ λ. now 0 6= f −1 (λ∗) ≤ f −1 (λ) .since f is some what fuzzy faintly semicontinuous from (x, t ) to (y, s) there exists a fuzzy semiopen set in (x, t ) say µ such that µ 6= 0 and µ ≤ f −1 (λ∗) . this means µ ≤ f −1 (λ) and so f is some what fuzzy faintly semicontinuous from (x, t ) to (y, s ′ ). 2 proposition 3.11 let (x, t ) and (y, s) be fuzzy topological spaces. suppose that f : (x, t ) → (y, s) is some what fuzzy faintly semicontinuous, t ′ and s′ are fuzzy topologies for x and y respectively such that t ′ is weakly semiequivalent to t and s′ is weakly θ-equivalent to s. then f : (x, t ′) → (y, s′) is some what fuzzy faintly semicontinuous. proof. proof follows from propositions 3.8 and 3.10. 2 definition 3.12 a fuzzy topological space (x, t ) is said to be fuzzy separable (semiseparable) if there exists a fuzzy dense (semidense) set λ in (x, t ) such that λ 6= 0 for atmost countably many points of x. proposition 3.13 if f : (x, t ) → (y, s) is a some what fuzzy faintly semicontinuous function and if x is fuzzy semiseparable, then y is fuzzy θ-separable. proof. since x is fuzzy semiseparable, there exists a fuzzy dense set λ such that λ 6= 0 for atmost countably many points of x. also since f is some what fuzzy faintly semicontinuous, it follows by proposition 3.5 that f (λ) is fuzzy θdense in (y, s) and since λ 6= 0 and λ is semidense for atmost countably many points, it follows that f (λ) 6= 0 for atmost countably many points. thus we find that (y, s) is fuzzy θ-separable. 2 cubo 10, 1 (2008) on some what fuzzy faintly semicontinuous functions 39 4 some what fuzzy faintly semiopen functions definition 4.1 let (x, t ) and (y, s) be fuzzy topological spaces. f : (x, t ) → (y, s) is called some what fuzzy faintly semiopen function if and only if for any fuzzy semiopen set λ, λ 6= 0 in (x, t ) implies that there exists a fuzzy θopen set µ in (y, s) such that µ 6= 0 and µ < f (λ).that is intθ [f (λ)] 6= 0. proposition 4.2 let (x, t ), (y, s) and (z, r) be fuzzy topological spaces. suppose that f : (x, t ) → (y, s) and g : (y, s) → (z, r) are some what fuzzy faintly semiopen functions. then g ◦ f : (x, t ) → (z, r) is some what fuzzy faintly semiopen. proof. let λ be a fuzzy semiopen set in t. since f is some what fuzzy faintly semiopen then there exists a fuzzy θ-open set µ in s such that µ ≤ f (λ). now intθf (λ) ∈ s and since g is some what fuzzy faintly semiopen, then there exists a fuzzy θ-open set γ in (z, r) such that γ < g [intf (λ)] . but g [intf (λ)] < g [f (λ)] . thus we find that there exists a fuzzy θ-open set γ in (z, r) such that γ < (g ◦ f ) (λ) . this proves g ◦ f is some what fuzzy faintly semiopen. 2 proposition 4.3 let (x, t ) and (y, s) be fuzzy topological spaces. let f : (x, t ) → (y, s) be an onto function. then the following are equivalent. (a) f is some what fuzzy faintly semiopen. (b) if λ is a fuzzy semiclosed set in x such that f (λ) 6= 1, then there exists a fuzzy θ-closed set µ in y such that µ 6= 1 and µ > f (λ). proof. (a) ⇒ (b). let λ be a fuzzy semiclosed set in x such that f (λ) 6= 1. then (1 − λ) is a fuzzy semiopen set such that f (1 − λ) = 1 − f (λ) 6= 0. since f is some what fuzzy faintly semiopen, there exists a fuzzy θ-open set σ in s such that σ 6= 0 and σ ≤ f (1 − λ). now 1 − σ is a fuzzy θ-closed set in y such that 1 − σ 6= 1 and σ < f (1 − λ). put 1 − σ = µ. then µ > 1 − f (1 − λ) = f (λ). this proves (a) ⇒ (b). (b) ⇒ (a). let λ in (x, t ) be a fuzzy semiopen set such that λ 6= 0. then 1 − λ is fuzzy semiclosed and 1 − λ 6= 1, f (1 − λ) = 1 − f (λ) 6= 1.. hence by (b) there exists a fuzzy θ-closed set µ in y such that µ 6= 1 and µ > f (1 − λ) = 1 − f (λ), that is, f (λ) > 1 − µ and let 1 − µ = γ (say). clearly γ is a fuzzy θ-open set in y such that γ < f (λ) and γ 6= 0. hence f is some what fuzzy faintly semiopen. this proves (b) ⇒ (a). 2 proposition 4.4 suppose (x, t ) and (y, s) be fuzzy topological spaces. let f : (x, t ) → (y, s) be any onto function. then the following are equivalent. 40 g. palanichetty and g. balasubramanian cubo 10, 1 (2008) (a) f is some what fuzzy faintly semiopen (b) if λ is a fuzzy θdense set in y, then f −1 (λ) is fuzzy semidense set in x. proof. (a) ⇒ (b). assume f is some what fuzzy faintly semiopen. suppose λ is fuzzy θ-dense set in y we have to show that f −1 (λ) is fuzzy semidense in x. suppose not, then there exists a fuzzy semiclosed set µ in x such that f −1 (µ) < µ < 1. now λ = f f −1 (λ) < f (µ) < f (1)(since f is onto). since f is some what fuzzy faintly semiopen by proposition 4.3, there exists a fuzzy θ-closed set δ in y such that f (µ) < δ.. thus we find λ < f (µ) < δ < 1, which is a contradiction to our hypothesis that λ is fuzzy θ-dense in x. hence f −1 (λ) must be fuzzy semidense set. this proves (a) ⇒ (b). (b) ⇒ (a). assume that f −1 (λ) is fuzzy semidense in x where λ is fuzzy θ-dense in y. we want to show that f is some what fuzzy faintly semiopen. assume that λ ∈ t and λ 6= 0, be a fuzzy semiopen set in (x, t ) we have to show that intθf (λ) 6= 0. suppose not, then intθf (λ) = 0 whenever λ ∈ t then clθ [1 − f (λ)] = 1−intθf (λ) = 1 − 0 = 1. that is, 1 − f (λ) is fuzzy θ-dense in y. therefore by assumption f −1 [1 − f (λ)] is fuzzy semidense in x. therefore 1 =s-cl [ f −1 (1 − f (λ)) ] =s-cl[1 − λ] = 1 − λ. this shows λ = 0 which is a contradiction and so s-intf (λ) 6= 0.this proves (b) ⇒ (a). 2 in [6], the concept of fuzzy d-space is defined as follows: definition 4.5 a fuzzy topological space (x, t ) is called a fuzzy d-space if every non-zero fuzzy open set λ of (x, t ) is dense in x. now we are ready to make the following. definition 4.6 a fuzzy topological space (x, t ) is called a fuzzy dsspace (dθspace) if every non-zero fuzzy semiopen set λ(non-zero fuzzy θopen set) in x is semidense (θ-dense) in x. proposition 4.7 let f : (x, t ) → (y, s) be some what fuzzy faintly semicontinuous. suppose x is a fuzzy dsspace. then y is a fuzzy dθspace. proof. let λ be a non-zero fuzzy θopen set in y. we want to show that λ is fuzzy θdense iny. suppose not. then there exists a fuzzy θclosed set µ in y such that λ < µ < 1. therefore f −1 (λ) < f −1 (µ) < f −1 (1) = 1.since λ 6= 0, f −1 (λ) 6= 0 and since f is some what fuzzy faintly semicontinuous, 0 6= s-intf −1 (λ) < f −1 (µ) < s-clf −1 (µ) < 1. this is a contradiction to the assumption that (x, t ) is a fuzzy ds space. hence y is a fuzzy dθspace. 2 cubo 10, 1 (2008) on some what fuzzy faintly semicontinuous functions 41 proposition 4.8 let x1, x2, y1 and y2 be fuzzy topological spaces such that y1 is a product related to y2 and x1 is a product related to x2. then the product f1 ×f2 : x1 ×x2 → y1 ×y2 of some what fuzzy faintly semicontinuous mapping f1 : x1 → y1 and f2 : x2 → y2 is some what fuzzy faintly semicontinuous. proof. let α = ∨ (αi × αj ) , where αi’s and αj ’s are fuzzy θ-open sets of y1 and y2 respectively be a fuzzy θ-open set of y1 × y2. then we have (f1 × f2) −1 (a) = ∨ ( f −1 1 (αi) × f −1 2 (αj ) ) since f1 and f2 are somewhat fuzzy faintly semicontinuous functions, there exists fuzzy semiopen sets σi in x1, fuzzy semiopen set σj in x2 such that σi ≤ f −1 1 (αi) ; σj ≤ f −1 2 (αj ) . therefore (f1 × f2) −1 (α) ≥ ∨ (σi × σj ) . since x1 and x2 are product related, it follows from [3] that ∨ (σi × σj ) is fuzzy semiopen. hence f1 × f2 is some what fuzzy faintly semicontinuous. 2 received: december 2005. revised: october 2007. references [1] a. mukherjee, fuzzy faintly continuous functions, fuzzy sets and systems, 59 (1993), 59–63. [2] a. mukherjee and s. debnath, on fuzzy faintly semicontinuous functions, indian j. pure appl. math., 34(2003), 1625–1630. [3] a.s. bin shahna, on fuzzy strong semicontinuity and fuzzy precontinunity, fuzzy sets and systems, 44 (1991), 303–308. [4] c.l. chang, fuzzy topological spaces, j.math. anal.appl., 141 (1968), 82–89. [5] k.r. gentry and h.b. holye, some what continuous functions, czech. math. j., 21(1971), 5–12. [6] g. thangaraj and g. balasubramanian, on some what fuzzy continuous functions, j. fuzzy math., 11 (2003), 1-12. [7] g. thangaraj and g. balasubramanian, on some what fuzzy semicontinuous functions, kybernetika, 37 (2001), 165–170. [8] l.a. zadeh, fuzzy sets, information and control, 8(1965), 338–353. semi.pdf cubo a mathematical journal vol.10, n o ¯ 04, (45–66). december 2008 fixed points for operators on generalized metric spaces adrian petruşel, ioan a. rus and marcel adrian şerban department of applied mathematics, babeş-bolyai university cluj-napoca, kogălniceanu 1, 400084, cluj-napoca, romania emails: petrusel{iarus, mserban}@math.ubbcluj.ro abstract the purpose of this paper is to present the fixed point theory for operators (singlevalued and multivalued) on generalized metric spaces in the sense of luxemburg. resumen el proposito de este art́ıculo es presentar la teoria de punto fijo para operadores (univariados y multivaluados) sobre espacios métricos generalizados en el sentido de luxemburg. key words and phrases: generalized metric in the sense of luxemburg, pompeiu-hausdorff generalized functional, weakly picard operator, fixed point, strict fixed point, generalized contraction, fibre generalized contraction, data dependence, pseudo-contractive multivalued operator. math. subj. class.: 47h10, 54h25. 46 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) 1. introduction let x be a nonempty set. a functional d : x × x → r+ ∪ {+∞} is said to be a generalized metric in the sense of luxemburg on x ([9], [13]) if: i) d(x, y) = 0 ⇔ x = y; ii) d(x, y) = d(y, x); iii) x, y, z ∈ x with d(x, z), d(z, y) < +∞ ⇒ d(x, y) ≤ d(x, z) + d(z, y). the pair (x, d) is called a generalized metric space. in a generalized metric space, the concepts of open and closed ball, cauchy sequence, convergent sequence, etc. are defined in a similar way to the case of a metric space. there are some contributions to fixed point theory for singlevalued operators (w.a.j. luxemburg [13], j.b. diaz and b. margolis [7], c.f.g. jung [9], s. kasahara [10], g. dezso [6],...) and multivalued operators (h. covitz and s.b. nadler [5], p.q. khanh [11],...) on a generalized metric space in the sense of luxemburg. the aim of this paper is to establish some new fixed point theorems for operators on a generalized metric space and, in this framework, to study the basic problems of the metrical fixed point theory. 2. generalized metric spaces in the sense of luxemburg we start our considerations by presenting some examples of generalized metric spaces. example 2.1 let x be a nonempty set and d : x × x → r+ ∪ {+∞}, given by d(x, y) = { 0, if x = y, +∞, otherwise. example 2.2 let x := c(r) and d : x × x → r+ ∪ {+∞} given by d(x, y) := sup t∈r |x(t) − y(t)|. example 2.3 let x := c(r) (the space of all continuous functions on r) and d : x × x → r+ ∪ {+∞} given by d(x, y) := sup t∈r (|x(t) − y(t)| · e−τ |t|), where τ > 0. example 2.4 (generic example) let (xi, di), i ∈ i be a family of metric spaces such that each two elements of the family are disjoint. denote x := ⋃ i∈i xi. if we define d(x, y) := { di(x, y), if x, y ∈ xi +∞, if x ∈ xi, y ∈ xj , i 6= j , then the pair (x, d) is a generalized metric space. cubo 10, 4 (2008) fixed points for operators ... 47 the following characterization theorem of a generalized metric space was given by jung. theorem 2.5 (jung [9]) let (x, d) be a generalized metric space. then there exists a partition x := ⋃ i∈i xi of x such that di := d|x i ×x i is a metric, for each i ∈ i. moreover, (x, d) is complete if and only if (xi, di) is complete, for each i ∈ i. notice that the above partition is induced by the following equivalence relation: x ∼ y ⇔ d(x, y) < +∞. let (x, d) be a generalized metric space. then, the partition x := ⋃ i∈i xi given by jung’s theorem is called the canonical decomposition of x into metric spaces. moreover, if x ∈ x, then there exists i(x) ∈ i such that x ∈ xi(x). we will denote bd(x0; r) := {x ∈ x|d(x0, x) < r} and ˜bd(x0; r) := {x ∈ x|d(x0, x) ≤ r}. if x ∈ xi, then ˜bd(x0; r) = ˜bdi (x0; r) and bd(x0; r) = bdi (x0; r). if (x, d) is a generalized metric space, then the metric topology induced on x is given by: τd := {y ⊆ x|y ∈ y ⇒ ∃r > 0 : bd(y, r) ⊂ y }. by this definition, it follows that: (xn)n∈n ⊂ x, x ∗ ∈ x, xn τ d → x∗ ⇔ d(xn, x ∗ ) → 0. a subset y of x is said to be d-closed (closed with respect to the topology induced by d) if and only if (yn)n∈n ⊂ y with d(yn, y) → 0, as n → +∞ implies y ∈ y . also, y is d-open if for each y ∈ y there exists a ball b(x0, r) := {x ∈ y |d(x0, x) < r} ⊂ y . let us remark that if x := ⋃ i∈i xi is the canonical decomposition of x, then xi is d-closed and d-open, for each i ∈ i. definition 2.6 two generalized metrics d1 and d2 on x are said to be: (a) topological equivalent if τd1 = τd2 ; (b) metric equivalent if there exist c1, c2 > 0 such that: i) d1(x, y) < +∞ implies d2(x, y) ≤ c1d1(x, y); ii) d2(x, y) < +∞ implies d1(x, y) ≤ c2d2(x, y). remark 2.7 if d1 is a generalized metric on x, then there exists a bounded metric d2 on x, topological equivalent to d1 (for example take d2(x, y) := min{d1(x, y), 1}). 3. functionals on generalized metric spaces throughout this section (x, d) will be a generalized metric space in the sense of luxemburg. 48 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) let us consider now the following families of subsets of the space (x, d): p (x) := {y ⊆ x| y 6= ∅} ; pb(x) := {y ∈ p (x)| y is bounded }; pcl(x) := {y ∈ p (x)| y is closed }; pb,cl(x) := {y ∈ p (x)| y is bounded and closed }. consider now some functionals on p (x) × p (x) (see also [3], [16]). (i) the gap functional dd defined by: dd : p (x) × p (x) → r+ ∪ {+∞} dd(a, b) := inf{d(a, b)| a ∈ a, b ∈ b}. (ii) the excess generalized functional ρd defined by: ρd : p (x) × p (x) → r+ ∪ {+∞}, ρd(a, b) := sup{d(a, b)| a ∈ a}. (iii) the pompeiu-hausdorff generalized functional hd defined by: hd : p (x) × p (x) → r+ ∪ {+∞}, hd(a, b) := max{ρ(a, b), ρ(b, a)}. (iv) the delta functional δd defined by: δd : p (x) × p (x) → r+ ∪ {+∞} δd(a, b) := sup{d(a, b)| a ∈ a, b ∈ b}. let a, b ∈ p (x). for the rest of the paper, we denote ai := a ∩ xi and bi := b ∩ xi, where xi are the sets from the characterization theorem 2.1. from (i), theorem 2.5 and example 2.4 we have: lemma 3.1 let (x, d) be a generalized metric space and a, b ∈ p (x). then: (i) d(a, b) = inf i∈i d(ai, bi); (ii) d(a, b) < +∞ if and only if there exists i ∈ i such that ai 6= ∅ and bi 6= ∅. a useful result is: lemma 3.2 let (x, d) be a generalized metric space x ∈ x and a ∈ p (x). then d(x, a) = 0 if and only if xi(x) ∩ a 6= ∅ and x ∈ a (where xi(x) denotes the unique element of the canonical decomposition of x where x belongs). cubo 10, 4 (2008) fixed points for operators ... 49 from (iv) and theorem 2.5 we obtain: lemma 3.3 let (x, d) be a generalized metric space and a, b ∈ p (x). then δ(a, b) < +∞ if and only if there exists i ∈ i such that a, b ∈ pb(xi). in particular, a ∈ pb(x) if and only if there exists i ∈ i such that a ∈ pb(xi). from (ii), theorem 2.5 and example 2.4 we have: lemma 3.4 let (x, d) be a generalized metric space and y, z ∈ p (x). then ρ(y, z) < +∞ if and only if there exists η > 0 such that for each y ∈ y there is z ∈ z such that d(y, z) < η. proof. if ρ(y, z) < +∞, then there is η > 0 such that ρ(y, z) < η. thus d(y, z) < η for each y ∈ y . hence there exists z ∈ z such that d(y, z) < η. suppose now there is η > 0 such that for each y ∈ y there exists z ∈ z with d(y, z) < η. then, y, z ∈ xi, where xi is an element of the partition of the generalized metric space x. hence d(y, z) ≤ η, for each y ∈ y . thus, ρ(y, z) ≤ η. 2 let (x, d) be a generalized metric space, y ∈ p (x) and ε > 0. an open neighborhood of radius ε for the set y is the set denoted vε(y ) and defined by: vε(y ) := {x ∈ x| d(x, y ) < ε}. let us remark that vε(y ) = ⋃ i∈i,yi 6=∅ vε(yi). in the usual case of a metric space (x, d) the following equivalent definitions of the pompeiuhausdorff functional are well-known. (iii) ′ hd(a, b) := inf{ε > 0|a ⊂ vε(b), b ⊂ vε(a)}, and (iii)′′ hd(a, b) := sup x∈x |d(x, a) − d(x, b)|. we have: lemma 3.5 let (x, d) be a generalized metric space. then, the definitions (iii), (iii)′ and (iii)′′ are equivalent. we can also prove the following result. lemma 3.6 let (x, d) be a generalized metric space and a, b ∈ p (x). then the following assertions are equivalent: (a) h(a, b) < +∞; (c) there exists η > 0 such that [for each a ∈ a there exists b ∈ b such that d(a, b) < η] and [for each b ∈ b there exists a ∈ a such that d(a, b) < η]. lemma 3.7 let (x, d) be a generalized metric space. then the following assertions hold: 50 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) i) let ε > 0 and y, z ∈ p (x) such that h(y, z) < +∞. then for each y ∈ y there exists z ∈ z such that d(y, z) ≤ h(y, z) + ε. ii) let q > 1 and y, z ∈ p (x) such that h(y, z) < +∞. then, for each y ∈ y there exists z ∈ z such that d(y, z) ≤ qh(y, z). proof. i) let y, z ∈ p (x) and ε > 0. suppose that h(y, z) < +∞. then, supposing, by contradiction, there is y ∈ y such that for every z ∈ z we have d(y, z) > h(y, z) + ε. if d(y, z) < +∞ then since h(y, z) ≥ d(y, z) ≥ h(y, z) + ε we get a contradiction. if d(y, z) = +∞ then, we get a contradiction to the supposition h(y, z) < +∞, since, by lemma 3.6, there is η > 0 such that for each y ∈ y there is z ∈ z with d(y, z) < η. 2 lemma 3.8 let (x, d) be a generalized metric space and a, b ∈ p (x). then: a) h(a, b) = sup i∈i h(a ∩ xi, b ∩ xi); b) a ∈ pcp(x) ⇔ card{i ∈ i|a ∩ xi 6= ∅} < +∞ and ai ∈ pcp(xi). remark 3.9 let (x, d) be a generalized metric space. then pcp(x) * pb(x). consider, for example, x, y ∈ x with d(x, y) = +∞, then {x, y} is compact but it is not bounded. 4. singlevalued operators on generalized metric spaces 4.1 general considerations let x be a nonempty set, s(x) := {(xn)n∈n|xn ∈ x, n ∈ n}, c(x) ⊂ s(x) and lim : c(x) → x an operator. by definition the triple (x, c(x), lim) is called an l-space if the following conditions are satisfied: (i) if xn = x, for all n ∈ n, then (xn)n∈n ∈ c(x) and lim(xn)n∈n = x. (ii) if (xn)n∈n ∈ c(x) and lim(xn)n∈n = x, then for all subsequences, (xni )i∈n, of (xn)n∈n we have that (xni )i∈n ∈ c(x) and lim(xni )i∈n = x. by definition an element of c(x) is convergent sequence and x := lim(xn)n∈n is the limit of this sequence and we write xn → x as n → ∞. in what follows we will denote an l-space by (x, →). actually, an l-space is any set endowed with a structure implying a notion of convergence for sequences. for example, hausdorff topological spaces, metric spaces, generalized metric spaces in perov’ sense (i.e., d(x, y) ∈ rm+ ), generalized metric spaces in luxemburg’ sense (i.e., d(x, y) ∈ r+ ∪ {+∞}), k-metric spaces (i.e., d(x, y) ∈ k, where k is a cone in an ordered banach space), gauge spaces, 2-metric spaces, d-r-spaces, probabilistic metric spaces, syntopogenous spaces, are such l-spaces. for more details see fréchet [8], blumenthal [4] and i.a. rus [22]. cubo 10, 4 (2008) fixed points for operators ... 51 let (x, d) and (y, ρ) be two generalized metric spaces and f : x → y . definition 4.1 the operator f : (x, d) → (y, ρ) is said to be: a) continuous, if xn → x ∗ implies f (xn) → f (x ∗ ); b) closed, if xn → x ∗ and f (xn) → y ∗ imply f (x∗) = y∗; c) α-lipschitz if α > 0 and d (x, y) < +∞ =⇒ ρ (f (x) , f (y)) ≤ α · d (x, y) . d) α-contraction if f is α-lipschitz with α < 1. 4.2 weakly picard operators on l-spaces let (x, →) be an l-space and f : x → x. we denote by f 0 := 1x , f 1 := f , f n+1 := f ◦ f n, n ∈ n the iterate operators of f . also: ff := {x ∈ x | f (x) = x}, i (f ) := {y ∈ p (x) | f (y ) ⊆ y } . definition 4.2 (i.a. rus [22]) let (x, →) be an l-space. then f : x → x is said to be 1) a picard operator if: i) ff = {x ∗}; ii) (f n (x)) n∈n → x∗ as n → +∞, for all x ∈ x. 2) a weakly picard (briefly wp) operator if the sequence (f n (x)) n∈n converges for all x ∈ x and the limit (which may depend on x) is a fixed point of f . if f : x → x is a weakly picard operator, then we define the operator f ∞ : x → x by: f ∞(x) := lim n→∞ f n(x). notice that f ∞(x) = ff . moreover, if f is a picard operator and we denote by x ∗ its unique fixed point, then f ∞(x) = x∗, for each x ∈ x. definition 4.3 let (x, →) be an l-space, c > 0 and d : x × x → r+. by definition, the operator f is called c-weakly picard with respect to d, if f is a weakly picard operator and d (x, f ∞ (x)) ≤ c · d (x, f (x)) , for all x ∈ x. if f is picard operator and the above condition holds, then f is said to be c-picard. theorem 4.4 (characterization theorem) (i.a. rus [25], [22]) let (x, →) be an l-space and f : x → x be an operator. then, f is a weakly picard operator if and only if there exists a partition of x, x = ⋃ λ∈λ xλ, such that: 52 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) a) xλ ∈ i (f ), for all λ ∈ λ; b) f |x λ : xλ → xλ is a picard operator, for all λ ∈ λ. 4.3 contractions on generalized metric spaces we present first some important auxiliary results. lemma 4.5 let (x, d) be a complete generalized metric space and f : x → x be an α-contraction. the following statements are equivalent: i) ff 6= ∅; ii) there exists x ∈ x such that d (x, f (x)) < +∞; iii) there exist x ∈ x and n (x) ∈ n such that d ( f n(x) (x) , f n(x)+1 (x) ) < +∞; iv) there exists i ∈ i such that xi ∈ i (f ). proof. i) =⇒ ii) let x∗ ∈ ff . we have d (x∗, f (x∗)) = d (x∗, x∗) = 0 < +∞. ii) =⇒ iii) we choose n (x) = 0; iii) =⇒ i) since f is an α-contraction we have that (f n (x)) is a cauchy sequence. this implies f n (x) → x∗, as n → +∞. from the continuity of f it follows that x∗ ∈ ff . ii) =⇒ iv) since d (x, f (x)) < +∞, there exists i ∈ i such that x ∈ xi. let y ∈ xi then d (x, y) < +∞. we have: d (x, f (y)) ≤ d (x, f (x)) + d (f (x) , f (y)) ≤ d (x, f (x)) + α · d (x, y) < +∞ which implies f (y) ∈ xi. iv) =⇒ ii) let x∈xi. since xi ∈i (f ), we get that f (x)∈xi. therefore d (x, f (x)) < +∞.2 lemma 4.6 let (x, d) be a complete generalized metric space and f : x → x be an α-contraction. we suppose that: i) there exists x ∈ x such that d (x, f (x)) < +∞; ii) if u, v ∈ ff then d (u, v) < +∞; then: a) ff = {x ∗}; b) f ∣ ∣ x i(x) : xi(x) → xi(x) is a picard operator. proof. from i) and lemma 4.5 we have that there exists i ∈ i such that xi ∈ i (f ), f n (x) ∈ xi for every n ∈ n, ff 6= ∅, f n (x) → x∗ ∈ ff ∩ xi. let u, v ∈ ff . then d (u, v) < +∞ and d (u, v) = d (f (u) , f (v)) ≤ α · d (u, v) . cubo 10, 4 (2008) fixed points for operators ... 53 therefore d (u, v) = 0, which implies u = v. hence ff = {x ∗}. since xi ∈ i (f ) then d (y, f (y)) < +∞ for every y ∈ xi and applying again lemma 4.5 we get that f ∣ ∣ x i(x) : xi(x) → xi(x) is a picard operator. 2 theorem 4.7 let (x, d) be a complete generalized metric space and f : x → x. we suppose that: i) f is an α-contraction; ii) for every x ∈ x there exists n (x) ∈ n such that d ( f n(x) (x) , f n(x)+1 (x) ) < +∞. then: a) f is a weakly picard operator. if in addition, for every x ∈ x we have d (x, f (x)) < +∞, then f is 1 1−α -weakly picard; b) if, in addition: b1) for every x ∈ x we have d (x, f (x)) < +∞; b2) u, v ∈ ff implies d (u, v) < +∞, then f is 1 1−α -picard. proof. a) the first part follows from lemma 4.5 and lemma 4.6. for the second conclusion, notice that for every x ∈ x such that d (x, f (x)) < +∞ and each n ∈ n we have: d (f n (x) , f ∞ (x)) ≤ αn 1 − α · d (x, f (x)) which implies d (x, f ∞ (x)) ≤ 1 1 − α · d (x, f (x)) . b) from b2) we obtain ff = {x ∗} and from a) we obtain that f is 1 1−α -picard operator. 2 theorem 4.8 let (x, d) be a complete generalized metric space and f, g : x → x two operators. we suppose that: i) f and g are α-contractions; ii) d (x, f (x)) < +∞ and d (x, g (x)) < +∞, for every x ∈ x; iii) there exists η > 0 such that d (f (x) , g (x)) ≤ η, for all x ∈ x. then: h (ff , fg) ≤ η 1 − α . proof. let x ∈ ff and y ∈ fg. from ii) and theorem 4.7 we have: d (x, g∞ (x)) ≤ 1 1 − α · d (x, g (x)) = 1 1 − α · d (f (x) , g (x)) ≤ η 1 − α . 54 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) since g∞ (x) ∈ fg then d (x, fg ) ≤ d (x, g ∞ (x)) ≤ η 1 − α . by taking the supremum over x ∈ ff we get ρ (ff , fg) ≤ η 1 − α . using the same technique we have: ρ (fg, ff ) ≤ η 1 − α which implies the conclusion. 2 theorem 4.9 (fibre contraction principle) let (x0, →) be an l-space and (xk, dk), k ∈ {0, 1, · · · , p} (where p ≥ 1) be complete generalized metric spaces. we consider the operators: fk : x0 × ... × xk → xk, k ∈ {0, 1, · · · , p}. we suppose that: i) f0 : x0 → x0 is a weakly picard operator; ii) fk (x0, ..., xk−1, ·) is an αk-contraction, k ∈ {1, 2, · · · , p}; iii) fk is continuous, k ∈ {1, 2, · · · , p}; iv) for every (x0, x1, ..., xk) ∈ x0 × ... × xk we have dk (xk, fk (x0, x1, ..., xk)) < +∞, k ∈ {1, 2, · · · , p}. then the operator gp : x0 × ... × xp → x0 × ... × xp gp (x0, x1, ..., xp) = (f0 (x0) , f1 (x0, x1) , ..., fp (x0, x1, ..., xp)) is weakly picard. proof. we will prove by induction. for p = 1 the conclusion follows by theorem 3.1 in m.a. şerban [31]. we suppose that conclusion holds for k ≤ p and we prove the conclusion for k + 1. we know that gk+1 = (gk, fk+1), gk are weakly picard and from ii) fk+1 (x0, ..., xk, ·) is an αk+1contraction, so we apply again theorem 3.1 from m.a. şerban [31] and we get that gk+1 is weakly picard. 2 theorem 4.10 let x be a nonempty set, α ∈]0; 1[ and f : x → x an operator. the following statements are equivalent: i) ff = ff n 6= ∅ for every n ∈ n; ii) there exists a complete generalized metric d on x such that: a) f : (x, d) → (x, d) is an α-contraction; cubo 10, 4 (2008) fixed points for operators ... 55 b) d (x, f (x)) < +∞ for every x ∈ x. proof. i) =⇒ ii) ff = ff n 6= ∅ for every n ∈ n implies that there exists a partition of x, x = ⋃ i∈i xi such that xi ∈ i (f ), card (ff ∩ xi) = 1 and f |xi is a bessaga operator (see i.a. rus [24]). from bessaga’s theorem [2] there exists a complete metric di on xi such that f |xi : xi → xi is an α-contraction for all i ∈ i. so, d : x × x → r+ ∪ {+∞} d (x, y) = { di (x, y) if x, y ∈ xi +∞ if x ∈ xi, y ∈ xi, i 6= j is the complete generalized metric on x that we are looking for. ii) =⇒ i) is theorem 4.7. 2 4.4 graphic contractions let (x, d) be a generalized metric space and f : x → x. definition 4.11 f : x → x is a graphic contraction if there exists α ∈ [0; 1[ such that: d ( f 2 (x) , f (x) ) ≤ α · d (x, f (x)) for all x ∈ x with d (x, f (x)) < +∞. theorem 4.12 let (x, d) be a complete generalized metric space and f : x → x. we suppose that: i) f is a closed graphic contraction; ii) for every x ∈ x there exists n (x) ∈ n such that d ( f n(x) (x) , f n(x)+1 (x) ) < +∞. then: a) f is a weakly picard operator. if, in addition, for every x ∈ x we have that d (x, f (x)) < +∞, then f is 1 1−α -weakly picard; b) if, in addition: b1) for every x ∈ x we have d (x, f (x)) < +∞; b2) if u, v ∈ ff implies d (u, v) < +∞, then f is 1 1−α -picard. proof. a) from i) and ii) we have that for each x ∈ x, the sequence (f n (x)) is cauchy. therefore there exists x∗ ∈ x such that f n (x) → x∗, as n → +∞ and d (f n (x) , x∗) ≤ αn−n(x) 1 − α · d ( f n(x) (x) , f n(x)+1 (x) ) , n ≥ n (x) . since f is closed we get that x∗ ∈ ff and f ∞ (x) = x∗. this means that f is a weakly picard operator. 56 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) if for every x ∈ x we have d (x, f (x)) < +∞, then n (x) = 0 and letting n = 0 in the above relation, we conclude that f is 1 1−α -weakly picard operator. b) if for u, v ∈ ff we have d (u, v) < +∞ then ff = {x ∗}, which means that f is a 1 1−α -picard operator. 2 4.5 meir-keeler operators let us consider now the case of meir-keeler operators on generalized metric spaces. definition 4.13 let (x, d) be a generalized metric space. then, f : x → x is called a meirkeeler type operator if for each ǫ > 0 there exists η = η(ǫ) > 0 such that for x, y ∈ x with ǫ ≤ d(x, y) < ǫ + η we have d(f (x), f (y)) < ǫ. by using an argument similar to the one in the meir-keeler fixed point theorem [14] we have: theorem 4.14 let (x, d) be a generalized complete metric space and f : x → x be a meir-keeler type operator. suppose there exists x0 ∈ x such that d(x0, f (x0)) < +∞. then ff 6= ∅. moreover, if additionally x, y ∈ ff implies d(x, y) < +∞, then ff = {x ∗}. proof. denote xn := f n (x0), n ∈ n. the proof of the theorem can be organized in five steps. step 1. we prove that d(f (x), f (y)) < d(x, y), for each x, y ∈ x with x 6= y and d(x, y) < +∞. let x, y ∈ x be such that x 6= y and d(x, y) < +∞. then by letting ǫ := d(x, y) in the definition of meir-keeler operators we get d(f (x), f (y)) < d(x, y). step 2. we can prove, by induction, that d(xn, xn+1) < +∞, for all n ∈ n. step 3. we prove that the sequence an := d(xn, xn+1) ց 0 as n → +∞. if there is n0 ∈ n such that an0 = 0 then xn0 ∈ ff . if an 6= 0, for each n ∈ n, then an = d(f (xn−1), f (xn)) < d(xn−1, xn) = an−1. hence the sequence (an)n∈n converges to a certain a ≥ 0. suppose that a > 0. then, for each ǫ > 0 there exists nǫ ∈ n such that ǫ ≤ an < ǫ + η, for all n ≥ nǫ. then, by the meir-keeler condition we obtain an+1 < ǫ, which is a contradiction with the above relation. step 4. we will prove that the sequence (xn) is cauchy. suppose, by contradiction, that (xn) is not a cauchy sequence. then, there exists ǫ > 0 such that lim sup d(xm, xn) > 2ǫ. for this ǫ there exists η := η(ǫ) > 0 such that for x, y ∈ x with ǫ ≤ d(x, y) < ǫ + η we have d(f (x), f (y)) < ǫ. choose δ := min{ǫ, η}. since an ց 0 as n → +∞ it follows that there is p ∈ n such that ap < δ 3 . let m, n ∈ n∗ with n > m > p such that d(xn, xm) > 2ǫ. for j ∈ [m, n] we have |d(xm, xj ) − d(xm, xj+1| ≤ aj < δ 3 . also, d(xm, xm+1 < ǫ and d(xm, xn) > ǫ+δ we obtain that there exists k ∈ [m, n] such that ǫ < ǫ+ 2δ 3 < d(xm, xk) < ǫ+δ. on the other hand, for any m, l ∈ n we have: d(xm, xl) ≤ d(xm, xm+1) + d(xm+1, xl+1) + cubo 10, 4 (2008) fixed points for operators ... 57 d(xl+1, xl) = am + d(f (xm), f (xl)) + al < δ 3 + ǫ + δ 3 . the contradiction proves that (xn) is cauchy. step 5. we prove that x∗ := lim n→+∞ xn is a fixed point of f . since f is continuous and xn+1 = f (xn), we get by passing to the limit that x ∗ = f (x∗). if x∗, y ∈ ff are two distinct fixed points of f then, by the contractive condition, we get the following contradiction: d(x∗, y) = d(f (x∗), f (y)) < d(x∗, y). this completes the proof. 2 4.6 caristi operators let (x, d) be a generalized metric space. definition 4.15 a space x is said to be sequentially complete in weierstrass’ sense (see [33]) if each sequence (xn)n∈n in x such that +∞ ∑ n=0 d(xn, xn+1) < +∞ is convergent in x. definition 4.16 let (x, d) be a generalized metric space. then, f : x → x is called a caristi operator if there exists a functional ϕ : x → r+ such that d (x, f (x)) ≤ ϕ (x) − ϕ (f (x)) , for every x ∈ x . theorem 4.17 let (x, d) be a sequentially complete (in weierstrass’ sense) generalized metric space and f : x → x be a closed caristi operator. then f is a weakly picard operator. proof. we remark that if f is a caristi operator, then d (x, f (x)) < +∞ for every x ∈ x. denote by xn := f n (x), for n ∈ n. then: +∞ ∑ n=0 d(xn, xn+1) = +∞ ∑ n=0 d(f n (x), f n+1 (x)). we will prove that the series +∞ ∑ n=0 d(f n(x), f n+1(x)) is convergent. for this purpose we need to show that the sequence of its partial sums is convergent in r+. denote by sn := n ∑ k=0 d(f k(x), f k+1(x)). then sn+1−sn = d(f n+1 (x), f n+2(x)) ≥ 0, for each n ∈ n. moreover sn = n ∑ k=0 d(f k(x), f k+1(x)) ≤ ϕ(x). hence (sn)n∈n is upper bounded and increasing in r+. then the sequence (sn)n∈n is convergent. it follows that the sequence (xn)n∈n is cauchy and, from the sequentially completeness of the space, convergent to a certain element x∗ ∈ x. the conclusion follows from the fact that f is closed. 2 58 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) 4.7 fixed point theorems in a set with two generalized metrics let x be a nonempty set and d, ρ : x × x → r+ ∪{+∞} be two generalized metrics on x. in this subsection we will present maia’s fixed point theorem for the case of a set with two generalized metrics. theorem 4.18 let x be a nonempty set, d, ρ : x × x → r+ ∪ {+∞} two generalized metrics on x and f : x → x. we suppose that: i) (x, d) is a complete generalized metric space; ii) there exists c > 0 such that d (x, y) ≤ c · ρ (x, y) for all x, y ∈ x with ρ (x, y) < +∞; iii) for every x ∈ x there exists n (x) ∈ n such that ρ ( f n(x) (x) , f n(x)+1 (x) ) < +∞; iv) f : (x, ρ) → (x, ρ) is an α-contraction. then f is weakly picard. proof. for each x ∈ x there exists n (x) ∈ n such that ρ ( f n(x) (x) , f n(x)+1 (x) ) < +∞. also, there exists i ∈ i such that xi ∈ i (f ) and f n (x) ∈ xi for all n ≥ n (x). since f : (x, ρ) → (x, ρ) is an α-contraction, the sequence (f n (x)) n∈n is cauchy in (x, ρ). using conditions ii), iii) and iv) we get d ( f n (x) , f n+p (x) ) ≤ c · ρ ( f n (x) , f n+p (x) ) ≤ c · αn−n(x) 1 − α ρ ( f n(x) (x) , f n(x)+1 (x) ) , n ≥ n (x) , so d (f n (x) , f n+p (x)) → 0 as n → +∞. thus (f n (x)) n∈n is cauchy sequence in (x, d), which implies that f n (x)→x∗ ∈xi. by condition iv) we have that x ∗ ∈ff . hence f is weakly picard.2 an improved version of maia’s theorem can be obtained by replacing the assumption ii) with a more useful condition (from an application point of view), see i.a. rus [20]. theorem 4.19 let x be a nonempty set, d, ρ : x × x → r+ ∪ {+∞} two generalized metrics on x and f : x → x. we suppose that: i) (x, d) is a complete generalized metric space; ii) there exists c > 0 such that d (f (x) , f (y)) ≤ c · ρ (x, y), for all x, y ∈ x with ρ (x, y) < +∞; iii) for every x ∈ x there exists n (x) ∈ n such that ρ ( f n(x) (x) , f n(x)+1 (x) ) < +∞; iv) f : (x, ρ) → (x, ρ) is an α-contraction. then f is a weakly picard operator. proof. the proof follows the method in theorem 4.18. 2 cubo 10, 4 (2008) fixed points for operators ... 59 5. multivalued operators in generalized metric spaces 5.1 general considerations let (x, d) be a generalized metric space. let y, z be two nonempty subsets of x and t : y → p (z) be a multivalued operator. by definition, t : y → z is a selection of t if t(x) ∈ t (x), for each x ∈ y . if t : x → p (x) is a multivalued operator, then x∗ ∈ x is a fixed point for t if and only if x∗ ∈ t (x∗). denote by ft the set of all fixed points for t . also, x ∗ ∈ x is called a strict fixed point for t if and only if {x∗} = t (x∗). we will denote by (sf )t the set of all strict fixed points of t . by graph(t ) := {(x, y) ∈ x × x|y ∈ t (x)} we denote the graph of the multivalued operator t and by t (y ) := ⋃ x∈y t (x) the image through t of the set y ∈ p (x). recall that if y ⊆ x, then t (y ) := ⋃ x∈y t (x). we also denote by t n := t ◦ t · · · ◦ t (the n times composition). recall that, if (x, d) is a metric space, then t : x → pcl(x) is said to be a multivalued a-contraction if a ∈ [0, 1[ and hd(t (x), t (y)) ≤ ad(x, y), for each x, y ∈ x. the following result is known as covitz-nadler fixed point principle. theorem 5.1 (covitz-nadler [5]) let (x, d) be a complete metric space and t : x → pcl(x) be a multivalued a-contraction. then, for each x0 ∈ x there exists a sequence (xn)n∈n in x with xn+1 ∈ t (xn) for all n ∈ n, which converges to a fixed point of t . remark 5.2 from the proof of the above result it follows that for each x ∈ x and each y ∈ t (x) there exists in x a sequence (xn)n∈n with the properties: a) x0 = x, x1 = y; b) xn+1 ∈ t (xn) for all n ∈ n ∗; c) (xn)n∈n converges to a fixed point of t . this principle gave rise to the following concept. definition 5.3 (rus-petruşel-ŝıntămărian [28], [29]) let (x, →) be an l-space. then t : x → p (x) is a multivalued weakly picard operator (briefly mwp operator) if for each x ∈ x and each y ∈ t (x) there exists a sequence (xn)n∈n in x such that: i) x0 = x, x1 = y ii) xn+1 ∈ t (xn), for all n ∈ n iii) the sequence (xn)n∈n is convergent and its limit is a fixed point of t . a sequence (xn)n∈n in x satisfying the conditions (i) and (ii) in definition 5.3 is called a sequence of successive approximations for t starting from (x, y). 60 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) the aim of this section is to establish some fixed point results for multivalued operators of contractive type on generalized metric space. 5.2 multivalued contractions on generalized metric spaces let us recall first some contractive-type conditions for multivalued operators. definition 5.4 let (x, d) be a generalized metric space. then t : x → pcl(x) is called a multivalued a-contraction if a ∈ [0, 1[ and hd(t (x), t (y)) ≤ ad(x, y), for each x, y ∈ x, with d(x, y) < +∞. let (x, d) be a generalized metric space. we denote by p(x) the set of all subsets of a nonempty set x. definition 5.5 let (x, d) be a generalized metric space. if t : x → p (x) is a multivalued operator, then we consider the following multivalued operators generated by t : ̂t : x → p(x), ̂t (x) := t (x) ∩ xi(x) (where xi(x) denotes the unique element of the canonical decomposition of x where x belongs), t̃ i : x → p(x), t̃ i(x) := t (x) ∩ xi (where xi denotes an arbitrary element of the canonical decomposition of x). then we have: lemma 5.6 ft = f ̂t . lemma 5.7 ft 6= ∅ ⇔ if there exists i ∈ i such that ft̃ i 6= ∅. the following result is a straightforward version of covitz and nadler alternative theorem in [5]. theorem 5.8 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a multivalued a-contraction. suppose that for each x ∈ x there is y ∈ t (x) such that d(x, y) < +∞. then there exists a sequence of successive approximations of t starting from any arbitrary x ∈ x which converges to a fixed point of t . the previous result gives rise to the following open question. open question. let t : x → pcl(x) be a multivalued a-contraction as in the above covitznadler fixed point result. is t a mwp operator ? theorem 5.9 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a multivalued a-contraction. suppose there exists x0 ∈ x and x1 ∈ t (x0) such that d(x0, x1) < +∞. cubo 10, 4 (2008) fixed points for operators ... 61 then there exists a sequence (xn)n∈n of successive approximations for t starting from x0 which converges to a fixed point of t . proof. let x := ⋃ i∈i xi be the canonical decomposition of x into metric spaces. recall that x is complete if and only if xi is complete for each i ∈ i. let j ∈ i such that x0 ∈ xj . for x ∈ x we successively have: d(x, t (x)) < +∞ ⇔ there exists y ∈ t (x) such that d(x, y) < +∞ ⇔ y ∈ t (x) ∩ xi(x). hence d(x, t (x)) < +∞ ⇔ t (x) ∩ xi(x) 6= ∅. consider now the multivalued operator t̃ j : x → p(x), t̃ j(x) := t (x) ∩ xj . we will prove that t̃ j |x j : xj → pcl(xj ). for this purpose, it is enough to show that d(x, t (x)) < +∞, for each x ∈ xj . for x ∈ xj we have: d(x, t (x)) ≤ d(x, t (x0)) + h(t (x0), t (x)) ≤ d(x, x0) + d(x0, t (x0)) + ad(x0, x) < +∞. hence t̃ j |x j : xj → pcl(xj ) is a multivalued a-contraction on the complete metric space (xj , d|x j ×x j ). the conclusion follows from lemma 5.7 and theorem 5.1. 2 an answer to the above problem is the following result. theorem 5.10 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a multivalued a-contraction. suppose that for each x ∈ x and y ∈ t (x) we have d(x, y) < +∞ (or equivalently, for each x ∈ x we have t (x) ⊂ xi(x)). then t is a mwp operator. proof. from the hypothesis we have that d(x, t (x)) < +∞, for each x ∈ x. hence, for each x ∈ x we have that t : xi(x) → pcl(xi(x)). since (xi(x), d|x i(x) ×x i(x) ) is a complete metric space, by theorem 5.1 and remark 5.2, we conclude that t is a mwp operator. 2 we introduce now the following concepts. definition 5.11 (rus-petruşel-ŝıntămărian [29]) let (x, →) be an l-space and t : x → p (x) be a mwp operator. define the multivalued operator t ∞ : graph(t ) → p (ft ) by the formula t ∞(x, y) = { z ∈ ft | there exists a sequence of successive approximations of t starting from (x, y) that converges to z }. definition 5.12 (see also rus-petruşel-ŝıntămărian [29]) let (x, d) be a generalized metric space and t : x → p (x) be a mwp operator such that for each x ∈ x and y ∈ t (x) we have that d(x, y) < +∞. then, t is called a c-multivalued weakly picard operator (briefly c-mwp operator) if there exists a selection t∞ of t ∞ such that d(x, t∞(x, y)) ≤ c d(x, y), for all (x, y) ∈ graph(t ). 62 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) as an example, we have: theorem 5.13 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a multivalued a-contraction, such that for each x ∈ x and y ∈ t (x) we have d(x, y) < +∞. then t is a 1 1−a -mwp operator. we present now an abstract data dependence theorem for the fixed point set of c-mwp operators on generalized metric spaces. theorem 5.14 let (x, d) be a generalized metric space and t1, t2 : x → p (x) be two multivalued operators. we suppose that: i) ti is a ci-mwp operator, for i ∈ {1, 2} ii) there exists η > 0 such that h(t1(x), t2(x)) ≤ η, for all x ∈ x. then h(ft1 , ft2 ) ≤ η max { c1, c2 }. proof. the proof follows in a similar way to rus-petruşel-ŝıntămărian [29]. for the sake of completeness we present it here. let ti : x → x be a selection of ti for i ∈ {1, 2}. let us remark that h(ft1 , ft2 ) ≤ max { sup x∈ft2 d(x, t∞1 (x, t1(x))), sup x∈ft1 d(x, t∞2 (x, t2(x))) } . let q > 1. then we can choose ti (i ∈ {1, 2}) such that d(x, t∞ 1 (x, t1(x))) ≤ c1qh(t2(x), t1(x)), for all x ∈ ft2 and d(x, t∞ 2 (x, t2(x)) ≤ c2qh(t1(x), t2(x)), for all x ∈ ft1 . thus we have h(ft1 , ft2 ) ≤ qη max{c1, c2}. letting q ց 1, the proof is complete. 2 notice that the above conclusions means that the data dependence phenomenon of the fixed point set for c-mwp operators holds. we also have: theorem 5.15 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a multivalued a-contraction. suppose: (i) (sf )t 6= ∅; (ii) if x, y ∈ ft then d(x, y) < +∞. then ft = (sf )t = {x ∗}. proof. we will prove first that (sf )t = {x ∗}. indeed, if z ∈ (sf )t with z 6= x ∗ , then d(z, x∗) < +∞ and d(z, x∗) = h(t (z), t (x∗)) ≤ ad(z, x∗), a contradiction. next we will prove that ft ⊆ cubo 10, 4 (2008) fixed points for operators ... 63 (sf )t . let y ∈ ft . then d(y, x ∗ ) < +∞. thus d(y, x∗) = d(y, t (x∗)) ≤ h(t (y), t (x∗)) ≤ ad(y, x∗), which implies y = x∗. this completes the proof. 2 5.3 pseudo-contractive multivalued operators on generalized metric spaces in d. azé and j.-p. penot [1] the following concept is introduced. definition 5.16 (azé-penot [1]) let (x, d) be a metric space. a multivalued operator t : x → p (x) is said to be pseudo-a-lipschitzian with respect to the subset u ⊂ x whenever, for all x, y ∈ u , we have ρd(t (x) ∩ u, t (y)) ≤ ad(x, y). also, the multivalued opeator t is called pseudo-a-contractive with respect to u if it is pseudo-alipschitzian with respect to u for some a ∈ [0, 1[. in azé-penot [1], the fixed point theory for multivalued pseudo-a-contractive operators with respect to the open ball bd(x0, r) of a complete metric space (x, d) is studied. the aim of this section is to give some fixed point results for multivalued pseudo-a-contractive operators in the setting of a generalized metric space. theorem 5.17 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a multivalued operator. let x := ⋃ i∈i xi be the canonical decomposition of x. suppose that there exists x0 ∈ x such that d(x0, t (x0)) < +∞ and t is pseudo a-contractive with respect to xi(x0). then ft 6= ∅. proof. since d(x0, t (x0)) < +∞ there exists b > 0 and x1 ∈ t (x0) such that d(x0, x1) < b < +∞. then x1 ∈ xi(x0) and thus x1 ∈ t (x0) ∩ xi(x0). hence we have d(x1, t (x1)) ≤ ρ(t (x0) ∩ xi(x0), t (x1)) ≤ ad(x0, x1) < ab. thus there exists x2 ∈ t (x1) such that d(x1, x2) < ab < +∞. thus x2 ∈ t (x1) ∩ xi(x0). in a similar way, we have d(x2, t (x2)) ≤ ρ(t (x1) ∩ xi(x0), t (x2)) ≤ ad(x1, x2) < a 2b < +∞. by induction, we obtain a sequence (xn)n∈n with the following properties: (a) xn+1 ∈ t (xn) ∩ xi(x0), for all n ∈ n; (b) d(xn, xn+1) < a nb, for all n ∈ n. from (b) we get that (xn)n∈n is cauchy and hence convergent in xi(x0). thus there exists x∗ ∈ xi(x0) (since xi(x0) is d-closed), such that xn → x ∗ as n → +∞. let us show now that x∗ ∈ ft . we have d(x ∗, t (x∗)) ≤ d(x∗, xn+1) + d(xn+1, t (x ∗ )) ≤ d(x∗, xn+1) + ρ(t (xn) ∩ xi(x0), t (x ∗ )) ≤ d(x∗, xn+1) + ad(x ∗, xn)) → 0 as n → +∞. hence x ∗ ∈ t (x∗). 2 a second answer to the open problem mentioned in section 3 is the following: theorem 5.18 let (x, d) be a generalized complete metric space and t : x → pcl(x) be a 64 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) multivalued operator such that for each x ∈ x and y ∈ t (x) we have d(x, y) < +∞. let x := ⋃ i∈i xi be the canonical decomposition of x. suppose that t is pseudo a-contractive with respect to xi(x), for each x ∈ x. then t is a mwp operator. proof. let x0 ∈ x and x1 ∈ t (x) such that d(x0, x1) 0. thus x1 ∈ t (x0) ∩ xi(x0). hence we have d(x1, t (x1)) ≤ ρ(t (x0) ∩ xi(x0), t (x1)) ≤ ad(x0, x1) < ab. we obtain that there exists x2 ∈ t (x1) such that d(x1, x2) < ab < +∞. thus x2 ∈ t (x1) ∩ xi(x0). in a similar way, we have d(x2, t (x2)) ≤ ρ(t (x1) ∩ xi(x0), t (x2)) ≤ ad(x1, x2) < a 2b < +∞. by induction, we obtain a sequence (xn)n∈n with the following properties: (a) xn+1 ∈ t (xn) ∩ xi(x0), for all n ∈ n; (b) d(xn, xn+1) < a nb, for all n ∈ n. from (b) we get that (xn)n∈n is cauchy and hence convergent in xi(x0) to a certain x ∗ . as before, we obtain x∗ ∈ t (x∗). since x0 ∈ x and x1 ∈ t (x0) were arbitrarily chosen, we get that t is a mwp operator. 2 received: february 2008. revised: february 2008. references [1] d. azé and j.-p. penot, on the dependence of fixed points sets of pseudo-contractive multifunctions. application to differential inclusions, nonlinear dyn. syst. theory, 6 (2006), 31–47. [2] c. bessaga, on the converse of the banach fixed point principle, colloq. math., 7 (1959), 41–43. [3] g. beer, topologies on closed and closed convex sets, kluwer acad. publ., dordrecht, 1994. [4] l.m. blumenthal, theory and applications of distance geometry, oxford university press, 1953. [5] h. covitz and s.b. nadler, multi-valued contraction mapping in generalized metric spaces, israel j. math., 8 (1970), 5–11. [6] g. dezso, fixed point theorems in generalized metric spaces, pure math. appl., 11 (2000), 183–186. [7] j.b. diaz and b. margolis, a fixed point theorem for the alternative for contractions on a generalized complete metric space, bull. amer. math. soc., 74 (1968), 305–309. [8] m. fréchet, les espaces abstraits, gauthier-villars, paris, 1928. cubo 10, 4 (2008) fixed points for operators ... 65 [9] c.f.k. jung, on generalized complete metric spaces, bull. a.m.s., 75 (1969), 113–116. [10] s. kasahara, on some generalizations of the banach contraction theorems, mathematics seinar notes, 3 (1975), 161–169. [11] p.q. khanh, remarks on fixed point theorems based on iterative approximations, polish acad. sciences, inst. of mathematics, preprint 361, 1986. [12] r. kopperman, all topologies come from generalized metrics, amer. math. monthly, 95 (1988), 89–97. [13] w.a.j. luxemburg, on the convergences of successive approximations in the theory of ordinary differential equations, indag. math., 20 (1958), 540–546. [14] a. meir and e. keeler, a theorem on contraction mappings, j. math. anal. appl., 28 (1969) 326–329. [15] s.b. nadler jr., multivalued contraction mappings, pacific j. math., 30 (1969), 475–488. [16] a. petruşel, multivalued weakly picard operators and applications, scientiae mathematicae japonicae, 59 (2004), 167–202. [17] a. petruşel and i.a. rus, multivalued picard and weakly picard operators, fixed point theory and applications (e. llorens fuster, j. garcia falset, b. sims-eds.), yokohama publishers, 2004, 207–226. [18] s. reich, some remarks concerning contraction mappings, canad. math. bull., 14 (1971), 121–124. [19] s. reich, fixed point of contractive functions, boll. u.m.i., 5 (1972), 26–42. [20] i.a. rus, metrical fixed point theorems, cluj-napoca, 1979. [21] i.a. rus, generalized contractions and applications, cluj university press, cluj-napoca, 2001. [22] i.a. rus, picard operators and applications, scientiae mathematicae japonicae, 58 (2003), 191–219. [23] i.a. rus, metric sapces with fixed point property with respect to contractions, studia univ. babeş-bolyai math., 51 (2006), 115–121. [24] i.a. rus, weakly picard mappings, comment. math. univ. carolinae, 34 (1993), 769–773. [25] i.a. rus, weakly picard operators and applications, seminar on fixed point theory, clujnapoca, 2 (2001), 41–58. [26] i.a. rus, the theory of a metrical fixed point theorem: theoretical and applicative relevances, fixed point theory, 9 (2008), to appear. 66 adrian petruşel, ioan a. rus and marcel adrian şerban cubo 10, 4 (2008) [27] i.a. rus, a. petruşel and g. petruşel, fixed point theory 1950–2000: romanian contributions, house of the book of science, cluj-napoca, 2002. [28] i.a. rus, a. petruşel and a. ŝıntămărian, data dependence of the fixed point set of multivalued weakly picard operators, studia univ. babeş-bolyai mathematica, 46 (2001), 111–121. [29] i.a. rus, a. petruşel and a. ŝıntămărian, data dependence of the fixed point set of some multivalued weakly picard operators, nonlinear analysis, 52 (2003), 1947–1959. [30] i.a. rus, a. petruşel and m.a. şerban, weakly picard operators: equivalent definitions, applications and open problems, fixed point theory, 7 (2006), 3–22. [31] m.a. şerban, fibre contraction theorem in generalized metric spaces, automation computers applied mathematics, 16 (2007), no. 1–2, 9–14. [32] s.-w. xiang, equivalence of completeness and contraction property, proc. amer. math. soc., 135 (2007), 1051–1058. [33] p.p. zabreiko, k-metric and k-normed linear spaces: survey, collect. math., 48 (1997), 825–859. n4-petrusel-rus-serban cubo a mathematical journal vol.10, n o ¯ 03, (133–136). october 2008 the fibonacci zeta-function is hypertranscendental jörn steuding department of mathematics, würzburg university, am hubland, 97 218 würzburg, germany email: steuding@mathematik.uni-wuerzburg.de abstract applying a theorem of reich on dirichlet series satisfying difference-differential equations, we show that the fibonacci zeta-function satisfies no algebraic differential equation. resumen aplicando el teorema de reich sobre series de dirichlet satisfaziendo ecuaciones diferenciales-diferencias, nosotros mostramos que la función zeta de fibonacci satisfaze una ecuación diferencial no algebraica. key words and phrases: hypertranscendence, fibonacci zeta-function. math. subj. class.: 11b39, 11m41, 34m15. 134 jörn steuding cubo 10, 3 (2008) 1 introduction the fibonacci numbers are recursively defined by f0 = 0, f1 = 1 and fn+2 = fn+1 + fn for n ∈ n. in number theory one can often obtain arithmetic information by studying a generating function of a given number theoretical object. in the case of fibonacci numbers this is usually the corresponding lambert series; however, in the recent past also the generating dirichlet series was studied; this function is more interesting with respect to its analytic properties. let s be a complex variable. for re s > 0 the fibonacci zeta-function is defined by ζf(s) = ∑ n∈n f −sn , and by analytic continuation throughout the complex plane except for simple poles at s = −2k + πi(2n + k)/ log ϕ for n ∈ z, k ∈ n0, where ϕ is the golden ratio; this was first proved by navas [6] (and relies mainly on binet’s formula). in [1], elsner et al. obtained several results on the algebraic independence of the values taken by ζf on the positive integers, e.g. ζf(2), ζf(4), ζf(6) are algebraically independent. in this note we show that the fibonacci zeta-function ζf(s) is hypertranscendental, i.e., it satisfies no non-trivial algebraic differential equation (that is no finite collection of derivatives of ζf) is algebraically dependent over the field of rational functions). actually, we shall prove a slightly stronger statement by applying reich’s theorem on dirichlet series satisfying holomorphic difference–differential equations. in order to state this result denote by d the set of all ordinary dirichlet series f (s) = ∑ ∞ n=1 ann −s satisfying the following two assumptions: • the abscissa of absoulte convergence is finite: σa(f ) < ∞, • the set of all divisors of indices n with an 6= 0 contains infinitely many prime numbers. furthermore, we introduce the following abbreviation: for a non-negative integer ν we write f [ν](s) = (f (s), f ′(s), . . . , f (ν)(s)). reich [9] proved the following theorem: assume that f ∈ d. let h0 < h1 < . . . < hm be any real numbers, ν0, ν1, . . . , νm be any non-negative integers, and let σ0 > σa(f ) − h0. put k := ∑m j=0 (νj + 1). if φ : c k → c is continuous and the difference-differential equation φ(f [ν0](s + h0), f [ν1](s + h1), . . . , f [νm](s + hm)) = 0 holds for all s with re s > σ0, then φ vanishes identically. to apply this result to the fibonacci zeta-function it suffices to show that the set of all fibonacci numbers fn is not generated by finitely cubo 10, 3 (2008) the fibonacci zeta-function is hypertranscendental 135 many primes. however, this follows immediately from lucas’ theorem gcd(fm, fn) = fgcd(m,n) (see [5]), since the right-hand side is equal to f1 = 1 for any pair m, n of relatively coprime integers. thus we obtain: theorem 1. given any real numbers h0 < h1 < . . . < hm, any non-negative integers ν0, ν1, . . . , νm, and any σ0 > −h0, if φ : c k → c is continuous and the difference-differential equation φ(ζf [ν0](s + h0), ζf [ν1](s + h1), . . . , ζf [νm](s + hm)) = 0 holds for all s with re s > σ0, then φ vanishes identically. notice that the proof does not use the meromorphic continuation of ζf(s) to c, obtained by navas. the statement of the theorem can easily be extended to other dirichlet series built from linear recursive sequences. here we only need that such sequences are divisible by infinitely many prime numbers which is true except for degenerate cases when the characteristic polynomial has two roots whose quotient is a root of unity; since roots are counted with multiplicities, this also includes the case of repeated roots. this was first shown by pólya [8] and has been rediscovered by several mathematicians (see [2, 10] for some history). we conclude with a few historical remarks on hypertranscendence and an interesting question. in 1887, hölder [4] proved that the gamma-function is hypertranscendental. in his challenging lecture at the international congress for mathematicians in paris 1900, hilbert [3] asked in problem 18 for a description of classes of functions definable by differential equations. in this context hilbert stated that the riemann zeta-function ζ(s) is hypertranscendental; the first published proof was written down by stadigh in his dissertation (cf. ostrowski [7]). the idea is to deduce the hypertranscendence of ζ(s) from hölder’s theorem and the fact that the gamma-function appears in the functional equation for zeta. besides, hilbert [3] asked for a proof of the hypertranscendence for the more general series ∑ ∞ n=1 xnn−s. this problem was solved by ostrowski [7] as a particular case of a more general theorem which also applies to the case when there is no functional equation at hand; his argument relies on a comparison of the differential independence with the linear independence of its frequencies. reich’s theorem [9], which we have used to prove theorem 1, may be regarded as the most general and powerful extension of this method. a different way for proving hypertranscendence was found by voronin. in [11], he developped a new technique to study the joint value distribution of dirichlet l-functions to pairwise inequivalent characters and their derivatives; in [12], he extended the method in order to prove his famous universality theorem for the riemann zeta-function: let 0 < r < 1 4 and suppose that g(s) is a non-vanishing continuous function on the disk |s| ≤ r which is analytic in the interior. then, for any ǫ > 0, lim inf t →∞ 1 t meas { τ ∈ [0, t ] : max |s|≤r ∣ ∣ζ ( s + 3 4 + iτ ) − g(s) ∣ ∣ < ǫ } > 0. 136 jörn steuding cubo 10, 3 (2008) voronin’s results imply the hypertranscendence of these dirichlet series. it is natural to ask whether the fibonacci zeta-function shares this or some other universality property: is it true or false that any (suitable) analytic function g(s) can be uniformly approximated by certain shifts of ζf(s)? acknowledgements. the author wants to thank the anonymous referee for her or his remarks and references to clarify to which extent the statement of the theorem can be generalized. received: april 2008. revised: june 2008. references [1] c. elsner, s. shimomura and i. shiokawa, algebraic relations for reciprocal sums of fibonacci numbers, acta arith., 130 (2007), 37–60. [2] g. everest, a. van der poorten, i. shparlinski and t. ward, recurrence sequences, ams mathematical surveys and monographs, 104 (2003). [3] d. hilbert, mathematische probleme, archiv f. math. u. physik, 1 (1901), 44–63, 213–317. [4] o. hölder, über die eigenschaft der gammafunktion keiner algebraischen differentialgleichung zu genügen, math. ann., 28 (1887), 1–13. [5] t. koshy, fibonacci and lucas numbers with applications, john wiley & sons, new york 2001. [6] l. navas, analytic continuation of the fibonacci dirichlet series, fibonacci q., 39 (2001), 409–418. [7] a. ostrowski, über dirichletsche reihen und algebraische differentialgleichungen, math. z., 8 (1920), 241–298. [8] g. pólya, arithmetische eigenschaften der reihenentwicklungen rationaler funktionen, j. reine angew. math., 151 (1921), 1–31. [9] a. reich, über dirichletsche reihen und holomorphe differentialgleichungen, analysis, 4 (1984), 27–44. [10] h. roskam, prime divisors of linear recurrences and artin’s primitive root conjecture for number fields, j. théo. nombres bordeaux, 13 (2001), 303–314. [11] s.m. voronin, on the functional independence of dirichlet l-functions, acta arith., 27 (1975), 493–503 (in russian). [12] s.m. voronin, theorem on the ‘universality’ of the riemann zeta-function, izv. akad. nauk sssr, ser. matem., 39 (1975) (in russian); engl. translation in math. ussr izv., 9 (1975), 443–445. n11 cubo a mathematical journal vol.10, n o ¯ 01, (43–66). march 2008 arc-wise essentially tangentially regular set-valued mappings and their applications to nonconvex sweeping process messaoud bounkhel king saud university, department of mathematics p.o. box 2455, riyadh 11451, saudi arabia. e-mail: bounkhel@ksu.edu.sa abstract recently, borwein and moors (1998) introduced a new class of tangentially regular sets in ir n (called arc-wise essentially smooth sets). they characterized the sets s of this class in terms of arc-wise essential smoothness of the distance function ds . very recently, the author (2002) gave an appropriate extension of this class to any banach space x and he extended the above characterization to any banach space x with a uniformly gâteaux differentiable norm. in this paper we extend the concept of arc-wise essentially smooth sets to set-valued mappings c : [0,t ]⇉x (t > 0) and we will use this concept to establish an important application to nonconvex sweeping process. resumen ricientemente borwein y moors (1998) introducem una nueva clase de conjuntos tangencialmente regulares en ir n chamados conjuntos essencialmente suaves por arcos). ellos caracterizan los conjuntos s de esta clase en terminos de la suavidad de la distancia por arco de la función ds . ricientemente, el autor (2002) dió una 44 messaoud bounkhel cubo 10, 1 (2008) extensión apropriada de esta clase en cualquer espacio de banach x y extiende la caractarización anterior a cualquer espacio de banach x y con una norma de gâteaux uniformemente diferenciable. en este art́ıculo extendemos el concepto de conjunto essencialmente suave por arcos para el conjunto de aplicaciones c : [0,t ]⇉x (t > 0) y usaremos este concepto para establecer una importante aplicación a procesos no convexos generales. key words and phrases: nonconvex sweeping process, tangentially regular sets. math. subj. class.: 34g25. 1 introduction in [9] borwein and moors introduced, in ir n , the concept of arc-wise essential smoothness for sets and for functions. they characterized the class of all sets s which are arc-wise essentially smooth in terms of arc-wise essential smoothness of the distance function ds . their definitions and results were strongly based on the finite dimensional structure. in [10] the author gave an appropriate extension of the arc-wise essentially smooth concept for sets and functions in any banach space and he extended the above characterization of the class of arc-wise essentially smooth sets in any banach space with a uniformly gâteaux differentiable norm. in this paper we intend to extend the concept of arc-wise essentially smooth sets to set-valued mappings and to give some applications of this new concept of regularity of set-valued mappings. the paper is organized as follows. in section two we recall some notations and preliminaries that are used in the paper. section three is devoted to introduce and to study the new concept of arc-wise essentially tangentially regular set-valued mappings. many examples of this class of set-valued mappings are given in this section. we prove in this section various characterizations of arc-wise essentially tangentially regular set-valued mappings. the main characterization is given in theorem 3.3 which establishes a relationship between arc-wise essentially tangential regularity of a set-valued mapping c and the arc-wise essentially smoothness of the distance function to the images of the set-valued mapping c. in the last section, we give an important application of this characterization to the nonconvex sweeping process. 2 preliminaries throughout, x will be a real banach space and x ∗ its topological dual. by 〈 ·, · 〉 we will denote the canonical pairing between these spaces. recall that a function f from x into ir is lipschitz around x0 ∈ x if there exist two real numbers k > 0 and δ > 0 such that |f(x′) − f(x)| ≤ k‖x′ − x‖ for all x′,x ∈ x0 + δib, cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 45 where ib denotes the closed united ball of x centered at the origin. we will say that f is locally lipschitz over x if it is lipschitz around any point of x. recall also that the usual directional derivative of f at x0 in the direction v is, f ′ (x0; v) := lim t→0 t −1 [ f(x0 + tv) − f(x0) ] , when this limit exists. for a locally lipschitz function f : x → ir, we recall that the clarke generalized directional derivative (resp. the lower dini directional derivative) of f at x0 ∈ x in the direction v is given by, f 0 (x0; v) := lim sup x→x0 t↓0 t −1 [ f(x + tv) − f(x) ] , ( resp. f − (x0; v) := lim inf t↓0 t −1 [ f(x0 + tv) − f(x0) ] . ) one always has f − (x0; v) ≤ f 0 (x0; v). the reverse inequality is not true in general (take for example f(x) = −‖x‖). the functions f satisfying the equality form in the last inequality are called directionally regular at x0 in the direction v. recall also that a locally lipschitz function f : x → ir is strictly differentiable (in short s.d.) at x0 in the direction v if f 0 (x0; v) = −f 0 (x0; −v). it is not difficult to check that, if f is s.d. at x0 in the direction v, then one has f 0 (x0; v) = f − (x0; v) = f ′ (x0; v) = −f 0 (x0; −v) and so it is directionally regular at x0 in the direction v. recall now, that the clarke subdifferential (resp. fréchet subdifferential ) of f at x0 ∈ x is defined by ∂ c f(x0) = {x ∗ ∈ x∗ : 〈 x ∗ ,v 〉 ≤ f0(x0; v), for all v ∈ x}, (resp. ∂ f f(x0) = {x ∗ ∈ x∗ : ∀ǫ > 0,∃δ > 0 : 〈 x ∗ ,x−x0 〉 ≤ f(x)−f(x0)+ǫ‖x−x0‖, ∀x ∈ x0+δib}). let s be a nonempty subset of x. we will let d(·,s) (or ds (·)) stand for the usual distance function to s, i.e., d(x,s) := inf u∈s ‖x − u‖. recall (see [20]) that the clarke tangent cone and the contingent cone of s at some point x ∈ s are given respectively by ts(x) = {v ∈ x : d 0 c (x; v) = 0}, (2.3) 46 messaoud bounkhel cubo 10, 1 (2008) and ks (x) = {v ∈ x : d − s (x; v) = 0}. (2.4) note that one always has ts(x) ⊂ ks (x). the sets s for which one has an equality in the last inclusion, will be called tangentially regular at x (see [20] for this definition). let us recall (see for instance [12]) that the clarke normal cone (resp. fréchet normal cone ) of s at x ∈ s is defined by n c s (x) = {x ∗ ∈ x∗ : 〈 x ∗ ,v 〉 ≤ 0, for all v ∈ ts (x)}, (resp. n f s (x) = {x ∗ ∈ x∗ : ∀ǫ > 0,∃δ > 0 : 〈 x ∗ ,x ′ − x 〉 ≤ ǫ‖x′ − x‖, ∀x′ ∈ x + δib}). the following proposition is needed in the sequel. it was proved for the first time by kruger [25] (see also iofee [26].) proposition 2.1 [12] let s be a nonempty closed subset in x and let x ∈ s. then ∂ f ds (x) = n f s (x) ∩ ib. let i be an interval and let ω be an open subset of x. by absolutely continuous mapping one means a mapping x : i → ω such that x(t) = x(a) + ∫ t a x ′ (s)ds, for all t ∈ i, with x ′ ∈ l1x (i) and a ∈ i. we will denote by ac(i, ω) the familly of all these mappings. remark 2.1 it is well known (see for instance [15]) that f ◦ x(·), the composition of a locally lipschitz mapping f : ω → y with an absolutely continuous mapping x : i → ω, is an absolutely continuous mapping, whenever the space y is reflexive. for more details concerning absolutely continuous mappings we refer the reader to brézis [15]. 3 arc-wise essentially tangentially regular set-valued mappings we start with the following definition of arc-wise essentially tangentially regular set-valued mappings: definition 3.1 let i :=]0, 1[ and let c : i⇉x be a set-valued mapping with nonempty closed values. we will say that c is arc-wise essentially tangentially regular and we will write c ∈ awet r(i,x), if for each x ∈ ac(i,x), the set {t ∈ i : x(t) ∈ c(t) and x′(t) or − x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))} has null measure. cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 47 in this paper we use the name arc-wise essential tangential regularity instead of arc-wise essential smoothness (used in [10] and [7, 8, 9]) because it seems for us that is more significant. remark 3.1 as one always has ks (x) = ts (x) = x, for each x ∈ ints (the topological interior of s), we can take x only in bd c(t) (the boundary of c(t)), in definition 3.1, that is, c is arc-wise essentially tangentially regular if and only if for each x ∈ ac(i,x) one has µ ( {t ∈ ] 0, 1 [ : x(t) ∈ bdc(t) and x′(t) or − x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))} ) = 0. example 3.1 1. it is easy to see that all set-valued mappings c : i⇉x with closed tangentially regular values are arc-wise essentially tangentially regular. 2. let s be a fixed set in x which is arc-wise essentially smooth in the sense of [9, 10]. then using proposition 4.1 in [10] we can check that the constant set-valued mapping c : i⇉x with c(t) = s is arc-wise essentially tangentially regular. 3. let s be a fixed set in x which is tangentially regular at each of its points except one point x0 ∈ s. define the set-valued mapping c as the translation of the set s in the direction v(t), that is, c(t) = s + v(t), with v ∈ ac(i,x). (1) assume now that v satisfies ±v′(t) 6∈ ks(x0) \ ts (x0), a.e. on i. then c is an arc-wise essentially tangentially regular set-valued mapping. indeed, for any x ∈ ac(i,x) we can easily check that µ ( {t ∈ i : x(t) ∈ bdc(t) and x′(t) or − x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))} ) = µ ({t ∈ i : x(t) = v(t) + x0 and v ′ (t) or − v′(t) ∈ ks (x0) \ ts (x0)}) = 0. take for example x = ir 2 , s1 is the epigraph of the absolute value function, and take s is the closure of the complement of s1. take v(t) = (t, 2t), for all t ∈ i \ n and v(t) = (t, 1) for all t ∈ n, where n is a subset of i with null measure. using what precedes we can easily check that the set-valued mapping c in (1) associated with the set s and v is arc-wise essentially tangentially regular. 4. more general and with the same manner we can prove that the set-valued mapping c in (1) is arc-wise essentially tangentially regular whenever the set s is tangentially regular at each of its points except on a countable set {xn} and with v satisfies ±v′(t) 6∈ ks (xn) \ ts (xn), for all n and a.e. on i. (2) 48 messaoud bounkhel cubo 10, 1 (2008) 5. the condition (2) on v cannot be removed in the last example. take for example s is the closure of the complement of the epigraph of the absolute value function and take v(t) = (t, 1), for all t ∈ i. the condition (2) is not satisfied and we can check that for some x ∈ ac(i,x) one has µ ( {t ∈ i : x(t) ∈ bdc(t) and x′(t) or − x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))} ) = 1, and so the set-valued mapping c in this case is not arc-wise essentially tangentially regular. from this example we can conclude that a set-valued mapping with values c(t) tangentially regular except on countable set is not necessarily arc-wise essentially tangentially regular. 6. let c0 be the cantor ternary set with 0 ∈ c0. let c(t) = c0 + t. we claim that c 6∈ awet r((0, 1),ir). let x(t) = t. then {t ∈ (0, 1) : x(t) ∈ c(t) with − x′(t) or x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))} = {t ∈ (0, 1) : 0 ∈ c0 with x ′ (t) 6= 0} = (0, 1). and so µ({t ∈ (0, 1) : x(t) ∈ c(t) with − x′(t) or x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))}) 6= 0, which ensures that c is not awet r((0, 1),ir). a first question, which arises naturally, is to ask whether the epigraph set-valued mapping t⇉c(t) := epift is arc-wise essentially tangentially regular, where epift := {(x,r) ∈ x × ir : f(t,x) ≤ r}. to give an answer to this question we introduce the following concepts. let f : ir×x → ir be a function from ir × x to ir. we define the following directional derivatives of f at (t0,x0) ∈ ir × x in a direction v ∈ x by f 0 ((t0,x0); v) := lim sup x→x0 (δ,t)↓(0,t0 ) δ −1 [ f(t,x + δv) − f(t,x) ] , and f − ((t0,x0); v) := lim inf (δ,t)↓(0,t0) δ −1 [ f(t,x0 + δv) − f(t,x0) ] . it is clear that if the two above limits exist then the clarke and the lower dini directional derivatives of ft0 (·) := f(t0, ·) at x0 in the direction v exist and equal respectively to f 0 ((t0,x0); v) and f − ((t0,x0); v). the converse is not true in general, take for example f(t,x) = f1(t)f2(x) with f1 is not right continuous. cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 49 definition 3.2 we will say that f is arc-wise essentially directionally regular and we will write f ∈ awedr(i × x), if for each x ∈ ac(i,x), the set {t ∈ i : f−((t,x(t)); x′(t)) 6= f0((t,x(t)); x′(t))} has null measure. example 3.2 1. any mapping f defined as follows f(t,x) = f1(t) + f2(x), is arc-wise essentially directionally regular whenever f2 is directionally regular and without any assumptions on f1. 2. any mapping f defined as follows f(t,x) = f1(t)f2(x), is arc-wise essentially directionally regular whenever f2 is directionally regular and f1 is continuous. recall that (see for instance [20]) for a function f : ir × x → ir one has for all t ∈ ir kepi ft ((x,ft(x)) = epif − t (x; ·), (3) and tepi ft ((x,ft(x)) = epif 0 t (x; ·). (4) now, we are ready to state the following result showing that the epigraph set-valued mapping c(t) := epift is arc-wise essentially tangentially regular whenever f is arc-wise essentially directionally regular. theorem 3.1 let i be an open interval and let f : i × x → ir be a locally lipschitz function from i ×x to ir. then the set-valued mapping c : i → x ×ir defined by t⇉epift is arc-wise essentially tangentially regular whenever the function f is arc-wise essentially directionally regular. proof. put c(t) = epift and suppose that f is arc-wise essentially directionally regular. let (x,r) ∈ ac(i,x × ir) and put j1 := {t ∈ i : f − ((t,x(t)); x ′ (t)) = f 0 ((t,x(t)); x ′ (t))}, and j2 := {t ∈ i : (x(t),r(t)) ∈ bdc(t) and (x ′ (t),r ′ (t)) ∈ kc(t)(x(t),r(t)) \ tc(t)(x(t),r(t))}. 50 messaoud bounkhel cubo 10, 1 (2008) first, we have µ(j1) = 1 because f is arc-wise essentially directionally regular. assume that there exists some t0 ∈ j1 ∩ j2. then f 0 ((t0,x(t0)); x ′ (t0)) and f − ((t0,x(t0)); x ′ (t0)) exist and coincide and they equal to f 0 t0 (x(t0); x ′ (t0)) = f − t0 (x(t0); x ′ (t0)). we also have x ′ (t0) and r ′ (t0) exist and such that (x(t0),r(t0)) ∈ bdc(t0) ( that is, r(t0) = ft0 (x(t0)), because the boundary of c(t0) is the graph of ft0 ) and (x ′ (t0),r ′ (t0)) ∈ kc(t)(x(t0),r(t0)) \ tc(t)(x(t0),r(t0)). (5) further, (3) and (4) yield r ′ (t0) ≥ f − t0 (x(t0); x ′ (t0)) = f 0 t0 (x(t0); x ′ (t0)) which means (x ′ (t0); r ′ (t0)) ∈ tc(t0)(x(t0),r(t0)) that is a contradiction with (5). therefore, we obtain j1 ∩j2 = ∅ and hence as µ(j1) = 1 we get µ(j2) = 0. so, c is arc-wise essentially tangentially regular. 2 the following theorem states a necessary condition on f for the arc-wise essential tangential regularity of the epigraph set-valued mapping epift. its proof follows some ideas from [9]. theorem 3.2 let i be an open interval and let f : i × x → ir be a locally lipschitz function from i × x to ir. if the set-valued mapping c : i → x × ir defined by t⇉epift is arc-wise essentially tangentially regular, then the function f is arc-wise essentially strictly differentiable function ( i.e., f ∈ awesd(x,ir)), in the following sense: for any x ∈ ac(i,x), one has µ ( {t ∈ i : f0t (x(t); −x ′ (t)) 6= −f0t (x(t); x ′ (t))} ) = 0. (6) proof. suppose that c is arc-wise essentially tangentially regular and fix any x ∈ ac(i,x). since f is lipschitz then by remark 2.1 the function θ(t) := f(t,x(t)) ∈ ac(i,x) and so θ ′ (t) exists a.e. on i. fix now any t ∈ i such that x′(t) and θ′(t) exist. then, by (4) one has (x ′ (t),f 0 (x(t); x ′ (t)) ∈ tc(t)(x(t),f(x(t))). by (3), one also has (x(t),ft(x(t))) ∈ c(t) and (x ′ (t),f ′ t(x(t); x ′ (t)) ∈ kc(t)(x(t),f(x(t))). put e := e1 ∪ e2 where e1 := {s ∈ e3 : (x ′ (s),θ ′ (s)) ∈ kc(s)(x(s),θ(s)) \ tc(s)(x(s),θ(s))}, e2 := {s ∈ e3 : (−x ′ (s),−θ′(s)) ∈ kc(s)(x(s),θ(s)) \ tc(s)(x(s),θ(s))}, and e3 := {s ∈ i : (x(s),θ(s)) ∈ c}. cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 51 as c is arc-wise essentially tangentially regular one gets by definition 3.1 that µ(e) = 0. if one assumes further that t /∈ e, we obtain (x ′ (t),θ ′ (t)) ∈ tc(t)(x(t),θ(t)) = epif 0 t (x(t); ·), and (−x′(t),−θ′(t)) ∈ tc(t)(x(t),θ(t)) = epif 0 t (x(t); ·). this means respectively f 0 t (x(t); x ′ (t)) ≤ θ′(t) and f 0 t (x(t); −x ′ (t)) ≤ −θ′(t), which yields f 0 t (x(t); x ′ (t)) ≤ −f0t (x(t); −x ′ (t)) and hence, because the reverse inequality always holds, one gets f 0 t (x(t); x ′ (t)) = −f0t (x(t); −x ′ (t)). thus, the set ẽ := {s ∈ i : f0t (x(s); x ′ (s)) 6= −f0t (x(s); −x ′ (s))}, is included in e and so µ(ẽ) = 0. the proof then is complete. 2 remark 3.2 combining theorems 3.1-3.2 we get the following inclusion: awedr(x,ir) ⊂ awesd(x,ir). (7) this means that any arc-wise essentially directionally regular is arc-wise essentially strictly differentiable in the sense of (6). this inclusion is strict. take for example the function f in example 3.2 part (2) with f1 is not continuous. the following lemma will be used in the sequel. lemma 3.1 let f be a locally lipschitz function defined from x into ir and let x0,v ∈ x. then f is s.d. at x0 in the direction v if and only if 〈 ∂ c f(x0),v 〉 = {f0(x0; v)} iff 〈 ∂ c f(x0),v 〉 is a singleton set. here 〈 ∂ c f(x),v 〉 := { 〈 x ∗ ,v 〉 : x ∗ ∈ ∂cf(x)}. proof. it is clear that it suffices to prove the following relation: 〈 ∂ c f(x0),v 〉 = [−f0(x0; −v),f 0 (x0; v)]. by (b) in proposition 2.1.2 in [20] one has f 0 (x0; v) = max 〈 ∂ c f(x0),v 〉 and hence −f0(x0; −v) = min 〈 ∂ c f(x0),v 〉 and as the set 〈 ∂ c f(x0),v 〉 is convex, one obtains the desired equality. 2 when the function f does not depend on t, that is, f : x → ir, it is easy to see (by proposition 3.1 in [10]) that the concept of arc-wise essential strict differentiability in the sense of theorem 3.2, is equivalent to the concept of arc-wise essential smoothness in the sense of [10]. the next corollary summarizes further characterizations of arc-wise essentially smooth functions. 52 messaoud bounkhel cubo 10, 1 (2008) corollary 3.1 let i be an open interval and let f : x → ir be a locally lipschitz function. then the following assertions are equivalent: 1. f is arc-wise essentially smooth in the sense of [10]; 2. f is arc-wise essentially strictly differentiable; 3. epif is arc-wise essentially tangentially regular; 4. f is arc-wise essentially directionally regular; 5. for each x ∈ ac( ] 0, 1 [ ,x) µ ( {t ∈ ] 0, 1 [ : 〈 ∂ c f(x(t); x ′ (t)) 〉 = {f0(x(t); x′(t))}}) = 1. 6. for each x ∈ ac( ] 0, 1 [ ,x) µ ( {t ∈ ] 0, 1 [ : f 0 (x(t); x ′ (t)) = (f ◦ x)′(t)} ) = 1; 7. for each x ∈ ac( ] 0, 1 [ ,x) µ ( {t ∈ ] 0, 1 [ : f 0 (x(t); x ′ (t)) = f ′ (x(t); x ′ (t))}) = 1. proof. the following equivalences follow from theorems 3.1-3.2, lemma 3.1 and by what precedes the corollary: (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (5). (6) ⇔ (7) : fix any x ∈ ac( ] 0, 1 [ ,x) and fix also any t ∈]0, 1[ where x′(t) exists. if we put δ = min{t, 1 − t}, the lipschitz behavior of f ensures for all s ∈] − δ,δ[ s −1 [(f ◦ x)(t + s) − (f ◦ x)(t)] = s−1[f(x(t) + sx′(t)] + ǫ(s) (8) with lim s→0 ǫ(s) = 0. therefore, for any such t , (f ◦ x)′(t) exists if and only if f′(x(t); x′(t)) exists. the equivalence then holds. (4) ⇔ (6) : fix any x ∈ ac( ] 0, 1 [ ,x) and t ∈]0, 1[ such that x′(t) and (f ◦x)′(t) exist. note that the set of such points t has 1 as lebesgue measure because x and f ◦ x are absolutely continuous, and note also that, by (8), for any such t (f ◦ x)′(t) = f−(x(t); x′(t)). so, the equivalence follows. 2 using theorem 3.1, we get the following examples of arc-wise essentially tangentially regular set-valued mappings: cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 53 1. the translation of the epigraph of directionally regular functions in any direction of y-axis, i.e., c(t) = epif2 + (0,f1(t)), with f2 is a directionally regular function and f1 is an arbitrary function. 2. the set-valued mapping c : ir⇉ir 2 c(t) = {(x,f1(t)r) : f2(t) ≤ r}, where f2 : ir → ir is directionally regular and f1 : ir → ir is continuous with f1 6≡ 0. now we are going to establish a characterization of the class of arc-wise essentially tangentially regular set-valued mappings c, in terms of the distance function to the images of the set-valued mapping c. its proof follows some ideas from [10]. it will be used to give an important application to nonconvex sweeping processes. in the proof of this theorem, we need the following characterization of the contingent cone kc (x). a vector v ∈ kc (x) if and only if there exist two sequences {tn}n∈in of positive real numbers converging to zero and {vn}n∈in in x converging to v such that x + tnvn ∈ c, for each n ∈ in. theorem 3.3 let c : i⇉x be a set-valued mapping with nonempty closed values. assume that dc(·)(·) is arc-wise essentially strictly differentiable. then c is arc-wise essentially tangentially regular. if, in addition, x is a banach space with uniformly gâteaux differentiable norm, then c is arc-wise essentially tangentially regular if and only if dc(·)(·) is arc-wise essentially strictly differentiable. proof. 1) assume that dc(·)(·) ∈ awesd(x,ir), i.e., for each x ∈ ac( ] 0, 1 [ ,x), the set a := {t ∈ ] 0, 1 [ : dc(t)(·) is not s.d. at x(t) in the direction x ′ (t) } has null measure. we will show that c is arc-wise essentially tangentially regular, i.e., µ(b) = 0 where b := {t ∈ ] 0, 1 [ : x(t) ∈ c(t) and x′(t) or − x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))}). it is enough to prove that b ⊂ a. let t0 /∈ a. if x(t0) 6∈ c(t0), then t0 6∈ b. so let us suppose that x(t0) ∈ c(t0). if x ′ (t) and −x′(t) 6∈ kc(t)(x(t))\tc(t)(x(t)), then t0 6∈ b. so, assume that x ′ (t) or − x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t)). this ensures d − c(t0) (x(t0); x ′ (t0)) = 0 or d − c(t0) (x(t0); −x ′ (t0)) = 0. since dc(t0) is s.d. at x(t0) ∈ c(t0) in the direction x ′ (t0), i.e., we have d 0 c(t0) (x(t0); x ′ (t0)) = −d 0 c(t0) (x(t0); −x ′ (t0)). on the other hand, the strict differentiability ensures the directional regularity, that is, d − c(t0) (x(t0); x ′ (t0)) = d 0 c(t0) (x(t0); x ′ (t0)) and d − c(t0) (x(t0); −x ′ (t0)) = d 0 c(t0) (x(t0); −x ′ (t0)) and hence d − c(t0) (x(t0); x ′ (t0)) =d 0 c(t0) (x(t0); x ′ (t0)) =−d 0 c(t0) (x(t0); −x ′ (t0)) =−d − c(t0) (x(t0); −x ′ (t0)). 54 messaoud bounkhel cubo 10, 1 (2008) so in both cases d − c(t0) (x(t0); x ′ (t0)) = 0 or d − c(t0) (x(t0); −x ′ (t0)) = 0, we obtain d 0 c(t0) (x(t0); x ′ (t0)) = d 0 c(t0) (x(t0); −x ′ (t0)) = 0, that is, x ′ (t0) ∈ tc(t0)(x(t0)) and −x ′ (t0) ∈ tc(t0)(x(t0)). thus, both directions x ′ (t0) and −x ′ (t0) lie in tc(t0)(x(t0)) and hence t0 /∈ b. consequently, each t0 /∈ a does not lie in b. this completes the proof of the inclusion b ⊂ a. 2) assume now that x is a banach space with uniformly gâteaux differentiable norm and assume that c is arc-wise essentially tangentially regular. then, for each fixed x in ac( ] 0, 1 [ ,x) by definition 3.1 we have µ(bx) = 0, where bx = b 1 x ∪ b 2 x, b 1 x := {t ∈ ] 0, 1 [ : x(t) ∈ c(t) and x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))} and b 2 x := {t ∈ ] 0, 1 [ : x(t) ∈ c(t) and −x′(t) ∈ kc(t)(x(t)) \ tc(t)(x(t))}. put a := {t ∈ ] 0, 1 [ : dc(t) is not s.d. at x(t) in the dir. x ′ (t) }. it is not difficult to check that a = {t ∈ ] 0, 1 [ : x(t) ∈ bdc(t), dc(t) is not s.d. at x(t) in the dir. x ′ (t) }. indeed, if t ∈ ] 0, 1 [ with x(t) ∈ (x \ c(t)) ∪ intc(t) and dc(t) is not s.d. at x(t) in the direction x ′ (t), then (−dc(t)) is not s.d. at x(t) in the direction x ′ (t) and so (−dc(t)) is not directionally regular at x(t) in the direction x ′ (t), which is impossible, because x(t) ∈ (x \c(t))∪intc(t), and theorem 8 in [2]. put now dx′ := {t ∈ ] 0, 1 [ : x ′ (t) exists }, hence µ(a \ dx′ ) = 0. (9) put also i := ir ∪ il with ir ( resp. il ) denotes the set of all isolated points in a ∩ dx′ relatively to the right topology ( resp. the left topology). it is not difficult to check that i is countable and hence µ(i) = 0. fix t0 ∈ (a ∩ dx′ ) \ i. then there exist two sequences of real positive numbers (λn)n and (ǫn)n converging to zero such that for n sufficiently large t0 + λn and t0 − ǫn lie in (a ∩ dx′ ) \ i and hence x(t0 + λn) ∈ bd c(t0 + λn) and x(t0 − ǫn) ∈ bd c(t0 − ǫn), for n sufficiently large. put vn := λ −1 n [ x(t0 + λn) − x(t0) ] and wn := ǫ −1 n [ x(t0 − ǫn) − x(t0) ] . clearly, vn → x ′ (t0) and wn → −x ′ (t0) and for n sufficiently large x(t0) + λnvn ∈ bd c(t0 + λn) ⊂ c(t0) + rcλnib ⊂ clc(t0) = c(t0) cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 55 and x(t0) + ǫnwn ∈ bd c(t0 − ǫn) ⊂ c(t0) + rcǫnib ⊂ cl c(t0) = c(t0). it follows (by the characterization given above of the contingent cone) that x ′ (t0) and −x ′ (t0) lie in kc(t0)(x(t0)). now, we distinguish two cases. firstly, if x ′ (t0) ∈ kc(t0)(x(t0)) \ tc(t0)(x(t0)), then t0 ∈ bx. secondly, if x ′ (t0) ∈ tc(t0)(x(t0)), then −x ′ (t0) ∈ kc(t0)(x(t0))\ tc(t0)(x(t0)) ( because, if −x′(t0) ∈ tc(t0)(x(t0)), we would have d 0 c(t0) (x(t0); x ′ (t0)) = −d0 c(t0) (x(t0); −x ′ (t0)) = 0, so dc(t0) would be s.d. at x(t0) in the direction x ′ (t0), which would contradict that t0 ∈ a ) . hence t0 ∈ bx. thus (dx′ ∩ a) \ i ⊂ bx and hence µ((dx′ ∩ a) \ i) = 0. (10) finally, according to (9) and (10), we obtain µ(a) = 0. this ensures that dc(t0) ∈ awesd(x,ir) and hence the proof is finished. 2 the following corollary follows from theorem 3.3 and lemma 3.1. it will be used in the next section. corollary 3.2 let h be a hilbert space. a set-valued mapping c : i⇉cl(h) is arc-wise essentially tangentially regular if and only if for each x ∈ ac(i,h) one has µ ({ t ∈ i : 〈 ∂ c dc(t)(x(t)),x ′ (t) 〉 6= {d0c(t)(x(t); x ′ (t))} }) = 0. (11) 4 applications to nonconvex sweeping process throughout this section, we will let h (resp. cl(h)) denote a separable hilbert space (resp. the collection of all nonempty closed sets in h). let f : h⇉h be a set-valued mapping from h to h. we will say that f is hausdorff upper semicontinuous (for more details on hausdorff upper semicontinuity see [23, 16]) if for any y ∈ h one has lim sup x→x̄ e(f(x),f(y)) ≤ e(f(x̄),f(y)), wheree e(a,b) := sup a∈a [ inf b∈b ‖b − a‖ ] = sup a∈a db(a). in all the sequel t > 0, i := [0,t ], and c : ir⇉cl(h) will denote a l ′ -lipschitz set-valued mapping (l ′ > 0) with nonempty closed values, i.e., for any y ∈ h and any t,s ∈ i |d(y,c(t)) − d(y,c(s))| ≤ l′|t − s|. 56 messaoud bounkhel cubo 10, 1 (2008) we prove in the following theorem our main application of the concept of arc-wise essentially tangentially regular set-valued mappings. it proves a stability result for nonconvex sweeping processes with nonconvex noncontinous perturbation. let us note that our assumption on f requiring the inclusion in the subdifferential of some function was introduced for the first time in the work by [14] and by many other authors (see for instance [1, 3, 4, 11, 28, 30]). theorem 4.1 assume that c : [0,t ]⇉h is arc-wise essentially tangentially regular and it has ball compact values. let f : h⇉h be hausdorff u.s.c. on h contained in the subdifferential of a directionally regular locally lipschitz function ψ : h → ir. let {xn(·)}n be a bounded sequence in ac(i,h) (that is, ‖xn(t)‖ ≤ m, for some m > 0, for any n and any t ∈ i) such that (nspp)        x ′ n(t) ∈ −n f c(t)(xn(t)) + fn(t) + bn(t) a.e. on [0,t ]; fn(t) ∈ f(xn(θn(t))) and bn(t) ∈ rn(t)ib a.e. on [0,t ] xn(t) ∈ c(t), ∀t ∈ [0,t ]; xn(0) = x0 ∈ c(0), where fn,bn ∈ l 2 (i,h) and rn(t) → 0 + uniformly on i, and θn(t) → t for all t ∈ [0,t ], and ‖x′n(t)‖ ≤ l ′′ a.e. on [0,t ]. then there exist b ∈]0,t ] and x ∈ ac([0,b],h) such that    x ′ (t) ∈ −nc c(t) (x(t)) + f(x(t)) a.e. on [0,t ]; x(t) ∈ c(t), ∀t ∈ [0,t ]; x(0) = x0 ∈ c(0), proof. let α > 0 such that ψ is lipschitz on x0+αib with ratio l > 0. put b := min{ α l′′ ,t} and i := [0,b]. let (fn)n and (bn)n in l 2 (i,h) such that fn(t) ∈ f(xn(θn(t))) and bn(t) ∈ rn(t)ib a.e. on i. so we have by (nspp) −x′n(t) + fn(t) + bn(t) ∈ n f c(t)(xn(t)) a.e. on i since {xn(·)}n is bounded sequence in ac(i,h) and c has ball compact values we get the set {xn(t) : n ≥ 1} is relatively strongly compact in h. thus, as ‖x ′ n(t)‖ ≤ l ′′ , we get by ascoli-arzela’s theorem xn → s x in ac(i,h), x ′ n → w x ′ in l 2 (i,h). since ‖xn(t) − x0‖ ≤ ∫ t 0 ‖x′n(s)‖ds ≤ l ′′ b ≤ α, for all t ∈ i we obtain fn(t) ∈ f(xn(θn(t))) ⊂ ∂ψ(xn(θn(t))) ⊂ lib, cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 57 and so we get for n0 large enough ‖ − x′n(t) + fn(t) + bn(t)‖ ≤ l ′′ + l + 1 n0 for all n ≥ n0. thus, proposition 2.1 ensures for σ := l ′ + l + 1 n0 and for a.e. t ∈ i −x′n(t) + fn(t) + bn(t) ∈ n f c(t)(xn(t)) ∩ σib = σ∂ f dc(t)(xn(t)). we can thus apply castaing techniques (see for instance [17]). the convergence of the sequences {rn}n and {xn}n to 0 and x respectively and the weak convergence of the sequences {x′n}n and {fn}n to x ′ and f, and using mazur’s lemma yield −x′(t) + f(t) ∈ σ∂cdc(t)(x(t)) and f(t) ∈ ∂ c ψ(x(t)). (12) now, since the function ψ is directionally regular we obtain, by corollary 3.1 (ψ ◦ x)′(t) = ψ0(x(t); x′(t)) = 〈 f(t),x ′ (t) 〉 and so ∫ b 0 ψ 0 (x(t); x ′ (t))dt = ∫ b 0 〈 f(t),x ′ (t) 〉 dt. on one hand, as fn(t) ∈ ∂ψ(xn(θn(t))) one has 〈 fn(t),x ′ n(t) 〉 ≤ ψ0(xn(θn(t)); x ′ n(t)), because ψ is regular. on the other hand, since ψ is directionally regular we get ψ 0 (xn(θn(t)); x ′ n(t)) = ψ ′ (xn(θn(t)); x ′ n(t)) and ψ 0 (x(t); x ′ (t)) = ψ ′ (x(t); x ′ (t)) and so by theorem 2.1 in [2] we obtain lim sup n ∫ b 0 ψ 0 (xn(θn(t)); x ′ n(t))dt = lim sup n ∫ b 0 ψ ′ (xn(θn(t)); x ′ n(t))dt ≤ ∫ b 0 ψ ′ (x(t); x ′ (t))dt = ∫ b 0 ψ 0 (x(t); x ′ (t))dt. consequently, we get lim sup n ∫ b 0 〈 fn(t),x ′ n(t) 〉 dt ≤ ∫ b 0 〈 f(t),x ′ (t) 〉 dt. coming back to (12) and using the fact c is arc-wise essentially tangentially regular and the fact that x(t) ∈ c(t) for all t ∈ i, we get (by corollary 3.2) for a.e. t ∈ i 〈 f(t) − x′(t),x′(t) 〉 = σ 〈 ∂ c dc(t)((x(t)),x ′ (t) 〉 = σd 0 c(t)(x(t); x ′ (t)) = 0 and 58 messaoud bounkhel cubo 10, 1 (2008) 〈 bn(t) + fn(t) − x ′ n(t),x ′ n(t) 〉 = σ 〈 ∂ c dc(t)((xn(t)),x ′ n(t) 〉 = σd 0 c(t)(xn(t); x ′ n(t)) = 0, which gives ‖x′(t)‖2 = 〈 f(t),x ′ (t) 〉 and ‖x′n(t)‖ 2 = 〈 bn(t) + fn(t),x ′ n(t) 〉 . therefore, ‖x′n‖ 2 l2 = ∫ b 0 〈 bn(t) + fn(t),x ′ n(t) 〉 dt and ‖x′‖2l2 = ∫ b 0 〈 f(t),x ′ (t) 〉 dt. finally, we have lim sup n ‖x′n‖ 2 l2 ≤ ∫ b 0 〈 f(t),x ′ (t) 〉 dt = ‖x′‖2l2. since x ′ n → w x ′ in l 2 (i,h) and using the weak l.s.c. of the norm, together with the last inequality we get ‖x′n‖l2 → ‖x ′‖l2. now, using the fact that l 2 (i,h) is a hilbert space we conclude the strong convergence of x ′ n to x ′ in l 2 (t,h). put now ζn(t) := −x ′ n(t) + bn(t) + fn(t), a.e. on i. we have d(ζn(t), f (x(t)) − x ′(t)) = d(ζn(t) + x ′(t), f (x(t))) ≤ ‖bn(t)‖ + ‖x ′ n (t) − x′(t)‖ + d (fn(t), f (x(t))) , ≤ ‖bn(t)‖ + ‖x ′ n (t) − x′(t)‖ + e (f (xn(θn(t))), f (x(t))) → 0 as n → +∞, because of the hausdorff u.s.c. of f and since xn(θn(t)) → x(t) on i and x ′ n(t) → x ′ (t) a.e. on i. so given ǫ > 0, we can find n0 ≥ 1 such that for all n ≥ n0 we have ζn(t) + x ′ (t) ∈ f(x(t)) + ǫib. since ǫ > 0 was arbitrary and f has closed values we get γ(t) := lim sup{ζn(t)}n≥1 ⊂ f(x(t)) − x ′ (t) a.e. on i. let ζ be a measurable selection of γ, i.e., ζ(t) ∈ γ(t) a.e. on i. then, we get for a.e. on i ζ(t) ∈ γ(t) ⊂ cow lim sup{ζn(t)}n≥1 ⊂ co w lim sup σ∂ f dc(t)(xn(t)) ⊂ σ∂ c dc(t)(xn(t)). therefore, we get for a.e. on i ζ(t) + x ′ (t) ∈ f(x(t)) and ζ(t) ∈ ncc(t)(x(t)), which ensures x ′ (t) ∈ −ncc(t)(x(t)) + f(x(t)) a.e. on i. cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 59 the proof then is complete. 2 using our stability result for sweeping processes, we prove a new existence result for nonconvex sweeping process with nonconvex and noncontinuous perturbation. first we recall the definition of r-prox-regularity (see [27]) (or equivalently r-proximal smoothness (see [21])) for subsets which is a generalization of convex subsets. definition 4.1 let s be a closed nonempty subset in h. we will say that s is r-proxregular (or r-proximally smooth) if ds is continuously gâteaux differentiable on the tube u(r) := {u ∈ h : 0 < ds (u) < r}. the following properties of uniformly prox-regular sets are necessary in the sequel. proposition 4.1 [13] let s be a r-prox-regular nonempty closed subset in h. then following holds 1. s is tangentially regular at each point x ∈ s. 2. for any x ∈ s and any ξ ∈ ∂f ds (x) one has 〈ξ,x′ − x〉 ≤ 2 r ‖x′ − x‖2 + ds (x ′ ) for all x′ ∈ h with ds (x ′ ) < r. 3. the clarke and the fréchet subdifferentials of the distance function ds coincide at each point x ∈ s, that is, ∂cds (x) = ∂ f ds (x) for all x ∈ s. therefore, in the sequel of all the paper we will denote ∂ds (x) for both subdifferentials for r-prox-regular sets. 4. the clarke and the fréchet normal cones coincide at each point x ∈ s, that is, n c s (x) = n f s (x) for all x ∈ s. hence, we will use the notation ns (x) for both normal cones for r-prox-regular sets. note that the converse in the second assertion (even in the finite dimensional setting) is not true in general. for more details and examples, we refer the reader to [13]. now, we are ready to state the following new existence result for prox-regular sweeping processes with nonconvex and noncontinuous perturbations. theorem 4.2 let r : i →]0, +∞] such that ∫ t 0 dt r(t) < ∞. assume that c : i⇉cl(h) has r(t)-prox-regular and ball compact values for almost every t in i. let f : h⇉h be hausdorff u.s.c. on h contained in the subdifferential of a directionally regular locally lipschitz function ψ : h → ir. then there exist b ∈]0,t ] such that the following nonconvex sweeping process with nonconvex noncontinuous perturbation (nspp)    x ′ (t) ∈ −nc(t)(x(t)) + f(x(t)) a.e. on [0,b]; x(t) ∈ c(t), ∀t ∈ [0,b]; x(0) = x0 ∈ c(0), 60 messaoud bounkhel cubo 10, 1 (2008) has at least one solution. to prove this theorem we need the following propositions: proposition 4.2 let r : i →]0, +∞] such that ∫ t 0 dt r(t) < ∞. assume that c : i⇉cl(h) has r(t)-prox-regular values and let h ∈ l2(i,h) with ‖h(t)‖ ≤ m a.e. on i. then the following sweeping process (sp)    x ′ (t) ∈ −nc(t)(x(t)) + h(t) a.e. on i; x(t) ∈ c(t), ∀t ∈ i; x(0) = x0 ∈ c(0), has one and only one solution satisfying ‖x′(t)‖ ≤ l′ + 2m a.e. on i. proof. put u(t) := x(t) + ∫ t 0 h(s)ds, k(t) := c(t) − ∫ t 0 h(s)ds. then (sp) is equivalent to (sp ′ )    u ′ (t) ∈ −nk(t)(u(t)) a.e. on i; u(t) ∈ k(t), ∀t ∈ i; u(0) = x0 ∈ k(0). by theorem 4.1 in [12] (sp ′ ) has one and only one solution u satisfying ‖u(t)‖ ≤ l′ + m a.e. on i. this completes the proof. 2 note that theorem 4.1 in [12] is given for set-valued mappings c with r-prox-regular values with r does not depend on t but an inspection of the proof of theorem 4.1 in [12] shows that it is also true if we take c(t) is r(t)-prox-regular for almost every t in i and with r satisfies ∫ t 0 dt r(t) < ∞. proposition 4.3 let r : i →]0, +∞] such that ∫ t 0 dt r(t) < ∞. assume that c : i⇉cl(h) has r(t)-prox-regular values for almost every t in i. let x0,y0 ∈ c(0), and f,g ∈ l 2 (i,h), and let x and y be two solutions of the two following problems, respectively (spf )    x ′ (t) ∈ −nc(t)(x(t)) + f(t) a.e. on i; x(t) ∈ c(t), ∀t ∈ i; x(0) = x0 ∈ c(0), and (spg)    y ′ (t) ∈ −nc(t)(y(t)) + g(t) a.e. on i; y(t) ∈ c(t), ∀t ∈ i; y(0) = y0 ∈ c(0), and satisfying ‖x′(t)‖ ≤ δf and ‖y ′ (t)‖ ≤ δg, for a.e. on i, cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 61 with δf,δg > 0. then for p(t) = ∫ t 0 2 r(τ ) max{δf + lf,δg + lg}dτ one has ‖x(t) − y(t)‖ ≤ ‖x0 − y0‖e p(t) + ∫ t 0 ‖f(τ) − g(τ)‖ep(t)−p(τ )dτ for all t ∈ i, where lf and lg are constants depending on f and g respectively. proof. by (spf ) and (spg) we have for a.e. t ∈ i −x′(t) + f(t) ∈ nc(t)(x(t)), with x(0) = x0 and −y′(t) + g(t) ∈ nc(t)(y(t)), with y(0) = y0 and ‖f(t) − x′(t)‖ ≤ δf + lf and ‖g(t) − y ′ (t)‖ ≤ δg + lg. so, by part (1) in proposition 2.1 we get −x′(t) + f(t) ∈ δ∂dc(t)(x(t)) and − y ′ (t) + g(t) ∈ δ∂dc(t)(y(t)), where δ := max{δf +lf,δg +lg}. now, by using the property of the uniform prox-regularity of the values of c recalled in part (3) in proposition 2.1, we obtain 〈 − x′(t) + f(t) + y′(t) − g(t),x(t) − y(t) 〉 ≥ −2δ r(t) ‖x(t) − y(t)‖2. hence 〈 x ′ (t) − y′(t),x(t) − y(t) 〉 ≤ 〈 f(t) − g(t),x(t) − y(t) 〉 + 2δ r(t) ‖x(t) − y(t)‖2, and hence 〈 x ′ (t) − y′(t),x(t) − y(t) 〉 ‖x(t) − y(t)‖ ≤ ‖f(t) − g(t)‖ + 2δ r(t) ‖x(t) − y(t)‖, (3.1) whenever x(t) 6= y(t). put s(t) := ‖x(t) − y(t)‖, a function which is lipschitz continuous on i, as the composition of two lipschitz mappings. let t be in the set of full measure in which s ′ (t), x ′ (t), and y ′ (t) exist and for which c(t) is r(t)-prox-regular. then s ′ (t) =        〈 x ′ (t) − y′(t),x(t) − y(t) 〉 ‖x(t) − y(t)‖ , if x(t) 6= y(t) 0, otherwise. thus, the relation (3.1) ensures for a. e. t ∈ i s ′ (t) ≤ ‖f(t) − g(t)‖ + 2δ r(t) s(t). 62 messaoud bounkhel cubo 10, 1 (2008) we rewrite this inequality in the form ( s ′ (t) − 2δ r(t) s(t) ) e −p(t) ≤ ‖f(t) − g(t)‖e−p(t), where p(t) = ∫ t 0 2δ r(τ) dτ. as the left side is the derivative of the function t 7→ s(t)e−p(t), we can write s(t)e −p(t) − s(0) ≤ ∫ t 0 ‖f(τ) − g(τ)‖e−p(τ )dτ and then ‖x(t) − y(t)‖ = s(t) ≤ ‖x(0) − y(0)‖ep(t) + ∫ t 0 ‖f(τ) − g(τ)‖ep(t)−p(τ )dτ. this completes the proof. 2 now, we are ready to prove theoerem 4.2. proof of theoerem 4.2. let α > 0 such that ψ is lipschitz on x0 + αib with ratio l > 0. put γ(t) = ∫ t 0 2(l′+l) r(τ ) dτ and b = min{ α 2(eγ(t )l+l′) ,t}. now, we consider a sequence of mappings defined on i := [0,b] and prove that a subsequence converges to a solution of (nspp). for very n ∈ in put ink := [0, t n k ], t n k := kb n , k ∈ {1, . . . ,n} and we are going to construct fn,xn : i → h. pick y n 0 ∈ f(x0) and define fn on i n 1 = [0, b n ] by fn(t) = y n 0 for all t ∈ i n 1 . then consider the problem (sppn,0)    x ′ (t) ∈ −nc(t)(x(t)) + fn(t) a.e. on i n 1 ; x(t) ∈ c(t), ∀t ∈ in1 ; x(0) = x0. by proposition 4.1, problem (spn) has a unique solution xn ∈ ac(i n 1 ,h) with ‖xn(t)‖ ≤ l ′ + 2l. let y0 ∈ ac(i n 1 ,h) be the unique solution of (spn,0)    x ′ (t) ∈ −nc(t)(x(t)) a.e. on i n 1 ; x(t) ∈ c(t), ∀t ∈ in1 ; x(0) = x0. then proposition 4.2 ensures ‖xn(t) − y0(t)‖ ≤ e γ(t) ∫ t 0 ‖fn(τ)‖dτ for all t ∈ i n 1 . also ‖y0(t) − x0‖ ≤ ∫ t 0 ‖y′0(τ)‖dτ ≤ l ′ t for all t ∈ in1 . cubo 10, 1 (2008) arc-wise essentially tangentially regular ... 63 therefore, we get ‖xn(t) − x0‖ ≤ ‖xn(t) − y0(t)‖ + ‖y0(t) − x0‖ ≤ eγ(t ) ∫ t 0 ‖fn(τ)‖dτ + l ′ t ≤ (eγ(t )l + l′)t ≤ α 2n , which ensures that ‖xn(t) − x0‖ < α n on i n 1 . assume now that fn and xn have defined on the interval i n k and we will extend these mappings to the interval i n k+1, for all k ∈ {1, . . . ,n}. taking y n k ∈ f(xn(t n k )), we define fn on (t n k, t n k+1] by fn(t) = y n k . again let xn ∈ ac(i n k+1,h) be the unique solution of (sppn,k)    x ′ (t) ∈ −nc(t)(x(t)) + fn(t) a.e. on i n k+1; x(t) ∈ c(t), ∀t ∈ ink+1; x(0) = xn(t n k ). then as above we have ‖xn(t) − x0‖ < α(k+1) n on i n k+1. indeed, let yn,k ∈ ac(i n k+1,h) be the unique solution of (spn,k).    x ′ (t) ∈ −nc(t)(x(t)) a.e. on i n k+1; x(t) ∈ c(t), ∀t ∈ ink+1; x(0) = xn(t n k ). by proposition 4.2 we have ‖xn(t) − yn,k(t)‖ ≤ e γ(t) ∫ t 0 ‖fn(τ)‖dτ ≤ e γ(t ) l ′ t for all t ∈ in1 . also, we have for all t ∈ ink+1 ‖yn,k(t) − x0‖ ≤ ‖yn,k(t) − yn,k(0)‖ + ‖xn(t n k ) − x0‖ < ∫ t 0 ‖y′n,k(τ)‖dτ + kα n < l ′ t + kα n . therefore, we get for all t ∈ ink+1 ‖xn(t) − x0‖ < (e γ(t ) l ′ + l ′ )t + kα n < (k + 1)α 2n + kα n < (k + 1)α n . so we have obtained two sequences of mappings (fn)n and (xn)n, defined on i. let θn : i → i be defined by θn(t) = t n k, if t ∈ (t n k, t n k+1] and θn(t n 0 ) = 0. then by our construction we have x ′ n(t) ∈ −nc(t)(xn(t)) + f(xn(θn(t))) and ‖x ′ n(t)‖ ≤ l ′ + 2l a.e. on i and θn(t) → t for all t ∈ i . furthermore we have ‖xn(t) − x0‖ < α, for all t ∈ i. thus, theorem 4.1 completes the proof. 2 64 messaoud bounkhel cubo 10, 1 (2008) we close the paper with a direct and important corollary of theorem 4.2. it establishes an existence result for the following differential inclusion: (∗) { x ′ (t) ∈ −nc (x(t)) + f(x(t)) a.e. on [0,b]; x(t) ∈ c, ∀t ∈ [0,b]; x(0) = x0, first, we recall that this type of differential inclusion has been introduced by henry [24] for studying some economic problems. in the case when f is an u.s.c set-valued mapping, he proved an existence result of (∗) under the convexity assumption on the set c and on the images of the set-valued mapping f . this result has been extended by cornet [22] by assuming the tangential regularity assumption on the set c and the convexity on the images of f with the u.s.c of f . thibault in [29], proved an existence result of (∗) for any closed subset c (without any assumption on c), which also required the convexity of the images of f and the u.s.c. of f . recently, the author proved in [11], without any assumption of convexity on the images of f , the existence of solutions of (∗), but a heavy price was payed for the absence of the convexity. the price is the continuity of f and a standard tangential condition. noting that all the results mentioned above in [11, 24, 22, 29] are given in the finite dimensional setting. the question arises whether we can drop the assumption of convexity of the images of f , without assuming any tangential condition and without the continuity of f , and if possible in the infinite dimensional setting. our next corollary establishes a positive answer to this question. corollary 4.1 let r ∈]0, +∞] and c be a uniformly prox-regular set in h which is ball compact. let f : h⇉h be hausdorff u.s.c. on h contained in the subdifferential of a directionally regular locally lipschitz function ψ : h → ir. then for any x0 ∈ c there exists b ∈]0,t ] such that the nonconvex sweeping process with nonconvex noncontinuous perturbation (∗) has at least one solution. acknowledgement: the author would like to thank the referees for their careful and thorough reading of the paper. received: november 2006. revised: march 2007. references [1] f. ancona and g. colombo, existence of solutions for a class of non convex differential inclusions, rend. sem. mat. univ. padova, vol. 83 (1990), pp.71–76. 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[32] m. valadier, lignes de descente de fonctions lipschitziennes non pathologiques, sem. d’anal. convexe montpellier (1988) exposé n 0 . 9. arcwise2007.pdf a mathematical journal vol. 7, no 2, (81 85). august 2005. a note on discrete monotonic dynamical systems dongsheng liu 1 department of applied mathematics, nanjing university of science & technology nanjing, 210094, jiangsu, peoples r china d.liu@lancaster.ac.uk abstract we give a upper bound of lebesgue measure v (s(f, h, ω)) of the set s(f, h, ω) of points q ∈ qdh for which the triple (h, q, ω) is dynamically robust when f is monotonic and satisfies certain condition on some compact subset ω ∈ rd. resumen damos una cota superior de la medida de lebesgue v (s(f, h, ω)) del conjunto s(f, h, ω) de puntos q ∈ qdh para los cuales el tŕıo (h, q, ω) es dinámicamente robusto cuando f es monótona y satisface ciertas condiciones en algunos subconjuntos compactos ω ∈ rd. key words and phrases: roundoff operator, dynamical robustness, dynamical system. math. subj. class.: 37c70, 37c75 1 introduction a discrete dynamical system on the state space rd is generated by the iteration of a mapping f : rd → rd, that is xn+1 = f(xn),n = 0, 1, 2, · · · . 1current address: department of physics, lancaster university, lancaster, la1 4yb, uk. 82 dongsheng liu 7, 2(2005) let qdh denote the h-cube in r d centered at origin, that is qdh = {x = (x1,x2, · · · ,xd) ∈ rd : − h 2 < xi ≤ h 2 , i = 1, 2, · · · ,d} and for each q ∈ qdh, let lh,q = {q + hz : z ∈ zd} be the uniform h-lattice in rd centered at q. for q ∈ qdh, we define the roundoff operator [.]h,q from rd into lh,q by [x]h,q = lh,q ∩ (x + qdh) for x ∈ rd, or equivalently by [x]h,q = ([x1 −q1]h + q1, · · · , [xd −qd]h + qd) where x = (x1,x2, · · · .xd),q = (q1,q2, · · · ,qd) and [y]h is scalar roundoff operator defined by [y]h = kh if (k − 1 2 )h ≤ y < (k + 1 2 )h. let f be a dynamical system in rd. the map fh,q : lh,q → lh,q defined by fh,q(x) = [f(x)]h,q, x ∈ lh,q is called lh,q-discretization of f. now we give the definition of dynamical robustness [2]: given h > 0, q ∈ qdh, and a compact set ω ⊂ rd, we say the triple (h,q, ω) is dynamically robust if the discretization fh,q has a single equilibrium xh,q = fh,q(xh,q) ∈ ω ∩lh,q and lim n→∞ |fnh,q(x) −xh,q| = 0 ∀x ∈ lh,q ∩ ω. in [2] the following question was raised: given f and a compact set ω ∈ rd, what is the lebesgue measure v (s(f,h, ω)) of the set s(f,h, ω) of points q ∈ qdh for which the triple (h,q, ω) is dynamically robust? they answered this question partially: when ω is a parallel-polyhedron in rd, f is monotonic on ω and satisfies some condition, they give a lower bound for v (s(f,h, ω)). in this paper we give a upper bound of v (s(f,h, ω)) for f is monotonic and satisfies certain condition on some compact subset ω ∈ rd. 2 main results we give the semi-ordering in rd: for x,y ∈ rd, we say x ≤ y if xi ≤ yi for i = 1, 2, · · · ,d and x < y if xi < yi for i = 1, 2, · · · ,d. we shall say f is monotonically increasing on a set s ∈ rd if f(x) ≤ f(y) for all x,y ∈ s with x ≤ y. in the following we restrict attention to monotonically increasing functions, noting that the monotonic decreasing case is handled similarly. by the definition of dynamically robust, we have proposition 1 let f be monotonic on the compact set ω ⊂ rd and suppose that (h,q, ω) is a dynamically robust, xh,q is the single equilibrium. ∀x ∈ lh,q ∩ ω, if x ≥ xh,q, fh,q(x) = xh,q; if x ≤ xh,q, there exists k ∈ n such that fkh,q(x) = xh,q. 7, 2(2005) a note on discrete monotonic dynamical systems 83 proof. because ω is a compact subset of rd, lh,q ∩ω is a finite set. for x ∈ lh,q ∩ω, if x ≤ xh,q, let fh,q(x) = x(1), then x(1) = fh,q(x) ≤ fh,q(xh,q) = xh,q. if x(1) = xh,q it is proved with k = 1. otherwise we consider x(2) := fh,q(x(1)) ≤ fh,q(xh,q) = xh,q. if x(2) = xh,q it is proved with k = 2. if x(2) = xh,q we can continue this process. but lh,q ∩ω is finite set there exists a k ∈ n such that fkh,q(x) = fh,q(x(k−1)) = xh,q. if x ≥ xh,q, and fh,q(x) = xh,q, by the monotonicty, fh,q(x) ≥ fh,q(xh,q) = xh,q. so |fnh,q(x)−xh,q| ≥ |fh,q(x)−xh,q| > 0 for any n ∈ n, it is contradiction to the definition of dynamically robust of (h,q, ω). so we have fh,q(x) = xh,q. in fact, ∀x ∈ lh,q ∩ ω there exists k ∈ n such that fkh,q(x) = xh,q. now we can estimate v (s(f,h, ω)). theorem 1 ω is a compact subset of rd and satisfies: ∀q ∈ qdh, there exist u1,q,u2,q ∈ lh,q ∩ω such that ∀x ∈ lh,q ∩ω, u1,q ≤ x ≤ u2,q. let f be monotonic on the compact set ω ⊂ rd and f(ω) ⊂ ω′ where ω′ ⊂ ω and satisfies ∀x = (x1, · · · ,xd) ∈ ∂ω, the boundary of ω, and ∀x′ = (x′1, · · · ,x′d) ∈ ω′, |xi −x′i| ≥ h2 , i = 1, 2, · · · ,d. we have v (s(f,h, ω)) ≤ l l− 1h d − 1 l− 1v ({x ∈ ω : x− h 2 ≤ f(x) < x + h 2 }) where l = l1×l2×···×ld and li is determined by following: let li = |max{xi : x = (x1, · · · ,xi, · · · ,xd) ∈ ω}−min{xi : x = (x1, · · · ,xi, · · · ,xd) ∈ ω}| and li = rh + p, 0 ≤ p < h then li = r + 1. proof. the method of this proof is following that in [2]. let f(h,q) = lh,q ∩{x : x− h 2 ≤ f(x) < x+ h 2 } and k(h,q) = #{f(h,q)}. in order to carry on proof, we need following lemma 1 [3]. ∫ qd h k(h,q)dq = v ({x : x− h 2 ≤ f(x) < x + h 2 }). we also need the following special case of the birkhoff-tarski theorem lemma 2 [1]. let g be a monotonic map of a finite set γ ∈ rd into itself. if g satisfies g(x) ≥ x or g(x) ≤ x for x ∈ γ, then the iterative sequence xn+1 = g(xn) with x0 = x converge to the fixed point g(x∗) = x∗ ∈ γ. remark : (1). we can get the fixed point by following: take any x ∈ γ with g(x) ≥ x or g(x) ≤ x and iterate xn+1 = g(xn) with x0 = x, because γ is finite, after a finite number of steps we can get a fixed point. (2). if fh,q has only one fixed point x∗, then x∗ = fkh,q(u1,q) and x ∗ = flh,q(u2,q) for some k,l ∈ n. since u1,q ≤ x ≤ u2,q it is easy to see fnh,q(x) = x∗, ∀x ∈ lh,q ∩ ω for large n ∈ n. the condition on f guarantees that fh,q is a mapping of lh,q ∩ ω into itself. the elements of fh,q are precisely the fixed points of fh,q. so it is easy to see k(h,q) ≥ 1 84 dongsheng liu 7, 2(2005) from the lemma 2 because fh,q(u1,q) ≥ u1,q. by the definition of dynamically robust and remark (2), we have q ∈ s(f,h, ω) if and only if k(h,q) = 1. so v (s(f,h, ω)) = v ({q : k(h,q) = 1}). but v ({q : k(h,q) = 1}) + v ({q : k(h,q) > 1}) = hd, and k(h,q) at most equal to l = l1 ×···×ld. by lemma 1, we have v ({x : x− h 2 ≤ f(x) < x + h 2 }) = ∫ qd h k(h,q)dq = v ({q : k(h,q) = 1}) + l∑ i=2 i×v ({q : k(h,q) = i}) ≤ v ({q : k(h,q) = 1}) + l×v ({q : k(h,q) > 1}) = v ({q : k(h,q) = 1}) + lhd −l×v ({q : k(h,q) = 1}) = lhd − (l− 1)v ({q : k(h,q) = 1}) = lhd − (l− 1)v (s(f,h, ω)). so, v (s(f,h, ω)) ≤ l l− 1h d − 1 l− 1v ({x ∈ ω,x− h 2 ≤ f(x) < x + h 2 }). under the condition of theorem 2, the result of theorem 1 in [2] still holds. combining with the theorem 1 in [2], we get corollary 1 under the condition of theorem 2 below we have max{0, 2hd −v ({x ∈ ω : x− h 2 ≤ f(x) < x + h 2 })}≤ v (s(f,h, ω)) ≤ l l−1h d − 1 l−1v ({x ∈ ω,x− h2 ≤ f(x) < x + h2}). remark : it is easy to see hd ≤ v ({x ∈ ω : x− h 2 ≤ f(x) < x + h 2 }) ≤ lhd. if f is not monotonic, the situation is complex. following we give a special example. for g is a map from ω into itself, we say x is a periodic point of g, if there exist n ∈ n such that gn(x) = x. the least n which satisfies gn(x) = x is called period of g at x. now we give the example. theorem 2 let f be a map from a compact set ω into ω′. where ω′ ⊂ ω and satisfies ∀x = (x1, · · · ,xd) ∈ ∂ω, the boundary of ω, and ∀x′ = (x′1, · · · ,x′d) ∈ ω′, |xi −x′i| ≥ h2 , i = 1, 2, · · · ,d. if ∀q ∈ qdh, fh,q has no periodic point with period more than 1, then v (s(f,h, ω)) ≤ l l− 1h d − 1 l− 1v ({x ∈ ω : x− h 2 ≤ f(x) < x + h 2 }) where l = l1×l2×···×ld and li is determined by following: let li = |max{xi : x = (x1, · · · ,xi, · · · ,xd) ∈ ω}−min{xi : x = (x1, · · · ,xi, · · · ,xd) ∈ ω}| and li = rh + p, 0 ≤ p < h then li = r + 1. 7, 2(2005) a note on discrete monotonic dynamical systems 85 proof. we note v ({q : k(h,q) = 0}) + v ({q : k(h,q) = 1}) + v ({q : k(h,q) > 1}) = hd, so v ({q : k(h,q) > 1}) ≤ hd −v ({q : k(h,q) = 1}). now we only need to prove following. lemma 3 q ∈ s(f,h, ω) if and only if k(h,q) = 1. proof. let q ∈ s(f,h, ω), but k(h,q) = 1. then k(h,q) = 0 or k(h,q) > 1. that means dynamical system fh,q has no equilibrium or has at least two distinct equilibria, it is contradition to q ∈ s(f,h, ω). if k(h,q) = 1, let xh,q is the unique fixed point of fh,q in lh,q ∩ω. ∀x1 ∈ lh,q ∩ω, the condition of f guarantee fh,q(x1) ∈ lh,q ∩ ω. let fh,q(x1) := x2, if x2 = x1, we consider fh,q(x2) := x3. if x3 = x2 then x3 = x1 since fh,q has no periodic point with period more than 1. we continue this process and get x1,x2, · · · ,∈ lh,q ∩ ω, which are pairwise distinct. but lh,q ∩ ω is finite, so after finite number of steps, say n steps, we have fnh,q(x1) = fh,q(xn ) = xn . but xh,q is the unique fixed point, we get xn = xh,q and fnh,q(x1) = xh,q. so f m h,q(x1) = xh,q for any m ≥ n, that is lim n→∞ fnh,q(x) = xh,q, ∀x ∈ lh,q ∩ ω. i.e., q ∈ s(f,h, ω). the next step of the proof is the same as that in theorem 2. received: june 2004. revised: august 2004. references [1] g. birkhoff, lattice theory, ams. colloq. publ. vol 25, amer.math. soc., providence (1967). [2] p. diamond, p. kloeden, v. kozyakin, and a. pokrovskii, monotonic dynamicl systems under spatial discretization, proc. of amer. math. soc, 126(7) (1998), 2169-2174. [3] m. g. kendall and p. a. p. moran, geometrical probability, c. griffin, london (1963). cubo a mathematical journal vol.11, no¯ 05, (39–49). december 2009 boundary stabilization of the transmission problem for the bernoulli-euler plate equation kaïs ammari département de mathématiques, faculté des sciences de monastir, 5019 monastir, tunisie email: kais.ammari@fsm.rnu.tn and georgi vodev département de mathématiques, umr 6629 du cnrs, université de nantes, 2 rue de la houssinière, bp 92208, fr-44322 nantes cedex 03, france email : vodev@math.univ-nantes.fr abstract in this paper we consider a boundary stabilization problem for the transmission bernoullieuler plate equation. we prove uniform exponential energy decay under natural conditions. resumen en este artículo consideramos un problema de estabilización en la frontera para la ecuación de bernoulli-euler plate. nosotros probamos decaimiento exponencial uniforme de la energia sobre condiciones naturales key words and phrases: transmission problem, boundary stabilization, bernoulli-euler plate equation. math. subj. class.: 35b05, 93d15, 93d20. 40 kaïs ammari and georgi vodev cubo 11, 5 (2009) 1 introduction and statement of results let ω1 ⊂ ω ⊂ r n, n ≥ 2, be strictly convex, bounded domains with smooth boundaries γ1 = ∂ω1, γ = ∂ω, γ1 ∩γ = ∅. then o = ω\ω1 is a bounded, connected domain with boundary ∂o = γ1 ∪γ. we are going to study the following mixed boundary value problem    (∂2t + c 2 ∆ 2 )u1(x,t) = 0 in ω1 × (0, +∞), (∂2t + ∆ 2 )u2(x,t) = 0 in o × (0, +∞), u1|γ1 = u2|γ1, ∂νu1|γ1 = ∂νu2|γ1, c∆u1|γ1 = ∆u2|γ1, c∂ν ∆u1|γ1 = ∂ν ∆u2|γ1, u2|γ = 0, ∆u2|γ = −a∂ν∂tu2|γ, u1(x, 0) = u 0 1(x), ∂tu(x, 0) = u 1 1(x) in ω1, u2(x, 0) = u 0 2(x), ∂tu(x, 0) = u 1 2(x) in o, (1.1) where c > 1 is a constant, ν denotes the inner unit normal to the boundary and a is a non-negative function on γ. we suppose that there exists a constant a0 > 0 such that a ≥ a0 on γ. (1.2) the controllability of the dynamical system determined by (1.1) without transmission (i.e. when ω1 = ∅) has been investigated by krabs, leugering and seidman [10], leugering [9], lasiecka and triggiani [5], ammari and khenissi [1]. with transmission the exact controllability has been established by liu and williams [11] in the case when the control is active on the part of boundary whereas the controlled part of the boundary is supposed to satisfy the lions geometric condition. in this paper we prove that, under (1.2), the solutions of (1.1) are exponentially stable in the energy space. the energy of a solution u = { u1 in ω1, u2 in o, of (1.1) at the time instant t is defined by e(t) = 1 2 ∫ ω ( |∂tu(x,t)| 2 + α2(x) |∆u(x,t)| 2 ) α(x)−1 dx, where α(x) = { c in ω1, 1 in o. the solution of (1.1) satisfies the energy identity e(t2) − e(t1) = − ∫ t2 t1 ∫ γ a |∂ν∂tu(x,t)| 2 dγ dt cubo 11, 5 (2009) boundary stabilization of the transmission ... 41 for all t2 > t1 ≥ 0, and therefore the energy is a nonincreasing function of the time variable t. introduce the hilbert space h = v × h, where h = l2(ω,α(x)−1dx) and the space v is definded as follows. on the hilbert space h consider the operator g defined by g ( u1 u2 ) = ( − c∆u1 − ∆u2 ) , ∀ ( u1 u2 ) ∈ d(g), with domain d(g) = { (u1,u2) ∈ h = l 2 (ω1,c −1dx) ⊕ l2(o) : u1 ∈ h 2 (ω1),u2 ∈ h 2 (o), u2|γ = 0, u1|γ1 = u2|γ1, ∂νu1|γ1 = ∂νu2|γ1 } . the operator g is a strictly positive self-adjoint one with a compact resolvent. set v = d(g) with norm ‖f‖v := ‖gf‖h . the solutions to (1.1) can be expressed by means of a semigroup on h as follows ( u ut ) = eita ( u0 u1 ) , where the operator a is defined by a ( u v ) = −i ( v − α2 ∆2u ) , ∀ ( u v ) ∈ d(a), with domain d(a) = { (u,v) ∈ h : (v, ∆2u) ∈ h, u|γ = 0, ∆u|γ = −a∂νv|γ, u1|γ1 = u2|γ1, ∂νu1|γ1 = ∂νu2|γ1, c ∆u1|γ1 = ∆u2|γ1, c∂ν ∆u1|γ1 = ∂ν ∆u2|γ1 } . using green’s formula it is easy to see that im 〈 a ( u v ) , ( u v )〉 h = ∫ γ a |∂νv| 2 dγ ≥ 0, ∀ ( u v ) ∈ d(a), (1.3) which in turn implies that a generates a continuous semigroup (e.g. see theorems 4.3 and 4.6 from [12, p.14-15]). let ρ(a) denote the resolvent set of a. since the resolvent of a is a compact operator on h, c \ ρ(a) is a discrete set of eigenvalues of a. it follows from (1.3) that c \ ρ(a) ⊂ {z ∈ c : im z ≥ 0}. moreover, using the carleman estimates of [3] one can conclude that, under the condition (1.2), the operator a has no eigenvalues on the real axis. our main result is the following theorem 1.1 assume (1.2) fulfilled. then there exist constants c,γ > 0 such that e(t) ≤ c e−γt e(0). (1.4) 42 kaïs ammari and georgi vodev cubo 11, 5 (2009) when ω1 = ∅ the estimate (1.4) follows from combining the results of [1] and [2] under the more natural assumptions that (1.2) holds only on some non-empty part γ0 of γ and that every generalized ray in ω hits γ0 at a non-diffractive point (see [2] for the definition and more details). therefore in the general case of transmission, the estimate (1.4) should hold true under less restrictive conditions, but to our best knoweledge no such results have been proved so far. one of the reasons for this is the fact that the classical flow for the transmission problem is quite complex. indeed, when a ray in o hits γ1 it splits into two rays a reflected one staying in o and another one entering into ω1. the picture is similar when a ray in ω1 hits γ1. in particular, there are rays which never reach the boundary γ where the dissipation is active. note that it is crucial for (1.4) to hold that c > 1. roughly speaking, what happens in this case is that the rays staying inside ω1 carry a negligible amount of energy. note that when c < 1 the estimate (1.4) does not hold anymore. indeed, in this case one can use the quasi-modes constructed in [13] (which are due to the existence of the so-called interior totally reflected rays and which are concentrated on γ1) to get quasi-modes for our problem, too. consequently, we conclude that when c < 1 there exist infinitely many eigenvalues {λj} of a such that 0 < im λj ≤ cn |λj| −n , ∀n ≫ 1, which is an obstruction to (1.4). to prove theorem 1.1 it suffices to show that (a − z)−1 : h → h is o(1) for z ∈ r, |z| ≫ 1, which implies that (a − z)−1 is analytic in im z ≤ c, c > 0. this is carried out in section 3 (see theorem 3.1) using the a priori estimates established in section 2. the advantage of this approach is that the problem of proving (1.4) is reduced to obtaining a priori estimates for the solutions to the corresponding stationary problem (see theorem 2.1). this in turn is easier than studying (1.1) directly because we can make use of the estimates for the stationary transmission problem obtained in [4] (under the assumptions that ω1 is strictly convex and c > 1). 2 a priori estimates let λ ≫ 1 and consider the problem    (−c2∆2 + λ4)u1 = v1 in ω1, (−∆2 + λ4)u2 = v2 in o, u1|γ1 = u2|γ1, ∂νu1|γ1 = ∂νu2|γ1, c∆u1|γ1 = ∆u2|γ1, c∂ν ∆u1|γ1 = ∂ν ∆u2|γ1, u2|γ = 0, ∆u2|γ = a ( −iλ2∂νu2|γ + f ) , (2.1) with functions v1 ∈ l 2 (ω1), v2 ∈ l 2 (o) and f ∈ l2(γ). in what follows in this section all sobolev spaces hs, s ≥ 0, will be equipped with the semi-classical norm (with a small parameter λ−1). this means that the semi-classical norm in hs is equivalent to the classical one times a factor λ−s. theorem 2.1 assume (1.2) fulfilled. then there exist constants c,λ0 > 0 so that for λ ≥ λ0 the solution to (2.1) satisfies the estimate ‖u1‖h3(ω1) + ‖u2‖h3(o) ≤ cλ −2 ‖v1‖l2(ω1) + cλ −2 ‖v2‖l2(o) + cλ −2 ‖f‖l2(γ) . (2.2) cubo 11, 5 (2009) boundary stabilization of the transmission ... 43 proof. we will first prove the following lemma 2.2 assume (1.2) fulfilled. then for every λ,µ > 0 we have the estimate ‖∂νu2|γ‖l2(γ) ≤ cµ‖u1‖l2(ω1) + cµ‖u2‖l2(o) +cλ−2µ−1 ‖v1‖l2(ω1) + cλ −2µ−1 ‖v2‖l2(o) + cλ −2 ‖f‖l2(γ) . (2.3) proof. applying green’s formula in ω1 and o, we get respectively im 〈 c−1v1,u1 〉 l2(ω1) = im 〈c∆u1|γ1,∂νu1|γ1〉l2(γ1) − im 〈c∂ν ∆u1|γ1,u1|γ1〉l2(γ1) , (2.4) im 〈v2,u2〉l2(o) = −im 〈∆u2|γ1,∂νu2|γ1〉l2(γ1) + im 〈∂ν ∆u2|γ1,u2|γ1〉l2(γ1) +im 〈∆u2|γ,∂νu2|γ〉l2(γ) . (2.5) summing up (2.4) and (2.5) and using the boundary conditions, we obtain the identity im 〈 c−1v1,u1 〉 l2(ω1) + im 〈v2,u2〉l2(o) = −λ2 〈a∂νu2|γ,∂νu2|γ〉l2(γ) + 〈f,a∂νu2|γ〉l2(γ) . (2.6) by (1.2) and (2.6), we conclude a0λ 2 ‖∂νu2|γ‖ 2 l2(γ) ≤ λ 2µ2 ‖u1‖ 2 l2(ω1) + λ2µ2 ‖u2‖ 2 l2(o) +λ−2µ−2 ‖v1‖ 2 l2(ω1) + λ−2µ−2 ‖v2‖ 2 l2(o) + cλ −2 ‖f‖ 2 l2(γ) , (2.7) which clearly implies (2.3). 2 it is easy to see that (2.2) follows from combining lemma 2.2 with the following proposition 2.3 there exist constants c,λ0 > 0 so that for λ ≥ λ0 the solution to (2.1) satisfies the estimate ‖u1‖h3(ω1) + ‖u2‖h3(o) ≤ cλ −2 ‖v1‖l2(ω1) + cλ −2 ‖v2‖l2(o) +c ‖∂νu2|γ‖l2(γ) + cλ −2 ‖f‖l2(γ) . (2.8) proof. clearly, the functions ũ1 = (λ 2 − c∆)u1, ũ2 = (λ 2 − ∆)u2 satisfy the equation    (c∆ + λ2)ũ1 = v1 in ω1, (∆ + λ2)ũ2 = v2 in o, ũ1|γ1 = ũ2|γ1, ∂νũ1|γ1 = ∂νũ2|γ1. (2.9) using the results of [4] we will prove the following 44 kaïs ammari and georgi vodev cubo 11, 5 (2009) proposition 2.4 there exist constants c,λ0 > 0 so that for λ ≥ λ0 the solution to (2.9) satisfies the estimate ‖ũ1‖h1(ω1) + ‖ũ2‖h1(o) ≤ cλ −1 ‖v1‖l2(ω1) + cλ −1 ‖v2‖l2(o) +c ‖ũ2|γ‖l2(γ) + cλ −1 ‖∂νũ2|γ‖l2(γ) . (2.10) proof. denote by d(x) the distance between x ∈ o and γ. choose a function χ ∈ c∞(rn), χ = 1 on ω1, χ = 0 on r n \ ω, so that supp [∆,χ] ⊂ {x ∈ o : d(x) ≤ δ}, where 0 < δ ≪ 1. choose also a function ϕ(t) ∈ c∞0 (r), 0 ≤ ϕ ≤ 1, ϕ(t) = 1 for |t| ≤ δ, ϕ(t) = 0 for |t| ≥ 2δ, dϕ(t)/dt ≤ 0 for t ≥ 0, and set ψ(x) = ϕ(d(x)). clearly, ψ = 1 on {x ∈ o : d(x) ≤ δ}. we have    (c∆ + λ2)ũ1 = v1 in ω1, (∆ + λ2)χũ2 = ṽ2 = χv2 + [∆,χ]ũ2 in r n \ ω1, ũ1|γ1 = χũ2|γ1, ∂νũ1|γ1 = ∂νχũ2|γ1. (2.11) by the results of [4] (see (4.2)) we have ‖ũ1‖h1(ω1) + ‖χũ2‖h1(rn\ω1) ≤ cλ −1 ‖v1‖l2(ω1) + cλ −1 ‖ṽ2‖l2(rn\ω1) ≤ cλ−1 ‖v1‖l2(ω1) + cλ −1 ‖v2‖l2(o) + c ‖ψũ2‖h1(o) . (2.12) since γ is strictly concave viewed from the interior, we have the following (see proposition 2.2 of [4]) proposition 2.5 there exist constants c,δ0 > 0 so that for 0 < δ ≤ δ0 we have the estimate ‖ψũ2‖h1(o) ≤ cλ −1 ∥∥(∆ + λ2)ũ2 ∥∥ l2(o) + c ‖ũ2|γ‖l2(γ) +cλ−1 ‖∂νũ2|γ‖l2(γ) + oδ(λ −1/2 ) ‖ũ2‖h1(o) . (2.13) combining (2.12) and (2.13), we get ‖ũ1‖h1(ω1) + ‖ũ2‖h1(o) ≤ cλ−1 ‖v1‖l2(ω1) + cλ −1 ‖v2‖l2(o) + cλ −1/2 ‖ũ2‖h1(o) +c ‖ũ2|γ‖l2(γ) + cλ −1 ‖∂νũ2|γ‖l2(γ) , (2.14) which clearly implies (2.10). 2 by green’s formula we have re 〈 c−1ũ1,u1 〉 l2(ω1) = c−1λ2 ‖u1‖ 2 l2(ω1) + ‖∇u1‖ 2 l2(ω1) + re 〈∂νu1|γ1,u1|γ1〉l2(γ1) , re 〈ũ2,u2〉l2(o) = λ 2 ‖u2‖ 2 l2(o) + ‖∇u2‖ 2 l2(o) − re 〈∂νu2|γ1,u2|γ1〉l2(γ1) cubo 11, 5 (2009) boundary stabilization of the transmission ... 45 +re 〈∂νu2|γ,u2|γ〉l2(γ) . summing up these identities and using that u2|γ = 0, we get re 〈 c−1ũ1,u1 〉 l2(ω1) + re 〈ũ2,u2〉l2(o) = c−1λ2 ‖u1‖ 2 l2(ω1) + ‖∇u1‖ 2 l2(ω1) + λ2 ‖u2‖ 2 l2(o) + ‖∇u2‖ 2 l2(o) . (2.15) it is easy to see that (2.15) implies the estimate ‖u1‖h2(ω1) + ‖u2‖h2(o) ≤ cλ −2 ‖ũ1‖l2(ω1) + cλ −2 ‖ũ2‖l2(o) . (2.16) applying the same arguments to the functions ∇u1 and ∇u2 we also get ∥∥λ−1∇u1 ∥∥ h2(ω1) + ∥∥λ−1∇u2 ∥∥ h2(o) ≤ cλ−2 ‖ũ1‖h1(ω1) + cλ −2 ‖ũ2‖h1(o) +ελ−1/2 ∥∥(λ−1∂ν )2u2|γ ∥∥ l2(γ) + cε−1λ−1/2 ∥∥λ−1∂νu2|γ ∥∥ l2(γ) , (2.17) for any ε > 0. on the other hand, by the trace theorem we have ∥∥λ−1∂νu2|γ ∥∥ l2(γ) ≤ cλ1/2 ‖u2‖h2(o) , (2.18) ∥∥(λ−1∂ν )2u2|γ ∥∥ l2(γ) ≤ cλ1/2 ‖u2‖h3(o) . (2.19) thus, combining (2.16)-(2.19) and taking ε small enough, independent of λ, we conclude ‖u1‖h3(ω1) + ‖u2‖h3(o) ≤ cλ −2 ‖ũ1‖h1(ω1) + cλ −2 ‖ũ2‖h1(o) . (2.20) by (2.10) and (2.20) we arrive at the estimate ‖u1‖h3(ω1) + ‖u2‖h3(o) ≤ cλ −3 ‖v1‖l2(ω1) + cλ −5/2 ‖v2‖l2(o) +cλ−1 ‖∂νu2|γ‖l2(γ) + cλ −2 ‖∆u2|γ‖l2(γ) + cλ −3 ‖∂ν ∆u2|γ‖l2(γ) . (2.21) it is easy to see now that (2.8) follows from combining (2.21) and the following lemma 2.6 we have the estimate λ−3 ‖∂ν ∆u2|γ‖l2(γ) ≤ cλ −1 ‖∂νu2|γ‖l2(γ) +cλ−2 ‖∆u2|γ‖l2(γ) + cλ −5/2 ‖v2‖l2(o) + cλ −1/2 ‖u2‖h3(o) . (2.22) proof. choose a function ψ ∈ c∞(rn) such that ψ = 1 on a small neighbourhood of rn \ ω, ψ = 0 outside another small neighbourhood of rn \ω. then the function w = (∆+λ2)ψu2 satisfies the equation    (−∆ + λ2)w = w̃ = −ψv2 − [∆ 2,ψ]u2 in ω, w|γ = ∆u2|γ, ∂νw|γ = ∂ν ∆u2|γ + λ 2∂νu2|γ. (2.23) 46 kaïs ammari and georgi vodev cubo 11, 5 (2009) clearly, to prove (2.22) it suffices to show the estimate λ−1 ‖∂νw|γ‖l2(γ) ≤ c ‖w|γ‖l2(γ) + cλ −3/2 ‖w̃‖l2(ω) . (2.24) by the trace theorem we have λ−1 ‖∂νw|γ‖l2(γ) + ‖w|γ‖l2(γ) ≤ cλ 1/2 ‖w‖h2(ω) . (2.25) on the other hand, by green’s formula re 〈w̃,w〉l2(ω) = λ 2 ‖w‖ 2 l2(ω) + ‖∇w‖ 2 l2(ω) + re 〈∂νw|γ,w|γ〉l2(γ) , we get ‖w‖h2(ω) ≤ cλ −2 ‖w̃‖l2(ω) + ελ −1/2 ∥∥λ−1∂νw|γ ∥∥ l2(γ) + cε−1λ−1/2 ‖w|γ‖l2(γ) (2.26) for any ε > 0. combining (2.25) and (2.26), and taking ε small enough, independent of λ, we obtain (2.24). 2 3 resolvent estimates we will show that theorem 2.1 implies the following theorem 3.1 under (1.2), we have the bound ∥∥(a − z)−1 ∥∥ h→h ≤ const, ∀z ∈ r. (3.1) proof. we will derive (3.1) from (2.2). clearly, it suffices to prove (3.1) for |z| ≫ 1. without loss of generality we may suppose that z > 0. let ( u v ) ∈ d(a) satisfy the equation (a − z) ( u v ) = ( f g ) ∈ h. clearly, this equation can be rewritten as follows ( −iv − zu iα2∆2u − zv ) = ( f g ) . hence the function u = { u1 in ω1, u2 in o, cubo 11, 5 (2009) boundary stabilization of the transmission ... 47 satisfies the equation    (−c2∆2 + z2)u1 = ig1 − izf1 in ω1, (−∆2 + z2)u2 = ig2 − izf2 in o, u1|γ1 = u2|γ1, ∂νu1|γ1 = ∂νu2|γ1, c∆u1|γ1 = ∆u2|γ1, c∂ν ∆u1|γ1 = ∂ν ∆u2|γ1, u2|γ = 0, ∆u2|γ = a (−iz∂νu2|γ + ∂νf2|γ) . (3.2) clearly, (3.1) is equivalent to the estimate ‖∆u‖l2(ω) + ‖v‖l2(ω) ≤ c ‖∆f‖l2(ω) + c ‖g‖l2(ω) . (3.3) to prove (3.3) we write u = ug + uf , v = vg + vf , where (a − z) ( ug vg ) = ( 0 g ) , (a − z) ( uf vf ) = ( f 0 ) . clearly, ug = (u g 1,u g 2) (resp. u f = (u f 1,u f 2 )) satisfies (3.2) with f ≡ 0 (resp. g ≡ 0). applying theorem 2.1 with λ = z1/2 and f ≡ 0 and recalling that vg = zug, we get ‖∆ug‖l2(ω) + ‖v g‖l2(ω) ≤ c ‖g‖l2(ω) , (3.4) with a constant c > 0 independent of z. it is easy to see that (3.3) would follow from (3.4) and the estimate ∥∥∆uf ∥∥ l2(ω) + ∥∥vf ∥∥ l2(ω) ≤ c ‖gf‖h , ∀f ∈ d(g), (3.5) with a constant c > 0 independent of z. in what follows we will derive (3.5) from theorem 2.1. choose a function φ ∈ c∞0 (r), φ = 1 on [1/2, 2], φ = 0 on (−∞, 1/3]∪[3, +∞). write f = f ♭ +f♮, where f♭ = φ(g/z)f, f♮ = (1 − φ)(g/z)f. we have uf = u♭ + u♮, vf = v♭ + v♮, where (a − z) ( u♭ v♭ ) = ( f♭ 0 ) , (a − z) ( u♮ v♮ ) = ( f♮ 0 ) . clearly, u♭ = (u♭1,u ♭ 2) (resp. u ♮ = (u ♮ 1,u ♮ 2)) satisfies (3.2) with g ≡ 0 and f = f ♭ (resp. f = f♮). applying theorem 2.1 with λ = z1/2 and f = ∂νf ♭ 2|γ leads to the estimate ∥∥∥∆u♭ ∥∥∥ l2(ω) + ∥∥∥v♭ ∥∥∥ l2(ω) ≤ ∥∥∥∆u♭ ∥∥∥ l2(ω) + z ∥∥∥u♭ ∥∥∥ l2(ω) + ∥∥∥f♭ ∥∥∥ l2(ω) ≤ cz ∥∥∥f♭ ∥∥∥ l2(ω) + c ∥∥∥∂νf♭2|γ ∥∥∥ l2(γ) ≤ cz ∥∥∥f♭ ∥∥∥ l2(ω) + c ∥∥∥f♭2 ∥∥∥ h2(o) ≤ cz ∥∥∥f♭ ∥∥∥ h + c ∥∥∥gf♭2 ∥∥∥ h ≤ c ∥∥zg−1φ(g/z) ∥∥ h→h ‖gf‖h +c ‖φ(g/z)‖h→h ‖gf‖h ≤ c ‖gf‖h , (3.6) 48 kaïs ammari and georgi vodev cubo 11, 5 (2009) with a constant c > 0 independent of z, where the sobolev space h2(o) is equipped with the classical norm and we have used the trace theorem together with the fact that the norms on h and l2(ω) are equivalent. we would like to get a similar estimate for the functions u♮ and v♮. set u = (z − g)−1(z + g)−1(−izf♮) = −iz(z − g)−1(z + g)−1(1 − φ)(g/z)f. since the operator g−1 is bounded on h, we have ‖u‖h ≤ c ‖gu‖h ≤ cz −1 ‖gf‖h (3.7) with a constant c > 0 independent of z. it is easy to see that the function u = (u1,u2) satisfies the equation    (−c2∆2 + z2)u1 = −izf ♮ 1 in ω1, (−∆2 + z2)u2 = −izf ♮ 2 in o, u1|γ1 = u2|γ1, ∂νu1|γ1 = ∂νu2|γ1, c∆u1|γ1 = ∆u2|γ1, c∂ν ∆u1|γ1 = ∂ν ∆u2|γ1, u2|γ = 0, ∆u2|γ = 0. hence the function w = u♮ − u satisfies the equation    (−c2∆2 + z2)w1 = 0 in ω1, (−∆2 + z2)w2 = 0 in o, w1|γ1 = w2|γ1, ∂νw1|γ1 = ∂νw2|γ1, c∆w1|γ1 = ∆w2|γ1, c∂ν ∆w1|γ1 = ∂ν ∆w2|γ1, w2|γ = 0, ∆w2|γ = a ( −iz∂νw2|γ + ∂νf ♮ 2|γ − iz∂νu2|γ ) . therefore, by theorem 2.1 we obtain ∥∥∆u♮ − ∆u ∥∥ l2(ω) + z ∥∥u♮ − u ∥∥ l2(ω) ≤ c ∥∥∥∂νf♮2|γ ∥∥∥ l2(γ) + cz ‖∂νu2|γ‖l2(γ) ≤ c ∥∥∥f♮2 ∥∥∥ h2(o) + cz ‖u2‖h2(o) ≤ c ∥∥gf♮ ∥∥ h + cz ‖gu‖h . (3.8) by (3.7) and (3.8) we conclude ∥∥∆u♮ ∥∥ l2(ω) + ∥∥v♮ ∥∥ l2(ω) ≤ ∥∥∆u♮ ∥∥ l2(ω) + z ∥∥u♮ ∥∥ l2(ω) + ∥∥f♮ ∥∥ l2(ω) ≤ c ‖gf‖h . (3.9) clearly, (3.5) follows from (3.6) and (3.9). 2 received: march, 2009. revised: april, 2009. cubo 11, 5 (2009) boundary stabilization of the transmission ... 49 references [1] ammari, k. and khenissi, m., decay rates of the plate equations, math. nachr., 278 (2005), 1647–1658. [2] bardos, c., lebeau, g. and rauch, j., sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, siam j. control optim., 305 (1992), 1024–1065. [3] bellassoued, m., carleman estimates and distribution of resonances for the transparent obstacle and applications to the stabilization, asymptot. analysis, 35 (2003), 257–279. [4] cardoso, f., popov, g. and vodev, g., distribution of resonances and local energy decay in the transmission problem. ii, math. res. lett., 6 (1999), 377–396. 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[13] popov, g. and vodev, g., resonances near the real axis for transparent obstacles, commun. math. phys., 207 (1999), 411–438. b3-a-v cubo a mathematical journal vol.10, n o ¯ 01, (11–18). march 2008 error inequalities for a taylor-like formula nenad ujević department of mathematics university of split teslina 12/iii, 21000 split – croatia email: ujevic@pmfst.hr abstract a taylor-like formula is derived. various error bounds for this formula are established. resumen se deduce una formula de tipo taylor. se establecen varias cotas de error para esta formula. key words and phrases: error inequalities, taylor formula. math. subj. class.: 26d10. 12 nenad ujević cubo 10, 1 (2008) 1 introduction in recent years a number of authors have considered the taylor and generalized taylor formulas from an inequalities point of view. for example, this topic is considered in [1], [2], [3], [4], [5], [6] and [8]. in [5] we can find the following generalization of taylor formula: f (x) = f (a) + n ∑ k=1 (−1)k+1 [ pk(x)f (k) (x) − pk(a)f (k) (a) ] + rn(f, a, x), (1) rn(f, a, x) = (−1) n ∫ x a pn(t)f (n+1) (t)dt, where{pk(t)} ∞ 0 is a harmonic (or appell) sequence of polynomials, that is p ′ k(t) = pk−1(t), p0(t) = 1. if we substitute pk(t) = (t − x)k k! in (1) then we get the classical taylor formula: f (x) = f (a) + n ∑ k=1 (x − a)k k! f (k) (a) + r c n (f, a, x), r c n (f, a, x) = 1 n! ∫ x a (x − t)nf (n+1)(t)dt. in this paper we derive a taylor-like formula. a way of obtaining this formula is similar to the way described in [5]. however, here we do not use an appell sequence of polynomials. we use functions of the form sn(t) = { pn(t), t ∈ [ a, a+x 2 ] qn(t), t ∈ ( a+x 2 , x ] , where pn(t) and qn(t) are appell-like sequences of polynomials. we also establish various error bounds for this formula. similar error inequalities are established in [7] for some quadrature rules. finally, we give an application of the mentioned taylor-like formula to logarithmic function. 2 main results theorem 1 let f : [a, x] → r be a function such that f (n) is absolutely continuous. then f (x) = f (a) − n ∑ k=1 (−1)k(x − a)k 4kk! (1 + k) [ f (k) (x) − (−1)kf (k)(a) ] (2) cubo 10, 1 (2008) error inequalities for a taylor-like formula 13 − n ∑ k=2 (−1)k(x − a)k 4kk! (1 − k) [ 1 − (−1)k ] f (k) ( a + x 2 ) + r(f ), where r(f ) = (−1)n ∫ x a sn(t)f (n+1) (t)dt (3) and sn(t) =    (t− 3a+x 4 )n−1 n! [ t + (n−3)a−(n+1)x 4 ] , t ∈ [ a, a+x 2 ] (t− a+3x )n−1 n! [ t + (n−3)x−(n+1)a 4 ] , t ∈ ( a+x 2 , x ] . (4) proof. we prove (2) by induction. we easily show that (2) holds for n = 1. now suppose that (2) holds for an arbitrary n. we have to prove that (2) holds for n → n + 1. to simplify the proof we introduce the notations pn(t) = (t − 3a+x 4 ) n−1 n! [ t + (n − 3)a − (n + 1)x 4 ] (5) qn(t) = (t − a+3x 4 ) n−1 n! [ t + (n − 3)x − (n + 1)a 4 ] . (6) we see that pn and qn form appell sequences of polynomials, that is p ′ n(t) = pn−1(t), q ′ n(t) = qn−1(t), p0(t) = q0(t) = 1. we have (−1)n+1 ∫ x a sn+1(t)f (n+2) (t)dt = (−1)n+1 ∫ a+x 2 a pn+1(t)f (n+2) (t)dt + (−1)n+1 ∫ x a+x 2 qn+1(t)f (n+2) (t)dt = (−1)n+1 [ pn+1( a + x 2 )f (n+1) ( a + x 2 ) − pn+1(a)f (n+1) (a) ] +(−1)n+1 [ qn+1(x)f (n+1) (x) − qn+1( a + x 2 )f (n+1) ( a + x 2 ) ] +(−1)n ∫ a+x 2 a pn(t)f (n+1) (t)dt + (−1)n ∫ x a+x 2 qn(t)f (n+1) (t)dt = (−1)n ∫ x a sn(t)f (n+1) (t)dt + (−1)n+1 [ pn+1( a + x 2 ) − qn+1( a + x 2 ) ] f (n+1) ( a + x 2 ) −(−1)n+1 [ pn+1(a)f (n) (a) − qn+1(x)f (n) (x) ] = − ∫ x a f ′ (t)dt + n ∑ k=1 (−1)k(x − a)k 4kk! [ f (k) (x) − (−1)kf (k)(a) ] 14 nenad ujević cubo 10, 1 (2008) + n ∑ k=2 (−1)k(x − a)k 4kk! (1 − k) [ 1 − (−1)k ] f (k) ( a + x 2 ) +(−1)n+1 [ pn+1( a + x 2 ) − qn+1( a + x 2 ) ] f (n+1) ( a + x 2 ) −(−1)n+1 [ pn+1(a)f (n) (a) − qn+1(x)f (n) (x) ] = − ∫ x a f ′ (t)dt + n+1 ∑ k=1 (−1)k(x − a)k 4kk! [ f (k) (x) − (−1)kf (k)(a) ] + n+1 ∑ k=2 (−1)k(x − a)k 4kk! (1 − k) [ 1 − (−1)k ] f (k) ( a + x 2 ), since (−1)n+1 [ pn+1( a + x 2 ) − qn+1( a + x 2 ) ] f (n) ( a + x 2 ) −(−1)n+1 [ pn+1(a)f (n) (a) − qn+1(x)f (n) (x) ] = (−1)n+1(x − a)n+1 4n+1(n + 1)! (1 − n − 1) [ 1 − (−1)n+1 ] f (n+1) ( a + x 2 ) + (−1)n+1(x − a)n+1 4n+1(n + 1)! [ f (n+1) (x) − (−1)n+1f (n+1)(a) ] . this completes the proof. lemma 2 the functions sn(t) satisfy: ∫ x a sn(t)dt = 0, if n is odd, (7) ∫ x a |sn(t)| dt = (4n + 4)(x − a)n+1 4n+1(n + 1)! , (8) max t∈[a,x] |sn(t)| = (n + 1)(x − a)n 4nn! . (9) proof. a simple calculation gives ∫ x a sn(t)dt = (x − a)n+1 4n(n + 1)! [ 1 − (−1)n+1 ] . from the above relation we see that (7) holds, since 1 − (−1)n+1 = 0 if n is odd. cubo 10, 1 (2008) error inequalities for a taylor-like formula 15 we now consider some properties of the appell sequences of polynomials pn(t) and qn(t), given by (5) and (6), respectively. since t + (n − 3)a − (n + 1)x 4 ≤ 0, t ∈ [ a, a + x 2 ] and t + (n − 3)x − (n + 1)a 4 ≥ 0, t ∈ ( a + x 2 , x ] we easily show that the following facts are valid. if n is odd then pn(t) ≤ 0 and qn(t) ≥ 0. furthermore, pn(t) is an increasing function for t ∈ [ a, 3a+x 4 ) and it is a decreasing function for t ∈ ( 3a+x 4 , a+x 2 ] . the function qn(t) is decreasing for t ∈ [ a+x 2 , a+3x 4 ) and it is increasing for t ∈ ( 3a+3x 4 , x ] . if n is even then pn(t) is a decreasing function and qn(t) is an increasing function. furthermore, pn(t) > 0 for t ∈ [ a, 3a+x 4 ) and pn(t) < 0 for t ∈ ( 3a+x 4 , a+x 2 ] , while qn(t) < 0 for t ∈ [ a+x 2 , a+3x 4 ) and qn(t) > 0 for t ∈ ( 3a+3x 4 , x ] . we use these properties to prove (8) and (9). if n is odd then we have ∫ x a |sn(t)| dt = ∫ a+x 2 a |pn(t)| dt + ∫ x a+x 2 |qn(t)| dt = ∣ ∣ ∣ ∣ ∣ ∫ a+x 2 a pn(t)dt ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∫ x a+x 2 qn(t)dt ∣ ∣ ∣ ∣ ∣ = (4n + 4)(x − a)n+1 4n+1(n + 1)! . if n is even then we find that the same result is valid. thus (8) holds. finally, we have max t∈[a,x] |sn(t)| = max { max t∈[a, a+x 2 ] |pn(t)| , max t∈[ a+x 2 ,x] |qn(t)| } = max { ∣ ∣ ∣ ∣ pn( a + x 2 ) ∣ ∣ ∣ ∣ , ∣ ∣ ∣ ∣ qn( a + x 2 ) ∣ ∣ ∣ ∣ , |pn(a)| , |qn(x)| } = (n + 1)(x − a)n 4nn! . 16 nenad ujević cubo 10, 1 (2008) we introduce the notation f (x, a) = f (a) − n ∑ k=1 (−1)k(x − a)k 4kk! (1 + k) [ f (k) (x) − (−1)kf (k)(a) ] − n ∑ k=2 (−1)k(x − a)k 4kk! (1 − k) [ 1 − (−1)k ] f (k) ( a + x 2 ). theorem 3 let f : [a, x] → r be a function such that f (n) is absolutely continuous and there exist real numbers γn, γn such that γn ≤ f (n+1) (t) ≤ γn, t ∈ [a, x]. then |f (x) − f (x, a)| ≤ γn − γn (n + 1)! (2n + 2)(x − a)n+1 4n+1 if n is odd (10) and |f (x) − f (x, a)| ≤ (4n + 4)(x − a)n+1 4n+1(n + 1)! ∥ ∥ ∥ f (n+1) ∥ ∥ ∥ ∞ if n is even. (11) proof. let n be odd. from (3) and (7) we get r(f ) = (−1)n ∫ x a sn(t)f (n+1) (t)dt = (−1)n ∫ x a sn(t) [ f (n+1) (t) − γn + γn 2 ] dt such that we have |r(f )| = |f (x) − f (x, a)| ≤ max t∈[a,x] ∣ ∣ ∣ ∣ f (n+1) (t) − γn + γn 2 ∣ ∣ ∣ ∣ ∫ x a |sn(t)| dt. (12) we also have max t∈[a,x] ∣ ∣ ∣ ∣ f (n+1) (t) − γn + γn 2 ∣ ∣ ∣ ∣ ≤ γn − γn 2 . (13) from (12), (13) and (8) we get |f (x) − f (x, a)| ≤ γn − γn (n + 1)! (2n + 2)(x − a)n+1 4n+1 . let n be even. then we have |r(f )| = |f (x) − f (x, a)| ≤ ∫ x a |sn(t)| dt ∥ ∥ ∥ f (n+1) ∥ ∥ ∥ ∞ = (4n + 4)(x − a)n+1 4n+1(n + 1)! ∥ ∥ ∥ f (n+1) ∥ ∥ ∥ ∞ . theorem 4 let f : [a, x] → r be a function such that f (n) is absolutely continuous and let n be odd. if there exists a real number γn such that γn ≤ f (n+1) (t), t ∈ [a, x] then |f (x) − f (x, a)| ≤ (tn − γn) (n + 1)(x − a)n+1 4nn! , (14) cubo 10, 1 (2008) error inequalities for a taylor-like formula 17 where tn = f (n) (x) − f (n)(a) x − a . if there exists a real number γn such that f (n+1) (t) ≤ γn, t ∈ [a, x] then |f (x) − f (x, a)| ≤ (γn − tn) (n + 1)(x − a)n+1 4nn! . (15) proof. we have |r(f )| = |f (x) − f (x, a)| = ∣ ∣ ∣ ∣ ∫ x a (f (n+1) (t) − γn)sn(t)dt ∣ ∣ ∣ ∣ , since (7) holds. then we have ∣ ∣ ∣ ∣ ∫ x a (f (n+1) (t) − γn)sn(t)dt ∣ ∣ ∣ ∣ ≤ max t∈[a,x] |sn(t)| ∫ x a (f (n+1) (t) − γn)dt = (n + 1)(x − a)n 4nn! [ f (n) (x) − f (n)(a) − γn(x − a) ] = (n + 1)(x − a)n+1 4nn! (tn − γn) . in a similar way we can prove that (15) holds. remark 5 note that we can apply the estimations (10) and (11) only if f (n+1) is bounded. on the other hand, we can apply the estimation (14) if f (n+1) is unbounded above and we can apply the estimation (15) if f (n+1) is unbounded below. 3 an application to logarithmic function we now apply the formula (2) to logarithmic function. we have f (j) (t) = (−1)j (j − 1)! (1 + t)j if f (t) = ln(1 + t). (16) from (2), (16) and a = 0, f (t) = ln(1 + t) we get f (x) = − n ∑ k=1 (−1)kxk 4kk [ (1 + k) ( (−1)k+1 (1 + x)k + 1 ) + (−1)k+1(1 − k)(1 − (−1)k) (1 + x 2 )k ] (17) ≈ ln(1 + x), x ∈ ( − 4 5 , 4 ) . 18 nenad ujević cubo 10, 1 (2008) the standard formula for this function is given by s(x) = m ∑ k=1 (−1)k+1xk k ≈ ln(1 + x), x ∈ (−1, 1) . (18) many numerical examples show that n can be much less than m if we wish to obtain a prior given accuracy and if x is close to 1 (x < 1). let us choose x = 0.99 and give the accuracy of order e − 14. the ”exact” value is ln(1 + 0.99) = 0.688134643528734. if we use (17) with n = 22 then we get f (0.99) ≈ 0.688134643528725. if we use (18) with m = 5000 then we get s(0.99) ≈ 0.688134643528737. all calculations are done in double precision arithmetic. the first approximate result is obtained faster than the second one. similar results are obtained when we chose x = 0.9, x = 0.95, etc. received: may 2006. revised: august 2006. references [1] g.a. anastassiou and s.s. dragomir, on some estimates of the remainder in taylor’s formula, j. math. anal. appl., 263 (2001), 246–263. [2] p. cerone, generalized taylor’s formula with estimates of the remainder, in inequality theory and applications, vol 2, y. j. cho, j. k. kim and s. s. dragomir (eds.), nova sicence publ., new york, (2003), 33-52. [3] s.s. dragomir, new estimation of the remainder in taylor’s formula using grüss type inequalities and applications, math. inequal. appl., 2(2) (1999), 183–193. [4] h. gauchman, some integral inequalities involving taylor’s remainder i, j. inequal. pure appl. math., 3(2), article 26, (2002). [5] m. matić, j. pečarić and n. ujević, on new estimation of the remainder in generalized taylor’s formula, math. inequal. appl., 2(3), (1999), 343–361. [6] e. talvila, estimates of the remainder in taylor’s theorem using the hentstockkurzweil integral, czechoslovak math. j., 55(4), (2005), 933–940. [7] n. ujević and a.j. roberts, a corrected quadrature formula and applications, anziam j., 45 (e), (2004), e41-e56. [8] n. ujević, a new generalized perturbed taylor’s formula, nonlin. funct. anal. appl., 7(2), (2002), 255-267. tslanje.pdf articulo 16.dvi cubo a mathematical journal vol.12, no¯ 02, (261–274). june 2010 real and stable ranks for certain crossed products of toeplitz algebras takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, nishihara, okinawa 903-0213, japan email: sudo@math.u-ryukyu.ac.jp abstract we consider the algebraic structure of certain crossed products of the toeplitz algebra and its tensor products. using the structure, we estimate the stable rank and real rank of those crossed products. in particular, we obtain a real rank estimate for extensions of c∗-algebras. resumen consideramos la estructura algebraica de ciertos productos cruzados de algebra de toeplitz y sus productos tensoriales. usando la estructura estimamos el rango estable y el rango real de estos productos cruzados. en particular, obtenemos una estimativa del rango real para extensiones de c∗-algebras. key words and phrases: c*-algebra, crossed products, stable rank, real rank, toeplitz algebra. 2000 math. subj. class.: primary 46l05, 46l80 1 introduction crossed products of c∗-algebras (by automorphisms) have been very interesting research objects in the c∗-algebra theory. see [9] as a reference. as well, crossed products of c∗-algebras by endomorphisms 262 takahiro sudo cubo 12, 2 (2010) have been studied (rather recently). a typical and important example is given by the rotation c∗algebra, that is defined as the crossed product of c(t) by the rotation action of the group z of integers, where c(t) is the c∗-algebra of all continuous functions on the 1-torus t, and is also the universal c∗-algebra generated by a unitary. more generally, noncommutative tori are defined as successive crossed products by z. on the other hand, another example is given by the group c∗-algebra of the semi-direct product zn ⋊ z, that is viewed as the crossed product of c(tn) by the adjoint action of z, where tn is the n-torus. more generally, the group c∗-algebras of successive semi-direct products by z are viewed as successive crossed products by z. our first motivation is to replace c(t) with the toeplitz algebra f, that is the universal c∗algebra generated by an isometry, and replace z with the semigroup n of natural numbers (with zero), and consider the crossed product of f by n. furthermore, we replace c(tn) with ⊗nf the nfold tensor product of f and consider the crossed product of ⊗nf by n. while the crossed products of c(tk) by z, that are viewed as noncommutative manifolds, have been studied well, the replacements by isometries: the crossed products of ⊗nf by n, have not been studied explicitly yet. under those circumstances, in this paper we study certain crossed products of the toeplitz algebra and its tensor products. the algebraic structure of those crossed products is given explicitly (and inductively) in section 1. it is found that the crossed products have quotients that are isomorphic to the group c∗-algebras of generalized discrete ax + b groups that are defined and studied in [14], so that they may be viewed as the c∗-algebras of generalized discrete ax + b semigroups in a sense. (our first effort was to find such an analogue to the group c∗-algebras of the heisenberg discrete group, but this has not been successful yet.) the k-theory groups of the crossed products are computed by using the pimsner-voiculescu exact sequence. using the structure (and in part the k-theory results) obtained, we estimate the stable rank and connected stable rank of the crossed products in section 1, and estimate their real rank as well as the real rank of the group c∗-algebras of the generalized discrete ax + b groups in section 2. note that the stable and real ranks are viewed as noncommutative complex and real dimensions respectively. for estimating the real rank, we obtain a new real rank estimate for extensions of c∗-algebras. it turns out that this estimate is quite useful for estimating and determining the real rank of extensions of c∗-algebras. the stable rank, connected stable rank, and real rank formulae obtained for those crossed products and the real rank formulae for those group c∗-algebras are new, and the ranks are estimated with the dimension of the spaces of 2-dimensional irreducible representations that correspond to certain subquotients of the group c∗-algebras. in addition, a partial duality result on crossed products of c∗-algebras by n is obtained, which may be of some independent interest and would be useful for further research in a direction. notation. we denote by sr(a) the stable rank of a (unital) c∗-algebra a, and by csr(a) its connected stable rank. by definition, sr(a) ≤ n if and only if ln(a) is dense in a n, where (aj ) ∈ ln(a) if there exists (bj) ∈ a n such that ∑n j=1 bj aj = 1 ∈ a. also, csr(a) ≤ n if and only if lm(a) is connected for any m ≥ n. refer to [10]. we denote by rr(a) the real rank of a. by definition, rr(a) ≤ n − 1 if and only if ln(a)sa is dense in (asa) n, where ln(a)sa and asa are the sets of all self-adjoint elements of ln(a) and a respectively. refer to [3]. recall from [14] that the generalized discrete ax + b group that is a semi-direct product zn ⋊ z cubo 12, 2 (2010) real and stable ranks for certain crossed products of toeplitz algebras 263 is defined by the following (n + 1) × (n + 1) matrices: ( ⊕neπit s 0n 1 ) ∈ gln+1(z), where ⊕neπit means the n × n diagonal matrix with diagonal entries eπit for t ∈ z, and s ∈ zn (a column vector), and 0n = (0, · · · , 0) ∈ z n (a row vector). 2 structure and stable rank let f be the toeplitz algebra, that is defined to be the universal c∗-algebra generated by a (proper) isometry s. write f = c∗(s). definition 2.1. we define the crossed product of f by an action of n to be the universal c∗-algebra generated by f and an isometry t such that the action α of n on f is given by α1(x) = txt ∗ for x ∈ f. denote it by c∗(h1,1) = f ⋊α n and call it the c ∗-algebra of the discrete ax + b semigroup h1,1 since f ∼= c∗(n) the c∗-algebra of n, so that we may write h1,1 = n ⋊ n just as a symbol like a semi-direct product. it is well known that f has the decomposition into the exact sequence: 0 → k → f → c(t) → 0, where k is the c∗-algebra of compact operators on a separable hilbert space. furthermore, this k is isomorphic to the commutator ideal of f. refer to [6]. theorem 2.2. the c∗-algebra c∗(h1,1) = f ⋊α n has the decomposition into the exact sequence: 0 → k ⋊α n → f ⋊α n → c(t) ⋊α n → 0, moreover, k ⋊α n ∼= k ⊗ c(t) and c(t) ⋊α n ∼= c(t) ⋊α z a crossed product of c(t) by a unitary action of z, which is isomorphic to the group c∗-algebra of the discrete ax + b group z ⋊ z. proof. let x ∈ f = c∗(s). since xx∗ − x∗x is a compact operator, t(xx∗ − x∗x)t∗ = (txt∗)(tx∗t∗) − (tx∗t∗)(txt∗) is also compact. therefore, k is invariant under the action α of n. hence we obtain the exact sequence. furthermore, we have k ⋊α n ∼= k ⋊α z ∼= k ⊗ c(t), where the first isomorphism follows from that the action α on k is an automorphism as discussed above, and the second isomorphism follows from that any automorphism on k is implemented by a unitary, i.e. an adjoint action by a unitary, so that k ⋊α z ∼= k ⊗ c ∗(z), where c∗(z) is the group c∗-algebra of z, that is isomorphic to c(t) by the fourier transform. also, we have c(t)⋊α n ∼= c(t)⋊α z since the action α on c(t) by n must be an automorphism implemented by a unitary, which is isomorphic to the group c∗-algebra of the discrete ax + b group z ⋊ z. 2 264 takahiro sudo cubo 12, 2 (2010) remark. we may view this extension property as the definition for c∗(h1,1) = f⋊αn. by universality, there is a quotient map from f ⋊α n to c(t) ⋊α z. the similar remark as this can be made for the structure results given below. proposition 2.3. the k-theory groups of c∗(h1,1) are obtained as: kj (c ∗(h1,1)) ∼= z (j = 0, 1). proof. since c∗(h1,1) = f ⋊α n, we have the pimsner-voiculescu exact sequence of k-groups for crossed products of c∗-algebras by n (as well as z): z (id−α)∗ −−−−−→ z −−−−→ k0(c ∗(h1,1)) x     y k1(c ∗(h1,1)) ←−−−− 0 ←−−−− 0 (see [12] and [2]), where k0(f) ∼= z and k1(f) ∼= 0. since the map (id−α)∗ is trivial, where id is the identity map on f, we obtain kj(c ∗(h1,1)) ∼= z for j = 0, 1. 2 remark. the six-term exact sequence for the exact sequence obtained above is z i∗ −−−−→ k0(c ∗(h1,1)) q∗ −−−−→ z2 x     y z 2 q∗←−−−− k1(c ∗(h1,1)) i∗ ←−−−− z where kj (k⊗c(t)) ∼= kj(c(t)) ∼= z for j = 0, 1, and kj(c(t) ⋊α z) ∼= z 2 (j = 0, 1) by the (usual) pimsner-voiculescu exact sequence. consequently, the maps i∗ induced by the inclusion i : k⊗c(t) → c∗(h1,1) are zero, so that the maps q∗ induced by the quotient map q : c ∗(h1,1) → c(t) ⋊α z are injective. theorem 2.4. the stable rank of c∗(h1,1) is 2. the connected stable rank of c ∗(h1,1) is 2. proof. by [10, theorems 4.3, 4.4, and 4.11], we have the following estimates: sr(c∗(h1,1)) ≤ max{sr(k ⊗ c(t)), sr(c(t) ⋊ z), csr(c(t) ⋊ z)}, and max{sr(k ⊗ c(t)), sr(c(t) ⋊ z)}≤ sr(c∗(h1,1)). furthermore, by [10, theorems 3.6 and 6.4 and proposition 1.7] sr(k ⊗ c(t)) = sr(c(t)) = 1. note that c(t) ⋊ z ∼= c∗(z ⋊ z). by the stable rank and connected stable rank formulae in [14, remark 3.4] with a correction (see the remark below) we have sr(c∗(z ⋊ z)) = 2, and csr(c∗(z ⋊ z)) ≤ 2. the same estimates (from above, ≤ 2) for c(t) ⋊ z are also obtained by using [10, theorem 7.1 and corollary 8.6]. on the other hand, by [13, theorem 3.9] we have csr(c∗(h1,1)) ≤ max{csr(k ⊗ c(t)), csr(c(t) ⋊ z)}. cubo 12, 2 (2010) real and stable ranks for certain crossed products of toeplitz algebras 265 by [13, theorem 3.10], csr(k ⊗ c(t)) ≤ 2. since k1-group of c ∗(h1,1) is not trivial as shown above, we have csr(c∗(h1,1)) ≥ 2 (cf. [4, corollary 1.6]). 2 remark. the stable rank estimate in [14, remark 3.4] after a correction is sr(c0(r n+1) ⊗ m2(c)) = ⌈⌊(n + 1)/2⌋/2⌉+ 1 ≤ sr(c∗(zn ⋊ z)) ≤ csr(c0(r n+1) ⊗ m2(c)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1, and the connected stable rank estimate in it after that is csr(c∗(zn ⋊ z)) ≤ csr(c0(r n+1) ⊗ m2(c)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1, where c∗(zn ⋊ z) is the group c∗-algebra of the generalized ax + b group defined in [14], and ⌊x⌋ means the maximum integer ≤ x, and ⌈y⌉ is the least integer ≥ y. in particular, if n is odd, then sr(c∗(zn ⋊ z)) = ⌈⌊(n + 1)/2⌋/2⌉+ 1. furthermore, if n = 4m, then the inequality does not become equality, but if n = 4m + 2, then the inequality becomes equality. as for the correction, in fact, c0(r n−j+1) in [14, theorem 3.3] should have been replaced with c0(r n−j+2) (1 ≤ j ≤ n). as a note, the toeplitz algebra f has stable rank 2 and connected stable rank ≤ 2. this follows from using [10], [8], and [13] as above, where the result of [8] says that if the index map in the six-term exact sequence of k-groups for a c∗-algebra extension e is nonzero, then e can not have stable rank 1. let f ⊗ f be the c∗-tensor product of f, which is also defined to be the universal c∗-algebra generated by ∗-commuting isometries s1, s2, which means that each sj commutes with both si and s∗i (i 6= j). definition 2.5. we define the c∗-algebra of the (generalized) ax + b semigroup h2,1 = n 2 ⋊ n (just as a symbol like a semi-direct product) to be the universal c∗-algebra generated by f ⊗ f and an isometry t ⊗ t such that the (product) action α ⊗ α of n on f ⊗ f is given by (α ⊗ α)1(x ⊗ y) = (t ⊗ t)(x ⊗ y)(t ⊗ t)∗ = txt∗ ⊗ tyt∗ for x ⊗ y ∈ f ⊗ f. denote it by c∗(h2,1) = (f ⊗ f) ⋊α⊗α n the crossed product of f ⊗ f by α ⊗ α of n. in what follows, we often omit the symbol for actions in crossed products. proposition 2.6. the c∗-algebra c∗(h2,1) = (f ⊗ f) ⋊α⊗α n has the structure as follows: 0 → (f ⊗ k) ⋊ n → c∗(h2,1) → (f ⊗ c(t)) ⋊ n → 0 and the quotient and closed ideal have the decompositions as follows: 0 → (k ⊗ c(t)) ⋊ n → (f ⊗ c(t)) ⋊ n → c(t2) ⋊ n → 0, and 0 → (k ⊗ k) ⋊ n → (f ⊗ k) ⋊ n → (c(t) ⊗ k) ⋊ n → 0. furthermore, (k⊗c(t)) ⋊ n ∼= k⊗(c(t) ⋊ z), (c(t)⊗k) ⋊ n ∼= (c(t) ⋊ z)⊗k, and (k⊗k) ⋊ n ∼= k⊗c(t), and c(t2)⋊n ∼= c(t2)⋊z, which is isomorphic to the group c∗-algebra of the (generalized) discrete ax + b group z2 ⋊ z. 266 takahiro sudo cubo 12, 2 (2010) proof. the quotient and closed ideal, and their decompositions are deduced from the invariance of the action α ⊗ α. note that (k ⊗ c(t)) ⋊ n ∼= (k ⊗ c(t)) ⋊ z, and (c(t) ⊗ k) ⋊ n ∼= (c(t) ⊗ k) ⋊ z. furthermore, (k ⊗ k) ⋊ n is isomorphic to the following: (k ⊗ k) ⋊ z ∼= (k ⊗ k) ⊗ c(t) ∼= k ⊗ c(t). 2 proposition 2.7. the k-theory groups of c∗(h2,1) are obtained as: kj (c ∗(h2,1)) ∼= z (j = 0, 1). proof. since c∗(h2,1) = (f ⊗ f) ⋊α n, we have the pimsner-voiculescu sequence: z (id−α)∗ −−−−−→ z −−−−→ k0(c ∗(h2,1)) x     y k1(c ∗(h2,1)) ←−−−− 0 ←−−−− 0 where k0(f ⊗ f) ∼= z and k1(f ⊗ f) ∼= 0 by the künneth formula (see [2]). since the map (id − α)∗ is trivial, where id is the identity map on ⊗2f, we obtain kj (c ∗(h2,1)) ∼= z for j = 0, 1. 2 theorem 2.8. the stable rank of c∗(h2,1) is 2. the connected stable rank of c ∗(h2,1) is 2. proof. by [10, theorems 4.3, 4.4, and 4.11], we have the following estimates: sr(c∗(h2,1)) ≤ max{sr((f ⊗ k) ⋊ n), sr((f ⊗ c(t)) ⋊ n), csr((f ⊗ c(t)) ⋊ n)}, and max{sr((f ⊗ k) ⋊ n), sr((f ⊗ c(t)) ⋊ n)}≤ sr(c∗(h2,1)), and moreover, sr((f ⊗ c(t)) ⋊ n) ≤ max{sr(k ⊗ (c(t) ⋊ z)), sr(c(t2) ⋊ z), csr(c(t2) ⋊ z)}, and max{sr(k ⊗ (c(t) ⋊ z)), sr(c(t2) ⋊ z)}≤ sr((f ⊗ c(t)) ⋊ n), and sr((f ⊗ k) ⋊ n) ≤ max{sr(k ⊗ c(t)), sr((c(t) ⋊ z) ⊗ k), csr((c(t) ⋊ z) ⊗ k)}, and max{sr(k ⊗ c(t)), sr((c(t) ⋊ z) ⊗ k)}≤ sr((f ⊗ k) ⋊ n). furthermore, by [10, theorems 3.6 and 6.4] sr(k ⊗ (c(t) ⋊ z)) = sr(c(t) ⋊ z) = 2. note that c(t2) ⋊ z ∼= c∗(z2 ⋊ z). by the stable rank and connected stable rank formulae in [14, remark 3.4] with a correction (see the remark above) we have sr(c∗(z2 ⋊ z)) = 2, and csr(c∗(z2 ⋊ z)) ≤ 2. cubo 12, 2 (2010) real and stable ranks for certain crossed products of toeplitz algebras 267 on the other hand, by [13, theorem 3.9] we have csr(c∗(h2,1)) ≤ max{csr((f ⊗ k) ⋊ n), csr((f ⊗ c(t)) ⋊ n)}, and moreover, csr((f ⊗ k) ⋊ n) ≤ max{csr(k ⊗ (c(t) ⋊ z)), csr(c(t2) ⋊ z)}≤ 2, and csr((f ⊗ c(t)) ⋊ n) ≤ max{csr(k ⊗ c(t)), csr((c(t) ⋊ z) ⊗ k)}≤ 2. hence, it follows that csr(c∗(h2,1)) ≤ 2. therefore, sr(c ∗(h2,1)) = 2 is obtained from the first part of this proof. since k1-group of c ∗(h2,1) is not trivial as shown above, we have csr(c ∗(h2,1)) ≥ 2 (cf. [4, corollary 1.6]). 2 let ⊗nf be the n-fold c∗-tensor product of f, which is also defined to be the universal c∗-algebra generated by n ∗-commuting isometries sj (1 ≤ j ≤ n), which means that each sj commutes with both si and s ∗ i for any i 6= j. definition 2.9. we define the c∗-algebra of the (generalized) ax + b semigroup hn,1 = n n ⋊ n (just as a symbol like a semi-direct product) to be the universal c∗-algebra generated by ⊗nf and an isometry ⊗nt such that the (product) action ⊗nα of n on ⊗nf is given by (⊗nα)1(⊗ n j=1xj ) = (⊗nt)(⊗nj=1xj )(⊗ nt)∗ = ⊗nj=1txj t ∗ for ⊗nj=1xj ∈ ⊗ nf. denote it by c∗(hn,1) = (⊗ nf) ⋊⊗nα n the crossed product of ⊗nf by ⊗nα of n. proposition 2.10. the c∗-algebra c∗(hn,1) = (⊗ nf) ⋊⊗nα n has the structure: 0 → ((⊗n−1f) ⊗ k) ⋊ n → c∗(hn,1) → ((⊗ n−1 f) ⊗ c(t)) ⋊ n → 0, the exact sequence at the level 1 (that we call so), and the quotient and closed ideal have the decompositions as follows: 0 → ((⊗n−2f) ⊗ k ⊗ c(t)) ⋊ n → ((⊗n−1f) ⊗ c(t)) ⋊ n → ((⊗n−2f) ⊗ c(t2)) ⋊ n → 0, and 0 → ((⊗n−2f) ⊗ (⊗2k)) ⋊ n → ((⊗n−1f) ⊗ k) ⋊ n → ((⊗n−2f) ⊗ c(t) ⊗ k) ⋊ n → 0, the exact sequences at the level 2. inductively, the exact sequences at the level k (1 ≤ k ≤ n) have quotients and closed ideals that are given by ((⊗n−kf) ⊗ (⊗lk) ⊗ c(tk−l)) ⋊ n (0 ≤ l ≤ k) where ⊗0k = c and c(t0) = c. in particular, the exact sequences at the level n have quotients and closed ideals that are given by ((kl) ⊗ c(tn−l)) ⋊ n ∼=      k ⊗ c(t) (l = n), k ⊗ (c(tn−l) ⋊ z) (1 ≤ l ≤ n − 1), c(tn) ⋊ z (l = 0), and c(tn−l) ⋊ z is isomorphic to the group c∗-algebra of the (generalized) discrete ax + b group z n−l ⋊ z. 268 takahiro sudo cubo 12, 2 (2010) proof. note that ((kl) ⊗ c(tn−l)) ⋊ n is isomorphic to the following: ((kl) ⊗ c(tn−l)) ⋊ z ∼= (k l) ⊗ (c(tn−l) ⋊ z) ∼= k ⊗ (c(t n−l) ⋊ z). 2 proposition 2.11. the k-theory groups of c∗(hn,1) are obtained as: kj (c ∗(hn,1)) ∼= z (j = 0, 1). proof. since c∗(hn,1) = (⊗ nf) ⋊α n, we have the pimsner-voiculescu sequence: z (id−α)∗ −−−−−→ z −−−−→ k0(c ∗(hn,1)) x     y k1(c ∗(hn,1)) ←−−−− 0 ←−−−− 0 where k0(⊗ nf) ∼= z and k1(⊗ nf) ∼= 0 by the künneth formula (see [2]). since the map (id − α)∗ is trivial, where id is the identity map on ⊗nf, we obtain kj(c ∗(hn,1)) ∼= z for j = 0, 1. 2 theorem 2.12. the stable rank of c∗(hn,1) is ⌈⌊(n + 1)/2⌋/2⌉+ 1 if n 6= 4m, and if n = 4m, then m + 1 ≤ sr(c∗(hn,1)) ≤ m + 2. the connected stable rank of c∗(hn,1) is estimated as: 2 ≤ csr(c∗(hn,1)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1. proof. using the structure obtained for c∗(hn,1) above and [10, theorems 4.3, 4.4, and 4.11] repeatedly as before for estimating the stable rank, we obtain that sr(c∗(hn,1)) is estimated by      sr(k ⊗ c(t)) = 1, sr(k ⊗ (c(tn−l) ⋊ z)) ≤ 2, csr(k ⊗ (c(tn−l) ⋊ z)) ≤ 2, sr(c(tn) ⋊ z), and csr(c(tn) ⋊ z) (1 ≤ l ≤ n − 1). note that c(tn) ⋊ z ∼= c∗(zn ⋊ z). by the stable rank and connected stable rank formulae in [14, remark 3.4] with a correction (see the remark above) we have ⌈⌊(n + 1)/2⌋/2⌉+ 1 ≤ sr(c∗(zn ⋊ z)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1, and csr(c∗(zn ⋊ z)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1 where the stable rank estimate becomes equality if n 6= 4m. therefore, we obtain the stable rank estimates as in the statement. on the other hand, using the structure obtained for c∗(hn,1) above and [13, theorem 3.9] repeatedly as before for estimating the connected stable rank, we obtain that csr(c∗(hn,1)) is estimated by      csr(k ⊗ c(t)) ≤ 2 csr(k ⊗ (c(tn−l) ⋊ z)) ≤ 2 (1 ≤ l ≤ n − 1), csr(c(tn) ⋊ z). cubo 12, 2 (2010) real and stable ranks for certain crossed products of toeplitz algebras 269 hence, it follows that csr(c∗(hn,1)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1. since k1-group of c ∗(hn,1) is not trivial as shown above, we have csr(c∗(hn,1)) ≥ 2 (cf. [4, corollary 1.6]). 2 remark. as a note, the stable rank and connected stable rank of ⊗nf are estimated as: max{2, sr(c(tn))}max{2,⌊n/2⌋ + 1}≤ sr(⊗nf) ≤ csr(c(tn)) ≤⌈(n + 1)/2⌉ + 1, and csr(⊗nf) ≤ csr(c(tn)) ≤⌈(n + 1)/2⌉ + 1 using the structure of ⊗nf as above. 3 real rank theorem 3.1. for an exact sequece of c∗-algebras: 0 → i → a → a/i → 0, we obtain the following real rank estimate: rr(a) ≤ max{rr(i), rr(a/i), csr(a/i) − 1}. proof. let n be the maximum given above. we may assume that n is finite since if it is infinite, the estimate is automatic. let (aj ) n j=0 be an element of a n+1 with aj = a ∗ j . let u be an open neighborhood of (aj ) n j=0. let π : a → a/i be the quotient map. write π for the map a n+1 → (a/i)n+1 extended by π. then there exists an element (b′j ) n j=0 of the intersection π(u ) ∩ ln+1(a/i)sa such that (π(aj )) n j=0 is approximated closely by (b ′ j ) n j=0. note that ln+1(a/i)sa is a subest of ln+1(a/i). since csr(a/i) ≤ n + 1, there exists an invertible matrix s′ of gln+1(a/i)0 the connected component of gln+1(a/i) with the identity matrix such that s ′(b′j ) n j=0 = (1, 0, · · · , 0). then there exists a lift s(bj ) n j=0 of s ′(b′j ) n j=0, where s ∈ gln+1(a)0 and (bj ) ∈ u such that s(bj ) n j=0 = (1 + c0, c1, · · · , cn) ∈ (i∼)n+1 with each cj ∈ i. set s(bj ) n j=0 + (s(bj) n j=0) ∗ = (dj ) n j=0 ∈ (i ∼)n+1sa . since rr(i) ≤ n, we may assume that (dj ) n j=0 ∈ ln+1(i ∼)sa, where i ∼ is the unitization of i. indeed, the set of the elements (dj ) n j=0 ∈ a n+1 sa such that s(bj) n j=0 is mapped by π to an open neighborhood of (1, 0, · · ·0) is open relative to (i∼sa) n+1, i.e., its intersection with (i∼sa) n+1 is open in (i∼sa) n+1 since any element of in+1 is mapped to (0)nj=0 by π. note that ln+1(i ∼)sa ⊂ ln+1(a ∼), where a∼ = a if a is unital and a∼ is the unitization of a if a is non-unital. note also that (bj ) n j=0 + s −1(s(bj) n j=0) ∗ = s−1(dj ) n j=0 ∈ ln+1(a ∼) that is invariant under multiplication by elements of gln+1(a)0. by taking a deformation of s (or s−1) to the identity matrix in gln+1(a)0, it is concluded that (bj ) n j=0 + (b ∗ j ) n j=0 is in ln+1(a)sa, and belongs to u , as desired. 2 remark. this real rank estimate for extensions of c∗-algebras, obtained above will be very useful for computing the real rank of the extensions, as shown below. the estimate corresponds to the following of rieffel [10, theorem 4.11]: sr(a) ≤ max{sr(i), sr(a/i), csr(a/i)} which is often used in section 1. theorem 3.2. the real rank of the toeplitz algebra f is 1. 270 takahiro sudo cubo 12, 2 (2010) proof. since 0 → k → f → c(t) → 0, the estimate obtained in the theorem above implies rr(f) ≤ max{rr(k), rr(c(t)), csr(c(t))} = max{0, 1, 1} = 1. on the other hand, by [5, theorem 1.4], rr(f) ≥ max{rr(k), rr(c(t))} = 1. 2 remark. the same result as above is obtained as a corollary of [5, theorem 1.2], which says that for an extension of c∗-algebras: 0 → k → a → a/i → 0, we have rr(a) = rr(a/i). also, the result [7, proposition 1.6] implies that for an extension of c∗-algebras: 0 → i → a → a/i → 0, we have rr(a) ≤ max{rr(m (i)), rr(a/i)}, where m (i) is the multiplier algebra of i. it follows from this estimate that f has real rank 1 since m (k) ∼= b the c∗-algebra of bounded operators has real rank 0 [3]. however, the above estimate of [7] is not always useful since it involves the multiplier algebra, and it is hard to know its structure in general so that it is difficult to estimate its real rank in general. theorem 3.3. the real rank of c∗(z ⋊ z) of the discrete ax + b group is 1. proof. it is shown in [14] that c∗(z ⋊ z) has a composition series {ij} 3 j=1 of closed ideals, with i3 = c ∗(z ⋊ z) such that i3/i2 ∼= c(t) ⊕ c(t), i2/i1 ∼= c0(r) ⊗ m2(c), and i1 ∼= c0(r 2) ⊗ m2(c). using the real rank estimate obtained above, rr(i3) ≤ max{rr(i2), rr(i3/i2), csr(i3/i2) − 1}, and rr(i2) ≤ max{rr(i1), rr(i2/i1), csr(i2/i1) − 1}. also, by [5], rr(ij ) ≥ max{rr(ij−1), rr(ij /ij−1)} for j = 2, 3. by [13], csr(i3/i2) = csr(c(t)) = 2. by [11, theorem 4.7], csr(c0(r) ⊗ m2(c)) ≤⌈(csr(c0(r)) − 1)/2⌉ + 1 = 2. by [3, proposition 1.1], rr(i3/i2) = rr(c(t)) = 1. by [1], rr(c0(r) ⊗ m2(c)) ≥⌈dim[0, 1]/(2 · 2 − 1)⌉ = 1, while rr(c0(r) ⊗ m2(c)) ≤⌈dim s 1/(2 · 2 − 1)⌉ = 1, and rr(c0(r 2) ⊗ m2(c)) ≥⌈dim[0, 1] 2/(2 · 2 − 1)⌉ = 1, while rr(c0(r 2) ⊗ m2(c)) ≤⌈dim s 2/(2 · 2 − 1)⌉ = 1, where c([0, 1]), c([0, 1]2) are quotients of c0(r), c0(r 2) respectively, and s1, s2 are the one-point compactifications of r, r2 (i.e., 1 and 2-dimensional spheres) respectively (see also [15] and [7, proposition 5.1]). therefore, it follows that rr(ij ) = 1 = rr(ij /ij−1) for j = 1, 2, 3. 2 cubo 12, 2 (2010) real and stable ranks for certain crossed products of toeplitz algebras 271 theorem 3.4. the real rank of c∗(h1,1) is 1. proof. using the real rank estimate, the structure for c∗(h1,1) obtained above, and the theorem above, we obtain rr(c∗(z ⋊ z)) = 1 ≤ rr(c∗(h1,1)) ≤ max{rr(k ⊗ c(t)), rr(c∗(z ⋊ z)), csr(c∗(z ⋊ z)) − 1} = 1, where rr(k ⊗ c(t)) ≤ 1 by [1]. 2 theorem 3.5. the real rank of ⊗nf is n. proof. by using the real rank estimate and the (n-fold) structure for ⊗nf, that can be obtained inductively as above from the structure for f, it follows that the real rank of ⊗nf is estimated by { rr(c(tn)) = n, csr(c(tn)) − 1 ≤⌈(n + 1)/2⌉, rr(c(tk) ⊗ k) ≤ 1, and csr(c(tk) ⊗ k) − 1 ≤ 1 (0 ≤ k ≤ n − 1), where c(t0) = c. the conclusion is deduced as before. 2 theorem 3.6. the real rank of c∗(zn ⋊ z) of the generalized discrete ax + b group is rr(c∗(zn ⋊ z)) = ⌈(n + 1)/3⌉. proof. it is shown in [14] that c∗(zn ⋊ z) has a composition series {ij} n+1 j=1 of closed ideals, with in+1 = c ∗(zn ⋊ z) such that in+1/in is isomorphic to the 2 n-fold direct sum of c(t), and each subquotient ij /ij−1 for 1 ≤ j ≤ n (with i0 = {0}) is isomorphic to the combination ncn−j−1 fold direct sum of the following extension ej : 0 → c0(r n−j+2) ⊗ (⊕n−j+1m2(c)) → ej →⊕ n−j+1m2(c) → 0. using the real rank estimate obtained above, we obtain rr(in+1) ≤ max{rr(in), rr(c(t)), csr(c(t)) − 1}, and rr(ij ) ≤ max{rr(ij−1), rr(ej ), csr(ej ) − 1}, and rr(ej ) ≤ max{rr(c0(r n−j+2) ⊗ m2(c)), rr(m2(c)), csr(m2(c)) − 1} = max{⌈(n − j + 2)/(2 · 2 − 1)⌉, 0, 0} = ⌈(n − j + 2)/3⌉. also, by [5], rr(in+1) ≥ max{rr(in), rr(c(t))}, and rr(ij ) ≥ max{rr(ij−1), rr(ej )}, and rr(ej ) ≥⌈(n − j + 2)/3⌉. by [13], csr(ej ) ≤ csr(c0(r n−j+2) ⊗ m2(c)) ≤⌈⌊(n − j + 3)/2⌋/2⌉+ 1. 272 takahiro sudo cubo 12, 2 (2010) furthermore, rr(ij ) ≥ rr(i1) ≥ rr(c0(r n+1) ⊗ m2(c)) ≥ rr(c([0, 1]n+1) ⊗ m2(c)). by [1], it follows that rr(c([0, 1]n+1) ⊗ m2(c)) = ⌈(n + 1)/(2 · 2 − 1)⌉ = ⌈(n + 1)/3⌉. it follows that rr(ij ) = ⌈(n + 1)/3⌉ for 1 ≤ j ≤ n + 1. 2 theorem 3.7. the real rank of c∗(hn,1) is ⌈(n + 1)/3⌉. proof. using the structure obtained for c∗(hn,1) above and the real rank esitame for extensions of c∗-algebras obtained above, we obtain that rr(c∗(hn,1)) is estimated by      rr(k ⊗ c(t)) ≤ 1, rr(k ⊗ (c(tn−l) ⋊ z)) ≤ 1, csr(k ⊗ (c(tn−l) ⋊ z)) − 1 ≤ 1, rr(c(tn) ⋊ z), and csr(c(tn) ⋊ z) − 1 (1 ≤ l ≤ n − 1). note that c(tn) ⋊ z ∼= c∗(zn ⋊ z). moreover, we have obtained above that rr(c∗(zn ⋊ z)) = ⌈(n + 1)/3⌉, and csr(c∗(zn ⋊ z)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1. therefore, we obtain the real rank formula as in the statement. 2 remark. more applications by using the real rank estimate for extensions of c∗-algebras, obtained above, could be expected, when the extensions are given, where the real ranks of their closed ideal and quotients are computable. furthermore, if a c∗-algebra has a composition series of closed ideals such that the real ranks of its subquotients are computable, then the real rank of the c∗-algebra can be estimated by using the real rank formula. 4 a partial duality definition 4.1. let a ⋊α n be the crossed product of a (unital) c ∗-algebra a by an action α of n by an isometry s1. define the second (or dual) crossed product of a ⋊α n to be the crossed product a ⋊α n ⋊β n by a (dual) action β of n such that β is trivial on a, and β on c ∗(s1) generated by s1 is implemented by an isometry s2. proposition 4.2. the second crossed product b = a ⋊α n ⋊β n has the following decomposition: 0 → (a ⋊α n) ⊗ k → b → (a ⊗ c(t)) ⋊α⊗β n → 0, where the action β on c(t) is the adjoint action implemented by a unitary. cubo 12, 2 (2010) real and stable ranks for certain crossed products of toeplitz algebras 273 proof. note that a ⋊α n ⋊β n ∼= (a ⋊id n) ⋊(α,β) n, where id is the identity action of the second n, and the action β on c∗(s1) is exchanged by the action α on c∗(s2) implemented by s1. since a ⋊id n ∼= a ⊗f, we have (α, β) = α ⊗ β a product action, so that there exists the following exact sequence: 0 → (a ⊗ k) ⋊ n → (a ⊗ f) ⋊ n → (a ⊗ c(t)) ⋊ n → 0 where the action β on c(t) becomes a unitary action. furthermore, we obtain (a ⊗ k) ⋊ n ∼= (a ⋊ n) ⊗ k. 2 remark. k-theory for the closed ideal and quotient in the above exact sequence for the second crossed product b by n can be computed by the pimsner-voiculescu exact sequence: d −−−−→ d −−−−→ d ⋊ n x     y d ⋊ n ←−−−− d ←−−−− d where d ⋊ n is the crossed product of a unital c∗-algebra d by an action of n by a corner endomorphism ([12]). furthermore, by using the six-term exact sequence for extensions of c∗-algebras, k-theory of b can be determined when k-theory for a is computable. example 4.3. let on be the cuntz algebra generated by n isometries sj such that ∑n j=1 sjs ∗ j = 1 (see [2] for instance). it is well known that on ∼= mn∞ ⋊α n where mn∞ is the uhf algebra of type n∞, that is an inductive limit of tensor products ⊗kmn(c) (∼= mnk (c)). then the second crossed product b = mn∞ ⋊ n ⋊β n ∼= on ⋊ n has the decomposition: 0 → on ⊗ k → b → (mn∞ ⊗ c(t)) ⋊α⊗β n → 0. note that sr(on ⊗ k) = 2 since sr(on) = ∞ by [10, proposition 6.5]. also, by [10, theorem 5.1], mn∞ ⊗c(t) has stable rank 1 because it can be viewed as an inductive limit of matrix algebras over c(t). however, b has stable rank ∞ since the quotient of b has stable rank ∞ because the quotient has on as a quotient. indeed, mn∞ ⊗ c1 is invariant under the action of n. this shows that the second crossed product b can not be stable. received: march 2009. revised: july 2009. references [1] e.j. beggs and d.e. evans, the real rank of algebras of matrix valued functions, internat. j. math. 2 (1991), 131-138. [2] b. blackadar, k-theory for operator algebras, second edition, cambridge, (1998). 274 takahiro sudo cubo 12, 2 (2010) [3] l.g. brown and g.k. pedersen, c∗-algebras of real rank zero, j. funct. anal. 99 (1991), 131-149. [4] n. elhage hassan, rangs stables de certaines extensions, j. london math. soc. 52 (1995), 605-624. [5] n. elhage hassan, rang réel de certaines extensions, proc. amer. math. soc. 123 (1995), 3067-3073. [6] g.j. murphy, c∗-algebras and operator theory, academic press, (1990). [7] m. nagisa, h. osaka and n.c. phillips, ranks of algebras of continuous c∗-algebra valued functions, canad. j. math. 53 (2001), 979-1030. [8] v. nistor, stable rank for a certain class of type i c∗-algebras, j. operator theory 17 (1987), 365-373. [9] g.k. pedersen, c∗-algebras and their automorphism groups, academic press (1979). [10] m.a. rieffel, dimension and stable rank in the k-theory of c∗-algebras, proc. london math. soc. 46 (1983), 301-333. [11] m.a. rieffel, the homotopy groups of the unitary groups of non-commutative tori, j. operator theory 17 (1987), 237-254. [12] m. rørdam, classification of certain infinite simple c∗-algebras, j. funct. anal. 131, (1995), 415-458. [13] a.j-l. sheu, a cancellation theorem for projective modules over the group c∗-algebras of certain nilpotent lie groups, canad. j. math. 39 (1987), 365-427. [14] t. sudo, the structure of group c∗-algebras of some discrete solvable semi-direct products, hokkaido math. j. xxxiii(33), no. 3, (2004), 587-606. [15] t. sudo, real rank estimate by hereditary c∗-subalgebras by projections, math. scand. 100 (2007), 361-367. cubo a mathematical journal vol.10, n o ¯ 04, (15–26). december 2008 a strong convergence theorem by a new hybrid method for an equilibrium problem with nonlinear mappings in a hilbert space rinko shinzato and wataru takahashi department of mathematical and computing sciences, tokyo institute of technology, ohokayama, meguro-ku, tokyo 152-8552, japan emails: shinzato.l.aa@m.is.titech.ac.jp, wataru@is.titech.ac.jp abstract in this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for a monotone mapping and the set of fixed points of a nonexpansive mapping in a hilbert space by using a new hybrid method. using this theorem, we obtain three new results for finding a solution of an equilibrium problem, a solution of the variational inequality for a monotone mapping and a fixed point of a nonexpansive mapping in a hilbert space. resumen en este art́ıculo, probamos un teorema de convergencia fuerte para encontrar un elemento común del conjunto de soluciones de un problema de equilibrio; del conjunto de soluciones de una desigualdad variacional para una aplicación monótona y del conjunto de punto fijos de una aplicación no expansiva en un espacio de hilbert mediante el uso 16 rinko shinzato and wataru takahashi cubo 10, 4 (2008) de un nuevo método h́ıbrido. usando nuestro teorema obtenemos tres nuevos resultados para encontrar una solución de un problema de equiĺıbrio; una solución de la desigualdad variacional para una aplicación monótona y un punto fijo para una aplicación no expansiva en un espacio de hilbert. key words and phrases: hilbert space, equilibrium problem, nonexpansive mapping, inversestrongly monotone mapping, iteration, strong convergence theorem. math. subj. class.: 47h05, 47h09, 47j25. 1 introduction let h be a real hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖ and let c be a nonempty closed convex subset of h. let f be a bifunction from c × c to r, where r is the set of real numbers. the equilibrium problem for f : c × c → r is to find x̂ ∈ c such that f (x̂, y) ≥ 0 (1.1) for all y ∈ c. the set of such solutions x̂ is denoted by ep (f ). the problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, nash equilibrium problem in noncoopetative games and others; see, for instance, [1] and [6]. a mapping s of c into h is called nonexpansive if ‖sx − sy‖ ≤ ‖x − y‖ for all x, y ∈ c. we denote by f (s) the set of fixed points of s. a mapping a : c → h is called inverse-strongly monotone if there exists α > 0 such that 〈x − y, ax − ay〉 ≥ α‖ax − ay‖2 for all x, y ∈ c. the variational inequality problem is to find a u ∈ c such that 〈v − u, au〉 ≥ 0 (1.2) for all v ∈ c. the set of such solutions u is denoted by v i(c, a). setting a = i−s, where s : c →h is nonexpansive, we have from [14] that a : c → h is a 1 2 -inverse-strongly monotone mapping. recently, tada and takahashi [9, 10] and takahashi and takahashi [11] obtained weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space. in particular, tada and takahashi [10] established a strong convergence theorem for finding a common element of such two sets by using the hybrid method introduced in nakajo and takahashi [7]. on the other hand, takahashi and toyoda [16] introduced an iterative method for finding a common element of the set of solutions of the variational inequality for an inverse-strongly monotone mapping and cubo 10, 4 (2008) a strong convergence theorem ... 17 the set of fixed points of a nonexpansive mapping. very recently, takahashi, takeuchi and kubota [15] proved the following theorem by a new hybrid method which is different from nakajo and takahashi’s hybrid method. we call such a method the shrinking projection method. theorem 1.1 (takahashi, takeuchi and kubota [15]). let h be a hilbert space and let c be a nonempty closed convex subset of h. let t be a nonexpansive mapping of c into h such that f (t ) 6= ∅ and let x0 ∈ h. for c1 = c and u1 = pc1 x0, define a sequence {un} of c as follows:      yn = αnun + (1 − αn)t un, cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖un − z‖}, un+1 = pcn+1 x0, n ∈ n, where 0 ≤ αn ≤ a < 1. then, {un} converges strongly to z0 = pf (t )x0, where pf (t ) is the metric projection of h onto f (t ). in this paper, motivated by tada and takahashi [10], takahashi and toyoda [16], and takahashi, takeuchi and kubota [15], we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a hilbert space by using the shrinking projection method. using this theorem, we obtain three new results for finding a solution of an equilibrium problem, a solution of the variational inequality for an inverse-strongly monotone mapping and a fixed point of a nonexpansive mapping in a hilbert space. 2 preliminaries let h be a real hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖. we denote by “→” strong convergence and by “⇀” weak convergence. we know from [14] that, for all x, y ∈ h and λ ∈ [0, 1], there holds ‖λx + (1 − λ)y‖2 = λ‖x‖2 + (1 − λ)‖y‖2 − λ(1 − λ)‖x − y‖2. let c be a nonempty closed convex subset of h. for any x ∈ h, there exists a unique nearest point in c, denoted by pc x, such that ‖x − pc x‖ ≤ ‖x − y‖ for all y ∈ c. pc is called the metric projection of h onto c. we know that pc satisfies ‖pc x − pc y‖ 2 ≤ 〈pc x − pc y, x − y〉 (2.1) for all x, y ∈ h. further, we have that 〈x − pc x, pc x − y〉 ≥ 0 (2.2) 18 rinko shinzato and wataru takahashi cubo 10, 4 (2008) for all x ∈ h and y ∈ c. a mapping a : c → h is called inverse-strongly monotone if there exists α > 0 such that 〈x − y, ax − ay〉 ≥ α‖ax − ay‖2 for all x, y ∈ c. the set of solutions of the variational inequality for a is denoted by v i(c, a). we know that, for all λ > 0, u ∈ v i(c, a) ⇐⇒ u = pc (u − λau). we also know that, for any λ with 0 < λ ≤ 2α, a mapping i − λa : c → h is nonexpansive; see [16, 14] for more details. it is also known that h satisfies opial’s condition, i.e., for any sequence {xn} with xn ⇀ x, the inequality lim inf n→∞ ‖xn − x‖ < lim inf n→∞ ‖xn − y‖ holds for every y ∈ h with y 6= x. a hilbert space h also has the kadec-klee property, i.e., if {xn} is a sequence of h with xn ⇀ x and ‖xn‖ → ‖x‖, then there holds xn → x. a set-valued mapping t : h → 2h is called monotone if for all x, y ∈ h, f ∈ t x and g ∈ t y imply 〈x − y, f − g〉 ≥ 0. a monotone mapping t : h → 2h is maximal if the graph g(t ) of t is not properly contained in the graph of any other monotone mapping. it is known that a monotone mapping t is maximal if and only if for (x, f ) ∈ h × h, 〈x − y, f − g〉 ≥ 0 for every (y, g) ∈ g(t ) implies f ∈ t x. let a be an inverse-strongly monotone mapping of c into h and let nc v be the normal cone to c at v ∈ c, i.e., nc v = {w ∈ h : 〈v − u, w〉 ≥ 0, ∀u ∈ c}, and define t v = { av + nc v, v ∈ c, ∅, v /∈ c. then t is maximal monotone and 0 ∈ t v if and only if v ∈ v i(c, a); see [8]. for solving an equilibrium problem for a bifunction f : c × c → r, let us assume that f satisfies the following conditions: (a1) f (x, x) = 0 for all x ∈ c; (a2) f is monotone, i.e. f (x, y) + f (y, x) ≤ 0 for all x, y ∈ c; (a3) for all x, y, z ∈ c, lim sup t↓0 f (tz + (1 − t)x, y) ≤ f (x, y); (a4) for all x ∈ c, f (x, ·) is convex and lower semicontinuous. the following lemma appears implicitly in blum and oettlli [1]. lemma 2.1 (blum and oettli). let c be a nonempty closed convex subset of h and let f be a bifunction of c × c into r satisfying (a1) − (a4). let r > 0 and x ∈ h. then, there exists z ∈ c such that f (z, y) + 1 r 〈y − z, z − x〉 ≥ 0 f or all y ∈ c. cubo 10, 4 (2008) a strong convergence theorem ... 19 the following lemma was also given in [2]. lemma 2.2. assume that f : c × c → r satisfies (a1) − (a4). for r > 0 and x ∈ h, define a mapping tr : h → c as follows: tr(x) = { z ∈ c : f (z, y) + 1 r 〈y − z, z − x〉 ≥ 0 for all y ∈ c } for all x ∈ h. then, the following hold: (1) tr is single-valued; (2) tr is a firmly nonexpansive mapping, i.e., for all x, y ∈ h, ‖trx − try‖ 2 ≤ 〈trx − try, x − y〉; (3) f (tr) = ep (f ); (4) ep (f ) is closed and convex. 3 strong convergence theorem in this section, using the shrinking projection method, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a hilbert space. theorem 3.1. let c be a nonempty closed convex subset of a real hilbert space h. let f be a bifunction from c × c to r satisfying (a1) − (a4) and let s be a nonexpansive mapping from c into h and let a be an α-inverse-strongly monotone mapping of c into h such that f (s) ∩ v i(c, a) ∩ ep (f ) 6= ∅. let {xn} be a sequence in c generated by x0 = x ∈ c, c0 = c and          un = trn (xn), yn = αnxn + (1 − αn)spc (un − λnaun), cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1 x, n ∈ n ∪ {0}, where 0 ≤ αn ≤ c < 1, 0 < d ≤ rn < ∞ and 0 < a ≤ λn ≤ b < 2α. then, {xn} converges strongly to pf (s)∩v i(c,a)∩ep (f )x. proof. from [7], we know that ‖yn − z‖ ≤ ‖xn − z‖ ⇐⇒‖yn − xn‖ 2 + 2〈yn − xn, xn − z〉 ≤ 0. 20 rinko shinzato and wataru takahashi cubo 10, 4 (2008) so, cn is a closed convex subset of h for all n ∈ n ∪{0}. next we show by mathematical induction that f (s) ∩ v i(c, a) ∩ ep (f ) ⊂ cn for all n ∈ n ∪ {0}. put zn = pc (un − λnaun) for all n ∈ n ∪ {0}. from c0 = c, we have f (s) ∩ v i(c, a) ∩ ep (f ) ⊂ c0. suppose that f (s)∩v i(c, a)∩ep (f ) ⊂ ck for some k ∈ n∪{0}. let u ∈ f (s)∩v i(c, a)∩ep (f ). since i − λka and tr k are nonexpansive and u = pc (u − λkau), we have ‖zk − u‖ = ‖pc (uk − λkauk) − pc (u − λkau)‖ ≤ ‖(i − λka)uk − (i − λka)u‖ ≤ ‖uk − u‖ = ‖tr k xk − tr k u‖ ≤ ‖xk − u‖. so, we have ‖yk − u‖ = ‖αkxk + (1 − αk)szk − u‖ ≤ αk‖xk − u‖ + (1 − αk)‖szk − u‖ ≤ αk‖xk − u‖ + (1 − αk)‖zk − u‖ ≤ αk‖xk − u‖ + (1 − αk)‖xk − u‖ = ‖xk − u‖. since u ∈ ck, we have u ∈ ck+1. this implies that f (s) ∩ v i(c, a) ∩ ep (f ) ⊂ cn for all n ∈ n ∪ {0}. so, {xn} is well-defined. from the definition of xn+1, we have ‖xn+1 − x‖ ≤ ‖u − x‖ for all u ∈ f (s) ∩ v i(c, a) ∩ ep (f ) ⊂ cn+1. then, {xn} is bounded. therefore, {yn}, {zn}, {un} and {szn} are also bounded. let us show that ‖xn+1 − xn‖ → 0. from xn+1 ∈ cn+1 ⊂ cn and xn = pcn x, we have ‖xn − x‖ ≤ ‖xn+1 − x‖ for all n ∈ n ∪ {0}. thus {‖xn − x‖} is nondecreasing. thus limn→∞ ‖xn − x‖ exists. since ‖xn+1 − xn‖ 2 = ‖xn+1 − x‖ 2 + ‖xn − x‖ 2 + 2〈xn+1 − x, x − xn〉 = ‖xn+1 − x‖ 2 − ‖xn − x‖ 2 − 2〈xn − xn+1, x − xn〉 ≤ ‖xn+1 − x‖ 2 − ‖xn − x‖ 2 cubo 10, 4 (2008) a strong convergence theorem ... 21 for all n ∈ n ∪ {0}, we have limn→∞ ‖xn+1 − xn‖ = 0. since xn+1 ∈ cn+1, we have ‖xn − yn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − yn‖ ≤ 2‖xn − xn+1‖. this together with ‖xn+1 − xn‖ → 0 implies that ‖xn − yn‖ → 0. we also show that ‖aun − au‖ → 0. for all u ∈ f (s) ∩ v i(c, a) ∩ ep (f ), we have ‖zn − u‖ 2 = ‖pc (un − λnaun) − pc (u − λnau)‖ 2 ≤ ‖(un − λnaun) − (u − λnau)‖ 2 = ‖un − u − λn(aun − au)‖ 2 = ‖un − u‖ 2 − 2λn〈un − u, aun − au〉 + λ 2 n‖aun − au‖ 2 ≤ ‖un − u‖ 2 − 2λnα‖aun − au‖ 2 + λ2n‖aun − au‖ 2 = ‖un − u‖ 2 + λn(λn − 2α)‖aun − au‖ 2 ≤ ‖un − u‖ 2 + a(b − 2α)‖aun − au‖ 2 . since ‖ · ‖2 is convex and ‖un − u‖ ≤ ‖xn − u‖, we have ‖yn − u‖ 2 ≤ αn‖xn − u‖ 2 + (1 − αn)‖szn − u‖ 2 ≤ αn‖xn − u‖ 2 + (1 − αn){‖un − u‖ 2 + a(b − 2α)‖aun − au‖ 2} ≤ ‖xn − u‖ 2 + a(b − 2α)‖aun − au‖ 2. therefore, we have −a(b − 2α)‖aun − au‖ 2 ≤ ‖xn − u‖ 2 − ‖yn − u‖ 2 = (‖xn − u‖ + ‖yn − u‖)(‖xn − u‖ − ‖yn − u‖) ≤ (‖xn − u‖ + ‖yn − u‖)‖xn − yn‖. since {xn} and {yn} are bounded and ‖xn − yn‖ → 0, we obtain ‖aun − au‖ → 0. further we show that ‖zn − un‖ → 0. for all u ∈ f (s) ∩ v i(c, a) ∩ ep (f ), we have from (2.1) that ‖zn − u‖ 2 = ‖pc (un − λnaun) − pc (u − λnau)‖ 2 ≤ 〈(un − λnaun) − (u − λnau), zn − u〉 = 1 2 {‖(un − λnaun) − (u − λnau)‖ 2 + ‖zn − u‖ 2 − ‖(un − λnaun) − (u − λnau) − (zn − u)‖ 2} ≤ 1 2 {‖un − u‖ 2 + ‖zn − u‖ 2 − ‖(un − zn) − λn(aun − au)‖ 2} = 1 2 {‖un − u‖ 2 + ‖zn − u‖ 2 − ‖un − zn‖ 2 + 2λn〈un − zn, aun − au〉 − λ 2 n‖aun − au‖ 2}, 22 rinko shinzato and wataru takahashi cubo 10, 4 (2008) and hence ‖zn − u‖ 2 ≤ ‖un − u‖ 2 − ‖un − zn‖ 2 + 2λn〈un − zn, aun − au〉. from this inequality and ‖un − u‖ ≤ ‖xn − u‖, we have ‖yn − u‖ 2 ≤ αn‖xn − u‖ 2 + (1 − αn)‖zn − u‖ 2 ≤ αn‖xn − u‖ 2 + (1 − αn){‖un − u‖ 2 − ‖un − zn‖ 2 + 2λn〈un − zn, aun − au〉} ≤ ‖xn − u‖ 2 − (1 − αn)‖un − zn‖ 2 + 2λn(1 − αn)〈un − zn, aun − au〉, and hence (1 − αn)‖un − zn‖ 2 ≤ ‖xn − u‖ 2 − ‖yn − u‖ 2 + 2λn(1 − αn)〈un − zn, aun − au〉 ≤ (‖xn − u‖ + ‖yn − u‖)‖xn − yn‖ + 2λn(1 − αn)〈un − zn, aun − au〉. since 0 ≤ αn ≤ c < 1, ‖xn − yn‖ → 0 and ‖aun − au‖ → 0, we have that ‖un − zn‖ → 0. let us show ‖xn − un‖ → 0. for all u ∈ f (s) ∩ v i(c, a) ∩ ep (f ), we have from lemma 2.2 and f (trn ) = ep (f ) that ‖un − u‖ 2 = ‖trn xn − trn u‖ 2 ≤ 〈trn xn − trn u, xn − u〉 = 〈un − u, xn − u〉 = 1 2 {‖un − u‖ 2 + ‖xn − u‖ 2 − ‖un − xn‖ 2}, and hence ‖un − u‖ 2 ≤ ‖xn − u‖ 2 − ‖un − xn‖ 2. from this inequality and ‖zn − u‖ ≤ ‖un − u‖, we have ‖yn − u‖ 2 ≤ αn‖xn − u‖ 2 + (1 − αn)‖zn − u‖ 2 ≤ αn‖xn − u‖ 2 + (1 − αn){‖xn − u‖ 2 − ‖un − xn‖ 2}, and hence (1 − αn)‖un − xn‖ 2 ≤ ‖xn − u‖ 2 − ‖yn − u‖ 2 ≤ (‖xn − u‖ + ‖yn − u‖)‖xn − yn‖. therefore, we obtain ‖un − xn‖ → 0. cubo 10, 4 (2008) a strong convergence theorem ... 23 since (1 − αn)(szn − zn) = αn(zn − xn) + (yn − zn), we have (1 − αn)‖szn − zn‖ ≤ ‖zn − xn‖ + ‖yn − zn‖ ≤ ‖zn − xn‖ + ‖yn − xn‖ + ‖xn − zn‖ = 2‖zn − xn‖ + ‖yn − xn‖ ≤ 2(‖zn − un‖ + ‖un − xn‖) + ‖yn − xn‖. therefore, we also obtain ‖szn − zn‖ → 0. since {zn} is bounded, there exists a subsequence {zni} of {zn} such that zni ⇀ z0. then, we can obtain that z0 ∈ f (s) ∩ v i(c, a) ∩ ep (f ). in fact, let us first show z0 ∈ f (s). assume that z0 /∈ f (s). by opial’s condition, lim inf i→∞ ‖zni − z0‖ < lim inf i→∞ ‖zni − sz0‖ = lim inf i→∞ ‖zni − szni + szni − sz0‖ = lim inf i→∞ ‖szni − sz0‖ ≤ lim inf i→∞ ‖zni − z0‖. this is a contradiction. therefore, we have z0 ∈ f (s). let us show z0 ∈ v i(c, a). define t v = { av + nc v, v ∈ c, ∅, v /∈ c. then t is maximal monotone and t −10 = v i(c, a); see [8]. let (v, u) ∈ g(t ). since u−av ∈ nc v and zn = pc (un − λnaun) ∈ c, we have 〈v − zn, u − av〉 ≥ 0. by the definition of zn, we also have 〈v − zn, zn − (un − λnaun)〉 ≥ 0, and hence 〈v − zn, zn − un λn + aun〉 ≥ 0. therefore, 〈v − zni , u〉 ≥ 〈v − zni , av〉 ≥ 〈v − zni , av − { zni − uni λni + auni }〉 = 〈v − zni , av − azni〉 + 〈v − zni , azni − auni〉 − 〈v − zni , zni − uni λni 〉 ≥ −‖v − zni‖‖azni − auni‖ − ‖v − zni‖‖ zni − uni λni ‖. since ‖zn − un‖ → 0 and a is lipschits continuous, we have 〈v − z0, u〉 ≥ 0. since t is maximal monotone, we have z0 ∈ t −1 0 and hence z0 ∈ v i(c, a). finally, we show that z0 ∈ ep (f ). by un = trn xn, we have f (un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0 24 rinko shinzato and wataru takahashi cubo 10, 4 (2008) for all y ∈ c. from (a2) we also have 1 rn 〈y − un, un − xn〉 ≥ f (y, un) and hence 〈y − uni , uni − xni rni 〉 ≥ f (y, uni ). since ‖un − zn‖ → 0 and zni ⇀ z0, we have uni ⇀ z0. since 0 < d ≤ rn < ∞ and ‖un − xn‖ → 0, we have from (a4) that 0 ≥ f (y, z0) for all y ∈ c. for t ∈ (0, 1] and y ∈ c, let yt = ty + (1 − t)z0. since y ∈ c and z0 ∈ c, we have yt ∈ c and hence f (yt, z0) ≤ 0. so, from (a1) and (a4) we have 0 = f (yt, yt) ≤ tf (yt, y) + (1 − t)f (yt, z0) ≤ tf (yt, y) and hence 0 ≤ f (yt, y). from (a3), we have 0 ≤ f (z0, y) for all y ∈ c and hence z0 ∈ ep (f ). therefore z0 ∈ f (s) ∩ v i(c, a) ∩ ep (f ). from z′ = pf (s)∩v i(c,a)∩ep (f )x, z0 ∈ f (s) ∩ v i(c, a) ∩ ep (f ) and ‖xn − x‖ ≤ ‖z ′ − x‖, we have ‖z′ − x‖ ≤ ‖z0 − x‖ ≤ lim inf i→∞ ‖zni − x‖ ≤ lim sup i→∞ ‖zni − x‖ ≤ lim sup i→∞ {‖zni − uni‖ + ‖uni − xni‖ + ‖xni − x‖} ≤ ‖z′ − x‖. thus, we have lim i→∞ ‖zni − x‖ = ‖z0 − x‖ = ‖z ′ − x‖. this implies z0 = z ′ . further, since a hilbert space has the kadec-klee property, we have that zni → z ′ . from ‖zn − xn‖ → 0, we also have xni → z ′ . therefore, xn → z ′ . this completes the proof. 4 applications in this section, using theorem 3.1, we prove three new results for finding a solution of an equilibrium problem, a solution of the variational inequality for an inverse-strongly monotone mapping and a fixed point of a nonexpansive mapping in a hilbert space. first, we obtain a result for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space. theorem 4.1. let c be a nonempty closed convex subset of a real hilbert space h. let f be a bifunction from c × c to r satisfying (a1) − (a4) and let s be a nonexpansive mapping from c cubo 10, 4 (2008) a strong convergence theorem ... 25 into h such that f (s)∩ep (f ) 6= ∅. let {xn} be a sequence in c generated by x0 = x ∈ c, c0 = c and          un = trn (xn), yn = αnxn + (1 − αn)s(un), cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1 x, n ∈ n ∪ {0}, where 0 ≤ αn ≤ c < 1 and 0 < d ≤ rn < ∞. then, {xn} converges strongly to pf (s)∩ep (f )x. proof. putting a = 0 in theorem 3.1, we obtain the desired result. next, we obtain a result for finding a common element of the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a hilbert space. theorem 4.2. let c be a nonempty closed convex subset of a real hilbert space h. let f be a bifunction from c × c to r satisfying (a1) − (a4) and let a be an α-inverse-strongly monotone mapping of c into h such that v i(c, a) ∩ ep (f ) 6= ∅. let {xn} be a sequence in c generated by x0 = x ∈ c, c0 = c and          un = trn (xn), yn = αnxn + (1 − αn)pc (un − λnaun), cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1 x, n ∈ n ∪ {0}, where 0 ≤ αn ≤ c < 1, 0 < d ≤ rn < ∞ and 0 < a ≤ λn ≤ b < 2α. then, {xn} converges strongly to pv i(c,a)∩ep (f )x. proof. putting s = i in theorem 3.1, we obtain the desired result. finally, we obtain a result for finding a common element of the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a hilbert space. theorem 4.3. let c be a nonempty closed convex subset of a real hilbert space h. let s be a nonexpansive mapping from c into h and let a be an α-inverse-strongly monotone mapping of c into h such that f (s) ∩ v i(c, a) 6= ∅. let {xn} be a sequence in c generated by x0 = x ∈ c, c0 = c and      yn = αnxn + (1 − αn)spc (xn − λnaxn), cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1 x, n ∈ n ∪ {0}, where 0 ≤ αn ≤ c < 1 and 0 < a ≤ λn ≤ b < 2α. then, {xn} converges strongly to pf (s)∩v i(c,a)x. proof. putting f = 0 in theorem 3.1, we obtain the desired result. received: january 2008. revised: february 2008. 26 rinko shinzato and wataru takahashi cubo 10, 4 (2008) references [1] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, math. student, 63 (1994), 123–145. 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[18] r. wittmann, approsimation of fixed points of nonexpansive mappings, arch. math., 58 (1992), 486–461. n2-shinn-tak cubo a mathematical journal vol.10, n o ¯ 04, (101–108). december 2008 browder convergence and mosco convergence for families of nonexpansive mappings tomonari suzuki* department of mathematics, kyushu institute of technology, tobata, kitakyushu 804-8550, japan email: suzuki-t@mns.kyutech.ac.jp abstract we study the relationship between browder’s strong convergence and mosco convergence of fixed-point set for families of nonexpansive mappings. resumen estudiamos la relación entre la convergencia fuerte de browder y la convergencia de mosco del conjunto de puntos fijos para familias de aplicaciones no espansivas. key words and phrases: nonexpansive mapping, nonexpansive semigroup, fixed point, browder convergence, mosco convergence. math. subj. class.: 47h10, 47h09, 47h20. *the author is supported in part by grants-in-aid for scientific research from the japanese ministry of education, culture, sports, science and technology. 102 tomonari suzuki cubo 10, 4 (2008) 1. introduction let c be a subset of a banach space e. a mapping t on c is called a nonexpansive mapping if ‖tx − ty‖ ≤ ‖x − y‖ for all x,y ∈ c. we denote by f(t ) the set of fixed points of t . using the results in gossez and lami dozo [6] and kirk [8], we can prove that f(t ) is nonempty in the case where c is weakly compact, convex and has the opial property. see also [1, 5, 7] and others. in 1967, browder [2] proved the following strong convergence theorem, theorem 1 (browder [2]). let c be a bounded closed convex subset of a hilbert space e and let t be a nonexpansive mapping on c. let {αn} be a sequence in (0, 1) converging to 0. fix u ∈ c and define a sequence {xn} in c by xn = (1 − αn) txn + αn u for n ∈ n. then {xn} converges strongly to pu, where p is the metric projection from c onto f(t ). a family of mappings {t (t) : t ≥ 0} is called a one-parameter strongly continuous semigroup of nonexpansive mappings (nonexpansive semigroup, for short) on c if the following are satisfied: (ns1) for each t ≥ 0, t (t) is a nonexpansive mapping on c. (ns2) t (s + t) = t (s) ◦ t (t) for all s,t ≥ 0. (ns3) for each x ∈ c, the mapping t 7→ t (t)x from [0,∞) into c is strongly continuous. suzuki [15] proved that ⋂ t f(t (t)) is nonempty provided c is bounded closed convex and every nonexpansive mapping on c has a fixed point. he also proved a semigroup version of browder’s convergence theorem in [11, 17]. theorem 2 ([11, 17]). let e be a smooth banach space with the opial property such that the normalized duality mapping j of e is weakly sequentially continuous at zero. let c be a weakly compact convex subset of e. let {t (t) : t ≥ 0} be a nonexpansive semigroup on c. let τ be a nonnegative real number. let {αn} and {tn} be sequences in r satisfying 0 < αn < 1, 0 ≤ τ + tn and tn 6= 0 for n ∈ n, and limn tn = limn αn/tn = 0. fix u ∈ c and define a sequence {xn} in c by xn = (1 − αn) t (τ + tn)xn + αn u for n ∈ n. then {xn} converges strongly to pu, where p is the unique sunny nonexpansive retraction from c onto ⋂ t f(t (t)). motivated by theorem 2, suzuki [19] considered the mosco convergence of {f(t (τ + tn))}. the following theorem is a corollary of the main result in [19]. theorem 3 ([19]). let e, c and {t (t) : t ≥ 0} be as in theorem 2. let τ be a nonnegative real number and let {tn} be a sequence in r satisfying 0 ≤ τ +tn and tn 6= 0 for n ∈ n, and limn tn = 0. then {f(t (τ + tn))} converges to ⋂ t f(t (t)) in the sense of mosco. therefore we can guess that browder convergence is strongly connected with mosco convergence. in this paper, we study the relationship between browder convergence and mosco convergence for families of nonexpansive mappings. cubo 10, 4 (2008) browder convergence and mosco convergence ... 103 2. preliminaries throughout this paper we denote by n the set of all positive integers and by r the set of all real numbers. let e be a banach space and let {an} be a sequence of subsets of e. define two sets s-liminf n→∞ an and w-limsup n→∞ an as follows: x ∈ s-liminfn an if and only if there exist a sequence {xn} in e and n0 ∈ n such that {xn} converges strongly to x and xn ∈ an for n ∈ n with n ≥ n0. x ∈ w-limsupn an if and only if there exists a sequence {xn} in e such that {xn} converges weakly to x and {n ∈ n : xn ∈ an} is an infinite subset of n. it is obvious that s-liminfn an ⊂ w-limsupn an holds. we say {an} converges to a subset a of e in the sense of mosco [9] if a = s-liminfn an = w-limsupn an. and we write a = m-lim n→∞ an. let e be a banach space. the normalized duality mapping j of e is defined by j(x) = {f ∈ e∗ : 〈x,f〉 = ‖x‖2 = ‖f‖2}. e is said to be smooth if and only if j(x) consists of one element for every x ∈ e. if e is smooth, then we can consider that j is a mapping from e into e∗. j is said to be weakly sequentially continuous at zero if for every sequence {xn} in e which converges weakly to 0 ∈ e, {j(xn)} converges weakly ∗ to 0 ∈ e∗. a nonempty subset c of a banach space e is said to have the opial property [10] if for each weakly convergent sequence {xn} in c with weak limit z0 ∈ c, lim inf n→∞ ‖xn − z0‖ < lim inf n→∞ ‖xn − z‖ holds for z ∈ c with z 6= z0. all nonempty compact subsets have the opial property. also, all hilbert spaces, ℓp(1 ≤ p < ∞) and finite dimensional banach spaces have the opial property. a banach space with a duality mapping which is weakly sequentially continuous also has the opial property [6]. we know that every separable banach space can be equivalently renormed so that it has the opial property [4]. let c and k be subsets of a banach space e. a mapping p from c into k is called sunny [3] if p ( px + t (x − px) ) = px for x ∈ c and t ≥ 0 with px + t (x − px) ∈ c. let {sn} be a sequence of nonexpansive mappings on a closed convex subset c of a banach space e and let {αn} be a sequence in (0, 1] with limn αn = 0. (e,c,{sn},{αn}) is said to have browder’s property [16] if for each u ∈ c, a sequence {xn} defined by xn = (1 − αn) snxn + αn u (1) 104 tomonari suzuki cubo 10, 4 (2008) for n ∈ n converges strongly. we note that {xn} is well defined because x 7→ (1 − αn) snx + αn u is contractive. we know the following. lemma 1 ([16]). let (e,c,{sn},{αn}) have browder’s property. for each u ∈ c, put pu = lim n→∞ xn, (2) where {xn} is a sequence in c defined by (1). then p is a nonexpansive mapping on c. using p , we can rewrite theorem 2 as follows. theorem 4. let e, c, {t (t) : t ≥ 0}, τ, {αn} and {tn} be as in theorem 2. then ( e,c,{t (τ + tn)},{αn} ) has browder’s property. moreover a mapping p defined by (2) is the unique sunny nonexpansive retraction from c onto ⋂ t f(t (t)). 3. main results in this section, we prove our main results. theorem 5. let (e,c,{sn},{αn}) satisfy browder’s property. assume that c has the opial property. define a mapping p on c by (2). then w-limsupn f(sn) ⊂ f(p) holds. proof. fix x ∈ w-limsupn f(sn). then there exist a subsequence {nk} of {n} and a sequence {uk} in c such that uk ∈ f(snk ) and {uk} converges weakly to x. we note that {uk} is bounded. define a sequence {vn} in c by vn = (1 − αn) snvn + αn x. then from the assumption, {vn} converges strongly to px. we have ‖uk − vnk ‖ ≤ (1 − αnk ) ‖uk − snkvnk ‖ + αnk ‖uk − x‖ ≤ (1 − αnk ) ‖uk − vnk ‖ + αnk ‖uk − x‖ and hence ‖uk − vnk ‖ ≤ ‖uk − x‖. so lim inf k→∞ ‖uk − px‖ ≤ lim inf k→∞ ( ‖uk − vnk ‖ + ‖vnk − px‖ ) ≤ lim inf k→∞ ‖uk − x‖. from the opial property, we obtain px = x. as a direct consequence of theorem 5, we obtain the following. theorem 6. let (e,c,{sn},{αn}) satisfy browder’s property. define a mapping p on c by (2). assume that c has the opial property and f(p) ⊂ f(sn) for n ∈ n. then m-limn f(sn) = f(p) holds. cubo 10, 4 (2008) browder convergence and mosco convergence ... 105 proof. from the assumption, f(p) ⊂ s-liminfn f(sn). so by theorem 5, we obtain the desired result. remark. using theorems 2 and 6, we can prove theorem 3. we next apply theorem 6 to infinite families of nonexpansive mappings. the following convergence theorem was proved in [12, 14]. theorem 7 ([12, 14]). let e and c be as in theorem 2. let {tn : n ∈ n} be an infinite family of commuting nonexpansive mappings on c. let {αn} and {tn} be sequences in (0, 1/2) satisfying limn tn = limn αn/tn ℓ = 0 for ℓ ∈ n. let {in} be a sequence of nonempty subsets of n such that in ⊂ in+1 for n ∈ n, and ⋃ n in = n. define a sequence {sn} of nonexpansive mappings on c by snx = (( 1 − ∑ k∈in tn k ) t1x + ∑ k∈in tn k tk+1x ) . then ( e,c,{sn},{αn} ) has browder’s property. moreover a mapping p defined by (2) is the unique sunny nonexpansive retraction from c onto ⋂ n f(tn). by theorem 6, we obtain the following. theorem 8. let e and c be as in theorem 2. let {tn : n ∈ n} be an infinite family of commuting nonexpansive mappings on c. let {tn} be a sequence in (0, 1/2) converging to 0. let {in} and {sn} be as in theorem 7. then {f(sn)} converges to ⋂ n f(tn) in the sense of mosco. proof. put αn = tn n . then it is obvious that limn αn/tn ℓ = 0 holds for ℓ ∈ n. by theorem 7, ( e,c,{sn},{αn} ) has browder’s property and a mapping p defined by (2) is the unique sunny nonexpansive retraction from c onto ⋂ n f(tn). since f(p) = ⋂ n f(tn), we have f(p) ⊂ f(tn). so by theorem 6, we obtain the desired result. we recall that a family of mappings {t (p) : p ∈ [0,∞)ℓ} is said to be an ℓ-parameter nonexpansive semigroup on a subset c of a banach space e if the following are satisfied: (ℓns1) for each p ∈ [0,∞)ℓ, t (p) is a nonexpansive mapping on c. (ℓns2) t (p + q) = t (p) ◦ t (q) for all p,q ∈ [0,∞)ℓ. (ℓns3) for each x ∈ c, the mapping p 7→ t (p)x from [0,∞)ℓ into c is continuous. we denote by q the set of all rational numbers. using the result in [13], we obtain the following. theorem 9. let e and c be as in theorem 2. let {t (p) : p ∈ [0,∞)ℓ} be an ℓ-parameter nonexpansive semigroup on c. let p1,p2, · · · ,pℓ ∈ [0,∞) ℓ such that {p1,p2, · · · ,pℓ} is linearly independent in the usual sense. let β1,β2, · · · ,βℓ ∈ r such that {1,β1,β2, · · · ,βℓ} is linearly 106 tomonari suzuki cubo 10, 4 (2008) independent over q. suppose p0 := β1p1 + β2p2 + · · · + βℓpℓ ∈ [0,∞) ℓ. let {tn} be a sequence in (0, 1/2) converging to 0. define a sequence {sn} of nonexpansive mappings on c by snx = ( 1 − ℓ ∑ k=1 tn k ) t (p0)x + ℓ ∑ k=1 tn k t (pk)x. then {f(sn)} converges to ⋂ p f(t (p)) in the sense of mosco. 4. additional results in this section, we observe browder’s property. proposition 1. let (e,c,{t},{αn}) satisfy browder’s property. define a mapping p on c by (2). then p is a nonexpansive retraction from c onto f(t ). proof. we first fix x ∈ c and define a sequence {un} in c by un = (1 − αn) tun + αn x. then since {un} converges strongly to px, we obtain px = tpx, which implies px ∈ f(t ). we next fix y ∈ f(t ) and define a sequence {vn} in c by vn = (1 − αn) tvn + αn y. then since y = (1 − αn) ty + αn y, we have vn = y and hence py = y. this completes the proof. remark. though it is not interesting, we have confirmed that m-limn f(t ) = f(p) holds. there is an example such that p is not a retraction. see also [18]. example 1. let e be the two dimensional real hilbert space and put c = e. for t ≥ 0, define a 2 × 2 matrices t (t) by t (t) = [ cos(t) − sin(t) sin(t) cos(t) ] . we can consider that {t (t) : t ≥ 0} is a linear nonexpansive semigroup on c. let {αn} and {tn} be sequences in r satisfying 0 < αn < 1 and 0 < tn for n ∈ n, limn αn = limn tn = 0 and η := limn tn/αn ∈ (0,∞). then (e,c,{t (tn)},{αn}) satisfies browder’s property. however, a mapping p defined by (2) is not a retraction. proof. for α ∈ (0, 1) and t ∈ (0,∞), we put a 2 × 2 matrix p(α,t) by p(α,t) = α 4 (1 − α) sin2(t/2) + α2 [ a −b b a ] , where a = α+ 2 (1−α) sin2(t/2) and b = (1−α) sin(t). it is easy to verify that for u ∈ c, p(α,t)u is the unique point satisfying x = (1 − α) t (t)x + αu. we have p := lim n→∞ p(αn, tn) = 1 η2 + 1 [ 1 −η η 1 ] = 1 √ η2 + 1 t (θ), cubo 10, 4 (2008) browder convergence and mosco convergence ... 107 where θ := arctan(η) ∈ (0,π/2). hence (e,c,{t (tn)},{αn}) satisfies browder’s property. however, p does not satisfy p 2 = p . we finally give an example such that m-limn f(sn) $ f(p). example 2. let t be a nonexpansive mapping on a bounded closed convex subset c of a banach space e. assume that t is not the identity mapping on c. define a sequence {sn} of nonexpansive mappings on c by snx = (1 − tn) x + tn tx, where {tn} is a sequence in (0, 1) converging to 0. let {αn} be a sequence in (0, 1) such that limn αn = 0 and limn αn/tn = ∞. then (e,c,{sn},{αn}) satisfies browder’s property, a mapping p defined by (2) is the identity mapping on c and m-limn f(sn) $ f(p) holds. proof. fix x ∈ c and define a sequence {un} in c by un = (1 − αn) snun + αn x. we have ‖un − x‖ = (1 − αn) ‖snun − x‖ ≤ (1 − αn) (1 − tn) ‖un − x‖ + (1 − αn) tn ‖tun − x‖ and hence lim n→∞ ‖un − x‖ ≤ lim n→∞ (1 − αn) tn αn + tn − αn tn ‖tun − x‖ = 0. thus, {un} converges strongly to x. therefore (e,c,{sn},{αn}) satisfies browder’s property and px = x holds. from the assumption, f(t ) $ c = f(p). since f(sn) = f(t ), we have m-limn f(sn) = f(t ) and hence m-limn f(sn) $ f(p). received: april 2008. revised: april 2008. references [1] f.e. browder, fixed-point theorems for noncompact mappings in hilbert space, proc. nat. acad. sci. usa, 53 (1965), 1272–1276. [2] , convergence of approximants to fixed points of nonexpansive nonlinear mappings in banach spaces, arch. ration. mech. anal., 24 (1967), 82–90. [3] r.e. bruck, nonexpansive retracts of banach spaces, bull. amer. math. soc., 76 (1970), 384–386. [4] d. van dulst, equivalent norms and the fixed point property for nonexpansive mappings, j. london math. soc., 25 (1982), 139–144. 108 tomonari suzuki cubo 10, 4 (2008) [5] j. garćıa falset, e. llorens fuster and e.m. mazcuñán navarro, uniformly nonsquare banach spaces have the fixed point property for nonexpansive mappings, j. funct. anal., 233 (2006), 494–514. [6] j.-p. gossez and e. lami dozo, some geometric properties related to the fixed point theory for nonexpansive mappings, pacific j. math., 40 (1972), 565–573. [7] m. kato and t. tamura, weak nearly uniform smoothness and worth property of ψ-direct sums of banach spaces, comment. math. prace mat., 46 (2006), 113–129. [8] w.a. kirk, a fixed point theorem for mappings which do not increase distances, amer. math. monthly, 72 (1965), 1004–1006. [9] u. mosco, convergence of convex sets and of solutions of variational inequalities, adv. math., 3 (1969), 510–585. [10] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc., 73 (1967), 591–597. [11] t. suzuki, on strong convergence to common fixed points of nonexpansive semigroups in hilbert spaces, proc. amer. math. soc., 131 (2003), 2133–2136. [12] , strong convergence theorems of browder’s type sequences for infinite families of nonexpansive mappings in hilbert spaces, bull. kyushu inst. technol., 52 (2005), 21–28. [13] , the set of common fixed points of an n-parameter continuous semigroup of mappings, nonlinear anal., 63 (2005), 1180–1190. [14] , browder’s type strong convergence theorems for infinite families of nonexpansive mappings in banach spaces, fixed point theory appl., 2006 (2006), article id 59692, 1–16. [15] , common fixed points of one-parameter nonexpansive semigroups, bull. london math. soc., 38 (2006), 1009–1018. [16] , moudafi’s viscosity approximations with meir-keeler contractions, j. math. anal. appl., 325 (2007), 342–352. [17] , browder’s type convergence theorems for one-parameter semigroups of nonexpansive mappings in banach spaces, israel j. math., 157 (2007), 239–257. [18] , some comments about recent results on one-parameter nonexpansive semigroups, bull. kyushu inst. technol., 54 (2007), 13–26. [19] , mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups, nonlinear anal. (2007), doi:10.1016/j.na.2007.04.026. n8-suzuki-accept corderonew.dvi cubo a mathematical journal vol.12, nβo03, (213–239). october 2010 strichartz estimates for the schrödinger equation elena cordero and davide zucco department of mathematics, university of torino, v. carlo alberto 10, torino, italy email: elena.cordero@unito.it email: davide.zucco@unito.it abstract the objective of this paper is to report on recent progress on strichartz estimates for the schrödinger equation and to present the state-of-the-art. these estimates have been obtained in lebesgue spaces, sobolev spaces and, recently, in wiener amalgam and modulation spaces. we present and compare the different technicalities. then, we illustrate applications to well-posedness. resumen el objetivo de este trabajo es reportar los progresos recientes sobre estimativas de strichartz para la ecuación de schrödinger y presentar el estado de arte. estas estimativas han sido obtenidas en espacios de lebesgue, espacios de sobolev, y recientemente, en espacios de wiener amalgamados y de modulación. presentamos y comparamos los diferentes aspectos técnicos envueltos. ilustramos los resultados con aplicaciones a buena colocación. key words and phrases: dispersive estimates, strichartz estimates, wiener amalgam spaces, modulation spaces, schrödinger equation. math. subj. class.: 42b35,35b65, 35j10, 35b40. 214 elena cordero & davide zucco cubo 12, 3 (2010) 1 introduction in this note, we focus on the cauchy problem for schrödinger equations. to begin with, the cauchy problem for the free schrödinger equation reads as follows { i∂t u +∆u = 0 u(0, x) = u0(x), (1) with t ∈ r and x ∈ rd , d ≥ 1. in terms of the fourier transform, we can write the solution as follows u(t, x) = ( e it∆ u0 ) (x) := ˆ rd e 2πix·ξ e −4π2 it|ξ|2 û0(ξ)dξ, (2) where the fourier multiplier eit∆ is known as schrödinger propagator. the corresponding inhomogeneous equation is { i∂tu +∆u = f(t, x) u(0, x) = u0(x), (3) with t > 0 and x ∈ rd , d ≥ 1. by duhamel’s principle and (2), the integral version of (3) has the form u(t, x) = eit∆u0(·) + ˆ t 0 e i(t−s)∆ f(s,·)ds. (4) the study of space-time integrability properties of the solution to (2) and (4) has been pursued by many authors in the last thirty years. the matter of fact is given by the strichartz estimates, that have become a fundamental and amazing tool for the study of pde’s. they have been studied in the framework of different function/distribution spaces, like lebesgue, sobolev, wiener amalgam and modulation spaces and have found applications to well-posedness and scattering theory for nonlinear schrödinger equations [3, 8, 9, 11, 19, 24, 26, 36, 38, 39, 49]. in this paper we exhibit these problems. first, in section 3, we introduce the dispersive estimates and show how can be carried out for these different spaces. the classical l p dispersive estimates read as follows ‖e it∆ u0‖lrx . |t| −d( 12 − 1 r ) ‖u0‖lr ′ x , 2 ≤ r ≤ ∞, 1 r + 1 r′ = 1. section 4 is devoted to the study of strichartz estimates. the nature of these estimates is highlighted and the results among different kinds of spaces are compared with each others. historically, the l p spaces [19, 24, 26, 39, 49] were the first to be looked at. the celebrated homogeneous strichartz estimates for the solution u(t, x) = ( eit∆u0 ) (x) read ‖e it∆ u0‖lqt l r x . ‖u0‖l2x , (5) cubo 12, 3 (2010) strichartz estimates ... 215 for q ≥ 2, r ≥ 2, with 2/q + d/r = d/2, (q, r, d) 6= (2,∞, 2), i.e., for (q, r) schrödinger admissible (see definition 4.1). here, as usual, we set ‖f‖ l q t lrx = (ˆ ‖f(t,·)‖ q lrx dt )1/q . in the sequel, the estimates for sobolev spaces were essentially derived from the lebesgue ones. recently, several authors ([1, 2, 6, 7, 8, 44, 45]) have turned their attention to fixed time and space-time estimates for the schrödinger propagator between spaces widely used in time-frequency analysis, known as wiener amalgam spaces and modulation spaces. the first appearance of amalgam spaces can be traced to wiener in his development of the theory of generalized harmonic analysis [46, 47, 48] (see [22] for more details). in this setting, cordero and nicola [6, 7, 8] have discovered that the pattern to obtain dispersive and strichartz estimates is similar to that of lebesgue spaces. the main idea is to show that the fundamental solution k t (see (20) below) lies in the wiener amalgam space w (l p, lq ) (see section 2 for the definition) which generalizes the classical l p space and, consequently, provides a different information between the local and global behavior of the solutions. beside the similar arguments, we point out also some differences, mainly in proving the sharpness of dispersive estimates and strichartz estimates. indeed, dilation arguments in wiener amalgam and modulation spaces don’t work as in the classical l p spaces. modulation spaces were introduced by feitchinger in 1980 and then were also redefined by wang [44] using isometric decompositions. the two different definitions allow to look at the problem in two different manners. as a result, in [45], a beautiful use of interpolation theory on modulation spaces allows to combine the estimates obtained by means of the classical definition in [1, 2] and the isometric definition in [44, 45], to obtain more general fixed time estimates in this framework. in order to control the growth of singularity at t = 0, we usually have the restriction d(1/2 − 1/ p) é 1; cf. [5, 26]. by using the isometric decomposition in the frequency space, as in [44, 45], one can remove the singularity at t = 0 and preserve the decay at t = ∞ in certain modulation spaces. the strichartz estimates can be applied, e.g., to the well-posedness of non-linear schrödinger equations or of linear schrödinger equations with time-dependent potentials. we shall show examples in the last section 5. notation. we define |x|2 = x·x, for x ∈ rd , where x· y = x y is the inner product on rd . the space of smooth functions with compact support is denoted by c ∞0 (r d), the schwartz class by s (rd ), the space of tempered distributions by s ′(rd ). the fourier transform is normalized to be f̂ (ξ) = f f (ξ) = ´ f (t)e−2πitξdt. translation and modulation operators (time and frequency shifts) are defined, respectively, by tx f (t) = f (t − x) and mξ f (t) = e 2πiξt f (t). 216 elena cordero & davide zucco cubo 12, 3 (2010) we have the formulas (tx f )̂ = m−x f̂ , (mξ f )̂ = tξ f̂ , and mξtx = e 2πixξtx mξ. the notation a . b means a ≤ cb for a suitable constant c > 0, whereas a ≍ b means c−1 a ≤ b ≤ c a, for some c ≥ 1. the symbol b1 ,→ b2 denotes the continuous embedding of the linear space b1 into b2. 2 function spaces and preliminaries in this section we present the function/distribution spaces we work with, and the properties used in our study. 2.1 lorentz spaces ([34, 35]). we recall that the lorentz space l p,q on rd is defined as the space of tempered distributions f such that ‖f ‖ ∗ pq = ( q p ˆ ∞ 0 [t1/p f ∗(t)]q dt t )1/q < ∞, when 1 ≤ p < ∞, 1 ≤ q < ∞, and ‖f ‖ ∗ pq = sup t>0 t 1/p f ∗(t) < ∞ when 1 ≤ p ≤ ∞, q = ∞. here, as usual, λ(s) = |{|f | > s}| denotes the distribution function of f and f ∗(t) = inf{s : λ(s) ≤ t}. one has l p,q1 ,→ l p,q2 if q1 ≤ q2, and l p,p = l p. moreover, for 1 < p < ∞ and 1 ≤ q ≤ ∞, l p,q is a normed space and its norm ‖·‖l p,q is equivalent to the above quasi-norm ‖·‖ ∗ pq . the function |x|−α lives in ld/α,∞, 0 < α < d but observe that this function doesn’t live in any l p, 1 é p é ∞. we now recall the following classical hardy-littlewood-sobolev fractional integration theorem (see e.g. [33, theorem 1, pag 119] and [34]), which will be used in the sequel proposition 2.1. let d ê 1, 0 < α < d and 1 < p < q < ∞ such that 1 q = 1 p − d −α d . (6) then the following estimate |||·| −α ∗ f ||q . ||f ||p (7) holds for all f ∈ l p(rd ). cubo 12, 3 (2010) strichartz estimates ... 217 potential and sobolev spaces. for s ∈ r, we define the fourier multipliers 〈∆〉s f = f −1((1 + | · |2)s/2 f̂ ), and |∆|s f = f −1(| · |s f̂ ). then, for 1 ≤ p ≤ ∞, the potential space [4] is defined by w p s = { f ∈ s ′, 〈∆〉s f ∈ l p} with norm ‖f ‖ w p s = ‖〈∆〉 s f ‖l p . the homogeneous potential space [4] is defined by ẇ p s = { f ∈ s ′, |〈∆〉|s f ∈ l p} with norm ‖f ‖ ẇ p s = ‖|∆| s f ‖l p . for p = 2 the previous spaces are called sobolev spaces h p s and homogeneous sobolev spaces ḣ p s , respectively. 2.2 wiener amalgam spaces ([12, 14, 15, 16, 17]). let g ∈ c ∞0 be a test function that satisfies ‖g‖l2 = 1. we will refer to g as a window function. for 1 ≤ p ≤ ∞, recall the f l p spaces, defined by f l p(rd) = { f ∈ s ′(rd ) : ∃ h ∈ l p(rd ), ĥ = f }; they are banach spaces equipped with the norm ‖f ‖f l p = ‖h‖l p , with ĥ = f . in the same way, for 1 < p < ∞, 1 ≤ q ≤ ∞, the banach spaces f l p,q are defined by f l p,q(rd) = { f ∈ s ′(rd ) : ∃ h ∈ l p,q (rd), ĥ = f }; equipped with the norm ‖f ‖f l p,q = ‖h‖l p,q , with ĥ = f . let b one of the following banach spaces: l p, f l p, 1 ≤ p ≤ ∞, f l p,q , 1 < p < ∞, 1 ≤ q ≤ ∞, valued in a banach space, or also spaces obtained from these by real or complex interpolation. let c be the l p space, 1 ≤ p ≤ ∞, scalar-valued. for any given function f which is locally in b (i.e. g f ∈ b, ∀g ∈ c ∞0 ), we set fb(x) = ‖f tx g‖b. the wiener amalgam space w (b, c) with local component b and global component c is defined as the space of all functions f locally in b such that fb ∈ c. endowed with the norm ‖f ‖w(b,c) = ‖fb‖c , w (b, c) is a banach space. moreover, different choices of g ∈ c ∞ 0 generate the same space and yield equivalent norms. if b = f l1 (the fourier algebra), the space of admissible windows for the wiener amalgam spaces w (f l1, c) can be enlarged to the so-called feichtinger algebra w (f l1, l1). recall that the schwartz class s is dense in w (f l1, l1). 218 elena cordero & davide zucco cubo 12, 3 (2010) we use the following definition of mixed wiener amalgam norms. given a measurable function f of the two variables (t, x) we set ‖f‖w(lq1 ,lq2 )tw(f lr1 ,lr2 )x = ‖‖f(t,·)‖w(f lr1 ,lr2 )x ‖w(lq1 ,lq2 )t . observe that [6] ‖f‖w(lq1 ,lq2 )tw(f lr1 ,lr2 )x = ‖f‖w ( l q1 t (w(f l r1 x ,l r2 x )),l q2 t ). the following properties of wiener amalgam spaces will be frequently used in the sequel. lemma 2.1. let bi, ci , i = 1, 2, 3, be banach spaces such that w (bi, ci ) are well defined. then, (i) convolution. if b1 ∗ b2 ,→ b3 and c1 ∗ c2 ,→ c3, we have w (b1, c1) ∗ w (b2, c2) ,→ w (b3, c3). (8) in particular, for every 1 ≤ p, q ≤ ∞, we have ‖f ∗ u‖w(f l p ,lq ) ≤ ‖f ‖w(f l∞ ,l1 )‖u‖w(f l p ,lq ). (9) (ii) inclusions. if b1 ,→ b2 and c1 ,→ c2, w (b1, c1) ,→ w (b2, c2). moreover, the inclusion of b1 into b2 need only hold “locally” and the inclusion of c1 into c2 “globally”. in particular, for 1 ≤ pi , qi ≤ ∞, i = 1, 2, we have p1 ≥ p2 and q1 ≤ q2 =⇒ w (l p1 , lq1 ) ,→ w (l p2 , lq2 ). (10) (iii) complex interpolation. for 0 < θ < 1, we have [w (b1, c1), w (b2, c2)][θ] = w ( [b1, b2][θ], [c1, c2][θ] ) , if c1 or c2 has absolutely continuous norm. (iv) duality. if b′, c′ are the topological dual spaces of the banach spaces b, c respectively, and the space of test functions c ∞0 is dense in both b and c, then w (b, c)′ = w (b′, c′). (11) the proof of all these results can be found in ([12, 14, 15, 22]). finally, let us recall the following lemma [8, lemma 6.1], that will be used in the last section 5. cubo 12, 3 (2010) strichartz estimates ... 219 lemma 2.2. let 1 ≤ p, q, r ≤ ∞. if 1 p + 1 q = 1 r′ , (12) then w (f l p ′ , l p)(rd ) ·w (f lq′, lq)(rd ) ⊂ w (f lr, lr ′ )(rd ) (13) with norm inequality ‖f h‖ w(f lr ,lr ′ ) . ‖f ‖w(f l p′ ,l p )‖h‖w(f lq′ ,lq ). 2.3 modulation spaces ([13, 21]). let g ∈ s (rd ) be a non-zero window function and consider the so-called short-time fourier transform (stft) vg f of a function/tempered distribution f with respect to the the window g: vg f (x,ξ) = 〈f , mξtx g〉 = ˆ e −2πiξy f ( y) g( y − x) d y, i.e., the fourier transform f applied to f tx g. for s ∈ r, we consider the weight function 〈x〉s = (1 +|x|2)s/2, x ∈ rd . if 1 ≤ p, q ≤ ∞, s ∈ r, the modulation space m p,q s (r d ) is defined as the closure of the schwartz class with respect to the norm ‖f ‖ m p,q s = ( ˆ r d (ˆ r d |vg f (x,ξ)| p dx )q/p 〈ξ〉 sq dξ )1/q (with obvious modifications when p = ∞ or q = ∞). among the properties of modulation spaces, we record that they are banach spaces whose definition is independent of the choice of the window g ∈ s (rd ), m 2,2 = l2, (m p,q s ) ′ = m p′,q′ −s , whenever p, q < ∞. another definition of these spaces uses the unite-cube decomposition of the frequency space, we address interested readers to [44]. finally we recall the behaviour of modulation spaces with respect to complex interpolation (see [14, corollary 2.3]). proposition 2.2. let 1 ≤ p1, p2, q1, q2 ≤ ∞, with q2 < ∞. if t is a linear operator such that, for i = 1, 2, ‖t f ‖m pi ,qi ≤ ai‖f ‖m pi ,qi ∀f ∈ m pi ,qi , then ‖t f ‖m p,q ≤ c a 1−θ 1 a θ 2‖f ‖m p,q ∀f ∈ m p,q , where 1/ p = (1 −θ)/ p1 +θ/ p2 , 1/q = (1 −θ)/q1 +θ/q2 , 0 < θ < 1 and c is independent of t. we observe that definition and properties of modulation spaces refer to the case p, q ê 1. for the quasi-banach case 0 < p, q < 1 see, e.g., [2, 44, 45]. 220 elena cordero & davide zucco cubo 12, 3 (2010) 2.4 t∗t method [19, 20] the t∗t method is an abstract tool of harmonic analysis, discovered by tomas in 1975. this method allows to know the continuity of a linear operator t (and thus of its adjoint t∗), simply by the boundedness of the composition operator t∗t. for any vector space d, we denote by d∗a its algebraic dual, by la(d, x ) the space of linear maps from d to some other vector space x , and by 〈 ϕ, f 〉 d the pairing between d∗a and d ( f ∈ d, ϕ ∈ d∗a ), taken to be linear in f and antilinear in ϕ. lemma 2.3. let h be a hilbert space, x a banach space, x ∗ the dual of x , and d a vector space densely contained in x . let t ∈ la(d, h ) and t ∗ ∈ la (h , d ∗ a ) be its adjoint, defined by 〈 t ∗ h, f 〉 d = 〈h, t f 〉 , ∀f ∈ d, ∀h ∈ h , where 〈,〉 is the inner product in h (antilinear in the first argument). then the following three conditions are equivalent. (1) there exists a, 0 é a < ∞ such that for all f ∈ d ‖t f ‖h é a‖f ‖x ; (14) (2) let h ∈ h . then t∗h can be extended to a continuous linear functional on x , and there exists a, 0 é a < ∞, such that for all h ∈ h ‖t ∗ h‖x ∗ é a‖h‖h . (15) (3) let f ∈ x . then t∗t f can be extended to a continuous linear functional on x, and there exists a, 0 é a < ∞, such that for all f ∈ d, ‖t ∗ t f ‖x ∗ é a 2 ‖f ‖x . (16) the constant a is the same in all the three cases. if one of (all) those conditions is (are) satisfied, the operators t and t∗t extend by continuity to bounded operators from x to h and from x to x ∗, respectively. proof. from the fact that d is densely contained in x , it follows that x ∗ is a subspace of d∗a . (1) ⇒ (2). let h ∈ h . then, for all f ∈ d | 〈 t ∗ h, f 〉 d | = |〈h, t f 〉| é ||h||h ||t f ||h é a||h||h ||f ||x . (2) ⇒ (1). let f ∈ d. then, for all h ∈ h |〈h, t f 〉| = | 〈 t ∗ h, f 〉 d | é ||t ∗ h||x ∗ ||f ||x é a||h||h ||f ||x cubo 12, 3 (2010) strichartz estimates ... 221 clearly (1) and (2) imply (3), and therefore (1) or (2) implies (3). (3) ⇒ (1). let f ∈ d. then ||t f || 2 = |〈t f , t f 〉| = | 〈 t ∗ t f , f 〉 d | é ||t ∗ t f ||x ∗ ||f ||x é a 2 ||f || 2 x . since d is a dense subspace of x , we see that t can be extended to a bounded linear functional from x to h . the following corollary is extremely useful. corollary 2.3. let h , d and two triplets (x i , ti, ai ), i = 1, 2, satisfy the conditions of lemma 2.3. then for all choices of i, j = 1, 2, r(t∗ i t j) ⊂ x ∗ i and for all f ∈ d, ‖t ∗ i t j f ‖x ∗ i é ai a j‖f ‖x j . (17) in particular, t∗ i t j extends by continuity to a bounded operator from x j to x ∗ i , and (17) holds for all f ∈ x j . ginibre and velo [19] applied lemma 2.3 and corollary 2.3 to the bounded operator t : l1(i, h ) → h , defined by t f = ˆ i u(−t) f (t)dt, (18) where i is an interval of r (possibly r itself) and u a unitary strongly continuous one parameter group in h . then its adjoint t∗ is the operator t ∗ h(t) = u(t)h from h to l∞(i, h ), where the duality is defined by the scalar products in h and in l2(i, h ), such that t∗t is the bounded operator from l1(i, h ) to l∞(i, h ) given by t ∗ t f = ˆ i u(t − t′) f (t′)dt′. clearly the conditions of lemma 2.3 are satisfied with x = l1(i, h ), the operator t defined in (18), the constant a = 1, and d any dense subspace of x . let us introduce the retarded operator (t∗t)r , defined by (t∗t)r f (t) = (ur ∗t f )(t) = ˆ i ur (t − t ′) f (t′)dt′ where ur (t) = χ+(t)u(t) := χ[0,∞)(t)u(t). we recall that a space x of distributions in space-time is said to be time cut-off stable if the multiplication by the characteristic function χj , of an interval j in time, is a bounded operator in x with norm uniformly bounded with respect to j. the spaces under our consideration are of the type x = l q t (i, y ), where y is a space of distribution in the space variable and for which that property obviously holds. 222 elena cordero & davide zucco cubo 12, 3 (2010) lemma 2.4. let h an hilbert space, let i be an interval of r, let x ⊂ s ′(i ×rd ) be a banach space, let x be time cut-off stable, and let the conditions of lemma 2.3 hold for the operator t defined in (18). then the operator (t∗t)r is (strictly speaking extends to) a bounded operator from l1t (i, h ) to x ∗ and from x to l∞t (i, h ). proof. we recall the proof for sake of clarity. it is enough to demonstrate the second property, from which the first one follows by duality. let f ∈ d. then, for each t ‖(t∗t)r f (t)‖h = ‖tχ+(t −·) f ‖h é a sup t {‖χ+(t −·)‖b(x )}‖f ‖x ≤ ca‖f ‖x , by the unitary of u, the estimate (14) of lemma 2.3, and the time cut-off stability of x . 3 fixed time estimates in this section we study estimates for the solution u(t, x) to the cauchy problem (1), for fixed t. since multiplication on the fourier transform side intertwines with convolution on the space side, formula (2) can be rewritten as u(t, x) = (k t ∗ u0)(x), (19) where k t is the inverse fourier transform of the multiplier e −4π2 it|ξ|2 , given by k t(x) = 1 (4πit)d/2 e i|x|2 /(4t). (20) first, we establish the estimates for lebesgue spaces. since eit∆ is a unitary operator, we obtain the l2 conservation law ‖e it∆ u0‖l2 (rd ) = ‖u0‖l2 (rd ). (21) furthermore, since k t ∈ l ∞ with ‖k t‖∞ ≍ t −d/2, applying young inequality to the fundamental solution (19) we obtain the l1 dispersive estimate ‖e it∆ u0‖l∞(rd ) . |t| −d/2 ‖u0‖l1 (rd ). (22) this shows that if the initial data u0 has a suitable integrability in space, then the evolution will have a power-type decay in time. using the riesz-thorin theorem (see, e.g., [35]), we can interpolate (21) and (22) to obtain the important l p fixed time estimates ‖e it∆ u0‖lr (rd ) . |t| −d ( 1 2 − 1 r ) ‖u0‖lr ′ (rd ) (23) cubo 12, 3 (2010) strichartz estimates ... 223 for all 2 é r é ∞, with 1/r + 1/r′ = 1. these estimates represent the complete range of l p to lq fixed time estimates available. in this setting, the necessary conditions are usually obtained by scaling conditions (see, for example, [39, exercise 2.35], and [29] for the interpretation in terms of gaussian curvature of the characteristic manifold). the following proposition ([50, page 45]) is an example of this technique in the case p = q′. proposition 3.1. let 1 é r é ∞ and α ∈ r such that ‖e it∆ u0‖lr (rd ) é ct α ‖u0‖lr′ (rd ), (24) for all u0 ∈ s(r d ), t 6= 0 and some c independent of t and u0. then α = −d( 1 2 − 1 r ), r′ é r (and thus 2 é r é ∞). proof. we can rescale the initial data u0 by a factor λ and use (24) for v(x) := u0(λx), λ > 0, u0 ∈ s (r d). the corresponding solution with v(x) as initial data is u(λ2 t,λx), where u(t, x) = eit∆u0. therefore, by (24) and the scaling property ‖f (λ·)‖r = λ −d/r ‖f (·)‖r one has λ −d/r ‖u(λ2 t,·)‖lr (rd ) é ct α λ −d/r′ ‖u0‖lr′ (rd ), for all λ > 0, t 6= 0 and u0 ∈ s(r d ). choosing t = λ−2, we obtain ‖u(1,·)‖lr (rd ) é cλ −2α− d r′ + d r ‖u0‖lr′ (rd ), for all λ > 0 and u0 ∈ s(r d ). since ‖u(1,·)‖lr (rd ) and ‖u0‖lr′ (rd ) are two positive constants, we have for λ → ∞, − 2α− d r′ + d r ê 0, for λ → 0, − 2α− d r′ + d r é 0 and then we obtain the necessary condition for α. moreover, since eit∆ is invariant under translation, by [23, theorem 1.1] we obtain r′ é r, i.e., 2 é r é ∞. by standard density argument we attain the desired result. for s ∈ r, consider the fourier multiplier 〈∆〉s, defined by 〈∆〉s f = f −1(〈·〉s f̂ ). then, from (23) and the commutativity property of fourier multipliers, one immediately obtains the w s,r fixed time estimates ‖e it∆ u0‖w s,r (rd ) . |t| −d ( 1 2 − 1 r ) ‖u0‖w s,r ′ (rd ) (25) 224 elena cordero & davide zucco cubo 12, 3 (2010) for all s ∈ r, 2 é r é ∞, 1/r + 1/r′ = 1. finally, we note that the conservation law (21) can be rephrased in this setting as the h s conservation law ‖e it∆ u0‖hs (rd ) = ‖u0‖hs (rd ). (26) the schrödinger propagator does not preserve any w s,r norm other than the h s norm. now, we focus on wiener amalgam spaces. k t in (20) lives in w (f l 1, l∞) ⊂ l∞, see [1, 6, 44]. this is the finest wiener amalgam space-norm for k t which, consequently, gives the worst behavior in the time variable. it is also possible to improve the latter, at the expense of a rougher x-norm, see [8]. indeed, since k t ∈ w (f l p, l∞) with norm (see [8, corollary 3.1]) ‖k t‖w(f l p ,l∞) ≍ |t| −d/p (1 + t2)(d/2)(1/p−1/2), (27) from the fundamental solution (19) and the convolution relations for wiener amalgam spaces in lemma 2.1(i), it turns out, for 2 ≤ q ≤ ∞, the w (f l p, lq ) dispersive estimates ‖e it∆ u0‖w(f lq ′ ,l∞) . |t| d(2/q−1)(1 + t2)d(1/4−1/q)‖u0‖w(f lq ,l1 ). (28) as well as for lebesgue spaces, we can use complex interpolation between the dispersive estimates (28) and the l2 conservation law (l2 = w (f l2, l2)) to obtain the following w (f l p, lq) fixed time estimates, that combine [6, theorem 3.5] and [8, theorem 3.3]. theorem 3.2. for 2 ≤ q, r, s ≤ ∞ such that 1 s = 1 r + 2 q ( 1 2 − 1 r ) , we have ‖e it∆ u0‖w(f ls ′ ,lr ) . |t| d ( 2 q −1 )( 1− 2 r ) (1 + t2) d ( 1 4 − 1 q )( 1− 2 r ) ‖u0‖w(f ls ,lr ′ ) (29) in particular, for s = 2, ‖e it∆ u0‖w(l2 ,lr ) . (1 + t 2)− d 2 ( 1 2 − 1 r ) ‖u0‖w(l2 ,lr′ ), (30) and, for s = r, ‖e it∆ u0‖w(f lr ′ ,lr ) . (|t| −2 +|t| −1 )d ( 1 2 − 1 r ) ‖u0‖w(f lr ,lr ′ ). (31) proof. let us sketch the proof for the sake of readers. estimate (29) follow by complex interpolation between estimate (28), which corresponds to r = ∞, and (21), which corresponds to r = 2. indeed, l2 = w (f l2, l2) = w (l2, l2). using lemma 2.1(iii), with θ = 2/r (observe that 0 < 2/r < 1), and 1/s′ = (1 − 2/r)/q′ + (2/r)/2, so that relation (29) holds, we obtain [ w (f lq ′ , l∞), w (f l2, l2) ] [θ] = w ( [f lq ′ , f l2][θ], [l ∞, l2][θ] ) cubo 12, 3 (2010) strichartz estimates ... 225 = w (f ls ′ , lr ) and [ w (f lq, l1), w (f l2, l2) ] [θ] = w ( [f lq, f l2][θ], [l 1, l2][θ] ) = w (f ls, lr ′ ). this yields the desired estimate (29). let us compare the previous results with the classical l p estimates. for 2 é r é ∞, f lr ′ ,→ lr, and the inclusion relations for wiener amalgam spaces (lemma 2.1 (ii)) yield w (f lr ′ , lr ) ,→ w (lr, lr ) = lr and lr ′ = w (lr ′ , lr ′ ) ,→ w (f lr, lr ′ ). thereby the estimate (31) is an improvement of (23) for every fixed time t 6= 0, and also uniformly for |t| > c > 0. moreover, in [8] cordero and nicola proved that the range r ≥ 2 in (31) is sharp, and the same for the decay t−d ( 1 2 − 1 r ) at infinity and the bound t−2d ( 1 2 − 1 r ) , when t → 0. modulation spaces are new settings inherited by time-frequency analysis where the fixed time estimates recently have been studied, see [1, 2, 44, 45]. here, instead of using the representation of the solution u(t, x) in (19), the solution is written in the form of fourier multiplier eit∆u0 as in (2), see [1, 2]. indeed, a sufficient condition for the boundedness of a fourier multiplier on modulation spaces is that its symbol is in w (f l1, l∞) ([1, lemma 8]). moreover, the schrödinger symbol σ = e−it|ξ| 2 lives in w (f l1, l∞) and its norm is ‖σ‖w(f l1 ,l∞) = sup x ˆ rd |vgσ(x,ω)|dω ≍ (1 + t2)−d/4 ˆ rd e − π t2+1 |ω|2 dω ≍ (1 + t2)d/4, where g(ξ) = e−π|ξ| 2 . then, by [2, lemma 2] (also for s = 0 [1, corollary 18]) one has that eit∆ extends to a bounded operator on m p,q s , i.e., the m p,q s fixed time estimates ‖u(t, x)‖ m p,q s . (1 +|t|)d/2‖u0‖m p,qs , (32) for all s ê 0 and 1 é p, q é ∞. in particular, modulation space properties are preserved by the time evolution of the schrödinger equation, in strong contrast with the case of lebesgue spaces. observe that (32), in the case s = 0, was also obtained using isometric decompositions in [44]. later, wang, zaho, guo in [45] obtain the following fixed time estimates ‖u(t, x)‖ m p,q s . (1 +|t|)−d(1/2−1/p)‖u0‖ m p′,q s , (33) for all s ∈ r, 2 é p é ∞ and 1 é q é ∞. comparing (23) with (32) and (33), we see that the singularity at t = 0 contained in (23) has been removed in (32) and (33) and the decay rate in (33) when t = ∞ is the same one as in (23). the estimate (33) also indicates that eit∆ is 226 elena cordero & davide zucco cubo 12, 3 (2010) uniformly bounded on m 2,q. the complex interpolation between the case p = 2 in (33), and p = ∞ in (32) yields ‖u(t, x)‖ m p,q s . (1 +|t|)d(1/2−1/p)‖u0‖m p,qs , (34) for all 2 é p é ∞, s ê 0. however, it is still not clear whether the growth order on time in the right-hand side of (34) is optimal. 4 strichartz estimates in many applications, especially in the study of well-posedness of pde’s, it is useful to have estimates for the solution both in time and space variables. in this direction, the main result is represented by the strichartz estimates. first, let us introduce the following definitions. definition 4.1. following [26], we say that the exponent pair (q, r) is schrödinger-admissible if d ê 1 and 2 é q, r é ∞, 1 q = d 2 ( 1 2 − 1 r ) , (q, r, d) 6= (2,∞, 2). definition 4.2. following [18], we say that the exponent pair (q, r) is schrödinger-acceptable if 1 é q < ∞, 2 é r é ∞, 1 q < d ( 1 2 − 1 r ) , or (q, r) = (∞, 2). the original version of strichartz estimates in l p spaces, closely related to restriction problem of fourier transform to surfaces, was elaborated by robert strichartz [36] in 1977(who, in turn, had precursors in [31, 41]). in 1995 a brilliant idea of ginibre and velo [20] was the use of the t∗t method (lemma 2.3) to detach the couple (q, r) from (q′, r′) (see also [49]). the study of the endpoint case (q, r) = (2, 2d/(d − 2)) is treated in [26], where keel and tao prove the estimate also for the endpoint when d ≥ 3 (for d = 2, the endpoint is (q, r) = (2,∞) and the estimate is false). we shall give a standard proof of the l p stichartz estimates in the nonendpoint cases [10, 50] (see also [39] where the following theorem is proved using an abstract lemma, the christ-kiselev lemma, which is very useful in establishing retarded strichartz estimates). theorem 4.3. for any schrödinger-admissible couples (q, r) and (q̃, r̃) one has the homogeneous strichartz estimates ‖e it∆ u0‖lq t lrx (r×r d ) . ‖u0‖l2x (rd ) , (35) the dual homogeneous strichartz estimates ∥∥∥ ˆ r e −is∆ f(s,·) ds ∥∥∥ l2x (r d ) . ‖f‖ l q̃′ t lr̃ ′ x (r×r d ) , (36) cubo 12, 3 (2010) strichartz estimates ... 227 and the inhomogenous (retarded) strichartz estimates ∥∥∥ ˆ s (d −2)/(2d). afterwards, for d > 2, foschi [18] improved this result by looking for the optimal range of lebesgue exponents for which inhomogeneous strichartz estimates hold (results almost equivalent have recently obtained by vilela [43]). actually, this range is larger than the one given by admissible exponents for homogeneous estimates, as was shown by the following result [18, proposition 24]. proposition 4.4. if v is the solution to (3), with zero initial data and inhomogeneous term f supported on r×rd , then we have the estimate ‖v‖lq t lrx (r×r d ) . ‖f‖lq̃ ′ t l r̃′ x (r×r d ) (39) whenever (q, r), (q̃, r̃) are schrödinger acceptable pairs which satisfy the scaling condition 1 q + 1 q̃ = d 2 ( 1 − 1 r − 1 r̃ ) , and either the conditions 1 q + 1 q̃ < 1, d − 2 r é d r̃ , d − 2 r̃ é d r or the conditions 1 q + 1 q̃ = 1, d − 2 r < d r̃ , d − 2 r̃ < d r , 1 r é 1 q , 1 r̃ é 1 q̃ . for a discussion about the sharpness of this proposition we refer to [18], where explicit counterexamples are constructed to show the necessary conditions for inhomogeneous strichartz estimates. since the schrödinger operator eit∆ commutes with fourier multipliers like |∆|s or 〈∆〉s, it is easy to obtain strichartz estimates for potential and sobolev spaces. in particular, if i is an interval containing the origin and u : i × rd → c is the solution to the inhomogeneous schrödinger equation with initial data u0 ∈ ḣ s x(r d ), given by the duhamel formula (4), then, applying |∆|s to both sides of the equation and using the estimate of theorem 4.3, one obtains ‖u‖ l q t ẇ s,r x (i×r d ) . ‖u0‖ḣsx (rd ) +‖f‖ l q̃′ t ẇ s,r̃′ x (i×r d ) cubo 12, 3 (2010) strichartz estimates ... 229 for all schrödinger admissible couples (q, r) and (q̃, r̃). in particular, if one considers the homogeneous case (i.e. f = 0), the sobolev embedding ẇ s,rx ,→ l r1 x , 0 < s < d/2 and 1/r1 = 1/r − s/d, yields the ḣ s stichartz estimates ‖u‖ l q t l r1 x (i×r d ) . ‖u0‖ḣsx (rd ) , 2 q + d r1 + s = d 2 . since s > 0 one has 2 q + n r1 < n 2 , hence, for any fixed value of s, the new schrödinger admissible couple (q, r1 ) lies on a parallel line below the corresponding case s = 0. strichartz estimates in wiener amalgam spaces enable us to control the local regularity and decay at infinity of the solution separately. for comparison, the classical estimates (35) can be rephrased in terms of wiener amalgam spaces as follows: ‖e it∆ u0‖w(lq ,lq )tw(lr ,lr )x . ‖u0‖l2x . (40) in this framework, cordero and nicola perform these estimates mainly in two directions. first, in [6], for q ê 4 they modify the classical estimate (40) by (conveniently) moving local regularity from the time variable to the space variable. indeed, f lr ′ ⊂ lr if r ê 2, but the bound in (31) is worse than the one in (23), as t → 0; consequently one has ‖e it∆ u0‖w(lq/2 ,lq )tw(f lr ′ ,lr ) . ‖u0‖l2x , (41) for 4 < q é ∞, 2 é r é ∞, with (q, r) schrödinger admissible. when q = 4 the same estimate holds with the lorentz space lr ′,2 in place of lr ′ . dual homogeneous and retarded estimates hold as well. thereby, the solution averages locally in time by the lq/2 norm, which is rougher than the lq norm in (35) or, equivalently, in (40), but it displays an f lr ′ behavior locally in space, which is better than lr . in [8] it is shown the sharpness of these strichartz estimates, except for the threshold q ≥ 4, which seems quite hard to obtain. secondly, in [8], a converse approach is performed, by showing that it is possible to move local regularity in (35) from the space variable to the time variable. as a result, new estimates involving the wiener amalgam spaces w (l p, lq), that generalize (35), are obtained, i.e., the following [8, theorem 1.1]. theorem 4.5. let 1 ≤ q1, r1 ≤ ∞, 2 ≤ q2, r2 ≤ ∞ such that r1 ≤ r2, 2 q1 + d r1 ≥ d 2 , (42) 2 q2 + d r2 ≤ d 2 , (43) 230 elena cordero & davide zucco cubo 12, 3 (2010) (r1, d) 6= (∞, 2), (r2, d) 6= (∞, 2) and, if d ≥ 3, r1 ≤ 2d/(d − 2). assume the same for q̃1, q̃2, r̃1, r̃2 . then, we have the homogeneous strichartz estimates ‖e it∆ u0‖w(lq1 ,lq2 )tw(lr1 ,lr2 )x . ‖u0‖l2x , (44) the dual homogeneous strichartz estimates ‖ ˆ e −is∆ f(s) ds‖l2 . ‖f‖ w(l q̃′ 1 ,l q̃′ 2 )tw(l r̃′ 1 ,l r̃′ 2 )x , (45) and the retarded strichartz estimates ‖ ˆ s q2. since there are no relations between the pairs (q1, r1) and (q2, r2) other than r1 ≤ r2, these estimates tell us, in a sense, that the analysis of the local regularity of the schrödinger propagator is quite independent of its decay at infinity. 1 r 1 q d−2 2d 1 2 1 1 2 1 i1 i2 2 q + d r = d 2 figure 1: when d ≥ 3, (44) holds for all pairs (1/q1, 1/r1 ) ∈ i1, (1/q2, 1/r2) ∈ i2, with 1/r2 ≤ 1/r1. cubo 12, 3 (2010) strichartz estimates ... 231 in [8] it is proved that, for d ≥ 3, all the constraints on the range of exponents in theorem 4.5 are necessary, except for r1 ≤ r2, r1 ≤ 2d/(d − 2), which is still left open. however, the following weaker result holds [8, proposition 5.3]: assume r1 > r2 and t 6= 0. then the propagator e it∆ does not map w (lr ′ 1 , lr ′ 2 ) continuously into w (lr1 , lr2 ). shows that the estimates (44) for exponents r1 > r2, if true, cannot be obtained from fixedtime estimates and orthogonality arguments. the arguments employed for the necessary conditions differ from the classical setting of lebesgue spaces, because the general scaling consideration does not work directly. indeed, the known bounds for the norm of the dilation operator f (x) 7−→ f (λx) between wiener amalgam spaces ([37, 40]), yield constraints which are weaker than the desired ones. so, the necessary conditions are obtained considering families of rescaled gaussians as initial data, for which the action of the operator eit∆ and the involved norms can be computed explicitly, see [8]. we end up this section with recalling stricharz estimates for modulation spaces. the main result in this framework is due to wang and hudzik [45]. they use the same arguments as in keel and tao [26], who point out that the ranges of exponents (q, r) in (23) could most likely be not optimal. in fact, keel and tao show that if the semigroup eit∆ satisfies the estimate ‖e it∆ u0‖l p . (1 +|t|) −d(1/2−1/p) ‖u0‖l p′ (47) then (35), (36) and (37) hold if one substitutes q and q̃ by any γ ≥ max(q, 2) and γ̃ ê max(q̃, 2), respectively. since the estimate (34) is similar to (47), they optimize (35), (36) and (37) in the function spaces m p,q s to cover the exponents (γ, q) and (γ̃, q̃) satisfying γ ê max(q, 2) and γ̃ ê max(q̃, 2). since the precise formulation of these results requires the introduction of other function spaces, we refer interested readers to [45, section 3]. 5 applications we start by focusing on the cauchy problem for the nonlinear schrödinger equation (nls) { i∂tu +∆u + n(u) = 0 u(0, x) = u0(x). (48) the nonlinearity n considered will be either power-like pk(u) = λ|u| 2k u, k ∈ n, λ ∈ r or exponential-like eρ(u) = λ(e ρ|u|2 − 1)u, λ,ρ ∈ r. 232 elena cordero & davide zucco cubo 12, 3 (2010) both nonlinearities are smooth. the corresponding equations having power-like nonlinearities pk are sometimes referred to as algebraic nonlinear schrödinger equations. the sign of the coefficient λ determines the defocusing, absent, or focusing character of the nonlinearity. we shall study the well-posedness of (48), in different spaces. recall that the problem (48) is locally well-posed in e.g. h sx(r d ) if, for any u∗0 ∈ h s x(r d ), there exists a time t > 0 and an open ball b in h sx(r d) containing u∗0 and a subset of c 0 t h sx([t, t] × r d) such that for each u0 ∈ b there exists a unique solution u ∈ x to the equation (48) and the map u0 7→ u is continuous from b (with the h sx topology) to x (with the c 0 t h sx([t, t] ×r d) topology). a fundamental tool in well-posedness theory is the contraction theorem. let us first work abstractly, viewing (48) as an instance of the more general u = ulin + d n(u) (49) where ulin := e it∆u0 is the linear solution, n is the nonlinearity and d is the duhamel operator df(t, x) := ˆ t 0 e i(t−s)∆ f(s,·)ds. the following abstract tool [39, proposition 1.38] then allows us to find the desired contraction map. proposition 5.1 (abstract iteration argument). let n , t be two banach spaces. let d : n → t be a bounded linear operator with the bound ||df||t é c0||f||n (50) for all f ∈ n and some constant c0 > 0, and let n : s → n , with n(0) = 0, be a nonlinear operator which is lipschitz continuous and obeys the bounds ||n(u) − n(v)||n é 1 2c0 ||u − v||t (51) for all u, v in the ball bǫ := {u ∈ s : ||u||t é ǫ}, for some ǫ > 0. then, for all ulin ∈ bǫ/2, there exists a unique solution u ∈ bǫ to the equation (49), with lipschitz map ulin 7→ u with constant at most 2. that is, we have ||u||t é 2||ulin||t (52) proof. observe that for v = 0 the estimate (51) becomes ||n(u)||n é 1 2c0 ||u||t (53) (since n(0) = 0 by hypothesis). then, fix ulin ∈ bǫ/2, and consider the map φ(u) := ulin + d n(u). cubo 12, 3 (2010) strichartz estimates ... 233 using (50) and (53) one has ||φ(u)||t = ||ulin + d n(u)||t é ǫ 2 + c0 2c0 ǫ = ǫ for all u ∈ bǫ, i.e., φ maps the ball bǫ into bǫ. moreover, φ is a contraction on bǫ, indeed by (50) and (51) one has ||φ(u) −φ(v)||t = ||d n(u) − d n(v)||t é c0||n(u) − n(v)||n é c0 1 2c0 ||u − v||t = 1 2 ||u − v||t , for all u, v ∈ bǫ. then, the contraction theorem asserts that there exists a unique fixed point u for φ and moreover the map ulin 7→ u is lipschitz with constant at most 2, that is (52). proposition 5.1 is the main ingredient of the results in [1, 7, 8, 9, 39, 45]. first, consider the nls (48) with n = pk, with the initial data u0 in the sobolev space h sx(r d ). to study this cauchy problem it is convenient to introduce a single space ss that recaptures all the strichartz norms at a certain regularity h sx(r d) simultaneously. for sake of simplicity, we reduce to the case s = 0 which corresponds to the case l2x, introducing the strichartz space s0(i ×rd ), for any time interval i, defined as the closure of schwartz class s with respect to the norm ||u|| s 0(i×rd ) := sup a ||u|| l q t lrx (i×r d ), where the set a is given by a := {(∞, 2), (q, r)}, with (q, r) schrödinger admissible. we define also the space n0(i × rd ) := l q′ t l r′ x . then, using proposition 5.1 and the l p strichartz estimates of theorem 4.3 one can prove the following [39, proposition 3.15] theorem 5.2 (l2x subcritical solution). let k be subcritical for l 2 x (that is, 0 < k < 2 d ) and let µ = ±1. then the nls (48) is locally well-posed in l2x in a subcritical sense. indeed, for any r > 0 there exists a time t > 0 such that for all u0 in the ball br := {u0 ∈ l 2 x(r d ) : ||u0||l2x < r} there exists a unique solution u in l2x of (48) in the space s 0([−t, t]×rd) ⊂ c0 t l2x([−t, t]×r d). moreover, the map u0 7→ u, from br to s 0([−t, t] ×rd), is lipschitz continuous. for results in the framework of modulation spaces we address to [2, 44, 45]. in particular, we examine [2]. the main result, obtained only with the m p,q s dispersive estimates (32), is the following. theorem 5.3. assume that u0 ∈ m p,1 s (r d ) and n ∈ { pk , eρ}. then, there exists t = t(‖u0‖m p,1s ) such that (48) has a unique solution u ∈ c0 m p,1 s ([0, t] × r d). moreover, if t < ∞, then lim supt→t ‖u(t,·)‖ = ∞. 234 elena cordero & davide zucco cubo 12, 3 (2010) proof. the proof is simply an application of the abstract iteration argument. let us write it for the nonlinearity n = pk. we choose the spaces t := c 0 m p,1 s ([0, t]×r), n := m p,1 s , and the duhamel operator d := ˆ t 0 e i(t−s)∆ · ds. then, it is sufficient to prove (50) and (51) in this setting. then, by the minkowsky integral inequality, m p,q s dispersive estimates (32) and [2, corollary 2] one has ∥∥∥ ˆ t 0 e i(t−τ)∆( pk(u))(τ)dτ ∥∥∥ m p,1 s é ˆ t 0 ‖e i(t−τ)∆( pk (u))(τ)‖m p,1s dτ é c1 t ct sup t∈[0,t] ‖pk (u)(t)‖m p,1s é c1 c2 ct t‖u(t)‖ 2k+1 m p,1 s where ct = supt∈[0,t)(1 +|t|) d/2. choosing t > 0 such that c1 c2 ct t é c0, it follows (50) and by pk (u)(τ) − pk (v)(τ) = λ(u − v)|u| 2k (τ) +λv(|u|2k −|v|2k)(τ), it follows (51). for wiener amalgam spaces there are no results for the nls. in [8] there is a result concerning linear schrödinger equations with time-dependent potentials. indeed, in [8] the well-posedness is proved in l2 of the following cauchy problem, for all d ≥ 1, { i∂tu +∆u = v (t, x)u, t ∈ [0, t] = it , x ∈ r d , u(0, x) = u0(x), (54) and for the class of potentials v ∈ l α(it ; w (f l p′ , l p )x), 1 α + d p ≤ 1, 1 ≤ α < ∞. d 4, r ≥ 2, the cauchy problem (54) has a unique solution (i) u ∈ c (it ; l 2(r))∩ lq/2(it ; w (f l r′ , lr )), if d = 1; (ii) u ∈ c (it ; l 2(rd )) ∩ lq/2(it ; w (f l r′ , lr ))∩ ∩ l 2(it ; w (f l 2d/(d+1),2, l2d/(d−1))), if d > 1. proof. it is enough to prove the case d = 1. indeed, for d ≥ 2, condition (55) implies p > 2, so that f l p ′ ,→ l p and the inclusion relations of wiener amalgam spaces yield w (f l p ′ , l p) ,→ cubo 12, 3 (2010) strichartz estimates ... 235 w (l p, l p) = l p. hence our class of potentials is a subclass of those of [6, theorem 6.1], for which the quoted theorem provides the desired result. we now turn to the case d = 1. the proof follows the ones of [9, theorem 1.1, remark 1.3] and [6, theorem 6.1] (see also [49]). first of all, since the interval it is bounded, by hölder’s inequality and by taking p large, we may assume 1/α+ d/ p = 1. we choose a small time interval j = [0,δ] and set, for q ≥ 2, q 6= 4, r ≥ 1, zq/2,r = l q/2(j; w (f lr ′ , lr )x). now, fix an admissible pair (q0, r0) with r0 arbitrarily large (hence (1/q0, 1/r0) is arbitrarily close to (1/4, 0)) and set z = c (j; l2)∩zq0/2,r0 , with the norm ‖v‖z = max{‖v‖c (j;l2),‖v‖zq0 /2,r0 }. we have z ⊂ zq/2,r for all admissible pairs (q, r) obtained by interpolation between (∞, 2) and (q0, r0). hence, by the arbitrary of (q0, r0 ) it suffices to prove that φ defines a contraction in z. consider now the integral formulation of the cauchy problem, namely u = φ(v), where φ(v) = eit∆u0 + ˆ t 0 e i(t−s)∆ v (s)v(s) ds. by the homogeneous and retarded strichartz estimates in [6, theorems 1.1, 1.2] the following inequalities hold: ‖φ(v)‖zq/2,r ≤ c0‖u0‖l2 + c0‖v v‖z(q̃/2)′,r̃′ , (56) for all admissible pairs (q, r) and (q̃, r̃), q > 4, q̃ > 4. consider now the case 1 ≤ α < 2. we choose ((q̃/2)′, r̃) = (α, 2 p/( p+2)). since v ∈ l∞(j; l2), applying (13) for q = 2 we get ‖v v‖ w(f lr̃ ,lr̃ ′ ) . ‖v ‖w(f l p′ ,l p )‖v‖l2 , whereas hölder’s inequality in the time-variable gives ‖v v‖z(q̃/2)′,r̃′ . ‖v ‖lα(j;w(f l p′ ,l p ))‖v‖l∞(j;l2). the estimate (56) then becomes ‖φ(v)‖zq/2,r ≤ c0‖u0‖l2 + c0‖v ‖lα(j;w(f l p′ ,l p ))‖v‖l∞(j;l2). (57) by taking (q, r) = (∞, 2) or (q, r) = (q0, r0) one deduces that φ : z → z (the fact that φ(u) is continuous in t when valued in l2x follows from a classical limiting argument [9, theorem 1.1, remark 1.3]). also, if j is small enough, c0‖v ‖lα t l p x < 1/2, and φ is a contraction. this gives 236 elena cordero & davide zucco cubo 12, 3 (2010) a unique solution in j. by iterating this argument a finite number of times one obtains a solution in [0, t]. the case 2 ≤ α < ∞ is similar. this result generalizes [6, theorem 6.1], by treating the one dimensional case as well and allowing the potentials to belong to wiener amalgam spaces with respect to the space variable x. other results on schrödinger equations with potentials in l p t l q x can be found in [9]. references [1] bényi, a., gröchenig, k., okoudjou, k.a. and rogers, l.g., unimodular fourier multipliers for modulation spaces, j. funct. anal., 246(2), 366–384, 2007. 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[50] zucco, d., stime dispersive e di strichartz per l’equazione di schrödinger, thesis, university of torino, 2008. cubo a mathematical journal vol.10, n o ¯ 04, (137–147). december 2008 a new version of fan’s theorem and its applications m. fakhar department of mathematics, university of isfahan, and institute for studies in theoretical physics and mathematics (ipm), isfahan, 81745-163, tehran, iran email: fakhar@math.ui.ac.ir and j. zafarani sheikhbahaee university and university of isfahan, isfahan, 81745-163, iran email: jzaf@zafarini.ir abstract in this article, using a generalized version of ky fan’s theorem, we deduce new proofs for some fixed point theorems and new existence theorems for equilibrium problems. resumen usando una versión generalizada del teorema de ky fan, deducimos nuevas demostraciones para algunos teoremas de punto fijo y nuevos teorema de existencia para problemas de equilibrio. key words and phrases: fan’s theorem, fixed point theorem, equilibrium problem, variational inequalities. math. subj. class.: 47h10; 49j53; 54h25. 138 m. fakhar and j. zafarani cubo 10, 4 (2008) 1 introduction many problems in nonlinear analysis can be solved by showing the nonemptyness of the intersection of certain family of subsets of an underlying set. each point of the intersection can be a fixed point, a coincidence point, an equilibrium point, a saddle point or an optimal. the first remarkable result on nonempty intersection was the celebrated knaster, kuratowski and mazurkiewicz in 1929 [15], which concerns with certain type of multimaps called the kkm maps later. fan [11] proved that the assertion of the kkm theorem for infinite dimensional topological vector space. brézis, nirenberg and stampacchia [4] improved fan’s kkm lemma [11] by assuming the closedness condition only on finite dimensional subspaces, with some topological pseudomonotone condition. chowdhury and tan [5], replacing finite dimensional subspaces by polytopes, restated the brézis, nirenberg and stampacchia result under weaker assumptions. ding and tarafdar [7] obtained the result of chowdhury and tan under weaker compactness condition. the chowdhury and tan’s result was also proved by kalmoun [14] for transfer closed-valued multi-valued mappings. our aim here is to derive a new version of brézis, nirenberg and stampacchia’s result and then apply it to obtain some fixed point theorems and established the existence solution of equilibrium problems and generalized variational inequalities. for the reader’s convenience, we review a few basic definitions and notations from the fixed point theory. let x be a hausdorff topological vector space and k be a nonempty subset of x, then we denote by < k > the family of all nonempty finite subsets of k. let k0 be a nonempty subset of k. a set-valued map γ : k0 ⇉ k is called a kkm map if for each a ∈< k0 >, conv(a) ⊆ ⋃ x∈a γ(x). let y be a nonempty set. then, γ : y ⇉ k is said to be transfer closedvalued if for any (y, x) ∈ y × k with x 6∈ γ(y) there exists y′ ∈ y such that x 6∈ clk γ(y ′ ). if y = k, then we will call γ transfer closed-valued on k. if k0 ⊆ k, then a map γ : k ⇉ k is called transfer closed-valued on k0 if the map y 7→ γ(y) ⋂ k0, y ∈ k0, is transfer closed-valued. a set-valued map γ : k ⇉ k is called transfer open-valued on k if the set-valued map γ̂ : k ⇉ k defined as follows: γ̂(x) := k \ γ(x) is transfer closed-valued on k. let us recall that a set-valued map γ : k ⇉ k has a maximal element, if there exists a point x̄ ∈ k such that γ(x̄) = ∅. suppose that f is a real-valued bifunction on y × k. then, we say that f is transfer lower semicontinuous(l.s.c.) in the second variable if for each (y, x) ∈ y × k with f (y, x) > 0 there exist y′ ∈ y and a neighborhood u (x) of x in k such that f (y′, z) > 0 for all z ∈ u (x). if y = k and a ⊆ k, then we call f transfer l.s.c. in the second variable on a, if f |a×a is transfer l.s.c. in the second variable. definition 1.1 let f : k × k → r. we recall that: (i) f is pseudomonotone if, for all (x, y) ∈ k × k, f (x, y) ≥ 0 implies f (y, x) ≤ 0; (ii) f is called 0-segmentary closed if ∀x, y ∈ k, when (yα) be a net on k converging to y, then the following implication holds, if f (u, yα) ≤ 0 for all u ∈ [x, y], then f (x, y) ≤ 0. cubo 10, 4 (2008) a new version of fan’s theorem and its applications 139 (iii) f (., y) is upper sign continuous if the following implication holds for every x ∈ k, f (u, y) ≥ 0, ∀u ∈ ]x, y[⇒ f (x, y) ≥ 0, we note that if f is hemicontinuous function, then f and −f both are upper sign continuous. 2 brézis, nirenberg and stampacchia type theorem in [8, 9], the authors refined the ding and tarafdar’s result [7] and the kalmoun’s result [14]. based on the remark 2 in [4], recently the authors obtain a short and direct proof of the following brézis, nirenberg and stampacchia version of fan’s kkm theorem [10] lemma 2.1 let k be a nonempty and convex subset of a hausdorff t.v.s. x. suppose that γ : k ⇉ k is a set-valued mapping such that the following conditions are satisfied: (i) γ is a kkm map; (ii) for all a ∈ < k >, γ is transfer closed-valued on conv(a); (iii) for all x, y ∈ k, clk ( ⋂ u∈[x,y] γ(u)) ∩ [x, y] = ( ⋂ u∈[x,y] γ(u)) ∩ [x, y]; (iv) there is a nonempty compact convex set b ⊆ k, such that clk ( ⋂ x∈b γ(x)) is compact. then, ⋂ x∈k γ(x) 6= ∅. based on the above lemma, here we obtain another new version of the above result. theorem 2.1. let k be a nonempty convex subset of a hausdorff t.v.s. x. suppose that γ : k ⇉ k is a set-valued mapping such that the following conditions are satisfied: (h1) γ is a kkm map; (h2) ∀a ∈ 〈k〉, γ is transfer closed-valued on conv(a); (h3) ∀x, y ∈ k, clk ( ⋂ u∈[x,y] γ(u)) ∩ [x, y] = ( ⋂ u∈[x,y] γ(u)) ∩ [x, y]; (h4) there exist a nonempty compact convex subset b of k and a nonempty compact subset d of k such that, for each y ∈ k \ d there exists x ∈ conv(b ∪ {y}) such that y 6∈ γ(x). then, ⋂ x∈k γ(x) 6= ∅. proof. suppose that a ∈< k > and la = conv(a ⋃ b), then la is compact. let γa : la ⇉ la be defined as γa(x) = γ(x) ∩ la. then, from lemma 2.1, we have ⋂ x∈la γa(x) 6= ∅. 140 m. fakhar and j. zafarani cubo 10, 4 (2008) now, we show that ⋂ x∈la γa(x) ⊆ d. suppose that this claim is not true, then there exists y ∈ ⋂ x∈la γa(x) such that y ∈ k \ d. but by assumption (h4) there exists x ∈ conv(b ∪ {y}) such that y 6∈ γ(x). therefore, x 6∈ la. but since y ∈ la, then conv(b ∪ {y}) ⊆ la which contradicts (h4). assume that ma = ⋂ x∈la γ(x) for any a ∈< k >, (1) then ma ⊆ d for all a ∈< k > . (2) if m = {ma : a ∈< k >}, then by (1) one can see that the class m has the finite intersection property. therefore, from (2), we have ⋂ a∈ clk ma 6= ∅. if x̄ ∈ ⋂ a∈ clk ma, x ∈ x and a0 = {x̄, x}, then conv(a0) = [x̄, x] and x̄ ∈ clk ma0 = clk   ⋂ u∈la0 γ(u)   ⊆ clk   ⋂ u∈[x̄,x] γ(u)   hence, by condition (h3) x̄ ∈ clk   ⋂ u∈[x̄,x] γ(u)   ∩ [x̄, x] =   ⋂ u∈[x̄,x] γ(u)   ∩ [x̄, x]. therefore, x̄ ∈ γ(x) for all x ∈ x and the proof is complete. remark 2.2. (a) by a similar proof as that of the above theorem, we can obtain some other versions of fan’s kkm theorem. let k0 be a nonempty subset of k and γ : k0 ⇉ k satisfying the following conditions: (i) γ is a kkm map, (ii) for each a ∈ f(k0), γ : a ⇉ conv(a) is transfer closed valued, (iii) for each x, y ∈ k0 , clk   ⋂ u∈[x,y]∩k0 γ(u)   ∩ [x, y] =   ⋂ u∈[x,y]∩k0 γ(u)   ∩ [x, y], (iv) there exists a nonempty convex compact subset b of k such that for each y ∈ k \ b there exists x ∈ conv(b ∪ {y}) ∩ k0 such that y 6∈ γ(x). cubo 10, 4 (2008) a new version of fan’s theorem and its applications 141 then, ⋂ x∈k0 γ(x) 6= ∅. (b) instead of assumptions (ii) and (iii) in part (a) we can assume that γ is transfer closed valued. furthermore, in this case, condition (iv) of part (a) can replaced by the following condition: (iv)′ there exist a nonempty compact convex subset b of k and a nonempty compact subset d of k such that, for each y ∈ k \ d there exists x ∈ conv(b ∪ {y}) ∩ k0 such that y 6∈ γ(x). 3 fixed point theorems in this section, we deduce slight generalizations of known fixed point theorems from theorem 2.1. theorem 3.1. let k be a nonempty convex subset of a (t.v.s.) x and s : k ⇉ k a set-valued map such that: (i) for each a ∈< k >, s− is transfer open valued on conv(a); (ii) for each x, y ∈ k; int   ⋃ z∈[x,y] s−(z)   ∩ [x, y] =   ⋃ z∈[x,y] s−(z)   ∩ [x, y]; (iii) there exist a nonempty convex compact subset b of k and a nonempty compact subset d of k such that, for each y ∈ k \ d there exists x ∈ conv(b ∪ {y}) such that x ∈ s(y). then, either s has a maximal element or convs has a fixed point. proof. suppose that s has no maximal element, then ⋃ y∈x s−(y) = k. if γ(x) = k \ s−(x) for all x ∈ x, then ⋂ x∈k γ(x) = ∅. therefore, one of the assumptions of theorem 2.1 does not hold for γ. by condition (i), γ is transfer closed-valued on conva for any a ∈< k >. now suppose that x, y ∈ k, z ∈ [x, y] and z 6∈ ∩u∈[x,y]γ(u). then z ∈ ( ⋃ u∈[x,y] s−(u)) ∩ [x, y] and so z ∈ int( ⋃ u∈[x,y] s−(u)) ∩ [x, y]. therefore, there exists an open neighborhood u of z in k such that u ⊆ ⋃ u∈[x,y] s−(u). hence, u ⋂ (∩u∈[x,y]γ(u)) = ∅. that is z 6∈ clk ( ⋂ u∈[x,y] γ(u) ) ∩[x, y] and so condition (h3) is satisfied. also condition (iii) implies condition (h4). therefore, γ is not kkm map. thus, there exists a finite subset a = {x1, ..., xn} of k such that conv(a) * n ⋃ i=1 γ(xi). 142 m. fakhar and j. zafarani cubo 10, 4 (2008) this implies that there is a point x ∈ conv(a) such that x ∈ s−(xi) for all i = 1, ..., n. therefore, for all i = 1, ..., n, xi ∈ s(x) and x ∈ convs(x). remark 3.2. when k is compact, then trivially condition (iii) holds. furthermore, if s− is transfer open valued on k, then we can replace s−(z) in condition (ii) by ints−(z). hence, in this case conditions (ii) and (iii) are fulfilled. now we deduce the following version of theorem 1.2 in ansari and yao [1] as a corollary of our theorem 3.1. corollary 3.3. let k be a nonempty convex subset of a hausdorff topological vector space x. suppose that s, t : k ⇉ k are two set-valued maps with nonempty values such that (a) for each x ∈ k, a ∈ 〈s(x)〉, conv(a) ⊂ t (x); (b) for each a ∈< k >, s transfer open valued on conv(a); (c) for each x, y ∈ k, int   ⋃ z∈[x,y] s − (z)   ∩ [x, y] =   ⋃ z∈[x,y] s − (z)   ∩ [x, y]; (e) there exist a nonempty convex compact subset b of k and a nonempty compact subset d of k such that, for each y ∈ k \ d there exists x ∈ conv(b ∪ {y}) such that x ∈ s(y). then t has a fixed point. from theorem 3.1, we also deduce theorem 1.1 of djafari-rouhani, tarafdar and watson [6]. corollary 3.4. let k be a nonempty convex subset of a hausdorff topological vector space x. suppose that s, t : k ⇉ k are two set-valued maps with nonempty values that (a) for each x ∈ k, a ∈ 〈s(x)〉, conv(a) ⊂ t (x). (b) for each y ∈ k, s−(y) contains an open set oy, which may be empty such that k = ∪{oy : y ∈ k} (c) there exist a nonempty convex compact subset b of k and a points {x̂1, x̂2, ..., x̂n} in k such that ⋂ x∈b ocx ⊆ n ⋃ i=1 ox̂i , where ocx is the complement of ox in k. then t has a fixed point. proof. from (b-c), we obtain that ⋂ x∈b {k \ s−(x)} ⊆ ⋂ x∈b ocx ⊆ n ⋃ i=1 ox̂i ⊆ n ⋃ i=1 s−(x̂i). cubo 10, 4 (2008) a new version of fan’s theorem and its applications 143 let c = b ∪ {x̂1, x̂2, ..., x̂n}. then k = ⋃ x∈c s−(x). let h = conv(c), then h is compact and convex and moreover, h = ⋃ x∈c s−(x) ∩ h ⊆ ⋃ x∈h s−(x). now we define the set-valued mapping γ : h ⇉ h as γ(x) = h \ s−(x). then from remark 3.2, we conclude the proof. as another consequence of theorem 3.1, we obtain the following fixed point theorem of ansari and lin [2]. corollary 3.5. let k be a nonempty convex subset of a hausdorff topological vector space x. suppose that s, t : k ⇉ k are two mutivalued maps such that (a) for each x ∈ k, a ∈ 〈s(x)〉, conv(a) ⊂ t (x); (b) there exist a nonempty convex compact subset b of k and a points {x̂1, x̂2, ..., x̂n} in k such that ⋂ x∈b (k \ intk s − (x)) ⊆ n ⋃ i=1 intk s − (x̂i). then t has a fixed point. proof. it is enough in the above corollary to set ox = intk s − (x) for all x ∈ k. remark 3.6. by the same argument as in the above corollary,one can also obtain a proof for theorem 2.1 in [2]. 4 equilibrium problems we now give some new applications of theorem 2.1 in obtaining existence results of equilibrium problem. 144 m. fakhar and j. zafarani cubo 10, 4 (2008) theorem 4.1. let k be a nonempty convex subset of a hausdorff (t.v.s.) x. suppose f is a pseudomomtone real-valued on k × k such that: (a1) f (x, x) = 0 for any x ∈ x; (a2) for each x, y, z ∈ x if f (x, y) < 0 and f (x, z) ≤ 0, then f (x, u) < 0 for all u ∈]y, z[; (a3) for each a ∈< k >, f is transfer l.s.c. in the second variable on conv(a); (a4) f is 0-segmentary closed; (a5) there exist a nonempty compact subset d ⊆ k and a nonempty convex compact subset b of k such that for each x ∈ k \ d, there exists y ∈ conv(b ∪ {x}) such that f (x, y) > 0. then, there exists x̄ ∈ x such that f (y, x̄) ≤ 0 for all y ∈ x. proof. assume that γ̂ : k ⇉ k, γ : k ⇉ k are defined by: γ̂(y) = {x ∈ x : f (x, y) ≥ 0}, γ(y) = {x ∈ x : f (y, x) ≤ 0}. then, as f is pseudomonotone, γ̂(y) ⊆ γ(y) for all y ∈ k. by (a1) and (a2) γ̂ is a kkm map, so γ is a kkm map. condition (a5) implies condition (h4). therefore, the conditions (h1) and (h4) are fulfilled by the set-valued map γ. the condition (a3) implies that the condition (h2) holds for γ. for condition (h3), suppose z ∈ clk   ⋂ u∈[x,y] γ(u)   ∩ [x, y]. then, there exists a net (zα) converging to z such that f (u, zα) ≤ 0 for all u ∈ [x, y]. since z ∈ [x, y], for each v ∈ [u, z], we have also f (v, zα) ≤ 0, hence by (a4), we obtain f (u, z) ≤ 0 for each u ∈ [x, y]. thus, we have z ∈   ⋂ u∈[x,y] γ(u)   ∩ [x, y]. hence γ satisfies also the condition (h3). therefore, from theorem 2.1, we have ⋂ y∈x γ(y) 6= ∅. thus, any point x̂ in this intersection is a solution for our problem. corollary 4.2. in theorem 4.1, if f (., y) is upper sign continuous for every y ∈ k, then there exists x̄ ∈ k such that f (x, x̄) ≥ 0 for all x ∈ k. proof. by theorem 4.1, suppose that x̄ ∈ k such that f (y, x̄) ≤ 0 for all y ∈ x. assume that there exists ȳ ∈ k such that f (x̄, ȳ) < 0. by our assumption on x̄ we have also f (ȳ, x̄) ≤ 0. we will show that f (u, ȳ) ≥ 0 for all u ∈ ]x̄, ȳ[. indeed, if f (u, ȳ) < 0 for some u ∈ ]x̄, ȳ[, then as cubo 10, 4 (2008) a new version of fan’s theorem and its applications 145 f (u, x̄) ≤ 0, we obtain from (a2) that f (u, u) < 0, which contradicts (a1). now by upper sign continuity of f, we have f (x̄, ȳ) ≥ 0, which is a contradiction. the following example shows that corollary 4.2 improves the corresponding results in [3] and [13]. example 4.3. assume that k = r and f : k × k → r is defined as follows: f (x, y) :=                  −y if x = 0, 1 − y if x = 1, 1 if x = 3 2 , 5 > |y| ≥ 3, −2 + y if x = 2, 0 otherwise. let d = [−1, 2] and b = [1, 2]. for y < −1, we have f (1, y) > 0. if y > 2, then f (2, y) > 0. therefore, f satisfies all of the condition of corollary 4.2. but for x = 3/2, the set {y ∈ r : f (3/2, y) ≤ 0} =] − 3, 3[∪] − ∞, −5[∪]5, ∞[, which is not convex and k is not compact. as a consequence of our results, we conclude a new version of theorem 15 in [12] and its corollary for existence of variational inequalities problem. corollary 4.3. let k be a nonempty convex subset of a hausdorff (t.v.s.) x and t : k ⇉ x∗. suppose that (i) t is upper semicontinuous from conv(a) of any a ∈< k > to x∗ endowed with w*-topology and for each x ∈ k, t (x) is convex w*-compact; (ii) f (x, y) := infy∗∈t (y)〈y ∗, y − x〉 is 0-segmentary closed; (iii) there exist a nonempty compact subset d and a nonempty convex compact subset b of k such that, for each y ∈ k\d, there exists x ∈ conv(b ∪ {y}) such that inf y∗∈t (y) 〈y∗, y − x〉 > 0. then, there exist ȳ ∈ k and y∗ 0 ∈ t (ȳ) such that 〈y∗ 0 , x − ȳ〉 ≥ 0, ∀x ∈ k. proof. let f (x, y) = inf y∗∈t (y) 〈y∗, y − x〉. 146 m. fakhar and j. zafarani cubo 10, 4 (2008) we will show that all of the conditions of theorem 4.1 and its corollary are fulfilled by f . lemma 2.2 in [7] implies that for each fixed x ∈ k, the function y → infy∗∈t (y)〈y ∗, y−x〉 is l.s.c. on conv(a) of any a ∈< k >, hence we trivially have condition(a3) of theorem 4.1. since f (x, x) = 0, for every x ∈ kand f is affine in the first argument, hence f satisfies conditions (a1) and (a2). trivially f (., y) is upper sign continuous, thus from corollary 4.2, there exists ȳ ∈ k such that inf y∗∈t (ȳ) 〈y∗, ȳ − x〉 ≤ 0, ∀x ∈ k. now, let g : k × t (ȳ) → r be defined as follows: g(x, y∗) = 〈y∗, ȳ − x〉, then since t (ȳ) is convex, by kneser’s minimax theorem, we have inf y∗∈t (ȳ) sup x∈k 〈y∗, ȳ − x〉 = sup x∈k inf y∗∈t (ȳ) 〈y∗, ȳ − x〉 ≤ 0. therefore, there exists a point y∗0 ∈ t (ȳ) such that sup x∈k 〈y∗0 , ȳ − x〉 ≤ 0. acknowledgment. the authors were partially supported by the center of excellence for mathematics (university of isfahan). the first author was partially supported by a grant from ipm (no. 86470016). received: may 2008. revised: may 2008. references [1] ansari, q.h. and yao, j.c., a fixed point theorem and its applications to a system of variational inequalities, bull. austral. math. soc., 59 (1999), no. 3, 433–442. [2] ansari, q.h. and lin, y.c., fixed point and maximal element theorems with application to abstract economy, nonlinear stud., 13 (2006), no. 1, 43–52. [3] bianchi, m. and pini,r., coercivity conditions for equilibrium problems, j. optim. theory appl., 124 (2005), 79–92. [4] brézis, h., nirenberg, l. and stampacchia, g., a remark on ky fan’s minimax principle, bolletino della unione matematica italiana, 6 (1972), 293–300. cubo 10, 4 (2008) a new version of fan’s theorem and its applications 147 [5] chowdhury, m.s.r. and tan, k.k., generalization of ky fan’s minimax inequality with applications to generalized variational inequalities for pseudomonotone operators and fixedpoint theorem, j. math. anal. appl., 204 (1996), 910–929. [6] djafari rouhani, b., tarafdar, e. and watson, p.j., existence of solutions to some equilibrium problems, j. optim. theory appl., 126 (2005), 97–107. [7] ding, x.p. and tarafdar, e., generalized variational-like inequalities with pseudomonotone set-valued mappings, archiv. math., 74 (2000), 302–313. [8] fakhar, m. and zafarani, j., generalized equilibrium problems for quasimonotone and pseudomonotone bifunctions, j. optim. theory appl., 123 (2004), 349–364. [9] fakhar, m. and zafarani, j., generalized vector equilibrium problems for pseudomonotone multi-valued bifunctions, j. optim. theory appl., 126 (2005), 109–124. [10] fakhar, m. and zafarani, j., on generalized variational inequalities, to appear in j. global optim. [11] fan, k., a generalization of tychonoff’s fixed-point theorem, math. ann., 142 (1961), 305– 310. [12] inoan, d. and kolumban, j., on pseudomonotone set-valued mappings, nonlinear anal., 68 (2008), 47–53. [13] iusem, a.n. and sosa, w., new existence results for equilibrium problems, nonlinear. anal., 52 (2003), 621–635. [14] kalmoun, e.m., on ky fan’s minimax inequalities, mixed equilibrium problems and hemivariational inequalities, jipam. j. inequal. pure appl. math., 2 (2001), 1–13. [15] knaster, b., kuratowski, c. and mazurkiewicz, s., ein bewies des fixpunktsatzes für n-dimensional simplexe, fund. math., 14 (1929), 132–137. n11-fans-faza cubo a mathematical journal vol.11, n o ¯ 02, (107–115). may 2009 a multiobjective model of oligopolies under uncertainty carl chiarella school of finance and economics, university of technology, sydney, p.o. box 123, broadway, nsw 2007, australia email: carl.chiarella@uts.edu.au and ferenc szidarovszky systems & industrial engineering department, the university of arizona, tucson, arizona, 85721-0020, usa email: szidar@sie.arizona.edu abstract it is assumed that in an n-firm single-product oligopoly without product differentiation the firms face an uncertain price function, which is considered random by the firms. at each time period each firm simultaneously maximizes its expected profit and minimizes the variance of the profit since it wants to receive as high as possible profit with the least possible uncertainty. it is assumed that the best response of each firm is obtained by the weighting method. we show the existence of a unique equilibrium, and investigate the local stability of the equilibrium. resumen es asumido que en un oligopolio de n-firmas “single-product” sin diferenciación producto firmas con función de precio variable, son consideradas randon por las firmas. 108 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) en todo peŕıodo de tiempo todo firma simultaneamente maximiza la utilidad esperada y minimiza la variación de utilidad desde que estan quisen obtener la utilidad mas alta posible con la menor incertidumbre posible. es asumido que la mejor respuesta de toda firma es obtenida por el metodo weighting. mostramos la existencia de un equilibrio único y investigamos la estabilidad local del equilibrio. key words and phrases: uncertainty, n-person games, multiobjective optimization. math. subj. class.: 91a06, 90c29. 1 introduction the uncertainty in the inverse demand functions of oligopoly models has been previously examined by many researches. cyert and degroot (1971, 1973) investigated mainly duopolies. kirman (1975, 1983) examined the case of differentiated products and linear demand functions and analysed how the resulting equilibria depend on the way the firms misspecify and try to assess the demand function. gates et al. (1982) also examined linear demand functions and differentiated products. leonard and nishimura (1999) assumed that the firms know the shape of the demand function but misspecify its scale, and investigated the asymptotic behavior of the equilibrium under discrete time scales. chiarella and szidarovszky (2001) have introduced the continuous counterpart of the leonard-nishimura model and in addition to equilibrium and local stability analysis, the destabilising effect of time delays, in obtaining and implementing information on the competitors’ output, was analysed. bischi et al. (2004) consider the situation in which the firms’ reaction functions are unimodal and analyse the various equilibria that may arise and their complicated basins of attraction. all these earlier studies assumed that the firms maximized their misspecified or expected payoffs at each time period depending on the type (deterministic or stochastic) of the model being used. in this paper we introduce a new approach. the uncertainty of the inverse demand function is treated here also with a stochastic model, but we assume that at each time period each firm maximize its expected profit and at the same time tends to reduce profit uncertainty by minimizing its variance. that is, at each time period the firms face a “pareto-game”, each with multiple payoffs (see for example szidarovszky et al., 1986). in our model, in each time period each firm uses a multiobjective optimization approach to find its best response. based on these best response functions a dynamic process develops. the subject of this paper is the properties of this dynamic process including equilibrium analysis and the investigation of the asymptotic behavior of the equilibrium. cubo 11, 2 (2009) a multiobjective model of oligopolies under uncertainty 109 2 the mathematical model and equilibrium analysis consider an n-firm single-product oligopoly without product differentiation. let qj be the output of firm j, and cj (qj ) the associated cost function. assume that firm j believes that the true price function is f (q) + ηj , where q = ∑n j=1 qj is the total output of the industry and ηj is a random variable such that e(ηj ) = 0 and v ar(ηj ) = σ 2 j . (2.1) using πj to denote profit, the expected profit of firm j is given as e(πj ) = qj f (q) − cj (qj ), (2.2) and the variance of profit is v ar(πj ) = q 2 j σ 2 j . (2.3) assume that firm j wants to maximize its expected profit and at the same time to minimize the variance of the profit. that is, the firm tends to obtain as high a profit as possible with minimum uncertainty. it is also assumed that firm j uses the weighting method (see for example, szidarovszky et al. 1986), therefore it maximizes a linear combination of the two objective functions, i.e. max [ e(πj ) − αj 2 v ar(πj ) ] , (2.4) where αj shows the relative importance of reducing uncertainty compared to the increase of the expected profit. in oligopoly theory it is usually assumed that the functions f and cj (j = 1, 2, · · · , n) are twice continuously differentiable, f is decreasing, cj is increasing, furthermore (a) f ′ (q) + qj f ′′ (q) ≤ 0, (b) f ′ (q) − c′′j (qj ) < 0, for all j and nonnegative qj and q. under conditions (a) and (b) the composite objective function (2.4) is strictly concave in qj with fixed value of qj = ∑ l 6=j ql. the derivative of the objective function (2.4) with respect to qj can be given as qj f ′ (q) + f (q) − c′j (qj ) − αj qj σ 2 j . notice that f ′ ≤ 0, c′ ≥ 0, f (q) ≤ f (0), therefore with positive αj and σj , this derivative converges to −∞ as qj → ∞ implying that there is a unique maximizing value, qj ≥ 0, with any fixed qj ≥ 0. the best response of firm j, rj (qj ), can be obtained in the following way. if f (qj ) − c ′ j (0) ≤ 0, (2.5) then rj (qj) = 0, otherwise it is the unique positive solution of the equation qj f ′ (qj + qj) + f (qj + qj ) − c ′ j (qj ) − αj qj σ 2 j = 0. (2.6) 110 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) since f is decreasing, from condition (2.5) we see that if rj (qj ) = 0 then for all q̄j > qj , rj (q̄j ) = 0. assume next that rj (qj) > 0. then equation (2.6) is satisfied with qj = rj (qj ). by implicit differentiation we have 1 r ′ j f ′ + rj f ′′ (1 + r ′ j ) + f ′ (1 + r ′ j ) − c ′′ j r ′ j − αj r ′ j σ 2 j = 0, from which we have r ′ j = − f ′ + rj f ′′ 2f ′ + rj f ′′ − c′′j − αj σ 2 j . (2.7) conditions (a) and (b) imply that −1 < r′j ≤ 0. (2.8) therefore rj is a decreasing function of qj. we can rewrite equation (2.6) as qj f ′ (qj + qj ) + f (qj + qj) − c ′ j (qj ) = αj qj σ 2 j . (2.9) the left hand side strictly decreases in qj , therefore the solution qj = rj (qj) decreases if αj and/or σj increases. we can also consider qj as a function of the total output level of the industry, qj = qj (q), which can be defined as follows. if f (q) − c′j (0) ≤ 0, (2.10) then qj (q) = 0, otherwise it is the unique positive solution of the equation qj f ′ (q) + f (q) − c′j (qj ) − αj qj σ 2 j = 0. (2.11) with fixed values of q, the left hand side is strictly decreasing in qj , it has a positive value at qj = 0 and converges to −∞ as qj → ∞. similarly to the previous case, condition (2.5) implies that if qj (q) = 0 then for all q̄ > q we have qj (q̄) = 0. if qj (q) > 0, then by letting q ′ j = d dq qj (q) we have q ′ j f ′ + qj f ′′ + f ′ − c′′j q ′ j − αj q ′ j σ 2 j = 0, implying that q ′ j = − f ′ + qj f ′′ f ′ − c′′j − αj σ 2 j ≤ 0, (2.12) so qj (q) is decreasing in q. we can rewrite equation (2.11) as qj f ′ (q) + f (q) − c′j (qj ) = αj qj σ 2 j . (2.13) the left hand side is decreasing in qj , therefore the solution qj (q) decreases if αj and/or σj increases. 1we use the notation r′ j = d dqj rj (qj ). cubo 11, 2 (2009) a multiobjective model of oligopolies under uncertainty 111 the total industry output q at the equilibrium is the solution of n ∑ j=1 qj (q) − q = 0. (2.14) at q = 0, all qj (q) ≥ 0, so at q = 0 the left hand side is nonnegative. since n ∑ j=1 qj (q) ≤ n ∑ j=1 qj (0), it follows that the left hand side converges to −∞ as q → ∞, furthermore it is strictly decreasing in q. consequently there is a unique nonnegative solution of equation (2.14) proving the existence and the uniqueness of the equilibrium. in summary, we have the following result. theorem 1 under conditions (a) and (b) there is a unique equilibrium of the modified nperson oligopoly with payoff functions (2.4). the unique equilibrium of theorem 1 is usually different from the cournot-nash equilibrium of the n-firm oligopoly under the assumption that all firms know the true price function f . let q ∗ (α, σ) denote the total industry output at the equilibrium with given parameters α = (α1, · · · , αn) and σ = (σ1, · · · , σn). we will prove the following result: theorem 2 the value of q∗(α, σ) decreases if any αj or σj increases. proof: assume that αj < ᾱj with all other αi and all σi values unchanged. let qi and q̄i denote the corresponding equilibrium outputs and let q = ∑ i qi and q̄ = ∑ i q̄i. contrary to the assertion assume that q < q̄. from the monotonicity of the functions qi(q) we have for all i 6= j, qi(q) ≥ qi(q̄) = q̄i(q̄), (2.15) since αi and σi do not change. however qj (q) ≥ q̄j (q) ≥ q̄j (q̄), (2.16) where we have used the monotonicity of the function qj (q) in αj . hence q = n ∑ i=1 qi(q) ≥ n ∑ i=1 q̄i(q̄) = q̄, (2.17) which is an obvious contradiction. the assertion of theorem 2 can be reformulated as follows: if any firm increases its weight αj of uncertainty and, or assumes larger level σ 2 j of uncertainty of the price function, then the total industry output decreases at the equilibrium. 3 dynamic models and local stability analysis we recall from the previous section that rj (qj) denotes the best response of firm j. in this section we consider dynamic processes with the firms’ adjustment of output based on their best responses. 112 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) considering continuous time scales first and assuming that each firm adjusts its output in the direction toward its best response, we obtain the system of ordinary differential equations q̇j = kj (rj (qj ) − qj ), (j = 1, 2, · · · , n), (3.1) where kj is a sign preserving function, i.e. kj (∆)    = 0, if ∆ = 0, > 0, if ∆ > 0, < 0, if ∆ < 0. theorem 3 the equilibrium is locally asymptotically stable under conditions (a) and (b) and by assuming that k′j (0) > 0 for all j. proof: the jacobian of system (3.1) at the equilibrium has the special form, j c =      −k′1 k ′ 1r ′ 1 · · · k ′ 1r ′ 1 k ′ 2r ′ 2 −k ′ 2 · · · k ′ 2r ′ 2 . . . . . . . . . k ′ nr ′ n k ′ nr ′ n · · · −k ′ n      , (3.2) where k ′ j = k ′ j(0), and r ′ j is the derivative of rj at the equilibrium. the jacobian (3.2) may be represented as j c = k + a · 1 t , (3.3) with k = diag(−k′1(1 + r ′ 1), · · · , −k ′ n(1 + r ′ n)), 1 t = (1, · · · , 1) and a = (k ′ 1r ′ 1, k ′ 2r ′ 2, · · · , k ′ nr ′ n) t . the characteristic polynomial of j c is given by det(k + a · 1t − λi) = det(k − λi)det(i + (k − λi)−1a · 1t ) =π n i=1(−k ′ i(1 + r ′ i) − λi) [ 1 + n ∑ i=1 k ′ ir ′ i −k′i(1 + r ′ i) − λ ] . (3.4) notice that relation (2.8) implies that 1 + r ′ j > 0 for all j, so all roots of the first product are real and negative. it is therefore sufficient to show that all roots of the equation n ∑ i=1 k ′ ir ′ i −k′i(1 + r ′ i) − λ = −1, (3.5) are also real and negative. we might assume that the denominators are different, otherwise the sum of terms with identical denominators can be represented as a single term where the numerators cubo 11, 2 (2009) a multiobjective model of oligopolies under uncertainty 113 are added. equation (3.5) is equivalent to a polynomial equation of degree n, so there are n real or complex roots. let g(λ) denote the left hand side, then clearly lim λ→±∞ g(λ) = 0, lim λ→−k′ i (1+r′ i )±0 g(λ) = ±∞, and g ′ (λ) = n ∑ i=1 k ′ ir ′ i (−k′i(1 + r ′ i) − λ) 2 < 0, so g is strictly decreasing locally. the graph of the function g is shown in figure 1. there are n negative poles at λ = −k′j(1 + r ′ j ) (j = 1, 2, · · · , n), there is a root between each pair of consecutive poles and there is an additional root before the first pole. we have demonstrated that there are n real negative roots, consequently all roots are real and negative. g(λ) λ −1 b b b b b b −k ′ 1 (1 + r ′ 1 ) −k ′ 2 (1 + r ′ 2 ) −k ′ n−2 (1 + r ′ n−2 ) −k ′ n−1 (1 + r ′ n−1 ) −k ′ n(1 + r ′ n) . . . figure 1: graph of the function g(λ) considering discrete time scales next, we assume again that the firms adjust their outputs into the direction toward their best responses, and so the outputs adjust according to qj (t + 1) = qj (t) + kj (rj (qj ) − qj ), (j = 1, 2, · · · , n), (3.6) where kj is a sign-preserving function for all j. 114 carl chiarella and ferenc szidarovszky cubo 11, 2 (2009) theorem 4 assume that conditions (a) and (b) hold, furthermore k′j(0) > 0 for all j. the equilibrium is locally asymptotically stable if k ′ j(0) < 2 1 + r ′ j (q ∗ j ) , (3.7) for all j, where q∗ = (q∗j ) is the equilibrium and q ∗ j = ∑ i6=j q ∗ i , and n ∑ j=1 k ′ j(0)r ′ j (q ∗ j ) 2 − k′j(0)(1 + r ′ j (q ∗ j )) > −1. (3.8) if any of these conditions is violated with strict inequality, then the equilibrium is unstable. proof: the jacobian of system (3.6) at the equilibrium can be written as j d = i + j c , (3.9) where i is the n × n identity matrix. all eigenvalues of j d are inside the unit circle if and only if all eigenvalues of j c are inside the interval (−2, 0). from the proof of theorem 3 we know that all eigenvalues of j c are negative, so the eigenvalues are larger than −2 if and only if −k′j(1 + r ′ j ) > −2, (3.10) for all j, and g(−2) > −1. (3.11) notice that inequality (3.10) can be rewritten as (3.7), and inequality (3.11) can also be rewritten as (3.8). if either (3.10) or (3.11) is violated with strict inequality, then at least one eigenvalue of j c becomes less than −2, so at least one eigenvalue of j d is below −1, demonstrating the instability of the equilibrium in this case. notice that all denominators of inequality (3.7) are positive because of the relation (2.8). condition (3.7) implies that all denominators on the left hand side of (3.8) are also positive. therefore condition (3.8) can be interpreted as stating that all derivatives k ′ j(0) must be sufficiently small in order to guarantee the local asymptotical stability of the equilibrium. received: april 22, 2008. revised: may 16, 2008. cubo 11, 2 (2009) a multiobjective model of oligopolies under uncertainty 115 references [1] bischi, g.i., chiarella, c. and m. kopel (2004), “the long run outcomes and global dynamics of a duopoly game with misspecified demand functions”, international game theory review, 6(3), 343–380. 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(1983), “on mistaken beliefs and resultant equilibria”, in individual forecasting and aggregate outcomes, (eds. r. frydman and e.s. phelps), cambridge university press, new york, 147–166. [8] leonard, d. and k. nishimura (1999), “nonlinear dynamics in the cournot model without full information”, annals of operations research, 89, 165–173. [9] szidarovszky, f., gershon, m. and l. duckstein (1986), techniques of multiobjective decision making in systems management, elsevier, amsterdam. n07-olgply_multiobj_v3 calculs&_nbsp_08.dvi cubo a mathematical journal vol.12, no¯ 03, (121–138). october 2010 calculations in new sequence spaces and application to statistical convergence bruno de malafosse lmah université du havre, bp 4006 iut le havre, 76610 le havre. france email: bdemalaf@wanadoo.fr and vladimir rakočević1 department of mathematics, university of niš, videgradska 33, 18000 niš, serbia email: vrakoc@bankerinter.net abstract in this paper we recall recent results that are direct consequences of the fact that (w∞ (λ) , w∞ (λ)) is a banach algebra. then we define the set wτ = dτw∞ and characterize the sets wτ (a) where a is either of the operators ∆, σ, ∆(λ), or c (λ). afterwards we consider the sets [a1, a2]wτ of all sequences x such that a1 (λ) (∣∣a2 ( µ ) x ∣∣) ∈ wτ where a1 and a2 are of the form c (ξ), c + (ξ), ∆(ξ), or ∆+ (ξ) and it is given necessary conditions to get [ a1 (λ) , a2 ( µ )] wτ in the form wξ. finally we apply the previous results to statistical convergence. so we have conditions to have xk → l (s ( a)) where a is either of the infinite matrices d1/τc (λ) c ( µ ) , d1/τ∆(λ)∆ ( µ ) , d1/τ∆(λ) c ( µ ) . we also give conditions to have xk → 0 (s ( a)) where a is either of the operators d1/τc + (λ)∆ ( µ ) , d1/τc + (λ) c ( µ ) , d1/τc + (λ) c+ ( µ ) , or d1/τ∆(λ) c + ( µ ) . 1supported by grant no. 144003 of the ministry of science, technology and development, republic of serbia. 122 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) resumen recordamos resultados recientes que son consecuencia directa del hecho de que (w∞(λ), w∞(λ)) es una algebra de banach. entonces nosotros definimos el conjunto wτ = dτw∞ y caracterizamos los conjuntos wτ (a) donde a es uno de los siguientes operadores ∆, σ, ∆(λ), o c (λ). después consideramos los conjuntos [a1, a2 ]wτ de todas las sucesiones x tal que a1 (λ) (∣∣a2 ( µ ) x ∣∣) ∈ wτ donde a1 y a2 son de la forma c (ξ), c+ (ξ), ∆(ξ), o ∆+ (ξ) y son dadas condiciones necesarias para obtener [ a1 (λ) , a2 ( µ )] wτ en la forma wξ. finalmente, aplicamos los resultados previos para tener xk → l (s ( a)) donde a es una de las matrices infinitas d1/τc (λ) c ( µ ) , d1/τ∆(λ)∆ ( µ ) , d1/τ∆(λ) c ( µ ) . nosotros también damos condiciones para tener xk → 0 (s ( a)) donde a es uno de los operadores d1/τc + (λ)∆ ( µ ) , d1/τc + (λ) c ( µ ) , d1/τc + (λ) c+ ( µ ) , o d1/τ∆(λ) c + ( µ ) . key words and phrases: banach algebra, statistical convergence, a−statistical convergence, infinite matrix. math. subj. class.: 40c05, 40f05, 40j05, 46a15. 1 introduction in this paper we consider spaces generalizing the well-known sets w0 and w∞ introduced and studied by maddox [12, 13]. recall that w0 and w∞ are the sets of strongly summable and strongly bounded sequences. in [15] malkowsky and rakočević gave characterizations of matrix maps between w0, w, or w∞ and w p ∞ and between w 0, w, or w∞ and l1. in [2] de malafosse defined the spaces wα (λ), w (c) α (λ) and w 0 α (λ) of all sequences that are α−strongly bounded, summable and summable to zero respectively. for instance recall that wα (λ) is the set of all sequences (xn)n such that 1/λn ∑n m=1 |xm| = αno (1) as n tends to infinity. it was shown that these spaces can be written in the form sξ, s (c) ξ and s0 ξ under some condition on α and λ. more recently in [5] it was shown that if λ is a sequence exponentially bounded then (w∞ (λ) , w∞ (λ)) is a banach algebra. this result led to consider bijective operators mapping between w∞ (λ). here we will use these results to study sets of the form wτ = dτw∞, wτ (∆(λ)), wτ (c (λ)) and wτ ( c+ (λ) ) generalizing the well-known set of strongly bounded sequences c∞ = w∞ ( ∆ ( µ )) where µn = n for all n. these results lead to the study of statistical convergence which was introduced by steinhaus in 1949, see [16], and studied by several authors such as fast [7], fridy, orhan [8-11] and connor. here we will deal with the notion of a− statistical convergence which generalizes the notion of statistical convergence, see [6], where a belongs to a special class of operators. cubo 12, 3 (2010) calculations in new sequence spaces ... 123 the paper is organized as follows. in section 2 among other things we recall a recent result on the operators ∆ρ and ∆ t ρ considered as map from w∞ (λ) to itself. in sections 3 and 4 our aim is to give necessary conditions to have wτ (a) in the form wξ when a is either one of the matrices ∆(λ), c (λ) or c+ (λ). then we consider spaces generalizing the wellknown set of all strongly bounded sequences [c,∆] = c∞ defined and studied by maddox. then we will define the sets [a1, a2]wτ of all sequences x with a1 (λ) (∣∣a2 ( µ ) x ∣∣) ∈ wτ where a1 and a2 are of the form c (ξ), c + (ξ), ∆(ξ), or ∆+ (ξ) and we will give necessary conditions to get [ a1 (λ) , a2 ( µ )] in the form wτ. in section 5 we apply these results to a− statistical convergence, where a is equal to d1/τ a1 a2 and a1, a2 are of the form c (ξ), ∆(ξ), ∆ ( µ ) , or c+ (ξ). 2 well known results for a given infinite matrix a = (anm)n,m≥1 we define the operators an for any integer n ≥ 1, by an (x ) = ∞∑ m=1 anm xm (1) where x = (xn)n≥1, the series intervening in the second member being convergent. so we are led to the study of the infinite linear system an (x ) = bn n = 1, 2, ... (2) where b = (bn)n≥1 is a one-column matrix and x the unknown, see [2-5]. the equations (2) can be written in the form a x = b, where a x = (an (x ))n≥1. in this paper we shall also consider a as an operator from a sequence space into another sequence space. we will write s for the set of all complex sequences and ℓ∞ for the set of all bounded sequences. let e and f be any subsets of s. when a maps e into f we write that a ∈ (e, f). so for every x ∈ e, a x ∈ f, (a x ∈ f means that for each n ≥ 1 the series defined by yn =∑ ∞ m=1 anm xm is convergent and ( yn)n≥1 ∈ f). body math for any subset e of s, we put body math ae = {y ∈ s : y = a x for some x ∈ e} . (3) if f is a subset of s, we shall denote f (a) = fa = {x ∈ s : y = a x ∈ f} . (4) 124 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) in all what follows we will use the set u + = { (un )n≥1 ∈ s : un > 0 for all n } and the notation e = (1, ..., 1, ...). so for λ = (λn)n≥1 ∈ u + we will consider the sets of strongly bounded and strongly summable sequences, respectively, that is w∞ (λ) = { x = (xn)n≥1 ∈ s : sup n 1 λn n∑ m=1 |xm| < ∞ } , w 0 (λ) = { x = (xn)n≥1 ∈ s : lim n→∞ 1 λn n∑ m=1 |xm| = 0 } and w (λ) = { x = (xn)n≥1 ∈ s : x − l e ∈ w 0 (λ) for some l ∈ c } were studied by malkowsky, with the concept of exponentially bounded sequences, see [3]. recall that maddox [12, 13], defined and studied the sets w∞ (λ) = w∞, w0 (λ) = w 0 and w (λ) = w where λn = n for all n. a banach space e of complex sequences with the norm ‖‖e is a bk space if each projection pn : x 7→ pn x = xn is continuous. a bk space e is said to have ak if every sequence x = (xn)n≥1 ∈ e has a unique representation x = ∑ ∞ n=1 xn e n where e n is the sequence with 1 in the n-th position and 0 otherwise. recall that a nondecreasing sequence λ = (λn)n≥1 ∈ u + is exponentially bounded if there is an integer m ≥ 2 such that for all non-negative integers ν there is at least one term λn ∈ i (ν) m = [ mν, mν+1 − 1 ] . it was shown (cf. [14, lemma 1]) that a non-decreasing sequence λ = (λn)n≥1 is exponentially bounded if and only if there are reals s ≤ t such that for some subsequence ( λni ) i≥1 0 < s ≤ λni λni+1 ≤ t < 1 for all i = 1, 2, ...; such a sequence is called an associated subsequence. consider now the norm ‖x‖λ = sup n ( 1 λn n∑ m=1 |xm| ) . in [5] it was shown that if λ = (λn)n≥1 ∈ u + is exponentially bounded the class (w∞ (λ) , w∞ (λ)) is a banach algebra with the norm ‖a‖(w∞(λ),w∞(λ)) = sup x 6=0 ( ‖a x‖λ ‖x‖λ ) . (5) cubo 12, 3 (2010) calculations in new sequence spaces ... 125 for ρ = ( ρn ) n≥1 consider now the following matrices ∆ + ρ =   1 −ρ1 . . 1 −ρn 0 . .   and ∆ρ =   1 −ρ1 1 0 . . −ρn−1 1 . . .   . it can easily be shown that if ρ = ( ρn ) n≥1 and (λn+1/λn)n≥1 ∈ ℓ∞ then ∆ + ρ ∈ (w∞ (λ) , w∞ (λ)). we also see that ∆ρ ∈ (w∞ (λ) , w∞ (λ)) for ρ, (λn−1/λn)n≥2 ∈ ℓ∞. recall the next result which is a direct consequence of [5, theorem 5.1 and theorem 5.12]. lemma 2.1. let λ ∈ u+ be a sequence exponentially bounded. (i) if lim n→∞ ( λn+1 λn ) < ∞ and lim n→∞ ∣∣ρn ∣∣ < 1 lim n→∞ ( λn+1 λn ) , (6) for given b ∈ w∞ (λ) the equation ∆ + ρ x = b has a unique solution in w∞ (λ). (ii) if lim n→∞ ∣∣ρn ∣∣ < 1 lim n→∞ ( λn−1 λn ) , (7) then for any given b ∈ w∞ (λ) the equation ∆ρ x = b has a unique solution in w∞ (λ). when λ is a strictly increasing sequence tending to infinity we obtain similar results on the banach algebra ( w0 (λ) , w0 (λ) ) with the norm ‖a‖(w∞(λ),w∞(λ)). 3 on the sets wτ (a) where a is either ∆(λ), c (λ) or c + (λ) in the following we will use the operators represented by c (λ) and ∆(λ). let u be the set of all sequences (un )n≥1 with un 6= 0 for all n. we define c (λ) for λ = (λn)n≥1 ∈ u, by [c (λ)]nm =    1 λn if m ≤ n, 0 otherwise. we will write c (λ)t = c+ (λ), c (e) = σ, σ+ = σt , and for λn = n, the matrix c1 = c ((n)n) is called the cesaro operator. if it can be proved that the matrix ∆(λ) with [∆(λ)]nm =    λn if m = n, −λn−1 if m = n − 1 and n ≥ 2, 0 otherwise, 126 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) is the inverse of c (λ), see [2, 3]. we will use the following sets γ = { x ∈ u + : lim n→∞ ( xn−1 xn ) < 1 } , γ + = { x ∈ u + : lim n→∞ ( xn+1 xn ) < 1 } . note that x ∈ γ+ if and only if 1/x ∈ γ. for given sequence τ = (τn)n≥1 ∈ u +, we write dτ for the diagonal matrix defined by [dτ]nn = τn for all n. for any subset e of s, we write dτe = { x = (xn)n≥1 ∈ s : ( xn τn ) n ∈ e } . we put wτ = dτw∞ for τ ∈ u +, that is wτ = { x : ‖x‖wτ = sup n ( 1 n ∞∑ m=1 |xm| τm ) < ∞ } . it can easily be seen that wτ = w∞ (d1/τ) is a bk space with norm ‖‖wτ , (cf. [17, theorem 4.3.6, p. 52]). in all that follows we will use the convention that the entries with subscripts strictly less than 1 are equal to zero. then we are interested in the study of the following sets where λ, τ ∈ u+. wτ (∆(λ)) = { x : sup n ( 1 n n∑ m=1 1 τm |λm xm −λm−1 xm−1| ) < ∞ } , wτ (c (λ)) = { x : sup n 1 n n∑ m=1 ( 1 λmτm m∑ k=1 |xk| ) < ∞ } , wτ ( c + (λ) ) = { x : sup n 1 n n∑ m=1 ( 1 τm ∞∑ k=m |xk| λk ) < ∞ } . note that for λn = n and τ = e, wτ (∆(λ)) is the well known set of all strongly and bounded sequences c∞. we obtain the following result that is a direct consequence of lemma 2.1. proposition 3.1. (i) if τ ∈ γ then the operators ∆ and σ are bijective from wτ into itself and wτ (∆) = wτ, wτ (σ) = wτ. (ii) a) if λτ ∈ γ then wτ (c (λ)) = wλτ. b) if τ ∈ γ then wτ (∆(λ)) = wτ/λ. cubo 12, 3 (2010) calculations in new sequence spaces ... 127 (iii) let τ ∈ γ+. then a) the operators ∆+ and σ+ are bijective from wτ into itself and wτ ( σ + ) = wτ. b) the operator c+ (λ) is bijective from wλτ into wτ and wτ ( c + (λ) ) = wλτ. proof. (i) by lemma 2.1 where ρn = τn−1/τn and λn = n for all n, we easily see that if lim n→∞ τn−1 τn < 1 limn→∞ ( n−1 n ) = 1, that is τ ∈ γ, then d1/τ∆dτ is bijective from w∞ to itself. this means that ∆ is bijective from dτw∞ to itself. since σ is also bijective from dτw∞ to itself, this shows wτ (∆) = wτ and wτ (σ) = wτ. (ii) we have x ∈ wτ (c (λ)) if and only if σx ∈ dλτw∞ = wλτ. this means that x ∈ wλτ (σ) and by (i) the condition λτ ∈ γ implies wλτ (σ) = wλτ. then wτ (c (λ)) = wλτ and c (λ) is bijective from wλτ to wτ. since ∆(λ) = c (λ) −1 we conclude ∆(λ) bijective from wτ to wλτ and wλτ (∆(λ)) = wτ. we deduce that for τ ∈ γ, wτ (∆(λ)) = wτ/λ. (iii) a) by lemma 2.1 with ρn = τn+1/τn and λn = n we have ∆ + ρ = d1/τ∆ +dτ and ∆ + is bijective from dτw∞ = wτ into itself for τ ∈ γ + and it is the same for σ+. now the equation σ + x = y for y ∈ wτ is equivalent to ∞∑ m=n xm = yn for all n. (8) we deduce (8) has a unique solution x = ( yn − yn+1)n≥1 = ∆ +y ∈ wτ and wτ ( σ + ) = wτ. b) we have wτ ( c + (λ) ) = { x : σ + d1/λ x ∈ wτ } = dλwτ ( σ + ) . now as we have seen above since τ ∈ γ+ we get wτ ( σ + ) = wτ and wτ ( c + (λ) ) = dλwτ ( σ + ) = dλwτ = wλτ. this gives the conclusion. 128 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) 4 calculations in new sequence spaces 4.1 the sets [c,∆]wτ , [c, c]wτ , [ c+,∆ ] wτ , [ c+, c ] wτ and [ c+, c+ ] wτ . in [4], were defined and studied the sets [a1, a2] = [ a1 (λ) , a2 ( µ )] = { x ∈ s : a1 (λ) (∣∣a2 ( µ ) x ∣∣) ∈ dτl∞ } where |x| = (|xn|)n≥1, a1 and a2 of the form c (ξ), c + (ξ), ∆(ξ), or ∆+ (ξ) for ξ ∈ u+. it was given necessary conditions to get [ a1 (λ) , a2 ( µ )] in the form sγ. similarly in the following we will put [a1, a2]wτ = [ a1 (λ) , a2 ( µ )] wτ = { x ∈ s : a1 (λ) (∣∣a2 ( µ ) x ∣∣) ∈ wτ } for λ, µ, τ ∈ u+. we can explicitly write the previous sets [a1, a2]wτ as follows. [c,∆]wτ = { x : sup n ( 1 n n∑ m=1 1 λmτm m∑ k=1 ∣∣µk xk −µk−1 xk−1 ∣∣ ) < ∞ } , [c, c]wτ = { x : sup n ( 1 n n∑ m=1 ( 1 λmτm m∑ k=1 1 µk ∣∣∣∣∣ k∑ i=1 xi ∣∣∣∣∣ )) < ∞ } , [ c + ,∆ ] wτ = { x : sup n ( 1 n n∑ m=1 ( 1 τm ∞∑ k=m 1 λk ∣∣µk xk −µk−1 xk−1 ∣∣ )) < ∞ } , [ c + , c ] wτ = { x : sup n ( 1 n n∑ m=1 ( 1 τm ∞∑ k=m 1 λk 1 µk ∣∣∣∣∣ k∑ i=1 xi ∣∣∣∣∣ )) < ∞ } , [ c + , c + ] wτ = { x : sup n ( 1 n n∑ m=1 ( 1 τm ∞∑ k=m 1 λk ∣∣∣∣∣ ∞∑ i=k xi µi ∣∣∣∣∣ )) < ∞ } . note that if λn = µn for all n we get the well known set of sequences that are strongly bounded [c,∆]we = c∞ (λ). we can state the following. theorem 4.1. let λ, µ,τ ∈ u+. (i) if λτ ∈ γ then [c,∆]wτ = wλτ/µ; (ii) if λτ, λµτ ∈ γ then [c, c]wτ = wλµτ; (iii) if τ ∈ γ+ and λτ ∈ γ then [ c + ,∆ ] wτ = wλτ/µ; cubo 12, 3 (2010) calculations in new sequence spaces ... 129 (iv) if τ ∈ γ+ and λµτ ∈ γ then [ c + , c ] wτ = wλµτ; (v) if τ, λτ ∈ γ+ then [ c + , c + ] wτ = wλµτ. proof. in the following we will use the fact that for any ξ ∈ u+ we have |x| ∈ wξ if and only if x ∈ wξ. (i) we have c (λ) (∣∣∆ ( µ ) x ∣∣) ∈ wτ if and only if ∣∣∆ ( µ ) x ∣∣ ∈ wτ (c (λ)) and by proposition 3.1, since λτ ∈ γ we get wτ (c (λ)) = wλτ. then by proposition 3.1 (ii) we have wλτ ( ∆ ( µ )) = wλτ/µ and we conclude ∆ ( µ ) x ∈ wλτ if and only if x ∈ wλτ ( ∆ ( µ )) = wλτ/µ, that is [c,∆]wτ = wλτ/µ. (ii) here we have c (λ) (∣∣c ( µ ) x ∣∣) ∈ wτ if and only if ∣∣c ( µ ) x ∣∣ ∈ wτ (c (λ)); and since λτ ∈ γ by proposition 3.1 we have wτ (c (λ)) = wλτ. so x ∈ [c, c]wτ if and only if c ( µ ) x ∈ wλτ, that is x ∈ wλτ ( c ( µ )) . then by proposition 3.1 (ii) a) λµτ ∈ γ implies wλτ ( c ( µ )) = wλµτ and we have shown (ii). (iii) for any given x ∈ [ c+,∆ ] wτ we have ∆ ( µ ) x ∈ wτ ( c+ (λ) ) and for τ ∈ γ+ we have wτ ( c+ (λ) ) = wλτ. now the condition λτ ∈ γ implies x ∈ [ c+,∆ ] wτ if and only if x ∈ wλτ ( ∆ ( µ )) = wλτ/µ and we have shown (iii). (iv) let x ∈ [ c+, c ] wτ . we have τ ∈ γ+ implies wτ ( c+ (λ) ) = wλτ and so x ∈ [ c+, c ] wτ if and only if c ( µ ) x ∈ wλτ. now since λµτ ∈ γ we have wλτ ( c ( µ )) = wλµτ and we conclude[ c+, c ] wτ = wλµτ. (v) as above x ∈ [ c+, c+ ] wτ if and only if c+ ( µ ) x ∈ wτ ( c+ (λ) ) and the condition τ ∈ γ+ implies wτ ( c+ (λ) ) = wλτ. since λτ ∈ γ + we conclude wλτ ( c+ ( µ )) = wλµτ that is [ c+, c+ ] wτ = wλµτ. now we are led to study sets of the form [∆, a2 ]wτ for a2 ∈ { ∆,∆, c+ } . 4.2 the sets [∆,∆]wτ , [∆, c]wτ and [ ∆, c+ ] wτ using the convention µ0 = 0, and the notation ∆ ( µ ) xm = µm xm − µm−1 xm−1 for m ≥ 1 we explicitly have [∆,∆]wτ = { x : sup n ( 1 n n∑ m=1 1 τm ∣∣λm ∣∣∆ ( µ ) xm ∣∣ −λm−1 ∣∣∆ ( µ ) xm−1 ∣∣∣∣ ) < ∞ } , [∆, c]wτ = { x : sup n ( 1 n n∑ m=1 1 τm ∣∣∣∣∣λm ∣∣∣∣∣ 1 µm m∑ k=1 xk ∣∣∣∣∣ −λm−1 ∣∣∣∣∣ 1 µm−1 m−1∑ k=1 xk ∣∣∣∣∣ ∣∣∣∣∣ ) < ∞ } , 130 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) [ ∆, c + ] wτ = { x : sup n ( 1 n n∑ m=1 1 τm ∣∣∣∣∣λm ∣∣∣∣∣ ∞∑ k=m xk µk ∣∣∣∣∣ −λm−1 ∣∣∣∣∣ ∞∑ k=m−1 xk µk ∣∣∣∣∣ ∣∣∣∣∣ ) < ∞ } . as a direct consequence of proposition 3.1 we also obtain the following results. theorem 4.2. let λ, µ, τ ∈ u+. then (i) if τ, τ/λ ∈ γ then [∆,∆]wτ = wτ/λµ. (ii) if τ, τµ/λ ∈ γ then [∆, c]wτ = wτµ/λ. (iii) if τ, τ/λ ∈ γ+ then [ ∆, c + ] wτ = wτµ/λ. proof. (i) let x ∈ [∆,∆]wτ . since τ ∈ γ we have wτ (∆(λ)) = wτ/λ and ∆(λ) ∣∣∆ ( µ ) x ∣∣ ∈ wτ means ∆ ( µ ) x ∈ wτ/λ. we conclude wτ/λ ( ∆ ( µ )) = wτ/λµ for τ/λ ∈ γ. (ii) reasoning as above since τ ∈ γ we have x ∈ [∆, c]wτ if and only if c ( µ ) x ∈ wτ/λ. we conclude since the condition τµ/λ ∈ γ implies wτ/λ ( c ( µ )) = wτµ/λ. (iii) here under the conditions τ, τ/λ ∈ γ+, we have x ∈ [ ∆, c+ ] wτ if and only if x ∈ wτ/λ ( c+ ( µ )) = wτµ/λ. the previous results can be applied to the case when w∞ is replaced by w 0. 4.3 the sets [a1, a2]w 0τ using the banach algebra ( w0 (λ) , w0 (λ) ) we get similar results to those given above replacing w∞ (λ) by w 0 (λ) and wτ by w 0 τ = dτw 0. note that x ∈ w 0τ if and only if 1 n n∑ m=1 |xm| τm → 0 (n → ∞) . by [17, theorem 4.3.6, p. 52] the set w 0τ is a bk space with ak normed by ‖‖wτ . so we can state the following. proposition 4.3. let λ, µ ∈ u+. (i) if λτ ∈ γ then [c,∆]w0τ = w 0 λτ/µ ; (ii) if λτ, λµτ ∈ γ then [c, c]w0τ = w 0 λµτ ; (iii) if τ ∈ γ+ and λτ ∈ γ then [ c+,∆ ] w0τ = w 0 λτ/µ ; cubo 12, 3 (2010) calculations in new sequence spaces ... 131 (iv) if τ ∈ γ+ and λµτ ∈ γ then [ c+, c ] w0τ = w 0 λµτ ; (v) if τ, λτ ∈ γ+ then [ c+, c+ ] w0τ = w 0 λµτ ; (vi) if τ, τ/λ ∈ γ then [∆,∆]w0τ = w 0 τ/λµ ; (vii) if τ, τµ/λ ∈ γ then [∆, c]w0τ = w 0 τµ/λ ; (viii) if τ, τ/λ ∈ γ+ then [ ∆, c+ ] w0τ = w 0 τµ/λ . we immediatly get the next remark. remark 4.4. it can easily be seen that in proposition 4.3 each of the sets [a1, a2]w0τ is equal to w 0τ (a1 a2). this result is a direct consequence of the previous proofs and of the fact that w 0 τ is of absolute type, that is |x| ∈ w 0τ if and only if x ∈ w 0 τ . these results can be applied to statistical convergence. 5 application to a−statistical convergence in this section we will give conditions to have xk → l (s (a)) where a is either of the infinite matrices d1/τc (λ) c ( µ ) , d1/τ∆(λ)∆ ( µ ) , or d1/τ∆(λ) c ( µ ) . then we give conditions to have xk → 0 (s (a)) where a is either of the operators d1/τc + (λ)∆ ( µ ) , d1/τc + (λ) c ( µ ) , d1/τc + (λ) c+ ( µ ) and d1/τ∆(λ) c + ( µ ) . the sequence x = (xn)n≥1 is said to be statiscally convergent to the number l if lim n→∞ 1 n |{k ≤ n : |xk − l| ≥ ε}| = 0 for all ε > 0, where the vertical bars indicate the number of elements in the enclosed set. in this case we will write xk → l (s) or st − lim x = l. let a ∈ (e, f) for given l ∈ c and for every ε > 0 we will use the notation iε (a) = {k ≤ n : |[a x ]k − l| ≥ ε} , (where we assume that every series [a x ]k = ak (x ) = ∑ ∞ m=1 akm xm for k ≥ 1 is convergent). we will say that x = (xn)n≥1 is a− statistically convergent to l if for every ε > 0, lim n→∞ 1 n |iε (a)| = 0. then we will write xk → l (s ( a)) and for a = i, xk → l (s (i)) means that st − lim x = l, (cf. [6]). now we require a lemma where we will put t−1 e = l̃ = (l n)n≥1 for given triangle t, that is t = (tnm)n,m≥1 with tnn 6= 0 and tnm = 0 if m > n for all n, m. 132 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) we can state the following. lemma 5.1. if x − ll̃ ∈ w0 (t) then xk is t− statistically convergent to l. proof. the condition x − ll̃ ∈ w0 (t) means that t ( x − ll̃ ) ∈ w0. since t x − le = t ( x − lt −1 e ) = t ( x − ll̃ ) for any ε > 0 we have yn = 1 n n∑ k=1 |[t x ]k − l| = 1 n n∑ k=1 ∣∣[t ( x − ll̃ )] k ∣∣ ≥ 1 n ∑ k∈iε(t) ∣∣[t ( x − ll̃ )] k ∣∣ ≥ 1 n ∑ k∈iε(t) ε ≥ ε n |{k ≤ n : |[t x ]k − l| ≥ ε}|. we conclude that x − ll̃ ∈ w0 (t) implies yn → 0 (n → ∞) and xk → l (s (t)). we are led to state the next results. theorem 5.2. (i) let λτ, λτµ ∈ γ. if lim n→∞ 1 n n∑ k=1 ∣∣xk − l [ λkµkτk + ( µk−1 +µk ) λk−1τk−1 −λk−2µk−2τk−2 ]∣∣ λkµkτk = 0 (9) then xk → l ( s ( d1/τc (λ) c ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 λkτk k∑ i=1 1 µi ( i∑ j=1 x j ) − l ∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (ii) let τ, τ/λ ∈ γ. if lim n→∞ 1 n n∑ k=1 λkµk τk ∣∣∣∣∣xk − l ( 1 µk k∑ i=1 1 λi i∑ j=1 τj )∣∣∣∣∣ = 0 then xk → l ( s ( d1/τ∆(λ)∆ ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣ { k ≤ n : ∣∣∣∣ 1 τk [ λk∆ ( µ ) xk −λk−1∆ ( µ ) xk−1 ] − l ∣∣∣∣ ≥ ε }∣∣∣∣ = 0. (iii) let τ, τµ/λ ∈ γ. if lim n→∞ 1 n n∑ k=1 λk µkτk ∣∣∣∣∣xk − l [( µk λk − µk−1 λk−1 ) k−1∑ i=1 τi + µk λk τk ]∣∣∣∣∣ = 0 cubo 12, 3 (2010) calculations in new sequence spaces ... 133 then xk → l ( s ( d1/τ∆(λ) c ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 τk [( λk µk − λk−1 µk−1 ) k−1∑ i=1 xi + λk µk xk ] − l ∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. proof. (i) first by proposition 4.3 (ii) and remark 4.4, we easily see that for λτ, λτµ ∈ γ we have w 0τ ( c (λ) c ( µ )) = w 0 λµτ . then putting t = d1/τc (λ) c ( µ ) we get w 0 (t) = w 0 τ ( c (λ) c ( µ )) = w 0 λµτ . (10) then l̃ = t−1 e = ∆ ( µ ) ∆(λ) dτ e for each n with l n = [ ∆ ( µ ) ∆(λ) dτ e ] n = λnµnτn + ( µn−1 +µn ) λn−1τn−1 −λn−2µn−2τn−2 (11) using (10) and (11) we see that condition (9) is equivalent x − ll̃ ∈ w0 (t). we conclude by lemma 5.1 that xk → l (s (t)). this completes the proof of (i). (ii) by proposition 4.3 (vi) and remark 4.4, since τ, τ/λ ∈ γ we have w 0τ ( ∆(λ)∆ ( µ )) = w 0 τ/λµ . then putting t′ = d1/τ∆(λ)∆ ( µ ) we get w 0 ( t ′ ) = w 0 τ ( ∆(λ)∆ ( µ )) = w 0 τ/λµ . (12) since l̃′ = t′−1 e = c ( µ ) c (λ) dτ e we have l ′ n = [ c ( µ ) c (λ) dτ e ] n = 1 µn n∑ i=1 1 λi ( i∑ j=1 τj ) for all n. by lemma 5.1 we conclude xk → l ( s ( d1/τ∆(λ)∆ ( µ ))) for all x with lim n→∞ 1 n n∑ k=1 ∣∣xk − ll′k ∣∣ λkµk τk = 0 this shows (ii). (iii) again by proposition 4.3 (vii) and remark 4.4, since τ, τµ/λ ∈ γ we have w 0τ ( ∆(λ) c ( µ )) = w 0 τµ/λ . then putting t ′′ = d1/τ∆(λ) c ( µ ) we get w 0 ( t ′′ ) = w 0 τ ( ∆(λ) c ( µ )) = w 0 τµ/λ . (13) writing l̃′′ = t ′′ −1 e = ∆ ( µ ) c (λ) dτ e we successively get dτ e = (τn)n≥1 , c (λ) dτ e = (( n∑ i=1 τi ) /λn ) n≥1 and ∆ ( µ ) c (λ) dτ e = ( µn λn n∑ i=1 τi − µn−1 λn−1 n−1∑ i=1 τi ) n≥1 . 134 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) so for each n we have l ′′ n = [ ∆ ( µ ) c (λ) dτ e ] n = ( µn λn − µn−1 λn−1 ) n−1∑ i=1 τi + µn λn xk. we conclude that for every x with lim n→∞ 1 n n∑ k=1 ∣∣xk − ll′′k ∣∣ λk µkτk = 0 then xk → l ( s ( t′′ )) . finally we easily get [ t ′′ x ] n = 1 τn ( λn µn n∑ i=1 xi − λn−1 µn−1 n−1∑ i=1 xi ) = 1 τn [( λn µn − λn−1 µn−1 ) n−1∑ i=1 xi + λn µn xn ] . this shows (iii). we are led to illustrate the previous results with some examples where we must have in mind that the condition xk/τk → 0 (k → ∞) implies x ∈ w 0 τ . example 5.3. the condition lim n→∞ 1 n n∑ k=1 ∣∣∣∣ xk 2k − 7 4 l ∣∣∣∣ = 0 for given l ∈ c implies xk → l ( s ( d(n/2n )n c1σ )) , that is, for each ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 2k k∑ i=1 i∑ j=1 x j − l ∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (14) indeed it is enough to apply theorem 5.2 (i) with λk = k, τk = 2 k/k and µk = 1 for all k. note that if xk/2 k → 7l/4 (k → ∞) then xk → l ( s ( d(n/2n )n c1σ )) . we can also state the next application. example 5.4. if limn→∞ (1/n) ∑n k=1 |xk|/k2 k = 0 then xk → l ( s ( d(2−n )n ∆c1 )) , that is for each ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 2k ( 1 k − 1 k − 1 ) k−1∑ i=1 xi + 1 k xk ∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. this result is a direct consequence of theorem 5.2 (iii) with λk = 1, τk = 2 k and µk = k for all k. again note that we have xk → l ( s ( d(2−n )n ∆c1 )) if xk/k2 k → 0 (k → ∞). cubo 12, 3 (2010) calculations in new sequence spaces ... 135 in the following we will use the previous proposition 4.3 and the expressions of w 0τ ( c+ (λ)∆ ( µ )) = [ c+,∆ ] w0τ , w 0τ ( c+ (λ) c ( µ )) = [ c+, c ] w0τ , w 0τ ( c+ (λ) c+ ( µ )) = [ c+, c+ ] w0τ and w 0τ ( ∆(λ) c+ ( µ )) = [ ∆, c+ ] w0τ . we now require a lemma which is a direct consequence of lemma 5.1. lemma 5.5. let a be an infinite matrix. if x ∈ w0 (a) then xk → 0 (s (a)) . we deduce the next results. theorem 5.6. (i) let τ ∈ γ+ and λτ ∈ γ. if lim n→∞ 1 n n∑ k=1 |xk| λkτk µk = 0 (15) then xk → 0 ( s ( d1/τc + (λ)∆ ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 τk ∞∑ i=k µi xi −µi−1 xi−1 λi ∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (16) (ii) let τ ∈ γ+ and λµτ ∈ γ. if lim n→∞ 1 n n∑ k=1 |xk| λkµkτk = 0 (17) then xk → 0 ( s ( d1/τc + (λ) c ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 τk ∞∑ i=k 1 λi ( 1 µi i∑ j=1 x j )∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (18) (iii) let τ, λτ ∈ γ+. if lim n→∞ 1 n n∑ k=1 |xk| λkµkτk = 0 (19) then xk → 0 ( s ( d1/τc + (λ) c+ ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣ 1 τk ∞∑ i=k 1 λi ( ∞∑ j=i x j µj )∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (20) (iv) let τ, τ/λ ∈ γ+. if lim n→∞ 1 n n∑ k=1 λk |xk| µkτk = 0 then xk → 0 ( s ( d1/τ∆(λ) c + ( µ ))) , that is for every ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : 1 τk ∣∣∣∣∣(λk −λk−1) ∞∑ i=k−1 xi µi + λk µk xk ∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (21) 136 bruno de malafosse & vladimir rakočević cubo 12, 3 (2010) proof. (i) condition (15) implies x ∈ w 0 λτ/µ and by proposition 4.3 and remark 4.4 since τ ∈ γ+ and λτ ∈ γ we have w 0 λτ/µ = w 0τ ( c+ (λ)∆ ( µ )) and x ∈ w 0τ ( c+ (λ) ∆ ( µ )) . now it can be easily seen that [ d1/τc + (λ)∆ ( µ )] n = 1 τn ∞∑ i=n µi xi −µi−1 xi−1 λi , so by lemma 5.5 with a = d1/τc + (λ)∆ ( µ ) we conclude xk → 0 ( s ( d1/τc + (λ)∆ ( µ ))) . this shows (i). (ii) here condition (17) means x ∈ w 0 λµτ and by proposition 4.3 and remark 4.4 since τ ∈ γ+ and λµτ ∈ γ we have w 0 λµτ = w 0τ ( c+ (λ) c ( µ )) and x ∈ w 0τ ( c+ (λ) c ( µ )) . now since [ d1/τc + (λ) c ( µ )] n = 1 τn ∞∑ i=n 1 λi ( 1 µi i∑ j=1 x j ) , by lemma 5.5 where a′ = d1/τc + (λ) c ( µ ) , we conclude xk → 0 ( s ( d1/τc + (λ) c ( µ ))) . so we have shown (ii). (iii) can be obtained reasoning as above with a′′ = d1/τc + (λ) c+ ( µ ) and so xk → 0( s ( d1/τc + (λ) c+ ( µ ))) . (iv) can also be obtained similarly. it is enough to put a′′′ = d1/τ∆(λ) c + ( µ ) . an elementary calculation gives [ a ′′′ x ] k = 1 τk [ (λk −λk−1) ∞∑ i=k−1 xi µi + λk µk xk ] and we conclude that xk → 0 ( s ( d1/τ∆(λ) c + ( µ ))) , that is (21). we can state the next example example 5.7. for each ε > 0 and for every x ∈ w 0 3/2 we have xk → 0 ( s ( d(2n )n σ +c ((3n)n) )) , that is lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣2 k ∞∑ i=1 1 3i ( i∑ j=1 x j )∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (22) it is enough to apply theorem 5.6 (ii) with τk = 2 −k, µk = 3 k and λk = 1 for all k. so if (2/3) k xk → 0 (k → ∞) then (22) holds. we also have the next example. example 5.8. from theorem 5.6 (iii) with λk = µk = k and τk = 2 −k the condition lim n→∞ 1 n n∑ k=1 2 k |xk| k2 = 0 cubo 12, 3 (2010) calculations in new sequence spaces ... 137 implies xk → 0 ( s ( d(2n )n c1c + 1 )) that is, for each ε > 0 lim n→∞ 1 n ∣∣∣∣∣ { k ≤ n : ∣∣∣∣∣2 k ∞∑ i=k 1 i ( ∞∑ j=i x j j )∣∣∣∣∣ ≥ ε }∣∣∣∣∣ = 0. (23) as in the previous cases (23) holds if 2k xk/k 2 → 0 (k → ∞). references [1] çolak, r., lacunary strong convergence of difference sequences with respect to a modulus, filomat, 17 (2003), 9–14. [2] de malafosse, b., on some bk space, int. j. of math. and math. sc., 28 (2003), 1783– 1801. [3] de malafosse, b., on the set of sequences that are strongly α-bounded and αconvergent to naught with index p, seminario matematico dell’università e del politecnico di torino, 61 (2003), 13–32. [4] de malafosse, b., calculations on some sequence spaces, int. j. of math. and math. sc., 31 (2004), 1653–1670. [5] de malafosse, b. and malkowsky, e., the banach algebra (w∞ (λ) , w∞ (λ)), in press far east journal math. [6] de malafosse, b. and rakočević, v., matrix transformations and statistical convergence, linear algebra and its applications, 420 (2007), 377–387. [7] fast, h., sur la convergence statistique, colloq. math., 2 (1951), 241–244. 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[17] wilansky, a., summability through functional analysis, north-holland mathematics studies, 85, 1984. a mathematical journal vol. 7, no 2, (39 55). august 2005. on the classical 2−orthogonal polynomials sequences of sheffer-meixner type boukhemis ammar 1 department of mathematics, faculty of sciences, university of annaba, b.p.12 annaba 23000, algeria aboukhemis@yahoo.com abstract the polynomial sequences of sheffer-meixner type designed by {sn}n≥0, are defined by the generating function g(x, t) = a(t)e xh(t) = � n≥0 sn(x) tn n! we are interested, in this work, in studying the sequences when they are 2−orthogonal. we will give the general properties of these sequences, and we study in details those which are classical. resumen las sucesiones polinomiales del tipo sheffer-meixner denotadas por {sn}n≥0 son definidas por la función generatriz g(x, t) = a(t)e xh(t) = � n≥0 sn(x) tn n! en este trabajo estamos interesados en estudiar aquellas sucesiones que son 2− ortogonales. mostraremos sus propiedades generales y estudiaremos en detalle aquellas que son clásicas. 1the author was partially supported by : l’agence nationale pour le développement de la recherche universitaire -andru-. 40 boukhemis ammar 7, 2(2005) key words and phrases: d−orthogonal polynomials, relations of recurrence, sheffer-meixner’s polynomials, generating function, classical polynomials, operator of hahn. math. subj. class.: 42c05, 33c45 1 introduction. let p be the vector space of polynomials with coefficients in c and p′ its algebraic dual. let us given d scalar linear forms γ1, γ2, · · · , γd defined from p into c. a monic sequence {pn}n≥0 ( i.e. pn(x) = xn + · · · , n ≥ 0) is said d−orthogonal with respect to γ = ( γ1, γ2, · · · , γd )t when it satisfies [7, 10, 12, 18, 19, 23] { 〈γα,xmpn(x)〉 = 0, n ≥ md + α, m ≥ 0 〈γα,xmpmd+α−1(x)〉 �= 0, m ≥ 0, (1.1) for every 1 ≤ α ≤ d, and where 〈 , 〉 is the dual bracket between p and p′. among the d−orthogonal sequences, we will be interested here by a particular class, but nevertheless important. indeed, theses sequences have many applications and have extensively investigated. this class consists of sequences of polynomials {sn}n≥0 defined by the generating function g(x,t) = a(t)exh(t) = ∑ n≥0 sn(x) tn n! (1.2) where a(t) = ∑ n≥0 ant n and h(t) = ∑ n≥1 hnt n with a(0) = 1, h(0) = 0 and h′(0) = 1 these sequences are said to be of sheffer-meixner type. the case d = 1 has been first studied by meixner [20] and sheffer [22] and then, completed by other authors [2, 13, 21]. meixner has shown that this class consists of 5 sequences, namely, hermite polynomials, laguerre polynomials, charlier polynomials, meixner polynomials and meixnerpollaczek polynomials. in the case d = 2 [5], we have shown that the functions h and a satisfy, respectively the equations ⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ h′(t) = 1 (1 −αt)(1 −βt)(1 −γt), α,β,γ ∈ c a′(t) a(t) = σ0 + σ1t + σ2t2 (1 −αt)(1 −βt)(1 −γt), σ0,σ1,σ2 ∈ c; σ2 �= 0. (1.3) 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 41 if we note by j the inverse function of h ( i.e. j(h(t) = t ), and by d = d dx , then we have [1, 20] j(d)sn+1(x) = (n + 1)sn(x), n ≥ 0. (1.4) moreover, the polynomials sn (n ≥ 0) are characterized by the four-term relation [5] sn+3(x) = [(x−σ0) + (n + 2)(α + β + γ)] sn+2(x) −(n + 2) [σ1 + (n + 1) (αβ + αγ + βγ)] sn+1(x) −(n + 1) (n + 2) (σ2 −nαβγ) sn(x), n ≥ 0 (1.5) we also have proved that this class is composed of 9 sequences, namely ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ (a) α = β = γ = 0, (b) α = β �= 0 and γ = 0, (c) α �= 0 and β = γ = 0, (d1) α �= β �= 0 and γ = 0, α , β ∈ r, (d2) α �= β �= 0 and γ = 0, α , β ∈ c, (e) α = β = γ �= 0, (f) α = β �= γ �= 0, (g1) α �= β �= γ �= 0 α,β , γ ∈ r, (g2) α �= β �= γ �= 0 α,β ∈ r and γ ∈ c. (1.6) we recall in paragraph 2, the principal properties of the d−orthogonal sequences [ 6, 11, 18, 19]. the paragraph 3 is denoted to the characterization of those which are in addition classical [3, 4, 12]. in paragraph 4, we show that the sequences (a), (b), (c) and (d) are classical sequences and we give certain of their properties. whereas in paragraph 5, we exhibit an integral representation of the forms with respect to which these sequences are 2−orthogonal in cases (a) and (b). 2 properties of d−orthogonal sequences. definition 2.1 let {pn}n≥0 be a sequence monic polynomials. we call dual sequence of the sequence {pn}n≥0, the sequence of linear forms {fn}n≥0 defined by 〈fn,pn(x)〉 = δn,m, n,m ≥ 0 (2.1) proposition 2.2 [11] if we denote by d the operator of derivation i.e. d = d dx and by {f̃n}n≥0 the dual sequence associated to the monic sequence {qn}n≥0 of the derivatives of {pn}n≥0, and defined by qn(x) = dpn+1(x) n + 1 , n ≥ 0, then df̃n = −(n + 1)fn, (2.2) 42 boukhemis ammar 7, 2(2005) with 〈 df̃n,r(x) 〉 = − 〈 f̃n,dr(x) 〉 , ∀r ∈p. proposition 2.3 [18, 19] let l∈p′ be and q an integer, in order that l satisfies 〈l,pn(x)〉 = 0 n ≥ q 〈l,pq−1(x)〉 �= 0 (2.3) it is necessary and sufficient that there exists λν ∈ c, 0 ≤ ν ≤ q − 1, λq−1 �= 0, such that l = q−1∑ ν=0 λνfν. (2.4) corollary 2.4 according to the preceding lemma, we have γα = α−1∑ ν=0 λανfν, λαα−1 �= 0, 1 ≤ α ≤ d, and in a equivalent manner fν = ν+1∑ α=0 τνα γ α; τνν+1 �= 0, 0 ≤ ν ≤ d− 1. consequently, every d−orthogonal sequence {pn}n≥0 with respect to γ = ( γ1, γ2, · · · , γd )t is also d−orthogonal with respect to f = (f0,f1, · · · ,fd−1)t . theorem 2.5 [18, 23] with the same notations as previously we have the following equivalences (a) the sequence {pn}n≥0 is d−orthogonal with respect to f = (f0,f1, · · · ,fd−1)t . (b) the sequence {pn}n≥0 satisfies a recurrence of order d + 1( d ≥ 1 ) pm+d+1(x) = (x−βm+d)pm+d(x) − d−1∑ ν=0 γd−1−νm+d−νpm+d−ν−1(x), m ≥ 0 (2.5) with the initial data{ p0(x) = 1, p1(x) = x−β0, and if d ≥ 2 pn(x) = (x−βn−1)pn−1(x) − ∑n−2 ν=0 γ d−1−ν n−1−νpn−2−ν (x), 2 ≤ n ≤ d (2.6) where γ0m+1 �= 0, m ≥ 0. ( regularity conditions ). (c) for every (n,v), n ≥ 0, 0 ≤ ν ≤ d− 1, there exists d polynomials v μ(n,ν), (0 ≤ μ ≤ d− 1) such that fnd+ν = d−1∑ μ=0 v μ(n,ν)fμ, n ≥ 0, 0 ≤ ν ≤ d− 1, (2.7) 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 43 where ⎧⎨ ⎩ deg v μ(n,μ) = n, 0 ≤ μ ≤ d− 1, deg v μ(n,ν) ≤ n, 0 ≤ μ ≤ ν − 1, if 1 ≤ ν ≤ d− 1, deg v μ(n,ν) ≤ n− 1, ν + 1 ≤ μ ≤ d− 1, if 0 ≤ ν ≤ d− 2. (2.8) theorem 2.6 [18] for every sequence {pn}n≥0 d−orthogonal with respect to f = (f0,f1, · · · ,fd−1)t , the following statements are equivalent (a) it exists l∈p′ and an integer s ≥ 1 such that { 〈l, pn(x)〉 = 0, n ≥ s, 〈l, ps−1(x)〉 �= 0. (2.9) (b) it exists l∈p′ and d polynomials φα, 0 ≤ α ≤ d− 1 such that l = d−1∑ α=0 φαfα, with the following properties if s− 1 = qd + r, 0 ≤ r ≤ d− 1, we have⎧⎨ ⎩ deg φr = q, 0 ≤ r ≤ d− 1, if d ≥ 2, deg φα ≤ q, 0 ≤ α ≤ r− 1, if 1 ≤ r ≤ d− 1, deg φα ≤ q − 1, r + 1 ≤ α ≤ d− 1, if 0 ≤ r ≤ d− 2. (2.10) 3 the d−orthogonal sequences and the finite differences operators δωand ∇ω. let us consider the progressive finite differences operators δω ( hahn’s operator) and regressive operator ∇ω, defined respectively by δωf(x) = f(x + ω) −f(x) ω , and ∇ωf(x) = f(x) −f(x−ω) ω = δ−ωf(x) these operators enjoy the following properties proposition 3.1 let f ∈p′ then we have 〈f, δωf(x)〉 = −〈∇ωf, f(x)〉 , ∀f ∈ c∞. (3.1) proof. we know that δωf(x) = eωd − 1 ω f(x), and that by definition we have 〈df, f(x)〉 = −〈f,df(x)〉 , 44 boukhemis ammar 7, 2(2005) therefore 〈f, δωf(x)〉 = 〈 f, ∑ k≥0 ωk (k + 1)! dk+1f(x) 〉 = 〈∑ k≥0 (−1)k+1ωk (k + 1)! dk+1f, f(x) 〉 = 〈 e−ωd − 1 ω f, f(x) 〉 = −〈∇ωf, f(x)〉 . proposition 3.2 let {qωn}n≥0 be the sequence of the monic polynomials defined by qωn(x) = δωpn+1(x) n + 1 = pn+1(x + ω) −pn+1(x) (n + 1)ω , n ≥ 0 (3.2) and {f̃n}n≥0 the dual sequence associated to the sequence {qωn}n≥0, then we have ∇ωf̃n=δ−ωf̃n = −(n + 1)fn+1; n ≥ 0. (3.3) proof. indeed, we have δn,m = 〈 f̃n,qm(x) 〉 = 1 m + 1 〈 f̃n, δωpm+1(x) 〉 = − 1 m + 1 〈 δ−ωf̃n,pm+1(x) 〉 , i.e. − 〈 δ−ωf̃n,pn+1(x) 〉 = (m + 1)δn,m but from the lemma (2.1), ∃ λν ∈ c, 0 ≤ ν ≤ n + 1, such that δ−ωf̃n = n+1∑ ν=0 λnνfν, with λnν = 0, 0 ≤ ν ≤ n and λnn+1 = n + 1. lemma 3.3 we have the following properties δω [(x−ω)mpn(x)] = xδω [ (x−ω)m−1pn(x) ] + (x−ω)m−1pn(x), m ≥ 0 (3.4) and xmδωpn(x) = δω [(x−ω)mpn(x)] − [mxm−1 − m(m− 1) 2 ωxm−2 + rωm−3(x)]pn(x), m ≥ 0, where rωm−3(x) is a polynomial of degree (m− 3) in x. (3.5) proof. clearly δω [(x−ω)mpn(x)] = xmpn(x + ω) − (x−ω)mpn(x) ω = x[xm−1pn(x + ω) − (x−ω)m−1pn(x)] + ω(x−ω)m−1pn(x) ω = xδω [ (x−ω)m−1pn(x) ] + (x−ω)m−1pn(x), m ≥ 0. 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 45 repeating m times the expression (3.4) we get δω [(x−ω)mpn(x)] = xmδωpn(x) + [ m−1∑ k=0 xm−k(x−ω)k ] pn(x), as ∑m−1 k=0 x m−k(x−ω)k = ∑m−1k=0 ∑kj=0 (kj ) (−1)jωjxm−1−j = ∑m−1 j=0 ∑m−1 k=j ( k j ) (−1)jωjxm−1−j = mxm−1 − m(m− 1) 2 ωxm−2 + rωm−3(x). from which we obtain (3.5). definition 3.4 [4, 11, 14, 15 ] a sequence of polynomials {pn}n≥0 d−orthogonal (d ≥ 1) with respect to f = (f0,f1, · · · ,fd−1)t ,those the monic sequence of finite differences {qωn}n≥0 defined by qωn(x) = δωpn+1(x) n + 1 , n ≥ 0 is also d−orthogonal (d ≥ 1) with respect to f̃ = ( f̃0,f̃1, · · · ,f̃d−1 )t is said to be classical. remark 3.5 in the case ω = 0, the operator δω becomes d = d dx . theorem 3.6 with the above hypothesis we have the following equivalence (a) the sequence {pn}n≥0 is classical d−orthogonal. (b) the functional f satisfies the vectorial functional equation ∇ω(φf) + ψf = 0, (3.6) where ψ and φ are 2 matrices d × d of polynomials ψ(x) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0 1 0 . . . 0 0 0 2 . . . 0 . . . . . . . . . . . . 0 0 0 . . . d− 1 ψ(x) ξ1 ξ2 . . . ξd−1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠ (3.7) and ψ is a polynomial of degree 1 and ξμ, 1 ≤ μ ≤ d− 1 are constants, φ(x) = ⎛ ⎜⎜⎜⎜⎜⎜⎝ φ00(x) φ 1 0(x) . . . φ d−1 0 (x) . . . . . . . . . φ0d−2(x) φ 1 d−2(x) . . . φ d−1 d−2(x) φ0d−1(x) φ 1 d−1(x) . . . φ d−1 d−1(x) ⎞ ⎟⎟⎟⎟⎟⎟⎠ (3.8) 46 boukhemis ammar 7, 2(2005) where φνα, 0 ≤ α,ν ≤ d− 1 are polynomials such that⎧⎨ ⎩ deg φνα ≤ 1, 0 ≤ ν ≤ α + 1 if 0 ≤ α ≤ d− 2 deg φνα = 0, α + 2 ≤ ν ≤ d− 1 if 0 ≤ α ≤ d− 3 deg φ0d−1 ≤ 2 and deg φνd−1 ≤ 1, 1 ≤ ν ≤ d− 1 (3.9) in addition, if we write ⎧⎨ ⎩ ψ(x) = e1x + e0, φ0d−1(x) = c2x 2 + c1x + c0 φα+1α (x) = kαx + lα, 0 ≤ α ≤ d− 2, then ⎧⎪⎪⎨ ⎪⎪⎩ c2 �= e1 m + 1 , m ≥ 0, e1 �= 0, kα �= α + 1 m + 1 , m ≥ 0, for 0 ≤ α ≤ d− 2. (3.10) remark 3.7 a) it is easy to show that : { f̃ = φf ∇ωf̃ = −ψf (3.11) (b) when ω = 0 the functional equation (3.6) may be written [11] ψf + d(φf) = 0 (3.12) and the conditions (3.7), (3.8), (3.9) and (3.10) remain unchanged. c) the proof of this theorem is the same as in the case ω = 0 [11], if we take into account the relation (3.4). 4 classification of the sequences 2−orthogonal of sheffer-meixner type. let us consider now the sequences of polynomials {sn}n≥0, sheffer-meixner type defined by the relation (1.5). we noted by {mωn}n≥0 and {mωn}n≥0 the sequences of monic polynomials defined respectively by mn(x) = dsn+1(x) n + 1 ; n ≥ 0, (4.1) and mωn (x) = δωsn+1(x) n + 1 ; n ≥ 0. (4.2) then we have 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 47 lemma 4.1 in the case (b) ( the case (a) if α = 0), the sequence of derivatives of monic polynomials defined by the relation (4.1) satisfies the following recurrence⎧⎨ ⎩ mn+3(x) = [(x−σ0) + (2n + 5) α] mn+2(x) −(n + 2) [ σ1 + (n + 2)α2 ] mn+1(x) − (n + 1)(n + 2)σ2mn(x); n ≥ 0 m0(x) = 1; m1(x) = x−σ0 + α; m2(x) = (x−σ0 + 3α) m1(x) − ( σ1 + α2 ) (4.3) proof. indeed, in the case (b) j is such that [5] j(d) = d 1 + αd , then by the relation (1.4) we have dsn+1(x) = (n + 1) [sn(x) + αdsn(x)] consequently sn+1(x) = mn+1(x) − (n + 1)αmn(x); n ≥ 0. differentiating the recurrence(1.5) and replacing sn+1 by {mν}n+1ν=n−1,we obtain the relation ( 4.3 ). lemma 4.2 in the case (d) ( the case (c) if β = 0), the sequence of finite differences of monic polynomials defined by the relation (4.2) satisfies the following recurrence⎧⎪⎪⎨ ⎪⎪⎩ m α−β n+3 (x) = [(x + α−σ0) + (n + 2) (α + β)] mα−βn+2 (x) −(n + 2) [σ1 + (n + 2)αβ] mα−βn+1 (x) − (n + 1)(n + 2)σ2mα−βn (x); n ≥ 0 m α−β 0 (x) = 1; m α−β 1 (x) = x−σ0 + α; mα−β2 (x) = (x−σ0 + 2α + β) mα−β1 (x) −σ1 −αβ (4.4) proof. indeed, in the case (d) the function j is such that [5] j(d) = δα−β 1 + αδα−β , i.e. by the relation (1.4). δα−βsn+1(x) = (n + 1) [αδα−βsn(x) + sn(x)] , consequently sn+1(x) = m α−β n+1 (x) − (n + 1)αmα−βn (x); n ≥ 0. by acting the operator δα−β on the recurrence (1.5) and replacing sn+1 by {mα−βn+1 }n+1ν=n−1, we obtain the relation (4.4). thus, we have the following classification. theorem 4.3 the sequences (a), (b), (c) and (d) are classical sequences and the 2− orthogonal polynomials sequences {mn}n≥0 and { mα−βn } n≥0 are “2−kernel” polynomial [8, 9, 17] for the 2− orthogonal polynomials sequences {sn}n≥0 . 48 boukhemis ammar 7, 2(2005) 5 integral representation of the functional f0 and f1. in this paragraph, we will be interested by the integral representation problem of the linear functional f0 and f1 in the cases (a) and (b). 5.1 properties of the functional f0 and f1. lemma 5.1 in the case (d) ( a fortiori the cases (a), (b) and (c)) we have φ(x) = [ 1 −α − α σ2 (x−σ0) 1 + α σ1 σ2 ] and ψ(x) = ⎡ ⎣ 0 11 σ2 (x−σ0) − σ1 σ2 ⎤ ⎦ proof. with the same notations as in theorem (3.1), we have deg φ00(x) ≤ 1, deg φ10(x) ≤ 1, deg φ01(x) ≤ 2, deg φ11(x) ≤ 1, and deg ψ(x) ≤ 1. putting ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ φ00(x) = d0 + d1x φ10(x) = e0 + e1x φ01(x) = a0 + a1x + a2x 2 φ11(x) = b0 + b1x ψ(x) = c0 + c1x the relations (3.11) may be written respectively { f̃0 = (d0 + d1x)f0 + (e0 + e1x)f1 f̃1 = ( a0 + a1x + a2x2 ) f0 + (b0 + b1x)f1 (r5.0) and { ∇α−βf̃0 = −f1 ∇α−βf̃1 = −(c0 + c1x)f0 − ξ1f1 (r5.1) by letting, firstly, the functional f̃0 and f̃1 act successively on s0(x),s1x),s2(x) s3(x), and s0(x),s1(x), · · · ,s4(x), respectively we determine the coefficients of the polynomials φji (x), (i,j = 0, 1), secondly, we let ∇α−βf̃1 act on s0(x),s1x) and s2(x) to determine the coefficients c0,c1 and ξ1. proposition 5.2 for α �= 0, the functional f0 is solution of the equation ∇α−β { ∇α−β [ ( α2x−σ2 −ασ1 −α2σ0 ) f0] − (2αx−σ1 − 2ασ0)f0 } + (x−σ0)f0 = 0, (5.1) proof. from the relation (3.6) we see that ⎧⎨ ⎩ ∇α−βf0 −α∇α−βf1 = −f1 − α σ2 ∇α−β [(x−σ0)f0] + ( 1 + α σ1 σ2 ) ∇α−βf1 = − 1 σ2 (x−σ0)f0 + σ1 σ2 f1 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 49 and by substitution we obtain the relation f1 = ∇α−β {[ α2 σ2 (x−σ0) − ( 1 + α σ1 σ2 )] f0 } − α σ2 (x−σ0)f0 (5.2) therefore, letting ∇α−β act on this last one and replacing ∇α−βf1 and f1 by there respective values with respect to ∇2α−βf0, ∇α−βf0 and f0 in the first relation, we find the expected result. remark 5.3 in the case (b) ( the case (a) if α = 0 ), the relations (5.1) and (5.2) may be written respectively d { d[ ( α2x−σ2 −ασ1 −α2σ0 ) f0] − (2αx−σ1 − 2ασ0)f0 } + (x−σ0)f0 = 0 (5.3) and f1 = α σ2 d {[ α(x−σ0) − (σ2 α + σ1 )] f0 } − α σ2 (x−σ0)f0 (5.4) 5.2 determination of weight functions in the cases (a) and (b). the problem consists now in representing the functional f0 and f1 as an integral by putting ⎧⎨ ⎩ 〈f0,p(x)〉 = ∫ c f0(x)p(x)dx, and 〈f1,p(x)〉 = ∫ c f1(x)p(x)dx, ∀p ∈p (5.5) where the weight functions f0(x) and f1(x) are supposed “booth regular ” and c is a contour to be determined. proposition 5.4 if f0 is a weight function representing the functional f0 and c the contour of this representation, then f0 and c must satisfy, respectively in the case (b) (the case (a) if α = 0) θ(x) d2f0(x,α) dx2 + [ ω(x) + 2α2 ] df0(x,α) dx + [π(x) − 2α] f0(x,α) = 0 (5.6) and [θ(x)f0(x,α)p ′(x) −{(θ(x)f0(x,α))′ + ω(x)f0(x,α)}p(x)]c = 0, ∀p ∈p (5.7) where ⎧⎨ ⎩ θ(x) = α2x− ( σ2 + ασ1 + α2σ0 ) ω(x) = −2αx + ( 2α2 + σ1 + 2ασ0 ) π(x) = x−σ0. proof. a solution of the equation (5.3) must satisfy 〈d{d[θ(x)f0] + ω(x)f0} + π(x)f,p(x)〉 = 0, ∀p ∈p , 50 boukhemis ammar 7, 2(2005) i.e. 〈f0, θ(x)p′′(x)〉−〈f0, ω(x)p′(x)) + 〈f, π(x)p(x)〉 = 0, as∫ c θ(x)f0(x,α)p ′′(x)dx− ∫ c ω(x)f0(x,α)p ′(x)dx + ∫ c π(x)f0(x,α)p(x)dx = 0, by an integration by parts we obtain [θ(x)f0(x,α)p′(x) −{(θ(x)f0(x,α))′ + ω(x)f0(x,α)}p(x)]c + ∫ c { [θ(x)f0(x,α)] ′′ + [ω(x)f0(x,α)] ′ + π(x)f0(x,α) } p(x)dx = 0, in particular if we take [θ(x)f0(x,α)] ′′ + [ω(x)f0(x,α)] ′ + π(x)f0(x,α) = 0 and [θ(x)f0(x,α)p ′(x) −{(θ(x)f0(x,α))′ + ω(x)f0(x,α)}p(x)]c = 0, ∀p ∈p. remark 5.5 in the case (a), the weight function f0 and the contour c must satisfy respectively −σ2 d2f0(x) dx2 + σ1 df0(x) dx + (x−σ0)f0(x) = 0 (5.8) and [ −σ2 {f0(x)p(x)}′ + σ1f0(x)p(x) ] c = 0, ∀p ∈p (5.9) theorem 5.6 when σ2 < 0, the differential equation (5.6) has a general solution f0(x,α) = (x + k) λ 2 e x α ⎧⎨ ⎩c1jλ ⎡ ⎣q(x + k) 1 2 ⎤ ⎦ + c2yλ ⎡ ⎣q(x + k) 1 2 ⎤ ⎦ ⎫⎬ ⎭ (5.10) where λ = ∣∣∣∣α 3 −ασ1 − 2σ2 α2 ∣∣∣∣ , k = −α 2σ0 + ασ1 + σ2 α2 and q = √−σ2 α2 and jλ and yλ are the bessel functions of first and second kind respectively. proof. the equation (5.6) can be written (x + k) d2f0(x,α) dx2 − 2 [x α − (σ0 α + σ1 α2 + 2 )] df0(x,α) dx + 1 α2 (x−σ0 − 2α) f0(x,α) = 0 (5.11) 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 51 let us denote by r(x) = x α − (σ0 α + σ1 α2 + 2 ) x + k and put f0(x) = w(x) exp [∫ r(x)dx ] , then equation (5.11) may be written (x + k)2 d2w(x) dx2 − σ2 α2 [ x−α−σ0 − σ1 2σ2 ( α2 − σ1 2 )] w(x) = 0. this last equation admits as a general solution w(x) = (x + k) 1 2 ⎧⎨ ⎩c1jλ ⎡ ⎣2q(x + k) 1 2 ⎤ ⎦ + c2yλ ⎡ ⎣2q(x + k) 1 2 ⎤ ⎦ ⎫⎬ ⎭ . as ∫ r(x)dx = (x + k) λ− 1 2 exp( x α ), we find (5.10). theorem 5.7 in the case (b), choosing, as a contour, the interval c =] − k,∞[ , then the function f b0 (x,α) = const.(x + k) λ 2 e x αjλ ⎡ ⎣2 √−σ2 α2 (x + k) 1 2 ⎤ ⎦ ,α < 0 and σ2 < 0 (5.12) is an integral representation of the functional f0, i.e. 〈f0,p(x)〉 = ∫ c f b0 (x,α)p(x)dx, ∀p ∈p. proof. we have ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ lim x→−k+ f b0 (x,α) = lim x→∞ f b0 (x,α) = 0 lim x→−k+ (x + k)f b0 (x,α) = lim x→−k+ (x + k) df b0 (x,α) dx = 0 lim x→∞ (x + k)f b0 (x,α) = lim x→∞ df b0 (x,α) dx = 0 consequently the condition (5.7) is satisfied. as f b0 (x,α) is a solution of the equation (5.6), with the choice of the interval c =] −k,∞[ as a contour, f b0 (x,α) may be an integral representation of f0. 52 boukhemis ammar 7, 2(2005) corollary 5.8 in the case (b), the function f b1 (x,α) defined by f b1 (x,α) = α σ2 {[ α (x−σ0) −σ1 − σ2 α ] df b0 (x,α) dx − (x−σ0 −α)f b0 (x,α) } (5.13) is an integral representation of f1. proof. from the relation (5.4) we have 〈f1,p(x)〉 = α σ2 〈d [ α (x−σ0) −σ1 − σ2 α ] f0 − (x−σ0)f0,p(x)〉 = α σ2 〈d [ α (x−σ0) −σ1 − σ2 α ] f0,p(x)〉− α σ2 〈(x−σ0)f0,p(x)〉 = − α σ2 ∫ c [ α (x−σ0) −σ1 − σ2 α ] f b0 (x,α)p ′(x)dx − α σ2 ∫ c (x−σ0) f b0 (x,α)p(x)dx, ∀p ∈p. integrating by parts the first term in the right hand side we find ∫ c f b1 (x,α)p(x)dx = α σ2 ∫ c [ α (x−σ0) −σ1 − σ2 α ] df b0 (x,α) dx p(x)dx − α σ2 ∫ c (x−σ0 −α)f b0 (x,α)p(x)dx − α σ2 [{ α (x−σ0) −σ1 − σ2 α } f b0 (x,α)p(x) ] c as the last term is zero we obtain the relation (5.13). theorem 5.9 when α = 0 ( the case (a) ), the equation (5.8) admits as general solution f0(x) = ( x−σ0 + σ21 4σ2 )1 2 exp( σ1 2σ2 x) ⎧⎪⎨ ⎪⎩k1j1 3 ⎡ ⎢⎣ 2 3 √−σ2 ( x−σ0 + σ21 4σ2 )3 2 ⎤ ⎥⎦ + k2y1 3 ⎡ ⎢⎣ 2 3 √−σ2 ( x−σ0 + σ21 4σ2 )3 2 ⎤ ⎥⎦ ⎫⎪⎬ ⎪⎭ (5.14) proof. the equation (5.8) may also be written as d2f0(x) dx2 − σ1 σ2 df0(x) dx − 1 σ2 (x−σ0)f0(x) = 0. (5.15) let us put f0(x) = v (x) exp( σ1 2σ2 ), 7, 2(2005) on the classical 2−orthogonal polynomials sequences of ... 53 by substitution, v must then satisfy d2v (x) dx2 − 1 σ2 (x−σ0 + σ21 4σ2 )v (x) = 0. this equation is of the type d2v (x) dx2 − 1 σ2 xv (x), where x = x−σ0 + σ21 4σ2 , the general solution of which is v (x) = x 1 2 ⎧⎨ ⎩k1j1 3 ⎛ ⎝ 2 3 √−σ2 x 3 2 ⎞ ⎠ + k2y1 3 ⎛ ⎝ 2 3 √−σ2 x 3 2 ⎞ ⎠ ⎫⎬ ⎭ . going back to the initial variable x and the function f0, we find (5.13). theorem 5.10 choosing as a contour the interval c =]σ0 − σ21 4σ2 ,∞[, the function f a0 (x) = const. ( x−σ0 + (σ1)2 4σ2 )1 2 exp ( σ1 2σ2 x)j1 3 [ 2 3 √−σ2 ( x−σ0 + (σ1)2 4σ2 )3 2 ] , σ2 < 0 is an an integral representation of functional f0 in the case (a). proof. as f a0 is a solution of (5.8) and⎧⎪⎨ ⎪⎩ lim x→a+ f a0 (x) = lim x→∞ f a0 (x) = 0, and lim x→a+ df b0 (x) dx = 0 = lim x→∞ df a0 (x) dx = 0, where a = σ21 4σ2 −σ0, the conditions of the proposition (5.2) are then satisfied and f a0 is an integral representation of the functional f0. corollary 5.11 in the case (a), the function f a1 (x) defined by f a1 (x) = − df a0 (x) dx (5.16) is an integral representation of the functional f1. proof. it suffices to note, according to (5.4), that f1 = −df0. remark 5.12 we just proved that the class of 2−orthogonal polynomials of sheffermeixner type consists of 9 sequences. 5 of which are classical and 2 of them have continuous weight functions. the investigation of the last 3 sequences (c), (d1) and (d2) will be the subject of another talk. received: april 2003. revised: november 2003. 54 boukhemis ammar 7, 2(2005) references [1] w. a. al-salam, characterization theorems for orthogonal polynomials, in : p. nevai ed., orthogonal polynomials: theory and practice, vol. c 294 (kluwer, dordrecht pub,1990), 1 − 23. 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[23] j. van iseghem, approximants de padé vextoriels. thèse d’état, univ. sci. tech. lille-flandre-artois, (1987). cubo a mathematical journal vol.11, no¯ 03, (79–100). august 2009 positive solutions for systems of three-point nonlinear boundary value problems with deviating arguments j. henderson department of mathematics, baylor university, waco, texas 76798-7328, usa email: johnny henderson@baylor.edu and s.k. ntouyas, i.k. purnaras department of mathematics, university of ioannina, 451 10, ioannina, greece emails: sntouyas@cc.uoi.gr, ipurnara@cc.uoi.gr abstract existence of eigenvalues yielding positive solutions for a system of two second order delay differential equations along with boundary conditons is established. the results are obtained by the use of a guo-krasnoselskii fixed point theorem in cones. resumen es establecida la existencia de autovalores produciendo soluciones positivas para un sistema de dos ecuaciones diferenciales de segundo orden con retardo, con condiciones de frontera. los resultados son obtenidos mediante el uso del teorema de punto fijo de guo-krasnoselskii en conos. 80 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) key words and phrases: three-point boundary value problem, system of differential equations, eigenvalue problem, positive solutions, deviating arguments. math. subj. class.: 34b18, 34a34. 1 introduction consider the three-point boundary value problem system consisting of the second order delay differential equations, u ′′ (t) + λa(t)f (u(σ1(t)), v(σ2(t))) = 0, 0 < t < 1, v ′′ (t) + µb(t)g (u(τ1(t)), v(τ2(t))) = 0, 0 < t < 1, (1) along with the conditions, u(0) = 0, u(1) = αu(η), v(0) = 0, v(1) = αv(η), u(t) = φ1(t), v(t) = φ2(t), −r ≤ t ≤ 0, (2) where 0 < η < 1, 0 < α < 1/η, −r = mint∈[0,1] σi(t) = mint∈[0,1] τi(t), i = 1, 2, and φ1, φ2 : [−r, 0] → r+ are continuous functions, with φ1(0) = φ2(0) = 0. our interest in this paper is to investigate the existence of eigenvalues λ and µ that yield positive solutions to the associated boundary value problem, (1), (2). we assume that (a) f, g ∈ c(r+ × r+, r+); (b) a, b ∈ c([0, 1], r+), and each does not vanish identically on any subinterval; (c) σi, τi : [0, 1] → [−r, 1], i = 1, 2 are continuous functions; (d) all of f0 := lim u+v→0+ f (u, v) u + v , g0 := lim u+v→0+ g(u, v) u + v f∞ := lim u+v→∞ f (u, v) u + v , g∞ := lim u+v→∞ g(u, v) u + v exist as positive real numbers; (e) there exist an η∗ ∈ [η, 1] such that σi(s), τi(s) ∈ [η, 1] for all s ∈ [η∗, 1], i = 1, 2. we say that a pair (u, v) ∈ c ([−r, 1]) is a solution of the boundary value problem (bvp for short) (1), (2) if, u and v are twice continuously differentiable on (0, 1), u(t) = φ1(t), v(t) = φ2(t), cubo 11, 3 (2009) positive solutions for three-point bvps 81 for −r ≤ t ≤ 0, (u, v) satisfies (1) for all t ∈ (0, 1), and u(0) = 0, u(1) = αu(η) and v(0) = 0, v(1) = αv(η). for several years now, there has been a great deal of activity in studying positive solutions of boundary value problems for ordinary differential equations. interest in such solutions is high from both a theoretical sense [4, 7, 10, 13, 20] and as applications for which only positive solutions are meaningful [1, 5, 14, 15]. these considerations are caste primarily for scalar problems, but good attention has been given to boundary value problems for systems of differential equations [11, 12, 17, 19, 21]. the existence of positive solutions for nonlocal three-point boundary value problems has been studied extensively in recent years. for some appropriate references we refer the reader to [17], [18]. recently, benchohra et al. [2] and henderson and ntouyas [8] studied the existence of positive solutions for systems of nonlinear eigenvalue problems, while henderson and ntouyas [9] obtained results for the case of systems with three-point nonlocal boundary conditions. the purpose of this paper is to extend the results given in [9] to the case where delays may appear in the equations of the system (1), (2). the main tool in this paper is an application of the guo-krasnosel’skii fixed point theorem for operators leaving a banach space cone invariant [7]. a green’s function plays a fundamental role in defining an appropriate operator on a suitable cone. since, in our problem, we cannot express system (1), (2) as a single operator equation, the method used for example in [9] is not applicable here. this difficulty can be overcome by employing a method proposed by dunninger and wang in [3]. 2 some preliminaries before we state and prove our main result, we recall some useful facts that will be used in the sequel. concerning the boundary value problem u ′′ (t) + y(t) = 0, 0 < t < 1, (3) u(0) = 0, u(1) = αu(η), (4) we have the following two lemmas. lemma 2.1. [6] let (a), (b) and (c) hold and assume that 0 < η < 1 and 0 < α < 1/η. then, for any y ∈ c[0, 1] the bvp (3), (4) has a unique solution, u(t) = ∫ 1 0 k(t, s)y(s)ds, t ∈ [0, 1], 82 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) where k(t, s) : [0, 1] × [0, 1] → r+ is the green function defined by k(t, s) =                          t(1 − s) 1 − αη − αt(η − s) 1 − αη − (t − s), 0 ≤ s ≤ t ≤ 1 and s ≤ η, t(1 − s) 1 − αη − αt(η − s) 1 − αη , 0 ≤ t ≤ s ≤ η, t(1 − s) 1 − αη , 0 ≤ t ≤ s ≤ 1 and η ≤ s, t(1 − s) 1 − αη − (t − s), η ≤ s ≤ t ≤ 1. (5) lemma 2.2. [16] let (a), (b) and (c) hold and assume that 0 < α < 1/η. then, the unique solution of the problem (3), (4) satisfies inf t∈[η,1] u(t) ≥ γ‖u‖, where γ := min { αη, η, α(1 − η) 1 − αη } . from lemma 2.1 and the analytical expression of k, it follows that u can be written as u(t) = 1 1 − αη ∫ 1 0 (1 − s)y(s)ds − αt 1 − αη ∫ η 0 (η − s)y(s)ds − ∫ t 0 (1 − s)y(s)ds from which it follows that u(t) ≤ 1 1 − αη ∫ 1 0 (1 − s)y(s)ds, for all t ∈ [0, 1], (6) and u(η) ≥ η 1 − αη ∫ 1 η (1 − s)y(s)ds. (7) we note that a pair (u(t), v(t)) is a solution of the eigenvalue problem (1), (2) if, and only if, u(t) = φ1(t), v(t) = φ2(t) for −r ≤ t ≤ 0, and u(t) = λ ∫ 1 0 k(t, s)a(s)f (u(σ1(s)), v(σ2(s)))ds, 0 ≤ t ≤ 1, v(t) = µ ∫ 1 0 k(t, s)b(s)g(u(τ1(s)), v(τ2(s)))ds, 0 ≤ t ≤ 1. the main tool in determining values of the parameters λ and µ, for which positive (with respect to a cone) solutions of the bvp (1), (2) exist, is the following fixed point theorem. theorem 2.1. [7] let b be a banach space, and let p ⊂ b be a cone in b. assume ω1 and ω2 are open subsets of b with 0 ∈ ω1 ⊂ ω1 ⊂ ω2, and let t : p ∩ (ω2 \ ω1) → p be a completely continuous operator such that, either cubo 11, 3 (2009) positive solutions for three-point bvps 83 (i) ||t u|| ≤ ||u||, u ∈ p ∩ ∂ω1, and ||t u|| ≥ ||u||, u ∈ p ∩ ∂ω2, or (ii) ||t u|| ≥ ||u||, u ∈ p ∩ ∂ω1, and ||t u|| ≤ ||u||, u ∈ p ∩ ∂ω2. then t has a fixed point in p ∩ (ω2 \ ω1). 3 positive solutions in a cone in this section, we apply theorem 2.1 to obtain solutions in a cone (that is, positive solutions) of (1), (2). for our construction, we let x = c([−r, 1], r+) × c([−r, 1], r+) with norm ‖(u, v)‖ = ‖u‖ + ‖v‖ where ‖u‖ = supt∈[−r,1] |u(t)|. then (x, ‖ · ‖) is a banach space. we will make use of the cone p ⊂ x defined by p = { (u, v) : (u, v) ∈ x : u, v ≥ 0 on [−r, 1], min t∈[η,1] [u(t) + v(t)] ≥ γ[‖u‖ + ‖v‖] } , where γ > 0 is the positive constant defined in lemma 2.2. for our first result, define positive numbers l1 and l2 by l1 := max { 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − r)a(r)f∞dr ]−1 , 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − r)b(r)g∞dr ]−1 } , and l2 := min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − r)a(r)f0dr ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − r)b(r)g0dr ]−1 } . theorem 3.1. assume that conditions (a), (b), (c), (d) and (e) hold. then, for each λ, µ satisfying l1 < λ, µ < l2, (8) there exists a pair (u, v) satisfying (1), (2) such that u(t) > 0 and v(t) > 0 on (0, 1). proof. let a, b : x → x and f : x → x be the integral operators defined by a(u, v)(t) =        φ1(t), −r ≤ t ≤ 0, λ ∫ 1 0 k(t, s)a(s)f (u(σ1(s)), v(σ2(s))) ds, 0 ≤ t ≤ 1, 84 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) b(u, v)(t) =        φ2(t), −r ≤ t ≤ 0, µ ∫ 1 0 k(t, s)b(s)g (u(τ1(s)), v(τ2(s))) ds, 0 ≤ t ≤ 1, f (u, v)(t) = (a(u, v)(t), b(u, v)(t)) , t ∈ [−r, 1]. then seeking solutions to our bvp (1), (2) is equivalent to looking for fixed points of the equation f (u, v) = (u, v) in the banach space x. choose some (u, v) ∈ p. then by lemma 2.2 we have inf t∈[η,1] a(u, v)(t) ≥ γ‖a(u, v)‖, inf t∈[η,1] b(u, v)(t) ≥ γ‖b(u, v)‖ and thus inf t∈[η,1] [a(u, v)(t) + b(u, v)(t)] ≥ inf t∈[η,1] a(u, v)(t) + inf t∈[η,1] b(u, v)(t) ≥ γ [‖a(u, v)‖ + ‖b(u, v)||] = γ‖(a(u, v), b(u, v)‖ which implies that f (p) ⊂ p for every (u, v) ∈ p. as a and b are integral operators, it is not difficult to see that using standard arguments we may conclude that both a and b are completely continuous; hence f is a completely continuous operator. let λ and µ be as in (8), and choose an ǫ > 0 such that max { 1 2 [ γη 1 − αη ∫ 1 η (1 − r)a(r)(f∞ − ǫ)dr ]−1 , 1 2 [ γη 1 − αη ∫ 1 η (1 − r)b(r)(g∞ − ǫ)dr ]−1 } ≤ λ, µ and λ, µ ≤ min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − r)a(r)(f0 + ǫ)dr ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − r)b(r)(g0 + ǫ)dr ]−1 } . from the definition of f0 and g0, there exists an h1 > 0 such that f (u, v) ≤ (f0 + ǫ)(u + v) for u, v ∈ p with 0 < u, v < h1, cubo 11, 3 (2009) positive solutions for three-point bvps 85 and g(u, v) ≤ (g0 + ǫ)(u + v) for u, v ∈ p with 0 < u, v < h1. set ω1 = {(u, v) ∈ x : ‖(u, v)‖ < h1} . now let (u, v) ∈ p ∩ ∂ω1, i.e., let (u, v) ∈ p with ‖ (u, v) ‖ = h1. then, in view of the inequality (6) we have a(u, v)(t) ≤ λ t 1 − αη ∫ 1 0 (1 − s)a(s)f (u(σ1(s), v(σ2(s))) ds ≤ λ 1 1 − αη ∫ 1 0 (1 − s)a(s)(f0 + ǫ) [u(σ1(s) + v(σ2(s))] ds ≤ λ 1 1 − αη ∫ 1 0 (1 − s)a(s)(f0 + ǫ)[‖u‖ + ‖v‖]ds ≤ 1 2 [‖u‖ + ‖v‖] = 1 2 ‖(u, v)‖, and so, ‖a(u, v)‖ ≤ 1 2 ‖(u, v)‖. similarily, we may take ‖b(u, v)‖ ≤ 1 2 ‖(u, v)‖. thus, for (u, v) ∈ p ∩ ∂ω1 it follows that ‖f (u, v)‖ = ‖ (a(u, v), b(u, v)) ‖ = ‖a(u, v)‖ + ‖b(u, v)‖ ≤ 1 2 ‖(u, v)‖ + 1 2 ‖(u, v)‖ = ‖(u, v)‖, that is, ‖f (u, v)‖ ≤ ‖(u, v)‖ for all (u, v) ∈ p ∩ ∂ω1. due to the definition of f∞ and g∞, there exists an h2 > 0 such that f (u, v) ≥ (f∞ − ǫ)(u + v) for all u, v ≥ h2, and g(u, v) ≥ (g∞ − ǫ)(u + v) for all u, v ≥ h2. set h2 = max { 2h1, h2 γ } and define ω2 = {(u, v) ∈ x : ‖(u, v)‖ < h2} . 86 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) as from our hypothesis on η∗ it follows that inf t∈[η∗,1] [u(σ1(t)) + v(σ2(t))] ≥ γ[‖u‖ + ‖v‖]. (9) by the use of (7), we have for (u, v) ∈ p ∩ ∂ω2, a(u, v)(η) ≥ λ η 1 − αη ∫ 1 η (1 − s)a(s)f (u(σ1(s)), v(σ2(s))) ds ≥ λ η 1 − αη ∫ 1 η∗ (1 − s)a(s)(f∞ − ǫ) (u(σ1(s)) + v(σ2(s))) ds ≥ λ η 1 − αη ∫ 1 η∗ (1 − s)a(s)(f∞ − ǫ)γ[‖u‖ + ‖v‖]ds ≥ 1 2 ‖(u, v)‖, that is, a(u, v)(t) ≥ 1 2 ‖(u, v)‖ for all t ≥ η and so, a(u, v)(t) ≥ 1 2 ‖(u, v)‖. similarily, we may take b(u, v)(t) ≥ 1 2 ‖(u, v)‖. thus, for (u, v) ∈ p ∩ ∂ω2 it follows that ‖f (u, v)‖ = ‖ (a(u, v), b(u, v)) ‖ = ‖a(u, v)‖ + ‖b(u, v)‖ ≥ 1 2 ‖(u, v)‖ + 1 2 ‖(u, v)‖ = ‖(u, v)‖, that is ‖f (u, v)‖ ≥ ‖(u, v)‖ for all (u, v) ∈ p ∩ ∂ω2. applying theorem 2.1, we obtain that f has a fixed point (u, v) ∈ p ∩ (ω2 \ ω1) such that h1 ≤ ‖(u, v)‖ ≤ h2, and so (1), (2) has a positive solution. the proof is complete. � for our next result we define the positive numbers l3 = max { 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − s)a(s)f0ds ]−1 , 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − s)b(s)g0ds ]−1 } and l4 = min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − s)a(s)f∞dr ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − s)b(s)g∞dr ]−1 } . we are now ready to state and prove our main result. cubo 11, 3 (2009) positive solutions for three-point bvps 87 theorem 3.2. assume that conditions (a), (b), (c), (d) and (e) hold. then for each λ, µ satisfying l3 < λ, µ < l4, (10) there exists a pair (u, v) satisfying (1), (2) such that u(t) > 0 and v(t) > 0 on (0, 1). proof. let λ and µ be as in (10) and choose a sufficiently small ǫ > 0 so that λ, µ ≤ min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − s)a(s)(f∞ + ǫ)ds ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − s)b(s)(g∞ + ǫ)ds ]−1 } and max { 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − s)a(s)(f0 − ǫ)dr ]−1 , 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − s)b(s)(g0 − ǫ)dr ]−1 } ≤ λ, µ. by the definition of f0 and g0, there exists an h1 > 0 such that f (u, v) ≥ (f0 − ǫ)(u + v) for all u, v with 0 < u, v ≤ h3, and g(u, v) ≥ (g0 − ǫ)(u + v) for all u, v with 0 < u, v ≤ h3. set ω1 = {(x, y) ∈ x : ‖(x, y)‖ < h3} and let (u, v) ∈ p ∩ ∂ω3. in view of (9) and by the use of (7), we find a(u, v)(η) ≥ λ η 1 − αη ∫ 1 η (1 − s)a(s)f (u(σ1(s)), v(σ2(s))) ds ≥ λ η 1 − αη ∫ 1 η∗ (1 − s)a(s)f (u(σ1(s)), v(σ2(s))) ds ≥ λ η 1 − αη ∫ 1 η∗ (1 − s)a(s)(f0 − ǫ) (u(σ1(s)) + v(σ2(s))) ds ≥ λ η 1 − αη ∫ 1 η∗ (1 − s)a(s)(f0 − ǫ)γ[‖u‖ + ‖v‖]ds ≥ 1 2 ‖(u, v)‖, 88 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) that is, ‖a(u, v)‖ ≥ 1 2 ‖(u, v)‖. in a similar manner ‖b(u, v)‖ ≥ 1 2 ‖(u, v)‖. thus, for an arbitrary (u, v) ∈ p ∩ ∂ω3 it follows that ‖f (u, v) ‖ = ‖ (a (u, v) , b (u, v)) ‖ = ‖a (u, v) ‖ + ‖b (u, v) ‖ ≥ 1 2 ‖ (u, v) ‖ + 1 2 ‖ (u, v) ‖ = ‖(u, v)‖, and so ‖f (u, v) ‖ ≥ ‖(u, v)‖ for all (u, v) ∈ p ∩ ∂ω3. now let us define two functions f ∗, g∗ : [0, ∞) → [0, ∞) by f ∗ (t) = max 0≤u+v≤t f (u, v) and g∗(t) = max 0≤u+v≤t g(u, v). it follows that f (u, v) ≤ f ∗(t) and g(u, v) ≤ g∗(t) for all (u, v) with 0 ≤ u + v ≤ t. it is clear that the functions f ∗ and g∗ are nondecreasing. also, there is no difficulty to see that lim t→∞ f ∗ (t) t = f∞ and lim t→∞ g ∗ (t) t = g∞. in view of the definitions of f∞ and g∞, there exists an h4 such that f ∗ (t) < (f∞ + ε) t for all t ≥ h4, and g ∗ (t) < (g∞ + ε) t for all t ≥ h4. set h4 = max { 2h3, h4 γ } , and ω4 = {(u, v) : (u, v) ∈ p and ‖(u, v)‖ < h4} . let (u, v) ∈ p ∩ ∂h4 and observe that, by the definition of f ∗, it follows that for any s ∈ [0, 1], we have f (u(σ1(s)), v(σ2(s))) ≤ f ∗ (‖u‖ + ‖v‖) = f ∗ (‖(u, v)‖) . cubo 11, 3 (2009) positive solutions for three-point bvps 89 in view of the above observation and by the use of inequality (6) we obtain for t ∈ [0, 1] a (u, v) (t) ≤ λ t 1 − αη ∫ 1 0 (1 − s)a(s)f (u(σ1(s)), v(σ2(s))) ds ≤ λ t 1 − αη ∫ 1 0 (1 − s)a(s)f ∗ (‖u‖ + ‖v‖) ds ≤ λ t 1 − αη ∫ 1 0 (1 − s)a(s)(f∞ + ε) (‖u‖ + ‖v‖) dr ≤ λ 1 1 − αη ∫ 1 0 (1 − s)a(s)(f∞ + ε)dr ‖(u, v)‖ ≤ 1 2 ‖(u, v)‖ , which implies ‖a (u, v)‖ ≤ 1 2 ‖(u, v)‖ . in a similar manner, we take ‖b (u, v) ‖ ≤ 1 2 ‖ (u, v) ‖. thus, for (u, v) ∈ p ∩ ∂ω4 it follows that ‖f (u, v)‖ = ‖ (a (u, v) , b (u, v)) ‖ = ‖a (u, v) ‖ + ‖b (u, v) ‖ ≤ 1 2 ‖ (u, v) ‖ + 1 2 ‖ (u, v) ‖ = ‖(u, v)‖, and so ‖f (u, v)‖ ≤ ‖(u, v)‖ for all (u, v) ∈ p ∩ ∂ω4 applying theorem 2.1, we obtain that f has a fixed point (u, v) ∈ p ∩ (ω4 \ ω3) such that h3 ≤ ‖(u, v)‖ ≤ h4, and so (1), (2) has a positive solution. the proof is complete. � 4 a general application in this section we apply theorems 3.1 and 3.2 to the case where each one of the functions f and g is the sum of two (nonlinear) functions of a single argument, i.e., we consider the three-point boundary value system u ′′ (t) + λa(t)[ ˜f1 (u(σ1(t))) + ˜f2 (v(σ2(t)))] = 0, 0 < t < 1, v ′′ (t) + µb(t) [g̃1 (u(τ1(t))) + g̃2 (v(τ2(t)))] = 0, 0 < t < 1, (11) u(0) = 0, u(1) = αu(η), v(0) = 0, v(1) = αv(η), u(t) = φ1(t), v(t) = φ2(t), −r ≤ t ≤ 0, (12) 90 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) where 0 < η < 1, 0 < α < 1/η, r is a positive number, φ1, φ2 : [−r, 0] → r+, with φ1(0) = φ2(0) = 0 and σi, τi : [0, 1] → [−r, 1], i = 1, 2 are continuous functions. we assume that (a1) ˜fi, g̃i ∈ c([0, ∞), [0, ∞)), i = 1, 2; (b1) a, b ∈ c([0, 1], [0, ∞)) and each function does not vanish on any subinterval of [0, 1]; (c1) all of ˜f0 := lim t→0+ ˜fi(t) t , ˜f∞ := lim t→∞ ˜f (t) t , i = 1, 2, g̃0 := lim t→0+ g̃i(t) t , g̃∞ := lim t→∞ g̃i(t) t , i = 1, 2 exist as positive real numbers. we say that a pair (u, v) ∈ c ([−r, 1]) is a solution of the bvp (11), (12) if (i) u(t) = φ1(t), v(t) = φ2(t), for −r ≤ t ≤ 0, (ii) (u, v) satisfies (11) for all t ∈ (0, 1), and (iii) u(0) = v(0) = 0, u(1) = αu(η) and v(1) = αv(η). before we state our existence results for the bvp (11), (12), we prove an elementary lemma. lemma 4.1. let hi : [0, ∞) → [0, ∞), i = 1, 2 be continuous functions for which lim t→0+ hi(t) t = k ∈ (0, ∞) and lim t→∞ hi(t) t = m ∈ (0, ∞) , i = 1, 2. then for the function ̂h : [0, ∞) × [0, ∞) → [0, ∞) with ̂h (u, v) = h1(u) + h2(v), it holds that lim u+v→0+ ̂h(u, v) u + v = k and lim u+v→∞ ̂h(u, v) u + v = m. proof. by lim t→0+ hi(t) t = k, i = 1, 2 for an arbitrary ε > 0, there exists a δ > 0 such that (k − ε) u ≤ h1(u) ≤ (k + ε) u for all u ∈ (0, δ) , (k − ε) v ≤ h2(v) ≤ (k + ε) v for all v ∈ (0, δ) , and so, for any (u, v) with u, v ∈ ( 0, δ 2 ) , we have k − ε = (k − ε) u + (k − ε) v u + v ≤ ̂h(u, v) u + v = h1(u) + h2(v) u + v ≤ (k + ε) u + (k + ε) v u + v = k + ε, cubo 11, 3 (2009) positive solutions for three-point bvps 91 i.e., it holds that ∣ ∣ ∣ ∣ ∣ ̂h(u, v) u + v − k ∣ ∣ ∣ ∣ ∣ ≤ ε for any u, v > 0 with u + v < δ which implies that lim u+v→0+ ̂h(u, v) u + v = k. now let us assume that lim t→∞ hi(t) t = m ∈ (0, ∞) , i = 1, 2. it follows that, for an arbitrarily small ε > 0, there exists an m0 > 0 such that (m − ε) u ≤ h1(u) ≤ (m + ε) u, for all u > m0, (m − ε) v ≤ h2(v) ≤ (m + ε) v, for all v > mo. let u, v ≥ 0 with u + v > 2m0. then either u > m0 and v > m0 or one of u, v is greater than m0 while the other is less than m0. if u > m0 and v > m0, then by the last two inequalities we have m − ε = (m − ε) u + (m − ε) v u + v ≤ ̂h(u, v) u + v = h1(u) + h2(v) u + v ≤ (m + ε) u + (m + ε) v u + v = m + ε, which implies that ∣ ∣ ∣ ∣ ∣ ̂h(u, v) u + v − m ∣ ∣ ∣ ∣ ∣ ≤ ε for any u, v ≥ 0 with u > m0 and v > m0. (13) now let us deal with the case that one of the arguments u and v is less than m0 and the other one is (necessarily) greater that m0. we consider only the case u ≤ m0 and v > m0, as the conclusion for the dual case u > m0 and v ≤ m0 follows by similar arguments. set m ∗ = supu∈[0,m] h1(u). then, as lim v→∞ m ∗ v = 0 and lim v→∞ mv m0 + v = m, and limv→∞ h2(v) v = m we may consider an m > 2m0 such that m ∗ v < ε 2 and m − ε < mv m0 + v and h2(v) v < ε 2 + m. then for any u, v ≥ 0 with u ≤ m0 and v > m , we find m − ε ≤ mv m0 + v ≤ ̂h(u, v) u + v = h1(u) + h2(v) u + v ≤ m ∗ + h2(v) v ≤ ε + m, which implies that ∣ ∣ ∣ ∣ ∣ ̂h(u, v) u + v − m ∣ ∣ ∣ ∣ ∣ ≤ ε for any u, v ≥ 0 with u ≤ m0, v > m. (14) 92 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) in view of (13) and (14) we see that for any arbitrarily small positive real number ε, we can always find an m > 0 such that ∣ ∣ ∣ ∣ ∣ ̂h(u, v) u + v − m ∣ ∣ ∣ ∣ ∣ ≤ ε for any u, v > 0 with u + v > 2m. consequently, it holds lim u+v→∞ ̂h(u, v) u + v = m, which completes the proof of the lemma. � applying our main results to the case of the bvp (11), (12), we obtain the following two theorems. theorem 4.1. assume that conditions (a1), (b1), (c1), (c) and (e) hold. then, for any λ, µ satisfiyng l1 < λ, µ < l2, (15) the bvp (11), (12) has at least one solution (u, v) such that u(t) > 0 and v(t) > 0 on (0, 1), where we have set l1 = max { 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − r) a(r) ˜f∞dr ]−1 , 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − r) b(r)g̃∞dr ]−1 } and l2 = min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − r) a(r) ˜f0dr ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − r) b(r)g̃0dr ]−1 } . theorem 4.2. assume that conditions (a1), (b1), (c1), (c) and (e) hold. then, for any λ, µ satisfiyng l3 < λ, µ < l4, (16) there exists a pair (u, v) satisfying the bvp (11), (12) such that u(t) > 0 and v(t) > 0 on (0, 1), where l3 = max { 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − s)a(s) ˜f0ds ]−1 , 1 2 [ γη 1 − αη ∫ 1 η∗ (1 − s)b(s)g̃0ds ]−1 } and l4 = min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − s)a(s) ˜f∞dr ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − s)b(s)g̃∞dr ]−1 } . cubo 11, 3 (2009) positive solutions for three-point bvps 93 5 examples in this section, we present some examples that illustrate the breadth of our results. in particular, we give two examples, from which the first one concerns our general application while the second one concerns theorems 3.1 and 3.2. example 5.1. for the sake of simplicity, we assume that a = b, f1 = f2 and g1 = g2, σ1 = σ2, τ1 = τ2, i.e., we consider the bvp u ′′ (t) + λa(t)[ ˜f (u(σ(t))) + ˜f (v(σ(t)))] = 0, 0 < t < 1, v ′′ (t) + µa(t) [g̃ (u(τ (t))) + g̃ (v(τ (t)))] = 0, 0 < t < 1, (17) u(0) = 0, u(1) = 2u ( 1 3 ) , v(0) = 0, v(1) = 2v ( 1 3 ) , u(t) = φ1(t), v(t) = φ2(t), −r ≤ t ≤ 0, (18) where ˜f (t) = p1(t) + q1 sin (t), t ∈ r, g̃(t) = p2(t) + q2 sin (t), t ∈ r, with pi, pi + qi > 0, i = 1, 2, φ1, φ2 : [−r, 0] → r+, and σ, τ : [0, 1] → [− 14 , 1] are given by σ(t) =            √ t, t ∈ [0, 1/4] , 1 2 , t ∈ [1/4, 1/2] , t, t ∈ [1/2, 3/4] , 1 2 ( t + 3 4 ) , t ∈ [3/4, 1] , and τ (t) = t − 1 4 , t ∈ [0, 1]. it is not difficult to see that the argument σ is advanced on the interval [0, 1/4] (nonconstant on [0, 1/4] and constant on [1/4, 1/2]), retarded on the interval [3/4, 1] while neither retarded nor advanced on the interval [1/4, 1/2]. by the definition of ˜f and g̃ we may verify that ˜f∞ = p1, g̃∞ = p2, ˜f0 = p1 + q1 g̃0 = p2 + q2. as α = 2 and η = 1 3 we find γ := min { αη, η, α(1 − η) 1 − αη } = min { 2 3 , 1 3 , 2 ( 1 − 1 3 ) 1 − 2 3 } = 1 3 . 94 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) note that σ(t) ≥ η = 1 3 , for all t ∈ [ 1 9 , 1 ] , while τ (t) = t − 1 4 ≥ η = 1 3 , for all t ∈ [ 7 12 , 1 ] , and so η ∗ = 7 12 . thus γη 1 − αη ∫ 1 η∗ (1 − r) a(r) ˜f∞dr = 1 3 · 1 3 1 − 2 3 ∫ 1 7 12 (1 − r) a(r)p1dr = 1 3 p1 ∫ 1 7 12 (1 − r) a(r)dr and γη 1 − αη ∫ 1 η∗ (1 − r) b(r)g̃∞dr = 1 3 p2 ∫ 1 7 12 (1 − r) a(r)dr. hence l1 = max    1 2 [ 1 3 p1 ∫ 1 7 12 (1 − r) a(r)dr ] −1 , 1 2 [ 1 3 p2 ∫ 1 7 12 (1 − r) a(r)dr ] −1    = 3 2 min {p1, p2} ∫ 1 7 12 (1 − r) a(r)dr and l2 = min { 1 2 [ 1 1 − αη ∫ 1 0 (1 − r) a(r) ˜f0dr ]−1 , 1 2 [ 1 1 − αη ∫ 1 0 (1 − r) a(r)g̃0dr ]−1 } = 1 6 min { [ ∫ 1 0 (1 − r) a(r) (p1 + q1) dr ]−1 , [ ∫ 1 0 (1 − r) a(r) (p2 + q2) dr ]−1 } = 1 6 max {p1 + q1, p2 + q2} ∫ 1 0 (1 − r) a(r)dr . therefore, assuming that p1, q1, p2, q2 have been chosen so that 3 2 < min {p1, p2} ∫ 1 7 12 (1 − r) a(r)dr and 9 max {p1 + q1, p2 + q2} ∫ 1 0 (1 − r) a(r)dr < min {p1, p2} ∫ 1 7 12 (1 − r) a(r)dr, it follows that l1 < l2, and from theorem 4.1, we derive that, for any λ, µ satisfiyng l1 < λ, µ < l2, the bvp (17), (18) has at least one solution (u, v) such that u(t) > 0 and v(t) > 0 on (0, 1). cubo 11, 3 (2009) positive solutions for three-point bvps 95 example 5.2. consider the bvp u ′′ (t) + λa(t)f (u(σ(t)), v(σ(t))) = 0, 0 < t < 1, v ′′ (t) + µb(t)g (u(τ (t)), v(τ (t))) = 0, 0 < t < 1, (19) u(0) = 0, u(1) = 2u ( 1 3 ) , v(0) = 0, v(1) = 2v ( 1 3 ) , u(t) = φ1(t), v(t) = φ2(t), −1 ≤ t ≤ 0, (20) where φ1, φ2 : [−1, 0] → r+ with φ1(0) = 0 = φ2(0), and σ, τ : [0, 1] → [−1, 1] are given by σ(t) =    − sin (3πt) , t ∈ [0, 1/3] , 9 4 ( −t2 + 2t − 5 9 ) , t ∈ [1/3, 1], and τ (t) = √ t, t ∈ [0, 1]. as α = 2 and η = 1 3 , we find γ := min { αη, η, α(1 − η) 1 − αη } = min { 2 3 , 1 3 , 2 ( 1 − 1 3 ) 1 − 2 3 } = 1 3 . since σ(t) ≥ η = 1 3 is equivalent to 9 4 ( −t2 + 2t − 5 9 ) ≥ 1 3 , from which σ(t) ≥ 1 3 for all t ∈ [ 1 − 2 3 √ 2 3 , 1 ] , while τ (t) = √ t ≥ η = 1 3 , for all t ∈ [ 1 9 , 1 ] , we conclude that η ∗ = 1 − 2 3 √ 2 3 . we mention that the argument τ is advanced while the argument σ is retarded on [ 0, 5 9 ] and delayed on [ 5 9 , 1 ] . 96 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) now we calculate the positive numbers l3 and l4. as in example 5.1, we have α = 2, η = 1 3 and γ = 1 3 as well as γη 1 − αη = 1 3 and 1 1 − αη = 3. we find l3 = 3 2 min { ∫ 1 1− 2 3 √ 2 3 (1 − s)a(s) ˜f0ds, ∫ 1 1− 2 3 √ 2 3 (1 − s)b(s)g̃0ds } and l4 = min          1, 1 6 max { ∫ 1 0 (1 − s)a(s) ˜f∞dr, ∫ 1 0 (1 − s)b(s)g̃∞dr }          . applying theorem 4.1, we find that if 3 2 < min { ∫ 1 1− 2 3 √ 2 3 (1 − s)a(s) ˜f0ds, ∫ 1 1− 2 3 √ 2 3 (1 − s)b(s)g̃0ds } and 9 max { ∫ 1 0 (1 − s)a(s) ˜f∞dr, ∫ 1 0 (1 − s)b(s)g̃∞dr } < min { ∫ 1 1− 2 3 √ 2 3 (1 − s)a(s) ˜f0ds, ∫ 1 1− 2 3 √ 2 3 (1 − s)b(s)g̃0ds } , then for any λ, µ satisfiyng l3 < λ, µ < l4, there exists a pair (u, v) satisfying the bvp (19), (20) such that u(t) > 0 and v(t) > 0 on (0, 1). 6 remarks (1) similar results to those of theorems 3.1 and 3.2 can be proved for the following system of two point boundary value problems with deviating arguments u ′′ (t) + λa(t)f (u(σ1(t)), v(σ2(t))) = 0, 0 < t < 1, v ′′ (t) + µb(t)g (u(τ1(t)), v(τ2(t))) = 0, 0 < t < 1, (21) αu(0) − βu′(0) = 0, γu(1) + δu′(1) = 0, αv(0) − βv′(0) = 0, γv(1) + δv′(1) = 0, u(t) = φ1(t), v(t) = φ2(t), −r ≤ t ≤ 0, (22) where α, β, γ, δ ≥ 0 with α + β + γ + δ > 0, ρ = γβ + αγ + αδ > 0. cubo 11, 3 (2009) positive solutions for three-point bvps 97 (2) nondecreasingness may be used to give a sufficient condition that yields the existence of a positive number η∗ such as the one described in theorem 3.1. it is not difficult to see that, if σi(t) ≤ t, τi(t) ≤ t, for all t ∈ [0, 1], σi and τi are nondecreasing and τi(1) > η, σi(1) > η, then there always exists an η∗ ∈ [η, 1] such that mins∈[η∗,1] {σi(s)} ∈ [η, 1], mins∈[η∗,1] {τi(s)} ∈ [η, 1], i = 1, 2. (3) a requirement equivalent to the one in theorems 3.1 and 3.2 is the following: there exists an η∗ ∈ [η, 1] such that mins∈[η∗,1] {σi(s)} ∈ [η, 1], mins∈[η∗,1] {τi(s)} ∈ [η, 1], i = 1, 2. (4) in the case of advanced arguments σi(t) > t, τi(t) > t for all t ∈ [0, 1], i = 1, 2, inequality (9) also holds, since inf t∈[η,1] u(σ(t)) ≥ inf t∈[η,1] u(t) ≥ γ‖u‖. consequently we can deduce similar results to those of theorems 3.1 and 3.2 for the case of advanced arguments. (5) we can easily find necessary conditions in order to have l1 < l2 and l3 < l4. for example, l1 < l2 gives max { [ ∫ 1 η∗ (1 − r) a(r)f∞dr ]−1 , [ ∫ 1 η∗ (1 − r) b(r)g∞dr ]−1 } < γη min { [ ∫ 1 0 (1 − r) a(r)f0dr ]−1 , [ ∫ 1 0 (1 − r) b(r)g0dr ]−1 } and 1 min { ∫ 1 η∗ (1 − r) a(r)f∞dr, ∫ 1 η∗ (1 − r) b(r)g∞dr } < γη max { ∫ 1 0 (1 − r) a(r)f0dr, ∫ 1 0 (1 − r) b(r)g0dr } or (i) max { ∫ 1 0 (1 − r) a(r)f0dr, ∫ 1 0 (1 − r) b(r)g0dr } < γη min { ∫ 1 η∗ (1 − r) a(r)f∞dr, ∫ 1 η∗ (1 − r) b(r)g∞dr } . in a similar manner from l3 < l4 it follows that (ii) max { ∫ 1 0 (1 − s)a(s)f∞dr, ∫ 1 0 (1 − s)b(s)g∞dr } < γη min { ∫ 1 η∗ (1 − s)a(s)f0ds, ∫ 1 η∗ (1 − s)b(s)g0ds } . 98 j. henderson, s.k. ntouyas and i.k. purnaras cubo 11, 3 (2009) in order that (i) holds, it is necessary that ∫ 1 0 (1 − r) b(r)g0dr < γη ∫ 1 η∗ (1 − r) b(r)g∞dr 1 < ∫ 1 0 (1 − r) b(r)dr ∫ 1 η∗ (1 − r) b(r)dr < γη g∞ g0 , and similarly, we take 1 < ∫ 1 0 (1 − r) a(r)dr ∫ 1 η∗ (1 − r) a(r)dr < γη f∞ f0 . from these relations it follows that 1 η2 ≤ 1 γη < g∞ g0 , f∞ f0 which gives a (first) estimation of the bound for η, i.e., √ g0 g∞ , √ f0 f∞ < η ≤ 1. clearly, from the above necessary inequalities it follows that: i) at most one of theorems 3.1 and 3.2 may be applicable. ii) it is possible that both theorems 3.1 and 3.2 may fail as both l1 < l2 and l3 < l4 may not be satisfied: if f∞ = g∞ = f0 = f0, then by the last inequality above, neither (i) nor (ii) holds. sufficient conditions so that (i) or (ii) hold may be easily obtained in terms of g∞, g0 f∞, f0, a, b, η. received: november 8, 2007. revised: april 29, 2008. references [1] agarwal, r.p., o’regan, d. and wong, p.j.y., positive solutions of differential, difference and integral equations, kluwer, dordrecht, 1999. 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[21] zhou, y. and xu, y., positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations, j. math. anal. appl., 320, (2006), 578–590. 11-hnp_finrev cubo a mathematical journal vol.10, n o ¯ 03, (211–222). october 2008 on two-sided centralizers of rings and algebras∗ joso vukman department of mathematics and computer sciences, faculty of natural sciences and mathematics, university of maribor, koroška 160, 2000 maribor, slovenia email: joso.vukman@uni-mb.si and irena kosi-ulbl faculty of education, university of maribor, koroška 160, 2000 maribor, slovenia email: irena.kosi@uni-mb.si abstract in this paper we prove the following result. let a be a semisimple h∗−algebra and let t : a → a be an additive mapping satisfying the relation (n + 1)t (xnm+1) = t (x)xnm + xmt (x)x(n−1)m + · · · + xnmt (x), for all x ∈ a and some fixed integers m ≥ 1, n ≥ 1. in this case t is a two-sided centralizer. resumen en este art́ıculo probamos el siguiente resultado. sea a una h∗−algebra semi-simple y t : a → a una aplicación aditiva satisfaziendo la relación (n + 1)t (xnm+1) = ∗this research has been supported by the research council of slovenia. 212 joso vukman and irena kosi-ulbl cubo 10, 3 (2008) t (x)xnm + xmt (x)x(n−1)m + · · · + xnmt (x), para todo x ∈ a y ciertos m ≥ 1, y n ≥ 1 enteros fixados. en este caso t es un centralizador “two-sided”. key words and phrases: prime ring, semiprime ring, banach space, standard operator algebra, h∗-algebra, left (right) centralizer, left (right) jordan centralizer, two-sided centralizer. math. subj. class.: 16w10, 46k15, 39b05. introduction throughout, r will represent an associative ring with center z(r). given an integer n ≥ 2, a ring r is said to be n−torsion free, if for x ∈ r, nx = 0 implies x = 0. as usual the commutator xy − yx will be denoted by [x, y] .let us recall that a ring r is prime if for a, b ∈ r, arb = (0) implies that either a = 0 or b = 0, and is semiprime in case ara = (0) implies a = 0. an additive mapping x 7−→ x∗ on a ring r is called involution in case (xy)∗ = y∗x∗ and x∗∗ = x hold for all pairs x, y ∈ r. a ring equipped with an involution is called a ring with involution or ∗−ring. we denote by qr and c martindale right ring of quotients and extended centroid of a semiprime ring r. for the explanation of qr and c we refer to [3] . an additive mapping t : r → r, where r is an arbitrary ring, is called a left centralizer in case t (xy) = t (x)y holds for all pairs x, y ∈ r. the concept appears naturally in c∗-algebras. in ring theory it is more common to work with module homomorphisms. ring theorists would write t : rr → rr of a right ring module r into itself. for a semiprime ring r all such homomorphisms are of the form t (x) = qx, for all x ∈ r, where q is some fixed element of qr (see chapter 2 in [3]). in case r has the identity element t : r → r is a left centralizer iff t is of the form t (x) = ax, for all x ∈ r, where a is some fixed element of r.an additive mapping t : r → r is called a left jordan centralizer in case t (x2) = t (x)x holds for all x ∈ r.the definitions of right centralizer and right jordan centralizer are self-explanatory. we call t : r → r a two-sided centralizer in case t is both a left and a right centralizer. in case t : r → r is a two-sided centralizer, where r is a semiprime ring with extended centroid c, then there exists an element λ ∈ c such that t (x) = λx, for all x ∈ r (see theorem 2.3.2 in [3]). one of the initial papers using the concept of centralizers (also called multipliers) is due to wendel [33] for group algebras. helgason [9] introduced centralizers for banach algebras. wang [32] studied centralizers of commutative banach algebras. johnson [11] introduced the concept of centralizers for rings. we refer to busby [7] for a study of socalled double centralizers in the extension of c∗−algebras. akemann, pedersen and tomiyama [1] have studied centralizers of c∗-algebras. several authors have also studied spectral properties of centralizers on banach algebras (see [15, 16]). johnson [12] has studied centralizers on some topological algebras. johnson [13] has studied the continuity of centralizers on banach algebras (see also [11]). husain [10] has also investigated centralizers on topological algebras with particular reference to complete metrizable locally convex algebras and topological algebras with orthogonal bases. khan, mohammad and thaheem [14] have studied centralizers and double centralizers on certain topological algebras. centralizers have also appeared in a variety, among which we cubo 10, 3 (2008) on two-sided centralizers of rings and algebras 213 mention representation theory of banach algebras, the study of banach modules, hopf algebras (see [18, 19]), the theory of singular integrals, interpolation theory, stohastic processes, the theory of semigroups of operators, partial differential equations and the study of approximation problems (see larsen [16] for more details). zalar [34] has proved that any left (right) jordan centralizer on a 2-torsion free semiprime ring is a left (right) centralizer. molnár [17] has proved that in case we have an additive mapping t : a → a, where a is a semisimple h∗−algebra, satisfying the relation t (x3) = t (x)x2 (t (x3) = x2t (x)) for all x ∈ a, then t is a left (right) centralizer. let us recall that a semisimple h∗-algebra is a semisimple banach ∗-algebra a whose norm is a hilbert space norm such that (x, yz∗) = (xz, y) = (z, x∗y) is fulfilled for all x, y, z ∈ a (see [2]). benkovič and eremita [4] have proved that in case there exists an additive mapping t : r → r,where r is a prime ring with suitable characteristic restrictions, satisfying the relation t (xn) = t (x)xn−1, for all x ∈ r and some fixed integer n > 1, then t is a left centralizer. vukman and kosi-ulbl [26] have proved that any additive mapping t , which maps a semisimple h∗−algebra a into itself and satisfies the relation 2t (xn+1) = t (x)xn + xnt (x), for all x ∈ a and some fixed integer n ≥ 1, is a two-sided centralizer (see also [5]). a result of vukman and kosi-ulbl [27] states that in case there exists an additive mapping t : r → r, where r is a 2−torsion free semiprime ∗−ring, satisfying the relation t (xx∗) = t (x)x∗ (t (x∗x) = x∗t (x)), for all x ∈ r,then t is a left (right) centralizer. for results concerning centralizers on prime and semiprime rings, operator algebras and h∗-algebras we refer to [8, 20 − 31] . let x be a real or complex banach space and let l(x) and f (x) denote the algebra of all bounded linear operators on x and the ideal of all finite rank operators in l(x), respectively. an algebra a(x) ⊂ l(x) is said to be standard in case f (x) ⊂ a(x). let us point out that any standard algebra is prime, which is a consequence of hahn-banach theorem. we denote by x∗ the dual space of a banach space x and by i the identity operator on x. vukman [20] has proved the following result. theorem a. let r be a 2-torsion free semiprime ring and let t : r → r be an additive mapping. suppose that 2t (x2) = t (x)x + xt (x) holds for all x ∈ r. in this case t is a two-sided centralizer. vukman and kosi-ulbl [23] have proved the result below. theorem b. let r be a 2-torsion free semiprime ring and let t : r → r be an additive mapping. suppose that 3t (xyx) = t (x)yx + xt (y)x + xyt (x) holds for all pairs x, y ∈ r. in this case t is of the form t (x) = λx, for all x ∈ r and some fixed element λ from the extended centroid c of r. 214 joso vukman and irena kosi-ulbl cubo 10, 3 (2008) motivated by theorem a and theorem b fošner and vukman [8] have proved the following theorem. theorem c. let r be a prime ring and let t : r → r be an additive mapping satisfying the relation nt (xn+1) = t (x)xn−1 + xt (x)xn−2 + ... + xn−1t (x), for all x ∈ r, where n ≥ 2 is some fixed integer. if char(r) = 0, then t is of the form t (x) = λx, for all x ∈ r and some fixed element λ from the extended centroid c of r. in the proof of theorem c fošner and vukman used as the main tool the theory of functional identities (beidar-brešar-chebotar theory). the theory of functional identities considers set-theoretic maps on rings that satisfy some identical relations. when threatening such relations one usually concludes that the form of the maps involved can be described, unless the ring is very special (see[6]). it this paper we consider the following more general relation (n + 1)t (xnm+1) = t (x)xnm + xmt (x)x(n−1)m + ... + xnmt (x), (1) where m ≥ 1, n ≥ 1 are some fixed integers. one can notice that the expression (1) for n = m = 1 is the same as hypothesis of theorem a. obviously, any two-sided centralizer on arbitrary ring satisfies the above relation. we proceed with the following conjecture. conjecture. let r be a semiprime ring with suitable torsion restrictions and let t : r → r be an additive mapping satisfying the relation (1) for all x ∈ r and some fixed integers m ≥ 1, n ≥ 1. in this case t is a two-sided centralizer. it is our aim in this paper to prove the above conjecture in semisimple h∗-algebras and in semiprime rings with the identity element. our methods differ from those used in [8]. theorem 1. let a be a semisimple h∗-algebra. suppose t : a → a is an additive mapping satisfying the relation (1) for all x ∈ a and some fixed integers m ≥ 1, n ≥ 1. in this case t is a two-sided centralizer. for the proof of theorem 1 we need the theorem below which is of independent interest. theorem 2. let x be a banach space over the real or complex field f, let a(x) ⊂ l(x) be a standard operator algebra. suppose t : a(x) → l(x) is an additive mapping satisfying the relation (n + 1)t (anm+1) = t (a)anm + amt (a)a(n−1)m + ... + anmt (a), cubo 10, 3 (2008) on two-sided centralizers of rings and algebras 215 for all a ∈ a(x) and some fixed integers m ≥ 1, n ≥ 1. in this case t is of the form t (a) = λa, for all a ∈ a(x) and some fixed λ ∈ f.in particular, t is continuous. proof. we have the relation (n + 1)t (anm+1) = t (a)anm + amt (a)a(n−1)m + ... + anmt (a). (2) let us first consider the restriction of t on f (x). let a be from f (x) and let p ∈ f (x), be a projection with ap = p a = a. from the above relation one obtains t (p ) = p t (p )p, which gives t (p )p = p t (p ). (3) putting a + p for a in the relation (2), we obtain (n + 1) nm+1 ∑ i=0 ( nm + 1 i ) t ( anm+1−ip i ) = (t (a) + b) ( nm ∑ i=0 ( nm i ) anm−ip i ) + ( m ∑ i=0 ( m i ) am−ip i ) (t (a) + b)   (n−1)m ∑ i=0 ( (n − 1) m i ) a(n−1)m−ip i   + ... + (4) ( nm ∑ i=0 ( nm i ) anm−ip i ) (t (a) + b) , where b stands for t (p ) . using (2) and rearranging the equation (4) in sense of collecting together terms involving equal number of factors of p we obtain nm ∑ i=1 fi (a, p ) = 0, (5) where fi (a, p ) stands for the expression of terms involving i factors of p. replacing a by a + 2p, a + 3p, . . . , a + nmp in turn in the equation (1), and expressing the resulting system of nm homogeneous equations of variables fi (a, p ), i = 1, 2, ..., nm, we see that the coefficient matrix of the system is a van der monde matrix        1 1 · · · 1 2 22 · · · 2nm ... ... ... ... nm (nm) 2 · · · (nm) nm        . 216 joso vukman and irena kosi-ulbl cubo 10, 3 (2008) since the determinant of the matrix is different from zero, it follows that the system has only the trivial solution. in particular, fnm−1 (a, p ) = (n + 1) ( nm+1 nm−1 ) t ( a2 ) − ( nm nm−1 ) t (a) a − ( nm nm−2 ) ba2− ( m m−2 )( (n−1)m (n−1)m ) a2b − ( m m−1 )( (n−1)m (n−1)m ) at (a) p − ( m m−1 )( (n−1)m (n−1)m−1 ) aba− ( m m )( (n−1)m (n−1)m−1 ) p t (a)a − ( m m )( (n−1)m (n−1)m−2 ) ba2 − · · · − ( m m )( (n−1)m (n−1)m−2 ) a2b − ( m m )( (n−1)m (n−1)m−1 ) at (a) p − ( m m−1 )( (n−1)m (n−1)m−1 ) aba− ( m m−1 )( (n−1)m (n−1)m ) p t (a) a − ( m m−2 )( (n−1)m (n−1)m ) ba2− ( nm nm−2 ) a2b − ( nm nm−1 ) at (a) = 0 and fnm (a, p ) = (n + 1) ( nm+1 nm ) t (a) − ( nm nm ) t (a) p − ( nm nm−1 ) ba− ( m m−1 )( (n−1)m (n−1)m ) ab − ( m m )( (n−1)m (n−1)m ) p t (a)p − ( m m )( (n−1)m (n−1)m−1 ) ba − · · · − ( m m )( (n−1)m (n−1)m−1 ) ab − ( m m )( (n−1)m (n−1)m ) p t (a)p − ( m m−1 )( (n−1)m (n−1)m ) ba− ( nm nm−1 ) ab − ( nm nm ) p t (a) = 0. the above equations reduce to 6 (n + 1) (nm + 1) t ( a2 ) = 12 (t (a)a + at (a)) + 6 (n − 1) (at (a) p + p t (a) a) + (n + 1) ((2n + 1) m − 3) ( a2b + ba2 ) + 2m (n − 1) (n + 1) aba, (6) and 2 (n + 1) (nm + 1) t (a) = 2 (t (a) p + p t (a)) + n (n + 1) m (ab + ba) + 2 (n − 1) p t (a) p. (7) right multiplication of the relation (7) by p gives 2 (n + 1) (nm + 1) t (a) p = 2 (t (a) p + p t (a)) + n (n + 1) m (ab + ba) + 2 (n − 1) p t (a) p. (8) similarly one obtains 2 (n + 1) (nm + 1) p t (a) = 2 (t (a) p + p t (a)) + cubo 10, 3 (2008) on two-sided centralizers of rings and algebras 217 n (n + 1) m (ab + ba) + 2 (n − 1) p t (a) p. (9) combining (8) with (9) we arrive at t (a)p = p t (a), which reduces the relation (6) to 6 (mn + 1) t ( a2 ) = 6 (t (a)a + at (a)) + ((2n + 1) m − 3) ( a 2 b + ba2 ) + 2m (n − 1) aba, (10) and the relation (7) to 2 (mn + 1) t (a) = 2t (a) p + mn (ab + ba) . (11) right multiplication of the above relation by p and combining the relation so obtained with (11) gives t (a) = t (a)p. according to the above relation the relation (11) reduces to 2t (a) = ab + ba. (12) from the above relation one obtains 2t ( a2 ) = a2b + ba2. (13) right and then left multiplication of the relation (12) by a gives 2t (a) a = aba + ba2 (14) and 2at (a) = a2b + aba, (15) respectively. using the relations (13), (14) and (15) in the relation (10) gives after some calculation a (m, n) ba2 + a (m, n) a2b − 2a (m, n) aba = 0, where a (m, n) stands for mn − m + 3. the above relation reduces to a 2 b + ba2 − 2aba = 0. (16) applying the relations (13) and (16) in the relation (10) one obtains 2t ( a2 ) = t (a)a + at (a). (17) from the relation (12) one can conclude that t maps f (x) into itself. we have therefore an additive mapping t : f (x) → f (x) satisfying the relation (17) for all a ∈ f (x). since f (x) 218 joso vukman and irena kosi-ulbl cubo 10, 3 (2008) is prime one can apply theorem a and conclude that t is a two-sided centralizer of f (x). we intend to prove that there exists an operator c ∈ l(x), such that t (a) = ca, for all a ∈ f (x). (18) for any fixed x ∈ x and f ∈ x∗ we denote by x ⊗ f an operator from f (x) defined by (x ⊗ f )y = f (y)x, for all y ∈ x. for any a ∈ l(x) we have a(x ⊗ f ) = ((ax) ⊗ f ).let us choose f and y such that f (y) = 1 and define cx = t (x ⊗ f )y. obviously, c is linear. using the fact that t is left centralizer on f (x) we obtain (ca)x = c(ax) = t ((ax) ⊗ f )y = t (a(x ⊗ f ))y = t (a)(x ⊗ f )y = t (a)x, x ∈ x. we have therefore t (a) = ca for any a ∈ f (x). since t right centralizer on f (x) we obtain c(ap ) = t (ap ) = at (p ) = acp, where a ∈ f (x) and p is arbitrary one-dimensional projection. we have therefore [a, c] p = 0. since p is arbitrary one-dimensional projection it follows that [a, c] = 0 for any a ∈ f (x). using closed graph theorem one can easily prove that c is continuous. since c commutes with all operators from f (x) one can conclude that cx = λx holds for any x ∈ x and some λ ∈ f, which gives together with the relation (17) that t is of the form t (a) = λa (19) for any a ∈ f (x) and some λ ∈ f. it remains to prove that the above relation holds on a(x) as well. let us introduce t1 : a(x) → l(x) by t1(a) = λa and consider t0 = t − t1. the mapping t0 is, obviously, additive and satisfies the relation (2). besides, t0 vanishes on f (x). it is our aim to prove that t0 vanishes on a(x) as well. let a ∈ a(x), let p be an one-dimensional projection and s = a+p ap −(ap +p a). note that s can be written in the form s = (i −p )a(i −p ), where i denotes the identity operator on x, since, obviously, s − a ∈ f (x), we have t0(s) = t0(a). besides, sp = p s = 0. we have therefore the relation (n + 1)t0(a nm+1) = t0(a)a nm + amt0(a)a (n−1)m + ... + anmt0(a), (20) for all a ∈ a(x). applying the above relation we obtain t0(s)s nm + smt0(s)s (n−1)m + ... + snmt0(s) = (n + 1)t0(s nm+1) = (n + 1)t0(s nm + p ) = (n + 1)t0((s + p ) nm+1) = t0(s + p )(s + p ) nm + (s + p )mt0(s + p )(s + p ) (n−1)m + ...+ (s + p )(n−1)mt0(s)(s + p ) m + (s + p )nmt0(s + p ) = t0(s)s nm+ smt0(s)s (n−1)m + ... + snmt0(s) + t0(s)p + s mt0(s)p + p t0(s)s (n−1)m + ... + s(n−1)mt0(s)p + p t0(s)s m + p t0(s) + (n − 1)p t0(s)p. cubo 10, 3 (2008) on two-sided centralizers of rings and algebras 219 we have therefore t0(a)p + s mt0(a)p + p t0(a)s (n−1)m + ... + s(n−1)mt0(a)p + p t0(a)s m + p t0(a) + (n − 1)p t0(a)p = 0. (21) multiplying the above relation from both sides by p we obtain p t0(a)p = 0. (22) now right multiplication of the relation (21) by p gives because of (22) t0(a)p + s mt0(a)p + ... + s (n−1)mt0(a)p = 0. (23) replacing a by 2a, 3a, . . . , na in turn in the equation (23), and expressing the resulting system of n homogeneous equations of variables t0(a)p, s imt0(a)p , i = 1, 2, ..., n− 1, we see that the coefficient matrix of the system is a matrix of the form        1 1 · · · 1 1 2m · · · 2(n−1)m ... ... ... ... 1 nm · · · n(n−1)m        . since the determinant of the matrix is different from zero, it follows that the system has only the trivial solution. we have therefore t0(a)p = 0. since p is an arbitrary one-dimensional projection, one can conclude that t0(a) = 0, for any a ∈ a(x), which completes the proof of the theorem. it should be mentioned that in the proof of theorem 2 we used some ideas similar to those used by molnár in [17]. let us point out that in theorem 2 we obtain as a result the continuity of t under purely algebraic assumptions concerning t, which means that theorem 2 might be of some interest from the automatic continuity point of view. proof of theorem 1. the proof goes through using the same arguments as in the proof of theorem in [17] with the exception that one has to use theorem 2 instead of lemma in [17]. we are ready for our last result. theorem 3. let n ≥ 1, m ≥ 1 be integers and let r be a 2, m, n, n + 1 and ((n − 1) m + 3)−torsion free semiprime ring with the identity element. suppose that we have an additive mapping t : r → r satisfying the relation (1) for all x ∈ r. in this case t is of the form t (x) = ax, for all x ∈ r and some fixed element a ∈ z(r). 220 joso vukman and irena kosi-ulbl cubo 10, 3 (2008) proof. we have the relation (1). using similar approach as in the proof of theorem 2, with the exception that we use the identity element e instead of a projection, we obtain from the above relation 6(nm + 1)t ( x2 ) = 6 (t (x)x + xt (x)) + ((2n + 1) m − 3) ( x2a + ax2 ) + 2 (n − 1) mxax, x ∈ r (24) and 2t (x) = xa + ax, x ∈ r, (25) where a stands for t (e). in the procedure mentioned above we used the fact that r is m, n and n + 1-torsion free. the substitution x2 for x in (25)gives 2t ( x2 ) = x2a + ax2, x ∈ r. (26) multiplying the relation (25) first from the right side then from the left side by x we obtain 2t (x) x = xax + ax2, x ∈ r (27) and 2xt (x) = x2a + xax, x ∈ r. (28) using (26), (27) and (28) in the relation (24)and applying the fact that r is (n − 1)m + 3−torsion free we obtain after some calculation x 2 a + ax2 − 2xax = 0, x ∈ r, which can be written in the form [[a, x] , x] = 0, x ∈ r. (29) putting x + y for x in the above relation we obtain [[a, x] , y] + [[a, y] , x] = 0, x, y ∈ r. (30) the substitution xy for y in relation (30) gives because of (29) and (30) 0 = [[a, x] , xy] + [[a, xy] , x] = = [[a, x] , x] y + x [[a, x] , y] + [[a, x] y + x [a, y] , x] = = x [[a, x] , y] + [[a, x] , x] y + [a, x] [y, x] + x [[a, y] , x] = [a, x] [y, x] , x, y ∈ r. thus we have [a, x] [y, x] = 0, x, y ∈ r. cubo 10, 3 (2008) on two-sided centralizers of rings and algebras 221 the substitution ya for y in the above relation gives [a, x] y [a, x] = 0, for all pairs x, y ∈ r. let us point out that so far we have not used the assumption that r is semiprime. since r is semiprime, it follows from the last relation that [a, x] = 0, for all x ∈ r. in other words, a ∈ z (r) , which reduces the relation (25) to t (x) = ax, x ∈ r, since r is 2−torsion free. the proof of the theorem is complete. received: may 2008. revised: august 2008. references [1] c.a. akemann, g.k. pedersen and j. tomiyama, multipliers of c∗−algebras, j. funct. anal., 13(1973), 277–301. 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[34] b. zalar, on centralizers of semiprime rings, comment. math. univ. carol., 32(1991), 609– 614. n17 a mathematical journal vol. 7, no 3, (27 37). december 2005. sufficiency of the maximum principle for time optimality h. o. fattorini department of mathematics, university of california los angeles, california 90095-1555 hof@math.ucla.edu abstract for infinite dimensional linear systems, pontryagin’s maximum principle is shown to be sufficient for time optimality with conditions on the initial condition and on the target. these conditions cannot be given up and are shown to be best possible by means of counterexamples. resumen para sistemas lineales en dimensión infinita, el principio del máximo de pontryagin es suficiente para alcanzar optimalidad en el tiempo con condiciones en el valor inicial y el final. estas condiciones no se pueden relajar y se muestra que son las mejores posibles, por medio de contraejemplos. key words and phrases: linear control systems in banach spaces, time optimal problem. math. subj. class.: 93e20, 93e25. 1 introduction. consider the time optimal problem of driving the solution y(t) of y′(t) = ay(t) + u(t) , y(0) = ζ (1.1) 28 h. o. fattorini 7, 3(2005) from the initial point ζ to a point target, y(t ) = ȳ (1.2) with maximum-norm bound ‖u(t)‖ ≤ 1 a. e. in 0 ≤ t ≤ t (1.3) in minimum time t ; a is the infinitesimal generator of a strongly continuous semigroup s(t) in a banach space e and the controls u(t) are strongly measurable (so that, in view of (1.3), belong to the unit ball of l∞(0, t ; e)). solutions or trajectories y(t) = s(t)ζ + ∫ t 0 s(t − σ)u(σ)dσ of (1.1) are named y(t) = y(t, ζ, u) and controls satisfying (1.3) are called admissible. let z be an arbitrary linear space with e∗ ⊆ z. we say that z is a multiplier space if (i) s(t)∗ is defined in z,(ii) s(t)∗z ⊆ e∗ for t > 0. a control ū(t) in the interval 0 ≤ t ≤ t satisfies the weak maximum principle if there exists z in a multiplier space z such that s(t)∗z is not identically zero in 0 < t ≤ t and 〈s(t − t)∗z, ū(t)〉 = max ‖u‖≤1 〈s(t − t)∗z, u〉 a. e. in 0 ≤ t ≤ t , (1.4) where 〈· , ·〉 is the duality of the space e and the dual e∗. the control satisfies the strong maximum principle if (1.4) holds and∫ t 0 ‖s(t)∗z‖e∗dt < ∞ . (1.5) the space z(t ) consists of all multipliers that satisfy 1 (1.5). in hilbert space, (1.4) is equivalent to ū(t) = s(t − t)∗z ‖s(t − t)∗z‖ (1.6) whenever the denominator is not zero.it is known [3] that if the control ū(t) drives ζ ∈ e to ȳ = y(t, ζ, ū) ∈ d(a) then the strong maximum principle (1.4)-(1.5) is a necessary condition for time optimality. it is also known [4] that(1.4)-(1.5) is a sufficient condition if ζ = 0 or ȳ = y(t, ζ, ū) = 0; then ū(t) drives ζ to ȳ time optimally.2 we prove below (theorem 1.2) that these conditions on the initial and final point of the trajectory can be relaxed to one of the two assumptions ζ ∈ d(a), ‖aζ‖ ≤ 1 or ȳ ∈ d(a), ‖aȳ‖ ≤ 1 , (1.7) 1the semigroup s(t)∗ may not be strongly continuous, but in all cases the norm ‖s(t)∗‖ is lower semicontinuous, thus theintegral (1.5) makes sense. the spaces z(t ) are the same for all t > 0; condition (1.5) only bears on the behavior of ‖s(t)∗‖ near zero. 2the weak maximum principle(1.4) is not a sufficient condition for time optimality; for a counterexample, see[6] 7, 3(2005) sufficiency of the maximum principle for time optimality 29 (the first with an additional condition on the adjoint semigroup). that restrictions on the initial condition ζ or the target ȳ cannot be completely given up is illustrated with several examples, two of which show that conditions (1.7) are the best possible of their kind. we also see (in example 4.2) that restrictions on ‖ζ‖, ‖ȳ‖ (rather than on ‖aζ‖, ‖aȳ‖) do not guarantee sufficiency of the maximum principle for time optimality. remark 1.1. if s(t) is a group or, more generally, if s(t )e = e (t > 0) then the condition ȳ ∈ d(a) is not required to show that the maximum principle is a necessary condition for time optimality; moreover, z(t ) = e∗. sufficiency of the maximum principle, however, requires the same conditions as those in the general case. 2 sufficiency of the maximum principle. let r∞(t ) ⊆ e be the space of all elements of the form y = y(t, 0, u) = ∫ t 0 s(t − σ)u(σ)dσ , u(·) ∈ l∞(0, t ; e) . (2.1) the norm ‖y‖r∞(t ) is the infimum of ‖u(·)‖l∞(0,t ;e) for all u(·) that satisfy (2.1); in other words, r∞(t ) is the quotient of l∞(0, t ; e) by the closed subspace characterized by y(t, 0, u) = 0. an element z ∈ z(t ) defines a bounded linear functional ξz in r∞(t ) through the formula 〈〈ξz, y〉〉 = ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ (2.2) where y and u(·) are related by (2.1) and 〈〈·, ·〉〉 indicates the duality of the space r∞(t ) and its dual r∞(t )?; the norm of ξz satisfies ‖ξz‖r∞(t )? = ∫ t 0 ‖s(t)∗z‖e∗dt . (2.3) theorem 2.1. assume that ū(t) satisfies (1.4) (1.5) and that either (a) ȳ = y(t, ζ, ū) ∈ d(a), ‖aȳ‖ < 1 , or (b) ζ ∈ d(a), ‖aζ‖ < 1 , s(t)∗z 6= 0 in 0 ≤ t < t . (2.4) then ū(t) drives ζ to ȳ time optimally in 0 ≤ t ≤ t. proof of case (a). assume ū(·) does not drive ζ to ȳ time optimally. then there exists δ > 0 and a control ũ(·) ∈ l∞(0, t − δ; e), ‖ũ(·)‖l∞(0,t−δ;e) ≤ 1, that drives ζ to ȳ in time t − δ. the control 30 h. o. fattorini 7, 3(2005) v(σ) = { ũ(σ) (0 ≤ σ < t − δ) −aȳ (t − δ ≤ σ ≤ t ) (2.5) satisfies ‖v(·)‖l∞(0,t ;e) ≤ 1 . we have ȳ − s(t − (t − δ))ȳ = − ∫ t−(t−δ) 0 s(σ)aȳ dσ = − ∫ t−(t−δ) 0 s(t − (t − δ) − σ)aȳ dσ = ∫ t t−δ s(t − σ)(−aȳ) dσ (t − δ ≤ t ≤ t ) (2.6) hence the trajectory y(t, ζ, v) starts at ζ, reaches ȳ at time t − δ and stays at ȳ for t − δ ≤ t ≤ t ; y(t, ζ, v) = ȳ (t − δ ≤ t ≤ t ) . (2.7) this can be also be seen noting that if y(t) = ȳ then we have y′(t) − ay(t) = −aȳ in t − δ ≤ t ≤ t. the maximum principle (1.4) is equivalent to∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ ≤ ∫ t 0 〈s(t − σ)∗z, ū(σ)〉dσ (u(·) ∈ l∞(0, t ; e), ‖u(·)‖l∞(0,t ;e) ≤ 1) . (2.8) in terms of the linear functional ξz in (2.2), this is〈〈 ξz, ∫ t 0 s(t − σ)u(σ)dσ 〉〉 ≤ 〈〈 ξz, ∫ t 0 s(t − σ)ū(σ)dσ 〉〉 (u(·) ∈ l∞(0, t ; e), ‖u(·)‖l∞(0,t ;e) ≤ 1) . (2.9) we have y(t, ζ, v) = y(t, ζ, ū), thus∫ t 0 s(t − σ)v(σ)dσ = ∫ t 0 s(t − σ)ū(σ)dσ , (2.10) and it follows from (2.9) that〈〈 ξz, ∫ t 0 s(t − σ)u(σ)dσ 〉〉 ≤ 〈〈 ξz, ∫ t 0 s(t − σ)v(σ)dσ 〉〉 (u(·) ∈ l∞(0, t ; e), ‖u(·)‖l∞(0,t ;e) ≤ 1) , (2.11) which, being equivalent to (1.4), gives 〈s(t − t)∗z, v(t)〉 = max ‖u‖≤1 〈s(t − t)∗z, u〉 a. e. in 0 ≤ t ≤ t . (2.12) 7, 3(2005) sufficiency of the maximum principle for time optimality 31 we have s(t − σ)∗z 6= 0 near3 t, hence (2.12) implies ‖v(σ)‖ = 1 near t. this is a contradiction, since by hypothesis ‖v(σ)‖ = ‖aȳ‖ < 1. proof of case (b). this time we define v(σ) = −aζ (0 ≤ σ ≤ δ) ũ(σ − δ) (δ ≤ σ ≤ t ) . (2.13) as in (2.6) we have ζ − s(t)ζ = ∫ t 0 s(t − σ)(−aζ)dσ , hence the trajectory y(t, ζ, v) stays at ζ for 0 ≤ t ≤ δ, y(t, ζ, v) = ȳ (0 ≤ t ≤ δ) , and then starts for the target ȳ, which hits at time t. the proof ends in the same way as that of (a) noting that s(t − σ)∗z 6= 0 in 0 ≤ σ ≤ t (in particular, in 0 ≤ σ ≤ δ) hence (2.12) implies ‖v(σ)‖ = 1 in 0 ≤ σ ≤ δ, in contradiction to the fact that ‖v(σ)‖ = ‖aζ‖ < 1 in 0 ≤ σ ≤ δ. 3 counterexamples, i. to see that (2.4) cannot be relaxed, we have example 3.1.consider the one dimensional system y′(t) = −ay(t) + u(t), y(0) = ζ (3.1) with a > 0. we have s(t) = e−at = s(t)∗, thus controls satisfying (1.4) with z 6= 0 are of one of the two forms ū(t) = { 1 if z > 0 , −1 if z < 0 . (3.2) for the initial condition and target ζ = 1/a, ȳ = 1/a we have∫ t 0 s(t − σ) · 1 dσ = ∫ t 0 e−a(t−σ)dσ = 1 − e−at a = ȳ − s(t )ζ so that the first control in (3.2) drives ζ to ȳ in any time t ≥ 0; in other words, y(t, ζ, ū) = ȳ for all t ≥ 0. none of these drives is time optimal except forthe one where t = 0. 3the semigroup equation for the adjoint semigroup s(t)∗ implies: if s(t − t)∗z = 0, then s(t − σ)∗z = s(t − σ)∗s(t − t)∗z = 0 for σ ≤ t. accordingly, unless s(t − t)∗z 6= 0 in an interval (ρ, t ), ρ < t,s(t − t)∗z will be identically zero in 0 < t ≤ t. 32 h. o. fattorini 7, 3(2005) example 3.2. for another counterexample (or, rather, family of counterexamples) we use an arbitrary unitary group s(t) in hilbert space. here we have z(t ) = e (see remark 1.1), s(t)∗ = s(−t) = s(t)−1, ‖s(t)y‖ = ‖y‖. controls satisfying the maximum principle are given by (1.6) (the denominator satisfies ‖s(t −t)∗z‖ = ‖z‖). assuming (as we may) that ‖z‖ = 1 we have∫ t 0 s(t − σ) s(t − σ)∗z ‖s(t − σ)∗z‖ dσ = ∫ t 0 s(t − σ)s(t − σ)∗z dσ = t z (3.3) so that the control (1.6) drives ζ to ȳ in time t if and only if t is a solution of the equation t z = ȳ − s(t )ζ . (3.4) this equation implies the scalar equation t = ‖s(t )ζ − ȳ‖ (3.5) and, conversely, if t > 0 is a solution of (3.5) it is clear that (3.4) will holdwith z = ȳ − s(t )ζ ‖ȳ − s(t )ζ‖ . (3.6) theorem 3.3. assume (3.5) has only one nonnegative solution t. then the control (1.6) drives ζ to ȳ in optimal time t. if (3.5) has multiple solutions, only the control (1.6) corresponding to the smallest t drives ζ to ȳ time optimally. proof. let t ≥ 0 be the smallest solution of (3.5). if t = 0 we don’t need to drive at all so that t is the optimal time. if t > 0 there exists an admissible control driving from ζ to ȳ, hence the standard existence theorem [2, theorem 1.2] provides a control ū(t) driving from ζ to ȳ in optimal time t . since s(t) is a group (remark 1.1) this control must satisfy the maximum principle (1.4) with a nonzero multiplier z ∈ z(t ) = e∗ = e. we are in a hilbert space, which means this control must of the form (1.6) (with ‖z‖ = 1), ū(t) = s(t − t)∗z ‖s(t − t)∗z‖ = s(t − t)∗z (the denominator cannot be zero since ‖s(t − t)∗z‖ = ‖z‖). as in (3.3) we then have∫ t 0 s(t − σ)ū(σ)dσ = t z = ȳ − s(t )ζ hence t is a solution of (3.5) and, as t is the optimal time, we must have t = t. corollary 3.4. assume that, either (a) ζ ∈ d(a), ‖aζ‖ > 1, or (b) ζ /∈ d(a). then there exists a control of theform (1.6) that drives ζ to ζ in time t > 0, thus is not time optimal. proof. we write (3.5) for ζ = ȳ as ‖s(t)ζ − ζ‖ t = 1 . (3.7) 7, 3(2005) sufficiency of the maximum principle for time optimality 33 in case (a) we have lim t→0+ ‖s(t)ζ − ζ‖ t = lim t→0+ ∥∥∥ s(t)ζ − ζ t ∥∥∥ → ‖aζ‖ > 1 , and we deduce that (3.7) has a positive solution, since the left side tends to 0 as t → ∞. in case (b), lim inf t→0+ ‖s(t)ζ − ζ‖ t = lim inf t→0+ ∥∥∥ s(t)ζ − ζ t ∥∥∥ = ∞ since a finite lim inf implies that ζ ∈ d(a) ([1, theorem 2.1.2. (c), p.88]). remark 3.5. corollary 3.4 has an interesting application. the equivalence s(t )ζ + ∫ t 0 s(t − σ)u(σ)dσ = ȳ ⇐⇒ ∫ t 0 s(t − σ)u(σ)dσ = ȳ − s(t )ζ says that u(t) drives ζ to ȳ in time t ⇐⇒ u(t) drives 0 to ȳ − s(t )ζ in time t. if “drives” is changed to “drives optimally”, the implication =⇒ remains. in fact, if ū(·) does not drive 0 to ȳ −s(t )ζ time optimally then there exists δ > 0 and acontrol u(·) with ‖u(·)‖l∞(0,t−δ;e) ≤ 1 and ∫ t−δ 0 s(t − δ − σ)u(σ)dσ = ȳ − s(t )ζ . then, if we define v(σ) = { 0 (0 ≤ σ < δ) u(σ − δ) (δ ≤ σ ≤ t ) we have ∫ t 0 s(t − σ)v(σ)dσ = ∫ t−δ 0 s(t − δ − σ)u(σ)dσ = ȳ − s(t )ζ , thus v(t) drives from ζ to ȳ in time t. if this drive were time optimal,the “bangbang” theorem 2.2 in [2] would say that ‖v(σ)‖ = 1 a. e., which is not the case since v(σ) = 0 in 0 ≤ σ ≤ δ. accordingly, the optimal driving time from ζ to ȳ is < t. the implication ⇐= is not true; in the setting of unitary semigroups in hilbert spaces it suffices to take ȳ = ζ ∈ d(a) with ‖aζ‖ > 1, and, applying corollary 3.4 construct a control ū(·) satisfying (1.6) and driving ζ to ζ in time t > 0. the same control drives 0 to ζ − s(t )ζ, but this drive is optimal since the initial condition satisfies (b) in theorem 2.1. 34 h. o. fattorini 7, 3(2005) 4 counterexamples, ii. the next example belongs to the family in example 3.2. example 4.1. consider e = ir2, a = [ 0 1 −1 0 ] . (4.1) the semigroup generated by a, s(t) = eat = [ cos t sin t − sin t cos t ] (4.2) is unitary. in polar coordinates, ζ = (r cos θ, r sin θ),ȳ = (s cos ϕ, s sin ϕ), and ‖s(t)ζ − ȳ‖2 = ∥∥∥∥s(t) [ r cos θ r sin θ ] − [ s cos ϕ s sin ϕ ] ∥∥∥∥2 = (r cos t cos θ + r sin t sin θ − s cos ϕ)2 +(−r cos θ sin t + r cos t sin θ − s sin ϕ)2 = r2 + s2 − 2rs cos(t − θ + ϕ) . we have ‖ay‖ = ‖y‖. for ζ = ȳ = (1.1, 0) (so that ‖aζ‖ = ‖aȳ‖ = 1.1) we have r = s = 1.1,θ = ϕ = 0. equation (3.5) (figure 1) has a positive solution t = 1.49797 (4.3) thus we can drive from ζ = (1.1, 0) back to ζ in time t with a control satisfying (1.6), ū(σ) = s(t − σ)z , (4.4) with z given by (3.6), z = (0.68090, 0.73238) . figure 2 below shows the drive (moving clockwise) whichis obviously not time optimal. 7, 3(2005) sufficiency of the maximum principle for time optimality 35 for ζ = (4, 0), ȳ = (−4, 0) we have r = s = 4, θ = 0, ϕ = π. equation (3.5) (figure 3) has three solutions, t0 = 2.50471 , t1 = 4.26666 , t2 = 7.19061 . (4.5) thus we can drive from (4, 0) to (−4, 0) with three different controls that satisfy (1.6), ūj (σ) = s(tj − σ)∗zj j = 0 , 1 , 2 , (4.6) where the zj are given by (3.6) for each tj , z0 = (−0.31308, 0.94972) , z1 = (−0.53333, −0.84590) , z2 = (−0.89882, 0.43830) . figure 2 shows the three trajectories, each plotted for 0 ≤ t ≤ tj ; only the first (thicker curve) is time optimal. remark 4.2. the strong maximum principle (1.4)-(1.5) is a sufficient condition for norm optimality [4] with no conditions on ζ or ȳ so that each of the controls ūj (t), 36 h. o. fattorini 7, 3(2005) j = 0, 1, 2 in (4.6) is norm optimal in its own interval; this means, if ζ = (4, 0) can be drivento ȳ = (−4, 0) in the interval 0 ≤ t ≤ tj by means of a control u(t) then ‖u(·)‖l∞(0,tj ;e) ≥ 1 = ‖ūj (·)‖l∞(0,tj ;e) . the same observation applies to the control (4.4); it drives ζ back to ζ norm optimally in the interval 0 ≤ t ≤ t. example 4.3. example 3.1 can be manipulated into evidence that restrictions on ‖ζ‖ or ‖ȳ‖ such as ‖ζ‖ ≤ � or ȳ ≤ � don’t guarantee sufficiency of the maximum principle for time optimality. to this end we consider thespace e = `2 of all sequences y = {y1, y2, . . . } such that ‖y‖2 = ∑ |yk|2 < ∞, equipped with the norm ‖ · ‖. the operator is ay = a{yk} = {−nyk} with maximal domain. it generates the semigroup s(t){yk} = {e−ktyk} = s(t)∗ . let ζn = 1 n {δnk} = ȳn , zn = {δnk} (δnk the kronecker delta). we have ‖ζn‖ = ‖ȳn‖ = 1 n , ‖aζn‖ = ‖aȳn‖ = 1 . if ūn(·) satisfies (1.6) in an interval 0 ≤ t ≤ t with z = zn then ūn(σ) = {δnk} and∫ t 0 s(t − σ)ū(σ)dσ = 1 − e−nt n δnk = ȳn − s(t )ζn for any t > 0. accordingly, ūn(σ) drives ζn to yn in an arbitrary interval 0 ≤ t ≤ t. the driveis not optimal unless t = 0. received: april 2004. revised: may 2004. references [1] p. l. butzer, h. berens, semi-groups of operators and approximation, springer, berlin 1967. [2] h. o. fattorini, time-optimal control of solutions of operational differential equations, siam j. control 2 (1964) 54-59. 7, 3(2005) sufficiency of the maximum principle for time optimality 37 [3] h. o. fattorini, the maximum principle in infinite dimension,discrete & continuous dynamical systems 6 (2000) 557-574. [4] h. o. fattorini, existence of singular extremals and singular functionals in reachable spaces, jour. evolution equations 1 (2001)325-347. [5] h. o. fattorini, a survey of the time optimal problem and the norm optimal problem in infinite dimension, cubo mat. educacional 3 (2001) 147-169. [6] h. o. fattorini, time optimality and the maximum principle in infinite dimension, optimization 50 (2001) 361-385. cubo a mathematical journal vol.10, n o ¯ 03, (103–114). october 2008 the flip crossed products of the c ∗ -algebras by almost commuting isometries takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, nishihara, okinawa 903-0213, japan email: sudo@math.u-ryukyu.ac.jp abstract we study the flip crossed products of the c∗-algebras by almost commuting isometries and obtain some results on their structure, k-theory, and continuity. resumen estudiamos el produto flip crossed de una c∗-algebra mediante isometrias casi commutando y obtenemos algunos resultados sobre su estructura, k-teoria, y continuidad. key words and phrases: c*-algebra, continuous field, k-theory, isometry. math. subj. class.: 46l05, 46l80. 104 takahiro sudo cubo 10, 3 (2008) introduction recall that the soft torus aε of exel [3] (for any ε ∈ [0, 2] the closed interval) is defined to be the universal c∗-algebra generated by almost commuting two unitaries uε,1 and uε,2 in the sense that ‖uε,2uε,1 − uε,1uε,2‖ ≤ ε. its k-theory is computed in [3] by showing that it can be represented as a crossed product by z and applying the pimsner-voiculescu six-term exact sequense for the crossed product. it is shown by exel [4] that there exists a continuous field of c∗-algebras on [0, 2] with fibers the soft tori varying continuously. furthermore, k-theory and continuity of the crossed products of aε by the flip (a z2-action) are considered by elliott, exel and loring [2]. on the other hand, we [8] began to study continuous fields of c∗-algebras by almost commuting isometries and obtained some similar results (but different in some senses) on their structure, ktheory and continuity as those by exel. in this paper we consider those properties for the flip crossed products of the c∗-algebras generated by almost commuting isometries. refer to [1], [5], and [9] for some basics in c∗-algebras and k-theory. 1 the flip crossed products by isometries the toeplitz algebra is defined to be the universal c∗-algebra generated by a (non-unitary) isometry, and it is denoted by f, which is also the semigroup c∗-algebra c∗(n) of the semigroup n of natural numbers. the c∗-algebra c(t) of all continuous functions on the 1-torus t is the universal c∗-algebra generated by a unitary, which is also the group c∗-algebra c∗(z) of the group z of integers. there is a canonical quotient map from f to c(t) by universality, whose kernel is isomorphic to the c∗-algebra k of all compact operators on a separable infinite dimensional hilbert space (cf. [5]). definition 1.1 for ε ∈ [0, 2], the soft toeplitz tensor product denoted by f ⊗ε f is defined to be the universal c∗-algebra generated by two isometries sε,1, sε,2 such that ‖sε,2sε,1 − sε,1sε,2‖ ≤ ε (ε-commuting). let π : f⊗ε f → aε be the canonical onto ∗-homomorphism sending the isometry generators to the unitary generators. remark. refer to [8], in which super-softness is further defined and assumed, but it should be unnecessary from the universality argument (as given below). instead, in fact, another norm estimate of the form ‖sε,2s ∗ ε,1 −s ∗ ε,1sε,2‖≤ ε (ε-∗-commuting) may be required, but we omit such an estimate in what follows. if not assuming the estimate, f⊗ε f should be replaced with c ∗(n2)ε, where c∗(n2) is the semigroup c∗-algebra of n2 (in what follows). definition 1.2 the flip on f ⊗ε f is the (non-unital) endomorphism σ defined by σ(sε,j ) = s ∗ ε,j for j = 1, 2. since σ2 is the identity on f ⊗ε f, we denote by (f ⊗ε f) ⋊σ z2 the crossed product of f ⊗ε f by the action σ of the order 2 cyclic group z2, i.e., a flip crossed product. cubo 10, 3 (2008) the flip crossed products of the c∗-algebras ... 105 definition 1.3 for ε ∈ [0, 2], we define eε to be the universal c ∗-algebra generated by an isometry t1 and the elements tn+1 = u nt1(u ∗)n for n ∈ n, where u is an isometry, such that ‖ut1−t1u‖≤ ε. let αε be the endomorphism of eε defined by αε(tn) = tn+1 = utnu ∗ for n ∈ n. let eε ⋊αε n be the semigroup crossed product of eε by the action αε of the additive semigroup n of natural numbers. remark. note that f⊗2 f (or c ∗(n2)2) is isomorphic to the unital full free product f∗c f, which is also isomorphic to the full semigroup c∗-algebra c∗(n ∗ n) of the free semigroup n ∗ n. as in the above remark, another estimate ‖ut∗ 1 − t∗ 1 u‖≤ ε may be required accordingly. it is shown in [8] that f⊗ε f ∼= eε ⋊αε n, where the map ϕ from f⊗ε f to eε ⋊αε n is defined by ϕ(sε,1) = t1 and ϕ(sε,2) = u, and its inverse ψ is given by ψ(tn+1) = s n ε,2sε,1(s ∗ ε,2) n for n ∈ n and n = 0 and ψ(u) = sε,2. proposition 1.4 for ε ∈ [0, 2], we have the following isomorphism: (f ⊗ε f) ⋊σ z2 ∼= eε ⋊αε∗β (n ∗ z2), where n ∗ z2 is the free product of n and z2, and the action β on eε is given by β(tn) = t ∗ n for n ∈ n. proof. the crossed product (f ⊗ε f) ⋊σ z2 is the universal c ∗-algebra generated by isometries sε,1, sε,2 and a unitary ρ such that ‖sε,2sε,1 − sε,1sε,2‖ ≤ ε and ρsε,jρ ∗ = sε,j (j = 1, 2) with ρ2 = 1, while eε ⋊αε∗β (n ∗ z2) is the c ∗-algebra generated by isometries t1, u and a unitary v such that ‖ut1 −t1u‖≤ ε and tn+1 = utnu ∗ = unt1(u ∗)n for n ∈ n, and vt1v ∗ = t∗ 1 and vuv∗ = u∗ with v2 = 1. the isomorphism between them is given by sending sε,1,sε,2, and ρ to t1,u, and v respectively (cf. [2]). 2 theorem 1.5 for 0 ≤ ε < 2, we obtain the k-theory isomorphisms: k0((f ⊗ε f) ⋊σ z2) ∼= z 9 , k1((f ⊗ε f) ⋊σ z2) ∼= 0. moreover, kj((f ⊗ε f) ⋊σ z2) ∼= kj((f ⊗ f) ⋊σ z2) for j = 0, 1. proof. since f ⊗ε f ∼= eε ⋊αε n and αε is a corner endomorphism on eε, note that eε ⋊αε n is isomorphic to a corner of (eε ⊗ k) ⋊ρ∧ ε ⊗id z, i.e., p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p for a certain projection p, where ρ∧ε is the dual action of the circle action on eε ⋊αε n and id is the identity action on k (this is a variation of [6], and see also [7]). hence, (eε ⋊αε n) ⋊σ z2 is isomorphic to 106 takahiro sudo cubo 10, 3 (2008) p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p ⋊σ z2. therefore, kj((eε ⋊αε n) ⋊σ z2) ∼= kj(p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p ⋊σ z2) ∼= k z2 j (p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p) ∼= k z2 j (p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p⊗ k) ∼= k z2 j (((eε ⊗ k) ⋊ρ∧ ε ⊗id z) ⊗ k) ∼= kj((eε ⊗ k) ⋊ρ∧ ε ⊗id z ⋊ z2), where kz2 j (·) is the equivariant k-theory, and note that p((eε⊗k) ⋊ρ∧ ε ⊗id z)p is stably isomorphic to (eε ⊗ k) ⋊ρ∧ ε ⊗id z, and (eε ⊗ k) ⋊ρ∧ ε ⊗id z ⋊ z2 ∼= (eε ⊗ k) ⋊σ′ ε ∗σ⊗id (z2 ∗ z2) ∼= (eε ⋊σ′ ε ∗σ (z2 ∗ z2)) ⊗ k since z ⋊ z2 ∼= z2 ∗ z2, where σ ′ ε(1) = ρ ∧ ε (1)σ(1) (cf. [2]). set fε = eε ⋊σ′ε∗σ (z2 ∗ z2). there exists the following six-term exact sequence (a) (cf. [2]): k0(eε) −−−−→ k0(eε ⋊σ′ ε z2) ⊕k0(eε ⋊σ z2) −−−−→ k0(fε) x     y k1(fε) ←−−−− k1(eε ⋊σ′ ε z2) ⊕k1(eε ⋊σ z2) ←−−−− k1(eε). consider the following exact sequence: 0 → iε → eε → π(eε) = b ′ ε → 0, where π is the canonical quotient map from eε to the quotient π(eε) = b ′ ε, where b ′ ε is the universal c ∗-algebra generated by unitaries un+1 = w nv(w∗)n for n ∈ n and n = 0, where π(tn+1) = π(u) nπ(t1)π(u ∗)n = un+1 with v = π(t1) and w = π(u). as shown in [8], k-theory groups of iε are the same as those of k. since this quotient is invariant under the action β = σ′ε or σ, we have the following exact sequence: (b) : 0 → iε ⋊β z2 → eε ⋊β z2 → π(eε) ⋊β z2 → 0 and iε ⋊β z2 ∼= iε ⊗c ∗(z2) and the group c ∗-algebra c∗(z2) is isomorphic to c 2 via the fourier transform. as shown in [2], it is deduced that π(eε) ⋊β z2 is homotopy equivalent to the crossed product c(t) ⋊β′ z2, where β ′(z) = z−1 for z ∈ t. it follows that kj (π(eε) ⋊β z2) is isomorphic to kj (c(t) ⋊β′ z2). since the points {±1} in t is fixed under the action β ′, we have 0 → c0(t \{±1}) ⋊β′ z2 → c(t) ⋊β′ z2 →⊕ 2c∗(z2) → 0, where c0(t\{±1}) is the c ∗-algebra of all continuous functions on t\{±1} vanishing at infinity, and c0(t \{±1}) ⋊β′ z2 ∼= c0(r) ⊗ (c 2 ⋊β′ z2) ∼= c0(r) ⊗m2(c) and c ∗(z2) ∼= c 2. hence the following six-term exact sequence is obtained: 0 −−−−→ k0(c(t) ⋊β′ z2) −−−−→ z 4 x     y 0 ←−−−− k1(c(t) ⋊β′ z2) ←−−−− z, cubo 10, 3 (2008) the flip crossed products of the c∗-algebras ... 107 where kj (c0(r) ⊗ m2(c)) ∼= kj+1(c) (mod 2) and kj(⊕ 2 c 2) ∼= ⊕4kj (c). it follows that k0(c(t) ⋊β′ z2) ∼= z 3 and k1(c(t) ⋊β′ z2) ∼= 0 (cf. [2]). therefore, for the above exact sequence (b), we obtain the diagram: z 2 −−−−→ k0(eε ⋊β z2) −−−−→ z 3 x     y 0 ←−−−− k1(eε ⋊β z2) ←−−−− 0, where kj(k ⊗c ∗(z2)) ∼= kj (c 2). hence we obtain k0(eε ⋊β z2) ∼= z 5 and k1(eε ⋊β z2) ∼= 0. this implies that the diagram (a) is z −−−−→ z5 ⊕ z5 −−−−→ k0(fε) x     y k1(f0) ←−−−− 0 ⊕ 0 ←−−−− 0 where it is shown in [8] that k0(eε) ∼= z and k1(eε) ∼= 0. it follows that k0(fε) ∼= z 9 and k1(fε) ∼= 0. it follows from this and the first part shown above that k0((f⊗ε f) ⋊σ z2) ∼= z 9 and k1((f ⊗ε f) ⋊σ z2) ∼= 0. the second claim follows from the case ε = 0 and the same argument as above. note that f ⊗ f ∼= f ⋊id n, where id is the trivial action. 2 corollary 1.6 for 0 ≤ ε < 2, the natural onto ∗-homomorphism ϕε,0 from (f ⊗ε f) ⋊σ z2 to (f ⊗ f) ⋊σ z2 sending sε,j to s0,j (j = 1, 2) induces the isomorphism between their k-groups. proposition 1.7 there exists a continuous field of c∗-algebras on the closed interval [0, 2] such that its fibers are (f ⊗ε f) ⋊σ z2 for ε ∈ [0, 2], and for any a ∈ (f ⊗2 f) ⋊σ z2, the sections [0, 2] ∋ ε 7→ ϕε(a) ∈ (f⊗ε f) ⋊σ z2 are continuous, where ϕε : (f⊗2 f) ⋊σ z2 → (f⊗ε f) ⋊σ z2 is the natural onto ∗-homomorphism sending s2,j to sε,j (j = 0, 1). proof. as shown before, (f ⊗ε f) ⋊σ z2 ∼= (eε ⋊αε n) ⋊σ z2. furthermore, this is isomorphic to p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p ⋊σ z2. hence it follows that ((eε ⋊αε n) ⋊σ z2) ⊗ k ∼= (p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p ⋊σ z2) ⊗ k ∼= (p((eε ⊗ k) ⋊ρ∧ ε ⊗id z)p⊗ k) ⋊σ⊗id z2 ∼= (((eε ⊗ k) ⋊ρ∧ ε ⊗id z) ⊗ k) ⋊σ⊗id z2 ∼= ((eε ⊗ k ⊗ k) ⋊ρ∧ ε ⊗id⊗id z) ⋊σ⊗id z2 ∼= ((eε ⊗ k) ⋊ρ∧ ε ⊗id z) ⋊σ z2 ∼= (eε ⊗ k) ⋊σ′ ε ∗σ⊗id (z2 ∗ z2). it is deduced from [2] that there exists a continuous field of c∗-algebras on [0, 2] such that its fibers are (eε ⊗ k) ⋊σ′ ε ∗σ⊗id (z2 ∗ z2) for ε ∈ [0, 2], and for any b ∈ (e2 ⊗ k) ⋊σ′ 2 ∗σ⊗id (z2 ∗ z2), the 108 takahiro sudo cubo 10, 3 (2008) sections [0, 2] ∋ ε 7→ ψε(b) ∈ (eε⊗k)⋊σ′ ε ∗σ⊗id (z2∗z2) are continuous, where ψε is the unique onto ∗-homomorphism from (e2 ⊗ k) ⋊σ′ 2 ∗σ⊗id (z2 ∗ z2) to (eε ⊗ k) ⋊σ′ ε ∗σ⊗id (z2 ∗ z2). cutting down this continuous field by cutting down the fibers from ((eε ⋊αε n) ⋊σ z2)⊗ k to (eε ⋊αε n) ⋊σ z2 by minimal projections, we obtain the desired continuous field. 2 2 the flip crossed products by n isometries the n-fold tensor product ⊗nf of f is the universal c∗-algebra generated by mutually commuting and ∗-commuting n isometries, while the universal c∗-algebra generated by mutually commuting n isometries is just the semigroup c∗-algebra c∗(nn) of the semigroup nn. the c∗-algebra c(tn) of all continuous functions on the n-torus tn is the universal c∗-algebra generated by mutually commuting n unitaries, which is also the group c∗-algebra c∗(zn) of the group zn. there is a canonical quotient map from ⊗nf to c(tn) ∼= ⊗nc(t) by universality, definition 2.1 for ε ∈ [0, 2], the soft toeplitz n-tensor product denoted by ⊗nε f is defined to be the universal c∗-algebra generated by n isometries sε,j (1 ≤ j ≤ n) such that ‖sε,ksε,j−sε,jsε,k‖≤ ε (1 ≤ j,k ≤ n). remark. note that, in fact, the norm estimates of the form ‖sε,ks ∗ ε,j −s ∗ ε,jsε,k‖≤ ε may be further required (and in what follows). if not assuming these estimates, ⊗nε f should be replaced with c∗(nn)ε in the same sense (and in what follows). definition 2.2 the flip on ⊗nε f is the (non-unital) endomorphism σ defined by σ(sε,j ) = s ∗ ε,j for 1 ≤ j ≤ n. since σ2 is the identity on ⊗nε f, we denote by (⊗ n ε f) ⋊σ z2 the crossed product of ⊗ n ε f by the action σ of z2. definition 2.3 for ε ∈ [0, 2], we define emε to be the universal c ∗-algebra generated by n isometries t (j) 1 (1 ≤ j ≤ m) and the partial isometries t (j) n+1 = u nt (j) 1 (u∗)n for n ∈ n, where u is an isometry such that ‖ut (j) 1 − t (j) 1 u‖ ≤ ε and ‖t (k) 1 t (j) 1 − t (j) 1 t (k) 1 ‖ ≤ ε (1 ≤ j,k ≤ m). let αε be the endomorphism of emε defined by αε(t (j) n ) = t (j) n+1 = ut (j) n u ∗ for n ∈ n. let emε ⋊αε n be the semigroup crossed product of emε by the action αε of n. remark. note that ⊗n 2 f (or c∗(nn)2) is isomorphic to the unital full free product ∗ n c f, which is also isomorphic to the full semigroup c∗-algebra c∗(∗nn) of the free semigroup ∗nn. as in the above remark, the additional estimates ‖u(t (j) 1 )∗ − (t (j) 1 )∗u‖≤ ε and ‖t (k) 1 (t (j) 1 )∗ − (t (j) 1 )∗t (k) 1 ‖ ≤ ε may be required accordingly. it is shown as in [8] that ⊗m+1ε f ∼= emε ⋊αε n as in the case in section 1. proposition 2.4 for ε ∈ [0, 2], we have (⊗m+1ε f) ⋊σ z2 ∼= e m ε ⋊αε∗β (n ∗ z2), cubo 10, 3 (2008) the flip crossed products of the c∗-algebras ... 109 where the action β on emε is given by β(t (j) n ) = (t (j) n ) ∗ for n ∈ n and 1 ≤ j ≤ m. proof. this is shown as in the proof of proposition 1.4 similarly. 2 theorem 2.5 for 0 ≤ ε < 2, we obtain (inductively) k0((⊗ m+1 ε f) ⋊σ z2) ∼= z 2 m+2 +3, k1((⊗ m+1 ε f) ⋊σ z2) ∼= 0. moreover, kj((⊗ m+1 ε f) ⋊σ z2) ∼= kj((⊗ m+1f) ⋊σ z2) for j = 0, 1. proof. since ⊗m+1ε f ∼= emε ⋊αε n, note that e m ε ⋊αε n is isomorphic to a corner of (e m ε ⊗k)⋊ρ∧ε ⊗idz, i.e., p((emε ⊗ k) ⋊ρ∧ε ⊗id z)p for a certain projection p, where ρ ∧ ε is the dual action of the circle action on emε ⋊αε n and id is the identity action on k (this is a variation of [6], and see also [7]). hence, (emε ⋊αε n) ⋊σ z2 is isomorphic to p((e m ε ⊗ k) ⋊ρ∧ε ⊗id z)p ⋊σ z2. therefore, kj((e m ε ⋊αε n) ⋊σ z2) ∼= kj(p((e m ε ⊗ k) ⋊ρ∧ε ⊗id z)p ⋊σ z2) ∼= k z2 j (p((emε ⊗ k) ⋊ρ∧ε ⊗id z)p) ∼= k z2 j (p((emε ⊗ k) ⋊ρ∧ε ⊗id z)p⊗ k) ∼= k z2 j (((emε ⊗ k) ⋊ρ∧ε ⊗id z) ⊗ k) ∼= kj((e m ε ⊗ k) ⋊ρ∧ε ⊗id z ⋊ z2), where p((emε ⊗ k) ⋊ρ∧ε ⊗id z)p is stably isomorphic to (e m ε ⊗ k) ⋊ρ∧ε ⊗id z, and (emε ⊗ k) ⋊ρ∧ε ⊗id z ⋊ z2 ∼= (e m ε ⊗ k) ⋊ρ∧ε ∗σ⊗id (z2 ∗ z2) ∼= (e m ε ⋊ρ∧ε ∗σ (z2 ∗ z2)) ⊗ k since z ⋊ z2 ∼= z2 ∗z2 (cf. [2]). set f m ε = e m ε ⋊ρ∧ε ∗σ (z2 ∗z2). there exists the following six-term exact sequence (a)m (cf. [2]): k0(e m ε ) −−−−→ k0(e m ε ⋊ρ∧ε z2) ⊕k0(e m ε ⋊σ z2) −−−−→ k0(f m ε ) x     y k1(f m ε ) ←−−−− k1(e m ε ⋊ρ∧ε z2) ⊕k1(e m ε ⋊σ z2) ←−−−− k1(e m ε ). we now have the following exact sequence: 0 → imε ⋊ z2 → e m ε ⋊ z2 → π(e m ε ) ⋊ z2 → 0, where the map π is sending isometries of emε to unitaries with the same norm estimates by universality, and imε is the kernel of π, and the action of z2 is given by ρ ∧ ε or σ. furthermore, it follows that imε ⋊ z2 ∼= imε ⊗c ∗(z2) and the k-theory of i m ε is the same as that of k. it is deduced that π(emε ) ⋊ z2 is homotopy equivalent to c(t m) ⋊σ z2, where β(zj ) = (z −1 j ) for (zj ) ∈ t m. since the points (±1, · · · ,±1) ∈ tm are fixed under α, we have 0 → c0(t m \ (±1, · · · ,±1)) ⋊ z2 → c(t m) ⋊ z2 →⊕ 2 m c ∗(z2) → 0, 110 takahiro sudo cubo 10, 3 (2008) where c0(x) is the c ∗-algebra of all continuous functions on a locally compact hausdorff space x vanishing at infinity (in what follows). set xm+1 = t m \ (±1, · · · ,±1). by considering invariant subspaces in xm+1 under β, we obtain a finite composition series {lj} m j=1 of c0(xm+1) ⋊ z2 such that l0 = {0}, lj = c0(xj ) × z2, and lj/lj−1 ∼= ⊕ mcm−j+1c0((t \{±1}) m−j+1) ⋊ z2, where mcm−j+1 mean the combinations. furthermore, c0((t \{±1}) m−j+1) ⋊ z2 ∼= c0(r m−j+1) ⊗ (c(πm−j+1{±i}) ⋊ z2) and c(πm−j+1{±i}) ⋊ z2 ∼= ⊕ m−j+1(c2 ⋊ z2) ∼= ⊕ m−j+1m2(c), where t\{±1} is homeomorphic to ir∪(−i)r so that the above isomorphisms are deduced from considering orbits under β in this identification. set c(m,j) = mcm−j+1(m− j + 1). thus, the following six-term exact sequences are obtained: k0(lj−1) −−−−→ k0(lj ) −−−−→ km−j+1(⊕ c(m,j) c) x     y km−j+2(⊕ c(m,j) c) ←−−−− k1(lj ) ←−−−− k1(lj−1). now consider the case m = 2. then 0 → c0(t 2 \ (±1,±1)) ⋊ z2 → c(t 2) ⋊ z2 →⊕ 2 2 c∗(z2) → 0. furthermore, 0 → c0(x1)⋊z2 → c0(x2)⋊z2 → c0(x2\x1)⋊z2 → 0, where x2 = t 2\(±1,±1), x1 = (t \ {±1}) 2, and c0(x2 \ x1) ⋊ z2 is isomorphic to ⊕ 2c0(t \ {±1}) ⋊ z2. we have the following six-term exact sequence: z 2 −−−−→ k0(c0(x2) ⋊ z2) −−−−→ 0 x     y z 2 ←−−−− k1(c0(x2) ⋊ z2) ←−−−− 0, which implies k0(c0(x2) ⋊ z2) ∼= 0 and k1(c0(x2) ⋊ z2) ∼= 0. thus, 0 −−−−→ k0(c(t 2) ⋊ z2) −−−−→ z 2 3 x     y 0 ←−−−− k1(c(t 2) ⋊ z2) ←−−−− 0, which implies k0(c(t 2) ⋊ z2) ∼= z 2 3 and k1(c(t 2) ⋊ z2) ∼= 0. therefore, z 2 −−−−→ k0(e 2 ε ⋊ z2) −−−−→ z 2 3 x     y 0 ←−−−− k1(e 2 ε ⋊ z2) ←−−−− 0. cubo 10, 3 (2008) the flip crossed products of the c∗-algebras ... 111 it follows that k0(e 2 ε ⋊ z2) ∼= z2 3 +2 and k1(e 2 ε ⋊ z2) ∼= 0. therefore, z −−−−→ z2 3 +2 ⊕ z2 3 +2 −−−−→ k0(f 2 ε ) x     y k1(f 2 ε ) ←−−−− 0 ⊕ 0 ←−−−− 0. hence, it follows that k0(f 2 ε ) ∼= z2 4 +3 and k1(f 2 ε ) ∼= 0. next consider the case m = 3. then 0 → c0(t 3 \ (±1,±1,±1)) ⋊ z2 → c(t 3) ⋊ z2 →⊕ 2 3 c∗(z2) → 0. furthermore, 0 → c0(x2) ⋊ z2 → c0(x3) ⋊ z2 → c0(x3 \ x2) ⋊ z2 → 0, where x3 = t 3 \ (±1,±1,±1), and 0 → c0(x1) ⋊ z2 → c0(x2) ⋊ z2 → c0(x2 \x1) ⋊ z2 → 0, where x1 = (t \{±1}) 3. we have the following six-term exact sequence: 0 −−−−→ k0(c0(x2) ⋊ z2) −−−−→ z 6 x     y 0 ←−−−− k1(c0(x2) ⋊ z2) ←−−−− z 3, which implies k0(c0(x2) ⋊ z2) ∼= z 3 and k1(c0(x2) ⋊ z2) ∼= 0. furthermore, z 3 −−−−→ k0(c0(x3) ⋊ z2) −−−−→ 0 x     y z 3 ←−−−− k1(c0(x3) ⋊ z2) ←−−−− 0, which implies k0(c0(x3) ⋊ z2) ∼= 0 and k1(c0(x3) ⋊ z2) ∼= 0. thus, 0 −−−−→ k0(c(t 3) ⋊ z2) −−−−→ z 2 4 x     y 0 ←−−−− k1(c(t 3) ⋊ z2) ←−−−− 0, which implies k0(c(t 3) ⋊ z2) ∼= z 2 4 and k1(c(t 2) ⋊ z2) ∼= 0. therefore, z 2 −−−−→ k0(e 3 ε ⋊ z2) −−−−→ z 2 4 x     y 0 ←−−−− k1(e 3 ε ⋊ z2) ←−−−− 0. it follows that k0(e 3 ε ⋊ z2) ∼= z2 4 +2 and k1(e 3 ε ⋊ z2) ∼= 0. therefore, z −−−−→ z2 4 +2 ⊕ z2 4 +2 −−−−→ k0(f 3 ε ) x     y k1(f 3 ε ) ←−−−− 0 ⊕ 0 ←−−−− 0. 112 takahiro sudo cubo 10, 3 (2008) hence, it follows that k0(f 3 ε ) ∼= z2 5 +3 and k1(f 3 ε ) ∼= 0. next consider the case m = 4. then 0 → c0(t 4 \ (±1,±1,±1,±1)) ⋊ z2 → c(t 4) ⋊ z2 →⊕ 2 4 c ∗(z2) → 0. furthermore, 0 → c0(x3) ⋊ z2 → c0(x4) ⋊ z2 → c0(x4 \ x3) ⋊ z2 → 0, where x4 = t 4 \ (±1,±1,±1,±1), and 0 → c0(x1) ⋊ z2 → c0(x2) ⋊ z2 → c0(x2 \x1) ⋊ z2 → 0, where x1 = (t \{±1}) 4. we have the following six-term exact sequence: z 4 −−−−→ k0(c0(x2) ⋊ z2) −−−−→ 0 x     y z 12 ←−−−− k1(c0(x2) ⋊ z2) ←−−−− 0, which implies k0(c0(x2) ⋊ z2) ∼= 0 and k1(c0(x2) ⋊ z2) ∼= z 8. furthermore, 0 −−−−→ k0(c0(x3) ⋊ z2) −−−−→ z 12 x     y 0 ←−−−− k1(c0(x3) ⋊ z2) ←−−−− z 8, which implies k0(c0(x3) ⋊ z2) ∼= z 4 and k1(c0(x3) ⋊ z2) ∼= 0. furthermore, z 4 −−−−→ k0(c0(x4) ⋊ z2) −−−−→ 0 x     y z 4 ←−−−− k1(c0(x4) ⋊ z2) ←−−−− 0, which implies k0(c0(x4) ⋊ z2) ∼= 0 and k1(c0(x4) ⋊ z2) ∼= 0. thus, 0 −−−−→ k0(c(t 4) ⋊ z2) −−−−→ z 2 5 x     y 0 ←−−−− k1(c(t 4) ⋊ z2) ←−−−− 0, which implies k0(c(t 4) ⋊ z2) ∼= z 2 5 and k1(c(t 4) ⋊ z2) ∼= 0. therefore, z 2 −−−−→ k0(e 4 ε ⋊ z2) −−−−→ z 2 5 x     y 0 ←−−−− k1(e 4 ε ⋊ z2) ←−−−− 0. it follows that k0(e 4 ε ⋊ z2) ∼= z2 5 +2 and k1(e 4 ε ⋊ z2) ∼= 0. therefore, z −−−−→ z2 5 +2 ⊕ z2 5 +2 −−−−→ k0(f 4 ε ) x     y k1(f 4 ε ) ←−−−− 0 ⊕ 0 ←−−−− 0. cubo 10, 3 (2008) the flip crossed products of the c∗-algebras ... 113 hence, it follows that k0(f 4 ε ) ∼= z2 6 +3 and k1(f 4 ε ) ∼= 0. the case for m general can be treated by the step by step argument as shown above. the argument for k-theory is inductive in a sense that it involves essentially suspensions and direct sums inductively. the second claim follows from considering the case ε = 0 and the same argument as above. 2 corollary 2.6 for 0 ≤ ε < 2, the natural onto ∗-homomorphism ϕε,0 from (⊗ m+1 ε f) ⋊σ z2 to (⊗m+1f) ⋊σ z2 sending sε,j to s0,j (1 ≤ j ≤ m + 1) induces the isomorphism between their k-groups. proposition 2.7 there exists a continuous field of c∗-algebras on the closed interval [0, 2] such that fibers are (⊗m+1ε f) ⋊σ z2 for ε ∈ [0, 2], and for any a ∈ (⊗ m+1 2 f) ⋊σ z2, the sections [0, 2] ∋ ε 7→ ϕε(a) ∈ (⊗ m+1 ε f) ⋊σ z2 are continuous, where ϕε : (⊗ m+1 2 f) ⋊σ z2 → (⊗ m+1 ε f) ⋊σ z2 is the natural onto ∗-homomorphism sending s2,j to sε,j (1 ≤ j ≤ m + 1). proof. as shown before, (⊗m+1ε f) ⋊σ z2 ∼= (emε ⋊αε n) ⋊σ z2. furthermore, this is isomorphic to p((emε ⊗ k) ⋊ρ∧ε ⊗id z)p ⋊σ z2. hence it follows that ((emε ⋊αε n) ⋊σ z2) ⊗ k ∼= (p((e m ε ⊗ k) ⋊ρ∧ε ⊗id z)p ⋊σ z2) ⊗ k ∼= (p((e m ε ⊗ k) ⋊ρ∧ε ⊗id z)p⊗ k) ⋊σ⊗id z2 ∼= (((e m ε ⊗ k) ⋊ρ∧ε ⊗id z) ⊗ k) ⋊σ⊗id z2 ∼= ((e m ε ⊗ k ⊗ k) ⋊ρ∧ε ⊗id⊗id z) ⋊σ⊗id z2 ∼= ((e m ε ⊗ k) ⋊ρ∧ε ⊗id z) ⋊σ z2 ∼= (e m ε ⊗ k) ⋊ρ∧ε ∗σ⊗id (z2 ∗ z2). it is deduced from [2] that there exists a continuous field of c∗-algebras on [0, 2] such that fibers are (emε ⊗ k) ⋊ρ∧ε ∗σ⊗id (z2 ∗ z2) for ε ∈ [0, 2], and for any b ∈ (e m 2 ⊗ k) ⋊ρ∧ 2 ∗σ⊗id (z2 ∗ z2), the sections [0, 2] ∋ ε 7→ ψε(b) ∈ (e m ε ⊗ k) ⋊ρ∧ε ∗σ⊗id (z2 ∗ z2) are continuous, where ψε is the unique onto ∗-homomorphism from (em 2 ⊗ k) ⋊ρ∧ 2 ∗σ⊗id (z2 ∗ z2) to (e m ε ⊗ k) ⋊ρ∧ε ∗σ⊗id (z2 ∗ z2). cutting down this continuous field by cutting down the fibers from ((emε ⋊αε n) ⋊σ z2) ⊗ k to (emε ⋊αε n) ⋊σ z2 by minimal projections, we obtain the desired continuous field. 2 received: may 2008. revised: june 2008. 114 takahiro sudo cubo 10, 3 (2008) references [1] b. blackadar, k-theory for operator algebras, second edition, cambridge, (1998). [2] g.a. elliott, r. exel and t.a. loring, the soft torus iii: the flip, j. operator theory 26 (1991), 333–344. [3] r. exel, the soft torus and applications to almost commuting matrices, pacific j. math. 160 no. 2, (1993), 207–217. [4] r. exel, the soft torus: a variational analysis of commutator norms, j. funct. anal. 126 (1994), 259–273. [5] g.j. murphy, c∗-algebras and operator theory, academic press, (1990). [6] w. paschke, the crossed product of a c∗-algebra by an endomorphism, proc. amer. math. soc. 80, no. 1, (1980), 113–118. [7] m. rørdam, classification of certain infinite simple c∗-algebras, j. funct. anal. 131, (1995), 415–458. [8] t. sudo, continuous fields of c∗-algebras by almost commuting isometries, int. j. pure appl. math. 44 (3) (2008), 425–438. [9] n.e. wegge-olsen, k-theory and c∗-algebras, oxford univ. press (1993). n09 cubo a mathematical journal vol.10, n o ¯ 02, (31–45). july 2008 a remark on the enclosure method for a body with an unknown homogeneous background conductivity masaru ikehata department of mathematics, graduate school of engineering, gunma university, kiryu 376-8515, japan email: ikehata@math.sci.gunma-u.ac.jp abstract previous applications of the enclosure method with a finite set of observation data to a mathematical model of electrical impedance tomography are based on the assumption that the conductivity of the background body is homogeneous and known. this paper considers the case when the conductivity is homogeneous and unknown. it is shown that, in two dimensions if the domain occupied by the background body is enclosed by an ellipse, then it is still possible to extract some information about the location of unknown cavities or inclusions embedded in the body without knowing the background conductivity provided the fourier series expansion of the voltage on the boundary does not contain high frequency parts (band limited) and satisfies a non vanishing condition of a quantity involving the fourier coefficients. resumen previas aplicaciones del método de cercamiento con un conjunto finito de datos de observación para un modelo matemático de tomografia electrica son basados en la suposición de que la conductividad del cuerpo es homogenea y conocida. este art́ıculo 32 masaru ikehata cubo 10, 2 (2008) considera el caso cuando la conductividad es homogenea y desconocida. es demostrado en dos dimensiones que si el domı́nio ocupado por el cuerpo es encerrado por una elipse, entonces es aún posible extraer alguna información acerca de la localización de las cavidades desconocidas o inclusiones inmersas en el cuerpo sin conocimiento de la conductividad con talque la expansión en series de fourier del voltaje sobre la frontera no contenga frecuencias altas (fajas acotadas) y satisfaga una condición de no nulidad de una cierta cantidad envolviendo los coeficientes de fourier. key words and phrases: enclosure method, inverse boundary value problem, cavity, inclusion, laplace equation, exponentially growing solution. math. subj. class.: 35r30 1 introduction the aim of this paper is to reconsider previous applications [5, 6] of the enclosure method with a finite set of observation data to inverse boundary value problems related to a continuum model of electrical impedance tomography [1, 2]. the point is: those applications are based on the assumption that the conductivity of the background body is homogeneous and known. however, from a mathematical point of view, the problem whether or not one can still extract some information about unknown discontinuity from the finite set of observation data without knowing the exact value of the conductivity is quite interesting. proofs of some previous known uniqueness results that employ a finite set of observation data, for example, [4] for cracks and [3, 9] for inclusions are based on the assumption that the conductivity of the background body is known. this is because they start with applying the uniqueness of the cauchy problem for elliptic equations. besides needless to say, we cannot know the exact value of the conductivity of the background body. the inaccurate value causes an error on the observation data and therefore on the indicator function in the enclosure method. in order to describe the problem more precisely let us start with recalling a typical application of the enclosure method with a single set of observation data. let ω be a bounded domain of r 2 with lipschitz boundary. let d be an open subset with lipschitz boundary of ω such that d ⊂ ω and ω \ d is connected. consider a non constant solution of the elliptic problem: △u = 0 in ω \ d, ∂u ∂ν = 0 on ∂d. (1.1) here ν = (ν1, ν2) denotes the unit outward normal vector field on ∂(ω\d). the d is a mathematical model of the union of cavities inside the body. cubo 10, 2 (2008) a remark on the enclosure method ... 33 in [5] we considered the problem of extracting information about the location and shape of d in two dimensions from the observation data that is a single set of cauchy data of u on ∂ω. assuming that d is given by the inside of a polygon with an additional condition on the diameter, we established an extraction formula of the convex hull of d from the data. the method uses a special exponential solution of the laplace equation. the solution takes the form e−τ teτ x·(ω+i ω ⊥) where τ (> 0) and t are parameters; both ω and ω⊥ are unit vectors and satisfy ω · ω⊥ = 0. the solution divides the whole plane into two half ones which have the line {x | x · ω = t} as the common boundary. in one part {x | x · ω > t} the solution is growing as τ −→ ∞ and in another part {x | x · ω < t} decaying. using this solution, we define the so-called indicator function i ω, ω⊥ (τ, t) of the independent variable τ with parameter t: i ω, ω⊥ (τ, t) = e−τ t ∫ ∂ω { − ∂ ∂ν eτ x·(ω+iω ⊥)u + ∂u ∂ν eτ x·(ω+iω ⊥) } ds. the enclosure method gives us information about the position of half plane x · ω > t relative to d by checking the asymptotic behaviour of the indicator function as τ −→ ∞. for the description of the behaviour we recall the support function hd(ω) = supx∈ d x · ω. moreover we say that ω is regular if the set {x | x · ω = hd(ω)} ∩ ∂d consists of only one point. what we established in [5] is: for regular ω there exist positive constants a and µ(> 1/2) such that, as τ −→ ∞ |i ω, ω ⊥ (τ, 0)| ∼ a τ µ eτ hd(ω) (1.2) provided diam d < dis (d, ∂ω). (1.3) this fact is the core of the enclosure method. since we have the trivial identity i ω, ω ⊥ (τ, t) = e−τ ti ω, ω ⊥ (τ, 0), from (1.2) one could conclude that: if t > hd(ω), then the indicator function is decaying exponentially; if t = hd(ω), then the indicator function is decaying truly algebraically; if t < hd(ω), then the indicator function is growing exponentially. moreover from (1.2), we immediately obtain also the one line formula lim τ −→∞ 1 τ log |i ω, ω ⊥ (τ, 0)| = hd(ω). however this is the case when the background conductivity is known. consider the case when the background conductivity is given by a positive constant γ. in this case the indicator function should be replaced with i ω, ω⊥ (τ, t) = e−τ t ∫ ∂ω { −γ ∂ ∂ν eτ x·(ω+iω ⊥)u + γ ∂u ∂ν eτ x·(ω+iω ⊥) } ds. needless to say we obtain the same result as above if γ is known. however, if γ is unknown, then the term e −τ t ∫ ∂ω γ ∂ ∂ν e τ x·(ω+iω⊥) uds 34 masaru ikehata cubo 10, 2 (2008) becomes unknown and therefore one can use only the term e −τ t ∫ ∂ω γ ∂u ∂ν e τ x·(ω+iω⊥) ds (1.4) if u = f on ∂ω is given. the purpose of this paper is to give a remark on the problem: can one still extract information about the location and shape of d from the quantity (1.4) in the case when f is given? in this paper we show that, in two dimensions if the domain occupied by the background body is enclosed by an ellipse, then it is still possible to extract some information about the location of unknown cavities or inclusions embedded in the body without knowing the background conductivity provided the fourier series expansion of the voltage on the boundary does not contain high frequency parts (band limited) and satisfies a non vanishing condition of a quantity involving the fourier coefficients. 2 extraction formulae let ω be the domain enclosed by an ellipse. by choosing a suitable system of orthogonal coordinates one can write ω = { (x1, x2) | ( x1 a )2 + ( x2 b )2 < 1 } where a ≥ b > 0. in what follows we always use this coordinates system. given ω = (ω1, ω2) ∈ s 1 set ω⊥ = (ω2, −ω1). then x · (ω + iω ⊥ ) = (x1 − ix2)(ω1 + iω2). let v = eτ x·(ω+iω ⊥) . 2.1 preliminary computation in this subsection first given f = u|∂ω we study the asymptotic behaviour of the integral ∫ ∂ω γ ∂u ∂ν vds. however, integration by parts yields ∫ ∂ω γ ∂u ∂ν vds = γ ∫ ∂ω u ∂v ∂ν ds − γ ∫ ∂d u ∂v ∂ν ds (2.1) and we have already studied the asymptotic behaviour of the second term as described in introduction (see (1.2)). therefore it suffices to study that of the first term. since ∫ ∂ω u ∂v ∂ν ds = τ (ω1 + iω2) ∫ ∂ω u v (ν1 − iν2)ds, (2.2) we compute the integral in the right hand side. cubo 10, 2 (2008) a remark on the enclosure method ... 35 write f (θ) = f (a cos θ, b sin θ) = 1 2 α0 + ∞∑ m=1 (αm cos mθ + βm sin mθ) where αm = 1 π ∫ 2π 0 f (a cos θ, b sin θ) cos mθdθ, βm = 1 π ∫ 2π 0 f (a cos θ, b sin θ) sin mθdθ. define γ0 = α0/2, γm = (αm − iβm)/2, γ−m = γm, m ≥ 1. lemma 2.1. we have: if a = b, then ∫ ∂ω u v (ν1 − iν2)ds = 2πa 2 ∞∑ m=0 {aτ (ω1 + iω2)} m m! γm+1; (2.3) if a > b, then ∫ ∂ω u v (ν1 − iν2)ds = 2πab ∞∑ m=0 i m jm(−i √ a2 − b2τ (ω1 + iω2))cm(f ) (2.4) where c0(f ) = a−γ1 + a+γ1, for m = 1, 2, · · · cm(f ) = (a−γm−1 + a+γm+1) (√ a + b a − b ) m + (a−γm+1 + a+γm−1) (√ a − b a + b ) m and a± = 1 2 ( 1 a ± 1 b ) . proof. set z = eiθ. since ν(a cos θ, b sin θ) = 1√( cos θ a )2 + ( sin θ b )2 ( cos θ a , sin θ b ) and ds = ab √( cos θ a )2 + ( sin θ b )2 dθ, we have (ν1 − iν2)ds = ab(a−z + a+z −1 ) dz iz . note also that f (a cos θ, b sin θ) = ∑ m γmz m 36 masaru ikehata cubo 10, 2 (2008) and x1 − ix2 = b−z + b+z −1 where b± = a ± b 2 . using those expressions, we can write ∫ ∂ω u v (ν1 − iν2)ds = ab i ∑ m γm ∫ |z|=1 (a−z + a+z −1 )zm−1 exp { τ (b−z + b+z −1 )(ω1 + iω2) } dz. define il(τ ) = ∫ |z|=1 zl exp { τ (b−z + b+z −1 )(ω1 + iω2) } dz. consider the case when a > b. using the generating function of the bessel functions, we have exp { τ (b−z + b+z −1 )(ω1 + iω2) } = ∑ n jn ( −i √ a2 − b2τ (ω1 + iω2) )( i √ a − b a + b ) n zn and therefore il(τ ) = 2πi(−1) l+1jl+1 ( −i √ a2 − b2τ (ω1 + iω2) )( −i √ a + b a − b ) l+1 . if a = b, then il(τ ) = 0, l ≤ −2; il(τ ) = 2πi {aτ (ω1 + iω2)} l+1 (l + 1)! , l ≥ −1. since ∫ ∂ω u v (ν1 − iν2)ds = ab i ∑ m γm (a−im(τ ) + a+im−2(τ )) , we obtain the desired conclusion. 2 2.2 main result we denote by e(ω) the set of all points on the segment that connects the focal points (− √ a2 − b2, 0) and ( √ a2 − b2, 0) of ω. it is easy to see that the support function of the set e(ω) is given by the formula he(ω)(ω) = √ a2 − b2|ω1|. we say that a function f (θ) = f (a cos θ, b sin θ) of θ is band limited if there exists a natural number n ≥ 1 such that, for all m ≥ n +1 the m-th fourier coefficients αm and βm of the function vanish. then we know that cm(f ) = 0 for all m ≥ n + 2. cubo 10, 2 (2008) a remark on the enclosure method ... 37 now we state the main result of this paper. theorem 2.1. let γ be a positive constant. assume that (1.3) is satisfied. let ω be regular with respect to d. let f be band limited and u be the solution of (1.1) with u = f on ∂ω. (1) let a > b. let ω satisfy ω1 6= 0. let f satisfy ∞∑ m=1 (sgn ω1) mm2cm(f ) 6= 0. (2.5) the formula lim τ −→∞ 1 τ log ∣∣∣∣ ∫ ∂ω γ ∂u ∂ν vds ∣∣∣∣ = max (hd(ω), he(ω)(ω)), (2.6) is valid. (2) let a = b. let f satisfy: for some n ≥ 1 αm = βm = 0 for all m with m ≥ n + 1 and α2 n + β2 n 6= 0. the formula lim τ −→∞ 1 τ log ∣∣∣∣ ∫ ∂ω γ ∂u ∂ν vds ∣∣∣∣ = max (hd(ω), 0), (2.7) is valid. • we say that a d is behind the line x · ω = t from the direction ω if the d is contained in the half plane x · ω < t. one important consequence of the formula (2.6) is: one can know whether the unknown cavity d is behind the line x · ω = he(ω)(ω) from the direction ω, however, in that case one cannot know the line x · ω = hd(ω) itself from the formula. this shows the limit to extract the whole convex hull of d without an additional assumption. • the assumption that f is band limited is just for a simplicity of the computation and can be relaxed. it is possible to apply directly the saddle point method to study the asymptotic behaviour of the integrals in lemma 2.1 for a f that is not band limited. moreover we want to point out that in a practical situation, one cannot produce highly oscillatory voltages on the boundary. this is due to the limit of numbers of electrodes attached on the boundary of the body. • a typical example of a band-limited f that satisfies (2.5) for all ω with ω1 6= 0 is the f given by f (θ) = a cos n θ + b sin n θ where n ≥ 1 and a2 + b2 6= 0. see remark 2.1 below for this explanation. in general we have to choose two f s corresponding to whether ω1 > 0 or ω1 < 0. proof of theorem 2.1. when a = b, the (2.7) is an easy consequence of (1.2), (2.1), (2.2) and (2.3). the problem is the case when a > b. we employ the compound asymptotic expansion (see 38 masaru ikehata cubo 10, 2 (2008) page 118 of [8] for the notion of the compound asymptotic expansion) of the bessel function due to hankel(see (9.09) and 9.3 of page 133 in [8]): jm(z) ∼ ( 2 πz )1/2 × { cos ( z − mπ 2 − π 4 ) ∞∑ s=0 (−1) s a2s(m) z2s − sin ( z − mπ 2 − π 4 ) ∞∑ s=0 (−1) s a2s+1(m) z2s+1 } (2.8) as z −→ ∞ in |arg z| ≤ π − δ for each fixed δ ∈ ]0, π[ where a0(m) = 1 and, for s = 1, 2, · · · as(m) = 1 s!8s (4m 2 − 1 2 )(4m 2 − 3 2 ) · · · (4m 2 − (2s − 1) 2 ). first we consider the case when ω1 > 0. from (2.8) in the case when z = −i √ a2 − b2τ (ω1 + iω2) we obtain jm(z) = ( 1 2πz )1/2 e iz (−i) m e −iπ/4 { 1 − 4m2 − 1 8iz + o ( 1 τ 2 )} (2.9) as τ −→ ∞. since f is band limited, one can find n ≥ 1 such that, for all m ≥ n + 1 the m-th fourier coefficients αm and βm of f vanish. then cm(f ) = 0 for m ≥ n + 2 and from (2.4) and (2.9) we obtain ∫ ∂ω uv(ν1 − iν2)ds = 2πab ( 1 2πz )1/2 eize−iπ/4 × {( 1 + 1 8iz ) n +1∑ m=0 cm(f ) + i 1 2z n +1∑ m=1 m2cm(f ) + o ( 1 τ 2 )} . (2.10) here we claim that n +1∑ m=0 cm(f ) = 0. (2.11) it suffices to prove the claim in the case when f (a cos θ, b sin θ) = αj cos jθ + βj sin jθ (2.12) for each fixed j = 1, 2, · · · , n . since ∑ ∞ m=0 cm(f ) = cj−1(f ) + cj (f ) + cj+1(f ) and we have cj+1(f ) = a−γj (√ a + b a − b ) j+1 + a+γj (√ a − b a + b ) j+1 , cj (f ) = 0, cj−1(f ) = a+γj (√ a + b a − b ) j−1 + a−γj (√ a − b a + b ) j−1 , cubo 10, 2 (2008) a remark on the enclosure method ... 39 we get ∞∑ m=0 cm(f ) = { a+ + a− ( a + b a − b )} γj (√ a + b a − b ) j−1 + γj (√ a − b a + b ) j+1    . since a+ + a− ( a + b a − b ) = 0, we see that the claim (2.11) is valid. therefore (2.10) becomes ∫ ∂ω uv(ν1 − iν2)ds = iπabz −1 ( 1 2πz )1/2 eize−iπ/4 { n +1∑ m=1 m2cm(f ) + o ( 1 τ )} . (2.13) set ω1 + iω2 = e iϑ with −π/2 < ϑ < π/2. then z1/2 = √ τ (a2 − b2)1/4ei(ϑ−π/2)/2. since eiz = eτ he(ω)(ω)eiτ √ a2−b2ω2 , from (1.2), (2.1), (2.2) and (2.13) we obtain the compound asymptotic formula: ∫ ∂ω γ ∂u ∂ν vds ∼ −γ √ π 2 ab(a2 − b2)−3/4e−iϑ/2τ −1/2eτ he(ω)(ω)eiτ √ a 2 −b 2 ω2 n +1∑ m=1 m2cm(f ) − γe τ hd(ω) a τ µ . from this we know that the quantity exp { −τ max (hd(ω), he(ω)(ω)) }∣∣∣∣ ∫ ∂ω γ ∂u ∂ν vds ∣∣∣∣ is truly algebraic decaying as τ −→ ∞. note that we have used the lower bound of µ: µ > 1/2. therefore we obtain the formula (2.6). next consider the case when ω1 < 0. write rω(τ ; f ) =∫ ∂ω f v(ν1 − iν2)ds. then we have rω (τ ; f ) = −r−ω(τ ; f ∗ ) where f ∗(x) = f (−x). since the m-th fourier coefficients of f ∗ are given by (−1)m times those of f and the first component of −ω is positive, we can derive the corresponding result in the case when ω1 < 0 from the result in the case when ω1 > 0 by replacing cm(f ) in the condition (2.5) with −(−1) mcm(f ). 2 remark 2.1. fix j = 1, 2, · · · , n and let f be given by (2.12). then a direct computation similar to the proof of the claim (2.11) yields ∞∑ m=1 m2cm(f ) = (j − 1) 2cj−1(f ) + (j + 1) 2cj+1(f ) = − 2 ab j(a2 − b2)−(j−1)/2{(a + b)j γj − (a − b) j γj}. 40 masaru ikehata cubo 10, 2 (2008) this yields also ∞∑ m=1 (−1) mm2cm(f ) = (−1) j−1 ∞∑ m=1 m2cm(f ) = (−1) j 2 ab j(a2 − b2)−(j−1)/2{(a + b)j γj − (a − b) j γj}. these yield: a f whose fourier coefficients αj and βj vanish for all j ≥ n + 1 with some n ≥ 1, satisfies the condition (2.5) if and only if n∑ j=1 (sgn ω1) j j(a2 − b2)−(j−1)/2{(a + b)j γj − (a − b) j γj} 6= 0. (2.14) it is clear that there are many f s satisfying the condition (2.14). remark 2.2. in (1) the case when ω1 = 0 is not treated. in this case ω2 = ±1. if ω2 = 1, then from (2.8) we have jm(z) = ( 1 2π √ a2 − b2τ )1/2 × { eiτ √ a 2 −b 2 (−i)me−iπ/4 ( 1 + i 4m2 − 1 8z ) + e−iτ √ a 2 −b 2 imeiπ/4 ( 1 − i 4m2 − 1 8z )} + o(τ −5/2) where z = −i √ a2 − b2 τ (ω1 + iω2). then from (1.2), (2.1), (2.2) and (2.4) the problem can be reduced to the study of the asymptotic behaviour of the quantity n +1∑ m=0 { eiτ √ a 2 −b 2 e−iπ/4 ( 1 + i 4m2 − 1 8z ) + (−1) me−iτ √ a 2 −b 2 eiπ/4 ( 1 − i 4m2 − 1 8z )} cm(f ) (2.15) as τ −→ ∞. this seems very complicated for general τ . however, if we choose τ = lπ √ a2 − b2 , l = 1, 2, · · · , (2.16) then (2.15) becomes (−1) l e −iπ/4 { n +1∑ m=0 ( 1 + i 4m2 − 1 8z ) cm(f ) + i n +1∑ m=0 (−1) m ( 1 − i 4m2 − 1 8z ) cm(f ) } = (−1)le−iπ/4i 2z n +1∑ m=1 m2{cm(f ) − icm(f ∗ )}. note that we have used the claim (2.11) for f and f ∗. therefore if f satisfies the condition ∞∑ m=1 m2{cm(f ) − icm(f ∗ )} 6= 0 (2.17) cubo 10, 2 (2008) a remark on the enclosure method ... 41 instead of (2.5), then for τ given by (2.16), the formula lim l−→∞ 1 τ log ∣∣∣∣ ∫ ∂ω γ ∂u ∂ν vds ∣∣∣∣ = max (hd(ω), 0), is valid. by replacing f with f ∗, we know also that: if ω2 = −1, then the same formula is valid provided ∞∑ m=1 m2{cm(f ) + icm(f ∗ )} 6= 0 (2.18) instead of (2.17). from the computation in remark 2.1 one can sum the conditions (2.17) and (2.18) up in the single form: n∑ j=1 { 1 + (−1) j (sgn ω2)i } j(a2 − b2)−(j−1)/2{(a + b)j γj − (a − b) j γj} 6= 0 where n ≥ 1 and chosen in such a way that, for all m ≥ n + 1 the m-th fourier coefficients of f vanish. 2.3 uniqueness as a corollary of theorem 2.1 we obtain a uniqueness theorem. corollary 2.1. let γ be a positive constant. assume that d satisfies (1.3). (1) let ω be a domain enclosed by an ellipse. let f+ and f− be band limited and satisfy ∞∑ m=1 (±) mm2cm(f±) 6= 0. let u± be the solution of (1.1) with u± = f± on ∂ω. then the neumann data γ∂u+/∂ν and γ∂u−/∂ν on ∂ω uniquely determine the convex hull of d ∪ e(ω). (2) let ω be a domain enclosed by a circle. let f be band limited and non constant. let u be the solution of (1.1) with u = f on ∂ω. then the neumann data γ∂u/∂ν uniquely determines the convex hull of d ∪ {0}. we emphasize that γ is unknown. this makes the situation difficult definitely. assume that we have two unknowns (d, γ) = (d1, γ1), (d2, γ2) and solutions u1 and u2 both satisfying (1.1) and the boundary condition u = f on ∂ω. the key point of a standard and traditional approach is to prove that if γ1∂u1/∂ν = γ2∂u2/∂ν on ∂ω, then u1 = u2 in a neighbourhood of ∂ω. if γ1 = γ2, then the conclusion is true because of the uniqueness of the cauchy problem for the laplace equation. however, if γ is unknown, i.e., the assumption γ1 = γ2 is dropped, one can not immediately get the conclusion (note that we are considering a finite set of observation data not the full dirichlet-to-neumann map). our approach skips this point by using an analytical formula that directly connects the data with unknown discontinuity. 42 masaru ikehata cubo 10, 2 (2008) the proof of corollary 2.1 is based on: given d the set of all directions that are not regular with respect to d is a finite set; the formulae (2.6) are valid for f = f± in (1); the formula (2.7) is valid for f in (2). therefore, for example, in (1) we see that the neumann data uniquely determine the values of max (hd(ω), he(ω)(ω)) which is the support function of the convex hull of d ∪ e(ω) at the directions ω except for a finite set of directions. since the support function hd and he(ω) are continues on the unit circle and so is max (hd( · ), he(ω)( · )). a density argument yields the desired uniqueness. remark 2.3. if ∂d is smooth, then (2) of corollary 2.1 does not hold. let ω be the unit open disc centered at the origin of the coordinates system and for 0 < r < 1 let d(r) be the open disc centered at the origin with the radius r. let 0 < r1, r2 < 1. fix an integer m ≥ 1. for each j = 1, 2 let uj be the weak solution of the problem (1.1) with d = d(rj ) and the dirichlet data uj (r, θ)|r=1 = cos mθ where (r, θ) denotes the usual polar coordinates centered at the origin. then we know that u1(r, θ) = 1 1 + r2m1 (rm + r2m1 r −m ) cos mθ, u2(r, θ) = 1 1 + r2m2 (rm + r2m2 r −m ) cos mθ. this yields 1 + r2m2 1 − r2m2 ∂u2 ∂ν = m cos mθ = 1 + r2m1 1 − r2m1 ∂u1 ∂ν on ∂ω. since r1 and r2 are arbitrary chosen, this means that one cannot uniquely determine d(r) from the single set of the dirichlet and neumann data f (θ) = cos mθ and γ∂u/∂ν on ∂ω in the case when γ = (1 + r2m)(1 − r2m). this suggests that the singularity of ∂d is essential for the validity of (2) in corollary 2.1. 3 an application to the inverse conductivity problem the idea in the proof of theorem 2.1 can be applied to the case when the unknown domain d is a model of an inclusion. we assume that the conductivity k = k(x) of the body that occupies ω is given by k(x) = γ if x ∈ ω \ d; k(x) = γ̃ if x ∈ d. it is assumed that the γ and γ̃ are positive constants and satisfy γ 6= γ̃. the voltage u inside the body satisfies the equation ∇ · k∇u = 0 in ω. given ω = (ω1, ω2) ∈ s 1 set ω⊥ = (ω2, −ω1). let τ > 0 and v = e τ x·(ω+iω⊥) . in [6] we have already proved that if u is not a constant function and d is polygonal and satisfies the condition (1.3), then for a given direction ω that is regular with respect to d the formula lim τ −→∞ 1 τ log ∣∣∣∣ ∫ ∂ω ( γ ∂u ∂ν v − γ ∂v ∂ν u ) ds ∣∣∣∣ = hd(ω), is valid. note that k = γ on ∂ω and we do not assume that the conductivity γ̃ of d is known. cubo 10, 2 (2008) a remark on the enclosure method ... 43 here we propose the same question as that of introduction. assume that we do not know k in the whole domain. given a non constant voltage f = u|∂ω on ∂ω is it possible to extract some information about the location of d from the corresponding current density k∂u/∂ν on ∂ω? the answer is yes in the case when the ω is enclosed by an ellipse. it starts with recalling the equation ∫ ∂ω γ ∂u ∂ν vds = ∫ ∂ω γ ∂v ∂ν uds − (γ − γ̃) ∫ ∂d u ∂v ∂ν ds. (3.1) recall key lemma in [6]: there exist positive constants b and λ(> 1/2) such that, as τ −→ ∞ ∣∣∣∣ ∫ ∂d u ∂v ∂ν ds ∣∣∣∣ ∼ b τ λ e τ hd(ω). (3.2) then from (2.2), (3.1), (3.2) and lemma 2.1 we see that the completely same statements as those in theorem 2.1, corollary 2.1 and remarks 2.1 and 2.2 are valid. remark 3.1. in [7] we employed the difference of the values of the voltage at arbitrary fixed two points on the boundary of a general two-dimensional bounded domain ω with smooth boundary. more precisely we introduced the operator λk(p, q) : g 7−→ u(p ) − u(q) where p and q are two arbitrary points on ∂ω; g satisfies ∫ ∂ω gds = 0; the u is a solution of the equation ∇ · k∇u = 0 in ω and satisfies the neumann boundary condition k∂u/∂ν = g on ∂ω. given ω = (ω1, ω2) ∈ s 1 set ω⊥ = (ω2, −ω1). let τ > 0 and v = e τ x·(ω+iω⊥) . what we have proved is: if g = ∂v/∂ν on ∂ω and d is polygonal and satisfies the condition (1.3), then for a given direction ω that is regular with respect to d the formula lim τ −→∞ 1 τ log | {λk(p, q) − λγ (p, q)} (g)| = hd(ω), (3.3) is valid. note that we have used the relationship {λk(p, q) − λγ (p, q)} (g) = 1 γ { λk/γ (p, q) − λ1(p, q) } (g). if γ is unknown, then one cannot use the term λγ (p, q)(g) in (3.3). however, that has the simple form λγ (p, q)(g) = 1 γ {v(p ) − v(q)} for g = ∂v/∂ν on ∂ω. using this form, proposition 3.1 and lemma 3.1 in [7], one immediately gets the following formulae provided d is polygonal and satisfies the condition (1.3) and ω is regular with respect to d: • if ω is not perpendicular to the line passing through p and q, then lim τ −→∞ 1 τ log |λk(p, q)(g)| = max ( hd(ω), h{p, q}(ω) ) ; 44 masaru ikehata cubo 10, 2 (2008) • if ω is perpendicular to the line passing through p and q, choose, for example, τ = π |p − q| ( 1 2 + 2l ) , l = 0, 1, 2, · · · , then lim l−→∞ 1 τ log |λk(p, q)(g)| = max ( hd(ω), h{p, q}(ω) ) . 4 conclusion we confirmed that: in the case when the background conductivity is homogeneous and unknown the enclosure method still works provided: • the domain that is occupied by a background body has a simple geometry; • the fourier series expansion of the voltage on the boundary does not contain high frequency parts (band limited) and satisfies a non vanishing condition of a quantity involving the fourier coefficients. however, the method yields a less information about the location and shape of unknown cavity or inclusion compared with the case when the conductivity is known. we found an explicit obstruction that depends on the geometry of the background body. acknowledgements this research was partially supported by grant-in-aid for scientific research (c)(no. 18540160) of japan society for the promotion of science. received: november 2007. revised: february 2008. references [1] l. borcea, electrical impedance tomography, inverse problems, 18 (2002), r99-r136. [2] l. borcea, addendum to “electrical impedance tomography”, inverse problems, 19 (2003), 997–998. [3] a. friedman and v. isakov, on the uniqueness in the inverse conductivity problem with one measurement, indiana univ. math. j., 38 (1989), 563–579. [4] a. friedman and m. vogelius, determining cracks by boundary measurements, indiana univ. math. j., 38 (1989), 527–556. cubo 10, 2 (2008) a remark on the enclosure method ... 45 [5] m. ikehata, enclosing a polygonal cavity in a two-dimensional bounded domain from cauchy data, inverse problems, 15 (1999), 1231–1241. [6] m. ikehata, on reconstruction in the inverse conductivity problem with one measurement, inverse problems, 16 (2000), 785–793. [7] m. ikehata, on reconstruction from a partial knowledge of the neumann-to-dirichlet operator, inverse problems, 17 (2001), 45–51. [8] f.w. olver, asymptotics and special functions, academic press, new york and london, 1974. [9] j.k. seo, on the uniqueness in the inverse conductivity problem, j. fourier anal. appl., 2 (1996), 227–235. n3 ppp.dvi cubo a mathematical journal vol.12, no¯ 01, (59–66). march 2010 weakly picard pairs of multifunctions valeriu popa department of mathematics, university of bacău, str. spiru haret nr. 8, 600114 bacău, romania email : vpopa@ub.ro abstract the purpose of this paper is to present a general answer for the following problem: let (x, d) be a metric space and t1, t2 : x → p (x) two multifunctions. determine the metric conditions which imply that (t1, t2) is a weakly picard pair of multifunctions and t1, t2 are weakly picard multifunctions , for multifunctions satisfying an implicit contractive condition, generalizing some results from [6] and [7]. resumen el proposito de este artículo es presentar una respuesta general para el siguiente problema: sea (x, d) un espacio métrico y t1, t2 : x → p (x) dos multifunciones. determine los condiciones metricas para las cuales (t1, t2) sea un par de multifunciones de picard debil y t1, t2 sean multifunciones satisfaziendo una condición contractiva implícita, generalizando algunos resultados de [6] y [7]. key words and phrases: multifunction, fixed point, implicit relation, weakly picard multifunction, weakly picard pair of multifunctions. math. subj. class.: 47h10, 54h25. 60 valeriu popa cubo 12, 1 (2010) 1 introduction and preliminaries let x be a nonempty set. we denote p(x) the set of all nonempty subsets of x, i.e. p (x) = {y : φ 6= y ⊂ x} . let t : x → p (x) a multifunction. we denote by ft the set of fixed points of t, i.e. ft = {x ∈ x : x ∈ t (x)}. let (x,d) be a metric space. we denote by pcl(x) the set of all nonempty and closed sets of x. we also recall the functional d : p (x) × p (x) → r+, defined by d(a, b) = inf{d(a, b) : a ∈ a, b ∈ b} for each a, b ∈ p (x) and generalized hansdorff-pompeiu metric h : p (x) × p (x) → r+ ∪ {+∞} defined by h(a, b) = max {sup[d(a, b), a ∈ a], sup[d(a, b), b ∈ b]} for a, b ∈ p (x). the following property of h is well-known. lemma 1.1. let (x,d) be a metric space, a, b ∈ p (x) and q > 1. then for every a ∈ a, there exists b ∈ b such that d(a, b) ≤ qh(a, b). definition 1.1. let (x,d) be a metric space and t : (x, d) → p (x) a multifunction. we say that t is a weakly picard multifunction [3],[4] if for each x ∈ x and for every y ∈ t (x), there exists a sequence (xn)n∈n such that: (i) x0 = x, x1 = y; (ii) xn+1 ∈ t (xn) , for each n ∈ n ∗ ; (iii) the sequence (xn)n∈n is convergent and its limit is a fixed point of t. definition 1.2. let (x,d) be a metric space and t1, t2 : x → p (x) two multifunctions. we say that (t1, t2) is a weakly picard pair of multifunctions if for each x ∈ x and for every y ∈ t1(x) ∪ t2(x), there exists a sequence (xn)n∈n such that (i) x0 = x, x1 = y; (ii) x2n+1 ∈ ti(x2n) and x2n+2 ∈ tj(x2n+1), for n ∈ n , where i, j ∈ {1, 2}, i 6= j ; (iii) the sequence (xn)n∈n is convergent and its limit is a common fixed point of t1 and t2. problem 1.1 [4]. let (x,d) be a metric space and t1, t2 : (x, d) → p (x) two multifunctions. determine the metric conditions which implies (t1, t2) is a weakly picard pair of multifunctions and t1, t2 are weakly picard multifunctions. partial answers to problem 1.1. was established by sintămărian in [4]-[7]. in [1] and [2] is introduced the study of fixed point for mappings satisfying implicit relations. the purpose of this paper is to prove two general fixed points theorems for multifunctions which satisfy a new type of implicit contractive relation which generalize some results from [6], [7]. cubo 12, 1 (2010) weakly picard pairs of multifunctions 61 2 implicit relations let f be the set of all continuous multifunctions f (t1, ..., t6) : r6+ → r satisfying the following conditions: (f1): f is increasing in variable t1 and nonincreasing in variables t3, ..., t6; (f2): there exists k > 1, h ∈ [0, 1) and g ≥ 0 such that for every u ≥ 0, v ≥ 0, w ≥ 0, such that (fa): u ≤ kt and f (t, v, v + w, u + w, u + v + w, w) ≤ 0, or (fb): u ≤ kt and f (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw. example 2.1. f (t1, ..., t6) = t1 − at2 − b(t3 + t4) − c(t5 + t6), when 0 < a + 2b + 2c < 1. (f1): obviously. (f2): f (t, v, v + w, u + w, u + v + w, w) = t − av − b(u + v + 2w) − c(u + v + 2w) ≤ 0, where 1 < k < 1 a+2b+2c . then u ≤ kt ≤ k[av+b(u+v+2w)+c(u+v+2w)]. hence u ≤ hv+gw, where 0 < h = k(a+b+c) 1−k(b+c) < 0 and g = 2k(b+c) 1−k(b+c) ≥ 0 similarly, f (t, v, u + w, w + v, w, u + v + w ≤ 0 implies u ≤ hv + gw. remark 2.1. if a + 4b + 4c < 1 and 1 < k < 1 a+4b+4c then h + g < 1. example 2.2. f (t1, ..., t6) = t1 − mmax{t2, t3, t4, 12 (t5 + t6)} where 0 < m < 1 2 . (f1): obviously. (f2): let u ≥ 0, v ≥ 0, w ≥ 0, 1 < k < 12m and f (t, v, v + w, u + w, u + v + w, w) = t − mmax{v, u + w, u + v, 1 2 (u + v + 2w)} ≤ 0 which implies t ≤ m(u + v + w). then u ≤ kt ≤ km(u + v + w). hence, u ≤ hv + gw where 0 < h = km 1−km < 1 and g = km 1−km ≥ 0. similarly, f (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw . remark 2.2. if 0 < m < 1 3 and 1 < k < 1 3m then h + g < 1. example 2.3. f (t1, ..., t6) = t 2 1 − mmax{t22, t3t4, t5t6}, where 0 ≤ m < 14 . (f1): obviously. (f2): let u ≥ 0, v ≥ 0, w ≥ 0, 1 < k < 12√m and f (t, v, v + w, u + w, u + v + w, w) = t2 − mmax{v2, (v + w)(u + w), w(u + v + w)) ≤ 0 which implies t2 ≤ m(u + v + w)2 and t ≤ √ m(u + v + w). then u ≤ kt ≤ k √ m(u + v + w). hence, u ≤ hv + gw, where 0 ≤ h ≤ k √ m 1=k √ m < 1 and g = k √ m 1=k √ m ≥ 0. similarly, f (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw. remark 2.3. if o ≤ m < 1 9 and 1 < k < 1 3 √ m then h + g < 1. example 2.4. f (t1, ..., t6) = t 3 1 + t 2 2 + 1 1+t5+t6 − m(t22 + t23 + t24), where 0 < m < 112 . (f1): obviously. (f2): let u ≥ 0, v ≥ 0, w ≥ 0 and 1 < k < 12√m and f (t, v, v + w, u + w, u + v + w, w) = t3 + t2 + t 1+u+v+w − m(v2 + (v + w)2 + (u + w)2) ≤ 0 which implies 62 valeriu popa cubo 12, 1 (2010) t2 ≤ m(v2 + (u + v)2 + (u + w)2) ≤ 3m(u + v + w)2 and t ≤ √ 3m(u + v + w). if u ≤ kt ≤ k √ 3m(u + v + w) then u ≤ hv + gw, where 0 ≤ h = k √ 3m 1−k √ 3m < 1 and g = k √ 3m 1−k √ 3m ≥ 0. similarly, f (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw. remark 2.4. if 0 < m < 1 27 and 1 < k < 1 3 √ 3m then h + g < 1. 3 main results theorem 3.1. let t1, t2 : (x, d) → p cl(x) be two multifunctions. if the inequality (1) φ(h(t1(x), t2(y)), d(x, y), d(x, t1(x)), d(y, t2(y)), d(x, t2(y)), d(y, t1(x)) ≤ 0 holds for all x, y ∈ x, where f ∈ f and ft1 6= φ or ft2 6= φ, then ft1 = ft2 . proof. let u ∈ ft1 , then u ∈ t1(u) and by (1) we have φ(h(t1(u), t2(u)), d(u, u), d(u, t1(u)), d(u, t2(u)), d(u, t2(u)), d(u, t1(u)) ≤ 0 by d(u, t2(u)) ≤ h(t1(u), t2(u)) it follows that φ(d(u, t2(u)), 0, 0, d(u, t2(u)), d(u, t2(u)), 0) ≤ 0 since d(u, t2(u)) ≤ kd(u, t2(u)) by (fa) we have that d(u, t2(u)) = 0. since t2(u) is closed we obtain u ∈ t2(u) i.e. u ∈ ft2 and ft1 ⊂ ft2 . similarly, by (fb) we obtain ft2 ⊂ ft1 . similarly, if u ∈ ft2 , then ft1 = ft2 . theorem 3.2. let (x,d) be a complete metric space and t1, t2 : (x, d) → p cl(x) two multifunctions. if (1) holds for all x, y ∈ x, where f ∈ f, then t1 and t2 have a common fixed point and ft1 = ft2 ∈ p cl(x). proof. let x0 ∈ x and x1 ∈ t1(x0). then there exists x2 ∈ t2(x1) so that d(x1, x2) ≤ kh(t1(x0), t2(x1)) suppose that x2, x3, ..., x2n−1, x2n, ... such that x2n−1 ∈ t1x2n−2, x2n ∈ t2x2n−1, n ∈ n ∗ and (2) d(x2n−1, x2n) ≤ kh(t1(x2n−2), t2(x2n−1)) , (3) d(x2n−2, x2n−1) ≤ kh(t1(x2n−2), t2(x2n−3)) . by (1) we have successively φ(h(t1(x2n−2), t2(x2n−1)), d(x2n−2, x2n−1), d(x2n−2, t1(x2n−2)), d(x2n−1, t2(x2n−1)), d(x2n−2, t2(x2n−1)), d(x2n−1, t1(x2n−2)) ≤ 0 φ(h(t1(x2n−2), t2(x2n−1)), d(x2n−2, x2n−1), d(x2n−1, x2n−2), d(x2n−1, x2n), d(x2n−2, x2n), 0) ≤ 0 (4) φ(h(t1(x2n−2), t2(x2n−1)), d(x2n−2, x2n−1), d(x2n−2, x2n−1), d(x2n−1, x2n), d(x2n−2, x2n−1) + d(x2n−1, x2n), 0) ≤ 0 since φ ∈ f then by (2),(4) and (fa) we obtain (5) d(x2n−1, x2n) ≤ hd(x2n−2, x2n−1) similarly, by (3) and (fb) we obtain (6) d(x2n−2, x2n−1) ≤ hd(x2n−2, x2n−3) cubo 12, 1 (2010) weakly picard pairs of multifunctions 63 then by a rutine calculation one can show that (xn)n∈n is a cauchy sequence and since (x,d) is complete we have limxn = x for some x ∈ x. now, if n ∈ n ∗, (1) implies φ(h(t1(x), t2(x2n−1)), d(x, x2n−1), d(x, t1(x)), d(x2n−1, t2(x2n−1)), d(x, t2(x2n−1)), d(x2n−1, t1x) ≤ 0 as d(x2n, t1(x)) ≤ h(t2(x2n−1), t1(x)) we have φ(d(x2n, t1(x)), d(x, x2n−1), d(x, t1(x)), d(x2n−1, x2n), d(x, x2n), d(x2n−1, t1(x)) ≤ 0 letting n tend to infinity we obtain φ(d(x, t1(x)), 0, d(x, t1(x)), 0, 0, d(x, t1(x)) ≤ 0 since d(x, t1(x)) ≤ kd(x, t1(x)) by (fb) we obtain d(x, t1(x)) = 0. since t1(x) is closed, x ∈ t1(x). hence x ∈ ft1 . by theorem 3.1 ft1 = ft2 . let us prove that ft1 = ft2 ∈ p cl(x). for this purpose that yn ∈ ft1 = ft2 for each n ∈ n such that yn → y∗ as n → ∞. for example yn ∈ t1(yn). then by lemma 1.1 there exists vn ∈ t2y∗ such that (7) d(yn, vn) ≤ kh(t1(yn), t2(y∗)) . by (1) and (f1) we have successively φ(h(t1(yn), t2(y ∗)), d(yn, y ∗), d(yn, t1(yn)), d(y ∗, t2(y ∗)), d(yn, t2(y ∗)), d(y∗, t1(yn)) ≤ 0 φ(h(t1(yn), t2(y ∗)), d(yn, y ∗), 0, d(y∗, vn), d(yn, vn), d(y ∗, yn)) ≤ 0 (8) φ(h(t1(yn), t2(y ∗)), d(yn, y ∗), d(yn, y ∗) + d(yn, y ∗), d(y∗, yn) + d(yn, vn), d(yn, vn) + d(yn, y ∗) + d(yn, y ∗), d(yn, y ∗)) ≤ 0 since φ ∈ f by (7) and (8) it follows that d(yn, vn) ≤ hd(yn, y∗) + gd(y∗, yn) using the triangle inequality we obtain d(y∗, vn) ≤ d(y∗, yn) + d(yn, vn) ≤ (1 + h + g)d(y∗, yn) letting n tend to infinity we obtain that limvn = y ∗. since vn ∈ t2(y∗), for each n ∈ n ∗ and t2(y ∗)is closed, it follows that y∗ ∈ t2(y∗), hence y∗ ∈ ft2 = ft1 and ft1 is closed. therefore, ft1 = ft2 ∈ p cl(x). theorem 3.3. let (x,d) be a complete metric space and t1, t2 : (x, d) → p cl(x). if (1) holds for all x, y ∈ x, where φ ∈ f, then ft1 = ft2 ∈ p cl(x) and (t1, t2) is a weakly picard pair of multifunctions. if in adition we have that h + g < 1, then t1 and t2 are weakly picard multifunctions. proof. the first part it follows from theorem 3.2. let x0 ∈ x and x1 ∈ t1(x0). there exists y1 ∈ t2(x1) such that (9) d(x1, y1) ≤ kh(t1(x0), t2(x1)) by (1) and (f1) we have successively φ(h(t1(x0), t2(x1)), d(x0, x1), d(x0, t1(x0)), d(x1, t2(x1)), d(x0, t2(x1)), d(x1, t1(x0)) ≤ 0 φ(h(t1(x0), t2(x1)), d(x0, x1), d(x0, x1), d(x1, y1), d(x0, y1), 0) ≤ 0 64 valeriu popa cubo 12, 1 (2010) (10) φ(h(t1(x0), t2(x1)), d(x0, x1), d(x0, x1), d(x1, y1), d(x0, x1) + d(x1, y1), 0) ≤ 0 since φ ∈ f by (9) and (10) it follows that d(x1, y1) ≤ hd(x0, x1) also, there exists x2 ∈ t1(x1) such that (11) d(x2, y1) ≤ kh(t1(x1), t2(x1)) by (1) we have successively φ(h(t1(x1), t2(x1)), 0, d(x1, t1(x1)), d(x1, t2(x1)), d(x1, t2(x1)), d(x1, t1(x1)) ≤ 0 φ(h(t1(x1), t2(x1)), 0, d(x1, x2), d(x1, y1), d(x1, y1), d(x1, x2)) ≤ 0 (12) φ(h(t1(x1), t2(x1)), 0, d(x1, x2), d(x1, x2) + d(x2, y1), d(x1, x2) + d(x2, y1), d(x1, x2)) ≤ 0 since φ ∈ f by (11) and (12) it follows that d(y1, x2) ≤ gd(x1, x2) using the triangle inequality we have d(x1, x2) ≤ d(x1, y1) + d(y1, x2) ≤ hd(x0, x1) + gd(x1, x2) which implies that d(x1, x2) ≤ h1−g d(x0, x1) now, there exists y2 ∈ t2(x2) such that (13) d(x2, y2) ≤ kh(t1(x1), t2(x2)) by (1) we have successively φ(h(t1(x1), t2(x2)), d(x1, x2), d(x1, t1(x1)), d(x2, t2(x2)), d(x1, t2(x2)), d(x2, t1(x1)) ≤ 0 φ(h(t1(x1), t2(x2)), d(x1, x2), d(x1, x2), d(x2, y2), d(x1, y2), 0) ≤ 0 (14) φ(h(t1(x1), t2(x2)), d(x1, x2), d(x1, x2), d(x2, y2), d(x1, x2) + d(x2, y2), 0) ≤ 0 since φ ∈ f by (13) and (14) it follows that d(x2, y2) ≤ hd(x1, x2) also, there exists x3 ∈ t1(x2) such that (15) d(x3, y2) ≤ kh(t1(x2), t2(x2)) by (1) we have successively φ(h(t1(x2), t2(x2)), 0, d(x2, t1(x2)), d(x2, t2(x2)), d(x2, t2(x2)), d(x2, t1(x2)) ≤ 0 φ(h(t1(x2), t2(x2)), 0, d(x2, x3), d(x2, y2), d(x2, y2), d(x2, x3)) ≤ 0 (16) φ(h(t1(x2), t2(x2)), 0, d(x2, x3), d(x2, x3) + d(x3y2), d(x2, x3) + d(x3, y2), d(x2, x3)) ≤ 0 since φ ∈ f by (15) and (16) it follows that d(x3, y2) ≤ gd(x2, x3) using again the triangle inequality we obtain d(x2, x3) ≤ d(x2, y2) + d(y2, x3) ≤ hd(x1, x2) + gd(x2, x3) and so cubo 12, 1 (2010) weakly picard pairs of multifunctions 65 d(x2, x3) ≤ h1−g d(x1, x2) by induction we obtain that there exists a sequence (xn)n∈n starting from x0, x1 with xn+1 ∈ t1(xn) such that d(xn, xn+1) ≤ h1−g d(xn−1, xn) for each n ∈ n ∗. since h 1−g < 1 it follows that (xn)n∈n is a convergent sequence, because (x,d) is a complete metric space. let x∗ = limxn. by (1) we have φ(t1(xn), t2(x ∗)), d(x∗, xn), d(xn, t1(xn)), d(x ∗, t2(x ∗)), d(xn, t2(x ∗)), d(x∗, t1(xn))) ≤ 0 since d(xn+1, t2(x ∗)) ≤ h(t1(xn), t2(x∗) we obtain φ(d(x2n+1), t2(x ∗)), d(x∗, xn), d(xn, xn+1), d(x ∗, t2(x ∗)), d(xn, t2(x ∗)), d(x∗, xn+1))) ≤ 0 letting n tend to infinity we obtain φ(d(x∗, t2(x ∗)), 0, 0, d(x∗, t2(x ∗)), d(x∗, t2(x ∗)), 0) ≤ 0 since d(x∗, t2(x ∗)) ≤ kd(x∗, t2(x∗)) and φ ∈ f we obtain d(x∗, t2(x∗)) = 0 and since t2(x∗) is closed we have that x∗ ∈ t2(x∗) and x∗ ∈ ft2 = ft1 . hence t1 is a weakly picard multifunction. the fact that t2 is a weakly picard multifunction is similar proved. remark 3.1. by theorems 2 and 3 and ex. 2.1 we obtain generalizations of the results from theorem 2.1 [6] and theorem 2.1 [7]. by ex. 2.2 -2.4 we obtain new results. received: may, 2008. revised: october, 2009. references [1] popa, v., some fixed point theorems for contractive mappings, stud.cerc.st.ser. mat. univ. bacău, 7(1997), 127–133. 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[7] sintămărian, a., some pairs of multivalued operators, carpathian j. of math. 21,1-2(2005), 115–125. a mathematical journal vol. 7, no 3, (49 63). december 2005. the ergodic measures related with nonautonomous hamiltonian systems and their homology structure. part 1 1 denis l.blackmore dept. of mathematical sciences at the njit, newark, nj 07102, usa deblac@m.njit.edu yarema a.prykarpatsky the agh university of science and technology, department of applied mathematics, krakow 30059 poland, and brookhaven nat. lab., cdic, upton, ny, 11973 usa yarchyk@imath.kiev.ua, yarpry@bnl.gov anatoliy m.samoilenko the institute of mathematics, nas, kyiv 01601, ukraine anatoliy k.prykarpatsky 2 department of applied mathematics, the agh university of science and technology applied mathematics, krakow 30059 poland pryk.anat@ua.fm, prykanat@cybergal.com abstract there is developed an approach to studying ergodic properties of time-dependent periodic hamiltonian flows on symplectic metric manifolds having applications in mechanics and mathematical physics. based both on j. mather’s [9] results about homology of probability invariant measures minimizing some lagrangian 1the authors are cordially indebted to profs. anthoni rosato (njit, nj,usa) and alexander s. mishchenko (moscow state university, russia) for useful comments on the article. they are also thankful to participants of the seminar ” nonlinear analysis” at the dept. of applied mathematics of the agh university of science and technology of krakow for valuable discussions. 2the fourth author was supported in part by a local agh grant. 50 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) functionals and on the symplectic field theory devised by a. floer and others [3-8,12,15] for investigating symplectic actions and lagrangian submanifold intersections, an analog of mather’s β-function is constructed subject to a hamiltonian flow reduced invariantly upon some compact neighborhood of a lagrangian submanifold. some results on stable and unstable manifolds to hyperbolic periodic orbits having applications in the theory of adiabatic invariants of slowly perturbed integrable hamiltonian systems are stated within the gromov-salamon-zehnder [3,5,12] elliptic techniques in symplectic geometry. resumen un método para estudiar propiedades ergódicas de flujos hamiltonianos que dependen del tiempo sobre variedades simplécticas es desarrollado. basados tanto en un trabajo de j. mather [9] sobre homoloǵıa de medidas invariantes de probabilidad que minimizan algunos funcionales lagrangianos, como en la teoŕıa de campos simplécticos, desarrollada por a. floer y otros [3-8,12,15] para investigar acciones simplécticas e intersecciones de subvariedades lagrangianas, se construye un análogo de la función β de mather sujeto a un flujo hamiltoniano reducido invariantemente sobre una vecindad compacta de una subvariedad lagrangiana. se plantean algunos resultados sobre variedades estables e intestables de órbitas hiperbólicas periódicas. estas tienen aplicaciones en la teoŕıa de sistemas hamiltonianos integrables con perturbaciones lentas, en el marco de las técnicas eĺıpticas de gromov-salamon-zehnder [3,5,12] en geometŕıa simpléctica. key words: ergodic measures, holonomy groups, dynamical systems, quasi-complex structures, symplectic field theory math. subj. class.: 37a05, 37b35, 37c40, 37c60, 37j10, 37j40, 37j45 introduction the past years have given rise to several exciting developments in the field of symplectic geometry and dynamical systems [3-12], which introduced new mathematical tools and concepts suitable for solving many before too hard problems. when studying periodic solutions to non-autonomous hamiltonian systems salamon & zehnder [3] developed a proper morse theory for infinite dimensional loop manifolds based on previous results on symplectic geometry of lagrangian submanifolds of floer [4, 6]. investigating at the same time ergodic measures related with lagrangian dynamical systems on tangent spaces to configuration manifolds, mather [9] devised a new approach to studying the correspondingly related invariant probabilistic measures based on a so called β-function. the latter made it possible to describe effectively the so called homology of these invariant probabilistic measures minimizing the correspond7, 3(2005) the ergodic measures related with nonautonomous hamiltonian ... 51 ing lagrangian action functional. as one can easily see, the mather approach doesn’t allow any its direct application to the problem of describing the ergodic measures related naturally with a given periodic non-autonomous hamiltonian system on a closed symplectic space. thereby, to overcome constraints to this task we suggest in the present work some new way to imbedding the non-autonomous hamiltonian case into the mather β-function theory picture, making use of the mentioned above salamon & zehneder and floer [3, 4, 6] loop space homology structures. based further on the gromov elliptic techniques in symplectic geometry, the latters make it possible to construct the invariant submanifolds of our hamiltonian system, naturally related with corresponding compact lagrangian submanifolds, and the related on them a β-function analog. 1 symplectic and analytic problem setting let (m2n,ω(2)) be a closed symplectic manifold of dimension 2n with a symplectic structure ω(2) ∈ λ(m2n) being weakly exact, that is ω(2)(π2(m2n)) = 0. every smooth enough time-dependent 2π-periodic function h : m2n × s1 → r gives rise to the non-autonomous vector field xh : m2n × s1 → t(m2n) defined by the equality ixh ω (2) = −dh, (1) where as usually [1], the operation ” ixh ” denotes the intrinsic derivation of the grassmann algebra λ(m2n) along the vector field xh. the corresponding flow on m2n × s1 takes the form: du/ds = xh (u; t), dt/ds = 1, (2) where u : r →m2n is an orbit, t ∈ r/2πz ' s1 and s ∈ r is an evolution parameter. we shall assume that solutions to (2) are complete and determine a one-parametric ψ-flow of diffeomorphisms ψs : m2n × s1 → m2n × s1 for all s ∈ r which are due to (1) evidently symplectic, that is ψs∗t0 ω (2) = ω(2) where ψst0 := ψ s|m 2n at any fixed t0 ∈ r/2πz ' s1. take now an (n + 1)-dimensional submanifold ln+1 ⊂ m2n × r, such that for any closed contractible curve γ with γ ⊂ ln+1 the following integral equality ∮ γ (α(1) − h(t)dt) = 0 (3) holds, where α(1) ∈ λ1(m 2n) is such a 1-form on m2n which satisfies the condition∫ d2 (ω(2) − dα(1)) = 0 for any compact two-dimensional disk d2 ⊂ m2n due to the weak exactness of the symplectic structure ω(2) ∈ λ2(m2n) and existing globally on ln+1 due to floer results [4, 6]. assume now also that for the flow of symplectomorphisms ψst0 : m 2n → m2n, s ∈ r, the condition {(ψst0l n t0 , t0 + s) : s ∈ r} ⊂ ln+1 (4) 52 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) holds for some compact lagrangian submanifold lnt0 ⊂ m 2n upon which ω(2) ∣∣ lnt0 = 0. the condition (4) in particular means [2] that the following expression α(1) − h(t)dt = da(t), (5) t = t0 + s(mod2π) ∈ r/2πz, holds in some vicinity of the lagrangian submanifold lnt0 ⊂ m 2n, where a mapping a : r/2πz → r is the so called [1, 2] generating function for the defined above continuous set of diffeomorphisms ψst0 ∈diff(m 2n), s ∈ r. the expression (5) makes it possible to define naturally the following poincarecartan type functional on a set of almost everywhere differentiable curves γ : [0,τ] → m 2n × s1 a(τ )t0 (γ) := 1 τ ∫ γ (α(1) − h(t)dt), (6) with end points { γ(τ) = ψτ (γ(0)) }, supp γ ⊂ u(lnt0 ) × s 1 for all τ ∈ r and u(lnt0 ) is some compact neighborhood of the lagrangian submanifold lnt0 ⊂ m 2n satisfying the condition ψst0u(l n t0 ) ⊂ u(lnt0 ) for all s ∈ r. let us denote by σt0 (h) the subset of curves γ with support in u(l n t0 ) × s1 and fixed end-points as before minimizing the functional (6). if the infimum is realized, one easily shows that any such curve γ ∈ σt0 (h) solves the system (2). for the above set of curves σt0 (h) to be specified more suitably, choose, following floer’s ideas [3-8,12], an almost complex structure j : m2n → end(t(m2n)) on the symplectic manifold m2n, where by definition j2 = −i, compatible with the symplectic structure ω(2) ∈ λ2(m2n). then the expexpression < ξ,η >:= ω(2)(ξ,jη), (7) where ξ,η ∈ t(m2n), naturally defines a riemannian metric on m2n. subject to the metric (7) our hamiltonian vector field xh : m2n ×s1 → t(m2n) is now represented as xh = j∇h, where ∇ : d(m2n) → t(m2n) denotes the usual gradient mapping with respect to this metric. consider now the space ω := ω(m2n × s1) of all continuous curves in m2n × s1 with fixed end-points. then one can similarly define the gradient mapping grad a(τ )t0 : ω → t(ω) as follows: (grad a(τ )t0 (γ),ξ) := 1 τ ∫ τ 0 ds < j(γt0 )γ̇t0 (s) + ∇h(γt0 ; s + t0),ξ >, (8) where γ = {(γt0 (s); t0 + s( mod 2π)) : s ∈ [0,τ]} ∈ ω as before, and ξ ∈ t(ω). since all critical curves γ ∈ σt0 (h) minimizing the functional (6) solve (2), this fact motivates a way of construction of an invariant subset ωh ⊂ ω, such that ωh := ω(u(lnt0 ) × s 1). namely, define a curve γ ∈ ωh (γ(−)) ⊂ ωh as satisfying [3] the following gradient flow in u(lnt0 ) × s 1 : ∂ut0/∂z = −grad a (τ ) t0 (u), ∂t/∂z = 0 (9) 7, 3(2005) the ergodic measures related with nonautonomous hamiltonian ... 53 for all z ∈ r and any τ ∈ r under the asymptotic conditions lim z→−∞ ut0 (s; z) = γ (−) t0 (s), lim z→∞ γt0 (s; z) = γt0 (s) (10) with the corresponding curves γ(−)t0 , γt0 : r →m 2n satisfying the system (2), and moreover, with the curve γ(−)t0 : r →m 2n being taken to be hyperbolic [1, 2] with supp γ (−) t0 ⊂ lnt0. now we can construct a so called [1] unstable manifold w u(γ(−)t0 ) to this hyperbolic curve γ(−)t0 defined for all τ ∈ r. thus due to the above construction, the functional manifold w u(γ(−)t0 ) when compact can be imbedded as a point submanifold into m2n thereby interpreting supports of all curves solving (9) and (10) where supp γt0 ⊂ lnt0, as a compact neighborhood l (−) t0 (h) ⊂ u(lnt0 ) of the compact lagrangian submanifold lnt0 ⊂ m 2n looked for above. the same construction can be done evidently for the case when the conditions (10) are changed either by lim z→+∞ γt0 (s; z) = γ (+) t0 (s), lim z→−∞ γt0 (s; z) = γt0 (s), (10a) or by lim z→−∞ γt0 (s; z) = γ (−) t0 (s), lim z→∞ γt0 (s; z) = γ (+) t0 (s), (10b) where γ(−)t0 : r →m 2n and γ(+)t0 : r →m 2n are some strictly different hyperbolic curves on m2n with supp γ(±)t0 ⊂ l n t0 and solving (2). based on (10a) one constructs similarly the stable manifold w s(γ(+)t0 (s)) to a hyperbolic curve γ (+) t0 and further the corresponding compact neighborhood l(+)t0 (h) ⊂ u(l n t0 ) of the compact lagrangian submanifold lnt0 ⊂ m 2n which is of crucial importance when studying intersection properties of stable w s(γ(+)t0 ) and unstable w u(γ(−)t0 ) manifolds. based similarly on (10b), one constructs the neighborhood lt0 (h) ⊂ u(lnt0 ) of the compact lagrangian submanifold lnt0 ⊂ m 2n being of interest when investigating so called adiabatic perturbations of integrable autonomous hamiltonian flows on the symplectic manifold m2n. now we make use of some statements [3, 5, 12] about the properties of the set ωh constructed above. for a generic choice of the hamiltonian function h : m2n×s1 → r the functional space of curves ωh is proved to be finite-dimensional what gives rise right away to hereditary finite-dimensionality of the neighborhood l(−)t0 (h) with the compact manifold structure. to see this linearize equation (9) in the direction of a vector field ξ ∈ t(ωh ). this leads to the linearized first-order differential operator: ft0 (u)ξ := ∇zξ + j(u)∇sξ + ∇ξj(u)∂u/∂s + ∇ξ∇h(u; t0 + s), (11) where u ∈ ωh satisfies the following equation stemming from (9) : ∂u/∂z + j(u)∂u/∂s + ∇h(u; s + t0) = 0 (12) 54 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) and ∇z, ∇s and ∇ξ denote here the corresponding covariant derivatives with respect to the metric (7) on m2n. if u ∈ ωh satisfies (12), the curve γt0 in m2n has supp γt0 ⊂ lnt0 and a curve γ (−) t0 in lnt0 is hyperbolic and nondegenerate [3], then the operator ft0 (u) : t(ωh ) → t(ωh ) defined by (11) is a fredholm operator [12] between appropriate sobolev spaces. the corresponding pair (h,j) with j : m2n → end(t(m2n)) satisfying (7) is called regular [3] if every hyperbolic solution to (2) is nondegenerate [1, 3] and the operator ft0 (u) is onto for u ∈ ωh. in general one can prove that the space (h,j )reg ⊂ (h,j ) of regular pairs (h,j) ∈ (h,j ) is dense with respect to the c∞-topology. thus, for the regular pairs it follows from an implicit function theorem [1] that the space ωh (γ (−) t0 ) is indeed for any curve γt0 with supp γt0 ⊂ lnt0 a finite-dimensional compact functional submanifold whose local dimension near u ∈ ωh (γ (−) t0 ) is exactly the fredholm index of the operator ft0 (u). as a simple inference from the finite-dimensionality of the set ωh (γ (−) t0 ) and its compactness one gets that the corresponding point set l(−)t0 (h) is finite-dimensional and compact submanifold smoothly imbedded into m2n. the same is evidently true for the point manifolds l(+)t0 (h) and lt0 (h) supplying us with compact neighborhoods of the compact lagrangian submanifold lnt0 ⊂ m 2n. let us specify the structure of the manifold l(−)t0 (h) more exactly making use of the floer type analytical results [3, 8, 12] about the space of solutions to the problem (9) and (10). one has that for any two curves γ(−), γ : [0,τ] → lnt0 × s 1 satisfying the system (2), the following functional φ(τ )t0 (u) := 1 τ ∫ τ 0 ds ∫ r dz(|∂u/∂z|2 + |∂u/∂s − xh (u; s + t0)| 2) (13) if bounded satisfies the characteristic equality φ(τ )t0 (u) = a (τ ) t0 (γ(−)) − a(τ )t0 (γ) (14) for any τ ∈ r. thereby, in the case when the right hand side of (14) doesn’t vanish, the functional space ωh (γ(−)) will be a priori nontrivial. similarly, for any u ∈ l (+) t0 (h) one finds that φ(τ )t0 (u) = a (τ ) t0 (γ) − a(τ )t0 (γ (+)), (14a) where the corresponding curve γ(+)t0 : [0,τ] → m 2n satisfies the system (2), is hyperbolic having supp γ(+)t0 ⊂ l n t0 , and the curve γt0 : [0,τ] → m2n also satisfies the system (2) having supp γt0 ⊂ lnt0, and at last, for u ∈ lt0 (h) φ(τ )t0 (u) = a (τ ) t0 (γ(−)) − a(τ )t0 (γ (+)), (14b) where γ(±) : [0,τ] → m2n × s1, τ ∈ r, are taken to be strictly different, hyperbolic and having supp γ(±) ⊂ lnt0. the case when γ (+) t0 = γ(−)t0 needs some modification of the construction presented above on which we shall not dwell here. thus we have constructed the corresponding neighborhoods l(±)t0 (h) and lt0 (h) of the compact lagrangian submanifold lnt0 ⊂ m 2n consisting of all bounded solutions to the 7, 3(2005) the ergodic measures related with nonautonomous hamiltonian ... 55 corresponding equations (9), (10) and (10a,b). based now on this fact and the analytical expressions (14) and (14a,b) one derives the following important lemma. lemma 1.1. all neighborhoods l(±)t0 (h) and lt0 (h) constructed via the scheme presented above are compact and invariant with respect to the hamiltonian flow of diffeomorphisms ψs ∈diff(m2n) × s1, s ∈ r. let us consider below the case of the neighborhood lt0 (h) ⊂ m2n. the preceding characterization of the space of curves ωh leads us following mather’s approach [9] to another important for applications description of the compact neighborhood lt0 (h) by means of the space of normalized probability measures mt0 (h) := m(t(lt0 (h))× s) with compact support and invariant with respect to our hamiltonian ψ-flow of diffeomorphisms ψs ∈diff(m2n) × s1, s ∈ r, naturally extended on t(lt0 (h)) × s. the hamiltonian ψ-flow due to lemma 1.1 can be reduced invariantly upon the compact submanifold lt0 (h) × s ⊂ m2n × s. for the behavior of this reduced ψflow upon lt0 (h) × s to be studied in more detail let us assume that our extended hamiltonian ψ∗-flow on t(lt0 (h))× s is ergodic, that is the limτ→∞ a (τ ) t0 (γ) doesn’t depend on initial points (u0, u̇0; t0) ∈ t(lt0 (h)) × s. recall now that the basic result [13] in functional analysis (the riesz representation theorem) states that the set of borel probability measures on a compact metric space x is a subset of the dual space c(x)∗ of the banach space c(x) of continuous functions on x. it is obviously a convex set and it is well known [13] to be metrizable and compact with respect to the weak topology on c(x)∗ defined by c(x), also called the weak (∗)-topology. the restriction of this topology to the set of borel measures is frequently called the vague topology on measures [9]. since the space pt0 := t(lt0 (h))×s is metrizable and can be as well compactified, it follows that the set of borel probability measures on pt0 is a metrizable, compact and convex subset of the dual to the banach space of continuous functions on pt0. the corresponding set mt0 (h) is then evidently a compact, convex subset of this set. the well known result of the kryloff and bogoliuboff [14] states that any ψ-flow on a compact metric space x has an invariant probability measure. this result one can suitably adapt [9] to our metric compactified space pt0 := t(lt0 (h)) × s as follows. take a trajectory γ ∈ ωh of the extended ψ∗-flow on pt0 with supp γ ⊂ lt0 (h) × s defined on a time interval [0,τ] ⊂ r and let a measure µτ on t(lt0 (h))×s be evenly distributed along the orbit γ. then evidently ||ψs∗µτ −µτ|| ≤ 2s/τ for s ∈ [0,τ]. denote by µ a point of accumulation of the set {µτ : τ ∈ r+} as τ → ∞ with respect to the before mentioned vague topology. for any continuous function f ∈ c(pt0 ), any s ∈ r and any τ0,ε > 0 there exists τ > τ0 such that | ∫ pt0 f ◦ψs̄∗dµ− ∫ pt0 f ◦ψs̄∗dµτ| < ε for s̄ ∈ {0,s}. then it follows from the above estimations | ∫ pt0 f ◦ ψs∗dµ − ∫ pt0 fdµ| ≤ | ∫ pt0 f ◦ ψs∗dµ −∫ pt0 f ◦ ψs∗dµτ| + | ∫ pt0 f ◦ ψs∗dµτ − ∫ pt0 fdµτ| + | ∫ pt0 fdµτ −∫ pt0 fdµ| ≤ 2ε + ||f|| ||ψs∗µτ − µτ|| ≤ 2ε + 2s||f||/τ, 56 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) that is | ∫ pt0 f ◦ ψs∗dµ − ∫ pt0 fdµ| = 0 since ε > 0 can be taken arbitrarily small and τ0 > 0 arbitrarily large. thereby one sees that the constructed measure µ ∈ mt0 (h), that is it is normalized and invariant with respect to the extended hamiltonian ψ∗-flow on pt0. thus, in the case of ergodicity of the ψ∗-flow on t(lt0 (h)) × s the mentioned above limit lim τ→∞ a(τ )t0 (γ) = ∫ pt0 (α(1) − h)dµ, (15) with 1-form α(1) ∈ λ1(m2n) being considered above as a function α(1) : pt0→ r, since the submanifold lt0 (h) by construction is compact and invariantly imbedded into m2n due to lemma 1.1. so, it is natural to study properties of the functional at0 (µ) := ∫ pt0 (α(1) − h)dµ (16) on the space mt0 (h), where we omitted for brevity the natural pullback of the 1-form α(1) ∈ λ1(m2n) upon the invariant compact submanifold lt0 (h) ⊂ m2n. being interested namely in ergodic properties of ψ∗-orbits on t(lt0 (h))×s), we shall develop below an analog of the j. mather lagrangian measure homology technique [9, 10] to a more general and complicated case of the reduced hamiltonian ψ-flow on the invariant compact submanifold lt0 (h) ⊂ m2n. in particular, we shall construct an analog of the so called mather β-function [9] on the homology group h1(lt0 (h); r) whose linear domains generate exactly ergodic components of a measure µ ∈ mt0 (h) minimizing the functional (16), being of great importance for studying regularity properties of ψ∗-orbits on t(lt0 (h))×s. the results can be extended further to adiabatically perturbed integrable hamiltonian systems depending on a small parameter ε ↓ 0 via the continuous dependence h(t) := h̃(εt), where h̃(τ + 2π) = h̃(τ) for all τ ∈ [0, 2π]. it makes also possible to state the existence of so called adiabatic invariants with compact supports in lt0 (h) having many applications in mathematical physics and mechanics. some of the results can be also applied to investigating the problem of transversal intersections of corresponding stable and unstable manifolds to hyperbolic curves or singular points, related closely with existence of highly irregular motions in a periodic time-dependent hamiltonian dynamical system under regard. 2 invariant measures and mather’s type β-function before studying the average functional (16) on the measure space mt0 (h), let us first analyze properties of the functional∮ σ a(1) :=≺ a(1),σ � (17) on h1(lt0 (h); r) at a fixed σ ∈ h1(lt0 (h); r). since the 1-form a(1) ∈ h1(lt0 (h); r) in (17) can be considered as a function a(1) : pt0→ r, in virtue of the riesz theorem 7, 3(2005) the ergodic measures related with nonautonomous hamiltonian ... 57 [13] there exists a borel measure µ : pt0 → r+ (still not necessary ψ-invariant), such that ≺ a(1),σ �= ∫ pt0 a(1)dµ. (18) the following lemma characterizing the right hand side of (18) holds. lemma 2.1. let a 1-form a(1) = dλ(0) ∈ λ1(lt0 (h)) be exact, that is the cohomology class [dλ(0)] = 0 ∈ h1(lt0 (h); r). then for any µ ∈ mt0 (h)∮ σ a(1) = 0. (19) c really, for a(1) = dλ(0), where λ(0) : lt0 (h) → r is an absolutely continuous mapping, the following holds due to the fubini theorem for any τ ∈ r+ : | ∫ pt0 dλ(0)dµ.| = |1 τ ∫ τ 0 ds ∫ pt0 dλ(0)(ψs∗dµ)| = |1 τ ∫ pt0 dµ ∫ τ 0 dsd(λ(0) ◦ ψs∗)/ds| = |1 τ ∫ pt0 dµ[λ(0) ◦ ψτ∗ − λ(0) ◦ ψ0∗]| ≤ 2||λ(0)||/τ. (20) the latter inequality as τ → ∞ gives rise to the wanted equality (19), that proves the lemma.b thus, the right hand side of (18) defines a true functional h1(lt0 (h); r) 3 a (1) → ∫ pt0 a(1)dµ ∈ r (21) on the cohomology space h1(lt0 (h); r). all the above can be formulated as the following theorem. theorem 2.2. let an element σ ∈ h1(lt0 (h); r) be fixed. then there exists a ψinvariant probability measure (not unique) µ ∈ mt0 (h), such that the representation (18) holds and vice versa, for any measure µ ∈ mt0 (h) there exists the homology class σ := ρt0 (µ) ∈ h1(lt0 (h); r), such that ≺ a(1),ρt0 (µ) �= ∫ pt0 a(1)dµ (22) for all a(1) ∈ h1(lt0 (h); r). definition 2.3. ([10]) for any measure µ ∈ mt0 (h) the homology class ρt0 (µ) ∈ h1(lt0 (h); r) is called its homology. corollary 2.4. the homology mapping ρt0 : mt0 (h) → h1(lt0 (h); r) defined within theorem 2.2 is surjective. c sketch of a proof of theorem 2.2. the fact that for each µ ∈ mt0 (h) there exists the unique homology class σ := ρt0 (µ) ∈ h1(lt0 (h); r) is based on the well known poincare duality theorem [1]. the inverse statement is about the surjectivity of the mapping ρt0 : mt0 (h) → h1(lt0 (h); r). for it to be stated, consider following [8-10] a covering space lt0 (h) over lt0 (h) defined by the condition 58 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) that π1(lt0 (h)) = ker ht0, where ht0 : π1(lt0 (h)) → h1(lt0 (h); r) denotes the hurewicz homomorphism [10]. since in reality the functional (22) is defined on the covering space lt0 (h), it is necessary to lift all curves γ ∈ ωh on lt0 (h)×s to curves γ̃ ∈∈ ω̃h on lt0 (h)×s. in the case when the homotopy group π1(lt0 (h)) is abelian, the covering space l̃t0 (h) becomes universal, but in general it is obtained as some universal covering of l̃t0 (h) quotioned further with respect to the action of the kernel of the corresponding hurewicz homomorphism ht0 : π1(lt0 (h)) → h1(lt0 (h); r). take now any element σ ∈ h1(lt0 (h); r) and construct a set of approximating it so called deck transformations τ−1στ ∈ im ht0 ⊂ h1(lt0 (h); r), τ ∈ r+, such that weakly limτ→∞ τ−1στ = σ holds. put further x̃τ := στ ◦ x̃0 ∈ lt0 (h) × s, τ ∈ r+, where x̃0 ∈ lt0 (h) × s is taken arbitrary and consider such a curve γ̃ : [0,τ] → lt0 (h) × s with end-points γ̃(0) = x̃0, γ̃(τ) = x̃τ whose projection on lt0 (h)×s is the curve γ ∈ σt0 (h), minimizing the functional (6). consider also a set {µτ : τ ∈ r+} of probability measures on pt0 evenly distributed along corresponding curves γ ∈ σt0 (h) for each τ ∈ r+ and denote by µ a point of its accumulation as τ → ∞. due to the uniform distribution of measures µτ, τ ∈ r+, along curves γ ∈ σt0 (h) having the end-points agreed with chosen above deck transformations στ ∈ h1(lt0 (h); r), τ ∈ r+, one gets right away from the birkhoff-khinchin ergodic theorem [1, 2] that ∫ pt0 a(1)dµτ =≺ a(1),τ−1στ ) � (23) for any a(1) ∈ h1(lt0 (h); r). passing now to the limit in (23) as τ → ∞ and taking into account that weakly limτ→∞ τ−1στ = σ, one gets right away that the equality (22) holds for some measure µ ∈ mt0 (h), such that ρt0 (µ) = σ ∈ h1(lt0 (h); r), thereby giving rise to the surjectivity of the mapping ρt0 : mt0 (h) → h1(lt0 (h); r) and proving the theorem. b return now to treating the average functional (16) subject to the space of all invariant measures mt0 (h). namely, consider the following β-function βt0 : h1(lt0 (h); r) → r defined as βt0 (σ) := inf µ {at0 (µ) : ρt0 (µ) = σ ∈ h1(lt0 (h); r)} (24) it will be further called a mather type β-function due to its analogy to the definition given in [9,10]. the following lemma holds. lemma 2.5. let a 1-form a(1) ∈ h1(lt0 (h); r) be taken arbitrary. then the mather type β-function β (a) t0 (σ) := inf µ {a(a)t0 (µ) : ρt0 (µ) = σ ∈ h1(lt0 (h); r)}, (25) where by definition a(a)t0 (µ) := ∫ pt0 (α(1) + a(1) − h)dµ, (26) satisfies the following equation: β (a) t0 (σ) = βt0 (σ)+ ≺ a (1),σ) � . (27) 7, 3(2005) the ergodic measures related with nonautonomous hamiltonian ... 59 c the proof easily stems from the definition (25) and the equality (22). b assume now that the infimum in (24) is attained at a measure µ(σ) ∈ mt0 (h). then evidently, ρt0 (µ(σ)) = σ for any homology class σ ∈ h1(lt0 (h); r). denote by m(σ)t0 (h) the set of all minimizing the functional (24) measures of mt0 (h). in the next chapter we shall proceed on study its ergodic and homology properties. 3 ergodic measures and their homologies consider the introduced above mather type β-function β(a)t0 : h1(lt0 (h); r) → r for any a(1) ∈ h1(lt0 (h); r). it is evidently a convex function on h1(lt0 (h); r), that is for any λ1,λ2 ∈ [0, 1], λ1 + λ2 = 1, and σ1,σ2 ∈ h1(lt0 (h); r) there holds the inequality β (a) t0 (λ1σ1 + λ2σ2) ≤ λ1β (a) t0 (σ1) + λ2β (a) t0 (σ2). (28) as usually dealing with convex functions, one says that an element σ ∈ h1(lt0 (h); r) is extremal point [13] if β(a)t0 (λ1σ1 + λ2σ2) < λ1β (a) t0 (σ1) + λ2β (a) t0 (σ2) for all λ1,λ2 ∈ (0, 1), λ1 + λ2 = 1, and σ = λ1σ1 + λ2σ2. correspondingly, we shall call a convex set zt0 (h) ⊂ h1(lt0 (h); r) by a linear domain of the mather type function (25) if β (a) t0 (λ1σ1 + λ2σ2) = λ1β (a) t0 (σ1) + λ2β (a) t0 (σ2) (29) for any σ1,σ2 ∈ zt0 (h) and λ1,λ2 ∈ r. it is easy to see now that if σ ∈ h1(lt0 (h); r) is extremal, then the set m(σ)t0 (h) contains [15] ergodic minimizing measure components. namely, following [9, 10] one states that if zt0 (h) is a linear domain and p(σ)t0 ⊂ pt0 is the closure of the union of the supports of measures µ(σ) ∈ m (σ) t0 (h) with σ ∈ zt0 (h), then the set p (σ) t0 is compact and the inverse mapping (pt0|p(σ)t0 )−1 : pt0 (p (σ) t0 ) → p(σ)t0 is lipschitzian, where pt0 : pt0 → lt0 (h) × s is the standard projection, being injective upon p(σ)t0 . moreover, one can show [9] that if a measure µ ∈ m(σ)t0 (h) is minimizing the functional (26), then its support supp µ ⊂ p (σ) t0 and all its ergodic components {µ̄} are minimizing this functional too, and the convex hull of the corresponding homologies conv{ρt0 (µ̄)} is a linear domain z (σ) t0 (h) of the mather type β-function (25). these results are of very interest concerning many applications in dynamics. especially, the ergodic measures, as is well known, possess the crucial property that every invariant borel set has measure either 0 or 1, giving rise to the following important equality: lim τ→∞ a(τ )t0 (γ) = at0 (µ̄)) (30) uniformly on (γt0,(0), γ̇t0 (0); t0) ∈ pt0∩ supp µ̄, where γ ∈ σt0 (h). all of the properties formulated above are inferred from the following theorem modeling the similar one in [10]. theorem 3.1. let a measure µ ∈ mt0 (h) be minimizing the functional (26) satisfying the condition β(a)t0 (ρt0 (µ)) = at0 (µ). then supp µ ⊂ σt0 (h) and the convex 60 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) hull of the set of homologies ρt0 (µ̄) ∈ h1(lt0 (h); r), where {µ̄} ⊂ mt0 (h) are the corresponding ergodic components of the measure µ ∈ mt0 (h), is a linear domain zt0 (h) of the mather type β-function (25). c sketch of a proof. let ht0 : π1(lt0 (h)) → h1(lt0 (h); r) be the corresponding hurewicz homomorphism and take some basis σk ∈ im ht0 ⊂ h1(lt0 (h); r), k = 1,r, where r = dim im ht0, being its dual basis a (1) j ∈ h 1(lt0 (h); r), j = 1,r. then for any points x̃, ỹ ∈ lt0 (h) × s one can define an element ξ(τ )(x̃, ỹ|γ̃) ∈ h1(lt0 (h); r) as the sum ξ(τ )(x̃, ỹ|γ̃) := 1 τ r∑ j=1 σj ∫ τ 0 ã (1) j (γ̃), (31) where γ : [0,τ] → lt0 (h) × s is any continuous arc joining these two chosen points x̃, ỹ ∈ lt0 (h) × s, and ã (1) j ∈ h 1(lt0 (h); r) are the corresponding liftings to lt0 (h) of 1-forms a(1)j ∈ h 1(lt0 (h); r), j = 1,r. one can show then that if µ ∈ mt0 (h) is ergodic and supp µ ⊂ σt0 (h), then the measure µ is minimizing the functional (26). put σ := ρt0 (µ) and let a set zt0 (h) ⊂ h1(lt0 (h); r) be a supporting domain containing this homology class σ ∈ h1(lt0 (h); r). thus, one can see that the extremal points of the convex set zt0 (h) are extremal points also of the mather type β-function (25). next expand the homology class σ = ρt0 (µ) as a convex combination of extremal points σ̄j ∈ zt0 (h), j = 1,m, for some m ∈ z+. then, since elements σ̄j ∈ zt0 (h), j = 1,m, are extremal, there exist ergodic measures µ̄j ∈ m (σj ) t0 (h), j = 1,m, such that ρt0 (µ̄j ) = σ̄j, j = 1,m. moreover, since z (σ) t0 (h) is a linear domain, one easily brings about that β (a) t0 (σ) = m∑ j=1 cjβ (a) t0 (σ̄j ) = m∑ j=1 cja (a) t0 (µ̄j ), (32) where σ = ∑m j=1 cjσ̄j with some real coefficients cj ∈ r, j = 1,m. due to the ergodicity of the measure µ ∈ mt0 (h) from the birkhoff-khinchin ergodic theorem [1] one derives that there exists an orbit γ̃ : [0,τ} → lt0 (h) × s with the supp γ ⊂ supp µ, such that the property (30) together with the equality σ := ρt0 (µ) = lim τ→∞ ξ(τ )(x̃, ỹ|γ̃) (33) hold. further, there exist curves γ̃j ∈ σt0 (h), supp γj ⊂ supp µ̄j, j = 1,m, such the expressions σ̄j := ρt0 (µ̄j ) = lim τ→∞ ξ(τ )(x̃, ỹ|γ̃j ) (34) as well as β(a)t0 (σ̄j ) = a (a) t0 (µ̄j ) = limτ→∞ a (τ ) t0 (γ̃j ) hold for every j = 1,m. under the conditions (14b) involved on the invariant neighborhood lt0 (h) one shows that for any measure µ ∈ mt0 (h) such that ρt0 (µ) = σ, the inequality a (a) t0 (µ) ≤ β(a)t0 (ρt0 (µ)) holds thereby proving its minimality. suppose now that the measure µ ∈ mt0 (h) has all its ergodic components with supports contained in σt0 (h) and the convex 7, 3(2005) the ergodic measures related with nonautonomous hamiltonian ... 61 hull of its homologies is a linear domain of the mather type function (25). one can approximate (in the weak topology) a measure µ ∈ mt0 (h) by means of a convex combination µ̂ := ∑m j=1 ĉjµ̄j, where ĉj ∈ r and µ̄j ∈ mt0 (h), j = 1,m, are ergodic components of the measure µ ∈ mt0 (h). then supp µ̄j ⊂ σt0 (h) implying that all µ̄j ∈ mt0 (h), j = 1,m, are minimizing (26), that is are minimal. therefore, since the convex hull of homologies {ρt0 (µ̄j ) ∈ h1(lt0 (h); r) : j = 1,m} is a linear domain due to its minimality, one gets that a(a)t0 (µ̂) = ∑m j=1 ĉja (a) t0 (µ̄j ) = ∑m j=1 ĉjβ (a) t0 (ρt0 (µ̄j )) = β(a)t0 (ρt0 ( ∑m j=1 ĉjµ̄j )) = β (a) t0 (ρt0 (µ), (35) meaning evidently that the measure µ̂ ∈ mt0 (h) is minimal too. making use now of the fact that limits of minimizing measures are minimizing too, one obtains finally that the measure µ ∈ mt0 (h) is minimizing the functional (26), thereby proving the theorem. b consider some properties of a so called [10] supporting domain z (a) t0 (h) := {σ ∈ h1(lt0 (h); r) : β (a) t0 (σ) =≺ a(1),σ � +c(a)t0 } (36) for the mather type β-function (25) at some fixed a(1) ∈ h1(lt0 (h); r) with c (a) t0 ∈ r properly defined by (27). define also by p(a)t0 := ∪σ∈z(a)t0 (h) supp µ(σ), where µ(σ) ∈ mt0 (h) and ρt0 (µ(σ)) = σ ∈ z (a) t0 (h). present now a supporting domain z (a) t0 (h) ⊂ h1(lt0 (h); r) due to the expression (27) as follows: z (a) t0 (h) = {σ ∈ h1(lt0 (h); r) : β (0) t0 (σ) = c(a)t0 }, (37) where the function β(0)t0 : h1(lt0 (h); r) being bounded from below is chosen in such a way that β(0)t0 (σ) ≥ c (a) t0 for all σ ∈ h1(lt0 (h); r). take now a measure µ ∈ mt0 (h) and suppose that supp µ ⊂ σt0 (h). since β (0) t0 (σ) ≥ c(a)t0 for all σ ∈ h1(lt0 (h); r) and due to (37) z(a)t0 (h) = (β (0) t0 )−1{c(a)t0 } at some fixed a (1) ∈ h1(lt0 (h); r), this evidently implies that the measure µ ∈ mt0 (h) is minimizing the functional (26) and ρt0 (µ) ∈ z (a) t0 (h). thereby the following theorem is stated. theorem 3.2. suppose that z(a)t0 (h) ⊂ h1(lt0 (h); r) is a supporting domain of the mather type function (27) and a measure µ ∈ mt0 (h) satisfies the condition supp µ ⊂ σt0 (h). then this measure µ ∈ mt0 (h) is minimizing and ρt0 (µ) ∈ z (a) t0 (h). the following corollaries from the theorem 3.2 as in [10] hold. corollary 3.3. the minimizing measure µ ⊂ mt0 (h) with supp µ ⊂ σt0 (h) satisfies the condition a(0)t0 (µ) = c (a) t0 . by means of choosing the element a(1) ∈ h1(lt0 (h); r) one can make the value c (a) t0 be zero, that is one can put c(a)t0 = 0. corollary 3.4. for any strictly extremal closed curve σ ∈ h1(lt0 (h); r) the following properties take place: i) there exists an ergodic measure µ̄(σ) ∈ mt0 (h) whose support is a minimal set and ρt0 (µ̄(σ)) = σ; 62 d.l.blackmore, y.a.prykarpatsky, a.m.samoilenko & a.k.prykarpatsky 7, 3(2005) ii) for every closed 1-form a(1) ∈ h1(lt0 (h); r) the equality ≺ a(1),σ �= limτ→∞ 1τ ∫ t0+τ t0 a(1)(γ̇)ds holds uniformly for all (γt0(0), γ̇t0(0); t0) ∈ pt0∩ supp µ̄(σ), ρt0 (µ̄(σ)) = σ and γ ∈ σt0 (h); iii) if (γt0(0), γ̇t0(0); t0) ∈ pt0∩ supp µ̄(σ), ρt0 (µ̄(σ)) = σ and γ ∈ σt0 (h) is the corresponding orbit in lt0 (h) × s, then β (a) t0 (σ) = limτ→∞ a (τ ) t0 (γ) uniformly. the statements formulated above can be effectively used for studying dynamics of many perturbed integrable hamiltonian flows and their regularity properties. as it is well known, they are strongly based on the intersection theory of stable and unstable manifolds related with hyperbolic either closed orbits or singular points of a hamiltonian system under regard. these aspects of our study of ergodic measure and homology properties of such hamiltonian flows are supposed to be treated in a proceeding article under preparation. received: june 2004. revised: january 2005. references [1] abraham r. and marsden j., foundations of mechanics, cummings, ny, 1978. 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[15] niemycki v.v. and stepanov v.v., qualitive theory of differential equations, princeton, univ. press, 1960. cubo a mathematical journal vol.11, no¯ 03, (41–53). august 2009 boundedness and global attractivity of solutions for a system of nonlinear integral equations bo zhang department of mathematics and computer science, fayetteville state university, fayetteville, nc 28301-4298, u.s.a. email: bzhang@uncfsu.edu abstract it is well-known that liapunov’s direct method has been used very effectively for differential equations. the method has not, however, been used with much success on integral equations until recently. the reason for this lies in the fact that it is very difficult to relate the derivative of a scalar function to the unknown non-differentiable solution of an integral equation. in this paper, we construct a liapunov functional for a system of nonlinear integral equations. from that liapunov functional we are able to deduce conditions for boundedness and global attractivity of solutions. as in the case for differential equations, once the liapunov function is constructed, we can take full advantage of its simplicity in qualitative analysis. resumen es conocido que el método directo de liapunov ha sido usado de manera efectiva en ecuaciones diferenciales. sin embargo este método no ha sido utilizado con mucho suceso en ecuaciones intergrales hasta ahora. la razón para esto reside en el hecho que 42 bo zhang cubo 11, 3 (2009) es difícil relacionar la derivada de una función escalar a la solución no diferenciable desconocida. en este artículo, construimos un funcional de liapunov para un sistema de ecuaciones integrales no lineales. usando tal funcional de liapunov somos capazes de deducir acotamiento y atractividad global de soluciones. como en el caso de ecuaciones diferenciales, una vez que el funcional de liapunov es construido, aprovechamos su simplicidad en el análisis cualitativo. key words and phrases: boundedness, global attractivity, integral equations. math. subj. class.: 45d05, 45g15, 45g99. 1 introduction this paper is concerned with a system of nonlinear integral equations x(t) = h(t,x(t)) − ∫ t 0 d(t,s)g(s,x(s))ds (1.1) where x(t) ∈ rn, h : r+ × rn → rn, d : r+ × r+ → rn×n, g : r+ × rn → rn are continuous, and r+ = [0,∞). the theory of integral equations has grown tremendously in the past several decades. the growth has been strongly promoted by the advanced technology in scientific computation and the large number of applications to models in biology, economics, engineering, and other applied sciences. it is the qualitative behavior of solutions of these models that is especially important to many investigators. for the historical background, basic theory, and discussion of applications, we refer the reader to, for example, the work of corduneanu [1], burton [6], gripenberg et al [7], levin ([10]-[12]), levin and nohel [13], maccamy [15], miller [17], and references therein. there is substantial literature on (1.1) and much of it can be traced back to the pioneering work of levin and nohel ([10]-[14]) in the study of asymptotic behavior of solutions of the scalar integral equation x(t) = a(t) − ∫ t 0 d(t,s)g(x(s))ds (1.2) and the integro-differential equation x ′ (t) = a(t) − ∫ t 0 d(t,s)g(x(s))ds. (1.3) these equations arise in problems related to evolutionary processes in biology, in nuclear reactors, and in control theory (see corduneanu [1], burton [5], levin and nohel [13], kolmanovskii and cubo 11, 3 (2009) boundedness and global attractivity of solutions ... 43 myshkis [9]). it is often required that a ∈ c(r+,r) and d(t,s) ≥ 0, ds(t,s) ≥ 0, and dst(t,s) ≤ 0. (1.4) with g(x) = ∫ x 0 g(s)ds → ∞ as |x| → ∞. when d(t,s) = d(t − s) is of convolution type, (1.4) represents d(t) ≥ 0, d′(t) ≤ 0, and d′′(t) ≥ 0. (1.5) levin’s work is based on (1.5) and the construction of a liapunov functional for (1.3). the method was further extended into a long line of investigation drawing together such different notions of positivity as liapunov functions, completely monotonic functions, and kernels of positive type (see corduneanu [1], gripenberg et al [7], levin and nohel [14], maccamy and wong [16]). in a series papers ([2]-[4]), burton obtains results on boundedness and attractivity of solutions for a scalar equation in form of (1.1) without asking the growth condition on g. liapunov functionals play an essential role in his proofs. a good summary for recent development of the subject may be found in burton [6]. many investigators mentioned above frequently use the fact that (1.3) can be put into the form of (1.2) by integration. we observe that the differentiability of x(t) in (1.2) is not required. for example, we may convert a system of neutral differential equations, say d dt [x(t) − g(t,x(s))] = ax(t) + g(t,x(t)) (1.6) into a system of integral equations in the form of (1.1) x(t) = g̃(t,x(t)) + ∫ t 0 e a(t−s) ˜g(s,x(s))ds (1.7) with a view of proving the existence and qualitative behavior of solutions by applying fixed point theorems. note that the initial functions for the differential equations are absorbed into the term g̃(t,x(t)). equation (1.7) often describes actual models calling for time-dependent feedback. the integral term here may be viewed as the assumption that the future state of the process depends not only on the present, but also on the past history. the object of this paper is to give conditions to ensure that all solutions x = x(t) of (1.1) are bounded and converge to zero as t → ∞ by constructing a liapunov functional. from that liapunov functional we are able to deduce conditions for boundedness and global attractivity of solutions. this will be done in section 2 and 3, respectively. we generalize some classical results on boundedness and attractivity of solutions for (1.2) without asking a growth condition on g and obtain theorems for (1.1) that are parallel to those of burton [2] for scalar equations. we notice that it is very difficult to relate the derivative of a scalar function to the unknown solution of (1.1) since the solution may not be differentiable, and this presents a significant challenge to 44 bo zhang cubo 11, 3 (2009) investigators finding a suitable liapunov function for (1.1). even if the function was found, it might not be positive definite or decreasing along the solutions of (1.1). however, the liapunov functional still provides us with a great deal of information on solutions of (1.1), and therefore, we can derive certain properties of solutions without actually solving the equation. for x ∈ rn, |x| denotes the euclidean norm of x. let c(x,y ) denote the space of continuous functions φ : x → y . for an n × n matrix b = (bij )n×n, we denote the norm of b by ‖b‖ = sup{|bx| : |x| ≤ 1}. if b is symmetric, we use the convention for self-adjoint positive operators to write b ≥ 0 whenever b is positive semi-definite. similarly, if b is negative semi-definite, then b ≤ 0. also, if b ≥ 0, we denote its square root by √ b. 2 boundedness in this section, we study the boundedness of solutions of (1.1). we shall focus on a priori bounds of solutions. an important application of an a priori bound is to establish the existence of a solution of (1.1) on r+. a well-known procedure for proving global existence of solutions calls for a local existence theorem, a continuation argument, and an a priori bound (see levin [12]). a continuous function x : r+ → rn is called a solution of (1.1) on r+ if it satisfies (1.1) on r +. it is to be understood that x(0) = h(0,x(0)). if x(t) is specified to be a certain initial function on an initial interval, say x(t) = φ(t) for 0 ≤ t ≤ t0, we are then looking for a solution of x(t) = h(t,x(t)) − ∫ t0 0 d(t,s)g(s,φ(s))ds − ∫ t t0 d(t,s)g(s,x(s))ds, t ≥ t0. (2.1) however, a change of variable y(t) = x(t + t0) will reduce the problem back to one of form (1.1). thus, the initial function on [0, t0] is absorbed into the forcing function, and hence, it suffices to consider (1.1) with the simple initial condition x(0) = h(0,x(0)). we shall prove the boundedness of solutions of system (1.1) by constructing a liapunov functional which has its roots in burton [2] and kemp [8]. the results here generalize the theorems in burton [2] for scalar equations. we require that (h1) d(t,s) is a symmetric matrix with d(t, 0) ≥ 0, ds(t,s) ≥ 0, dt(t, 0) ≤ 0, and dst(t,s) ≤ 0 with ds(t,s) and dst(t,s) continuous for all t ≥ s ≥ 0. (h2) g(t,x) is bounded for x bounded and there exists a constant k > 0 such that g t (t,x)[x − h(t,x)] ≥ |g(t,x)| for all |x| ≥ k and t ≥ 0 (2.2) cubo 11, 3 (2009) boundedness and global attractivity of solutions ... 45 where gt is the transpose of g. (h3) there exists a continuous function ψ : r + → r+ such that |h(t,x)| ≤ ψ(|x|) for all t ≥ 0 and x ∈ rn with limr→∞(r − ψ(r)) = ∞. (h4) sup t≥0 [ ‖d(t, 0)‖t2 + ∫ t 0 ‖ds(t,s)‖ ( 1 + (t − s)2 ) ds ] < ∞. we observe that some of these conditions are interconnected. for example, (h3) nearly implies (h2) if x t g(t,x) ≥ 0 for |x| ≥ k. in many applications, h(t,x) is a nonlinear contraction in which ψ in (h3) is a nondecreasing function with ψ(r) < r for all r > 0. theorem 2.1. suppose that (h1) (h4) hold. then every solution of (1.1) on r + is bounded. proof. we first define some constants to simplify notations. let sup t≥0 ∫ t 0 ‖ds(t,s)‖(t − s)2ds = j2 (2.3) and define sup t≥0 ∫ t 0 ‖ds(t,s)‖ds = j1 and sup t≥0 ‖d(t, 0)‖t2 = d2. (2.4) we now let x = x(t) be a solution of (1.1) on r+ and define a liapunov functional v (t,x(·)) = ∫ t 0 ( ∫ t s g(v,x(v))dv )t ds(t,s) ( ∫ t s g(v,x(v))dv ) ds + ( ∫ t 0 g(v,x(v))dv )t d(t, 0) ∫ t 0 g(v,x(v))dv. (2.5) differentiate v (t,x(·)) with respect to t to obtain v ′ (t,x(·)) = ∫ t 0 ( ∫ t s g(v,x(v))dv )t dst(t,s) ( ∫ t s g(v,x(v))dv ) ds + 2 g t (t,x(t)) ∫ t 0 ds(t,s) ( ∫ t s g(v,x(v))dv ) ds + ( ∫ t 0 g(v,x(v))dv )t dt(t, 0) ∫ t 0 g(v,x(v))dv + 2 g t (t,x(t)) d(t, 0) ∫ t 0 g(v,x(v))dv. (2.6) 46 bo zhang cubo 11, 3 (2009) we integrate the third to last term by parts to obtain 2 g t (t,x(t)) [ d(t,s) ∫ t s g(v,x(v))dv ∣ ∣ ∣ s=t s=0 + ∫ t 0 d(t,s)g(s,x(s))ds ] = 2 g t (t,x(t)) [ −d(t, 0) ∫ t 0 g(s,x(s))ds + ∫ t 0 d(t,s)g(s,x(s))ds ] . cancel terms, use the sign conditions, and use (1.1) in the last step of the process to unite the liapunov functional and the equation obtaining v ′ (t,x(·)) = ∫ t 0 ( ∫ t s g(v,x(v))dv )t dst(t,s) ( ∫ t s g(v,x(v))dv ) ds + ( ∫ t 0 g(v,x(v))dv )t dt(t, 0) ∫ t 0 g(v,x(v))dv + 2g t (t,x(t)) ∫ t 0 d(t,s)g(s,x(s))ds ≤ 2 gt (t,x(t)) ∫ t 0 d(t,s)g(s,x(s))ds = 2 g t (t,x(t))[−x(t) + h(t,x(t))]. (2.7) by (h2), we see that if |x(t)| ≥ k, then v ′ (t,x(·)) ≤ −|g(t,x(t))| (2.8) it is clear that v ′(t,x(·)) is bounded above for 0 ≤ |x(t)| ≤ k since g(t,x) is bounded for x bounded and |h(t,x)| ≤ ψ(|x|), and hence, there exists a constant l > 0 depending on k such that v ′ (t,x(·)) ≤ −|g(t,x(t))| + l (2.9) by the schwarz inequality, we have ∣ ∣ ∣ ∣ ∫ t 0 ds(t,s) ∫ t s g(v,x(v))dvds ∣ ∣ ∣ ∣ 2 = ∣ ∣ ∣ ∣ ∫ t 0 √ ds(t,s) [ √ ds(t,s) ∫ t s g(v,x(v))dv ] ds ∣ ∣ ∣ ∣ 2 ≤ ∫ t 0 ‖ √ ds(t,s)‖2ds ∫ t 0 ∣ ∣ ∣ ∣ √ ds(t,s) ∫ t s g(v,x(v))dv ∣ ∣ ∣ ∣ 2 ds = ∫ t 0 ‖ √ ds(t,s)‖2ds ∫ t 0 ( ∫ t s g(v,x(v))dv )t ds(t,s) ( ∫ t s g(v,x(v))dv ) ds ≤ ∫ t 0 ‖ds(t,s)‖ds v (t,x(·)) ≤ j1v (t,x(·)) (2.10) cubo 11, 3 (2009) boundedness and global attractivity of solutions ... 47 where j1 is defined in (2.4). we have just integrated the left-hand side by parts, obtaining ∣ ∣ ∣ ∣ ∫ t 0 ds(t,s) ∫ t s g(v,x(v))dvds ∣ ∣ ∣ ∣ 2 = ∣ ∣ ∣ ∣ −d(t, 0) ∫ t 0 g(s,x(s))ds + ∫ t 0 d(t,s)g(s,x(s))ds ∣ ∣ ∣ ∣ 2 so that by (2.10) and (1.1) we now have j1v (t,x(·)) ≥ ∣ ∣ ∣ ∣ ∫ t 0 ds(t,s) ∫ t s g(v,x(v))dvds ∣ ∣ ∣ ∣ 2 = ∣ ∣ ∣ ∣ −d(t, 0) ∫ t 0 g(s,x(s))ds + ∫ t 0 d(t,s)g(s,x(s))ds ∣ ∣ ∣ ∣ 2 = ∣ ∣ ∣ ∣ x(t) − h(t,x(t)) + d(t, 0) ∫ t 0 g(s,x(s))ds ∣ ∣ ∣ ∣ 2 ≥ 1 2 |x(t) − h(t,x(t))|2 − ∣ ∣ ∣ ∣ d(t, 0) ∫ t 0 g(s,x(s))ds ∣ ∣ ∣ ∣ 2 ≥ 1 2 |x(t) − h(t,x(t))|2 − ‖d(t, 0)‖ ∣ ∣ ∣ ∣ √ d(t, 0) ∫ t 0 g(s,x(s))ds ∣ ∣ ∣ ∣ 2 ≥ 1 2 |x(t) − h(t,x(t))|2 − d1v (t,x(·)) where d1 = supt≥0 ‖d(t, 0)‖. here we have used the inequality 2(a2 + b2) ≥ (a + b)2. it is now clear that |x(t) − h(t,x(t))|2 ≤ 2(j1 + d1)v (t,x(·)). (2.11) we now show that v (t,x(·)) is bounded. if v (t,x(·)) is not bounded, then there exists a sequence {tn} ↑ ∞ with v (tn,x(·)) ≥ v (s,x(·)) for 0 ≤ s ≤ tn. it then follows from (2.9) that 0 ≤ v (tn,x(·)) − v (s,x(·)) ≤ − ∫ tn s |g(v,x(v))|dv + l(tn − s). this implies ∫ tn s |g(v,x(v))|dv ≤ l(tn − s). (2.12) 48 bo zhang cubo 11, 3 (2009) substitute (2.12) into v (tn,x(·)) to obtain v (tn,x(·)) ≤ ∫ tn 0 ‖ds(tn,s)‖ ∣ ∣ ∣ ∣ ∫ tn s g(v,x(v))dv ∣ ∣ ∣ ∣ 2 ds + ‖d(tn, 0)‖ ∣ ∣ ∣ ∣ ∫ tn 0 g(v,x(v))dv ∣ ∣ ∣ ∣ 2 ≤ ∫ tn 0 ‖ds(tn,s)‖ [ l 2 (tn − s)2 ] ds + ‖d(tn, 0)‖l2(tn)2 ≤ (j2 + d2)l2. this implies that v (t,x(·)) ≤ (j2 + d2)l2 for all t ≥ 0, and therefore, by (2.11) we have |x(t) − h(t,x(t))|2 ≤ 2(j1 + d1)(j2 + d2)l2. (2.13) observe that |x(t) − h(t,x(t))| ≥ |x(t)| − ψ(|x(t)|). (2.14) since r −ψ(r) → ∞ as r → ∞ by (h3), there exists a constant b > 0 such that r ≥ b implies r − ψ(r) > 0 and [r − ψ(r)]2 > 2(j1 + d1)(j2 + d2)l2. (2.15) we now claim that |x(t)| < b for all t ≥ 0. suppose there exists t∗ ≥ 0 with |x(t∗)| ≥ b. then by (2.13)-(2.15), we have 2(j1 + d1)(j2 + d2)l 2 < [|x(t∗)| − ψ(|x(t∗)|)]2 ≤ |x(t∗) − h(t∗,x(t∗))|2 ≤ 2(j1 + d1)(j2 + d2)l2 a contradiction, and thus, |x(t)| < b for all t ≥ 0, whenever x is a solution of (1.1) on r+. this completes the proof. 3 attractivity in this section, we study the global attractivity of solutions of (1.1). we shall show that every solution of (1.1) on r+ tends to zero as t → ∞ regardless of its initial condition. we may view h(t,x) = u(t,x),u ∈ g, as a perturbation term (or control) of the system where g is a pre-described class of functions. the project is to characterize g so that the stability property (x(t) → 0 as t → ∞) is independent of the special choice of u ∈ g. to arrive at these conclusions, we assume that (p1) there exists a function ψ̃ ∈ c(r+,r+) with ψ̃(0) = 0 and ψ̃(r) > 0 for r > 0 such that g t (t,x)[ x − h(t,x) + h(t, 0) ] ≥ |g(t,x)| ψ̃(|x|) for all t ≥ 0, x ∈ r, cubo 11, 3 (2009) boundedness and global attractivity of solutions ... 49 (p2) for each µ > 0 and α > 0, there exists β > 0 such that |x| ≤ µ implies |g(t,x)| ≤ α + β ψ̃(|x|)) for all t ≥ 0, (p3) h(t, 0) → 0 as t → ∞, (p4) ‖d(t, 0)‖ t2 → 0 as t → ∞, (p5) ∫ p 0 ‖ds(t,s)‖(t − s)2ds → 0 as t → ∞ for each fixed p > 0, one may notice from (p1) and (p2) that g(t,x) is almost independent of h(t,x) and it can be highly nonlinear. we will discuss these conditions in details later after presenting the main theorem of this section. theorem 3.1. suppose that (h1) (h4) and (p1) (p5) hold. then every solution of (1.1) on r + tends to zero as t → ∞. proof. let x = x(t) be a solution of (1.1) on r+. since (h1) (h4) hold, by theorem 2.1, all solutions of (1.1) on r+ are bounded. there exists a constant µ > 0 such that |x(t)| ≤ µ for all t ≥ 0. for this fixed solution, let v (t,x(·)) be defined in (2.5). then by (2.7), we have v ′ (t,x(·))) ≤ 2 gt (t,x(t))[−x(t) + h(t,x(t))] = −2 gt (t,x(t))[x(t) − h(t,x(t)) + h(t, 0)] + 2 gt (t,x(t))h(t, 0) ≤ −2 |g(t,x(t))| ψ̃(|x(t)|) + mµ(t) (3.1) where mµ(t) = sup{ 2 g∗µ|h(s, 0)| : s ≥ t } with g∗µ = sup{ |g(t,x)| : t ∈ r+, |x| ≤ µ }. note that mµ(t) is decreasing and converges to zero as t → ∞ by (p3). we first claim that v (t,x(·)) → 0 as t → ∞. observe that v (t,x(·)) is bounded since x is bounded. now let lim sup t→∞ v (t,x(·)) = p ≥ 0. then for any ε > 0, there exists a positive constant k > 0 and a sequence {tn} ↑ ∞ with v (tn,x(·)) ≥ v (s,x(·)) − ε for k ≤ s ≤ tn. (3.2) in fact, by the definition of lim supt→∞ v (t,x(·)), for any ε > 0, there exists k > 0 such that t ≥ k implies −ε 2 p − ε 2 = ( p + ε 2 ) − ε > v (s,x(·)) − ε for all k ≤ s ≤ tn and for n = 1, 2, · · ·. by (3.1) and (3.2), we now see that −ε ≤ v (tn,x(·)) − v (s,x(·)) ≤ − ∫ tn s |g(s,x(v))| ψ̃(|x(v)|)dv + mµ(k)(tn − s) or ∫ tn s |g(s,x(v))| ψ̃(|x(v)|)dv ≤ ε + mµ(k)(tn − s) (3.3) cubo 11, 3 (2009) boundedness and global attractivity of solutions ... 51 for all k ≤ s ≤ tn. apply (p2) and (3.3) in the following argument to obtain v (tn,x(·)) ≤ ∫ k 0 ‖ds(tn,s)‖ ∣ ∣ ∣ ∣ ∫ tn s g(v,x(v))dv ∣ ∣ ∣ ∣ 2 ds + ∫ tn k ‖ds(tn,s)‖ ∣ ∣ ∣ ∣ ∫ tn s g(v,x(v))dv ∣ ∣ ∣ ∣ 2 ds + ‖d(tn, 0)‖ ∣ ∣ ∣ ∣ ∫ tn 0 g(v,x(v))dv ∣ ∣ ∣ ∣ 2 ≤ (g∗µ)2 ∫ k 0 ‖ds(tn,s)‖(tn−s)2ds + (g∗µ)2 ‖d(tn, 0)‖t2n + ∫ tn k ‖ds(tn,s)‖ [ (tn−s) ∫ tn s |g(v,x(v))|2dv ] ds ≤ (g∗µ)2 ∫ k 0 ‖ds(tn,s)‖(tn−s)2ds + (g∗µ)2 ‖d(tn, 0)‖t2n + ∫ tn k ‖ds(tn,s)‖ { (tn−s) ∫ tn s |g(v,x(v))| [ α + βψ̃(|x(v)|) ] dv } ds ≤ (g∗µ)2 { ∫ k 0 ‖ds(tn,s)‖(tn−s)2ds + ‖d(tn, 0)‖t2n } + αg ∗ µ ∫ tn 0 ‖ds(tn,s)‖(tn−s)2ds + β ∫ tn k ‖ds(tn,s)‖(tn−s) [ ∫ tn s |g(v,x(v))| ψ̃(|x(v)|)dv ] ds ≤ (g∗µ)2 { ∫ k 0 ‖ds(tn,s)‖(tn−s)2ds + ‖d(tn, 0)‖t2n } + αg ∗ µj2 + β ∫ tn k ‖ds(tn,s)‖(tn−s)[ε + mµ(k)(tn − s)]ds ≤ (g∗µ)2 { ∫ k 0 ‖ds(tn,s)‖(tn−s)2ds + ‖d(tn, 0)‖t2n } + αg ∗ µj2 + εβ(j1 + j2) + mµ(k)βj2 (3.4) where j1 and j2 are defined in (2.4) and (2.3), respectively. now, for a given δ > 0, choose k > 0 so large, ε > 0 and α > 0 so small that αj2 g ∗ µ + εβ(j1 + j2) < δ, and mµ(k)βj2 < δ. since ∫ k 0 ‖ds(tn,s)‖(tn−s)2ds + ‖d(tn, 0)‖t2n → 0 as n → ∞ by (p4) and (p5), so that as δ → 0, we see that v (tn,x(·)) → 0 as n → ∞. this implies that p = 0, and therefore, v (t,x(·)) → 0 as t → ∞. we now show that x(t) → 0 as t → ∞. observe (p1) implies that |x − h(t,x) + h(t, 0)| ≥ ψ̃(|x|). 52 bo zhang cubo 11, 3 (2009) it then follows from (2.11) that 2(j1 + d1)v (t,x(·)) ≥ |x(t) − h(t,x(t))|2 = |x(t) − h(t,x(t)) + h(t, 0) − h(t, 0)|2 ≥ 1 2 |x(t) − h(t,x(t)) + h(t, 0)]2 − |h(t, 0)|2 ≥ 1 2 ψ̃ 2 (|x(t)|) − |h(t, 0)|2. this yields that ψ̃ 2 (|x(t)|) ≤ 4(j1 + d1)v (t,x(·)) + 2 |h(t, 0)|2 → 0 as t → ∞. (3.5) by (p1), we see that x(t) → 0 as t → ∞. this completes the proof. remark 3.1. we point out that (p1) and (p2) are quite mild conditions which allow g(t,x) to be highly nonlinear and nearly independent of h(t,x). note also that (p5) is a fading memory condition of the integral ∫ t 0 ‖ds(t,s)‖(t − s)2ds. if d(t,s) = d(t − s) is of convolution type, then (h4) implies (p5). example 3.1. let g(t,x) = (x3 1 ,x 3 2 ) t for x = (x1,x2) t ∈ r2 , and let h(t,x) be continuous satisfying |h(t,x) − h(t, 0)| ≤ φ(|x|)|x| (3.6) where φ ∈ c(r+,r+) with φ(r) < 1/2 . then (p1) and (p2) hold. proof. we will use the following inequalities for x = (x1,x2) t |g(t,x)| = √ |x1|6 + |x2|6 ≤ |x1|3 + |x2|3 (3.7) and ( |x1|3 + |x2|3 ) |x| ≤ ( |x1|3 + |x2|3 ) (|x1| + |x2|) ≤ 2 ( |x1|4 + |x2|4 ) . (3.8) cubo 11, 3 (2009) boundedness and global attractivity of solutions ... 53 note that we are not seeking the best estimate here. use these inequalities to obtain g t (t,x)[x − h(t,x) + h(t, 0)] = x t g(t,x) − gt (t,x)[h(t,x) − h(t, 0)] ≥ ( |x1|4 + |x2|4 ) − |g(t,x)||h(t,x) − h(t, 0)| ≥ ( |x1|4 + |x2|4 ) − ( |x1|3 + |x2|3 ) φ(|x|)|x| ≥ ( |x1|3 + |x2|3 ) [ 1 2 |x| − φ(|x|)|x| ] ≥ |g(t,x)|ψ̃(|x|) where ψ̃(r) = 1 2 r[1 − 2φ(r)]. thus, (p1) holds. to prove (p2), let µ > 0 and α > 0 be given. if |x| ≤ µ, then |g(t,x)| ≤ |x1|3 + |x2|3 ≤ |x|2µ ≤ α + ( µ 2 α ) |x|4 = α + ( µ 2 α ) 2 |x|3 1 − 2 φ(|x|) ψ̃(|x|)) ≤ α + β ψ̃(|x|)) where β = 2(µ2/α)µ3/φ∗µ with φ ∗ µ = inf { 1 − 2φ(r) : 0 ≤ r ≤ µ }. thus, (p2) holds and the proof is complete. received: october 12, 2008. revised: october 31, 2008. references [1] corduneanu, c., integral equations and stability of feedback systems, academic press, orlando, florida, 1973. 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[17] miller, r.k., nonlinear volterra integral equations, benjamin, menlo park, ca, 1971. 08-boundedness a mathematical journal vol. 7, no 2, (223 236). august 2005. a survey on the oscillation of solutions of first order delay difference equations l. k. kikina 1 department of mathematics, university of gjirokastra gjirokaster, albania i.p. stavroulakis 1 department of mathematics, university of ioannina 451 10 ioannina, greece ipstav@cc.uoi.gr abstract in this paper, a survey of the most interesting results on the oscillation of all solutions of the first order delay difference equation of the form xn+1 − xn + pnxn−k = 0, n = 0, 1, 2, ..., where {pn} is a sequence of nonnegative real numbers and k is a positive integer is presented, especially in the case when neither of the well-known oscillation conditions lim sup n→∞ n� i=n−k pi > 1 and lim inf n→∞ 1 k n−1� i=n−k pi > kk (k + 1)k+1 is satisfied. resumen en este art́ıculo, hacemos una revisión de los resultados más interesantes sobre oscilaciones de las soluciones de la ecuación en diferencias de primer orden 1the authors would like to express many thanks to professor yuri domshlak for useful discussions concerning this paper. also many thanks to the referee for some helpful comments. 224 l. k. kikina and i.p. stavroulakis 7, 2(2005) con retardo, de la forma xn+1 − xn + pnxn−k = 0, n = 0, 1, 2, ..., en donde {pn} es una sucesión de números reales no negativos, k es un entero positivo, en especial cuando ni siquiera se satisfacen las conocidas condiciones de oscilación lim sup n→∞ n� i=n−k pi > 1 y lim inf n→∞ 1 k n−1� i=n−k pi > kk (k + 1)k+1 key words and phrases: oscillation, nonoscillation, delay difference equation. math. subj. class.: 39a 11. 1 introduction in the last few decades the oscillation theory of delay differential equations has been extensively developed. the oscillation theory of discrete analogues of delay differential equations has also attracted growing attention in the recent few years. the reader is referred to [1-12,14-16, 21, 22, 24-26, 29-46] and the references cited therein. in particular, the problem of establishing sufficient conditions for the oscillation of all solutions of the delay difference equation xn+1 −xn + pnxn−k = 0, n = 0, 1, 2, ..., (1.1) where {pn} is a sequence of nonnegative real numbers and k is a positive integer, has been the subject of many recent investigations. see, for example, [2-12, 14, 21, 22, 24-26, 29-39, 42-46] and the references cited therein. strong interest in (1.1) is motivated by the fact that it represents a discrete analogue of the delay differential equation (see [13, 17-20, 23, 27, 28] and the references cited therein) x′(t) + p(t)x(t− τ) = 0, p(t) ≥ 0, τ > 0. (1.2) by a solution of (1.1) we mean a sequence {xn} which is defined for n ≥−k and which satisfies (1.1) for n ≥ 0. a solution {xn} of (1.1) is said to be oscillatory if the terms xn of the solution are not eventually positive or eventually negative. otherwise the solution is called nonoscillatory. for convenience, we will assume that inequalities about values of sequences are satisfied eventually for all large n. in this paper, our main purpose is to present the state of the art on the oscillation of solutions to (1.1) especially in the case that the oscillation conditions (see below) lim sup n→∞ n∑ i=n−k pi > 1 and lim inf n→∞ 1 k n−1∑ i=n−k pi > kk (k + 1)k+1 are not satisfied. 7, 2(2005) a survey on the oscillation of solutions of first order ... 225 2 oscillation criteria for eq. (1.1) in 1981, domshlak [7] was the first who studied this problem in the case where k = 1. then, in 1989, erbe and zhang [14] established the following oscillation criteria for (1.1). theorem 2.1.([14]) assume that β := lim inf n→∞ pn > 0 and lim sup n→∞ pn > 1 −β (c1) then all solutions of (1.1) oscillate. theorem 2.2.([14]) assume that lim inf n→∞ pn > kk (k + 1)k+1 . (c2) then all solutions of (1.1) oscillate. theorem 2.3.([14]) assume that a := lim sup n→∞ n∑ i=n−k pi > 1. (c3) then all solutions of (1.1) oscillate. in the same year 1989 ladas, philos and sficas [22] proved the following theorem. theorem 2.4.([22]) assume that lim inf n→∞ 1 k n−1∑ i=n−k pi > kk (k + 1)k+1 . (c4) then all solutions of (1.1) oscillate. therefore they improved the condition (c2) by replacing the pn of (c2) by the arithmetic mean of the terms pn−k, ..., pn−1 in (c4). concerning the constant k k (k+1)k+1 in (c2) and (c4) it should be empasized that, as it is shown in [14], if sup pn < kk (k + 1)k+1 , (n1) then (1.1) has a nonoscillatory solution. in 1990, ladas [21] conjectured that eq. (1.1) has a nonoscillatory solution if 1 k n−1∑ i=n−k pi ≤ kk (k + 1)k+1 holds eventually. however this conjecture is false and a counterexample was given in 1994 by yu, zhang and wang [43]. 226 l. k. kikina and i.p. stavroulakis 7, 2(2005) it is interesting to establish sufficient conditions for the oscillation of all solutions of (1.1) when (c3) and (c4) are not satisfied. (for the equation (1.2) this question has been investigated by many authors, see, for example, [13, 17-20, 23, 27, 28] and the references cited therein.) in 1993, yu, zhang and qian [42] and lalli and zhang [24], trying to improve (c3), established the following (false) sufficient oscillation conditions for (1.1) 0 < α := lim inf n→∞ n−1∑ i=n−k pi ≤ ( k k + 1 )k+1 and a > 1 − α 2 4 (f1) and n∑ i=n−k pi ≥ d > 0 for large n and a > 1 − d4 8 ( 1 − d 3 4 + √ 1 − d 3 2 )−1 (f2) respectively. unfortunately, the above conditions (f1) and (f2) are not correct. this is due to the fact that they are based on the following (false) discrete version of koplatadzechanturia lemma. (see [6] and [10]). lemma a (false). assume that {xn} is an eventually positive solution of (1.1) and that n∑ i=n−k pi ≥ m > 0 for large n. (1.3) then xn > m2 4 xn−k for large n. as one can see, the erroneous proof of lemma a is based on the following (false) statement. (see [6] and [10]). statement a (false). if (1.3) holds, then for any large n, there exists a positive integer n such that n−k ≤ n ≤ n and n∑ i=n−k pi ≥ m 2 , n∑ i=n pi ≥ m 2 . it is obvious that all the oscillation results which have made use of the above lemma a or statement a are incorrect. for details on this problem see the paper by cheng and zhang [6]. here it should be pointed out that the following statement (see [22], [31]) is correct and it should not be confused with the statement a. statement 2.1.([22], [31]) if n−1∑ i=n−k pi ≥ m > 0 for large n, (1.4) 7, 2(2005) a survey on the oscillation of solutions of first order ... 227 then for any large n, there exists a positive integer n∗ with n−k ≤ n∗ ≤ n such that n∗∑ i=n−k pi ≥ m 2 , n∑ i=n∗ pi ≥ m 2 . in 1995, stavroulakis [31], based on this correct statement 2.1, proved the following theorem. theorem 2.5.([31]) assume that 0 < α ≤ ( k k + 1 )k+1 and lim sup n→∞ pn > 1 − α2 4 . (c5) then all solutions of (1.1) oscillate. in 1999, domshlak [10] and in 2000, cheng and zhang [6] established the following lemmas, respectively, which may be looked upon as (exact) discrete versions of koplatadze-chanturia lemma. lemma 2.1.([10]) assume that {xn} is an eventually positive solution of (1.1) and that the condition (1.4) holds. then xn > m2 4 xn−k for large n. (1.5) lemma 2.2.([6]) assume that {xn} is an eventually positive solution of (1.1) and that the condition (1.4) holds. then xn > m kxn−k for large n. (1.6) based on these lemmas the following theorem was established in [32]. theorem 2.6.([32]) assume that 0 < α ≤ ( k k + 1 )k+1 . then either one of the conditions lim sup n→∞ n−1∑ i=n−k pi > 1 − α2 4 (c6) or lim sup n→∞ n−1∑ i=n−k pi > 1 −αk (c7) 228 l. k. kikina and i.p. stavroulakis 7, 2(2005) implies that all solutions of (1.1) oscillate. remark 2.1.([32]) from the above theorem it is now clear that 0 < α := lim inf n→∞ n−1∑ i=n−k pi ≤ ( k k + 1 )k+1 and lim sup n→∞ n−1∑ i=n−k pi > 1 − α2 4 is the correct oscillation condition by which the (false) condition (f1) should be replaced. remark 2.2.([32]) observe the following: (i) when k = 1, 2, αk > α2 4 , (since, from the above mentioned conditions, it makes sense to investigate the case when α < ( k k+1 )k+1 ) and therefore condition (c6) implies (c7). (ii) when k = 3, α3 > α2 4 when α > 1 4 while α3 < α2 4 when α < 1 4 . so in this case the conditions (c6) and (c7) are independent. (iii) when k ≥ 4, αk < α2 4 , and therefore condition (c7) implies (c6). (iv) when k < 12 condition (c6) or (c7) implies (c3). (v) when k ≥ 12 condition (c6) may hold but condition (c3) may not hold. we illustrate these by the following examples. example 2.1.([32]) consider the equation xn+1 −xn + pnxn−3 = 0, n = 0, 1, 2, ..., where p2n = 1 10 , p2n+1 = 1 10 + 64 95 sin2 nπ 2 , n = 0, 1, 2, .... here k = 3 and it is easy to see that α = lim inf n→∞ n−1∑ i=n−3 pi = 3 10 < ( 3 4 )4 and lim sup n→∞ n−1∑ i=n−3 pi = 3 10 + 64 95 > 1 −α3. 7, 2(2005) a survey on the oscillation of solutions of first order ... 229 thus condition (c7) is satisfied and therefore all solutions oscillate. observe, however, that condition (c6) is not satisfied. if, on the other hand, in the above equation p2n = 8 100 , p2n+1 = 8 100 + 746 1000 sin2 nπ 2 , n = 0, 1, 2, ..., then it is easy to see that α = lim inf n→∞ n−1∑ i=n−3 pi = 24 100 < ( 3 4 )4 and lim sup n→∞ n−1∑ i=n−3 pi = 24 100 + 746 1000 > 1 − α 2 4 . in this case condition (c6) is satisfied and therefore all solutions oscillate. observe, however, that condition (c7) is not satisfied. example 2.2.([32]) consider the equation xn+1 −xn + pnxn−16 = 0, n = 0, 1, 2, ..., where p17n = p17n+1 = = p17n+15 = 2 100 , p17n+16 = 2 100 + 655 1000 , n = 0, 1, 2, .... here k = 16 and it is easy to see that α = lim inf n→∞ n−1∑ i=n−16 pi = 32 100 < ( 16 17 )17 and lim sup n→∞ n−1∑ i=n−16 pi = 32 100 + 655 1000 = 0.975 > 1 − α 2 4 . we see that condition (c6) is satisfied and therefore all solutions oscillate. observe, however, that a = lim sup n→∞ n∑ i=n−16 pi = 34 100 + 655 1000 = 0.995 < 1; that is, condition (c3) is not satisfied. in 1995, chen and yu [2], following the above mentioned direction, derived a condition which formulated in terms of α and a says that all solutions of (1.1) oscillate if 0 < α ≤ kk+1 (k+1)k+1 and a > 1 − 1 −α− √ 1 − 2α−α2 2 . (c8) 230 l. k. kikina and i.p. stavroulakis 7, 2(2005) in 1998, domshlak [9], studied the oscillation of all solutions and the existence of nonoscillatory solution of (1.1) with r -periodic positive coefficients {pn},pn+r = pn. it is very important that in the following cases where {r = k},{r = k + 1}, {r = 2},{k = 1,r = 3} and {k = 1,r = 4} the results obtained are stated in terms of necessary and sufficient conditions and it is very easy to check them. in 2000, tang and yu [38] improved condition (c8) to the condition a > λk2 (1 −k ln λ2) − 1 −α− √ 1 −α−α2 2 , (c9) where λ2 is the greater root of the algebraic equation kλk(1 −λ) = α. in 2000, shen and stavroulakis [30], using new techniques, improved the previous results. lemma 2.3.([30]) let the number m ≥ 0 be such that k∑ i=1 pn−i ≥ m for large n. assume that (1.1) has an eventually positive solution {xn}. then m ≤ kk+1/(k + 1)k+1 and lim sup n→∞ xn−k xn k∏ i=1 k∑ j=1 pn−i+j ≤ [d(m)]k, where d(m) is the greater real root of the algebraic equation dk+1 −dk + mk = 0, on [0, 1]. note that from this lemma we obtain a better and perhaps optimal bound which essentially improves (1.6). theorem 2.7.([30]) assume that 0 ≤ α ≤ kk+1/(k + 1)k+1 and that there exists an integer l ≥ 1 such that lim sup n→∞ ⎧⎨ ⎩ k∑ i=1 pn−i + [d(α)] −k k∏ i=1 k∑ j=1 pn−i+j + l−1∑ m=0 [d(α/k)]−(m+1)k k∑ i=1 m+1∏ j=0 pn−jk−i ⎫⎬ ⎭ > 1, (c10) where d(α) and d(α/k) are the greater real roots of the equations dk+1 −dk + αk = 0 and dk+1 −dk + α/k = 0, 7, 2(2005) a survey on the oscillation of solutions of first order ... 231 respectively. then all solutions of (1.1) oscillate. notice that when k = 1, d(α) = d(α) = (1+ √ 1 − 4α)/2 (see [30]), and so condition (c10) reduces to lim sup n→∞ ⎧⎨ ⎩cpn + pn−1 + l−1∑ m=0 cm+1 m+1∏ j=0 pn−j−1 ⎫⎬ ⎭ > 1, (c11) where c = 2/(1 + √ 1 − 4α), α = lim infn→∞ pn. therefore, from theorem 2.7, we have the following corollary. corollary 2.1.([30]) assume that 0 ≤ α ≤ 1/4 and that (c11) holds. then all solutions of the equation xn+1 −xn + pnxn−1 = 0 (1.7) oscillate. a condition derived from (c11) and which can be easier verified, is given in the next corollary. corollary 2.2.([30]) assume that 0 ≤ α ≤ 1/4 and that lim sup n→∞ pn > ( 1 + √ 1 − 4α 2 )2 . (c12) then all solutions of (1.7) oscillate. remark 2.2.([30]) observe that when α = 1/4, condition (c12) reduces to lim sup n→∞ pn > 1/4 which can not be improved in the sense that the lower bound 1/4 can not be replaced by a smaller number. indeed, by condition (n1) (theorem 2.3 in [14]), we see that (1.7) has a nonoscillatory solution if sup pn < 1/4. note, however, that even in the critical state where limn→∞ pn = 1/4, (1.7) can be either oscillatory or nonoscillatory. for example, if pn = 14 + c n2 then (1.7) will be oscillatory in case c > 1/4 and nonoscillatory in case c < 1/4 (the kneser-like theorem, [8]). example 2.2.([30]) consider the equation xn+1 −xn + ( 1 4 + a sin4 nπ 8 ) xn−1 = 0, where a > 0 is a constant. it is easy to see that lim inf n→∞ pn = lim inf n→∞ ( 1 4 + a sin4 nπ 8 ) = 1 4 , 232 l. k. kikina and i.p. stavroulakis 7, 2(2005) lim sup n→∞ pn = lim sup n→∞ ( 1 4 + a sin4 nπ 8 ) = 1 4 + a. therefore, by corollary 2.2, all solutions oscillate. however, none of the conditions (c1) − (c9) is satisfied. the following corollary concerns the case when k > 1. corollary 2.3.([30]) assume that 0 ≤ α ≤ kk+1/(k + 1)k+1 and that lim sup n→∞ n−1∑ i=n−k pi > 1 − [d(α)]−kαk − k[d(α/k)]−kβ2 1 − [d(α/k)]−kβ, (c13) where d(α),d(α/k) are as in theorem 2.7. then all solutions of (1.1) oscillate. in 2000, shen and luo [29] proved the following theorems. theorem 2.8.([29]) assume that there exists some positive integer l such that lim sup n→∞ ⎧⎨ ⎩ k∑ i=0 pn−i + k∏ i=0 k∑ j=1 pn−i+j + l−1∑ m=0 k∑ i=1 m+1∏ j=0 pn−jk−i ⎫⎬ ⎭ > 1. (c14) then all solutions of (1.1) oscillate. theorem 2.9.([29]) assume that there exists some positive integer l such that lim sup n→∞ ⎧⎨ ⎩ k∑ i=1 pn−i + k∏ i=1 k∑ j=1 pn−i+j + l−1∑ m=0 k∑ i=1 m+1∏ j=0 pn−jk−i ⎫⎬ ⎭ > 1. (c15) then all solutions of (1.1) oscillate. from theorem 2.8 and theorem 2.9 the following corollaries are derived. corollary 2.4. ([29]) assume that a > 1 −αk+1 − kβ 2 1 −β. (c16) then all solutions of (1.1) oscillate. corollary 2.5. ([29]) assume that lim sup n→∞ n−1∑ i=n−k pi > 1 −αk − kβ2 1 −β. (c17) then all solutions of (1.1) oscillate. following this historical (and chronological) review we also mention that in the case where 1 k n−1∑ i=n−k pi ≥ kk (k + 1)k+1 and lim n→∞ 1 k n−1∑ i=n−k pi = kk (k + 1)k+1 , 7, 2(2005) a survey on the oscillation of solutions of first order ... 233 the oscillation of (1.1) has been studied in 1994 by domshlak [8] and in 1998 by tang [33] (see also tang and yu [35]). in a case when pn is asymptotically close to one of the periodic critical states, unimprovable results about oscillation preperties of the equation xn+1 −xn + pnxn−1 = 0 were obtained by domshlak in 1999 [11] and in 2000 [12]. received: june 2003. revised: october 2003. references [1] r. p. agarwal and p. j. y. wong, advanced topics in difference equations, kluwer academic publishers, 1997. 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[46] b. g. zhang and yong zhou, comparison theorems and oscillation criteria for difference equations, j. math. anal. appl. , 247 (2000), 397-409. cubo a mathematical journal vol.10, n o ¯ 03, (65–82). october 2008 equilibrium cycles in a two-sector economy with sector specific externality miki matsuo, kazuo nishimura institute of economic research, kyoto university email: nishimura@kier.kyoto-u.ac.jp tomoya sakagami department of economics, kumamoto gakuen university and alain venditti† cnrs-greqam abstract in this paper, we study the two-sector ces economy with sector-specific externality (feedback effects). we characterize the equilibrium paths in the case that allows negative externality, and show how the degree of externality may generate equilibrium cycles around the steady state. resumen en este artculo estudiamos economia de dos-sector ces con externalidad de sectorespecifico (efecto de retroalimentacin). nosotros caracterizamos la trajectoria de equi†this paper has been written while alain venditti was visiting the institute of economic research of kyoto university. he thanks professor kazuo nishimura and all the staff of the institute for their kind invitation. 66 miki matsuo et al. cubo 10, 3 (2008) librio en el caso que permite externalidad negativa, e demonstramos como el grado de externalidad puede generar ciclos de equilibrio alrededor del estado regular. key words and phrases: difference equations, nonlinear dynamics, bifurcation, two-periodic cycle, multiple equilibria. math. subj. class.: 37g10, 39a11, 91b50, 91b62, 91b64, 91b66. 1 introduction the aim of this paper is to show the existence of equilibrium cycles around the steady state in the two-sector model with ces production function and sector specific externality.1 a representative agent has concrete expectations on the level of externality and make a decision assuming that the externality is not affected by his own choice of decision variables. however, externalities come from the average values of capital and labor on the market. therefore, if a representative agent chooses values of decision variables, externalities also vary as everybody also takes the same decision. over the last decade, an important literature has focused on the existence of locally indeterminate equilibria in dynamic general equilibrium economies with technological external effects. local indeterminacy means that there exists a continuum of equilibria starting from the same initial condition, all of which converging to the same steady state. it is now well-known that local indeterminacy is a sufficient condition for the existence of endogenous fluctuations generated by purely extrinsic belief shocks which do not affect the fundamentals, i.e. the preferences and technologies.2 indeed, in presence of local indeterminacy, by randomizing beliefs over the continuum of equilibrium paths, one may construct equilibria defined with respect to shocks on expectations, and thus provide an alternative to technology or taste shocks to get propagation mechanisms and to explain macroeconomic volatility. benhabib and nishimura [3, 4] proved that indeterminacy may arise in a continuous time economy in which the production functions from the social perspective have constant return to scale. benhabib, nishimura and venditti [5] studied the two-sector model with sector specific external effects in discrete time framework. they provided conditions in which indeterminacy may occur even if the production function is decreasing return to scale from the social perspective. nishimura and venditti [7] study the interplay between the elasticity of capital-labor substitution and the rate of depreciation of capital, and its influence on the local behavior of equilibrium paths in a neighborhood of the steady state. however, in all these contributions, the existence of local bifurcations as the degree of externalities is modified is not discussed. in this paper, we study the model in nishimura and venditti [7], focusing on the external effect of capital-labor ratio in the pure capital good sector and characterize the equilibrium paths 1external effects are feedbacks from the other agents in the economy who also face identical maximizing problems. see benhabib and farmer [2] for a survey. 2see cass and shell [6]. cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 67 in the case that allows negative externality, which was not discussed in their paper. we will focus on the existence of flip bifurcation, i.e. of period-two equilibrium cycles, through the existence of local indeterminacy. in section 2 we describe the model. we discuss the existence of a steady state and give the local characterization of the equilibrium paths around the steady state in section 3. section 4 contains some concluding comments. 2 the model we consider a two-sector model with an infinitely-lived representative agent. we assume that its single period linear utility function is given by u (ct) = ct. we assume that the consumption good, c, and capital good are produced with a constant elasticity of substitution (ces) production functions. ct = [ α1k −ρ c ct + α2l −ρ c ct ]− 1 ρc (1) yt = [ β 1 k −ρ 2 yt + β2l −ρ 2 yt + et ]− 1 ρy (2) where ρc, ρy > −1 and et represents the time-dependent externality (feedback effects) in the capital good sector. let the elasticity of capital-labor substitution in each sector be σc = 1 1+ρ c ≥ 0 and σy = 1 1+ρ y ≥ 0. we assume that the externalities are as follows: e = bk̄ −ρ y yt − bl̄ −ρ y yt , (3) where b > 0, and k̄y and l̄y represents the economy-wide average values. the representative agent takes these economy-wide average values as given. definition 1 we call y = [ β 1 k −ρ y y + β2l −ρ y y + e ] − 1 ρy the production function from the private perspective, and y = [ (β 1 + b) k −ρ y y + (β2 − b) l −ρ y y ]− 1 ρy the production function from the social perspective. in the rest of the paper we will assume that α1 + α2 = β1 + β2 = 1 so that the consumption good sector does not have externalities. notice then that denoting β̂ 1 = β 1 + b and β̂ 2 = β 2 − b, we get also β̂ 1 + β̂ 2 = 1. the investment good sector has externalities but the technology is linearly homogeneous, i.e. has constant returns, from the social perspective. 68 miki matsuo et al. cubo 10, 3 (2008) remark 1 notice that the externality (3) may be expressed as follows e = bl̄ −ρ y 2 [ ( k̄y l̄y )−ρ y − 1 ] . (4) now consider the production function from the social perspective as given in definition 1. dividing both sides by ly, we get denoting ky = ky/ly and ỹ = y/ly ỹ = [ (β 1 + b) k −ρ y y + (β2 − b) ]− 1 ρy . (5) from equations (4) and (5) we derive that the externality is given in terms of the capital-labor ratio in the investment good sector. the aggregate capital is divided between sectors, kt = kct + kyt, and the labor endowment is normalized to one and divided between sectors, lct + lyt = 1. the capital accumulation equation is kt+1 = yt, as the capital depreciates completely in one period. to simplify we assume that both technologies are characterized by the same properties of substitution, i.e. ρc = ρy = ρ. the consumer optimization problem will be given by max ∞ ∑ t=0 δ t [ α1k −ρ ct + α2l −ρ ct ]− 1 ρ s.t. yt = [ β 1 k −ρ yt + β2l −ρ yt + et ]− 1 ρ 1 = lct + lyt kt = kct + kyt yt = kt+1 k0, {et} ∞ t=0 given (6) where δ ∈ (0, 1) is the discount factor. pt, rt, and wt respectively denote the price of capital goods, the rental rate of the capital goods and the wage rate of labor at time t ≥ 03. for any sequences {et} ∞ t=0 of external effects that the representative agent considers given, the lagrangian at time 3we normalize the price of consumption goods to one. cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 69 t ≥ 0 is defined as follows: lt = [ α1k −ρ ct + α2l −ρ ct ]− 1 ρ + pt [ [ β 1 k −ρ yt + β2l −ρ yt + et ]− 1 ρ − kt+1 ] + rt (kt − kct − kyt) + wt (1 − lct − lyt) . (7) then the first order conditions derived from the lagrangian are as follows: ∂lt ∂kct = α1 ( ct kct ) − rt = 0, (8) ∂lt ∂lct = α2 ( ct lct ) − wt = 0, (9) ∂lt ∂kyt = ptβ1 ( yt kyt ) − rt = 0, (10) ∂lt ∂lyt = ptβ2 ( yt lyt ) − wt = 0. (11) from the above first order conditions, we derive the following equation, ( α1�α2 β 1 �β 2 ) = ( kct�lct kyt�lyt )1+ρ . (12) if α1/α2 > (<) β1/β2, the consumption (capital) good sector is capital intensive from the private perspective. for any value of (kt, yt) , solving the first order conditions with respect to kct, kyt, lct, lyt gives these inputs as functions of capital stock at time t and t + 1, and external effect, namely: kct = kc (kt, yt, et) , lct = lc (kt, yt, et) , kyt = ky (kt, yt, et) , lyt = ly (kt, yt, et) . for any given sequence {et} ∞ t=0 , we define the efficient production frontier as follows: t ∗ (kt, kt+1, et) = [ α1kc (kt, yt, et) −ρ + α2lc (kt, yt, et) −ρ ]− 1 ρ . using the envelope theorem we derive the equilibrium prices,4 t2 (kt, kt+1, et) = −pt, (13) t1 (kt, kt+1, et) = rt. (14) 4see takayama for the envelope theorem, pp160-165. using the envelope theorem, we get ∂lt ∂kt = ∂t ∂kt and ∂lt ∂kt+1 = ∂t ∂kt+1 . this is equivalent to (13) and (14). 70 miki matsuo et al. cubo 10, 3 (2008) next we solve the intertemporal problem (6). in this model, lifetime utility function becomes u = ∞ ∑ t=0 δ t t ∗ (kt, kt+1, et) . from the first order conditions with respect to kt+1, we obtain the euler equation t2 (kt, kt+1, et) + δt1 (kt+1, kt+2, et+1) = 0. (15) the solution of equation (15) also has to satisfy the following transversality condition lim t→+∞ δ t ktt1 (kt, kt+1, et) = 0. (16) we denote the solution of this problem {kt} ∞ t=0. this path depends on the choice of sequence {et} ∞ t=0 . if the sequence {et} ∞ t=0 satisfies et = bky (kt, yt, et) −ρ − bly (kt, yt, et) −ρ , (17) then {k̂t} ∞ t=0 is called an equilibrium path. along an equilibrium path, the expectations of the representative agent on the externalities {et} ∞ t=0 are realized. definition 2 {kt} ∞ t=0 is an equilibrium path if {kt} ∞ t=0 satisfies (15), (16) and (17). solving the equation (17) for et, we derive et that is given as a function of (kt, kt+1), namely et = ê (kt, kt+1). let us substitute ê (kt, kt+1) into equations (13) and (14) and define the equilibrium prices as pt = pt (kt, kt+1) , rt = rt (kt, kt+1) . then the euler equation (15) evaluated at {kt} ∞ t=0 is −p (kt, kt+1) + δr (kt+1, kt+2) = 0. (18) we have the following lemma. lemma 1 the partial derivatives of t (kt, kt+1, et) with respect to kt and kt+1 are given by t1 (kt, kt+1, ê (kt, kt+1)) = α1 [ α1 + α2 ( α1β2 α2β1 ) 1+ρ ρ ( ( g k t+1 ) ρ β 2 −b − (β 1 +b) β 2 −b )ρ]− 1+ρ ρ t2 (kt, kt+1, ê (kt, kt+1)) = t1(kt, kt+1,ê(kt,kt+1)) β 1 ( g kt+1 ) 1+ρ where g = g (kt, kt+1) = { kyt ∈ [0, kt] | α1β2 α2β1 = ( kt−kyt 1−lyt(kyt,kt+1) )1+ρ ( lyt(kyt ,kt+1) kyt )1+ρ } cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 71 and lyt (kyt, kt+1) = ( k −ρ t+1 −(β 1 +b)k −ρ yt β 2 −b ) − 1 ρ . 3 steady state definition 3 a steady state is defined by kt = kt+1 = yt = k ∗ and is given by the solution of t2 (k ∗, k∗, e∗) + δt1 (k ∗, k∗, e∗) = 0 with e∗ = ê (k∗, k∗) . in the rest of the paper we assume the following restriction on parameters’ values that guarantees all the steady state values are positive. assumption 1 the parameters δ, β 1 , b and ρ satisfy (δβ 1 ) −ρ 1+ρ < β 1 + b. we obtain the steady state value. proposition 1 in this model, there exists a unique stationary capital stock k∗ such that: k∗ = { 1 + ( α1β2 α2β1 ) −1 1+ρ (δβ 1 ) −1 1+ρ [ 1 − (δβ 1 ) 1 1+ρ ] }−1 [ 1−β̂ 1 (δβ 1 ) −ρ 1+ρ β̂ 2 ] 1 ρ . (19) to study local behavior of the equilibrium path around the steady state k∗, we linearize the euler equation (15) at the steady state k∗ and obtain the following characteristic equation δt12λ 2 + [δt11 + t22] λ + t21 = 0, or δλ 2 + [ δ t11 t12 + t22 t12 ] λ + t21 t12 = 0. (20) as shown in nishimura and venditti [7], the expressions of the characteristic roots are as follows: proposition 2 the characteristic roots of equation (20) are λ1 = 1 (δβ 2 ) 1 1+ρ [ ( β 1 β 2 ) 1 1+ρ − ( α1 α2 ) 1 1+ρ ], (21) λ2(b) = (δβ 2 ) 1 1+ρ [ β 1 +b β 1 ( β 1 β 2 ) 1 1+ρ − β 2 −b β 2 ( α1 α2 ) 1 1+ρ ] δ . 72 miki matsuo et al. cubo 10, 3 (2008) the roots of the characteristic equation determine the local behavior of the equilibrium paths. the sign of λ1 is determined by factor intensity differences from the private perspective. 5 we now characterize the equilibrium paths in this model. in particular we can show that the local behavior of equilibrium path around the steady state changes according to the degree of external effect in the capital good sector. definition 4 a steady state k∗ is called locally indeterminate if there exists ε such that for any k0 ∈ (k ∗ − ε, k∗ + ε) , there are infinitely many equilibrium paths converging to the steady state. as there is one pre-determined variable, the capital stock, local indeterminacy occurs if the stable manifold has two dimension, i.e. if the two characteristic roots are within the unit circle. we will also present conditions for local determinacy (for saddle-point stability) in which there exists a unique equilibrium path. such a configuration occurs if the stable manifold has one dimension, i.e. if one root is outside the unit circle while the other is inside. when the investment good is capital intensive, local indeterminacy and flip bifurcation cannot occur. proposition 3 suppose that the capital good sector is capital intensive from the private perspective, i.e. α2β1 > α1β2. then the characteristic roots λ1 and λ2(b) are positive with λ1 > 1. next we present our results assuming that the capital good is labor intensive from the private perspective, i.e. α2β1 − α1β2 < 0. equilibrium period-two cycles may occur in this case through a flip bifurcation. we will also get local indeterminacy of equilibria. by rewriting equation (21), the characteristic roots are λ1 = − 1 (δβ 2 ) 1 1+ρ [ ( α1 α2 ) 1 1+ρ − ( β 1 β 2 ) 1 1+ρ ], (22) λ2(b) = − (δβ 2 ) 1 1+ρ [ β 2 −b β 2 ( α1 α2 ) 1 1+ρ − β 1 +b β 1 ( β 1 β 2 ) 1 1+ρ ] δ . to get λ1 ∈ (−1, 0), we need however to suppose a slightly stronger condition than simply ensuring the capital good sector to be labor intensive from the private perspective. the capital intensity difference α1β2 − α2β1 needs to be large enough and the discount factor has to be close enough to 1. proposition 4 assume that (α1β2) 1 1+ρ − (α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (δ3, 1) with δ3 = β −1 2 [ (β 1 /β 2 ) 1 1+ρ − (α1/α2) 1 1+ρ ]−1−ρ < 1. 5if α2β1 − α1β2 > 0, the capital good sector is capital intensive from the private perspective. cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 73 then there exist b(δ) > 0 and b(δ) > b(δ) such that the steady state is saddle point for b ∈ (0, b (δ)), undergoes a flip bifurcation when b = b (δ), becomes locally indeterminate for b ∈ ( b (δ) , b (δ) ) and is again saddle-point stable for (b (δ) , +∞). generically, there exist period-two cycles in a left (right) neighborhood of b (δ) that are locally indeterminate (saddle-point stable). next we still assume that the capital good is labor intensive from the private perspective with α2β1 − α1β2 < 0, but make λ1 an unstable root, i.e. λ1 < −1. as a result local indeterminacy cannot occur but period-two cycles may still exist through a flip bifurcation. two cases need to be considered: (α1β2) 1 1+ρ − (α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (0, δ3), as well as (α1β2) 1 1+ρ − (α2β1) 1 1+ρ < α 1 1+ρ 2 . the following result is proved along the same lines as proposition 4. proposition 5 suppose that the capital goods sector is labor intensive from the private perspective and let δ4 = β 1 ρ 2 [ (β 1 /β 2 ) 1 1+ρ − (α1/α2) 1 1+ρ ] 1+ρ ρ . assume also that one of the following sets of conditions hold: i) (α1β2) 1 1+ρ − (α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (0, δ∗) with δ∗ = min{δ3, δ4}, ii) (α1β2) 1 1+ρ − (α2β1) 1 1+ρ < α 1 1+ρ 2 , ρ > 0 and δ ∈ (0, δ4), then there exist b(δ) > 0 and b(δ) > b(δ) such that the steady state is totally unstable for b ∈ (0, b (δ)), undergoes a flip bifurcation when b = b (δ), becomes saddle-point stable for b ∈ ( b (δ) , b (δ) ) and is again totally unstable for (b (δ) , +∞). generically, there exist period-two cycles in a left (right) neighborhood of b (δ) that are locally saddle-point stable (unstable). remark 2 consider the production function from the social perspective as given in definition 1 and recall from (5) that we can write it as follows ỹ = [ (β 1 + b) k −ρ y y + (β2 − b) ]− 1 ρy . (23) according to b ≷ β 2 , the following inequality holds: for any η > 1, [ (β 1 + b) (ηky) −ρ + (β 2 − b) ]− 1 ρ ≷ [ (β 1 + b) (ηky) −ρ + η−ρ (β 2 − b) ]− 1 ρ = η [ (β 1 + b) k−ρy + (β2 − b) ]− 1 ρ . if b is larger than β 2 , the function ỹ exhibits increasing returns while if b is smaller than β 2 the function ỹ exhibits decreasing returns. as we consider in proposition 5 values of δ close to zero, the role of b on the local stability properties of the steady state is multiple. indeed, starting from a low amount of externalities, an increase of b contributes to saddle-point stability and the existence of cycles through a flip 74 miki matsuo et al. cubo 10, 3 (2008) bifurcation. but then if b is increased too much, total instability occurs since the returns to scale becomes increasing as shown in the previous remark. 4 concluding remarks in this paper we have characterized the local dynamics of equilibrium paths depending on the size of external effects b. we have shown that when the consumption good is capital intensive, the effect of b on the local dynamics of equilibrium path depends on the value of the discount factor. if the discount factor is close enough to one and the capital intensity difference is large enough, local indeterminacy occurs for intermediary values of b while saddle-point stability is obtained when b is low enough or large enough. on the contrary, if the discount factor is low enough, local indeterminacy cannot occur. but the existence of equilibrium cycles and saddle-point stability require intermediary values of b while total instability is obtained when b is low enough or large enough. 5 appendix 5.1 proof of lemma 1 we shall derive the first partial derivatives of t (kt, kt+1, et) along an equilibrium path. the first order conditions derived from the lagrangian are as below: α1 ( ct kct ) − rt = 0, (a1.1) α2 ( ct lct ) − wt = 0, (a1.2) ptβ1 ( yt kyt ) − rt = 0, (a1.3) ptβ2 ( yt lyt ) − wt = 0. (a1.4) in the equilibrium the equation (2) is rewritten as lyt = ( y −ρ t − (β1 + b) k −ρ yt β 2 − b )− 1 ρ . (a1.5) from the first order conditions (a1.1)-(a1.4), α1β2 α2β1 = ( kct lct ) 1+ρ ( lyt kyt ) 1+ρ . cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 75 substituting kct = kt − kyt, lyt = 1 − lct into the equation, α1β2 α2β1 = ( kt − kyt 1 − lyt )1+ρ ( lyt kyt )1+ρ . (a1.6) by solving equations (a1.5) and (a1.6) with respect to kyt and substituting yt = kt+1, we have kyt = g (kt, kt+1) . from the equation (a1.1), rt = α1 [ α1 + α2 ( kct lct )ρ]− 1+ρ ρ . using the equation (a1.6) we have rt = α1 [ α1 + α2 ( α1β2 α2β1 ) 1+ρ ρ ( g (kt, kt+1) lct )ρ ]− 1+ρ ρ . and then from (a1.5) rt can be rewritten as the following equation by substituting ( g(kt,kt+1) lyt )ρ = ( g(kt ,kt+1) yt ) ρ β 2 −b − (β 1 +b) β 2 −b 6 , rt = α1    α1 + α2 ( α1β2 α2β1 ) 1+ρ ρ   ( g(kt,kt+1) yt )ρ β 2 − b − (β 1 + b) β 2 − b   ρ    − 1+ρ ρ . (a1.7) moreover from the equation (a1.3), we have pt = rt β 1 ( g (kt, kt+1) yt )1+ρ . (a1.8) therefore we get t1 and t2 from the envelope theorem which gives t1 = rt, t2 = −pt. � 5.2 proof of proposition 1 by definition k∗ satisfies t2 (k ∗, k∗, e∗) + δt1 (k ∗, k∗, e∗) = 0 with e∗ = ê (k∗, k∗) . in the steady state, g∗ = g (k∗, k∗) and y∗ = k∗. using lemma 1, the euler equation is − r β 1 ( g∗ y∗ )1+ρ + δr = 0. 6substitute the equation (a1.5) into ( g lyt )ρ . 76 miki matsuo et al. cubo 10, 3 (2008) thus, g∗ = (δβ 1 ) 1 1+ρ k∗. (a2.1) as y∗ = k∗ at the steady state, the equation (a1.5) becomes, l ∗ y = k ∗ ( 1 − (β 1 + b) (δβ 1 ) −ρ 1+ρ β 2 − b )− 1 ρ . (a2.2) using k∗c = k ∗ − k∗y and l ∗ c = 1 − l ∗ y, l∗c = 1 − k ∗ ( 1 − (β 1 + b) (δβ 1 ) −ρ 1+ρ β 2 − b )− 1 ρ , (a2.3) k∗c = k ∗ ( 1 − (δβ 1 ) 1 1+ρ ) . (a2.4) then equation (a1.6) can be rewritten as follows; ( k∗c l∗c )1+ρ ( l∗y g∗ )1+ρ = α1β2 α2β1 . (a2.5) substituting these input demand functions into the above equation and solving with respect to k∗, we can get k∗ = [ 1 + ( α1β2 α2β1 ) −1 1+ρ (δβ 1 ) −1 1+ρ ( 1 − (δβ 1 ) 1 1+ρ ) ]−1 [ 1 − (β 1 + b) (δβ 1 ) −ρ 1+ρ β 2 − b ] 1 ρ . then k∗ is well defined if and only if (δβ 1 ) −ρ 1+ρ < 1 β 1 + b . � 5.3 proof of proposition 2 we give some lemmas in order to derive the characteristic roots. lemma 2 at the steady state the following holds g1 = 1 + ρ ∆kc , g2 =   (1 + ρ) lρy + (1 + ρ) l 1+ρ y lc ∆   y−1−ρ β 2 − b , cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 77 where ∆ = 1 + ρ g + 1 + ρ kc + ( (1 + ρ) lρy + (1 + ρ) l1+ρy lc ) β 1 + b β 2 − b g−1−ρ. proof. from equation (a2.5) we get α1β2 α2β1 = g−1−ρ (k − g) 1+ρ ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1+ρ ρ { 1 − ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1 ρ }−1−ρ . totally differentiating this equation, we have the following relationship, [(1 + ρ) g−1 + (1 + ρ) (k − g) −1 + (1 + ρ) ( y−ρ − (β 1 + b) g−ρ β 2 − b )−1 β 1 + b β 2 − b g−1−ρ + (1 + ρ) { 1 − ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1 ρ }−1 ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1+ρ ρ β 1 + b β 2 − b g−1−ρ]dg = (1 + ρ) (k − g) −1 dk + (1 + ρ) ( y−ρ − (β 1 + b) g−ρ β 2 − b )−1−1 y−1−ρ β 2 − b dy + (1 + ρ) { 1 − ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1 ρ }−1 ( y−ρ − (β 1 + b) g−ρ β 2 − b )− 1+ρ ρ y−1−ρ β 2 − b dy. (a3.1.1) notice from equation (a1.5) l −ρ y = y−ρ − (β 1 + b) g−ρ β 2 − b (a3.1.2) and (a2.5) ( α1β2 α2β1 ) 1 1+ρ = k∗c l∗c ( g∗ l∗y )−1 . (a3.1.3) then substituting these equations and dyt = dkt+1 into (a3.1.1) gives rhs = 1 + ρ kc dkt + ( (1 + ρ) lρy + (1 + ρ) l1+ρy lc ) y1−ρ β 2 − b dkt+1, lhs = [ 1 + ρ g + 1 + ρ kc + ( (1 + ρ) lρy + (1 + ρ) l1+ρy lc ) β 1 + b β 2 − b g−1−ρ ] dg, where we denote ∆ ≡ 1 + ρ g + 1 + ρ kc + ( (1 + ρ) lρy + (1 + ρ) l1+ρy lc ) β 1 + b β 2 − b g−1−ρ, 78 miki matsuo et al. cubo 10, 3 (2008) and we derive ∆dg = 1 + ρ kc dkt + [ (1 + ρ) lρy + (1 + ρ) l1+ρy lc ] y1−ρ β 2 − b dkt+1. therefore dg = 1 + ρ ∆kc dkt + [(1 + ρ) lρy + (1 + ρ) l 1+ρ y lc ] ∆ y1−ρ β 2 − b dkt+1. lemma 3 at the steady state the following holds g1y = (g − g2y) y g ( 1 − kc lc ly ky ) −1 with g, kc, ly, lc respectively given by equations (a2.1) − (a2.4). proof. from equation (a1.5) we get ( l−ρy + β 1 + b β 2 − b k−ρy ) y−1 = y−ρ β 2 − b y−1. substituting this equation into g2, g2 = [(1 + ρ) lρy + (1 + ρ) l 1+ρ y lc ](l−ρy + β 1 +b β 2 −b k−ρy )y −1 ∆ . using the expression of ∆ we derive g2y = g + (1 + ρ) ∆ ly lc − (1 + ρ) ∆ g kc . then, g − g2y = (1 + ρ) ∆kc g ( 1 − ly lc kc g ) , (a3.2.1) g1y = 1 + ρ ∆kc y. (a3.2.2) from equations (a3.2.1) and (a3.2.2), we finally get g1y = (g − g2y) y g ( 1 − kc lc ly ky )−1 . lemma 4 under assumption 1, at the steady state, kt = kt+1 = yt = k ∗ and the following holds v11 (k ∗, k∗) v12 (k∗, k∗) = − y g ( 1 − kc lc ly ky ) −1 , cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 79 v22 (k ∗, k∗) v12 (k∗, k∗) = − g β 1 y [ β 1 (β 2 − b) β 2 kc lc ly g + (β 2 − b) ( g ly )ρ − ( g y )ρ] , v21 (k ∗, k∗) v12 (k∗, k∗) = v22 (k ∗, k∗) v12 (k∗, k∗) v11 (k ∗, k∗) v12 (k∗, k∗) , where g, kc, ly, lc are given by equations (a2.1) − (a2.4), respectively. proof. let v (kt, kt+1) denote ti (kt, kt+1, ê (kt, kt+1)) for i = 1, 2. by definition, v ∗ 11 = ∂t1 ∂kt = ∂r ∂kt , v ∗ 12 = ∂t1 ∂kt+1 = ∂r ∂kt+1 , v ∗ 21 = ∂t2 ∂kt = − ∂p ∂kt , v ∗ 22 = ∂t2 ∂kt+1 = − ∂p ∂kt+1 . computing the these equations, we have v ∗ 11 = ∂r ∂kt = − (1 + ρ) α − ρ 1+ρ 1 r 1+2ρ 1+ρ α2 β 2 − b ( α1β2 α2β1 ) ρ 1+ρ ( g y )ρ g1 g , v ∗ 12 = ∂r ∂kt+1 = − (1 + ρ) α − ρ 1+ρ 1 r 1+2ρ 1+ρ α2 β 2 − b ( α1β2 α2β1 ) ρ 1+ρ ( g y )ρ ( g2y − g yg ) , ∂p ∂kt = 1 β 1 ∂r ∂kt ( g y )1+ρ + (1 + ρ) r β 1 ( g y )1+ρ g1 g , ∂p ∂kt+1 = 1 β 1 ∂r ∂kt+1 ( g y ) 1+ρ + (1 + ρ) r β 1 ( g y ) 1+ρ ( g2y − g yg ) . from equation (a1.7), ( r α1 ) ρ 1+ρ = [ α1 + α2 ( α1β2 α2β1 ) ρ 1+ρ ( g lc )ρ ] . substituting the above equation into v ∗ 11 , and using (a3.1.2) and (a3.1.3) we obtain v ∗ 11 = − (1 + ρ) r ( g y )ρ g1 g [ α1β̂2 α2 ( α2β1 α1β2 ) k∗c l∗c l∗y g∗ + β̂ 2 ( y ly )ρ ]−1 , where a ≡ α1β̂2 α2 ( α2β1 α1β2 ) k∗c l∗c l∗y g∗ + β̂ 2 ( y ly )ρ . 80 miki matsuo et al. cubo 10, 3 (2008) we can calculate v ∗ 21 , v ∗ 12 , and v ∗ 22 as we did previously, v ∗ 21 = − (1 + ρ) r β 1 ( g y ) 1+ρ g1 g [ 1 − ( g y )ρ a−1 ] , v ∗ 12 = − (1 + ρ) r ( g y )ρ ( g2y − g yg ) a−1, v ∗ 22 = − (1 + ρ) r β 1 ( g y )1+ρ ( g2y − g yg )[ 1 − ( g y )ρ a−1 ] . then we get v ∗ 11 v ∗ 12 = g1y g2y − g , v ∗ 22 v ∗ 12 = g β 1 y [ a− ( g y )ρ] , we shall now prove proposition 2. from lemma 4 the characteristic polynomial may be rewritten as g (λ) = ( λ + v ∗ 11 v ∗ 12 )( δλ + v ∗ 22 v ∗ 12 ) . then the characteristic roots are λ1 = − v ∗ 11 v ∗ 12 , λ2 = − v ∗ 22 δv ∗ 12 . (a3.3.1) we can calculate v ∗ 11 v ∗ 12 and v ∗ 22 v ∗ 12 by substituting the following relationship kc lc ly g = ( α1β2 α2β1 ) 1 1+ρ , g y = (δβ 1 ) 1 1+ρ , ( g ly )ρ = (δβ 1 ) ρ 1+ρ − β̂ 1 β̂ 2 , g − g2y = (1 + ρ) ∆kc g ( 1 − ly lc kc g ) , and we obtain the first root by substituting all the above equations into the expressions given in lemma 4 λ1 = − 1 (δβ 2 ) 1 1+ρ [ ( α1 α2 ) 1 1+ρ − ( β 1 β 2 ) 1 1+ρ ]. cubo 10, 3 (2008) equilibrium cycles in a two-sector economy ... 81 moreover we can rewrite a by using these equations, a =β̂ 2 α1 α2 ( α2β1 α1β2 )− ρ 1+ρ + (δβ 1 ) ρ 1+ρ − β̂ 1 . from lemma 4, we finally have the second characteristic root, λ2 = − (δβ 2 ) 1 1+ρ [ β 2 −b β 2 ( α1 α2 ) 1 1+ρ − β 1 +b β 1 ( β 1 β 2 ) 1 1+ρ ] δ . � 5.4 proof of proposition 3 notice from (21) that λ1 > 0. denoting 7 δ1 ≡ β −1 2 [ (β 1 /β 2 ) 1 1+ρ − (α1/α2) 1 1+ρ ]−1−ρ > 1 then we obtain λ1 = (δ1/δ) 1 1+ρ > 1 for 0 < δ < 1. since (β 1 + b)/β 1 > 1 and (β 2 − b)/β 2 < 1, λ2(b) is always positive. � 5.5 proof of proposition 4 if (α1β2) 1 1+ρ −(α2β1) 1 1+ρ > α 1 1+ρ 2 and δ ∈ (δ3, 1), then −1 < λ1 < 0. the size of λ2(b) is determined in the following way. notice that λ2(b) is increasing in b. for b = 0, λ2 (0) = 1/δλ1 < −1 by the above hypothesis and for b = β 2 , λ2 (β2) = (δβ1) −ρ 1+ρ . (i) if −1 < ρ < 0, λ2 (β2) < 1. therefore there exist b (δ) ∈ (0, β2) and b (δ) > β2 such that λ2 < −1 for b ∈ (0, b (δ)) , −1 < λ2 < 1 for any b ∈ ( b (δ) , b (δ) ) and λ2 > 1 for any b > b (δ) . (ii) if ρ = 0, λ2 (β2) = 1. therefore λ2 (b) < −1 for b ∈ (0, β2 − 2α2) , −1 < λ2 (b) < 1 for b ∈ (β 2 − 2α2, β2) and λ2 (b) > 1 for b > β2. (iii) if ρ > 0, λ2 (β2) > 1. therefore there exist b (δ) and b (δ) in (0, β2) such that λ2 (b) < −1 for b ∈ (0, b (δ)) , −1 < λ2 (b) < 1 for b ∈ ( b (δ) , b (δ) ) , and λ2 (b) > 1 for b > b (δ). in each of these three cases, when b = b(δ), λ2(b) = −1 and λ ′ 2 (b)|b=b(δ) > 0. it follows that b = b(δ) is a flip bifurcation value. the result follows from the flip bifurcation theorem (see ruelle [8]). � received: april 2008. revised: may 2008. 7note that δ1 = α2 [ (α2β1) 1 1+ρ − (α1β2) 1 1+ρ ]−1−ρ > α2 (α2β1) = 1 β 1 > 1. 82 miki matsuo et al. cubo 10, 3 (2008) references [1] j. benhabib and r.e. farmer, indeterminacy and increasing returns, journal of economic theory, 20 (1994), 19–41. [2] j. benhabib and r.e. farmer, indeterminacy and sunspots in macroeconomics, in handbook of macroeconomics, edited by j.b. taylor and m. woodford, north-holland, amsterdam, holland, (1999), pp. 387–448. [3] j. benhabib and k. nishimura, indeterminacy and sunspots with constant returns, journal of economic theory, 81 (1998), 58–96. [4] j. benhabib and k. nishimura, indeterminacy arising in multisector economies, japanese economic review, 50 (1999), 485–506. [5] j. benhabib, k. nishimura, and a. venditti, indeterminacy and cycles in two-sector discrete-time model, economic theory, 20 (2002), 217–235. [6] d. cass and k. shell, do sunspots matter ?, journal of political economy, 91 (1983), 193–227. [7] k. nishimura and a. venditti, capital depreciation, factor substitutability and indeterminacy, journal of difference equations and applications, 10 (2004), 1153–1169. [8] d. ruelle, elements of differentiable dynamics and bifurcation theory, academic press, san diego, 1989. n06 a mathematical journal vol. 6, no 4, (33 51). december 2004. the forced korteweg–de vries equation as a model for waves generated by topography paul a. milewski 1 department of mathematics, university of wisconsin 480 lincoln dr., madison, wi 53706 milewski@math.wisc.edu abstract this is a brief survey article discussing simple yet very rich models for free surface flows over topographical features. we consider the most interesting case where the flow is near critical (the froude number is near 1). we derive kortewegde vries and burgers’ equations, and consider both steady configurations and time-dependent numerical solutions. the topography is taken to be either a localized bump or hole or a more extended plateau or depression. the calculations involving extended topography are new and in some cases show surprisingly complex dynamics. 1 introduction in this paper we consider a model to describe the evolution of the free-surface when a fluid flows over an obstacle. we shall see that the free surface dynamics will depend on the speed of the flow and the characteristics of the obstacle. we shall focus on the resonant case, where the most interesting and largest free-surface response occurs. the problem of flow over an obstacle has applications in many physical situations, from the flow of water over rocks to atmospheric and oceanic stratified flows encountering topographic obstacles [1]. (equations similar to those we study 1paul milewski is partially supported by nsf-dms 34 paul a. milewski 6, 4(2004) here can be derived in the stratified flow case.) in addition, the equations we discuss can also be obtained when one considers a pressure distribution moving over a free surface and generating waves. physical examples of this are models of ship waves, and of ocean waves generated by storms. −20 −15 −10 −5 0 5 10 15 20 −1 −0.5 0 0.5 z=η(x,t) z=−h+h(x,t) → u x z figure 1: sketch of physical problem. figure 1 shows the physical problem we are considering. we shall discuss only the onedimensional free surface case and derive an approximate governing equation for the free surface evolution η over a topography h: ητ + f ηξ − 1 6 ηξξξ − 3 2 ηηξ = 1 2 hξ. (1) this equation is known as the forced korteweg-de vries (kdv) equation (see [12], [7], [4] for early references) and includes the effects of nonlinearity, dispersion, topography and flow when these effects are comparable to each other in importance. the unforced equation (h = 0) has been extensively studied both for its mathematical properties (integrability and inverse scattering theory was first studied on the kdv equation) and its approximation to free nonlinear waves. at smaller scales, when surface tension is comparable to gravity, one can obtain similar models with a fifth derivative term (see [9]). the important parameter in this problem is the froude number f = u√ gh , which is the ratio of the flow speed to the linear shallow water wave speed. the parameter f appearing in (1) is proportional to f − 1. flows that are faster than the linear wave speed (f > 0) are called supercritical and f < 0 is called subcritical. we shall consider three main variations on (1): 1. the case where the free surface is steady. 2. the “shallow water” limit where the equation is nondispersive (the ηξξξ term is ommitted). 3. the full equation (1). these cases will be studied for two main types of topography: localized bumps and holes, and extended plateaus or 6, 4(2004) waves generated by topography 35 depressions. previous studies of this problem have focussed only on the flow over a localized bumps, not on holes or extended topography. this is possibly due to the fact that when water flows over a hole there is a stronger possibility of flow separation and recirculation in the hole, effects that are beyond the present theory. however, if the hole is not too abrupt (certainly in the case of an extended depression), the current model is still valid. wherever possible we construct exact solutions, but for most cases our solutions are numerical. 2 the forced korteweg-de vries equation for the purposes of the derivation, instead of considering a uniform flow over a fixed topographic obstacle, it is convenient to view the obstacle moving at a fixed speed u under a fluid at rest. then, using the typical wavelength l as the horizontal length scale, the far field depth h as the vertical length scale and a as the scale for typical free surface displacements and the topography, we define the nonlinearity parameter � = a/h and the long wave (or dispersive) parameter as μ = h/l. these will be the small parameters on which our asymptotic theory is based. the case μ = 0 is usually called the shallow water limit whereas μ small but finite is called the long wave limit. using aμ √ gh as the velocity potential scale, l/ √ gh as the time scale, the dimensionless equations for an irrotational fluid with a free surface can be written in terms of the velocity potential φ(x, z, t), the bottom boundary −h + h(x + u t), and the free surface displacement η(x, t) as μ2φxx + φzz = 0, −1 + �h(x + f t) < z < �η(x, t), (2) μ2(f hx + �φxhx) − φz = 0, z = −1 + �h(x + f t), (3) ηt + �ηxφx − 1 μ2 φz = 0, z = �η, (4) φt + � ( φ2x 2 + 1 μ2 φ2z 2 ) + η = 0, z = �η. (5) here, f = u/ √ gh is the froude number. laplace’s equation (2) follows from the incompressibility of the fluid, (3) is the condition of no fluid flux through the bottom, (4) is the statement that η is carried with the flow, and (5) is bernoulli’s equation stating that the pressure at the free surface is a constant (taken as zero). expanding the two surface boundary conditions (5,4) about z = 0 and eliminating η leads to a single boundary condition in φ at z = 0 correct to o(�): φtt + 1 μ2 φz + � [ 1 2 ( φ2x + 1 μ2 φ2z ) t − (φtφtz )t + (φtφx)x ] = 0, z = 0. (6) next, we write the solution to laplace’s equation (2) in the vertical as φ(x, z, t) = cos [μ(z + 1)∂x] φ(x, t) + μ sin [μ(z + 1)∂x] ∂ −1 x ψ(x, t). (7) 36 paul a. milewski 6, 4(2004) the trigonometric functions appearing above should be thought as representing their taylor expansions. note that φ = φ(x, −1, t) and that ψ = μ−2φz (x, −1, t). the leading order velocity is depth independent and is given by φx and the free surface displacement can be recovered from (5) and (7) as η ≈ −φt ≈ −φt (8) we now substitute this solution in the free surface condition (6) and the bottom boundary condition (3), yielding two equations for our unknowns φ and ψ, respectively: φtt − φxx + ψ + μ2 [ 1 6 φxxxx − 1 2 φxxtt + ψtt − 1 2 ψxx ] (9) + � [ 1 2 (φ2x)t + (φtφx)x ] = 0 μ2f hx + μ sin [μ�h∂x] φx − μ2 cos [μ�h∂x] ψ = 0. (10) equation (9) has been truncated to o(�) and o(μ2) and a similar truncation on (10) yields: f hx − ψ + � [hφxx] = 0. (11) we first consider the simplest case of free surface dynamics, the linear shallow water case obtained by setting � = μ2 = 0 in (9,11) φtt − φxx = −f hx(x + f t). (12) we are interested here in the surface displacements due to the topography so we always consider cases with zero initial data. the particular solution of (12) is φ = f 1 − f 2 ∫ h(x + f t) dx. (13) therefore, the surface displacement η ≈ f 2 f 2 − 1 h(x + f t), (14) is steady relative to the obstacle and follows the bottom shape for f > 1. the homogeneous solution (often called the free waves) travels with speed ±1 and therefore, for f not close to ±1, propagates away from a localized bottom obstacle. the approximation of (12) becomes invalid when f ≈ ±1, because the free surface displacement becomes large, and because the free waves may remain over the obstacle for long enough times to be affected by it. in fact, when f 2 − 1 = o(�), we need to rescale the bottom to size o(�2) for the surface displacement to remain o(�) as required by the expansion. (in obtaining (12) the bottom was assumed to be o(�).) this is called the resonant case because a small topography generates a relatively large free-surface response. 6, 4(2004) waves generated by topography 37 there is another physical situation in which there is a possible resonance between the waves and the topography. bragg scattering occurs when periodic waves scatter over periodic topography. a resonant form of bragg scattering will take place when the scattered waves are themselves free modes of the system. this limit is modeled by coupled kdv equations in [2]. now, considering the distinguished limit obtained by balancing the nonlinear, topographic and dispersive effects, we put � = μ2, f = 1 + �f , and h → �h into (9,11). these become φtt − φxx + � [ 1 6 φxxxx − 1 2 φxxtt + 1 2 (φ2x)t + (φtφx)x ] = −�hx(x + t + �f t). (15) we now consider waves traveling only to the left at a speed close to −1. this is because the waves traveling at speeds close to +1 will be far from resonant and relatively small. we do this by moving to the frame of the obstacle by setting ξ = x + (1 + �f )t, τ = �t and η(ξ, τ ) ≡ −φξ. a final truncation of the equation leads to ητ + f ηξ − 1 6 ηξξξ − 3 2 ηηξ = 1 2 hξ(ξ). (16) this is the forced korteweg de-vries equation that we study in the remainder of the paper. it governs the slow dynamics of the free surface shown in figure 1 when the flow velocity u is close to √ gh. we shall always assume an infinite domain −∞ < ξ < ∞ and that η(ξ, 0) = 0. this corresponds to a quiescent flow which at t = 0 is impulsively started over the topography. we will mainly be interested in the behavior after initial transients have propagated away. the equation (16) conserves mass, with: d dτ ∫ ∞ −∞ ηdξ = 0, (17) and the rate of change of momentum is given by the topographical stress: d dτ ∫ ∞ −∞ η2dξ = ∫ ∞ −∞ ηhξdξ = − ∫ ∞ −∞ ηξhdξ, (18) where the last equality is obtained by integration by parts. in subsequent sections we shall find several types of steady solutions near the localized topography, and they yield different topographical stresses. these are: 1. localized free surface deflections. these solutions have zero stress, and the topography does no work on the fluid; 2. hydraulic “falls”: monotonic solutions tending to different values up and downstream from the topography. these solutions do add energy to the flow since ηξ has one sign. 3. shocks. these solutions also do work on the fluid proportional to the height of the topography under the shock. for h = 0 (16) is the well known korteweg-de vries equation with solitary wave solutions η = asech2(κξ − cτ ), a = 4 3 κ2, c = f + 1 2 a. (19) 38 paul a. milewski 6, 4(2004) the parameter f defines how close the flow is to exact resonance (f = 1). in the remainder of the paper we shall consider two main cases with h �= 0: the case when h a highly localized bump or hole, and the case when h corresponds to a long depression or plateau. 3 localized topography and steady solutions in this section we study steady solutions of (16) when the topography is a localized bump, first with a phase plane analysis and h = αδ(ξ) (where δ is a dirac mass), and then with the hydraulic approximation for a smoother bump. η s (a) η s (b) figure 2: phase plane of the unforced problem for e = 0. (a) f < 0, (b) f > 0. circles indicate centers and crosses indicate saddles. the steady equation from (16) is f η − 1 6 ηξξ − 3 4 η2 = 1 2 h − e, (20) where e is a constant that enforces zero mean free surface elevation lim l→∞ 1 2l ∫ l −l ηdξ = 0. (21) thus, we have e = lim l→∞ 1 2l ∫ l −l 3 4 η2dξ ≥ 0. (22) if η tends to zero at both ξ = ±∞, then e = 0. there are physically important steady solutions for which the free surface is constant (but not zero) upstream of the bump and either constant or periodic downstream. in these cases, e �= 0. this state is reached as t → ∞ in (16) only if work is continually done by the topography on the fluid (compare (18) and (22)). 6, 4(2004) waves generated by topography 39 3.1 flat bottom for h = 0, the solutions η can be described in the phase plane with the equations for s ≡ ηξ and η ηξ = s, sξ = 6f η − 9 2 η2 + 6e. (23) when e = 0, the system has, for f > 0, a saddle at η = 0 and a center at η = 4 3 f , and, for f < 0, a center at η = 0 and a saddle at η = 4 3 f . (there is a transcritical bifurcation at f = 0). the system (23) is integrable, with the integral curves satisfying h(η, s) = f η2 − 1 6 s2 − 1 2 η3 + 2eη = constant. (24) the phase plane for e = 0 is shown in figure 2. the homoclinic orbit for f > 0 corresponds to the solitary wave of elevation (19), which has zero mean surface elevation. the maximum slope of the solitary wave is smax = 4 √ 2 3 f 3/2 and its amplitude is umax = 2f . note that for f < 0 the homoclinic orbit has limξ→±∞ η �= 0, meaning that the mean surface elevation is no longer zero, and, therefore, that this solution is unphysical. the periodic orbits in figure 2 correspond to nonlinear periodic waves which are also called cnoidal waves from the cn elliptic functions that describe them. these periodic waves also have nonzero mean for the case e = 0, but we shall see that they are important and will require us to consider e �= 0. 3.2 dirac bump we consider now the case of a localized bump of the form h = αδ(ξ). in this case, integrating (20) through ξ = 0 one obtains s+ − s− = −3α, where s± = lim ξ→0± ηx. (25) thus the free surface has a corner over the extremum of the topography. a positive bump (α > 0) corresponds to a negative jump in the slope of the free surface, and, conversely, a depression on the bottom gives a positive jump in the free surface slope. for f > 0, we show in figure 3 how to construct steady bounded solutions with mean zero and the appropriate jump in s over the bump. solutions shown include a vertical jump of size 3α connecting to the smooth solutions away from the bump. for α < 0 there is a simple solution where the free surface has a localized depression (see the η < 0 trajectory) with a corner at the trough (see figure 4 (c)). this solution exists for all α < 0. there are other more complicated solutions obtained by following the homoclinic orbit for more than half of its length and jumping up to either the upper branch of the orbit (yielding a wave of elevation with a dimple at the crest see figure 4 (d)) or to a periodic solution (which would require e �= 0). these more 40 paul a. milewski 6, 4(2004) η s 3 α figure 3: solutions to the steady problem for f > 0 with dirac topography h = αδ(x). the free-surface has a jump in slope of −3α over the bump. there are two solutions shown for α > 0 and one for α < 0. other possible solutions are discussed in the text. complicated steady configurations exist for − 8 9 √ 2f 3/2 < α < 0 but are not usually observed for the full problem. for α > 0 there are two e = 0 solutions, both with corners at the crest from the two possible equal jumps in ηx for η > 0 trajectories: a small amplitude solution (see figure 4 (a)), and a large amplitude solution similar to the solitary wave (see figure 4 (b)). these solutions exist only for 0 < α < 8 9 √ 2f 3/2. for larger bumps there are no steady solutions. alternatively, for a fixed bump size there is a fcrit > 0 below which there is no steady solution and above which there are two steady solutions. the behavior is that of a saddle-node bifurcation as f is increased through fcrit. in addition to these, jumps with 0 < α < 8 9 √ 2f 3/2, may also connect to periodic solutions if e �= 0. for f < 0 there are no localized solutions satisfying η → 0 since the η = 0 fixed point is now a center. figure 5 however, shows the possibility of a steady horizontal free surface on one side of the obstacle and a periodic wave on the other side. to understand whether these waves appear upstream or downstream, we consider an argument [5] based on a linearization of (16). the behavior of the linear waves is obtained by substituting η = exp(ikξ − ωτ ) into the linearized and unforced (16) and obtaining ω = f k + 1 6 k3, cp = f + 1 6 k2, cg = f + 1 2 k2. (26) here ω is the frequency, k the wavenumber (inversely proportional to wavelength), and cp, cg, the phase and group velocity, respectively. for f < 0, steady waves of 6, 4(2004) waves generated by topography 41 −5 0 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 (a) −5 0 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 (c) −5 0 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 (b) −5 0 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 (d) figure 4: sketches of various solutions obtained from figure 3 for e = 0. (a) small amplitude wave over bump; (b) large amplitude wave over bump; (c) localized depression over hole; (d) dimpled wave over hole. solutions with periodic waves (e �= 0) can be found in [3]. η s 3 α figure 5: solutions to the steady problem for f < 0 with dirac topography h = αδ(x). the free-surface has a jump in slope of −3α over the bump. the solution shown for α > 0 connects an upstream flat free surface to periodic waves downstream. if the jump is sufficiently large then the solution is a hydraulic fall to a downstream horizontal free surface at the level of the saddle. the solutions for α < 0 are similar. 42 paul a. milewski 6, 4(2004) wavenumber k = √ −6f (27) (obtained at cp = 0) are expected. these waves will form downstream from the obstacle since they propagate at the group speed and cg > 0 for these waves. these flat-periodic solutions, however, do not have mean zero and require a nonzero e. the effect of e > 0 on figure 5 is to move the center to the right (η > 0 upstream) and the saddle to the left. however, e is a priori unknown, and therefore the simple arguments for the scaling between α and f valid for f > 0 do not apply. in the limiting case the solution can jump from the center η > 0 upstream to the homoclinic orbit, asymptoting to η < 0 downstream. we shall describe this as a “hydraulic fall” over the bump. note that the solutions here are similar for α > 0 and α < 0. there are several authors that have computed steady solutions, similar to those described above, to the full potential flow problem with a free surface. see, for example, [10]. the solutions obtained are qualitatively similar to those discussed here. throughout the discussion of steady configurations, one must keep in mind that the solutions found may not be stable, and, therefore, may not be observed in time-dependent solutions. 4 localized topography and time dependent solutions 4.1 hydraulic approximation another possible approximation of (16) is the hydraulic, or shallow water approximation [4]. in this case, we neglect the dispersive term and obtain ητ + f ηξ − 3 2 ηηξ = 1 2 hξ(ξ). (28) although our results here are independent of the precise shape of the bump we may think of h as a gaussian: h(x) = α exp(−ξ2). this “forced” burger’s equation can be solved by the method of characteristics for ξ(τ, ξ0) and u(τ, ξ0) dξ dτ = f − 3 2 u, ξ(0) = ξ0, (29) du dτ = 1 2 hξ, u(0) = 0. (30) from these, one can obtain an equation for the characteristics dξ dτ = ±f ( 1 − 3 2 h(ξ) − h0 f 2 )1 2 , (31) where h0 = h(ξ0). the characteristics will change directions over the topography if 3 2 δh > f 2, where δh is the maximum height change of the topography. from (31) 6, 4(2004) waves generated by topography 43 figure 6: free surface evolution of a flow over a bump according to the forced burgers (28) equation for the case 3 2 δh > f 2 (here f = 0, δh = 0.1). hydraulic fall over a bump (top) and a shock trapped in a hole (bottom). flow is from left to right. and (29) one obtains, for smooth parts, u = 2 3 f ( 1 ∓ √ 1 − 3 2 h − h0 f 2 ) (32) when the characteristics turn, the sign in (31) changes, and u = 2 3 f . when characteristics intersect, a shock forms, and from (28) the shock speed s is s = f − 3 4 (ul + ur), (33) where ul, ur are the left and right values of u at the shock. we discuss below the main cases that occur, and illustrate them with time-dependent calculations based on a godunov scheme [6]. for a positive bump and f > 0, there will always be a shock forming downstream since characteristics starting over the bump will accelerate descending it and cross the characteristics from the flat part of the bottom. since ul < 0 and ur is zero, for f > 0 this shock will be carried downstream. 44 paul a. milewski 6, 4(2004) if 3 2 δh > f 2, upstream characteristics can not cross over the bump. since they must turn around, there will be an upstream shock with ul = 0, ur > 0 and s < 0. thus, after long times, and in the vicinity of the bump, the solution will be a hydraulic fall: a smooth transition between an elevated flat free surface upstream and a lower flat surface downstream (see figure 6 top). if 3 2 δh < f 2, upstream characteristics can travel over the bump and continue downstream. the downstream shock propagates away, thus leaving a steady localized elevation of the free surface over the bump (see figure 7 top). figure 7: free surface evolution of a flow over a bump according to the forced burgers (28) equation for the case 3 2 δh < f 2 (here f = 0.5, δh = 0.1. flow over a bump (top) and over a hole (bottom). flow is from left to right. for a positive bump and f < 0, the solutions are similar to those above since the equation (28) is invariant under f → −f , u → −u, and ξ → −ξ. the solutions change considerably for a negative bump (hole). here, for f = 0 the solution will form a shock which remains trapped in the hole (see figure 6 bottom). for f > 0 a shock will form either in the hole or downstream of the hole and propagate downstream, leaving a localized depression in the free surface over the hole (see figure 7 bottom). similarly, the behavior for f < 0 can be obtained by the invariances of the equations. there is never a hydraulic fall in this case. 6, 4(2004) waves generated by topography 45 a complete characterization of solutions to the forced burgers equation in a periodic domain can be found in [6]. 4.2 time dependent korteweg-de vries solutions figure 8: free surface evolution of a supercritical flow over a bump according to the forced kdv (16) equation with h = 1 2 √ 2 π exp ( − x2 2 ) , and f = 0.5 (top), f = 0.25 (bottom). flow is from left to right. with the understanding of steady forced kdv solutions and of the shallow water burgers’ limit, we now discuss typical time dependent solutions for the forced kdv equation. all solutions shown are computed using a pseudo-spectral numerical scheme [8]. the domain on which the solution was computed is periodic, however, only the neighborhood of the obstacle is shown. the initial data, for all cases, is zero. we show first a sequence of distinct cases with a localized bump h > 0. the amplitude of the bump is fixed in the four pictures, but the froude number f changes from strongly supercritical in figure 8 (top) to strongly subcritical in figure 9 (bottom). the flow is always from left to right.the four distinct cases are: 1. figure 8 (top): the flow is strong enough that all transients are swept downstream and the free surface in the vicinity of the bump is steady with an elevation over the bump. this steady solution near the bump is the smooth version of the case described in 46 paul a. milewski 6, 4(2004) figure 4 (a), and the wave transient carried downstream is an ”undular bore”: a wave connection between the two different free-surface levels on either side of the downstream shock in figure 7 top. 2. figure 8 (bottom): the froude number is now reduced so that there is no possible steady solution (f < fcrit in figure 3), and, in addition to the transient bore downstream there is a periodic generation of upstream waves. figure 9: free surface evolution of a subcritical flow over a bump according to the forced kdv (16) equation with h = 1 2 √ 2 π exp ( − x2 2 ) , and f = −0.25 (top), f = −0.5 (bottom). flow is from left to right. the upstream waves are free solitary waves of the kdv equation and are periodically shed when the free surface response over the bump becomes too large. this is a new, genuinely unsteady nonlinear phenomenon, not captured by simpler models. there is also a net hydraulic fall over the obstacle as in figure 6 top. 3. figure 9 (top): the flow is now slightly subcritical. there is a steady solution consisting of a “hydraulic fall” over the bump, as in figure 6 top, and there are transient upstream and downstream undular bores. this case is related to figure 5. 4. figure 9 (bottom): the flow is now strongly subcritical, and the steady solution corresponds to an upstream surface elevation with a downstream periodic wave train (also called lee waves) which is steady in the frame of the bump. this case is related to figure 5. 6, 4(2004) waves generated by topography 47 next, we consider a similar sequence for h < 0 (hole). 1. figure 10 (top): the flow is sufficiently strong that all transients are swept downstream and the free surface in the vicinity of the bump is a steady depression. this is a smooth version of the case described in figure 4 (c). figure 10: free surface evolution of a supercritical flow over a hole according to the forced kdv (16) equation with h = − 1 2 √ 2 π exp ( − x2 2 ) , f = 0.25 (top), f = 0 (bottom). flow is from left to right. 2. figure 10 (bottom): the froude number is now reduced and there is a periodic generation of upstream waves, and a hydraulic fall over the hole. the unsteady flow directly over the topography is more dramatic than in the bump case with large free surface variations. these probably arise from the hydraulic tendency to trap the shock. 3. figure 11 (top): the flow is now slightly subcritical, and the flow is similar to the previous case, but the waves are of smaller amplitude. the hydraulic fall over the bump can be predicted from figure 5 for sufficiently large topography (α) relative to f . 4. figure 11 (bottom): the flow is now strongly subcritical, and there are steady lee waves which also can be explained from figure 5 for smaller α relative to f . solutions of the full time-dependent free-surface navier-stokes equations have also been computed for the flow over a bump [11]. there is qualitative agreement between 48 paul a. milewski 6, 4(2004) those computations and those of kdv. figure 11: free surface evolution of a subcritical flow over a hole according to the forced kdv (16) equation with h = − 1 2 √ 2 π exp ( − x2 2 ) , f = −0.25 (top), f = −0.5 (bottom). flow is from left to right. 5 extended topography we now discuss some of the phenomena for extended topographical features such as plateaus (β > 0 below) and wide valleys (β < 0). consider the topography given by h = β [h(x + a) − h(x − a)], where h(x) is the unit step function and a 1. then, seeking steady solutions, one would replace (20) by f η − 1 6 ηξξ − 3 4 η2 = 1 2 β [h(ξ + a) − h(ξ − a)] − e. (34) thus, at ξ = ±a the solutions and are continuous and have continuous derivatives, but have discontinuous second derivatives. they satisfy the equation with an additional constant term β for −a < ξ < a. as an example, let us consider the subcritical case (where the steady problem has the phase plane as in figure 2 (a)) of flow over a plateau (β > 0). if e = 0, over the 6, 4(2004) waves generated by topography 49 plateau the fixed points move to η = 2 3 ( f ± √ f 2 − 3β ) . thus a continuous solution starting upstream on the center at the origin in figure 2 (a) will start a wave over the plateau since the new center will move to the left. this wave will have a negative mean surface elevation. then, at the end of the plateau, the center moves back to the origin, and the wave will continue as a trajectory about the origin. (it is possible to construct a steady solution with no waves downstream of the plateau for particular lengths of the plateau a nonlinear eigenvalue problem however these would not be physically relevant.) figure 5 shows a subcritical flow’s free surface elevation in the vicinity of a plateau and we see that the solution matches the description above. the wavelength over the plateau is longer than the wavelength downstream because the “effective” froude number (u/ √ gh) over a plateau is larger and, from (27) this yields a longer wave. in numerical calculations we smoothen the step functions by using 1 + 1 2 tanh(x). figure 12: grayscale plot of the free surface elevation of subcritical flow over a plateau situated at −20 < ξ < 20. in this calculation, β = 0.25 and f = −0.85. flow is from left to right. if f is large one can argue that there is a steady free surface displacement with the same shape as the topography (and this is confirmed by computations), however for flows that are close to critical the behavior can be quite striking, especially in the case of a wide valley. figure 13 shows the time dependent evolution of flow over a valley. first, solitary upstream propagating waves are formed at the downstream wall of the valley. when they reach the upstream wall they either reflect back downstream or continue to propagate upstream away from the valley. only sufficiently large waves can leave the valley upstream since they encounter an “effectively” larger froude number as they exit the valley and therefore they must slow down. the reflected waves then appear to “perturb” the apparently unstable flow at the downstream wall and generate new upstream waves. 50 paul a. milewski 6, 4(2004) figure 13: grayscale plot of the free surface elevation of supercritical flow over a wide valley situated at −20 < ξ < 20. in this calculation, β = −0.25 and f = 0.25. flow is from left to right. 6 conclusion we have reviewed many aspects of a simple model for the free-surface flow over topography. although the model is simple, it contains many of the phenomena of the full problem such as hydraulic falls, lee waves and upstream nonlinear wave propagation. for flows over a wide valley there are particularly striking transient flows with combinations of multiple reflections of trapped waves and escaping solitary waves. some natural extensions to consider are the two-dimensional free surface case, and models for stratified fluid that allow for vertical propagation of wave energy. references [1] baines, p., 1995: topographic effects in stratified flows, cambridge university press [2] choi, j.; milewski, p. a. resonances between surface waves and periodic topography. to appear in studies in appl. math. (2002) [3] dias, f.; vanden-broeck, j.-m. . to appear in phil. trans. r. soc. lond. a (2002). [4] grimshaw, r. h. j.; smith, n., 1986 resonant flow of a stratified fluid over topography. j. fluid mech. 169, 429–464. [5] lighthill, j., 1978: waves in fluids, cambridge university press 6, 4(2004) waves generated by topography 51 [6] menzaque, f.; rosales, r. r.; tabak, e. g.; turner, c. v., 2001: the forced inviscid burgers equation as a model for nonlinear interactions among dispersive waves. contemporary mathematics 283, 51–82. [7] mielke, a., 1986: steady flows of inviscid fluids under localized perturbations. j. diff. eq. 65, 85–116. [8] milewski, p. a.; tabak, e. g., 1999: a pseudo-spectral algorithm for the solution of nonlinear wave equations, siam j. sci. comp. 21, 1102–1114. [9] milewski, p. a.; vanden-broeck, j.-m., 1999: time-dependent gravity-capillary flows past an obstacle, wave motion 29, 63–79. [10] vanden-broeck, j.-m., 1988: free surface flow over a semi-circular obstruction in a channel, phys. fluids 30 2315–2317. [11] nadiga, b. t.; margolin, l. g.; smolarkiewicz, p. k., 1996: different approximations of shallow fluid flow over an obstacle, phys. fluids 8 2066–2077. [12] wu, t. y., 1987: generation of upstream advancing solitons by moving disturbances, j. fluid. mech. 184 75–99. cubo a mathematical journal vol.11, no¯ 04, (29–48). september 2009 towards accurate artificial boundary conditions for nonlinear pdes through examples xavier antoine, institut elie cartan nancy, nancy-université, cnrs, inria corida team, boulevard des aiguillettes b.p. 239 f-54506 vandoeuvre-lès-nancy, france. email: xavier.antoine@iecn.u-nancy.fr christophe besse projet simpaf-inria futurs, laboratoire paul painlevé, unité mixte de recherche cnrs (umr 8524), ufr de mathématiques pures et appliquées, université des sciences et technologies de lille, cité scientifique, 59655 villeneuve d’ascq cedex, france. email: christophe.besse@math.univ-lille1.fr and jérémie szeftel 1 department of mathematics, princeton university, fine hall, washington road, princeton nj 08544-1000, usa. and cnrs, mathématiques appliquées de bordeaux, université bordeaux 1, 351 cours de la libération, 33405 talence cedex france. email: jszeftel@math.princeton.edu abstract the aim of this paper is to give a comprehensive review of current developments related to the derivation of artificial boundary conditions for nonlinear partial differential equations. the essential tools to build such boundary conditions are based on pseudodifferential and paradifferential calculus. we present various derivations and compare 1partially supported by nsf grant dms-0504720. 30 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) them. some numerical results illustrate their respective accuracy and analyze the potential of each technique. resumen la meta de este artículo es entregar una revisión comprensiva de los desarrollos actuales relacionados con la derivación de condiciones de borde artificiales para ecuaciones diferenciales parciales nolineales. las herramientas esenciales para construir tales condiciones de borde se basan en el cálculo pseudodiferencial y paradiferencial. presentamos varias derivaciones y las comparamos. algunos resultados numéricos ilustran su precisión respectiva y se analiza el potencial de cada técnica. key words and phrases: nonlinear pdes, wave equation, schrödinger equation, artificial boundary conditions for nonlinear pdes, numerical schemes. math. subj. class.: 35a21, 35a27, 35l05, 35q55, 35s50, 65m99. 1 introduction the subject of designing artificial boundary conditions for linear scalar and systems of partial differential equations (pdes) has been studied since more than thirty years now. essentially, the goal of these boundary conditions is to truncate an infinite domain into a finite one for computational considerations. to this end, a fictitious boundary γ is introduced delimiting therefore a finite domain ω. all the difficulty is then to build an accurate, robust and easy-to-implement approximate boundary condition on this fictitious boundary. these boundary conditions can be found in the literature under different names (which in fact have different subtle meanings) like artificial boundary conditions, absorbing boundary conditions, non-reflecting or transparent boundary conditions and sometimes dirichlet-to-neumann operators. among the major contributions written on the topic and without being exhaustive, let us quote e.g. the papers by engquist and majda [9, 10], bayliss and turkel [4], mur [15] or also bérenger [5]. a few review papers have also been published to establish the current state-of-the-art on the subject (see e.g. [2, 11, 12, 22]). while many improvements have been achieved over the past years, most of them are developed for linear equations. practically, most available methods for linear equations do not apply to nonlinear equations since they often rely on the explicit computation of the transparent boundary condition by using the fourier or laplace transforms (note however that in the particular case of integrable equations, one may use the inverse scattering transform to explicitly compute the transparent boundary condition, see e.g. [23]). now, nonlinear problems have recently received some increasing special care because of their importance in applications like e.g. in wave propagation, quantum mechanics, fluid mechanics,... the aim of this paper is to give a comprehensive cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 31 introduction to possible solutions to handle such nonlinear situations. they are mainly based on pseudodifferential [21] and paradifferential operators techniques [6]. the paper is organized as follows. in section 2, we analyze in detail the way of constructing approximate artificial boundary conditions for a general one-dimensional wave equation with variable coefficients using pseudodifferential calculus. then, we test numerically these absorbing boundary conditions on a model problem showing that they yield accurate computations at least for small times. in a third section, we consider a general one-dimensional nonlinear schrödinger equation. we present several ways to extend the method of section 2 to this nonlinear equation depending on the kind of nonlinearity involved in the equation. the various types of absorbing boundary conditions are obtained using either the pseudodifferential or the paradifferential calculus. in a fourth section, some numerical comparisons are developed to test the accuracy of the various absorbing boundary conditions. the last section draws a conclusion and suggests some future directions of research. 2 artificial boundary conditions for linear variable coefficients equations: the case of the wave equation 2.1 the case of the constant coefficients wave equation before directly going to the case of the wave equation with variable coefficients, let us first consider the simple wave equation (∂ 2 t − ∂2x)u = 0, (2.1) with initial data u(0,x) = u0(x) and ∂tu(0,x) = u1(x), where the field u = u(t,x) is defined on the whole space (t,x) ∈ [0; +∞[×r. for simulation purposes, it is standard to introduce a bounded spatial computational domain setting now (t,x) ∈ [0; +∞[×[xℓ; xr], where xℓ (respectively xr) is a left (respectively right) fictitious endpoint introduced to get a bounded domain ω = [xℓ; xr]. let us assume that the initial data of our problem, that is u0 and u1, are compactly supported in ω. then, we can define the extension of u (which is still denoted by u) for negative times by fixing its value to zero so that u is a solution to (2.1) for all times t as long as x ∈ ωc, where ωc = r−ω. let us denote by ût(τ,x) = ft(u)(τ,x) the partial time-fourier transform of u, where τ ∈ r. applying ft to (2.1) for (t,x) ∈ r × ωc leads to the helmholtz-type constant coefficients equation (∂ 2 x + τ 2 )ût(τ,x) = 0, (2.2) where the wavenumber is τ. the solution of this equation can be written as the superposition of two waves ût(τ,x) = a + e iτx + a − e −iτx , (2.3) where a± are two smooth functions depending on τ. computing the derivative ∂xût, we obtain (∂x − iτ)ût = −2iτe−iτxa−, (2.4) 32 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) and (∂x + iτ)ût = 2iτe iτx a + , (2.5) and we obviously check that: (∂x + iτ)(∂x − iτ)ût = 0. we also have the following operator factorization ∂ 2 t − ∂2x = −(∂x + ∂t)(∂x − ∂t). (2.6) equation (2.4) (respectively (2.5)) gives a characterization of the right (respectively left) traveling solution to (2.1) by setting: a− = 0 (respectively a+ = 0). therefore, the following boundary condition (2.4) (∂ n − iτ)ût = 0, at γ, (2.7) acts as a filter in the time-fourier domain and translates the property that there is no reflection back into the computational domain ω, where n is the unit normal vector to γ = {xℓ; xr}, outwardly directed to ω. this is a constraint which forces the wave to be outgoing to ω. in the time-space domain, the corresponding boundary condition writes down (∂ n − ∂t)u = 0, at [0; +∞[×γ. (2.8) since there is no reflection, this boundary condition is usually called transparent or non-reflecting boundary condition (tbc). let us remark at this point that another interpretation of writing a transparent boundary condition is that we require u ∈ ker(∂ n − b), setting b(x,t,∂t) = ∂t. 2.2 the case of the variable coefficients wave equation let us consider that α, β and γ are three c∞ functions. writing a tbc for a variable coefficient model wave equation (∂ 2 t + β(t,x)∂t − ∂2x + γ(t,x)∂x + α(t,x))u = 0, (2.9) is much more complicate than in the constant coefficients case. indeed, in such a situation i) directly applying a time-fourier transform to the equation (2.9) leads to a convolution equation which is extremely difficult to write down explicitly, ii) and even if it is possible to write an inhomogeneous helmholtz-type equation, solving this equation for general functions associated with α, β and γ cannot be expected. building an accurate boundary condition which approximates the tbc can however be expected since the pioneering work of engquist & majda [9, 10] in the middle of the seventies using the theory of reflection of singularities at the boundary [14] and pseudodifferential calculus (see for example [21]). let us develop the main ideas. like in the previous situation with constant coefficients, we assume that u as been extended by zero for negative times t and that the initial data u0 cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 33 and u1 are compactly supported in ω. then, engquist and majda prove that there exist two classical pseudodifferential operator a and b of ops1 such that we get the following nirenberg-like factorization [16] −∂2x + ∂2t + β(t,x)∂t + γ(t,x)∂x + α(t,x) = −(∂x − a(x,t,dt))(∂x − b(x,t,dt)) + r, (2.10) where r is a smoothing operator of ops−∞. this approximate factorization can be considered as the extension to the variable coefficients case of the exact form (2.6). actually, the smoothing operator r accounts for the fact that the factorization is now true only at high frequencies. the two pseudodifferential operators a(x,t,dt) and b(x,t,dt), with dt = −i∂t, have respective associated symbols a(x,t,τ) and b(x,t,τ) of s1 admitting the following asymptotic expansions in homogeneous symbols a(x,t,τ) ∼ ∑ j≥0 a1−j(x,t,τ) and b(x,t,τ) ∼ ∑ j≥0 b1−j(x,t,τ), (2.11) with classical homogeneous symbols a1−j and b1−j of order 1 − j. this means that we have e.g. a1−j(x,t,λτ) = λ 1−j a1−j(x,t,τ), ∀λ > 0. developing the factorization (2.10), we get −∂2x + γ(t,x)∂x + ∂2t + β(t,x)∂t + α(t,x) = −∂2x + (a + b)∂x − ab + op(∂xb) + r (2.12) since ∂x(bu) = op(∂xb)u + b∂xu. in the above equation, we designate by op(σ) the pseudodifferential operator with symbol σ. if it is possible to compute a and b then, it can be proved that the tbc for equation (2.9) is given by (∂ n − b(x,t,dt))u = 0, at [0; +∞[×γ. (2.13) indeed, the results in [14] imply that (2.13) annihilates the wave reflected back in the computational domain. generally speaking, this tbc, which extends (2.8), cannot be directly implemented since b is given by an infinite expansion, but it can however be approximated by a k-th order artificial boundary condition by truncating the series (2.11) to the first k symbolic terms and considering (∂ n − k−1∑ j=0 b1−j(x,t,dt))u = 0, at [0; +∞[×γ. (2.14) the computation of the terms {b1−j}k−1j=0 is therefore needed. to this end, we identify the operators on both sides of equality (2.12) and we obtain at the symbolic level the system { a + b = γ −a#b + ∂xb = −τ2 + iβτ + α, (2.15) which can also be rewritten as b#b − γb + ∂xb = −τ2 + iβτ + α, (2.16) by eliminating a. in the above notations, a#b designates the symbol of the composition operator ab which admits the following expansion (see for example [21]): b#b ∼ ∞∑ m=0 (−i)m m! ∂ m τ b ∂ m t b. (2.17) 34 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) since b is developable in terms of homogeneous symbols using relation (2.11), we can extract each term b1−j from (2.16) by identifying the decaying order terms starting from 2 to 2 −j using (2.17). beginning with b1, we get that b1(x,t,τ) = iτ, (2.18) fixing also the uniqueness of the expansion (2.11). computing the two next terms gives    b0 = β + γ 2 , b−1 = (γ 2 − β2) + 4α − 2(∂x + ∂t)(γ + β) 8iτ , (2.19) at the right point xr. it directly gives the following proposition. proposition 2.1. let u be the solution to the generalized wave equation (2.9) with c∞ variable coefficients α, β and γ. then, the artificial boundary conditions of order k, for k = 1, 2, 3, are respectively given at the right endpoint xr by ∂xu − ∂tu = 0, at [0; +∞[×{xr}, ∂xu − ∂tu − β + γ 2 u = 0, at [0; +∞[×{xr}, ∂xu − ∂tu − β + γ 2 u − (γ 2 − β2) + 4α − 2(∂x + ∂t)(γ + β) 8 itu = 0, at [0; +∞[×{xr}, (2.20) where it is defined by itu(t) = ∫ t 0 u(s)ds. similar formulas can be derived at xℓ. 2.3 short-time vs long-time behavior let us now consider that we wish to compute a numerical solution to the problem    (∂ 2 t + β(t,x)∂t − ∂2x + γ(t,x)∂x + α(t,x))u = 0, (t,x) ∈]0; t [×ω, u(0,x) = u0(x), x ∈ ω, ∂tu(0,x) = u1(x), x ∈ ω, (∂ n − k−1∑ j=0 b p 1−j(x,t,dt))u = 0, at [0; t ] × {xp}, (2.21) for a maximal time of computation t and where the k-th artificial boundary condition is defined by operators {bp 1−j} k−1 j=0 , for p = ℓ,r, at the left or right fictitious point xp (see e.g. proposition 2.1). introducing n intervals of discretization in time, we denote by ∆t the time step defined by ∆t = t/n. we next seek to compute an approximate solution un(x) ≈ u(tn,x) to system (2.21), with tn = n∆t, for n ∈ {1, ...,n}. we have seen before that the derivation of the artificial boundary conditions at the continuous level is made under the high frequency assumption |τ| ≫ 1. since we consider discrete times tn, for n = 1, ...,n, discrete time frequencies τn = π/tn are then associated and lie in the interval [π/t ; π/∆t]. hence, the artificial boundary conditions which work well for high frequencies will be accurate if t ≪ 1. this means that an artificial boundary cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 35 condition is accurate as long as it is used for small times of computation. they may fail for large computational times which is a known problem of artificial boundary conditions techniques (see e.g. [8] [13]). as an illustration, we compare in figure 1 the performances of the artificial boundary conditions of order 1, 2 and 3 in the case of the wave equation (∂2t + ∂t − ∂2x)u = 0. we give the relative error in the l2(ω)-norm for times between 0 and 10. as predicted by the theory, we notice indeed an improvement for small times by increasing the order. the second order condition is more efficient than the first order condition for all computed times but the third order condition is more efficient than the second order condition only for t ≤ 5.2. 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 figure 1: (∂2t + ∂t − ∂2x)u = 0. relative error in l2 norm in function of time. abc −− order 1, −− order 2 and order 3 · − ·. 3 different approaches for nonlinear equations: the case of nonlinear schrödinger equations 3.1 nonlinear and linear schrödinger equations we have seen in section 2 that it is possible to build accurate artificial boundary conditions using techniques based on pseudodifferential calculus in the model case of the linear wave equation 36 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) with variable coefficients. the aim of the present section is to develop some applications of the pseudodifferential calculus method and its nonlinear version, the paradifferential calculus strategy, to obtain accurate artificial boundary conditions for nonlinear equations. as a model equation, we consider the time dependent nonlinear schrödinger equation { i∂tu + ∂ 2 xu + α(u)∂xu + β(u)u = 0, (t,x) ∈ [0; +∞[×r, u(0,x) = u0(x), x ∈ r, (3.1) where we assume again that the initial condition u0 has compact support in ω and that α and β are two c∞ functions. let us now consider the following associated variable coefficients linear schrödinger equation { i∂tu + ∂ 2 xu + a(t,x)∂xu + b(t,x)u = 0, (t,x) ∈ [0; +∞[×r, u(0,x) = u0(x), x ∈ r. (3.2) extending the previous strategy presented for the variable coefficients wave equation in section 2.2 to equation (3.2), one can prove that there exist two pseudodifferential operators a(x,t,dt) and b(x,t,dt) such that we have ∂ 2 x + i∂t + a∂x + b = (∂x − a(x,t,dt))(∂x − b(x,t,dt)) + r, (3.3) where again r ∈ ops−∞. the operators a and b are elements of ops1/2 admitting the following expansion in homogeneous symbols a(x,t,τ) ∼ ∞∑ j=0 a(1−j)/2(x,t,τ) and b(x,t,τ) ∼ ∞∑ j=0 b(1−j)/2(x,t,τ), (3.4) where a(1−j)/2 and b(1−j)/2 are homogeneous symbols of order (1 − j)/2. this means that ∀λ > 0, we have: a(1−j)/2(x,t,λτ) = λ (1−j)/2 a(1−j)/2(x,t,τ), b(1−j)/2(x,t,λτ) = λ (1−j)/2 b(1−j)/2(x,t,τ). (3.5) if one considers e.g. the right fictitious point xr, then by fixing b1/2(x,t,τ) = − √ τ, where √ τ is defined by: √ τ =    √ τ if τ ≥ 0, −i √ −τ if τ < 0, (3.6) it can be shown (see [17, 20]) that the tbc is given by (∂ n − b(x,t,dt))u = 0, at [0; +∞[×{xr}, (3.7) and that an approximate artificial boundary condition of order k is (∂ n − k−1∑ j=0 b(1−j)/2(x,t,dt))u = 0, at [0; +∞[×{xr}, (3.8) cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 37 with the convention that the artificial boundary condition of order zero corresponds to the neumann boundary condition. the required inhomogeneous symbols can be obtained by adapting relation (2.16) b#b + ab + ∂xb = τ − b, (3.9) using the suitable substitutions. then, using the leibniz symbolic rule we get the four first symbols b1/2 = − √ τ,b0 = − a 2 ,b−1/2 = − 1 2 √ τ ( a 2 4 − b + ∂xa 2 ), b−1 = − 1 8τ (a∂xa + ∂ 2 xa − 2∂xb + i∂ta). (3.10) to explain the different strategies which can be considered, we propose now to investigate first the case α = 0 and next to detail the situation when β = 0, where α and β are the functions in (3.1). 3.2 case i: α = 0 3.2.1 potential strategy the point of view adopted in this strategy considers that α(u) and β(u) act as potential functions independent of u. more specifically, they have respectively corresponding functions a and b in equation (3.2). if we assume that a = 0, then, the symbols in (3.10) simplify as b1/2 = − √ τ,b0 = 0,b−1/2 = b 2 √ τ ,b−1 = ∂xb 4τ . (3.11) using the definition (3.8), we obtain the following artificial boundary conditions of order k at [0; +∞[×{xr}    ∂ n u + e −i π 4 ∂ 1/2 t u = 0, for k = 1, 2, ∂ n u + e −i π 4 ∂ 1/2 t u − ei π 4 b 2 i 1/2 t u = 0, for k = 3, (3.12) and finally ∂ n u + e −i π 4 ∂ 1/2 t u − ei π 4 b 2 i 1/2 t u − i ∂ n b 4 itu = 0, for k = 4. (3.13) in the above equations, the fractional half-order derivative operator ∂ 1/2 t , with symbol √ −iτ, is given by ∂ 1/2 t f(t) = 1√ π ∂t ∫ t 0 f(s)√ t − s ds, (3.14) and the half-order integration operator i 1/2 t (with symbol (−iτ)−1/2) is defined by i 1/2 t f(t) = 1√ π ∫ t 0 f(s)√ t − s ds. (3.15) 38 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) following our strategy, we replace b by the nonlinearity β(u) which gives the three following artificial boundary conditions of order k    ∂ n u + e −i π 4 ∂ 1/2 t u = 0, for k = 1, 2, ∂ n u + e −i π 4 ∂ 1/2 t u − ei π 4 β(u) 2 i 1/2 t u = 0, for k = 3, ∂ n u + e −i π 4 ∂ 1/2 t u − ei π 4 β(u) 2 i 1/2 t u − i ∂ n β(u) 4 itu = 0, for k = 4. (3.16) these conditions will be denoted by abc β 1,k in the sequel of the paper. 3.2.2 gauge change strategy let us remark that the artificial boundary condition (3.13) is not a transparent boundary condition even when b is a constant potential. now, in the case of a time-dependent potential b(x,t) = b(t), one can get the transparent boundary condition by using the gauge change v(x,t) = e −ib(t) u(x,t), (3.17) where b(t) = itb(t), and noticing that v is now solution to the free schrödinger equation i∂tv + ∂ 2 xv = 0. (3.18) then, the transparent boundary condition ∂ n v + e −i π 4 ∂ 1/2 t v = 0 (3.19) holds for v and, coming back to the initial unknown u, we obtain the transparent boundary condition for u ∂ n u + e −i π 4 e ib(t) ∂ 1/2 t (e −ib(t) u(x,t)) = 0. (3.20) this boundary condition is clearly not exact if b also depends on x. nevertheless, we can use a similar change of gauge, that is v(x,t) = e −ib(t,x) u(x,t), (3.21) with b(t,x) = itb(t,x). then, v is sought as the solution to the variable coefficients schrödinger equation i∂tv + ∂ 2 xv + (2i∂xb)∂xv + (i∂2xb − (∂xb)2)v = 0, (3.22) which is of the general form (3.2) with initial condition v(x, 0) = u0(x). we can therefore apply the previous general derivation of artificial boundary conditions of section 3.1 to this equation of unknown v for suitably defined variable coefficients. as a consequence, if { b(1−j)/2 } j≥0 designates the symbolic asymptotic expansion of the transparent boundary condition associated with v solution to (3.22), then an artificial boundary condition of order k is given for u as ∂ n u − k−1∑ j=0 e ib b(1−j)/2(x,t,dt)(e −ib u) − i∂ n bu = 0, at [0; +∞[×{xr}. (3.23) cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 39 more precisely, using (3.10), the computation of the first four symbols gives b1 = − √ τ, b0 = −i∂nb, b−1/2 = 0, b−1 = ∂ n b 4τ . (3.24) finally, we obtain [3] the secondand fourth-order artificial boundary conditions given respectively by ∂ n u + e −i π 4 e ib ∂ 1/2 t (e −ib u) = 0, at [0; +∞[×{xr}. (3.25) and ∂ n u + e −i π 4 e ib ∂ 1/2 t (e −ib u) − i∂nb 4 e ib it(e −ib u) = 0, at [0; +∞[×{xr}. (3.26) replacing b by β(u), the associated nonlinear artificial boundary conditions are then given by ∂ n u + e −i π 4 e ib(u) ∂ 1/2 t (e −ib(u) u) = 0, at [0; +∞[×{xr}. (3.27) and ∂ n u + e −i π 4 e ib(u) ∂ 1/2 t (e −ib(u) u) − i∂nβ(u) 4 e ib(u) it(e −ib(u) u) = 0, at [0; +∞[×{xr}, (3.28) setting b(u)(x,t) = it(β(u))(x,t). these conditions will be referred to as abc β 2,j in the sequel, for j = 2, 4. let us develop the connection existing between the artificial boundary conditions abc β m,j, for m = 1, 2 and j = 2, 4. to this end, let us recall the following leibniz formula for computing the fractional derivative of the product of two functions ∂ p t (fg) = ∞∑ k=0 γs(p + 1) k!γs(p − k + 1) ∂ k t f∂ p−k t g, (3.29) for p > 0. the real-valued function f is supposed to be c∞ and g is a continuous function. the notation γs designates the gamma special function. for p = 1/2, we obtain ∂ 1/2 t (fg) = f∂ 1/2 t g + 1 2 ∂tfi 1/2 t g + r, (3.30) where r is an error operator in ops−3/2. using a similar formula for the integral operator gives it(fg) = fitg + s (with s ∈ ops−2). using these two relations to approximate the half-order operator in (3.26) by setting f = e−ib and g = u, we see that (3.26) exactly corresponds to (3.13) up to an operator in ops−3/2. this error may be not negligible since is involves time derivatives of the potential, and, in the nonlinear case, of β(u). this difference can therefore be significant between the two kinds of artificial boundary conditions. this will be more deeply investigated during the numerical simulations. 40 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) 3.3 case ii: β = 0 3.3.1 potential strategy let us now consider the second case where β = 0. then, the potential strategy consists in replacing b = 0 in (3.10) leading to b1/2 = − √ τ,b0 = − a 2 ,b−1/2 = − 1 2 √ τ ( a 2 4 + ∂xa 2 ), b−1 = − 1 8τ (a∂xa + ∂ 2 xa + i∂ta). (3.31) this gives the following artificial boundary conditions    ∂xu + e −i π 4 ∂ 1/2 t u = 0, ∂xu + e −i π 4 ∂ 1/2 t u + a 2 u = 0, ∂xu + e −i π 4 ∂ 1/2 t u + a 2 u + e i π 4 2 ( a 2 4 + ∂xa 2 )i 1/2 t u = 0, (3.32) and ∂xu + e −i π 4 ∂ 1/2 t u + a 2 u + e i π 4 2 ( a 2 4 + ∂xa 2 )i 1/2 t u + i 8 (a∂xa + ∂ 2 xa + i∂ta)itu = 0, (3.33) at [0; +∞[×{xr}. again, replacing a by α(u) yields the nonlinear artificial boundary conditions of order k    ∂xu + e −i π 4 ∂ 1/2 t u = 0, for k = 1, ∂xu + e −i π 4 ∂ 1/2 t u + α(u) 2 u = 0, for k = 2, ∂xu + e −i π 4 ∂ 1/2 t u + α(u) 2 u + e i π 4 2 ( α(u) 2 4 + ∂xα(u) 2 )i 1/2 t u = 0, for k = 3, ∂xu + e −i π 4 ∂ 1/2 t u + α(u) 2 u + e i π 4 2 ( α(u) 2 4 + ∂xα(u) 2 )i 1/2 t u + i 8 (α(u)∂xα(u) + ∂ 2 xα(u) + i∂tα(u))itu = 0, for k = 4. (3.34) the set of above j-th order artificial boundary conditions will be called abcα 1,j is the sequel, for j = 1, ..., 4. 3.3.2 linearization strategy we linearize equation (3.1) with β = 0 around a mean state u and call v its linearization. we obtain i∂tv + ∂ 2 xv + α(u)∂xv + α ′ (u)∂xuv = 0, (3.35) which is of the form (3.2) with a = α(u) and b = α′(u)∂xu. equations (3.10) give b1/2 = − √ τ,b0 = − α(u) 2 ,b−1/2 = − 1 2 √ τ ( α(u) 2 4 − α ′ (u)∂xu 2 ) , b−1 = − 1 8τ (α(u)∂xα(u) − ∂2xα(u) + i∂tα(u)). (3.36) cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 41 this yields absorbing boundary conditions for the linearized problem. to go back to the original problem, we now need to "unlinearize" these boundary conditions. the first order absorbing boundary condition for v does not involve u and we immediately obtain for u ∂xu + e −i π 4 ∂ 1/2 t u = 0, for k = 1. (3.37) the second order absorbing boundary condition for v reads ∂xv + e −i π 4 ∂ 1/2 t v + 1 2 α(u)v = 0, (3.38) where α(u)v is the linearization of γ(u), where γ is the primitive of α vanishing at 0. thus, the unlinearization of (3.38) is: ∂xu + e −i π 4 ∂ 1/2 t u + 1 2 γ(u) = 0, for k = 2. (3.39) the unlinearization of the third and fourth order absorbing boundary conditions of v are far more challenging. we have to unlinearize: ∂xv + e −i π 4 ∂ 1/2 t v + 1 2 α(u)v + e i π 4 α(u) 2 8 i 1/2 t (v) − e i π 4 α ′ (u)∂xu 4 i 1/2 t (v) = 0, (3.40) and ∂xv + e −i π 4 ∂ 1/2 t v + 1 2 α(u)v + e i π 4 α(u) 2 8 i 1/2 t (v) − e i π 4 α ′ (u)∂xu 4 i 1/2 t (v) + i(α(u)∂xα(u) − ∂2xα(u) + i∂tα(u)) 8 it(v). (3.41) to achieve this goal, we rely on the paradifferential operators of j. m. bony [6] which are generalization of pseudodifferential operators well-suited to nonlinear problems. we refer to [17] [20] for details about these operators and about the rigorous unlinearization of (3.40) and (3.41). we finally obtain the following nonlinear artificial boundary conditions of order k for u at [0; +∞[×{xr}:    ∂xu + e −i π 4 ∂ 1/2 t u = 0, for k = 1, ∂xu + e −i π 4 ∂ 1/2 t u + γ(u) 2 = 0, for k = 2, ∂xu + e −i π 4 ∂ 1/2 t u + γ(u) 4 − e i π 4 4 i 1/2 t (α(u)∂xu) = 0, for k = 3, ∂xu + e −i π 4 ∂ 1/2 t u − e i π 4 2 i 1/2 t (α(u)∂xu) + i 8 it ( −α′(u)(∂xu)2 + α(u)2∂xu ) = 0, for k = 4, (3.42) where γ is the primitive of α vanishing at 0. from now, these j-th order artificial boundary conditions will be referred to as abcα 2,j, for j = 1, ..., 4. remark 1. the unlinearization step of the linearization strategy has been successful not only in the case of (3.1) with β = 0, but also in the case of the semilinear wave equation with various nonlinearities (see [19]). however, the unlinearization step of the linearization strategy is not always successful, as shown by the case of the cubic nonlinear schrödinger equation (see [20]). 42 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) 4 numerical examples we consider the model schrödinger equation (3.1) for the two specific cases i (α = 0) and ii (β = 0) of section 3. more specifically, we choose to present results in case i for the nonlinearity β(u) = |u|2, which corresponds to the well-known one-dimensional cubic nonlinear schrödinger equation, and in case ii for α(u) = u. we therefore will focus on systems (4.1) and (4.2) respectively given by { i∂tu + ∂ 2 xu + |u|2u = 0, (t,x) ∈ [0; +∞[×r, u(0,x) = u0(x), x ∈ r, (4.1) and { i∂tu + ∂ 2 xu + u∂xu = 0, (t,x) ∈ [0; +∞[×r, u(0,x) = u0(x), x ∈ r. (4.2) in both cases, we have to our disposal explicit solutions. concerning system (4.1), we consider the so-called soliton solution computed by using the inverse scattering theory and given by uex,α=0(x,t) = √ 2a sech( √ a(x − ct)) exp(ic 2 (x − ct)) exp(i(a + c 2 4 )t). (4.3) concerning system (4.2), adapting the cole-hopf transform [20] , we have the explicit solution uex,β=0(x,t) = ∫ 2 0 exp ( i (x − y)2 4t ) u0(y) exp( ∫ y 0 u0(s) 2 ds)dy × (√ iπt − ∫ x 0 exp(i y 2 4t )dy + ∫ 2 0 exp(i (x − y)2 4t ) exp( ∫ y 0 u0(s) 2 ds)dy + exp( ∫ 2 0 u0(s) 2 ds)( √ iπt − ∫ 2−x 0 exp(i y 2 4t )dy) )−1 , (4.4) which has a compact support in [0, 2] at time t = 0. in the two situations, we have to solve a nonlinear equation coupled with nonlinear boundary conditions. the schrödinger equations are discretized at time tn+1/2 = (tn+1 + tn)/2 by a secondorder approximation. in the sequel, if δt designates the time step, then tn = nδt stands for the n-th time step, where n ∈ n. the crank-nicolson schemes are adapted from the one proposed by durán and sanz-serna [7] and are given by i u n+1 − un δt + ∂ 2 x u n+1 + u n 2 + ∣∣∣∣ u n+1 + u n 2 ∣∣∣∣ 2 u n+1 + u n 2 = 0 (4.5) and i u n+1 − un δt + ∂ 2 x u n+1 + u n 2 + u n+1 + u n 2 ∂x ( u n+1 + u n 2 ) = 0 (4.6) respectively for cases i and ii. we denote here by un the approximate value of u at time tn. in order to reduce the computational time, we set 2vn+1 = un+1 +un, and the schemes read for n ≥ 0 2i v n+1 − un δt + ∂ 2 xv n+1 + |vn+1|2vn+1 = 0, (4.7) cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 43 (a) case i, a = 2, c = 15, θ = 0 (b) case ii figure 2: exact solutions representations. and 2i v n+1 − un δt + ∂ 2 xv n+1 + v n+1 ∂xv n+1 = 0, (4.8) imposing in both cases that v0 = 0. clearly, a general form of the previous schemes is 2i v n+1 − un δt + ∂ 2 xv n+1 + v n+1 f(v n+1 ) = 0, (4.9) where f designates the map |·|2 or ∂x according to the equation. the crank-nicolson approximation must be coupled to the boundary conditions (3.16), (3.27), (3.28), (3.34) and (3.42). since the jacobian of the maps associated to these nonlinear problems is difficult to obtain, we choose to use a classical fixed-point method based on the semi-discrete crank-nicolson schemes. the choice of a variational approximation method is thus obvious. here, we specifically use a p1 linear lagrange finite element approximation. the bounded computational domain is the open set ω =]xl,xr[. the fictitious boundary is limited to the two endpoints γ = {xl,xr}. at this point, let us note that the boundary conditions for the case β = 0 have been given explicitly only at the right endpoint. the left boundary conditions differ. concerning the potential strategy, the system of equations (3.34) is transformed on [0; +∞[×{xl} as:    ∂xu − e−i π 4 ∂ 1/2 t u = 0, for k = 1, ∂xu − e−i π 4 ∂ 1/2 t u + α(u) 2 u = 0, for k = 2, ∂xu − e−i π 4 ∂ 1/2 t u + α(u) 2 u − e i π 4 2 ( α(u) 2 4 + ∂xα(u) 2 )i 1/2 t u = 0, for k = 3, ∂xu − e−i π 4 ∂ 1/2 t u + α(u) 2 u − e i π 4 2 ( α(u) 2 4 + ∂xα(u) 2 )i 1/2 t u + i 8 (α(u)∂xα(u) + ∂ 2 xα(u) + i∂tα(u))itu = 0, for k = 4. (4.10) 44 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) concerning the linearization strategy, we get on [0; +∞[×{xl}:    ∂xu − e−i π 4 ∂ 1/2 t u = 0, for k = 1, ∂xu − e−i π 4 ∂ 1/2 t u + γ(u) 2 = 0, for k = 2, ∂xu − e−i π 4 ∂ 1/2 t u + γ(u) 4 + e i π 4 4 i 1/2 t (α(u)∂xu) = 0, for k = 3, ∂xu − e−i π 4 ∂ 1/2 t u + e i π 4 2 i 1/2 t (α(u)∂xu) + i 8 it ( −α′(u)(∂xu)2 + α(u)2∂xu ) = 0, for k = 4, (4.11) where γ is the primitive of α vanishing at 0. the boundary conditions are of memory-type and involve half-order fractional derivatives and integrals. to preserve the second-order approximation and the unconditional stability of the crank-nicolson schemes, the operators ∂ 1/2 t and i 1/2 t are approximated through quadrature rules which are well suited to the crank-nicolson schemes. namely, we choose the quadrature formulas derived in [1], which read for the sequence of complex values {fn}n∈n approximating {f(tn)}n∈n, i 1/2 t f(tn) ≈ √ 2δt 2 n∑ k=0 αkf n−k and ∂ 1/2 t f(tn) ≈ 2√ 2δt n∑ k=0 βkf n−k , (4.12) where (αk)k∈n and (βk)k∈n designate the sequences defined by    (α0,α1,α2,α3,α4,α5, · · · ) = ( 1, 1, 1 2 , 1 2 , 1 · 3 2 · 4, 1 · 3 2 · 4, · · · ) , βk = (−1)kαk, ∀k ≥ 0. the composition of the approximation of i 1/2 t with itself gives the approximation of it by the trapezoidal rule, which is coherent with the underlying crank nicolson scheme. using these quadratures formulas, the numerical versions of the boundary conditions (3.16), (3.27), (3.28), (3.34) and (3.42) are discrete convolutions which may be represented by the following formulation ∂ n v n+1 + e −i π 4 √ 2 δt v n+1 + g(v n+1 ,v n ,v n−1 , · · · ,v0) = 0, where g is a function giving information on the construction of the approximation. for example, the approximation of abc β 1,3 is given by ∂ n v n+1 + e −i π 4 √ 2 δt n+1∑ k=0 βkv n+1−k − e i π 4 2 |vn+1| √ 2δt 2 n+1∑ k=0 αk|vn+1−k|vn+1−k = 0. the other approximations of abc α,β j,k , j = 1, 2, k = 1, 2, 3 can be found in [3] and [20]. the complete crank nicolson scheme with fixed point procedure therefore takes the form given in table 1. cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 45 let w0 = un s = 0 while ‖ws+1 − ws‖l2(ωi) > ε do solve the linear boundary-value problem ∫ ω 2i δt w s+1 ψdx − ∫ ω ∂xw s+1 ∂xψdx − ∫ γ e −i π 4 √ 2 δt w s+1 ψdγ = − ∫ ω f(w s )w s ψdx + ∫ ω 2i δt u n ψdx + ∫ γ g s ψdγ, setting g s = g(w s ,v n ,v n−1 , · · · ,v0) and ψ is one of the basis functions of the p1 finite element set. end while v n+1 = w s+1 u n+1 = 2v n+1 − un table 1: fixed-point algorithm for solving the nonlinear schrödinger equation with nonlinear abc. we present below some numerical experiments to show the effectiveness of the different boundary conditions. since we have an exact solution in cases i and ii, we choose to evaluate the schemes on the solutions with initial data (4.3) and (4.4) respectively. concerning case i, the computational domain is limited to the open set (−10, 10) discretized with 4000 points. the time step is fixed to δt = 10 −3. in case ii, the finite domain is (0, 2) discretized with 2000 points. the time step δt is equal to 2 · 10−3. to analyze the accuracy of the different boundary conditions, we compute the relative error for the l2(ω)-norm ‖uex − unum‖0,ω(t) ‖uex‖0,ω(t = 0) , where unum denotes the numerical solution. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 time l 2 e r r o r n o r m abc β 1,1 abc β 1,3 abc β 1,4 abc β 2,2 abc β 2,4 figure 3: evolution of the relative error for case i, a = 2, c = 15, θ = 0. 46 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) for case i, we compare in figure 3 the abcs obtained with the potential strategy and the gauge change strategy for various orders. we can see that increasing the order improves the accuracy. moreover, the gauge change strategy (abc β 2,k) provides better accuracy compared to the potential strategy (abc β 1,k). finally, let us also notice that the long-time behaviour of the various abcs seems correct. to analyze the accuracy behaviour of the computed solution with respect to the velocity parameter, we plot on fig. 4 the evolution of the relative error with respect to c for the most accurate abc: abc β 2,2. we see that the relative error increases with lower velocities. this is in agreement with the theory developed in the paper since the boundary conditions are constructed under a high-frequency assumption. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 c=5 c=8 c=12 c=15 time l 2 e r r o r n o r m figure 4: evolution of the relative error for the simulation of the soliton solution with respect to the velocity c for abc β 2,2. in our last experiment, we focus on case ii. we compare in figure 5 the abcs obtained with the potential strategy and the linearization strategy for various orders. generally, the linearization strategy leads to the most accurate solutions. increasing the order of the abc improves the accuracy for small times (t ≤ 2.5 in the experiments). after this time, the relative errors cross and the best results are obtained for abcα 2,2. the potential strategy is accurate and competitive for abcα 1,3 but only for sufficiently large computational times. this shows that each strategy has his own strengths and weaknesses. finally, let us mention that abcα 2,2 has been shown to give optimal results within a large class of abcs (see [18]). 5 conclusion we presented an analysis of the construction and some numerical validations of abcs for nonlinear pdes considering the example of the nonlinear schrödinger equation. the methods are mainly based on pseudoand paradifferential operator techniques. we show that each strategy can lead to powerful solutions. however, much work remains to be done. in particular, developing rigorous extensions for higher dimensions, coupled problems and systems is a complete open problem. cubo 11, 4 (2009) towards accurate artificial boundary conditions for ... 47 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.005 0.01 0.015 0.02 0.025 time l 2 e r r o r n o r m abc α 1,1 abc α 1,2 abc α 1,3 abc α 2,2 abc α 2,3 abc α 2,4 figure 5: relative error for case ii. received: april 2008. revised: july 2008. references [1] antoine, x. and besse, c., unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional schrödinger equation, j. comput. phys. 181 (1) (2003), pp. 157-175. [2] antoine, x. arnold, a. besse, c. ehrhardt, m. and schädle, a., a review of transparent and artificial boundary conditions techniques for linear and nonlinear schrödinger equations, communications in computational physics 4 (4) (2008), pp. 729-796 (open access online article). [3] antoine, x. besse, c. and descombes, s., artificial boundary conditions for onedimensional cubic nonlinear schrödinger equations, siam j. numer. anal. 43 (6), (2006), pp.2272-2293. [4] bayliss, a. and turkel, e., radiation boundary-conditions for wave-like equations, comm. pure appl. math. 33 (6), (1980), pp.707-725. [5] bérenger, j.p., a perfectly matched layer for the absorption of electromagnetic-waves, j. comp. phys. 114 (2), (1994), pp. 185-200. [6] bony, j. m., calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, ann. sci. ec. norm. sup (4 ème série). 14, (1981), pp.209-246. [7] durán, a. and sanz-serna, j. m., the numerical integration of relative equilibrium solutions. the nonlinear schrödinger equation, ima j. numer. anal. 20 (2) (2000), pp. 235-261. 48 xavier antoine, christophe besse and jérémie szeftel cubo 11, 4 (2009) [8] engquist, b. and halpern, l., long-time behavior of absorbing boundary conditions, math. meth. appl. sci. 13, (1990), pp.189-203. [9] engquist, b. and majda, a., absorbing boundary conditions for the numerical simulation of waves, math. comp. 31, (1977), pp.629-651. [10] engquist, b. and majda, a., radiation boundary conditions for acoustic and elastic wave calculations, comm. pure appl. math. 32, (1979), pp.313-357. [11] givoli, d., nonreflecting boundary conditions, j. comp. phys. 94 (1), (1991), pp.1-29. [12] hagstrom, t., radiation boundary conditions for the numerical simulation of waves, acta numerica, (1999), pp. 47-106. [13] hagstrom, t. hariharan, s.i. and maccamy, r.c., on the accurate long-time solution of the wave equation in exterior domains: asymptotic expansions and corrected boundary conditions, math. comp. 63, (1994), pp.507-539. [14] majda, a. and osher, s., reflection of singularities at the boundary, comm. pure appl. math. 28, (1975), pp.479-499. [15] mur, g., absorbing boundary-conditions for the finite-difference approximation of the timedomain electromagnetic-field equations, ieee trans. electromagnetic compatibility 23 (4), (1981), pp. 377-382. [16] nirenberg, l., pseudodifferential operators and some applications, cbms regional conf. ser. in math. ams 17 (1973), lectures on linear partial differential equations, pp.19-58. [17] szeftel, j., calcul pseudodifférentiel et paradifférentiel pour l’étude de conditions aux limites absorbantes et de propriétés qualitatives d’équations aux dérivées partielles non linéaires, ph.d. thesis, université paris 13, 2004. [18] szeftel, j., absorbing boundary conditions for nonlinear scalar partial differential equations, comput. methods appl. mech. engrg. 195, (2006), pp.3760-3775. [19] szeftel, j., a nonlinear approach to absorbing boundary conditions for the semilinear wave equation, math. comp. 75, (2006), pp.565-594. [20] szeftel, j., absorbing boundary conditions for nonlinear schrödinger equations, numer. math. 104, (2006), pp.103-127. [21] taylor, m. e., pseudodifferential operators, princeton university press, nj, 1981. [22] tsynkov, s.v., numerical solution of problems on unbounded domains. a review, appl. numer. math. 27 (4), (1998), pp. 465-532. [23] zheng, c., exact nonreflecting boundary conditions for one-dimensional cubic nonlinear schrödinger equations, j. comput. phys., 215, (2006), pp.552-565. articulo 3 cubo a mathematical journal vol.11, no¯ 01, (21–54). march 2009 dirac type gauge theories – motivations and perspectives jürgen tolksdorf tu bergakademie freiberg, freiberg/sachsen, germany. email: juergen.tolksdorf@math.tu-freiberg.de abstract we summarize the geometrical description of a specific class of gauge theories, called “of dirac type”, in terms of dirac type first order differential operators on twisted clifford bundles. we show how these differential operators may be geometrically considered as being the images of sections of a specific principal fibering naturally associated with twisted clifford bundles. based on the notion of real hermitian vector bundles, we discuss the most general real dirac type operator on “particle-anti-particle” modules over an arbitrary (orientable) semi-riemannian manifold of even dimension. this setting may be appropriate for a common geometrical description of both the dirac and the majorana equation. resumen nosotros resumimos la descripción geométrica de una clase específica de teoría gauge, llamada "de tipo dirac", en términos del tipo de dirac de operadores diferenciales de primer orden sobre fibrados de clifford twisted. mostramos como esos operadores pueden ser geométricamente considerados como siendo imágenes de secciones de una fibra principal específica naturalmente asociada con el fibrado de clifford twisted. basado en la noción de fibrado vectorial hermitiano real, discutimos el más general operador de tipo dirac real sobre módulos "partícula-anti-partícula" sobre una variedad semi-riemanniana (orientable) arbitraria de dimensión par. este contexto puede ser apropiado para una descripción geométrica común para las ecuaciones de dirac y de majorana. 22 jürgen tolksdorf cubo 11, 1 (2009) key words and phrases: dirac type differential operators, real clifford modules, general relativity, gauge theories, majorana equation math. subj. class.: 53c05, 53c07, 70s05, 70s15, 83c05. 1 synopsis in a nutshell, dirac type gauge theories are based on the following “universal (dirac) action functional”: id : ∫ m [〈ψ, /dψ〉e + trγcurv( /d)] dvolm . (1) here, /d ∈ d(e) denotes the most general dirac type first order differential operator, acting on the c∞(m)−module of smooth sections ψ ∈ sec(m,e) on a hermitian clifford module bundle πe : e −→ m over a smooth orientable semi-riemannian manifold (m,gm) of even dimension n = 2k ≥ 2. the notation 〈·, ·〉e denotes a chosen hermitian form on e and curv( /d) ∈ ω2(m, end(e)) (2) is the curvature of /d. a detailed general discussion of the geometrical background of the functional (1) and how it is related to the well-known general lichnerowicz decomposition (c.f. [5] and [3]) /d 2 = −△b + vd (3) can be found in [22] and [23]. note that in contrast to what has been stated in the latter reference, however, the “dirac potential” vd actually reads: vd = γ(curv( /d)) − evg(ω 2 d ) + evg(∂dωd) (4) where the “dirac form” ωd ∈ ω 1 (m, end(e)) is a certain one-form canonically associated with /d and “evg” means evaluation with respect to the metric gm (c.f. below). in our discussion presented here we shall omit the quadratic term. the third term in (4) only contributes as a boundary term. as a consequence, the functional (1) differs from the “universal dirac action” that is discussed in reference [22] by the quadratic term (and the boundary term) in (4). to calculate the curvature of a general dirac type operator is generally more involved than for a connection. in contrast, explicit formulae are available for the corresponding dirac potential. hence, (4) may be used to also obtain the curvature of a general /d via (neglecting the boundary term) γ(curv( /d)) = vd + evg(ω 2 d ) . (5) we follow this line of reasoning to calculate the curvature of the most general dirac type operator on “particle-anti-particle” modules in section four. cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 23 in the following we focus on a summary of some of the basic features of the universal dirac action (1). a detailed discussion of its motivation is presented, whereby we put emphasize to its “universality” and its relation to various partial differential equations well-known from physics and geometry. in the particular case of twisted clifford bundles we discuss how dirac type first order differential operators can be geometrically considered as being images of sections of a principal fibering that is naturally associated with the geometry of clifford modules. finally, we discuss a specific class of real hermitian clifford modules. for these we present an explicit formula for the universal dirac action. our work is organized as follows: the second section is addressed to present some detailed discussion of the motivation for the dirac action and how it is related to various well-known “field equations”, like yang-mills and einstein’s equation of gravity. in the third section we discuss how the dirac action may be regarded as a functional of the metric and (endomorphism valued) super fields. the fourth section is addressed to the dirac action on the geometrical background of (a specific class of) real hermitian clifford modules which may allow to incorporate the geometrical description of the majorana equation in terms of the universal dirac action. finally, in the fives section we present some outlook. 2 motivation: four equations and one action to get started, let us call in mind that the two most profound equations in classical physics are provided by the maxwell equations of electrodynamics: df = 0 , (6) d∗f = jelm (7) and the einstein equation of gravity: ric(gm) − 1 2 scal(gm) = λgravτ . (8) here, the “electromagnetic field strength” is geometrically represented by a (closed) two-form f ∈ ω2(m) on a given four-dimensional, orientable semi-riemannian manifold (m,gm) with index of gm equals ±2. accordingly, the two-form ∗f denotes the hodge-dual of f with respect to gm and a chosen orientation of m. moreover, the differential operator d is the usual exterior derivative. we stress, that in the case of the maxwell equations (6–7) the metric structure gm on the manifold m is supposed to be fixed. in contrast, in the case of einstein’s theory of gravity the gravitational field is supposed to be fully described in terms of the metric structure gm on m. however, only those metric structures are physically admissible which satisfy einstein’s field equation (8). the tensor ric ∈ sec(m, endtm) denotes the “ricci-tensor” and scal ∈ c∞(m) its trace the so-called “ricci-scalar”. for tm ։ m 24 jürgen tolksdorf cubo 11, 1 (2009) being the tangent bundle of m, the bundle endtm ։ m is the associated bundle of endomorphisms on tm (over the identity on m). the right-hand side of the maxwell equation (7) denotes the “electrically charged matter current”. similarly, the right-hand side of the einstein equation (8) is the “energy-momentum current” associated with any form of energy and matter. the numerical constant λgrav is called the “gravitation coupling constant”. it carries a physical dimension in contrast to the electromagnetic coupling constant which is purely numerical (approximately 1/137). in classical physics these source terms for non-trivial electromagnetic field strength and gravitational fields are supposed to be given objects, reflecting the physical situation at hand. of course, as a special case one may consider the physical situation where (part of) space-time (m,gm) is filled only with an electromagnetic field that is generated by electrically charged matter whose support is outside the considered region of space-time. then, within this region jelm vanishes identically and τ is a unique function of f such that the pair (gm,f) is physically admissible provided it is a solution of the coupled einstein-maxwell equations. in a so-called “semi-classical” description of matter (i.e. within a certain approximation of a full quantum description), the classical maxwell and einstein equations are supplemented by the (gauge covariant) dirac equation (i/∂ a − m)ψ = 0 . (9) in particular, the electromagnetic current ∗jelm = 3∑ µ,ν=0 qelm〈ψ,γ(e µ )ψ〉e gm(eµ,eν)e ν ≡ qelm〈ψ,γ(e µ )ψ〉e gm(eµ,eν)e ν (10) becomes a (quadratic) function of ψ ∈ sec(m,e) such that the triple (gm,f,ψ) is physically admissible if and only if it fulfills the now coupled einstein-maxwell-dirac equation (6–9). here, when appropriate units are used the parameters (m,qelm) ∈ r+ × z are physically interpreted as “mass” and “electric charge” of the matter described in terms of the “matter field” ψ (e.g. an electron). in (10), e0, . . . ,e3 ∈ sec(m,tm) is a local orthonormal frame with respect to gm and e0, . . . ,e3 ∈ sec(m,t∗m) its (local) dual frame. note that henceforth we will make use of einstein’s summation convention whenever local expressions come up like in (10). geometrically, the matter field ψ is usually considered as a section of a twisted spinor bundle πe : e = s ⊗ w −→ m (11) over a semi-riemannian spin-manifold (m,gm, λspin) with λspin being a chosen spin structure on m. the hermitian vector bundle w = p ×ρ v ։ m is an associated vector bundle of a given cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 25 principal g-bundle g →֒ p ։ m that represents the so-called “internal gauge degrees of freedom” of matter. here, ρ : g → gl(v ) is a unitary representation of g on a hermitian vector space v which servers a the typical fiber of the twisting bundle w ։ m. in the case of electromagnetism the (semi-simple real) lie group g equals the unitary group u(1) with lie-algebra lieg = ir. accordingly, the gauge covariant dirac operator i/∂ a ≡ iγ ◦ (∂a)iγ ◦ (∂ s ⊗ idw + ids ⊗ ∂ w ) (12) is given by • the covariant derivative with respect to the spin connection on the spinor bundle s ։ m : ∂s µ loc. = ∂µ + 1 4 [γa,γb] ωlc µab (13) with ωlc ∈ ω1(m,so(p,q)) being the levi-civita form that is determined by gm; • the gauge covariant derivative on the hermitian vector bundle w ։ m : ∂w µ loc. = ∂µ + ρ ′ (aµ) (14) with ρ′(a) ≡ ρ′ ◦ a ∈ ω1(m, end(w)) being a (local) u(1)−gauge potential represented on w . the lie-algebra representation ρ′ : lieg → end(v ) is the derived representation with respect to the underlying group representation ρ. hence, locally the gauge covariant dirac operator reads: i/∂ a loc. = iγµ ( ∂µ + 1 4 ωlc µab [γa,γb] ⊗ idw + ids ⊗ ρ ′ (aµ) ) . (15) here and in the expression (10) the symbol “γ” denotes a clifford mapping, i.e. γ : t∗m −→ end(e) ω 7→ γ(ω) (16) satisfying γ(ω)2gm(ω,ω) ide. in the following we will suppress the identity mappings ide, ids, idw on e,s,w whenever this will not cause any confusion. also, we will not make a distinction in our notation with respect to the metric on the tangent and the co-tangent bundle t∗m ։ m. finally, γa ≡ γ(ea) are the usual “gamma matrices” and [·, ·] is the ordinary commutator. note that the lie algebra so(p,q) is isomorphic to the lie algebra spin(p,q) of the spin group and 1 2 [γa,γb] (0 ≤ a 6= b ≤ 3) are the corresponding generators of the “spinor representation” of so(p,q). indeed, the clifford action γ on a twisted spinor bundle (11) is simply given by the regular left action of the clifford bundle clm ։ m (17) 26 jürgen tolksdorf cubo 11, 1 (2009) on s ⊂ clm. here, the clifford bundle is the algebra bundle over (m,gm) whose typical fiber is given by the clifford algebra clp,q that is generated by the minkowski space r p,q ≡ (r4,η), where η(eµ, eν) :    ±1 for µ = 0 , ∓1 for 1 ≤ µ = ν ≤ 3 , 0 for 0 ≤ µ 6= ν ≤ 3 (18) and e0, . . . , e3 ∈ r 4 the standard basis. therefore, on a twisted spinor bundle the clifford action γ is uniquely determined by the metric gm. if γ denotes the clifford action with respect to the metric gm, then iγ is the clifford action with respect to the metric −gm. likewise, if the clifford action is “even”, i.e. γ(ω) t = −γ(ω) for all ω ∈ t∗m, then the clifford action given by iγ is “odd”: iγ(ω)t = iγ(ω) and vice verse. apparently, the geometrical background of the equations (6– 9) seems quite different like the equations themselves. to summarize: the geometrical background of the maxwell equations is given by the grassmann bundle λm ։ m over a given orientable semi-riemannian manifold (m,gm). in contrast, the geometrical background of the einstein equation is provided by so-called so(p,q)−reductions of the frame bundle fm ։ m. that is, the geometrical background is given by the fiber bundle eeh := fm ×gl(4) gl(4)/so(p,q) −→ m . (19) in fact, a section of this associated bundle with typical fiber gl(4)/so(p,q) is in one-to-one correspondence to a semi-riemannian structure gm of signature (p,q). we thus do not make a distinction between a section of the einstein-hilbert bundle (19) and the corresponding metric. we denote both by the same symbol. finally, the geometrical background of the dirac equation is provided by a clifford module (e,γ) over a given orientable semi-riemannian (spin-)manifold (m,gm). apparently, maxwell’s equations, einstein’s equation and dirac’s equation are rather different equations. nonetheless, one may ask for a common geometrical root of these three equations which play such a fundamental role in physics and mathematics. an appropriate hint is provided by the gauge covariant dirac equation (9) and the geometrical interpretation of the maxwell equations (6–7) in terms of gauge theory. for this one may regard the electromagnetic field strength f as a section of the complexified grassmann bundle λm ⊗r c ։ m (20) which corresponds to the curvature of a u(1)−connection on u(1) →֒ p ։ m. we emphasize that with respect to this geometrical interpretation of the electromagnetic field strength the maxwell equation (6) becomes an identity (the “bianci-identity”). if a denotes a local gauge potential of the curvature, then (7) is read as the u(1)−yang-mills equation: da∗fa = j . (21) cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 27 here, respectively, fa = if ∈ ω 2 (m,ir) is, again, the curvature of a u(1)−connection and da its gauge covariant exterior derivative, locally given by the first order differential operator d + a and a ∈ ω1(m,ir). clearly, the adjoint action is trivial, for u(1) is abelian. hence, da∗fa loc. = d∗fa + [a,∗fa] = d∗fa . (22) if j ≡ ijelm, then the yang-mills equation (21) is equivalent to (7). note that fa = d 2 a loc. = da. that is, the square of the first order differential operator da is a zero order differential operator taking values in ω2(m,ir). let δa be the formal adjoint of da with respect to the pairing ∫ m αc ∧ ∗β for all compactly supported α,β ∈ ω(m, c) ≡ sec(m, λm ⊗r c). by α c we denote the complex conjugate of α with respect to the canonical real structure on λm ⊗r c that is given by α c := eµ ⊗ λµ for α loc. = eµ ⊗ λµ ∈ ω 1 (m, c). it follows that δa = ± ∗ da∗, where the sign depends on the signature of gm and the degree of the form the operator acts on. then, the equation (21) may be rewritten as δafa = ±j (23) and thus the original maxwell equations become equivalent to ( da + δa)fa = ±j . (24) the point to be stressed here is, that the (complexified) grassmann bundle serves as a canonical clifford module with respect to the clifford action γ : t∗m −→ end(λm ⊗r c) ω 7→ { λm ⊗r c −→ λm ⊗r c α 7→ −i(extω(α) − intω(α)) . (25) here, extω(α) := ω ∧ α and intω(α) := α(ω ♯, ·) with ω♯ ∈ tm is the metric dual with respect to gm : β(ω ♯ ) := gm(ω,β) for all β ∈ t ∗m. as consequence, da + δai/∂a , (26) with i/∂ a loc. = iγµ(∂µ + 1 4 ωlc µab [γa,γb] + aµ) . (27) the maxwell equations for purely imaginary fa ∈ sec(m, λm ⊗r c) can thus be brought into a form analogous to the dirac equation for ψ ∈ sec(m,e) : i/∂ a fa = ±j . (28) 28 jürgen tolksdorf cubo 11, 1 (2009) the similarity between the dirac equation (9) and (28) can be made even more close by noting that λm ⊗r c ≃ clm ⊗r c ≃ end(sc), where sc ≡ s ⊗r c. hence, λm ⊗ c ≃ sc ⊗ s ∗ c and the (complexified) spinor bundle sc ։ m (with respect to a chosen spin structure) can be regarded as a sub-vector bundle of the grassmann bundle: sc →֒ λm ⊗ c ։ m . (29) geometrically, the complexified grassmann bundle λm⊗r c is but a special twisted grassmann bundle λm ⊗ l −→ m (30) with l := m × c → m being the trivial complex line bundle over m. the hermitian clifford module πλ, e : eλ, e ≡ λm ⊗ e −→ m , (31) with e := l ⊕ w ։ m being the whitney sum of the two hermitian vector bundles l ։ m and e ։ m, actually provides a common geometrical setting for the dirac and maxwell equations. obviously, all of this can be immediately generalized to arbitrary twisted grassmann bundles parameterized by arbitrary hermitian vector bundles e ։ m over (m,gm). in this case, one only has to replace the covariant derivative ∂s that corresponds to a chosen spin structure on m by the covariant derivative ∂λ of the levi-civita connection on λm ։ m with respect to the induced metric gλm. then, (12) is replaced by the twisted gauss-bonnet like operator i/∂ a = iγ ◦ (∂λ ⊗ ide + idλ ⊗ ∂ e ) = da + δa . (32) note that locally there is no distinction between the first order operators (32) and (12). this is, because the bundle of homomorphisms end(e) ։ m of any clifford module (e,γ) over an even dimensional semi-riemannian manifold (m,gm) globally decomposes as end(e) ≃ (clm ⊗r c) ⊗ endcl(e) . (33) here, endcl(e) denote the sub-algebra of γ−invariant endomorphisms on e ։ m. the fundamental isomorphism (33) can be inferred from the two wedderburn theorems about “invariant linear mappings” (c.f. [9] and [3]). in fact, the use of this global decomposition forces the dimension of m to be even such that clp,q is simple. finally, nothing basically chances even in the case the maxwell equations are replaced by general yang-mills equations, i.e. the abelian structure group u(1) is replaced by an arbitrary (semi-simple, real and compact) lie group g. in this case, one only has to replace the (trivial) line bundle l ։ m by the adjoint bundle ad(p) := p ×ad lieg ։ m. like in the particular case of a spinor bundle, the clifford action (25) is uniquely determined by the metric gm. actually, both clifford actions coincide on their common domain. hence, with cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 29 respect to twisted grassmann bundles we may consider γ and gm as being basically the same. accordingly, the einstein equation is seen to provide a physical constraint on the possible clifford module structures to which (31) refers to. note the change of the meaning of the metric when the maxwell equations are written similar to the dirac equation. once we have established a common geometrical setup for the dirac and the maxwell (resp. yang-mills) equations we proceed to show that this common setup also provides an appropriate geometrical background for the einstein equation. for this we remark that on the one hand side the maxwell and dirac equations make use of a given metric gm (i.e. a fixed clifford module structure of the underlying twisted grassmann bundle). on the other hand, the einstein equation are considered as differential equations determining gm. in particular, the (levi-civita) connection which fixes the first order operator ∂λ is fully determined by gm. in contrast to the maxwell equations (resp. yang-mills equations), the gravitational gauge potential has thus an underlying geometrical structure given by the metric gm from which the connection is derived. for this matter the einstein-hilbert functional, from which the einstein equation can be derived as eulerlagrange equation, is linear in the curvature. in contrast, the yang-mills functional, which yields the (homogeneous) maxwell equation (7) in the case g = u(1), is quadratic in the curvature: ieh(gm) := λ −1 grav ∫ m scal(gm) dvolm , (34) iym(gm; a) := λ −1 elm ∫ m gλm(fa,fa) dvolm . (35) note that the variation of iym(gm; a) with respect to the metric gm gives rise to the energymomentum current τ ∈ sec(m, end(tm)) as a function of fa as mentioned before. to get a relation between these seemingly different functionals (34) and (35) we notice that in contrast to the square d2 a = fa of the first order operator da, the square of the associated dirac operator i/∂ a has the well-known lichnerowicz decomposition into a specific second order differential operator and a specific zero order operator: i/∂ 2 a = ( da + δa) 2 g=u(1) = d ◦ δ + δ ◦ d = −△b + vd (36) with △b loc. = −gµν(∇µ◦∇ν−γ σ µν ∇σ) being the bochner-laplacian and ∇µ ≡ ∂µ+ 1 4 ωlc µab [γa,γb]+aµ. the local functions γσ µν are the usual christoffel symbols with respect to gm and a chosen coordinate frame. the “dirac potential” has the specific form: vd = 1 4 scal(gm) + γ(fa) ∈ sec(m, end(eλ, e )) (37) where locally γ(fa) 1 2 γµγν ⊗fµν. the tensor product refers to the fundamental decomposition (33). as a consequence, the zero order operator γ(fa) is always a trace-less operator: tre (γ(fa)) ≡ 0, where the trace is taken in end(eλ, e ). 30 jürgen tolksdorf cubo 11, 1 (2009) therefore, the einstein-hilbert functional may be expressed in terms of i/∂ a as ieh(gm) = λ ′−1 grav ∫ m trevd dvolm . (38) note that the dirac potential is uniquely determined by i/∂ a . we notice that the trace-less zero order operator γ(fa) ∈ sec(m, end(eλ, e )) is indeed the “square root” of the yang-mills lagrangian, for iym(gm; a) = λ ′−1 elm ∫ m tre (γ(fa) 2 ) dvolm . (39) however, also in this form the yang-mills action is still quadratic in the curvature in contrast to the einstein-hilbert action. the question then is whether the yang-mills lagrangian can be “linearized” such that it becomes most similar to the einstein-hilbert lagrangian without violating the second order character of the yang-mills equations. note that both the einstein and the yang-mills equations are of second order. hence, one cannot simply try to square the einstein-hilbert lagrangian to bring it into a form similar to the yang-mills lagrangian without obtaining higher order differential equations for gm. in order to appropriately linearize the integrand of (39) one may take into account that i/∂ a also determines a specific curvature on the bundle (31), denoted by curv(i/∂ a ) ∈ ω2(m, end(eλ, e )) (c.f. [22] and [23]). explicitly it reads curv(i/∂ a ) = rg ⊗ ide + idλ ⊗ fa ≡ rg + fa . (40) again, this is due to the fundamental decomposition (33). here, rg ∈ ω 2 (m, end(eλ, e )) is the riemannian curvature with respect to the induced metric gλm on the grassmann bundle over m. locally, it reads: rg loc. = 1 2 eµ ∧ eν ⊗ 1 4 [γa,γb]rabµν, where the local functions rabµν ≡ gm(ea,ec) e c (riem(eµ,eν)eb) and riem ∈ ω 2 (m, end(tm)) denotes the (semi-)riemann curvature tensor with respect to gm. note again that einstein’s summation convention is employed in local formulae. therefore, the yang-mills curvature (especially the electromagnetic field strength) may be expressed in terms of i/∂ a . in fact, it is but the “relative curvature” of the curvature of i/∂ a (again, neglecting the identity mappings): fa = curv(i/∂a) − rg ∈ ω 2 (m, endcl(eλ, e )) . (41) this geometrical interpretation of fa in terms of i/∂a yields a different interpretation of the yang-mills (resp. of the maxwell) equations. the latter are considered to yield a constraints for i/∂ a . of course, this simply means constraints for ∂e and thus does not yield anything new in cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 31 comparison with the usual description of yang-mills type gauge theories in terms of g principal bundles. however, the strength of the presented geometrical viewpoint of the yang-mills equations in terms of i/∂ a has a powerful potential for a straightforward generalization. this is, because the geometrical point of view can be immediately generalized to arbitrary dirac type first order differential operators. in other word, there are much more general dirac type operators on (31) than those given by i/∂ a . in fact, the latter are only very specific dirac type operators. they are fully characterized by the decomposition (37) and the fact that fa ∈ ω 2 (m, endcl(eλ, e )) is γ−invariant. this may provide a sufficient motivation to consider the form (38) of the einstein-hilbert function as more profound than the form (34). in fact, the former should be considered as a functional of a specific class of dirac type operators on (31) and hence as a specific restriction of a much more general functional (c.f. our discussion in the next section). as discussed in ([23]), the trace of the dirac potential (37) can be recast into the geometrical form (neglecting boundary terms): trevd = trγcurv(i/∂a) . (42) therefore, ieh(gm) ≡ ieh(i/∂a) = λ ′−1 grav ∫ m trγ(curv(i/∂a)) dvolm ≡ λ′ −1 grav ∫ m trγ(curv( da + δa)) dvolm (43) where trγ(curv(i/∂a)) ≡ tre [γ(curv(i/∂a))] ∈ c ∞ (m). the form (43) of the einstein-hilbert action makes it most explicit how the metric gm can be replaced by (a specific class of) dirac type operators and hence how the einstein-hilbert functional determines a clifford action γ on a twisted grassmann bundle (31). note that, despite of its appearance, (43) is actually independent of the connection on the twisting part e ։ m of (31). in other words, it is independent of the gauge potential a that (locally) determines the first order operator ∂e. the functional (43) thus yields a constraint only on how the vector bundle (31) can be actually regarded as a specific clifford module. it thus determines γ as stated before. in fact, since i/∂ a is fully characterized by (37), it is straightforward to prove that these dirac type operators provide the biggest class of dirac type first order differential operators on a twisted grassmann bundle such that the universal dirac action (1) is proportional to the einstein-hilbert action and thus only depends on gm. note that there is only a canonical choice for ∂ e if the hermitian vector bundle e ։ m equals the trivial bundle m × v ։ m. only in this case, there is a natural choice for i/∂ a given the gauss-bonnet like operator d + δ. in the general case, however the latter operator is not gauge covariant. for this matter one has to chose some ∂e to obtain an appropriate gauge covariant generalization da +δa of d+δ. again, the functional (43) is independent of this arbitrary choice. we are still left with the question whether it is possible to find a dirac type operator i/d a , say, 32 jürgen tolksdorf cubo 11, 1 (2009) on a certain twisted grassmann bundle such that the yang-mills functional can be expressed in terms of the universal dirac action (1). the answer to this question turns out to be affirmative, actually, and has been discussed in some detail in [22] (c.f. also the appropriate references cited therein, in particular [2] in the case of a closed compact riemannian manifold). in general, the yang-mills action may be written as iym(gm; a) = λ ′ ym (id(i/da) − id(i/∂a)) = λ′ym ∫ m trγ(curv(i/da) − curv(i/∂a)) dvolm . (44) where the corresponding dirac type operator reads i/d a = i/∂ a + i ⊗ γ(fa) . (45) here, i ≡ off − diag(−1, 1) is an additional complex structure on the doubled twisted grassmann bundle 2eλ, e ≡ eλ, e ⊕ eλ, e λm ⊗r (e ⊕ e) −→ m . (46) the thus defined class of first order differential operators (45) are called dirac operators of “pauli-type”. the reason for this chosen terminology is that first order differential operators of the form i/∂ a + iγ(fa) (47) have been introduced in physics in order to describe the so-called “magnetic moment” of the proton long before it has been realized that the proton is a composite of more fundamental elementary particles (the “quarks”). in this context, the additional term γ(fa) = i 2 γµγν⊗fµν, with f ∈ ω 2 (m) being the electromagnetic field strength, is named “pauli-term” after the famous physicist w. pauli. note that the first order operator (47), however, is not a dirac type first order operator. this is because the pauli-term γ(fa) is an even operator in the sense that it commutes with the canonical z2−grading provided by the riemannian volume form: γm = iγ(dvolm) (called “γ5” in the physics literature). indeed, a first order differential operator is said to be of dirac type provided it is odd with respect to a given z2−grading of the underlying clifford module and the principal symbol of its square is given by the underlying metric. only the latter feature is shared by the first order operator (47). in contrast, the first order operator (45) is both odd and its square is a “generalized laplacian”. it is thus of dirac type. note that dirac’s original first order operator (or its gauge covariant generalization) i/∂ a − m (48) is also not of dirac type for exactly the same reason as (47) is not of dirac type. we shall come back to this in our third section where we discuss a specific class of hermitian clifford modules and the most general dirac type operators thereof. cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 33 concerning dirac type operators of the form (45) the “square root” of the yang-mills lagrangian becomes most obvious. it is not simply given by the traceless zero order operator γ(fa) itself but, instead, by dirac operators of pauli type. basically, this is because of the additional grading one obtains from the doubling of (31). this additional grading also allows to express the “fermionic part” of the universal dirac action (1) as 〈ψ, i/d a ψ〉2e = 2〈ψ,i/∂a ψ〉e . (49) at least, this holds true for those sections ψ ∈ sec(m, 2eλ, e ) that are given by ψ = (ψ,ψ) and hence are determined by a section ψ ∈ sec(m,eλ, e ). in other words, the “pauli-term” does not contribute to the fermionic action but only to the bosonic action. this is a very desirable feature of this class of pauli type dirac operators, for it is well-known that the pauli-term in the fermionic action yields a generalized dirac equation that is not compatible with “quantization”. we shall return to the pauli type dirac operators when considering a specific class of hermitian clifford modules and the corresponding most general dirac type operators thereof. the underlying structure of this class of clifford modules is basically motivated by our fourth equation: the majorana equation: i/∂ψ = mψc (50) where ψc denotes the “charge conjugate” of ψ (c.f. below). we call in mind that the einstein-hilbert functional may be expressed in terms of dirac type operators of the form (32) with an arbitrary choice of ∂e. in contrast, when restricted to pauli type dirac operators i/d a , the universal dirac action (38) yields the combined einstein-hilbert-yangmills functional. it reduces to the pure yang-mills functional only if (m,gm) is fixed to be (ricci) flat. this is consistent with the einstein equation, however, only with respect to the physical approximation that the gravitational field produced by the energy-momentum of the yang-mills field can be neglected to some extend. in general, however, (1) yields the coupled einstein-yangmills-weyl equations as the corresponding euler-lagrange equations if (1) is restricted to pauli type dirac operators (45). in this case, the right-hand side of the yang-mills equation is similar to (10) and the energy-momentum current is a well-determined function of (gm,fa,ψ). we stress that the pauli type dirac operators are more general than those given by i/∂ a . in particular, the relative curvature of i/d a : fd := curv(i/da) − rg (51) is not γ−invariant, i.e. fd /∈ ω 2 (m, endcl(2eλ, e )) . (52) for that matter, γ(fd) ∈ ω 0 (m, end(e)) is not a traceless operator. we close our motivation for the universal dirac action (1) with the remark that the underlying invariance group of this functional is provided by the full diffeomorphism group diff(eλ, e ) of (31). 34 jürgen tolksdorf cubo 11, 1 (2009) this (infinite) gauge group decompose into the semi-direct product (c.f. [22]): diff(eλ, e )autm(eλ, e ) ⋉ diff(m) (53) with autm(eλ, e ) consisting of all (bundle) automorphisms of (31) over the identity mapping on the base manifold m. moreover, this group decomposes further into the direct sum of to sub-groups: autm(eλ, e )auteh(eλ, e ) × autym(eλ, e ) . (54) here, the “yang-mills” sub-group autym(eλ, e ) ⊂ autm(eλ, e ) consists of all automorphisms of (31) being isomorphic to the gauge transformations on the frame bundle that is induced by the vector bundle (31). it is thus a normal sub-group of autm(eλ, e ) and auteh(eλ, e ) : autm(eλ, e )/autym(eλ, e ) . (55) locally, the “einstein-hilbert” sub-group auteh(eλ, e ) consists of all so(p,q) rotations of orthonormal frames of tm ։ m and autym(eλ, e ) consists of all ordinary gauge transformations encountered in the usual geometrical description of yang-mills gauge theories in terms of principal g-bundles g →֒ p ։ m. thus, the universal dirac action (1) contains all the physical symmetries which are usually imposed on physical field theories. to enlarge this symmetry to “super-symmetry” transformations, however, is still an open issue. having presented a detailed discussion of the motivation for the universal dirac action (1) and how it is related to well-known field equations, we may proceed with a discussion in what sense the dirac functional is more general than the ordinary yang-mills functional. in other words, in the following section we want to discuss the precise domain of dependence of the universal dirac functional. 3 the dirac action as a functional of “super fields” in this section we discuss in more detail the domain of dependence of the (bosonic part of the) universal dirac action: id,bos : ∫ m trγcurv( /d) dvolm . (56) in the foregoing section we have shown how this functional covers both the einstein-hilbert and the yang-mills functional. in fact, the dirac functional may be considered as a natural generalization of the einstein-hilbert functional of the form (43). hence, when restricted to certain “sub-domains” on the “set of all dirac type operators” (c.f. below), the universal functional (56) becomes a functional on (an appropriate subset of) sec(m,eeh) in the case of the einstein-hilbert action, or a functional on the affine manifold of all linear connections a(e) in the case of the yang-mills action. accordingly, when restricted to pauli type dirac operators the (bosonic part of cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 35 the) universal dirac action becomes a functional on the smooth manifold sec(m,eeh) × a(e). in general, (1) is considered as a functional on the smooth manifold d(eλ, e ) × sec(m,eλ, e ) (57) of all dirac type first order operators on a twisted grassmann bundle (31) and the module of smooth sections therein. note that the “fermionic part” of (1) id,ferm : ∫ m 〈ψ, /dψ〉e dvolm , (58) is viewed simply as a quadratic form on sec(m,eλ, e ). this quadratic form is fully determined by (symmetric) elements of d(eλ, e ). for this reason, it suffices to focus on the affine manifold of all dirac type operators on a given grassmann bundle. the aim of this section is to make this more precise and to show how the above two cases of the einstein-hilbert and the yang-mills functional are special cases of the more general functional (56). basically, the reason is provided by the following (highly non-canonical) isomorphisms: d(eλ, e ) ≃ ω 0 (m, end(eλ, e )) ≃ ω ∗ (m, endcl(eλ, e )) , (59) which holds true for a fixed clifford module structure on (31) (i.e. metric on m). the second isomorphism of (59) is implied by (33), where the abbreviation ω ∗ (m, endcl(eλ, e )) ≡ ⊕ p∈z ω p (m, endcl(eλ, e )) (60) has been used. consequently, any dirac type operator on a clifford module is determined by differential forms of all degrees. this is in strong contrast to connections on a vector bundle which are determined by one-forms, only. to make this more precise, let again m be a smooth orientable manifold of even dimension n = 2k ≥ 2. also, let again e ։ m be a smooth hermitian vector bundle over m and eλ, e ։ m the corresponding twisted grassmann bundle. we call the smooth fiber bundle ed : eeh × end(eλ, e ) −→ m (61) the “dirac bundle” associated to the twisted grassmann bundle. so far (31) is considered as a vector bundle over m. there is no given clifford structure at all, for m is not yet supposed to be endowed with a metric. we call in mind that a metric on m is in one-to-one correspondence with a section of the dirac bundle given by σd : m −→ ed x 7→ (gm(x), 0) (62) where gm ∈ sec(m,eeh). 36 jürgen tolksdorf cubo 11, 1 (2009) we consider the following equivalence relation on the manifold of smooth sections sec(m,ed) : σ′ d ≡ (g′, φ′) ∼ σd ≡ (g, φ) ∈ sec(m,ed) (63) iff g′ = g ∈ sec(m,eeh) and there exists an α ∈ ω 1 (m, end(e)) →֒ ω1(m, endcl(eλ, e )), such that φ′ = φ + γ(α) ∈ ω0(m, end(eλ, e )). we put sd := sec(m,ed)/ ∼ . (64) there are various equivalent definitions available for dirac type first order differential operators, depending on the appropriate focus (see, for example, [1], [5], [3], [4]). we present a different one which is most adopted to our purpose. definition 1. let d(eλ, e ) be the set of all first order differential operators acting on sec(m,eλ, e ), such that for /d ∈ d(eλ, e ) there exists a section gm ∈ sec(m,eeh) with t∗m γ −→ end(eλ, e ) df 7→ [ /d,f] . (65) here, the gm−induced clifford action γ is defined by (25). a first order differential operator /d ∈ d(eλ, e ) is called a “dirac type operator" provided it is odd with respect to the z2−grading that is given by an involution τe := γm ⊗ τe. the set of all dirac type operators on eλ, e carries a natural action of the translational group te ≡ ω 1 (m, end + (e)) →֒ ω1(m, end+(eλ, e )) (66) that is given by d(eλ, e ) × te µ −→ d(eλ, e ) ( /d,α) 7→ /d + γ(α) . (67) clearly, this action is free and the corresponding orbit space d(eλ, e )/µ can be identified with sd. furthermore, with respect to this identification πd : d(eλ, e ) −→ sd /d 7→ s ≡ [(gm, φ)] (68) is a principal fibering with structure group te. this principal fibering is actually trivial. however, every bijection χa : d(eλ, e ) ≃ −→ sd × te /d 7→ (s,α) (69) strongly depends on the choice of ∂e. this holds true unless e ։ m is trivial. cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 37 indeed, for every choice of ∂e one may define d(eλ, e ) ∋ /d ≡ χ −1 a (s,α) := /∂ a + φ̂a + γ(α) ≡ /∂ a + φa . (70) here, /∂ a ∈ d(eλ, e ) is given by (32) and φ̂a ∈ sec(m, end(eλ, e )) ≃ sec(m, λm ⊗ endcl(eλ, e )) , (71) which does not contain a one-form part. note that φ̂a has to have odd total degree. it is uniquely defined as follows: every /d ∈ d(eλ, e ) can be decomposed (in a highly non-unique way) as /d = /∂ a + φa with φa ≡ /d − /∂a . then, φa =: φ̂a + γ(α) and /∂a + φ̂a + γ(α) is equivalent to /∂ a + φ̂a. it follows that πd(χ −1 a (s,α)) = pr1(s,α) = s, if and only if [ /d] ∈ d(e)/µ corresponds to s ∈ sd. proposition 1. let ed ։ m be the dirac bundle associated with a twisted grassmann bundle eλ, e ։ m. the functional (56) can be considered as a canonical functional on sec(m,ed) : proof: since the value of the integral ∫ m trγcurv(i/∂a) dvolm (72) is independent of the choice of ∂e, it follows that (56) is constant along the fibers of (68). hence, it descents to a well-defined functional on sd. for this matter id,bos can be considered as a functional of (gm, φ) that constitutes a general section of the dirac bundle (61). 2 as a consequence id = i (gm, φ,ψ) (73) with φ ∈ sec(m, λm ⊗ endcl(eλ, e )) ⊕ 0≤l≤n sec(m, λlt∗m ⊗ endcl(eλ, e )) . (74) being a “super-field” of odd total degree that takes values in the γ−invariant endomorphisms on eλ, ee + λ, e ⊕ e− λ, e ։ m. especially, for φ = 0, the action (1) reduces to the sum of the usual (massless) dirac functional and the einstein-hilbert functional: ∫ m [〈ψ, i/∂ a ψ〉e + trγcurv(i/∂a)] dvolm . (75) 38 jürgen tolksdorf cubo 11, 1 (2009) in this case, the appropriate euler-lagrange equations are given by the combined einstein-weyl equations with the energy-momentum current τ being defined by ψ ∈ sec(m,eλ, e ). likewise, to obtain the combined einstein-yang-mills-weyl equations one considers φ = fa, with the requirement that fa ∈ sec(m, λ 2t∗m ⊗ endcl(2eλ, e )) being defined by the curvature fa ∈ ω 2 (m, end(e)) with respect to the chosen ∂e. in other words, one restrict the right-hand side of (1) to pauli type dirac operators on 2eλ, e ։ m. in this case, the energy-momentum current τ is fully determined as a function of (ψ,fa), whereas the electromagnetic current is given by (10) (or an appropriate generalization thereof if g 6= u(1)). of course, this reduces to pure yang-mills theory when one restricts to ψ = 0 and gm (ricci) flat. eventually, one can also recover the full action functional of the so-called standard model of elementary particles including the famous higgs potential. for this one has to consider even more general super fields φ for appropriate twisted grassmann bundles. interestingly, the structure of this bundle is determined by the topology of m and the choice of the “ground-state” of the still to find “higgs boson” (c.f. [22] and the corresponding references cited therein). the above mentioned examples may suffice to exhibit the generality of the dirac action (1) and how it covers important classes of coupled partial differential equations as euler-lagrange equations of a natural generalization of the einstein-hilbert lagrangian of einstein’s theory of gravity (43). once one has the universal functional (1) one may ask for the corresponding form of the euler-lagrange equations. this, however, depends on the choice of the (twisting part of the) underlying twisted grassmann bundle and is still a major challenge to exhibit in full generality. in the case, where the bundle is fixed and endowed with sufficient structure one may determine the most general dirac type operator that is compatible with the endowed structure. basically, this amounts to determine the most general super-field that is compatible with the given structure and then rewriting the universal dirac action in terms of this super-field. as a specific example, this will be demonstrated in the next section in terms of a specific class of “real, hermitian clifford modules”, called “particle-anti-particle modules”. before, however, we want to briefly comment on “spin versus non-spin manifolds”. so far, we concentrated on twisted grassmann bundles and one may ask what does it give more than twisted spinor bundles. also, one may ask how the latter fits with the geometrical frame of twisted grassmann bundles. first of all, if m is a spin-manifold (i.e. it has vanishing second stiefel-whitney classes) and s ։ m is the (complexified) spinor bundle with respect to a chosen spin-structure, then λm ⊗ c ≃ s ⊗ s ∗ −→ m (76) and hence eλ, e ≃ s ⊗ w −→ m (77) where w ≡ s∗ ⊗ e ։ m. cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 39 moreover, if s ≃ clme ≡ {ae ∈ clm | a ∈ clm} with e ∈ sec(m,clm) being an appropriately global primitive idempotent and s ≃ s ⊗ c, then s ⊗ e →֒ eλ, e s ⊗ y 7→ s ⊗ e∗ ⊗ y (78) yields a canonical inclusion of the twisted spinor bundle ee := s ⊗ e −→ m (79) into the twisted grassmann bundle (31). here, e∗ is the idempotent that yields the dual spinor module s∗ := e clm and s ∗ ≃ s∗ ⊗ c. note that we only consider complex modules. in this way, the slightly more general situation of a twisted grassmann bundles also covers the geometrical situation where m is supposed to be a spin-manifold. on the other hand, by a famous theorem due to r. geroch, a non-compact four-dimensional lorentzian manifold possesses a spin-structure if and only if its frame bundle is trivial (c.f. [7]). apparently, to propose that m is a spin-manifold is thus a very strong assumption about the topology of m. note that locally every clifford module looks like a twisted spinor bundle according to the fundamental decomposition (33). therefore, the geometrical setup of twisted grassmann bundles is slightly more general than twisted spinor modules and much less restrictive (actually, twisted grassmann bundles always exist). on the other hand, to consider arbitrary clifford modules seems far too general. in particular, the metric gm does not fix the clifford module structure γ, in general, like (25) does in the case of a twisted grassmann bundle. for that matter it becomes difficult to fix the domain of the universal dirac action for general clifford modules. only in the case of twisted grassmann bundles, the einstein-hilbert functional may interpret to provide restrictions also with respect to the module structure of the vector bundle (31). we close this section by two remarks: first, one obtains for (m,gm) denoting a closed compact and orientable riemannian manifold of even dimension that there exists real constants α,β such that ∫ m trevd dvolm = αieh(i/∂a) + β wres ( /d 2−2k ) (80) independent of the chosen ∂e. here, “wres” is the “wodzicki residue”, i.e. the trace functional on the algebra of classical pseudo-differential operators acting on sec(m,eλ, e ) (see, for example, [20] and the given references therein; also see [2] and [21]). therefore, in the case of dimm = 4, the universal dirac action is basically equal (up to a shift and the quadratic term in (4)) to the trace of the “propagator” (i.e. the greens operator) of /d 2 . this may demonstrate once again how natural the functional (1) actually is. second, the dirac-like form (28) has been studied since from the beginning of the last century, c.f. [19], [13], [15], [14], [16], [10], [12], [11] and [17]. apparently, this form of the maxwell equations 40 jürgen tolksdorf cubo 11, 1 (2009) has a natural generalization: /dfd = 0 (81) where, again, fd := curv( /d) − rg is the relative curvature with respect to /d ∈ d(eλ, e ). accordingly, solutions of this generalized maxwell equation like, for example, (anti-) self dual solutions may provide interesting restrictions to d(eλ, e ) and hence to the dirac action (1). note that, when expressed in terms of the super field φ ∈ sec(m, λm ⊗ endcl(eλ, e )) the generalized maxwell equation (81) actually becomes a system of nonhomogeneous partial differential equations. we finally mention that generalizations of the dirac type operator i/∂ a also play a fundamental role in a. connes’s noncommutative geometry (c.f., for example, [6]) and in the case of the proof of the family index theorem, (c.f., for example, in [18], [4]). 4 particle-anti-particle modules and dirac operators of pauli type in this section we discuss another specific class of clifford modules. these modules are mainly motivated by the structure that underpins the majorana equation (50). these “particle-antiparticle” modules will also provide us with a better geometrical understanding of pauli type dirac operators. in particular, these modules will yield a geometrical motivation for the restriction of “diagonal sections” ψ = (ψ,ψ), such that the pauli term appears in the bosonic part of the universal dirac action (1) but drops out in fermionic part (58) of (1). to get started let, again, (m,gm) be a given orientable, semi-riemannian manifold of even dimension n = 2k ≥ 2. also, let τcl ≡ (clm,m) collect the data of the clifford bundle clm ։ m associated with (m,gm). definition 2. by a “real hermitian clifford module (bundle)” we understand a collection of data (e,〈·, ·〉e,τe,je,γe ) (82) where, respectively, e is the total space of a complex vector bundle ξe ≡ (e,m,πe ) over m, 〈·, ·〉e a fiber-wise hermitian product turning ξe into a hermitian vector bundle over m, τe ∈ end(e) is an involution giving rise to a z2−grading of ξe and je : e → e denotes a real structure, i.e. an anti-linear involution on e that allows to identify ξe with its conjugate complex vector bundle ξē := ξ̄e ≡ (ē,m,π̄e ) over m. finally, γe : t ∗m −→ end(e) ν 7→ γe (ν) (83) is a clifford mapping such that all mappings are “quasi-hermitian” (i.e. either hermitian or skewhermitian) and τe and γe are “quasi real” (i.e. either real or purely imaginary with respect to cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 41 je ) : je ◦ τe ◦ je = ±τe je ◦ γe ◦ je = ±γe . (84) here, a real structure is called “quasi hermitian” provided it fulfills 〈je (z),je (w)〉e ± 〈w,z〉e (85) for all z,w ∈ e. similar to complex linear mappings this is denoted by jt e = ±je, where, in general, “ t” means hermitian transpose with respect to 〈·, ·〉e. if j t e = + je, the real structure is also called an “anti-unitary involution”. in the following we are interested in a specific class of real hermitian clifford modules, called “particle modules”. definition 3. a real hermitian clifford module over τcl is called a “particle module” if 1. the involution is skew-hermitian and purely imaginary; 2. the clifford mapping is skew-hermitian and real. the corresponding conjugate complex module is called an “anti-particle module” over τcl. we denote a particle module by ξp ≡ (p,〈·, ·〉p,τp,jp,γp) . (86) a particle-anti particle module over m is a real hermitian clifford module (bundle) over τcl ξpp̄ ≡ (pp̄,〈·, ·〉pp̄,τpp̄,jpp̄,γpp̄) (87) where, respectively 1. pp̄ := p ⊕m p̄ ; 2. 〈(z1,w1), (z2,w2)〉pp̄ := 1 2 (〈z1,z2〉p + 〈w1,w2〉p) ; 3. τpp̄(z,w) : (τp(z),−τp(w)) ; 4. jpp̄(z,w) : (jp(w),jp(z)) ; 5. γpp̄(z,w) : (γp(z),γp(w)) for all z,w,. . . ,w2 ∈ p. it follows that for all ν ∈ t∗m : 42 jürgen tolksdorf cubo 11, 1 (2009) 1. jt pp̄ = ±jpp̄ ⇔ j t p = ±jp ; 2. τt pp̄ = ±τpp̄ ⇔ τ t p = ±τp ; 3. γt pp̄ (ν) = ±γpp̄(ν) ⇔ γ t p (ν) = ±γp(ν) ; 4. jpp̄ ◦ τpp̄ = ±τpp̄ ◦ jpp̄ ⇔ jp ◦ τp = ∓τp ◦ jp 5. jpp̄ ◦ γpp̄(ν) = ±γpp̄(ν) ◦ jpp̄ ⇔ jp ◦ γp(ν) = ±γp(ν) ◦ jp ; 6. τpp̄ ◦ γpp̄(ν) = ±γpp̄(ν) ◦ τpp̄ ⇔ τp ◦ γp(ν) = ±γp(ν) ◦ τp . theorem 1. the most general real dirac type operator on a particle-anti-particle module ξpp̄ reads /d pp̄ ( /d p jp ◦ φp ◦ jp φp jp ◦ /dp ◦ jp ) ≡ ( /d p φ c p φp /d c p ) , (88) with /d p : sec(m,p) −→ sec(m,p) (89) being a general dirac type operator on the underlying particle module ξp and φp ∈ sec(m, end(p)) being a zero order operator that is even with respect to the z2−grading on p proof: to prove the statement we mention that an odd first order differential operator on a z2−graded vector bundle ξw ≡ (w,m,πw) over a (semi-)riemannian manifold (m,gm) is of dirac type if and only if for all f ∈ c∞(m) the mapping γw : t ∗m −→ end(w) df 7→ [ /d,f] ≡ /d ◦ f − f ◦ /d (90) yields a clifford action on ξw. here, the ring c ∞ (m) acts multiplicatively on the (sheave) of sections sec(m,w). likewise, if (ξw,γw) denotes a clifford module, then an odd first order differential operator /d : sec(m,w) −→ sec(m,w) (91) is of dirac type (that is compatible with the given module structure) if and only if [ /d,f] = γw(df) . (92) let ξpp̄ be a particle-anti-particle module and /d := ( d1 d2 d3 d4 ) (93) cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 43 be a general first order differential operator acting on sec(m,pp̄) : dk : sec(m,p) −→ sec(m,p) (94) for k = 1, . . . , 4. the operator /d is odd with respect to τpp̄ if and only if τpp̄ ◦ dk − dk ◦ τpp̄ (95) for k = 1, 4 and τpp̄ ◦ dk + dk ◦ τpp̄ (96) for k = 2, 4. then, [ /d,f] = γpp̄(df) (97) for all f ∈ c∞(m) if and only if [dk,f] = γp(df) (98) for k = 1, 4 and [dk,f] = 0 (99) for k = 2, 3. therefore, the first order differential operators d1 ≡ /d1 and d4 ≡ /d2 are of dirac type on the underlying particle module ξp. in contrast, the operators d2 ≡ φ2 and d3 ≡ φ1 are of zero order. next, we consider the conditions on the dirac type operator /d := ( /d 1 φ2 φ1 /d2 ) (100) such that /d is real with respect to jpp̄. it follows that jpp̄ ◦ /d ◦ jpp̄ /d ⇔ { /d 2 = jp ◦ /d1 ◦ jp , φ2 = jp ◦ φ1 ◦ jp . (101) this finally proves the statement. 2 note that neither /d p , nor φp are supposed to be real, in general. 44 jürgen tolksdorf cubo 11, 1 (2009) let mpp̄ : {(z,z c ) ∈ pp̄ |z,zc ≡ jp(z) ∈ p} (102) be the real subspace defined by jpp̄ such that pp̄ = mpp̄ ⊗ c . (103) the corresponding real vector bundle is denoted by ξm ≡ (mpp̄,m,πm) with the projection πm being given by the restriction of πpp̄ to mpp̄ ⊂ pp̄. note that ξm ⊂ ξpp̄ is a real τcl submodule. clearly, the latter itself contains a distinguished real sub-module given by z ∈ p fulfilling zc = z. that is, it is given by the real sub (bundle) space mp ⊕ mp : {(z,z) ∈ pp̄ |jp(z) = z ∈ p} ⊂ mpp̄ , (104) where mp := {z ∈ p |z = jp(z)} ⊂ p, such that p = mp ⊗ c. on a particle-anti-particle module, the first order differential operator (88) is the most general real dirac type operator. hence, one may restrict the universal dirac action (1) to this type of dirac operators: id,real : 1 2 ∫ m [〈ψpp̄, /dpp̄ψpp̄〉pp̄ + trγcurv( /dpp̄)] dvolm (105) with ψpp̄ = (ψp, ψ c p ) ∈ sec(m,mpp̄) and /dpp̄ any real dirac operator on the particle-anti-particle module ξpp̄. proposition 2. when boundary terms are neglected, the dirac action (105) decomposes as follows: id,realid,ferm( /dpp̄) + id,bos( /dpp̄) (106) where 2id,ferm( /dpp̄) ≡ ∫ m [〈ψp, /dpψp〉p + 〈ψ c p , /d c p ψ c p 〉p + 〈ψp, φ c p ψ c p 〉p + 〈ψ c p , φpψp〉p] dvolm ; (107) 2id,bos( /dpp̄) ≡ ∫ m [trγcurv( /dp) + trγcurv( /d c p ) + 2 tr(φ c p ◦ φp) + 8 (tr ◦ evg)(α c p ◦ αp) + 2 (tr ◦ evg)(β c p ◦ βp)] dvolm (108) where 2αp(v) : φp ◦ γp(v ♭ ) ∈ end(p) and v♭(u) = gm(v,u) for all u,v ∈ tm. accordingly, 2αc p (v) : jp ◦ 2αp(v) ◦ j −1 p φ c p ◦ γp(v ♭ ) ∈ end(p). furthermore, βp : extθ(φp − 2 /αp) with θ ∈ ω 1 (m, end(p)) being the canonical one-form that exists on every clifford module (c.f. [22]) and βc p : jp ◦ βp ◦ j −1 p extθ(φ c p − 2 /α c p ). cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 45 in the sequel, we make use of the common “dagger” abbreviation: /α ≡ γ(α) ∈ ω0(m, end(p)) for any α ∈ ω∗(m, end(p)). for example, for α = ek ⊗ λk one has /α = γ k ⊗ λk, etc. proof: the fermionic part is straightforward to prove. the bosonic part of the dirac action is proved in several steps. first, we prove the following lemma 1. let /d 1 and /d 2 be two dirac type first order differential operators on an arbitrary clifford module (ξe,γe ) ≡ (e,m,πe,γe ) with πe : e = e1⊕e2 → m being a z2−graded (hermitian) vector bundle over (m,gm). the zero-order term vh of the generalized laplacian h : sec(m,e) −→ sec(m,e) ψ 7→ /d 1 ( /d 2 ψ) (109) has the explicit form: vh = vd + φ ◦ /ωd + evg(α 2 h ) + u . (110) here, respectively, vd and ωd are the dirac potential and dirac form of /d ≡ /d2 (c.f. [23]). moreover, φ := /d 1 − /d 2 ∈ sec(m, end−(e)) and u := evg ( ∇t ∗m ⊗end(e) h αh ) (111) with ∇e h being the covariant derivative that defines the connection laplacian of h : △h : −evg ( ∇t ∗m ⊗e h ◦ ∇e h ) (112) and αh ∈ ω 1 (m, endm(e)) is given by 2 αh(gradgf) := [φ ◦ /d,f] = φ ◦ γe (df) (113) for all f ∈ c∞(m). here and henceforth we make use of the following notation: “evg” means “evaluation/contraction” with respect to gm. for instance, evg(α 2 ) loc. = evg(e µ ⊗ eν ⊗ αµ ◦ αν) : gm(e µ,eν) αµ ◦ αν ∈ end(e) for all α ∈ ω1(m, end(e)) etc. proof: with /d 1 = /d 2 + φ ≡ /d + φ it becomes sufficient to consider laplace type operators of the form h = /d 2 + φ ◦ /d . (114) 46 jürgen tolksdorf cubo 11, 1 (2009) every generalized laplacian h decomposes as (see, for instance, in [3]) h = −△h + vh (115) with ∇e h being given by 2 ev(f0 gradf1,∇ e h ψ) := f0 ([h,f1] + △gf1) ψ (116) for all f0,f1 ∈ c ∞ (m) and ψ ∈ sec(m,e). here, △g denotes the usual laplace-beltrami operator restricted to zero-forms on m. it follows that ∇e h ∇e d + αh (117) with ∇e d being the covariant derivative that defines the bochner-laplacian of /d 2 . as a consequence, the connection laplacian of h may be expressed in terms of the bochnerlaplacian1 of /d 2 : △h = △d + 2 evg (αh ◦ ∇ e d ) + evg(α 2 h ) + u . (118) the statement then follows by comparison of the general lichnerowicz decomposition (115), taking into account that /d 2 = −△d + vd and 2 evg(αh ◦ ∇ e d ) = φ ◦ /∇e d . (119) 2 note that treu = divg ξh (120) with ξh := (treαh) ♯ ∈ sec(m,tm) . (121) hence, trevh dvolm [ trevd + tre (φ ◦ /ωd) + (tre ◦ evg)(α 2 h ) ] dvolm + £ξhdvolm (122) which demonstrates that [∗trevh][∗tre (vd + φ ◦ /ωd + evg(α 2 h ))] ∈ hn der (m) (123) with “∗” being the hodge map induced by gm and the orientation defined by dvolm. 1i would like to thank m. schneider for appropriate comments. cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 47 clearly, for /d 1 = /d 2 = /d one has vh = vd . (124) next, we present a bochner-lichnerowicz-weizenböck type formula for a slightly more general laplace type second order differential operator h′. corollary 2. let again (ξe,γe ) be a clifford module over (m,gm). also, let dk = /dk + φk (k = 1, 2) be two first order differential operators acting on sec(m,e). the zero-order term vh′ of the generalized laplacian h′ : sec(m,e) −→ sec(m,e) ψ 7→ d1(d2ψ) (125) reads: vh′ = vh + v (126) where vh is given by (110) with the replacement φ := d1 − d2 + 2φ2 (127) and v := (φ − φ2) ◦ φ2 + [ /d2, φ2] . (128) proof: we put d1 = /d2 + φ0 + φ1 ≡ /d + φ01 and rewrite h ′ as h′ = /d 2 + φ ◦ /d + v = h + v . (129) hence, the connection laplacian △h′ of h ′ is the same as the connection laplacian △h of h. one may thus apply the former lemma 1 to prove the statement. 2 as a consequence, one obtains explicitly (neglecting boundary terms): trevh′ trevd + tre (φ ◦ φ2 − φ 2 2 ) + tre [ /d2, φ2] + tre (φ ◦ /ωd) + (tre ◦ evg)(α 2 h ) . (130) we are now in the position to prove the bosonic part of proposition 2. in fact, this will be an immediate consequence of the following more general 48 jürgen tolksdorf cubo 11, 1 (2009) proposition 3. let (ξe,γe ) be a clifford module over (m,gm). also, let (ξ2e,γ2e ) be the clifford module that is defined by the corresponding whitney sum: ξ2e : ξe ⊕ ξe , τ2e : τe ⊕ (−τe ) , γ2e : γe ⊕ γe (131) with τe being the grading involution on ξe. the zero order term vd ∈ sec(m, end(2e)) associated with the (square of the) most general dirac type first order differential operator /d ≡ ( /d 1 φ2 φ1 /d2 ) : sec(m,e) ⊕ sec(m,e) −→ sec(m,e) ⊕ sec(m,e) (132) reads: vd = ( v1 + φ2 ◦ φ1 + 4 evg(α2 ◦ α1) [ /d1 , φ2] + φ2 ◦ ( /d1 − /d2) + 2 φ2 ◦ /ω2 [ /d 2 , φ1] + φ1 ◦ ( /d2 − /d1) + 2 φ1 ◦ /ω1 v2 + φ1 ◦ φ2 + 4 evg(α1 ◦ α2) ) (133) where, respectively, vk and ωk denote the dirac potential and the dirac form of /dk (k = 1, 2) and αk is defined in terms of /d 2 k , similar to αh of lemma (1). proof: we may write /d = /d + φ (134) with /d : ( /d 1 0 0 /d 2 ) , φ : ( 0 φ2 φ1 0 ) . (135) then, we simply make use of the preceding corollary 2 and apply the corresponding bochnerlichnerowicz-weizenböck type formula to h′ := /d 2 . note that the bochner-laplacian of /d is given by ∇2e d ∇2e d + βd (136) with βd ∈ ω 1 (m, end(2e)) being βd : ( 0 2α2 2α1 0 ) . (137) here, again 2 αk(gradgf) := [φk ◦ /dk ,f] = φk ◦ γe (df) (138) cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 49 for all f ∈ c∞(m) and k = 1, 2. then, similar to the results presented before vd = vd + u + 2 φ ◦ /ωd + evg(β 2 d ) (139) where /ω d : γ2e (ωd) ≡ ( /ω 1 0 0 /ω 2 ) , u := evg ( ∇t ∗m ⊗end(2e) d βd ) . (140) 2 therefore, tr2evdtrev1 + trev2 + 2 tre (φ1 ◦ φ2) + 8 (tre ◦ evg)(α1 ◦ α2) + divgξd (141) with ξd : (tr2eβd) ♯ . (142) the bosonic part of the proposition (2) is then implied by (again, omitting all boundary terms): trγ(curv( /d)) = tr2evd + (tr2e ◦ evg)(ω 2 d ) = trγ(curv( /d1)) + trγ(curv( /d2)) + + 2 tre (φ1 ◦ φ2) + 8 (tre ◦ evg)(α1 ◦ α2) + + 2 (tre ◦ evg)(σ1 ◦ σ2) , (143) where σk := extθ(φk − 2 /αk) ∈ ω 1 (m, end(e)) and 2αk(v)φk ◦ γe (v ♭ ) for all v ∈ tm and k = 1, 2. this finally ends the proof of proposition (2). 2 the functionals (107–108) may look complicated at first glance. however, they yield a straightforward generalization of the usual action of the standard model of particle physics as discussed in [22], which allows to also include majorana mass terms. the latter feature will be discussed in detail elsewhere. proposition 4. let jp be anti-unitary and ψpp̄(ψp, ψp) ∈ sec(m,mp ⊕ mp). also, let /dp be real with respect to jp and both /dp and the real part of φp be formally self-adjoint. then, the dirac action id,realid,ferm( /dpp̄) + id,bos( /dpp̄) (144) 50 jürgen tolksdorf cubo 11, 1 (2009) reads: id,ferm( /dpp̄) = ∫ m [〈ψp, ( /dp + yp)ψp〉p] dvolm , id,bos(/dpp̄) = ∫ m [ trγcurv( /dp) + (tr ◦ evg′ )(y 2 p ) − (tr ◦ evg′ )(f 2 p ) ] dvolm (145) where yp, fp ∈ ω ∗ (m, endcl(p)). proof: this is a simple application of proposition 2 taking into account that the (endomorphism valued) one-forms αp, βp ∈ ω 1 (m, end(p)) are linearly determined by the zero order operator φp ∈ sec(m, end(p)). furthermore, due to the fundamental decomposition (33) every zero order operator locally reads: φpγ i ⊗ φi (146) where i = (i1, i2, . . . , il) is a multi-index (1 ≤ ik ≤ n for k = 0, 1, . . . ,n), γ i ≡ γi1γi2 · · ·γil and φi are local sections of sec(m, endcl(p)) which are totally antisymmetric with respect to the multi index i. to avoid double counting the summation is thus take only for the ordered indices: i1 < i2 < ... < il , l = 0, 1, , . . . ,n. in other words, the zero order section φp ∈ ω(m, end(p)) is in one-to-one correspondence with a general section φp ∈ ω ∗ (m, endcl(p)). then, φp = e i ⊗ φi with {ei ≡ ei1 ∧ ei2 ∧ · · · ∧ eil | l = 0, 1, . . . ,n} being a local basis of λm ։ m. hence, the bosonic part of (105) reduces to 2id,bos( /dpp̄) ≡ ∫ m [trγcurv( /dp) + trγcurv( /d c p ) + 2(tr ◦ evg′ )(φ c p ◦ φp)] dvolm (147) where the evaluation map on the right hand side refers to the re-scaled (fiber) metric g′ λm on the grassmann bundle λm ։ m that is defined by g′ ij ≡ λ′ gλm(e i,ej) := trγiγj + 1 4 gij trγ iγiγjγj + + 1 n2 gij tr ( γiγi − gab γ iγaγiγb )( γjγj − gcd γ jγcγjγd ) . (148) indeed, one explicitly has φp ◦ φ c p = γiγj ⊗ φi ◦ φ c j , evg(αp ◦ α c p ) = 1 4 gij γ iγiγjγj ⊗ φi ◦ φ c j , evg(βp ◦ β c p ) = 1 n2 gij ( γiγi − gab γ iγaγiγb )( γjγj − gcd γ jγcγjγd ) ⊗φi◦φ c j (149) where gij ≡ gm(ei,ej) etc., j = (j1,j2, . . . ,jk) is again a multi-index and einstein’s summation convention is applied. let us call in mind that we do not distinguish between the metric on the tangent and the co-tangent space of m. cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 51 as a consequence, tr(φ c p ◦ φp) + 4 (tr ◦ evg)(α c p ◦ αp) + (tr ◦ evg)(β c p ◦ βp)(tr ◦ evg′ )(φ c p ◦ φp) . (150) furthermore, every real dirac type first order differential operator on a particle-anti-particle module may be rewritten as /d pp̄ ( /d p yp − fp yp + fp /d c p ) . (151) here, yp := 1 2 (φp + φ c p ) ≡ rejφp , (152) fp := 1 2 (φp − φ c p ) ≡ iimjφp , (153) such that yp is the real and −ifp is the imaginary part of the zero order operator φp with respect to the real structure jp. according to the general case, we put yp = γ i ⊗ yi , fp = γ j ⊗ fj . (154) the statement then follows from tr(φ c p ◦ φp)tr(y 2 p ) − tr(f 2 p ) , (155) which is analogous to the case of complex numbers (remember that fc p − fp). 2 we note that /d pp̄ leaves the real submodule sec(m,mp ⊕ mp) ⊂ sec(m,mpp̄) invariant if and only if /d c p = /d p and fp = 0. clearly, for yp = 0 and fp the curvature of /dp := i/∂a we get back the pauli type dirac operators as specific real dirac type operators on the real hermitian clifford module ξpp̄. moreover, the “diagonal sections” are motivated by the distinguished real submodule sec(m,mp ⊕ mp) ⊂ sec(m,pp̄) (156) of ξpp̄. note that it is the doubling of mp (which we may identify with our former twisted grassmann bundle eλ, e ) which allows to add the pauli-term to i/∂a such that the resulting first order operator is still of dirac type. also note that the additional complex structure encountered in the definition of (45) corresponds to the assumption that the zero order part of (88) is purely imaginary. for the same matter it has to drop out in the fermionic part of the universal dirac action since it would yield a non-real contribution. 52 jürgen tolksdorf cubo 11, 1 (2009) 5 outlook we presented a detailed motivation of “dirac type gauge theories” which are gauge theories that are based on the universal dirac action (1). in particular, we have exhibit how the dirac action covers well-known differential equations, like the maxwell and the einstein equation. indeed, the dirac action turns out to be a natural generalization of the einstein-hilbert functional. to also obtain the yang-mills functional, one has to introduce a specific class of dirac type operators and we discussed their geometrical origin in terms of real hermitian clifford modules. we also discussed the domain of the dirac action from a geometrical point of view. we thereby proved several lichnerowicz type formulae for decomposable laplace type operators which generalize the corresponding result presented in [23]. it is well-known that there is a one-to-one correspondence between dirac type operators on a clifford module and clifford super-connections (see, for instance, in [3]). for this reason, the domain of dependence of the dirac action may not come as a surprise, especially because of the isomorphisms (59). however, the latter hold true only when the module structure is fixed from the outset. this, of course, does not permit interpreting the einstein-hilbert functional as a constraint on the module structure. moreover, our discussion clearly demonstrates that there exists a natural functional on the dirac bundle provided by the dirac action. the presented results clearly exhibit in what sense dirac type operators and clifford modules provide a more general geometrical setting to describe gauge theories than connections and principal bundles. indeed, dirac type gauge theories allow to describe different types of gauge theories, like yang-mills theory, einstein’s theory of gravity and spontaneously broken yang-mills gauge theories, in a geometrically unified setting based on the same universal dirac functional. this is independent of whether the base manifold (“space-time”) m is supposed to be spin or not. in order to gain more insight, however, one has to deal with the “moduli space of dirac operators” m(e) ≡ d(e)/diff(e) (157) on which the dirac functional descents. of course, this set is probably far too wild and thus has to be restricted to appropriate subsets like the solutions of ∗fd = ±fd , (158) similar to the moduli space of (anti-) self-dual solutions of the ordinary yang-mills equation: ∗fa ±fa. for this, however, the domain of the dirac functional has to be discussed more seriously, in particular from an analytical point of view. in contrast to the ordinary yang-mills equations one obtains still another reasonable constrains to dirac type operators, similar to the (anti-) self-duality condition as, for instance, the “unimodularity” condition: £ξddvolm = 0 . (159) cubo 11, 1 (2009) dirac type gauge theories – motivations and perspectives 53 finally, one may pose the question to what extent is there a relation between the stationary points of the dirac action (1) and the generalized maxwell equation /dfd = 0 . (160) again, in full generality this seems a hopeless task. however, it might be reasonable to discuss this question using appropriate simple geometrical settings. this will be done in a forthcoming work. received: february 2008. revised: august 2008. references [1] atiyah m. f. and bott r. and shaprio a., clifford modules, topology 3, (1964), 3 38. [2] ackermann t. and tolksdorf j., the generalized lichnerowicz formula and analysis of dirac operators, j. reine angew. math., (1996). [3] berline n. and getzler e. and vergne m., heat kernels and dirac operators, springer verlag, (1996). [4] bismut j. m., the atiyah-singer index theorem for families of dirac operators: two heat equation proofs, mathematicae, springer verlag, (1986). 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[19] riesz m., clifford numbers and spinors, the inst. for fluid dynamics and applied mathematics, lecture series no. 38, university of maryland, 1958. reprinted by e. f. bolinder, p. lounesto (eds.), kluwer, dordrecht, the netherlands, 1993. [20] e. schrohe, noncommutative residues, dixmier’s trace, and heat trace expansions on manifolds with boundary, b. booss-bavnbek, k. wojciechowski (eds) geometric aspects of partial differential equations, proceedings of a minisymposium on spectral invariants, heat equation approach, september 18-19, 1998, roskilde, denmark. contemporary mathematics vol. 242, amer. math. soc., providence, r. i. (1999), 161 186. [21] tolksdorf, j., the einstein-hilbert-yang-mills-higgs action and the dirac-yukawa operator, j. math. phys. 39, (1998). [22] tolksdorf j. and thumstädter t., gauge theories of dirac type, j. math. phys., 47, 082305, (2006). 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[25] m. wodzicki, noncommutative residue, k-theory, arithmetic and geometry, i. manin (ed), lecture notes in mathematics 1289, springer-verlag berlin, 1987, 320 399. final&_nbsp_version.faoa.dvi cubo a mathematical journal vol.12, no¯ 01, (219–230). march 2010 strong vector equilibrium problems in topological vector spaces via kkm maps a.p. farajzadeh department of mathematics, razi university, kermanshah, 67149, iran email : ali-ff@sci.razi.ac.ir a. amini-harandi ∗ department of mathematics, university of shahrekord, shahrekord, 88186-34141, iran email : aminih a@yahoo.com d. o’regan department of mathematics, national university of ireland, galway, ireland email : donal.oregan@nuigalway.ie and r.p. agarwal department of mathematical sciences, florida institute of technology, melbourne, fl 32901, usa email : agarwal@fit.edu abstract in this paper, we establish some existence results for strong vector equilibrium problems (for short, svep) in topological vector spaces. the solvability of the svep is presented using the ∗the second author was in part supported by a grant from ipm (no. 85470015) 220 a.p. farajzadeh et. al. cubo 12, 1 (2010) fan-kkm lemma. these results give a positive answer to an open problem proposed by chen and hou and generalize many important results in the recent literature. resumen en este art́ıculo, establecemos algunos resultados de existencia para problemas de equilibrio strong vector en espacios vectoriales topológicos (abreviadamente, svep). la salubilidad del svep es presentada usando el lema de fan-kkm. estos resultados dan una respuesta positiva a problemas abiertos propuestos por chen y hon y generalizan varios resultados importantes en la literatura reciente. key words and phrases: strong vector equilibrium, upper sign continuity, pseudomonotone bifunction, quasimonotone bifunction. math. subj. class.: please inform 1 introduction it is well known that the vector equilibrium problem provides a unified model for several classes of problems, for example, vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems ( see [1,8] ). let x and y be two real hausdorff topological vector spaces, and k a nonempty subset of x. let f : k × k → y be a given bifunction and w : k → 2y be a set valued mapping. let c ⊆ y . c is said a convex cone subset of y if, λc ⊆ c for each λ ≥ 0 and c + c ⊆ c. the convex cone c is called pointed if, c ∩ (−c) = {0}. in this paper, we consider the following problem : strong vector equilibrium problem (for short, svep) which consists in finding x ∈ k such that f (x, y) ∈ w (x), ∀y ∈ k. if x and y be two banach spaces and f (x, y) = 〈t (x), y − x〉, where t : k → l(x, y ) and l(x, y ) denotes the set of all continuous linear mappings from x into y , then svep reduces to the strong vector variational inequality svvi which was considered by fang and huang [7]. in [7] fang and huang obtained some existence results for svvi which gave a positive answer to the open problem proposed by chen and hou [3] in a real banach space. the main purpose of this work is to establish some existence results for svep in a real hausdorff topological vector spaces. our results extend and improve the corresponding results of fang and huang [6], iusem and sosa [9] and many others. the paper is organized as follows. in the rest of this section we recall some notation,definitions, and the fan-kkm lemma which are used in the next section. in section 2, we first present the solvability of svep without monotonicity by using the fan-kkm lemma and then obtain an existence result for svep with generalized monotonicity. cubo 12, 1 (2010) strong vector equilibrium problems ... 221 throughout the paper, unless otherwise specified, let x and y be two real hausdorff topological vector spaces. for given bifunction f : k × k → y and f ∈ y ∗ = l(y, r) consider f ◦ f : k × k → r as 〈f ◦ f, (x, y)〉 = f (〈f, (x, y)〉). we denote the duality pairing between x∗ and x, by 〈., .〉, and the open line segment joining between x, y ∈ k by ]x, y[. let a be a nonempty subset of a topological space x. we denote by 2a the family of all subsets of the set a and by f(a) the family of all nonempty finite subsets of a. in a topological vector space x, let int, cl, and co denote the interior, closure and convex hull respectively. let k be a nonempty convex subset of x and let k0 be a subset of k. a multivalued map γ : k0 → 2 k is said to be a kkm map if coa ⊆ ⋃ x∈a γ(x), ∀ a ∈ f(k0). let x and y be two topological spaces. the set-valued mapping t : x → 2y is called upper semi-continuous (u.s.c.) at x ∈ x if for each open set v containing t (x) there is an open set u containing x such that for each t ∈ u , t (t) ⊆ v ; t is said to be u.s.c. on x if it is u.s.c. at all x ∈ x. the map t is said to be closed if the set gr(t ) = {(x, y) : x ∈ x, y ∈ t (x)} is a closed set in x × y. proposition 1.1. if t is closed and t (x) = ⋃ x∈k t (x) is compact, then t is u.s.c. we need the following theorems which are special cases of [5, theorem 3.1] and [12, theorem 2.1 ], respectively, in the sequel. theorem 1.2. ([5]). let k be a nonempty subset of a topological vector space x and f : k → 2x be a kkm mapping with closed values. assume that there exist a nonempty compact convex subset b of k such that d = ⋂ x∈b f (x) is compact. then ⋂ x∈k f (x) 6= ∅. 2 the main results the following theorem provides sufficient conditions in order to guarantee nonemptyness and compactness of the solution set of svep. theorem 2.1. let x and y be real hausdorff topological vector spaces, k a nonempty closed subset of x, f : k × k → y and w : k → 2y be two set valued mappings. assume the following hypotheses hold (a) for all finite subsets a of k and for all x ∈ coa there exists y ∈ a such that f (x, y) ∈ w (x); 222 a.p. farajzadeh et. al. cubo 12, 1 (2010) (b) for each y ∈ k, the set {x ∈ k : f (x, y) ∈ w (x)} is closed (c) there exist a nonempty compact subset b of k and a nonempty convex compact subset d of k such that, for each x ∈ k\b there exists y ∈ d such that f (x, y) 6∈ w (x). then, the solution set of svep is nonempty and compact. proof. we define γ : k → 2k as follows γ(y) = {x ∈ k : f (x, y) ∈ w (x)}. by (a) , γ is a kkm mapping. applying (b) and (c), we deduce that ⋂ y∈d γ(y) is a closed subset of b. now, γ satisfies all of the assumptions of theorem 1.2 and hence ⋂ x∈k γ(x) 6= ∅. this means that svep has a solution. by (b), the solution set of svep is closed and by (c) it is subset of the compact set b. thus the solution set of svep is compact and the proof is complete. remark 2.2. (i) if k is convex, f (x, x) ∈ w (x), c(x) = y \w (x) is a convex cone, for all x ∈ k, and the mapping f is concave in y with respect to c(x), that is, f (x, ty1 + (1 − t)y2) − (tf (x, y1) + (1 − t)f (x, y2)) ∈ c(x), for each x, y1, y2 ∈ k and t ∈]0, 1[ then condition (a) of theorem 2.1 holds. to see this let a = {y1, ..., yn} be a finite subset of k and x = σ n i=1λiyi ∈ co(a) where λi ≥ 0 and ∑n i=1 λi = 1. on the contrary, we suppose that (a) does not hold. then for each 1 ≤ i ≤ n, f (x, yi) 6∈ w (x). thus f (x, yi) ∈ c(x), for each 1 ≤ i ≤ n. (2.1) since f is concave in y with respect to c(x) we get f (x, σni=1λiyi) − n ∑ i=1 λif (x, yi) ∈ c(x). (2.2) by (2.1), note c(x) is convex, we have n ∑ i=1 λif (x, yi) ∈ c(x). (2.3) now since c(x) is a convex cone, by (2.2) and (2.3) we obtain f (x, x) = f (x, σni=1λiyi) ∈ c(x), which contradicts our assumption. for example if we let x = y = r, w (x) = (−∞, 0], and define f (x, y) = x − y then f satisfies condition (a). (ii) it is clear that condition (a) is different from diagonally quasiconvex, in the single valued case, defined in [4, page 114]. cubo 12, 1 (2010) strong vector equilibrium problems ... 223 (iii) if the mapping f is continuous in x, for all fixed y ∈ k and the graph of the mapping w is closed then d = {x ∈ k : f (x, y) ∈ w (x)} is closed and so condition (b) holds. to see this let xn ∈ d and xn → x ∈ k. then for each n ∈ n, f (xn, y) ∈ w (xn). since f (., y) is continuous and the graph of w is closed then f (x, y) ∈ w (x) which shows that x ∈ d. finally if k is compact then condition (c) trivially holds. in the following, as an application of theorem 2.1, we give a topological vector space version of theorem 2.1 of fang and huang [7] for a family of moving closed pointed convex cones {c(x) : x ∈ k}. corollary 2.3. let k be a nonempty closed convex subset of x and t : k → l(x, y ) be a mapping such that (a′) for each y ∈ k, the set {x ∈ k : 〈t x, y − x〉 ∈ −c(x)\{0}} is open in k, (b′) there exist a nonempty compact subset b of k and a nonempty convex compact subset d of k such that, for each x ∈ k\b there exists y ∈ d such that 〈t x, y − x〉 ∈ −c(x)\{0}. then the set {x ∈ k : 〈t x, y − x〉 6∈ −c(x)\{0}, ∀ y ∈ k} is a nonempty and compact subset of k. proof. let f (x, y) = 〈t (x), y − x〉 and w (x) = y \(−c(x)\{0}). we claim that f and w satisfy all of the assumptions of theorem 2.1. indeed, by (a′), the set {x ∈ k : f (x, y) ∈ w (x)} is closed in k (so closed in x), for all y ∈ k. now if for a finite subset a = {y1, y2, ..., yn} of k there exists x ∈ coa such that f (x, y) 6∈ w (x), for all y ∈ a. then f (x, yi) ∈ −c(x)\{0}, for each1 ≤ i ≤ n. since c(x) is closed pointed convex cone then −c(x)\{0} is a convex cone and so from the definition of f we get (here x = σni=1λiyi with λi ≥ 0 and ∑n i=1 λi = 1) 0 = f (x, x) = f ( x, n ∑ i=1 λiyi ) = n ∑ i=1 λi〈t x, yi − x〉 ∈ −c(x)\{0}, which is a contradiction.this proves (a) of theorem 2.1. finally (b′) guarantees condition (c) of theorem 2.1 and so the proof is complete. in what follows we give another application of theorem 2.1 in order to provide sufficient conditions that a function defined on a nonempty convex subset of a normed space has a fixed point. corollary 2.4. let k be a nonempty, closed convex subset of a real normed vector space x and let f : k → k be a continuous mapping. if there is a nonempty compact subset b of k and a compact convex subset d of k such that for every x ∈ k there exists y ∈ d satisfying ‖ x − f (x) ‖>‖ y − f (x) ‖, then there exists x ∈ k such that x = f (x). furthermore, the set of all such elements is a compact subset of b. 224 a.p. farajzadeh et. al. cubo 12, 1 (2010) proof. define f : k × k → r as f (x, y) = − ‖ y − f (x) ‖ + ‖ x − f (x) ‖, ∀x, y ∈ k, and let w (x) = [−∞, 0) ⊂ y = r, for all x ∈ k. the function f is concave in the second variable with respect to the set (y \w (x)) = [0, ∞) which is a convex subset of y. hence by remark 2.2 (i), f satisfies condition (a) of theorem 2.1. by the continuity of f in the first variable and the closedness of graph of w (see remark 2.2 (iii)) we deduce that f satisfies condition (b) of theorem 2.1. finally f satisfies condition (c) of theorem 2.1 by assumption. thus there exists x ∈ k such that f (x, y) = − ‖ y − f (x) ‖ + ‖ x − f (x) ‖≤ 0, for all y ∈ k. now, if, in the previous relation, we take y = f (x) then ‖ x − f (x) ‖≤ 0 which implies the result requested. remark 2.5. corollary 2.4 generalizes theorem 3.2 in [12] for infinite dimensional normed spaces. now using a scalarization method, we establish an existence theorem for svep. in particular, we prove the following theorem which is an extension of theorem 2.2 in [7]. theorem 2.6. let k be a nonempty, closed convex subset of a real hausdorff topological vector space x, w be a nonempty subset of x. let f : k × k → y be a bifunction . suppose that there exists x0 ∈ k and f ∈ {y ∗ ∈ y ∗ : 〈y∗, c〉 < 0, ∀ c ∈ y \w } such that (a) f (x, x) = 0, for all x ∈ k; (b) for each x, y, z ∈ k, if f of (x, y) ≤ 0 and f of (x, z) < 0, then f of (x, u) < 0, ∀u ∈]y, z[; (c) for each y ∈ k, the set {x ∈ k : f of (x, y) ≥ 0} is closed in k; (d) there exist a nonempty compact subset b of k such that, ∀ x ∈ k\b, f of (x, x0) < 0. then there exists x ∈ k such that f (x, y) ∈ w, for all y ∈ k. proof. define g : k → 2k by g(y) = {x ∈ k : f of (x, y) ≥ 0}. by (c), g(y) is closed for each y ∈ k. now we show that g is a kkm map. let a = {y1, y2, ..., yn} be a finite subset of k and x = σni=1λiyi ∈ co(a) where λi ≥ 0 and ∑n i=1 λi = 1. we prove that x ∈ ∪ n i=1g(yi). on the contrary, suppose for each i = 1, ..., n, x 6∈ g(yi). consequently f ◦ f (x, yi) < 0, for each i = 1, ..., n. (2.4) cubo 12, 1 (2010) strong vector equilibrium problems ... 225 notice that using induction, we can deduce from (b) that for each x, y1, ..., yn ∈ k, if for each i = 1, ..., n, f ◦ f (x, yi) < 0 then f ◦ f (x, u) < 0 for each u ∈ co(y1, y2, ..., yn). thus by (2.4) we get 0 = f ◦ f (x, x) < 0, which is a contradiction. let d = {x0}, then ∩y∈dg(y) = g(x0) ⊆ b by (d). thus theorem 1.2. implies ⋂ y∈k g(y) 6= ∅. let x ∈ ⋂ y∈k g(y), then f of (x, y) ≥ 0, for all y ∈ k. (2.5) now, if f (x, y) 6∈ w, for some y ∈ k, then from f ∈ {y∗ ∈ y ∗ : 〈y∗, c〉 < 0, ∀ c ∈ y \w }, we get f of (x, y) < 0, which is a contradiction by (2.5). as an application of theorem 2.6 we now present two corollaries. the first is a topological version of theorem 2.2 in [7] with mild assumptions and the second improves theorem 3.12 in [10]. corollary 2.7. let k be a nonempty closed convex subset of a topological vector space x, c be a nonempty closed pointed convex cone of x and let t : k → l(x, y ) be a nonlinear mapping. suppose that there exist x0 ∈ k, f ∈ {y ∗ ∈ y ∗ : 〈y∗, c〉 > 0, ∀ c ∈ c\{0}}, and a nonempty compact subset b of k such that ∀ x ∈ k\b, f ot (x)(x − x0) < 0. then there exists x ∈ k such that 〈t x, y − x〉 6∈ −c\{0}, for all y ∈ k. proof. define f : k × k → y by f (x, y) = 〈t x, y − x〉 and w = y \ − c. now the result follows of theorem 2.6. corollary 2.8. suppose that (x, ‖.‖) is a real reflexive banach space and k is a nonempty closed convex subset of x. let f : k × k → r be a bifunction such that: (a) f (x, x) = 0, for each x ∈ k, (b) for each x, y, z ∈ k, if f (x, y) ≥ 0 and f (x, z) > 0, then f (x, u) > 0, ∀u ∈]y, z[; (c) for each y ∈ k, the set {x ∈ k : f (x, y) ≥ 0} is closed; (d) there exists r0 > 0 such that for each x ∈ k\kr0 there exists y ∈ kr0 with f (x, y) ≤ 0, where kr0 = {x ∈ k : ‖x‖ ≤ r0}. then, the solution set of the equilibrium problem,i.e., {x ∈ k : f (x, y) ≤ 0, ∀y ∈ k}, is nonempty and compact. 226 a.p. farajzadeh et. al. cubo 12, 1 (2010) proof. set y = r, w (x) = [0, ∞) and define f : r → r by f (x) = x for each x ∈ k. by (b), note f ◦ f = f , so we get condition (b) of theorem 2.6. pick b = d = kr0 , since kr0 is weakly compact convex, then condition (d) of the theorem 2.6 trivially holds and so theorem 2.6 implies that there is x̄ ∈ k such that f (x̄, y) ≥ 0, for all y ∈ k. moreover, by (b) and (d), the set {x ∈ k : f (x, y) ≥ 0, ∀x ∈ k}, is a closed subset of kr0 , respectively. this completes the proof. definition 2.9. let f : k × k → y, be a given bifunction and c = {c(x) : x ∈ k} is a family of closed convex cone proper subsets of y. we say f is c−psudomonotone if the following implication holds f (x, y) 6∈ −c(x)\{0} ⇒ f (y, x) ∈ −c(y). a nonlinear mapping t : k → l(x, y ) is c-psudomonotone if the bifunction f (x, y) = 〈t x, y −x〉 is c-psudomonotone. remark 2.10. for the single-valued bifunction f , our definition of pseudomonotonicity reduces to that in [11]. moreover, if we let f (x, y) = 〈t (x), y − x〉, where t : k → l(x, y ) is a nonlinear mapping, and c(x) = c for all x ∈ k, c is a convex cone of y, then the previous definition reduces to definition 2.1 in [7]. definition 2.11. let f : k × k → y, and c = {c(x) : x ∈ k} is a family of convex cone proper subsets of y. we say that f is c−upper sign continuous if the following implication holds for every x, y ∈ k, f (u, y) 6∈ −c(u)\{0}, ∀u ∈]x, y[⇒ f (x, y) 6∈ −c(x)\{0}. a nonlinear mapping t : k → l(x, y ) is c-upper sign continuous if the bifunction f (x, y) = 〈t x, y − x〉 is c-upper sign continuous. remark 2.12. if y = r and c(x) = [0, ∞), for all x ∈ r, then our definition of upper sign continuity, reduces to the definition of upper sign continuity introduced by bianchi and pini in [2]. also, if the graph of the set-valued map w : k → 2y defined by w (x) = y \ − c(x)\{0} is closed and the function t → f (xt, y) is continuous at t = 0, where xt = (1 − t)x + ty, then f is c− upper sign continuous. this shows that if we set f (x, y) = 〈t (x), y − x〉, where t is nonlinear mapping, and c(x) = c for all x ∈ k, c is a closed convex cone of y, then hemicontinuity of t (definition 2.2 in [6]) implies c−upper sign continuity of f. lemma 2.13. let k be a nonempty closed convex subset of x, and c = {c(x) : x ∈ k} is a family of convex cone proper subsets of y . let f : k × k → y be a c−pseudomonotone and c−upper sign continuous function. assume that the following assumptions hold, (a) f (x, x) = 0, for each x ∈ k, (b) for each x, y, z ∈ k, if f (x, y) ∈ −c(x)\{0} and f (x, z) ∈ −c(x), then f (x, u) ∈ −c(x)\{0}, for all u ∈]y, z[, cubo 12, 1 (2010) strong vector equilibrium problems ... 227 then for every x0 ∈ k, f (x0, y) 6∈ −c(x0)\{0}, ∀y ∈ k if and only if f (y, x0) ∈ −c(y), ∀y ∈ k. proof. let x0 ∈ k be such that f (x0, y) 6∈ −c(x0)\{0}, ∀y ∈ k. now the c−pseudomonotonicity of f, implies that f (y, x0) ∈ −c(y), ∀y ∈ k. conversely, suppose that f (y, x0) ∈ −c(y), ∀y ∈ k. we first show that for each y ∈ k, u ∈]x0, y[⇒ f (u, y) 6∈ −c(u)\{0}. (2.6) on the contrary, we suppose that there exist y ∈ k and u ∈]x0, y[ such that f (u, y) ∈ −c(u)\{0}. (2.7) by our assumption, f (u, x0) ∈ −c(u). (2.8) from (2.7), (2.8) and (b) we get 0 = f (u, u) ∈ −c(u)\{0}, which is a contradiction. thus, (2.6) holds. since f is c−upper sign continuous, (2.6) implies that f (x0, y) 6∈ −c(x0)\{0}. the following result improves proposition 2.5 in [10]. corollary 2.14. let k be a nonempty convex subset of x and f : k × k → r be a pseudomonotone bifunction satisfying the following conditions: 228 a.p. farajzadeh et. al. cubo 12, 1 (2010) (a) for each x ∈ k, f (x, x) = 0, (b) for each x, y, z ∈ k, if f (x, y) < 0 and f (x, z) ≤ 0, then f (x, u) < 0, for all u ∈]y, z[, (c) for every x and y in k the following implication holds: f (u, y) ≥ 0, ∀ u ∈]x, y[⇒ f (x, y) ≥ 0. let x0 ∈ k, and then f (x0, y) ≥ 0, ∀y ∈ k ⇔ f (y, x0) ≤ 0, ∀y ∈ k. proof. in the previous lemma, let y = r and c(x) = [0, ∞), for every x ∈ k. obviously, f is c− pseudomonotone by (b) and c−upper sign continuous by (c). now, the result follows from lemma 2.13. theorem 2.15. let k be a nonempty convex subset of x and c = {c(x) : x ∈ k} is a family of convex cone proper subsets of y. let f : k × k → y be a c−pseudomonotone and c−upper sign continuous bifunction such that: (a) for each x ∈ k, f (x, x) = 0, (b) for each y ∈ k, the set {x ∈ k : f (y, x) ∈ −c(y)} is closed in k; (c) for each x, y, z ∈ k, if f (x, y) ∈ −c(x)\{0} and f (x, z) ∈ −c(x), then f (x, u) ∈ −c(x)\{0}, for all u ∈]y, z[, (d) there exist a nonempty compact subset b of k and a nonempty convex compact subset d of k such that, for each x ∈ k\b there exists y ∈ d such that f (y, x) 6∈ −c(y). then, the solution set of svep with respect to the family w (x) = y \(−c(x)\{0}), is nonempty and compact. proof. we define γ, γ̂ : k → 2k by γ̂(y) = {x ∈ k : f (x, y) 6∈ −c(x)\{0}}, γ(y) = {x ∈ k : f (y, x) ∈ −c(y)}. by (a), γ(y) and γ̂(y) are nonempty for each y ∈ k. by the c-pseudomonotoncity of f we get, γ̂(y) ⊂ γ(y), ∀y ∈ k. now, we show that γ̂ is a kkm mapping. indeed, assume that γ̂ is not a kkm mapping, then there exist y1, y2, ..., yn in k and z ∈ co{y1, y2, ..., yn} such that z 6∈ ⋃n i=1 γ̂(yi). hence, we have cubo 12, 1 (2010) strong vector equilibrium problems ... 229 f (z, yi) ∈ −c(z)\{0}, ∀ i = 1, 2, ..., n. now, it follows from (c) that f (z, z) ∈ −c(z)\{0}, which contradicts (a). thus γ̂ is a kkm mapping. now since for each y ∈ k, γ̂(y) ⊆ γ(y) we deduce (use the same reasoning as above with in this case if z 6∈ ∪ni=1γ(yi) then z 6∈ γ(yi) so z 6∈ γ̂(yi) for each 1 ≤ i ≤ n) that γ is a kkm mapping. the other conditions of theorem 1.2 are fulfilled by (b) and (d) and hence, ⋂ x∈k γ(x) 6= ∅. (2.9) also, lemma 2.13 implies that ⋂ x∈k γ(x) = ⋂ x∈k γ̂(x). (2.10) from (2.9) and (2.10), svep has a solution. since the solution set of svep is ⋂ x∈k γ̂(x) = ⋂ x∈k γ(x) 6= ∅, then is is closed and a subset of the compact set b. this completes the proof. the next corollary generalize theorem 2.3 in [7]. corollary 2.16. let k be a nonempty pointed closed convex subset of x and c = {c(x) : x ∈ k} is a family of proper pointed closed convex cone subsets of y . let t : k → l(x, y ) be a cpseudomonotone and c-upper sign continuous bifunction such that there exist a nonempty compact subset b of k and a nonempty convex compact subset d of k such that for each x ∈ k\b there exists y ∈ d such that 〈t y, y − x〉 6∈ c(y). then there exists an x ∈ k such that 〈t x, y − x〉 6∈ −c(x\{0}), ∀y ∈ k. proof. let f (x, y) = 〈t (x), y − x〉. we show that f satisfies the conditions of theorem 2.15. note (a) obviously holds. for (b) notice for each y ∈ k, the set {x ∈ k : f (y, x) ∈ −c(y)} = {x ∈ k : 〈t (y), x − y〉 ∈ −c(y)} is closed in k, since c(y) is a closed set and t (y) ∈ l(x, y ) for each y ∈ k. to show (c) let x, y, z ∈ k and t ∈]0, 1[. if f (x, y) = 〈t (x), y − x〉 ∈ −c(x)\{0} and f (x, z) = 〈t (x), z − x〉 ∈ −c(x), then 〈t (x), ty + (1 − t)z − x〉 = 〈t (x), t(y − x) + (1 − t)(z − x)〉 = t〈t (x), y − x〉 + (1 − t)〈t (x), z − x〉 ∈ −tc(x)\{0} + (1 − t)(−c(x)) ⊆ −c(x)\{0}, for all t ∈]0, 1[ since −c(x)\{0} is convex, note c(x) is a pointed convex cone. to show (d) note for all x ∈ k\b there exists y ∈ d such that 〈t (y), y − x〉 6∈ c(y) which implies f (y, x) 6∈ −c(y). now apply theorem 2.15 so there exists an x̄ ∈ k such that f (x̄, y) = 〈t x̄, y − x̄) 6∈ −c(x̄) for all y ∈ k. received: january, 2009 . revised: march, 2009. 230 a.p. farajzadeh et. al. cubo 12, 1 (2010) references [1] ansari, q.h., vector equilibrium problems and vector variational inequalities, in: f. giannessi(ed) vector variational inequalities and vector equilibria, mathematical theories, kluwer, dordrecht, (2000), 1–16 [2] bianchi, m. and pini, r., coercivity conditions for equilibrium problems, j. optim. theory appl., 124 (2005), 79–92. 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[13] yang, f., wu, c. and he, q., applications of ky fan’s inequality on σ−compact set to variational inclusion and n− person game theory, j. math. anal. appl., 319 (2006), 177–186. articulo 7.dvi cubo a mathematical journal vol.12, no¯ 02, (97–121). june 2010 the tree of primes in a field wolfgang rump institute for algebra and number theory, university of stuttgart, pfaffenwaldring 57, d-70550 stuttgart, germany email: rump@mathematik.uni-stuttgart.de dedicated to b. v. m. abstract the product formula of algebraic number theory connects finite and infinite primes in a stringent way, a fact, while not hard to be checked, that has never ceased to be tantalizing. we propose a new concept of prime for any field and investigate some of its properties. there are algebraic primes, corresponding to valuations, such that every prime contains a largest algebraic one. for a number field, this algebraic part is zero just for the infinite primes. it is shown that the primes of any field form a tree with a kind of self-similar structure, and there is a binary operation on the primes, unexplored even for the rationals. every prime defines a topology on the field, and each compact prime gives rise to a unique haar measure, playing an essential part in the product formula. resumen la fórmula producto de la teoŕıa de números algebraicos conecta primos finitos e infinitos de una formula estricta, un hecho no dif́ıcil de ser verificado, es que nunca cesa de ser estudiado. nosotros proponemos un nuevo concepto de primos para cualquier cuerpo e investigamos algunas de sus propiedades. hay primos algebraicos, correspondientes a valuaciones, talque todo primo contiene un primo algebraico mayor. para un número de cuerpos, esta parte algebraica es cero solamente para primos infinitos. es demostrado que los primos de cualquier cuerpo forman un árbol con una clase de estructura auto-similar, y 98 wolfgang rump cubo 12, 2 (2010) hay una operación binaria sobre los primos inexplorada incluso para los racionales. todo primo define una topoloǵıa sobre el cuerpo, y todo primo compacto da origen a una única medida de haar, jugando rol esencial en la fórmula producto. key words and phrases: prime, valuation, product formula. ams (mos) subj. class.: primary: 11s15, 11n80, 12j20, 13a18. secondary: 28c10 let k be a field, and let p 6= 0 be a polynomial in k[x]. assume first that p is normed and decomposes into powers of distinct linear factors: p = (x − b1) e1 · · · (x − br) er . (1) then p is completely determined by the function e: k → z with e(bi) := ei and e(b) := 0 for all b /∈ {b1, . . . ,br}. furthermore, e(b) > 0 for all b ∈ k. the same holds for rational functions p = g h ∈ k(x) if g,h ∈ k[x] are normed and decompose into linear factors. here e may also attain negative values in z. to complete the picture, replace k by the projective line p1(k) := k ∪ {∞}. if k = ir, we can calculate the limit of p for x → ∞: in case that deg p := deg g − deg h < 0, we have lim x→∞ p(x) = 0, and −deg p can be regarded as the multiplicity of ∞ ∈ p1(k) as a zero of p. if deg p > 0, then p has a pole in ∞, i. e. a zero with negative multiplicity −deg p. therefore, we set e(∞) := −deg p, (2) and this definition makes sense for an arbitrary field k. the function e vanishes almost everywhere, and ∑ b∈p1(k) e(b) = 0. (3) in general, instead of (1), a polynomial p is of the form p = a · pe11 · · ·p er r (4) with a ∈ k× and different irreducible polynomials p1, . . . ,pr. then the pi take the rôle of the points bi ∈ p 1(k). however, k can be extended such that every pi splits into deg pi linear factors. to take this into account, a weight f(pi) := deg pi is attached to the point pi, i. e. pi counts f(pi)-fold. at infinity ∞ we still have e(∞) = −deg p, and f(∞) := 1. so eq. (3) turns into r∑ i=1 e(pi)f(pi) + e(∞)f(∞) = 0. (5) in other words, every point pi has two invariants: the multiplicity e(pi), that is, the ramification order, and the weight f(pi), called the inertial degree. historically, the analogy between function fields and the field q of rationals became more and more apparent by the work of gauß, dedekind and artin [11]. in this vein, it was natural to ask whether a formula like (5) holds for the field q. in fact, every n ∈ q× is of the form n = ±pe11 · · ·p er r with different primes p1, . . . ,pr and integers ei. the unit ±1 corresponds to the unit a in formula (4). to get a multiplicative analogue of (5), we cubo 12, 2 (2010) the tree of primes in a field 99 set e(pi) := ei and f(pi) := p −1 i . for the infinite prime, we set e(∞) := 1 and f(∞) := |n|. so we get the product formula for q: r∏ i=1 f(pi) e(pi) · f(∞)e(∞) = 1. (6) an analogous product formula holds for algebraic number fields (see, e. g., [8], iii.1.3), or more generally, for global fields [10]. artin and whaples [1, 2] have shown that the product formula for algebraic number fields follows by formal arguments from its validity for q, and that a similar statement holds for function fields. for a global field k, (finite or infinite) primes can be conceived as equivalence classes of absolute values v : k → ir, normalized in a natural way using the modulus function of the locally compact completion k̂v (see [4], vi.9, proposition 1). in this way, the product formula admits a natural formulation, but we feel that it remains to be surprising: it just happens to be true still a mystery! it deeply touches the relationship between the additive and the multiplicative structure of q. to reveal this mystery, attempts have been made to elaborate on the concept of “prime” in a field [6, 7, 5, 13]. in the present paper, we give a new definition, partly anticipated by krull [7], which might also shed some more light upon the distinction between finite and infinite primes of an algebraic number field. namely, we define a prime in a field k to be a subset p ⊂ k with the following properties: (p1) 1 /∈ p . (p2) a,b ∈ p ⇒ ab ∈ p . (p3) ab ∈ p ⇒ a ∈ p or b ∈ p . (p4) ∃d ∈ k× : p + p ⊂ dp . as a consequence, the subsets ap of k with a ∈ k× form a chain. there are two types of primes we call them algebraic and transcendental such that the algebraic primes correspond to valuation rings in k. for every prime p , there is a largest algebraic prime q ⊂ p (proposition 7.2); we denote it by p◦. if p◦ = 0, we call p purely transcendental. for an algebraic number field k, the purely transcendental primes just correspond to the infinite primes. there are unexpected properties of our concept of prime. as is well-known, the valuation rings of a field form a dual tree under inclusion. more generally, we show that all primes of a field k form a tree which is almost self-similar (section 2). namely, we introduce a binary operation p ∗ q on the primes which satisfies p ⊂ p ∗q (proposition 2.4). thus even for k = q, there are a lot more primes p ⊂ q than rational prime numbers. a further investigation of this operation might be desirable. two primes define the same topology on k if and only if their intersection is non-zero (proposition 3.3). up to a constant, we associate a unique pseudo-valuation v : k → [0,∞] to any prime of k (proposition 3.4). this means that v(a) = 0 may also occur for non-zero a ∈ k. for a minimal (non-zero) prime p , the corresponding pseudo-valuation is of the form vt with t > 0 for some absolute value v, and then p can be recovered from v. every prime p makes k into a separated topological field such that the completion k̂p is again a topological field. if p is open in its topology, then p is induced by a prime of k̂p (theorem 3.1). we introduce compact primes, which are in a sense totally bounded (definition 4.1). they always contain a minimal prime which is again compact. for example, all the primes of q not only the minimal ones are compact. to any compact prime in any field, we associate a haar integral and 100 wolfgang rump cubo 12, 2 (2010) show that it admits a natural normalization (theorem 6.1). for the minimal primes of q or fp[x], this normalization corresponds to the standard normalization of absolute values. so the product formula becomes a natural statement on the minimal primes of certain fields. a special case of our concept of prime already occurs in a paper of w. krull [7]. such primes are open in their topology, and we call them krull primes. we show that every minimal prime is a krull prime, and that every prime p of a field k contains a largest krull prime p◦ (proposition 3.1). there is a close relationship between algebraic primes and krull primes. namely, we show that p◦ ⊂ p ◦, so that there is no other prime between p◦ and p ◦. therefore, every compact krull prime is either algebraic or purely transcendental (proposition 7.5). furthermore, a non-zero purely transcendental prime is a krull prime if and only if it is minimal. the same holds for compact primes (corollary of proposition 7.6). 1 primes the two examples of the introduction show that up to units, the elements of certain fields are given by their values at minimal “primes”, where a relation between these values is given by a product formula (6). let us first give a general definition of a prime. for subsets p,q of a field k and c ∈ k, define p + q := {a + b | a ∈ p,b ∈ q} and pq := {ab | a ∈ p,b ∈ q}. if p = {a}, we simply write a + q := {a} + q and aq := {a}q. the unit group of a ring r will be denoted by r×. definition 1.1. let k be a field. we call a non-empty subset p ⊂ k a prime of k if the following are satisfied (∀a,b ∈ k): (p1) 1 /∈ p . (p2) a,b ∈ p ⇒ ab ∈ p . (p3) ab ∈ p ⇒ a ∈ p or b ∈ p . (p4) ∃d ∈ k× : p + p ⊂ dp . from (p1) and (p2), we get −1 /∈ p , since (−1)(−1) = 1. therefore, (−1)(−a) = a gives a ∈ p ⇔ −a ∈ p (7) by virtue of (p3). by (p4), every a ∈ p satisfies 0 = a − a ∈ p + p ⊂ dp for some d ∈ k×. hence 0 ∈ p. (8) it is easily checked that p = 0 := {0} is a prime. we call it the trivial prime. by (p1) and (p2), a finite field has only the trivial prime. for a prime p , we call p̃ := {a ∈ k | ap ⊂ p} (9) the hull and ∂p := p̃ r p (10) the boundary of p . the trivial prime 0 satisfies 0̃ = k and ∂0 = k×. cubo 12, 2 (2010) the tree of primes in a field 101 proposition 1.1. let k be a field with a prime p. the ap̃ with a ∈ k× form a chain. proof. for a,b ∈ k×, assume that ap̃ 6⊂ bp̃ . then ac /∈ bp̃ for some c ∈ p̃ . for every p ∈ p , this gives acb−1 · ba−1p = cp ∈ p , hence ba−1p ∈ p by (p3). therefore, ba−1 ∈ p̃ , and thus bp̃ = a · ba−1p̃ ⊂ ap̃ . � proposition 1.2. let k be a field with a prime p. then ∂p = {a ∈ k× | ap = p} = {a ∈ k× | a,a−1 /∈ p}. (11) proof. assume that a ∈ ∂p . then a 6= 0 and ap ⊂ p . for b ∈ p , we have a · a−1b = b. thus (p3) gives a−1b ∈ p . therefore, b ∈ ap , which yields p ⊂ ap ⊂ p . now let a ∈ k× with ap = p be given. then aa−1 = 1 /∈ p implies a−1 /∈ p . similarly, a−1p = p , hence a /∈ p . thus {a ∈ k |ap = p} ⊂ {a ∈ k× |a,a−1 /∈ p}. finally, assume that a ∈ k× with a,a−1 /∈ p . for b ∈ p , this gives a−1 · ab = b, hence ab ∈ p by (p3). consequently, a ∈ p̃ r p = ∂p . � note that (11) implies that ∂p is a subgroup of k×, and a /∈ p̃ ⇔ a−1 ∈ p (12) for all a ∈ k×, that is, p and p̃ determine each other. furthermore, −1 ∈ k rp , and thus −1 ∈ ∂p . definition 1.2. we call a prime p of a field k algebraic if p + p ⊂ p , that is, (p4) holds for d = 1, otherwise transcendental. if p is algebraic, then (p̃ + p̃)p ⊂ p̃p + p̃p ⊂ p + p ⊂ p , which gives p̃ + p̃ ⊂ p̃. thus p̃ is a subring of k, and by (12), it follows that k is the quotient field of p̃ . by proposition 1.1, the principal ideals of p̃ form a chain. therefore, all ideals of p̃ form a chain. namely, if i 6⊂ j are ideals of p̃ , there is some a ∈ i r j. for all b ∈ j, this gives ap̃ 6⊂ bp̃, hence bp̃ ⊂ ap̃ ⊂ i. furthermore, p is an ideal of p̃ , and by (11) and (12), ∂p = (p̃)×. thus p is the unique maximal ideal of p̃ . in other words, p̃ is a valuation domain. example. for k = q, every rational prime p gives rise to an algebraic prime pp = { pa b ∈ q | a,b ∈ z, p ∤ b}. (13) by (12), we get p̃p = zp, whence p̃p/pp = zp/pzp ∼= z/pz. (14) moreover, q has a transcendental prime p∞ := (−1, 1) ∩ q = {a ∈ q | |a| < 1}. (15) here (p4) holds for d = 2. 2 the primes of q by (12), two primes p ⊃ q of a field k satisfy q ⊂ p ⊂ p̃ ⊂ q̃, (16) hence cp ⊂ cq̃ ⊂ q for all c ∈ q. more precisely: 102 wolfgang rump cubo 12, 2 (2010) proposition 2.1. let k be a field with a prime p. for any submonoid u of k× with p̃ ⊂ u, q := {0} ∪ {a ∈ k× | a−1 /∈ u} (17) is a prime with q ⊂ p and q̃ = u. every prime q ⊂ p is of this form. proof. let u be a submonoid of k×, and p̃ ⊂ u. by (12), this implies that q ⊂ p . assume that a,b ∈ q ∩ k×. then a−1,b−1 /∈ u. so b−1 /∈ p̃ , and therefore, b ∈ p ⊂ u. now ab /∈ q would yield a−1b−1 ∈ u, whence a−1 = a−1b−1 · b ∈ u, which is impossible! thus (p1) and (p2) hold for q. to prove (p3), we have to verify the implication a,b /∈ q ⇒ ab /∈ q, that is, a−1,b−1 ∈ u ⇒ a−1b−1 ∈ u. this follows since u is multiplicatively closed. finally, q + q ⊂ p + p ⊂ dp holds for some d ∈ k×. to prove (p4), we can assume that q 6= 0. for arbitrary a ∈ k r u and b ∈ p ∩ k×, we have a /∈ p̃ . thus a−1 ∈ p . so b−1a ∈ u would give a ∈ bu ⊂ u, a contradiction. hence b−1a /∈ u, and thus ba−1 ∈ q. so we get p ⊂ aq, i. e. q + q ⊂ daq, which completes the proof of (p4). conversely, let q ⊂ p be a prime. then (12) shows that q is of the form (17) with a submonoid u := q̃ of k×. � corollary. for each pair p,q of primes of a field k, there is an infimum p ∧ q. proof. by proposition 2.1, the submonoid p̃q̃ corresponds to a prime p ∧ q with the desired property. � definition 2.1. we say that two primes p and q of a field k are dependent if p ∧ q 6= 0, otherwise independent. the minimal among the non-zero primes are called minimal primes. minimal primes can be characterized as follows. proposition 2.2. a non-trivial prime p of a field k is minimal if and only if for every pair a,b ∈ p ∩ k×, there exists some n ∈ in with an ∈ bp. proof. assume that a,b ∈ p ∩ k×. then u := ⋃ n∈in a −np is a submonoid of k×, and p̃ ⊂ a−1p ( u. suppose that an /∈ bp for all n ∈ in. then b−1 /∈ u. by proposition 2.1, there is a non-trivial prime q ( p with q̃ = u. conversely, let q be a prime with 0 6= q ( p . choose a ∈ p r q and b ∈ q ∩ k×. then a /∈ q implies that a−1 ∈ q̃, hence a−n ∈ q̃ for all n ∈ in. furthermore, bp ⊂ bp̃ ⊂ bq̃ ⊂ q, and thus a−nbp ⊂ a−nq ⊂ q ⊂ p . hence a−nb ∈ p̃ , that is, anb−1 /∈ p for all n. thus an /∈ bp holds for n ∈ in. � now we are able to give an explicit description of the minimal primes of q. to this end, we need two auxiliary results. lemma 2.1. let p be a prime of q, such that p + p ⊂ dp for some d ∈ q×. then np ⊂ dkp holds for all k ∈ in and n ∈ z with |n| 6 2k. proof. for k = 0, this is trivial. if the assertion holds for some k ∈ in, then every n ∈ z with |n| 6 2k+1 can be written as a sum n = r + s in z such that |r|, |s| 6 2k. hence np ⊂ rp + sp ⊂ dkp + dkp ⊂ dk+1p . � cubo 12, 2 (2010) the tree of primes in a field 103 lemma 2.2. let p be a prime of q, and let a > b be relatively prime in in, such that a(p +p) ⊂ bp. then naap ⊂ ba+k−1p (18) holds for all k ∈ in and n ∈ z with |n| < ak. proof. we proceed by induction. for k = 0, the statement (18) is trivial. suppose that (18) is shown for some k ∈ in. we prove (18) for k + 1. assume that |n| < ak+1. since a and b are relatively prime, there are m,r ∈ z with n = ma + rbk. if r is replaced by r + a, the equation holds for m − bk instead of m. so we can assume that |r| < a. for r 6= 0, there are still two possibilities. we choose r such that the signs of n and r are not different. then |m| = 1 a |n − rbk| < ak. so the inductive hypothesis gives maap ⊂ ba+k−1p , hence naap ⊂ ma · aap + rbk · aap ⊂ ba+k−1ap + rbkaap . since |r| 6 a − 1 < 2a−1, lemma 2.1 implies that rp ⊂ ( b a )a−1 p . therefore, we get naap ⊂ ba+k−1ap + ba+k−1ap ⊂ ba+k−1bp = ba+kp . � as a consequence of lemma 2.2, we have lemma 2.3. let p be a prime of q, and assume that 1 < c ∈ p. then there exists an integer e > 1 with ze ⊂ p and e(p + p) ⊂ p. proof. by (p4), there is some d ∈ q× with p + p ⊂ dp . choose i ∈ in such that ci > |d|. then ci · |d|−1 = a b with relatively prime a,b ∈ in. thus a > b and a(p + p) = bcid−1(p + p) ⊂ bcip ⊂ bp . for every m ∈ z, there exists some k ∈ in with |m| · ba < ( a b )k . we set n := m · ba+k. then |n| = |m| · ba+k < ak. by lemma 2.2, this implies that naap ⊂ ba+k−1p , that is, maabp ⊂ p for all m ∈ z, and thus zaa+1 = zaab · a b ⊂ p . furthermore, aa+1(p + p) = aa · a(p + p) ⊂ aabp ⊂ p . � now we show that the list of minimal primes of q given in (13) and (15) is complete. the first part of the following theorem is essentially due to ostrowski [9] (cf. [12]). theorem 2.1. every minimal prime p of q is of the form p = pp with a rational prime p or p = ∞, and every prime of q contains a minimal prime. proof. by proposition 2.2, the prime p∞ is minimal. therefore, if |c| < 1 for all c ∈ p , then p ⊂ p∞, and thus p = p∞. otherwise, there exists some c ∈ p with c > 1. by lemma 2.3, there is an integer e > 1 with ze ⊂ p and e(p + p) ⊂ p . assume that e = p1 · · ·pr with rational primes pi. suppose that ep −r i ∈ p holds for all i. then 1 = (ep −r 1 ) · · · (ep −r r ) ∈ p , a contradiction. hence ep−r /∈ p for some prime p|e. this implies that e−1pr ∈ p̃ . for every n ∈ z, we thus get (np)r = nre · e−1pr ∈ pp̃ ⊂ p . therefore, np ∈ p , and thus zp ⊂ p . now let p ∤ n for some n ∈ z. then there are u,v ∈ z with un + vpr+1 = 1. so the residue class n + pr+1z is invertible in z/pr+1z. therefore, we find some k ∈ in r 0 with nk ≡ 1 (pr+1). with s := (r + 1)k, we also have ns ≡ 1 (pr+1), and thus 1 = ns + mpr+1 for some m ∈ z. suppose that ns ∈ prp . then 1 ∈ prp + prp = e−1pr · e(p + p) ⊂ p̃p ⊂ p , a contradiction. hence nsp−r /∈ p , and thus n−spr ∈ p̃ . so we get (n−1p)s = n−sp(r+1)k ∈ n−sprp ⊂ p . this gives n−1p ∈ p , whence pp ⊂ p . � remark. there exist non-minimal primes in q. for example, p := 2z2 ∪ (z2 ∩ 3z3) 104 wolfgang rump cubo 12, 2 (2010) is a prime. in fact, this is a general phenomenon. we show first that the primes form a tree with the trivial prime as a root: proposition 2.3. let k be a field with a prime p. then the primes q ⊂ p form a chain. proof. let q,q′ be primes with q,q′ ⊂ p and q′ 6⊂ q. then there is an element a ∈ q′ r q. for b ∈ q, the inclusions (16) give bp̃ ⊂ bq̃ ⊂ q. hence ap̃ 6⊂ bp̃. therefore, proposition 1.1 yields b ∈ bp̃ ⊂ ap̃ ⊂ aq̃′ ⊂ q′, and thus q ⊂ q′. � moreover, the tree of primes has a strong symmetry: proposition 2.4. let k be a field with two primes p and q. then p ∗ q := p ∪ (p̃ ∩ q) is a prime of k. proof. obviously, 1 /∈ p ∗ q. assume that a,b ∈ p ∗ q. if a ∈ p or b ∈ p , then ab ∈ p ⊂ p ∗ q since a,b ∈ p̃ . for a,b ∈ p̃ ∩ q, however, we have ab ∈ p̃ ∩ q. this proves (p2). to verify (p3), let a,b ∈ k with ab ∈ p ∗ q be given. if ab ∈ p , then a ∈ p or b ∈ p . therefore, assume that ab ∈ p̃ ∩q and a,b /∈ p . then a−1,b−1 ∈ p̃ . hence a = ab ·b−1 ∈ p̃ , and similarly, b ∈ p̃ . therefore, a ∈ p̃ ∩q or b ∈ p̃ ∩q. finally, assume that d ∈ k× with p +p ⊂ dp . if p = 0, we have p ∗q = q. otherwise, there exists some a ∈ p r 0. now (p̃ + p̃)p ⊂ p + p ⊂ dp , hence (p̃ + p̃)d−1 ⊂ p̃ . consequently, (p ∗ q) + (p ∗ q) ⊂ p̃ + p̃ ⊂ dp̃ = da−1ap̃ ⊂ da−1(p ∗ q). � we call p ∗ q the lexicographic product of p and q. thus for a fixed p , every prime q yields a prime p ∗ q ⊃ p which may coincide with p . apart from this exception, the tree of primes is self-similar, i. e. the whole tree sits above each prime. 3 the topology of a prime let k be a field with a prime p . for a,b ∈ k, we define a 6 b : ⇔ ap̃ ⊂ bp̃ (19) cubo 12, 2 (2010) the tree of primes in a field 105 and a b. then a 6 b 6 c ⇒ a 6 c a 6 b ⇒ ac 6 bc (20) holds for a,b,c ∈ k, and proposition 1.1 yields a 6 b oder b 6 a. (21) in particular, this implies the reflexity a 6 a for all a ∈ k. by (11), we have ap = bp ⇔ b−1ap = p ⇔ b−1a ∈ ∂p for a,b ∈ k×. therefore, (19) induces a linear order on k×/∂p . from (19) and (12), we get p̃ = {a ∈ k | a 6 1} ; p = {a ∈ k | a < 1}. (22) for a ∈ k and c ∈ k×, we define the c-neighbourhood of a as follows: uc(a) = u p c (a) := a + cp. (23) a subset u ⊂ k is open with respect to p if for every a ∈ u, there is an element c ∈ k× with uc(a) ⊂ u. the p-open sets define a topology on k, but it is not clear in advance that the cneighbourhoods (23) are neighbourhoods in this topology. the following proposition shows that this is in fact the case. proposition 3.1. let k be a field with a prime p. the set p◦ := {a ∈ p | ∃n ∈ in: an(p + p) ⊂ p} (24) is open. if p is minimal, then p◦ = p. proof. assume that a ∈ p◦, say, an(p + p) ⊂ p . then a2n(p + p + p + p) ⊂ an(p + p) ⊂ p , and by induction, we get amn(p + · · · + p) ⊂ p for m ∈ in, where p + · · · + p may have at most 2m summands. we set b := a2n(n+1). now let p ∈ p be arbitrary. expanding (a + bp)2n gives 22n summands, namely, a2n plus 22n − 1 summands in bp . so we get (a + bp)2n(p + p) ⊂ (a2n + ba−2n 2 p)(p + p) ⊂ (a2n + a2np)(p + p) ⊂ a2n(p + p) + a2n(p + p) ⊂ anp + anp ⊂ p. this yields a + bp ⊂ p◦. finally, let p be minimal and d ∈ k× with p + p ⊂ dp . then p ⊂ dp , hence d−1 ∈ p̃ . for a ∈ p , proposition 2.2 yields an n ∈ in with an ∈ ad−1p , whence an(p + p) ⊂ ad−1(p + p) ⊂ ap ⊂ p . thus a ∈ p◦. � in section 7, we will show that p◦ is again a prime. corollary. let k be a field with a minimal prime p. for every a ∈ p, there are b,c ∈ p ∩ k× such that ap̃ + bp ⊂ cp̃. proof. assume that p + p ⊂ dp with d ∈ k× and a ∈ p . by proposition 3.1, there is some b ∈ d−1p ∩ k× with a + bdp ⊂ p . we show first that ap̃ + bp ⊂ p . to this end, assume that e ∈ p̃ . 106 wolfgang rump cubo 12, 2 (2010) case i: e ∈ d−1p . then ae + bp ⊂ d−1p + d−1p ⊂ p . case ii: e /∈ d−1p . then de /∈ p , hence d−1e−1 ∈ p̃ . this gives ae + bp = e(a + bdd−1e−1p) ⊂ e(a + bdp) ⊂ p . if ap̃ = p , we can set c := a. otherwise, there exists some c ∈ p r ap̃ . then ca−1 /∈ p̃ , hence c−1a ∈ p . by the above, we find an element b ∈ d−1p ∩ k× ⊂ p ∩ k× with c−1ap̃ + bp ⊂ p . whence ap̃ + cbp ⊂ cp ⊂ cp̃. � lemma 3.1. let k be a field with a prime p. assume that d ∈ k× with p + p ⊂ dp. then every a,b ∈ k× with b ∈ ad−1p satisfies ∀p ∈ p : (a + bp)−1 − a−1 ∈ da−2bp. proof. assume that p ∈ p . then bp ∈ ap implies that a+bp 6= 0. suppose that (1+a−1bp)d ∈ p . then d = (1 + a−1bp)d − a−1bpd ∈ p + p ⊂ dp , a contradiction. hence (1 + a−1bp)−1d−1 ∈ p̃ . from (1 + a−1bp)−1(1 + a−1bp) = 1 we get (1 + a−1bp)−1 + (1 + a−1bp)−1a−1bp = 1. multiplying by a−1 gives (a + bp)−1 + (1 + a−1bp)−1a−2bp = a−1, i. e. (a + bp)−1 − a−1 = −(1 + a−1bp)−1a−2bp ∈ dp̃ · a−2bp ⊂ da−2bp . � theorem 3.1. let k be a field with a prime p. the topology of p makes k into a separated topological field, and the completion k̂p of k with respect to p is a topological field. if p is open, then p̂ := ⋃ a∈p ap̃ (25) is a prime in k̂p with p̂ ∩ k = p, and p̂ defines the topology of k̂p . proof. by (p4), there is some d ∈ k with p +p ⊂ dp . this implies that k is a topological ring. hence k is a topological field by lemma 3.1. to show that the same holds for the completion k̂ of k, we have to verify that the image under the mapping x 7→ x−1 of every cauchy filter c in k which does not have a cluster point at 0, is a cauchy filter (see [3], iii.6.8, proposition 7). thus let c ∈ k× and c ∈ c be such that c∩cp = ∅. we can assume that a−b ∈ cd−1p for all a,b ∈ c. choose any a ∈ c. suppose that b ∈ c ∩cd−1p . then a = (a−b) +b ∈ cd−1p +cd−1p ⊂ cp , a contradiction. hence c ∩cd−1p = ∅. this gives ac−1d /∈ p , and thus a−1cd−1 ∈ p̃ , i. e. a−1 ∈ c−1dp̃ . for an arbitrary e ∈ k×, we find some c′ ⊂ c in c with a − b ∈ c2d−2ep for all a,b ∈ c′. this gives a−1 − b−1 = a−1b−1(b − a) ∈ c−2d2p̃ · c2d−2ep ⊂ ep for a,b ∈ c′. now let p be open. we show first that for a ∈ p , ap̃ ⊂ p (26) holds in k. for a = 0, this is trivial. so we assume that a ∈ p ∩ k×. let b /∈ p be given. then b−1 ∈ p̃, which gives (a−1b)−1 = b−1a ∈ p . since p is open, the continuity of x 7→ x−1 implies that (a−1b + cp)−1 ⊂ p for some c ∈ k×. hence (a−1b + cp) ∩ p̃ = ∅, and thus (b + acp) ∩ ap̃ = ∅. this proves (26). together with (25), we infer that p̂ ∩ k = p . cubo 12, 2 (2010) the tree of primes in a field 107 furthermore, (26) shows that k̂p satisfies (p1). if b,c ∈ p̂ , then b,c ∈ ap̃ for some a ∈ p . hence bc ∈ ap̃ · ap̃ ⊂ a2p̃ ⊂ p̂ , which gives (p2). similarly, (p4) is easily verified. to prove (p3), assume that b,c ∈ k̂p satisfy bc ∈ p̂ and b /∈ p̂ , say, bc ∈ ap̃ for some a ∈ p . then b /∈ pp̃ for all p ∈ p . for any p ∈ p , we find a neighbourhood u of 0 in k̂p such that (b + u) ∩pp̃ = ∅. hence (bc + uc) ∩pcp̃ = ∅. on the other hand, (bc + uc) ∩ ap̃ 6= ∅. thus for any p ∈ p , we get ap̃ 6⊂ pcp̃. if p 6= 0, this implies that p−1c−1a /∈ p̃ . hence pca−1 ∈ p for all p ∈ p . consequently, ca−1 ∈ p̃ , and thus c ∈ ap̃ ⊂ p̂ . it remains to verify that p̂ defines the topology of k̂p . for a ∈ p , we have ap ⊂ p̂ . conversely, (26) implies that p̂ ⊂ p . this completes the proof. � in particular, theorem 3.1 implies that the limit of a sequence in k with respect to a prime p is unique. a minimal prime p is uniquely determined by its topology: proposition 3.2. let k be a field with a minimal prime p. then p = {a ∈ k | lim n→∞ an = 0}. (27) proof. let a ∈ p . by proposition 2.2, every c ∈ k× gives rise to some n ∈ in with an ∈ cp . therefore, an ∈ cp for all n > n, and thus lim n→∞ an = 0. on the other hand, if a /∈ p , then a−1 ∈ p̃ , which gives a−n ∈ p̃ for all n ∈ in. but this implies that an /∈ p for all n. � if a prime contains a minimal prime p , then p can be determined topologically by means of proposition 3.2. the following proposition decides whether two primes define the same topology. proposition 3.3. for two primes p,q 6= 0 of a field k, the following are equivalent. (a) pq 6= k. (b) p̃q̃ 6= k. (c) p and q are dependent (see definition 2.1). (d) there is some a ∈ k× with ap ⊂ q. (e) p and q define the same topology. proof. (a) ⇒ (b): suppose that p̃q̃ = k. for a ∈ p ∩ k× and b ∈ q ∩ k×, this implies that k = abp̃q̃ ⊂ pq, which is impossible. the implication (b) ⇒ (c) follows by proposition 2.1. (c) ⇒ (d): for every a ∈ (p ∧ q) ∩ k×, we have ap ⊂ p ∧ q ⊂ q. (d) ⇒ (a): from ap ⊂ q with a ∈ k×, we get apq ⊂ q2 ⊂ q, hence pq ⊂ a−1q 6= k. therefore, condition (d) is symmetric. it states that every q-open set with respect to p is open. hence (d) ⇔ (e). � let k be a field with a prime p . a function f : k → [0,∞] is said to be monotonous (with respect to p) if a 6 b ⇒ f(a) 6 f(b) (28) holds for all a,b ∈ k. we call f strictly monotonous if (28) can be replaced by an equivalence. for the following definition, we set ∞ · a = a · ∞ := ∞ (29) for 0 < a 6 ∞. 108 wolfgang rump cubo 12, 2 (2010) definition 3.1. let k be a field with a prime p . we call v : k → [0,∞] a pseudo-valuation with respect to p if v is monotonous with v(0) = 0 and v(1) = 1, such that v(ab) = v(a)v(b) (30) holds for all a,b ∈ k with v(a) /∈ {0,∞}. we call v finite if v(a) < ∞ for all a ∈ k. if v is finite and strictly monotonous, we say that v is a p-valuation. for a ∈ k×, the definition yields v(a) = 0 ⇔ v(a−1) = ∞. (31) note that by (22), the prime p of a p-valuation v is given by p = {a ∈ k | v(a) < 1}. by [4], vi.6.1, a p-valuation is of the form vt, where t > 0 and v is an absolute value on k. proposition 3.4. let k be a field with a prime p. for every c ∈ k r p̃ and every real number r > 1, there is exactly one pseudo-valuation v : k → [0,∞] with v(c) = r. if p is minimal, then v is a p-valuation. proof. first, let v be a pseudo-valuation with v(c) = r, and let a ∈ k be given. if a 6 cn for all n ∈ z, then v(a) 6 rn for all n, and thus v(a) = 0. if, however, a > cn for all n ∈ z, we get v(a) = ∞. otherwise, there exists some n ∈ in with c−n 6 a 6 cn, and then r−n 6 v(a) 6 rn, hence v(a) /∈ {0,∞}. for such an a and an integer n > 0, there exists some m ∈ z with cm−1 6 an 6 cm, which gives rm−1 6 v(a)n 6 rm, i. e. r m−1 n 6 v(a) 6 r m n . thus v is unique: v(a) = inf{rm/n | m,n ∈ z, n > 0, an 6 cm}. (32) this formula extends to v(a) ∈ {0,∞}. conversely, we show that (32) defines a pseudo-valuation. again, v(a) = 0 respectively v(a) = ∞ holds if a 6 cn resp. a > cn for all n ∈ z. therefore, assume that v(a) /∈ {0,∞}. then for every n > 0 in in, there is an integer m with cm−1 6 an 6 cm. this implies (30) for v(b) ∈ {0,∞}. assume that ck−1 6 bn 6 ck. then we get r m−1 n 6 v(a) 6 rm/n and r k−1 n 6 v(b) 6 rk/n. hence r m+k−2 n 6 v(a)v(b) 6 r m+k n . on the other hand, ck+m−2 6 (ab)n 6 cm+k, and thus r m+k−2 n 6 v(ab) 6 r m+k n . for n → ∞, this gives (30). furthermore, (32) implies that v is monotonous, and v(0) = 0, v(1) = 1 holds together with v(c) = r. hence v is a pseudo-valuation. if p is minimal, proposition 2.2 shows that for every a ∈ p , the inequality v(a) < 1 holds. hence v is a p-valuation. � examples. 1. for the transcendental prime p∞ = (−1, 1) ∩ q of q, the unique p∞-valuation v : q → [0,∞] with r = c > 1 (proposition 3.4) is given by v(a) := |a|. so the completion is ir, and p̂∞ = (−1, 1). for a prime p and p := pzp, every a ∈ q × admits a unique representation a = pn · b c with p ∤ b,c and n ∈ z. then v(a) = |a|p := p −n with |0|p := 0 is a p-valuation v : q → [0,∞], the p-adic absolute value of q. the c-neighbourhoods (23) of 0 are then of the form pnzp, n ∈ z. as usual, we write q̂p for the completion of q with respect to p = pzp. here we have p̂ = pẑp, where ẑp denotes the ring of p-adic numbers. 2. for the field c of complex numbers, p := {a ∈ c | |a| < 1} is a prime, and the absolute value is a p-valuation v : c → [0,∞]. here we have ĉp = c. cubo 12, 2 (2010) the tree of primes in a field 109 4 compact primes and the haar integral in the sequel, we deal with primes like those occuring in algebraic number fields. definition 4.1. we call a prime p of a field k compact if for every b ∈ k× there exist a1, . . . ,an ∈ k with p ⊂ (a1 + bp) ∪ · · · ∪ (an + bp). (33) example. the primes of q are compact: for the prime p∞ this is trivial; for a rational prime p, we have pzp = ⋃pn−1 i=1 ip + p nzp. so the minimal primes of q are compact. (34) by the following result, the other primes are compact, too: proposition 4.1. let k be a field. the primes q which are dependent of a non-trivial compact prime p are compact. proof. assume that b ∈ k×. since p ∧ q 6= 0, there exists some a ∈ (p ∧ q) ∩ k×. as p is compact, there are a1, . . . ,an ∈ k with p ⊂ (a1 + a 2bp) ∪ · · · ∪ (an + a 2bp). hence aq ⊂ p ⊂⋃n i=1(ai + a 2bp) ⊂ ⋃n i=1(ai + abq), and thus q ⊂ ⋃n i=1(a −1ai + bq). � every compact prime p of a field k leads to a concept of integral. let cp (k) denote the set of real functions f : k → ir such that f vanishes on krcp for some c ∈ k×, and is uniformly continuous with respect to p , i. e. for every ε > 0 there exists some b ∈ k× such that |f(a) −f(a′)| < ε holds for all a,a′ ∈ k with a− a′ ∈ bp . clearly, cp (k) is an ir-vector space. we write f 6 g for f,g ∈ cp (k) if f(a) 6 g(a) holds for all a ∈ k. if, in addition, f 6= g, we write f < g. the subset of the f > 0 in cp (k) will be denoted by c + p (k). we show first that c + p (k) 6= ∅. proposition 4.2. let k be a field with a prime p. for every b ∈ k×, there exists a function f ∈ c+p (k) which satisfies f(a) = 1 for all a ∈ bp. proof. since a ∈ bp ⇔ ab−1 ∈ p , we can assume that b = 1. furthermore, we can assume that p 6= 0. by (p4), there is an element d ∈ k r p̃ with p + p ⊂ dp , and by proposition 3.4, there exists a pseudo-valuation v : k → [0,∞] with v(d) = 2. we show first that for every n ∈ in there exists some c ∈ k× such that d−1 6 (1 + cp)n 6 d (35) holds for all p ∈ p . from (p4) we get by induction p + · · ·+p ⊂ dnp , if the sum p + · · ·+p contains at most 2n summands. for c ∈ p , we therefore have (1 + cp)n ∈ 1 + cp + · · · + cp ⊂ 1 + cdnp . replacing c by cd−n, the inequality (35) turns into d−1 6 1 + cp 6 d for some c ∈ k× and p ∈ p . by lemma 3.1 (with a = 1), this reduces to the single inequality 1 + cp 6 d. this is equivalent to (1 + cp)d−1 ∈ p̃ , i. e. (1 + cp)d−1p ⊂ p . by (p4), the latter is satisfied for c = 1. 110 wolfgang rump cubo 12, 2 (2010) using v, we define f : k → ir by f(a) :=    1 for a ∈ p 2 − v(a) for a ∈ dp r p 0 for a /∈ dp and prove the uniform continuity of f. let ε > 0 be given. we have to find c ∈ k× so that |f(a) − f(a′)| < ε holds for all a,a′ ∈ k with a − a′ ∈ cp . by symmetry, we can assume that a ∈ dp r p . case i: a′ ∈ p . for c ∈ d−1p , this implies that a′ /∈ d−1p , since otherwise, a = (a − a′) + a′ ∈ d−1p + d−1p ⊂ p . hence if n ∈ in, the inequality (35) yields an element c ∈ k× with (1 + c a′ p)n 6 d for all p ∈ p . for a− a′ = cp, we thus have a < a a′ , hence d−1 < 1 6 an 6 ( a a′ )n = (1 + c a′ p)n 6 d. so we get 2−1 6 v(an) 6 2, and thus 2− 1 n 6 v(a) 6 2 1 n . consequently, |f(a) − f(a′)| = |(2 −v(a)) − 1| = |v(a) − 1| < ε holds for sufficiently large n. case ii: a′ /∈ dp . for any n ∈ in, the inequality (35) shows that there exists some c ∈ k× with d−1 6 (1 + c a′ p)n for all p ∈ p . for a − a′ = cp, this gives d−1 6 (1 + c a′ p)n = ( a a′ )n 6 ( a d )n 6 1 < d. therefore, dn−1 6 an 6 dn+1, hence 2n−1 6 v(a)n 6 2n+1, and thus 2 n−1 n 6 v(a) 6 2 n+1 n . for sufficiently large n, we get |f(a) − f(a′)| = |v(a) − 2| < ε. case iii: a′ ∈ dp r p . again by (35), for any n ∈ in, there exists c ∈ k× with d−1 6 (1 + c a′ p)n 6 d for all p ∈ p . for a − a′ = cp, we get d−1 6 ( a a′ )n 6 d for all p ∈ p . this gives 2−1 6 v(a)nv(a′)−n 6 2, hence 2− 1 n 6 v(a) v(a′) 6 2 1 n . for sufficiently large n, it follows that |f(a) − f(a′)| = |v(a) − v(a′)| < ε. � note that f ∈ cp (k) implies that |f| ∈ cp (k). therefore, every f ∈ cp (k) admits a decomposition f = f+ − f− (36) with f+ := 1 2 (|f| + f) and f− := 1 2 (|f| − f), so that |f| = f+ + f− and f+,f− > 0. for f ∈ cp (k) and a ∈ k, we define fa ∈ cp (k) by fa(b) := f(b − a). (37) definition 4.2. let k be a field with a compact prime p . a linear form i : cp (k) → ir is said to be a haar integral if i(f) > 0 holds for f > 0, and i(fa) = i(f) for all a ∈ k. so the haar integral is linear (like any integral) and monotonous (f < g ⇒ i(f) < i(g)), and invariant under translations. we will show first that a haar integral exists and is unique up to a constant. our proof is similar to [14], but does not assume that the topology of k given by the compact prime p is locally compact. in section 6, we will show that the constant can be chosen in a canonical way. for f,g ∈ c+p (k), let f : g be the infimum of all sums r1 + · · · + rn with r1, . . . ,rn > 0 in ir, such that f 6 r1g a1 + · · · + rng an holds for suitable a1, . . . ,an ∈ k. proposition 4.3. let k be a field with a compact prime p. every function f ∈ cp (k) is bounded. for f,g ∈ c+p (k), we have 0 < f : g < ∞. proof. first let f ∈ cp (k) and b ∈ k × be given such that f(a) = 0 holds for a /∈ bp . for every ε > 0, there exists some c ∈ k× such that |f(a)−f(a′)| < ε holds for a−a′ ∈ cp . since p is compact, there are a1, . . . ,an ∈ k with bp ⊂ (a1 + cp) ∪ · · · ∪ (an + cp). hence f is bounded. cubo 12, 2 (2010) the tree of primes in a field 111 next let f,g ∈ c+p (k) be given. choose a0 ∈ k with g(a0) > 0. then we find ε > 0 in ir and c ∈ k× with g(a) > ε for a − a0 ∈ cp . assume that f(a) = 0 for a /∈ bp and a1, . . . ,an ∈ k with bp ⊂ (a1 + cp) ∪ · · · ∪ (an + cp). then there exists some m ∈ in with f(a) 6 mε for all a ∈ bp . hence f 6 mga1−a0 + · · · + mgan−a0 , which gives f : g < ∞. furthermore, g(a) 6 r for all a ∈ k and a real r > 0. now if f(a) > 0, we obtain f : g > f(a)r−1 > 0. � lemma 4.1. let k be a field with a compact prime p. let b ∈ k× and f,g ∈ c+p (k) satisfy f(a) = 0 for a /∈ bp and g(a) > ε for some ε > 0 in ir and a ∈ b(p + p). then h := f g ∈ c+p (k), where h(a) := 0 for a /∈ bp. proof. for all a,a′ ∈ k, we have: |h(a) − h(a′)| = |f(a)g(a′) − g(a)f(a′)| g(a)g(a′) = ∣∣(f(a) − f(a′) ) g(a′) + f(a′) ( g(a′) − g(a) )∣∣ g(a)g(a′) 6 |f(a) − f(a′)| g(a) + f(a′) g(a′) · |g(a) − g(a′)| g(a) . for a,a′ /∈ bp , this expression vanishes. otherwise, if a − a′ ∈ bp , then a,a′ ∈ bp + bp . hence |h(a) − h(a′)| 6 ε−1|f(a) − f(a′)| + ε−2f(a′) · |g(a) − g(a′)| for all a,a′ ∈ k. since f is bounded by proposition 4.3, this gives h ∈ c+p (k). � proposition 4.4. let k be a field with a compact prime p. for every f ∈ c+p (k) and c ∈ k ×, there are f1, . . . ,fn ∈ c + p (k) and a1, . . . ,an ∈ k such that fi(a) = 0 holds for a − ai /∈ cp, so that f = f1 + · · · + fn. proof. assume that f(a) = 0 holds for a /∈ bp , where b ∈ k×. by proposition 4.2, there exists some f′ ∈ c+p (k) which satisfies f ′(a) = 1 for a ∈ b(p + p). furthermore, there exists some g ∈ c+p (k) that vanishes outside cp . by proposition 4.3, we have (f + f ′) : g < ∞. hence there are real numbers r1, . . . ,rn > 0 and a1, . . . ,an ∈ k with f + f ′ 6 r1g a1 + · · · + rng an . by lemma 4.1, h := f r1g a1 +···+rngan ∈ c+p (k), whence f = r1hg a1 + · · · + rnhg an . � 5 existence and uniqueness now we are ready to prove existence and uniqueness of the haar integral. proposition 5.1. for every compact prime p of a field k, there exists a haar integral. proof. by (36), it is enough to find a map i : c+p (k) → (0,∞) which satisfies i(rf + sg) = ri(f) + si(g) for r,s > 0 and i(fa) = i(f) for a ∈ k. for r > 0 in ir and f,g ∈ c+p (k), we have (rf : g) = r(f : g). furthermore, for f,g,h ∈ c+p (k), (f : g)(g : h) > f : h (38) 112 wolfgang rump cubo 12, 2 (2010) in fact, assume that f 6 r1g a1 + · · · + rng an and g 6 s1h b1 + · · · + smh bm . then f 6 ∑n i=1 ri (∑m j=1 sjh bj )ai = ∑n i=1 ∑m j=1 risjh bj +ai , and thus f : h 6 (∑n i=1 ai )(∑m j=1 bj ) . to determine i(f), let f0 ∈ c + p (k) be fixed. then f : f0 can be regarded as a first approximation of i(f) : i(f0). an improvement is given by the function ig : c + p (k) → (0,∞) with ig(f) := f : g f0 : g (39) for small g ∈ c+p (k). by (38), we have (f0 : f) −1 6 ig(f) 6 f : f0. (40) for b ∈ k×, let vb be the set of all ig with g(a) = 0 for a /∈ bp . by proposition 4.2, vb 6= ∅ for all b ∈ k×. by (40), we can regard ig as a point of the compact space x := ∏ f∈c + p (k) [(f0 : f) −1,f : f0]. (41) as the closed sets vb ⊂ x form a chain, their intersection is non-empty. hence there exists some i ∈ ⋂ b∈k× vb. consequently, for every b ∈ k × and arbitrary f1, . . . ,fn ∈ c + p (k) and ε > 0 in ir, there exists some g ∈ c+p (k) with g(a) = 0 for a /∈ bp , so that for i ∈ {1, . . . ,n}, |i(fi) − ig (fi)| < ε. (42) we will prove that i is a haar integral. firstly, (f1 + f2) : g 6 (f1 : g) + (f2 : g) implies that ig(f1 + f2) 6 ig(f1) + ig (f2) holds for f1,f2,g ∈ c + p (k). hence (42) gives i(f1 + f2) 6 i(f1) + i(f2) (43) for all f1,f2 ∈ c + p (k). for 0 < r < ∞ and f,g ∈ c + p (k), we have (rf : g) = r(f : g), i. e. ig(rf) = rig (f), and thus i(rf) = ri(f), again by (42). similarly, ig(f a) = ig(f) for all a ∈ k, hence i(fa) = i(f). from (41) and (42), it follows that i is monotonous. thus it remains to show that the inequality (43) is an equation. let f1,f2,g ∈ c + p (k) be given. then there exists some b ∈ k × such that f1 +f2 vanishes outside bp . by proposition 4.2, we find f3 ∈ c + p (k) with f3(a) = 1 for a ∈ b(p + p). for an arbitrary ε > 0, define f := f1 + f2 + εf3 and hj := fj f for j ∈ {1, 2}. by lemma 4.1, this gives hj ∈ c + p (k). hence there exists some c ∈ k× such that |hj (a) − hj (a ′)| < ε holds for a − a′ ∈ cp and j ∈ {1, 2}. choose g ∈ c+p (k) with g(a) = 0 for a /∈ cp , and f 6 r1g a1 + · · · + rng an. then fj(a) = f(a)hj(a) 6 ∑n i=1 rig ai (a)(hj (ai) + ε), hence fj : g 6 n∑ i=1 ri(hj (ai) + ε). this implies (f1 : g) + (f2 : g) 6 ∑n i=1 ri(1 + 2ε) since h1 + h2 6 1. so we get (f1 : g) + (f2 : g) 6 (f : g)(1 + 2ε), and thus ig(f1) + ig(f2) 6 ig (f) · (1 + 2ε). by (42), this gives i(f1) + i(f2) 6 i(f)(1 + 2ε). using (43), we get i(f1)+i(f2) 6 (i(f1 +f2)+εi(f3))(1+2ε) for all ε, hence i(f1)+i(f2) 6 i(f1 +f2). � cubo 12, 2 (2010) the tree of primes in a field 113 proposition 5.2. let k be a field with a compact prime p. for every f0 ∈ c + p (k), there exists a unique haar integral i : cp (k) → ir with i(f0) = 1. proof. existence follows by proposition 5.1. therefore, let i be a haar integral, and f0 ∈ c + p (k). then ri is a haar integral for any r > 0 in ir. assume first that f,g ∈ c+p (k) and f 6 ∑n i=1 rig ai with ri > 0 and ai ∈ k. then i(f) 6 ∑n i=1 rii(g), hence i(f) i(g) 6 f : g. (44) for f ∈ c+p (k) and ε > 0, there exists some b ∈ k × which satisfies |f(a) − f(a′)| < ε for a − a′ ∈ bp . choose g ∈ c+p (k) with g(a) = 0 for a /∈ bp . passing from g(a) to g(a) + g(−a), we can assume that g(a) = g(−a) for all a ∈ k. therefore, if a,a′ ∈ k, we have f(a′)ga(a′) > (f(a) − ε)ga(a′), hence i(fga) > (f(a) − ε)i(g), and thus f(a) 6 ε + i(fga) i(g) . (45) for every δ > 0, there is some c ∈ k× with |g(a) − g(a′)| < δ for a − a′ ∈ cp . by proposition 4.4, there are elements f1, . . . ,fn ∈ c + p (k) and a1, . . . ,an ∈ k with f = f1 + · · · + fn and fi(a ′) = 0 for a′ − ai /∈ cp . this gives fi(a ′)g(a′ − a) 6 fi(a ′)(g(ai − a) + δ) for all a,a ′ ∈ k, hence i(fga) = n∑ i=1 i(fig a) 6 n∑ i=1 i(fi)(g a(ai) + δ) = n∑ i=1 i(fi)(g ai (a) + δ). (46) by proposition 4.2, there exists some h ∈ c+p (k) such that h(a ′) = 1 holds for all a′ with f(a′) + f0(a ′) > 0. then (45) and (46) yield f 6 εh + n∑ i=1 i(fi) i(g) (gai + δh) = (ε + i(f) i(g) δ)h + n∑ i=1 i(fi) i(g) gai. so we get f : g 6 ε(h : g) + i(f ) i(g) , and by virtue of (38) and (44), f : g f0 : g 6 ε · h : g f0 : g + i(f) i(g)(f0 : g) 6 ε(h : f0) + i(f) i(f0) . similarly, f0 : g f : g 6 ε(h : f) + i(f0) i(f) = i(f0) i(f) ( 1 + ε(h : f) i(f ) i(f0) ) , and thus i(f) i(f0) ( 1 + ε(h : f) i(f ) i(f0) )−1 6 f : g f0 : g 6 ε(h : f0) + i(f) i(f0) . this implies that the quotient f :g f0:g converges to i(f ) i(f0) for b → 0, which shows that i(f) is unique for i(f0) = 1. � 6 the normalized haar integral let k be a field with a compact prime p . by proposition 5.2, there exists a haar integral i : cp (k) → ir, unique up to a real constant. furthermore, the proof of proposition 5.2 yields an explicit determination of the haar integrals. in fact, for any b ∈ k×, let l(b) be the set of g ∈ c+p (k) which vanish 114 wolfgang rump cubo 12, 2 (2010) outside bp . moreover, we set ls(b) := {g ∈ l(b) | ∀a ∈ k : g(−a) = g(a)}. (47) for f,f0 ∈ c + p (k) and every ε > 0 in ir, there exists some b ∈ k × such that ∣∣∣∣ i(f) i(f0) − f : g f0 : g ∣∣∣∣ < ε (48) for all g ∈ ls(b). in this way, the quotient i(f ) i(f0) can be approximated by the expressions (39) with g ∈ ls(b). for any c ∈ k×, consider the ir-linear map ρc : cp (k) → cp (k) (49) with ρc(f)(a) := f(c −1a) for all f ∈ cp (k) and a ∈ k. since f(k r bp) = 0 ⇔ ρc(f)(k r cbp) = 0, we have ρc(l(b)) = l(cb). (50) similarly, the uniform continuity is preserved under the map ρc. furthermore, ρc is monotonous, i. e. for f,g ∈ cp (k), f 6 g ⇐⇒ ρc(f) 6 ρc(g). (51) since ρc is invertible, it induces a bijection ρc : c + p (k) −→ ∼ c+p (k). (52) for a,b ∈ k and f ∈ cp (k), we finally have ρc(f a)(b) = fa(c−1b) = f(c−1b − a) = f(c−1(b − ca)) = ρc(f)(b − ca) = ρc(f) ca(b), hence ρc(f a) = ρc(f) ca. (53) consequently, for any haar integral i : cp (k) → cp (k) and c ∈ k ×, it follows that iρc is again a haar integral. so there is a real function vp : k → [0,∞] with vp (0) := 0 and iρc = vp (c)i (54) for c 6= 0,which does not depend on the choice of the haar integrals i. now we prove that vp is a pseudo-valuation. using vp , the haar integral can be normalized in a natural way. proposition 6.1. let k be a field with a compact prime p. then the uniquely defined function vp : k → [0,∞] in (54) is a pseudo-valuation with respect to p. proof. from (54) we infer that vp is multiplicative: vp (bc) = vp (b)vp (c) for all b,c ∈ k. in fact, for b,c ∈ k× and a haar integral i, vp (bc)i = iρbc = iρbρc = vp (b)iρc = vp (b)vp (c)i. similarly, vp (1) = 1. to prove that vp is monotonous, we use i to define a function vi : k → [0,∞) which is related to vp . (in contrast to vp , however, vi depends on i.) for b ∈ k, we set u(b) := {f ∈ c + p (k) | ∀a ∈ bp̃ : f(a) = 1}, and vi (b) := inf f∈u(b) i(f). (55) cubo 12, 2 (2010) the tree of primes in a field 115 although the function (55) is not multiplicative, in general, it is always monotonous. in fact, if b,c ∈ k with b 6 c, then u(b) ⊃ u(c), and therefore, vi (b) 6 vi (c). before we compare vp with vi , we prove, in analogy to (50), that ρc(u(b)) = u(cb) (56) holds for b,c ∈ k×. this follows by the equivalence ρc(f) ∈ u(cb) ⇔ f ∈ u(b) for all f ∈ c + p (k). for b,c ∈ k×, we have the equation vp (c) · inff∈u(b) i(f) = inff∈u(b) i(ρc(f)) = inff∈u(cb) i(f), which gives vi (cb) = vp (c) · vi (b). (57) in particular, this gives vi (c) = vp (c) · vi (1), i. e. vi = vi (1) · vp . (58) by proposition 4.2, we find a function g ∈ l(1) with g 6 1. for every f ∈ u(1), we get f > g, hence i(f) > i(g), and thus vi (1) > i(g) > 0. by (58), the monotonous property of vi carries over to vp , i. e. vp is a pseudo-valuation. � the fact that vi depends on i enables us to normalize the haar integral: theorem 6.1. let k be a field with a compact prime p. there exists a unique haar integral j with vj = vp . proof. let i be a haar integral, and j = ri with r > 0 in ir. then vri (1) = rvi (1). by eq. (58), the condition vj = vp is equivalent to vj (1) = 1, that is, r = vi (1) −1. � in the sequel, we simply call j the integral and vp = vj the normed pseudo-valuation with respect to p . then (55) shows that vp (b) measures the content of bp̃ . our normalization thus fixes the content of p̃ to 1. next we determine the pseudo-valuations of a compact prime. first, let p be an arbitrary prime. we call a pseudo-valuation v : k → [0,∞] with respect to p trivial if the values of v belong to {0, 1,∞}. proposition 6.2. let k be a field with a prime p. there is a natural bijection between the primes q ⊂ p and the trivial pseudo-valuations v : k → [0,∞] with respect to p. proof. first, let v : k → [0,∞] be a trivial pseudo-valuation with respect to p . by proposition 2.1, the primes q ⊂ p correspond to the multiplicatively closed subsets u ⊂ k with p̃ ⊂ u, where u = q̃. for such a q, we define v by v(a) :=    0 for a ∈ q 1 for a ∈ ∂q ∞ for a ∈ k r q̃. (59) then v is monotonous with v(0) = 0 and v(1) = 1. for a,b ∈ k with v(a) = 1, we have b ∈ q ⇔ ab ∈ q and b ∈ q̃ ⇔ ab ∈ q̃. hence v is a pseudo-valuation by definition 3.1. conversely, let v : k → [0,∞] be a trivial pseudo-valuation with respect to p . we set u := {a ∈ k |v(a) < ∞}. then u is multiplicatively closed with p̃ ⊂ u. the corresponding prime q ⊂ p satisfies a ∈ q̃ ⇔ v(a) 6 1, and by (31), a ∈ q ⇔ a−1 /∈ q̃ ⇔ v(a−1) = ∞ ⇔ v(a) = 0 for a ∈ k×, in accordance with (59). � 116 wolfgang rump cubo 12, 2 (2010) proposition 6.3. let k be a field with a finite, non-trivial pseudo-valuation v : k → [0,∞] with respect to a prime p. then q := {a ∈ k | v(a) < 1} ⊂ p is a minimal prime. proof. for a,b ∈ q ∩ k×, we have v(ab) = v(a)v(b) < 1, hence ab ∈ q. if a,b /∈ q, then v(a),v(b) > 1, which gives v(ab) = v(a)v(b) > 1, i. e. ab /∈ q. this proves (p1), (p2), and (p3). thus q ⊂ p implies that q is a prime. now assume that a,b ∈ q ∩ k×. then there exists some n ∈ in with v(an) = v(a)n < v(b). hence an < b, and thus an ∈ bp . so q is minimal by proposition 2.2. � corollary. every compact prime p 6= 0 of a field k contains a minimal prime. proof. proposition 6.1 implies that vp is a finite pseudo-valuation, and vp is non-trivial since p 6= 0. now proposition 6.3 completes the proof. � 7 krull primes by proposition 3.4, every minimal prime p of a field k gives rise to a family of p-valuations. conversely, proposition 7.1. let k be a field with a prime p 6= 0 and a p-valuation v : k → [0,∞]. then p is minimal. proof. let a,b ∈ p ∩ k× be given. then v(a−1) < ∞ implies that v(a) 6= 0, and a < 1 yields 0 < v(a) < 1. so there is an n ∈ in with v(an) = v(a)n 6 v(b). hence an 6 b. by proposition 2.2, we infer that p is minimal. � in algebraic number theory, the distinction between finite and infinite primes is crucial. for an arbitrary field, we have to distinguish between algebraic and transcendental primes. first, we prove lemma 7.1. a prime p of a field k is algebraic if and only if p̃ is a subring of k. proof. if p is algebraic and a,b ∈ p̃ , then (a + b)p ⊂ p + p ⊂ p , hence a + b ∈ p̃ . therefore, p̃ is a subring. conversely, assume that p̃ + p̃ ⊂ p̃ and a,b ∈ p , say, a 6 b. then ab−1 ∈ p̃ , which implies that ab−1 + 1 ∈ p̃ . hence a + b = (ab−1 + 1)b ∈ p . so we get p + p ⊂ p , and thus p is algebraic. � proposition 7.2. for every prime p of a field k, there is a largest algebraic prime p◦ ⊂ p. proof. the subring of k generated by p̃ is u := {a1 + · · · + an | ai ∈ p̃}. (60) by proposition 2.1, u corresponds to a prime p◦ ⊂ p with p̃◦ = u. by lemma 7.1, this prime meets the requirement. � there is a close relationship between p◦ and the open subset p ◦ of p given by proposition 3.1. first, we prove proposition 7.3. if p is a prime of a field k, then p◦ is a prime, and p◦◦ = p◦. cubo 12, 2 (2010) the tree of primes in a field 117 proof. clearly, p◦ satisfies (p1) and (p2). let a,b /∈ p◦ be given. for any n ∈ in, there are elements pn,qn,rn,sn ∈ p with a n(pn+qn) /∈ p and b n(rn+sn) /∈ p . hence a nbn(pn+qn)(rn+sn) /∈ p , and thus anbn(p + p + p + p) 6⊂ p for all n ∈ in. if ab ∈ p◦, there exists some n ∈ in with (ab)n(p + p) ⊂ p , whence (ab)2n(p + p + p + p) ⊂ (ab)n(p + p) ⊂ p , a contradiction. this proves (p3) for p◦. to verify (p4), choose d ∈ k× with p + p ⊂ dp . we show that p◦ + p◦ ⊂ dp◦. to this end, assume that a,b ∈ p◦, say, an(p + p) ⊂ p and bn(p + p) ⊂ p . then (a + b)2n(p + p) =∑2n i=0 ( 2n i ) aib2n−i(p + p) ⊂ ∑2n i=0 ( 2n i ) p ⊂ ∑22n i=1 p ⊂ d 2np , which gives (a + b)d−1 ∈ p◦. whence p◦ + p◦ ⊂ dp◦. finally, assume that a ∈ p◦, say, an(p + p) ⊂ p . then p + p ⊂ a−np , and by the above, this implies that p◦ + p◦ ⊂ a−np◦. hence an(p◦ + p◦) ⊂ p◦, and thus a ∈ p◦◦. � primes p with p = p◦ were first considered by krull [7]. we call them krull primes. by proposition 3.1, every minimal prime is a krull prime. the connection to algebraic primes is given by proposition 7.4. let k be a field with a prime p. then every prime q ( p◦ of k is algebraic. proof. choose a ∈ p◦ r q. so there is an n ∈ in with an(p + p) ⊂ p . let b ∈ q be arbitrary. then a /∈ q implies that a−1 ∈ q̃, hence a−n−1b ∈ q. this gives b(p̃ + p̃) ⊂ an+1q(p̃ + p̃) ⊂ an(ap̃ + ap̃)q ⊂ an(p + p)q ⊂ pq ⊂ q. assume that a,b ∈ q, say ap̃ ⊂ bp̃ , and b 6= 0. then a + b = (ab−1 + 1)b ∈ (p̃ + p̃)b ⊂ q. so we get q + q ⊂ q. � corollary 1. let k be a field with a prime p. then p◦ ⊂ p ◦, and there is no other prime between p◦ and p ◦. proof. since p◦(p + p) ⊂ p◦ + p◦ = p◦ ⊂ p , we have p◦ ⊂ p ◦. for every prime q ( p◦, proposition 7.4 gives q ⊂ p◦. � corollary 2. let k be a field with a prime p. a prime q ⊂ p is algebraic if and only if q ⊂ p◦. proof. assume that q ⊂ p◦. if q 6= p◦, then q ( p◦ ⊂ p ◦, hence q is algebraic by proposition 7.4. the converse is trivial. � we call a prime p purely transcendental if p◦ = 0. if p 6= 0, then (p4) implies that p ◦ 6= 0. by corollary 1, this shows that p◦ is minimal if p 6= 0 is purely transcendental. thus proposition 3.1 gives corollary 3. a purely transcendental prime p 6= 0 is minimal if and only if p is a krull prime. in the paragraph that follows definition 1.2, we have shown that algebraic primes p of a field k correspond to the valuation rings p̃ with quotient field k. if p is principal, say, p = pp̃ 6= 0, and⋂∞ n=0 p np̃ = 0, then p̃ is a discrete valuation domain. now we turn our attention to compact primes. proposition 7.5. a compact krull prime p of a field k is either algebraic or purely transcendental. proof. assume that p is not purely transcendental. then p◦ 6= 0. choose a ∈ p◦ r 0. then ap̃◦ ⊂ p◦. by proposition 4.1, p◦ is compact. so there are a1, . . . ,an ∈ k with p◦ ⊂ ⋃n i=1(ai + ap◦), hence p̃◦ ⊂ a −1p◦ ⊂ ⋃n i=1(a −1ai + p◦). thus |p̃◦/p◦| < ∞. by (60), this gives p̃◦ ⊂ p̃ + · · · + p̃ 118 wolfgang rump cubo 12, 2 (2010) with finitely many summands. suppose that p 6= p◦. then there exist b ∈ p r p◦ and n ∈ in with bn(p + p) ⊂ p . this gives bn+1(p̃ + p̃) ⊂ bn(bp̃ + bp̃) ⊂ p . hence, for a suitably large m ∈ in, we get bmp̃◦ ⊂ b m(p̃ + · · · + p̃) ⊂ p . if b−m ∈ p̃◦, then 1 = b m · b−m ∈ bmp̃◦ ⊂ p , a contradiction. hence b−m /∈ p̃◦, and thus b m ∈ p◦. so we obtain b ∈ p◦, contrary to our assumption. thus p = p◦. � proposition 7.6. for an algebraic prime p 6= 0 of a field k, the following are equivalent. (a) p is compact and minimal. (b) p is a compact krull prime. (c) p̃ is a discrete valuation domain with finite resudue field. proof. the implication (a) ⇒ (b) follows by proposition 3.1. (b) ⇒ (c): assume that a ∈ p r 0. then there exist a1, . . . ,an ∈ k with p ⊂ ⋃n i=1(ai + a2p), hence p̃ ⊂ a−1p ⊂ ⋃n i=1(a −1ai + ap). thus |p̃/ap | < ∞. so we get |p̃/p | < ∞, and by proposition 1.1, p is a principal ideal, say p = pp̃ . since |p̃/ap | < ∞ holds for all a ∈ p r 0, it follows that ⋂ ∞ n=0 p n = 0. (c) ⇒ (a): assume that a,b ∈ p ∩k×. then there is some n ∈ in with b /∈ p n. since an ∈ p n, we have bp̃ 6⊂ anp̃ , and thus anp̃ ⊂ bp̃ by proposition 1.1. hence an+1 ∈ bp . thus, by proposition 2.2, p is minimal. assume that p = pp̃ . then |p̃/p n| = |p̃/p |n, which implies that p is compact. � corollary. a compact prime p 6= 0 of a field k is a krull prime if and only if p is minimal. proof. assume that p is a compact krull prime. if p is purely transcendental, corollary 1 of proposition 7.4 implies that p is minimal. otherwise, p is algebraic by proposition 7.5, hence minimal by proposition 7.6. the converse follows by proposition 3.1. � as already shown in section 6, proposition 4.1 implies that every non-trivial compact prime p contains a minimal prime. 8 the product formula in this section, we focus our attention upon the minimal compact primes p of a field k. if p is algebraic, then proposition 7.6 implies that p̃ is a discrete valuation domain with finite residue field. so we have a chain · · · ( p 2 ( p ( p 0 ( p−1 ( p−2 · · · , (61) with p 0 = p̃ and p n := pnp̃. for a ∈ k×, let e(a) denote the greatest integer n with a ∈ p n. thus ap̃ = p e(a) (62) holds for any a ∈ k×. to get the normalized p-valuation, we define a function χn : k → [0,∞) for any n ∈ z by χn(a) := { 1 for a ∈ p n 0 for a /∈ p n. (63) cubo 12, 2 (2010) the tree of primes in a field 119 for a,b ∈ k with a − b ∈ p n, we have |χn(a) − χn(b)| = 0, which yields χn ∈ c + p (k). we set rp := |p̃/p |. then p n/p n+1 = pnp̃/pn+1p̃ ∼= p̃/pp̃ , hence |p n/p n+1| = rp , and thus |p n/p n+m| = rmp for m ∈ in. this gives χn : χn+m = r m p , hence χn : χn+m χ0 : χn+m = rmp rn+mp = r−np . therefore, by (48), there is a haar integral i with i(χ0) = 1 and i(χn) = r −n p . for b ∈ k ×, eq. (62) shows that χe(b) is the smallest function in c + p (k), which takes the value 1 on bp̃ . now (55) gives vi (b) = r −e(b) p , hence vi (1) = 1. by (58), we get a normalized p-valuation vp (a) = r −e(a) p . (64) let us return to the product formula (6) of the introduction. the normalized pp-valuations vp := vpp for the primes pp of q are obtained by (64): vp(a) = p −e(a), (65) where e(a) is the exponent of p in a prime factorization of a. thus vp coincides with the p-adic absolute value (cf. section 3, example 1). the topology of the prime p∞ = (−1, 1) ∩ q of q is the usual one. therefore, a function f : q → ir belongs to c∞(q) := cp∞ (q) if and only if it is uniformly continuous (in the usual sense) and vanishes outside a bounded interval. the haar integral with respect to p∞ is just the riemann-integral. therefore, the normalized p∞-valuation v∞ can be obtained from (54). in fact, for f ∈ c∞(q) and c ∈ q ×, we get by the substitution b = c−1a: ∫ q fρc(a)da = ∫ q f(c−1a)da = |c| ∫ q f(b)db. so the normalized p∞-valuation is the absolute value: v∞(a) = |a|. (66) in the sequel, we denote the set of minimal primes of a field k by p(k). from (65) and (66) we thus get for all a ∈ q× the product formula: ∏ p ∈p(q) vp (a) = 1. (67) lemma 8.1. let k be a field with a minimal prime p, and let a,b ∈ k× be such that an ⊂ bp̃ for all n ∈ in. then a ∈ p̃. proof. suppose that a /∈ p̃ . then a−1 ∈ p . by proposition 2.2, there is an n ∈ in with a−n ∈ b−1p . hence 1 = ana−n ∈ bp̃b−1p ⊂ p , a contradiction! � proposition 8.1. every minimal prime of a field k of characteristic p > 0 is algebraic. proof. let f be the prime field of k. suppose that a ∈ f rp̃ . then a−1 ∈ p . thus |f×| = p−1 implies that 1 = (a−1)p−1 ∈ p , a contradiction. therefore, f ⊂ p̃ . choose d ∈ k× with p̃ +p̃ ⊂ dp̃ . 120 wolfgang rump cubo 12, 2 (2010) for a,b ∈ p̃ and an arbitrary n ∈ in we find some m ∈ in with mn < 2m. then (a + b)mn =∑mn i=0 ( mn i ) aibmn−i ∈ ∑2m i=1 p̃ ⊂ d mp̃ . hence ((a + b)nd−1)m ∈ p̃ , and thus (a + b)nd−1 ∈ p̃ . by lemma 3.1, this gives a + b ∈ p̃ . so p̃ + p̃ ⊂ p̃ . now lemma 2.4 implies that p is algebraic. � for a rational prime p, let fp ∼= z/pz denote the field with p elements. consider the quotient field fp(x) of fp[x]. by proposition 8.1, every minimal prime of fp(x) is algebraic. for any normed irreducible polynomial q ∈ fp[x], consider the subring fp[x]q := { f g ∈ fp(x) | f,g ∈ fp[x],q ∤ g} (68) of fp(x). then fp[x]q is a discrete valuation ring with residue class field fp. hence pq := qfp(x) is a prime of fp(x) with p̃q = fp[x]q. for all these primes, p̃q ⊃ fp[x]. furthermore, p∞ := x −1fp[x −1]x−1 (69) is a prime with x /∈ p̃∞, hence p̃∞ 6⊃ fp[x]. in analogy with (15), we write p∞, which does not mean, however, that p∞ is transcendental (which is impossible by proposition 8.1. the primes of the form pq with q irreducible, or q = ∞, are compact krull primes by proposition 7.6. in fact, if m = deg q is the degree of an irreducible polynomial q ∈ fp[x], then dimfp (p̃q/pq) = dimfp (fp[x]/qfp[x]) = m, hence |p̃q/pq| = p m. for p∞, we have |p̃∞/p∞| = p. the normalized pqvaluation vq := vpq results from (64): vq(f) = p −eq (f )deg q, (70) where eq(f) denotes the multiplicity of q in f. for example, ex−1(x 3 − 2x2 + x) = 2. moreover, we get v∞(f) = p deg f. (71) proposition 8.2. let p be a rational prime. every minimal compact prime of fp(x) is of the form pq with q irreducible, or q = ∞. proof. let p be a minimal compact prime of fp(x). as in the proof of proposition 8.1, we have fp ⊂ p̃ . assume first that x ∈ p̃ . then fp[x] ⊂ p̃ . since p is a prime ideal of p̃, it follows that p ∩ fp[x] is a prime ideal of fp[x]. if p ∩ fp[x] = 0, then a /∈ p for all a ∈ fp[x] r 0. hence a −1 ∈ p̃ , and thus fp(x) = p̃ , i. e. p = 0. consequently, p ∩ fp[x] is a maximal ideal of fp[x]. as fp[x] is a principal ideal domain, there is an irreducible polynomial q ∈ fp[x] with p ∩ fp[x] = qfp[x]. for every f ∈ fp[x] with q ∤ f, we have f /∈ p , hence f −1 ∈ p̃ . therefore, pq = qfp[x]q ⊂ p . as p is minimal, we obtain p = pq. finally, assume that x /∈ p̃ . then x −1 ∈ p . applying the preceding argument to fp[x −1] instead of fp[x], we get p∞ = x −1fp[x −1]x−1 ⊂ p . thus p∞ = p . � from (70) and (71) we again obtain the product formula by proposition 8.2. note that in both cases, the product formla arises in a natural way: the minimal primes p are all compact, and their normalized p-valuations are unique. so the formula describes an intrinsic relation among the minimal primes. received: november 2008. revised: march 2009. cubo 12, 2 (2010) the tree of primes in a field 121 references [1] e. artin and g. whaples, axiomatic characterization of fields by the product formula for valuations, bull. amer. math. soc. 51 (1945), 469-492 [2] e. artin and g. whaples, a note on axiomatic characterization of fields, bull. amer. math. soc. 52 (1946), 245-247 [3] n. bourbaki, general topology, chapters 1-4; elements of mathematics, springer-verlag, berlin, 1998 [4] n. bourbaki, commutative algebra, hermann, paris 1972 [5] d. k. harrison, finite and infinite primes for rings and fields, mem. amer. math. soc. 68 (1966) [6] w. krull, über geschlossene bewertungssysteme, j. reine angew. math. 190 (1952), 75-92 [7] w. krull, ordnungsfunktionen und 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december 2004. turbulent mixing of stratified flows esteban g. tabak courant institute of mathematical sciences new york university falta e-mail fabio a. tal 1 courant institute of mathematical sciences new york university falta e-mail abstract a mathematical model for turbulent mixing of stratified, sheared flows is developed and explored. two applications are emphasized, one to the dynamics of the ocean well–mixed layer, and another to the analysis of the stability of equilibrium profiles away from boundaries. 1 introduction the ocean is a highly inhomogeneous medium, characterized by spatial and temporal contrasts in temperature and salinity, as well as in chemical composition and in the distribution of biological agents. the inhomogeneity of the ocean, however, is not static, but follows from a dynamical equilibrium, in which contrasts are permanently being created and attenuated. local processes that generate contrasts include evaporation and rain, freezing and melting of sea–ice, river inflows, and volcanic activity. attenuation is mainly due to mixing processes, such as turbulent diffusion, breaking waves, and stirring by surface winds, ocean currents, and planetary tides interacting 1the work of fabio a. tal was supported by grant # 200800/09-1 from cnpq–brasil. 98 esteban g. tabak and fabio a. tal 6, 4(2004) with the ocean’s bottom and lateral morphology. the dynamical equilibrium emerging from the balance of these processes is a determining factor to the earth’s climate: slight changes in the properties of the upper layers of the ocean, in particular, can lead to significant variations of its ice–coverage, of local patterns of convection and rain and, ultimately, to dramatic changes in the global patterns of surface temperature, humidity and prevailing winds. yet the quantification, and even the identification of some of the critical processes involved in this dynamical balance, remain to a large degree incomplete. the mixing side of the balance is particularly elusive, due to its vast distribution over whole basins, to the difficulties inherent to its observation and measurement, to its highly anisotropic nature, and to the incompletely understood physics of its underlying processes, such as turbulent diffusion, shear and convective instabilities and wave overturning. state-of-the-art computational ocean circulation models typically parameterize these processes, introducing empirical closures designed to fit as well as possible the [sparse] available experimental and observational data. such an approach, driven by necessity, may yield large errors in the prediction of climatic changes, since the adjustment of parameters to match features of today’s climate may fail to capture those of tomorrow’s. a mathematical criterion for the stability of a sheared, stratified flow, based on the richardson number, was developed in [10, 5]. physical experiments and dimensional analysis were used in [2, 12] to develop a closure for entrainment and mixing of ambient fluid into a plume of buoyant fluid. this closure was improved in [3] to better represent entrainment into oceanic dense overflows. in [9] and more recently [1], a closure for mixing was proposed based on turbulent diffusion, the approach that we explore in this article. examples of observations, physical experiments, theory and parameterizations in general circulation models, of mixing of stratified flows, can be found in [7, 11, 4, 6, 8], and references therein. a first, striking feature of oceanic mixing, is its highly anisotropic character. a tracer deposited in the water will diffuse much more rapidly along isopycnals –surfaces of constant density, typically close to horizontal– than across them. this is due, to a large extent, to the relative freedom that fluid parcels experiment to move along isopycnals, as contrasted to the relatively high rigidity that a stratified flow in a rotating environment opposes to vertical, diapycnal motion. this leads to the formation of horizontal eddies covering a wide range of scales, which work as very effective enhancers of mixing. another way of understanding this disparity of mixing rates is through energetics: isopycnal mixing is energetically free, while diapycnal mixing is costly, since vertical mixing involves raising and sinking respectively heavy and light parcels of fluid, thus increasing the total amount of potential energy in the system. our focus in this work is on vertical, diapycnal mixing, which in the long term determines the properties of the water at the sea–surface, by bringing up denser waters from beneath. mixing of waters of different properties occurs throughout the ocean and at all scales. however, there are highly localized areas of intense mixing, which play a fundamental role in establishing the main properties of large water masses. a locus 6, 4(2004) turbulent mixing of stratified flows 99 of intense stirring and mixing, with a direct influence on the surface waters, is the upper mixed layer, occupying typically the first fifty to one hundred meters below the ocean’s surface. this layer is characterized by its vertically nearly uniform properties, in high contrast with the ocean’s interior relatively strong stratification. the main stirring agent for this layer is the wind which, through the generation of surface and internal waves and turbulence, leads to entrainment of water from the ocean’s interior into the well–mixed layer, and subsequent mixing throughout it. despite its obvious significance for the weather and climate –this is the only part of the ocean that communicates directly with the atmosphere–, the upper well–mixed layer is not properly resolved in current general circulation models. this is due not only to its small vertical scale, which makes it difficult to resolve in the relatively coarse grids of general circulations models, but also to many lacoons in our understanding of its basic dynamics and driving physical processes. in this paper, we describe some mathematical and physical tools useful for the description of mixing in the ocean. we focus on conceptual, idealized models that, though lacking the richness, complexity and diversity of the real ocean, may help shed light on some of its underlying processes. in particular, we explore some of the properties of a model of mixing based on turbulent diffusion, i.e. a diffusive process supported not in the microscopic scale of brownian motion, but in the intermediate, imprecise scales of turbulence. the plan of the paper is the following: after this introduction, in section 2, we describe a mathematical model of mixing basin on nonlinear, turbulent diffusion. in section 3, we apply this model to the dynamics of the well–mixed layer. in section 4, we show how the model sheds light on subtle issues arising in the shear instability of stratified flows. finally, in 5, we make some closing remarks. 2 a mathematical model for turbulent diffusion how do ocean waters with different properties mix? there is not a simple answer to this question: a plurality of mixing scenarios exist, each shedding a distinct light on the mixing process. when a dense mass of water is placed above a ligther mass – as when the ocean surface layers are cooled by very cold winds –, a convective instability occurs. when localized currents give rise to a marked shear, either horizontal or vertical, this shear may go unstable and shed mixing eddies. pronounced internal waves may nonlinearly deform and break, leading to intense, localized mixing bursts. dense overflows descending into the ocean may generate internal hydraulic jumps, with high rates of localized entrainment of lighter ambient waters. diverse as these scenarios are, they all share a common feature: flow instabilities or hydraulic constraints give rise to highly turbulent bursts, which rapidly homogenize the fluid properties. it is therefore attractive to treat them all under the common umbrella of turbulence driven diffusion. models of this kind are currently used in general circulation models [9]. in this section, we describe in some detail a candidate model of mixing based on turbulent diffusion. for simplicity, we shall restrict our attention to flows which are 100 esteban g. tabak and fabio a. tal 6, 4(2004) horizontally uniform, so that the problem becomes one–dimensional. this is not to say, however, that we are modeling a vertical slab of fluid, since horizontal velocities will be allowed, though depending only on the vertical coordinate z. since density variations in the ocean are very small, typically ranging bellow 3%, we will adopt in our model the boussinnesq approximation, whereby only the buoyancy effect of density variations is retained, while their effect on the fluid’s inertia is neglected. our variables are the buoyancy b = g ρ−ρ0 ρ0 , where ρ is the fluid’s density and ρ0 some reference value, the horizontal velocity u, and the turbulent kinetic energy per unit of mass e. we assume that both the buoyancy and the momentum are turbulently diffused, so that the equations of mass and momentum conservation are bt = (kbbz)z , (1) ut = (kuuz)z , (2) where kb and ku are the turbulent buoyancy and shear diffusivities, which we model bellow. since we anticipate that the diffusivities kb and ku will depend on the local amount of turbulence in the system, characterized for instance by a typical value of the turbulent velocity field w = √ 2e, still another equation is needed in order to close the system. our choice is an equation for the diffusion of the turbulent energy itself, which reads et = (keez)z + kbbz + ku(uz) 2. (3) the first term on the right-hand side represents diffusion; the other two are required so that the total energy ∫ ( b z + u2 2 + e ) dz is preserved by the flow. the physical interpretation of these two terms is straightforward. the first, a sink, represents the energetic cost of mixing a stratified fluid, raising heavy and bringing down light parcels of fluid. the second, a source, accounts for the energy surplus provided by mixing a shear flow; it follows from the mathematical fact that the mean of a square is always bigger than the square of the mean. finally, we must determine the turbulent diffusivities. one reasonable assumption is that each diffusivity must be proportional to the mean turbulent velocity. in order to simplify our approach, we will assume this simplest scenario, and set kj = le 1 2 sj (4) for j ∈ {b, u, e}, where l is some fixed lenghtscale –the typical size of a mixing eddy– and the sj ’s are dimensionless parameters accounting for the possibly different rates of mixing for the various physical quantities of the system. other models may be produced by treating l not as a constant, but as a function of the dependent variables. dimensional analysis suggests the possibilities l = √ e bz , 6, 4(2004) turbulent mixing of stratified flows 101 l = √ u2 bz , and l = √ e (uz)2 . still another possibility is to let l evolve dynamically, following a dynamic equation modeling the cascade of turbulent energy across scales. in this note, however, we concentrate on the simple choice that has l fixed at some externally provided value. the arbitrariness of this value should work as a reminder that treating turbulent mixing as a diffusive process is not a first–principled approach, but a convenient, often deceptively convincing closure. 3 an application to the dynamics of the well–mixed layer one of the most relevant feature of the oceans is the existence of a turbulent and well–mixed top layer, typically occupying the top 50–100 meters. in this layer the temperature and salt content of the water are almost independent of depth. at the bottom of the layer there is a shallow region where both salinity and temperature change very rapidly, appearing at times to be nearly discontinuous. the mixed layer plays a fundamental role in regulating our climate, since it is through it that all heat and momentum exchanges between the ocean and the atmosphere take place. its depth is largely regulated by the atmosphere, thus generating a nontrivial feedback mechanism. we describe three ways by which this regulation may happen: 1. storms over the ocean may stir the waters directly bellow it, adding to the total turbulent energy at the top of the mixed layer. this extra turbulence is nonlinearly diffused through the layer and enhances mixing at its bottom, thus increasing its depth. 2. the daily variations of atmospheric temperature causes a periodic warming and cooling of the ocean’s surface, reflected in a periodic change of buoyancy. when the top water becomes colder than its surroundings, it also becomes heavier, an unstable situation that causes convection. this, in turn, releases potential energy that is transformed into turbulence, inducing further mixing. 3. winds in the atmosphere transmit their momentum to the ocean. the surface then acquires a velocity distinct from its surroundings, developing a marked shear. this velocity shear may develop instabilities and mix, thus releasing kinetic energy that also becomes turbulent. as an application of our turbulent mixing model, we show numerically how these three scenarios may generate a mixed layer. in our numerical runs, the atmospheric influence will be represented by boundary fluxes. our numerical scheme uses finite differencing in conservation form. the conserved quantities b, u and e are represented by their averages over numerical cells, and sit at their centers, while the fluxes are computed at the interfaces between cells. these fluxes are further limited so as to proscribe negative turbulent energies, that numerical inaccuracies might otherwise 102 esteban g. tabak and fabio a. tal 6, 4(2004) produce. in all runs, the eddy mixing length l is set to 1/4, the diffusivities sb and se are set to one, as is su, except for the results displayed in figure 9, where su = −0.3. the initial vertical profile for the buoyancy is linear, representing a background stratification, and the velocity profile is initially depth independent. finally, the turbulent energy is initialized as zero everywhere except for some small initial turbulence close to the surface, necessary in our model to start the boundary fluxes. in the first run (see figure 1), a boundary flux of turbulent energy represents the input of turbulence into the ocean by the storm, while the fluxes of buoyancy and momentum are set to zero. without an initial shear or a mechanism to generate it, the momentum equation 2 is satisfied trivially, so we have run our model without it. 0 2 4 6 8 10 12 −120 −100 −80 −60 −40 −20 0 evolution driven by stirring at the top b (solid) and e (dashed) z figure 1: formation of a well–mixed layer in a stratified fluid by a constant flux of turbulent energy from the top. the solid line represents the buoyancy profile, and the dashed line the turbulent energy. the snapshots plotted are 100 time units apart. the energy flux through the top surface is given by keez = 0.025. one remarkable feature is the large gradients at the bottom of the mixed layer. our model is diffusive, and diffusion is usually associated with attenuation of disparities, but in this case the strongly nonlinear nature of this diffusion yields the inverse phenomenon. in fact, if the storm stops, setting the energy flux at the surface to zero, the base of the well mixed layer rapidly becomes discontinuous. this counterintuitive and mathematically appealing feature is in good agreement with physical reality. it is also worth noticing that, as time evolves, the rate of growth of the mixed layer slows down considerably. this can be understood by a simple energetic argument, where the mixed layer is taken to be completely homogeneous, with the same buoyancy throughout its depth. if b = g′z is the linear background stratification of the ocean, and the mixed layer thickness is h, mass conservation implies that the buoyancy in the layer is b̄ = g ′h 2 . then, if the mixed layer is deepened to h + δh, the potential 6, 4(2004) turbulent mixing of stratified flows 103 energy increase is: δp.e. = ∫ 0 −h ( g′ (h + δh) 2 − g ′(h) 2 ) z dz + + ∫ h h+δh ( g′h + δh 2 − g′z ) z dz = = δhh2 4 + o((δh2)). so we see that the work per unit time required for the growth of the mixed layer increases as the square of the depth, explaining the reduced velocity observed in the numerical runs. 0 1 2 3 4 5 6 7 −70 −60 −50 −40 −30 −20 −10 0 evolution driven by heating and cooling at the top b (solid) and e (dashed) z figure 2: formation of a well–mixed layer in a stratified fluid by a periodic cycle of heating and cooling at the surface. the solid line represents the buoyancy profile, and the dashed line the turbulent energy. the times of the snapshots plotted correspond to the beginning of cooling periods. the buoyancy flux at the top is given by kbbz = 0.0015 sin ( 2πt 100 ) . in order to trigger convection, there is a small constant flux of turbulent energy through the top as well, given by keez = 0.0003. in the second run (see figure 2), a daily periodic, sinusoidal boundary flux of buoyancy mimics the cooling and heating of the ocean’s surface. the times shown correspond to the beginning of cooling periods. again, the momentum equation has not been used, since there is no external source of momentum in this experiment. two features are particularly noticeable: the sharp discontinuity at the base of the mixed layer, and the presence, in the first day plotted, of an inversion layer of warm water next to the surface. in later days, this layer diffuses rapidly throughout the mixed layer, due to the persistence of significant amounts of turbulence from previous cooling events. in the third run (figures 3 and 4), a boundary flux of momentum accounts for the action of the wind. we see a substantial horizontal velocity developing near the surface and diffusing rapidly, due to the turbulence generated by shear instability, throughout the mixed layer. hence the mixed layer decouples from the bulk of the ocean, developing a mean velocity of its own. the base of the mixed layer is smoother 104 esteban g. tabak and fabio a. tal 6, 4(2004) here than in the previous runs, since turbulence is more effectively generated precisely at this interface, which has the maximum shear. hence diffusion is locally enhanced, and the potential discontinuity at the base is smoothed away. 0 2 4 6 8 10 12 −120 −100 −80 −60 −40 −20 0 evolution driven by wind stress at the top b z figure 3: buoyancy profile corresponding to the evolution of a well–mixed layer driven by wind stress, represented by a constant flux of horizontal momentum through the surface: kuuz = 0.15. as in figure 2, there is a turbulent energy flux as well, given by keez = 0.0003. the snapshots are displayed 700 time units apart. 0 2 4 6 8 10 12 −120 −100 −80 −60 −40 −20 0 evolution driven by wind stress at the top u z figure 4: horizontal velocity profile for the same wind stress driven evolution of figure 3. 4 flow stability and the richardson number in this section we analyze what is the qualitative behavior of the solutions to the equations far from boundaries and close to equilibrium. the relevant parameter here is the richardson number, defined as ri = − bz (uz)2 , which measures the relative stabilizing influence of the stratification versus the unstabilizing influence of the shear. in this section we shall consider scenarios with no or little initial turbulent energy. there are two critical values for ri. the first one arises from considerations involving the total energy of the system, while the second follows from the details of the dynamics. 6, 4(2004) turbulent mixing of stratified flows 105 if one would replace a stably stratified and sheared fluid at a given layer z0 −δz ≤ z ≤ z0 + δz by a homogeneous fluid with the same mass and momentum, there would be an increase in the potential energy of the layer, but a decrease in the kinetic energy of the flow. the difference in the potential and kinetic energy would be, to leading order in δz, δp.e. = ∫ δz −δz b(z0)(z0 + s)ds − ∫ δz −δz (b(z0) + bz(z0)s)(s + z0)ds = ∫ δz −δz −bz(z0)s2ds δk.e. = ∫ δz −δz (u(z0))2 2 ds − ∫ δz −δz (u(z0) + uz(z0)s)2 2 ds = ∫ δz −δz (uz(z0)2s2) 2 ds. so we see that locally, if the richardson number is smaller than 1 2 , then the kinetic energy of the shear is larger than the potential energy necessary to completely mix the stratification, and one could expect a final state where the fluid is neither stratified nor sheared, and all of the extra energy has been converted into turbulence. the other critical value follows from the turbulent energy equation. if ri < su sb , where the sj ’s are the coefficients defining the diffusivities in 4, then the input of kinetic energy, kbbz + ku(uz)2 = l e 1 2 (su − ri sb)uz 12 , is positive; i.e., the potential energy sink due to mixing is smaller than the kinetic energy gain produced by the suppression of shear. this gives rise to instability. interestingly, this instability is related in this case to a problem of non–uniqueness: neglecting diffusion, the energy equation 3 can be written as et = l e 1 2 (su − ri sb)uz 1 2 . (5) if e is initially zero but the richardson number ri is smaller than su sb , this equation has a similar nature to the classical textbook example for non–uniqueness, xt = cx 1 2 . in other words, a profile with no initial turbulence gives rise to a large family of solutions, included but not limited to the trivial equilibrium. we shall prove below that, consequently, the equilibrium is indeed unstable. this leaves us with two situations to consider, depending on whether the critical value su sb is larger or smaller than 1 2 . in the numerical examples that illustrate the discussion that follows, we have always used as initial profile linear backgrounds of buoyancy and horizontal velocity, and a profile of turbulent energy that is zero everywhere except for a small bump included to trigger potential instabilities (see figure 5). a) su sb > 1 2 106 esteban g. tabak and fabio a. tal 6, 4(2004) −6 −4 −2 0 2 4 6 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 initial profiles z e b u figure 5: qualitative initial buoyancy, velocity and turbulent energy profile for all numerical experiments on shear instability. in this situation we can expect three different behaviors. first, if the value of ri is larger then the critical value su sb , then small disturbances to the main flow will have little effect. any sufficiently small initial turbulent energy added to the flow will be consumed and transferred mainly to potential energy. if the initial turbulent energy is confined to a portion of the domain, say some layer between the depths a and b, then the mixing will take place only in a somewhat broader layer, but it will still be localized. a numerical run of this situation can be seen in figure 6. −3 −2 −1 0 1 2 3 4 5 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 dynamically and energetically stable profile u/4, b and e z u / 4 be figure 6: evolution of a profile that is stable, both on dynamic and energetic grounds. a small patch of turbulence added to the flow yields a localized and moderate amount of mixing. in this run, sb = su = se = 1, and initially bz = −0.1 and uz = 0.25. in this and in all remaining figures, the dashed and solid lines correspond to the initial and final profiles respectively. second, if 1 2 < ri < su sb , we expect any small initial turbulent energy to grow and to produce more mixing. on the other hand, since the total energy is not sufficient to completely mix the fluid, this process must end at some point, which can only happen if ri grows beyond su sb . so, at the final state, we expect the value of the richardson number to be everywhere larger than the dynamical critical value. even 6, 4(2004) turbulent mixing of stratified flows 107 though the initial turbulent kinetic energy is confined to a small layer, the mixing will spread through the full depth of the fluid, and the final state will still be stratified and sheared, but the mean absolute values of bz and uz will decrease. this is indeed the case, as the numerical experiment displayed in figure 7 shows. this scenario corresponds to the double diffusive instability, where a seemingly (energetically) stable profile can grow unstable due to disparities in the diffusivities of two quantities involved. −4 −3 −2 −1 0 1 2 3 4 5 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 dynamically unstable and energetically stable profile u/4, b and e z e b u / 4 figure 7: evolution of a profile that is dynamically unstable, yet lacks enough kinetic energy to fully mix. part of the energy in the shear is used for mixing, but the final state has both shear and stratification. in this run, sb = su = se = 1, and initially bz = −0.1 and uz = 0.35. finally, if ri < 1 2 , then the mixing will be able to completely overcome the stratification, and the final state will be homogeneous with uniform buoyancy and velocity, with all the excess energy converted into turbulence. the results of a numerical experiment confirming this scenario are plotted in figure 8. −6 −4 −2 0 2 4 6 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 dynamically and energetically unstable profile u/4, b and e z u / 4 b e 0 e f figure 8: evolution of a profile that is dynamically and energetically unstable. the final profile is fully homogeneous, with neither shear nor stratification, and all extra energy converted into turbulence. in this run, sb = su = se = 1, and initially bz = −0.1 and uz = 0.5. 108 esteban g. tabak and fabio a. tal 6, 4(2004) b) su sb < 1 2 in this situation there are only two relevant cases. if the initial richardson number is smaller than the critical dynamical value, than mixing will spread throughout the depth of the fluid and the final state will again be one of a homogeneous fluid, much as in the last scenario discussed above. if, on the other hand, ri > su sb , then the initial turbulent energy will be insufficient to trigger a large mixing process. the most interest case is when ri < 1 2 . even though the kinetic energy of the shear is sufficient in principle to overcome the stratification completely, the dynamics do not allow this to happen, at least for small perturbations. the existing turbulent energy will be transformed into potential energy faster than it can collect kinetic energy from the shear, and the turbulence will eventually disappear, not allowing for any further mixing to occur. figure 9 displays this interesting behavior. this scenario, where the state of maximal entropy is not dynamically reachable, is reminiscent of other geophysical situations, such as the high potential energy states in geostrophic balance with zonal winds prevailing in the atmosphere, that can only acquire entropy by eliminating some potential energy through violent nonlinear instabilities, yielding mid–latitude storms. −6 −4 −2 0 2 4 6 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 energetically unstable yet dynamically stable profile u/4, b and e z b u / 4 e figure 9: evolution of a profile that is dynamically stable, even though the kinetic energy in the shear is sufficient to fully homogenize the buoyancy. small perturbations are not enough to trigger nonlinear instabilities, and so yield only small, localized mixing. in this run, sb = se = 1, su = 0.3, and initially bz = −0.1 and uz = 0.5. 4.1 stability analysis we analyze now the equilibrium states for the system of equations 1 to 3. of course, if no–flux boundary conditions are applied, the only equilibrium states are those in which e = 0. on the other hand, in the ocean interior, fluxes of buoyancy and horizontal momentum coming from the upper and lower boundaries do exist: the buoyancy flux arises mainly from heating and cooling of the ocean surface, while the momentum flux has a more diverse source: wind stress, bottom roughness, and inhomogeneous buoyancy effects coupled with rotation, though the thermal wind effect. if we allow boundary fluxes to exist, there are other relevant equilibrium states, in 6, 4(2004) turbulent mixing of stratified flows 109 which ri = su sb , the buoyancy and velocity gradients are constant, and e may assume any constant positive value. we will first analyze this case, where b̄z = b, ūz = u , sbb = −su(u )2 and ē = e. let b = b̄ + b′, u = ū + u′ and e = e(1 + e′). the linearization of the equations read: bt = (le 1 2 sb)(bzz + ezb 2 ), ut = (le 1 2 su)(uzz + ezu 2 ), (6) et = (le 1 2 se)ezz + (le − 12 sb)(bz ) + (le − 12 suu )(2uz), where we have dropped the primes in the new variables, and made use of the fact that sbb = −suu 2. let us now propose solutions of the form (b, u, e) = (b0, u0, e0) ei(kz−wt), where (b0, u0, e0) is a constant. it is easy to see that, in this case, the system 6 adopts the form a v = 0, where a = le 1 2 ⎡ ⎢⎢⎣ iw le 1 2 − k2sb 0 ik sbb2 0 iw le 1 2 − k2su ik suu2 iksb e 2iksu u e iw le 1 2 − k2se ⎤ ⎥⎥⎦ and v = ⎡ ⎣ b0u0 e0 ⎤ ⎦ . clearly, this system has nontrivial solutions only when det(a) = 0. if we set x = iw le 1 2 , α = −k2sb, β = −k2su, γ = −k2se, then det(a)(x) = (le 1 2 )3((x − α)(x − β)(x − γ) + (x − α) βsuu 2 e + (x − β) αsbb 2e ). at x = 0 we see that det(a)(x) = (le 1 2 )3(−αβγ − αβ suu2 2e ) < 0. since limx→∞ det(a) = +∞, there is a positive real x̄ such that the determinant is null. this corresponds to a frequency w̄ with positive imaginary part and so the linear system is unstable. in fact, since the frequency w̄ is purely imaginary, the disturbances will grow in fixed locations, without moving. this is in fact verified by numerical experiments in this regime. the analysis of the system when e = 0 is somewhat different. here b and u can be any regular functions in the domain. if we perturb an equilibrium state b = ¯b(z), u = ¯u(z), e = 0, then the equations, to leading order, are b′t = (l(e ′) 1 2 sbb̄z)z u′t = (l(e ′) 1 2 suūz)z (7) e′t = l(e ′) 1 2 (sbb̄z + su(ūz) 2). we first note that the right–hand side of the system is independent of b′ and u′. one can see that, if the richardson number of the equilibrium state is everywhere 110 esteban g. tabak and fabio a. tal 6, 4(2004) larger than the critical dynamical value, then sbb̄z + su(ūz)2 < 0 and we may find a positive c such that e′t < −c(e′) 1 2 everywhere in the domain. this implies that e′t(z, t) ≤ (e′(z, 0)2 − c2 t)2, at any time t and any depth z, so the solutions to the system 7 will reach e′ = 0 in a time smaller than max−h≤z≤0 2(e′(z,0))2 c . this shows that this equilibrium is stable. finally, if the richardson number is smaller than su sb at some point in the domain, e′ will grow at this depth until the terms of larger order begin to matter, and so the equilibrium is unstable. 5 conclusions fluid mixing is a problem of high scientific and practical significance, and full of mathematical and physical challenges. here we have concentrated on one possible mathematical description of mixing, probably the simplest, based on the assumption that turbulent mixing can be conceptualized as a nonlinear diffusive process. independently of its range of validity, this model has a number of appealing features: • it yields a phenomenology very much in agreement with physical reality, such as the formation of well–mixed layers with sharp interfaces. • it provides a description of the mixing process easy to grasp intuitively, helping clarify complex concepts, such as the stability of sheared and stratified flows, and the distinction between dynamic (local) and energetic (global) stability. • it is mathematically treatable, though far from trivial. it should be remembered, however, that this class of models is phenomenological, and not based on a first–principled approach. a reminder of this is the free parameter l, the “eddy mixing length”, which needs to be provided externally. in this paper, we have concentrated on one–dimensional scenarios, where the flow is assumed to be horizontally homogeneous. straightforward extensions of the model can be applied to two and three dimensional situations, shedding light on phenomena such as the anisotropy of mixing rates in the ocean, and the effects of breaking waves. such extensions will be explored in further work. references [1] balmforth, n. j. , smith, s. g. l. and young, w.r., dynamics of interface and layers in a stratified turbulent fluid , j. fluid mech., 355, pp. 329–358, 1998. [2] ellison, t. h., and turner, j. s., 1959: turbulent entrainment in stratified flows, j. fluid mech., 99, 423–448. [3] hallberg, r. w., 2000: time integration of diapycnal diffusion and richardson number dependent mixing in isopycnal coordinate ocean models, mon. wea. rev., 128, 1402–1419. 6, 4(2004) turbulent mixing of stratified flows 111 [4] holland, d. m., rosales, r. r., stefanica, d., and tabak, e. g., 2002: internal hydraulic jumps and mixing in two–layer flows, j. fluid mech., 470, 63–83. [5] howard, l. n., 1961: note on a paper of john w. miles, j. fluid mech., 10, 509–512. [6] kantha, l.h. and clayson,c.a., an improved mixed layer model for geophysical applications, j. geoph. research, 99, pp. 25235–25266, 1994. [7] ledwell, j. r., montgomery, e. t., polzin, k. l., st laurent, l. c., schmitt, r. w., and toole, j. m., 2000: evidence for enhanced mixing over rough topography in the abyssal ocean, nature, 403, 179–182. [8] makinson, k., modeling tidal current profiles and vertical mixing beneath filchnerronne ice shelf, antartica , j. phys. ocean., 32, pp. 202215, 2002. [9] mellor, g.l. and yamada, t., a hierarchy of turbulent closure models for planetary boundary layers, j. atmos. sci., 31, pp. 1791-1806, 1974. [10] miles, j. w., 1961: on the stability of heterogeneous shear flows, j. fluid mech.,10, 496–508. [11] pawlak, g. p. and armi, l., 2000: mixing and entrainment in developing stratified currents, j. fluid mech., 424, 45–73. [12] turner, j. s., 1986: turbulent entrainment – the development of the entrainment assumption, and its application to geophysical flows, j. fluid mech., 173, 431– 471. cubo a mathematical journal vol.10, n o ¯ 03, (21–41). october 2008 multiple solutions for doubly resonant elliptic problems using critical groups ravi p. agarwal department of mathematical sciences, florida institute of technology, melbourne 32901-6975, fl, u.s.a email: agarwal@fit.edu michael e. filippakis department of mathematics, national technical university, zografou campus, athens 15780, greece email: mfilip@math.ntua.gr donal o’regan department of mathematics, national university of ireland, galway, ireland email: donal.oregan@nuigalway.ie and nikolaos s. papageorgiou department of mathematics, national technical university, zografou campus, athens 15780, greece email: npapg@math.ntua.gr abstract we consider a semilinear elliptic equation, with a right hand side nonlinearity which may grow linearly. throughout we assume a double resonance at infinity in the spectral interval [λ1,λ2]. in this paper, we can also have resonance at zero or even double 22 ravi p. agarwal et al. cubo 10, 3 (2008) resonance in the order interval [λm,λm+1], m ≥ 2. using morse theory and in particular critical groups, we prove two multiplicity theorems. resumen nosotros consideramos una ecuación semilinear eliptica con una no-linealidad la cual puede crecer linealmente. asumimos una doble resonancia en infinito en el intervalo espectral [λ1,λ2]. en este art́ıculo, podemos también tener resonancia en cero o incluso doble resonancia en el intervalo ordenado [λm,λm+1], m ≥ 2. usando teoria de morse y en particular grupos cŕıticos, provamos dos teoremas de mulplicidad. key words and phrases: double resonance, c-condition, critical groups, critical point of mountain pass-type, poincare-hopf formula. math. subj. class.: 35j20, 35j25. 1 introduction let z ⊆ rn be a bounded domain with a c2-boundary ∂z. we consider the following semilinear elliptic problem: { −△x(z) = λ1x(z) + f(z,x(z)) a.e. on z, x|∂z = 0. } (1.1) here λ1 > 0 is the principal eigenvalue of (−△,h 1 0 (z)). assume that lim |x|→∞ f(z,x) x = 0 uniformly for a.a. z ∈ z. (1.2) the problem (1.1) is resonant at infinity with respect to the principal eigenvalue λ1 > 0. resonant problems, were first studied by landesman-lazer [7], who assumed a bounded nonlinearity and introduced the well-known sufficient asymptotic solvability conditions, which carry their name (the ll-conditions for short). we can be more general and instead of (1.2), assume only that lim inf |x|→∞ f(z,x) x and lim sup |x|→∞ f(z,x) x belong in the interval [0,λ2 − λ1] uniformly for a.a. z ∈ z, with λ2 (λ2 > λ1) being the second eigenvalue of (−△,h1 0 (z)). in this more general setting, the nonlinearity f(z,x) need not be bounded. this more general situation was examined by berestycki-de figueiredo [2], landesman-robinson-rumbos [8], nkashama [11], robinson [13],[14], rumbos [15] and su [16]. from these works, berestycki-de figueiredo [2], nkashama [11], robinson [13] and rumbos [15], prove existence theorems in a double resonance setting (i.e. asymptotically at ±∞, we have cubo 10, 3 (2008) multiple solutions for doubly ... 23 complete interaction of the ”slope” f (z,x) x with both ends of the spectral interval [0,λ2 − λ1]; see berestycki-de figueiredo [2] who coined the term ”double resonance” and robinson [13]) or in a one-sided resonance setting (i.e. the ”slope” f (z,x) x is not allowed to cross λ2 − λ1; see nkashama [11] and rumbos [15]). multiplicity results were proved by landesman-robinson-rumbos [8] (onesided resonant problems) and by robinson [14] and su [16] (doubly resonant problems). in this paper, we extend the work of landesman-robinson-rumbos [8] and partially extend and complement the works of robinson [14] and su [16], by covering cases which are not included in their multiplicity results. 2 mathematical background we start by recalling some basic facts about the following weighted linear eigenvalue problem: { −△u(z) = ̂λm(z)u(z) a.e. on z, u|∂z = 0, ̂λ ∈ r. } (2.1) here m ∈ l∞(z)+ = {m ∈ l ∞(z) : m(z) ≥ 0 a.e. on z}, m 6= 0 (the weight function). by an eigenvalue of (2.1), we mean a real number ̂λ, for which problem (2.1) has a nontrivial solution u ∈ h1 0 (z). it is well-known (see for example gasinski-papageorgiou [5]), that problem (2.1) (or equivalently that (−△,h1 0 (z),m)), has a sequence {̂λk(m)}k≥1 of distinct eigenvalues, ̂λ1(m) > 0 and ̂λk(m) → +∞ as k → +∞. moreover, ̂λ1(m) > 0 is simple (i.e. the corresponding eigenspace e(̂λ1) is one-dimensional). also we can find an orthonormal basis {un}n≥1 ⊆ h 1 0 (z) ∩ c∞(z) for the hilbert space l2(z) consisting of eigenfunctions corresponding to the eigenvalues {̂λk(m)}k≥1. note that {un}n≥1 is also an orthogonal basis for the hilbert space h 1 0 (z). moreover, since by hypothesis ∂z is a c2-manifold, then un ∈ c 2(z) for all n ≥ 1. for every k ≥ 1, by e(̂λk) we denote the eigenspace corresponding to the eigenvalue ̂λk(m). this space has the so-called ”unique continuation property”, namely, if u ∈ e(̂λk) is such that it vanishes on a set of positive measure, then u(z) = 0 for all z ∈ z. we set hk = k ⊕ i=1 e(̂λi) and ̂hk+1 = ⊕ i≥k+1 e(λi) = h ⊥ k , k ≥ 1. we have the orthogonal direct sum decomposition h 1 0 (z) = hk ⊕ ̂hk+1. using these spaces, we can have useful variational characterizations of the eigenvalues {̂λk(m)}k≥1 using the rayleigh quotient. namely we have: ̂λ1(m) = min [ ‖du‖2 2 ∫ z mu2dz : u ∈ h1 0 (z),u 6= 0 ] . (2.2) 24 ravi p. agarwal et al. cubo 10, 3 (2008) in (2.2) the minimum is attained on e(̂λ1)\{0}. by u1 ∈ c 2 0 (z), we denote the principal eigenfunction satisfying ∫ z mu2 1 dz = 1. for k ≥ 2, we have ̂λk(m) = max [ ‖du‖2 2 ∫ z mu2dz : u ∈ hk,u 6= 0 ] (2.3) = min [ ‖dû‖2 2 ∫ z mû2dz : û ∈ ̂hk, û 6= 0 ] . (2.4) in (2.3) (resp.(2.4)), the maximum (resp.minimum) is attained on e(̂λk). from these variational characterizations of the eigenvalues and the unique continuation property of the eigenspaces e(̂λk), we see that the eigenvalues {̂λk(m)}k≥1 have the following strict monotonicity property: ”if m1,m2 ∈ l ∞(z)+, m1(z) ≤ m2(z) a.e. on z and m1 6= m2, then ̂λk(m2) < ̂λk(m1) for all k ≥ 1.” if m ≡ 1, then we simply write λk for all k ≥ 1 and we have the full-spectrum of (−△,h 1 0 (z)). let h be a hilbert space and ϕ ∈ c1(h). we say that ϕ satisfies the ”cerami condition” (the c-condition for short), if the following is true:”every sequence {xn}n≥1 ⊆ h such that |ϕ(xn)| ≤ m1 for some m1 > 0, all n ≥ 1 and (1 + ‖xn‖)ϕ ′(xn) → 0 in h ∗ as n → ∞, has a strongly convergent subsequence”. this condition is a weakened version of the well-known palais-smale condition (ps-condition for short). bartolo-benci-fortunato [1], showed that the c-condition suffices to prove a deformation theorem and from this produce minimax expressions for the critical values of the functional ϕ. for every c ∈ r, let ϕc = {x ∈ x : ϕ ≤ c} (the sublevel set at c of ϕ), k = {x ∈ x : ϕ′(x) = 0} (the set of critical points of ϕ) and kc = {x ∈ k : ϕ(x) = c} (the critical points of ϕ at level c). if x is a hausdorff topological space and y a subspace of it, for every integer n ≥ 0, by hn(x,y ) we denote the n th-relative singular homology group with integer coefficients. the critical groups of ϕ at an isolated critical point x0 ∈ h with ϕ(x0) = c, are defined by cn(ϕ,x0) = hn(ϕ c ∩ u, (ϕc ∩ u)\{x0}), where u is a neighborhood of x0 such that k ∩ϕ c ∩u = {x0}. by the excision property of singular homology theory, we see that the above definition of critical groups, is independent of u (see for example mawhin-willem [10]). suppose that −∞ < inf ϕ(k). choose c < inf ϕ(k). the critical groups at infinity, are defined by ck(ϕ,∞) = hk(h,ϕ c) for all k ≥ 0. cubo 10, 3 (2008) multiple solutions for doubly ... 25 if k is finite, then the morse-type numbers of ϕ, are defined by mk = ∑ x∈k rankck(ϕ,x). the betti-type numbers of ϕ, are defined by βk = rankck(ϕ,∞). by morse theory (see chang [4] and mawhin-willem [10]), we have m ∑ k=0 (−1)m−kmk ≥ m ∑ k=0 (−1)m−kβk and ∑ k≥0 (−1)kmk = ∑ k≥0 (−1)kβk. from the first relation, we deduce that βk ≤ mk for all k ≥ 0. therefore, if βk 6= 0 for some k ≥ 0, then ϕ must have a critical point x ∈ h and the critical group ck(ϕ,x) is nontrivial. the second relation (the equality), is known as the ”poincare-hopf formula”. finally, if k = {x0}, then ck(ϕ,∞) = ck(ϕ,x0) for all k ≥ 0. 3 multiplicity of solutions the hypotheses on the nonlinearity f(z,x) are the following: h(f): f : z × r → r is a function such that f(z, 0) = 0 a.e. on z and (i) for all x ∈ r, z → f(z,x) is measurable; (ii) for almost all z ∈ z, f(z, ·) ∈ c1(r); (iii) |f′x(z,x)| ≤ c(1 + |x| r), r < 4 n−2 ,c > 0. (iv) 0 ≤ lim inf |x|→∞ f (z,x) x ≤ lim sup |x|→∞ f (z,x) x ≤ λ2 − λ1 uniformly for a.a. z ∈ z; (v) suppose that ‖xn‖ → ∞, (i) if ‖x 0 n ‖ ‖xn‖ → 1, xn = x 0 n + x̂n with x 0 n ∈ e(λ1) = h1, x̂n ∈ ̂h2, then there exist γ1 > 0 and n1 ≥ 1 such that ∫ z f(z,xn(z))x 0 n(z)dz ≥ γ1 for all n ≥ n1; (ii) if ‖x 0 n ‖ ‖xn‖ → 1, xn = x 0 n + x̂n with x 0 n ∈ e(λ2), x̂n ∈ w = e(λ2) ⊥, then there exist γ2 > 0 and n ≥ 1 such that ∫ z (f(z,xn(z)) − (λ2 − λ1)xn(z))x 0 n(z)dz ≤ −γ2 for all n ≥ n2; 26 ravi p. agarwal et al. cubo 10, 3 (2008) (vi) if f(z,x) = ∫ x 0 f(z,s)ds, then there exist η ∈ l∞(z) and δ > 0, such that η(z) ≤ 0 a.e. on z with strict inequality on a set of positive measure and f(z,x) ≤ η(z) 2 x 2 for a.a. z ∈ z and all |x| ≤ δ. remark 3.1. hypothesis h(f)(iv) implies that asymptotically at ±∞, we have double resonance. hypothesis h(f)(v) is a generalized ll-condition. similar conditions can be found in the works of landesman-robinson-rumbos [8], robinson [13],[14] and su [16]. consider a c2-function x → f(x) which in a neighborhood of zero equals x4 − sinx2, while for |x| large (say |x| ≥ m > 0), f(x) = c|x| 3 2 ,c > 0. if f(x) = f ′(x), then f ∈ c1(r) satisfies hypothesis h(f) above. to verify the generalized ll-condition in hypothesis h(f)(v), we use lemma 2.1 of su-tang [17]. similarly we can consider if near the origin, f(x) = 1 2 x2 − tan−1x2 or f(x) = −cosx2. this second case is interesting because then f(x) = 2xsinx2 and f′(x) = 2sinx2+4x2cosx2. so f′(0) = 0. this example, which is covered by hypotheses h(f), illustrates that our framework of analysis incorporates also problems with resonance at zero with respect to λ1 > 0 (double-double resonance). this is not possible in the setting of landesman-robinson-rumbos [8] (see theorem 2 in [8]). also such a potential function is not covered by the multiplicity results of robinson [14] (theorem 2) and su [16] (theorem 2). we consider the euler functional for problem (1.1), ϕ : h1 0 (z) → r defined by ϕ(x) = 1 2 ‖dx‖2 2 − λ1 2 ‖x‖2 2 − ∫ z f(z,x(z))dz for all x ∈ h1 0 (z). it is well-known that ϕ ∈ c2(h1 0 (z)) and if by 〈·, ·〉 we denote the duality brackets for the pair (h1 0 (z),h−1(z) = h1 0 (z)∗), we have 〈ϕ′(x),y〉 = ∫ z (dx,dy)rndz − λ1 ∫ z xydz − ∫ z f(z,x(z))y(z)dz and ϕ′′(x)(u,v) = ∫ z (du,dv)rndz − λ1 ∫ z uvdz − ∫ z f ′(z,x(z))u(z)v(z)dz for all x,y,u,v ∈ h1 0 (z). proposition 3.2. if hypotheses h(f) hold then ϕ satisfies the c-condition. proof. let {xn}n≥1 ⊆ h 1 0 (z) be a sequence such that (1 + ‖xn‖)ϕ ′(xn) → 0 as n → ∞. we will show that {xn}n≥1 ⊆ h 1 0 (z) is bounded. we argue indirectly. suppose that {xn}n≥1 ⊆ h 1 0 (z) is unbounded. we may assume that ‖xn‖ → ∞. let yn = xn ‖xn‖ , n ≥ 1. by passing to a suitable subsequence if necessary, we may assume that yn w → y in h1 0 (z), yn → y in l 2(z), yn(z) → y(z) a.e. on z and |yn(z)| ≤ k(z) a.e. on z, for all n ≥ 1, with k ∈ l 2(z)+. cubo 10, 3 (2008) multiple solutions for doubly ... 27 hypotheses h(f)(iii) and (iv), imply that |f(z,x)| ≤ a(z) + c|x| for a.a. z ∈ z, all x ∈ r, with a ∈ l∞(z)+,c > 0, ⇒ |f(z,xn(z))| ‖xn‖ ≤ a(z) ‖xn‖ + c|yn(z)| for a.a. z ∈ z, all n ≥ 1, (3.1) ⇒ { f(·,xn(·)) ‖xn‖ } n≥1 ⊆ l2(z) is bounded. thus we may assume that f(·,xn(·)) ‖xn‖ w → h in l2(z) as n → ∞. for every ε > 0 and n ≥ 1, we set c+ε,n = {z ∈ z : xn(z) > 0, −ε ≤ f(z,xn(z)) xn(z) ≤ λ2 − λ1 + ε} and c−ε,n = {z ∈ z : xn(z) < 0, −ε ≤ f(z,xn(z)) xn(z) ≤ λ2 − λ1 + ε} note that xn(z) → +∞ a.e. on {y > 0} and xn(z) → −∞ a.e. on {y < 0}. then by virtue of hypothesis h(f)(iv), we have χ c + ε,n (z) → χ{y>0}(z) and χc−ε,n (z) → χ{y<0}(z) a.e. on z. using the dominated convergent theorem, we see that ‖(1 − χ c + ε,n ) f(·,xn(·)) ‖xn‖ ‖l2({y>0}) → 0 and ‖(1 − χ c − ε,n ) f(·,xn(·)) ‖xn‖ ‖l2({y<0}) → 0 as n → ∞. it follows that χ c + ε,n (·) f(·,xn(·)) ‖xn‖ w → h in l({y > 0}) and χ c − ε,n (·) f(·,xn(·)) ‖xn‖ w → h in l({y < 0}) as n → ∞. from the definitions of the sets c+ε,n and c − ε,n we have −εyn(z) ≤ f(z,xn(z)) ‖xn‖ = f(z,xn(z)) xn(z) yn(z) ≤ (λ2 − λ1 + ε)yn(z) a.e. on c + ε,n and −εyn(z) ≥ f(z,xn(z)) ‖xn‖ = f(z,xn(z)) xn(z) yn(z) ≥ (λ2 − λ1 + ε)yn(z)a.e. on c − ε,n. 28 ravi p. agarwal et al. cubo 10, 3 (2008) passing to the limit as n → ∞, using mazur’s lemma and recalling that ε > 0 is arbitrary, we obtain 0 ≤ h(z) ≤ (λ2 − λ1)y(z) a.e. on {y > 0} (3.2) and 0 ≥ h(z) ≥ (λ2 − λ1)y(z) a.e. on {y < 0}. (3.3) moreover, from (3.1) it is clear that h(z) = 0 a.e. on {y = 0}. (3.4) from (3.2), (3.3) and (3.4), it follows that h(z) = g(z)y(z) a.e. on z, where g ∈ l∞(z)+, 0 ≤ g(z) ≤ λ2 − λ1 a.e. on z. recall that by 〈·, ·〉 we denote the duality brackets for the pair (h1 0 (z),h−1(z)). let a ∈ l(h1 0 (z),h−1(z)) be defined by 〈a(x),y〉 = ∫ z (dx,dy)rndz for all x,y ∈ h 1 0 (z). also let n : l2(z) → l2(z) be the nemitskii operator corresponding to the nonlinearity f(z,x), i.e. n(x)(·) = f(·,x(·)) for all x ∈ l2(z). because of (3.1), by krasnoselskii’s theorem, we know that n is continuous and bounded. moreover, exploiting the compact embedding of h1 0 (z) into l2(z), we see that n is completely continuous (hence compact too) as a map from h1 0 (z) into l2(z) (see for example gasinskipapageorgiou [5], pp.267-268). we have ϕ′(xn) = a(xn) − λ1xn − n(xn) for all n ≥ 1. from the choice of the sequence {xn}n≥1 ⊆ h 1 0 (z), we know that |〈ϕ′(xn),v〉| ≤ εn for all v ∈ h 1 0 (z) with εn ↓ 0, ⇒ ∣ ∣ ∣ ∣ 〈a(yn) − λ1yn − n(xn) ‖xn‖ ,v〉 ∣ ∣ ∣ ∣ ≤ εn ‖xn‖ for all n ≥ 1. (3.5) let v = yn − y ∈ h 1 0 (z), n ≥ 1. then ∣ ∣ ∣ ∣ 〈a(yn),yn − y〉 − λ1 ∫ z yn(yn − y)dz − ∫ z n(xn) ‖xn‖ (yn − y)dz ∣ ∣ ∣ ∣ ≤ εn ‖xn‖ for all n ≥ 1. (3.6) cubo 10, 3 (2008) multiple solutions for doubly ... 29 evidently ∫ z yn(yn − y)dz → 0 and ∫ z n(xn) ‖xn‖ (yn − y) → 0 as n → ∞. so from (3.6), we infer that 〈a(yn),yn − y〉 → 0. (3.7) we have a(yn) w → a(y) in h−1(z). from (3.7) it follows that 〈a(yn),yn〉 → 〈a(y),y〉, ⇒‖dyn‖2 → ‖dy‖2. also dyn w → dy in l2(z, rn). since the hilbert space l2(z, rn) has the kadec-klee property, we deduce that dyn → dy in l 2(z, rn) ⇒ yn → y in h 1 0 (z), i.e. ‖y‖ = 1, y 6= 0. we return to (3.5) and we pass to the limit as n → ∞. we obtain 〈a(y) − λ1y − gy,v〉 = 0 for all v ∈ h 1 0 (z), ⇒a(y) = (λ1 + g)y in h −1(z), ⇒ − △y(z) = (λ1 + g(z))y(z) a.e. on z, y|∂z = 0. (3.8) we distinguish three cases for problem (3.8) depending on where the function g ∈ l∞(z)+ stands in the interval [0,λ2 − λ1]. case 1: g(z) = 0 a.e. on z. then from (3.8), we have − △y(z) = λ1y(z) a.e. on z, y|∂z = 0, ⇒y ∈ e(λ1), y 6= 0. we consider the orthogonal direct sum decomposition h1 0 (z) = e(λ1) ⊕ ̂h2, ̂h2 = e(λ1) ⊥. then for every n ≥ 1, we have xn = x 0 n + x̂n and x 0 n ∈ e(λ1), x̂n ∈ ̂h2. we have yn = y 0 n + ŷn, with y0n = x0n ‖xn‖ ∈ e(λ1) and ŷn = x̂n ‖xn‖ ∈ ̂h2 for all n ≥ 1. 30 ravi p. agarwal et al. cubo 10, 3 (2008) since y ∈ e(λ1), ‖y‖ = 1, we have ‖x0n‖ ‖xn‖ → 1 as n → ∞. recall that ∣ ∣ ∣ ∣ 〈a(xn),v〉 − λ1 ∫ z xnvdz − ∫ z n(xn)vdz ∣ ∣ ∣ ∣ ≤ εn for all v ∈ h 1 0 (z). let v = x0n ∈ h 1 0 (z). we have ∣ ∣ ∣ ∣ ‖dx0n‖ 2 2 − λ1‖x 0 n‖ 2 2 − ∫ z f(z,xn(z))x 0 n(z)dz ∣ ∣ ∣ ∣ ≤ εn, ⇒ ∫ z f(z,xn(z))x 0 n(z)dz ≤ εn (see (2.2)) for all n ≥ 1. (3.9) but by virtue of hypothesis h(f)(v) 0 < γ1 ≤ ∫ z f(z,x(z))x0n(z)dz for all n ≥ n1. (3.10) comparing (3.9) and (3.10), we reach a contradiction. case 2: g(z) = λ2 − λ1 a.e. on z. in this case, from (3.8) we have − △y(z) = λ2y(z) a.e. on z, y|∂z = 0, ⇒y ∈ e(λ2), y 6= 0. now we consider the orthogonal direct sum decomposition h1 0 (z) = e(λ2) ⊕ w, with w = e(λ2) ⊥. then xn = x 0 n + x̂n with x 0 n ∈ e(λ2), x̂n ∈ w, n ≥ 1. since y ∈ e(λ2), ‖y‖ = 1, we have ‖x0n‖ ‖xn‖ → 1 as n → ∞. (3.11) we have |〈a(xn),v〉 − λ1 ∫ z xnvdz − ∫ z f(z,xn(z))v(z)dz| ≤ εn for all v ∈ h1 0 (z), with εn ↓ 0. cubo 10, 3 (2008) multiple solutions for doubly ... 31 let v = x0n. then ∣ ∣ ∣ ∣ ‖dx0n‖ 2 2 − λ1‖x 0 n‖ 2 2 − ∫ z f(z,xn(z))x 0 n(z)dz ∣ ∣ ∣ ∣ ≤ εn, ⇒ ∣ ∣ ∣ ∣ ‖dx0n‖ 2 2 − λ2‖x 0 n‖ 2 2 − ∫ z (f(z,xn(z)) − (λ2 − λ1)xn(z))x 0 n(z)dz ∣ ∣ ∣ ∣ ≤ εn, ⇒ ∫ z (f(z,xn(z)) − (λ2 − λ1)xn(z))x 0 n(z)dz ≥ −εn (see (2.3) and (2.4)). (3.12) but again hypothesis h(f)(v) implies 0 > −γ2 ≥ ∫ z (f(z,xn(z)) − (λ2 − λ1)xn(z))x 0 n(z)dz for all n ≥ n2. (3.13) comparing (3.12) and (3.13) we reach a contradiction. case 3: 0 ≤ g(z) ≤ λ2 − λ1 a.e. on z with g 6= 0, g 6= λ2 − λ1. note that λ1 ≤ λ1 + g(z) ≤ λ2 a.e. on z and the inequalities are strict on sets (in general different) of positive measure. exploiting the strict monotonicity property of the eigenvalues of (−△,h1 0 (z), m) on the weight function m (see section 2), we have ̂λ1(λ1 + g) < ̂λ1(λ1) = 1 and ̂λ2(λ1 + g) > ̂λ2(λ2) = 1. combining this with (2.2), we see that y = 0, a contradiction to the fact that ‖y‖ = 1. so in all these cases we have reached a contradiction. this means that {xn}n≥1 is bounded and so we may assume (at least for a subsequence) that xn w → x in h1 0 (z), xn → x in l 2(z), xn(z) → x(z) a.e. on z and |xn(z)| ≤ k(z) a.e. on z for all n ≥ 1, with k ∈ l 2(z)+. recall that ∣ ∣ ∣ ∣ 〈a(xn),xn − x〉 − λ1 ∫ z xn(xn − x)dz − ∫ z f(z,xn(z))(xn − x)dz ∣ ∣ ∣ ∣ ≤ εn. since ∫ z xn(xn − x)dz → 0 and ∫ z f(z,xn(z))(xn − x)(z)dz → 0 as n → ∞, we obtain 〈a(xn),xn − x〉 → 0 as n → ∞. we know that a(xn) w → a(x) in h−1(z). so as before, via the kadec-klee property of h1 0 (z), we conclude that xn → x in h 1 0 (z). this proves that ϕ satisfies the c-condition. 32 ravi p. agarwal et al. cubo 10, 3 (2008) in the sequel, we will need the following simple lemma: lemma 3.3. if β ∈ l∞(z), β(z) ≤ λ1 a.e. on z and the inequality is strict on a set of positive measure, then there exists ξ1 > 0 such that ψ(x) = ‖dx‖2 2 − ∫ z β(z)x(z)2dz ≥ ξ1‖dx‖ 2 2 for all x ∈ h1 0 (z). proof. from (2.2), we see that ψ ≥ 0. suppose that the lemma is not true. exploiting the 2homogeneity of ψ, we can find {xn}n≥1 ⊆ h 1 0 (z) such that ‖dxn‖2 = 1 for all n ≥ 1 and ψ(xn) ↓ 0 as n → ∞. by poincare’s inequality {xn}n≥1 ⊆ h 1 0 (z) is bounded. so we may assume that xn w → x in h1 0 (z), xn → x in l 2(z), xn(z) → x(z) a.e. on z and |xn(z)| ≤ k(z) a.e. on z for all n ≥ 1, with k ∈ l 2(z)+. from the weak lower semicontinuity of the norm functional, we have ‖dx‖2 2 ≤ lim inf n→∞ ‖dxn‖ 2 2 , while from the dominated convergence theorem, we have ∫ z β(z)xn(z) 2dz → ∫ z β(z)x(z)2dz as n → ∞. hence ψ(x) ≤ lim inf n→∞ ψ(xn) = 0, (3.14) ⇒‖dx‖2 2 ≤ ∫ z β(z)x(z)2dz ≤ λ1‖x‖ 2 2 , ⇒‖dx‖2 2 = λ1‖x‖ 2 2 (see (2.2)), ⇒x = 0 or x = ±u1 with u1 ∈ e(λ1). if x = 0, then ‖dxn‖2 → 0, a contradiction to the fact that ‖dxn‖2 = 1 for all n ≥ 1. if x = ±u1, then |x(z)| > 0 for all z ∈ z and so from the first inequality in (3.9) and the hypothesis on β, we have ‖dx‖2 2 < λ1‖x‖ 2 2 , a contradiction to (2.2). using this lemma, we prove the following proposition. proposition 3.4. if hypotheses h(f) hold, then the origin is a local minimizer of ϕ. cubo 10, 3 (2008) multiple solutions for doubly ... 33 proof. let δ > 0 be as in hypothesis h(f)(vi) and consider the closed ball b c 1 0 δ = {x ∈ c 1 0 (z) : ‖x‖ c1 0 (z) ≤ δ}. by virtue of hypothesis h(f)(vi), for every x ∈ b c 1 0 δ , we have f(z,x(z)) ≤ η(z) 2 x(z)2 for a.a. z ∈ z. (3.15) thus, for all x ∈ b c 1 0 δ , we have ϕ(x) = 1 2 ‖dx‖2 2 − λ1 2 ‖x‖2 2 − ∫ z f(z,x(z))dz ≥ 1 2 ‖dx‖2 2 − 1 2 ∫ z (λ1 + η(z))x(z) 2 dz (see (3.15)) ≥ ξ1 2 ‖dx‖2 2 (apply lemma 3.3 with g = λ1 + η ∈ l ∞(z)) ≥ 0 = ϕ(0). (3.16) from (3.16) we see that x = 0 is a local c1 0 (z)-minimizer of ϕ. but then from brezis-nirenberg [3], we have that x = 0 is a local h1 0 (z)-minimizer of ϕ. we may assume that the origin is an isolated critical point of ϕ or otherwise we have a sequence of nontrivial solutions for problems (1.1). then from the description of the critical groups at an isolated local minimizer (see chang [4], p.33 and mawhin-willem [10], p.175), we have: corollary 3.5. if hypotheses h(f) hold, then ck(ϕ, 0) = δk,0z for all k ≥ 0. in the next proposition, we produce the first nontrivial solution for problem (1.1). proposition 3.6. if hypotheses h(f) hold then problem (1.1) has a nontrivial solution x0 ∈ c 1 0 (z) and x0 is a critical point of ϕ of mountain pass-type. proof. recall that x = 0 is an isolated local minimum of ϕ. so we can find ρ0 > 0 such that ϕ|∂bρ0 > 0. (3.17) let u1 ∈ c 1 0 (z) be the l2(z)-normalized principal eigenfunction of (−△, h1 0 (z)) and let t > 0. for 0 < β0 < t, via the mean value theorem, we have f(z,tu1(z)) = f(z,β0u1(z)) + ∫ t β0 f(z,µu1(z))u1(z)dµ a.e. on z. (3.18) integrating over z and using fubini’s theorem, we obtain ∫ z f(z,tu1(z))dz = ∫ z f(z,β0u1(z))dz + ∫ t β0 1 µ ∫ z f(z,µu1(z))µu1(z)dzdµ. 34 ravi p. agarwal et al. cubo 10, 3 (2008) choosing β0 > 0 large, because of hypothesis h(f)(v), we have ∫ z f(z,µu1(z))µu1(z)dz ≥ γ1 > 0 for all µ ∈ [β0, t]. (3.19) from (3.18) and (3.19), we obtain ∫ z f(z,tu1(z))dz ≥ ∫ z f(z,β0u1(z))dz + ∫ t β0 γ1 µ dµ for β0 > 0 large, ⇒ ∫ z f(z,tu1(z))dz ≥ ∫ z f(z,β0u1(z))dz + γ1(lnt − lnβ0). (3.20) so from (3.20) it follows that − ∫ z f(z,tu1(z))dz → −∞ as t → +∞. hence ϕ(tu1) = − ∫ z f(z,tu1(z))dz → −∞ as t → +∞ (see (2.2)). therefore for t > 0 large, we have ϕ(tu1) < ϕ(0) = 0 < inf ∂bρ0 ϕ = c. this fact together with proposition 3.2, permit the use of the mountain pass theorem (see bartolobenci-fortunato [1]), which gives x0 ∈ h 1 0 (z) such that ϕ ′(x0) = 0 and ϕ(0) = 0 < c ≤ ϕ(x0). (3.21) from (3.21), we deduce that x0 6= 0. from the equality in (3.21), we have a(x0) = λ1x0 + n(x0), ⇒ − △x0(z) = λ1x0(z) + f(z,x0(z)) a.e. on z, x0|∂z = 0. thus x0 ∈ h 1 0 (z) is a nontrivial solution of problem (1.1) and from regularity theory (see for example gasinski-papageorgiou [5], pp.737-738), we have x0 ∈ c 1 0 (z). let d = ϕ(x0) and assume without loss of generality that kd is discrete (otherwise we have a whole sequence of nontrivial solutions for problem (1.1)). then invoking theorem 1 of hofer [6], we can say that x0 ∈ c 1 0 (z) is a critical point of ϕ which is of mountain pass-type. from the description of the critical groups for a critical point of a mountain pass-type (see chang [4], p.91 and mawhin-willem [10], pp.195-196), we have: corollary 3.7. if hypotheses h(f) hold and x0 ∈ c 1 0 (z) is the nontrivial solution of (1.1) obtained in proposition 3.6, then ck(ϕ,x0) = δk,1z for all k ≥ 0. cubo 10, 3 (2008) multiple solutions for doubly ... 35 in the next proposition, we determine the critical groups of ϕ at infinity. to do this, we will need the following slight generalization of lemma 2.4 of perera-schechter [12]. lemma 3.8. if h is a hilbert space, {ϕt}t∈[0,1] is a one-parameter family of c 1(h)-functions such that ϕ′t and ∂tϕt are both locally lipschitz in u ∈ h and there exists r > 0 such that inf[(1 + ‖u‖)‖ϕ′t(u)‖ : t ∈ [0, 1],‖u‖ > r] > 0 and inf[ϕt(u) : t ∈ [0, 1],‖u‖ ≤ r] > −∞, then ck(ϕ0,∞) = ck(ϕ1,∞) for all k ≥ 0. proof. let ξ < inf[ϕt(u) : t ∈ [0, 1],‖u‖ ≤ r]. let h(t; u) (t ∈ [0, 1],u ∈ ϕ ξ 0 ) be the flow generated by the cauchy problem · h(t) = − ∂tϕt(h(t)) ‖ϕ′t(h(t))‖ 2 ϕ′t(h(t)) a.e. on r+, h(0) = u. we have d dt ϕt(h(t)) = 〈ϕ ′ t(h(t)), · h(t)〉 + ∂tϕt(h(t)) = 0 for all t ≥ 0, ⇒ϕt(h(t)) = ϕ0(u) for all t ≥ 0. since u ∈ ϕa 0 , we have ϕt(h(t)) ≤ ξ and so ‖h(t)‖ > r for all t ≥ 0. this then by virtue of the hypothesis of the lemma, implies that this flow exists for all t ≥ 0 (see bartolo-benci-fortunato [1]). it can be reversed, if we replace ϕt with ϕ1−t. therefore h(1) is a homeomorphism of ϕ ξ 0 and ϕ ξ 1 and so ck(ϕ0,∞) = hk(h,ϕ ξ 0 ) ∼= hk(h,ϕ ξ 1 ) = ck(ϕ1,∞). proposition 3.9. if hypotheses h(f)(i) → (v) hold, then ck(ϕ,∞) = δk,1z for all k ≥ 0. proof. let 0 < σ < λ2 − λ1 and consider the following one-parameter c 2-functions on the hilbert space h1 0 (z) : ϕt(x) = 1 2 ‖dx‖2 2 − λ1 + σ 2 ‖x‖2 2 − t ∫ z (f(z,x(z)) − σx(z))dz for all x ∈ h1 0 (z). we claim that we can find r > 0 such that inf[(1 + ‖u‖)‖ϕ′t(u)‖ : t ∈ [0, 1], ‖u‖ > r] > 0. (3.22) 36 ravi p. agarwal et al. cubo 10, 3 (2008) suppose that this is not possible. then we can find tn → t ∈ [0, 1] and ‖un‖ → ∞ such that ϕ′tn (un) → 0 in h −1(z) as n → ∞. let yn = un ‖un‖ , n ≥ 1. by passing to a suitable subsequence if necessary, we may assume that yn w → y in h−1(z), yn → y in l 2(z), yn(z) → y(z) a.e. on z, and |yn(z)| ≤ k(z) for a.a. z ∈ z, all n ≥ 1, with k ∈ l 2(z). we have ∣ ∣ ∣ ∣ 〈 ϕ′tn (un) ‖un‖ ,v〉 ∣ ∣ ∣ ∣ ≤ εn for all v ∈ h ( 0 z), with εn ↓ 0 (see (3.22)) ⇒ ∣ ∣ ∣ ∣ 〈a(yn),v〉 − (λ1 + σ) ∫ z ynvdz − tn ∫ z n(un) ‖un‖ vdz + tnσ ∫ z ynvdz ∣ ∣ ∣ ∣ ≤ εn (3.23) from the proof of proposition 3.2, we know that n(un) ‖un‖ w → h = gy in l2(z) with g ∈ l∞(z)+, 0 ≤ g(z) ≤ λ2 − λ1 a.e. on z. moreover, arguing as in that proof, we can also show that yn → y in h 1 0 (z), hence ‖y‖ = 1, i.e. y 6= 0. so, if we pass to the limit as n → ∞ in (3.23), we obtain 〈a(y),v〉 = (λ1 + σ) ∫ z yvdz + t ∫ z (g + σ)yvdz for all v ∈ h1 0 (z), ⇒a(y) = (λ1 + (1 − t)σ + tg)y. (3.24) as in the proof of proposition 3.2, we consider three distinct possibilities for the weight function m = λ1 + (1 − t)σ + tg ∈ l ∞(z)+. case 1: t = 1 and g = 0. from (3.24), we have a(y) = λ1(y), ⇒ − △y(z) = λ1y(z) a.e. on z, y|∂z = 0, ⇒y ∈ e(λ1), y 6= 0. so, if un = u 0 n + ûn with u 0 n ∈ e(λ1), ûn ∈ ̂h2 = e(λ1) ⊥, n ≥ 1, then ‖u0n‖ ‖un‖ → 1 as n → ∞. (3.25) cubo 10, 3 (2008) multiple solutions for doubly ... 37 we have ∣ ∣ ∣ ∣ 〈a(un),v〉 − (λ1 + σ) ∫ z unvdz − tn ∫ z n(un)vdz + tnσ ∫ z unvdz ∣ ∣ ∣ ∣ ≤ εn for all v ∈ h1 0 (z). let v = u0n ∈ e(λ1). we obtain ∣ ∣ ∣ ∣ ‖du0n‖ 2 2 − (λ1 + σ)‖u 0 n‖ 2 2 − tn ∫ z f(z,un(z))u 0 n(z)dz + tnσ‖u 0 n‖ 2 2 ∣ ∣ ∣ ∣ ≤ εn. (3.26) since u0n ∈ e(λ1), we know that ‖du 0 n‖ 2 2 = λ1‖u 0 n‖ 2 2 . also because of (3.25) and hypothesis h(f)(v), we have ∫ z f(z,un(z))u 0 n(z)dz ≥ γ1 for all n ≥ n1. then from (3.26), we obtain (1 − tn)σ‖u 0 n‖ 2 2 + tnγ1 ≤ εn for all n ≥ n1, ⇒tnγ1 ≤ εn for all n ≥ n1. since tn → t = 1 and εn ↓ 0, in the limit as n → ∞, we obtain 0 < γ1 ≤ 0, a contradiction. case 2: t = 1 and g = λ2 − λ1. from (3.24), we have a(y) = λ2y, ⇒ − △y(z) = λ2y(z) a.e. on z, y|∂z = 0, ⇒y ∈ e(λ2), y 6= 0. now we write un = u 0 n + ûn with u 0 n ∈ e(λ2) and ûn ∈ w = e(λ2) ⊥. we have ‖u0n‖ ‖un‖ → 1 as n → ∞. (3.27) recall that ∣ ∣ ∣ ∣ 〈a(un),v〉 − (λ1 + σ) ∫ z unvdz − tn ∫ z n(un)vdz + tnσ ∫ z unvdz ∣ ∣ ∣ ∣ ≤ εn for all v ∈ h1 0 (z). 38 ravi p. agarwal et al. cubo 10, 3 (2008) let v = u0n ∈ e(λ2). we obtain ∣ ∣ ∣ ∣ ‖du0n‖ 2 2 − tnλ2‖u 0 n‖ 2 2 − (1 − tn)(λ1 + σ)‖u 0 n‖ 2 2 − tn ∫ z (f(z,un(z)) − (λ2 − λ1)un(z))u 0 n(z)dz ∣ ∣ ∣ ∣ ≤ εn. (3.28) note that tnλ2 + (1 − tn)(λ1 + σ) < λ2 and so 0 < ‖du0n‖ 2 2 − (tnλ2 + (1 − tn)(λ1 + σ))‖u 0 n‖ 2 2 . (3.29) in addition because of (3.27) and hypothesis h(f)(v), we have ∫ z (f(z,un(z)) − (λ2 − λ1)un(z))u 0 n(z)dz ≤ −γ2 < 0 for all n ≥ n2. (3.30) using (3.29) and (3.30) in (3.28), we obtain tnγ2 ≤ εn for all n ≥ n2. passing to the limit as n → ∞ and recalling that tn → 1 and ε ↓ 0, we get 0 < γ2 ≤ 0, again a contradiction. case 3: t 6= 1 or 0 ≤ g(z) ≤ λ2 − λ1 a.e. on z with g 6= 0 and g 6= λ2 − λ1. from (3.24), we have a(y) = (λ1 + ̂ξ)y, y 6= 0 with ̂ξ = (1 − t)σ + tg ∈ l ∞(z)+, ⇒ − △y(z) = (λ1 + ̂ξ(z))y(z) a.e. on z, y|∂z = 0. (3.31) note that since t 6= 1 or (g 6= 0 and g 6= λ2 − λ1), we have λ1 ≤ λ1 + ̂ξ(z) ≤ λ2 a.e. on z, λ1 6= λ1 + ̂ξ and λ2 6= λ1 + ̂ξ. hence from the strict monotonicity of the eigenvalues on the weight function, we infer that ̂λ1(λ1 + ̂ξ) < ̂λ1(λ1) = 1 and ̂λ2(λ1 + ̂ξ). (3.32) using (3.32) in (3.31), we infer that y = 0, a contradiction to the fact that ‖y‖ = 1. so in all three cases we have reached a contradiction and this means that there exists r > 0 for which (3.22) is valid. also it is clear, that due to hypotheses h(f)(iii), (iv), we have inf[ϕt(u) : t ∈ [0, 1], ‖u‖ ≤ r] > −∞. cubo 10, 3 (2008) multiple solutions for doubly ... 39 so we can apply lemma 3.8 and have that ck(ϕ0,∞) = ck(ϕ,∞) for all k ≥ 0. (3.33) note that ϕ0(x) = 1 2 ‖dx‖2 2 − λ1 + σ 2 ‖x‖2 2 and ϕ1(x) = ϕ(x) for all x ∈ h 1 0 (z). since 0 < σ < λ2 − λ1, the only critical point of ϕ0 is u = 0. hence ck(ϕ0,∞) = ck(ϕ, 0) for all k ≥ 0. (3.34) moreover, from proposition 2.3 of su [16], we have ck(ϕ0, 0) = δk,1z for all k ≥ 0. (3.35) from (3.33), (3.34) and (3.35), we conclude that ck(ϕ,∞) = δk,1z for all k ≥ 0. now we are ready for the first multiplicity theorem. theorem 3.10. if hypotheses h(f) hold, then problem (1.1) has at least two nontrivial solutions x0,v0 ∈ c 1 0 (z). proof. one nontrivial solution x0 ∈ c 1 0 (z), exists by virtue of proposition 3.6. suppose that {0,x0} are the only critical points of ϕ. then using corollaries 3.5, 3.7, 3.9 and the poincare-hopf formula, we have (−1)0 + (−1)1 = (−1)1, a contradiction. so there exists a third critical point v0 6= x0, v0 6= 0. evidently v0 is a solution of (1.1) and by regularity theory, we have v0 ∈ c 1 0 (z). we have another multiplicity result by modifying hypothesis h(f)(vi). so the new hypotheses on the nonlinearity f(z,x) are the following: h(f)′: f : z × r → r is a function such that f(z, 0) = 0 a.e. on z, hypotheses h(f)′(i) → (v) are the same as hypotheses h(f)(i) → (v) respectively and (vi) there exist m ≥ 2 and δ > 0 such that λm − λ1 ≤ f(z,x) x ≤ λm+1 − λ1 for a.a. z ∈ z and all 0 < |x| ≤ δ. 40 ravi p. agarwal et al. cubo 10, 3 (2008) remark 3.11. hypotheses h(f)′(iv) and (vi) imply that we can have double resonance both at infinity and at zero. a double-double resonance situation. theorem 3.12. if hypotheses h(f)′ hold, then problem (1.1) has at least two nontrivial solutions x0,v0 ∈ c 1 0 (z). proof. because of hypothesis h(f)′(vi) and proposition 1.1 of li-perera-su [9], we have ck(ϕ, 0) = δk,dz, (3.36) where d =sum of multiplicities of {λk} m k=1 = dimhm ≥ 2, since m ≥ 2. also from proposition 3.9, we know that ck(ϕ,∞) = δk,1z. (3.37) so there exists a critical point x0 of ϕ such that c1(ϕ,x0) 6= 0. (3.38) comparing this with (3.36), we infer that x0 6= 0. moreover, due to (3.38) x0 is of mountain pass type and so c1(ϕ,x0) = δk,1z. (3.39) if {0,x0} are the only critical points of ϕ, then from (3.36), (3.37) and (3.39) and the poincare-hopf formula, we have (−1)d + (−1)1 = (−1)1, ⇒(−1)d = 0, a contradiction. so there exists a second nontrivial critical point v0 of ϕ. evidently x0,v0 ∈ h 1 0 (z) are nontrivial solutions of problem (1.1). from regularity theory, we conclude that x0,v0 ∈ c 1 0 (z). remark 3.13. theorem 3.12 above partially extends theorem 3 of robinson [14] and also theorem 2 of su [16]. received: february 2008. revised: april 2008. references [1] p. bartolo, v. benci and d. fortunato, abstract critical point theorems to some nonlinear problems with strong resonance at infinity, nonlin. anal., 7 (1983), 981–1012. 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[15] a. rumbos, a semilinear elliptic boundary value problem at resonance where the nonlinearity may grow linearly, nonlin. anal., 16 (1991), 1159–1168. [16] j-b. su, semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, nonlin. anal., 48 (2002), 881–895. [17] j-b.su and c.l. tang, multiplicity results for semilinear eliptic equations with resonance at higher eigenvalues, nonlin. anal., 44 (2001), 311–321. n03 cubo a mathematical journal vol.11, n o ¯ 02, (37–54). may 2009 the dynamic evolution of industrial clusters ferenc szidarovszky systems & industrial engineering department, the university of arizona, tucson, arizona, 85721-0020, usa email: szidar@sie.arizona.edu and jijun zhao complexity science institute qingdao university, qingdao, shandong, 266071, china email: jijunzhao@yahoo.com abstract a mathematical model and its implementation are presented to analyze the evolution of industrial clusters. the static model is a combination of basic ideas of oligopoly and oligopsony theory as well as fundamental economic results describing the effects of technology development. the dynamic model is based on gradient adjustment in which each decision variable is adjusted in proportion to the corresponding partial derivative of the profit function of the firm controlling this variable. the complexity of the model makes it analytically intractable, therefore computer simulation is used. a sensitivity analysis is performed to examine how the trajectories of the main characteristics of the firms depend on model parameters, and how the entire cluster develops in time. resumen un modelo matemático y su implementación son presentados para analizar la evolución de grupos industriales. el modelo estático es una combinación de ideas básicas de oligopolios y teoria oligoposonia bien como resultados de economia fundamental describiendo los efectos de desarrollo tecnológico. el modelo dinámico es basado en el gradiente de ajustamiento en el cual toda decisión variable es ajustada en proporsión a la correspondiente derivada parcial de la función de utilidad 38 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) de la firma controlando esta variable. la complejidad del modelo hace este analiticamente muy dificil, por lo tanto es usada simulación computacional. un análisis de sensibilidad es realizado para examinar como las trajecgtorias de la caracteŕıstica principal de las firmas dependem del modelo de parametros, y como el grupo entero se desarrolla con el tiempo. key words and phrases: dynamic model, industrial clusters, simulation. math. subj. class.: 91b26, 91b70. introduction the classical oligopoly theory dates back to the work of cournot [3]. it examines an industry in which several firms produce identical product or offer identical service to a homogeneous market. since then a significant number of researchers focused on the different extensions and generalizations of cournot’s classical model. a comprehensive summary of the earlier works is given in [5] and multi-product models with some applications are discussed in [6]. in the early stages, oligopolies were considered as noncooperative games in which the firms are the players, their output levels are the strategies, and the profit functions are the payoffs. the existence and uniqueness of the equilibrium was first the main issue, under certain monotonicity and convexity assumptions the existence and uniqueness of equilibrium was proved. this important result was later extended to more realistic model variants including single product models with product differentiation, multiproduct oligopolies, labor-managed and rent-seeking games. the main focus of the studies in oligopoly theory has later turned into dynamic extensions. models were developed with discrete and continuous time scales and the resulting difference and differential equation systems were investigated. the main issue was the asymptotical stability of the equilibrium, conditions were derived to guarantee that the output trajectories converge to the equilibrium in the long run. most models were linear, where local and global stability are equivalent and very little attention was given to nonlinear dynamics until the late 80s. in developing dynamic models there are usually two alternative concepts. in the case of best response dynamics it is assumed that each firm adjusts its output into the direction toward its best response. this approach requires the knowledge of the best response functions of the firms, which needs the solution of usually nonlinear optimization problems based on global information on the payoff functions. in the case of gradient adjustments it is assumed that the firms adjust their outputs in proportion to their marginal profits. this idea has a lot of sense, since in the case of positive (negative) gradient value the firm’s interest is to increase (decrease) output level. this concept requires only local information about the payoff functions, so it is much more realistic than the use of best response dynamics. a comprehensive summary of the recent developments in this area can be found in [7] and [1]. most studies in oligopoly theory considered only the market as a link between the firms; the unit price was always a function of the total output level of the industry due to the demand-supply balance. however in realistic economies the firms are linked together in much more complicated ways. first, they use common supply of energy, raw material, labor, capital etc., and therefore they also compete on this secondary market in addition to the market of their product. this idea was elaborated in the studies of oligopsonies [10]. in multiproduct oligopolies on the other hand, the firms might buy and use the products of other firms, so a network of firms develops. network oligopolies were introduced and some elementary results were reported in [9]. cubo 11, 2 (2009) the dynamic evolution of industrial clusters 39 in most models analytic results could be derived under only very special conditions, which are not the case in realistic economies. instead of investigating very limited cases theoretically, it is much more important and useful to use computer simulation under realistic conditions and examine the evolution of more advanced production systems such as the industrial clusters. the industry of a certain region consists of several types of firms. some produce final products which are sold directly to the market, while others (maybe in addition to final products) produce material, parts, components what other firms buy and build in their final products. therefore there is a complicated inputoutput relation between the firms. they get their work force, energy and capital from the same secondary markets, where they also compete with each other. the r&d investment of any firm spills over to others who can also benefit from this innovation. in the literature discussing industrial clusters the authors focus on mainly descriptive and statistical issues and very little attention is given to the evolution of the clusters in time [8]. in this paper we will examine the way how an industrial cluster develops in time and how important economic characteristics depend on model parameters. our simulation methodology is similar to agent-based techniques. some initial attempts combining industrial clusters with agent-based simulation were presented in [2], [4], [11]. this paper develops as follows. in section 2 the mathematical model will be outlined, and the simulation methodology and numerical results will be presented in section 3. final conclusions will be drawn in section 4. 1 the mathematical model for the sake of simplicity we assume that there are two types of firms in the cluster: suppliers and producers. producers sell their products to an open market, while suppliers sell their products to the producers. there are altogether total of m suppliers and n producers in the system. they are linked together in the open market, in the energy and labor markets and by innovation spillovers. in addition, the system has to satisfy the product-balance conditions. for any supplier i, we denote its output by si, the price of its product by p s i , its labor usage by l s i and its profit by ϕ s i . for any producer j, we denote its production level by zj , the price of its product by p p j , its labor usage by l p j , its innovation development by ij and the total cumulative innovation level by ĩj , the impact of innovation level on sale price by f (ĩj ), the cost function of innovation by dj(ij ). the profit of this producer is ϕ p j . the price function of labor in the whole cluster is denoted by p l , which depends on the total demand of labor. the profit of a supplier can be obtained as ϕ s i = si · p s i (s1, . . . , sm) − l s i (si)p l   m ∑ i=1 l s i (si) + n ∑ j=1 l p j (zj, ĩj )   , (1.1) which is the difference of its revenue and labor cost. we set all other costs to zero. the profit function of the producers is the following: ϕ p j = zj ·p p j (z1, . . . , zn)fj (ĩj )−l p j (zj, ĩj )p l   m ∑ i=1 l s i (si)+ n ∑ j=1 l p j (zj , ĩj )  − m ∑ i=1 xij p s i (s1, . . . , sm)−dj (ij ), (1.2) 40 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) where xij is the amount of the product of supplier i purchased by producer j. the total output of supplier i is therefore si = n ∑ j=1 xij . (1.3) in our study we select special function forms. we assume that p s i (s1, . . . , sm) = ai − bisi − ∑ l 6=i bilsl, (1.4) that is, the price of any supply decreases if the output by any supplier increases; l s i (si) = γi + δisi, (1.5) that is, more production requires more labor. the output of producer j is given by the linear function zj = m ∑ i=1 aij xij + a0j . (1.6) the prices of the final products are also linear: p p j = aj − bj zj − ∑ l 6=j bjlzl, (1.7) which assumes that the final products are substitutes. the innovation development and spillover are modeled as ĩj (t + 1) = ĩj (t) + ij + ∑ l 6=j kjlil, (1.8) that is, each producer invests in innovation and each of them can utilize the innovation development of the competitors. the price of any final product is affected by the innovation dependent factor fj ( ĩj ) = 1 + ( f max j − 1 ) · ( 1 − e−ωj ĩj ) . (1.9) in this function form we model the fact that with higher technological level better final products are produced, so their prices become higher. if ĩj = 0, then this factor equals 1, it increases in ĩj and converges to a maximum value f max j as ĩj tends to infinity. the need of labor of producer j depends on its production and innovation levels l p j (zj, ĩj ) = (γj + δj zj ) · e −ωj ĩj , (1.10) that is, innovation decreases the labor need of producers. the innovation development cost is also assumed to be linear dj (ij ) = uj + vj ij (1.11) and finally the price of labor is a linear function of the total labor usage: p l = c − d ·   ∑ i l s i + ∑ j l p j   . (1.12) in this decreasing function form we model the fact that in higher labor force the ratio of skilled workers becomes less, so the average wage decreases. cubo 11, 2 (2009) the dynamic evolution of industrial clusters 41 in this model the decision variables of the suppliers are their output levels si, while those of the producers are the xij flows from the suppliers to them. we assume in our model that extra supplies can be sold outside the cluster for the same price, and in the case of supply shortages they can be purchased from sources outside the cluster. we also assume for the sake of simplicity that the firms increase their innovation levels at each time period by a constant innovation step. 2 simulation and analysis of the results the evolution of the cluster in time is simulated by using gradient adjustments. we start with a set of initial values of all decision variables, and then we adjust them in the following way. we compute first the derivatives of the payoff functions with respect to all decision variables. then the value of each variable is adjusted in proportion to the value of the partial derivative of the profit function of the associated firm. this step is then repeated at each time period t ≥ 1. this dynamism can be generated for a long period of time with many different selected values of model parameters, so we can gain insight into the dependence of the firms’ outputs on certain important model characteristics. in this paper, we consider the simple situation when m = 3 and n = 2. in the following simulation experiments, a sensitivity analysis is performed, in which we will alter the value of one of the parameters in a given range, while the values of all other parameters are kept constant. in this way we can see how the output trajectories depend on this altered parameter. we will examine first the effect of the maximum prices ai, aj and c defined in equations (1.4), (1.7), and (1.12). we will also alter the value of parameter bjl in price function (1.7), and also the innovation development step ij . the values of the other parameters are fixed as follows. for suppliers i = 1, 2, 3, we have bi = 1, bil = 0.1 for l 6= i (to represents the low level interaction between the suppliers in their prices), furthermore γi = 10 and δi = 0.4. for producers j = 1, 2, we have a0j = 50, aij = 0.3, bj = 1, kjl = 0.1 for l 6= j, f max j = 2, γj = 50, δj = 0.3 or 0.5, uj = 50, vj = 0.1, ωj = 0.1, and ωj = 0.05. for the labor market, we have d = 0.6. inside an experiment group only one parameter value is varied, all others are left constant. at the beginning of the simulation process, the initial values xij (0) are generated randomly using uniform distribution from the interval [0, 40]. the values of zj and si are calculated according to equations (1.3) and (1.6). the same set of the xij (0) values are selected in the same simulation group for comparison purposes. the initial value of innovation of all producers is chosen as 1. the numerical results can be summarized in detail as follows. 2.1 effect of changing ai in this group of experiments, we have δj = 0.5, aj = 700, c = 300, bjl = 0.1, innovation increment is selected as 0.001 and the value of variable ai changes from 100 to 400. the generated xij (0) values in this group are x11(0) = 32, x12(0) = 36, x21(0) = 36, x22(0) = 25, x31(0) = 5 and x32(0) = 3. figure 1 shows the behavior patterns of the suppliers and producers. since the initial values for the three suppliers and those of the two producers are different, they have different trajectories. in this group of simulation, when ai ≤ 135 (see figure 1(a)), the profits of the suppliers always start 42 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) 0 50 100 150 200 0 10 20 30 40 50 60 70 output of suppliers 0 50 100 150 200 20 30 40 50 60 70 80 price of suppliers 0 50 100 150 200 −4000 −3500 −3000 −2500 −2000 −1500 −1000 −500 profit of suppliers 0 50 100 150 200 60 80 100 120 140 160 output of final products 0 50 100 150 200 520 540 560 580 600 620 640 price of final products 0 50 100 150 200 3 3.5 4 4.5 5 5.5 6 6.5 7 x 10 4profit of producers (a) with ai = 100 0 50 100 150 200 0 10 20 30 40 50 60 70 output of suppliers 0 50 100 150 200 80 90 100 110 120 130 140 price of suppliers 0 50 100 150 200 −1000 −500 0 500 1000 profit of suppliers 0 50 100 150 200 60 70 80 90 100 110 120 130 output from final products 0 50 100 150 200 560 570 580 590 600 610 620 630 price of final products 0 50 100 150 200 2.8 3 3.2 3.4 3.6 3.8 4 4.2 x 10 4profit of producers (b) with ai = 155 cubo 11, 2 (2009) the dynamic evolution of industrial clusters 43 0 50 100 150 200 0 50 100 150 200 output of suppliers 0 50 100 150 200 150 200 250 300 350 400 price of suppliers 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 x 10 4 profit of suppliers 0 50 100 150 200 50 55 60 65 70 75 output from final products 0 50 100 150 200 620 625 630 635 640 645 price of final products 0 50 100 150 200 1 1.5 2 2.5 3 3.5 x 10 4profit of producers (c) with ai = 400 figure 1: results with varying values of ai from a negative value and converge to a negative value with relatively small absolute value. depending on the initial values xij (0), this limit could be a little different. in such cases government subsidies are needed. the suppliers’ productivities decrease (or increase) from the initial values and then converge. the labor of the suppliers is a linear function of the supplier’s productivity; hence it has the same pattern as that of their productivity. the prices of the supplies increase or decrease, but also converge. there are some oscillations at the first few iterations in the labor price, and this results in similar oscillations of the profits of the suppliers at the beginning. to react to the higher price of supplies and higher labor price, the producers cut their production levels and labor usage. even though their price increases, their profit is still decreasing. some oscillations can be observed in the producers’ behaviors: in the productivity, in the price, in the profit, and in the labor usage. since the labor usage of the producers is approximately a linear function of their output, its pattern is very similar to that of the producer’s output. therefore, we will not discuss the labor usages of the producers and suppliers in later analysis. interval (135, 165) for ai is a pattern transition domain (see figure 1(b)). depending on the actual initial values xij (0), this transition domain could be slightly different. in this range, patterns transfer gradually. when ai > 135, the profit of the suppliers start converging to a positive value. the long term behavior of the output and the profit of producers changes from convex trajectory to concave trajectory; similarly, that of the price of final products changes from a concave trajectory to a convex one; and the trajectory of the labor price transfers from concave into convex. 44 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) for ai > 165 (see figure 1(c)), except the profit of the suppliers, all other trajectories reverse their shape from that observed with ai < 135. output, profit and labor usage of suppliers increase before they converge, while the final product prices decrease before converging. when ai becomes larger, the limit values increase accordingly. the labor, output and profit of the producers increase fast at the beginning and then slow down, while their product price and labor price decrease fast at the beginning and then decrease slowly. when ai is increased, the price of supplies also increases. if ai > 300, then as the reaction to high supply price, the producers decide not to purchase from local suppliers. producers have constant output level of 50 at the first few iterations, until supply prices and labor usage decrease to a certain value when the producers start having positive marginal profits. then the producers increase their output levels and purchase from the suppliers. this can be observed in the case of ai = 300, when zj drops down to 50 and labor usage of the producers decrease down to around 71 and stays there for a few more iterations. the price function of the supplies not only affects the supplier’s own behavior, but also the behavior of producers. a low maximum price implies that the suppliers and also the producers lose money, and it results in job reductions. this situation won’t change if the suppliers would increase their prices later on. a high maximum supply price could increase their labor usage, the output and the profit of the suppliers, however the producers’ output and labor need could even decrease. this situation can lead to the departure of skilled workers who can find jobs in other regions and this would lead to the increase of the price of labor, so oscillation and hence instability could occur in the system. the other consequences of decreasing labor force, such as offering incentives to people to move into the cluster, hiring immigrants etc. are not considered in this paper. we will return to these issues in a future paper. 2.2 effect of changing aj in this group of simulation, we select δj = 0.5, ai = 200, c = 300, bjl = 0.1 and innovation increment 0.001. the value of aj changes from 300 to 1000 (figure 2). the generated initial xij (0) values in this group are x11(0) = 38, x12(0) = 19, x21(0) = 9, x22(0) = 35, x31(0) = 24 and x32(0) = 30. all figures show convergent trajectories indicating stability of the cluster in this selected range of parameter values. the changing value of aj does not alter the patterns of the suppliers’ behavior, only the limit values vary. for example, the limit value of the output of the suppliers changes from around 67 for aj = 300 to around 80 as aj = 1000; the limit of the profit of suppliers changes from around 2496 to around 5000. as the result of convergent supply trajectories, the labor need of suppliers also converges and the limit changes from around 37 to around 42. in the case of aj = 300 (see figure 2(a)), with z1(0) = 71.3 and z2(0) = 75.2, both trajectories drop down to 50 at the first iteration. the prices of final products are initially 221.18 and 217.67, but both increase to 245 and then keep this price. the initial profits of the producers are -4133.6 and -5794.8, but both increase to the same value of 2164.4 and then increase gradually later on. the impact of aj on the producers’ behavior is relatively strong. in the above group of simulations, when aj is less than 460, the outputs (and therefore the labor) of the producers drop down from their initial values to relatively small values in the first iteration, then keep them for a certain number of iterations, and then increase gradually before they converge. accordingly, the profit and price of the producers have a sudden increase in the first iteration, then change slowly afterwards. when aj becomes larger, the tendency of the increase in the profit, labor and productivity and that of the decrease in the price of labor become cubo 11, 2 (2009) the dynamic evolution of industrial clusters 45 0 50 100 150 200 250 300 10 20 30 40 50 60 70 output of suppliers 0 50 100 150 200 250 300 110 120 130 140 150 160 170 180 190 price of suppliers 0 50 100 150 200 250 300 −500 0 500 1000 1500 2000 2500 3000 profit of suppliers 0 50 100 150 200 250 300 50 52 54 56 58 60 62 64 output from final products 0 50 100 150 200 250 300 232 234 236 238 240 242 244 246 price of final products 0 50 100 150 200 250 300 −6000 −4000 −2000 0 2000 4000 profit of producers (a) with aj = 300 0 50 100 150 200 250 300 10 20 30 40 50 60 70 80 output of suppliers 0 50 100 150 200 250 300 100 120 140 160 180 200 price of suppliers 0 50 100 150 200 250 300 −1000 0 1000 2000 3000 4000 5000 profit of suppliers 0 50 100 150 200 250 300 40 60 80 100 120 140 160 output from final products 0 50 100 150 200 250 300 820 840 860 880 900 920 940 price of final products 0 50 100 150 200 250 300 4 5 6 7 8 9 10 x 10 4profit of producers (b) with aj = 1000 figure 2: results with varying value of aj 46 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) stronger. we can also observe that oscillations start to emerge gradually in the behavior of the producers, in the profit of the suppliers and in the price of labor. when aj varies from 460 to 1000 (see figure 2(b)), the behavior of the producers remains similar, however the productivity of the producers at iteration 300 changes from 59.958 and 61.537 to 159.26 and 161.97; the profit of the producers at iteration 300 changes from 12168 and 12048 to 101770 and 102820; and the price of final products increases from 393.89 and 392.47 to 824.54 and 822.1, respectively. similarly to the previous case, the value of the maximum final product price impacts both producers’ and suppliers’ behavior, however with much stronger impact on the producers. lower aj value prohibits the producers from earning more profit; too high aj value generates higher profits and leads to the expansion of the final product segment. if higher profit encourages more producers to join the cluster, then the situation might change. we will consider the impact of entry in our later study. the complex impact of labor decrease will be also considered later. 2.3 effect of changing c in this group of simulations, we select δj = 0.3, ai = 200, aj = 700, bjl = 0.1, innovation increment 0.001 and the value of c changes from 100 to 500. the generated initial xij (0) values in this group are x11(0) = 19, x12(0) = 25, x21(0) = 35, x22(0) = 32, x31(0) = 32 and x32(0) = 36. with changing value of c, we can observe a pattern change in the behavior of the suppliers and the producers and also in the labor market. for c < 155 (see figure 3(a)), the supplier’s productivity, the labor and the profit all increase and then converge, however the final product price decreases and then converges. however, the price of the labor keeps a constant level of 10 (see figure 3(f)), and the productivity and the labor of the producers increase exponentially. the final product price decreases to 0 in the first few iterations. the profit of the producers decreases from 0 to a negative value rapidly showing that in this case some government intervention is required. starting from c = 155 (see figure 3(b)), oscillations emerge in the first few iterations in the suppliers’ behavior and these oscillations shift gradually to the later iterations when the value of c becomes larger. the final product price remains at around 600 during these initial iterations, and the price of labor drops to 0 then increases suddenly to 10 after a few oscillations between 0 and 4. the patterns of the other characteristics do not change. at c = 180 (figure 3(c)), there are oscillations in all characteristics of the suppliers with similar main tendencies as observed before, while the patterns of the productivity, profit and the labor of the producers change significantly. they increase first, and then converge to an oscillating pattern, the profit of the producers starts now at a positive value and then converges to a higher value with an oscillating pattern. the price of labor decreases gradually until iteration 110 then starts oscillating between 0 and 4. as the value of c increases to 190, these oscillations disappear, except at the beginning in the producers’ behavior, in the profit of suppliers and in the price of labor. until c < 420, the patterns remain similar but the limiting values have small changes. the case of c = 200 is shown in figure 3(d). when c = 350, the profit of the producer looks like a linear function of time. the patterns of other behavioral trajectories are the same as in previous cases. when c becomes larger, all figures gradually start reversing their shapes, with the difference that the profit of the suppliers now starts from negative values and then converge to an identical negative limit. cubo 11, 2 (2009) the dynamic evolution of industrial clusters 47 0 50 100 150 200 250 300 40 50 60 70 80 90 output of suppliers 0 50 100 150 200 250 300 90 100 110 120 130 140 150 price of suppliers 0 50 100 150 200 250 300 6000 6500 7000 7500 8000 8500 profit of suppliers 0 50 100 150 200 250 300 0 1 2 3 4 5 6 x 10 6 output from final products 0 50 100 150 200 250 300 0 100 200 300 400 500 600 700 price of final products 0 50 100 150 200 250 300 −20 −15 −10 −5 0 5 x 10 8 profit of producers (a) with c = 100 0 50 100 150 200 250 300 40 50 60 70 80 90 output of suppliers 0 50 100 150 200 250 300 90 100 110 120 130 140 150 price of suppliers 0 50 100 150 200 250 300 5500 6000 6500 7000 7500 8000 8500 profit of suppliers 0 50 100 150 200 250 300 0 0.5 1 1.5 2 2.5 x 10 6 output from final products 0 50 100 150 200 250 300 0 100 200 300 400 500 600 700 price of final products 0 50 100 150 200 250 300 −8 −6 −4 −2 0 2 x 10 8 profit of producers (b) with c = 160 48 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) 0 50 100 150 200 250 300 40 50 60 70 80 90 100 output of suppliers 0 50 100 150 200 250 300 90 100 110 120 130 140 150 price of suppliers 0 50 100 150 200 250 300 5000 5500 6000 6500 7000 7500 8000 8500 profit of suppliers 0 50 100 150 200 250 300 70 80 90 100 110 120 130 output from final products 0 50 100 150 200 250 300 560 570 580 590 600 610 620 price of final products 0 50 100 150 200 250 300 3.5 4 4.5 5 5.5 6 x 10 4profit of producers (c) with c = 180 0 50 100 150 200 250 300 40 50 60 70 80 90 100 output of suppliers 0 50 100 150 200 250 300 80 90 100 110 120 130 140 150 price of suppliers 0 50 100 150 200 250 300 4500 5000 5500 6000 6500 7000 7500 profit of suppliers 0 50 100 150 200 250 300 70 80 90 100 110 120 130 output from final products 0 50 100 150 200 250 300 560 570 580 590 600 610 620 price of final products 0 50 100 150 200 250 300 3.5 4 4.5 5 5.5 6 x 10 4profit of producers (d) with c = 200 cubo 11, 2 (2009) the dynamic evolution of industrial clusters 49 0 50 100 150 200 250 300 20 30 40 50 60 70 output of suppliers 0 50 100 150 200 250 300 120 130 140 150 160 170 180 price of suppliers 0 50 100 150 200 250 300 −5000 −4500 −4000 −3500 −3000 profit of suppliers 0 50 100 150 200 250 300 60 65 70 75 80 85 90 output from final products 0 50 100 150 200 250 300 600 605 610 615 620 625 630 price of final products 0 50 100 150 200 250 300 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 10 4profit of producers (e) with c = 500 0 50 100 150 200 250 300 0 10 20 30 40 50 60 price of labor c=100 c=160 c=180 c=200 (f) patterns of price of labor when c changed figure 3: results with varying value of c 50 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) different values of c can generate numerous different patterns of the suppliers and the producers, especially in the range between 150 and 190. the short-term behavior of the firms is very sensitive to the value of c. this instability could be controlled by the local government in stabilizing the system with higher profit for the firms which will then attract other firms to join the cluster, and simultaneously by keeping relatively higher labor price to attract skilled workers. 2.4 effects of changing bjl this group of simulations studies how the similarity of the final products affects the behavior of the firms in the cluster. in the simulation, we select δj = 0.3, ai = 200, aj = 700, c = 300 and innovation increment 0.001. the value of bjl changes from 0.1 to 1. the generated xij (0) values in this group are x11(0) = 33, x12(0) = 28, x21(0) = 22, x22(0) = 21, x31(0) = 14 and x32(0) = 17. 0 50 100 150 200 250 300 30 35 40 45 50 55 60 65 70 output of suppliers 0 50 100 150 200 250 300 110 120 130 140 150 160 price of suppliers 0 50 100 150 200 250 300 1000 1500 2000 2500 3000 profit of suppliers 0 50 100 150 200 250 300 65 70 75 80 85 90 95 100 output from final products 0 50 100 150 200 250 300 580 585 590 595 600 605 610 615 620 price of final products 0 50 100 150 200 250 300 2.6 2.8 3 3.2 3.4 3.6 x 10 4profit of producers figure 4: results with value of bjl = 0.2 regardless of the value of bjl , the patterns are the same for all characteristics. there is a very minor affect on the suppliers’ behavior. a typical case (bjl = 0.2) is shown in figure 4. with changing the value of bjl , the labor price at iteration 300 increases from 144 to 148; the labor of producers decreases from 74 to around 70, and the profit of producers from 35101 and 35014 to 26673 and 26556, respectively. the final product price decreases from around 582 to around 527, and the productivity of the producers from 98 to around 86. if the final products are more similar to each other, then there is more competition among the producers, they have less profit and less productivity, and in addition, the price in the labor market slightly increases. cubo 11, 2 (2009) the dynamic evolution of industrial clusters 51 0 100 200 300 10 20 30 40 50 60 70 output of suppliers 0 100 200 300 110 120 130 140 150 160 170 180 price of suppliers 0 100 200 300 0 500 1000 1500 2000 2500 3000 profit of suppliers 0 100 200 300 140 150 160 170 180 price of labor 0 100 200 300 50 60 70 80 90 100 110 output from final products 0 100 200 300 580 590 600 610 620 630 640 price of final products 0 100 200 300 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 x 10 4 profit of producers 0 100 200 300 64 66 68 70 72 74 76 labor of producers (a) with ij = 0.001 0 100 200 300 10 20 30 40 50 60 70 output of suppliers 0 100 200 300 110 120 130 140 150 160 170 180 price of suppliers 0 100 200 300 0 500 1000 1500 2000 2500 3000 profit of suppliers 0 100 200 300 145 150 155 160 165 170 175 180 price of labor 0 100 200 300 50 60 70 80 90 100 110 output from final products 0 100 200 300 580 590 600 610 620 630 640 price of final products 0 100 200 300 2.5 3 3.5 4 4.5 x 10 4profit of producers 0 100 200 300 64 66 68 70 72 74 labor of producers (b) with ij = 0.005 52 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) 0 100 200 300 10 20 30 40 50 60 70 output of suppliers 0 100 200 300 110 120 130 140 150 160 170 180 price of suppliers 0 100 200 300 0 500 1000 1500 2000 2500 3000 profit of suppliers 0 100 200 300 145 150 155 160 165 170 175 180 price of labor 0 100 200 300 50 60 70 80 90 100 110 120 output from final products 0 100 200 300 570 580 590 600 610 620 630 640 price of final products 0 100 200 300 2.5 3 3.5 4 4.5 5 5.5 x 10 4 profit of producers 0 100 200 300 64 66 68 70 72 74 labor of producers (c) with ij = 0.009 figure 5: results with varying value of ij 2.5 effects of innovation step ij in the simulation, the innovation step, ij , changes from 0.001 to 0.01. we select the other parameters as δj = 0.3, ai = 200, aj = 700, c = 300 and bjl = 0.1 (figure 5). the generated xij (0) values in this group are x11(0) = 11, x12(0) = 39, x21(0) = 18, x22(0) = 23, x31(0) = 2 and x32(0) = 16. the changes in the innovation step ij affect the limit values of the variables and alter the shape of the trajectories slightly. it is interesting to notice that the price of labor follows a convex trajectory with its minimum occurring earlier as ij increases. in contrast, the labor usage of the producers is concave, its maximum shifts to earlier periods as well with increasing value of ij . the profit of the producers gradually changes into a linear shape in time. the limit values of the price, labor and productivity of the suppliers are not affected much. the output of the producers at time period 300 increases with ij , and so the price of the final product decreases if ij increases. the profit of the producers at iteration 300 also shows an increasing tendency with ij . 3 conclusions this paper first introduced a static model of describing the mechanism of industrial clusters including suppliers and final product producers. the competition of these firms was modeled by using the major concepts cubo 11, 2 (2009) the dynamic evolution of industrial clusters 53 of the theory of oligopolies and oligopsonies as well as basic economic principles of technology development. for mathematical simplicity we assumed that supply surplus can be sold outside the cluster and in the case of supply shortages the firms are able to purchase their needed supply from outside sources. we also assumed a constant increase in the innovation level of the firms. in the dynamic extension we used gradient adjustment, in which the firms adjust the values of their decision variables in proportion to the corresponding partial derivatives of their profit functions. instead of analytic methods computer simulation was used to see the evolution and dynamic development of the firms and therefore those of the entire cluster. with this simplified model we were able to gain insight into the dependence of the limit values and the shapes of the trajectories of important characteristics of the firms on model parameters such as maximum prices, similarity of final products, and the innovation step. in our future research we plan to add more realistic features to the model by considering innovation as decision variables, including input-output balances between the firms, entry of firms, attracting and changing the structure of the labor force, and the effect of import and export to mention only a few. acknowledgment the authors are grateful for the support of this research by the national science foundation of china (grant 70771052). received: march 03, 2008. revised: april 17, 2008. references [1] g.i. bischi, c. chiarella, m. kopel and f. szidarovszky, nonlinear oligopolies: stability and bifurcations, springer-verlag, berlin/new york (in press), 2008. [2] t. brenner, simulating the evolution of localised industrial clusters-an identification of the basic mechanism, journal of artificial societies and social simulation, 4(3)(2001), . [3] a. cournot, recherches sur les principes mathématiques de la théorie de richesses, hachett, paris (english translation, 1960. researches into the mathematical principles of the theory of wealth. kelley, new york), 1838. [4] f. giardini, g.d. tosto and r. conte, a model for simulating reputation dynamics in industrial districts, simulation modelling practice and theory, 16(2008), 231–241. [5] k. okuguchi, expectations and stability in oligopoly models, springer-verlag, berlin/new york, 1976. [6] k. okuguchi and f. szidarovszky, the theory of oligopoly with multi-product firms, springer-verlag, berlin/new york, 1999. 54 ferenc szidarovszky and jijun zhao cubo 11, 2 (2009) [7] t. puu and i. sushko, oligopoly dynamics, springer-verlag, berlin/new york, 2002. [8] g.m.p. swann, m. prevezer and d. stout, the dynamics of industrial clustering, oxford university press, new york, 1998. [9] f. szidarovszky, network oligopolies, pure mathematics and applications, 8(1)(1997), 117–123. [10] f. szidarovszky and k. okuguchi, dynamic analysis of oligopsony under adaptive expectations, southwest journal of pure and applied mathematics, 2(2001), 53–60. [11] j. zhang, growing silicon valley on a landscape: an agent-based approach to high-tech industrial clusters, journal of evolutionary economics, 13(2003), 529–548. n04-clusters cubo a mathematical journal vol.10, n o ¯ 02, (83–106). july 2008 the hilbert transform on a smooth closed hypersurface f. brackx and h. de schepper clifford research group, department of mathematical analysis, faculty of engineering, ghent university, galglaan 2, b–9000 gent, belgium email: fb@cage.ugent.be abstract in this paper a condensed account is given of results connected to the hilbert transform on the smooth boundary of a bounded domain in euclidean space and some of its related concepts, such as hardy spaces and the cauchy integral, in a clifford analysis context. clifford analysis, also known as the theory of monogenic functions, is a multidimensional function theory, which is at the same time a generalization of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. it offers a framework which is particularly suited for the integrated treatment of higher dimensional phenomena, without having to rely on tensorial approaches. resumen en este art́ıculo damos un relato condensado de los resultados conectados con la transformada de hilbert sobre dominios acotados con frontera suave en espacios euclideanos y también damos conceptos relacionados, tales como espacios de hardy y la integral de cauchy en el contexto del análisis de clifford. el análisis de clifford, también conocido 84 f. brackx and h. de schepper cubo 10, 2 (2008) como la teoria de funciones monogénicas, es una teoŕıa de funciones multidimensionales, la cual es al mismo tiempo una generalización de la teoŕıa de funciones holomorfas en el plano complejo y un refinamiento del análisis armónico clásico. el art́ıculo ofrece un referencial que es particularmente conveniente para el tratamiento integrado de fenómenos en dimensiones altas, sin tener que recurrir a un abordaje tensorial. key words and phrases: hilbert transform, hardy space, cauchy integral. math. subj. class.: 30g35, 44a15 1 introduction in one–dimensional signal processing the hilbert transform is an indispensable tool for both global and local descriptions of a signal, yielding information on various independent signal properties. the instantaneous amplitude, phase and frequency are estimated by means of so–called quadrature filters, which allow for distinguishing different frequency components and in this way locally refine the structure analysis. these filters are essentially based on the notion of analytic signal, which consists of the linear combination of a bandpass filter, selecting a small part of the spectral information, and its hilbert transform, the latter basically being the result of a phase shift by π 2 on the original filter (see e.g. [18]). mathematically, if f(x) ∈ l2(r) is a real valued signal of finite energy, and h[f] denotes its hilbert transform given by the cauchy principal value h[f](x) = 1 π pv ∫ +∞ −∞ f(y) x − y dy then the corresponding analytic signal is the function 1 2 f + i 2 h[f], which belongs to the hardy space h2(r) and arises as the l2 non–tangential boundary value (ntbv) for y → 0+ of the holomorphic cauchy integral of f in the upper half of the complex plane. though initiated by hilbert, the concept of a conjugated pair (f,h[f]), nowadays called a hilbert pair, was developed mainly by titchmarch and hardy. the multidimensional approach to the hilbert transform usually is a tensorial one, considering the so–called riesz transforms in each of the cartesian variables separately. as opposed to these tensorial approaches clifford analysis is particularly suited for a treatment of multidimensional phenomena encompasssing all dimensions at the same time as an intrinsic feature. during the last fifty years clifford analysis has gradually developed to a comprehensive theory offering a direct, elegant and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. in its most simple but still useful setting, flat m–dimensional euclidean space, clifford analysis focusses on so–called monogenic functions, i.e. null solutions of the clifford– vector valued dirac operator ∂ = ∑ m j=1 ej∂xj where (e1, . . . ,em) forms an orthogonal basis for the quadratic space r m underlying the construction of the clifford algebra r0,m. monogenic functions cubo 10, 2 (2008) the hilbert transform ... 85 have a special relationship with harmonic functions of several variables in that they are refining their properties. the reason is that, as does the cauchy–riemann operator in the complex plane, the rotation–invariant dirac operator factorizes the m–dimensional laplace operator. this has, a.o., allowed for a nice study of hardy spaces of monogenic functions, see [7, 24, 8, 9, 1, 11]. in this context the hilbert transform, as well as more general singular integral operators have been studied in higher dimensional euclidean space (see [14, 24, 31, 19, 10, 12]), in particular on lipschitz hypersurfaces (see [25, 21, 20, 22]) and also on smooth closed hypersurfaces, in particular the unit sphere (see [11, 3, 6]). the subject of this paper is the hilbert transform, within the clifford analysis context, on the smooth boundary of a bounded domain in euclidean space of dimension at least three. for the two–dimensional case we refer to the inspiring book [2]. we have gathered the relevant results spread over the literature and have moulded them together with some new results and insights into a comprehensive text. 2 clifford analysis: the basics in this section we briefly present the basic definitions and some results of clifford analysis which are necessary for our purpose. for an in–depth study of this higher dimensional function theory and its applications we refer to e.g. [4, 13, 14, 15, 16, 17, 26, 27, 28, 29, 30]. let r 0,m be the real vector space r m , endowed with a non–degenerate quadratic form of signature (0,m), let (e1, . . . ,em) be an orthonormal basis for r 0,m , and let r0,m be the universal clifford algebra constructed over r 0,m . the non–commutative multiplication in r0,m is governed by the rules eiej + ejei = −2δi,j, i,j ∈ {1, . . . ,m}. for a set a = {i1, . . . , ih} ⊂ {1, . . . ,m} with 1 ≤ i1 < i2 < ... < ih ≤ m, let ea = ei1ei2 . . .eih . moreover, we put e ∅ = 1, the latter being the identity element. then (ea : a ⊂ {1, . . . ,m}) is a basis for the clifford algebra r0,m. any a ∈ r0,m may thus be written as a = ∑ a aa ea with aa ∈ r or still as a = ∑ m k=0[a]k where [a]k = ∑ |a|=k aa ea is the so–called k–vector part of a (k = 0, 1, . . . ,m). if we denote the space of k–vectors by rk0,m, then the clifford algebra r0,m decomposes as ⊕ m k=0 r k 0,m. we will identify an element x = (x1, . . . ,xm) ∈ r m with the one–vector (or vector for short) x = ∑ m j=1 xj ej . the multiplication of any two vectors x and y is given by xy = x ◦ y + x ∧ y 86 f. brackx and h. de schepper cubo 10, 2 (2008) with x ◦ y = − m∑ j=1 xjyj = 1 2 (xy + yx) = −〈x,y〉 x ∧ y = ∑ i ε } . by a classical argument involving cauchy’s theorem it is found that lim ε→0+ ∫ ∂ωε e(ζ − ξ) dσζ = 1 2 leading to lim x→ξ c[u](x) = 1 2 u(ξ) + lim ε→0+ ∫ ∂ωε e(ζ − ξ) dσζ u(ζ) 88 f. brackx and h. de schepper cubo 10, 2 (2008) or finally lim ω+∋x→ξ c[u](x) = 1 2 u(ξ) + 1 2 h[u](ξ), ξ ∈ ∂ω (3.1) where we have put for ξ ∈ ∂ω: h[u](ξ) = 2 lim ε→0+ ∫ ∂ωε e(ζ − ξ) dσζ u(ζ) = 2 am pv ∫ ∂ω ζ − ξ |ζ − ξ|m dσζ u(ζ) = 2 am pv ∫ ∂ω ξ − ζ |ξ − ζ|m ν(ζ) u(ζ) ds(ζ) this integral transform h is mostly called the hilbert transform; it is sometimes denoted by s∂ω, a notation we will not use any further in this paper. note that, in view of the example given above h[1](ξ) = 2 am pv ∫ ∂ω ξ − ζ |ξ − ζ|m dσζ = 1, ξ ∈ ∂ω. similarly we find for the exterior ntbv of the cauchy integral: lim ω−∋ x→ξ c[u](x) = − 1 2 u(ξ) + 1 2 h[u](ξ), ξ ∈ ∂ω. (3.2) the obtained results (3.1)–(3.2) are the so–called plemelj–sokhotzki formulae, leading to the cauchy transforms defined on c∞(∂ω) by c + [u] = 1 2 u + 1 2 h[u], c−[u] = − 1 2 u + 1 2 h[u]. it follows that u = c + [u] − c − [u], h[u] = c + [u] + c − [u], expressing the function u ∈ c∞(∂ω) as the jump of its cauchy integral over the boundary ∂ω. in section 5 the operators h and c± will be extended to operators on l2(∂ω). 4 the double–layer potential there is a nice connection between the cauchy integral and the related operators h and c± on the one side and the double–layer potential on ∂ω on the other. indeed, the splitting of the product of two clifford vectors into the scalar valued dot product and the bivector valued wedge product allows for rewriting the cauchy integral of the function u ∈ c∞(∂ω) as c[u](x) = 1 am ∫ ∂ω (ζ − x) ◦ ν(ζ) |ζ − x|m u(ζ)ds(ζ) + 1 am ∫ ∂ω (ζ − x) ∧ ν(ζ) |ζ − x|m u(ζ)ds(ζ) (4.1) denoting by −→ x the geometric vector associated with the clifford vector x, we have that (ζ − x) ◦ ν(ζ) = 〈 −→ ζ − −→ x , −→ ν (ζ)〉, where 〈., .〉 here denotes the standard euclidean scalar product. as −→ ∇−→ ζ 1 | −→ ζ − −→ x |m−2 = − (m − 2) −→ ζ − −→ x | −→ ζ − −→ x |m cubo 10, 2 (2008) the hilbert transform ... 89 we have (ζ − x) ◦ ν(ζ) |ζ − x|m = − 1 m − 2 〈 −→ ∇ζ 1 | −→ ζ − −→ x |m−2 , −→ ν (ζ)〉 = − 1 m − 2 ∂ ∂ −→ ν ( 1 | −→ ζ − −→ x |m−2 ) this first term in the cauchy integral (4.1) then takes the form − 1 m − 2 1 am ∫ ∂ω ∂ ∂ −→ ν ( 1 | −→ ζ − −→ x |m−2 ) u(ζ) ds(ζ) in which one recognizes, up to constants, the double–layer potential with density u(ζ) on ∂ω. note that for the constant density u = 1 on ∂ω, one has ∫ ∂ω (ζ − x) ∧ ν(ζ) |ζ − x|m ds(ζ) = 0, x ∈ ω + ∪ ω − (4.2) whence ∫ ∂ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) ds(ζ) = −(m − 2)am, x ∈ ω + (4.3) ∫ ∂ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) ds(ζ) = 0, x ∈ ω− (4.4) confirming known results about gauß’s integral (see e.g. [23, p.360]). it is well–known from classical potential theory that the double–layer potential is harmonic in ω + ∪ ω−. under the assumptions made on the region ω and the function u, the double–layer potential is even defined for ξ ∈ ∂ω; the value at ξ ∈ ∂ω is called its direct value and denoted by w̃(ξ). this function w̃(ξ) is a continuous function on ∂ω and moreover one has (see e.g. [23, p.360]): lim ω+∋ x→ξ ∫ ∂ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) u(ζ)ds(ζ) = − 1 2 (m − 2)amu(ξ) + w̃(ξ) lim ω−∋ x→ξ ∫ ∂ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) u(ζ)ds(ζ) = 1 2 (m − 2)amu(ξ) + w̃(ξ) with w̃(ξ) = −(m − 2) ∫ ∂ω (ξ − ζ) ◦ ν(ζ) |ξ − ζ|m u(ζ) ds(ζ), ξ ∈ ∂ω. it follows that the hilbert transform of a scalar valued function contains a scalar and a bivector part: h[u](ξ) = − 1 m − 2 2 am w̃(ξ) + 2 am pv ∫ ∂ω (ξ − ζ) ∧ ν(ζ) |ξ − ζ|m u(ζ)ds(ζ), ξ ∈ ∂ω, 90 f. brackx and h. de schepper cubo 10, 2 (2008) and that the principal value has to be taken only of the bivector part. as we know from the above example that h[1] = 1, we obtain the following formulae completing (4.2)–(4.4): pv ∫ ∂ω (ζ − ξ) ∧ ν(ζ) |ζ − ξ|m ds(ζ) = 0, ξ ∈ ∂ω, (4.5) ∫ ∂ω ∂ ∂ −→ ν ( 1 |ζ − ξ|m−2 ) ds(ζ) = − 1 2 (m − 2)am, ξ ∈ ∂ω, (4.6) again confirming well–known properties of gauß’s integral. 5 the hardy spaces h± 2 (∂ω) by m∞(ω + ) we denote the space of left–monogenic functions in ω + which are moreover c∞(ω +). similarly m∞(ω − ) denotes the space of left–monogenic functions in ω − , moreover being c∞(ω −) ánd vanishing at infinity. the cauchy integral operator c maps c∞(∂ω) into m∞(ω + ) as well as into m∞(ω − ), while the operators h and c± map c∞(∂ω) into itself. we call m ± ∞ (∂ω) the spaces of functions on ∂ω which are the ntbvs of the functions in m∞(ω ± ) respectively, and we define the hardy spaces h ± 2 (∂ω) as the closure in l2(∂ω) of m ± ∞ (∂ω). it should be emphasized that the usual notation for h + 2 (∂ω) is h 2 (∂ω), and that h − 2 (∂ω) is mostly not considered. our notation however reflects the symmetry in the properties of both hardy spaces. the operators c,h and c± may be extended, through a density argument, to operators defined on l2(∂ω). introducing the hardy spaces h2(ω ± ) of left–monogenic functions in ω ± , which do have ntbvs in l2(∂ω), and, in the case of ω − , also vanish at infinity, we have the following properties of those operators. theorem 5.1. (i) the cauchy integral operator c maps l2(∂ω) into h2(ω ± ) and the ntbvs of c[f],f ∈ l2(∂ω), are given by c ± [f] = ± 1 2 f + 1 2 h[f]; (ii) the cauchy transforms c± are bounded linear operators from l2(∂ω) into h ± 2 (∂ω); (iii) the hilbert transform h is a bounded linear operator from l2(∂ω) onto l2(∂ω); (iv) h2 = 1 or h−1 = h on l2(∂ω); (v) the adjoint operator h∗ of h is given by h∗ = νhν and h∗2 = 1 on l2(∂ω); (vi) h±2 (∂ω) are eigenspaces of h with respective eigenvalues ±1. it is important to note that a function g ∈ l2(∂ω) belongs to the hardy space h + 2 (∂ω) if and only if c + [g] = g, which is equivalent with c−[g] = 0 and still with h[g] = g. thus a function g ∈ h + 2 (∂ω) may be identified with its left–monogenic extension c[g] ∈ h2(ω + ), which is tacitly done most of the time. on the other hand, due to cauchy’s theorem, c[g] = 0 in ω− for each g ∈ h + 2 (∂ω). similarly a function g̃ ∈ l2(∂ω) belongs to h − 2 (∂ω) if and only if c − [g̃] = −g̃ or c + [g̃] = 0 or still h[g̃] = −g̃. a function g̃ ∈ h − 2 (∂ω) may thus be identified with its left– monogenic extension c[−g̃] ∈ h2(ω − ), while here c[g̃] = 0 in ω+ for all g̃ ∈ h − 2 (∂ω). cubo 10, 2 (2008) the hilbert transform ... 91 clearly the cauchy transforms ±c ± are (skew) projection operators on l2(∂ω), sometimes called the hardy projections, since (±c ± ) 2 [f] = 1 4 (1 ± h)2[f] = 1 2 (1 ± h)[f] = (±c±)[f] and c + (−c − )[f] = 1 4 (1 + h)(1 − h)[f] = 0 = (−c − )c + [f] by means of the hardy projections a skew direct sum decomposition of l2(∂ω) is obtained at once: l2(∂ω) = h + 2 (∂ω) ⊕ h − 2 (∂ω) with f = c + [f] + (−c − )[f] = 1 2 (1 + h)[f] + 1 2 (1 − h)[f] and h[f] = c+[f] + c−[f] = 1 2 (1 + h)[f] − 1 2 (1 − h)[f] naturally we have h + 2 (∂ω) = im c + = ker c − and h − 2 (∂ω) = im c − = ker c + expressing the fact that c± are projections parallel to h∓2 (∂ω) (see also figure 1). figure 1 92 f. brackx and h. de schepper cubo 10, 2 (2008) although this decomposition of l2(∂ω) is rather immediate, it is an important result. in fact it states that an l2(∂ω)–function f may be split into a sum of a function c + [f] ∈ l2(∂ω) with left–monogenic extension to ω + and a function (−c−)[f] ∈ l2(∂ω) with left–monogenic extension to ω − vanishing at infinity. this result is the general counterpart of the classical result in clifford analysis stating that each spherical harmonic may be split into the sum of an inner and an outer spherical monogenic (see e.g. [13]). we conclude this section by giving two examples. as already mentioned before h[1] = 1, which means that the constant function 1 belongs to h + 2 (∂ω) with c[1] = 1 ∈ h2(ω + ), c[1] = 0 in ω− and 〈1, 1〉 = area(∂ω). the function x |x| m is left–monogenic in r m \ {0} and vanishes at infinity. its restriction to ∂ω, given by x |x|m |∂ω = ζ |ζ|m , belongs to h − 2 (∂ω) with c[ ζ |ζ|m ] = 1 am ∫ ∂ω x − ζ |x − ζ|m dσζ ζ |ζ|m =    x |x|m , x ∈ ω−, 0 , x ∈ ω+. and also h[ ζ |ζ|m ] = − ζ |ζ|m , ζ ∈ ∂ω. 6 the orthogonal decomposition of l2(∂ω) as the hardy space h + 2 (∂ω) is a closed subspace of l2(∂ω), it is itself a hilbert space and it induces the following orthogonal direct sum decomposition of l2(∂ω): l2(∂ω) = h + 2 (∂ω) ⊕ h + 2 (∂ω) ⊥. the orthogonal projections p and p ⊥ on h + 2 (∂ω) and h + 2 (∂ω) ⊥ respectively are called the szegö projections. the hilbert space h + 2 (∂ω) possesses a reproducing kernel s(ζ,x), ζ ∈ ∂ω, x ∈ ω + , the so–called szegö kernel, for which 〈s(ζ,x),g(ζ)〉 = c[g](x), x ∈ ω+ for all g ∈ h + 2 (∂ω). stricly speaking the reproducing character is only obtained by identifying the function g ∈ h + 2 (∂ω) with its left–monogenic extension c[g] to ω + . note that the szegö kernel cubo 10, 2 (2008) the hilbert transform ... 93 s(ζ,x) is only defined for x ∈ ω+. it is the kernel function of the integral transform expressing the projection p of l2(∂ω) on h + 2 (∂ω): 〈s(ζ,x),f(ζ)〉 = p[f](x), f ∈ l2(∂ω), x ∈ ω +. there is an intimate relationship between the cauchy and szegö kernels as established in the following proposition. proposition 6.1. the szegö kernel is the szegö projection of the cauchy kernel on the hardy space h+2 (∂ω), i.e. for all x ∈ ω + holds s(ζ,x) = p[c(ζ,x)] = p[ 1 am ν(ζ) ζ − x |ζ − x|m ], ζ ∈ ∂ω. proof. take g ∈ h + 2 (∂ω) and x ∈ ω + . then 〈s(ζ,x),g(ζ)〉 = c[g](x) = 〈c(ζ,x),g(ζ)〉 = 〈p[c(ζ,x)],g(ζ)〉. proposition 6.2. the szegö kernel is hermitean symmetric, i.e. for all x,y ∈ ω+ it holds that s(x,y) = s(y,x). proof. take x,y ∈ ω+. then, with ζ ∈ ∂ω, 〈s(ζ,x),s(ζ,y)〉 = 〈s(ζ,x), p[c(ζ,y)]〉 = 〈s(ζ,x),c(ζ,x)〉 = 〈c(ζ,x),s(ζ,x)〉 = c[s(ζ,x)](y) the result follows in view of the identifications c[s(ζ,y)](x) ≈ s(x,y), c[s(ζ,x)](y) ≈ s(y,x). proposition 6.3. one has s(x,x) > 0 for all x ∈ ω+. proof. first observe that it is impossible that s(ζ,x) = 0 for a.e. ζ ∈ ∂ω, since for all x ∈ ω+: ∫ ∂ω s(ζ,x) ds(ζ) = 〈s(ζ,x), 1ζ〉 = c[1](x) = 1. as a consequence of proposition 6.2 one has for all x ∈ ω+: s(x,x) = s(x,x) = 〈s(ζ,x),s(ζ,x)〉 = ∫ ∂ω s(ζ,x)s(ζ,x)ds(ζ) 94 f. brackx and h. de schepper cubo 10, 2 (2008) or, as the szegö kernel is parabivector valued (i.e. the sum of a scalar and a bivector): s(x,x) = s(x,x) = ∫ ∂ω |s(ζ,x)|2ds(ζ) from which the result follows. using the szegö kernel the cauchy integral of a function f ∈ l2(∂ω) may now be expressed as follows: c[f](x) = 〈c(ζ,x),f(ζ)〉 = 〈p[c(ζ,x)], p[f]〉 + 〈p ⊥ [c(ζ,x)], p ⊥ [f]〉. (6.1) for x ∈ ω− this reduces to c[f](x) = 〈p⊥[c(ζ,x)], p⊥[f]〉 since c[p[f]] = 0 in ω−. in particular for a function g ∈ h + 2 (∂ω), and still with x ∈ ω − , we obtain c[g] = 〈c(ζ,x),g(ζ)〉 = 0 showing that for x ∈ ω− the cauchy kernel c(ζ,x) = ν(ζ)e(ζ −x) belongs to h + 2 (∂ω) ⊥ and hence coincides with p ⊥ [c(ζ,x)], while p[c(ζ,x)] = 0. this is confirmed by the fact that for x ∈ ω− the fundamental solution e(ζ − x) ∈ h + 2 (∂ω), since it may be extended left–monogenically to ω + by the function e(y − x). for x ∈ ω+ the expression (6.1) for the cauchy integral reduces to c[f](x) = 〈s(ζ,x), p[f]〉 + 〈p⊥[c(ζ,x)], p⊥[f]〉 which in general cannot be simplified further. from the previous section we know that for f ∈ l2(∂ω) the hardy projection p[f] ∈ h + 2 (∂ω) possesses a left–monogenic extension c[p[f]] ∈ h2(ω + ) with c[p[f]] = 0 in ω−, and also that p[f] = h[p[f]] = c+[p[f]], while c−[p[f]] = 0. now we search for similar properties of the other hardy projection p ⊥ [f] ∈ h + 2 (∂ω) ⊥ . in any case its cauchy integral c[p⊥[f]], though left–monogenic in ω+ and in ω−, is not an extension to ω − of p ⊥ [f]. proposition 6.4. for a function h ∈ l2(∂ω) to belong to h + 2 (∂ω) ⊥ it is necessary and sufficient that h∗[h] = −h. proof. if h ∈ h + 2 (∂ω) ⊥ then 〈g,h〉 = 〈h[g],h〉 = 0 for all g ∈ h + 2 (∂ω) and conversely. this is equivalent with 〈g,h∗[h]〉 = 0 for all g ∈ h + 2 (∂ω) and so h ∗ [h] ∈ h + 2 (∂ω) ⊥ . we also have that for all f ∈ l2(∂ω): 〈c + [f],h〉 = 1 2 〈f + h[f],h〉 = 0 cubo 10, 2 (2008) the hilbert transform ... 95 or 〈f,h〉 + 〈f,h∗[h]〉 = 0 = 〈f,h + h∗[h]〉 which means that h + h∗[h] = 0. conversely, if h∗[h] = −h, then for all g ∈ h + 2 (∂ω): 〈g,h〉 = 〈h[g],h〉 = 〈g,h∗[h]〉 = 〈g,−h〉 and hence 〈g, 2h〉 = 0, which means that h ∈ h + 2 (∂ω) ⊥ . note that for a non–zero function h ∈ h + 2 (∂ω) ⊥ there cannot exist a left–monogenic function in ω + such that its ntbv is h. however there is a one–to–one correspondence between h + 2 (∂ω) and h + 2 (∂ω) ⊥ , which is easily expressed by means of the unit normal vector ν(ξ), ξ ∈ ∂ω. proposition 6.5. a function g ∈ l2(∂ω) belongs to h + 2 (∂ω) if and only if νg ∈ h + 2 (∂ω) ⊥, and vice–versa. proof. if g ∈ h + 2 (∂ω) then h[g] = g and so h ∗ [νg] = νhν[νg] = −νh[g] = −νg, from which it follows that νg ∈ h + 2 (∂ω) ⊥ , and conversely. if h ∈ h + 2 (∂ω) then −ννh ∈ h + 2 (∂ω) ⊥ and so νh ∈ h + 2 (∂ω), and conversely. corollary 6.6. the orthogonal direct sum decomposition of l2(∂ω) takes the form l2(∂ω) = h + 2 (∂ω) ⊕ ν h + 2 (∂ω) = ν h + 2 (∂ω) ⊥ ⊕ h + 2 (∂ω) ⊥. 7 the kerzman–stein operator the hilbert operator h on l2(∂ω) is not self–adjoint. the so–called kerzman–stein operator, defined by a = 1 2 (h − h∗) measures the ”non–selfadjointness” of the hilbert transform. we will find alternative expressions for this operator at the end of this section. to this end, we first introduce four self–adjoint bounded operators on l2(∂ω), by means of the unit normal function ν on ∂ω. proposition 7.1. the operators hν,νh,νh∗ and h∗ν are self–adjoint bounded operators on l2(∂ω) moreover satisfying (i) (νh)2 = (h∗ν)2 = h∗h; (ii) (νh∗)2 = (hν)2 = hh∗; (iii) (νh)(hν) = −1 = (hν)(νh); (iv) (νh∗)(h∗ν) = −1 = (h∗ν)(νh∗); (v) 〈hνf,hνg〉 = 〈h∗f,h∗g〉 = 〈νh∗f,νh∗g〉, f,g ∈ l2(∂ω); (vi) 〈h∗νf,h∗νg〉 = 〈hf,hg〉 = 〈νhf,νhg〉, f,g ∈ l2(∂ω). 96 f. brackx and h. de schepper cubo 10, 2 (2008) note that the function ν belongs to h + 2 (∂ω) ⊥ since the constant function 1 ∈ h + 2 (∂ω). it thus follows that h∗[ν] = −ν. moreover one has (i) ||ν||2 = 〈ν,ν〉 = 〈1, 1〉 = area(∂ω); (ii) 〈ν,hν〉 = 〈h∗ν,ν〉 = −〈ν,ν〉 = −area(∂ω); (iii) ||hν||2 = 〈hν,hν〉 = 〈h∗1,h∗1〉 = ||h∗1||2. as we have seen in the previous section the unit normal vector function ν allows for an alternative form of the orthogonal decomposition of l2(∂ω). take f ∈ l2(∂ω) then also νf ∈ l2(∂ω) and we have on the one side f = p[f] + p⊥[f] so that νf = νp⊥[f] + νp[f] and on the other νf = p[νf] + p ⊥ [νf]. hence p[νf] = νp⊥[f] and p⊥[νf] = νp[f] while also p[f] = −νp⊥[νf] and p⊥[f] = −νp[νf]. this leads to f = p[f] − νp[νf] = −νp⊥[νf] + p⊥[f] νf = p[νf] + νp[f] = νp⊥[f] + p⊥[νf]. taking the hilbert transform into account we obtain h[f] = h[p[f]] + h[p⊥[f]] = p[f] + h[p⊥[f]] from which it follows that 1 2 (1 − h)[f] = 1 2 (1 − h)[p⊥[f]]. by a similar argument we find 1 2 (1 + h∗)[f] = 1 2 (1 + h∗)[p[f]]. in the operator 1 2 (1 − h) we clearly recognize the cauchy transform (−c−) for which indeed( 1 2 (1 − h) )2 = 1 2 (1 −h) and 1 2 (1 −h) 1 2 (1 + h) = 0, with 1 2 (1 + h) = c+. on grounds of analogy we put 1 2 (1 + h∗) = d− and 1 2 (−1 + h∗) = d+ defining in this way two bounded linear operators on l2(∂ω) which moreover satisfy (i) (d − ) 2 = d − ; (ii) (−d+)2 = (−d+); (iii) d+d− = d−d+ = 0. cubo 10, 2 (2008) the hilbert transform ... 97 in a similar way as the hardy space h + 2 (∂ω) is characterized by c ± , we may now characterize its orthogonal complement h + 2 (∂ω) ⊥ by means of the operators d±: h ∈ h + 2 (∂ω) ⊥ ⇐⇒ d − [h] = 0 ⇐⇒ d+[h] = −h ⇐⇒ h∗[h] = −h. notice that these newly introduced operators are the adjoints of the cauchy transforms, i.e. (c+)∗ = d − and (c − ) ∗ = d + , and for each function f ∈ l2(∂ω) 〈c + [f],d + [f]〉 = 0 and 〈c − [f],d − [f]〉 = 0 meaning that d+[f] belongs to h + 2 (∂ω) ⊥ , while d−[f] belongs to h − 2 (∂ω) ⊥ . to the authors’ knowledge no integral transform, similar to the cauchy integral, has d± as its ntbv. figure 2 the four operators c ± and d ± are really fundamental; they are the building blocks of the operators 1, h and h∗ (see figure 2): 1 = c+ − c− = d− − d+, while h = c+ + c− and h∗ = d+ + d−. 98 f. brackx and h. de schepper cubo 10, 2 (2008) figure 3 moreover coming back now to the kerzman–stein operator, we observe that a = 1 2 (h − h ∗ ) = 1 2 (1 + h) − 1 2 (1 + h ∗ ) = c + − (c + ) ∗ = c + − d − as well as a = 1 2 (h − h ∗ ) = 1 2 (−1 + h) − 1 2 (−1 + h ∗ ) = c − − (c − ) ∗ = c − − d + . in a similar way we define the operator b by c + + d + = c − + d − = 1 2 (h + h ∗ ) = b which clearly is a self–adjoint bounded operator on l2(∂ω) as well. next 1 + a = 1 2 (1 + h) + 1 2 (1 − h ∗ ) = c + − d + , −1 + a = − 1 2 (1 − h) − 1 2 (1 + h ∗ ) = c − − d − , and 1 + b = 1 2 (1 + h) + 1 2 (1 + h ∗ ) = c + + d − , −1 + b = − 1 2 (1 − h) − 1 2 (1 − h ∗ ) = c − + d + . cubo 10, 2 (2008) the hilbert transform ... 99 it follows that p(1 + a) = pc + = c + = pb, p⊥(1 + a) = −p⊥d+ = −d+ = −p⊥b since c+[f] ∈ h + 2 (∂ω) and d + [f] ∈ h + 2 (∂ω) ⊥ for all f ∈ l2(∂ω), which means that (1 + a)[f] and b[f] lie ”symmetric” w.r.t h + 2 (∂ω) (see figure 3). it should be noted that one of the above formulae relating the hardy and szegö projections to each other is the famous kerzman–stein formula p(1 + a) = c + which in fact allows for proving the boundedness of the cauchy operator c+ on l2(ω), since the kerzman–stein operator a may be expressed as an integral operator which is no longer singular: a[f](ξ) = 1 2 (h − h∗)[f](ξ) = 1 am pv ∫ ∂ω ξ − ζ |ξ − ζ|m ν(ζ)f(ζ)ds(ζ) − 1 am pv ∫ ∂ω ν(ξ) ξ − ζ |ξ − ζ|m ν(ζ)ν(ζ)f(ζ)ds(ζ) = 1 am ∫ ∂ω (ξ − ζ)ν(ζ) + ν(ξ)(ξ − ζ) |ξ − ζ|m f(ζ)ds(ζ) = 1 am ∫ ∂ω (ξ − ζ) ◦ (ν(ζ) + ν(ξ)) |ξ − ζ|m f(ζ)ds(ζ) + 1 am ∫ ∂ω (ξ − ζ) ∧ (ν(ζ) − ν(ξ)) |ξ − ζ|m f(ζ)ds(ζ). moreover we have ap = c − p − d + p = −d + p bp = c − p + d − p = d − p implying that for each function g ∈ h + 2 (∂ω), a[g] belongs to h + 2 (∂ω) ⊥ and b[g] belongs to h − 2 (∂ω) ⊥ , and similarly ap ⊥ = c + p ⊥ − d − p ⊥ = c + p ⊥ = (1 + b)p ⊥ bp ⊥ = c − p ⊥ + d − p ⊥ = c − p ⊥ = (−1 + a)p ⊥ implying that for each function h ∈ h + 2 (∂ω) ⊥ , a[h] = (1 + b)[h] belongs to h + 2 (∂ω) and b[h] = (−1 + a)[h] belongs to h − 2 (∂ω). 100 f. brackx and h. de schepper cubo 10, 2 (2008) 8 a second orthogonal decomposition of l2(∂ω) making use of the hardy space h − 2 (∂ω), introduced in section 5, a second orthogonal direct sum decomposition of l2(∂ω) is at hand: l2(∂ω) = h − 2 (∂ω) ⊕ h − 2 (∂ω) ⊥. both components may be characterized in a similar way as was done for h + 2 (∂ω) and h + 2 (∂ω) ⊥ . proposition 8.1. (i) a function g̃ belongs to h−2 (∂ω) if and only if h[g̃] = −g̃ or c + [g̃] = 0, or still c−[g̃] = −g̃. (ii) a function h̃ belongs to h−2 (∂ω) ⊥ if and only if h∗[h̃] = h̃ or d+[h̃] = 0, or still d−[h̃] = h̃. note that for a non–zero function h̃ ∈ h − 2 (∂ω) ⊥ there cannot exist a left–monogenic function in ω − , vanishing at infinity, such that its ntbv is h̃. however there again is a one–to–one correspondence, now between h − 2 (∂ω) and h − 2 (∂ω) ⊥ , established by means of the unit normal vector ν(ξ), ξ ∈ ∂ω. proposition 8.2. a function g̃ ∈ l2(∂ω) belongs to h − 2 (∂ω) if and only if νg̃ ∈ h − 2 (∂ω) ⊥, and vice–versa. proof. similar to the proof of proposition 6.5. corollary 8.3. the second orthogonal direct sum decomposition of l2(∂ω) takes the form l2(∂ω) = h − 2 (∂ω) ⊕ ν h − 2 (∂ω) = ν h − 2 (∂ω) ⊥ ⊕ h − 2 (∂ω) ⊥ . we denote the orthogonal projections on h − 2 (∂ω) and h − 2 (∂ω) ⊥ by q and q ⊥ respectively, and we put for x ∈ ω− t (ζ,x) = −q[c(ζ,x)], ζ ∈ ∂ω clearly the counterpart of the szegö kernel for the hilbert space h − 2 (∂ω). the function t (ζ,x), x ∈ ω − , possesses the reproducing property since for each g̃ ∈ h − 2 (∂ω) 〈t (ζ,x), g̃(ζ)〉 = 〈−q[c(ζ,x)], g̃(ζ)〉 = 〈c(ζ,x),−g̃(ζ)〉 = c[−g̃](x) where at the utmost right hand side the functions g̃ ∈ h − 2 (∂ω) and c[−g̃] are identified. in the same order of ideas it is also the kernel function of the integral transform expressing the projection q of l2(∂ω) on h − 2 (∂ω): 〈t (ζ,x),f(ζ)〉 = q[f](x), f ∈ l2(∂ω), x ∈ ω −. on the other hand for x ∈ ω+ we obtain for the cauchy integral of an arbitrary function f ∈ l2(∂ω), c[f] = 〈c(ζ,x),f(ζ)〉 = 〈q[c(ζ,x)], q[f(ζ)]〉 + 〈q⊥[c(ζ,x)], q⊥[f(ζ)]〉 = c[q[f]] + 〈q⊥[c(ζ,x)], q⊥[f(ζ)]〉 = 〈q⊥[c(ζ,x)], q⊥[f(ζ)]〉 cubo 10, 2 (2008) the hilbert transform ... 101 since the cauchy integral of q[f] ∈ h − 2 (∂ω) vanishes in ω + . for a function g̃ ∈ h − 2 (∂ω) this leads in particular to 0 = c[g̃] = 〈c(ζ,x), g̃(ζ)〉, x ∈ ω + which means that for x ∈ ω+ the cauchy kernel c(ζ,x) = ν(ζ)e(ζ − x) belongs to h − 2 (∂ω) ⊥ , which is confirmed by the fact that for x ∈ ω+ the fundamental solution e(ζ −x) ∈ h − 2 (∂ω), since it may be extended left–monogenically to ω − by the function e(y − x). returning to the kerzman–stein operator a and the related operator b, it also follows that q(−1 + a) = qc − = c − = qb, q⊥(−1 + a) = −q⊥d− = −d− = −q⊥b since c−[f] ∈ h − 2 (∂ω) and d − [f] ∈ h − 2 (∂ω) ⊥ for all f ∈ l2(∂ω), which means that (−1 + a)[f] and b[f] lie ”symmetric” w.r.t h − 2 (∂ω) (see again figure 3). moreover aq = c + q − d − q = −d − q bq = c + q + d + q = d + q implying that for each function g̃ ∈ h − 2 (∂ω), a[g̃] belongs to h − 2 (∂ω) ⊥ and b[g̃] belongs to h + 2 (∂ω) ⊥ , and similarly aq ⊥ = c − q ⊥ − d + q ⊥ = c − q ⊥ = −(1 − b)q ⊥ bq ⊥ = c + q ⊥ + d + q ⊥ = c + q ⊥ = (1 + a)q ⊥ implying that for each function h̃ ∈ h − 2 (∂ω) ⊥ , a[h̃] = (−1 + b)[h̃] belongs to h − 2 (∂ω) and b[h̃] = (1 + a)[h̃] belongs to h + 2 (∂ω). 9 extension of the unit normal function ν as the boundary ∂ω is assumed to be c∞–smooth, it is always possible to introduce the vector function n(x) in an open neighbourhood ∂̃ω of ∂ω such that (i) n(x) is a smooth function (ii) |n(x)| = 1 for all x ∈ ∂̃ω (iii) the restriction of n(x) to ∂ω is precisely ν(ξ), ξ ∈ ∂ω. if the closed surface ∂ω has a defining c∞–function ρ(x), i.e. ∂ω = {x : ρ(x) = 0}, while ω + = {x : ρ(x) < 0} and ω− = {x : ρ(x) > 0}, then ν(ξ) = ∂ρ(ξ) |∂ρ(ξ)| 102 f. brackx and h. de schepper cubo 10, 2 (2008) for all ξ ∈ ∂ω and the function n(x) = ∂ρ(x) |∂ρ(x)| , x ∈ ∂̃ω satisfies all above requirements. note that certainly |∂ρ(ξ)| 6= 0 for all ξ ∈ ∂ω due to the supposed smoothness of ∂ω, so that |∂ρ(x)| 6= 0 in an appropriate open neighbourhood ∂̃ω of ∂ω. for a given function f ∈ l2(∂ω) we consider in ∂̃ω the function f(x) = c[p[f]](x) − n(x)c[p[νf]](x). the first term f1(x) = c[p[f]](x) is left–monogenic in ω + while vanishing in ω − and moreover for ξ ∈ ∂ω it holds that lim ω+∋x→ξ f1(x) = p[f](ξ) lim ω−∋x→ξ f1(x) = 0. the function f2(x) = n(x) c[p[νf]](x) apparently is not left–monogenic in ∂̃ω + = ∂̃ω ∩ ω+, but still vanishes in ∂̃ω − = ∂̃ω ∩ ω−, and for ξ ∈ ∂ω it holds that lim ∂̃ω + ∋x→ξ f2(x) = ν(ξ)p[νf](ξ) = −p ⊥ [f](ξ) lim ∂̃ω − ∋x→ξ f2(x) = 0. it follows that f(x) is not left–monogenic in ∂̃ω, but for ξ ∈ ∂ω it holds that lim ∂̃ω + ∋x→ξ f(x) = p[f](ξ) − ν(ξ)p[νf](ξ) = f(ξ) lim ∂̃ω − ∋x→ξ f(x) = 0. we will now show that f(x) is harmonic in ∂̃ω\∂ω. to that end consider the operator ∂ ∗ = n ∂ n for which also holds ∂ ∗ n = −n∂ and n∂ ∗ = −∂n. for this operator, the following lemmata are easily proved. lemma 9.1. if f is sufficiently smooth then ∂f = 0 if and only if ∂∗(nf) = 0. lemma 9.2. the operator ∂∗ factorizes the laplace operator: (∂ ∗ ) 2 = −∆. we then arrive at the desired result. proposition 9.3. in ∂̃ω \ ∂ω one has: cubo 10, 2 (2008) the hilbert transform ... 103 (i) f1(x) = c[p[f]](x) is left–monogenic with f1 = 0 in ∂̃ω − ; (ii) f2(x) = n(x) c[p[νf]](x) is a null solution of ∂ ∗ with f2 = 0 in ∂̃ω − ; (iii) f(x) = f1(x) − f2(x) is harmonic with f = 0 in ∂̃ω − . proof. (i) this is a property of the cauchy integral in r m \ ∂ω. (ii) follows from lemma 9.1 since a cauchy integral is left–monogenic in r m \ ∂ω. (iii) ∆f = ∆f1 − ∆f2 = ∂ 2 f1 − ∂ ∗2 f2 = 0. we thus have proved that, given a function f ∈ l2(∂ω), there exists a function f in ∂̃ω \ ∂ω, namely f(x) = c[p[f]](x) − n(x)c[p[νf]](x), which is harmonic in ∂̃ω \ ∂ω, vanishes in ∂̃ω − and for which one has lim ∂̃ω + ∋x→ξ f(x) = f(ξ). remark 9.4. unfortunately the function n(x) is only defined in an open neighbourhood of ∂ω. solving the dirichlet problem and constructing the associated poisson kernel by means of the szegö projections and the cauchy integral, remains an open problem. this problem can be reformulated as follows. let h2(ω + ) be the hardy space of harmonic functions in ω+ with a ntbv in l2(∂ω). clearly h2(ω + ) is a closed subspace of h2(ω + ) leading to the direct sum decomposition h2(ω + ) = h2(ω + ) ⊕ h2(ω + ) ⊥ the orthogonal complement being taken in h2(ω + ). the question now is: what is h2(ω + ) ⊥? in the specific case where ω is the unit ball the answer is known (see e.g. [11]), in general it is not. finally the unit vector function n may also be used in the construction of a reproducing kernel for the hilbert space h + 2 (∂ω) ⊥ . indeed, take h ∈ h + 2 (∂ω) ⊥ , then νh ∈ h + 2 (∂ω) and by means of the szegö kernel we obtain for x ∈ ω+: c[νh](x) = 〈s(ζ,x),ν(ζ)h(ζ)〉 and hence for x ∈ ∂̃ω + : n(x) c[νh](x) = −n(x) 〈ν(ζ)s(ζ,x),h(ζ)〉 or −n(x) c[νh](x) = n(x) 〈l(ζ,x),h(ζ)〉 where we have introduced to so–called garabedian kernel for h + 2 (∂ω) ⊥ : l(ζ,x) = ν(ζ)s(ζ,x), ζ ∈ ∂ω, x ∈ ω+. this garabedian kernel is reproducing for h + 2 (∂ω) ⊥ in the sense that for h ∈ h + 2 (∂ω) ⊥ and for x ∈ ∂̃ω + , the function n(x) 〈l(ζ,x),h(ζ)〉 equals −n(x) c[νh](x) which in ∂̃ω + is a null solution 104 f. brackx and h. de schepper cubo 10, 2 (2008) of the operator ∂ ∗ and has ntbv h(ξ) for x ∈ ∂̃ω + tending to ξ ∈ ∂ω. as s(ζ,x) = p[c(ζ,x)] we also have for x ∈ ω+ and ζ ∈ ∂ω that l(ζ,x) = ν(ζ)p[c(ζ,x)] = p⊥[ν(ζ)c(ζ,x)] = p⊥[e(x − ζ)]. note that the translated fundamental solution e(x−ζ) = ν(ζ)c(ζ,x) of the dirac operator belongs to h − 2 (∂ω). 10 conclusions the central notion in this paper is the hilbert transform on the smooth boundary ∂ω of a bounded domain ω in euclidean space, which has been defined quite naturally as a part of the inner and outer ntbvs of the cauchy integral of an l2–function on ∂ω, the success of this approach being entirely due to the powerful concept of monogenic function in clifford analysis. at the same time we have devoted some attention to the concept of hardy space, to which the hilbert transform is closely related. in this we have treated the inner and the outer region determined by the considered closed hypersurface ∂ω on equal footing, enabling us to obtain new bounded linear operators on ∂ω, similar to the cauchy transforms, as well as to derive new relations in between those operators, similar to the traditional kerzman–stein formula. finally, we have also paid attention to the dirichlet problem which, in its turn, is intimately related to the concepts of hilbert transform and hardy space. we have succeeded in constructing a harmonic function in a neighborhood of the boundary ∂ω, tending to the given function on ∂ω itself, but we have not obtained an expression for the poisson kernel in this general setting. it goes without saying that the study of the triptych hilbert transform – hardy space – dirichlet problem in the particular case of the unit sphere (see [5]) has much more concrete results to offer, in particular w.r.t. this last issue. however, on the unit sphere, some interesting concepts, features and insights are inevitably lost, since the hilbert transform becomes a self–adjoint operator. received: march 2007. revised: april 2008. cubo 10, 2 (2008) the hilbert transform ... 105 references [1] s. bernstein, l. lanzani, szegö projections for hardy spaces of monogenic functions and applications, ijmms, 29 (10) 2002, 613–624. 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[31] m.v. shapiro, n.l. vasilevski, quaternionic ψ–holomorphic functions, singular integral operators and boundary value problems, parts i and ii, complex variables: theory and application, 27 1995, 14–46 and 67–96. n7 articulo 11.dvi cubo a mathematical journal vol.12, no¯ 02, (169–187). june 2010 a new solution algorithm for skip-free processes to the left claus bauer dolby laboratories, san francisco, 94103, usa email: cb@dolby.com abstract this paper proposes a new solution algorithm for steady state models describing skip-free processes to the left where each level has one phase. the computational complexity of the algorithm is independent of the number of levels of the system. if the skip parameter of the skip-free process is significantly smaller than the number of levels of the system, our algorithm numerically outperforms existing algorithms for skip-free processes. the proposed algorithm is based on a novel method for applying generalized fibonacci series to the solution of steady state models. resumen este art́ıculo propone un nuevo algoritmo solución para modelos estado-steady describiendo procesos libres-salto para la izquierda donde todo nivel tiene una fase. la complejidad computacional del algoritmo es independiente del número de niveles del sistema. si el parámetro de salto de los procesos libre-salto es significativamente pequeño respecto del número de niveles del sistema, nuestro algoritmo numérico supera algoritmos existentes para procesos libre-salto. el algoritmo propuesto se basa en un método reciente para aplicar series de fibonacci generalizados para la solución de modelos-steady. key words and phrases: skip-free processes, markovian environment, stationary distribution ams 2000 subj. class.: 60j10, 60j99 170 claus bauer cubo 12, 2 (2010) 1 introduction skip-free processes have been widely researched as a a way to study packet based communication systems. in particular, skip-free processes to the left have been applied to model the dynamics of finite buffers of switches in data networks that experience the arrival of several data packets at a time while they are only being able to forward one packet per time slot. using a discrete time model and common queueing theoretic terminology, a skip-free process to the left defines a state model where the (queue) occupancy can move down by one level in a time slot, but might move up by several levels in a time slot. the dynamics of these finite queues are commonly modeled as m/g/1/k queues where k denotes the size of the queue. similarly, buffers that experience only one packet arrival at a time, but can forward more than one packet per time slot can be modeled as skip-free processes to the right. algorithms to solve steady state models for skip-free processes were first investigated in [14], [15] by neuts. in these papers, skip-free processes are considered as a subset of a more general class of steady state models and the application of matrix-geometric methods to solve this class of steady state models is investigated. a steady state analysis that takes explicitly into account the special structure of skip-free processes is proposed in [13, chap. 13]. combining methods from [5] and [6], skip-free processes are modeled as a special case of quasi-birth-and-death processes (qbds). following the terminology introduced in [13], we define a qbd process as a skip-free process where the system can not move more than one level both downwards or upwards. several other variations of skip-free processes have been researched in [1], [3], [4], [20], [22], [23]. in this paper, we investigate homogeneous finite skip-free processes to the left, i.e., we assume a finite number of levels and we assume that all levels have the same number of phases. in particular, we assume that each level has one phase. we first give a result of primarily theoretical interest by presenting a new way to derive a closed-form solution solution of a steady state model for skip-free processes to the left. in a second step, we show that the structure of this closed-form solution can be characterized by specific linear recurrent equations. we then apply the theory of general fibonacci sequences [19] to these linear recurrent equations,. this allows us to derive the main contribution of this paper which is a new solution algorithm for skip-free processes to the left. this solution algorithm has a complexity that is independent of the number of levels of the system. previous solution algorithms for skip-free models have complexities that also depend on the number of levels of the system. prior to our work, solution algorithms for steady state models that have a complexity independent of the number of levels of the system were only known for specific classes of qbds [13, chap. 10.4], [7], [18]. these classes of qbds do not contain the qbds used to model skip-free processes in [13, chap. 13]. finally, we perform numerical experiments to compare the numerical complexity of our algorithm with the numerical complexity of previously known algorithms. our experiments show that the complexity of our method is lower than the complexity of previous algorithm if the skip parameter is small compared to the overall number of levels of the system. the skip parameter is defined as the maximum number of levels that the queue occupancy can increase in a time slot. we note without presenting any details in this paper that our methods can also be applied to derive corresponding results for skip-free processes to the right. in the next section, we provide the exact problem definition. in sec. 3 5, we prove and analyze our theorem 3.1 which gives a closed-form solution for skip-free processes to the left. in sec. 6, we show that the obtained solution can be described by a set of linear recurrent equations. based cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 171 on this observation, we apply the theory of generalized fibonacci sequences to obtain a new solution algorithm for skip-free processes to the left. in sec. 7 and 8, we compare the computational complexity of the solution algorithm with previous techniques. we summarize our results in sec. 9. 2 problem formulation we consider a discrete time markov chain xt, t ∈ n on the two-dimensional state space {(n, i) : 0 ≤ n ≤ n, 1 ≤ i ≤ p}, which we partition as ⋃ n≥1 l(n), where l(n) = {(n, 1), (n, 2), ..., (n, p)} for 0 ≤ n ≤ n. the first coordinate n is called the level, and the second coordinate j is called the phase of the state (n, j). in this paper, we only consider the case p = 1. we suppose that the discrete matrix chain is described by a transition matrix t = (ti,j )0≤i,j≤n , where ti,j is the transition probability from the level i to level j. we assume t to be of the following form: j 0 → n t = 0 i ↓ n                         e b2 b3 .. bm b0 b1 b2 .. bm−1 bm b0 b1 .. .. bm−1 bm b0 .. .. .. .. bm .. .. .. .. .. .. b0 bm−1 cm b0 bm−2 cm−1 .. .. .. .. .. .. .. .. b0 b1 c2 b0 c1                         (2.1) here the expressions bj , 0 ≤ j ≤ m, e = b0 + b1, ci = m ∑ j=i bj , 1 ≤ i ≤ m are real numbers in the open interval (0, 1). we see that the structure of the matrix t depends essentially on the skip parameter m as the parameter defines which levels can be reached from a current level in a one-step transition. any level i can be reached from all stages i − j, −1 ≤ j ≤ m − 1 if m − 1 ≤ i ≤ n − 1. by the definition of a transition matrix, we have n ∑ j=1 ti,j = 1, ∀i, 0 ≤ i ≤ n, 172 claus bauer cubo 12, 2 (2010) which implies that m ∑ l=0 bl = 1. (2.2) defining the steady state vector as π = (π0, .., πn ), we can write the steady state equation π = πt as π1b0 = π0(1 − e), (2.3) πkb0 = πk−1(1 − b1) − k ∑ l=2 πk−lbl, 2 ≤ k ≤ m − 1, (2.4) πkb0 = πk−1(1 − b1) − m ∑ l=2 πk−lbl, m ≤ k ≤ n − 1, (2.5) πn (c1 − 1) = − m−1 ∑ l=1 πn−lcl+1. (2.6) we now define new variables c = b0, (2.7) am−l = 1 − l ∑ n=0 bn, 1 ≤ l ≤ m − 1. (2.8) using eqn. (2.2), we see that a1 = bm = cm. using the definitions (2.7) and (2.8), we can rewrite the eqn. (2.3) (2.5) as follows: π1c = π ∗ 0 am−1, π ∗ 0 = π0(1 − e)a −1 m−1 (2.9) πkc = πk−1(am−1 + c) + k ∑ l=2 πk−l(am−l − am−(l−1)), 2 ≤ k ≤ m − 1, (2.10) πkc = πk−1(am−1 + c) + m−1 ∑ l=2 πk−l(am−l − am−(l−1)) − πk−ma1, (2.11) m ≤ k ≤ n − 1, πn (c1 − 1) = − m−1 ∑ l=1 πn−lcl+1. (2.12) 3 closed form solution of the steady state model in this section, we give a closed form solution of the steady state model defined via the transition matrix t in (2.1) or equivalently in (2.9) (2.12). we show the following theorem: theorem 3.1. π∗0 = ( 1 + n−1 ∑ k=1 ∑ bl k e(bm−1, .., b1) + y (c0 − 1) −1 )−1 , (3.1) πk = π ∗ 0 ∑ bl k e(bm−1, ..., b1), 1 ≤ k ≤ n − 1, (3.2) πn = π ∗ 0 y (c1 − 1) −1, (3.3) cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 173 where e(bm−1, ..., b1) = c −d ( d cm−1 cm−2 .... c1 ) m−1 ∏ l=1 acll , (3.4) cl = bl − 2bl−1 + bl−2, 3 ≤ l ≤ m − 1, (3.5) c2 = b2 − 2b1, (3.6) c1 = b1, (3.7) d = bm−1 − bm−2. (3.8) the summation ∑ bl k runs over all bl ≥ 0, 1 ≤ l ≤ m − 1 such that cl ≥ 0 for 1 ≤ l ≤ m − 1, in this summation, the variable bm−1 only takes the fixed value bm−1 = k. further, y = − m−1 ∑ l=1 cl+1 ∑ bl n−l e(bm−1, ..., b1). (3.9) remarks: we state some remarks for later use in this paper: 1. by definition, m−1 ∑ l=1 cl = d. (3.10) 2. for any fixed bl+1 over which is summed in the summation ∑ bl k , the number of bl over which is summed in ∑ bl k is limited by bl ≤ l l + 1 bl+1. (3.11) this follows from the summation condition c2 ≥ 0 and the eqn. (3.6) for l = 1. for l ≥ 2, we apply the induction principle. assuming that eqn. (3.11) holds for l, then by the summation condition cl+2 ≥ 0 2bl+1 ≤ bl + bl+2 ≤ l l + 1 bl+1 + bl+2, which implies eqn. (3.11) for l + 1. 3. the eqn. (3.11) implies that d ≥ 1. (3.12) 4 proof of theorem 3.1 we first recall a well-known identity for multinomial coefficients. for any set of strictly positive integers a1, .., ak with n = k ∑ j=1 aj , there is ( n a1 a2 ... ak ) = k ∑ j=1 ( n − 1 a1 .. (aj − 1) ... ak ) . (4.1) 174 claus bauer cubo 12, 2 (2010) for any integer x, 1 ≤ x ≤ m − 1 and a given set of variables b1, ..., bm−1 as defined in (3.4), we introduce the additional variables bxl and c x l defined as follows: bxl = bl − max(0, x − (m − 1 − l)), l ≥ 1.. (4.2) cxl = b x l − 2b x l−1 + b x l−2, 3 ≤ l ≤ m − 1, (4.3) cx2 = b x 2 − 2b x 1 , (4.4) cx1 = b x 1 , (4.5) dx = bxm−1 − b x m−2. (4.6) we now state two lemmas which we will need for the proof of theorem 3.1. lemma 4.1. for 1 ≤ x ≤ m − 1, cxl = cl − { 1 if l = m − x, 0 else . } (4.7) dx = d − 1. (4.8) proof of lemma 4.1: we prove the eqn. (4.7) by considering different ranges of the variables l, m and x : range 1: for l ≥ m − x + 1, l ≥ 3, cxl = cl − x + m − 1 − l + 2x − 2m + 2 + 2l − 2 − x + m − 1 − l + 2 = cl. range 2: for l = m − x, l ≥ 2, cxl = cl − x + m − 1 − l + 2x − 2m + 2 + 2l − 2 = cl + x − m + l − 1 = cl − 1. range 3: for l = m − x = 1 cxl = cl − 1. range 4: for l ≤ m − x − 1, l ≥ 1, cxl = cl. we now prove the eqn. (4.8): dx = bm−1 − x − (bm−2 − max(0, (x − 1)) = d − 1. for later usage, we note that the eqn. (3.10), (4.7), and (4.8) imply that for 1 ≤ x ≤ m − 1, there is m−1 ∑ l=1 cxl = d x. (4.9) lemma 4.2. e(bm−1, .., b1) = m−1 ∑ x=1, cm−x 6=0 ex(bxm−1, ..., b x 1 )am−xc −1. cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 175 proof of lemma 4.2: we first note that in the definition (3.4), if for any x there is cm−x = 0, then c−d ( d cm−1 cm−2 cm−x+1 cm−x cm−x−1.... c1 ) m−1 ∏ l=1 acll = c−d ( d cm−1 cm−2 cm−x+1 cm−x−1.... c1 ) m−1 ∏ l=1, l 6=m−x a cl l , i.e., we can neglect the contribution of the quantity cm−x. now, lemma 4.2 follows from the definition (3.4) and the eqn. (3.12), (4.1), (4.7), and (4.8). proof of theorem 3.1: first, we prove eqn. (3.2) by induction over k. for k = 1, we see bm−1 = cm−1 = d = 1, whereas bl = cl = 0 for l ≤ m − 2. thus, we obtain from eqn. (3.2) that π1 = π ∗ 0 am−1c −1 as required by eqn. (2.9). for the inductive step, we now assume that eqn. (3.2) holds for πi, 1 ≤ i ≤ k − 1, and prove it for i = k < n. in view of the eqn. (2.10) and (2.11), we see that in order to show that the eqn. (3.2) holds for πk, it is sufficient to prove the following two equations: πk = min(m−1,k) ∑ x=1 πk−xam−xc −1, (4.10) πk−1 = min(m−1,k−1) ∑ x=1 πk−1−xam−xc −1. (4.11) for the proof of the eqn. (4.10), we first consider the case m − 1 < k. replacing the probabilities πl, k − m + 1 ≤ l ≤ k, in eqn. (4.10) with the right hand side of eqn. (3.2), we obtain ∑ bl k e(bm−1, ..., b1) = m−1 ∑ x=1 am−xc −1 ∑ b̃l k−x e(b̃m−1, ..., b̃1). (4.12) applying lemma 4.2 to the left hand side of eqn. (4.12), we see that in order to prove eqn. (4.10) we have to show the following identity: m−1 ∑ x=1, cm−x 6=0 am−xc −1 ∑ bl k e(bxm−1, ..., b x 1 ) = m−1 ∑ x=1 am−xc −1 ∑ b̃l k−x e(b̃m−1, ..., b̃1). (4.13) for the proof of eqn. (4.13), it is sufficient to prove the following two claims: 1. for each integer x, 1 ≤ x ≤ m − 1, and for each set bxm−1, ..., b x 1 derived via the relation (4.2) from a set bm−1, ..., b1 over which is summed in the sum ∑ bl k , and for which cm−x 6= 0, there exists a set b̃m−1, ..., b̃1 over which is summed in the sum ∑ b̃l k−x for which holds cxl = c̃l for all l, 1 ≤ l ≤ m − 1, and dx = d̃. 2. for each integer x, 1 ≤ x ≤ m − 1, and for each set b̃m−1, ..., b̃1 over which is summed in the sum ∑ b̃l k−x , there exists a set bxm−1, ..., b x 1 derived via the relation (4.2) from a set bm−1, ..., b1 176 claus bauer cubo 12, 2 (2010) over which is summed in the sum ∑ bl k , for which cm−x 6= 0, and c̃l = c x l for all l, 1 ≤ l ≤ m − 1, and d̃ = dx. in order to show claim 2, we set eb̃l = b̃l + max(0, x − (m − 1 − l)). further, we define e c̃ l and e d̃ l via e b̃ l in the same way cl and d are defined via bl as in (4.3) (4.6). by definition, we see that (e b̃ l ) x defined as in (4.2) equals b̃l which implies that c̃l = (e c̃ l ) x and d̃l = (e d̃ l ) x. now, in order to prove claim 2, it remains to show that the set eb̃m−1, ..., e b̃ 1 belongs to the summation ∑ bl k , i.e., eb̃m−1 = k, e c̃ m−l ≥ 0 if l 6= x and ec̃m−x > 0. the first claim is obvious by the definition of e b̃ m−1 = b̃m−1 + x = k − x + x = k. we note that reversing the proof of the relation (4.7), we can show that ec̃l = c̃l + { 1 l = m − x 0 else. } (4.14) as c̃l ≥ 0, the eqn. (4.14) implies that e c̃ l ≥ 0, 1 ≤ l ≤ m − 1, l 6= m − x, and e c̃ m−x > 0, q.e.d. in order to show claim 1, we have to show that the set bxm−1, ..., b x 1 belongs to the summation ∑ bl k−x . for this purpose, we have to show that bxm−1 = k − x and c x l ≥ 0, 1 ≤ x ≤ m − 1. the first inequality follows from eqn. (4.2). the second relation follows from eqn. (4.7) and the fact that on the left-hand side of eqn. (4.13) we only sum over cm−x 6= 0. in the case m − 1 ≥ k, we argue as above and have to show that k ∑ x=1, cm−x 6=0 am−xc −1 ∑ bl k e(bxm−1, ..., b x 1 ) = k ∑ x=1 am−xc −1 ∑ b̃l k−x e(b̃m−1, ..., b̃1). (4.15) eqn. (4.15) is shown in the same way as eqn. (4.13). eqn. (4.11) is shown in the same way as eqn. (4.10). eqn. (3.3) follows from (2.12) and (3.2). eqn. (3.1) follows from (3.2), (3.3), and the eqn. n ∑ i=0 πi = 1. 5 complexity of the calculation of πk using theorem 3.1 the inequality (3.11) gives an upper bound for the number of bl over which is summed in the summation ∑ bl k . using bm−1 = k, eqn. (3.11), and the well-known relation, n ∑ k=1 km = 1 m + 1 n m+1 + o(n m), we obtain ⌊ (m−2)k m−1 ⌋ ∑ bm−2=1 ⌊ (m−3)bm−2 m−2 ⌋ ∑ bm−3=1 ..... ⌊ b2 2 ⌋ ∑ b1=1 1 = 1 (m − 1)! m−1 ∏ l=2 ( l − 1 l )l km−1 + o(km−2). cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 177 thus, the complexity of the calculation of the steady state probability of π∗0 using the relation (3.1) is ∼ 1 (m−1)! m−1 ∏ l=2 ( l−1 l )l n−1 ∑ k=1 km−1 ∼ 1 m! m−1 ∏ l=2 ( l−1 l )l n m. here, we assume that the values of the multinomial coefficients have been pre-computed as they do not depend on the actual values of the transition matrix t. once π∗0 is calculated, the calculation of πk for any 2 ≤ k ≤ n − 1 requires o(1) steps if we assume that the values of the sums ∑ bl k already determined for the calculation of π∗0 have been stored. the calculation of πn requires o(m) steps. in summary, we see that the overall computational complexity of theorem 3.1 is ∼ 1 m! m−1 ∏ l=2 ( l−1 l )l n m. for m > 3 and large values of n, this complexity is too high for any practical applications. in the next section, we will show how the complexity of theorem 3.1 can be significantly reduced. 6 a reduction of the complexity of theorem 3.1 in this section, we apply the theory of linear recurrence equations to theorem 3.1. this will allows to find a computationally very efficient way of computing the sum ∑ bl k e(bm−1, .., b1). first, we recall some facts from the theory of linear recurrent equations: lemma 6.1. we consider a sequence of real numbers xn, n ≥ 1 defined via a set of real numbers a1, .., an−1 and a set of strictly positive real numbers b1, .., bn−1 as follows: xi = ai, 1 ≤ i ≤ n − 1, (6.1) xi = n−1 ∑ k=1 bkxi−k, i ≥ n. (6.2) then, we have: xn = w ∑ u=1 fu−1 ∑ v=0 cu,vn vrnu . (6.3) the numbers ru, fu, and w are defined via the characteristic polynomial yn−1 + n−1 ∑ k=1 bky n−1−k. (6.4) ru are the roots, fu are the multiplicities of the roots ru, and w is the number of different roots of the characteristic polynomial (6.4). obviously, w ∑ u=1 fu = n − 1. the numbers cu,v are the solutions of the (n − 1)− equations ai = w ∑ u=1 fu−1 ∑ v=0 cu,vi vriu, 1 ≤ i ≤ n − 1. (6.5) proof: the proof can be found in [10]. we now apply lemma 6.1 to the eqn. (4.12). setting xk = ∑ bl k e(bm−1, .., b1) in (6.1) and (6.2), we obtain from lemma 6.1: 178 claus bauer cubo 12, 2 (2010) lemma 6.2. ∑ bl k e(bm−1, .., b1) = w ∑ u=1 gu−1 ∑ a=0 tu,ak ahku, here, the numbers gu denote the respective multiplicities of the w different roots hu of the characteristic polynomial xm−1 − m−1 ∑ u=1 am−uc −1xm−1−u. (6.6) the numbers tu,h are the solutions of the (m − 1)linear equations ∑ bl k e(bm−1, .., b1) = m−1 ∑ u gu−1 ∑ h=0 tu,hv hhvu, 1 ≤ v ≤ m − 1. we now use lemma 6.2 to simplify theorem 3.1 as follows: theorem 6.1. π∗0 = ( 1 + n−1 ∑ k=1 fk + y ∗(1 − c0) )−1 , (6.7) πk = π ∗ 0 fk, 0 ≤ k ≤ n − 1, (6.8) πn = π ∗ 0 y ∗(c1 − 1) −1, (6.9) where fk = w ∑ u=1 gu−1 ∑ a=0 tu,ak ahku. (6.10) the numbers w, hu, gu, tu,h are defined as in lemma 6.2, and y ∗ = − m−1 ∑ l=1 cl+1fn−l. proof: the eqn. (6.7) (6.10) follow directly from theorem 3.1 and lemma 6.2. 7 the complexity of theorem 6.1 7.1 preliminaries in this section, we investigate the computational complexity of theorem 6.1. we see from the formulation of theorem 6.1 that the complexity of the calculation of a steady state probability πk mainly derives from the calculation of the sum n−1 ∑ k=1 fk, the solution of the (m − 1) eqn. (6.5), and the calculation of the roots of the characteristic polynomial (6.6). it is well-known that the solution of the system of equations (6.5) using gaussian elimination requires o(m3) operations. we consider the complexity of the other two operations separately in the next two sections. cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 179 7.2 the calculation of the sum n−1 ∑ k=1 fk we see from eqn. (6.10) that the sum n−1 ∑ k=1 fk can be rewritten as follows: n−1 ∑ k=1 fk = w ∑ u=1 gu−1 ∑ a=0 tu,a n−1 ∑ k=1 kahku := w ∑ u=1 gu−1 ∑ a=0 tu,ahn−1,a,hu , (7.1) where hn,q,x := n ∑ k=1 kqxk. for the calculation of hn,q,x we prove the following lemma: lemma 7.1. hn,q,1 = q+1 ∑ k=1 (−1)q−k+1beq−k+1 p + 1 ( p + 1 k ) n k, (7.2) hn,0,x = x xn − 1 x − 1 , (7.3) hn,q,x = x (n + 1)qxn + (n + 1)q − 2 x − 1 − x x − 1 q−1 ∑ b=0 hn,b,x ( q b ) , x 6= 1, q > 1. (7.4) in eqn. (7.2), bek denotes the k − th bernoulli number [2]. proof: the relation (7.2) is explained in [2] and the relation (7.3) is a known formula for geometric sums. for the proof of eqn. (7.4), we will use the technique of partial summation defined in [17] as follows: for any complex series ak, bk, m < n ≤ n, there is ∑ m 4 no general exact solutions exists. thus, the solution of polynomials of higher degree requires the application of approximative methods. an overview of these methods is given in [8]. among these methods, laguerre’s method has been widely researched [16]. whereas its convergent behavior is well understood, the speed of its convergence is not well described in the literature. therefore, we performed numerical experiments in order to understand the computational complexity of the calculation of the roots of the characteristic polynomial (6.6). for each degree m, we evaluated 10000 polynomials of degree m, and for each polynomial we used a random generator to generate the coefficients of the polynomial. for each polynomial, we measured the execution time s(m) measured in milliseconds needed to determine all roots of the polynomial on a pentium iv, 2.4 ghz pc. the graph in fig. 1 shows the logarithm to the base 10 of s(m) as a function of the logarithm to the base 10 of the degree m of the polynomial. an least mean square analysis of the simulation data shows that log(s(m)) can be described as a linear function of log m as follows: log(s(m)) = 1.99 log m + c for some constant c. thus, the execution time s(m) can be approximately described as a second order polynomial of the polynomial degree m. 2 2.5 3 3.5 4 4.5 5 5.5 6 0 1 2 3 4 5 6 7 logarithm of degree m of polynomial lo ga rit hm o f e xe cu tio n tim e in m s figure 1: speed of convergence of the calculation of the roots of the characteristic polynomial 7.4 combined complexity of theorem 6.1 the combinations of the results of sec. (7.1) (7.3) shows that the overall computational complexity required to calculate a specific steady state probability πk using theorem 6.1 is o(m 3). 8 comparison of theorem 6.1 with previous work in this section, we compare the computational complexity of theorem 6.1 with known methods to solve steady state models as defined via the transition matrix (2.1). first, we present a well-known recursive way of solving the system described by the matrix (2.1). 182 claus bauer cubo 12, 2 (2010) then, we follow an idea from [6] and model the markov process defined via (2.1) as a qbd process with u := ⌈ n +1 m−1 ⌉ levels. we discuss two known methods to solve qbd models due to gaver et. al [5], [13, chap. 13] and ye and li [21], respectively. finally, we compare the computational complexity of these approaches with the complexity of theorem 6.1. 8.1 recursive calculation the equations (2.3) (2.6) allow a recursive calculation of the steady state probabilities πk, 0 ≤ k ≤ n. we see from these equations that each πk can be expressed as a linear function of all πl, 0 ≤ l ≤ k − 1. thus, using a recursive argument each πl can be expressed as the product of π0 and a real number kl. as n ∑ l=0 kl = π −1 0 , we can can calculate all steady state probabilities πk. these calculations require o(n 2) calculations. we note that a similar approach can be used to derive a closed form solution for πk. in particular, we consider the unconstrained random walk corresponding to (2.1) without a reflecting barrier on the right, i.e., with an infinite number of levels. in order to define this random walk formally, we generalize the transition probabilities bn defined in (2.1) to dn = { bn, if 0 ≤ n ≤ m, 0, n ∈ z\[0, m]. } . (8.6) we define the transition probabilities from state i to state j of the unconstrained random walk as pij = { d0 + d1, if i = j = 0, dj−i+1, else. } . we denote by {mj : j ≥ 0} the invariant measure of this markov chain and set m0 = 1. then, mj = ∑ i≥0 pij mi. using the definition (8.6), we see m1 = 1 − d0 − d1 d0 , d0mj+1 = mj − j ∑ i=0 midj−i+1 (j ≥ 1). (8.7) the first n columns of the transition matrix defined by (8.6) are identical to the first n columns of the transition matrix t defined in (2.1). thus, there exists a constant k such that πj = kmj if j < n. there is πn = k m−1 ∑ i=1 cm−i+1mn−m+i + πn c1, and πn c1 = πn (1 − d0) = πn − d0 + d0k n−1 ∑ j=0 mj , (8.8) which implies k = d0 d0 n−1 ∑ i=0 mi + m−1 ∑ i=1 cm−i+1mn−m+i . cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 183 let d(s) := ∑ k≥0 skdk be the generating function of the series dk. using standard manipulations with generating functions yields m (s) := ∑ k≥0 mks k = d0 1 − s b(s) − s , (8.9) valid in the disc |s| < q, where q is the least positive solution of d(s) = s. as dn = 0 for n > m, b(s) is a polynomial and therefore m (s) is meromorphic function with a finite number of poles. using a partial fraction expansion of m (s) and picking the coefficients of sk on the right hand side of (8.9) gives a representation of the mk. the calculation of the coefficients uses the recursive relations (8.7) and relation (8.8). thus, it has a complexity of o(n 2). 8.2 modeling of the markov process as a qbd process with u := ⌈ n +1 m−1 ⌉ levels first, we describe how the matrix t can be redefined as a block matrix that describes a qbd process. for the sake of simplicity, we first assume that n + 1 = u(m − 1) for some integer u. then, we can rewrite t as defined in (2.1) as a u × u block matrix as follows: t =                    p1,1 a0 0 0 .. .. .. .. 0 a2 a1 a0 .. .. .. .. .. 0 0 a2 a1 .. .. .. .. .. .. 0 .. .. .. .. .. .. .. .. 0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. a1 a0 0 .. .. .. .. .. .. a2 a1 a0 0 0 .. .. .. .. .. a2 pu,u                    , (8.10) a0 =                    bm+1 0 0 .. .. .. .. .. .. bm bm+1 0 .. .. .. .. .. .. bm−1 bm bm+1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. b5 .. .. .. .. .. bm+1 0 0 b4 b5 .. .. .. .. bm bm+1 0 b3 b4 .. .. .. .. .. bm bm+1                    , 184 claus bauer cubo 12, 2 (2010) a1 =                    b2 b3 b4 .. .. .. .. .. bm b1 b2 b3 .. .. .. .. .. bm−1 0 b1 b2 .. .. .. .. .. bm−2 0 0 b1 b2 .. .. .. .. bm−3 .. .. .. b1 b2 .. .. .. .. .. .. .. .. b1 b2 .. .. .. .. .. .. .. .. b1 b2 .. .. .. .. .. .. .. 0 b1 b2 b3 0 0 0 0 0 0 0 b1 b2                    , (8.11) a2 =                    .. .. .. .. .. .. .. 0 b1 .. .. .. .. .. .. .. 0 0 .. .. .. .. .. .. .. .. 0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..                    , p1,1 =                    e b3 b4 .. .. .. .. .. bm b1 b2 .. .. .. .. .. .. bm−1 0 b1 b2 .. .. .. .. .. bm−2 0 0 b1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. b1 b2                    , (8.12) pu,u =                    b2 b3 b4 .. .. .. .. bm−1 cm−1 b1 b2 b3 .. .. .. .. bm−2 cm−2 0 b1 b2 .. .. .. .. bm−3 cm−3 0 0 b1 b2 .. .. .. bm−4 cm−4 .. .. .. b1 b2 .. .. .. .. .. .. .. .. b1 b2 .. .. .. .. .. .. .. .. b1 b2 .. .. .. .. .. .. .. 0 b1 b2 c2 0 0 0 0 0 0 0 b1 c1                    . cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 185 table 1: comparison of complexity of different solution algorithms method complexity recursive calculation o(n 2) linear level reduction o(n m2) method of folding o(m3log n m ) theorem 6.1 o(m3) the matrices a0, a1, a2, p1,1 and pu,u are (m − 1) × (m − 1) matrices. in the case that n + 1 is not an integer multiple of m − 1, i.e, n + 1 = u(m − 1) − c, 0 < c < m − 1, the presentation (8.10) changes as follow. the matrix pu,u is replaced by the matrix (m − 1 − c) × m matrix p ∗ n,n which consists of the m− 1 − c right columns of pn,n. similarly, the matrix a0 in right-most column of (8.10) is replaced by the (m − c) × m matrix a∗0 which consists of the m − c right columns of a2. gaver et. al [5], [13, chap. 13] propose the linear level reduction method to solve qbd systems. applied to (8.10), the complexity of this solution algorithm is o(um3) = o(n m2). ye and li [21] propose the method of folding. the application of this algorithm to solve (8.10) requires o(m3log2 u) = o(m3log2 n m ) computational steps. 8.3 comparison of complexities we see from table 1 that the complexity of theorem 6.1 is lower than the complexity of the recursive calculations if m = o(n 2/3). its complexity is lower than the complexity of the linear reduction based method if m = o(n ). theorem 6.1 also outperforms the method of folding if n is sufficiently larger than m. however, the performance gain is only of the order (log n m )−1, whereas theorem 6.1 achieves a performance gain of the order m/n compared to the linear level reduction method. we note that in contrast to the recursive calculation, the linear level reduction approach and the folding method, the number of computations required to calculate a specific steady state probability πk using theorem 6.1 does not depend on the number of phases n + 1. in summary, we see that theorem 6.1 outperforms the other approaches presented in this section if n is significantly larger than m. thus, the applicability of theorem 6.1 to the efficient solution of markov models for skip-free processes depends on the specific choice of the parameters n and m describing the skip-free process. 9 conclusions this paper proposes a new solution algorithm for skip-free processes. the complexity of the algorithm is independent of the number of levels of the system. in consequence, it numerically outperforms previous algorithms if the skip parameter is small compared to the number of system levels. it is based on a novel method for deriving a closed-form solution for skip-free processes and analyzing this solution using fibonacci sequences. 186 claus bauer cubo 12, 2 (2010) received: july 2008. revised: april 2009. references [1] a. adhikari, skip free processes. ph.d thesis, berkeley, 1986. [2] c. b. boyer, pascal’s formula for the sums of powers of the integers. scripta math. 9, 237-244, 1943. [3] p.j. brockwell, the extinction time of a birth, death and catastrophe process and of a related diffusion model. advances applied probability, vol. 17, 1985, 17 42. [4] m.f. chen, single birth processes, chinese ann. math. ser. a, 20:77-82, 1999. [5] d.p. gaver, p.a. jacobs and g. latouche, finite birth-and-death models in randomly changing environments. advances in applied probability, 16:715-731, 1984. [6] w.k grassmann and d.a. stanford, matrix analytic methods. in: computational probability, wk grassmann (ed.), kluwer, 2000, pp 153-202. [7] l. guen and a.m makowski, matrix-geometric solution for finite capacity queues with phasetype distributions. performance’87, edited by courtois, p.j.; latouche, g.; amsterdam, 1987, p. 269 282. [8] e. hansen, m. patrick and j. rusnack, some modifications of laguere’s method. bit, 17(1977), 409 -417. [9] t. w. hungerford, algebra. springer publishing house, 1974. [10] w.g. kelley and a.c. peterson, difference equations, an introduction with applications, academic press, inc. 1991. [11] p.a. jacobs, d.p. gaver and g. latouche, finite markov chain models skip free in one direction. naval research logistics quarterly, 31, 1984, pp. 571-588. [12] e. kuntz, algebra., vieweg verlag, braunschweig, 1991. [13] g. latouche and v. ramaswami, introduction to matrix analytic methods in stochastic modeling. society for industrial and applied mathematics, 1999. [14] m.f. neuts, matrix-geometric solutions in stochastic models. an algorithmic approach. the john hopkins university press, baltimore, md, 1981. [15] m.f. neuts, structured stochastic matrices of m/g/1 type and their applications. marcel dekker, new york, 1989. [16] y.v. pan, solving a polynomial equation : some history and recent progress. siam rev., vol. 39, no. 2, pp. 187 -220, june 1997. [17] k. prachar, primzahlverteilung., berlin, heidelberg, new york, springer verlag, 1978. cubo 12, 2 (2010) a new solution algorithm for skip-free processes to the left 187 [18] w.j. stewart, on the use of numerical methods for atm model. modeling and performance evaluation of atm technology, edited by perros, h.; pujolle, g.; takahashi, p. 375 396. [19] d.a. wolfram, solving generalized fibonacci recurrences. the fibonacci quarterly 36.2. may 1998, 129-45. [20] s.j. yan and m.f. chen, multidimensional q-processes. chin. ann. math. ser. a,7:90-110, 1986 [21] j. ye and s.q. li, folding algorithm: a computational method for finite qbd processes with level dependent transitions. ieee transactions on communications., 42:625-639, 1994. [22] j.k. zhang, generalized birth-death processes. acta mathematica sinica, vol. 46, 1984, 241 259. [23] y.h. zhang, strong ergodicity for single-birth processes. journal of applied probability, vol. 38, 2001, 207 277. cubo a mathematical journal vol.11, no¯ 01, (101–121). march 2009 wrap groups of fiber bundles and their structure s.v. ludkovsky department of applied mathematics, moscow state technical university mirea, av. vernadsky 78, moscow 119454, russia. email: sludkowski@mail.ru abstract this article is devoted to the investigation of wrap groups of connected fiber bundles. these groups are constructed with mild conditions on fibers. their examples are given. it is shown, that these groups exist and for differentiable fibers have the infinite dimensional lie groups structure, that is, they are continuous or differentiable manifolds and the composition (f,g) 7→ f−1g is continuous or differentiable depending on a class of smoothness of groups. moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. nevertheless, it is proved that these groups does not necessarily satisfy the campbell-hausdorff formula even locally. resumen este artículo es dedicado a la investigación de grupos wrap de fibrados conexos. estos grupos son construidos con condiciones blandas sobre las fibras, ejemplos son dados. es demostrado que estos grupos existen y para fibras diferenciables tienen una estructura de grupo de lie infinito dimensional, es decir, son variedades continuas o diferenciables y la composición (f,g) 7→ f−1g es continua o diferenciable dependiendo de la clase de suavidad de los grupos. además es demostrado que en el caso de variedades real, compleja, cuaternion y octonion esos grupos tienen una estructura de variedad real, compleja, cuaternion o octonion respectivamente. también es probado que estos grupos no necesariamente satisfacen la fórmula de campbell-hausdorff incluso localmente. 102 s.v. ludkovsky cubo 11, 1 (2009) 1 introduction. wrap groups of fiber bundles considered in this paper are constructed with the help of families of mappings from a fiber bundle with a marked point into another fiber bundle with a marked point over the fields r, c, h and the octonion algebra o. conditions on fibers supplied with parallel transport structures are rather mild here. therefore, they generalize geometric loop groups of circles, spheres and fibers with parallel transport structures over them. a loop interpretation is lost in their generalizations, so they are called here wrap groups. this paper continues previous works of the author on this theme, where generalized loop groups of manifolds over r, c and h were investigated, but neither for fibers nor over octonions [15, 23, 21, 22]. loop groups of circles were first introduced by lefshetz in 1930-th and then their construction was reconsidered by milnor in 1950-th. lefshetz has used the c0-uniformity on families of continuous mappings, which led to the necessity of combining his construction with the structure of a free group with the help of words. later on milnor has used the sobolev’s h1-uniformity, that permitted to introduce group structure more naturally [27]. iterations of these constructions produce iterated loop groups of spheres. then their constructions were generalized for fibers over circles and spheres with parallel transport structures over r or c [4]. wrap groups of quaternion and octonion fibers as well as for wider classes of fibers over r or c are defined and investigated here for the first time. holomorphic functions of quaternion and octonion variables were investigated in [19, 20, 17]. there specific definition of super-differentiability was considered, because the quaternion skew field has the graded algebra structure. this definition of super-differentiability does not impose the condition of right or left super-linearity of a super-differential, since it leads to narrow class of functions. there are some articles on quaternion manifolds, but practically they undermine a complex manifold with additional quaternion structure of its tangent space (see, for example, [28, 39] and references therein). therefore, quaternion manifolds as they are defined below were not considered earlier by others authors (see also [17]). applications of quaternions in mathematics and physics can be found in [6, 9, 10, 14]. in this article wrap groups of different classes of smoothness are considered. henceforth, we consider not only orientable manifolds m and n, but also nonorientable manifolds. in particular, geometric loop groups have important applications in modern physical theories (see [11, 24] and references therein). groups of loops are also intensively used in gauge theory. wrap groups defined below with the help of families of mappings from a manifold m into another manifold n with a dimension dim(m) > 1 can be used in the membrane theory which is the generalization of the string (superstring) theory. section 2 is devoted to the definitions of topological and manifold structures of wrap groups. the existence of these groups is proved and that they are infinite dimensional lie groups not satisfying even locally the campbell-hausdorff formula (see theorems 3, 6, 12, corollaries 5, 8, 9 cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 103 and examples 10). in the cases of complex, quaternion and octonion manifolds it is proved that they have structures of complex, quaternion and octonion manifolds respectively. all main results of this paper are obtained for the first time. 2 wrap groups of fibers. to avoid misunderstandings we first give our definitions and notations. 1.1. note. denote by ar the cayley-dickson algebra such that a0 = r, a1 = c, a2 = h is the quaternion skew field, a3 = o is the octonion algebra. henceforth we consider only 0 ≤ r ≤ 3. 1.2. definition. a canonical closed subset q of the euclidean space x = rn or of the standard separable hilbert space x = l2(r) over r is called a quadrant if it can be given by the condition q := {x ∈ x : qj(x) ≥ 0}, where (qj : j ∈ λq) are linearly independent elements of the topologically adjoint space x∗. here λq ⊂ n (with card(λq) = k ≤ n when x = r n) and k is called the index of q. if x ∈ q and exactly j of the qi’s satisfy qi(x) = 0 then x is called a corner of index j. if x is an additive group and also left and right module over h or o with the corresponding associativity or alternativity respectively and distributivity laws then it is called the vector space over h or o correspondingly. in particular l2(ar) consisting of all sequences x = {xn ∈ ar : n ∈ n} with the finite norm ‖x‖ < ∞ and scalar product (x,y) := ∑∞ n=1 xny ∗ n with ‖x‖ := (x,x)1/2 is called the hilbert space (of separable type) over ar, where z ∗ denotes the conjugated cayley-dickson number, zz∗ =: |z|2, z ∈ ar. since the unitary space x = a n r or the separable hilbert space l2(ar) over ar while considered over the field r (real shadow) is isomorphic with xr := r 2 r n or l2(r), then the above definition also describes quadrants in an r and l2(ar). in the latter case we also consider generalized quadrants as canonical closed subsets which can be given by q := {x ∈ xr : qj(x + aj) ≥ 0,aj ∈ xr,j ∈ λq}, where λq ⊂ n (card(λq) = k ∈ n when dimrxr < ∞). 1.2.2. definition. a differentiable mapping f : u → u′ is called a diffeomorphism if (i) f is bijective and there exist continuous mappings f′ and (f−1)′, where u and u′ are interiors of quadrants q and q′ in x. in the ar case with 1 ≤ r ≤ 3 we consider bounded generalized quadrants q and q ′ in an r or l2(ar) such that they are domains with piecewise c ∞-boundaries. we impose additional conditions on the diffeomorphism f in the 1 ≤ r ≤ 3 case: (ii) ∂̄f = 0 on u, (iii) f and all its strong (frechét) differentials (as multi-linear operators) are bounded on u, where ∂f and ∂̄f are differential (1, 0) and (0, 1) forms respectively, d = ∂ + ∂̄ is an exterior derivative, for 2 ≤ r ≤ 3 ∂ corresponds to super-differentiation by z and ∂̃ = ∂̄ corresponds to 104 s.v. ludkovsky cubo 11, 1 (2009) super-differentiation by z̃ := z∗, z ∈ u (see [19, 20]). the cauchy-riemann condition (ii) means that f on u is the ar-holomorphic mapping. 1.2.3. definition and notation. an ar-manifold m with corners is defined in the usual way: it is a metric separable space modelled on x = an r or x = l2(ar) respectively and is supposed to be of class c∞, 0 ≤ r ≤ 3. charts on m are denoted (ul,ul,ql), that is, ul : ul → ul(ul) ⊂ ql is a c∞-diffeomorphism for each l, ul is open in m, ul ◦ uj −1 is biholomorphic for 1 ≤ r ≤ 3 from the domain uj(ul ∩ uj) 6= ∅ onto ul(ul ∩ uj) (that is, uj ◦ u −1 l and ul ◦ u −1 j are holomorphic and bijective) and ul ◦ u −1 j satisfy conditions (i − iii) from §1.2.2, ⋃ j uj = m. a point x ∈ m is called a corner of index j if there exists a chart (u,u,q) of m with x ∈ u and u(x) is of index indm(x) = j in u(u) ⊂ q. a set of all corners of index j ≥ 1 is called a border ∂m of m, x is called an inner point of m if indm(x) = 0, so ∂m = ⋃ j≥1 ∂ jm, where ∂jm := {x ∈ m : indm(x) = j}. for a real manifold with corners on the connecting mappings ul ◦ u −1 j ∈ c∞ of real charts only condition 1.2.2(i) is imposed. 1.2.4. terminology. in an ar-manifold n there exists an hermitian metric, which in each analytic system of coordinates is the following ∑ n j,k=1 hj,kdzjdz̄k, where (hj,k) is a positive definite hermitian matrix with coefficients of the class c∞, hj,k = hj,k(z) ∈ ar, z are local coordinates in n. as real manifolds we shall consider riemann manifolds. in accordance with the definition above for internal points of n it is supposed that they can belong only to interiors of charts, but for boundary points ∂n it may happen that x ∈ ∂n belongs to boundaries of several charts. it is convenient to choose an atlas such that ind(x) is the same for all charts containing this x. 1.3.1. remark. if m is a metrizable space and k = km is a closed subset in m of codimension codimr n ≥ 2 such that m \ k = m1 is a manifold with corners over ar, then we call m a pseudo-manifold over ar, where km is a critical subset. two pseudo-manifolds b and c are called diffeomorphic, if b \ kb is diffeomorphic with c \ kc as for manifolds with corners (see also [4, 26]). take on m a borel σ-additive measure ν such that ν on m \ k coincides with the riemann volume element and ν(k) = 0, since the real shadow of m1 has it. the uniform space ht p (m1,n) of all continuous piecewise h t sobolev mappings from m1 into n is introduced in the standard way [21, 22], which induces ht p (m,n) the uniform space of continuous piecewise ht sobolev mappings on m, since ν(k) = 0, where r ∋ t ≥ [m/2] + 1, m denotes the dimension of m over r, [k] denotes the integer part of k ∈ r, [k] ≤ k. then put h∞ p (m,n) = ⋂ t>m ht p (m,n) with the corresponding uniformity. for manifolds over ar with 1 ≤ r ≤ 3 take as h t p (m,n) the completion of the family of cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 105 all continuous piecewise ar-holomorphic mappings from m into n relative to the h t p uniformity, where [m/2] + 1 ≤ t ≤ ∞. henceforth we consider pseudo-manifolds with connecting mappings of charts continuous in m and ht ′ p in m \ km for 0 ≤ r ≤ 3, where t ′ ≥ t. 1.3.2. note. since the octonion algebra o is non-associative, we consider a non-associative subgroup g of the family matq(o) of all square q × q matrices with entries in o. more generally g is a group which has a ht p manifold structure over ar and group’s operations are h t p mappings. the g may be non-associative for r = 3, but g is supposed to be alternative, that is, (aa)b = a(ab) and a(a−1b) = b for each a,b ∈ g. as a generalization of pseudo-manifolds there is used the following (over r and c see [4, 34]). suppose that m is a hausdorff topological space of covering dimension dim m = m supplied with a family {h : u → m} of the so called plots h which are continuous maps satisfying conditions (d1 − d4): (d1) each plot has as a domain a convex subset u in an r , n ∈ n; (d2) if h : u → m is a plot, v is a convex subset in al r and g : v → u is an ht p mapping, then h ◦ g is also a plot, where t ≥ [m/2] + 1; (d3) every constant map from a convex set u in an r into m is a plot; (d4) if u is a convex set in an r and {uj : j ∈ j} is a covering of u by convex sets in a n r , each uj is open in u, h : u → m is such that each its restriction h|uj is a plot, then h is a plot. then m is called an ht p -differentiable space. a mapping f : m → n between two ht p -differentiable spaces is called differentiable if it continuous and for each plot h : u → m the composition f ◦ h : u → n is a plot of n. a topological group g is called an ht p -differentiable group if its group operations are ht p -differentiable mappings. let e, n, f be ht ′ p -pseudo-manifolds or ht ′ p -differentiable spaces over ar, let also g be an ht ′ p group over ar, t ≤ t ′ ≤ ∞. a fiber bundle e(n,f,g,π, ψ) with a fiber space e, a base space n, a typical fiber f and a structural group g over ar, a projection π : e → n and an atlas ψ is defined in the standard way [4, 26, 35] with the condition, that transition functions are of ht ′ p class such that for r = 3 a structure group may be non-associative, but alternative. local trivializations φj ◦ π ◦ ψ −1 k : vk(e) → vj(n) induce the h t ′ p -uniformity in the family w of all principal ht ′ p -fiber bundles e(n,g,π, ψ), where vk(e) = ψk(uk(e)) ⊂ x 2 (g), vj(n) = φj(uj(n)) ⊂ x(n), where x(g) and x(n) are ar-vector spaces on which g and n are modelled, (uk(e), ψk) and (uj(n),φj) are charts of atlases of e and n, ψk = ψ e k , φj = φ n j . if g = f and g acts on itself by left shifts, then a fiber bundle is called the principal fiber bundle and is denoted by e(n,g,π, ψ). as a particular case there may be g = a∗ r , where a∗ r denotes the multiplicative group ar \ {0}. if g = f = {e}, then e reduces to n. 2. definitions. let m be a connected ht p -pseudo-manifold over ar, 0 ≤ r ≤ 3 satisfying the following conditions: 106 s.v. ludkovsky cubo 11, 1 (2009) (i) it is compact; (ii) m is a union of two closed subsets over ar a1 and a2, which are pseudo-manifolds and which are canonical closed subsets in m with a1 ∩ a2 = ∂a1 ∩ ∂a2 =: a3 and a codimension over r of a3 in m is codimra3 = 1, also a3 is a pseudo-manifold; (iii) a finite set of marked points s0,1, ...,s0,k is in ∂a1 ∩ ∂a2, moreover, ∂aj are arcwise connected j = 1, 2; (iv) a1\∂a1 and a2\∂a2 are h t p -diffeomorphic with m\[{s0,1, ...,s0,k}∪(a3\int(∂a1∩∂a2))] by mappings fj(z), where j = 1 or j = 2, ∞ ≥ t ≥ [m/2] + 1, m = dimrm such that h t ⊂ c0 due to the sobolev embedding theorem [25], where the interior int(∂a1 ∩ ∂a2) is taken in ∂a1 ∪ ∂a2. instead of (iv) we consider also the case (iv′) m, a1 and a2 are such that (aj \ ∂aj) ∪ {s0,1, ...,s0,k} are c0([0, 1],ht p (aj,aj))-retractable on x0,q ∩ aj, where x0,q is a closed arcwise connected subset in m, j = 1 or j = 2, s0,q ∈ x0,q, x0,q ⊂ km, q = 1, ...,k, codimr km ≥ 2. let m̂ be a compact connected ht p -pseudo-manifold which is a canonical closed subset in al r with a boundary ∂m̂ and marked points {ŝ0,q ∈ ∂m̂ : q = 1, ..., 2k} and an h t p -mapping ξ : m̂ → m such that (v) ξ is surjective and bijective from m̂ \ ∂m̂ onto m \ ξ(∂m̂) open in m, ξ(ŝ0,q) = ξ(ŝ0,k+q)s0,q for each q = 1, ...,k, also ∂m ⊂ ξ(∂m̂). a parallel transport structure on a ht ′ p -differentiable principal g-bundle e(n,g,π, ψ) with arcwise connected e and g for ht p -pseudo-manifolds m and m̂ as above over the same ar with t′ ≥ t + 1 assigns to each ht p mapping γ from m into n and points u1, ...,uk ∈ ey0 , where y0 is a marked point in n, y0 = γ(s0,q), q = 1, ...,k, a unique h t p mapping pγ̂,u : m̂ → e satisfying conditions (p1 − p5): (p1) take γ̂ : m̂ → n such that γ̂ = γ ◦ ξ, then pγ̂,u(ŝ0,q) = uq for each q = 1, ...,k and π ◦ pγ̂,u = γ̂ (p2) pγ̂,u is the h t p -mapping by γ and u; (p3) for each x ∈ m̂ and every φ ∈ difht p (m̂,{ŝ0,1, ..., ŝ0,2k}) there is the equality pγ̂,u(φ(x)) = pγ̂◦φ,u(x), where difh t p (m̂,{ŝ0,1, ..., ŝ0,2k}) denotes the group of all h t p homeomorphisms of m̂ preserving marked points φ(ŝ0,q) = ŝ0,q for each q = 1, ..., 2k; (p4) pγ̂,u is g-equivariant, which means that pγ̂,uz(x) = pγ̂,u(x)z for every x ∈ m̂ and each z ∈ g; (p5) if u is an open neighborhood of ŝ0,q in m̂ and γ̂0, γ̂1 : u → n are h t ′ p -mappings such that γ̂0(ŝ0,q) = γ̂1(ŝ0,q) = vq and tangent spaces, which are vector manifolds over ar, for γ0 and γ1 at vq are the same, then the tangent spaces of pγ̂0,u and pγ̂1,u at uq are the same, where q = 1, ...,k, u = (u1, ...,uk). cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 107 two ht ′ p -differentiable principal g-bundles e1 and e2 with parallel transport structures (e1, p1) and (e2, p2) are called isomorphic, if there exists an isomorphism h : e1 → e2 such that p2,γ̂,u(x) = h(p1,γ̂,h−1(u)(x)) for each h t p -mapping γ : m → n and uq ∈ (e2)y0 , where q = 1, ...,k, h−1(u) = (h−1(u1), ...,h −1 (uk)). let (sme)t,h : (s m,{s0,q :q=1,...,k}e; n,g, p)t,h be a set of h t p -closures of isomorphism classes of ht p principal g fiber bundles with parallel transport structure. 3. theorems. 1. the uniform space (sme)t,h from §2 has the structure of a topological alternative monoid with a unit and with a cancelation property and the multiplication operation of hl p class with l = t′ −t (l = ∞ for t′ = ∞). if n and g are separable, then (sme)t,h is separable. if n and g are complete, then (sme)t,h is complete. 2. if g is associative, then (sme)t,h is associative. if g is commutative, then (s me)t,h is commutative. if g is a lie group, then (sme)t,h is a lie monoid. 3. the (sme)t,h is non-discrete, locally connected and infinite dimensional for dimr(n × g) > 1. proof. if there is a homomorphism θ : g → f of ht ′ p -differentiable groups, then there exists an induced principal f fiber bundle (e×θf)(n,f,πθ, ψθ) with the total space (e×θf)(e×f)/y, where y is the equivalence relation such that (vg,f)y(v,θ(g)f) for each v ∈ e, g ∈ g, f ∈ f . then the projection πθ : (e ×θ f) → n is defined by πθ([v,f]) = π(v), where [v,f] := {(w,b) : (w,b)y(v,f),w ∈ e,b ∈ f} denotes the equivalence class of (v,f). therefore, each parallel transport structure p on the principal g fiber bundle e(n,g,π, ψ) induces a parallel transport structure pθ on the induced bundle by the formula pθ γ̂,[u,f] (x) = [pγ̂,u(x),f]. define multiplication with the help of certain embeddings and isomorphisms of spaces of functions. mention that for each two compact canonical closed subsets a and b in al r hilbert spaces ht(a, rm) and ht(b, rm) are linearly topologically isomorphic, where l,m ∈ n, hence ht p (a,n) and ht p (b,n) are isomorphic as uniform spaces. let ht p (m,{s0,1, ...,s0,k}; w,y0) := {(e,f) : e = e(n,g,π, ψ) ∈ w,f = pγ̂,y0 ∈ h t p : π ◦ f(s0,q) = y0∀q = 1, ...,k; π ◦ f = γ̂,γ ∈ h t p (m,n)} be the space of all ht ′ p principal g fiber bundles e with their parallel transport ht p -mappings f = pγ̂,y0 , where w is as in §1.3.2. put ω0 = (e0, p0) be its element such that γ0(m) = {y0}, where e ∈ g denotes the unit element, e0 = n × g, π0(y,g) = y for each y ∈ n, g ∈ g, pγ̂0,u = p0. the mapping ξ : m̂ → m from §2 induces the embedding ξ ∗ : ht p (m,{s0,1, ...,s0,k}; w,y0) →֒ h t p (m̂,{ŝ0,1, ..., ŝ0,2k}; w,y0), where m̂ and â1 and â2 are retractable into points. let as usually a∨b := ρ(z) be the wedge sum of pointed spaces (a,{a0,q : q = 1, ...,k}) and (b,{b0,q : q = 1, ...,k}), where z := [a × {b0,q : q = 1, ...,k} ∪ {a0,q : q = 1, ...,k} × b] ⊂ a × b, ρ is a continuous quotient mapping such that ρ(x) = x for each x ∈ z \ {a0,q × b0,j; q,j = 1, ...,k} and ρ(a0,q) = ρ(b0,q) for each q = 1, ...,k, where a and b are topological spaces with marked 108 s.v. ludkovsky cubo 11, 1 (2009) points a0,q ∈ a and b0,q ∈ b, q = 1, ...,k. then the wedge product g ∨ f of two elements f,g ∈ ht p (m,{s0,1, ...,s0,k}; n,y0) is defined on the domain m ∨ m such that (f ∨ g)(x × b0,q) = f(x) and (f ∨ g)(a0,q × x) = g(x) for each x ∈ m, where to f,g there correspond f1,g1 ∈ ht p (m̂,{ŝ0,1, ..., ŝ0,2k}; n,y0) such that f1f ◦ ξ and g1 = g ◦ ξ. let (ej, pγ̂j,uj ) ∈ h t p (m,{s0,1, ...,s0,k}; w,y0), j = 1, 2, then take their wedge product pγ̂,u1 := pγ̂1,u1 ∨ pγ̂2,v on m ∨ m with vq = uqg −1 2,q g1,q+k = y0 × g1,q+k for each q = 1, ...,k due to the alternativity of g, γ = γ1 ∨γ2, where pγ̂j,uj (ŝj,0,q)y0 ×gj,q ∈ ey0 for every j and q. for each γj : m → n there exists γ̃j : m → ej such that π ◦ γ̃j = γj. denote by m : g × g → g the multiplication operation. the wedge product (e1, pγ̂1,u1 ) ∨ (e2, pγ̂2,u2 ) is the principal g fiber bundle (e1 × e2) × m g with the parallel transport structure pγ̂1,u1 ∨ pγ̂2,v. the uniform space ht p (j,a3; w,y0) := {(e,f) ∈ h t p (j,w) : π ◦ f(a3) = {y0}} has the h t p manifold structure and has an embedding into ht p (m,{s0,1, ...,s0,k}; w,y0) due to conditions 2(i − iii), where either j = a1 or j = a2. this induces the following embedding χ∗ : ht p (m ∨m,{s0,q ×s0,q : q = 1, ...,k}; w,y0) →֒ h t p (m,{s0,q : q = 1, ...,k}; w,y0). analogously considering ht p (m,{x0,q : q = 1, ...,k}; w,y0) = {f ∈ h t (m,w) : f(x0,q) = {y0},q = 1, ...,k} and h t p (j,a3 ∪ {x0,q : q = 1, ...,k}; w,y0) in the case (iv ′ ) instead of (iv) we get the embedding χ∗ : ht p (m ∨ m,{x0,q × x0,q : q = 1, ...,k}; w,y0) →֒ h t p (m,{x0,q : q = 1, ...,k}; w,y0). therefore, g◦f := χ ∗ (f∨g) is the composition in ht p (m,{s0,q : q = 1, ...,k}; w,y0). there exists the following equivalence relation rt,h in h t p (m,{x0,q : q = 1, ...,k}; w,y0): frt,hh if and only if there exist nets ηn ∈ difh t p (m,{x0,q : q = 1, ...,k}), also fn and hn ∈ ht p (m,{x0,q : q = 1, ...,k}; w,y0) with limn fn = f and limn hn = h such that fn(x) = hn(ηn(x)) for each x ∈ m and n ∈ ω, where ω is a directed set and convergence is considered in ht p (m,{x0,q : q = 1, ...,k}; w,y0). henceforward in the case 2(iv) we get s0,q instead of x0,q in the case 2(iv ′ ). thus there exists the quotient uniform space ht p (m,{x0,q : q = 1, ...,k}; w,y0)/rt,h =: (s me)t,h. in view of [30, 31] difh t p (m) is the group of diffeomorphisms for t ≥ [m/2] + 1. the lebesgue measure λ in the real shadow of m̂ by the mapping ξ induces the measure λξ on m which is equivalent to ν, since ξ is the ht p -mapping from the compact space onto the compact space, λ(∂m̂) = 0 and ξ : m̂ \ ∂m̂ → m is bijective. due to conditions (p1 − p5) each element f = pγ̂,u up to a set qm of measure zero, ν(qm) = 0, is given as f ◦ ξ −1 on m \ qm, where π ◦ f = γ̂, γ̂ = γ ◦ ξ. denote f ◦ ξ −1 also by f. thus, for each (e,f) ∈ ht p (m,{s0,q : q = 1, ...,k}; w,y0) the image f(m) is compact and connected in e. therefore, for each partition z there exists δ > 0 such that for each partition z∗ with sup i infj dist(mi,m ∗ j) < δ and (e,f) ∈ h t (m,w ; z), f(s0,q) = uq, there exists (e,f1) ∈ ht(m,w ; z∗) with f1(s0,q) = uq for each q = 1, ...,k such that frt,hf1, where mi and m ∗ j are canonical closed pseudo-submanifolds in m corresponding to partitions z and z∗, ht(m,w ; z) denotes the space of all continuous piecewise ht-mappings from m into w subordinated to the cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 109 partition z such that z and z∗ respect ht p structure of m. hence there exists a countable subfamily {zj : j ∈ n} in the family of all partitions υ such that zj ⊂ zj+1 for each j and limj d̃iamzj = 0. then (i) str − ind{ht(m,{s0,q : q = 1, ...,k}; w,y0; zj); h zi zj ; n}/rt,h = (s me)t,h is separable if n and g are separable, since each space ht p (m,{s0,q : q = 1, ...,k}; w,y0; zj) is separable. the space str−ind{ht(m,{s0,q : q = 1, ...,k}; w,y0; zj); h zi zj ; n} is complete due to theorem 12.1.4 [29], when n and g are complete. each class of rt,h-equivalent elements is closed in it. then to each cauchy net in (sme)t,h there corresponds a cauchy net in str−ind{h t (m ×[0, 1],{s0,q × e× 0; w,y0; zj ×yj); h zi×yi zj ×yj ; n} due to theorems about extensions of functions [25, 33, 38], where yj are partitions of [0, 1] with limj d̃iam(yj) = 0, zj × yj are the corresponding partitions of m × [0, 1]. hence (sme)t,h is complete, if n and g are complete. if f,g ∈ ht(m,x) and f(m) 6= g(m), then (ii) infψ∈difhtp(m,{s0,q :q=1,...,k}) ‖f ◦ ψ − g‖ht(m,x) > 0. thus equivalence classes < f >t,h and < g >t,h are different. the pseudo-manifold m̂ is arcwise connected. take η : [0, 1] → m̂ an ht p -mapping with η(0) = ŝ0,q and η(1) = ŝ0,k+q, where 1 ≤ q ≤ k. choose in m̂ h t p -coordinates one of which is a parameter along η. therefore, for each gq,gk+q ∈ g there exists pγ̂,u with pγ̂,u(s0,q) = y0 × gq and pγ̂,u(s0,k+q) = y0 × gk+q for each q = 1, ...,k. since e and g are arcwise connected, then n is arcwise connected and (sme)t,h is locally connected for dimrn > 1. thus, the uniform space (sme)t,h is non-discrete. the tangent bundle tht p (m,e) is isomorphic with ht p (m,te), where te is the ht ′−1 p fiber bundle, t′ ≥ t + 1. there is an infinite family of fα ∈ h t p (m,te) with pairwise distinct images in te for different α such that fα(m) is not contained in ⋃ β<α fβ(m), α ∈ λ, where λ is an infinite ordinal. therefore, t(sme)t,h is an infinite dimensional fiber bundle due to (ii) and inevitably (sme)t,h is infinite dimensional. evidently, if f ∨ g = h ∨ g or g ∨ f = g ∨ h for {f,g,h} ⊂ ht p (m,{s0,q : q = 1, ...,k}; w,y0), then f = h. thus χ∗(f ∨ g) = χ∗(h ∨ g) or χ∗(g ∨ f) = χ∗(g ∨ h) is equivalent to f = h due to the definition of f ∨ g and the definition of equal functions, since χ∗ is the embedding. using the equivalence relation rt,h gives < f >t,h ◦ < g >t,h=< h >t,h ◦ < g >t,h or < g >t,h ◦ < f >t,h=< g >t,h ◦ < h >t,h is equivalent to < h >t,h=< f >t,h. therefore, (sme)t,h has the cancelation property. since g is alternative, then a2,q[a −1 2,q (a2,q+k(a −1 2,q a1,q+k))]a2,q+k(a −1 2,q a1,q+k), hence p1 ∨ (p2 ∨ p2) = (p1 ∨p2)∨p2; also a2,q[a −1 2,q (a1,q+k(a −1 1,q a1,q+k))]a1,q+k(a −1 1,q a1,q+k), consequently, p1 ∨(p1 ∨ p2) = (p1 ∨ p1) ∨ p2 and inevitably for equivalence classes (aa)b = a(ab) and b(aa) = (ba)a for each a,b ∈ (sme)t,h. thus (s me)t,h is alternative. if g is associative, then the parallel transport structure gives (f ∨ g) ∨ h = f ∨ (g ∨ h) on m ∨ m ∨ m for each {f,g,h} ⊂ ht p (m,{s0,q : q = 1, ...,k; w,y0). applying the embedding χ ∗ and the equivalence relation rt,h we get, that (s me)t,h is associative < f >ξ ◦(< g >ξ ◦ < h >ξ) = 110 s.v. ludkovsky cubo 11, 1 (2009) (< f >ξ ◦ < g >ξ)◦ < h >ξ. in view of conditions 2(i − iv) there exists an ht p -diffeomoprhism of (a1 \ a3) ∨ (a2 \ a3) with (a2 \a3) ∨ (a1 \a3) as pseudo-manifolds (see §1.3.1). for the measure ν on m naturally the equality ν(a3) = 0 is satisfied. if m ′ is the submanifold may be with corners or pseudo-manifold, accomplishing the partition z = zf of the manifold m, then the codimension m ′ in m is equal to one and ν(m′) = 0. for the point s0,q in (m \ a3) ∪ {s0,q} there exists an open neighborhood u having the ht p -retraction f : [0, 1] × u → {s0,q}. hence it is possible to take a sequence of diffeomorphisms ψn ∈ difh t p (m,{s0,q : q = 1, ...,k}) such that limn→∞ diam(ψn(u)) = 0. let w0 be a mapping w0 : m → w such that w0(m) = {y0 ×e}. consider w0 ∨(e,f) for some (e,f) ∈ ht p (m,{s0,q : q = 1, ...,k}; w,y0). if (e,f) ∈ h t p (m,{s0,q : q = 1, ..,k}; w,y0) with the natural positive t ∈ n, then f is bounded relative to the uniformity of the uniform space ht p (m; e). if un is a sequence of bounded open or canonical closed subsets in m such that limn diam(un) = 0, then limn→∞ ν(vn) = 0 for the sequence of ν-measurable subsets vn such that vn ⊂ un. therefore, for each bounded sequence {gn : gn ∈ h t p (m; e); n ∈ n} there exists the limit limn→∞ gn|un = 0 relative to the ht p uniformity, where un is subordinated to the partition of m into h t submanifolds. then if {gn : gn ∈ h t p (m,{s0,q : q = 1, ...,k}; e,y0); n ∈ n} is a bounded sequence such that gn converges to g ∈ ht p (m,{s0,q : q = 1, ...,k}; n,y0) on m \ wk for each k relative to the h t p uniformity, the given open wk in m, where k,n ∈ n and limn→∞ ν(wn △ un) = 0, then gn converges to g in the uniform space ht p (m,{s0,q : q = 1, ...,k}; e,y0). mention that for each marked point s0,q in m there exists a neighborhood u of s0,q in m such that for each γ1 ∈ h t p (m,{s0,q : q = 1, ...,k}; n,y0) there exists γ2 ∈ h t p such that they are rt,h equivalent and γ2|u = y0. therefore, if c is an arcwise connected compact subset in m of codimension codimrc ≥ 1 such that s0,q ∈ c, then the standard proceeding shows that for each γ1 ∈ h t p there exists γ2 ∈ h t p such that γ1rt,hγ2 and γ2|c = y0. since c is compact, then each its open covering has a finite subcovering and hence (y0) there exists an open neighborhood u of c in m such that for each γ1 there exists γ2 such that γ1rt,hγ2 and γ2|u = y0. there exists a sequence ηn ∈ difh t p (m,{s0,q : q = 1, ...,k}) such that limn→∞ diam(ηn(a2 \ ∂a2)) = 0 and wn,fn ∈ h t p (m,{s0,q : q = 1, ...,k}; e,y0) with (iii) limn→∞ fn = f, limn→∞ wn = w0 and limn→∞ χ ∗ (fn ∨wn)(η −1 n ) = f due to π◦f(s0,q) = s0,q in the formula of differentiation of compositions of functions (over h and o see it in [19, 20, 17]). in more details, the sequence ηn as a limit of ηn(a2) produces a pseudo-submanifold b in m of codimension not less than one such that b can be presented with the help of the wedge product of spheres and compact quadrants up to ht p -diffeomorphism with marked points {s0,q : q = 1, ...,k}, but as well b may be a finite discrete set also. then by induction the procedure can be continued lowering the dimension of b. particularly there may be circles and curves in the case of the unit dimension. two quadrants up to an ht p quotient mapping gluing boundaries produce a sphere. thus the consideration reduces to the case of the wedge product of spheres. the case cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 111 of spheres reduces to the iterated construction with circles, since the reduced product s1 ∧ sn is ht p homeomorphic with sn+1 (see lemma 2.27 [37] and [4]). for the particular case of the n-dimensional sphere mn = s n take m̂n = d n, where dn is the unit ball (disk) in rn or in a n dimensional over r subspace in al r , d1 = [0, 1] for n = 1. but s n \ s0 has the retraction into the point in sn, where s0 ∈ s n, n ∈ n. therefore, w0 ∨ (e,f) and (e,f) belong to the equivalence class < (e,f) >t,h: {g ∈ ht p (m,{s0,q : q = 1, ...,k}; w,y0) : (e,f)rt,hg} due to (iii) and (y0). thus, < w0 >t,h ◦ < g >t,h=< g >t,h. the pseudo-manifold m ∨ m \ {s0,q × s0,j : q,j = 1, ...,k} has the h t p -diffeomorphism ψ (see definition in §1.3.1) such that ψ(x,y) = (y,x) for each (x,y) ∈ (m × m \ {s0,q × s0,j : q,j = 1, ...,k}). suppose now, that g is commutative. then (f ∨ g) ◦ ψ|(m×m\{s0,q×s0,j :q,j=1,...,k}) = g ∨ f|(m×m\{s0,q×s0,j :q,j=1,...,k}). on the other hand, < f ∨ w0 >t,h=< f >t,h=< f >t,h ◦ < w0 >t,h=< w0 >t,h ◦ < f >t,h, hence, < f ∨ g >t,h=< f >t,h ◦ < g >t,h=< f ∨ w0 >t,h ◦ < w0 ∨ g >t,h=< (f ∨ w0) ∨ (w0 ∨ g) >t,h=< (w0 ∨ g) ∨ (f ∨ w0) >t,h due to the existence of the unit element < w0 >t,h and due to the properties of ψ. indeed, take a sequence ψn as above. therefore, the parallel transport structure gives (g ∨ f)(ψ(x,y)) = (g ◦ f)(y,x) for each x,y ∈ m, consequently, (f ◦ g)rt,h(g ◦ f) for each f,g ∈ h t p (m,{s0,q : q = 1, ...,k}; w,y0). the using of the embedding χ∗ gives that (sme)t,h is commutative, when g is commutative. the mapping (f,g) 7→ f ∨ g from ht p (m,{s0,q : q = 1, ...,k}; w,y0) 2 into ht p (m ∨ m \ {s0,q × s0,j : q,j = 1, ...,k}; w,y0) is of class h t p . since the mapping χ∗ is of class ht p , then (f,g) 7→ χ∗(f∨g) is the ht p -mapping. the quotient mapping from ht p (m,{s0,q : q = 1, ...,k}; w,y0) into (sme)t,h is continuous and induces the quotient uniformity, t b (sme)t,h has embedding into (smtbe)t,h for each 1 ≤ b ≤ t ′ − t, when t′ > t is finite, for every 1 ≤ b < ∞ if t′ = ∞, since e is the ht ′ p fiber bundle, tbe is the fiber bundle with the base space n. hence the multiplication (< f >t,h,< g >t,h>) 7→< f >t,h ◦ < g >t,h=< f ∨ g >t,h is continuous in (s me)t,h and is of class hl p with l = t′ − t for finite t′ and l = ∞ for t′ = ∞. 4. definition. the (sme)t,h from theorem 3.1 we call the wrap monoid. 5. corollary. let φ : m1 → m2 be a surjective h t p -mapping of ht p -pseudo-manifolds over the same ar such that φ(s1,0,q) = s2,0,a(q) for each q = 1, ...,k1, where {sj,0,q : q = 1, ...,kj} are marked points in mj, j = 1, 2, 1 ≤ a ≤ k2, l1 ≤ k2, l1 : card φ({s1,0,q : q = 1, ...,k1}). then there exists an induced homomorphism of monoids φ∗ : (sm2e)t,h → (s m1e)t,h. if l1 = k2, then φ ∗ is the embedding. proof. take ξ1 : m̂1 → m1 with marked points {ŝ1,0,q : q = 1, ..., 2k1} as in §2, then take m̂2 the same m̂1 with additional 2(k2 − l1) marked points {ŝ2,0,q : q = 1, ..., 2k3} such that ŝ1,0,q = ŝ2,0,q for each q = 1, ..,k1, k3 = k1 + k2 − l1, then φ ◦ ξ1 := ξ2 : m̂2 → m2 is the desired mapping inducing the parallel transport structure from that of m1. therefore, each γ̂2 : m̂2 → n induces γ̂1 : m̂1 → n and to pγ̂2,u2 there corresponds pγ̂1,u1 with additional conditions in extra marked points, where u1 ⊂ u2. the equivalence class < (e2, pγ̂2,u2 ) >t,h∈ (s m2e)t,h 112 s.v. ludkovsky cubo 11, 1 (2009) gives the corresponding elements < (e1, pγ̂1,u1 ) >t,h∈ (s m1e)t,h, since difh t p (m̂1,{ŝ0,q : q = 1, ..., 2k2}) ⊂ difh t p (m̂1,{ŝ0,q : q = 1, ..., 2k3}). then φ ∗ (< (e2, pγ̂2,u2 ) ∨ (e1, pη̂2,v2 ) >t,h ) = φ∗(< (e2, pγ̂2,u2 ) >t,h)φ ∗ (< (e1, pη̂2,v2 ) >t,h), since f2 ◦ φ(x) for each x ∈ ξ1(m̂1 \ ∂m̂1) coincides with f1(x), where fj corresponds to pγj,y0×e (see also the beginning of §3). if l1 = k2, then m̂1 = m̂2 and the group of diffeomorphisms difh t p (m̂1,{ŝ0,q : q = 1, ..., 2k1}) is the same for two cases, hence φ∗ is bijective and inevitably φ∗ is the embedding. 6. theorems. 1. there exists an alternative topological group (wme)t,h containing the monoid (sme)t,h and the group operation of h l p class with l = t′ − t (l = ∞ for t′ = ∞). if n and g are separable, then (wme)t,h is separable. if n and g are complete, then (w me)t,h is complete. 2. if g is associative, then (wme)t,h is associative. if g is commutative, then (w me)t,h is commutative. if g is a lie group, then (wme)t,h is a lie group. 3. the (wme)t,h is non-discrete, locally connected and infinite dimensional for dimr(n × g) > 1. moreover, if there exist two different sets of marked points s0,q,j in a3, q = 1, ...,k, j = 1, 2, then two groups (wme)t,h,j, defined for {s0,q,j : q = 1, ...,k} as marked points, are isomorphic. 4. the (wme)t,h has a structure of an h t p -differentiable manifold over ar. proof. if γ ∈ ht p (m,{s0,q : q = 1, ...,k}; n,y0), then for u ∈ ey0 there exists a unique hq ∈ g such that pγ̂,u(ŝ0,q+k) = uqhq, where hq = g −1 q gq+k, y0 × gq = pγ̂,u(ŝ0,q), gq ∈ g. due to the equivariance of the parallel transport structure h depends on γ only and we denote it by h(e,p)(γ) = h(γ) = h, h = (h1, ...,hk). the element h(γ) is called the holonomy of p along γ and h(e,p)(γ) depends only on the isomorphism class of (e, p) due to the use of difht p (m̂; {ŝ0,q : q = 1, ..., 2k}) and boundary conditions on γ̂ at ŝ0,q for q = 1, ..., 2k. therefore, h(e1,p1)(e2,p2)(γ) = h(e1,p1)(γ)h(e2,p2)(γ) ∈ gk, where gk denotes the direct product of k copies of the group g. hence for each such γ there exists the homomorphism h(γ) : (sme)t,h → g k, which induces the homomorphism h : (sme)t,h → c 0 (ht p (m,{s0,q : q = 1, ...,k}; n,y0),g k ), where c0(a,gk) is the space of continuous maps from a topological space a into gk and the group structure (hb)(γ) = h(γ)b(γ) (see also [4] for sn). thus, it is sufficient to construct (wmn)t,h from (s mn)t,h. for the commutative monoid (smn)t,h with the unit and the cancelation property there exists a commutative group (w mn)t,h. algebraically it is the quotient group f/b, where f is the free commutative group generated by (smn)t,h, while b is the minimal closed subgroup in f generated by all elements of the form [f + g] − [f] − [g], f and g ∈ (smn)t,h, [f] denotes the element in f corresponding to f (see also about such abstract grothendieck construction in [13, 36]). by the construction each point in (smn)t,h is the closed subset, hence (s mn)t,h is the topological t1-space. in view of theorem 2.3.11 [7] the product of t1-spaces is the t1-space. on the other hand, for the topological group g from the separation axiom t1 it follows, that g is the cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 113 tychonoff space [7, 32]. the natural mapping η : (smn)t,h → (w mn)t,h is injective. we supply f with the topology inherited from the topology of the tychonoff product (smn)z t,h , where each element z in f has the form z = ∑ f nf,z[f], nf,z ∈ z for each f ∈ (s mn)t,h, ∑ f |nf,z| < ∞. by the construction f and f/b are t1-spaces, consequently, f/b is the tychonoff space. in particular, [nf] − n[f] ∈ b, hence (wmn)t,h is the complete topological group, if n and g are complete, while η is the topological embedding, since η(f + g) = η(f) + η(g) for each f,g ∈ (smn)t,h, η(e) = e, since (z + b) ∈ η(smn)t,h, when nf,z ≥ 0 for each f, and inevitably in the general case z = z+ − z−, where (z+ + b) and (z− + b) ∈ η(smn)t,h. using plots and ht ′ p transition mappings of charts of n and e(n,g,π, ψ) and equivalence classes relative to difht p (m,{s0,q : q = 1, ...,k}) we get, that (w me)t,h has the structure of the ht p -differentiable manifold, since t′ ≥ t. the rest of the proof and the statements of theorems 6(1-4) follows from this and theorems 3(1-3) and [21, 22]. 7. definition. the (wme)t,h = (w m,{s0,q :q=1,...,k}e; n,g, p)t,h from theorem 6.1 we call the wrap group. 8. corollary. there exists the group homomorphism h : (wme)t,h → c 0 (ht p (m,{s0,q : q = 1, ...,k}; n,y0),g k ). proof follows from §6 and putting hf −1 (γ)(hf (γ))−1. 9. corollary. if m1 and m2 and φ satisfy conditions of corollary 5, then there exists a homomorphism φ∗ : (wm2e)t,h → (w m1e)t,h. if l1 = k2, then φ ∗ is the embedding. 10. remarks and examples. consider examples of m which satisfy sufficient conditions for the existence of wrap groups (wme)t,h. take m, for example, d n r , sn r \ v with s0 ∈ ∂v , dn r \ int(dn b ) with s0 ∈ ∂d n b and 0 1 over r and radius r, v is ht p -diffeomorphic with the interior int(dn r ) of the n-dimensional ball dn r := {x ∈ rn : ∑ n k=1 x2 k ≤ r} or in n dimensional over r subspace in al r and is the proper subset in sn r := {x ∈ rn+1 : ∑ n+1 k=1 x2 k = r}. instead of sphere it is possible to take an ht p pseudo-manifold qn homeomorphic with a sphere or a disk, particularly, milnor’s sphere. indeed, divide m by the equator {x1 = 0} into two parts a1 and a2 and take a3 = {x ∈ m : x1 = 0} ∪p , where s0 ∈ ∂a1 ∩ ∂a2, while p = ∅, p = ∂v , p = ∂d n b correspondingly. then take also v and dn b such that their equators would be generated by the equator {x1 = 0} in s n r or dn r respectively or more generally qn. take then m = qn\ ⋃ l k=1 vk, where vk are h t p -diffeomorphic to interiors of bounded quadrants in rn or in n dimensional subspace in aa r , where l > 1, l ∈ n, ∂vk ∩ ∂vj = {s0} and vk ∩ vj = ∅ for each k 6= j, diam(vk) ≤ b < r/3. in more details it is possible make a specification such that if l is even, then [l/2] − 1 among vk are displayed above the equator and the same amount below it, two of vk have equators, generated by equators {x1 = 0} in q n. if l odd, then [(l− 1)/2] among vk are displayed above and the same amount below it, one of vk has equator generated by that of {x1 = 0} in q n, s0 ∈ ⋂ k ∂vk ∩ {x ∈ m : x1 = 0}. 114 s.v. ludkovsky cubo 11, 1 (2009) divide m by the equator {x1 = 0} into two parts a1 and a2 and let a3 = {x ∈ m : x1 = 0}∪p , where p = ⋃ l k=1 ∂vk. then either a1 \ a3 and a2 \ a3 are h t p diffeomorphic as pseudo-manifolds or manifolds with corners and ht p diffeomorphic with m \ [{s0} ∪ (a3 \ int(∂a1 ∩ ∂a2))] =: d or 2(iv′) is satisfied, since the latter topological space d is obtained from qn by cutting a non-void connected closed subset, n > 1, consequently, d is retractable into a point. in a case of a usual manifold m the point s0 ∈ ∂m (for ∂m 6= ∅) may be a critical point, but in the case of a manifold with corners this s0 is the corner point from ∂m, since for x ∈ ∂m there is not less than one chart (u,u,q) such that u(x) ∈ ∂q, m \ ∂m = ⋃ k u −1 k (int(qk)), ∂m ⊂ ⋃ k u −1 k (∂qk). further, if m satisfies conditions 2(i−v) or (i−iii, iv ′,v), then m ×dm r = p also satisfies them for m ≥ 1, since dm r is retractable into the point, taking as two parts aj(k) = aj(m) × d m r of p , where j = 1, 2, aj(m) are pseudo-submanifolds of m. then a1(p) ∩ a2(p) = (a1(m) ∩ a2(m)) × d m r and it is possible to take a3(p) = a3(m) × d m r , s0(p) ∈ s0(m) × {x ∈ d m r : x1 = 0}. in particular, for m = s 1 and m = 1 this gives the filled torus. this construction can be naturally generalized for non-orientable manifolds, for example, the möbius band l, also for m := l \ ( ⋃ β j=1 vj) with the diameter bj of vj less than the width of l, where each vj is h t p diffeomorphic with an interior of a bounded quadrant in r2, s0,q ∈ ∂l ∩ ( ⋂ a1+...+aq j=a1+...+aq−1+1 ∂vj), a0 := 0, a1 + ... + ak = β, q = 1, ...,k, since ∂l is diffeomorphic with s1, also s1 \ {s0,q} is retractable into a point, consequently, a1 and a2 are retractable into a point. for l take m̂ = i2, then take a connected curve η̂ consisting of the left side {0} × [0, 1] joined by a straight line segment joining points {0, 1} and {1, 0} and then joined by the right side {1} × [0, 1]. this gives the proper cutting of m̂ which induces the proper cutting of l and of m with a3 ⊃ η ∪ ∂l up to an h t p diffeomorphism, where η := ξ(η̂), hence the möbius band l and m satisfy conditions 2(i − iii, iv′,v). take a quotient mapping φ : i2 → s1 such that φ({s0,1,s0,2}) = s0 ∈ s 1, s0,1 = (0, 0), s0,2 = (0, 1) ∈ i 2, where i = [0, 1], hence there exists the embedding φ∗ : (ws 1 ,s0e)t,h →֒ (wi 2 ,{s0,1,s0,2}e)t,h. the klein bottle k has m̂ = i2 with twisting equivalence relation on ∂i2 so it satisfies sufficient conditions. moreover, k is the quotient φ : z → k of the cylinder z with twisted equivalence relation of its ends s1 using reflection relative to a horizontal diameter. thus a3 ⊃ φ(s1). therefore, there exists the embedding φ∗ : (wk,{s0}e)t,h → (w z,{s0,1,s0,2}e)t,h, where s0,1,s0,2 ∈ ∂z, φ({s0,1,s0,2}) = s0. take a pseudo-manifold qn ht p -diffeomorphic with sn for n ≥ 2, cut from it β non-intersecting open domains v1, ...,vβ h t p -diffeomorphic with interiors of bounded quadrants in rn, s0,q ∈⋂ a1+...+aq j=a1+...+aq−1+1 ∂vj, a0 := 0, a1 + ... + ak = β, q = 1, ...,k. then glue for v1, ...,vl, 1 ≤ l ≤ β, by boundaries of slits ht p -diffeomorphic with sm−1 the reduced product l ∨ sn−2, since ∂l = s1, s1 ∧ sn−2 is ht p -diffeomorphic with sn−1 [37]. we get the non-orientable ht p -pseudo-manifold m, satisfying sufficient conditions. cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 115 since the projective space rpn is obtained from the sphere by identifying diametrically opposite points. then take m ht p -diffeomorphic with rpn for n > 1 also m with cut v1, ...,vβ ht p -diffeomorphic with open subsets in rpn, s0,q ∈ ( ⋂ a1+...+aq j=a1+...+aq−1+1 ∂vj) ∩ {x ∈ m : x1 = 0}, vj ∩ vl = ∅ for each j 6= l, a0 := 0, a1 + ... + ak = β, q = 1, ...,k. then conditions 2(i − v) or (i − iii, iv′,v) are also satisfied for rpn and m. in view of proposition 2.14 [37] about h-groups [x,x0; k,k0] there is not any expectation or need on rigorous conditions on a class of acceptable m for constructions of wrap groups (wme)t,h. if m1 is an analytic real manifold, then taking its graded product with generators {i0, ..., i2r−1} of the cayley-dickson algebra gives the ar manifold (see [19, 17, 18]). particularly this gives l2 r dimensional torus in al r for the l dimensional real torus t2 = (s 1 ) l as m1. consider t2. it can be slit along a closed curve (loop) c h ∞ p -diffeomorphic with s1 and marked points s0,q ∈ c ⊂ t2 such that c rotates on the surface of t2 = s 1 r × s1 b on angle π around s1 b while c rotates on 2π around s1 r , such that c rotates on 4π around s1 r that return to the initial point on c, where 0 < b < r < ∞, q = 1, ...,k, k ∈ n. therefore, the slit along c of t2 is the non-orientable band which inevitably is the möbius band with twice larger number of marked points {sl 0,j : j = 1, ..., 2k} ⊂ ∂l. therefore, for m = t2 as m̂ take a quadrant in r 2 with 2k pairwise opposite marked points ŝ0,q and ŝ0,q+k on the boundary of m̂, q = 1, ...,k, k ∈ n. suitable gluing of boundary points in ∂m̂ gives the mapping ξ : m̂ → t2, ξ(ŝ0,q) = ξ(ŝ0,q+k) = s0,q, q = 1, ...,k. proper cutting of m̂ into âj, j = 1, 2, or of l induces that of t2. thus we get a pseudo-submanifold a3(t2) =: a3 ⊃ c, while a1 and a2 are retractable into a marked point s0,q ∈ c for each q, hence t2 satisfies conditions 2(i − iii, iv ′,v). in view of corollary 9 there exists the embedding φ∗ : (wt2,{s0,q :q=1,...,k}e)t,h → (w l,{sl0,q :q=1,...,2k}e)t,h, where φ : l → t2 is the quotient mapping with φ({sl 0,q ,sl 0,q+k }){s0,q}, q = 1, ...,k. for the n-dimensional torus tn in a a r with n > 2 take a n−1-dimensional surface b such that each its projection into t2 is h t p -diffeomorphic with c for a loop c as above. therefore, the slit along b up to a ht p -diffeomorphism gives m0 := l × i n−2 for even n or m0 := s 1 × in−1 for odd n, where i = [0, 1]. since im is retractable into a point, where m ≥ 1. thus we lightly get for tn a pseudo-submanifold a3 ⊃ b and two a1 and a2 retractable into points and satisfying sufficient conditions 2(i − iii, iv′,v), where m̂ = in up to a ht p -diffeomorphism, s0,q ∈ b ⊂ a3 := a3(tn), {sm0 0,q ,s m0 0,q+k } ⊂ ∂m0, q = 1, ...,k, k ∈ n. proper cutting of m̂ into âj, j = 1, 2, induces that of tn. thus there exists an h t p quotient mapping φ : m0 → tn with φ({s m0 0,q ,s m0 0,q+k }) = {s0,q} and the embedding φ∗ : (wtn,{s0,q :q=1,...,k}e)t,h →֒ (w m0,{s m0 0,q :q=1,...,2k}e)t,h due to corollary 9. more generally cut from tn open subsets vj which are h t p diffeomorphic with interiors of bounded quadrants in rn embedded into al r , j = 1, ...,β, such that s0,q ∈ b∩( ⋂ a1+...+aq j=a1+...+aq−1+1 ∂vj), vj∩vi = ∅ for each j 6= i, vj∩b = ∅ for each j, where b is defined up to an h t p diffeomorhism, a0 := 0, a1 + ... + ak = β, q = 1, ...,k, that gives the manifold m2. then from m0 cut analogously corresponding vj,b, such that s0,q ∈ b ∩ ( ⋂ a1+...+aq j=a1+...+aq−1+1 ∂vj,1), s0,q+k ∈ b ∩ ( ⋂ a1+...+aq j=a1+...+aq−1+1 ∂vj,2), 116 s.v. ludkovsky cubo 11, 1 (2009) vj,b1 ∩ vi,b2 = ∅ for each j 6= i or b1 6= b2, a0 := 0, a1 + ... + ak = β, q = 1, ...,k, j = 1, ...,β, b = 1, 2, that produces the manifold m1. we choose vj,b such that for the restriction φ : m1 → m2 of the mapping φ there is the equality φ(vj,1 ∪ vj,2) = vj for each j, φ({s m1 0,q ,s m1 0,q+k }) = {s0,q}. this gives the embedding φ∗ : (wm2,{s0,q :q=1,...,k}e)t,h →֒ (w m1,{s m1 0,q :q=1,...,2k}e)t,h. another example is m3 obtained from the previous m2 with 2k marked points and 2β cut out domains vj, when s0,q is identified with s0,q+k and each ∂vj is glued with ∂vj+β for each j ∈ λq ⊂ {d : a1 + ... + aq−1 + 1 ≤ d ≤ a1 + ... + aq}, q = 1, ...,k, k ∈ n, by an equvalence relation υ. such m3 is obtained from the torus tn,m with m holes instead of one hole in the standard torus tn,1 = tn cutting from it vj with j ∈ {1, ..., 2β} \ ( ⋃ q=1,...,k λq), where m = m1 +...+mk, mq := card(λq). for tn and m2 the surface b is h t p diffeomorphic with (∂l)×in−2 for even n or s1 × in−1 for odd n. take a3 ⊃ b ∪ ( ⋃ j∈λq υ(∂vj)), it is arcwise connected and contains all marked points. therefore, m3 satisfies conditions of §2 and there exists the embedding υ∗ : (wm3,{s m3 0,q :q=1,...,k}e)t,h →֒ (w m2,{s m2 0,q :q=1,...,2k}e)t,h. this also induces the embedding (wtn,m,{s tn,m 0,q :q=1,...,k}e)t,h →֒ (w tn,{stn0,q :q=1,...,2k−1}e)t,h such that each element g ∈ (wtn,m,{s tn,m 0,q :q=1,...,k}e)t,h can be presented as a product g = (..(g1g2)...gm) of m elements gj ∈ (w tn,{stn0,q :q=1,...,2k−1}e)t,h, gj =< fj >t,h, supp(π ◦ fj) ⊂ bj, b1 ∪ ... ∪ bm = tn, bi ∩ bj = ∂bi ∩ ∂bj for each i 6= j, each bj is a canonical closed subset in tn, s0,1 ∈ b1, s0,2q,s2q+1 ∈ bd for m1 + ... + m0 + 1 ≤ d ≤ m1 + ... + mq, q = 1, ...,k − 1, where m0 := 0. evidently, in the general case for different manifolds m and n wrap groups may be non isomorphic. for example, as m1 take a sphere s n of the dimension n > 1, as m2 take m1 \ k, where k is up to an ht p -diffeomorphism the union of non intersecting interiors bj of quadrants of diameters d1, ...,ds much less, than 1, k = b1 ∪ ... ∪ bl, l ∈ n. let n be a δ-enlargement for m2 in r n+1 relative to the metric of the latter euclidean space, where 0 < δ < min(d1, ...,dl)/2. then the groups (wm1n)t,h and (w m2n)t,h are not isomorphic. this lightly follows from the consideration of the element b :=< f >t,h∈ (w m2n)t,h, where f : m2 → n is the identity embedding induced by the structure of the δ-enlargement. recall, that for orientable closed manifolds a and b of the same dimension m the degree of the continuous mapping f : a → b is defined as an integer number deg(f) ∈ z such that f∗[a] = deg(f)[b], where [a] ∈ hm(a) or [b] ∈ hm(b) denotes a generator, defined by the orientation of a or b respectively [5]. consider mappings fj : s n → n such that vj ⊃ ∂bj ∩ n, where vj is a domain in r n+1 bounded by the hyper-surface fj(bj), fj is w0 on each bi with i 6= j, while the degree of the mapping fj from s n onto fj(s n ) is equal to one. if there would be an isomorphism θ : (wm2n)t,h → (w m1n)t,h, then θ(b) would have a non trivial decomposition into the sum of non canceling non zero additives, which is induced by mappings fj : s n → n. nevertheless, an element b in (wm2n)t,h has not such decomposition. if two groups g1 and g2 are not isomorphic, then certainly (w me; n,g1, p)t,h and (wme; n,g2, p)t,h are not isomorphic. the construction of wrap groups can be spread on locally compact non compact m satisfying cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 117 conditions 2(ii − iv) or (ii, iii, iv′) changing (v) such that m̂ is locally compact non-compact ht p domain in al r , its boundary ∂m̂ may happen to be void. for this it is sufficient to restrict the family of functions to that of with compact supports f : m → w relative to w0 : m → w , that is suppw0 (f) := clm{x ∈ m : f(x) 6= y0 × e} is compact, clma denotes the closure of a subset a in m. then classes of equivalent elements are given with the help of closures of orbits of the group of all ht p diffeomorphisms g with compact supports preserving marked points difht p,c (m,{s0,q : q = 1, ...,k}) that is suppid(g) := clm{x ∈ m : g(x) 6= x} are compact, where id(x) = x for each x ∈ m. then wrap groups (wme)t,h for manifolds m such as hyperboloid of one sheet, one sheet of two-sheeted hyperboloid, elliptic hyperboloid, hyperbolic paraboloid and so on in larger dimensional manifolds over ar. for non compact locally compact manifolds it is possible also consider an infinite countable discrete set of marked points or of isolated singularities. these examples can be naturally generalized for certain knotted manifolds arising from the given above. milnor and lefshetz have used for m = s1 and g = {e} the diffeomorphism group preserving an orientation and a marked point of s1. so their loop group l(s1,n) may be non-commutative. the iterated loop group l(s1,l(sn−1,n)) is isomorphic with l(sn,n), where the latter group is supplied with the uniformity from the iterated loop group, so n times iterated loop group of s1 gives loop group of sn [4]. for dimrm > 1 orientation preservation loss its significance. here above it was used the diffeomorphism group without any demands on orientation preservation of m such that two copies of m in the wedge product already are not distinguished in equivalence classes and for commutative g it gives a commutative wrap group. mention for comparison homotopy groups. the group πq(x) for a topological space x with a marked point x0 in view of proposition 17.1 (b) [2] is commutative for q > 1. for q = 1 the fundamental group π1(x) may be non-commutative, but it is always commutative in the particular case, when x = g is an arcwise connected topological group (see §49(g) in [32]). 11. proposition. let l(s1,n) be an h1 p loop group in the classical sense. then the iterated loop group l(s1,l(s1,n)) is commutative. proof. consider two elements a,b ∈ l(s1,l(s1,n)) and two mappings f ∈ a, g ∈ b, (f(x))(y) = f(x,y) ∈ n, where x,y ∈ i = [0, 1] ⊂ r, e2πx ∈ s1. an inverse element d−1 of d ∈ l(s1,n) is defined as the equivalence class d−1 =< h− >, where h ∈ d, h−(x) : h(1 − x). then (1) f(x, 1 −y) = (f(x))(1 −y) ∈ a−1 and g(x, 1 −y) = (g(x))(1 −y) ∈ b−1 for l(s1,l(s1,n)) and symmetrically (2) (f(y))(1 − x) = f(1 − x,y) ∈ a−1 and (g(y))(1 − x) = g(1 − x,y) ∈ b−1. on the other hand, f ∨ g corresponds to ab, and g ∨ f corresponds to ba, where the reduced product s1 ∧ s1 is ht p -diffeomorphic with s2 in the sense of pseudo-manifolds up to critical subsets of codimension not less than two. consider (s1 ∨s1) ∧ (s1 ∨s1) and (f ∨w0) ∨ (w0 ∨g) and (g ∨w0) ∨ (w0 ∨f) and the iterated 118 s.v. ludkovsky cubo 11, 1 (2009) equivalence relation r1,h. this situation corresponds to m̂ = i 2 divided into four quadrats by segments {1/2} × [0, 1] and [0, 1] × {1/2} with the corresponding domains for f, g and w0 in the considered wedge products, where < f ∨ w0 >=< w0 ∨ f >=< f > is the same class of equvalent elements. since g = {e}, (ab)−1 = b−1a−1, then g(1−x,y)∨f(1−x,y) is in the same class of equivalent elements as g(x, 1 − y) ∨ f(x, 1 − y). but due to inclusions (1, 2) < g(1 − x,y) ∨ f(1 − x,y) >=< f(x,y)∨g(x,y) >−1 and < f(x,y)∨g(x,y) >=< g(x, 1−y)∨f(x, 1−y) >−1 and < h(x,y) >−1=< h(x, 1 − y) >=< h(1 − x,y) > for h ∈ ab, consequently, < h(x,y) >=< h(1 − x, 1 − y) > and < (f ∨ g)(x, 1 − y) >< f(x, 1 − y) ∨ g(x, 1 − y) >∈ (ab)−1, since (x,y) 7→ (1 − x, 1 − y) interchange two spheres in the wedge product s2 ∨ s2. hence a−1b−1 = b−1a−1 and inevitably ab = ba. 12. theorem. let m and n be connected both either c∞ riemann or ar holomorphic manifolds with corners, where m is compact and dimm ≥ 1 and dimn > 1. then (wmn)t,h has no any nontrivial continuous local one parameter subgroup gb for b ∈ (−ǫ,ǫ) with ǫ > 0. proof. suppose the contrary, that {gb : b ∈ (−ǫ,ǫ)} with ǫ > 0 is a local nontrivial one parameter subgroup, that is, gb 6= e for b 6= 0. then to gδ for a marked 0 < δ < ǫ there corresponds f = fδ ∈ h ∞ p such that < f >t,h= g δ, where f ∈ ht p . if f(u) = {y0 × e} for a sufficiently small connected open neighborhood u of s0,q in m, then there exists a sequence f ◦ψn in the equivalence class < f >t,h with a family of diffeomorphisms ψn ∈ difh t p (m; {s0,q : q = 1, ...,k}) such that limn→∞ diamψn(u) = 0 and ⋂∞ n=1 ψn(u) = {s0,q}. if h(x) 6= y0, then in view of the continuity of h there exists an open neighborhood p of x in m such that y0 /∈ h(p). consider the covariant differentiation ∇ on the manifold m (see [12]). the set sh of points, where ∇ kh is discontinuous is a submanifold of codimension not less than one, hence of measure zero relative to the riemann volume element in m. for others points x in m, x ∈ m \ sh, all ∇ kh are continuous. take then open v = v (f) in m such that v ⊃ u and ∇k ν f|∂v 6= 0 for some k ∈ n, where ∇νf(x) := limz→x,z∈m\v ∇νf(z), ν is a normal (perpendicular) to ∂v in m at a point x in the boundary ∂v of v in m. practically take a minimal k = k(x) with such property. since m is compact and ∂v := cl(v ) ∩ cl(m \ v ) is closed in m, then ∂v is compact. the function x 7→ k(x) ∈ n is continuous, since f and ∇lf for each l are continuous. but n is discrete, hence each ∂qv := {x ∈ ∂v : k(x) = q} is open in v . therefore, ∂v is a finite union of ∂qv , 1 ≤ q ≤ qm, where qm : maxx∈∂v k(x) < ∞ for f = fδ, since ∂v is compact. thus, there exists a subset λ ⊂ {1, ...,qm} such that ∂v = ⋃ q∈λ ∂qv and ∂qv 6= ∅ for each q ∈ λ. if ∇ lf(x) = 0 for l = 1, ...,k(x) − 1 and ∇k(x)f(x) 6= 0, then ∇k(x)f(ψ(y)) = ∇k(x)(ψ(y)).(∇ψ(y))⊗k(x) 6= 0 for y ∈ m such that ψ(y) = x, since ∇ψ(y) 6= 0, where ψ ∈ difh∞ p (m; {s0,q : q = 1, ...,k}). we can take ǫ > 0 such that {gb : b ∈ (−ǫ,ǫ)} ⊂ u, where u = −u is a connected symmetric open neighborhood of e in (wmn)t,h. since g b1 + gb2 = gb1+b2 for each b1,b2,b1 + b2 ∈ (−ǫ,ǫ), then limt→0 g b = e for the local one parameter subgroup and in particular limm→∞ g 1/m = e, where m ∈ n. take δ = δm = 1/m and f = fm ∈ h ∞ p such that < fm >t,h= g 1/m. on the other hand, jg1/m = gj/m for each j < mǫ, j ∈ n, hence fj/m(m) = f1/m(m) for each j < mǫ, since f ∨ h(m ∨ m) = f(m) ∨ h(m) and using embedding η of (smn)t,h into (w mn)t,h. cubo 11, 1 (2009) wrap groups of fiber bundles and their structure 119 the function |∇ k(x) ν fδ(x)| for x ∈ ∂v is continuous by δ due to the sobolev embedding theorem [25], 0 < δ < ǫ, consequently, infx∈∂v |∇ k(x) ν fδ(x)| > 0, since ∂v is compact. we can choose a family fδ such that z (l) (δ,x) := ∇lfδ(x) is continuous for each 0 ≤ l ≤ k0 by (δ,x) ∈ (−ǫ,ǫ) × m, since {gb : b ∈ (−ǫ,ǫ)} is the continuous by b one parameter subgroup, where k0 := qm(δ0). therefore, for this family there exists a neighborhood [−ǫ + c,ǫ − c] such that δ0 ∈ [−ǫ+c,ǫ−c] ⊂ (−ǫ,ǫ) with 0 < c < ǫ/3 such that qm(δ) ≤ k0 for each δ ∈ [−ǫ+c,ǫ−c] with a suitable choice of v (fδ), since n is discrete. on the other hand, supx∈∂v (fδ),0<δ≤ǫ−c |∇ k(x) ν fδ(x)| ≤ sup x∈m,0<δ≤ǫ−c |∇ k(x) ν fδ(x)| =: b < ∞, since m and [−ǫ + c,ǫ − c] are compact. therefore, for this family there exists a neighborhood [−ǫ+c,ǫ−c] such that δ0 ∈ [−ǫ+c,ǫ−c] ⊂ (−ǫ,ǫ) with 0 < c < ǫ/3 such that qm(δ) ≤ k0 for each δ ∈ [−ǫ + c,ǫ − c] with a suitable choice of v (fδ), since n is discrete. then lim δ→0,δ>0|∇ k(x) ν fδ(x)| =: b > 0 for x ∈ ∂v with a suitable choice of v = v (fδ), since m is connected, dimm ≥ 1 and infm∈n diamfj/m(m) > 0 for a marked δ0 = j/m0 < ǫ with j,m > m0 ∈ n mutually prime, (j,m) = 1, (j,m0) = 1. to < fl/m >t,h there corresponds < f1/m >t,h ∨...∨ < f1/m >t,h=:< f1/m > ∨l t,h which is the l-fold wedge product. thus there exists c = const > 0 for m such that |∇ k(x) ν fl/m(x)| ≥ cl infy∈∂v (f1/m) |∇ k(y) ν f1/m(y)| ≥ clb, where c > 0 is fixed for a chosen atlas at(m) with given transition mappings φi ◦ φ −1 j of charts. consider δ0 ≤ l/m < ǫ − c and m and l tending to the infinity. then this gives b ≥ clb for each l ∈ n, that is the contradictory inequality, hence (wmn)t,h does not contain any non trivial local one parameter subgroup. received: january 2008. revised: august 2008. 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[39] k. yano, m. ako, an affine connection in almost quaternion manifolds. j. differ. geom. 3 (1973), 341-347. cubo a mathematical journal vol.10, n o ¯ 03, (171–194). october 2008 rosso-yamane theorem on pbw basis of uq(an ) ∗ yuqun chen, hongshan shao school of mathematical sciences, south china normal university, guangzhou 510631, p. r. china emails: yqchen@scnu.edu.cn, shaohongshan118@163.com and k.p. shum department of mathematics, the university of hong kong, pokfulam road, hong kong, china (sar) email: kpshum@maths.hku.hk abstract let uq(an ) be the drinfeld-jimbo quantum group of type an . in this paper, by using gröbner-shirshov bases, we give a simple (but not short) proof of the rosso-yamane theorem on pbw basis of uq(an ). resumen sea uq(an ) el grupo cuántico de drinfel-jimbo de tipo an . en este art́ıculo, usando bases de gröbner-shirshov damos una demonstración simple (pero no corta) del teorema de rosso-yamane sobre bases pbw de uq(an ). key words and phrases: quantum group, quantum enveloping algebra, gröbner-shirshov basis. math. subj. class.: 20g42, 16s15, 13p10. ∗supported by the nnsf of china (no.10771077) and the nsf of guangdong province (no.06025062). 172 yuqun chen et al. cubo 10, 3 (2008) 1 introduction since any algebra (commutative, associative, lie), as well as any module over an algebra, can be presented by generators and defining relations, it is important to have a general method to deal with these presentations. such a method now exists and is called the gröbner bases method (due to b. buchberger [18], [19]), or standand bases method (due to h. hironaka [21]), or gröbnershirshov bases method (due to a. i. shirshov [35]). the original shirshov’s paper [35] is for lie algebra presentations, but it can be easily adopted for associative algebra presentations as well, see l. a. bokut [3] and g. bergman [1]. let, for example, l = lie(x|[xixj ] − ∑ αkij xk, i > j, xi, xj , xk ∈ x) be a lie algebra over a field (or a commutative ring) k presented by a k-basis x and the multiplication table. then s = {[xixj ] − ∑ αkij xk| i > j, xi, xj , xk ∈ x} is a gröbner-shirshov basis (subset) of the free lie algebra lie(x) over k. on the other hand, the universal enveloping algebra u (l) = k〈x|xixj − xjxi − ∑ αkij xk, i > j, xi, xj , xk ∈ x〉 is the associative algebra presented by the same set x and the defining relations s(−) (we rewrite s using [xy] = xy − yx). there is a general but not difficult result that for any s ⊂ lie(x), s is a gröbner-shirshov basis in the sense of lie algebras if and only if s(−) ⊂ k〈x〉 is a gröbner-shirshov basis in the sense of associative algebras (see, for example, [9] and [7]). this means that in our case, s(−) is a gröbner-shirshov basis (subset) in k〈x〉. by composition-diamond lemma (see below), the s-irreducible words on x, irr(s) = {xi1 . . . xik , i1 ≤ . . . ≤ ik, k ≥ 0} form a k-basis of u (l). this is a conceptional proof of the pbw-theorem by using gröbner-shirshov bases theory. there are many results on gröbner-shirshov bases for associative and lie algebras, as well as for semigroups, groups, conformal algebras, dialgebras, and so on, see, for example, surveys [14], [15], [25] and [8]. let us mention those for simple lie algebras and lie superalgebras via serre’s presentations ([10], [11], [12], [13], [9]), for modules over simple lie algebras and iwahori-hecke algebras ([23], [24], [25]), for kac-moody algebras of types a (1) n , b (1) n , c (1) n , d (1) n ([31], [32], [33]), for coxeter groups ([17]), for braid groups via artin-burau, artin-garside and briman-ko-lee presentations ([4], [5] and [6]). drinfeld-jimbo ([20], [22]) presentations for quantized enveloping algebras uq(g), where g is a semisimple lie algebra, are a natural source of associative presentations. m. rosso [34] and i. yamane [36] found the pbw-basis of uq(an ). g. lusztig [29] and [30], and m. kashiwara [26] and [27] found the bases of uq(g) for any semisimple algebra g, as well as for their representations. their approach work equally well for quantized enveloping algebras associated with arbitrary symmetrizable cartan matrix, not just those corresponding to finite dimensional lie algebras. v. k. kharchenko [28] found the approach to linear bases of quantized enveloping algebras via the so called character hopf algebras. it the paper [16], gröbner-shirshov bases approach was applied to study uq(g) for any symmetrizable cartan matrix. using this approach, they got a new proof of the triangular decomposicubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 173 tion of uq(g) (see, for example, jantzen [37]). for uq(an ), it was proved by bokut and malcolmson [16] that the jimbo relations (see [36]) of u +q (an ) constitute a gröbner-shirshov basis of u + q (an ) in jimbo generators xij , 1 ≤ i, j ≤ n + 1 (see below). in this paper, we give an elementary proof that jimbo relations s is a gröbner-shirshov basis of u +q (an ). for such a purpose, we just check all possible compositions of polynomials from s and proved that all them can be resolved. also in §1 in this paper, we are giving necessary definitions and composition-diamond lemma following shirshov [35]. 2 preliminaries we first cite some concepts and results from the literature which are related to the gröbnershirshov bases for associative algebras. let k be a field, k〈x〉 the free associative algebra over k generated by x and x∗ the free monoid generated by x, where the empty word is the identity which is denoted by 1. for a word w ∈ x∗, we denote the length of w by deg(w). let x∗ be a well ordered set. let f ∈ k〈x〉 with the leading word f̄ . then we call f monic if f̄ has coefficient 1. definition 2.1. ([35], see also [2], [3]) let f and g be two monic polynomials in k〈x〉 and < a well ordering on x∗. then, there are two kinds of compositions: (i) if w is a word such that w = f̄ b = aḡ for some a, b ∈ x∗ with deg(f̄ )+deg(ḡ) >deg(w), then the polynomial (f, g)w = f b − ag is called the intersection composition of f and g with respect to w. (ii) if w = f̄ = aḡb for some a, b ∈ x∗, then the polynomial (f, g)w = f − agb is called the inclusion composition of f and g with respect to w. definition 2.2. ([2], [3], cf. [35]) let s ⊂ k〈x〉 such that every s ∈ s is monic. then the composition (f, g)w is called trivial modulo (s, w) if (f, g)w = ∑ αiaisibi, where each αi ∈ k, ai, bi ∈ x ∗, si ∈ s and aisibi < w. if this is the case, then we write (f, g)w ≡ 0 mod(s, w). in general, for p, q ∈ k〈x〉, we write p ≡ q mod(s, w) which means that p − q = ∑ αiaisibi, where each αi ∈ k, ai, bi ∈ x ∗, si ∈ s and aisibi < w. definition 2.3. ([2], [3], cf. [35]) we call the set s with respect to the well ordering < a gröbner-shirshov set (basis) in k〈x〉 if any composition of polynomials in s is trivial modulo s. if a subset s of k〈x〉 is not a gröbner-shirshov basis, then we can add to s all nontrivial compositions of polynomials of s, and by continuing this process (maybe infinitely) many times, we eventually obtain a gröbner-shirshov basis sc. such a process is called the shirshov algorithm. 174 yuqun chen et al. cubo 10, 3 (2008) a well ordering > on x∗ is called a monomial order if it is compatible with the multiplication of words, that is, for u, v ∈ x∗, we have u > v ⇒ w1uw2 > w1vw2, f or all w1, w2 ∈ x ∗. a standard example of monomial order on x∗ is the deg-lex order to compare two words first by degree and then lexicographically, where x is a well ordered set. the following lemma was first proved by shirshov [35] for free lie algebras (with deg-lex order) in 1962 (see also bokut [2]). in 1976, bokut [3] specialized the approach of shirshov to associative algebras (see also bergman [1]). for the case of commutative polynomials, this lemma is known as the buchberger’s theorem in [18] and [19]. lemma 2.4. (composition-diamond lemma) let k be a field, k〈x|s〉 = k〈x〉/id(s) and < a monomial order on x∗, where id(s) is the ideal of k〈x〉 generated by s. then the following statements are equivalent: (i) s is a gröbner-shirshov basis. (ii) f ∈ id(s) ⇒ f̄ = as̄b for some s ∈ s and a, b ∈ x∗. (iii) irr(s) = {u ∈ x∗|u 6= as̄b, s ∈ s, a, b ∈ x∗} is a basis of the algebra k〈x|s〉. � 3 rosso-yamane theorem on pbw basis of uq(an ) let k be a field, a = (aij ) an integral symmetrizable n × n cartan matrix so that aii = 2, aij ≤ 0 (i 6= j) and there exists a diagonal matrix d with diagonal entries di which are nonzero integers such that the product da is symmetric. let q be a nonzero element of k such that q4di 6= 1 for each i. then the quantum enveloping algebra is (see [20], [22]) uq(a) = k〈x ∪ h ∪ y |s + ∪ k ∪ t ∪ s−〉, where cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 175 x = {xi}, h = {h±1 i }, y = {yi}, s+ = { 1−aij ∑ ν=0 (−1)ν ( 1 − aij ν ) t x 1−aij −ν i xj x ν i , where i 6= j, t = q 2di}, s− = { 1−aij ∑ ν=0 (−1)ν ( 1 − aij ν ) t y 1−aij −ν i yj y ν i , where i 6= j, t = q 2di}, k = {hihj − hj hi, hih −1 i − 1, h−1 i hi − 1, xjh ±1 i − q∓diaij h±1xj , h ±1 i yj − yj h ±1}, t = {xiyj − yjxi − δij h2i − h −2 i q2di − q−2di } and ( m n ) t =    n ∏ i=1 t m−i+1 −t i−m−1 ti−t−i m > n > 0, 1 n = 0 or m = n. let a = an =          2 −1 0 · · · 0 −1 2 −1 · · · 0 0 −1 2 · · · 0 · · · · · 0 0 0 · · · 2          and q8 6= 1. it is reminded that in this case, the diagonal matrix d is identity. we introduce some new variables defined by jimbo (see [36]) which generate uq(an ): ˜x = {xij , 1 ≤ i < j ≤ n + 1}, where xij = { xi j = i + 1, qxi,j−1xj−1,j − q −1xj−1,j xi,j−1 j > i + 1. we now order the set ˜x in the following way. xmn > xij ⇐⇒ (m, n) >lex (i, j). 176 yuqun chen et al. cubo 10, 3 (2008) let us recall from yamane [36] the following notation: c1 = {((i, j), (m, n))|i = m < j < n}, c2 = {((i, j), (m, n))|i < m < n < j}, c3 = {((i, j), (m, n))|i < m < j = n}, c4 = {((i, j), (m, n))|i < m < j < n}, c5 = {((i, j), (m, n))|i < j = m < n}, c6 = {((i, j), (m, n))|i < j < m < n}. let the set ˜s+ consist of jimbo relations: xmnxij − q −2xij xmn ((i, j), (m, n)) ∈ c1 ∪ c3, xmnxij − xij xmn ((i, j), (m, n)) ∈ c2 ∪ c6, xmnxij − xij xmn + (q 2 − q−2)xinxmj ((i, j), (m, n)) ∈ c4, xmnxij − q 2xij xmn + qxin ((i, j), (m, n)) ∈ c5. it is easily seen that u +q (an ) = k〈 ˜x|˜s+〉. the following theorem is from [16]. theorem 3.1. ([16] theorem 4.1) let the notation be as before. then, with the deg-lex order on ˜x∗, ˜s+ is a gröbner-shirshov basis for k〈 ˜x|˜s+〉 = u +q (an ). proof. we will prove that all compositions in ˜s+ are trivial modulo ˜s+. we consider the following cases. case 1. f = xmnxij − q −2xij xmn, g = xij xkl − q −2xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −q −2 xij xmnxkl + q −2 xmnxklxij . there are four subcases to consider. ((i, j), (m, n)) ∈ c1 ((i, j), (m, n)) ∈ c3 ((k, l), (i, j)) ∈ c1 1.1. ((k, l), (m, n)) ∈ c1 1.3. ((k, l), (m, n)) ∈ c4, c5 or c6 ((k, l), (i, j)) ∈ c3 1.2. ((k, l), (m, n)) ∈ c4 1.4. ((k, l), (m, n)) ∈ c3 1.1. ((i, j), (m, n)) ∈ c1, ((k, l), (i, j)) ∈ c1 and ((k, l), (m, n)) ∈ c1. then, we have (f, g)w ≡ −q −4xij xklxmn + q −4xklxmnxij ≡ −q−6xklxij xmn + q −6 xklxij xmn ≡ 0. cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 177 1.2. ((i, j), (m, n)) ∈ c1, ((k, l), (i, j)) ∈ c3 and ((k, l), (m, n)) ∈ c4. then, we have (i, j) = (m, l), ((k, n), (i, j)) ∈ c2 and (f, g)w ≡ −q −2xij [xklxmn − (q 2 − q−2)xknxml] + q −2[xklxmn − (q 2 − q−2)xknxml]xij ≡ −q−4xklxij xmn + q −2(q2 − q−2)xknxij xml +q−4xklxij xmn − q −2(q2 − q−2)xknxij xml ≡ 0. 1.3. ((i, j), (m, n)) ∈ c3, ((k, l), (i, j)) ∈ c1 and ((k, l), (m, n)) ∈ c4, c5 or c6. 1.3.1. if ((k, l), (m, n)) ∈ c4 (m < l), then (k, n) = (i, j), ((i, j), (m, l)) ∈ c2 and (f, g)w ≡ −q −2 xij [xklxmn − (q 2 − q−2)xknxml] + q −2[xklxmn − (q 2 − q−2)xknxml]xij ≡ −q−4xklxij xmn + q −2(q2 − q−2)xij xknxml + q −4 xklxij xmn −q−2(q2 − q−2)xknxij xml ≡ 0. 1.3.2. if ((k, l), (m, n)) ∈ c5 (m = l), then (k, n) = (i, j) and (f, g)w ≡ −q −2xij (q 2xklxmn − qxkn) + q −2(q2xklxmn − qxkn)xij ≡ −xij xklxmn + q −1xij xkn + xklxmnxij − q −1xknxij ≡ −q−2xklxij xmn + q −2xklxij xmn ≡ 0. 1.3.3. if ((k, l), (m, n)) ∈ c6 (m > l), then (f, g)w ≡ −q −2 xij xklxmn + q −2 xklxmnxij ≡ −q−4xklxij xmn + q −4xklxij xmn ≡ 0. 1.4. ((i, j), (m, n)) ∈ c3, ((k, l), (i, j)) ∈ c3 and ((k, l), (m, n)) ∈ c3. this case is similar to 1.1. case 2. f = xmnxij − q −2xij xmn, g = xij xkl − xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −q −2xij xmnxkl + xmnxklxij . there are also four subcases to consider. ((i, j), (m, n)) ∈ c1 ((i, j), (m, n)) ∈ c3 ((k, l), (i, j)) ∈ c2 2.1. ((k, l), (m, n)) ∈ c2, c3 or c4 2.3. ((k, l), (m, n)) ∈ c2 ((k, l), (i, j)) ∈ c6 2.2. ((k, l), (m, n)) ∈ c6 2.4. ((k, l), (m, n)) ∈ c6 178 yuqun chen et al. cubo 10, 3 (2008) 2.1. ((i, j), (m, n)) ∈ c1, ((k, l), (i, j)) ∈ c2 and ((k, l), (m, n)) ∈ c2, c3 or c4. 2.1.1. if ((k, l), (m, n)) ∈ c2 (n < l), then (f, g)w ≡ −q −2xij xklxmn + xklxmnxij ≡ −q−2xklxij xmn + q −2xklxij xmn ≡ 0. 2.1.2. if ((k, l), (m, n)) ∈ c3 (n = l), then (f, g)w ≡ −q −4 xij xklxmn + q −2 xklxmnxij ≡ −q−4xklxij xmn + q −4 xklxij xmn ≡ 0. 2.1.3. if ((k, l), (m, n)) ∈ c4 (n > l), then ((k, n), (i, j)) ∈ c2, ((i, j), (m, l)) ∈ c1 and (f, g)w ≡ −q −2xij [xklxmn − (q 2 − q−2)xknxml] + [xklxmn − (q 2 − q−2)xknxml]xij ≡ −q−2xklxij xmn + q −2(q2 − q−2)xknxij xml + q −2xklxij xmn −q−2(q2 − q−2)xknxij xml ≡ 0. for the cases 2.2, 2.3 and 2.4, the proofs are similar to 2.1.1. case 3. f = xmnxij − q −2xij xmn, g = xij xkl − xklxij + (q 2 − q−2)xkj xil, w = xmnxij xkl. in the case, we have (f, g)w = −q −2 xij xmnxkl + xmnxklxij − (q 2 − q−2)xmnxkj xil. there are two subcases to consider. ((i, j), (m, n)) ∈ c1 ((i, j), (m, n)) ∈ c3 3.1. 3.2. ((k, l), (i, j)) ∈ c4 ((k, l), (m, n)), ((k, j), (m, n)) ∈ c4 ((k, l), (m, n)) ∈ c4, c5 or c6 ((k, j), (m, n)) ∈ c3 3.1. ((i, j), (m, n)) ∈ c1, ((k, l), (i, j)) ∈ c4 and (k, l), (m, n)), ((k, j), (m, n)) ∈ c4. then, we have ((k, n), (i, j)) ∈ c2, ((i, l), (m, n)) ∈ c1, ((i, l), (m, j)) ∈ c1, ((m, l), (i, j)) cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 179 ∈ c1 and (f, g)w ≡ −q −2xij [xklxmn − (q 2 − q−2)xknxml] + [xklxmn − (q 2 − q−2)xknxml]xij −(q2 − q−2)[xkj xmn − (q 2 − q−2)xknxmj ]xil ≡ −q−2[xklxij − (q 2 − q−2)xkj xil]xmn + q −2(q2 − q−2)xknxij xml + q −2xklxij xmn −(q2 − q−2)xknxmlxij − q −2(q2 − q−2)xkj xilxmn + q −2(q2 − q−2)2xknxilxmj ≡ q−4(q2 − q−2)xknxmlxij − (q 2 − q−2)xknxmlxij + q −2(q2 − q−2)xknxmlxij ≡ 0. 3.2. ((i, j), (m, n)) ∈ c3, ((k, l), (i, j)) ∈ c4, (k, l), (m, n)) ∈ c4, c5 or c6 and ((k, j), (m, n)) ∈ c3. 3.2.1. if ((k, l), (m, n)) ∈ c4 (l > m) and ((k, j), (m, n)) ∈ c3, then ((k, n), (i, j)) ∈ c3, ((i, j), (m, l)) ∈ c2, ((i, l), (m, n)) ∈ c4 and (f, g)w ≡ −q −2xij [xklxmn − (q 2 − q−2)xknxml] + [xklxmn − (q 2 − q−2)xknxml]xij −q−2(q2 − q−2)xkj xmnxil ≡ −q−2[xklxij − (q 2 − q−2)xkj xil]xmn + q −4(q2 − q−2)xknxij xml + q −2xklxij xmn −(q2 − q−2)xknxij xml − q −2(q2 − q−2)xkj [xilxmn − (q 2 − q−2)xinxml] ≡ 0. 3.2.2. if ((k, l), (m, n)) ∈ c5 (l = m) and ((k, j), (m, n)) ∈ c3, then ((k, l), (i, j)) ∈ c4, ((k, n), (i, j)) ∈ c3, ((i, l), (m, n)) ∈ c5 and (f, g)w ≡ −q −2xij (q 2xklxmn − qxkn) + (q 2xklxmn − qxkn)xij − q −2(q2 − q−2)xkj xmnxil ≡ −[xklxij − (q 2 − q−2)xkj xil]xmn + q −3 xknxij + xklxij xmn − qxknxij −q−2(q2 − q−2)xkj [q 2 xilxmn − qxin] ≡ q−3xknxij − qxknxij + q −1(q2 − q−2)xknxij ≡ 0. 3.2.3. if ((k, l), (m, n)) ∈ c6 (l < m) and ((k, j), (m, n)) ∈ c3, then ((i, l), (m, n)) ∈ c6 and (f, g)w ≡ −q −2xij xklxmn + xklxmnxij − q −2(q2 − q−2)xkj xmnxil ≡ −q−2[xklxij − (q 2 − q−2)xkj xil]xmn + q −2xklxij xmn − q −2(q2 − q−2)xkj xilxmn ≡ 0. case 4. f = xmnxij − q −2xij xmn, g = xij xkl − q 2xklxij + qxkj , w = xmnxij xkl. in the case, we have (f, g)w = −q −2 xij xmnxkl + q 2 xmnxklxij − qxmnxkj . 180 yuqun chen et al. cubo 10, 3 (2008) there are two subcases to consider. ((i, j), (m, n)) ∈ c1 ((i, j), (m, n)) ∈ c3 4.1. 4.2. ((k, l), (i, j)) ∈ c5 ((k, l), (m, n)) ∈ c5 ((k, l), (m, n)) ∈ c6 ((k, j), (m, n)) ∈ c4 ((k, j), (m, n)) ∈ c3 4.1. ((i, j), (m, n)) ∈ c1, ((k, l), (i, j)) ∈ c5, ((k, l), (m, n)) ∈ c5 and ((k, j), (m, n)) ∈ c4. then, we have ((k, n), (i, j)) ∈ c2 (m = i) and (f, g)w ≡ −q −2 xij (q 2 xklxmn − qxkn) + q 2(q2xklxmn − qxkn)xij −q[xkj xmn − (q 2 − q−2)xknxmj ] ≡ −(q2xklxij − qxkj )xmn + q −1xknxij + q 2xklxij xmn −q3xknxij − qxkj xmn + q(q 2 − q−2)xknxmj ≡ 0. 4.2. ((i, j), (m, n)) ∈ c3, ((k, l), (i, j)) ∈ c5, ((k, l), (m, n)) ∈ c6 and ((k, j), (m, n)) ∈ c3. then, we have (f, g)w ≡ −q −2xij xklxmn + q 2xklxmnxij − q −1xkj xmn ≡ −q−2(q2xklxij − qxkj )xmn + xklxij xmn − q −1xkj xmn ≡ 0. case 5. f = xmnxij − xij xmn, g = xij xkl − q −2xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + q −2xmnxklxij . there are four subcases to consider. ((i, j), (m, n)) ∈ c2 ((i, j), (m, n)) ∈ c6 ((k, l), (i, j)) ∈ c1 5.1. ((k, l), (m, n)) ∈ c2, c3, c4, c5 or c6 5.3. ((k, l), (m, n)) ∈ c6 ((k, l), (i, j)) ∈ c3 5.2. ((k, l), (m, n)) ∈ c2 5.4. ((k, l), (m, n)) ∈ c6 5.1. ((i, j), (m, n)) ∈ c2, ((k, l), (i, j)) ∈ c1, and ((k, l), (m, n)) ∈ c2, c3, c4, c5 or c6. 5.1.1. if ((k, l), (m, n)) ∈ c2 (l > n), then we have ((k, l), (i, j)) ∈ c1 and (f, g)w ≡ −xij xklxmn + q −2xklxmnxij ≡ −q−2xklxij xmn + q −2 xklxij xmn ≡ 0. cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 181 5.1.2. if ((k, l), (m, n)) ∈ c3 (l = n), then (f, g)w ≡ −q −2xij xklxmn + q −4xklxmnxij ≡ −q−4xklxij xmn + q −4xklxij xmn ≡ 0. 5.1.3. if ((k, l), (m, n)) ∈ c4 (m < l < n), then we have ((k, l), (i, j)) ∈ c1, ((k, n), (i, j)) ∈ c1, ((i, j), (m, l)) ∈ c2 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + q −2[xklxmn − (q 2 − q−2)xknxml]xij ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + q −2 xklxmnxij − q −2(q2 − q−2)xknxmlxij ≡ −q−2xklxij xmn + q −2(q2 − q−2)xknxij xml + q −2 xklxij xmn −q−2(q2 − q−2)xknxij xml ≡ 0. 5.1.4. if ((k, l), (m, n)) ∈ c5 (m = l), then we have ((k, n), (i, j)) ∈ c1 and (f, g)w ≡ −xij (q 2 xklxmn − qxkn) + q −2(q2xklxmn − qxkn)xij ≡ −q2xij xklxmn + qxij xkn + xklxmnxij − q −1xknxij ≡ −xklxij xmn + q −1xknxij + xklxij xmn − q −1xknxij ≡ 0. 5.1.5. if ((k, l), (m, n)) ∈ c6 (l < m), the proof is similar to 5.1.1. for the cases of 5.2, 5.3 and 5.4, the proofs are also similar to 5.1.1. case 6. f = xmnxij − xij xmn, g = xij xkl − xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + xmnxklxij . there are four subcases to consider. ((i, j), (m, n)) ∈ c2 ((i, j), (m, n)) ∈ c6 ((k, l), (i, j)) ∈ c2 6.1. ((k, l), (m, n)) ∈ c2 6.3. ((k, l), (m, n)) ∈ c2, c3, c4, c5 or c6 ((k, l), (i, j)) ∈ c6 6.2. ((k, l), (m, n)) ∈ c6 6.4. ((k, l), (m, n)) ∈ c6 6.1. ((i, j), (m, n)) ∈ c2, ((k, l), (i, j)) ∈ c2 and ((k, l), (m, n)) ∈ c2. then, we have (f, g)w ≡ −xij xklxmn + xklxmnxij ≡ −xklxij xmn + xklxij xmn ≡ 0. 182 yuqun chen et al. cubo 10, 3 (2008) 6.2. ((i, j), (m, n)) ∈ c2, ((k, l), (i, j)) ∈ c6 and ((k, l), (m, n)) ∈ c6. this case is similar to 6.1. 6.3. ((i, j), (m, n)) ∈ c6, ((k, l), (i, j)) ∈ c2 and ((k, l), (m, n)) ∈ c2, , c3, c4, c5 or c6. 6.3.1. if ((k, l), (m, n)) ∈ c2 (l > n), the proof is similar to 6.1. 6.3.2. if ((k, l), (m, n)) ∈ c3 (l = n), then (f, g)w ≡ −q −2xij xklxmn + q −2xklxmnxij ≡ −q−2xklxij xmn + q −2xklxij xmn ≡ 0. 6.3.3. if ((k, l), (m, n)) ∈ c4 (m < l < n), then we have ((k, n), (i, j)) ∈ c2, ((i, j), (m, n)) ∈ c6 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + [xklxmn − (q 2 − q−2)xknxml]xij ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + xklxmnxij − (q 2 − q−2)xknxmlxij ≡ −xklxij xmn + (q 2 − q−2)xknxij xml + xklxij xmn − (q 2 − q−2)xknxij xml ≡ 0. 6.3.4. if ((k, l), (m, n)) ∈ c5 (m = l), then we have ((k, n), (i, j)) ∈ c2 and (f, g)w ≡ −xij (q 2 xklxmn − qxkn) + (q 2 xklxmn − qxkn)xij ≡ −q2xij xklxmn + qxij xkn + q 2 xklxmnxij − qxknxij ≡ −q2xklxij xmn + qxknxij + q 2xklxij xmn − qxknxij ≡ 0. 6.3.5. if ((k, l), (m, n)) ∈ c6 (l < m), the proof is similar to 6.1. 6.4. ((i, j), (m, n)) ∈ c6, ((k, l), (i, j)) ∈ c6 and ((k, l), (m, n)) ∈ c6. this case is also similar to 6.1. case 7. f = xmnxij − xij xmn, g = xij xkl − xklxij + (q 2 − q−2)xkj xil, w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + xmnxklxij − (q 2 − q−2)xmnxkj xil. there are two subcases to consider. ((i, j), (m, n)) ∈ c2 ((i, j), (m, n)) ∈ c6 7.1. 7.2. ((k, l), (i, j)) ∈ c4 ((k, l), (m, n)) ∈ c2, c3, c4, c5 or c6 ((k, l), (m, n)), ((k, j), (m, n)) ∈ c2 ((k, j), (m, n)) ∈ c6 cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 183 7.1. ((i, j), (m, n)) ∈ c2, ((k, l), (i, j)) ∈ c4, ((k, l), (m, n)) ∈ c2, c3, c4, c5 or c6 and ((k, j), (m, n)) ∈ c2. 7.1.1. if ((k, l), (m, n)) ∈ c2 (n < l) and ((k, j), (m, n)) ∈ c2, then we have ((i, l), (m, n)) ∈ c2 and (f, g)w ≡ −xij xklxmn + xklxmnxij − (q 2 − q−2)xkj xmnxil ≡ −[xklxij − (q 2 − q−2)xkj xil]xmn + xklxij xmn − (q 2 − q−2)xkj xilxmn ≡ 0. 7.1.2. if ((k, l), (m, n)) ∈ c3 (n = l) and ((k, j), (m, n)) ∈ c2, then ((i, l), (m, n)) ∈ c3 and (f, g)w ≡ −q −2 xij xklxmn + q −2 xklxmnxij − (q 2 − q−2)xkj xmnxil ≡ −q−2[xklxij − (q 2 − q−2)xkj xil]xmn + q −2 xklxij xmn − q −2(q2 − q−2)xkj xilxmn ≡ 0. 7.1.3. if ((k, l), (m, n)) ∈ c4 (m < l < n) and ((k, j), (m, n)) ∈ c2, then we obtain ((k, n), (i, j)) ∈ c4, ((i, j), (m, l)) ∈ c2, ((i, l), (m, n)) ∈ c4 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + [xklxmn − (q 2 − q−2)xknxml]xij −(q2 − q−2)xkj xmnxil ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + xklxmnxij − (q 2 − q−2)xknxmlxij −(q2 − q−2)xkj [xilxmn − (q 2 − q−2)xinxml] ≡ −[xklxij − (q 2 − q−2)xkj xil]xmn + (q 2 − q−2)[xknxij − (q 2 − q−2)xkj xin]xml +xklxij xmn − (q 2 − q−2)xknxij xml ≡ 0. 7.1.4. if ((k, l), (m, n)) ∈ c5 (m = l) and ((k, j), (m, n)) ∈ c2, then ((k, n), (i, j)) ∈ c4, ((i, l), (m, n)) ∈ c5 and (f, g)w ≡ −xij (q 2xklxmn − qxkn) + (q 2xklxmn − qxkn)xij − (q 2 − q−2)xkj xmnxil ≡ −q2xij xklxmn + qxij xkn + q 2xklxmnxij − qxknxij −(q2 − q−2)xkj (q 2xilxmn − qxin) ≡ −q2[xklxij − (q 2 − q−2)xkj xil]xmn + q[xknxij − (q 2 − q−2)xkj xin] +q2xklxij xmn − qxknxij − q 2(q2 − q−2)xkj xilxmn + q(q 2 − q−2)xkj xin ≡ 0. 7.1.5. if ((k, l), (m, n)) ∈ c6 (l < m) and ((k, j), (m, n)) ∈ c2, then ((i, l), (m, n)) ∈ c6. this case is similar to 7.1.1. 184 yuqun chen et al. cubo 10, 3 (2008) 7.2. ((i, j), (m, n)) ∈ c6, ((k, l), (i, j)) ∈ c4, ((k, l), (m, n)), ((k, j), (m, n)) ∈ c6. this case is also similar to 7.1.1. case 8. f = xmnxij − xij xmn, g = xij xkl − q 2xklxij + qxkj , w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + q 2xmnxklxij + qxmnxkj . there are two subcases to consider. ((i, j), (m, n)) ∈ c2 ((i, j), (m, n)) ∈ c6 8.1. 8.2. ((k, l), (i, j)) ∈ c5 ((k, l), (m, n)) ∈ c6 ((k, l), (m, n)), ((k, j), (m, n)) ∈ c6 ((k, j), (m, n)) ∈ c2 8.1. ((i, j), (m, n)) ∈ c2, ((k, l), (i, j)) ∈ c5, ((k, l), (m, n)) ∈ c6 and ((k, j), (m, n)) ∈ c2. then, we have (f, g)w ≡ −xij xklxmn + q 2xklxmnxij + qxkj xmn ≡ −(q2xklxij − qxkj )xmn + q 2xklxij xmn + qxkj xmn ≡ 0. 8.2. ((i, j), (m, n)) ∈ c6, ((k, l), (i, j)) ∈ c5, ((k, l), (m, n)), ((k, j), (m, n)) ∈ c6. this case is similar to 8.1. case 9. f = xmnxij − xij xmn + (q 2 − q−2)xinxmj , g = xij xkl − q −2xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + (q 2 − q−2)xinxmj xkl + q −2xmnxklxij . there are two subcases to consider. ((i, j), (m, n)) ∈ c4 ((k, l), (i, j)) ∈ c1 9.1. ((k, l), (m, n)), ((k, l), (m, j)) ∈ c4, c5 or c6 ((k, l), (i, j)) ∈ c3 9.2. ((k, l), (m, n)) ∈ c4 ((k, l), (m, j)) ∈ c3 9.1. ((i, j), (m, n)) ∈ c4, ((k, l), (i, j)) ∈ c1 and ((k, l), (m, n)), ((k, l), (m, j)) ∈ c4, c5 or c6. cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 185 9.1.1. if ((k, l), (m, n)), ((k, l), (m, j)) ∈ c4 (l > m), then we have ((i, j), (k, n)) ∈ c1, ((k, n), (m, l)) ∈ c2, ((k, j), (i, n)) ∈ c1, ((k, l), (i, n)) ∈ c1, ((i, j), (m, l)) ∈ c2 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + (q 2 − q−2)xin[xklxmj − (q 2 − q−2)xkj xml] +q−2[xklxmn − (q 2 − q−2)xknxml]xij ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + (q 2 − q−2)xinxklxmj −(q2 − q−2)2xinxkj xml + q −2xklxmnxij − q −2(q2 − q−2)xknxmlxij ≡ −q−2xklxij xmn + (q 2 − q−2)xij xknxml + q −2(q2 − q−2)xklxinxmj −q−2(q2 − q−2)2xkj xinxml + q −2xkl[xij xmn − (q 2 − q−2)xinxmj ] −q−2(q2 − q−2)xknxij xml ≡ (q2 − q−2)xij xknxml − q −2(q2 − q−2)2xkj xinxml − q −4(q2 − q−2)xij xknxml ≡ 0. 9.1.2. if ((k, l), (m, n)), ((k, l), (m, j)) ∈ c5 (l = m), then we have ((i, j), (k, n)) ∈ c1, ((k, l), (i, n), ((k, j), (i, n)) ∈ c1 and (f, g)w ≡ −xij (q 2xklxmn − qxkn) + (q 2 − q−2)xin(q 2xklxmj − qxkj ) +q−2(q2xklxmn − qxkn)xij ≡ −q2xij xklxmn + qxij xkn + q 2(q2 − q−2)xinxklxmj − q(q 2 − q−2)xinxkj +xklxmnxij − q −1xknxij ≡ −xklxij xmn + qxij xkn + (q 2 − q−2)xklxinxmj − q −1(q2 − q−2)xkj xin +xkl[xij xmn − (q 2 − q−2)xinxmj ] − q −3 xij xkn ≡ qxij xkn − qxkj xin + q −3 xkj xin − q −3 xij xkn ≡ 0. 9.1.3. if ((k, l), (m, n)), ((k, l), (m, j)) ∈ c6 (l < m), then we have ((k, l), (i, n)) ∈ c1 and (f, g)w ≡ −xij xklxmn − (q 2 − q−2)xinxklxmj + q −2 xklxmnxij ≡ −q−2xklxij xmn − q −2(q2 − q−2)xklxinxmj + q −2xkl[xij xmn − (q 2 − q−2)xinxmj ] ≡ 0. 9.2. ((i, j), (m, n)) ∈ c4, ((k, l), (i, j)) ∈ c3, ((k, l), (m, n)) ∈ c4 and ((k, l), (m, j)) ∈ c3. 186 yuqun chen et al. cubo 10, 3 (2008) then, we have ((k, n), (i, j)) ∈ c2, ((k, l), (i, n)) ∈ c4, ((i, j), (m, l)) ∈ c3 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + q −2(q2 − q−2)xinxklxmj +q−2[xklxmn − (q 2 − q−2)xknxml]xij ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + q −2(q2 − q−2)xinxklxmj + q −2 xklxmnxij −q−2(q2 − q−2)xknxmlxij ≡ −q−2xklxij xmn + (q 2 − q−2)xknxij xml + q −2(q2 − q−2)[xklxin −(q2 − q−2)xknxil]xmj + q −2xkl[xij xmn − (q 2 − q−2)xinxmj ] −q−4(q2 − q−2)xknxij xml ≡ (q2 − q−2)xknxij xml − q −2(q2 − q−2)xknxilxmj − q −4(q2 − q−2)xknxij xml ≡ 0. case 10. f = xmnxij − xij xmn + (q 2 − q−2)xinxmj , g = xij xkl − xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + (q 2 − q−2)xinxmj xkl + xmnxklxij . there are two subcases to consider. ((i, j), (m, n)) ∈ c4 ((k, l), (i, j)) ∈ c2 10.1. ((k, l), (m, n)) ∈ c2, c3 or c4 ((k, l), (m, j)) ∈ c2 ((k, l), (i, j)) ∈ c6 10.2. ((k, l), (m, n)), ((k, l), (m, j)) ∈ c6 10.1. ((i, j), (m, n)) ∈ c4, ((k, l), (i, j)) ∈ c2, ((k, l), (m, n)) ∈ c2, c3 or c4 and ((k, l), (m, j)) ∈ c2. 10.1.1. if ((k, l), (m, n)) ∈ c2 (l > n), then we have ((k, l), (i, n)) ∈ c2 and (f, g)w ≡ −xij xklxmn + (q 2 − q−2)xinxklxmj + xklxmnxij ≡ −xklxij xmn + (q 2 − q−2)xklxinxmj + xkl[xij xmn − (q 2 − q−2)xinxmj ] ≡ 0. 10.1.2. if ((k, l), (m, n)) ∈ c3 (l = n), then we have ((k, l), (i, n)) ∈ c3 and (f, g)w ≡ −q −2xij xklxmn + (q 2 − q−2)xinxklxmj + q −2xklxmnxij ≡ −q−2xklxij xmn + q −2(q2 − q−2)xklxinxmj + q −2 xkl[xij xmn − (q 2 − q−2)xinxmj ] ≡ 0. cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 187 10.1.3. if ((k, l), (m, n)) ∈ c4 (l < n), then we have ((k, n), (i, j)) ∈ c2, ((k, l), (i, n)) ∈ c4, ((i, j), (m, l)) ∈ c4 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + (q 2 − q−2)xinxklxmj +[xklxmn − (q 2 − q−2)xknxml]xij ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + (q 2 − q−2)xinxklxmj +xklxmnxij − (q 2 − q−2)xknxmlxij ≡ −xklxij xmn + (q 2 − q−2)xknxij xml + (q 2 − q−2)[xklxin − (q 2 − q−2)xknxil]xmj +xkl[xij xmn − (q 2 − q−2)xinxmj ] − (q 2 − q−2)xkn[xij xml − (q 2 − q−2)xilxmj ] ≡ 0. 10.2. ((i, j), (m, n)) ∈ c4, ((k, l), (i, j)) ∈ c6, ((k, l), (m, n)), (k, l), (m, j)) ∈ c6. this case is similar to 10.1. case 11. f = xmnxij −xij xmn + (q 2 −q−2)xinxmj , g = xij xkl −xklxij + (q 2 −q−2)xkj xil, w = xmnxij xkl. in the case, we have (f, g)w = −xij xmnxkl + (q 2 − q−2)xinxmj xkl + xmnxklxij − (q 2 − q−2)xmnxkj xil, with ((i, j), (m, n)) ∈ c4 ((k, l), (i, j)) ∈ c4 ((k, l), (m, n)), ((k, l), (m, j)) ∈ c4, c5 or c6 11.1. if ((k, l), (m, n)), ((k, l), (m, j)) ∈ c4 (l > m), then we have ((k, n), (i, j)) ∈ c2, ((k, l), (i, n)) ∈ c4, ((k, j), (i, n)) ∈ c4, ((i, j), (m, l)) ∈ c2, ((i, l), (m, n)) ∈ c4, ((i, l), (m, j)) ∈ c4 and (f, g)w ≡ −xij [xklxmn − (q 2 − q−2)xknxml] + (q 2 − q−2)xin[xklxmj − (q 2 − q−2)xkj xml] +[xklxmn − (q 2 − q−2)xknxml]xij − (q 2 − q−2)[xkj xmn − (q 2 − q−2)xknxmj ]xil ≡ −xij xklxmn + (q 2 − q−2)xij xknxml + (q 2 − q−2)xinxklxmj −(q2 − q−2)xinxkj xml + xklxmnxij − (q 2 − q−2)xknxmlxij −(q2 − q−2)xkj xmnxil + (q 2 − q−2)2xknxmj xil ≡ −[xklxij − (q 2 − q−2)xkj xil]xmn + (q 2 − q−2)xknxij xml + (q 2 − q−2)[xkj xin −(q2 − q−2)xknxil]xmj − (q 2 − q−2)[xkj xin − (q 2 − q−2)xknxij ]xml +xkl[xij xmn − (q 2 − q−2)xinxmj ] − (q 2 − q−2)xknxij xml −(q2 − q−2)xkj [xilxmn − (q 2 − q−2)xinxml] +(q2 − q−2)2xkn[xilxmj − (q 2 − q−2)xij xml] ≡ 0. 188 yuqun chen et al. cubo 10, 3 (2008) 11.2. if ((k, l), (m, n)), ((k, l), (m, j)) ∈ c5 (l = m), then we have ((k, n), (i, j)) ∈ c2, ((k, l), (i, n)) ∈ c4, ((k, j), (i, n)) ∈ c4, ((i, l), (m, n)) ∈ c5, ((i, l), (m, j)) ∈ c5 and (f, g)w ≡ −xij (q 2xklxmn − qxkn) + (q 2 − q−2)xin(q 2xklxmj − qxkj ) + (q 2xklxmn − qxkn)xij −(q2 − q−2)[xkj xmn − (q 2 − q−2)xknxmj ]xil ≡ −q2xij xklxmn + qxij xkn + q 2(q2 − q−2)xinxklxmj − q(q 2 − q−2)xinxkj +q2xklxmnxij − qxknxij − (q 2 − q−2)xkj xmnxil + (q 2 − q−2)2xknxmj xil ≡ −q2[xklxij − (q 2 − q−2)xkj xil]xmn + qxknxij + q 2(q2 − q−2)[xklxin −(q2 − q−2)xknxil]xmj − q(q 2 − q−2)[xkj xin − (q 2 − q−2)xknxij ] +q2xkl[xij xmn − (q 2 − q−2)xinxmj ] − qxknxij −(q2 − q−2)xkj [q 2xilxmn − qxin] + (q 2 − q−2)2xkn[q 2xilxmj − qxij ] ≡ 0. 11.3. if ((k, l), (m, n)), ((k, l), (m, j)) ∈ c6 (l < m), then ((k, j), (m, n)) ∈ c4, ((k, l), (i, n)) ∈ c4, ((i, l), (m, n)), ((i, l), (m, j)) ∈ c6 and (f, g)w ≡ −xij xklxmn + (q 2 − q−2)xinxklxmj + xklxmnxij −(q2 − q−2)[xkj xmn − (q 2 − q−2)xknxmj ]xil ≡ −[xklxij − (q 2 − q−2)xkj xil]xmn + (q 2 − q−2)[xklxin − (q 2 − q−2)xknxil]xmj +xkl[xij xmn − (q 2 − q−2)xinxmj ] − (q 2 − q−2)xkj xilxmn + (q 2 − q−2)2xknxilxmj ≡ 0. case 12. f = xmnxij −xij xmn+(q 2−q−2)xinxmj , g = xij xkl−q 2xklxij +qxkj , w = xmnxij xkl, with ((i, j), (m, n)) ∈ c4 ((k, l), (i, j)) ∈ c5 ((k, l), (m, n)), ((k, l), (m, j)) ∈ c6 ((k, j), (m, n)) ∈ c4 ((k, l), (i, n)) ∈ c5 in the case, we can deduce that (f, g)w = −xij xmnxkl + (q 2 − q−2)xinxmj xkl + q 2 xmnxklxij − qxmnxkj ≡ −xij xklxmn + (q 2 − q−2)xinxklxmj + q 2xkj xmnxij −q[xkj xmn − (q 2 − q−2)xknxmj ] ≡ −(q2xklxij − qxkj )xmn + (q 2 − q−2)(q2xklxin − qxkn)xmj +q2xkl[xij xmn − (q 2 − q−2)xinxmj ] − qxkj xmn + q(q 2 − q−2)xknxmj ≡ −q2xklxij xmn + qxkj xmn + q 2(q2 − q−2)xklxinxmj − q(q 2 − q−2)xknxmj +q2xklxij xmn − q 2(q2 − q−2)xklxinxmj − qxkj xmn + q(q 2 − q−2)xknxmj ≡ 0. cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 189 case 13. f = xmnxij − q 2xij xmn + qxin, g = xij xkl − q −2xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −q 2xij xmnxkl + qxinxkl + q −2xmnxklxij . there are two subcases to consider. ((i, j), (m, n)) ∈ c5 ((k, l), (i, j)) ∈ c1 13.1. ((k, l), (m, n)) ∈ c6 ((k, l), (i, n)) ∈ c1 ((k, l), (i, j)) ∈ c3 13.2. ((k, l), (m, n)) ∈ c5 ((k, l), (i, n)) ∈ c4 13.1. ((i, j), (m, n)) ∈ c5, ((k, l), (i, j)) ∈ c1, ((k, l), (m, n)) ∈ c6 and ((k, l), (i, n)) ∈ c1. then, we have (f, g)w = −q 2xij xklxmn + q −1xklxin + q −2xklxmnxij ≡ −xklxij xmn + q −1xklxin + q −2xkl(q 2xij xmn − qxin) ≡ −xklxij xmn + q −1xklxin + xklxij xmn − q −1xklxin ≡ 0. 13.2. ((i, j), (m, n)) ∈ c5, ((k, l), (i, j)) ∈ c3, ((k, l), (m, n)) ∈ c5 and ((k, l), (i, n)) ∈ c4. then, we have ((k, n), (i, j)) ∈ c2 and (f, g)w ≡ −q 2 xij (q 2 xklxmn − qxkn) + q[xklxin − (q 2 − q−2)xknxil] −q−2(q2xklxmn − qxkn)xij ≡ −q4xij xklxmn + q 3xij xkn + qxklxin − q(q 2 − q−2)xknxil + xklxmnxij −q−1xknxij ≡ −q2xklxij xmn + q 3xknxij + qxklxin − q 3xknxil +q−1xknxil + q 2xklxij xmn − qxklxin − q −1xknxij ≡ 0. case 14. f = xmnxij − q 2xij xmn + qxin, g = xij xkl − xklxij , w = xmnxij xkl. in the case, we have (f, g)w = −q 2 xij xmnxkl + qxinxkl + xmnxklxij . there are two subcases to consider. ((i, j), (m, n)) ∈ c5 ((k, l), (i, j)) ∈ c2 14.1. ((k, l), (m, n)), ((k, l), (i, n)) ∈ c2, c3 or c4 ((k, l), (i, j)) ∈ c6 14.2. ((k, l), (m, n)), ((k, l), (i, n)) ∈ c6 190 yuqun chen et al. cubo 10, 3 (2008) 14.1. ((i, j), (m, n)) ∈ c5, ((k, l), (i, j)) ∈ c2 and ((k, l), (m, n)), ((k, l), (i, n)) ∈ c2, c3 or c4. 14.1.1. if ((k, l), (m, n)) and ((k, l), (i, n)) ∈ c2 (l > n), then (f, g)w = −q 2xij xklxmn + qxklxin + xklxmnxij ≡ −q2xklxij xmn + qxklxin + xkl(q 2xij xmn − qxin) ≡ 0. 14.1.2. if ((k, l), (m, n)) and ((k, l), (i, n)) ∈ c3 (l = n), then (f, g)w = −xij xklxmn + q −1xklxin + q −2xklxmnxij ≡ −xklxij xmn + q −1xklxin + xklxij xmn − q −1xklxin ≡ 0. 14.1.3. if ((k, l), (m, n)), ((k, l), (i, n)) ∈ c4 (l < n), then we have ((k, n), (i, j)) ∈ c2, ((i, j), (m, l)) ∈ c5 and (f, g)w ≡ −q 2xij [xklxmn − (q 2 − q−2)xknxml] + q[xklxin − (q 2 − q−2)xknxil] +[xklxmn − (q 2 − q−2)xknxml]xij ≡ −q2xij xklxmn + q 2(q2 − q−2)xij xknxml + qxklxin − q(q 2 − q−2)xknxil +xklxmnxij − (q 2 − q−2)xknxmlxij ≡ −q2xklxij xmn + q 2(q2 − q−2)xknxij xml + qxklxin − q(q 2 − q−2)xknxil +xkn(q 2xij xmn − qxin) − (q 2 − q−2)xkn(q 2xij xmn − qxil) ≡ 0. 14.2. ((i, j), (m, n)) ∈ c5, ((k, l), (i, j)) ∈ c6 and ((k, l), (m, n)), ((k, l), (i, n)) ∈ c6. this case is similar to 14.1.1. case 15. f = xmnxij −q 2xij xmn +qxin, g = xij xkl −xklxij +(q 2 −q−2)xkj xil, w = xmnxij xkl, with ((i, j), (m, n)) ∈ c5 ((k, l), (i, j)) ∈ c4 ((k, l), (m, n)) ∈ c6 ((k, l), (i, n)) ∈ c4 ((k, j), (m, n)) ∈ c5 ((i, l), (m, n)) ∈ c6 cubo 10, 3 (2008) rosso-yamane theorem on pbw basis of uq(an ) 191 then, we have (f, g)w = −q 2 xij xmnxkl + qxinxkl + xmnxklxij − (q 2 − q−2)xmnxkj xil ≡ −q2xij xklxmn + q[xklxin − (q 2 − q−2)xknxil] + xklxmnxij −(q2 − q−2)(q2xkj xmn − qxkn)xil ≡ −q2[xklxij − (q 2 − q−2)xkj xil]xmn + qxklxin − q(q 2 − q−2)xknxil +xkl(q 2xij xmn − qxin) − q 2(q2 − q−2)xkj xmnxil + q(q 2 − q−2)xknxil ≡ −q2xklxij xmn + q 2(q2 − q−2)xkj xilxmn + qxklxin − q(q 2 − q−2)xknxil +q2xklxij xmn − qxklxin − q 2(q2 − q−2)xkj xilxmn + q(q 2 − q−2)xknxil ≡ 0. case 16. f = xmnxij − q 2xij xmn + qxin, g = xij xkl − q 2xklxij + qxkj , w = xmnxij xkl, with ((i, j), (m, n)) ∈ c5 ((k, l), (i, j)) ∈ c5 ((k, l), (m, n)) ∈ c6 ((k, l), (i, n)), ((k, j), (m, n)) ∈ c5 in the case, we have (f, g)w = −q 2xij xmnxkl + qxinxkl + q 2xmnxklxij − qxmnxkj ≡ −q2xij xklxmn + q(q 2xklxin − qxkn) + q 2xklxmnxij − q(q 2xkj xmn − qxkn) ≡ −q2(q2xklxij − qxkj )xmn + q 3xklxin − q 2xkn + q 2xkl(q 2xij xmn − qxin) −q3xkj xmn + q 2xkn ≡ 0. thus, ˜s+ is a gröbner-shirshov basis. this completes the proof of theorem 3.1. � similarly, with the deg-lex order on ˜y ∗, ˜s− is a gröbner-shirshov basis for u −q (an ) = k〈˜y |˜s−〉. we now use the same notation as before. order the generators by: xi > xj , hi > h −1 i > hj > h −1 j , yi > yj if i > j, and xi > h ±1 j > ym for all i, j, m. then we obtain a well ordering (deg-lex) on ˜x ∪ h ∪ ˜y . thus, by theorem 3.1, we re-obtain the following theorem in [16]. theorem 3.2. ([16] theorem 2.7) let the notation be as before. then with the deg-lex order on { ˜x ∪ h ∪ ˜y }∗, ˜s+ ∪ t ∪ k ∪ ˜s− is a gröbner-shirshov basis for uq(an ) = k〈 ˜x ∪ h ∪ ˜y |˜s + ∪ t ∪ k ∪ ˜s−〉. 192 yuqun chen et al. cubo 10, 3 (2008) acknowledgement: the authors would like to express their deepest gratitude to professor l. a. bokut for his kind guidance, useful discussions and enthusiastic encouragement. received: june 2008. revised: august 2008. references [1] g.m. bergman, the diamond lemma for ring theory, adv. in math., 29(1978), 178–218. [2] l.a. bokut, unsolvability of the word problem, and subalgebras of finitely presented lie algebras, izv. akad. nauk. sssr ser. mat., 36(1972), 1173–1219. [3] l.a. bokut, imbeddings into simple associative algebras, algebra i logika, 15(1976), 117– 142. [4] l.a. bokut, gröbner-shirshov bases for the braid groups in the briman-ko-lee generators, sumbitted. 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[37] j.c. jantzen, lectures on quantum groups, graduate texts in mathematics, vol. 6, ams, 1996. n15 cubo a mathematical journal vol.11, no¯ 05, (71–97). december 2009 optical tomography for media with variable index of refraction stephen mcdowall department of mathematics, western washington university, bellingham, washington, usa. email : stephen.mcdowall@wwu.edu abstract optical tomography is the use of near-infrared light to determine the optical absorption and scattering properties of a medium m ⊂ r n . if the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the euclidean metric and photons which do not get absorbed or scatter travel along straight lines. in this expository article we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a riemannian metric; in the absence of interaction, photons follow the geodesics of this metric. the data one has is the measurement of the out-going flux of photons leaving the body at the boundary. this may be knowledge of both the locations and directions of the exiting photons (fully angularly resolved measurements) or some kind of average over direction (angularly averaged measurements). we discuss the results known for both types of measurements in all spatial dimensions. resumen tomografia óptica es el uso de luz infrarroja próxima para determinar la absorción óptica y las propriedades de dispersión de un medio m ⊂ r n . si el indice de refración es constante a través del medio, el caso estado “steady” es modelado por la ecuación de transporte linear 72 stephen mcdowall cubo 11, 5 (2009) estacionaria en terminos de la métrica euclideana y fatores los cuales não são obsorvidos o dispersados viajando a lo largo de lineas retas. en este artículo expositorio consideramos el caso de indice de refración variable donde las dinámicas son modeladas escribiendo la ecuación de transporte e terminos de la metrica riemanniana; en la ausencia de interacción, fotones siguen las geodesicas de esta métrica. los datos que uno tiene es la medida del flujo out-going de fotones dejando el cuerpo en la frontera. esto puede ser conocido de la localización y las direciones de los fotones existentes (mediciones completamente angulares) o alguna clase de promedio sobre dirección ( mediciones angularmente promediadas). nosotros discutimos los resultados conocidos para ambos tipos de mediciones en todas las dimensiones espaciales. key words and phrases: boltzmann equation, integral geometry, optical tomography, riemannian metric, transport equation. math. subj. class.: 35r35, 35q99. 1 introduction the stationary linear transport equation models, among other things, the propagation of the energy density of waves in heterogeneous media [b3, cha, rpk, vanr-nh], neutrons in nuclear reactors [mo], and near-infrared photons in tissue. the term “linear” refers to the fact that the equation models only scattering of particles from the material and assumes that the density of particles is low enough that particle-to-particle interaction may be neglected. recently, photon propagation has been applied in optical tomography for use in medical imaging [a, ch-al, rbh]. in the absence of scattering, propagation is usually assumed to be along straight lines, as would be the case for light propagation in a medium with constant index of refraction. this amounts to writing the transport equation in terms of the euclidean metric. here we consider the situation where, in the absence of scattering, propagation is along geodesics of a riemannian metric. we consider this a model for optical tomography in a medium with continuously varying refractive index. when prescribed in-going, and measured out-going, fluxes are allowed to depend on both position and direction (in other words, measurements are fully “angularly resolved”), the problem of optical tomography in the euclidean setting has been well studied. in many instances restrictions are placed on the scattering kernel, for example that it be small, or that it be independent of direction. for recent results see [b1, su, t1, t2]. for dimensions three and greater, these restrictions are relaxed in [cs2] in the stationary case, and [cs1] in the time-dependent case. analogous results are proven in the riemannian setting in [m1, m2]; see also [klu]. stability of the reconstruction of material parameters has been studied in [bj, su, w] for the euclidean case. such measurements are, however, unrealistic since angular resolution of out-going flux is difficult to achieve. in [l], langmore introduces an angularly averaged measurement operator and proves that angularly averaged measurements taken at every point on the boundary of the medium enables reconstruction of the extinction coefficient σ; once σ is known, measurements made at a cubo 11, 5 (2009) optical tomography for media ... 73 single boundary point allow unique determination of the scattering kernel under some additional assumptions. in particular, it is assumed that scattering is supported away from the boundary of the body, the scattering phase function is known, and the scattering kernel is small. it is also assumed that σ and the phase function are “close to” real-analytic. these results are extended to the riemannian case in [lm]. in this article we shall present results for the inverse problem of optical tomography in the presence of a riemannian metric. these results have appeared in [lm, m1, m2]. let m ⊂ rn be an open bounded set with smooth boundary ∂m and let g be a smooth riemannian metric on m̄. we shall make geometric assumptions on the riemannian manifold (m,g) in due course. if u(x,v) represents the density of particles at position x with velocity v in the unit tangent sphere at x, ωxm, then the stationary linear transport equation is tu(x,v) = −du(x,v) − σ(x,v)u(x,v) + ∫ ωxm k(x,v′,v)u(x,v′) dv′ = 0. (1.1) the functions σ and k are the material specific parameters we seek to determine from boundary measurements. the first of these, σ is the extinction coefficient and σ(x,v)u(x,v) accounts for loss of energy in direction v due to both absorption at x as well as scattering into other directions. the second, k(x,v′,v) is the scattering kernel and is proportional to the probability of a particle with position x and velocity v′ ∈ ωxm being scattered to velocity v ∈ ωxm. the operator d is the derivative along the geodesic flow (see (2.2) below) which in the case of g being euclidean is simply du(x,v) = v · ∇xu(x,v). the measure dv ′ is the volume form on ωxm induced from the euclidean volume form on txm determined by g at x; here txm is the full tangent space to m at x. while u is strictly speaking a density, for large numbers of particles it is reasonable to represent u as, say, an l1 function. in [b2] a transport equation is derived as a limiting case of maxwell’s equations with nonconstant, but isotropic, permeability resulting in a varying refractive index n(x). if the metric of (1.1) is conformal to the euclidean metric, gij(x) = c −2 0 n(x) 2δij with c0 the speed of light in a vacuum, then (1.1) can be shown to correctly describe energy density propagation in isotropic material. while it has not been shown that a general anisotropic index of refraction leads to a limiting transport equation of the form (1.1), we consider it as a model for this case. we point out that the same model appears in [sh2]. define the “incoming” and “outgoing” bundles on ∂m, the boundary of m, γ± = {(x,v) : x ∈ ∂m, and ± 〈v,νx〉 > 0} where νx is the unit outer normal vector to the boundary ∂m at x and 〈·, ·〉 is the inner product, each with respect to g at x. if u− is an incoming flux of particles on γ− let u be the solution, should it exist, to tu = 0 with the boundary condition u|γ− = u−. denote this solution operator by t −1, i.e. u = t −1u−. the measurement operator corresponding to fully angularly dependent measurements is the albedo 74 stephen mcdowall cubo 11, 5 (2009) operator a : u− 7→ u|γ+. it is from knowledge of a that we seek to determine uniquely the material parameters σ and k. in section 5 we will present the corresponding operator for angularly averaged measurements. determination of σ and k relies upon invertibility of certain geodesic ray transforms (integrals along geodesics). to ensure injectivity of these transforms we assume that the metric is “simple”: assumption 1.1. (m,g) is simple: m is strictly convex, and for any x ∈ m̄ the exponential map expx : exp −1 x (m̄) → m̄ is a diffeomorphism (and consequently m is diffeomorphic to a ball). we must assume that σ depends only on position. if k ≡ 0 and g is euclidean then determination of σ from a is simply the usual x-ray transform. if σ depends on direction v, then it is not uniquely determined from its x-ray transform (see the introduction of [cs2]). characterization of non-uniqueness when σ depends on both x and v has recently been investigated in [st]. we will also assume that 0 ≤ σ ∈ l∞(m) and 0 ≤ k ∈ l∞(ω2m) are bounded functions. here we introduce the somewhat unconventional notation ω 2m := {(x,v,w) : x ∈ m, v,w ∈ ωxm}. the forward problem is not necessarily well-posed without some subcriticality assumptions (see [rs]). such conditions will be satisfied by smallness assumptions on k to be stated precisely in the following section. the content of this article has appeared in [lm, m1, m2]. the purpose here is to collect together these results into a single summarizing exposition. in section 2 we describe the solution to the forward problem, thus enabling the definition of the measurement operator a, and expressing it as a series. this series can be understood intuitively as contributions from ballistic particles which do not scatter, singularly scattered particles, and multiply scattered particles. in section 3 we prove unique identifiability of σ for all dimensions and of quite general k for dimensions three and higher. in section 4 a finer analysis in two dimensions provides determination of scattering kernels k which are a-priori known to be small relative to σ. the results of sections 3 and 4 assume knowledge of angularly resolved measurements. in section 5 we restrict to angularly averaged measurements and prove identifiability of σ and a restricted class of k in arbitrary dimensions. 2 the forward problem and the albedo operator the approach utilized in the results of this article is to obtain a series expansion for the distribution kernel of the boundary measurement operator. in this section we prove solveability of the forward problem, and in the process obtain such a series expansion for a. cubo 11, 5 (2009) optical tomography for media ... 75 if (x,v) ∈ ωm we shall denote by γ(x,v)(t) the geodesic satisfying γ(x,v)(0) = x and γ̇(x,v)(0) = v; we shall also use the compressed notation ~γ(x,v)(t) = ( γ(x,v)(t), γ̇(x,v)(t) ) . define the “distance-to-boundary” functions τ± : ωm → r + by τ±(x,v) = min{t > 0 : γ(x,v)(±t) ∈ ∂m} and set τ = τ− + τ+. the volume forms on ωxm and on γ± are the following: on m we have the naturally defined volume form of the metric. at any x ∈ m, the volume form dv on ωxm is the form induced from the euclidean volume form on txm defined by the metric g at x. the resulting form on ωm is the liouville form and is preserved under the geodesic flow of g. we denote by dµ the induced volume form on γ± which has the property that dt dµ(x,v) is the pull-back of the liouville form by the geodesic flow. equivalently, we have the induced volume form of ∂m included in m̄; if (x,v) are local coordinates for ∂m and dx is this volume form on ∂m, then it holds that dµ(x,v) = |〈v,νx〉| dv dx. the operator d in (1.1) is the derivative along the geodesic flow and is defined by du(x,v) = ∂ ∂t ∣∣∣ t=0 u(γ(x,v)(t), γ̇(x,v)(t)). (2.2) if (xi,yi)ni=1 are local coordinates for ωm with the (y i ) with respect to the natural basis ( ∂ ∂xi ) then in these coordinates df = ∂f ∂xi yi + ∂f ∂yi (−yjykγijk) where γijk are the christoffel symbols of the levi-civita connection of g. if x,y ∈ m then simplicity of (m,g) ensures there exists a unique geodesic from x to y; let d(x,y) be the geodesic distance between x and y, and let v(x,y) be the tangent vector at x of this geodesic. define e(x,y) := exp { − ∫ d(x,y) 0 σ ( γ(x,v(x,y))(t) ) dt } . note, using the fact γ(y,v(y,x))(d(y,x) − s) = γ(x,v(x,y))(s) we have e(x,y) = e(y,x). define t0u = −du − σu, t1u(x,v) = ∫ ωxm k(x,v′,v)u(x,v) dv′, so that t = t0 + t1. we begin by re-writing this as an integral equation. in the absence of scattering, the homogeneous boundary value problem t0u = 0, u|γ− = u− has solution ju− where ju−(x,v) = e ( x,γ(x,v)(−τ−(x,v)) ) u− ( γ(x,v)(−τ−(x,v)) ) . 76 stephen mcdowall cubo 11, 5 (2009) the inhomogeneous dirichlet problem t0u = f, u|γ− = 0 has solution u(x,v) = t −10 f(x,v) = ∫ τ−(x,v) 0 e ( x,γ(x,v)(t − τ−(x,v)) ) u ( ~γ(x,v)(t − τ−(x,v)) ) dt. defining ku(x,v) = ∫ τ−(x,v) 0 e ( x,γ(x,v)(t − τ−(x,v)) ) t1u ( ~γ(x,v)(t − τ−(x,v)) ) dt, we obtain the solution to tu = 0, u|γ− = u− satisfies (i − k)u = ju−. (2.3) we shall make repeated use of the following immediate lemma. lemma 2.1 (change of variables). if u ∈ l1(ωm) then ∫ m ∫ ωxm u(x,v) dvx dx = ∫ γ± ∫ τ∓(x′,v′) 0 u(γ(x′,v′)(t), γ̇(x′,v′)(t)) dt dµ(x ′,v′). assume the following smallness condition on k: with σp(x,v ′ ) := ∫ ωxm k(x,v′,v) dv, ∥∥τσp ∥∥ l∞(ωm,dv dx) < 1. (2.4) this subcriticality condition will ensure well-posedness of the boundary value problem (see [rs]). proposition 2.1. the operator k is bounded on l1(ωm,τ−1 dv dx) with operator norm bounded by ‖τσp‖ < 1 and so (i −k) is invertible on this space. equation (2.3) and hence (1.1) is uniquely solvable for u− ∈ l 1 (γ−,dµ) and the solution u has a well-defined trace u|γ− . the albedo operator a : l1(γ−,dµ) → l 1 (γ+,dµ) is a bounded map. since we have ‖τσp‖ < 1 we may express the solution to (1.1) as a neumann series, u = ∞∑ j=0 kjju− = ju− + kju− + (i − k) −1k2ju−. (2.5) we seek the solution, in the sense of distributions, to (1.1) with singular boundary condition u|γ− = δ{x0,v0}(x ′,v′); the right hand side is the distribution on γ− defined by (δ{x0,v0},ϕ) = ∫ γ− δ{x0,v0}(x ′,v′)ϕ(x′,v′) dµ(x′,v′) = ϕ(x0,v0) for ϕ ∈ c∞0 (γ−). it is convenient to use parallel translation in what follows: for x,y ∈ m define p : ωxm → ωym, p(v; x,y) to be the parallel translation of v from x to y (along the unique geodesic joining x to y). cubo 11, 5 (2009) optical tomography for media ... 77 proposition 2.2. the three terms in (2.5) can be written kjju−(x,v) = ∫ γ− uj(x,v,x ′,v′)u−(x ′,v′) dµ(x′,v′) with u0(x,v,x ′,v′) = ∫ τ+(x′,v′) 0 e ( x,γ(x,v)(−τ−(x,v)) ) δ(x,v)(~γ(x′,v′)(t)) dt u1(x,v,x ′,v′) = ∫ τ+(x′,v′) 0 ∫ τ−(x,v) 0 e(x,y(s))e(x′,z(t))k ( z(t), ż(t),p(ẏ(s); y(s),z(t)) ) δ{y(s)}(z(t)) ds dt y(s) = γ(x,v)(s − τ−(x,v)), z(t) = γ(x′,v′)(t) u2 ∈ l ∞ (γ−,w), w = {f : df ∈ l 1 (ωm), τ−1f ∈ l1(ωm)}. proof. we present the straight-forward proof for u0 and refer the reader to [m1] for the more involved proofs for u1 and u2. if ϕ ∈ c ∞ 0 (ωm) then (ju−,ϕ) = ∫ γ− u−(x ′,v′) ∫ τ+(x′,v′) 0 e(x′,γ(x′,v′)(t))ϕ(~γ(x′,v′)(t)) dt dµ(x ′,v′) = ∫ m ∫ ωxm ∫ γ− u−(x ′,v′) ∫ τ+(x′,v′) 0 δ(x,v)(~γ(x′,v′)(t)) dt dµ(x ′,v′)ϕ(x,v) dv dx. from proposition 2.2 we obtain the distribution kernel α(x,v,x′,v′) of the albedo operator a. theorem 2.1. [m1] the distribution kernel α(x,v,x′,v′) of a is α = α0 + α1 + α2 with α0 = e ( x,γ(x,v)(−τ−(x,v))bigr)δ{~γ(x,v) (−τ−(x,v))}(x ′,v′), α1 = u1 ∣∣ (x,v)∈γ+ , α2 ∈ l ∞ (γ−; l 1 (γ+,dµ)). proof. if ϕ− ∈ c ∞ 0 (γ−) then changing variables to (y,w) = ~γ(x′,v′)(t), ∫ γ− u0(x,v,x ′,v′)ϕ−(x ′,v′) dµ(x′,v′) = ∫ m ∫ ωym e ( x,γ(x,v)(−τ−(x,v)) ) δ(x,v)(y,w)ϕ− ( ~γ(y,w)(−τ−(y,w)) ) dw dy = e ( x,γ(x,v)(−τ−(x,v)) ) ϕ− ( ~γ(x,v)(−τ−(x,v)) ) = ∫ γ− α0(x,v,x ′,v′)ϕ−(x ′,v′) dµ(x′,v′). the claim for α2 follows from [m1] theorem 2.3 which shows that the trace on γ± is continuous from w into l1(γ±,dµ). 78 stephen mcdowall cubo 11, 5 (2009) 3 full measurements in arbitrary dimensions we show here that both the extinction coefficient and the scattering kernel are uniquely determined in dimensions three and greater. this is achieved by isolating terms in the kernel of the albedo operator a that differ in strength of singularity. briefly, α0 determines σ and α1 determines k. in dimension two, however, α2 is in fact a locally l 1 function and is not immediately distinguishable from α2. thus, the method demonstrated in this section fails to determine k in dimension two. we address the solution to this problem in the next section. 3.1 determination of σ we construct an appropriate approximate identity. let ψ ∈ c∞0 ([0,∞)) be such that ψ(0) = 1 and ∫ ∞ 0 ψ(t) dt = 1; define ψε(x) = ψ(x/ε). proposition 3.1. the following limit holds in l1(γ+,dµ(x,v)): lim ε→0 ∫ γ− α(x,v,x′,v′)ψε ( d ( x′,γ(x,v)(−τ−(x,v)) )) dµ(x′,v′) = e ( x,γ(x,v)(−τ−(x,v)) ) . proof. when α is replaced by α0 the result is immediate. when α is replaced by α1, 0 ≤ ∫ γ+ ∫ γ− ∫ τ+(x′,v′) 0 ∫ τ−(x,v) 0 k ( z(t), ż(t),p(ẏ(s); y(s),z(t)) ) δ{y(s)}(z(t))ψε ( d ( x′,γ(x,v)(−τ−(x,v)) )) × ds dt dµ(x′,v′) dµ(x,v) = ∫ m ∫ ω2ym k(y,w′,w)ψε ( d ( γ(y,w′)(−τ−(y,w ′ )),γ(y,w)(τ+(y,w)) )) dw′ dw dy where (y,w′) = ~γ(x′,v′)(t), (y,w) = ~γ(x,v)(s − τ−(x,v)). now there exists constant c such that supp ψε ( d ( γ(y,w′)(−τ−(y,w ′ )),γ(y,w)(τ+(y,w)) )) ⊂ wε = {(y,w ′,w) ∈ ω2m : ‖w′ − w‖g < cε} and so 0 ≤ ∫ γ+ ∫ γ− α1(x,v,x ′,v′)ψε ( d ( x′,γ(x,v)(−τ−(x,v)) )) dµ(x′,v′) dµ(x,v) = ∫ wε k(y,w′,w) dw′ dw dy → 0 as ε → 0 since k ∈ l1(ω2m) and the measure of wε → 0 as ε → 0. finally, for α2 0 ≤ ∫ γ+ ∣∣∣ ∫ γ− α1(x,v,x ′,v′)ψε ( d ( x′,γ(x,v)(−τ−(x,v)) )) dµ(x′,v′) ∣∣∣ dµ(x,v) ≤ ∫ vε ∣∣α2(x,v,x′,v′) ∣∣ dµ(x′,v′) ∣∣∣ dµ(x,v) → 0 as ε → 0. here supp ψε(x,v,x ′,v′) ⊂ vε = {(x,v,x ′,v′) ∈ γ+ × γ− : ‖w ′ − w‖g < cε} and the limit above holds since by theorem 2.1 α2 ∈ l 1 (γ+ × γ−) and the measure of vε → 0 as ε → 0. cubo 11, 5 (2009) optical tomography for media ... 79 proposition 3.1 enables us to obtain from a the integral of σ along the geodesic between any two points in the boundary of m. that is, we may determine the geodesic x-ray transform of σ. for simple riemannian manifolds this transform is known to be invertible (see [sh1]). we have thus proven the following theorem. theorem 3.1. [m1] let m ⊂ rn, n ≥ 2, be a bounded domain with smooth boundary and let g be a known simple riemannian metric on m. let 0 ≤ σ(x) ∈ l∞(m) depend only on x, and 0 ≤ k ∈ l∞(ω2m) satisfy (2.4). then we may determine σ from a. 3.2 determination of k toward determining the scattering kernel k, fix (y,w,w′) ∈ ω2m, w 6= w′. let expw′ : tw′ ωym → ωym be the exponential map of the unit tangent sphere based at w ′ ∈ ωym. denote by v̂(v) = expw′ (v), and let j(y,w′)(v̂) be the determinant of the jacobian of this change of variables. let ϕ1 ∈ c ∞ 0 (r n ) be such that 0 ≤ ϕ1 ≤ 1, ϕ1(0) = 0, ϕ1(v̂) = 0 for |v̂| > ε0 for sufficiently small ε0, and ∫ rn−1 ϕ1(v̂) dv̂ = 1. now define ψε : ωym → r by ψε(v) = 1 εn−1 ϕ1 ( exp −1 w′ (v) ε ) . note that if f : ωym → r is continuous at w ′ then ∫ ωym f(v)ψε(v) dv = ∫ rn−1 f ( exp −1 w′ (v̂) ) 1 εn−1 ϕ1 (v̂ ε ) j(y,w′)(v̂) dv̂ → f ( exp −1 w′ (0) ) j(y,w′)(0) = f(w ′ ) as ε → 0. define y(s) = γ(y,w)(s − τ−(y,w)), −τ+(y,w) ≤ s ≤ τ−(y,w). define β(s) ∈ ∂m to be the unique point in the boundary for which it holds that γ(β(s),p(w′;y,β(s)))(·) contains the point y(s), and define v(s) ∈ ωy(s)m to be its tangent vector there. since w and w ′ are independent, β′(0) 6= 0. now let ϕ ∈ c∞0 (r) be such that 0 ≤ ϕ ≤ 1, ϕ(0) = 1, ∫ r ϕ(x) dx = ‖β′(0)‖−2g ; let ϕη(x) = 1 η ϕ (x η ) . define h1 : ∂m → r by h1(x ′ ) = 〈exp−1 β(0) (x′),β′(0)〉. note that, restricted to the curve β(s), for sufficiently small s, h1(β(s)) = 0 if and only if s = 0. furthermore, d ds ∣∣∣ s=0 h1(β(s)) = ‖β ′ (0)‖2. the construction of β(s) for fixed (y,w,w′) is now repeated for arbitrary (z,ξ,ξ′) ∈ ω2m and we denote this by β(z,ξ,ξ′)(s). define h2 : ∂m × ω 2m by h2(x ′,z,ξ,ξ′) = dist ( x′,β(z,ξ,ξ′)(·) ) . let µ ∈ c∞0 (r) with µ(0) = 1 and define µδ(t) = µ(t/δ). let w = {(z,ξ,ξ′) ∈ ω2m : ξ 6= ξ′}. 80 stephen mcdowall cubo 11, 5 (2009) proposition 3.2. if n ≥ 3 and (y,w,w′) ∈ ω2m with w 6= w′ then lim η→0 lim δ→0 lim ε→0 ∫ γ− ψε(p(v ′ ; x′,y))ϕη(h1(x ′ ))µδ(h2(x ′,y,w,w′))α(~γ(y,w)(τ+(y,w)),x ′,v′) dµ(x′,v′) = e ( y,γ(y,w)(τ+(y,w)) ) e ( y,γ(y,w′)(−τ−(y,w ′ )) ) k(y,w′,w) the limit holding in l1(w). proof. replacing α by α0 and integrating with respect to dµ(x ′,v′), the integrand gets evaluated at (x′,v′) = ~γ(y,w)(−τ−(y,w)) and so we obtain a multiple of ψε ( p ( ~γ(y,w)(−τ−(y,w)) ) ; γ(y,w)(−τ−(y,w)),y ) = ψε(w) = 0 for all sufficiently small ε since w 6= w′. replacing α by α1, i1 : = ∫ γ− ψε(p(v ′ ; x′,y))ϕη(h1(x ′ ))µδ(h2(x ′,y,w,w′))α1(~γ(y,w)(τ+(y,w)),x ′,v′) dµ(x′,v′) = ∫ τ(y,w) 0 ∫ ωy(s)m ψε ( p(γ̇(y(s),v̂)(−τ−(y(s), v̂)); γ(y(s),v̂)(−τ−(y(s), v̂)),y) ) × ϕη ( h1 ( γ(y(s),v̂)(−τ−(y(s), v̂)) )) µδ ( h2(γ(y(s),v̂)(−τ−(y(s), v̂)),y,w,w ′ ) ) × e ( y(s),γ(y,w)(τ+(y,w)) ) e ( y(s),γ(y(s),v̂)(−τ−(y(s), v̂)) ) k(y(s), v̂, ẏ(s)) dv̂ ds now define ṽ(v̂,s) = p ( γ̇(y(s),v̂)(−τ−(y(s), v̂)); γ(y(s),v̂)(−τ−(y(s), v̂)),y ) ∈ ωym and denote by dv̂ dṽ the change of volume element of this change of variables. we obtain i1 = ∫ τ(y,w) 0 ∫ ωym ψε(ṽ)ϕη ( h1 ( γ(y(s),v̂(ṽ,s))(−τ−(y(s), v̂(ṽ,s))) )) × µδ ( h2(γ(y(s),v̂(ṽ,s))(−τ−(y(s), v̂(ṽ,s))),y,w,w ′ ) ) e(·)e(·)k(y(s), v̂(ṽ,s), ẏ(s)) dv̂ dṽ dṽ ds → i2 := ∫ τ(y,w) 0 ϕη ( h1(β(y,w,w′)(s)) ) µδ ( h2(β(y,w,w′)(s),y,w,w ′ ) ) e(·)e(·) × k(y(s),v(s), ẏ(s)) dv̂ dṽ ∣∣∣ v̂=v(s) ds as ε → 0. note that µδ ( h2(β(y,w,w′)(s),y,w,w ′ ) ) = 1. now define s̃(s) = h1(β(y,w,w′)(s)). then s̃ = 0 if and only if s = 0 (for sufficiently small s), and ds̃ ds ∣∣ s=0 = ‖β′ (y,w,w′) (0)‖2. so, for sufficiently small s̃0, i2 = ∫ s̃0 −s̃0 ϕη(s̃)e(·)e(·)k ( y(s(s̃)),v(s(s̃)), ẏ(s(s̃)) )dv̂ dṽ ∣∣∣ v̂=v(s) ds ds̃ ds̃ → e ( y,γ(y,w)(τ+(y,w)) ) e ( y,γ(y,w′)(−τ−(y,w ′ )) ) k(y,w′,w) as η → 0 since ∫ r ϕ2(x) dx = ‖β′(y,w,w′)(0)‖ −2 g . cubo 11, 5 (2009) optical tomography for media ... 81 finally, we must show that the integral vanishes when we replace α by α2. let χ(z,ξ,ξ ′ ) ∈ c∞0 (w). then lim ε→0 ∫ m ∫ ω2 z m ∣∣∣ ∫ γ− ψε(p(v ′ ; x′,z))ϕη(h1(x ′ ))µδ(h2(x ′,z,ξ,ξ′))α2(~γ(z,ξ)(τ+(z,ξ)),x ′,v′)χ(z,ξ,ξ′) × dµ(x′,v′) ∣∣∣ dξ′ dξ dz ≤ 1 η ∫ γ+ ∫ τ−(x,v) 0 ∫ ωγ (x,v) (s−τ − (x,v))m ∫ ∂m µδ(h2(x ′,~γ(x,v)(s − τ−(x,v)),ξ ′ )) × |χ(~γ(x,v)(s − τ−(x,v)),ξ ′ )||〈p(ξ′;~γ(x,v)(s − τ−(x,v)),x ′ ),νx′〉| dx ′ dξ′ ds dµ(x,v) → 0 as δ → 0 since the support of µδ is a (3n − 1)-dimensional variety in the (4n − 3)-dimensional domain of integration, and since α2 ∈ l ∞ (γ−; l 1 (γ+,dµ)). combining proposition 3.2 with theorem 3.1 we first determine the function e, and then obtain k. we thus have the following. theorem 3.2. [m1] let m ⊂ rn, n ≥ 3, be a bounded domain with smooth boundary and let g be a known simple riemannian metric on m. let 0 ≤ σ(x) ∈ l∞(m) depend only on x, and 0 ≤ k ∈ l∞(ω2m) satisfy (2.4). then we may determine k from a. 4 full measurements in dimension two in this section we present the results of [m2] where it is proven that knowledge of the albedo operator on a simple riemannian surface uniquely determines the unknown metric g, and the coefficients σ and k. we shall refer the reader to [m2] for most of the proofs. we impose two additional conditions on the manifold (m,g). assumption 4.1. if the maximal sectional curvature κ0 of (m,g) is positive then we assume that there are no focal points. that is, for every geodesic γ : [a,b] → m and every non-zero jacobi field j(t) along γ with j(a) = 0 it holds that ‖j(t)‖ is strictly increasing on [a,b]. assumption 4.2. in the case that κ0 > 0 we assume that the diameter of (m,g) satisfies diam(m,g) < π/(2 √ κ0). theorem 4.1. [m2] let (m,g) be a two-dimensional riemannian manifold satisfying assumptions 1.1 and 4.1. then the albedo operator a determines the metric g up to a diffeomorphic change of coordinates which is the identity on ∂m. proof. from theorem 3.1 we see that a determines the so-called scattering relation of (m,g), that is, the set {(x,v,γ(x,v)(−τ−(x,v)), γ̇(x,v)(−τ−(x,v))}. it is a result of [pu1] (see also [pu2]) that for simple manifolds with no focal points, g is uniquely determined by this scattering relation. 82 stephen mcdowall cubo 11, 5 (2009) we now present the precise statement for the determination of σ and k. let uς,ε = {(σ(x),k(x,w ′,w)) : ‖σ‖l∞ ≤ σ, ‖k‖l∞ ≤ ε}. theorem 4.2. [m2] let (m,g) be a two-dimensional riemannian manifold satisfying assumptions 1.1 and 4.2. given σ > 0 there exists ε > 0 such that any pair (σ,k) ∈ uς,ε is uniquely determined, within (σ,k) ∈ uς,ε, by the associated albedo operator a. one may take ε = ce −2 diam(m,g)σ where c depends only on (m,g). from theorem 3.1 we may assume that σ (and hence e) is known. the main starting point for determination of k is a more precise expression for α1. throughout we are restricting to the case n = 2. we will frequently omit the exact points of evaluation of e writing instead “e(·).” we will not need the omitted information. proposition 4.1. the second term α1 in the series expansion for the kernel of a has the expression α1(x,v,x ′,v′) = χ(x,v,x′,v′)e(·)e(·)j (x,v,x′,v′) × k ( ~γ(x′,v′)(s(x ′,v′)), γ̇(x,v)(t(x ′,v′) − τ−(x,v)) ) | sin ψ| where χ = 1 if the geodesics γ(x′,v′)(s(x ′,v′)) and γ(x,v)(t(x ′,v′) − τ−(x,v)) intersect for s = s(x′,v′) > 0 and t = t(x′,v′) > 0, and χ = 0 otherwise, and where ψ is the angle between the tangent vectors at this point of intersection. the function j is uniformly bounded 0 < m1 ≤ j ≤ m2. proof. (sketch) by definition, if ϕ− ∈ c ∞ 0 (γ−) then kjϕ−(x,v) = ∫ τ−(x,v) 0 e(·) ∫ ωy(t)m k(y(t),w, ẏ(t))e(·)ϕ− ( ~γ(y(t),w)(−τ−(y(t),w)) ) dw dt where y(t) = γ(x,v)(t − τ−(x,v)). we re-write this integral in terms of an integral over γ−. define the family of indicator functions χ : ωm → {0, 1}, parameterized by (x,v), by χ(x,v,x′,v′) = { 1 if γ(x′,v′)(s) = γ(x,v)(t − τ−(x,v)) = y(t) for some s > 0, t > 0, 0 otherwise. on the support of χ we have well-defined functions s(x′,v′), t(x′,v′); on this support the change of variables (t,w) = φ(x′,v′) = ( t(x′,v′), γ̇(x′,v′)(s(x ′,v′)) ) is well-defined and smooth. in [m2] it is shown that if jφ is the jacobian of this change of variables then | det jφ| = |〈v′,νx′〉| | sin ψ| j (x,v,x′,v′) where j 6= 0 and j is bounded as in the statement of the proposition. the expression for α1 follows immediately. cubo 11, 5 (2009) optical tomography for media ... 83 suppose now that we have two identical manifolds (m,g) with material parameters (σ,k) and σ,k̃). let a and ã be their respective albedo operators and let αj, α̃j be the terms in their series expansions. then it follows that (α2 − α1)(x,v,x ′,v′) = χ(x,v,x′,v′)e(·)e(·)j (·) (k̃ − k) ( ~γ(x′,v′)(s(x ′,v′)), γ̇(x,v)(t(x ′,v′) − τ−(x,v)) ) | sin ψ| , and this in turn implies χ|(k̃ − k)(y,w,ŵ)| ≤ ce2 diam(m,g)σχ| sin ψ||(α2 − α̃2)(x,v,x ′,v′)| a.e., (4.6) where y is the point of intersection of the geodesics γ(x′,v′) and γ(x,v) and w, w ′ are their tangent vectors there. in what follows we outline the proof of another estimate of α2 − α̃2 of the form ‖ sin ψ(α2 − α̃2)‖l∞ ≤ cε‖k − k̃‖l∞. (4.7) once this is established, we combine it with (4.6) to obtain ‖k − k̃‖l∞ ≤ cε‖k − k̃‖l∞ so that for sufficiently small ε we have k = k̃. this completes the proof of theorem 4.2. toward proving (4.7) let ϕ− be the dirac delta distribution on γ− with respect to the measure dµ(x′,v′): ϕ−(x ′,v′) = 1 |〈v′,νx′〉| δ(x0,v0)(x ′,v′). then α2 − α̃2 = (i − k) −1k2jϕ− − (i − k̃) −1k̃2jϕ− = (i − k)−1 [ k(k − k̃) + (k − k̃)k̃ ] jϕ− + (i − k̃) −1 (k − k̃)(i − k)−1k̃2jϕ−. (4.8) from this we see that we must obtain estimates for k2 and k3. these are obtained in [m2] and we present them here without proof. proposition 4.2. for almost every (x,v,x′0,v ′ 0) ∈ γ+ × γ− such that there exist s > 0, t > 0 with γ(x,v)(s − τ−(x,v)) = γ(x′ 0 ,v′ 0 )(t) |k̃kjϕ−(x,v,x ′ 0,v ′ 0)| ≤    c‖k̃‖l∞ ‖k‖l∞ cos( √ κ0 diam(m,g)) ( 1 + log 1 | sin ψ| ) κ0 > 0, c‖k̃‖l∞‖k‖l∞ ( 1 + log 1 | sin ψ| ) κ0 ≤ 0, where κ0 is the maximal sectional curvature of (m,g) and where ψ is the angle between the tangent vectors of the intersecting geodesics at the point of intersection. 84 stephen mcdowall cubo 11, 5 (2009) although the proof of this proposition is not presented here, we point out that a recurring theme in the proof is comparison of geodesic triangles in (m,g) to triangles with (for example) the same side-angle-side property on either the sphere or the hyperbolic plane of constant curvature κ0. such constant curvature manifolds represent the “worst case” scenario with regard to how intersecting geodesics might get close to intersecting a second time. the final estimate needed is the following. proposition 4.3. it holds that k3jϕ− ∈ l ∞ (γ+ × γ−) with norm bounded by c‖k‖ 3 l∞. we now complete the proof of (4.7) via (4.8). lemma 4.1. for almost every (x,v,x′0,v ′ 0) ∈ γ+ × γ− such that γ(x,v) and γ(x′0,v ′ 0 ) we have |(i − k)−1(k2 − k̃2)jϕ−(x,v,x ′ 0,v ′ 0)| ≤ c‖k − k̃‖l∞ (‖k‖l∞ + ‖k̃‖l∞ ) ( 1 + log 1 | sin ψ| ) and |(i − k̃)−1(k − k̃)(i − k)−1k̃2jϕ−(x,v,x ′ 0,v ′ 0)| ≤ c‖k − k̃‖l∞‖k̃‖ 2 l∞ (1 + ‖k‖l∞ ). proof. for the first estimate, we re-write (i − k)−1 [ k(k−k̃) + (k − k̃)k̃ ] jϕ− = (i + (i − k) −1k) [ k(k − k̃) + (k − k̃)k̃ ] jϕ− = [ k(k − k̃) + (k − k̃)k̃ ] jϕ− + (i − k) −1 [ k2(k − k̃) + k(k − k̃)k̃ ] jϕ−. the contribution from the first term is estimated by proposition 4.2 (with k or k̃ replaced by k −k̃). for the final term, k2(k −k̃) + k(k −k̃)k̃jϕ− is an l ∞ function by proposition 4.3, and (i − k)−1 preserves this space. for the second estimate, express (i − k)−1k̃2 = k̃2 + (i − k)−1kk̃2 and apply proposition 4.3. combining the estimates of lemma 4.1 with (4.8) we obtain cχ| sin ψ||(α2 − α̃2)(x,v,x ′,v′)| ≤ c′ε‖k − k̃‖l∞, and this together with (4.6) yields ‖k − k̃‖l∞(ω2m) ≤ cε‖k − k̃‖l∞(ω2m). (4.9) thus, for sufficiently small ε, it must hold that k = k̃. although it has not been carefully tracked in this article, in [m2] it is shown that ε can be taken to be ε = ce−2 diam(m,g)σ with c depending only on (m,g). cubo 11, 5 (2009) optical tomography for media ... 85 5 angularly averaged measurements we now consider the problem of parameter determination with less information. the results presented here appear in [lm]. instead of knowing the angularly resolved measurement u(x,v) on γ+ we will assume only knowledge of an average over outgoing directions of u at x ∈ ∂m. this is motivated in part by the fact that in practice it is very difficult and perhaps impossible to measure the angular resolution of exiting photons. to be more precise, for x ∈ ∂m, define ω ± x m = {v ∈ ωxm : ±〈v,νx〉 > 0} (and so γ± are the disjoint unions of the ω ± x m over x ∈ ∂m). definition 5.1. the data of which we will be assuming knowledge consists of angularly averaged measurements on ∂m, weighted with respect to a prescribed function m(x,v). specifically, given u− on γ− and u = t −1u− we define m : l 1 (γ−,dµ) → l 1 (∂m) by mu−(x) := ∫ ω + x m u(x,v)m(x,v) dv. we require that m ∈ l∞(γ+), that m does not vanish on γ+, and that |m(x,v)/〈v,νx〉| be bounded on γ+. the function m corresponds to the limitations of the measurement apparatus. it may represent a limited aperture or, for example, when m(x,v) = 〈v,νx〉, the measurement is power flux exiting the boundary. we assume here that the metric g is known. with this limited data, we are still able to determine the extinction coefficient σ, but are only able to determine scattering kernels of a more restricted class. we assume that k is of the form k(x)θ(x,w,w′) where θ is assumed to be known. this corresponds to knowing the angular behavior of the scattering events, but not the density of scatterers (which is quantified by k(x)). there are also assumptions of analyticity of various coefficients. to be precise we present the main theorems. if h ⊂ γ−, then by γ(h) = {γ(x′,v′)(t) : (x ′,v′) ∈ h, 0 ≤ t ≤ τ+(x ′,v′)} we mean the set of geodesics with initial data in h. with constants cκm and cκm to be defined later, we have the following. theorem 5.1. [lm] suppose that ‖k‖l∞ < min { [(cκmcκm ) n−1 diam m|sn−1|]−1, [diam m|sn−1|]−1 } , and that hσ ⊂ γ− is open and such that the geodesic x-ray transform restricted to γ(hσ) is injective. suppose that σ = σ(x) depends on position only. then σ is uniquely determined by {(u−,mu−) : u− ∈ l 1 (hσ)}. fixing d > 0 and making the definition kdε := {k ∈ l ∞ (m) : dist(supp(k),∂m) > d,‖k‖l∞ ≤ ε} we also have the following uniqueness result for k. 86 stephen mcdowall cubo 11, 5 (2009) theorem 5.2. suppose the scattering kernel has the form k(x)θ(x,v′,v), with (g,m,σ, θ) known and real analytic, and with both m and θ non-vanishing. let hk ⊂ γ− be open, and assume that for every (x,v) ∈ tm \ {0}, there exists γ ∈ γ(hk) through x and normal to v at x. then there exits ε sufficiently small such that for a.e. x ∈ ∂m, knowledge of {(u−,mu−(x)) : u− ∈ l 1 (hk)} uniquely determines k within the class kdε . furthermore, ε may be chosen such that this result holds in some c2 neighborhood of (m,σ), and some c3n neighborhood of (θ,g). we begin by deriving an expression for the averaged albedo operator m as an infinite sum of the operators mi (see definition 5.2), and where we give expressions for the schwartz kernel of mi, i ≥ 1. in order to simplify the presentation of what follows, we introduce some notation. if x1, . . . ,xj are points in m, then e(x1,x2, . . . ,xj) := e(x1,x2)e(x2,x3) · · ·e(xj−1,xj) = j−1∏ i=1 e(xi,xi+1). (5.10) if y ∈ m, consider z = zy(t,v) = γ(y,v)(t) defined from the “polar” coordinates (t,v) ∈ r × ωym. let jy(z) denote the jacobian determinant | det ∂z/∂(t,v)| −1 of this change of variables. for a given z, let (t,v) be its polar coordinate expression, and let {v1, . . . ,vn} be an orthonormal basis for tym with v 1 = v. let yy,z,i be the jacobi field along γ(y,v)(·) with yy,z,i(0) = 0, ẏy,z,i = v i, 2 ≤ i ≤ n. then jy(z) is given by the expression jy(z) = n∏ i=2 |yy,z,i(d(y,z))| −1. (5.11) analogous to (5.10) we introduce the notation j (y1, . . . ,yj) := j∏ i=2 jyi (yi−1). (5.12) by considering comparison theorems for jacobi fields, one easily obtains the following. lemma 5.1. with jy(z) defined as above, there exists a constant cκm , such that for all y,z ∈ m jy(z) ≤ cκm d(y,z)n−1 . (5.13) further, there exists a constant cκm such that if y is any jacobi field along a geodesic γ(t) with y (0) = 0 and ẏ (0) ∈ γ̇⊥(0) ⊂ ωγ(0)m, then |y (t)| t ≤ cκm (5.14) for all 0 ≤ t ≤ τ+(γ(0), γ̇(0)). cubo 11, 5 (2009) optical tomography for media ... 87 the notation cκm and cκm is chosen because these constants are determined in terms of minimal and maximal sectional curvatures of (m,g). definition 5.2. for each i ≥ 0 we define the operator mi by miu−(x) := ∫ ω + x m kiju−(x,v)m(x,v) dv. theorem 5.3. [lm] let ‖k‖l∞(ω2m) < ( |sn−1| diam m )−1 . then mi,m : l 1 (γ−,dµ) → l1(∂m) continuously, and for almost every x ∈ ∂m, m has the expansion mu−(x) = ∞∑ j=0 mju−(x) = ∞∑ j=0 ∫ γ− αj(x,x ′,v′)u−(x ′,v′) dµ(x′,v′) where α0(x, ·, ·) is a distribution supported on a manifold of dimension n − 1, and for j ≥ 1, αj(x,x ′,v′) are the following schwartz kernels: when j = 1, α1(x,x ′,v′) = ∫ τ+(x′,v′) 0 k(~γ(x′,v′)(t), w̄1)e(x ′,γ(x′,v′)(t),x)m(x,ŵ1)jx(γ(x′,v′)(t)) dt (5.15) where w̄1, ŵ1 are the initial and final tangent vectors of the geodesic from γ(x′,v′)(t) to x, and e and jx are defined in (5.10), and (5.11); for j ≥ 2, αj(x,x ′,v′) = ∫ τ+(x′,v′) 0 ∫ m · · · ∫ m k(~γ(x′,v′)(t), w̄1) j∏ i=2 k(yi, ŵi−1, w̄i) e(x′,γ(x′,v′)(t),y2, . . . ,yj,x)j m(x,ŵj) dyj · · · dy2 dt (5.16) where: • w̄1, ŵ1 are the initial and final tangent vectors of the geodesic joining γ(x′,v′)(t) to y2, • w̄i, ŵi are the initial and final tangent vectors of the geodesic joining yi to yi+1 for i = 2, . . . ,j − 1, • w̄j, ŵj are the initial and final tangent vectors of the geodesic joining yj to x, and • j = j (γ(x′,v′)(t),y2, . . . ,yj,x). proof. we refer the reader to [lm] for the full proof of this theorem. to give a flavor of the proof, 88 stephen mcdowall cubo 11, 5 (2009) we present the derivation of m1 here. let φ− be a function on γ−. we have m1φ−(x) = ∫ ω + x m kjφ−(x,v) dv = ∫ ω + x m ∫ τ−(x,v) 0 e ( x,γ(x,v)(t − τ−(x,v)) ) t1jφ−(~γ(x,v)(t − τ−(x,v)) dt m(x,v) dv = ∫ ω + x m ∫ τ−(x,v) 0 e ( x,γ(x,v)(t − τ−(x,v)) ) × ∫ ωym k(y,ŵ,w̄1)e ( γ(x,v)(t − τ−(x,v)),γ(y,ŵ)(−τ−(y,ŵ)) ) × φ−(~γ(y,ŵ)(−τ−(y,ŵ)) dŵ dt m(x,v) dv where (y,w̄1) = ~γ(x,v)(t − τ−(x,v)) = ∫ m ∫ ωym e ( x,y,γ(y,ŵ)(−τ−(y,ŵ)) ) k(y,ŵ,w̄1)φ− ( ~γ(y,ŵ)(−τ−(y,ŵ)) ) dŵ × m(x,ŵ1)jx(y) dy where w̄1, ŵ1 are the initial and final tangent vectors (respectively) of the geodesic from y to x, = ∫ γ− ∫ τ+(x′,v′) 0 k(~γ(x′,v′)(t), w̄1)e(x ′,γ(x′,v′)(t),x)m(x,ŵ1)jx(γ(x′,v′)(t)) dt × φ−(x ′,v′) dµ(x′,v′). this proves (5.15). as seen in the (partial) proof of theorem 5.3, there appear weakly singular integrals due to the presence of the functions jx (see also (5.13)). definition 5.3. let t be the operator with kernel d(x, ·)n−1, that is t f(x) := ∫ m f(y) d(x,y)n−1 dy. (5.17) from (5.13) one has t : lp(m) → lp(m) continuously, 1 ≤ p ≤ ∞ with ‖t ‖ ≤ (cκm ) n−1|sn−1| diam m (see [t], prop. 5.1, appendix a). we also define the analogous operator t̃ , t̃ f(x) := ∫ m f(y)jy(x) dy (5.18) for which it holds t̃ : lp(m) → lp(m), 1 ≤ p ≤ ∞, with ‖t̃ ‖ ≤ |sn−1| diam m. we have seen that α0 is more singular than the remaining αj. in the following proposition we estimate the contribution from the terms for j ≥ 1. cubo 11, 5 (2009) optical tomography for media ... 89 proposition 5.1. let p = (n − µ)/(n − 1), 0 < µ < 1, q = (1 − 1/p)−1. with cκm,cκm defined as in lemma 5.1, let ‖k‖l∞(ω2m) < [ (cκmcκm ) n−1 diam m|sn−1| ]−1 . then for almost every x ∈ ∂m, ∣∣∣ ∞∑ j=1 ∫ γ− αj(x,x ′,v′)f(x′,v′) dµ(x′,v′) ∣∣∣ ≤ c0‖f‖lq(γ−,dµ) (5.19) where c0 > 0 depends on κm, κm, ‖k‖l∞(ω2m), ‖m‖l∞(ωm) and diam m. proof. when j = 1, ∣∣∣ ∫ γ− α1(x,x ′,v′)f(x′,v′) dµ(x′,v′) ∣∣∣ = ∣∣∣ ∫ γ− ∫ τ+(x′,v′) 0 k(~γ(x′,v′)(t), w̄)e(·)m(x,ŵ1)jx(γ(x′,v′)(t)) dt f(x ′,v′) dµ(x′,v′) ∣∣∣ ≤ ‖k‖∞‖m‖∞(cκm ) n−1 ∫ m ∫ ωym |f ( ~γ(y,v)(−τ−(y,v)) ) | d(x,y)n−1 dv dy by (5.13) ≤ (cκm ) n−1‖k‖∞‖m‖∞ (∫ ωm 1 d(x,y)(n−1)p dv dy ) 1 p × (∫ ωm ∣∣f ( ~γ(y,v)(−τ−(y,v)) )∣∣q dv dy ) 1 q by hölder’s inequality ≤ (cκm ) n−1‖k‖∞‖m‖∞|s n−1|1/pc 1/p d (diam m) 1/q‖f‖lq(γ−,dµ) see (5.17) = c‖k‖∞‖m‖∞(diam m) 1/q‖f‖lq(γ−,dµ), say. the computation for the terms j ≥ 2 are slightly different from the above. we refer the reader to [lm] for the details, where it is also proven that the infinite series converges in l1. 5.1 determination of σ the determination of σ is attained via a limiting argument exactly as in proposition 3.1. the construction of the approximate identity is slightly different but the spirit is the same. the details are in [lm]. let hσ ⊂ γ− be such that the geodesic x-ray transform restricted to geodesics in γ(hσ) is invertible. theorem 5.4. [lm] let (σ,k,m) satisfy the hypothesis of theorem 5.1. for almost every (x∗,v∗) ∈ hσ lim η→0 mfη(x ∗ ) = e ( γ(x∗,v∗)(−τ−(x ∗,v∗)),x∗ ) m(x∗,v∗) where fη concentrates to (x ′,v′) = ~γ(x∗,v∗)(−τ−(x ∗,v∗)) as η → 0. since m is known we thus know the integrals of σ along every geodesic in γ(hσ) and hence σ is uniquely determined. 90 stephen mcdowall cubo 11, 5 (2009) 5.2 determination of k throughout this section, the measurement point x will be fixed. as mentioned earlier we assume that the scattering kernel is of the form k(x)θ(x,v′,v), where θ(x,v′,v) is a-priori known. we prove in this setting that the spatial distribution k(x) is uniquely determined by the averaged albedo operator m from measurements at the single point x. definition 5.4. given a complete riemannian manifold (m,g) with geodesics γ(x,v)(t), and functions η : ωm → r, and β ∈ c∞(γ−) we may define the weighted geodesic transform by, for f : m → r, iη,βf(x ′,v′) := β(x′,v′) ∫ τ+(x′,v′) 0 f(γ(x′,v′)(t))η(~γ(x′,v′)(t)) dt. (5.20) we also have the l2 adjoint, for f : γ− → r, i∗η,βf(x) = ∫ ωxm f ( ~γ(x,v)(−τ−(x,v)) ) β ( ~γ(x,v)(−τ−(x,v)) ) η(x,v) dv. (5.21) let χ ∈ c∞0 (m) with χ ≡ 1 on a neighborhood of {y ∈ m : dist(y,∂m) ≥ d}. if x1 ∈ m, let v̄ = v̄(x1) ∈ ωx1m and v = v(x1) ∈ ω + x m be the initial and final tangent vectors, respectively, of the geodesic joining x1 to x. then for v1 ∈ ωx1m define the weight function w(x1,v1) := θ(x1,v1, v̄)e ( γ(x1,v1)(−τ−(x1,v1)),x1,x ) m(x,v)jx(x1)χ(x1). (5.22) from (5.15) and (5.20) we see that α1(x,x ′,v′) = iw,1k(x ′,v′). if g, θ, σ and m are real-analytic, then w is a real-analytic, non-vanishing weight function in {y ∈ m : dist(y,∂m) ≥ d}. it is injectivity of this weighted ray transform which will allow determination of k(x). we make use of injectivity results of [fsu]. these results allow a possibly proper subsets of geodesics on which the transform is known. it is the inclusion of the factor β in definition (5.20) that restricts the transform to a such a subset geodesics. definition 5.5. we say that γ is a regular family of curves (for the metric g) if for any (x,v) ∈ t ∗m\{0} there exists γ ∈ γ through x, normal to v, and such that γ has no conjugate points (automatically satisfied for simple metrics). we say that β ∈ c∞(γ−) is regular if there exists a set h ⊂ {(x ′,v′) : β(x′,v′) 6= 0} such that γ(h) is a regular family. suppose that k(x)θ(x,v,v′) and k̃(x)θ(x,v,v′) are two scattering kernels with k, k̃ ∈ kdε ; let m, m̃, and α, α̃ be the averaged albedo operators and schwarz kernels associated to k and k̃ respectively. we set ∆k = k − k̃ and ∆αj = αj − α̃j, j = 1, 2, . . . . lemma 5.2. [fsu] let (m, θ,σ,g) be fixed and real analytic, and suppose β ∈ c∞(γ−) is a regular function for g. then there exits c > 0, independent of k, k̃ ∈ kdε such that ‖∆k‖l2(m) ≤ c‖i ∗ w,βiw,β∆k‖h1(m), (5.23) with the above estimate holding in a c2 neighborhood of (m, θ,σ), and a c3 neighborhood of g. cubo 11, 5 (2009) optical tomography for media ... 91 proposition 5.2. let (m, θ,σ,g) be fixed and real analytic, and suppose β ∈ c∞(γ−) is a regular function for g. furthermore, suppose hk ⊂ {(x ′,v′) : β(x′,v′) 6= 0} is such that γ(hk) is a regular set of geodesics. suppose that m = m̃ on l1(hk,dµ). then there exists c > 0 such that for all k, k̃ ∈ kdε ‖∆k‖l2(m) ≤ c‖i ∗ w,βiw,β∆k‖h1(m) = c‖i ∗ w,ββ∆α1‖h1(m) = c ∥∥∥i∗w,ββ ∞∑ j=2 ∆αj ∥∥∥ h1(m) , with the above estimate holding in a c2 neighborhood of (m,σ, θ) and a c3 neighborhood of g. proof. the first inequality is (5.23); next, βiw,1 = iw,β and w has been defined so that iw,1∆k = ∆α1; finally, since σ = σ̃, α0 = α̃0 and so m = m̃ implies that ∑ ∞ j=1 ∆αj = 0 which proves the final equality. analogous to (4.9) we proceed to prove that ‖∆k‖l2(m) ≤ εc‖∆k‖l2(m), and so for sufficiently small ε > 0, we will have ∆k = 0. the following proposition is an extension of proposition 4 of [fsu] to more general weight functions ηj and restrictions βj. the proof is essentially the same and further details are given in [lm]. proposition 5.3. there is c > 0 such that for all f ∈ l2(m) with supp f ⊂ {y ∈ m : d(x,∂m) > d}, ‖i∗η1,β1iη2,β2f‖h1(m) ≤ c‖η1‖c2(ωm)‖η2‖c2(ωm)‖β1β2‖c2(γ−)‖f‖l2(m), (5.24) with c depending continuously on the c4 norm of g. we wish to apply (5.24) to each ∆αi, i > 1, and to do so we express each as a sum of weighted x-ray transforms. this is achieved by expanding one instance of the kernel θ occurring in ∆αi in a manner based on spherical harmonic expansions of functions on sn−1. this expansion is the content of lemma 5.3. the proof is postponed to the end of this section. lemma 5.3. let θ(x,v′,v) ∈ c3n(ω2m) and g ∈ c3n(m). there exist θj(x,v ′ ), ϕj(x,v) ∈ c3n(ωm) such that θ(x,v′,v) = ∞∑ j=1 θj(x,v ′ )ϕj(x,v) with ‖θj‖c2(ωm) ≤ c 1 + j2 , and ‖ϕj(x,v)‖l∞(ωm) ≤ 1, the above estimate holding in a c3n neighborhood of (θ,g). 92 stephen mcdowall cubo 11, 5 (2009) proposition 5.4. fix (m,g, θ,σ) with m,σ ∈ c2, and g, θ ∈ c3n. suppose that ‖k‖∞,‖k̃‖∞ < [‖θ‖∞ diam m|s n−1|]−1, k, k̃ ∈ kdε , and β ∈ c ∞ (γ−). then there is c > 0 such that ∥∥∥i∗w,ββ ∞∑ i=2 ∆αi ∥∥∥ h1(m) ≤ cε‖∆k‖l2(m), with the above estimate holding in a c2 neighborhood of (m,σ), and a c3n neighborhood of (θ,g). proof. for (x,v) ∈ ωm we define e(x,v) = e(γ(x,v)(−τ−(x,v)),x). from theorem 5.3, ∆α2(x,x ′,v′) = ∫ τ+(x′,v′) 0 e(x′,γ(x′,v′)(t)) ∫ m [ ∆k(γ(x′,v′)(t))k(y2) − k̃(γ(x′,v′)(t))∆k(y2) ] × θ(~γ(x′,v′)(t), w̄1)θ(y2, ŵ1, w̄2)e(γ(x′,v′)(t),y2)e(y2,x)j (γ(x′,v′)(t),y2,x) dy2 dt and from lemma 5.3 we have (formally at least), ∆α2(x,x ′,v′) = ∞∑ l=1 ∫ τ+(x′,v′) 0 e(~γ(x′,v′)(t))θl(~γ(x′,v′)(t)) × ∫ m [ ∆k(γ(x′,v′)(t))k(y2) + k̃(γ(x′,v′)(t))∆k(y2) ] φl(γ(x′,v′)(t), w̄1)θ(y2, ŵ1, w̄2) × e(γ(x′,v′)(t),y2)e(y2,x)j (γ(x′,v′)(t),y2,x)m(x,ŵ2) dy2 dt = ∞∑ l=1 iθle,1 [ ψ2,1,l + ψ2,2,l ] (x′,v′) where ψ2,1,l(z) = ∫ m ∆k(z)k(y2)ϕl(z,w̄1(z,y2))θ(y2, ŵ1(z,y2), w̄2(y2,x))e(z,y2)e(y2,x) × j (z,y2,x)m(x,ŵ2(y2,x)) dy2, ψ2,2,l(z) = ∫ m k̃(z)∆k(y2)ϕl(z,w̄1(z,y2))θ(y2, ŵ1(z,y2), w̄2(y2,x))e(z,y2)e(y2,x) × j (z,y2,x)m(x,ŵ2(y2,x)) dy2. cubo 11, 5 (2009) optical tomography for media ... 93 in a similar manner, with y1 = γ(x′,v′)(t), ∆αj(x,x ′,v′) = ∞∑ l=1 ∫ τ+(x′,v′) 0 e(~γ(x′,v′)(t)θl(~γ(x′,v′)(t)) × ∫ m · · · ∫ m ( j∑ i=1 k̃(y1) · · · k̃(yi−1)∆k(yi)k(yi+1) · · ·k(yj) ) ϕl(γ(x′,v′)(t), w̄1) × ( j∏ i=2 θ(yi, ŵi−1, w̄i)e(yi−1,yi) ) e(yj,x)j (γ(x′,v′)(t),y2, . . . ,yj,x) × m(x,ŵj) dyj · · ·dy2 dt = ∞∑ l=1 j∑ i=1 iθle,1ψj,i,l(x ′,v′). explicitly, ψj,i,l(y1) = ∫ m · · · ∫ m k̃(y1) · · · k̃(yi−1)∆k(yi)k(yi+1) · · ·k(yj)ϕl(y1, w̄1) × ( j∏ i=2 θ(yi, ŵi−1, w̄i)e(yi−1,yi) ) e(yj,x)j (·) m(x,ŵj) dyj · · ·dy2. one may estimate ‖ψj,i,l‖l2(m) (see [lm]): ‖ψj,i,l‖l2(m) ≤ cε(ε‖θ‖∞(diam m)|s n−1|)j−2‖∆k‖l2(m). (5.25) now, using the relation βiθj e,1 = iθj e,β, we have ∥∥∥i∗w,ββ ∞∑ j=2 ∆αj ∥∥∥ h1(m) ≤ ∞∑ j=2 j∑ i=1 ∞∑ l=1 ‖i∗w,βiθj e,βψj,i,l‖h1(m) ≤ ∞∑ j=2 j∑ i=1 ∞∑ l=1 c 1 + l2 ‖ψj,i,l‖l2(m) ≤ ∞∑ j=2 j∑ i=1 ∞∑ l=1 c′ 1 + l2 ε(ε‖θ‖∞(diam m)|s n−1|)j−2‖∆k‖l2(m) ≤ εc′′‖∆k‖l2(m) ∞∑ j=2 j(ε‖θ‖∞(diam m)|s n−1|)j−2. this follows from proposition 5.3 and lemma 5.3, with c depending continuously on the c3n norm of g and θ, and the c2 norms of σ and m as well. we see that for sufficiently small ε this series converges (thus justifying the formal computations performed above). proof of theorem 5.2. fix real analytic (m,g,σ, θ). given the hypothesis of theorem 5.2 we are ensured that both proposition 5.2 and proposition 5.4 hold. combining them, we have the existence 94 stephen mcdowall cubo 11, 5 (2009) of a constant c, depending continuously on the c2 norms of (m,σ), and the c3n norms of (θ,g) such that ‖∆k‖l2(m) ≤ cε‖∆k‖l2(m) and so for 0 ≤ ε < c−1, we must have k = k̃. proof of lemma 5.3. in a fixed coordinate system for m we define a smooth bijection ωxm → s n−1 and define θ̃ ∈ c∞(ωm × sn−1) θ̃(x,v′,θ) = θ(x,v′,v(θ)). for f ∈ hk, the space of spherical harmonics of order k, the laplacian ∆s on s n−1 is ∆sf = −k(k+n−2)f. denote by zxk (θ) the so-called zonal harmonics for which f(x) = 〈f,z x k (θ)〉l2(sn−1) for all f ∈ hk. then one has (see, for example, [f]), dim(hk) = dk ≤ cn(k n−2 + 1) ‖zxk‖l2(sn−1) = c ′ n √ dk for all x ∈ s n−1. let {ψ̃kl} dk l=1 ∈ l ∞ (s n−1 ) be an l2(sn−1) orthonormal basis for hk (so z x k = ∑dk l=1 ψ̃kl(θ)ψ̃kl(x)). define ψkl := ‖ψ̃kl‖ −1 l∞(sn−1) ψ̃kl. then for each (x,v ′ ) ∈ ωm, θ̃(x,v′,θ) = ∞∑ k=0 dk∑ l θkl(x,v ′ )ψkl(θ) with θkl(x,v ′ ) = ‖ψ̃kl‖l∞(sn−1) ∫ sn−1 θ̃(x,v′,θ)ψ̃kl(θ) dθ. for each θ, |ψ̃kl(θ)| = |〈ψ̃kl,z θ k〉| ≤ ‖ψ̃kl‖l2(sn−1)‖z θ k‖l2(sn−1) = c ′ n √ dk. next, since ψ̃kl ∈ hk, for any n ∈ n and k ≥ 1, |θkl(x,v ′ )| = ‖ψ̃kl‖l∞(sn−1) ∣∣∣ ∫ sn−1 θ̃(x,v′,θ)ψ̃kl(θ) dθ ∣∣∣ = ‖ψ̃kl‖l∞(sn−1) ∣∣∣ ∫ sn−1 θ̃(x,v′,θ) (∆s) nψ̃kl(θ) (−k(k + n − 2))n dθ ∣∣∣ ≤ c′n √ dk (k(k + n − 2))n ∣∣∣ ∫ sn−1 ψ̃kl(∆s) n θ̃(x,v′,θ) dθ ∣∣∣ ≤ c′′nk (n−2)/2 (k(k + n − 2))n ‖(∆s) n θ̃(x,v′, ·)‖l2(sn−1), cubo 11, 5 (2009) optical tomography for media ... 95 where c′′n depends only on the dimension n. thus for sufficiently large n (in fact n ≥ 5n/2), there is cn,n such that |θkl(x,v ′ )| ≤ cn,n 1 + k2n ‖(∆s) n θ̃(x,v′, ·)‖l2(sn−1). now renumber the collection of coefficient functions and define new basis functions as follows: with j = d0 + d1 + · · · + dk−1 + l, set θj(x,v ′ ) = θkl(x,v ′ ), and ϕj(x,v) := ψkl(θx(v)). then θ(x,v′,v) = ∞∑ j=1 θj(x,v ′ )ϕj(x,v). now j ≤ k∑ m=0 dm ≤ cn(k + 1)(k n−2 + 1) ≤ ĉnk n so |θj(x,v ′ )| = |θkl(x,v ′ )| ≤ cn,n 1 + k2n ‖(∆s) n θ̃(x,v′, ·)‖l2(sn−1) ≤ c̃n,n 1 + j2 ‖(∆s) n θ̃(x,v′, ·)‖l2(sn−1). if one applies the same decomposition to ∂αx θ(x,v ′,v), α ∈ {0, 1, 2}n, one finds that the coefficients are nothing more than ∂αx θj(x,v ′ ) and these satisfy |∂αx θj(x,v ′ )| ≤ c̃n,n 1 + j2 ‖(∆s) n∂αx θ̃(x,v ′, ·)‖l2(sn−1) ≤ c 1 + j2 . note that c depends on 2n derivatives of θ̃ in the last variable, and hence 2n derivatives of g due to the change of variables introduced earlier. this proves the claim of the lemma. received: december, 2008. revised: april, 2009. references [a] arridge, s., optical tomography in medical imaging, inverse problems, 15 (1999), r41–r93. 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[w] wang, j.n., stability estimates of an inverse problem for the stationary transport equation, ann. inst. henri poincare, 70 (1999), 473–495. b6-ot-varying-ir cubo 8 , 87-89 {1992) s~ptlma jornada de matemática de la zo n" sur. formas bilineales asociativas en una algebra bárica • r. baeza v. y r. benavides g.t 1 introducción. sea j( un cuerpo de caraclerfsbica difert::nle de clos y a una álgebra sobre 1(, conmlltntiva pero no necesariamen~e asociabiva. para x en a y k ~ 1 un n(1mero entero, la potencia p lena de :i: es deanida inclue bivamente por xl!j = x y xlkl = :¡:fkllxlk-11, k ~ 2 sea ahora (a,w) una k-álgebr.a bárica (o álgebra ponderada), es decir, una kálgebra conmutativa a y un homom0t1fismo uo tir ivial de k-á1gebras w: a i<. diremos que (a,w) es una álgebra de bernsbein de orden k, si para todo x de a se ve11iñca (!) siendo k d menor en tero para el cual esba identidad es válida. la definción de á lgeb ras de bernsoein ele orden k y algunos ejemplos para k = 2 fue dada por abraham 1980. cuando k = 1, a es llamada álgebra de bernslein. eslas álgebras han sido estiudiada.s exlensivamenle, ver por ejemplo holgale 1975 , ljubic 1978, 1987 y hentzel · peresi 1988. se sabe que si ( a ,w) es una álgebra de bernslein de o rdeu k , enlonces · la ponderación w es (mica ·ella adm ite un e\emenlo idempoten te e tal que w( e: ) = 1 · si t : n n, con n = ke.rw, denota la mu\liplicnció n a izquierda por d idempolente e: y u = lm-r~ , v = /( er-rk, enlonccs a = ke.e u e v (d escompoiiición de pierce re lali \-'b a l id empotent e e), u = {n en: ch= ~n} co n u2 ~v. sea w un s ubespocio com plemcntnrio d e u2 e n v , o.sí que u1 e w = \!. consideremos b : a x a /( uno forma biliueal so bre .a , di remos que b es asociat iva si se \"e rificn que 'flnri.n cin.d o po i pto~'f'c lo fondecyt 105-91 y diuf"ro 9103 1e1q10liito r 87 88 cubos r. baeza v., r. benavides g. b(zy, z) = b(z, yz) vz, y, z , e a. 2 re!tuftados. procederemos a realizar algunos cálculos referente a una forma bilineal asociativa b sobre a. sean e n,b(e,n) = b(e',n) = b(e,en) = b(en,e) = b(n,e') = b(n,e) : .b(e, n) = b(n, e) (2) por otro lado 'r/11 e u, tenemos b(e, u) = b(e', u) = b(e, eu) = b(e, ju) = jb(e, u) :.b(e, u) =o . para v e v, se obtie11e b(e , v) = b(e'+l, v) = b(e', ev) = b(e', rv) =b(e'1, r 2v) = ... = b(e, r'v) =o así, tenem os que b( e, n) = b(n,e) =o para u e u y 11 en, tenemos b(u, 11) = 2b{4u, n) = 2b(eu, n) = 2b(e, m1) =o (n ideal de a). análogamente se tiene b(n, u) = o. así b(n, u)= b(u, n) =o para cada u.¡u; generador de u2 y n e n. (3) (4) b(u¡u;, n) = b(u¡, u;n) =o, de la misma forma b(n , u;u; } =o entonces tenemos b(u'. n) = b(n, u') =o así la matriz asociada a b, es : /( e u ( b(e,e) o (b) = o o o u' o o o w d ke u u' w de esta forma matricial, válida para toda dimensión de a, se concluye (5) (6) proposición 2.1 cualquier forma bilineai 11.3ocialiva s obre unn cilge brn de bt!rn.$lein de orden k, con u .j; {o}, es degenem da. demostración. es inmediata desde (6) • proposición 2.2 cualquier fonna. bilinea.1 asocia.tiua b de una. .álgcbro de benutein a de orden k, está totalmente det enninada por su acción en k ce w. cub o 8 89 d emostraclón. es obvio, porque si :i: = ,\e+ u+ v1 +va, y = ).1e + u' + 11í + 11~ con..\,).' e k¡u ,u' e u¡v¡,u; e 1j'l y 112,t12: e w; eulonces b(x,y) = !.!.'b(e,e) + 8("2.v;) • p l'opoaición 2.3 si existe una. forma. bilineal a.sociativo no degenerada. b sobre una rlgebra de bernslein a de orden k, entonces a es ca.si constante v el elemento idempolente e es 1lnieo. demostración. si bes no degenerada, u= {o}. teorema 5.10 c. ma1lol est.ablcce que a es una á lgebra casi constante y el elemento idempolenle es tínico. • referencias ¡n.j r. baeza, a. catalán, r. cos~a, bernstein algebra8 wilh msocia.live bilinear /omn...,.,arch. mat!h.58, 234-238 (1992) [2] l. ne nlzel, l. peresi, ph. holgal.e 1 on kth-onler bernsteln oj.gebms n.ncl slabilily at tlle k+l genemtion in polyploi~, lma. j. malh. app l. biol.7,33-40 (1990). [3] c. mallo!, a propos eles algebres de. bernsle.in,tb ese de doclioraclo, montpellier-france. {1990) d i.r.ección· cle los autores: departamento de matemática y estadística unive rsidad de la frontera casilla 54-d . te.muco revista de matemáticas_0103 revista de matemáticas_0104 revista de matemáticas_0105 a mathematical journal vol. 7, no 2, (139 169). august 2005. differential forms and/or multi-vector functions f. brackx department of mathematical analysis ghent university. galglaan 2, b-9000 gent, belgium fb@cage.ugent.be r. delanghe department of mathematical analysis ghent university. galglaan 2, b-9000 gent, belgium rd@cage.ugent.be f. sommen department of mathematical analysis ghent university. galglaan 2, b-9000 gent, belgium fs@cage.ugent.be ”... on peut se poser la question: quel est le théorème mathématique le plus profond, le plus difficile, dont il existe une interprétation physique concrète et indubitable? (...) pour moi, c’est le théorème de stokes qui est le candidat numéro un. et cela témoigne d’un fait: la différentielle extérieure est une notion très mystérieuse, dont la véritable nature, je crois, recèle encore bien des énigmes, et cela en dépit de la simplicité de sa définition formelle.” rené thom, la science malgré tout ... abstract similarities are shown between the algebras of differential forms and of clifford algebra-valued multi-vector functions in an open region of euclidean space. the poincaré lemma and the dual poincaré lemma are restated and proved in a refined version. in the case of real-analytic differential forms an alternative proof of the poincaré lemma is given using the euler operator. a position is taken in 140 f. brackx, r. delanghe and f. sommen 7, 2(2005) the debate on the redundancy of either of the two algebras. resumen se muestran similitudes entre las álgebras de formas diferenciales y las de funciones multivectoriales valuadas de una álgebra de clifford en una región abierta del espacio euclidiano. el lema de poincaré y lema de poincaré dual son presentados y probados en una versión refinada. en el caso de formas diferenciales reales anaĺıticas una prueba alternativa del lema de poincaré es dada usando el operador de euler. una posición es tomada en el debate en redundancia de cualquiera de las dos álgebras. key words and phrases: differential forms, multi-vector functions, poincaré lemma math. subj. class.: 58a10, 30g35 1 introduction in this paper two mathematical languages are confronted with each other: the language of differential forms and the one of clifford algebra-valued multi-vector functions. the cartan algebra ∧ (ω) of smooth differential forms on an open subset ω of euclidean space rm+1, endowed with exterior multiplication, is of course well-known. a fundamental operator on ∧ (ω) is the exterior derivative d with its important property that for any differential form ω, d(dω) = 0. introducing the hodge co-derivative d∗ leads to the differential operator d = d + d∗, by means of which the so-called ”harmonic” r-forms (0 < r < m + 1) are characterized as smooth differential r-forms ωr satisfying dωr = 0. the algebra e(ω) of smooth multi-vector functions is less well-known. multi-vector functions arise in a natural way when considering functions defined in ω and taking values in the universal real clifford algebra r0,m+1 constructed over r0,m+1, i.e. rm+1 equipped with an anti-euclidean metric. if rr0,m+1 (0 ≤ r ≤ m + 1) denotes the space of r-vectors, then the clifford algebra r0,m+1 is precisely the graded associative algebra r0,m+1 = ∑m+1 r=0 ⊕ rr0,m+1, and an r-vector function fr is a map fr : ω → rr0,m+1. it was william kingdon clifford who introduced his so-called geometric algebra in the 1870s, building on earlier work of hamilton and grassmann. a fundamental operator on the space of smooth multi-vector functions, is the dirac operator ∂, by means of which the so-called monogenic functions are characterized as the smooth functions f satisfying ∂f = 0. note that the monogenic functions are 7, 2(2005) differential forms and/or multi-vector functions 141 at the core of so-called clifford analysis, a function theory which developed extensively during the last decades, offering a direct and elegant generalization to higher dimension of the theory of holomorphic functions in the complex plane. note also that the above mentioned dirac equation may be expressed in the language of systems of partial differential equations by modelling clifford algebra through its matrix representation. the spaces of smooth differential forms on the one hand, and of smooth multi-vector functions on the other, are shown to be isomorphic in a natural way: a smooth r-form is identified with a smooth r-vector function, the action of the differential operator d = d + d∗ on the space ∧r(ω) of smooth r-forms, is identified with the action of the dirac operator ∂ on the space er(ω) of smooth r-vector functions, and the counterparts in the space of multi-vectors of the exterior derivative d and the co-derivative d∗ are pinpointed. this isomorphism is moreover fully exploited in that proofs can be given in either of both languages and that the results obtained are mutually exchangeable (section 4). in fact the paper also focusses on two well-known theorems on differential forms: the poincaré lemma and the dual poincaré lemma. they are restated in a refined version which, to the authors’ knowledge, rarely appears in the literature. combining these two theorems, a structure theorem for monogenic multi-vector functions and its counterpart in the space of smooth differential forms is given (section 5). in proving these structure theorems, we heavily rely on the classical poincaré lemma and the classical dual poincaré lemma. in section 6 an alternative proof of those lemmata are given in the special case of real-analytic differential forms in an open ball centred at the origin. we wish to emphasize that the present paper may not be seen as a pleading to substitute one of the languages for the other, nor to prefer one language above the other. on the contrary, we are convinced that differential forms and multi-vector functions, despite the natural identification given, are quite different mathematical objects, the use of which is very much imposed by the mathematical context. this in-depth difference between and context-dependence of differential forms and multi-vector functions will be fully discussed in a forthcoming paper by one of the authors. 2 multi-vector functions: preliminaries in this section we recall some basic notions and results from clifford algebra and clifford analysis. for a detailed account we refer the reader to [10] and [2]; the recent book [3] gives a nice and broad overview of the intrinsic value and usefulness of clifford algebra and clifford analysis for mathematical physics. the construction of the universal real clifford algebra is well-known; we restrict ourselves to a schematic approach. let r0,m+1 be the real vector space rm+1 (m ≥ 1) endowed with a non-degenerate symmetric bilinear form b of signature (0,m + 1), and let (e0,e1, · · · ,em) be an associated orthonormal basis: b(ei,ej ) = { −1 if i = j 0 if i �= j (0 ≤ i,j ≤ m). 142 f. brackx, r. delanghe and f. sommen 7, 2(2005) the anti-euclidean metric on r0,m+1 is induced by the scalar product < ei,ej >= −b(ei,ej) = δij, 0 ≤ i,j ≤ m. introduce the anti-symmetric outer product by the rules: ei ∧ ei = 0, 0 ≤ i ≤ m ei ∧ ej + ej ∧ ei = 0, 0 ≤ i �= j ≤ m. for each a = {i1, i2, · · · , ir} ⊂ m = {0, 1, · · · ,m}, ordered in the natural way: 0 ≤ i1 < i2 < · · · < ir ≤ m, put ea = ei1 ∧ ei2 ∧ ·· · ∧ eir and eφ = 1. then for each r = 0, 1, · · · ,m + 1, the set {ea : a ⊂ m and |a| = r} is a basis for the space rr0,m+1 of so-called r-vectors. introducing the inner product by ei • ej = − < ei,ej >, 0 ≤ i,j ≤ m leads to the so-called geometric product in the clifford algebra, given by eiej = ei • ej + ei ∧ ej, 0 ≤ i,j ≤ m. the respective definitions of the inner product, the outer product and the (geometric) product are then extended to r-vectors by the formulae: ej • ea = ej • (ei1 ∧ ·· · ∧ eir ) = ∑ k (−1)kδjik ea\{ik} where ea\{ik} = ei1 ∧ ·· · ∧ eik−1 ∧ [eik ∧] eik+1 ∧ ·· · ∧ eir and{ ej ∧ ea = ej ∧ (ei1 ∧ ·· · ∧ eir ) = ej ∧ ei1 ∧ ·· · ∧ eir, if j /∈ a ej ∧ ea = 0, if j ∈ a and finally ejea = ej • ea + ej ∧ ea. the inner and outer products are distributive over addition, and so is the (geometric) product. the universal real clifford algebra r0,m+1 is the graded associative algebra r0,m+1 = m+1∑ r=0 ⊕ rr0,m+1. 7, 2(2005) differential forms and/or multi-vector functions 143 if [ . ]r : r0,m+1 → rr0,m+1 denotes the projection operator from r0,m+1 onto rr0,m+1, then each clifford number a ∈ r0,m+1 may be written as a = m+1∑ r=0 [a]r. note that in particular for a 1-vector u and an r-vector vr , one has u vr = u • vr + u ∧ vr with u • vr = [u vr]r−1 = 1 2 ( u vr − (−1)rvr u ) and u ∧ vr = [u vr]r+1 = 1 2 ( u vr + (−1)rvr u ) . usually r and rm+1 are identified with r00,m+1 and r 1 0,m+1 respectively. an element x = (x0,x1, · · · ,xm) ∈ rm+1 is thus identified with the 1-vector x = ∑m j=0 xj ej . now let ω be an open region in rm+1. a smooth r-vector function fr is a map fr : ω → rr0,m+1, x → ∑ |a|=r fr,a(x) ea where for each a, fr,a is a smooth real-valued function in ω. we denote by er(ω) the space of smooth r-vector functions in ω, and we put e(ω) = m+1∑ r=0 ⊕ er(ω). the projection operator from e(ω) onto er(ω) is denoted by [ . ]r. for the linear operator t : er(ω) →e(ω) we denote by r ker t the kernel of t in er(ω), while r im t stands for the image of er(ω) under t . a fundamental operator in clifford analysis is the so-called dirac operator, a vector differential operator given by ∂ = m∑ j=0 ej ∂xj . due to the non-commutativity of the multiplication in the clifford algebra, it can act from the left or from the right on a function. for f = ∑ a eafa ∈e(ω) these actions are given by ∂f = ∑ j ∑ a ejea ∂xj fa and f∂ = ∑ j ∑ a eaej ∂xj fa. 144 f. brackx, r. delanghe and f. sommen 7, 2(2005) a function f ∈e(ω) is called left (resp. right) monogenic in ω iff it satisfies ∂f = 0 (resp. f∂ = 0) in ω. restricting the dirac operator ∂ to the space er(ω) we find for an r-vector function fr, that ∂fr and fr∂ split up into an (r− 1)-vector and an (r + 1)-vector function: ∂fr = m∑ j=0 ej ∂xj fr = ∑ j ej • ∂xj fr + ∑ j ej ∧ ∂xj fr and fr∂ = m∑ j=0 ∂xj fr ej = ∑ j ∂xj fr • ej + ∑ j ∂xj fr ∧ ej. it readily follows that [∂fr]r−1 = ∑ j ej • ∂xj fr = (−1)r+1 ∑ j ∂xj fr • ej = (−1)r+1[fr∂]r−1 [∂fr]r+1 = ∑ j ej ∧ ∂xj fr = (−1)r ∑ j ∂xj fr ∧ ej = (−1)r[fr∂]r+1. consequently, for an r-vector function fr , the notions of left monogenicity and right monogenicity coincide. moreover, if for f ∈e(ω) we put fe = ∑ |a|=even ea fa and fo = ∑ |a|=odd ea fa, then f is monogenic in ω iff both fe and fo are monogenic in ω. commonly one introduces the notations: ∂ • fr = [∂fr]r−1 , ∂ ∧ fr = [∂fr]r+1 fr • ∂ = [fr∂]r−1 , fr ∧ ∂ = [fr∂]r+1. the action of the dirac operator ∂ on er(ω) thus gives rise to two auxiliary differential operators: ∂− : er(ω) → er−1(ω) : fr → ∂−fr = ∂ • fr = [∂fr]r−1 and ∂+ : er(ω) → er+1(ω) : fr → ∂+fr = ∂ ∧ fr = [∂fr]r+1. symbolically these operators may be written as: ∂− = (∂ • ) = ∑ j (ej • )∂xj and ∂+ = (∂ ∧) = ∑ j (ej ∧)∂xj . 7, 2(2005) differential forms and/or multi-vector functions 145 their action on er(ω) is two-fold in the sense that they act on the multi-vector by means of the inner and outer product with basis vectors, and at the same time on the function coefficients by partial differentiation. as on er(ω) holds: ∂ = ∂− + ∂+ we obtain that a smooth r-vector function fr is left monogenic (as well as right monogenic) in ω iff in ω ∂fr = 0 ⇐⇒ fr∂ = 0 ⇐⇒ { ∂−fr = 0 ∂+fr = 0 . (i) as the dirac operator ∂ splits the laplace operator: ∂2 = ∂ • ∂ + ∂ ∧ ∂ = ∂ • ∂ = − < ∂,∂ > = − a monogenic function in ω is also harmonic in ω, but the converse clearly is not true. as moreover (∂−)2 = (∂+)2 = 0 we have − = (∂− + ∂+)2 = ∂−∂+ + ∂+∂−. the second order differential operators ∂−∂+ and ∂+∂− are scalar operators in the sense that they keep the order of the multi-vector function, but the function coefficients, while being differentiated, are interchanged w.r.t. the basis multi-vectors. now observe that the system (i), expressing the monogenicity of an r-vector function, is also equivalent to ∂̃fr = (∂+ −∂−)fr = 0 or fr∂̃ = fr (∂+ −∂−) = 0 where we have introduced the modified dirac operator ∂̃ = ∂+ −∂−. we directly have the basic formulae: ∂∂̃ = ∂−∂+ −∂+∂− ∂̃∂ = ∂+∂− −∂−∂+ ∂̃∂̃ = −∂+∂− −∂−∂+ = 146 f. brackx, r. delanghe and f. sommen 7, 2(2005) which leads to the modified laplace operator ̃ = ∂−∂+ −∂+∂− which clearly is a scalar operator in the sense that it keeps the order of the multivector function on which it acts. taking into account the main involution, also called inversion, of the clifford algebra, for which (ei1 · · · eir )∗ = (ei1 ∧ ·· · ∧ eir )∗ = (−1)r ei1 ∧ ·· · ∧ eir we get the formulae: ∂fr = f∗r ∂̃ and ∂̃fr = f ∗ r ∂ − fr = ∂∂fr = ∂f∗r ∂̃ = (−1)r ∂fr∂̃ fr = ∂̃∂̃fr = ∂̃f∗r ∂ = (−1)r ∂̃fr∂ ̃fr = ∂∂̃fr = ∂f∗r ∂ = (−1)r ∂fr∂ − ̃fr = ∂̃∂fr = ∂̃f∗r ∂̃ = (−1)r ∂̃fr∂̃. 3 differential forms: preliminaries this section is also introductory; there is a vast literature on differential forms; we may refer to e.g. [8], [15]. let rm+1 be endowed with the standard euclidean metric. denoting by ∧r rm+1 the space of alternating (or skew-multilinear) real-valued rforms (0 ≤ r ≤ m + 1), the grassmann algebra or exterior algebra over rm+1 is the graded associative algebra ∧ rm+1 = m+1∑ r=0 ⊕ ∧r rm+1 endowed with the exterior multiplication. a basis for ∧r rm+1 is obtained as follows. let {dx0,dx1, · · · ,dxm} be a basis for the dual space (rm+1)∗ of rm+1. if again the set a = {i1, . . . , ir} ⊂ m = {0, 1, · · · ,m} is ordered in the natural way, put dxa = dxi1 ∧ dxi2 ∧ ·· · ∧ dxir and dxφ = 1. 7, 2(2005) differential forms and/or multi-vector functions 147 then for each r = 0, 1, · · · ,m + 1, the set {dxa : a ⊂ m and |a| = r} is a basis for∧r rm+1. note that in particular dxi ∧ dxi = 0, i = 0, 1, · · · ,m + 1 and dxi ∧ dxj + dxj ∧ dxi = 0, 0 ≤ i �= j ≤ m. a smooth r-form in an open region ω of rm+1 is a map ωr : ω → ∧r rm+1, x → ∑ |a|=r ωra(x) dx a where for each a, ωra is a smooth real-valued function in ω. we denote by ∧r(ω) the space of smooth r-forms in ω and we put ∧ (ω) = m+1∑ r=0 ⊕ ∧r (ω). the projection operator from ∧ (ω) onto ∧r(ω) is denoted by [ . ]r, and the notations of the foregoing section are kept for the kernel and the image of a linear operator t : ∧r(ω) −→ ∧(ω). a fundamental linear operator on the space of smooth forms is the exterior derivative d. it is first defined on ∧r(ω) (r < m + 1) by d : ∧r (ω) −→ ∧r+1 (ω) ωr = ∑ |a|=r ωra dx a −→ dωr = ∑ a ∑ j ∂xj ω r a dx j ∧ dxa and this definition is then extended to ∧ (ω) by linearity. the kernel of the exterior derivative d , r ker d = {ωr ∈ ∧r (ω) : dωr = 0} consists of the so-called closed r-forms in ω, while its image of ∧r−1(ω) in ∧r(ω) r−1 im d = {dωr−1 : ωr−1 ∈ ∧r−1 (ω)} consists of the so-called exact r-forms in ω. the quotient space hr(ω) = r ker d / r−1 im d 148 f. brackx, r. delanghe and f. sommen 7, 2(2005) is the so-called de rham r-th cohomology space. the well-known poincaré lemma (see also section 5) asserts that if ω is contractible to a point, then for each r > 0, hr(ω) = 0, in other words: if ω is contractible to a point and ωr ∈ ∧r(ω) is closed, then ωr is exact. the converse, i.e. that any exact r-form in an open region of rm+1 is also closed, follows at once from the observation that d(dω) = 0. a second fundamental linear operator on the space of smooth forms is the hodge co-derivative d∗. for a = {ii, · · · , ir}⊂ m we denote dxa\{ij } = dxi1 ∧ ·· · ∧ dxij−1 ∧ [dxij ∧] dxij+1 ∧ ·· · ∧ dxir and in a first step we put: d∗(ωadx a) = r∑ j=1 (−1)j ∂xj ωa dxa\{ij }. then d∗ is defined on ∧r(ω) (r > 0) by d∗ : ∧r (ω) −→ ∧r−1 (ω) ωr = ∑ |a|=r ωra dx a −→ d∗(ωr) = ∑ |a|=r d∗(ωra dx a) and this definition is extended to ∧ (ω) by linearity. the kernel of the co-derivative d∗ acting on ∧ (ω): r ker d∗ = {ωr ∈ ∧r (ω) : d∗ωr = 0} consists of the so-called co-closed r-forms in ω, while its image of ∧r+1(ω) in ∧r(ω) r+1 im d∗ = {d∗ωr+1 : ωr+1 ∈ ∧r+1 (ω)} consists of the so-called co-exact r-forms in ω. by observing that for any smooth form in ω, d∗(d∗ω) = 0, it follows that each co-exact r-form in ω is also co-closed. the quotient space hr(ω) = r ker d∗ / r+1 im d∗ is the so-called de rham r-th homology space. it could be confusing to use the term ”homology” here, since it usually refers to the complex associated with the algebra of chains subject to the action of the boundaryoperator; in the space of currents however there is a connection (see [8], p.313). 7, 2(2005) differential forms and/or multi-vector functions 149 by virtue of the weyl duality we have for a region ω which is contractible to a point, and for each r < m + 1, that hr(ω) = 0, in other words: if ω is contractible to a point, then each co-closed r-form in ω is also co-exact; this is dealt with in the so-called dual poincaré lemma (see section 5). a smooth r-form in ω which is at the same time closed and co-closed is called harmonic in ω (in the sense of hodge). introducing the operator d = d + d∗, a necessary and sufficient condition for a smooth r-form ωr in ω to be harmonic in ω thus reads: dωr = (d + d∗)ωr = 0 ⇐⇒ { dωr = 0 d∗ωr = 0 . (ii) the system (ii) is called the hodge-de rham system. note that if ωr is harmonic in an open region ω of rm+1 then automatically ωr satisfies ωr = 0 in ω, since d2 = (d + d∗)2 = d d∗ + d∗ d = − . the converse is however not true. the action of the operators d and d∗ on differential forms is two-fold in the sense that they act on the form itself as well as on the function coefficients by partial differentiation. in order to explicit this double action we introduce the following symbolic notations for the operators d and d∗. for d the following notation is rather obvious: d = m∑ j=0 (dxj∧) ∂xj . we then indeed have dωr = ⎛ ⎝∑ j (dxj∧)∂xj ⎞ ⎠ ⎛ ⎝ ∑ |a|=r ωra dx a ⎞ ⎠ = ∑ j ∑ a ∂xj ω r a dx j ∧dxa illustrating the above mentioned double action and the fact that d acts in an ”exterior” way. but this raises the question whether there exists a differential operator on forms acting in an ”inner” way, to which end an ”inner product” in the grassmann algebra should be defined. inspired by the inner product in the clifford algebra, we put by definition: dxi • dxj = − < dxi,dxj > = −δij, 0 ≤ i,j ≤ m. in fact this scalar product in the grassmann algebra already tacitly exists. indeed, as rm+1 is endowed with the standard euclidean metric, there is a canonical isomorphism between the tangent space txω ∼= rm+1 and its dual t ∗x ω ∼= (rm+1)∗, given 150 f. brackx, r. delanghe and f. sommen 7, 2(2005) by ej iso←→ < ej, • > = e∗j = dxj and hence < dxi,dxj > = < e∗i ,e ∗ j > = < ei,ej > = δij, 0 ≤ i,j ≤ m. so we introduce the operator m∑ j=0 (dxj • ) ∂xj clearly an operator with a double action. in a next step we put dxj • dxa = dxj • (dxi1 ∧ ·· · ∧ dxir ) = r∑ k=1 (−1)k δjik dxa\{ik}. we then get, by linearity, for a smooth r-form ωr: ⎛ ⎝ m∑ j=0 (dxj • )∂xj ⎞ ⎠ ⎛ ⎝ ∑ |a|=r ωradx a ⎞ ⎠ = r∑ k=1 ∑ |a|=r (−1)k (∂xik ω r a) dx a\{ik } in which we recognize the action of the co-derivative d∗ on ωr. consequently this co-derivative may be written as: d∗ = m∑ j=0 (dxj • )∂xj also nicely illustrating the double action of d∗. from this point of view the coderivative d∗ might as well have been called ”interior derivative”. finally for the operator d = d + d∗ we obtain the expressions d = d + d∗ = m∑ j=0 (dxj∧ )∂xj + m∑ j=0 (dxj • )∂xj = m∑ j=0 (dxj ∧ + dxj • )∂xj = m∑ j=0 (dxj∨ )∂xj where dxj∨ = dxj • + dxj∧ 7, 2(2005) differential forms and/or multi-vector functions 151 is the so-called ”vee-product”-operator, which was introduced in e.g. [7] and [12] in the more general context of a metric with (p,q)-signature on rm+1. in the sequel we will deal with the operators d and d∗ on the same footing and systematically mention the properties of d∗ next to those of d, for the sake of aesthetical symmetry. however, from the mathematical point of view this is superfluous; considering the operator d∗ only leads to new results when it appears in connection with the operator d. note in this context the interesting operators dd∗ and d∗d, which are the ”components” of the laplace operator (− ). 4 differential forms and multi-vector functions: an identification in becomes clear from sections 2 and 3 that the world of differential forms in an open region ω of rm+1 and the world of multi-vector functions in ω, may be identified in a natural way. if for each a ⊂ m, fa is a smooth real-valued function in ω, then the following correspondence table may already be drawn (see next page). this identification is now further developed. first one may wonder what the counterpart is of the hodge ∗ (star) operator. on the one hand one has ∗ ( dxj1 ∧ ·· · ∧ dxjr ) = σ dxjr+1 ∧ ·· · ∧ dxjm+1 where j1 < · · · < jr , jr+1 < · · · < jm+1 , {j1, · · · ,jr}∪{jr+1, · · · ,jm+1} = m = {0, 1, · · · ,m} and σ is the signature of the permutation (jr+1, · · · ,jm+1,j1, · · · ,jr). this corresponds, for a = {j1, · · · ,jr}⊂ m to ∗ea = (−1)r eme†a where em = e0 ∧ e1 ∧ ·· · ∧ em is the so-called pseudoscalar and † stands for the main anti-involution of the clifford algebra, also called reversion, given by e † a = (ej1 ∧ ·· · ∧ ejr ) † = ejr ∧ ·· · ∧ ej1 = (−1) r(r−1) 2 ea. next we identify some differential operators and establish similar formulae in both worlds. to start with, the euler operator e = m∑ j=0 xj ∂xj 152 f. brackx, r. delanghe and f. sommen 7, 2(2005) dxj ej dxi ∧dxj ei ∧ej dxi • dxj ei • ej ωr = ∑ |a|=r fa dx a fr = ∑ |a|=r fa ea d = m∑ j=0 (dxj∧)∂xj ∂+ = m∑ j=0 (ej∧)∂xj d∗ = m∑ j=0 (dxj •)∂xj ∂ − = m∑ j=0 (ej •)∂xj d = d + d∗ = m∑ j=0 (dxj∨)∂xj ∂ = ∂+ + ∂− = m∑ j=0 ej ∂xj ωr harmonic in ω ⊂ rm+1 fr monogenic in ω ⊂ rm+1 d2 = dd = 0 ∂+2 = ∂+∂+ = 0 d∗2 = d∗d∗ = 0 ∂−2 = ∂−∂− = 0 dd∗ ∂+∂− d∗d ∂−∂+ d2 = (d + d∗)2 = dd∗ + d∗d = − ∂2 = (∂+ + ∂−)2 = ∂+∂− + ∂−∂+ = − d̃ = d−d∗ ∂̃ = ∂+ −∂− d̃2 = (d−d∗)2 = −dd∗ −d∗d = ∂̃2 = −∂+∂− −∂−∂+ = dd̃ = −d̃d = d∗d−dd∗ = ̃ ∂∂̃ = −∂̃∂ = ∂−∂+ −∂+∂− = ̃ 7, 2(2005) differential forms and/or multi-vector functions 153 defined by eωr = m∑ j=0 xj ∂xj ω r = ∑ |a|=r dxa m∑ j=0 xj ∂xj ω r a and efr = m∑ j=0 xj ∂xj fr = ∑ |a|=r ea m∑ j=0 xj ∂xj fr,a is a scalar operator, measuring the degree of homogenicity of a function, and not affecting the order of a differential form or a multi-vector function. the euler operator thus has the same defining expression in both worlds. from the world of differential forms we now focus on the contraction operators ∂xj�, j = 0, 1, · · ·m, acting only on the basis elements of the differential form, but not on the function coefficients, and given by ∂xj�dxa = ∂xj� ( dxi1 ∧ ·· · ∧ dxir ) = r∑ k=1 (−1)k−1 δjik dxa\{ik } . apparently the contraction operator ∂xj� is, up to a minus sign, nothing else but the ”inner product”-operator (dxj • ) : ∂xj� = (−dxj • ) , j = 0, 1, · · · ,m. however bear in mind that contractions are more fundamental than dot products. indeed, they can be introduced independently of a scalar product, and their behaviour is invariant under diffeomorphisms, which is not the case for the dot product. for a first order operator v = m∑ j=0 vj (x) ∂xj , vj being a scalar-valued smooth function, mostly called a vector field, one may consider the associated contraction operator v� = m∑ j=0 vj (x) ∂xj� which also takes the form v� = m∑ j=0 vj (x)(−dxj • ). this inspires an associated ”inflation” operator v� = m∑ j=0 vj (x) ∂xj� = ∑ j=0 vj (x) ( −dxj∧ ) where the action of ∂xj� = (−dxj∧) is given by ∂xj� dxa = −dxj ∧ dxa. 154 f. brackx, r. delanghe and f. sommen 7, 2(2005) so from the euler operator e we deduce the operators e� = ∑ j xj ∂xj� = ∑ j xj (−dxj • ) and e� = ∑ j xj (−dxj ∧ ) which are in a sense complementary to the operators d and d∗ — think of replacing xj by dxj and dxj by xj . so the operators e� and e� must show properties similar to those of the operators d en d∗, which they indeed do, as shown in the next lemma. lemma 4.1 the operators e� and e� enjoy the following fundamental properties: (i) (e�)2 = 0 (ii) (e�)2 = 0 (iii) e� + e� = − m∑ j=0 xj (dxj ∨) (iv) (e� + e�)2 = e�e� + e�e� = −|x|2 the counterparts in the clifford setting of the operators (−dxj •) and (−dxj∧) clearly are (−ej •) and (−ej∧). the properties of the operators m∑ j=0 xj (−ej •) = (−x •) and m∑ j=0 xj (−ej∧) = (−x∧) corresponding to the ones in lemma 4.1, are then straightforward: (i) (−x • )(−x • ) = 0 (ii) (−x∧ )(−x∧ ) = 0 (iii) (−x • ) + (−x∧ ) = −x (clifford product understood) (iv) ((−x • ) + (−x∧ ))2 = (−x • )(−x∧ ) + (−x∧ )(−x • ) = −|x|2. note that the operators (ej •) and (ej∧), j=0,1, . . ., m, coincide with the socalled de witt basis of the algebra of endomorphisms on the clifford algebra r0,m+1. indeed, if ej and εj, j = 0, 1, . . . ,m denote the endomorphisms, given for an arbitrary clifford number a, by ej : a −→ eja εj : a −→ εja = ãej then the witt basis is formed by fj = 1 2 (ej −εj) , f′j = 1 2 (ej + εj ), j = 0, 1, . . . ,m 7, 2(2005) differential forms and/or multi-vector functions 155 and apparently fj = (ej •) and f′j = (ej∧). in the same order of ideas and starting from the operators d and d∗, we introduce the contraction and ”inflation” operators d� = m∑ j=0 (dxj∧) ∂xj� = m∑ j=0 (dxj∧)(−dxj •) d∗� = m∑ j=0 (dxj •) ∂xj� = m∑ j=0 (dxj •)(−dxj∧) the operators d� and d∗� have er(ω) as an eigenspace since d�ωr = r ωr and d∗�ωr = (m + 1 −r) ωr. in other words: they measure the order of a differential form. they are sometimes called fermionic euler operators. in the clifford analysis setting we get ∂+� = m∑ j=0 (ej∧)(−ej •) and ∂−� = m∑ j=0 (ej •)(−ej∧) for which indeed: ∂+� fr = r fr and ∂−� fr = (m + 1 −r) fr. now we turn our attention, still in the world of differential forms, to a so-called lie-derivative of differential forms. for a given scalar vector field v = ∑ j vj∂xj we define lvω = d v� ω + v�d ω. it is clear that the operators lv and d, as well as lv and v�, commute, since d lv = d v� d = lv d and v�lv = v� d v� = lv v�. this implies that closedness and exactness of differential forms are preserved under ”lie-derivation”. we now prove a fundamental formula about the lie-derivative of the euler operator. lemma 4.2 for any smooth differential form ω ∈ ∧(ω) one has le ω = (e�d + d e�) ω = (e + d�) ω. 156 f. brackx, r. delanghe and f. sommen 7, 2(2005) proof. first we have e� d ω = ∑ j xj (−dxj •) (∑ k (dxk∧)∂xkω ) = ∑ j ∑ k xj δjk ∂xk ω + ∑ j ∑ k xj dx k ∧ ∂xk dxj • ω = ∑ j xj ∂xj + ∑ j ∑ k xj dx k ∧ ∂xk (dxj • ω) while d e� ω = ∑ k (dxk ∧) ∂xk ∑ j xj (−dxj • ω) = ∑ j ∑ k dxk ∧ (−dxj • ω) δjk − ∑ j ∑ k dxk ∧ xj (dxj • ∂xk ω) = − ∑ j dxj ∧ (dxj • ω) − ∑ j ∑ k xj dx k ∧ ∂xk (dxj • ω). hence e� d ω = e ω + ∑ j ∑ k xj dx k ∧ ∂xk (dxj • ω) while d e� ω = d� ω − ∑ j ∑ k xj dx k ∧ ∂xk (dxj • ω) and the desired result follows. � by transposing the identity of lemma 4.1 into clifford analysis language we get corollary 4.3. for any smooth multi-vector function f ∈e(ω) one has ( (−x •) ∂+ + ∂+(−x •) ) f = (e + ∂+�) f. corollary 4.4. (i) for ωr ∈ ∧r(ω) one has le ωr = (e� d + d e�) ωr = (e + r) ωr. (ii) for fr ∈er(ω) one has ( (−x •) ∂+ + ∂+(−x •) ) fr = (e + r) fr. 7, 2(2005) differential forms and/or multi-vector functions 157 corollary 4.5. (i) if ωrk ∈ ∧r(ω) is homogeneous of degree k, then le ωrk = (e� d + d e�) ωrk = (k + r) ωr. (ii) if fr,k ∈er (ω) is homogeneous of degree k, then( (−x •) ∂+ + ∂+(−x •) ) fr,k = (k + r) fr,k. the similar fundamental identity involving the operators e� and d∗ is now proven in the language of multi-vector functions. corollary 4.6. for any smooth multi-vector function f ∈e(ω) one has ( (−x ∧) ∂− + ∂−(−x ∧) ) f = (e + ∂−�) f. proof. on the one hand we have x ∧ (∂−f) = ∑ j xj (ej ∧) ∑ k (ek •) ∂xkf = ∑ j xj (ej∧)(ej •) ∂xj f + ∑ j �=k xj (ej∧)(ek •) ∂xkf = − ∑ j xj ∂xj (j) f + ∑ j �=k xj (ej∧)(ek •) ∂xkf where (j) f denotes that part of f containing the basis vector ej . on the other hand we have ∂−(x∧f) = ∑ k (ek •) ∂xk ∑ j xj ej ∧f = ∑ j (ej •)(ej∧)f + ∑ j ∑ k xj (ek •)(ej∧) ∂xkf = −∂−� f + ∑ j xj (ej •)(ej∧) ∂xj f + ∑ j �=k xj (ek •)(ej∧) ∂xkf = −∂−� f − ∑ j xj ∂xj co(j) f + ∑ j �=k xj (ek •)(ej∧) ∂xkf 158 f. brackx, r. delanghe and f. sommen 7, 2(2005) where co(j) f denotes that part of f not containing the basis vector ej . adding both expressions yields the desired result. � corollary 4.7. for any smooth differential form ω ∈ ∧(ω) one has (e� d∗ + d∗ e�) ω = (e + d∗�) ω. corollary 4.8. (i) for ωr ∈ ∧r(ω) one has (e� d∗ + d∗e�) ωr = (e + m + 1 −r) ωr. (ii) for fr ∈er(ω) one has ( (−x ∧) ∂− + ∂− (−x∧) ) fr = (e + m + 1 −r) fr. corollary 4.9. (i) if ωrk ∈ ∧ (ω) is homogeneous of degree k, then (e� d∗ + d∗e�) ωrk = (k + m + 1 −r) ωrk. (ii) if fr,k ∈er (ω) is homogeneous of degree k, then ( (−x ∧) ∂− + ∂− (−x ∧) ) fr,k = (k + m + 1 −r) fr,k. the above considerations lead to the completion of our identification table set up at the beginning of this section. e = ∑ j xj ∂xj e = ∑ j xj ∂xj ∂xj� = −dxj • −ej • ∂xj� = −dxj • −ej ∧ e� = ∑ j xj (−dxj •) ∑ j xj (−ej •) = −x • e� = ∑ j xj (−dxj ∧) ∑ j xj (−ej ∧) = −x∧ 7, 2(2005) differential forms and/or multi-vector functions 159 e� + e� = ∑ j xj (dx j ∨) (−x •) + (−x∧) = −x clifford product understood d� = ∑ j (dxj ∧)(−dxj •) ∂+� = ∑ j (ej ∧)(−ej •) d∗� = ∑ j (dxj •)(−dxj ∧) ∂−� = ∑ j (ej •)(−ej ∧) le = d e� + e� d = e + d� ∂+ (−x •) + (−x •) ∂+ = e + ∂+� l∗e = d∗ e� + e� d∗ = e + d∗� ∂− (−x∧) + (−x∧) ∂− = e + ∂−� 5 the poincaré and the dual poincaré lemmata revisited in this section we formulate refinements of the classical poincaré lemma and its dual, both in the language of differential forms and in the one of multi-vector functions, exploiting the identification established in the previous section. as it appears to us that these refinements are rarely cited in the literature, we add their proofs. we start with a classical result, which in the language of three dimensional vector fields is usually called the helmholtz decomposition. proposition 5.1. for each r-form ωr ∈ ∧r(ω) (0 < r < m + 1) there exist ar+1 ∈ ∧r+1(ω) and br−1 ∈ ∧r−1(ω) such that (i) d ar+1 = 0 ; (ii) d∗ br−1 = 0 ; (iii) ωr = d∗ ar+1 + d br−1 . proposition 5.2. for each r-vector function fr ∈ er(ω) (0 < r < m + 1) there exist ar+1 ∈ er+1(ω) and br−1 ∈er−1(ω) such that (i) ∂+ ar+1 = 0 ; (ii) ∂− br−1 = 0 ; 160 f. brackx, r. delanghe and f. sommen 7, 2(2005) (iii) fr = ∂− ar+1 + ∂+ br−1 . proof. as the laplace operator : er(ω) −→ er(ω) is surjective (see e.g. [14], there ought to exist gr ∈er(ω) such that (− ) gr = fr or (∂− ∂+ + ∂+∂−) gr = fr . put ar+1 = ∂+ gr and br−1 = ∂− gr to obtain the desired result. � note that dωr = 0 iff the (r + 1)-form ar+1 in the above helmholtz decomposition is harmonic (in the sense of hodge), while d∗ωr = 0 iff br−1 is harmonic. similarly, we have that ∂+fr = 0 iff ar+1 is monogenic, while ∂−fr = 0 iff br−1 is monogenic. but there is more. the poincaré lemma and the dual poincaré lemma will assert that one of those harmonic forms ar+1 and br−1, respectively one of those monogenic multi-vector functions ar+1 and br−1, is absorbed in the other remaining term. lemma 5.3. (poincaré) let r ≥ 1 and let ω be an open region contractible to a point. then r ker d = d ( r−1 ker d∗ ) i.e. the following are equivalent: (i) dωr = 0 (ii) there exists ωr−1 ∈ ∧r−1(ω) such that d∗ωr−1 = 0 and ωr = dωr−1. lemma 5.4. (poincaré) let r ≥ 1 and let ω be an open region contractible to a point. then r ker ∂+ = ∂+ ( r−1 ker ∂− ) i.e. the following are equivalent: (i) ∂+fr = ∂ ∧ fr = 0 (ii) there exists fr−1 ∈er−1(ω) such that ∂−fr−1 = ∂ • fr−1 = 0 and fr = ∂+fr−1 = ∂ ∧fr−1 . proof. we prove lemma 5.3. (i) =⇒ (ii) from the classical poincaré lemma follows the existence of αr−1 ∈ ∧r−1(ω) such that ωr = dαr−1. as : ∧r−1(ω) −→ ∧r−1(ω) is surjective, there ought to exist βr−1 ∈ ∧r−1(ω) 7, 2(2005) differential forms and/or multi-vector functions 161 such that βr−1 = αr−1. put ωr−1 = αr−1 + dd∗βr−1. then clearly dωr−1 = dαr−1 = ωr. moreover d∗ωr−1 = d∗αr−1 + d∗dd∗βr−1 = d∗αr−1 + d∗ (dd∗ + d∗d) βr−1 = d∗ αr−1 −d∗ βr−1 = 0. (ii) =⇒ (i) trivial. � lemma 5.5. (dual poincaré lemma) let r < m + 1 and let ω be an open region contractible to a point. then r ker d∗ = d∗ ( r+1 ker d ) i.e. the following are equivalent: (i) d∗ωr = 0 (ii) there exists ωr+1 ∈ ∧r+1(ω) such that dωr+1 = 0 and ωr = d∗ωr+1. lemma 5.6. (dual poincaré lemma) let r < m + 1 and let ω be an open region contractible to a point. then r ker ∂− = ∂− ( r+1 ker ∂+ ) i.e. the following are equivalent: (i) ∂−fr = ∂ • fr = 0 (ii) there exists fr+1 ∈er+1(ω) such that ∂+fr+1 = 0 and fr = ∂− fr+1. proof. we prove lemma 5.6. (i) =⇒ (ii) for each fr ∈ er(ω), fr em = fr eoe1 . . .em = gm+1−r belongs to em+1−r (ω). as ∂ gm+1−r = (∂ fr ) em , we get: ∂− gm+1−r = ∂ • gm+1−r = [∂ gm+1−r]m−r = [∂ fr]r+1 em = (∂ + fr) em and also ∂+ gm+1−r = ∂ ∧ gm+1−r = [∂ gm+1−r]m+2−r = [∂ fr ]r−1 em = (∂− fr) em. 162 f. brackx, r. delanghe and f. sommen 7, 2(2005) hence fr will satisfy ∂− fr = 0 iff ∂+ gm+1−r = 0. lemma 5.4 then asserts the existence of gm−r ∈em−r (ω) such that ∂− gm−r = 0 and gm+1−r = ∂+ gm−r. as e2m = εm, εm = ± 1, we get, putting gm−r em εm = fr+1 : fr = gm+1−r em εm = (∂+ gm−r) em εm = [∂ gm−r]m+1−r em εm = [∂ fr+1]r = ∂− fr+1 while ∂+ fr+1 = ∂ − gm−r = 0. (ii) =⇒ (i) trivial. � corollary 5.7. if the open region ω is contractible to a point, then the differential operators: (i) ∂−∂+ : r ker ∂− −→ r ker ∂− (ii) ∂+∂− : r ker ∂+ −→ r ker ∂+ (iii) ∼ : er(ω) −→er(ω) are surjective. proof. (i) take fr ∈ r ker ∂−. by lemma 5.6 there exists fr+1 ∈er+1 ω such that ∂+ fr+1 = 0 and ∂− fr+1 = fr. so by lemma 5.4 there exists gr ∈er(ω) such that ∂−gr = 0 and ∂+ gr = fr+1. it follows that ∂−∂+ gr = ∂− fr+1 = fr with gr ∈ r ker ∂−. (ii) similar to the proof of (i). (iii) take fr ∈ er (ω). by proposition 5.2 there exist ar+1 ∈ er+1(ω) and br−1 ∈ er−1(ω) such that ∂+ar+1 = 0, ∂−br−1 = 0 and fr = ∂−ar+1 + ∂+br−1. by (i) and (ii) there exist gr ∈ r ker ∂− and hr ∈ r ker ∂+ such that ∂−∂+ gr = ∂−ar+1 ∈ r ker ∂− and ∂+∂− hr = −∂+br−1 ∈ r ker ∂+. hence ∂−∂+(gr + hr) = ∂−ar+1 and ∂+∂−(gr + hr) = −∂+br−1, and thus also ∼ (gr + hr) = (∂−∂+ − ∂+∂−)(gr + hr) = ∂−ar+1 + ∂+br−1 = fr. � now combining the poincaré lemma and the dual poincaré lemma, we obtain the following structure theorem on monogenic multi-vector functions and its counterpart on harmonic differential forms. 7, 2(2005) differential forms and/or multi-vector functions 163 theorem 5.8. if the open region ω is contractible to a point, then for each ωr ∈ ∧r(ω) (0 < r < m + 1) the following are equivalent: (i) ωr is harmonic in ω, i.e. dωr = (d + d∗) ωr = 0 in ω (ii) there exists ωr−1 ∈ ∧r−1(ω) such that d∗ ωr−1 = 0, ωr−1 = 0 and ωr = dωr−1 (ii’) there exists ωr−1 ∈ ∧r−1(ω) such that d∗ ωr−1 = 0, ∼ ωr−1 = 0 and ωr = dωr−1 (iii) there exists ωr+1 ∈ ∧r+1(ω) such that d ωr+1 = 0, ωr+1 = 0 and ωr = d∗ωr+1 (iii’) there exists ωr+1 ∈ ∧r+1(ω) such that d ωr+1 = 0, ∼ ωr+1 = 0 and ωr = d∗ωr+1. theorem 5.9. if the open region ω is contractible to a point, then for each fr ∈er(ω) (0 < r < m + 1) the following are equivalent: (i) fr is monogenic in ω, i.e. ∂ fr = (∂+ + ∂−) fr = 0 in ω (ii) there exists fr−1 ∈er−1(ω) such that ∂−fr−1 = 0, fr−1 = 0 and fr = ∂+fr−1 (ii’) there exists fr−1 ∈er−1(ω) such that ∂−fr−1 = 0, ∼ fr−1 = 0 and fr = ∂+fr−1 (iii) there exists fr+1 ∈er+1(ω) such that ∂+fr+1 = 0, fr+1 = 0 and fr = ∂−fr+1 (iii’) there exists fr+1 ∈er+1(ω) such that ∂+fr+1 = 0, ∼ fr+1 = 0 and fr = ∂−fr+1. proof. (ii) ⇒ (i) and (ii′) ⇒ (i) : trivial (iii) ⇒ (i) and (iii′) ⇒ (i) : trivial (i) ⇒ (ii) and (i) ⇒ (ii′) if fr is monogenic in ω then ∂+fr = 0 and ∂−fr = 0 in ω. by lemma 5.4 there exists fr−1 ∈er−1(ω) such that ∂− fr−1 = 0 and ∂+ fr−1 = fr. it follows that in ω: ∂ fr−1 = ∂ + fr+1 = fr 164 f. brackx, r. delanghe and f. sommen 7, 2(2005) and (− ) fr−1 = ∂ (∂ fr−1) = ∂ fr = 0. it also follows that in ω ∼ ∂ fr−1 = (∂ + −∂−) fr−1 = fr and ∼ fr−1 = ∂ ( ∼ ∂ fr−1) = ∂ fr = 0. (i) ⇒ (iii) and (i) ⇒ (iii′) by lemma 5.6 there exists fr+1 ∈er+1 (ω) such that ∂+ fr+1 = 0 and ∂− fr+1 = fr. it follows that in ω : ∂ fr+1 = ∂ − fr+1 = fr and (− ) fr+1 = ∂ (∂ fr+1) = ∂ fr = 0. it also follows that in ω ∼ ∂ fr+1 = (∂ + −∂−) fr+1 = −∂− fr+1 = −fr and ∼ fr+1 = ∂ ( ∼ ∂ fr+1) = −∂ fr = 0. � remarks 5.10. (i) the above theorems 5.8. and 5.9 may be rephrased as follows. if the open region ω is contractible to a point and 0 < r < m + 1, then r ker d = d ( r−1 ker d∗ ∩ r−1 ker (d∗d) ) = d ( r−1 ker ∩ r−1 ker d∗) = d ( r−1 ker ∼ ∩ r−1 ker d∗) r ker d = d∗ ( r+1 ker d ∩ r+1 ker (dd∗) ) = d∗ ( r+1 ker ∩ r+1 ker d) = d∗ ( r+1 ker ∼ ∩ r+1 ker d) r ker ∂ = ∂+ ( r−1 ker ∂− ∩ r−1 ker (∂−∂+) ) = ∂+ ( r−1 ker ∩ r−1 ker ∂−) = ∂+ ( r−1 ker ∼ ∩ r−1 ker ∂−) r ker ∂ = ∂− ( r+1 ker ∂+ ∩ r+1 ker (∂+∂−) ) = ∂− ( r+1 ker ∩ r+1 ker ∂+) = ∂− ( r+1 ker ∼ ∩ r+1 ker ∂+). 7, 2(2005) differential forms and/or multi-vector functions 165 (ii) for the equivalence (i) ⇐⇒ (ii) of theorem 5.8 we also refer to [4]. 6 from the euler operator to the poincaré lemma the proof of lemma 5.3. heavily relies on the classical poincaré lemma. in this section we reflect upon the proof of this classical poincaré lemma and we present an alternative proof, however restricted to real-analytic differential forms in an open ball. the essence of the proof of the poincaré lemma for one-forms is easily grasped. indeed, one-forms may be integrated along curves and the integral of a closed oneform from a fixed point to a variable endpoint, in a homologically trivial domain such as a ball, only depends on this endpoint; in other words: for closed one-forms there is a natural notion of primitive. for higher-order forms the integral operators in the proof of the poincaré lemma, are still one-dimensional. how is it possible that such a kind of method is still successful? the answer to this question, at least for the case of a ball, lies in considering the euler operator e (see also section 4). let p be the algebra of polynomials generated by {x0,x1, . . . ,xm} and let pk be the subspace of homogeneous polynomials of degree k, k ∈ n. then it is clear that p = +∞∑ k=0 ⊕ pk is the eigenspace decomposition of p associated with the euler operator e. next consider the algebra φ of polynomial differential forms, i.e. the free associative algebra generated by {x0,x1, . . . ,xm,dx0,dx1, . . . ,dxm}. if φrk denotes the subspace of r-forms with function coefficients in pk, then one has the decomposition φ = m+1∑ r=1 +∞∑ k=0 ⊕ φrk and the question arises with which operator this decomposition is associated. the answer to this question is given by corollary 4.5.(i): for each ϕrk ∈ φrk we indeed have le ϕrk = (e�d + de�) ϕrk = (k + r) ϕrk , showing that φrk is an eigenspace of the operator le , which, for r ≥ 1, has only positive eigenvalues. the injective linear operator le : φ −→ φ thus has a left inverse l−1e , given by l−1e ϕ = m+1∑ r=1 ∑ k l−1e (ϕrk) = m+1∑ r=1 ∑ k 1 k + r ϕrk , 166 f. brackx, r. delanghe and f. sommen 7, 2(2005) which is also a right inverse: l−1e le ϕ = le l−1e ϕ = ϕ, for all ϕ ∈ φ. moreover, as in the case for le , the operator l−1e commutes with the operators d and e� : l−1e d = dl−1e and l−1e e� = e� l−1e . for any polynomial differential form ϕ, not containing a scalar part, we thus have ϕ = l−1e e�d ϕ + l−1e d e�ϕ = l−1e e�d ϕ + d l−1e e�ϕ and, in particular, for any closed polynomial differential form ϕclosed we find ϕclosed = d (l−1e e� ϕclosed) = d (e�l−1e ϕclosed). this proves the poincaré lemma for closed polynomial differential forms in any open region of rm+1. finally, let ωr be a closed real-analytic r-form in a ball centred at the origin, say ◦ b(0,r). then the series ωr(x) = ∞∑ k=0 ωrk (x) , ω r k ∈ φrk together with all its derived series, converges uniformly on the compact subsets of ◦ b(0,r). as for each k, l−1e ωrk = 1 k + r ωrk and as the series ∞∑ k=0 1 k + r ωrk (x) together with all its derived series, also converges uniformly on the compact subsets of ◦ b(0,r), we may define l−1e ωr = ∞∑ k=0 l−1e ωrk. hence ωr = ∞∑ k=0 ωrk = ∞∑ k=0 d (e� l−1e ωrk) = d (e� ∞∑ k=0 l−1e ωrk) = d (e� l−1e ωr) which concludes the proof of the poincaré lemma for closed real-analytic r-forms in an open ball centred at the origin. 7, 2(2005) differential forms and/or multi-vector functions 167 remark 6.1. in a similar way the dual poincaré lemma for co-closed real-analytic differential forms in an open ball may be proved. the key steps in the proof are (i) corollary 4.8.(i) stating that for each ϕrk ∈ φrk : l∗e ϕrk = (e�d∗ + d∗e�) ϕrk = (k + m + 1 −r) erk ; (ii) the commutation rules: d∗l∗e = d∗e�d∗ = l∗e d∗ e�l∗e = e�d∗e� = l∗e e� ; (iii) the inversion formula for a polynomial differential form ϕ : ϕ = (l∗−1e e�d∗ + l∗−1e d∗e�) ϕ ; (iv) and in particular for a co-closed polynomial differential form ϕco−closed : ϕco−closed = l∗−1e d∗e�ϕco−closed = d∗ (l∗−1e e�) ϕco−closed. 7 differential forms versus multivector functions in the previous sections we established and illustrated a ”natural” isomorphism between on the one hand the cartan algebra of differential forms (extended with the hodge star operator and the inner product or dot product), with the underlying structure of the grassmann algebra, and on the other hand the algebra of multivector functions in clifford analysis with the underlying structure of clifford algebra. this could easily lead to the conclusion that either one of both is redundant. indeed it is true that the equations of clifford analysis may often be rewritten using vector calculus or more generally differential forms. this is nicely illustrated by the correspondence table of section 4 and in particular by the correspondence between the action of the dirac operator ∂ on multi-vector functions and the action of the operator d = d + d∗ on differential forms. historically this redundancy issue has led to a long and repeated discussion between those who advocate the use of differential forms and those who consider differential forms as an intermediate concept that can be fully replaced by clifford algebra. examples of papers where clifford algebra is realized by means of grassmann algebra are [7], [9] and [12]. a typical construct in these is the so-called ”vee-product” or clifford product of differential forms (see e.g. [2]). the dirac operator d on the cartan algebra ∧ (ω) may then be defined by dω = d ∨ω , ω ∈ ∧ (ω) . 168 f. brackx, r. delanghe and f. sommen 7, 2(2005) it turns out that d = d + d∗. on the other hand, in their book [6] hestenes and sobczyk recover most of the theory and calculus of differential forms by interpreting them as alternating tensors which may be represented by means of linear functions on the subspaces of r-vectors in a clifford algebra, an approach which was made more explicit in [5]. strictly speaking both points of view are mathematically correct. what we do not agree with is the conclusion that either the use of an extra clifford basis (e0,e1, . . . ,em) next to (dx0,dx1, . . . ,dxm) or the use of the differential forms dx0,dx1, . . . ,dxm as basic elements of calculus, is redundant. despite the similarities as depicted in this paper, the dxj and ej are different calculus objects with a different calculus behaviour, which will be fully demonstrated and illustrated in the forthcoming paper [13]. many examples illustrating the falsity of the ”redundancy idea” could be given, but the main counter-argument relies in the success and the richness of the results obtained by considering both the basic differential forms dxj and the clifford algebra generators ej as independent calculus elements. this is nicely demonstrated in e.g. [11] where chapter 9 focusses on the interplay between complex differential forms and complex clifford algebras and its usefulness for classical several complex variables theory is shown. received: june 2004. revised: august 2004. references [1] f. brackx, r. delanghe and f. sommen, clifford analysis, pitman advanced publishing program, boston london melbourne, 1982 [2] r. delanghe, f. sommen and v. souček, clifford algebra and spinor-valued functions, kluwer academic publishers, dordrecht boston london, 1992 [3] ch. doran and a. lasenby, geometric algebra for physicists, cambridge university press, cambridge, 2003 [4] j. e. gilbert, j. a. hogan and j. d. lakey, frame decompositions of form-valued hardy spaces, in: clifford algebras in analysis and related topics, j. ryan (ed.), crc press, boca raton, new york, 1996, 239-259 [5] d. hestenes, differential forms in geometric calculus, in: clifford algebras and their applications in mathematical physics, f. brackx, r. delanghe 7, 2(2005) differential forms and/or multi-vector functions 169 and h. serras (eds.), kluwer academic publishers, dordrecht boston london, 1993, 269-285 [6] d. hestenes and g. sobczyk, clifford algebra to geometric calculus, d. reidel publishing company, dordrecht, 1984 [7] e. kähler, der innere differentialkalkül, rend. mat.(roma), 21, 1962, 425-523 [8] k. maurin, analysis, part ii, d. reidel publishing company, dordrecht boston london, pwn-polish scientific publishers, warszawa, 1980 [9] z. oziewicz, from grassmann to clifford, in: clifford algebras and their applications in mathematical physics, j.s.r. chisholm and a.k. common (eds.), nato asi series, d. reidel publishing company, dordrecht boston lancaster tokyo, 1986, 245-255 [10] i. r. porteous, topological geometry, van nostrand reinhold company, london new york toronto melbourne, 1969 [11] r. rocha-chávez, m. shapiro and f. sommen, integral theorems for functions and differential forms in cm, research notes in mathematics 428, chapman & hall / crc, boca raton london new york washington, d.c., 2002 [12] n. a. salingaros and g. p. wene, the clifford algebra of differential forms, acta appl. math., 4, 1985, 271-292 [13] f. sommen, differential forms in clifford analysis, to appear [14] f. trèves, linear partial differential equations with constant coefficients, gordon and breach, new york, 1966 [15] c. von westenholz, differential forms in mathematical physics, stud. math. appl., vol 3, north-holland, amsterdam, 1978 cubo 8 , 25-29 (1992) recjbldo· odubre 1992. acerca de álgebras báricas satisfaciendo (x2 ) 2 = w(x )3x • a. catalán y r. costa abstract. let. (a,w) be a baric algebra, we define the e-ideal associated to the train polynomial p(x) = xn + -y1w(x)x"1 + .. . +"'yn-iw(x)"1x, by the ideal e,.,(.p) de a generated by ali p(a), a e a. different. train polynomials may give rise to the same e-ideal. two t rain polynomials p(x) and q(x) are equivalen t when ea(p) = e,.,(q). we prove tbat for baric algebras satisfying (x2) 2 = w(x) 3x there are 3 equivalence das.ses of train polynomials. 1 introducción sean f un cuerpo de característica cero, a una álgebra conmutativa y no necesariamente asociativa sobre f. si w : a_. fes un bomorfismo no nulo, entonces el par (a,w) se llama una álgebra bárica. (conmutativa) sobre f,w es su función peso, y para cada x € a,w(x) es su peso. los elementos de a de peso cero forman un ideaj n de codimensión 1. el concepto de álgebra bá.rica fue iotroducido por l.m.h. etherington [2j. en lo que sigue describimos brevemente dos ejem plos de álgebras bá.ricas: l.sean "yi, .•. , "yn1 elementos de f tal que 1 + 'yi + ... + 'yn-1 = o. la expresión forma l (1) es llamada un tren polinomio con coeficientes -y1, .•. ,'yn-i de grado n. 'fi n&.dcill.do por p royecto dlufro 9204 25 26 cubos a. catalán y r. costa si los elementos de una álgebra bárica (a, w) satisfacen la identidad p(a) =a" + 'y1w(a)a"-1 + ... + 'yn-1w(a)"1a = o (2) se die.e que a es una tren álgebra. el elemento ale está definido por a 1 = a y al = a11a para k :?: 2. si p(x} tiene grado mínimo n de entre todos los t ren polinomios que se anulan en a, entonces p(x) = o se llama tren ecuación de a y n es el rango de a. par.a estas álgebras n es un nilideal. 11.recientemente, walcher [3] obtuvo resultados acerca de álgebras báricas que satisfacen la igualdad {a2) 2 ~ w(a)3 a va e a (3) ellas admilen la existencia de idempotentes y para uno de ellos fijo, a saber e, tienen una descomposición de pierce a= fe$ nt eb n_t, donde n t = {a e n/ea =~a} y n_t ={a en/ea= -~a} . no es dificil probar que las dimensiones de estos subespacios son independientes de e, así podemos definir el invariante tipo como el par (1 + dimn1¡2,dimn_ 1fl )· más aún nl;2 ~ n-1¡2, n1fln-1fl ~ n1¡2 y !f'!. 112 ~ n-1¡2 (4) si u e n 1¡2 y v e n-1¡2, valen las siguientes identidades: i) u' o u) (u')' ii) 2u(uv) u2v vi) u2(uv) iii) 2(uv)v uv2 vii) 2(uv)2 + u2v 2 iv) v' viii) (uv)v' (5) ix) (v')' para obtener esta.s identidades ba.sta analizar la expresión polinomial nula en las variables j.,µ e f, ((w(x)e + j.u+ µv) 2)2 = w(x) 3 (w(x)e+ ).u+µv) que resultan de {3), haciendo a= w(x)e + j.u +µv. sea ahora x = ae+u+v un elemento de a, con a= w(x) . entonces para k ~ l donde a.ti ... , ík son números racionales. monomios en u y v de grado 4 no aparee.en debido a las relaciones (5). como xk+i = xkx, obtenenos el sistema de ecuaciones de diferencias con valor inicial a.1 = 1, b1 = c1 = d 1 = e1 = ji = o 2bk+i + bk 2 = 0 ¡ 2»+1 + ª' + l = o 2ck+i ck 2ak 2 = o 2dk+i + dk 2ak = 0 2ek+l +ek-2bk cj: =o 2/j:+i íj, ck 2dk = 0 es fáci l probar por inducción que { ' acerca de ... cubos 27 deonlclón 1.1 el ideal generalizado de etherington (o .simplernente e-ideal} de una álgebra bórica (a,w), baociado al tre n polinomio p(:z:} , e.sel ideal de a generado por lo.s elementoa p(a) = a"+'l'iw(a)a"1 + ... +'l'n-1w(a)"1a va e a . este ideal lo denotaremo.s por e..t( l ,")'¡, ... ,")',._ ¡)oe..t(p). ethe ri ngton en 12] define el ideal generado por jos elementos a2 w(a)a , que según nuestra notación es el eideal e..t( l , -1 ). observamos que e..t(p) ~ n,a/e..t(p) satisface p(x) =o y e..t(p) es el menor ideal j ~ n tal que a / / satisface p(x) =o. más aún, (a,w) es una tren álge bra c uand o a lguno de sus e-ideales es cero. proposición 1.2 paro toda álgeb ra bárica (a,w) y todo tren polinomio p(x) se ti ene e,,.(p) i;_ e,,.(i, 1}. demostración : sea e,,.(n, k) el e-ideal asociado a l tren po li nomio x"-w(x)".\:xk, 1 :s k :s n k. para t.odo tren polinomio (1) tenemos que va e a. p(a) =a"+ ")'1w(a)a"1 + ... + 'l'n iw(a)"1a =a" (1 + ")'2 + ... + 'l'ndw(a)a"1 + ... + 'l'n-1w(a)" 1a =(a" -w (a)a"1)-'ylw(a)(a"1 w(a)a"-')... ~.-1w(a)(a"1 -w(a)"-1a). es decir p(a) e e..t(n, n 1.) + e,,.(n 1, n 2) + ... + e..t (n 1 , 1) 1 enton ces ea(p) <;; e, (n , n -1) + ¿;;: ¡ ea(nl ,k) analogament.e para k = 1,2 1 ... ,n 2, e..t(n l,k) s:: e..t(n 1,n 2) + 2:~.:~e..t(n-2 , r). aplicando ésto reiteradament.e, obtenemos e,,.(p) s:: 2:~ e..t(k , k 1). pero cada uno de los ideales e..t(k,k 1} está cont.enido en e..t (l , -1 ) debido a que a' w(a)a•1 = ( ... (a' -w(a)a) ... )a e ea(l, -1 ). • en general , tren polinomios diferentes pueden generar el mismo e-ideal, as í, introducimos la siguient.e definición: d e finición 1.3 sea fl una clase de álgebras bárica.s. los tren polinomio s p(x) y q(x ) son equivalentes m6dulo n si e,(p) = e,(q)va en. cuando n ={a} las clases de equivalencias co rres ponden a los e-ideales de a. 2 e-ideales para á lgebras báricas satisfaciendo (x2 ) 2 = w(x) 3 x. denotemos a , .. , f las sucesiones (a .1:).1: e1n , ... , (f.1:h.e1 n, d e (7) tenemos a+ c = l , a +28 = l ,2d +2e+ a= 1, e= f (7') dondel = (1, 1, ... , 1). parak 2:, 1 seaa.1: =(a.1: , at1, ... , a 1 ) e f.\: y simila rm ente 8.1:, c.1:, d.1: , e.1: , f.1: 1 1.1: las relaciones (7' ) va len también para estos vecto res de f.1:. un t. ren polinomio p(x ) = x"+")'1w(x)x" 1 + ... + 'l'n-1w(:t)"1x puede ser iden· tiñc ado co n el vecto r p = (1,,.1 , ... ")'n1' e f". así el conjunto de todos los tren polinomios d e grado n es identificad o co n la variedad lineal de f'" defi nid a por las ecuaciones :r:1 + ... +:tn = o y x 1 =l. sea <, > la fo rma bilineal usual en f" , c uand o reemplazamos cada potencia x .\: d e x dada po r (6), ob t.e nemoo 28 cubos a. cata/in y r. c..la p(x) =< p,ln > a"e+ a"1(< p, l n >u+< a.,p> v) + a"2(< bn1p > u 2+ < cn,p > tw+ < d,.,p > v2 ) + a"-3(< en,p > u2v+ < f,. , p > uv2). como= l+"'y1+ ..• + "yn1 = o, los dos primeros sumandos desaparecen. por (7'), podemos expresar an, cn, d,. y f,. como combinaciones lineales de 8,., e.. y 1,.. entonces p(x) = q'n-j < 1,. 2bn 1p >v +a"2 ( < b,.,p > u2+ < 28,.,p > uv+ < bn en. p > v'l) +crn-3 < en,p > (u2v + uv2) -2an-l < bn,p >v +a"2 (< bn,p > u 2 + 2 < bn1p > ut1+ < bn en,p > ti2 ) +a"3 < en 1 p > (u2v + uv2 ) (8) ahora est.amos en condiciones de probar que sólo existen tres clases de equiva· lencia.s de tren polinomios, descritos geometricamente por las siguient.es propoai· ciones. proposició n 2 .1 si p y 8,. no son ortogonales, entonces ea(p) = n1¡2n-1ri 11+ < bn en,p > v 2 y p(e v} = 2 < bn,p > v+ < bn en , p > v 2, mi p(ev) p(e +v) = 4 < b",p >y v = l < b",p >-1 (p(e-v)-p(e+v)) e e,(p) . 1 pro posición 2.2 si p es ortogonal a bn y p no ortogonal a en, entoncu e,(p) = n1ri~1 /'l e n:,1,. d e mostración: como < bn,p > =o, la igualdad (8) se reduce a p{:z:) = -crn-2 v2 +an-3 (u2 v +u112 ) = a"3 < enip > (u211 av2 + u112 ) e ni12n"_ 1¡2 $ n'!.1¡ 2 (9) usando (5) es fácil probar que éste espacio es un ideal. esto implica que ea(p) ~ n1/'l1v'!._ 1/2 $ n~ 1 12 • para la otra inclusión, es suficiente probar que n:.. 112 ~ ea(p) cuando p y en no son ortogonales . en efecto, s upongamos inicialmente que 112 e n"_ 1fl. entonces p(e +v) = < en,p > v 2, así 112 =< e.n,p >-1 p(e + v ) e ea(p). para un generador aditivo v 1112 de n'!.. 112 , es suficiente recorda r que 211 1v-i = (111 + v,)2 v1 ,;¡ e e,(p). 1 proposición 2 . 3 si p e8 ortogonal a bn y en, entonces ea(p) =o demos tración : es suficiente ver de (9) que p(:z:) =o cuando p y e,. son ortogonales. • observación: los e-ideales para álgebras que satisfacen (x 2) 2 == w(:z:)3x están det.ermfoados por :t2 -w(x)x, x 3 4w(x)x 2 4w(x)2x y x 4 ~w(:z:)2x2 tw(:z:)3x respectiwmente y cuyos grados son los núnimos. en particula r todas .las álgebras que satisfacen (:z:2)2 = w(:z:)3:z: son tren álgebras d e rango 4 , como fue observado po' wakhec [3, ec( l l )]. ( cubo 8 29 referencias [l] costa r.: principal tr:ain algebras ofriank 3 and dimeosión:::; 5. proc. edimb. mabh. soc. 33, 61-io ( 1 990~ . [2] etiherington l. m. h.: genetic algebras. proc. r.oy. soc. edim 59, 242-258 (1939~ . [3] walcher s.: algebras wich satisf.y a train equat.ion for the firts three plenary powers. arch. matb. 561 547-sm (1991). [4] wor.z a.: algebras in genetics, lectiure notes in biomathematics, vol. 36, springer-verlag1 1980. dirección de los auteres: abdón catalán depar:tamento de matemática y estadística universidad de la frontera. casilla 54-d. temuco. roberto cos l'la instituto de matemática e estatística u ni versidade de 88.o paulo caixa postal 20570. cep 01498 sio paulo. brasil. ( '\ revista de matemáticas_0037 revista de matemáticas_0038 revista de matemáticas_0039 revista de matemáticas_0040 revista de matemáticas_0041 revista de matemáticas_0042 cubo a mathematical journal vol.10, n o ¯ 01, (117–142). march 2008 converse fractional opial inequalities for several functions george a. anastassiou department of mathematical sciences university of memphis, memphis, tn 38152 u.s.a. e-mail address: ganastss@memphis.edu tel: 901-678-3144 – fax: 901-678-2480 abstract a variety of very general lp(0 < p < 1) form converse opial type inequalities ([8]) is presented involving generalized fractional derivatives ([3],[6]) of several functions in different orders and powers. from the established results derive other particular results of special interest. resumen una variedad general de desigualdades inversas de tipo opial en lp(0 < p < 1) son presentadas, las cuales envuelven deridadas fraccionarias generalidades ([3],[6]) de varias funciones con diferentes ordenes y potencias. deducimos algunos casos particulares de especial interes. key words and phrases: opial type inequality, fractional derivative. math subj. class.: 26a33, 26d10, 26d15. 118 george a. anastassiou cubo 10, 1 (2008) 1 introduction opial inequalities appeared for the first time in [8] and then many authors dealt with them in different directions and for various cases. for a complete recent account on the activity of this field see [1], and still it remains a very active area of research. one of their main attractions to these inequalities is their applications, especially to proving uniqueness and upper bounds of solution of initial value problems in differential equations. the author was the first to present opial inequalities involving fractional derivatives of functions in [2], [3] with applications to fractional differential equations. fractional derivatives come up naturally in a number of fields, especially in physics, see the recent books [7], [9]. here the author continues his study of fractional opial inequalities now involving several different functions and produces a wide variety of converse results. to give an idea to the reader of the kind of inequalities we are dealing with, briefly we mention a specific one (see corollary 15). ∫ x x0   m ∑ j=1 ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv   dw ≥ c (x)   ∫ x x0   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw   ( λα+λv p ) , (∗) all x0 ≤ x ≤ b. in (∗) , c (x) is a constant that depends on x0, x, and the involved parameters, γ1 ≥ 0, 1 ≤ v − γ1 < 1 p , 0 p. here f (i) j (x0) = 0, i = 0, 1, . . . , n − 1, n := [v] (integral part); j = 1, . . . , m. and d γ1 x0 fj , d v x0 fj are the generalized (of canavati) type [6], [2] fractional derivatives of fj of orders γ1, v respectively. 2 preliminaries in the sequel we follow [6]. let g ∈ c ([0, 1]) . let v be a positive number, n := [v] and α := v − n (0 < α < 1) . define (jvg) (x) := 1 γ (v) ∫ x 0 (x − t) v−1 g (t) dt, 0 ≤ x ≤ 1, (1) cubo 10, 1 (2008) converse fractional opial inequalities ... 119 the riemann-liouville integral, where γ is the gamma function. we define the subspace c v ([0, 1]) of c n ([0, 1]) as follows: c v ([0, 1]) := { g ∈ cn ([0, 1]) : j1−αd n g ∈ c1 ([0, 1]) } , where d := d dx . so for g ∈ cv ([0, 1]) , we define the v−fractional derivative of g as d v g := dj1−αd n g. (2) when v ≥ 1 we have the taylor’s formula g (t) = g (0) + g ′ (0) t + g” (0) t 2 2! + . . . + g (n−1) (0) t n−1 (n − 1)! + (jvd v g) (t) , for all t ∈ [0, 1] . (3) when 0 < v < 1 we find g (t) = (jvd v g) (t) , for all t ∈ [0, 1] . (4) next we carry above notions over to arbitrary [a, b] ⊆ r (see[3]). let x, x0 ∈ [a, b] such that x > x0, where x0 < b is fixed. let f ∈ c ([a, b]) and define (j x0 v f ) (x) := 1 γ (v) ∫ x x0 (x − t) v−1 f (t) dt, x0 ≤ x ≤ b, (5) the generalized riemann-liouville integral. we define the subspace cvx0 ([a, b]) of c n ([a, b]) : c v x0 ([a, b]) := { f ∈ cn ([a, b]) : j x01−αd n f ∈ c1 ([x0, b]) } , clearly c 0 x0 ([a, b]) = c ([a, b]) , also c n x0 ([a, b]) = c n ([a, b]) , n ∈ n. for f ∈ cvx0 ([a, b]) , we define the generalized v−fractional derivative of f over [x0, b] as d v x0 f := dj x0 1−αf (n) ( f (n) := d n f ) . (6) notice that ( j x0 1−αf (n) ) (x) = 1 γ (1 − α) ∫ x x0 (x − t) −α f (n) (t) dt exists for f ∈ cvx0 ([a, b]) . we recall the following generalization of taylor’s formula (see [6], [3]). 120 george a. anastassiou cubo 10, 1 (2008) theorem 1. let f ∈ cvx0 ([a, b]) , x0 ∈ [a, b] fixed. (i) if v ≥ 1 then f (x) = f (x0) + f ′ (x0) (x − x0) + f ” (x0) (x − x0) 2 2 + . . . + f (n−1) (x0) (x − x0) n−1 (n − 1)! + ( j x0 v d v x0 f ) (x) , for all x ∈ [a, b] : x ≥ x0. (7) (ii) if 0 < v < 1 then f (x) = ( j x0 v d v x0 f ) (x) , for all x ∈ [a, b] : x ≥ x0. (8) we make remark 2. 1) ( d n x0 f ) = f (n) , n ∈ n. 2) let f ∈ cvx0 ([a, b]) , v ≥ 1 and f (i) (x0) = 0, i = 0, 1, . . . , n − 1, n := [v] . then by (7) f (x) = ( j x0 v d v x0 f ) (x) . i.e. f (x) = 1 γ (v) ∫ x x0 (x − t) v−1 ( d v x0 f ) (t) dt, (9) for all x ∈ [a, b] with x ≥ x0. notice that (9) is true, also when 0 < v < 1. we need from [3] lemma 3. let f ∈ c ([a, b]) , µ, v > 0. then j x0 µ (j x0 v f ) = j x0 µ+v (f ) . (10) we also make remark 4. let v ≥ γ + 1, γ ≥ 0, so that γ < v. call n := [v] , α := v − n; m := [γ] , ρ := γ−m. note that v−m ≥ 1 and n−m ≥ 1. let f ∈ cvx0 ([a, b]) be such that f (i) (x0) = 0, i = 0, 1, . . . , n − 1. hence by (7) f (x) = ( j x0 v d v x0 f ) (x) , for all x ∈ [a, b] : x ≥ x0. therefore by leibnitz’s formula and γ (p + 1) = pγ (p) , p > 0, we get that f (m) (x) = ( j x0 v−md v x0 f ) (x) , for all x ≥ x0. (11) cubo 10, 1 (2008) converse fractional opial inequalities ... 121 it follows that f ∈ cγx0 ([a, b]) and thus ( d γ x0 f ) (x) := ( dj x0 1−ρf (m) ) (x) exists for all x ≥ x0. (12) really by the use of (11) we have on [x0, b] j x0 1−ρ ( f (m) ) = j x0 1−ρ ( j x0 v−md v x0 f ) = ( j x0 1−ρ ◦ j x0 v−m )( d v x0 f ) = j x0 v−m+1−ρ ( d v x0 f ) = j x0 v−γ+1 ( d v x0 f ) , by (10). that is, ( j x0 1−ρf (m) ) (x) = 1 γ (v − γ + 1) ∫ x x0 (x − t) v−γ ( d v x0 f ) (t) dt. therefore ( d γ x0 f ) (x) = d (( j x0 1−ρf (m) ) (x) ) = 1 γ (v − γ) · ∫ x x0 (x − t) (v−γ)−1 ( d v x0 f ) (t) dt; (13) hence ( d γ x0 f ) (x) = ( j x0 v−γ ( d v x0 f )) (x) and is continuous in x on [x0, b] . in particular when v ≥ 2 we have ( d v−1 x0 f ) (x) = ∫ x x0 ( d v x0 f ) (t) dt, x ≥ x0. (14) that is ( d v−1 x0 f )′ = d v x0 f, ( d v−1 x0 f ) (x0) = 0. 3 main results 3.1 results involving two functions we present our first main result theorem 5. let γj ≥ 0, 1 ≤ v − γj < 1/p, 0 < p < 1, j = 1, 2, and f1, f2 ∈ c v x0 ([a, b]) with f (i) 1 (x0) = f (i) 2 (x0) = 0, i = 0, 1, . . . , n − 1, n := [v] . here x, x0 ∈ [a, b] : x ≥ x0. we assume here that d v x0 fj is of fixed sign on [x0, b] , j = 1, 2. consider also p (t) > 0 and q (t) > 0 continuous functions on [x0, b] . let λv > 0 and λα, λβ ≥ 0 such that λv > p. 122 george a. anastassiou cubo 10, 1 (2008) set pk (w) := ∫ w x0 (w − t) (v−γk−1)p p−1 (p (t)) − 1 p−1 dt, k = 1, 2, x0 ≤ x ≤ b; (15) a (w) := q (w) · (p1 (w)) λα( p−1 p ) · (p2 (w)) λβ( p−1 p ) (p (w)) − λv p (γ (v − γ1)) λα (γ (v − γ2)) λβ , (16) a0 (x) := ( ∫ x x0 (a (w)) p p−λv dw ) p−λv p , (17) and δ1 := 2 1−( λα+λv p ) (18) if λβ = 0, we obtain that, ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα · ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα · ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ ( a0 (x) |λβ =0 ) · ( λv λα + λv ) λv p · δ1 · [ ∫ x x0 p (w) [ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p] dw ]( λα+λv p ) . (19) proof. from (13) and assumption we have ∣ ∣ ( d γk x0 fj ) (w) ∣ ∣ = 1 γ (v − γk) ∫ w x0 (w − t) v−γk−1 ∣ ∣ ( d v x0 fj ) (t) ∣ ∣dt, for k = 1, 2, j = 1, 2 and for all x0 ≤ w ≤ b. next applying hölder’s inequality with indices p, p p−1 we get ∣ ∣ ( d γk x0 fj ) (w) ∣ ∣ = 1 γ (v − γk) ∫ w x0 (w − t) v−γk−1 (p (t)) − 1 p (p (t)) 1 p ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ dt ≥ 1 γ (v − γk) ( ∫ w x0 ( (w − t) v−γk−1 (p (t)) − 1 p ) p p−1 dt ) p−1 p ( ∫ w x0 p (t) ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ p dt ) 1 p cubo 10, 1 (2008) converse fractional opial inequalities ... 123 = 1 γ (v − γk) (pk (w)) p−1 p ( ∫ w x0 p (t) ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ p dt ) 1 p . i.e., it holds ∣ ∣ ( d γk x0 fj ) (w) ∣ ∣ ≥ 1 γ (v − γk) (pk (w)) p−1 p ( ∫ w x0 p (t) ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ p dt ) 1 p . (20) put zj (w) := ∫ w x0 p (t) ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ p dt, thus, z ′ j (w) = p (w) ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p , zj (x0) = 0; j = 1, 2. hence, we have ∣ ∣ ( d γk x0 fj ) (w) ∣ ∣ ≥ 1 γ (v − γk) (pk (w)) p−1 p (zj (w)) 1 p , and ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv = p (w) − λv p ( z ′ j (w) ) λv p , j = 1, 2. therefore we obtain q (w) ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv ≥ q (w) 1 (γ (v − γ1)) λα (p1 (w)) λα( p−1 p ) (z1 (w)) λα p 1 (γ (v − γ2)) λβ (p2 (w)) λβ ( p−1 p ) (z2 (w)) λβ p (p (w)) − λv p ( z ′ 1 (w) ) λv p = a (w) (z1 (w)) λα p (z2 (w)) λβ p ( z ′ 1 (w) ) λv p . consequently, by another hölder’s inequality application, we find (by p λv < 1) ∫ x x0 q (w) ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv dw ≥ a0 (x) [ ∫ x x0 (z1 (w)) λα λv (z2 (w)) λβ λv z ′ 1 (w) dw ] λv p . (21) similary one finds ∫ x x0 q (w) ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv dw 124 george a. anastassiou cubo 10, 1 (2008) ≥ a0 (x) [ ∫ x x0 (z1 (w)) λβ λv (z2 (w)) λα λv z ′ 2 (w) dw ] λv p . (22) taking λβ = 0 and adding (21) and (22) we obtain ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα · ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα · ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ ( a0 (x) |λβ =0 ) { [ ∫ x x0 (z1 (w)) λα λv z ′ 1 (w) dw ] λv p + [ ∫ x x0 (z2 (w)) λα λv z ′ 2 (w) dw ] λv p } = ( a0 (x) |λβ =0 ) { (z1 (x)) (λα+λv ) p + (z2 (x)) (λα+λv ) p } ( λv λα + λv ) λv p = ( a0 (x) |λβ =0 ) ( λv λα + λv ) λv p      ( ∫ x x0 p (t) ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ p dt ) (λα+λv ) p + ( ∫ x x0 p (t) ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ p dt ) (λα+λv ) p      =: (∗) . in this article we are using frequently the basic inequalities 2 r−1 (a r + b r ) ≤ (a + b) r ≤ ar + br, a, b ≥ 0, 0 ≤ r ≤ 1, (23) a r + b r ≤ (a + b) r ≤ 2r−1 (ar + br) , a, b ≥ 0, r ≥ 1. (24) finally using (23) , (24) and (18) we get (∗) ≥ ( a0 (x) |λβ =0 ) · ( λv λα + λv ) λv p · δ1 cubo 10, 1 (2008) converse fractional opial inequalities ... 125 { ∫ x x0 p (t) [ ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ p] dt } (λα+λv ) p . inequality (19) has been established. here we see that ( p p−1 ) (v − γi − 1) + 1 > 0, − 1 (p−1) > 0 and p (t) ∈ c ([x0, b]) , thus (see (15)) pi (w) ∈ r for every w ∈ [x0, b] , also pi (w) is continuous and bounded on [x0, b] for i = 1, 2. by λv > p > 0, we have 0 0 on (x0, b] , 1 a(x0) = 0. therefore 0 < a0 (x) < ∞, and all we have done in this proof are valid. � it follows the counterpart of the last theorem. theorem 6. all here as in theorem 5. further assume λβ ≥ λv. denote δ2 := 2 1−(λβ /λv ), δ3 := (δ2 − 1) 2 −(λβ /λv ). (25) if λα = 0, then it holds ∫ x x0 q (w) [ ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λβ · ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ · ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ (a0 (x) |λα=0) 2 p−λv p ( λv λβ + λv ) λv p δ λv p 3 · ( ∫ x x0 p (w) [ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p] dw ) ( λv +λβ p ) , all x0 ≤ x ≤ b. (26) proof. when λα = 0 from (21) and (22) we obtain ∫ x x0 q (w) ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv dw 126 george a. anastassiou cubo 10, 1 (2008) ≥ (a0 (x) |λα=0) [ ∫ x x0 (z2 (w)) λβ λv z ′ 1 (w) dw ] λv p , (27) and ∫ x x0 q (w) ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv dw ≥ (a0 (x) |λα=0) [ ∫ x x0 (z1 (w)) λβ λv z ′ 2 (w) dw ] λv p , (28) all x0 ≤ x ≤ b. adding (27) and (28) we get ∫ x x0 q (w) [ ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λβ · ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ · ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ (a0 (x) |λα=0) { [ ∫ x x0 (z2 (w)) λβ λv z ′ 1 (w) dw ] λv p + [ ∫ x x0 (z1 (w)) λβ λv z ′ 2 (w) dw ] λv p } ≥ (a0 (x) |λα=0) · 2 p−λv p · (m (x)) λv p =: (∗) , (29) by λv p > 1 and (24) , where m (x) := ∫ x x0 (z2 (w)) λβ λv z ′ 1 (w) + (z1 (w)) λβ λv z ′ 2 (w) dw. (30) next we work on m (x) . we have that m (x) = ∫ x x0 ( (z1 (w)) λβ λv + (z2 (w)) λβ λv ) ( z ′ 1 (w) + z ′ 2 (w) ) dw − ∫ x x0 [ (z1 (w)) λβ λv z ′ 1 (w) + (z2 (w)) λβ λv z ′ 2 (w) ] dw (by (24)) ≥ δ2 ∫ x x0 (z1 (w) + z2 (w)) λβ λv (z1 (w) + z2 (w)) ′ dw − ( λv λβ + λv )[ (z1 (x)) ( λv +λβ λv ) + (z2 (x)) ( λv +λβ λv ) ] = δ2 (z1 (x) + z2 (x)) ( λv +λβ λv ) ( λv λv + λβ ) − ( λv λβ + λv ) cubo 10, 1 (2008) converse fractional opial inequalities ... 127 [ (z1 (x)) ( λv +λβ λv ) + (z2 (x)) ( λv +λβ λv ) ] = ( λv λβ + λv )[ δ2 (z1 (x) + z2 (x)) ( λv +λβ λv ) − ( (z1 (x)) ( λv +λβ λv ) + (z2 (x)) ( λv +λβ λv ) )] (24) ≥ ( λv λβ + λv )[ δ2 ( (z1 (x)) ( λv +λβ λv ) + (z2 (x)) ( λv +λβ λv ) ) − ( (z1 (x)) ( λv +λβ λv ) + (z2 (x)) ( λv +λβ λv ) )] = ( λv λβ + λv ) (δ2 − 1) [ (z1 (x)) ( λv +λβ λv ) + (z2 (x)) ( λv +λβ λv ) ] (24) ≥ ( λv λβ + λv ) δ3 (z1 (x) + z2 (x)) ( λv +λβ λv ) . i.e. we present that m (x) ≥ ( λv λβ + λv ) δ3 (z1 (x) + z2 (x)) ( λv +λβ λv ) . (31) consequently, by (29) and (31) we get (∗) ≥ (a0 (x) |λα=0) 2 p−λv p ( λv λβ + λv ) λv p δ λv p 3 (z1 (x) + z2 (x)) ( λv +λβ p ) = (a0 (x) |λα=0) 2 p−λv p ( λv λβ + λv ) λv p δ λv p 3 ( ∫ x x0 p (t) [ ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ p] dt ) ( λv +λβ p ) . we have established (26) . � a special important case follows. theorem 7. let v ≥ 2 and γ1 ≥ 0 such that 2 ≤ v − γ1 < 1 p , 0 < p < 1. let f1, f2 ∈ c v x0 ([a, b]) with f (i) 1 (x0) = f (i) 2 (x0) = 0, i = 0, 1, . . . , n − 1, n := [v] . here x, x0 ∈ [a, b] : x ≥ x0. we assume here that d v x0 fj is of fixed sign on [x0, b] , j = 1, 2. consider also p (t) > 0 and q (t) > 0 continuous functions on [x0, b] . let λα ≥ λα+1 > 1. 128 george a. anastassiou cubo 10, 1 (2008) denote θ3 := ( 2 1−(λα /λα+1) − 1 ) 2 −λα/λα+1 , (32) l (x) := ( 2 ∫ x x0 (q (w)) ( 1 1−λα+1 ) dw )(1−λα+1) ( θ3λα+1 λα + λα+1 )λα+1 , (33) and p1 (x) := ∫ x x0 (x − t) (v−γ1−1)p p−1 (p (t)) − 1 p−1 dt, (34) t (x) := l (x) · ( p1 (x) ( p−1 p ) γ (v − γ1) )(λα+λα+1) , ω1 := 2 ( p−1 p )(λα+λα+1), (35) and φ (x) := t (x) ω1. (36) then ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ1+1 x0 f2 ) (w) ∣ ∣ λα+1 + ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d γ1+1 x0 f1 ) (w) ∣ ∣ λα+1 ] dw (37) ≥ φ (x) [ ∫ x x0 p (w) ( ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p) dw ] (λα+λα+1) p , all x0 ≤ x ≤ b. proof. for convenience we set γ2 := γ1 + 1. from (13) and assumption we obtain ∣ ∣ ( d γk x0 fj ) (w) ∣ ∣ = 1 γ (v − γk) ∫ w x0 (w − t) v−γk−1 ∣ ∣ ( d v x0 fi ) (t) ∣ ∣dt =: gj,γk (w) , (38) where j = 1, 2, k = 1, 2, all x0 ≤ x ≤ b. we observe that (( d γ1 x0 fj ) (x) )′ = ( d γ1+1 x0 fj ) (x) = ( d γ2 x0 fj ) (x) , (39) cubo 10, 1 (2008) converse fractional opial inequalities ... 129 all x0 ≤ x ≤ b. and also (gj,γ1 (w)) ′ = gj,γ2 (w) ; gj,γk (x0) = 0. (40) notice that if v − γ2 = 1, then gj,γ2 (w) = ∫ w x0 ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ dt. next we apply hölder’s inequality with indices 1 λα+1 < 1, 1 (1−λα+1) < 0, we obtain ∫ x x0 q (w) ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ1+1 x0 f2 ) (w) ∣ ∣ λα+1 dw = ∫ x x0 q (w) (g1,γ1 (w)) λα ( (g2,γ1 (w)) ′)λα+1 dw (41) ≥ ( ∫ x x0 (q (w)) ( 1 1−λα+1 ) dw )(1−λα+1) ( ∫ x x0 (g1,γ1 (w)) λα λα+1 (g2,γ1 (w)) ′ dw ) λα+1 . similarly we get ∫ x x0 q (w) ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d γ1+1 x0 f1 ) (w) ∣ ∣ λα+1 dw ≥ ( ∫ x x0 (q (w)) ( 1 1−λα+1 ) dw )(1−λα+1) ( ∫ x x0 (g2,γ1 (w)) λα λα+1 (g1,γ1 (w)) ′ dw ) λα+1 . (42) adding (41) and (42) we observe ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ1+1 x0 f2 ) (w) ∣ ∣ λα+1 + ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d γ1+1 x0 f1 ) (w) ∣ ∣ λα+1 ] dw ≥ ( ∫ x x0 (q (w)) ( 1 1−λα+1 ) dw )(1−λα+1) 130 george a. anastassiou cubo 10, 1 (2008) [ ( ∫ x x0 (g1,γ1 (w)) λα λα+1 (g2,γ1 (w)) ′ dw ) λα+1 + ( ∫ x x0 (g2,γ1 (w)) λα λα+1 (g1,γ1 (w)) ′ dw ) λα+1 ] (24) ≥ ( 2 ∫ x x0 (q (w)) ( 1 1−λα+1 ) dw )(1−λα+1) · [ ∫ x x0 [ (g1,γ1 (w)) λα λα+1 (g2,γ1 (w)) ′ + (g2,γ1 (w)) λα λα+1 (g1,γ1 (w)) ′ ] dw ]λα+1 (notice (30) and the proof of (31) , accordingly here we have) ≥ ( 2 ∫ x x0 (q (w)) ( 1 1−λα+1 ) dw )(1−λα+1) ( λα+1θ3 λα + λα+1 )λα+1 (g1,γ1 (x) + g2,γ1 (x)) (λα+λα+1) = l (x) (g1,γ1 (x) + g2,γ1 (x)) (λα+λα+1) = l (x) (γ (v − γ1)) (λα+λα+1) { ∫ x x0 (x − t) v−γ1−1 (p (t)) − 1 p (p (t)) 1 p [ ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ + ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ ] dt }(λα+λα+1) (applying hölder’s inequality with indices p p − 1 and p we find) ≥ l (x) (γ (v − γ1)) (λα+λα+1) · ( ∫ x x0 (x − t) (v−γ1−1)p p−1 (p (t)) − 1 p−1 dt )( p−1 p )(λα+λα+1) ( ∫ x x0 p (t) [ ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ + ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ ]p dt ) ( λα+λα+1 p ) cubo 10, 1 (2008) converse fractional opial inequalities ... 131 = t (x) · [ ∫ x x0 p (t) ( ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ + ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ )p dt ] ( λα+λα+1 p ) ≥ φ (x) · [ ∫ x x0 p (t) ( ∣ ∣ ( d v x0 f1 ) (t) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (t) ∣ ∣ p) dt ] ( λα+λα+1 p ) . we have proved (37) . � next we treat the case of exponents λβ = λα + λv. theorem 8. all here as in theorem 5. consider the special case of λβ = λα + λv. assume here for j = 1, 2 that zj (x) := ∫ x x0 p (t) ∣ ∣ ( d v x0 fj ) (t) ∣ ∣ p dt ∈ [h, ψ] , 0 < h < ψ, h := ψ h > 1, mh (1) := (h − 1) h 1 h−1 e ln h . (43) denote t̃ (x) := a0 (x) ( λv λα + λv ) λv p 2 p−2λα−3λv p (mh (1)) −2(λα+λv)/p (44) then ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λα+λv ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λα+λv ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ t̃ (x) ( ∫ x x0 p (w) ( ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p) dw )2( λα+λv p ) , (45) all x0 ≤ x ≤ b. proof. we apply (21) and (22) for λβ = λα + λv and add to get ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λα+λv ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv 132 george a. anastassiou cubo 10, 1 (2008) + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λα+λv ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ a0 (x) { [ ∫ x x0 (z1 (w)) λα λv (z2 (w)) λα λv +1 z ′ 1 (w) dw ] λv p + [ ∫ x x0 (z1 (w)) λα λv +1 (z2 (w)) λα λv z ′ 2 (w) dw ] λv p } (24) ≥ a0 (x) 2 1− λv p { ∫ x x0 [ (z1 (w)) λα λv (z2 (w)) λα λv +1 z ′ 1 (w) + (z1 (w)) λα λv +1 (z2 (w)) λα λv z ′ 2 (w) ] dw } λv p = a0 (x) 2 1− λv p { ∫ x x0 (z1 (w) z2 (w)) λα λv [z2 (w) z ′ 1 (w) + z1 (w) z ′ 2 (w)] dw} λv p = a0 (x) 2 1− λv p { ∫ x x0 (z1 (w) z2 (w)) λα λv (z1 (w) z2 (w)) ′ dw } λv p = a0 (x) 2 1− λv p ( (z1 (x) z2 (x)) λα λv +1 λα λv + 1 ) λv p = a0 (x) 2 p−λv p ( λv λα + λv ) λv p (z1 (x) z2 (x)) (λα+λv ) p (see [10]) ≥ a0 (x) 2 p−λv p ( λv λα + λv ) λv p ( z1 (x) + z2 (x) 2mh (1) ) 2(λα+λv ) p = t̃ (x) (z1 (x) + z2 (x)) 2(λα+λv ) p = t̃ (x) ( ∫ x x0 p (w) ( ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p) dw )2( λα+λv p ) . cubo 10, 1 (2008) converse fractional opial inequalities ... 133 we have established (45) . � next follow special cases of the above theorems. corollary 9. (to theorem 5; λβ = 0, p (t) = q (t) = 1). then ∫ x x0 [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ1 x0 f2 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ c1 (x) · ( ∫ x x0 ( ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p) dw )( λα+λv p ) , (46) all x0 ≤ x ≤ b, where c1 (x) := ( a0 (x) |λβ =0 ) · ( λv λα + λv ) λv p · δ1, (47) δ1 := 2 1−( λα+λv p ) (48) we have that ( a0 (x) |λβ =0 ) = {( (p − 1)( λαp−λα p ) (γ (v − γ1)) λα (vp − γ1p − 1) ( λαp−λα p ) ) (49) · ( (p − λv) ( p−λv p ) (λαvp − λαγ1p − λα + p − λv) ( p−λv p ) )} · (x − x0) ( λαvp−λαγ1p−λα+p−λv p ) . proof. by theorem 5. the constant ( a0 (x) |λβ =0 ) was calculated in [4]. � corollary 10. (to theorem 6; λα = 0, p (t) = q (t) = 1, λβ ≥ λv). then ∫ x x0 [ ∣ ∣ ( d γ2 x0 f2 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ λv ] dw ≥ c2 (x) ( ∫ x x0 [ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 f2 ) (w) ∣ ∣ p] dw ) ( λv +λβ p ) , (50) all x0 ≤ x ≤ b, where 134 george a. anastassiou cubo 10, 1 (2008) c2 (x) := (a0 (x) |λα=0) 2 p−λv p ( λv λβ + λv ) λv p δ λv p 3 . (51) we have that (a0 (x) |λα=0) =      (p − 1) ( λβ p−λβ p ) (γ (v − γ2)) λβ (vp − γ2p − 1) ( λβ p−λβ p )   (52) · ( (p − λv) ( p−λv p ) (λβ vp − λβ γ2p − λβ + p − λv) ( p−λv p ) )} · (x − x0) ( λβ vp−λβ γ2p−λβ +p−λv p ) . proof. by theorem 6. the constant (a0 (x) |λα=0) was calculated in [4]. � 3.2 results involving several functions here we use the following basic inequality. let α1, ..., αn ≥ 0, n ∈ n, then a r 1 + . . . + a r n ≤ (a1 + . . . + an) r ≤ nr−1 ( n ∑ i=1 a r i ) , r ≥ 1, (53) we present theorem 11. let γ1, γ2 ≥ 0 such that 1 ≤ v − γi < 1 p , 0 < p < 1, i = 1, 2, and fj ∈ c v x0 ([a, b]) with f (i) j (x0) = 0, i = 0, 1, . . . , n − 1, n := [v] , j = 1, . . . , m ∈ n. here x, x0 ∈ [a, b] : x ≥ x0. we assume that d v x0 fj is of fixed sign on [x0, b] , j = 1, . . . , m. consider also p (t) > 0, and q (t) > 0 continuous functions on [x0, b] . let λv > 0 and λα, λβ ≥ 0 such that λv > p. set pk (w) := ∫ w x0 (w − t) (v−γk −1)p p−1 (p (t)) − 1 p−1 dt, k = 1, 2; x0 ≤ w ≤ b; (54) a (w) := q (w) (p1 (w)) λα( p−1 p ) (p2 (w)) λβ( p−1 p ) (p (w)) − λv p (γ (v − γ1)) λα (γ (v − γ2)) λβ ; (55) cubo 10, 1 (2008) converse fractional opial inequalities ... 135 a0 (x) := ( ∫ x x0 (a (w)) p p−λv dw ) p−λv p . (56) call ϕ1 (x) := ( a0 (x) |λβ =0 ) · ( λv λα + λv ) λv p , (57) δ ∗ 1 := m 1−( λα+λv p ). (58) if λβ = 0, we obtain that ∫ x x0 q (w)   m ∑ j=1 ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv   dw ≥ δ∗1 · ϕ1 (x) ·   ∫ x x0 p (w)   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw   ( λα+λv p ) , (59) all x0 ≤ x ≤ b. proof. by theorem 5 we get ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ1 x0 fj+1 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ] dw (60) ≥ δ1ϕ1 (x) [ ∫ x x0 p (w) [ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ p] dw ]( λα+λv p ) , j = 1, 2, . . . , m − 1. hence by adding all the above we find ∫ x x0 q (w)   m−1 ∑ j=1 [ ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ1 x0 fj+1 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ]) dw (61) 136 george a. anastassiou cubo 10, 1 (2008) ≥ δ1ϕ1 (x) ·   m−1 ∑ j=1 [ ∫ x x0 p (w) [ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ p] dw ]( λα+λv p )   . also it holds ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ1 x0 fm ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ λv ] dw (62) ≥ δ1ϕ1 (x) [ ∫ x x0 p (w) [ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ p] dw ]( λα+λv p ) . adding (61) and (62) , and using (53) we have 2 ∫ x x0 q (w)   m ∑ j=1 ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv  dw ≥ δ1ϕ1 (x)       m−1 ∑ j=1 [ ∫ x x0 p (w) [ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p (63) + ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ p] dw ]( λα+λv p ) } + { ∫ x x0 p (w) [ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ p] dw }( λα+λv p ) } ≥ m 1−( λα+λv p )δ1ϕ1 (x)    ∫ x x0 p (w)  2 m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw    ( λα+λv p ) . (64) we have proved ∫ x x0 q (w)   m ∑ j=1 ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv   dw ≥ (65) m 1−( λα+λv p )δ1 ( 2 ( λα+λv p )−1 ) ϕ1 (x) ·    ∫ x x0 p (w)   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw    ( λα+λv p ) . cubo 10, 1 (2008) converse fractional opial inequalities ... 137 that is proving (59) . � next we give theorem 12. all here as in theorem 11. assume λβ ≥ λv. denote ϕ2 (x) := (a0 (x) |λα=0) 2 (p−λv ) p ( λv λβ + λv ) λv p δ λv p 3 . (66) if λα = 0, then ∫ x x0 q (w)       m−1 ∑ j=1 [ ∣ ∣ ( d γ2 x0 fj+1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 fj ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ]} + [ ∣ ∣ ( d γ2 x0 fm ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ λv ]} dw ≥ m 1− ( λv +λβ p ) 2 ( λv +λβ p ) ϕ2 (x) · { ∫ x x0 p (w)   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw    ( λv +λβ p ) , x ≥ x0. (67) proof. from theorem 6 we have ∫ x x0 q (w) [ ∣ ∣ ( d γ2 x0 fj+1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 fj ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ] dw ≥ ϕ2 (x) ( ∫ x x0 p (w) [ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ p] dw ) ( λv +λβ p ) , (68) for j = 1, . . . , m − 1. hence by adding all of the above we get 138 george a. anastassiou cubo 10, 1 (2008) ∫ x x0 q (w)   m−1 ∑ j=1 [ ∣ ∣ ( d γ2 x0 fj+1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 fj ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ]) dw ≥ ϕ2 (x)    m−1 ∑ j=1 ( ∫ x x0 p (w) [ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ p] dw ) ( λv +λβ p )} . (69) similarly it holds ∫ x x0 q (w) [ ∣ ∣ ( d γ2 x0 fm ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ λv ] dw ≥ ϕ2 (x) ( ∫ x x0 p (w) [ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ p] dw ) ( λv +λβ p ) . (70) adding (69) , (70) and using (53) we derive (67) . � we continue with theorem 13. let v ≥ 2 and γ1 ≥ 0 such that 2 ≤ v − γ1 < 1/p, 0 < p < 1. let fj ∈ c v x0 ([a, b]) with f (i) j (x0) = 0, i = 0, 1, . . . , n − 1, n := [v] , j = 1, . . . , m ∈ n. here x, x0 ∈ [a, b] : x ≥ x0.assume that d v x0 fj is of fixed sign on [x0, b] , j = 1, . . . , m. consider also p (t) > 0, and q (t) > 0 continuous functions on [x0, b] . let λα ≥ λα+1 > 1, φ is as in theorem 7. then ∫ x x0 q (w)       m−1 ∑ j=1 [ ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 fj+1 (w) ∣ ∣ λα+1 + ∣ ∣ ( d γ1 x0 fj+1 ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 fj (w) ∣ ∣ λα+1 ]} + [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 fm (w) ∣ ∣ λα+1 + ∣ ∣ ( d γ1 x0 fm ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 f1 (w) ∣ ∣ λα+1 ]} dw ≥ cubo 10, 1 (2008) converse fractional opial inequalities ... 139 m 1− ( λα+λα+1 p ) 2 ( λα+λα+1 p ) φ (x)   ∫ x x0 p (w)   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw   ( λα+λα+1 p ) , (71) all x0 ≤ x ≤ b. proof. from theorem 7 we get ∫ x x0 q (w) m−1 ∑ j=1 [ ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 fj+1 (w) ∣ ∣ λα+1 + ∣ ∣ ( d γ1 x0 fj+1 ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 fj (w) ∣ ∣ λα+1 ] dw ≥ φ (x) m−1 ∑ j=1 [ ∫ x x0 p (w) ( ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ p) dw ] ( λα+λα+1 p ) (72) all x0 ≤ x ≤ b. also it holds ∫ x x0 q (w) [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 fm (w) ∣ ∣ λα+1 + ∣ ∣ ( d γ1 x0 fm ) (w) ∣ ∣ λα ∣ ∣d γ1+1 x0 f1 (w) ∣ ∣ λα+1 ] dw ≥ φ (x) [ ∫ x x0 p (w) ( ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ p + ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ p) dw ] ( λα+λα+1 p ) , (73) all x0 ≤ x ≤ b. adding (72) and (73) , along with (53) we derive (71) . � next it comes theorem 14. all here as in theorem 11. consider the special case of λβ = λα + λv. here t̃ (x) as in (44) . assume here for j = 1, . . . , m that zj (x) := ∫ x x0 p (t) ∣ ∣d v x0 fj (t) ∣ ∣ p dt ∈ [h, ψ] , 0 < h < ψ. then 140 george a. anastassiou cubo 10, 1 (2008) ∫ x x0 q (w)       m−1 ∑ j=1 [ ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d γ2 x0 fj+1 ) (w) ∣ ∣ λα+λv ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 fj ) (w) ∣ ∣ λα+λv ∣ ∣ ( d γ1 x0 fj+1 ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ]} + [ ∣ ∣ ( d γ1 x0 f1 ) (w) ∣ ∣ λα ∣ ∣ ( d γ2 x0 fm ) (w) ∣ ∣ λα+λv ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λα+λv ∣ ∣ ( d γ1 x0 fm ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ λv ]} dw ≥ (74) m (1− 2(λα+λv ) p )2 2( λα+λv p )t̃ (x)   ∫ x x0 p (w)   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw   (2( λα+λv p )) , all x0 ≤ x ≤ b. proof. based on theorem 8. the rest as in the proof of theorem 13. � we continue with corollary 15. (to theorem 11, λβ = 0, p (t) = q (t) = 1). then ∫ x x0   m ∑ j=1 ∣ ∣ ( d γ1 x0 fj ) (w) ∣ ∣ λα ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv   dw ≥ δ∗1 ϕ1 (x)   ∫ x x0   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p  dw   ( λα+λv p ) (75) all x0 ≤ x ≤ b. in (75) , ( a0 (x) |λβ =0 ) of ϕ1 (x) is given by (49) . proof. based on theorem 11. � corollary 16. (to theorem 12, λα = 0, p (t) = q (t) = 1). it holds cubo 10, 1 (2008) converse fractional opial inequalities ... 141 ∫ x x0       m−1 ∑ j=1 [ ∣ ∣ ( d γ2 x0 fj+1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 fj ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fj+1 ) (w) ∣ ∣ λv ]} + [ ∣ ∣ ( d γ2 x0 fm ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 f1 ) (w) ∣ ∣ λv + ∣ ∣ ( d γ2 x0 f1 ) (w) ∣ ∣ λβ ∣ ∣ ( d v x0 fm ) (w) ∣ ∣ λv ]} dw ≥ ( m 1− ( λv +λβ p ) ) 2 ( λv +λβ p ) ϕ2 (x)    ∫ x x0   m ∑ j=1 ∣ ∣ ( d v x0 fj ) (w) ∣ ∣ p   dw    ( λv +λβ p ) , (76) all x0 ≤ x ≤ b. in (76) , (a0 (x) |λα=0) of ϕ2 (x) is given by (52) . proof. based on theorem 12. � received: october 2007. revised: december 2007. references [1] r.p. agarwal and p.y.h. pang, opial inequalities with applications in differential and difference equations, kluwer academic publishers,dordrecht, boston, london, 1995. [2] g.a. anastassiou, general fractional opial type inequalities, acta applicandae mathematicae, 54 (1998), 303–317. [3] g.a. anastassiou, opial type inequalities involving fractional derivatives of functions, nonlinear studies, 6, no.2 (1999), 207–230. [4] g.a. anastassiou, opial-type inequalities involving fractional derivatives of two functions and applications, computers and mathematics with applications, vol. 48 (2004), 1701–1731. [5] g.a. anastassiou and j.a. goldstein, fractional opial type inequalities and fractional differential equations, result. math., 41 (2002), 197–212. 142 george a. anastassiou cubo 10, 1 (2008) [6] j.a. canavati, the riemann-liouville integral, nieuw archief voor wiskunde 5, no. 1 (1987), 53–75. [7] r. hilfer(editor), applications of fractional calculus in physics, volume published by world scientific, singapore, 2000. [8] z. opial, sur une inégalite, ann. polon. math., 8 (1960), 29–32. [9] i. podlubny, fractional differential equations, academic press, san diego, 1999. [10] w. specht, zur theorie der elementaren mittel, math. z., 74 (1960), 91–98. [11] e.t. whittaker and g.n. watson, a course in modern analysis, cambridge university press, 1927. gaconversefractopial_modified3.pdf os1.dvi cubo a mathematical journal vol.12, no¯ 01, (133–148). march 2010 well-posedness results for anisotropic nonlinear elliptic equations with variable exponent and l1-data stanislas ouaro laboratoire d’analyse mathématique des equations (lame), ufr. sciences exactes et appliquées, université de ouagadougou 03 bp 7021 ouaga 03 ouagadougou, burkina faso email : souaro@univ-ouaga.bf abstract we study the anisotropic boundary value problem − n ∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) = f in ω, u = 0 on ∂ω, where ω is a smooth open bounded domain in rn (n ≥ 3) and f ∈ l1(ω). we prove the existence and uniqueness of an entropy solution for this problem. resumen estudiamos el problema de valores en la frontera anisotropico − n ∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) = f en ω, u = 0 sobre ∂ω, donde ω es un dominio abierto suave do rn (n ≥ 3) y f ∈ l1(ω). proveamos la existencia y unicidad de una solución de entroṕıa para este problema. key words and phrases: anisotropic; variable exponent; entropy solution; electrorheological fluids. math. subj. class.: 35j20, 35j25, 35d30, 35b38, 35j60. 134 stanislas ouaro cubo 12, 1 (2010) 1 introduction let ω be an open bounded domain of rn (n ≥ 3) with smooth boundary. our aim is to prove the existence and uniqueness of an entropy solution to the anisotropic nonlinear elliptic problem            − n ∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) = f in ω u = 0 on ∂ω, (1.1) where the right-hand side f ∈ l1(ω). we assume that for i = 1, ...,n the function ai : ω × r → r is carathéodory and satisfies the following conditions: ai(x,ξ) is the continuous derivative with respect to ξ of the mapping ai : ω × r → r, ai = ai(x,ξ), i.e. ai(x,ξ) = ∂ ∂ξ ai(x,ξ) such that the following equality and inequalities holds ai(x, 0) = 0, (1.2) for almost every x ∈ ω. there exists a positive constant c1 such that |ai(x,ξ)| ≤ c1(ji(x) + |ξ| pi(x)−1) (1.3) for almost every x ∈ ω and for every ξ ∈ r, where ji is a nonnegative function in l p′i(.)(ω), with 1/pi(x) + 1/p ′ i(x) = 1. there exists a positive constant c2 such that (ai(x,ξ) − ai(x,η)) . (ξ − η) ≥ { c2 |ξ − η| pi(x) if |ξ − η| ≥ 1 c2 |ξ − η| p − i if |ξ − η| < 1, (1.4) for almost every x ∈ ω and for every ξ,η ∈ r, with ξ 6= η, and |ξ| pi(x) ≤ ai(x,ξ).ξ ≤ pi(x)ai(x,ξ) (1.5) for almost every x ∈ ω and for every ξ ∈ r. we also assume that the variable exponent pi(.) : ω → [2,n) are continuous functions for all i = 1, ...,n such that: p̄(n − 1) n(p̄ − 1) < p−i < p̄(n − 1) n − p̄ , n ∑ i=1 1 p−i > 1 and p+i − p − i − 1 p−i < p̄ − n p̄(n − 1) , (1.6) where 1 p̄ = 1 n n ∑ i=1 1 p−i , p−i := ess inf x∈ω pi(x) and p + i := ess sup x∈ω pi(x). we introduce the numbers q = n(p̄ − 1) n − 1 , q∗ = nq n − q = n(p̄ − 1) n − p̄ . (1.7) cubo 12, 1 (2010) well-posedness results for anisotropic ... 135 a prototype example that is covered by our assumptions is the following anisotropic (p1(.), ...,pn (.))harmonic equation: set ai(x,ξ) = (1/pi(x)) |ξ| pi(x), ai(x,ξ) = |ξ| pi(x)−2 ξ where pi(x) ≥ 2. then we get the following equation: − n ∑ i=1 ∂ ∂xi ( ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ ∣ pi(x)−2 ∂ ∂xi u ) = f which, in the particular case when pi = p for any i ∈ {1, ...,n} is the anisotropic p(.)-laplace equation. for the proof of existence of entropy solutions of (1.1), we follow [2] and derive a priori estimates for the approximated solutions un and the partial derivatives ∂un ∂xi in the marcinkiewicz spaces mq ∗ and mp − i q/p̄ respectively (see section 2 or [2] for definition of marcinkiewicz spaces). the study of nonlinear elliptic equations involving the p−laplace operator is based on the theory of standard sobolev spaces w m,p(ω) in order to find weak solutions. for the nonhomogeneous p(.)-laplace operators, the natural setting for this approach is the use of the variable exponent lebesgue and sobolev spaces lp(.)(ω) and w m,p(.)(ω). variable exponent lebesgue spaces appeared in the literature for the first time in a 1931 article by orlicz[21]. after [21], orlicz abandoned the study of variable exponent spaces to concentrate on the theory of function spaces that now bears his name (orlicz spaces). after orlicz’s work (cf. [21]), h. hudzik [14], and j. musielak [20] investigated the variable exponent sobolev spaces. variable exponent lebesgue spaces on the real line have been independently developed by russian researchers, notably sharapudinov[26] and tsenov [27]. the next major step in the investigation of variable exponent lebesgue and sobolev spaces was the comprehensive paper by o. kovacik and j. rakosnik in the early 90’s [16]. this paper established many of basic properties of lebesgue and sobolev spaces with variables exponent. variable sobolev spaces have been used in the last decades to model various phenomena. in [5], chen, levine and rao proposed a framework for image restoration based on a laplacian variable exponent. another application which uses nonhomogeneous laplace operators is related to the modelling of electrorheological fluids. the first major discovery in electrorheological fluids was due to willis winslow in 1949 (cf. [28]). these fluids have the interesting property that their viscosity depends on the electric field in the fluid. they can raise the viscosity by as much as five orders of magnitude. this phenomenon is known as the winslow effect. for some technical applications, consult pfeiffer et al [22]. electrorheological fluids have been used in robotics and space technology. the experimental research has been done mainly in the usa, for instance in nasa laboratories. for more information on properties, modelling and the application of variable exponent spaces to these fluids, we refer to diening [6], rajagopal and ruzicka [22], and ruzicka [24]. in this paper, the operator involved in (1.1) is more general than the p(.)−laplace operator. thus, the variable exponent sobolev space w 1,p(.)(ω) is not adequate to study nonlinear problems of this type. this leads us to seek entropy solutions for problems (1.1) in a more general variable exponent sobolev space which was introduced for the first time by mihäılescu et al [18]. 136 stanislas ouaro cubo 12, 1 (2010) as the right-hand side in (1.1) is in l1(ω), the suitable notion of solution is a notion of entropy solution (cf. [3]). the remaining part of this paper is organized as follows: section 2 is devoted to mathematical preliminaries including, among other things, a brief discussion of variable exponent lebesgue, sobolev, anisotropic spaces and marcinkiewicz spaces. existence of weak energy solution for (1.1) where f ∈ l∞(ω) was proved in [14]; we will also briefly recall the results of [14] in section 2. the main existence and uniqueness result is stated and proved in section 3. 2 mathematical preliminaries in this section, we define lebesgue, sobolev and anisotropic spaces with variable exponent and give some of their properties. roughly speaking, anistropic lebesgue and sobolev spaces are functional spaces of lebesgue’s and sobolev’s type in which different space directions have different roles. given a measurable function p(.) : ω → [1,∞). we define the lebesgue space with variable exponent lp(.)(ω) as the set of all measurable function u : ω → r for which the convex modular ρp(.)(u) := ∫ ω |u| p(x) dx is finite. if the exponent is bounded, i.e., if p+ < ∞, then the expression |u|p(.) := inf { λ > 0 : ρp(.)(u/λ) ≤ 1 } defines a norm in lp(.)(ω), called the luxembourg norm. the space (lp(.)(ω), |.|p(.)) is a separable banach space. moreover, if p− > 1, then l p(.)(ω) is uniformly convex, hence reflexive, and its dual space is isomorphic to lp ′(.)(ω), where 1 p(x) + 1 p′(x) = 1. finally, we have the hölder type inequality: ∣ ∣ ∣ ∣ ∫ ω uvdx ∣ ∣ ∣ ∣ ≤ ( 1 p− + 1 p′− ) |u|p(.) |v|p′(.) , (2.1) for all u ∈ lp(.)(ω) and v ∈ lp ′(.)(ω). now, let w 1,p(.)(ω) := { u ∈ lp(.)(ω) : |∇u| ∈ lp(.)(ω) } , which is a banach space equipped with the norm ‖u‖1,p(.) := |u|p(.) + |∇u|p(.) . an important role in manipulating the generalized lebesgue-sobolev spaces is played by the modular ρp(.) of the space l p(.)(ω). next, we define w 1,p(.) 0 (ω) as the closure of c ∞ 0 (ω) in w 1,p(.)(ω) under the norm ‖u‖1,p(.). set c+(ω) = { p(.) ∈ c(ω) : min x∈ω p(x) > 1 } . furthermore, if p(.) ∈ c+(ω) is logarithmic hölder continuous, then c ∞ 0 (ω) is dense in w 1,p(.) 0 (ω), cubo 12, 1 (2010) well-posedness results for anisotropic ... 137 that is h 1,p(.) 0 (ω) = w 1,p(.) 0 (ω) (cf. [13]). since ω is an open bounded set and p(.) ∈ c+(ω) is logarithmic hölder, the p(.)−poincaré inequality |u|p ≤ c |∇u|p(.) holds for all u ∈ w 1,p(.) 0 (ω), where c depends on p, |ω|, diam(ω) and n (see [13]), and so ‖u‖ := |∇u|p(.) , is an equivalent norm in w 1,p(.) 0 (ω). of course also the norm ‖u‖p(.) := n ∑ i=1 ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ ∣ p(.) is an equivalent norm in w 1,p(.) 0 (ω). hence the space w 1,p(.) 0 (ω) is a separable and reflexive banach space. let us present a natural generalization of the variable exponent sobolev space w 1,p(.) 0 (ω) (cf. [18]) that will enable us to study the problem (1.1) with sufficient accuracy. first of all, we denote by −→p (.) : ω → rn the vectorial function −→p = (p1(.), ...,pn (.)). the anisotropic variable exponent sobolev space w 1,−→p (.) 0 (ω) is defined as the closure of c ∞ 0 (ω) with respect to the norm ‖u‖−→p (.) := n ∑ i=1 ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ ∣ pi(.) . the space ( w 1,−→p (.) 0 (ω),‖u‖−→p (.) ) is a reflexive banach space (cf. [18]). let us introduce the following notations: −→ p + = (p + 1 , ...,p + n ), −→ p − = (p − 1 , ...,p − n ), p ++ = max { p+1 , ...,p + n } ,p +− = max { p−1 , ...,p − n } ,p−− = min { p−1 , ...,p − n } , and p∗− = n n ∑ i=1 1 p−i − 1 , p−,∞ = max { p +− ,p ∗ − } . we have the following result (cf. [18]): theorem 2.1. assume ω ⊂ rn (n ≥ 3) is a bounded domain with smooth boundary. assume relation (1.6) is fulfilled. for any q ∈ c(ω) verifying 1 < q(x) < p−,∞ for all x ∈ ω, 138 stanislas ouaro cubo 12, 1 (2010) then the embedding w 1,−→p (.) 0 (ω) →֒ l q(.)(ω) is continuous and compact. we also recall the result of the study of problem (1.1) for the right-hand side f ∈ l∞(ω) (cf. [15]). we first recall the definition of weak energy solution of (1.1) for the right-hand side more regular i.e f ∈ l∞(ω). definition 2.2. let f ∈ l∞(ω); a weak energy solution of (1.1) is a function u ∈ w 1,−→p (.) 0 (ω) such that ∫ ω n ∑ i=1 ai(x, ∂ ∂xi u). ∂ ∂xi ϕdx = ∫ ω f(x)ϕdx, for all ϕ ∈ w 1,−→p (.) 0 (ω). (2.2) we have proved among other results in [15] the following theorem theorem 2.3. assume (1.2)-(1.6) and f ∈ l∞(ω). then there exists a unique weak energy solution of (1.1). finally, in this paper, we will use the marcinkiewicz spaces mq(ω)(1 < q < ∞) with constant exponent. note that the marcinkiewicz spaces mq(.)(ω) in the variable exponent setting was introduced for the first time by sanchon and urbano (see [25]). marcinkiewicz spaces mq(ω)(1 < q < ∞) contain the measurable functions g : ω → r for which the distribution function λg(γ) = |{x ∈ ω : |g(x)| > γ}| , γ ≥ 0, satisfies an estimate of the form λg(γ) ≤ cγ −q, for some finite constant c > 0. the space mq(ω) is a banach space under the norm ‖g‖ ∗ mq (ω) = sup t>0 t1/q ( 1 t ∫ t 0 g∗(s)ds ) , where g∗ denotes the nonincreasing rearrangement of f: g∗(t) = inf {γ > 0 : λg (γ) ≤ t} . we will use the following pseudo norm ‖g‖mq (ω) = inf { c : λg(γ) ≤ cγ −q, ∀γ > 0 } , which is equivalent to the norm ‖g‖ ∗ mq (ω) (see [2]). we have the following lemma (for the proof, see [2, proof of lemma 2.2]). lemma 2.4. let g be a nonnegative function in w 1, −→ p − 0 (ω). assume p̄ < n, and that there exists a constant c such that n ∑ i=1 ∫ {|g|≤γ} ∣ ∣ ∣ ∣ ∂g ∂xi ∣ ∣ ∣ ∣ p − i dx ≤ c(γ + 1),∀γ > 0. (2.3) cubo 12, 1 (2010) well-posedness results for anisotropic ... 139 then there exists a constant c, depending on c, such that ‖g‖ m n (p̄−1) n −p̄ (ω) ≤ c. 3 existence and uniqueness of entropy solution in this section, we study the problem (1.1) for a right-hand side f ∈ l1(ω). in the l1 setting, the suitable notion of solution for the study of (1.1) is the notion of entropy solution (cf. [3]). we first define the troncation function tt by tt(s) := max{−t,min{t,s}}. definition 3.1. a measurable function u is an entropy solution to problem (1.1) if, for every t > 0, tt(u) ∈ w 1,−→p (.) 0 (ω) and ∫ ω n ∑ i=1 ai(x, ∂ ∂xi u). ∂ ∂xi tt(u − ϕ)dx ≤ ∫ ω f(x)tt(u − ϕ)dx, (3.1) for all ϕ ∈ w 1,−→p (.) 0 (ω) ∩ l ∞(ω). remark 3.2. a function u such that tt(u) ∈ w 1,−→p (.) 0 (ω) for all t > 0 does not necessarily belong in w 1,1 0 (ω). however, it is possible to define its weak gradient, still denoted by ∇u. our main result in this section is the following: theorem 3.3. assume (1.2)-(1.6) and f ∈ l1(ω). then there exists a unique entropy solution u to problem (1.1). proof. the proof of this theorem will be done in three steps. ∗step 1. a priori estimates. we start with the existence of the weak gradient for every measurable function u such that tt(u) ∈ w 1,−→p (.) 0 (ω) for all t > 0. proposition 3.4. if u is a measurable function such that tt(u) ∈ w 1,−→p (.) 0 (ω) for all t > 0, then there exists a unique measurable function v : ω → rn such that vχ{|u| 0, where χa denotes the characteristic function of a measurable set a. moreover, if u belongs to w 1,1 0 (ω), then v coincides with the standard distributional gradient of u. proof. as tt(u) ∈ w 1,−→p (.) 0 (ω) →֒ w 1, −→ p − 0 (ω) ⊂ w 1,1 0 (ω) for all t > 0 since 1 < p − i for all i = 1, ...,n, then by theorem 1.5 in [1], the result follows. proposition 3.5. assume (1.2)-(1.6) and f ∈ l1(ω). let u be an entropy solution of (1.1). if there exists a positive constant m such that n ∑ i=1 ∫ {|u|>t} tqi(x)dx ≤ m, for all t > 0, (3.2) then n ∑ i=1 ∫ { ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ αi(.) >t } tqi(x)dx ≤ ‖f‖1 + m for all t > 0, 140 stanislas ouaro cubo 12, 1 (2010) where αi(.) = pi(.)/(qi(.) + 1)., for all i = 1, ...,n. proof. take ϕ = 0 in (3.1), we have n ∑ i=1 ∫ ω ai(x, ∂ ∂xi tt(u)). ∂ ∂xi tt(u)dx ≤ ∫ ω f(x)tt(u)dx. we deduces from inequality above that n ∑ i=1 ∫ ω ∣ ∣ ∣ ∣ ∂ ∂xi tt(u) ∣ ∣ ∣ ∣ pi(x) dx ≤ t‖f‖1 , for all t > 0. therefore, defining ψ := tt(u)/t, we have, for all t > 0, n ∑ i=1 ∫ ω tpi(x)−1 ∣ ∣ ∣ ∣ ∂ ∂xi ψ ∣ ∣ ∣ ∣ pi(x) dx = 1 t ∫ ω ∣ ∣ ∣ ∣ ∂ ∂xi tt(u) ∣ ∣ ∣ ∣ pi(x) dx ≤ ‖f‖1 . from the above inequality, the definition of αi(.) and (3.2), we have n ∑ i=1 ∫ { ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ αi(.) >t } tqi(x)dx ≤ ∫ { ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ αi(.) >t } ∩{|u|≤t} tqi(x)dx + ∫ {|u|>t} tqi(x)dx ≤ ∫ {|u|≤t} tqi(x)    ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ αi(x) t    pi(x) αi(x) dx + m ≤ ‖f‖1 + m, for all t > 0. proposition 3.6. assume (1.2)-(1.6) and f ∈ l1(ω). let u be an entropy solution of (1.1), then 1 h n ∑ i=1 ∫ {|u|≤h} ∣ ∣ ∣ ∣ ∂ ∂xi th(u) ∣ ∣ ∣ ∣ pi(x) dx ≤ m for every h > 0, with m a positive constant. more precisely, there exists d > 0 such that meas{|u| > h} ≤ dp − − 1 + h hp − − . proof. taking ϕ = 0 in the entropy inequality (3.1) and using (1.5), we obtain n ∑ i=1 ∫ {|u|≤h} ∣ ∣ ∣ ∣ ∂ ∂xi th(u) ∣ ∣ ∣ ∣ pi(x) dx ≤ h‖f‖1 ≤ mh for all h > 0. next, n ∑ i=1 ∫ {|u|≤h} ∣ ∣ ∣ ∣ ∂ ∂xi th(u) ∣ ∣ ∣ ∣ pi(x) dx ≤ mh ⇒ n ∑ i=1 ∫ {|u|≤h} ∣ ∣ ∣ ∣ ∂ ∂xi th(u) ∣ ∣ ∣ ∣ p − − dx ≤ c(1 + h). cubo 12, 1 (2010) well-posedness results for anisotropic ... 141 we can write the above inequality as n ∑ i=1 ∥ ∥ ∥ ∥ ∂ ∂xi th(u) ∥ ∥ ∥ ∥ p − − p − − ≤ c(1 + h) or ‖th(u)‖ w 1,p − − 0 (ω) ≤ [c(1 + h)] 1 p − − . by the poincaré inequality in constant exponent, we obtain ‖th(u)‖ l p − − (ω) ≤ d(1 + h) 1 p − − . the above inequality imply that ∫ ω |th(u)| p − − dx ≤ dp − − (1 + h), from which we obtain meas{|u| > h} ≤ dp − − 1 + h hp − − . ∗ step 2. uniqueness of entropy solution. the proof of uniqueness of entropy solutions follow the technics by bénilan et al [3] (see also [25]). let h > 0 and u,v two entropy solutions of (1.1). we write the entropy inequality (3.1) corresponding to the solution u, with thv as test function, and to the solution v, with thu as test function. upon addition, we get                              ∫ {|u−thv|≤t} n ∑ i=1 ai(x, ∂ ∂xi u). ∂ ∂xi (u − thv)dx+ ∫ {|v−thu|≤t} n ∑ i=1 a(x, ∂ ∂xi v). ∂ ∂xi (v − thu)dx ≤ ∫ ω f(x)(tt(u − thv) + tt(v − thu))dx. (3.3) define e1 := {|u − v| ≤ t, |v| ≤ h} , e2 := e1 ∩ {|u| ≤ h} , and e3 := e1 ∩ {|u| > h} . 142 stanislas ouaro cubo 12, 1 (2010) we start with the first integral in (3.3). by (1.5), we have                                                                                                                                              ∫ {|u−thv|≤t} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − thv)dx = ∫ {|u−thv|≤t}∩({|v|≤h}∪{|v|>h}) n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − thv)dx = ∫ {|u−thv|≤t,|v|≤h} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − thv)dx+ ∫ {|u−thv|≤t,|v|>h} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − thv)dx = ∫ {|u−v|≤t,|v|≤h} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx + ∫ {|u−thv|≤t,|v|>h} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi udx ≥ ∫ {|u−v|≤t,|v|≤h} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx = ∫ e1 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx = ∫ e1∩({|u|≤h}∪{|u|>h}) n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx = ∫ e2 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx + ∫ e3 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx = ∫ e2 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx+ ∫ e3 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi udx − ∫ e3 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi vdx ≥ ∫ e2 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx − ∫ e3 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi vdx. (3.4) cubo 12, 1 (2010) well-posedness results for anisotropic ... 143 using (1.3) and (2.1), we estimate the last integral in (3.4) as follows                  ∣ ∣ ∣ ∣ ∣ ∫ e3 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi vdx ∣ ∣ ∣ ∣ ∣ ≤ c1 ∫ e3 n ∑ i=1 ( ji(x) + ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ ∣ pi(x)−1 ) ∣ ∣ ∣ ∣ ∂ ∂xi v ∣ ∣ ∣ ∣ dx ≤ c n ∑ i=1  |j|p′ i (.) + ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ ∣ pi(x)−1 ∣ ∣ ∣ ∣ ∣ p′ i (.),{h<|u|≤h+t}   ∣ ∣ ∣ ∣ ∂ ∂xi v ∣ ∣ ∣ ∣ pi(.),{h−t<|v|≤h} , (3.5) where ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ pi(x)−1 ∣ ∣ ∣ ∣ p′ i (.),{h<|u|≤h+t} = ∥ ∥ ∥ ∥ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ pi(x)−1 ∥ ∥ ∥ ∥ l p′ i (.) ({h<|u|≤h+t}) . for all i = 1, ...,n, the quantity ( |ji|p′ i (.) + ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∂ ∂xi u ∣ ∣ ∣ pi(x)−1 ∣ ∣ ∣ ∣ p′ i (.),{h<|u|≤h+t} ) is finite, since u ∈ w 1,−→p (.) 0 (ω) and ji ∈ l p′i(.)(ω); then by proposition 3.6, the last expression converges to zero as h tends to infinity. therefore, from (3.4) and (3.5), we obtain ∫ {|u−thv|≤t} n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − thv)dx ≥ ih + ∫ e2 n ∑ i=1 a(x, ∂ ∂xi u). ∂ ∂xi (u − v)dx, (3.6) where ih converges to zero as h tends to infinity. we may adopt the same procedure to treat the second term in (3.3) to obtain ∫ {|v−thu|≤t} n ∑ i=1 a(x, ∂ ∂xi v). ∂ ∂xi (v − thu)dx ≥ jh − ∫ e2 n ∑ i=1 a(x, ∂ ∂xi v). ∂ ∂xi (u − v)dx, (3.7) where jh converges to zero as h tends to infinity. next, consider the right-hand side of inequality (3.3). noting that tt(u − thv) + tt(v − thu) = 0 in {|u| ≤ h, |v| ≤ h} ; we obtain ∣ ∣ ∣ ∣ ∫ ω f(x)(tt(u − thv) + tt(v − thu))dx ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∫ {|u|>h} f(x)(tt(u − thv) + tt(v − thu))dx + ∫ {|u|≤h} f(x)(tt(u − thv) + tt(v − thu))dx ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∫ {|u|>h} f(x)(tt(u − thv) + tt(v − thu))dx + ∫ {|u|≤h,|v|>h} f(x)(tt(u − thv) + tt(v − thu))dx ∣ ∣ ∣ ∣ ∣ ≤ 2t ( ∫ {|u|>h} |f|dx + ∫ {|v|>h} |f|dx ) . according to proposition 3.6, both meas{|u| > h} and meas{|v| > h} tend to zero as h goes to infinity, then by the inequality above, the right-hand side of inequality (3.3) tends to zero as h 144 stanislas ouaro cubo 12, 1 (2010) goes to infinity. from this assertion, (3.3), (3.6) and (3.7), we obtain, letting h → +∞, ∫ {|u−v|≤t} n ∑ i=1 (a(x, ∂ ∂xi u) − a(x, ∂ ∂xi v)). ∂ ∂xi (u − v)dx ≤ 0, for all t > 0. by assertion (1.4), we conclude that ∂ ∂xi u = ∂ ∂xi v, for all i = 1, ..,n a.e. in ω. we deduce that ‖u − v‖−→p (.) = n ∑ i=1 ∣ ∣ ∣ ∣ ∂ ∂xi u − ∂ ∂xi v ∣ ∣ ∣ ∣ pi(.) = 0, and hence u = v, a.e. in ω. ∗ step 3. existence of entropy solutions. let (fn)n be a sequence of bounded functions, strongly converging to f ∈ l1(ω) and such that ‖fn‖1 ≤ ‖f‖1 , for all n. (3.8) we consider the problem            − n ∑ i=1 ∂ ∂xi ai(x, ∂ ∂xi un) = fn in ω un = 0 on ∂ω. (3.9) it follows from theorem 2.3 (cf. [15]) that problem (3.9) has a unique weak energy solution un ∈ w 1,−→p (.) 0 (ω) since fn ∈ l ∞(ω). our interest is to prove that these approximated solutions un tend, as n goes to infinity, to a measurable function u which is an entropy solution of the limit problem (1.1). to this end, we derive a priori estimates for un and ∂ ∂xi un in the marcinkiewicz spaces mq ∗ and mp − i q/p̄ where q∗ and p̄ are defined in (1.6) and (1.7). let γ > 0, denote tγ the corresponding truncation function. note that dtγ (r) =        1 if |r| < γ 0 if |r| > γ. in particular, we have ai(x,ξ)dtγ (r)ξ ≥ ai(x,ξ)ξχ{|r|<γ}, for all i = 1, ...,n. (3.10) lemma 3.7. there exists a constant c, not depending on n, such that n ∑ i=1 ∫ {|un|≤γ} ∣ ∣ ∣ ∣ ∂un ∂xi ∣ ∣ ∣ ∣ p − i dx ≤ c(γ + 1), for all γ > 0. (3.11) cubo 12, 1 (2010) well-posedness results for anisotropic ... 145 proof. inserting ϕ = tγ (un) into (2.2), we have ∫ ω n ∑ i=1 ai(x, ∂ ∂xi un)dtγ (un) ∂ ∂xi undx = ∫ ω fntγ (un)dx. using (3.10) and the coercivity condition (1.5), we obtain (3.11). lemma 3.8. there exists a constant c, not depending on n, such that ‖un‖mq∗ (ω) ≤ c and ∥ ∥ ∥ ∥ ∂un ∂xi ∥ ∥ ∥ ∥ m p − i q/p̄ (ω) ≤ c, for all i = 1, ...,n. proof. the result of lemma 3.8 is a direct consequence of lemma 3.7 (cf. [2, proof of lemma 3.3]). in view of lemma 3.8 and following [2] ( see also [4, lemma a.2]), we deduce that un is uniformly bounded in lk0 (ω) for some k0 < q ∗ with k0 > p − i q/p̄ and ∂un ∂xi is uniformly bounded in lki (ω) for some ki > 1 with p − i − 1 < ki < p − i q/p̄, for all i = 1, ...,n. from this, we get that un is uniformly bounded in the isotropic sobolev space w 1,kmin 0 (ω), where kmin = min(k1, ...,kn ). consequently, we can assume without loss of generality (see also [2, lemma 3.4]) that as n → 0,                                un → u a.e in ω and in l kmin (ω) un ⇀ u in w 1,kmin 0 (ω) ∂un ∂xi → ∂u ∂xi in l1(ω), for all i = 1, ...,n. ai(x, ∂un ∂xi ) → ai(x, ∂u ∂xi ) a.e in ω and in l1(ω), for all i = 1, ...,n. (3.12) now, fix t > 0, ϕ ∈ w 1,−→p (.) 0 (ω) ∩ l ∞(ω), and choose tt(un − ϕ) as a test function in (2.2), with u replaced by un to obtain ∫ ω n ∑ i=1 ai(x, ∂ ∂xi un). ∂ ∂xi tt(un − ϕ)dx = ∫ ω fn(x)tt(un − ϕ)dx. note that this choice can be made using a standard density argument. we now pass to the limit in the previous identity. for the right-hand side, the convergence is obvious since fn converges strongly in l1 to f and tt(un − ϕ) converges weakly-* in l ∞, and a.e., to tt(u − ϕ). next, we write the left hand side as ∫ {|un−ϕ|≤t} n ∑ i=1 ai(x, ∂ ∂xi un). ∂ ∂xi undx − ∫ {|un−ϕ|≤t} n ∑ i=1 ai(x, ∂ ∂xi un). ∂ ∂xi ϕdx (3.13) 146 stanislas ouaro cubo 12, 1 (2010) by (3.12), the second integral of (3.13) converges to ∫ {|u−ϕ|≤t} n ∑ i=1 ai(x, ∂ ∂xi u). ∂ ∂xi ϕdx. for the first integral in (3.13), as n ∑ i=1 ai(x, ∂ ∂xi un). ∂ ∂xi un is nonnegative by (1.5), we obtain by using (3.12) and fatou’s lemma ∫ {|u−ϕ|≤t} n ∑ i=1 ai(x, ∂ ∂xi u). ∂ ∂xi udx ≤ lim n→∞ inf ∫ {|un−ϕ|≤t} n ∑ i=1 ai(x, ∂ ∂xi un). ∂ ∂xi undx. gathering results, we obtain ∫ ω n ∑ i=1 ai(x, ∂ ∂xi u). ∂ ∂xi tt(u − ϕ)dx ≤ ∫ ω f(x)tt(u − ϕ)dx, i.e., u is an entropy solution of (1.1). received: june, 2008. revised: november, 2008. references [1] alvino, a., boccardo, l., ferone, v., orsina, l. and trombetti, g., existence results for non-linear elliptic equations with degenerate coercivity, ann. mat. pura appl., 182 (2003), 53–79. 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[28] winslow, w.m., induced fibration of suspensions, j. applied physics, 20 (1949), 1137– 1140. cubo a mathematical journal vol.10, n o ¯ 04, (67–72). december 2008 a fixed point theorem for certain operators b.e. rhoades department of mathematics, indiana university, bloomington, in 47405-7106 email: rhoades@indiana.edu abstract we obtain a fixed point theorem for a class of operators. this result is an extension of a similar theorem of constantin (1994). resumen obtenemos un teorema de punto fijo para una clase de operadores. este resultado es una extensión de un teorema similar devido a constantin (1994). key words and phrases: fixed point theorem. math. subj. class.: 47h10. most fixed point theorems are proved either by examining the successive iterates of the operator, or by constructing an iteration scheme, such as that of mann or ishikawa. in this paper we consider the situation in which the operator is used on the successive iterates of a sequence. 68 b.e. rhoades cubo 10, 4 (2008) in a recent paper, constantin [3] obtained a fixed point theorem for a class of operators which are selfmaps of a banach space x, and which satisfy the condition ‖t x − t y‖ ≤ g(‖x − y‖, ‖x − t x‖, ‖y − t y‖) (1) for all x, y ∈ x, where g : r3+ → r+, g is continuous, nondecreasing in each variable, and is such that, if h(r) := g(r, r, r), then r − h(r) is nonnegative and strictly increasing on r+. a natural extension of (1) would be: let λ denote the set of all continuous functions g : r5 + → r+, nondecreasing in each variable and such that, if h(r) := g(r, r, r, r, r), then h(r) < r for each r > 0. define t from x to x satisfying ‖t x − t y‖ ≤ g(‖x − y‖, ‖x − t x‖, ‖y − t y‖, (2) ‖x − t y‖, ‖y − t x‖) for all x, y ∈ x, some g ∈ λ. however, a slightly more general extension of (1) is the following. let x be a banach space, g : r3+ → r+, g continuous, nondecreasng, and satisfying g(t) < t for each t > 0. let t be a selfmap of x satisfying ‖t x − t y‖ ≤ g(m (x, y)), for all x, y ∈ x, (3) where m (x, y) := max{‖x − y‖, ‖x − t x‖, ‖y − t y‖, ‖x − t y‖, ‖y − t x‖). theorem 1. let a satisfy (3) and {xn} ⊂ x. then the following are equivalent: (i) t xn − xn → 0 as n → ∞, (ii) {t xn − xn} is bounded, and {xn} converges to a point p which is the unique fixed point of t . proof. (i) ⇒ (ii). define yn = t xn − xn, αn = supn{‖xm − xn‖ : m ≥ n}, and βn = supn{‖ym‖ : m ≥ n}. then {αn} and {βn} are nonincreasing nonnegative sequences. hence lim αn = α ≥ 0 and, from the hypotheses, lim yn = 0 and {yn} is bounded. assume that α > 0. from (3), with m ≥ n, ‖xm − xn‖ ≤ ‖t xm − ym − (t xn − yn)‖ ≤ ‖t xm − t xn‖ + ‖ym − yn‖ ≤ g(max{‖xm − xn‖, ‖ym‖, ‖yn‖, ‖xm − t xn‖, ‖xn − t xm‖}) + 2βn ≤ g(max{αn, βn, βn, αn + βn, αn + βn}) + 2βn. cubo 10, 4 (2008) a fixed point theorem for certain operators 69 thus, αn ≤ g(αn +βn)+2βn. taking the limit as n → ∞ yields α ≤ g(α) < α, a contradiction. therefore α = 0 and {xn} is cauchy, hence convergent to some point p in x. since t xn − xn → 0 and xn → p, it follows that t xn → p. again using (3), ‖t p − t xn‖ ≤ g(max{‖p − xn‖, ‖p − t p‖, ‖yn‖, ‖p − t xn‖, ‖xn − t p‖}). taking the limit as n → ∞ yields ‖t p − p‖ ≤ g(max{0, ‖p − t p‖, 0, 0, ‖p − t p‖}) = g(‖p − t p‖), which implies that ‖p − t p‖ = 0, or t p = p. to prove uniqueness, suppose that q is also a fixed point of t . then, from (3), ‖p − q‖ = ‖t p − t q‖ ≤ g(max{‖p − q‖, 0, 0, ‖p − q‖, ‖q − p‖}) = g(‖p − q‖), which implies that p = q. (ii) ⇒ (i) using (3), ‖t xn − xn‖ = ‖t xn − t p + t p − xn‖ ≤ ‖t xn − t p‖ + ‖p − xn‖ ≤ g(max{‖xn − p‖, ‖xn − t xn‖, ‖p − t p‖, ‖xn − t p‖, ‖p − t xn‖}) + ‖p − xn‖. taking the lim sup of both sides, since {yn} is bounded, one obtains, with the identification γ = lim sup ‖yn‖, γ ≤ g(max{0, γ, 0, 0, lim sup ‖p − t xn‖}) + 0. but lim sup ‖p − t xn‖ ≤ lim sup(‖p − xn‖ + ‖xn − t xn‖) = γ. therefore we have γ ≤ g(γ), which implies that γ = 0. the special case of (3) with g(t) := kt for some 0 ≤ k < 1, and x a metric space is that of ćirić [2], which was shown in [7] to be one of the most general contractive definitions for which a unique fixed point exists. in order to prove that a map satisfying (3) has a fixed point, it would be necessary to show that the orbit of some x ∈ x is bounded, which cannot be implied from (3). however, the following is true. theorem 2. let x be a complete metric space, t a selfmap of x satisfying d(t x, t y) ≤ g(m (x, y)), for each x, y ∈ x, (4) 70 b.e. rhoades cubo 10, 4 (2008) where m (x, y) = max{d(x, y), d(x, t x), d(y, t y), d(x, t y), d(y, t x)}. if there exists a point x0 ∈ x with bounded orbit, then t has a unique fixed point in x. proof. for any n ∈ n, o(x, n) := {x, t x, t 2x, . . . , t nx}, and δ(a) denotes the diameter of a set a. let m, n ∈ n, n < m. then, from (4), with x = x0, d(t nx, t mx) = d(t (t n−1x), t (t m−1x)) ≤ g(max{d(t n−1x, t m−1x), d(t n−1x, t nx), d(t m−1x, t mx), d(t n−1 x, t m x), d(t m−1 x, t n x)} ≤ g(δ[o(t n−1x, n − m + 1)]) ≤ g(g(δ[o(t n−2x, n − m + 2)]) · · · ≤ gn(δ[o(x, m)]). (5) it is well known that the hypotheses on g imply that lim gn(t) = 0 for each t ≥ 0. since the orbit of x = x0 is bounded, (5) implies that {t nx} is cauchy, hence convergent to a point p ∈ x. suppose that p 6= t p. then, from (4), d(p, t p) ≤ d(p, t n+1x) + d(t n+1x, t p) ≤ d(p, t n+1x) + g(max{d(t nx, p), d(t nx, t n+1x), d(p, t p), d(t nx, p), d(p, t n+1x)}. taking the limit of both sides of the above inequality as n → ∞ yields d(p, t p) ≤ g(d(p, t p)) < d(p, t p), a contradiction, and p = t p. suppose that p and q are fixed points of t , with p 6= q. then, using (4), d(p, q) = d(t p, t q) ≤ g(max({d(p, q), 0, 0, d(p, q), d(q, p)} = g(d(p, q)) < d(p, q), a contadiction. therefore p = q. if t is continuous, then, even with x unbounded, theorem 2 is a special case of theorem 3.3 of [4] cubo 10, 4 (2008) a fixed point theorem for certain operators 71 if one replaces m (x, y) with m(x, y) := max{d(x, y), d(x, t x), d(y, t y), [d(x, t y) + d(y, t x)]/2}, in theorem 2, then theorem 2 is true without the boundedness assumption. see, e.g., theorem 2.2 of [1]. most of the recent papers on fixed ont theory, which do not involve fixed point iterations, deal with four maps. for a survey of these results the reader may wish to consult [6] and the references therein. let f (t ) denote the fixed point set of a mapping t . in [5] it was conjectured that f (t n) = f (t ) for every map t which satisfies a contractive condition that does not include nonexpansive maps. that conjecture was verified in [5] for many such maps. we shall now show that the same is true for maps satisfying (4). theorem 3. let x be a metric space, t a selfmap of x satisfying (4) with f (t ) 6= ∅. then f (t n) = f (t ) for every integer n ≥ 1. proof. since f (t ) 6= ∅. f (t n) 6= ∅. clearly f (t ) ⊆ f (t n). suppose that p ∈ f (t n), for some positive integer n. we shall assume that n > 1, since the case for n = 1 is trivial. let i, j be integers, 0 ≤ i < j ≤ n. then, using (4), d(t ip, t jp) ≤ g(m (t i−1p, t j−1p)) ≤ g(δ[(o(p, n)]). suppose that δ[(o(p, n)] > 0. then the above inequality implies that δ[(o(p, n)] ≤ g(δ[(o(p, n)]) < δ[(o(p, n)], a contradiction. therefore δ[(o(p, n)] = 0, and p ∈ f (t ). received: february 2008. revised: march 2008. references [1] r.p. agarwal, donal o’regan, and m. sambandham, random and deterministic fixed point theory for generalized contractive maps, applicable analysis 83 (2004), 711–725. [2] lj. b. ćirić, a generalization of banach’s contraction principle, proc. amer. math. soc. 45 (1974), 27–273. [3] a. constantin, on the approximation of fixed points of operators, bull. calcutta math. soc. 86 (1994), 323–326. 72 b.e. rhoades cubo 10, 4 (2008) [4] j. jachymski, on common fixed point theorems for some families of maps, int. j. pure & appl. math. 25 (1994), 925–937. [5] g.s. jeong and b.e. rhoades, maps for which f (t ) = f (t n), fixed point theory and appl. 6 (2003), 87–131. [6] donal o’regan, naseer shahzad, and ravi p. agarwal, common fixed point theory for compatible maps, nonlinear analysis forum 8 (2003), 179–22. [7] b.e. rhoades, a comparison of various definitions of contractive mappings, trans. amer. math. soc. 226 (1977), 257–290. n5-fixedpointtheorembcms cubo a mathematical journal vol.11, no¯ 05, (129–172). december 2009 scattering theory on geometrically finite quotients with rational cusps colin guillarmou département de mathématiques j.a. dieudonné, umr cnrs no. 6621, université de nice sophia-antipolis parc valrose, 06108, nice, france email : cguillar@math.unice.fr abstract we study eisenstein functions and scattering operator on geometrically finite hyperbolic manifolds with infinite volume and ‘rational’ non-maximal rank cusps. for both we prove the meromorphic extension and we show that the scattering operator belongs to a certain class of pseudo-differential operators on the conformal infinity which is a manifold with fibred boundaries. resumen estudiamos funciones de eisenstein y el operador de dispersión sobre variedades hiperbólicas geometricamente finitas con volumen infinito y puntas de rango no maximos racionales. para ambos probamos las extensiones meromorficas y mostramos que el operador de dispersión pertence a cierta clase de operadores pseudo-diferenciales sobre la variedade conforme infinita con fibrados en la frontera. key words and phrases: scattering theory, geometrically finite quotients. math. subj. class.: 58j50, 35p25. 130 colin guillarmou cubo 11, 5 (2009) 1 introduction and results the purpose of this work is to study the eisenstein functions and scattering operator on a class of geometrically finite hyperbolic quotients γ\hn+1 with non-maximal rank cusps. such problems involving spectral and scattering theory on geometrically finite hyperbolic quotients have been studied probably since selberg and lead to many important results. however, most of the results known are obtained when the group has no parabolic subgroups of non-maximal rank, in other words when the quotient x = γ\hn+1 of hyperbolic space hn+1 has no cusps of non-maximal rank. as far as we know, the only results concerning meromorphic extension of the resolvent or scattering operator for this cases were due, until recently, to froese-hislop-perry [3] in dimension 3. however, in a preprint, bunke and olbrich [1] deal with the meromorphic extension of the scattering operator in all generality using a very different approach; in particular they do not study the (pseudo-differential) structure of this operator. we refer the reader to the introduction of [8] for a more detailed review of works on meromorphic extension of the resolvent for the laplacian through the essential spectrum, resonances (i.e. the poles of this extension), meromorphic continuation of eisenstein functions and scattering operator for geometrically finite hyperbolic manifolds, though we do not claim to be complete about references therein. we consider an infinite volume hyperbolic quotient x := γ\hn+1 where γ is a discrete group of isometries of hn+1 which admits a fundamental domain with finitely many sides, x is said geometrically finite, and such that each rank k parabolic subgroup of γ fixing a point p ∈ sn is generated by k independent translations in the horospheres centered at p. we shall say that the cusps are rational cusps. for exemple, this last condition is always satisfied in dimension n + 1 = 3. in general, a rank k parabolic subgroup γp fixing a point p ∈ s n gives rise to a model manifold γp\h n+1 which is isometric to r+ × m where m is a flat bundle with basis a flat compact manifold and with fibers rn−k; then if the holonomy representation of this bundle has finite image in o(n−k), there is a finite cover which satisfies our assumptions, in which case the resolvent, scattering operator and eisenstein functions are obtained as a finite sum on the cover. similarly, elliptic elements of γ can also be excluded by passing to a finite cover, x is then a smooth manifold, and since the presence of maximal-rank cusps do not add difficulties, we will avoid them for simplicity of exposition. the laplacian on such manifolds have been studied by froese-hislop-perry [3] in dimension 3 and by perry [23] in higher dimension. the manifold x equipped with the hyperbolic metric is complete and the spectrum of the laplacian ∆x splits into continuous spectrum [ n 2 4 ,∞) and a finite number of l2 eigenvalues included in (0, n 2 4 ) which form the point spectrum σpp(∆x ) (see lax-phillips [14]). in [8] we proved that the modified resolvent r(λ) := (∆x −λ(n−λ)) −1 extends from {ℜ(λ) > n 2 } to c meromorphically with poles of finite multiplicity (i.e. the rank of the polar part in the laurent expansion at each pole is finite) from l2comp(x) to l 2 loc(x), these cubo 11, 5 (2009) scattering theory on geometrically finite quotients 131 poles are called resonances. in the present work, we define a poisson operator, eisenstein functions, a scattering operator and we show that they extend meromorphically to c. to explain the main theorems, we recall briefly the structure at infinity of the manifold x but in any case, we refer the reader to section 2 of mazzeo-phillips [19] for a comprehensive description of geometrically finite quotients γ\hn+1 (see also [2, 23, 8]). the first approach is to see x as the interior of a smooth compact manifold with boundary x̄. if ρ is a boundary defining function of the boundary ∂x̄ and if g is the hyperbolic metric on x, then ρ2g extends as a smooth non-negative tensor on x̄ which is positive definite outside some submanifolds of the boundary ∂x̄ where it becomes degenerate. each one of these submanifolds arises from a cusp point of x, i.e. a fixed point at infinity of hn+1 for a parabolic subgroup of γ, and is diffeomorphic to a k-dimensional torus t k if the parabolic subgroup has rank k. if we note c the union of these submanifolds, b = ∂x̄ \ c is a non-compact manifold which can be thought as the infinity of x; actually b = γ\ω where ω ⊂ sn is the domain of discontinuity of γ. after a real blow-up of these submanifolds in x̄, we obtain a manifold x̄c with corners of codimension 2 which is the compactification of x defined by mazzeo-phillips [19] in the general case. the topological boundary of x̄c splits into two kind of smooth hypersurfaces with boundaries, the regular ones whose union is a compactification b̄ of b and the cusp ones which are diffeomorphic to sn−k+ ×t k, sn−k+ being an n−k dimensional half-sphere with boundary. it turns out that b has ends diffeomorphic to (rn−ky \ {|y| < 1}) × t k, each end arising from a rank-k parabolic subgroup of γ fixing a point at infinity of hn+1. the compactification b̄ of b corresponds to the radial compactification in the y variable in each end thus b̄ is a fibred boundary manifold in the sense of mazzeo-melrose [18], the fibrations being the projections sn−k−1 ×t k → sn−k−1. when equipped with the metric h0 := ρ 2g|b, (b,h0) is conformal to an ‘exact φ-type metric’ near its infinity as defined in [18], the conformal factor decreasing enough to make the volume of b finite the vanishing rate is even stronger than the fibred cusp metrics (see figure 1 for illustration). we construct poisson and scattering operators p(λ),s(λ) by solving a poisson problem in a way similar to that introduced on euclidean manifolds by melrose and on many other settings by various authors (see [21] for review). however, in view of the sensitive structure of the metric near the cusps c, it appears that p(λ),s(λ) do not act naturally on c∞(∂x̄) but on subspaces related to this structure. we then define the subalgebra c∞ acc (x̄) of c∞(x̄) of functions which are asymptotically constant in the cusps, these are the f ∈ c∞(x̄) such that z(f|c) = 0, z((x1 . . .xnf)|c) = 0 for all smooth vector fields x1, . . . ,xn on x̄ (∀n ∈ n) and all smooth vector fields z on c. in other words, these are the functions whose restrictions at the cusp submanifolds are locally constant and similarly for all derivatives. it is actually possible to find a boundary defining function ρ in this 132 colin guillarmou cubo 11, 5 (2009) subalgebra. then the volume form dvolg of g can be expressed by ρ −n−1r2cµx̄ for a function rc which is smooth positive in x̄ \ c with r2c ∈ c ∞ acc (x̄) vanishing at order 2k at each k-dimensional component of c and where µx̄ is a smooth volume density on x̄. the functions rc and ρ are not uniquely determined but we show that the set r−1c c ∞ acc (x̄) is independent of the choice of r2c,ρ in c∞ acc (x̄) (but it certainly depends on the metric). then we define c∞ acc (∂x̄) and r−1c c ∞ acc (∂x̄) by restriction of c∞ acc (x̄) and r−1c c ∞ acc (x̄) at ∂x̄ and b = ∂x̄ \c (here we use the same notation for rc and its restriction rc|∂x̄ ). for any boundary defining function ρ ∈ c ∞ acc (x̄), one can define the poisson operator p(λ) by showing that if ℜ(λ) ≥ n 2 and λ /∈ n 2 + n, then for all f ∈ r−1c c ∞ acc (∂x̄) there exists a unique solution p(λ)f of the following poisson problem    (∆x −λ(n−λ))p(λ)f = 0 p(λ)f = ρn−λf(λ,f) + ρλg(λ,f) f(λ,f),g(λ,f) ∈ r−1c c ∞ acc (x̄) f(λ,f)|ρ=0 = f . the construction of the solution is a consequence of an indicial equation for ∆x and the following precise mapping property of the meromorphically extended resolvent r(λ) : ċ∞(x̄) → ρλr−1c c ∞ acc (x̄). where ċ∞(x̄) is the set of functions in c∞(x̄) vanishing at all order at ∂x̄. next we analyze eisenstein functions. the metric h0 induces an l 2 (b) hilbert space on b and we prove theorem 1.1. if r(λ; w; w′) denotes the schwartz kernel of the extended resolvent, then the eisenstein function e(λ; b; w′) := lim w→b [ρ(w)−λr(λ; w; w′)], b ∈ b,w′ ∈ x is a smooth function on b×x if λ is not a resonance. there exists c > 1 such that, for all n > 0, e(λ; ., .) is the schwartz kernel of a meromorphic operator e(λ) : ρnl2(x) → l2(b) in ℜ(λ) > n 2 − c−1n with poles of finite multiplicity, satisfying p(λ) = (2λ − n)te(λ) on r−1c c ∞ acc (∂x̄). except possibly at {λ;ℜ(λ) < n 2 ,λ(n − λ) ∈ σpp(∆x )}, the set of poles of e(λ) coincides with the set of resonances. using the asymptotic expression of p(λ)f, the scattering operator is then defined (with the same notations) by s(λ) : { r−1c c ∞ acc (∂x̄) → r−1c c ∞ acc (∂x̄) f → f(λ,f)|ρ=0 . for ℜ(λ) = n 2 , s(λ) can be extended to l2(b) as a unitary operator and it gives, as usual in scattering theory, a parametrization of the absolutely continuous spectrum of ∆x . then, we prove cubo 11, 5 (2009) scattering theory on geometrically finite quotients 133 the following result which is expressed in more details in theorem 6.5, lemma 6.1, corollary 6.3 and proposition 7.1: theorem 1.2. the scattering operator s(λ) extends meromorphically to c as a family of pseudodifferential operators in the full φ-calculus on the manifold with fibred boundary b̄ in the sense of mazzeo-melrose [18]. in {ℜ(λ) ≤ n 2 ,λ(n−λ) /∈ σpp(∆x )}, λ0 is a pole of s(λ) if and only if λ0 is a resonance and it has finite multiplicity. in {ℜ(λ) > n 2 }, s(λ) has only first order poles whose residue is resλ0s(λ) = { − (−1)j+12−2j j!(j−1)! pj + πλ0 if λ0 = n 2 + j,j ∈ n πλ0 if λ0 /∈ n 2 + n where pj is the j-th gjms conformal laplacian of [6] on (b,h0) and πλ0 is an operator with rank dim kerl2 (∆x −λ0(n−λ0)). note that the gjms conformal laplacians pj in [6] are well-defined for all j if n ≥ 3 (resp. for j ≤ 1 if n = 2) if the manifold is locally conformally flat (it is actually done in the compact setting but they can be extended for non-compact manifolds by using the same local expression in the curvature tensor), which is the case for b. the general case of irrational cusps is more technically involved and it is not clear if such precise results can be obtained, at least the meromorphic extension of the resolvent is carried out in a forthcoming paper. it is also important to add that this analysis could be used to study the divisors of selberg’s zeta function as patterson-perry [22] did for convex co-compact hyperbolic manifolds. the paper is organized as follows: we first introduce in section 2 the geometric setting, discuss the compactification x̄ of the manifold x and analyze its infinity b; then in section 3 we define the class of pseudo-differential operators on b which contains the scattering operator and in section 4 we study the mapping properties and the structure of the resolvent for the laplacian. in section 5, we construct the poisson operator and eisenstein functions using section 4 and in section 6 we define and describe the scattering operator. to conclude we investigate the relation between the conformal geometry of b and the scattering theory on x. along the paper, we will identify operators with their schwartz kernel and we consider operators acting on functions for simplicity of exposition though the correct approach would be to use half-densities. consequently the kernels of pseudo-differential operators have to be understood as tensorized by appropriate half-densities. aknowledgements: we thank rafe mazzeo, robin graham and jared wunsch for helpful discussions. this work was written at purdue university in 2005 but we are also grateful to the 134 colin guillarmou cubo 11, 5 (2009) mathematics department of nantes where it was completed. research was partially supported by nsf grant dms0500788. 2 geometry of the manifold 2.1 assumptions on the group we describe here with more details the assumptions about the cusps discussed roughly in the introduction; we strongly use section 2 of mazzeo-phillips [19]. let γ a discrete subgroup of orientation preserving isometries of the hyperbolic space hn+1. recall that γ acts also on the natural compactification h̄n+1 = {m ∈ rn+1; ||m|| ≤ 1} of hn+1 and on its boundary sn; an element γ is called hyperbolic if it fixes exactly two points on sn and no point in hn+1, parabolic if it fixes one point on sn and no point in hn+1, then γ is elliptic if it fixes at least a point of hn+1. if γ contains elliptic elements (other than the identity), there exists a subgroup γ0 of finite index of γ without elliptic elements, thus x is finitely covered by γ0\h n+1, the latter being a smooth manifold. since we study resolvent of the laplacian and other related objects, we can always pass to a finite cover without difficulties: objects on x can indeed be obtained by summing on a finite set objects on the finite cover. thus we exclude elliptic elements in γ. we suppose that γ is geometrically finite, which means here that it admits a fundamental domain f with finitely many sides. each fixed point p ∈ sn of a parabolic element of γ is called a cusp point, and for each cusp point p, let γp be the subgoup of γ fixing p. actually γp contains only parabolic elements and it can be shown that there is a γp invariant neighbourhood up of p such that γ\(f ∩up) is isometric to a neighbourhood of p in γp\(f ∩ up). the subgroup γp has a maximal free abelian subgoup γa with rank k, the rank of the cusp p is defined to be the integer k. we suppose that k ≤ n− 1 for each p since this case is well known in term of scattering theory. using now conjugation, it suffices to look at the case where p = ∞ in the upper half model hn+1 = r+ × rn. section 2 of [19] (the arguments come from bieberbach’s analysis of discrete groups of isometries of euclidean space) shows that there is an affine subspace rk ⊂ rn globally preserved by γ∞ on which γa acts as a group of k translations. this allows to see that every γ ∈ γ∞ acts as γ(y,z) = (ry,az + b) on rn−ky ⊕ r k z for some a ∈ o(k),r ∈ o(n−k) and b ∈ rk; elements in γa have a = id. there is a flat compact manifold n = γ∞\r k such that γ∞\r n is a flat vector bundle with basis n and t k := γa\r k such that γa\r n is a flat bundle over t k. assuming that the holonomy representation of these bundles γ → o(n−k) has finite image implies that the elements r decompose into rotations with rational angles pπ/q for some p,q ∈ n, then there is a finite cover of this bundle which is t k × rn−k, t k being a flat torus. thus, as we mentionned before, it suffices somehow to study the case where each rotation r is the identity to get a good description of the analytic objects conidered in the paper. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 135 2.2 neighbourhoods of infinity, models. from previous discussions and assumptions on the cusps and using [2, 23, 8] we obtain a covering of the manifold x by model charts. there exists a compact k of x such that x \k is covered by a finite number of charts isometric to either a regular neighbourhood (mr,gr) or a rank-k cusp neighbourhood (mk,gk) where mr := {(x,y) ∈ (0,∞) × r n ; x2 + |y|2 < 1,}, gr = x −2 (dx2 + dy2), mk := {(x,y,z) ∈ (0,∞) × r n−k ×t k; x2 + |y|2 > 1}, gk = x −2 (dx2 + dy2 + dz2) for k = 1, . . . ,n− 1 with (t k,dz2) a k-dimensional flat torus. note that we could allow maximal rank cusps as in [8] without difficulties but since these cases are well-known, we restrict ourselves to the non-maximal rank cusps cases for simplicity of exposition. to avoid too many indices in the exposition, we will assume for simplicity that the manifold has only one neighbourhood of each type, it will be clear from the analysis which will follow that it does not change anything in the proofs; we then note ir, (ik)k the corresponding chart isometries. one can also choose the covering such that i−1k (mk) ∩ i −1 j (mj) = ∅ for k 6= j, possibly by adding regular neighbourhoods. the model mk can be considered as a subset of the quotient xk = γk\h n+1 of hn+1 by a rank-k parabolic subgroup γk of γ which fixes a single point at infinity of h n+1. indeed, modulo conjugation by an isometry, one can suppose that the fixed point is the point at infinity of hn+1 in the half-space model (0,∞) × rn. the group γk is generated by k independent translations acting on rn, therefore it is the image of the lattice zk by a map ak ∈ glk(r) and the flat torus t k := γk\r k is well defined. then xk is isometric to r + x × r n−k y ×t k z equipped with the metric gk = dx2 + dy2 + dz2 x2 dz2 being the flat metric on a k-dimensional torus t k. therefore mk is the subset of xk with x2 + |y|2 > 1. as a matter of fact it will be often useful to consider r+ × rn−k as the n−k + 1dimensional hyperbolic space hn−k+1. hence xk can be compactified into the compact manifold with boundary x̄k = h̄ n−k+1 ×t k where h̄n−k+1 is the ball {|w| ≤ 1} in rn−k+1. then ρk(x,y,z) := x |y|2 + x2 + 1 = (2 cosh(dhn−k+1 (x,y; 1, 0))) −1 is a natural boundary defining function in x̄k (∂x̄k = {ρk = 0} and dρk 6= 0 on ∂x̄k). let us define the new coordinates t := x x2 + |y|2 , u := −y x2 + |y|2 (2.1) which induce an isometry from (mk,gk) to {(t,u,z) ∈ (0,∞) × rn−k ×t k; t2 + |u|2 < 1} 136 colin guillarmou cubo 11, 5 (2009) equipped with the metric dt2 + du2 + (t2 + |u|2)2dz2 t2 (2.2) and ρk(t,u) = ρk(x,y). these coordinates can be thought as compactification coordinates for mk, since t and u extend smoothly to x̄k\{x = y = 0}. the infinity of x in the chart mk is then given by {ρk = 0} or equivalently {t = 0}. also we will call cusp submanifold the submanifold {t = u = 0} of x̄k it will be denoted by ck and we remark that ck ≃∞×t k ≃ t k in x̄k where ∞ is the point at infinity in the half-space model of hn−k+1. we also have mk = {w ∈ xk; t(w) 2 + |u(w)|2 < 1} which is a subset of x̄k and we will denote m̄k := {w ∈ x̄k; t 2 (w) + |u(w)|2 < 1}. at last we define the manifold yk := r n−k ×t k which can be viewed as (x̄k \ ck) ∩{x = 0}. the model mr is simpler and can be considered as a subset of h n+1. we define as before m̄r := {(x,y) ∈ [0,∞) × r n ; x2 + |y|2 < 1}. there exist some smooth functions χ,χr,χ1, . . . ,χn−1 on respectively x,mr,m1, . . . ,mn−1 which, through the isometric charts ir,i1, . . . ,in, satisfy i∗r χ r + n−1∑ k=1 i∗kχ k + χ = 1 with χ having compact support in x. note that it is possible to choose χk which does not depend on the variable z ∈ t k. for what follows we will consider mk,mr,m̄k,m̄r as neighbourhoods in x̄ instead of using the notations i−1k (mk),i −1 r (mr)... 2.3 compactification, volume densities. using the previous discussion about the compactification of the cusp neighbourhoods, one obtains an obvious compactification of x as a smooth compact manifold with boundary x̄. moreover, we can choose a boundary defining function ρ which is equal to the function t in each neighbourhood m̄k. the boundary ∂x̄ is covered by some charts b1, . . . ,bn−1,br induced by m1, . . . ,mn−1,mr by taking bk := m̄k ∩∂x̄ ≃{(u,z) ∈ r n−k ×t k; |u|2 < 1} br := m̄r ∩∂x̄ ≃{y ∈ r n ; |y|2 < 1}. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 137 from the discussion above, we see that the metric on x can be expressed by g = h ρ2 with h a smooth non-negative symmetric 2-tensor on x̄ which degenerates at the cusps submanifolds (ck)k=1,...,n−1. let us define c := (∪kck) ⊂ ∂x̄ ⊂ x̄, and b := ∂x̄ \ c, then the restriction h0 := h|b = (ρ 2g)|b (2.3) is a smooth metric on the non-compact manifold b. we will also need to use functions representing the distance to the cusps submanifolds as follows: for k = 1, . . . ,n − 1, let rck be a continuous non-negative function in x̄, smooth and positive in x̄ \ ck which satisfies ik∗(rck ) = √ t2 + |u|2 in m̄k and is equal to 1 in mj when j 6= k. then we define the functions rc := n−1∏ k=1 rck, rc := n−1∏ k=1 (rck ) k (2.4) on x̄ and we will also denote by rck , rc and rc their restriction to ∂x̄. it can easily be checked that b equipped with the metric h0 of (2.3) has a volume density dvolh0 which is of the form dvolh0 = r 2 cµ∂x̄ (2.5) with µ∂x̄ a smooth non-vanishing density (volume density) on ∂x̄. similarly the volume density dvolg on x can be expressed by dvolg = ρ −n−1r2cµx̄ (2.6) for a smooth volume density µx̄ on x̄. in what follows, we will write l 2 (x) and l2(b) for the hilbert spaces of square integrable functions on x and b with respect to the volume densities dvolg and dvolh0 . 2.4 class of functions. for a compact manifold m̄ with boundary ∂m̄, we denote by ċ∞(m̄) the set of smooth functions on m̄ which vanish at all orders at ∂m̄. its topological dual is the set of extendible distribution on m̄, denoted c−∞(m̄) (note that a correct definition would include density bundles). there will be a special set of smooth functions on x̄,∂x̄ which will play an important role for what follows, these are the functions which are “asymptotically constant in the cusp variables”. to give a precise definition we begin by introducing the sets c(tx̄), c(t∂x̄) and c(tc) of smooth vector fields on x̄,∂x̄,c. then we set c∞ acc (x̄) := {f ∈ c∞(x̄);∀x1, . . . ,xn ∈ c(tx̄),∀z ∈ c(tc),z(f|c) = 0,z(x1 . . .xnf|c) = 0} 138 colin guillarmou cubo 11, 5 (2009) and c∞ acc (∂x̄),c∞ acc (x̄k),c ∞ acc (∂x̄k) are defined similarly by replacing x̄ by ∂x̄,x̄k,∂x̄k. these functions are constant on each cusp submanifold ck and their derivatives too. in local coordinates (t,u,z) near the cusp ck = {t = u = 0}, one can check by a taylor expansion at (0, 0,z) ∈ ck and borel lemma that a function f ∈ c∞ acc (x̄) can be decomposed locally as a sum f(t,u,z) = f0(t,u) + o((t 2 + |u|2)∞) = f0(t,u) + o(r ∞ c ) (2.7) for some f0 smooth. we remark the following properties, the proofs of which are straightforward: lemma 2.1. the set c∞ acc (x̄) is a subalgebra of c∞(x̄) which is stable under the action of c(tx̄), and stable by composition with smooth real functions on r. observe also that r2c and r 2 c defined by (2.4) are in c ∞ acc (x̄). actually this implies that if ρ̂ ∈ c∞ acc (x̄) is a boundary defining function of ∂x̄ and r̂2c ∈ c ∞ acc (x̄) is a non-negative function vanihing at order 2k at each ck such that dvolg = ρ̂ −n−1r̂2cµ̂x̄ for a smooth volume form on x̄, then ρ̂ = f1ρ, r̂ 2 c = f2r 2 c, µ̂x̄ = f3µx̄ for some functions f1,f2 ∈ c ∞ acc (x̄) and f3 ∈ c ∞ (x̄) satisfying f−n−11 f2f3 = 1 and f1 > 0, f3 > 0. then necessarily f3 ∈ c ∞ acc (x̄) and f2 > 0 which shows that r −1 c c ∞ acc (x̄) = r̂−1c c ∞ acc (x̄) and this space does not depend on the choices of ρ,r2c in c ∞ acc (x̄). actually the map f → f|dvolg| 1 2 naturally identifies r−1c c ∞ (x̄) with the space of smooth half-densities c∞(x̄, γ 1 2 0 ) defined in the 0-calculus of mazzeo-melrose [17] (depending only on the c∞ structure of x̄) and the space r−1c c ∞ acc (x̄) could then be considered as a subspace of c∞(x̄, γ 1 2 0 ) (depending on the metric) if we worked with densities. we also define the set of smooth functions on x̄k (resp. x̄) vanishing at all order at the cusps ċ∞c (x̄) := {f ∈ c ∞ (x̄);∀x1, . . . ,xn ∈ c(tx̄),f|c = 0, (x1 . . .xnf)|c = 0} and ċ∞c (∂x̄), ċ ∞ c (∂x̄k), ċ ∞ c (∂x̄k) similarly. remark that there is a natural identification between ċ∞c (∂x̄) and ċ ∞ (b̄) if b̄ is defined as the real blow-up of ∂x̄ around c. by similar arguments, the spaces c∞ acc (∂x̄), ċ∞c (∂x̄), r −1 c c ∞ acc (∂x̄) can be defined (here we note again rc instead of rc|b) an they coincide with the restriction of c ∞ acc (x̄), ċ∞c (x̄), and r −1 c c ∞ acc (x̄) at b = ∂x̄ \ c. to conclude this part, remark the following inclusions ċ∞(x̄) ⊂ ċ∞c (x̄) ⊂ c ∞ acc (x̄). and the same for their restriction at b. 2.5 model form for the metric. to use the same ideas than for asymptotically hyperbolic manifolds, we need to choose boundary defining functions of ∂x̄ in x̄ which induce product decompositions of the metric near infinity. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 139 the different choices of boundary defining functions induce a conformal class of smooth tensors on ∂x̄ which are metrics on b, this is the conformal class [h0] of h0 := ρ 2g|∂x̄ . however, in view of the presence of the cusps, we need to consider the following smaller class of conformal metrics on b [h0]acc := {fh0; f ∈ c ∞ acc (∂x̄),f > 0}. lemma 2.2. for all ĥ0 ∈ [h0]acc, there exists a boundary defining function ρ̂ ∈ c ∞ acc (x̄) of ∂x̄ in x̄ such that |dρ̂|ρ̂2g − 1 ∈ ċ ∞ (x̄) in a collar neighbourhood of ∂x̄ and ρ̂2g|b = ĥ0. moreover, ρ̂ is uniquely determined modulo ċ∞(x̄) by ĥ0. proof : for ĥ0 ∈ [h0], the construction of a boundary defining function ρ̂ = ρe ω which satisfies |dρ̂|ρ̂2g = 1 and ρ̂ 2g|b = h0 is equivalent to solving the pde 2(∇ρ2gρ)(ω) + ρ|dω| 2 ρ2g = 1 −|dρ|2ρ2g ρ (2.8) with initial condition ω|∂x̄ = ω0 where ĥ0 = e 2ω0h0 (see [5, lem. 2.1]). the construction of a solution is possible in regular neighbourhoods m̄r and is unique since the equation is noncharacteristic there. in m̄k, we write the equation in coordinates and this gives 2∂tω + t ( (∂tω) 2 + |∂uω| 2 + (t2 + |u|2)−2|∂zω| 2 ) = 0 in view of the form of the metric (2.2) there (recall that ρ = t in m̄k). taking this equation at t = 0, we see that ∂tω|t=0 = 0 and by differentiating it n times with respect to t and setting t = 0 we see by induction that all the values ∂ j t ω|t=0 in {u 6= 0} are determined by ω|t=0 for j ≤ n + 1. in particular when j is odd this is 0 (see again [5] for a similar study). since w0 ∈ c ∞ acc (∂x̄), we can write it locally under the form (2.7) which shows by induction that ∂ j t ω|t=0 ∈ c ∞ acc (∂x̄); the essential arguments to use are that the singular term in the equation is killed by |∂zω| = o((t 2 +|u|2)∞) and the properties of c∞ acc (∂x̄) discussed previously. by using borel lemma, we can construct a smooth function ω in a neighbourhood of ∂x̄ in x with those derivatives, thus ω satisfies (2.8) modulo o(ρ∞) and this proves that there exists a function ρ̂ which satisfies the lemma, the uniqueness of its taylor expansion with respect to ρ at ∂x̄ is clear from the construction. � we will now use this function to obtain a certain model form of the metric near ∂x̄. using again the same arguments than [5, 9], it suffices to consider the collar neighbourhood [0,ǫ)s ×∂x̄ of ∂x̄ induced by the flow ϕs(m) of the gradient ∇ρ̂2gρ̂ with initial condition ϕ0(m) = m for m ∈ ∂x̄, that is the diffeomorphism ϕ : (s,m) → ϕs(m) from [0,ǫ) × ∂x̄ to its image. we consider the function ω constructed in the proof of previous lemma (thus ρ̂ = ρeω) and since ∂sρ̂(ϕs(m)) = 1 + o(ρ ∞ ) = 1 + o(s∞), we deduce ρ = se−ω + o(s∞). 140 colin guillarmou cubo 11, 5 (2009) now, we remark that the identity |∇ρ̂2gρ̂|ρ̂2g = 1 + o(s ∞ ) implies that s2g can be expressed by s2ϕ∗g = ds2 + ĥ(s) + o(s∞) in [0,ǫ) ×∂x̄ where ĥ(s) is a smooth family of tensors on ∂x̄ which are positive for s > 0, with ĥ(0) = ĥ0 positive on b. we have seen in the proof of last lemma that, in m̄k, ω is an even function of ρ = t, thus s is an odd function of t and t is an odd function of s. let (v,ζ) ∈ rn−k ×t k some coordinates on ∂x̄ near ck. we have ϕ0(v,ζ) = (v,ζ) and using the form (2.2) of g ∂sϕs(v,ζ) = ∇ρ̂2gρ̂ = e −ω (1 + t∂tω)∂t + te −ω∂uω.∂u + te−ω (t2 + |u|2)2 ∂zω.∂z then the function ϕ(s,v,ζ) = ϕs(v,ζ) can be locally written near ck (in coordinates (t,u,z)) ϕ(s,v,ζ) = ( t = se−ω + t1,u = v + su1,z = ζ + sz1 ) (2.9) t1 ∈ ċ ∞ (x̄), u1 ∈ c ∞ acc (x̄), z1 ∈ ċ ∞ c (x̄). using that ω is even in s and t odd in s, it is straightforward to verify that u,z are even in s. we deduce that locally dt = l1(s,v,ds,dv) + o(r ∞ c ), du = l2(s,v,ds,dv) + o(r ∞ c ), dz = dζ + o(r ∞ c ). (2.10) for some smooth tensors l1, l2, even in s. we want now to write the metric g in these coordinates (s,v,ζ). by looking at the expression (2.2) and using (2.9), (2.10) with the properties of c∞ acc (x̄) discussed in previous section, we obtain that ĥ(s) = h1(s,v,dv) + h2(s,v,z,dv,dζ) + e 2ωr4cdζ 2 + o(s∞) (2.11) where h1,h2 are smooth tensors, even in s, such that h2 = o(r ∞ c ). since ρ̂ − s = o(ρ̂ ∞ ), we can replace s by ρ̂ in (2.11) and we have the same expression for the metric. now in a regular neighbourhood mr, there exists coordinates (x,y) ∈ (0,ǫ)×r n such that g = x−2(dx2 +dy2), thus by writing ρ̂ = xeθ for some θ smooth, we have by mimicking last lemma that (from (2.8)) 2∂xθ + x((∂xθ) 2 + |∂yθ| 2 ) = o(x∞) with θ|x=0 = θ0 satisfying ĥ0 = e 2θ0dy2. exactly as before for mk, this gives that ρ̂ is odd in x, thus x is odd in s and y even in s, which easily implies that ĥ(s) has an even taylor expansion in s at s = 0. this discussion proves that there exists a collar neighbourhood (0,ǫ)ρ̂ ×∂x̄ of ∂x̄ in x̄ such that g = dρ̂2 + ĥ(ρ̂) ρ̂2 + o(ρ̂∞) (2.12) for a smooth family of symmetric tensors ĥ(ρ̂) on ∂x̄ with an even taylor expansion in ρ̂ at ρ̂ = 0, positive for ρ̂ > 0, ĥ(0) = ĥ0 being positive on b and with the local expression (2.11) near the cubo 11, 5 (2009) scattering theory on geometrically finite quotients 141 cusps ck. actually, the evenness of the metric in ρ̂ is a consequence of the constant curvature of x and is studied in details in [9] more generally for asymptotically hyperbolic manifolds. is is quite direct and similar to a result of graham [5] to check that for two functions ρ̂1, ρ̂2 satisfying lemma 2.2, then for all j ∈ n ∂ 2j ρ̂1 ρ̂2|∂x̄ = 0, ∂ 2j ρ̂2 ρ̂1|∂x̄ = 0 which will be useful to define renormalized volume in an invariant way. there is however a very special case of boundary defining function ρ̂ which can be chosen to put the metric into a simpler form. it is obtained by taking ρ̂ = t in the neighbourhood m̄k of the cusp ck and extending it to a neighbourhood of ∂x̄ so that it satisfies |dρ̂|ρ̂2g = 1 in this neighbourhood and ρ̂2g|∂x̄ = h0. to prove the existence of such an extension, it suffices to go back to the proof of lemma 2.2 and we see that this amounts to solve the pde (2.8) without the error term o(ρ∞) and with initial condition ω|∂x̄ = 0. since the equation is non-characteristic out of the cusp c, there exists a unique solution ω in some neighbourhood {ρ < ǫ,δ < rc} (for some δ,ǫ > 0) of the boundary ∂x̄ avoiding the cusp c, and it is clear that ω = 0 satisfies the equation in m̄k. for what follows, we will often work with this boundary defining functions ρ̂ and by convention we will note it ρ, forgetting the previous choice of function ρ. then we have in some collar neighbourhood (0,ǫ)ρ ×∂x̄ of ∂x̄ g = dρ2 + h(ρ) ρ2 (2.13) for some smooth family of symmetric tensors h(ρ) on ∂x̄, depending smoothly on ρ, positive for ρ > 0, with h(0) = h0 positive on b and satisfying h(ρ) = du2 + (ρ2 + |u|2)2dz2 in each m̄k. 2.6 geometry of b to study the scattering operator and to define the class of pseudo-differential operators which contains it, we can consider the manifold b as the union of a compact manifold er (covered by the charts br) and n− 1 ends e1, . . . , ek with ek diffeomorphic to {(y,z) ∈ rn−k ×t k; |y| > 1}⊂ yk = r n−k ×t k. for simplicity, we will consider ek as this last subset of yk. by using the radial compactification in the y variable in each end ek we see that the manifold b compactifies in a smooth compact manifold with boundary b̄, the boundary ∂b̄ being a disjoint union on k = 1, . . . ,n−1 of products 142 colin guillarmou cubo 11, 5 (2009) ∂ek := s n−k−1 ×t k. a boundary defining function of ∂ek is given by v = rck = rc = |y| −1 and rc is a boundary defining function of ∂b̄. note that b̄ 6= ∂x̄ but b̄ is actually the blow-up of ∂x̄ around the cusps submanifolds c1, . . . ,cn−1. the structure of the compactified manifold b̄ near ∂ek is [0, 1)v ×∂ek and ∂ek fibers by the projection φk : s n−k−1 ×tk → s n−k−1. (2.14) the metric h0 on b is not exactly a fibred cusp metric since too much decreasing at infinity h0 = dv 2 + v2dω2 + v4dz2. for following purposes, it is also quite natural to consider b with the metric h̃0 := r −4 c h0 conformal to h0 since this is the flat metric dy 2 + dz2 on each end ek. note that h̃0 in (0, 1)v ×s n−k−1 ω ×t k z is h̃0 = dv2 v4 + dω2 v2 + dz2 which is an “exact φ-metric” in the sense of mazzeo-melrose [18]. the volume induced by the metric h0 on b is finite whereas the volume of b with the metric h̃0 is not finite. 3 pseudo-differential operators at infinity there is a natural way to define pseudo-differential operators on b using the euclidean structure of each end ek. recall first from schwartz theorem that for any continuous linear operator a : ċ∞(b̄) → c−∞(b̄) there exists a unique extendible distribution a ∈ c∞(b̄×b̄) (we dropped the density factor for simplicty), called schwartz kernel, such that 〈aφ,ψ〉 = 〈a,ψ ⊗φ〉, ∀φ,ψ ∈ ċ∞(b̄). thus we will identify schwartz kernel with its associated operator. we can define the space ψm,l(b) of pseudo-differential operators of order (m,l) ∈ r2 as the set of linear operators a : ċ∞(b̄) → c−∞(b̄) (3.1) such that in each compact coordinate patch on b (those are the br of previous section), a has a distributional schwartz kernel of the type a(w; w′) = ∫ rn eiξ.(w−w ′)a(w,ξ)dξ (3.2) with a(w,ξ) a symbol in the coordinate patch, i.e. a(w,ξ) is smooth and |∂αw∂ β ξ a(w,ξ)| ≤ cα,β (1 + |ξ|) m−|β|, whereas on the end ek with coordinates w = (y,z) ∈ r n−k ×t k, the distributional kernel of a is of the form (3.2) but with a(w; ξ) smooth and satisfying |∂αy ∂ β z ∂ γ ξ a(y,z,ξ)| ≤ cα,β,γ (1 + |y|) −l−|α| (1 + |ξ|)m−|γ|. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 143 the cusp c = s1 b with the metric h0 the manifold b̄ the manifold ∂x̄ d1 d2 d4 d3 figure 1: the infinity b of the quotient x = γ\h3 where γ is a schottky group gluing d3 ←→ d4 and d1 ←→ d2; b̄ is a manifold with fibred boundary. it is not hard to check the mapping property (3.1). one can also define classical (or polyhomogeneous) pseudo-differential operators of order m,l ∈ c as operators in ψℜ(m),ℜ(l)(b) with the symbol in (3.2) satisfying (for all k) a(y,z,ξ) = |y|−l|ξ|mã(|y|−1,y/|y|,z, |ξ|−1,ξ/|ξ|) for |ξ| > 1 for some ã ∈ c∞([0, 1) × sn−k−1 ×t k × [0, 1) ×sn−k−1), we will use the notation ψ m,l cl (b). in each end ek, this corresponds in a sense to the class of pseudo-differential treated by hörmander in the y ∈ rn−k variable (or the scattering calculus of melrose [21]) but with the additional compact variable z ∈ t k. in particular, an operator a ∈ ψm,l(b) can be defined in term of its distributional kernel lifted from b̄ × b̄ to a blown-up version of this product. this is a standard way due to melrose to describe in details the various singularities of the kernel: we always have the usual conormal singularity at the diagonal of x̄ × x̄ (like in the compact setting) but for 144 colin guillarmou cubo 11, 5 (2009) non-compact manifolds, it is important to include informations in the symbol about the behaviour at infinity, these can be interpreted as conormal singularities for the kernel on the boundaries of the compactification x̄ × x̄ (boundary of the compactification = infinity of the manifold). since singularities with different nature intesects at the diagonal of the corner ∂x̄×∂x̄, it is convenient to define a bigger manifold, the blow-up, where the kernel is more readable. the blow-up here is slightly different from that of scattering calculus, it is in a sense the scattering blow-up defined in [21] but only in y variable. this blow-up corresponding to manifolds with fibred boundaries is explained in generality by mazzeo-melrose in [18], it is achieved in two essential steps. the principle is to start with the manifold with corners x̄ ×x̄ and to construct a larger manifold with corners where the phase of (3.2) defines a smooth submanifold (“the diagonal”) intersecting transversally the boundary of this larger manifold at only one hypersurface. for what follows, we will use part of the notations of [18]. the manifold b̄ × b̄ has 2n − 2 boundary hypersurfaces lk := ∂ek×b̄, rk = b̄×∂ek for k = 1, . . . ,n−1 and we have lk∩lj = ∅ if j 6= k, the same with rk and finally lk ∩rj = ∂ek ×∂ej is a corner of codimension 2. we need to define the first blow-up of b̄ × b̄ by taking the “b”blow-up b̄ ×b b̄ := [b̄ × b̄; ∂e1 ×∂e1; . . . ; ∂en−1 ×∂en−1] which means that we blow-up successively each corner ∂ek × ∂ek of ek × ek ⊂ b̄ × b̄. this is done by replacing in b̄ × b̄ the submanifold ∂ek ×∂ek by its spherical normal interior pointing bundle in b̄ × b̄. the blow-down map is denoted βb : b̄ ×b b̄ → b̄ × b̄. the manifold b̄×b b̄ has 3n−3 boundary hypersurfaces, the first 2n−2 are the top and bottom faces b ′ k := β −1 b (b ×∂ek), t ′ k := β −1 b (∂ek ×b), k = 1, . . . ,n− 1. the new ones are called front faces (f′k)k=1,...,n−1 for the b blow-up and f ′ k is the spherical normal interior pointing bundle of ∂ek ×∂ek in b̄× b̄ and is mapped by βb on ∂ek ×∂ek. note that f ′ k is diffeomorphic to [−1, 1]τ ×∂ek ×∂ek using the function τ = v−v′ v+v′ (see melrose [20]), thus we will identify them. the closure db := β −1 b (db) of the diagonal db of b×b meets the boundary of b̄×b b̄ only at the (interior of the) hypersurfaces f′k and it does transversally at a submanifold denoted ∂db. the blow-up of b̄×bb̄ along ∂db would give the blow-up associated to the scattering calculus but it turns out that the second kind of blow-up we need for our purpose are the successive blow-ups of b̄ ×b b̄ along the submanifolds φk = {(0,m,m ′ ) ∈ f′k = [−1, 1]τ ×∂ek ×∂ek; φk(m) = φk(m ′ )}, with φk the fibration of (2.14), this gives the manifold with corners b̄ ×φ b̄ := [b̄ ×b b̄; φ1; . . . ; φn−1]. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 145 the blow-down maps are b̄ ×φ b̄ βφ−b −→ b̄ ×b b̄ βb −→ b̄ × b̄, βφ := βb ◦βφ−b. the boundaries of b̄ ×φ b̄ are the top and bottom faces bk = β −1 φ (b ×∂b ′ k), tk = β −1 φ (∂b ′ k ×b) the front faces of the b blow-up fk := β −1 φ−b(f ′ k \ φk) and the front face of the φ blow-up is the normal spherical interior pointing bundle of φk in b̄×b b̄ ik := sn+(φk; b̄ ×b b̄). we will denote by ρtk,ρbk,ρfk,ρik some functions which define the respective hypersurfaces: {ρtk = 0} = tk, {ρbk = 0} = bk, {ρfk = 0} = fk, {ρik = 0} = ik. the closure dφ := β −1 φ (db ) meets the topological boundary of b̄ ×φ b̄ only at (the interior of) the hypersurfaces ik and it does transversally. one can thus define (using extension through the boundary hypersurface) the set im(b̄ ×φ b̄; dφ) of distributions classically conormal of order m to the submanifold dφ. tk bk dφ z −z′ ik fk figure 2: the blow-up of φk in b̄ ×b b̄ the important point is that β∗φ is a one-to-one map between ċ ∞ (b̄ × b̄) and ċ∞(b̄ ×φ b̄), this induces a one-to-one map between their respective duals, which allows to indentify continuous 146 colin guillarmou cubo 11, 5 (2009) operators (3.1) with their schwartz kernel lifted to b̄×φ b̄. with this identification, we define the space ψ m,l φ (b̄) := {k ∈ ρ l ik im(b̄ ×φ b̄; dφ);∀k,k ≡ 0 at fk, tk, bk} for m,l ∈ c, where ≡ means equality of taylor series. this forms the (classical) “small φ-calculus” and it is not difficult to check that ψ m,l cl (b) = ψ m,l φ (b̄) with the notations introduced before for the standard pseudo-differential operators on b. we sketch the proof of the sense ψ m,l cl (b) ⊂ ψ m,l (b̄). recall that v = |y|−1,ω = y |y| ,v′ = |y|′,ω′ = y′ |y′| ,z,z′ give some local coordinates near the corner ∂ek ×∂ek on b̄ × b̄ and s = v v′ ,v′,ω,ω′,z,z′ with |ω| = |ω′| = 1 give some coordinates on b̄×b b̄ near the front face f ′ k (valid out of b ′ k), in particular φk = {v ′ = 0; s = 1; ω = ω′}. if a ∈ ψ m,l cl (b), the expression (3.2) with w = (y,z),w ′ = (y′,z′) can be put in these coordinates a(w; w′) = ∫ ei( 1 v′ ( ω s −ω′).ξ1+(z−z ′).ξ2)a ( ω v′s ,z; ξ1,ξ2 ) dξ1dξ2. (3.3) it can be checked that ωi s −ω′i,ω ′ i,v ′,z,z′ for i = 1, . . . ,n−k give some coordinates near f′k ∩ φk and φk = { ω s −ω′ = 0}. the functions (ωi−sω ′ i)/(sv ′ ) lift under βφ−b to some functions wi which are smooth near ik \ (ik ∩ fk) and we have near dφ ∩ ik dφ = {w1 = · · · = wn−k = 0; z = z ′}, ik = {v ′ = 0} in coordinates w := (w1, . . . ,wn−k),ω ′,v′,z,z′ with ∑ i ω ′ i 2 = 1. this gives in (3.3) a(w; w′) = ∫ ei(w.ξ1+(z−z ′).ξ2)a ( w + ω′ v′ ,z; ξ1,ξ2 ) dξ1dξ2 with {w = 0} = dφ. this last expression shows that a(w; w ′ ) has a classical conormal singularity at dφ of order m. near the front face ik, that is when v ′ → 0, then v′ −l a( w +ω ′ v′ ,z; ξ) is a smooth function near dφ ∩ ik. using other systems of coordinates covering ik ∩ fk one easily see that β∗φ(a) vanishes at all order at fk (using integration by parts in oscillating integrals and the “polynomial growth” of a(w,ξ) in |w|) and that ρ−l ik β∗φ(a) ∈ i m (b̄ ×φ b̄; dφ). the vanishing of (3.3) at {v′ = 0; |ω − sω′| > ǫ; 1 > s} comes by integration by parts and shows the vanishing of β∗φ(a) at all order at the boundaries near fk ∩ tk and the behaviour near fk ∩ bk is similar. finally the vanishing at tk and bk far from fk is again a consequence of non-stationary phase (3.2). the converse ψ m,l φ (b̄) ⊂ ψ m,l cl (b) is essentially similar. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 147 now one can define the “full φ-calculus” by considering the set of operators (identifying lifted kernels and operators) ψ m,l,e φ (b̄) := ψ m,l φ (b̄) + ∏ f =f,i,t,b k=1,...,n−1 (ρfk ) e(fk )c∞(b̄ ×φ b̄) (3.4) e = {e(t1),e(b1),e(f1),e(i1), . . . ,e(tn−1),e(bn−1),e(fn−1),e(in−1)}, e(fk) ∈ c i.e. we allow some classically conormal singularities at all faces. for operators we deal with, the conormal singularity at the front faces ik will be of the same order for both terms, that is l = e(i1) = · · · = e(in−1), hence we will write ψ m,e φ (b̄) instead of ψ m,l,e φ (b̄). finally, a subclass with much more regularity will appear as error terms in the expression of the scattering operator, those are operators with kernels of the form ∏ k (rck ) ak (r′ck ) bkc∞(∂x̄ ×∂x̄). where ak,bk ∈ c and rck (w,w ′ ) := rck (w), r ′ ck (w,w′) := rck (w ′ ). recall again that ∂x̄ can be viewed as the smooth compact manifold without boundary obtained from b̄ by collapsing each ∂ek ≃ s n−k−1 ×t k to φk(∂ek) = ck ≃ t k. actually, since we forgot the density factors for the kernels, the orders of such pseudodifferential operators depend on the density we use to pair two fonctions in ċ∞(b̄), thus it will be necessary to precise it. 4 resolvent in this section we analyze the meromorphic extension of the modified resolvent r(λ) := (∆x −λ(n−λ)) −1 and more precisely the necessary informations we shall need to define eisenstein functions, poisson operator and scattering operator. the meromorphic extension of the resolvent is proved in [8] by parametrix construction. using also spectral theorem, this can be summarized as follows: theorem 4.1. there exists c > 1 such that for all n > 0, the modified resolvent r(λ) on x extends meromorphically with poles of finite multiplicity from {ℜ(λ) > n 2 } to {ℜ(λ) > n 2 −cn} with values in the bounded operators from ρnl2(x) to ρ−nl2(x). the only poles of r(λ) in {ℜ(λ) > n 2 } are first order poles at each λ0 such that λ0(n−λ0) ∈ σpp(∆x ) and with residue resλ0r(λ) = (2λ0 −n) −1 r∑ j=1 φj ⊗φj, φj ∈ ρ λ0r−1c c ∞ acc (x̄) ⊂ l2(x) where (φj )j=1,...,r is an orthonormal basis of kerl2 (∆x −λ0(n−λ0)). 148 colin guillarmou cubo 11, 5 (2009) actually the form of φj is a consequence of (4.20) which will be proved in this section. to construct the poisson operator, we need more precise information about the mapping properties of r(λ) and about its schwartz kernel structure near infinity. one of the main points is to analyze the schwartz kernel of the meromorphic extension of the resolvent rxk (λ) = (∆xk −λ(n−λ)) −1 for the laplacian ∆xk on the model spaces xk = γk\h n+1, and its mapping properties. recall that x̄ is a compact manifold with boundary ∂x̄, hence x̄ × x̄ is a manifold with corners on which we define the functions ρ(w,w′) := ρ(w), ρ′(w,w′) := ρ(w′), rc(w,w ′ ) := rc(w), r ′ c(w,w ′ ) := rc(w ′ ). (4.1) since ρ,rc are well defined on m̄k via ik, the functions (4.1) can also be defined on m̄k ×m̄k. lemma 4.2. let θ,θ′ ∈ c∞(x̄k) be functions with support in m̄k and constant near ck, then the extended resolvent rxk (λ) satisfies θrxk (λ)θ ′ : ċ∞(x̄k) → ρ λr−1c c ∞ acc (x̄k) (4.2) for λ /∈ ( k 2 −n0) if n−k + 1 is odd and for λ ∈ c otherwise. if moreover θ,θ ′ are chosen satisfying supp(θ) ∩ ck = ∅ and θθ ′ = 0 then θ′rxk (λ)θ ∈ ρ λρ′ λ r−1c c ∞ (x̄k × x̄k), θrxk (λ)θ ′ ∈ ρλρ′ λ r′c −1 c∞(x̄k × x̄k) (4.3) proof : clearly, it is enough to show the lemma with θ,θ′ which are independent of the variable z ∈ t k. we recall from [8] that the explicit formula for the resolvent on xk can be obtained by fourier analysis on the z ∈ t k variable, rxk (λ) admits a meromorphic continuation to c and its schwartz kernel can be written rxk (λ) = ∑ m∈zk eiωm.(z−z ′)rm(λ) (4.4) for λ /∈ ( k 2 − n0) if n−k + 1 is odd and for λ ∈ c otherwise, with rm(λ; x,y; x ′,y′) := ck ∫ rk eiωm.zrhn+1 (λ; x,y,z; x ′,y′, 0)dz (4.5) where ck is a constant, rhn+1 (λ) is the kernel of the resolvent of the laplacian on h n+1 and ωm := 2π t (a−1k )m. note that rm(λ) can be considered as an operator -a resolventon h n−k+1. we have seen in [8] that if τ := xx′ r2 + |z|2 , r2 := |y −y′|2 + x2 + x′ 2 , d := xx′ r2 cubo 11, 5 (2009) scattering theory on geometrically finite quotients 149 then for all n ∈ n ∪∞ there exists a function fn (λ,τ) smooth in τ ∈ [0, 1 2 ) with a conormal singularity at τ = 1 2 such that rhn+1 (λ; x,y,z; x ′,y′, 0) = τλ n−1∑ j=0 αj (λ)τ 2j + τλ+2n fn (λ,τ) for some αj (λ) meromorphic in λ (with only poles at −n0 if n + 1 is even) and if n = ∞, f∞(λ,τ) = 0 and the sum converges locally uniformly if τ 6= 1 2 (see also [12] and [23, appendix a]). thus by a change of variable w = z/r in (4.5), one has as in [8, sect. 3.1] rm(λ) = d λrk n−1∑ j=0 d2jfj,λ(r|ωm|) + d λ+2n rk ∫ rk e−irωm.z fn (λ,d(1 + |z| 2 ) −1 ) (1 + |z|2)λ+2n dz (4.6) with fj,λ(u) := ck,j (λ)|u| λ− k 2 +2jk −λ+ k 2 −2j (|u|), fj,λ(0) := dk,j (λ) ks(z) = ∫ ∞ 0 cosh(st)e−z cosh(t)dt being the modified bessel function, ck,j (λ) some holomorphic functions and dk,j (λ) some meromorphic functions in c with only first order poles at k 2 − n0 if n−k + 1 is even (in fact we have r0(λ) = (xx ′ ) k 2 rhn−k+1 (λ− k 2 )). the sum (4.6) with n = ∞ is locally uniformly convergent in {d < 1 2 , 0 < r}. we first show (4.3) using these explicit formulae. we will better use the compactification coordinates (t,u) on mk, the functions r and d become d = tt′ |u−u′|2 + t2 + t′2 , r2 = t2 + t′ 2 + |u−u′|2 (t2 + |u|2)(t′2 + |u′|2) . (4.7) on the support of θrxk (λ)θ ′ we have t2 + t′ 2 + |u−u′|2 > ǫ and d ≤ 1 2 − ǫ for some ǫ > 0 since θθ′ = 0, thus (4.6) with n = ∞ is absolutely convergent there and r → +∞ when t2 + |u|2 → 0, that is when we approach the cusp submanifold ck with respect to variables (t,u). since bessel’s function ks(x) = k−s(x) and all its derivatives with respect to x vanish exponentially when x →∞, the kernel ∑ m 6=0 θrm(λ)e iωm .(z−z ′)θ′ is in ρλρ′ λ rc −1c∞({x̄k \ ck}× x̄k) and can be extended to x̄k × x̄k with ∑ m 6=0 θrm(λ)θ ′eiωm.(z−z ′) ∈ ρλρ′ λ c∞(x̄k × x̄k) vanishing at all order at (ck ×x̄k)∪(x̄k ×ck). note that we have used that ρ = t in mk. for the term r0(λ), it is clear, using (4.6) and (4.7) that θr0(λ)θ ′ ∈ ρλρ′ λ r′c −1 c∞(x̄k × x̄k) 150 colin guillarmou cubo 11, 5 (2009) which concludes the proof of (4.3) using the symmetry of the resolvent kernel. the property (4.2) is more technical since it involves the singularity of rxk (λ) near the diagonal. let f ∈ ċ∞(x̄k), with support in m̄k. we first study for m 6= 0 the function θrm(λ)θ ′fm in m̄k where fm = 〈f,e iωm.z〉tk is the m-th fourier mode on t k of f. we clearly have fm ∈ ċ∞(h̄n−k+1) with ∀l ∈ n, |∂αfm| ≤ cα,l|ωm| −l with cα,l uniform in m. for simplicity, we consider (4.6) with n = 0 and decompose f0(λ,τ) = χ(τ)f0(λ,τ) + (1 −χ(τ))f0(λ,τ) =: f0,1(λ,τ) + f0,2(λ,τ) with χ a c∞0 ([0, 1/4)) which is equal to 1 near τ = 0. the integral θ(t,u)θ′(t′,u′)rkdλ ∫ rn−k e−irωm.z(1 + |z|2)−λf0,1(λ,d(1 + |z| 2 ) −1 )dz is well defined for ℜ(λ) > k 2 and is equal by integration by parts to κ1 := θ(t,u)θ ′ (t′,u′)(r|ωm|) −2nrkdλ ∫ rn−k e−irωm.z∆nz ( f0,1(λ,d(1 + |z| 2 ) −1 ) (1 + |z|2)λ ) dz (4.8) for all n > 0. in view of the smoothness of f0,1(λ,τ) for τ ∈ r +, it is straightforward to see that the integrand in (4.8) satisfies ∣∣∣∣∆ n z ( f0,1(λ,d(1 + |z| 2 ) −1 ) (1 + |z|2)λ )∣∣∣∣ ≤ cn (1 + |z| 2 ) −ℜ(λ)−n and is a smooth function of d for λ ∈ c\−n0, now integrable with respect to z ∈ r k if ℜ(λ)+n > k 2 . now since fm(t ′,u′) = o(t′ ∞ ), we have in h̄n−k+1 × h̄n−k+1 |∂αt,u(d/t)∂ βfm| ≤ cα,β,l|ωm| −l, |∂αt,ud∂ βfm| ≤ cα,β,l|ωm| −l |∂αt,ur∂ βfm| ≤ cα,β,l(t 2 + |u|2)−(1+|α|)/2|ωm| −l, |∂αt,u(r √ t2 + |u|2)∂βfm| ≤ cα,β,l|ωm| −l by looking at the expression of d,r in (4.7). for λ /∈−n0 fixed, we take n ≫ 2|ℜ(λ)|, this proves that t−λ(t2 + |u|2)−m ∫ hn−k+1 dλκ1fm(t ′,u′)t′ −n+k−1 (t′ 2 + |u′|2) k 2 dt′du′ is cn in (t,u) ∈ h̄n−k+1 for 2m ≪ n and all its derivatives of order α with |α| < n are bounded by cl,n|ωm| −l for all l,n,m. thus for m fixed, by taking n → ∞ we see that this function is smooth in h̄n−k+1 and its derivatives are rapidly decreasing in |ωm|. we now have to deal with the integral kernel κ2 := θ(t,u)θ ′ (t′,u′)rkdλ ∫ rn−k e−irωm.z(1 + |z|2)−λf0,2(λ,d(1 + |z| 2 ) −1 )dz cubo 11, 5 (2009) scattering theory on geometrically finite quotients 151 and we will show that f′m(t,u) := ∫ hn−k+1 κ2fm(t ′,u′)t′ −n+k−1 (t′ 2 + |u′|2) k 2 dt′du′ satisfies f′m ∈ ċ(h̄ n−k+1 ), |∂αt,uf ′ m| ≤ cα,l|ωm| −l. (4.9) first remark that, since d < 1 2 , we have 1 − χ(d(1 + |z|2)−1) = 0 if |z| > c for some c > 0 depending on χ. we use the change of variables s = t/t′,v = (u−u′)/t′ in this last integral. by elementary computations, it turns out that d = (2 cosh(dhn−k+1 (t,u; t ′,u′)))−1 = (2 cosh(dhn−k+1 (1, 0rn−k ; s,v))) −1 but f0,2(λ,d(1 + |z| 2 ) −1 ) is supported in {d > ǫ} for some ǫ > 0 depending on χ thus it is supported in {(s,v) ∈ k} where k is a euclidean ball included in hn−k+1 (thus a compact of h n−k+1). moreover in the variables (t,u,s,v), κ2 = θ(t,u)θ ′ (t s ,u− t s v ) rkdλ ∫ |z| 0 large in (4.6) and essentially the same arguments than for n = 0. � now we briefly review the construction of a parametrix for r(λ) in [8, prop 3.1 and 3.5] which can be continued to infinite order (at least formally, the problem of convergence will be discussed later). this is obtained by localizing in the neighbourhoods mk and mr near infinity. one can construct some operators ek ∞ (λ) on mk (k = 1, . . . ,n− 1) and e r ∞ (λ) on mr such that (∆mk −λ(n−λ))e k ∞ (λ) = χk + kk ∞ (λ), (∆mr −λ(n−λ))e r ∞(λ) = χ r + k r ∞(λ) with kk ∞ (λ), kr ∞ (λ) having smooth schwartz kernels kk ∞ (λ; w,w′) and kr ∞ (λ; w,w′)) which vanish at all order when ρ(w) → 0. the first step of the parametrix construction of ek ∞ (λ) is to take a smooth function χkl with support in mk which is equal to 1 in {x 2 + |y|2 > 4} such that χklχ k = χk and 1 − χkl can be chosen as a product (see the construction in [8]) 1 −χkl(x,y,z) = ψ k l(y)φl(x) (4.10) independent of the variable on t k; then set ek0 (λ) := χ k lrxk (λ)χ k, kk0 (λ) = [∆xk,χ k l]rxk (λ)χ k and we obtain (∆mk −λ(n−λ))e k 0 (λ) = χ k +kk0 (λ) as a first parametrix in the neighbourhood mk of ∂x̄ in x̄. the next steps of the construction in [8, prop.3.1] involve only some operators with schwartz kernels of the same type than kk0 (λ) but with additional decay at ∂x̄×x̄ in x̄×x̄. the part of the parametrix on mr is done as in the work of guillopé-zworski [12] (and more generally [17]) by using at first step er0 (λ) := χ r lrhn+1 (λ)χ r, kr0 (λ) = [∆hn+1,χ r l]rhn+1 (λ)χ r with a function χrl which is equal to 1 on the support of χ r and which can be expressed as a product χrl(x,y) = φ r l(x)ϕ r l(y) in mr. the other steps of the construction in mr do not make more singular kernels than kr0 (λ) appear. the previous lemma allows to deduce the following proposition 4.3. let θ,θ′ ∈ c∞(x̄) be constant near c and such that supp(θ′)∩c = ∅ and θθ′ = 0. then for λ not a resonance, we have θr(λ)θ′ ∈ rc −1ρ′ λ ρλc∞(x̄ × x̄), θ′r(λ)θ ∈ r′c −1 ρλρ′ λ c∞(x̄ × x̄) and r(λ) has the mapping property r(λ) : ċ∞(x̄) → r−1c ρ λc∞ acc (x̄). (4.11) cubo 11, 5 (2009) scattering theory on geometrically finite quotients 153 proof : if we carefully look at the expression of k∞(λ) following [8, prop. 3.1 and 3.5] and we use previous lemma, it is not difficult to check that (ik) ∗ k k ∞(λ)(ik )∗ ∈ ρ ∞ρ′ λ r′c −1 c∞(x̄ × x̄), (4.12) (ik) ∗ k r ∞(λ)(ir )∗ ∈ ρ ∞ρ′ λ c∞(x̄ × x̄). (4.13) the second statement is essentially well-known (see [8, 12] for instance) and is a direct consequence of the explicit formula of rhn+1 (λ). to prove the first one, we essentially use lemma 4.2. it is not difficult to check (see again [8]) that [∆xk ,χ k l] is a first order operator with smooth coefficients supported in {1 < x2 + |y|2 ≤ 4, 0 ≤ x} and vanishing at second order at x = 0. using the compactification coordinates (t,u) of (2.1), it is also a first order operator with smooth coefficients supported in {ǫ < t2 + |y|2 ≤ 1, 0 ≤ t} for some ǫ > 0 and vanishing at second order at t = 0, moreover its support does not intersect the support of χk. therefore, using (4.3) in lemma 4.2 we easily deduce that (ik) ∗ [∆xk,χ k l]rxk (λ)χ k (ik)∗ ∈ ρ λ+2ρ′ λ r′c −1 c∞(x̄ × x̄). (4.14) now the iterative construction of [8, prop. 3.1] corresponds to capture the taylor expansion of this term at ρ = 0 and the remaining error terms at each step are like (4.14) but with more decay in ρ; this finally implies (4.12). the terms appearing in the expression of ek ∞ (λ) in [8, prop. 3.1], are thus χklrxkχ k plus some operators whose schwartz kernels are in ρλ+2ρ′ λ r′c −1 c∞(x̄k ×x̄k). therefore ek∞(λ) satisfies exactly the same properties than rxk (λ) described in lemma 4.2. by standard pseudo-differential calculus on compact manifolds, we can obtain the compact part of the parametrix ei∞(λ) so that (∆x −λ(n−λ))e i ∞(λ) = χ + k i ∞(λ) with ki ∞ (λ) having a smooth kernel with compact support in x ×x and ei ∞ (λ) being a pseudodifferential operator of order −2 supported in a compact set of x ×x. thus we obtain (∆x −λ(n−λ))e∞(λ) = 1 + k∞(λ) with e∞(λ) := e i ∞ (λ) + ∑ α=1,...,n−1,r (iα) ∗ e α ∞ (λ)(iα)∗, k∞(λ) := k i ∞ + ∑ α=1,...,n−1,r (iα) ∗ k α ∞(λ)(iα)∗. using lemma 4.2, (4.12), (4.13) and the explicit formulae of the regular terms in er ∞ (λ) in [8, 12] it is straightforward to see that k∞(λ) ∈ ρ ∞ρ′ λ r′c −1 c∞(x̄ × x̄) (4.15) θe∞(λ)θ ′ ∈ rc −1ρλρ′ λ c∞(x̄ × x̄), θ′e∞(λ)θ ∈ ρ λρ′ λ r′c −1 c∞(x̄ × x̄). (4.16) 154 colin guillarmou cubo 11, 5 (2009) moreover using lemma 4.2 for the mapping properties of the cusps terms and [7, prop. 3.1] for the mapping properties of the regular terms, we have e∞(λ) : ċ ∞ (x̄) → ρλr−1c c ∞ acc (x̄). (4.17) we can then write r(λ) = e∞(λ) − e∞(λ)k∞(λ) + e∞(λ)k∞(λ)(1 + k∞(λ)) −1 k∞(λ) (4.18) and (1 + k∞(λ)) −1 = 1 + f(λ) with f(λ) = −k∞(λ) − k∞(λ)f(λ). this proves that f(λ) is hilbert-schmidt on ρnl2(x) for ℜ(λ) > n−1 2 and n large, since k∞(λ) is. using that ρ′ n r′c −1 is bounded, the composition k∞(λ)f(λ)k∞(λ) has a schwartz kernel in the same class than k∞(λ) (and k∞(λ) 2 too). in view of its construction, we see that the range of k∞(λ) is composed of functions with support in x̄ \ c, thus we can find a smooth function θ′ ∈ c∞(x̄) with supp(θ′)∩c = ∅ such that θ′k∞(λ) = k∞(λ). thus if θ is a function in c ∞ (x̄) such that θ = 1 near c and θθ′ = 0 we have from (4.16), (4.15) that θe∞(λ)k∞(λ) ∈ ρ λρ′ λ r−1c r ′ c −1 c∞(x̄ × x̄). (4.19) now we can for example use mazzeo’s composition results in [15] to deal with the regular terms (e i ∞ (λ) + (ir ) ∗ e r ∞ (λ)(ir )∗)k∞(λ) ∈ ρ λρ′ λ c∞(x̄ × x̄). then (1 − θ)(ik) ∗ e k ∞(λ)(ik)∗k∞(λ) can be studied exactly with the same method than for the proof of (4.2) in lemma 4.2 and we see that (1 −θ)(ik) ∗ e k ∞ (λ)(ik )∗k∞(λ) ∈ ρ λρ′ λ r′c −1 c∞(x̄ × x̄) and we conclude, using (4.19), that e∞(λ)k∞(λ) ∈ ρ λρ′ λ r−1c r ′ c −1 c∞(x̄ × x̄) and the same holds for e∞(λ)k∞(λ)(1 + f(λ))k∞(λ). we have completed the proof in view of (4.18) and the symmetry of the resolvent kernel. moreover we have also proved that r(λ) − e∞(λ) ∈ (ρρ ′ ) λ (rcr ′ c) −1c∞(x̄ × x̄). (4.20) the mapping property of r(λ) is then easily deduced from (4.18) and (4.17) since k(λ) maps ρnl2(x) to ċ∞(x̄) if n ≫|ℜ(λ)| in view of the form (4.15) of its kernel. � remark: we did not study the convergence problem of the infinite order parametrix e∞(λ) but to avoid this problem, it suffices to take the parametrix en (λ) of [8] for large n and the same proof actually would show the same results for r(λ) but with cm regularity for some m > n −c|ℜ(λ)| (with c > 0) instead of c∞ regularity. since it is true for all n, we get the same results. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 155 5 poisson operator, eisenstein function 5.1 poisson operator using the product decomposition of the metric in lemma 2.2, an indicial equation for the laplacian and the mapping property of the resolvent, we can construct a poisson operator following the method of graham-zworski [7]. actually, we now work with the special boundary defining function ρ but every other choice of boundary defining function ρ̂ ∈ c∞ acc (x̄) defined in lemma 2.2 would induce an equivalent construction for the poisson operator. we will simply add the necessary arguments when the generalization is not transparent. with the metric under the form (2.13), the laplacian is ∆x = −(ρ∂ρ) 2 + nρ∂ρ − 1 2 tr(h−1(ρ).∂ρh(ρ))ρ 2∂ρ + ρ 2 ∆h(ρ). (5.1) in the neighbourhood mk of the cusp ck this gives ∆x = −(ρ∂ρ) 2 + nρ∂ρ − 2k(ρ 2 + |u|2)−1ρ3∂ρ + ρ 2 ∆h(ρ) with h(ρ) = du2 +(ρ2 +|u|2)2dz2 a metric on {0 < |u| < 1}×t kz , and by an elementary computation we obtain rc∆xr −1 c = −(ρ∂ρ) 2 + nρ∂ρ + ρ 2 (∆u + (ρ 2 + |u|2)−2∆z ) (5.2) where ∆u, ∆z are the flat laplacians on r n−k u ,t k z . similarly with a function ρ̂ of lemma 2.2 we have ∆x = −(ρ̂∂ρ̂) 2 + nρ̂∂ρ̂ − 1 2 tr(ĥ−1(ρ̂).∂ρ̂ĥ(ρ̂))ρ̂ 2∂ρ̂ + ρ̂ 2 ∆h(ρ̂) + o(ρ̂ ∞ ). and in coordinates (ρ̂,v,ζ) near ck, we see from (2.11) that rc∆xr −1 c = −(ρ̂∂ρ̂) 2 + nρ̂∂ρ̂ + p1 + p2 + ρ̂ 2e−2ωr−4c ∆ζ + o(ρ̂ ∞ ) for some differential operators p1 = p1(ρ̂,v, ρ̂ 2∂ρ̂, ρ̂∂v), p2 = p2(ρ̂,v,ζ, ρ̂∂v, ρ̂∂ζ ) = o(r ∞ c ) of order 2, with p2 (resp. p1) having smooth coefficents on x̄ (resp. smooth outside ck). by making the same change of coordinates (2.9) in (5.2), it would give some differential operators with smooth coefficients at ck except the term with ∆ζ thus p1 has to be smooth at ck. we now use graham-zworski’s construction [7] and we refer the reader to their paper for additional details. if f ∈ c∞ acc (∂x̄) we deduce from (5.1) and (5.2) the indicial equation in {ρ < ǫ} (∆x −λ(n−λ))ρ n−λ+jr−1c f − j(2λ−n− j)ρ n−λ+jr−1c f ∈ ρ n−λ+j+1r−1c c ∞ acc (x̄). (5.3) 156 colin guillarmou cubo 11, 5 (2009) here, the key fact is that the singular term r−4c ∆z applied to f ∈ c ∞ acc(∂x̄) gives a functions in ċ∞c (x̄) by (2.7). therefore for all f ∈ r −1 c c ∞ acc (∂x̄) one can construct by induction and borel lemma (see again [7]) a function φ(λ)f ∈ ρn−λr−1c c ∞ acc (x̄) for λ ∈ c \ 1 2 (n + n) such that (∆x −λ(n−λ))φ(λ)f ∈ ċ ∞ (x̄), ρλ−nφ(λ)f|ρ=0 = f. by construction, we have the formal taylor expansion φ(λ)f ∼ ρn−λ ∞∑ j=0 ρ2jcj,λpj,λf, ∀f ∈ c ∞ acc (∂x̄) (5.4) where pj,λ is a differential operator on b which is polynomial in λ and cj,λ := (−1) j γ(λ− n 2 − j) 22jj!γ(λ− n 2 ) . now we can set for λ /∈ 1 2 (n + n) and λ not a resonance p(λ)f = φ(λ)f −r(λ)(∆x −λ(n−λ))φ(λ)f (5.5) which satisfies    (∆x −λ(n−λ))p(λ)f = 0 p(λ)f = ρn−λf(λ,f) + ρλg(λ,f) f(λ,f),g(λ,f) ∈ r−1c c ∞ acc (x̄) f(λ,f)|ρ=0 = f (5.6) using proposition 4.3. we have defined a family of operators p(λ) : r−1c c ∞ acc (∂x̄) → ρn−λr−1c c ∞ acc (x̄) + ρλr−1c c ∞ acc (x̄) and we will now prove the uniqueness of an operator satisfying (5.6) in {ℜ(λ) ≥ n 2 }. the principle is the same than in [7]: if ℜ(λ) > n 2 , λ not a resonance and p1(λ)f, p2(λ)f are two solutions of (5.6), then the previous indicial equation shows that p1(λ)f − p2(λ)f ∈ ρ λr−1c c ∞ (x̄) but this function is in l2(x) using (2.6) so this must be 0; to treat the case ℜ(λ) = n 2 , we use a boundary pairing lemma like proposition 3.2 of [7]: lemma 5.1. for i = 1, 2, let ui = ρ n−λfi + ρ λgi some functions satisfying (∆x −λ(n−λ))ui = ri ∈ ċ ∞ (x̄) with fi,gi ∈ r −1 c c ∞ (x̄), then we have for ℜ(λ) = n 2 and λ 6= n 2 ∫ x (u1r2 −r1u2) dvolg = (2λ−n) ∫ b (f1|bf2|b −g1|bg2|b) dvolh0 cubo 11, 5 (2009) scattering theory on geometrically finite quotients 157 proof : we apply green lemma in xǫ = {ρ ≥ ǫ} ∫ xǫ (u1r̄2 −u2r̄1) dvolg = ǫ −n+1 ∫ ρ=ǫ (u1∂ρū2 − ū2∂ρu1) dvolh(ǫ) (5.7) and we will take the limit as ǫ → 0. using the asymptotics of u1,u2 we get u1∂ρū2 − ū2∂ρu1 = (2λ−n)ρ n−1 (f1f2 −g1g2) + ρ n (g1∂ρg2 −g2∂ρg1 + f1∂ρf2 −f2∂ρf1). recall from (2.5) that dvolh(ǫ) = rc(ǫ) 2µ∂x̄ with rc(ǫ) = (|u| 2 + ǫ2) 1 2 in the neighbourhood bk of the cusp submanifold ck, so the only terms in the right hand side of (5.7) for which the limit are not apparent are ǫ ∫ ρ=ǫ (g1∂ρg2 −g2∂ρg1) dvolh(ǫ), ǫ ∫ ρ=ǫ (f1∂ρf2 −f2∂ρf1) dvolh(ǫ). the study of both terms when ǫ → 0 is the same and can be clearly reduced to the limit of ∫ t k ∫ |u|≤1 g1(ǫ,u,z)ǫ∂ǫg2(ǫ,u,z)(|u| 2 + ǫ2)kdurn−kdzt k (5.8) when ǫ → 0, gi(ρ,u,z) being the function gi in the coordinates of the neighbourhood bk of ck. using that on gi ∈ r −1 c c ∞ (x̄), it suffices to show that the limit of ∫ |u|≤1 ǫ∂ǫ[(|u| 2 + ǫ2)− k 2 ](|u|2 + ǫ2) k 2 durn−k is 0 when ǫ → 0 to prove that the limit of (5.8) is 0. now this last integral is equal to c ∫ 1 0 ǫ2(r2 + ǫ2)−1rn−k−1dr ≤ cǫ ∫ ∞ 0 (1 + r2)−1dr for a constant c, this finally proves the lemma. � now using this lemma with u2 = r(n−λ)ϕ for ϕ ∈ ċ ∞ (x̄) and u1 = p1(λ)f − p2(λ)f this proves that 〈u1,ϕ〉 = 0 for all ϕ ∈ ċ ∞ (x̄), thus u1 = 0. as a conclusion, we have proposition 5.2. for ℜ(λ) ≥ n 2 , λ /∈ 1 2 (n + n0), λ(n−λ) /∈ σpp(∆x ) there exists a unique linear operator p(λ) : r−1c c ∞ acc (∂x̄) → ρn−λr−1c c ∞ acc (x̄) + ρλr−1c c ∞ acc (x̄) analytic in λ and solution of the poisson problem (5.6). it is given by (5.5) and called poisson operator. by (5.5) it admits a meromorphic continuation with poles of finite multiplicity to c\1 2 (n+n0). 158 colin guillarmou cubo 11, 5 (2009) 5.2 eisenstein functions in this part, we define eisenstein functions as a weighted restriction of the schwartz kernel of the resolvent at b×x and we prove that they are the schwartz kernel of the transpose of the poisson operator. as a consequence of proposition 4.3 and (4.20) we first obtain the corollary 5.3. the eisenstein function e(λ) := (ρ−λr(λ))|b×x is well defined, meromorphic in λ ∈ c and satisfies e(λ) ∈ rc −1c∞(∂x̄ ×x). (5.9) moreover, if emod(λ) is the ‘model eisenstein function’ defined by emod(λ) := (ρ −λ e∞(λ))|b×x then e(λ) −emod(λ) ∈ ρ ′λ (rcr ′ c) −1c∞(∂x̄ × x̄). (5.10) let exk (λ) be the eisenstein function for the model space xk obtained from (4.4) and (4.6) (recall that ρ = t = x x2+|y|2 with our choice in lemma 2.2) exk (λ; y,z; x ′,y′,z′) = |y|2λx′ λ r−2λ+k ∑ m∈zk eiωm.(z−z ′)f0,λ(r|ωm|) for y 6= 0, where by convention r = (|y−y′|2 + x′ 2 ) 1 2 denotes here the restriction of r to x = 0. in the compactification coordinates (t,u) of (2.1) this gives exk (λ; u,z; t ′,u′,z′) = t′ λ r−2λ+k|u|−2λ(t′ 2 + |u′|2)−λ ∑ m∈zk eiωm.(z−z ′)f0,λ(r|ωm|) (5.11) and r is expressed in these coordinates by r2 = t′ 2 + |u−u′|2 |u|2(t′2 + |u′|2) . (5.12) similarly let ehn+1 (λ) be the eisenstein function on h n+1 ehn+1 (λ; y; x ′,y′) = π− n 2 γ(λ) (2λ−n)γ(λ− n 2 ) x′ λ (|y −y′|2 + x′2)λ . (5.13) using the construction of the parametrix for the resolvent, we can deduce an expression for the model eisenstein function emod(λ) = ∑ α=1,...,n−1,r (ια) ∗eαmod(λ)(iα)∗ (5.14) cubo 11, 5 (2009) scattering theory on geometrically finite quotients 159 with ια := iα|ρ=0 and in mk,mr ekmod(λ; y,z; w ′ ) := ψkl(y)exk (λ; y,z; w ′ )χk(w′), ermod(λ; y; w ′ ) := ψrl(y)γr(y) −λehn+1 (λ; y; w ′ )χr(w′). (5.15) with ρ(x,y) = xγr(y) + o(x) in mr for some positive smooth function γr in br and ψ α l defined in (4.10). we show that the eisenstein functions can be viewed as a schwartz distributional kernel of an operator, that we also denote e(λ), mapping ċ∞(x̄) to c−∞(b̄), actually with weighted l2 continuity results. lemma 5.4. there exists c > 1 such that for |ℜ(λ) − n 2 | ≤ c−1n, e(λ) : ρnl2(x) → l2(b) is a meromorphic family of hilbert-schmidt operators with poles of finite multiplicity, included in the set of resonances. moreover for ℜ(λ) < 0 and λ not a resonance, (b,w) → ρ(w)−λe(λ; b; w) is a continuous function on b × (x̄ \ c). proof : the terms e(λ) − emod(λ) and (ιr ) ∗ermod(λ)(ir )∗ in e(λ) clearly satisfy those two properties, we thus only have to deal with ekmod(λ) in xk. from (5.11) and (5.12) we have |t′ n exk (λ; u,z; t ′,u′,z′)| ≤ t′ ℜ(λ)+n (|u−u′|2 + t′ 2 ) k 2 −ℜ(λ) |u|k|u′|k ∑ m∈zk |f0,λ(r|ωm|)|. when r|ωm| > 1, the classical estimate |ks(z)| ≤ ce −cℜ(z) for ℜ(z) > 1 (with c > 0 depending on s) on mac donald’s function shows that |f0,λ(r|ωm|)| ≤ e −cr|ωm| thus ∑ |ωm|>1/r |f0,λ(r|ωm|)| ≤ cr −k ≤ ct′ −k where c depends on λ. therefore we get for n > 4|ℜ(λ)| |t′ n exk (λ)| ≤ ct ′ n 2 |u|−k|u′|−k + t′ ℜ(λ)+n (|u−u′|2 + t′ 2 ) k 2 −ℜ(λ) |u|k|u′|k ∑ |ωm|≤1/r |f0,λ(r|ωm|)|. (5.16) now for r|ωm| ≤ 1 we use the definition (6.4) of mac donald function ks(z) to decompose f0,λ(r|ωm|) under the form f0,λ(r|ωm|) = c(λ)(ϕ−λ+ k 2 (r2|ωm| 2 ) + r2λ−k|ωm| 2λ−kϕλ− k 2 (r2|ωm| 2 )) with ϕs(x) smooth on x ∈ [0,∞) and c(λ) constant depending on λ. the term coming from ϕ−λ+ k 2 is treated exactly as before (the part with r|ωm| > 1) and for the term coming from ϕλ− k 2 we have ∑ |ωm|<1/r (r|ωm|) 2ℜ(λ)−k|ϕλ− k 2 (r2|ωm| 2 )| ≤ { c(r−k + r2ℜ(λ)−2k ) if ℜ(λ) − k 2 ≤ 0 cr−k if ℜ(λ) − k 2 > 0 160 colin guillarmou cubo 11, 5 (2009) for some c > 0 depending on |λ|. in view of (5.16), we conclude that for n > 4|ℜ(λ)| + 2k |(ιk) ∗t′ n exk (λ)(ik )∗| ≤ cρ ′ n 2 r−1c r ′ c −1 and this function is in l2(b×x) if n is large enough using (2.6) (here rc denotes the restriction of rc to b × x). the meromorphic property and the finiteness of the poles multiplicity comes from the discussion before the lemma, using the formulae for the model eisenstein functions and the fact that the poles of the resolvent have finite multiplicity. the second statement of the lemma is essentially treated in the same way. using that for ℜ(λ) < 0 r−2λ+kf0,λ(r|ωm|) = c(λ)(r −2λ+kϕ −λ+ k 2 (r2|ωm| 2 ) + |ωm| 2λ−kϕλ− k 2 (r2|ωm| 2 )) is continuous in (u,t′,u′) ∈ {u 6= 0,u′ 6= 0, t′ 2 + |u′|2 < 1, |u| < 1} (the power in r−2λ+k being negative) and that the sum ∑ m r −2λ+kf0,λ(r|ωm|) is locally uniformly convergent in the same set by previous estimates, we deduce that t′ −λ exk (λ; u,z; t ′,u′,z′) is also continous there and this achieves the proof. � the transpose te(λ) is then well-defined from from l2(b) to ρ−nl2(x) for some n depending on λ and its kernel is e(λ; w,b). let ϕ ∈ ċ∞(x̄) and f ∈ ċ∞c (∂x̄) ≃ ċ ∞ (b̄), then for ℜ(λ) = n 2 we use lemma 5.1, identity r(λ) = tr(λ) = r(n−λ)∗ and lemma 5.4 to deduce ∫ x ϕ̄(p(λ)f) dvolg = (2λ−n) ∫ b f(ρλ−nr(n−λ)ϕ)|b dvolh0 = (2λ−n) ∫ b f(ρ−λr(λ)ϕ̄)|b dvolh0 = (2λ−n) ∫ b f(e(λ)ϕ̄) dvolh0 which proves lemma 5.5. the schwartz kernel of p(λ) is (2λ−n)e(λ; w; b) ∈ c∞(x ×b). this also implies that p(λ) admits a meromorphic continuation to c with poles of finite multiplicity, and in particular it is analytic in {ℜ(λ) > n 2 } except a finite number of poles at points λ0 such that λ0(n−λ0) ∈ σpp(∆x ). by mimicking the proof of graham-zworski [7, prop. 3.5] it is straightforward to see that, for f ∈ r−1c c ∞ acc (∂x̄), p( n 2 + k)f has log(ρ) terms in the asymptotic expansion and it is the unique solution of the problem    (∆x − n2 4 + k2)p( n 2 + k)f = 0 p( n 2 + k)f = ρ n 2 −kfk(f) + ρ n 2 +k log(ρ)gk(f) fk(f),gk(f) ∈ r −1 c c ∞ acc (x̄) fk(f)|ρ=0 = f (5.17) cubo 11, 5 (2009) scattering theory on geometrically finite quotients 161 the eisenstein functions are linked to the spectral projectors (via stone’s formula) of ∆x in the following sense proposition 5.6. if ℜ(λ) = n 2 and λ 6= n 2 then r(λ; w; w′) −r(n−λ; w; w′) = (n− 2λ) ∫ b e(λ; b; w′)e(n−λ; b; w) dvolh(b) (5.18) where h = (ρ2g)|b. moreover there exists c > 1 such that for n large, we have r(λ) −r(n−λ) = (2λ−n)te(n−λ)e(λ) in the strip |ℜ(λ)| ≤ c−1n as operators from ρnl2(x) to ρ−nl2(x). proof : the proof of (5.18) contains nothing more than the proof of theorem 1.3 of [3] or proposition 2.1 of [11] in a simpler case. note that the convergence of the integral in (5.18) is insured by (5.9) and (2.5). the second part of the proposition is a consequence of the mapping properties of r(λ),e(λ) proved before. � combined with lemma 5.4, this relation implies that e(λ) and r(λ) have same poles, except possibly at the points λ such that λ(n−λ) ∈ σpp(∆x ). 6 scattering operator using notations of (5.6), we can define the scattering operator as the linear operator s(λ) : { r−1c c ∞ acc (∂x̄) → r−1c c ∞ acc (∂x̄) f → g(λ,f)|b (6.1) for ℜ(λ) ≥ n 2 , λ /∈ 1 2 (n + n) and λ not a resonance. with (5.5), one obtains a meromorphic continuation of s(λ) to c. like p(λ), the scattering operator certainly depends on the choice of boundary defining function (here ρ), but any other choice ρ̂ = eωρ ∈ c∞ acc (x̄) of lemma 2.2 induces an equivalent construction and two corresponding scattering operators s(λ) and ŝ(λ) are related by the covariant rule ŝ(λ) = e−λω0s(λ)e(n−λ)ω0, ω0 = ω|∂x̄, this is a trivial consequence of uniqueness of solution of poisson problem. therefore it suffices in this section to deal with the special boundary defining function ρ. from lemma 5.5, (5.5) and (6.1), we deduce that for f ∈ ċ∞c (∂x̄) ≃ ċ ∞ (b̄) and ℜ(λ) < 0 s(λ)f = lim ρ→0 [ρ−λ((2λ−n)te(λ)f − φ(λ)f)] = (2λ−n) lim ρ→0 [ρ−λ(te(λ)f)] (6.2) 162 colin guillarmou cubo 11, 5 (2009) which is well defined in view of the continuity of e(λ; b; w′) proved in lemma 5.4. as a consequence the distributional kernel of s(λ) on b is s(λ; b; b′) = (2λ−n) lim w′→b′ (ρ(w′)−λe(λ; b; w′)) which can be rewritten using the symmetry of the resolvent kernel as the restriction s(λ) = (2λ−n)(ρ−λρ′ −λ r(λ))|ρ=ρ′ =0 (6.3) for ℜ(λ) < 0 and λ not resonance. moreover we deduce from (4.20) that s(λ) − (ρ−λρ′ −λ e∞(λ))|ρ=ρ′ =0 ∈ r −1 c r ′ c −1 c∞(∂x̄ ×∂x̄) which is easily seen to be compact on l2(b) in view of (2.5), and this term extends meromorphically to c with poles of finite multiplicity. we want to study the structure of the extendible distribution (6.3) on b̄×b̄, which continues meromorphically to c; it suffices actually to describe the singular part (ρ−λρ′ −λ e∞(λ))|ρ=ρ′ =0 of s(λ). to analyze this singular part of s(λ) in the neighbourhood of the cusp submanifolds, it turns out to be more convenient to work in the neighbourhood mk with the coordinates (x,y,z) than in their compactified version (t,u,z). indeed we will see that, up to conformal factors, the scattering operator for the model xk = γk\h n+1 is ∆ λ− n 2 yk where again yk = r n−k ×t k with the flat metric. this is what froese-hislop-perry used in [3] in dimension 3. using fourier transform in the (y,z) variable on xk we see that the laplacian on xk is transformed into the one dimensional operator pξm = −x 2∂2x + (n− 1)x∂x + x 2|ξm| 2 with ξm = (ξ,ωm). we easily deduce that the resolvent can be expressed by rxk (λ; w,w ′ ) = −(xx′) n 2 ∑ m∈z ∫ rn−k eiξm.(y−y ′,z−z′)gξm (λ; x,x ′ )dξ gξm (λ; x,x ′ ) := kλ− n 2 (|ξm|x)iλ− n 2 (|ξm|x ′ )h(x−x′) + kλ− n 2 (|ξm|x ′ )iλ− n 2 (|ξm|x)h(x ′ −x) with h the heaviside function, (w; w′) = (x,y,z; x′,y′,z′) the coordinates on xk × xk and iν (z),kν (z) the modified bessel functions. therefore using that ρ = x x2+|y|2 and iν (z) = 2 −νzν νγ(ν) + o(zℜ(ν+2)), kν (z) = − ν 2 γ(ν)γ(−ν)(iν (z) − i−ν (z)) (6.4) as z → 0, we obtain for ℜ(λ) < 0 (using {ρ = 0} = {x = 0} on b) exk (λ; y ′,z′; w) = −|y′|2λ2 n 2 −λ γ(λ− n 2 + 1) x n 2 ∑ m∈z ∫ rn−k eiξm .(y−y ′,z−z′)|ξm| λ− n 2 kλ− n 2 (|ξm|x)dξ (6.5) cubo 11, 5 (2009) scattering theory on geometrically finite quotients 163 and sxk (λ; y,z; y ′,z′) := (2λ−n)[ρ(x,y)−λexk (λ; y ′,z′; x,y,z)]|x=0 = 2 n−2λ γ( n 2 −λ) γ(λ− n 2 ) |y|2λ|y′|2λ ∑ m∈z ∫ rn−k eiξm.(y−y ′,z−z′)|ξm| 2λ−ndξ where this last sum-integral is understood (by splitting the term with ωm = 0 and the terms with ωm 6= 0) as the function on r n−k y ×t k z × r n−k y′ ×t k z′ 2 2λ−nπ− n−k 2 γ(λ− k 2 ) γ( n 2 −λ) |y −y′|−2λ+k + ∑ m 6=0 ∫ rn−k eiξm .(y−y ′,z−z′)|ξm| 2λ−ndξ which is continuous on {y 6= 0,y′ 6= 0}. this last function continues meromorphically to λ ∈ c in the distribution sense thus skmod(λ; y,z; y ′,z′) := [ρ(x′,y′) −λ ekmod(λ; y,z; x ′,y′,z′)]|x=0 = ψ k l(y)sxk (λ; y,z; y ′,z′)ψk(y′) (6.6) continues meromorphically to c as a distribution. note that the measure dvolh0 on yk is dvolh0 = |y| −2ndydz. to work on yk = r n−k y ×t k z with the natural measure dydz corresponding to the flat metric h̃0, we have to multiply the kernel of sxk (λ) by |y| −n|y′|−n, thus (6.6) can be rewritten, acting on l2(yk,dydz) skmod(λ) = c(λ)ψ k l|y| 2λ−n ∆ λ− n 2 yk |y|2λ−nψk with c(λ) := 2n−2λ γ( n 2 −λ) γ(λ− n 2 ) . (6.7) note that it has poles at λ = n 2 + j (with j ∈ n) with residue the differential operator on yk res n 2 +j (s k mod(λ)) = (−1)j+12−2j j!(j − 1)! ψkl|y| 2j ∆ j yk |y|2jψk on l2(yk,dydz). for the singularity of the kernel of s(λ) in the regular neighbourhood br on l 2 (br, dvolh0 ) (to see it acting on l2(br, dvolh̃0 ) it suffices to multiply the kernel by (rcr ′ c) n) we define the model scattering operator using (5.13) shn+1(λ; y; y ′ ) := (2λ−n)[x′ −λ ehn+1 (λ; y; x ′,y′)]|x′=0 = π− n 2 γ(λ) γ(λ− n 2 ) |y −y′|−2λ and we get from (5.15) srmod(λ; y; y ′ ) := [ρ(x′,y′) −λ ermod(λ; y; x ′,y′)]|x′=0 = ψrl(y)ψ r (y′) γr(y)λγr(y′)λ shn+1 (λ; y; y ′ ), (6.8) which continues meromorphically to c with poles at n 2 + j (with j integers) and residue res n 2 +j (s r mod(λ)) = (−1)j+12−2j j!(j − 1)! ψrlγ − n 2 −j r ∆ j rn γ − n 2 −j r ψ r. 164 colin guillarmou cubo 11, 5 (2009) with notations of (6.8), (6.6) we can now define the model scattering operator smod(λ) := ∑ α=1,...,n−1,r (ια) ∗sαmod(λ)(ια)∗ (6.9) and we have s(λ) −skmod(λ) ∈ r −1 c r ′ c −1 c∞(∂x̄ ×∂x̄) which is a compact operator on l2(b). from this study, it is straightforward to check that s(λ) is a bounded operators on l2(b) in {ℜ(λ) ≤ n 2 } (and λ not resonance). we summarize this discussion in the following lemma 6.1. s(λ) is meromorphic in c as an operator acting on r−1c c ∞ acc (∂x̄), with schwartz kernel the meromorphic continuation from {ℜ(λ) < 0} to c of the distribution (2λ−n)(ρ−λρ′ −λ r(λ))|b×b ∈ c −∞ (x̄ × x̄). its poles in {ℜ(λ) ≤ n 2 } are included in the set of resonances and have finite multiplicity, whereas the poles in {ℜ(λ) > n 2 } are first order poles with residue resλ0s(λ) = { − (−1)j+12−2j j!(j−1)! pj + πλ0 if λ0 = n 2 + j,j ∈ n πλ0 if λ0 /∈ n 2 + n where pj is the differential operator on (b,h0) with principal symbol σ0(pj ) = |ξ| 2j h0 , defined by [res n 2 +jρ −λ φ(λ)]|ρ=0 = (−1)j2−2j j!(j − 1)! pj and πλ0 is a finite-rank operator with schwartz kernel 2j ( (ρρ′)−λ0 resλ0r(λ) ) |b×b satisfying rank πλ0 = dim kerl2 (∆x −λ0(n−λ0)). proof : the meromorphic property of s(λ) and its schwartz kernel have been discussed, the statement about the poles outside {ℜ(λ) ≤ n 2 } is also clear by (5.5) . for the case of a pole λ0 with ℜ(λ0) > n 2 , the proof is the same than [7, prop 3.6]. the fact about the rank of πλ0 is quite straightforward by mimicking the proof of [10, th. 1.1]: we only need the indicial equation (5.3) and that there is no solution of (∆x − λ0(n − λ0))u = 0 with u ∈ ċ ∞ (x), this last fact being already proved by mazzeo [16]. � note that this lemma also holds for any boundary defining function ρ̂ ∈ c∞ acc (x̄). the operators pj will be discussed in next section. we now give functional relations for eisenstein functions and scattering operator: cubo 11, 5 (2009) scattering theory on geometrically finite quotients 165 proposition 6.2. if ℜ(λ) < 0, we have for w ∈ x, b′ ∈ b, e(λ; b′; w) = − ∫ b s(λ; b′; b)e(n−λ; b; w) dvolh0 (b) and there exists c > 1 such that for n large the meromorphic identity e(λ) = −s(λ)e(n−λ) (6.10) holds true in the strip −c−1n < ℜ(λ) ≤ n 2 as operators from ρnl2(x) to l2(b). proof : if for w ∈ x fixed and ℜ(λ) < 0 we multiply (5.18) by ρ(w′)−λ and take the limit w′ → b′ ∈ b, then we obtain the first result using the symmetry of the resolvent kernel (which also induces the symmetry of the kernel of s(λ)). the next part is just a meromorphic continuation using mapping properties of e(λ) and s(λ). � we deduce easily from this proposition and proposition 5.6 the corollary 6.3. if λ0 is such that ℜ(λ0) ≤ n 2 , λ0(n−λ0) /∈ σpp(∆x ) and s(λ) holomorphic at λ0, then λ0 is not a resonance. here is another inmportant property of s(λ): proposition 6.4. for ℜ(λ) = n 2 , s(λ) is invertible on l2(b) and we have s(λ)−1 = s(n−λ) = s(λ)∗ proof : the unitarity of s(λ) on the critical line comes directly from the density of ċ∞(b̄) ⊂ c∞ acc (∂x̄) in l2(b) and lemma 5.1 whereas the equation s(λ)−1 = s(n−λ) is a consequence of the definition of s(λ) and again the density of c∞ acc (∂x̄) in l2(b). � we give a description of the scattering operator as a pseudo differential in the class defined in section 3 and characterized by the type of singularity of its schwartz kernel on the blown-up manifold b̄ ×φ b̄. theorem 6.5. let λ 6∈ n 2 + n and λ not a resonance, then with definition (3.4), the scattering operator s(λ) is a φ-pseudo-differential operator on b̄ of order s(λ) ∈ ψ 2λ−n,eλ φ (b̄) + (rcr ′ c) −1c∞(∂x̄ ×∂x̄) with respect to volume density dvolh0 , where for k = 1, . . . ,n− 1 eλ(fk) = −2λ−k, eλ(ik) = −4λ, eλ(tk) = eλ(bk) = −k. 166 colin guillarmou cubo 11, 5 (2009) proof : for technical reasons, we begin by working with the density dvol h̃0 and it will suffice to multiply by the correct factors at the end. if η ∈ c∞0 ([0,∞)) is a function which is equal to 1 in a small neighbourhood of 0, we can decompose (6.7) as skmod(λ) = c(λ)ψ k l|y| 2λ−n ( η(∆y)∆ λ− n 2 y + (1 −η(∆yk ))∆ λ− n 2 yk ) ψk|y|2λ−n on l2(yk,dydz). the first term has a kernel ψkl(y)ψ k (y′)|y|2λ−n|y′|2λ−n ∫ rn−k eiξ.(y−y ′)|ξ|2λ−nη(|ξ|)dξ which is smooth for y,y′ in rn−k and since it is the fourier transform of a distribution classically conormal to 0, it is straightforward to check that it can be expressed by ψkl(y)ψ k (y′)|y|2λ−n|y′|2λ−nfλ( √ 1 + |y −y′|2) (6.11) with fλ(x) smooth on [0,∞) and having an expansion fλ(x) ∼ x −2λ+k ∞∑ j=0 aj (λ)x −j (6.12) when x → ∞. to describe the singularity of this kernel on the manifold b̄, we use near infinity the polar coordinates v = |y|−1,ω = y/|y|,v′ = |y′|−1,ω′ = y′/|y′|. since |y − y′| = | ω v′ − ω ′ v | we deduce that the kernel (6.11) ψkl( ω v )ψk( ω′ v′ )v−2λ+nv′ −2λ+n fλ   √ 1 + ∣∣∣∣ ω v′ − ω′ v ∣∣∣∣ 2   . first, it is clearly smooth in b ×b. by lifting | ω v′ − ω ′ v |,v,v′ on b̄ ×φ b̄ we have that βφ ∗   √ 1 + ∣∣∣∣ ω v′ − ω′ v ∣∣∣∣ 2  ρtkρbkρfk ∈ c ∞ (b̄ ×φ b̄) (6.13) does not vanish on fk, bk, tk and βφ ∗ (vv′)ρ−1 tk ρ−1 bk ρ−2 fk ρ−2 ik ∈ c∞(b̄ ×φ b̄) (6.14) does not vanish on tk, bk, fk, ik. from this and (6.12) it is straightforward to check that ψkl|y| 2λ−nη(∆y)∆ λ− n 2 y ψ k|y|2λ−n ∈ (ρtkρbk ) n−kρ2n−2λ−k fk ρ−4λ+2n ik c∞(b̄ ×φ b̄). (6.15) to deal with the term ψkl|y| 2λ−n (1 −η(∆yk ))∆ λ− n 2 yk ψk|y|2λ−n, we first analyze the operator a(λ) := ψkl|y| 2λ−n (1 + ∆yk ) λ− n 2 ψk|y|2λ−n. cubo 11, 5 (2009) scattering theory on geometrically finite quotients 167 for that we can begin to use a partition of unity (θi)i associated to a covering by some euclidian ball on t k and some functions θ′i ∈ c ∞ 0 (t k ) such that θ′i = 1 on the support of θi, then it is standard to see that for s ∈ c \ [0,∞) (∆y k + 1 −s) −1 = ∑ i θ′i(∆rn + 1 −s) −1θi + κ(s) (6.16) κ(s) := (∆y k + 1 −s) −1 ∑ i [∆z,θ ′ i](∆rn + 1 −s) −1θi. the kernel κ(s; y,z; y′,z′) of κ(s) can be written as the composition κ(s; y,z; y′′,z′′) = (∆yk + 1 −s) p ∫ yk κ1(s; y −y ′,z −z′)κ2(s; y ′ −y′′,z′,z′′)dy′dz′ (6.17) with κ1(s; y,z) := ∑ m∈z ∫ rn−k ei(ξ.y +ωm.z)(1 + |ξ|2 + |ωm| 2 ) −1−pdξ κ2(s; y ′ −y′′,z′,z′′) := ∑ i [∆z′,θ ′ i(z ′ )](∆rn + 1 −s) −1 (y′,z′; y′′,z′′)θi(z ′′ ). since for some ǫ > 0 we have [∆z′,θ ′ i(z ′ )]θi(z ′′ ) = 0 for |z − z′′| < ǫ, it suffices to use the explicit formula of the resolvent kernel of ∆rn with bessel functions to see that κ2(s) is smooth and satisfies the estimate |∂αy,z′,z′′κ2(s; y,z ′,z′′)| ≤ cα exp(−cα √ ℜ(s)(1 + |y |2)) for ℜ(s) ≥ 1 2 and some constant cα > 0. the kernel κ1(s) is continuous and uniformly bounded if p is large enough, moreover it satisfies for all n > 0 the estimate |∂αy κ2(s; y,z)| ≤ cα,n (1 + |y |) −n for some constant cα,n > 0. therefore, using all these estimates and change of variables y ′ = u+y in (6.17), it is straightforward to check that κ(s; w; w′) is smooth and satisfies the estimate for all n > 0 |∂αw,w′κ(s; w; w ′ )| ≤ cα,ne −c′ α ℜ(s) (1 + |y −y′|)−n. (6.18) for some constant cα,n,c ′ α > 0 and using the notation w = (y,z),w ′ = (y′,z′). let γ be the oriented contour in c defined by γ = { 1 2 + rei π 4 ;∞ > r > 0}∪{ 1 2 re−i π 4 ; 0 < r < ∞}. as a consequence of (6.16) and using cauchy formula, the kernel of a(λ) is (with the notation w = (y,z),w′ = (y′,z′)) a(λ; w; w′) = a1(λ; w,w ′ ) + a2(λ; w; w ′ ), a1(λ; w; w ′ ) := ψkl(y)|y| 2λ−nψk(y′)|y′|2λ−n ∑ i θ′i(z)θi(z ′ ) ∫ rn eiξ.(w−w ′) (1 + |ξ|2)λ− n 2 dξ, 168 colin guillarmou cubo 11, 5 (2009) a2(λ; w; w ′ ) := ψkl(y)|y| 2λ−nψk(y′)|y′|2λ−n ∫ γ sλ− n 2 κ(s; w; w′)ds. to analyze a1(λ), we use the polar coordinates v = |y| −1,ω = y/|y|,v′ = |y′|−1,ω′ = y′/|y′| in the y,y′ variables and we have w −w′ = ( ω v′ − ω ′ v ,z −z′) which vanishes only (and at first order) on the lifted interior diagonal dφ of b̄ ×φ b̄. from the fourier representation of a1(s; w; w ′ ), we deduce that a1(s; w; w ′ ) is a distribution which is polyhomogeneous conormal to dφ of order 2λ−n, vanishes at all order on the boundaries tk, bk, fk of b̄×φ b̄ and has a conormal singularity of order −4λ + 2n at ik (this last one coming from the term |y| 2λ−n|y′|2λ−n as before): β∗φa1(λ) ∈ ρ −4λ+2n ik i2λ−n(b̄ ×φ b̄; dφ). the behaviour of a2(λ) comes directly from (6.18) using the polar coordinates and (6.13) and (6.14) as before: we see that β∗φa2(λ) ∈ ρ ∞ tk ρ∞ bk ρ∞ fk ρ−4λ+2n ik c∞(b̄ ×φ b̄) thus β∗φa(λ) ∈ ρ −4λ+2n ik i2λ−n(b̄ ×φ b̄; dφ). (6.19) for n > ℜ(λ) − n 2 , we have skmod(λ) = c(λ)ψ k l|y| 2λ−n ( η(∆y )∆ λ− n 2 y + (1 + ∆yk ) λ− n 2 + (1 + ∆yk ) nϕ(1 + ∆yk ) ) ψk|y|2λ− n 2 with ϕ(x) = x−n ( (1 −η(x− 1))(x− 1)λ− n 2 − (1 −η(x))xλ− n 2 ) which is a symbol in (0,∞) of order λ− n 2 −n −1 in the sense that it has a support in [ǫ,∞) for some ǫ > 0, it is smooth and satisfies |∂lxϕ(x)| ≤ cl(1 + x) ℜ(λ)− n 2 −1−n−l. hence following the method of helffer-robert [13], we have ϕ(1 + ∆yk ) = 1 2πi ∫ i∞ −i∞ m[ϕ](s)(1 + ∆yk ) −sds where m[ϕ](s) is the mellin transform of ϕ defined by m[ϕ](s) := ∫ ∞ 0 ts−1ϕ(t)dt and which is rapidly decreasing on ir. from the previous study of (1+∆yk ) λ− n 2 and using mellin’s transform, we deduce that if b(λ) is the operator b(λ) := ψkl|y| 2λ−n (1 + ∆yk ) nϕ(1 + ∆yk )ψ k|y|2λ−n cubo 11, 5 (2009) scattering theory on geometrically finite quotients 169 then its kernel satisfies b(λ; w; w′) = b1(λ; w; w ′ ) + b2(λ; w; w ′ ) b1(λ; w,w ′ ) := ψkl(y)|y| 2λ−nψk(y′)|y′|2λ−n ∑ i θ′i(z)θi(z ′ ) ∫ rn eiξ.(w−w ′) (1 + |ξ|2)nϕ(1 + |ξ|2)dξ b2(λ; w; w ′ ) := ψkl(y)|y| 2λ−nψk(y′)|y′|2λ−n (1 + ∆w) n 2πi ∫ i∞ −i∞ m[ϕ](s) ∫ γ τs− n 2 κ(τ; w,w′)dτds. in view of the estimate (6.18) on κ(τ; w; w′) and its smoothness, we easily obtain that the kernel b2(λ; w; w ′ ), when lifted on b̄ ×φ b̄, has exactly the same properties than a2(λ; w,w ′ ). for the term b1(λ; w; w ′ ) we can proceed as for a1(λ; w,w ′ ) and it finally shows that βφ ∗b(λ) ∈ ρ−4λ+2n ik i2λ−n−1(b̄ ×φ b̄; dφ). combined with (6.15), (6.19), this proves the theorem after multiplying by the lift of (rcr ′ c) −n to return with the correct density. � remark: as a consequence, we can obtain quite general mapping properties for s(λ) (i.e. the actions of s(λ) on extendible distributions on b̄ conormal to ∂b̄) using general theory for those operators, see for exemple vaillant [26, section 2.2]. 7 conformal operators on the boundary as explained by graham-zworski [7], there is a strong connection between scattering theory on einstein conformally compact manifolds (in particular convex co-compact hyperbolic quotients) and conformal theory of its boundary. here similar results hold in this degenerate case. first recall from lemma 2.2 that for any ĥ0 := e 2ω0h0 ∈ [h0]acc, there exists a boundary defining function ρ̂ = eωρ ∈ c∞ acc (x̄), unique up to ċ∞(x̄), such that ω|∂x̄ = ω0 and which put the metric under the almost product form (2.12). this gives a way to identify special boundary defining functions of lemma 2.2 with representatives of the subconformal class [h0]acc. moreover we saw that the scattering operators s(λ), ŝ(λ) obtained by solving poisson problem respectively with ρ and ρ̂ (i.e. for conformal representatives h0 and ĥ0) are related by ŝ(λ)f = e−λω0s(λ)e(n−λ)ω0f. (7.1) in this sense, s(λ) is a conformally covariant operator and by looking at the residues we have the rule p̂j = e (− n 2 −j)ω0pje ( n 2 −j)ω0 which also makes this differential operator being conformally covariant. 170 colin guillarmou cubo 11, 5 (2009) let us now give a few words about conformal gjms laplacians. in [6], graham-jennemanson-sparling defined, on any n-th dimensional riemannian compact manifold (m,h0), a family of “natural” conformally covariant differential operators (pj )j with principal symbol ∆ j h0 . we call pj the j-th gjms laplacian. they are defined for j ∈ n if n is odd and for j ≤ n/2 integer if n is even and natural in the sense that they can be written in terms of covariant derivatives and curvature of h0 and conformally covariant in the sense that the operator p̂j obtained with the same expression than pj but with a conformal metric ĥ0 = e 2ω0h0 is related to pj by the identity p̂j = e −( n 2 +j)ω0pje ( n 2 −j)ω0. moreover p1 is yamabe’s laplacian and p2 is paneitz operator. if h0 is locally conformally flat and n > 2 is even, it is also proved in [6] that the pj can be constructed without obstruction for any j ∈ n, this is the case in particular of the conformal infinity of a convex co-compact hyperbolic quotients. note that, since the expression of pj is local with respect to the metric, these operators can also be defined on non-compact riemannian manifolds. graham and zworski [7] show that on asymptotically einstein manifolds (x,g) of dimension n + 1 (with x̄ the conformal compactification), the residue res n 2 +js(λ) of the scattering operator obtained by solving the poisson problem with boundary defining function x is pj on the conformal infinity (∂x̄,x 2g|t ∂x̄ ) for any j integer if n is odd (resp. for j ≤ n 2 if n is even). actually, we learnt from robin graham that this also holds for any j if n > 2 is even and if (x,g) has negative constant curvature outside a compact set, where in this case the conformal infinity is locally conformally flat. the reason, given in [4], which makes this special case working is that there is no obstruction to construct a hyperbolic conformally compact metric g on (0,ǫ]x × m with conformal infinity (m ≃ {x = 0},h0) for any (m,h0) locally conformally flat compact manifold, and actually g is necessarily given by g = x−2(dx2 + h0 −x 2p + x4( 1 4 ph−10 p)) (7.2) where p = (n− 2)−1(ric − (2n− 2)−1kh0) is the schouten tensor of h0, with k, ric the scalar and ricci curvatures of h0. this is a consequence of the constant curvature equation. since in our case the metric on x = γ\hn+1 is also hyperbolic, the curvature equation (which is local) implies again that the tensor ĥ(ρ̂) in (2.12) has all its taylor expansion with respect to ρ̂ at ρ̂ = 0 determined by ĥ0 = ĥ(0) if n > 2: the expression of ĥ(ρ̂) is explicit and, like (7.2), ĥ(ρ̂) = ĥ0 − ρ̂ 2p + ρ̂4( 1 4 pĥ−10 p) with p is the schouten tensor of ĥ0. if n > 2, we saw that the expression of res n 2 +js(λ) is obtained from the construction of φ(λ) exactly like in the convex co-compact case (the construction is local in term of ĥ(ρ̂) thus in term of ĥ0). by equivalence of the construction of φ(λ) in [7] and in our case, it is clear that cubo 11, 5 (2009) scattering theory on geometrically finite quotients 171 proposition 7.1. the operator pj of lemma 6.1 is the j-th conformal gjms laplacian defined in [6] on locally conformally flat compact manifolds in the sense that it has the same local expression in term of the metric h0. received: march, 2009. revised: may, 2009. references [1] bunke, u. and olbrich, m., scattering theory for geometrically finite groups, geometry, analysis and topology of discrete groups, 40-136, adv. lect. math. 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[26] vaillant, b., index and spectral theory for manifolds with generalized fibred cusps, phd. thesis. b9-cubo cubo a mathematical journal vol.10, n o ¯ 03, (137–144). october 2008 on some bitopological γ-separation axioms m. lellis thivagar department of mathematics, arul anandar college, karumathur, madurai (dt.)-625514, tamilnadu, india email: mlthivagar@yahoo.co.in s. athisaya ponmani department of mathematics, jayaraj annapackiam, college for women, periyakulam, theni (dt.)-625601, tamilnadu, india email: athisayaponmani@yahoo.co.in r. raja rajeswari department of mathematics, sri parasakthi college, courtalam, tirunelveli (dt.)-627802, tamilnadu, india email: raji−arul2000@yahoo.co.in and erdal ekici department of mathematics, canakkale onsekiz mart university, terzioglu campus, 17020 canakkale, turkey email: e.mail:eekici@comu.edu.tr abstract the aim of this paper is to introduce the notions of (i, j)-γ-t1, (i, j)γ-r1, (i, j)-γ-t2 and (i, j)-γ-us spaces, (i, j)-γ-open mappings and (i, j)-γ-irresolute mappings. 138 m. lellis thivagar et al. cubo 10, 3 (2008) resumen el objetivo de este art́ıculo es introducir las nociones de espacios (i, j)-γ-t1, (i, j)γ-r1, (i, j)-γ-t2 y (i, j)-γ-us, aplicaciones (i, j)-abiertas y (i, j)-γ-irresolutas. key words and phrases: (i, j)-γ-open set, (i, j)-γ-t1 space, (i, j)-γ-r1space, (i, j)-γ-us space, (i, j)-γ-open mapping, and (i, j)-γ-irresolute mapping. math. subj. class.: 54c55. 1 introduction in 1982, mashhour et al. [11] introduced the notion of preopen sets, also called locally dense sets by corson and michael [4]. the class of preopen sets properly contains the class of open sets. as the intersection of two preopen sets may fail to be preopen, the class of preopen sets does not always form a topology. in a submaximal space i.e. a topological space x in which every dense subset is open, collection of all preopen sets form a topology. indeed, many notions in topology can be defined in terms of preopen sets (see [3], [5], [8], [12] and [13]). in 1987, andrijevic [2] offered a new class of open sets called γ-open sets by utilizing preopen sets. recently, abd el monsef et al. [1] have applied preopen sets in connection with the topological applications of rough set measures in information systems. moreover, it has been shown in [6] that the notion preopen sets is important with respect to the digital topology. many researchers also used the notion of preopen sets in fuzzy topological spaces which professor el-naschie has recently shown in [7] the importance of the notion of fuzzy topology which may be relevent to quantam particle physics in connection with string theory and ǫ∞theory. in a bitopological space (x, τ 1, τ 2), the γ-open set is generalized in the form of (i, j)-γ-open set, i, j = 1, 2 and i 6= j [14] and these sets are used to define the separation axiom (i, j)-γ-t0 [14]. in this paper we define (i, j)γ-t1, (i, j)-γ-r1, (i, j)-γ-t2 and (i, j)-γ-u s spaces and show that (i, j)-γ-u s axiom is stronger than (i, j)-γ-t1 axiom and is weaker than (i, j)-γ-t2 axiom. we recall some definitions and concepts which are useful in the following sections. 2 preliminaries in a topological space (x, τ ), the interior and the closure of a subset a are denoted by int(a) and cl(a), respectively. definition 1 a subset a of x is called pre-open set [11] if a ⊂ int(cl(a)). cubo 10, 3 (2008) on some bitopological γ-separation axioms 139 definition 2 a subset a of a topological space (x, τ ) is called a γ-set [2] if a ∩ s ∈ p o(x) for each s ∈ p o(x). in the above definition, p o(x) is the family of all pre-open sets in x. the family of all γ-sets in x is denoted by γo(x). in the following sections by a space x, we mean a bitopological space (x, τ 1, τ 2). definition 3 a subset a of x is called (i, j)-pre-open [9] if a ⊂ τ i-int(τ j -cl(a)). definition 4 a subset a of x is called (i, j)-γ-open [14], if a ∩ b is (i, j)-pre-open for every (i, j)-pre-open set b in x. we denote the family of (i, j)-γ-open sets in x by (i, j)-γo(x). theorem 5 [14] the family of all (i, j)-γ-open sets in x forms a topology on x. definition 6 a subset a ⊂ x is called (i, j)-γ-closed [14] if its complement, ac in x is (i, j)-γopen. definition 7 for any a ⊂ x (i) (i, j)-γ-closure of a [14] is the intersection of all the (i, j)-γ-closed sets containing a and is written as (i, j)-γ-cl(a). (ii) (i, j)-γ-kernal of a [14] is the intersection of all the (i, j)-γ-open sets containing a and is written as (i, j)-γ-ker(a). definition 8 a space x is called (i, j)-γ-t0[14] if for x, y ∈ x, x 6= y, there exists u ∈ (i, j)γo(x) such that u contains only one of x and y but not the other where i, j = 1, 2, i 6= j. definition 9 a map f :(x, τ 1, τ 2) → (y, σ1, σ2) is called pairwise γ-continuous (briefly p.γ-continuous)[14] if the inverse image of each σi-open set of y is (i, j)-γ-open in x for i, j = 1, 2 and i 6= j. in the following section the (i, j)-γ-open sets are used to define some separation axioms. 140 m. lellis thivagar et al. cubo 10, 3 (2008) 3 some separation axioms in this section we define the (i, j)-γ-t1, (i, j)-γ-r1, (i, j)-γ-t2 and (i, j)-γ-u s spaces and study some characterizations. definition 10 a space x is called (i, j)-γ-t1 if for x, y in x, x 6= y, there exist u , v ∈ (i, j)γo(x) such that x ∈ u , y /∈ v and y ∈ v , x /∈ v . definition 11 a space x is said to be (i, j)-γ-r1 if for x, y in x, x 6= y with (i, j)-γ-cl({x}) 6= (i, j)-γcl({y}), there exist disjoint (i, j)-γ-open sets u , v such that (i, j)-γ-cl({x}) ⊂ u and (i, j)γ-cl({y}) ⊂ v . theorem 12 a space x is (i, j)-γ-t1 if and only if the singletons in x are (i, j)-γ-closed sets. proof. proof is evident since the family (i, j)-γo(x) is a topology. theorem 13 a space x is (i, j)-γ-r1 if and only if (i, j)-γ-ker({x}) 6= (i, j)-γ-ker({y}) for any x, y in x , there exist disjoint (i, j)-γ-open sets u and v such that γ-cl({x}) ⊂ u and γ-cl({y}) ⊂ v . definition 14 a space x is said to be (i, j)-γ-t2 if for any two distinct points x, y in x, there exist disjoint (i, j)-γ-open sets u , v such that x ∈ u and y ∈ v . theorem 15 a space x is (i, j)-γ-t2 if and only if it is (i, j)-γ-t0 and (i, j)-γ-r1. proof. necessity. if x is (i, j)-γ-t2 then it is (i, j)-γ-t1 and then (i, j)-γ-t0. since x is (i, j)γ-t1, by theorem 12, (i, j)-γ-cl({x}) = {x} and (i, j)-γ-cl({y}) = {y} for any two distinct points x, y in x. therefore, (i, j)-γ-cl({x}) 6= (i, j)-γ-cl({y}) for any two distinct points x, y in x and hence x is (i, j)-γ-r1. sufficiency. if x is (i, j)-γ-t0 and if x, y are two distinct points in x, there exists an (i, j)γ-open set u containing only one of x and y but not the other. let x ∈ u and y /∈ u , say. then y /∈ (i, j)-γ-ker({x}) and so (i, j)-γ-ker({x}) 6= (i, j)γ-ker({y}) for any two distinct points x, y in x. since x is (i, j)-γ-r1, by theorem 13, there exist disjoint (i, j)-γ-open sets u and v such that (i, j)-γcl({x}) ⊂ u and (i, j)-γ-cl({y}) ⊂ v . thus x ∈ u and y ∈ v and u ∩ v = ∅. hence x is (i, j)-γ-t2. definition 16 a net {xα:α ∈ d, ≥} is said to be bitopologically converges to a point x ∈ x, denoted by {xα:α ∈ d, ≥} γ → x if the net is eventually in every (i, j)-γ-open set containing x, i, j = 1, 2, i 6= j. cubo 10, 3 (2008) on some bitopological γ-separation axioms 141 theorem 17 if a map f :(x, τ 1, τ 2) → (y, σ1, σ2) is p.γ-continuous then for each x ∈ x and each net {xα:α ∈ d, ≥} in x, bitopologically γ-converging to x the image net {f (xα):α ∈ d, ≥} is bitopologically γ-convergent to f (x) in y . proof. let v ⊂ y be σi-open in y containing f (x), i = 1, 2. the bitopologically γ-convergence of the net {xα:α ∈ d, ≥} in x implies that there exists α0 ∈ d such that for all α ≥ α0, xα ∈ f −1(v ). therefore, f (xα) ∈ v for all α ≥ α0. hence the net {f (xα):α ∈ d, ≥} γ → f (x). definition 18 a space x is said to be (i, j)-γ-u s if every bitopologically γ-convergent net {xα:α ∈ d, ≥} in x is bitopologically γ-convergent to a unique pioint in x. proposition 19 every (i, j)γ-t2 space is (i, j)-γ-u s. proof. if possible, let the net {xα:α ∈ d, ≥} in a (i, j)γ-t2 space x be bitopologically γ-convergent to two distinct points x, y in x. then the net is eventually in every (i, j)-γ-open set containing x and also in every (i, j)-γ-open set containing y. this contradicts that x is (i, j)-γ-t2. proposition 20 every (i, j)-γ-u s space is (i, j)-γ-t1. proof. let x, y ∈ x, x 6= y. if xn = x for every x in the net {xα:α ∈ d, ≥} then it is evident that the net is bitopologically γ-convergent to x. since x is (i, j)-γ-u s, the net {xα:α ∈ d, ≥} cannot be bitopologically γ-convergent to y and hence there exists an (i, j)-γ-open set containing y but not x. a similar argument gives an (i, j)-γ-open set containing x but not y. hence x is (i, j)-γ-t1. remark 21 the following diagram holds for a space x as shown in the proposition 19 and 20. (i, j)-γ-t2 space ⇒ (i, j)-γ-u s space ⇒ (i, j)-γ-t1 space theorem 22 a space x is (i, j)-γ-t2 if and only if it is (i, j)-γ-r1 and (i, j)-γ-u s. proof. if x is (i, j)-γ-t2, then it is (i, j)-γ-r1, by theorem 15 and by proposition 19, x is (i, j)-γ-u s. conversely, if x is (i, j)-γ-r1 and (i, j)-γ-u s then by proposition 20, x is (i, j)-γ-t1. thus x is (i, j)γ-t1 and (i, j)-γ-r1. hence by theorem 15, x is (i, j)-γ-t2. 142 m. lellis thivagar et al. cubo 10, 3 (2008) 4 some bitopolgical γmappings in this section we define (i, j)-γ-open mappings and (i, j)-γ-irresolute mappings. definition 23 a map f :(x, τ 1, τ 2) → (y, σ1, σ2) is called (i, j)-γ-open if the image of each τ iopen set in x is (i, j)-γ-open in y , i, j = 1, 2, i 6= j. recall that a map f :(x, τ 1, τ 2) → (y, σ1, σ2) is called (i, j)-pre-open if for each τ i-open set in x, f (u ) is (i, j)-pre-open, i, j = 1, 2, i 6= j. remark 24 every (i, j)-γ-open map is (i, j)-pre-open but the converse is not true in general as shown in the following example. example 25 let x = {a, b, c}, τ 1 = {∅, {b, c}, x}, y = {a, b, c}, σ1 = {∅, {a}, x} and σ2 = {∅, {a, b}, x}. define a map f :x → y as follows f (a) = b, f (b) = a, f (c) = c. then f is (1, 2)pre-open but not (1, 2)-γ-open since f ({b, c}) = {a, c} which is (1, 2)-pre-open but not (1, 2)-γ-open. recall that a space x is said to be pairwise hausdorff[12] if for x, y ∈ x, x 6= y, there exist open sets u , v , u ∈ τ 1, v ∈ τ 2 such that x ∈ u and y ∈ v . theorem 26 let f :(x, τ 1, τ 2) → (y, σ1, σ2) be a bijective (i, j)-γ-open map.if x be pairwise hausdorff, then y is (i, j)-γ-t2. proof. let y1 and y2 be two distinct points in y . since f is bijective there exist x1 and x2 in x such that f (x1) 6= f (x2). the space x is pairwise hausdorff and so there exist disjoint sets u , v , u ∈ τ 1 and v ∈ τ 2 such that x1 ∈ u and x2 ∈ v . then f (x1) ∈ f (u ) and f (x2) ∈ f (v ), f (u ) and f (v ) are (i, j)-γ-open sets and f (u ) ∩ f (v ) = ∅. thus y is (i, j)-γ-t2. definition 27 a map f :(x, τ 1, τ 2) → (y, σ1, σ2) is called (i, j)-γ-irresolute if the inverse image of every (i, j)-γ-open set in y is (i, j)-γ-open in x, i, j = 1, 2, i 6= j. theorem 28 if f :(x, τ 1, τ 2) → (y, σ1, σ2) is an (i, j)-γ-irresolute bijective mapping and if y is a (i, j)-γ-t2 space then, x is (i, j)-γ-t2. proof. let x1, x2 be two distinct points in x. then there exist y1, y2 in y such that f (x1) = y1 and f (x2) = y2 and y1 6= y2. since y is (i, j)-γ-t2, there exist disjoint (i, j)-γ-open sets u , v such that y1 ∈ u and y2 ∈ v . as f is (i, j)-γ-irresolute, f −1(u ) and f −1(v ) are (i, j)-γ-open sets in x containing x1 and x2 respectively. hence x is (i, j)-γ-t2. cubo 10, 3 (2008) on some bitopological γ-separation axioms 143 theorem 29 let f :(x, τ 1, τ 2) → (y, σ1, σ2) and g:(y, σ1, σ2) → (z, ̺1, ̺2) be two maps.then (i) if f is (i, j)-γ-irresolute and g is p.γ-continuous then, g ◦ f is p.γ-continuous (ii) if both f and g are (i, j)-γ-irresolute then, g ◦ f is (i, j)-γ-irresolute. proof. obvious. acknowledgement the second and third authors wish to acknowledge the support of university grants commission, new delhi, india, fip-x plan. received: june 2008. revised: august 2008. references [1] m.e. abd el-monsef, b.m. taher and a.s. salama, topological applications of rough set measures in information systems, preprint. 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[14] saeid jafari, m. lellis thivagar, s. athisaya ponmani and r. raja rajeswari, a bitopological view of γ-open sets, communicated. n12 cubo a mathematical journal vol.10, n o ¯ 01, (67–75). march 2008 somewhat fuzzy semi α-irresolute functions v. seenivasan p.g. department of mathematics, jawahar science college neyveli – 607 803, tamilnadu, india email: krishnaseenu@rediffmail.com g. balasubramanian ramanujan institute for advanced study in mathematics university of madras, chennai, 600 005, tamilnadu, india. and g. thangaraj p.g. department of mathematics, jawahar science college neyveli – 607 803, tamilnadu, india abstract in this paper the concepts of somewhat fuzzy semi α-irresolute functions and strongly somewhat fuzzy semi-open functions are introduced and studied. besides giving characterizations of these functions, some interesting properties of these functions are also given. resumen en este art́ıculo son introducidos y estudiados lon conceptos de funciones semi fuzzy α-irresoluta y funciones fuertemente fuzzy semi-abiertas. propiedades de esta clase de funciones son dadas. 68 v. seenivasan, g. balasubramanian and g. thangaraj cubo 10, 1 (2008) key words and phrases: somewhat fuzzy semi α-irresolute, fuzzy semi α-irresolute, fuzzy α-irresolute, somewhat fuzzy semi-open functions. math. subj. class.: 54a40. 1 introduction the fuzzy concept has invaded almost all branches of mathematics ever since the introduction of fuzzy sets by l.a.zadeh [12]. fuzzy sets have applications in many fields such as information [6] and control [8]. the theory of fuzzy topological spaces was introduced and developed by c.l.chang [3] and since then various notions in classical topology have been extended to fuzzy topological spaces. the concept of somewhat continuous functions was introduced by karl r.gentry and hughes b.hoyle iii in [4] and this concept was studied in connection with the idea of feeble continuous function and feebly open function introduced and by zdenek frolik in [13]. the concept of semi α-irresolute functions was introduced and studied in [11]. the concepts of somewhat fuzzy continuous functions and somewhat fuzzy semi-continuous functions was introduced and studied by g. thangaraj and g. balasubramanian in [9] and [10] respectively. in this paper we introduce the concepts of somewhat fuzzy semi α-irresolute functions and strongly somewhat fuzzy semi-open functions and study their properties. 2 preliminaries by a fuzzy topological space we shall mean a non-empty set x together with fuzzy topology t [3] and denote it by (x, t ). a fuzzy point in x with support x ∈ x and value p (0 < p ≤ 1) is denoted by xp. the complement µ ′ of a fuzzy set µ is 1 − µ, defined by µ ′ (x) = (1x −µ)(x) = 1 − µ(x), for all x ∈ x [3]. if λ is a fuzzy set in x and µ is a fuzzy set in y , then λ × µ is a fuzzy set in x × y , defined by (λ × µ)(x, y) = min(λ(x), µ(y)), for every (x, y) in x × y [1]. a fuzzy topological space x is product related to a fuzzy topological space y [1] if for fuzzy sets γ in x and ξ in y whenever λ ′ (= 1 − λ) � γ and µ ′ ( = 1 − µ) � ξ (in which case (λ ′ × 1) ∨ (1 × µ ′ ) ≥ (γ × ξ)) where λ is a fuzzy open set in x and µ is a fuzzy open set in y , there exists a fuzzy open set λ1 in x and a fuzzy open set µ1 in y such that λ1 ′ ≥ γ or µ1 ′ ≥ ξ and (λ1 ′ × 1) ∨ (1 × µ ′ 1) = (λ ′ × 1) ∨ (1 × µ ′ ). let f : x → y be a mapping from x to y . if λ is a fuzzy set in x, f (λ) is defined by f (λ) (y) = { sup λ(x), x∈f −1(y) if f −1 (y) 6= φ 0, otherwise, for each y ∈ y ; and if µ is a fuzzy set in y , f −1(µ) is defined by f −1(µ) (x) = µ f (x), for cubo 10, 1 (2008) somewhat fuzzy semi α-irresolute functions 69 each x ∈ x [3]. let f be a mapping from x to y . then the graph g of f is mapping from x to x × y sending x in x to (x, f (x)). for two mappings f1 : x1 → y1 and f2 : x2 → y2 , we define the product f1 × f2 of f1 and f2 to be a mapping from x1×x2 to y1 ×y2 sending (x1, x2) in x1× x2 to (f1(x1), f2 (x2)). for any fuzzy set λ in a fuzzy topological space, it is shown in [1] that (i) 1 – cl λ = int(1 − λ), (ii) cl(1 − λ) = 1−int λ. a fuzzy set λ in fuzzy topological space (x, t ) is called fuzzy α-dense if there exists no fuzzy α-closed set µ such that λ < µ < 1. definition 2.1 [5]: a function f : (x, t ) → (y , s) is said to be fuzzy α-irresolute if f −1 (λ) is fuzzy α-open set in (x, t ) for every fuzzy α-open set λ of (y , s). definition 2.2 [7]: a function f from a fuzzy topological space (x, t ) to a fuzzy topological space (y , s) is said to be fuzzy semi α-irresolute if f −1(λ) is fuzzy semi-open set in (x, t ) for each fuzzy α-open set λ in (y , s). definition 2.3 [10]: let (x, t ) and (y, s) be fuzzy topological spaces. a function f : (x, t ) → (y, s) is called somewhat fuzzy semi-open function if and only if for all λ ∈ t , λ 6= 0 there exists a fuzzy semi-open set µ in y such that µ 6= 0 and µ ≤ f (λ). definition 2.4 let (x, t ) be a fuzzy topological space and λ be any fuzzy set in x. 1. λ is called fuzzy α-open set [2] if λ ≤ int cl intλ 2. λ is called fuzzy semi-open set [1] if λ ≤ cl intλ the complement of fuzzy α-open (fuzzy semi-open) set is called fuzzy α-closed ( fuzzy semi-closed) set. 3 somewhat fuzzy semi α-irresolute functions the concept of fuzzy semi α-irresolute functions was introduced and studied in [7]. in this section we shall introduce the concept of somewhat fuzzy semi α-irresolute functions and study their properties. definition 3.1 let (x,t ) and (y, s) be any two fuzzy topological spaces. a function f : (x,t ) → (y, s) is said to be somewhat fuzzy semi α-irresolute if for any non-zero fuzzy α-open set λ in y and f −1(λ ) 6= 0 then there exists a fuzzy semi-open set µ 6= 0 in x such that µ ≤ f −1(λ ). 70 v. seenivasan, g. balasubramanian and g. thangaraj cubo 10, 1 (2008) clearly every fuzzy semi α-irresolute function is somewhat fuzzy semi α-irresolute, but the converse is not true as the following example shows:example 3.1 let µ1, µ2 and µ3 be fuzzy sets on i = [0, 1] µ1(x) = { 0, 0 ≤ x ≤ 1 2 2x − 1, 1 2 ≤ x ≤ 1, µ2(x) =    1, 0 ≤ x ≤ 1 4 −4x + 2, 1 4 ≤ x ≤ 1 2 0, 1 2 ≤ x ≤ 1, µ3(x) = { 0, 0 ≤ x ≤ 1 4 1 3 (4x − 1), 1 4 ≤ x ≤ 1. let s1 = {0, µ2, µ3, (µ2 ∨ µ3), (µ2 ∧ µ3), 1} and s2 = { 0, µ2, 1}. then s1 and s2 are fuzzy topologies on i. let f : (i, s1) → (i,s2) be defined by f (x) = x/2 for each x ∈ i. let λ be fuzzy set such that 0 < λ < µ2. then λ is not fuzzy α-open set in (i, s2). therefore the only non-zero fuzzy α-open sets in (i, s2) are 1, µ2 and fuzzy sets ρ such that µ2 < ρ < 1. now f −1(1) = 1; f −1(µ2) = µ ′́ 1 and f −1 (ρ) = 1. the fuzzy semi-open set µ2 in (i, s1) is contained in f −1(1), f −1(µ2) and f −1 (ρ). this proves f is somewhat fuzzy semi α-irresolute function from (i, s1) to (i, s2). it can be easily seen that int µ1 ′ = µ2 and cl µ2 = µ3 ′ in (i, s1). now µ2 is a fuzzy α-open set in (i, s2). since µ1 ′ � cl int µ1 ′ in (i, s1), µ1 ′ is not fuzzy semi-open set in (i, s1). then f −1 (µ2) = µ1 ′, which is not a fuzzy semi-open set in (i, s1). hence f is not fuzzy semi α-irresolute function. theorem 3.1 let f : (x,t ) → (y, s) and g : (y, s) → (z,q) be any two functions. if f is somewhat fuzzy semi α-irresolute and g is fuzzy α-irresolute, then g ◦ f is somewhat fuzzy semi α-irresolute. proof: let λ be non-zero fuzzy α-open set in (z, q). since g is fuzzy α-irresolute, g−1(λ) 6= 0 is fuzzy α-open set in (y , s). now (g ◦ f )−1(λ ) = f −1(g−1(λ)) 6= 0. since g−1(λ) is fuzzy α-open in (y , s) and f is somewhat fuzzy semi α-irresolute, we conclude that there exists a fuzzy semi-open set µ 6= 0 in (x, t ) such that µ ≤ f −1( g−1(λ)) = (g ◦ f )−1(λ ). hence g ◦ f is somewhat fuzzy semi α-irresolute. in above theorem 3.1 if f is either fuzzy α-irresolute or fuzzy semi α-irresolute and g is somewhat fuzzy semi α-irresolute, then it is not necessarily true that g ◦ f is somewhat fuzzy semi α-irresolute as the following example shows:example 3.2 define f : i → i by f (x) = x/2. let µ1, µ2 and µ3 be fuzzy sets in i described in example 3.1. let t1 = {0, µ1, µ2, µ1∨µ2, 1}; t2 = { 0, µ ′ 2, 1} and t3 = {0, cubo 10, 1 (2008) somewhat fuzzy semi α-irresolute functions 71 µ3 ′ , 1}. then t1, t2 and t3 are fuzzy topologies on i. consider the mapping f : (i, t3) → (i, t1). it can be easily seen that int µ ′ 1 = µ ′ 3; cl µ ′ 3 = 1; cl µ ′ 1 = 1 in (i, t3). since int µ ′ 1 = µ ′ 3 and cl µ ′ 3= 1, µ ′ 1 is fuzzy semi-open set and also fuzzy α-open set in (i, t3). let λ, ρ and δ be fuzzy sets such that 0 < λ < µ1, µ1 < ρ < µ2 and µ2 < δ < (µ1 ∨ µ2). then λ , ρ and δ are not fuzzy α-open sets in (i, t1). therefore the only fuzzy α-open sets in (i, t1) are 0, 1 , µ1, µ2, µ1∨µ2 and fuzzy sets µ such that (µ1 ∨ µ2) < µ < 1. now f −1 (0) = 0; f −1 (1) = 1; f −1(µ1) = 0; f −1 (µ2) = µ ′ 1 = f −1 (µ1 ∨ µ2) and f −1 (µ) = 1 are fuzzy α-open sets in (i, t3) and also fuzzy semi-open sets in (i,t3). therefore f is fuzzy α-irresolute from (i,t3) to (i,t1) and also f is fuzzy semi α-irresolute from (i, t3) to (i, t1). let g : (i, t1) → (i, t2) be defined by g(x) = x, for each x ∈ i. let λ be fuzzy set such that 0 < λ < µ ′ 2. then λ is not fuzzy α-open set in (i, t2). therefore the only non-zero fuzzy α-open set are 1, µ ′ 2 and fuzzy sets ρ such that µ ′ 2 < ρ < 1. now g −1 (1) = 1; g−1( µ ′ 2) = µ ′ 2 and g −1 (ρ) = 1. the fuzzy semi-open set µ1 in (i, t1) is contained in g −1 (1), g−1( µ ′ 2) and g −1 (ρ). therefore g is somewhat fuzzy semi α-irresolute function. now consider the functions (g ◦ f ) : (i,t3) → (i, t2). then (g ◦ f ) −1 (µ ′ 2) = f −1 (g −1 (µ ′ 2)) = f −1 (µ ′ 2) = µ1 and (g ◦ f ) −1 (1) = f −1 (g −1 (1)) = f −1 (1) = 1. but (g ◦ f ) −1 (µ ′ 2) = µ1 and there is no non-zero fuzzy semi-open set in (i, t3) such that it is contained in (g ◦ f )−1(µ ′ 2) = µ1. therefore (g ◦ f ) is not somewhat fuzzy semi α-irresolute function. definition 3.2 [10]: a fuzzy set λ in fuzzy topological space (x, t ) is called fuzzy semidense if there exists no fuzzy semi-closed set µ such that λ < µ < 1. theorem 3.2 : let (x,t ) and (y, s) be any two fuzzy topological spaces and f : (x,t ) → (y, s) be a function. then the following assertions are equivalent. (1) f is somewhat fuzzy semi α-irresolute. (2) if λ is a fuzzy α-closed set in y such that f −1(λ ) 6= 1, then there exists a fuzzy semi-closed set µ 6= 1 in x such that µ ≥ f −1(λ ). (3) if λ is a fuzzy semi-dense set in x, then f (λ) is fuzzy α-dense set in y . proof: (1) ⇒(2): suppose f is somewhat fuzzy semi α-irresolute and λ is a fuzzy α-closed set in y such that f −1 (λ) 6= 1. therefore clearly 1 − λ is fuzzy α-open set in y , and f −1 (1 − λ) = 1 − f −1(λ) 6= 0 (since f −1(λ) 6= 1). by (1), there exists a fuzzy semi-open set η in x such that η ≤ f −1(1 − λ). that is, η ≤ 1 − f −1(λ) which implies that f −1(λ) ≤ 1 − η. clearly 1 − η is fuzzy semi-closed set and taking µ = 1 − η, we have therefore f −1(λ) ≤ µ. thus we find that (1) ⇒(2) is proved. 72 v. seenivasan, g. balasubramanian and g. thangaraj cubo 10, 1 (2008) (2) ⇒ (3): let λ be a fuzzy semi-dense set in x and suppose f (λ) is not fuzzy α-dense in y . then there exists a fuzzy α-closed set η(say) in y such that f (λ) < η < 1. (a) since η < 1, f −1 (η) 6= 1 and so by (2) there exists a fuzzy semi-closed set δ (δ 6= 1) such that δ ≥ f −1(η) > f −1(f (λ)) ≥ λ ( from (a)). that is, there exists a fuzzy semi-closed set δ such that δ > λ which is contradiction to the assumption on λ .therefore (2) ⇒ (3) is proved. (3) ⇒(1): suppose λ is fuzzy α-open set in y and f −1(λ) 6= 0 and therefore λ 6= 0. suppose there exists no fuzzy semi-open set µ in x such that µ ≤ f −1(λ ). then 1 − f −1(λ) is fuzzy set in x such that there is no fuzzy semi-closed set δ in x with 1−f −1(λ) < δ < 1(otherwise 1 − f −1(λ) < δ ⇒ 1 − δ ≤ f −1(λ) and 1 − δ is fuzzy semi-open set, which is a contradiction). this means 1−f −1(λ) is fuzzy semi-dense in x. then by (3), f (1−f −1(λ)) is fuzzy α-dense in y . but f (1 − f −1(λ)) = f ( f −1(1 − λ)) < 1 −λ < 1(since λ 6= 0). this is contradiction to the fact that f (1 − f −1(λ)) is fuzzy α-dense. therefore, there exists a fuzzy semi-open set µ in x such that µ ≤ f −1(λ). hence f is somewhat fuzzy semi α-irresolute function. theorem 3.3 let (x1, t1), (x2, t2), (y1, s1) and (y2, s2) be fuzzy topological spaces such that x1 is product related to x2 and y1 is product related to y2. let f1 : x1 → y1 and f2 : x2 → y2 be somewhat fuzzy semi α-irresolute functions. then f1 × f2 : x1 × x2→ y1 × y2 is somewhat fuzzy semi α-irresolute function. proof: let λ = ∨ i,j (λ i × µj ) be fuzzy α-open set in y1 ×y2 (where λi and µj are fuzzy α -open sets in y1 and y2, respectively). we can assume that λi’s and µj’s are not all zeros. if any one is zero, that factor can be omitted. now (f1 × f2) −1(λ) = (f1 × f2) −1( ∨ i,j (λi×µj)) = ∨ i,j (f1 × f2) −1 (λi ×µj ) = ∨ i,j (f −11 (λi) × f −1 2 (µj)). since f1 : x1 → y1 is somewhat fuzzy semi α-irresolute and λi is fuzzy α-open set in y1 and f −1 1 (λi) 6= 0, there exists a fuzzy semi-open set δi in x1 such that δi ≤ f −1 1 (λi). also since f2 : x2 → y2 is somewhat fuzzy semi α-irresolute and µj is fuzzy α-open set in y2 and f −1 2 (µj ) 6= 0, there exists a fuzzy semi-open set ηj in x2 such that ηj ≤ f −1 2 (µj ). therefore δi×ηj ≤ f −1 1 (λi) × f −1 2 (µj ) = (f1 ×f2) −1 (λi ×µj). then by theorem 4.3 and theorem 4.6 in [1] ∨ i,j (δi×ηj) is a fuzzy semi-open set and ∨ i,j (δi×ηj) ≤ ∨ i,j (f1 × f2) −1 (λi ×µj ) = (f1 × f2) −1( ∨ i,j (λi×µj )) = (f1 × f2) −1(λ). this proves f1 × f2 is somewhat fuzzy semi α-irresolute. cubo 10, 1 (2008) somewhat fuzzy semi α-irresolute functions 73 the following lemma which is established in [1] is required to prove theorem 3.4. lemma 3.1 [1]: let g : x → x ×y be the graph of a function f : x → y . if λ is a fuzzy set in x and µ is a fuzzy set in y , then g−1(λ ×µ) = λ ∧ f −1(µ). theorem 3.4 let f : (x,t ) → (y, s) be a function from fuzzy topological space (x,t ) to another fuzzy topological space (y, s).if the graph g : x → x ×y of f is somewhat fuzzy semi α-irresolute, then f is somewhat fuzzy semi α-irresolute. proof: let λ be a non zero fuzzy α-open set in y . then, by lemma 3.1, we have f −1(λ) = 1 ∧ f −1(λ) = g−1(1 ×λ). since g is somewhat fuzzy semi α-irresolute and 1 ×λ 6= 0 is a fuzzy α-open set in x ×y , there exists a fuzzy semi-open set µ(6= 0) (say) in x such that µ ≤ g−1(1 ×λ) = f −1(λ). this proves that f is somewhat fuzzy semi α-irresolute function. 4 strongly somewhat fuzzy semi-open functions the concept of somewhat fuzzy semi-open function was introduced and studied in[10]. in this section we shall introduce a strongly notion as follows:definition 4.1 : let (x, t ) and (y , s) be any two fuzzy topological spaces. a function f : (x, t ) → (y , s) is called strongly somewhat fuzzy semi-open if and only if for each non-zero fuzzy α-open set λ in (x,t ), there exists a fuzzy semi-open set µ in (y ,s) such that µ 6= 0 and µ < f (λ ). clearly every strongly somewhat fuzzy semi-open function is somewhat fuzzy semi-open function. however the converse is not true as the following example shows:example 4.1 let µ1, µ2 and µ3 be fuzzy sets in i described in example 3.1. clearly t1 = {0, µ1, 1} and t2 = {0, µ2, 1} are fuzzy topologies on i. let f : (i, t1) → (i, t2) be defined by f (x) = min{2x, 1} for each x ∈ i. it can be easily seen that int µ3 = µ1; cl µ1 = 1 in ( i, t1). simple computations gives f (0) = 0; f (1) = 1; f (µ1) = 0. thus f is somewhat fuzzy semi-open function. since µ3≤ int cl int µ3 in (i, t1), µ3 is fuzzy α-open set in (i, t1). but f (µ3)(y) = { 0, 0 ≤ y ≤ 1 2 , 1 3 , 1 2 ≤ y ≤ 1. let λ be any non-zero fuzzy set such that λ ≤ f (µ3) in (i, t2). then cl int λ = 0, this shows that λ is not fuzzy semi-open set. thus there is no non-zero fuzzy semi-open set such that it is contained in f (µ3). hence f is not strongly somewhat fuzzy semi-open functions. 74 v. seenivasan, g. balasubramanian and g. thangaraj cubo 10, 1 (2008) theorem 4.1 suppose (x, t ) and (y , s) be fuzzy topological spaces. let f : (x, t ) → (y , s) be an onto function. if f is strongly somewhat fuzzy semi-open function and λ is a fuzzy semi-dense set in y , then f −1(λ) is fuzzy α-dense in x. proof: suppose λ is a fuzzy semi-dense set in y . we want to show that f −1(λ) is a fuzzy α-dense set in x. suppose not. then there exists a fuzzy α-closed set µ in x such that f −1 (λ) < µ < 1. then 1 − f −1(λ) > 1 − µ > 0. f (1 − f −1(λ)) > f (1 − µ) which implies that f ( f −1 (1 − λ) > f (1 − µ). that is, f (1 − µ) < f (f −1(1 − λ)) = 1 − λ . now µ is fuzzy α-closed set ⇒ 1 − µ is fuzzy α-open set in x. since f is strongly somewhat fuzzy semi-open, 1 − µ is fuzzy α-open in x ⇒there exists a fuzzy semi-open set δ 6= 0 in y such that δ < f (1 − µ). therefore δ < f (1 − µ) < 1 − λ ⇒ δ < 1 − λ ⇒ λ < 1 − δ. now 1 − δ is fuzzy semi-closed set and λ < 1 − δ ⇒ λ is not a fuzzy semi-dense set in y , which is a contradiction to our hypothesis. therefore f −1 (λ) must be a fuzzy α-dense in (x, t ). theorem 4.2 suppose (x, t ) and (y , s) be fuzzy topological spaces. let f : (x, t ) → (y , s) be a 1-1 and onto function. then the following conditions are equivalent. (1) f is strongly somewhat fuzzy semi-open. (2) if λ is a fuzzy α-closed set in x such that f (λ) 6= 1, then there exists a fuzzy semiclosed set µ in y such that µ 6= 1 and f (λ) < µ . proof: (1)⇒(2). let λ be a fuzzy α-closed set in x such that f (λ) 6= 1. then 1 − λ is fuzzy α-open set and since f is 1-1 and onto f (1 − λ) = 1 − f (λ) 6= 0 [3]. as f is strongly somewhat fuzzy semi-open, there exists a fuzzy semi-open set η in y such that η 6= 0 and η < f (1 − λ) = 1 − f (λ). that is f (λ) < 1 − η = µ (say) and µ is fuzzy semi-closed set. this proves (1)⇒(2). (2)⇒(1). let λ be a fuzzy α-open set in x such that λ 6= 0. then 1 − λ is fuzzy α-closed set and 1 − λ 6= 1. now f (1 − λ) = 1 − f (λ) 6= 1 (for, if 1 − f (λ) = 1, then f (λ) = 0 ⇒ λ = 0). hence by (2) there exists a fuzzy semi-closed set µ in y such that µ > f (1 − λ). then µ > 1 − f (λ ). that is f (λ) > 1 − µ = δ (say). clearly δ is fuzzy semi-open set in y such that δ < f (λ) and δ 6= 0 (since µ 6= 1). this completes the proof of (2)⇒(1). received: november 2006. revised: august 2007. cubo 10, 1 (2008) somewhat fuzzy semi α-irresolute functions 75 references [1] k.k. azad, on fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, j. math. anal. appl., 82 (1981), 14–32. [2] a.s. bin shahna, on fuzzy strong semicontinuity and fuzzy pre-continuity, fuzzy sets and systems, 44 (1991), 303–308. [3] c.l. chang, fuzzy topological spaces, j. math. anal.appl., 24 (1968), 182–190. 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[13] zdenek frolik, remarks concerning the invariance of baire spaces under mappings, czech. math. j., 11 (1961), 381–385. fuzzy1.pdf a mathematical journal vol. 7, no 2, (1-20). august 2005. gröbner and diagonal bases in orlik-solomon type algebras raul cordovil 1 departamento de matemática, instituto superior técnico av. rovisco pais. 1049-001 lisboa portugal cordovil@math.ist.utl.pt david forge laboratoire de recherche en informatique. batiment 490 universite paris sud 91405 orsay cedex france forge@lri.fr abstract the orlik-solomon algebra of a matroid m is the quotient of the exterior algebra on the points by the ideal �(m) generated by the boundaries of the circuits of the matroid. there is an isomorphism between the orlik-solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. in this article a generalization of the orlik-solomon algebras, called χ-algebras, are considered. these new algebras include, apart from the orlik-solomon algebras, the orlik-solomon-terao algebra of a set of vectors and the cordovil algebra of an oriented matroid. to encode an important property of the “no broken circuit bases” of the orlik-solomon-terao algebras, andrás szenes has introduced a particular type of bases, the so called “diagonal bases”. this notion extends naturally to the χ-algebras. we give a survey of the results obtained by the authors concerning the construction of gröbner bases of �χ(m) and diagonal bases of orlik-solomon type algebras and we present the combinatorial analogue of an “iterative residue formula” introduced by szenes. 1the first author’s research was supported in part by fct (portugal) through program pocti and the project sapiens/36563/99. 2 raul cordovil and david forge 7, 2(2005) resumen el álgebra de orlik-solomon de una matroide m es el cuociente del álgebra exterior en los puntos por el ideal �(m) generado por los acotamientos de los circuitos de la matroide. existe un isomorfismo entre el álgebra de orlik-solomon de una matroide compleja y la cohomoloǵıa del complemento de un arreglo complejo de hiperplanos. en este art́ıculo se considera una generalización de las algebras de orlik-solomon, llamadas χ-algebras. estas nuevas álgebras incluyen, además de las álgebras de orlik-solomon, el álgebra de orlik-solomon-terao de un conjunto de vectores y el álgebra de cordovil de una matroid orientada. para recalcar una importante propiedad de las ”bases de circuitos no quebrados” de las álgebras de orlik-solomon-terao, andrás szenes ha introducido un particular tipo de bases, llamadas ”bases diagonales”. este concepto se extiende naturalemente a la χalgebras. damos una mirada a los resultados obtenidos por los autores referentes a la construcción de las bases de gröbner de �χ(m) y bases diagonales de los tipos de algebras de orlik-solomon, y presentamos el análogo combinatorio de una ”fórmula de residuos iterativa” introducida por szenes. key words and phrases: arrangement of hyperplanes, broken circuit, cohomology algebra, matroid, oriented matroid, orlik-solomon algebra, gröbner bases. math. subj. class.: primary: 05b35, 52c35; secondary: 14f40. 1 introduction let m = m([n]) be a matroid on the ground set [n] := {1, 2, . . . ,n}. the orliksolomon algebra of a matroid m is the quotient of the exterior algebra on the points by the ideal �(m) generated by the boundaries of the circuits of m. the isomorphism between the orlik-solomon algebra of complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes was established in [12]. the orlik-solomon algebras have been then intensively studied. a general reference on hyperplane arrangements and orlik-solomon algebras is [14]. descriptions of developments from the early 1980’s to the end of 1999, together with the contributions of many authors, can be found in [9, 21]. in this article a generalization of the orlik-solomon algebras, called χ-algebra, is considered. these new algebras include, apart from the orlik-solomon algebras, the orlik-solomon-terao algebra of a set of vectors [15] and the cordovil algebra of an oriented matroid [7]. we will survey recent results concerning this family of orliksolomon type algebras (see [8, 10, 11]). in this introduction, we will recall the origin of the orlik-solomon algebra and we will develop the different notions used in the next sections like matroids and oriented matroids, the orlik-solomon algebra and its generalizations, its diagonal bases and the gröbner bases of the defining ideal. let v be a vector space of dimension d over some field k. a (central) arrangement (of hyperplanes) in v, ak = {h1, . . . ,hn}, is a finite listed set of codimension one 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 3 vector subspaces. given an arrangementak we always suppose fixed a family of linear forms { θhi ∈ v ∗ : hi ∈ak, ker(θhi ) = hi } , where v ∗ denotes the dual space of v . let l(ak) be the intersection lattice of ak: i.e., the set of intersections of hyperplanes in ak, partially ordered by reverse inclusion. there is a matroid m(ak) on the ground set [n] determined by ak: a subset d ⊆ [n] is a dependent set of m(ak) iff there are scalars ζi ∈ k, i ∈ d, not all nulls, such that ∑ i∈d ζiθhi = 0. a circuit is a minimal dependent set with respect to inclusion. if k is an ordered field an additional structure is obtained: to every circuit c, ∑ i∈c ζiθhi = 0, we associate a partition (determined up to a factor ±1) c+ := {i ∈ c : ζi > 0},c− := {i ∈ c : ζi < 0}. with this new structure m(ak) is said a (realizable) oriented matroid and denoted by m(ak). oriented matroids on a ground set [n], denoted m([n]), are a very natural mathematical concept and can be seen as the theory of generalized hyperplane arrangements, see [3]. set m(ak) := v \ ⋃ h∈ak h. the manifold m(ac) plays an important role in the aomoto-gelfand theory of multidimensional hypergeometric functions (see [16] for a recent introduction from the point of view of arrangement theory). let k be a commutative ring. in [12, 13, 14] the determination of the cohomology kalgebra h∗ ( m(ac); k ) from the matroid m(ac) is accomplished by first defining the orlik-solomon k-algebra os(ac) in terms of generators and relators which depends only on the matroid m(ac), and then by showing that this algebra is isomorphic to h∗ ( m(ac); k ) . aomoto suggested the study of the (graded) k-vector space ao(ak), generated by the basis {q(bi)−1}, where i is an independent set of m(ak), bi := {hi ∈ak : i ∈ i}, and q(bi) = ∏ i∈i θhi denotes the corresponding defining polynomial. to answer a conjecture of aomoto, orlik and terao have introduced in [15] a commutative k-algebra, ot(ak), called the orlik-solomon-terao algebra. the algebra ot(ak) is isomorphic to ao(ak) as a graded k-vector space in terms of the equations {θh : h ∈ ak}. a “combinatorial analogue” of the algebra of orlik-solomon-terao was introduced in [7]: to every oriented matroid m was associated a commutative z-algebra, denoted by a(m) and called the cordovil algebra. the χ-algebras generalizes the three just mentioned algebras: orlik-solomon, orlik-solomon-terao and the cordovil algebras, see [11] or example 2.4 below. in section two we will give the definition of a χ-algebra and recall the principal examples. in general a χ-algebra, denoted aχ(m), is defined as the quotient of some kind of a finite k-algebra a by an ideal �χ(m) of a whose generators are defined from the circuits of m and are depending of the map χ, see definition 2.2. in particular the first important result is that like for the original orlik-solomon algebra we get nbc-bases of the χ-algebra (as a module) from the “no broken circuit” sets of the matroid and corresponding basis for the ideal �χ(m). in section three, we construct the reduced gröbner basis of the ideal �χ(m) for any term order ≺ on the set of the monomials t(a) of the algebra a. this result gives as a corollary a universal gröbner basis (a gröbner basis who works for every term order) which is shown to be minimal. finally we remark that the nbc-bases are in some sense the bases corresponding to the gröbner bases for the different term orders. 4 raul cordovil and david forge 7, 2(2005) in section four, following szenes [17], we define a particular type of basis of aχ, the so called “diagonal basis”, see definition 4.7. the nbc-bases are an important examples of diagonal bases. we construct the dual bases of these bases, see theorem 4.8. our definitions make also use of an “iterative residue formula” based on the matroidal operation of contraction, see equation (4.6). this formula can be seen as the combinatorial analogue of an “iterative residue formula” introduced by szenes, [17]. as applications we deduce nice formulas to express a pure element in a diagonal basis. we prove also that the χ-algebras verify a splitting short exact sequence, see theorem 4.4. this theorem generalizes for the χ-algebras previous similar theorems of [7, 14]. we use [19, 20] as a general reference in matroid theory. we refer to [3] and [14] for good sources of the theory of oriented matroids and arrangements of hyperplanes, respectively. 2 χalgebras let ind�(m) ⊆ ( [n] � ) [resp. dep�(m) ⊆ ( [n] � ) ] be the family of independent [resp. dependents] sets of cardinality � of the matroid m and set ind(m) := ⋃ �∈n ind�(m), dep(m) := ⋃ �∈n dep�(m). we denote by c = c(m) the set of circuits of m. for shortening of the notation the singleton set {x} is denoted by x. when the smallest element α of a circuit c, |c| > 1, is deleted, the remaining set, c\α, is said to be a broken circuit. (note that our definition is slightly different from the standard one. in the standard definition c\α can be empty.) a no broken circuit set of a matroid m is an independent subset of [n] which does not contain any broken circuit. let nbc�(m) ⊆ ( [n] � ) be the set of the no broken circuit sets of cardinal � of m and set nbc(m) := ⋃ �∈n nbc�(m). let l(m) be the lattice of flats of m. ( we remark that the lattice map φ : l(ak) → l(m(ak)), determined by the one-to-one correspondence φ′ : hi ←→{i}, i = 1, . . . ,n, is a lattice isomorphism. ) for an independent set i ∈ ind(m), let c�(i) be the closure of i in m. for every permutation σ ∈ sm, let xσ be the ordered set xσ := iσ(1) ≺ ···≺ iσ(m) = (iσ(1), iσ(2), . . . , iσ(m)). when necessary we also see the set x = {i1, . . . , im}, as the ordered set xid = (i1, . . . , im). 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 5 set xσ \x := (iσ(1), . . . , x̂, . . . , iσ(m)). if y β = (jβ(1), . . . ,jβ(m′)) and x ∩y = ∅, let xσ ◦y β be the concatenation of xσ and y β , i.e., the ordered set xσ ◦y β := (iσ(1), . . . , iσ(m),jβ(1), . . . ,jβ(m′)). definition 2.1 let χ be a mapping χ : 2[n] → k. let us also define χ for ordered sets by χ(xσ) = sgn (σ)χ(x), where sgn (σ) denotes the sign of the permutation σ. fix a set e = {e1, . . . ,en}. let a := k ⊕ a1 ⊕···⊕ an be the graded algebra over the field k generated by the elements 1,e1, . . . ,en and satisfying the relations: ◦ 1ei = ei1 = ei, for all ei ∈ e, ◦ e2i = 0, for all ei ∈ e, ◦ ej ·ei = βi,jei ·ej with βi,j ∈ k∗ for all i < j. by definition the χ-boundary of an element ex ∈ a, x �= ∅, is given by the formula ∂ex := p=m∑ p=1 (−1)pχ(x \ ip)ex\ip. we set ∂ei = 1, for all ei ∈ e. we extend ∂ to the k-algebra a by linearity. let x = (i1, i2, . . . , im). in the sequel we will denote by ex the (pure) element of the k-algebra a, ex := ei1 ·ei2 · · · · ·eim. by convention we set e∅ := 1. both the exterior k-algebra, ∧ e, (take βi,j = −1) and the polynomial algebra k[e1, . . . ,en]/〈e2i〉 with squares zero (take βi,j = 1) considered in [7, 15], are such k-algebras a and will be the only ones to be used in the examples. it is clear that for any x �∈ x, ±∂ex∪x = (−1)m+1χ(x)ex + p=m∑ p=1 (−1)pχ(x \ ip ◦x)ex\ip∪x. from the equality χ(xσ) = sgn (σ)χ(x), it is easy to see that for σ ∈ s|x| we have ∂ex = sgn (σ) p=m∑ p=1 (−1)pχ(xσ \ iσ(p))ex\iσ(p). given an independent set i, an element a ∈ c�(i) \ i is said active in i if a is the minimal element of the unique circuit contained in i ∪ a. we say that a subset u ⊆ [n] is a unidependent set of m([n]) if it contains a unique circuit, denoted c(u). note that u is unidependent iff rk(u) = |u| − 1. we say that a unidependent set u is an inactive unidependent if min(c(u)) is the the minimal active element of u \ min(c(u)). we will denote by uni�(m) for the sets of inactive unidependent sets of size � and set uni(m) := ⋃ �∈n uni�(m). 6 raul cordovil and david forge 7, 2(2005) let us remark that u is a unidependent set of m iff for some (or every) x ∈ u, rk(x) �= 0, u \x is a unidependent set of m/x. definition 2.2 ([11]) let χ be a mapping χ : 2[n] → k. let �χ(m([n])) be the (right) ideal of a generated by the χ-boundaries {∂ec : c ∈ c(m), |c| > 1} and the set of the loops of m, {ei : {i} ∈ c(m)}. we say that aχ(m) := a/�χ(m) is a χ-algebra if χ satisfies the following two properties: (2.2.1) χ(i) �= 0 if and only if i is independent. (2.2.2) for any two unidependents u and u′ of m with u′ ⊆ u there is a scalar ε u,u′ ∈ k∗, such that ∂eu = εu,u′ (∂eu′ )eu\u′ . note that {ec : c ∈ c(m)}⊆�χ(m([n])). for every x ⊆ [n], we denote by [x]a or shortly by ex when no confusion will result, the residue class in aχ(m) determined by the element ex . since �χ(m) is a homogeneous ideal, aχ(m) inherits a grading from a. more precisely we have aχ(m) = k ⊕ a1 ⊕···⊕ ar, where a� = a�/a� ∩�χ(m) denotes the subspace of aχ(m) generated by the elements { [i]a : i ∈ ind�(m) } . set nbc � := { [i]a : i ∈ nbc �(m) } and nbc := ⋃ �=0 nbc �, dep� := { [d]a : d ∈ dep�(m) } and dep := ⋃ �=0 dep �, uni� := { [u]a : u ∈ uni�(m) } and uni := ⋃ �=0 uni �. remark 2.3 from (2.2.1) and (2.2.2) we conclude that �χ(m) has the basis dep∪ ∂uni and that nbc := { [i]a : i ∈ nbc(m) } is a basis of the vector space a = aχ(m). we also have that nbc� is a basis of the vector space a�. this fundamental property was first discovered for the orlik-solomon algebras [14], and then also for the other classical χ-algebras, see [7, 15] and the following example for more details. note also that this implies that [x]a �= 0 iff x is an independent set of m. example 2.4 recall the three usual χ-algebras aχ(m). ◦ let a = ∧e be the exterior k-algebra (taking βi,j = −1). setting χ(iσ) = sgn(σ) for every independent set i of a matroid m and every permutation σ ∈ s|i|, we obtain the orlik-solomon algebra, os(m). ◦ let ak = {hi : hi = ker(θi), i = 1, 2, . . . ,n} be an hyperplane arrangement and m(ak) its associated matroid. for every flat f := {f1, . . . ,fk} ⊆ [n] of m(ak) we choose a bases bf of the vector subspace of (kd)∗ generated by {θf1, . . . ,θfk}. by taking a = k[e1, . . . ,en]/〈e2i〉 the polynomial algebra with squares null (taking βi,j = 1) and taking for any {i1, . . . , i�} = i ∈ ind�, χ(i) = det(θi1, . . . ,θi� ), where the vectors are expressed in the basis bc�(i), we obtain the orlik-solomon-terao algebra ot(ak), defined in [15]. 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 7 ◦ let m([n]) be an oriented matroid. for every flat f of m([n]), we choose (determined up to a factor ±1) a bases signature in the restriction of m([n]) to f . we define a signature of the independents of an oriented matroid m([n]) as a mapping, sgn : ind(m) →{±1}, where sgn (i) is equal to the basis signature of i in the restriction of m([n]) to c�(i). by taking a = q[e1, . . . ,en]/〈e2i〉 the polynomial algebra over the rational field q with squares zero (take βi,j = 1) and taking χ(i) = sgn (i) (resp. χ(x) = 0) for every independent (resp. dependent) set of the matroid, we obtain the algebra a(m)⊕z q, where a(m) denotes the cordovil z-algebra defined in [7]. 3 gröbner bases of χ-ideals for general details on gröbner bases of an ideal, see [1, 2]. we begin by adapting some definitions to our context. consider the k-algebra a introduced in definition 2.1. note that there are monomials ey ,ez ∈ a, such that ey · ez = 0. in the standard case where a is replaced by the polynomial ring k[e1, . . . ,en], this is not possible. so the the following definitions are slightly different from the standard corresponding ones given in [1, 2]. let m = m([n]) be a matroid, �χ(m) and aχ(m) the χ-ideal and χ-algebra as defined in the previous section. we will denote for shortness a(m) for aχ(m). definition 3.1 let t = t(a) be the set of the monomials of the k-algebra a, i.e., t(a) := {ex : x = (ei1, . . . ,eim )}. a total ordering ≺ on the monomials t is said a term order on t if e∅ = 1 is the minimal element and ≺ is compatible with the multiplication in a, i.e., ∀ex,ey ,ez ∈ t, (ex ≺ ey )&(ex ·ez �= 0)&(ey ·ez �= 0) =⇒ ex ·ez ≺ ey ·ez. given a term order ≺ on t and a non-null polynomial f ∈ a, we may write f = a1ex1 + a2ex2 + · · · + amexm, where ai ∈ k∗ and exm ≺ ··· ≺ ex1 . we say that the aiexi [resp. exi ] are the terms [resp. monomials] of f. we say that lp≺(f) := ex1 [resp. lt≺(f) := a1ex1 ] is the leading monomial [resp. leading term] of f (with respect to ≺). we also define lp≺(0) = lt≺(0) = 0. note that in general we have lp≺(hg) �= lp≺(h)lp≺(g), contrarily to the cases considered in [1, 2]. example 3.2 a permutation π ∈ σn defines a linear reordering of the set [n]: π−1(1) <π π−1(2) <π · · · <π π−1(n). consider the ordering of the set e eπ−1(1) ≺π eπ−1(2) ≺π · · · ≺π eπ−1(n). the corresponding degree lexicographic ordering on the monomials t, also denoted ≺π, is a term order on t. 8 raul cordovil and david forge 7, 2(2005) for a subset s, s ⊆ a and a term order ≺ on t(a), we define the leading term ideal of s, denoted lt≺(s), as the ideal generated by the leading monomials of the polynomial in s, i.e., lt≺(s) := 〈lp≺(f) : f ∈ s〉. in the remaining of this section we suppose that m([n]) is a loop free matroid. definition 3.3 let m be a matroid. let ≺ be a term order on t(a). consider the ideal �χ(m) of a a family g of non-null polynomials of the ideal �χ(m) is called a gröbner basis of the ideal �χ(m) with respect to ≺ iff lt≺(g) = lt≺(�χ(m)). the gröbner basis g is called reduced if, for every element g ∈ g we have lt≺(g) = lp≺(g), and for every two distinct elements g,g ′ ∈ g, no term of g′ is divisible by lp≺(g). the gröbner basis g is called a universal gröbner basis if it is a gröbner basis with respect to all term orders on t(a) simultaneously. if u is a universal gröbner basis, minimal for inclusion with this property, we say that u is a minimal universal gröbner basis. from definition 3.3 we conclude: proposition 3.4 let g≺ be a gröbner basis of the ideal �χ(m) with respect to the term order ≺ on t(a). then bg≺ := { ex + �χ(m) : ex �∈ lt≺(g) } is a basis of the module aχ(m). we say that the well determined basis bg≺ is the canonical basis of the χ-algebra aχ(m) for the gröbner basis g of the ideal �χ(m), with respect to the term order ≺ on t(a). consider the partition t(a) = ti(a) ⊎ td(a) where: ti(a) := { ei : i ∈ ind(m) } and td(a) := { ed : d ∈ dep(m) } . let k[ti] and k[td] be the k-vector subspaces of a generated by the basis ti and td, respectively. so a = k[ti] ⊕ k[td]. we also see the set k[td] ⊆�χ(m) as the ideal of a generated by the set of monomials {ec : c ∈ c(m)}. let pi : a → k[ti] be the first projection. we define the term orders on the set of monomials ti in a similar way to the corresponding definition on t. it is clear that the restriction of every term order of t to the subset ti is also a term order on ti. we can also add to k[ti] a structure of k-algebra with the product : k[ti]×k[ti] → k[ti], determined by the equalities ex ex′ = pi(exex′ ) for all x,x ′ ∈�χ(m). note that if ex ex′ �= 0, then ex ex′ = exex′ . we remember that exex′ �= 0 iff x ∩x′ = ∅ and x ∪x′ ∈ ind(m). so �χi (m) := pi ( �χ(m) ) is an ideal of k[ti]. 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 9 proposition 3.5 let ≺ be a term order on t(a). then the leading term ideals of a, lt≺(pi(�χ(m))) and lt≺(�χ(m)) are equal. in particular a gröbner basis of the ideal �χi (m) of k[ti] with respect to term order ≺ on ti is also a gröbner basis of the ideal �χ(m) of a with respect to the term order ≺ on t. proof. note first that if we see �χ(m) as a k-vector space it is clear that �χ(m) = �χi (m) ⊕ k[td]. pick a non-null polynomial f ∈ �χ(m) and let ex1 := lp≺(f). so ex1 ∈ �i(m) if x1 ∈ ind(m), or ex1 ∈ k[td] \ 0 if x1 is a dependent set of m. if x1 ∈ ind(m) then ex1 ∈ lt≺(�χ(m)). suppose now that x1 is a dependent set of m. then there is a circuit c ⊆ x1. from definition 2.2 we know that ∂ec ∈�χ(m). it is clear that ec ∈ lt≺(pi(�χ(m))) and so we have also ex1 ∈ lt≺(pi(�χ(m))). remark 3.6 it is well known that a term order ≺ of t(a) determines also a unique reduced gröbner basis of �χ(m) denoted (gr )≺. from the definitions we can deduce also that, for every pair of term orders ≺ and ≺′ on t(a), bg≺ = bg≺′ ⇔ (gr)≺ = (gr )≺′ ⇔ lt≺ ( �χ(m) ) = lt≺′ ( �χ(m) ) . definition 3.7 for a term order ≺ on t(a) we say that π≺ ∈ sn, is the permutation compatible with ≺ if, for every pair i,j ∈ [n], we have ei ≺ ej iff i <π≺ j ( ⇔ π≺−1(i) < π≺−1(j) ) . let cπ≺ be the subset of circuits of m such that: ◦ c ∈ cπ≺ iff inf<π≺ (c) = απ(c) ( = inf<π≺ (cl(c)\c) ) and c\απ(c) is inclusion minimal with this property. in the following we replace “π≺” by “π” if no mistake can results. we recall that given a unidependent set u of the matroid m([n]), c(u) denotes the unique circuit of m contained in u. theorem 3.8 let ≺ be a term order on t(a) compatible with the permutation π ∈ sn. then the family gr := { ∂ec(u) : u ∈ cπ≺ (m) } form a reduced gröbner basis of �χi (m) with respect to the term order ≺. proof. from proposition 3.5 it is enough to prove that (gr)≺ is a reduced gröbner of �χi (m). let f be any element of �χi (m), we have from theorem 2.3 that f = ∑ u∈uπ ξu∂eu, ξu ∈ k�. let now remark that lp≺ ( ∂eu ) = eu\απ (u) and that these terms are all different. we have then clearly that lp≺(f) = sup≺ { lp≺(∂eu ) : u ∈ uπ } . 10 raul cordovil and david forge 7, 2(2005) given an arbitrary u′ ∈ uπ(m) it is clear that απ(c(u′)) = απ(u′). so, c(u′) \απ(c(u′)) ⊆ u′ \απ(u′). let c′ be a circuit of cπ such that c′ \απ(c′) ⊆ c(u) \απ(c(u)). so we have that lp≺(∂ec′ ) divides lp≺(∂eu ), and (gr )≺ is a gröbner basis. suppose for a contradiction that (gr)≺ is not a reduced gröbner basis: i.e., there exists two circuits c and c′ in cπ and an element c ∈ c such that ec′\απ (c′) divides ec\c ( ⇔ c′ \ απ(c′) ⊆ c \ c ) . first we can say that c �= απ(c) because the sets c′ \ απ (c′) and c \ απ(c) are incomparable. this in particular implies that απ(c) ∈ c′ \ απ (c′), and απ(c′) ≺ απ(c). on the other hand we have απ(c′) ∈ cl ( c′ \απ(c′) ) ⊆ cl(c \ c) = cl(c \απ(c)), so απ(c) ≺ απ(c′), a contradiction. corollary 3.9 the set gu := { ∂ec : c ∈ c(m)} is a minimal universal gröbner basis of the ideal �χ(m). proof. from theorem 3.8, the reduced gröbner bases constructed for the different orders ≺ are all contained in gu. we prove the minimality by contradiction. let c0 = {i1, . . . , im} be a circuit of m and let π ∈ sn be a permutation such that π−1(ij ) = j, j = 1, . . . ,m. then g′u := {∂ec : c ∈ c \c0} it is not a gröbner basis because lp≺π (∂ec0 ) = ec0\i1 is not in lt≺π (g′u). to finish this section we give an important characterization of the no broken circuit bases of the χ-algebras in terms of the gröbner bases of their ideals. definition 3.10 consider a permutation π ∈ sn and the associated re-ordering <π of [n]. when the <π-smallest element inf<π (c) of a circuit c ∈ c(m), |c| > 1, is deleted, the remaining set, c \ inf<π (c), is called a π-broken circuit of m. we say that π-nbc(m) := {ex : x ⊆ [n] contains no π-broken circuit of m} is the π-no broken circuit bases of aχ(m). as the algebra aχ(m) does not depend of the ordering of the elements of m it is clear that π-nbc(m) is a no broken circuit bases of aχ(m). corollary 3.11 let b be a basis of the module aχ(m). then are equivalent: (3.11.1) b is the canonical basis b≺, for some term order ≺ on t(a). (3.11.2) b is the π-no broken circuit bases π-nbc(m), for some permutation π ∈ sn. (3.11.3) b is the canonical basis bgr, for some reduced gröbner basis gr of the ideal �χ(m). proof. (3.11.1) ⇒ (3.11.2) let ≺ be a term order of t(a). since gu is a universal gröbner basis of �χ(m) (see corollary 3.9) it is trivially a gröbner basis relatively to ≺. we have already remarked that the leading term of ∂ec is ec\c where c = inf<π≺ (c). from proposition 3.4 we conclude that b≺ = π≺-nbc(m). 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 11 (3.11.2) ⇒ (3.11.3) suppose that b = π-nbc(m). let ≺π be the degree lexicographic order of t determined by the permutation π ∈ sn. note that π≺π = π. ¿from theorem 3.8 we know that (gr )≺π = { ∂ec : c ∈ c≺π} is the reduced gröbner basis of �χ(m) with respect to the term order ≺π. then b is the canonical basis for the reduced gröbner basis (gr )≺π . (3.11.3) ⇒ (3.11.1) it is a consequence of proposition 3.4 and remark 3.6. 4 diagonal bases of χ-algebras proposition 4.1 let aχ(m) be a χ-algebra with the associated map χ : 2[n] → k. for any non loop element x of m([n]), we define the two maps: χm\x : 2 [n]\x → k by χm\x(x) = χ(x) and (4.1) χm/x : 2 [n]\x → k by χm/x(x) = χ(x ◦x). (4.2) there are two χ-algebras, aχm/x (m/x) and aχm\x (m\x), associated to the maps χm\x and χm/x, respectively. proof. from (2.2.1) we know that χ(x) �= ∅ iff x ∈ ind(m). the deletion case being trivial, we will just prove the contraction case. we have to show that χm/x verifies properties (2.2.1) and (2.2.2). the first property is verified since a set i is independent in m/x iff i ∪x is independent in m. to see that the second property is also verified, let u and u′ be two unidependents sets of m/x. i.e., iff u ∪ x and u′ ∪x are two unidependents sets of m. from (2.2.1) we know that ∂eu∪x = εu∪x,u′∪x (∂eu′∪x)eu\u′ where εu∪x,u′∪x ∈ k∗. let ∂′ be the χm/x-boundary, i.e., the linear mapping ∂ ′ : a/〈ex〉 → a/〈ex〉 such that for ever ei ∈ e\x we have ∂′ei = 1, ∂′e∅ = 1 and for every monomial ex, x �∈ x and x �= ∅, ∂′ex = p=m∑ p=1 (−1)pχm/x(x \ ip)ex\ip = p=m∑ p=1 (−1)pχ(x \ ip ◦x)ex\ip. to finish the proof we will show that there is a scalar ε̃ u,u′ ∈ k∗ such that ∂′eu = ε̃u,u′ (∂ ′eu′ )eu\u′. let x,x′ ⊆ [n] be two disjoint subsets. from definition 2.1 we known that ex ·ex′ = βx,x′ex∪x′, where βx,x′ = ∏ βi,j, (ei ∈ x,ej ∈ x′ and i > j). so we have with u = (i1, . . . , im) and u′ = (j1, . . . ,jk), u ∩u′ = ∅, x �∈ u ∪u′: ±∂eu∪x = p=m∑ p=1 (−1)pχ(u \ ip ◦x)eu∪x\ip + (−1)m+1χ(u)eu, 12 raul cordovil and david forge 7, 2(2005) ∂′eu = p=m∑ p=1 (−1)pχ(u \ ip ◦x)eu\ip , ±(∂eu′∪x)eu\u′ = p=k∑ p=1 (−1)pχ(u′ \ jp ◦x) ·β ·eu∪x\jp + (−1)k+1χ(u′) ·β′ ·eu, where β = β u′∪x\jp ,u\u′ and β′ = β u′,u\u′ . (∂′eu′ )eu\u′ = p=k∑ p=1 (−1)pχ(u′ \ jp ◦x) ·βu′\jp ,u\u′ ·eu\jp. after remarking that β u′∪x\jp ,u\u′ β−1 u′\jp ,u\u′ = β x,u\u′ does not depend on jp, we can deduce that ∂′eu = ε̃u,u′ (∂ ′eu′ )eu\u′ with ε̃u,u′ = ±εu∪x,u′∪x ·β−1x,u\u′. proposition 4.2 for every non loop element x of m([n]), there is a unique monomorphism of vector spaces, ix : a(m \ x) → a(m), such that such that for every i ∈ ind(m\x), we have ix(ei ) = ei . proof. by a reordering of the elements of the matroid m we can suppose that x = n. it is clear that nbc(m\x) = { x : x ⊆ [n− 1] and x ∈ nbc(m) } , so the proposition is a consequence of equation (4.1). proposition 4.3 for every non loop element x of m([n]), there is a unique epimorphism of vector spaces, px : a(m) → a(m/x), such that, for every ei, i ∈ ind(m), we have px(ei ) := ⎧⎪⎪⎨ ⎪⎪⎩ ei\x if x ∈ i, χ(i\y,x) χ(i\y,y) ei\y if there is y ∈ i parallel to x, 0 otherwise. (4.3) proof. from remark 2.3, it is enough to prove that px(∂eu ) = 0, for all unidependent u = (i1, . . . , im). we recall that if x ∈ u then u \x is a unidependent set of m/x. there are only the following four cases: ◦ if u contains x but no y parallel to x then: ±px(∂eu ) = px((−1)mχ(u \x)eu\x + ∑ ip∈u\x (−1)pχ(u \{ip,x}◦x)eu\ip )) = ∑ ip∈u\x (−1)pχ(u \{ip,x}◦x)eu\{ip ,x} = 0 from proposition 4.1. 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 13 ◦ if u does not contain x but contains a y parallel to x then: ±px(∂eu ) = px ( (−1)mχ(u \y)eu\y + ∑ ip∈u\y (−1)pχ(u \{ip,y}◦y)eu\ip ) = ∑ ip∈u\y (−1)pχ(u \{ip,y}◦y) χ(u \{ip,x}◦x) χ(u \{ip,y}◦y) eu\{ip,y} = 0 like previously since u \y is again a unidependent of m/x. ◦ if u contains x and a y parallel to x then: ±px(∂eu ) = px(χ(u \{x,y}◦y)eu\x −χ(u \{x,y}◦x)eu\y ) = χ(u \{x,y}◦y)χ(u \{x,y}◦x) χ(u \{x,y}◦y)eu\{x,y} −χ(u \{x,y}◦x)eu\{x,y} = 0. ◦ if u does not contain x nor a y parallel to x then: px(∂eu ) = px ( ∑ ip∈u (−1)pχ(u \ ip)eu\ip ) = 0. theorem 4.4 for every element x of a simple m([n]), there is a splitting short exact sequence of vector spaces 0 → a(m\x) ix−→ a(m) px−→ a(m/x) → 0. (4.4) proof. from the definitions we know that the composite map px◦ ix, is the null map so im(ix) ⊆ ker(px). we will prove the equality dim(ker(pn)) = dim(im(in)). by a reordering of the elements of [n] we can suppose that x = n. the minimal broken circuits of m/n are the minimal sets x such that either x or x ∪{n} is a broken circuit of m (see the proposition 3.2.e of [5]). then nbc(m/n) = { x : x ⊆ [n− 1] and x ∪{n}∈ nbc(m) } and nbc(m) = nbc(m\n) ⊎{ i ∪n : i ∈ nbc(m/n) } . (4.5) so dim(ker(pn)) = dim(im(in)). there is a morphism of modules p−1n : a(m/n) → a, where p−1n ([i]a(m/n)) := [i ∪ n]a, ∀i ∈ nbc(m/n). it is clear that the composite map pn◦ p−1n is the identity map. from equation (4.5) we conclude that the exact sequence (4.4) splits. similarly to [17] (see also [4]), we now construct, making use of iterated contractions, the dual bases nbc∗� = (b ∗ i ) of the bases nbc � := (bj ) of the vector space a�. more precisely nbc∗� is the basis of a ∗ � the vector space of the linear forms such that 〈b∗i ,bj〉 = δij (the kronecker delta). 14 raul cordovil and david forge 7, 2(2005) we associate to the ordered independent set iσ := (iσ(1), . . . , iσ(p)) of m the linear form on a�, piσ : a� → k, defined as the composite of the maps peiσ(p) , peiσ(p−1) , . . . , peiσ(1) , i.e., piσ := peiσ(1) ◦ peiσ(2) ◦ · · · ◦ peiσ(p) . (4.6) we call piσ the iterated residue with respect to the ordered independent set i σ. we remark that the map piσ depends on the order chosen on i σ and not only on the underlying set i. we associate to iσ the flag of flats of m, flag(iσ) := c� ( {iσ(p)} ) � c� ( {iσ(p), iσ(p−1)} ) � · · · � c� ( {iσ(p), . . . , iσ(1)} ) . proposition 4.5 let j ∈ ind�(m) then we have piσ (ej ) �= 0 iff there is a unique permutation τ ∈ s� such that flag (jτ ) = flag (iσ). and in this case we have piσ (ej ) = χ(i σ)/χ(jτ ). in particular we have piσ (ei ) = 1 for any independent set i and any permutation σ. proof. the first equivalence is easy to prove in both direction. to obtain the expression of piσ (ej ) we just need to iterate � times the residue. this gives: piσ (ej ) = χ(j \ jτ (�) ◦ iσ(�)) χ(j \ jτ (�) ◦ jτ (�)) × χ(j \{jτ (�),jτ (�−1)}◦ iσ(�−1) ◦ iσ(�)) χ(j \{jτ (�),jτ (�−1)}◦ jτ (�−1) ◦ iσ(�)) ×··· · · ·× χ(i σ) χ(jτ (1) ◦ iσ \ iσ(1)) . after simplification we obtain the announced formula. the last result is clear. remark 4.6 the fact that piσ (ej ) is null depends on the permutation σ. for example, for any simple matroid of rank 2 we have p13(e12) = 0 and p31(e12) �= 0. but if piσ (ej ) �= 0 then its value does not depend on σ. we mean by this that if there are two permutations σ and σ′ such that piσ (ej ) �= 0 and piσ′ (ej ) �= 0 then piσ (ej ) = piσ′ (ej ). definition 4.7 ([17]) we say that the subset i� ⊆ { [i]a : i ∈ ind�(m)} is a diagonal basis of a� if and only if the following three conditions hold: (4.7.1) for every [i]a ∈ i� there is a fixed permutation of the set i denoted σi ∈ s�; (4.7.2) ∣∣i�| ≥ dim(a�); (4.7.3) for every [i]a, [j]a ∈ i� and every permutation τ ∈ s�, the equality flag (jτ ) = flag (iσi ) implies j = i. theorem 4.8 suppose that i� is a diagonal basis of a�. then i� is a basis of a� and i∗� := {piσi : [i]a ∈ i�} is the dual basis of i�. 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 15 proof. pick two elements [i]a, [j]a ∈ i�. note that piσi (ej ) = δij (the kronecker delta), from condition (4.7.2) and proposition 4.5. the elements of i� are linearly independent: suppose that [j] = ∑ ζj [ij ], ζj ∈ k \ 0; then 1 = pj σj ([j]) = pj σj (∑ ζj [ij ] ) = 0, a contradiction. it is clear also that i∗� is the dual basis of i�. the following result gives an interesting explanation of results of [6, 7]. corollary 4.9 nbc�(m) is a diagonal basis of a� where σi is the identity for every [i]a ∈ nbc�(m). for a given [j]a ∈ a�, suppose that (4.9.2) [j]a = ∑ ξ(i,j)[i]a, where [i]a ∈ nbc�(m) and ξ(i,j) ∈ k. then are equivalent: ◦ ξ(i,j) �= 0, ◦ flag (i) = flag (jτ ) for some permutation τ. if ξ(i,j) �= 0 we have ξ(i,j) = χ(i) χ(j τ ) . in particular if a is the orlik-solomon algebra then ξ(i,j) = sgn (τ). proof. by hypothesis (4.7.1) and (4.7.2) are true. we claim that nbc �(m) verifies (4.7.3). suppose for a contradiction that j �= i, [j]a, [i]a ∈ nbc �(m) and there is τ ∈ s�, such that flag (jτ ) = flag (i). set i = (i1, . . . , i�) and j = (jτ (1), . . . ,jτ (�)), and suppose that jτ (m+1) = im+1, . . . ,jτ (�) = i� and im �= jτ (m). then there is a circuit c of m such that im,jτ (m) ∈ c ⊆{im,jτ (m), im+1, im+2, . . . , i�}. if jτ (m) < im [resp. im < jτ (m)] we conclude that i �∈ nbc�(m) [resp. j �∈ nbc�(m)] a contradiction. so nbc �(m) is a diagonal basis of a�. from theorem 4.8 we conclude that nbc∗� := { pi : [i]a ∈ nbc} is the dual basis of nbc. suppose now that [j]a = ∑ ξi [i]a, where [i]a ∈ nbc�(m) and ξi ∈ k. then ξi = pi (ej ) and the remaining follows from proposition 4.5. making full use of the matroidal notion of iterated residue, see equation (4.6), we are able to prove the following result very close to proposition 2.1 of [18]. proposition 4.10 consider the set of vectors v := {v1, . . . ,vk} in the plane xd = 1 of kd. set ak := {hi : hi = ker(vi) ⊆ (kd)∗, i = 1, . . . ,k} and let ot(ak) be its orlik-solomon-terao corresponding algebra. fix a diagonal basis i� ⊆ {[i]a : i ∈ ind�(m)} of a� and let i∗� = {piσi : [i]a ∈ i�} be the corresponding dual basis. then, for any ej ∈ a� \ 0, we have ∑ i∈i� piσi (ej ) = ∑ i∈i� 〈 piσi ,ej 〉 = 1. 16 raul cordovil and david forge 7, 2(2005) proof. we have for any � + 1-subset of v, ∑p=�+1p=1 (−1)pχ(u \ ip) = 0. (this is the development of a determinant with two lines of 1.) for any rank � unidependent u = {i1, . . . , i�+1} of the matroid m(ak), we have ∂eu = p=�+1∑ p=1 (−1)pχ(u \ ip)eu\ip. since the sum of the coefficients in these relations is 0 and that these relations are generating, see remark 2.3, we can deduce that the sum of the coefficients in any relation in ot(ak) is also equal to 0 which concludes the proof. 5 examples in this section we will show on a small example the different results of the three previous sections. consider the the set of 6 points {p1, . . . ,p6} in the affine plane z = 1 of three dimensional real vector space r3, whose coordinates are indicated in figure 1. set vi := −−−→ (0,pi), i = 1, . . . , 6. and let a be the corresponding hyperplane arrangement of (r3)∗, a := {hi = ker(vi), i = 1, . . . , 6}. let m(a) [resp. m(a)] be the corresponding rank three [resp. oriented] matroid. so like in example 2.4, the arrangement a defines the three classical orlik-solomon type algebras: the original orlik-solomon algebra os(m(a)) through m(a), the orlik-solomon-terao algebra ot(a) directly from the vi and the cordovil algebra a(m(a)) from m(a). p1 p3 p2 p4 p5 p6 � � � � � �(0,0,1) (0, 1 2 , 1) (0,1,1) ( 1 2 , 0, 1) (1,0,1) ( 1 3 , 1 3 , 1) �� figure 1: the rank 3 matroid on the set {p1, . . . ,p6}. let aχ be a χ-algebra on m(a). we know that nbc 3 = {e124,e125,e126,e134,e135,e136} together with σ124 = σ125 = σ134 = σ135 = σ136 = σ156 = id is a diagonal basis of a3, from corollary 4.9. directly from the definition 4.7 we see that b3 = {e124,e125,e134,e135,e136,e156} with σ124 = σ134 = σ135 = σ136 = σ156 = id and 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 17 σ125 = (132) is also a diagonal basis of a3. we will look at expressions on the basis nbc3 (resp. b3) of the vector space a3, of some elements of the type eb, b basis of m(a), for the three χ-algebras of example 2.4. especially, we will verify as stated in remark 4.6 that p125id (e235) = p125(132) (e235). let also point out that for the orlik-solomon-terao algebra, we will have have ∑ i∈b piσ (ej ) = 1 as proved in proposition 4.10. finally recall that t is set of the monomials of a and set t� := {ex ∈ t : |x| = �}. (a) let us first take the orlik-solomon algebra os(m(a)) : from remark 2.3 , the basis of os(m(a)) is simply the nbc-bases: nbc(m) = t0 ∪ t1 ∪ nbc2 ∪ nbc3, with nbc2 = {e12,e13,e14,e15,e16,e24,e25,e26,e34,e35,e36}, and nbc3 = {e124,e125,e126,e134,e135,e136}. the basis of �χ(m(a)) is the union of the dependents and of the boundaries of the inactive unidependents: ∂uni3 ∪ dep3 ∪∂uni4 ∪ t4 ∪ t5 ∪ t6 where ∂uni3 = {∂e123,∂e145,∂e256,∂e346}, dep3 = {e123,e145,e256,e346} and ∂uni4 is the set {∂e1234,∂e1235,∂e1236,∂e1245,∂e1246,∂e1256,∂e1345,∂e1346,∂e1356,∂e1456}. note that we have |nbc2| + |∂uni3| = 11 + 4 = 15 = dim(a2) and |nbc3| + |∂uni4| + |dep3| = 6 + 10 + 4 = 20 = dim(a3). take first on [n] the natural order. we have then for the leading term ideal lt<(g) = 〈ebc : bc broken circuit〉. we obtain explicitly: lt<(g) = 〈e23,e45,e56,e46,e246,e345,e356〉. always for the natural order, from theorem 3.8, we obtain for the reduced gröbner basis: gr = { ∂e123,∂e145,∂e256,∂e346 } . if we take now the term order ≺π on t(a), defined by the permutation π := (234561), we get now: lt≺(g) = 〈e13,e15,e56,e46,e146,e345,e165〉, 18 raul cordovil and david forge 7, 2(2005) and then for the corresponding reduced gröbner basis: gr = { ∂e123,∂e145,∂e256,∂e346,∂e2345 } . finally from corollary 3.9, we get the minimal universal gröbner basis gu = { ∂ec : c ∈ c(m)}. we obtain explicitly: gu = {∂e123,∂e145,∂e256,∂e346,∂e1246,∂e1356,∂e2345}. now we will use the results of section 4 to express pure elements in different diagonal bases. consider the diagonal basis nbc3 of the k-vector space os(m(a))3. so we have: e156 = sgn(165)e125 + sgn(156)e126 = −e125 + e126 and e235 = sgn (325)e125 + sgn (235)e135 = −e125 + e135. for the diagonal basis b3 of the k-vector space os(m(a))3, we have: e126 = sgn(162)sgn(152)e125 + sgn(126)e156 = e125 + e156 and e235 = sgn(152)sgn(352)e125 + sgn(235)e135 = −e125 + e135. (b) let us take the orlik-solomon-terao algebra ot(a) : for the different bases and gröbner bases we obtain formally the same results. there is in fact differences which are hidden by the operator ∂ (indeed ∂ is function of χ). for the diagonal basis nbc3 of the k-vector space ot(a)3 we have: e156 = det(125) det(165) e125 + det(126) det(156) e126 = 3 2 e125 − 1 2 e126 and e235 = sgn (325)e125 + sgn (235)e135 = −e125 + e135 for the diagonal basis b3 of the k-vector space ot(a)3 we have: e126 = det(152) det(162) e125 + det(156) det(126) e156 = 3e125 − 2e156. and e235 = det(152) det(352) e125 + det(135) det(235) e135 = −e125 + 2e135. 7, 2(2005) gröbner and diagonal bases in orlik-solomon type algebras 19 (c) let us take the cordovil z-algebra a(m(a)) : for the diagonal basis nbc3 of the k-vector space a(m(a))3 we have: e156 = χ(125)χ(165)e125 + χ(126)χ(156)e126 = e125 −e126 and e235 = sgn (325)e125 + sgn (235)e135 = −e125 + e135. for the diagonal basis b3 of the k-vector space a(m(a))3 we have: e126 = χ(152)χ(162)e125 + χ(156)χ(126)e156 = e125 −e156 and e235 = sgn(152)sgn(352)e125 + sgn(235)e135 = −e125 + e135. received: september 2003. revised: january 2004. references [1] adams william w., loustaunau philippe, an introduction to gröbner bases. graduate studies in mathematics 3. amer. math. soc., providence, ri, 1994. [2] becker thomas, weispfenning volker, gröbner bases. a computational approach to commutative algebra. in cooperation with heinz kredel. graduate texts in mathematics 141. springer-verlag, new york, 1993. [3] björner a., las vergnas m., sturmfels b., white n., ziegler g. m., oriented matroids. second edition. encyclopedia math. appl. 46, cambridge university press, cambridge, 1999. [4] brion michel, vergne michèle, arrangement of hyperplanes. i. rational functions and jeffrey-kirwan residue. ann. sci. école norm. sup. 32 (1999), 715–741. [5] brylawski t., the broken-circuit complex. trans. amer. math. soc. 234 (1977), 417–433. [6] cordovil r., etienne g., a note on the orlik-solomon algebra. european j. combin. 22 (2001), 165–170. [7] cordovil r., a commutative algebra for oriented matroids. discrete and comput. geometry 27 (2002), 73–84. 20 raul cordovil and david forge 7, 2(2005) [8] cordovil r., forge d., diagonal bases in orlik-solomon type algebras. ann. comb. 7 (2003), 25-32. [9] falk michael j., combinatorial and algebraic structure in orliksolomon algebras. combinatorial geometries (luminy, 1999). european j. combin. 22 (2001), 687–698. [10] forge d., bases in orlik-solomon type algebras. european j. combin. 23 (2002), 567–572. [11] forge d., las vergnas m., orlik-solomon type algebras. european j. combin. 22 (2001), 699–704. [12] orlik peter, solomon louis, unitary reflection groups and cohomology. invent. math. 59 (1980), 77–94. [13] orlik peter, solomon louis, combinatorics and topology of complements of hyperplanes. invent. math. 56 (1980), 167–189. [14] orlik peter, terao hiroaki, arrangements of hyperplanes. grundlehren der mathematischen wissenschaften [fundamental principles of mathematical sciences] 300. springer-verlag, berlin, 1992. [15] orlik peter, terao hiroaki, commutative algebras for arrangements. nagoya math. j. 134 (1994), 65–73. [16] orlik peter, terao hiroaki, arrangements and hypergeometric integrals. msj memoirs, 9. mathematical society of japan, tokyo, 2001 [17] szenes a., iterated residues and multiple bernoulli polynomials. internat. math. res. notices 18 (1998), 937–956. [18] szenes a., a residue theorem for rational trigonometric sums and verlinde’s formula. duke math. j. 118 (2003), 189–227. [19] white neil (ed.), theory of matroids. encyclopedia of mathematics and its applications 26. cambridge university press, cambridge-new york, 1986. [20] white neil (ed.), combinatorial geometries. encyclopedia of mathematics and its applications 29. cambridge university press, cambridge-new york, 1987. [21] yuzvinsky sergey, orlik-solomon algebras in algebra and topology. russian math. surveys, 56 (2001), 293–364. cubo a mathematical journal vol.19, no¯ 03, (43–55). october 2017 weak homoclinic solutions to discrete nonlinear problems of kirchhoff type with variable exponents aboudramane guiro, idrissa ibrango and stanislas ouaro laboratoire de mathématiques et informatique (lami) 1 ufr, sciences et techniques, universit nazi boni 01 bp 1091 bobo-dioulasso, 01 bobo dioulasso, burkina faso 2 ufr. sciences exactes et appliques, universit ouaga i pr joseph ki-zerbo, 03 bp 7021 ouaga 03, ouagadougou, burkina faso abouguiro@yahoo.fr, ibrango2006@yahoo.fr, ibrango2006@yahoo.fr, ouaro@yahoo.fr abstract in this paper, we prove the existence of weak homoclinic solutions for discrete nonlinear problems of kirchhoff type. the proof of the main result is based on a minimization method. as extension, we prove the existence result of weak homoclinic solutions for more general data depending on the solutions. resumen en este art́ıculo, probamos la existencia de soluciones homocĺınicas débiles para problemas discretos no-lineales de tipo kirchhoff. la demostración del resultado principal está basado en un método de minimización. como extensión, probamos la existencia de soluciones homocĺınicas débiles para datos más generales dependiendo de las soluciones. 2010 ams mathematics subject classification: 47a75, 35b38, 35p30, 34l05, 34l30. 44 aboudramane guiro, idrissa ibrango and stanislas ouaro cubo 19, 3 (2017) 1 introduction in this paper, we study the following nonlinear discrete anisotropic problem (p)    −m(a(k − 1, ∆u(k − 1)))∆(a(k − 1, ∆u(k − 1))) + |u(k)|p(k)−2u(k) = f(k), k ∈ z lim |k|→∞ u(k) = 0, (1.1) where ∆u(k) = u(k + 1) − u(k) is the forward difference operator. note that difference equations can be seen as a discrete counterpart of partial difference equations and are usually studied in connection with numerical analysis. in this way, the operator in problem (1.1) ∆(a(k − 1, ∆u(k − 1))) can be seen as a discrete counterpart of the anisotropic operator n∑ i=1 ∂ ∂xi a ( x, ∂ ∂xi u ) . the first study in that direction for constant exponents was done by cabada et al. [2] and for variable exponent by mihailescu et al. [10] (see also [3]). in [3], the authors studied the following problem    −∆ ( a ( k − 1, ∆u(k − 1) )) + |u(k)|p(k)−2u(k) = f(k), k ∈ z lim |k|→+∞ u(k) = 0. (1.2) they proved an existence result of weak homoclinic solution of (1.2). in this paper we consider the same boundary conditions as in [3], but the function m(a(k − 1, ∆u(k − 1))) which appear in the left-hand side of problem (1.1) is more general than the one which appear in [3]. indeed, if we take m(t) = 1 in the problem (1.1), we obtain the probem studied by guiro et als in [3]. to prove an existence result of problem (1.1), we define other new spaces and new associated norms and we adapt the classical minimization methods used for the study of anisotropic pdes. the idea is to transfer the problem of the existence of solutions for (1.1) into the problem of the existence of a minimizer for some associated energy functional. the remaining part of this paper is organized as follows: section 2 is devoted to mathematical preliminaries. the main existence result is stated and proved in section 3. finally, in section 4, we discuss some extensions. cubo 19, 3 (2017) weak homoclinic solutions to discrete nonlinear problems . . . 45 2 preliminaries we use the notations p+ = sup k∈z p(k) and p− = inf k∈z p(k). for the data f and a, we assume the following: { a(., .) : z × r −→ r and a(., .) : z × r −→ r with a(k, ξ) = ∂ ∂ξ a(k, ξ) and a(k, 0) = 0, ∀k ∈ z. (2.1) |ξ|p(k) ≤ a(k, ξ)ξ ≤ p(k)a(k, ξ), ∀ k ∈ z and ξ ∈ r. (2.2) there exists a positive constant c1 such that |a(k, ξ)| ≤ c1(j(k) + |ξ| p(k)−1), (2.3) for all k ∈ z and ξ ∈ r, where j ∈ lp ′ (.) with 1 p(k) + 1 p′(k) = 1. let us take f ∈ lp ′ (.). (2.4) for ξ, η ∈ r with ξ 6= η, and for almost every k ∈ z, ( ai(x, ξ) − ai(x, η) ) (ξ − η) ≥ 0. (2.5) moreover, in this paper, we assume that the function p : z → (1, +∞) such that 1 < p− < p(.) < p+ < +∞. (2.6) we also assume that the function m : (0, +∞) → (0, +∞) is continuous, non decreasing and there exist two positive reals number b1, b2 such that b1 ≤ b2 and α ≥ 1 with b1t α−1 ≤ m(t) ≤ b2t α−1, for t > 0. (2.7) let us define the functional spaces, lp(.) = { u : z −→ r such that ρp(.)(u) := ∑ k∈z |u(k)|p(k) < ∞ } and w1,p(.)α = { u : z −→ r such that ρ1,α,p(.)(u) := ∑ k∈z |u(k)|p(k) + ( ∑ k∈z |∆u(k)|p(k) )α < ∞ } . we introduce in lp(.) the luxemburg norm ||u||p(.) := inf { λ > 0 such that ∑ k∈z ∣∣∣∣ u(k) λ ∣∣∣∣ p(k) ≤ 1 } , and we define, on the space w 1,p(.) α , the norm ||u||1,α,p(.) := inf { λ > 0 such that ∑ k∈z ∣∣∣∣ u(k) λ ∣∣∣∣ p(k) + ( ∑ k∈z ∣∣∣∣ ∆u(k) λ ∣∣∣∣ p(k))α ≤ 1 } . 46 aboudramane guiro, idrissa ibrango and stanislas ouaro cubo 19, 3 (2017) example 2.1. as example of functions which satsfies above assumptions, we have the following. • a(k, ξ) = 1 p(k) |ξ|p(k), where a(k, ξ) = |ξ|p(k)−2ξ, ∀k ∈ z and ξ ∈ r, • a(k, ξ) = 1 p(k) (( 1+ |ξ|2 )p(k)/2 −1 ) , where a(k, ξ) = ( 1+ |ξ|2 )(p(k)−2)/2 ξ, ∀ k ∈ z, ξ ∈ r and • b(t) = ctα−1, for t > 0, where α ≥ 1 and c > 0. remark 2.2. if u ∈ lp(.), then lim |k|→+∞ u(k) = 0. indeed, if u ∈ lp(.), then ∑ k∈z |u(k)|p(k) < +∞. let s1 = {k ∈ z; |u(k)| < 1} and s2 = {k ∈ z; |u(k)| ≥ 1} s2 is necessary a finite set and |u(k)| < +∞ for any k ∈ s2 since u ∈ lp(.). as s2 is a finite set, then ∑ k∈s2 |u(k)|p + < +∞. therefore ∑ k∈z |u(k)|p + = ∑ k∈s1 |u(k)|p + + ∑ k∈s2 |u(k)|p + < +∞. thus, lim |k|→+∞ u(k) = 0. � in the sequel, we will use the following result: proposition 2.3. ([3], proposition 2.3). if u ∈ lp(.) and p+ < ∞, then the following properties hold: (1) ||u||p(.) < 1 =⇒ ||u|| p+ p(.) ≤ ρp(.)(u) ≤ ||u|| p− p(.) ; (2) ||u||p(.) > 1 =⇒ ||u|| p− p(.) ≤ ρp(.)(u) ≤ ||u|| p+ p(.) ; (3) ||u||p(.) < 1 (= 1; > 1) ⇐⇒ ρp(.)(u) < 1 (= 1; > 1). theorem 2.4. ([3], theorem 2.1). let u ∈ lp(.) and v ∈ lq(.) such that 1 p(k) + 1 q(k) = 1 for any k in z. then ∑ k∈z |uv| ≤ ( 1 p− + 1 q− ) ||u||p(.)||v||q(.). (2.8) cubo 19, 3 (2017) weak homoclinic solutions to discrete nonlinear problems . . . 47 proposition 2.5. ([10], proposition 2.1). let q : z −→ r such that 1 < p− ≤ p+ < q− ≤ q+. (2.9) then lp(.) ⊂ lq(.). (2.10) as in [3], we have the following result. proposition 2.6. (1) ρ1,α,p(.)(u + v) ≤ 2 αp+−1(ρ1,α,p(.)(u) + ρ1,α,p(.)(v)), ∀ u, v ∈ w 1,p(.) α . (2) let u in w 1,p(.) α . i) if λ > 1, we have λp − ρ1,α,p(.)(u) ≤ ρ1,α,p(.)(λu) ≤ λ αp+ρ1,α,p(.)(u). (2.11) ii) if 0 < λ < 1, we have λαp + ρ1,α,p(.)(u) ≤ ρ1,α,p(.)(λu) ≤ λ p−ρ1,α,p(.)(u). (2.12) theorem 2.7. let u ∈ w 1,p(.) α \ {0}. then ||u||1,α,p(.) = a if and only if ρ1,α,p(.)(u/a) = 1. proposition 2.8. if u ∈ w 1,p(.) α and p + < ∞, then the following properties hold: (1) ||u||1,α,p(.) < 1 =⇒ ||u|| αp+ 1,α,p(.) ≤ ρ1,α,p(.)(u) ≤ ||u|| p− 1,α,p(.) ; (2) ||u||1,α,p(.) > 1 =⇒ ||u|| p− 1,α,p(.) ≤ ρ1,α,p(.)(u) ≤ ||u|| αp+ 1,α,p(.) ; (3) ||u||1,α,p(.) < 1 (= 1; > 1) ⇐⇒ ρ1,α,p(.)(u) < 1 (= 1; > 1). 3 existence of weak homoclinic solutions in this section we investigate the existence of weak homoclinic solutions of problem (1.1). the energy functional corresponding to problem (1.1) is defined as j : w 1,p(.) α −→ r such that j(u) = m̂ ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) + ∑ k∈z 1 p(k) |u(k)|p(k) − ∑ k∈z f(k)u(k), (3.1) where m̂(t) = ∫t 0 m(s) ds. 48 aboudramane guiro, idrissa ibrango and stanislas ouaro cubo 19, 3 (2017) definition 3.1. a weak homoclinic solution of problem (1.1) is a function u ∈ w 1,p(.) α such that    m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z a(k − 1, ∆u(k − 1))∆v(k − 1) + ∑ k∈z |u(k)|p(k)−2u(k)v(k) = ∑ k∈z f(k)v(k), (3.2) for any v ∈ w 1,p(.) α . hence the critical points of functionnal j are the weak solutions for problem (1.1). the main result is the following: theorem 3.2. assume that hypotheses (2.1)-(2.7) hold. then, there exists at least one weak homoclinic solution of problem (1.1). proof. we first present some basic properties of the functional j. proposition 3.3. the functional j is well defined on w 1,p(.) α and is of class c 1(w 1,p(.) α , r) with the derivative given by    〈j′(u), v〉 = m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z a(k − 1, ∆u(k − 1))∆v(k − 1) + ∑ k∈z |u(k)|p(k)−2u(k)v(k) − ∑ k∈z f(k)v(k), (3.3) for all u, v ∈ w 1,p(.) α . indeed, we denote by i(u) = m̂ ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) , l(u) = ∑ k∈z 1 p(k) |u(k)|p(k) and λ(u) = ∑ k∈z f(k)u(k). we have, by using (2.7), that |i(u)| = ∣∣∣∣ ∫ ∑ k∈z a ( k − 1, ∆u(k − 1) ) 0 m(t)dt ∣∣∣∣ ≤ b2 ∣∣∣∣ ∫ ∑ k∈z a ( k − 1, ∆u(k − 1) ) 0 tα−1dt ∣∣∣∣ ≤ b2 α ( ∑ k∈z ∣∣a(k − 1, ∆u(k − 1)) ∣∣ )α . cubo 19, 3 (2017) weak homoclinic solutions to discrete nonlinear problems . . . 49 according to (2.1), (2.3) and the discrete hlder type inequality, we write ∑ k∈z ∣∣a(k − 1, ∆u(k − 1)) ∣∣ ≤ ∑ k∈z ∫∆u(k−1) 0 |a(k − 1, t)|dt ≤ c1 ∑ k∈z ( j(k − 1) + 1 p(k − 1) |∆u(k − 1)|p(k−1)−1 ) ∆u(k − 1) ≤ c1 ∑ k∈z j(k − 1)|∆u(k − 1)| + c1 p− ∑ k∈z |∆u(k − 1)|p(k−1) ≤ c1 ( 1 q− + 1 p− ) ||j||q(.)||∆u||p(.) + c1 p− ||∆u||p(.) < +∞, and we deduce that |i(u)| < +∞. we have |l(u)| = ∣∣∣∣ ∑ k∈z 1 p(k) |u(k)|p(k) ∣∣∣∣ ≤ 1 p− ∣∣∣∣ ∑ k∈z |u(k)|p(k) ∣∣∣∣ < +∞, and |λ(u)| = ∣∣∣∣ ∑ k∈z f(k)u(k) ∣∣∣∣ ≤ ∑ k∈z |f(k)||u(k)| < +∞. clearly, the functionals i, l and λ are in c1(w 1,p(.) α , r). therefore, the functional j is well defined on w 1,p(.) α and is of class c 1(w 1,p(.) α , r). in what follows we prove (3.3). let u, v ∈ w 1,p(.) α . we have    lim h→0+ i(u + hv) − i(u) h = lim h→0+ m̂ ( ∑ k∈z a ( k − 1, ∆u(k − 1) + h∆v(k − 1) )) − m̂ ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) h = m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z a ( k − 1, ∆u(k − 1) ) ∆v(k − 1). let us denote gh = |u(k) + hv(k)|p(k) − |u(k)|p(k) p(k)h . we have ∑ k∈z |gh| ≤ 1 p−h ∑ k∈z |u(k)|p(k) + 1 p−h ∑ k∈z |u(k) + hv(k)|p(k) < +∞. 50 aboudramane guiro, idrissa ibrango and stanislas ouaro cubo 19, 3 (2017) thus lim h→0+ l(u + hv) − l(u) h = lim h→0+ ∑ k∈z |u(k) + hv(k)|p(k) − |u(k)|p(k) p(k)h = ∑ k∈z lim h→0+ |u(k) + hv(k)|p(k) − |u(k)|p(k) p(k)h = ∑ k∈z |u(k)|p(k)−2u(k)v(k). and lim h→0+ λ(u + hv) − λ(u) h = lim h→0+ ∑ k∈z f(k) ( u(k) + hv(k) ) − f(k)u(k) h = ∑ k∈z f(k)v(k). � proposition 3.4. the functional j is weakly lower semi-continuous. indeed, by (2.1), (2.5) and (2.7) we have that j is convex. thus, it is enough to show that j is lower semi-continuous. for this, we fix u ∈ w 1,p(.) α and ǫ > 0. since j is convex, we deduce that for any v ∈ w 1,p(.) α , j(v) ≥ j(u) + 〈j′(u), v − u〉 ≥ j(u) + r(u, v) + s(u, v) + t(u, v), with r(u, v) = m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z a(k − 1, ∆u(k − 1)) ( ∆v(k − 1) − ∆u(k − 1) ) and s(u, v) = ∑ k∈z |u(k)|p(k)−2u(k) ( v(k) − u(k) ) and t(u, v) = ∑ k∈z f(k) ( u(k) − v(k) ) . using the hlder type inequality there exist tree non negative constants c3, c4 and c5 such that r(u, v) ≥ −m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z |a(k − 1, ∆u(k − 1))||∆v(k − 1) − ∆u(k − 1)| ≥ −c3||∆u − ∆v||p(.) ≥ −c3||u − v||1,α,p(.) (3.4) and t(u, v) ≥ −c4||u − v||1,α,p(.). (3.5) cubo 19, 3 (2017) weak homoclinic solutions to discrete nonlinear problems . . . 51 also s(u, v) ≥ − ∑ k∈z |u(k)|p(k)−1|v(k) − u(k)| ≥ − ( 1 p− + 1 (p′)− )∣∣∣∣|u|p(.)−1 ∣∣∣∣ p′(.) ||u − v||p(.) ≥ −c5||u − v||1,α,p(.). (3.6) then, combining (3.4), (3.5) and (3.6), we get j(v) ≥ j(u) − c6||u − v||1,α,p(.) (3.7) with c6 = c3 + c4 + c5. finally for all v ∈ w 1,p(.) α with ||v − u||1,α,p(.) < δ = ǫ c6 , we get j(v) ≥ j(u) − ǫ. then j is lower semi-continuous and by corollary iii.8 in [1], we conclude that j is weakly lower semi-continuous. � proposition 3.5. the functional j is coercive and bounded from below. indeed, we have j(u) = m̂ ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) + ∑ k∈z 1 p(k) |u(k)|p(k) − ∑ k∈z f(k)u(k) ≥ b1 α ( ∑ k∈z a ( k − 1, ∆u(k − 1) ))α + 1 p+ ∑ k∈z |u(k)|p(k) − c7||f||p′(.)||u||p(.) ≥ b1 α ( ∑ k∈z 1 p(k − 1) ∣∣∆u(k − 1) ∣∣p(k−1) )α + 1 p+ ∑ k∈z |u(k)|p(k) − c′7||u||1,α,p(.) ≥ b1 α(p+)α ( ∑ k∈z ∣∣∆u(k) ∣∣p(k) )α + 1 p+ ∑ k∈z |u(k)|p(k) − c′7||u||1,α,p(.) ≥ min ( b1 α(p+)α ; 1 p+ )(( ∑ k∈z ∣∣∆u(k) ∣∣p(k) )α + ∑ k∈z |u(k)|p(k) ) − c′7||u||1,α,p(.) ≥ min ( b1 α(p+)α ; 1 p+ ) ρ1,α,p(.)(u) − c ′ 7||u||1,α,p(.). to prove the coerciveness of the functional j, we may assume that ||u||1,α,p(.) > 1 and, using proposition 2.8, we deduce from the above inequality that j(u) ≥ min ( b1 α(p+)α ; 1 p+ ) ||u|| p − 1,α,p(.) − c′7||u||1,α,p(.). thus, j(u) −→ +∞ as ||u||1,α,p(.) −→ +∞, 52 aboudramane guiro, idrissa ibrango and stanislas ouaro cubo 19, 3 (2017) namely j is coercive. besides, for ||u||1,α,p(.) ≤ 1, we have j(u) ≥ min ( b1 α(p+)α ; 1 p+ ) ρ1,α,p(.)(u) − c ′ 7||u||1,α,p(.) ≥ −c′7||u||1,α,p(.) > −∞. thus j is bounded below. � since j is proper, weakly lower semi-continuous and coercive on w 1,p(.) α , using the relation between critical points of j and problem (1.1), we deduce that j has a minimizer which is a weak homoclinic solution of (1.1). � 4 an extension in this section we consider the following problem (pu)    −m(a(k − 1, ∆u(k − 1)))∆(a(k − 1, ∆u(k − 1))) +|u(k)|p(k)−2u(k) = f(k, u(k)), k ∈ z lim |k|→∞ u(k) = 0, (4.1) where we assume that    f(k, .) : r −→ r is a continuous function for all k in z; c : z −→ [0; +∞) is such that c+ < +∞ and |f(k, t)| ≤ c(k)|t|β(k)−1, with β : z −→ r such that 1 < p− ≤ p+ < β− ≤ β+. (4.2) for any u ∈ w 1,p(.) α , the energy functional corresponding to (4.1) is j(u) = m̂ ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) + ∑ k∈z 1 p(k) |u(k)|p(k) − ∑ k∈z f ( k, u(k) ) , (4.3) where f ( k, u(k) ) = ∫u(k) 0 f(k, t)dt. cubo 19, 3 (2017) weak homoclinic solutions to discrete nonlinear problems . . . 53 definition 4.1. a weak homoclinic solution of problem (4.1) is a function u ∈ w 1,p(.) α such that    m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z a(k − 1, ∆u(k − 1))∆v(k − 1) + ∑ k∈z |u(k)|p(k)−2u(k)v(k) = ∑ k∈z f ( k, u(k) ) v(k), (4.4) for any v ∈ w 1,p(.) α . we have the following result: theorem 4.2. under assumptions (2.1)-(2.7) and (4.2), the problem (4.1) has at least one weak solution. proof. let h(u) = ∑ k∈z f ( k, u(k) ) , therefore h′ : w 1,p(.) α −→ w 1,p(.) α is completly continuous and thus, h is weakly lower semi-continuous. then j ∈ c1 ( w 1,p(.) α ; r ) and is weakly lower semicontinuous. on the other hand, for all u, v ∈ w 1,p(.) α , we have lim δ→0+ h(u + δv) − h(u) δ = lim δ→0+ ∑ k∈z f ( k, u(k) + δv(k) ) − f ( k, u(k) δ , and since ∑ k∈z ∣∣∣∣ f ( k, u(k) + δv(k) ) − f ( k, u(k) ) δ ∣∣∣∣ ≤ c+ δ ∑ k∈z ( ∫u(k)+δv(k) 0 |t|β(k)−1dt + ∫u(k) 0 |t|β(k)−1dt ) ≤ c+ δβ− ∑ k∈z ( |u(k) + δv(k)|β(k) + |u(k)|β(k) ) < +∞, we obtain lim δ→0+ h(u + δv) − h(u) δ = ∑ k∈z lim δ→0+ f ( k, u(k) + δv(k) ) − f ( k, u(k) δ = ∑ k∈z f ( k, u(k) ) v(k). consequently,    〈j′(u), v〉 = m ( ∑ k∈z a ( k − 1, ∆u(k − 1) )) ∑ k∈z a(k − 1, ∆u(k − 1))∆v(k − 1) + ∑ k∈z |u(k)|p(k)−2u(k)v(k) − ∑ k∈z f ( k, u(k) ) v(k), (4.5) 54 aboudramane guiro, idrissa ibrango and stanislas ouaro cubo 19, 3 (2017) for all u, v ∈ w 1,p(.) α . a critical point to j, i.e. a point u ∈ w 1,p(.) α such that 〈j′(u), v〉 = 0 for all v ∈ w1,p(.)α is a weak solution to (4.1). to end the proof of theorem 4.2, we have to prove that j is coercive and bounded from below. for u ∈ w 1,p(.) α such that ||u||1,α,p(.) > 1, we have j(u) ≥ min ( b1 α(p+)α ; 1 p+ ) ||u|| p− 1,α,p(.) − ∑ k∈z f ( k, u(k) ) ≥ min ( b1 α(p+)α ; 1 p+ ) ||u|| p− 1,α,p(.) − ∑ k∈z ∫u(k) 0 |f(k, t)|dt ≥ min ( b1 α(p+)α ; 1 p+ ) ||u|| p− 1,α,p(.) − c+ β− ∑ k∈z |u(k)|β(k) ≥ min ( b1 α(p+)α ; 1 p+ ) ||u|| p− 1,α,p(.) − c+k β− where k is a positive constant. as p− > 1, then j is coercive. on the other hand, for u ∈ w 1,p(.) α such that ||u||1,α,p(.) < 1, we have j(u) ≥ min ( b1 α(p+)α ; 1 p+ ) ||u|| αp+ 1,α,p(.) − c+k β− ≥ − c+k β− > −∞. therefore, j is bounded from below. � references [1] h. brezis; analyse fonctionnelle: theorie et applications. paris, masson, 1983. [2] a. cabada, c. li, s. tersian; on homoclinic solutions of a semilinear p-laplacian difference equation with periodic coefficients. adv. difference equ. 2010, art. id 195376, 17 pp. [3] a. guiro, b. kon and s. ouaro; weak homoclinic solutions of anisotropic difference equation with variable exponents. adv. difference equ. 154 (2013), 13 pp. [4] a. guiro, b. kon and s. ouaro; weak heteroclinic solutions of anisotropic difference equation with variable exponents. electron. j. diff. equ. 225 (2013), 1-9. cubo 19, 3 (2017) weak homoclinic solutions to discrete nonlinear problems . . . 55 [5] b. kon and s. ouaro; weak solutions for anisotropic discrete boundary value problems. j. difference equ. appl. 17, n.10, (2011), 1537-1547. [6] b. kon, s. ouaro and s. traor; weak solutions for anisotropic nonlinear elliptic equations with variable exponent. electron. j. diff. equ. 144 (2009), 1-11. [7] m. mihailescu, p. pucci and v. radulescu; nonhomogeneous boundary value problems in anisotropic sobolev spaces. c. r. acad. sci. paris, ser. i 345 (2007), 561-566. [8] m. mihailescu, p. pucci and v. radulescu; eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. j. math. anal. appl. 340 (2008), 687-698. [9] m. mihailescu, v. radulescu and s. tersian; eigenvalue problems for anisotropic discrete boundary value problems. j. difference equ. appl. 15(6) (2009), 557-567. [10] m. mihailescu, v. radulescu and s. tersian; homoclinic solutions of difference equations with variable exponents. topol. methods nonlinear anal. 38 (2011) 2, 277-289. introduction preliminaries existence of weak homoclinic solutions an extension a mathematical journal vol. 6, no 4, (209 257). december 2004. matrix liapunov’s functions method and stability analysis of continuous systems a.a. martynyuk stability of processes departament of mechanics s.p. timoshenko institute nesterov str.3, 03057, kiev-57. ukraine anmart@stability.kiev.ua 1 introduction this paper contains main results of qualitative analysis of motions for large scale dynamical systems described by ordinary differential equations in terms of matrixvalued liapunov’s functions. the paper is arranged as folows. section 2 deals with stability problems for continuous large scale dynamical systems. the definitions and sufficient conditions for various types of motion stability of nonautonomous and nonlinear systems are presented. the main theorems of the section are supplied with corollaries which illustrate the generality of the results obtained and indicate the sources for the assertions. in section 3 general theorems of section 2 are suplied with a constructive algorithm of constructing the liapunov functions in terms of matrix-valued auxiliary function. the conditions for various types of stability of zero solution for a wide class of largescale systems are formulated in terms of the property of having a fixed sign of special matrices. section 4 sets out conditions of exponential stability with respect to a part of variables. these conditions are established in terms of matrix-valued function constructed by the method proposed in section 3. liapunov functions for linear nonautonomous and autonomous systems in section 5 are constructed by adapting general algorithm from section 3. 210 a.a. martynyuk 6, 4(2004) section 6 presents a discussion of the algorithm and a numerical example which demonstrate the efficiency of the proposed method of constructing the liapunov’s functions in terms of matrix-valued auxiliary function as compared with the bellman– bailey approach based on the vector liapunov’s functions. in final section 7 some unsolved problems of the method of matrix liapunov’s functions are presented. thus, this paper provides a development of the direct liapunov method consisting in both the establishment of general theorems and proposition of a new method of constructing of appropriate liapunov functions for some classes of linear and nonlinear dynamical systems. 2 the direct liapunov’s method via matrix-valued functions in this section the notions of motion stability corresponding to the motion properties of nonautonomous systems are presented being necessary in subsequent presentation. basic notions of the method of matrix-valued liapunov functions are discussed and general theorems and some corollaries are set out. throughout this section, real systems of ordinary differential equations will be considered. notations will be used. 2.1 stability concept in the sense of liapunov we consider systems which can appropriately be described by ordinary differential equations of the form dyi dt = yi(t, y1, . . . ,yn), i = 1, 2, . . . ,n, (2.1) or in the equivalent vector form dy dt = y (t,y), (2.2) where x ∈ rn, y (t,y) = (y1(t,y), . . . ,yn(t,y))t, y t × rn → rn. in the present section we will assume that the right-hand part of ( 2.2) satisfies the solution existence and uniqueness conditions of the cauchy problem dy dt = y (t,y), y(t0) = y0, (2.3) for any (t0,y0) ∈ t × ω, where ω ⊂ rn, 0 ∈ ω and ω is an open connected subset of r.. it is clear that the solution of problem ( 2.3) may not exist on r (on r+), even if the right-hand part y (t,y) of system ( 2.3) is definite and continuous for all (t,y) ∈ t × rn. 6, 4(2004) matrix liapunov’s functions method ... 211 for example, the cauchy problem for the equation dy dt = 1 + y2, y(0) = 0, (2.4) where y is a scalar, has a unique solution y(t) = tg t, existing on the interval (−π 2 , π 2 ), only, while the right-hand part of equation ( 2.4) is definite on the whole plane (t,y). let y(t) = ψ(t; t0,y0) be the solution of system ( 2.2), definite on the interval [t0,τ) and noncontinuable behind the point τ, i.e˙ y(t) is not definite for t = τ. then lim ‖y(t)‖ = +∞ as t → τ − 0. (2.5) using solution y(t) and the right-hand part of system ( 2.2) we construct the vectorfunction f(t,x) = y (t, x + ψ(t)) − y (t,ψ(t)) (2.6) and consider the system dx dt = f(t,x). (2.7) it is easy to verify that the solutions of systems ( 2.2) and ( 2.7) are correlated as x(t) = y(t) − ψ(t) on the general interval of existence of solutions y(t) and ψ(t). it is clear that system ( 2.7) has a trivial solution x(t) ≡ 0. this solution corresponds to the solution y(t) = ψ(t) of system ( 2.2). obviously, the reduction of system ( 2.2) to system (2.7) is possible only when the solution y(t) = ψ(t) is known. qualitative investigation of solutions of system ( 2.2) relatively solution ψ(t) is reduced to the investigation of behaviour of solution x(t) to system ( 2.7) which differs ”little” from the trivial one for t = t0 . in case when stability of unperturbed motion is discussed with respect to some continuously differentiable functions qs(t,ψ1, . . . ,ψn) the perturbed motion equations are found by the system of equations dxi dt = ∂qi ∂t + n∑ s=1 ∂qi ∂ψs ys(t,ψ1, . . . ,ψn) − ḟ(t), where xi = qi(t,ψ1(t), . . . ,ψn(t)) − fs(t), and fi(t) = qi(t, ψ1(t), . . . , ψn(t)) are some known time functions. the system of equations of ( 2.7) type obtained hereat satisfies the condition f(t, 0) = 0 for all t ∈ t and therefore these system has a trivial solution in this case as well. in motion stability theory system ( 2.7) is called the system of perturbed motion equations. since equations ( 2.7) can generally not be solved analytically in closed from, the qualitative properties of the equilbrium state are of great practical interest. we begin with a series of definitions. 212 a.a. martynyuk 6, 4(2004) a very large number of definitions of stability exist for the system ( 2.7). of course, the various definitions of stability can be broadly classified as those which deal with the trajectory, or a motion, or the equilibrium of the null solution of free or unforced systems and those which consider the dynamic response of systems subject to various classes of forcing functions or inputs. in the following, the equilibrium state of ( 2.7) can always be set equal to zero by a linear state transformation, so that the equilibrium state and the null solution to ( 2.7) are considered throughout as equivalent. definition 2.1 the equilibrium state x = 0 of the system ( 2.7) is: (i) stable iff for every t0 ∈ ti and every ε > 0 there exists δ(t0,ε) > 0, such that ‖x0‖ < δ(t0,ε) implies ‖x(t; t0,x0)‖ < ε for all t ∈ t0; (ii) uniformly stable iff both (i) holds and for every ε > 0 the corresponding maximal δm obeying (i) satisfies inf [δm (t,ε) t ∈ ti] > 0; (iii) stable in the whole iff both (i) holds and δm (t,ε) → +∞ as ε → +∞ for all t ∈ r; (iv) uniformly stable in the whole iff both (ii) and (iii) hold; (v) unstable iff there are t0 ∈ ti, ε ∈ (0, +∞) and τ ∈ t0, τ > t0, such that for every δ ∈ (0, +∞) there is x0, ‖x0‖ < δ, for which ‖x(τ; t0,x0)‖ ≥ ε. definition 2.2 the equilibrium state x = 0 of the system ( 2.7) is: (i) attractive iff for every t0 ∈ ti there exists ∆(t0) > 0 and for every ζ > 0 there exists τ(t0; x0,ζ) ∈ [0, +∞) such that ‖x0‖ < ∆(t0) implies ‖x(t; t0,x0)‖ < ζ for all t ∈ (t0 + τ(t0; x0,ζ), +∞); (ii) x0-uniformly attractive iff both (i) is true and for every t0 ∈ r there exists ∆(t0) > 0 and for every ζ ∈ (0, +∞) there exists τu[t0, ∆(t0),ζ] ∈ [0, +∞) such that sup [τm(t0; x0,ζ) x0 ∈ b∆(t0)] = τu(t0,x0,ζ); (iii) t0-uniformly attractive iff both (i) is true, there is ∆ > 0 and for every (x0,ζ) ∈ b∆ × (0, +∞) there exists τu(r,x0,ζ) ∈ [0, +∞) such that sup [τm(t0); x0,ζ) t0 ∈ ti] = τu(ti,x0,ζ); 6, 4(2004) matrix liapunov’s functions method ... 213 (iv) uniformly attractive iff both (ii) and (iii) hold, that is, that (i) is true, there exists ∆ > 0 and for every ζ ∈ (0, +∞) there is τu(r, ∆,ζ) ∈ [0, +∞) such that sup [τm(t0; x0,ζ) (t0,x0) ∈ ti × b∆] = τ(ti, ∆,ζ); the properties (i) – (iv) hold “in the whole” iff (i) is true for every ∆(t0) ∈ (0, +∞) and every t0 ∈ ti. definition 2.3 the equilibrium state x = 0 of the system ( 2.7) is: (i) asymptotically stable iff it is both stable and attractive; (ii) equi-asymptotically stable iff it is both stable and x0-uniformly attractive; (iii) quasi-uniformly asymptotically stable iff it is both uniformly stable and t0uniformly attractive; (iv) uniformly asymptotically stable iff it is both uniformly stable and uniformly attractive; (v) the properties (i) – (iv) hold “in the whole” iff both the corresponding stability of x = 0 and the corresponding attraction of x = 0 hold in the whole; (vi) exponentially stable iff there are ∆ > 0 and real numbers α ≥ 1 and β > 0 such that ‖x0‖ < ∆ implies ‖x(t; t0,x0)‖ ≤ α‖x0‖ exp[−β(t − t0)], for all t ∈ t0, for all t0 ∈ ti. this holds in the whole iff it is true for ∆ = +∞. let g : rn → rn define the time invariant system dx dt = g(x), (2.8) where g(0) = 0 and the components of g are smooth functions of the components of x for x near zero. every stability property of x = 0 of ( 2.11) is uniform in t0 ∈ r. note that the nonperturbed motion equations of the time invariant system can be reduced to the time invariant system ( 2.11) iff the solution ψ(t) = const. otherwise, i.ei̇f ψ(t) 6= const , equations ( 2.11) can be nonstationary. in the investigation of both system ( 2.2) and ( 2.11) the solution x(t) is assumed to be definite for all t ∈ t (for all t ∈ t0). 2.2 classes of liapunov’s functions presently the liapunov direct method (see liapunov [1]) in terms of three classes of auxiliary functions: scalar, vector and matrix ones is intensively applied in qualitative theory. in this point we shall present the description of the above mentioned classes of functions. 214 a.a. martynyuk 6, 4(2004) 2.2.1 matrix-valued liapunov function for the system ( 2.7) we shall consider a continuous matrix-valued function u(t,x) = [vij (t,x)], i,j = 1, 2, . . . ,m, (2.9) where vij ∈ c(tτ ×rn,r) for all i,j = 1, 2, . . . ,m. we assume that next conditions are fulfilled (i) vij (t,x), i,j = 1, 2, . . . ,m, are locally lipschitzian in x; (ii) vij (t, 0) = 0 for all t ∈ r+ (t ∈ tτ ), i,j = 1, 2, . . . ,m; (iii) vij (t,x) = vji(t,x) in any open connected neighbourhood n of point x = 0 for all t ∈ r+ (t ∈ tτ ). definition 2.4 all function of the type v(t,x,α) = αtu(t,x)α, α ∈ rm, (2.10) where u ∈ c(tτ × n , rm×m), are attributed to the class sl. here the vector α can be specified as follows: (i) α = y ∈ rm, y 6= 0; (ii) α = ξ ∈ c(rn, rm+ ), ξ(0) = 0; (iii) α = ψ ∈ c(tτ × rn, rm+ ), ψ(t, 0) = 0; (iv) α = η ∈ rm+ , η > 0. note that the choice of vector α can influence the property of having a fixed sign of function ( 2.13) and its total derivative along solutions of system ( 2.7). 2.2.2 comparison functions comparison functions are used as upper or lower estimates of the function v and its total time derivative. they are usually denoted by ϕ, ϕ : r+ → r+. the main contributor to the investigation of properties of and use of the comparison functions is hahn [2]. what follows is mainly based on his definitions and results. definition 2.5 a function ϕ, ϕ : r+ → r+, belongs to (i) the class k[0,α), 0 < α ≤ +∞, iff both it is defined, continuous and strictly increasing on [0,α) and ϕ(0) = 0; (ii) the class k iff (i) holds for α = +∞, k = k[0,+∞); (iii) the class kr iff both it belongs to the class k and ϕ(ζ) → +∞ as ζ → +∞; 6, 4(2004) matrix liapunov’s functions method ... 215 (iv) the class l[0,α) iff both it is defined, continuous and strictly decreasing on [0,α) and lim [ϕ(ζ) : ζ → +∞] = 0; (v) the class l iff (iv) holds for α = +∞, l = l[0,+∞). let ϕ−1 denote the inverse function of ϕ, ϕ−1[ϕ(ζ)] ≡ ζ. the next result was established by hahn [2]. proposition 2.1 1. if ϕ ∈ k and ψ ∈ k then ϕ(ψ) ∈ k; 2. if ϕ ∈ k and σ ∈ l then ϕ(σ) ∈ l; 3. if ϕ ∈ k[0,α) and ϕ(α) = ξ then ϕ−1 ∈ k[0,ξ); 4. if ϕ ∈ k and lim [ϕ(ζ) : ζ → +∞] = ξ then ϕ−1 is not defined on (ξ, +∞]; 5. if ϕ ∈ k[0,α), ψ ∈ k[0,α) and ϕ(ζ) > ψ(ζ) on [0,α) then ϕ−1(ζ) < ψ−1(ζ) on [0,β], where β = ψ(α). definition 2.6 a function ϕ, ϕ : r+ × r+ → r+, belongs to: (i) the class kk[0;α,β) iff both ϕ(0,ζ) ∈ k[0,α) for every ζ ∈ [0,β) and ϕ(ζ, 0) ∈ k[0,β) for every ζ ∈ [0,α); (ii) the class kk iff (i) holds for α = β = +∞; (iii) the class kl[0;α,β) iff both ϕ(0,ζ) ∈ k[0,α) for every ζ ∈ [0,β) and ϕ(ζ, 0) ∈ l[0,β) for every ζ ∈ [0,α); (iv) the class kl iff (iii) holds for α = β = +∞; (v) the class ck iff ϕ(t, 0) = 0, ϕ(t,u) ∈ k for every t ∈ r+; (vi) the class m iff ϕ ∈ c(r+ × rn,r+), inf ϕ(t,x) = 0, (t,x) ∈ r+ × rn; (vii) the class m0 iff ϕ ∈ c(r+ × rn,r+), inf x ϕ(t,x) = 0 for each t ∈ r+; (viii) the class φ iff ϕ ∈ c(k,r+): ϕ(0) = 0, and ϕ(w) is increasing with respect to cone k. definition 2.7 two functions ϕ1, ϕ2 ∈ k (or ϕ1, ϕ2 ∈ kr) are said to be of the same order of magnitude if there exist positive constants α, β, such that αϕ1(ζ) ≤ ϕ2(ζ) ≤ βϕ1(ζ) for all ζ ∈ [0,ζ1] (or for all ζ ∈ [0,∞)). 216 a.a. martynyuk 6, 4(2004) 2.2.3 properties of matrix-valued functions for the functions of the class sl we shall cite some definitions which are applied in the investigation of dynamics of system ( 2.7). definition 2.8 the matrix-valued function u : tτ × rn → rm×m is: (i) positive semi-definite on tτ = [τ, +∞), τ ∈ r, iff there are time-invariant connected neighbourhood n of x = 0, n ⊆ rn, and vector y ∈ rm, y 6= 0, such that (a) v(t,x,y) is continuous in (t,x) ∈ tτ × n × rm; (b) v(t,x,y) is non-negative on n , v(t,x,y) ≥ 0 for all (t,x,y 6= 0) ∈ tτ × n × rm, and (c) vanishes at the origin: v(t, 0,y) = 0 for all t ∈ tτ × rm; (d) iff the conditions (a) – (c) hold and for every t ∈ tτ , there is w ∈ n such that v(t,w,y) > 0, then v is strictly positive semi-definite on tτ ; (ii) positive semi-definite on tτ × g iff (i) holds for n = g; (iii) positive semi-definite in the whole on tτ iff (i) holds for n = rn; (iv) negative semi-definite (in the whole) on tτ (on tτ × n) iff (−v) is positive semi-definite (in the whole) on tτ (on tτ × n) respectively. the expression “on tτ ” is omitted iff all corresponding requirements hold for every τ ∈ r. definition 2.9 the matrix-valued function u : tτ × rn → rm×m is: (i) positive definite on tτ , τ ∈ r, iff there are a time-invariant connected neighbourhood n of x = 0, n ⊆ rn and a vector y ∈ rm, y 6= 0, such that both it is positive semi-definite on tτ ×n and there exists a positive definite function w on n , w : rn → r+, obeying w(x) ≤ v(t,x,y) for all (t,x,y) ∈ tτ ×n ×rm; (ii) positive definite on tτ × g iff (i) holds for n = g; (iii) positive definite in the whole on tτ iff (i) holds for n = rn; (iv) negative definite (in the whole) on tτ (on tτ × n × rm) iff (−v) is positive definite (in the whole) on tτ (on tτ × n × rm) respectively; (v) weakly decrescent if there exists a ∆1 > 0 and a function a ∈ ck such that v(t,x,y) ≤ a(t,‖x‖) as soon as ‖x‖ < ∆1; (vi) asymptotically decrescent if there exists a ∆2 > 0 and a function b ∈ kl such that v(t,x,y) ≤ b(t,‖x‖) as soon as ‖x‖ < ∆2. the expression “on tτ ” is omitted iff all corresponding requirements hold for every τ ∈ r. 6, 4(2004) matrix liapunov’s functions method ... 217 proposition 2.2 the matrix-valued function u : r × rn → rm×m is positive definite on tτ , τ ∈ r, iff it can be written as ytu(t,x)y = ytu+(t,x)y + a(‖x‖), where u+(t,x) is a positive semi-definite matrix-valued function and a ∈ k. definition 2.10 (cf grujić, et al. [1]) set vζ (t) is the largest connected neighborhood of x = 0 at t ∈ r which can be associated with a function u : r × rn → rm×m so that x ∈ vζ (t) implies v(t,x,y) < ζ, y ∈ rm. definition 2.11 the matrix-valued function u : r × rn → rs×s is: (i) decreasing on tτ , τ ∈ r, iff there is a time-invariant neighborhood n of x = 0 and a positive definite function w on n , w : rn → r+, such that ytu(t,x)y ≤ w(x) for all (t,x) ∈ tτ × n ; (ii) decreasing on tτ × g iff (i) holds for n = g; (iii) decreasing in the whole on tτ iff (i) holds for n = rn. the expression “on tτ ” is omitted iff all corresponding conditions still hold for every τ ∈ r. proposition 2.3 the matrix-valued function u : r × rn → rm×m is decreasing on tτ , τ ∈ r, iff it can be written as ytu(t,x)y = ytu−(t,x)y + b(‖x‖), (y 6= 0) ∈ rm, where u−(t,x) is a negative semi-definite matrix-valued function and b ∈ k. definition 2.12 the matrix-valued function u : r × rn → rm×m is: (i) radially unbounded on tτ , τ ∈ r, iff ‖x‖→∞ implies ytu(t,x)y → +∞ for all t ∈ tτ , y ∈ rm, y 6= 0; (ii) radially unbounded, iff ‖x‖ → ∞ implies ytu(t,x)y → +∞ for all t ∈ tτ for all τ ∈ r, y ∈ rm, y 6= 0. proposition 2.4 the matrix-valued function u : tτ × rn → rm×m is radially unbounded in the whole (on tτ ) iff it can be written as ytu(t,x)y = ytu+(t,x)y + a(‖x‖) for all x ∈ rn, where u+(t,x) is a positive semi-definite matrix-valued function in the whole (on tτ ) and a ∈ kr. according to liapunov function ( 2.17) is applied in motion investigation of system ( 2.7) together with its total derivative along solutions x(t) = x(t; t0,x0) of system 218 a.a. martynyuk 6, 4(2004) ( 2.7). assume that each element vij (t,x) of the matrix-valued function ( 2.16) is definite on the open set tτ × n , n ⊂ rn, i.e. vij (t,x) ∈ c(tτ × n , r). if γ(t; t0,x0) is a solution of system ( 2.7) with the initial conditions x(t0) = x0, i.e. γ(t0; t0,x0) = x0, the right-hand upper derivative of function ( 2.17) for α = y, y ∈ rm, with respect to t along the solution of ( 2.7) is determined by the formula d+v(t,x,y) = ytd+u(t,x)y, (2.11) where d+u(t,x) = [d+vij (t,x)], i,j = 1, 2, . . . ,m, and d+vij (t,x) = lim sup { sup γ(t,t,x)=x [vij (t + σ, γ(t + σ, t,x)) − vij (t,x)] σ−1 : σ → 0+ } , i,j = 1, 2, . . . ,m. (2.12) in case when system ( 2.7) has a unique solution for every initial value of x(t0) = x0 ((t0,x0) ∈ tτ × n), the expression ( 2.19) is equivalent to d+vij (t,x) = lim sup{[vij (t + σ, γ(t + σ, t,x)) − vij (t,x)] σ−1 : σ → 0+}, i,j = 1, 2, . . . ,m. (2.13) further we assume that for all i,j = 1, 2, . . . ,m the functions vij (t,x) are continuous and locally lipschitzian in x, i.e. for every point in n there exists a neighbourhood ∆ and a positive number l = l(∆ ) such that |vij (t,x) − vij (t,y)| ≤ l‖x − y‖, i,j = 1, 2, . . . ,m, for any (t,x) ∈ tτ × ∆, (t,y) ∈ tτ × ∆. besides, the expression ( 2.12) is equivalent to d+vij (t,x) = lim sup {[vij (t + σ, x + σf(t,x)) − vij (t,x)] σ−1 : σ → 0+}, i,j = 1, 2, . . . ,m. (2.14) if the matrix-valued function u(t,x) ∈ c1(tτ × n , rm×m), i.eȧll its elements vij (t,x) are functions continuously differentiable in t é x, then the expression ( 2.14) is equivalent to dvij (t,x) = ∂vij ∂t (t,x) + n∑ s=1 ∂vij ∂xs (t,x) fs(t,x), (2.15) where fs(t,x) are components of the vector-function f(t,x) = (f1(t,x), . . . ,fn(t,x))t. function ( 2.15) has the euler derivative ( 2.10) at point (t,x) along solution x(t; t0,x0) of system ( 2.7) iff d+v(t,x,y) = d+v(t,x,y) = d −v(t,x,y) = d−v(t,x,y) = dv(t,x,y). (2.16) note that the application of any of the expressions ( 2.12), ( 2.13) or ( 2.15) in ( 2.11) is admissible. 6, 4(2004) matrix liapunov’s functions method ... 219 2.2.4 vector liapunov function a vector-valued liapunov function v (t,x) = (v1(t,x),v2(t,x), . . . ,vm(t,x)) t (2.17) can be obtained via matrix-valued function ( 2.9) in several ways. definition 2.13 all vector functions of the type l(t,x,b) = au(t,x)b, (2.18) where u ∈ c(tτ × rn, rs×s), a is a constant matrix s × s, and vector b is defined according to (i) – (iv) similarly to the definition of the vector α, are attributed to the class vl. if in two-index system of functions ( 2.9) for all i 6= j the elements vij (t,x) = 0, then v(t,x) = diag (v11(t,x), . . . , vmm(t,x)) t, where vii ∈ c(tτ × rn, r), i = 1, 2, . . . ,m, , is a vector-valued function. besides, the function ( 2.18) has the components lk(t,x,b) = m∑ i=1 akibivii(t,x), k = 1, 2, . . . ,m. the methods of application of liapunov’s vector functions in motion stability theory are presented in a number of monographs some of which are mentioned in the end of this section. 2.2.5 scalar liapunov function the simplest type of auxiliary function for system ( 2.7) is the function v(t,x) ∈ c(t0 × rn, r+), v(t, 0) = 0, (2.19) for which (a) v(t, 0) = 0 for all t ∈ tτ ; (b) v(t,x) is locally lipschitzian in x; (c) v(t,x) ∈ c(tτ × n , r). in stability theory both sign-definite in the sense of liapunov and semi-definite functions (see hahn [2]) are applied. we shall set out some examples. 220 a.a. martynyuk 6, 4(2004) example 2.1 (i) the function v(t,x) = (1 + sin2 t)x21 + (1 + cos 2 t)x22 is positive definite and decreasing, while the function v(t,x) = (x21 + x 2 2) sin 2 t is decreasing and positive semi-definite. (ii) the function v(t,x) = x21 + (1 + t)x 2 2 is positive definite but not non-decreasing, while the function v(t,x) = x21 + x22 1 + t is decreasing but not positive definite. (iii) the function v(t,x) = (1 + t)(x1 − x2)2 is positive semi-definite and non-decreasing. among the variety of the liapunov functions the quadratic forms v(t,x) = xtp(t)x, p t(t) = p(t), (2.20) are of special importance, where p(t) is n × n -matrix with continuous and bounded elements for all t ∈ tτ . proposition 2.5 for the quadratic form ( 2.20) to be positive definite it is necessary and sufficient that∣∣∣∣∣∣ p11(t) . . . p1s(t) . . . . . . . . . . . . . . . . . . . ps1(t) . . . pss(t) ∣∣∣∣∣∣ > k > 0, s = 1, 2, . . . ,n, (2.21) for all t ∈ tτ . note that if conditions ( 2.21) are satisfied, the ”quasi-quadratic” form v(t,x) = xtp(t)x + ψ(t,x), ψ(t, 0) = 0, (2.22) is positive definite, provided that some constants a,b (a > 0, b ≥ 2) exist, such that |ψ(t,x)| ≤ arb, where r = (xtx)1/2. to calculate total derivative of function ( 2.22) along solutions of system ( 2.7) either ( 2.12) – ( 2.15) is applied for i = j = 1, depending on the assumptions on system ( 2.7) and function ( 2.22). 6, 4(2004) matrix liapunov’s functions method ... 221 it is well known (see yoshizawa [1]) that if d+v(t,x) ≤ 0 and consequently d+v(t,x(t)) ≤ 0, then the function v(t,x) is nonincreasing function of t ∈ tτ . further, if d+v(t,x) ≥ 0, then v(t,x(t)) is nondecreasing along any solution of (2.7) and vice versa. we shall formulate these observations as follows. proposition 2.6 suppose m(t) = v(t,x(t)) is continuous on (a,b). then m(t) is nondecreasing (nonincreasing) on (a,b) if and only if d+m(t) ≥ 0 (≤ 0) for every t ∈ (a,b), where d+m(t) = lim sup {[m(t + σ) − m(t)] σ−1 : σ → 0+}. further all auxiliary functions allowing the solution of the problem on stability (instability) of the equilibrium state x = 0 of system ( 2.7) are called the liapunov functions. the construction of the liapunov functions still remains one of the central problems of stability theory. 2.3 liapunov’s like theorems in general functions ( 2.10), ( 2.18) and ( 2.20) together with their total derivatives ( 2.11) along solutions of system ( 2.7) allow to establish existence conditions for the motion properties of system ( 2.7) of various types such as stability, instability, boundedness, etc. below we shall set out some results in the direction. theorem 2.1 let the vector-function f in system ( 2.7) be continuous on r × n (on tτ × n). if there exist 1. an open connected time-invariant neighborhood g ⊂ n of the point x = 0 ; 2. a matrix-valued function u ∈ c (r × n ,rm×m) and a vector y ∈ rm such that the function v(t,x,y) = ytu(t,x)y is locally lipschitzian in x for all t ∈ r (t ∈ tτ ); 3. functions ψi1, ψi2, ψi3 ∈ k, ψ̃i2 ∈ ck, i = 1, 2, . . . ,m; 4. m × m matrices aj (y), j = 1, 2, 3, ã2(y) such that (a) ψt1 (‖x‖)a1(y)ψ1(‖x‖) ≤ v(t,x,y) ≤ ψ̃t2 (t,‖x‖)ã2(y)ψ̃2(t,‖x‖) for all (t,x,y) ∈ r × g × rm (for all (t,x,y) ∈ tτ × g × rm); (b) ψt1 (‖x‖)a1(y)ψ1(‖x‖) ≤ v(t,x,y) ≤ ψt2 (‖x‖)a2(y)ψ2(‖x‖) for all (t,x,y) ∈ r × g × rm (for all (t,x,y) ∈ tτ × g × rm); (c) d+v(t,x,y) ≤ ψt3 (‖x‖)a3(y)ψ3(‖x‖) for all (t,x,y) ∈ r × g × rm (for all (t,x,y) ∈ tτ × g × rm). then, if the matrices a1(y), a2(y), ã2(y), (y 6= 0) ∈ rm are positive definite and a3(y) is negative semi-definite, then 222 a.a. martynyuk 6, 4(2004) (a) the state x = 0 of system ( 2.7) is stable (on tτ ), provided condition (4)(a) is satisfied; (b) the state x = 0 of system ( 2.7) is uniformly stable (on tτ ), provided condition (4)(b) is satisfied. corollary 2.1 let 1. condition (1) of theorem 2.1 be satisfied; 2. there exist at least one couple of indices (p,q) ∈ [1,m] for which (vpq(t,x) 6= 0) ∈ u(t,x) and function v(t,x,e) = etu(t,x)e = v(t,x) for all (t,x) ∈ r×g (for all (t,x) ∈ tτ × g) satisfy the conditions (a) ψ1(‖x‖) ≤ v(t,x); (b) v(t,x) ≤ ψ2(‖x‖); (c) d+v(t,x)|(2.7) ≤ 0, where ψ1,ψ2 are some functions of the class k. then, the state x = 0 of system ( 2.7) is stable (on tτ ) under conditions (a) and (c), and uniformly stable (on tτ ) under conditions (a) – (c). theorem 2.2 let the vector-function f in system ( 2.7) be continuous on r × rn (on tτ × rn). if there exist 1. a matrix-valued function u ∈ c (r × rn,rm×m) (u ∈ c(tτ × rn,rm×m)) and a vector y ∈ rm such that the function v(t,x,y) = ytu(t,x)y is locally lipschitzian in x for all t ∈ r (t ∈ tτ ); 2. functions ϕ1i, ϕ2i, ϕ3i ∈ kr, ϕ̃2i ∈ ckr, i = 1, 2, . . . ,m; 3. m × m matrices bj (y), j = 1, 2, 3, b̃2(y) such that (a) ϕt1 (‖x‖)b1(y)ϕ1(‖x‖) ≤ v(t,x,y) ≤ ϕ̃t2 (t,‖x‖)b̃2(y)ϕ̃2(t,‖x‖) for all (t,x,y) ∈ r × rn × rm (for all (t,x,y) ∈ tτ × rn × rm); (b) ϕt1 (‖x‖)b1(y)ϕ1(‖x‖) ≤ v(t,x,y) ≤ ϕt2 (‖x‖)b2(y)ϕ2(‖x‖) for all (t,x,y) ∈ r × rn × rm (for all (t,x,y) ∈ tτ × rn × rm); (c) d+v(t,x,y) ≤ ϕt3 (‖x‖)b3(y)ϕ3(‖x‖) for all (t,x,y) ∈ r× rn × rm (for all (t,x,y) ∈ tτ × rn × rm). then, provided that matrices b1(y), b2(y) and b̃2(y) for all (y 6= 0) ∈ rm are positive definite and matrix b3(y) is negative semi-definite, (a) under condition 3(a) the state x = 0 of system ( 2.7) is stable in the whole (on tτ ); 6, 4(2004) matrix liapunov’s functions method ... 223 (b) under condition 3(b) the state x = 0 of system ( 2.7) is uniformly stable in the whole (on tτ ). corollary 2.2 let for function v(t,x,e) = v(t,x), mentioned in condition 2 of corollary 2.1 for all (t,x) ∈ r×rn (for all (t,x) ∈ tτ ×rn) the following conditions hold (a) ϕ1(‖x‖) ≤ v(t,x); (b) v(t,x) ≤ ϕ2(‖x‖), for some function ϕ2; (c) d+v(t,x)|(2.7) ≤ 0, where ϕ1, ϕ2 are of class kr. then the state x = 0 of system ( 2.7) is stable in the whole (on tτ ) under conditions (a) and (c) and uniformly stable in the whole (on tτ ) under conditions (a) – (c). theorem 2.3 let the vector-function f in system ( 2.7) be continuous on r × n (on tτ × n). if there exist 1. an open connected time-invariant neighborhood g ⊂ n of the point x = 0 ; 2. a matrix-valued function u ∈ c (r × n , rm×m) (u ∈ c(tτ × n ,rm×m)) and a vector y ∈ rm such that the function v(t,x,y) = ytu(t,x)y is locally lipschitzian in x for all t ∈ r (t ∈ tτ ); 3. functions η1i, η2i, η3i ∈ k, η̃2i ∈ ck, i = 1, 2, . . . ,m; 4. m × m matrices cj (y), j = 1, 2, 3, c̃2(y) such that (a) ηt1 (‖x‖)c1(y)η1(‖x‖) ≤ v(t,x,y) ≤ η̃t2 (t,‖x‖)c̃2(y)η̃2(t,‖x‖) for all (t,x,y) ∈ r × g × rm (for all (t,x,y) ∈ tτ × g × rm); (b) ηt1 (‖x‖)c1(y)η1(‖x‖) ≤ v(t,x,y) ≤ ηt2 (‖x‖)c2(y)η2(‖x‖) for all (t,x,y) ∈ r × g × rm (for all (t,x,y) ∈ tτ × g × rm); (c) d∗v(t,x,y) ≤ ηt3 (‖x‖)c3(y)η3(‖x‖) + m (t, η3(‖x‖)) for all (t,x,y) ∈ r×g×rm (for all (t,x,y) ∈ tτ ×g×rm), where function m(t, ·) satisfies the condition lim |m (t,η3(‖x‖)) | ‖η3‖ = 0 as ‖η3‖ → 0 uniformly in t ∈ r (t ∈ tτ ). then, provided the matrices c1(y), c2(y), c̃2(y) are positive definite and matrix c3(y) (y 6= 0) ∈ rm is negative definite, then (a) under condition 4(a) the state x = 0 of the system ( 2.7) is asymptotically stable (on tτ ); 224 a.a. martynyuk 6, 4(2004) (b) under condition 4(b) the state x = 0 of the system ( 2.7) is uniformly asymptotically stable (on tτ ). corollary 2.3 let 1. condition 1 of theorem 2.2 be satisfied; 2. for function v(t,x,e) = v(t,x), mentioned in condition 2 of corollary 2.1 for all (t,x) ∈ r × g (for all (t,x) ∈ tτ × g) ) (a) ψ1(‖x‖) ≤ v(t,x) ≤ ψ2(‖x‖); (b) d+v(t,x)|(2.7) ≤ −ψ3(‖x‖), where ψ1,ψ2,ψ3 are of class k. then the state x = 0 of system ( 2.7) is uniformly asymptotically stable (on tτ ). theorem 2.4 let the vector-function f in system ( 2.7) be continuous on r × rn (on tτ × rn) and conditions 1 – 3 of theorem 2.2 be satisfied. then, provided that matrices b1(y), b2(y) and b̃2(y) are positive definite and matrix b3(y) for all (y 6= 0) ∈ rm is negative definite, (a) under condition 3(a) of theorem 2.2 the state x = 0 of system ( 2.7) is asymptotically stable in the whole (on tτ ); (b) under condition 3(b) of theorem 2.2 the state x = 0 of system ( 2.7) is uniformly asymptotically stable in the whole (on tτ ). corollary 2.4 for function v(t,x,e) = v(t,x), mentioned in condition 2 of corollary 2.1 for all (t,x) ∈ r × rn (for all (t,x) ∈ tτ × rn) let (a) ϕ1(‖x‖) ≤ v(t,x) ≤ ϕ2(‖x‖); (b) d+v(t,x)|(2.7) ≤ −ψ3(‖x‖), where ϕ1,ϕ2 are of class kr and ψ3 is of class k. then the state x = 0 of system ( 2.7) is uniformly stable in the whole (on tτ ). theorem 2.5 let the vector-function f in system ( 2.7) be continuous on r × n (on tτ × n). if there exist 1. an open connected time-invariant neighborhood g ⊂ n of the point x = 0 ; 2. a matrix-valued function u ∈ c (r × n , rm×m) and a vector y ∈ rm such that the function v(t,x,y) = ytu(t,x)y is locally lipschitzian in x for all t ∈ r (t ∈ tτ ); 3. functions σ2i, σ3i ∈ k, i = 1, 2, . . . ,m, a positive real number ∆1 and positive integer p, m × m matrices f2(y), f3(y) such that 6, 4(2004) matrix liapunov’s functions method ... 225 (a) ∆1‖x‖p ≤ v(t,x,y) ≤ σt2 (‖x‖)f2(y)σ2(‖x‖) for all (t,x,y 6= 0) ∈ r × g × rm (for all (t,x,y 6= 0) ∈ tτ × g × rm); (b) d+v(t,x,y) ≤ σt3 (‖x‖)f3(y)σ3(‖x‖) for all (t,x,y 6= 0) ∈ r× g×rm (for all (t,x,y 6= 0) ∈ tτ × g × rm). then, provided that the matrices f2(y), (y 6= 0) ∈ rm are positive definite, the matrix f3(y), (y 6= 0) ∈ rm is negative definite and functions σ2i, σ3i are of the same magnitude, then the state x = 0 of system ( 2.7) is exponentially stable (on tτ ). corollary 2.5 let 1. condition (1) of theorem 2.1 be satisfied; 2. for function v(t,x,e) = v(t,x), mentioned in condition (2) of corollary 2.1 for all (t,x) ∈ r × g (for all (t,x) ∈ tτ × g) (a) c1‖x‖p ≤ v(t,x) ≤ ϕ1(‖x‖), (b) d+v(t,x)|(2.7) ≤ −ϕ2(‖x‖). then, if the functions ϕ1,ϕ2 are of class k and of the same magnitude, the state x = 0 of system ( 2.7) is exponentially stable (on tτ ). theorem 2.6 let the vector-function f in system ( 2.7) be continuous on r × rn (on tτ × rn). if there exist 1. a matrix-valued function u ∈ c (r × rn, rm×m) (u ∈ c(tτ × rn,rm×m)) and a vector y ∈ rm such that the function v(t,x,y) = ytu(t,x)y is locally lipschitzian in x for all t ∈ r (for all t ∈ tτ ); 2. functions ν2i, ν3i ∈ kr, i = 1, 2, . . . ,m, a positive real number ∆2 > 0 and a positive integer q; 3. m × m matrices h2, h3 such that (a) ∆2‖x‖q ≤ v(t,x,y) ≤ νt2 (‖x‖)h2(y)ν2(‖x‖) for all (t,x,y 6= 0) ∈ r×rn × rm (for all (t,x,y) ∈ tτ × rn × rm); (b) d+v(t,x,y) ≤ νt3 (‖x‖)h3(y)ν3(‖x‖) for all (t,x,y 6= 0) ∈ r × rn × rm (for all (t,x,y 6= 0) ∈ tτ × rn × rm). then, if the matrix h2(y) for all (y 6= 0) ∈ rm is positive definite, the matrix h3(y) for all (y 6= 0) ∈ rm is negative definite and functions ν2i, ν3i are of the same magnitude, the state x = 0 of system ( 2.7) is exponentially stable in the whole (on tτ ). corollary 2.6 for function v(t,x,e) = v(t,x), mentioned in condition (2) of corollary 2.1 for all (t,x) ∈ r × rn (for all (t,x) ∈ rn × g) let (a) c2‖x‖q ≤ v(t,x) ≤ ψ1(‖x‖), 226 a.a. martynyuk 6, 4(2004) (b) d+v(t,x)|(2.7) ≤ −ψ2(‖x‖), where ψ1,ψ2 ∈ kr–class and are of the same magnitude. then the state x = 0 of system ( 2.7) is exponentially stable in the whole (on tτ ). theorem 2.7 let the vector-function f in system ( 2.7) be continuous on r × n (on tτ × n). if there exist 1. an open connected time-invariant neighborhood g ⊂ n of the point x = 0 ; 2. a matrix-valued function u ∈ c1 (r × n , rm×m) (u ∈ c1(tτ × n ,rm×m)) and a vector y ∈ rm; 3. functions ψ1i,ψ2i,ψ3i ∈ k, i = 1, 2, . . . ,m, m × m matrices a1(y), a2(y), g(y) and a constant ∆ > 0 such that (a) ψt1 (‖x‖)a1(y)ψ1(‖x‖) ≤ v(t,x,y) ≤ ψt2 (‖x‖)a2(y)ψ2(‖x‖) for all (t,x,y) ∈ r × g × rm (for all (t,x,y) ∈ tτ × g × rm); (b) d+v(t,x,y) ≥ ψt3 (‖x‖)g(y)ψ3(‖x‖) for all (t,x,y) ∈ r× g × rm (for all (t,x,y) ∈ tτ × g × rm); 4. point x = 0 belongs to ∂g; 5. v(t,x,y) = 0 on t0 × (∂g ∩ b∆), where b∆ = {x : ‖x‖ < ∆}. then, if matrices a1(y), a2(y) and g(y) for all (y 6= 0) ∈ rm are positive definite, the state x = 0 of system ( 2.7) is unstable (on tτ ). corollary 2.7 let 1. condition (1) of theorem 2.7 be satisfied; 2. there exist at least one couple of indices (p,q) ∈ [1,m] such that (vpq(t,x) 6= 0) ∈ u(t,x) and a function v(t,x,e) = v(t,x) ∈ c1(r × b∆, r+), b∆ ⊂ g, such that on t0 × g (a) 0 < v(t,x) ≤ a < +∞, for some a > 0; (b) d+v(t,x)|(2.7) ≥ ϕ(v(t,x)) for some function ϕ of class k; (c) point x = 0 belongs to ∂g; (d) v(t,x) = 0 on t0 × (∂g ∩ b∆). then the state x = 0 of the system ( 2.7) is unstable. we shall pay our attention to some specific features of the functions applied in corollary 2.7. function v(t,x) specifies the domain v(t,x) > 0, which is changing for t ∈ tτ . clearly this domain may cease its existence before the instability of motion is discovered. 6, 4(2004) matrix liapunov’s functions method ... 227 if the function v(t,x) is positive definite (strictly positive semi-definite), then the domain v(t,x) > 0 exists for all t ∈ tτ . if the function v(t,x) is constant negative, the domain v(t,x) > 0 does not exist. example 2.2 (i) function v(t,x) = sin tx1x2 is of variable sign and domain v(t,x) > 0 exists but not for all t ∈ tτ . (ii) for the function v(t,x) = (cos t − 2) x21x2 the domain v(t,x) > 0 exists for all t ∈ tτ . (iii) for the function v(t,x) = ( 1 t − a ) x1x2 − x22, a > 0, the domain v(t,x) > 0 exists for all t ≥ t0, and for t0 > 1/a. corollary 2.8 let condition (1) of theorem 2.7 be satisfied. if there exist t0 ∈ t0, ∆ > 0, (b∆ ⊂ n) and an open set g ⊂ b∆ and the function v(t,x,e) = v(t,x) ∈ c1(t0 × b∆, r), mentioned in corollary 2.7 such that on t0 × g (a) 0 < v(t,x) ≤ ϕ1(‖x‖); (b) d+v(t,x)|(2.7) ≥ ϕ2(‖x‖) for some ϕ1,ϕ2 of class k; (c) point x = 0 belongs to ∂g; (d) v(t,x) = 0 on t0 × (∂g ∩ b∆). then the state x = 0 of ( 2.7) is unstable. corollary 2.9 if in corollary 2.8 condition (b) is replaced by (b′) d+v(t,x)|(2.7) ≥ kv(t,x) + w(t,x) on t0 ×g, where k > 0 and function w(t,x) ≥ 0 is continuous on t0 ×g, then the state x = 0 of system ( 2.7) is unstable. 3 formulas of liapunov matrix-valued functions the two-index system of functions ( 2.9) being suitable for construction of the lyapunov functions allows to involve more wide classes of functions as compared with those ussually applied in motion stability theory. for example, the bilinear forms 228 a.a. martynyuk 6, 4(2004) prove to be natural non-diagonal elements of matrix-valued functions. another peculiar feature of the approach being of importance is the fact that the application of the matrix-valued function in the investigation of multidimensional systems enables to allow for the interconnections between the subsystems in their natural form, i.eṅot necessarily as the destabilizing factor. finally, for the determination of the property of having a fixed sign of the total derivative of auxiliary function along solutions of the system under consideration it is not necessary to encorporate the estimation functions with the quasimonotonicity property. naturally, the awkwardness of calculations in this case is the price. 3.1 a class of large-scale systems we consider a system with finite number of degrees of freedom whose motion is described by the equations ( 3.1) dxi dt = fi(xi) + gi(t,x1, . . . ,xm), i = 1, 2, . . . ,m (3.1 where xi ∈ rni , t ∈ tτ , tτ = [τ, +∞), fi ∈ c(rni,rni ), gi ∈ c(tτ × r\∞ × ··· × r\m,r\〉). introduce the designation gi(t,x) = gi(t,x1, . . . ,xm) − m∑ j=1, j 6=i gij (t,xi,xj ), (3.2) where gij (t,xi,xj ) = gi(t, 0, . . . ,xi, . . . ,xj, . . . , 0) for all i 6= j; i, j = 1, 2, . . . ,m. taking into consideration ( 3.2) system ( 3.1) is rewritten as dxi dt = fi(xi) + m∑ j=1, j 6=i gij (t,xi,xj ) + gi(t,x). (3.3) actually equations ( 3.3) describe the class of large-scale nonlinear nonautonomously connected systems. it is of interest to extend the method of matrix liapunov functions to this class of equations in view of the new method of construction of nondiagonal elements of matrix-valued functions. 3.2 formulae for non-diagonal elements of matrix-valued function in order to extend the method of matrix liapunov functions to systems ( 3.3) it is necessary to estimate variation of matrix-valued function elements and their total derivatives along solutions of the corresponding systems. such estimates are provided by the assumptions below. assumption 3.1 there exist open connected neighborhoods ni ⊆ rni of the equilibriums state xi = 0, functions vii ∈ c1(rni,r+), the comparison functions ϕi1, ϕi2 and ψi of class k(kr) and real numbers c¯ii > 0, c̄ii > 0 and γii such that 6, 4(2004) matrix liapunov’s functions method ... 229 1. vii(xi) = 0 for all (xi = 0) ∈ ni; 2. c ¯ii ϕ2i1(‖xi‖) ≤ vii(xi) ≤ c̄iiϕ 2 i2(‖xi‖); 3. (dxivii(xi)) tfi(xi) ≤ γiiψ2i (‖xi‖) for all xi ∈ ni, i = 1, 2, . . . ,m. it is clear that under conditions of assumption 3.1 the equilibrium states xi = 0 of nonlinear isolated subsystems dxi dt = fi(xi), i = 1, 2, . . . ,m (3.4) are (a) uniformly asymptotically stable in the whole, if γii < 0 and (ϕi1, ϕi2, ψi) ∈ kr-class; (b) stable, if γii = 0 and (ϕi1, ϕi2) ∈ k-class; (c) unstable, if γii > 0 and (ϕi1, ϕi2, ψi) ∈ k-class. the approach proposed in this section takes large scale systems ( 3.3) into consideration, subsystems ( 3.4) having various dynamical properties specified by conditions of assumption 3.1 assumption 3.2 there exist open connected neighborhoods ni ⊆ rni of the equilibrium states xi = 0, functions vij ∈ c1,1,1(tτ ×r\〉×r\|,r), comparison functions ϕi1, ϕi2 ∈ k(kr), positive constants (η1, . . . ,ηm)t ∈ rm, ηi > 0 and arbitrary constants c ¯ij , c̄ij , i, j = 1, 2, . . . ,m, i 6= j such that 1. vij (t,xi,xj ) = 0 for all (xi,xj ) = 0 ∈ ni × nj , t ∈ tτ , i, j = 1, 2, . . . ,m, (i 6= j); 2. c ¯ij ϕi1(‖xi‖)ϕj1(‖xj‖) ≤ vij (t,xi,xj ) ≤ c̄ijϕi2(‖xi‖)ϕj2(‖xj‖) for all (t,xi,xj ) ∈ tτ × n〉 × n|, i 6= j; 3. dtvij (t,xi,xj ) + (dxivij (t,xi,xj )) tfi(xi) + (dxj vij (t,xi,xj )) tfj (xj ) + ηi 2ηj (dxivii(xi)) tgij (t,xi,xj ) + ηj 2ηi (dxj vjj (xj )) tgji(t,xi,xj ) = 0; ( 3.5) it is easy to notice that first order partial equations ( 3.5) are a somewhat variation of the classical liapunov equation proposed for determination of auxiliary function in the theory of his direct method of motion stability investigation. in a particular case these equations are transformed into the systems of algebraic equations whose solutions can be constructed analytically. assumption 3.3 there exist open connected neighbourhoods ni ⊆ rni of the equilibrium states xi = 0, comparison functions ψ ∈ k(kr), i = 1, 2, . . . ,m, real numbers α1ij , α 2 ij , α 3 ij , ν 1 ki, ν 1 kij , µ 1 kij and µ 2 kij , i, j, k = 1, 2, . . . ,m, such that 230 a.a. martynyuk 6, 4(2004) 1. (dxivii(xi)) tgi(t,x) ≤ ψi(‖xi‖) m∑ k=1 ν1kiψ(‖xk‖) + r1(ψ) for all (t,xi,xj ) ∈ tτ × n〉 × n|; 2. (dxivij (t, ·))tgij (t,xi,xj ) ≤ α1ijψ 2 i (‖xi‖)+α 2 ijψi(‖xi‖)ψj (‖xj‖)+α 3 ijψ 2 j (‖xj‖)+ r2(ψ) for all (t,xi,xj ) ∈ tτ × n〉 × n|; 3. (dxivij (t, ·))tgi(t,x) ≤ ψj (‖xj‖) m∑ k=1 ν2ijkψk(‖xk‖) + r3(ψ) for all (t,xi,xj ) ∈ tτ × n〉 × n|; 4. (dxivij (t, ·))tgik(t,xi,xk) ≤ ψj (‖xj‖)(µ1ijkψk(‖xk‖) + µ2ijkψi(‖xi‖)) + r4(ψ) for all (t,xi,xj ) ∈ tτ × n〉 × n|. here rs(ψ) are polynomials in ψ = (ψ1(‖x1‖, . . . ,ψm(‖xm‖)) in a power higher than three, rs(0) = 0, s = 1, . . . , 4. under conditions (2) of assumptions 3.1 and 3.2 it is easy to establish for function v(t,x,η) = ηtu(t,x)η = m∑ i,j=1 vij (t, ·)ηiηj (3.6) the bilateral estimate ut1 h tc ¯ hu1 ≤ v(t,x,η) ≤ ut2 h tc̄hu2, (3.7) where u1 = (ϕ11(‖x1‖, . . . ,ϕm1(‖xm‖))t, u2 = (ϕ12(‖x1‖, . . . ,ϕm2(‖xm‖))t which holds true for all (t,x) ∈ tτ × n , n = n1 × ··· × nm. based on conditions (3) of assumptions 3.1, 3.2 and conditions (1) – (4) of assumption 3.3 it is easy to establish the inequality estimating the auxiliary function variation along solutions of system ( 3.3). this estimate reads dv(t,x,η) ∣∣ (2.1) ≤ ut3 mu3, (3.8) where u3 = (ψ1(‖x1‖), . . . ,ψm(‖xm‖) and holds for all (t,x) ∈ tτ × n . elements σij of matrix m in the inequality ( 3.8) have the following structure σii = η2i γii + η 2 i νii + m∑ k=1, k 6=i (ηkηiν2kii + η 2 i ν 2 kii) + 2 m∑ j=1, j 6=i ηiηj (α1ij + α 3 ji); σij = 12 (η 2 i ν 1 ji + η 2 j ν 1 ij ) + m∑ k=1, k 6=j ηkηjν 2 kij + m∑ k=1, k 6=i ηiηjν 2 kij + ηiηj (α 2 ij + α 2 ji) + m∑ k=1, k 6=i,k 6=j (ηkηjµ1kji + ηiηjµ 2 ijk + ηiηkµ 1 kij + ηiηjµ 2 jik), i = 1, 2, . . . ,m, i 6= j. 6, 4(2004) matrix liapunov’s functions method ... 231 3.3 theorems on stability sufficient criteria of various types of stability of the equilibrium state x = 0 of system ( 3.3) are formulated in terms of the sign definiteness of matrices c ¯ , c̄ and m from estimates ( 3.7), ( 3.8). we shall show that the following assertion is valid. theorem 3.1 assume that the perturbed motion equations are such that all conditions of assumptions 3.1 – 3.3 are fulfilled and moreover 1. matrices c and c in estimate ( 3.7) are positive definite; 2. matrix m in inequality ( 3.8) is negative semi-definite (negative definite). then the equilibrium state x = 0 of system ( 3.3) is uniformly stable (uniformly asymptotically stable). if, additionally, in conditions of assumptions 3.1 – 3.3 all estimates are satisfied for ni = rni , rk(ψ) = 0, k = 1, . . . , 4 and comparison functions (ϕi1,ϕi2) ∈ krclass, then the equilibrium state of system ( 3.3) is uniformly stable in the whole (uniformly asymptotically stable in the whole). proof if all conditions of assumptions 3.1 – 3.2 are satisfied, then it is possible for system ( 3.3) to construct function v(t,x,η) which together with total derivative dv(t,x,η) satisfies the inequalities ( 3.7) and ( 3.8). condition (1) of theorem 4.1 implies that function v(t,x,η) is positive definite and decreasing for all t ∈ tτ . under condition (2) of theorem 4.1 function dv(t,x,η) is negative semi-definite (definite). therefore all conditions of theorem 2.3.1, 2.3.3 from martynyuk [/] are fulfilled. the proof of the second part of theorem 4.1 is based on theorem 2.3.4 from the same monograph. an example of non-linear systems consider the non-linear system dxi dt = aiixi + n∑ j=1,j 6=i aij (xj )xj, (3.9) where xi ∈ r, aii < 0 for i = 1, 2, . . . ,n. we assume on functions aij (x) as follows. assumption 3.4 there exist constants ∆ > 0, ε > 0 and q > 0 such that 1. aij (x) ∈ c(r \ (−ε,ε), r) 2. |aij (x)| < q|τ|γij +∆ for all τ ∈ (−ε,ε) i,j = 1, 2, . . . ,n, i 6= j, where γji = −(aii + ajj )/aii, γij = −(aii + ajj )/ajj . for each scalar subsystem dxi dt = aiixi, i = 1, 2, . . . ,n, (3.10) 232 a.a. martynyuk 6, 4(2004) we take an auxiliary function in the form vii = x2i . non-diagonal elements of matrixvalued function u(x) are found as pseudo-quadratic forms vij (xi,xj ) = p(xi,xj )xixj . basing on equation ( 3.5) of assumption 3.2 for η = (1, 1, . . . , 1)t we get aiixi ∂vij ∂xi + ajjxj ∂vij ∂xj = −[aij (xj ) + aji(xi)]xixj. (3.11) in view that the partial derivatives of functions vij (xi,xj ) are ∂vij ∂xi = pij (xi,xj )xj + ∂pij ∂xi xixj, ∂vij ∂xj = pij (xi,xj )xi + ∂pij ∂xj xixj, we find from equations ( 3.11) aiixi ∂pij ∂xi + ajjxj ∂pij ∂xj + (aii + ajj )pij (xi,xj ) = −aij (xj ) − aji(xi). (3.12) further function pij (xi,xj ) is found as a sum of two functions pij (xi,xj ) = q1(xi) + q2(xj ). besides equation ( 3.12) becomes aiixi dq1 dxi +(aii +ajj )q1(xi)+aji(xi) = −ajjxj dq2 dxj −(aii +ajj )q2(xj )−aij (xj ). (3.13) the right-hand part of ( 3.13) depends on xi, while the left-hand part of ( 3.13) depends on xj , therefore the right-hand and the left-hand parts equal to a constant which is set equal to zero aiixi dq1 dxi + (aii + ajj )q1(xi) + aji(xi) = 0, ajjxj dq2 dxj + (aii + ajj )q2(xj ) + aij (xj ) = 0. (3.14) the corresponding homogeneous equations aiixi dq̃1 dxi + (aii + ajj )q̃1(xi) = 0, ajjxj dq̃2 dxj + (aii + ajj )q̃2(xj ) = 0 (3.15) have general solutions lclq̃1(xi) = c1|xi|γji, q̃2(xj ) = c2|xj|γij respectively. to find partial solutions to equations ( 3.14) the method of variation of a constant is applied. if these solutions are presented as q1(xi) = c1(xi)|xi|γji, q2(xj ) = c2(xj )|xj|γij 6, 4(2004) matrix liapunov’s functions method ... 233 with the initial conditions c1(0) = c2(0) = 0, it is easy to find that q1(xi) = −|xi|γji xi∫ 0 aji(τ) sign τ aii|τ|1+γji dτ, q2(xj ) = −|xj|γij xj∫ 0 aij (τ) sign τ ajj|τ|1+γij dτ, (3.16) where sign τ ,   −1, for τ < 0, ∈ [−1, 1], for τ = 0, 1, for τ > 0. in view of the assumption on functions aij (x) it is easy to show that the functions q1(xi) and q2(xj ) are determined over the whole numerical axis and are differentiable there. thus, we can choose pij (xi,xj ) = −|xi|γji xi∫ 0 aji(τ) sign τ aii|τ|1+γji dτ − |xj|γij xj∫ 0 aij (τ) sign τ ajj|τ|1+γij dτ (3.17) and setting pii = 1 we present function v(x,η) as v(x,η) = ηtu(x)η = xtp(x)x, (3.18) where p(x) = [pij (xi,xj )], i,j = 1, 2, . . . ,n. calculating the corresponding total derivatives of the components of matrix-valued function u(x) we find dv(x,η) ∣∣∣ (3.9) = xts(x)x, (3.19) where s(x) = [σij (x)]ni,j=1 is a matrix whose elements have the following structure 234 a.a. martynyuk 6, 4(2004) σii = 2aii + 2 n∑ j=1,j 6=i ( aii a2jj |xj|γij xj∫ 0 aij (τ) sign τ |τ|1+γij dτ −|xi|γji xi∫ 0 aji(τ) sign τ aii|τ|1+γji dτ − aij (xj ) ajj ) aji(xi), i = 1, 2, . . . ,n, σij (x) = n∑ k=1, k 6=i,k 6=j [( aii a2kk |xk|γik xk∫ 0 aik(τ) sign τ |τ|1+γik dτ −|xi|γki xi∫ 0 aki(τ ) sign τ aii|τ|1+γki dτ − aik(xk) akk ) akj (xj ) + ( ajj a2kk |xk|γjk xk∫ 0 ajk(τ) sign τ |τ|1+γjk dτ −|xj|γkj xj∫ 0 akj (τ) sign τ ajj|τ|1+γkj dτ − ajk(xk) akk ) aki(xi) ] , i 6= j, i,j = 1, 2, . . . ,n. using theorem 3.1 and estimates ( 3.18) and ( 3.19) one can formulate the sufficient conditions of stability, asynptotic stability and asymptotic stability in the whole of system ( 3.9). theorem 3.2 let system of equations ( 3.9) be such that 1. matrix p(x) is positive definite; 2. matrix s(x) is negative semi-definite (negative definite). then the equilibrium state x = 0 of system ( 3.9) is stable (asymptotically stable). if in addition to conditions (1) and (2) one more condition is satisfied, namely 1. there exist constants r > 0, ε > 0 and l > 0 such that ‖p−1(x)‖ > l ‖x‖2−ε for ‖x‖ > r, then the equilibrium state x = 0 of syste ( 3.9) is asymptotically stable in the whole. 4 4 on polystability of motion analysis consider the nonlinear system of differential equations dx dt = a(t,x)x + b(t,y)y + f(t,x,y), dy dt = d(x)x + g(t,x,y), (4.1) 6, 4(2004) matrix liapunov’s functions method ... 235 where x ∈ rn1 , y ∈ rn2 . assume that functions a(x), b(t,y), d(x), f(t,x,y) and g(t,x,y) are definite and continuous in the domain d = {(t,x,y)|t ≥ 0, ‖x‖ ≤ h, ‖y‖ ≤ h}, and functions f(t,x,y) and g(t,x,y) satisfy the inequalities ‖f‖ ≤ c1(x,y)‖x‖γ1, ‖g‖ ≤ c2‖x‖γ2 for all (t,x,y) ∈ d. here function c1(x,y) → 0 as ‖x‖ + ‖y‖ → 0, f(t, 0, 0) = 0, g(t, 0, 0) = 0 for all t ∈ j+t . according to [10, 11] we introduce the following definition. definition 4.1 the equilibrium state x = 0 of system ( 3.9) is called 1. x-polystable, iff it is stable and asymptotically x-stable; 2. uniformly x-polystable, if it is uniformly stable and uniformly asymptotically x-stable; assumption 4.1 the pseudo-linear system dx dt = a(t,x)x (4.2) satisfies the following conditions 1. the equilibrium state x = 0 of system ( 4.2) is uniformly asymptotically stable; 2. there exists a function v(t,x) continuously differentiable in the domain h = {(t,x) : t ≥ 0, ‖x‖ ≤ h}, positive definite and such that c(‖x‖)‖x‖2 ≤ v(t,x) ≤ c̄(‖x‖)‖x‖2, dv dt ∣∣∣ (4.2) ≤ −α(‖x‖)‖x‖x2,∥∥∥∥∂v∂x ∥∥∥∥ ≤ ρ‖x‖α, ρ > 0 α > 0, where c, c̄,α ∈ c(r+,r+). consider a pseudo-linear approximation of system ( 4.1) dx dt = a(x)x + b(t,y)y, dy dt = d(x)x, and construct a matrix-valued function u(t,x,y). the diagonal elements of this function are taken as v11(x) = v(t,x), v22(y) = y ty. 236 a.a. martynyuk 6, 4(2004) to construct the non-diagonal elements v12(t,x,y) of the matrix-valued function we consider the equation dtv12 + (dxv12) ta(t,x)x = − η1 2η2 (dxv(x)) tb(t,y)y − η2 η1 ytd(x)x (4.3) for some η = (η1,η2)t. applying function u(t,x,y) and vector η we construct a scalar function v(t,x,y) = ηtu(t,x,y)η. theorem 4.1 assume that the perturbed motion equations are such that 1. all conditions of assumption 4.1 are satisfied; 2. equation ( 4.3) has a solution in the form of a continuously differentiable function v12(t,x,y) admitting the estimates c12(x,y)‖x‖‖y‖ ≤ v12(t,x,y) ≤ c̄12(x,y)‖x‖‖y‖ ‖dxv12‖ ≤ ρ1‖x‖α1‖y‖β1 ; ‖dxv12‖ ≤ ρ2‖x‖α2‖y‖β2, ρ1,ρ2 > 0, where c12 ∈ c(rn1 × rn2, r), c̄12 ∈ c(rn1 × rn2, r); 3. matrices c(x,y) = ( c11(x) c12(x,y) c12(x,y) 1 ) , c11(x) = c(‖x‖), c(x,y) = ( c̄11(x) c̄12(x,y) c̄12(x,y) 1 ) , c̄11(x) = c̄(‖x‖), satisfy in the domain d = {(x,y) : ‖x‖ ≤ h, ‖y‖ ≤ h} the generalized silvester conditions; 4. there exists a constant κ > 0 such that −a(‖x‖)η21 + sup ‖x‖=1 (dyv12)td(x)x + xtdt(x)dyv12 ‖x‖2 < −κ for all (t,x,y) ∈ d; 5. sup ‖y‖=1 (dxv12)tb(t,y)y + ytbt(t,y)dxv12 ‖y‖2 ≤ 0 for all t ≥ 0 and ‖x‖ ≤ h; 6. α + γ1 ≥ 2, α1 + γ1 ≥ 2, α2 + γ2 ≥ 2, β1 ≥ 0, β2 > 0, β1 > 0. 6, 4(2004) matrix liapunov’s functions method ... 237 then the equilibrium state x = y = 0 of system ( 4.1) is unoformly x-polystable proof conditions (2) from assumption 4.1 and theorem 4.1 for the components of matrix-valued function u(t,x,y) allow to estimate the scalar function v(t,x,y,η) = ηtu(t,x,y)η as uthtc(x,y)hu ≤ v(t,x,y,η) ≤ uthtc(x,y)hu, where u = (‖x‖,‖y‖)t, h = diag (η1,η2). under condition (3) the function v(t,x,y,η) is positive definite and decreasent. we shall estimate the total derivative of function v(t,x,y,η) along solutions of system ( 4.1) taking unto account conditions (4) and (5) dv dt ∣∣∣ (4.1) ≤ −κ‖x‖2 + η21(dxv)tf(t,x,y) + η22ytg(t,x,y) + 2η1η2(dxv12)tf(t,x,y) + 2η1η2(dyv12)tg(t,x,y) ≤ −κ‖x‖2 + ρη21‖x‖αc1(x,y)‖x‖γ1 + η22ρ2c2‖x‖α2‖y‖β2‖x‖γ2 +2η1η2ρ1c1(x,y)‖x‖α1‖y‖β1‖x‖γ1 + 2η1η2ρ2c2‖x‖α2‖y‖β2‖x‖γ2 ≤ −κ‖x‖2 + c(x,y)‖x‖2, where the function c(x,y) → 0 as ‖x‖+‖y‖ → 0. therefore there exists a magnitude h1 ≤ h such that c(x,y) < κ/2 for ‖x‖ + ‖y‖ ≤ h1. thus in the domain d̃ = {(t,x,y) : t ≥ 0, ‖x‖+‖y‖ ≤ h1} the derivative of function v(t,x,y) along solutions of system ( 4.1) is estimated by the inequality dv dt ∣∣∣ (4.1) ≤ − κ 2 ‖x‖2. in terms of theorem 2.5.2 from martynyuk [9] we conclude on uniform asymptotic stability of the equilibrium state x = y = 0 of system ( 4.1). the assertion on uniform asymptotic x-stability follows from theorem 2.6.1 by martynyuk [9] (see also theorem 6.1 from rumyantsev and oziraner [1]). 5 large-scale linear systems linear systems of perturbed motion equations are of an essential interest in the description of various phenomena in physical and technical systems. general theory of such systems is developed well because in some cases such systems can be integrated precisely. on the other hand systems of the type are the first approximation of quasilinear equations in the investigation of which the information on the properties of the first approximation system is encorporated. for this class of systems of equations the construction of the liapunov functions remains in the focus of attention of many researchers. 238 a.a. martynyuk 6, 4(2004) 5.1 non-autonomous linear systems consider a large-scale system whose motion is described by the equations dxi dt = aiixi + m∑ j=1,j 6=i aij (t)xj, i = 1, 2, . . . ,m. (5.1) here the state vectors xi ∈ rni and aii ∈ rni×ni are constant matrices for all i = 1, 2, . . . ,m; aij (t) ∈ c(r,rni×nj ), i,j = 1, 2, . . . ,m, i 6= j, n = m∑ i=1 ni. for the independent subsystems dxi dt = aiixi, i = 1, 2, . . . ,m, (5.2) the auxiliary functions vii(xi) are constructed as the quadratic forms vii(xi) = x t i piixi, i = 1, 2, 3, (5.3) whose constant matrices pii are determined by the algebraic liapunov equations atiipii + piiaii = −gii, i = 1, 2, . . . ,m, (5.4) where gii are pre-assigned matrices of constant sign. for the construction of nondiagonal elements vij (t,xi,xj ) of the matrix-valued function u(t,x) we apply equation ( 3.5). note that for the system ( 5.1) fi(xi) = aiixi, fj (xj ) = ajjxj, gij (t,xi,xj ) = aij (t)xj, gji(t,xi,xj ) = aji(t)xj, gi(t,x) = 0. suppose that at least one of the matrices aij or aji is not equal to constant. then we determine function vij (t,xi,xj ) as vij (t,xi,xj ) = vji(t,xj,xi) = x t i pij (t)xj, (5.5) where pij ∈ c1(r,rni×nj ). since for the bilinear forms ( 5.5) dtvij (t,xi,xj ) = xi dpijdt xj, dxivij (t,xi,xj ) = x t j pij (t) t, dxj vij (t,xi,xj ) = x t i pij (t) the equation ( 3.5) becomes xti ( dpij dt + atiipij + pijajj + ηi ηj piiaij (t) + ηj ηi atji(t)pjj ) xj = 0. 6, 4(2004) matrix liapunov’s functions method ... 239 for determination of matrices pij this correlation yields a system of matrix differential equations dpij dt + atiipij + pijajj = − ηi ηj piiaij (t) − ηj ηi atji(t)pjj i, j = 1, 2, . . . ,m, i 6= j. (5.6) note that equations ( 5.6) can be solved in the explicit form. to this end we consider a linear operator (general information on linear operators can be found, for example, in daletskii and krene [1]) fij : r ni×nj → rni×nj , fijx = atiix + xajj. equation ( 5.6) can be represented as dpij dt + fijpij = − ηi ηj piiaij (t) − ηj ηi atji(t)pii, i 6= j. consider the homogeneous equations dpij dt + fijpij = 0, (5.7) whose general solution is presented as pij (t) = exp{−fijt}cij, where cij is a constant ni × nj matrix and exp{−fijt} = ∞∑ k=0 (−1)kf kijt k k! is an operator exponent. to find the solution of equation ( 5.6) the method of variation of a constant is applied. solution of equation ( 5.6) is presented in the form pij (t) = exp{−fijt}cij (t), (5.8) where cij ∈ c1(r,rn1×n2 ) and cij (0) = 0. substituting by ( 5.8) into ( 5.6) yields dcij dt = − exp{fijt} ( ηi ηj piiaij + ηj ηi atjipii ) , i 6= j. integrating the last correlation from 0 to t we determine a partial solution of equation ( 5.6) pij (t) = − t∫ 0 exp{−fij (t − τ)} ( ηi ηj piiaij (τ) + ηj ηi atji(τ)pjj ) dτ, i 6= j. (5.9) we establish estimates for the function v(t,x,η) = ηtu(t,x)η = m∑ i,j=1 vij (t, .)ηiηj, 240 a.a. martynyuk 6, 4(2004) where u(t,x) =   v11(x1) · · · v1m(t,x1,xm)... . . . ... v1m(t,x1,xm) · · · vmm(xm)   . introduce the designations c̄ii = λm (pii) and cii = λm(pii) and assuming sup t≥0 ‖pij (t)‖ < ∞ denote c̄ij = sup t≥0 ‖pij (t)‖, cij = −c̄ij . since for the forms ( 5.3) and ( 5.5) the estimates λm(pii)‖xi‖2 ≤ vii(xi) ≤ λm (pii)‖xi‖2, xi ∈ rni ; −c̄ij‖xi‖‖xj‖ ≤ vij (t,xi,xj ) ≤ c̄ij‖xi‖‖xj‖, (xi,xj ) ∈ rni × rnj , (5.10) are valid, for the function v(t,x,η) = ηtu(t,x)η wthtchw ≤ v(t,x,η) ≤ wthtchw for all x ∈ rn, (5.11) where w = (‖x1‖, . . . ,‖xm‖)t, h = diag (η1,η2, . . . ,ηm), c = [c̄ij ]mi,j=1, c = [cij ] m i,j=1. in order to estimate the derivative of function v(t,x,η) along solutions of system ( 5.1) we calculate the constants from assumption 3.3 α1ij = α 2 ij = 0, α 3 ij (t) = λm (a t ij (t)pij (t) + p t ij (t)aij (t)), ν1ki = ν 2 ijk = 0, ν 1 ijk(t) = λ 1/2 m [(p t ij (t)aik(t))(p t ij (t)aik(t))], µ 2 ijk = 0. therefore the elements σij of matrix m(t) in estimate ( 3.8) for system ( 5.1) have the structure σii(t) = −η2i λm(gii) + 2 m∑ j=1,j 6=i ηiηjα 3 ij, i = 1, . . . ,m, σij (t) = m∑ k=1,k 6=i,k 6=j (ηkηjν 1 ijk + ηiηkν 1 kij ), i,j = 1, . . . ,m, i 6= j. consequently, the variation of function dv(t,x,η) along solutions of system ( 5.1) is estimated by the inequality dv(t,x,η) ∣∣∣ (5.1) ≤ wtm(t)w (5.12) for all (x1, . . . ,xm) ∈ rn1 × ··· × rnm . remark 5.1 in the partial case when matrices aij and aji do not depent on t it is reasonable to choose pij (t) = const. then equation ( 5.6) becomes aiipij + pijajj = − ηi ηj piiaij − ηj ηi atjipjj (5.13) 6, 4(2004) matrix liapunov’s functions method ... 241 or in the operator form fijpij = − ηi ηj piiaij − ηj ηi atjipjj. therefore for the equation ( 5.13) to have a unique solution it is necessary and sufficient that the operator fij be nondegenerate. it is known (see daletskii and krene [1]) that the set of eigenvalues of the operator fij consists of the numbers λk(aii) + λl(ajj ), where λk(·) is an eigenvalue of the corresponding matrix. basing on these speculations one can formulate the following result. for the equation ( 5.13) to have a unique solution it is necessary and sufficient that λk(aii) + λl(ajj ) 6= 0 for all k, l, and this solution can be presented as pij = −f−1ij ( ηi ηj piiaij + ηj ηi atjipjj ) . this result is summed up as follows. theorem 5.1 assume that for system ( 5.1) the following conditions are satisfied 1. the sign-definite matrices pii, i = 1, 2, 3, are the solution of algebraic equations ( 5.4); 2. the bounded matrices pij (t) for all i,j = 1, 2, . . . ,m, i 6= j, are the solution of matrix differential equations ( 5.6); 3. matrices c é c in estimate ( 5.11) are positive definite; 4. matrix m(t) in estimate ( 5.12) is negative semi-definite (negative definite). then the equilibrium state x = 0 of system ( 5.1) is uniformly stable in the whole (uniformly asymptotically stable in the whole). 5.2 time invariant linear systems assume that in the system dx1 dt = a11x1 + a12x2 + a13x3, dx2 dt = a21x1 + a22x2 + a23x3, dx3 dt = a31x1 + a32x2 + a33x3, (5.14) the state vectors xi ∈ rni , i = 1, 2, 3, and aij ∈ rni×nj are constant matrices for all i,j = 1, 2, 3. 242 a.a. martynyuk 6, 4(2004) for the independent systems dxi dt = aiixi, i = 1, 2, 3, (5.15) we construct auxiliary functions vii(xi) as the quadratic forms vii(xi) = x t i piixi, i = 1, 2, 3, (5.16) whose matrices pii are determined by atiipii + piiaii = −gii, i = 1, 2, 3, (5.17) where gii are prescribed matrices of definite sign. in order to construct non-diagonal elements vij (xi,xj ) of matrix-valued function u(x) we employ equation ( 3.5). note that for system ( 5.14) fi(xi) = aiixi, fj (xj ) = ajjxj, gij (xi,xj ) = aijxj, gi(t,x) = 0, i = 1, 2, 3. since for the bilinear forms vij (xi,xj ) = vji(xj,xi) = x t i pijxj (5.18) the correlations dxivij (xi,xj ) = x t j p t ij , dxj vij (xi,xj ) = x t i pij are true, equation ( 3.5) becomes xti ( atiipij + pijajj + ηi ηj piiaij + ηj ηi atjipii ) xj = 0. ¿from this correlation for determining matrices pij we get the system of algebraic equations aiipij + pijajj = − ηi ηj piiaij − ηj ηi atjipii, i 6= j, i,j = 1, 2, 3. (5.19) since for ( 5.16) and ( 5.18) the estimates vii(xi) ≥ λm(pii)‖xi‖2, xi ∈ rni ; vij (xi,xj ) ≥ −λ 1/2 m (pijp t ij )‖xi‖‖xj‖, (xi,xj ) ∈ r ni × rnj , hold true, for function v(x,η) = ηtu(x)η the inequality wthtchw ≤ v(x,η) (5.20) 6, 4(2004) matrix liapunov’s functions method ... 243 is satisfied for all x ∈ rn, w = (‖x1‖,‖x2‖,‖x3‖)t and the matrix c =   λm(p11) −λ 1/2 m (p12p t 12) −λ 1/2 m (p13p t 13) −λ1/2m (p12p t 12) λm(p22) −λ 1/2 m (p23p t 23) −λ1/2m (p13p t 13) −λ 1/2 m (p23p t 23) λm(p33)   . for system ( 5.14) the constants from assumption 3.3 are: α1ij = α 2 ij = 0; α 3 ij = λm (a t ijpij + p t ij aij ), ν1ki = ν 2 ijk = 0; ν 1 ijk = λ 1/2 m [(p t ij aik)(p t ij aik)], µ 2 ijk = 0. therefore the elements σij of matrix m in ( 5.12) for system ( 5.14) have the structure σii = −η2i λm(gii) + 2 3∑ j=1, j 6=i ηiηjα 3 ij, i = 1, 2, 3, σij = 3∑ k=1, k 6=i,k 6=j (ηkηjν1ijk + ηiηkν 1 kij ), i,j = 1, 2, 3, i 6= j. consequently, the function dv(x,η) variation along solutions of system ( 5.14) is estimated by the inequality dv(x,η) ∣∣ (5.14) ≤ wtmw (5.21) for all (x1,x2,x3) ∈ rn1 × rn2 × rn3 . we summarize our presentation as follows. corollary 5.1 assume for system ( 5.14) the folowing conditions are satisfied: 1. algebraic equations ( 5.17) have the sign-definite matrices pii, i = 1, 2, 3, as their solutions; 2. algebraic equations ( 5.19) have constant matrices pij , for all i,j = 1, 2, 3, i 6= j, as their solutions; 3. matrix c in ( 5.20) is positive definite; 4. matrix m in ( 5.21) is negative semi-definite (negative definite). then the equilibrium state x = 0 of system ( 5.14) is uniformly stable (uniformly asymptotically stable). this corollary follows from theorem 3.1 and hence its proof is obvious. 244 a.a. martynyuk 6, 4(2004) example 5.3 we study the motion of two non-autonomously connected oscillators whose behaviour is described by the equations dx1 dt = γ1x2 + v cos ωty1 − v sin ωty2, dx2 dt = −γ1x1 + v sin ωty1 + v cos ωty2, dy1 dt = γ2y2 + v cos ωtx1 + v sin ωtx2, dy2 dt = −γ2y2 + v cos ωtx2 − v sin ωtx1, (5.22) where γ1, γ2, v, ω, ω + γ1 − γ2 6= 0 are some constants. for the independent subsystems dx1 dt = γ1x2, dx2 dt = −γ1x1 dy1 dt = γ2y2, dy2 dt = −γ2y1 (5.23) the auxiliary functions vii, i = 1, 2, are taken in the form v11(x) = xtx, x = (x1,x2)t, v22(y) = yty, y = (y1,y2)t. (5.24) we use the equation ( 3.5) (see assumption 3.2) to determine the non-diagonal element v12(x,y) of the matrix-valued function u(t,x,y) = [vij (·)], i,j = 1, 2. to this end set η = (1, 1)t and v12(x,y) = xtp12y, where p12 ∈ c1(tτ,r∈×∈). for the equation dp12 dt + ( 0 −γ1 γ1 0 ) p12 + p12 ( 0 γ2 −γ2 0 ) +2v ( cos ωt − sin ωt sin ωt cos ωt ) = 0, (5.25) the matrix p12 = − 2v ω + γ1 − γ2 ( sin ωt cos ωt − cos ωt sin ωt ) is a partial solution bounded for all t ∈ tτ . thus, for the function v(t,x,y) = ηtu(t,x,y)η it is easy to establish the estimate of ( 3.7) type with matrices c and c in the form c = ( c11 c12 c12 c22 ) , c = ( c̄11 c̄12 c̄12 c̄22 ) , where c̄11 = c11 = 1, c̄22 = c22 = 1, c̄12 = −c12 = |2v| |ω + γ1 − γ2| . besides, the vector ut1 = (‖x‖,‖y‖) = ut2 , since the system ( 5.22) is linear. 6, 4(2004) matrix liapunov’s functions method ... 245 for system ( 5.22) the estimate ( 5.12) becomes dv(t,x,y) ∣∣ (5.1) = 0 for all (x,y) ∈ r2 × r2 because m = 0. due to theorem 3.1 the motion stability conditions for system ( 5.22) are established basing on the analysis of matrices c and c property of having fixed sign. it is easy to verify that the matrices c and c are positive definite, if 1 − 4v2 (ω + γ1 − γ2)2 > 0. consequently, the motion of nonautonomously connected oscillators is uniformly stable in the whole, if |v| < 1 2 |ω + γ1 − γ2|. 5.3 discussion and numerical example to start to illustrate the possibilities of the proposed method of liapunov function construction we consider a system of two connected equations that was studied earlier by the bellman-bailey approach (see barbashin [1], voronov and matrosov [1], etc.). partial case of system ( 6.14) is the system dx1 dt = ax1 + c12x2 dx2 dt = bx2 + c21x1, (6.1) where x1 ∈ rn1 , x2 ∈ rn2 , and a, b, c12 and c21 are constant matrices of corresponding dimensions. for independent subsystems dx1 dt = ax1, dx2 dt = bx2 (6.2) the functions v11(x1) and v22(x2) are constructed as the quadratic forms v11 = x t 1 p11x1, v22 = x t 2 p22x2, (6.3) where p11 and p22 are sign-definite matrices. function v12 = v21 is searched for as a bilinear form v12 = xt1 p12x2 whose matrix is determined by the equation atp12 + p12b = − η1 η2 p11c12 − η2 η1 ct21p22, η1 > 0, η2 > 0. (6.4) according to lankaster [1] equation ( 6.4) has a unique solution, provided that matrices a and −b have no common eigenvalues. 246 a.a. martynyuk 6, 4(2004) matrix c in (/././) for system ( 6.4) reads c =   λm(p11) −λ 1/2 m (p12p t 12) −λ1/2m (p12p t 12) λm(p22)   . (6.5) here λm(·) are minimal eigenvalues of matrices p11, p22, and λ 1/2 m (·) is the norm of matrix p12p t12. estimate ( 5.15) for function dv(x,η) by virtue of system ( 6.1) is dv(x,η)|(6.1) ≤ wtξw, (6.6) where w = (‖x1‖, ‖x2‖)t, ξ = [σij ], i,j = 1, 2; σ11 = λ1η21 + η1η2α22, cσ22 = λ2η22 + η1η2β22, v σ12 = σ21 = 0. the notations are λ1 = λm (atp11 + p11a), λ2 = λm (btp22 + p22b), α22 = λm (ct12p12 + p t 12c12), β22 = λm (ct21p t 12 + p12c21), λ(·) is a maximal eigenvalue of matrix (·). partial case of assumption 3.1 is as follows. corollary 6.1 for system ( 6.1) let functions vij (·), i,j = 1, 2, be constructed so that matrix c for system ( 6.1) is positive definite and matrix ξ in inequality (/././) is negative definite. then the equilibrium state x = 0 of system ( 6.1) is uniformly asymptotically stable. we consider the numerical example. let the matrices from system ( 6.1) be of the form a = ( −2 1 3 −2 ) , b = ( −4 1 2 −1 ) , (6.7) c12 = ( −0.5 −0.5 0.8 −0.7 ) , c21 = ( 1 0.5 −0.6 −0.3 ) . (6.8) functions vii for subsystems ẋ = ax, x = (x1,x2)t, ẏ = bx, y = (y1,y2)t 6, 4(2004) matrix liapunov’s functions method ... 247 are taken as the quadratic forms v11 = 75x21 + x1x2 + 5x 2 2, v22 = 0.35y21 + 0.9y1y2 + 0.95y 2 2. (6.9) let η = (1, 1)t. then λ1 = λ2 = −1, p12 = ( −0.011 0.021 −0.05 −0.022 ) , α22 = 0.03, β22 = −0.002. it is easy to verify that σ11 < 0 and σ22 < 0, and hence all conditions of corollary 6.1 are fulfilled in view that λ 1/2 m (p12p t 12) ≤ (λm(p11)λm(p22)) 1/2, for the values of λ1/2m (p12p t 12) = 0.06, λm(p11) = 08, λm(p22) = 0.115. this implies uniform asymptotic stability in the whole of the equilibrium state of system ( 6.1) with matrices ( 6.7) and ( 6.8). let us show now that stability of system ( 6.1) with matrices ( 6.7) and ( 6.8) can not be studied in terms of the bailey [1] theorem. we recall that in this theorem the conditions of exponential stability of the equilibrium state are 1. for subsystems ( 6.2) functions ( 6.3) must exist satisfying the estimates (a) ci1‖xi‖2 ≤ vi(t,xi) ≤ ci2‖xi‖2, (b) dvi(t,xi) ≤ −ci3‖xi‖2, (c) ‖∂vi/∂xi‖ ≤ ci4‖xi‖ for xi ∈ rni , where cij are some positive constants, i = 1, 2, j = 1, 2, 3, 4; 2. the norms of matrices cij in system (/././) must satisfy the inequality (see voronov and matrosov [1], p. 106) ‖c12‖‖c21‖ < ( c11c21 c12c22 )1/2( c13c23 c14c24 ) . (6.10) we note that this inequality is refined as compared with the one obtained firstly by bailey [1]. the constants c11, . . . ,c24 for functions (/././) and system ( 6.1) with matrices (/././) and (/././) take the values c11 = 1.08, c21 = 0.115, c12 = 2.14, c22 = 2.14, c22 = 1.135 , c13 = c23 = 1, c14 = 4.83, c24 = 2.4. 248 a.a. martynyuk 6, 4(2004) condition (/././) requires that ‖c12‖‖c21‖ < 0.0184 (6.11) whereas for system (/././), (/././), and (/././) we have ‖c12‖‖c21‖ = 75. thus, the bailey theorem turns out to be nonapplicable to this system and the condition ( 6.11) is “super-sufficient” for the property of stability. 6 problems for investigations 7.1 to obtain existence conditions for solutions to system ( 3.5) which satisfy bilinear estimates (condition (2) of assumption 3.2) or other similar conditions allowing to establish algebraic conditions of sign-definiteness and decrease (radial unboundedness) of function ( 3.6). 7.2 to construct an algorithm of approximate solution of system ( 3.5) in terms of the method of perturbed nonlinear mechanics. 7.3 to obtain criterion for exponential stability of system ( 3.3) in terms of function ( 3.6) provided that the independent subsystems ( 3.4) are not exponentially stable. 7.4 to investigate other than stability in the sense of liapunov dynamical properties of system ( 3.3) or its partial cases such as stability, boundedness, uniform boundedness in terms of two measures. 7.5 in terms of liapunov function ( 3.6) to construct algorithms for estimation of domains of stability, attraction and asymptotic stability of system ( 3.3) and its partial cases in the phase space or/and in the parameter space. hint. for the initial definitions of the corresponding domains of stability, attraction and asymptotic stability see grujic, et al. [1], krasovskii [1], and martynyuk [7]. 7.6 for the class of autonomous systems dx dt = x(x(t)), x(t0) = x0, (7.1) where x ∈ rn, x ∈ c1(rn, rn), x(0) = 0, admitting decomposition to ( 3.3) form, to establish conditions of global asymptotic stability under condition that the origin for system ( 7.1) is an asymptotic attractor. 7 brief outline of the references and remarks section 2 nonlinear dynamics of continuous systems is a traditional domain of intensive investigations starting with the works by galilei, newton, euler, lagrange, etc. the problem of motion stability arises whenever the engineering or physical problem 6, 4(2004) matrix liapunov’s functions method ... 249 is formulated as a mathematical problem of qualitative analysis of equations. poincaré and liapunov laid a background for the method of auxiliary functions for continuous systems which allow not to integrate the motion equations for their qualitative analysis. the ideas of poincare and liapunov were further developed and applied in many branches of modern natural sciences. the results of liapunov [1], chetaev [1], persidskii [1], malkin [1], ascoli [1], barbasin and krasovskii [1], massera [1], and zubov [1], were base for the definitions 2.1 – 2.3 (ad hoc see grujić et al. [1], pp. 8 – 12) and cfṙao mohana rao [1], yoshizawa [1], rouche et al. [1], antosiewicz [1], lakshmikantham and leela [1], hahn [2], etc. for the definitions 2.4 – 2.7, and 2.13 see hahn [2], and martynyuk [9]. definitions 2.8 – 2.12 are based on some results by liapunov [1], hahn [2], barbashin and krasovskii [1] (see and cfḋjordjevic [1], grujić [2], martynyuk [3 – 6]). the proofs of proposalls 2.1 – 2.5 are in hahn [2], kuz’min [1], martynyuk [9], zubov [2], etc. theorems 2.1 – 2.7 are set out according to martynyuk [10] (see also martynyuk [13]). for the proof of corollary 2.1 see liapunov [1], and chetaev [1]; for the proof of corollary 2.2 see barbashin and krasovskii [1]; for the proof of corollary 2.3 – 2.4 see liapunov [1], massera [1], yoshizava [1], halanay [1], etc; for the proof of corollary 2.5 – 2.6 see he and wang [1], krasovskii [1], and hahn [2]; for the proof of corollary 2.8 see chetaev [1], rouche, et al. [1]; and for the proof of corollary 2.9 – 2.10 see liapunov [1], and rouche, et al. [1]. further results obtained via the liapunov’s methods can be found in burton [1], galperin [1], gruyitch [1], rama mohana rao [1], coppel [1], cesari [1], lakshmikantham and leela [2], martynyuk [14], sivasundaram [1], vincent [1], vorotnikov [1], zubov [3] (see also cd rom by kramer and hofmann [1] for references), etc. section 3 the problem of constructing the liapunov functions for nonlinear nonautonomous system of general type remains still unsolved though its more than onehundred existence. meanwhile the efforts of many mathematicians and mechanical scientists have resulted in the efficient approach of constructing the appropriate auxiliary functions for specific classes of systems of equations with reference to many applications. the approach proposed in this section is based on the idea of matrix-valued function as an appropriate medium for liapunov function construction. this approach has been developed since 1984 and some of the obtained results are published and summarized by martynyuk [9, 12], and kats and martynyuk [1]. actually, the problem of constructing the liapunov functions for the class of nonlinear systems of ( 3.3) type is reduced to the solution of systems of first order partial equations ( 3.5) which are more simple than the liapunov equation for the initial system proposed by in 1892 in his famous dissertation paper. this section is based on some results by martynyuk and slyn’ko [1, 2, 3], and slyn’ko [2]. besides, some results by djordjevic [1, 2], hahn [1], krasovskii [1], lankaster [1], etcȧre used. section 4 the phenomenon of motion polystability has been investigated in nonlinear dynamics since 1987. as noticed by aminov and sirazetdinov [1], and martynyuk [16] 250 a.a. martynyuk 6, 4(2004) this phenomenon was discovered while developing the notion of stability with respect to a part of variables. in monographs by martynyuk [9, 12] some results are presented obtained in the development of the theory of motion polystability including sufficient conditions for exponential polystability in the first approximation (see also martynyuk [14, 15], and slyn’ko [1]). this section encorporated the results by martynyuk and slyn’ko [3]. section 5 linear nonautonomous system of ( 5.1) type or autonomous system ( 5.14) is of essential interest in context with the problem of constructing the liapunov function since this allows to investigate stability of the equilibrium state of some quasilinear systems. in spite of the seeming simplicity of linear systems the problem of constructing the appropriate liapunov function remains open in this case es well (see, e.gḃarbashin [1], zhang [1], etc.). in this section for the above-mentioned systems we adopt the algorithm of liapunov function construction presented in section 3. since in this case systems of equations ( 3.5) turns to be linear differential or algebraic, their exact solutions can sometimes be found. the section is based on the results by martynyuk and slyn’ko [1 – 3]. section 6 the bellman–bailey approach (see bellman [1] and bailey [1]) to stability investigation of large-scale systems has been developed considerably in many papers. in monographs by barbashin [1], michel and miller [1], siljak [1], grujic, martynyuk and ribbens-pavella [1], voronov and matrosov [1], etcȧlongside the original results the results of many investigations of dynamics of linear and nonlinear systems in terms of vector liapunov functions are summarized. an essential deficiency of this approach is the supersufficiency of stability conditions for the systems of motion equations under consideration (see piontkovskii and rutkovskaya [1], and martynyuk and slyn’ko [1]). the application of the matrix-valued liapunov function for the same classes of systems of equations provides wider conditions of stability. the reasons for this were scrutinized by martynyuk [9,/,12]. in this section by the example of linear system it is shown how supersufficient the stability conditions obtained via the bellman–bailey approach are as compared with those obtained via the application of matrix-valued function. section 7 the problems set out in this section are addressed first of all to the young researchers in the area of motion stability theory and its application. solution of any of the problems will be not only a subject of a significant paper but an essential contribution to the development of the method of matrix liapunov functions which was called by profv̇. lakshmikantham (in moscow, 2001) one of three outstanding achivements of stability theory in the 20th century. received: march 2003. revised: june 2003. 6, 4(2004) matrix liapunov’s functions method ... 251 references aminov, a.b., and sirazetdinov, t.k. 1. the method of liapunov functions in problems of multistability of motion, prikl. mat. mekh. 51 (1987) 553-558. 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[russian] cubo a mathematical journal vol.19, no¯ 03, (01–14). october 2017 the solvability and fractional optimal control for semilinear stochastic systems surendra kumar department of mathematics, university of delhi, delhi-110007, india. mathdma@gmail.com abstract this paper deals with fractional optimal control for a class of semilinear stochastic equation in hilbert space setting. to ensure the existence and uniqueness of mild solution, a set of sufficient conditions is constructed. the existence of fractional optimal control for semilinear stochastic system is also discussed. finally, an example is included to show the applications of the developed theory. resumen este art́ıculo estudia el control óptimo fraccional para una clase de ecuaciones estocásticas semilineales en un contexto de espacios de hilbert. para asegurar la existencia y la unicidad de soluciones blandas, construimos un conjunto de condiciones suficientes. la existencia del control óptimo fraccional para sistemas estocásticos semilineales también es discutido. finalmente, incluimos un ejemplo para mostrar la aplicabilidad de la teoŕıa desarrollada. keywords and phrases: fractional calculus, semilinear stochastic system, mild solution, optimal control, fixed point theory 2010 ams mathematics subject classification: 26a33, 34a08, 34a12, 47h10, 49j99, 93c23 2 surendra kumar cubo 19, 3 (2017) 1 introduction fractional calculus is about to generalization of the integer order integral and derivative to arbitrary order. the potential applications of fractional calculus are in many fields of science and engineering including fluid flow, electrical networks, diffusive transport, rheology, control theory, electromagnetic theory, and probability [13, 10, 9, 5, 7, 8, 15]. it is well known that many real world problems in science and engineering are modeled as stochastic differential equations [4]. since fractional stochastic differential equations describe a physical dynamical system more accurately, it seems necessary to discuss the qualitative properties for such systems. if a fractional differential equation describes the performance index and system dynamics, then an optimal control problem is known as a fractional optimal control problem. using the fractional variational principle and lagrange multiplier technique, agrawal [1] discussed the general formulation and solution scheme for riemann-liouville fractional optimal control problems. it is remarkable thathe fixed point technique, which is used to establish the existence result for abstract fractional differential equations, could be extended to address the fractional optimal control problems. wang et. al. [19, 20] discussed the existence of local and global solutions for fractional semilinear systems, and extended the results for fractional optimal control. using fractional powers of the linear operator, the existence of fractional optimal controls for the lagrange problem investigated in infinite dimensional reflexive banach space [21, 22, 24]. wang et al. [23] studied the solvability and the existence of optimal controls for fractional integro-differential systems with infinite delay via contraction mapping principle. pan et al. [14] constructed a set of sufficient conditions that guarantees the existence of optimal control to the riemann-liouville fractional control systems in the banach space. li and liu [11] presented the optimal control for nonlinear impulsive differential equations in banach space setting. yan and lu [25] investigated the optimal control problems for fractional stochastic neutral integro-differential equations with infinite delay in a hilbert space by using the fractional calculus, stochastic analysis theory, and fixed point theorem. using the lerayschauder fixed point theorem, balasubramaniam and tamilalagan [2] studied the solvability and optimal controls for impulsive fractional stochastic integro-differential equations. recently, tamilalagan and balasubramaniam [18] investigated the solvability and optimal controls for fractional stochastic differential equations driven by poisson jumps in hilbert space by using analytic resolvent operators and classical banach contraction mapping principle. by motivated from the above work, the main objective of this paper is to obtain sufficient conditions for existence and uniqueness of mild solution of fractional stochastic semilinear system of fractional order via contraction mapping principle. to prove the existence and uniqueness of mild solution it is assumed that nonlinear functions satisfying lipschitz continuity and linear growth conditions. next, we introduce a formulation for fractional optimal control governed by the fractional stochastic semilinear systems, where the fractional derivative is defined in the sense of the caputo. finally, the existence of fractional optimal controls for the lagrange problem cubo 19, 3 (2017) the solvability and fractional optimal control 3 is investigated. the obtained results are new and considered as a contribution to the theory of stochastic fractional optimal control. the rest of the paper is organized as follows: in sect. 2, we present some basic definitions, notations and lemmas as preliminaries. in sect. 3, the existence and uniqueness of mild solution are proved. existence of fractional optimal control is shown in sect. 4. in sect. 5, an example is given to illustrate the theory. 2 preliminaries this section contains basic definitions, notations and preliminary results, which help us to obtain desired results. throughout this paper, we use the following notations: let h and k be separable hilbert spaces. for convenience, we denote the inner products and norms in all spaces by 〈·, ·〉 and ‖ · ‖, respectively. let (ω,f,p) be a complete probability space furnished with complete family of right continuous increasing sub σ-algebras {ft : 0 ≤ t ≤ τ} satisfying ft ⊂ f. let {en} ∞ n=1 be a complete orthonormal system in k, and {βn} ∞ n=1 a sequence of independent brownian motions such that ω(t) := ∞∑ n=1 √ λnenβn(t), t ∈ [0,τ], where {λn} ∞ n=1 is a bounded sequence of non-negative real numbers such that qen = λnen, n = 1,2, · · · with tr(q) = ∑ ∞ n=1 λn < ∞ (tr(q) denotes the trace of the operator q). then the above k-valued stochastic process ω(t) is called a wiener process. the normal filtration ft is the sigma algebra generated by {ω(s) : 0 ≤ s ≤ t} and fτ = f. denoted by l(k,h) the space of all bounded operators from k into h equipped with the usual operator norm. for ψ ∈ l(k,h), we define ‖ψ‖2q = tr(ψqψ ∗ ) = ∞∑ n=1 ‖ √ λnψen‖ 2. if ‖ψ‖2q < ∞, then ψ is called a q-hilbert schmidt operator. let lq(k,h) be the space of all q-hilbert schmidt operators ψ : k → h. the completion lq(k,h) of l(k,h) with respect to the topology induced by the norm ‖·‖q, where ‖ψ‖ 2 q = 〈ψ,ψ〉 is a hilbert space with the above norm topology. the space of strongly measurable, h-valued, square integrable random variables, denoted by l2(ω,h), is a banach space equipped with the norm topology ‖x(·)‖ = (e‖x(t)‖ 2)1/2, where e(·) is the expectation with respect to the measure p. let c([0,τ],l2(ω,h)) be the banach space of continuous maps from [0,τ] into l2(ω,h) satisfying sup0≤t≤τ e‖x(t)‖ 2 < ∞. let h2 be the closed subspace of c([0,τ],l2(ω,h)) consisting 4 surendra kumar cubo 19, 3 (2017) of measurable, ft-adapted, h-valued processes x ∈ c([0,τ],l2(ω,h)) equipped with the norm ‖x‖h2 := ( sup 0≤t≤τ e‖x(t)‖2 )1/2 . consider the following integral cost functional j(x,u) := e {∫τ 0 ℓ(t,xu(t),u(t))dt } , (2.1) subject to the state equation cdαt x(t) = ax(t) + b(t)u(t) + f(t,x(t)) + g(t,x(t)) dω(t) dt , t ∈]0,τ], x(0) = x0 ∈ h, } (2.2) where the integrand ℓ : [0,τ] × h × u → r ∪ {∞} is specified latter; cdαt is the caputo fractional derivative of order 0 < α < 1, the state x(·) is h-valued stochastic process; the control function u(·) takes its values from a separable reflexive hilbert space u; a : d(a) ⊆ h → h is the infinitesimal generator of a resolvent sα(t), t ≥ 0 on h; {b(t) : t ≥ 0} is a family of linear operators from u to h; the functions f : [0,τ] × h → h and g : [0,τ] × h → l(k,h) are nonlinear, l(k,h) denotes the space of all bounded linear operators form e into h, x0 is f0-measurable h-valued random variable independent of ω. define the admissible set uad, the set of all v(·) : [0,τ] × ω → u such that v is ft adapted stochastic process and e ∫τ 0 ‖v(t)‖pdt < ∞. clearly uad 6= ∅ and uad ⊂ l p([0,τ];u) (1 < p < +∞) is bounded, closed, and convex. denoted by the set of all admissible state-control pairs (x,u) by aad, where x is the mild solution of the system (2.2) corresponding to the control u ∈ uad. the main objective of this paper is to find a pair (x0,u0) ∈ aad such that j(x0,u0) := inf{j(x,u) : (x,u) ∈ aad} = ε. we now recall the following well known definitions related to fractional order differentiation and integration. for more details on fractional calculus one can see [13, 10]. definition 2.1 the riemann-liouville fractional integral operator of order α > 0 of a function f : [0,∞) → r with the lower limit 0 is defined as iαf(t) = 1 γ(α) ∫t 0 (t − s)α−1f(s)ds, where γ is the euler gamma function. definition 2.2 the caputo fractional derivative of order α > 0 for the function f ∈ cm([0,τ],r) is defined by cdαt f(t) = 1 γ(m − α) ∫t 0 (t − s)m−α−1f(m)(s)ds, m − 1 < α < m ∈ n. cubo 19, 3 (2017) the solvability and fractional optimal control 5 if f is an abstract function with values in h, then the integrals appearing in definition 2.1 and definition 2.2 are taken in the bochner sense. moreover, the caputo derivative of a constant is always zero. the two-parameter function of the mittag-leffler type is defined by the series expansion eα,β(z) = ∞∑ j=1 zj γ(αj + β) = 1 2πi ∫ c eλ λα−β λα − z dλ; α,β > 0, z ∈ c, where c is a contour that start and end at −∞ and encircles the disc ‖λ‖ ≤ |z|1/2 counterclockwise (see [6] for more results on mittag-leffler function). definition 2.3 [16, 17] a closed and linear operator a is said to be sectorial of type µ if there exist π/2 ≤ θ ≤ π, m̃ > 0 and µ ∈ r such that the following conditions are satisfied: ρ(a) ⊂ ∑ (θ,µ) := {λ ∈ c : λ 6= µ, |arg(λ − µ)| < θ}, and ‖r(λ,a)‖ := ‖(λ − a) −1‖ ≤ m̃ |λ−µ| , λ ∈ ∑ (θ,µ) . lemma 2.1 [17] for 0 < α < 2, a linear closed densely defined operator a belongs to aα(θ0,µ0) if and only if λα ∈ ρ(a) for each λ ∈ σ(θ0+π/2,µ) and for any µ > µ0, θ < θ0 there is a constant c0 = c0(θ,µ) such that ‖λα−1r(λα,a)‖ ≤ c0 |λ − µ| , for λ ∈ σ(θ+π/2,µ). lemma 2.2 [17] if f satisfies the uniform hölder condition with the exponent 0 < γ ≤ 1 and a is a sectorial operator, then the unique solution of the cauchy problem cdαt y(t) = ay(t) + f(t), 0 < α < 1, t ∈ (0,τ], y(0) = y0, } (2.3) is given by y(t) = sα(t)y0 + ∫t 0 tα(t − s)f(s)ds, where sα(t) = eα,1(at α) = 1 2πi ∫ b̂ρ eλt λα−1 λα − a dλ, tα(t) = t α−1eα,α(at α ) = 1 2πi ∫ b̂ρ eλt 1 λα − a dλ, b̂ρ is the bromwich path, tα(t) is called the α-resolvent family, and sα(t) is the solution operator generated by a. an operator a is said to belong to cα(m̃,µ) if problem (2.3) with f = 0 has a solution operator sα(t) satisfying ‖sα(t)‖ ≤ m̃e µt. denote cα(µ) := ∪{cα(m̃,µ) : m̃ ≥ 1}, cα := {cα(µ) : µ ≥ 0}, and aα(θ0,µ0) = {a ∈ c α : a generates analytic solution operators sα(t) of type (θ0,µ0)}. 6 surendra kumar cubo 19, 3 (2017) if 0 < α < 1 and a ∈ aα(θ0,µ0), then we have ‖sα(t)‖ ≤ m̃e µt and ‖tα(t)‖ ≤ ce µt(1 + tα−1), t > 0, µ > µ0. if ms := sup 0≤t≤τ ‖sα(t)‖, mt := sup 0≤t≤τ ceµt(1 + t1−α), then, we have ‖sα(t)‖ ≤ ms, ‖tα(t)‖ ≤ t α−1mt. by lemma 2.2, a mild solution of the system (2.2) is defined as definition 2.4 an ft-adapted stochastic process x(t) ∈ c([0,τ],l 2(ω,f,h) is called a mild solution of system (2.2) if for each u(·) ∈ lp([0,τ];u), x(t) is measurable and the following stochastic integral equation is satisfied: x(t) = sα(t)x0 + ∫t 0 tα(t − s)[b(s)u(s) + f(s,x(s))]ds + ∫t 0 tα(t − s)g(s,x(s))dω(s). (2.4) lemma 2.3[12] a measurable function χ : [0,τ] → v is bochner integrable, if ‖χ‖ is lebesgue integrable. 3 existence and uniqueness of mild solution to prove the existence and uniqueness of mild solution of the system (2.2), we impose the following conditions to the system parameter: [h0] for any x ∈ h, the function t → f(t,x(t)) and t → g(t,x(t)) are ft-measurable. [h1] the functions f : [0,τ] ×h → h, g : [0,τ] ×h → l(k,h) are continuous, and satisfying linear growth and lipschitz conditions. moreover, without loss of generality, we may assume that there are positive constants lf and lg such that ‖f(t,x) − f(t,y)‖2 ≤ lf‖x − y‖ 2, ‖f(t,x)‖2 ≤ lf ( 1 + ‖x‖2 ) , ‖g(t,x) − g(t,y)‖2q ≤ lg‖x − y‖ 2, ‖g(t,x)‖2q ≤ lg ( 1 + ‖x‖2 ) . [h3] the operator b ∈ l∞([0,τ]; l(u,h)) and ‖b‖∞ stand for the norm of operator b in the banach space l∞([0,τ]; l(u,h)). [h4] the multi-valued map u(·) : [0,τ] ⇒ 2u \ {∅} has closed, convex and bounded values; u(·) is graph measurable and u(·) ⊆ ξ, where ξ is a bounded subset of u. first we show that the system (2.4) has at least one solution. theorem 3.1 under assumptions [h0]–[h3] the system (2.4) admits a unique mild solution on [0,τ] for each control function u(·) ∈ uad and for some p such that pα > 1. cubo 19, 3 (2017) the solvability and fractional optimal control 7 proof. define an operator im : h2 → h2 as (im x)(t) = sα(t)x0 + ∫t 0 tα(t − s)[b(s)u(s) + f(s,x(s))]ds + ∫t 0 tα(t − s)g(s,x(s))dω(s). to show that (2.4) is the mild solution of the system (2.4) on [0,τ], it is enough to prove that im has a fixed point in the space h2. for this purpose, we will employ the classical fixed point theorem for contractions. we first show that im(h2) ⊂ h2. let x ∈ h2, then we have e‖(im x)(t)‖2 ≤ 4[i0 + i1 + i2 + i3] (3.1) clearly i0 ≤ m 2 se‖x0‖ 2. next, using the cauchy-schwartz inequality, we have i1 = ‖ ∫t 0 tα(t − s)b(s)u(s)ds‖ 2 ≤ m2t ‖b‖ 2 ∞ [∫t 0 (t − s)α−1‖u(s)‖ds ]2 ≤ m2t ‖b‖ 2 ∞   (∫t 0 (t − s) p(α−1) p−1 ds ) p−1 p (∫t 0 ‖u(s)‖ p uds ) 1 p   2 ≤ m2t ‖b‖ 2 ∞ ‖u‖2lp([0,τ];u)τ 2( pα−1p ) ( p − 1 pα − 1 ) 2(p−1) p the cauchy-schwartz inequality, and hypothesis (h1) imply that i2 = e ∥ ∥ ∥ ∥ ∫t 0 tα(t − s)ef(s,x(s))ds ∥ ∥ ∥ ∥ 2 ≤ (∫t 0 ‖tα(t − s)‖e‖f(s,x(s))‖ds )2 ≤ (∫t 0 mt (t − s) α−1 2 (t − s) α−1 2 e‖f(s,x(s))‖ds )2 ≤ m2t (∫t 0 (t − s)α−1ds ) (∫t 0 (t − s)α−1‖e‖f(s,x(s))‖2ds ) ≤ m2tlf τα α ∫t 0 (t − s)α−1(1 + e‖x(s)‖2)ds ≤ m2tlf τ2α α2 (1 + ‖x‖2h2), 8 surendra kumar cubo 19, 3 (2017) and i3 = e ∥ ∥ ∥ ∥ ∫t 0 tα(t − s)eg(s,x(s))dω(s) ∥ ∥ ∥ ∥ 2 ≤ m2ttr(q) (∫t 0 (t − s)α−1ds ) (∫t 0 (t − s)α−1e‖g(s,x(s))‖2qds ) ≤ m2ttr(q)lg τ2α α2 (1 + ‖x‖2h2). thus 3.1 becomes e‖(im x)(t)‖2 ≤ a + b‖x‖2h2, where a and b are suitable positive constants. this implies that im map h2 into itself. next, we show that im is a contraction map. for x, y ∈ h2, the cauchy-schwartz inequality, and hypothesis (h1) yield that e‖(im x)(t) − (im y)(t)‖2 ≤ 2e‖ ∫t 0 tα(t − s)[f(s,x(s)) − f(s,y(s))]ds‖ 2 +2e‖ ∫t 0 tα(t − s)[g(s,x(s)) − g(s,y(s))]dω(s)‖ 2 ≤ 2m2t (lf + lgtr(q)) τ2α α2 ‖x − y‖2h2. consequently if 2m2t (lf + lgtr(q)) τ2α α2 < 1, (3.2) then the operator im has a unique fixed point in h2, which is a solution of the system (2.2). the extra condition on τ can be easily removed by considering the equation on intervals [0, τ̃], [τ̃,2τ̃], · · · with τ̃ satisfying (3.2). we now obtain a priori estimate of mild solution for the system (2.2), that helps us to obtain our main result. lemma 3.1 (a priori estimate). assuming that system (2.4) is the mild solution of system (2.2) on [0,τ] corresponding to the control u. then there exists a constant m > 0 such that e‖x(t)‖2 ≤ m, ∀ t ∈ [0,τ]. cubo 19, 3 (2017) the solvability and fractional optimal control 9 proof. using condition [h1] and hölder’s inequality, we obtain e‖x(t)‖2 ≤ 4e‖sα(t)x0‖ 2 + 4e‖ ∫t 0 tα(t − s)b(s)u(s)ds‖ 2 +4e‖ ∫t 0 tα(t − s)f(s,x(s))ds‖ 2 + 4e‖ ∫t 0 tα(t − s)g(s,x(s))dω(s)‖ 2 ≤ 4m2s‖x0‖ 2 + 4m2t ‖b‖ 2 ∞ [∫t 0 (t − s)α−1‖u(s)‖ds ]2 +4m2t (lf + lgtr(q)) (∫t 0 (t − s)α−1ds ) ∫t 0 (t − s)α−1{1 + e‖x(s)‖2}ds ≤ 4m2s‖x0‖ 2 + 4m2t ‖b‖ 2 ∞   (∫t 0 (t − s) p(α−1) p−1 ds ) p−1 p (∫t 0 ‖u(s)‖ p uds ) 1 p   2 +4m2t (lf + lgtr(q)) τα α ∫t 0 (t − s)α−1{1 + e‖x(s)‖2}ds ≤ 4m2s‖x0‖ 2 + 4m2t ‖b‖ 2 ∞ ‖u‖2lp([0,τ];u)τ 2( pα−1p ) ( p − 1 pα − 1 ) 2(p−1) p +4m2t (lf + lgtr(q)) τ2α α2 + 4m2t (lf + lgtr(q)) τα α ∫t 0 (t − s)α−1e‖x(s)‖2ds. now using the gronwall inequality, one can easily obtain the boundedness of x in h2. 4 existence of fractional optimal control in this section, we prove the existence of fractional optimal control under the hypothesis: [hl] following conditions are imposed on the integrand ℓ : [0,τ] × h × u → r ∪ {∞} such that (i) the integrand ℓ : [0,τ] × h × u → r ∪ {∞} is ft-measurable. (ii) the integrand ℓ(t, ·, ·) is sequentially lower semicontinuous on h × u for almost all t ∈ [0,τ]; (iii) the integrand ℓ(t,x, ·) is convex on u for each x ∈ h and almost all t ∈ [0,τ]. (iv) there exist constants d ≥ 0, e > 0, µ0 is nonnegative and µ0 ∈ l 1([0,τ]; r) such that µ0(t) + de‖x‖ 2 + ee‖u‖ p u ≤ ℓ(t,x,u). 10 surendra kumar cubo 19, 3 (2017) theorem 4.1 suppose hypothesis of theorem 3.1 and [hl] hold, then lagrange problem (2.1) admits at least one optimal pair, that is, there exists an admissible state-control pair (x0,u0) ∈ aad such that j(x0,u0) := e {∫τ 0 ℓ(t,x0(t),u0(t))dt } ≤ j(x,u), ∀ (x,u) ∈ aad. proof. if inf{j(x,u)|(x,u) ∈ aad} = +∞, then there is nothing to prove. without any loss of generality, we may assume that inf{j(x,u)|(x,u) ∈ aad} = ε < +∞. now assumption [hl] implies that ε > −∞. by definition of infimum, there is a minimizing sequence of feasible pairs (xm,um) ∈ aad, such that j(x m,um) → ε as m → +∞. since {um} ⊆ uad, m = 1,2, · · · , {u m} is a bounded subset of the separable reflexive banach space lp([0,τ];u), there exists a subsequence, relabeled as {um} and u0 ∈ lp([0,τ];u) such that um w −→ u0 (um → u0 weakly as m → +∞) in lp([0,τ];u). since uad is closed and convex, the mazur lemma forces us to conclude that u 0 ∈ uad. let {xm} be the sequence of solutions of the system (2.2) corresponding to {um}, that is xm(t) : = sα(t)x0 + ∫t 0 tα(t − s)[b(s)u m(s) + f(s,xm(s))]ds + ∫t 0 tα(t − s)g(s,x m(s))dω(s). by lemma 3.1, it is easy to see that there exists δ > 0 such that e‖xm‖2 ≤ δ, m = 0,1,2, · · · , where x0 is the mild solution of the system (2.2) corresponding to the control u0 ∈ uad given by x0(t) : = sα(t)x0 + ∫t 0 tα(t − s)[b(s)u 0(s) + f(s,x0(s))]ds + ∫t 0 tα(t − s)g(s,x 0(s))dω(s). for all t ∈ [0,τ], using condition [h1], the cauchy-schwartz inequality and the hölder inequality, we obtain e‖xm(t) − x0(t)‖2 ≤ 3e‖ ∫t 0 tα(t − s)[b(s)u m(s) − b(s)u0(s)]ds‖2 +3e‖ ∫t 0 tα(t − s)[f(s,x m(s)) − f(s,x0(s))]ds‖2 +3e‖ ∫t 0 tα(t − s)[g(s,x m (s)) − g(s,x0(s))]dω(s)‖2 ≤ 3m2t ( p − 1 pα − 1 ) 2p−2 p τ 2α− 2 p (∫t 0 ‖b(s)um(s) − b(s)u0(s)‖pds ) 2 p +3m2t τα α (lf + lgtr(q)) ∫t 0 (t − s)α−1e‖xm(s) − x0(s)‖2ds. cubo 19, 3 (2017) the solvability and fractional optimal control 11 by the well known singular version of gronwall inequality, there exists a constant k∗(α) independent of u, m and t such that e‖xm(t) − x0(t)‖2 ≤ k∗(α) (∫τ 0 ‖b(s)um(s) − b(s)u0(s)‖pds ) 2 p ≤ k∗(α)‖bum − bu0‖2lp([0,τ];u). (4.1) since b is strongly continuous, we get ‖bum − bu0‖2lp([0,τ];u) s −→ 0 as m → ∞. (4.2) from (4.1) and (4.2), we conclude that e‖xm(t) − x0(t)‖2 → 0 as m → ∞. this implies that e‖xm − x0‖2 → 0 in c([0,τ];l2(ω,h)) as m → ∞. note that [hl] implies the assumptions of balder (see theorem 2.1, [3]). hence, by balder’s theorem, we can conclude that (x,u) → e ∫τ 0 l(t,x(t),u(t))dt is sequentially lower semicontinuous in the strong topology of l1([0,τ];h) and weak topology of lp([0,τ];u) ⊂ l1([0,τ];u). hence, j is weakly lower semicontinuous on lp([0,τ];u), and since by [hl] (iv), j > −∞, j attains its infimum at u0 ∈ uad, that is, ε := lim m→∞ e ∫τ 0 ℓ(t,xm(t),um(t))dt ≥ e ∫τ 0 ℓ(t,x0(t),u0(t))dt = j(x0,u0) ≥ ε. this completes the proof. 5 applications let ω1 ⊂ r 3 be a bounded domain and ∂ω1 ∈ c 3. further let h = u := l2(ω1), ω(t) is a standard cylindrical wiener process in h defined on a stochastic space (ω,f,p). suppose d(a) := h2(ω1) ∩ h 1 0(ω1) and for z ∈ d(a), az := ( ∂ 2 ∂z2 1 + ∂ 2 ∂z2 2 + ∂ 2 ∂z2 3 ) z. the admissible control set uad := {u ∈ u : ‖u‖lp([0,1]:u) ≤ 1}. consider the following fractional stochastic equation    cd 2 3 t x(t,z) = ∆x(t,z) + ∫1 0 k(z,s)u(s,t)ds + ∫1 0 ν(z,s) sin(x,s)ds + (x(t,z))2 1+(x(t,z))2 dω(t), x(0,z) = x0(z), z ∈ ω1, x(t,z)|z∈∂ω = 0, t > 0, (5.1) 12 surendra kumar cubo 19, 3 (2017) define x(t)(z) = x(t,z), (bu)(t)(z) = ∫1 0 k(z,s)u(s,t)ds, f(t,x(t))(z) = f(t,x(t,z)) = ∫1 0 ν(z,s) sin(x,s)ds, g(t,x(t))(z) = g(t,x(t,z)) = (x(t,z)) 2 1+(x(t,z))2 , and x(0)(z) = x(0,z) = x0(z). moreover, we assume that k : ω1 × [0,1] → r is continuous. the function ν is measurable and ∫ ω1 ∫1 0 ν(z,s)dsdz < ∞. the one-dimensional standard brownian motion is denoted by ω(t). thus, for α = 2/3 the problem (5.1) can be written as the abstract form of system (2.2) with the cost function j(x,u) := e {∫1 0 ℓ(t,x(t),u(t))dt } , where ℓ(t,x(t),u(t))(z) = ∫ ω1 |x(t,z)|2dz + ∫ ω1 |u(t,z)|2dz. it is easy to see that the assumptions [h0]–[h4] are satisfied. if [hl] is satisfied, then there exists an optimal pair (x0,u0) ∈ l2([0,1] × ω1) ×l 2([0,1] ×ω1) such that j(x 0,u0) ≤ j(x,u) for all (x,u) ∈ l2([0,1] ×ω1) ×l 2([0,1] ×ω1). 6 conclusions this paper considers fractional optimal control of stochastic semilinear dynamic systems with the caputo fractional derivative in a hilbert space. the existence and uniqueness of mild solution are studied using the fixed point technique. it is also shown that, under some natural assumptions the lagrange problem admits at least one optimal pair of state-control. references [1] agrawal, o.p.: a general formulation and solution scheme for fractional optimal control problems. nonlinear dyn. 38, 323–337 (2004) [2] balasubramaniam, p., tamilalagan, p.: the solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators. j. optim. theory appl. 174, 139–155 (2017) [3] balder, e.: necessary and sufficient conditions for l1-strong-weak lower semicontinuity of integral functional. nonlinear anal. tma 11, 1399–1404 (1987) [4] da 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jaipur-302022, rajasthan, india. prof−sky16@yahoo.com 2section of mathematics, department of information technology, shinas college of technology, shinas, p.o. box 77 postal code 324, oman. sk22−math@yahoo.co.in, sudhakar.chaubey@shct.edu.om 3department of mathematics wollo university, p. o. box: 1145, dessie, south wollo, amhara region, ethiopia. dlsuthar@gmail.com abstract in the context of para-contact hausdorff geometry η−ricci solitons and gradient ricci solitons are considered on manifolds. we establish that on an (lcs)n−manifold (m, φ, ξ, η, g), the existence of an η−ricci soliton implies that (m, g) is quasi-einstein. we find conditions for ricci solitons on an (lcs)n−manifold (m, φ, ξ, η, g) to be shrinking, steady and expanding. at the end we show examples of such manifolds with η−ricci solitons. resumen en el contexto de geometŕıa para-contacto hausdorff, consideramos η−ricci solitones y ricci solitones gradientes en variedades. establecemos que en una (lcs)n−variedad (m, φ, ξ, η, g), la existencia de un η−ricci solitón implica que (m, g) es casi-einstein. encontramos condiciones para que los ricci solitones en una (lcs)n−variedad (m, φ, ξ, η, g) sean contractivos, estables o expansivos. al concluir, mostramos ejemplos de dichas variedades con η−ricci solitones. keywords and phrases: η−ricci solitons, gradient ricci solitons, (lcs)n−manifold. 2010 ams mathematics subject classification: 53c25, 53c15, 53c21. 34 s. k. yadav, s. k. chaubey & d. l. suthar cubo 19, 2 (2017) 1 introduction in 2003, shaikh [34] introduced the notion of lorentzian concircular structure manifolds (briefly, (lcs)n−manifolds) with an example, which generalize the notion of lp-sasakian manifolds introduced by matsumoto [27] and also by mihai and rosca [26]. then shaikh and baishya ([32], [33]) investigated the application of (lcs)n−manifolds to the general theory of relativity and cosmology. the (lcs)n−manifolds are also studied by atceken et al. ([1], [2], [21]), d. narain and s. yadav [30], s. yadav et al. ([39]-[42]), shaikh and his co-authors ([35], [36]) and many others. ricci solitons represent a natural generalization of einstein metrics on a riemannian manifold, being generalized fixed points of hamilton’s ricci flow ∂ ∂t g = −2s [24]. the evolution equation defining the ricci flow is a kind of nonlinear diffusion equation, an analogue of heat equation for metrics. under ricci flow, a metric can be improved to evolve into more canonical one by smoothing out its irregularities, depending on the ricci curvature of the manifold, it will expand in the directions of negative ricci curvature and shrink in the positive case. ricci solitons have been studied in many contexts: on kähler manifolds [18], on contact and lorentzian manifolds ([3], [15], [25], [31], [38]), on sasakian ([19], [20]), α−sasakian [25] and k−contact manifolds [31], on kenmotsu ([4], [28]) and f−kenmotsu manifolds [15] and by ([16], [43]) etc. in paracontact geometry, ricci soliton firstly appeared in the paper of g. calvoruso and d. perrone [13]. c. l. bejan and m. crasmarean studied ricci solitons on 3−dimensional normal paracontact manifolds [5]. a more general notion is that of η−ricci soliton introduced by j. t. cho and m. kimura [6], which was treated by c. calin and m. crasmareanu on hopf hypersurfaces in complex space forms [14]. 2 (lcs)n−manifolds (m, φ, ξ, η, g) an n−dimensional lorentzian manifold m is a smooth connected paracontact hausdorff manifold with lorentzian metric g, that is, m admits a smooth symmetric tensor field g of type (0, 2) such that for each point p ∈ m the tensor gp : tpm × tpm → re is a non degenerate inner product of signature (−, +, ..., +), where tpm denotes the tangent space of m at p and re is the real number space. a non-zero tangent vector v ∈ tpm is said to be timelike (resp., non-spacelike, null and spacelike) if it satisfies gp(v, v) < 0 (resp., < 0, =, > 0) [29]. definition 2.1. in a lorentzian manifold (m, g) a vector field ρ defined by g(x, ρ) = a(x), for any x ∈ χ(m) is said to be a concircular vector field [44] if (∇xa) (y) = α {g(x, y) + ω(x)a(y)} , where α is a non-zero scalar and ω is a closed 1−form. here ∇ denotes the operator of covariant differentiation with respect to the lorentzian metric g. cubo 19, 2 (2017) some geometric properties of η− ricci solitons ... 35 let m be a lorentzian manifold admitting a unit timelike concircular vector field ξ, called the generator of the manifold. then we have g(ξ, ξ) = −1. (2.1) since ξ is the unit concircular vector field, then there exists a non-zero 1−form η such that for g(x, ξ) = η(x), (2.2) the following equation holds (∇xη) (y) = α {g(x, y) + η(x)η(y)} (α 6= 0), (2.3) that is, ∇xξ = α{x + η(x)ξ}, for all vector fields x, y on m, where α is a non-zero scalar function satisfies ∇xα = (xα) = dα(x) = ρη(x), (2.4) ρ being a certain scalar function given by ρ = −(ξα). if we put φx = 1 α ∇xξ, (2.5) then from (2.3) and (2.5), we have φx = x + η(x)ξ, (2.6) from which it follows that φ is a (1, 1) tensor and called the structure tensor of the manifold. thus the lorentzian manifold m together with the unit timelike concircular vector field ξ, its associated 1−form η and (1, 1) tensor field φ, is said to be a lorentzian concircular structure manifold (briefly (lcs)n−manifold) [34]. specially, if we take α = 1, then we can obtain the lp-sasakian structure of matsumoto [27]. in a (lcs)n−manifold (n > 2), the following relations hold ([32]-[35]): a) η(ξ) = −1, b) φξ = 0, c) φ2x = x + η(x)ξ, (2.7) d) η(φx) = 0, e) g(φx, φy) = g(x, y) + η(x)η(y), η(r(x, y)z) = (α2 − ρ) {g(y, z)η(x) − g(x, z)η(y)} , (2.8) r(x, y)ξ = (α2 − ρ) {η(y)x − η(x)y } , (2.9) r(ξ, x)y = (α2 − ρ) {g(x, y)ξ − η(y)x} , (2.10) r(ξ, x)ξ = (α2 − ρ) {η(x)ξ + x} , (2.11) (∇xφ) (y) = α {g(x, y)ξ + 2η(x)η(y)ξ + η(y)x} , (2.12) s(x, ξ) = (n − 1)(α2 − ρ)η(x), (2.13) s(φx, φy) = s(x, y) + (n − 1)(α2 − ρ)η(x)η(y), (2.14) (xρ) = dρ(x) = βη(x), (2.15) for any vector fields x, y, z on m and β = −(ξρ) is a scalar function, where r is the curvature tensor and s is the ricci tensor of the manifold. 36 s. k. yadav, s. k. chaubey & d. l. suthar cubo 19, 2 (2017) 3 η−ricci solitons on (lcs)n−manifolds let (m, φ, ξ, η, g) be a (lcs)n−manifold. we follow the equation lξg + 2s + 2λg + 2µη ⊗ η = 0, (3.1) where lξ is the lie-derivative operator along the vector field ξ, s is the ricci tensor field of the metric g, λ and µ are real constants. we write lξg in term of the levi-civita connection ∇, we obtain (lξg)(x, y) = g(∇yξ, x) + g(y, ∇xξ) = 2α[g(x, y) + η(x)η(y)]. (3.2) in view of (3.1) and (3.2), we get qx = −(α + λ)x − (α + µ)η(x)ξ, (3.3) r = −nλ − (n − 1)α + µ, (3.4) s(x, y) = −(α + λ)g(x, y) − (α + µ)η(x)η(y), (3.5) s(x, ξ) = s(ξ, x) = (µ − λ)η(x), (3.6) µ − λ = (n − 1)(α2 − ρ), (3.7) for any x, y ∈ χ(m). here r is the scalar curvature and q denotes the ricci operator corresponding to s, that is, s(x, y) = g(qx, y), for all x, y on m. the structure (g, ξ, λ, µ) that follows the equation (3.1) is said to be an η−ricci soliton to (m, g) [6]. in particular, if µ = 0, (g, ξ, λ) is a ricci soliton [24] and it is called shrinking, steady, or expanding according as λ is negative, zero or positive, respectively [12]. proposition 3.1. on a (lcs)n−manifold (m, φ, ξ, η, g) the following relations hold (i) η(∇xξ) = 0, (ii) ∇ξξ = 0, (iii) ∇η = α{g + η ⊗ η}, (iv) ∇ξη = 0, (v) lξφ = 0, (vi) lξη = 0, (vii) lξ(η ⊗ η) = 0, (viii) lξg = 2α(g + η ⊗ η), where ∇ is the levi-civita connection associated to g. also η is closed, the distribution is involutive and tensor field of φ vanishes identically, i. e., the structure is normal. proof. since (∇xφ)(y) = α{g(x, y)ξ + 2η(x)η(y)ξ + η(y)x}, this indicates that ∇xφy − φ(∇xy) = α{g(x, y)ξ + 2η(x)η(y)ξ + η(y)x}. taking y = ξ in above equation, we have φ(∇xξ) = αφx. applying φ both sides, we get ∇xξ + η(∇xξ) = α{x + η(x)ξ}. cubo 19, 2 (2017) some geometric properties of η− ricci solitons ... 37 since ∇xξ = αφx and x(g(ξ, ξ)) = 2g(∇xξ, ξ), therefore η(∇xξ) = 0 and ∇ξξ = 0. also (∇xη)(y) = αg(y, φx) = α{g(x, y) + η(x)η(y)}, this implies that ∇η = α{g + η ⊗ η}, i.e., ∇ξη = 0. in view of definition of lie-derivative, we get (lξφ)(x) = [ξ, φx] − φ([ξ, x]) = ∇ξφx − φ(∇ξx) = (∇ξφ)(x) = 0, i.e., lξφ = 0. also, ( lξη)(x) = ξ(η(x) − η([ξ, x]) = g(x, ∇ξξ) + g(∇xξ, ξ) = 0, i.e., lξη = 0. further we compute (lξ(η ⊗ η))(x, y) = ξ(η(x)η(y)) − η([ξ, x])η(y) − η(x)η([ξ, y]), which implies that (lξ(η ⊗ η))(x, y) = η(x)g(y, ∇ξξ) − η(y)g(x, ∇ξξ) = 0, i.e., lξ(η ⊗ η) = 0. again (lξg)(x, y) = ξ g(x, y) − g([ξ, x], y) − g(x, [ξ, y]), implies that (lξg)(x, y) = α[g(φx, y) + g(x, φy)]. using (2.6), we get lξg = 2α(g + η ⊗ η). at last (dη)(x, y) = x(η(y)) − y(η(x)) − η([x, y]), that implies (dη)(x, y) = g(y, ∇xξ) − g(x, ∇yξ) = α{g(y, x) + η(x)η(y)} − α{g(x, y) + η(x)η(y)} = 0. finally, nφ(x, y) = φ 2[x, y] + [φx, φy] − φ[φx, y] − φ[x, φy]. this yields that nφ(x, y) = φ 2(∇xy) − φ 2(∇yx) − φ(∇xφy) + φ(∇yφx) +∇φxφy − φ(∇φxy) − ∇φyφx + φ(∇φyx) = 0. thus the structure is normal. in [17], s. chandra et al. proved that a second order parallel symmetric tensor on a (lcs)n− manifold with α2 − ρ 6= 0, is a constant multiple of the ricci tensor. thus we apply this concept for η−ricci soliton and we prove the following result: 38 s. k. yadav, s. k. chaubey & d. l. suthar cubo 19, 2 (2017) theorem 3.2. let (m, φ, ξ, η, g) be a (lcs)n−manifold. assume that the symmetric (0, 2) tensor field h = lξg + 2s + 2µη ⊗ η is parallel with respect to the levi-civita connection associated to g, then (g, ξ, λ) yields an η−ricci soliton on m. proof. since h(ξ, ξ) = (lξg)(ξ, ξ) + 2s(ξ, ξ) + 2µη(ξ)η(ξ) = 2λ, implies that λ = 1 2 h(ξ, ξ). (3.8) in [17], we have h(x, y) = −h(ξ, ξ)g(x, y), x, y ∈ χ(m). (3.9) therefore, lξg + 2s + 2µη ⊗ η = 2λg. our theorem is proved. if µ = 0, it follows that lξg + 2s + 2(n − 1)(α 2 − ρ)g = 0. thus we conclude that corollary 3.3. on a (lcs)n−manifold (m, φ, ξ, η, g) with the property that the symmetric (0, 2) tensor field h = lξg+2s is parallel with respect to the levi-civita connection associated to g, then the equation (3.1), for µ = 0 and λ = −[(n − 1)(α2 − ρ)], defines a ricci soliton. as a consequence of the existence of η−ricci soliton on a (lcs)n−manifold. from (3.1), we state that corollary 3.4. if the equation (3.1) define an η−ricci soliton on a (lcs)n− manifold, then (m, g) is quasi-einstein. since the manifold is quasi-einstein, if the ricci tensor field s is a linear combination (with real scalar λ and µ, respectively, with µ 6= 0) of g and the tensor product of a non-zero 1−form η satisfying (2.2) and for an einstein if s is co-linear with g ([13], [23]). theorem 3.5. if (m, φ, ξ, η, g) be a (lcs)n−manifold and equation (3.1) define an η−ricci soliton on (m, g), then (i) q ◦ φ = φ ◦ q, (ii) q and s are parallel along ξ. proof. the prove of (i) follows by direct computation. for (ii) we have (∇ξq)x = ∇ξqx − q(∇ξx) and (∇ξs)(x, y) = ξ(s(x, y)) − s(∇ξx, y) − s(x, ∇ξy). in view of (3.3) and (3.5), above equation leads the result. in a particular case if the manifold is φ−ricci symmetric, then φ2 ◦ ∇q = 0, therefore we state the following proposition as: cubo 19, 2 (2017) some geometric properties of η− ricci solitons ... 39 proposition 3.6. if a (lcs)n−manifold (m, φ, ξ, η, g) is φ−ricci symmetric and equation (3.1) leads to η−ricci soliton, then µ = −α, λ = −[(n − 1)(α2 − ρ) + α] and the manifold reduces to einstein. proof. we compute (∇xq)y = ∇xqy − q(∇xy). in view of (3.3) above equation takes the form (µ + α){x + η(x)ξ}η(y) = 0, for x, y ∈ χ(m). from above it follows that µ = −α, λ = −[(n − 1)(α2 − ρ) + α], and s = (n − 1)(α2 − ρ)g. as a weaker version of local symmetry, the notion of local-symmetric sasakian manifold was introduced by takahashi [37]. chaubey ([7][11]) studied the properties of symmetric spaces in different extent. shaikh et al. ([35], [36]) studied locally φ−symmetric and locally φ−recurrent (lcs)n−manifolds. hui [22] studied φ−pseudosymmetric (lcs)n−manifolds and obtained the form of ricci tensor s as s(x, y) = { α(n − 1)(α2 − ρ) α + a(ξ) } g(x, y) + { (n − 1)(α2 − ρ)a(ξ) α + a(ξ) } η(x)µ(y), (3.10) provided α + a(ξ) 6= 0. theorem 3.7. if the tensor field lξg + 2s on a φ−pseudo ricci symmetric (lcs)n− manifold with α2 − ρ 6= 0 is parallel with respect to levi-civita connection associated to g, then for µ = 0 the ricci soliton (g, ξ, λ) is shrinking, steady and expanding according as (α 2 −ρ){a(ξ)−α} α+a(ξ) < 0, a(ξ) = α and (α 2 −ρ){a(ξ)−α} α+a(ξ) > 0 respectively. proof. let h is a (0, 2) symmetric parallel tensor field on (lcs)n−manifold. in view of (3.1), we obtain h(x, y) = (lξg)(x, y) + 2s(x, y). (3.11) using (3.2) and (3.10), equation (3.11) reduces to h(x, y) = 2α[g(x, y) + η(x)η(y)] + 2 { α(n−1)(α 2 −ρ) α+a(ξ) } g(x, y) +2 { (n−1)(α 2 −ρ)a(ξ) α+a(ξ) } η(x)µ(y). (3.12) replacing x = y = ξ in (3.12), we get h(ξ, ξ) = { 2(n − 1)(α2 − ρ){a(ξ) − α} α + a(ξ) } . (3.13) in view of (3.8) and (3.13), we obtain λ = { (n − 1)(α2 − ρ){a(ξ) − α} α + a(ξ) } . 40 s. k. yadav, s. k. chaubey & d. l. suthar cubo 19, 2 (2017) since n > 1, α2 − ρ 6= 0 and α + a(ξ) 6= 0 , we conclude that λ > 0 if (α 2 −ρ){a(ξ)−α} α+a(ξ) > 0, λ = 0 if a(ξ) = α and λ < 0 if (α 2 −ρ){a(ξ)−α} α+a(ξ) < 0. our theorem is proved. corollary 3.8. if the tensor field lξg + 2s on a φ−pseudo riccisymmetric (lcs)n− manifold with α2 −ρ 6= 0 is parallel with respect to levi-civita connection associated to g, then for µ = 0 the ricci soliton (g, ξ, λ)is shrinking and expanding according as α2−ρ > 0 and α2−ρ < 0 respectively. let (lcs)n−manifold admits a ricci soliton defined by (3.1) for µ = 0. it is known that ∇g = 0. we consider λ constant, so ∇λg = 0. thus lvg + 2s is parallel. hence lvg + 2s is a constant multiple of metric tensors g, i.e. lvg + 2s = ag, where a is constant. thus lvg + 2s + 2λg reduces to (a + 2λ)g, we get λ = − a 2 . in view of above statement we state the result as the proposition. proposition 3.9. in (lcs)n−manifold the ricci soliton (v, ξ, λ) is shrinking or expanding according as a is positive or negative. theorem 3.10. if in a (lcs)n− manifold, the metric g is a ricci soliton and v is a point-wise co-linear with ξ, then v is a constant multiple of g provided λ = −(n − 1)(α2 − ρ). proof. suppose that v is pointwise colinear with ξ, i.e., v = c ξ, where c is a smooth function on (m, g). then (lvg + 2s + 2λg)(x, y) = 0 implies that cg(∇xξ, y) + (xc)η(y) + cg(∇yξ, x) + (yc)η(x) +2s(x, y) + 2λg(x, y) = 0. with the help of (2.5), the above equation takes the form cαg(φx, y) + (xc)η(y) + cαg(φy, x) + (yc)η(x) +2s(x, y) + 2λg(x, y) = 0. (3.14) substituting y = ξ in (3.14) and using (2.7) and (2.13) in it, we get (xc) = 2 [ λ + (n − 1)(α2 − ρ) ] η(x). (3.15) since η is closed, i. e., dη = 0 on (lcs)n−manifold. from (3.15) we yield xc = 0, provided λ = −(n − 1)(α2 − ρ). our theorem is proved theorem 3.11. if in a lp-sasakian manifold the metric g is a ricci soliton and v is a point-wise co-linear with ξ, then the manifold is an η−einstein manifold provided c 6= −1. proof. particularly if v = ξ, then in view of that equation (lvg + 2s + 2λg)(x, y) = 0, we have αg(φx, y) + s(x, y) + λg(x, y) = 0. (3.16) putting x = ξ in (3.16), we get λ = −(n−1)(α2 −ρ). since n > 1, α2 −ρ 6= 0. therefore the ricci soliton is shrinking or expanding as α2 < ρ or α2 > ρ respectively. specially, if we take α = 1, cubo 19, 2 (2017) some geometric properties of η− ricci solitons ... 41 then (m, g) reduces to a lp-sasakian structure of matsumoto [27]. then in view of (3.14) and (3.15), equation (3.16) reduces to s(x, y) = ( 2λ 1 + c ) g(x, y) + ( 2 1 − c ) (−λ − (n − 1)(1 − ρ)η(x)η(y), (3.17) provided c 6= −1. our theorem is proved. corollary 3.12. if in a (lcs)n−manifold, the metric g is a ricci soliton (v, ξ, λ) and v is a point-wise co-linear with ξ, then the ricci solution (v, ξ, λ) is shrinking or expanding according as α2 − ρ > 0 or α2 − ρ < 0. in [31], sharma proved that a compact ricci soliton of constant scalar curvature is einstein. on contracting (3.17), we get r = ( 2 1+b ) [λ(n + 1) + (1 − ρ)] =constant. thus we state the result as: corollary 3.13. a lp-sasakian manifold equipped with a compact ricci soliton is an einstein manifold. theorem 3.14. if (lcs)n−manifold is η−einstein of the form s = δ g + γ η ⊗ η with δ, γ = constant, then the manifold is equipped a ricci soliton (g, ξ, −(δ + α)). proof. let (m, g) be an η−einstein (lcs)n−manifold, then s(x, y) = δg(x, y) + γη(x)η(y), (3.18) where δ, γ = constants. taking v = ξ in (3.1) (for µ = 0) and using (3.18), we get (lξg)(x, y) + 2s(x, y) + 2λg(x, y) = 2(α + δ + λ) + 2(α + γ)η(x)η(y). (3.19) it is clear from (3.19) that (m, g) admits a ricci soliton (g, ξ, λ) if α + δ + λ = 0 and α + γ = 0 it implies that γ = −α = constant. also from (3.18) we have δ = −α + (n − 1)(α2 − 1) = constant. thus λ = −(α + δ) = constant. our theorem is proved. corollary 3.15. if an η−einstein (lcs)n−manifold with the form s = δ g + γ η ⊗ η admits a compact ricci soliton (g, ξ, −(δ + α)) then it leads to an einstein. 4 gradient ricci solitons in this section we consider gradient ricci soliton on (lcs)n− manifold and prove the following results theorem 4.1. if an η−einstein (lcs)n−manifold equipped with a gradient ricci soliton then manifold reduces to an einstein provided λ = (n − 1)(α2 − ξ) within the frame field ξf = 0. 42 s. k. yadav, s. k. chaubey & d. l. suthar cubo 19, 2 (2017) proof. let the vector field v be the gradient of a potential function f, is called gradient ricci soliton. thus (3.1) takes the form ∇∇f = s + λg, (4.1) that implies ∇ydf = qy + λy, (4.2) where ∇ is the gradient operator of g. from above we notice that g(r(ξ, y)df, ξ) = g((∇ξq)y, ξ) − g((∇yq)ξ, ξ). (4.3) in view of (2.5) and (3.18), equation (4.3) yields g(r(ξ, y)df, ξ) = 0. (4.4) using (2.10) in (4.4), we get df = (α2 − ρ){−η(df)ξ} = −(α2 − ρ)(g(df, ξ)ξ) = −(α2 − ρ)(ξf)ξ. (4.5) from (4.2) and (4.5), we obtain s(x, y) + λg(x, y) = −y(ξf)(α2 − ρ)η(x) − (ξf)(α2 − ρ)g(φy, x). (4.6) replacing x = ξ in (4.6) and using (3.15), we yield y(ξf)(α2 − ρ) = {λ − (n − 1)(α2 − ρ)}η(y). it implies that if λ = (n − 1)(α2 − ρ) then ξf =constant and therefore from (4.5), we have df = −(α2 − ρ)(ξf)ξ = ωξ, ω = −(α2 − ρ)(ξf). if we consider ξf = 0, then (4.5) implies that f = constant. thus (4.1) yields that s = (n−1)(α2 − ρ)g. our theorem is proved. 5 examples of an η−ricci solition on (lcs)n−manifolds example 5.1. let m = { (x, y, z) ∈ re3 : z 6= 0 } be a 3−dimensional smooth manifold, where (x, y, z) are the standard coordinates in re3. let {e1, e2, e3} be linearly independent global frame on m given by e1 = e z ( x ∂ ∂x + y ∂ ∂y ) , e2 = e z ∂ ∂y , e3 = e 2z ∂ ∂z . let g be the lorentzian metric defined by g(e1, e3) = g(e2, e3) = g(e1, e2) = 0, g(e1, e1) = g(e2, e2) = 1, g(e3, e3) = −1. cubo 19, 2 (2017) some geometric properties of η− ricci solitons ... 43 and η be the 1-form defined by η(v) = g(v, e3) for any v ∈ χ(m). let φ be the (1, 1) tensor field defined by φe1 = e1, φe2 = e2, φe3 = 0. then using the linearity of φ and g we have η(e3) = −1, φv = v + η(v)e3, g(φv, φw) = g(v, w) + η(v)η(w), for any v, w ∈ χ(m). let ∇ be the levi-civita connection with respect to the lorentzian metric g and r be the curvature tensor of g. then we obtain [e1, e2] = −e ze2, [e1, e3] = −e 2ze1, [e2, e3] = −e 2ze2. taking e3 = ξ and using koszul’s formula for the lorentzian metric g, we have ∇e1e3 = −e 2ze1, ∇e1e1 = −e 2ze3, ∇e1e2 = 0, ∇e2e3 = −e 2ze2, ∇e3e2 = 0, ∇e2e1 = −e 2ze2, ∇e3e3 = 0, ∇e2e2 = −e 2ze3 − e ze1, ∇e3e1 = 0. from the above it can be easily see that e3 = ξ is a unit timelike concircular vector field and hence (φ, ξ, η, g) is a (lcs)3-structure on m. consequently m 3(φ, ξ, η, g) is a (lcs)3-manifold with α = −e2z 6= 0 such that (xα) = ρη(x) where ρ = 2e4z. using the above relations, we can easily calculate the non-vanishing components of the curvature tensor r and ricci tensor s as follows: r(e2, e3)e3 = e 4ze2, r(e1, e3)e3 = e 4ze1, r(e1, e2)e2 = {e 4z − e2z}e1, r(e2, e3)e2 = e 4ze3, r(e1, e3)e1 = e 4ze3, r(e1, e2)e1 = {−e 4z − e2z}e2, s(e1, e1) = 0, s(e2, e2) = 0, s(e3, e3) = 2e 4z. also from (3.5), we calculated that s(e1, e1) = −(α + λ), s(e2, e2) = −(α + λ), s(e3, e3) = (λ − µ). thus we conclude that from (3.5) for α = −e2z, λ = e2z and µ = e2z −e4z, the structure (g, ξ, λ, µ) is an η−ricci soliton on m3(φ, ξ, η, g). example 5.2. let a 3−dimensional manifold m = { (x, y, z) ∈ re3 : z 6= 0 } , where (x, y, z) are the standard coordinates in re3 . let{e1, e2, e3} be linearly independent global frame on m given by e1 = e −z ( x ∂ ∂x + y ∂ ∂y ) , e2 = e −z ∂ ∂y , e3 = e −2z ∂ ∂z . let g be the lorentzian metric defined by g(e1, e3) = g(e2, e3) = g(e1, e2) = 0, g(e1, e1) = g(e2, e2) = 1, g(e3, e3) = −1. let η be the 1−form defined by η(v) = g(v, e3) for any v ∈ χ(m). let φ be the (1, 1) tensor field defined by φe1 = e1, φe2 = e2, φe3 = 0. then using the linearity of φ and g we have η(e3) = −1, φv = v + η(v)e3, g(φv, φw) = g(v, w) + η(v)η(w), 44 s. k. yadav, s. k. chaubey & d. l. suthar cubo 19, 2 (2017) for any v, w ∈ χ(m). let ∇ be the levi-civita connection with respect to the lorentzian metric g and r be the curvature tensor of g. then we obtain [e1, e2] = −e −ze2, [e1, e3] = −e −2ze1, [e2, e3] = −e −2ze2. taking e3 = ξ and using koszul’s formula for the lorentzian metric g, we have ∇e1e3 = e −2ze1, ∇e1e1 = e −2ze3, ∇e1e2 = 0, ∇e2e3 = e −2ze2, ∇e3e2 = 0, ∇e2e1 = e −2ze2, ∇e3e3 = 0, ∇e3e2 = e −2ze3 − e −ze1, ∇e3e1 = 0. from the above it can be easily seen that e3 = ξ is a unit timelike concircular vector field and hence (φ, ξ, η, g) is a (lcs)3-structure on m. consequently m 3(φ, ξ, η, g) is a (lcs)3-manifolds with α = e−2z 6= 0 such that (xα) = ρη(x) where ρ = 2e−4z using the above relations, we can easily calculate the non-vanishing components of the curvature tensor r and ricci tensor s as follows: r(e2, e3)e3 = e −4ze2, r(e1, e3)e3 = e −4ze1, r(e1, e2)e2 = {e −4z − e−2z}e1, r(e2, e3)e2 = e −4ze3, r(e1, e3)e1 = e −4ze3, r(e1, e2)e1 = {−e −4z − e−2z}e2, s(e1, e1) = 2e −4z − e−2z, s(e2, e2) = 2e −4z − e−2z, s(e3, e3) = 2e −4z. also from (3.5), we calculated that s(e1, e1) = −(α + λ), s(e2, e2) = −(α + λ), s(e3, e3) = (λ − µ). we summaries that from (3.5) for α = e2z, λ = −2e−4z and µ = −4e−4z, the data (g, ξ, λ, µ) admits an η−ricci soliton on m3(φ, ξ, η, g). example 5.3. we consider the 4−dimensional manifold m = { (x, y, z, u) ∈ re4 : u 6= 0 } , where (x, y, z, u) are the standard coordinates in re4 . let{e1, e2, e3, e4} be linearly independent global frame on m given by e1 = u ( x ∂ ∂x + y ∂ ∂y ) , e2 = u ∂ ∂y , e3 = u ( ∂ ∂y + ∂ ∂z ) , e4 = (u) 3 ∂ ∂u . let g be the lorentzian metric defined by g(e1, e1) = g(e2, e2) = g(e3, e3) = 1, g(e4, e4) = −1, g(ei, ej) = 0, i 6= j, i, j = 1, 2, 3, 4. let η be the 1−form defined by η(v) = g(v, e4), ξ = (u) 4 ∂ ∂u for any v ∈ χ(m). let φ be the (1, 1) tensor field defined by φe1 = e1, φe2 = e2, φe3 = e3, φe4 = 0. then using the linearity of φ and g we have η(e4) = −1, φv = v + η(v)e3, g(φv, φw) = g(v, w) + η(v)η(w), cubo 19, 2 (2017) some geometric properties of η− ricci solitons ... 45 for any v, w ∈ χ(m). let ∇ be the levi-civita connection with respect to the lorentzian metric g and r be the curvature tensor of g. then we obtain [e1, e2] = −ue2, [e1, e4] = −(u) 4e1, [e2, e4] = −(u) 4e2, [e3, e4] = −(u) 4e3. taking e4 = ξ and using koszul’s formula for the lorentzian metric g, we have ∇e1e4 = −(u) 2e1, ∇e2e1 = ue3, ∇e1e1 = −(u) 4e4, ∇e3e4 = −(u) 4e3, ∇e3e3 = −(u) 4e4, ∇e2e2 = −(u) 2e4 − ue1. from the above it can be easily seen that the structure (φ, ξ, η, g) is an (lcs)4− structure on m. consequently m4(φ, ξ, η, g) is an(lcs)4-manifold with α = −(u) 4 6= 0 such that (xα) = ρη(x) where ρ = 2(u)4. using the above relations, we can easily calculate the non-vanishing components of the curvature tensor r and ricci tensor s as follows: r(e1, e4)e1 = (u) 4e4, r(e2, e4)e2 = (u) 4e4, r(e3, e4)e3 = (u) 4e4, r(e1, e3)e3 = (u) 4e1, r(e1, e3)e1 = −(u) 4e3, r(e2, e3)e2 = −(u) 4e3, r(e1, e4)e4 = (u) 4e1, r(e2, e4)e4 = (u) 4e2, r(e1, e2)e2 = [(u) 4 − (u)2]e1, r(e2, e3)e3 = (u) 4e2, r(e3, e4)e4 = (u) 4e3, r(e1, e2)e1 = −[(u) 4 − (u)2]e2, s(e1, e1) = 3(u) 4 − (u)2, s(e2, e2) = 3(u) 4 − (u)2, s(e3, e3) = 3(u) 4, s(e4, e4) = 3(u) 4. also from (3.5), we calculated that s(e1, e1) = −(α + λ), s(e2, e2) = −(α + λ), s(e3, e3) = (λ − µ), s(e4, e4) = (λ − µ). we conclude that from (3.5) for α = −(u)4, λ = −3(u)4 + 2(u)2 and µ = −6(u)4 + 2(u)2 the data (g, ξ, λ, µ) admits an η−ricci soliton on m4(φ, ξ, η, g). references [1] m. atceken, on geometry of submanifold of (lcs)n−manifolds, int. j. math. sci., (2012), art. id304647. 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[44] k. yano, concircular geometry i. concircular transformations, proc. imp. acad. tokyo, 16, (1940), 195-200. introduction (lcs)n-manifolds (m, , , , g) -ricci solitons on (lcs)n-manifolds gradient ricci solitons examples of an -ricci solition on (lcs)n-manifolds cubo a mathematical journal vol.10, n o ¯ 01, (77–92). march 2008 regular and strongly regular time and norm optimal controls h. o. fattorini department of mathematics, university of california los angeles, california 90095-1555 email: hof@math.ucla.edu abstract pontryagin’s maximum principle in its infinite dimensional version provides (separate) necessary and sufficient conditions for both time and norm optimality for the system y ′ = ay + u (a the infinitesimal generator of a strongly continuous semigroup). among controls that satisfy the maximum principle, a smoothness distinction can be defined in terms of smoothness of the final value of the costate. this paper addresses some issues related to this distinction. resumen el principio del máximo de pontryagin, en su version de dimension infinita, proporciona condiciones necesarias y suficientes (separadamente) para optimalidad 78 h. o. fattorini cubo 10, 1 (2008) en el tiempo y en la norma para el sistema y ′ = ay + u (a el generador infinitesimal de un semigrupo fuertemente cont́ınuo). entre los controles que satisfacen el principio del máximo se puede establecer una jerarqúıa de regularidad en términos de la regularidad del valor final del co-estado. este art́ıculo considera algunas cuestiones relacionadas con ésta jerarqúıa. key words and phrases: linear control systems in banach spaces, time optimal problem, norm optimal problem math. subj. class.: 93e20, 93e25. 1 introduction. we consider the control system y ′ (t) = ay(t) + u(t) , y(0) = ζ (1.1) with controls u(·) ∈ l∞(0, t ; e), where a is the infinitesimal generator of a strongly continuous semigroup s(t) in a banach space e. in the norm optimal problem we drive the initial point ζ to a point target, y(t ) = ȳ in a fixed time interval 0 ≤ t ≤ t minimizing ‖u(·)‖l∞(0,t ;e), while in the time optimal problem we drive to the target with a bound on the norm of the control (say ‖u(·)‖l∞(0,t ;e) ≤ 1) in optimal time t. solutions or trajectories y(t) = s(t)ζ + ∫ t 0 s(t − σ)u(σ)dσ of the initial value problem (1.1) are continuous and denoted by y(t) = y(t, ζ, u). for the time optimal problem, controls in l ∞ (0, t ; e) with norm ‖u(·)‖l∞(0,t ;e) ≤ 1 are named admissible. separate necessary and sufficient conditions for both norm and time optimality can be given in terms of the maximum principle, which requires the construction of spaces of multipliers (final values of the costates). we summarize [5] or [7, 2.3]. when the infinitesimal generator a has a bounded inverse, we define the space e ∗ −1 as the completion of e ∗ in the norm ‖y∗‖e∗ −1 = ‖(a−1)∗y∗‖e∗ . each s(t) ∗ can be extended to an (equally named) operator s(t) ∗ : e ∗ −1 → e ∗ −1, and the cubo 10, 1 (2008) regular and strongly regular ... 79 space z 1 (t ) consists of all z ∈ e∗−1 such that s(t) ∗ z ∈ e∗ and1 ‖z‖z1(t ) = ∫ t 0 ‖s(t)∗z‖dt < ∞ . (1.2) equipped with ‖ ·‖z1(t ), z 1 (t ) is a banach space. all spaces z 1 (t ) coincide and all norms ‖ · ‖z1(t ) are equivalent for t > 0. z 1 (t ) is an example of a multiplier space, defined as an arbitrary linear space z ⊇ e∗ to which s(t)∗ can be extended in such a way that s(t) ∗z ⊆ e∗. when a does not have a bounded inverse, the construction of the spaces above is modified as follows. since a is a semigroup generator, (λi − a)−1 exists for λ > ω and e ∗ −1 is the completion of e ∗ in any of the equivalent norms ‖y∗‖e∗ −1 ,λ = ‖((λi − a) −1 ) ∗ y ∗‖e∗ , (λ > ω) . the definition of z 1 (t ) (and of multiplier spaces) is the same. see [8, 2.3] for details. a control u(·) ∈ l∞(0, t ; e) satisfies pontryagin’s maximum principle if 〈s(t − t)∗z, ū(t)〉 = max ‖u‖≤ρ 〈s(t − t)∗z, u〉 a. e. in 0 ≤ t < t , (1.3) 〈· , ·〉 the duality of the space e and the dual e∗, with ρ = ‖u(·)‖l∞(0,t ;e) and z in some multiplier space z. we call z the multiplier and s(t − t)∗z the costate corresponding to the control ū(t). we work under the standing assumption that (1.3) is nonempty; this means s(t −t)∗z is not identically zero in the interval 0 ≤ t < t, although we don’t mind s(t −t)∗ vanishing in part of the interval (in which part (1.3) provides no information on ū(t)). the assumption that (1.3) is nonempty implies in particular that z 6= 0. the maximum principle takes a simple form when e is a hilbert space; if fact, it reduces to ū(t) = ρ s(t − t)∗z ‖s(t − t)∗z‖ a. e. in 0 ≤ t < t , (1.4) where s(t − t)∗z 6= 0 in 0 ≤ t < t. a large part of the theory of optimal controls for the system (1.1) deals with the relation between optimality and the maximum principle (1.3), a relation which is elementary in finite dimension but becomes rather involved in an infinite dimensional space e. all one has (at present) are separate necessary and sufficient conditions for optimality based on the maximum principle (theorem 1.1 below). we call an optimal control ū(t) regular if it satisfies (1.3) with z ∈ z1(t ). 1at this level of generality, the semigroup s(t)∗ may not be strongly continuous, or even strongly measurable (consider, for instance, the translation semigroup s(t)y(x) = y(x − t) in e = l1(∞, ∞)). however, s(t)∗ is always e-weakly continuous, which guarantees that ‖s(t)∗‖ is lower semicontinuous, hence measurable. this gives sense to the integral (1.2). note also that in existing literature (for instance, [7]) z1(t ) is called z(t ) (sometimes zw(t ) for “weak” to emphasize that s(t) ∗ may not be strongly continuous). we use the superindex 1 since spaces zp(t ) (p 6= 1) will be introduced later. 80 h. o. fattorini cubo 10, 1 (2008) theorem 1.1. assume ū(t) drives ζ ∈ e to ȳ = y(t, ζ, ū) time or norm optimally in the interval 0 ≤ t ≤ t and that ȳ − s(t )ζ ∈ d(a) . (1.5) then u(t) is regular. conversely, let ū(t) be a regular control. then ū(t) drives ζ ∈ e to ȳ = y(t, ζ, ū) norm optimally in the interval 0 ≤ t ≤ t ; if ρ = 1 the drive is time optimal. for the proof see [5, theorem 5.1], [7, theorem 2.5.1]; we note that in the sufficiency half of theorem 1.1 no conditions of the type of (1.5) are put on the initial value ζ or the target ȳ. 2 a control u(·) is called strongly regular if it satisfies (1.3) with z ∈ e∗. the notion of strongly regular control adds nothing to the two implications in theorem 1.1, but it is of interest in applications. in fact, if e ∗ is a hilbert space then (1.4) shows that a strongly regular control is (at least) continuous in 0 ≤ t ≤ t, whereas a merely regular control may “oscillate” at the endpoint t of the control interval. this makes a difference, for instance, in numerical approximations of the optimal control. 3 the question addressed in this paper is, characterize the control systems (1.1) for which all (time, norm) optimal controls are strongly regular. part of the answer to this question is known; a sufficient condition for all optimal controls being strongly regular is s(t)e = e (t > 0) . (1.6) this condition is valid in any banach space (theorem 2.1 below). the main contribution of this paper is the opposite implication, which we only prove under special assumptions on e (corollary 4.8). we also show (remark 4.9) that if these special assumptions are dropped, the implication ceases to be true. 2 reversible semigroups. semigroups satisfying (1.6) we call reversible. in this section, no restrictions are placed upon the banach space e. theorem 2.1. let s(t) be a reversible semigroup. then all optimal controls for (1.1) are strongly regular, that is, they satisfy (1.3) with z ∈ e∗. 2the statement on time optimality, however, needs additional assumptions on the initial condition ζ and the target ȳ. these conditions are satisfied if either ζ = 0 or ȳ = 0 [6], [7, theorem 2.5.7]. we point out that the conditions are on the “size” of ζ ȳ, not on their smoothness like (1.5); for instance, for ζ = 0, ȳ may be an arbitrary element of e. we also need to assume that s(t)∗z 6= 0 in the entire interval 0 ≤ t ≤ t. 3piermarco cannarsa has pointed out situations involving optimal controls for semilinear equations, where strong regularity of (linear) optimal controls is actually needed; plain regularity is not enough. cubo 10, 1 (2008) regular and strongly regular ... 81 the proof of theorem 2.1 requires some auxiliary results. lemma 2.2. let the e∗-valued, e-weakly continuous function f (t) satisfy ∫ t 0 〈f (t), u(t)〉dt ≤ c ( ∫ t 0 ‖u(t)‖p )1/p dt (u(·) ∈ l∞(0, t ; e)) (2.1) for some p, 1 ≤ p < ∞. then, if p > 1 and 1/p + 1/q = 1 we have ( ∫ t 0 ‖f (t)‖q )1/q ≤ c , (2.2) with equality in (2.2) if c is the smallest constant satisfying (2.1). if p = 1, ‖f (t)‖ ≤ c (0 ≤ t ≤ t ) , (2.3) with equality in (2.3) if c is the smallest constant satisfying (2.1). for p > 1 the proof of lemma 2.2 is essentially similar to that of [7, lemma 2.2.1 and lemma 2.2.10] thus we omit it. for p = ∞, assume (2.3) fails. then there exists y ∈ e and a nontrivial interval e such that 〈f (t), y〉 ≥ (c + ǫ)‖y‖. setting u(t) = { y t ∈ e 0 t /∈ e we obtain ∫ t 0 〈f (t), u(t)〉dt = ∫ e 〈f (t), y〉dt ≥ (c + ǫ)|e|‖y‖ = (c + ǫ) ∫ e ‖u(t)‖dt , contradicting (2.1). this completes the proof. given t > 0 and 1 ≤ p ≤ ∞, the reachable space rp(t ) (at time t ) of the system (1.1) consists of all y = y(t, 0, u) = ∫ t 0 s(t − σ)u(σ)dσ u(·) ∈ lp(0, t ; e) , and is equipped with the norm ‖y‖rp(t ) = inf { ‖u‖lp(0,t ;e); ∫ t 0 s(t − σ)u(σ)dσ = y } , which makes r p (t ) a banach space, isometrically isomorphic to the quotient space l p (0, t ; e)/n p 82 h. o. fattorini cubo 10, 1 (2008) where n p is the closed subspace of lp(0, t ; e) of all u(·) with ∫ t 0 s(t − σ)u(σ)dσ = 0 . we note in passing that all spaces r ∞ (t ) coincide (with equivalent norms) for t > 0. this is proved in [2], [7, 2.1] and can be extended to p < ∞, but is not particularly relevant here. if r > p hölder’s inequality gives ∫ t 0 ‖u(σ)‖pdσ = ∫ t 0 1 · (‖u(σ)‖r)p/rdt ≤ t (r−p)/r ( ∫ t 0 ‖u(σ)‖rdr )p/r , thus ‖y‖rp(t ) ≤ t (r−p)/pr‖y‖rr(t ) , and it follows that r r (t ) →֒ rp(t ). (the symbol →֒ means “is imbedded in”. another application of hölder’s inequality produces ‖y‖ ≤ ‖s(t − ·)‖lp/(p−1)(0,t )‖y‖rp(t ) ≤ t (p−1)/p‖s(t − σ)‖l∞(0,t )‖y‖rp(t ) , so that r p (t ) →֒ e. finally, if y ∈ d(a), integration by parts gives y = ∫ t 0 s(t − σ) y − σay t dσ thus, if we equip d(a) with its customary graph norm, we have d(a) →֒ r∞(t ). putting all the imbeddings together, d(a) →֒ r∞(t ) →֒ rr(t ) →֒ rp(t ) →֒ e (p < r) . (2.4) all imbeddings except the first are dense in the norm of the bigger space. for the imbeddings r ∞ (t ) →֒ rr(t ) →֒ rp(t ) this follows from denseness of l∞(0, t ; e) (thus of lr(0, t ; e)) in l p (0, t ; e), and for r p (t ) →֒ e from denseness of d(a) in e (for these results and more details about function spaces of e-valued functions see [1, chapter iii] or [9, chapter iii]. whether or not d(a) is dense in r ∞ (t ) in the norm of the latter space is one of the main themes of [7, chapters 2 and 3]. the following result follows immediately from denseness of r ∞ (t ) in e and from the open mapping principle. lemma 2.3. we have r∞(t ) = e (with equivalent norms ) if and only if ‖y‖r∞(t ) ≤ c‖y‖ (y ∈ r ∞ (t )) . lemma 2.4 below is proved in [7, theorem 2.2.3 and beginning of 2.2]: lemma 2.4. r∞(t ) = e (with equivalent norms) if and only if s(t) is reversible. cubo 10, 1 (2008) regular and strongly regular ... 83 theorem 2.5. assume s(t) is reversible. then z 1 (t ) = e ∗ (t > 0) (2.5) with equivalent norms. proof. assume s(t) is reversible. then, by lemma 2.4, r∞(t ) = e with equivalent norms. it follows that r ∞ (t ) ∗ = e ∗ with equivalent norms as well. let z ∈ z1(t ). we can define a bounded linear functional ξz on r ∞ (t ) by 〈ξz , y〉 = 〈 ξz , ∫ t 0 s(t − σ)u(σ)dσ 〉 = ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ . (2.6) it can be easily seen that (2.6) pays heed to the equivalence relation in r ∞ (t ) = l ∞ (0, t ; e) /n ∞ [5], [7, lemma 2.3.5] and it follows from (1.2) that ξz is bounded in the norm of r ∞ (t ), precisely ‖ξz‖ = ∫ t 0 ‖s(t − σ)∗z‖dσ = ∫ t 0 ‖s(σ)∗z‖dσ . (2.7) the inequality ≤ in (2.7) is obvious; for the equality, see [5] or [7, 2.3]. by lemma 2.4, ξz is as well bounded in the norm of e. accordingly, ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ ≤ c ∥ ∥ ∥ ∥ ∫ t 0 s(t − σ)u(σ)dσ ∥ ∥ ∥ ∥ . this implies ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ ≤ c ∫ t 0 ‖u(σ)‖dσ (u(·) ∈ l∞(0, t ; e)) and lemma 2.2 shows that ‖s(t)∗z‖ ≤ c (0 < t ≤ t ) . (2.8) we show that (2.8) implies z ∈ e∗. let {tn} be a positive decreasing sequence with tn → 0. then |〈s(tn) ∗ z − s(tm)z, y〉| = |〈s(tm) ∗ z, s(tn − tm)y − y〉| ≤ c‖s(tn − tm)y − y‖ for n < m, so that {s(tn)z} is cauchy in the e-weak topology of e ∗ . accordingly, it converges e-weakly to y ∗ ∈ e∗, and we have 〈s(t)∗z, y〉 = 〈s(t − tn) ∗ s(tn) ∗ z, y〉 = 〈s(tn) ∗ z, s(t − tn)y〉 → 〈s(t) ∗ y ∗ , y〉 , thus s(t) ∗ z = s(t) ∗ y ∗ (t > 0) . (2.9) 84 h. o. fattorini cubo 10, 1 (2008) the operator (a −1 ) ∗ : e ∗ −1 → e ∗ is 1-1 (and onto). applying (a −1 ) ∗ to both sides of (2.9) and applying the functionals on both sides to an element y ∈ e we obtain 〈(a−1)∗z, s(t)y〉 = 〈(a−1)∗y∗, s(t)y〉 (t > 0) . letting t → 0 we obtain (a−1)∗z = (a−1)∗y∗, thus z = y∗ as claimed. this ends the proof. we note the following interesting byproduct of the proof of theorem 2.5 (in particular, of the lines following (2.8)). define z ∞ (t ) as the space of all z ∈ e∗−1 such that s(t) ∗ z is bounded in 0 ≤ t ≤ t equipped with the norm ‖z‖z∞(t ) = max 0≤t≤t ‖s(t)∗z‖ . then (with no conditions on the space e or the semigroup s(t)), lemma 2.6. we have z ∞ (t ) = e ∗ with equivalent norms. 3 regular implies strongly regular, i. the first question is this. assume that (2.5) fails, that is, that the inclusion z 1 (t ) ⊃ e∗ (3.1) is strict. does this mean that there are regular controls which are not strongly regular? to attempt to answer this question is complicated by lack of uniqueness of z in the maximum principle (1.3) as in the following example, which is taken from [8]. example 3.1. consider the space e = ℓ0 consisting of all numerical sequences y = {yn} = {y1, y2, . . . } such that limn→∞ yn = 0 , equipped with the norm ‖y‖0 = maxn≥1 |yn|. the dual is e ∗ = ℓ 1 , the space of all numerical sequences y ∗ = {y∗n} such that ‖y ∗‖1 = ∑∞ n=1 |y ∗ n| < ∞, the duality of both spaces given by 〈y ∗ , y〉 = ∑∞ n=1 y ∗ nyn. the semigroup and generator are s(t){yn} = {e −nt yn} , a{yn} = −{nyn} , (3.2) a with maximal domain (limn→∞ n|yn| = 0). the space e ∗ −1 consists of all sequences {yn} with ‖(a−1)∗{y∗n}‖ = ∞ ∑ n=1 |y∗n| n < ∞ . (3.3) cubo 10, 1 (2008) regular and strongly regular ... 85 if {y∗n} ∈ e ∗ −1 we have ∫ t 0 ‖s(t)∗z‖dt = ∥ ∥ ∥ ∥ { ∫ t 0 e −nt y ∗ n } ∥ ∥ ∥ ∥ = ∞ ∑ n=1 |y∗| 1 − e−nt n ≤ ‖(a−1)∗{y∗n}‖ , thus e ∗ −1 = z 1 (t ) and the inclusion (3.1) is strict. due to existence requirements for optimal controls for (1.1) with this choice of space and generator, controls are taken in l ∞ w (0, t ; ℓ ∞ ) rather than in l ∞ (0, t ; ℓ 0 ), where ℓ ∞ is the space of all bounded numerical sequences y = {yn} equipped with the norm ‖y‖∞ = maxn≥1 |yn|. this means the u in the maximum principle (1.3) belongs to ℓ ∞ rather than in ℓ 0 . see [8] for additional details. we also take the following result from [8]. theorem 3.2. an admissible control ū(t) = {ūn(t)} satisfies the maximum principle (1.3) with z = {zn} in any multiplier space if and only if ūm(t) = 1 (0 ≤ t ≤ t ) or um(t) = −1 (0 ≤ t ≤ t ) for at least one m ≥ 1. proof. we take ρ = 1. the maximum principle for this space and generator is 〈s(t − t)∗{zn}, {ūn(t)}〉 = ∞ ∑ n=1 e −n(t −t) znūn(t) = max ‖{un}‖ℓ∞ ≤1 〈s(t − t)∗{zn}, {un}〉 = max |un|≤1 ∞ ∑ n=1 e −n(t −t) znun = ∞ ∑ n=1 e −n(t −t)|zn| , so that we must have ūm(t) = sign zm whenever zm 6= 0. conversely, if the assumptions of theorem 3.2 are satisfied for {ūn(t)} we obtain the maximum principle (1.3) with {zn} = δmn (δmn the kronecker delta). this ends the proof. strictness of the inclusion (3.1) and uniqueness of z in the maximum principle (1.3)4 do imply the existence of optimal controls that are regular but nor strongly regular. we just take z ∈ z1(t ) \ e∗ and use the sufficiency statement in theorem 1.1. uniqueness of z holds (for instance) in hilbert spaces. if ū(t) satisfies the maximum principle with two different z, ζ ∈ z1(t ) then, assuming (as always) that the maximum principle is nonempty and taking ρ = 1 for simplicity, ū(t) = s(t − t)∗z ‖s(t − t)∗z‖ = s(t − t)∗ζ ‖s(t − t)∗ζ‖ 4“uniqueness of z” obviously means “uniqueness up to multiplication by a constant”; if ū(t) satisfies the maximum principle (1.3) with two different z, ζ then ζ = αz, α 6= 0. the condition that α 6= 0 is required by the assumed nontriviality of (1.3). 86 h. o. fattorini cubo 10, 1 (2008) in some interval t −ǫ ≤ t ≤ t. multiplying by the product of the denominators and applying (a −1 ) ∗ to both sides we obtain ‖s(t − t)∗ζ‖s(t − t)∗(a−1)∗z = ‖s(t − t)∗z‖s(t − t)∗(a−1)∗ζ where (a −1 ) ∗ z, (a −1 ) ∗ ζ ∈ e∗, thus, if {tn} is a decreasing sequence with tn → 0 we have s(t − tn) ∗ (a −1 ) ∗ ζ = αns(t − tn) ∗ (a −1 ) ∗ z . (3.4) now, if y ∈ e is such that 〈y, (a−1)∗z〉 6= 0 we apply the functionals on both sides of (3.4) to y and obtain 〈(a−1)∗ζ , s(t − tn)y〉 = αn〈(a −1 ) ∗ z , s(t − tn)y〉 , (3.5) which shows that αn → α, thus we can take limits in (3.5), now written for arbitrary y ∈ e, obtaining 〈(a−1)∗ζ , y〉 = α〈(a−1)∗z , y〉 , thus ζ = αz (3.6) where α 6= 0 due to the requirement that (1.3) be nonempty (see comments after (1.3)). 4 regular implies strongly regular, ii. we show in this section the converse of theorem 2.5. the first result is on one of the imbeddings in (2.4), d(a) →֒ rp(t ) . (4.1) lemma 4.1. if 1 ≤ p < ∞ the imbedding (4.1) is dense. proof. let {λn} be an increasing sequence with λn → ∞. it follows from the dominated convergence theorem that if u(·) ∈ lp(0, t ; e) then λnr(λn; a)u(·) → u(·) in the norm of l p (0, t ; e) thus λnr(λn; a) ∫ t 0 s(t − σ)u(σ)dσ → ∫ t 0 s(t − σ)u(σ)dσ in the norm of r p (t ). this ends the proof. given 1 ≤ q < ∞, the space zq(t ) ⊆ z1(t ) consists of all z ∈ z1(t ) such that ∫ t 0 ‖s(t)∗z‖qdt < ∞ (4.2) cubo 10, 1 (2008) regular and strongly regular ... 87 equipped with the norm ‖s(·)∗z‖lq(0,t ). for q = ∞, the space was defined at the end of 2 (and shown to coincide with e ∗ ). theorem 4.2. the dual space rp(t )∗, 1 ≤ p < ∞ is algebraically and metrically isomorphic to zq(t ), 1/q + 1/p = 1. the proof is based on the calculation of the dual for p = ∞, which we outline below. bounded functionals ξz on r ∞ (t ) of the form (2.6) are called regular, and r(t ) ⊆ r∞(t )∗ is the subspace of all regular functionals. bounded functionals ξs on r ∞ (t ) that vanish in d(a) ⊆ r∞(t ) are called singular; the space of all such functionals is s(t ) ⊆ r∞(t )∗. application of the hahn banach theorem gives s(t ) = {0} ⇐⇒ d(a) is dense in r∞(t ) (in the norm of r∞(t )) . theorem 4.3. [7, theorem 2.4.1]. we have5 r ∞ (t ) ∗ = r(t ) ⊕ s(t ) (banach direct sum ) . proof of theorem 4.2. let ξ be a bounded linear functional in rp(t ). then (due to the second imbedding (2.4)) ξ is a bounded linear functional in r ∞ (t ) as well, hence, due to theorem 4.3. we have ξ = ξz + ξs with ξz regular and ξs singular. if u(·) ∈ l ∞ (0, t ; e) and ∫ t 0 s(t − σ)u(σ)dσ ∈ d(a) (4.3) we have 〈 ξ , ∫ t 0 s(t − σ)u(σ)dσ 〉 = 〈 ξz , ∫ t 0 s(t − σ)u(σ)dσ 〉 = ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ. (4.4) now, d(a) is dense in r p (t ) and r ∞ (t ) is dense in r p (t ), thus (4.4) can be extended to all elements (4.3) of r p (t ) whether or not they belong to d(a). since ξ is bounded in r p (t ) we have ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ ≤ ‖ξ‖rp(t )∗ ∥ ∥ ∥ ∥ ∫ t 0 s(t − σ)u(σ)dσ ∥ ∥ ∥ ∥ rp(t ) . (4.5) for the case p > 1 this implies ∫ t 0 〈s(t − σ)∗z, u(σ)〉dσ ≤ ‖ξ‖rp(t )∗ ( ∫ t 0 ‖u(σ)‖pdσ )1/p (u(·) ∈ lp(0, t ; e)) 5“banach direct sum” means algebraic direct sum plus bounded projections from the space into each of the two subspaces. 88 h. o. fattorini cubo 10, 1 (2008) and it follows from lemma 2.2 that s(·)∗z ∈ zq(t ) , ‖z‖zq(t ) = ( ∫ t 0 ‖s(t − σ)‖qdσ )1/q = ‖ξ‖rp(t )∗ , equality coming from the fact that ‖ξ‖rp(t )∗ is the least constant that does the job in (4.5). that an element of z q (t ) produces a functional in r p (t ) through (2.6) is a consequence of hölder’s inequality. in the case p = 1, (4.5) implies s(·)∗z ∈ z∞(t ) , ‖z‖z∞(t ) = sup 0≤t≤t ‖s(t)∗z‖ = ‖ξ‖rp(t )∗ . this ends the proof of theorem 4.2. we note that a proof of theorem 4.2 which is independent of the (rather involved) identification of r ∞ (t ) ∗ can be given using the equality l p w(0, t ; e) ∗ = l q (0, t ; e ∗ ) (4.6) for 1/p + 1/q = 1 ([10], [4]) where the subindex w means “e-weakly measurable”. however, the description of the spaces l p w(0, t ; e) ∗ (and the definition of the norms) is rather involved as well. corollary 4.4. we have r 1 (t ) = e (4.7) with equivalent norms. proof. we have r1(t ) →֒ e and r1(t ) is dense in e. on the other hand, by lemma 2.6 we have r 1 (t ) ∗ = z ∞ (t ) = e ∗ with equivalent norms. this is easily seen to imply equivalence of the norms of r 1 (t ) and e. in fact, it suffices to note that, as a consequence of the hahn banach theorem we have ‖y‖ = sup ‖y∗‖≤1 〈y∗, y〉 . (4.8) for any banach space e and its dual e ∗ . equivalence of the norms and the fact that r 1 (t ) is dense in e implies (4.7). so far, all results in this section have been proved for an arbitrary banach space e and strongly continuous semigroup s(t). our objective below is the proof of the converse of theorem 2.5, thus we assume z 1 (t ) = e ∗ . by intercession of lemma 2.6, this is the same as z 1 (t ) = z ∞ (t ), which in turn is equivalent to ∫ t 0 ‖s(t)∗z‖dt < ∞ =⇒ ‖s(t)∗z‖ is bounded in 0 ≤ t ≤ t . (4.9) cubo 10, 1 (2008) regular and strongly regular ... 89 it follows that all z p (t ) coincide, 1 ≤ p ≤ ∞. lemma 4.5. under (4.9) all zp(t ) norms are equivalent to the norm in z1(t ), with constants that do not depend on p. proof. independently of (4.9) we have ‖z‖zp(t ) = ( ∫ t 0 ‖s(t)∗z‖pdt )1/p ≤ t 1/p‖s(·)z‖l∞(0,t ) = t 1/p‖z‖z∞(t ) . (4.10) if (4.9) holds then, since z 1 (t ) = z ∞ (t ) is a banach space under the two norms, by the open mapping principle these norms have to be equivalent: hence ‖z‖z∞(t ) ≤ c‖z‖z1(t ) which, combined with (4.10) gives ‖z‖zp(t ) ≤ ct 1/p‖z‖1(t ) . on the other hand, and independently of (4.9), ‖z‖z1(t ) = ∫ t 0 ‖s(t)∗z‖dt ≤ t (p−1)/p ( ∫ t 0 ‖s(t)∗z‖pdt )1/p = ct (p−1)/p‖z‖zp(t ) . theorem 4.6. assume (4.9) holds. then r p (t ) = e (1 ≤ p < ∞) , all norms equivalent to the norm of e with constants that do not depend on p. proof. since rp(t )∗ = zp/(p−1)(t ) algebraically and metrically, lemma 4.5 says that r p (t ) ∗ = z ∞ (t ) = e ∗ , all norms equivalent to the norm of e ∗ with constants than do not depend of p. the corresponding statement for r p (t ), e is a consequence of denseness of r p (t ) in e and (4.8). this completes the proof. theorem 4.7. let e be reflexive and separable. if z 1 (t ) = e ∗ . then s(t) is reversible, that is, (1.6) holds. proof. we shall show that s(t) is reversible by proving that r∞(t) = e and using lemma 2.4. if e is reflexive and separable then x = e ∗ is reflexive and separable as well and l q (0, t ; e) = l p (0, t ; x) ∗ 90 h. o. fattorini cubo 10, 1 (2008) (1/p + 1/q = 1) where, due to the assumptions (and unlike in the generality of (4.6)) the space on the left is described exactly in the same form as the space on the right. since x is separable, the space l p (0, t ; x) is separable as well for 1 ≤ p < ∞. this implies that the l p (0, t ; x)-weak topology in any bounded subset of l q (0, t ; e) is defined by a metric ([1, theorem 3, p. 434]), which justifies the “passing to a subsequence” arguments below. under the assumptions, given y ∈ e we may avail ourselves of theorem 4.6, and construct a sequence {un(·)}, un(·) ∈ l n (0, t ; e) such that ∫ t 0 s(t − σ)un(σ)dσ = y , (‖un(·)‖ln(0,t ;e) ≤ c‖y‖ , n = 2, 3, 4, . . . ) (4.11) where c does not depend on n. since l 2 (0, t ; e) = l 2 (0, t ; x) ∗ we can select a subsequence of {un(·)} l 2 (0, t ; x)-weakly convergent in l 2 (0, t ; e); since l 3 (0, t ; x) = l 3/2 (0, t ; x) ∗ we can select a subsequence of the previous subsequence that is l 3/2 (0, t, x)-weakly convergent in l 3 (0, t, x) (thus l 2 (0, t ; x)-weakly convergent in l 2 (0, t ; e)); since l 4 (0, t ; x) = l 4/3 (0, t ; x) ∗ we can select a subsequence of the previous subsequence that is l 4/3 (0, t ; x)-weakly convergent in l 4 (0, t ; x) (thus l 3/2 (0, t ; x)-weakly convergent in l 3 (0, t ; e)), l 2 (0, t ; x)weakly convergent in l 2 (0, t ; e)), . . . and so on. picking the diagonal sequence, we finally obtain a sequence {un(·)} such that, eventually, it belongs to every l m (0, t ; e) and such that un(·) → ū(·) ∈ l m (0, t ; e) l m/(m−1) (0, t ; x)-weakly in l m (0, t ; e) . (the fact that the limit ū(·) is the same in all spaces is elementary). now, it follows from the norm estimation in (4.11) that ‖ū(·)‖lm(0,t ;e) ≤ c (2, 3, . . . ) with c independent of m, hence ū(·) ∈ l∞(0, t ; e). the first relation (4.11) implies 〈 y ∗ , ∫ t 0 s(t − σ)un(σ)dσ 〉 = ∫ t 0 〈s(t − σ)∗y∗, un(σ)〉dσ → ∫ t 0 〈s(t − σ)∗y∗, ū(σ)〉dσ = 〈 y ∗ , ∫ t 0 s(t − σ)ū(σ)dσ 〉 (y ∈ e∗) cubo 10, 1 (2008) regular and strongly regular ... 91 so that y = ∫ t 0 s(t − σ)ū(σ)dσ and the proof of theorem 4.7 is finished. corollary 4.8. let e be reflexive and separable. assume all regular controls for (1.1) are strongly regular and that z in the maximum principle (1.3) depends uniquely on ū(t) (as in the comments preceding (3.6)). then s(t) is reversible. proof. if the inclusion z1(t ) ⊃ e∗ is strict, taking z ∈ z1(t ) \ e∗ and using the sufficiency statement in theorem 1.1 we can construct a regular ū(t) which is not strongly regular. accordingly, we must have z 1 (t ) = e ∗ and theorem 4.7 applies. remark 4.9. theorem 3.2. shows that the conclusion of corollary 4.8 collapses if we drop the assumptions that e be reflexive and that z be unique. in fact, if e = ℓ 1 and a is given by (3.2) every control ū(t) that satisfies the maximum principle (1.3) with any multiplier z = {zn} 6= 0 satisfies (1.3) as well with {zn} = {δmn} ∈ ℓ 1 = e ∗ , thus it qualifies as strongly regular. however, the semigroup (3.1) is far from reversible. received: january 2007. revised: september 2007. references [1] n. dunford and j. schwartz, linear operators, part i, wiley interscience 1958. [2] h.o. fattorini, time-optimal control of solutions of operational differential equations, siam j. control, 2 (1964), 54–59. [3] h.o. fattorini, the time optimal control problem in banach spaces, applied math. optimization, 1 (1974), 163–188. [4] h.o. fattorini, infinite dimensional optimization and control theory, cambridge university press, cambridge, 1999. [5] h.o. fattorini, existence of singular extremals and singular functionals in reachable spaces, jour. evolution equations, 1 (2001), 325–347. [6] h.o. fattorini, sufficiency of the maximum principle for time optimality, cubo; a mathematical journal 7, (2005) 27–37. [7] h.o. fattorini, infinite dimensional linear control systems, north-holland mathematical studies 201, elsevier, amsterdam 2005. 92 h. o. fattorini cubo 10, 1 (2008) [8] h.o. fattorini, linear control systems in sequence spaces, g. lumer memorial volume [9] e. hille and r.s. phillips, functional analysis and semi-groups, american mathematical society, providence 1957. [10] a.i. tulcea and c.i. tulcea, topics in the theory of lifting, springer, berlin 1969. paper.pdf a mathematical journal vol. 7, no 2, (89 110). august 2005. global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations hassan sedaghat department of mathematics, virginia commonwealth university richmond, virginia, 23284-2014, usa hsedagha@vcu.edu abstract the asymptotic properties of a class of nonlinear second order difference equations are studied. sufficient conditions that imply the types of behavior mentioned in the title are discussed, in some cases within the context of the macroeconomic business cycle theory. we also discuss less commonly seen types of behavior, such as the equilibrium being simultaneously attracting and unstable, or the occurrence of oscillations away from a unique equilibrium. resumen se estudian las propiedades asintóticas de una clase de ecuaciones en diferencia no lineales de segundo orden. se discuten condiciones suficientes que implican los tipos de comportamiento mencionados en el t́ıtulo, en algunos casos dentro del contexto de la teoŕıa del ciclo de los negocios macroeconómicos. además se discuten tipos de comportamiento menos vistos comúnmente, tales como el equilibrio siendo simultáneamente atractivo e inestable, o la ocurrencia de oscilaciones lejos de un equilibrio único. key words and phrases: global attractivity, monotonic convergence, persistent oscillations, absorbing intervals, chaos, off-equilibrium oscillations, unstable global attractors math. subj. class.: 39a10, 39a11. 90 hassan sedaghat 7, 2(2005) introduction few known difference equations display the wide range of dynamic behaviors that the equation xn+1 = cxn + f (xn − xn−1), 0 ≤ c ≤ 1, n = 0, 1, 2, . . . (1) exhibits with even a limited selection of function types for f (assumed continuous throughout this paper). the nonlinear, second order difference equation (1) has its roots in the early macroeconomic models of the business cycle. indeed, a version of (1) in which f (t) = αt + β is a linear-affine function first appeared in samuelson (1939). various nonlinear versions of (1) subsequently appeared in the works of many other authors, notably in hicks (1950) and puu (1993). for more details, some historical remarks and additional references see sedaghat (2003a). the mapping f in (1) includes as a special case the sigmoid-type map first introduced into business cycle models (in continuous time) in goodwin (1951). these classical models provide an intuitive context for the interpretation of the many varied results about (1). in this paper we discuss several mathematical results that have been obtained about the asymptotic behavior of (1). these results include sufficient conditions for the global attractivity of the fixed point and conditions that imply the occurrence of persistent oscillations of solutions of (1). historically, the latter, endogenously driven oscillatory behavior was one of the main attractions of (1) in the economic literature. the case c = 1 which in puu (1993) models full consumption of savings, is substantially different from 0 ≤ c < 1; we discuss both cases in some detail. we also show that under certain conditions, solutions of (1) exhibit strange and complex behavior. these conditions include a case where the fixed point is globally attracting yet unstable. also seen as possible is the occurrence of persistent, offequilibrium oscillations; i.e., oscillations which do not occur about a fixed point. we also state various conjectures and open problems pertaining to (1). the essential background required for understanding the results of this paper is minimal beyond elementary real analysis and some mathematical maturity. however, some readers may benefit from a look at helpful existing texts and monographs such as elaydi (1999), kocic and ladas (1993), lasalle (1986), sedaghat (2003a). 1 oscillations in this first section of the paper, we consider the problem of oscillations for solutions of (1). in addition to being of interest mathematically, from a historical point of view this was the main attraction of business cycle models based on (1). in fact, in those economic models the kind of non-decaying, nonlinear oscillation that is discussed next was of particular interest. 7, 2(2005) global attractivity, oscillations and chaos in ... 91 1.1 persistent oscillations consider the general n-th order autonomous difference equation xn+1 = f (xn, xn−1, . . . , xn−m+1) (2) this clearly includes (1) as a special case with m = 2. definition 1. (persistent oscillations) a bounded solution {xn} of (2) is said to be persistently oscillating if the set of limit points of the sequence {xn} has two or more elements. persistent oscillations are basically a nonlinear phenomenon because nonlinearity is essential for the occurrence of robust or structurally stable persistent oscillations. indeed, if f is linear then its persistently oscillating solutions can occur only when a root of its characteristic polynomial has magnitude one; i.e., for linear maps persistent oscillations do not occur in a structurally stable fashion. next we quote a fundamental result on persistent oscillations; for a proof which uses standard tools such as the implicit function theorem and the hartman-grobman theorem, see sedaghat (2003a). theorem 1. assume that f in eq.(2) satisfies the following conditions: (a) the equation f (x, . . . , x) = x has a finite number of real solutions x̄1 < . . . < x̄k; (b) for i = 1, . . . , m, the partial derivatives ∂fi . = ∂f/∂xi exist continuously at x̄j = (x̄j , . . . , x̄j ), and every root of the characteristic polynomial λm − m∑ i=1 ∂fi(x̄j )λ m−i has modulus greater than 1 for each j = 1, . . . , k; (c) for every j = 1, . . . , k, f (x̄j , . . . , x̄j , x) 6= x̄j if x 6= x̄j . then all bounded solutions of (2) except the trivial solutions x̄j , j = 1, . . . , k, oscillate persistently. if only (a) and (b) hold, then all bounded solutions that do not converge to some x̄j in a finite number of steps oscillate persistently. the next result is the second-order (and sharper) version of theorem 1. corollary 1. consider eq.(2) with m = 2 and f = f (x, y). assume that the following conditions hold : (a) the equation f (x, x) = x has a finite number of solutions x̄1 < . . . < x̄k; (b) fx = ∂f/∂x and fy = ∂f/∂y both exist continuously at (x̄j , x̄j ) for all j = 1, . . . , k, with: |fy(x̄j , x̄j )| > 1 , |fy(x̄j , x̄j ) − 1| > |fx(x̄j , x̄j )| . (c) for every j = 1, . . . , k, f (x̄j , y) 6= x̄j if y 6= x̄j . then all non-trivial bounded solutions oscillate persistently. definition 2. (absorbing intervals) equation (2) has an absorbing interval [a, b] if for every set x0, x−1, . . . x−m+1 of initial values, the corresponding solution {xn} is 92 hassan sedaghat 7, 2(2005) eventually in [a, b]; that is, there is a positive integer n = n (x0, . . . x−m+1) such that xn ∈ [a, b] for all n ≥ n. we may also say that f (or its standard vectorization) has an absorbing interval. in the special case where a > 0, (2) is said to be permanent. remarks. 1. if f in (2) is bounded, then obviously (2) has an absorbing interval. also, if (2) has an absorbing interval, then obviously every solution of (2) is bounded. the converses of these statements are false; the simplest counter-examples are provided by linear maps which are typically unbounded. a straightforward consideration of eigenvalues shows that if f is linear, then an absorbing interval exists if and only if the origin is attracting. on the other hand, if all eigenvalues have magnitude at most one with at least one eigenvalue having magnitude one, then every solution is bounded although there can be no absorbing intervals. an example of a nonlinear mapping that has no absorbing intervals, yet all of its solutions are bounded is the well-known lyness map f (x, y) = (a + x)/y, a > 0; also see theorem 6 below. 2. an absorbing interval need not be invariant, as trajectories may leave it and then re-enter it (to eventually remain there); see corollary 6 below and the remark following it. also an invariant interval may not be absorbing since some trajectories may never reach it. 3. the importance of the concept of permanence in population biology (commonly referred to as “persistence” there) has led to a relatively larger body of results than is available for absorbing intervals in general. these results are also of interest in social science models where the state variable is often required to be positive and in some cases, also bounded away from zero. see kocic and ladas (1993) and sedaghat (2003a) for more examples and details. our next result requires the following lemma which we quote from sedaghat (1997). lemma 1 refers to the first order equation vn+1 = f (vn), v1 = x1 − x0 (3) which with the given initial value relates naturally to (1). lemma 1. let f be non-decreasing. (a) if {xn} is a non-negative solution of (1), then xn ≤ cn−1x0 + n∑ k=1 cn−kvk for all n, where {vn} is a solution of (3). (b) if {xn} is a non-positive solution of (1), then xn ≥ cn−1x0 + n∑ k=1 cn−kvk . theorem 2. let f be non-decreasing and bounded from below on r, and let c < 1. if there exists α ∈ (0, 1) and u0 > 0 such that f (u) ≤ αu for all u ≥ u0, then (1) has a nontrivial absorbing interval. in particular, every solution of (1) is bounded. 7, 2(2005) global attractivity, oscillations and chaos in ... 93 proof. if we define wn . = f (xn − xn−1) for n ≥ 1, then it follows inductively from (1) that xn = c n−1x1 + c n−2w1 + . . . + cwn−2 + wn−1. (4) for n ≥ 2. let l0 be a lower bound for f (u), and without loss of generality assume that l0 ≤ 0. as wk ≥ l0 for all k, we conclude from (4) that xn ≥ cn−1x1 + ( 1 − cn−1 1 − c ) l0 for all n, and therefore, {xn} is bounded from below. in fact, it is clear that there is a positive integer n0 such that for all n ≥ n0, xn ≥ l . = l0 1 − c − 1. we now show that {xn} is bounded from above as well. define zn . = xn+n0 − l for all n ≥ 0, so that zn ≥ 0 for all n. now for each n ≥ 1 we note that zn+1 = cxn+n0 + f (xn+n0 − xn+n0−1) − l = czn + f (zn − zn−1) − l(1 − c) . define g(u) . = f (u) − l(1 − c), and let δ ∈ (α, 1). it is readily verified that g(u) ≤ δu for all u ≥ u1 where u1 . = max { u0, −l(1 − c) δ − α } . if {rn} is a solution of the first order problem rn+1 = g(rn) , r1 = z1 − z0 then since f is bounded from below by l0 − (1 − c)l = 1 − c, we have rn = g(rn−1) ≥ 1 − c for all n ≥ 2. thus {rn} is bounded from below. also, if rk ≥ u1 for some k ≥ 1, then rk+1 = g(rk) ≤ δrk < rk . if rk+1 ≥ u1 also, then δrk ≥ rk+1 ≥ u1 and since g is non-decreasing, rk+2 = g(rk+1) ≤ g(δrk) ≤ δ2rk . it follows inductively that rk+l ≤ δlrk as long as rk+l ≥ u1. clearly there is m ≥ k such that rm < u1. then rm+1 = g(rm) ≤ g(u1) ≤ δu1 < u1 94 hassan sedaghat 7, 2(2005) by the definition of u1. by induction rn < u1 for all n ≥ m. now lemma 1(a) implies that for all such n, zn ≤ cn−1z0 + cn−1r1 + · · · + cn−m+1rm−1 + ∑n k=m c n−krk < cn−m+1(z0cm−2 + · · · + rm−1) + u1 ∑n−m k=0 c k = cn−m+1k0 + u1(1 − c)−1(1 − cn−m+1) . thus there exists n1 ≥ m such that zn ≤ u1 1 − c + 1 for all n ≥ n1. hence, for all n ≥ n0 + n1 we have xn ∈ [l, m ] where m . = u1 1 − c + 1 − l . it follows that [l, m ] is an absorbing interval. there is also the following more recent result which we quote from kent and sedaghat (2003). in contrast to theorem 1, f is not assumed to be increasing in the next theorem, and if f is unbounded from below then it is also unbounded from above. theorem 3. let c < 1 and assume that constants 0 ≤ a < 1 and b > 0 exist such that a 6= (1 − √ 1 − c)2 and |f (t) − at| ≤ b for all t. then (1) has a non-trivial absorbing interval. in particular, all solutions of (1) are bounded. remark. it is noteworthy that both theorems 2 and 3 exclude non-increasing functions, except when f is bounded (at least from above). this is not a coincidence; for example, if f (t) = −at then (1) is linear and all solutions are unbounded if c + 1 2 < a < 1. the next corollary concerns the persistent oscillations of trajectories of (1). corollary 2. in addition to the conditions stated in either theorem 2 or theorem 3, assume that f is continuously differentiable at the origin with f ′(0) > 1. then for all initial values x0, x−1 that are not both equal to the fixed point x̄ = f (0)/(1 − c), the corresponding solution of (1) oscillates persistently, eventually in an absorbing interval [l, m ]. proof. to verify condition (b) in corollary 1, we note that fx(x̄, x̄) = c + f ′(0), fy(x̄, x̄) = −f ′(0) which together with the fact that f ′(0) > 1 > c imply the inequalities in (b). 7, 2(2005) global attractivity, oscillations and chaos in ... 95 as for condition (c) in corollary 1, since f is strictly increasing in a neighborhood of 0, if there is y such that x̄ = f (x̄, y) = cx̄ + f (x̄ − y) then f (0) = f (x̄ − y), so that x̄ − y = 0, as required. we now consider an application to the goodwin-hicks model of the business cycle. this model is represented by the following generalization of samuelson’s linear equation yn = cyn−1 + i(yn−1 − yn−2) + a0 + c0 + g0 (5) where i : r → r is a non-decreasing induced investment function. the terms yn give the output (gdp or national income) in period n and the constants a0, c0, g0 are, respectively, the autonomous investment, the minimum consumption and government input. we assume in the sequel that a0 + c0 + g0 ≥ 0. the number c here is the “marginal propensity to consume” or mpc. it gives the fraction of output that is consumed in the current period. if we define the function f (t) . = i(t) + a0 + c0 + g0, t ∈ r, we see that (5) is a special case of (1). the next definition gives more precise information about that function. definition 3. a goodwin investment function is a mapping g ∈ c1(r) that satisfies the following conditions: (i) g(0) = 0 and g(t) + a0 + c0 + g0 ≥ 0 for all t ∈ r; (ii) g′(t) ≥ 0 for all t ∈ r and g′(0) > 0; (iii) there are constants t0 > 0, 0 < a < 1 such that g(t) ≤ at for all t ≥ t0. the next result is an immediate consequence of corollary 2. it gives specific criteria for persistent oscillations of output trajectories, as is expected of a business cycle. corollary 3. (persistent oscillations) consider the equation yn = cyn−1 + g(yn−1 − yn−2) + a0 + c0 + g0, 0 ≤ c < 1, (6) where g is a goodwin investment function. if g′(0) > 1, then all non-trivial solutions of (6) oscillate persistently, eventually in the absorbing interval [l, t1/(1 − c) + 1], where t1 ≥ t0 is large enough that g(t) + a0 + c0 + g0 ≤ at for t ≥ t1 if g(t) ≤ at for t ≥ t0 and where l = lim t→−∞ g(t) + a0 + c0 + g0 ≥ 0. 96 hassan sedaghat 7, 2(2005) 1.2 other oscillatory behavior here we approach the oscillation problem for (1) at a more general level, without requiring that the oscillatory behavior to be persistent. we begin with the following lemma; it gives conditions that imply a more familiar type of oscillatory behavior than that seen in the preceding sub-section. note that if tf (t) ≥ 0 for all t, then by continuity f (0) = 0 and the origin is the unique equilibrium of (1). lemma 2. if tf (t) ≥ 0 for all t then every eventually non-negative and every eventually non-positive solution of (1) is eventually monotonic. proof. suppose that {xn} is a solution of (1) that is eventually non-negative, i.e., there is k > 0 such that xn ≥ 0 for all n ≥ k. either xn ≥ xn−1 for all n > k in which case {xn} is eventually monotonic, or there is n > k such that xn ≤ xn−1. in the latter case, xn+1 = cxn + f (xn − xn−1) ≤ cxn ≤ xn so that by induction, {xn} is eventually non-increasing, hence monotonic. the argument for an eventually non-positive solution is similar and omitted. the preceding lemma and the first part of the next theorem are taken from sedaghat (2003b). theorem 4. let 0 ≤ c < 1. (a) if tf (t) ≥ 0 for all t, then (1) has no solutions that are eventually periodic with period two. (b) let b = ( 1 − √ 1 − c )2 . if β ≥ α > b and α|t| ≤ |f (t)| ≤ β|t| for all t, then every solution of (1) oscillates about the origin. proof. let {xn} be a solution of (1). we claim that if c > 0 then for all k ≥ 1, xk > 0 > xk+1 implies xk+2 < 0 xk < 0 < xk+1 implies xk+2 > 0 for suppose that xk > 0 > xk+1 for some k ≥ 1. then xk+2 = cxk+1 + f (xk+1 − xk) ≤ xk+1 < 0. the argument for the other case is similar and omitted. now by lemma 2, if a solution {xn} eventually has period 2, then for all sufficiently large n, there is xn > 0, xn+1 ≤ 0 and xn+2 = xn > 0. if c > 0 then this contradicts the above claim. if c = 0 then 0 < xn = xn+2 ≤ f (xn+1 − xn) ≤ 0 which is again a contradiction. hence, no solution of (1) can eventually have period two. (b) see kent and sedaghat (2003) for a proof. 7, 2(2005) global attractivity, oscillations and chaos in ... 97 2 the case c=1 we now consider the case c = 1. this case is substantially different from the case 0 ≤ c < 1 and it is informative to contrast these two cases. also puu’s equation below reduces to this case. first, we note that with c = 1, eq.(1) may be put in the form xn+1 − xn = f (xn − xn−1) (7) from this it is evident that the standard vectorization of (7) is semiconjugate to the real factor f relative to the link map h(x, y) . = x − y; see sedaghat (2003a). further, the solutions {xn} of (1) are none other than the sequences of partial sums of solutions {vn} of (3), since the difference sequence {∆xn} satisfies (3). theorem 5. let {xn} be a solution of the second order equation (7) and let {vn} be the corresponding solution of the first order equation (3). (a) if v∗ is a fixed point of (3) then xn = x0 + v∗n is a solution of (7). (b) if {v1, . . . , vp} is a periodic solution of (3) with period p, then xn = x0 − ωn + vn (8) is a solution of (7) with v = p−1 ∑p i=1 vi the average solution, and ωn = vρn − ρn∑ j=0 vj , (v0 . = 0) where ρn is the remainder resulting from the division of n by p. the sequence {ωn} is periodic with period at most p. proof. part (a) follows immediately from the identity xn = x0 + n∑ i=1 vi (9) which also establishes the fact that solutions of the second order equation are essentially the partial sums of solutions of the first order equation. to prove (b), observe that in (9), after every p iterations we add a fixed sum∑p i=1 vi to the previous total. therefore, since n may generally take on any one of the values pk + ρn, where 0 ≤ ρn ≤ p − 1, we have xn = x0 + k p∑ i=1 vi + ρn∑ j=0 vj . (10) now substituting k = n/p − ρn/p in (10) and rearranging terms we obtain (8). also ωn is periodic since ρn is periodic, and the period of ωn cannot exceed p, since ωpk = 0 for each non-negative integer k. corollary 4. if {vn} is periodic with period p ≥ 1, then the sequence {xn − vn} is also periodic with period at most p. in particular, {xn} is periodic (hence bounded) if and only if v = 0. 98 hassan sedaghat 7, 2(2005) the next result in particular shows that unlike the case c < 1, under conditions implying boundedness of all solutions, (1) typically does not have an absorbing interval. theorem 6. assume that there exists a constant α ∈ (0, 1) such that |f (u)| ≤ α |u| for all u. then every {vn} converges to zero and every {xn} is bounded and converges to a real number that is determined by the initial conditions x0, x1. proof. note that |vn+1| = |f (vn)| ≤ α |vn| for all n ≥ 1. it follows inductively that |vn| ≤ αn |v1|, and hence, n∑ k=1 |vk| ≤ |v1| n∑ k=1 αk ≤ α |v1| 1 − α which implies that the series ∑∞ n=1 |vn| converges. it follows at once that {vn} must converge to zero and that {xn} is bounded and in fact converges to the real number x0 + ∑∞ n=1 vn. 3 complex behavior under the conditions of corollary 3, the unique equilibrium x̄ = a0 + c0 + g0 1 − c of (6) is repelling (or expanding) but it is not a snap-back repeller (see marotto, 1978 or sedaghat, 2003a for a definition). this is due to condition (c) in corollary 1. indeed, with a goodwin function numerical simulations tend to generate quasiperiodic rather than chaotic trajectories. however, if we do not assume that f is increasing, more varied and complex types of behavior are possible. this is the case in puu’s model, which we describe next. 3.1 chaos and puu’s model the number s = 1 − c ∈ (0, 1] is called the marginal propensity to save, or mps for short. in each period n, a percentage of income syn is saved in the samuelson-hicksgoodwin models and is never consumed in future periods hence, savings are said to be “eternal.” at the opposite extreme, we have the case where the savings of a given period are consumed entirely within the next period (puu, 1993, chapter 6). puu suggested an investment function in the form of a cubic polynomial q seen in the following type of difference equation yn = (1 − s)yn−1 + syn−2 + q(yn−1 − yn−2) (11) where q is the cubic polynomial q(t) . = at(1 − bt − t2), b > 0, a > s. puu took b = 0 (which makes q symmetric with respect to the origin); but as we will see later, this 7, 2(2005) global attractivity, oscillations and chaos in ... 99 restriction is problematic (see the remarks on growth and viability below). equation (11) may alternatively be written as follows yn = yn−1 + p (yn−1 − yn−2) (12) in which we call the (still cubic) function p (t) . = q(t) − st = t(a − s − abt − at2) puu’s (asymmetric) investment function. note that (12) is of the form (1) with c = 1 as in the preceding section. therefore, each solution {yn} of (12) is expressible as the series yn = y0 + ∑n−1 k=0 zk where {zn} is a solution of the first order initial value problem zn = p (zn−1), z0 . = y1 − y0. (13) each term zn is just the forward difference yn+1 − yn, and a solution of (13) gives the sequence of output or income differences for (12). in order to study the dynamics of equations (12) and (13), we gather some basic information about p. using elementary calculus, it is easily found that the real function p has two critical points ξ± = 1 3 [ −b ± √ b2 + 3 ( 1 − s a )] with ξ− < 0 < ξ+. similarly, p has three zeros, one at the origin and two more given by ζ± = 1 2 [ −b ± √ b2 + 4 ( 1 − s a )] . further, if a > s + 1, then for all b > 0, p has three fixed points, one at the origin and two more given by t± = 1 2 [ −b ± √ b2 + 4 ( 1 − s + 1 a )] . remarks. (growth and viability criteria) assume that the following inequalities hold: 0 < p 2(ξ−) ≤ p (ξ+). (14) then it is not hard to see that p (ξ−) < ζ− and that the interval i . = [p (ξ−), max{ζ+, p (ξ+)}] is invariant under p . we refer to inequalities (14) as the viability criteria for puu’s model as they prevent undesirable outcomes such as negative income. see figure 1. next, suppose that p (ξ+) ≤ ζ+, or equivalently, p 2(ξ+) ≥ 0. (15) 100 hassan sedaghat 7, 2(2005) figure 1: a viable puu investment function in this case, the right half i+ . = [0, ζ+] of i is invariant under p , and it follows that the income sequence {yn} is eventually increasing. for this reason, we refer to condition (15) as the steady growth condition. the function depicted in figure 1 satisfies both the steady growth and the viability criteria. the proofs of (a) and (b) in the next corollary follow from theorem 5. for a proof of the rest and some examples, see sedaghat (2003a). corollary 5. (steady growth) assume that inequalities (14) and (15) hold. also suppose that a > s + 1 and y0 − y−1 ∈ i+. then the following statements are true: (a) if p ′(t+) < 1, then each non-constant solution {yn} of (12) is increasing and the difference |yn − t+n| approaches a constant as n → ∞. (b) if p ′(t+) > 1 and {v1, . . . , vk} is a limit cycle of (13), then each non-constant solution {yn} of (12) is increasing and the difference |yn − v̄n| approaches a periodic sequence {ωn} of period at most k, where v̄ . = 1 k k∑ i=1 vi, ωn . = α + v̄ρn − ρn∑ i=0 vi, (v0 . = 0) with α a constant, and ρn the remainder resulting from the division of n by k. (c) if p has a snap-back repeller (e.g., if it has a 3-cycle) then each non-constant solution {yn} of (12) is increasing, the corresponding difference sequence {∆yn} is bounded, and for an uncountable set of initial values, chaotic. 7, 2(2005) global attractivity, oscillations and chaos in ... 101 remarks. 1. the preceding result shows that unlike the hicks-goodwin model, puu’s model (and hence, (1) with c = 1) is capable of generating endogenous growth (i.e., without external input). under the conditions of corollary 5(c), this growth occurs at an unpredictable rate. the implication that the existence of a 3-cycle implies chaotic behavior was first established in the well-known paper li and yorke (1975). the existence of 3-cycles implies the existence of snap-back repellers (marotto, 1978). figure 2 shows a situation where the fixed point is a snap-back repeller because it is unstable yet a nearby point t0 moves into it. figure 2: a snap-back repeller in puu’s investment chaotic behavior may be observed in the output trajectory {yn} itself and not just in its rate sequence. for instance, if inequalities (14) hold but (15) does not, then i is invariant but not i+. hence, ∆yn is negative (and positive) infinitely often, and sustained growth for {yn} either does not occur, or if it occurs over longer stretches of time, it will not be steady or strict. see sedaghat (2003a) for an example of this situation. 3.2 strange behavior going in a different direction, note that by (i) and (ii) in definition 3 a goodwin function can exist only if a0 + c0 + g0 > 0. to study the consequences of the equality a0 + c0 + g0 = 0, we replace (ii) in definition 3 by: (ii)′ h is non-decreasing everywhere on r, and it is strictly increasing on an interval (0, δ) for some δ > 0; here we are using h rather than g to denote the more general type of investment function that (ii)′ allows. the next corollary identifies an important difference between the smooth and non-smooth cases. 102 hassan sedaghat 7, 2(2005) corollary 6. (economic ruin) assume that a0 + c0 + g0 = 0. then every solution of yn = cyn−1 + h(yn−1 − yn−2) (16) converges to zero, eventually monotonically. moreover, the following is true: (a) if h ∈ c1(r), then h′(0) = 0 and the origin is locally asymptotically stable. thus, the income trajectory stays near the origin if the initial income difference is sufficiently small. (b) if h(t) = bt on an interval (0, r) for some r > 0 and b ≥ ( 1 + √ 1 − c )2 (17) then the origin is not stable. if 0 = y−1 < y0 < r then the income trajectory {yn} is increasing, moving away from the origin until yn − yn−1 > r, no matter how close y0 is to zero. proof. if a0 + c0 + g0 = 0, then h(t) = 0 for t ≤ 0. now, there are two possible cases: (i) some solution {yn} of (16) is strictly increasing as n → ∞, or (ii) for every solution there is k ≥ 1 such that yk−1 ≥ yk. case (i) cannot occur for positive solutions, since by theorem 2 the increasing trajectory has a bounded limit ỹ with ỹ = lim n→∞ [cyn−1 + h(yn−1 − yn−2)] = cỹ + h(0) = cỹ, which implies that ỹ = 0. for y−1, y0 < 0 the sequence {yn} is increasing since yn+1 = cyn + h(yn−1 − yn−2) ≥ cyn > yn as long as yn remains negative. thus either yn → 0 as n → ∞, or yn must become positive, in which case the preceding argument applies. in case (ii), we find that yk+1 = cyk < yk so that, proceeding inductively, {yn} is strictly decreasing for n ≥ k. since the origin is the only fixed point of (16), it follows that yn → 0 as n → ∞. next, suppose that (a) holds. if h(t) is constant for t ≤ 0, then h′(t) = 0 for t < 0, and thus, h′(0) = 0 if h′ is continuous. thus the linearization of (16) at the origin has eigenvalues 0 and c, both with magnitude less than 1. now assume that (b) is true. on the interval (0, r), a little algebraic manipulation shows that due to condition (17), the eigenvalues λ1 and λ2 of the linear equation yn+1 = cyn + b(yn − yn−1) = (b + c)yn − byn−1 (18) are real and that 0 < λ1 = b + c − √ (b + c)2 − 4b 2 < 1 < λ2 = b + c + √ (b + c)2 − 4b 2 . 7, 2(2005) global attractivity, oscillations and chaos in ... 103 with initial values y−1 = 0 and y0 ∈ (0, r), the corresponding solution of the linear equation (18) is yn = y0√ (b + c)2 − 4b ( λn+12 − λ n+1 1 ) which is clearly increasing exponentially away from the origin, at least until yn (hence also the difference yn −yn−1 since y0 −y−1 = y0) exceeds r and h assumes a possibly different form. the instability of the origin is now clear. remark. (unstable global attractors) under conditions of corollary 6(b), the origin is evidently a globally attracting equilibrium which however, is not stable. this is a consequence of the non-smoothness of the hicks-goodwin map at the origin, since in part (a), where the map h is smooth, the origin is indeed stable. remark. (off-equilibrium oscillations) note that the mapping h(t) of corollary 6(b) has its minimum value at the origin, which is also the unique fixed point or equilibrium of the system. if the mapping f in (1) is characterized by this property, then solutions of (1) are in general capable of exhibiting other types of strange behavior that do not occur with non-decreasing maps of type h. suppose that f has a global (though not necessarily unique) minimum at the origin and without loss of generality, assume that f (0) = 0. then the origin is the unique fixed point of (1). clearly, if (1) exhibits oscillatory behavior in this case, then such oscillations occur off-equilibrium, i.e., they do not occur about the equilibrium or fixed point. in particular, if xn is a solution exhibiting such oscillations, then its limit superior is distinct from its limit inferior. as a very simple example of this sort of oscillation, it is easy to verify that with f (t) = min{|t|, 1}, c = 0 (1) has a periodic solution {0, 1, 1} exhibiting off-equilibrium oscillations with limit superior 1 and limit inferior 0. however, off-equilibrium oscillations can generally be quite complicated (and include chaotic behavior) with non-monotonic f ; see the example of non-monotonic convergence after conjecture 1 below. for examples not involving convergence see sedaghat (2003a). 4 global attractivity in this section we consider various conditions that imply the global attractivity of the unique fixed point of (1), namely, x̄ = f (0)/(1 − c). throughout this section it is assumed that 0 ≤ c < 1. we begin with a condition on f under which the origin is globally asymptotically stable. we need a result from sedaghat (1998) which we quote here as a lemma. lemma 3. let g : rm → r be continuous and let x̄ be an isolated fixed point of xn+1 = g(xn, xn−1, . . . , xn−m). 104 hassan sedaghat 7, 2(2005) also, assume that for some α ∈ (0, 1) the set aα = {(u1, . . . , um) : |g(u1, . . . , um) − x̄| ≤ α max{|u1 − x̄|, . . . , |um − x̄|} has a nonempty interior (i.e., g is not very steep near x̄) and let r be the largest positive number such that [x̄−r, x̄ + r]m ⊂ aα. then x̄ is exponentially stable relative to the interval [x̄ − r, x̄ + r]. the function g in lemma 3 is said to be a weak contraction on the set aα; see sedaghat (2003a) for further details about weak contractions and weak expansions. as a corollary to lemma 3 we have the following simple, yet general fact about equation (1). theorem 7. if |f (t)| ≤ a|t| for all t and 0 < a < (1−c)/2 then the origin is globally attracting in (1). proof. the inequality involving f in particular implies that f (0) = 0, so that the origin is the unique fixed point of (1). define g(x, y) = cx + f (x − y) and notice that |g(x, y)| ≤ c|x| + a|x − y| ≤ (c + a)|x| + a|y| ≤ (c + 2a) max{|x|, |y|}. since c + 2a < 1 by assumption, it follows that g is a weak contraction on the entire plane and therefore, lemma 3 implies that the origin is globally attracting (in fact, exponentially stable) in (1). remark. if f (0) = 0 and f is continuously differentiable with derivative bounded in magnitude by a or more generally, if f satisfies the lipschitz inequality |f (t) − f (s)| ≤ a|t − s| then in particular (with s = 0), |f (t)| ≤ a|t| for all t. however, if f satisfies the conditions of theorem 7 then it need not satisfy a lipschitz inequality. theorems 8 and 9 below improve the range of values for a in theorem 7 with the help of extra hypotheses. 4.1 when f is minimized at the origin in this sub-section we look at the case where f has a global minimum (not necessarily unique) at the origin. these types of maps were noted in the previous section when remarking on off-equilibrium oscillations. it will not be any loss of generality to assume that f (0) = 0 in the sequel, so that x̄ = 0. this will simplify the notation. we begin with a simple result about the non-positive solutions. lemma 4. if f (t) ≥ 0 for all t and f (0) = 0 then every non-positive solution of (1) is nondecreasing and converges to zero. given lemma 4 and the fact that if xk > 0 for some k ≥ 0 then xn > 0 for all n ≥ k, it is necessary to consider only the positive solutions. before stating the main 7, 2(2005) global attractivity, oscillations and chaos in ... 105 result of this section, we need another version of lemma 3 above which we quote here as a lemma. see sedaghat (1998) or sedaghat (2003a) for a proof. lemma 5. let g : rm → r be continuous and let x̄ be an isolated fixed point of xn+1 = g(xn, xn−1, . . . , xn−m). let vg(u1, . . . , um) = (g(u1, . . . , um), u1, . . . , um−1) and for α ∈ (0, 1) define the set aα = {(u1, . . . , um) : |g(u1, . . . , um) − x̄| ≤ α max{|u1 − x̄|, . . . , |um − x̄|} if s is a subset of aα such that vg(s) ⊂ s and (x̄, . . . , x̄) ∈ s, then (x̄, . . . , x̄) is asymptotically (in fact, exponentially) stable relative to s. the next theorem is from sedaghat (2003c). theorem 8. let 0 ≤ f (t) ≤ a|t| for all t. (a) if a < 1 − c, then every positive solution {xn} of (1) converges to zero. (b) if a < c then every positive solution {xn} of (1) eventually decreases monotonically to zero. (c) if a < max{c, 1 − c} then the origin is globally attracting. proof. (a) assume that a < 1 − c. define g(x, y) = cx + f (x − y) and for x, y ≥ 0 notice that g(x, y) ≤ cx + a|x − y| ≤ cx + a max{x, y} ≤ (c + a) max{x , y}. since c + a < 1 by assumption, it follows that g is a weak contraction on the non-negative quadrant, i.e., [0, ∞)2 ⊂ aa+c. since [0, ∞)2 is invariant under g, lemma 5 implies that the origin is exponentially stable relative to [0, ∞)2. thus every positive solution {xn} of (1) converges to zero. (b) let {xn} be a positive solution of (1). then the ratios rn = xn xn−1 , n ≥ 0 are well defined and satisfy rn+1 = c + f (xn − xn−1) xn ≤ c + a|xn − xn−1| xn = c + a ∣∣∣∣1 − 1rn ∣∣∣∣ . since it is also true that rn+1 = c + f (xn − xn−1)/xn ≥ c we have c ≤ rn+1 ≤ c + a ∣∣∣∣1 − 1rn ∣∣∣∣ , n ≥ 0. 106 hassan sedaghat 7, 2(2005) if r1 ≤ 1 then since r1 ≥ c, we have c ≤ r2 ≤ c − a + a r1 ≤ c − a + a c < 1 where the last inequality holds because a < c < 1. inductively, if for k ≥ 2, c ≤ rn < 1, n < k then c ≤ rk ≤ c − a + ac < 1 so that r1 ≤ 1 ⇒ rn < 1 for all n > 1. (19) now suppose that r1 > 1. then c ≤ r2 ≤ c + a − a r1 < c + a. if c + a ≤ 1, then r2 < 1 and (19) holds. assume that a + c > 1 and r2 > 1. then r3 ≤ c + a − a r2 < r2. the last inequality holds because for every r > 1, c + a − a/r < r if and only if r2 − (c + a)r + a > 0. (20) inequality (20) is true because the quadratic on its left side can have zeros only for r ≤ 1. now, if r3 < 1, then (19) holds for n > 2. otherwise, using (20) we can show inductively that r1 > r2 > r3 > · · · so there is k ≥ 1 such that rk ≤ 1 and (19) applies with n > k. hence, we have shown that for any choice of r0, the sequence rn is eventually less than 1; i.e., xn < xn−1 for all n sufficiently large and the proof is complete. (c) immediate from parts (a) and (b) above and lemma 4. figure 3: global attractivity regions when f is minimized at 0 figure 3 shows the parts (shaded) of the unit square in the (c, a) parameter space for which global attractivity is established so far.the diagonal lines represent a = 1−c 7, 2(2005) global attractivity, oscillations and chaos in ... 107 and a = c. the horizontal line a = b in the middle of the diagram comes from sedaghat (2003c) where it is shown that if 1 − b < a, c < b, b = 2/( √ 5 + 1), then the origin is globally asymptotically stable. numerical simulations indicate that the origin is possibly attracting for points in the unshaded region of figure 3 also. therefore, we state the following: conjecture 1. if 0 ≤ f (t) ≤ a|t| for all t, then the origin is globally attracting for all points of the unit square in the (c, a) parameter space with a, c 6= 1. the next example shows that convergence in part (a) of theorem 8 need not be monotonic. example. let c < 1/2 and let f (t) = a|t| with c < a < 1 − c. by theorem 8 every solution {xn} of (1) converges to zero. let rn be the ratio defined in the proof of theorem 8(b) and define the mapping φ(r) = c + a ∣∣∣∣1 − 1r ∣∣∣∣ , r > 0. then rn+1 = φ(rn) for all n ≥ 0 and φ has a unique positive fixed point r̄ = 1 2 [√ (a − c)2 + 4a − (a − c) ] < √ a which is unstable because |φ′(r̄)| = a/ r̄2 > 1. suppose that r0 < 1, i.e., x0 < x−1 but r0 6= r̄. then r1 = φ(r0) = φ1(r0) where φ1 is the decreasing function φ1(r) = c − a + a r . since φ1(r) > 1 for r ∈ (0, r∗) where r∗ = a/(1 + a − c), it follows that either r1 > 1 or some iterate rk = φk1 (r0) > 1. this means that xk > xk−1 while x1 > x2 > · · · > xk−1. next, rk+1 = φ2(rk) where φ2 is the increasing function φ2(r) = a + c − a r . since φ2(r) < a + c < 1 for all r we see that rk+1 < 1 and so the preceding process repeats itself ensuring that there are infinitely many terms xkj , j = 1, 2, . . . where the inequality xkj +1 > xkj holds. the magnitude of the up-jump depends on the parameters; since c is the absolute minimum value of φ for r > 0, we see that rn ≤ φ(c) = φ1(c) = a c − (a − c) for all n ≥ 1. thus xkj < xkj +1 < [ a c − (a − c) ] xkj , j = 1, 2, . . . the difference kj+1−kj need not be a constant function of j. indeed, if a+c > 1/2, then it can be shown that φ1 has a snap-back repeller at r̄ for a certain range of values of a (see sedaghat, 2003c). in such a case the ratios rn exhibit chaotic behavior. 108 hassan sedaghat 7, 2(2005) 4.2 when tf (t) ≥ 0 the situation we discuss in this sub-section is, in a sense, complementary to the one we considered in the preceding sub-section. however, as seen below there are some interesting parallels between the two cases. the condition tf (t) ≥ 0 encountered in the section other oscillatory behavior above, has significant consequences in the case of convergence too. we note that if tf (t) ≥ 0, then by continuity f (0) = 0 so the origin is the unique fixed point of (1) in this case. the following is proved in kent and sedaghat (2003). theorem 9. (a) assume that there is a > 0 such that |f (t)| ≤ a|t| and that tf (t) ≥ 0 for all t. if a < 2 − c 3 − c or a ≤ 1 − c then every solution of (1) converges to zero; i.e., the origin is globally attracting. (b) let b = ( 1 − √ 1 − c )2 . if a ≤ b in part (a), then every solution of (1) is eventually monotonic and converges to zero. the conditions of theorem 9 specify the shaded region of the (c, a) parameter space shown in figure 4 below. figure 4: global attractivity regions when tf (t) ≥ 0 the line a = 1 − c and the curve a = (2 − c)/(3 − c) are readily identified (the latter clearly by its intercepts with c = 0 and c = 1). the third curve represents a = b where b is defined in theorem 9(b). figure 4 is analogous to figure 3 and it leads to the following conjecture which is analogous to conjecture 1. conjecture 2. if f (t) ≤ a|t| and tf (t) ≥ 0 for all t, then the origin is globally attracting for all points of the unit square(a, c 6= 1) in the (c, a) parameter space. 7, 2(2005) global attractivity, oscillations and chaos in ... 109 5 concluding remarks and open problems the preceding sections shed some light on the problem of classifying the solutions of (1). however, there are also many unresolved issues, some of which are listed below as open problems and conjectures (they all refer to the case 0 ≤ c < 1): conjecture 3. all solutions of (1) are bounded if and only if (1) has an absorbing interval. problem 1. find sufficient conditions on f that imply (1) has an absorbing interval when f is minimized at the origin. conjecture 4. related to problem 1 where f (t) ≥ f (0), (1) has an absorbing interval if there is a ∈ (0, 1) and t0 > 0 such that f (t) ≤ a|t| for |t| > t0 problem 2. find sufficient conditions on f for the fixed point x̄ = f (0)/(1 − c) to be a snap-back repeller. together with theorem 3, this establishes the occurrence of chaotic behavior in a compact set. conjecture 5. related to problem 2, if f (t) is non-decreasing then every solution of (1) is either periodic or almost (or quasi) periodic. problem 3. find either sufficient conditions on f, or specify some classes of functions f for which every solution of (1) is eventually periodic or every solution approaches a periodic solution for a range of values of c. problem 4. investigate the consequences of f being an even function, i.e., f (−t) = f (t) in the case where f is minimized at zero. similarly, when tf (t) ≥ 0, what is the significance of the oddness of f (i.e., f (−t) = −f (t)) for the asymptotic behavior of solutions? problem 5. extend the results of this paper, where possible, to the more general difference equation xn+1 = cxn + dxn−1 + f (xn − xn−1), c, d ∈ [0, 1]. puu’s general model is a special case of this equation with c + d ≤ 1. received: march 2003. revised: july 2003. references [1] elaydi, s.n., an introduction to difference equations, 2nd ed., springer, new york.(1999). 110 hassan sedaghat 7, 2(2005) [2] goodwin, r.m., the nonlinear accelerator and the persistence of business cycles, econometrica, 19(1951), 1-17. [3] hicks, j.r., a contribution to the theory of the trade cycle, 2nd ed.(1950), clarendon press, oxford 1965(first ed. oxford university press, [4] kent, c.m. and h. sedaghat, boundedness and global attractivity in xn+1 = cxn + f (xn − xn−1), j. diff. eq. appl., to appear. [5] kocic, v.l. and g. ladas, global behavior of nonlinear difference equations of higher order with applications, kluwer, dordrecht. (1993). [6] lasalle, j.p., stability and control of discrete processes, springer, new york (1986). [7] li, t.y. and j.a. yorke, period three implies chaos, amer. math. monthly 82(1975), 985-992. [8] marotto, f.r., snap-back repellers imply chaos in rn, j. math. anal. appl. 63(1978), 199-223. [9] puu, t., nonlinear economic dynamics, 3rd ed., springer, new york. (1993). [10] samuleson, p.a., interaction between the multiplier analysis and the principle of acceleration, rev. econ. stat. 21(1939), 75-78. [11] sedaghat, h., a class of nonlinear second order difference equations from macroeconomics, nonlinear anal. tma 29(1997), 593-603. [12] sedaghat, h., geometric stability conditions for higher order difference equations, j. math. anal. appl. 224(1998), 255-272. [13] sedaghat, h., (2003a) nonlinear difference equations: theory with applications to social science models, kluwer, dordrecht. [14] sedaghat, h. , (2003b) on the equation xn+1 = cxn + f (xn − xn−1), proceedings of the 7th international conference on difference equations and applications, changsha, (changsha, pr china), fields institute communications, 42 (2004), 323-326. [15] sedaghat, h., (2003c) the global stability of equilibrium in a nonlinear second order difference equation, international journal of pure and applied mathematics, 8 (2003), 209-223. articulo 17.dvi cubo a mathematical journal vol.12, no¯ 02, (275–298). june 2010 the maxwell problem and the chapman projection1 v. v. palin, e. v. radkevich 2 department of mech.-math., moscow state university, moscow 119899, vorobievy gory, russia. email: evrad07@gmail.com abstract we study the large-time behavior of global smooth solutions to the cauchy problem for hyperbolic regularization of conservation laws. an attracting manifold of special smooth global solutions is determined by the chapman projection onto the phase space of consolidated variables. for small initial data we construct the chapman projection and describe its properties in the case of the cauchy problem for moment approximations of kinetic equations. the existence conditions for the chapman projection are expressed in terms of the solvability of the riccati matrix equations with parameter. resumen nosotros estudiamos el comportamiento temporal de soluciones globales suaves del problema de cauchy para regularización hiperbólica de leyes de conservación. una variedad atractora de soluciones globales suaves es determinada por la proyección de chapman sobre el espacio de fase de las variables consolidadas. para datos iniciales pequeños nosotros construimos la proyección de chapman y descubrimos sus propiedades en el caso del problema de cauchy para aproximación de momentos en ecuaciones kineticas. las condiciones de existencia para la proyección de chapman son expresadas en términos de la solubilidad de las ecuaciones matriciales de riccati con parámetros. 1this work was supported by the russian foundation of basic researches (grant no. 09-01-00288) 2corresponding author 276 v. v. palin and e. v. radkevich cubo 12, 2 (2010) key words and phrases: closure, the state equation, the chapman projection, matrix equation, dynamic separation, inertional manifold ams (mos) subj. class.: udc 517.9 1 introduction 1.1 the state equation. closure this paper is devoted to mathamatical aspects of the maxwell problem [2] about the derivation of the navier-stokes equation from kinetics. following [1] we study the behavior of solutions to the cauchy problem for hyperbolic regularizations of conservation laws or (in another terminology) for systems of conservation laws with relaxation. consider m− conservation laws (1.1) and n − m conservation laws with relaxation (1.2): ∂tui + divx f i(u, v) = 0, i = 1, . . . , m, (1.1) ∂tvk + divx g k(u, v) + bk(u)v = 0, k = m + 1, . . . , n. (1.2) then we have m conservative variables u(x, t) : rd × r+ → rm and n − m co-called nonequilibrium variables v(x, t) : rd × r+ → rn−m, where x ∈ rd, b is the relaxation (n − m) × (n − m)-matrix, f i(u, v) ∈ rd, i = 1, . . . , m; gk(u, v) ∈ rd, k = 1, . . . , n − m, are currents. the leading part of the system (1.1) is nonstrictly hyperbolic in the sense of the following definition. definition 1.1. a system is nonstrictly hyperbolic if the characteristic matrix τ e + ξ · ( fu(u, v) fv(u, v) gu(u, v) gv(u, v) ) (1.3) has only real (possibly multiple) roots τ = τj (ξ, u, v), j = 1, . . . , n . the condition in definition 1.1 is satisfied if the system (1.1) is simmetrizable. examples of such systems are the following: moment approximations of kinetic equations and the dirac-schwinger extension of the maxwell equations [3]. hyperbolic regularizations of conservation laws (or systems of conservation laws with relaxation) were considered by many authors. first of all, this concerns the study of the relaxation phenomenon, in particular, the stability and singular limit as the relaxation time tends to zero (cf., for example, [4]-[7]). the so-called ”intermediate attractor” for (1.1-1.2) was studied in connection with the maxwell problem (cf. [8,9]). to derive equations of hydrodynamics from the kinetic gas theory, it is important to find a simple functional dependence of the transport coefficients on the interaction potential and thereby to simplify the analysis of the equations under consideration. we intereste in the chapman conjecture [1], [9], about the existence problem of state equation v = qu, (1.4) (so-called the chapman state equation or the chapman projection) expressing the nonequillibrium variables in terms of the conservative variables (the projection into the phase space of conservative cubo 12, 2 (2010) the maxwell problem and the chapman projection 277 variables), where q is an operator with respect to space variables x. this equation completes the system of conservation laws ∂tw + ∂xf (w, qu(w)) = 0. (1.5) so that the solutions w to the caushy problem of the corresponding closer (1.5) define the set of the special solutions uchens = {u = w, v = qw} to the cauchy problem for the system (1.1-1.2) form an invariant attracting manifold mchens, called an intermediate attractor. in other words, for any solution u = (u, v) to the cauchy problem for the system (1.1-1.2) with the initial data u|t=0 = (u0, v0) it is possible to choose initial data w0 = t (u0, v0) for the closure (1.5) in such a way that some norm of the difference u − uchens between u and the special solution uchens = (w, qw) tends to zero as t → ∞. moreover, if, in the phase space of conservative variables, w → 0, when t → ∞, then uh tends to zero faster than uchens. we can say that in this case the influence of nonequilibrium veriables is inessential(we have the separation of dynamics) [9]. now we can define the approximation of the state equation and corresponding closure(so-called navier-stokes approximation). due to physical point of view [9] we assume that derivatives of nonequilibrium variables are small, then we find the following relation v = −b−1 divx g(u, 0) (1.6) and the corresponding closure ∂tu + ∂xf (u, −b−1 divx g(u, 0)) = 0 (1.7) (so-called navier-stokes approximation to (1.1-1.2)), where det b(u) 6= 0. for thirteen-moment grad system to the boltzman kinetic equation the navier-stokes approximation (1.7) is the navier-stokes equations exactly. considering conservation laws with stiff relaxation ∂tu + divx f (u, v) = 0, ∂tv + divx g(u, v) + 1 ε b(u)v = 0, (1.8) we find that the navier-stokes approximation v = −εb−1 divx g(u, 0), ∂tu + divx f (u, −εb−1 divx g(u, 0)) = 0, (1.9) is the first approch to so-called local equilibrium approch(see [1]) ∂tu + divx f (u, 0) = 0 2 linear analysis. reduction to a quadratic matrix equation. 2.1 reduction to a quadratic matrix equation we consider the cauchy problem for the first order linear hyperbolic system with constant coefficients and with relaxation [1] ∂tu + aj ∂xj u + bu = 0 (2.1) 278 v. v. palin and e. v. radkevich cubo 12, 2 (2010) where x ∈ rn, u ∈ rn , aj and b are constant matrices. in the case of the system (2.1), the chapman conjecture [1,9] of the state equation existence asserts that u = π uc = (uc, π21uc), where uc = (u1, .., um, 0, .., 0) t and π is a zero order pseudodifferential matrix operator. suppose following to [1] that the matrix of operator π corresponding to the chapman-enskog projection into m equations of the system (2.1) has the form π = ( π11 π12 π21 π22 ) where π11 = em is the unit matrix of order m and π22 = 0n−m is the zero square matrix of order n − m. we denote by λ(ξ) the resolvent matrix ∑n j=1 aiξj + b and represent it in the block form: λ = ( λ11 λ12 λ21 λ22 ) . since π is a projection, π∂tuc + aπ∂xuc + bπ uc = 0, (2.2) since π2 = π, π∂tuc + πaπ∂xuc + π bπ uc = 0. (2.3) subtracting (2.3) from (2.2), we find (e − π)(a∂x + b)π uc = 0 we denote by p the fourier image of π with respect to x. after the fourier transform with respect to x, the last equality takes the form (e − p )λ p vc = 0, i.e. λ p vc ∈ ker(e − p ). we note that for ∀v ∈ ker(e − p ) admitting the representation vt = (vtm, vtn−m), with vk ∈ ck the following equality holds: vn−m = p21vm. hence we find the system of equations for p21 which completely determines the projection π: p21(λ11 + λ12p21) = λ21 + λ22p21. after transformations this equation takes the form p21λ12(ξ)p21 − λ22(ξ)p21 + p21λ11(ξ) − λ21(ξ) = 0, (2.4) i.e., we obtain a riccati type matrix equation. this object is nontrivial object. for example, we will consider two special 2 × 2 cases (2.4): x2 = 0, x2 = ( 0 1 0 0 ) there are infinitely many such matrices in the first case, and they form two-dimensional cone in c4(det x = 0, trx = 0). there are no solutions to the second equation, since a matrix has only cubo 12, 2 (2010) the maxwell problem and the chapman projection 279 the zero eigenvalue if the squared matrix possesses this property, i.e. x is nilpotent and the squared nilpotent matrix of second order vanishes. lemma 3.1. for any γ ∈ r the set of solutions to the matrix equation (2.4) with a matrix λ coincides with the set of solutions to the same equation (2.4) with the matrix è λ + γ e. proof. indeed, with λ + γ e we associate the matrix equation p21λ12p21 − (λ22 + γ e)p21 + p21(λ11 + γ e) − λ21 = 0, where the left-hand side differs from the left-hand side of (2.4) by −γ ep21 + p21γ e = 0. it is obvious that the sets of solutions to these equations coincide. thus, to study the matrix equation (2.4), we can assume without loss of generality that det(λ) 6= 0. 2.2 solutions to the quadratic matrix equation in the case |λ| 6= 0 this section is devoted to the solvability condition for the matrix equation. proposition 3.1. assume that |λ| 6= 0 and p = ( p11 p12 p21 p22 ) , (2.5) where p11 is the unit matrix of order m, p22 is the zero square matrix of order n − m, and p12 is the zero matrix. then the quadratic matrix equation (2.4) is solvable if and only if there exists a matrix p of the form (2.5) such that p is a solution to the quadratic matrix equation (e − p )λ p = 0. (2.6) proof. we first assume that the matrix equation (2.4) is solvable. taking p of the form (2.5) and representing the product (e − p )λ p in the block form, we see that p is a solution to the matrix equation (2.6). conversely, let p of the form (2.5) be a solution to the matrix equation (2.6). representing m = (e − p )λ p via blocks of the same size as the blocks of p , we see that the blocks m11, m12, and m22 are zero. the equation for m21 coincides with (2.4) up to a sign, i.e., the matrix equation (2.4) is solvable. as a consequence it follows theorem 3.1. let a matrix π21 be a solution π21λ12π21 − λ22π21 + π21λ11 − λ21 = 0 (2.7) and x = λ π, where π = ( π11 π12 π21 π22 ) is a quadratic matrix of order n , π11 is the identity matrix of order m, and π12, π22 are zero matrices. then x is a solution to the quadratic matrix equation x2 − λ x = 0. (2.8) 280 v. v. palin and e. v. radkevich cubo 12, 2 (2010) the matrix equation (2.8) is simpler than the general matrix equation and it’s not difficulty to describe one completely. solutions of the matrix equation (2.4) correspond to a part of the set of solutions for the equation (2.8) only. so that we must to define the selection rule. theorem 3.2. let |λ| 6= 0. then the quadratic matrix equation (2.4) is solvable if and only if there are two solutions x1 and x2 to the quadratic matrix equation x2 − λ x = 0, (2.9) such that 1. x1ej = 0 for all j > m. 2. etj x2 = e t j λ for all j ≤ m. 3. λ x2 = x1λ. proof. assume that the matrix equation (2.4) is solvable. then the matrix equation (2.6) is also solvable. we note that a matrix p belongs to the above class if and only if p ej = 0 ∀j > m and etj p = e t j ∀j ≤ m. multiplying the matrix equation (2.6) by λ from the left and making the change of variables x1 = λ p , we see that the matrix x1 is a solution to (2.9) and satisfies condition 1). similarly, multiplying (2.6) by λ from the right and making the change of variables x2 = p λ, we find that the matrix x2 is a solution to (2.9) and satisfies condition 2). since x1 = λ p and x2 = p λ condition 3) is also valid. assume that there exist two solutions x1 and x2 to the matrix equation (2.9) satisfying conditions 1)-3). we set p = λ−1x1 = x2λ −1. then the matrix p has the required form because of conditions 1) and 2). substituting x1 = λ p into (2.9) and multiplying by λ −1 from the left, we find that p is a solution to (2.6). theorem 3.3. let |λ| 6= 0. then the quadratic matrix equation (2.4) is solvable if and only if there is a solution x1 to the quadratic matrix equation (2.9) such that 1. x1ej = 0 for all j > m. 2. etj λ −1x1 = e t j for all j ≤ m. proof. we set x2 = λ −1x1λ. it is obvious that x2 is a solution to the matrix equation (2.9). furthermore, x2 satisfies condition 2) of theorem 3.2 because of condition 2) of theorem 3.3. since x2 = λ −1x1λ we have λ x2 = x1λ, i.e., condition 3) of theorem 3.2 is also satisfied. the proof details of next results look for in [14,17] lemma 3.1. suppose that det(λ) 6= 0, x is a solution to the matrix equation (2.9), and vectors h1, . . . , hn form the jordan basis for x. then there exists k ≥ 0 such that h1, . . . , hk belong to the jordan basis for λ(moreover, if xhj = λ hj + hj−1, then λ hj = λ hj + hj−1) and hk+1, . . . , hn are the eigenvectors corresponding to the eigenvalue 0. lemma 3.2. let det(λ) 6= 0. for k ≥ 0 we denote by x a matrix with the jordan basis h1, . . . , hn , where the vectors h1, . . . , hk form the jordan basis for λ. (listed in such as way that if xhj = λ hj + hj−1, then λ hj = λ hj + hj−1) and hk+1, . . . , hn are the eigenvectors corresponding to the eigenvalue 0. then x is a solution to the matrix equation (2.9). bring one more the geometrical formulation of the necessary and sufficient conditions of the solvability of the quadratic matrix equation (2.7) cubo 12, 2 (2010) the maxwell problem and the chapman projection 281 theorem 3.4. let |λ| 6= 0, and let vectors v1, . . . , vm satisfy the following conditions: 1. v = lin{vj}m1 is an eigenspace of the matrix λ, i.̊a. λ v = v . 2. v1, .., vm, em+1, .., en form a basis. then the quadratic matrix equation (2.4) is solvable. the inverse assertion is also true. 2.3 explicit formula now, we discuss a possible explicit formula for solutions to the riccati matrix equation. theorem 4.1. suppose that vectors v1, . . . , vm form a basis for a linear λ-invariant subspace v and v1, . . . , vm, em+1, . . . , en is a basis for r n. we regard these vectors as columns of a matrix ( c11 c21 ) . then the solution to the matrix equation (2.4), associated with these vectors listed in the above order, is represented in the form p21 = c21c −1 11 (2.10) proof. since we can assume that det(λ) 6= 0, for the solution to the matrix equation (2.4) we have ( e 0 p21 0 ) = λ−1 ( c11 0 c21 e )( j1 0 0 0 )( c−111 0 −c21c−111 e ) , where j1 is a block from the jordan form of the matrix λ corresponding to the space v . hence ( e 0 p21 0 ) = λ−1 ( c11j1c −1 11 0 c21j1c −1 11 0 ) . multiplying both sides of the last equality by λ from the left, we find ( λ11 + λ12p21 0 λ21 + λ22p21 0 ) = ( c11j1c −1 11 0 c21j1c −1 11 0 ) , which implies p21c11j1c −1 11 − c21j1c −1 11 = 0. in view of (2.4). since λ is invertible, the matrix j1 is also invertible. hence we can multiply the last equality by c11j −1 1 from the right. then p21c11 = c21, which implies (2.10). 2.4 the number of solutions corollary 5.1. with every m-dimensional eigenspace v of the matrix λ at most one solution to the matrix equation (2.4) is associated. proof. indeed, either v does not provide any solution to (2.4) (if lin{v1, . . . , vm, em+1, . . . , en} = rn, where v = lin{v1, . . . , vm}) or v can be associated with a solution to (2.4) by formula (2.10). in the second case, we show that the solution is independent of the choice of the basis for the space v . let w1, . . . , wm be another basis for v . we write the vectors v1, . . . , vm as columns of a matrix w 0 282 v. v. palin and e. v. radkevich cubo 12, 2 (2010) and the vectors w1, . . . , wm as columns of a matrix w 1. since these bases generate the same linear space v , there exists a nonsingular matrix k such that w 1 = w 0k or, in the block form, ( w 11 w 12 ) = ( w 01 w 02 ) k, which implies w 1j = w 0 j k, j = 1, 2. hence the solution of the form (2.10) corresponding to the basis for w 1 can be written as p21,w = w 1 2 (w 1 1 ) −1 = w 02 kk −1(w 01 ) −1 = p21,v . thus, the solutions defined by the bases v1, . . . , vm and w1, . . . , wm coincide. next results we bring for the information(details look in [19,22]) theorem 5.2. let the matrix equation (2.4) have infinite number of solutions. then there exists λ ∈ c such that dim(ker(λ − λ e)) ≥ 2. theorem 5.2. the set of solutions to the matrix equation (2.4) is infinite then and only then if there exists the eigenspaces v and w of the matrix λ, satisfying the following conditions: 1. v defines the solution to the equation (2.4). 2. w is a eigenspace of the matrix λ, corresponding to a eigenvalue λ. 3. w contain two incollinear eigenvectors. 4. v ∩ w 6= {0}. 5. w \ v 6= ∅. 2.5 continuity of solutions to the quadratic matrix equation in this section, we study the continuity of the constructed solutions with respect to the parameter ξ. we begin with auxiliary assertions(see [17]). lemma 6.1. let a matrix a(x) be a continuous function of the parameter x in some neighborhood u (x0) of a point x0. denote by λ(x) an eigenvalue of a(x) that continuously depends on x in u (x0) and is simple in a punctured neighborhood of x0. then the corresponding eigenvector vλ(x) is also continuous with respect to x in the same neighborhood. lemma 6.2. suppose that a matrix-valued function a(x) is continuous with respect to x in some neighborhood u (x0) of a point x0 and its kernel has constant dimension k in u (x0). let vectors v1(x), . . . , vm(x) . ker(a(x)) be continuous in u (x0) and linearly independent in the corresponding punctured neighborhood. then there exists a basis for the invariant subspace v (x) = lin{v1(x), . . . , vm(x)}, i.e., w1(x), . . . , wm(x), that is continuous in u (x0) and is linearly independent in u (x0). theorem 6.1. suppose that a matrix a(x) continuously depends on x in some neighborhood u (x0) of a point x0 and λ1(x), . . . , λk(x) are continuous eigenvalues of a(x) such that each of them is simple in a punctured neighborhood of x0, λ1(x0) = ... = λk(x0) = λ0. let v (x) be the eigenspace of a(x) corresponding to λ1(x), ..., λk(x). if there are no eigenvalue λ(x) of a(x), different from cubo 12, 2 (2010) the maxwell problem and the chapman projection 283 λj (x), j = 1, . . . , k, that is continuous and λ(x0) = λ0, then there exists a basis for v (x) continuously depending on x in u (x0). proof. in a punctured neighborhood of x0, for a basis for v (x) we take the eigenvectors vj (x) corresponding to the eigenvalues λj (x) of the matrix a(x). by the above lemma, vj (x) are continuous in u (x0). further, let us introduce the matrix m (x) = π k j=1(a(x)−λj (x)e).. under the assumptions of the theorem, m (x) is continuous in u (x0), dim(ker(m (x))) = k and v (x) = ker(m (x)). hence the matrix m (x) and subspace v (x) satisfy the assumptions of lemma 5.2 with m = k, which implies the required assertion. theorems 5.1 and 4.1 lead to the following assertion concerning the continuity of solutions to the quadratic matrix equation (2.4) with respect to the parameter ξ theorem 6.2. assume that a matrix λ is continuously depends on the parameter ξ and is invertible for all ξ ∈ ξ0; moreover, the eigenvalues of λ are simple for all ξ /∈ ξ∗, where the set ξ∗ is finite. then the matrix equation (2.4) with the matrix λ has a solution continuously depending on the parameter ξ if and only if there exists an m-dimensional eigenspace v satisfying the assumptions of theorem 4.1 for all ξ ∈ ξ0. proof. indeed, by theorem 4.1 and its consequences, the subspace v determines a solution to the quadratic matrix equation (2.4) in the form p21 = c21c −1 11 . since the matrix λ satisfies the assumptions of theorem 5.1, the basis for the space v continuously depends on ξ. therefore, the invertibility of c11 immediately implies the required assertion. 2.6 the lyapunov equation. separation of dynamics the lyapunov matrix equation −m11q12 + q12m22 − m12 = 0 (2.11) is a special case of the quadratic matrix equation (2.4) with vanishing quadratic term. the following assertion is proved in [20]. theorem 7.1. suppose that det(m11) 6= 0 and det(m22) 6= 0. assume that the matrix m = ( m11 m12 0 m22 ) has no eigenvalues λ such that, in the block form, the corresponding eigenvector has the form v0 = ( v0,1 0 ) and the corresponding associated eigenvector has the form v1 = ( v1,1 v1,2 ) , where v1,2 6= 0. then there exists a solution q12 to the lyapunov matrix equation (2.11) with the matrices m11, m12, and m22. we construct the canonical form of (2.1). lemma 7.1. suppose that a matrix s is invertible and can be written in the block form with blocks sij , i, j = 1, 2, where s11 and s22 are square matrices. assume also that f s = sf = e, 284 v. v. palin and e. v. radkevich cubo 12, 2 (2010) where the matrix f can be represented by blocks of the same size. in this case, if s11 = e then the matrix f22 is invertible. proof. assume the contrary. since f s = e, we have f21 + f22s21 = 0. (2.12) since f22 is noninvertible, there is a row h 6= 0 such that hf22 = 0. using (2.12), we find hf21 = 0. but, in this case, the last rows of the matrix f are linearly dependent: there is a row v such that v 6= 0 and vf = 0. thus, the matrix f is noninvertible. on the other hand, the matrix f is the inverse of s. we arrive at a contradiction. theorem 7.2. suppose that λ is divided into blocks λij , i, j, = 1, 2. then the quadratic matrix equation (2.4) is solvable if and only if there exists a matrix s satisfying the following conditions: 1. s is invertible, 2. s11 = e. 3. (s−1λ s)21 = 0. proof. assume that there exists a matrix s satisfying conditions 1) -3). for f = s−1 we have f21 + f22s21 = 0, f21(λ11 + λ12s21) + f22(λ21 + λ22s21) = 0. expressing f21 from the first equation and substituting into the second equation, we find f22(−s21(λ11 + λ12s21)) + f22(λ21 + λ22s21) = 0. we note that the matrix s satisfies the assumptions of lemma 6.1. hence (−s21(λ11 + λ12s21)) + (λ21 + λ22s21) = 0, i.e., the matrix s21 satisfies the quadratic matrix equation (2.4). assume that the quadratic matrix equation (2.4) is solvable. we set s11 = e, s12 = 0, s21 = p21, s22 = e. it is easy to verify that the inverse matrix exists: s−1 = 2e − s. we see that the matrix s satisfies conditions 1) and 2). computing (s−1λ s)21, we find (s−1λ s)21 = f21(λ11 + λ12s21) + f22(λ21 + λ22s21) = = (−p21)(λ11 + λ12p21) + (λ21 + λ22p21) = 0, since p21 is a solution to the quadratic matrix equation (2.4). thus, the matrix s also satisfies condition 3). the theorem is proved. thus, the existence of a chapman-enskog projection is equivalent to the possibility to represent the original system in the block form such that (s−1λ s)21 = 0, which allows us to separate dynamics. the following theorem (cf. the proof in [16]) provides us with conditions under which a matrix . can be reduced to the block-diagonal form. theorem 7.3. assume that a matrix λ is invertible and v1, . . . , vm is a basis for its eigenspace v such that lin{v1, . . . , vm, em+1, . . . , en } = rn . we also assume that v cannot be extended to an cubo 12, 2 (2010) the maxwell problem and the chapman projection 285 m + 1-dimensional eigenspace of the matrix λ by extending the basis v1, . . . , vm with an associated eigenvector of λ. then there exist matrices p21 and q12 such that ( e −q12 0 e )( e 0 −p21 e ) λ ( e 0 p21 e )( e q12 0 e ) = ( m11 0 0 m22 ) . now, we consider the representation of the solution as the sum of three terms and introduce the notion of the l2-well-posedness in the sense of chapman-enskog. suppose that a matrix λ satisfies the assumptions of theorem 7.3. we make the change of variables u = s−1u. then a solution to the cauchy problem (2.1) with the initial data u|t=0 = ( u0 v0 ) can be written in terms of the fourier images as follows: u = e−mt ( u0 v0 ) , where m = s−1λ s. by theorem 7.3, the matrix m takes the form m = ( e q12 0 e )( m11 0 0 m22 )( e −q12 0 e ) , which implies u = ( e q12 0 e ) exp ( − ( m11 0 0 m22 ) t )( e −q12 0 e )( u0 v0 ) = uch + ucor + uh , where each of the terms is a solution to the system (2.1) with some initial data: uch = e −mt ( u0 0 ) , ucor = e −mt ( −q12v0 0 ) , uh = ( q12e −m22tv0 e−m22tv0 ) . the first term uch corresponds to the projection onto the phase space of consolidated variables, the second term ucor is a corrector describing the influence of the initial data relative to nonequillibrium variables, and the third term uh is a remainder. definition 7.1. we say that a projection p satisfies the chapman l2-well-posedness condition for a class of initial data h = {(u0, v0)} if for any initial data (u0, v0) ∈ h there is a constant t0 > 0 such that for all t > t0 ||uh||(t) ||uch||(t) ≤ ke−δ t, t > t0, (2.13) where k and δ > 0 are constants. 286 v. v. palin and e. v. radkevich cubo 12, 2 (2010) 2.7 crack condition and the existence of an attracting manifold we find conditions that guarantee the validity of the estimate ||uh|| = o(||uch||), t → ∞, where ‖f‖ denotes the norm of f in the space l2. for this purpose, we prove several technical auxiliary assertions(see [21]): lemma 8.1. suppose that a matrix λ polynomially depends on ξ and there exists k0 > 0 such that for all ξ : |ξ| > k0, all the eigenvalues λ(ξ) of λ are algebraically simple and |λ(ξ)| ≤ c1(1+|ξ|)d1 , where c1 and d1 are constants. let v be an eigenvector of λ. then for |ξ| > k0 : max{|eti v|} min{|eti v| 6= 0} ≤ c2(1 + |ξ|)d2 , (2.14) where c2 and d2 are constants. lemma 8.2. suppose that a matrix λ is defined for all ξ ∈ r and satisfies the assumptions of theorem 6.3 for all ξ ∈ ξ, where ξ = r\ξ− and the set ξ− is finite. then p21 and q12 are defined on ξ. assume that the matrices p21(ξ) and q12(ξ) can be defined by continuity on the set ξ−. we also assume that the matrix λ. polynomially depends on ξ and there is k0 > 0 such that for all ξ : |ξ| > k0, all the eigenvalues of the matrix λ are algebraically simple and satisfy the following estimate: |λ(ξ)| ≤ c1(1 + |ξ|)d1 , where c1 and d1 are constants. then there is d ∈ n such that for all ξ ∈ r |p21| ≤ k1(1 + |ξ|)d, |q12| ≤ k2(1 + |ξ|)3d, where k1 and k2 are constants and |a| is the matrix norm of a in l∞. notation 8.1. the minimal number d ∈ n satisfying the assumptions of lemma 8.2 is denoted by dλ. we also need a two-sided estimate for |e−mtv|, where |.| — denotes the l∞(r). for the sake of brevity, we introduce the following notation. notation 8.2. suppose that a square matrix m continuously depends on the parameter ξ. let λj , j = 1, ..., s, be eigenvalues of m . we denote by dj the maximal size of the jordan cell corresponding to the eigenvalue λj . let the eigenvalues λj be listed in ascending order of the real part. let l(m ) and l(m ) denote the minimal and maximal eigenvalues respectively, i.e. l(m ) = re λ1 ≤ re λ2 ≤ · · · ≤ re λs = l(m ). we set d(m ) = d1. we will use one technical lemma still(see [19]: lemma 8.3. let a square matrix m continuously depend on the parameter ξ. then for any ε > 0 there is t0 > 0 such that for all t > t0 the following estimate holds: e−l(m)t|v| ≤ |e−mtv| ≤ 1 + ε (d(m ) − 1)!|m| d(m)−1e−l(m)ttd(m)−1|v|, (2.15) where |a| denotes the matrix norm of a in l∞(r). cubo 12, 2 (2010) the maxwell problem and the chapman projection 287 notation 8.3. let γ(ξ) be a finite set of continuous functions γ1(ξ), . . . , γs(ξ) of the parameter ξ. introduce the notation l(ξ, γ(ξ)) = infs{re γs(ξ) | γs(ξ) ∈ γ(ξ)}, l0(γ) = infξ l(ξ, γ(ξ)), l(ξ, γ(ξ)) = sups{re γs(ξ) | γs(ξ) ∈ γ(ξ)}, l0(γ) = supξ l(ξ, γ(ξ))). condition 8.1. a pair of sets γ1(ξ) and γ2(ξ) satisfy the strong crack condition if ∃γ > 0 : l0(γ2) − l0(γ1) ≥ γ. (2.16) now, we formulate the conditions for the existence of an attracting manifold. theorem 8.1. let the matrix λ in the problem (2.1) satisfy the assumptions of lemma 8.2. suppose that γ1 is the set of all those eigenvalues of λ that determine the separation of dynamics for the eigenspace v and γ2 is the set of all the remaining eigenvalues of λ. assume that γ1 and γ2 satisfy the strong crack condition. let the fourier images of initial data (u0, v0) belong to the set h = {(u0, v0) : ||u0|| 6= 0, (1 + |ξ|)3dλ |m22|d(m22)−1v0 ∈ l2(r)}, then the projection p corresponding to the separation of dynamics satisfies the chapman-enskog l2-well-posedness condition (definition 6.1) for the class of initial data h with constants k and δ such that (i) k depends on ||u0||, ||v0||, (ii) δ depends on δ and some properties of the matrix m . proof. indeed, ||uh (t)|| =   ∫ r ∣ ∣ ∣ ∣ ∣ ( q12e −m22tv0 e−m22tv0 )∣ ∣ ∣ ∣ ∣ 2 dξ   1 2 ≤ ( ∫ r |1 + |q12|2||e−m22tv0|2dξ ) 1 2 . using lemmas 8.2 and 8.3, we find ||uh (t)||2 ≤ ∫ r (1 + k22 (1 + |ξ|)10dλ )( 1 + ε (d(m22) − 1)! )2|m22|2d(m22)−2e−2l(m22)tt2d(m22)−2|v0|2dξ. from (2.16) it follows that e−l(m22)t ≤ e−l0(γ2)t ≤ e−γte−l0(γ1)t; e−l(m11)t ≥ e−l0(γ1)t. by lemma 8.3, ||uch(t)|| ≥ ( ∫ r e−2l(m11)t|u0|2dξ ) 1 2 . combining the last four inequalities, we find ( ||uh||(t) ||uch||(t) )2 ≤ e−2γtt2d(m22)−2 ∫ r e−2l0(γ1)t(1 + k22 (1 + |ξ|)10dλ ) ( 1+ε (d(m22)−1)! )2 |v0|2dξ ∫ r e−2l0(γ1)t|u0|2dξ , which implies the required estimate (2.13) because l0(γ1) is independent of ξ. 288 v. v. palin and e. v. radkevich cubo 12, 2 (2010) 3 nonlinear analysis. chapman projection 3.1 statement of the problem and auxiliaries we consider the nonlinear system of equations ∂tu + n ∑ j=1 aj ∂xj u + bu = f (u), (3.1) with the initial condition u|t=0 = φ, where u is an n -dimensional vector, aj and b are constant matrices, n ≤ 3, and f (u) is a vector-valued polynomial, i.e. f (u) = ∑n j=1 ( ∑ σ∈θj k(j, σ)uσ ) ej , where σ ∈ (n ∪ {0})n , uσ = ∏n j=1 u σj j . we set α = min{|σ| : σ ∈ ∪nj=1θj}, α + β = max{|σ| : σ ∈ ∪nj=1θj}. assume that f (u) contains no terms of zero or first order, i.e. α ≥ 2. we denote by || · || the norm in l2 with respect to the variable x. let |u| = √ ut u and let |u|0 denote the norm of u in c. following [10], we denote by ∂ the vector consisting of all first order derivatives and by ∂x the vector consisting of first order derivatives with respect to the spatial variables, i.e. ∂ = (∂x, ∂t). for the sake of brevity, we write ∂j instead of ∂xj . we begin with the following auxiliary assertion generalizing lemma 8.3. lemma 9.1. let m be a square matrix. then there are constants cm ∈ r and dm ∈ z such that for any vector v and a number t ≥ 0 |e−mtv| ≤ cm (1 + tdm )e−l(m)t|v|. (3.2) the following assertion concerns estimates for the norms of f (u) and its derivatives is tru: lemma 9.2. for a vector-valued function u(x, t) ∈ c([0, t ), h2) ∩ c1([0, t ), h1) with t > 0 and s ∈ {1, 2}, j ∈ {1, 2, 3} the following estimates hold: ||f (u)|| ≤ c0,0|u|α−10 (1 + |u| β 0 )||u||, (3.3) ||∂sj f (u)|| ≤ cs,0|u|α−10 (1 + |u| β 0 )||∂sj u||, (3.4) the following assertion concerning the norm of a vector-valued polynomial is a consequence of the above lemma. lemma 9.3. consider a vector-valued function u(x, t) ∈ c([0, t ), h2) ∩ c1([0, t ), h1) with t > 0, x ∈ rn, n ≤ 3. let g(u) be a vector-valued polynomial with α ≥ 1. then there are κ ∈ (0, 1) and cg > 0 such that for all u(x, t) such that ||g(u)||h2 ≤ 2cg||u||αh2 . (3.5) cubo 12, 2 (2010) the maxwell problem and the chapman projection 289 proof. indeed, from the inequalities (3.3) and (3.4) it follows that ||g(u)||h2 ≤ const |u|α−10 (1 + |u| β 0 )||u||h2 . by the embedding theorem, ||g(u)||h2 ≤ cg||u||αh2 (1 + ||u|| β h2 ). hence the required inequality (3.5) holds for sufficiently small κ. whence we obtain lemma 9.4. let u(x, t) and v(x, t) be vector-valued functions such that u(x, t), v(x, t) ∈ c([0, t ), h2) ∩ c1([0, t ), h1) for some t > 0. assume that x ∈ rn and n ≤ 3, f (u) is a vector-valued polynomial with α ≥ 2. then there are κ ∈ (0, 1) and c∗ > 0 such that for all u(x, t), v(x, t) the inequalities ||u||h2 < κ, ||v||h2 < κ imply the inequality ||f (u) − f (v)||h2 ≤ c∗(||u||α−1h2 + ||v|| α−1 h2 )||u − v||h2 . (3.6) lemma 9.5. suppose that t > 0 and p (τ ) is a continuous function such that the inequality p (τ ) ≥ 0 for all τ ∈ [0, t]. let d > 0. then there is a constant cp > 0 such that ∫ t 0 (1 + (t − τ )d)2p (τ )dτ ≤ cp (1 + td)2 ∫ t 0 (1 + τ d)2p (τ )dτ. (3.7) proof. we have ∫ t 0 (1 + (t − τ )d)2p (τ )dτ ≤ c1 ∫ t 0 (1 + τ 2d + t2d)p (τ )dτ ≤ ≤ c1 ( t2d ∫ t 0 p (τ )dτ + ∫ t 0 (1 + τ 2d)p (τ )dτ ) ≤ c1(1 + t2d) ∫ t 0 (1 + τ 2d)p (τ )dτ ≤ ≤ c1(1 + td)2 ∫ t 0 (1 + τ d)2p (τ )dτ. 3.2 method of successive approximations we look for a solution to the system (3.1) with the initial data u|t=0 = φ(x) for small φ(x) by the method of successive approximations. we set u0 = 0, ∂tuk + n ∑ j=1 aj ∂j uk + buk = f (uk−1), uk|t=0 = φ(x). (3.8) introduce the notation λ = ∑n j=1 aj iξj + b, l1 = infξ minλ∈σ(λ) re λ. we denote by f(·) the fourier transform with respect to the spatial variables. we estimate from above the solution uk to the problem (3.8). 290 v. v. palin and e. v. radkevich cubo 12, 2 (2010) lemma 10.1. let l1 > 0. then there exist constants κ ∈ (0, 1) and c∗1 > 0, c∗2 > 0 such that the solution uk to the problem (3.8) with the initial data φ such that ||φ||h2 < κ satisfies the following inequality for any t ≥ 0 : ||uk||h2 ≤ c∗1 (1 + tdλ )e−l1t(||φ||h2 + c∗2 √ t||φ||αh2 ). (3.9) proof. we first prove that for sufficiently small initial data ||uk||h2 ≤ cλ(1 + tdλ )e−l1t(||φ||h2 + ck √ t||φ||αh2 ), (3.10) where the constants ck depend on k. we write an explicit expression for ck. for this purpose, we use the method of mathematical induction. let k = 1. then the problem (3.8) takes the form ∂tu1 + n ∑ j=1 aj ∂j u1 + bu1 = 0, u1|t=0 = φ. the solution to this problem is written in terms of the fourier images as follows: f(u1) = e−λtf(φ). by lemma 10.1 ||u1||2 = ||f(u1)||2 = ∫ rn |e−λtf(φ)|2dξ ≤ c2λ ∫ rn (1 + tdλ )2e−2l1t|f(φ)|2dξ = = c2λ(1 + t dλ )2e−2l1t||f(φ)||2 = c2λ(1 + tdλ )2e−2l1t||φ||2. a similar inequality holds for the derivatives of u1. thus, ||u1||h2 ≤ cλ(1 + tdλ )e−l1t||φ||h2 , and the inequality (3.10) is true with c1 = 0. now, we write an explicit formula for the solution to the problem (3.8) in terms of the fourier images: f(uk) = e−λtf(φ) + ∫ t 0 eλ(τ −t)f(f (uk−1(τ )))dτ. (3.11) we set iσ,k = (iξ) σ ∫ t 0 eλ(τ −t)f(f (uk−1(τ )))dτ and find ||iσ,k|| ≤ cλck √ t(1 + tdλ )e−l1t||φ||αh2 , where the constant ck is independent of σ. the proof of this assertion is similar to that of the inequality (3.10) for all k. indeed, we have the auxiliary estimates ||iσ,k||2 ≤ t ∫ t 0 ||(iξ)σeλ(τ −t)f(f (uk−1(τ )))||2dτ ≤ ≤ tc2λ ∫ t 0 (1 + (t − τ )dλ )2e2l1(τ −t) ∫ rn |(iξ)σf(f (uk−1(τ )))|2dξdτ ≤ ≤ tc2λ ∫ t 0 (1 + (t − τ )dλ )2e2l1(τ −t)||f (uk−1(τ ))||2h2 dτ. cubo 12, 2 (2010) the maxwell problem and the chapman projection 291 using lemmas 9.3 and 9.5, we find ||iσ,k||2 ≤ 4c2λc2f cp t(1 + tdλ )2e−2l1t ∫ t 0 (1 + τ dλ )2e2l1τ ||uk−1(τ )||2αh2 dτ. (3.12) let k = 2. using the estimate (3.10) for k = 1, we find ||iσ,2||2 ≤ 4c2+2αλ c 2 f cp t(1 + t dλ )2e−2l1t ∫ t 0 e2l1(1−α)τ (1 + τ dλ )2+2α||φ||2αh2 dτ ≤ ≤ 4c2+2αλ c 2 f cp t(1 + t dλ )2e−2l1t ∫ +∞ 0 e2l1(1−α)τ (1 + τ dλ )2+2α||φ||2αh2 dτ. note that for α ≥ 2 the integral is convergent. setting c22 = 4c 2α λ c 2 f cp ∫ +∞ 0 e2l1(1−α)τ (1 + τ dλ )2+2αdτ, we obtain an inequality of the required form for ||iσ,2||. assume that the inequality (3.10) is valid for all k ≤ r, where r ≥ 2. then for k = r + 1, by the inequality (3.12) ||iσ,r+1||2 ≤ c2λt(1 + tdλ )2e−2l1t||φ||2αh2 ( c′c22 + +4c2αλ c 2 f cp c ′c2αr ∫ +∞ 0 e2l1(1−α)τ (1 + τ dλ )2+2ατ α||φ||2α(α−1) h2 dτ ) . setting j = ∫ +∞ 0 e2l1(1−α)τ (1 + τ dλ )2+2ατ αdτ ), we obtain the required estimate (3.10) with c2r+1 = c 2 2 c ′ + 4c2αλ c 2 f cp c ′j||φ||2α(α−1) h2 c2αr we note that c′ > 1. we choose κ > 0 such that for ||φ|| < κ 4c2αλ c 2 f cp c ′j||φ||2α(α−1) h2 < 1 (c22 c ′ + 1)α . let qr = c 2 r . then q2 < c 2 2 c ′ + 1. we note that for κ, as above, qr < c 2 2 c ′ + 1 for all r ≥ 2. indeed, qr+1 = c 2 2 c ′ + 4c2αλ c 2 f cp c ′j||φ||2α(α−1) h2 qαr < c 2 2 c ′ + 1 (c22 c ′ + 1)α qαr < c 2 2 + 1. thus, cr < √ c22 c ′ + 1 and the inequality (3.10) with small κ implies (3.9). from this result it follows lemma 10.2. suppose that l1 > 0 and ||φ||h2 < κ in (3.8) with sufficiently small κ. then the solutions uk to the system (3.8) converge in c((0, +∞); h2). 292 v. v. palin and e. v. radkevich cubo 12, 2 (2010) 3.3 construction of a nonlinear chapman projection 11.1. weak nonlinearity. we consider the system ∂tu + a11∂xu + a12∂xv + b11u + b12v = 0, ∂tv + a21∂xu + a22∂xv + b21u + b22v = g(u)v (3.13) with the initial data u|t=0 = φ1(x), v|t=0 = φ2(x). we set φ = ( φ1 φ2 ) . suppose that u(x, t) : r × r+ → rm and v(x, t) : r × r+ → rn−m. assume that the data of the problem (3.13) satisfy all the assumptions of lemma 10.2. we also assume that the following condition is satisfied. condition 11.1. the linearized part of the problem (3.1) and the initial data satisfy all the assumptions of theorem 7.1. moreover, lj = infξ minλ∈γj re λ and l1 > 0, l2 − αl1 < 0. we denote by p21 the symbol of the chapman-enskog projection for the linearized problem (3.13). if the initial data φ are sufficiently smooth and ||φ||2h2 + ||p21(∂x)φ||2h2 < κ ≪ 1 then, according to the method of successive approximations, there exists a solution ( w z ) to the problem (3.13) with the initial data w|t=0 = υ(φ1, φ2), z|t=0 = p21(∂x)υ(φ1, φ2), where υ is the operator of the initial data corresponding to the sum of uch and ucor in the linear case. the goal of this section is to construct a nonlinear operator p21(w, ∂x) such that z = p21(w, ∂x)w. let m = sλs−1, where s = ( e 0 −p21 e ) . we write the system (3.13) in terms of the fourier images and use the fact that p21 is the symbol of the chapman projection for the linearized problem. then ∂tf(w) + m11f(w) + m12f(v′) = 0, ∂tf(v′) + m22f(v′) = f(g(w)z), where z = p21w + v ′. based on this fact, we look for z in the form z = p21w + ∞ ∑ j=1 vj , (3.14) where vj is a solution to the equation ∂tf(vj ) + m22f(vj ) = f(g(w)vj−1) (3.15) with the initial data vj|t=0 = 0 and v0 = p21w. using the method of variation of constants, we find vj = f−1 ( e−m22t ∫ t 0 em22τ f(g(w(τ ))vj−1 (τ ))dτ ) . cubo 12, 2 (2010) the maxwell problem and the chapman projection 293 this representation shows that vj = πj (w, ∂x)w. it remains to prove that for small φ the series (3.14) is convergent. from lemma 10.1 and the method of successive approximations it follows that ||w||h2 ≤ c0e−l1t(1 + td)(||φ||h2 + ||p21φ||h2 ). furthermore, for p21w we have the similar estimate ||p21w|| ≤ c1e−l1t(1 + td)||φ||hs with some s. based on these two inequalities and the embedding theorem, we find the following estimate for v1: ||v1||2 ≤ t ∫ t 0 ||em22(τ −t)f(g(w(τ ))p21 w(τ ))||2dτ ≤ te−2l2t ∫ t 0 e2l2τ q0(t, τ )|w|2α−2∞ ||p21w(τ )||2dτ, where q0(t, τ ) is a polynomial depending only on the structure of m . further, ||v1||2 ≤ c2te−2l2t ∫ t 0 e2(l2−αl1)τ q0(t, τ )(1 + τ d)2α(||φ||h2 + ||p21φ||h2 )2α−2||φ||2hs dτ. hence, under the above conditions on the system (3.13), there are constants d1 ≥ 0 and k1 > 0 such that ||v1||2 ≤ k1t(1 + td1 )e−2l2t||φ||2αhs . (3.16) moreover, d1 depends only on the structure of the matrix m and k1 is independent of φ. similarly, for v2 we find ||v2||2 ≤ t ∫ t 0 ||em22(τ −t)f(g(w(τ ))v1 (τ ))||2dτ ≤ ≤ c3te−2l2t ∫ t 0 e2l2τ q0(t, τ )(1 + τ d)2α−2e−(2α−2)l1τ ||v1(τ )||2(||φ||h2 + ||p21φ||h2 )2α−2dτ. using the above estimate for v1, we find ||v2||2 ≤ k2te−2l2t(1 + td1 )||φ||4α−2hs , where the constant k2 is independent of φ, because ∫ t 0 τ re−γτ dτ ≤ ∫ ∞ 0 τ re−γτ dτ = const. arguing in the same way, it is easy to obtain the inequality ||vj||2 ≤ kjt(1 + td1 )e−2l2t||φ||2jα−2j+2hs , where the constants kj are independent of φ and kj ≤ kj0, k0 = const. hence for sufficiently small φ the series (3.14) is convergent. we note that the smallness of the norm of the initial data in some space hs and the estimate |g(w)|∞ ≤ cg|w|α−1∞ (1 + |w|β∞) 294 v. v. palin and e. v. radkevich cubo 12, 2 (2010) imply |g(w)|∞ ≤ c′g|w|α−1∞ , where c′g > cg. furthermore, d1 is the degree of the polynomial q0(t, τ ) in the variable t. since q0 is a polynomial, there exist constants i1 and i2 such that ∫ +∞ 0 e2(l2−αl1)τ q0(t, τ )(1 + τ d)2αdτ ≤ i1(1 + td1 ), and ∫ +∞ 0 e−(2α−2)l1τ q0(t, τ )(1 + τ d)2α−2(1 + τ d1 )τ dτ ≤ i2(1 + td1 ). indeed, both integrals on the left-hand sides of these inequalities are polynomials in t of degree d1, which implies the required estimates. to prove the assertions concerning the constants kj , we need the following lemma. lemma 11.1. assume that all the assumptions of lemma 10.2 and condition 11.1 are satisfied. then the solution vj to the problem (3.15) with the initial data vj|t=0 = 0 for j ≥ 1 satisfies the inequality ||vj||2 ≤ kjt(1 + td1)e−2l2t||φ|| 2jα−2j+2 hs , where kj ≤ (c′g)j c 2jα−2j+2 w i j−1 2 i1 and cw = max{c0, c1}. proof. we use the method of mathematical induction. as was already shown, the required estimate is valid for ||v1||2. furthermore, it is easy to see that k1(1 + t d1 ) = c2 ∫ +∞ 0 e2(l2−αl1)τ q0(t, τ )(1 + τ d)2αdτ ≤ c2i1(1 + td1 ), where c2 = c ′ gc 2α w . thus, the corresponding inequality for k1 also holds. assume that the assertion holds for j ≤ k. then for j = k + 1 we have ||vk+1||2 ≤ t ∫ t 0 ||em22(τ −t)f(g(w(τ ))vk (τ ))||2dτ ≤ ≤ te−2l2t ∫ t 0 e2l2τ q0(t, τ )|w(τ )|2α−2∞ ||vk(τ )||2dτ ≤ kk+1t(1 + td1 )e−2l2t||φ|| 2(k+1)α−2k hs , where c3 = c ′ gc 2α−2 w , and from the inequality ∫ t 0 q0(t, τ )(1 + τ d)2α−2e−(2α−2)l1τ (1 + τ d1 )τ dτ ≤ ≤ ∫ +∞ 0 q0(t, τ )(1 + τ d)2α−2e−(2α−2)l1τ (1 + τ d1 )τ dτ ≤ i2(1 + td1 ) we obtain the required estimate for kk+1. 11.2. general case. we consider the system ∂tu + a11∂xu + a12∂xv + b11u + b12v = g11(u)u, ∂tv + a21∂xu + a22∂xv + b21u + b22v = g21(u)u + g22(u)v (3.17) cubo 12, 2 (2010) the maxwell problem and the chapman projection 295 with the same initial data, as above. then we construct a nonlinear operator p21(∂x, w) that determines the solution ( w z ) . we look for z in the form (3.14), where vj are solutions to the problem ∂tf(v1) + m22f(v1) = f(p21(g11(w)w) + g21(w)w + g22(w)p21w), f(v1)|t=0 = 0, ∂tf(vj ) + m22f(vj ) = f(g22(w)vj−1), f(vj )|t=0 = 0, j ≥ 2. then we can estimate ||v1|| as follows: ||v1||2 ≤ t ∫ t 0 c2m (1 + (t − τ )dm )2e2l2(τ −t)||p21(g11(w)w) + g21(w)w + g22(w)p21w||2(τ )dτ ≤ ≤ const t(1 + tdm )2e−2l2t||φ||2αh2 ∫ +∞ 0 e2(l2−αl1)τ (1 + τ dm )2(1 + τ dλ )2α(1 + √ τ )2αdτ = = k′1t(1 + t dm )2e−2l2t||φ||2αh2 . thus, for v1 we have an estimate of the form (3.16). we note that the equation for vj , j > 1, is the same as in the previous subsection. furthermore, for estimating from above vj , j > 1, we used the estimate (3.16), but not an explicit form of v1. consequently, lemma 11.1 remains valid. therefore, the series (3.14) converges in the l2-norm for small initial data, which means the existence of a nonlinear projection p21. 3.4 properties of nonlinear projections we study properties of the nonlinear operator p21 constructed in the previous section. lemma 12.1. let the data of the problem (3.17) satisfy all the assumptions of lemma 10.2 and condition 11.1. assume that φ ∈ h3, |p21|0 ≤ const(1 + |ξ|s), s ≤ 2. then for every term vj of the series (3.14) the following inequality holds: ||vj||2h1 ≤ k j 0 t(1 + t dm )2e−2l2t||φ||2jα−2j+2 h3 . (3.18) proof. we estimate each ||∂kvj||. for this purpose, we note that ||∂kvj|| satisfies the problem ∂tf(∂kvj ) + m22f(∂kvj ) = f(∂k(fj (w, vj−1))), f(∂kvj )|t=0 = 0, where f1(w, v0) = f1(w) = −p21(g11(w)w) + g12(w)w + g22(w)p21w, fj (w, vj−1) = g22(w)vj−1, j ≥ 2. using lemmas 9.1 and 9.5, we find ||∂kv1||2 ≤ const t(1 + tdm )2e−2l2t ∫ +∞ 0 e2l2τ (1 + τ dm )2||f1(w)||2h1 dτ, using lemma 9.2 and the inequality |p21| ≤ const(1 + |ξ|s), s ≤ 2 we finally find ||∂kv1||2 ≤ const t(1 + tdm )2e−2l2t||φ||2αh3 . 296 v. v. palin and e. v. radkevich cubo 12, 2 (2010) futher ||∂kvj||2 ≤ cm t(1 + tdm )2e−2l2t ∫ +∞ 0 e2l2τ (1 + τ dm )2||∂k(g22(w)vj−1)||2dτ, taking into account that g22 is a matrix polynomial and arguing as in lemma 11.1, we obtain the required estimates. for the sake of brevity, we introduce the notation l1 = supξ maxλ∈γ1 re λ. theorem 12.1. let ( w z ) be a solution to the system (3.17) with the initial data w|t=0 = φ1, z|t=0 = p21φ1. let φ ∈ h3, and let all the assumptions of lemma 12.1 be satisfied. denote by ( w0 z0 ) the solution to the linearized problem (3.17) with the same initial data. if α > l1 l1 , ||φ||h3 < κ ≪ 1, then the following estimate holds: ||em11tf(w − w0)||2 ≤ 1 γ const(||φ||2αh2 + ||φ|| α+1 h3 ), where 0 < γ < min{ l2 − l1 2 , 2αl1 − 2l1}. proof. we note that the fourier images satisfy the equality ∂tf(w) + m11f(w) + m12f(z − p21w) = f(g11(w)w). hence f(w) = e−m11t ( f(φ1) + ∫ t 0 em11τ (f(g11(w)w) − m12f(z − p21w))dτ ) . thus, ||em11t(w − w0)|| ≤ || ∫ t 0 em11τ f(g11(w)w)dτ|| + || ∫ t 0 em11τ m12f(z − p21w)dτ||. further, || ∫ t 0 em11τ f(g11(w)w)dτ||2 ≤ const γ ∫ +∞ 0 e(2l1+γ)τ (1 + τ d)2||w||2αh2 dτ. using lemma 10.1, we find || ∫ t 0 em11τ f(g11(w)w)dτ||2 ≤ const γ ∫ +∞ 0 e(2l1+γ−2αl1)τ (1+τ d)2(1+τ dλ )2α(||φ||h2 +||φ||αh2 √ τ )2αdτ, by the conditions on α, γ, and φ, it follows that || ∫ t 0 em11τ f(g11(w)w)dτ||2 ≤ constγ ||φ|| 2α h2 . estimating the second term, we find || ∫ t 0 em11τ m12f(z − p21w)dτ||2 ≤ ∫ +∞ 0 e−γτ dτ ∫ +∞ 0 e(2l1+γ)τ (1 + τ d)2||m12f(z − p21w)||2dτ. cubo 12, 2 (2010) the maxwell problem and the chapman projection 297 we note that m12 = λ12. thus, ||m12f(z − p21w)|| = ||z − p21w||h1 . using lemma 14.1 and taking φ with sufficiently small h3-norm, we find || ∫ t 0 em11τ m12f(z − p21w)dτ||2 ≤ const γ ||φ||α+1 h3 . which implies the required assertion. remark 12.1. applications of the obtained results to models of continuum mechanics can be found in [11, 13, 14, 16] acknowledgments. received: july 2009. revised: august 2009. references [1] g. q. chen, c. d. levermore, and t.-p. lui, hyperbolic conservation laws with stiff relaxation terms and entropy, commun. pure appl. math. 47, no. 6, 787-830 (1994). 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[19] e. v. radkevich, problems with insufficient informationabout initial-boundary data, advances in mathematical fluid mechanics (amfm), special amfm volume in honour of professor kazhikhov volume editor(s): a. fursikov, g.p.galdi, v. pukhnachov, birkhauser verlag(2009), to appear [20] l. hsiao and t.-p. liu, convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, commun. math. phys. 143(1992), pp. 599-605 [21] w. dreyer and h. struchtrup, heat pulse experiments revisted, continuum mech. thermodyn. 5 (1993), pp. 3-50 [22] v. v. palin, dynamics separation in conservation laws with relaxation [in russian], vestn. mgu, ser. 1(2009), to appear. «jllllloo a l\lojhcma ti col .l o ur,1t1l \ 'ol 9, ,\' :j. '5). 1'rrrmbtt 1!007, co n c r ete a lge br a ic co h omo log for t h e gro u p (ir,+) o r h ow to o lv t h funct io n a l cq uat io n f (.r + y) f (x) f (y ) = g(x, y ) mi l iai p n111 · u ll o 1m'<'kt•r sof!wm't'l' nt wic'kluug, f'rl'ibur~. gnm1rny lt mlllllt' of ~latllt'lllll! lc's o f t lw n o mn11i nn ¡.\ cndrmy, dudum .. "sl, no 111 011ia . ~ lihni . prn1 1l'~t ll q.111 01 h.uni·frl'ibtirr.dt.• adst llact llw funr11uunl l"< 1uulio u f(.1· +y) j (.r) /(ji) 9(.r.lf) hm n lio lut.iou f th.11 im"lonlh 10 cº(r) if 11 11d o ul y if 1 ht' sy nunt'lnr roc\·ch9 ix'kmg.-t to cº(r'). lf tlw ~\muwtlic rory rl(• y ih ll'<·mi;ilvdy n1>prc-)(1111nblt•. thr«" t"xl""t 11 so l111l ou f v.h1~h l .. 11..:-t1,...1h•ly 11p p1u xh11111jl e 11 lso. ir 9 bdon1ts lo c 1\rj) then t l 1('1~ cxl~l~ tu• 111h'1't11tl t·xpu'"s.'lio u in !i for 11 so luti ou f tlmt bdon~ to c 1(r), oud th1: s11 m t• h.1p1wru for tlw cln.,...._~ ('~, ('"", 111111l y t1 c 11nd po))iu'llllmi re ' 11en l\ t"("u.-.c1ón funrio nnl f (.~ 1 y) /(.r) /(ji) 9(.1:. ji) tjt·•h' unn so lu ció n f c¡u• p. rl<"llt'c"\' 11 c'°(r) t1l .v 11610 s i el codrlo mllltlnco 9 ¡>1:rh'llc('l' n cº(r2). s1 c-1 coi lo ""lm1•1 1 ito f1 ('!! np rox i11111hlt• rl.'l'ul"lll\'i\11\l'lllt'. t'xl!ill.' unn !>0l11ci ó 11 f 111 tll•"'i l.lmhll'n 1 nproxi nmbl l' n ·c urslvn rn l' ntl.'. s1 9 p<'rtl.'nl"n' ft ('1 (r 1 ), l' ui o il cl's t'u"'"" una 4'"\:jltt""ión inl('~rn l l' ll g 1mrn 111111 sa luc10n f qut" pcrl<'n (' 11 c' 1(r) y lo> llll'iukt •uc('(lt• ¡mm ln.i dm•l's : ('4, ('....,, 11nulh1cn, 1><'hnom1al k t')' w rd ,. 11 11rl p h rn 11c11: 11/grl>mtc c'ohomoi091,f\iru:-1tono/ 8quot1011, j\11ol¡ir1c prophc• \imh . s ubj . ' l11 1u1.: 39805, 39821 iu i\ lil ml prunr-<11 l in t r o l u ct i o 11 tlw 1•xi. .. h.'uc'\' oí,, f1111 c li o 11 j(.1') su t b¡ fyiu~ 1lw íuuc'lionul lot1uo1\011: f(.•· 1 y ) j(r ) /(u) g(r. y) is ich..>ntical wi1h ! lu.· 2.1·o bo1 111 d11r y co 11d ition ror lht• funrtion g{r,y), fl.'i dcfin1.:d in t lw 1ll1itehrnicrohom log,v o f nlw l i o1i g ro 11ps. thi:; th rory ~i\'rs in gl'11ernl no1h·o ns t 1ucti w ¡>roof" for the exb tt •!lt'c o f 1 lt e so l 11t io 11 , nnd s111tfü.~ thc' oru.111cle~ fo r 1 lw funnionnl t'<¡muion to bt• :-0111\ik· (ii1 t..lil' so·cidl rd cohomolof!,i<' no n· trivinl c'n .. -:cs). i\ ly gon l h(·r~· i~ to study .rn11 lytk p r o pe r1í t's o f lill' :so lu1i o n rrnd its l'xf)n.._,_..._._¡bility i11 !ht~ r ~·ril rn .. ~· thl' word ro11<:rtlc' tt sl'd i 11 lli e lil,ll' c:fln he nlso undel'!i tood ns t. he co 111hi nntion of cont111uou .. • md d1.scmle. in ¡v'nc'ral, ic:'l /\" 11 11cl l ... \w 1.wo fl lw li nn group:s nnd le1 g: l .. x l ¡,· ben fun rlion tr thnt' i-;. ,, fonctio 11 f : /., _, ¡,· ver ifyi 11g lhe íunc1 ionnl ec:¡u11tio11 for 1\!l .r, !j e /_, t ht1 n g(.r.y) tull:>l \"er i(v 1hc fo ll owi 11 µ, co ndit ions: g(.r. y) must bc' s.)'lt lll l('1,r ic, t,lrnt. is: g(:t ,y) =g(y,:r), g(.r.y) m11 s t bl· n 2-cocycll: nrcordiu g to the trivinl nct ion of l 1~ /\" , llmt is: 'j(.•·, y) 1 .'!("· + 11· :) = g(.•·, !i ... :) + 9(y, :). \\"l• ob:-.l"' rvc t lrnt. if fu : ij f ¡,· i:i n purtic11h1 r solulion o f the fun c1 iono l ~1quntion, then the se t of nll so l11t.io 11:; is {!0 +-ó1 ó e l lo m (/_,, /\' )} = /o+ 11 0 111 (/_,, /\' ). thc> cocycle co mlit. io11 for y = o givc.s g(.r,o) = g(o . .:) = g(o,o). 11 c c11 11 11lw11ys sup1~ tlmt g(o,o) = o. l11 deed, if f is 11 soluti on o f tlw funct io na l r.>qm1tio n, th t•n /(o) -g(o, o). 1 f y(o, o) io t,l1~n we re plncc g(.r, y) b~· g(:r. y) !j(o, o). tht• 1ww t''<(uotion hn."' c'xnct ly tlw so lu1.i ons f (.r) + g (o.o ), whcre / (.r) me tlw m> lu tions for g(r.y). tlw íollowi ng fn ct s 1n c provvd iu !=ji. pg. 2:n 2j9_ th res ult s go bnck 10 eill~nhcrg ,uui ~ll1c l.amt', . ..;vv [2]. lfg lx / .. -· h' isnsy 111 111 0l l'i l'cocyck·wi1hg(o.o) o th1111tht•!>t'ig: /\" x /_, with tlw o¡wrnt ion ( 11 .. r) o (v,y) :-=( u + ti+g(.r,u).r + 11) i!\ an nb>hn n g ro up ~uch dtllt tlw nbl.'lio n g rou ps /( 1 g' 1111d l.. forni n ~hort t'xllct -..'quenn•: 1m:ordini;t to thl~ t'mht· ddi11 g 1: u /\' ..... (u.o) e g 1\ml to .ju• projl'<"ti 11 ¡1: (11,.r) e-.. .r l. in 1h is s it u t1 ti o 11 olw :mys tlmt ' i!; nn f"xl' il .. íoll of "' by l ... two t'xll'n'ion ..... g 1111d g' 1if f\' by /_, nrc called ~"(1ui,'í\l1•m 1( th• 11• is 011 iso morphis 111 of 1\bdi.m ~toul)s t.¡! : e . (,'' sur h tl \l\t 1' ~·1 nnrl ,,. ,,_ lt:t lls dt'lloh' i; impl y lw /, l ilw trh•ill l l'x k11 s io11 of ¡,· h~· / .. , rom~pamlmit 10 1lw s.v um1l'lric coc,•dc' !j(i,ji) o. th1· t'x h •ns io 11 gis 1-'<1ui \o¡\l1•nt with k lf l 1f .u1d u nl y if tlwn• j¡.¡ nn l..'()lluuj)hi-.m ' : c • /\' x /., of tlll' forlll 1,•(u,.r) /(r}, r) if lilld o nl y lf wllo 11. ll~l>\hl ( 2 1) •¡(l. 1)) "c¡11.lhtw,/u{1+ l) /n(1) fo( ! ) f/ (1, \ ) íor1 lton l o1w gds thc t'x p rrssio11 fnr n o. on tlw o tlwr htnid /(o) o nnd /( l ) /(l) g( 1, ! ) 9( 1, 1). ·-· thls \'•lhl• l .. !h. -.ion. 1tnd 0 11 the ot her h1111d \\'i' kuow !h.,1 1lwr1· 1·xbt¡.; 11 soj111 k.1 11. 1 e. nm1>ll' l...t ~i z >< z • z bl' giwn by y{r.11) ry_ tlds funcl ion is 11 !()'111111nrir ('ol:\'fh'. 1\ sol111io11 f: z · z b ~iwn hy /{n) " 1"l' ij. tht•:.t• lll't' lllt' l111ml(ulu nmnhn~. t'x lt•ml1•d owr tht• whok• z. ~ow lt•t q r '< r • r kivt•n 11¡.;11i 11 hr 11(r 1,1 ~.\11 íu nrtio11s f: r • r ~in.•n by /(r) ; +-11• , ,.,,,, / r · r 1/ 111ul 011/y 1/ 1/1(' -'jl"1r11ctnr cot"ydr g : r'l · r 18 n 12 ~ liluu pr m ... u rv11lmuo j 1"• lrnrj / ri 1111 .~ rww / or 11jl jo 1o /u • rl tju:: jollo1rmq l .\ t n u ' /"t 11tl. 11 'r tllf~ r rnt~ 1j·1" ·t111 1w1' 1·1111 f 111r wu\ \ulul1m1 j r • r , ,u-h tlwt /( .ru) proor h f ¡..;ton; i nuo us, t ht· 11 nis() y . 11p1 1h, 1t g l" 1' co111in110 11 -. s y 11111 wtri1 l()( n j,,_ "•lhout tt'sl rk ti11 µ; ! lit• µ;1 · 1 u •mllt~· w1lh g(o.u) u. cm1 .. 1rm1 tlw .. 110rl t•.x, 1t t ""1111'111 •'uf lopolo¡.i, kn l µ; 1oup.'i: · r • -r ·ll ll 1·n• ¡_.. (; r x r with tht· l'lwlldi 11 11 topo lqky .u to ló~i c';\i gro111 >: tl 1t' i11 \'t·1" ' (u,.r) 1 • ( " !j(.r , .r ) , .r ) i:; n l ~o ll ro nt111 111> u-. •'plllit,hió n . tl w (• m ht"t ldin r 1 ". ¡, . (u.o) e e 111111 tlu· pro jl't'li oll fl: (11 . .r) t g .r e l .. lllt' ho mnmor ph bm .. u f 1 opul~1l,tl p,ro u ps. ;\('1'0l"di11 ¡.i; 10 u fundum t• nt .11 th t'o rt• m o f ,\ hn·ko ff (:.t'i.' 1·1jl 11 1opologir.1l jtruup i:-; iso 111 o rp l1 k wi1h son tt' t•ur lid in n group {r". +. o) if nml on lr if 11 i:,,bdi.\n. 1-1 ,m~do rfl", locn ll.y ro ih]j flc' t , c'o ll llt'<"l t') .rz. w hich i~ unlxm nd c'ci. lf .r o tlt t·11 7r 1(< (u,0) >) uz w h k h is al-.o 1mho 11 11 d.-'ioj!;ir.,1 ~roup~ i.p: (.' • ir.2 . lni m .;1{r) '' 1111('l'lo1 ·. /i11r . l ndt,'(i. ,;1(qr) qj,t11(.r ). [f t ht• dn~>cl sub~roup ,:1{r) c'oll lllins r· lh w11r h• indt • 11':'mlrnl t•l1·1111·111i. 11 1 .111d !j'l , 1 ht• 11 i1 w~m ld ro nl a in 1 lw .,.., o f llll rn l io nnl rnmb iuo t ion !i !ji • qy1 .lnd it i. t'l os u n., so i1 \\'\"l til d ht• llw w holt• r1. whic h is 11 r o ut rndit 1iu 11 . so .;1(r) l .. 1h1• to1w lo,11.k nl clos 11 n • of q.,_"1( 1). whi r h is,\ n · \i \'i'<" l <'l r· lhw . 1 0 1w r.111 i.up pnh• t lt nt ip1(1r.) / {o} x r; if not , 'n' :-; ul~lltuh'.; wit h to{), wh t• n • t i!i ,\ '>llmll tot,1• ion c'o ns ick r 1lu• 11 p pl itatio n 15: r · r giwn hy j(.r ): p..; 1(0 , .r ). j ¡... '' honmmot)l his 111 o f !o po lo g k rd al'oups, so i s nddiliw .u u l ro nl inuous. t his 1111 •1111 s th.n ,¡ ¡., ,\ ro n1in11 ous :-;o l111i o 11 lo r 1hl' fun r tio n11 l l'<¡muio11 o f c'a ur hy 6(r +y) d(rl · j(y) owr r. ll1 •11< 't' 11t t' l'l' is 1m 11 r !'1 11 h thl\l d(r) a.r, ,111 d a ¡ o h1't' í\11 -.1• t" 1 {o} x r) "/. kl•111 p . \\',, ro1i..tn11 1 11 11 11 1ipli rn ti o 11 () : r 2 · rl s at is í\·ing tlll' foll owin j:; nindl l io n:; : o(;<(r)) r x {o} ; o.,,.¡ 1) (l.o) ; o ' 101 a (r1 r). u thl"' '-" 1lo1w lw 1lw liiw111 nppl icn ti o n o s11d1 th nl 0(~1(1)) tj o} .111d fj( o, 1) (0 , 11 ). 11 l. ... 11\ 1somorphis 111 o f 1opoloj.1 ir11 i p; rnups. \,ji" fl.;. t' : 1/11 nud ¡/ : fj\.'. 1 th11 11 t'(u) (fl. 0 ) und 1l{ u, r ) z, m jmthrul111 ,,,,, o. lt fo llows 111111 \.' ii. 11 11 1 morphi.."111 lw tw ('(' ll tl w r xn r t .. bnrt 11!11("(111• nct-s of l 11pulog if'n l p; rot tps (r,1, g.p. r) .111<1 r. 1'. rl , p ', r) , so t h1•n • i!'i ll runtmuou .. fun1 1lo 11 f : r • r !l uc" h 1h 1l for 1l ll (u . ..r) e r 2 11 hold s \.1( 11 ,r) {u / .r) \ flo lrll11 ¡i. 10 lh l' tl's ult s c111oh'fl in 1 h~· hi11odu1 11011 , tlw <' 1'h l'o l'l' 111 2.2 jf tl1(' ro 11tn111n11.'i s1111w1ftr'lc rocr¡clt' g r~ r u rrc11r . .,11rcly o¡i1irnrir11t1hlr tju11 1111 rr nn· ra11ltr11w11.~ mjlullori. / • r • r uluch ort nl'lo ry'('tir.vwcly fll'/lnliltllllbh proor ld / r • r ht• auy so\ 111 io11 of tlw f1111 ct1c"11mi c-qu1uion. as wt' kuow, / fo + ts. wht•t\' / o is 1\ c-o1h inuous ik>lutio u nnd 6 nu .ulchtin· hom()lnorphism of r. lt fo\lo,,-s tlmt f lo / o lo + o.r lo fo t wmt' fl e r th b nw.ul.! t lutt / lo : q · r is 11lw11~·14 c-ontmuous ou 1 lu• o t ht'i' l111ml, l'onlinuo us soluuon .... dt·fuwd ow•r q or ow1· r llrt' unh11ll'ly dt1h •rmhll'd hy fl vn llll' in sollll' jo--/: o. ror t·:i...'lmplt• h y f( l ). l.a·i n z lw 11 ndk .. ul>an.l ujl ()f r. onsidl'rlng tlh' :olmllnr íunc;tion,\i t-qmu ion co11t'spo 11d i11 ¡:. to ,,¡..,7 n1 tuul lh<' form for t lw solution gi\·l'll in tfl(' rf'nmrk 1.2 writtt: n in inll'¡.tl'r 11111lu11h-.. oí o "'' ""' tlmt t l li!sl' dist·11·ll' 1'0lu1ions m<' ill."o uniqu<'iy clt•tt·rmin('d by /(1) l1•111111n 2.3 ú t q r x r · r. /j1• a s!111w1(n) n m l / 't hy thc vnlm' / 1 (~l but / 1 1 .. z is n solu tiou for tlw ... \m• lll'ohlt-•m 11s fn· l h.•jll't', lhl· o n\_\' thintt lo do¡_ .. lo choo~t· /11(;) sud1 llml f i (n) /,... (n ). oy so lvlng t lw l''t ll íll io n f 0 (0) 2/ '1 ( ~) g(cl, ~) o m · g t'i s 1 he \'1\l1w /~( ~) / n_(_o) _ _!l_(_o, ~-) so. wh.u '"" imw 10 d o, í!'i 'º conslruc't 1lu.• ~'qllt'lll"t.' or di..<;,('h'll' í111 1{' tioml /1 , /~ . ft· • j. , wl1 h th(• p ro pc•rty thnt n ll /r}n 1 .z. f.;.,· t lwy iht' nll n·~1rk1 io1l" of tlw ro1ttin110111! solu1ion /: r · r dt•tt•rmim t b\' /( 1) / 1( 1), w hirh 1111.11 to lw aj,;.. n ,\ ft'('\lrsin• n·nl. tlw 1111io n o r nll tht"'4' domoiml f\rt' llll' d ynclic 1111111lwr 111·h1dt .u1• dl'n~ i11 r. 11t1cl tlw u11ion of nll grnph_.. i.-. dt·nm' 111 tl 1t• gropl1 o f / . sn fe m b· u'cut!->iwl~· 11pp1oximntt·d. 1 3 las c 1 a n d m o r e lm u m n 3.1 !al g rl • r /ir o .~ymmrtr1r roryc/( o/ ria.~~ c 1 t/1rr1 //ir /oflowm9 11frnm1n he d 11 ~ lll1111 pnu.ll • i g{r o) g(u, .) •¡(o, ll) . e. (1j1g)(u.1•} {ljj.fj){p,11) . j. (ij,g)(r.y) (ll,11)(.•· 1 y, o) (i),11)(y.o} (llogl r.y) (ij,y)(o,.•· 1 11) (81y){o.r). proor poi111 1 lms ht'~'11 w ow d i11 l h<' introdun ion point 2 fol\ow~ by ~yn111w1n· point 1 follow~ fro 111 2 11 11d :1. to prov1..• :}. con:-id<'r th(' following rcfo1mul111io11s for 1 lu.' l'ocyrh.'-1o:ic'l111 1 fo r .::: / o: y(.•·, /1 1 z ) y(.•·, y) g(r +y.:) g(y. :) '!_(r-j1_+•l _y_é•·, 111 y(.•· i·y.:)-g{r+y,o) g{y,zl y.(11.0l ~lnki• now ; o 1111d n.•t·1t!l t,lmt, fj e (' 1. h follow~ (8,y)(.•·, 11) = (8,y)(r +y. o) {ij,g)(y.o). thcor e m 3.2 tl1r fr111 f'l,irm ol eq1111l.i1.· 11 s)' ltl lllt'lric co(')'c k· o r c)n.._,, c 1 wit h 9 (0,0) o. tnk1• f to bt• lhl• func-t ion µ; iw 11 i11 tlw sll\l ('ltwlll 1\nd ron!-ickr tht• íu nct i 11: ll{r.11) / (i + !j) j(.r) f(y) , üí ('o lll's<.', /¡ is(\ :<)'llllllt'lric cocyclt>, hlld fl íu !lc'l ioll o f dll'-" c 1 by apj>lying lr11w 111 :1. 1 st'\'t'ro l l imcs, onl' rom¡>tth°!' (iloh){r. y) (0211)(...+ 11• o) (8,<¡)(r,o) (il«¡)(o.r ~y) (il1g)(o, r) (il1 •¡)(r, y) (cj l<)(..-,y) (8,•¡ )(., +y.o) (111•/){y.o) • (8,y)(r,y) l.'l't 110\\ l(r,11): (11 y )(.r,y) c 1 ll1'<'0u~· (éj1ll(.r.y) o 1md (c-jil)(r,y) o, llw íuu,11011 l(r,y) mus ! b~· co 11:rlngcr vcrlng, 2000. cubo a mathematical journal 2007 v9 nª3_0047 cubo a mathematical journal 2007 v9 nª3_0048 cubo a mathematical journal 2007 v9 nª3_0049 cubo a mathematical journal 2007 v9 nª3_0050 cubo a mathematical journal 2007 v9 nª3_0051 cubo a mathematical journal 2007 v9 nª3_0052 cubo a mathematical journal 2007 v9 nª3_0053 cubo a mathema tical jo1wnal \'ol.05/nf103 october 2003 proofs for the limit of r a tios o f consec utive te rms in fibo nacci seque nce c /1110-piug c he11 dcp"rtr11e11t of tl pplitd m" themat11.:s m1d jnfom1atlcs. jwo;uo fris t1t11te of tedm ology. jwoz uo c:ity, f fo11a11 454000. clwia a i-qi liu smu11e11xi.a collegc o/ voc(l f101wf tedmology. s"mnena:ia oit y, he11 afl 472000, clww (lll(/ fe ug q i depa rt me11t of a pplicd matl1wwt1cs a11d lnformatics. jiaowo l mtit rd e o/ teclmology, jiaozlllj cit y, llenan 454000, clima anstll ac't. in thc short note, six proofs fo r l hl• limit of rnt ios of consecu tivc tcrms in f' ibonacci .sequcncc nrc providcd , includ ing usi ng wcicrst rass.bolzano theorcm , series nll'thod . by dl'fin it ion oí lim it , diffcrcncc cquat ion mcthod, matrix mcthod , a lgebraic 11wtl1u(i. :moo 1\/n1h,-,,.n11r• s ub1u l cln.t~tjicntion. primnry 11839. j..·y u'onl• rrnd phrcur:s. llnlio, fibonncci numbcto 1 fund o f .j111ozuo l n~1itul f' o í tcchm1ioll.,\'. ('h1111• 211 l.,imit o( 1j1e ra tios of co11secutive ter.ms in fibonacci sequence introd uct ion lt is well -kn owu that t he fibonacci sequence {f,,};;",,, 1 is defined by the following recurren ce formul a { fu+2 = fn+t +f." ' f1=1 , f2 = 1 we al so cal\ fn t he f ibonacci number. its general term can be expressed as _ 1 [(l+,/5)" (i-./5)"] f,,-js -,-,for 11 en. (1) (2) th e f ih onncci mnnbe rs give the numbe r of pairs of rabbits n mont hs after a singl e pnir begi 11 s br ecd in g (a nd ue wly born bunnies are a.ssumed to begin breeding when they are two 111 0 111 h!' o ld ). defin e f .. +1 x,.=---¡;:, (3) t he n t he seq ue nce { :z:., } ~= i converges , this limit (•1) i!' ca ll cd go ld e n rat io. thc rnt ios o f nltcmat.c fibouacci umuben; are givcu by the conve rgeut.s t.o efi 2 uml urc said to mensure t he frnction of a tum between successive lea.ves 011 the ·sta\k of a plaut ( phy ll ot a.xis): 4 fo r elm and linden, & for beech aud hazel , ~ for oak a.nd apple , ~ for poplar nu d rosf'. fj fa r wil! ow and 11.lmoncl, etc .. the fibonacci numbers are sometimes call ed pinc c'u 111 · nuru hcrs. the ro le of t.h e fibouacci muuhers iu botany is som etimes callcd ludwig 's l. nw . to prove t hc convc rgcuce all(] t.o salve its limit, oí the sequeuce {x.,} ~:= l is ll stnudnrd 1•x1• 1-ci~· or cxmu plc' i11 calc nlus aud 111111,hematical una!ysi:-; íor grnc.hmt c stu{'l"i l's 1111•! hod. b,\" 1101 11101101ouic . using fonnu la (5) agaiu giw:s lb .i:,.2 = l + :c.,l-1 = 1 + 1 +! t, = 2it·: : .. ! . (7) 2:i:,, + l 2x,, _2 + 1 .e,. .cn-2 .c,._2 i · ., = ~ l + :i:,.2 = (.c.,+ i )(.c,,_2 + ! )' (8) sincc (x ,, + 1)(.c11 _ 2 + j) > o, t hen (x,,+2 :i.: 11 )(:1:,, x,, _2) ?: o. this implies that the scq uences {.i:2,. _ 1 ):c= i a nd (x2 ,, }i.'::: i are al! monotonic. in fact , by iu duct ion, we can prove l b a t {:i.:2,,d i,= 1 is increasi11g a n.: 1_ 1 !x.¡ x2j, lx2k-4-l -x2k-1i s 4k. 1_, lx3-x1i, (12) (13) where k e n. by propertoies of series wi th positive terms , it is deduced thato the sepies l;;o=1 (x2 k+2 x2k) and ¿:;;o= 1 (x2>.:+1 x2k·-d a11e all absolutely convergent. since ' xu· = x2 + ¿(x2; x2(i-1j), (14) i=2 ' x2k-i =x¡+ ¿(x2i-l x2;-:i) , (15) i=2 uhc sequcuccs {x2d f'= 1 and {.t:2l·-dr.; 1 c011verige. the rcst is sam e as t.hat in the fi.rst 1~ro0f. o the third proof: by definition af limit. note tihat the equation x = l + ~ corresponding to .t,.+ 1 = 1 + -!_: for n en has unique positiive root ~· by defü1itio11 of limit, using the result l ::; x .. ::; 2 for n e n, we have thch l. 1+v'5 1 (j5 l)"-'1·· l+v'51-(j5-l)" o .i:,.--2~ -2:j,¡-~ ~ (17) n~" ce. tl1crf'fon• li n1,,_«>:r., = 1±f· o the fo nrlh proo f: diffc r e nc e equal ion method. fr1>hi v .. ~ = ¡. ~·1 f r ... \\' i ' nl11111 11 cjrno-pi11g c j1e11 , ai·qi uu & feng qi bhc linear differencc cquat.ion o f second order with constnnt cocfficients: fn+2 f .. + 1 f., =o. ( 18) lts cigen<•quut.iou is ,\2 ,\ 1 =o. it.s e igenvalues ni-e ,, = 1 + j5 '" = 1 j5 2 1 • 2 the gencrnl solutio11 of t.hc diffcrence equntio n is (l+ js)" (' -js)" f,, c1 2+c, 2 . by the iu it inl condi t.i011s f1 = l llltd fi = l. we o bt nin heuce f =_!__ [(1 +j5)" (1 -j5) "] " j5 2 2 nml [1.±..>d] .... [1=1] "+' f .. + 1 2 2 _ 1 + /5 x,,=t.= r~rr~r 2 o the fiíth p rooí: matrix m cthod. i!: is ~ns.v t.o sf'e that (f~·-') = (' ') (f•.•+ •) = (' 1)" (f') = (¡ 1)" (¡) (lg) l 11_ 1 1 o f11 l o fi 1 o 1 for 11 e n. ll't a= (: fi). thc11 t he eigcuvalucs o f t hc squnre mntrix a are ,\1 = ~ aud ,\1 = 1-=:p. t lu·ir corrcsponding eigc11vect0rs are ( '=f!i ) ( 1=/f' ) 111 = . 0 2 = 1 1 (20) 27 28 umit of rlie rnr.ios of co11secutive terms in fibonacci sequence tnking ( !.±>d ~) p , , ' 1 1 t hcn ~his imp\ies a"= p 2 " pl ([!±.i!j" o ) o ['-=f'] , , 1 ( [1±.il]""' [~]"" ~ 7s [ "i']" [ '-=f']" ~[~r+l -~[~r+l) ~[~]"-~[~]" . tll ld thus __!__ [(l + v's)"" (1 v's)""] f,.+ 1 j5 2 2 ' thnt is . ~ __!__ [( l + v's)" (1 v's)"] f,, v's 2 2 for 11 en. tht• rcsl is snme ns ~hnt, in the tlhird proof. the sixth proof: algebrnic met hod. le t us rewri!.c f .. +2 = f11 1 + f,. ns tlm! i:(21) (22) (23) (25) (2g) o clwo-pi11g c /ie11. ai-qi liu & fe11g qi 29 lc1 { p+q = l. /)(/= l. (29) lhcn wp obtam solu11011~ of (29) as follows { p= ll;f', cr, t.hrcc prooís for an identity involving deriva t i ves o f a poti.iti\'c dcfinit.c mntrix nncl its dct.erminant. a.re given using tcchniquc of hnca.r algebra. t hc idcnlity is bn.sic in diffcrential geomctry 2000 molhcmatia sub1ect clas:iification: isa is, i5a24. keu worw ond phrwes: l clcntity, positi vo clefimtc malrix, d et.erminant , deri va t.i ve 1 int roductio n let iw be an n-di me ns iona \1 n .$ 11 co nnected, c 1 rie.mannia n ma n ifold . for d efin ition of manifold , piense re~ r t o s tandard texts fl, 4]. the riema nnian metric on m associates t o each p e m an inner product o n mp , which we d enote by ( , ). t he associat ed no rm will be deaoted by 1 i· t he rieman nia n metn c ls c m 1.he s nse lha l if x 1 y a re c vector fields on m , l hen (x, y} is n re.al-\'al ued funct iou on m . "th~ au'bor wm s uppor tcd in pnrt by nnsf (# 10001016) of china., s f for lhc promincnt youth ol hcnan provlnce, ns f of hcnan provincc (#~051800), f fo r p urc llcscnrdt of nutur&i so0>ct ol the educnt lon dcpmtmcnt oí hcnan provioce (# 1999 11 000•1), doctor f\md ol j..oauo lnft.itulc of tcchnology, china.. 225 226 bai-ni guo let u be an open set in m 1 and x ; u --t r" a diffeomorphism of u into ir" 1 lbat is , a cbart on m. then associated to tbe chart are n coordinate vector 6elds , written as 8/8:.r) oras 8;, j = 1, ... 1 n . for lhe given ri emannian metric, define 9;> = (8;,8>), g = detg, g = (g;>h9.> $•• a-' = (~'h9.> $• • where j , k = 1, . . . , n , det g and a1 denote tbe determinant and t he inverse oí g respectively. it is well-known that g is a positive definite mat rix . see {2 1 pp. 37j . the íollowing id entity involving de ri vatives of a posit ive defin ite matrix and its determfoant is fundamental in differentiaj geometry. theorem 1 por 1 :s j ::; n, we have tr(g18;g) = 8;( 1og). (1) in th is s ho rl note , we wi ll g ive t hree proo fs of t he identit.y ( 1) us ing different technique oí linear a lge bra. fa r concepta of linear a lgebra, picase refer to l3j. 2 three proofs of identity (1) firs t proo í. si.nce lhe metr ic ma trix g = (9ij) is a pos itive definite matrix 1 then we can assume its eigenvalues oí g are >.¡ > 0 1 i = 1, ... , n . from t heory o f linear a lge bra1 we ha.ve g = det g = igi = i1 >. ;, (2) i= i lng = ¿in >.¡, (3) i = l " 8 ·>. · a;(lng) = i: t· i = l 1 (4) where j :=:: 1, . .. , n . fu rlher, t here is a n orthogonal matrix p such t hat ) = a, >-n (5) t hree proofs of an identity involviog ... therefore, ,.-e have g = pap1 1 c1 = pa 1 p 1 1 and 8, g = 8,(p/\p1 ) = (81p)/\p1 + p(8; a)p-1 + p/$8,(p1)), c-1(8;g) = (p/\ 1 p ' )(8;p)/\p-1 + (p/\1 p 1)(p(8,a)p1) + (p/\ 1 p ')(p/\(8;(p1))) = pf1 1(p18¡p)/\p1 + p(/\ 18, ¡ p • + p8;(p1 ). l'rom p • p = e, it follows t hat (8,(p '))p + p 1(8, p) = o, thus (6) (7) g 1(/j,g) = (pr1)[(8;( p1))pj(pa 1 ) 1 + p( 18, a)p1 + p8;(p1) = (pa1 p ')p[(8;(p1 ))pjp1(pa 1 p -•¡-• + p(/\ 18; /$p' + p8,(p1 ) = g(p8,(p1 ))g1 + p(a18,a)p' + p8,(p' ). (8) sing 1he formula o, it.s element a,1 u o /uncl 1on o/ x, lhe r1 d( ln iai) = tr[a ' da] = tr( d;! a-1) . dx dx dx (19) t hree proofs of an identity invol ving ... r e m a.rk 3 lct a(t ) ia an in ve rtible differentiable molru-, lh en (d el a)' = (d el a ) lr(a1 a'). where a' dt nott:j th e derivati ue o/ m atrix a unth rupo;t to t . 229 (20) t bird pro of. let e· = (g11 ) d enote the adj oint of the positive d efi nit e mat rix g 1 lhen g,1 = g¡¡, nnd _, _ g' 8¡g _ t r(g' 81g) _ ~ ~ .. tr(g 8,g) t r igi igi ¿_, c ,,(8,g,,), 9 1.1=i (21} 8 g 8 igi l n l n 8,{ln g ) = :e. = =.! = 8; l y11 g11 = l: [{8,g,,)g 11 + 9118;g11]. 9 g g l = l g l=i (22} the prooí red uces lo p rove t hnl (23) l -= 1 •-=2 1= 1 lu íact 1 wc have ¿g.,(8, g.,) = l l (8,g,¡)g,, , k = l , 2, ... , n. (2~ ) i.-=i 1j lc l= l t his compleles t he prooí. • r efe r e n ces [ij m p do carmo1 oifferential geometry o/ curves and s urfaces, prc nticellnll. loe., englewood c liffs, ncw jersey, 1976. {2j 1 cha\-el, e 19cnvalu es in riema nruon gcornctry, acadcmic press, 1984. [3j a ramachandra h.ao and p. bhi masankarnm, lmeor algebm, 2nd edition, tcxls aod readi ngs in mathematics 19 , hindustan book agency, new delh1, lodi•, 2000. 141 f w \\'aro r , foundations o/ d1fferent1able mantfolds and lie groups, cradua~ thxts in mothemnt ics 9 4, pringer·verlag, 1983. c hina academte pubhshcrs, bcij ing, 19 3. cubo 11, 01-06 (1995) recibido: mayo 1995. a propos des algebres pondérables.* cristián mallo! abstract. we present the simple notion of coding t hc product of an a.lgebrs. in arder to construct "canonical" embeddings, by adjuction of an element, of any algebra into a weighted algebra. a descriptien of tihe possible ex-tensions is given. 1 introduction pour des ra.isons de simplification d'écriture ce travai! est présenté da.ns un cadre commutatif; les idées exposées par ja suite ma.rchent aussi bien sans cetite condition. soienl k un corps commutatif infini, car {k):¡f 2, et a une k-alg(::bre commutative. on (a ) car l'ap plication w(ae + x) = ,\ ou ,\ e k et x e e, est une pondération. • da.ns les a lgcbres pondérables il est souvent nécessai re de voir si elles so nt uniqucmcnt po ndérées [7]; la proposition qui suit nous fourn it un critcre simple. propositioo 2 . 2 soient w et w' deux pondémtions. les a.ffirmation.s suivan· tes so nl équiva/en.tes: (a)w = w' (b) ker (w) e k er (w'). démonstration (b) =>(a): de 2 . l.d , il existe e e a, w(e)-::¡:. o t e! que e2 -e est dans ker (w), d'oú a = k e(fj k er (w) e t forcé mcnt w'(e) #o (sinon a= k er(w') et done w' =o). ii s'ensuit que w(e) = w'(e) = l et k er(w) = i< er(w'). • soit a une e..lgcbre pondéra.blcct noti;:ins b(a) c a' ]'ensemble de ses pondé rat ions. propositlon 2.3 s oit {w;/l :::;: i:::;: p} e b(a). ( a) pour lo ut j, il. exmte e; e a tel que w;(e;) = ó;j pour tout 1. (b) 11 existe e e a tel que w¡(e) = 1 pour tout i.. démons t r ntio n a) par réc urrence montrons qu ' il existe e.1 e a,w.1:(e1) = 5,1:1. s upposo ns que w 1(e) = 1 e t w2( e) = · = w.1: i(e) = o; si w1:(e) #o com.mme a propo.~ d es ... cubo 11 k er wi) #=k er (w¡.) {proposilion 2.2) on pose e 1 = c-w(:z:)1:z:e a.vec x e k cr(w1)k er(w~) qusnt á. (b), il suffü de prendre e= c 1 + e2 + ··· +e, avec w;(e;) = 6¡; . • cor o llairc 2.4 si a c3l d e dimen.sion fin.ic , toul ensemble de pondémtions ••l ubre el d;m(nb(a¡ker(w)) = dim(a)1 b( a ) [. démonstration. (c. r. !ij. §7.5, cor. l e t 2 du thm . 7) • on rspcll e qu ' une algcbre s c a est une sous-&lgcbre pondérée de a s i elle csl pondérée par is restriction d 'une pondération d e a. le rés ultat qui suit montrc que si 1 b(a) i> 1, l'exislence de sous-algebres pondérées propres est garantic: proposition 2.5 soit w,)¡ une /amilie de b(a ). 11 y a équivtjence entre: a) 11 ez13te un e sous-algebre pondérée (s , "y) l elle. qu e 'y= wij.~ · b) // exist e e e a, tel qu e w;(c) = 1, pour tout i. dómonslration. (a) ~ (b) 11 s uffi t de décomposer s = k c(f) k er('y) avcc 7(e) = l. (b) ~(a) 11 suffit de prendre s = k ee n,kcr (w, ) avece e a . w,(e) =l. • cor o llairo 2.6 soit a de dimension fin.ie. s i (w,), e.,, l une famiuc de b( a} le.lle que n, k er(w;) = {o} , alm·s a admet un il = 2e:.-w(z)e'. soil y e ker(w)' on a(e +y)'= m(e+y) +q(e +y); mais m (e} =e! et q(c + v) = u2 d'oú m{y) = 2ey. le résu ltat vienten cajculant m s ur :z: = .\e+y. • proposition 3 .3 dan.s un e-coda.ge wm +q,w est u ne pondérotion si et seulement .riwom = w et woq= o. dómonstration si w csl multiplicative, de w(m(x)) = w(zez-w(x}e.2 ) on oblient w o m = w¡ puis de w(x2 ) = w(w(x)m(x) + q(x)), il e n rémlte w o q = o. la rcciproque ne pose aucun probléme. • thóoromo 3.4 toute alg ebre a pcut se plo nger da.ns u ne alg ebre pondérée ( e , w) j.a,i.c ct l' i1ijt..'ll.:ll u 11 u1. 11 0 11iquc a e e-j l un rnorpl.c:in~ j 'alk~i e.le plw:i , a propos d es .. cullo 11 co rn me les identit.és w o m = w el woq =o sonl vérifiécs, w est une pondérat.ion . • la démonstralion ci-d~u~ donne la mesure des cxtensiom pc:>ssiblcs h. reáji.,cr. défini r une cxtens ion k e ed a de l 'algcbrc a équivaut á définir une mult.iplication par e; notons exte(a) l'cnscmblc des possib\cs mu ltiplications le. proposit.ion 3.5 /., 'application ext6 (a ) axend( a ), /_,e (e2 e , 2l6 ), c3l bijective. dómonstr ntio n en cffct , son inversc cst l'application (=, m ) le, d éfinic par /, ,(e)= e+= el l,(y) = jm(y), y e a. • notons a(z,m) une telle exlcnsion . re m a r que 3.6 a) le théorémc 3.5 admet une r éciproque dan.s le 3ens que toute atgebre pondérée (e,w) est isomorphe a l 'en.tension k er(w)(c2 e , 2le) avec w(e) = 1. b) u" morpli~me d 'cxtcn.sion.s c3l un. mor-pliisme j : a (=, m) a(y , n) tel qu e /(e) = e et f ( a) = a; un calcul simple montre que / (:)= y et /o m = no / . ccci veut dú-e que le groupe aut (aj opére / .. a (.:, m ) = a(f(z)) , f o m o ¡ 1 ; l 'o rbitc de a (z, m) est / orm ée de tout.e.s 3 1!3 exten3ion...'i isomorphes, l 'application / étoblit t 'iso morphisme entre a (z ,m) et a(/( :), f o m o ¡ 1). r:) (.,es cztcn .. "io n.s ii irle mpoten.t sont celie.s de type j\ (o, m }; tous les itlempot.ents de poid.' non n.ul sont donnés par· e+ !u v(m + q) , ou / no( m + q) est l 'ensemble dw poinuj ftxe.s de m + q. p r oposition 3. 7 si a e3t n.on pondémblc alo,.s a ( : ,m ) c3l un.iquement pondérée. d ó monstr nlion pour toute 1>0ndérl\lion w' de a(z, m), forcément wía = o car si non wí a· scrait une ¡>0ndérntio n de a. 11 s'cnsuit que a = ker(w) c k er(omega') , done w = w' ( prop. 2.2). • commc ntairc 3.9 p lus ic\1rs &\gcbrcs pondérércs classiquemcnt étudiées, sont des cx t cnsiorui a(z,m) d'l\lgcbrcs no n pondérablcs. ains i par cxamplc, il y a équivalencc entre les ext ensions a(z,m) d es zéroa lgcbrcs el les alg brcs de mu tation, ccllcs ad mcttant un codage wm (c.f l4j e t [71); les extensio ns de type a(o, i d) étant les j a lgcbrcs gamétiqucs (o u el e berns tcin d órclre o). par 8illeurs, les extensions a(o,m ) des a lgcbra.s vérifiant l'identité d e ja.cobi oi'1 2f\.·1 est 1111 pro jectcm, cont ic nnent les a..l gcbrcs ele bern.,t.einjoral tmin alg ebm.s of mnk 3 nnd d.im en.~ion::; 5 , proc. of thc gdinb ur gh malh. soc., 33, 61-70 (1990). !3j ethcrington 1. m.h. gonot.ic a lgobrns, proc . roy. soc. &iinburgh, 59 242258 ( 19 39). l•ij mall o! c. , varro r. lc.s algé br'c.'l de mutn.lion, co mmuni cation au third ln tcrnat.io nnl confercn cc on non u.ssociat iv c algcbra, ovie0r ejemplo , parit ih:i, flc t iene que { x = i'... ;~· = i'... x.!'..;z =.!'.. } ax 8y {j= (}:. es bft..;;c de 113 y {o = d:1;; f3 = dy; w = d.:: + xdy} es base dual y ¡>0 r t.r n to las m ét.ricil'< in"nrirntcs 1\ i7..c¡uicrda sobre ll·l3 son ec¡ui va lc n tcs bajo isomctrin a. u n " de las s ic:uicntcs : clli.!. sea ade-m á3 , {en) } tam bié n es ba.je d e g. se a < , >! produ ct o i r1.t en 10 en g tal que 8 2 e~ ort on. ormal 11 <,>' m ét n ca in. variant e a i:quienfo en. c, dctenninn.d.a por < . > ~. en tonce,, (c.<, >1 ) c3 i.so m étrico n {g, <. >' ). demosu sció n . supo ngtimos pri mero qu e e es simp lemenle co nexo, mi d e (2. 3) ex iste ({' : g c automo rfis mo de jt?'u pqs de lic t al qllc d1p" = l/j, dond e 76 cubo 11 j. j.,cim 1) ___. (c,<,> 1 ) es unrisomctrín. ose>\ {d1p,,. (11), dj.p7 (v)}!(:i:) =={u, u)!, v'j: e g,'r/11 , u e 't;,.g . scj\n x = e (el e le mento ncut>r o de g'), u = e, , u= e,; e., e, e bi; se tiene (d(e,),'!> (e,)); ó,, (c,,c,}! (2) (pues b1 y eh so n or t.onor males respecto a <,>! y <. >~ rcspcclivamcnte). sean, a hora, x e e' u, v e t:i:g cll /\lcsquicr a. como l,. es di fcomorfismo, entonces: es isomorfis mo y por t.uinto existen 11 ', u' e t.,c tales que a sí tenemos : (d))~¡,¡ j.1u-go rp rs hn f\ isn rnr.llrín .. / j = (d j.,,,.)., 11 1 ,11 = {d/_,7),.1/. ((dl ¡.¡ -•) d'); • •í"i •(•\ {d ( f.,op(:i:)' o rp o/.,,.)., u', d ( l<.,:.(r)-' o cp o /.,z ),, u')~ {dvi., 11' , dip.,u')~. pues l....,(:r)' o i.p o l,. = <¡:> {111, v')! . ele (2) y bilincalidad ((dl,); ' u , (dl,); ' u)! (1i,11)! , p ues ja métricn es in,·arian tc 8 izq uie rda. ahora supon amos que g' no es si mpl emente co nexo. de ( 2, l ), cxis l e e grupo de lie simplcmcm c co nexo , c::ipacio el e c ubrimi ento ele g , tal que la aplicació n ele cnbrimie nlo r. : (; c es homomorfis mo de gru pos de lie. además, de (2.2)sc tiene que d'ti; : !j g es iso morfis mo (donde !j es el n1o?ebra de líe de g y c es 16 idcnticltu:i ele g· ). 0/&.1 y <. >2 m éfric n.'i i n.va,-in.nlc:j rt i ::t¡ 11ie r'(/n ctl c . .. 'ú (c:.<.> 1)c.! ,,,.omél r i'co n (g» <. >2) c 11 lo ru·e ... e.r1~d. t o nut o morfi.smo tft f nlgcbrn d e l u .. g in.! t/ll c s·i 8 1 = (c 1 , .•• , cn) c ... bn3c rh. _q .o r·to11or-r 11al 1u pect o ni protl uclo inlt..rno < . >~ 1m.g,8c l.ic n e q11 c 8 1 = {q(i 1) .... o(c,, )} e.'1 bn.:1c rh y. orton ormal r"t-'1p ec, f,o n.l protlw;lv i11 t.tw11 0 <. >'! de m ostració n sea 1p : (c , <, > 1 ) (c, <, > 2)isomctrío. , enloces o= d.tpe : g g e<: i:;om o rfi:-m o d e cspnci o~ vector iales. sea 8 1 = { t 1, •••• r n} ba..<:.e d e g ,ortono r mnl res¡>t'cto al 1>roduct o inte rn o < . >!en g. 8 1 d e te rm in a 11nn ba...:;c d e cam pos invnr irlllc:'i 1\ izq u ierd l\ en c, l e 11 .. . ,en}, dcflnidog por c.nto nce::1 t rmbicn es base de cnmpog \nvariiultcs a izq uierda. como dvj,,cs isomorfonno, 11 t1n 1hlo log s ímbolos el e c hris to íel , se obtiene que : d1>nclf' '(' <"" hl c"fl lw:.: ió n llif'1111umin1111 l' ll c . ,_\¡':f t " llf'lllo" el<" (5) y ele in !limctd n , (e.jd~,, (/~:j \ •• ·.4&.>"?. en t'íccto · cubo 11 79 (~(e,)'~ (e;))~ (d<,o, (e;)' d of tloe pol:ynomial tf>(t) is defin ed formally (w it hout a ny reference to limi ts ) fuy tille ns1!1 a l form ula . jf f (x ) is a poly nomi al ma trix t hen , setting x equal to d, we get a ma trix operator f(d) a li of whose en tri es a re poly nom ials f(d ) in d . for example, if i is the ident it.y matri x t hen xi corresponds t o tloe cli fferen ti ation opera to r (also d enot ed by d) which has d 1s clown the d iagolll al a nd zeros elsewhere. clearl y, for a ny f(x) , one has f(d)d = df(d ). therefo re, if f ( d )x = o t h ern f(d)dx = o as we!j. a basis (alias íltnd amental m11trix ) for f(d) is a poly nomiaj ma t rix (t ) s uch t ha t a ny polynomi al solu t.io n x (t ) of f ( d )x = o is a unique linear combina li o n (wi t h r ea l coeffi oien ts) of t he columns of (in other words 1 t. he columns of a re a basis fo r t.he space of solu t io ns x of t he syst.em f ( d)x =o). since each column of d is t he d eri vati ve of a olu t ion it is a li near combina tion of t. he columns of
= m. in other words 1 a/ is t he mat.rix of d relative to t he basis . simi la rly, if el> a nd .v are two bases for f ( d) t hen t. her is a constant invertib le mat rix p s uch that. l!j = p. applying d, we have d -11 = d(p) = d() p = m p = '1/ p1/lf p, dcrck hacon 6 1 80 thnt l hc mnt.rires oí d rclativ lo any t.wo bases far f(d) are simila r. thl'r ar two cas wh re bases are w 11 known. first, if a is any 11 il pol c nt ·onstant mntr i x th n 1 6 = l +ta+2t'a'+ .. is a polynomial ma l rix whi h is asily n to be o basi for d a. lcarly oe = ea. ndly, if a d iagona l rn ntrix 'd(d) has a basi then nonc of its d iagonal c'ntnes can b zcro, fo r o t.h rwi t.h ·re would b infi ni tcly mauy !i n arly in lrp<'nd nt olut ions. onvcrscly, 1 t / (d) be a diagona l ent ry f 'd(d) of 1 hr fonn dlr+ high r pow r f d wh r k > o. th n t.hc row vect.or (1 t ... (k¡:li )i) '"n ""'"' for f(d). 11ch bases mny be as.cmbled 11110 r basis 6. for 'd(d). clt·arly d = d. j wlwn• j 1s d 1rn't s11111 of jorclan blotk ·, arh ro1t.-.ist111g of 011cs jusi abovc 111(' dt&ronal and z r s i:; wh r . in li:l'lltrnl , r "'"is f r f ( d ) may ll<' obta1111~l by r l11ci ng f t,o diagonaj form by row nnd c' olu mn opcra1 ions. th1 ml.'i hod, whid 1 is brs 1 on long d1,'1mon of poly noll\inl:;, is xplo.incd m varions tcx t.bo ks, for c'xtun¡ ic' [j orchm (\lol 3, s<'<'li n 14 1) a nd gocs as follows. l...t·1 g bí.' a nonzcro cnlry oí f . 13y • dmngmg rows oí p and t.hcn c·olumn: oí f w<.' 111ay u.ss11mr t hat g is in lht· upprr 1 ft hand com er f f' (i t' m rh firs1 row nnd c·ol11 mn). lí g d1vicl~ rv<·ry nt.ry in ils row nncl <·olnmn th n , by row n11cl c' lumn perations. all tht• rntri~ in t.h first. row ami 1ht-" firs1 rolumn of f cxrcpt g may be r<.'dul''-'tl to zero. in olh r word p ma~· be r!'d11('('(l to 11 © g (say). therwi , f 1s red uccd lo n matri x wit,h nonzt--ro entry of 1 w n el grrc ami th pro<'cdurr 1s l hen r p at.ed. sine r hr dcgrre rannol be rocluecd bcyond í'c'ro, f will ev nt.11 nlly b rccl uccd to h ji (sny). rx t., // is a simila rly redurt'd and o on. in thc cnd , a diagonal matnx 'd is blni nc'd . tlms therr urc poly norni o.i mul.rircs u(.c) nnd \ "(r) (w11h polynorn il\i invrr ') s11rh 1ha1 upv= d 62 jordan normal form vio ode's since u and v are invertible, is a basis for 'd(d) if and only if v(d) is a basis for f(d). 3. jordan normal form for nilpotent matrices let a be a nilpotent matrix. by reducing xi a to diagonal form , one can calculate explicitly u, v and 'd such that u(d)(d a)v(d) = 'd(d) now d a has a basis, aarnely e. so 'd(d) also has a basis. hence ó., as defined above, is a basis for 'd(d). thus v(d)ó. is a basis for d a. therefore v(d)ó. = 8p where pisan invertible c0fj.sta111t matrix. applying d , we have 8ap = d (8p) = dv (d) ó. =v (d) dó. = (v(d)ó.) j = 8pj since e is a basis, this implies that ap = pj, as required. lf needed , the matrix p may be easily calculated , because it is t he constant term of 8p and hence ajso of v (d)ó.. 4. the general case here we switch from poly nomial solut.ions to formal power series solut ions with d defined formally by the usual formul a. many textbooks (for example j ordan 's cours d'nnalyse) prove the basic result (due to euler) [eulerj that the polynomial /(d) has a basis consisling of f,tie"' for each factor (x a)" of f and j¡t1 eªtsin bt and fit'eª'cosbt for each factor ( (x a)'+ b2) ' where o ,,,; j < k . for diagona l 'd such bases may be asse mbled 111 lhe obvious way into a basis ó. for 'd( d ). again , dó. = ó.j where j is o jordan matrix. the res!. of t.he argument. goes t hrough as in t.he n.ilpotent case.. derek hacon 63 111 . jordn.n, cours d 'analysc, nu1 hicr. v11lrus l2j l . eul r , de integmttone aequat1orum d1ffcren1talmm altionmi groduum, opero omnin vol. xx ii 1> 10 149 32 cqbo vol. 3. octubre 1987 p!ga: 32-36 solucion de la c~artica l.i<·>mel henríquez b. resumen: este trabajo presenta um m·uev0 mét0do sim1pli<;:ado para resalver la cuártica definida sobre ~. mediant~ elementas b! sicos de matrices se encuentra la res0lvente c6bica y c0n tjraa de sus raíces, p o r med i o de álgebra elemental se 0btienen dos polinomios de 2° grad e , cuyas raíi;;e1s s0,n las d ,e ·ta ctjá,rti·ca dada. el objetiv o de este artícu l 0 es prese mtar una nueva met~ dología p ara re s ol v er la c uártica, esencialmente distinta de la pr e se n tada p o r yang yu-cheng (yu cheng, 19§§ 877 879) ,ya que e~te nu evo mé t o d o só l o ut i liza elemento s básicos de álgeel trabaj o citado. * institut o d e mat e má ti c a , un i versida d austra l d e chile l. henrlquez 3 3 p (x) 4 + + 2 1. sea e x º3 ' º2' + c 1 x + ºo = o 4 2. p (x) q (x) r ( x) ' con gr ( q (x ) ) gr ( r (x)) 2. 3. hag a mos -t q a .. lj ) = a . don de q 1 ª2 ª1 •o> y con l os coeficie n tes de q(x) y r(x) (h e n ríquez, 1966 p.62) 4. en to n ce s : r ª2b2 ª2b1 ª2bo ] a = ª1b 2 ª1b 1 ª1b o ªob2 ªob1 ªobo 5.sea 2a 2 b 2 ª2b 1 + ª1b2 ., ...... , l a+at= [ .,., + ª1b2 2a 1 b 1 ª1bo + ªob1 ª2bo + ªob2 ª1bo + ª ob1 2a 0 b 0 34 6. se a soluci6n de la cul rtica raíz de ibi z 3 e2z2 + ( c3c1 + 1 4 c 4 c 2 c 0 e~ 4e 4 c 0 ) 2 e 0 e 4 c 1 z + = o con este valor de r se de t er minam las ecuaciones: = o c u yas ra íces so n las de p(x). 7 . e n e f ecto, hagamos u 1 ª2b1 , " 2 ª 1 b 2, t, ª 2b0 , t2 = ª0 6 2· y1 ª 1bo • y2 = ª0"' 1 8.o sea : [ c4 u 1 t 1 l a = " 2 c 2 y1 t2 y2 º o 9.t 1 t2 se obt i e n en de 2 ¡:: 4 k kr + e© o , ya que t2 = k º4• k t, = ºo t 1 = r t 2' l. henriquez 35 1 0.ade más se determinara de: u1 + cjz 1 1 .final mente: cuyas raíces s0n las de p(x) . n0ta: y 1 , y 2 pueclen ser también 0~tenidos, ya que per0 es in0ficios0, puesto que se obtenclrían ecuaciones equivalentes a las dadas en (11). e j em¡;.i 10: 2x 4 + 7x 3 + 17x 2 + 19x + 15 = o primer0 encontramos r, resolvie~do la ecuaci0n z 3 17z 2 + 13z + 583 = o por tanteo obtenemos z = 11 36 soluci6n de la cuártica a ho ra p o r (9) , ten em10 s 2k 2 11k + 1 5 = o , o btenié n d 0s e k = 3 a s í t 1 12. p o r o t ro la d o co n ( 1 0 ), se tie r.i e q u e : 2h 2 7 h + = © o btenién dose h 2, va l or eo n el cua l deter minam os qu e y 1 = 6 "t = 8 . fina l men t e d e ( 11 ) , ob t er.. em 0s : -1 + -y-; bibliqgrafia ' i , 3 :: '-! 31 4 1. yang yu cheng , classroo m notes. the ame r ica n ma them a t ical mon t h ly . 73 ( 8) ' 877 8 7 9 , 1 96 6 . 2 . ~en rí quez l i one l , c u bo , re vi sta de mate má ti ca . vo l . 2 : 5 7 §5, 1986. uni v ersidad de l a fr o n t e r a a mathematical journal vol. 7, no 2, (57 67). august 2005. quaternionic analysis and maxwell’s equations wolfgang sprössig institute of applied analysis tu bergakademie freiberg prüfer-strasse 9 09596 freiberg germany sproessig@math.tu-freiberg.de abstract methods of quaternionic analysis are used to obtain solutions of maxwell’ s equations. by the help of time-discretisation maxwell’s equations are reduced to an equation of yukawa type. initial value and boundary value conditions are realized by a representation formula in each time step. approximation and stability is proved. resumen se usan los métodos de análisis quaternionico para obtener soluciones de las ecuaciones de maxwell. con la ayuda de la discretización del tiempo, las ecuaciones de maxwell son reducidas a una ecuación del tipo yukawa. valores iniciales y condiciones de valores en la frontera son realizadas por una fórmula de representación en cada paso de tiempo. se demuestra la aproximación y estabilidad. key words and phrases: maxwell equations, quaternionic analysis, operator calculus. math. subj. class.: 35f10, 30g35 58 wolfgang sprössig 7, 2(2005) 1 introduction in 1873 j. c. maxwell’ s fundamental paper a treatise on electricity and magnetism was published. since this time generations of physicists and mathematicians felt facsinated from the deepness and beauty of these equations. from the very beginning scientists tried to give maxwell’s equations a more simply algebraical structure maybe in the form du + au = f with a suitable derivation operator d and a potential operator a. in this connection we should mention people like l. silberstein (the theory of relativity, 1914), h. weyl (raum -zeit-materie, 1921) and m. mercier expression des équations de electromagnetique au moyen des nombres de clifford, 1946). new algebraical notions were introduced and used for the description of maxwell’s equations (for instance: c. lanczos (1929): spinors, a. proca (1930): clifford numbers, a. einstein/a. mayer (1932): semi-vectors, f. bolinder (1957): 4-d forms, g. kron (1959): skew-symmetric tensors and d. hestenes (1968): multivector calculus). we will use in our conception real and complex quaternionic analysis, more exactly a quaternionic operator calculus, which also contains a corresponding quaternionic numerical analysis. we intent to apply a time-discretisation method (rothe’ s method) in order to reduce maxwell’s equations to a disturbed yukawa equation. the latter one is considered under realization of initial and boundary values by means of a suited quaternionic calculus. this paper belongs to a series of papers where we use rothe’s method to involve time-dependent problems in our quaternionic calculus. this paper can also be seen as an alternative supplement to latest papers by v.v. kravchenko et al (cf. [3], [8] and [9]. 2 maxwell’ s equations in a chiral medium let g ⊂ ir3 a bounded domain with a sufficiently smooth boundary γ. in mks maxwell’s equations read as follows: div d = ρ ε0 , (1) rot e = −∂tb, (2) div b = 0, (3) rot h = μ0j + ε0μ0∂td, (4) where μ0 is the permeability of the free space, ε0 the permittivity of the free space, e the imposed electric field, b the magnetic field, ρ the (free) charge density, h the effective magnetic field inside the dielectric medium, d the effective electric field 7, 2(2005) quaternionic analysis and maxwell’s equations 59 inside the dielectric and j the charge density. in a homogeneous chiral medium g the operators b and d have to fulfil the drude-born-feodorov constitutive relations: b : = μh + μβrot h, (5) d : = εe + εβrot e, (6) where β is the chirality measure of the medium. further, we assume: initial value conditions: e(x, 0) = e0(x) and h(x, 0) = h0(x). (7) boundary condition: e(x,t) = g(x,t) (x ∈ γ). (8) good references for maxwell’s equations in chiral media are the book by a. lakhtakia [11] in 1994 as well as the article [1]. 3 preliminaries let us now introduce notations and operators needed in our approach. let ih be the set of real quaternions. each quaternion permits the representation a = 3∑ k=0 akek (ak ∈ ir; k = 0, 1, 2, 3), where e0 = 1 and e1,e2,e3 are the so-called imaginary units. by definition these units ek obey the following arithmetic rules: e20 = 1,e1e2 = −e2e1 = e3,e2e3 = −e3e2 = e1 and e3e1 = −e1e3 = e2. addition and multiplication in ih turn it into a non-commutative number field. the main-involution in ih is called quaternionic conjugation and defined by e0 = e0, ek = −ek (k = 1, 2, 3), which can be extented onto ih by ir-linearity. therefore, we have a = a0 − 3∑ k=1 akek. note that aa = aa = 3∑ k=0 a2k =: |a|2ih. 60 wolfgang sprössig 7, 2(2005) if a ∈ ih \{0} then the quaternion a−1 := a |a|2 is the inverse to a. for a,b ∈ ih we have ab = ba. the set of complex quaternions, which we also need, is denoted by ih(c) and consist of all elements of the form a = 3∑ k=0 akek (ak ∈ c; k = 0, 1, 2, 3). by definition we state: iek = eki, k = 0, 1, 2, 3. here i denotes the usual imaginary unit in c. elements of ih(c) can also be represented in the form a = a1 + ia2 (ak ∈ ih; k = 1, 2). notice that the quaternionic conjugation acts only on the quaternionic units and not on i. let now g ⊂ ir3 be a bounded domain with sufficient smooth boundary γ. assume that all function spaces b(g, ih(c)) =: b(g) have the usual componentwise meaning. let u ∈ c1(g). the dirac-operator d is defined by du = 3∑ k=1 ek∂ku. the operator d is right-linear with respect to complex numbers. on c2(g)) the 3-dimensional laplacian permits the factorization δ = −d2. we consider the disturbed laplacian δ −ν2, (ν ∈ ir). which is called yukawa operator and acts on c2(g). this operator has the factorization property δ −ν2 = (iν + d)(iν −d). the factors iν + d and iν − d are called mutually generalized dirac type operators. functions u ∈ ker(d + iν) are called left-(iν)-hyperholomorphic. the fundamental solution of the yukawa operator in ir3 is given by θν := − 1 4π 1 |x|e −ν|x|. then the corresponding fundamental solution of the operator iν + d is given by eiν (x) := (iν −d)θν (x) = ( iν + x |x|2 + ν x |x| ) θν (x) 7, 2(2005) quaternionic analysis and maxwell’s equations 61 4 a time discretisation method for simplicity we introduce the following abbreviations: f := ρ ε0ε , a := −μ, b := −μβ, c := ε0μ0ε, d := ε0μ0εβ. a simply calculation shows that maxwell’s equations in a homogeneous chiral medium transform into de = −f + ∂th + b∂trot h, (9) dh = μ0j + c∂te + d∂trot e. let t > 0. the equations (9) are considered in the time-intervall [0,t ]. a decomposition of [0,t )into n equal parts yields t = nτ, where τ is called the meshwidth of the decomposition. we briefly write for k = 0, 1, ...,n : ek := e(kτ,x), hk := h(kτ,x), fk := fk(kτ,x) and jk := j(kτ,x). we want to approximate the time derivatives ∂te and ∂th by the finite forward differences: ek+1 −ek τ and hk+1 −hk τ . more detailed we will consider the case β = 0. from (9) we obtain for k = 0, 1, ...,n−1: dek+1 = −fk + a τ (hk+1 −hk), (10) dhk+1 = μ0jk + c τ (ek+1 −ek). (11) setting now ν2 = −ca τ2 and l := − √ −a c , then we have dek+1 = −fk + lν(hk+1 −hk), (12) dhk+1 = μ0jk − 1 l ν(ek+1 −ek). (13) applying the dirac operator d from the left we get ddek+1 = −dfk + lνdhk+1 −lνdh, = −dfk −lνdhk + lνμ0jk −ν2ek+1 + ν2ek and so (δ −ν2)ek+1 = dfk + lνdhk −lνμ0jk −ν2ek. (14) 62 wolfgang sprössig 7, 2(2005) with the same actions we obtain the dual relation (δ −ν2)hk+1 = −μ0djk − 1 l νfk − 1 l νdek −ν2hk. (15) we intent to consider equations (14) and (15) in the hypercomplex setting in (cf.[4], [12]). main idea is to factorize the yukawa operator on the left hand side. this leads to (d + iν)(d − iν)ek+1 = −dfk + lνμ0jk −lνdhk + ν2ek. (16) 5 borel-pompeiu’s formula let g be a bounded domain in ir3 with the liapunov boundary γ and let n = (n1,n2,n3) be the unit vector of the outward pointing normal at the point y ∈ γ. the kernel eiμ(x) function generates two important integrals: teodorescu transforms, which are defined by: (t±iνu)(x) = ∫ g e±iν (y −x)u(y)dy as well as the cauchy-bizadse operators: (f±iνu)(x) = ∫ γ e±iμ(x−y)n(y)u(y)dγy. these operators are well studied in several papers, see e.g. [6],[10] and [12]. in [7] was obtained the following borel-pompeiu formula: u = (d ± iν)t±iνu = t±iν (d ± iν)u + f±iνu in g, (17) where u ∈ c1(g) ∪ c(g). notice that borel-pompeiu’ s formula is also valid for u ∈ w 12 (g). on the boundary γ we have trγu ∈ w 1/22 (γ). 6 representations applying teodorescu transforms t±iν to formula (16) we get the iteration procedures: ek+1 = t−iνtiν [ −dfk + lνμ0jk −lνdhk + ν2ek ] + t−iν φ+ + φ− (18) and hk+1 = t−iνtiν [ μ0djk + 1 l νμ0fk + 1 l νdek −ν2hk ] + t−iν ψ+ + ψ−, where φ± and ψ± belong to the kernel of the operators d±iν. notice also that holds hk+1 = hk + 1 l νrot ek. 7, 2(2005) quaternionic analysis and maxwell’s equations 63 the unknown functions φ+ and φ− have to be determined now. in [12] is shown that holds (d ± iμ)t±iμu = 0 and therefore also f±iμt±iμu = 0 for any function u ∈ w 12 .this leads to φ− = f−iνek+1 = f−iνtrγek+1 = f−iνgk+1 where gk := g(kτ,x). the determination of φ+ is more complicated. using fiν φ+ = φ+, which is a consequence of borel-pompeiu’s formula and trγuk+1 = gk+1 we have qγ,−iνgk+1 = trγt−iνtiν [ −dfk + lνμ0jk −lνdhk + ν2ek ] + t−iνfiν φ+, where qγ,−iν is one of the so-called plemelj projections, which are defined by n.− t.− lim x′∈ω± x′→x∈γ f±iνu(x ′) =: { (pγ,±iνu)(x), x′ ∈ ω+ = g (qγ,±iνu)(x), x′ ∈ ω− = ir3 \g . in [6] is shown that trγt−iνfiν : im pγ,iν ∩w k−1/22 (γ) → imqγ,−iν ∩w k+1/2 2 (γ) (k > 1) is an isomorphism. notice that the pair of plemelj projections act within corresponding hardy spaces (cf. [12],[6]). further, we obtain φ+ = fiν (trγt−iνfiν ) −1 [trγt−iνtiν (dfk −lνμ0jk + lνdhk −ν2ek)] + +fiν (trγt−iνfiν ) −1qγ,−iνgk+1. replacing φ+ in (18) we get ek+1 = t−iν [tiν ( −dfk + lνμ0jk −lνdhk + ν2ek ) +fiν (trγt−iνfiν ) −1trγt−iνtiν (dfk −lνμ0jk + νdhk −ν2ek) +fiν (trγt−iνfiν ) −1qγ,−iνgk+1] + fiνgk+1 = t−iν [ i −fiν (trγt−iνfiν )−1trγt−iν ] tiν (−dfk + lνμ0jk −lνdhk + ν2ek) +t−iνfiν (trγt−iνfiν ) −1trγt−iν(d − iν)g̃k+1 + fiνgk+1, where g̃k+1 is a smooth extension of gk+1 into the domain g. we now introduce the orthoprojections qiν := i − ipiν , where ipiν := fiν (trγt−iνfiν )−1t−iν is a modified bergman projection onto the subspace ker(d + iν) and qiν a projection onto the subspace (d− iν)ẇ 12 . finally, we find ek+1 = t−iνqiνtiν [ −dfk + lνμ0jk −lνdhk + ν2ek ] (19) + t−iνp(d− iν)g̃k+1 + f−iνgk+1. (20) 64 wolfgang sprössig 7, 2(2005) 7 realization of boundary conditions the realization of boundary conditions for the imposed electric field makes use of the follwing proposition: proposition 7.1 let u ∈ l2(g, ih(c)). the condition trγt−iνu = 0 is sufficient and necessary that u ∈ imqiν ∩l2(g, ih(c)). proof. at first let u ∈ imqiν ∩l2(g, ih(c)). then u permits the representation u = (d − iν)w with w ∈ ẇ 12 (g, ih(c)) and therefore trγt−ivu = 0. on the other hand, if trγt−iνu = 0 then immediately we obtain from hodge’s decomposition of the hilbert space l2(g, ih(c))and the representation of the generalized bergman projection: u = fγ,iν (trγt−iνfγ,iν ) −1trγt−iνu + qiνu. the first term vanishes and we have u = qiνu. # let ẽk+1 := t−iνp(d− iν)g̃k+1 + f−iνgk+1, borel-pompeiu’ s formula yields (−δ + ν2)ẽk+1 = (d + iν)(d − iν)ẽk+1 = 0. furthermore, we get from proposition (7.1) trγẽk+1 = trγt−iνpiν (d − iν)g̃k+1 + pγ,−iνgk+1 = trγt−iν (d − iν)g̃k+1 − trγt−iνqiν (d − iν)g̃k+1 + pγ,−iνgk+1 = gk+1 −pγ,−iνgk+1 + pγ,−iνgk+1 − trγt−iνqiν (d − iν)g̃k+1 = gk+1. 8 approximation and stability from (15) we obtain by setting fk := ek + ilhk the representation fk+1 = tiνt−iν (iν−d)(fk−iμ0ljk)−iνtiν t−iν (iν −d)fk + tiνχ− + χ+, where χ± belongs to the sets ker (d±iν). these functions can be defined by boundary conditions. further, we abbreviate mk := fk−iμ0jk. using borel-pompeiu’ s formula t−iν (iν −d)u = −t−iν (d − iν)u = fγ,−iνu−u 7, 2(2005) quaternionic analysis and maxwell’s equations 65 we get fk+1 = −t−iνmk + t−iνfγ,−iνmk + iνtiνfk − iνt−iνfγ,−iνfk + tiνχ− + χ+. because the image of the cauchy-bizadse-type operators fγ,±iν belongs to the kernels ker (d ± iν) eventually we achieve the formula: fk+1 = iνt−iνfk −tiνmk + tiνχ∗− + χ∗+. the operator iνtiν is bounded in l2(g, ih(c)) which follows from ([2] corollary 2.5). there it is deduced the estimation for the generalized teodorescu transform ‖t±iν‖[l2,l2] ≤ d ν , where d depends on the diameter of the domain g and tends to zero for diam g → 0. now it remains to analyze the approximation property of our semi-discretisation procedure for convergence. put l1 := c∂te(t,x) −dh(t,x) and l2 := a∂th(t,x) −de(t,x). furthermore, we introduce the operators: l1τ = c e(t + τ,x) −e(t,x) τ + dh(t + τ,x), l2τ = a h(t + τ,x) −h(t,x) τ + de(t + τ,x). we have to estimate the differences lj −ljτ for j = 1, 2. with t = kτ it follows |(l1 −l1τ )| ≤ c τ |(ek+1 −ek − τ∂tek) + d(hk+1 −hk)|, |(l2 −l2τ )| ≤ a τ |(hk+1 −hk − τ∂thk) + dek+1 −ek|. we intent to continue with the first estimate. it is easy to show that |l1 −l1τ| ≤ cτ|∂tte(kτ + θτ,x)| + μ0τ|∂tjk(kτ + θ′τ,x)| + cτ|∂tte(kτ + θ′′τ,x)| ≤ τc1k (e,j) with θ,θ′,θ′′ ∈ (kτ, (k + 1)τ). analogously, we obtain for the second estimation |l2 −l2τ| ≤ τc2k (h,f). on this way the truncation error is estimated for sufficient smooth e, h, j and ρ 66 wolfgang sprössig 7, 2(2005) remark 8.1 with the same principle also the chiral case can be considered. one obtains with similar calculations the following iteration procedure: ( d + iν 1+βiν ) fk+1 = mk 1+βiν + ( d + iν 1+βiν )( βiν 1+βiν ) fk − ( iν 1+βiν )( βiν 1+βiν ) fk. received: june 2004. revised: november 2004. references [1] c. athanasiatis, p.a. martin and i. g. stratis, electromagnetic scattering by homogeneous chiral obstacle: boundary integral equations and low-chirality approximations, siam j. appl. math. vol. 59, no. 5, 1745 – 1762. [2] h. bahmann, k. gürlebeck, m. shapiro and w. sprössig, on a modified teodorescu transform , integral transforms and special functions, vol. 12(2001), number 3, 213-226. [3] s. m. grudsky, k. v. khmelnytskaya and v. v. kravchenko, on a quaternionic maxwell equation for the time-dependent electromagnetic field in a chiral medium j. phys. a. math.gen. 37(2004), 4641–4647. [4] k. gürlebeck, hypercomplex factorization of the helmholtz equation, zaa, 5 (1986), 125-131. [5] k. gürlebeck and w. sprössig, quaternionic analysis and boundary value problems, birkhäuser verlag, basel.(1990). [6] k. gürlebeck and w. sprössig, quaternionic and clifford calculus for physicists and engineers , john wiley, chichester, sydney, new york. (1998). [7] k. gürlebeck and w. sprössig, on a teodorescu transform for a class of metaharmonic functions. j. nat. geopmetry, 21(2002), no.1-2, 17– 38. [8] k. v. khmelnytskaya, v. v. kravchenko and v. s. rabinovich, quaternionic fundamental solutions for electromagnetic scattering problems and application, zaa, vol 22(2003), no. 1, 147–166. [9] v. v. kravchenko, applied quaternionic analysis, research and exposition in mathematics vol. 28(2003), heldermann-verlag,lemgo. 7, 2(2005) quaternionic analysis and maxwell’s equations 67 [10] v. kravchenko and m. shapiro, integral representations for spatial models of mathematical physics, pitman research notes in mathematics 351, longman,, harlow.(1996). [11] a. lakhtakia, beltrami fields in chiral media, world scientific, singapore. (1994). [12] w. sprössig, on the decomposition of the clifford valued hilbert space and their applications to boundary value problems, advances in applied clifford algebras, 5(1995), 167 – 186. a mathematical journal vol. 7, no 3, (39 48). december 2005. the exact solution of the potts models with external magnetic field on the cayley tree nasir ganikhodjaev 1 centre for computational and theoretical sciences,faculty of science, international islamic university malaysia,53100 kuala lumpur,malaysia and department of mechanics and mathematics, national university of uzbekistan,vuzgorodok 700095,tashkent ,uzbekistan nasirgani@hotmail.com seyit temir harran university, department of mathematics sanliurfa-turkey temirseyit@harran.edu.tr hasan akin harran university, department of mathematics sanliurfa-turkey akinhasan@harran.edu.tr abstract the exact solution is found for the problem of phase transition in the potts model and the potts model with competing ternary and binary interactions with external magnetic field. 1this research was supported in part by the grant uz.r.ftm f-2.1.56. the first named author (n.g.) thanks nato-tubitak for providing financial support and harran university for kind hospitality and providing all facilities. 40 nasir ganikhodjaev, seyit temir and hasan akin 7, 3(2005) resumen se encuentran soluciones exactas para los problemas de transisión de fases en el modelo de pott y también para el modelo de pott con interacciones binarias y ternarias en un campo magnético externo. key words and phrases: cayley tree, potts model, competing interactions, external magnetic field. math. subj. class.: 82b20 secondary 82b26 1 introduction the potts model was introduced as a generalization of the ising model. the idea came from the representation of the ising model as interacting spins which can be either parallel or antiparallel. an obvious generalization was to extend the number of directions of the spins. such a model was proposed by c.domb as a phd thesis for his student r.potts in 1952. at present the potts model encompasses a number of problems in statistical physics and lattice theory. it has been a subject of incresing intense research interest in recent years. it includes the ice-rule vertex and bond percolation models as special cases. we consider a semi-infinite cayley tree j k for order k ≥ 2, i.e., a graph having no cycles, from each vertex of which, except on vertex x0, emanates exactly k + 1 edges and from vertex x0, which is the root of the tree, emanates k edges. the vertices x and y are called nearest neighbors, which is denoted by < x, y >, if there exists an edge connecting them. the vertices x,y and z are called a triple of neighbors ,which is denoted by < x, y, z >,if < x, y > and < y, z > are nearest neighbors and x 6= z let v be the set of vertices in j k. we set wn = {x ∈ v |d(x, x0) = n}, vn = ∪nm=0wm = {x ∈ v |d(x, x 0) ≤ n}. where the distance d(x, y), x, y, ∈ v is given by the formula, d(x, y) = min{d|x = x0, x1, x2, ..., xd−1, xd = y ∈ v such that the pairs < x0, x1 >, ..., < xd−1, xd > are nearest neighbors. the set wn is called n-th level of j k and the set vn is called n-storeyed home with root x0. we consider models where the spin takes values in the set φ = {0, 1, 2, ..., q}, q ≥ 2 and assigned to the vertices of the tree.a configuration σ on v is then defined as a 7, 3(2005) the exact solution of the potts models ... 41 function x ∈ v → σ(x) ∈ φ;the set of all configurations coincides with ω = φv . the potts model on the cayley tree is defined by the hamiltonian h(σ) = −j ∑ δσ(x)σ(y) − h ∑ x∈v δ0σ(x) (1) where the first sum is taken over all nearest neighbors, δ in the first and second sums is the kroneker’s symbol,j, h ∈ r are coupling constants and σ ∈ ω. along with this model, we will consider the potts model with competing interactions on the cayley tree which is defined by the hamiltonian below h(σ) = −j1 ∑ δσ(x)σ(y)σ(z) − j2 ∑ δσ(x)σ(y) − h ∑ x∈v δ0σ(x) (2) where the first sum is taken over all neighbors tripples, and δ in this sum is the generalized kroneker’s symbol (see [1]-[4] for models with competing interactions). such model was investigated in [3], where for the neighbors tripple < x, y, z >the generalized kroneker’s symbol δ had a form δσ(x)σ(y)σ(z) = { 1 if σ(x) = σ(y) = σ(z), 0 else. for the neighbors tripple < x, y, z >, we assume δσ(x)σ(y)σ(z) =   1 if σ(x) = σ(y) = σ(z), 1 2 if σ(x) = σ(y) 6= σ(z) or σ(x) 6= σ(y) = σ(z); 0 else. (3) where x, z ∈ wn for some n and y ∈ wn−1. this definition is well coordinated with the theory of quadratic stochastic operators, where the quadratic stochastic operator corresponding to the generalized kroneker’s symbol (4) is the identity transformation [5]. let sm−1 = {x = (x1, · · · , xm) ∈ rm : m∑ i=1 xi = 1} xi ≥ 0 ∀i = 1, · · · , m} be the (m − 1)-dimensional simplex in rm. the transformation v : sm−1 → sm−1 is called quadratic stochastic operator , if (v x)k = m∑ i,j=1 pij,kxixj where pij,k ≥ 0 , pij,k = pji,k and ∑m k=1 pij,k = 1 for arbitrary i, j, k ∈ {1, · · · , m} . such operator have applications in mathematical biology, namely theory of heredity, 42 nasir ganikhodjaev, seyit temir and hasan akin 7, 3(2005) where the coefficients pij,k are interpreted as coefficients of heredity. assume pij,k = δijk,where the generalized kroneker’s symbol δ has a form δijk =   1 if i = j = k, 1 2 if i = k 6= j or i 6= j = k; 0 else. (4) then it is easy to show that the corresponding quadratic stochastic operator is the identity transformation . 2 recurrent equations for partition function there are several approaches to derive the equation or a system of equations describing the limiting gibbs measures for lattice models on a cayley tree. one approach is based on the properties of the markov random fields on a cayley tree [6, 7]. another approach is based on recurrent equations for partition functions(see for example [8]). naturally both approaches lead to the same equation(for example [9]). the second approach, however, is more suitable for models with competing interactions. let λ be a finite subset of v. assume σ(λ) and σ(v \ λ) are the restriction of σ to λ and v \ λ respectively. let σ(v \ λ) be a fixed boundary configuration. the total energy of configuration σ(λ) under condition σ(v \ λ) is defined as h(σ(λ)|σ(v \ λ)) = −j ∑ < x, y > x, y ∈ λ δσ(x)σ(y) − j ∑ < x, y > x ∈ λ, y /∈ λ δσ(x)σ(y) − h ∑ x∈λ δ0σ(x). in the first case and h(σ(λ)|σ(v \ λ)) = −j1 ∑ < x, y, z > x, y, z ∈ λ δσ(x)σ(y)σ(z) − j2 ∑ < x, y > x, y ∈ λ δσ(x)σ(y) − h ∑ x∈λ δ0σ(x) −j1 ∑ < x, y, z > x /∈ λ, y ∈ λ, z /∈ λ δσ(x)σ(y)σ(z) − j2 ∑ < x, y > x ∈ λ, y /∈ λ δσ(x)σ(y) . for the second hamiltonian respectively. the partition function zλ(σ(v \ λ)) in volume λ under boundary condition σ(v \ λ) is defined as zλ = ∑ σ(λ)∈ω(λ) exp(−βh(σ(λ))|σ(v \ λ)), (5) where ω(λ) is the set of all configuration on λ, and β = 1 t is the inverse temperature. we consider the configurations σ(vn), the partition functions zvn in the volume vn and for brevity we denote it as σn, z(n) respectively. 7, 3(2005) the exact solution of the potts models ... 43 let us first consider the model (1).we decompose the partition function z(n) into the following summands z(n) = q∑ i=1 z (n) i , where z (n) i = ∑ σn∈ω(vn):σn(x0)=i exp(−βhn(σn)). (6) let θ = exp (βj), θ3 = exp (βh) . from (5) and (6), the following system of recurrent equations can be easily derived z (n) 0 = θ3 [ θz (n−1) 0 + z (n−1) 1 + z (n−1) 2 + ... + z (n−1) q ]k z (n) i = [ z (n−1) 0 + ... + z (n−1) i−1 + θz (n−1) i + z (n−1) i+1 ...z (n−1) q ]k (7) for i=1,2,...,q , where z(n−1)i is a partition function in (n − 1)-storeyed home with root located a vertex x ∈ w1 for which σ(x) = i. after replacing u(n)i = z (n) i z (n) 0 , we have the following system of recurrent equations u (n) i = 1 θ3 ( 1 + (θ − 1)u(n−1)i + q∑ j=1 u (n−1) j θ + q∑ j=1 u (n−1) j )k; (8) for i=1,2,...,q and n=2,3,... we describe the fixed points of this system recurrent equation (5). for this, it suffices to solve the system of equations ui = 1 θ3 ( 1 + (θ − 1)ui + q∑ j=1 uj θ + q∑ j=1 uj )k; i = 1, 2, ..., q. (9) before we begin to solve this system of equations, we turn to the model (2). here we consider a slight modification of the hamiltonian (2) definition 1 a triple of neighbours < x, y, z > is said to be two-level and is denoted by ¯< x, y, z > if the vertices x and z belong to wn for some n, i.e. they are located on the same level and y ∈ wn−1. we consider the hamiltonian h(σ) = −j1 ∑ ¯ δσ(x)σ(y)σ(z) − j2 ∑ δσ(x)σ(y) − h ∑ x∈v δ0σ(x) (10) 44 nasir ganikhodjaev, seyit temir and hasan akin 7, 3(2005) where j1 6= 0 and in contrast to (2), the first sum inlcudes only the two-level triples of neighbours. such a model is called a two-level model (see [3], [4] and the references there for the physical motivation underlying the study of these model). it is not hard to derive, in this case, the system of recurrent equations is as the following z (n) 0 = θ3 [ θ1θ2z (n−1) 0 + z (n−1) 1 + z (n−1) 2 + ... + z (n−1) q ]k z (n) i = [ z (n−1) 0 + ... + z (n−1) i−1 + θ1θ2z (n−1) i + z (n−1) i+1 ...z (n−1) q ]k for i = 1, 2, ..., q where θ1 = exp(βj1), θ2 = exp(βj2) and θ3 = exp(βh). thus both models (1) and (10) are described by the same system of recurrent equations. 3 the proof of existence of phase transitions for zero external field in this section, we let j > 0 for model (1), that is we consider model (1) as a ferromagnetic potts model and j1 + j2 > 0 for model (10). then θ > 1 in the first case and θ1θ2 > 1 for second case. we consider the system of equations (9).assume ui = exp hi, i = 1, 2, ..., q then h′i = k ln 1 θ3 ( 1 + (θ − 1)hj + q∑ j=1 exp(hj ) θ + q∑ j=1 exp hj ); i = 1, 2, ..., q (11) is the transformation rq into rq. evidently the line l0 : h1 = h2 = ... = hq in rq is invarinat with respect to transformation (11) and the restriction of (11) on the line l0 has the following form h′ = k ln 1 θ3 ( (θ + q − 1) exp(h) + 1 q exp(h) + θ ) where h ∈ r. again, after renaming u = exp(h), we have u = 1 θ3 ( (θ + q − 1)u + 1 qu + θ )k. the following lemma is a generalization of the proposition 10.7 from [8]. lemma 1 the equation θ3u = ( (θ + q − 1)u + 1 qu + θ )k (12) 7, 3(2005) the exact solution of the potts models ... 45 (with u > 0, k ≥ 2, q ≥ 2) has a single solution if 1 < θ < θcr = −(k − 1)(q − 1) + √ (k − 1)2(q + 1)2 + 8q(k − 1) 2(k + 1) if θ > θcr then there are numbers η1(θ, q, k), η2(θ, q, k) with 0 < η1(θ, q, k) < η2(θ, q, k) such that equation (12) has three roots, when 0 < η1(θ, q, k) < θ3 < η2(θ, q, k) and it has two roots if either θ3 = η1(θ, q, k) or θ3 = η2(θ, q, k) or θ = θcr. the numbers ηi, i = 1, 2 are defined from the formula ηi(θ, q, k) = 1 ui ( (θ + q − 1)ui + 1 qui + θ )k (13) where u1 and u2 are the solution of the equation (θ + q − 1)qu2 − [k(θ − 1)(θ + q) − θ(θ + q − 1) − q]u + θ = 0 (14) proof. assume f (u) = ( (θ+q−1)u+1 qu+θ )k. it is easy to check that equation (12) has more than one root if and only if the equation uf′ = f (u) has more than one solution. the equation uf′ = f (u) is no other than just equation (14). although there are three solutions for the system of equations (9) for θ > θcr, one cannot claim that there is a phase transition. among these solutions, only one of them is a stable solution. it is necessary to find other stable solutions. this problem is rather complete for arbitrary k and q when θ 6= 1. the case with θ 6= 1 will be considerd separately when k = 2 and q = 2. we shall now solve this problem for θ3 = 1, that is, h = 0. then, the system of equation (9) has the following form ui = ( 1 + (θ − 1)ui + q∑ j=1 uj θ + q∑ j=1 uj )k i = 1, 2, ..., q (15) and the transformation rq into rq (11) has the following form h′i = k ln( 1 + (θ − 1) exp hj + q∑ j=1 exp(hj ) θ + q∑ j=1 exp hj ); i = 1, 2, ..., q. (16) then, apart from the invariant line l0 we can find other q invariant lines, namely the line lj : h1 = ... = hj−1 = hj+1 = ... = hq = 0, j = 1, 2, ..., q. the transformation (11) reduces to the following transformation of r: h′ = k ln( θ exp h + q exp h + θ + q − 1 ) on each invariant line lj , j = 1, 2, ..., q. 46 nasir ganikhodjaev, seyit temir and hasan akin 7, 3(2005) now we will solve this simpler equation u = ( θu + q u + θ + q − 1 )k (17) let us consider the function φ(u) = ( θu + q u + θ + q − 1 )k. with the help of the lemma, it is not hard to show that the equation (17) has three solutions when θ > θ∗cr = k + 2q − 1 k − 1 . in this case only one of these roots is stable, namely, largest of them. for equation (12), when θ3 = 1, we showed above that it has three solutions when θ > θ∗cr (see lemma) and only one of them is stable. it is easy to check that θ∗cr > θcr . as uj = p (x0=j) p (x0=0) for some limiting gibbs measure p with θ > θ∗cr = k+2q−1 k−1 , we have q + 1 differences translated invariant limiting gibbs measures. the same way as in [9], it is possible to prove that all of them are extremal. theorem 1 for potts model (1) with null external field, a phase transition occurs when, θ > k + 2q − 1 k − 1 . similar assertation is also valid for the two-level potts model with competing ternary and binary potentials with null external field. theorem 2 for the two-level potts model (10) with competing ternaty and binary potentials with null external magnetic field, a phase transition occurs when θ1θ2 > k+2q−1 k−1 . 4 the case of non-zero external magnetic field when k = q = 2 here we consider potts models both (1) and (10) with external magnetic field h 6= 0, when k = q = 2 and θ > 1 for model (1) and θ1θ2 > 1 for model (10) respectively (the case h = 0 was considered in [9]for model (1) and for model (10) in [10] ). then the system of equations (9) reduces to the following x = 1 θ3 ( θx + y + 1 x + y + θ )2 y = 1 θ3 ( x + θy + 1 x + y + θ )2 (18) where x = u1, y = u2 for brevity. as x − y = 1 θ3 (θ1)(x − y)[(θ + 1)(x + y) + 1] (x + y + θ2) , 7, 3(2005) the exact solution of the potts models ... 47 then some solutions of (18) can be found from equation u = 1 θ3 ( (θ + 1)u + 1 2u + θ )2 (19) where x = y = u and other solutions can be found from equation θ3z 2 − (θ2 − 2θ3θ − 1)z + θ3θ2 − 2θ + 2 = 0 (20) where z = x + y. first of all, let us consider equation (19).then the equation (13)(see lemma)has the following form 2(θ + 1)u2 − (θ2 + θ − 6)u + θ = 0. (21) this equation has two roots u1, u2 if θ > √ 73−1 2 . then by lemma, equation (19)have three roots if θ > √ 73−1 2 and η1(θ) < θ3 < η2(θ), where ηi = 1ui ( (θ+1)ui+2 3ui+θ )2, i = 1, 2. now we consider the equation (20). again with the help of elementary analysis it is not hard to show that the equation has two solutions for θ3 > 12 with θ > 2θ3 − 1 + 2 √ θ3(θ3 + 1) and for 0 < θ3 < 12 , with θ > 1+ √ 1−2θ3 θ3 . by virtue of symmetry of equations (18) we have two stable solutions. assume a = {(θ3, θ) : η1(θ) < θ3 < η2(θ) ; θ > √ 73 − 1 2 } where η1(θ) and η2(θ) as above and b = {(θ3, θ): 0 < θ3 < 12 ; θ > 1+ √ 1−2θ3 θ3 } ∪ {(θ3, θ): θ3 > 12 ; θ > 2θ3 − 1 + 2 √ θ3(θ3 + 1) }. then for arbitrary (θ3, θ) ∈ a ∩ b there are three stable solutions of the equations (18). we have thus proved the following theorems. theorem 3 for potts model (1) with q = k = 2 and non-zero external magnetic field, a phase transition occurs when (θ3, θ) ∈ a ∩ b a similar result is valid for model (10). theorem 4 for the two-level potts model (10) with competing ternary and binary potentials q = k = 2 and non-zero external magnetic field a phase transition occurs when (θ3, θ1θ2) ∈ a ∩ b. received: july 2005. revised: august 2005. references [1] m. mariz ,c.tsalis, a.l.albuquerque, phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field, j. stat. phys., 40,(1985), 577-592. 48 nasir ganikhodjaev, seyit temir and hasan akin 7, 3(2005) [2] c.r. da silca, s.coutinho, ising model on the bethe lattice with competing interactions up to the third-nearest-neighbor generation, phys. review b, 34, (1986), 7975-7985. [3] j.l. monroe, phase diagrams of ising models on husimi trees ii. pure multisite interactions systems, j. stat. phys. 67, (1992), 1185-2000. [4] j.l. monroe, a new criterion for the location of phase transitions for spin systems on a recursive lattices, phys. lett. a 188, (1994), 80-84. [5] s.n. bernstein, the solution of a mathematical problem concerning the theory of heredity, uchnye zapiski naucho-issled. kaf. ukr. otd. mat. 1, (1924), 83-115.(russian) [6] f. spitzer, markov random field on infinite tree, ann. prob., 3, (1975), 387398. [7] k. preston, gibbs states on countable sets, cambridge, london (1974). [8] r. kindermann and j.l. snell, markov random fields and their applications, contemporary mathematics 1, (1980). [9] n.n.ganikhodjaev (ganikhodzhaev), on pure phases of the ferromagnetic potts model with three states on the bethe lattice order two, theor. math. phys. 82(2), (1990), 163-175. [10] n.n. ganikhodjaev, s. temir, h. akin, the exact solution of the three-state potts model with competing interactions on the cayley tree, uzbek math. journal 3-4, (2002), 37-40. cubo 7, 16-21 (1991) modelos del mal de chagas poc ana hada naturana c. re•umen . se an a liz a n do s modelos continuos compartimentados del ha l d~ cha gas , co n el ob j e llvo de s e r uotlllzados po steriormente en poll tl cas d e contro l . lo s modelos se red uce n a un problema de cauchy y son re sue1 tos utilizando las t éc n i cas us ua l es d e aná l isis de la estabilidad d e l o s flunlo s d e equ i libri os (es l ud lo local y gl o bal). e l c 11 ogas · (tripallosdhia s i s ame:lli canal e s una e nf e rm e dad de c on l og l o i nd i r ec t o , tras mitida por c o nt a mina ció n d e l a he ri da por la mord ed ura d e l a vi ncllu c a" (tíl!atoha jnfest an s {protozoos ) ) , e s la ln f e cc l ón s e pr od uce po r mt?d l o d e l a s d e í ccacl o nes qu e e s t e in sec t o e x puls o. a medida que se a lime n t a, y la pu e r t a d e e ntr a d a s o n la s h e rid as d e e s l as pi caduras. la e nfer medad p rod uce d c b ill ta ml c n t o d e l indi v i d uo , oc a s i o nando a f e cc i o nes resp i r a t o ri as , d i ges tiv as (m e g ac ol o n), c ar d i a c a s, cutá nea s y ha s l a l a a ucrte e n c a sos e xt re mo s . es l o lns c c lo v ive asoc i ad o a l hombre , e n la s he ndid u r as de jos l e c hos , fi s uras de l as par e d es . hu ecos d e los p i sos o en ga l l ln c r os y pa l omare s p r 6 xl111o s a l as v l v lcmda s de am é ri c a la ti na qu e , en g e ne r al, so n d e ti po preca r i o y co ns tr uid a s de mate rial inade c uad o. e u e o pag . t7 no ex l ste t ra t a mt e nt o sa ll síactor l o para el 11.tl de cll rgrs , s i endo esta e níe rmedad de ll po c r ó ni co. asl, la un\ c a í orma de comba ti rla, es a nivel prevenllvo. llab ltuaimc n te para esto se usa n pul ver i zaciones de l as v i v l e ndas para e liminar al vec t or, pero este mé t od o llene problemas, pues es t e vaclo es ll e nado por l os vec t o res s il vestres ya que las pulve rizac i o nes sólo se remiten a l as viv i endas llj . el objeti vo de este a rti c ulo es una p r i mera aproxl111ac l 6 n n l as mode l ac l ones del ha l de ch agas con la in tenc i ó n a futuro de deter minar l as pol l llcas de contro l má s ap r op i adas . en es t e traba j o es tudiare mos d os modelos del h1t j de cj1rg11s. en el prlmcr mod e l o, e l má s s impl e, usaremos las s igui e ntes a notac i o nes. n: pob l ac i ó n tolal huma na . s: susceptibl es (fracc i ón d e lnd l v l duos que podr l a n se r lnfectados por la e nf e rmedad). i: inf ecc i osos (frac ción d e ind i viduos que han s ido contagiados). µ: tasa de nac imi e ntos y tasa de muertes . ;\: tasa de contac to. y: nu111e ro de v lnc hucas por uni dades d e 100 . cons i deraremos e n e s te mod e l o l a poblac i ó n cons tante, n y po r esle r1ollvo l a lasa d e nac lml e n tos y l a tasa de muertes se as umen i g uales . el d i a grama para este modelo compar timentado es e l s i guiente: 1, •• s u •c;epllbl ea µns que e n términos de ecuac l.o nes dlferenc l ales, se reduce a l p r ob l ema de cauchy ( 5 1: { (ns)' = µn ;\v(l)ns ¡jns (ni)' • ;\v(t)ns µ ni el cual es equ i va l e nt e a: { s ' "' µ ;\v( t )5 µ s i' = ;\v{t) ¡¡! d onde 5(0)•50 >0, 1 (0) • 10 >0 y s+ ( z l. ( ' ) pag. 1 8 g u b o la d i námica de la vinch u ca la su pon e mos (en este caso ) regida po r in ecuac16n l og lstl c a 14 1: qu e tien e como soluc i ón: v'(l) = olv( t )(l v( t )) v(t)=--1 ---t 1 + 4>e rp ( .. ) .u mo se co ns i deró u n a po b l aci ó n constante tenemos que s'+i'=o , e s dec tr, e l s i stema ( " ) se r ed u c e a la e c uación diferenc ial o rdinaria: s ' = ¡j -µs-~v(t)s qu e l1ene co mo soluc l 6 n [ 3 ) s (t) cl a r amente se lle n o que s(l)~s· y i(t) _. ¡ cuando t---+ai, donde es a s l qu e (5 · , !• )so lució n del sist e ma(•) es, globa lmente aslnt6tlca•enle estable [ 2 ] . el modelo que a conll n u a c 16n p 1· os pvs ~ ~ pac. 19 en t é rmlno de s l s l e ma d e ecuo.c l ones diferenciales o rd ina ria s , a hor a tenemos e l s i g ui e nt e p robl e ma d e cauchy (si : 1 s ' • rn .ws µ s !' "'a( l p)vs ¡d s s 1 a • apvs ( µ + c5) 1 _., d onde sio)•s0 >0: j 9 (0) • 19 >o: t_., (0)• 1.., >o y 5•1 9 • 1_., • n. o o la reglón f ac t.ibl e ep lde mlol 6g l camen le hablando es e l pr i mer octanle de r3 . 1 >o • 1 >o ) s ' los puntos de equ lllbrl o deben sat i s facer el sistema : ( r av ~l)s • rl 5 • rl/" o a(t p)vs µis= o( ••• ) apvs ( µ • c5) 1a = o el cual llene co mo lml c a soluc l 6 n, la lr l v l al (s, js' 1" ) = ( 0 , 0,0 ) 16 1. este sistema line a l l l e ne co1110 polinom i o caracter l s ll co a l pol lno"' l o pcx )• x3 • ( 311 • 6 ·•;\ v) x2 + ( 3µ 2 • 2¡1a v • 2c5µ-211r-c5r • c5av-rav )x+..µ 3 •µ 2c5 ... avµ 2rµ2q.1c5•avµl5 r av( µ •6-c5p) que se obtiene desde el ja cob l ano d e l campo vectorial c.. ) e n e l pun t o de e quili br i o. el punto de e qull l br l o ser á globalmente, aslnt6 ll c a me nle estable si los coe fl c l e nlcs d e l po linom i o p(x) sallsfacen l as condi c i o nes de huruitz (4), es d ec i r: a 1>0 , ªlº· ª/º y a 1 a 2-a/0· d onde p (x) ... x3,.a 1x 2+a 2x+a 3. para efectos d e s l mpllfl cac l 6 n denola r e11es c•r-µ y anali zaremos para qu6 valores d e e se llene la establ l l dad globa l. c:aso 1 : 51 e~ se tendrá que e l punto de equi librio 10. 0,0 ) es un atrnctor j>ac . 20 c u b o global, ya qu e n 1'" 2µ+6 +>..y+ c > o n 2 • µ 2 +2µ c +6µ+ 6 c +µ.\ y+>.. yc +6>..y > o a 3• r>.. y6 p+>.. y6 c+µ 2c +6 µc +>..vµc > o a1 n2 a 3 bµ 2 c +2µ 2 >..y +-6µ>.. vc+ 2µ 2 6+6µc5c+ '1µ ó>.. v+c5avc+µc5 2 +c5 2 c+µ>.. 2 v2 +>.. 2v2 c+ cs 2 >..y +6>.. 2 v2 +2µ r 2 +·ó1 ·2 +ó.\vµ+r 2.\y+ó.\vr(l p) > o caso 2: s1 c >o se t e nd r é la e stabil i dad global s i c <µ+c5 (esto viene dado por la s co nd i c i o nes d e lfljr\.lltz ). conclu s 1 ones . como se ve, l a es labl l ldad en e l primer mod e l o no está afectada por al cre c i mi ento de la p ob l a c 16n de la vln c hu c a ya que la s o lucl 6 n ealnble d e csle ltodc l o no r e fl e ja e l efecto d e la mag nitud del pará •elro ~ d e l a ecuacl6n ( ••j . es a s l co•o s e pla n tea e l segundo mod~ lo pe n s and o q ue la p o bla c l 6 n de l a v l n c h uca e s ta al n l ve 1 d e s a tu r a c h n y po r· e so se t o ma c onslan l e . en e l .ado lo co n pob l acl6 n varlnble, el pr oceso de na c imientos y muertes controla el crec imi e nt o de la po bla ció n (e n egallvo ). pe r o e s to c r ecl•lenlo de la p ob la c l 6n pu e d e se r co n t r o lado d eb i d o a la mo rt a li dad causad a por la e nf er med ad ( e pos i t i vo) . la separac i ó n e n s .l nl o11allcos y as lnlolllá ll cos no es r e fl e j a da e n l a din á mi ca e n c uant o al c r cc l • l enlo de l a pob l ac l 6 n . referenc lf\ s. l!j enclclopcdjabarsa , to mo xv, (1 962 ). l 2 l ll c lhco lc , 11 . u .. asymp t o fj c dellllvl o r in a de t c rmi nisl/ c ep i de ml c hodcj. bull. halh . bl o l ogy 35 :607-6 11\ (1 97j ) . fjj lll rs c h , h. v. o.nd s majo , s. , ou f c re n tla l eqwll l o ns . . dyna mj c al systc&s. and une.ar al geb rd, ac adc ml c press , nc w york (1 97 11 1. hodelos otl . e u e o pag. 21 141 jord a n , d.w. and smllh, p. nonllne ar or dina r y dlfferentlal equatl on . oxfo rd, (1983). ! sl hen a, j ., es tnbljjdad e n epldemlo l ogl.a hat.erná t lc11. apunles de l a semana de l a male máll c o, un l ve r s l dad cal6l l ca de va lpar· aiso (1988). ,. ( 6 j so l omay or , j., l~oes d e equa cs dlferenclals o r d/nárl1s , impa ( 1979). direcc ion del autor ana harta halurana c. ins tituto de ha te hati cas uhtvells ioao ca-tol ica dt: valpalla.i so cas ll la ~ 059 valpallfd soffc lll le. cubo a mathematical journal vol.20, no¯ 3, (31–36). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300031 postulation of general unions of lines and +lines in positive characteristic e. ballico 1 department of mathematics, university of trento, 38123 povo (tn), italy ballico@science.unitn.it abstract a +line is a scheme r ⊂ pr with a line as its reduction l = rred which is the union of l and a tangent vector v * l with vred ∈ l. here we prove in arbitrary characteristic that for r ≥ 4 a general union of lines and +lines has maximal rank. we use the case r = 3 proved by myself in a previous paper and adapt the characteristic zero proof of the case r > 3 given in the same paper. resumen una +ĺınea es un esquema r ⊂ pr con una ĺınea como su reducción l = rred que es la unión de l y un vector tangente v * l, con vred ∈ l. acá demostramos que para r ≥ 4 una unión general de ĺıneas y +ĺıneas tiene rango máximo en caracteŕıstica arbitraria. usamos el caso r = 3 demostrado por el autor en un art́ıculo anterior y adaptamos la demostración en caracteŕıstica cero del caso r > 3 dado en el mismo art́ıculo anterior. keywords and phrases: hilbert function; decorated line; disjoint unions of lines. 2010 ams mathematics subject classification: 14n05. 1the author was partially supported by miur and gnsaga of indam (italy). http://dx.doi.org/10.4067/s0719-06462018000300031 32 e. ballico cubo 20, 3 (2018) 1 introduction the aim of this note is to extend to the positive characteristic case a results in [1]. this extension is sufficient to extend [2, 3] to the positive characteristic case. a scheme x ⊂ pr is said to have maximal rank if h0(ix(t)) · h 1(ix(t)) = 0 for all t ∈ n. fix a line l ⊂ pr, r ≥ 2, and p ∈ l. a tangent vector of pr with p as its support is a zero-dimensional scheme z ⊂ pr such that deg(z) = 2 and zred = {p}. the tangent vector z is uniquely determined by p and the line 〈z〉 spanned by z. conversely, for each line d ⊂ pr with p ∈ d there is a unique tangent vector v with vred = p and 〈v〉 = d. a +line m ⊂ p r supported by l and with nilradical supported by p is the union v ∪ l of l and a tangent vector v with p as its support and spanning a line 〈v〉 6= l. the set of all +lines of pr supported by l and with a nilradical at p is an irreducible variety of dimension r − 1 (the complement of l in the (r − 1)-dimensional projective space of all lines of pr containing p). hence the set of all +lines of pr supported by l is parametrized by an irreducible variety of dimension r. for any +line r and every integer k > 0 we have h0(or(k)) = k + 2 and h 1(or(k)) = 0. for any integers r ≥ 3, t ≥ 0, c ≥ 0 with (t, c) 6= (0, 0) let l(r, t, c) be the set of all schemes x ⊂ pr which are the disjoint union of t lines and c +lines. every element of l(r, c, t) has the map k 7→ (k + 1)t + (k + 2)c as its hilbert function. consider the following statement. theorem 1.1. for all integers r ≥ 3, a ≥ 0 and b ≥ 0, (a, b) 6= (0, 0), a general union x ⊂ pr of a lines and b +lines has maximal rank, this statement was proved in [1] when either r = 3 or r ≥ 4 and the algebraically closed base field has characteristic zero. the aim of this note is to prove theorem 1.1 in positive characteristic (using the case r = 3 proved in [1]). hence we may assume r ≥ 4. we also use numerical lemmas and elementary remarks contained in [1]. we only need to change all parts which quote [4, lemma 1.4] or [6], the only characteristic zero tool used in [1]. we recall that the case c = 0 is due to r. hartshorne and a. hirschowitz ([7]). 2 proof of theorem 1.1 for all integers r ≥ 3 and k ≥ 0 let hr,k denote the following statement: assertion hr,k, r ≥ 3, k ≥ 0: fix (t, c) ∈ n 2 \ {(0, 0)} and take a general x ∈ l(r, t, c). if (k+1)t+(k+2)c ≥ ( r+k k ) , then h0(ix(k)) = 0. if (k+1)t+(k+2)c ≤ ( r+k k ) , then h1(ix(k)) = 0. for all integers r ≥ 3 and k ≥ 0 define the integers mr,k and nr,k by the relations (k + 1)mr,k + nr,k = ( r + k r ) , 0 ≤ nr,k ≤ k (2.1) cubo 20, 3 (2018) postulation of general unions of lines and +lines in positive . . . 33 from (2.1) for the pairs (r, k) and (r, k − 1) we get mr,k−1 + (k + 1)(mr,k − mr,k−1) + nr,k − nr,k−1 = ( r + k − 1 r − 1 ) (2.2) for all k > 0. for all integers r ≥ 3 and k ≥ 0 set ur,k := ⌈ ( r+k r ) /(k + 2)⌉ and vr,k := (k + 2)ur,k − ( r+k r ) . we have (k + 2)(ur,k − vr,k) + (k + 1)vr,k = ( r + k r ) , 0 ≤ vr,k ≤ k + 1 (2.3) as in [1] we need the following assumption br,k: assumption br,k, r ≥ 4, k > 0. fix a hyperplane h ⊂ p r. there is x ∈ l(r, mr,k − nr,k, nr,k) such that the support of the nilradical sheaf of x is contained in h and h 0(ix(k)) = 0. for all x ∈ l(r, mr,k − nr,k, nr,k) we have h 0(ox(k)) = ( r+k r ) and so h1(ix(k)) = h 0(ix(k)). lemma 2.1. we have mr,k − mr,k−1 ≥ nr,k−1 + nr,k if r ≥ 4 and k ≥ 2. proof. assume mr,k − mr,k−1 ≤ nr,k−1 + nr,k − 1. from (2.1) we get mr,k−1 + knr,k−1 + (k + 2)nr,k − k − 1 ≥ ( r + k − 1 r − 1 ) since nr,k−1 ≤ k − 1 and nr,k ≤ k, we get mr,k−1 ≥ ( r+k−1 r−1 ) − 2k2 + 1. since kmr,k−1 ≤ ( r+k−1 r ) and k ( r+k−1 r−1 ) − ( r+k−1 r ) = (r − 1) ( r+k−1 r ) , we get 2k3 − k ≥ (r − 1) ( r + k − 1 r ) (2.4) this inequality is false if r = 4 and k ≥ 2, because it is equivalent to the inequality k(2k2 − 1) ≥ (k+3)(k+2)(k+1)k/8. since the right hand side of (2.4) is an increasing function of r, we conclude for all r ≥ 5 and k ≥ 2. lemma 2.2. fix an integer r ≥ 4 and assume that theorem 1.1 is true in pr−1. then br,k is true for all k > 0. proof. since the case k = 1 is true ([1, remark 3]), we may assume k ≥ 2 and use induction on k. by lemma 2.1 we have mr,k − mr,k−1 ≥ nr,k−1 + nr,k. fix a solution x ∈ l(r, mr,k−1 − nr,k−1, nr,k−1) of br,k−1, say x = a⊔b with a ∈ l(r, mr,k−1 −nr,k−1, 0), b ∈ l(r, 0, nr,k−1) and the tangent vectors of b have support s ⊂ h. by semicontinuity we may assume that no irreducible component of xred is contained in h, that no tangent vector associated to the nilradical of b is contained in h and that s is a general subset of h with cardinality nr,k−1. let c1 ⊂ h be a general union of mr,k − mr,k−1 − nr,k − nr,k−1 lines. let c2 ⊂ h be a general union of nr,k−1 lines, each of them containing a different point of s. let e ⊂ h be a general union of nr,k +lines. since s is 34 e. ballico cubo 20, 3 (2018) general, c1 ∪c2 ∪e is a general element of l(r−1, mr,k −mr,k−1 −nr,k, rn,k). since theorem 1.1 is true in pr−1, by (2.2) we get h1(h, ic1∪c2∪f(k)) = 0 and h 0(h, ic1∪c2∪e(k)) = mr,k−1 −nr,k−1. deforming a with b ∪ c1 ∪ c2 ∪ e fixed, we may assume a ∩ (b ∪ c1 ∪ c2 ∪ e) = ∅ and that hi(h, ic1∪c2∪e∪(a∩h)(k)) = 0, i = 0, 1. since a∩(b∪c1 ∪c2 ∪e) = ∅, y := a∪b∪c1 ∪c2 ∪e is a disjoint union of nr,k +lines with support in h (even contained in h), mr,k −2nr,k−1 −nr,k lines and nr,k−1 sundials in the sense of [5]. hence y is a flat limit of a family of elements l(r, mr,k−1 − nr,k−1, nr,k−1) whose nilpotent sheaf is contained in h ([7], [5]). by the semicontinuity theorem to prove br,k it is sufficient to prove that h 0(iy(k)) = 0. since no tangent vector of b is contained in h, then resh(y) = x and y ∩ h = c1 ∪ c2 ∪ e ∪ (a ∩ h). since h 0(ix(k − 1)) = 0 and h0(h, ic1∪c2∪e∪(a∩h)(k)) = 0, a residual exact sequence gives h 0(iy(k)) = 0. lemma 2.3. assume r ≥ 4 and that theorem 1.1 is true in h = pr−1. fix an integer k ≥ 2 and assume that hr,k−1 is true. fix integers a ≥ 0, b ≥ 0, e ≥ 0 such that e ≤ 2⌊(k + 2)/2⌋, (k + 2)a + (k + 1)b + 4⌊(k + 2)/2⌋ ≤ ( r+k−1 r−1 ) . let x ⊂ h be a general union of a +lines, b lines and e tangent vectors. then h1(h, ix(k)) = 0. proof. it is sufficient to do the case e = ⌊(k + 2)/2⌋. let a ⊂ h be a general union of a lines and b 2-lines. first assume that k is even. let l1, l2 ⊂ h be general lines. fix a general si ⊂ li with ♯(si) = k/2 and a general pi ∈ li, i = 1, 2. let vi ⊂ h be a general tangent vector of h with pi as its support; in particular we assume vi * li. let ei ⊂ li be the union of the k/2 tangent vectors of li with (ei)red = si. set y := a ∪ e1 ∪ v1 ∪ e2 ∪ v2. let ri the +lines with li as their supports and with vi as the tangent vectors associated to their nilpotent sheaf. we have h0(oa∪e1∪e2∪v1∪v2(k)) = h 0(oa∪r1∪r2(k)), h 1(oa∪e1∪e2∪v1∪v2(k)) = h 1(oa∪r1∪r2(k)) and h0(ia∪e1∪e2∪v1∪v2(k)) = h 0(ia∪r1∪r2(k)). therefore we have h 1(ia∪e1∪e2∪v1∪v2(k)) = h1(ia∪r1∪r2(k)). since (k + 2)a + (k + 1)b + 2(k + 2) ≤ ( r+k−1 r−1 ) and theorem 1.1 is true in pr−1, we have h1(ia∪r1∪r2(k)) = 0. hence h 1(ia∪e1∪e2∪v1∪v2(k)) = 0. the semicontinuity theorem gives h1(h, ix(k)) = 0. now assume that k is even. let fi ⊂ li be any disjoint union of (k + 1)/2 tangent vectors. we have h0(oa∪f1∪f2(k)) = h 0(oa∪l1∪l2(k)), h 1(oa∪f1∪f2(k)) = h 1(oa∪l1∪l2(k)) and h0(ia∪f1∪f2(k)) = h 0(ia∪l1∪l2(k)). therefore we obtain h 1(ia∪f1∪f2(k)) = h 1(ia∪l1∪l2(k)). since (k + 2)a + (k + 1)b + 2(k + 1) ≤ ( r+k−1 r−1 ) and theorem 1.1 is true in pr−1, we have h1(ia∪l1∪l2(k)) = 0. therefore h 1(ia∪f1∪f2(k)) = 0. the semicontinuity theorem gives h1(h, ix(k)) = 0. proof of theorem 1.1: by [1] we may assume r ≥ 4. by induction on r we may also assume that theorem 1.1 is true in pr−1. by [1, remark 3] it is sufficient to prove hr,k for all integers k ≥ 1. hr,1 is true ([1, lemma 3]). hence we may assume k ≥ 2 and that hr,k−1 is true. by [1, remark 4] it is sufficient to prove hr,k for the pairs (t, c) such that either t = 0 and ( r+k r ) − k − 1 ≤ c(k + 2) ≤ ( r+k r ) or t(k + 1) + (k + 2)c = ( r+k r ) and c > 0; in the former case either cubo 20, 3 (2018) postulation of general unions of lines and +lines in positive . . . 35 vr,k = 0 and c = ur,k or vr,k > 0 and c = ur,k − 1; in the latter case we have t + c ≥ ur,k. if c < nr,k−1, then we use step (b) of the proof of theorem 1 in [1], because we gave a characteristic free proof of br,k (lemma 2.2). the case c ≥ nr,k−1 and t ≥ mr,k−1 − nr,k−1 was proved as step (a1) without using the characteristic zero assumption. hence we may assume c ≥ nr,k−1 and t < mr,k−1 − nr,k−1, i.e. the case of step (a2) of the proof in [1]. (i) assume t = 0 and hence either vr,k = 0 and c = ur,k or vr,k > 0 and c = ur,k − 1. fix a general u ∈ l(r, 0, vr,k−1, ur,k−1 −vr,k−1), say u = a⊔b with a the union of the vr,k−1 lines. by hr,k−1 we have h i(iu(k−1)) = 0, i = 0, 1. it is easy to check using (2.3) that ur,k > ur,k−1. hence c ≥ ur,k−1. let e ⊂ h be a general union of c − ur,k−1 +lines. we may assume e ∩ (h ∩ u)) = ∅. let g ⊂ h be a general union of vr,k−1 tangent vectors of h with the only restriction that gred = a ∩ h. for general a (and hence a general a ∩ h)) the scheme e ∪ g is a general union inside h of ur,k − uk−1 +lines and vr,k−1 tangent vectors. we have vr,k−1 ≤ k. using (2.3) for the integer k − 1 is easy to check that if vr,k−1 > 0, then ur,k−1 − vr,k−1 ≥ 2(k + 2) − 2vr,k−1. hence lemma 2.3 gives h1(h, ie∪g(k)) = 0. since b ∩ h is a general union of (ii) assume t > 0, c > 0, t(k+1)+(k+2)c = ( r+k r ) and t < mr,k−1 −nr,k−1. first assume t ≤ 2⌊(k + 2)/2⌋. in this case we may use the proof given in [1] (step (a2)) quoting lemma 2.3 instead of [4, lemma 1.4] for the postulation of the t tangent vectors, because mr,k−1 − t ≥ 2k + 2 in this case. therefore we may assume t ≥ k + 1. since t < mr,k−1 − nr,k−1, we have k ≥ 3 and kt < ( r+k−1 r ) . set d := ⌊( ( r+k−1 r ) − kt)/(k + 1)⌋ and z := (k + 1)d + kt − ( r+k r ) . we have 0 ≤ z ≤ k + 1. fix a general w ∈ l(r, t, d). since hr,x holds for x = k − 1, k − 2, we have h0(iw(k − 2)) = 0 and h 1(iw(k)) = 0 and h 0(iw(k)) = z. since s is general in h and ♯(s) = z, we get hi(iw∪s(k−1)) = 0, i = 0, 1. since kt+(k+1)t+z = ( r+k−1 r ) and t(k+1)+(k+2)c = ( r+k r ) , we get t + d + (k + 2)(c − d − z) + (k + 1)z = ( r + k − 1 r − 1 ) (2.5) claim 1: we have c ≥ d + z. proof of claim 1: assume c ≤ d+z−1. from (2.5) we get t+d+(k+1)z−(k+1) ≥ ( r+k−1 r−1 ) and hence k(t + d) + (k + 1)kz − (k + 1)k ≥ k ( r+k−1 r−1 ) . since kt + (k + 1)d + z = ( r+k−1 r ) and z ≤ k, we get (k + 1)k2 − k(k + 1) − k ≥ k ( r+k−1 r−1 ) − ( r+k−1 r ) , i.e. k3 − 2k ≥ (r − 1) ( r+k−1 r ) . call φ(r, k) the difference between the right hand side and the left hand side of this inequality. we have φ(r, k) = (r − 1) ( r+k−1 r ) − k3 + 2k, which is positive if r ≥ 4 and k ≥ 2. let m ⊂ h be a general union of c − d − z +lines of h. let n ⊂ h be z general lines of h, each of them containing a different point of z. since s is general, m ∪ n has the hilbert function of a general element of l(r − 1, z, c − d − z) and hence it has maximal rank. by (2.5) we have h1(h, im∪n(k)) = 0 and h 0(im∪n(k)) = t + d. let z ⊂ p r be a general union of z +lines of pr with n as their support. we have g ∩ h = n and resh(z) = s. since w ∪ m ∪ z ∈ l(r, t, c), it is sufficient to prove that hi(iw∪m∪z(k)) = 0, i = 0, 1. since resh(w ∪ m ∪ z) = w ∪ s, we have hi(iresh(w∪m∪z)(k − 1)) = 0. since w ∩ h is a general union of d + c points of h and (w ∪ m ∪ z) = (w ∩ h) ∪ m ∪ n as schemes, (2.5) gives hi(h, ih∩(w∪m∪z(k)) = 0. apply the 36 e. ballico cubo 20, 3 (2018) castelnuovo’s lemma. references [1] e. ballico, postulation of general unions of lines and decorated lines, note mat. 35 (2015), no. 1, 1–13. [2] e. ballico, postulation of disjoint unions of lines and a multiple point ii, mediterr. j. math. 13 (2016), no. 4, 1449–1463. [3] e. ballico, postulation of general unions of lines and double points in a higher dimensional projective space, acta math. vietnam. 41 (2016), no. 3, 495–504. [4] a. bernardi, m. v. catalisano, a. gimigliano and m. idà, secant varieties to osculating varieties of veronese embeddings of pn,j. algebra 321 (2009), no. 3, 982–1004. [5] e. carlini, m. v. catalisano and a. v. geramita, 3-dimensional sundials, cent. eur. j. math. 9 (2011), no. 5, 949–971. [6] c. ciliberto and r. miranda, interpolations on curvilinear schemes, j. algebra 203 (1998), no. 2, 677–678. [7] r. hartshorne and a. hirschowitz, droites en position générale dans pn, algebraic geometry, proceedings, la rábida 1981, 169–188, lect. notes in math. 961, springer, berlin, 1982. introduction proof of theorem 1.1 cubo a mathematical journal vol.20, no¯ 01, (31–39). march 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000100031 pre-regular sp-open sets in topological spaces p. jeyanthi and p. nalayini research centre, department of mathematics, govindammal aditanar college for women, tiruchendur-628 215, tamil nadu, india. jeyajeyanthi@rediffmail.com,nalayini4@gmail.com t. noiri shiokita cho, hinagu, yatsushiro shi, kuvempu university kumamoto ken, 869-5142 japan t.noiri@nifty.com abstract in this paper, a new class of generalized open sets in a topological space, called preregular sp-open sets, is introduced and studied. this class is contained in the class of semi-preclopen sets and cotains all pre-clopen sets. we obtain decompositions of regular open sets by using pre-regular sp-open sets. resumen en este art́ıculo se introduce y estudia una nueva clase de conjuntos abiertos generalizados en un espacio topológico, llamados conjuntos pre-regulares sp-abiertos. esa clase está contenida en la clase de conjuntos semi-preclopen y contiene todos los conjuntos pre-clopen. obtenemos descomposiciones de conjuntos abiertos regulares usando conjuntos pre-regulares sp-abiertos. keywords and phrases: generalized open sets, preopen, regular open, pre-regular sp-open, decompositions of complete continuity. 2010 ams mathematics subject classification: 54a05. http://dx.doi.org/10.4067/s0719-06462018000100031 ignacio castillo ignacio castillo ignacio castillo ignacio castillo 32 p. jeyanthi, p. nalayini and t. noiri cubo 20, 1 (2018) 1 introduction in general topology, by repeated applications of interior (int) and closure (cl) operators several different new classes of sets are defined in the following way. definition 1. a subset a of a space x is said to be i) semi-open [10] if a ⊆ cl(inta). ii) preopen [11] if a ⊆ int(cla). iii) semi-preopen [2] or β-open [1] if a ⊆ cl(int(cla)). iv) α-open [12] if a ⊆ int(cl(inta)). v) regular open [13] if a = int(cla). vi) b-open [3] if a ⊆ cl(inta) ∪ int(cla). vii) pre-regular p-open [9] if a = pint(pcla). the complements of the above open sets are called their respective closed sets. definition 2. a subset a of a space x is called a q-set [14] or δ-set [5] if int(cla) ⊆ cl(inta). in this paper, we introduce and study a new class of sets, called pre-regular sp-open sets using pre-interior and semi-preclosure operators. this class is contained in the class of semi-preclopen sets and cotains all pre-clopen sets. moreover, we investigate the relationship between this class of sets and other class of open sets. by using pre-regular sp-open sets, we obtain decompositions of regular open sets. in the last section, we obtain decompositions of complete continuity. throughout this paper (x, τ) (briefly x) denotes a topological space on which no separation axioms are assumed, unless explicity stated. we recollect some of the relations that, together with their duals, we shall use in the sequel. proposition 1. [2] let a be a subset of a space x. then i) pcla = a ∪ cl(inta) and pinta = a ∩ int(cla). ii) spcla = a ∪ int(cl(inta)) and spinta = a ∩ cl(int(cla)). iii) pint(spcla) = (a ∩ int(cla)) ∪ int(cl(inta)). iv) pcl(spinta) = (a ∪ cl(inta)) ∩ cl(int(cla)). definition 3. a function f : x → y is called completely continuous [4] (resp. α-continuous [8],semi-continuous [10], q-continuous [14] ) if the inverse image of every open subset of y is a regular open (resp. α-open, semi-open, a q-set) subset of x. cubo 20, 1 (2018) pre-regular sp-open sets in topological spaces 33 2 pre-regular sp-open sets in this section, we define and characterize pre-regular sp-open sets and study some of their properties. definition 4. a subset a of a topological space (x, τ) is said to be pre-regular sp-open if a = pint(spcla). the complement of a pre-regular sp-open set is said to be pre-regular sp-closed. we denote the collection of all pre-regular sp-open (resp. preopen, preclosed, pre-semiopen, pre-semiclosed, pre-clopen, pre-semiclopen) sets of (x, τ) by prspo(x) (resp. po(x), pc(x), pso(x), psc(x), pco(x), psco(x)). theorem 2.1. let (x, τ) be a topological space and a, b subsets of x. then the following hold: i) if a ⊆ b, then pint(spcla) ⊆ pint(spclb). ii) if a ∈ po(x, τ), then a ⊆ pint(spcla). iii) if a ∈ spc(x, τ), then pint(spcla) ⊆ a. iv) we have pint(spcl(pint(spcla))) = pint(spcla). v) if a ∈ spc(x, τ), then pinta is a pre-regular sp-open set. proof. i) suppose that a ⊆ b. then pint(spcla) ⊆ pint(spclb). ii) suppose that a ∈ po(x, τ). since a ⊆ spcla, we have a ⊆ pint(spcla). iii) suppose that a ∈ spc(x, τ). since pinta ⊆ a, we have pint(spcla) ⊆ a. iv) we have pint(spcl(pint(spcla))) ⊂ pint(spcl(spcla)) = pint(spcla) and pint(spcl(pint(spcla))) ⊃ pint(pint(spcla)) = pint(spcla). hence pint(spcl(pint(spcla))) = pint(spcla). v) suppose that a ∈ spc(x, τ). by (i), we have pint(spcl(pinta)) ⊆ pint(spcla) = pinta. on the other hand, we have pinta ⊆ spcl(pinta). therefore pinta ⊆ pint(spcl(pinta)) and hence pint(spcl(pinta)) = pinta. remark 2.2. the family of pre-regular sp-open sets is not closed under finite union as well as finite intersection. it will be shown in the following example. example 2.3. let x = {a, b, c, d} and τ = {∅, {a, b}, {a, b, c}, {a, b, d}, x}. then {a} and {b} are pre-regular sp-open sets but their union {a, b} is not a pre-regular sp-open set. moreover, {a, c, d} and {b, c, d} are pre-regular sp-open but their intersection {c, d} is not a pre-regular sp-open set. theorem 2.5 and 2.6 give the characterizations of pre-regular sp-open sets. 34 p. jeyanthi, p. nalayini and t. noiri cubo 20, 1 (2018) theorem 2.4. let (x, τ) be a topological space. for a subset a of x, the following are equivalent: i) a is pre-regular sp-open. ii) a = spcla ∩ int(cla). iii) a = pinta ∪ int(cl(inta)). proof. it follows form proposition 1.3. theorem 2.5. let (x, τ) be a topological space. a subset a of x is pre-regular sp-open if and only if it is preopen and semi-preclosed . proof. let a be pre-regular sp-open. then a = pint(spcla). hence pinta = pint(pint(spcla)) = pint(spcla) = a. thus a is preopen. by theorem 2.5, a = pinta ∪ int(cl(inta)) and int(cl(inta)) ⊆ a. therefore, a is semi-preclosed. conversely assume that a is both preopen and semi-preclosed. then a = pinta and a = spcla. now pint(spcla) = pinta = a. hence a is pre-regular sp-open. corolary 1. for a topological space (x, τ), we have po(x) ∩ pc(x) ⊆ prspo(x) ⊆ spo(x) ∩ spc(x). proof. this is obvious. remark 2.6. the converse inclusions in corollary 2.7 need not be true as the following examples show. example 2.7. let x = {a, b, c, d} and τ = {∅, {a}, {b}, {a, b}, {b, c}, {a, b, c}, x}. then {a, d} is semi-preclopen but not pre-regular sp-open. example 2.8. let x = {a, b, c, d} and τ = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {c, d}, {a, c, d}, {a, b, c}, {b, c, d}, x}. then {c} is pre-regular sp-open but it is not pre-clopen. theorem 2.9. in any space (x, τ), the empty set is the only subset which is nowhere dense and pre-regular sp-open. proof. suppose a is nowhere dense and pre-regular sp-open. then by theorem 2.5, a = pint(spcla) = spcla ∩ int(cla) = spcla ∩ ∅ = ∅. remark 2.10. the notions of pre-regular sp-open sets and open sets (hence α-open sets, semiopen sets, q-sets) are independent of each other. it is shown in [5] and [14] that every semi-open set is a q-set, that is, a δ-set. cubo 20, 1 (2018) pre-regular sp-open sets in topological spaces 35 example 2.11. let x = {a, b, c} and τ = {∅, {a, b}, x}. then {a, b} is open hence α-open, semiopen, a q-set but it is not pre-regular sp-open. also, {a} is pre-regular sp-open but it is not a q-set. theorem 2.12. every regular open set is pre-regular sp-open. proof. let a be regular open. then a = int(cla). by proposition 1.3, pint(spcla) = (spcla) ∩ int(cl(spcla)) = spcla ∩ int(cl[a ∪ int(cl(inta))]) = spcla ∩ int(cla) = spcla ∩ a = a. this shows that a is pre-regular sp-open. the above disscusion can be summarized in the following diagram: diagram regular open ⇒ open ⇒ α-open ⇒ semi-open ⇒ q-set ⇓ ⇓ ⇓ pre-regular sp-open ⇒ preopen ⇒ b-open ⇒ semi-preopen remark 2.13. a q-set and a semi-preopen set are independent by example 2.13 and the following example. example 2.14. let r be the real numbers with the usual topology. then for each x ∈ r, cl(int(cl{x})) = ∅ and it does not contain {x}. hence {x} is not semi-preopen. but int(cl{x}) = cl(int{x}) = ∅ and {x} is a q-set. theorem 2.15. every pre-regular p-open set is pre-regular sp-open. proof. let a be pre-regular p-open. then a = pint(pcla) and a is preopen. since spcla ⊆ pcla, we have pint(spcla) ⊆ pint(pcla) = a. on the other hand, we have a ⊆ spcla. since a is preopen, a = pinta ⊆ pint(spcla). hence a = pint(spcla). theorem 2.16. for a subset a of a space x, the following are equivalent: i) a is regular open. ii) a is pre-regular sp-open and a q-set. iii) a is α-open and semi-preclosed. proof. i) ⇒ ii). let a be regular open. then, by theorem 2.14 a is pre-regular sp-open and also by diagram, a is a q-set. ii) ⇒ i). since a is a q-set, int(cla) ⊂ cl(inta) and int(cla) ⊂ int(cl(inta)) ⊂ int(cla). therefore, we have int(cla) = int(cl(inta)). by using theorem 2.5, we obtain int(cla) = [a ∪ int(cla)] ∩ int(cla) = [a ∪ int(cl(inta))] ∩ int(cla) = spcla ∩ int(cla) = a. i) ⇒ iii). let a be regular open. then a is open and a = int(cla) = int(cl(inta)). therefore, every regular open set is α-open and semi-preclosed. 36 p. jeyanthi, p. nalayini and t. noiri cubo 20, 1 (2018) iii) ⇒ i). let a be α-open and semi-preclosed. then int(cl(inta)) ⊂ a ⊂ int(cl(inta)). therefore, a = int(cl(inta)) and hence int(cla) = int(cl(int(cl(inta)))) = int(cl(inta)) = a. hence a is regular open. corolary 2. suppose a is pre-regular sp-open. then the following are hold: i) if a is open, then a is regular open. ii) if a is closed, then a is clopen. iii) if a is semi-open, then a is regular open. iv) if a is semi-closed, then a is α-open and semi-preclosed. proof. since a is pre-regular sp-open, by theorem 2.5 a = spcla∩int(cla) = pinta∪int(cl(inta)). i) suppose a is open. then by diagram, a is a q-set and by theorem 2.18, we have a is regular open. ii) suppose a is closed. now a = spcla ∩ int(cla) = spcla ∩ inta = inta. hence a is open and hence clopen. iii) since every semi-open set is a q-set, by theorem 2.18 a is regular open. iv) suppose a is semiclosed. then int(cla) ⊆ a. this implies int(cla) ⊂ inta ⊂ cl(inta). hence a is a q-set and by theorem 2.18, a is α-open and semi-preclosed. remark 2.17. in a partition space (x, τ), a subset a of x is preopen if and only if a is pre-regular sp-open. theorem 2.18. if a space (x, τ) is submaximal, then any finite intersection of pre-regular sp-open sets is pre-regular sp-open. proof. let {ai|i ∈ i} be a finite family of pre-regular sp-open sets. then {ai|i ∈ i} is a finite family of preopen sets. since x is submaximal, ⋂ i∈i ai is pre open. therefore by theorem 2.2 (ii), ⋂ i∈i ai ⊆ pint(spcl( ⋂ i∈i ai). on the other hand, for each i ∈ i, we have ⋂ i∈i ai ⊆ ai and by theorem 2.2 (i) pint(spcl( ⋂ i∈i ai))) ⊆ pint(spclai). since pint(spclai) = ai, we have pint(spcl( ⋂ i∈i ai))) ⊆ ⋂ i∈i ai. hence pint(spcl( ⋂ i∈i ai))) = ⋂ i∈i ai. theorem 2.19. if a is pre-regular sp-closed and a rare set of a space (x, τ), then a is semipreopen. cubo 20, 1 (2018) pre-regular sp-open sets in topological spaces 37 proof. since a is pre-regular sp-closed, by theorem 2.5 a = pcl(spinta) = spinta ∪ cl(inta). since a is a rare set, inta = ∅. thus a = spinta. hence a is semi-preopen. recall that a space (x, τ) is said to be an extremally disconnected if the closure of every open subset of x is open. moreover, it is shown in [7] (x, τ) is extremally disconnected if and only if spo(x) = po(x). theorem 2.20. for an extremally disconnected space (x, τ), the following are equivalent: i) a is pre-regular sp-open. ii) a is pre-regular sp-closed. iii) a is pre-clopen. iv) a is semi-preclopen. proof. (i) ⇔ (iii). suppose that a is pre-regular sp-open. then by theorem 2.6, a is preopen and semi-preclosed. since x is extremally disconnected, a is pre-clopen. hence a is pre-closed. the converse is obvious by theorem 2.6. (ii) ⇔ (iv). let a be pre-regular sp-closed. then x\a is pre-regular sp-open and by (i) ⇔ (iii) x\a is pre-clopen. therefore, a is semi-preclopen. the converse is obvious. (iii) ⇔ (iv). this is obvious. recall that a space (x, τ) has the property q [10] if int(cla) = cl(inta) for all subset a of x. theorem 2.21. let (x, τ) be a space with the property q. for a subset a ⊆ x, the following properties are equivalent: i) a is pre-regular sp-open. ii) a is pre-regular sp-closed. iii) a is regular open. iv) a is regular closed. proof. (i) ⇔ (iii). by proposition 1.3, pint(spcla) = [a ∩ int(cla)] ∪ int(cl(inta)) = [a ∩ int(cla)] ∪ int(int(cla)) = int(cla). this completes the proof. (ii) ⇔ (iv). by proposition 1.3, pcl(spinta) = [a ∪ cl(inta)] ∩ cl(int(cla)) = [a ∪ cl(inta)] ∩ cl(cl(inta)) = cl(inta). this completes the proof. (iii) ⇔ (iv). this is obvious. 38 p. jeyanthi, p. nalayini and t. noiri cubo 20, 1 (2018) 3 decompositions of complete continuity in this section, the notion of pre-regular sp-continuous functions is introduced and the decompositions of complete continuity are discussed. definition 5. a function f : x → y is said to be pre-regular sp-continuous (briefly, prspcontinuous) if f−1(v) is pre-regular sp-open in x for each open subset v of y. by theorems 2.18 and daigram, we have the following main theorem theorem 3.1. for a function f : x → y, the following properties are equivalent: i) f is completely continuous. ii) f is prsp-continuous and continuous. iii) f is prsp-continuous and α-continuous. iv) f is prsp-continuous and semi-continuous. v) f is prsp-continuous and q-continuous. remark 3.2. as shown by the following examples, prsp-continuity and continuity (hence αcontinuity, semi-continuity, q-continuity) are independent of each other. example 3.3. let x = {a, b, c}, τ = {∅, {a}, x} and σ = {∅, {a, b}, x}. then i) the identity function f : (x, τ) → (x, τ) is continuous but it is not prsp-continuous since f−1({a}) = {a} is open but it is not pre-regular sp-open. ii) consider the function f : (x, σ) → (x, τ) defined by f(a) = a, f(b) = c and f(c) = b. then f is prsp-continuous but it is not q-continuous, since f−1({a}) = {a} is pre-regular sp-open but it is not a q-set in (x, σ). references [1] m. e. abd el-monsef, s. n. el-deeb and r. a. mahmoud, β-open and β-continuous mappings, bull. fac. sci. assiut univ., 12 (1983), 77-90. [2] d. andrijević, semi-preopen sets, mat. vesnik., 38 (1986), 24-32. [3] d. andrijević, on b-open sets, mat. vesnik., 48 (1996), 59-64. [4] s. p. arya and r. gupta, on strongly continuous mappings, kyungpook math. j., 14 (1947), 131-143. cubo 20, 1 (2018) pre-regular sp-open sets in topological spaces 39 [5] c. chattopadhyay and c. bandyopadhyay, on structure of δ-sets, bull. calcutta math. soc., 83 (2011), 281-290. [6] j. dugundji, topology, allyn and bacon, boston (1966). [7] m. ganster and d. andrijević, on some questions concerning semi-preopen sets, j. inst. math. comp. sci. (math. ser.) 1 (2) (1988), 65-75. [8] i. a. hasanein, m. e. abd elmonsef and s. n. el-deep, α-continuity and α-open mappings, acta. math. hungar., 41 (1983), 213-218. [9] s. jafari, on certain types of notions via preopen sets , tamkang j. math., 37 (4) (2006), 391-398. [10] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly, 70 (1963), 36-41. [11] a. s. mashhour, m. e. abd el-monsef and s. n. el-deep, on precontinuous and weak precontinuous mappings, proc. math. phys. soc. egypt., 53 (1982), 47-53. [12] o. nj̊astad, on some classes of nearly open sets, pacific j. math. 15 (1965), 961-970. [13] m. stone, applications of the theory of boolean ring to general topology , trans. amer. math. soc., 41 (1937), 374. [14] p. thangavelu and k. c. rao, q-sets in topological spaces , prog of maths., 36 (1-2) (2002), 159-165. introduction pre-regular sp-open sets decompositions of complete continuity a mathematical journal vol. 7, no 3, (65 73). december 2005. conjectures in inverse boundary value problems for quasilinear elliptic equations ziqi sun department of mathematics and statistics wichita state university wichita, ks 67226, usa ziqi.sun@wichita.edu abstract inverse boundary value problems originated in early 80’s, from the contribution of a.p. calderon on the inverse conductivity problem [c], in which one attempts to recover the electrical conductivity of a body by means of boundary measurements on the voltage and current. since then, the area of inverse boundary value problems for linear elliptic equations has undergone a great deal of development [u]. the theoretical growth of this area contributes to many areas of applications ranging from medical imaging to various detection techniques [b-b][che-is]. in this paper we discuss several conjectures in the inverse boundary value problems for quasilinear elliptic equations and their recent progress. these problems concern anisotropic quasilinear elliptic equations in connection with nonlinear materials and the nonlinear elasticity system. resumen problemas inversos a valores en la frontera se desarrollaron a comienzos de la década de los 80, a partir de contribuciones de a.p. calderon en el problema de conductividad inversa [c], en el cual se intenta recuperar las conductividad eléctrica de un cuerpo mediante mediciones de voltaje y corriente en la frontera. desde entonces, el área de problemas a valores en la forntera inversos para ecuaciones lineales eĺıpticas ha sido objeto de mucho desarrollo [u]. el crecimiento de la teoŕıa en esta área tiene aplicaciones en muchas aplicaciones, las que vaŕıan desde imagenoloǵıa médica, hasta diversos métodos de detección [bb], [che-is]. en este art́ıculo, discutimos varias conjeturas en problemas inversos de valores en 66 ziqi sun 7, 3(2005) la frontera para ecuaciones eĺıpticas quasi-lineales y sus progresos recientes. estos problemas dicen relación con ecuaciones eĺıpticas quasilineales anisotrópicas en conexión con materiales nolineales y sistemas de elasticidad no lineal. key words and phrases: inverse boundary value problem. dirichlet to neumann map math. subj. class.: 35r30 1 anisotropic quasilinear conductivity equations consider the quasilinear elliptic equation lau = n∑ i,j=1 (aij (x, u)uxi )xj = 0, u|γ = f ∈ c 2,α(γ) (1) on a bounded domain ω ⊂ rn, n ≥ 2, with smooth boundary γ. here a(x, t) = (aij (x, t))n×n is the quasilinear coefficient matrix which is assumed to be in the c1,α class with 0 < α < 1. the nonlinear dirichlet to neumann map λa : f → ν · a(x, f )∇u|γ is an operator from c2,α(γ) to c1,α(γ), which carries essentially all information about the solution u which can be measured on the boundary. here we denote ν to be the unit outer normal of ω. the inverse problem under discussion is to recover information about the quasilinear coefficient matrix a from the knowledge of λa. this problem was raised by r. kohn and m. vogelius [kv] in mid 80’s as a nonlinear analogue of the well known inverse conductivity problem posed by a.p. calderon [c]. physically, the problem is connected to electrical impedance tomography in nonlinear media. it has been shown in [su1] that, in the isotropic case of the problem, i.e., when a is a scalar matrix, the dirichlet to neumann map λa gives full information about a. in other words, λa determines a uniquely as a function on ω × r. this generalizes to the quasilinear case the well known uniqueness theorems of the linear case (i.e., when a is scalar and is indenpendent on t)[su1,2][suu2] [n]. in the anisotropic case, however, one only expects to recover a module the group g = {all c3,α diffeomorphisms φ: ω̄ → ω̄ with φ ∣∣ ∂ω = identity}. in fact, λa is invariant under g: for any a and φ ∈ g, λa = λhφa. here hφa is the pull back of a under φ: hφa(x, t) = (|detdφ|−1(dφ)t a(x, t)(dφ)) ◦ φ−1 (2) 7, 3(2005) conjectures in inverse boundary value problems for ... 67 where dφ is the jacobian matrix of φ. one should observe that (2) holds only when φ is independent on t. thus, the following conjecture is natural: conjecture 1: assume that λa1 = λa2 . then there exists a unique diffeomorphism φ ∈ g so that a2 = hφa1. in [suu1] we have verified this conjecture in the c2,α category for dimension n = 2 and in the real analytic category for dimension n ≥ 3. these results extend all known results regarding this conjecture in the case of linear coefficient matrices (i.e. when a is independent of t), obtained earlier in the works of sylvester [s], nachman [n] and lee-uhlmann [lu]. we mention that in the two dimensional case the unique diffeomorphism φ in the result belongs to the c3,α class, which is one order smoother than a1 and a2 and in the case n ≥ 3, φ is in the real analytic category. assuming holder smoothness for the coefficient seems quite essential to assure that φ is one order smoother than the coefficient matrices. as explained in [suu1], this is closely related to the elliptic regularity theory. the proof is based on a well known linearization technique introduced in [i1] and further developed in [i2][is][in][su1,3] which reduces the nonlinear problem to a linear one. let t ∈ r and g ∈ c2,α(γ). from λa one determines two linear operators: k (1) a,t : g → d/dsλa(t + sg)|s=0 k (2) a,t : g → d 2/ds2(s−1λa(t + sg))|s=0 (3) one observes that k(1)a,t = λat , the dirichlet to neumann map corresponding to the linear coefficient matrix at(x) = a(x, t) for a fixed t. so, if λa1 = λa2 for two quasilinear coefficient matrices a1 and a2, then λat1 = λat2 , ∀t ∈ r, and since the conjecture is true in the linear case, one obtains a family of diffeomorphisms φt ∈ g, depending on the parameter t, so that hφt a t 1 = a t 2, ∀t ∈ r. (4) the mathematical difficulty is to show that φt is actually independent on t, which would imply the result. it has been verified in [suu1] that φt is smooth in t. for dimension n ≥ 3, this was achieved by studying a related geometrical problem in which φt becomes a family of isometries between two families of riemannian metrics |ati| 1/(n−2)(ati) −1 on ω̄, i = 1, 2. for n = 2, one can transform it to a similar problem where φt becomes a family of conformal diffeomorphisms between riemannian metrics (ati) −1, i = 1, 2. in the latter case, the smoothness is verified via the standard theory of the beltrami equation [ab]. so, the task is to show that φ̇|t=0, where dot means differentiation in t variable. we only give a very brief description of the proof. one only needs to show φ̇0 = φ̇t|t=0 = 0 (5) 68 ziqi sun 7, 3(2005) since the same argument works for t 6= 0. by a transformation one may assume that φ0 = identity map. the proof of (5) is then based on the information obtained from (3): k (2) a1,t = k(2)a2,t. (6) a crucial step of the proof is to show that one can recover from k(2)a,t information about ∂a/∂t(x, 0). so (6) implies ∂ ∂t a1(x, 0) = ∂ ∂t a2(x, 0), ∀x ∈ ω. (7) one views (7) as a certain control over the flows at1 and a t 2 at t = 0. actually, the assumption φ0 =id. together with (7) give a01 = a 0 2 and ȧ 0 1 = ȧ 0 2. consider now the solution flows uti,f for the linear equations lati (u t i,f ) = 0 with u t i,f |γ = f , i = 1, 2. one observes that the control over the flows of coefficient matrices translates to a control over the solution flows. in fact, for every f , u01,f = u 0 2,f and u̇ 0 1,f = u̇ 0 2,f . since the transformation in (4) links ut1,f to u t 2,f via the relation u̇ t 1,f = u̇ t 2,f ◦ φ t, one differentiates it in t at t = 0 to get φ̇0 ·∇u01,f = 0 for all boundary value f , from which (5) follows by an argument based on runge approximation. see [suu1] for details. the above result obtained in [suu1] covers the two dimensional case and the real analytic case in dimension three or higher. however, the remaining case in dimension n ≥ 3 is essentially open even when the equation (1) is linear. an interesting problem for further study in this direction is whether one can reduce the conjecture in the quasilinear case directly to the conjecture in the linear case. in other words, one would like to verify conjecture 1 under the assumption that conjecture 1 holds in the linear case. such a full reduction has already been obtained in the scalar case (where a is a scalar matrix) [su1]. it is possible that the same reduction also hold in the anisotropic case. one possible approach to attack this problem is to further study the relation between (6) and (7) in the general case, which is the heart of proof in [suu1]. the main issue is how to avoid the use of the property of completeness of products of solutions which is currently available only in the two dimensional case and the case of real analytic coefficient matrices. 2 quasilinear equations in connection with nonlinear elastic materials consider the quasilinear elliptic equation ∇ · a(x, ∇u) = 0, u|γ = f ∈ c3,α(γ), (8) on a bounded domain ω ⊂ rn, n ≥ 2, with smooth boundary γ. here a(x, p) = (a1(x, p), a2(x, p), ..., an(x, p)) is the quasilinear coefficient vector. we assume that a and ap (which is assumed to be symmetric) are both in c2,α(ω̄×r) with 0 < α < 1, a(x, 0) = 0 and the structure conditions which guarantee the unique solvability in the c3,α class [hsu]. 7, 3(2005) conjectures in inverse boundary value problems for ... 69 the nonlinear dirichlet to neumann map λa : f → ν · a(x, ∇u)|γ, (9) is an operator from c3,α(γ) to c2,α(γ), which carries essentially all information about the solution u observable on the boundary. one verifies that λa is invariant under the group g: λa = λhφa for all φ ∈ g. here the transformation hφ is defined as hφa(x, p) = (|detdφ|−1(dφ)t a(x, (dφ)p)) ◦ φ−1. the main problem is whether the converse is true. conjecture 2: assume that λa1 = λa2 . then there exists a unique diffeomorphism φ ∈ g so that a2 = hφa1. the equation (8) can be considered as a simple scalar model of the nonlinear elasticity system, which takes the form ∇{σ(x, e) + (∇u)σ(x, e)} = 0, (10) where u is the displacement vector function resulting from a deformation of an elastic body and the matrix function σ is the constitutive relation with the strain tensor e = 1 2 (∇ut + ∇u + ∇ut ∇u). in [hsu], we developed a mathematical framework towards proving this conjecture in the case of two dimensions. in the discussion below, we assume λa1 = λa2 for two quasilinear coefficient vectors a1 and a2 in dimension two. by linearizing (9) one obtains, as in the case of conjecture 1, a family of diffeomorphisms {φf }⊂ g which transforms a1,p(x, ∇u1,f ) to a2,p(x, ∇u2,f ): a2,p(x, ∇u2,f ) = hφf a1,p(x, ∇u1,f ), and the main problem is to show that φf is independent on f . here we denote by ui,f solution of (11) with a replaced by ai, i = 1, 2. one notices that {φf , f ∈ c2,α(γ)} is an infinite dimensional family rather than an one dimensional family in the case of conjecture 1. also, contrary to (3), any further linearization on (9) would not provide any new information about φf . so, technically, the task in this case is much harder to accomplish. for a f ∈ c3,α(γ), let gi,f be the riemannian metric (on ω̄) generated by the metrix a−1i,p (x, ∇ui,f ), i = 1, 2. one verifies that φf is a family of conformal diffeomorphisms sending (ω̄, g1,f ) to (ω̄, g2,f ). if one uses φ∗f g to denote the pullback of a tensor g under φf , then (15) can be rewritten as φ∗f g2,f = |dφf |g1,f . given f , h ∈ c3,α(γ), let’s denote by ġi,f,h the frechet derivative of gi,f at the reference point f in the direction h, i = 1, 2. once again, one can show that φf 70 ziqi sun 7, 3(2005) is smooth in f (parallel to those in conjecture 1) and we denote by x = φ̇f,h the corresponding derivative of φf in the direction h (viewed as a vector field). for a fixed f , we may once again assume that φf = identity and set g1,f = g2,f =: gf and u1,f = u2,f =: uf . in order to prove the conjecture by showing x = φ̇f,h = 0, ∀h ∈ c3,α(γ), (11) let us take a deep look at the relation φ∗f g2,f =| dφf | g1,f by differentiating it in f with a direction h ∈ c3,α(γ). we get ġ1,f,h − ġ2,f,h = lx gf − (eσ∇gf · (e −σx))gf . (12) where lx gf stands for lie derivative of gf under the vector field x and σ = log √ det(g). equation (12) implies that x is connected to the inhomogeneous conformal killing field equation (with respect to the metric gf ) with the boundary condition x |γ= 0. however, this equation has no real consequence if one just considers one direction. the main observation made in [hsu] is that if one considers a pair of directions, then one can use the theory of conformal killing field to obtain useful consequences leading to (11). indeed, when one is given a pair of directions h1, h2 ∈ c2,α(γ), one can show that the following symmetric relation ġf,h1 lf,h2 = ġf,h2 lf,h1 holds for ġf,h1 = ġ1,f,h1 or ġ2,f,h1 and lf,h = ∇gf u̇f,h = g −1 f ∇u̇f,h. this is proven in [hsu] using the special structure of the linearized coefficient matrix. combining this symmetric relation together with (12) one gets lf,h2c(lx1 gf − (e σ∇gf · (e −σx1))gf ) = lf,h1c(lx2 gf − (e σ∇gf · (e −σx2))gf ), (13) where xi = φ̇f,hi , i = 1, 2. equation (13) implies that both xi, i = 1, 2, satisfy the inhomogeneous conformal killing field equation of the type lc(lx (g) − (eσ∇ · (e−σx))g) = f (14) with the same inhomogeneous term f , which is a 1-form. the equation (14) is the crucial equation for the proof. we have proven that if x and l satisfy the equation (14) with x |γ= 0, then both inner products 〈 l, x 〉 g and 〈 l⊥, x 〉 g are uniquely determined by f, where l⊥ stands for the unique vector perpendicular to l with ∥∥l⊥∥∥ = ∥∥l∥∥ in the counterclockwise direction under the metric g [su2], base on this result, one concludes from (13) that the vector fields xi and lf,hi must satisfy the following system of equations: { 〈x1,lf,h2 〉gf = 〈x2, lf,h1 〉gf〈 x1,l⊥f,h2 〉 gf = 〈 x2, l ⊥ f,h1 〉 gf , (15) 7, 3(2005) conjectures in inverse boundary value problems for ... 71 to understand (15) better, consider now a two-parameter family of conformal diffeomorphisms φf +η1h1+η2h2 ⊂ g with parameters η1 and η2 in r. for a fixed point x ∈ ω, define ω(η1, η2) = φf +η1h1+η2h2 (x) : r 2 → ω̄ as a function from (η1, η2) to the image of x under φf +η1h1+η2h2 . one checks that ωη1 = φ̇f +η1h1+η2h2,h1 (x), ωη2 = φ̇f +η1h1+η2h2,h2 (x). by replacing f by f + η1h1 + η2h2 one can shows from (15) that the function ω satisfies the following first order system: { 〈ωη1 , l2〉g = 〈ωη2 , l1〉g〈 ωη1 , l ⊥ 2 〉 g = 〈 ωη2 , l ⊥ 1 〉 g , (16) where lj = lf +η1h1+η2h2,hj ◦ φf +η1h1+η2h2 , j = 1, 2. here the additional term φf +η1h1+η2h2 is needed once one removes the assumption φf = identity. system (16) can be viewed as a generalized cauchy-riemann system under the vector fields l1 and l2. the proof of (11) with h = h1 and h2 is now reduced to showing that system (16) admits no bounded nonconstant solution ω. note that ω is always bounded. in order to do that, one way is to apply liouville’s type theorems to the system (16). however, one must choose the directions h1 and h2 in a way that the gradients of the solution l1 and l2 are uniformly independent. once (11) is proven with two independent directions, one can show that (11) holds for all directions. this is proven in [hsu] using the geometric argument developed in [su2]. in [hsu] the above framework has been successfully to two important special cases: the case in which a(x, p) is independent of x and the case in which ap(x, p) is independent of p. in both cases one is allowed to construct the needed independent directions h1 and h2. see [hsu] for details. to verify the conjecture completely, the main difficulty is the construction of special directions. the construction of special directions in the known cases has been completed by using techniques of exponentially growing solutions, which is not available in the general case. one possible way to overcome this difficulty is to replace the two-parameter family of conformal diffeomorphisms φf +η1h1+η2h2 by φf (η1,η2), where f (η1, η2) is a two dimensional nonlinear variety in c3,α(γ) passing through f . the nonlinearity of f (η1, η2) should correspond to the quasilinear nature of a(x, p). once one identifies the correct form of f (η1, η2), the rest of the argument can be modified to cover the general case. received: april 2004. revised: may 2004. 72 ziqi sun 7, 3(2005) references [ab] l. ahlfors and l. bers, riemann’s mapping theorem for variable metrics, ann. of math. 72 (1960), 385-404. [bb] d. c. barber and b. h. brown applied potential tomography j. phys. e. 17 (1984), 723-733. [c] a.p.calderon, on an inverse boundary value problem, seminar on numerical analysis and its applications to continuum physics, soc. brasileira de matematica, rio de janeiro, (1980), 65-73. [cheis] m. cheney and d. isaacson, an overview of inversion algorithm for impedance imaging, contemporary math. 122 (1991), 29-39. [hsu] . hervas and z. sun, an inverse boundary value problem for quasilinear elliptic equations, comm. in pde 27 (2002), 2449-2490. [i1] v. isakov, on uniqueness in inverse problems for semilinear parabolic equations, arch. rat. mech.anal. 124 (1993), 1-12. [i2] v. isakov, uniqueness of recovery of some systems of semilinear partial differential equations, inverse problems 17 (2001) 607-618. [in] v. isakov and a. nachman, global uniqueness for a two-dimensional semilinear elliptic inverse problem, trans, of ams 347 (1995), 3375-3390 [is] v. isakov and j. sylvester, global uniqueness for a semilinear elliptic inverse problem, comm. pure appl. math. 47 (1994), 1403-1410. [kv] r. kohn and m. vogelius, identification of an unknown conductivity by means of measurements ii, inverse problems, d. w. mclaughlin, ed., siamams proc. 14 (1984), 113-123. [lu] j. lee and g. uhlmann, determining anisotropic real-analytic conductivity by boundary measurements, comm. pure appl. math, 42 (1989), 1097-1112. [n] a. nachman, global uniqueness for a two-dimensional inverse boundary value problem, ann. of math, 143 (1996), 71-96. [s] j. sylvester, an anisotropic inverse boundary value problem, comm. pure appl. math. 43 (1990), 201-232. [su1] j. sylvester and g. uhlmann, a global uniqueness theorem for an inverse boundary value problem, ann. of math. 125 (1987), 153-169. [su2] j. sylvester and g. uhlmann, inverse problems in anisotropic media, contemporary math. 122 (1991), 105-117. [su1] z. sun, on a quasilinear inverse boundary value problem, math. z. 221 (1996), 293-305. 7, 3(2005) conjectures in inverse boundary value problems for ... 73 [su2] z. sun, an inverse problem for inhomogeneous conformal killing field equations, proc. amer. math. soc. 131 (2003), 1583-1590. [su3] z. sun, inverse boundary value problems for a class of semilinear elliptic equations, to appear in advances in applied math. [suu1] z. sun and g. uhlmann, inverse problems in quasilinear anisotropic media, amer. j. of math. 119 (1997), 771-797. [suu2] z. sun and g. uhlmann, anisotropic inverse problems in two dimensions, inverse problems 19 (2003), 1-10. [u] g. uhlmann, developments in inverse problems since calderon’s foundational paper, harmonic analysis and pde, university of chicago press, 1999. cubo 8, 71. 74 ( 1992) saxln jgrnmla de m11tm4t1g. ds !1 m n1, sur. trayectorias de un sistema de control lineal en ir.n sobre variedades lineales de codimensión 1 y n-1. * víctor delgado a. introducción. si un in descompo:iición de j orduu d e uno mulr iz a cada pwde bloques t.lcmm, rcspect.ivnmcnt.e, \"a lores pro pios dlstint.os, e nto nces e l sis t.e ma lineal co1u.rolob lc :r= a:r.+ 811 , x e n nn. u e n ir"' puede trnns formerse e n uu si!lwn n controla ble con n un ir e l cunl, o s u vez, es equivnlcntc al si::it.crna cnn6 11ico ((31) : ihll· ··· (1 ) dnclo una \-ariedad lineal m, se planten el p rob lem a d e dct.cnniua.r un control 11 de lul n1odo que la respectiva solució n x (t , u), con condición inicial :z;(o, u) = z 0, purmnnczcn e u m duranle, a lo men os, un int.er vnlo de t.icmpo ¡o, t) con t posit ivo. 2 r esultados . p ropot1lcl6 n 2.1 i m e~ 1m uecl.or no rrnal a unn unn.cd, do nde < ., . > ea el 11rotluct a inl,crno 1..,u.al en dt " y n es una m a t ri= n ilpotcnt.c . domos troción. sin perder gcncrulidod puede suponerse q ue "'• = t ~ lo úllimu compone nte no nulu de m . si se d enota x(t , u) = (r i.·· · ,z,.) , 2.l{0, 11) = (:r.o1 ,· ·· .:ro..)= %0 cnt.onces lo rcloci6 11 < :r(t , u) :ro, m >=o pn.r111 t cu !o,'i'\, t 1>0.sitivo, implica que: (:r, ro1)rn 1 + ·· · + (:z:1 :roi )( 1} =o {2) ~~~~~~~~~~ 7 1 72 cubos al deri\v (2) y ree mplazar en (1) se obtiene: x1 m 1 + · ··+:i:•(-t) -= o :i:2m1 + ···+itt1(-l) = o :i:n-k+2 m 1 + · · ·+ :i:" mt-1+ z.. (-1) pero z,. =< a, z > +u cloncle a= (a.1, ··.a.-). v. dcrlpjo a. en consecuencia u= < m, (znk+'li · · ,z,., , 0, ··, o) >,por lo tanlo 1 o 1 ll o o observación. si cu la proposición anteri or k = n, es deci r la última componente de m e:; -l , ento nces u= < m , ax >.esta expresión aparece implícita en [ij para el caso en que el sistema conside rado es z = az + bu y sin restricciones sob re la matriz a. ejomplo e n ir3 : sea m = (m 1, 1, 0) normal a una variedad lineal m de codimensión 1 , entonces u=< (m,,l , o),n(a(x,y,z)') > = < (m¡, 1,0),(:,< a,x' >, 0 > = m1:< o,z• >,donde a.= (a.1,a.;¡,a.:¡,). z • = {z,y,.::) reemplazando u en el rcsp eclivo sist ema canónico se o btiene :i:= 11 , li= .: .i= m1.:: cuya solució n es z = zo ~ + (yo ;:; )t + ~e11111 y=~ ~ + ~e"'i' ;: = .:oern,r. ent.onces < (x ,y,.:) (z o.t/o ,zo),m > =o si y solo s i .=o= m 1yo es d eci r lo condición inicial debe pertenecer ni plano= = m 1y (\.'et figura 1) figura l. 'ih.ytetorla.s d~ ... cubos 73 p r o p o1lcl60 2 .l . si m et un vector que determino lo cltft:iccidn de uno va.n'edod lineal m de codimenaión n 1 11 :i;(t, u) u uno •oluc:ión n.o trivial del aiatcma ( i ) que penn.cmc::ce en m durante un tiempo .fi.nilaj, en.ton.cea el control 11 u de tipo rcti.lim.entaclo y lu direccionu 1/ con.jicionea lnicitlle• f octiblea aon, rupcctivamente, del tipo m = (1,m1.~1 ..... ,m;-1) , :c0 = (:o1,zo"j,m2:r02 •..•.• m;-2:1:o2) donde m2 o (3) si la ké:t imo compo nente d e m es no nulo entonces la componcntic an t.crior c8 no nulo. en efeclo, s i m.t ,po y m 11 _1 =o entonces al reemplazar (3) en (1) se obllene qu e 8(1) = o y, con e ll o, lo solución lrivio.l :z::( t, 11) = :r0 = (zo¡, 0 , ... ,0) co n u = -a.1zo1· si m:i: # o puede suponerse, sin perder gencro.lidad, que rn 1 = 1, e ntonce5 de 2: 1 = :zio 1 + o(t) y z2 = to:i: + o(t)m2 se obtiene z 1=ij= :c:i: = :co2 + om:i: ¡ o(l) = ~(e'"'11 1) rcc mplnznndo (4) cu (1) resu lto: %, z02em~1 :i::o:i: + zo'j(c•i' t ) %, zo2m2 e11111 :i::os + ~%ol(ea:i 1 1) zn-1 z 02 rrii-2e"l1'" :r.0n + ~zqj(e"'i' l ) .. :i::02m;'1e""'' +u en co n::1t.-cuencir u= m.l 1x 2< a,x >. ad e más: el e d o nd e m!'-l= !!!a .. -, "'l m = (l,m2 1 ~,· •• ,m;1) :to= (:r.01,zo2.m1:r:02,· ·,m;-2.:tm) (4) fino.lmentc, s:i m 2 = 01 es d ecir m = ( 1, 0, ··, 0) 1 resulta que 9(t,) = :r.o:i:t y \u co nd ición inicial es de in formu xo (2lo1, :r.02 10, ··, o). obsorvn clóo. la facti bilidad d e 111 y d e :ro de la pro p«\ición 2.2 se ob tuvo un 121 co mo co i\:llcc.ucnci& de un tcorc nrn qu e re lociono la permanencia de solu c iones de un 8ls te mn un cal r = a:.z: +b u so bre vnricdod e:5 lineales m, c.or1 lknica.s de !lubcspn~los vcctoria.lo (;\ , 8) -in~wiant.cs dctollad m en {4j. 74 cubos ejemplo en r 3 : las dl recclo ues factibles m • (t, m 2 , ~) rest rin1e n lu condj. ciones iniciajes a ja forma z 0 = (a, b, 17126"), es decir deben estar en el plano .= r "'111 (ver figura 2) figura 2. referencias jlj delgado v., conlro l n. bili 1l n. d direccional tmiparnmitrica. e11 aiate ui.m fi. nea/u, anales 111 cla io, t o m o 1 23 1239 ( 1988). l2 j henrlquez h., delgodo v.1 san mart ín m.e ., propiedades de in vari a nza de lrayect.orias para sist emas de co nt ro l lineales, (enviado a revise.e proyeccio nes u. c . del nor t.e). jjj lee e.b., mark us l. , fom~datio~ of optimai contro l t heo '1/, j o hn wil ey, (1967). [4j wonham w .m. , linea r m.ullíva n"able control, springer verl ag, (1985). dire cció n d e l a uto r : instit u to de matemáticas uni vcr!idad aust ral casilla 567va ldivia revista de matemáticas_0087 revista de matemáticas_0088 revista de matemáticas_0089 revista de matemáticas_0090 cubo a mathematical journal vol.19, no¯ 02, (01–09). june 2017 on some recurrent properties of three dimensional k-contact manifolds venkatesha and r.t. naveen kumar department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india. vensmath@gmail.com, rtnaveenkumar@gmail.com abstract in this paper we characterize some recurrent properties of three dimensional k-contact manifolds. here we study ricci η-recurrent, semi-generalized recurrent and locally generalized concircularly φ-recurrent conditions on three dimensional k-contact manifolds. resumen en este paper caracterizamos algunas propiedades recurrentes de variedades k-contacto tridimensionales. estudiamos las condiciones de ricci η-recurrencia, recurrencia semigeneralizada y φ-recurrencia concircular localmente generalizada en variedades k-contacto tridimensionales. keywords and phrases: k-contact manifold, ricci η-recurrent, semi-generalized recurrent, locally generalized concircularly φ-recurrent, scalar curvature, einstein manifold. 2010 ams mathematics subject classification: 53c25, 53d15. 2 venkatesha & r.t. naveen kumar cubo 19, 2 (2017) 1 introduction in 1950, walker [17] introduced the notion of recurrent manifolds. in the last five decades, recurrent structures have played an important role in the geometry and the topology of manifolds. in [3], the authors de and guha introduced the idea of generalized recurrent manifold with the nonzero 1-form a and another non-zero associated 1-form b. if the associated 1-form b becomes zero, then the manifold reduces to a recurrent manifold given by ruse [11]. as a generalization of recurrency, khan [6] introduced the notion of generalized recurrent sasakian manifold. semigeneralized recurrent manifolds were first introduced and studied by prasad [10]. the notion of recurrency in a riemannian manifold has been weakened by many authors in several ways to different extent viz., [1, 8, 12] etc., a k-contact manifold is a differentiable manifold with a contact metric structure such that ξ is a killing vector field [2, 13]. these are studied by several authors like [4, 9, 14, 15] and many others. it is well known that every sasakian manifold is k-contact, but the converse ia not true, in general. however a three-dimensional k-contact manifold is sasakian [5]. motivated by the above studies, in this study we consider some recurrent properties of three dimensional k-contact manifolds. the paper is organized in the following way: in section 2, we give the definitions and some results concerning the k-contact manifolds that will be needed hereafter. in section 3, we discuss the ricci η-recurrent property of three dimensional k-contact manifold. in particular, we obtain the 1-form a is η parallel and give the expression for ricci tensor. the section 4 is devoted to three dimensional semi-generalized recurrent k-contact manifolds. here we prove some interesting results, such as the facts that a specific linear combination of the 1-forms a and b is always zero and that the manifold is einstein. in section 5, we consider three dimensional locally generalized concircularly φ-recurrent k-contact manifolds. in this case the manifold is a space of constant curvature. 2 preliminaries a riemannian manifold m is said to admit an almost contact metric structure (φ, ξ, η, g) if it carries a tensor field φ of type (1, 1), a vector field ξ, 1-form η and compatible riemannian metric g on m, such that φ2x = −x + η(x)ξ, φξ = 0, η(φx) = 0, (2.1) η(ξ) = 1, g(x, ξ) = η(x), (2.2) g(φx, φy) = g(x, y) − η(x)η(y), (2.3) g(φx, y) = −g(x, φy), g(φx, x) = 0. (2.4) cubo 19, 2 (2017) on some recurrent properties of three dimensional . . . 3 if moreover ξ is killing vector field, then m is called a k-contact manifold [2, 13]. a k-contact manifold is called sasakian [2], if the relation (∇xφ)(y) = g(x, y)ξ − η(y)x, (2.5) holds on m, where ∇ denotes the operator of covariant differentiation with respect of metric g. in a k-contact manifold, the following relations hold: ∇xξ = −φx, (2.6) (∇xη)(y) = g(∇xξ, y). (2.7) also in a three dimensional k-contact manifold, the curvature tensor is given by r(x, y)z = r − 4 2 [g(y, z)x − g(x, z)y] − r − 6 2 [g(y, z)η(x)ξ (2.8) − g(x, z)η(y)ξ + η(y)η(z)x − η(x)η(z)y], s(x, y) = 1 2 [(r − 2)g(x, y) − (r − 6)η(x)η(y)], (2.9) qx = 1 2 [(r − 2)x − (r − 6)η(x)ξ], (2.10) s(φx, φy) = s(x, y) − 2η(x)η(y), (2.11) where r, s and q are the scalar curvature, ricci tensor and ricci operator respectively. definition 1. a k-contact manifold is said to be einstein if the ricci tensor s is of the form s(x, y) = ag(x, y), where a is constant. 3 on three dimensional ricci η-recurrent k-contact manifold definition 2. the ricci tensor of an three dimensional k-contact manifold is said to be η-recurrent if its ricci tensor satisfies the following: (∇xs)(φ(y), φ(z)) = a(x)s(φ(y), φ(z)), (3.1) for all vector fields x, y, z ∈ tm, where a(x) = g(x, ρ), ρ is called the associated vector field of 1-form a. in particular, if the 1-form a vanishes then the ricci tensor is said to be η-parallel and this notion for sasakian manifold was first introduced by kon [18]. 4 venkatesha & r.t. naveen kumar cubo 19, 2 (2017) now consider three dimensional ricci η-recurrent k-contact manifold. from (3.1), it follows that ∇zs(φ(x), φ(y)) − s(∇zφx, φy) − s(φx, ∇zφy) = a(z)s(φ(x), φ(y)). (3.2) by using (2.5), (2.6) and (2.11) in (3.2), yields (∇zs)(x, y) = −η(x)[2g(φz, y) + s(z, φy)] − η(y)[2g(φz, x) + s(φx, z)] (3.3) + a(z)[s(x, y) − 2η(x)η(y)]. hence we can state the following: theorem 3.1. in a three dimensional k-contact manifold, the ricci tensor is η-recurrent if and only if (3.3) holds. by virtue of (3.3), let {ei} is an local orthonormal basis of the tangent space at each point of the manifold and taking summation over i, 1 ≤ i ≤ 3, we have dr(z) = [r − 2]a(z). (3.4) if the manifold has a constant scalar curvature r (r 6= 2 because the 1-form a is definite), then from (3.4) it follows that a(z) = 0, ∀ z. this leads to the following: theorem 3.2. in a three dimensional ricci η-recurrent k-contact manifold m if the scalar curvature is constant then the 1-form a is η-parallel. again putting x = z = ei in (3.3), and taking summation over i, 1 ≤ i ≤ 3, we get 1 2 dr(y) + µη(y) = s(y, ρ) − 2η(ρ)η(y), (3.5) where µ = σ3 i=1s(φei, ei). by using (3.4) in (3.5), we obtain 1 2 a(y)[r − 2] + µη(y) = s(y, ρ) − 2η(ρ)η(y), (3.6) putting y = ξ in (3.6), yields µ = ( 1 − r 2 ) η(ρ). (3.7) considering (3.7) in (3.6), we get s(y, ρ) = ( r 2 − 1 ) g(y, ρ) + ( 3 − r 2 ) η(ρ)η(y). (3.8) thus we have the following result: cubo 19, 2 (2017) on some recurrent properties of three dimensional . . . 5 theorem 3.3. if the ricci tensor in a three dimensional k-contact manifold is η-recurrent, then its ricci tensor along the associated vector field of the 1-form is given by (3.8). substituting y = φy in (3.8) and by virtue of (2.1), we obtain s(y, l) = kg(y, l), (3.9) where l = φρ, k = r 2 − 1. hence we can state the following: theorem 3.4. if the ricci tensor in a three dimensional k-contact manifold is η-recurrent, then k = r 2 − 1 is an eigen value of the ricci tensor corresponding to the eigen vector φρ. 4 on three dimensional semi-generalized recurrent k-contact manifolds definition 3. a riemannian manifold is said to be semi-generalized recurrent manifold if its curvature tensor r satisfies the relation (∇xr)(y, z)w = a(x)r(y, z)w + b(x)g(z, w)y, (4.1) where a and b are two 1-forms, b is non-zero, ρ1 and ρ2 are two vector fields such that g(x, ρ1) = a(x), g(x, ρ2) = b(x), (4.2) for any vector field x and ∇ be the covariant differentiation operator with respect to the metric g. definition 4. a riemannian manifold m is said to be three dimensional semi-generalized ricci recurrent manifold if: (∇xs)(y, z) = a(x)s(y, z) + 3b(x)g(y, z). (4.3) taking cyclic sum of (4.1) with respect to x, y, z, and using second bianchi’s identity, we get 0 = a(x)r(y, z)w + a(y)r(z, x)w + a(z)r(x, y)w (4.4) + b(x)g(z, w)y + b(y)g(x, w)z + b(z)g(y, w)x. on contracting above equation with respect to y, yields 0 = a(x)s(z, w) − g(r(z, x)ρ1, w) − a(z)s(x, w) (4.5) + 3b(x)g(z, w) + g(x, w)g(ρ2, z) + b(z)g(x, w). again putting z = w = ei in (4.5), and taking summation over i, 1 ≤ i ≤ 3, we obtain ra(x) + 11b(x) − 2s(x, ρ1) = 0. (4.6) 6 venkatesha & r.t. naveen kumar cubo 19, 2 (2017) putting x = ξ in (4.6) and by virtue of (4.2) and (2.11), we get r = 1 η(ρ1) [4η(ρ1) − 11η(ρ2)]. (4.7) since for a contact metric manifold η(ρ1) 6= 0. hence we can state the following: theorem 4.1. in a three dimensional semi-generalized recurrent k-contact manifold, the scalar curvature r takes the form (4.7). again taking z = ξ in (4.3), we get (∇xs)(y, ξ) = a(x)s(y, ξ) + 3b(x)g(y, ξ). (4.8) left hand side of the above equation can be written as (∇xs)(y, ξ) = ∇xs(y, ξ) − s(∇xy, ξ) − s(y, ∇xξ). (4.9) in view of (2.2), (2.9) and (4.9) in (4.8), gives −2g(φx, y) + s(φx, y) = 2a(x)η(y) + 3b(x)η(y). (4.10) plugging y = ξ in (4.10), we obtain 2a(x) + 3b(x) = 0. this leads to the following: theorem 4.2. in a three dimensional semi-generalized ricci recurrent k-contact manifold, the linear combination 2a + 3b is always zero. replace y by φy in (4.10), we get s(x, y) = 2g(x, y). thus we have the following result: theorem 4.3. a three dimensional semi-generalized ricci recurrent k-contact manifold is einstein manifold. 5 on three dimensional locally generalized concircularly φrecurrent k-contact manifolds definition 5. a three dimensional k-contact manifold is called the locally generalized concircularly φ-recurrent if its concircular curvature tensor c̃ c̃(x, y)z = r(x, y)z − r 6 [g(y, z)x − g(x, z)y], (5.1) cubo 19, 2 (2017) on some recurrent properties of three dimensional . . . 7 satisfies the condition φ2((∇wc̃)(x, y)z) = a(w)c̃(x, y)z + b(w)[g(y, z)x − g(x, z)y], (5.2) for all x, y, z and w orthogonal to ξ. taking covariant differentiation of (2.8) with respect to w, we get (∇wr̃)(x, y)z = dr(w) 2 [g(y, z)x − g(x, z)y] − dr(w) 2 [g(y, z)η(x)ξ (5.3) −g(x, z)η(y)ξ − η(y)η(z)x − η(x)η(z)y] − r − 6 2 [g(y, z)(∇wη)(x)ξ −g(x, z)(∇wη)(y)ξ + (∇wη)(y)η(z)x + η(y)(∇wη)(z)x −(∇wη)(x)η(z)y − η(x)(∇wη)(z)y]. again taking x, y, z and w orthogonal to ξ, we obtain (∇wr̃)(x, y)z = dr(w) 2 [g(y, z)x − g(x, z)y] − r − 6 2 [g(y, z)g(φx, w)ξ (5.4) − g(x, z)g(φy, w)ξ]. from above equation it follows that φ2((∇wr̃)(x, y)z) = dr(w) 2 [g(x, z)y − g(y, z)x]. (5.5) taking covariant differentiation of (5.1) with respect to w, we get (∇w ˜̃ c)(x, y)z = (∇wr̃)(x, y)z − dr(w) 6 [g(y, z)x − g(x, z)y], (5.6) from which it follows that φ2((∇w ˜̃ c)(x, y)z) = φ2((∇wr̃)(x, y)z) (5.7) − dr(w) 6 [g(y, z)φ2x − g(x, z)φ2y]. by virtue of (2.1), (5.2), (5.5) in (5.7), yields r(x, y)z = [ r 6 − ( b(w) a(w) + dr(w) 3a(w) )] [g(y, z)x − g(x, z)y]. (5.8) since in a locally generalized concircularly φ-recurrent k-contact manifold a(w) 6= 0. on contracting above equation over w, we get r(x, y)z = µ[g(y, z)x − g(x, z)y], (5.9) where µ = r 6 − ( b(ei) a(ei) + dr(ei) 3a(ei) ) is a scalar. then by schur’s theorem [7] µ will be constant on the manifold. thus we have the following result: theorem 5.1. a three dimensional locally generalized concircularly φ-recurrent k-contact manifold is a space of constant curvature. 8 venkatesha & r.t. naveen kumar cubo 19, 2 (2017) references [1] archana singh, j.p. singh and rajesh kumar, on a type of semi generalized recurrent psasakian manifolds, facta universitatis, ser. math. inform, 31 (1), (2016), 213-225. [2] d.e. blair, contact manifolds in riemannian geometry. lecture notes in math., no. 509, springer, 1976. [3] u.c. de and n. guha, on generalized recurrent manifold, j. nat. acad. math., 9 (1991), 85-92. [4] u.c. de and avik de, on some curvature properties of k-contact manifolds, extracta mathematicae, 27 (1), (2012), 125-134. [5] j.b. jun and u.k. kim, on 3-dimensional almost contact metric manifolds, kyungpook math. j., 34 (2), (1994), 293-301. [6] q. khan, on generalized recurrent sasakian manifolds, kyungpook math. j., 44 (2004),167172. [7] s. kobayashi and k. nomizu, foundations of differential geometry, interscience publishers, vol. 1, (1963), p. 202. [8] e. peyghan, h. nasrabadi and a. tayebi, on φ-recurrent contact metric manifolds, math. j. okayama univ., 57 (2015), 149-158. [9] k.t. pradeep kumar, c.s. bagewadi and venkatesha, projective φ-symmetric k-contact manifold admitting quarter-symmetric metric connection, differential geometry-dynamical systems, 13, (2011), 128-137. [10] b. prasad, on semi-generalized recurrent manifold, mathematica balkanica, new series, 14 (2000), 77-82. [11] h.s. ruse, a classification of k4 space, london mathematical society, 53 (1951), 212-229. [12] a.a. shaikh and h. ahmed, on generalized φ-recurrent sasakian manifolds, applied mathematics, 2, (2011), 1317-1322. [13] s. sasaki, lecture note on almost contact manifolds, tohoku university, tohoku, japan, 1965. [14] d. tarafdar and u.c. de, on k-contact manifolds, bull. math. soc. sci. math. roumanie, 37 (85), (3-4), (1993), 207-215. [15] m.m. tripathi and m.k. dwivedi, the structure of some classes of k-contact manifolds, proceedings of the indian academy of sciences: mathematical sciences, 118 (3), (2008), 371-379. cubo 19, 2 (2017) on some recurrent properties of three dimensional . . . 9 [16] s. tanno, locally symmetric k-contact riemannian manifolds, proc. japan acad., 43, 581 (1967). [17] a.g. walker, on ruses spaces of recurrent curvature, proc. london math. soc., 52 (1976), 36-64. [18] k. yano and m. kon, structures on manifolds, vol. 3 of series in pure mathematics, world scientific publishing co., singapore, (1984). introduction preliminaries on three dimensional ricci -recurrent k-contact manifold on three dimensional semi-generalized recurrent k-contact manifolds on three dimensional locally generalized concircularly -recurrent k-contact manifolds cubo a mathematical journal vol.21, no02, (65–76). august 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000200065 generalized trace pseudo-spectrum of matrix pencils aymen ammar, aref jeribi and kamel mahfoudhi department of mathematics faculty of sciences of sfax, university of sfax route de soukra km 3.5, b.p. 1171, 3000, sfax, tunisia ammar aymen84@yahoo.fr, aref.jeribi@fss.rnu.tn, kamelmahfoudhi72@yahoo.com abstract the objective of the study was to investigate a new notion of generalized trace pseudospectrum for an matrix pencils. in particular, we prove many new interesting properties of the generalized trace pseudo-spectrum. in addition, we show an analogue of the spectral mapping theorem for the generalized trace pseudo-spectrum in the matrix algebra. resumen el objetivo de este estudio es investigar una nueva noción de pseudo-espectro traza generalizado para pinceles de matrices. en particular, demostramos variadas propiedades nuevas e interesantes del pseudo-espectro traza generalizado. adicionalmente, mostramos un análogo del teorema espectral de aplicaciones para el pseudo-espectro traza generalizado en el álgebra de matrices. keywords and phrases: pseudo-spectrum, condition pseudo-spectrum, trace pseudo-spectrum. 2010 ams mathematics subject classification: 15a09, 15a86, 65f40, 15a60, 65f15. http://dx.doi.org/10.4067/s0719-06462019000200065 66 aymen ammar, aref jeribi and kamel mahfoudhi cubo 21, 2 (2019) 1 introduction let mn(c) (mn(r)) denote the algebra of all n × n complex (real) matrices, i denotes the n × n identity matrix and the conjugate transpose of u is denoted by u∗. we denote by tr, (resp. det) the trace (resp. determinant) map on mn(c). in the present paper, we study the problem of finding the eigenvalues of the generalized eigenvalue problem ux = λvx. next, let λ ∈ c and sn(λv − u) ≤ . . . ≤ s2(λv − u) ≤ s1(λv − u) be the singular values of the matrix pencils λv −u where s1(λv −u) is the smallest and sn(λv −u) is largest singular values of the matrix pencil. let u, v ∈ mn(c), then the set of all eigenvalues of the matrix pencils of the form λv − u is denoted by σ(u, v) and is defined as σ(u, v) = { λ ∈ c : λv − u is not invertible } , and its spectral radius by r(u, v) = sup { |λ| : λ ∈ σ(u, v) } . for an n×n complex matrices u and v and a non-negative real number ε, the pseudo-spectrum of the matrix pencils of the form λv − u is defined as the following closed set in the complex plane σε(u, v) = { λ ∈ c : sn(λv − u) ≤ ε } . let u, v ∈ mn(c) and 0 < ε < 1. the condition pseudo-spectrum of the matrix pencils λv − u is denoted by σε(u, v) and is defined as σε(u, v) = { λ ∈ c : sn(λv − u) ≤ ε s1(λv − u) } . let ε be a small positive number. for an operator u, v ∈ mn(c), recall that the determinant spectrum of matrix pencils of the form λv − u is the set dε(u, v) and is defined as dε(u, v) = { λ ∈ c : |det(λv − u)| ≤ ε } . the analysis of eigenvalues and eigenvectors has had a great effect on mathematics, science, engineering, and many other fields. then, there are countless applications for this type of analysis. the study of matrix pencils is by now a very thoughtful subject, with the notion of pseudospectrum playing a key role in the theory. however, matrix pencils play an important role in numerical linear algebra, perturbation theory, generalized eigenvalue problems. in this paper, we interest by a generalization of eigenvalues called generalized trace pseudo-spectrum for an element in the matrix cubo 21, 2 (2019) generalized trace pseudo-spectrum of matrix pencils 67 algebra to give more information about the matrix pencils of the form λv−u. for more information on various details on the above concepts, properties and applications of pseudo-spectrum [2, 3, 6, 7, 9], condition spectrum [1, 4, 5] and determinant spectrum [8]. now, we introduce the new concept of the generalized trace pseudo-spectrum in the following definition. definition 1.1. for ε > 0, the generalized trace pseudo-spectrum of the matrix pencils of the form λv − u ∈ mn(c) is denoted by trε(u, v) and is defined as trε(u, v) = σ(u, v) ⋃ { λ ∈ c : |tr(λv − u)| ≤ ε } . the generalized trace pseudoresolvent of the matrix pencils of the form λv − u is denoted by trρε(u, v) and is defined as trρε(u, v) = ρ(u, v) ⋂ { λ ∈ c : |tr(λv − u)| > ε } . the singular values of a the matrix pencil are important not only for their role in diagonalization but also for their utility in a variety of applications. since trε(u, v) use all the singular values of λv −u to get defined, it is expected to give more information about u, v than pseudo-spectrum and condition spectrum. since the definition use idea of ”trace” the generalization of eigenvalues defined above is named as generalized trace pseudo-spectrum. it is easily seen that the map u → tr(u) is continuous linear functional. here, some important properties of the trace of u, b ∈ mn(c) are tr(ub) = tr(bu), tr(αu) = αtr(u) with α ∈ c, tr(u + b) = tr(u) + tr(b). an outline of this paper is the following. in section 2, we focuses on a new description of the generalized trace pseudo-spectra. not only do we give a characterization of the generalized trace pseudo-spectrum in the matrix algebra. but also we investigate the connection between generalized trace pseudo-spectrum and algebraic multiplicity of the eigenvalues. in section 3, we give an analogue of the spectral mapping theorem for the generalized trace pseudo-spectrum in the matrix algebra. 2 generalized trace pseudo-spectrum. in this section, some relevant properties of the generalized trace pseudo-spectrum are discussed in detail. for u, v ∈ mn(c) and ε > 0, the generalized trace pseudo-spectrum of the matrix pencils of the form λv − u is denoted by trε(u, v) and is defined as trε(u, v) = σ(u, v) ⋃{ λ ∈ c : |tr(λv − u)| ≤ ε } . 68 aymen ammar, aref jeribi and kamel mahfoudhi cubo 21, 2 (2019) the generalized trace pseudo-resolvent of the matrix pencils of the form λv − u is denoted by trρε(u, v) and is defined as trρε(u, v) = ρ(u, v) ⋂{ λ ∈ c : |tr(λv − u)| > ε } while the generalized trace pseudo-spectral radius of the matrix pencils of the form λv − u is defined as trrε(u, v) := sup { |λ| : λ ∈ trε(u, v) } . remark 2.1. let u, v ∈ mn(c). then, if v is nonsingular, then it is possible to reduce the generalized trace pseudo-spectrum to a standard trace pseudo-spectrum for the matrices v−1u or uv−1. i.e. trε(u, v) = σ(v −1u, i) ⋃ { λ ∈ c : |tr(λ − v−1u)| ≤ ε } , or trε(u, v) = σ(uv −1, i) ⋃ { λ ∈ c : |tr(λ − uv−1)| ≤ ε } . the following theorem gives some properties of the generalized trace pseudo-spectrum that follow in a straightforward manner from the definition of the generalized trace pseudo-spectrum. theorem 2.1. let u, v ∈ mn(c) and ε > 0. then, (i) tr0(u, v) = ⋂ ε>0 trε(u, v). (ii) if 0 < ε1 < ε2, then trε1(u, v) ⊂ trε2(u, v). (iii) trε(u, v) is a non-empty compact subset of c. (iv) if α ∈ c and β ∈ c\{0}, then trε(βu + αv, v) = βtr ε |β| (u, v) + α. (v) trε(αv, v) = { λ ∈ c : |λ − α| ≤ ε |tr(v)| } for all λ, α ∈ c. proof. the proofs of items (i) and (ii) are clear from the definition of generalized trace pseudospectrum. (iii) using the continuity from c to [0, ∞[ of the map λ → |tr(λv − u)|, we get that trε(u, v) is a compact set in the complex plane containing the eigenvalues of the matrix pencils λv − u. cubo 21, 2 (2019) generalized trace pseudo-spectrum of matrix pencils 69 (iv) in fact, it is well know trε(βu + αv, v) = { λ ∈ c : |tr(λv − βu − αv)| ≤ ε } = { λ ∈ c : |β| ∣ ∣ ∣ ∣ tr ( λ − α β v − u ) ∣ ∣ ∣ ∣ ≤ ε } = { λ ∈ c : ∣ ∣ ∣ ∣ tr ( λ − α β v − u ) ∣ ∣ ∣ ∣ ≤ ε |β| } . then, λ ∈ trε(βu + αv, v). thus, λ − α β ∈ tr ε |β| (u, v). hence, λ ∈ βtr ε |β| (u, v) + α. (v) let λ ∈ trε(αv, v), then |tr(λv − αv)| = |λ − α||tr(v)| ≤ ε. this means that trε(αv, v) = { λ ∈ c : |λ − α| ≤ ε |tr(v)| } for all λ, α ∈ c. q.e.d. theorem 2.2. let u, v ∈ mn(c) and ε > 0. then, (i) if u = zbz−1 and zv = vz for all nonsingular matrix z ∈ mn(c) we have, trε(u, v) = trε(b, v). (ii) if u = zbz−1 and v = zkz−1 for all nonsingular matrix z ∈ mn(c) we have, trε(u, v) = trε(b, k). (iii) the map t → trε(u, v) is an upper semi continuous function from mn(c) to compact subsets of c. ♦ proof. (i) let λ ∈ trε(b, v), then |tr(λv − b)| = |tr(λv − z−1uz)|, = |tr(λz−1zv − z−1uz)| = |tr(z−1(λzv − uz)| = |tr(z−1(λv − u)z| = |tr(λv − u)| ≤ ε. it follows that, λ ∈ trε(u, v). the proofs of items (ii) and (iii) follows immediately from definition 1.1. q.e.d. the following example shows that the converse of the assertion (i) is not true. 70 aymen ammar, aref jeribi and kamel mahfoudhi cubo 21, 2 (2019) example 2.1. let u = ( 1 2 0 1 ) , b = ( 1 0 0 1 ) and v = ( 0 0 0 1 ) . then, u and b are not similar and for ε > 0, we have trε(u, v) = trε(b, v) = { λ ∈ c : |λ − 2| ≤ ε } . in the following, we obtain additional results on trε(u, v) that are useful in our analysis. theorem 2.3. let u, v ∈ mn(c), λ ∈ c, and ε > 0. then, there is d ∈ mn(c) such that |tr(d)| ≤ ε and tr(λv − u − d) = 0 if, and only if, λ ∈ trε(u, v). ♦ proof. to see this, we suppose that there exists d ∈ mn(c) such that |tr(d)| ≤ ε and tr(λv − u − d) = 0. then, |tr(λv − u)| = |tr(d)| ≤ ε. thus, λ ∈ trε(u, v). conversely, let λ ∈ trε(u, v). then, we will discuss these two cases: 1st case : if λ ∈ tr0(u, v), then it is sufficient to take (d = 0n×n). 2nd case : λ ∈ trε(u, v)\tr0(u, v). then, |tr(λv − u)| ≤ ε. now, we consider d = tr(λv − u) n i. it is easy to verify that, d ∈ mn(c) and |tr(d)| = ∣ ∣ ∣ ∣ tr ( tr(λv − u) n i ) ∣ ∣ ∣ ∣ = |tr(λv − u)| n tr(i) ≤ ε. also, we have tr(λv − u − d) = tr ( λv − u − tr(λv − u) n i ) = 0. q.e.d. theorem 2.4. let u, v ∈ mn(c) and ε > 0. then, trδ(u, v) + θε ⊆ trε+δ(u, v), (1) holds for δ, ε > 0 with θε, denoting the closed disk in the complex plane centered at the origin with radius ε |tr(v)| . if we take δ = 0, we obtain an inner bound for trε(u, v), namely tr0(u, v) + θε ⊆ trε(u, v). (2) cubo 21, 2 (2019) generalized trace pseudo-spectrum of matrix pencils 71 proof. let λ ∈ trδ(u, v) + θε. then, there exists there exists λ1 ∈ trδ(u, v) and λ2 ∈ θε such that λ = λ1 + λ2. therefore, |tr(λ1v − u)| ≤ δ and |tr(λv − u)| = |tr((λ1 + λ2)v − u)| = |tr(λ2v) + tr(λ1v − u)| ≤ |λ2||tr(v)| + |tr(λ1v − u)| ≤ |tr(v)||λ2| + |tr(λ1v − u)| ≤ ε + δ, so that (1) holds. finally, let δ = 0, then the desired inclusion (2) is obtained. q.e.d. theorem 2.5. let u, v ∈ mn(c) such that ub = bu and ε > 0. if u is normal, then trε(u + b, v) ⊆ σ(u, v) + trε(b, v). proof. we assume that u is normal, so there exists a unitary matrix z ∈ mn(c) such that z∗uz = λ1in1 ⊕ λ2in2 ⊕ . . . ⊕ λkink. the condition ub = bu implies that z∗bz = u1 ⊕ u2 . . . ⊕ uk where, ui ∈ mnk(c), i = 1, . . . , k. then, trε(u + b, v) = trε(z ∗uz + z∗bz, v) = trε((λ1in1 + u1) ⊕ . . . ⊕ (λkink + uk), v) = k ⋃ i=1 trε(λiini + ui, v) = k ⋃ i=1 λi + trε(ui, v) ⊆ σ(u, v) + trε(b, v). the proof is thus complete. q.e.d. remark 2.2. let u, b and v ∈ mn(c) and ε > 0. then, using theorem 2.5, we obtain the following inequality, trrε(u + b, v) ⊆ r(u, v) + trrε(b, v). ♦ 72 aymen ammar, aref jeribi and kamel mahfoudhi cubo 21, 2 (2019) theorem 2.6. let u, b and v ∈ mn(c) and ε > 0. then, (i) trε(ub, v) = trε(bu, v). (ii) tr ε 2 (u, v) + tr ε 2 (b, v) ⊆ trε(u + b, v). proof. (i) let λ ∈ trε(ub, v), then ε ≥ |tr(λv − ub)| = |tr(λv) + tr(−ub)| = |tr(λv) + tr(−bu)| = |tr(λv − bu)|. hence, λ ∈ trε(bu, v). thus, trε(ub, v) ⊆ trε(bu, v). the conclusion can be obtained similarly to the first inclusion, then we deduce that trε(bu, v) = trε(ub, v). (ii) let λ ∈ tr ε 2 (u, v) + tr ε 2 (b, v). then, there exists λ1 ∈ tr ε 2 (u, v) and λ1 ∈ tr ε 2 (b, v) such that λ = λ1 + λ2. therefore, tr(λ1v − u) ≤ ε 2 and tr(λ2v − b) ≤ ε 2 . on the other hand, |tr(λv − u − b)| = |tr(λ1v − u + λ2v − b)| ≤ |tr(λ1v − u)| + |tr(λ2v − b)| ≤ ε then, λ ∈ trε(u + b, v). q.e.d. theorem 2.7. let u, v ∈ mn(c) and n ∈ mn(c) is a nilpotent matrix and ε > 0. then, trε(u + n , v) = trε(u, v). ♦ proof. " ⊆ " let λ ∈ trε(u + n , v), then |tr(λv − u − n)| ≤ ε. since |tr(λv − u) − tr(n)| ≤ ε. cubo 21, 2 (2019) generalized trace pseudo-spectrum of matrix pencils 73 using the fact that the matrix trace vanishes on nilpotent matrices, therefore λ ∈ trε(u, v). hence, trε(u + n , v) ⊆ trε(u, v). " ⊇ " let λ ∈ trε(u, v), then |tr(λv − u)| ≤ ε. now, we can write for any λ ∈ c |tr(λv − u)| = |tr(λv − u − n + n)| = |tr(λv − u − n) + tr(n)|. because, tr(n) = 0, it follows that |tr(λv − u − n)| ≤ ε. consequently, trε(u, v) ⊆ trε(u + n , v). q.e.d. 3 trace pseudospectral mapping theorem let u, v ∈ mn(c) and f be an analytic function defined on d, an open set containing tr0(u, v). for each ε > 0, we define ϕ(ε) = sup λ∈trε(u,v) |tr ( f(λ)v − f(u) ) | and suppose there exists ε0 > 0 such that trε0(f(u), v) ⊆ f(d). then, for 0 < ε < ε0 we define φ(ε) = sup µ ∈ f−1(trε(u, v)) ∩ d |tr(µv − u)|. lemma 3.1. let u, v ∈ mn(c) and ε > 0, then ϕ(ε) and φ(ε) are well defined, lim ε→0 ϕ(ε) = 0 and lim ε→0 φ(ε) = 0. proof. in the order to prove that ϕ(ε) is well defined, we define h : c → r+ h(λ) = |tr ( f(λ)v − f(u) ) | since h(λ) is continuous and trε(u, v) is a compact subset of c, then it is clear that ϕ(ε) = sup { h(λ) : λ ∈ trε(u, v) } . we conclude, ϕ(ε) is well defined. now, let assume that there exists ε0 > 0 such that trε0(f(u), v) ⊆ f(d). 74 aymen ammar, aref jeribi and kamel mahfoudhi cubo 21, 2 (2019) we show that for 0 < ε < ε0, φ(ε) is well defined. define g : c → r+, g(µ) = |tr(µv − u)|. since g is continuous for all µ ∈ c, then φ(ε) is well defined. it is also clear that ϕ(ε) and φ(ε) are a monotonically non-decreasing function, ϕ(ε) and φ(ε) goes to zero as ε goes to zero. q.e.d. theorem 3.1. let u, v ∈ mn(c) and let f be an analytic function defined on d, an open set containing tr0(u, v). then, for each f(trε(u, v)) ⊆ trϕ(ε)(f(u), v), where ϕ(ε) defined above. proof. let λ ∈ trε(u, v). then, using lemma 3.1 we obtain that ϕ(ε) is well defined and lim ε→0 ϕ(ε) = 0. therefore, h(λ) ≤ ϕ(ε). hence |tr ( f(λ)v − f(u) ) | := h(λ) ≤ ϕ(ε). thus, f(λ) ∈ trϕ(ε)(f(u), v). this means that f(trε(u, v)) ⊆ trϕ(ε)(f(u), v). q.e.d. theorem 3.2. let u, v ∈ mn(c) and let f be an analytic function defined on d, an open set containing tr0(u, v). then, for each trε(f(u), v) ⊆ f(trφ(ε)(u, v)). where φ(ε) defined above. proof. let λ ∈ trε(f(u), v). then, using lemma 3.1 we obtain the existence of ε0 > 0 such that trε(f(u), v) ⊆ trε0(f(u), v) ⊆ f(d). consider µ ∈ d such that λ = f(µ). then µ ∈ f−1(trε(u, v)), hence g(µ) ≤ φ(ε). therefore, |tr ( µv − u ) | := g(µ) ≤ φ(ε) thus, µ ∈ trφ(ε)(u, v). then, λ = f(µ) ∈ f(trφ(ε)(u, v)). this means that trε(f(u), v) ⊆ f(trφ(ε)(u, v)). q.e.d. cubo 21, 2 (2019) generalized trace pseudo-spectrum of matrix pencils 75 corollary 3.1. combining the two inclusions in theorems 3.1 and 3.2, we get f(trε(u, v)) ⊆ trϕ(ε)(f(u), v) ⊆ f(trφ(ϕ(ε))(u, v) and trε(f(u), v) ⊆ f(trφ(ε)(u, v)) ⊆ trϕ(φ(ε))(f(u), v). here are some remarks. remark 3.1. (i) it will be clear from the proofs of theorems 3.1 and 3.2 that the the functions ϕ and φ measure the sizes of the trace pseudo-spectra are optimal. (ii) from the definitions of ϕ and φ, the set inclusions are sharp in the sense that the functions cannot be replaced by smaller functions. (iii) in general, the spectral mapping theorem is not true for generalized trace pseudo-spectrum. example 3.1. let α, β ∈ c with α 6= β 6= 0 and let u = ( α 1 0 β ) , v = ( 2 0 0 0 ) and f(λ) = λ2. then f(u) = ( α2 α + β 0 β2 ) . a direct computation shows that trε(f(u), v) = { λ ∈ c : |2λ − α2| ≤ ε − β2 } , f(trε(u, v)) = { λ2 ∈ c : |2λ − α2| ≤ ε − β2 } . we can see for all ε > 0 that trε(f(u), v) 6= f(trε(u, v)). 76 aymen ammar, aref jeribi and kamel mahfoudhi cubo 21, 2 (2019) references [1] a. ammar, a. jeribi and k. mahfoudhi, a characterization of the essential approximation pseudospectrum on a banach space, filomath 31, (11), 3599-3610 (2017). [2] a. ammar, a. jeribi and k. mahfoudhi, a characterization of the condition pseudospectrum on banach space, funct. anal. approx. comput. 10 (2) (2018), 13–21. [3] ammar, a., jeribi, a., mahfoudhi, k., the essential condition pseudospectrum and related results, j. pseudo-differ. oper. appl., (2018) 1-14. [4] ammar, a., jeribi, a., mahfoudhi, k., the essential approximate pseudospectrum and related results, filomat, 32, 6, (2018), 2139-2151. [5] ammar, a., jeribi, a.,mahfoudhi, k., a characterization of browder’s essential approximation and his essential defect pseudospectrum on a banach space, extracta math. ,34(1) (2019), 29-40. [6] a. jeribi. spectral theory and applications of linear operators and block operator matrices, springer-verlag, new-york, (2015). [7] r.a. horn, c.r. johnson, topics in matrix analysis, cambridge university press, (1991). [8] krishna kumar. g, determinant spectrum: a generalization of eigenvalues, funct. anal. approx. comput. 10 (2) (2018), 1–12. [9] l. n. trefethen and m. embree, spectra and pseudospectra: the behavior of nonnormal matrices and operators. prin. univ. press, princeton and oxford, (2005). introduction generalized trace pseudo-spectrum. trace pseudospectral mapping theorem articulo 8.dvi cubo a mathematical journal vol.12, no¯ 02, (123–126). june 2010 a note on generalized topological spaces and preorder saeid jafari college of vestsjaelland south, herrestraede 11, 4200 slagelse, denmark, email: jafari@stofanet.dk raja mohammad latif department of mathematics and statistics, king fahd university of petroleum and minerals, dhahran 31261, saudi arabia. email: raja@kfupm.edu.sa and seithuti p. moshokoa department of mathematical sciences, university of south africa, p.o. box 392, pretoria 0003, south africa. email: moshosp@unisa.ac.za abstract this paper deals with the notion of continuity in ordered generalized topological spaces. we also characterize general topological spaces satisfying the hausdorff property via the graph of the preorder assigned to the general topological space. resumen este art́ıculo trata de continuidad en espacios topológicos ordenados. también caracterizamos espacios topológicos generalizados satisfaciendo la propiedad de hausdorff v́ıa el grafo designado de un espacio topológico general preordenado. 124 s. jafari. r. m. latif and s. p. moshokoa cubo 12, 2 (2010) key words and phrases: topological spaces, hausdorff, preorder. ams subject classification (2000): 54d10 1 introduction by a generalized topology ( briefly gt) on x [1], we mean a subset µ of the power set exp x, of x such that ∅ ∈ µ and arbitrary union of elements of µ belongs to µ. the elements of µ are said to be µ-open, their compliments are said to be µ-closed. clearly, each topology is a gt, but not conversely. as a generalization of the concept of a hausdorff topology, we shall say that a gt, written (x, µ) is hausdorff if for distinct points x and y in x there exists disjoint µ-open sets u and v containing x and y, respectively. by (x, µ, ≤) we mean a gt together with a preorder. let a be a subset of x, let i(a) = ∪x∈ax ↑, where x↑ = {y ∈ x : x ≤ y}. let d(a) = ∪x∈ax ↓, where x↓ = {y ∈ x : y ≤ x}. clearly, we have a ⊆ i(a) and a ⊆ d(a) for all a ⊆ x. we say that a subset a of x is increasing (decreasing) if i(a) = a(a = d(a)). if the preorder is discrete, then (x, µ, ≤) is just a gt, as in the literature, see for example [1] [2] . we shall simply write ogt, for a ordered generalized topological space (x, µ, ≤). now let g = {(x, y) : x ≤ y}. then g is a subset of x × x, we shall refer to g as the graph of the preorder. we shall say that a subset g is closed if x × x − g is open in the product generalized topology, denoted by µ × µ. that is, a subset b of x × x is (µ × µ)-open if for every (x, y) ∈ b there exist µ-open sets u and v, such that (x, y) ∈ u × v ⊂ b, where u and v contain x and y, respectively. we shall say that the preorder on x is (µ × µ)-closed if its graph g is (µ × µ)-closed in x × x. 2 results let a be a subset of x. we denote the µ-closure of a by cµa, and recall: lemma 2.1. [2] if µ is a gt on x, a ⊂ x, x ∈ x, then x ∈ cµa if and only if x ∈ u ∈ µ implies u ∩ a 6= ∅. proof. define b = {x ∈ x : x ∈ u ∈ µ ⇒ u ∩ a 6= ∅}. clearly a ⊂ b. further b is µ-closed, that is x − b ∈ µ because x ∈ x − b if and only if there is ux ∈ µ such that x ∈ ux and ux ∩ a 6= ∅ and then y ∈ ux implies that y ∈ x − b so that x − b = ∪x∈x−bux ∈ µ. finally, if a ⊂ q and q is µ-closed, then x − q ∈ µ and (x − q) ∩ a = ∅ so that x − q ⊂ x − b, b ⊂ q. therefore b is the smallest µ-closed set containing a, i.e. b = cµa. clearly, for a subset a of (x, µ) we have a ⊂ cµa. also, cµa is µ-closed. to see this just note that the µ-closure of a is the intersection of all µ-closed supersets of a in x. therefore, the complement of the µ-closure of a is µ-open, being an arbitrary union of members of µ. example 2.2. let x = {1, 2, 3} and define µ = {∅, {1, 2}, {2, 3}, {1, 2, 3}}. then (x, µ, ≤) is an ogt , where the preorder is induced from n but it is not an ordered topological space. also, for the subset a = {3} we have cµa = a. we shall need the following: cubo 12, 2 (2010) a note on generalized topological spaces and preorder 125 theorem 2.3. let µ be a gt on x, if a ⊆ b in x, then cµa ⊆ cµb. proof. let (x, µ) be a gt, and a and b be subsets of x such that a ⊆ b. if x ∈ cµa, then by lemma 2.1, there exists ux ∈ µ containing x such that ux ∩ a 6= ∅. since a ⊆ b, we have ux ∩ b 6= ∅. therefore x ∈ cµb. this shows that cµa ⊆ cµb. we will say that (x, µ, ≤) is µ-continuous (µ-anticontinuous) is and only if for each µ-open subset g (µ-closed subset f )i(g) and d(g) are µ-open (i(f ) and d(f ) are µ-closed). also, we say that (x, µ, ≤) is bicontinuous if it is both continuous and anti-continuous. example 2.4. let the ogt (x, µ, ≤) be that of example 2.1. then for every µ-open set a and µ-closed set b in x, the sets: i(a) and d(a) are µ-open and the sets i(b) and d(b) are µ-closed. it follows that the ogt (x, µ, ≤) is bicontinuous. theorem 2.5. the space (x, µ, ≤) is anti-continuous if and only if (i) cµ(i(a)) ⊆ i(cµ(a)) and cµd(a) ⊆ d(cµa) for every a ⊆ x. (ii) (x, µ, ≤) is continuous if and only if i(cµa) ⊆ cµ(i(a)) and d(cµa) ⊆ cµd(a) for every a ⊆ x. (iii) (x, µ, ≤) is bicontinuous if and only if i(cµa) = cµ(i(a)) and d(cµa) = cµd(a) for every a ⊆ x. proof. (i) suppose that (x, µ, ≤) is anticontinuous and a be a subset of x. note that a ⊆ cµa, and therefore i(a) ⊆ i(cµa). we apply theorem 2.3, to obtain cµ(i(a)) ⊆ cµ(i(cµa)). note that i(cµa) is µ-closed as (x, µ, ≤) is anticontinuous, we then have cµ(i(cµa)) = i(cµa). hence cµ(i(a)) ⊆ i(cµ(a)). dually, we can show cµd(a) ⊆ d(cµa). conversely, suppose that cµ(i(a)) ⊆ i(cµ(a)) holds for all a ⊆ x. let f be a µ-closed subset of x. first, we note that i(f ) ⊆ cµi(f ) this holds even when f is not necessarily µ-closed. on the other hand, let f be µ-closed and cµ(i(f )) ⊆ i(cµ(f )) holds. we have cµi(f ) ⊆ i(f ). hence cµi(f ) = i(f ). this shows that i(f ) is µ-closed. dually, d(f ) is µ-closed. (ii) let (x, µ, ≤) be continuous and a be a subset of x. then cµi(a) is µ-closed, so is that x − cµi(a) is µ-open. hence, x − d(x − cµi(a)) is µ-closed. note that x − d(x − cµi(a)) is increasing and contains a. it follows that cµa ⊆ x − d(x − cµi(a)). since a ⊆ d(a) for all a ⊆ x, we have x − clµi(a) ⊆ d(x − cµi(a)). this implies that x − d(x − cµi(a)) ⊆ cµi(a). therefore, cµi(a) = x − d(x − cµi(a)). hence i(cµa) ⊆ cµi(a). dually, we have d(cµa) ⊆ cµd(a) conversely, suppose that i(cµa) ⊆ cµi(a) and d(cµa) ⊆ cµd(a) holds for all a ⊆ x. let g be a µ-open subset of x. then x − g is µ-closed and cµ(x − i(g)) ⊆ x − g, hence g ⊂ x − cµ(x − i(g)). it follows that d(cµ(x − i(g))) ⊆µ d(x − i(g)) = cµ(x − i(g)) so that cµ(x − i(g)) is decreasing. thus i(g) ⊂ x − cµ(x − i(g)). on the other hand, it is clear that x − i(g) ⊆ cµ(x − i(g)) and hence x − cµ(x − i(g)) ⊆ i(g). 126 s. jafari. r. m. latif and s. p. moshokoa cubo 12, 2 (2010) consequently, i(g) = x − cµ(x − i(g)) is µ-open. dually, d(g) is µ-closed. (iii) suppose that (x, µ, ≤) is bicontinuous. then (x, µ, ≤) is both anticontinuous and continuous. so by (i) we have cµ(i(a)) ⊆ i(cµ(a)) and cµd(a) ⊆ d(cµa) for every a ⊆ x. also by (ii), we have i(cµa) ⊆ cµ(i(a)) and d(cµa) ⊆ cµd(a). together, we get i(cµa) = cµ(i(a)) and d(cµa) = cµd(a) for every a ⊆ x. the converse is clear from (i) and (ii). theorem 2.6. let (x, µ, ≤) be an ogt. then the preorder ≤ is closed if and only if, for every two points a, b ∈ x such that a ≤ b is false, it is possible to determine a increasing µ-open set u that contains a and a decreasing set µ-open set v containing b, such that u ∩ v = ∅. proof. let a, b ∈ such that a ≤ b is false, that is, (a, b) /∈ g. since g is (µ × µ)-closed, we have that x × x − g is (µ × µ)-open and (a, b) ∈ x × x − g. now find µ-open sets ú and v́ containing a and b, respectively such that ú × v́ ∩ g = ∅. now if x ∈ ú and y ∈ v́ , then x ≤ y is false. we set u = i(ú ) and v = d(v́ ). now, ú ⊂ u and v́ ⊂ v. then u is µ-open increasing and contains a, and v is µ-open decreasing and contains b. next, we show that u ∩ v = ∅. suppose that u ∩ v 6= ∅, let z ∈ u ∩ v. since z ∈ u , there exists a point x ∈ ú such that x ≤ z. also, z ∈ v gives a point y ∈ v́ , such that z ≤ y. hence, x ≤ y follows from the fact that the ”≤” is a preorder on x. on the other hand x ∈ ú and y ∈ v́ implies that x ≤ y is false. so we arrive at a contradiction. we conclude that u ∩ v = ∅. conversely, assume that g is not closed, then find (a, b) in the µ-closure of g, such that (a, b) /∈ g. then consider the sets u and v containing a and b, increasing and decreasing, respectively. then u ∩ v contains (a, b) and satisfy u × v ∩ g 6= ∅. observe that u × v is (µ × µ)-open. so, find u ∈ u and v ∈ v such that (u, v) ∈ g and u ≤ v. from u ∈ u and u ≤ v results that v ∈ u , since u is increasing; thus v belongs to both u and v , that is u and v are not disjoint. theorem 2.7. let (x, µ, ≤) be an ogt. if the preorder ≤ is closed, then for every point a ∈ x, the sets d(a) and i(a) are µ-closed. proof. suppose that the preorder is closed that is its graph g is closed. let a ∈ x. if b ∈ x −i(a), then a ≤ b is false. by theorem 2.6, there exists an increasing µ-open set u containing a and a decreasing µ-open set v containing b such that u and v are disjoint. now i(a) ⊂ u implies that w ∩ i(a) 6= ∅ this shows that i(a) is closed. acknowledgement. the second author is highly and greatly indebted to the king fahd university of petroleum and minerals, for providing necessary research facilities during the preparation of this paper. the third author has been supported by the south african national research foundation. received: january 2009. revised: march 2009. references [1] a. csaszar, generalized topology, generalized continuity, acta math. hungar., 96 (2oo2), 351357. [2] a. csaszar, δand θmodifications of generalized topologies, acta. math. hungar., to appear. cubo a mathematical journal vol.10, n o ¯ 02, (47–59). july 2008 green function for a two-dimensional discrete laplace-beltrami operator volodymyr sushch koszalin university of technology, sniadeckich 2, 75-453 koszalin, poland, email: volodymyr.sushch@tu.koszalin.pl abstract we study a discrete model of the laplacian in r 2 that preserves the geometric structure of the original continual object. this means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the riemannian structure. we consider this structure on the appropriate combinatorial analog of differential forms. self-adjointness and boundness for a discrete laplacian are proved. we define the green function for this operator and also derive an explicit formula of the one. resumen nosotros estudiamos un modelo discreto del laplaciano en r 2 que preserva la estructura geométrica del objeto continuo original. esto significa hablando de un modelo discreto, que nosotros no tenemos la intención de remplazar directamente el operador diferencial por uno en diferencias como también un análogo discreto de la estructura riemanniana. nosostros consideramos esta estructura sobre un apropriado análogo combinatório de formas diferenciables. provamos para un laplaceano discreto que es auto adjunto y 48 volodymyr sushch cubo 10, 2 (2008) acotado. definimos la función de green de tal operador e conseguimos una formula explicita para este. key words and phrases: discrete laplacian, difference equations, green function. math. subj. class.: 39a12, 39a70 1 introduction we begin with a brief review of some well known definitions that are related to the contents of this paper. denote by λ r (r 2 ) the set of all differentiable complex-valued r-forms on r2, where r = 0, 1, 2. let ∗ : λr(r2) → λ2−r(r2) be the hodge star operator. an inner product for r-forms with compact support is defined by (ϕ, ψ) = ∫ r2 ϕ ∧ ∗ψ, (1.1) where the bar over ψ denotes complex conjugation. let l2λr(r2) denote the completion of λr(r2) with respect to the norm generated by the inner product (1.1). let the exterior derivative d : λ r (r 2 ) → λr+1(r2) be defined as usual. we define the operator d : l 2 λ r (r 2 ) → l 2 λ r+1 (r 2 ) (1.2) as the closure in the l2-norm the corresponding operation specified on smooth forms. the adjoint of d, denoted by δ, is given by (dϕ, ω) = (ϕ, δω), ϕ ∈ l2λr(r2), ω ∈ l2λr+1(r2). note that δ : l2λr+1(r2) → l2λr(r2). the following relations hold among ∗, d and δ. see for instance [15]. ∗ 2 = (−1) r(2−r)id, δ = (−1)r ∗−1 d ∗ . (1.3) the laplacian is defined to be −∆ = dδ + δd : l2λr(r2) → l2λr(r2). (1.4) in this paper we develop some combinatorial structures that are analogs of objects in differential geometry. we are interested in finding of a natural discrete analog of the laplacian on cochains. speaking of a discrete model, we mean not only the direct replacement of differential operators by difference ones but also discrete analogs of all essential ingredients of the riemannian structure over a properly introduced combinatorial object. our approach bases on the formalism proposed by dezin [5]. we adapt the combinatorial constructions from [5, 11, 12] and define discrete analogs of operators (1.2)–(1.4) in a similar cubo 10, 2 (2008) green function for a two-dimensional discrete ... 49 way. in [5], dezin study discrete laplace operators in finite-dimensional hilbert spaces, i.e. on cochains given in domains with boundary. in this paper we extend these results on an infinite complex of complex-valued cochains. we prove self-adjointness and boundness for the discrete laplace-beltrami operator in infinite hilbert spaces that are associated with l2λr(r2). spectral properties are discussed. we define the green function for the discrete laplacian and derive the one in an explicit form. there are other geometric approaches to discretisation of the hodge theory of harmonic forms presented in [6, 7, 8]. in all these papers discrete models are given on the simplicial cochains of triangulated closed riemannian manifolds. see also [3, 4, 9] and references given there. classical references on second order difference equations are the books by berezanski [2, ch. 7], atkinson [1] and the most recent monograph by teschl [14]. 2 combinatorial structures let us denote by c(2) the two-dimensional complex. this complex is defined by c(2) = c 0 ⊕ c 1 ⊕ c 2, where c r is a real linear space of r-dimensional chains. we follow the notation of [5, 12]. let {xk,s}, {e 1 k,s , e2 k,s }, {ωk,s}, k,s ∈ z, be the sets of basis elements of c 0, c1, c2 respectively. it is convenient to introduce shift operators τk = k + 1, σk = k − 1 in the set of indices. the boundary operator ∂ is defined by the rule ∂xk,s = 0, ∂ωk,s = e 1 k,s + e 2 τ k,s − e 1 k,τ s − e 2 k,s , ∂e 1 k,s = xτ k,s − xk,s, ∂e 2 k,s = xk,τ s − xk,s. (2.1) the definition of ∂ is linearly extended to arbitrary chains. we call the complex c(2) a combinatorial model of r 2 . on the other hand, we can consider c(2) as the tensor product c(2) = c ⊗ c of the onedimensional complex c (combinatorial model of a real line). then basis elements of c(2) can be written as follows xk ⊗ xs = xk,s, ek ⊗ xs = e 1 k,s , ek ⊗ es = ωk,s, xk ⊗ es = e 2 k,s , where xk, ek are the basis elements of c. let us introduce an object dual to c(2). namely, the complex of complex-valued functions over c(2). the dual complex k(2) we can consider as the set of complex-valued cochains and it 50 volodymyr sushch cubo 10, 2 (2008) has the same structure as c(2), i.e. k(2) = k0 ⊕ k1 ⊕ k2. in other words, k(2) is a linear complex space with basis elements {xk,s, e k,s 1 , e k,s 2 , ω k,s }. the pairing (chain-cochain) operation is defined by the rule: < xk,s, x p,q >=< ωk,s, ω p,q >=< e 1 k,s , e p,q 1 >=< e 2 k,s , e p,q 2 >= δ p,q k,s , (2.2) where δ p,q k,s is kronecker symbol. we call elements of the complex k(2) forms, emphasizing their closeness to the corresponding continual objects, differential forms. then the 0-, 1-, 2-forms ϕ, ω = (u,v), η can be written as ϕ = ∑ k,s ϕk,sx k,s, η = ∑ k,s ηk,sω k,s, ω = ∑ k,s (uk,se k,s 1 + vk,se k,s 2 ), (2.3) where ϕk,s, uk,s, vk,s, ηk,s ∈ c for any k,s ∈ z. operation (2.2) is extended to arbitrary forms (2.3) by linearity. the boundary operator (2.1) in c(2) induces the dual operation dc in k(2): < ∂a, α >=< a, dcα >, (2.4) where a ∈ c(2), α ∈ k(2). we assume that the coboundary operator dc : kr → kr+1 is a discrete analog of the exterior differentiation operator d (1.2). if ϕ ∈ k0 and ω = (u,v) ∈ k1, then we have the following difference representations for dc: < e 1 k,s , d c ϕ >= ϕτ k,s − ϕk,s ≡ ∆kϕk,s, < e 2 k,s , d c ϕ >= ϕk,τ s − ϕk,s ≡ ∆sϕk,s < ωk,s, d cω >= vτ k,s − vk,s − uk,τ s + uk,s ≡ ∆kvk,s − ∆suk,s. (2.5) note that if η ∈ k2, then dcη = 0. let us now introduce in k(2) a multiplication which is an analog of the exterior multiplication ∧ for differential forms. we denote this operation by ∪ and define it according to the rule: xk,s ∪ xk,s = xk,s, e k,s 2 ∪ e k,τ s 1 = −ω k,s, xk,s ∪ e k,s 1 = e k,s 1 ∪ x τ k,s = e k,s 1 , xk,s ∪ e k,s 2 = e k,s 2 ∪ x k,τ s = e k,s 2 , xk,s ∪ ωk,s = ωk,s ∪ xτ k,τ s = e k,s 1 ∪ e τ k,s 2 = ω k,s, (2.6) supposing the product to be zero in all other cases. the ∪-multiplication is extended to discrete forms by linearity. in terms of the theory of homologies, this is the so-called whitney multiplication. for arbitrary forms α,β ∈ k(2) we have the following relation dc(α ∪ β) = dcα ∪ β + (−1)rα ∪ dcβ, (2.7) cubo 10, 2 (2008) green function for a two-dimensional discrete ... 51 where r is the degree of α. the proof of this can be found in dezin [5, p. 147]. relation (2.7) is an analog of the corresponding continual relation for differential forms (see [15]). define a discrete analog of the hodge star operator. let εk,s denote an arbitrary basis element of k(2). we introduce the operation ∗ : kr → k2−r by setting εk,s ∪ ∗εk,s = ωk,s. (2.8) using (2.6) we get ∗xk,s = ωk,s, ∗e k,s 1 = e τ k,s 2 , ∗e k,s 2 = −e k,τ s 1 , ∗ω k,s = xτ k,τ s. the operation ∗ is extended to arbitrary forms by linearity. let α ∈ kr is an arbitrary r-form: α = ∑ k,s αk,sε k,s. (2.9) denote by kr0 the set of all discrete r-form with compact support on c(2). let ω be the following ”domain” ω = ∑ k,s ωk,s, k,s ∈ z, (2.10) where ωk,s is a two-dimensional basis element of c(2). note that if the sum (2.10) is finite and let −n ≤ k,s ≤ n, n ∈ n, then we will write ω = ωn . the relation (α, β) =< ω, α ∪ ∗β >, (2.11) where α,β ∈ kr0 , gives a correct definition of inner product in k(2). using (2.2), (2.6) and (2.8), this definition can be rewritten as follows (α, β) = ∑ k,s αk,sβk,s. (2.12) for ω = ωn we will write (α, β)n =< ωn, α ∪ ∗β >= n∑ k,s=−n αk,sβk,s. let α ∈ kr, β ∈ kr+1. the relation (dcα, β)n =< ∂ωn, α ∪ ∗β > +(α, δ cβ)n (2.13) defines the operator δc : kr+1 → kr, δcβ = (−1)r ∗−1 dc ∗ β, 52 volodymyr sushch cubo 10, 2 (2008) which is the formally adjoint operator to dc (see [5] for more details). it is obvious that the operator δc can be regarded as a discrete analog of the codifferential δ (cf. (1.3)). equation (2.13) is an analog of the green formula for the formally adjoint differential operators d and δ. it is easy to check that for α ∈ kr0, β ∈ k r+1 0 we obtain (d c α, β) = (α, δ c β). (2.14) according to (2.5), we have δcϕ = 0 and < xk,s, δ cω >= −∆kuσk,s − ∆svk,σs, (2.15) < e 1 k,s , δ c η >= ∆sηk,σs, < e 2 k,s , δ c η >= −∆kησk,s, (2.16) where ϕ ∈ k0, ω ∈ k1 and η ∈ k2. therefore a discrete analog of the laplace-beltrami operator (1.4) can be defined as follows −∆ c = δcdc + dcδc : kr → kr. (2.17) obviously, if ϕ ∈ k0, then we have −∆ c ϕ = δ c d c ϕ. (2.18) combining (2.15) with (2.5) we can rewrite (2.18) as < xk,s, −∆ cϕ >= 4ϕk,s − ϕτ k,s − ϕk,τ s − ϕσk,s − ϕk,σs. (2.19) the same difference form of (2.17) can be drawn for the components ηk,s of η ∈ k 2 and for the two components uk,s vk,s of ω ∈ k 1 . 3 discrete laplacian let us now introduce the linear space h r = {α ∈ kr : ∑ k,s |αk,s| 2 < +∞, k,s ∈ z}, (3.1) where r = 0, 1, 2. clearly, hr is a hilbert space with inner product (2.11) (or (2.12)) and with the following norm ‖α‖ = √ (α, α) = ( ∑ k,s |αk,s| 2 ) 1 2 . (3.2) note that if α ∈ hr, then the set of complex-valued sequences (αk,s) is ℓ 2 (z 2 ). from now on we regard dc, δc and −∆c as the following operators d c : h r → h r+1 , δ c : h r → h r−1 , −∆ c : h r → h r , where r = 0, 1, 2. it is convenient to suppose that h−1 = h3 = 0. cubo 10, 2 (2008) green function for a two-dimensional discrete ... 53 theorem 1. the operators −∆ c : h r → h r, r = 0, 1, 2, (3.3) are bounded and self-adjoint. moreover, ‖ − ∆c‖ = 8, where ‖ − ∆c‖ denotes the operator norm of −∆c. proof. we begin by proving self-adjointness of −∆c for the case r = 0. let ϕ,ψ ∈ h0 and ω = (u,v) ∈ h1. then the green formula (2.13) can be rewritten as (dcϕ, ω)n = n∑ k=−n (ϕk,τ nvk,n − ϕk,−n vk,−τ n )+ + n∑ s=−n (ϕτ n,sun,s − ϕ−n,su−τ n,s) + (ϕ, δ cω)n (3.4) the substitution of dcψ for ω in (3.4) gives (dcϕ, dcψ)n = n∑ k=−n ( ϕk,τ n (ψk,τ n − ψk,n ) − ϕk,−n (ψk,−n − ψk,−τ n ) ) + + n∑ s=−n ( ϕτ n,s(ψτ n,s − ψn,s) − ϕ−n,s(ψ−n,s − ψ−τ n,s) ) + (ϕ, δcdcψ)n . (3.5) letting n → +∞ we get (dcϕ, dcψ) = (ϕ, δcdcψ). (3.6) it follows immediately that −∆c : h0 → h0 is self-adjoint. the same proof remains valid for the case r = 2. now we have −∆c = dcδc and the analog of relation (3.5) results from the inner product (δcη, δcζ)n , where η,ζ ∈ h 2 . a trivial verification shows that properties of (δcη, δcζ)n are completely similar to those of (d cϕ, dcψ)n . hence (δ c η, δ c ζ) = (η, d c δ c ζ). (3.7) finally, let r = 1. in this case we have −∆c = dcδc + δcdc and we must study the sum (dcω, dcϑ) + (δcω, δcϑ), where ω = (u,v) ∈ h1, ϑ = (f,g) ∈ h1. taking in (2.13) α = ω and β = dcϑ we obtain the analog of relation (3.5) for 1-forms (dcω, dcϑ)n ≡ n∑ k,s=−n (∆kvk,s − ∆suk,s)(∆kgk,s − ∆sfk,s) = = n∑ k=−n [ uk,−n (∆kgk,−τ n − ∆nfk,−τ n ) − uk,τ n (∆kgk,n − ∆n fk,n ) ] + + n∑ s=−n [ vτ n,s(∆ngn,s − ∆sfn,s) − v−n,s(∆ng−τ n,s − ∆sf−τ n,s) ] + + (ω, δcdcϑ)n. 54 volodymyr sushch cubo 10, 2 (2008) letting n → +∞ we obtain equation (3.6) for the 1-forms ω, ϑ ∈ h1. in the same manner we can see that equation (3.7) holds for ω, ϑ ∈ h1. adding we obtain (dcω, dcϑ) + (δcω, δcϑ) = (ω, −∆cϑ). (3.8) thus it follows that (−∆ cω, ϑ) = (ω, −∆cϑ). for the rest of the proof let α ∈ hr be an arbitrary r-form. substituting (2.19) into (2.12) we get |(−∆ cα, α)| = ∣∣∣ ∑ k,s (4αk,s − ατ k,s − αk,τ s − ασk,s − αk,σs)αk,s ∣∣∣ ≤ ≤ 4 ∑ k,s |αk,s| 2 + ∑ k,s |ατ k,sαk,s| + ∑ k,s |αk,τ sαk,s|+ + ∑ k,s |ασk,sαk,s| + ∑ k,s |αk,σsαk,s| ≤ 8‖α‖ 2. from this we conclude that ‖−∆c‖ ≤ 8. since −∆c is self-adjoint, it follows easily that ‖−∆c‖ = 8 (see for instance [10, ch. 3]). corollary 2. the operators (3.3) are positive, i.e. for any non-trivial r-form α ∈ hr we have (−∆ cα, α) > 0. proof. this follows from (3.6), (3.7) and (3.8). corollary 3. for any r, r = 0, 1, 2, we have σ(−∆c) = [0, 8], where σ(−∆c) denotes the spectrum of −∆c. proof. straightforward. 4 discrete analog of the green function let ̺(−∆c) = c \ σ(−∆c) denotes the resolvent set of −∆c. in this section we try to describe the resolvent operator (−∆c − λ)−1, λ ∈ ̺(−∆c), of the operator −∆c : h0 → h0. let us introduce a discrete form g(x,x̃) on c0 × c0 as follows g(x,x̃) = ∑ k,s gk,s(x̃)x k,s, where gk,s(x̃) = ∑ m,n gk,s,m,nx m,n cubo 10, 2 (2008) green function for a two-dimensional discrete ... 55 and gk,s,m,n ∈ c for any k,s,m,n ∈ z. hence we have g(x,x̃) = ∑ k,s ∑ m,n gk,s,m,nx k,sxm,n. (4.1) this is a so-called discrete double form (for details see [11]). note that in the continual case (for differential forms) this construction is due to de rham [13]. it is obvious that the basis elements xk,s and xm,n in (4.1) commute and g(x,x̃) = g(x̃,x). by analogy with (2.2), the double 0-form g(x,x̃) can be written pointwise as < xk,sxm,n, g(x,x̃) >= gk,s,m,n. (4.2) let ϕ ∈ h0. then we define a 0-form δm,n, m,n ∈ z, by setting (ϕ, δm,n) = ∑ k,s ϕk,sδ m,n k,s = ϕm,n, (4.3) where δ m,n k,s is the kronecker delta. by analogy with the continual case, the 0-form δm,n defined by (4.3) will be called a discrete analog of dirac’s δ-function at the point xm,n. we can write δ m,n as δ m,n = ∑ k,s δ m,n k,s x k,s = x m,n . we need also the following double form δ(x,x̃) = ∑ k,s ∑ m,n δ m,n k,s xk,sxm,n = ∑ m,n δm,nxm,n = ∑ m,n xm,nxm,n. (4.4) definition 4. the double form (4.1) is called the green function for the operator −∆c : h0 → h0 if gk,s,m,n(λ) = (δ k,s , (−∆ c − λ) −1 δ m,n ) (4.5) for any k,s,m,n ∈ z. of course, (−∆ c − λ)−1δm,n = gm,n(x,λ). it follows easily that (−∆ c − λ)xg(x,x̃,λ) = δ(x,x̃), (4.6) where (−∆c − λ)x is the operator −∆ c − λ that acts with respect to x. recall that in our abbreviation x corresponds to k,s. we have (−∆ c − λ)−1ϕ = ∑ k,s ∑ m,n gk,s,m,n(λ)ϕm,nx k,sxm,n, ϕ ∈ h0, λ ∈ ̺(−∆c). 56 volodymyr sushch cubo 10, 2 (2008) indeed, applying (−∆ c − λ)x to the right-hand side gives (−∆ c − λ)x ∑ m,n ϕm,ngm,n(x,λ)x m,n = ∑ m,n ϕm,n(−∆ c − λ)xgm,n(x,λ)x m,n = = ∑ m,n ϕm,nδ m,nxm,n = ∑ m,n ϕm,nx m,nxm,n = ϕ. we now try to write the green function for −∆ c in a somewhat more explicit way. for this we construct a solution of the equation −∆ cϕ = λϕ, λ ∈ c. (4.7) the following construction is adapted from [14], where the green function is studied for jacobi operators. by (2.19), equation (4.7) can be written pointwise (at the point xk,s) as 4ϕk,s − ϕτ k,s − ϕk,τ s − ϕσk,s − ϕk,σs = λϕk,s. (4.8) applying the transformation λ = −4µ + 4 we reduce (4.8) to the equation 1 4 (ϕτ k,s + ϕk,τ s + ϕσk,s + ϕk,σs) = µϕk,s. (4.9) an easy computation shows that, substituting the ansatz ϕk,s = p k+s into (4.9), we obtain ϕ ± k,s (µ) = (µ ± r(µ)) k+s , (4.10) where r(µ) = − √ µ2 − 1 and √ · denotes the standard brunch of the square root. it follows that we can write the solutions ϕ±(λ) of (4.7) in the form ϕ±(λ) = ∑ k,s ϕ ± k,s (µ)xk,s, (4.11) where ϕ ± k,s (µ) are given by (4.10) and µ = 1 − λ 4 . obviously, λ ∈ [0, 8] if and only if µ ∈ [−1, 1]. by corollary 3, λ ∈ ̺(−∆c) leads to µ ∈ c \ [−1, 1]. it is convenient to write ϕ ± k,s (µ) as ϕ ± k,s (µ) = ϕ ± k (µ) · ϕ± s (µ), (4.12) where ϕ ± k (µ) = (µ ± r(µ))k and k,s ∈ z. suppose µ ∈ c \ [−1, 1]. then one has to examine that the sequences (ϕ ± k (µ))k∈z are square summable near ±∞ respectively, i.e these are ℓ 2 (z) near ±∞. let h0 ± denotes the set of 0-forms whose restriction to k0 ± belongs to h0. here k0+ (k 0 − ) denotes the set of 0-cochains (2.9) with k + s > 0 (k + s < 0). proposition 5. let λ ∈ ̺(−∆c). then ϕ±(λ) ∈ h0 ± . proof. this follows immediately from (4.12). cubo 10, 2 (2008) green function for a two-dimensional discrete ... 57 lemma 6. for any k,s ∈ z the component ϕ+ k (µ) · ϕ− s (µ) is a solution of (4.9) proof. it is easy to check that ϕ ± 2 − 2µϕ ± 1 + 1 = 0. it follows that ϕ + k (µ) · ϕ− s (µ) satisfies equation (4.9). indeed, putting this in (4.9), we have ϕ + σk (µ)ϕ− σs (µ) [ (ϕ + 2 − 2µϕ + 1 + 1)ϕ − 1 + (ϕ − 2 − 2µϕ − 1 + 1)ϕ + 1 ] = 0. theorem 7. let λ ∈ ̺(−∆c). then the components (4.5) of the green function are given by gk,s,m,n(λ) = −1 4r(µ)    ( µ + r(µ) ) |τ k−m|+|τ s−n| for k = m, s > n or k > m, s = n,( µ + r(µ) ) |σk−m|+|σs−n| for k = m, s < n or k < m, s = n,( µ + r(µ) ) |k−m|+|s−n| for the all others, (4.13) where µ = 1 − λ 4 . proof. we must prove that < xk,sxm,n, (−∆ c − λ)xg(x,x̃) >= δ m,n k,s , (4.14) where g(x,x̃) is given by (4.1). using (2.19) and (4.1), we can rewrite the left-hand side of (4.14) as (4 − λ)gk,s,m,n − gτ k,s,m,n − gσk,s,m,n − gk,τ s,m,n − gk,σs,m,n = = 4µgk,s,m,n − (gτ k,s,m,n + gσk,s,m,n + gk,τ s,m,n + gk,σs,m,n). (4.15) the proof falls naturally into three parts. fix xm,n. first, let k = m and s = n. substituting (4.13) into (4.15) and using (4.12), we obtain < xk,sxm,n, (−∆ c − λ)xg(x,x̃) > = 1 r(µ) (−µϕ + 0 (µ)ϕ + 0 (µ) + ϕ + 1 (µ)ϕ + 0 (µ)) = = −µ + ϕ + 1 r(µ) = 1. note that ϕ ± 0 (µ) = (µ ± r(µ)) 0 = 1. now we show that (4.13) satisfies equation (4.8) for the cases k = m, s 6= n and k 6= m, s = n. check the case k = m, s > n. in this case the left-hand side of (4.14) is equal to −1 4r(µ) [ 4µϕ + 1 (µ)ϕ + s−n+1(µ) − ϕ + 2 (µ)ϕ + s−n+1(µ) − ϕ + 0 (µ)ϕ + s−n+1(µ)− − ϕ + 1 (µ)ϕ + s−n+2(µ) − ϕ + 1 (µ)ϕ + s−n (µ) ] = ϕ + s−n+1(µ) 2r(µ) (ϕ + 2 − 2µϕ + 1 + 1) = 0. 58 volodymyr sushch cubo 10, 2 (2008) the proof for the case k = m, s < n (or k 6= m, s = n) is similar. finely, let k 6= m, s 6= n. we give the proof only for the case k > m, s < n. for the other cases the proof runs similarly. in this case we have gk,s,m,n(λ) = 1 −4r(µ) ϕ + k−m (µ)ϕ + n−s (µ) = 1 −4r(µ) ϕ + k (µ)ϕ− m (µ)ϕ+ n (µ)ϕ− s (µ). here we use that ϕ + −k = ϕ − k for any k ∈ z. by lemma 6, ϕ + k (µ)ϕ− s (µ) is a solution of (4.9) and so is gk,s,m,n(λ) at the point xk,s. thus, since g(x,x̃,λ) with components given by (4.13) satisfies (4.6) and, by proposition 5, g(x,x̃,λ) ∈ h0 with respect to x, it must be the green function of −∆c : h0 → h0. it should be noted that the consideration of −∆c : h2 → h2 does not differ from that carried out for the 0-forms ϕ ∈ h0. in this case we consider (4.1) as a double form on c2 × c2 with basis elements ω k,s ω p,q . then we define the green function as above and its components are given by (4.13). the situation with −∆c : h1 → h1 is more difficult. in this case, having written equation (4.7) pointwise (at the elements e1 k,s and e2 k,s ), we obtain a pair of equations (4.8) which corresponds to the components uk,s, vk,s of ω ∈ h 1 . roughly speaking, here we must describe the green function for each component of the 1-form ω = (u,v). the reader can verify that the components gk,s,m,n(λ) of the green function have again the form (4.13). received: december 2007. revised: february 2008. references [1] f. atkinson, discrete and continuous boundary problems, academic press, new york, 1964. 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[9] t. mantuano, discretization of compact riemannian manifolds applied to the spectrum of laplacian, ann. global anal. geom., 27 (2005), 33-46. [10] s. mizohata, the theory of partial differential equations, cambridge university press, 1973. [11] v. sushch, a discrete analogue of the cauchy integral, differential equations, 29 (1993), no. 8, 1242–1249. [12] v. sushch, on some discrete model of magnetic laplacian, ukr. math. bull., 2 (2005), no. 4, 591–607. [13] g. de rham, variétés différentiables, hermann, paris, 1955. [14] g. teschl, jacobi operators and completely integrable nonlinear lattices, math. surveys and monographs, vol. 72, amer. math. soc., providence, r.i., 2000. [15] c. von westenholz, differential forms in mathematical physics, north-holland publishing company, amsterdam, 1981. n4 cubo a mathematical journal vol.19, no¯ 03, (69–77). october 2017 totally degenerate extended kleinian groups rubén a. hidalgo 1 departamento de matemática y estad́ıstica universidad de la frontera, temuco, chile ruben.hidalgo@ufrontera.cl abstract the theoretical existence of totally degenerate kleinian groups is originally due to bers and maskit. in fact, maskit proved that for any co-compact non-triangle fuchsian group acting on the hyperbolic plane h2 there is a totally degenerate kleinian group algebraically isomorphic to it. in this paper, by making a subtle modification to maskit’s construction, we show that for any non-euclidean crystallographic group f, such that h2/f is not homeomorphic to a pant, there exists an extended kleinian group g which is algebraically isomorphic to f and whose orientation-preserving half is a totally degenerate kleinian group. moreover, such an isomorphism is provided by conjugation by an orientation-preserving homeomorphism φ : h2 → ω, where ω is the region of discontinuity of g. in particular, this also provides another proof to miyachi’s existence of totally degenerate finitely generated kleinian groups whose limit set contains arcs of euclidean circles. resumen la existencia teórica de grupos kleinianos totalmente degenerados se debe originalmente a bers y maskit. de hecho, maskit demostró que para cualquier grupo fuchsiano co-compacto y no-triangular actuando en el plano hiperbólico h2 existe un grupo kleiniano totalmente degenerado algebraicamente isomorfo a él. en este art́ıculo, haciendo una modificación sutil a la construcción de maskit, mostramos que para cualquier grupo cristalográfico no-euclidiano f tal que h2/f no es homeomorfo a un pantalón, existe un grupo kleiniano extendido g que es algebraicamente isomorfo a f y cuya mitad que preserva orientación es un grupo kleiniano totalmente degenerado. más aún, un tal isomorfismo está dado por la conjugación por un homeomorfismo que preserva orientación φ: h2 → ω, donde ω es la región de discontinuidad de g. en particular, esto 1 partially supported by project fondecyt 1150003 and anillo act 1415 pia-conicyt 70 rubén a. hidalgo cubo 19, 3 (2017) también entrega otra demostración del resultado de miyachi acerca de la existencia de grupos kleinianos totalmente degenerados finitamente generados cuyo conjunto ĺımite contiene arcos de circunferencias euclidianas. keywords and phrases: kleinian groups, nec groups 2010 ams mathematics subject classification: 30f40, 30f50 cubo 19, 3 (2017) totally degenerate extended kleinian groups. 71 1 introduction the classical uniformization theorem asserts that every non-exceptional riemann surface s, i.e., non-isomorphic to either the riemann sphere ĉ, the complex plane c, the puncture plane c − {0} or a torus, is conformally equivalent to a quotient h2/γ, where γ ∼= π1(s,∗) is a discrete group of conformal automorphisms of the hyperbolic plane h2. similarly, every klein surface (i.e., a real surface where the local change of coordinates are either conformal or anti-conformal) which are non-exceptionals (i.e., those whose orientation-preserving doble covers are non-exceptional riemann surfaces) is di-analytically equivalent to a quotient x = h2/f, where f ∼= πorb1 (x) is a discrete group of conformal and anti-conformal automorphisms of h2 (necessarilly containing anticonformal ones). if x is compact and f+ is the index two subgroup of f consisting of its conformal elements, then f is called a non-euclidean crystallographic (nec) group of algebraic genus equal to the genus of the closed riemann surface x+ = h2/f+. if x is homeomorphic to the closure of the complement of three disjoint closed discs on the riemann sphere, then it is called a (compact) pant. a finitely generated non-elementary kleinian group with a non-empty connected and simplyconnected region of discontinuity is called totally degenerate. the theoretical existence of such groups, in the boundary of teichmüller spaces of co-compact non-triangle fuchsian groups, is by now well-know due to bers [1] and maskit [4], but it seems that there is no explicit example of such type of groups in the literature. there is a nice construction in [3, ix.g] due to maskit, as an application of the klein-maskit combination theorems [5, 6, 7] to (theoretically) obtain a totally degenerate kleinian group isomorphic to a given co-compact fuchsian group f+, different from a triangle one (i.e., the uniformized orbifold is not the sphere with exactly there cone points). the idea was to observe the existence of some ρ0 ∈ (0,+∞) (which can be chosen in the complement of a suitable countable subset), an open arc t of the circle centered at the origin and radius ρ0, with ρ0 ∈ t, such that: (i) for each t ∈ t, there is a quasifuchsian group g+(t) and there is an isomorphism θt : f + → g+(t), (ii) g+(ρ0) is a fuchsian group, and (iii) for one of the end points t# of t, there is a totally degenerate kleinian group g+(t#), with region of discontinuity ω, so that there is an orientation-preserving homeomorphism φ : h2 → ω inducing, by conjugation, an isomorphism between f+ and g+(t#). an extended kleinian group is a group g of conformal and anticonformal automorphisms of the riemann sphere, necessarily containing anticonformal elements, whose index two orientationpreserving half g+ is a kleinian group. in the case that g+ is totally degenerate, we say that g is a totally degenerate extended kleinian group. the existence of totally degenerate extended kleinian groups should not be a surprise as, from a general abstract point of view, this follows 72 rubén a. hidalgo cubo 19, 3 (2017) from the non-triviality of the teichmuller space of an orbifold with mirrored boundary. in this note we indicate the subtle modifications in maskit’s construction from [3, ix.g], to observe that for given a nec group there is a totally degenerate kleinian group isomorphic to it. theorem 1.1. let f be a nec group so that h2/f is not homeomorphic to a pant. then there exists a totally degenerate extended kleinian group g, with region of discontinuity ω, and there exists an orientation-preserving homeomorphism φ : h2 → ω inducing, by conjugation, an isomorphic between f and g. if for the nec group f it holds that h2/f has non-empty boundary, then theorem 1.1 implies the following result due to miyachi. corollary 1.2 (miyachi [8]). there are totally degenerate kleinian groups for which there exist arcs of euclidean circles inside the limit set; these arcs connect fixed points of distinct parabolic transformations and/or connect the fixed points of the same hyperbolic transformation. these arcs of circles are dense in the limit set. 2 preliminaries 2.1 möbius and extended möbius transformations the conformal automorphisms of the riemann sphere ĉ are the möbius transformations and its anti-conformal ones are the extended möbius transformations (the composition of the standard reflection j(z) = z with a möbius transformation). we denote by m the group of möbius transformations and by m̂ the group generated by m and j. clearly, m is an index two subgroup of m̂. if k is a subgroup of m̂, then we set k+ := k ∩ m. möbius transformations are classified into parabolic, loxodromic (including hyperbolic) and elliptic transformations. first, we need to observe that a non-trivial möbius transformation has at least one fixed point and at most two of them. the parabolic ones are those having exactly one fixed point, elliptic ones are conjugated to rotations and loxodromic are conjugated to transformations of the form z 7→ reiθz, where r ∈ (0,1) ∪ (1,+∞) (if eiθ = 1, then we call it hyperbolic). similarly, extended möbius transformations are classified into pseudo-parabolic (the square is parabolic), glide-reflection (the square is hyperbolic), pseudo-elliptic (the square is elliptic), reflection (of order two admitting a circle of fixed points on ĉ) and imaginary reflection (of order two and having no fixed points on ĉ) [3]. each möbius transformation γ can be identified with a projective linear transformation γ = [ a b c d ] ∈ psl2(c) and the square of its trace tr(γ)2 = (a + d)2 is well defined. if γ is different from the identity transformation, then the following hold: cubo 19, 3 (2017) totally degenerate extended kleinian groups. 73 (1) γ is parabolic if and only if tr(γ)2 = 4. (2) γ is elliptic if and only if tr(γ)2 ∈ [0,4). (3) γ is loxodromic if and only if tr(γ)2 /∈ [0,4] (hyperbolic ones correspond to tr(γ)2 ∈ (4,+∞)). 2.2 kleinian and extended kleinian groups a kleinian group is a discrete subgroup of m, and an extended kleinian group is a discrete subgroup of m̂ containing extended möbius transformations. if g is either a kleinian or an extended kleinian group, then its region of discontinuity ω(g) is the open subset of ĉ (which might be empty) formed by those points p ∈ ĉ with finite g-stabilizer gp = {γ ∈ g : γ(p) = p} and for which there is an open set up, p ∈ up, such that γ(up)∩up = ∅, for γ ∈ g − gp. the complement λ(g) = ĉ − ω(g) is called the limit set of g. if the limit set is finite, then g is called elementary; otherwise, it is called non-elementary. if g is an extended kleinian group, then both g and g+ have the same region of discontinuity. basic examples of kleinian groups are the following ones. a function group is a finitely generated kleinian group g+ with an invariant connected component ∆ of its region of discontinuity.; in this case, λ(g+) = ∂∆. a quasifuchsian group is a function group whose limit set is a jordan loop. a b-group is a finitely generated function group with a simply connected invariant component of its region of discontinuity. if g+ is a b-group, say with the simply-connected invariant connected component ∆, then the riemann mapping’s theorem ensures the existence of a biholomorphism f : h2 → ∆. then f−1g+f is a discrete group of automorphisms of h2 (a fuchsian group). a parabolic transformation l ∈ g+ is called accidental if f−1 ◦ l ◦ f is hyperbolic. examples of extended kleinian groups are the following ones. an extended function group is a finitely generated extended kleinian group g with an invariant connected component ∆ of its region of discontinuity. in this case, λ(g) = ∂∆ and g+ is a function group (the converse of this last fact is not in general true). an extended quasifuchsian group is an extended function group whose limit set is a jordan loop (so its orientation-preserving half is a quasifuchsian group). an extended b-group is a finitely generated extended kleinian group with a simply connected invariant connected component of its region of discontinuity. we observe that the orientation-preserving half of an extended b-group is a b-group, but the converse is in general not true (see part (2) of the next result). lemma 2.1. if g is an extended kleinian group whose orientation-preserving half g+ is a bgroup, then either (1) g is an extended b-group, or (2) g+ is a quasifuchsian group and there is an element of g − g+ permuting both components of its region of discontinuity. proof. since g+ is finitely generated, so is g. let ∆ be a simply connected invariant component of g+. if g is not an extended b-group, then there is some γ ∈ g−g+ so that ∆′ = γ(∆) is another 74 rubén a. hidalgo cubo 19, 3 (2017) different simply connected component of g+. it follows that g+ is necessarily a quasifuchsian group. remark 2.2. let g be an extended b-group. then the following properties are easy to see (just from the previous definitions). (1) g is an extended quasifuchsian group if and only if g+ is a quasifuchsian group. (2) g is an extended totally degenerate group if and only if g+ is totally degenerate; (3) g has accidental parabolic transformations if and only if g+ has accidental parabolic transformations. the above remark permits us to see the following fact. lemma 2.3. let g be a non-elementary extended b-group. then either (1) g is an extended quasifuchsian group, or (2) g is an extended totally degenerate group, or (3) g contains accidental parabolic transformations. proof. as g+ is a non-elementary b-group, then either g+ is a quasifuchsian group, or g+ is a totally degenerate group, or g+ contains accidental parabolic transformations. the result now follows from remark 2.2. 2.3 the klein-maskit’s combination theorem we next state a simple version of klein-maskit’s combination theorems which is enough for us in this paper. theorem 2.4 (klein-maskit’s combination theorem [5, 6, 7]). (1) (free products) let kj be a (extended) kleinian group with region of discontinuity ωj, for j = 1,2. let fj be a fundamental domain for kj and assume that there is a simple closed loop σ, contained in the interior of f1 ∩ f2, bounding two discs d1 and d2, so that, for j = 1,2, the set σ∪dj ⊂ ω3−j is precisely invariant under the identity in k3−j. then k = 〈k1,k2〉 is a (extended) kleinian group, with fundamental domain f1 ∩ f2, which is the free product of k1 and k2. every finite order element in k is conjugated in k to a finite order element of either k1 or k2. moreover, if both k1 and k2 are geometrically finite, then k is so. (2) (hnn-extensions) let k0 be a (extended) kleinian group with region of discontinuity ω, and let f be a fundamental domain for k0. assume that there are two pairwise disjoint simple closed loops σ1 and σ2, both of them contained in the interior of f0, so that σj bounds a disc dj such that cubo 19, 3 (2017) totally degenerate extended kleinian groups. 75 (σ1 ∪d1)∩(σ2 ∪d2) = ∅ and that σj ∪dj ⊂ ω is precisely invariant under the identity in k0. if t is either a loxodromic transformation or a glide-reflection so that t(σ1) = σ2 and t(d1)∩d2 = ∅, then k = 〈k0,f〉 is a (extended) kleinian group, with fundamental domain f1 ∩ (d1 ∪d2) c, which is the hnn-extension of k0 by the cyclic group 〈t〉. every finite order element of k is conjugated in k to a finite order element of k0. moreover, if k0 is geometrically finite, then k is so. 3 proof of theorem 1.1 we proceed to describe the main points of the arguments done in [4, ix.g], for the fuchsian groups case, and the corresponding adaptation to the nec groups case. let f be a nec group acting on the upper-half plane h2, such that h2/f is not a pant, and let π : h2 → h2/f be a di-analytic regular branched covering map induced by the action of f. as h2/f is not a pant, we may choose a simple loop w ⊂ h2/f so that each of its lifted arcs in h 2, under π, has as f-stabilizer a cyclic group generated by a hyperbolic element being primitive (that is, it is not a non-trivial power of an element of f). let a ⊂ h2 be one of the arcs in σ := π−1(w) and let j = 〈j〉 be its f-stabilizer (so j is a primitive hyperbolic transformation). note that the connected components of h2 − σ are planar regions. let e1 and e2 be the two of these regions containing a on their borders and, for m ∈ {1,2}, let fm be the f-stablizer of em. as em is precisely invariant under fm in f, it follows that em/fm is embedded in h 2/f. moreover, h2/f is the union of e1/f1, e2/f2 and w. as em/fm is topologically finite, fm is finitely generated. as a consequence of the first klein-maskit combination theorem [5] it holds that f = 〈f1,f2〉 = f1 ∗j f2. we may normalize f so that a is contained in the imaginary line, that is, j has its fixed points at 0 and ∞. we assume that e1 contains positive real points on its border. if t ∈ c − {0} and kt(z) = tz, then let g(t) = 〈f1,f2(t)〉, where f2(t) = ktf2k −1 t . for each t there is a natural surjective homomorphism φt : f → g(t), defined as the identity on f1, and as the isomorphism f 7→ kt ◦f◦k −1 t on f2. clearly, this restricts to a surjective homomorphism φt : f + → g(t)+. we should note that, for t ∈ (1,+∞), the group g(t) is a group of conformal and anticonformal automorphisms of h2, in particular, asserting that g(t) is a nec group topologically conjugated to f. in [4, lemma g.6]) it was noted that, for every f ∈ f+, the fixed points, and the square of the trace of φt(f), are holomorphic functions of t ∈ c − {0}. as a consequence, for each hyperbolic f ∈ f+, there are only countable many complex numbers t so that φt(f) ∈ g(t) + is parabolic. since f+ is countable, there are only countably many ρ > 0 for which there is a t = ρeiθ so that for some hyperbolic f ∈ f+, φt(f) is parabolic. so, we may find ρ = ρ0 so that tr 2(φt(f)) 6= 4 for all hyperbolic f ∈ f+ and for all t = ρ0e iθ. 76 rubén a. hidalgo cubo 19, 3 (2017) we know from the above that g(ρ0) is still a nec group and, for θ small, the group g(ρ0e iθ) is an extended quasifuchsian group. similarly as done for the fuchsian situation, we let t be the set of complex number of the form t = ρ0e iθ for which there is a loop w(t) dividing ĉ into two closed discs, b1(t) and b2(t), where b1(t) is a (j,f1)-block (see [4, vii.b.4]) and b2(t) is precisely invariant under j in f2(t). note that: (i) ρ0 ∈ t, in this case b1 = b1(ρ0) is the left half-plane and b2 = b2(ρ0) is the right half-plane, and (ii) for small values of θ, ρ0e iθ ∈ t. if t ∈ t, then the groups f1 and f2(t) satisfy the hypothesis to use the first klein-maskit combination theorem, so g(t) = f1 ∗j f2(t) and g(t) is an extended quasifuchsian group so that φt : f → g(t) is in an isomorphism. also, there is a homeomorphism ψt : ω(f) → ω(g(t)) inducing the isomorphism φt : f → g(t) and so that w(t) is the image under ψt the circle given as the union of the imaginary line with ∞. note that b1(t) is the disc containing ψt(e2) and b2(t) is the disc containing ψt(e1). the limit set λ(g(t)) is a simple close curve passing through 0 and ∞; the complement, ω(g(t)) has two components, the upper component ∆(t) and the lower component ∆′(t). as a consequence of [3, proposition ix.g.8], applied to the fuchsian group f+ and the quasifuchsian group g(t)+, one obtains that, if t ∈ t and im(t) > 0, then h2 ⊂ ∆(t). also, as observed in [4, ix.g.9], by interchanging the roles of f1 and f2(t) it permits to observe that ∆(t) contains the half-plane {arg(t) < arg(z) < arg(t) + π}. applying [3, lemma ix.g.10] to f+ and g(t)+, it can be seen that t is an open arc of the circle of radius ρ0 with centre at the origin. next, we follow [3, ix.g.12]. we start at the point t0 = ρ0 ∈ t and traverse counterclockwise to reach some first point t# not in t; and we set t0 be the arc of t between t0 and t # (as noted in there, as t traverses t0 counterclockwise, from t0 to t #, the upper component ∆(t) gets larger and the lower component ∆′(t) gets smaller). next, we fix a fundamental polygon p1 ⊂ h 2 for f1 and a fundamental polygon p2 ⊂ h 2 for f2. we choose p1 and p2 so that they are both contained in some fundamental polygon e for j. leave p1 fixed and define p2(t) = kt(p2). so p2(t) is a fundamental polygon for f2(t) in the appropriate half-plane. the union p1 ∪ p2(t) bounds a fundamental domain d(t) ⊂ ∆(t) for g(t) acting on ∆(t) and there is a homeomorphism ψt : h 2 → ∆(t) so that ψt(d(t0)) = d(t) inducing the isomorphism φt : f → g(t). it follows from the construction that, as t ∈ t0 approaches t #, φt : f → g(t) converges to a homomorphism φ : f → g(t#). by our choice on ρ0, it follows from [2] that φ : f + → g(t#)+ is type-preserving isomorphism. lemma 3.1. φ : f → g(t#) is an isomorphism. proof. otherwise, there should be some f ∈ f − f+ so that φ(f) = 1. but, as f2 ∈ f+ and that φ : f+ → g(t#)+ is an isomorphism, then f2 = 1. as f is an anticonformal involution, φt(f) ∈ g(t) is an anticonformal involution; so φt(f) cannot approach the identity as t approaches t#; a contradiction. cubo 19, 3 (2017) totally degenerate extended kleinian groups. 77 also, as t ∈ t0 approaches t #, ψt : d(t0) → ψt(d(t0)) ⊂ ∆(t) converges to a homeomorphism from d(t0) onto its image d. now, the same proof as [3, lemma ix.g.13] permits to obtain that g = g(t#) is an extended kleinian group and that d is precisely invariant under the identity in g. working with f+, g(t)+ and g+, we obtain (from [3, ix.g.14]) that g+ is a b-group with a simply connected invariant component ∆, where ∆/g+ is a finite riemann surface homeomorphic to h2/f+. as consequence of [3, proposition ix.g.15] it follows that g+ is not quasifuchsian. combining [3, ix. g.14], lemma 2.1 and the above, one obtains that g is an extended b-group, different from a quasifuchsian one. as ρ0 was constructed so that, for every f ∈ f + hyperbolic and every t = ρ0e iθ, the element φt(f) ∈ g +(t) is not parabolic, it follows that φ(f) is neither parabolic. hence the only elements of g+(t) that are parabolic are conjugates of the parabolic elements of f+1 or f2(t) +. but, as seen in [4, ix.g.14], they represent punctures on ∆/g+(t), so they are not accidental. now, as consequence of all the above, together lemma 2.3, we obtain that the group g(t#) is an extended totally degenerate group as desired. references [1] l. bers. on boundaries of teichmüller spaces and on kleinian groups: i. ann. of math. 91 (1970), 570-600. [2] v. chuckrow. on schottky groups with applications to kleinian groups. ann. of math. 88 (1968), 47-61. [3] b. maskit, kleinian groups, gmw, springer-verlag, 1987. [4] b. maskit. on boundaries of teichmüller spaces and on kleinian groups: ii. ann. of math. 91 (1970), 607-639. [5] b. maskit. on klein’s combination theorem trans. of the amer. math. soc. 120, no. 3 (1965), 499–509. [6] b. maskit. on klein’s combination theorem iii. advances in the theory of riemann surfaces (proc. conf., stony brook, n.y., 1969), ann. of math. studies 66 (1971), princeton univ. press, 297-316. [7] maskit, b. on klein’s combination theorem. iv. trans. amer. math. soc. 336 (1993), 265-294. [8] h. miyachi. quasi-arcs in the limit set of a singly degenerate group with bounded geometry. in kleinian groups and hyperbolic 3-manifolds (eds. y.komori, v.markovic c.series) lms. lec. notes 299 (2003), 131-144. introduction preliminaries möbius and extended möbius transformations kleinian and extended kleinian groups the klein-maskit's combination theorem proof of theorem 1.1 cubo a mathematical journal vol.21, no¯ 03, (63–74). december 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000300063 beta-almost ricci solitons on sasakian 3-manifolds pradip majhi and debabrata kar department of pure mathematics, university of calcutta, 35, ballygaunge circular road, kolkata 700019, west bengal, india mpradipmajhi@gmail.com, debabratakar6@gmail.com abstract in this paper we characterize the sasakian 3-manifolds admitting β-almost ricci solitons whose potential vector field is a contact vector field. among others we prove that a βalmost ricci soliton whose potential vector field is a contact vector field on a sasakian 3-manifold is shrinking, einstein and non-trivial. moreover, we prove that this type of manifolds are isometric to a sphere of radius √ 7. resumen en este art́ıculo caracterizamos las 3-variedades sasakianas que admiten solitones βcasi ricci cuyo campo de vectores potencial es un campo de vectores de contacto. entre otros, probamos que un solitón β-casi ricci cuyo campo de vectores potencial es un campo de vectores de contacto en una 3-variedad sasakiana se contrae, es einstein y no trivial. más aún, probamos que este tipo de variedades son isométricas a una esfera de radio √ 7. keywords and phrases: ricci soliton, β-almost ricci soliton, sasakian 3-manifolds, einstein. 2010 ams mathematics subject classification: 53c15, 53c25. http://dx.doi.org/10.4067/s0719-06462019000300063 64 pradip majhi and debabrata kar cubo 21, 3 (2019) 1 introduction in 1982, r. s. hamilton [17] introduced the notion of ricci flow to find a canonical metric on a smooth manifold. the ricci flow is an evolution equation for metrics on a riemannian manifold defined as follows: ∂ ∂t g = −2s, (1) where s denotes the ricci tensor of g. ricci solitons are special solutions of the ricci flow equation (1) of the form g = σ(t)ψ∗tg with the initial condition g(0) = g, where ψt are diffeomorphisms of m and σ(t) is the scaling function. a ricci soliton is a generalization of an einstein metric. we recall the notion of ricci soliton according to [5]. on the manifold m, a ricci soliton is a triple (g,v,λ) with g, a riemannian metric, v a vector field, called the potential vector field and λ a real scalar such that £vg + 2s + 2λg = 0, (2) where £ is the lie derivative. metrics satisfying (2) are interesting and useful in physics and are often referred as quasi-einstein ([6],[7]). compact ricci solitons are the fixed points of the ricci flow ∂ ∂t g = −2s projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings, and often arise blow-up limits for the ricci flow on compact manifolds. theoretical physicists have also been looking into the equation of ricci soliton in relation with string theory. the initial contribution in this direction is due to friedan [14] who discusses some aspects of it. recently, the notion of almost ricci soliton has been introduced in [24] by piagoli, riogoli, rimoldi and setti. the ricci soliton is said to be shrinking, steady or expanding according as λ is negative, zero or positive respectively. ricci solitons have been studied by several authors ([8], [9], [18], [19], [20], [27], [28], and many others). recently, gomes, wang and xia [26] generalized almost ricci soliton to h -almost ricci soliton as follows: definition 1.1. a complete connected riemannian manifold (m2n+1,g) is said to be a β-almost ricci soliton, denoted by (m2n+1,g,v,β,λ), if there exist a smooth vector field v on m2n+1 such that s + β 2 £vg + λg = 0, (3) where λ and β are smooth functions on m2n+1. λ is called soliton function and v is called the potential vector field. cubo 21, 3 (2019) beta-almost ricci solitons on sasakian 3-manifolds 65 a β-almost ricci soliton is said to be shrinking, steady or expanding according as λ is negative, zero or positive respectively. a β-almost ricci soliton is called β-ricci soliton if λ is constant. a β-almost ricci soliton is said to be trivial, that is, einstein if the flow vector field v is homothetic, that is, £vg = cg, for some constant c. otherwise, it is called non-trivial. a β-almost ricci soliton is said to be β-almost gradient ricci soliton if the potential vector field v is the gradient of a smooth function f on m2n+1, that is, v = df, where d is the gradient operator of g on m2n+1. for convenience, we denote (m2n+1,g,df,β,λ) as a β-almost gradient ricci soliton with potential function f. in particular, a ricci soliton is a 1-almost ricci soliton with constant soliton λ and an almost ricci soliton is nothing but a 1-almost ricci soliton. recently, ghosh and patra studied [16] the k-almost ricci solitons on contact geometry. in [1], barros and ribeiro proved that a compact almost ricci soliton with constant scalar curvature is isometric to an euclidean sphere. in this connection, a theorem has also been proved by gomes, wang and xia in [26] for β-almost ricci soliton which is given as follows: theorem 1.1. [26] let (mn,g,v,β,λ),n ≥ 3 be a non-trivial β-almost ricci soliton with constant scalar curvature r. if mn is compact, then it is isometric to a standard sphere sn(c) of radius c = √ 2n(2n+1) r . the above theorem will be used in later to prove our results. the paper is organized as follows: after preliminaries in section 2, we study β-almost ricci solitons on a sasakian 3-manifold. among others we prove that β-almost ricci solitons whose potential vector field is a contact vector field on sasakian 3-manifolds are shrinking and einstein. beside these, we prove that this type of manifolds are isometric to a sphere of radius √ 7. also we prove that a β-almost ricci soliton whose potential vector field is a contact vector field on a sasakian 3-manifold is non-trivial. 2 preliminaries an odd dimensional smooth manifold m2n+1 (n ≥ 1) is said to admit an almost contact structure, sometimes called a (φ,ξ,η)-structure, if it admits a tensor field φ of type (1,1), a vector field ξ and a 1-form η satisfying ([2], [3]) φ2 = −i + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0. (4) 66 pradip majhi and debabrata kar cubo 21, 3 (2019) the first and one of the remaining three relations in (4) imply the other two relations in (4). an almost contact structure is said to be normal if the induced almost complex structure j on mn ×r defined by j(x,f d dt ) = (φx − fξ,η(x) d dt ) (5) is integrable, where x is tangent to m, t is the coordinate of r and f is a smooth function on mn × r. let g be a compatible riemannian metric with the (φ,ξ,η)-structure, that is, g(φx,φy) = g(x,y) − η(x)η(y) (6) or equivalently, g(x,φy) = −g(φx,y) (7) and g(x,ξ) = η(x), (8) for all vector fields x, y tangent to m. then m becomes an almost contact metric manifold equipped with an almost contact metric structure (φ,ξ,η,g). an almost contact metric structure becomes a contact metric structure if g(x,φy) = dη(x,y), (9) for all x, y tangent to m. the 1-form η is then a contact form and ξ is its characteristic vector field. given the contact metric manifold (m,η,ξ,φ,g), we define a symmetric (1,1)-tensor field h as h = 1 2 lξφ, where lξφ denotes lie differentiation in the direction of ξ. we have the following identities ([2], [3]): hξ = 0, hφ + φh = 0, (10) ∇xξ = −φx − φhx, (11) ∇ξφ = 0, (12) r(ξ,x)ξ − φr(ξ,φx)ξ = 2(h2 + φ2)x, (13) (∇ξh)x = φx − h2φx + φr(ξ,x)ξ, (14) s(ξ,ξ) = 2n − trh2, (15) r(x,y)ξ = −(∇xφ)y + (∇yφ)x − (∇xφh)y + (∇yφh)x. (16) here, ∇ is the levi-civita connection and r is the riemannian curvature tensor of (m,g) with the sign convention defined by r(x,y)z = ∇x∇yz − ∇y∇xz − ∇[x,y]z (17) cubo 21, 3 (2019) beta-almost ricci solitons on sasakian 3-manifolds 67 for vector fields x, y, z on m. the tensor l = r(.,ξ)ξ is the jacobi operator with respect to the characteristic field ξ. if the characteristic vector field ξ is a killing vector field, the contact metric manifold (m,η,ξ,φ,g) is called k-contact manifold. this is the case if and only if h = 0. the contact structure on m is said to be normal if the almost complex structure on m × r defined by (5), is integrable. a normal contact metric manifold is called a sasakian manifold. sasakian metrices are k-contact and k-contact metrics on 3-manifolds are sasakian. for a sasakian manifold, the following hold ([2], [3]): ∇xξ = −φx, (18) (∇xφ)y = g(x,y)ξ − η(y)x, (19) r(x,y)ξ = η(y)x − η(x)y, (20) qξ = 2nξ, (21) where q denotes the (1,1)-tensor metrically equivalent to the ricci tensor of g. the curvature tensor of a 3-dimensional riemannian manifold is given by r(x,y)z = [s(y,z)x − s(x,z)y + g(y,z)qx − g(x,z)qy] − r 2 [g(y,z)x − g(x,z)y], (22) where s and r are the ricci tensor and scalar curvature respectively and q is the ricci operator defined by g(qx,y) = s(x,y). it is known that the ricci tensor of a sasakian 3-manifold is given by [4] s(x,y) = 1 2 {(r − 2)g(x,y) + (6 − r)η(x)η(y)} (23) where r is the scalar curvature which need not be constant, in general. so, g is einstein (hence has constant curvature 1) if and only if r = 6. as a consequence of (23), we have s(x,ξ) = 2η(x). (24) contact metric manifolds have also been studied by several authors ([4], [10], [11], [12], [13], [21], [22], [23], [25], and many others). 68 pradip majhi and debabrata kar cubo 21, 3 (2019) definition 2.1. ([16]) a vector field v on a contact manifold is said to be a contact vector field if it preserves the contact form η, that is £vη = ψη, (25) for some smooth function ψ on m. when ψ = 0 on m, the vector field v is called a strict contact vector field. lemma 2.1. ([15]) if a vector field x leaves the structure tensor φ of the contact metric manifold m invariant, then there exists a constant c such that £xg = c(g + η ⊗ η). 3 β-almost ricci solitons on sasakian 3-manifolds in this section we characterize sasakian 3-manifolds m3 admitting β-almost ricci solitons whose potential vector field v is a contact vector field. then the equations (3) and (25) hold good. the equation (3) can be exhibited as s(x,y) + β 2 {g(∇xv,y) + g(x,∇yv)} + λg(x,y) = 0. (26) using (23) in the above equation we get β{g(∇xv,y) + g(x,∇yv)} = −(r + 2λ − 2)g(x,y) +(r − 6)η(x)η(y). (27) tracing the equation (27) we obtain βdivv = −(r + 3λ). (28) with the help of (25) we have £vdη = d£vη = (dψ) ∧ η + ψ(dη). (29) let us consider ω as the volume form of the manifold m3, that is, ω = η ∧ dη 6= 0. (30) taking lie derivative of the preceding equation along the potential vector field v and using (25) and (29) we have £vω = 2ψω, and hence divv = 2ψ. (31) cubo 21, 3 (2019) beta-almost ricci solitons on sasakian 3-manifolds 69 using the foregoing equation in (28) we infer r = −2ψβ − 3λ. (32) the soliton equation (3) also can be represented as s(x,y) + β 2 (£vg)(x,y) + λg(x,y) = 0. (33) substituting x = y = ξ in (33) we get βg(£vξ,ξ) = λ + 2. (34) putting y = ξ in (33) and using (24), β 2 (£vη)(x) − β 2 g(x,£vξ) + (λ + 2)η(x) = 0. (35) making use of (25) we obtain β£vξ = (ψβ + 2λ + 4)ξ. (36) by the virtue of (34) and (36) we have ψβ = −λ − 2. (37) using (37), (36) entails β£vξ = (λ + 2)ξ. (38) from (9) we deduce that (£vdη)(x,y) = (£vg)(x,φy) + g(x,(£vφ)y). (39) multiplying both sides of (39) by β and then using (33) we infer β(£vdη)(x,y) = −2s(x,φy) − 2λg(x,φy) + βg(x,(£vφ)y). (40) in view of (23) and (40) we get β(£vdη)(x,y) = −(r + 2λ − 2)g(x,φy) + βg(x,(£vφ)y). (41) from (29) we derive (£vdη)(x,y) = 1 2 {dψ(x)η(y) − dψ(y)η(x)} + ψg(x,φy). (42) comparing (41) and (42), after simplification we obtain 2β(£vφ)y = 2(r + 2λ − 2)φy + βη(y)dψ − β(yψ)ξ + 2ψβφy. (43) 70 pradip majhi and debabrata kar cubo 21, 3 (2019) replacing y by ξ we get 2β(£vφ)ξ = βdψ − β(ξψ)ξ. (44) with the help of (4) and (36) we find that β(£vφ)ξ = 0. (45) applying (45) on (44) we have dψ = (ξψ)ξ. (46) taking inner product of (46) with x gives dψ(x) = (ξψ)η(x), (47) or equivalently, dψ = (ξψ)η. (48) taking exterior derivative we get d(ξψ) ∧ η + (ξψ)dη = 0. (49) taking wedge product of (49) with η we have (ξψ)η ∧ dη = 0, (50) from which it follows that ξψ = 0, (51) since η ∧ dη 6= 0, and by (48), dψ = 0 (52) and hence ψ is constant. integrating (31) and then using divergence theorem we infer ψ = 0. (53) thus the potential vector field v becomes a strict contact vector field and hence we have the following: theorem 3.1. let (m3,g,v,β,λ) be a non-trivial β-almost ricci soliton whose potential vector field is a contact vector field on a sasakian 3-manifold. then the potential vector field is a strict cubo 21, 3 (2019) beta-almost ricci solitons on sasakian 3-manifolds 71 contact vector field. by the virtue of (37) and (53) we find λ = −2. (54) therefore, the β-almost ricci soliton is shrinking. thus we are in a position to state that theorem 3.2. a non-trivial β-almost ricci soliton (m3,g,v,β,λ) whose potential vector field is a contact vector field on a sasakian 3-manifold is shrinking. making use of (53) and (54), from (32) we get r = 6. (55) then we can conclude that theorem 3.3. the scalar curvature of a non-trivial β-almost ricci soliton (m3,g,v,β,λ) whose potential vector field is a contact vector field on a sasakian 3-manifold is 6. with the help of (55) from (23) we have s(x,y) = 2g(x,y). (56) hence we can state the following: theorem 3.4. a non-trivial β-almost ricci soliton (m3,g,v,β,λ) whose potential vector field is a contact vector field on sasakian 3-manifold is einstein. from (55) we can say that r is constant. then in view of theorem 1.1 we can conclude the following: theorem 3.5. let (m3,g,v,β,λ) be a non-trivial β-almost ricci soliton whose potential vector field is a contact vector field on a sasakian 3-manifolds. then (m3,g,v,β,λ) is isometric to a sphere s3(c) of radius c = √ 7. using (53), (54) and (55) in (43) we infer (£vφ)y = 0, (57) as we have considered β as positive, that is, v leaves the structure tensor φ of the sasakian 3-manifold invariant. then, by lemma 2.2, exists a constant a such that £vg = a(g + η ⊗ η), (58) 72 pradip majhi and debabrata kar cubo 21, 3 (2019) which shows that the β-almost ricci solitons are non-trivial. thus our next theorem can be stated as follows: theorem 3.6. let (m3,g,v,β,λ) be a non-trivial β-almost ricci soliton whose potential vector field is a contact vector field on a sasakian 3-manifolds. then the β-almost ricci solitons are non-trivial. acknowledgement: the authors are thankful to the referee for his/her valuable suggestions and comments towards the improvement of the paper. the author debabrata kar is supported by the council of scientific and industrial research, india (file no : 09/028(1007)/2017-emr-1). cubo 21, 3 (2019) beta-almost ricci solitons on sasakian 3-manifolds 73 references [1] barros, a. and ribeiro, jr., some characterizations for compact almost ricci solitons, proc. amer. math. soc., 140(3) (2012), 1033-1040. 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[28] wang, y., ricci solitons on 3-dimensional cosympletic manifolds, math. slovaca, 67(4) (2017), 979-984. introduction preliminaries -almost ricci solitons on sasakian 3-manifolds cubo 8 , 07-09 (1992) recibido: junio 1992. una condición suficiente para que un punto encontrado por multiplicadores de lagrange sea un extremo. gustavo avello j. a} se probará el siguiente teorema: teorema .1 sea a un abierto de ir",/,g¡, ... ,911'1(-•) funciones de a en ir de clase c 2 , g = (g¡, ... , 9m), m = g(o). si para. todo x e m, los vectores gradiente..¡, .. , >.m) e irm tal que 'v j(a) = f; >.;'v g;(a). j=i sea q la forma cuadrática a.sociaáa a la función auxiliar f = f l >.;g;, es ¡,,,,¡ decir g(h) = j ¿: ~h;h;;q(h) = jf"(a)(h,h) •,j=i se tiene: 1} si q es definida positiva sobre el espacio tangente ta.(m) = ñ k ergj(a) entonces a es un mínimo relativo de j sobre m. 2} si q es definida negativa sobre t.,(m), a es un máximo relativo de f !m. 3) si q es no definida, a no es ni máximo ni mínimo de f 1 m. b) aplicaremos el siguiente teorema teorema .2 sea[} un abierto irri,v!: [l r de clase cfl, si b e [l es un punto críticn de ![r tal que para todo z e r,z #o: !l'11 (b)(z,z) >o. entonces b es un mínimo de t/j, demostración: h. cartan pág. 100, teorema 8.33. • cubos g. a vello c) la hessiana: sea muna subvariedad de r", me a,j: a e r" r y a e m. diremos que a es un punto crítico de f sobre m si la diferencial j'(a) se anu la sobre el espacio tangente t0 (m). supongamos ahora que f y m sean de clase c 2 , y sea ip : n m una parametrización de clase c2 de m en una vecindad de a, tal que rp(o) = a, es decir ip : fl 'p(d) = un m, con u vecindad abierta de a, i.p inmersión, inyectiva y homeomorfismos; luego o, entonces a es un punto de mínimo de f sobre m. d e mostración: se aplica el teorema 2 a 1/j = fo r.p en el punto b = o. • d) probemos el teorema enunciado en (a) en el caso de un mínimo, loo otros casos, con ligeras modificaciones son similares. basta probar que: 'v(x1 y) e ta(m) x ta(m) se tiene: hes0 /(x , y) = [/"(a) .1 o g"(a)j(x, y) donde .l. : rm ___. r es la aplia.ción lineal tal que .l.(ej) = .áj , ei base canónica d e rm. sea r.p : n ___. m una c 2 parametrización de m en una vecindad de a tal que ; 0 for all k ∈ [1, n], φk(s) = |s| p−2s with 1 < p < +∞ and f : [1, n] × r −→ r is a continuous function. the authors obtain a suitable interval of cubo 19, 3 (2017) existence of solutions for discrete boundary value . . . 59 parameters for which problem (1.3) admits constant-sign solutions which are local minimizers of the corresponding euler-lagrange functional. motivated by the paper [6] and the ideas in [10], we used the mountain pass lemma to prove an existence of a non trivial solution under some hypothesis. with additional conditions we prove the uniqueness of the non trivial solution. in order to bring evidence our main result, in the section 2 we deal with some preliminary materials. in section 3, using the mountain pass lemma, we prove the existence of an non trivial solution and adding some conditions, we study the uniqueness of solution. 2 preliminaries in the n-dimensional hilbert space x = {x : [0, n + 1] −→ r; such that ∆x(0) = 0 = ∆x(n)} with the inner product (x, y) = n+1∑ k=1 ∆x(k − 1)∆y(k − 1), ∀ x, y ∈ x, we consider the norm ‖x‖ = ( n+1∑ k=1 |∆x(k − 1)|2 ) 1 2 . (2.1) let the function p : [0, n] −→ [2, +∞) (2.2) and denoted by p− = min k∈[0,t] p(k) and p+ = max k∈[0,t] p(k). for the data a, f and δ, we assume the following: (h0). { a(k, .) : r → r, k ∈ [0, n] and there exists a(., .) : [0, n] × r → r which satisfies a(k, ξ) = ∂ ∂ξ a(k, ξ) and a(k, 0) = 0, for all k ∈ [0, n]. (h1). for any k ∈ [0, n], ξ ∈ r there exist a constant c0 > 0 such that |a(k, ξ)| ≤ c0 ( 1 + |ξ|p(k)−1 ) . (2.3) (h2). for any k ∈ [0, n], ξ ∈ r, we have a(k, ξ) ≥ 1 p(k) |ξ|p(k). (2.4) 60 aboudramane guiro and idrissa ibrango cubo 19, 3 (2017) (h3). for each k ∈ [0, n], the function f(k, ., .) : r × w −→ r is jointly continuous and there exists a constant c1 > 0 and a function r : [0, n] −→ (1, +∞) such that |f(k, ξ, w)| ≤ c1 ( 1 + |ξ|r(k)−1 ) (2.5) where the space w is some topological space. we denoting f(k, ξ, w) = ∫ξ 0 f(k, s, w)ds for (k, ξ, w) ∈ [0, n] × r × w (2.6) and we deduce that there exist a constant c2 > 0 such that |f(k, ξ, w)| ≤ c2 ( 1 + |ξ|r(k) ) . (2.7) (h4). lim ξ→0 f(k, ξ, w) |ξ|p +−1 = 0 uniformly for all k ∈ [1, n], w ∈ w. (h5). δ : [0, n] −→ (0, +∞) such that for all k ∈ [0, n], 0 < δ = inf k∈[0,n] (δ(k)) ≤ δ(k) ≤ δ̄ = sup k∈[0,n] (δ(k)) < +∞. (2.8) (h6). there exists constants c3, c4 > 0 and µ > p + such that f(k, ξ, w) ≥ c3|ξ| µ − c4, for all k ∈ [0, n], ξ ∈ r, w ∈ w. (2.9) example 2.1. we give the following functions where a non trivial solution can be estimate as in [10]: • a(k, ξ) = 1 p(k) |ξ|p(k), where a(k, ξ) = |ξ|p(k)−2ξ, ∀ k ∈ [0, n] and ξ ∈ r, • a(k, ξ) = 1 p(k) ( ( 1 + |ξ|2 )p(k)/2 − 1 ) , where a(k, ξ) = ( 1 + |ξ|2 )(p(k)−2)/2 ξ, ∀ k ∈ [0, n], ξ ∈ r, • f(k, ξ, w) = 1 + ∣ ∣ξ ∣ ∣ p(k)−1 , ∀ k ∈ [0, n] and ξ ∈ r, • δ(k) = 1, ∀ k ∈ [0, n]. the conditions (h0) − (h6) are fulfilled. moreover we may consider x with the following norm |x|m = ( n∑ k=1 |x(k)|m ) 1 m , ∀ x ∈ x and m ≥ 2. (2.10) we have the following inequalities (see [3, 9]) n(2−m)/(2m)|x|2 ≤ |x|m ≤ n 1/m |x|2, ∀ x ∈ x and m ≥ 2. (2.11) cubo 19, 3 (2017) existence of solutions for discrete boundary value . . . 61 we defined the convex modular ρ : x −→ r by ρ(x) = n+1∑ k=1 |∆x(k − 1)|p(k−1) and we introduce the luxembourg norm ‖x‖p(.) = inf { λ > 0 : ρ(x/λ) ≤ 1 } . (2.12) since x has a finite dimension there exist constants c5 > 0, c6 > 1 such that c5‖x‖p(.) ≤ ‖x‖ ≤ c6‖x‖p(.). (2.13) the followings inequalities holds: min { ‖x‖ p− p(.) ; ‖x‖ p+ p(.) } ≤ ρ(x) ≤ max { ‖x‖ p− p(.) ; ‖x‖ p+ p(.) } . (2.14) we need the following auxiliary results throughout our paper (see [5]). lemma 2.2. (1) for every x ∈ x and for every m ≥ 2 we have n+1∑ k=1 |∆x(k − 1)|m ≤ 2m n∑ k=1 |x(k)|m (2.15) and n+1∑ k=1 |∆x(k − 1)|m ≥ (n + 1) 2−m 2 ‖x‖m. (2.16) (2) for every x ∈ x and for every m > 1 we have n∑ k=1 |x(k)|m ≤ n(n + 1)m−1 n+1∑ k=1 |∆x(k − 1)|m. (2.17) (3) for every x ∈ x and for every m ≥ 1 we have n+1∑ k=1 |∆x(k − 1)|m ≤ (n + 1)‖x‖m. (2.18) (4) for every x ∈ x we have n+1∑ k=1 |∆x(k − 1)|p(k−1) ≤ (n + 1)‖x‖p + + (n + 1). (2.19) 62 aboudramane guiro and idrissa ibrango cubo 19, 3 (2017) definition 2.3. a c1 functional j satisfies the palais-smale condition (in short (ps) condition) if any sequence {un} ⊂ x such that {j(un)} is bounded and j′(un) −→ 0, as n −→ ∞ has a convergent subsequence. from (ps) condition, it follows that the set of critical points for a bounded functional is compact. this condition was a base for the modern development of critical point theory. it is needed for the mountain pass lemma (see [11]). lemma 2.4. let x be a real reflexive banach space. assume that j ∈ c1(x, r) and j satisfies the (ps) condition. suppose also that: (1) j(0) = 0, (2) there exist ρ > 0 and α > 0 such that j(u) ≥ α for all u ∈ x with ‖u‖ = ρ, (3) there exists u1 ∈ x with ‖u1‖ > ρ such that j(u1) < α. then, j has a critical value c ≥ α. moreover, c can be characterized as inf g∈γ ( max u∈g([0,1]) j(u) ) , where γ = {g ∈ c([0, 1], x) : g(0) = 0, g(1) = u1}. lemma 2.5 ([10], lemma 2.2). let x be a finite dimensional banach space and let j ∈ c1(x, r) be an anti-coercive functional (namely, lim ‖u‖−→+∞ j(u) = −∞). then j satisfies the (ps) condition. 3 existence of nontrivial solutions we define the energy functional j : x −→ r by j(u) = n+1∑ k=1 a ( k − 1, ∆u(k − 1) ) − n∑ k=1 δ(k)f ( k, u(k), w ) (3.1) and we put i(u) = n+1∑ k=1 a ( k − 1, ∆u(k − 1) ) and λ(u) = n∑ k=1 δ(k)f ( k, u(k), w ) . lemma 3.1. the functional j is well defined on x and is of class c1 ( x, r ) with the derivative given by 〈j′(u), v〉 = n+1∑ k=1 a(k − 1, ∆u(k − 1))∆v(k − 1) − n∑ k=1 δ(k)f ( k, u(k), w ) v(k), (3.2) for all u, v ∈ x. cubo 19, 3 (2017) existence of solutions for discrete boundary value . . . 63 proof we have |j(u)| ≤ |i(u)| + |λ(u)|. since a(k, .) is continuous for all k ∈ [0, n], then |i(u)| = ∣ ∣ ∣ ∣ n+1∑ k=1 a ( k − 1, ∆u(k − 1) ) ∣ ∣ ∣ ∣ < +∞. also from (2.7) and holder’s inequality we have |λ(u)| = ∣ ∣ ∣ ∣ n∑ k=1 δ(k)f ( k, u(k), w ) ∣ ∣ ∣ ∣ ≤ n∑ k=1 |δ(k)||f ( k, u(k), w ) | ≤ c2δ̄ n∑ k=1 ( 1 + |u(k)|r(k) ) ≤ nc2δ̄ + c2δ̄ n∑ k=1 |u(k)|r(k) ≤ 2nc2δ̄ + c2δ̄ n∑ k=1 |u(k)|r + ≤ 2nc2δ̄ + c2δ̄ ( n∑ k=1 ∣ ∣|u(k)|r +∣ ∣ 2 r+ ) r+ 2 ( n∑ k=1 |1| 1 1− r + 2 )1− r + 2 ≤ 2nc2δ̄ + c2δ̄n 2−r+ 2 ( n∑ k=1 |u(k)|2 ) r+ 2 < +∞. then, the energy functional j is well defined on x. as in [6], lemma 3.4, we can prove that the functional i derivative is given by 〈i′(u), v〉 = n+1∑ k=1 a ( k − 1, ∆u(k − 1) ) ∆v(k − 1). (3.3) 64 aboudramane guiro and idrissa ibrango cubo 19, 3 (2017) on the other hand, for all u, v ∈ x, we have 〈λ′(u), v〉 = lim h→0+ λ(u + hv) − λ(u) h = lim h→0+ n∑ k=1 δ(k) f(k, u(k) + hv(k), w) − f(k, u(k), w) h = n∑ k=1 δ(k) lim h→0+ f(k, u(k) + hv(k), w) − f(k, u(k), w) h = n∑ k=1 δ(k)f(k, u(k), w). the functional j is clearly of class c1 � lemma 3.2. the functional j is anti-coercive. proof let ‖u‖p(.) > 1. according to (2.3) and the inequality (2.14) there exist a constant c7 > 0 such that i(u) ≤ c7(n + 1) + c7 n+1∑ k=1 |∆u(k − 1)|p(k−1) ≤ c7(n + 1) + c7‖u‖ p+ p(.) . (3.4) according to (2.9), (2.15) and (2.16) we have −λ(u) = − n∑ k=1 δ(k)f(k, u(k), w) ≤ − n∑ k=1 δ(k) ( c3|u(k)| µ − c4 ) ≤ −δc3 n∑ k=1 |u(k)|µ + c4δn ≤ −δc32 −µ n+1∑ k=1 |∆u(k − 1)|µ + c4δn ≤ −δc32 −µ(n + 1) 2−µ 2 ‖u‖µ + c4δn. (3.5) consequently, combining (3.4) and (3.5) we get j(u) ≤ −δc32 −µ(n + 1) 2−µ 2 ‖u‖µ + c4δn + c7‖u‖ p+ p(.) + c7(n + 1). (3.6) cubo 19, 3 (2017) existence of solutions for discrete boundary value . . . 65 since the norms ‖.‖ and ‖.‖p(.) are equivalents and since µ > p + then the functional j is anticoercive � we can deduce from lemma 2.5 that the function j satisfies the (ps) condition. theorem 3.3. for any w ∈ w, the problem (1.2) has at least one non trivial solution. proof according to (h4), for any ε > 0 there exists β > 0 such that for all ξ ∈ [−β, β], |f(k, ξ, w)| ≤ ε|ξ|p + −1 for all k ∈ [1, n], w ∈ w. as in [10], it is obvious to see that for any ε ∈ ( 0, (n+1) 2−p+ 2 δ̄n(n+1)p + ) there exists β > 0 such that for all x ∈ [−β, β] we have |f(k, ξ, w)| ≤ ε |ξ|p + p+ , ∀ k ∈ [1, n], w ∈ w. (3.7) let u ∈ x ∩ { |u(k)| ≤ β, ∀ k ∈ [1, n] } ∩ { ‖u‖ ≤ 1 } . we have ‖u‖ ≤ 2β(n + 1) 1 2 and according to (2.16) we have n+1∑ k=1 |∆u(k − 1)|p(k−1) ≥ n+1∑ k=1 |∆u(k − 1)|p + ≥ (n + 1) 2−p+ 2 ‖u‖p + . (3.8) if we put η = min{1; 2β(n + 1) 1 2 }, then for any u ∈ x with ‖u‖ ≤ η, by (3.7), (3.8), (h2), (2.17) and (2.18) it follows that j(u) ≥ 1 p+ n+1∑ k=1 |∆u(k − 1)|p(k−1) − δ̄ε 1 p+ n∑ k=1 |u(k)|p + (3.9) ≥ 1 p+ (n + 1) 2−p+ 2 ‖u‖p + − δ̄ε 1 p+ n(n + 1)p + ‖u‖p + (3.10) ≥ ‖u‖p + 1 p+ ( (n + 1) 2−p+ 2 − δ̄εn(n + 1)p + ) . (3.11) remark that ε is such that (n + 1) 2−p+ 2 − δ̄εn(n + 1)p + > 0. so there exists positive numbers 0 < ρ = ‖u‖ ≤ η and α = ρp + p+ ( (n + 1) 2−p+ 2 − δ̄εn(n + 1)p + ) > 0 such that j(u) ≥ α for all u ∈ x with ‖u‖ = ρ. also we have j(0) = 0 and since j is anti-coercive, there exists u1 ∈ x with ‖u1‖ > ρ such that j(u1) < α. by the mountain pass lemma (see lemma 2.4), the functional j has a critical value c∗ > 0, i.e, there exists u∗ ∈ x such that j(u∗) = c∗ and 〈j′(u∗), v〉 = 0, ∀ v ∈ x. since j(0) = 0 it is obvious to see that u∗ 6= 0. the critical value c∗ can be characterized as c∗ = inf v∈γ ( max t∈[0,1] j(v(t)) ) , 66 aboudramane guiro and idrissa ibrango cubo 19, 3 (2017) where γ = {v ∈ c([0, 1], x) : v(0) = 0, v(1) = u1}. then, the proof of the existence of solution to problem (1.2) for any parameter w ∈ w is complete � now, in what follow, we examine conditions under which our problem (1.2) has a unique non trivial solution. assume that for any k ∈ [0, n], ξ1, ξ2 ∈ r and w ∈ w the followings additional conditions are satisfies: (h7). there exist a constant c8 > 0 such that ( a(k, ξ1) − a(k, ξ2) ) ≥ c8|ξ1 − ξ2| p+, (3.12) (h8). there exist a constant 0 < d < c8 δ̄n(n+1) 3p+−2 2 such that |f(k, ξ1, w) − f(k, ξ2, w)| ≤ d|ξ1 − ξ2| p+−1. (3.13) theorem 3.4. if u ∈ x is a non trivial solution of problem (1.2), if (h7) and (h8) are holds, then for every fixed parameter w ∈ w, the solution u is unique. proof let u, v ∈ x two nonzero solutions of problem (1.2). then n+1∑ k=1 a(k − 1, ∆u(k − 1))∆(u − v)(k − 1) = n∑ k=1 δ(k)f ( k, u(k), w ) (u − v)(k) (3.14) and n+1∑ k=1 a(k − 1, ∆v(k − 1))∆(v − u)(k − 1) = n∑ k=1 δ(k)f ( k, v(k), w ) (v − u)(k). (3.15) upon addition, we get n+1∑ k=1 [ a(k − 1, ∆u(k − 1)) − a(k − 1, ∆v(k − 1)) ] ∆(u − v)(k − 1) = n∑ k=1 δ(k) [ f ( k, u(k), w ) − f ( k, v(k), w ) ] (u − v)(k). (3.16) therefore, according to (h7) and (h8) we have c8 n+1∑ k=1 |∆(u − v)(k − 1)|p + ≤ dδ̄ n∑ k=1 |(u − v)(k)|p + . (3.17) since p+ ≥ 2, using the inequalities (2.16)-(2.18) we can write c8(n + 1) 2−p+ 2 ‖u − v‖p + ≤ dδ̄n(n + 1)p + −1 n+1∑ k=1 |∆(u − v)(k − 1)|p + ≤ dδ̄n(n + 1)p + ‖u − v‖p + , cubo 19, 3 (2017) existence of solutions for discrete boundary value . . . 67 namely [ c8 δ̄n(n + 1) 3p+−2 2 − d ] ‖u − v‖p + ≤ 0. (3.18) recall that the constant d is such that c8 δ̄n(n + 1) 3p+−2 2 − d > 0. consequently, ‖u − v‖p + = 0, thus u = v � references [1] r. p. agarwal, k. perera and d. o’regan; multiple positive solutions of singular and nonsingular discrete problems via variational methods, nonlinear anal. 58 (2004), 69-73. [2] a. cabada, a. iannizzoto and s. tersian; multiple solutions for discrete boundary value problems. j math anal appl. 356 (2009), 418-428. [3] x. cai and j. yu; existence theorems for second-order discrete boundary value problems, j. math. anal. appl. 320 (2006), 649-661. 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[13] g. zhang and s. liu; on a class of semipositone discrete boundary value problem. j math anal appl. 325 (2007), 175-182. introduction preliminaries existence of nontrivial solutions cubo a mathematical journal vol.21, no¯ 01, (21–35). april 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000100021 some new simple inequalities involving exponential, trigonometric and hyperbolic functions yogesh j. bagul1, christophe chesneau2 1department of mathematics, k. k. m. college manwath, parbhani(m.s.) 431505, india yjbagul@gmail.com 2lmno, university of caen normandie, france christophe.chesneau@unicaen.fr abstract the prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x2), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). the bounds obtained for hyperbolic cosine are very sharp. new proofs, refinements as well as new results are offered. some graphical and numerical results illustrate the findings. resumen el objetivo principal de este art́ıculo es establecer cotas inferiores y superiores precisas para funciones útiles tales como las funciones exponenciales, con énfasis especial en exp(−x2), las funciones trigonométricas (coseno y seno) y las funciones hiperbólicas (coseno y seno). las cotas obtenidas para el coseno hiperbólico son muy precisas. se presentan, tanto nuevas demostraciones y refinamientos, como resultados nuevos. algunos resultados numéricos y gráficos ilustran los resultados encontrados. keywords and phrases: exponential function; trigonometric function; hyperbolic function. 2010 ams mathematics subject classification: 26d07, 33b10, 33b20. http://dx.doi.org/10.4067/s0719-06462019000100021 22 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) 1 introduction sharp bounds for useful functions play a central role in many areas of mathematics and theoretical physics. they aim to provide some properties of functions of interest, possibly complex, by dealing with more tractable functions (in the context). the literature on the bounds dealing with the special functions such as e−x 2 , cos(x), sin(x), sinc(x), cosh(x), sinh(x) and tanh(x), is very vast. recent developments can be found in [10, 11, 7, 5, 1, 20, 17, 4, 15, 6, 21, 16, 3, 8, 14, 13, 18, 19] and the references therein. in this paper, we offer new simple tight (lower and upper) bounds involving these functions, with a high potential of interest for many researchers in mathematics or theoretical physics. some proofs of our results are based on the so-called l’hospital’s rule of monotonicity, the others used recent results with a new approach. the sharpness of our bounds are highlighted by some graphics and numerical studies using a global l2 error as benchmark. the result below shows bounds for e−x 2 defined with the cosine function and well-chosen constants. proposition 1.1. for x ∈ (0, π/2), the best possible constants α and β in the following inequalities cos(x) − 1 + α α 6 e−x 2 6 cos(x) − 1 + β β (1.1) are 1/2 and ≈ 1.092663 respectively. the interest of proposition 1.1 is the simplicity of the bounds, with very tractable expressions. it can be useful to evaluate complex functions depending on e−x 2 (gaussian probability density function, error function etc.). the bounds of proposition 1.1 are illustrated in figure 1. we see that the lower bound is sharp for small values for x. 0.0 0.5 1.0 1.5 0 .2 0 .4 0 .6 0 .8 1 .0 exp(− x2) (cos(x) − 1 + α) α (cos(x) − 1 + β) β cubo 21, 1 (2019) some new simple inequalities involving exponential . . . 23 figure 1: graphs of the functions of the bounds (1.1) for x ∈ (0, π/2). note: using exponential and cosine series, proposition 1.1 can be expressed in terms of alternating series as follows. for x ∈ (−π/2, π/2), we have 1 α ∞∑ k=1 (−1)kx2k (2k)! 6 ∞∑ k=1 (−1)kx2k k! 6 1 β ∞∑ k=1 (−1)kx2k (2k)! , where α and β are as defined above. now let us recall that the sinc function is defined by sinc(x) = { sin(x) x x 6= 0, 1 x = 0. (1.2) it is of importance due to it’s frequent occurrence in fourier analysis. so the interest of finding the bounds of this type of functions is increasing. in the next proposition, we give new bounds to sinc function using hyperbolic tangent. proposition 1.2. for x ∈ (0, π/2), we have ( tanh(x) x )δ < sin(x) x < ( tanh(x) x )η (1.3) with the best possible constants δ = 0.839273 and η = 1/2. in the following propositions, the inequalities presented are somewhat cusa-huygen’s type [13, 18]. proposition 1.3 below provides bounds for the sinc function using e−x 2 or hyperbolic cosine. proposition 1.3. for x ∈ (0, π/2), the inequalities ( 2 + e−x 2 3 )a < sin(x) x < ( 2 + e−x 2 3 )b (1.4) and ( 3 2 + cosh(x) )c < sin(x) x < ( 3 2 + cosh(x) )d (1.5) are true with the best possible constants a ≈ 1.240827, b = 1/2, c ≈ 1.108171 and d = 1. 24 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) in view of propositions 1.2 and 1.3, it is natural to address the following question: which bounds for sinc are the best ? we provide the answer by doing a numerical study. we investigate the global l2 error defined by e(u) = ∫π/2 0 ( sin x x − u(x) )2 dx, where u(x) denotes bound (lower or upper) in (1.3), (1.4) and (1.5). the results are summarized in table 1. table 1: global l2 errors e(u) for sinc(x) and the functions u(x) in the bounds of (1.3), (1.4) and (1.5) for x ∈ (0, π/2). inequality (1.3) u(x) lower upper e(u) ≈ 0.001421437 ≈ 0.003648618 inequality (1.4) u(x) lower upper e(u) ≈ 0.006242974 ≈ 0.008628254 inequality (1.5) u(x) lower upper e(u) ≈ 6.53313 × 10−5 ≈ 0.0001542441 it follows from table 1 that the bounds (1.5) are more sharp. this sharpness is illustrated in figure 2. 0.0 0.5 1.0 1.5 0 .7 0 .8 0 .9 1 .0 sin(x) x (3 (2 + cosh(x)))c (3 (2 + cosh(x)))d figure 2: graphs of the functions of the bounds (1.5) for x ∈ (0, π/2). cubo 21, 1 (2019) some new simple inequalities involving exponential . . . 25 the next result provides bounds for x/ sinh(x) using cosine function. proposition 1.4. if x ∈ (0, π/2) then we have ( 2 + cos(x) 3 )m < x sinh(x) < ( 2 + cos(x) 3 )n (1.6) with the constants m ≈ 1.014227 and n ≈ 0.928648. the obtained bounds are illustrated in figure 3. 0.0 0.5 1.0 1.5 0 .7 0 0 .7 5 0 .8 0 0 .8 5 0 .9 0 0 .9 5 1 .0 0 x sinh(x) ((2 + cos(x)) 3)m ((2 + cos(x)) 3)n figure 3: graphs of the functions of the bounds (1.6) for x ∈ (0, π/2). note: the inequality 2 + cos(x) 3 < x sinh(x) is more sharp version of left inequality of (1.6). it is appeared in [19, theorem 6]. proposition 1.5 below presents sharp bounds for sinh(x)/x using hyperbolic cosine. proposition 1.5. for x ∈ (0, π/2) one has ( 2 + cosh(x) 3 )p < sinh(x) x < ( 2 + cosh(x) 3 )q (1.7) with the constants p ≈ 0.928648 and q ≈ 1.009155. the bounds are illustrated in figure 4. 26 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) 0.0 0.5 1.0 1.5 1 .0 1 .1 1 .2 1 .3 1 .4 sinh(x) x ((2 + cosh(x)) 3)p ((2 + cosh(x)) 3)q figure 4: graphs of the functions of the bounds (1.7) for x ∈ (0, π/2). note: the hyperbolic cusa-huygen’s inequality[16] sinh(x) x < 2 + cosh(x) 3 is however more sharp than right inequality of (1.7). the rest of the study is devoted to new bounds for cosh(x), with discussion. a well-known upper bound for cosh(x) is given by ex 2/2. this result was recently completed by yogesh bagul[3, theorem 2.1] who finds a sharp lower bound, i.e. eax 2 < cosh(x) < ex 2/2, x ∈ (0, 1), (1.8) with the best possible constants a ≈ 0.433781 and 1/2. we now aim to refine the inequalities of (1.8) in proposition 1.6 below. proposition 1.6. for x ∈ (0, 1), we have exp ( 3 2 ( 1 − e−x 2/3 ) ) 6 cosh(x) 6 exp ( 1 2θ ( 1 − e−θx 2 ) ) (1.9) with θ ≈ 0.272342. note: using the well-known inequality ey > 1+y for y ∈ r, we obtain exp (( 1 − e−θx 2 ) /(2θ) ) 6 ex 2/2. this proves that the upper bound in (1.9) is sharper to the one in (1.8). alternative bounds are given in proposition 1.7 below, with discussion. cubo 21, 1 (2019) some new simple inequalities involving exponential . . . 27 proposition 1.7. for x ∈ (0, 1), we have ( 1 + x2 3 )3/2 6 cosh(x) 6 ( 1 + x2 ξ )ξ/2 (1.10) with ξ ≈ 3.194528. note: again, using the well-known inequality ey > 1 + y for y ∈ r, we get ( 1 + x2/ξ )ξ/2 6 ex 2/2. this shows that the upper bound in (1.10) is sharper to the one in (1.8). we now claim that the bounds obtained in (1.10) are better than those in (1.8) and (1.9). numerical results support this claim. indeed, by considering the global l2 error defined by e∗(u) = ∫1 0 (cosh(x) − u(x)) 2 dx, where u(x) denotes bound (lower or upper) in (1.8), (1.9) and (1.10), table 1 indicates that (1.10) are the best. table 2: global l2 errors e∗(u) for cosh(x) and the functions u(x) in the bounds of (1.8), (1.9) and (1.10) for x ∈ (0, 1). inequality (1.8) u(x) lower upper e∗(u) ≈ 0.0001352084 ≈ 0.001139289 inequality (1.9) u(x) lower upper e∗(u) ≈ 1.335929 × 10 −5 ≈ 7.004029 × 10−6 inequality (1.10) u(x) lower upper e∗(u) ≈ 9.456552 × 10 −7 ≈ 6.895902 × 10−7 the sharpness of the obtained bounds is illustrated in figures 5 and 6 (for a zoom on the interval (0.95, 1), where the hierarchy of the bounds is more clear). 28 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) 0.0 0.2 0.4 0.6 0.8 1.0 1 .0 1 .1 1 .2 1 .3 1 .4 1 .5 cosh(x) (1 + x2 3)(3 2) (1 + x2 ξ)(ξ 2) figura 5: graphs of the functions of the bounds (1.10) for x ∈ (0, 1). 0.95 0.96 0.97 0.98 0.99 1.00 1 .4 9 1 .5 0 1 .5 1 1 .5 2 1 .5 3 1 .5 4 cosh(x) (1 + x2 3)(3 2) (1 + x2 ξ)(ξ 2) figura 6: graphs of the functions of the bounds (1.10) for x ∈ (0.95, 1). note: to prove the inequalities (1.5), (1.6) and (1.7), we will simply use the results of [7, 5, 12]. we stress on the fact that it is not difficult to verify that all the results in [5] are also true in (0, π/2) with the respective best possible constants obtained accordingly (see [12]). propositions 1.6 and 1.7 will be proved by the techniques of integration on some known results[4, 6]. for proving proposition 1.1, proposition 1.2 and proposition 1.3, we need the lemmas presented in the next section. cubo 21, 1 (2019) some new simple inequalities involving exponential . . . 29 2 lemmas the following lemma is known as l’hospital’s rule of monotonicity. the details are given in [9] and [2]. lemma 2.1. ([2]) let f, g be two real valued functions which are continuous on [a, b] and differentiable on (a, b), where −∞ < a < b < ∞ and g′(x) 6= 0, for ∀x ∈ (a, b). let, a(x) = f(x) − f(a) g(x) − g(a) and b(x) = f(x) − f(b) g(x) − g(b) . then, i) a(x) and b(x) are increasing on (a, b) if f′/g′ is increasing on (a, b) and ii) a(x) and b(x) are decreasing on (a, b) if f′/g′ is decreasing on (a, b). the strictness of the monotonicity of a(x) and b(x) depends on the strictness of monotonicity of f′/g′. lemma 2.2. h(x) = sin(x)−x cos(x) x2 sin(x) is strictly positive increasing in (0, π/2). proof: h(x) is positive as cos(x) < sin(x) x on (0, π/2). consider, h(x) = sin(x) − x cos(x) x2 sin(x) = h1(x) h2(x) , where h1(x) = sin(x) − x cos(x) and h2(x) = x 2 sin(x) are such that h1(0) = 0 and h2(0) = 0. by differentiating h′1(x) h′2(x) = sin(x) x cos(x) + 2 sin(x) = h3(x) h4(x) , where h3(x) = sin(x) and h4(x) = x cos(x) + 2 sin(x) with h3(0) = 0 and h4(0) = 0. again differentiating we get h′3(x) h′4(x) = cos(x) −x sin(x) + 3 cos(x) = 1 −x tan(x) + 3 . now, it is well known that −x tan(x) is decreasing in (0, π/2) and so is −x tan(x) + 3. by lemma 1, h(x) is a strictly increasing function in (0, π/2). 3 proofs of the main results this section is devoted to the proofs of our main results. 30 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) proof of proposition 1.1: clearly, the equalities hold at x = 0. consider f(x) = cos(x) − 1 e−x 2 − 1 = f1(x) f2(x) , where f1(x) = cos(x) − 1 and f2(x) = e −x2 − 1 with f1(0) = 0 and f2(0) = 0. by differentiation, we obtain f′1(x) f′2(x) = sin(x)ex 2 2x = f3(x) f4(x) , where f3(x) = sin(x)e x2 and f4(x) = 2x with f3(0) = 0 and f4(0) = 0. again differentiating we get f′3(x) f′4(x) = ex 2 2 [cos(x) + 2x sin(x)] = ex 2 2 f(x), where f(x) = cos(x) + 2x sin(x). differentiation gives f′(x) = 2x cos(x) + sin(x) > 0 in (0, π/2), which implies that f(x) is increasing. thus f′ 3 (x) f′ 4 (x) being a product of two positive increasing functions is a positive increasing. by lemma 2.1, f(x) is also increasing in (0, π/2). so α = f(0+) = 1/2 and β = f(π/2−) = −1/[e−(π/2) 2 − 1] ≈ 1.092663. proof of proposition 1.2: let us set h(x) = log(sin(x)/x) log(tanh(x)/x) = h1(x) h2(x) , where h1(x) = log(sin(x)/x) and h2(x) = log(tanh(x)/x) with h1(0+) = 0 and h2(0+) = 0. differentiating we get h′1(x) h′2(x) = sin(x) − x cos(x) x2 sin(x) x2 tanh(x) tanh(x) − x sech2(x) = h(x) j(x), where h(x) = sin(x)−x cos(x) x2 sin(x) and j(x) = x2 tanh(x) tanh(x)−x sech2(x) . now set j(x) = j1(x) j2(x) , where j1(x) = x 2 tanh(x) and j2(x) = tanh(x) − x sech 2 (x) with j1(0) = 0 and j2(0) = 0. differentiation gives j′1(x) j′2(x) = x sech2(x) + 2 tanh(x) 2 sech2(x) tanh(x) = 1 2 x tanh(x) + cosh2(x), cubo 21, 1 (2019) some new simple inequalities involving exponential . . . 31 which is clearly increasing as both x/ tanh(x) and cosh2(x) are increasing. by lemma 2.1, j(x) is also increasing in (0, π/2). moreover, j(x) is positive as x/ sinh(x) < cosh(x). by lemma 2.2, h(x) is strictly positive increasing in (0, π/2). h′1(x)/h ′ 2(x), being product of two positive increasing functions is positive increasing. again by lemma 2.1, h(x) is strictly increasing in (0, π/2). so δ = log(2/π)/ log(2 tanh(π/2)/π) ≈ 0.839273 and η = f(0+) = 1/2, by l’hospital’s rule. this completes the assertion. proof of proposition 1.3: • proof of (1.4). let f(x) = log (sin(x)/x) log ( 2 + e−x 2 ) − log 3 = f1(x) f2(x) , where f1(x) = log (sin(x)/x) and f2(x) = log ( 2 + e−x 2 ) − log 3 such that f1(0+) = 0 and f2(0) = 0. differentiation gives f′1(x) f′2(x) = 1 2 (sin(x) − x cos(x)) x2 sin(x) (2ex 2 + 1) = 1 2 h(x) g(x), where h(x) = sin(x)−x cos(x) x2 sin(x) is strictly positive increasing in (0, π/2) by lemma 2.2 and g(x) = 2ex 2 + 1 is also clearly positive increasing. therefore h(x) g(x) is strictly increasing. by making use of lemma 2.1, we conclude that f(x) is strictly increasing in (0, π/2). so f(0+) < f(x) < f(π/2); x ∈ (0, π/2). hence, a = f(π/2) = log(2/π)/[log(2 + e−(π/2) 2 ) − log 3] ≈ 1.240827 and b = f(0+) = 1/2 by l’hospital’s rule. • proof of (1.5). utilizing [5, theorem 2], [12, proposition 3] we have e−kx 2 < sin(x) x < e−x 2/6, where k = − log(2/π) (π/2)2 . after rearrangement, it can be written as ( sin(x) x )6 < e−x 2 < ( sin(x) x )1/k . (3.1) by virtue of [7, theorem 2] we write ( 3 2 + cosh(x) )γ < e−x 2 < ( 3 2 + cosh(x) )6 , (3.2) 32 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) where γ = (π/2)2 log[(2+cosh(π/2))/3] . combining (3.1) and (3.2), we get ( 3 2 + cosh(x) )c < sin(x) x < ( 3 2 + cosh(x) ) , where c = kγ = − log(2/π) log[(2+cosh(π/2))/3] ≈ 1.108171. proof of proposition 1.4: according to [5, theorem 3] and [12] we have e−x 2/6 < x sinh(x) < e−tx 2 , x ∈ (0, π/2), where t = − log[π/(2 sinh(π/2))] (π/2)2 . it is equivalent to ( x sinh(x) )1/t < e−x 2 < ( x sinh(x) )6 . (3.3) similarly, using [7, theorem 1] we have ( 2 + cos(x) 3 )λ < e−x 2 < ( 2 + cos(x) 3 )6 , (3.4) where λ = −(π/2)2 log(2/3) . combining (3.3) and (3.4) we get ( 2 + cos(x) 3 )m < x sinh(x) < ( 2 + cos(x) 3 )n , where m = λ 6 = −(π/2)2 6 log(2/3) ≈ 1.014227 and n = 6t = −6 log[π/(2 sinh(π/2))] (π/2)2 ≈ 0.928648. proof of proposition 1.5: the proof follows easily by combining inequalities (3.2) and (3.3) to get p = −6 log[π/(2 sinh(π/2))] (π/2)2 ≈ 0.928648 and q = (π/2)2 6 log[(2+cosh(π/2))/3] ≈ 1.009155. proof of proposition 1.6: for x = 0 equalities hold obviously. rearranging [4, theorem 5], for any t ∈ (0, 1), we have te−t 2/3 < tanh(t) < te−θt 2 with θ ≈ 0.272342. therefore by integration, for x ∈ (0, 1), we get ∫x 0 te−t 2/3dt < ∫x 0 tanh(t)dt < ∫x 0 te−θt 2 dt, which yields 3 2 ( 1 − e−x 2/3 ) < log(cosh(x)) < 1 2θ ( 1 − e−θx 2 ) . cubo 21, 1 (2019) some new simple inequalities involving exponential . . . 33 by composing with the exponential function, we get the required result. proof of proposition 1.7: clearly, the equalities hold at x = 0. rearranging [6, theorem 4], for any t ∈ (0, 1), we have 3t 3 + t2 < tanh(t) < ξt ξ + t2 with ξ ≈ 3.194528. on integration, for x ∈ (0, 1), we have ∫x 0 3t 3 + t2 dt < ∫x 0 tanh(t)dt < ∫x 0 ξt ξ + t2 dt which implies that 3 2 log ( 1 + x2 3 ) < log(cosh(x)) < ξ 2 log ( 1 + x2 ξ ) . the desired result follows by composing with the exponential function. acknowledgments: we would like to thank the referee for the thorough comments which have helped the presentation of the paper. 34 yogesh j. bagul and christophe chesneau cubo 21, 1 (2019) references [1] h. alzer and m. k. kwong, on jordan’s inequality, period math hung, volume 77, number 2, pp. 191-200, 2018, doi: 10.1007/s10998-017-0230-z. 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[online]. available: https://doi.org/10.1016/j.camwa.2009.07.076 introduction lemmas proofs of the main results cubo a mathematical journal vol.20, no¯ 02, (23–39). june 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000200023 new approach to prove the existence of classical solutions for a class of nonlinear parabolic equations svetlin g. georgiev 1 and khaled zennir 2 1sorbonne university, paris, france, university of sofia, faculty of mathematics and informatics, department of differential equations. 2department of mathematics, college of sciences and arts, al-ras. qassim university, kingdom of saudi arabia. svetlingeorgiev1@gmail.com, sgg2000bg@yahoo.com, khaledzennir2@yahoo.com abstract in this article, we consider a class of nonlinear parabolic equations. we use an integral representation combined with a sort of fixed point theorem to prove the existence of classical solutions for the initial value problem (1.1), (1.2). we also obtain a result on continuous dependence on the initial data. we propose a new approach for investigation for existence of classical solutions of some classes nonlinear parabolic equations. resumen en este art́ıculo, consideramos una clase de ecuaciones parabólicas nolineales. usamos una representación integral combinada con una especie de teorema de punto fijo para probar la existencia de soluciones clásicas para el problema de valor inicial (1.1), (1.2). también obtenemos un resultado sobre la dependencia continua de la data inicial. proponemos una estrategia nueva para la investigación de la existencia de soluciones clásicas de algunas clases de ecuaciones parabólicas nolineales. keywords and phrases: parabolic equation, existence, differentiability with respect to the initial data 2010 ams mathematics subject classification: 35k55, 35k45. http://dx.doi.org/10.4067/s0719-06462018000200023 24 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) 1 introduction here, we consider the cauchy problem ut − uxx = f(t, x, u, ux) in (0, ∞) × r, (1.1) u(0, x) = φ(x) in r, (1.2) where φ ∈ c2(r), f : [0, ∞) × r × r × r 7−→ c is a given continuous function, u : [0, ∞) × r 7−→ c is the main unknown. our main results are as follows. theorem 1.1. let f ∈ c([0, ∞) × r × r × r), φ ∈ c2(r). then there exists m ∈ (0, 1) such that the problem (1.1), (1.2) has a solution u ∈ c1([0, m], c2([0, 1])). theorem 1.2. let f ∈ c([0, ∞) × r × r × r), φ ∈ c2(r). then there exists m ∈ (0, 1) such that the problem (1.1), (1.2) has a solution u ∈ c1([0, m], c2(r)). for o1, o2 ⊂ r with c 1(o1, c 2(o2)) we denote the space of all continuous functions u on o1 ×o2 such that ut, ux and uxx exist and are continuous on o1 × o2. example 1.3. let p > 1 and a ∈ c be chosen so that ap−1 = − 1 p − 1 . consider the cauchy problem ut − uxx = u p in (0, ∞) × r u(0, x) = a in r. then u(t, x) = a(t + 1)− 1 p−1 is its solution. actually, ut(t, x) = − a p − 1 (t + 1) − p p−1 , and uxx(t, x) = 0, and (u(t, x))p = − a p − 1 (t + 1) − p p−1 . therefore ut(t, x) − uxx(t, x) = (u(t, x)) p in (0, ∞) × r and u(0, x) = a in r. cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 25 to prove our main result we propose new integral representation of the solutions of the initial value problem (1.1), (1.2). many works have been devoted to the investigation of initial value problems for parabolic equations and systems (see, for example, [13]-[16] and the references therein). we note that in the references the ivp (1.1), (1.2) is connected with the dimension n, fujita exponent, sobolev critical exponents, bounded and unbounded domain. in this article we propose new idea which tell us that the local existence of classical solutions of the ivp is connected with the integral representation of the solutions, it is not connected with the dimension n and if the domain is bounded or not. as an application of our new integral representation we deduce some results connected with the continuous dependence on the initial data and parameters of the problem (1.1), (1.2). theorem 1.4. let f ∈ c([0, ∞)×r×r×r), ∂f ∂u , ∂f ∂ux exist and are continuous in [0, ∞)×r×r×r, φ ∈ c2(r). let also, u(t, x, φ) ∈ c1([0, m], c2([c, d])) be a solution to the problem (1.1), (1.2) for some m ∈ (0, 1) and for some [c, d] ⊂ r. then u(t, x, φ) is differentiable with respect to φ and v(t, x) = ∂u ∂φ (t, x, φ) satisfies the following initial value problem vt − vxx = ∂f ∂u (t, x, u(t, x, φ), ux(t, x, φ))v + ∂f ∂ux (t, x, u(t, x, φ), ux(t, x, φ))vx in [0, m] × [c, d], (1.3) v(0, x) = 1 in [c, d]. (1.4) 2 auxiliary results we will start with the following useful lemma. lemma 2.1. let f ∈ c([a, b]×[c, d]×r×r), g ∈ c2([c, d]). then the function u ∈ c1([a, b], c2([c, d])) is a solution to the problem ut − uxx = f(t, x, u, ux) in (a, b] × [c, d], (2.1) u(a, x) = g(x) in [c, d], (2.2) if and only if it is a solution to the integral equation ∫x c ∫y c (u(t, z) − g(z)) dzdy − ∫t a (u(τ, x) − u(τ, c) − (x − c)ux(τ, c)) dτ = ∫t a ∫x c ∫y c f(τ, z, u(τ, z), ux(τ, z))dzdydτ, x ∈ [c, d], t ∈ [a, b]. (2.3) proof. (1) let u ∈ c1([a, b], c2([c, d])) is a solution to the problem (2.1), (2.2). we integrate the equation (2.1) with respect to x and we get ∫x c ut(t, z)dz − ∫x c uxx(t, z)dz = ∫x c f(t, z, u(t, z), ux(t, z))dz, x ∈ [c, d], t ∈ [a, b], 26 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) or ∫x c ut(t, z)dz − ux(t, x) + ux(t, c) = ∫x c f(t, z, u(t, z), ux(t, z))dz, x ∈ [c, d], t ∈ [a, b]. now we integrate the last equation with respect to x and we find ∫x c ∫y c ut(t, z)dzdy − ∫x c (ux(t, z) − ux(t, c)) dz = ∫x c ∫y c f(t, z, u(t, z), ux(t, z))dzdy, x ∈ [c, d], t ∈ [a, b], or ∫x c ∫y c ut(t, z)dzdy − u(t, x) + u(t, c) + (x − c)ux(t, c) = ∫x c ∫y c f(t, z, u(t, z), ux(t, z))dzdy, x ∈ [c, d], t ∈ [a, b]. we integrate the last equality with respect to t and we obtain ∫t a ∫x c ∫y c ut(s, z)dzdyds − ∫t a (u(s, x) − u(s, c) − (x − c)ux(s, c)) ds = ∫t a ∫x c ∫y c f(s, z, u(s, z), ux(s, z))dzdyds, x ∈ [c, d], t ∈ [a, b], or ∫x c ∫y c (u(t, z) − g(z)) dzdy − ∫t a (u(s, x) − u(s, c) − (x − c)ux(s, c)) ds = ∫t a ∫x c ∫y c f(s, z, u(s, z), ux(s, z))dzdyds, x ∈ [c, d], t ∈ [a, b], i.e., u satisfies the equation (2.3). (2) let u ∈ c1([a, b], c2([c, d])) be a solution to the integral equation (2.3). we differentiate the equation (2.3) with respect to x and we get ∫x c (u(t, z) − g(z)) dz − ∫t a (ux(s, x) − ux(s, c)) ds = ∫t a ∫x c f(s, z, u(s, z), ux(s, z))dzds, x ∈ [c, d], t ∈ [a, b]. again we differentiate with respect to x and we find u(t, x) − g(x) − ∫t a uxx(s, x)ds = ∫t a f(s, x, u(s, x), ux(s, x))ds, x ∈ [c, d], t ∈ [a, b]. (2.4) now we put t = a in the last equation and we find u(a, x) = g(x), x ∈ [c, d], cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 27 i.e., the function u satisfies (2.2). now we differentiate the equation (2.4) with respect to t and we find ut(t, x) − uxx(t, x) = f(t, x, u(t, x), ux(t, x)), x ∈ [c, d], t ∈ [a, b]. the proof of the existence results are based on the following theorem. theorem 2.2 ([14]). let x be a nonempty closed convex subset of a banach space y. suppose that t and s map x into y such that (1) s is continuous and s(x) contained in a compact subset of y. (2) t : x 7−→ y is expansive and onto. then there exists a point x∗ ∈ x such that sx∗ + tx∗ = x∗. definition 2.3. let (x, d) be a metric space and m be a subset of x. the mapping t : m 7−→ x is said to be expansive if there exists a constant h > 1 such that d(tx, ty) ≥ hd(x, y) for any x, y ∈ m. 3 proof of theorem 1.1 let b > ‖φ‖c2([0,1]) be arbitrarily chosen. since φ ∈ c([0, 1]), f ∈ c([0, 1]×[0, 1]×[−b, b]×[−b, b]) we have that there exists a constant m11 > 0 such that |φ(x)| ≤ m11 in [0, 1], |f(t, x, y, z)| ≤ m11 in [0, 1] × [0, 1] × [−b, b] × [−b, b]. we take l, m ∈ (0, 1) so that lb + l(b + m11) + 3lbm + lm11m ≤ b l(5b + 2m11) ≤ b. (3.1) let e11 = c 1([0, m], c2([0, 1])) be endowed with the norm ||u|| = max { max (t,x)∈[0,m]×[0,1] |u(t, x)|, max (t,x)∈[0,m]×[0,1] |ut(t, x)|, max (t,x)∈[0,m]×[0,1] |ux(t, x)|, max (t,x)∈[0,m]×[0,1] |uxx(t, x)| } . 28 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) by k̃11 we denote the set of all equi-continuous families in e11, i.e., for every ǫ > 0 there exists δ = δ(ǫ) > 0 such that |u(t1, x1) − u(t2, x2)| < ǫ, |ut(t1, x1) − ut(t2, x2)| < ǫ, |ux(t1, x1) − ux(t2, x2)| < ǫ, |uxx(t1, x1) − uxx(t2, x2)| < ǫ whenever |t1 − t2| < δ, |x1 − x2| < δ. let also, k′11 = k̃11, k11 = {u ∈ k ′ 11 : ||u|| ≤ b} and l11 = {u ∈ k ′ 11 : ||u|| ≤ (1 + l)b}. we note that k11 is a closed convex subset of l11. for u ∈ l11 we define the operators t11(u)(t, x) = (1 + l)u(t, x), s11(u)(t, x) = −lu(t, x) + l ∫x 0 ∫y 0 (u(t, z) − φ(z))dzdy −l ∫t 0 (u(τ, x) − u(τ, 0) − xux(τ, 0))dτ −l ∫t 0 ∫x 0 ∫y 0 f(τ, z, u(τ, z), ux(τ, z))dzdydτ. we will prove that the problem ut − uxx = f(t, x, ux) in [0, m] × [0, 1], (3.2) u(0, x) = φ(x) in [0, 1], (3.3) has a solution u ∈ c1([0, m], c2([0, 1])). a)s11 : k11 7−→ k11. let u ∈ k11. then s11(u) ∈ c 1([0, m], c2([0, 1])) and for (t, x) ∈ [0, m] × cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 29 [0, 1], using the first inequality of (3.1), we get |s11(u)(t, x)| = ∣ ∣ ∣ −lu(t, x) + l ∫x 0 ∫y 0 (u(t, z) − φ(z))dzdy −l ∫t 0 (u(τ, x) − u(τ, 0) − xux(τ, 0))dτ −l ∫t 0 ∫x 0 ∫y 0 f(τ, z, u(τ, z), ux(τ, z))dzdydτ ∣ ∣ ∣ ≤ l|u(t, x)| + l ∫x 0 ∫y 0 (|u(t, z)| + |φ(z)|) dzdy +l ∫t 0 (|u(τ, x)| + |u(τ, 0)| + x|ux(τ, 0)|) dτ +l ∫t 0 ∫x 0 ∫y 0 |f(τ, z, u(τ, z), ux(τ, z))|dzdydτ ≤ lb + l(b + m11) + 3lbm + lm11m ≤ b. note that s11(u)t(t, x) = −lut(t, x) + l ∫x 0 ∫y 0 ut(t, z)dzdy −l(u(t, x) − u(t, 0) − xux(t, 0)) −l ∫x 0 ∫y 0 f(t, z, u(t, z), ux(t, z))dzdy, (t, x) ∈ [0, m] × [0, 1]. 30 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) then, using the second inequality of (3.1), we obtain |s11(u)t(t, x)| = ∣ ∣ ∣ −lut(t, x) + l ∫x 0 ∫y 0 ut(t, z)dzdy −l(u(t, x) − u(t, 0) − xux(t, 0)) −l ∫x 0 ∫y 0 f(t, z, u(t, z), ux(t, z))dzdy ∣ ∣ ∣ ≤ l|ut(t, x)| + l ∫x 0 ∫y 0 |ut(t, z)|dzdy +l (|u(t, x)| + |u(t, 0)| + x|ux(t, 0)|) +l ∫x 0 ∫y 0 |f(t, z, u(t, z), ux(t, z))|dzdy ≤ lb + lb + 3lb + lm11 = l(5b + m11) ≤ b, (t, x) ∈ [0, m] × [0, 1]. also, s11(u)x(t, x) = −lux(t, x) + l ∫x 0 (u(t, z) − φ(z))dz −l ∫t 0 (ux(τ, x) − ux(τ, 0))dτ −l ∫t 0 ∫x 0 f(τ, z, u(τ, z), ux(τ, z))dzdτ, (t, x) ∈ [0, m] × [0, 1]. cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 31 hence, using the first inequality of (3.1), |s11(u)x(t, x)| = ∣ ∣ ∣ −lux(t, x) + l ∫x 0 (u(t, z) − φ(z))dz −l ∫t 0 (ux(τ, x) − ux(τ, 0))dτ −l ∫t 0 ∫x 0 f(τ, z, u(τ, z), ux(τ, z))dzdτ ∣ ∣ ∣ ≤ l|ux(t, x)| + l ∫x 0 (|u(t, z)| + |φ(z)|) dz +l ∫t 0 (|ux(τ, x)| + |ux(τ, 0)|) dτ +l ∫t 0 ∫x 0 |f(τ, z, u(τ, z), ux(τ, z))|dzdτ ≤ lb + l(b + m11) + 2lbm + lm11m ≤ b, (t, x) ∈ [0, m] × [0, 1]. for (t, x) ∈ [0, m] × [0, 1] we have s11(u)xx(t, x) = −luxx(t, x) + l(u(t, x) − φ(x)) −l ∫t 0 uxx(τ, x)dτ −l ∫t 0 f(τ, x, u(τ, x), ux(τ, x))dτ, 32 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) from where, using the first inequality of (3.1), |s11(u)xx(t, x)| = ∣ ∣ ∣ −luxx(t, x) + l(u(t, x) − φ(x)) −l ∫t 0 uxx(τ, x)dτ −l ∫t 0 f(τ, x, u(τ, x), ux(τ, x))dτ ∣ ∣ ∣ ≤ l|uxx(t, x)| + l (|u(t, x)| + |φ(x)|) +l ∫t 0 |uxx(τ, x)|dτ +l ∫t 0 |f(τ, x, u(τ, x), ux(τ, x))|dτ ≤ lb + l(b + m11) + lbm + lm11m ≤ b. we note that {s11(u) : u ∈ k11} is an equi-continuous family in e11. consequently s11 : k11 7−→ k11. also, s11(k11) ⊂ k11 ⊂ l11, i.e., s11(k11) resides in a compact subset of l11. b) s11 : k11 7−→ k11 is a continuous operator. we note that if {un} ∞ n=1 be a sequence of elements of k11 such that un −→ u in k11 as n −→ ∞, then s11(un) −→ s11(u) in k11 as n −→ ∞. therefore s11 : k11 7−→ k11 is a continuous operator. c) t11 : k11 7−→ l11 is an expansive operator and onto. for u, v ∈ k11 we have that ||t11(u) − t11(v)|| = (1 + l)||u − v||, i.e., t11 : k11 7−→ l11 is an expansive operator with constant 1 + l. let v ∈ l11. then v 1+l ∈ k11 and t11 ( v 1 + l ) = v, i.e., t11 : k11 7−→ l11 is onto. from a), b), c) and from theorem 2.2, it follows that there is u11 ∈ k11 such that t11u11 + s11u11 = u11 cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 33 or (1 + l)u11(t, x) − lu11(t, x) + l ∫x 0 ∫y 0 (u11(t, z) − φ(z))dzdy −l ∫t 0 (u11(τ, x) − u11(τ, 0) − xu11x(τ, 0))dτ −l ∫t 0 ∫x 0 ∫y 0 f(τ, z, u11(τ, z), u11x(τ, z))dzdydτ = u11(t, x), or ∫x 0 ∫y 0 (u11(t, z) − φ(z))dzdy − ∫t 0 (u11(τ, x) − u11(τ, 0) − xu11x(τ, 0))dτ − ∫t 0 ∫x 0 ∫y 0 f(τ, z, u11(τ, z), u11x(τ, z))dzdydτ = 0, (t, x) ∈ [0, m] × [0, 1], whereupon, using lemma 2.1, we conclude that u11 ∈ c 1([0, 1], c2([0, 1])) is a solution to the problem (3.2), (3.3). 4 proof of theorem 1.2 now we consider the problem ut − uxx = f(t, x, u(t, x), ux(t, x)) in (0, m] × [1, 2], (4.1) u(0, x) = φ(x) in [1, 2]. (4.2) let e12 = c 1([0, m], c2([1, 2])) be endowed with the norm ||u|| = max { max (t,x)∈[0,m]×[1,2] |u(t, x)|, max (t,x)∈[0,m]×[1,2] |ut(t, x)|, max (t,x)∈[0,m]×[1,2] |ux(t, x)|, max (t,x)∈[0,m]×[1,2] |uxx(t, x)| } . by k̃12 we denote the set of all equi-continuous families in e12. let k′12 = k̃12, k12 = {u ∈ k ′ 12 : ||u|| ≤ b}. 34 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) since φ ∈ c([1, 2]), f ∈ c([0, m] × [1, 2] × [−b, b] × [−b, b]) we have that there exists a constant m12 > 0 such that |φ(x)| ≤ m12 in [1, 2], |f(t, x, y, z)| ≤ m12 in [0, m] × [1, 2] × [−b, b] × [−b, b]. let l1 > 0 be chosen so that l1(5b + 2m12) ≤ b l1b + l1(b + m12) + 3l1bm + l1m12m ≤ b let also, l12 = {u ∈ k ′ 12 : ||u|| ≤ (1 + l1)b}. we note that k12 is a closed convex subset of l12. for u ∈ l12 we define the operators t12(u)(t, x) = (1 + l1)u(t, x), s12(u)(t, x) = −l1u(t, x) + l1 ∫x 1 ∫y 1 (u(t, z) − φ(z))dzdy −l1 ∫t 0 (u(τ, x) − u11(τ, 1) − (x − 1)u11x(τ, 1))dτ −l1 ∫t 0 ∫x 1 ∫y 1 f(τ, z, u(τ, z), ux(τ, z))dzdydτ. as in the previous section one can prove that there is u12 ∈ c 1([0, 1], c2([1, 2])) which is a solution to the problem (4.1), (4.2). this solution u12 satisfies the integral equation ∫x 1 ∫y 1 (u12(t, z) − φ(z))dzdy − ∫t 0 (u12(τ, x) − u11(τ, 1) − (x − 1)u11x(τ, 1))dτ − ∫t 0 ∫x 1 ∫y 1 f(τ, z, u12(τ, z), u12x(τ, z))dzdydτ = 0, (t, x) ∈ [0, m] × [1, 2]. (4.3) now we put x = 1 in (4.3) and we find ∫t 0 (u12(τ, 1) − u11(τ, 1))dτ = 0, cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 35 which we differentiate with respect to t and we get u12(t, 1) = u11(t, 1) in [0, m]. (4.4) now we differentiate (4.3) with respect to x and we find ∫x 1 (u12(t, z) − φ(z))dz − ∫t 0 (u12x(τ, x) − u11x(τ, 1))dτ − ∫t 0 ∫x 1 f(τ, z, u12(τ, z), u12x(τ, z))dzdτ = 0, (t, x) ∈ [0, m] × [1, 2]. in the last equation we put x = 1 and we become ∫t 0 (u12x(τ, x) − u11x(τ, 1))dτ = 0, (t, x) ∈ [0, m] × [1, 2], which we differentiate with respect to t and we find u12x(t, 1) = u11x(t, 1) in [0, m]. (4.5) now we differentiate (4.4) with respect to t and we get u12t(t, 1) = u11t(t, 1) in [0, m]. hence, (4.4), (4.5) and f(t, 1, u11(t, 1), u11x(t, 1)) = f(t, 1, u12(t, 1), u12x(t, 1)), we find u12xx(t, 1) = u12t(t, 1) − f(t, 1, u12(t, 1), u12x(t, 1)) = u11t(t, 1) − f(t, 1, u11(t, 1), u11x(t, 1)) = u11xx(t, 1) in [0, m]. consequently the function u(t, x) =    u11(t, x) in [0, m] × [0, 1] u12(t, x) in [0, m] × [1, 2], is a c1([0, m], c2([0, 2]))-solution to the problem ut − uxx = f(t, x, u(t, x), ux(t, x)) in (0, m] × [0, 2], u(0, x) = φ(x) in [0, 2]. 36 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) then we consider the problem ut − uxx = f(t, x, u(t, x), ux(t, x)) in (0, m] × [2, 3] u(0, x) = φ(x) in [2, 3]. (4.6) as in above there is u13 ∈ c 1([0, m], c2([2, 3])) which is a solution to the problem (4.6) and satisfies the integral equation ∫x 2 ∫y 2 (u13(t, z) − φ(z))dzdy − ∫t 0 (u13(τ, x) − u12(τ, 2) − (x − 2)u12x(τ, 2))dτ − ∫t 0 ∫x 2 ∫y 2 f(τ, z, u13(τ, z), u13x(τ, z))dzdydτ = 0, t ∈ [0, m], x ∈ [2, 3]. the function u(t, x) =    u11(t, x) in [0, m] × [0, 1] u12(t, x) in [0, m] × [1, 2] u13(t, x) in [0, m] × [2, 3] is a c1([0, m], c2([0, 3]))-solution to the problem ut − uxx = f(t, x, u(t, x), ux(t, x)) in [0, m] × [0, 3], u(0, x) = φ(x) in [0, 3]. an so on. we construct a solution u1 ∈ c 1([0, m], c2(r)) which is a solution to the problem ut − uxx = f(t, x, u(t, x), ux(t, x)) in (0, m] × r, u(0, x) = φ(x) in r. cubo 20, 2 (2018) new approach to prove the existence of classical solutions . . . 37 5 proof of theorem 1.4 we have that the solution u(t, x, φ) satisfies the following integral equation q(φ) = ∫x c ∫y c (u(t, z, φ(z)) − φ(z))dzdy − ∫t 0 (u(τ, x, φ(x)) − u(τ, c, φ(c)) − (x − c)ux(τ, c, φ(c)))dτ − ∫t 0 ∫x c ∫y c f(τ, z, u(τ, z, φ(z)), ux(τ, z, φ(z)))dz = 0, t ∈ [0, m], x ∈ [c, d]. then q(φ) − q(φ1) = ∫x c ∫y c (u(t, z, φ(z)) − u(t, z, φ1(z)) − (φ(z) − φ1(z)))dzdy − ∫t 0 (u(τ, x, φ(x)) − u(τ, x, φ1(x)))dτ + ∫t 0 (u(τ, c, φ(c)) − u(τ, c, φ1(c)))dτ + ∫t 0 (x − c)(ux(τ, c, φ(c)) − ux(τ, c, φ1(c)))dτ − ∫t 0 ∫x c ∫y c ( f(τ, z, u(τ, z, φ(z)), ux(τ, z, φ(z))) −f(τ, z, u(τ, z, φ1(z)), ux(τ, z, φ1(z))) ) dzdydτ = ∫x c ∫y c ( ∂u ∂φ (t, z, φ(z)) − 1 ) dzdy − ∫t 0 ∂u ∂φ (τ, x, φ(x))dτ + ∫t 0 ∂u ∂φ (τ, c, φ(c))dτ + ∫t 0 (x − c) ( ∂u ∂φ ) x (τ, c, φ(c))dτ − ∫t 0 ∫x c ∫y c ∂f ∂u (τ, z, u(τ, z, φ(z)), ux(τ, z, φ(z))) ∂u ∂φ (τ, z, φ(z))dzdydτ − ∫t 0 ∫x c ∫y c ∂f ∂ux (τ, z, u(τ, z, φ(z)), ux(τ, z, φ(z))) ( ∂u ∂φ ) x (τ, z, φ(z))dzdydτ +δ{φ, φ1}, 38 svetlin g. georgiev and khaled zennir cubo 20, 2 (2018) where δ{φ, φ1} −→ 0 as φ(x) −→ φ1(x) for every x ∈ [c, d]. hence, when φ(x) −→ φ1(x) for every x ∈ [c, d], we get 0 = ∫x c ∫y c (v(t, z) − 1)dzdy − ∫t 0 v(τ, x)dτ + ∫t 0 v(τ, c)dτ + ∫t 0 xvx(τ, c)dτ − ∫t 0 ∫x c ∫y c ∂f ∂u (τ, z, u(τ, z, φ(z)), ux(τ, z, φ(z)))v(τ, z)dzdydτ − ∫t 0 ∫x c ∫y c ∂f ∂ux (τ, z, u(τ, z, φ(z)), ux(τ, z, φ(z)))vx(τ, z)dzdydτ, (5.1) which we differentiate twice in x and once in t and we get that v satisfies (1.3). now we put t = 0 in (5.1) and then we differentiate twice in x, and we find that v satisfies (1.4). acknowledgments the authors would like to thank the anonymous referees for their helpful comments and suggestions. references [1] a. braik, a. beniani and kh. zennir polynomial stability for system of 3 wave equations with infinite memories, math. meth. appl. sci. 2017; 1–15. doi: 10.1002/mma.4599 [2] a. braik, y. miloudi and kh. zennir a finite-time blow-up result for a class of solutions with positive initial energy for coupled system of heat equations with memories, math. meth. appl. sci. 2017; 1–9. doi: 10.1002/mma.4695 [3] k.deng, comparison principle for some nonlocal problems, quart. appl. math. 50 (1992), no. 3, 517–522. 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[16] j. zhou and d.yang blowup for a degenerate and singular parabolic equation with nonlocal source and nonlocal boundary, appl. math. comput. 256 (2015), 881-884. introduction auxiliary results proof of theorem 1.1 proof of theorem 1.2 proof of theorem 1.4 c ubo matemática edu cacional vol. 5., n° 1, enero 2003 a n e w expansion formula g e orge a . anastassiou dcvm·tmenl o/ mathematu;al s cien ces uniuc 1·sby o/ m emphis mcmvhis, tn 38 152 u.s. rt. c-ma·íl: ganaslss@mcmphis. ed1j 2000 ams s ubject c lassilication : 26a42, 26a99, 26d 15 key words and phr ases: t aylo r li ke expan ion , rie ma nnslieltjes integral , inequalily íor integ ral ancl s\!im. a bstract a new tay lor like exp1:1insion íormula is est.ablished . rere t!he juemannst.ieltjes i.ntegral of a funcllion is expanded into a finit.e sum for m wh ich involves the deriva.tives ar llhe funct ion evaluated al t.he rigbt end point of t he int.erwl of integrnllimi. thc error of t hc appro.ximation is given in un integral fo rm involv ing tlh~ nllh deriva.ti ve of thc íunction. lmp!icullions and applicntio ns o f tbe fo r mu la follmv. l. r esults we give our first end mai n resull: t beorem l. let go be a lebesgue integrable and of bou.nded variation function on [a, bj, a < b. we fonn g¡ (x) :=f.' go(t.)dt,. ( 1) j ' (x t)•1 g.(x) := 0 --¡;;-::-¡y-oo(l)dl, n en, x e [a, b[ . (2) 26 george a. anastassiou let f be such that ¡cni) is a absolutely continuous function on [a, bj. then [ fdgo = ~(!)• ¡<•l( b)gk(b) f(a)g0(a) k:::o + (-1)" l 9n-1(t)j(n)(t)dt. proof. we apply integration by parts repeatedly (see [lj, p. 195): t fdgo = f(b)go(b) /(a)go(a) l gof'dt, and 1• gof'dt furthermore 1· j"g¡dt so far we have got 1• j'dg1 = j'(b)g1(b) /'(a)g1(a) [ 91/"dt ¡'(b)g1(b) t 91/"dt. 1• j"d92 = j"(b)92 (b) /"(a)g2(a) -1• 92/"'dt j"(b)92(b) -1· g2j'"dt. t fdgo f(b)9o(b) f( a)go(a) j'(b)g1(b) + !"(b)92(b) -1· 92/"'dt. similarly we find j.' 92/"'dt = ¡· f"'dg, = f"' (b)g3(b) -1· g,¡c4ldt. t hat is, t fd!jo = f(b)go(b) /(a)g0 (a) ¡'(b)91(b) + /"(b)92(b) j"'(b)g3(b) + t g,¡c4ldt. (3) a ncw expnnsion fbrmu/a 27 'fhc validit.y of (3) is now cleer. o on theorem 1 wc have coro llnry l. additionally a.ssurne that ¡c11> exists and rs boun ded. 1'lien ll fdyo ~(1 )•¡ 01 we want t o prove that ~ --+ o as n--+ +oo. set see that a" xn := nf' n = 1,2, . xn+ 1 = ___.:'.'.!__·xn , n = l,2,. n+l but there exists no en: n > a 1, i.e., a< 1l{) + 1, that is , r := n:+l < 1 (in fac t lake no := ra il + 1, where r·l is t he ceijj ng of the number) . thus xno+ l = re, where c := x,..0 > 0. t herefore a no w expansion fbrmula 29 likcwi we gct and in g ncral wc obtain o < x,.0 ¡ k < rk · e e· · r 010 tk, k e n whcrc e· : *°· that is inccrn-oasn · ¡ , wcobtainthatx,.,.-oasnl . l .c.,~ -o, ~n-1 o re mark l. (o n thooro1n 1) t;u rthcrmorc w 1ha1 (hcrc f e c"(la,bl)) lt fd!jo 1 /(a)9o(a) i c3> lf:( 1j' ¡c'>¡¡ ). (9) thcn 11' fdgo 1 / (a)9o(a)i :s /, . { ~¡9,(b)l i f 19n -1(l) jd1} . n el'i fi xcd. (10) 2. applications (i} lcl {ttn.}nien be a scquo11cc of 13orcl finitcsigncd mcasu rcs on [a, bl. consiclcr thc dislribution fun tions g0 ,,,.(x) : ''"'¡a,xl, x e a, bji m. en, which 1:1rc of boundl' is a absoluwly co ntinuous íunclion on [0 1 q. then by thoorem j we find 9u{x) = ~, ali n e n. fu rlhermore we gel 1 ,,_ , ¡ i• l( ) ( )" 1' f fdt = l::hl•--1+ ...=.!__ t." ! 1" >(t )dt. lo k = o ( k + 1 )! n! o ( 16) one can d eri ve o t her formulas like ( 16} ror various basic ga 1s. references 111 apostol, t. m .1 11mathemat1cai auajy 1s", ad d isonwesley, rea.ding, m a1 1960. (21 hogoas, g. 1 11characterizal'ion o/ weak conuergen.ce of si91ied measures on jo, q•, math. sean st \l dir1d . t l1 t' 1l 1t 'qt\' uf b uudt,.f lull"a.t frr('rfl.l or:for whkl1 ( l . l ) liolds. 11( ;\ ) < . u{.'\ ")< . but r1 {a) 1111 1\ hl' 110 1 fttuid 10 "(a "). tlms frcrl(x. y) is n propcr s\lbsct of t hc :'\oeth cr opcrntors. th1 • ncwd1n opttllll oft(; are coll('(i 111 honor o f f . :'\oc thcr. who wns 1lu;• firsl l o si ud ,y n • lnf>.~ of 111 11¡.¡ulnr 1otiti;.111i rqmu1011s with op('rntors of 1h1 s dnss iu 1921 jjj. 92 a cj11ll'a cterizl.ltior1 o( u11bounded f'rcdl1olm operntoffl in f11j " si mpl e nnd s hort proof of t he fredho lm ahernat1 \-e and n chnratl riuuon ol fredholm opero t.ors ar e g:iven for boundt..>d linear opem1ors. rccall thnt " lincnr boundtd opern tor f ls ca ll ed a finitcrank operntor ií dim r{f ) < , whe re r(f ) ili th e rftng~ of f in t he presc nt. papcr ti he resu lts of (4] are generalized to thc en.se of closcd uuboundi!d linear opern1ors. niuncl y, t he following rt.>s ult is pro\•ed: theo r em 1.1 lf a is r1 frcdholm opcmtor, then a = 8 f, (131 u!hr:rt: bu a /mear c/o.rnrl 071emlor, d(b ) = d( a ), r ( b ) = )~. n(b) = {o) , 1md fu cr fimlc-ronk opem tor. comrnr.~ely, if (1. 3) lwlds. whcnr 13 ·x )' .... a l111eor closc.d dc.n,,rlr defi11ed oparo t-o,.. r {/3) = i' , n(b) = {o) . arid f r.s u fimtc-ronj.: opttm tor, tl1r; 11 a u rü,jcj /j(.-1) = 0(8). and (1. 1) rmd (l. !!} hofd. so a 1s a fredholm opcmtor. lu !>ce t1011 2 11 proof of t heo rti m 1 is givc n. 111 th c htcraturc 1lu.' co.:.c of unboundtd frcdholm opcrators is usun lly no! d iscusscd dírcc tly. in jsj tmd m [6j. pp.5i· gi. s mgulllfllu"' of 1hc pn.r1u nct er-dcpc11dc11!, frcd holm opcrntors nrc s tud icd. 1111(! iu !7j nppll cn11011?i of tlwo frcdho hn oj>crn to rs in brnn chiu g tlhco ry nrc prcsc nt cd thcorcm 1 1 is ubc ítll. for cx1u11plr 111 lhc 1hoory of c l\ipt,ic boundnry vnlu c problcms, but "-"t' do no t go in10 fur1h1•r dl•ltlll (~ c.g .. (11. (21. l;j). 2 proof 1 ;\ssi1111c 1hnt a : x )" i~ lin cnr , closecra 1or, und ( 1 1) nnd (12) hold l..c t ll~ provc 1h111, t1hc11 ( ! .:.!) holds, d ( /j ) = d ( a ). r( b ) = l '.n( b} = fdt . 8111rl~ 11ml f 1:, fim1c11111k opt·1·ntor . lc1 ~;,}i'5'.,'fr1 be n bnsis of n(i\) nnd fl •j }i -5'.j°f" be a bas11> of n(ji º ) h lb k111rrm 1h8' /1 (j\)j. = n (a ' ). (2 1) whcre r( ..t)j. 1s !lw set of liticur fun c1io11nl1t h'11 111 1 ~ · sud1 1lia1 (i.-•1, j\11 ) = o v11 1)(11). whcrc (-.·,. /) is tl1 c vn lu c of 11 li11 cn r íu11 c110 11nl v, e l .. on lht' e l<'1m·111 f }' clr1ul~·. t•1 e ,\ '( a") . 1 :s: ; $ 11 . dc-fmt' /ju := au + l ( /1 j.u)11, ·=( a + f 1u. ,,, e } ". '' 1221 a .c. h.nmm 93 "'hnro f in n flnllc-ra.nk opcrnlor, {vj }i sj.s n is n aot. of cloments of y , biorthogo nul to the { o,j ¡é m . le ~ ! "1j h sj 5 111 (~,. u.,.) .. 6jm := l ,j = m , and {lij }t sj~n is th e: set of elemcnta of x·, ulorthogo11nl 10 t ho set (1pj)1 ~j.sn • (l1 j,¡p ,., ) = ój•n · existence of .!!ets bio rt hogonal to flnlt o.ly mnr.\)· hncml.y lndo pcmdc.nt ole.menta of n bnnnch spnce follows from t.he jfo.lm b11 nool1 lhoorom. an ar bilrnry olomcmt 'll e x can be un iquely represc.nt.ed a.s 1j = u 1 + i:;'. 1 l°j 'pj• cj m t'odsi, rutd (ji¡, u¡)= q, 1 :5 j $ 11. lct ue chedt thft t n(b) = {o} and /l(b) = y . ass um e b u = o, t.hnt is au + 'f..j'. 1(111 , u}vj • o. apply l/'m t.o tlhia oquntiou, uso (rb,,, , a-u)= o, and get n " o• l(~m,v¡)(h¡, u) = l; 6,,,;(h¡, u) = (h,,,, u), 1 $ m $ n. ,.. j• l 1'lwruforc . f and u,...-+ u , then bun fu" -o f, and the above argument shows that bu fu= f so au =f. thus a is closed. finall y, !et us prove (1.2). let au =o, i.e. bu fu= o. app lying the bounded linear injective eperator a1, ene ge t s an equivalent equation u tu=o , t:=b 1f, t:x--x, (2 .6) with a finite-rank o perator t. it is an elementary fact (see {4 ]) that dim n(j t) := n < oo if t is a fini te-rank operator. since n(a) = n(i -t), ene has dim n( a ) = n < oo. now jet aºv =o. then b ºv-f•v =o. (2.7) since (b º ) 1 = (b1 )º is a bounded and injective linear operater, the e lements v are in one-to-o ne corresp ond ence with the elements w := b ºv , aud (2.7) is equiva.lent to w t ' w =o, t' ~ f ' (b ' )-' , (2.8) so that t' is the a.djoint to operator t := s -1 f . since t is a finite-rank operator, it is an elementary fact (see [4)) that dim n(l t ' ) = dim(l -t) = n < oo. since n(a') = n(i -t'), property (1.2) is preved . theerem 1. 1 is preved . o an immediate conseque nce of theorem 1.1 is the fredholm alternative (see theorcm 1.1 in [4]) for unbounded operators a e fred(x, y ). a.g. ramm references [111] kantorovjch, l. anill akil0v, g., f'unctional analysis in normed spaces, macmillan, new york, 1964. [2] knro, t., perturbation theory for linear operut o tal que para cada a e f , la] ';?: 6 implica x e as . d e finició n 1.3 sea (f, l· i~ un anillo div isión valuado. un s ubconjun to s de e de un espacio vectorial e sobre f se llama balanceado s i as es su bconjun t o de s para ca.da jal $. l . c ubo 10 a . figueroa c. de8nicló n 1.4 s oo e un. espacio 11ectorial sobre. tm anillo de d i visión f . uri subcnnjunto s de e se llama n o-arqulmodiano si s + s e.s s ubcnnju nto de s . de8nición 1.5 s ea e un espaci o vec;to ribl s o bre. ( f, 1 1). un a seminorma sobre e es un a func ión p : e ir t al que: ( 1) para todo x en e' p(x ) ~ o {2} para todo x en. e y t odo a en f : p(ax) = la lp(z ) ( .9) pam t odo :r:, y en e , t od o a en f : p(:r: +y) .5 p(x ) + p( y ) s'i además ( ,1) paro todo :r:, y en e: p(x+y) .5 máx(p(x) , p(y ) ) entonces diremos que p es una sem i n o mna no-an1uimediana so bre e. definició n 1.6 { choquet , (2/ ) s ea e un espacio vectorial s obre ( f, i · 1) . l la maremos subpro d ucto escalar en e a toda fu nción b d e ex e en r+ u {o} t al que: ( 1) paro. todo x , y en e : b (x , y ) = b ( y , x ) (2) p a ra t odo x , y e n e y tod o a en f : b (ax, y) = la lb(:r:, y ) (9) pam todo x , y , = en e ' b (x + y, z),;; b(x , =) + b(y , z) (,1) para tod o x , y en e: b'(x, y) ,;; b (x ,x)b(y , y) 2 prelimina res p~oposición 2. 1 sea e un espacio vectorial s obre un. anillo divi.3ión v tiluado ( f, 1 · 1) . las siguientes s on equivalen t es paro cualquier .mbconj v. nto s n o vacío d e e : ( a) s e3 f -convezo y o es; {b} s e.,, balanceado y n oa,vuúnediano; {e} x , y en s y a . b en f co n lal .5 1, jbf .5 1 implican qu e ax+ by e s dem ost ración : proll a !3j, pag. 8 1. • proposició n 2.2 sen e un. e3pacio vectorial sobre u n anillo div isión no tn·vialmen te valuado ( f , i · 1) . s i exist e un 3ubconj unto v de e con o e v, nbsove n.te y f -con uu.o; ent.on~s existe u na semino ,.mn nonrqu imediana p sobre e ta l que: v,(o, i) e v c \ip(d, 1). s i el valo r abso luto de f e.,, di.scre.to . e ntonce.' \1 = vi o, 1) subprod uctos esc&lares ... cubo 10 demostración : prolla [3), pag. 85. • observación: en proposición 2.2 c uando ( f , l · i) es no trivialmente value.do y v es abierto, entonces le. seminorme. p definida allí es continua. demostración: prolle. [3], pag 87. • 3 resultados principales teorem a 3.1 sea ( f, l · i) un anillo divisi6n no trivialmente valuado y sea e un espacio vectorial {no trivial) sobre f. si existe un. subconjt.mto v de e con o en v, f convexo, absoniente, entonces erist e b: e x e w u {o} , no trivial, no constante, con la.s siguientes propiedades: ( a) paro todo x,y en e' 8(x,y) = 8 (y , x ) ( b) paro todo x en e.' 8(0,x) = 8(x,o) =o {c) para todo x , y,u en e: b(x+y, u ) :s b(x,u) + b(y,u) {d) (condici6n de no arquimedianidad) para. todo x , y , u en e: 8 (x + y, u) s máx(8(x, u), 8(y, u)) ( e) paro todo x,y en e ' 8 2 (x, y) s 8(x, x )8(y,y) (f ) 8(0, 0) = o ( g) paro todo x, y en e y a en p, 8(ax, y) = lal8(x, y). toorema 3.2 sea (f, 1 • 1) un anillo división no trivialmente valuado y sea (e , t) un espacio vectorial topol6gico {no trivial} sobre f . si existe una vecindad v del o en e, f-convexa, entonces existe b: ex e jr+ u {o} , continua, no trivial no constante, con las siguientes propiedades: (a) paro todo x,y en e, 8(x, y) = 8(y , x ) ( b) paro todo x en e .. 8(0 , x) = 8(x,o) =o (e) paro todo x , y , u en e ' 8(x + y, u) s 8 (x , u) + 8 (y , u) {d) {condición de no arquimedianidad) pam todo x , y , u en e: 8 (x +y, u) 5 máx(8(x, u) , 8(y, u)) (e} paro todo x,y en e.' 8 2 (x , y ) 5 8(x ,x)8(y , y ) (f ) 8 (0, 0) =o (g) paro todo x , y en e y a en p , 8(ax , y ) = lal8(x, y). cubo 10 4 demostración de los teoremas demostración: (teorema 3.1) sea (f, 1 · 1) un anillo división no trivialmente valuado y sea e un espacio vectorial (no t ri vial) so bre f. si existe un subconjunto v de e con o en v , f-convexo, absorvente , entonces se puede aplicar la proposición 2.2 de este trabajo y se concluye que: existe un a seminorma p: e lft+-u{o} tal que x .._. p(x) (no trivial ) . definimos b' e x e ir:' u {o} ?0' b(x, y) '= p(x)p(y) (a) sean x , y en e' b(x, y) ,= p(x)p(y) = p(y)p(x ) = ' b(y, x ) (b) sea x en e, b (x , o) '= p(x)p(o) = p(x)o = o b(o, x ) '= p(o)p(x) = op(x) = o (e) sean x , y , u en e, b(x + y, u) ,= p(x + y)p(u) s (p(x) + p(y))p(u) = p(x)p(u ) + p(y)p(u) =' b(x, u)+ b(y, u) (d) sean x,y , u e n e, b(x + y, u) = p(x + y)p(u) s (máx(p(x),p(y))p(u) = máx(p(x)p(u), p(y)p(u)) = máx(b(x, u), b(y, u)) (e) sean x , y en e' b'(x , y) = (p(x)p(y))' = (p(x))'(p(y))' = (p(x)p(x))(p(y)p(y) ) = b(x,x)b(y , y) luego b'(x , y) s b(x,x)b(y,y) (f) 8(0, o) = p(o)p(o) = o · o = o (g) sean x , y en e y a en p, b(ax , y) '= p(ax)p( y ) = \a\p(x)p(y) =' la lb(x , y ) . además, por ser p una semi norma no trivial, existe un xo en e tal que p(x0 ) =fo. luego , b (x0 , x 0 ) = p(xo)p{xo) -:/; o por lo tanto, b es no constante y no trivial. • demostració n: (teorema 3.2) .f'alta por demostrar que bes continua. teniendo presem.e la observación de este tra bajo se tiene que si v es abierto, entonces la seminorma pes oont inua en e. luego, b (x , y) = p(x)p(y) es con tinua.. • referencias [ij figueroa. a. conj untos r-convexos y subproducto8 escalares. comunicación presenta.da a. la. j oma.da de matemática de ja zo na s ur, universidad del bíobío, concepció n, mayo d e 1991. [2j choq uet c. topología. toray~ma.s.son 1971. cuiio 10 (3] prolla j .b. tapies in functional ana.lysi.s over v a.i.ued divi.sion rings, north holla.nd,mat>hema.tfrs studies 77, notas mateá.tica, ed. leopoldo na.€hbin, (1982). [4] van rooi~ a.c.m. non archimedean, functional analysis, marcel dekker, inc., new yo11k and basel, 197.8. dirección del autor: alejandro figueroa cor.tés departamento lile matemátiica y física universidad de maga\lanes ca.silla 113-d. punta arenas cubo , vol. 2 , marzo 1986 pág . 73-74 probl ema s prop est os 73 baj o e s te títul o incluiremos a lgun os eje r cicios p r oblemas se n ci llo s e n l as áreas de mate má t ica , ya sea pura o aplicada , el obj eto de acrecentar el diálo g o co n n u estros estudian t es . dese am os qu e ésta se a otra ma n e r a d e acercamiento entre este de p a r ta me n to y ust ed . es p e r a mos mu chas so lu ciones a lo s prob l e ma s planteado s . las c uales p u e d e n ser enviadas al c o mité editor de la revi sta . para s u poste r ior publicac1ón . ~~~~~~~~ _ _! : s ea f: la,b] ... jr . demostrar qu e sj v x 1 , x 2 , x 3 c(a,b• e on se s ati s f ace la relació n 7 4 prob l e ma s propue stos f( '2) (• 1 1 o< ------2---~ (x2x 1) f(•3l f(,2) ---------;¡--( x jx2) e n tonces la f un ció n fe s co n stante sob r e ~ , bl pro bl ema 2 • ---------sea f una func1ó n c ontinua sobr"e { 0 , l) con valo r es e n ffi, de r 1vable en (0 . 1) . s up ong a mos que 1 x f' (x ) f (x) + f (o) 1 < x 2 m v x e (o , l) e on m f i j o . ¿ex i s t e f ' (o)?' probl e ma 3 : ----------el n umero co mple j o (a,b) pu ede descr i birse med ia nte la mat riz real de orde n d o s ( a -b) b a . l a s u ma y p r od u cto d e numer as co mplejos se de fine c omo l a s um a y p r od u ctos d e ma tr ~ ces , re spectivamente . obte nga la forma mat ric ia l de l a r ~i~ c uadra da d el complejo problema 4: ---------dem u estre que l a s uma de l as qui nt a s pote n c i as d e l os c e r o s del pol1no m1 0 x 4 + ax 2 + b x + e es 5ab . a mathematical journal vol. 6, no 4, (53-72). december 2004. on the scale up problem for two-phase flow in petroleum reservoirs1 frederico furtado department of mathematics, university of wyoming laramie, wyoming 82071-3036, u.s.a. furtado@uwyo.edu felipe pereira instituto politécnico, universidade do estado do rio de janeiro rua alberto rangel, s/n, nova friburgo, rj, 28601-970, brazil pereira@iprj.uerj.br abstract the basic goal of scale up procedures is to simulate on coarse grids, with modified equations, multiphase reservoir flow and transport problems defined on fine grids. scaling up is in general difficult. difficulties arise both from the highly nonlinear fluid-fluid interactions in the flow of multiphase fluid mixtures and from the complex interactions between heterogeneities and nonlinearities. our recent results show that several different flow regimes occur, depending on the relative strengths of flow nonlinearity and medium heterogeneity, as well as on the spatial structure of such heterogeneity. in the present study such results are used to investigate the applicability of a simplifying assumption made in several studies in the development of theories for the scale up problem for immiscible (water-oil) displacement. our results indicate that such assumption is not appropriate to describe a typical mixing regime encountered in petroleum reservoirs, namely, the nonlinear unstable regime, in which nonlinearities in the governing equations dominate the fluid mixing process. key words and phrases: porous media, scale up, multiphase flow, heterogeneity, fractals, scaling laws 1this work was supported in part by the national science foundation (nsf) under grant int0104529 and anp/cnpq under the “plano nacional de ciência e tecnologia do setor petróleo e gás natural (ctpetro)”. 54 frederico furtado and felipe pereira 6, 4(2004) 1 introduction stochastic partial differential equations arise in theories which model complex (random) physical systems at small (microscopic) length scales, while their goal is to provide a description of the behavior of the system at large (macroscopic) length scales [28]. thus, an essential step in these theories is the derivation of effective (or averaged) equations along with scaling laws governing their solutions, capturing essential features, at the macroscopic level, of the random microscopic structure. in mathematical studies of the transport of pollutants in groundwater and of oil recovery processes one is faced with the situation described above, where a system of stochastic partial differential equations models the two-phase flow in a porous medium. the system is composed of two equations, a transport equation for the saturation (the relative volume of one of the two fluids) coupled to an equation for the velocity field, which is given by darcy’s law and the incompressibility condition for the flow. the randomness enters the problem through the unknown properties of the rocks, especially the permeability tensor. a question of a major importance in applications concerns the growth of the region where mixture of two fluids occurs, which is produced by multi length scale rock heterogeneity as well as by fluid instabilities. we focus our attention on the length of the mixing region (the mixing length). we are specially interested in the study of anomalous (non fickean) diffusive mixing. in the case of oil recovery, anomalous diffusion leads to early breakthrough of the water drive and reduced recovery, while in groundwater flow anomalous diffusion leads to the rapid growth of contaminant plumes. statistical models have been proposed to describe rock heterogeneity properties which are incompletely known, especially at length scales below the interwell spacing. we consider fractal (or self similar) models for geological variability, which were introduced in earlier theoretical work [29, 4, 36, 27]. the use of fractal models is a basic advance over earlier modeling based on fixed length scale heterogeneities. the self-similar hypothesis is very powerful. it allows reconstruction of heterogeneity on all length scales from measurements on a small range of length scales, e.g., laboratory and pilot-scale field studies, and provides improved agreement with the observed scale dependent dispersivities in the field data and with the observed geological heterogeneities on all length scales. there is, however, no known law of geology which relates variability on distinct length scales. we view self similarity as a convenient tool to create simulated data which encompasses many significant length scales. as geological data appears to be nonuniversal and slowly varying as a function of length scale, the multi fractal (rather than pure fractal) approach [46] has been introduced. the basic assumption of the multi fractal approach is that data varies smoothly in log-log variables, having well defined tangents with nonuniversal (scale dependent) slopes. such an assumption provides a very general description of variability, in contrast to the self similarity assumption, and may provide better models for specific reservoirs or aquifers. in order to investigate the effect of rock heterogeneity alone, the authors and collaborators have considered the passive tracer flow problem (linear transport). the 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 55 determination of the dynamics governing the growth of the mixing region for such model is an open problem as far as full rigorous proofs are concerned. however, a deep insight into this problem and evidence of the existence of asymptotic scaling laws governing the fluid mixing can be achieved through perturbative methods and high resolution computations, used to solve the stochastic system of differential equations. a characterization of the long time asymptotic growth of the mixing length was obtained in refs. [27, 46] within distinct analytic approximations at the level of perturbation theory. these theories, developed in the eulerian picture, give the time asymptotic growth rate of the mixing length as a function of the distance asymptotic exponent of the permeability field. the finite time transient effect was also studied in ref. [46]. these theoretical predictions are restricted to the regime of small fluctuations of either porous media heterogeneity [27] or velocity field [46]. high resolution numerical simulations are not limited to the small fluctuations regime. in the absence of an exact solution for the problem of determining the timedependent behavior of the mixing length, numerical simulations provide the only available tool in placing limits on the validity of the perturbative analysis and in the prediction of complex fluid flow behavior outside the regime governed by perturbation theory. the tracer flow problem was carefully investigated numerically in refs. [38, 25, 26, 18, 19, 20] and the predictions of perturbation theory were fully validated. to add more references to these, in the context of linear flows, we refer the reader to ref. [2] for a renormalization group approach, to ref. [44] for a perturbation expansion analysis, and to ref. [1] for a homogenization treatment. see also refs. [13, 12, 10, 11, 8, 23, 35, 47, 48] and the references cited therein. although still an area of active research, the body of work described above lead to a very good understanding about the scale up problem for linear transport problems in heterogeneous formations. for nonlinear problems the study of the scale up problem is considerably more difficult, due to the complex interactions between heterogeneities and nonlinearities. consequently, nonlinear flows have not been so extensively investigated. in most existing approaches for scaling up multiphase (nonlinear) flow systems only the first order (hyperbolic) transport terms are modified. this is known as hyperbolic renormalization, or “pseudoization” of the laboratory scale relative permeability curves. for a review of some standard procedures of this type we refer the reader to refs. [5, 7, 16]. effective procedures have been proposed recently in the literature [40, 41, 42]. alternative scale-up methods, not based on a hyperbolic renormalization, can be found in refs. [17, 32, 9, 14, 45, 30, 31]. recent results of the authors (see ref. [22]) show that several different flow regimes occur, depending on the relative strengths of flow nonlinearity and medium heterogeneity, as well as on the spatial structure of such heterogeneity. see also refs. [3, 24, 33, 43] for related work. it is then plausible to expect that different scale-up methods (different types of coarse-scale models), tailored to the specifics of the distinct flow regimes, might provide a better coarse-grained description of multiphase flow in heterogeneous petroleum reservoirs. thus we believe that an improved understanding of the interplay between heterogeneity and nonlinearity, as we have 56 frederico furtado and felipe pereira 6, 4(2004) pursued, will furnish essential insight about existing techniques for scaling up and their limits. in particular, in this paper we investigate the validity of a simplifying assumption made in the theoretical developments reported in refs. [30, 31, 45, 32, 14] through high resolution numerical simulations. in these studies it is assumed that the transport equation for the saturation is not coupled to the equation for the velocity field. our results indicate that such assumption is not appropriate to describe the flow regime dominated by nonlinearities (or, equivalently, the regime characterized by weak heterogeneities). this paper is organized as follows. the stochastic model for two-phase immiscible displacement considered in this work is described in section 2. in section 3 we discuss our strategy for solving numerically the model introduced in section 2. the method we have developed to perform a quantitative analysis of the macroscopic water-oil mixing process is presented in section 4. the recent effort of the authors in identifying a new family of mixing regimes which were investigated through a high resolution numerical study of the problem at hand is reviewed in section 5. we then turn to the main result of this work in section 6, with a discussion of the applicability of recently proposed theories for the scale up problem for immiscible (water-oil) displacement. our conclusions along with a discussion of important open problems related to the material discussed here will appear in section 7. 2 stochastic modeling of two-phase flow in this paper we consider the scale up problem for two-phase, immiscible, incompressible displacement in petroleum reservois. we neglect the effects of gravity, compressibility and capillarity and set the porosity equal to a constant. the two phases will be referred to as water and oil, and indicated by the subscripts w and o, respectively. the governing system of equations in the laboratory scale (see ref. [37]) can be described as follows. darcy’s law for each phase takes the form: ui = − kri(s) μi k(�x)∇p, where ui is the phase velocity, k is the absolute permeability, kri is the relative permeability and μi is the viscosity, each with respect to phase i, s is the water saturation, and p is the pressure. darcy’s law combined with the conservation laws for the phases can be expressed in the familiar form of “pressure” and “saturation” equations: u = −λ(s)k∇p, ∇ · u = 0, (1) ∂s ∂t + ∇ · (f (s)u) = 0. (2) (the constant porosity has been scaled out by a change of the time variable.) here, λ is the total mobility, f is the fractional flow of water, and u is the total velocity. 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 57 these parameters are given as λ(s) = krw(s) μw + kro(s) μo , f (s) = krw(s)/μw λ(s) , u = uw + uo. we consider eqs. (1)–(2) in a two-dimensional rectangle ω = (0, lx) × (0, ly), with the boundary conditions u · n = −q, on x = 0, p = 0, on x = lx, u · n = 0, on y = 0, ly, (3) where n is the outward-pointing normal vector to ∂ω, and a uniform initial condition s(�x, 0) = s0. (4) water is injected uniformly (at a constant rate q) through the left vertical boundary (x = 0) of ω, no flow conditions are imposed along the horizontal boundaries (y = 0, ly), and fluid is produced from a well kept at constant (zero) pressure at the right vertical boundary (x = lx). the relative permeabilities are assumed to be: kro(s) = (1 − (1 − sro)−1s)2, krω(s) = (1 − srw)−2(s − srw)2, where sro and srw are is the residual oil and water saturations, respectively. in our studies we consider scalar, log-normal permeability fields, k(�x) = k0e ρξ(�x), (5) where ξ is a stationary gaussian random field, characterized by its mean 〈ξ〉 = 0 (angle brackets denote ensemble averaging) and its covariance function c(�x, �y) =< ξ(�x)ξ(�y) > . the mean 〈k〉 and variance σ2k of the log-normal field k are set by the coefficients k0 and ρ. changing ρ varies the coefficient of variation, cvk ≡ σk < k > , (6) of the permeability field. we use the coefficient of variation as a dimensionless measure of the heterogeneity of the permeability field. we take the field ξ to be isotropic and, to introduce variability over all length scales, fractal, or self-similar. thus its covariance function is given by a power law: c(�x, �y) = |�x − �y|β, β < 0. (7) the scaling exponent β, known as the hurst exponent, controls the degree of multiscale heterogeneity: as it increases, the heterogeneities concentrated in the larger length scales are emphasized and the field becomes more regular (locally). 58 frederico furtado and felipe pereira 6, 4(2004) 3 numerical approximation of two-phase flow 3.1 computational solutions for flow and transport an operator splitting technique is employed. the saturation equation (2) and the pressure equation (1) are solved sequentially with distinct time steps. tipically, for computational efficiency, larger time steps are used for the pressure calculation. figure 1: saturation surface plots displayed as a function of the viscosity ratio m for flow in a β = −0.5 permeability field with coefficient of variation cvk = 0.99. the values of m , from top to bottom, are m = 1, 2.657, 5. a detailed description of the numerical method that we employ for the solution of eqs. (2)–(1) is given in [49] (see also [15] for an earlier version of this procedure). this 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 59 method has proved to be computationally efficient in producing accurate numerical solutions for two-phase flow problems. a second order accurate finite difference scheme (see ref. [34]) is used for the discretization of the saturation equation (2). this method can accurately resolve sharp fronts in the fluid saturations without introducing spurious oscillations or excessive numerical diffusion. for the global pressure solution, we use a (locally conservative) hybridized mixed finite element discretization equivalent to cell-centered finite differences [15], which effectively treats the rapidly changing permeabilities that arise from stochastic geology and produces accurate velocity fields. important features of the simulation code described in [49] include mass conservation of fluid phases, absence of grid orientation bias and reduced numerical diffusion. we illustrate the numerical solutions obtained with this simulation code in figure 1. saturation surface plots displayed as a function of the viscosity ratio m ≡ μo/μw for flow in a β = −0.5 permeability field with coefficient of variation cvk = 0.99. the values of m , from top to bottom, are m = 1, 2.657, 5. 3.2 computational generation of fractal fields the (log) permeability field (a correlated gaussian random process) is the image under a convolution mapping of a white noise (uncorrelated gaussian) field. in more detail, let η = η(�x) be the white noise random field. the correlation function of η is < η(�x)η(�y) >= δ(�x − �y). (8) let f ∗ g denote the convolution product of f and g, ( f ∗ g ) (�x) = ∫ f (�x − �y)g(�y)d�y, (9) and let f ∨(�x) = f (−�x). the permeability field fluctuation is given by ξ = f ∗ η, (10) where f is the convolution kernel to be specified below. by direct computation, the correlation function (covariance) of the field ξ is given by < ξ(�x)ξ(�y) >= ∫ f (�x − �z)f (�y − �z)d�z = ( f ∗ f ∨ ) (�x − �y). (11) to get the desired asymptotic behavior given by (7) the convolution kernel is taken to be f (�x) = c1 (c2 + |�x|)τ · c3 (c3 + |�x|)d/2 , (12) where τ = (d − β)/2, d is the number of space dimensions (d = 2 in our case), and c1, c2 and c3 are constants which set the overall strength of the correlation function, regularize its behavior near the origin, and regularize its behavior near infinity. for 60 frederico furtado and felipe pereira 6, 4(2004) τ < d (and x �= y), the convolution integral (11) is convergent at z = x and z = y, and we can take the limit c2 → 0 directly in (10) and (11). similarly for d < 2τ , (11) is convergent at infinity, and we can take the limit c3 → ∞ in (10) and (11). since we are interested in the range β < 0, we have d < 2τ and c3 = ∞ in all cases, while τ < d corresponds to the restriction β > −d. either β > −d or c2 > 0 is required for the covariance to be locally integrable, and because the covariance is positive, this condition is also required for the covariance to be a distribution, as is necessary in the framework of generalized random fields [39]. thus we require β > −d or c2 > 0. we now discuss the spatial discretization of the procedure just explained. the basic white noise gaussian field η is now defined on a discrete level, i.e. is taken to be piecewise constant on the mesh blocks of a finite lattice. we distinguish two discretization errors, namely the local ones due to the finite mesh size of the η field, and the long range ones due to the finite extent of the η field computational grid. our first results were to understand and then to minimize these errors. to do this, we studied the covariance of the discrete random field ξ = f ∗ η, which is just the discrete convolution product f ∗ f ∨, as compared to (7). figure 2: two examples of realizations of permeability fields used in the fluid flow simulations. the top picture refers to an uncorrelated gaussian field (β = −∞) and the bottom one to a fractal field with β = −0.5. cutoff values are 0.3 md for black and 4.0 md for white. we restrict our attention to β ∈ (−2, 0). in this case, the convolution (10) is not singular, and we can set c2 = 0, c3 = ∞ in (12). however we found that nearly singular integrals cause slow convergence, a fact which is further complicated by the 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 61 requirement that to analyze scaling laws over multiple length scales, very small effects are being observed. to accelerate convergence, we use nonuniform grids (local mesh refinement) to allow greater resolution of the local and long distance behavior of the convolution defining ξ and its covariance. to overcome the local and grid boundary errors, we introduce variable sized grids, allowing for finer local resolution and greater separation from the boundary. the spatial grids depend on β. as β → 0, long range blocks (large blocks) are used, while as β → −2, local refinement is important. we refer the reader to [38] for more details. the computed correlated field is illustrated in figure 2 where two realizations of random permeability fields are displayed, corresponding to β = −∞ (uncorrelated gaussian), top picture, and β = −0.5 (correlated fractal), bottom picture. 4 quantitative analysis of the mixing process a quantitative analysis of the mixing process is afforded by analysis of the growth rate of the mixing region as a function of time or, equivalently, travel distance. for this purpose, we introduce a time dependent length scale—the mixing length—characteristic of the extent of the mixing region and study its growth as a function of time. figure 3: saturation level curves for two-phase flow with viscosity ratio m = 5 in a β = −∞ permeability field (ragged contours). superposed is the planar saturation profile for the corresponding homogeneous flow. 62 frederico furtado and felipe pereira 6, 4(2004) since the bulk incompressible flow is driven by a constant injection rate, the fluctuations in u around its mean 〈u〉 = (q, 0) result in the spreading of the heterogeneous saturation fronts around their mean position, which is specified by the location of the planar homogeneous front. see figure 3. an estimate of the spreading in any realization kr of the permeability field is provided by �r = �r(t): �r(t) ≡ 1 (s− − s+) ∫ lx 0 |s̄r(x, t) − sh (x, t)|dx. (13) here, s̄r is the average in the direction transverse to the mean flow of the saturation solution corresponding to k = kr; sh is the homogeneous saturation solution corresponding to the constant permeability kh ≡ 〈k(�x)〉; s− (respectively s+) is the saturation value immediately behind (resp. ahead) the saturation front in sh . in the stochastic context, �r is a random variable, specified in terms of its statistical moments. in particular, the expected value �(t) ≡ 〈�r(t)〉 (14) provides the best estimate of the spreading of the heterogeneous fronts and is adopted as the definition of mixing length. we refer to ref. [22] for a detailed discussion. 5 scaling laws and mixing regimes both nonlinearity of the flow equations and permeability heterogeneity can cause dispersive mixing of the fluid transport. to understand their combined effect on the mixing process, we first clarify the effect of each separately. 5.1 the effect of nonlinearity a linear stability analysis of the water/oil front in a homogeneous porous medium is carried out in, e.g., ref. [6]. this analysis reveals the existence of two distinct flow regimes, characterized in terms of the frontal mobility ratio λ = λ(s−)/λ(s+): the front is stable when λ < 1 and unstable when λ > 1. for our choice of constitutive functions (f and λ), λ < 1 (respectively λ > 1) when μo/μw < 2.657 (resp. μo/μw > 2.657). the linearized stability theory predicts that for λ > 1 small front perturbations will grow exponentially. the long-time nonlinear development of front perturbations is studied numerically in ref. [21]. in brief, to a first approximation, in an unstable flow process �(t) = o(t), as t → ∞. (15) 5.2 the effect of heterogeneity we consider linear displacement (tracer flow) in heterogeneous porous media. in tracer flows, the two fluids are miscible and have equal viscosities. in this case, the frontal 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 63 mobility ratio λ ≡ 1 and the flow is neutrally stable. as a result, the mixing in tracer flows is solely the effect of velocity dispersion caused by heterogeneity. for tracer flow in fractal permeability fields, the theory of fluid mixing discussed in ref. [26] provides the following scaling laws: �(t) = o(tγ ), with γ = max { 1 2 , 1 2 + 1 + β 2 } . (16) two qualitatively distinct regimes are seen. if β ≤ −1, then γ = 1/2, and the mixing process is fickian. if −1 < β < 0, then γ > 1/2, and the mixing is anomalous (i.e., the diffusivity increases with time or, equivalently, with travel distance). these scaling laws were consistently derived from leading order results of primitive and renormalized perturbation theory. they were also verified for large perturbation parameters by numerical simulations in ref. [22]. 5.3 the combined effect of nonlinearity and heterogeneity the analysis of fluid mixing dynamics in heterogeneous, two-phase flow is via computational experiments. the computations involved (1) the construction of the various ensembles of random permeability fields, characterized by different values of the exponent β and the coefficient of variation cvk, and (2) the subsequent determination of the resultant mixing region for the fluid flows through these ensembles. both numerical and statistical convergence were verified by successive mesh refinements and increase in ensemble size, respectively. all results reported are for a flow region with lx/ly = 4. all log-permeability fields were drawn from a gaussian distribution with the power law covariance (7). two permeability exponents, β = −∞, −0.5, and several values for the permeability coefficient of variation, cvk, were considered. we use the latter as a dimensionless measure of heterogeneity. the rationale for the choice of exponents is the desire to explore the effect of short length (β = −∞) and long length scale (β = −0.5) heterogeneities in two-phase flow dispersion. table 1 compiles our results. this table displays the values of m , cvk, and β used in each study, and the corresponding mixing regimes. the regimes are classified according to the asymptotic scaling of �(t) = o(tδ), as t → ∞: n u (nonlinear unstable) if δ = 1; n s (nonlinear stable) if δ = 0; l (linear) if δ = γ, the scaling exponent for the linear flow problem given in (16); l+ (superlinear) if γ < δ < 1; l− (sublinear) if 0 < δ < γ. table 1: mixing regimes for two-phase flow cvk/m 0.5 1 2.657 5 10 20 β 0.54 n s l− l n u n u n u −∞ 1.33 l− l l+ n u n u −∞ 2.93 l− l l l+ n u −∞ 0.49 l− l− l+ n u n u −0.5 0.99 l− l n u n u n u −0.5 1.83 l− l l n u −0.5 64 frederico furtado and felipe pereira 6, 4(2004) our findings can be summarized as follows: • distinct mixing regimes occur depending on whether one of the driving mechanisms (nonlinearity and heterogeneity) dominates the mixing dynamics or not. • if nonlinearity dominates, then two distinct mixing regimes are possible, according to whether the nonlinear effects are stabilizing (n s regime) or destabilizing (n u regime). • if heterogeneity dominates, then the linear regime l is observed. in this case, mixing regions in linear and nonlinear flows grow at essentially identical asymptotic rates. • intermediate mixing regimes (sublinear and superlinear) occur when nonlinearity and heterogeneity compete for the dominance of the mixing dynamics. figure 4 is a pictorial representation of these results in terms of a “phase-diagram” in the m vs. cvk plane. the results suggest that for a given value of β there is a critical value for m (mcritical in the figure). at nearby values, the two-phase flow system behaves according to the several mixing regimes discussed above. the location of mcritical seems to depend on β, being smaller for larger values of β (c.f. table 1). figure 4: a pictorial representation of the distinct mixing regimes that result from the interplay between nonlinearity and heterogeneity: nonlinearity controlled regimes n s (nonlinear stable) and n u (nonlinear unstable); heterogeneity controlled regime l (linear); intermediate regimes l− (sublinear) and l+ (superlinear). figure 5 illustrates the crossover from heterogeneity controlled to nonlinearity controlled mixing as the heterogeneity strength is varied while the nonlinearity strength is kept fixed. the figure refers to a study in which β = −∞, m = 5 were kept fixed and the strength of heterogeneity was increased by increasing cvk. 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 65 figure 5: large time mixing length curves for two-phase flows (m = 5.0) in β = −∞ permeability fields corresponding to distinct mixing regimes. from top to bottom the cvk values are 0.54, 1.33 and 2.93 and the mixing regimes are nonlinear unstable n u , superlinear l+ and linear l, respectively. 66 frederico furtado and felipe pereira 6, 4(2004) each picture in this figure shows the log-log plot of the mixing length �(t) as a function of the travel distance (the irregular curve). also plotted are two straight lines with vertical positions fixed to agree with the last plotted point on the mixing length curves. the slopes of these lines are 0.5, the asymptotic scaling exponent for the linear mixing regime l for this value of β, and 1, the asymptotic scaling exponent of the nonlinear unstable mixing regime n u . it is clear that as the cvk is increased the mixing length curves bend continuously from the n u regime towards the l regime. we close this section with a few remarks on the scale up problem. for weak heterogeneities the mixing process is ‘convective’, with the mixing region growing linearly as a function of time. this behavior is typical of mixing processes driven by the nonlinear instability of the flow and should be properly modeled with a hyperbolic renormalization of the flow equations. however, for sufficiently strong heterogeneities with short correlation length, the mixing process is fickian and a dispersive renormalization, like the one developed recently for linear flow [26], should perhaps be more appropriate. 6 computational solutions: the decoupling hypothesis the studies [30, 31, 45, 32, 14] assume that the transport and saturation equations are not coupled. under this assumption distinct theoretical developments for the scale up of two-phase flow systems have been reported in these papers. here we test such assumption by computing directly the mixing length growth in a fixed (in time) velocity field. we consider the nonlinear unstable n u mixing regime characterized by weak heterogeneities. new ensemble averages have been performed using the sets of parameters (cvk,m ,β) displayed in table 1 which produced the nonlinear unstable mixing regime n u . in these new ensemble average studies the velocity field was computed at the first time step of the simulations (with the given intial condition for the saturation) and was kept fixed in time. thus, the computations were restricted to the solution of the saturation equation (2). all the new ensemble average studies fall in the linear mixing regime l. the asymptotic scaling exponents for the linear mixing regimes which we found were 0.5 for the ensemble averages with β = −∞ and 0.75 for the ensemble averages having β = −0.5 for the hurst exponent. figures 6 and 7 illustrate our new results for β = −∞ and β = −0.5, respectively. consider figure 6 first. it refers to the case cvk = 0.54 and m = 5.0. this figure illustrates the change in the mixing regime as the velocity field is kept fixed in time. each picture in this figure shows the log-log plot of the mixing length �(t) as a function of the travel distance (the irregular curve). also plotted are two straight lines with vertical positions fixed to agree with the last plotted point on the mixing length curves. the slopes of these lines are 0.5, the asymptotic scaling exponent for the linear mixing regime l for β = −∞, and 1, the asymptotic scaling exponent of the 6, 4(2004) on the scale up problem for two-phase flow in petroleum reservoirs 67 figure 6: the effect of decoupling saturation and pressure equations. if the coupling is taken into account we have determined that the mixing regime is the nonlinear unstable n u (curve labeled with variable velocity) for the parameters cvk = 0.54, m = 5.0 and β = −∞. once the equations are decoupled, the computed mixing regime is the linear l (curve labeled with constant velocity). clearly the dark solid line would lead to incorrect predictions for the behavior of the mixing process as a function of time. figure 7: the effect of decoupling saturation and pressure equations. if the coupling is taken into account we have determined that the mixing regime is the nonlinear unstable n u (curve labelled with variable velocity) for the parameters cvk = 0.99, m = 10.0 and β = −0.5. once the equations are decoupled, the computed mixing regime is the linear l (curve labelled with constant velocity). predictions made with the constant (in time) velocity field would be very inaccurate. 68 frederico furtado and felipe pereira 6, 4(2004) nonlinear unstable mixing regime n u . it is clear that as the velocity field is kept fixed in time the mixing length curves change from the n u regime to the l regime. it is also clear from these results that the new mixing length curves would lead to incorrect predictions for the behavior of the mixing process as a function of time. figure 7 refers to a study for the parameters cvk = 0.99 and m = 10.0. the same behavior for the mixing length growth described by figure 6 was observed here. as the velocity field is kept fixed the nonlinear unstable mixing regime n u changes to the linear mixing regime l. consequently predictions made with the constant (in time) velocity field would be very inaccurate. the authors are now investigating the validity of the decoupling hypothesis in the description the linear mixing regime l. 7 conclusions in this paper we reviewed our recent effort to characterize fluid mixing regimes in multiphase flow in multiscale heterogeneous porous formations. we believe such a characterization is an essential step in the scientific program leading to an improved understanding of the scale up problem for porous media flow. our recent results on the characterization of mixing regimes for two-phase, immiscible, incompressible displacement in petroleum reservoirs have been used to investigate a hypothesis that has been used in the literature in the development of theoretical predictions for the scale up problem. according to these studies the saturation and pressure equations are not coupled. highly resolved numerical simulations were conducted to study this hypothesis and we determined that for the mixing regime characterized by weak heterogeneities 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[49] j. zhu. a numerical study of stochastic dispersion for two-phase immiscible flow in porous media. m.sc. thesis, university of wyoming, 2001. cubo a mathematical journal vol.20, no¯ 01, (79–94). march 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000100079 anti-invariant ξ⊥-riemannian submersions from hyperbolic β-kenmotsu manifolds mohd danish siddiqi department of mathematics, faculty of science, jazan university, jazan-kingdom of saudi arabia. anallintegral@gmail.com, msiddiqi@jazanu.edu.sa mehmet akif akyol department of mathematics, faculty of arts and sciences, bingöl university, 12000 bingöl, turkey, mehmetakifakyol@bingol.edu.tr abstract in this paper, we introduce anti-invariant ξ⊥-riemannian submersions from hyperbolic β-kenmotsu manifolds onto riemannian manifolds. necessary and sufficient conditions for a special anti-invariant ξ⊥-riemannian submersion to be totally geodesic are studied. moreover, we obtain decomposition theorems for the total manifold of such submersions. resumen en este art́ıculo se introducen las submersiones ξ⊥-riemannianas anti-invariantes desde variedades hiperbólicas β-kenmotsu sobre variedades riemannianas. se estudian condiciones necesarias y suficientes para que ciertas submersiones ξ⊥-riemannianas antiinvariantes especiales sean totalmente geodésicas. más aún, se obtienen teoremas de descomposión para la variedad total de dichas submersiones. keywords and phrases: riemannian submersion anti-invariant ξ⊥-riemannian submersions, hyperbolic β-kenmotsu manifolds, integrability conditions. geometry. 2010 ams mathematics subject classification: 53c25, 53c20, 53c50, 53c40. http://dx.doi.org/10.4067/s0719-06462018000100079 ignacio castillo ignacio castillo ignacio castillo ignacio castillo 80 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) 1 introduction the geometry of riemannian submersions between riemannian manifolds has been intensively studied and sevral results has been pulished (see o’neill [7] and gray [4]). in [11] waston defined almost hermitian submersion between almost hermitian manifolds and in most cases he show that the base manifold and each fiber has the same kind of structure as the total space. he also show that the vertical and horizontal distributions are invariant. on the other hand, the geometry of anti-invariant riemannian submersions is different from the geometry of almost hermitian submersions. for example, since every holomorphic map between kahler manifolds is harmonic [2], it follows that any holomorphic submersion between kahler manifolds is harmonic. however, this result is not valid for anti-invariant riemannian submersions, which was first studied by sahin in [8]. similarly, ianus and pastore [5] shows φ-holomorphic maps between contact manifolds are harmonic. this implies that any contact submersion is harmonic. however, this result is not valid for anti-invariant riemannian submersions. in [1], chinea defined almost contact riemannian submersion between almost contact metric manifolds. in [6], lee studied the vertical and horizontal distribution are φ-invariant. moreover, the characteristic vector field ξ is horizontal. we note that only φ-holomorphic submersions have been consider on an almost contact manifolds [3]. it was 1976, upadhyay and dube [10] introduced the notion of almost hyperbolic contact (f, g, η, ξ)structure. some properties of cr-submanifolds of trans hyperbolic sasakian manifold were studied in [9]. in this paper, we consider a riemannian submersion from a hyperbolic β-kenmotsu manifolds under the assumption that the fibers are anti-invariant with respect to the tensor field of type (1, 1) of almost hyperbolic contact manifold. this assumption implies that the horizontal distribution is not invariant under the action of tensor field of the total manifold of such submersions. in other words, almost hyperbolic contact are useful for describing the geometry of base manifolds, anti-invariant submersion are however served to determine the geometry of total manifold. the paper is organized as follows: in section 2, we present the basic information needed for this paper. in section 3, we give the definition of anti-invariant ξ⊥-riemannian submersions. we also introduce a special anti-invariant ξ⊥-riemannian submersions and obtain necessary and sufficient conditions for such submersions to be totally geodesic or harmonic. in section 4, we give decomposition theorems by using the existence of anti-invariant ξ⊥-riemannian submersions and observe that such submersions put some restrictions on the geometry of the total manifold. 2 preliminaries in this section, we define almost hyperbolic contact manifolds, recall the notion of riemannian submersion between riemannian manifolds and give a brife review of basic facts if riemannian submersion. let m be an almost hyperbolic contact metric manifold with an almost hyperbolic contact metric structure (φ, ξ, η, gm), where φ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form and cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 81 gm is a compatible riemannian metric on m such that φ2 = i − η ⊗ ξ, φξ = 0, η ◦ φ = 0, η(ξ) = −1, (2.1) gm(φx, φy) = −gm(x, y) − η(x)η(y) (2.2) gm(x, φy) = −gm(φx, y), gm(x, ξ) = η(x) (2.3) an almost hyperbolic contact metric structure (φ, ξ, η, gm) on m is called trans-hyperbolic sasakian [9] if and only if (∇xφ)y = α(g(x, y)ξ − η(y)φx) + β(g(φx, y) − η(y)φx) (2.4) for all x, y tangent to m, α and β are smooth functions on m and we say that the trans-hyperbolic sasakian structure of type (α, β). from the above condition it follows that ∇xξ = −α(φx) + β(x − η(x)ξ), (2.5) (∇xη)y = −αg(φx, y) + βg(φx, φy), (2.6) where ∇ is the riemannian connection of levi-civita covariant differentiation. more generally one has the notion of a hyperbolic β-kenmotsu structure which be defined by (∇xφ)y = β(g(φx, y)ξ − η(y)φx), (2.7) where β is non-zero smooth function. also we have ∇xξ = β[x − η(x)ξ]. (2.8) thus α = 0 and therefore a trans-hyperbolic sasakian structure of type (0, β) with a non-zero constant is always hyperbolic β-kenmotsu manifold. let (mm, gm) and (n n, gn) be riemannian manifolds, where dimm = m, dimn = n and m > n. a riemannian submersion f : m → n is a map from m onto n satisfying the following axioms: (1) (s1) f has maximal rank (2) (s2) the differential f∗ preserves the lengths of horizontal vectors. for each q ∈ n, f−1(q) is an (m − n)-dimensional submanifold of m. the submanifold f−1(q) are called fibers. a vector field on m is called vertical if it is always tangent to fibers. a vector field on m is called horizontal if it is always orthogonal to fibers. a vector field x on m is called basic if x is horizontal and f-related to a vector field x∗ on n, i.e., f∗xp = x∗f(p) for all p ∈ m. note that we denote the projection morphisms on the distributions kerf∗ and (kerf∗) by v and h, respectively. we recall the following lemma from o’neill [7]. 82 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) lemma 2.1. let f : m → n be a riemannian submersion between riemannian manifolds and x, y be basic vector fields of m. then (1) (1) gm(x, y) = gn(x∗, y∗) ◦ f. (2) (2) the horizontal part [x, y]h of [x, y] is a basic vector field and corresponds to [x∗, y∗], i.e., f∗([x, y]) = [x∗, y∗]. (3) (3) [v, x] is vertical for any vector field v of kerf∗. (4) (4) ((∇)mx y)h is the basic vector field corresponding to ∇nx∗y∗. the geometry of riemannian submersion is characterized by o’neill’s tensor t and a defined for vector fields e, f on m by aef = h∇hevf + v∇hehf (2.9) tef = h∇vevf + v∇vehf (2.10) where ∇ is the levi-civita connection of gm. it is easy to see that a riemannian submersion f : m → n has totally geodesic fibers if and only if t vanishes identically. for any e ∈ (tm), tc = tvc and a is horizontal, a = ahe. we note that the tensor t and a satisfy tuw = twu, u, w ∈ (kerf∗) (2.11) axy = −ayx = 1 2 v[x, y], x, y ∈ (kerf∗)⊥ (2.12) on the other hand, from (2.6) and (2,7), we have ∇vw = tvw + ∇̄vw (2.13) ∇vx = h∇vx + tvx (2.14) ∇xv = axv + v∇xv (2.15) ∇xy = h∇xy + axv (2.16) for x, y ∈ (kerf∗)⊥ and v, w ∈ (kerf∗), where ∇̄vw = v∇vw. if x is basic then h∇vx = axv. finally, we recall the notion of harmonic maps between riemannian manifolds. let (m, gm) and (n, gn) be riemannian manifolds and supposed that φ : m → n is a smooth map. then the differential φ∗ of φ can be viewed a section of the bundle hom(tm, φ −1tn) → m, where φ−1tn is the pullback bundle which has fibers (φ−1tn)p = tφ(p)n, p ∈ m. hom(tm, φ−1tn) cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 83 has a connection ∇ induced from the levi-civita connection ∇m and the pullback connection ∇φ. then the second fundamental form of φ is given by (∇φ∗)(x, y) = ∇φ x φ ∗ (y) − φ ∗ (∇mx y) (2.17) for x, y ∈ tm. it is known that the second fundamental form is symmetric. a smooth map φ : (m, gm) → (n, gn) is said to be harmonic if trace(∇φ∗) = 0. on the other hand, the tensor field of φ is the section τ(φ) of (φ−1tn) defined by τ(φ) = divφ∗ = m∑ i=1 (∇φ∗)(ei, ei), (2.18) where {e1, .....em} is the orthogonal frame on m. then it follows that φ is harmonic if and only if τ(φ) = 0 (see [7]). 3 anti-invariant ξ⊥riemannian submersions in this section, we define anti-invariant ξ⊥riemannian submersion from hyperbolic β-kenmotsu manifold onto a riemannian manifold and investigate the integrability of distributions and obtain a necessary and sufficient condition for such submersions to be totally geodesic map. we also investigate the harmonicity of a special riemannian submersion. definition 3.1. let (m, gm, φ, ξ, η) be a hyperbolic β-kenmotsu manifold and (n, gn) a riemannian manifold. suppose that there exists a riemannian submersion f : m → n such that ξ is normal to kerf∗ and kerf∗ is anti-invariant with respect to φ, ie., φ(kerf∗) ⊂ (kerf∗)⊥. then we say that f is an anti-invariant ξ⊥-riemannian submersion. now, we assume that f : (m, gm, φ, ξ, η) → (n, gn) is an anti-invariant ξ⊥-riemannian submersion. first of all, from definition 3.1, we have (kerf∗) ⊥ ∩ (kerf∗) 6= 0. we denote the complementary orthogonal distribution to φ(kerf∗) in (kerf∗) ⊥ by µ. then we have (kerf∗) ⊥ = φ(kerf∗) ⊕ µ, (3.1) where φ(µ) ⊂ µ. hence µ contains ξ. thus, for x ∈ (kerf∗)⊥, we have φx = bx + cx, (3.2) where bx ∈ (kerf∗) and cx ∈ (µ). on the other hand, since f∗(kerf∗)⊥ = tn and f is a riemannian submersion, using (3.2), we have gn (f∗φv, f∗φcx) = 0 for any x ∈ (kerf∗)⊥ and v ∈ (kerf∗), which implies tn = f∗(φ((kerf∗)) ⊕ f∗(µ). 84 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) example 3.2. let us consider a 5-dimensional manifold m̄ = { (x1, x2, x3, x4, z) ∈ r5 : z 6= 0 } , where (x1, x2, x3, x4, z) are standard coordinates in r 5. we choose the vector fields e1 = e −z ∂ ∂x1 , e2 = e −z ∂ ∂x2 , e3 = e −z ∂ ∂x3 , e4 = e −z ∂ ∂x4 , e5 = e −z ∂ ∂x1 , which are linearly independent at each point of m̄. we define g by g = e2zg, where g is the euclidean metric on r5. hence {e1, e2, e3, e4, e5} is an orthonormal basis of m̄. we consider an 1-form η defined by η = ezdz, η(x) = g(x, e5), ∀x ∈ tm̄. we defined the (1, 1) tensor field φ by φ { 2∑ i=2 ( xi ∂ ∂xi + xi+2 ∂ ∂xi+2 + z ∂ ∂z ) } = 2∑ i=2 ( xi ∂ ∂xi+2 − xi+2 ∂ ∂xi ) . thus, we have φ(e1) = e3, φ(e2) = e4, φ(e3) = −e1, φ(e4) = −e2, φ(e5) = 0. the linear property of g and φ yields that η(e5) = −1, φ 2(x) = x − η(x)e5 g(φx, φy) = −g(x, y) − η(x)η(y), for any vector fields x, y on m̄. thus, m̄ (φ, ξ, η, g) defines an almost hyperbolic contact metric manifold with ξ = e5. moreover, let ∇̄ be the levi-civita connection with respect to metric g. then we have [e1, e2] = 0. similarly [e1, ξ] = e −ze1, [e2, ξ] = e −ze2, [e3, ξ] = e −ze3, [e4, ξ] = e −ze4, [ei, ej] = 0, 1 ≤ i 6=≤ 4. the riemannian connection ∇̄ of the metric g is given by 2g(∇̄xy, z) = xg(y, z) + yg(z, x) − zg(x, y) − g(x, [y, z]) − g(y, [x, z]) + g(z, [x, y]), by koszul’s formula, we obtain the following equations ∇̄e1e1 = −e−zξ, ∇̄e2e2 = −e−zξ, ∇̄e3e3 = −e−zξ, ∇̄e4e4 = −e−zξ, ∇̄ξξ = 0, ∇̄ξei = 0, ∇̄eiξ = e−zei, 1 ≤ i ≤ 4 and ∇̄eiei = 0 for all 1 ≤ i, j ≤ 4. thus, we see that m is a trans-hyperbolic sasakian manifold of type (0, e−z), which is hyperbolic β-kenmotsu manifold. here α = 0 and β = e−z. now, we define (1, 1) tensor field as follows φ(x1, x2, x3, x4, z) = (−x3, −x4, x1, x3, z). now, we can give the following example. cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 85 example 3.3. let (m1, g1 = e 2zg, φ, ξ, η) be an almost hyperbolic contact manifolds and m2 be r 3. the riemannian metric tensor field g2 is defined by g2 = e 2z(dy1 ⊗ dy1 +dy2 ⊗ dy2 +dy3 ⊗ dy3) on m2. let φ be a submersion defined by φ : r5 −→ r3 (x1, x2, x3, x4, z) ( x1 + x3√ 2 , z, x1 + x2√ 2 ) then it follows that kerφ∗ = span {v1 = ∂x1 − ∂x3, v2 = ∂x2 − ∂x2} and (kerφ∗) ⊥ = span {x1 = ∂x1 + ∂x3, x2 = ∂x2 + ∂x2, x3 = z = ξ} hence we have φv1 = x1 and φv2 = x2. it means that φ(kerφ) ⊂ (kerφ)⊥. a straight computations, we get φ∗x1 = ∂y1, φ∗x2 = ∂y3 and φ∗x3 = ∂y2. hence, we have g1(xi, xi) = g2(φ∗xi, φ∗xi), for i = 1, 2, 3. thus φ is a anti-invariant ξ⊥ riemannian submersion. lemma 3.4. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn). then we have gm(cy, φv) = 0, (3.3) gm(∇xcy, φv) = −gm(cy, φaxv) (3.4) for x, y ∈ ((kerf∗)⊥) and v ∈ (kerf∗). proof. for y ∈ ((kerf∗)⊥) and v ∈ (kerf∗), using (2.2), we have gm(cy, φv) = gm(φy − by, φv) = gm(φy, φv) = −gm(y, v) − η(y)η(v) = −gm(y, v) = 0 since by ∈ (kerf∗) and φv, ξ ∈ ((kerf∗)⊥). differentiating (3.3) with respect to x, we get gm(∇xcy, φv) = − gm(cy, ∇xφv) =gm(cy, (∇xφ)v) − gm(cy, φ(∇xv)) = − gm(cy, φ(∇xv)) = − gm(cy, φaxv) − gm(cy, φν∇xv) = − gm(cy, φaxv) due to φν∇xv ∈ (kerf∗)). our assertion is complete. 86 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) we study the integrability of the distribution (kerf∗) ⊥ and then we investigate the geometry of leaves of kerf∗ and (kerf∗) ⊥. we note it is known that the distribution (kerf∗) is integrable. theorem 3.5. let f be an anti-invaraint ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn). the followings are equivalent. (1) (kerf∗) ⊥ is integrable, (2) gn((∇f∗)(y, bx), f∗φv) = gn((∇f∗)(x, by), f∗φv) +gm(cy, φaxv) − gm(cx, φayv) +βη(y)gm(x, v) − βη(x)gm(y, v), (3) gm(axby − ayby, φv) = gm(cy, φaxv) − gm(cx, φayv) +βη(y)gm(x, v) − βη(x)gm(y, v). for x, y ∈ (kerf∗)⊥ and v ∈ (kerf∗). proof. for y ∈ (kerf∗)⊥ and v ∈ (kerf∗), from definition 3.1, φv ∈ (kerf∗)⊥ and φy ∈ (kerf∗)⊕ µ. using (2.2) and (2.4), we note that for x ∈ (kerf∗)⊥, gm(∇xy, v) = gm(∇xφy, φv) − βη(y)gm(x, v) (3.5) −(α + β)η(x)η(y)η(v). therefore, from (3.5), we get gm([x, y], v) = gm(∇xφy, φv) − gm(∇yφx, φv) = βη(x)gm(y, v) − βη(y)gm(x, v) = gm(∇xby, φv) + gm(∇xcy, φv) −gm(∇ybx, φv) − gm(∇ycx, φv) −βη(y)gm(x, v) + βη(x)gm(y, v). since f is a riemannian submersion, we obtain gm([x, y], v) = gn(f∗∇xby, f∗φv) + gm(∇xcy, φv) −gn(f∗∇ybx, f∗φv) − gm(∇ycx, φv) −βη(y)gm(x, v) + βη(x)gm(y, v). cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 87 thus, from (2.15) and (3.4), we have gm([x, y], v) = gn(−(∇f∗(x, by) + (∇f∗)(y, bx), f∗φv) −gm(cy, φaxv + gm(cx, φayv) −βη(y)gm(x, v) + βη(x)gm(y, v). which proves (1) ⇐⇒ (2). on the other hand, using (2.14), we obtain (∇f∗)(y, bx) − (∇f∗)(x, by) = −f∗(∇ybx − ∇xby) = −f∗(aybx − axby), which shows that (2) ⇐⇒ (3) corollary 3.6. let f be an anti-invaraint ξ⊥-riemannian submersion from a hyperbolic βkenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn) with (kerf∗) ⊥ = φ(kerf∗)⊕ < ξ >. then the following are equivalent: (1) (kerf∗) ⊥ is integrable (2) (∇f∗)(x, φy) + βη(x)f∗y = (∇f∗)(y, φx) + βη(y)f∗x (3) axφy + βη(x)y = ayφx + βη(y)x, for x, y ∈ (kerf∗)⊥. theorem 3.7. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn). the following are equivalent: (1) (kerf∗) ⊥ defines a totally geodesic foliation on m. (2) gm(axby, φv) = gm(cy, φaxy) − βη(x)gm(x, v) − βη(x)gm(y, v), (3) gn((∇f∗)(y, φx), f∗φv) = gm(cy, φaxv) − βη(x)gm(x, v) − βη(x)gm(y, v), for x, y ∈ (kerf∗) ⊥ and v ∈ (kerf∗). proof. for x, y ∈ (kerf∗)⊥ and v ∈ (kerf∗), from (3.5), we have gm(∇xy, v) = gm(axby, φv) + gm(∇xcy, φv) − βη(y)gm(x, v) − βη(x)η(y)η(v) then from (3.4), we have gm(∇xy, v) = gm(axby, φv) + gm(cy, φaxv) − βη(y)gm(x, v) − βη(x)η(y)η(v) which shows (1) ⇐⇒ (2). on the other hand, from (2.12) and (2.14), we have gm(axby, φv) = gn(−(∇f∗)(x, by), f∗φv), which proves (2) ⇐⇒ (3). 88 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) corollary 3.8. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic βkenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn) with (kerf∗) ⊥ = φ(kerf∗)⊕ < ξ >. then the following are equivalent: (1) (kerf∗) ⊥ defines a totally geodesic folition on m (2) axφy = βη(y)x − (α + β)η(x)y (3) (∇f∗)(y, φx) = βη(y)f∗x − β)η(x)f∗y for x, y ∈ (kerf∗)⊥. theorem 3.9. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn). the following are equivalent: (1) kerf∗ defines a totally geodesic folition on m (2) −gn(∇f∗)(v, φx, f∗φw) = 0 (3) tvbx + acxv ∈ (µ), for x, ∈ (kerf∗)⊥ and v, w ∈ (kerf∗) proof. for x, ∈ (kerf∗)⊥ and v, w ∈ (kerf∗), gm(w, ξ) = 0 implies that from (2.4) gm(∇vw, ξ) = −gm(w, ∇vξ) = gm(w, β(v − η(v)ξ)) = 0. thus we have gm(∇vw, x) = −gm(φ∇vw, φx) − η((∇vw)η(x) = −gm(φ∇v w, φx) = −gm(∇vφw, φx) + gm((∇v φ)w, φx) = gm(φw, ∇v φx). since f is riemannian submersion, we have gm(∇vw, x) = gn(f∗φw, f∗∇vφx) = −gn(f∗φw, (∇f∗)(vφx)), which proves (1) ⇐⇒ (2). by direct calculation, we derive −gn(f∗φw, (∇f∗)(vφx)) = gm(φw, ∇vφx) = gm(φw, ∇v bx + ∇vcx) = gm(φw, ∇vbx + [v, cx] + ∇cxv). cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 89 since [v, cx] ∈ (kerf∗), from (2.10) and (2.12), we obtain −gn(f∗φw, (∇f∗)(vφx)) = gm(φw, tvbx + acxv), which proves (2) ⇐⇒ (3). as an analouge of a lagrangian riemannian submersion in [11], we have a similar result; corollary 3.10. let f be an anti-invaraint ξ⊥-riemannian submersion from a hyperbolic βkenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn) with (kerf∗) ⊥ = φ(kerf∗)⊕ < ξ >. then the following are equivalent: (1) (kerf∗) ⊥ defines a totally geodesic folition on m (2) −(∇f∗)(v, φx) = 0 (3) tvφw = 0, x, ∈ (kerf∗)⊥ and v, w ∈ (kerf∗). proof. from theorem 3.6, it is enough to show (2) ⇐⇒ (3). using (2.14) and (2.11), we have −gn(f∗φw, (∇f∗)(vφx)) = gm(∇vφw, φx) = gm(tvφw, φx). since tvφw ∈ (kerf∗), the proof is complete. we note that a differentiable map f between two riemannian manifolds is called totally geodesic if ∇f∗ = 0. for the special riemannian submersion, we have the following characterization. theorem 3.11. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic βkenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn) with (kerf∗) ⊥ = φ(kerf∗)⊕ < ξ >. then f is a totally geodesic map if and only if tvφw = 0, v, w ∈ (kerf∗) (3.6) and axφw = 0, x ∈ (kerf⊥∗ ). (3.7) proof. first of all, we recall that the second fundamental form of a riemannian submersion satisfies (∇f∗)(x, y) = 0 ∀ x, y ∈ (kerf⊥∗ ). (3.8) for v, w ∈ (kerf∗), we get 90 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) (∇f∗)(x, y) = f∗(φtvφw). (3.9) on the other hand, from (2.1), (2.2) and (2.14), we get (∇f∗)(x, w) = f∗(φaxφw), x ∈ (kerf⊥∗ ). (3.10) therefore, f is totally geodesic if and only if φ(tvφw) = 0 ∀ v, w ∈ (kerf⊥∗ ). (3.11) and φ(axφw) = 0 ∀ x ∈ (kerf⊥∗ ). (3.12) from (2.2), (2.6) and (2.7), we have tvφw = 0 ∀ v, w ∈ (kerf∗). (3.13) and axφw = 0 ∀ x ∈ (kerf⊥∗ ). from (2.4), f is totally geodesic if and only the equation (3.6) and (3.7) hold finally, in this section, we give a necessary and sufficient condition for a special riemannian submersion to be harmonic as an analouge of lagrangian riemannian submersion in [11]. theorem 3.12. let f be an anti-invaraint ξ⊥-riemannian submersion from a hyperbolic βkenmotsu manifold (m, gm, φ, ξ, η) onto a riemannian manifold (n, gn) with (kerf∗) ⊥ = φ(kerf∗)⊕ < ξ >. then f is harmonic if and only if trace(φtv ) = 0 for v ∈ (kerf∗). proof. from [5], we know that f is harmonic if and only if f has minimal fibers. thus f is harmonic if and only if ∑m1 i=1 teiei = 0. on the other hand, from (2.4), (2.11) and (2.10), we have tvφw = φtvw (3.14) due to ξ ∈ (kerf⊥ ∗ ) for any v, w ∈ (kerf∗). using (3.14), we get m1∑ i=1 gm(teiφei, v) = m1∑ i=1 gm(φteiφei, v) = − m1∑ i=1 gm(teiei, φv) for any v ∈ (kerf∗). thus skew-symmetric t implies that m1∑ i=1 gm(φteiφei, v) = − m1∑ i=1 gm(teiei, φv). using (2.8) and (2.2), we have m1∑ i=1 gm(ei, φtvei) = − m1∑ i=1 gm(φei, tvei) = − m1∑ i=1 gm(teiei, φv) which shows our assertion. cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 91 4 decomposition theorems in this section, we obtain decomposition theorems by using the existence of anti-invariant ξ⊥riemannian submersions. first, we recall the following. theorem 4.1. [10] let g be a riemannian metric on the manifold b = m × n and assume that the canonical foliations dm and dn intersect perpendicular every where. then g is the metric tensor of (1) (i) a twisted product m ×f n if and only if dm is totally geodesic foliation and dn is a totally umbilical foliation. (2) (ii) a warped product m ×f n if and only if dm is totally geodesic foliation and dn is a spheric foliation, i.e., it is umbilical and its mean curvature vector field is parallel. (3) (iii) a usual product of riemannian manifold if and only if dm and dn are totally geodesic foliations. our first decomposition theorem for anti-invariant ξ⊥-riemannian submersion comes from theorem 3.4 and 3.6 in terms of the second fundamental forms of such submersions. theorem 4.2. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) on to a riemannian manifold (n, gn). then m is locally product manifold if and only if −gn((∇f∗)(y, φx), f∗φv) = gm(cy, φaxv) − βη(y)gm(x, v) and −gn((∇f∗)(v, φx), f∗φw) = 0 for x, y ∈ (kerf⊥ ∗ ) and v, w ∈ (kerf∗). from corollary 3.5 and 3.7, we have the following decomposition theorem: theorem 4.3. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) on to a riemannian manifold (n, gn) with (kerf ⊥ ∗ )⊕ < ξ >. then m is a locally product manifold if and only if axφy = (α+β)η(y)x and tvφw = 0, for x, y ∈ (kerf⊥∗ ) and v, w ∈ (kerf∗). next we obtain a decomposition theorem which is related to the notion of a twisted product manifold. theorem 4.4. let f be an anti-invariant ξ⊥-riemannian submersion from a hyperbolic β-kenmotsu manifold (m, gm, φ, ξ, η) on to a riemannian manifold (n, gn) with (kerf ⊥ ∗ )⊕ < ξ >. then m is locally twisted product manifold of the form mkerf⊥ ∗ ×f mkerf∗ if and only if 92 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) tvφx = −gm(x, tvv) ‖v‖−2 − βη(y)gm(φx, φv). and axφy = βη(y)x for x, y ∈ (kerf⊥ ∗ ) and v ∈ (kerf∗), where m(kerf⊥ ∗ ) and m(kerf∗) are integrable manifolds of the distributions (kerf⊥ ∗ ) and (kerf∗). proof. for x ∈ (kerf⊥ ∗ ) and v ∈ (kerf∗), from (2.4) and (2.11), we obtain gm(∇vw, x) = gm(tvφw, φx) = −gm(φw, tvφx) since tv is skew-symmetric. this implies that kerf∗ is totally umbilical if and only if tvφx − βη(v)gm(φx, φv) = −x(λ)φv, where λ is a function on m. by direct computation, tvφx = −gm(x, tvv) ‖v‖−2 − βη(y)gm(φx, φv). then the proof follows from corollary 3.5 however, in the sequel, we show that the notion of anti-invariant ξ⊥-riemannian submersion puts some restrictions on the source manifold. theorem 4.5. let (m, gm, φ, ξ, η) be a hyperbolic β-kenmotsu manifold and (n, gn) be a riemannian manifold . then there does not exist an anti-invariant ξ⊥-riemannian submersion from m to n with (kerf∗) ⊥ = φ(kerf∗) ⊥⊕ < ξ > such that m is a locally proper twisted product manifold of the form mkerf∗ ×f m(kerf∗)⊥. proof. suppose that f : (m, gm, φ, ξ, η) −→ (n, gn) is an anti-invaraiant ξ⊥-riemannian submersion with (kerf∗) ⊥ = φ(kerf∗) ⊥⊕ < ξ > and m is a locally twisted product of the form mkerf∗ ×f m(kerf∗)⊥ .then mkerf∗ is a totally geodesic foliation and m(kerf⊥∗ ) is a totally umbilical foliation. we denote the second fundamental form of m(kerf⊥ ∗ ) by h. then we have gm(∇xy, v) = gm(h(x, y), v) x, y ∈ ((kerf∗)⊥, v ∈ (kerf∗). (4.1) since m(⊥kerf∗ ) is a totally umbilical foliation, we have gm(∇xy, v) = gm(h, v)gm(x, y), where h is the mean curvature vector field of m(kerf∗)⊥. on the other hand, from (3.5), we derive gm(∇xy, v) = −gm(φy, ∇xφv) − βη(y)g(x, v) − βη(x)η(y)η(v). (4.2) cubo 20, 1 (2018) anti-invariant ξ⊥-riemannian submersions . . . 93 using (2.13), we obtain gm(∇xy, v) = gm(φy, axφv) − βη(y)g(x, v) − βη(x)η(y)η(v) (4.3) = gm(y, axφv) − βg(x, v) − βη(x)η(v)ξ) therefore, from (4.1), (4.3) and (2.2), we have axφv = gm(h, v)φx + η(axφv)ξ. since axφv ∈ (kerf∗), η(axφv) = gm(axφv, ξ) = 0. thus, we have axφv = gm(h, v)φx. hence, we derive gm(axφv, φx) − βη(x)η(v)g(y, φx) = −gm(h, v) { ‖x‖2 − η2(x) } gm(∇xφv, φx) = −gm(h, v) { ‖x‖2 − η2(x) } + βη(x)η(v)g(y, φx) gm(∇xy, v) + βη(y)g(x, v) − βη(x)η(y)η(v) = −gm(h, v) { ‖x‖2 − η2(x) } + βη(x)η(v)g(y, φx). thus using (2.9), we have axx = 0, which implies βη(x)gm(x, v) = −gm(h, v) { ‖x‖2 − η2(x) } + βη(x)η(y)[η(v) − gm(y, φx)] for every x ∈ ((kerf⊥ ∗ ), v ∈ (kerf∗). choosing x which is orthogonal to ξ gm(h, v) ‖x‖2 = 0. since gm is the riemannian metric and h ∈ (kerf∗), we conclude that h = 0, which shows kerf⊥∗ is totally geodesic, so m is usual product of riemannian manifolds. references [1] chinea, c. almost contact metric submersions, rend. circ. mat. palermo, 43(1), 89-104, 1985. [2] eells, j., sampson, j. h. harmonic mappings of riemannian manifolds, amer. j. math., 86, 109-160. 1964. [3] falcitelli, m., ianus, s., pastore, a. m. riemannian submersions and related topics, (world scientific, river edge, nj, 2004. [4] gray, a. pseudo-riemannian almost product manifolds and submersion, j. math. mech., 16, 715-737, 1967. 94 mohd danish siddiqi and mehmet akif akyol cubo 20, 1 (2018) [5] ianus, s., pastore, a. m., harmonic maps on contact metric manifolds, ann. math. blaise pascal, 2(2), 43-53, 1995. [6] lee, j. w., anti-invariant ξ⊥− riemannian submersions from almost contact manifolds, hacettepe j. math. stat. 42(2), 231-241, 2013. [7] o’neill , b. the fundamental equations of a submersions, mich. math. j., 13, 458-469, 1996. [8] sahin, b. anti-invariant riemannian submersions from almost hermition manifolds, cent. eur. j. math., 8(3), 437-447, 2010. [9] siddiqi, m. d., ahmed, m and ojha, j.p., cr-submanifolds of nearly-trans hyperbolic sasakian manifolds admitting semi-symmetric non-metric connection, afr. diaspora j. math. (n.s.), vol 17(1), 93-105, 2014. [10] upadhyay, m. d, dube., k. k., almost contact hyperbolic (f, g, η, ξ) structure, acta. math. acad. scient. hung., tomus, 28, 1-4, 1976. [11] watson, b. almost hermitian submersions, j. differential geometry, 11(1), 147-165, 1976. introduction 0.2cmpreliminaries 0.2cm anti-invariant riemannian submersions 0.2cmdecomposition theorems cubo a mathematical journal vol.21, no¯ 01, (61–75). april 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000100061 on fractional integro-differential equations with state-dependent delay and non-instantaneous impulses khalida aissani1, mouffak benchohra21 and nadia benkhettou2 1 laboratory of mathematics, djillali liabes university of sidi bel-abbès, po box 89, 22000, sidi bel-abbès, algeria. 2 university of bechar po box 417, 08000, bechar, algeria benchohra@yahoo.com, aissani − k@yahoo.fr abstract in this paper, we prove the existence of mild solution of the fractional integro-differential equations with state-dependent delay with not instantaneous impulses. the existence results are obtained under the conditions in respect of kuratowski’s measure of noncompactness. an example is also given to illustrate the results. resumen en este art́ıculo, demostramos la existencia de soluciones mild de ecuaciones integrodiferenciales fraccionarias con retardo dependiente del estado e impulsos no instantáneos. los resultados de existencia se obtienen bajo condiciones respecto de la medida de kuratowski de no compacidad. también se entrega un ejemplo para ilustrar los resultados. keywords and phrases: non-instantaneous impulsive conditions, fractional integro-differential equations, caputo fractional derivative, mild solution, fixed point, state-dependent delay. 2010 ams mathematics subject classification: 26a33, 34a12, 34a37, 34g20. 1corresponding author http://dx.doi.org/10.4067/s0719-06462019000100061 62 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) 1 introduction fractional differential equations play the crucial and significant role in the field of science and engineering. most importantly non-integer order differential equations have ability to describe the real behavior and memory effects of the system and processes. for more details about fractional differential equations and its applications refer the monographs of abbas et al. [1, 2, 3], baleanu et al. [12], diethelm [18], hilfer [24], kilbas et al. [26], miller and ross [32], samko et al. [37], tarasov [38], and zhou [39] and the references therein. most of the research papers deal with the existence of solutions for differential equations with instantaneous impulsive conditions see [6, 7, 10, 11, 14, 28, 31]. but many times it has seen that certain dynamics of evolution processes cannot describe by instantaneous impulses, for instance: pharmacotherapy, high or low levels of glucose, this situation can be interpreted as an impulsive action which starts abruptly at certain point of time and continue with a finite time interval. such type of systems are known as non-instantaneous impulsive systems which are more suitable to study the dynamics of evolution processes [4]. this theory of a new class of impulsive differential equation was initiated by hernández et al. [23]. afterwards, pierri et al. [35] continued the work in this field and extend the theory of [23] in a pcα normed banach space. the existence of solutions for non-instantaneous impulsive fractional differential equations have also been discussed in [8, 19, 27, 29, 34]. recently, benchohra et al. [15] investigated the existence and uniqueness of solutions on a compact interval for non-linear fractional integro-differential equations with state-dependent delay and noninstantaneous impulses. anguraj and kanjanadevi [9] studied the existence and uniqueness of fractional neutral differential equations with state-dependent delay subject to non-instantaneous impulsive conditions. motivated by the papers cited above, in this paper, we consider the existence of mild solutions for fractional integro-differential equations with state-dependent delay and non instantaneous impulses described by the form c d q t x(t) + ax(t) = ∫t 0 a(t,s)f(s,xρ(s,xs),x(s))ds, a.e. t ∈ (si, ti+1] ⊂ j,i = 0, . . . ,n, x(t) = hi(t,xρ(t,xt),x(t)), t ∈ (ti,si], i = 1, . . . ,n, x0 = φ ∈ b, (1.1) where cd q t is the caputo fractional derivative of order 0 < q < 1, a : d(a) ⊂ x → x is the infinitesimal generator of an analytic semigroup {s(t)}t≥0 of uniformly bounded linear operators on x, f : j × b × x −→ x, j = [0,t], t > 0, and ρ : j × b → (−∞,t] are appropriate functions, a : d → r (d = {(t,s) ∈ j × j : t ≥ s}). here 0 = t0 = s0 < t1 ≤ s1 ≤ t2 < ... < tn−1 ≤ sn ≤ tn ≤ tn+1 = t are pre-fixed numbers, and hi ∈ c((ti,si] × b × x,x), for all i = 1,2, . . . ,n. for cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 63 any continuous function x defined on (−∞,t] and any t ∈ j, we denote by xt the element of b defined by xt(θ) = x(t + θ), θ ∈ (−∞,0]. here xt represents the history of the state up to the present time t and φ ∈ b to be specified later. 2 preliminaries let (x,‖ · ‖) be a real banach space. c = c(j,x) be the space of all x-valued continuous functions on j. l(x) be the banach space of all linear and bounded operators on x. l1(j,x) the space of x−valued bochner integrable functions on j with the norm ‖y‖l1 = ∫t 0 ‖y(t)‖dt. l∞(j,r) is the banach space of measurable functions which are essentially bounded, normed by ‖y‖l∞ = inf{d > 0 : |y(t)| ≤ d, a.e. t ∈ j}. we need some basic definitions of the fractional calculus theory which are used in this paper. definition 2.1. let α > 0 and f : r+ → x be in l1(r+,x). then the riemann–liouville integral is given by: iαt f(t) = 1 γ(α) ∫t 0 f(s) (t − s)1−α ds, where γ(·) is the euler gamma function. for more details on the riemann–liouville fractional derivative, we refer the reader to [17]. definition 2.2. [36] the caputo derivative of order α for a function f : [0,+∞) → x can be written as dαt f(t) = 1 γ(n − α) ∫t 0 f(n)(s) (t − s)α+1−n ds = in−αf(n)(t), t > 0, n − 1 ≤ α < n. if 0 ≤ α < 1, then dαt f(t) = 1 γ(1 − α) ∫t 0 f(1)(s) (t − s)α ds. obviously, the caputo derivative of a constant is equal to zero. definition 2.3. a function f : j×b ×x −→ x is said to be an carathéodory function if it satisfies : (i) for each t ∈ j the function f(t, ·, ·) : b × x −→ x is continuous; 64 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) (ii) for each (v,w) ∈ b × x the function f(·,v,w) : j → x is measurable . next we give the concept of a measure of noncompactness [13]. definition 2.4. let b be a bounded subset of a banach space y. the kuratowski measure of noncompactness of b is defined as α(b) = inf{d > 0 : b has a finite cover by sets of diameter ≤ d}. we note that this measure of noncompactness satisfies the properties ([13]). lemma 2.5. 1. if a ⊆ b then α(a) ≤ α(b), 2. α(a) = α(a), where a denotes the closure of a, 3. α(a) = 0 ⇔ a is compact (a is relatively compact), 4. α(λa) = |λ|a, with λ ∈ r, 5. α(a ∪ b) = max{α(a),α(b)}, 6. α(a + b) ≤ α(a) + α(b), where a + b = {x + y : x ∈ a,y ∈ b}, 7. α(a + a) = α(a) for any a ∈ x, 8. α(conva) = α(a), where conva is the closed convex hull of a. for h ⊂ c(j,x), we define ∫t 0 h(s)ds = {∫t 0 u(s)ds : u ∈ h } for t ∈ j, (2.1) where h(s) = {u(s) ∈ x : u ∈ h}. lemma 2.6. [13] if h ⊂ c(j,x) is a bounded, equicontinuous set, then αc(h) = sup t∈j α(h(t)). (2.2) lemma 2.7. [21] if {un} ∞ n=1 ⊂ l 1(j,x) and there exists m ∈ l1(j,r+) such that ‖un(t)‖ ≤ m(t), a.e. t ∈ j, then α({un(t)} ∞ n=1) is integrable and α ({∫t 0 un(s)ds }∞ n=1 ) ≤ 2 ∫t 0 α({un(s)} ∞ n=1)ds. (2.3) cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 65 in this paper, we will employ an axiomatic definition for the phase space b which is similar to those introduced by hale and kato [20]. specifically, b will be a linear space of functions mapping (−∞,0] into x endowed with a seminorm ‖ · ‖b, and satisfies the following axioms: (a1) if x : (−∞,t ] −→ x is continuous on j and x0 ∈ b, then xt ∈ b and xt is continuous in t ∈ j and ‖x(t)‖ ≤ c‖xt‖b, (2.4) where c ≥ 0 is a constant. (a2) there exist a continuous function c1(t) > 0 and a locally bounded function c2(t) ≥ 0 in t ≥ 0 such that ‖xt‖b ≤ c1(t) sup s∈[0,t] ‖x(s)‖ + c2(t)‖x0‖b, (2.5) for t ∈ [0,t] and x as in (a1). (a3) the space b is complete. remark 2.8. condition (2.4) in (a1) is equivalent to ‖φ(0)‖ ≤ c‖φ‖b, for all φ ∈ b. example 2.9. the phase space cr × l p(g,x). let r ≥ 0,1 ≤ p < ∞, and let g : (−∞,−r) → r be a nonnegative measurable function which satisfies the conditions (g − 5),(g − 6) in the terminology of [25]. briefly, this means that g is locally integrable and there exists a nonnegative, locally bounded function λ on (−∞,0], such that g(ξ + θ) ≤ λ(ξ)g(θ), for all ξ ≤ 0 and θ ∈ (−∞,−r)\nξ, where nξ ⊆ (−∞,−r) is a set with lebesgue measure zero. the space cr × l p(g,x) consists of all classes of functions ϕ : (−∞,0] → x, such that ϕ is continuous on [−r,0], lebesgue-measurable, and g‖ϕ‖p on (−∞,−r). the seminorm in ‖.‖b is defined by ‖ϕ‖b = sup θ∈[−r,0] ‖ϕ(θ)‖ + (∫−r −∞ g(θ)‖ϕ(θ)‖pdθ ) 1 p . the space b = cr ×l p(g,x) satisfies axioms (a1), (a2), (a3). moreover, for r = 0 and p = 2, this space coincides with c0 × l 2(g,x),h = 1,m(t) = λ(−t) 1 2 ,k(t) = 1 + (∫0 −r g(τ)dτ ) 1 2 , for t ≥ 0 (see [25], theorem 1.3.8 for details). for our purpose we will only need the following fixed point theorems. theorem 2.10. [5, 33] let u be a bounded, closed and convex subset of a banach space, and let n be a continuous mapping of u into itself. if the implication v = convn(v) or v = n(v) ∪ {0} =⇒ α(v) = 0 holds for every subset v of u, then n has a fixed point. 66 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) a continuous map n : d ⊆ e → e is said to be a α-contraction if there exists a constant ν ∈ [0,1) such that α(n(c)) ≤ να(c) for any bounded closed subset c ⊆ d. theorem 2.11. (darbo–sadovskii)[13] let e be a banach space. if d ⊆ e is bounded closed and convex, the continuous map n : d → d is a α-contraction, then the map n has at least one fixed point in d. consider the space pc(j,x) = { x : j → x,x ∈ c ( j ∩ ( ∪nk=0 (tk,sk] ) ,x ) , and x(t+k ),x(s − k ) exist with, x(s − k ) = x(sk),k = 1, . . . ,n } . obviously, pc(j,x) is a banach space with the norm ‖x‖pc = sup t∈j ‖x(t)‖. 3 existence results in this section, we prove the existence of mild solution of (1.1). definition 3.1. a function x : (−∞,t] → x is said to be a mild solution of the equation (1.1) if x0 = φ on (−∞,t],x|[0,t] ∈ pc([0,t],x) and x satisfies x(t) =                                  q(t)φ(0) + ∫t 0 ∫s 0 r(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ [0,t1], hi(t,xρ(t,xt),x(t)), t ∈ (ti,si], i = 1,2, . . . ,n, q(t − si)hi(si,xρ(si,xsi ) ,x(si)) + ∫t 0 ∫s 0 r(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ (si,ti+1], (3.1) where q(t) = ∫ ∞ 0 ξq(σ)s(t qσ)dσ, r(t) = q ∫ ∞ 0 σtq−1ξq(σ)s(t qσ)dσ and ξq is a probability density function defined on (0,∞) such that ξq(σ) = 1 q σ −1−( 1 q ) ̟q(σ − 1 q ) ≥ 0, cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 67 where ̟q(σ) = 1 π ∞∑ k=1 (−1)k−1σ−qk−1 γ(kq + 1) k! sin(kπq), σ ∈ (0,∞). remark 3.2. note that {s(t)}t≥0 is a uniformly bounded i.e there exists a constant m > 0 such that ‖s(t)‖l(x) ≤ m for all t ≥ 0. remark 3.3. according to [30], direct calculation gives that ‖r(t)‖ ≤ cq,mt q−1, t > 0, (3.2) where cq,m = qm γ(1 + q) . set r(ρ−) = {ρ(s,ϕ) : (s,ϕ) ∈ j × b,ρ(s,ϕ) ≤ 0}. we always assume that ρ : j × b → (−∞,t] is continuous. additionally, we introduce following hypothesis: (hϕ) the function t → ϕt is continuous from r(ρ−) into b and there exists a continuous and bounded function lφ : r(ρ−) → (0,∞) such that ‖φt‖b ≤ l φ(t)‖φ‖b for every t ∈ r(ρ −). remark 3.4. condition (hϕ), is frequently verified by the continuous and bounded functions. for more details see [25]. remark 3.5. in the rest of this section, c∗1 and c ∗ 2 are the constants c∗1 = sup s∈j c1(s) and c ∗ 2 = sup s∈j c2(s). lemma 3.6. [22] if x : r → x is a function such that x0 = φ, then ‖xs‖b ≤ (c ∗ 2 + l φ)‖φ‖b + c ∗ 1 sup{|x(θ)|;θ ∈ [0,max{0,s}]}, s ∈ r(ρ −) ∪ j, where lφ = sup t∈r(ρ−) lφ(t). let us introduce the following hypotheses: (h1) f : j × b × x −→ x satisfies the carathéodory conditions. (h2) there exist functions µ,µ∗ ∈ l1(j,r+) and continuous nondecreasing functions ψ,ψ∗ : r+ → (0,+∞) such that ‖f(t,x,y)‖ ≤ µ(t)ψ(‖x‖b + ‖y‖) , (t,x,y) ∈ j × b × x, ‖hi(t,x,y)‖ ≤ µ ∗ (t)ψ∗ (‖x‖b + ‖y‖) , (t,x,y) ∈ j × b × x, 68 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) (h3) for any bounded sets d1 ⊂ b,d2 ⊂ x, and 0 ≤ s ≤ t ≤ t, there exists an integrable positive function η such that α(r(t − s)f(τ,d1,d2)) ≤ ηt(s,τ) ( α(d2) + sup −∞<θ≤0 α(d1(θ)) ) , where ηt(s,τ) = η(t,s,τ) and sup t∈j ∫t 0 ∫s 0 ηt(s,τ)dτds = η ∗ < ∞. (h4) there exists a constant l > 0 such that, for each bounded sets d1 ⊂ b,d2 ⊂ x, α(hi(τ,d1,d2)) ≤ l ( α(d2) + sup −∞<θ≤0 α(d1(θ)) ) . (h5) for each t ∈ j, a(t,s) is measurable on [0,t] and a(t) = ess sup{|a(t,s)|,0 ≤ s ≤ t} is bounded on j. the map t → at is continuous from j to l∞(j,r), here, at(s) = a(t,s). set a = sup t∈j a(t). our first result is based on the mönch fixed point theorem. theorem 3.7. suppose that the assumptions (hϕ),(h1) − (h5) hold, and if 2ml + 16 a η∗ < 1, (3.3) then the problem (1.1) has at least one mild solution. proof. let y = {u ∈ pc(x) : u(0) = φ(0) = 0} endowed with the uniform convergence topology and define the operator p : y → y by p(x)(t) =                                  q(t)φ(0) + ∫t 0 ∫s 0 r(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ [0,t1], hi(t,xρ(t,xt),x(t)), t ∈ (ti,si], i = 1,2, . . . ,n, q(t − si)hi(si,xρ(si,xsi ) ,x(si)) + ∫t si ∫s 0 r(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ (si,ti+1], where x : (−∞,t] → x is such that x0 = φ and x = x on j. let φ : (−∞,t] −→ x be the extension of φ to (−∞,t] such that φ(θ) = φ(0) = 0 on j. choose r ≥ m‖µ∗‖l1ψ ∗ ( (c∗2 + l φ )‖φ‖b + (c ∗ 1 + 1)r ) +acq,m‖µ‖l1 tq q ψ ( (c∗2 + l φ )‖φ‖b + (c ∗ 1 + 1)r ) , and define the set br = {x ∈ y : ‖x‖pc ≤ r} , cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 69 then br is a bounded, closed-convex subset in y. step 1: p is continuous. let {xk}k∈n be a sequence such that x k → x in br as k → ∞. case 1. for each t ∈ [0,t1], we have ‖p(xk)(t) − p(x)(t)‖ ≤ ∫t 0 ∫s 0 ‖r(t − s)‖‖a(s,τ)‖‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ acq,m ∫t 0 ∫s 0 (t − s)q−1‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds. case 2. for each t ∈ [ti,si), i = 1,2, . . . ,n, we have ‖p(xk)(t) − p(x)(t)‖ = ‖hi(t,x k ρ(t,xkt ) ,xk(t)) − hi(t,xρ(t,xt),x(t))‖ → 0 k → ∞. case 3. for each t ∈ (si,ti+1], i = 1,2, . . . ,n, we obtain ‖p(xk)(t) − p(x)(t)‖ ≤ ‖q(t − si)‖‖hi(si,x k ρ(si,x k si ) ,xk(si)) − hi(si,xρ(si,xsi ) ,x(si))‖ + ∫t si ∫s 0 ‖r(t − s)‖‖a(s,τ)‖‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ m‖hi(si,x k ρ(si,x k si ) ,xk(si)) − hi(si,xρ(si,xsi ) ,x(si))‖ + acq,m ∫t si ∫s 0 (t − s)q−1‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds. since the function hi is continuous and f is of carathéodory type, we have by the lebesgue dominated convergence theorem that ‖p(xk)(t) − p(x)(t)‖ → 0 as k → ∞, which shows the operator p is continuous. step 2: p maps br into itself. 70 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) case 1. for all t ∈ [0,t1], we get ‖p(x)(t)‖ ≤ ‖q(t)φ(0)‖ + ∫t 0 ∫s 0 ‖r(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ mc‖φ‖b + acq,m ∫t 0 ∫s 0 (t − s)q−1µ(τ)ψ(‖xρ(τ,xτ)‖b + ‖x‖)dτds ≤ mc‖φ‖b + acq,m ∫t 0 ∫s 0 (t − s)q−1µ(τ) × ψ ( (c∗2 + l φ)‖φ‖b + c ∗ 1r + r ) dτds ≤ mc‖φ‖b + acq,m‖µ‖l1 tq q ψ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) ≤ r. case 2. for all t ∈ [ti,si), i = 1,2, . . . ,n, we have ‖p(x)(t)‖ ≤ ‖hi(t,xρ(t,xt),x(t))‖ ≤ µ∗(t)ψ∗ ( ‖xρ(t,xt)‖b + ‖x‖ ) ≤ ‖µ∗‖l1ψ ∗ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) ≤ r. case 3. for all t ∈ (si,ti+1], i = 1,2, . . . ,n, we obtain ‖p(x)(t)‖ ≤ ‖q(t − si)hi(si,xρ(si,xsi ) ,x(si))‖ + ∫t si ∫s 0 ‖r(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))‖dτds, ≤ m‖µ∗‖l1ψ ∗ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) + acq,m‖µ‖l1 tq q ψ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) ≤ r. step 3: p(br) is bounded and equicontinuous. case 1. for each t ∈ [0,t1],0 ≤ τ2 ≤ τ1 ≤ t1, and x ∈ br. then we have ‖p(x)(τ1) − p(x)(τ2)‖ ≤ i1 + i2 + i3, where i1 = ‖q(τ1) − q(τ2)‖‖φ(0)‖ i2 = ∥ ∥ ∥ ∥ ∫τ2 0 ∫s 0 [r(τ1 − s) − r(τ2 − s)]a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds ∥ ∥ ∥ ∥ i3 = ∥ ∥ ∥ ∥ ∫τ1 τ2 ∫s 0 r(τ1 − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds ∥ ∥ ∥ ∥ . cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 71 i1 tends to zero as τ2 → τ1, since s(t) is uniformly continuous operator. for i2, using (3.2) and (h2), we have i2 ≤ aψ ( (c∗2 + l φ )‖φ‖b + (c ∗ 1 + 1)r ) ‖µ‖l1 ∫τ2 0 [r(τ1 − s) − r(τ2 − s)]ds ≤ aψ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) ‖µ‖l1 × ∫τ2 0 [ q ∫ ∞ 0 σ(τ1 − s) q−1ξq(σ)s((τ1 − s) qσ)dσ −q ∫ ∞ 0 σ(τ2 − s) q−1ξq(σ)s((τ2 − s) qσ)dσ ] ds ≤ aψ ( (c∗2 + l φ )‖φ‖b + (c ∗ 1 + 1)r ) ‖µ‖l1 × [ q ∫τ2 0 ∫ ∞ 0 σ‖[(τ1 − s) q−1 − (τ2 − s) q−1]ξq(σ)s((τ1 − s) qσ) +q ∫τ2 0 ∫ ∞ 0 σ(τ2 − s) q−1ξq(σ)‖s((τ1 − s) qσ) − s((τ2 − s) qσ)‖ ] ≤ aψ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) ‖µ‖l1 × [cq,m ∫τ2 0 ∣ ∣(τ1 − s) q−1 − (τ2 − s) q−1 ∣ ∣ds + q ∫τ2 0 ∫ ∞ 0 σ(τ2 − s) q−1ξq(σ)‖s((τ1 − s) qσ) − s((τ2 − s) qσ)‖dσds]. clearly, the first term on the right-hand side of the above inequality tends to zero as τ2 → τ1. from the continuity of s(t) in the uniform operator topology for t > 0, the second term on the right-hand side of the above inequality tends to zero as τ2 → τ1. in view of (h2), we have i3 ≤ a cq,m ∫τ1 τ2 ∫s 0 (τ1 − s) q−1‖f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ a cq,mψ ( (c∗2 + l φ)‖φ‖b + (c ∗ 1 + 1)r ) ‖µ‖l1 ∫τ1 τ2 (τ1 − s) q−1ds. as τ2 → τ1, i3 tends to zero. case 2. for each t ∈ [ti,si), i = 1,2, . . . ,n,ti ≤ τ2 ≤ τ1 ≤ si, and x ∈ br. then we have ‖p(x)(τ1) − p(x)(τ2)‖ = ‖hi(τ1,xρ(τ1,xτ1 ) ,x(τ1)) − hi(τ2,xρ(τ2,xτ2 ) ,x(τ2))‖ → 0 as τ2 → τ1. case 3. for each t ∈ (si,ti+1], i = 1,2, . . . ,n,si ≤ τ2 ≤ τ1 ≤ ti+1, and x ∈ br. then we have ‖p(x)(τ1) − p(x)(τ2)‖ ≤ ‖q(τ1 − si) − q(τ2 − si)‖‖hi(si,xρ(si,xsi ) ,x(si))‖ + i1 + i2 + i3. since s(t) is uniformly continuous operator, so lim τ2→τ1 ‖q(τ1 − si) − q(τ2 − si)‖ = 0, i = 1, . . . ,n. 72 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) consequently lim τ2→τ1 ‖p(x)(τ1) − p(x)(τ2)‖ = 0. thus, p(br) is equicontinuous. now let v be a subset of br such that v ⊂ conv(p(v) ∪ {0}). moreover, for any ε > 0 and bounded set d, we can take a sequence {vn} ∞ n=1 ⊂ d such that α(d) ≤ 2α({vn}) +ε ([16], p. 125). thus, for {vn} ∞ n=1 ⊂ v, and using lemmas 2.5-2.7 and (h3), we have, for t ∈ [0,t1], α(pv) ≤ 2α({pvn}) + ε = 2sup t∈j α({pvn(t)}) + ε = 2sup t∈j α ({∫t 0 r(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 4sup t∈j ∫t 0 α ({ r(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 8sup t∈j ∫t 0 ∫s 0 α({r(t − s)a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 8 asup t∈j ∫t 0 ∫s 0 α({r(t − s)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 8 a sup t∈j ∫t 0 ∫s 0 ηt(s,τ) [ α(vn(τ)) + sup −∞<θ≤0 α(vn(θ + τ)) ] dτds + ε ≤ 8 a sup t∈j ∫t 0 ∫s 0 ηt(s,τ) [ α(vn) + sup 0<µ≤τ α(vn(µ)) ] dτds + ε ≤ 16 a α(vn) sup t∈j ∫t 0 ∫s 0 ηt(s,τ)dτds + ε ≤ 16 a η∗α(v) + ε. for any t ∈ [ti,si), i = 1,2, . . . ,n, we get α(pv) = α ( hi(t,xρ(t,xt),x(t)) ) ≤ l ( α(vn(t)) + sup −∞<θ≤0 α(vn(θ + t)) ) ≤ l ( α(vn) + sup 0<µ≤τ α(vn(µ)) ) ≤ 2lα(vn) ≤ 2lα(v). cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 73 in the same way, for any t ∈ (si,ti+1], i = 1,2, . . . ,n, we obtain α(pv) ≤ 2α({pvn}) + ε = 2sup t∈j α({pvn(t)}) + ε = 2sup t∈j α ( q(t − si)hi(si,xρ(si,xsi ) ,x(si)) ) + 2sup t∈j α ({∫t si r(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 2mlα(vn) + 4sup t∈j ∫t si α ({ r(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 2mlα(vn) + 8sup t∈j ∫t si ∫s 0 α({r(t − s)a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 2mlα(vn) + 8 asup t∈j ∫t si ∫s 0 α({r(t − s)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 2mlα(vn) + 8 asup t∈j ∫t si ∫s 0 ηt(s,τ) [ α(vn(τ)) + sup −∞<θ≤0 α(vn(θ + τ)) ] dτds + ε ≤ 2mlα(vn) + 8 a sup t∈j ∫t si ∫s 0 ηt(s,τ) [ α(vn) + sup 0<µ≤τ α(vn(µ)) ] dτds + ε ≤ 2mlα(vn) + 16 a α(vn) sup t∈j ∫t 0 ∫s 0 ηt(s,τ)dτds + ε ≤ 2mlα(v) + 16 a η∗α(v) + ε ≤ (2ml + 16 a η∗)α(v) + ε. therefore, in view of lemma 2.5, we have α(v) ≤ α(pv) ≤ (2ml + 16 a η∗)α(v) + ε, since ε is arbitrary we obtain that α(v) ≤ (2ml + 16 a η∗)α(v). this means that α(v) (1 − (2ml + 16 a η∗)) ≤ 0. by (3.3) it follows that α(v) = 0. in view of the ascoli-arzelà theorem, v is relatively compact in br. applying now theorem 2.10, we conclude that p has a fixed point which is a solution of the problem (1.1). the second result is established using the darbo’s fixed point theorem. 74 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) theorem 3.8. assume that (h1)−(h5) are satisfied, then the problem (1.1) has at least one mild solution. proof. in what follows we show that the operator p : y → y is a strict set contraction. we know that p : y → y is bounded and continuous, we need to prove that there exists a constant 0 ≤ ν < 1 such that α(pv) ≤ να(v) for v ⊂ br. using the same method as the proof of theorem 3.7, for t ∈ [0,t], we have α(pv) ≤ (2ml + 16 a η∗)α(v) + ε, since ε is arbitrary we obtain that α(pv) ≤ να(v). hence p is a set contraction. according to theorem 2.11 the operator p has at least one fixed point which is obviously a mild solution of the problem (2.4). this completes the proof. 4 an example we consider the fractional integro-differential equations with state-dependent delay and noninstantaneous impulses of the form ∂ q t ∂tq v(t,ζ) + ∂2 ∂ζ2 v(t,ζ) = ∫t 0 (t − s)2 ∫s −∞ γ(τ − s)v(τ − ρ1(s)ρ2(|v(s,ζ)|),ζ)dτds + ∫t 0 (t − s)2 cos |v(s,ζ)|ds, (t,x) ∈ n ∈ ∪ni=1[si,ti+1] × [0,π], v(t,0) = v(t,π) = 0, t ∈ [0,t], v(τ,ζ) = v0(θ,ζ), θ ∈ (−∞,0],x ∈ [0,π] v(t,ζ) = hi(t,v(t − ρ1(t)ρ2(|v(t,ζ)|),ζ),ζ), (t,x) ∈ (ti,si] × [0,π], i = 1,2, . . . ,n, (4.1) where 0 < q < 1,0 = t0 = s0 < t1 ≤ s1 ≤ t2 < ... < tn−1 ≤ sn ≤ tn ≤ tn+1 = t are prefixed real numbers and the functions γ : r → r,ρi : [0,+∞) → [0,+∞), i = 1,2 are continuous functions. let x = l2([0,π]) and define the operator a : d(a) ⊂ x → x by aω = ω′′ with domain d(a) = {ω ∈ e : ω,ω′ are absolutely continuous, ω′′ ∈ e,ω(0) = ω(π) = 0}. then aω = ∞∑ n=1 n2(ω,ωn)ωn, ω ∈ d(a), cubo 21, 1 (2019) on fractional integro-differential equations with state-dependent . . . 75 where ωn(x) = √ 2 π sin(nx),n ∈ n is the orthogonal set of eigenvectors of a. it is well known that a is the infinitesimal generator of an analytic semigroup {s(t)}t≥0 in x and is given by s(t)ω = ∞∑ n=1 e−n 2 t(ω,ωn)ωn, ∀ω ∈ x, and every t > 0. from these expressions, it follows that {s(t)}t≥0 is a uniformly bounded compact semigroup on x. for the phase space, we choose b = c0 × l 2(g,x), see example 2.9 for details. set x(t)(ζ) = v(t,ζ), φ(θ)(ζ) = v0(θ,ζ), a(t,s) = (t − s)2 f(t,ϕ,x(t))(ζ) = ∫0 −∞ γ(t)ϕ(t,ζ)ds + cos |x(t)(ζ)|, hi(t,ϕ,x(t))(ζ) = hi(t,v(t − ρ1(t)ρ2(|x(t)|),ζ),ζ) ρ(t,ϕ) = t − ρ1(t)ρ2(|ϕ(0)|). under the above conditions, we can represent the problem (4.1) by the abstract problem (1.1). proposition 4.1. let ϕ ∈ b be such that (hϕ) holds, and let t → ϕt be continuous on r(ρ−). then there exists a mild solution of (4.1). 76 khalida aissani, mouffak benchohra and nadia benkhettou cubo 21, 1 (2019) references [1] s. abbas, m. benchohra, j. graef and j. henderson, implicit fractional differential and integral equations; existence and stability, de gruyter, berlin, 2018. 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[39] y. zhou, fractional evolution equations and inclusions: analysis and control, academic press elsevier, 2016. introduction preliminaries existence results an example cubo a mathematical journal vol.10, n o ¯ 02, (01–14). july 2008 semi-classical dispersive estimates for the wave and schrödinger equations with a potential in dimensions n ≥ 4 f. cardoso universidade federal de pernambuco, departamento de matemática, av. prof. luiz freire, s/n, cid. universitária, cep. 50.540-740 – recife-pe, brazil email: fernando@dmat.ufpe.br and g. vodev université de nantes, département de mathématiques, umr 6629 du cnrs, 2, rue de la houssinière, bp 92208, 44332 nantes cedex 03, france email: georgi.vodev@math.univ-nantes.fr abstract we expand the operators |t|(n−1)/2eit √ −∆+v ϕ(h √ −∆ + v ) and |t|n/2eit(−∆+v ) ψ(h2(−∆ + v )), 0 < h ≪ 1, modulo operators whose l1 → l∞ norm is on (h n ), ∀n ≥ 1, where ϕ,ψ ∈ c∞0 ((0, +∞)) and v ∈ l ∞ (r n ), n ≥ 4, is a real-valued potential satisfying v (x) = o ( 〈x〉−δ ) , δ > (n + 1)/2 in the case of the wave equation and δ > (n + 2)/2 in the case of the schrödinger equation. as a consequence, we give sufficent conditions in order that the wave and the schrödinger groups satisfy dispersive estimates with a loss of ν derivatives, 0 ≤ ν ≤ (n − 3)/2. roughly speaking, we reduce this problem to estimating the l1 → l∞ norms of a finite number of operators with 2 f. cardoso and g. vodev cubo 10, 2 (2008) almost explicit kernels. these kernels are completely explicit when 4 ≤ n ≤ 7 in the case of the wave group, and when n = 4, 5 in the case of the schrödinger group. resumen en este trabajo son expandidos los operadores |t|(n−1)/2eit √ −∆+v ϕ(h √ −∆ + v ) y |t|n/2eit(−∆+v )ψ(h2(−∆+v )), 0 < h ≪ 1, modulo operadores cuja l1 → l∞ norma es on (h n ), ∀n ≥ 1, donde ϕ,ψ ∈ c∞0 ((0, +∞)) y v ∈ l ∞ (rn), n ≥ 4, es un potencial real satisfaziendo v (x) = o ( 〈x〉−δ ) , δ > (n + 1)/2 en el caso de la ecuación de la onda y δ > (n + 2)/2 en el caso de la ecuación de schrödinger. como consequencia presentamos condiciones suficientes a fin de que los grupos de la onda y schrödinger cumplan estimativas dispersivas con una perdida de ν derivadas 0 ≤ ν ≤ (n − 3)/2. rigurosamente hablando, reduzimos este problema a estimar las normas l1 → l∞ de un número finito de operadores con nucleos casi explicitos. estos nucleos son completamente explicitos cuando 4 ≤ n ≤ 7 en el caso del grupo de la onda y cuando n = 4, 5 en el caso del grupo de schrödinger. key words and phrases: potential, dispersive estimates. math. subj. class.: 35l15, 35b40, 47f05 1 introduction and statement of results denote by g the self-adjoint realization of the operator −∆ + v on l2(rn), n ≥ 4, where v ∈ l∞(rn) is a real-valued potential satisfying |v (x)| ≤ c〈x〉−δ, ∀x ∈ rn, (1.1) with constants c > 0, δ > (n + 1)/2. it is well known that g has no strictly positive eigenvalues and resonances. we will also denote by g0 the self-adjoint realization of the operator −∆ on l2(rn). it is well known that the free wave group satisfies the following semi-classical dispersive estimate ∥∥∥eit √ g0ϕ(h √ g0) ∥∥∥ l1→l∞ ≤ ch−(n+1)/2|t|−(n−1)/2, ∀t 6= 0, h > 0, (1.2) where ϕ ∈ c∞0 ((0, +∞)). the natural question is to find the bigest possible class of potentials for which we have an analogue of (1.2) for the perturbed wave group. it is proved in [16] that under the assumption (1.1) only, we have such an estimate but with a significant loss in h for 0 < h ≪ 1, namely ∥∥∥eit √ gϕ(h √ g) ∥∥∥ l 1 →l ∞ ≤ ch−n+1|t|−(n−1)/2, ∀t 6= 0, 0 < h ≤ 1, (1.3) cubo 10, 2 (2008) semi-classical dispersive estimates for the wave ... 3 and this seems hard to improve without extra assumptions on the potential. this estimate is then used in [16] to obtain dispersive estimates with a loss of (n − 3)/2 derivatives for eit √ gχa( √ g), ∀a > 0, where χa ∈ c ∞ ((−∞, +∞)), χa(λ) = 0 for λ ≤ a, χa(λ) = 1 for λ ≥ 2a. in the present work we will expand eit √ gϕ(h √ g) modulo remainders whose l1 → l∞ norm is upper bounded by cmh m−n+1|t|−(n−1)/2, 0 < h ≤ h0 ≪ 1, for every integer m ≥ 0. in order to state the precise result we need to introduce some notations. let ϕ1 ∈ c ∞ 0 ((0, +∞)) be such that ϕ1 = 1 on supp ϕ, and set ϕ̃(λ) = λϕ(λ), ϕ̃1(λ) = λ −1ϕ1(λ). under (1.1) there exists a constant h0 > 0 so that for 0 < h ≤ h0, the operator t (h) := ( id + ϕ1(h √ g0) − ϕ1(h √ g) ) −1 = id + o(h2) is uniformely bounded on lp, 1 ≤ p ≤ +∞, as well as on weighted l2 spaces (see lemma 2.3 of [16] and lemma a.1 of [11]). set u0(t,h) = ϕ̃1(h √ g0) sin(t √ g0), e 0 0 (t,h) = e it √ g0ϕ(h √ g0), e0(t,h) = ϕ1(h √ g0) cos(t √ g0)ϕ(h √ g) + iϕ̃1(h √ g0) sin(t √ g0)ϕ̃(h √ g). furthermore, given any integer j ≥ 1, define the operators ej (t,h) = −h ∫ t 0 u0(t − τ,h)v t (h)ej−1(τ,h)dτ, e0 j (t,h) = −h ∫ t 0 u0(t − τ,h)v e 0 j−1(τ,h)dτ. theorem 1.1 let v satisfy (1.1). then, there exists a constant h0 > 0 so that for all 0 < h ≤ h0, t 6= 0, we have the estimate ∥∥∥∥∥∥ eit √ gϕ(h √ g) − t (h) m∑ j=0 ej (t,h) ∥∥∥∥∥∥ l 1 →l ∞ ≤ cmh m−n+1 |t|−(n−1)/2, (1.4) for every integer m ≥ 0 with a constant cm > 0 independent of t and h. moreover, the operators ej satisfy the estimates ‖e0(t,h)‖l1→l∞ ≤ ch −(n+1)/2 |t|−(n−1)/2, (1.5) ‖ej (t,h)‖l1→l∞ ≤ cjh j−n |t|−(n−1)/2, j ≥ 1, (1.6) ∥∥ej (t,h) − e0j (t,h) ∥∥ l1→l∞ ≤ cjh j+2−n |t|−(n−1)/2, j ≥ 1. (1.7) it follows from this theorem that to improve the estimate (1.3) in h, it suffices to improve the estimate (1.6). we also have the following 4 f. cardoso and g. vodev cubo 10, 2 (2008) corollary 1.2 let v satisfy (1.1) and suppose in addition that there exists 0 ≤ k ≤ (n − 3)/2 such that the operators ej satisfy the estimate ‖ej (t,h)‖l1→l∞ ≤ ch k−n+1 |t|−(n−1)/2, (1.8) for all integers 1 ≤ j < k + 1. then, for every a > 0, 0 < ǫ ≪ 1, we have the estimate ∥∥∥eit √ g ( √ g)k−n+1−ǫχa( √ g) ∥∥∥ l1→l∞ ≤ cǫ|t| −(n−1)/2, ∀t 6= 0, (1.9) while for every 0 ≤ q ≤ (n − 3)/2 − k, 2 ≤ p < 2(n−1−2q−2k) n−3−2q−2k , we have ∥∥∥eit √ g ( √ g) −α((n+1)/2+q) χa( √ g) ∥∥∥ l p ′ →l p ≤ c|t| −α(n−1)/2 , ∀t 6= 0, (1.10) where 1/p + 1/p′ = 1, α = 1 − 2/p. moreover, when 4 ≤ n ≤ 7 the estimates (1.9) and (1.10) hold true if we suppose (1.8) fulfilled with ej replaced by e 0 j . the estimate (1.8) with k > 0 seems hard to establish (even if we replace ej by e 0 j ) and the proof would probably require some regularity condition on the potential. note that when n = 2 and n = 3 the estimates (1.9) and (1.10) (with k = (n − 3)/2, q = 0) are proved in [2] under (1.1) only. in the case of n = 2 these estimates are proved (for a large enough) in [10] for a much larger class of potentials satisfying sup y∈r2 ∫ r2 |v (x)|dx |x − y|1/2 ≤ c < +∞. (1.11) when n = 3 these estimates are proved in [4] for a quite large subclass of potentials satisfying sup y∈r3 ∫ r3 |v (x)|dx |x − y| ≤ c < +∞. (1.12) when n ≥ 4 optimal dispersive estimates (that is, without loss of derivatives) are proved in [1] for potentials belonging to the schwartz class. when n ≥ 4, as mentioned above, the estimates (1.9) and (1.10) with k = 0 are proved in [16] under (1.1) only. the proof of theorem 1.1 and corollary 1.2, which will be given in section 2, is based very much on the analysis developed in [16]. a similar analysis as above can be carried out for the schrödinger group as well. the free one satisfies the following dispersive estimate ∥∥eitg0ψ(h2g0) ∥∥ l 1 →l ∞ ≤ c|t| −n/2 , ∀t 6= 0, h > 0, (1.13) where ψ ∈ c∞0 ((0, +∞)). on the other hand, it is proved in [15] that under the assumption (1.1) with δ > (n + 2)/2 only, the perturbed schrödinger group satisfies ∥∥eitgψ(h2g) ∥∥ l1→l∞ ≤ ch−(n−3)/2|t|−n/2, ∀t 6= 0, 0 < h ≤ 1. (1.14) this estimate is used in [15] to obtain dispersive estimates with a loss of (n − 3)/2 derivatives for eitgχa(g), ∀a > 0. cubo 10, 2 (2008) semi-classical dispersive estimates for the wave ... 5 in this work we will also expand eitgψ(h2g) modulo remainders whose l1 → l∞ norm is upper bounded by cmh m−(n−2)/2−ǫ |t|−n/2, 0 < h ≤ h0 ≪ 1, for every integer m ≥ 0, similarly to the wave group above. to this end, choose a function ψ1 ∈ c ∞ 0 ((0, +∞)) such that ψ1 = 1 on supp ψ, and set t (h) := ( id + ψ1(h 2g0) − ψ1(h 2g) ) −1 = id + o(h2), f 00 (t,h) = e itg0ψ(h2g0), f0(t,h) = ψ1(h 2g0)e itg0ψ(h2g), w0(t,h) = e itg0ψ1(h 2g0), fj (t,h) = i ∫ t 0 w0(t − τ,h)v t (h)fj−1(τ,h)dτ, j ≥ 1, f 0 j (t,h) = i ∫ t 0 w0(t − τ,h)v f 0 j−1(τ,h)dτ, j ≥ 1. theorem 1.3 let v satisfy (1.1) with δ > (n + 2)/2. then, there exists a constant h0 > 0 so that for all 0 < h ≤ h0, t 6= 0, 0 < ǫ ≪ 1, we have the estimate ∥∥∥∥∥∥ eitgψ(h2g) − t (h) m∑ j=0 fj (t,h) ∥∥∥∥∥∥ l 1 →l ∞ ≤ cmh m−(n−2)/2−ǫ |t|−n/2, (1.15) for every integer m ≥ 0 with a constant cm > 0 independent of t and h. moreover, the operators fj satisfy the estimates ‖f0(t,h)‖l1→l∞ ≤ c|t| −n/2, (1.16) ‖fj (t,h)‖l1→l∞ ≤ cjh j−n/2−ǫ |t|−n/2, j ≥ 1, (1.17) ∥∥fj (t,h) − f 0j (t,h) ∥∥ l1→l∞ ≤ cjh j+2−n/2−ǫ |t|−n/2, j ≥ 1. (1.18) thus, to improve the estimate (1.14) in h, it suffices to improve the estimate (1.17). we also have the following corollary 1.4 let v satisfy (1.1) with δ > (n + 2)/2 and suppose in addition that there exists 0 ≤ k ≤ (n − 3)/2 such that the operators fj satisfy the estimate ‖fj (t,h)‖l1→l∞ ≤ ch k−(n−3)/2 |t|−n/2, (1.19) for all integers 1 ≤ j ≤ k + 3/2. then, for every a > 0, 0 < ǫ ≪ 1, we have the estimate ∥∥∥eitggk/2−(n−3)/4−ǫχa(g) ∥∥∥ l1→l∞ ≤ cǫ|t| −n/2, ∀t 6= 0, (1.20) while for every 0 ≤ q ≤ (n − 3)/2 − k, 2 ≤ p < 2(n−1−2q−2k) n−3−2q−2k , we have ∥∥∥eitgg−αq/2χa(g) ∥∥∥ l p ′ →l p ≤ c|t|−αn/2, ∀t 6= 0, (1.21) where 1/p + 1/p′ = 1, α = 1 − 2/p. moreover, if there exists an operator fk(t), independent of h, such that the following estimates hold ∥∥∥fk(t)gk/2−(n−3)/40 ∥∥∥ l1→l∞ ≤ c|t|−n/2, (1.22) 6 f. cardoso and g. vodev cubo 10, 2 (2008) ∥∥f1(t,h) − fk(t)ψ(h2g0) ∥∥ l 1 →l ∞ ≤ chk−(n−3)/2+ε|t|−n/2, (1.23) ‖fj (t,h)‖l1→l∞ ≤ ch k−(n−3)/2+ε |t| −n/2 , (1.24) for 2 ≤ j ≤ k + 3/2 with some ε > 0, then we have ∥∥∥eitggk/2−(n−3)/4χa(g) ∥∥∥ l 1 →l ∞ ≤ c|t|−n/2, ∀t 6= 0. (1.25) furthermore, when n = 4, 5 the estimates (1.20), (1.21) and (1.25) hold true if we suppose (1.19), (1.23) and (1.24) fulfilled with fj replaced by f 0 j . as in the case of the wave group above, the estimates (1.19), (1.22), (1.23) and (1.24) with k > 0 seem hard to establish (even if we replace fj by f 0 j ) and the proof would certainly require some regularity condition on the potential. indeed, it follows from the results in [5] that there exist compactly supported potentials v ∈ cν (rn), ∀ν < (n− 3)/2, for which these estimates with k = (n − 3)/2 fail to hold. therefore, it is naural to expect that they hold true for potentials v ∈ c(n−3)/2−k(rn). we also conjecture that the statements of theorem 1.3 and corollary 1.4 hold true for potentials satisfying (1.1) with δ > (n + 1)/2 as for the wave group above. note that when n = 2 the estimate (1.25) without loss of derivatives (that is, with k = (n − 3)/2) is proved in [12] under (1.1) with δ > 2. in this case this estimate is proved (for a large enough) in [10] for potentials satisfying (1.11). when n = 3 this estimate is proved in [6] for potentials v ∈ l3/2−ǫ ∩l3/2+ǫ, 0 < ǫ ≪ 1, and in particular for potentials satisfying (1.1) with δ > 2. in this case it is also proved in [13] for potentials satisfying (1.12) with c < 4π. when n ≥ 4 the optimal dispersive estimate (that is, without loss of derivatives) is proved in [9] for potentials satisfying (1.1) with δ > n as well as v̂ ∈ l1. this result has been recently extended in [11] to potentials satisfying (1.1) with δ > n − 1 as well as v̂ ∈ l1. when n ≥ 4, as mentioned above, the estimates (1.21) and (1.25) with k = 0 are proved in [15] under (1.1) with δ > (n + 2)/2 only. the proof of theorem 1.3 and corollary 1.4, which will be given in section 3, relies very much on the analysis developed in [15]. acknowledgements. a part of this work was carried out while f. c. was visiting the university of nantes in may 2007 with the support of the agreement brazil-france in mathematics proc. 69.0014/01-5. the first author has also been partially supported by the cnpq-brazil. 2 semi-classical expansion of eit √ gϕ(h √ g) we keep the same notations as in the introduction. our starting point is the following identity which can be derived easily from duhamel’s formula (see [16]) ( id + ϕ1(h √ g0) − ϕ1(h √ g) ) eit √ gϕ(h √ g) = e0(t,h) − h ∫ t 0 u0(t − τ,h)v e iτ √ g ϕ(h √ g)dτ. (2.1) cubo 10, 2 (2008) semi-classical dispersive estimates for the wave ... 7 we rewrite (2.1) as follows eit √ gϕ(h √ g) = ẽ0(t,h) + ∫ t 0 ũ0(t − τ,h)v e iτ √ gϕ(h √ g)dτ, (2.2) where ẽ0(t,h) = t (h)e0(t,h), ũ0(t,h) = −ht (h)u0(t,h). iterating (2.2) m times leads to the identity e it √ g ϕ(h √ g) = m∑ j=0 ẽj (t,h) + rm+1(t,h), (2.3) where the operators ẽj , j ≥ 1, are defined by ẽj (t,h) = ∫ t 0 ũ0(t − τ,h)v ẽj−1(τ,h)dτ, while the operators rm, m ≥ 0, are defined as follows r0(t,h) = e it √ gϕ(h √ g), rm+1(t,h) = ∫ t 0 ũ0(t − τ,h)v rm(τ,h)dτ. it is clear from (2.3) that the estimate (1.4) follows from the following proposition 2.1 under the assumptions of theorem 1.1, for all 0 < h ≤ h0, t 6= 0, 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1, we have the estimates ‖rm+1(t,h)‖l1→l∞ ≤ cmh m−n+1 |t| −(n−1)/2 , (2.4) ∥∥〈x〉−s−ǫrm+1(t,h) ∥∥ l1→l2 ≤ cmh m−n/2+1 |t|−s, (2.5) for every integer m ≥ 0. proof. for m = 0 the estimate (2.4) is proved in [16] (see (4.10)). we will now derive (2.4) for m ≥ 1 from (2.5) and the following estimate proved in [16] (see (2.4)): ∫ ∞ −∞ |t|2s ∥∥∥〈x〉−1/2−s−ǫeit √ g0ϕ(h √ g0)f ∥∥∥ 2 l 2 dt ≤ ch−n ‖f‖ 2 l 1 , ∀f ∈ l 1, (2.6) for 0 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. by (2.5) and (2.6), we have |t|(n−1)/2 |〈rm+1(t,h)f,g〉| ≤ c ∫ t t/2 |τ|(n−1)/2 ∥∥∥〈x〉−1−ǫũ0(t − τ,h)∗g ∥∥∥ l2 ∥∥∥〈x〉−(n−1)/2−ǫrm(τ,h)f ∥∥∥ l2 dτ 8 f. cardoso and g. vodev cubo 10, 2 (2008) +c ∫ t t/2 |τ| (n−1)/2 ∥∥∥〈x〉−n/2−ǫũ0(τ,h)∗g ∥∥∥ l 2 ∥∥∥〈x〉−1/2−ǫrm(t − τ,h)f ∥∥∥ l 2 dτ ≤ ch m−n/2 ‖f‖l1 (∫ ∞ −∞ 〈τ ′ 〉 1+ǫ/2 ∥∥∥〈x〉−1−ǫũ0(τ′,h)∗g ∥∥∥ 2 l 2 dτ ′ )1/2 +c (∫ ∞ −∞ |τ|n−1 ∥∥∥〈x〉−n/2−ǫũ0(τ,h)∗g ∥∥∥ 2 l2 dτ )1/2 (∫ ∞ −∞ ∥∥∥〈x〉−1/2−ǫrm(τ′,h)f ∥∥∥ 2 l2 dτ′ )1/2 ≤ chm+1−n‖f‖l1‖g‖l1. we will now prove (2.5) by induction in m. for m = 0 it is proved in [16] (see (4.6)) with s = (n − 1)/2 but the proof for general s is the same. we will show that (2.5) for rm+1 follows from (2.5) for rm and the following estimate proved in [16] (see (2.1)): ∥∥∥〈x〉−seit √ g0ϕ(h √ g0)〈x〉 −s ∥∥∥ l2→l2 ≤ c〈t〉−s, ∀t, 0 < h ≤ 1. (2.7) consider first the case 1 ≤ s ≤ (n − 1)/2. we have |t| s ∥∥〈x〉−s−ǫrm+1(t,h) ∥∥ l 1 →l 2 ≤ c ∫ t t/2 |τ|s ∥∥∥〈x〉−s−ǫũ0(t − τ,h)〈x〉−1−ǫ ∥∥∥ l 2 →l 2 ∥∥〈x〉−s−ǫrm(τ,h) ∥∥ l 1 →l 2 dτ +c ∫ t t/2 |τ|s ∥∥∥〈x〉−s−ǫũ0(τ,h)〈x〉−s−ǫ ∥∥∥ l2→l2 ∥∥〈x〉−1−ǫrm(t − τ,h) ∥∥ l1→l2 dτ ≤ chm+1−n/2 ∫ ∞ −∞ 〈τ′〉−1−ǫdτ′ + ch ∫ ∞ −∞ ∥∥〈x〉−1−ǫrm(τ′,h) ∥∥ l 1 →l 2 dτ′ ≤ chm+1−n/2. let now 1/2 − ǫ/4 ≤ s ≤ 1. we have |t| s ∥∥〈x〉−s−ǫrm+1(t,h) ∥∥ l 1 →l 2 ≤ c ∫ t t/2 |τ|s ∥∥∥〈x〉−s−ǫũ0(t − τ,h)〈x〉−1/2−ǫ ∥∥∥ l2→l2 ∥∥∥〈x〉−s−1/2−ǫrm(τ,h) ∥∥∥ l1→l2 dτ +c ∫ t t/2 |τ|s ∥∥∥〈x〉−s−ǫũ0(τ,h)〈x〉−s−ǫ ∥∥∥ l2→l2 ∥∥〈x〉−1−ǫrm(t − τ,h) ∥∥ l1→l2 dτ ≤ ch (∫ ∞ −∞ 〈τ′〉−1−ǫdτ′ )1/2 (∫ ∞ −∞ |τ|2s ∥∥∥〈x〉−s−1/2−ǫrm(τ,h) ∥∥∥ 2 l1→l2 dτ )1/2 +ch ∫ ∞ −∞ ∥∥〈x〉−1−ǫrm(τ′,h) ∥∥ l 1 →l 2 dτ ′ ≤ ch m+1−n/2 . 2 to prove (1.6) observe first that in the same way as in the proof of (2.5) one can show that the operator ej satisfies the estimate ∥∥〈x〉−s−ǫej (t,h) ∥∥ l 1 →l 2 ≤ cjh j−n/2 |t| −s , j ≥ 1, (2.8) cubo 10, 2 (2008) semi-classical dispersive estimates for the wave ... 9 for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. on the other hand, proceeding as in the proof of (2.4), one can easily see that (2.8) implies (1.6). to prove (1.7) we decompose ej − e 0 j as follows ej (t,h) − e 0 j (t,h) = −h ∫ t 0 u0(t − τ,h)v (t (h) − id)ej−1(τ,h)dτ +h ∫ t 0 u0(t − τ,h)v (ej−1(τ,h) − e 0 j−1(τ,h))dτ := e 1 j (t,h) + e2 j (t,h). (2.9) using (2.8), in the same way as in the proof of (1.6), one gets ∥∥e1 j (t,h) ∥∥ l1→l∞ ≤ cjh j+2−n |t|−(n−1)/2. (2.10) on the other hand, it is easy to see from (2.9) by induction in j that we have the estimate ∥∥〈x〉−s−ǫ(ej (t,h) − e0j (t,h)) ∥∥ l 1 →l 2 ≤ cjh j+2−n/2 |t|−s, j ≥ 0, (2.11) for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. it follows from (2.11) that the operator e2 j satisfies the estimate ∥∥e2 j (t,h) ∥∥ l 1 →l ∞ ≤ cjh j+2−n |t|−(n−1)/2. (2.12) now (1.7) follows from (2.9), (2.10) and (2.12). proof of corollary 1.2. following [16] we set φ(t,h) = eit √ gϕ(h √ g) − eit √ g0ϕ(h √ g0). it follows from (1.4) and (1.8) that the operator φ satisfies the estimate ‖φ(t,h)‖ l 1 →l ∞ ≤ ch k−n+1 |t| −(n−1)/2 . (2.13) on the other hand, we have (see theorem 3.1 of [16]) ‖φ(t,h)‖ l2→l2 ≤ ch, ∀t. (2.14) by interpolation between (2.13) and (2.14) we conclude ‖φ(t,h)‖ l p ′ →l p ≤ ch 1−α(n−k) |t| −α(n−1)/2 , (2.15) for every 2 ≤ p ≤ +∞, where 1/p + 1/p′ = 1, α = 1 − 2/p. now we will make use of the identity σ−α((n+1)/2+q)χa(σ) = ∫ 1 0 ϕ(θσ)θα((n+1)/2+q)−1dθ, where ϕ(σ) = σ1−α((n+1)/2+q)χ′ a (σ) ∈ c∞0 ((0, +∞)). by (2.15) we get ∥∥∥eit √ g ( √ g)−α((n+1)/2+q)χa( √ g) − eit √ g0 ( √ g0) −α((n+1)/2+q)χa( √ g0) ∥∥∥ lp ′ →lp 10 f. cardoso and g. vodev cubo 10, 2 (2008) ≤ ∫ 1 0 ‖φ(t,θ)‖ lp ′ →lp θα((n+1)/2+q)−1dθ ≤ c|t|−α(n−1)/2 ∫ 1 0 θ−α((n−1)/2−k−q)dθ ≤ c|t|−α(n−1)/2, (2.16) provided α((n − 1)/2 − k − q) < 1, that is, for 2 ≤ p < 2(n−1−2q−2k) n−3−2q−2k . now (1.10) follows from (2.16) and the fact that it holds for the free operator. similarly, by (2.13) we get ∥∥∥eit √ g ( √ g) k−n+1−ǫ χa( √ g) − e it √ g0 ( √ g0) k−n+1−ǫ χa( √ g0) ∥∥∥ l 1 →l ∞ ≤ ∫ 1 0 ‖φ(t,θ)‖ l1→l∞ θn−k−2+ǫdθ ≤ c|t|−(n−1)/2 ∫ 1 0 θ−1+ǫdθ ≤ cǫ|t| −(n−1)/2. (2.17) now (1.9) follows from (2.17) and the fact that it holds for the free operator. 2 3 semi-classical expansion of eitgψ(h2g) we keep the same notations as in the introduction. we will make use of the following identity which can be derived easily from duhamel’s formula (see [15]) ( id + ψ1(h 2g0) − ψ1(h 2g) ) eitgψ(h2g) = f0(t,h) + i ∫ t 0 w0(t − τ,h)v e iτ gψ(h2g)dτ. (3.1) we rewrite (3.1) as follows e itg ψ(h 2 g) = f̃0(t,h) + ∫ t 0 w̃0(t − τ,h)v e iτ g ψ(h 2 g)dτ, (3.2) where f̃0(t,h) = t (h)f0(t,h), w̃0(t,h) = it (h)w0(t,h). iterating (3.2) m times leads to the identity eitgψ(h2g) = m∑ j=0 f̃j (t,h) + rm+1(t,h), (3.3) where the operators f̃j , j ≥ 1, are defined by f̃j (t,h) = ∫ t 0 w̃0(t − τ,h)v f̃j−1(τ,h)dτ, while the operators rm, m ≥ 0, are defined as follows r0(t,h) = e itg ψ(h 2 g), cubo 10, 2 (2008) semi-classical dispersive estimates for the wave ... 11 rm+1(t,h) = ∫ t 0 w̃0(t − τ,h)v rm(τ,h)dτ. it is clear from (3.3) that the estimate (1.15) follows from the following proposition 3.1 under the assumptions of theorem 1.2, for all 0 < h ≤ h0, t 6= 0, 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1, we have the estimates ‖rm+1(t,h)‖l1→l∞ ≤ cmh m−(n−2)/2−ǫ |t| −n/2 , (3.4) ∥∥∥〈x〉−1/2−s−ǫrm+1(t,h) ∥∥∥ l1→l2 ≤ cmh m+s−(n−3)/2−ǫ/6 |t|−s−1/2, (3.5) for every integer m ≥ 0. proof. for m = 0 these estimates are proved in section 4 of [15]. we will now derive (3.4) for m ≥ 1 from (3.5) and the following estimate proved in [15] (see (2.1)): ∥∥∥eitg0ψ(h2g0)〈x〉−1/2−s−ǫ ∥∥∥ l2→l∞ ≤ chs−(n−1)/2|t|−s−1/2, t 6= 0, 0 < h ≤ 1, (3.6) for 0 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. we have |t|n/2 ‖rm+1(t,h)‖l1→l∞ ≤ c ∫ t t/2 |τ|n/2 ∥∥∥w̃0(t − τ,h)〈x〉−1−ǫ ∥∥∥ l2→l∞ ∥∥∥〈x〉−n/2−ǫrm(τ,h) ∥∥∥ l1→l2 dτ +c ∫ t t/2 |τ| n/2 ∥∥∥w̃0(τ,h)〈x〉−n/2−ǫ ∥∥∥ l 2 →l ∞ ∥∥〈x〉−1−ǫrm(t − τ,h) ∥∥ l 1 →l 2 dτ ≤ chm−ǫ/6 ∫ ∞ −∞ ∥∥∥w̃0(τ′,h)〈x〉−1−ǫ ∥∥∥ l 2 →l ∞ dτ′ +c ∫ ∞ −∞ ∥∥〈x〉−1−ǫrm(τ′,h) ∥∥ l 1 →l 2 dτ ′ ≤ ch m−(n−2)/2−ǫ/3 . we will now show that (3.5) for rm+1 follows from (3.5) for rm and the following estimate proved in [15] (see (2.2)): ∥∥〈x〉−seitg0ψ(h2g0)〈x〉−s ∥∥ l 2 →l 2 ≤ c〈t/h〉−s, ∀t, 0 < h ≤ 1. (3.7) we have |t|s+1/2 ∥∥∥〈x〉−1/2−s−ǫrm+1(t,h) ∥∥∥ l1→l2 ≤ c ∫ t t/2 |τ| s+1/2 ∥∥∥〈x〉−1/2−s−ǫw̃0(t − τ,h)〈x〉−1−ǫ ∥∥∥ l 2 →l 2 ∥∥∥〈x〉−1/2−s−ǫrm(τ,h) ∥∥∥ l 1 →l 2 dτ +c ∫ t t/2 |τ|s+1/2 ∥∥∥〈x〉−1/2−s−ǫw̃0(τ,h)〈x〉−1/2−s−ǫ ∥∥∥ l2→l2 ∥∥〈x〉−1−ǫrm(t − τ,h) ∥∥ l1→l2 dτ 12 f. cardoso and g. vodev cubo 10, 2 (2008) ≤ chm+s−(n−1)/2−ǫ/6 ∫ ∞ −∞ 〈τ′/h〉−1−ǫ/2dτ′ +chs+1/2 ∫ ∞ −∞ ∥∥〈x〉−1−ǫrm(τ′,h) ∥∥ l1→l2 dτ′ ≤ chm+s−(n−3)/2−ǫ/6. 2 to prove (1.17) observe that in the same way as in the proof of (3.5) one can show that the operator fj satisfies the estimate ∥∥∥〈x〉−1/2−s−ǫfj (t,h) ∥∥∥ l 1 →l 2 ≤ cjh j+s−(n−1)/2−ǫ/6 |t| −s−1/2 , j ≥ 1, (3.8) for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. on the other hand, proceeding as in the proof of (3.4), one can easily see that (3.8) implies (1.17). to prove (1.18) we decompose fj − f 0 j as follows fj (t,h) − f 0 j (t,h) = i ∫ t 0 w0(t − τ,h)v (t (h) − id)fj−1(τ,h)dτ −i ∫ t 0 w0(t − τ,h)v (fj−1(τ,h) − f 0 j−1(τ,h))dτ := n 1 j (t,h) + n2 j (t,h). (3.9) using (3.8), in the same way as in the proof of (1.17), one gets ∥∥n1 j (t,h) ∥∥ l1→l∞ ≤ cjh j+2−n/2−ǫ |t|−n/2. (3.10) on the other hand, it is easy to see from (3.9) by induction in j that we have the estimate ∥∥∥〈x〉−1/2−s−ǫ(fj (t,h) − f 0j (t,h)) ∥∥∥ l 1 →l 2 ≤ cjh j+2+s−(n−1)/2−ǫ/6 |t| −s−1/2 , j ≥ 0, (3.11) for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. it follows from (3.11) that the operator n2 j satisfies the estimate ∥∥n2 j (t,h) ∥∥ l1→l∞ ≤ cjh j+2−n/2−ǫ |t|−n/2. (3.12) now (1.18) follows from (3.9), (3.10) and (3.12). proof of corollary 1.4. following [15] we set ψ(t,h) = eitgψ(h2g) − eitg0ϕ(h2g0). it follows from (1.15) and (1.19) that the operator ψ satisfies the estimate ‖ψ(t,h)‖ l 1 →l ∞ ≤ ch k−(n−3)/2 |t|−n/2. (3.13) on the other hand, we have (see theorem 3.1 of [15]) ‖ψ(t,h)‖ l 2 →l 2 ≤ ch, ∀t. (3.14) cubo 10, 2 (2008) semi-classical dispersive estimates for the wave ... 13 by interpolation between (3.13) and (3.14) we conclude ‖ψ(t,h)‖ l p ′ →l p ≤ ch 1−α((n−1)/2−k) |t| −αn/2 , (3.15) for every 2 ≤ p ≤ +∞, where 1/p + 1/p′ = 1, α = 1 − 2/p. now we will make use of the identity σ−αq/2χa(σ) = ∫ 1 0 ψ(θσ)θαq/2−1dθ, where ψ(σ) = σ1−αq/2χ′ a (σ) ∈ c∞0 ((0, +∞)). by (3.15) we get ∥∥∥eitgg−αq/2χa(g) − eitg0g−αq/20 χa(g0) ∥∥∥ l p ′ →l p ≤ ∫ 1 0 ∥∥∥ψ(t, √ θ) ∥∥∥ lp ′ →lp θαq/2−1dθ ≤ c|t| −αn/2 ∫ 1 0 θ −1/2−α((n−1)/2−k−q)/2 dθ ≤ c|t| −αn/2 , (3.16) provided 1/2 + α((n − 1)/2 − k − q)/2 < 1, that is, for 2 ≤ p < 2(n−1−2q−2k) n−3−2q−2k . now (1.21) follows from (3.16) and the fact that it holds for the free operator. similarly, by (3.13) we get ∥∥∥eitggk/2−(n−3)/4−ǫχa(g) − eitg0gk/2−(n−3)/4−ǫ0 χa(g0) ∥∥∥ l1→l∞ ≤ ∫ 1 0 ∥∥∥ψ(t, √ θ) ∥∥∥ l 1 →l ∞ θ−k/2+(n−3)/4−1+ǫdθ ≤ c|t|−n/2 ∫ 1 0 θ−1+ǫdθ ≤ cǫ|t| −n/2. (3.17) now (1.20) follows from (3.17) and the fact that it holds for the free operator. by (1.15), (1.19), (1.23) and (1.24), we have ∥∥ψ(t,h) − fk(t)ψ(h2g0) ∥∥ l 1 →l ∞ ≤ chk−(n−3)/2+ε|t|−n/2. (3.18) proceeding as above with a suitably chosen function ψ, we obtain from (3.18) ∥∥∥eitggk/2−(n−3)/4χa(g) − eitg0gk/2−(n−3)/40 χa(g0) − fk(t)g k/2−(n−3)/4 0 ∥∥∥ l 1 →l ∞ ≤ ∫ 1 0 ∥∥∥ψ(t, √ θ) − fk(t)ψ(θg0) ∥∥∥ l 1 →l ∞ θ−k/2+(n−3)/4−1dθ ≤ c|t| −n/2 ∫ 1 0 θ −1+ε/2 dθ ≤ c|t| −n/2 . (3.19) now (1.25) follows from (3.19), (1.22) and the fact that it holds for the free operator. 2 received: october 2007. revised: december 2007. 14 f. cardoso and g. vodev cubo 10, 2 (2008) references [1] m. beals, optimal l∞ decay estimates for solutions to the wave equation with a potential, commun. partial diff. equations 19 (1994), 1319–1369. [2] f. cardoso, c. cuevas and g. vodev, dispersive estimates of solutions to the wave equation with a potential in dimensions two and three, serdica math. j. 31 (2005), 263–278. [3] f. cardoso, c. cuevas and g. vodev, weighted dispersive estimates for solutions of the schrödinger equation, serdica math. j., 34 (2008), 39–54. [4] p. d’ancona and v. pierfelice, on the wave equation with a large rough potential, j. funct. analysis 227 (2005), 30–77. [5] v. georgiev and n. visciglia, decay estimates for the wave equation with potential, commun. partial diff. equations 28 (2003), 1325–1369. [6] m. goldberg, dispersive bounds for the three dimensional schrödinger equation with almost critical potentials, gafa 16 (2006), 517–536. [7] m. goldberg and w. schlag, dispersive estimates for schrödinger operators in dimensions one and three, commun. math. phys. 251 (2004), 157–178. [8] m. goldberg and m. visan, a conterexample to dispersive estimates for schrödinger operators in higher dimensions, commun. math. phys. 266 (2006), 211–238. [9] j.-l. journé, a. soffer and c. sogge, decay estimates for schrödinger operators, commun. pure appl. math. 44 (1991), 573–604. [10] s. moulin, high frequency dispersive estimates in dimension two, submitted. [11] s. moulin and g. vodev, low frequency dispersive estimates for the schrödinger group in higher dimensions, asymptot. anal., 55 (2007), 49–71. [12] w. schlag, dispersive estimates for schrödinger operators in two dimensions, commun. math. phys. 257 (2005), 87–117. [13] i. rodnianski and w. schlag, time decay for solutions of schrödinger equations with rough and time-dependent potentials, invent. math. 155 (2004), 451–513. [14] g. vodev, dispersive estimates of solutions to the schrödinger equation, ann. h. poincaré 6 (2005), 1179–1196. [15] g. vodev, dispersive estimates of solutions to the schrödinger equation in dimensions n ≥ 4, asymptot. anal. 49 (2006), 61–86. [16] g. vodev, dispersive estimates of solutions to the wave equation with a potential in dimensions n ≥ 4, commun. partial diff. equations 31 (2006), 1709–1733. n1 cubo 11, 07-11 (1996) recibido: abril 1995. harniltonety and automorphirns group of graph preserved by substitution. • eduardo montenegro v. abstrae t . thc sube~itutlion is a graph opcration. this o pcrn.tion con!'{ists in rcplacing a vertex by a gtaph. the a..im of ti-his work is to analizc thc pr cscrvation o í ccr tnn prop<:rtics in thc substlit.utiion of o. graph. spccificall y, thcsc propcrtics a re: (i) hnmiltoncty o.nd (ii) group o í aut.omorphis ms of n givc n graph c. lntroduction t hc graphs to be cons idercd will be in general si mple a.nd finitc, wit.h o. nonempty set o f cdgcs. for a grnph g , v(g) deno te thc set of n : r tice5 and e(g') denote thc ~t of cdgcs. t hc cardinafüy of v(g) is crllccl o rdc r of g a.ne! thc cnrdinajity of b(c) i5 cn.ucd s izc of g. a (p , q) graph has p ordc r and q s izc. two vcrticcs u 1.m d u iuc callcd n c ighbor.9 if { 111 u} is (ul edge oí g. far l\o)' vcr t.cx u oí (,' , de note by n ., thc set o f ncighbors of 11. to sim plify the nott1.lion , l\jl cdgc {x, y} is writ.tcn m :r:y (or yx). ot.hcr concepts uscd i11 lhis work 1rnd no t dcfincd cxplicitily can be found in lhc rcforenccs. ' pmtw ~i. by dgi u placeo, proj«:l 1vo 1394~. cubo 11 e. mont~tgro \f. 2 the subst it uti o n assu me lhal e and k are two di sjoint graphs by \'c rli ces. fo r a vcrlex v in v(g) a nd a fun clion s: n,, v(k) it will be deflned th e s ubs titut ion (9] of the vert ex v by the graph k , as the gmp h m = g(v ,s) k s uch that: ( i) v ( m ) ~ ( v (g) u v(k)) {v ) and (2) e(m ) ~ ( e(g)(vx 'x e n0 } ) u (x•(x ) 'x e n0 ) thc ve r tex u is s aid to be t hc ve r tex s u bstit utcd by k in g undcr t he fun ction ·' and thi s function is cal led of s ubst it ut ion . now let v1 ,· ,vn be thc ve rt iccs of a graph g ami h 1 , , fl,, a seq uence of graphs with no co mmon ve rt iccs runong themselves o r with c . lt will be de not ed by 1\h = 1\ij,_1(v1:,s1:) flt t he graph which is obtaincd by s u b6t it ut ion of k ve r tices of c by graphs h., 1 :s i :s k, whcre mo =c. in o th c r wo r ds, m 1 denotes a grap h obtained by s ubstitution of on ly one ver tcx of g, m 2 denotes a grap h obtained by s ubstitution of on ly one ver t ex of m 1 , and so on. note that cve ry vertcx substit u t cd musl bclong to v( c). lt can be said t hat a n e 2, whos c uerticc.'j 11re lobelcd 111, • · , u,.. !/ {s;} is a .sequ en.ce o/ grnph.3 wi1h n.o common vcrticc.'j amo ng t.hcm.3elves n or wil.h c , wh crc cn ch s , ~ k ""'(~). t.h cn ( 1) .h,.( g ) '·' hn..millommn nnd ( ii) aw ( m,(c))" aut.(c). 10 cubo 11 proo r (i) since g is a h1ut1iltonea11 graph lhe n it has a ge ne rator cycle de notcd by c(g ). also cl.\ch s; havc a gcnerator cycle denoted by c<1>(g) . through a su itable sclection of the s ubstitution functio ns, th e cycl e c (mp(g ) ) defi ned by c(m,.(c)):::::: u~~ 1 c(i)(g) is generator o f mp(c ). (ii) the o nly one admissible m overnc nl in mp(g), t hroug h a symmetry of its vcrtices that preserve edgcs, a.re th e induoed by symmetry of th c verticcs of g that preser ve edge. in fa.et the interna! edge of a block of mp(g) may be intercha.nged only by internal cd ges of other b lock of m,,{g) [1 2]. f'or this reason g and mp(g) have the same group of automorphisms. • thoorcm 3.4 let e be a hamiltonean gmph , r-regular, t > 2, whose verticc.s are la beled 11 1 , · , vp. lf { s;} is a sequence of grop/i.00. ti o mt\t.h. , 6 239-2 5 0 {1938) . {4) &b idu &q i g . c: mpm 111ith give n group and ginn gru.11 h th eo ri ca.l properties, c ""nad . . l. m1\t h ., 9 515-552 (1 95 7) . l!>j abiduss i c. et al. ( k oll lu' , f'rmkl , babai ) l/nrmftonion cub ic grn ph.s and centmlúflrs of in.uoluli on_, , l\ nd . j . m&lh ., 31 458-464 (1979) . n'aíso c nsilln 4 05 9. vlllpuraíso cubo a mathematical journal vol.20, no¯ 3, (65–79). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300065 ball comparison between jarratt’s and other fourth order method for solving equations ioannis k. argyros 1 and santhosh george 2 1department of mathematical sciences, cameron university, lawton, ok 73505, usa. 2department of mathematical and computational sciences, national institute of technology karnataka, india-575 025. iargyros@cameron.edu, sgeorge@nitk.edu.in abstract the convergence order of iterative methods is determined using high order derivatives and taylor series, and without providing computable error bounds, uniqueness of the solution results or information on how to choose the initial point. we address all these problems by using hypotheses only on the first derivative. moreover, to achieve all these we present our technique using a comparison between the convergence radii of jarratt’s fourth order method and another method of the same convergence order. resumen el orden de convergencia de métodos iterativos es determinado usando derivadas de orden alto y series de taylor, y sin poder entregar cotas de error calculables, resultados de unicidad de soluciones o información de cómo elegir el punto inicial. tratamos estos problemas usando hipótesis sólo en la primera derivada. más aún, para responder todos los anteriores, presentamos una técnica que usa una comparación entre el radio de convergencia del método de cuarto orden de jarratt y otro método con el mismo orden de convergencia. http://dx.doi.org/10.4067/s0719-06462018000300065 66 ioannis k. argyros and santhosh george cubo 20, 3 (2018) keywords and phrases: jarratt method; banach space; ball convergence. 2010 ams mathematics subject classification: 65d10, 65d99. cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 67 1 introduction let b1 and b2 stand for banach spaces, with ω ⊆ b1 being nonempty, open and convex. consider an equation f(x) = 0, (1.1) where f : ω −→ b2 is a differentiable in the of fréchet-sense. the task of finding a solution p of equation (1.1) is very difficult in general. it is even harder to find a solution p in closed form, since this can be achieved in some special cases. that explains why most authors develop iterative methods, to generate a sequence approximating p under some initial conditions. notice that, solution methods for equation (1.1) is an important area of research, since a plethora of problems from diverse disciplines such that mathematics, optimization, mathematical programming, chemistry, biology, physics, economics, statistics, engineering and other disciplines can be modeled into an equation of the form (1.1) using mathematical modeling [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. the most popular method is without a doubt newton’s method (nm) xn+1 = xn − f ′(xn) −1f(xn), x0 ∈ ω, and all n = 0, 1, 2, . . . . (1.2) nm converges quadratically to p for x0 sufficiently close to p [10]. to increase the convergence order numerous methods have been proposed [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. the order of these methods is almost exclusively been obtained using taylor series, and hypotheses on high order derivatives. no computable error bounds or uniqueness results are given, and the choice of the initial point is a shot in the dark. iterative methods are usually studied based on: semi-local and local convergence. the semilocal convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. a radius of convergence about p determines a ball such that if an initial point is selected from that ball convergence of the method to p is guaranteed. to deal with all these problems we have selected two popular fourth order methods. in particular, we compare the radii of convergence of fourth order jarratt’s iterative method defined [9, 12] for n = 0, 1, 2, . . . , as yn = xn − 2 3 f′(xn) −1f(xn) xn+1 = xn − 1 2 [(3f′(yn) − f ′(xn)) −1(3f′(yn) + f ′(xn))] ×f′(xn) −1f(xn), (1.3) 68 ioannis k. argyros and santhosh george cubo 20, 3 (2018) to the fourth order sharma’s method [13] defined for n = 0, 1, 2, . . . , as yn = xn − 2 3 f′(xn) −1f(xn) xn+1 = xn − 1 2 [−i + 9 4 f′(yn) −1f′(xn) + 3 4 f′(xn) −1f′(yn)] ×f′(xn) −1f(xn). (1.4) earlier convergence analysis of these methods, in the special case when b1 = b2 = r k used, assumptions of the fréchet derivatives of f of order up to five [9, 12, 13]. but these assumptions limit the applicability of methods (1.3) and (1.4). let as an example, b1 = b2 = r, ω = [− 1 2 , 3 2 ]. define f on ω as f(x) = x3 log x2 + x5 − x4 then, we have p = 1, and f′(x) = 3x2 log x2 + 5x4 − 4x3 + 2x2, f′′(x) = 6x log x2 + 20x3 − 12x2 + 10x, f′′′(x) = 6 log x2 + 60x2 = 24x + 22. clearly, f′′′(x) is not bounded on ω. so, methods (1.3) and (1.4) cannot be applied to solve the above example, if we use the analysis in the earlier studies. in this study, our analysis uses only the assumptions on the first fréchet derivative of f. moreover, we provide computable upper estimates on ‖xn −p‖, a radius of convergence as well as uniqueness results based on generalized lipschitz conditions. hence, we extend the applicability of these methods. our technique can be used to extend the applicability of other high order methods along the same lines. the rest of the study is organized as follows. in section 2 , the local convergence analysis is given and numerical examples are given in the last section 4. 2 local convergence it is convenient for the local convergence analysis of method (1.3) and method (1.4) to introduce some fucntions and parameters. first for method (1.3): let ω0 : s −→ s be a continuous and increasing function with w0(0) = 0, where s = [0, ∞). suppose that equation ω0(t) = 1 (2.1) has at least one positive solution. denote by ρ0 the smallest such solution. set s0 = [0, ρ0). let also ω : s0 −→ s and ω1 : s0 −→ s be continuous and increasing functions with ω(0) = 0. define cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 69 functions ϕ1 and ϕ̄1 on the interval s0 by ϕ1(t) = ∫1 0 ω((1 − θ)t)dθ + 1 3 ∫1 0 ω1(θt)dθ 1 − ω0(t) and ϕ̄1(t) = ϕ1(t) − 1. suppose that ω0(0) < 3. (2.2) then, we get by (2.2) that ϕ̄1(0) < 0 and ϕ̄1(t) −→ ∞ as t −→ ρ − 0 . the intermediate value theorem guarantees that equation ϕ̄1(t) = 0 has at least one solution in (0, ρ0). denote by r1 the smallest such solution. suppose that equation ω0(ϕ1(t)t) = 1 (2.3) has at least one positive solution. denote by ρ1 the smallest such solution. set ρ = min{ρ0, ρ1} and s1 = [0, ρ). define functions ϕ2 and ϕ̄2 on s1 by ϕ2(t) = ∫1 0 ω((1 − θ)t)dθ 1 − ω0(t) + 3 8 [ (ω0(ϕ1(t)t) + ω0(t)) 2 (1 − ω0(t))(1 − ω0(ϕ1(t)t)) +2 w0(ϕ1(t)t) + ω0(t) 1 − ω0(ϕ1(t)t) ] ∫1 0 ω1(θt)dθ 1 − ω0(t) and ϕ̄2(t) = ϕ2(t) − 1. we get that ϕ̄2(0) = −1 and ϕ̄2(t) −→ ∞ as t −→ ρ −. denote by r2 the smallest such solution of equation ϕ̄2(t) = 0. moreover, define a radius of convergence r by r = min{r1, r2}. (2.4) it follows that for each t ∈ [0, r) 0 ≤ ω0(t) < 1 (2.5) 0 ≤ ω0(ϕ1(t)t) < 1 (2.6) 0 ≤ ϕ1(t) < 1 (2.7) and 0 ≤ ϕ2(t) < 1. (2.8) let us introduce conditions (a): (a1) f : ω −→ b2 is continuously differentiable in the sense of fréchet and there exists p ∈ ω such that f(p) = 0 and f′(p)−1 ∈ l(b2, b1). 70 ioannis k. argyros and santhosh george cubo 20, 3 (2018) (a2) there exists function ω0 : s −→ s continuous and increasing with ω0(0) = 0 and for each x ∈ ω ‖f′(p)−1(f′(x) − f′(p))‖ ≤ ω0(‖x − p‖) and (2.2) holds. set ω0 = ω ∩ u(p, ρ0), where ρ0 is given in (2.1). (a3) there exist functions ω : s0 −→ s, ω1 : s0 −→ s continuous and increasing with ω(0) = 0 such that for each x, y ∈ ω0 ‖f′(p)−1(f′(y) − f′(x))‖ ≤ ω(‖y − x‖) and ‖f′(p)−1f′(x)‖ ≤ ω1(‖x − p‖). (a4) ū(p, r) ⊂ ω, ρ0, ρ1 exist and are given by (2.1) and (2.3), respectively. (a5) there exists r∗ ≥ r such that ∫1 0 ω0(θr ∗ )dθ < 1. set ω1 = ω ∩ ū(p, r ∗). next, the local convergence analysis is given for method (1.3) based on the conditions (a) and the preceding notation. theorem 2.1. suppose that the conditions (a) hold. then, sequence {xn} generated by (1.3), starting at x0 ∈ u(p, r) − {p} is well defined, remains in u(p, r) for each n = 0, 1, 2, 3, . . . and converges to p. moreover, the following error bounds hold ‖yn − p‖ ≤ ϕ1(‖x − p‖)‖x − p‖ ≤ ‖x − p‖ < r (2.9) and ‖xn+1 − p‖ ≤ ϕ2(‖x − p‖)‖x − p‖ ≤ ‖x − p‖, (2.10) where functions ϕ1 and ϕ2 are given previously and r is defined in (2.4). furthermore, the limit point p is the only solution of equation f(x) = 0 in the set ω1, which is defined in (a5). proof. mathematical induction is utilized to show (2.9) and (2.10). let x ∈ u(p, r) − {p}. then, by (a1), (a2), (2.1), (2.4) and (2.5), we obtain in turn that ‖f′(p)−1(f′(x) − f′(p))‖ ≤ ω0(‖x − p‖) ≤ ω0(r) < 1. (2.11) in view of (2.11) and the banach lemma on invertible operators [7, 8, 10], f′(x)−1 ∈ l(b2, b1) and ‖f′(x)−1f′(p)‖ ≤ 1 1 − ω(‖x − p‖) . (2.12) cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 71 the point y0 is well defined by the first substep of method (1.3) and (2.12) for x = x0. we can write by (a1) f(x) = f(x) − f(p) = ∫1 0 f′(p + θ(x − p))(x − p)dθ. (2.13) then, by the second hypothesis in (a3), we get by (2.13) that ‖f′(p)−1f′(p)‖ ≤ ∫1 0 ω1(θ‖x − p‖)dθ‖x − p‖. (2.14) using the first substep of method (1.3) for n = 0, (a3), (2.4), (2.7), (2.12) (for x = x0) and (2.14), we have in turn from y0 − p = x0 − p − f ′(x0) −1f(x0) + 1 3 f′(x0) −1f(x0) that ‖y0 − p‖ ≤ ‖f ′(x0) −1f′(p)‖‖ ∫1 0 f′(p)−1(f′(p + θ(x0 − p)) − f ′(x0))dθ(x − p)‖ 1 3 ‖f′(x0) −1f′(p)‖‖f′(p)−1f(x0)‖ ≤ [ ∫1 0 ω((1 − θ)‖x0 − p‖)dθ + 1 3 ∫1 0 ω1(θ‖x0 − p‖)dθ] 1 − ω0(‖x0 − p‖) ×‖x0 − p‖ = ϕ1(‖x0 − p‖)‖x0 − p‖ ≤ ‖x0 − p‖ < r, (2.15) which implies that (2.9) holds for n = 0 and y0 ∈ u(p, r). moreover, f ′(y0) −1 ∈ l(b2, b1), so x1 is well defined by the second substep of method (1.3) for n = 0 and (2.6). furthermore, by (2.4), (2.8), (2.12) (for x = y0), (2.14) (for x = y0) and the estimate x1 − p = x0 − p − f ′ (x0) −1f(x0) − 1 2 [−3i + 9 4 f′(y0) −1f′(x0) + 3 4 f′(x0) −1f′(y0)]f ′(x0) −1f(x0) = x0 − p − f ′ (x0) −1f(x0) − 3 2 [−i + 3 4 f′(y0) −1f′(x0) + 1 4 f′(x0) −1f′(y0)]f ′(x0) −1f(x0) = x0 − p − f ′(x0) −1f(x0) − 3 8 [f′(x0) −1(f′(y0) − f ′(x0))f ′(y0) −1(f′(y0) − f ′(x0)) −2f′(y0) −1 (f′(y0) − f ′ (x0)]f ′ (x0) −1f(x0), (2.16) 72 ioannis k. argyros and santhosh george cubo 20, 3 (2018) we have in turn that ‖x1 − p‖ ≤ ‖x0 − p − f ′ (x0) −1f(x0)‖ + 3 8 [‖f′(x0) −1f′(p)‖(‖f′(p)−1(f′(y0) − f ′(x0))‖ +‖f′(p)−1(f′(x0) − f ′ (p))‖)2 ‖f′(y0) −1f′(p)‖ +2‖f′(y0) −1f′(p)‖(‖f′(p)−1(f′(y0) − f ′(x0)‖ +‖f′(p)−1(f′(x0) − f ′(p))‖)] ‖f′(x0) −1f′(p)‖‖f′(p)−1f(x0)‖ ≤ {∫1 0 ω((1 − θ)‖x0 − p‖)dθ 1 − ω0(‖x0 − p‖) 3 8 [ (ω0(‖y0 − p‖) + ω0(‖x0 − p‖)) 2 (1 − ω0(‖x0 − p‖))(1 − ω0(‖y0 − p‖)) +2 ω0(‖x0 − p‖) + ω0(‖y0 − p‖) 1 − ω0(‖y0 − p‖) ] ∫1 0 ω1(θ‖x0 − p‖)dθ 1 − ω0(‖x0 − p‖) } ‖x0 − p‖ ≤ ϕ2(‖x0 − p‖)‖x0 − p‖ ≤ ‖x0 − p‖, (2.17) so (2.10) holds for n = 0 and x1 ∈ u(p, r), where we also used the following estimates in the derivativation of (2.16): −i + 3 4 f′(y0) −1f′(x0) + 1 4 f′(x0) −1f′(y0) (2.18) = − 3 4 i + 3 4 f′(yn) −1f′(xn) − 1 4 i + 1 4 f′(x0) −1f′(y0) = 3 4 [f′(y0) −1f′(x0) − i] + 1 4 [f′(x0) −1f′(y0) − i] = 3 4 f′(y0) −1(f′(x0) − f ′(y0)) + 1 4 f′(x0) −1(f′(y0) − f ′(x0)) = 1 4 f′(x0) −1 (f′(y0) − f ′ (x0)) − 1 4 f′(y0) −1 (f′(y0) − f ′ (x0)) − 2 4 f′(y0) −1(f′(y0) − f ′(x0)) = 1 4 (f′(x0) −1 − f′(y0) −1)(f′(y0) − f ′(x0)) − 1 2 f′(y0) −1 (f′(y0) − f ′ (x0)) = 1 4 f′(x0) −1(f′(y0) − f ′(x0))f ′(y0) −1(f′(y0) − f ′(x0)) − 1 2 f′(y0) −1(f′(y0) − f ′(x0)). (2.19) cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 73 the induction for (2.9) and (2.10) is completed, if xm, ym, xm+1 replace x0, y0, x1 in the preceding estimations. then, from the estimate ‖xm+1 − p‖ ≤ r‖xm − p‖ < r, r = ϕ2(‖x0 − p‖) ∈ [0, 1) (2.20) we conclude that limm−→∞ xm = p and xm+1 ∈ u(p, r). finally, let g = ∫1 0 f′(p1 + θ(p − p1))dθ for p1 ∈ ω1 with f(p1) = 0. then, by (a2), we get that ‖f′(p)−1(g − f′(p))‖ ≤ ∫1 0 ω0(θ‖p − p1‖)dθ ≤ ∫1 0 ω0(θr ∗)dθ < 1 (2.21) leading to g−1 ∈ l(b2, b1). then, from the identity 0 = f(p) − f(p1) = g(p − p1), we deduce that p1 = p. next, we study the local convergence analysis of method (1.4) in an analogous way. let ω0, ω, ω1, ρ0, ϕ1 and ϕ̄1 are previously. suppose that equation q(t) = 1 (2.22) where q(t) = 1 2 (3ω0(ϕ1(t)t)+ω0(t)) has at least one positive solution. denote by ρ1 the smallest such solution. set d1 = [0, ρ) where ρ = min{ρ0, ρ1}. define functions ϕ3 and ϕ̄3 on d1 by ϕ3(t) = ∫1 0 ω((1 − θ)t)dθ 1 − ω0(t) + 3 2 (ω0(t) + ω0(ϕ1(t)t)) ∫1 0 ω1(θt)dθ (1 − q(t))(1 − ω0(t)) and ϕ̄3 = ϕ3 − 1. we get ϕ̄3(t) = −1 and ϕ̄3(t) −→ ∞ as t −→ ρ −. denote by r3 the smallest solution of equation ϕ̄3 = 0 in (0, ρ). define a radius of convergence r by r = min{r1, r3}. (2.23) consider the conditions (a) again but with r given in (2.23) and ρ1 given in (2.22). call these conditions (a)’. then, for each t ∈ [0, r), we have 0 ≤ ω0(t) < 1 (2.24) 0 ≤ q(t) < 1 (2.25) 0 ≤ ϕ1(t) < 1 (2.26) and 0 ≤ ϕ3(t) < 1. (2.27) 74 ioannis k. argyros and santhosh george cubo 20, 3 (2018) theorem 2.2. suppose that the conditions (a) hold. then, sequence {xn} defined by (1.4), starting at x0 ∈ u(p, r) − {p} is well defined, remains in u(p, r) for each n = 0, 1, 2, 3, . . . and converges to p. moreover, the following error bounds hold ‖yn − p‖ ≤ ϕ1(‖x − p‖)‖x − p‖ ≤ ‖x − p‖ < r (2.28) and ‖xn+1 − p‖ ≤ ϕ3(‖x − p‖)‖x − p‖ ≤ ‖x − p‖, (2.29) where functions ϕ1 and ϕ3 are given previously and r is defined in (2.23). furthermore, the limit point p is the only solution of equation f(x) = 0 in the set ω1, which is defined previously. proof. it follows as in theorem 2.1 but notice ‖(2f′(p))−1(3f′(y0) − f ′(x0) − 3f ′(p) + f′(p))‖ ≤ 1 2 (3‖f′(p)−1(f′(y0) − f ′ (p))‖ +‖f′(p)−1(f′(x0) − f ′(p))‖) ≤ 1 2 (3ω0(‖y0 − p‖) + ω0(‖x0 − p‖)) ≤ 1 2 (3ω0(ϕ1(‖x0 − p‖)‖x0 − p‖) + ω0(‖x0 − p‖) = q(‖x0 − p‖) < 1 (2.30) and x1 − p = x0 − p − 1 2 [(3f′(y0) − f ′(x0)) −1(3f′(y0) − f ′(x0)) +2f′(x0)]f ′(x0) −1f(x0) = x0 − p − f ′(x0) −1f(x0) − 1 2 [−i + 2(3f′(y0) − f ′ (x0)) −1 ]f′(x0) −1f(x0) = x0 − p − f ′(x0) −1f(x0) − 3 2 (3f′(y0) − f ′(x0)) −1 ×(f′(x0) − f ′(y0))f ′(x0) −1f(x0), (2.31) where for the derivation of (2.31), we also used the estimate −i + 2(3f′(y0) − f ′(x0)) −1f′(x0) = (3f′(y0) − f ′ (x0)) −1 [−(3f′(y0) − f ′ (x0)) + 2f ′ (x0)] = 3(3f′(y0) − f ′ (x0)) −1 [f′(x0) − f ′ (y0)], cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 75 so we get by (2.31) ‖x1 − p‖ ≤ ‖x0 − p − f ′(x0) −1f(x0)‖ + 3 2 ‖(3f′(y0) − f ′(x0)) −1f′(p)‖ ×[‖f′(p)−1(f′(y0) − f ′(p))‖ + ‖f′(p)−1(f′(x0) − f ′(p))‖] ×‖f′(x0) −1f′(p)‖‖f′(p)−1f(p)‖ ≤ [∫1 0 ω0((1 − θ)‖x0 − p‖)dθ 1 − ω0(‖x0 − p‖) + 3 2 (ω0(‖x0 − p‖) + ω0(‖y0 − p‖)) ∫1 0 ω1(θ‖x0 − p‖)dθ (1 − q(‖x0 − p‖))(1 − ω0(‖x0 − p‖)) ] ‖x0 − p‖ ≤ ϕ3(‖x0 − p‖)‖x0 − p‖ ≤ ‖x0 − p‖ < r, (2.32) which shows (2.29) for n = 0 and x1 ∈ u(p, r). the rest of the proof as identical to the one in theorem 2.1 is omitted. ✷ remark 2.3. (a) let ω0(t) = l0t, ω(t) = lt. then, the radius ra = 2 2l0+l was obtained by argyros in [4] as the convergence radius for newton’s method under condition (2.12)-(2.14). notice that the convergence radius for newton’s method given independently by rheinboldt [14] and traub [16] is given by ρ = 2 3l < ra, where ω1(t) = l1t replaces ω(t), and l1 is the lipschitz constant on ω. notice that ω0 ⊆ ω, so l0 ≤ l1 and l ≤ l1. as an example, let us consider the function f(x) = e x − 1. then p = 0. set d = u(0, 1). then, we have that l0 = e − 1 < l = e 1 e−1 < l1 = e, so ρ = 0.24252961 < ra = 0.3827. moreover, the new error bounds [4, 5, 6, 7, 8] are: ‖xn+1 − p‖ ≤ l 1 − l0‖xn − p‖ ‖xn − p‖ 2, whereas the old ones [10, 14, 16] ‖xn+1 − p‖ ≤ l 1 − l‖xn − p‖ ‖xn − p‖ 2. clearly, the new error bounds are more precise, if l0 < l. then, the radius of convergence of method (1.3) or method (1.4) cannot be larger than ra. (b) the local results can be used for projection methods such as arnoldi’s method, the generalized minimum residual method(gmrem), the generalized conjugate method(gcm) for combined newton/finite projection methods and in connection to the mesh independence principle in order to develop the cheapest and most efficient mesh refinement strategy [4, 5, 6, 7, 8, 10]. 76 ioannis k. argyros and santhosh george cubo 20, 3 (2018) (c) the results can be also be used to solve equations where the operator f′ satisfies the autonomous differential equation [4, 5, 6, 7, 8, 10]: f′(x) = p(f(x)), where p is a known continuous operator. since f′(p) = p(f(p)) = p(0), we can apply the results without actually knowing the solution p. let as an example f(x) = ex − 1. then, we can choose p(x) = x + 1 and p = 0. (d) it is worth noticing that method (1.3) or method (1.4) are not changing, if we use the new instead of the old conditions [9, 12, 13]. moreover, for the error bounds in practice we can use the computational order of convergence (coc) ξ = ln ‖xn+2−xn+1‖ ‖xn+1−xn‖ ln ‖xn+1−xn‖ ‖xn−xn−1‖ , for all n = 1, 2, . . . or the approximate computational order of convergence (acoc) ξ∗ = ln ‖xn+2−p‖ ‖xn+1−p‖ ln ‖xn+1−p‖ ‖xn−p‖ , for all n = 0, 1, 2, . . . instead of the error bounds obtained in theorem 2.1. notice that these formulae do not require high order derivatives and in the case of acoc not even knowledge of p. the convergence radii are optimum under conditions (a). (e) in view of (a2) and the estimate ‖f′(p)−1f′(x)‖ = ‖f′(p)−1(f′(x) − f′(p)) + i‖ ≤ 1 + ‖f′(p)−1(f′(x) − f′(p))‖ ≤ 1 + l0‖x − p‖ second condition in (a3) can be dropped and m can be replaced by m(t) = 1 + l0t or m(t) = m = 2, since t ∈ [0, 1 l0 ). 3 numerical examples example 3.1. let b1 = b2 = r 3, ω = ū(0, 1), x∗ = (0, 0, 0)t . define function f on ω for u = (x, y, z)t by f(u) = (ex − 1, e − 1 2 y2 + y, z)t . cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 77 then, the fréchet-derivative is given by f′(v) =     ex 0 0 0 (e − 1)y + 1 0 0 0 1     . notice that using the (2.8)-(2.12), conditions, we get ω0(t) = (e−1)t, ω(t) = e 1 e−1 t, ω1(t) = e 1 e−1 . then using the definition of r, we have that r1 = 0.15440695135715407082521721804369 = r and r2 = 0.17352535186531112265662102345232. example 3.2. let b1 = b2 = c[0, 1], the space of continuous functions defined on [0, 1] and be equipped with the max norm. let ω = u(0, 1). define function f on ω by f(ϕ)(x) = ϕ(x) − 5 ∫1 0 xθϕ(θ)3dθ. (3.1) we have that f′(ϕ(ξ))(x) = ξ(x) − 15 ∫1 0 xθϕ(θ)2ξ(θ)dθ, for each ξ ∈ ω. then, we get that x∗ = 0, ω0(t) = 7.5t, ω(t) = 15t, ω1(t) = 2. this way, we have that r1 = 0.022222222222222222222222222222222 = r and r2 = 0.18929637111931424398036938328005. example 3.3. let b1 = b2 = r, ω = [− 1 2 , 3 2 ]. define f on ω by f(x) = x3 log x2 + x5 − x4 then f′(x) = 3x2 log x2 + 5x4 − 4x3 + 2x2, then, we get that ω0(t) = ω(t) = 147t, ω1(t) = 2. so, we obtain r1 = 0.0015117157974300831443688586545729 = r and r2 = 0.01297295712377562193484692443235. example 3.4. let b1 = b2 = c[0, 1], ω = ū(x ∗, 1) and consider the nonlinear integral equation of the mixed hammerstein-type [1, 2, 3, 5, 11] defined by x(s) = ∫1 0 g(s, t)(x(t)3/2 + x(t)2 2 )dt, 78 ioannis k. argyros and santhosh george cubo 20, 3 (2018) where the kernel g is the green’s function defined on the interval [0, 1] × [0, 1] by g(s, t) = { (1 − s)t, t ≤ s s(1 − t), s ≤ t. the solution x∗(s) = 0 is the same as the solution of equation (1.1), where f : c[0, 1] −→ c[0, 1]) is defined by f(x)(s) = x(s) − ∫1 0 g(s, t)(x(t)3/2 + x(t)2 2 )dt. notice that ‖ ∫1 0 g(s, t)dt‖ ≤ 1 8 . then, we have that f′(x)y(s) = y(s) − ∫1 0 g(s, t)( 3 2 x(t)1/2 + x(t))dt, so since f′(x∗(s)) = i, ‖f′(x∗)−1(f′(x) − f′(y))‖ ≤ 1 8 ( 3 2 ‖x − y‖1/2 + ‖x − y‖). then, we get that ω0(t) = ω(t) = 1 8 (3 2 t1/2 + t), ω1(t) = 1 + ω0(t). so, we obtain r1 = 1.2 and r2 = 0.82757632634917221992054692236707 = r. cubo 20, 3 (2018) ball comparison between jarratt’s and other fourth order method . . . 79 references [1] amat, s., busquier, s., plaza, s., on two families of high order newton type methods, appl. math. comput., 25, (2012), 2209-2217. 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[14] rheinboldt, w.c., an adaptive continuation process for solving systems of nonlinear equations, polish academy of science, banach ctr. publ. 3 (1978), no. 1, 129–142. 80 ioannis k. argyros and santhosh george cubo 20, 3 (2018) [15] sharma, j.r., guha , r. k., sharma, r., an efficient fourth order weighted newton method for systems of nonlinear equations, numer. algorithm, 62 (2013), 307–323. [16] j.f. traub, iterative methods for the solution of equations, prenticehall series in automatic computation, englewood cliffs, n. j., 1964. introduction local convergence numerical examples cubo matemática educacional vol. l. junio rngg el@m€mtos mat@máticos l:llil la civilización incaiea f. castro gut!1j;rrez apartade 2@.4 maiturín, ed:o. menagas venezuela fax .58-91-41804.2 e-mail: fe-ricasg,w@j.fo tmai'l. eem en es:t:e trabaj.q se presend:alr.i algrum..es &/rof;eceden:f;es hi·stórices del qv.:ip.u ineaic@. se mwestra su use y signifocaei@n en esa civilizaei6n1 se g/fu:ljliza su es-tructura y gj.ilg.une-s a·sped@s m@f;emáüc@s implícif;es en su funcienam-ien.fo. l@s eóm•pl!lt©s y el 11eg;ist!f0 de ci:arntidad·es 11eamzad0s c©n la a:yl!l.0.a cl'e e-ae110.as afjil!l.daifilias es-tám. pmesentes a tra:vés de l'0s tiemp0s em dij·vefsas cmlitmir.as tle la rumarniidad [%~. e:m. efecu0 1 hary a-r1 .. neee@entes clel us0 @e cuericl.1as airnl!l.d•aclas en ch .. ina 1 hawa.i, .las lisias ca:rnlifoaas> af.ri,ca oeeidelil.ita:l y euu0pa. lars cuerdas a.:mmd!aclas junt0 a las mueseais en mad:era © t .1mes0 11epr.esentaim. um.0 de 10s estacl.0i0s cle la ev0lil!lci0n d'e uro sistema de m.r11i:rneraciófj. haista fo!las cl.e amériga del $ur -ajl:gl!l.:füas cl.e las cl!lajles jll0stefi0frnente i0m.teg;¡-ai110m el im¡:>eri0 cl.e 10s ie.cas¡def0 ta:l vez die.id.o feel!lrs0 tuv0 su ai(!>l~ca.cúólil. m·ás in.tensa en este vast0 ím¡deri0 a j!>airte @e lo j0. so for any generic g, {(1, ȧα(j)i,j) : j < λ ++}[g] and ȧ′ πα(i) [g] differ in at most |j0| elements. the lemma follows. � lemma 5.10. for each element i of a and each neat map α, ȧπ(α)(i)[αg] = ȧi[g]. in particular ȧ⋆[αg] = ȧ⋆[g]. proof. by lemma 5.9, ȧπ(α)(i)[αg] = (αȧi)[αg]. then by lemma 5.4 and the fact that αȧi lies in mp, (αȧi)[αg] = ȧi[g]. this shows that ȧ⋆[αg] = ȧ⋆[g]. 20 wilfrid hodges and saharon shelah cubo 21, 3 (2019) we write ε̇−1 for a p-name such that ε̇−1[g] = (ε̇[g])−1 for all generic g. lemma 5.11. suppose α is a neat map and g is p-generic over m. then ḃ∗[α−1g] = ḃ∗[g], and the map (ε̇−1 ◦ αε̇)[g] is an automorphism of b which extends π(α). proof. since m[α−1g] = m[g] and ȧ∗[α−1g] = ȧ∗[g], statement (5.1) (before lemma 5.7) tells us that ė[α−1g](i) = ȧi[α −1g] for each i ∈ dom(a), and that ḃ∗[α−1g] = ḃ∗[g] and ε̇[g]−1◦ε̇[α−1g] extends ė[g]−1 ◦ ė[α−1g]. now using lemma 5.10, ė[g]−1 ◦ ė[α−1g](i) = ė[g]−1(ȧi[α −1g]) = ė[g]−1(ȧπ(α)(i)[g]) = π(α)(i). lemma 5.12. for every neat map α and all p ∈ p there are p′ 6 p and g ∈ autb extending π(α), such that p′ ⊢p ε̇ −1 ◦ α(ε̇) = ǧ. proof. let f be π(α). by lemma 5.11 we have 1 = ||ε̇−1 ◦ αε̇ is an automorphism of b extending f̌||p = ∑ g ||ε̇−1 ◦ αε̇ = ǧ||p where g ranges over the automorphisms of b that extend f. definition 5.13. (a) for each p ∈ p and each i < λ++, define tp,i to be the set of all pairs (f,g), with f ∈ aut(a) and g ∈ aut(b), such that for some α ∈ ni, π(α) = f and p ⊢p ε̇ −1 ◦ αε̇ = ǧ. (b) clearly if p′ 6 p then tp′,i ⊇ tp,i. the number of possible values for f and g is 6 λ by choice of λ, and p is λ+-closed; so there is pi such that for all p ′ 6 pi, tp′,i = tpi,i. we fix a choice of pi for each i, and we write ti for the resulting value tpi,i. (c) for each i and each (f,g) in ti we choose α in ni with π(α) = f so that pi ⊢p ε̇ −1 ◦ αε̇ = ǧ. we write αif,g for this α. cubo 21, 3 (2019) naturality and definability ii 21 lemma 5.14. for each i < λ++, ti is a subset of aut(a) × aut(b) such that (a) for each (f,g) in ti, g|a = f; (b) for each f in aut(a) there is g with (f,g) in ti. (so ti(−,−) is a first attempt at a lifting of the restriction map from aut(b) to aut(a).) proof. by lemma 5.12 and the surjectivity of π. lemma 5.15. there is a stationary subset s of λ++ such that: (a) for each i ∈ s and j < i, the domain of pj is a subset of i × i × i; (b) for each i ∈ s and j < i, every map α j f,g : λ++ → aut(a) is constant on [i,λ++); (c) for all i,j ∈ s, ti = tj; (d) there is a condition p⋆ ∈ p such that for all i ∈ s, pi ↾ (i × i × i) = p ⋆. proof. first, there is a club c ⊆ λ++ on which (a) and (b) hold. let sη be {δ < λ ++ : cf(δ) = λ+}. clearly sν = sη ∩ c is stationary; and for each i ∈ sν, pi ↾ (i × i × i) has domain ⊆ j × j × j for some j = ji < i. then by fődor’s lemma there is a stationary subset s of sν on which (c) and (d) hold. 6 the weak lifting continuing section 5, we use the notation s, p⋆ from lemma 5.15. we write t for the constant value of ti (i ∈ s) from clause (c) of lemma 5.15, and t − for the set of all g such that (1,g) ∈ t. we write ν : aut(b) → aut(a) for the restriction map. if x is a subset of aut(b), we write 〈x〉 for the subgroup of aut(b) generated by x. lemma 6.1. the relation t is a subset of aut(a) × aut(b) that projects onto aut(a), and if (f,g) is in t then ν(g) = f. proof. this repeats lemma 5.14 (a) and (b). lemma 6.2. if (f1,g1) and (f2,g2) are both in t then (f1f2,g1g2) is in t. 22 wilfrid hodges and saharon shelah cubo 21, 3 (2019) proof. take any i,j ∈ s with i < j. put α1 = α j f1,g1 , α2 = α i f2,g2 and α3 = α1α2. note that α1α2 is in ni since i < j. trivially we have pj ⊢ ε̇ −1 ◦ α3(ε̇) = ε̇ −1 ◦ α1(ε̇) ◦ (α1(ε̇)) −1 ◦ α3(ε̇) and by assumption pj ⊢ ε̇ −1 ◦ α1(ε̇) = ǧ1. so pj ⊢ ε̇ −1 ◦ α3(ε̇) = ǧ1 ◦ (α1(ε̇)) −1 ◦ α1(α2ε̇). also by assumption pi ⊢ ε̇ −1 ◦ α2(ε̇) = ǧ2. acting on this last formula by α1 gives α1pi ⊢ α1ε̇ −1 ◦ α1α2ε̇ = α1ǧ2. now α1ǧ2 = ǧ2. also α1pi = pi since the support of pi lies entirely below j (by lemma 5.15(a)), and α1 = α j f1,g1 is the identity in this region since it lies in nj. so we have shown that pi ⊢ α1ε̇ −1 ◦ α1α2ε̇ = ǧ2. now we note that pi ∪ pj is a condition in p, by (a), (d) of lemma 5.15. hence we have that pi ∪ pj ⊢ ε̇ −1 ◦ α3ε̇ = ǧ1ǧ2. since α3 is in ni, this shows that (f1f2,g1g2) ∈ tpi∪pj,i. then by the maximality property of pi, (f1f2,g1g2) ∈ tpi,i so that (f1f2,g1g2) is in t. corollary 6.3. if (f,g1) and (f,g2) are in t then g1g −1 2 is in 〈t −〉. cubo 21, 3 (2019) naturality and definability ii 23 proof. by lemma 6.1 there is some g′ ∈ aut(b) such that (f−1,g′) is in t. then by lemma 6.2, (1,g1g ′) and (1,g2g ′) are in t and so g1g ′, g2g ′ are in t−. hence the element g1g −1 2 = (g1g ′)(g2g ′)−1 lies in 〈t−〉. lemma 6.4. every element of t− is central in aut(b). proof. suppose g2 ∈ t −, so that (1,g2) ∈ t. consider (f1,g2) ∈ t, and apply the notation of the proof of lemma 6.2 with f2 = 1. in that notation, α1 is the identity below j and α2 is the identity below i (since i,j ∈ s). but also g2 lies in t −, so α2 is the identity on [j,λ +). in particular α1 commutes with α2. as in the proof of lemma 6.2 we have pi ⊢ ε̇ −1 ◦ α3ε̇ = ε̇ −1 ◦ α2ε̇ ◦ α2ε̇ −1 ◦ α3ε̇. as before, we have that pi ⊢ ε̇ −1 ◦ α2ε̇ = ǧ2 and α2pj ⊢ α2ε̇ −1 ◦ α2α1ε̇ = α2ǧ1. now the support of pj lies below i or within [j,λ +) × doma, and α2 is the identity in both these regions, and so α2(pj) = pj. thus, since α1 commutes with α2, pj ⊢ α2ε̇ −1 ◦ α3ε̇ = ǧ1. so as before, pi ∪ pj ⊢ ε̇ −1 ◦ α3ε̇ = ǧ2ǧ1. recalling that in the proof of lemma 6.2 we showed that pi ∪ pj ⊢ ε̇ −1 ◦ α3(ε̇) = ǧ1ǧ2, we deduce that pi ∪ pj ⊢ ǧ1ǧ2 = ǧ2ǧ1. but the equation g1g2 = g2g1 is about the ground model, and hence it is true. 24 wilfrid hodges and saharon shelah cubo 21, 3 (2019) now in m choose a map s : aut(a) → aut(b) so that for each f ∈ aut(a), s(f) is some g with (f,g) ∈ t. this is possible by lemma 6.1. lemma 6.5. in m the map s is a weak splitting of ν : aut(b) → aut(a). proof. trivially νs is the identity on aut(a). write s′ : aut(a) → z(aut(b)) for the composite of s and nat : aut(b) → z(aut(b)). we show that s′ is a homomorphism as follows. suppose f1f2 = f3 in aut(a). put gi = s(fi) for each i (1 6 i 6 3). then by lemma 6.2, (f3,g1g2) is in t, so by corollary 6.3 and lemma 6.4, g1g2g −1 3 is in 〈t −〉 ⊆ z(aut(b)). then s′(f1)σ ′(f2) = g1z(aut(b)).g2z(aut(b)) = g1g2.z(aut(b)) = g3z(aut(b)) = s ′(f3) as required. � this completes the proof of theorem 5.1. 7 answers to questions the results above answer most of the problems stated in [6]. in that paper we showed: theorem 3 of [6] if c is a small natural construction in a model of zfc, then c is uniformisable with parameters. we asked (problem a) whether it is possible to remove the restriction that c is small. the answer is no: theorem 7.1. there is a transitive model of zfc in which some ∅-representable construction is natural but not uniformisable (even with parameters). proof. let n be the model of theorem 5.1. let c be some construction ∅-representable in n which is not weakly natural (cf. example one in section 3). then by theorem 5.1, c is not uniformisable. the rigidifying construction cr of section 3 is ∅-representable, natural and not uniformisable. problem b asked whether in theorem 3 of [6] the formula defining c can be chosen so that it has only the same parameters as the formulas chosen to represent c. the answer is no: theorem 7.2. there is a transitive model n of zfc with the following property: cubo 21, 3 (2019) naturality and definability ii 25 for every set y there are a set x and a unitype rigid-based (hence small natural) x-representable construction that is not x ∪ y -uniformisable. proof. take n to be the model given by theorem 5.1. let y be any set in n. if n and y are not as stated above, then for every set x and every unitype rigid-based x-representable construction in n, x is x∪y -uniformisable. so the hypothesis of theorem 4.4 holds, and by that theorem there is in n a small {c̄}-uniformisable construction that is not weakly natural. but this contradicts the choice of n. problem c asked whether there are transitive models of zfc in which every uniformisable construction is natural. theorem 5.1 is the best answer we have for this; the problem remains open. in [4] one of us asked whether there can be models of zfc in which the algebraic closure construction on fields is not uniformisable. theorem 7.3. there are transitive models of zfc in which: (a) no formula without parameters defines for each field a specific algebraic closure for that field, and (b) no formula without parameters defines for each abelian group a specific divisible hull of that group. proof. let the model n be as in theorem 5.1. in n the constructions of example one in section 3 are not uniformisable, since they are not weakly natural. so these two examples prove (a) and (b) respectively. we close with some remarks on related notions in other papers. one result in [4] was that there is no primitive recursive set function which takes each field to an algebraic closure of that field. this is an absolute result which applies to every transitive model of zfc, and so it is not strictly comparable with the consistency results proved above. in this context we note that garvin melles showed [8] that there is no “recursive set-function” (he gives his own definition for this notion) which finds a representative for each isomorphism type of countable torsion-free abelian group. the paper [1] of adámek et al. gives a simple universal algebraic sufficient condition for injective hull constructions not to be natural, and notes that two of their examples are also in 26 wilfrid hodges and saharon shelah cubo 21, 3 (2019) [6]. the comparison between our notions and theirs is a little tricky. for both adámek et al. and us, ‘natural’ is as in ‘natural transformation’ in the categorical sense. but we work in different categories. in this paper and [6], the relevant morphisms are isomorphisms; but for [1] they include embeddings. hence the notion of naturality in [1] is stricter than ours. for example their condition implies that the macneille completion of posets, which embeds every poset in a lattice, is not natural. but it is natural in our sense, since isomorphisms between posets lift functorially to isomorphisms between their macneille completions. in fact this is clear from the standard definition of macneille completions ([2] p. 40ff), which also provides a uniformisation of this construction in any model of zfc. it seems very unlikely that the condition in [1] adapts to give a sufficient condition for failure of weak naturality in the sense above. in a related context harvey friedman [3] used the term ‘naturalness’ in a weaker sense than ours. cubo 21, 3 (2019) naturality and definability ii 27 references [1] jiř́ı adámek, horst herrlich, jiř́ı rosický and walter tholen, ‘injective hulls are not natural’, algebra universalis 48 (2002) 379–388. [2] b. a. davey and h. a. priestley, introduction to lattices and order, cambridge university press, cambridge 1990. [3] h. friedman, ‘on the naturalness of definable operations’, houston j. math. 5 (1979) 325– 330. [4] w. hodges, ‘on the effectivity of some field constructions’, proc. london math. soc. (3) 32 (1976) 133–162. [5] w. hodges, ’definability and automorphism groups’, in proceedings of international congress in logic, methodology and philosophy of science, oviedo 2003, ed. petr hájek et al., king’s college publications, london 2005, pp. 107–120; isbn 1-904987-21-4. [6] w. hodges and s. shelah, ‘naturality and definability i’, j. london math. soc. 33 (1986) 1–12. [7] t. jech, set theory (academic press, new york, 1978). [8] g. melles, ‘classification theory and generalized recursive functions’, d.phil. dissertation, university of california at irvine, 1989. introduction constructions up to isomorphism splitting, naturality and weak naturality uniformisability the set theory the weak lifting answers to questions cubo, a mathematical journal vol.22, no¯ 01, (125–136). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100125 on katugampola fractional order derivatives and darboux problem for differential equations djalal boucenna1, abdellatif ben makhlouf2 and mohamed ali hammami3 1 laboratory of advanced materials, faculty of sciences, badji mokhtar-annaba university, p.o. box 12, annaba, 23000, algeria. 2 department of mathematics, college of science, jouf university, aljouf, saudi arabia department of mathematics, faculty of sciences of sfax, route soukra, bp 1171, 3000 sfax, tunisia benmakhloufabdellatif@gmail.com 3 department of mathematics, faculty of sciences of sfax, route soukra, bp 1171, 3000 sfax, tunisia abstract in this paper, we investigate the existence and uniqueness of solutions for the darboux problem of partial differential equations with caputo-katugampola fractional derivative. resumen en este art́ıculo investigamos la existencia y unicidad de soluciones para el problema de darboux de ecuaciones diferenciales parciales con derivada fraccional de caputokatugampola. keywords and phrases: darboux problem, fractional differential equations, caputo-katugampola derivative. 2010 ams mathematics subject classification: 34a34, 34a08, 65l20. http://doi.org/10.4067/s0719-06462020000100125 126 djalal boucenna, abdellatif ben makhlouf & mohamed ali hammami cubo 22, 1 (2020) 1 introduction to investigate many different fields of science and engineering, the fractional calculus represents a powerful tool, with many applications in mathematical physics, hydrology, finance, astrophysics, thermodynamics, statistical mechanics, biophysics, control theory, cosmology, bioengineering and so on, [5, 6]. in recent years, there has been an important works in ordinary and partial fractional differential equations. for the caputo fractional-order ordinary differential equations case, see kilbas et al. [7], miller and ross [8]. in addition, yunru bai and hua kong have treated the existence of solution for nonlinear caputo-hadamard fractional differential equations in [9]. for the caputo fractional-order partial differential equations case, see the work of tian liang guo and kanjian zhang in [10]. furthermore, xianmin zhang has investigated the caputo-hadamard partial fractional differential equations in [11]. the choice of an appropriate fractional derivative (or integral) depends on the considered system, and for this reason there are a large number of works devoted to different fractional operators. recently, u. katugampola presented new types of fractional operators, which generalize both the riemann-liouville and hadamard fractional operators [4]. although the katugampola fractional integral operator is an erdélyi-kober type operator [13] author in [14] argued that is not possible to obtain hadamard equivalence operators from erdélyi-kober type operators. in this sense, almeida, malinowska and odzijewicz [2] introduced a new fractional operator, called the caputo-katugampola derivative, which generalizes the concept of caputo and caputo-hadamard fractional derivatives. it turns out that, the new operator is the left inverse of the katugampola fractional integral and keeps some of the fundamental properties of the caputo and caputohadamard fractional derivatives. such derivative is the generalization of the caputo and caputohadamard fractional derivative. the existence and uniqueness of the solution of the ordinary caputo-katugampola differential equations is given in [3]. a. cernea in [12] studied a darboux problem associated to a fractional hyperbolic integro-differential inclusion defined by caputokatugampola fractional derivative and several existence results for this problem are proved. in this paper, we study the existence and uniqueness of solutions of the following partial differential equation with caputo-katugampola fractional derivative cd α,ρ a+ u (x,y) = f (x,y,u (x,y)) ,(x,y) ∈ j = [a1,b1] × [a2,b2] , (1.1) u (x,a2) = ϕ(x) , x ∈ [a1,b1] , u (a1,y) = ψ (y) , y ∈ [a2,b2] , ϕ(a1) = ψ (a2) , (1.2) cubo 22, 1 (2020) on katugampola fractional order derivatives and darboux . . . 127 where f : j × r → r, ϕ : [a1,b1] → r and ψ : [a2,b2] → r are given continuous functions. the rest of the paper is organized as follows. some definitions and preliminaries are presented in sect. 2. finally, the existence and uniqueness results, is given in sect. 3. 2 preliminaries in this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. definition 1. [2, 3, 4] given α > 0, ρ > 0 and an interval [a,b] of r, where 0 < a < b. the katugampola fractional integral of a function u ∈ l1([a,b]) is defined by i α,ρ a+ u (t) = ρ1−α γ (α) t ∫ a sρ−1u (s) (tρ − sρ) 1−α ds, where γ is the gamma function. definition 2. [2, 3, 4] given α > 0, ρ > 0 and an interval [a,b] of r, where 0 < a < b. the katugampola fractional derivative is defined by d α,ρ a+ u (t) = ρα γ (1 − α) t 1−ρ d dt t ∫ a sρ−1u (s) (tρ − sρ) α ds. definition 3. [2, 3, 4] given 0 < α < 1, ρ > 0 and an interval [a,b] of r, where 0 < a < b. the caputo-katugampola fractional derivative is defined by c d α,ρ a+ u (t) =d α,ρ a+ [u (t) − u (a)] = ρα γ (1 − α) t1−ρ d dt t ∫ a sρ−1[u (s) − u(a)] (tρ − sρ) α ds. definition 4. let 0 < ai < bi, i = 1,2 reals numbers, a = (a1,a2) and u : [a1,b1] × [a2,b2] → r be an integrable function. the mixed katugampola fractional integrals of order α = (α1,α2) , and parameter ρ = (ρ1,ρ2) is defined by i α,ρ a+ u (x,y) = ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 sρ1−1tρ2−1 (xρ1 − sρ1) 1−α1 (yρ2 − tρ2) 1−α2 u (s,t)dtds. where α1,α2,ρ1 and ρ2 are strictly positives. definition 5. let 0 < ai < bi, i = 1,2 reals numbers, a = (a1,a2) and u : [a1,b1] × [a2,b2] → r be a function. the mixed katugampola fractional derivative of order α = (α1,α2) , and parameter 128 djalal boucenna, abdellatif ben makhlouf & mohamed ali hammami cubo 22, 1 (2020) ρ = (ρ1,ρ2) is defined by d α,ρ a+ u (x,y) = x1−ρ1y1−ρ2d2x,yi 1−α,ρ a+ u (x,y) = x1−ρ1y1−ρ2ρ α1 1 ρ α2 2 γ (1 − α1) γ (1 − α2) d 2 x,y ∫ x a + 1 ∫ y a + 2 sρ1−1tρ2−1 (xρ1 − sρ1) α1 (yρ2 − tρ2) α2 ×u (s,t)dtds. where (α1,α2) ∈ (0,1) 2 , d2x,y = ∂ 2 ∂x∂y and ρ1, ρ2 are strictly positives. definition 6. let 0 < ai < bi, i = 1,2 reals numbers, a = (a1,a2) and u : [a1,b1] × [a2,b2] → r be a function. the mixed caputo-katugampola fractional derivative of order α = (α1,α2) , and parameter ρ = (ρ1,ρ2) is defined by cd α,ρ a+ u (x,y) = d α,ρ a+ (u (x,y) − u (x,a2) − u (a1,y) + u (a1,a2)) where (α1,α2) ∈ (0,1) 2 and ρ1, ρ2 are strictly positives. lemma 2.1. let 0 < ai < bi, i = 1,2 reals numbers, a = (a1,a2) and u : [a1,b1] × [a2,b2] → r is an absolutely continuous function. the mixed caputo-katugampola fractional derivative of order α = (α1,α2) , and parameter ρ = (ρ1,ρ2) is given by c d α,ρ a+ u (x,y) = i 1−α,ρ a+ ( x 1−ρ1y 1−ρ2d 2 x,yu (x,y) ) = ρ α1 1 ρ α2 2 γ (1 − α1) γ (1 − α2) ∫ x a + 1 ∫ y a + 2 d2s,tu (s,t) (xρ1 − sρ1) α1 (yρ2 − tρ2) α2 dtds almost everywhere, where (α1,α2) ∈ (0,1) 2 , d2s,t = ∂ 2 ∂s∂t and ρ1,ρ2 are strictly positives. lemma 2.2. let 0 < ai < bi, i = 1,2 reals numbers, a = (a1,a2) and u : [a1,b1] × [a2,b2] → r be an integrable function. then i α,ρ a+ i β,ρ a+ u (x,y) = i α+β,ρ a+ u (x,y) (2.1) almost everywhere, where α = (α1,α2) , β = (β1,β2) and parameter ρ = (ρ1,ρ2). if additionally u is a continuous function, then the identity (2.1) holds everywhere. cubo 22, 1 (2020) on katugampola fractional order derivatives and darboux . . . 129 proof. using fubini’s theorem we get i α,ρ a+ i β,ρ a+ u (x,y) = ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 s ρ1−1 1 s ρ2−1 2 i β,ρ a+ u (s1,s2) (xρ1 − s ρ1 1 ) 1−α1 (yρ2 − s ρ2 2 ) 1−α2 ds2ds1 = ρ 1−β1 1 ρ 1−β2 2 γ (β1) γ (β2) ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 s ρ1−1 1 s ρ2−1 2 (xρ1 − s ρ1 1 ) 1−α1 (yρ2 − s ρ2 2 ) 1−α2 × ∫ s1 a + 1 ∫ s2 a + 2 t ρ1−1 1 t ρ2−1 2 (s ρ1 1 − t ρ1 1 ) 1−β1 (s ρ2 2 − t ρ2 2 ) 1−β2 u (t1, t2)dt2dt1ds2ds1 = ρ 1−β1 1 ρ 1−β2 2 γ (β1) γ (β2) ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 t ρ1−1 1 t ρ2−1 2 u (t1, t2) × (2.2) ∫ x t1 ∫ y t2 s ρ1−1 1 s ρ2−1 2 ds2ds1dt2dt1 (xρ1 − s ρ1 1 ) 1−α1 (yρ2 − s ρ2 2 ) 1−α2 (s ρ1 1 − t ρ1 1 ) 1−β1 (s ρ2 2 − t ρ2 2 ) 1−β2 . using the change of variables x = (s ρ1 1 − t ρ1 1 ) 1−β1 (xρ1 − t ρ1 1 ) 1−α1 and y = (s ρ2 2 − t ρ2 2 ) 1−β2 (yρ2 − t ρ2 2 ) 1−α2 , we get ∫ x t1 ∫ y t2 s ρ1−1 1 s ρ2−1 2 (xρ1 − s ρ1 1 ) 1−α1 (yρ2 − s ρ2 2 ) 1−α2 1 (s ρ1 1 − t ρ1 1 ) 1−β1 (s ρ2 2 − t ρ2 2 ) 1−β2 ds2ds1 = ∫ x t1 s ρ1−1 1 (xρ1 − s ρ1 1 ) 1−α1 (s ρ1 1 − t ρ1 1 ) 1−β1 ds1 × ∫ y t2 s ρ2−1 2 (yρ2 − s ρ2 2 ) 1−α2 (s ρ2 2 − t ρ2 2 ) 1−β2 ds2 = (xρ1 − t ρ1 1 ) ρ1 (yρ2 − t ρ2 2 ) ρ2 ∫ 1 0 (1 − x) α1−1 x β1dx ∫ 1 0 (1 − y) α1−1 y β1dy = (xρ1 − t ρ1 1 ) ρ1 (yρ2 − t ρ2 2 ) ρ2 b (α1,β1)b (α2,β2) = (xρ1 − t ρ1 1 ) ρ1 (yρ2 − t ρ2 2 ) ρ2 γ (α1) γ (β1) γ (α1 + β1) γ (α2) γ (β2) γ (α2 + β2) . (2.3) from (2.2) and (2.3) we obtain (2.1). lemma 2.3. let 0 < ai < bi, i = 1,2 reals numbers, a = (a1,a2) and u : [a1,b1] × [a2,b2] → r be an integrable function. then d α,ρ a+ i α,ρ a+ u (x,y) = u (x,y) almost everywhere, where α = (α1,α2) ∈ (0,1) 2 and parameter ρ = (ρ1,ρ2). proof. from lemma (2.2), we get d α,ρ a+ i α,ρ a+ u (x,y) = x1−ρ1y1−ρ2d2x,yi 1−α,ρ a+ i α,ρ a+ u (x,y) = x1−ρ1y1−ρ2d2x,yi 1,ρ a+ u (x,y) = u (x,y) . 130 djalal boucenna, abdellatif ben makhlouf & mohamed ali hammami cubo 22, 1 (2020) 3 existence and uniqueness results for the existence and uniqueness of solutions for the problem (1.1)-(1.2) we need the following lemma. lemma 3.1. the function u ∈ c (j) is a solution of fractional order problem (1.1)-(1.2) if and only if u (x,y) = ϕ(x) + ψ (y) − ϕ(a1) + i α,ρ a+ f (x,y,u (x,y)) . (3.1) proof. first suppose that u is a solution of the integral equation (3.1). applied cd α,ρ a+ and using lemma 2.3 we obtain that u solves the the equation (1.1). since the integral is zero when x = a1, or y = a2, then the initial conditions in (1.2) are satisfied. hence u solves the problem (1.1)-(1.2). conversly, if u is a solution of the problem (1.1)-(1.2). let h(x,y) = f (x,y,u (x,y)) = d α,ρ a+ (u (x,y) − u (x,a2) − u (a1,y) + u (a1,a2)) = x1−ρ1y1−ρ2d2x,yi 1−α,ρ a+ [u (x,y) − u (x,a2) − u (a1,y) + u (a1,a2)] . (3.2) applying the operator i 1,ρ a+ to (3.2), we get i 1,ρ a+ h(x,y) = i 1−α,ρ a+ [u (x,y) − u (x,a2) − u (a1,y) + u (a1,a2)] . applying the operator d 1−α,ρ a+ to this equation we find [u (x,y) − u (x,a2) − u (a1,y) + u (a1,a2)] = d 1−α,ρ a+ i 1,ρ a+ h(x,y) = ( x1−ρ1y1−ρ2 ) d2x,yi α,ρ a+ i 1,ρ a+ h(x,y) = i α,ρ a+ h(x,y) . hence, the proof is complete. 3.1 existence of solutions in this subsection we study the existence of solutions for the problem (1.1)-(1.2). theorem 3.1. let k > 0,h∗1 > a1 and h ∗ 2 > a2. define g = {(x,y,u) : (x,y) ∈ [a1,h ∗ 1] × [a2,h ∗ 2] , |u − ϕ(x) − ψ (y) + ϕ(a1)| ≤ k} , m = sup (x,y,u)∈g |f (x,y,u)| cubo 22, 1 (2020) on katugampola fractional order derivatives and darboux . . . 131 and (h1,h2) =      (h∗1,h ∗ 2) if m = 0, ( min ( h∗1, ( k 1 2 ρ α1 1 γ(α1+1) m 1 2 ) 1 α1 ) ,min ( h∗2, ( k 1 2 ρ α2 2 γ(α2+1) m 1 2 ) 1 α2 )) otherwise. then, there exists a function u ∈ c [a1,h1] × [a2,h2] that solves the problem (1.1)-(1.2). proof. if m = 0 then f (x,y,u) = 0, for all (x,y,u) ∈ g. in this case it is clear that the function u : [a1,h1] × [a2,h2] → r with u (x,y) = ϕ(x) + ψ (y) − ϕ(a1) is a solution of the problem (1.1)(1.2). for m 6= 0, using lemma 3.1 we obtain that the problem (1.1)-(1.2) is equivalent to the volterra integral equation (3.1). define the function t by t (x,y) = ϕ(x) + ψ (y) − ϕ(a1) . (3.3) and the set u by u = {u ∈ c ([a1,h1] × [a2,h2]) ,‖u − t‖∞ ≤ k} . (3.4) the set u is nonempty since t ∈ u. it is clear that u is a closed and convex subset of the banach space of all continuous functions on [a1,h1] × [a2,h2]. we define the operator a on this set u by (au) (x,y) = t (x,y) + ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 sρ1−1tρ2−1f (s,t,u (s,t)) (xρ1 − sρ1) 1−α1 (yρ2 − tρ2) 1−α2 dtds. (3.5) we have to show that a has a fixed point. this is done through the schauder’s fixed point theorem. it is easy to see that a is continuous. now we show that a is defined to u into itself, let u ∈ u and (x,y) ∈ [a1,h1] × [a2,h2] then |(au) (x,y) − t (x,y)| = ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 sρ1−1tρ2−1 |f (s,t,u (s,t))| (xρ1 − sρ1) 1−α1 (yρ2 − tρ2) 1−α2 dtds ≤ mρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x a + 1 ∫ y a + 2 sρ1−1tρ2−1 (xρ1 − sρ1) 1−α1 (yρ2 − tρ2) 1−α2 dtds ≤ m γ (α1 + 1) γ (α2 + 1) ( xρ1 − a ρ1 1 ρ1 )α1 ( yρ2 − a ρ2 2 ρ2 )α2 ≤ m ρ α1 1 ρ α2 2 γ (α1 + 1) γ (α2 + 1) h ρ1α1 1 h ρ2α2 2 ≤ m ρ α1 1 ρ α2 2 γ (α1 + 1) γ (α2 + 1) h α1 1 h α2 2 ≤ m ρ α1 1 ρ α2 2 γ (α1 + 1) γ (α2 + 1) kρ α1 1 ρ α2 2 γ (α1 + 1) γ (α2 + 1) m ≤ k. 132 djalal boucenna, abdellatif ben makhlouf & mohamed ali hammami cubo 22, 1 (2020) thus, we have au ∈ u if u ∈ u. we will now show that au = {au : u ∈ u} is relatively compact. this is done by the using arzela-ascoli theorem. firstly, we show that a(u) is uniformly bounded. indeed, let u ∈ u and (x,y) ∈ [a1,h1] × [a2,h2] and from the previous step we get ‖au‖ ∞ ≤ ‖t‖ ∞ + k. secondly, we show that a(u) is equicontinuous. indeed, let (x1,y1) ∈ [a1,h1] × [a2,h2] ,(x2,y2) ∈ [a1,h1] × [a2,h2] such that x1 < x2 and y1 < y2, we have |(au) (x1,y1) − (au) (x2,y2)| ≤ |t (x1,y1) − t (x2,y2)| + mρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x1 a + 1 ∫ y1 a + 2 sρ1−1tρ2−1 (x ρ1 1 − s ρ1) 1−α1 (y ρ2 1 − t ρ2) 1−α2 − sρ1−1tρ2−1 (x ρ1 2 − s ρ1) 1−α1 (y ρ2 2 − t ρ2) 1−α2 dtds + mρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x1 a + 1 ∫ y2 y1 sρ1−1tρ2−1 (x ρ1 2 − s ρ1) 1−α1 (y ρ2 2 − t ρ2) 1−α2 dtds + mρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x2 x1 ∫ y1 a + 2 sρ1−1tρ2−1 (x ρ1 2 − s ρ1) 1−α1 (y ρ2 2 − t ρ2) 1−α2 dtds + mρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ x2 x1 ∫ y2 y1 sρ1−1tρ2−1 (x ρ1 2 − s ρ1) 1−α1 (y ρ2 2 − t ρ2) 1−α2 dtds ≤ |t (x1,y1) − t (x2,y2)| + 3m ρ α1 1 ρ α2 2 γ (α1) γ (α2) [ (x ρ1 2 − a ρ1 1 ) α1 (y ρ2 2 − y ρ2 1 ) α2 + (y ρ2 2 − a ρ2 2 ) α2 (x ρ1 2 − x ρ1 1 ) α1 ] hence, a(u) is equicontinous, since t is uniformly continuous in [a1,h1]×[a2,h2]. as a consequence of the schauder’s fixed point theorem, we deduce that a has a fixed point u in u. this fixed point is the required solution of the problem (1.1)-(1.2). hence, the proof is complete. 3.2 uniqueness of solutions in this subsection we discuss the uniqueness results for the problem (1.1)-(1.2). let u1,u2 ∈ c ([a1,h1] × [a2,h2]) , and (x,y) ∈ [a1,h1] × [a2,h2]. suppose there exists a constant l > 0 independent of x,y,u1, and u2 such that |f (x,y,u1) − f (x,y,u2)| ≤ l |u1 − u2| , (3.6) then we have ‖(au1) − (au2)‖c([a1,x]×[a2,y]) ≤ l ‖u1 − u2‖c([a1,x]×[a2,y]) γ (α1 + 1) γ (α2 + 1) ( xρ1 ρ1 )α1 ( yρ2 ρ2 )α2 . (3.7) cubo 22, 1 (2020) on katugampola fractional order derivatives and darboux . . . 133 indeed, let u1,u2 ∈ c ([a1,h1] × [a2,h2]) , (x,y) ∈ [a1,h1] × [a2,h2] and (v,w) ∈ [a1,x] × [a2,y], we have |(au1) (v,w) − (au2) (v,w)| = ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ v a + 1 ∫ w a + 2 sρ1−1tρ2−1 |f (s,t,u1 (s,t)) − f (s,t,u2 (s,t))| (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 dtds ≤ lρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ v a + 1 ∫ w a + 2 sρ1−1tρ2−1 (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 |u1 (s,t) − u2 (s,t)|dtds ≤ lρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ‖u1 − u2‖c([a1,x]×[a2,y]) ∫ v a + 1 ∫ w a + 2 sρ1−1tρ2−1 (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 dtds ≤ l γ (α1 + 1) γ (α2 + 1) ‖u1 − u2‖c([a1,x]×[a2,y]) ( vρ1 ρ1 )α1 ( wρ2 ρ2 )α2 ≤ l γ (α1 + 1) γ (α2 + 1) ‖u1 − u2‖c([a1,x]×[a2,y]) ( xρ1 ρ1 )α1 ( yρ2 ρ2 )α2 . from the above inequality we get (3.7). ‖(au1) − (au2)‖c([a1,x]×[a2,y]) ≤ l ‖u1 − u2‖c([a1,x]×[a2,y]) γ (α1 + 1) γ (α2 + 1) ( xρ1 ρ1 )α1 ( yρ2 ρ2 )α2 . next, we have the following result theorem 3.2. suppose that the assumptions of theorem 3.1 are satisfied. also let j ∈ n,(x,y) ∈ [a1,h1]×[a2,h2] and u1,u2 ∈ u. suppose f satisfies the lipschitz condition with respect to the third variable with the lipschitz constant l. then ∥ ∥aju1 − a ju2 ∥ ∥ c([a1,x]×[a2,y]) ≤ ( x ρ1 ρ1 )α1j ( y ρ2 ρ2 )α2j γ (1 + α1j) γ (1 + α2j) ‖u1 − u2‖c([a1,x]×[a2,y]) . (3.8) proof. we will prove (3.8) by induction. in the case j = 0, the inequality holds. assume (3.8) is 134 djalal boucenna, abdellatif ben makhlouf & mohamed ali hammami cubo 22, 1 (2020) true for j − 1 ∈ n0 then for all (x,y) ∈ [a1,h1] × [a2,h2] and (v,w) ∈ [a1,x] × [a2,y] we have ∣ ∣ ( aju1 ) (v,w) − ( aju2 ) (v,w) ∣ ∣ = ∣ ∣ ( aa j−1 u1 ) (v,w) − ( aa j−1 u2 ) (v,w) ∣ ∣ = ρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ v a + 1 ∫ w a + 2 sρ1−1tρ2−1 ∣ ∣f ( s,t,aj−1u1 (s,t) ) − f ( s,t,aj−1u2 (s,t) ) ∣ ∣ (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 dtds ≤ lρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ v a + 1 ∫ w a + 2 sρ1−1tρ2−1 ∣ ∣aj−1u1 (s,t) − a j−1u2 (s,t) ∣ ∣ (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 dtds ≤ lρ 1−α1 1 ρ 1−α2 2 γ (α1) γ (α2) ∫ v a + 1 ∫ w a + 2 sρ1−1tρ2−1 ∥ ∥aj−1u1 − a j−1u2 ∥ ∥ c([a1,s]×[a2,t]) (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 dtds ≤ ljρ 1−α1j 1 ρ 1−α2j 2 γ (α1) γ (α2) γ (1 + α1 (j − 1)) γ (1 + α2 (j − 1)) ‖u1 − u2‖c([a1,x]×[a2,y]) ∫ v a + 1 ∫ w a + 2 sρ1+α1ρ1(j−1)−1tρ2+α2ρ2(j−1)−1 (vρ1 − sρ1) 1−α1 (wρ2 − tρ2) 1−α2 dtds ≤ ljρ 1−α1j 1 ρ 1−α2j 2 γ (α1) γ (α2) γ (1 + α1 (j − 1)) γ (1 + α2 (j − 1)) ‖u1 − u2‖c([a1,x]×[a2,y]) × γ (α1) γ (α2) γ (1 + α1 (j − 1)) γ (1 + α2 (j − 1)) γ (1 + α1j) γ (1 + α2j) xρ1α1j ρ1 yρ2α2j ρ2 ≤ ( x ρ1 ρ1 )α1j ( y ρ2 ρ2 )α2j γ (1 + α1j) γ (1 + α2j) ‖u1 − u2‖c([a1,x]×[a2,y]) . hence, the proof is complete. theorem 3.3. let k,h∗1 and h ∗ 1 are positive numbers, define the set g as in theorem 3.1 and assume that the function f : g → r satisfies a lipschitz condition with respect to the third variable with the lipschitz constant l. then, there exists a unique solution u ∈ c ([a1,h1] × [a2,h2]) for the problem (1.1)-(1.2). where h1,h2 are the same as in theorem 3.1. proof. according to theorem 3.1, the problem (1.1)-(1.2) has a solution. to prove the uniqueness, we adopt theorem 3.2, we use the operato a as defined in (3.5), the function t as defined in (3.3) and the set u as defined in (3.4). we will apply weissinger’s fixed point theorem to prove that a has a unique fixed point. let j ∈ n and u1,u2 ∈ c ([a1,h1] × [a2,h2]) . from (3.8) and taking the norms on [a1,h1]×[a2,h2], we get ∥ ∥aj−1u1 − a j−1u2 ∥ ∥ c([a1,h1]×[a2,h2]) ≤ ( x ρ1 ρ1 )α1j ( y ρ2 ρ2 )α2j γ (1 + α1j) γ (1 + α2j) ‖u1 − u2‖c([a1,h1]×[a2,h2]) . cubo 22, 1 (2020) on katugampola fractional order derivatives and darboux . . . 135 let ωj = ( x ρ1 ρ1 ) α1j ( y ρ2 ρ2 ) α2j γ(1+α1j)γ(1+α2j) . it is clear that ∞ ∑ j=0 ωj = ∞ ∑ j=0 (( x ρ1 ρ1 )α1 ( y ρ2 ρ2 )α2 )j γ (1 + α1j) γ (1 + α2j) = e ( (αi,1)i=1,2 ; ( ( xρ1 ρ1 )α1 ( yρ2 ρ2 )α2 ) ) , hence the series converges. this completes the proof. 4 conclusion here we have studied the existence and uniqueness of the solutions for the darboux problem of partial differential equations with caputo-katugampola fractional derivative. 136 djalal boucenna, abdellatif ben makhlouf & mohamed ali hammami cubo 22, 1 (2020) references [1] h . j. haubold, a. m. mathai, and r. k. saxena, mittag-leffler functions and their applications, jour of app math volume 2011, article id 298628 , 51 pages. 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[14] katugampola, u. n., “a new approach to generalized fractional derivative,”bull. math. anal. appl., vol. 6, pp. 1-15 (2014). introduction preliminaries existence and uniqueness results existence of solutions uniqueness of solutions conclusion a mathematical journal vol. 7, no 3, (1 13). december 2005. fuzzy taylor formulae george a. anastassiou department of mathematical sciences university of memphis memphis, tn 38152 u.s.a. ganastss@memphis.edu abstract we produce fuzzy taylor formulae with integral remainder in the univariate and multivariate cases, analogs of the real setting. resumen se presentan versiones fuzzy análogas a las reales de fórmulas de taylor con resto integral en el caso univariado y multivariado. key words and phrases: fuzzy taylor formula, fuzzy–riemann integral remainder, h-fuzzy derivative, fuzzy real analysis. 2000 ams subj. class.: 26e50. 1 background we need the following definition a (see [10]). let µ : r → [0, 1] with the following properties. (i) is normal, i.e., ∃x0 ∈ r; µ(x0) = 1. 2 george a. anastassiou 7, 3(2005) (ii) µ(λx + (1 − λ)y) ≥ min{µ(x), µ(y)}, ∀x, y ∈ r, ∀λ ∈ [0, 1] (µ is called a convex fuzzy subset). (iii) µ is upper semicontinuous on r, i.e., ∀x0 ∈ r and ∀ε > 0, ∃ neighborhood v (x0): µ(x) ≤ µ(x0) + ε, ∀x ∈ v (x0). (iv) the set supp(µ) is compact in r (where supp(µ) := {x ∈ r; µ(x) > 0}). we call µ a fuzzy real number. denote the set of all µ with rf. e.g., x{x0} ∈ rf, for any x0 ∈ r, where x{x0} is the characteristic function at x0. for 0 < r ≤ 1 and µ ∈ rf define [µ]r := {x ∈ r: µ(x) ≥ r} and [µ]0 := {x ∈ r : µ(x) > 0}. then it is well known that for each r ∈ [0, 1], [µ]r is a closed and bounded interval of r. for u, v ∈ rf and λ ∈ r, we define uniquely the sum u ⊕ v and the product λ � u by [u ⊕ v]r = [u]r + [v]r, [λ � u]r = λ[u]r, ∀r ∈ [0, 1], where [u]r + [v]r means the usual addition of two intervals (as subsets of r) and λ[u]r means the usual product between a scalar and a subset of r (see, e.g., [10]). notice 1 � u = u and it holds u ⊕ v = v ⊕ u, λ � u = u � λ. if 0 ≤ r1 ≤ r2 ≤ 1 then [u]r2 ⊆ [u]r1 . actually [u]r = [u(r)− , u (r) + ], where u (r) − ≤ u (r) + , u (r) − , u (r) + ∈ r, ∀r ∈ [0, 1]. for λ > 0 one has λu(r)± = (λ � u) (r) ± , respectively. define d : rf × rf → r+ by d(u, v) := sup r∈[0,1] max{|u(r)− − v (r) − |, |u (r) + − v (r) + |}, where [v]r = [v(r)− , v (r) + ]; u, v ∈ rf. we have that d is a metric on rf. then (rf, d) is a complete metric space, see [10], with the properties d(u ⊕ w, v ⊕ w) = d(u, v), ∀u, v, w ∈ rf, d(k � u, k � v) = |k|d(u, v), ∀u, v ∈ rf, ∀k ∈ r, d(u ⊕ v, w ⊕ e) ≤ d(u, w) + d(v, e), ∀u, v, w, e ∈ rf. let f, g : r → rf be fuzzy number valued functions. the distance between f, g is defined by d∗(f, g) := sup x∈r d(f (x), g(x)). on rf we define a partial order by “≤”: u, v ∈ rf, u ≤ v iff u (r) − ≤ v (r) − and u (r) + ≤ v (r) + , ∀r ∈ [0, 1]. we mention 7, 3(2005) fuzzy taylor formulae 3 lemma 2.2 ([5]). for any a, b ∈ r : a, b ≥ 0 and any u ∈ rf we have d(a � u, b � u) ≤ |a − b| · d(u, õ), where õ ∈ rf is defined by õ := x{0}. lemma 4.1 ([5]). (i) if we denote õ := x{0}, then õ ∈ rf is the neutral element with respect to ⊕, i.e., u ⊕ õ = õ ⊕ u = u, ∀u ∈ rf. (ii) with respect to õ, none of u ∈ rf, u 6= õ has opposite in rf. (iii) let a, b ∈ r : a · b ≥ 0, and any u ∈ rf, we have (a + b) � u = a � u ⊕ b � u. for general a, b ∈ r, the above property is fale. (iv) for any λ ∈ r and any u, v ∈ rf, we have λ � (u ⊕ v) = λ � u ⊕ λ � v. (v) for any λ, µ ∈ r and u ∈ rf, we have λ � (µ � u) = (λ · µ) � u. (vi) if we denote ‖u‖f := d(u, õ), ∀u ∈ rf, then ‖·‖f has the properties of a usual norm on rf, i.e., ‖u‖f = 0 iff u = õ, ‖λ � u‖f = |λ| · ‖u‖f, ‖u ⊕ v‖f ≤ ‖u‖f + ‖v‖f, ‖u‖f − ‖v‖f ≤ d(u, v). notice that (rf, ⊕, �) is not a linear space over r, and consequently (rf, ‖ ·‖f) is not a normed space. we need definition b (see [10]). let x, y ∈ rf. if there exists a z ∈ rf such that x = y + z, then we call z the h-difference of x and y, denoted by z := x − y. definition 3.3 ([10]). let t := [x0, x0 + β] ⊂ r, with β > 0. a function f : t → rf is h-differentiable at x ∈ t if there exists a f′(x) ∈ rf such that the limits (with respect to metric d) lim h→0+ f (x + h) − f (x) h , lim h→0+ f (x) − f (x − h) h exist and are equal to f′(x). we call f′ the derivative or h-derivative of f at x. if f is h-differentiable at any x ∈ t , we call f differentiable or h-differentiable and it has h-derivative over t the function f′. the last definition was given first by m. puri and d. ralescu [9]. we need also a particular case of the fuzzy henstock integral (δ(x) = δ 2 ) introduced in [10], definition 2.1. that is, 4 george a. anastassiou 7, 3(2005) definition 13.14 ([6], p. 644). let f : [a, b] → rf. we say that f is fuzzy-riemann integrable to i ∈ rf if for any ε > 0, there exists δ > 0 such that for any division p = {[u, v]; ξ} of [a, b] with the norms ∆(p ) < δ, we have d (∑ p ∗(v − u) � f (ξ), i ) < ε, where ∑∗ denotes the fuzzy summation. we choose to write i := (f r) ∫ b a f (x)dx. we also call an f as above (f r)-integrable. we mention lemma 1 ([3]). if f, g : [a, b] ⊆ r → rf are fuzzy continuous functions, then the function f : [a, b] → r+ defined by f (x) := d(f (x), g(x)) is continuous on [a, b], and d ( (f r) ∫ b a f (x)dx, (f r) ∫ b a g(x)dx ) ≤ ∫ b a d(f (x), g(x))dx. lemma 2 ([3]). let f : [a, b] → rf fuzzy continuous (with respect to metric d), then d(f (x), õ) ≤ m , ∀x ∈ [a, b], m > 0, that is f is fuzzy bounded. equivalently we get χ−m ≤ f (x) ≤ χm , ∀x ∈ [a, b]. lemma 3 ([3]). let f : [a, b] ⊆ r → rf be fuzzy continuous. then (f r) ∫ x a f (t)dt is a fuzzy continuous function in x ∈ [a, b]. lemma 5 ([4]). let f : [a, b] → rf have an existing h-fuzzy derivative f′ at c ∈ [a, b]. then f is fuzzy continuous at c. we need theorem 3.2 ([7]). let f : [a, b] → rf be fuzzy continuous. then (f r) ∫ b a f (x)dx exists and belongs to rf, furthermore it holds[ (f r) ∫ b a f (x)dx ]r = [∫ b a (f )(r)− (x)dx, ∫ b a (f )(r)+ (x)dx ] , ∀r ∈ [0, 1]. (1) clearly f (r)± : [a, b] → r are continuous functions. we also need theorem 5.2 ([8]). let f : [a, b] ⊆ r → rf be h-fuzzy differentiable. let t ∈ [a, b], 0 ≤ r ≤ 1. (clearly [f (t)]r = [ (f (t))(r)− , (f (t)) (r) + ] ⊆ r.) (2) 7, 3(2005) fuzzy taylor formulae 5 then (f (t))(r)± are differentiable and [f′(t)]r = [ ((f (t))(r)− ) ′, ((f (t))(r)+ ) ′]. (3) the last can be used to find f′. here cn([a, b], rf), n ≥ 1 denotes the space of n-times fuzzy continuously hdifferentiable functions from [a, b] ⊆ r into rf. by above theorem 5.2 of [8] for f ∈ cn([a, b], rf) we obtain [f (i)(t)]r = [ ((f (t))(r)− ) (i), ((f (t))(r)+ ) (i) ] , (4) for i = 0, 1, 2, . . . , n and in particular we have (f (i)± ) (r) = (f (r)± ) (i), ∀r ∈ [0, 1]. (5) definition 1. let a1, a2, b1, b2 ∈ r such that a1 ≤ b1 and a2 ≤ b2. then we define [a1, b1] + [a2, b2] = [a1 + a2, b1 + b2]. (6) let a, b ∈ r such that a ≤ b and k ∈ r, then we define, if k ≥ 0, k[a, b] = [ka, kb], if k < 0, k[a, b] = [kb, ka]. (7) here we use lemma 1. let f : [a, b] → rf be fuzzy continuous and let g : [a, b] → r+ be continuous. then f (x) � g(x) is fuzzy continuous function ∀x ∈ [a, b]. proof. the same as of lemma 2 ([1]), using lemma 2 of [3]. 2 main results we present the following fuzzy taylor theorem in one dimension. theorem 1. let f ∈ cn([a, b], rf), n ≥ 1, [α, β] ⊆ [a, b] ⊆ r. then f (β) = f (α) ⊕ f′(α) � (β − α) ⊕ ··· ⊕ f (n−1)(α) � (β − α)n−1 (n − 1)! ⊕ 1 (n − 1)! � (f r) ∫ β α (β − t)n−1 � f (n)(t) dt. (8) the integral remainder is a fuzzy continuous function in β. proof. let r ∈ [0, 1]. we have here [f (β)]r = [f (r)− (β), f (r) + (β)], and by theorem 5.2 ([8]) f (r)± is n-times continuously differentiable on [a, b]. by (5) we get (f (i)± (α)) (r) = (f (r)± (α)) (i), all i = 0, 1, . . . , n, (9) 6 george a. anastassiou 7, 3(2005) and [f (i)(α)]r = [ (f (r)− (α)) (i), (f (r)+ (α)) (i) ] . thus by taylor’s theorem we obtain f (r) ± (β) = f (r) ± (α) + (f (r) ± (α)) ′(β − α) + · · · + (f (r)± (α)) (n−1) (β − α) n−1 (n − 1)! + 1 (n − 1)! ∫ β α (β − t)n−1(f (r)± ) (n)(t)dt. furthermore by (9) we have f (r) ± (β) = f (r) ± (α) + (f ′ ±(α)) (r)(β − α) + · · · + (f (n−1)± (α) (r) (β − α) n−1 (n − 1)! + 1 (n − 1)! ∫ β α (β − t)n−1(f (n)± ) (r)(t)dt. here it holds β − α ≥ 0, β − t ≥ 0 for t ∈ [α, β], and (f (i)− (t)) (r) ≤ (f (i)+ (t)) (r), ∀t ∈ [a, b] all i = 0, 1, . . . , n, and any r ∈ [0, 1]. we see that[ f (r) − (β), f (r) + (β)] = [f (r) − (α) + (f ′ −(α)) (r)(β − α) + · · · + (f (n−1)− (α)) (r) (β − α) n−1 (n − 1)! + 1 (n − 1)! ∫ β α (β − t)n−1(f (n)− ) (r)(t)dt, , f (r)+ (α) + (f′+(α)) (r)(β − α) + · · · + (f (n−1)+ (α)) (r) (β − α) n−1 (n − 1)! + 1 (n − 1)! ∫ β α (β − t)n−1(f (n)+ ) (r)(t) dt ] . to split the above closed interval into a sum of smaller closed intervals is where we use β − α ≥ 0. so we get [f (β)r] = [f (r)− (β), f (r) + (β)] = [f (r) − (α), f (r) + (α)] + [(f ′ −(α)) (r), (f′+(α)) (r)](β − α) + · · · + [(f (n−1)− (α))(r), (f (n−1) + (α)) (r)] (β−α) n−1 (n−1)! + 1 (n−1)! [∫ β α (β − t)n−1(f (n)− )(r)(t)dt, ∫ β α (β − t)n−1(f (n)+ )(r)(t)dt ] = [f (α)]r + [f′(α)]r(β − α) + · · · + [f (n−1)(α)]r (β−α) n−1 (n−1)! + 1 (n−1)! [∫ β α ((β − t)n−1 � f (n)(t))(r)− dt, ∫ β α ((β − t)n−1 � f (n)(t))(r)+ dt ] . 7, 3(2005) fuzzy taylor formulae 7 by theorem 3.2 ([7]) we next get [f (β)]r = [f (α)]r + [f′(α)]r(β − α) + · · · + [f (n−1)(α)]r (β − α)n−1 (n − 1)! + 1 (n − 1)! [ (f r) ∫ β α (β − t)n−1 � f (n)(t)dt ]r . finally we obtain [f (β)]r = [ f (α) ⊕ f′(α) � (β − α) ⊕ ··· ⊕ f (n−1)(α) � (β − α)n−1 (n − 1)! ⊕ 1 (n − 1)! � (f r) ∫ β α (β − t)n−1 � f (n)(t)dt ]r , all r ∈ [0, 1]. by theorem 3.2 of [7] and lemma 1 we get that the remainder of (8) is in rf, and by lemma 3 ([3]) is a fuzzy continuous function in β. the theorem has been proved. next we present a multivariate fuzzy taylor theorem. we need the following multivariate fuzzy chain rule. here the h-fuzzy partial derivatives are defined according to the definition 3.3 of [10], see section 1, and the analogous way to the real case. theorem 3 ([2]). let φi : [a, b] ⊆ r → φi([a, b]) := ii ⊆ r, i = 1, . . . , n, n ∈ n, are strictly increasing and differentiable functions. denote xi := xi(t) := φi(t), t ∈ [a, b], i = 1, . . . , n. consider u an open subset of rn such that ×ni=1ii ⊆ u . consider f : u → rf a fuzzy continuous function. assume that fxi : u → rf, i = 1, . . . , n, the h-fuzzy partial derivatives of f , exist and are fuzzy continuous. call z := z(t) := f (x1, . . . , xn). then dzdt exists and dz dt = n∑∗ i=1 dz dxi � dxi dt , ∀t ∈ [a, b] (10) where dz dt , dz dxi , i = 1, . . . , n are the h-fuzzy derivatives of f with respect to t, xi, respectively. the interchange of the order of h-fuzzy differentiation is needed too. theorem 4 ([2]). let u be an open subset of rn, n ∈ n, and f : u → rf be a fuzzy continuous function. assume that all h-fuzzy partial derivatives of f up to order m ∈ n exist and are fuzzy continuous. let x := (x1, . . . , xn) ∈ u . then the h-fuzzy mixed partial derivative of order k, dx`1 ,...,x`k f (x) is unchanged when the indices `1, . . . , `k are permuted. each `i is a positive integer ≤ n. here some or all of `i’s can be equal. also k = 2, . . . , m and there are nk partials of order k. we give 8 george a. anastassiou 7, 3(2005) theorem 2. let u be an open convex subset of rn, n ∈ n and f : u → rf be a fuzzy continuous function. assume that all h-fuzzy partial derivatives of f up to order m ∈ n exist and are fuzzy continuous. let z := (z1, . . . , zn), x0 := (x01, . . . , x0n) ∈ u such that xi ≥ x0i, i = 1, . . . , n. let 0 ≤ t ≤ 1, we define xi := x0i + t(zi − z0i), i = 1, 2, . . . , n and gz(t) := f (x0 + t(z − x0)). (clearly x0 + t(z − x0) ∈ u .) then for n = 1, . . . , m we obtain g(n)z (t) =  ( n∑∗ i=1 (zi − x0i) � ∂ ∂xi )n f  (x1, x2, . . . , xn). (11) furthermore it holds the following fuzzy multivariate taylor formula f (z) = f (x0) ⊕ m−1∑∗ n=1 g (n) z (0) n ! ⊕ rm(0, 1), (12) where rm(0, 1) := 1 (m − 1)! � (f r) ∫ 1 0 (1 − s)m−1 � g(m)z (s)ds. (13) comment. (explaining formula (11)). when n = n = 2 we have (zi ≥ x0i, i = 1, 2) gz(t) = f (x01 + t(z1 − x01), x02 + t(z2 − x02)), 0 ≤ t ≤ 1. we apply theorems 3 and 4 of [2] repeatedly, etc. thus we find g′z(t) = (z1 − x01) � ∂f ∂x1 (x1, x2) ⊕ (z2 − x02) � ∂f ∂x2 (x1, x2). furthermore it holds g′′z (t) = (z1 − x01) 2 � ∂2f ∂x21 (x1, x2) ⊕ 2(z1 − x01) · (z2 − x02) (14) � ∂2f (x1, x2) ∂x1∂x2 ⊕ (z2 − x02)2 � ∂2f ∂x22 (x1, x2). when n = 2 and n = 3 we obtain g′′′z (t) = (z1 − x01) 3 � ∂3f ∂x31 (x1, x2) ⊕ 3(z1 − x01)2(z2 − x02) � ∂3f (x1, x2) ∂x21∂x2 ⊕ 3(z1 − x01)(z2 − x02)2 · ∂3f (x1, x2) ∂x1∂x 2 2 ⊕ (z2 − x02)3 � ∂3f ∂x32 (x1, x2). (15) 7, 3(2005) fuzzy taylor formulae 9 when n = 3 and n = 2 we get (zi ≥ x0i, i = 1, 2, 3) g′′z (t) = (z1 − x01) 2 � ∂2f ∂x21 (x1, x2, x3) ⊕ (z2 − x02)2 � ∂2f ∂x22 (x1, x2, x3) ⊕ (z3 − x03)2 � ∂2f ∂x23 (x1, x2, x3) ⊕ 2(z1 − x01)(z2 − x02) � ∂2f (x1, x2, x3) ∂x1∂x2 ⊕ 2(z2 − x02)(z3 − x03) � ∂2f (x1, x2, x3) ∂x2∂x3 ⊕ 2(z3 − x03)(z1 − x01) � ∂2f ∂x3∂x1 (x1, x2, x3), (16) etc. proof of theorem 2. let z := (z1, . . . , zn), x0 := (x01, . . . , x0n) ∈ u , n ∈ n, such that zi > x0i, i = 1, 2, . . . , n. we define xi := φi(t) := x0i + t(zi − x0i), 0 ≤ t ≤ 1; i = 1, 2, . . . , n. thus dxi dt = zi − x0i > 0. consider z := gz(t) := f (x0 + t(z − x0)) = f (x01 + t(z1 − x01), . . . , x0n + t(zn − x0n)) = f (φ1(t), . . . , φn(t)). since by assumptions f : u → rf is fuzzy continuous, also fxi exist and are fuzzy continuous, by theorem 3 (10) of [2] we get dz(x1, . . . , xn) dt = n∑∗ i=1 ∂z(x1, . . . , xn) ∂xi � dxi dt = n∑∗ i=1 ∂f (x1, . . . , xn) ∂xi � (zi − x0i). thus g′z(t) = n∑∗ i=1 ∂f (x1, . . . , xn) ∂xi � (zi − x0i). next we observe that d2z dt2 = g′′z (t) = d dt ( n∑∗ i=1 ∂f (x1, . . . , xn) ∂xi � (zi − x0i) ) = n∑∗ i=1 (zi − x0i) � d dt ( ∂f (x1, . . . , xn) ∂xi ) = n∑∗ i=1 (zi − x0i) �   n∑∗ j=1 ∂2f (x1, . . . , xn) ∂xj ∂xi � (zj − x0j )   = n∑∗ i=1 n∑∗ j=1 ∂2f (x1, . . . , xn) ∂xj ∂xi � (zi − x0i) · (zj − x0j ). 10 george a. anastassiou 7, 3(2005) that is g′′z (t) = n∑∗ i=1 n∑∗ j=1 ∂2f (x1, . . . , xn) ∂xj ∂xi � (zi − x0i) · (zj − x0j ). the last is true by theorem 3 (10) of [2] under the additional assumptions that fxi ; ∂2f ∂xj ∂xi , i, j = 1, 2, . . . , n exist and are fuzzy continuous. working the same way we find d3z dt3 = g′′′z (t) = d dt   n∑∗ i=1 n∑∗ j=1 ∂2f (x1, . . . , xn) ∂xj ∂xi � (zi − x0i) · (zj − x0j )   = n∑∗ i=1 n∑∗ j=1 (zi − x0i) · (zj − x0j ) d dt ( ∂2f (x1, . . . , xn) ∂xj ∂xi ) = n∑∗ i=1 n∑∗ j=1 (zi − x0i) · (zj − x0j ) [ n∑∗ k=1 ∂3f (x1, . . . , xn) ∂xk∂xj ∂xi � (zk − x0k) ] = n∑∗ i=1 n∑∗ j=1 n∑∗ k=1 ∂3f (x1, . . . , xn) ∂xk∂xj ∂xi � (zi − x0i) · (zj − x0j ) · (zk − x0k). therefore, g′′′z (t) = n∑∗ i=1 n∑∗ j=1 n∑∗ k=1 ∂3f (x1, . . . , xn) ∂xk∂xj ∂xi � (zi − x0i) · (zj − x0j ) · (zk − x0k). that last is true by theorem 3 (10) of [2] under the additional assumptions that ∂3f (x1, . . . , xn) ∂xk∂xj ∂xi , i, j, k = 1, . . . , n do exist and are fuzzy continuous. etc. in general one obtains that for n = 1, . . . , m ∈ n, g(n)z (t) = n∑∗ i1=1 n∑∗ i2=1 · · · n∑∗ in =1 ∂n f (x1, . . . , xn) ∂xin ∂xin−1 · · · ∂xi1 � n∏ r=1 (zir − x0ir ), which by theorem 4 of [2] is the same as (11) for the case zi > x0i, see also (14), (15), and (16). the last is true by theorem 3 (10) of [2] under the assumptions that all h-partial derivatives of f up to order m exist and they are all fuzzy continuous including f itself. next let tm̃ → t̃, as m̃ → +∞, tm̃, t̃ ∈ [0, 1]. consider xim̃ := x0i + tm̃(zi − x0i) and x̃i := x0i + t̃(zi − x0i), i = 1, 2, . . . , n. 7, 3(2005) fuzzy taylor formulae 11 that is xm̃ = (x1m̃, x2m̃, . . . , xnm̃) and x̃ = (x̃1, . . . , x̃n) in u . then xm̃ → x̃, as m̃ → +∞. clearly using the properties of d-metric and under the theorem’s assumptions, we obtain that g(n)z (t) is fuzzy continuous for n = 0, 1, . . . , m. then by theorem 1, from the univariate fuzzy taylor formula (8), we find gz(1) = gz(0) ⊕ g′z(0) ⊕ g′′z (0) 2! ⊕ ··· ⊕ g (m−1) z (0) (m − 1)! ⊕ rm(0, 1), where rm(0, 1) comes from (13). by theorem 3.2 of [7] and lemma 1 we get that rm(0, 1) ∈ rf. that is we get the multivariate fuzzy taylor formula f (z) = f (x0) ⊕ g′z(0) ⊕ g′′z (0) 2! ⊕ ··· ⊕ g (m−1) z (0) (m − 1)! ⊕ rm(0, 1), when zi > x0i, i = 1, 2, . . . , n. finally we would like to take care of the case that some x0i = zi. without loss of generality we may assume that x01 = z1, and zi > x0i, i = 2, . . . , n. in this case we define z̃ := g̃z(t) := f (x01, x02 + t(z2 − x02), . . . , x0n + t(zn − x0n)). therefore one has g̃′z(t) = n∑∗ i=2 ∂f (x01, x2, . . . , xn) ∂xi � (zi − x0i), and in general we find g̃(n)z (t) = n∑∗ i2=2,...,in =2 ∂n f (x01, x2, . . . , xn) ∂xin ∂xn−1 · · · ∂xi2 � n∏ r=2 (zir − x0ir ), for n = 1, . . . , m ∈ n. notice that all g̃(n)z , n = 0, 1, . . . , m are fuzzy continuous and g̃z(0) = f (x01, x02, . . . , x0n), g̃z(1) = f (x01, z2, z3, . . . , zn). then one can write down a fuzzy taylor formula, as above, for g̃z. but g̃ (n) z (t) coincides with g(n)z (t) formula at z1 = x01 = x1. that is both taylor formulae in that case coincide. at last we remark that if z = x0, then we define z∗ := g∗z (t) := f (x0) =: c ∈ rf a constant. since c = c + õ, that is c − c = õ, we obtain the h-fuzzy derivative (c)′ = õ. consequently we have that g∗(n)z (t) = õ, n = 1, . . . , m. 12 george a. anastassiou 7, 3(2005) the last coincide with the g(n)z formula, established earlier, if we apply there z = x0. and, of course, the fuzzy taylor formula now can be applied trivially for g∗z . furthermore in that case it coincides with the taylor formula proved earlier for gz. we have established a multivariate fuzzy taylor formula for the case of zi ≥ x0i, i = 1, 2, . . . , n. that is (11)–(13) are true. note. theorem 2 is still valid when u is a compact convex subset of rn such that u ⊆ w , where w is an open subset of rn. now f : w → rf and it has all the properties of f as in theorem 2. clearly here we take x0, z ∈ u . received: march 2003. revised: july 2003. references [1] george a. anastassiou, fuzzy wavelet type operators, submitted. [2] george a. anastassiou, on h-fuzzy differentiation, mathematica balkanica, new series, vol. 16 volumen fasc. 1-4 (2002), 153-193. [3] george a. anastassiou, rate of convergence of fuzzy neural network operators, univariate case, journal of fuzzy mathematics, 10, no. 3 (2002), 755–780. [4] george a. anastassiou, univariate fuzzy-random neural network approximation operators, submitted. [5] george a. anastassiou and sorin gal, on a fuzzy trigonometric approximation theorem of weierstrass-type, journal of fuzzy mathematics, 9, no. 3 (2001), 701–708. [6] s. gal, approximation theory in fuzzy setting. chapter 13 , handbook of analytic computational methods in applied mathematics (edited by g. anastassiou), chapman & hall crc press, boca raton, new york, 2000, pp. 617–666. [7] r. goetschel , jr. and w. voxman, elementary fuzzy calculus, fuzzy sets and systems, 18 (1986), 31–43. [8] o. kaleva,fuzzy differential equations, fuzzy sets and systems, 24 (1987), 301–317. [9] m. l. puri and d. a. ralescu, differentials of fuzzy functions, j. of math. analysis & appl., 91 (1983), 552–558. 7, 3(2005) fuzzy taylor formulae 13 [10] congxin wu and zengtai gong, on henstock integral of fuzzy number valued functions (i), fuzzy sets and systems, 120, no. 3, 2001, 523–532. [11] l. a. zadeh, fuzzy sets, information and control, 8, 1965, 338–353. (olulioo j\ ¡\foth c mo t.i co/ j our nol \lol. 9, ft.'"!. j, (39 • -4 5). a ¡wil p.007. note s on the isomorphism and splitting proble ms for commutative modular group alge bras p e t e r d a n c h e v 13, ccncrnl l(utuzov st.rcct, bl. 7, hoor 2 , npn.rl. <1 t\003 plovcli v, bulgaria pvdu11ch cv@ynl100.co1n; pvdanc h cv m ail.bg a bstract h is provee! lhat. t.hc p-splitti ug croup on.sis p roblcm fo r a rbi trury ubclin n grou ps nnd thc s pli tting croup busis problcm for p-mixed groups rrc, in foc t., c c¡ui\'n lc'.nt in lhe com1m1 1.ativc g rou p nlgcbrn o f chnrnclcri:slic p. in nddi t.ion , n ncw schcmc o f o btni11ing t hc complete set of invnrinnts for such n g ro up nlgcb ru of n p-splíhing grou p is givou. in pnrticu lnr, ns npplicmion , thc full systom of invnrinnt.s fo r a gro u1> nlgobru whcn thc group is p-s plitling warficld is cstnblit1hcd ns wc ll. this s upplics own rcccnt resulls in (bol. soc. mal. t\l cxic111111 , 20011) nnd ( acln t\lalh. si nicn, 2005). resume sc' pruebo. que el problonm de doses ngru1>ndns p-dh•idid n p arn grnpos nbclianm nrbi t rurlos y ol prnblcmn de brscs ngrnpndns divididn ¡mm grnpos rnl'zl:lndm 11 es, c u cít..."clo, cqui v11lcntc e n el filgcbrn de gtupo coun111t11tiv11 d e cnrsclerfst icn 11. adcm án, ho e nt rega un l'tiqucmn nuevo parn obt.encr el conj unto completo de invnrin nt cs parn e:m filgcbrn d e grupo de un grupo 1>-dividido. eu porticulnr. romo nplknción, se c:;tnhll>co ndcmfi.s el s islemn comploto de invnrinnt~ paro un álgcbrn ele grupo cmmdo e l gru¡>0 es \\'l\tfield ,,..div idido. 8sto e nt ttga rcsult.ndos p roplo11 y novcd o!kls e n (bol. soc. mal. mcxicntm, 200•1) y ( acln mnth. sinicn , 2005). •10 pctcr danchcv k cy words and phrosos: i.~omorplu.m1 problem. p-6jlf1fhng group.f, p -mued ymu71/j, ll'arfidd group.~ • .f•mply prc_,c11tc:d grou¡11. muth. subj. c loss.: p rinwry roc01, 1653,, eok ei : secondo ry j6u60. 1 introd u ction throughout lhis brief puper wc shall m;c, in a traditional mnnnc r, thc lcttcr fg ns thc b'toup algcbra o f an abclian g:rou p c writtcn multiplicati vcly ovcr a ficlcl f. por such a group c , 1hc symbols g,. a nd g1, urc rcscrvcd for its ma ximal to rs ion subgroup (= torsio n pnrt in othcr tcrms) a nd 1>-primary cornponent, rcspcctively. the splitti11g problem for arbitrnry mixcd g roups1 going fro m i\lny [10] ru; a spccial pa rt of the / somor¡1hism problem, as ks of whethcr the separntion of g, as a dircct factor of g can invarifmtly be rotricvcd from the group algcbra fg. unfortmrnt.cly, this qucstion has a ncgative scttling in general firstly provcn aga.in by li.·lay {scc, for cxa mple, ¡t2j). r..lajor advantngc in various aspects on t hc prescntcd t.hcmc was done in ll j {sec [5] t oo for commu1;ntivc grou p ulgcbras ovcr ali fielcls). as was firs t.ly obscrvcd and conjccturcd in [13], t.hc wca k \•crs ion of t.hc split· ting pro blcm fo r 1>-mixed g roups can , cvcnt ually, t o have a posit ivo solut.ion ovcr thc commulativc group algcbru with prime charnct eristic 7j. f'or this purposc, li.foy, howcvcr, assumcd t hat thc direct factor problem fo r p-mixcd g rou ps is true. in whut follows, wc s hall s how thut. thc :;nmc cnn nlso be cxpccted undcr t.hc vnlidity of thc lsomorphis m problem for 7>-mixcd groups. thc difficu ltics cncountercd in thc thcory o f mixcd s plitting abelia n g roups can bccomc decidcd ly lcss complcx, if it is po&;iblc to reduce thc qu ·t.ion t.o s plit.t.ing mixcd abclia n groups whosc nu.1.ximal t.ors ion s ubg roup is ¡>-primary, t.hat. is e, = e,,. call such an a b cli a n grou p a v-mi:i:e-splitting groups for which lhc com m ut ati \•c modula r g roup algcbrn with chnract eris tic p posscsscs a complclc set. of irwarinnt.s. 2 !jai n theore ms as u.sunl. thc mixed abclion group i:s sflid lo be ¡1-svlitting, rcspecti vcly .sj1litting, if gp, rcspcctivcly g,, is n dircct fact or of c . bcfor p rcscnt ing t hc ce nt.mi rc:sults, wc n cccl lhc following tcchnicnl nsscr tion which is thc crucial point. lc mnm. la!t c be g / u g q is spltttmg. .... pro of. first of ali, not e tlmt (g / 11 cq)1 = g,/ u gq = (c,. x 11 gq)/ u c q ~ 1¡i¡t11 '11-p qr¡i.p qfl p not.cs o n t.hl! lsomorphhm l nud split.ling p roblcms for . •11 e:,,. l. º=;-" : writc c = c 1, x m fo r somc m :5 g. thcrcforc. it is self-cvident t hn l g/ ll e,,= (g, / ll c,,)(m( ll e:., )/ ll g,,) = (g/ ll g,).(m(ll g,,)/ ll g,,). ,,,.¡.,, ,,,,,_,, <¡ji:p •1 rf.:.1• ,,.,¡,,, '*"''' 'l~l' wlml. rcnm ins lo dcmons lrnlc is llllll lhc int.crsccliou bet.wccn thc t,wo fnctors is equul lo ohc. this, ccrt11 iu ly, is uccomplis hcd by s howing thnl e, n (/1/ lle,,) = il c:,1. l ndccd , wit.h t.hc nid of t.hc modular law, wo cnlculntc thn1. 1¡rf.¡1 e, n (m il e,,) = = tai ning thc i n vnrinnls o f s pl it.li ng com m utntivc group nlgcbrns wcrc fir~t givc11 in [2[. t hcor c m 1 ( l n va r ia n t.s). suppo.~e llwt g is a f)·spl1tt111g warfirld f1bclio11 group oltd lhat !( i.s an algebmicolly clohed fidtl of cjw,.(i< ) = 1' >o. tlic r1 /( 11 ~ f( c ru /( -olgcbms /01· rmothc r ,qroup 11 ·if tm.f-p why, lhc mnin thcorcm npplics lo d e ri ve thc cle;ircd clnim. t h cor c m 2 ( l nvu ri n n ts) . s npposc lltat e 1$ a s11litt111g warfield abelian fjivup wiu1 11 jitiile factor g,/c1, cmd thal. f is rw llrb1lrnry firld o/ chor( f) = p > o. th c11 /·' 11 ~ fc as f -11l9cbm.~ for nnolher gro11¡1 11 1/ and only 1/ lh~ foffowi ng c011ditio118 o,.c 1"cal1zcd: { i} 11 1s ,:,;p/1ttrng nbrlian; (2) 11, 9! g,; (9) 11/ h, 9! g/ g,; (4) ihf / //,[ = !gf / g,,i, jm· all ¡16mes q #/> aad i e sq(f) u (o). p r oor. bccnu.sc in vir luo o f [g] (cf. 17] too) w(' hn\'(' dcduccd thnt c ¡ lj g'1, ~ q t/;¡> ¡.¡ f li llq, wc cmploy ni o m.:c lhc rvlniu throrcm to gct thc '''fmtcd c lnim . . ,-;.,, wc l('rminnt th(' pnpcr by t.hc hclpíul obm-rvnlions thel although (,' und g'/g'1, urc both wo..rlicld. // nnd ll / 111, urc nol 11cccssnrily from lhi.s grou p cl n.ss. also, s iucc (;1, bcing an -group (:.ce o.g. [o] for the ,rlocal cose or ll5j for thc g lobn l 0110) ncccl uol be simply pre;cn1ccl, lho ln!"!l two thror ms nrc nl firsl look indcpcndonl fro m [3j. llom·vcr. bocatl..c cp f\8 bci11g h dirc.•ct fo(_·t or oí lhc wnrlicld grou p e is nl:so \ovnrfic ld, ho ucc simply p~nlccl , theorc ms 1 nnd 2 nrc d educible from l3j. r o n mrk in fá, ¡>. 160. lin(' 8(-)] thcrc is n misprint, 1mmt'ly thc farnu d n t(v p [g'j) = v p[t(c')l ,hould be writtrn "" t (i ' p[c'j) = 7'(\i p[t(c')i):""" also [•i/. morcovcr, in [2, p . 11. line l l{+)j t lw " ifr' 11111st b r<'nd ns "if'. 11 pctc r dnn chcv recei ved: oc t 2005. revisad: oc t 2005 . r e fe r e n ces [t] z. c llatz idal\i s 1\n d p. p 1\pp1-\ s, on tli e spli tlt ng gro 11p bosi.s pro blt> rii for abelra n 9m1qj rings, .l. purc ap p\. algc bra (1 ) 78 ( 1992) , 1526 . [2 j p . üan cll ev, lsonw17jhism o/ commutat.ive group algebms of m ixe. iay , com.11ut l. lat. h. soc. ( 1) 136 ( 1969), 139 1,19. ji lj \v . tl. iay , ln v m·úm.ls f 01· co 11 mwtative gro up algebrn.'i, llli nois .j. mnt.h . (3 ) 1 5 (1 97 1), 52:> 531. [12j \v . lw , /so m.0 1·ph:ü m. of gmup olgt>brns, .j. algcb ra ( i} <10 ( 1976), 10 18. !i jj \\" . ~iay , t he di.rnct f actor probfem f or modular abcli o n gro 11p algebros, contemp. t-. l nlh. 93 { 19 9), 30330 . p .1¡ j . üppe l.t, ali.xetl nbelilmgro 1111s, can .. j. mnth . (6) 19 (1967), 1259 1262. l15j r . stanton, wmfre/rl groujj s tmd -gro111m, (prc print.). ,«ffl!!!u ot~ 0 11 t lw b o111nrpl1is111 nncl spli lting p roblenl-. for . [ig] \\'. lle!t'i' , a ron;crtwy' rcfot1119 lo /11(' 1somorph1sm j1ry1bfcm fo1· comm:utatwc grou¡' algebm.t, in croup nnd scmigro up rinp.. north110111111<\ t-. lnth. studi~, no. 126, o rtl1l loll1111d, an1slc'rdnm , 19sg, 2·1i 252. cubo a mathematical journal 2007 v9 nª1_0044 cubo a mathematical journal 2007 v9 nª1_0045 cubo a mathematical journal 2007 v9 nª1_0046 cubo a mathematical journal 2007 v9 nª1_0047 cubo a mathematical journal 2007 v9 nª1_0048 cubo a mathematical journal 2007 v9 nª1_0049 cubo a mathematical journal 2007 v9 nª1_0050 cubo a mathematical journal vol.11, no¯ 01, (55–71). march 2009 discrete clifford analysis: an overview fred brackx1, hennie de schepper, frank sommen and liesbet van de voorde clifford research group, department of mathematical analysis, faculty of engineering, ghent university, galglaan 2, 9000 gent, belgium. email: freddy.brackx@ugent.be abstract we give an account of our current research results in the development of a higher dimensional discrete function theory in a clifford algebra context. on the simplest of all graphs, the rectangular zm grid, the concept of a discrete monogenic function is introduced. to this end new clifford bases, involving so–called forward and backward basis vectors and introduced by means of their underlying metric, are controlling the support of the involved operators. as our discrete dirac operator is seen to square up to a mixed discrete laplacian, the resulting function theory may be interpreted as a refinement of discrete harmonic analysis. after a proper definition of some topological concepts, function theoretic results amongst which cauchy’s theorem and a cauchy integral formula are obtained. finally a first attempt is made at creating a general model for the clifford bases used, involving geometrically interpretable curvature vectors. resumen nosotros damos un relato de los resultados de investigación actual en el desarrollo de la teoría de funciones discretas de dimensión grande en un álgebra de clifford. sobre el mas simple de todos los gráficos, la red de rectangulos zm, el concepto de función monogénica discreta es presentado. con esta finalidad nuevas bases de clifford, envolviendo las bases de vectores llamadas forward and backward, son introducidas mediante su métrica fundamental, estas controlan el soporte de los operadores envueltos. como nuestro operador de dirac discreto puede ser visto como un operador 1corresponding author: freddy.brackx@ugent.be 56 f. brackx, h. de schepper, f. sommen and l. van de voorde cubo 11, 1 (2009) laplaciano discreto mixto, la teoría de funciones resultante puede ser interpretada como refinamiento de análisis armónico discreto. después de definir algunos conceptos topológicos, resultados de teoría de funciones entre los cuales el teorema de cauchy y la fórmula de cauchy integral son obtenidos. finalmente, una primera tentativa es hacer uso de un modelo general de bases de clifford envolviendo vectores de curvatura geométricamente interpretables. key words and phrases: discrete clifford analysis, discrete function theory, discrete cauchy formula. math. subj. class.: 30g35. 1 introduction to the clifford analysis setting clifford analysis (see e.g. [3, 4, 14]) is a higher dimensional function theory centred around the notion of monogenic functions, i.e. null solutions of the rotation invariant vector valued dirac operator ∂x, defined below. it is a popular viewpoint to consider this function theory both as a higher dimensional analogue of the theory of holomorphic functions in the complex plane and as a refinement of classical harmonic analysis. in order to clarify these statements, let us introduce the underlying framework. to this end, let r0,m be endowed with a non–degenerate quadratic form of signature (0,m), let (e1, . . . ,em) be an orthonormal basis for r 0,m and let r0,m be the real clifford algebra constructed over r0,m, see e.g. [22]. the non–commutative multiplication in r0,m is governed by ejek + ekej = −2δjk, j,k = 1, . . . ,m (1) a basis for r0,m is obtained by considering for each set a = {j1, . . . ,jh} ⊂ {1, . . . ,m} the element ea = ej1 . . .ejh , with 1 ≤ j1 < j2 < ... < jh ≤ m. for the empty set ∅ one puts e∅ = 1, the identity element. any clifford number a in r0,m may thus be written as a = ∑ a eaaa, aa ∈ r. when allowing for complex constants, the same set of generators (e1, . . . ,em), still satisfying the anti–commutation rules (1), also produces the complex clifford algebra cm, as well as all real clifford algebras rp,q of any signature (p + q = m). the euclidean space r0,m is embedded in r0,m by identifying (x1, . . . ,xm) with the clifford vector x = m∑ j=1 ejxj cubo 11, 1 (2009) discrete clifford analysis: an overview 57 the multiplication of two vectors x and y is given by xy = x • y + x ∧ y with x • y = − m∑ j=1 xjyj = 1 2 (xy + yx) x ∧ y = ∑ i = −|x|2. conjugation in r0,m is defined as the anti-involution for which ēj = −ej, j = 1, . . . ,m. in particular for a vector x we have x̄ = −x. the fourier dual of the vector x is the vector valued first order differential operator ∂x = m∑ j=1 ej∂xj called dirac operator. it is precisely this dirac operator which underlies the notion of monogenicity of a function, a notion which may be considered as the higher dimensional counterpart of holomorphy in the complex plane. a function f defined and differentiable in an open region ω of rm and taking values in r0,m is called left–monogenic in ω if ∂x[f] = 0. in what follows, we will use the concept of inner spherical monogenics; these are homogeneous polynomials pk(x) of degree k (k ∈ n), which are moreover monogenic, i.e. for which it holds that ∂x[pk](x) = 0. since the dirac operator factorizes the laplacian, ∆ = −∂2 x , monogenicity may also be regarded as a refinement of harmonicity; in this sense, spherical monogenics can be seen as refinements of spherical harmonics. the fundamental group leaving the dirac operator ∂x invariant is the special orthogonal group so(m), doubly covered by the spin(m) group of the clifford algebra r0,m. for this reason, the dirac operator is called a rotation invariant operator. in the present context, we will refer to this setting as the continuous case, as opposed to the discrete setting treated in this paper. recently, several authors have shown interest in finding an appropriate framework for the development of discrete counterparts of the basic notions and concepts of clifford analysis, see a.o. [15, 16, 9, 10, 12]. some, yet not all, of these contributions are explicitly oriented towards the numerical treatment of problems from potential theory and boundary value problems, rather than towards discrete function theoretic results, see also [17, 18]. in this paper, however, we will abandon the path of possible applications in order to focus on the fundamental features of a concrete model for a clifford algebra framework in which discrete dirac operators and the corresponding discrete function theories can be developed, see also [5, 6]. seen the above mentioned connection between continuous clifford analysis and complex analysis in the plane, special attention should 58 f. brackx, h. de schepper, f. sommen and l. van de voorde cubo 11, 1 (2009) be paid to the important property of the discrete dirac operator factorizing a discrete laplacian. this was also the case in the study of holomorphic functions on z2, see e.g. [13, 19, 8] and, more recently [20, 21]. discrete mathematics always involve graphs; here, we will only consider the simplest of all graphs in euclidean space, namely the one corresponding to the rectangular zm grid. 2 definition of a discrete dirac operator as announced above, we will consider the natural graph corresponding to the equidistant grid zm; thus a clifford vector x as introduced above will now only show integer co–ordinates. for the pointwise discretization of the partial derivatives ∂ ∂xj we then introduce the traditional one–sided forward and backward differences, respectively given by ∆ + j [f](x) = f(. . . ,xj + 1, . . .) − f(. . . ,xj, . . .) = f(x + ej) − f(x), j = 1, . . . ,m ∆ − j [f](x) = f(. . . ,xj, . . .) − f(. . . ,xj − 1, . . .) = f(x) − f(x − ej), j = 1, . . . ,m we then first introduce a discrete laplacian by its usual definition for an arbitrary connected graph. definition 1. let f be a function defined on the vertices of a connected graph and let x be such an arbitrary vertex. then the action of the discrete laplace operator on f at x is defined by ∆f(x) = ∑ y∼x ( f(y) − f(x) ) = ∑ y∼x f(y) − ( #nx ) f(x) where the notation y ∼ x means that there is an edge in the graph under consideration which links the vertex y to x, and where nx stands for the neighbourhood of x with respect to the graph, i.e. the set of all points y ∼ x. in the present case, with respect to the zm neighbourhood of x, the above definition explicitly reads ∆ ∗ [f](x) = m∑ j=1 [ ∆ + j [f](x) − ∆− j [f](x) ] = m∑ j=1 [f(x + ej) + f(x − ej)] − 2mf(x) (2) where we have denoted the corresponding discrete laplacian by ∆∗; it is usually called the star laplacian and involves the values of the considered function at the midpoints of the faces of the unit cube centred at x. clearly, with respect to the same grid, but changing the graph, other discrete laplacians may be defined, involving e.g. the function values at the vertices of the cube (the cross laplacian), or at the midpoints of the ”edges”. cubo 11, 1 (2009) discrete clifford analysis: an overview 59 for now, we restrict ourselves to the star laplacian (2); note that it can also be written as ∆ ∗ [f](x) = m∑ j=1 ∆ + j ∆ − j [f](x) = m∑ j=1 ∆ − j ∆ + j [f](x) when passing to the dirac operator, we cannot simply combine each discretized partial derivative, be it forward or backward, with the corresponding basis vector ej, j = 1, . . . ,m, since such attempts do not serve our aim at developing a discrete function theory in which the notion of discrete monogenicity implies discrete harmonicity, as has been shown in [5]. instead, an alternative approach is followed, in which the basis vectors will carry an orientation, just like the forward and backward differences do. to this end, we need to embed the clifford algebra r0,m into a bigger one, with an underlying vector space of the double dimension, e.g. c2m, where we consider 2m vectors e+ j and e− j , j = 1, . . . ,m, satisfying the following anti–commutator relations: e + j e + k + e + k e + j = −2g+ jk , e − j e − k + e − k e − j = −2g− jk , e + j e − k + e − k e + j = −2mjk where the symmetric tensors (g+ jk ), (g− jk ) and the general tensor (mjk) determine the corresponding metric, see also [12]. three subsequent assumptions on this metric will now significantly reduce the degrees of freedom in the choice of the metric scalars. assumption 1. the forward and the backward basis vector in each particular cartesian direction add up to the traditional basis vector in that direction, i.e. e+ j + e − j = ej, j = 1, . . . ,m. assumption 2. there are no preferential cartesian directions, or: all cartesian directions play the same role in the metric. this assumption will be referred to as the principle of dimensional democracy and may be seen as a kind of rotational invariance. assumption 3. the positive and negative orientations of any cartesian direction play an equivalent role. this assumption may be interpreted as a kind of reflection invariance. on the basis of the second and third assumptions, one may put g+ 11 = g + 22 = . . . = g+ mm = g − 11 = g − 22 = . . . = g− mm = λ, where g± jj = −(e± j ) 2, j = 1, . . . ,m, and m11 = m22 = . . . = mmm = µ, where 2mjj = −(e + j e − j + e − j e + j ), j = 1, . . . ,m. furthermore, also g± jk and mjk, for j 6= k, should be independent of their subscripts, whence we put g± jk = g and mjk = mkj = m, j,k = 1, . . . ,m, j 6= k. the first assumption, combined with the traditional clifford multiplication rules, then leads to the additional conditions λ + µ = 1 2 and g + m = 0. summarizing, the forward and backward basis vectors e+ j and e− j , j = 1, . . . ,m, will submit to the following multiplication rules: • e+ j e + k + e + k e + j = e − j e − k + e − k e − j = −2g, j 6= k • e+ j e − k + e − k e + j = 2g, j 6= k • (e+ j ) 2 = (e − j ) 2 = −λ, j = 1, . . . ,m • e+ j e − j + e − j e + j = 2λ − 1, j = 1, . . . ,m 60 f. brackx, h. de schepper, f. sommen and l. van de voorde cubo 11, 1 (2009) we are now led to the definition of our discrete dirac operator. definition 2. the discrete dirac operator ∂ is the first order, clifford vector valued difference operator given by ∂ = ∂+ + ∂− where the forward and backward discrete dirac operators ∂+ and ∂− are respectively given by ∂+ = m∑ j=1 e + j ∆ + j and ∂− = m∑ j=1 e − j ∆ − j we obtain, using the above multiplication rules, that ∂2 = −λ m∑ j=1 (∆ + j ∆ + j + ∆ − j ∆ − j ) + (2λ − 1) m∑ j=1 ∆ + j ∆ − j + g ∑ j 6=k (2∆ + j ∆ − k − ∆− j ∆ − k − ∆+ j ∆ + k ) if we require the support of ∂2 to remain at least in the unit cube centred at x, the isotropy of the forward and backward basis vectors needs to be imposed, i.e. we have to put λ = (e+ j ) 2 = (e − j ) 2 = 0 as in [12], whence in our case it follows in addition that µ = 1 2 , or e+ j e − j +e − j e + j = −1, j = 1, . . . ,m. one thus finally arrives at • e+ j e + k + e + k e + j = e − j e − k + e − k e − j = −2g, j 6= k • e+ j e − k + e − k e + j = 2g, j 6= k • (e+ j ) 2 = (e − j ) 2 = 0, j = 1, . . . ,m • e+ j e − j + e − j e + j = −1, j = 1, . . . ,m see also [5]. these relations completely determine the metric of the underlying 2m–dimensional space in terms of one free scalar parameter g, the metric tensor being given by mjk =    e + j • e+ k , j,k = 1, . . . ,m e + j • e− k , j = 1, . . . ,m, k = m + 1, . . . , 2m e − j • e+ k , j = m + 1, . . . , 2m, k = 1, . . . ,m e − j • e− k , j,k = m + 1, . . . , 2m cubo 11, 1 (2009) discrete clifford analysis: an overview 61 or explicitly: m =   0 −g · · · −g − 1 2 g · · · g −g 0 . . . ... g − 1 2 . . . ... ... . . . 0 −g ... . . . − 1 2 g −g · · · −g 0 g · · · g − 1 2 − 1 2 g · · · g 0 −g · · · −g g − 1 2 . . . ... −g 0 . . . ... ... . . . − 1 2 g ... . . . 0 −g g · · · g − 1 2 −g · · · −g 0   its determinant reads det m = (−1)m (1 + 4g)m−1(1 − 4(m − 1)g) 4m whence it should hold that g 6= − 1 4 and g 6= 1 4(m−1) , since these specific values would induce a collapse of dimension; for a further discussion of this phenomenon we refer to section 7. under the above conditions, ∂2 takes the form ∂2 = − m∑ j=1 ∆ + j ∆ − j + g ∑ j 6=k (∆ + j ∆ − k + ∆ + k ∆ − j − ∆− j ∆ − k − ∆+ j ∆ + k ) = (4(m − 1)g − 1)∆∗ − 2g ∑ j,x̄ ∈ rn}, is called a subdifferential of the function g at the point x and is denoted by ∂g(x), there symbol < .,. > denotes scalar product. it is known that when g(x) is convex, the given definition coincides with the usual definition of the subdifferential.(see [1]) 3) the mapping a∗(y∗; z) = {x∗ : (−x∗,y∗) ∈ k∗a(z)} is called a locally conjugate mapping (lcm) to the convex mapping a at the point z. theorem 1.1: let a : r n → 2rn be convex-valued closed bounded continuous mapping such that the function wa(x,y∗) = inf y∈a(x) < y,y∗ > is continuous differentiable on x. let us suppose that the vector z̄1 = (x̄1, ȳ1) satisfies the inequality < x̄1, ∂wa(x0,y∗) ∂x > − < ȳ1,y∗ >< 0. then the following statements are true for a point z0 = (x0,y0),y0 ∈ a(x0,y∗): i)the cone ka(z0) = { z̄ :< x̄, ∂wa(x0,y∗) ∂x > − < ȳ,y∗ >< 0 } is the smooth local tent, which is the cone of tangent directions to gfa(graph of a) at the point z0. ii) lcm a∗ corresponding to the cone ka(z0) may be given by the formula a∗(y∗; z0) = { ∂wa(x0,y∗) ∂x } proof. if sa(x,y∗), y∗ ∈ rn, is the support function to a(x) , then by the theory of convex analysis it is known that y ∈ a(x) if and only if < y,y∗ >≤ sa(x,y∗) for all y∗ ∈ rn. since sa(x,y∗) = −wa(x,−y∗), the preceding inequality means that < y,y∗ >≥ wa(x,y∗). thus a(x) is given by a(x) = {y : wa(x,y∗)− < y,y∗ >≤ 0}, y∗ ∈ rn. (5) suppose fy∗ (z) = wa(x,y ∗)− < y,y∗ >, (6) then by the lemma 3.1[1,p.225], fy∗ (z) is continuous on y∗ and is continuous differentiable on z. by the theorem 2.2[1,p.211], uca(upper convex approximation) hy∗ (z̄,z) of the function fy∗ (z) is hy∗ (z̄,z) =< z̄, ∂wa(x,y∗) ∂x ×{−y∗} > . (7) 24 e. n. mahmudov and g. çiçek 7, 2(2005) furthermore fy∗ (z0) = 0 and fy∗ (z) has an uca hy∗ (z̄,z0), which is continuous on z̄ and by the condition on the vector z̄1, we have hy∗ (z̄1,z0) < 0. then applying theorem 3.3[1,p.234] by (7) we see that i) of the theorem follows. since in this case −con∂fy∗ (z0) = con { −∂wa(x0,y ∗) ∂x ,y∗ } , then by the same theorem 3.3[1,p.234] the equality a∗(y∗; z0) = { ∂wa(x0,y∗) ∂x } holds. this, in turn, implies that ii) is correct. the theorem is proved. let o+(gfa) be the recession cone[2] to a convex function a in the space z = x ×y , i.e. o+(gfa) = {z̄ : z + λz̄ ∈ gfa,λ ≥ 0,∀z ∈ gfa}. (8) for such convex function a, let us define ωa(x ∗,y∗) = inf{− < x,x∗ > + < y,y∗ >: (x,y) ∈ gfa}. (9) it is evident that ωa(x ∗,y∗) = inf x {− < x,x∗ > +wa(x,y∗)}. (10) definition 1.2: the function a∗(y∗) = {x∗ : (−x∗,y∗) ∈ (o+gfa)∗} is called conjugate function to a convex function a. it is clear that if mapping a is superlinear[5], i.e. gfa is a cone, then this definition coincides with the definition of b.h.pshenichnyi [1]. conjugate function can be used in different problems connected with duality theorems. definition 1.3: multivalued mapping a is called quasisuperlinear if its graph is in the form of gfa = m + k, where m is a convex compactum, k is a closed convex cone. lemma 1.1: for a convex mapping a we have domωa = {(−x∗,y∗) : ωa(x∗,y∗) > −∞}⊆ (o+gfa)∗. if a is a quasisuperlinear mapping then domωa = k ∗. 7, 2(2005) optimization of differential inclusions ... 25 proof. let us assume the contrary: let (−x∗0,y∗0 ) ∈ domωa, but (−x∗0,y∗0 ) �∈ (o+gfa)∗ . it means that there exists a pair (x̄0, ȳ0) ∈ o+gfa, for which − < x∗0, x̄0 > + < y∗0, ȳ0 >< 0. by the definition of o+gfa, we have (x,y) + λ(x̄0, ȳ0) ∈ gfa, (x,y) ∈ gfa, λ > 0. then − < x + λx̄0,x∗0 > + < y + λȳ0,y∗0 >= − < x∗0,x > + < y∗0,y > + +λ{− < x̄0,x∗0 > + < ȳ0,y∗0 >}→−∞ for λ → +∞, which contradicts the fact that (−x∗0,y∗0 ) ∈ domωa. this proves the first statement of the lemma. furthermore, when a is a quasisuperlinear mapping, applying result 9.1.2[2] and lemma3.6.1[1],we get (o+gfa)∗ = [o+(m + k)]∗ = (o+m)∗ ∩ (o+k)∗ = rn ∩k∗ = k∗. on the other hand domωa = dom(ωm + ωk ) = domωm ∩domωk = domωk = k∗. hence domωa = k ∗. lemma is proved. the following example shows that the inverse inclusion generally is not true. in fact, let a : x → 2y (x,y one-dimensional axises) is given as: a(x) = {y : y ≥ x2} , gfa = {(x,y) : y ≥ x2}. check that o+gfa = {0}×y +, where y + is the positive y-axis. therefore (o+gfa)∗ = {(−x∗,y∗) : x∗ ∈ x,y∗ ∈ y +}. then it is clear that (−x∗0,y∗0 ) ∈ (o+gfa)∗, x∗0 = 1,y ∗ 0 = 0, but (−x∗0,y∗0 ) �∈ domωa. lemma 1.2: let a be a quasisuperlinear mapping and wa(.,y∗) be proper closed function. then the relation sup x∗∈a∗(y∗) {< x,x∗ > +ωm (x∗,y∗)} = inf y∈a(x) < y,y∗ > holds. proof. from lemma 1.1, we have domωa = (o +gfa)∗ = k∗. 26 e. n. mahmudov and g. çiçek 7, 2(2005) therefore with regard to theorem 4.1.iii[1] we find the relation sup x∗ {ωa(x∗,y∗)+ < x,x∗ >} = sup x∗ {< x,x∗ > +ωm (x∗,y∗) : x∗ ∈ a∗(y∗)} = wa(x,y∗). remark 1.2.1: if m = {0}, then ωm = 0 and so the result of the above lemma coincides with the result of the theorem 4.5.iii[1,p.129]. lemma 1.3: let a be a convex mapping. then the point x0 is a solution of the problem inf x {− < x,x∗ > +wa(x,y∗)}, x∗,y∗ ∈ rn if and only if x∗ ∈ a∗(y∗,z0), y0 ∈ a(x0,y∗). proof. by the theorem 2.1.iv[1], x0 is a minimum point of the convex function − < x,x∗ > +wa(x,y∗) if and only if 0 ∈ ∂x[− < x0,x∗ > +wa(x0,y∗)], i.e. x∗ ∈ ∂xwa(x0,y∗). and, therefore by the definition of ωa it is evident that y0 ∈ a(x0,y∗). then by the theorem 2.1.iii[1], we find the required result. theorem 1.2: let a be a convex-valued closed bounded continuous mapping, satisfying the lipschitz condition, and let the function waz (x̄,y ∗) be closed, where az (x̄) = {ȳ : (x̄, ȳ) ∈ ka(z)}. then for arbitrary y ∈ a(x,y∗),z = (x,y) ∈ gfa, the function waz (.,y∗) is an uca for wa(.,y∗) and, besides, a∗(y∗; z) = ∂xwa(x,y ∗). proof. if z̄ = (x̄, ȳ) ∈ ka(z),z = (x,y),y ∈ a(x), then by the definition of the cone of tangent directions, there is a function τ(λ), λ−1τ(λ) → 0, λ ↓ 0 (τ(λ) ∈ z = x ×y ) such that z + λz̄ + τ(λ) ∈ gfa for a sufficiently small λ ≥ 0. that means y + λȳ + τy(λ) ∈ a(x + λx̄ + τx(λ)),τ = (τx,τy),τx(λ) ∈ x,τy(λ) ∈ y. 7, 2(2005) optimization of differential inclusions ... 27 since a satisfies the lipschitz condition, wa(x,y∗) also satisfies the same condition by lemma 3.2.v[1,p.226]. for such functions we have f(x̄,x) = lim sup λ↓0 1 λ (wa(x + λx̄,y ∗) −wa(x,y∗)). it is easily shown that f(x̄,x) = lim sup λ↓0 1 λ (wa(x + λx̄ + τx(λ),y ∗) −wa(x,y∗)) holds independently from the choice of τ(λ). from the definition of wa(x,y∗) and from the condition y ∈ a(x,y∗) it follows that 1 λ (wa(x + λx̄ + τx(λ),y ∗) −wa(x,y∗)) ≤ 1 λ (< y + λȳ + τy(λ),y ∗ > − < y,y∗ >) = < ȳ,y∗ > + < τy (λ) λ ,y∗ > . then we have f(x̄,x) = lim sup λ↓0 1 λ (wa(x + λx̄ + τx(λ),y ∗) −wa(x,y∗)) ≤ limsupλ↓0[< ȳ,y∗ > + < λ−1τy (λ),y∗ >] =< ȳ,y∗ > . it means that f(x̄,x) ≤ inf ȳ {< ȳ,y∗ >: ȳ ∈ az(x̄)}. in addition, given x̄ �∈ domaz let us put waz (x̄,y∗) = +∞. then, by applying lemma 1.2 to az, we get waz (x̄,y ∗) = sup x∗ {< x̄,x∗ >: x∗ ∈ a∗z(y∗)}. but on the other hand, by the definition, a∗(y∗; z) = a∗z(y ∗). hence f(x̄,x) ≤ waz (x̄,y∗) = sup x∗ {< x̄,x∗ >: x∗ ∈ a∗(y∗; z)}, where waz (x̄,y ∗) is positive homogenous convex closed function of x̄, i.e. waz (x̄,y ∗) is an uca function of wa(.,y∗) at the point x. now to conclude the proof, it remains only to apply theorem 3.2.ii[1], thus we find ∂wa(x,y ∗) = ∂h(0,x) = a∗(y∗; z). let us investigate the relation between conjugate function and lcm(locally conjugate mapping). we need the following two theorems. let km (z) be the cone of tangent directions to a convex set m ⊆ z = x ×y at 28 e. n. mahmudov and g. çiçek 7, 2(2005) a point z ∈ m, i.e. km (z) = con(m −z) = {z̄ : z̄ = λ(z1 −z),λ > 0,z1 ∈ m}. (11) theorem 1.3: let o+m be the recession cone of a convex closed set m ⊂ z. then we have ⋂ z∈m km (z) = o +m. proof. let us show that m = ⋂ z∈m (z + km (z)). (12) in fact, let z0 ∈ m be an arbitrary fixed point. it is evident that all vectors as z̄ = z0 − z (in definition (11) they corresponds to λ = 1) belong to the cone km (z), i.e. z0 ∈ z + km (z),z ∈ m, then z0 ∈ ⋂ z∈m (z + km (z)). conversely, if we have the last inclusion then z0 ∈ z + km (z) or there are such z1 ∈ m and a number γ > 0, that z0 − z = γ(z1 − z) ∈ km (z). hence z0 = γz1 + (1 − γ)z ∈ m. formula (12) follows. on the other hand, we easily show that o+[ ⋂ z∈m (z + km (z))] = ⋂ z∈m [o+(z + km (z))]. in fact if z is an arbitrary point of closed convex set m = ⋂ z∈m (z + km (z)) then by the definition of the recession cone, it is evident that, directed ray z + λz̄, ∀λ ≥ 0, is contained in any cone z + km (z),z ∈ m. but it means that z̄ ∈ ⋂ z∈m [o+(z + km (z))]. therefore o+m = o+[ ⋂ z∈m (z + km (z))] = ⋂ z∈m [o+(z + km (z))] = ⋂ z∈m km (z). theorem is proved. remark 1.3.1: in the statement of the above theorem, the closedness of m is essential. proof. actually, let m = {(x,y) : x > 0,y > 0}∪{(0, 0)}⊂ r2. clearly, o+m = m. the set m contains points (x0,y0) + λ(0,y0), where x0 > 0, y0 > 0 are fixed. but (0,y0) �∈ o+m. theorem 1.4: let m be a closed convex set and let k∗m (z) be the conjugate cone to the cone of tangent directions km (z),z ∈ m. then⋃ z∈m k∗m (z) = (o +m)∗, 7, 2(2005) optimization of differential inclusions ... 29 where the bar denotes closure. proof. it is sufficient to show that ⋃ z∈m k∗m (z) = ( ⋂ z∈m km (z)) ∗. (13) get any fixed point z∗0 ∈ ⋃ z0∈m k∗m (z). then there exists a sequence z ∗ n → z∗0 , z∗n ∈⋃ z∈m k∗m (z). let us define sequence {zn} by the relation z∗n ∈ k∗m (zn). note that z∗n ∈ ⋃ z∈m k∗m (z) implies the existence of zn ∈ m such that z∗n ∈ k∗m (zn). on the other hand, since km (zn) ⊇ ⋂ z∈m km (z) it is evident that k∗m (zn) ⊆ ( ⋂ z∈m km (z))∗. so that z∗n ∈ ( ⋂ z∈m km (z))∗, and therefore z∗0 ∈ ( ⋂ z∈m km (z))∗. let us prove the converse inclusion in (13). let us z∗1 ∈ ( ⋂ z∈m km (z))∗ be arbitrary fixed point and let us assume the contrary i.e. let z∗1 �∈ ⋃ z∈m k∗m (z). then z ∗ 1 �∈ k∗m (z) for any z ∈ m. in other words, there exists a vector z̄1(z̄1 �= 0) such that < z∗1, z̄1 >< 0, z̄1 ∈ km (z), ∀z ∈ m or < z∗1, z̄1 >< 0, z̄1 ∈ ⋂ z∈m km (z), i.e. z ∗ 1 �∈ ( ⋂ z∈m km (z)) ∗. this contradiction shows that ( ⋂ z∈m km (z)) ∗ ⊆ ⋃ z∈m k∗m (z). the proof of the theorem is over now. theorem 1.5: let a be a closed convex mapping. then the conjugate function a∗(y∗) and the lcm of a implies the following relation a∗(y∗) = ⋃ z∈gf a a∗(y∗; z), y ∈ a(x,y∗). proof. setting m = gfa as in the previous theorem, we obtain a∗(y∗) = ⋃ z∈gf a a∗(y∗; z). by the theorem 2.1.iii[1] z = (x,y), y �∈ a(x,y∗), implies a∗(y∗; z) = ∅. 30 e. n. mahmudov and g. çiçek 7, 2(2005) 2 sufficient conditions of the optimization. according to [1], the lcm (locally conjugate mapping) a∗ of the multi-valued mapping a at a point z = (x,y) ∈ gfa(., t), t ∈ [t0, t1], is defined as follows: a∗(y∗, (x,y), t) = {x∗ : (−x∗,y∗) ∈ k∗a (z,t)},y∗ ∈ r n , where k∗a(z,t) is the conjugate cone to the cone of tangent directions ka(z,t). let us define wa(x,y∗, t) = { inf{< y,y∗ >: y ∈ a(x,t)}, a(x,t) �= ∅ +∞ a(x,t) = ∅, a(x,y∗, t) = {y ∈ a(x,t) :< y,y∗ >= wa(x,y∗, t)} and wm (x∗) = inf y∈m < x∗,y > . note that for a convex mapping a, the lcm coincides with the subdifferential[1] ∂xwa(x̃,y∗, t) of the function wa(.,y∗, t) at the point x̃.it is known that a∗(y∗, (x̃, ỹ), t) = { ∂xwa(x̃,y∗, t), ỹ ∈ a(x̃,y∗, t) ∅, ỹ �∈ a(x̃,y∗, t). let x̃(t), t ∈ [t0, t1], x̃(t0) = x0, be any admissible solution of the problem (1)-(4). let us construct the conjugate differential inclusion of the conjugate variable x∗(t) by a) −ẋ∗(t) ∈ a∗(x∗(t); (x̃(t), ˙̃x(t)), t) + ∂g(x̃, t), t ∈ [t0, t1], a.e; ˙̃x(t) ∈ a(x̃(t),x∗(t), t), t ∈ [t0, t1] a.e; which should be fulfilled for all x ∈ f(t). the solution x∗(t), t ∈ [t0, t1], satisfies the conjugate differential inclusion a) almost everywhere and is in the form of the sum of absolutely continuous functions and jump functions. let us denote points of jumps and values of jumps x∗(t) by τi(i = 1, 2, . . .), t0 < τi < t1, x∗i = x ∗(τi + 0) −x∗(τi − 0) (i = 1, 2, . . .), respectively. if the following condition − < x∗(t), x̃(t) >< wm∩f (t)(−x∗(t)), t0 ≤ t < t1 holds, the admissible trajectory x̃(t) would be called strictly transversal on the set m. note that this definition guarantees that point x̃(t) /∈ m for every t ∈ [t0, t1). if the inequality i(x(.),θ) < i(x(.),θ ′ ) holds for any θ,θ ′ ∈ [t0, t1] with θ < θ ′ and for any admissible trajectory of the differential inclusion (2) with initial condition x(t0) = x0, then the function i(x(.), t) is called monotone increasing with respect to argument t. theorem 2.1: let x̃(t), t ∈ [t0, t1], be any admissible trajectory of the problem 7, 2(2005) optimization of differential inclusions ... 31 (1)-(4) and let there exists absolutely continuous function x∗(t) which satisfies the inclusion a). furthermore assume that i(x(.), t) is monotone increasing with respect to argument t for any admissible trajectory x(t), t ∈ [t0, t1], of the differential inclusion (2) and the following conditions are satisfied: 1) x∗(t1) ∈ ∂ϕ(x̃(t1), t1),x∗(t1) ∈ k∗m (x̃(t1)); 2) the jumps x∗i satisfy < x̃(τi),x ∗ i >= wf (τi)(x ∗ i ); 3) x̃(t) is strictly transversal on m. then trajectory x̃(t) is optimal. proof. let x(t) ∈ f(t) be an arbitrary admissible trajectory, realising the transition from the interval [t0,θ] to the set m. let us show that i(x(.),θ) ≥ i(x̃(.), t1). using ∂xwa(.,x∗(t), t) as the representation of lcm and by the moreau-rockafellar theorem[4] we can rewrite the inclusion a) as follows: −ẋ∗(t) ∈ ∂x[wa(x̃(t),x∗(t), t) + g(x̃(t), t)], i.e. wa(x(t),x ∗(t), t) −wa(x̃(t),x∗(t), t) + g(x(t), t) −g(x̃(t), t) ≥ < −ẋ∗(t),x(t) − x̃(t) >, t ∈ [t0, t1], (14) wa(x̃(t),x∗(t), t) =< ˙̃x(t),x∗(t) > . since wa(x(t),x∗(t), t) ≤< ẋ(t),x∗(t) >, from (14) we have dψ(t)/dt ≥ g(x̃(t), t) −g(x(t), t) (15) for almost every t ∈ [t0, t1], where ψ(t) =< x(t) − x̃(t),x∗(t) > . then integrating (15) we find ∫ t1 t0 ψ̇(t)dt =< x(t1) − x̃(t1),x∗(t1) >≥ ∫ t1 t0 [g(x̃(t), t) −g(x(t), t)]dt. (16) x(t), x̃(t) are absolutely continuous, therefore ψ(t) can be represented by the sum of absolutely continuous functions and jump functions (see [9]). ψ(θ) = ψ(t0) + ∫ θ t0 ψ̇(t)dt + ∑ i∈j(θ) [ψ(τi + 0) −ψ(τi − 0)], j(t) = {i : τi ∈ [t0, t]}. (17) let us compute the values of the jumps of the function ψ(t) at points τi(i = 1, 2, . . .). using the condition 2) of the theorem, we find ψ(τi + 0) −ψ(τi − 0) =< x(τi) − x̃(τi),x∗i >=< x(τi),x∗i > −wf (τi)(x∗i ). 32 e. n. mahmudov and g. çiçek 7, 2(2005) then by the relation x(τi) ∈ f(τi), it is evident that ψ(τi + 0) −ψ(τi − 0) ≥ 0 ∀τi ∈ [t0,θ], i.e. ∑ i∈j(θ) [ψ(τi + 0) −ψ(τi − 0)] ≥ 0. by the condition 1) of the theorem and definition of dual cone the inequality < x(t1) − x̃(t1),x∗(t1) >≥ 0 holds. since t1 is free the last inequality is correct for any t1 = θ. obviously, the inequality (16) is correct for any t1 = θ. therefore from (17), it is evident that ψ(θ) ≥ ψ(t0) i.e. < x(θ) − x̃(θ),x∗(θ) >≥< x(t0) − x̃(t0),x∗(t0) >= 0. from the last inequality and condition 3) of the theorem − < x(θ),x∗(θ) > ≤− < x̃(θ),x∗(θ) > < wm∩f (θ)(−x∗(θ)). (18) let �i = i(x(.),θ) − i(x̃(.), t1) be the increment of the target functional i, obtained by the transition from the trajectory x̃(t) to the trajectory x(t). then �i = ϕ(x(θ),θ) + ∫ θ t0 g(x(t), t)dt−ϕ(x̃(t1), t1) − ∫ t1 t0 g(x̃(t), t)dt = ϕ(x(θ),θ) + ∫ θ t0 g(x(t), t)dt−ϕ(x(t1), t1) − ∫ t1 t0 g(x(t), t)dt + ϕ(x(t1), t1)+ + ∫ t1 t0 g(x(t), t)dt−ϕ(x̃(t1), t1) − ∫ t1 t0 g(x̃(t), t)dt. on the other hand from the inequality (16) and by condition 1) of the theorem we obtain: ∫ t1 t0 [g(x(t), t) −g(x̃(t), t)]dt + ϕ(x(t1), t1) −ϕ(x̃(t1), t1) ≥ 0. since (16) is correct for any t ∈ [t0, t1], the last relation implies �i ≥ ϕ(x(θ),θ) + ∫ θ t0 g(x(t), t)dt−ϕ(x(t1), t1) − ∫ t1 t0 g(x(t), t)dt. (19) to prove the optimality of x̃(t) let us assume the contrary, i.e. let for any admissible trajectory x(t), t ∈ [t0,θ],x(t0) = x0,x(θ) ∈ m, �i < 0, i.e. i(x(.),θ) < i(x̃(.), t1). then by the inequality (19), we have i(x(.),θ) < i(x(.), t1). since i(x(.), t) is monotone we conclude that θ < t1. (20) 7, 2(2005) optimization of differential inclusions ... 33 thus by the inequalities (18) and (20) we have x(θ) /∈ m∩f(θ). hence x(θ) /∈ m, i.e. the trajectory x(t) cannot realize the transition from the interval [t0,θ] to the set m. it means that, x̃(t) is the optimal trajectory. remark 2.1.1: if t1 is fixed then θ = t1, and then �i ≥ 0 (see (19)), i.e. x̃(t) is optimal. moreover, in that case the condition of monotone increasingness of i(x(.), t) on t is superfluous . remark 2.1.2: condition of monotonicity of i(x(.), t) on t for any admissible trajectory x(t) is not very restrictive and we can verify it. for example it is fulfilled for high speed problems and for problems with quadratic criteria of quality and in case when ϕ(x,t) ≡ 0,g(x,t) ≥ 0. remark 2.1.3: suppose a is a convex-valued closed bounded continuous mapping and wa(x,y∗) is continuous differentiable on x. theorem 1.1 and the condition a) of theorem 2.1 imply −ẋ∗(t) ∈ ∂wa(x̃(t),x ∗(t), t) ∂x + ∂g(x̃(t), t). 3 duality. let us reconsider the problem (1)-(4), given in the introduction. this problem is called a convex problem if the functions, multivalued mapping and the set are convex and the function f is convex-valued. now consider (1)-(4) as a convex problem. let us denote by ϕ∗(., t1) and g∗(., t) conjugate functions [1,10] to functions ϕ(., t1) and g(., t), respectively. let us recall the equality ωa(x ∗,y∗, t) = inf{− < x,x∗ > + < y,y∗ >: (x,y) ∈ gfa}. evidently ωa(x∗,y∗, t) = inf x {− < x,x∗ > +wa(x,y∗, t)}. the following problem is called the dual problem to (1)-(4): sup x∗(t),ξ∗(t),u∗(t),v∗(t1) {−ϕ∗(v∗(t1) − ξ∗(t1), t1) − ∫ t1 t0 g∗(u∗(t), t)dt + < x(t0),x ∗(t0) > + ∫ t1 t0 ωa(−ξ∗(t) − ẋ∗(t) −u∗(t),x∗(t), t)dt − ∫ t1 t0 wf (t)(ξ ∗(t))dt + wm (v ∗(t1) −x∗(t1) − ξ∗(t1))}. (21) here x∗(t),ξ∗(t),u∗(t) and v∗(t1), are absolutely continuous functions. let us denote the expression in curly brackets by i∗(x∗(t),ξ∗(t),u∗(t),v∗(t1), t1). theorem 3.1. for any admissible solutions x(t) and {x∗(t),ξ∗(t),u∗(t),v∗(t1)} of the direct problem (1)-(4) and the dual problem (21), respectively, the relation i(x(t), t1) ≥ i∗(x∗(t),ξ∗(t),u∗(t),v∗(t1), t1) 34 e. n. mahmudov and g. çiçek 7, 2(2005) holds. proof. by the definitions of the conjugate function, ωa,wm and wf (t), we have i∗(x ∗(t),ξ∗(t),u∗(t),v∗(t1), t1) ≤− < x(t1),v∗(t1) − ξ∗(t1) > +ϕ(x(t), t) −∫ t1 t0 [< x(t),u∗(t) > −g(x(t), t)]dt+ < x(t0),x∗(t0) > + + ∫ t1 t0 [− < x(t),−ξ∗(t) − ẋ∗(t) −u∗(t) > + < ẋ(t),x∗(t) >]dt− ∫ t1 t0 < x(t),ξ∗(t) > dt+ < x(t1),v ∗(t1) −x∗(t1) − ξ∗(t1) >= i(x(t), t1)+ < x(t0),x ∗(t0) > + ∫ t1 t0 d < x(t),x∗(t) > − < x(t1),x∗(t1) >= i(x(t), t1). (22) theorem is proved. theorem 3.2: let the trajectory x(t), t ∈ [t0, t1], be a solution of the direct convex problem (1)-(4). further, let x∗(t), ξ∗(t),u∗(t) and v∗(t1) be functions such that x∗(t) satisfies the dual differential inclusion a), u∗(t) ∈ ∂g(x̃(t), t), v∗(t1) − ξ∗(t1) ∈ ∂ϕ(x̃(t1), t1), ξ∗(t) ∈ k∗f (t)(x̃(t)) and v∗(t1) − x∗(t1) − ξ∗(t1) ∈ k∗m (x̃(t1)). then {x∗(t),ξ∗(t),u∗(t),v∗(t1)} is a solution of the dual problem and in this case, the values of the two problems coincide. proof. by the definitions of locally conjugate mapping and conjugate cone we have < ξ∗(t) + ẋ∗(t) + u∗(t),x− x̃(t) > + < x∗(t),y − ˙̃x(t) >≥ 0 at almost every t ∈ [t0, t1] and all x ∈ f(t), (x,y) ∈ gfa(., t). it means that (−ξ∗(t) − ẋ∗(t) −u∗(t),x∗(t)) ∈ domωa, t ∈ [t0, t1]. if we consider ∂xg(x,t) ⊂ domg∗(., t) and ∂xϕ(x,t1) ⊂ domϕ∗(., t1) then we may conclude that {x∗(t),ξ∗(t),u∗(t),v∗(t1)} is an admissible solution of the dual problem. further, by lemma 1.3 and from the conjugate differential inclusion a) it is clear that ωa(ξ ∗(t) − ẋ∗(t) −u∗(t),x∗(t), t) = − < x̃(t),−ξ∗(t) − ẋ∗(t) −u∗(t) > +wa(x̃(t),x∗(t), t), t ∈ [t0, t1]. (23) from conditions of the theorem and from the fact that ˙̃x(t) ∈ a(x̃(t),x∗(t), t), t ∈ [t0, t1], it follows that g∗(u∗(t), t) =< x̃(t),u∗(t) > −g(x̃(t), t), ϕ∗(v∗(t1) − ξ∗(t1), t1) =< x̃(t1),v∗(t1) − ξ∗(t1) > −ϕ(x̃(t1), t1), wf (t)(ξ ∗(t)) =< ξ∗(t), x̃(t) >, t ∈ [t0, t1], (24) wm (v ∗(t1) −x∗(t1) − ξ∗(t1)) =< v∗(t1) −x∗(t1) − ξ∗(t1), x̃(t1) >, wa(x̃(t),x ∗(t), t) =< ˙̃x(t),x∗(t) >, t ∈ [t0, t1]. 7, 2(2005) optimization of differential inclusions ... 35 from relations (23), (24) and the proof of theorem 3.1(see (22)), we get the required result. 4 examples about the construction of the dual problem. let consider the following problem i(x(.), t1) = ϕ(x(t1), t1) → inf ẋ(t) = f(x(t),u(t)),u(t) ∈ u ⊂ rn, t ∈ [t0, t1], (25) x(t0) = x0, x(t1) ∈ m = {x1}, where f(x,u) is differentiable on x and a(x) = f(x,u) is convex. let us replace the problem (25) with the following: i(x(.), t1) → inf ẋ(t) ∈ a(x(t)) (26) x(t0) = x0, x(t1) = x1. it is obvious that wa(x,y ∗) = inf u∈u < y∗,f(x,u) > . (27) then,if ũ is a solution of the problem (27), and x̃ is a solution of the problem which is formulated in lemma 1.3, then the following relation is valid x∗ = f ′∗ x (x̃, ũ)y ∗, (28) where f ′∗ x is matrix conjugate to the matrix f ′ x. when f(t) ≡ r n and m = r n , wf (t) and wm in (24) show that ξ∗(t) = 0,v∗(t1) = x ∗(t1). (29) since g(x,t) ≡ 0 in the problem (25), then g∗(u∗, t) = { 0 ,u∗ = 0 ∞ ,u∗ �= 0. (30) considering ωa in various intervals (see(21)) and using (29) and (30), we obtain sup x∗(t) {−ϕ∗(x∗(t1), t1) + ∫ t1 t0 ωa(−ẋ∗(t),x∗(t), t)dt}. (31) from relations (28)-(30) we have −ẋ∗(t) = f ′∗x (x̃(t), ũ(t))x∗(t), t ∈ [t0, t1], (32) 36 e. n. mahmudov and g. çiçek 7, 2(2005) wa(x̃(t),x ∗(t)) =< x∗(t),f(x̃(t), ũ(t)) > . thus the dual problem is defined by the formulas (31) and (32). let us consider the problem with polyhedral mapping[1] a(x) = {y : ax−by ≤ d}, where a,b are (m×n) matrices and d is an m-dimensional column-vector. we compute easily that the lcm is given by a∗(y∗; (x̃, ỹ)) = {a∗λ : y∗ = b∗λ,λ ≥ 0,< ax̃−bỹ −d,λ >= 0}. using the last formula it is easy to show that the dual problem consists of the following: i∗(x ∗(.), t1) → sup, −ẋ∗(t) = a∗λ(t), t ∈ [t0, t1], x∗(t) = b∗λ(t), t ∈ [t0, t1], < ax̃(t) −b ˙̃x(t) −d,λ(t) >= 0, λ(t) ≥ 0, t ∈ [t0, t1]. received: april 2003. revised: december 2003. references [1] b. n. pshenichnyi, convex analysis and extremal problems, m., nauka, (1980). [2] r. t. rockafellar, convex analysis, princeton university press, new jersey, (princeton 1972). 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[10] a. d. ioffe, v. m. tihomirov, theory of extremal problems, m. : nauka, (1974). [11] r. vinter, optimal control systems & control foundations & application, birkhäuser boston inc. , ma. , (boston 2000). [12] b. s. mordukhovich, optimal control of difference, differentialdifference inclusions, j. math. sci. , (new york)100(2000), no. 6, 2613-2632. [13] f. h. clarke, yu. s. ledyaev, m. l. radulesev, approximate invariance and differential inclusions in hilbert spaces, j. dynam. control systems, 3(1997), no. 4, 493-518. [14] e. n. mahmudov, optimization of discrete inclusions with distributed parameters, optimization, 21(1990), no. 2, 197-207, berlin. [15] yu. g. borisovich et. al. , multivalued mappings, itogi nauki i tekniki: math. anal. , vol. 19, viniti moscow, (1982), 127-230, english transl. in j. soviet mat. , 26(1984), no. 4 [16] j. v. outrata, minimization of nonsmooth nonregular functions. application to discrete-time optimal control problems, problems of control and information theory, 13(1984), no. 6, 413-424. [17] r. vinter, h. h. zheng, necessary conditions for optimal control problems with state constraints, transactions of the american mathematical society, 350(3)(1998), 1181-1204. [18] r. vinter, h. h. zheng, necessary conditions for free end-time measurably time dependent optimal control problems with state constraints, setvalued anal. 8(2000), no. 1-2, 11-29. [19] b. s. mordukhovich, optimal control of nonconvex discrete and differential inclusions, sociedad matematica mexicana, mexico, (1998), vi+324. [20] f. h. clarce, optimization and nonsmooth analysis, (1983), john wiley & sons inc. [21] p. d. loewen, r. t. rockafellar, bolza problems with general time constraints, siam j. control optim. , 35(6)(1997), 2050-2069. 38 e. n. mahmudov and g. çiçek 7, 2(2005) [22] p. d. loewen, r. t. rockafellar, new necessary conditions for the generalized problem of bolza, siam j. control optim. , 34(5)(1996), 14961511. [23] p. d. loewen, r. t. rockafellar, optimal-control of unbounded differential inclusions, siam j. control optim. , 32(2)(1994), 442-470. [24] a. v. kryazhimskii, convex optimization via feedbacks, siam j. control optim. , 37(1)(1998), 37(1), 278-302. [25] e. n. mahmudov, duality in the problems of optimal control for systems described by convex differential inclusions with delay, problems of control and information theory, 16(6), (1987), 411-422, budapest. cubo, a mathematical journal vol.22, no¯ 01, (55–70). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100055 super-halley method under majorant conditions in banach spaces shwet nisha and p. k. parida department of applied mathematics, central university of jharkhand, ranchi-835205, india. shwetnisha1988@gmail.com, pkparida@cuj.ac.in abstract in this paper, we have studied local convergence of super-halley method in banach spaces under the assumption of second order majorant conditions. this approach allows us to obtain generalization of earlier convergence analysis under majorizing sequences. two important special cases of the convergence analysis based on the premises of kantorovich and smale type conditions have also been concluded. to show efficacy of our approach we have given three numerical examples. resumen en este art́ıculo, hemos estudiado la convergencia local del método super-halley en espacios de banach, asumiendo condiciones mayorantes de segundo orden. este punto de vista nos permite obtener generalizaciones de análisis de convergencia bajo sucesiones mayorantes obtenidos anteriormente. también se han concluido dos casos especiales del análisis de convergencia basados en las premisas de condiciones tipo kantorovich y smale. para mostrar la eficacia de nuestro enfoque, damos tres ejemplos numéricos. keywords and phrases: nonlinear equations; super-halley method; majorant conditions; local convergence; semilocal convergnce; smale-type conditions; kantorovich-type conditions. 2010 ams mathematics subject classification: 65d10, 65g99, 65k10, 47h17, 49m15, 47h99. http://doi.org/10.4067/s0719-06462020000100055 56 shwet nisha & p. k. parida cubo 22, 1 (2020) 1 introduction let f be a given operator that maps from some nonempty open convex subset ω of a banach space x to another banach space y. approximating a locally unique solution x̄ of a nonlinear equation f(x) = 0 (1.1) is widely studied in both theoretical and applied areas of mathematics. generally, this is done by using some iterative processes. an iterative process is a mathematical procedure that, from one or several initial approximations of a solution of (1.1), a sequence of iterates {xn}n∈n is constructed so that each subsequent iterate of the sequence is a better approximation to the previous approximation to the solution of (1.1); that is, the sequence {‖xn − x̄‖}n∈n is convergent to zero. usually, in order to study convergence analysis of the method, we could consider the study of local and semilocal convergence analysis. if the convergence analysis seeks assumptions around a solution x̄, then it is called local convergence and this type of analysis estimates the radii of convergence balls, where as we could also consider assumptions around an initial point x0 to study convergence analysis of the method. in that case the convergence analysis is called semilocal one and it gives criteria ensuring the convergence. it is also very important to give convergence ball of an iterative method, because that shows the extent to which we can choose an initial guesses for that method. one of the most important iterative methods to solve this problem is newton’s method given by xn+1 = xn − f ′(xn) −1f(xn), k = 0, 1, 2, . . . (1.2) where x0 ∈ ω is an initial point. as anybody can recall that one of the most famous results to study convergence of newton’s method (1.2) is the well known kantorovich method[16], which guarantees convergence of the method to a solution, using semilocal conditions. it does not require a priori existence of a solution, proving instead the existence of the solution and its uniqueness on some region. many researches have also done works related to kantorovich-like method (for details see [4, 8, 10, 26, 27, 29] and references there in). also, smale’s point theory [21] assumes that the nonlinear operator is analytic at the initial point, which is an important result concerning newton’s method. wan and han[25, 22] has discussed the generalization and the particular cases of smale’s point estimate theory. for a positive number α and x ∈ x, we consider b(x, α) to stand for the open ball with radius α and center x and b̄(x, α) is the corresponding close ball. in [7, 6], ferreira and svaiter had studied the local convergence of newton’s method (1.2) under the following majorant conditions: ‖f′(x̄)−1[f′(y) − f′(x)]‖ ≤ h′(‖y − x‖ + ‖x − x̄‖) − h′(‖x − x̄‖), (1.3) for x, y ∈ b(x̄, r), r > 0, where ‖y−x‖ +‖x− x̄‖ < r and h : (0, r) → r is a continuously differentiable, convex and strictly increasing function that satisfies h(0) > 0, h′(0) = −1 and has a zero in cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 57 (0, r). note that by this study the assumptions for guaranteeing q-quadratic convergence of the respective iterative methods has been relaxed and a new estimate of the q-quadratic convergence has been obtained. recently, inspired by these ideas, ling and xu[17] have presented a new convergence analysis of halley’s method which makes a relationship of the majorizing function h and the nonlinear operator f under majorant conditions similar to given in (1.3). argyros and ren[1] also presented a local convergence of halley’s method which gives a ball convergence of the method under assumptions similar to (1.3). on the other hand, one of the famous third order iterative method to solve nonlinear equation (1.1) in banach space is the super-halley method denoted by xn+1 = xn − [ i + 1 2 lf(xn)[i − lf(xn)] −1 ] f′(xn) −1f(xn), (1.4) where for x ∈ x, lf(x) is the linear operator defined as lf(x) = f ′(x)−1f′′(x)f′(x)−1f(x). the results concerning the convergence of this method have been studied in [2, 9, 20] under different types of assumptions by using recurrence relations. on the other hand ezquerro and hernández[5] and gutiérrez and hernández[11] have studied semilocal convergence of this method by using majorizing sequences. now, if the nonlinear operator f is analytic at the initial point then motivated by the ideas of argyros and ren[1] and ling and xu[17], we have studied local convergence of super-halley method (1.4) using second order majorant condition. this majorant condition generalizes the earlier results on super-halley method[5, 11] using majorizing sequences. two particular cases namely results based of affine invariant lipschitz-type condition and smaletype condition have also been derived. numerical efficacy of the method has also been derived by way of a number of numerical examples. rest of the paper is organized as follows. some preliminaries results are contained in section 2. in section 2.1, we studied local convergence analysis of super-halley method. two special cases of main result are presented in section 3. in section 4, we have shown a number of numerical examples to show efficacy of our study. section 5 forms the conclusion part of the paper. 2 preliminaries in this section we provide some basic results which is required for our convergence analysis of the method. assume a > 0 and φ : (0, a) → r be a twice continuously differentiable function. let x, y ∈ b(x̄, a) ⊂ ω, with ‖y − x‖ + ‖x − x̄‖ < a. we say that the operator f satisfy a second order 58 shwet nisha & p. k. parida cubo 22, 1 (2020) majorizing function φ at x̄ if the following conditions hold on f: ‖f′(x̄)−1[f′′(y) − f′′(x)]‖ ≤ φ′′(‖y − x‖ + ‖x − x̄‖) − φ′′(‖x − x̄‖), (2.1) with the assumptions: (m1) φ(0) > 0, φ′′(0) > 0, φ′(0) = −1, (m2) φ′′ is convex and strictly increasing in (0, a), (m3) φ has atleast one zero in (0, a) with t∗ as the smallest zero and φ′(t∗) < 0.    (2.2) and ‖f′(x̄)−1f(x̄)‖ ≤ φ(0), ‖f′(x̄)−1f′′(x̄)‖ ≤ φ′′(0). (2.3) in this paper we assume that φ is the majorizing function of f. note that if we define θf(x) := x − [ i + 1 2 lf(x)[i − lf(x)] −1 ] f′(x)−1f(x) (2.4) where lf(x) = f ′(x)−1f′′(x)f′(x)−1f(x), then θf(x) can be taken as the iterative function of superhalley method as it can be written as xn+1 = θf(xn). also the scalar sequence {tn} can be generated by applying the method to φ(t). in this case we can write tn+1 = θφ(tn) with θφ(t) := t − [ 1 + lφ(t) 2(1 − lφ(t)) ] φ(t) φ′(t) , lφ(t) = φ(t)φ′′(t) φ′(t)2 , t ∈ (0, a). (2.5) now we can easily establish some basic properties of the majorizing function φ, the iterative functions θf(x) and θφ(t) which are described in the following lemmas. lemma 2.1. let φ satisfies assumptions (m1) − (m3). then (i) φ′ is strictly convex and strictly increasing on (0, a). (ii) φ is strictly convex on (0, a), φ(t) > 0 for t ∈ (0, t∗) and equation φ(t) = 0 has at most one root in (t∗, a). (iii) −1 < φ′(t) < 0 for t ∈ (0, t∗). (iv) 0 ≤ lφ(t) ≤ 1 2 for t ∈ [0, t∗]. proof. the proof is similar to one given in [17], so omitted. lemma 2.2. let φ satisfies assumptions (m1)−(m3). then for all t ∈ (0, t∗), t < θφ(t) < t∗. moreover, φ′(t∗) < 0 if and only if there exist t ∈ (t∗, a) such that φ(t) < 0. cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 59 proof. it is not to be mentioned that by using lemma 2.1, one can have φ(t) > 0, −1 < φ′(t) < 0 and 0 ≤ lφ(t) ≤ 12 for t ∈ (0, t ∗). this gives [ 1 + lφ(t) 2(1 − lφ(t)) ] φ(t) φ′(t) < 0 and hence t < θφ(t). also, for any t ∈ (0, t∗), from the definition of directional derivative and assumption (m2) it follows that since φ′′(t) is increasing in (0, a) and t < t∗ < a, we have φ′′(t) < φ′′(t∗) and φ′′(t) > 0 which implies that left directional derivative d−φ′′(t) > 0. as φ′′(t)φ(t)−2φ′(t)2 ≤ −4φ′′(t)φ(t), we obtain d−θφ(t) = 1+ 2φ′(t)φ(t)d−φ′′(t) (φ′′(t)φ(t) − 2φ′(t)2)φ′′(t) > 0 for t ∈ (0, t∗). and this implies that θφ(t) < θφ(t ∗) = t∗ for any t ∈ (0, t∗). so the first part of this lemma is complete. now, if φ′(t∗) < 0, then it is obvious that there exists t ∈ (t∗, a) such that φ(t) < 0. conversely, noting that φ′(t∗) = 0, then we have φ(t) > φ(t∗)+φ′(t)(t∗ −t) for t ∈ (t∗, a), which implies that φ′(t∗) < 0. this completes the proof. remark following properties are implied by the condition φ′(t∗) < 0 in (m3): • φ(t∗∗) = 0 for some t∗∗ ∈ (t∗, a). • φ(t) < 0 for some t ∈ (t∗, a). lemma 2.3. let φ satisfies assumptions (m1) − (m3). then t∗ − θφ(t) ≤ [ 1 2 φ′′(t∗)2 φ′(t∗)2 + 1 3 d−φ′′(t∗) −φ′(t∗) ] (t∗ − t)3, t ∈ (0, t∗). (2.6) proof. we can derive the following relation, by using the definition of θφ in (2.5) t∗ − θφ(t) = 1 1 − lφ(t) [ (1 − lφ(t))(t ∗ − t) + φ(t) 2φ′(t) (1 − lφ(t)) + φ(t) 2φ′(t) ] = − 1 φ′(t)(1 − lφ(t) ∫1 0 [φ′′(t + σ(t∗ − t)) − φ′′(t)](t∗ − t)2(1 − σ)dσ + (t∗ − t)φ′′(t) 2(1 − lφ(t))φ ′(t)2 ∫1 0 φ′′(t + σ(t∗ − t))(t∗ − t)2(1 − σ)dσ − (t∗ − t)φ′′(t) 2(1 − lφ(t))φ ′(t) ( φ(t) + φ(t)2 φ′(t)(t∗ − t) ) since φ′′ is convex and t < t∗, it follows from lemma 2.1 that φ′′(t + σ(t∗ − t)) − φ′′(t) ≤ [φ′′(t∗) − φ′′(t)]σ(t ∗ − t) (t∗ − t) 60 shwet nisha & p. k. parida cubo 22, 1 (2020) so by noting that φ′′ is strictly increasing, we have t∗ − θφ(t) ≤ − φ′′(t∗) − φ′′(t) 6φ′(t)(1 − lφ(t)) (t∗ − t)2 + φ′′(t∗)φ′′(t) 4φ′(t)2(1 − lφ(t)) (t∗ − t)3 − φ(t)φ′′(t) 2(1 − lφ(t))φ ′(t) (t∗ − t) − φ(t)2φ′′(t) 2(1 − lφ(t))φ ′(t)2 since φ′(t) < 0, φ′′(0) > 0 and φ′, φ′′ are strictly increasing on (0, t∗) and 0 ≤ lφ(t) ≤ 12 for t ∈ [0, t∗] by lemma 2.1, so we have t∗ − θφ(t) ≤ [ 1 2 φ′′(t∗)2 φ′(t∗)2 + 1 3 d−φ′′(t∗) −φ′(t∗) ] (t∗ − t)3. (2.7) as φ′ is increasing, φ′(t∗) < 0 and φ′(t) < 0 t in (0, t∗), we have φ′′(t∗)) − φ′′(t) −φ′(t) ≤ φ ′′(t∗) − φ′′(t) −φ′(t∗) = 1 −φ′(t∗) φ′′(t∗)) − φ′′(t) t∗ − t (t∗ − t) ≤ d −φ′′(t∗) −φ′(t∗) (t∗ − t) where the last inequality follows from definitions of directional derivative. combining the above inequality with (2.7), we conclude that (2.6) holds. this completes the proof. � let {tk} denote the majorizing sequence generated by t0 = 0, tk+1 = θφ(tk) = tk − [ i + lφ(tk) 2(1 − lφ(tk)) ] φ(tk) φ′(tk) , k = 0, 1, 2, . . . (2.8) we arrive at the following theorem using lemma 2.3 that theorem 2.4. let the sequence {tk} be defined by (2.8). then {tk} is well defined, strictly increasing and is contained in (0, t∗). moreover, {tk} satisfies (2.6) and converges to t∗ with q − cubic, i.e.: t∗ − tk+1 ≤ [ 1 2 φ′′(t∗)2 φ′(t∗)2 + 1 3 d−φ′′(t∗) −φ′(t∗) ] (t∗ − tk) 3, tk ∈ (0, t∗). 2.1 local convergence results for super-halley method this section is devoted to giving the local convergence analysis of (1.4). for that the following lemmas will play important role. lemma 2.5. assume ‖x − x̄‖ ≤ t < t∗. if φ : (0, t∗) → r is a twice continuously differentiable function which majorizes f at x̄, then (i) f′(x) is nonsingular and ‖f′(x)−1f′(x̄)‖ ≤ − 1 φ′(‖x − x̄‖) ≤ − 1 φ′(t) . (2.9) cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 61 (ii) ‖f′(x̄)−1f′′(x)‖ ≤ φ′′(‖x − x̄‖) ≤ φ′′(t). proof. let us take x ∈ b(x̄, t), 0 ≤ t < t∗. since f′(x) = f′(x̄) + ∫1 0 [f′′(x̄ + σ(x − x̄)) − f′′(x̄)](x − x̄)dσ + f′′(x̄)(x − x̄), we get ‖i − f′(x̄)−1f′(x)‖ ≤ ∫1 0 ‖f′(x̄)−1[f′′(x̄ + σ(x − x̄)) − f′′(x̄)]‖‖x − x̄‖dσ +‖f′(x̄)−1f′′(x̄)‖‖x − x̄‖ ≤ ∫1 0 [φ′′(σ(‖(x − x̄)‖)) − φ′′(0)]‖x − x̄‖dσ + φ′′(0)‖x − x̄‖ = φ′(‖(x − x̄)‖) − φ′(0). so, we conclude that ‖i − f′(x̄)−1f′(x)‖ ≤ φ′(t) − φ′(0) < 1 as φ′(0) = −1 and −1 < φ′(t) < 0 for (0, t∗) using lemma 2.1. therefore, it follows from banach lemma that f′(x̄)−1f′(x) is nonsingular and (2.9) holds as ‖f′(x)−1f′(x̄)‖ ≤ 1 1 − φ′(‖x − x̄‖) + φ′(0) = − 1 φ′(‖x − x̄‖) ≤ − 1 φ′(t) . thus we conclude that, f′ is nonsingular in b(x̄, t∗). by using majorant conditions, we have ‖f′(x̄)−1f′′(x)‖ ≤ ‖f′(x̄)−1[f′′(x) − f′′(x̄)]‖ + ‖f′(x̄)−1f′′(x̄)‖ ≤ φ′′(‖x − x̄‖) − φ′′(0) + φ′′(0) = φ′′(‖x − x̄‖) ≤ φ′′(t). the last inequality holds true because of φ′′ is strictly increasing. this completes the proof. ✷ now the main local convergence result for the super-halley method (1.4) is presented as follows. theorem 2.6. let f satisfies the second order majorant conditions (2.1)-(2.3). then, the sequence of iterates {xn} generated by super-halley method (1.4) is well defined, contained in b(x̄, t∗) and converges to the unique solution x̄ of (1.1). moreover, the following error estimate hold ‖x̄ − xk+1‖ ≤ (t∗ − tk+1) ( ‖x̄ − xk‖ t∗ − tk )3 , k = 0, 1, 2, . . . (2.10) thus the sequence {xn} generated by super-halley method (1.4) converges q−cubic as follows ‖x̄ − xk+1‖ ≤ [ 1 2 h′′(t∗)2 h′(t∗)2 + 1 3 d−h′′(t∗) −h′(t∗) ] ‖x̄ − xk‖3, k = 0, 1, 2, . . . (2.11) 62 shwet nisha & p. k. parida cubo 22, 1 (2020) proof. by using f(x̄) = 0 and some standard analytic techniques, one can have x̄ − xk+1 = −γf(xk)f ′ (xk) −1 [−f′(xk)(x̄ − xk) − f(xk)] − γf(xk)lf(xk)(x̄ − xk) + [ 1 2 f′(xk) −1f(xk) − 1 2 γf(xk)f ′(xk) −1f(xk) ] = −γf(xk)f ′(xk) −1 ∫1 0 (1 − σ)[f′′(xk + σ(x̄ − xk)) − f ′′(xk)](x̄ − xk) 2dσ +f′′(xk)γf(xk)f ′(xk) −1 [ f′(xk) −1 ∫1 0 (1 − σ)f′′(xk + σ(x̄ − xk)) ×(x̄ − xk)2dσ ] (x̄ − xk) − 1 2 f′′(xk)γf(xk)f ′ (xk) −1 [ f′(xk) −1 ∫1 0 (1 − σ)f′′(xk + σ(x̄ − xk)) ×(x̄ − xk)2dσ ]2 , where γf(x) = (i − lf(x)) −1. using majorant condition, we can get ∫1 0 ‖f′(x̄)−1[f′′(xk + σ(x̄ − xk)) − f′′(xk)]‖(1 − σ)dσ ≤ ∫1 0 ‖[φ′′(σ‖x̄ − xk‖ + ‖xk − x̄‖) −φ′′(‖xk − x̄‖)]‖(1 − σ)dσ. also, we know that, if u, v, w ∈ (0, a) and u < v < w, then because of convexity[7] of φ(x) in (0, a), we have φ(v) − φ(u) ≤ [φ(w) − φ(u)] v − u w − u . therefore, φ′′(σ‖x̄ − xk‖ + ‖xk − x̄‖) − φ′′(‖xk − x̄‖) ≤ φ′′(σ‖x̄ − xk‖ + tk) − φ′′(tk) ≤ [φ′′(σ(t∗ − tk) + tk) − φ′′(tk)] ‖x̄ − xk‖ t∗ − tk . thus lemma 2.5 and above inequality implies ‖x̄ − xk+1‖ ≤ − 1 (1 − lφ(tk))φ ′(tk) [ ∫1 0 [φ′′(σ(t∗ − tk) + tk) − φ ′′(tk)](1 − σ)dσ ]‖x̄ − xk‖3 t∗ − tk + φ′′(tk) (1 − lφ(tk))φ ′(tk) 2 [ ∫1 0 [φ′′(σ(t∗ − tk) + tk)(1 − σ)dσ ] ‖x̄ − xk‖3 + 1 2 φ′′(tk) (1 − lφ(tk))φ ′(tk) 2 [ ∫1 0 [φ′′(σ(t∗ − tk) + tk)(1 − σ)dσ ]2 ‖x̄ − xk‖4 ≤ φ(tk) (1 − lφ(tk))φ ′(tk) (‖x̄ − xk‖ t∗ − tk )3 = (t∗ − tk+1) (‖x̄ − xk‖ t∗ − tk )3 . finally, we want to show that the solution x̄ of (1.1) is unique in b̄(x̄, t∗). for that assume ȳ be another solution in b̄(x̄, t∗). then proceeding similarly as above we get ‖ȳ − xk+1‖ ≤ (t∗ − tk+1) (‖ȳ − xk‖ t∗ − tk )3 . cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 63 since the sequence {xk} converges to x̄ and {tk} converges to t ∗, we conclude that ȳ = x̄. therefore, x̄ is the unique zero of (1.1) in b̄(x̄, t∗). ✷ remark 2.7. it is to be noted that if we replace x̄ with the initial approximation x0 in (2.1), then after some manipulations we can obtain a semilocal convergence analysis of our iteration method. this analysis approach enables us to drop out the assumption of existence of a second root for the majorizing function, still guarantee q−cubic convergence rate. thus the semilocal convergence theorem for the iteration method (1.4) is as follows: theorem 2.8. suppose f : ω ⊆ x → y be a twice continuously differentiable nonlinear operator, and ω is open and convex. consider that for a given initial guess x0 ∈ ω, f′(x0) is nonsingular that is f′(x0) −1 exists and that φ is the majorizing function to f at x0 and φ satisfies the assumptions (m1) − −(m3). then sequence {xk} generated by the method (1.4) for solving equation (1.1) with a starting point x0 is well defined, contained in b(x0, t ∗) and converges to a solution x̄ ∈ b̄(x0, t∗) of the eq.(1.1). the solution is unique in b(x0, σ), where σ is defined as σ := sup{t ∈ (t∗, r) : φ(t) ≤ 0}. for k = 0, 1, 2, . . . , a priori error estimate and a posteriori error estimate are given respectively as ‖x̄ − xk+1‖ ≤ (t∗ − tk+1) (‖x̄ − xk‖ t∗ − tk )3 , and ‖x̄ − xk+1‖ ≤ (t∗ − tk+1) (‖xk+1 − xk‖ tk+1 − tk )3 . also the method converges q−cubically as ‖x̄ − xk+1‖ ≤ [ 1 2 φ′′(t∗)2 φ′(t∗)2 + 1 3 d−φ′′(t∗) −φ′(t∗) ] (‖x̄ − xk‖)3. 3 special cases and applications this section consists of two special cases of the local convergence results obtained in previous section. namely, convergence results under affine covariant kantorovich-type condition and the smale-type γ-condition. 3.1 kantorovich-type suppose that f satisfies the affine covariant lipschitz condition (see han and wang[12]) as given by: ‖f′(x̄)−1[f′′(y) − f′′(x)]‖ ≤ λ1‖y − x‖, x, y ∈ ω. (3.1) and the following initial conditions ‖f′(x̄)−1f(x̄)‖ ≤ β, (3.2) 64 shwet nisha & p. k. parida cubo 22, 1 (2020) ‖f′(x̄)−1f′′(x̄)‖ ≤ λ2. (3.3) consider the scalar valued function φ(t) = λ1 6 t3 + λ2 2 t2 − t + β. (3.4) this function was considered as majorizing function in [28, 5, 11] for establishing convergence of super-halley method. if we choose the above cubic polynomial as the majorizing function φ in (2.1), then the majorant condition (2.1) reduced to the lipschitz condition (3.1) and in this way the results based on lipschitz condition have been generalized by our assumptions of majorant conditions. the assumptions (m1) and (m2) are satisfied for f if the following criterion holds β ≤ 2(λ2 + 2(λ 2 2 + 2λ1) 1/2) 3(λ2 + 2(λ 2 2 + 2λ1) 1/2)2 . (3.5) therefore, theorem 2.6 reduces to the following form: theorem 3.1. suppose that f satisfies the conditions (3.1)-(3.3) with the assumptions given in (3.5). then, the sequence {xk} generated by super-halley method (1.4) for solving equation (1.1) with a starting point x0 is well defined, contained in b(x̄, t ∗) and converges to a solution x̄ ∈ b̄(x̄, t∗) of the eq.(1.1). note that t∗ is the smallest positive root of φ defined by (3.4) in [0, r1] where r1 = (−λ2 + (λ 2 2 + 2λ1) 1/2)/λ1 is the positive root of φ ′. the limit x̄ of the sequence {xk} is the unique zero of eq.(1.1) in b(x̄, t∗∗), where t∗∗ is the root of φ in the interval (r1, +∞). moreover, the following error estimates holds ‖x̄ − xk+1‖ ≤ (t∗ − tk+1) (‖x̄ − xk‖ t∗ − tk )3 , k = 0, 1, 2, . . . and the sequence generated by super-halley method (1.4) converges q-cubic as follows ‖x̄ − xk+1‖ ≤ [ 3(λ1 + λ2t ∗)2 + 2λ2(1 − λ1t ∗ − λ2t ∗2/2) 6(1 − λ1t ∗ − λ2t ∗2/2)2 ] (‖x̄ − xk‖)3, k = 0, 1, 2, . . . 3.2 smale-type this subsection contains the local convergence results for the super-halley method (1.4) under the γ-condition. in [21], smale has studied the convergence and error estimation of newton’s method under the hypotheses that f is analytic and satisfies ‖f′(x̄)−1f(n)(x̄)‖ ≤ n!γn−1, n > 2 cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 65 where γ := sup k>1 ‖f ′(x̄)−1f(n)(x̄) k! ‖ 1 k−1 smale’s result is completely improved by wang and han[24, 25] by introducing a majorizing function φ(t) = β − t + γt2 1 − γt , t ∈ [ 0, 1 γ ) (3.6) where ‖f′(x̄)−1f(x̄)‖ ≤ β (3.7) and ‖f′(x̄)−1f′′(x̄)‖ ≤ 2γ (3.8) if we choose this function as the majorizing function φ in (2.1), then it reduces to the following condition: ‖f′(x̄)−1[f′′(y) − f′′(x)]‖ ≤ 2γ (1 − γ‖y − x‖ − γ‖x − x̄‖)3 − 2γ (1 − γ‖x − x̄‖)3 , γ > 0 (3.9) where ‖y − x‖ + ‖x − x̄‖ < 1 γ , and the assumptions (m1) and (m2) are satisfied for φ. also, if α := βγ < 3 − 2 √ 2, then assumption (m3) is satisfied for φ. therefore, the concrete form of theorem 2.6 is given as follows. theorem 3.2. suppose f satisfies (3.7)-(3.9). if α := βγ < 3 − 2 √ 2, then the sequence {xk} generated by the super-halley method (1.4) for solving the equation (1.1) with a starting point x0 is well defined, is contained in b(x̄, t∗) and converges to a solution x̄ ∈ b̄(x̄, t∗) of the eq.(1.1). the limit x̄ of the sequence {xk} is unique in b(x̄, t∗∗), where t∗ and t∗∗ are given as t∗ = α + 1 − √ (α + 1)2 − 8α 4γ and t∗∗ = α + 1 + √ (α + 1)2 − 8α 4γ respectively. moreover, the following error bound holds: for all k ≥ 0, we have ‖x̄ − xk+1‖ ≤ (t∗ − tk+1) (‖x̄ − xk‖ t∗ − tk )3 , k = 0, 1, 2, . . . and the sequence {xk} converges q-cubic as follows ‖x̄ − xk+1‖ ≤ 2γ2 [2(1 − γt∗)2 − 1]2 (‖x̄ − xk‖)3, k = 0, 1, 2, . . . 66 shwet nisha & p. k. parida cubo 22, 1 (2020) 4 numerical examples this section is devoted to illustrate the above theoretical results by a number of numerical examples. example 4.1. let x = y = r with ω = b(0, 1) and the function f on ω is f(x) = ex − 1 (4.1) and for x̄ = 0, f′(x̄) = 1, f′′(x̄) = 1. also, we obtain that λ1 = e, λ2 = 1, β = 0 therefore, the convergence criterion (3.5) holds and the theorem 3.1 is applicable to conclude that the sequence generated by super-halley method (1.4) with initial point x0 = 0.25 converges to a root of (4.1). in this case, we have t∗ = 0 and t∗∗ = 1.03304078, that is the existence and uniqueness ball are b(0.25, 0) and b̄(0.25, 1.03304078) respectively and the error bound is 1.525807581. example 4.2. let x = c[0, 1] the space of continuous functions defined on interval [0, 1] equipped with max norm and let ω = u[0, 1] and the function f on ω is . f(x)(s) = x(s) − 2λ ∫1 0 g(s, t)x(t)3dt (4.2) therefore we have f′(x)u(s) = u(s) − 6λ ∫1 0 g(s, t)x(t)2u(t)dt, u ∈ ω, and f′′(x)[uv](s) = −λ ∫1 0 g(s, t)x(t)(uv)(t)dt, u, v ∈ ω, now, let m = max s∈[0,1] ∫1 0 |g(s, t)|dt. then m = 1 8 .. also, for any x, y ∈ ω, we have ‖f′(x̄)−1[f′′(x) − f′′(x̄)]‖ ≤ 3|λ| 2 ‖x − x̄‖. so, we obtain the values of β, λ2 and λ1 in as follows β = 0, λ2 = 0, λ1 = 3|λ| 2 . therefore, the convergence criterion (3.5) holds and the theorem 3.1 is applicable to conclude that the sequence generated by super-halley method (1.4) with initial point x0 converges to a zero of f defined by (4.2). cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 67 for the different values of λ i.e. for λ = 0.0625, 0.125, 0.25, 0.5, 1 and the initial point x0 = 0.25 the corresponding domain of existence and uniqueness of solution, are given in table-4.2. table-4.2: domains of existence and uniqueness of solution for super-halley’s method λ convergence ball in our work existence uniqueness 0.0625 b̄(0.25, 0) b(0.25, 8) 0.125 b̄(0.25, 0) b(0.25, 5.656854249) 0.25 b̄(0.25, 0) b(0.25, 4) 0.5 b̄(0.25, 0) b(0.25, 2.828427125) 1 b̄(0.25, 0) b(0.25, 2) now, we present a numerical example to illustrate the smale-type conditions. example 4.3. let x = y = r with ω = u[0, 1] and the function f on ω is f(x) = ex + 2x2 − 1 (4.3) for x̄ = 0, f′(x̄) = 1, f′′(x̄) = 5 and we obtain that β = 0, γ = 2.5 therefore, the convergence criterion (3.9) holds (which can be seen from the above graph in case y > x) and the theorem 3.2 is applicable to conclude that the sequence generated by superhalley method (1.4 ) converges to a zero of f defined by (4.3) with t∗ = 0 and t∗∗ = 0.2. 68 shwet nisha & p. k. parida cubo 22, 1 (2020) 5 conclusions in this paper, the local convergence of super-halley method has been studied under majorant conditions on second derivative of f. convergence ball of the method has been included. two special cases: one kantorovich-type conditions and another smale-type conditions have also been studied. a number of numerical examples also given to illustrate our study. cubo 22, 1 (2020) super-halley method under majorant conditions in banach spaces 69 references [1] i. k. argyros and h. ren. ball convergence theorem for halley’s method in banach spaces. j. appl. math. comp. 38(2012)453-465. 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[29] t.j. ypma. affine invariant convergence results for newton’s method. bit numer. math. 22(1982)108-118. introduction preliminaries local convergence results for super-halley method special cases and applications kantorovich-type smale-type numerical examples conclusions cubo a mathematical journal vol.19, no¯ 03, (15–29). october 2017 periodicity and stability in neutral nonlinear differential equations by krasnoselskii’s fixed point theorem bouzid mansouri 1, abdelouaheb ardjouni1,2 and ahcene djoudi3 1faculty of sciences, department of mathematics univ annaba, p.o. box 12, annaba 23000, algeria 2faculty of sciences and technology, department of mathematics and informatics univ souk ahras, p.o. box 1553, souk ahras, 41000, algeria 3applied mathematics lab, faculty of sciences, department of mathematics univ annaba, p.o. box 12, annaba 23000, algeria mansouri.math@yahoo.fr, abd ardjouni@yahoo.fr abstract the nonlinear neutral functional differential equation with variable delay d dt u(t) − q(t) d dt g(u(t − r(t))) = p(t) − a(t)u(t) − a(t)q(t)g(u(t − r(t))) − b(t)f(u(t)) + b(t)q(t)f(u(t − r(t))). is investigated. by using krasnoselskii’s fixed point theorem we obtain the existence and the asymptotic stability of periodic solutions. sufficient conditions are established for the existence and the stability of the above equation. our results extend some results obtained in the work [19]. resumen la ecuación diferencial funcional no-lineal neutral con retardo variable d dt u(t) − q(t) d dt g(u(t − r(t))) = p(t) − a(t)u(t) − a(t)q(t)g(u(t − r(t))) − b(t)f(u(t)) + b(t)q(t)f(u(t − r(t))). es investigada. usando el teorema del punto fijo de krasnoselskii obtenemos la existencia y la estabilidad asintótica de las soluciones periódicas. se establecen condiciones suficientes para la existencia y la estabilidad de soluciones de la ecuación anterior. nuestros resultados extienden algunos de los resultados obtenidos en [19]. keywords and phrases: fixed point, periodic solutions, stability, neutral differential equations 2010 ams mathematics subject classification: primary 34k13, 34a34; secondary 34k30, 34l30 16 b. mansouri, a. ardjouni, a. djoudi cubo 19, 3 (2017) 1 introduction delay differential equations have received increasing attention during recent years since these equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering, see the monograph [8, 22] and the papers [1]-[7], [9]-[21], [23]-[26], [28]-[31] and the references therein. ding and li [19] discussed the existence and the stability of periodic solutions for the following neutral functional differential equation d dt u(t) − q d dt u(t − r) = p(t) − au(t) − aqu(t − r) − bf(u(t)) + bqf(u(t − r)). (1.1) by employing krasnoselskii’s fixed point theorem, the authors obtained existence and asymptotic stability results for periodic solutions. in this paper, we are interested on the existence and the asymptotic stability of periodic solutions of the following nonlinear neutral differential equation d dt u(t) − q(t) d dt g(u(t − r(t))) = p(t) − a(t)u(t) − a(t)q(t)g(u(t − r(t))) − b(t)f(u(t)) + b(t)q(t)f(u(t − r(t))), (1.2) where r is positive differentiable function, and q, p, a, b, f and g are continuously differentiable functions. to show the existence and the asymptotic stability of periodic solutions, we transform (1.2) into integral equation and then use krasnoselskii’s fixed point theorem. the obtained integral equation split in the sum of two mappings, one is a contraction and the other is compact. it is easy to see that (1.2) reduce to (1.1) when, q(t) = q, a(t) = a, b(t) = b, r(t) = r are constants and g(u(t − r(t))) = u(t − r) with |q| < 1. then, the results obtained here extend some results of the work of ding and li [19]. 2 existence of periodic solutions in this section, c1(r) or c(r) denotes the set of all continuously differentiable functions or all continuous functions φ : r → r respectively. cω = {φ ∈ c(r), φ(t + ω) = φ(t)} is with the supremum norm ‖.‖0 and c 1 ω = c 1(r) ∩ cω is with the norm ‖φ‖1 = ‖φ‖0 + ‖φ ′‖0 in a period interval. since we are searching for the existence of periodic solutions for (1.2), it is natural to assume that q(t + t) = q(t), p(t + t) = p(t), a(t + t) = a(t), b(t + t) = b(t), r(t + t) = r(t). function g(x) is globally lipschitz continuous. that is, there is a positive constant k such that |g(x) − g(y)| ≤ k‖x − y‖0, cubo 19, 3 (2017) periodicity and stability of neutral nonlinear differential equations 17 we also assume that g is continuously differentiable with ‖g′‖0 = l1. the next two lemmas will be used in the sequel. lemma 2.1. if a(t) 6= 0, f ∈ cω, then the scalar equation x ′(t) + a(t)x(t) = f(t) has a unique ω-periodic solution x(t) = ( 1 − e− ∫ t+ω t a(u)du )−1 ∫t+ω t e ∫ s t+ω a(u)duf(s)ds. proof. it is easy to prove. we can find it in many ode textbooks. lemma 2.2 ([27]). let m be a closed convex nonempty subset of a banach space (s,‖ · ‖). suppose that a and b map m into s such that (i) x,y ∈ m, implies ax + by ∈ m, (ii) a is compact and continuous, (iii) b is a contraction mapping. then there exists z ∈ m with z = az + bz. by applying lemmas 2.1 and 2.2, we obtain in this section the existence of periodic solution of (1.2). for a sufficiently small positive l, (1.2) can be transformed as d dt v(t) − lq1(t) d dt g(v(t − τ(t))) = lp1(t) − la1(t)v(t) − la1(t)q1(t)g(v(t − τ(t))) − lb1(t)f(v(t)) + lb1(t)q1(t)f(v(t − τ(t))), (2.1) where v(t) = u(lt), τ(t) = r(lt) l , q1(t) = q(lt), p1(t) = p(lt), a1(t) = a(lt), b1(t) = b(lt) and ω = t l . theorem 2.3. suppose that f ∈ c1(r) and q1,p1,a1,b1 ∈ c 1 ω, we also assume that ‖q ′ 1 ‖ = β, and ‖1 − τ′(t)‖ = l2, if there exist constants ρ ∈ (0,1) and h > 0 such that lk(‖q1‖0(1 + l2h) + β) ≤ ρ, sup |u|≤h |f(u)| < θ1h − θ3 ‖b1‖0 , ‖q1‖0 < θ1 − θ3+‖b1‖0 sup|u|≤h|f(u)| h θ2 + ‖b1‖0 sup|u|≤h|f(u)| h , (2.2) and ‖p1‖0 < (θ1 − θ2 ‖q1‖0)h − ‖b1‖0 (1 + ‖q1‖0) sup |u|≤h |f(u)| − θ3, (2.3) where θ1 = 1 l − kβ 1 + αωm(1 + l‖a1‖0) , θ2 = k + l1l2 1 + αωm(1 + l‖a1‖0) + 2k‖a2‖0 , θ3 = (‖q1‖0 + β) |g(0)| 1 + αωm(1 + l‖a1‖0) + 2‖a2‖0 ‖q1‖0 |g(0)| , 2a2(s)q1(s) = (l + 1)a1(s)q1(s) + q ′ 1(s), 18 b. mansouri, a. ardjouni, a. djoudi cubo 19, 3 (2017) and m = sup t∈r ∣ ∣ ∣ ∣ ∫t+ω t e ∫ s t+ω la1(u)duds ∣ ∣ ∣ ∣ , α = ( 1 − e− ∫ t+ω t la1(u)du )−1 . then (1.2) has a t-periodic solution. proof. according to the conditions (2.2) and (2.3) we get (1 + αωm(1 + l‖a1‖0))[l‖p1‖0 + l‖b1‖0(1 + ‖q1‖0) sup |u|≤h |f(u)| + 2l‖a2‖0‖q1‖0(kh + |g(0)|)] + l((k + l1l2)‖q1‖0 + kβ)h + l(‖q1‖0 + β)|g(0)| ≤ h. (2.4) we need to prove that (2.1) has a ω-periodic solution. let s = { φ ∈ c1(r), ‖φ‖1 = ‖φ‖0 + ‖φ ′‖0 < +∞ } , and m = {φ ∈ c1ω, ‖φ‖1 ≤ h}, then m is a bounded closed convex set of the banach space s. for all φ ∈ m, consider the nonhomogeneous equation d dt v(t) + la1(t)v(t) = lp1(t) − la1(t)q1(t)g(v(t − τ(t))) − lb1(t)f(v(t)) + lb1(t)q1(t)f(v(t − τ(t))) + lq1(t) d dt g(v(t − τ(t))). (2.5) according to the lemma 2.1, this equation has a unique ω-periodic solution v(t) = ( 1 − e− ∫ t+ω t la1(u)du )−1 ∫t+ω t e ∫ s t+ω la1(u)du[lp1(s) − la1(s)q1(s)g(v(s − τ(s))) − lb1(s)f(v(s)) + lb1(s)q1(s)f(v(s − τ(s))) + lq1(s) d ds g(v(s − τ(s)))]ds, performing an integration by part, we obtain v(t) = ( 1 − e− ∫ t+ω t la1(u)du )−1 ∫t+ω t e ∫ s t+ω la1(u)du[lp1(s) − l((l + 1)a1(s)q1(s) + q ′ 1(s))g(v(s − τ(s))) − lb1(s)f(v(s)) + lb1(s)q1(s)f(v(s − τ(s)))]ds + lq1(t)g(v(t − τ(t))), where 2a2(s)q1(s) = (l + 1)a1(s)q1(s) + q ′ 1 (s). then v(t) = ( 1 − e− ∫ t+ω t la1(u)du )−1 ∫t+ω t e ∫ s t+ω la1(u)du[lp1(s) − 2la2(s)q1(s)g(v(s − τ(s))) − lb1(s)f(v(s)) + lb1(s)q1(s)f(v(s − τ(s)))]ds + lq1(t)g(v(t − τ(t))). cubo 19, 3 (2017) periodicity and stability of neutral nonlinear differential equations 19 define operators a and b by (aφ)(t) = ( 1 − e− ∫ t+ω t la1(u)du )−1 ∫t+ω t e ∫ s t+ω la1(u)du[lp1(s) − 2la2(s)q1(s)g(φ(s − τ(s))) − lb1(s)f(φ(s)) + lb1(s)q1(s)f(φ(s − τ(s)))]ds, and (bφ)(t) = lq1(t)g(φ(t − τ(t))). in order to prove that (2.1) has a periodic solution, we shall make sure that a and b satisfy the conditions of lemma 2.2. for all x,y ∈ m, we have x(t+ω) = x(t), y(t+ω) = y(t) and ‖x‖1 ≤ h, ‖y‖1 ≤ h. now let us discuss ax + by. (ax)(t + ω) = ( 1 − e− ∫ t+2ω t+ω la1(u)du )−1 ∫t+2ω t+ω e ∫ s t+2ω la1(u)du[lp1(s) − 2la2(s)q1(s)g(x(s − τ(s))) − lb1(s)f(x(s)) + lb1(s)q1(s)f(x(s − τ(s)))]ds = ( 1 − e− ∫ t+ω t la1(u)du )−1 ∫t+ω t e ∫ s t+ω la1(u)du[lp1(s) − 2la2(s)q1(s)g(x(s − τ(s))) − lb1(s)f(x(s) + lb1(s)q1(s)f(x(s − τ(s)))]ds = (ax)(t), and (by)(t + ω) = lq1(t + ω)g(y(t + ω − τ(t + ω))) = lq1(t)g(y(t − τ(t)) = (by)(t), therefore (ax + by)(t + ω) = (ax + by)(t). meanwhile, we get (ax)′(t) = −la1(t) ( 1 − e− ∫ t+ω t la1(u)du )−1 ∫t+ω t e ∫ s t+ω la1(u)du[lp1(s) − 2la2(s)q1(s)g(x(s − τ(s))) − lb1(s)f(x(s)) + lb1(s)q1(s)f(x(s − τ(s)))]ds + [lp1(t) − 2la2(t)q1(t)g(x(t − τ(t))) − lb1(t)f(x(t)) + lb1(t)q1(t)f(x(t − τ(t)))], and (by)′(t) = lq′1(t)g(y(t − τ(t))) + lq1(t)(1 − τ ′ (t))y′(t − τ(t))g′(y(t − τ(t))). 20 b. mansouri, a. ardjouni, a. djoudi cubo 19, 3 (2017) thus, ‖ax‖1 = ‖ax‖0 + ‖(ax) ′‖0 ≤ (1 + αωm(1 + l‖a1‖0))[l‖p1‖0 + l‖b1‖0(1 + ‖q1‖0) sup |u|≤h |f(u)| + 2l‖a2‖0‖q1‖0(kh + |g(0)|)], and ‖by‖1 = ‖by‖0 + ‖(by) ′‖0 ≤ l((k + l1l2)‖q1‖0 + kβ)h + l(‖q1‖0 + β)|g(0)|, therefore ‖ax + by‖1 ≤ ‖ax‖1 + ‖by‖1 ≤ (1 + αωm(1 + l‖a1‖0))[l‖p1‖0 + l‖b1‖0(1 + ‖q1‖0) sup |u|≤h |f(u)| + 2l‖a2‖0‖q1‖0(kh + |g(0)|)] + l((k + l1l2)‖q1‖0 + kβ)h + l(‖q1‖0 + β)|g(0)|, by (2.4), ‖ax + by‖1 ≤ h. accordingly, ax + by ∈ m. for all x ∈ m, ‖ax‖0 ≤ h, ‖(ax) ′‖0 ≤ h. according to ascoli-arzila lemma, the subset am of cω is a precompact set, therefore for all sequence {xn} of m, there exists the subsequence {xnk} of {xn} such that axnk → x0 ∈ cω as k → +∞. meanwhile, we get, (ax)′′(t) = (l2a21(t) − la ′ 1(t)) ( 1 − e− ∫ t+w t la1(u)du )−1 ∫t+w t e ∫ s t+w la1(u)du[lp1(s) − 2la2(s)q1(s)g(x(s − τ(s))) − lb1(s)f(x(s)) + lb1(s)q1(s)f(x(s − τ(s)))]ds − la1(t)[lp1(t) − 2la2(t)q1(t)g(x(t − τ(t))) − lb1(t)f(x(t)) + lb1(t)q1(t)f(x(t − τ(t)))] + [lp′1(t) − 2la ′ 2(t)q1(t)g(x(t − τ(t))) − 2la2(t)q ′ 1(t)g(x(t − τ(t))) − 2la2(t)q1(t)(1 − τ ′(t))x′(t − τ(t))g′(x(t − τ(t))) − lb′1(t)f(x(t)) − lb1(t)x ′(t)f′(x(t)) + lb′1(t)q1(t)f(x(t − τ(t))) + lb1(t)q ′ 1(t)f(x(t − τ(t))) + lb1(t)q1(t)(1 − τ ′ (t))x′(t − τ(t))f′(x(t − τ(t)))]. cubo 19, 3 (2017) periodicity and stability of neutral nonlinear differential equations 21 thus sup t∈r |(ax) ′′ (t)| ≤ [ (l2‖a21‖0 + l‖a ′ 1‖0)αωm + l‖a1‖0 ] [l‖p1‖0 + l‖b1‖0(1 + ‖q1‖0) sup |u|≤h |f(u)|+2l‖a2‖0‖q1‖0(kh + |g(0)|)] + l‖p′1‖0 + l(‖b ′ 1‖0(1 + ‖q1‖0) + β‖b1‖0) sup |u|≤h |f(u)| + l(‖b1‖‖f ′‖0(1 + l2‖q1‖0) + 2l1l2‖a2‖0‖q1‖0)h + 2l(‖a′2‖0‖q1‖0 + β‖a2‖0)(kh + |g(0)|). therefore there is a constant h1 > 0 such that supt∈r |(ax) ′′(t)| ≤ h1 and {(ax) ′ : x ∈ m} ⊂ cω. according to ascoli-arzela lemma, for all sequence {xn} of m, there exists the subsequence {xnk} of {xn}, such that (axnk) ′ → z0 ∈ cω. since d dt is a closed operator, z0 = (x0) ′. hence, x0 ∈ c 1 ω and {axn} is contained in a compact set. a is a compact operator. suppose that {xn} ⊂ m, x ∈ s, xn → x, then ‖xn − x‖0 → 0 and ‖x ′ n − x ′‖0 → 0 as n → ∞, and we get ‖axn − ax‖0 ≤ αωm[2l‖a2‖0‖q1‖0k‖xn(t) − x(t)‖0 + l‖b1‖0(1 + ‖q1‖0) sup t∈[0,ω] |f(xn(t)) − f(x(t))|], and ‖(axn) ′ − (ax)′‖ 0 ≤ (lαωm‖a1‖0 + 1)[2l‖a2‖0‖q1‖0k‖xn(t) − x(t)‖0 + l‖b1‖0(1 + ‖q1‖0) sup t∈[0,ω] |f(xn(t)) − f(x(t))|], when ‖xn − x‖1 → 0 as n → ∞ and ‖xn(t) − x(t)‖0 → 0 for t ∈ [0,ω] uniformly. and since f is continuous, ‖axn − ax‖0 → 0, ‖(axn) ′ − (ax)′‖0 → 0. consequently, a is continuous. for all x,y ∈ m ‖bx − by‖1 = ‖bx − by‖0 + ‖(bx) ′ − (by)′‖0 ≤ lk(‖q1‖0(1 + l2h) + β)‖x − y‖0 ≤ ρ‖x − y‖0. therefore b is a contraction operator. thus, the conditions of lemma 2.2 are satisfied and there is a φ ∈ m, such that φ = aφ+bφ. it is ω-periodic solution for (2.1). since v(t) = u(lt), p1(t) = p(lt), q1(t) = q(lt), b1(t) = b(lt) and a1(t) = a(lt), (1.2) has a t-periodic solution. 22 b. mansouri, a. ardjouni, a. djoudi cubo 19, 3 (2017) 3 asymptotic stability of periodic solutions this section concerned with the asymptotic stability of periodic solutions. let u∗ is the equilibrium of (1.2). let v = u − u∗ then (1.2) is transformed as d dt v(t) − q(t) d dt (g(v(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) = −a(t)v(t) − a(t)q(t)(g(v(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) − b(t)(f(v(t) + u∗(t)) − f(u∗(t))) + b(t)q(t)(f(v(t − r(t)) + u∗(t − r(t))) − f(u∗(t − r(t)))). (3.1) obviously (3.1) has the zero solution. now we use krasnoselskii’s fixed point theorem to prove that the zero solution of (3.1) is asymptotically stable. we set s as the banach space of bounded continuous function φ : [m(0),∞) → r with the supremum norm ‖·‖ and m(0) = inf{t − r(t), t ≥ 0}. also, given the initial function ψ, denote the norm of ψ by ‖ψ‖ = supt∈[m(0),0] |ψ(t)|, which should not cause confusion with the same symbol for the norm in s. theorem 3.1. if all conditions of theorem 2.3 are satisfies, f satisfies the locally lipschitz condition. further assume that ∫t 0 a(u)du > 0, e− ∫ t 0 a(u)du → 0, t − r(t) → ∞ as t → ∞, and k‖q‖ < 1, and there exists r > h such that ‖b‖ sup |u|≤h+r |f(u)| < ( 1 δ − kβ ) r − (2kβ + θ1)h − θ3, (3.2) and that ‖q‖ < (1 − kδβ)r − 2kδβh − δ‖b‖ sup|u|≤h+r |f(u)| − δ(θ1h + θ3) (2δ‖a‖ + 1)kr + 4kδ‖a‖h + δ‖b‖ sup|u|≤h+r |f(u)| + δ(θ1h + θ3) , (3.3) and ‖ψ‖ ≤ {r − (kδ(2‖a‖ ‖q‖ + β) + k‖q‖)r − 2kδ(2‖a‖ ‖q‖ + β)h − δ‖b‖ (1 + ‖q‖) sup |u|≤h+r |f(u)| − δ(1 + ‖q‖)(θ1h + θ3)}/(1 + k‖q‖), (3.4) where sup t≥0 ∣ ∣ ∣ ∣ ∫t 0 e ∫ t −s a(u)duds ∣ ∣ ∣ ∣ ≤ δ. then the solution of (3.1) v(t) → 0 as t → ∞. cubo 19, 3 (2017) periodicity and stability of neutral nonlinear differential equations 23 proof. according to the conditions (3.2), (3.3) and (3.4), we have 2kδ(2‖a‖‖q‖ + β)h + [kδ(2‖a‖‖q‖ + β) + k‖q‖]r + δ‖b‖(1 + ‖q‖) sup |u|≤h+r |f(u)|+δ(1 + ‖q‖)(θ1h + θ3) + (1 + k‖q‖)‖ψ‖ ≤ r. (3.5) given the initial function ψ, there exists a unique solution v for (3.1). let mψ = {φ ∈ s, ‖φ‖ ≤ r, φ(t) = ψ(t) for t ∈ [m(0),0], |φ(t)| → 0 as t → ∞}, then mψ is a bounded convex closed set of s. we write (3.1) as d dt v(t) + a(t)v(t) = −a(t)q(t)(g(v(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) + q(t) d dt (g(v(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) − b(t)(f(v(t) + u∗(t)) − f(u∗(t))) + b(t)q(t)(f(v(t − r(t)) + u∗(t − r(t))) − f(u∗(t − r(t)))). (3.6) according to lemma 2.1 this equation has a unique solution written as v(t) = v(0)e− ∫ t 0 a(u)du + ∫t 0 e− ∫ t s a(u)du [−a(s)q(s)(g(v(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) + q(t) d ds (g(v(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) − b(s)(f(v(s) + u∗(s)) − f(u∗(s))) + b(s)q(s)(f(v(s − r(s)) + u∗(s − r(s))) − f(u∗(s − r(s))))]ds. performing an integration by part, we obtain v(t) = (v(0) − q(0)(g(v(−r(0)) + u∗(−r(0))) − g(u∗(−r(0))))e− ∫ t 0 a(u)du + q(t)(g(v(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) + ∫t 0 e− ∫ t s a(u)du[−(2a(s)q(s) + q′(s)) × (g(v(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) − b(s)(f(v(s) + u∗(s)) − f(u∗(s))) + b(s)q(s)(f(v(s − r(s)) + u∗(s − r(s))) − f(u∗(s − r(s))))]ds. 24 b. mansouri, a. ardjouni, a. djoudi cubo 19, 3 (2017) let v(t) = ψ(t), t ∈ [m(0),0], and (lφ)(t) = (ψ(0) − q(0)(g(ψ(−r(0)) + u∗(−r(0))) − g(u∗(−r(0))))e− ∫ t 0 a(u)du + q(t)(g(φ(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) + ∫t 0 e− ∫ t s a(u)du[−(2a(s)q(s) + q′(s)) × (g(φ(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) − b(s)(f(φ(s) + u∗(s)) − f(u∗(s))) + b(s)q(s)(f(φ(s − r(s)) + u∗(s − r(s))) − f(u∗(s − r(s))))]ds. for all φ ∈ mψ, define the operators a and b by (aφ)(t) =    0, t ∈ [m(0),0], ∫t 0 e− ∫ t s a(u)du[−(2a(s)q(s) + q′(s)) ×(g(φ(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) −b(s)(f(φ(s) + u∗(s)) − f(u∗(s))) +b(s)q(s)(f(φ(s − r(s)) + u∗(s − r(s)))]ds, t ≥ 0, (bφ)(t) =    ψ(t), t ∈ [m(0),0], (ψ(0) − q(0)(g(ψ(−r(0)) + u∗(−r(0))) − g(u∗(−r(0))))e− ∫ t 0 a(u)du +q(t)(g(φ(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))), t ≥ 0. (i) for all x,y ∈ mψ, x(t) → 0 and y(t) → 0 as t → ∞ then (by)(t) = (ψ(0) − q(0)(g(ψ(−r(0)) + u∗(−r(0))) − g(u∗(−r(0))))e− ∫ t 0 a(u)du + q(t)(g(y(t − r(t)) + u∗(t − r(t)) − g(u∗(t − r(t)))) → 0 as t → ∞, and lim t→∞ (ax)(t) = lim t→∞ {∫t 0 e ∫ s 0 a(u)du [−(2a(s)q(s) + q′(s)) × (g(x(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) − b(s)(f(x(s) + u∗(s)) − f(u∗(s))) +b(s)q(s)(f(x(s − r(s)) + u∗(s − r(s))) − f(u∗(s − r(s))))]ds}/e ∫ t 0 a(u)du = 0, cubo 19, 3 (2017) periodicity and stability of neutral nonlinear differential equations 25 therefore limt→∞(ax + by)(t) = 0, and ‖ax‖ ≤ { (2‖a‖‖q‖ + ‖q′‖)[ sup |x|≤h+r |g(x)| + sup |x|≤h |g(x)|] +‖b‖(1 + ‖q‖)[ sup |x|≤h+r |f(x)| + sup |x|≤h |f(x)] } sup t≥0 ∣ ∣ ∣ ∣ ∫t 0 e− ∫ t s a(u)duds ∣ ∣ ∣ ∣ ≤ 2kδ(2‖a‖‖q‖ + β)h + kδ(2‖a‖‖q‖ + β)r + δ‖b‖(1 + ‖q‖) sup |x|≤h+r |f(x)| + δ(1 + ‖q‖)(θ1h + θ3), and ‖by‖ = sup t≥m(0) |(by)(t)| = max{‖ψ‖,sup t≥0 |(ψ(0) − q(0)(g(ψ(−r(0)) + u∗(−r(0))) − g(u∗(−r(0))))e− ∫ t 0 a(u)du + q(t)(g(y(t − r(t)) + u∗(t − r(t)) − g(u∗(t − r(t))))|} ≤ (1 + k‖q‖)‖ψ‖ + k‖q‖r. thus ‖ax + by‖ ≤ ‖ax‖ + ‖by‖ ≤ 2kδ(2‖a‖‖q‖ + β)h + [kδ(2‖a‖‖q‖ + β) + k‖q‖]r + δ‖b‖(1 + ‖q‖) sup |x|≤h+r |f(x)| + δ(1 + ‖q‖)(θ1h + θ3) + (1 + k‖q‖)‖ψ‖. according to the condition (3.5), ‖ax + by‖ ≤ r. thus ax + by ∈ mψ. (ii) for all x ∈ mψ, ‖x‖ ≤ r, |(ax) ′(t)| = 0, t ∈ [m(0),0], |(ax)′(t)| ≤ (1 + ‖a‖δ)[k(2‖a‖‖q‖ + β)(2h + r) + ‖b‖(1 + ‖q‖) sup |x|≤h+r |f(x)| + (1 + ‖q‖)(θ1h + θ3)], here, the derivative of (ax)′(t) at zero means the left hand side derivative when t ∈ [m(0),0] and the right hand side derivative when t ≥ 0. we can see |(ax)′(t)| is bounded for all ψ ∈ mψ and amψ is a precompact set of s. therefore a is compact. let (xn) ⊂ mψ, x ∈ s, xn → x as n → ∞, then |xn(t) − x(t)| → 0 uniformly for t ≥ m(0) as n → ∞. since ‖axn − ax‖ ≤ δ[k(2‖a‖‖q‖ + β)‖xn − x‖ + k1‖b‖(1 + ‖q‖)‖xn − x‖], and f is continuous therefore, ‖axn − ax‖ → 0 as n → ∞ and a is continuous. 26 b. mansouri, a. ardjouni, a. djoudi cubo 19, 3 (2017) (iii) for all x,y ∈ mψ, ‖bx − by‖ ≤ k‖q‖‖x − y‖, therefore b is a contraction operator. according to krasnoselskii’s fixed point theorem, there is a φ ∈ mψ such that (a+b)φ = φ and φ is a solution for (3.1). because the solution through ψ for the equation is unique, the solution v(t) → 0 as t → ∞. when f and g satisfy the locally lipschitz condition, h in theorem 2.3 and r in theorem 3.1 exist, there are constant p,k > 0 such that |f(v(t) + u∗(t)) − f(u∗(t))| < p|v(t)| and |g(v(t) + u∗(t)) − g(u∗(t))| < k|v(t)|. since φ satisfies φ(t) = (ψ(0) − q(0)(g(ψ(−r(0)) + u∗(−r(0))) − g(u∗(−r(0))))e− ∫ t 0 a(u)du + q(t)(g(φ(t − r(t)) + u∗(t − r(t))) − g(u∗(t − r(t)))) + ∫t 0 e− ∫ t s a(u)du[−(2a(s)q(s) + q′(s)) × (g(φ(s − r(s)) + u∗(s − r(s))) − g(u∗(s − r(s)))) − b(s)(f(φ(s) + u∗(s)) − f(u∗(s))) + b(s)q(s)(f(φ(s − r(s)) + u∗(s − r(s))) − f(u∗(s − r(s))))]ds, then ‖φ‖ ≤ (1 + k‖q‖)‖ψ‖ + k‖q‖‖φ‖ + δ[k(2‖a‖‖q‖ + β)‖φ‖ + ‖b‖(1 + ‖q‖)p‖φ‖], that is [1 − k‖q‖ − δk(2‖a‖‖q‖ + β) − δ‖b‖(1 + ‖q‖)p]‖φ‖ ≤ (1 + k‖q‖)‖ψ‖. then there clearly exists a σ > 0 for each ǫ > 0 such that |φ(t)| < ǫ for all t ≥ m(0) if ‖ψ‖ < σ. thus we have the following theorem. theorem 3.2. if p and k satisfy 1 − k‖q‖ − δk(2‖a‖‖q‖ + β) − δ‖b‖(1 + ‖q‖)p > 0. then the zero solution for (3.1) is stable. 4 conclusion in this paper, we provided the existence and asymptotic stability of periodic solutions with sufficient conditions for nonlinear neutral differential equations. the main tool of this paper is the method of fixed points. however, by introducing new fixed mappings, we get new existence and stability conditions. the obtained results have a contribution to the related literature, and they improve cubo 19, 3 (2017) periodicity and stability of neutral nonlinear differential equations 27 and extend the results in [19] from the cases of linear neutral term with constant delay to that general nonlinear cases with variable delay. acknowledgement. the authors would like to thank the anonymous referee for his/her valuable comments and good advice. references [1] a. ardjouni and a. djoudi, existence of periodic solutions for a second-order nonlinear neutral differential equation with variable delay, palestine journal of mathematics, vol. 3(2) (2014), 191–197. 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[31] g. zhang, s. cheng, positive periodic solutions of non autonomous functional differential equations depending on a parameter, abstr. appl. anal. 7 (2002) 279–286. introduction existence of periodic solutions asymptotic stability of periodic solutions conclusion cubo, a mathematical journal vol.22, n◦03, (289–297). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300289 received: 05 september, 2019 | accepted: 15 august, 2020. topological algebras with subadditive boundedness radius m. sabet 1 and r. g. sanati 2 1department of mathematics payame noor university, tehran, iran. 2institute of higher education of acecr (academic center of education and culture research), rasht branch, iran. sabet.majid@gmail.com, reza sanaaty@yahoo.com abstract let a be a topological algebra and β a subadditive boundedness radius on a. in this paper we show that β is, under certain conditions, automatically submultiplicative. then we apply this fact to prove that the spectrum of any element of a is non-empty. finally, in the case when a is a normed algebra, we compare the initial normed topology with the normed topology τβ, induced by β on a, where β −1(0) = 0. resumen sea a un álgebra topológica y β un radio de acotamiento subaditivo en a. en este art́ıculo mostramos que β es, bajo ciertas condiciones, automáticamente submultiplicativo. luego aplicamos este hecho para probar que el espectro de cualquier elemento de a es no-vaćıo. finalmente, en el caso cuando a es una álgebra normada, comparamos la topoloǵıa normada inicial con la topoloǵıa normada τβ, inducida por β en a, donde β−1(0) = 0. keywords and phrases: topological algebra, strongly sequential algebra, boundedness radius. 2020 ams mathematics subject classification: 46h05; 46h20 c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300289 290 m. sabet and g. sanati cubo 22, 3 (2020) 1 introduction and preliminaries a topological algebra is an algebra a which is a topological vector space in such a way that the ring multiplication in a is separately continuous. (i.e., continuous in each one of the two variables the latter operation being a map of a × a into a.) if the multiplication of a given topological algebra a is in both variables (i.e., jointly) continuous, we say that a is a topological algebra with a continuous multiplication. (see e.g., [9, p. 4. definition 1.1.]) in what follows the topological algebras cosidered are supposed with a continuous multiplication and to be hausdorff. among topological algebras, the normed ones have been studied intensively by many mathematicians where the norm is used as a useful tool in measuring the distance between two elements. but, since 1940, there has been a considerable interest in investigating other topological algebras in absence of any norm and this led to the introduction of some famous topological algebras such as locally bounded algebras, locally convex algebras, locally multiplicatively convex algebras, et cetera. the common idea to study a non-normed topological algebra, say a, is to substitute the role of a norm on a, which determines the topology, in an appropriate way. for instance, aoki [2] proved that on a locally bounded algebra a, there is a p-norm generating the original topology on a. later, zelazko [12] applied this fact on locally bounded algebras, and obtained many classical results, known in the context of normed algebras. for example, in a complete locally bounded topological algebra a, the operator x 7→ x−1 on inv(a) (the group of all invertible elements of a) is continuous. also, the cohen factorization theorem holds whenever a has a bounded approximate identity. the role of norm in a locally convex topological algebra a is played by a separating family of submultiplicative seminorms generating the topology on a. recall that a seminorm is a non-negative real-valued function p on a such that (i) p(x + y) ≤ p(x) + p(y) (ii) p(αx) = |α| p(x) for all x and y in a and α in c. we say that p is submultiplicative seminorm if in addition p(xy) ≤ p(x)p(y) for all x and y in a. hence, there is a good motivation for mathematicians to extend notions and theorems from normed algebras to other topological algebras. a locally convex algebra is a topological algebra a whose the underlying topological vector space is a locally convex space. the topology of a cubo 22, 3 (2020) topological algebras with subadditive boundedness radius 291 such topological algebra is determined by a family of (non-zero) seminorms. in the case when the seminorms are submultiplicative, a is called a locally m-convex algebra. let a be a unital topological algebra with the unit e and x ∈ a. the spectrum of x, denoted by σ(x), is defined σ(x) = {λ ∈ c : λe − x /∈ inv(a)}. the spectral radius of x is the defined as ρ(x) = sup{|λ| : λ ∈ σ(x)}. it is well known that ρ(x) = inf{‖xn‖ 1 n : n ∈ n} for every element x in a banach algebra. using this fact, allan [1], introduced the notion of boundedness radius in a topological algebra a as follows β(x) = inf{r > 0 : (( xn rn ))n → 0} , (inf φ = +∞). allan attempted to compare β(x) with ρ(x) and in one of the obtained results, shows that β(x) is equal to ρ(x) in a complete locally convex algebra [1, theorem 3.12], specially, in a banach algebra we have ρ(x) = lim n→∞ ‖xn‖ 1 n = inf{‖xn‖ 1 n : n ∈ n} = β(x). oubbi [11] investigated on ρ and β and compared them together. in a topological algebra a, he showed that ρ ≤ β if and only if ∑ xn is convergent whenever x ∈ a and β(x) < 1. also, he showed β(xn) = β(x)n for all n ∈ n and β(xy) ≤ β(x)β(y) whenver x, y ∈ a, xy = yx. in any topological algebra, it is clear that β(λx) = |λ|β(x) (x ∈ a, λ ∈ c). let a b a topological algebra. the boundedness radius β is said to be subadditive, if for each x, y ∈ a, β(x + y) ≤ β(x) + β(y). moreover, β is called submultiplicative whenever β(xy) ≤ β(x)β(y). kinani, oubbi and oudadess [8], proved that in a unital and commutative locally convex algebra, β is subadditive and submultiplicative. [8, proposition ii.9.] in this paper, we show that in a topological algebra, if β is finite and subadditive, then it is submultiplicative. (see corollary 2.6.) also, we refer to topological algebras in which β defines a norm on them. (see theorem 2.10 and theorem 2.11.) actually, we consider a topological algebra a with boundedness radius β such that β satisfies the following conditions: (1) β−1(0) = 0 (2) ∀x ∈ a, β(x) < ∞ (3) ∀x, y, β(x + y) ≤ β(x) + β(y). 292 m. sabet and g. sanati cubo 22, 3 (2020) 2 the normed topology τβ induced by the boundedness radius β in a locally convex algebra, amongst the important results is the following. theorem 2.1. let a be a unital and commutative locally convex algebra, then β is subadditive. proof. see [8, proposition ii.9]. in this section, we first give an example to show that we can not drop commutativity in theorem 2.1 and we also prove that locally convexity of an algebra is not sufficient for β to be subadditive. finally, we consider topological algebras in which β is subadditive and we show that in such algebras, β is automatically submultiplicative. example 2.2. let a = ( 0 0 1 0 ) and b = ( 0 1 0 0 ) be elements of the noncommutative algebra a = m2(c). since a and b are nilpotent elements of a, β(a) = β(b) = 0, on the other hand β(a + b) = 1 because a + b = ( 0 1 1 0 ) and so (a + b)2 = ( 1 0 0 1 ) = i, now we have (β(a + b))2 = β((a + b)2) = β(i) = 1. which implies that β(a + b) = 1. hence β(a + b) � β(a) + β(b). this shows that we can not drop commutativity in theorem 2.1. let a be a topological algebra and b = a × c is equipped with the following multiplication (x1, α1)(x2, α2) = (α1x2 + α2x1, α1α2) (x1, x2 ∈ a, α1, α2 ∈ c). the unitization b of a is a topological algebra under the product topology of b = a × c and we have: theorem 2.3. the boundedness radius β is subadditive in b. proof. it is enough to show that for every z = (x, α) ∈ b , β(z) = |α|. let z = (x, α) ∈ b. if α = 0 then (x, 0) is a nilpotent element of b and β(z) = 0 = |α|. for α 6= 0, suppose that n ∈ n, ε > 0. let r ∈ (|α|, |α| + ε) then 1 rn αn−1 → 0 and so 1 rn αn−1x → 0. on the other hand, (x, α)n = (nxαn−1, αn). therefore 1 (r + ε)n (x, α)n = 1 (r + ε)n (nxαn−1, αn) = (n( r r + ε )n xαn−1 rn , αn (r + ε)n ) → (0, 0). cubo 22, 3 (2020) topological algebras with subadditive boundedness radius 293 this shows that β(z) ≤ r + ε < |α| + 2ε, since ε is arbitrary, β(z) ≤ |α|. for the converse, note that there exists r > 0 such that r < β(x, α) + ε and (x, α)n rn = ( nxαn−1 rn , αn rn ) → (0, 0). hence, α n rn convergent to zero. so |α| < r < β(x, α)+ ε, it follows that β(α) = |α| < β(z)+ ε. thus |α| ≤ β(z). definition 2.4. let a be an algebra. we say that a seminorm p on a has square property, if it is square-preserving, namely, p(x2) = p(x)2 for all x ∈ a. dedania in 1998 [6] proved the following theorem. theorem 2.5. let a be an algebra and a seminorm p on a which has the square property. then p is submultiplicative. according to the terminology in [4, p. 437, (6)] and due to theorem 2.5, a seminorm p. as in the latter theorem, is finally a uniform seminorm. corollary 2.6. let a be a topological algebra, such that β(x) < ∞ and β(x + y) ≤ β(x) + β(y) for all x, y ∈ a. then β is submultiplicative. proof. since β(x2) = β(x)2, the assertion follows from theorem 2.5. let a be a topological algebra such that β satisfies conditions (1)-(3) then β is a norm on a and by corollary 2.6, β is a submultiplicative norm. through this section, τβ denotes the topology, induced by β on a. now we face the following questions. question 1. is there any topological algebra for which β satisfies the conditions (1)-(3)? question 2. let (a, τ) be a topological algebra for which β satisfies the conditions (1)-(3). what is then the relation between τ and τβ? question 3. let a be a complete topological algebra such that β satisfies the conditions (1)-(3). does the normed algebra (a, τβ) is a banach algebra? in what follows, we are going to answer to these questions. theorem 2.7. let a be a unital and commutative semisimple banach algebra, then β satisfies (1) − (3). proof. since a is a commutative locally convex algebra, then, by [8, lemma 2.9] β(x + y) ≤ β(x) + β(y) for all x, y in a. 294 m. sabet and g. sanati cubo 22, 3 (2020) since a is a banach algebra then a is locally bounded and using [3, lemma 3.4], β(x) ≤ ‖x‖ for all x in a, hence β satisfies condition (2). a is a commutative semisimple banach algebra, using corollary 7.(iv) in [5] we have ρ−1(0) = 0. because a is a banach algebra, β(x) = ρ(x) for each x ∈ a. so β−1(0) = 0. it follows that β satisfies (1)-(3). theorem 2.7 gives a category of algebras satisfying conditions (1)-(3) and an affirmative answer to question 1. definition 2.8. a topological algebra a is called strongly sequential if there is a neighborhood u of zero such that, for all x ∈ u, (xk)k∈n converges to zero. lemma 2.9. let a be a topological algebra. then β is continuous at zero if and only if a is strongly sequential. proof. see [3, proposition 3.1]. in order to answer question 2, we give the following theorem. theorem 2.10. let (a, τ) be a topological algebra. suppose that β satisfies (1)-(3). then (a, τ) is strongly sequential if and only if τβ ⊆ τ. proof. suppose (a, τ) is strongly sequential. by lemma 2.9, β is continuous at zero. since β is subadditive, it is continuous on a. on the other hand, {β−1(0, 1 n ) : n ∈ n} is a local base for the normed topology τβ. therefore τβ ⊆ τ. conversely, let xα → 0 in τ. since τβ ⊆ τ, xα converges to zero in τβ which implies that β(xα) → 0. hence β is continuous at zero and, again by lemma 2.9, a is strongly sequential. in order to answer question 3, first we characterize a topological algebra for which (a, τβ) is a banach algebra and then we apply this characterization to give a negative answer to question 3. it is well known (see e.g., [10, p. 41]) that in a commutative c∗-algebra, the unique c∗-norm is the spectral norm i.e. ‖x‖ = ρ(x) (x ∈ a) (2.1) on the other hand, for each x ∈ a , ρ(x) = β(x) and so β is indeed the c∗-norm on a. thus (a, τβ) is a complete normed algebra whenever a is a c ∗-algebra. the following theorem gives a more general characterization of topological algebras for which (a, τβ) is complete. in the sequel, by an f-algebra we mean a completely metrizable topological algebra. cubo 22, 3 (2020) topological algebras with subadditive boundedness radius 295 theorem 2.11. let (a, τ) be a strongly sequential f-algebra. suppose that β satisfies (1)-(3). then (a, τβ) is a banach algebra if and only if τβ = τ. proof. if τβ = τ, then (a, τβ) is a banach algebra, trivially. for the converse, suppose that (a, τβ) is a banach algebra. by the assumption, (a, τ) is a strongly sequential f-algebra, so, theorem 2.10 implies, τβ ⊆ τ. from the open mapping theorem, one immediately gets that τβ = τ. the following example shows that the answer of question 3 is not affirmative. example 2.12. let a be the set of c1-functions on the interval [0, 1] and f ∈ a. then a is a semisimple commutative banach algebra where the norm on a is given by ‖f‖ = ‖f‖∞ + ‖f ′‖∞ (see [10, p. 10, example 1.2.6]). thus β satisfies (1)-(3) and so (a, τβ) is normed algebra. if (a, τβ) is complete, then a is a banach algebra, also it is a strongly sequential algebra (see e.g., [7, p. 58, example 3.26]). now, by theorem 2.11, τβ = τ. let x be the identity map on [0, 1]. since β(x) = 1 , the sequence ( xn n )n convergence to zero in τβtopology. but xn n 9 0 in the original topology τ. this is a contradiction and so (a, τβ) is not complete. as we mentioned, if β satisfies (1)-(3), then it is an algebraic norm on a. but in the absence of one of the properties (1)-(3), β is not a norm necessarily. in what follows, we concentrate to study topological algebras for which β is not a norm. lemma 2.13. let (a, ‖ ‖) be a normed algebra such that, ‖a‖2 = ‖a2‖ for all a ∈ a. then a is commutative. proof. see [5, p. 77, corollary 8, see also the comments after it]. theorem 2.14. let a be a topological algebra such that (1) β(x) < ∞ (x ∈ a) (2) β(x + y) ≤ β(x) + β(y) (x, y ∈ a). then β(xy − yx) = 0 for all x, y in a. proof. let n = {x ∈ a, β(x) = 0}. it is clear that n is an ideal in a and a/n is a normed algebra with the norm ‖x + n‖ = β(x). since β has the square property, ‖(x + n)2‖ = ‖x + n‖2 for all x in a. now, according to lemma 2.13, the normed algebra a/n is commutative. hence, (x + n)(y + n) = (y + n)(x + n) for all x, y in a which means that β(xy − yx) = 0. corollary 2.15. let a be a topological algebra and β satisfies (1)-(3), then a is commutative. 296 m. sabet and g. sanati cubo 22, 3 (2020) proof. since β(x2) = β(x)2, the assertion follows from lemma 2.13 or theorem 2.14. lemma 2.16. let a be a banach algebra and x, y ∈ a. if xy = yx, xox = yoy (xoy = x+y −xy) and β(x + y) < 2, then x = y. proof. see [5, p. 44, lemma 12]. lemma 2.17. let a be a normed algebra and x, y ∈ a. if xy = yx, xox = yoy and β(x + y) < 2, then x = y. proof. let ã be the completion of a. suppose that t is an isometric isomorphism of a onto ã. since (t x)(t y) = (t y)(t x), (t x)o(t y) = (t y)o(t x) and βã(t x+t y) = βã(t (x+y)) = βa(x+y) < 2 by lemma 2.16, t x = t y and so x = y. theorem 2.18. let a be a topological algebra and x, y ∈ a such that xox = yoy and β(x+y) < 2. if β is finite and subadditive, then β(x − y) = 0. proof. assume that n = {x ∈ a, β(x) = 0}. by the proof of theorem 2.14, a/n is a normed algebra under the norm ‖x + n‖ = β(x). then (x + n)o(x + n) = (y + n)o(y + n), and, βa/n((x + n) + (y + n)) = βa/n((x + y) + n) ≤ ‖(x + y) + n‖ = β(x + y) < 2. applying lemma 2.17, x + n = y + n which means that β(x − y) = 0. theorem 2.19. let a be a topological algebra such that (i) β(x + y) ≤ β(x) + β(y) (x, y ∈ a) (ii) β(x) < ∞ (x ∈ a). then σa(x) 6= ∅ for all x ∈ a. proof. let n = {x ∈ a, β(x) = 0}. since a/n is a normed algebra, the spectrum of any of its elements is non-empty. 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[12] w. żelazko, on the locally bounded and m-convex topological algebras, studia math. vol. 19, pp. 333–356, 1960. introduction and preliminaries the normed topology induced by the boundedness radius cubo a mathematical journal vol.11, no¯ 01, (123–143). march 2009 regular quaternionic functions and conformal mappings alessandro perotti 1 department of mathematics, university of trento, via sommarive, 14, i–38050 povo trento, italy. email: perotti@science.unitn.it abstract in this paper we study the action of conformal mappings of the quaternionic space on a class of regular functions of one quaternionic variable. we consider functions in the kernel of the cauchy-riemann operator d = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 − k ∂ ∂x3 , a variant of the cauchy–fueter operator. this choice is motivated by the strict relation existing between this type of regularity and holomorphicity w.r.t. the whole class of complex structures on h. for every imaginary unit p ∈ s2, let jp be the corresponding complex structure on h. let holp(ω, h) be the space of holomorphic maps from (ω,jp) to (h,lp), where lp is defined by left multiplication by p. every element of holp(ω, h) is regular, but there exist regular functions that are not holomorphic for any p. these properties can be recognized by computing the energy quadric of a function. we show that the energy quadric is invariant w.r.t. three–dimensional rotations of h. as an application, we obtain that every rotation of the space h can be generated by biregular rotations, invertible regular functions with regular inverse. moreover, we prove that the energy quadric of a regular function can always be diagonalized by means of a three–dimensional rotation. 1work partially supported by miur (prin project “proprietà geometriche delle varietà reali e complesse") and gnsaga of indam 124 alessandro perotti cubo 11, 1 (2009) resumen en este artículo estudiamos la acción de aplicaciones conforme del espacio de cuaterniones sobre la clase de funciones regulares de una variable cuaternionica. nosotros consideramos funciones en el kernel del operador de cauchy–riemann d = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 − k ∂ ∂x3 , una variante del operador de cauchy–fueter. esta elección es motivada por la relación estricta existente entre este tipo de regularidad y holomorficidad w.r.t. de la clase entera de estructuras complejas sobre h. para todo imaginario unitario p ∈ s2, sea jp la correspodiente estructura compleja sobre h. sea holp(ω, h) el espacio de aplicaciones holomórficas de (ω,jp) a (h,lp), donde lp es definido por multiplicación a la izquierda por p. todo elemento de holp(ω, h) es regular, pero existen funciones regulares que no son holomórficas para cualquer p. estas propiedades pueden ser reconocidas mediante el cálculo de la energía cuadrica de una función. nosotros mostramos que la energía cuadrica es invariante w.r.t. por rotaciones tres–dimensionales de h. como aplicación, obtenemos que toda rotación del espacio h puede ser generada por rotaciones bi regulares, funciones regulares invertibles con inversa regular. además mostramos que la energía cuadrica de una función regular siempre puede ser diagonalizada por una rotación tres–dimensional. key words and phrases: quaternionic regular functions, hyperholomorphic functions, conformal mappings, möbius transformations. math. subj. class.: primary 30g35; secondary 30a30 1 introduction. the aim of this paper is to analyze the action of the conformal group of the one–point compactification h∗ of h on a class of regular functions of one quaternionic variable. let ω be a smooth bounded domain in c2. let h be the space of real quaternions q = x0 +ix1 +jx2 +kx3, where i,j,k denote the basic quaternions. we identify h with c 2 by means of the mapping that associates the quaternion q = z1 +z2j with the pair (z1,z2) = (x0 +ix1,x2 +ix3). we consider the class r(ω) of left–regular (also called hyperholomorphic) functions f : ω → h in the kernel of the cauchy–riemann operator d = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 − k ∂ ∂x3 . this differential operator is a variant of the original cauchy–riemann–fueter operator (cf. for cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 125 example [19] and [5, 5]) ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 + k ∂ ∂x3 . hyperholomorphic functions have been studied by many authors (see for instance [1, 7, 11, 12, 14, 17, 18]). many of their properties can be easily deduced from known properties satisfied by fueter–regular functions, since a function f is regular on ω if and only if f ◦ γ is fueter–regular on γ(ω) = γ−1(ω), where γ is the reflection of c2 defined by γ(z1,z2) = (z1, z̄2). however, regular functions in the space r(ω) have some characteristics that are more intimately related to the theory of holomorphic functions of two complex variables. in particular, the space r(ω) contains the spaces of holomorphic maps with respect to any constant complex structure. this is no longer true if we adopt the original definition of fueter regularity (see section 2 for more details). let j1,j2 be the complex structures on the tangent bundle th ≃ h defined by left multiplication by i and j. let j∗ 1 ,j∗ 2 be the dual structures on the cotangent bundle t∗h ≃ h and set j∗ 3 = j∗ 1 j∗ 2 . for every complex structure jp = p1j1 + p2j2 + p3j3 (p a imaginary unit in the unit sphere s2), let ∂p = 1 2 ( d + pj∗ p ◦ d ) be the cauchy–riemann operator with respect to the structure jp. let us define holp(ω, h) = ker ∂p, the space of holomorphic maps from (ω,jp) to (h,lp), where lp is the complex structure defined by left multiplication by p. then every element of holp(ω, h) is regular. these subspaces do not fill the whole space of regular functions (cf. [13]). this result is a consequence of a criterion of jp–holomorphicity, based on the concept of energy quadric of a regular function (cf. section 3.2 for exact definitions). in section 4 we come to conformal transformations. >from a theorem of liouville, the general conformal mapping of h∗ is the composition of a sequence of translations, dilations, rotations and inversions. it can be written as a quaternionic möbius transformation, i.e. a fractional linear map of the form la(q) = (aq + b)(cq + d) −1, with a ∈ gl(2, h). for properties of these maps, see for example [2], [5]§6.2, [11] and [19] and the references cited in those papers. given a function f ∈ c1(ω) and a conformal transformation la, let f a be the function fa(q) = (cγ(q) + d)−1 |cγ(q) + d|2 f(l′ γ(a) (q)), where l′ γ(a) (q) = γ ◦ la ◦ γ(q). in theorem 3, we prove that f is regular on ω if and only if f a is regular on ω′ = (l′ γ(a) ) −1 (ω). moreover, (fa)b = fab for every a,b ∈ gl(2, h). the first property can be deduced from theorem 6 of sudbery [19] using the reflection γ. we are interested also in the action of conformal mappings on the energy quadric and on the holomorphicity properties of the maps. for a general conformal transformation la, the energy 126 alessandro perotti cubo 11, 1 (2009) and, a fortiori, the energy quadric of a regular function is not conserved. however, we show that three–dimensional rotations of h (those which fix the real numbers) conserve the energy quadric (for translations this it is a trivial fact). let a ∈ h, a 6= 0. let rota(q) = aqa −1 be the three–dimensional rotation of h defined by a. in theorem 4, we prove that the function fa = rotγ(a) ◦ f ◦ rota is regular on ωa = rot−1 a (ω) if and only if f is regular on ω. moreover, the energy density of fa is e(fa) = e(f) ◦ rota and the matrix function m(f) (for f regular m(f) is the energy quadric, cf. section 3) transforms in the following way m(fa) = qa(m(f) ◦ rota)q t a , where qa ∈ so(3) is the orthogonal matrix associated to the rotation rotγ(a) of the space r 3 = 〈i,j,k〉. this formula has many consequences. it allows to obtain (corollary 3) that fa is jp– holomorphic if and only if f is jp′ –holomorphic, with p ′ = rot −1 γ(a) (p). moreover, we get (corollary 4) that the energy quadric of a regular function can always be diagonalized by means of a three– dimensional rotation. finally, we obtain a biregularity result about rotations (proposition 2 and corollary 5). we prove that every three-dimensional rotation is the composition of (at most) two three-dimensional biregular rotations, and that every four-dimensional rotation is the composition of two biregular rotations. 2 notations and definitions 2.1 fueter regular functions we identify the space c2 with the set h of quaternions by means of the mapping that associates the pair (z1,z2) = (x0 +ix1,x2 +ix3) with the quaternion q = z1 +z2j = x0 +ix1 +jx2 +kx3 ∈ h. a quaternionic function f = f1 + f2j ∈ c 1 (ω) is (left) regular (or hyperholomorphic) on ω if df = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂f ∂x0 + i ∂f ∂x1 + j ∂f ∂x2 − k ∂f ∂x3 = 0 on ω. we will denote by r(ω) the space of regular functions on ω. with respect to this definition of regularity, the space r(ω) contains the identity mapping and every holomorphic mapping (f1,f2) on ω (with respect to the standard complex structure) defines a regular function f = f1 + f2j. we recall some properties of regular functions, for which we refer to the papers of sudbery[19], shapiro and vasilevski[17] and nōno[12]: 1. the complex components are both holomorphic or both non–holomorphic. cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 127 2. every regular function is harmonic. 3. if ω is pseudoconvex, every complex harmonic function is the complex component of a regular function on ω. 4. the space r(ω) of regular functions on ω is a right h–module with integral representation formulas. 5. f is regular ⇔ ∂f1 ∂z̄1 = ∂f2 ∂z2 , ∂f1 ∂z̄2 = − ∂f2 ∂z1 . we note that a function f = f1 + f2j is regular on ω if and only if its jacobian matrix has the form j(f) = ( ∂(f1,f2, f̄1, f̄2) ∂(z1,z2, z̄1, z̄2) ) =   a1 −b̄2 −c̄2 −c1 a2 b̄1 c̄1 −c2 −c2 −c̄1 ā1 −b2 c1 −c̄2 ā2 b1   at every z ∈ ω, where a = ( ∂f1 ∂z1 , ∂f2 ∂z1 ) , b = ( ∂f̄2 ∂z̄2 ,− ∂f̄1 ∂z̄2 ) , c = ( ∂f̄2 ∂z1 ,− ∂f̄1 ∂z1 ) = − ( ∂f1 ∂z̄2 , ∂f2 ∂z̄2 ) . we shall call a matrix of this form a regular matrix. note that a regular matrix can have rank 0, 2, 3 or 4 but not rank 1. a definition equivalent to regularity has been given by joyce[6] in the setting of hypercomplex manifolds. joyce introduced the module of q–holomorphic functions on a hypercomplex manifold. a hypercomplex structure on the manifold h is given by the complex structures j1,j2 on th ≃ h defined by left multiplication by i and j. let j∗ 1 ,j∗ 2 be the dual structures on t∗h ≃ h. in complex coordinates    j∗ 1 dz1 = idz1, j ∗ 1 dz2 = idz2 j∗ 2 dz1 = −dz̄2, j ∗ 2 dz2 = dz̄1 j∗ 3 dz1 = idz̄2, j ∗ 3 dz2 = −idz̄1 where we make the choice j∗ 3 = j∗ 1 j∗ 2 , which is equivalent to j3 = −j1j2. in real coordinates, the action of the structures is the following    j1dx0 = −dx1, j1dx2 = −dx3, j2dx0 = −dx2, j2dx1 = dx3, j3dx0 = dx3, j3dx1 = dx2. a function f is regular if and only if f is q–holomorphic, i.e. df + ij∗ 1 (df) + jj∗ 2 (df) + kj∗ 3 (df) = 0. in complex components f = f1 + f2j, we can rewrite the equations of regularity as ∂f1 = j ∗ 2 (∂f2). 128 alessandro perotti cubo 11, 1 (2009) the original definition of regularity given by fueter (cf. [19] or [5]) differs from that adopted here by a real coordinate reflection. let γ be the transformation of c2 defined by γ(z1,z2) = (z1, z̄2). then a c 1 function f is regular on the domain ω if and only if f ◦ γ is fueter–regular on γ(ω) = γ−1(ω), i.e. it satisfies the differential equation ( ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 + k ∂ ∂x3 ) (f ◦ γ) = 0 on γ−1(ω). 2.2 biregular functions a quaternionic function f ∈ c1(ω) is called biregular if f is invertible and f, f−1 are regular. if this property holds locally, f is called locally biregular. these functions were studied in [8], [9] and [15]. the class br(ω) of biregular functions is closed with respect to right multiplication by a nonzero quaternion, but it is not closed with respect to composition or sum: even if f + g is invertible and f,g ∈ br(ω), the sum can be not biregular. 2.2.0.1 examples 1. every biholomorphic map (f1,f2) on ω defines a biregular function f = f1 + f2j. 2. the identity function is biregular on h. more generally, the affine functions f(q) = qa + b, a ∈ h∗, b ∈ h, are biregular on h. 3. f = z̄1 + z̄2j ∈ r(h), f −1 = f ∈ br(h). 4. the function f = z1 + z2 + z̄1 + (z1 + z2 + z̄2)j is regular, but the inverse map f−1 = 1 3 (z1 + z2 + z̄1 − 2z̄2 + (z1 + z2 − 2z̄1 + z̄2)j) is not regular. note that in this case the jacobian determinant is negative. this cannot happen for a biregular function (cf. [15]). 2.3 holomorphic functions w.r.t. a complex structure jp let jp = p1j1 +p2j2 +p3j3 be the orthogonal complex structure on h defined by a unit imaginary quaternion p = p1i + p2j + p3k in the sphere s 2 = {p ∈ h | p2 = −1}. in particular, j1 is the standard complex structure of c2 ≃ h. let cp = 〈1,p〉 be the complex plane spanned by 1 and p and let lp be the complex structure defined on t∗cp ≃ cp by left multiplication by p. if f = f 0 + if1 : ω → c is a jp–holomorphic cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 129 function, i.e. df0 = j∗ p (df1) or, equivalently, df + ij∗ p (df) = 0, then f defines a regular function f̃ = f0 + pf1 on ω. we can identify f̃ with a holomorphic function f̃ : (ω,jp) → (cp,lp). we have lp = jγ(p), where γ(p) = p1i + p2j − p3k. more generally, we can consider the space of holomorphic maps from (ω,jp) to (h,lp) holp(ω, h) = {f : ω → h of class c 1 | ∂pf = 0 on ω} = ker ∂p where ∂p is the cauchy–riemann operator with respect to the structure jp ∂p = 1 2 ( d + pj∗ p ◦ d ) . these functions will be called jp–holomorphic maps on ω. for any positive orthonormal basis {1,p,q,pq} of h (p,q ∈ s2), let f = f1 + f2q be the decomposition of f with respect to the orthogonal sum h = cp ⊕ (cp)q. let f1 = f 0 + pf1, f2 = f 2 + pf3, with f0,f1,f2,f3 the real components of f w.r.t. the basis {1,p,q,pq}. then the equations of regularity can be rewritten in complex form as ∂pf1 = j ∗ q (∂pf2), where f2 = f 2 − pf3 and ∂p = 1 2 ( d − pj∗ p ◦ d ) . therefore every f ∈ holp(ω, h) is a regular function on ω. remark 1. 1. the identity map belongs to the space holi(ω, h)∩holj(ω, h) but not to holk(ω, h). 2. for every p ∈ s2, hol−p(ω, h) = holp(ω, h). 3. every cp–valued regular function is a jp–holomorphic function. 4. if f ∈ holp(ω, h) ∩ holq(ω, h), with p 6= ±q, then f ∈ holr(ω, h) for every r = αp+βq ‖αp+βq‖ (α,β ∈ r) in the circle of s2 generated by p and q. if the almost complex structure jp is not constant, i.e. not compatible with the hyperkähler structure of h, we get a similar result. note that in this case the structure is not necessarily integrable. let f ∈ c1(ω) and assume that p = p(z) ∈ s2 varies continuously with z in ω. we will say that p is f-equivariant if f(z) = f(z′) implies p(z) = p(z′) (z,z′ ∈ ω). this property allows to define p∗ : f(ω) → s2 such that p∗ ◦ f = p on ω. in [15], the following result was proved. proposition 1. if f ∈ c1(ω) satisfies the equation ∂p(z)f = 1 2 [ df(z) + p(z)j∗ p(z) ◦ df(z) ] = 0 (1) at every z ∈ ω, then f is a regular function on ω. if, moreover, the structure p is f-equivariant and p∗ admits a continuous extension to an open set u ⊇ f(ω), then f is a (pseudo)holomorphic map from (ω,jp) to (u,lp∗ ). 130 alessandro perotti cubo 11, 1 (2009) example 1. f(z) = z̄1 + z 2 2 + z̄2j is regular on h. on ω = h \ {z2 = 0} f is holomorphic w.r.t. the almost complex structure jp, where p(z) = 1√ |z2|2+|z2|4 ( |z2| 2i + (im z2)j − (re z2)k ) . note that p(z) can not be continued to h as a continuous map. also the inverse map f−1(z) = z̄1 − z 2 2 + z̄2j is regular on h. then f is biregular on h. but f is also (pseudo)biholomorphic on ω: f(ω) = ω and f−1 : (ω,jp′ ) → (h,lp′◦f) is holomorphic, where p′(z) = 1√ |z2|2+|z2|4 ( |z2| 2i − (im z2)j + (re z2)k ) . note that lp∗ = lp◦f−1 = jp′ at f(z) and lp′◦f = jp at z ∈ ω. 3 a criterion for holomorphicity 3.1 energy and regularity in [13] it was proved that on every domain ω there exist regular functions that are not jpholomorphic for any p. a similar result was obtained by chen and li[3] for the larger class of q-maps between hyperkähler manifolds. the criterion for holomorphicity is based on an energy-minimizing property of holomorphic maps. the energy density (w.r.t. the euclidean metric) of a function f : ω → h, of class c1(ω), is given by e(f) = 1 2 ‖df‖2 = 1 2 tr(j(f)j(f) t ). after integration on ω, we get the energy of f ∈ c1(ω): eω(f) = 1 2 ∫ ω e(f)dv. using ideas from [10] and [3], it was proved in [13] that a regular function f ∈ c1(ω) minimizes energy in the homotopy class constituted by maps u with u|∂ω = f|∂ω which are homotopic to f relative to ∂ω: now we introduce the lichnerowicz invariants. let a(f) = (aαβ) be the 3 × 3 matrix with entries the real functions aαβ = −〈jα,f ∗liβ 〉, where (i1, i2, i3) = (i,j,k). for f ∈ c 1 (ω), we set aω(f) = ∫ ω a(f) dv and mω(f) = 1 2 ((tr aω(f))i3 − aω(f)) , where i3 denotes the identity matrix. we recall the criterion for regularity and holomorphicity proved in [13]. cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 131 theorem 1. 1. mω(f) is a relative homotopy invariant of f. 2. f is regular on ω if and only if eω(f) = tr mω(f). 3. if f ∈ r(ω), then mω(f) is symmetric and positive semidefinite. 4. if f ∈ r(ω), then f belongs to some space holp(ω, h) (for a constant structure jp) if and only if det mω(f) = 0. 5. f ∈ holp(ω, h) if and only if xp = (p1,p2,p3) is a unit vector in the kernel of mω(f). >from the criterion it can be seen that almost all regular functions are not holomorphic with respect to any constant complex structure jp. example 2. f = z̄1 + z2 + z̄2j is jp-holomorphic, with p = 1√ 5 (i − 2k), since on the unit ball b (with normalized unit volume) eb(f) = 3 and mb(f) =   2 0 1 0 1 2 0 1 0 1 2   . example 3. f = z1 + z2 + z̄1 + (z1 + z2 + z̄2)j is regular, but not holomorphic: eb(f) = 6 and mb(f) =   2 0 0 0 2 0 0 0 2   . example 4. f = z̄1 + z̄2j is regular and has matrix mb(f) =   2 0 0 0 0 0 0 0 0   of rank one. this means that f ∈ holj(h, h) ∩ holk(h, h). example 5. the identity mapping belongs to the space holi(h, h) ∩ holj(h, h) = ⋂ p∈ holp(h, h). example 6 (nonlinear case). f = |z1| 2 − |z2| 2 + z̄1z̄2j has energy eb(f) = 2 on the unit ball. the matrix mb(f) is mb(f) =   4 3 0 0 0 1 3 0 0 0 1 3   . therefore f is regular but not holomorphic w.r.t. any constant complex structure jp. 132 alessandro perotti cubo 11, 1 (2009) 3.2 the energy quadric in [15], a pointwise version of the criterion for holomorphicity was established. theorem 2. let ω be connected and f ∈ c1(ω). consider the matrix of real functions on ω m(f) = 1 2 ((tr a(f))i3 − a(f)) . 1. f is regular on ω if and only if e(f) = tr m(f) at every point z ∈ ω. 2. if f ∈ r(ω), then m(f) is symmetric and positive semidefinite. 3. if f ∈ r(ω), then det m(f) = 0 on ω if and only if there exists an open, dense subset ω ′ ⊆ ω on which f satisfies equation (1) for some function p(z) : ω′ → s2. moreover, if det m(f) = 0 and p(z) is f-equivariant, p∗ ◦ f = p and p∗ extends continuously to an open set u ⊇ f(ω), then f is a (pseudo)holomorphic map from (ω′,jp) to (u,lp∗ ). let a = ( ∂f1 ∂z1 , ∂f2 ∂z1 ) , b = ( ∂f̄2 ∂z̄2 ,− ∂f̄1 ∂z̄2 ) , c = ( ∂f̄2 ∂z1 ,− ∂f̄1 ∂z1 ) , d = − ( ∂f1 ∂z̄2 , ∂f2 ∂z̄2 ) . then the energy density is given by e(f) = |a|2+|b|2+|c|2+|d|2. a lengthy but straightforward computation gives the following expression for the matrix m(f): m(f) =   |c|2 + |d|2 im(〈d,a〉 − 〈c,b〉) re(〈d,a〉 + 〈c,b〉) im(〈c,a〉 − 〈d,b〉) 1 2 |a − b|2 + 1 2 |c − d|2 − im(〈a,b〉 + 〈c,d〉) re(〈c,a〉 + 〈d,b〉) − im(〈a,b〉 − 〈c,d〉) 1 2 |a + b|2 + 1 2 |c − d|2   . then e(f) = tr m(f) if and only if c = d, i.e. f is regular. in this case the matrix m(f) becomes m(f) =   2|c|2 im〈c,a − b〉 re〈c,a + b〉 im〈c,a − b〉 1 2 |a − b|2 − im〈a,b〉 re〈c,a + b〉 − im〈a,b〉 1 2 |a + b|2   . definition 1. for a regular function f on ω, the family of positive semi-definite quadrics with matrices {m(f)(z) |z ∈ ω} will be called the energy quadric of f. remark 2. if f is invertible, then every p(z) is f-equivariant. if p is a constant complex structure, then p is f-equivariant for every f. remark 3. if f is (real) affine, m(f) is a constant matrix. if f is not affine, det m(f) = 0 on ω does not imply that det mω(f) = 0, but theorems 1 and 2 imply that the converse is true. cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 133 example 7. the function f(z) = z̄1 + z 2 2 + z̄2j is regular (also biregular, cf. example 1) on h. we have e(f) = 2 + 4|z2| 2, m(f) = 2   1 − im z2 re z2 − im z2 |z2| 2 0 re z2 0 |z2| 2   . then the energy quadric of f is singular on h. on the domain ω′ = h \{z2 = 0}, where m(f) has maximum rank 2, the kernel of m(f) is spanned by the vector x = (|z2| 2, im z2,− re z2). then f is jp-holomorphic on ω ′, with p(z) = 1√ |z2|2+|z2|4 ( |z2| 2i + (im z2)j − (re z2)k ) . on the unit ball b, eb(f) = 10 3 and the matrix mb(f) = ∫ b m(f)dv =   2 0 0 0 2 3 0 0 0 2 3   is non-singular. therefore, f is not jq-holomorphic for any constant complex structure jq. example 8. the function f = |z1| 2 − |z2| 2 + z̄1z̄2j introduced in example 6 has energy density 3|z|2 and energy quadric with matrix m(f) =   2|z|2 0 0 0 1 2 |z|2 0 0 0 1 2 |z|2   . therefore f is regular but not holomorphic w.r.t. any almost complex structure jp. note that det m(f) = 1 2 |z|6 vanishes only at the origin. in [15], it was shown that if f ∈ br(ω) is a biregular function, then there exists an open, dense subset ω′ ⊆ ω, and an almost complex structure p(z) on ω′, such that f : (ω′,jp) → (f(ω ′ ),lp∗ ) is a holomorphic map, with holomorphic inverse f−1 : (f(ω′),jp′ ) → (ω ′,lp′◦f). here p = p1i + p2j + p3k : ω ′ → s2, p∗ = p ◦ f−1 and p′ = p1i + p2j − p3k. in particular, any such map f preserves orientation. 4 regular functions and conformal mappings in this section we are going to analyze the action of the conformal group of h on regular functions. some of the results we describe can be deduced from [19] theorem 6 using the reflection γ(z1,z2) = 134 alessandro perotti cubo 11, 1 (2009) (z1, z̄2) introduced in §2.1, but here we want to investigate also the action on the energy quadric and the holomorphicity properties of the maps. we recall some definitions and properties of conformal and orientation preserving mappings of the one–point compactification ĥ of h, for which we refer to [2], [5]§6.2, [11] and [19] and to the references cited in those papers. the dieudonné determinant of a quaternionic matrix a = [ a b c d ] is the real non–negative number deth(a) = √ |a|2|d|2 + |b|2|c|2 − 2re(cābd̄). it satisfies binet property deth(ab) = deth(a)deth(b) and a 2 × 2 matrix a is (left and right) invertible if and only if detha 6= 0. then we can consider the general linear group gl(2, h) = { a = [ a b c d ] quaternionic matrix of order 2 | detha 6= 0 } . a theorem of liouville tells that the general conformal transformation of h∗ is a quaternionic möbius transformation, i.e. a fractional linear map of the form la(q) = (aq + b)(cq + d) −1, for a ∈ gl(2, h). the matrix a is determined by la up to a real scalar multiple. for every pair of matrices a,b ∈ gl(2, h), la ◦ lb = lab. we have also the alternative representation of conformal mappings l′ a (q) = (qc + d)−1(qa + b), dethā 6= 0. theorem 3. given a function f ∈ c1(ω) and a conformal transformation la(q) = (aq + b)(cq + d)−1, let fa be the function fa(q) = (cγ(q) + d)−1 |cγ(q) + d|2 f(l′ γ(a) (q)), where γ(a) = [ γ(a) γ(b) γ(c) γ(d) ] . then f is regular on ω if and only if fa is regular on ω′ = (l′ γ(a) ) −1 (ω). moreover, (fa)b = fab for every a,b ∈ gl(2, h). proof. we deduce the first statement from the result of sudbery (cf. [19] theorem 6), since f ∈ r(ω) iff f = f ◦ γ is fueter–regular on γ(ω). this last condition is equivalent to the fueter– regularity of the transformed function fa(p) = (cp + d)−1 |cp + d|2 f(la(p)) cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 135 on (la) −1 (γ(ω)). note that this function differs from the one given by sudbery by a real constant factor. we then obtain that f is regular iff fa ◦ γ is regular. we have fa ◦ γ(q) = (cγ(q) + d)−1 |cγ(q) + d|2 f ◦ γ ◦ la ◦ γ(q) = f a (q), since γ◦la◦γ(q) = l ′ γ(a) (q). now we come to the last statement of the theorem. let b = [ a′ b′ c′ d′ ] and c = ab = [ a′′ b′′ c′′ d′′ ] . then (fa)b(q) = (c′γ(q) + d′)−1 |c′γ(q) + d′|2 fa(l′ γ(b) (q)) (c′γ(q) + d′)−1 |c′γ(q) + d′|2 (cγ(l′ γ(b) (q)) + d)−1 |cγ(l′ γ(b) (q)) + d|2 f((l′ γ(a) ◦ l′ γ(b) )(q)) let q′ = γ(q). the last statement of the theorem follows from the equalities l′ γ(a) ◦ l′ γ(b) = (γ ◦ la ◦ γ) ◦ (γ ◦ lb ◦ γ) = γ ◦ lab ◦ γ = l ′ γ(ab) and (c′q′ + d′) (cγ(l′ γ(b) (q)) + d) = (q′c′ + d′) ((q′c′ + d′)−1(q′a′ + b′)c̄ + d̄) (q′a′ + b′)c̄ + (q′c′ + d′)d̄ = q′(a′c̄ + c′d̄) + b′c̄ + d′d̄ c′′q′ + d′′ remark 4. if t is a non–zero real number, fta = t−3fa. then fa depends only for a real scalar multiple on the matrix chosen to represent the conformal transformation la. we can also restrict the choice of the matrix to the subgroup sl(2, h) = {a ∈ gl(2, h) | deth(a) = 1}. in this case, the same conformal transformation gives rise to two functions, fa and f−a = −fa. every conformal transformation is the composition of a sequence of translations, dilations, rotations and inversions. in order to illustrate the preceding theorem, we now apply it to these basic cases. example 9. the inversion q 7→ q−1 corresponds to the matrix a = [ 0 1 1 0 ] (up to a real scalar multiple) and transforms a regular f ∈ r(ω) into finv(q) = γ(q)−1 |q|2 f(q−1), regular on ω′ = {q ∈ h | q−1 ∈ ω}. 136 alessandro perotti cubo 11, 1 (2009) example 10. in particular, the inverted function of the constant function f = 1 2π2 is the cauchy– fueter kernel for the module of regular functions g(q) = g(z1 + z2j) = 1 2π2 z̄1 − z̄2j |z|4 . example 11. a translation q 7→ q + b corresponds to the matrix a = [ 1 b 0 1 ] . the transformed function is fa(q) = f(l′ γ(a) (q)) = f(q + γ(b)). example 12. a dilation q 7→ aq, a 6= 0 real, has matrix a = [ a 0 0 1 ] . a function f transforms into fa(q) = f(qa). example 13. given two unit quaternions a,d ∈ h, the diagonal matrix a = [ a 0 0 d ] induces the four–dimensional rotation q 7→ aqd−1. given a regular function f on ω, the function fa(q) = d−1f(γ(d)−1qγ(a)) is regular on ω′ = γ(d)ωγ(a)−1. example 14. the quaternionic cayley transformation ψ(q) = (q + 1)(1 − q)−1 maps diffeomorphically the unit ball b to the right half–space h+ = {q ∈ h | re(q) > 0} (see [2] for geometric properties of ψ). it transforms regular functions f on h+ into fψ(q) = 23/2 (1 − γ(q))−1 |1 − γ(q)|2 f(ψ(q)), regular on b. the inverse mapping ψ−1(q) = (q − 1)(1 + q)−1 transforms f ∈ r(b) into fψ −1 (q) = 23/2 (1 + γ(q))−1 |1 + γ(q)|2 f(ψ−1(q)) ∈ r(h+). the factor 23/2 in the formulas has been chosen to get (fψ)ψ −1 = f. if we take the identity map, which is regular on h, as f, from theorem 3 we get the following: corollary 1. for every conformal transformation la(q) = (aq + b)(cq + d) −1, the function (cγ(q) + d)−1 |cγ(q) + d|2 l′ γ(a) (q), is regular on {q ∈ h | cγ(q) + d 6= 0}. cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 137 4.1 the quadric energy of rotated regular functions a unit quaternion d defines the three–dimensional rotation q 7→ rotd(q) := dqd −1, which gives rise to the function (cf. example 13) fa(q) = d−1f(γ(d)−1qγ(d)), where a is the scalar matrix a = [ d 0 0 d ] . taking d = γ(a)−1 and multiplying by γ(a)−1 on the right, we obtain the function fa = rotγ(a) ◦ f ◦ rota. from theorem 3 we immediately get: corollary 2. let f ∈ c1(ω) and let a ∈ h, a 6= 0. let rota(q) = aqa −1 be the three–dimensional rotation of h defined by a. then the function fa = rotγ(a) ◦ f ◦ rota is regular on ωa = rot−1 a (ω) = a−1ωa if and only if f is regular on ω. remark 5. the rotated function fa has the following properties: 1. (fa)b = fab and (f + g)a = fa + ga. 2. (fa)a −1 = f. 3. f−a = fa. 4. if b ∈ h, then (fb)a = fa rotγ(a)(b). now we analyze the action of rotations on the energy quadric. we obtain in this way a new proof of the preceding result and we get new holomorphicity properties of rotated regular functions. theorem 4. let f ∈ c1(ω) and let a ∈ h, a 6= 0. let fa = rotγ(a) ◦ f ◦ rota. then the energy density of fa is e(fa) = e(f) ◦rota and the matrix function m(f) defined in section 3 transforms in the following way m(fa) = qa(m(f) ◦ rota)q t a , where qa is the orthogonal matrix in so(3) associated to the rotation rotγ(a) of the space 〈i,j,k〉. before coming to the theorem, we prove a simple result about holomorphicity of rotations. lemma 1. for every p ∈ s2, the three-dimensional rotation rota(q) = aqa −1 is a holomorphic map from (h,jγ(p)) to (h,lrota(p)). proof. let b = {p,p′,pp′} be a positive orthonormal base of r3 = 〈i,j,k〉. let xp = (p1,p2,p3), xp′ = (p ′ 1 ,p′ 2 ,p′ 3 ), xr = (r1,r2,r3), with r = pp ′ = r1i + r2j + r3k. given the transition matrix a 138 alessandro perotti cubo 11, 1 (2009) with columns xp,xp′,xr, the coordinates x ′ α (α = 1, 2, 3) of q = x0 + x1i + x2j + x3k w.r.t. b are given by the product (x′ 1 x′ 2 x′ 3 ) t = at (x1 x2 x3) t . then x′ 1 = ∑ α pαxα, x ′ 2 = ∑ α p′ α xα, x ′ 3 = ∑ α rαxα. >from this we get that the functions g1 = x0 +x ′ 1 rota(p) and g2 = x ′ 2 +x′ 3 rota(p) are holomorphic from (h,jγ(p)) to (h,lrota(p)), since jγ(p)(dx0) = (p1j1 + p2j2 − p3j3)(dx0) = − ∑ α pαdxα = −dx ′ 1 and jγ(p)(dx ′ 2 ) = ∑ α p′ α (p1j1 + p2j2 − p3j3)(dxα) = ∑ α pαp ′ α dx0 − (p2p ′ 3 − p3p ′ 2 )dx1 − (p3p ′ 1 − p1p ′ 3 )dx2 − (p1p ′ 2 − p2p ′ 1 )dx3 = −r1dx1 − r2dx2 − r3dx3 = −dx ′ 3 . the lemma now follows from the equality rota(q) = a(x0 + x ′ 1 p + x′ 2 p′ + x′ 3 r)a−1 = (x0 + x ′ 1 rota(p)) + (x ′ 2 + x′ 3 rota(p))rota(p ′ ) = g1 + g2 rota(p ′ ) if in the preceding lemma p is replaced by γ(p), we get that the map rota(q) is holomorphic also from (h,jp) to (h,lrota(γ(p))) = (h,jp′ ), where p ′ = γ(rota(γ(p))) = γ(a) −1pγ(a) = rot−1 γ(a) (p). replacing a with γ(a) we also get that rotγ(a) is holomorphic from (h,lp′ ) = (h,jrota(γ(p))) to (h,lrotγ(a)(p′)) = (h,lp). then we can draw a commutative diagram with holomorphic vertical maps (h,jp′ ) f // (h,lp′ ) rotγ(a) �� (h,jp) rota oo f a // (h,lp) (2) proof of theorem 4. let j be the real jacobian matrix of f ◦rota. then the real jacobian matrix of fa is the product qaj. it follows that e(f a ) = 1 2 tr(qajj tqt a ) = 1 2 tr(jjt ) = e(f ◦ rota). a similar computation gives e(f ◦ rota) = e(f) ◦ rota. for the second statement of the theorem, it is sufficient to prove the equality a(fa) = qa(a(f) ◦ rota)q t a , (3) cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 139 for the matrix functions a(f) and a(fa) defined in section 3, since then the matrices a(fa) and a(f) ◦ rota have the same trace and therefore qa(m(f) ◦ rota)q t a = 1 2 (tr a(f) ◦ rota) i3 − 1 2 a(fa) = 1 2 (tr a(fa)i3 − a(f a )) = m(fa). it remains to prove (3). let p = p1i + p2j + p3k ∈ s 2 and p′ = rot−1 γ(a) (p). let us define the p–holomorphic energy of f ip(f) = 1 2 ‖df + lp ◦ df ◦ jp‖ 2 = 1 2 ‖df + pdf ◦ jp‖ 2 = 2‖∂pf‖ 2. then we obtain, as in [3], e(f) + 〈jp,f ∗lp〉 = 1 4 ip(f). if x = (p1,p2,p3), then xa(fa)xt = ∑ α,β pαpβaαβ = −〈 ∑ α pαjα, (f a ) ∗ ∑ β pβliβ 〉 = −〈jp, (f a ) ∗lp〉 = e(f a ) − 1 4 ip(f a ). now let x′ = (p′ 1 ,p′ 2 ,p′ 3 ) = xqa. a similar computation gives xqaa(f ◦ rota)q t a xt = x′a(f ◦ rota)x ′t = e(f) ◦ rota − 1 4 ip′ (f) ◦ rota. >from the first statement of the theorem and the arbitrariness of x, equation (3) is equivalent to the equality, for any p ∈ s2, of the holomorphic energies ip′ (f) ◦ rota = ip(f a ). (4) >from lemma 1 (cf. diagram (2)) and rotational invariance of the norm we get 2ip(f a ) = ‖dfa + lp ◦ df a ◦ jp‖ 2 = ‖rotγ(a) ◦ df ◦ drota + lp ◦ rotγ(a) ◦ df ◦ drota ◦ jp‖ 2 = ‖rotγ(a) ◦ df ◦ drota + rotγ(a) ◦ lp′ ◦ df ◦ jp′ ◦ drota‖ 2 = ‖df + lp′ ◦ df ◦ jp′‖ 2 ◦ rota = 2ip′ (f) ◦ rota. then the equality (4) is true and the theorem is proved. corollary 3. let f ∈ c1(ω) and let a ∈ h, a 6= 0. let fa = rotγ(a) ◦ f ◦ rota. let qa ∈ so(3) be associated to the rotation rotγ(a) of the space 〈i,j,k〉. then 1. f is regular on ω if and only if fa is regular on ωa = rot−1 a (ω) = a−1ωa. 140 alessandro perotti cubo 11, 1 (2009) 2. fa is jp–holomorphic if and only if f is jp′ –holomorphic, with p ′ = rot −1 γ(a) (p). 3. if f ∈ c1(ω), then (cf. theorem 1) mωa (f a ) = qamω(f)q t a . proof. 1) from theorem 4 we get that tr m(fa) = tr m(f) ◦ rota and e(f a ) = e(f) ◦ rota. the first statement follows from theorem 2, which tells that f is regular iff e(f) = tr m(f). 2) it is an immediate consequence of equality (4), since a function is jp–holomorphic iff its p–holomorphic energy vanishes. 3) it follows easily by integration of m(fa) on ωa. corollary 4. for every f ∈ r(ω), there exists a ∈ h, a 6= 0, such that the matrices m(fa) and mωa (f a ) are diagonal, with non–negative entries. proof. it follows immediately from theorems 4 and 2, since when f is regular m(f) is symmetric and positive semidefinite. remark 6. for a general conformal transformation la, the energy and, a fortiori, the energy quadric of a regular function is not conserved. for example, the constant function 1 has zero energy, while e(2π2g) 6= 0 and 1inv = 2π2g (cf. example 10). the same happens for jp–holomorphicity. for example, the identity function is in the spaces holi(h) and holj(h), while idinv(q) = γ(q)−1q−1 |q|2 ∈ r(h \ {0}) is not holomorphic w.r.t. any structure jp. this can be seen by computing the energy quadric m(idinv). since det m(idinv) = 32/|q|30 is always non–zero, it follows from theorem 2 that idinv is not jp–holomorphic, for any p (even non–constant). the rank of id inv is three, because its image is contained in the space 〈1, i,j〉, and the function can not have rank less than three, otherwise its quadric energy would have zero determinant (cf. [15] theorem 7). a simpler example is given again by the function 1inv, since the energy quadric of the kernel g is m(g) = 2/|q|8i3. 4.2 biregular rotations if in theorem 4 and its corollaries we take as f the identity map we get the following: proposition 2. for every a ∈ h, a 6= 0, the three–dimensional rotation rotγ(a)a is a biregular function on h, with energy quadric m(rotγ(a)a) of rank 1. this means that rotγ(a)a is holomorphic w.r.t. a circle of structures p ∈ s2. more precisely, rotγ(a)a ∈ holp(h) for every p ∈ 〈rotγ(a)(i),rotγ(a)(j)〉 ∩ s 2. cubo 11, 1 (2009) regular quaternionic functions and conformal mappings 141 proof. we have rotγ(a)a = id a (cf. theorem 4). then m(rotγ(a)a) = qam(id)q t a = qa   0 0 0 0 0 0 0 0 2  q t a has rank 1. its kernel gives the structures with respect to which the rotation is holomorphic. from corollary 3(2), these structures are generated by rotγ(a)(i) and rotγ(a)(j), since id ∈ holi(h) ∩ holj(h). biregularity follows from (γ(a)a)−1 = a−1γ(a−1), which implies the equality (ida)−1 = idγ(a −1 ) ∈ r(h). remark 7. not every rotation is a regular function, since the quaternion γ(a)a is a reduced quaternion, with fourth component zero. these quaternion numbers correspond to rotations of r 3 = 〈i,j,k〉 with axis orthogonal to the k axis. however, every quaternion is the product of two reduced quaternions and the map a 7→ γ(a)a is surjective from h to the space hr of reduced quaternions. the surjectivity of a 7→ γ(a)a can be seen explicitly, or can be deduced from a property of the regular function idinv (cf. remark 6). its restriction to the unit sphere s3 is the map q 7→ γ(q̄)q̄ ∈ s3 ∩ hr. it is surjective since id inv has rank three. corollary 5. 1. the left–multiplication map la′ (q) = a ′q is biregular for every reduced quaternion a′ = γ(a)a 6= 0. 2. every three-dimensional rotation is the composition of two three-dimensional biregular rotations. 3. every four-dimensional rotation is the composition of two biregular rotations. proof. 1) la′ (q) = γ(a)aq = rotγ(a)a(q)(a −1γ(a)−1) has the same regularity and holomorphicity properties of rotγ(a)a, since r(ω) is a right h–module for every ω and ∂p(fb) = (∂pf)b for every f and every b ∈ h. 2) it follows from what has been said in the above remark: if c = a′b′, with a′ = γ(a)a, b′ = γ(b)b ∈ hr, then rotc = rota′ ◦ rotb′ = rotγ(a)a ◦ rotγ(b)b. 3) a four–dimensional rotation rotc,d(q) = cqd −1, with |cd−1| = 1, can be decomposed as rotc,d(q) = cqc −1 (cd−1) = rotc(q) (cd −1 ) = (rota′ ◦ rotb′ )(q) (cd −1 ), where c = a′b′ as before. let f(q) = rota′ (q) (cd −1 ) ∈ br(h). then rotc,d = f ◦ rotb′ . the pair of biregular functions in the corollary can be chosen in the same space holp(h). this comes from proposition 2, because the two great circles of complex structures in s2 coincide 142 alessandro perotti cubo 11, 1 (2009) or intersect in two antipodal points defining a space holp(h). note that this space is not closed under composition, unless jp = lp, which happens only when p = γ(p) is a reduced quaternion. received: july 2008. revised: august 2008. references [1] r. abreu-blaya, j. bory-reyes, m. shapiro, on the notion of the bochner-martinelli integral for domains with rectifiable boundary. complex anal. oper. theory 1 (2007), no. 2, 143–168. 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[19] a. sudbery, quaternionic analysis, mat. proc. camb. phil. soc. 85 (1979), 199–225. cubo, a mathematical journal vol.22, no¯ 01, (01–21). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100001 bounds for the generalized (φ, f)-mean difference silvestru sever dragomir 1,2 1mathematics, college of engineering & science, victoria university, po box 14428, melbourne city, mc 8001, australia. sever.dragomir@vu.edu.au, http://rgmia.org/dragomir 2school of computer science & applied mathematics, university of the witwatersrand, private bag 3, johannesburg 2050, south africa abstract in this paper we establish some bounds for the (φ, f)-mean difference introduced in the general settings of measurable spaces and lebesgue integral, which is a two functions generalization of gini mean difference that has been widely used by economists and sociologists to measure economic inequality. resumen en este art́ıculo establecemos algunas cotas para la (φ, f)-diferencia media introducida en el contexto general de espacios medibles e integral de lebesgue, que es una generalización a dos funciones de la diferencia media de gini que ha sido ampliamente utilizada por economistas y sociólogos para medir desigualdad económica. keywords and phrases: gini mean difference, mean deviation, lebesgue integral, expectation, jensen’s integral inequality. 2010 ams mathematics subject classification: 26d15; 26d10; 94a17. http://doi.org/10.4067/s0719-06462020000100001 2 silvestru sever dragomir cubo 22, 1 (2020) 1. introduction let (ω, a, ν) be a measurable space consisting of a set ω, a σ -algebra a of subsets of ω and a countably additive and positive measure ν on a with values in r ∪ {∞} . for a ν-measurable function w : ω → r, with w (x) ≥ 0 for ν-a.e. (almost every) x ∈ ω and ∫ ω w (x) dν (x) = 1, consider the lebesgue space lw (ω, ν) := {f : ω → r, f is ν-measurable and ∫ ω w (x) |f (x)| dν (x) < ∞}. let i be an interval of real numbers and φ : i → r a lebesgue measurable function on i. for f : ω → i a ν-measurable function with φ ◦ f ∈ lw (ω, ν) we define the generalized (φ, f)-mean difference rg (φ, f; w) by rg (φ, f; w) := 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (x) − (φ ◦ f) (y)| dν (x) dν (y) (1.1) and the generalized (φ, f)-mean deviation md (φ, f; w) by md (φ, f; w) := ∫ ω w (x) |(φ ◦ f) (x) − e (φ, f; w)| dν (x) , (1.2) where e (φ, f; w) := ∫ ω (φ ◦ f) (y) w (y) dν (y) the generalized (φ, f)-expectation. if φ = e, where e (t) = t, t ∈ r is the identity mapping, then we can consider the particular cases of interest, the generalized f-mean difference rg (f; w) := rg (e, f; w) = 1 2 ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) (1.3) and the generalized f-mean deviation md (f; w) := md (e, f; w) = ∫ ω w (x) |f (x) − e (f; w)| dν (x) , (1.4) where e (f; w) := ∫ ω f (y) w (y) dν (y) is the generalized f-expectation. if ω = [−∞, ∞] and f = e then we have the usual mean difference rg (w) := rg (f; w) = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ w (x) w (y) |x − y| dxdy (1.5) and the mean deviation md (w) := md (f; w) = ∫ ω w (x) |x − e (w)| dx, (1.6) cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 3 where w : r →[0, ∞) is a density function, this means that w is integrable on r and ∫ ∞ −∞ w (t) dt = 1, and e (w) := ∫ ∞ −∞ xw (x) dx (1.7) denote the expectation of w provided that the integral exists and is finite. the mean difference rg (w) was proposed by gini in 1912 [21], after whom it is usually named, but was discussed by helmert and other german writers in the 1870’s (cf. h. a. david [13], see also [26, p. 48]). it has a certain theoretical attraction, being dependent on the spread of the variate-values among themselves and not on the deviations from some central value ([26, p. 48]). further, its defining integral (1.5) may converge when that of the variance σ (w) , σ (w) := ∫ ∞ −∞ (x − e (w)) 2 w (x) dx, (1.8) does not. it is, however, more difficult to compute than the standard deviation. for some recent results concerning integral representations and bounds for rg (w) see [5], [6], [8] and [9]. for instance, if w : r →[0, ∞) is a density function we define by w (x) := ∫x −∞ w (t) dt, x ∈ r its cumulative function. then we have [5], [6]: rg (w) = 2 cov (e, w) = ∫ ∞ −∞ (1 − w (y)) w (y) dy = 2 ∫ ∞ −∞ xw (x) w (x) dx − e (w) = 2 ∫ ∞ −∞ (x − e (w)) (w (x) − γ) w (x) dx = 2 ∫ ∞ −∞ (x − δ) ( w (x) − 1 2 ) w (x) dx (1.9) for any γ, δ ∈ r and [6]: rg (w) = ∫ ∞ −∞ ∫ ∞ −∞ (x − y) (w (x) − w (y)) w (x) w (y) dxdy. (1.10) with the above assumptions, we have the bounds [5]: 1 2 md (w) ≤ rg (w) ≤ 2 sup x∈r |w (x) − γ| md (w) ≤ md (w) , (1.11) 4 silvestru sever dragomir cubo 22, 1 (2020) for any γ ∈ [0, 1] , where w (·) is the cumulative distribution of w and md (w) is the mean deviation. consider the n-tuple of real numbers a = (a1, ..., an) and p = (p1, ..., pn) a probability distribution, i.e. pi ≥ 0 for each i ∈ {1, ..., n} with ∑n i=1 pi = 1, then by taking ω = {1, ..., n} and the discrete measure, we can consider from (1.1) and (1.2) that (see [7]) rg (a; p) := 1 2 n∑ i=1 n∑ j=1 pipj |φ (ai) − φ (aj)| , (1.12) and md (a; p) := 1 2 n∑ i=1 pi ∣ ∣ ∣ ∣ ∣ ∣ φ (ai) − n∑ j=1 pjφ (aj) ∣ ∣ ∣ ∣ ∣ ∣ (1.13) where a ∈ in := i × ... × i and φ : i → r. the quantity rg (a; p) has been defined in [7] and some results were obtained. in the case when φ = e, then we get the special case of gini mean difference and mean deviation of an empirical distribution that is particularly important for applications, rg (a; p) := 1 2 n∑ i=1 n∑ j=1 pipj |ai − aj| , (1.14) and md (a; p) := 1 2 n∑ i=1 pi ∣ ∣ ∣ ∣ ∣ ∣ ai − n∑ j=1 pjaj ∣ ∣ ∣ ∣ ∣ ∣ . (1.15) the following result incorporates an upper bound for the weighted gini mean difference [7]: for any a ∈ rn and any p a probability distribution, we have the inequality: 1 2 md (a; p) ≤ rg (a; p) ≤ ı́nf γ∈r [ n∑ i=1 pi |ai − γ| ] ≤ md (a; p) . (1.16) the constant 1 2 in the first inequality in (1.16) is sharp. for some recent results for discrete gini mean difference and mean deviation, see [7], [11], [14] and [15]. cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 5 2. general bounds we have: theorem 1. let i be an interval of real numbers and φ : i → r a lebesgue measurable function on i. if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν-a.e. (almost every) x ∈ ω and ∫ ω w (x) dν (x) = 1 and if f : ω → i is a ν-measurable function with φ ◦ f ∈ lw (ω, ν) , then 1 2 md (φ, f; w) ≤ rg (φ, f; w) ≤ i (φ, f; w) ≤ md (φ, f; w) , (2.1) where i (φ, f; w) := ı́nf γ∈r ∫ ω w (x) |(φ ◦ f) (x) − γ| dν (x) . (2.2) demostración. using the properties of the integral, we have rg (φ, f; w) = 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (x) − (φ ◦ f) (y)| dν (x) dν (y) ≥ 1 2 ∫ ω w (x) ∣ ∣ ∣ ∣ (φ ◦ f) (x) ∫ ω w (y) dν (y) − ∫ ω w (y) (φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) = 1 2 ∫ ω w (x) ∣ ∣ ∣ ∣ (φ ◦ f) (x) − ∫ ω w (y) (φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) = 1 2 md (φ, f; w) and the first inequality in (2.1) is proved. by the triangle inequality for modulus we have |(φ ◦ f) (x) − (φ ◦ f) (y)| = |(φ ◦ f) (x) − γ + γ − (φ ◦ f) (y)| (2.3) ≤ |(φ ◦ f) (x) − γ| + |(φ ◦ f) (y) − γ| for any x, y ∈ ω and γ ∈ r. 6 silvestru sever dragomir cubo 22, 1 (2020) now, if we multiply (2.3) by 1 2 w (x) w (y) and integrate, we get rg (φ, f; w) = 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (x) − (φ ◦ f) (y)| dν (x) dν (y) ≤ 1 2 ∫ ω ∫ ω w (x) w (y) [|(φ ◦ f) (x) − γ| + |(φ ◦ f) (y) − γ|] dν (x) dν (y) = 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (x) − γ| dν (x) dν (y) + 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (y) − γ| dν (x) dν (y) = 1 2 ∫ ω w (x) |(φ ◦ f) (x) − γ| dν (x) + 1 2 ∫ ω w (y) |(φ ◦ f) (y) − γ| dν (y) = ∫ ω w (x) |(φ ◦ f) (x) − γ| dν (x) (2.4) for any γ ∈ r. taking the infimum over γ ∈ r in (2.4) we get the second part of (2.1). since, obviously i (φ, f; w) = ı́nf γ∈r ∫ ω w (x) |(φ ◦ f) (x) − γ| dν (x) ≤ ∫ ω w (x) ∣ ∣ ∣ ∣ (φ ◦ f) (x) − ∫ ω w (y) (φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) = md (φ, f; w) , the last part of (2.1) is thus proved. by the cauchy-bunyakowsky-schwarz (cbs) inequality, if (φ ◦ f) 2 ∈ lw (ω, ν) , then we have [∫ ω w (x) ∣ ∣ ∣ ∣ (φ ◦ f) (x) − ∫ ω w (y) (φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) ]2 ≤ ∫ ω w (x) [ (φ ◦ f) (x) − ∫ ω w (y) (φ ◦ f) (y) dν (y) ]2 dν (x) = ∫ ω w (x) (φ ◦ f) 2 (x) dν (x) − 2 ∫ ω w (y) (φ ◦ f) (y) dν (y) ∫ ω w (x) (φ ◦ f) (x) dν (x) + [∫ ω w (y) (φ ◦ f) (y) dν (y) ]2 ∫ ω w (x) dν (x) = ∫ ω w (x) (φ ◦ f) 2 (x) dν (x) − [∫ ω w (x) (φ ◦ f) (x) dν (x) ]2 . cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 7 by considering the generalized (φ, f)-dispersion σ (φ, f; w) := (∫ ω w (x) (φ ◦ f) 2 (x) dν (x) − [∫ ω w (x) (φ ◦ f) (x) dν (x) ]2 )1/2 , then we have md (φ, f; w) ≤ σ (φ, f; w) (2.5) provided (φ ◦ f) 2 ∈ lw (ω, ν). if there exists the constants m, m so that − ∞ < m ≤ φ (t) ≤ m < ∞ for almost any t ∈ i (2.6) then by the reverse cbs inequality σ (φ, f; w) ≤ 1 2 (m − m) , (2.7) by (2.1) and by (2.5) we can state the following result: corollary 1. let i be an interval of real numbers and φ : i → r a lebesgue measurable function on i satisfying the condition (2.6) for some constants m, m. if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and if f : ω → i is a ν-measurable function with (φ ◦ f) 2 ∈ lw (ω, ν) , then we have the chain of inequalities 1 2 md (φ, f; w) ≤ rg (φ, f; w) ≤ i (φ, f; w) ≤ md (φ, f; w) ≤ σ (φ, f; w) ≤ 1 2 (m − m) . (2.8) we observe that, in the discrete case we obtain from (2.1) the inequality (1.16) while for the univariate case with ∫ ∞ −∞ w (t) dt = 1 we have 1 2 md (w) ≤ rg (w) ≤ i (w) ≤ md (w) ≤ σ (φ, f; w) (2.9) where i (w) := ı́nf γ∈r ∫ ∞ −∞ w (x) |x − γ| dx. (2.10) if w is supported on the finite interval [a, b] , namely ∫b a w (x) dx = 1, then we have the chain of inequalities 1 2 md (w) ≤ rg (w) ≤ i (w) ≤ md (w) ≤ σ (φ, f; w) ≤ 1 2 (m − m) . (2.11) 8 silvestru sever dragomir cubo 22, 1 (2020) 3. bounds for various classes of functions in the case of functions of bounded variation we have: theorem 2. let φ : [a, b] → r be a function of bounded variation on the closed interval [a, b] . if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and if f : ω → [a, b] is a ν-measurable function with φ ◦ f ∈ lw (ω, ν) , then rg (φ, f; w) ≤ 1 2 b ∨ a (φ) , (3.1) where ∨b a (φ) is the total variation of φ on [a, b] . demostración. using the inequality (2.4) we have rg (φ, f; w) ≤ ∫ ω w (x) |(φ ◦ f) (x) − γ| dν (x) (3.2) for any γ ∈ r. by the triangle inequality, we have ∣ ∣ ∣ ∣ (φ ◦ f) (x) − 1 2 [φ (a) + φ (b)] ∣ ∣ ∣ ∣ ≤ 1 2 |φ (a) − φ (f (x))| + 1 2 |φ (b) − φ (f (x))| (3.3) for any x ∈ ω. since φ : [a, b] → r is of bounded variation and d is a division of [a, b] , namely d ∈ d ([a, b]) := {d := {a = t0 < t1 < ... < tn = b}} , then b ∨ a (φ) = sup d∈d([a,b]) n−1∑ i=0 |φ (ti+1) − φ (ti)| < ∞. taking the division d0 := {a = t0 < t < t2 = b} we then have |φ (t) − φ (a)| + |φ (b) − φ (t)| ≤ b ∨ a (φ) for any t ∈ [a, b] and then |φ (f (x)) − φ (a)| + |φ (b) − φ (f (x))| ≤ b ∨ a (φ) (3.4) for any x ∈ ω. cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 9 on making use of (3.3) and (3.4) we get ∣ ∣ ∣ ∣ (φ ◦ f) (x) − 1 2 [φ (a) + φ (b)] ∣ ∣ ∣ ∣ ≤ 1 2 b ∨ a (φ) (3.5) for any x ∈ ω. if we multiply (3.5) by w (x) and integrate, then we obtain ∫ ω w (x) ∣ ∣ ∣ ∣ (φ ◦ f) (x) − 1 2 [φ (a) + φ (b)] ∣ ∣ ∣ ∣ ≤ 1 2 b ∨ a (φ) . (3.6) finally, by choosing γ = 1 2 [φ (a) + φ (b)] in (3.2) and making use of (3.6) we deduce the desired result (3.1). in the case of absolutely continuous functions we have: theorem 3. let φ : [a, b] → r be an absolutely continuous function on the closed interval [a, b] . if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and if f : ω → [a, b] is a ν-measurable function with φ ◦ f ∈ lw (ω, ν) , then rg (φ, f; w) ≤    ‖φ′‖ [a,b],∞ rg (f; w) if φ ′ ∈ l ∞ ([α, β]) , 1 21/p ‖φ′‖ [a,b],p r 1/q g (f; w) if φ ′ ∈ lp ([α, β]) , p > 1, 1 p + 1 q = 1, (3.7) where the lebesgue norms are defined by ‖g‖ [α,β],p :=    essupt∈[α,β] |g (t)| if p = ∞, (∫β α |g (t)| p dt )1/p if p ≥ 1 and lp ([α, β]) := { g| g measurable and ‖g‖ [α,β],p < ∞ } , p ∈ [1, ∞] . demostración. since f is absolutely continuous, then we have φ (t) − φ (s) = ∫t s φ′ (u) du for any t, s ∈ [a, b] . using the hölder integral inequality we have |φ (t) − φ (s)| = ∣ ∣ ∣ ∣ ∫t s φ′ (u) du ∣ ∣ ∣ ∣ ≤    ‖φ′‖ [a,b],∞ |t − s| if p = ∞, ‖φ′‖ [a,b],p |t − s| 1/q if p > 1, 1 p + 1 q = 1 (3.8) 10 silvestru sever dragomir cubo 22, 1 (2020) for any t, s ∈ [a, b] . using (3.8) we then have |(φ ◦ f) (x) − (φ ◦ f) (y)| ≤    ‖φ′‖ [a,b],∞ |f (x) − f (y)| if p = ∞, ‖φ′‖[a,b],p |f (x) − f (y)| 1/q if p > 1, 1 p + 1 q = 1 (3.9) for any x, y ∈ ω. if we multiply (3.9) by 1 2 w (x) w (y) and integrate, then we get 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (x) − (φ ◦ f) (y)| dν (x) dν (y) ≤    1 2 ‖φ′‖[a,b],∞ ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) if p = ∞, 1 2 ‖φ′‖ [a,b],p ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| 1/q dν (x) dν (y) if p > 1, 1 p + 1 q = 1. (3.10) this proves the first branch of (3.7). using jensen’s integral inequality for concave function ψ (t) = ts, s ∈ (0, 1) we have for s = 1 q < 1 that ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| 1/q dν (x) dν (y) ≤ (∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q , which implies that 1 2 ‖φ′‖ [a,b],p ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| 1/q dν (x) dν (y) ≤ 1 2 ‖φ′‖ [a,b],p (∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q = ‖φ′‖ [a,b],p ( 1 2q ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q = ‖φ′‖ [a,b],p ( 1 2q−1 1 2 ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q = 1 2 q−1 q ‖φ′‖ [a,b],p (rg (f; w)) 1/q = 1 21/p ‖φ′‖ [a,b],p r 1/q g (f; w) and the second part of (3.7) is proved. the function φ : [a, b] → r is called of r-h-hölder type with the given constants r ∈ (0, 1] and h > 0 if |φ (t) − φ (s)| ≤ h |t − s| r cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 11 for any t, s ∈ [a, b] . in the case when r = 1, namely, there is the constant l > 0 such that |φ (t) − φ (s)| ≤ l |t − s| for any t, s ∈ [a, b] , the function φ is called l-lipschitzian on [a, b] . we have: theorem 4. let φ : [a, b] → r be a function of r-h-hölder type on the closed interval [a, b] . if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν-a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and if f : ω → [a, b] is a ν-measurable function with φ ◦ f ∈ lw (ω, ν) , then rg (φ, f; w) ≤ 1 21−r hrrg (f; w) . (3.11) in particular, if φ is l-lipschitzian on [a, b] , then rg (φ, f; w) ≤ lrg (f; w) . (3.12) demostración. we have |(φ ◦ f) (x) − (φ ◦ f) (y)| ≤ h |f (x) − f (y)| r (3.13) for any x, y ∈ ω. if we multiply (3.13) by 1 2 w (x) w (y) and integrate, then we get 1 2 ∫ ω ∫ ω w (x) w (y) |(φ ◦ f) (x) − (φ ◦ f) (y)| dν (x) dν (y) ≤ 1 2 h ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| r dν (x) dν (y) . (3.14) by jensen’s integral inequality for concave functions we also have ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| r dν (x) dν (y) ≤ (∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )r . (3.15) therefore, by (3.14) and (3.15) we get rg (φ, f; w) ≤ 1 2 h (∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )r = 1 21−r h ( 1 2 ∫ ω ∫ ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )r = 1 21−r hrrg (f; w) and the inequality (3.11) is proved. 12 silvestru sever dragomir cubo 22, 1 (2020) we have: theorem 5. let φ, ψ : [a, b] → r be continuos functions on [a, b] and differentiable on (a, b) with ψ′ (t) 6= 0 for t ∈ (a, b) . if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and if f : ω → [a, b] is a ν-measurable function with φ ◦ f ∈ lw (ω, ν) , then ı́nf t∈(a,b) ∣ ∣ ∣ ∣ φ′ (t) ψ′ (t) ∣ ∣ ∣ ∣ rg (ψ, f; w) ≤ rg (φ, f; w) ≤ sup t∈(a,b) ∣ ∣ ∣ ∣ φ′ (t) ψ′ (t) ∣ ∣ ∣ ∣ rg (ψ, f; w) . (3.16) demostración. by the cauchy’s mean value theorem, for any t, s ∈ [a, b] with t 6= s there exists a ξ between t and s such that φ (t) − φ (s) ψ (t) − ψ (s) = φ′ (ξ) ψ′ (ξ) . this implies that ı́nf τ∈(a,b) ∣ ∣ ∣ ∣ φ′ (τ) ψ′ (τ) ∣ ∣ ∣ ∣ |ψ (t) − ψ (s)| ≤ |φ (t) − φ (s)| ≤ sup τ∈(a,b) ∣ ∣ ∣ ∣ φ′ (τ) ψ′ (τ) ∣ ∣ ∣ ∣ |ψ (t) − ψ (s)| (3.17) for any t, s ∈ [a, b] . therefore, we have ı́nfτ∈(a,b) ∣ ∣ ∣ ∣ φ′ (τ) ψ′ (τ) ∣ ∣ ∣ ∣ |ψ (f (x)) − ψ (f (y))| ≤ |φ (f (x)) − φ (f (y))| ≤ supt∈(a,b) ∣ ∣ ∣ ∣ φ′ (τ) ψ′ (τ) ∣ ∣ ∣ ∣ |ψ (f (x)) − ψ (f (y))| (3.18) for any x, y ∈ ω. if we multiply (3.18) by 1 2 w (x) w (y) and integrate, we get the desired result (3.16). corollary 2. let φ : [a, b] → r be a continuos function on [a, b] and differentiable on (a, b) . if w is as in theorem 5, then we have ı́nf t∈(a,b) |φ′ (t)| rg (f; w) ≤ rg (φ, f; w) ≤ sup t∈(a,b) |φ′ (t)| rg (f; w) . (3.19) we also have: theorem 6. let φ : [a, b] → r be an absolutely continuous function on the closed interval [a, b] . if w : ω → r is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 13 and if f : ω → [a, b] is a ν-measurable function with φ ◦ f ∈ lw (ω, ν) , then rg (φ, f; w) ≤    ‖φ′‖ [a,b],∞ m (f; w) if p = ∞, ‖φ′‖ [a,b],p m 1/q (f; w) if p > 1, 1 p + 1 q = 1 ≤    1 2 (b − a) ‖φ′‖ [a,b],∞ if p = ∞, 1 21/q (b − a) 1/q ‖φ′‖ [a,b],p if p > 1, 1 p + 1 q = 1, (3.20) where m (f; w) is defined by m (f; w) := ∫ ω w (x) ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ dν (x) . (3.21) demostración. from the inequality (3.8) we have ∣ ∣(φ ◦ f) (x) − φ ( a+b 2 ) ∣ ∣ ≤    ‖φ′‖ [a,b],∞ ∣ ∣f (x) − a+b 2 ∣ ∣ if p = ∞, ‖φ′‖[a,b],p ∣ ∣f (x) − a+b 2 ∣ ∣ 1/q if p > 1, 1 p + 1 q = 1 (3.22) for any x ∈ ω. now, if we multiply (3.22) by w (x) and integrate, then we get ∫ ω w (x) ∣ ∣ ∣ ∣ (φ ◦ f) (x) − φ ( a + b 2 ) ∣ ∣ ∣ ∣ dν (x) ≤    ‖φ′‖ [a,b],∞ ∫ ω w (x) ∣ ∣f (x) − a+b 2 ∣ ∣dν (x) if p = ∞, ‖φ′‖ [a,b],p ∫ ω w (x) ∣ ∣f (x) − a+b 2 ∣ ∣ 1/q dν (x) if p > 1, 1 p + 1 q = 1. (3.23) by jensen’s integral inequality for concave functions we have ∫ ω w (x) ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ 1/q dν (x) ≤ (∫ ω w (x) ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ dν (x) )1/q . (3.24) on making use of (3.2), (3.23) and (3.24) we get the first inequality in (3.20). the last part of (3.20) follows by the fact that ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ ≤ 1 2 (b − a) for any x ∈ ω. 14 silvestru sever dragomir cubo 22, 1 (2020) 4. bounds for special convexity when some convexity properties for the function φ are assumed, then other bounds can be derived as follows. theorem 7. let w : ω → r be a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and f : ω → [a, b] be a ν-measurable function with φ ◦ f ∈ lw (ω, ν) . assume also that φ : [a, b] → r is a continuous function on [a, b] . (i) if |φ| is concave on [a, b] , then rg (φ, f; w) ≤ |φ (e (f; w))| , (4.1) (ii) if |φ| is convex on [a, b] , then rg (φ, f; w) ≤ 1 b − a [(b − e (f; w)) |φ (a)| + (e (f; w) − a) φ |(b)|] . (4.2) demostración. (i) if |φ| is concave on [a, b] , then by jensen’s inequality we have ∫ ω w (x) |(φ ◦ f) (x)| dν (x) ≤ ∣ ∣ ∣ ∣ φ (∫ ω w (x) f (x) dν (x) ) ∣ ∣ ∣ ∣ . (4.3) from (3.2) for γ = 0 we also have rg (φ, f; w) ≤ ∫ ω w (x) |(φ ◦ f) (x)| dν (x) . (4.4) this is an inequality of interest in itself. on utilizing (4.3) and (4.4) we get (4.1). (ii) since |φ| is convex on [a, b] , then for any t ∈ [a, b] we have |φ (t)| = ∣ ∣ ∣ ∣ φ ( (b − t) a + b (t − a) b − a ) ∣ ∣ ∣ ∣ ≤ (b − t) |φ (a)| + (t − a) φ |(b)| b − a . this implies that |(φ ◦ f) (x)| ≤ (b − f (x)) |φ (a)| + (f (x) − a) φ |(b)| b − a (4.5) for any x ∈ ω. if we multiply (4.5) by w (x) and integrate, then we get ∫ ω w (x) |(φ ◦ f) (x)| dν (x) ≤ 1 b − a [( b ∫ ω w (x) dν (x) − ∫ ω w (x) f (x) dν (x) ) |φ (a)| + (∫ ω w (x) f (x) dν (x) − a ∫ ω w (x) dν (x) ) φ |(b)| ] , which, together with (4.4), produces the desired result (4.2). cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 15 in order to state other results we need the following definitions: definition 1 ([19]). we say that a function f : i → r belongs to the class p (i) if it is nonnegative and for all x, y ∈ i and t ∈ [0, 1] we have f (tx + (1 − t) y) ≤ f (x) + f (y) . it is important to note that p (i) contains all nonnegative monotone, convex and quasi convex functions, i.e. functions satisfying f (tx + (1 − t) y) ≤ máx {f (x) , f (y)} for all x, y ∈ i and t ∈ [0, 1] . for some results on p-functions see [19] and [28] while for quasi convex functions, the reader can consult [18]. definition 2 ([3]). let s be a real number, s ∈ (0, 1]. a function f : [0, ∞) → [0, ∞) is said to be s-convex (in the second sense) or breckner s-convex if f (tx + (1 − t) y) ≤ tsf (x) + (1 − t) s f (y) for all x, y ∈ [0, ∞) and t ∈ [0, 1] . for some properties of this class of functions see [1], [2], [3], [4], [16], [17], [25], [27] and [29]. theorem 8. let w : ω → r be a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1 and f : ω → [a, b] be a ν-measurable function with φ ◦ f ∈ lw (ω, ν) . assume also that φ : [a, b] → r is a continuous function on [a, b] . (i) if |φ| belongs to the class p on [a, b] , then rg (φ, f; w) ≤ |φ (a)| + φ |(b)| ; (4.6) (ii) if |φ| is quasi convex on [a, b] , then rg (φ, f; w) ≤ máx {|φ (a)| , φ |(b)|} ; (4.7) (iii) if |φ| is breckner s-convex on [a, b] , then rg (φ, f; w) ≤ 1 (b − a) s [ |φ (a)| ∫ ω w (x) (b − f (x)) s dν (x) +φ |(b)| ∫ ω w (x) (f (x) − a) s dν (x) ] ≤ 1 (b − a) s [ |φ (a)| (b − e (f; w)) s dν (x) +φ |(b)| (e (f; w) − a) s dν (x) ] . (4.8) 16 silvestru sever dragomir cubo 22, 1 (2020) demostración. (i) since |φ| belongs to the class p on [a, b] , then for any t ∈ [a, b] we have |φ (t)| = ∣ ∣ ∣ ∣ φ ( (b − t) a + b (t − a) b − a ) ∣ ∣ ∣ ∣ ≤ |φ (a)| + φ |(b)| . this implies that |(φ ◦ f) (x)| ≤ |φ (a)| + φ |(b)| (4.9) for any x ∈ ω. if we multiply (4.9) by w (x) and integrate, then we get ∫ ω w (x) |(φ ◦ f) (x)| dν (x) ≤ |φ (a)| + φ |(b)| , (4.10) which, together with (4.4), produces the desired result (4.6). (ii) goes in a similar way. (iii) by breckner s-convexity we have |φ (t)| = ∣ ∣ ∣ ∣ φ ( (b − t) a + b (t − a) b − a ) ∣ ∣ ∣ ∣ ≤ ( b − t b − a )s |φ (a)| + ( t − a b − a )s φ |(b)| for any t ∈ [a, b] . this implies that |(φ ◦ f) (x)| ≤ 1 (b − a) s [ (b − f (x)) s |φ (a)| + (f (x) − a) s φ |(b)| ] (4.11) for any x ∈ ω. if we multiply (4.11) by w (x) and integrate, then we get ∫ ω w (x) |(φ ◦ f) (x)| dν (x) ≤ 1 (b − a) s [ |φ (a)| ∫ ω w (x) (b − f (x)) s dν (x) +φ |(b)| ∫ ω w (x) (f (x) − a) s dν (x) ] , (4.12) which, together with (4.4), produces the first part of (4.8). the last part follows by jensen’s integral inequality for concave functions, namely ∫ ω w (x) (b − f (x)) s dν (x) ≤ ( b − ∫ ω w (x) f (x) dν (x) )s and ∫ ω w (x) (f (x) − a) s dν (x) ≤ (∫ ω w (x) f (x) dν (x) − a )s , where s ∈ (0, 1) . cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 17 5. some examples let f : ω → [0, ∞) be a ν-measurable function and w : ω → r a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1. we define, for the function φ (t) = tp, p > 0, the generalized (p, f)-mean difference rg (p, f; w) by rg (p, f; w) := 1 2 ∫ ω ∫ ω w (x) w (y) |fp (x) − fp (y)| dν (x) dν (y) (5.1) and the generalized (p, f)-mean deviation md (p, f; w) by md (p, f; w) := ∫ ω w (x) |fp (x) − e (p, f; w)| dν (x) , (5.2) where e (p, f; w) := ∫ ω fp (y) w (y) dν (y) (5.3) is the generalized (p, f)-expectation. if f : ω → [a, b] ⊂ [0, ∞) is a ν-measurable function, then by (3.1) we have rg (p, f; w) ≤ 1 2 (bp − ap) . (5.4) by (3.7) we have rg (p, f; w) ≤ pδp (a, b) rg (f; w) , (5.5) where δp (a, b) :=    bp−1 if p ≥ 1, ap−1 if p ∈ (0, 1) and rg (p, f; w) ≤ p 21/α [ bα(p−1)+1 − aα(p−1)+1 α (p − 1) + 1 ]1/α r 1/β g (f; w) , (5.6) where α > 1, 1 α + 1 β = 1. from (3.20) we also have rg (p, f; w) ≤    δp (a, b) m (f; w) , p ( bα(p−1)+1−aα(p−1)+1 α(p−1)+1 )1/α m1/β (f; w) if α > 1, 1 α + 1 β = 1 ≤    1 2 (b − a) δp (a, b) , 1 21/β (b − a) 1/β p ( b α(p−1)+1 −a α(p−1)+1 α(p−1)+1 )1/α if α > 1, 1 α + 1 β = 1, (5.7) 18 silvestru sever dragomir cubo 22, 1 (2020) where m (f; w) is defined by (3.21). if p ∈ (0, 1) , then the function |φ (t)| = tp is concave on [a, b] ⊂ [0, ∞) and by (4.1) we have rg (p, f; w) ≤ e p (f; w) . (5.8) for p ≥ 1 the function |φ (t)| = tp is convex on [a, b] ⊂ [0, ∞) and by (4.2) we have rg (p, f; w) ≤ 1 b − a [(b − e (f; w)) ap + (e (f; w) − a) bp] . (5.9) let f : ω → [0, ∞) be a ν-measurable function and w : ω → r a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ ω and ∫ ω w (x) dν (x) = 1. we define, for the function φ (t) = ln t, the generalized (ln, f)-mean difference rg (ln, f; w) by rg (ln, f; w) := 1 2 ∫ ω ∫ ω w (x) w (y) |ln f (x) − ln f (y)| dν (x) dν (y) (5.10) and the generalized (p, f)-mean deviation md (ln, f; w) by md (ln, f; w) := ∫ ω w (x) |ln f (x) − e (ln, f; w)| dν (x) , (5.11) where e (ln, f; w) := ∫ ω w (y) ln f (y) dν (y) (5.12) is the generalized (ln, f)-expectation. if f : ω → [a, b] ⊂ [0, ∞) is a ν-measurable function, then by (3.1) we have rg (ln, f; w) ≤ 1 2 (ln b − ln a) . (5.13) by (3.7) we have rg (ln, f; w) ≤    1 a rg (f; w) , 1 21/p ( bp−1−ap−1 (p−1)bp−1ap−1 )1/p r 1/q g (f; w) if p > 1, 1 p + 1 q = 1. (5.14) by (3.20) we have rg (ln, f; w) ≤    1 a m (f; w) , ( bp−1−ap−1 (p−1)bp−1ap−1 )1/p m1/q (f; w) if p > 1, 1 p + 1 q = 1 ≤    1 2 ( b a − 1 ) , 1 21/q (b − a) 1/q ( b p−1 −a p−1 (p−1)bp−1ap−1 )1/p if p > 1, 1 p + 1 q = 1. (5.15) cubo 22, 1 (2020) bounds for the generalized (φ, f)-mean difference 19 now, observe that the function |φ (t)| = |ln t| is convex on (0, 1) and concave on [1, ∞). if f : ω → [a, b] ⊂ (0, 1) is a ν-measurable function, then by (4.2) we have rg (ln, f; w) ≤ 1 b − a [(b − e (f; w)) |ln a| + (e (f; w) − a) |ln b|] (5.16) and if f : ω → [a, b] ⊂ [1, ∞), then by (4.1) we have rg (ln, f; w) ≤ ln (e (f; w)) . 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[29] e. set, m. e. özdemir and m. z. sarikaya, new inequalities of ostrowski’s type for s-convex functions in the second sense with applications. facta univ. ser. math. inform. 27 (2012), no. 1, 67–82. introduction general bounds bounds for various classes of functions bounds for special convexity some examples a mathematical journal vol. 7, no 2, (69 79). august 2005. two-phase structures as singular limit of a one-dimensional discrete model thomas blesgen max-planck-institute for mathematics in the sciences inselstraße 22-26, d-04103 leipzig, germany e-mail: blesgen@mis.mpg.de abstract a one-dimensional energy functional that models the elastic free energy of a monatomic chain of atoms occupying a bounded real domain is discussed and the γ-limit of this functional when the number of particles becomes infinite is derived. the particular ansatz allows for the first time the presence of two coexisting phases in the singular limit and thus can be used as a prototype towards modeling three dimensional cases of physical relevance. resumen se discute una funcional de enerǵıa unidimensional que modela la enerǵıa elástica liberada po una cadena monotómica de átomos ocupando un dominio real acotado y se deduce el ĺımite γ de esta funcional cuando el número de part́ıculas llega a ser infinito. el particular ansatz permite por primera vez la presencia de dos fases coexistentes en el ĺımite singular y de esta forma puede ser utilizado como un prototipo para modelar casos tridimensionales de relevancia en f́ısica. key words and phrases: singular limits, phase transitions, elasticity math. subj. class.: 35e99, 49j45, 74m25, 74n99 70 thomas blesgen 7, 2(2005) 1 introduction in this article we study the behaviour of a one-dimensional monatomic lattice comprising of (n + 1) particles that interact via three and four body potentials that represent the interatomic forces. we are interested in the description of the γ-limit when n → ∞. this leads to a continuum theory of regular crystals under the idealising assumption that the interatomic distance vanishes in the limit. the notion of γ-limit goes back to work by de giorgi, [5]. γ-limits can be regarded as a convergence related not only to one lattice, but provide a natural framework to formulate convergence with respect to an entire family of lattices, depending on the parameter n. a comprehensive discussion of this issue can be found in [1]. the derivation of a continuum theory relying on atomistic descriptions has only recently become a focus of research after the article [2] studied softening in fracture mechanics. in a similar direction goes [3]. in [6] a stochastic framework is considered, in [8] the topic of phase transitions is discussed by considering oscillations of lattices. in this article, we can extend the analysis in these articles and will discuss a prototype of a free energy functional whose minimisers give rise to two different lattice structures. this energy functional consists of one part that accounts for the local lattice symmetry, a second part that represents the surface energy and a third contribution that stands for the elastic energy. this work is organised in the following way. in section 2, a one-dimensional discrete free energy functional w n(un) for deformations un is introduced that describes the elastic energy of the atomic chain. in section 3, the γ-limit n → ∞ of w n(un) is identified. the article is ended by a short discussion and outlook. 2 the energy functional let ω := (0, 1) ⊂ r be the domain that contains a regular monatomic chain. we suppose that the undeformed discrete reference configuration of ω is given by a system of n + 1 atoms with equal distance located at points rni ∈ r, where rni := ihn 0 ≤ i ≤ n. here, the setting hn := 1/n defines for given number n ∈ n the interatomic distance. the limit n →∞ corresponds to hn ↘ 0. the index n is always used to indicate the dependency on the number of subdivisions. by r̂ni , 0 ≤ i ≤ n we denote the position of atom i after the deformation. finally, by uni , 0 ≤ i ≤ n we denote the two-dimensional displacement vector of atom i, given by the relationship uni = r̂ n i −rni , 0 ≤ i ≤ n. for shortness we introduce the numbers s1 := 1, s2 := 2 and s3 := 12 and let dnuni := uni+1 −uni hn . 7, 2(2005) two-phase structures as singular limit of a ... 71 this is a forward difference quotient that approximates (uni ) ′ for small hn. we will study the behaviour of the following energy functional. w n(un) := { +∞ if uni+1 = uni for some i,∑3 k=1 w n k (u n) else where w n1 (u n) := n−2∑ i=0 nα 3∏ k=1 ∣∣∣sk − u n i+2 −uni+1 uni+1 −uni ∣∣∣2, w n2 (u n) := n−3∑ i=0 ∣∣∣1 − uni+3 −uni+2 uni+1 −uni ∣∣∣2, w n3 (u n) := hn n−2∑ i=0 [(uni+2 −uni 2hn −α1 )2 βni + (uni+2 −uni 2hn −α2 )2 γni ] and βni := [ 1 −nα ∣∣∣1 − uni+2 −uni+1 uni+1 −uni ∣∣∣2 ] + , γni := [ 1 −nα ∣∣∣2 − uni+2 −uni+1 uni+1 −uni ∣∣∣2 ∣∣∣1 2 − u n i+2 −uni+1 uni+1 −uni ∣∣∣2 ] + . here, 0 < α < 1 and [·]+ denotes the positive part, this is [x]+ = x for x ≥ 0 and [x]+ = 0 for x < 0. the concept behind the ansatz for w n is the following. a minimiser of w n1 either fulfils dnuni+1 dnuni which specifies one lattice order that is in the sequel referred to as phase 1, or dnuni+1 2dnuni resp. dnuni+1 12dnuni which characterises phase 2. w n2 represents a surface energy. it counts(and limits) the number of transitions between the two phases, as within a phase one asymptotically has dnuni+2 = d nuni . finally, w n3 represents an elastic energy. we will show below that β n i converges in l1(ω) to the indicator function of phase 1 and γni to the indicator function of phase 2 as n →∞. αk corresponds to the elastic constant to phase k. the functional w n1 represents the electrostatic energy due to interatomic potentials that force the atoms to positions of a certain given lattice order. the given example of a doubling (halfing) of the period of dnun is only a first simple example. energy functionals motivated by real physical applications are higher dimensional and much more complicated. for the analysis we extend the discrete deformation values {uni }i, to piecewise linear functions un in l2(ω)∩an, where an denotes the space of piecewise linear functions, see [2]. in this article, discrete quantities are always specified by subscript i. 72 thomas blesgen 7, 2(2005) 3 identification of the γ-limit for w n now we can state the main result. it characterises the γ-limit of w n as n tends to infinity. we use the notation χ1 := χ, χ2 := 1−χ. for u ∈ h1,2(ω), χ ∈ bv (ω,{0, 1}) we define e(u,χ) := 1 4 ∫ ω |∇χ| + 2∑ k=1 ∫ ω χk (u ′ −αk)2. additionally we introduce w : l2(ω) → r by w(u) := { infχ∈bv (ω,{0,1}) e(u,χ) if u ∈ h1,2(ω) is strictly monotone, +∞ else. theorem 3.1 (characterisation of the γ-limit of w n) the following statements are valid: (i) the boundedness of the energy functional w n(un) implies the boundedness of(∫ ω |(un)′|2 ) n uniformly in n. (ii) w is the γ-limit of w n as n →∞ with respect to the convergence in l2(ω). proof of (i): step 1: construction of the characteristic function χ: by c we denote various positive constants that may change from line to line. let (un) ⊂ l2(ω) be a sequence with w n(un) ≤ c. we set dik := ∣∣∣uni+2 −uni+1 uni+1 −uni −sk ∣∣∣, ki0 := argmin { k �→ dik ∣∣ 1 ≤ k ≤ 3}. the boundedness of w n1 (u n) implies n−1∑ i=0 nα 3∏ k=1 ( sk − uni+2 −uni+1 uni+1 −uni )2 ≤ c. therefore there exists a constant c > 0 such that sup i diki0 ≤ chα/2n . (1) for sufficiently large n we can thus define an indicator function χn to phase 1 by χn(x) := ⎧⎨ ⎩ 0 if x ∈ [ihn, (i + 1)hn), i ≤ n− 2, ki0 = 1, 1 if x ∈ [ihn, (i + 1)hn), i ≤ n− 2, ki0 = 1, χn(1 − 2hn) if x ∈ [1 −hn, 1]. next we show that χn ∈ bv (ω; {0, 1}), i.e. ∫ ω |∇χn| ≤ c. (2) 7, 2(2005) two-phase structures as singular limit of a ... 73 this follows from the boundedness of w n2 (u n). since for large n uni+2 −uni+1 uni+1 −uni = sk + o(1) for some k ∈{1, 2, 3}, we see that if χn(x) jumps in x = (i + 1)hn between 0 and 1, then ( 1 − u n i+3 −uni+2 uni+1 −uni )2 ≥ 1 4 + o(1) which shows w n2 (u n) ≥ ( 1 4 + o(1) )∫ ω |∇χn| and proves (2). here we adapted the landau notation and denote by o(1) terms that tend to 0 as n →∞. with (2), well-known compactness results imply the existence of a subsequence (again denoted by) χn and a χ ∈ bv (ω, {0, 1}) such that χn → χ in l1(ω). step 2: convergence of βn, γn in l1(ω): we extend the discrete quantities {βni }i, {γni }i to piecewise constant functions in l1(ω) by the definition βn(x) := { βni if x ∈ [ihn, (i + 1)hn) and i ≤ n− 2, 0 if x ∈ [1 −hn, 1]. in the same manner, the extension γn of {γni }i is defined. now we show that βn → χ, γn → (1 −χ) in l1(ω) for n →∞, (3) where the function χ ∈ bv (ω, {0, 1}) is the limit of χn found in step 1. without loss of generality we may restrict to i ≤ n− 2. case 1: χn(ihn) = 1. fix a small ε > 0. from the boundedness of w n(un) together with (1) we see that there exists a n0 ∈ n such that∣∣∣1 − uni+2 −uni+1 uni+1 −uni ∣∣∣2nα ≤ ε for all n ≥ n0. so we find 1 ≥ βni ≥ 1 −ε for large n. similarly, ∣∣∣2 − uni+2 −uni+1 uni+1 −uni ∣∣∣2 ∣∣∣1 2 − u n i+2 −uni+1 uni+1 −uni ∣∣∣2 ≥ hαn for all n ≥ n0 and thus γni = 0 for sufficiently large n. case 2: χn(ihn) = 0. analogous to case 1 we see that for large n ∣∣∣2 − uni+2 −uni+1 uni+1 −uni ∣∣∣2 ∣∣∣1 2 − u n i+2 −uni+1 uni+1 −uni ∣∣∣2nα ≤ ε, 74 thomas blesgen 7, 2(2005) so 1 ≥ γni ≥ 1 −ε for large n. similarly, ∣∣∣1 − uni+2 −uni+1 uni+1 −uni ∣∣∣2 ≥ hαn and βni = 0 for sufficiently large n. the discussion of these two cases yields the pointwise convergence of βn to χ and of γn to 1 −χ. together with lebesgue’s dominated convergence theorem the proof of (3) is finished. step 3: boundedness of ∫ ω |(un)′|2 uniformly in n: we choose constants a ∈ r+, b ∈ r such that min{(x−α1)2, (x−α2)2}≥ ax2 − b. due to the boundedness of w n3 (u n) we thus find that there exist constants c1, c2 > 0 such that c1 ≥ h2nc2 n−2∑ i=0 (uni+2 −uni 2hn )2( βni + γ n i ) . since dnuni+1 = skd nuni + o(1) for one k ∈{1, 2, 3} and large n we find that (uni+2 −uni 2hn )2 ≥ ( 1 + 1 2 + o(1) )(uni+1 −uni 2hn )2 . the term ( βni + γ n i ) can for large n be estimated from below by a constant. so we find the existence of a constant c > 0 with c ≥ h2n n−2∑ i=0 (uni+1 −uni 2hn )2 . (4) due to the estimate (dnunn−1) 2 ≤ (2 + o(1))dnunn−2 the sum in (4) can be extended to i = n− 1 and the estimate still holds. the sum ∑ i(d nuni ) 2 is directly related to ∫ ω |(un)′|2 where un is the piecewise affine linear extension of {uni }i. with (4) extended to i = n− 1 this finally yields sup n ∫ ω |(un)′|2 = sup n hn n−1∑ i=0 (dnuni ) 2 ≤ c. (5) proof of (ii): we assume that the reader is familiar with the concept of γ-convergence. step 4: lower semicontinuity inequality along the sequence w n: we have to show: for every sequence (un)n∈n with un → u in l2(ω) there exists a subsequence (unk )k∈n with w(u) ≤ lim inf k→∞ w nk (unk ). 7, 2(2005) two-phase structures as singular limit of a ... 75 if w n(un) is unbounded, there is nothing to show. hence we may assume w.l.o.g. w n(un) ≤ c for all n. from (5) follows un, u ∈ h1,2(ω) for all n ∈ n. because of the reflexivity of the hilbert space h1,2(ω) we know that there exists a subsequence (again denoted by) un such that un ⇀ u, in h1,2(ω) for n →∞. from step 2 we know that χn → χ, βn → χ, γn → 1 −χ in l1(ω) for n →∞. because of uni+2−uni+1 un i+1−uni ≥ 1 2 +o(1), for n ≥ n0 we find that un is monotone for sufficiently large n. now we estimate w n(un) from below. we claim lim inf n→∞ w n(un) ≥ e(u,χ) ≥ w(u). (6) in order to prove (6), let us estimate every component of w n(un) separately. 1. w n1 (u n) ≥ 0. 2. estimate of the surface energy: lim inf n→∞ w n2 (u n) ≥ lim inf n→∞ (1 4 + o(1) )∫ ω |∇χn| ≥ 1 4 ∫ ω |∇χ|. 3. estimate of the elastic energy: for x ∈ ω let ũn be given by ũn(x) := { un(x+2hn)−un(x) 2hn if x ∈ (0, 1 − 2hn), 0 if x ∈ [1 − 2hn, 1), using this notation we may rewrite w n3 , w n3 (u n) = ∫ ω [ (ũn −α1)2βn + (ũn −α2)2γn ] . next we show ũn → u′ in l2(ω) for n →∞. (7) we observe ∣∣∣un(x + 2hn) −un(x) 2hn ∣∣∣ ≤ 1∫ 0 |(un)′(x + shn)|ds (8) and due to the boundedness of un in h1,2(ω), an application of hölder’s inequality yields the boundedness of the left hand side of (8) uniformly in n. with lebesgue’s dominated convergence theorem, ũn → u′ in l2(ω) follows. in the same way, the other convergence results in (7) can be derived. exploiting (7), with the help of theorem 3.4, p.74 in [4] we obtain lim inf n→∞ w n3 (u n) ≥ 2∑ k=1 ∫ ω (u′ −αk)2χk. 76 thomas blesgen 7, 2(2005) combining the estimates for w nl (u n), 1 ≤ l ≤ 3, (6) is shown. step 5: existence of a ”recovery sequence”: we have to find a sequence (un) ⊂ l2(ω) converging to u in l2(ω) with w(u) ≥ lim sup n→∞ w n(un). if w(u) = +∞, there is nothing to show. due to the monotonicity properties of u demonstrated above we know that the functional χ �→ e(u,χ) is bounded from below in the bv-norm. using the compactness properties of bv (ω) and the coercivity of e, it is clear that e(u, ·) attains its minimum, i.e. w(u) = e(u,χ) for some χ ∈ bv (ω,{0, 1}). next we show that for piecewise affine, strictly monotone u there exists a sequence un with un → u and w n(un) → e(u,χ). we start with special cases, then generalise. case 1: u′ ≡ a1 > 0, χ ≡ const in ω: (a) χ ≡ 1 in ω: we simply set un := u for all n. (b) χ ≡ 0 in ω: for x > 0 choose un such that dnuni is alternating between 2 3 a1 and 43a1. furthermore un satisfies un(x = 0) = u(x = 0). case 2: u′ ≡ a1 > 0, χ ≡ 1 for 0 ≤ x ≤ 12 , χ ≡ 0 for x > 12 . the treatment of this case is more difficult. it is not possible to directly combine the two ansatz functions for un of case 1 because for one index i this would mean dnuni = a1hn and either dnuni+1 = 2 3 a1hn or dnuni+1 = 4 3 a1hn, leading to limn→∞ w n1 (u n) = ∞. therefore we have to introduce a transition layer of width hsn between the two phases, where s > 0 is a small constant to be chosen later. for convenience we introduce ϕn(x) := ⎧⎨ ⎩ a1 for 0 ≤ x ≤ 12, a1 + a1 3 ns(x− 1 2 ) for 1 2 < x ≤ 1 2 + hsn, 4 3 a1 for 12 + h s n < x ≤ 1. we set un such that un(x = 0) = u(x = 0) and dnuni := { ϕn(ihn) for ihn ≤ 12, 1 2 ϕn(ihn), ϕn(ihn) alternating for ihn > 12. let us now analyse what this means for the limit of w n(un). we notice uni+2 −uni+1 uni+1 −uni = sk if (i + 1)hn ≤ 1 2 or ihn > 1 2 . (9) if 1 2 < (i + 1)hn ≤ 12 + hsn and ihn ≤ 12 we have uni+2 −uni+1 uni+1 −uni = sk ϕn((i + 1)hn) ϕn(ihn) = sk a1 + a1 3 h1−sn a1 = sk ( 1 + 1 3 h1−sn ) (10) 7, 2(2005) two-phase structures as singular limit of a ... 77 with sk = 1 or sk = 12 . to guarantee the convergence to 0 of the corresponding expressions in w n1 which are weighted with a factor n α we require 0 < s < 1−α 2 . if 1 2 < ihn ≤ 12 + hsn and (i + 1)hn > 12 + hsn we have uni+2 −uni+1 uni+1 −uni = sk ϕn((i + 1)hn) ϕn(ihn) = sk 4 3 a1 a1 + a1 3 h−sn hsn = sk (11) where sk = 2 or sk = 12 . (9), (10) and (11) cover all possible cases and demonstrate the convergence to 0 of the dnuni -terms in w n 1 . hence we find w n 1 (u n) → 0 as n →∞. for the estimation of the functional w n2 (u n) we have ∣∣∣1 − uni+3 −uni+2 uni+1 −uni ∣∣∣2 = ∣∣∣1 − 1 2 ϕn((i + 2)hn) ϕn(ihn) ∣∣∣2 = ∣∣∣1 2 − 1 2 ϕn(ihn) −ϕn((i + 2)hn) ϕn(ihn) ∣∣∣2. for i := ϕ n(ihn)−ϕn((i+2)hn) ϕn(ihn) simple computations yield i = ⎧⎪⎪⎨ ⎪⎪⎩ 0 if (ihn > 12 ) or ((i + 1)hn ≤ 12 ) or ( 1 2 < ihn ≤ 12 + hsn and (i + 2)hn > 12 + hsn), −skh1−sn if (ihn > 12 and (i + 2)hn ≤ 12 + hsn) or (ihn ≤ 12 and 12 < (i + 1)hn ≤ 12 + hsn). and for 0 < s < 1 the convergence of w n2 (u n) to 1 4 can be assured. for the estimation of w n3 (u n), it is clear that outside the strip of width hsn the summands in w n3 (u n) are exactly equal to hsn [ χ(a1 −α1)2 + (1 −χ)(a1 −α2)2 ] . inside the strip, we have approximately hs−1n summands, where each summand is of the form hnc. thus, for s > 0 the part inside the strip tends to 0 for n →∞. case 3: general χ ∈ bv (ω; {0, 1}), u piecewise affine, monotone, continuous: the construction of un can be done by iteratively applying the construction given in case 2. 78 thomas blesgen 7, 2(2005) case 4: general monotone u ∈ h1,2(ω): let u be a generic monotone function in h1,2(ω) and let {un} be a sequence in an such that un → u in h1,2(ω). for every n we can apply case 3 to find a sequence {wnl }l such that wnl → un in l2(ω) as n → ∞ and lim supl w l(wnl ) ≤ w(un). then we have lim sup n→∞ lim sup l→∞ w l(wnl ) ≤ lim sup n→∞ w(un) = w(u), (12) where (12) holds because of the strong convergence of un to u in h1,2(ω). by diagonalisation, we find a sequence ũn := wn l(n) such that ũn → u in l2(ω) and lim supn→∞ w n(ũn) ≤ w(u,v). � 4 discussion and outlook the present article analysed the γ-limit of a one-dimensional lattice as the number of particles tends to infinity in a particular case. the discussed free energy functional represents a first example that gives rise to two different lattice orders. it seems very likely that this concept can be generalised to more than two phases and higher space dimensions although the energy is even more artificial in this case. physically relevant formulas for the elastic energy replacing (u′ − αk)2 can for instance be found in [7] and are mostly non-linear. finally it is important to realize that the concept of γ-convergence is only partly suitable for the understanding of the static behaviour of solids. this is because the γ-limit is bound to global minimisers whereas the evolution in nature may get stuck in a local minimum as the activation energy needed to pass an energy barrier is not available. unfortunately, at this point no appropriate mathematical instruments seem to be at hand to deal with formulations that arise from this consideration. received: june 2004. revised: august 2004. references [1] a. braides, approximation of free-discontinuity problems, lecture notes in mathematics 1694, springer 1998 [2] a. braides, g. dal maso and a. garroni,variational formulation of softening phenomena in fracture mechanics: the one-dimensional case, arch. rat. mech. anal.146(1999), 23–58 [3] a. braides and m. s. gelli, limits of discrete systems with long-range interactions, journal of convex analysis 9(2002), 363–399 7, 2(2005) two-phase structures as singular limit of a ... 79 [4] b. dacorogna, direct methods in the calculus of variations, springer 1989 [5] e. de giorgi, sulla convergenza di alcune successioni di integrali del tipo dell’area, rendiconti di matematica 4(1955), 95–113 [6] o. iosefescu, c. licht and g. michaille, variational limit of a one-dimensional discrete and statistically homogeneous system of material points, asympt. anal. 28(2001), 309–329 [7] j. f. nye, physical properties of crystals, clarendon press, oxford 1964 [8] s. pagano, r. paroni, a simple model for phase transitions: from the discrete to the continuum problem, quart.appl.math. 61(2003), 89–109 cubo, a mathematical journal vol.22, n◦02, (233–255). august 2020 http://dx.doi.org/10.4067/s0719-06462020000200233 received: 23 may, 2020 | accepted: 20 july, 2020 hyers-ulam stability of an additive-quadratic functional equation vediyappan govindan1, choonkil park2, sandra pinelas3 and themistocles m. rassias4 1 department of mathematics, sri vidya mandir arts & science college, katteri, india. 2 research institute of natural sciences, hanyang university, seoul-04763, korea. 3 departamento de ciências exatas e engenharia, academia militar, portugal. 4 department of mathematics, national technical university of athens, greece. govindoviya@gmail.com, baak@hanyang.ac.kr, sandra.pinelas@gmail.com, trassias@math.ntua.gr abstract in this paper, we introduce the following (a, b, c)-mixed type functional equation of the form g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2) where a, b, c are positive integers with a > 1, and investigate the solution and the hyers-ulam stability of the above functional equation in banach spaces by using two different methods. resumen en este art́ıculo introducimos la siguiente ecuación funcional de tipo (a, b, c)-mixta de la forma g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2) donde a, b, c son enteros positivos con a > 1, e investigamos la solución y la estabilidad de hyers-ulam de la ecuación funcional anterior en espacios de banach usando dos métodos diferentes. keywords and phrases: hyers-ulam stability, mixed type functional equation, banach space, fixed point. 2020 ams mathematics subject classification: 39b52, 32b72, 32b82. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000200233 234 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) 1 introduction the stability problem of functional equations originated form a question of ulam [28] concerning the stability of group homomorphisms. hyers [12] gave a first affirmative partial answer to the question of ulam [28] for banach spaces. hyers theorem was generalized by aoki [3] for additive mappings and rassias [12] for quadratic mappings. during the last three decades the stability theorem of rassias [26] provided a lot of influence for the development of stability theory of a large variety of functional equations (see [1, 2, 4, 7, 9, 11, 14, 17, 18, 21, 22, 23, 27]). one of the most famous functional equations is the following additive functional equation g(x + y) = g(x) + g(y) (1.1) in 1821, it was first solved by cauchy in the class of continuous real-valued functions. it is often called cauchy additive functional equation in honour of cauchy. the theory of additive functional equations is frequently applied to the development of theories of other functional equations. moreover, the properties of additive functional equations are powerful tools in almost every field of natural and social science ([6, 24, 26]). every solution of the additive functional equation (1.1) is called an additive mapping. the function g(x) = x2 satisfies the functional equation g(x + y) + g(x − y) = 2g(x) + 2g(y) (1.2) and therefore, the functional equation (1.2) is called quadratic functional equation. the hyersulam stability theorem for the quadratic functional equation (1.2) was proved by skof [25] for the mapping g : e1 → e2, where e1 is a normed space and e2 is a banach space. moslehian and rassias [20] studied the hyers-ulam stability problem in non-archimedean normed spaces. mirzavaziri and moslehian [19] studied the hyers-ulam stability of a quadratic functional equation in banach spaces by using the fixed point method and ciepliński [5] surveyed the hyers-ulam stability of functional equations by using the fixed point method. ebadian, ghobadipour and eshaghi gordji [8] proved the hyers-ulam stability of bimultipliers and jordan bimultipliers in c∗-ternary algebras by using the fixed point method for a three variable additive functional equation. motivated by ebadian et al. [8], we introduce the following three variable generalized additivequadratic functional equation of the form dg(x1, x2, x3) := g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 235 − [4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2)] = 0 (1.3) where a, b, c are positive integers with a > 1, and investigate the solution and the hyers-ulam stability of the three variable generalized additive-quadratic functional equation (1.3) in banach spaces by using the direct method and the fixed point method. 2 solution of the functional equation (1.3): when g is odd in this section, we investigate the solution of the functional equation (1.3) for an odd mapping case. throughout this section, let x and y be real vector spaces. theorem 1. if an odd mapping g : x → y satisfies the functional equation (1.1) if and only if g : x → y satisfies the functional equation (1.3). proof. assume that g : x → y satisfies the functional equation (1.1). since g is odd, g(0) = 0. replacing (x, y) by (x, x) and by (x, 2x) respectively in (1.1), we obtain g(2x) = 2g(x) and g(3x) = 3g(x) (2.1) for all x ∈ x. in general for any positive integer d, we have g(dx) = dg(x) (2.2) for all x ∈ x. it is easy to verify from (1.1) that g(d2x) = d2g(x) and g(d3x) = d3g(x) (2.3) for all x ∈ x. replacing (x, y) by (ax1 + bx2, cx3) in (1.1), we get g(ax1 + bx2 + cx3) = g(ax1 + bx2) + g(cx3) (2.4) for x1, x2, x3 ∈ x. replacing x1 by −x1 in (2.4), we get g(−ax1 + bx2 + cx3) = g(−ax1 + bx2) + g(cx3) (2.5) for x1, x2, x3 ∈ x. replacing x2 by −x2 in (2.4), we have g(ax1 − bx2 + cx3) = g(ax1 − bx2) + g(cx3) (2.6) for x1, x2, x3 ∈ x. replacing x3 by −x3 in (2.4), we obtain g(ax1 + bx2 − cx3) = g(ax1 + bx2) + g(−cx3) (2.7) 236 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) for x1, x2, x3 ∈ x. by (2.4), (2.5), (2.6), (2.7), (1.1) and (2.3), we get g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) = 2ag(x1) − 2bg(x2) + 2cg(x3) (2.8) for x1, x2, x3 ∈ x. adding 2ag(x1) − 2bg(x2) + 2cg(x3) + 2a 2g(x1) + 2b 2g(x2) + 2c 2g(x3) to both sides and using the oddness of g, we get (1.3). conversely, assume that g satisfies (1.3). letting x3 = 0 in (1.3), we have g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 2g(ax1 − bx2) + 2ag(x1) + 2bg(x2) for all x1, x2 ∈ x, since g is odd. so 2g(ax1 − bx2) + 2ag(x1) + 2bg(x2) = 4g(ax1) (2.9) for all x1, x2 ∈ x. letting x2 = 0 in (2.9), we have 2g(ax1) + 2ag(x1) = 4g(ax1) and so g(ax1) = ag(x1) for all x1 ∈ x. letting x1 = 0 in (2.9), we have −2g(bx2) + 2bg(x2) = 0 and so g(bx2) = bg(x2) for all x2 ∈ x. it follows from (2.9) that 2g(ax1 − bx2) + 2g(ax1) + 2g(bx2) = 4g(ax1) for all x1, x2 ∈ x and so g(x − y) + g(y) = g(x) for all x, y ∈ x. letting z = x − y in the above equation, we get g(z) + g(y) = g(z + y) for all z, y ∈ x. 3 solution of the functional equation (1.3): when g is even in this section, we investigate the solution of the functional equation (1.3) for an even mapping case. throughout this section, let x and y to be real vector spaces. theorem 2. if an even mapping g : x → y satisfies the functional equation (1.2) if and only if g : x → y satisfies the functional equation (1.3). proof. assume that g : x → y satisfies the functional equation (1.2). setting x = y = 0 in (1.2), we get g(0) = 0. cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 237 replacing (x, y) by (x, x) and by (x, 2x), respectively, in (1.2), we obtain g(2x) = 4g(x) and g(3x) = 9g(x) (3.1) for all x ∈ x. in general for any positive integer d, we have g(dx) = d2g(x) (3.2) for all x ∈ x. it is easy to verify from (1.2) that g(d2x) = d4g(x) and g(d3x) = d6g(x) (3.3) for all x ∈ x. replacing (x, y) by (ax1, cx3) in (1.2), we get g(ax1 + cx3) + g(ax1 − cx3) = 2g(ax1) + 2g(cx3) (3.4) for x1, x2, x3 ∈ x. multiplying 2 on both sides and using (3.3), we get 2g(ax1 + cx3) + 2g(ax1 − cx3) = 4a 2g(x1) + 4c 2g(x3) (3.5) for x1, x2, x3 ∈ x. adding 2g(ax1 + cx3) to (3.5) on both sides and using (3.3), we obtain 2g(ax1 + cx3) + 2g(ax1 − cx3) + 2g(ax1 + cx3) = 4a 2g(x1) + 4c 2g(x3) + 2g(ax1 + cx3) (3.6) for x1, x2, x3 ∈ x. so 4g(ax1 + cx3) = 4a 2 g(x1) + 4c 2 g(x3) + 2g(ax1 + cx3) − 2g(ax1 − cx3). (3.7) adding and subtracting 2g(bx2) to (3.7), we get 4g(ax1 + cx3) = 4a 2g(x1) + 4c 2g(x3) + g(ax1 + cx3 + bx2) + g(ax1 + cx3 − bx2) − g(ax1 − cx3 + bx2) − g(ax1 − cx3 − bx2) (3.8) for x1, x2, x3 ∈ x. adding 4g(bx2) to (3.8) on both sides, we obtain 4g(ax1 + cx3) + 4g(bx2) = 4a 2g(x1) + 4c 2g(x3) + g(ax1 + cx3 + bx2) + g(ax1 + cx3 − bx2) − g(ax1 − cx3 + bx2) − g(ax1 − cx3 − bx2) + 4g(bx2) (3.9) for x1, x2, x3 ∈ x. by (3.9) and (3.3), we get 4g(ax1 + cx3) + 4g(bx2) = 4a 2 g(x1) + 4c 2 g(x3) + 4b 2 g(x2) + g(ax1 + cx3 + bx2) + g(ax1 + cx3 − bx2) − g(ax1 − cx3 + bx2) − g(−ax1 + cx3 + bx2) (3.10) 238 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) for x1, x2, x3 ∈ x. using (3.10), (3.3) and the evenness of g, we get g(ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) − g(−ax1 + bx2 + cx3) + 4a2g(x1) + 4b 2g(x2) + 4c 2g(x3) = 4g(ax1 + cx3) + 4g(bx2) (3.11) for all x1, x2, x3 ∈ x. conversely, assume that g : x → y satisfies the functional equation (1.3). replacing (x1, x2, x3) by ( x a , 0, y c ) in (1.3), we get g(x − y) − g(−x + y) + g(x + y) − g(x − y) + 4g(x) + 4g(y) = 4g(x + y) (3.12) for all x, y ∈ x. using (1.3) and the evenness of g, we get g(x + y) + g(x − y) = 2g(x) + 2g(y), which is quadratic. 4 stability results for (1.3): odd case and direct method in this section, we present the hyers-ulam stability of the functional equation (1.3) for an odd mapping case. theorem 3. let j ∈ {−1, 1} and α : x3 → [0, ∞) be a function such that ∞ ∑ k=0 α(akjx1, a kjx2, a kjx3) akj < ∞ for all x1, x2, x3 ∈ x. let g : x → y be an odd mapping satisfying the inequality ‖dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (4.1) for all x1, x2, x3 ∈ x. there exists a unique additive mapping a : x → y which satisfies the functional equation (1.3) and ‖g(x) − a(x)‖ ≤ 1 2 ∞ ∑ k= 1−j 2 α(akjx1, 0, 0) akj (4.2) for all x1 ∈ x. the mapping a(x) is defined by, a(x) = lim k→∞ g(akjx1) akj for all x ∈ x proof. assume that j = 1. replacing (x1, x2, x3) by (x, 0, 0) in (4.2) and using the oddness of g, we get ‖2g(ax) − 2ag(x)‖ ≤ α(x, 0, 0) (4.3) cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 239 for all x ∈ x. it follows from (4.3) that ∥ ∥ ∥ ∥ g(ax) a − g(x) ∥ ∥ ∥ ∥ ≤ 1 2a α(x, 0, 0) (4.4) for all x ∈ x. replacing x by ax in (4.4) and dividing by a, we obtain ∥ ∥ ∥ ∥ g(a2x) a2 − g(ax) a ∥ ∥ ∥ ∥ ≤ 1 2a2 α(ax, 0, 0) (4.5) for all x ∈ x. it follows from (4.4) and (4.5) that ∥ ∥ ∥ ∥ g(a2x) a2 − g(x) ∥ ∥ ∥ ∥ ≤ 1 2a [ α(x, 0, 0) + α(ax, 0, 0) a ] (4.6) for all x ∈ x. similarly, for any positive integer n, we have ∥ ∥ ∥ ∥ g(x) − g(anx) an ∥ ∥ ∥ ∥ ≤ 1 2a n−1 ∑ k=0 α(akx, 0, 0) ak ≤ 1 2a ∞ ∑ k=0 α(akx, 0, 0) ak (4.7) for all x ∈ x. in order to prove convergence of the sequence { g(akx) ak } , replacing x by amx and dividing am in (4.7) for any m, n > 0, we get ∥ ∥ ∥ ∥ g(amx) am − g(am+nx) am+n ∥ ∥ ∥ ∥ = 1 2am ∥ ∥ ∥ ∥ g(amx) − g(amanx) an ∥ ∥ ∥ ∥ ≤ 1 2a n−1 ∑ m=0 α(am+nx, 0, 0) am+n ≤ 1 2a n−1 ∑ m=0 α(am+nx, 0, 0) am+n → 0 as m → ∞. hence the sequence { g(anx) an } is a cauchy sequence. since y is complete, there exists a mapping a : x → y such that a(x) = lim n→∞ g(anx) an , ∀x ∈ x. (4.8) letting n → ∞ in (4.8), we see that (4.8) holds for x ∈ x. to prove that a satisfies (1.3), replacing (x1, x2, x3) by (a nx, anx, anx) and dividing an in (4.1), we obtain 1 an ‖dg(anx, anx, anx)‖ ≤ 1 an α(anx, anx, anx) for all x1, x2, x3 ∈ x. letting m → ∞ in the above inequality and using the definition of a(x), we see that da(x1, x2, x3) = 0. hence a satisfies (1.3) for all x1, x2, x3 ∈ x. to show that a is unique, let b(x) be another additive mapping satisfying (4.2). then ‖a(x) − b(x)‖ = 1 an ‖a(anx) − b(anx)‖ ≤ 1 an {‖a(anx) − g(anx)‖ + ‖g(anx) − b(anx)‖} → 0 as n → ∞. 240 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) hence a is unique. assume that j = −1. replacing x by x a in (4.3), we get ∥ ∥ ∥ ag(x) − a2g ( x a ) ∥ ∥ ∥ ≤ α ( x a , 0, 0 ) (4.9) for all x ∈ x. the rest of the proof is similar to the proof of the case j = 1. this completes the proof of the theorem. the following corollary is an immediate consequence of theorem 3 concerning the stability of (1.3). corollary 1. let ǫ and p be nonnegative real numbers. let g : x → y be an odd mapping satisfiying the inequality ‖dg(x1, x2, x3)‖ (4.10) ≤        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; p > 1 or p < 1 ǫ ( ‖x1‖ p + ‖x2‖ p + ‖x3‖ p + ‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p ) ; p > 1 3 or p < 1 3 for all x1, x2, x3 ∈ x. then there exists a unique additive mapping a : x → y such that ‖g(x) − a(x)‖ ≤        ǫ 2|a−1| ; ǫ‖x‖p 2|a−ap| ; p > 1 or p < 1 ǫ‖x‖3p 2|a−a3p| ; p > 1 3 or p < 1 3 (4.11) for all x ∈ x. proof. letting α(x1, x2, x3) =        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; ǫ ( ‖x1‖ p + ‖x2‖ p + ‖x3‖ p + ‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p ) for all x1, x2, x3 ∈ x, we can get the result. 5 stability results for (1.3): even case and direct method in this section, we discuss the hyers-ulam stability of the functional equation (1.3) for an even mapping case by using the direct method. cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 241 theorem 4. let j ∈ {−1, 1} and α : x3 → [0, ∞) be a function such that ∞ ∑ k=0 α(akjx1, a kjx2, a kjx3) akj < ∞ (5.1) for all x1, x2, x3 ∈ x. let g : x → y be an even mapping satisfying g(0) = 0 and the inequality ‖dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (5.2) for all x1, x2, x3 ∈ x. there exists a unique additive mapping q : x → y which satisfies the functional equation (1.3) and ‖g(x) − q(x)‖ ≤ 1 4a2 ∞ ∑ k= 1−j 2 α(akjx, 0, 0) a2kj (5.3) for all x ∈ x. the mapping q(x) is defined by q(x) = lim n→∞ g(akjx) a2kj (5.4) for all x ∈ x. proof. assume that j = 1. replacing (x1, x2, x3) by (x, 0, 0) in (5.2), we get ‖4g(ax) − 4a2g(x)‖ ≤ α(x, 0, 0) (5.5) for all x ∈ x. it follows from (5.5) that ∥ ∥ ∥ ∥ g(ax) a2 − g(x) ∥ ∥ ∥ ∥ ≤ 1 4a2 α(x, 0, 0) (5.6) for all x ∈ x. replacing x by ax in (5.6) and dividing by a2, we obtain ∥ ∥ ∥ ∥ g(a2x) a4 − g(ax) a2 ∥ ∥ ∥ ∥ ≤ 1 4a4 α(ax, 0, 0) (5.7) for all x ∈ x. it follows from (5.6) and (5.7) that ∥ ∥ ∥ ∥ g(a2x) a4 − g(x) ∥ ∥ ∥ ∥ ≤ 1 4a2 [ α(x, 0, 0) + α(ax, 0, 0) a2 ] (5.8) for all x ∈ x. inductively, we have ∥ ∥ ∥ ∥ g(x) − g(anx) a2n ∥ ∥ ∥ ∥ ≤ 1 4a2 n−1 ∑ k=0 α(akx, 0, 0) a2k ≤ 1 a3 ∞ ∑ k=0 α(akx, 0, 0) a2k (5.9) 242 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) for all x ∈ x. in order to prove convergence of the sequence { g(akx) a2k } , replacing x by amx and dividing am in (5.9) for any m, n > 0, we get ∥ ∥ ∥ ∥ g(amx) a2m − g(am+nx) a2(m+n) ∥ ∥ ∥ ∥ = 1 a2m ∥ ∥ ∥ ∥ g(amx) − g(amanx) a2n ∥ ∥ ∥ ∥ ≤ 1 a3 n−1 ∑ m=0 α(am+nx, 0, 0) a2(m+n) ≤ 1 a3 n−1 ∑ m=0 α(am+nx, 0, 0) a2(m+n) → 0 as m → ∞. hence the sequence { g(anx) a2n } is a cauchy sequence. since y is complete, there exists a mapping q : x → y such that q(x) = lim n→∞ g(anx) a2n , ∀x ∈ x. (5.10) letting n → ∞ in (5.10) we see that (5.10) holds for x ∈ x. to prove that q satisfies (1.3), replacing (x1, x2, x3) by (a nx, anx, anx) and dividing a2n in (5.2), we obtain 1 a2n ‖dg(anx, anx, anx)‖ ≤ 1 a2n α(anx, anx, anx) for all x1, x2, x3 ∈ x. letting n → ∞ in the above inequality and using the definition of q(x), we see that dq(x1, x2, x3) = 0. hence q satisfies (1.3) for all x1, x2, x3 ∈ x. to show that q is unique, let b(x) be another quadratic mapping satisfying (5.4). then ‖q(x) − b(x)‖ = 1 a2n ‖q(anx) − b(anx)‖ ≤ 1 a2n {‖q(anx) − g(anx)‖ + ‖g(anx) − b(anx)‖} → 0 as n → ∞. hence q is unique. assume that j = −1. replacing x by x a in (5.5), we get ∥ ∥ ∥ ag(x) − a2g ( x a ) ∥ ∥ ∥ ≤ 1 4 α ( x a , 0, 0 ) (5.11) for all x ∈ x. the rest of the proof is similar to the proof of the case j = 1. this completes the proof of the theorem. the following corollary is an immediate consequence of theorem 4 concerning the stability of (1.3). cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 243 corollary 2. let ǫ and p be nonnegative real numbers. let gq : x → y be an even mapping satisfiying g(0) = 0 and the inequality ‖dg(x1, x2, x3)‖ (5.12) ≤        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; p > 2 or p < 2 ǫ ( ‖x1‖ p‖x2‖ p‖x3‖ p + {‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p} ) ; p > 2 3 or p < 2 3 for all x1, x2, x3 ∈ x. then there exists a unique quadratic mapping q : x → y such that ‖g(x) − q(x)‖ ≤        ǫ 4|a2−1| ǫ‖x‖p 4|a2−ap| ǫ‖x‖3p 4|a2−a3p| (5.13) for all x ∈ x. proof. letting α(x1, x2, x3) =        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; ǫ ( ‖x1‖ p‖x2‖ p‖x3‖ p + {‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p} ) ; for all x1, x2, x3 ∈ x, we get the result. 6 stability results of (1.3): mixed case in this section, we establish the hyers-ulam stability of the functional equation(1.3) for a mixed mapping case. theorem 5. let j ∈ {−1, 1} and α : x3 → [0, ∞) be a function satisfying (1.3) for all x1, x2, x3 ∈ x. let g : x → y be a mapping satisfying the inequality ‖dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (6.1) for all x1, x2, x3 ∈ x. there exist a unique additive mapping a : x → y and a unique quadratic mapping q : x → y which satisfies the functional equation (1.3) and ‖f(x) − a(x) − q(x)‖ ≤ 1 2 {[ 1 2a ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) akj + α(−akjx, 0, 0) akj ] ] + 1 4n2 [ ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) a2kj + α(−akjx, 0, 0) a2kj ] ]} for all x ∈ x. the mapping a(x) and q(x) are defined in (4.2) and (5.10), respectively. 244 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) proof. let go(x) = ga(x)−ga(−x) 2 for all x ∈ x. then go(0) = 0 and go(−x) = −go(x) for all x ∈ x. hence ‖dgo(x1, x2, x3)‖ ≤ 1 2 { ‖dga(x1, x2, x3)‖ + ‖dga(−x1, −x2, −x3)‖ } ≤ α(x1, x2, x3) 2 + α(−x1, −x2, −x3) 2 for all x1, x2, x3 ∈ x. by theorem 3, we have ‖go(x) − a(x)‖ ≤ 1 4a ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) akj + α(−akjx, 0, 0) akj ] (6.2) for all x ∈ x. let ge(x) = gq(x)+gq(−x) 2 for all x ∈ x. then ge(0) = 0 and ge(−x) = ge(x) for all x ∈ x. hence, ‖dge(x1, x2, x3)‖ ≤ 1 2 { ‖dgq(x1, x2, x3)‖ + ‖dgq(−x1, −x2, −x3)‖ } ≤ α(x1, x2, x3) 2 + α(−x1, −x2, −x3) 2 for all x1, x2, x3 ∈ x. by theorem 4, we have ‖ge(x) − q(x)‖ ≤ 1 8a2 ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) a2kj + α(−akjx, 0, 0) a2kj ] (6.3) for all x ∈ x. then g(x) = ge(x) + go(−x) (6.4) for all x ∈ x. it follows from (6.2), (6.3) and (6.4) that ‖g(x) − a(x) − q(x)‖ = ‖ge(x) + go(−x) − a(x) − q(x)‖ ≤ ‖go(−x) − a(x)‖ + ‖ge(x) − q(x)‖ ≤ 1 4a ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) akj + α(−akjx, 0, 0) akj ] + 1 8a2 ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) a2kj + α(−akjx, 0, 0) a2kj ] for all x ∈ x. hence the theorem is proved. using corollaries 1 and 2, we have the following corollary concerning the stability of (1.3). cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 245 corollary 3. let λ and s be a nonnegative real numbers. let gq : x → y be a mapping satisfiying the inequality ‖dg(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); s 6= 1, 2 λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s}; s 6= 1 3 , 2 3 (6.5) for all x1, x2, x3 ∈ x. then there exist a unique additive function a : x → y and a unique quadratic mapping q : x → y such that ‖g(x) − a(x) − q(x)‖ ≤          λ 2 [ 1 |a−1| + 1 2|a2−1| ] λ‖x‖s 2 [ 1 |a−as| + 1 2|a2−as| ] λ‖x‖3s 2 [ 1 |a−a3s| + 1 2|a2−a3s| ] (6.6) for all x ∈ x. 7 fixed point stability of (1.3): odd mapping case the following theorems are useful to prove our fixed point stability results. theorem 6. [12] (banach contraction principle) let (x, d) be a complete metric space and consider a mapping t : x → x which is strictly contractive mapping. (a1) d(t x, t y) ≤ ld(x, y) for some (lipschitz constant) l < 1. (i) the mapping t has one and only fixed point x∗ = t (x∗); (ii) the fixed point for each given element x∗ is globally contractive, that is, (a2) limn→∞ t nx = x∗ for any starting point x ∈ x; (iii) one has the following estimation inequalities (a3) d(t nx, x∗) ≤ 1 1−l d(t nx, t n+1x), ∀n ≥ 0, ∀x ∈ x; (a4) d(x, x∗) = 1 1−l d(x, x∗), ∀x ∈ x. theorem 7. [12] (alternative fixed point theorem) suppose that for a complete generalized metric space (x, d) and a strictly contractive mapping t : x → x with lipschitz constant l. then for each given element x ∈ x, (b1) d(t nx, t n+1x) = ∞, ∀n ≥ 0; (b2) there exists a natural number n0 such that 246 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) (i) d(t nx, t n+1x) < ∞, ∀n ≥ 0; (ii) the sequence {t nx} is convergent to a fixed point y∗ of t ; (iii) y∗ is the unique fixed point of t in the set y = {y ∈ y : d(t n0, y) < ∞}; (iv) d(y∗, y) ≤ 1 1−l d(y, t y) for all y ∈ y . in this method, we investigate the hyers-ulam stability of the functional equation (1.3) for an odd mapping case by using fixed point method. theorem 8. let g : w → b be an odd mapping for which there exists a function α : w 3 → [0, ∞) with the condition lim n→∞ α(aki x1, a k i x2, a k i x3) aki = 0, (7.1) for ai =      a i = 0 1 a i = 1, such that the functional inequality ‖dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (7.2) for all x1, x2, x3 ∈ w. if there exists l = l(i) such that the function x → β(x) = 1 2 α ( x a , 0, 0 ) has the property 1 ai β(aix) = l (β(x)) (7.3) for all x ∈ w. then there exists a unique additive function a : w → b satisfying the functional equation (1.3) and ‖g(x) − a(x)‖ ≤ l1−i 1 − l β(x) (7.4) for all x ∈ w. proof. consider the set x = {p |p : w → b, p(0) = 0} and introduce the generalized metric on x. d(p, q) = inf{k ∈ (0, ∞) : ‖p(x) − q(x)‖ ≤ β(x), x ∈ w} it is easy to see that (x, d) is complete. define t : x → x by tp(x) = 1 ai p(aix) for all x ∈ w . now p, q ∈ x, d(p, q) ≤ k ⇒ ‖p(x) − q(x)‖ ≤ kβ(x), x ∈ w. ⇒ ∥ ∥ ∥ ∥ 1 ai p(aix) − 1 ai q(aix) ∥ ∥ ∥ ∥ ≤ 1 ai kβ(aix), ∀x ∈ w ⇒ ‖tp(x) − tq(x)‖ ≤ lkβ(x), ∀x ∈ w ⇒ d(tp, tq) ≤ lk. cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 247 this implies d(tp, tq) ≤ ld(p, q) for all p, q ∈ x. that is, t is a strictly contractive mapping on x with lipschitz constant l. it follows from (4.3) that ‖2g(ax) − 2ag(x)‖ ≤ α(x, 0, 0) (7.5) for all x ∈ w . it follows from (7.5) that, ∥ ∥ ∥ ∥ g(x) − g(ax) a ∥ ∥ ∥ ∥ ≤ 1 2a α(x, 0, 0) (7.6) for all x ∈ w . using (6.2), for this case i = 0, it reduces to ∥ ∥ ∥ ∥ g(x) − g(ax) a ∥ ∥ ∥ ∥ ≤ 1 a β(x) (7.7) for all x ∈ w . thus d(ga, t ga) ≤ 1 a = l = l1 < ∞. again replacing x by x a in (7.5), we get ∥ ∥ ∥ g(x) − ag ( x a ) ∥ ∥ ∥ ≤ 1 2 α ( x a , 0, 0 ) (7.8) for all x ∈ w . by using (7.3) for the case i = 1, it reduces to ∥ ∥ ∥ g(x) − ag ( x a ) ∥ ∥ ∥ ≤ β(x). (7.9) that is, d(g, t g) ≤ 1 ⇒ d(g, t g) ≤ 1 = l0 < ∞. in the above case, we have d(g, t g) ≤ l1−i. therefore (b2(i)) holds. from (b2(ii)), it follows that there exists a fixed point a of t in x such that a(x) = lim i→∞ ga(a k i x) aki , ∀x ∈ w. (7.10) in order to prove a : w → b is additive, replacing (x1, x2, x3) by (a k i x1, a k i x2, a k i x3) in (7.2) and dividing aki , it follows from (7.3) and (7.10) that a satisfies (1.3) for all x1, x2, x3 ∈ w . by (b2(iii)), a is the unique fixed point of t in the set, y = {g ∈ x : d(t g, a) < ∞}. using the fixed point alternative result, a is the unique function such that ‖g(x) − a(x)‖ ≤ kβ(x) for all x ∈ w and k > 0. finally, by (b2(iv)), we obtain d(g, a) ≤ 1 1 − l d(g, t g). that is, d(g, a) ≤ l 1−i 1−l . hence we conclude that ‖g(x) − a(x)‖ ≤ l1−i 1 − l β(x) for all x ∈ w . this completes the proof of the theorem. 248 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) corollary 4. let g : w → b be an odd mapping and assume that there exist real numbers λ and s such that ‖dga(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s} (7.11) for all x1, x2, x3 ∈ x. then there exists a unique additive mapping a : w → b such that ‖g(x) − a(x)‖ ≤        λ 2|a−1| ; λ‖x‖s 2|a−as| ; s 6= 1 λ‖x‖3s 2|a−a3s| ; s 6= 1 3 (7.12) for all x ∈ x. proof. let α(x1, x2, x3) =        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {λ(‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s)}; for all x1, x2, x3 ∈ w . now, α(aki x1, a k i x2, a k i x3) aki =                λ aki ; λ aki (‖aki x1‖ s + ‖aki x2‖ s + ‖aki x3‖ s); λ ak i (‖aki x1‖ s + ‖aki x2‖ s + ‖aki x3‖ s) + {‖aki x1‖ 3s + ‖aki x2‖ 3s + ‖aki x3‖ 3s} =        → 0 as k → ∞ → 0 as k → ∞ → 0 as k → ∞. (7.13) that is, (7.1) holds. but we have β(x) = 1 2 α ( x a , 0, 0 ) . hence β(x) = 1 2 α ( x a , 0, 0 ) =        λ 2 λ 2as (‖x‖s) λ 2as (‖x‖s). cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 249 also 1 ai β(ai, x) =        λ 2ai λ 2ai (‖aix‖ s) λ 2ai (‖aix‖ s) =        a −1 i β(x) a s−1 i β(x) a 3s−1 i β(x). hence the inequality (7.7) holds. either l = a−1 for s = 0 if i = 0 and l = 1 a−1 for s = 0 if i = 1. either l = as−1 for s < 1 if i = 0 and l = 1 as−1 for s > 1 if i = 1. either l = a3s−1 for s < 1 if i = 0 and l = 1 a3s−1 for s > 1 if i = 1. now from (7.2), we prove the following cases: case: 1 l = a−1, i = 0 ‖ga(x) − a(x)‖ ≤ l1−i 1 − l β(x) = (a−1)1−0 1 − a−1 λ 2 = λ 2(a − 1) . (7.14) case: 2 l = ( 1 a )−1 , i = 1 ‖ga(x) − a(x)‖ ≤ l1−i 1 − l β(x) = (a)1−1 1 − a λ 2 = λ 2(1 − a) . (7.15) case: 3 l = as−1, s < 1, i = 0 ‖ga(x) − a(x)‖ ≤ l1−i 1 − l β(x) = (as−1)1−0 1 − as−1 λ 2as ‖x‖s = λ‖x‖s 2|a − as| . (7.16) case: 4 l = ( 1 a )s−1 , s > 1, i = 1 ‖ga(x) − a(x)‖ ≤ l1−i 1 − l β(x) = (a1−s)1−1 1 − a1−s λ 2as ‖x‖s = λ‖x‖s 2(as − a) . (7.17) case: 5 l = a3s−1, s < 1 3 , i = 0 ‖ga(x) − a(x)‖ ≤ l1−i 1 − l β(x) = (a3s−1)1−0 1 − a3s−1 λ 2a3s ‖x‖s = λ‖x‖s 2(a − a3s) . (7.18) case: 6 l = ( 1 a )−1 , i = 1 ‖ga(x) − a(x)‖ ≤ l1−i 1 − l β(x) = (a1−3s)1−1 1 − a1−3s λ 2a3s ‖x‖s = λ‖x‖s 2(a3s − a) . (7.19) hence the proof of the corollary is completed. 8 fixed point stability of (1.3): even mapping case in this method, we investigate the hyers-ulam stability of the functional equation (1.3) for an even case mapping by using fixed point method. 250 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) theorem 9. let g : w → b be an even mapping for which there exists a function α : w 3 → [0, ∞) with the condition lim n→∞ α(aki x1, a k i x2, a k i x3) a2ki = 0 (8.1) for ai =      a i = 0 1 a i = 1, such that the functional inequality ‖dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (8.2) for all x1, x2, x3 ∈ w. if there exists l = l(i) such that the function x → β(x) = 1 2 α ( x a , 0, 0 ) (8.3) has the property 1 a2i β(aix) = l (β(x)) (8.4) for all x ∈ w, then there exists a unique quadratic mapping q : w → b satisfying the functional equation (1.3) and ‖g(x) − q(x)‖ ≤ l1−i 1 − l β(x) (8.5) for all x ∈ w. proof. consider the set x = {p |p : w → b, p(0) = 0} and introduce the generalized metric on x. d(p, q) = inf{k ∈ (0, ∞) : ‖p(x) − q(x)‖ ≤ β(x), x ∈ w} it is easy to see that (x, d) is complete. define t : x → x by tp(x) = 1 a2 i p(aix) for all x ∈ w . now p, q ∈ x, d(p, q) ≤ k ⇒ ‖p(x) − q(x)‖ ≤ kβ(x), x ∈ w. ⇒ ∥ ∥ ∥ ∥ 1 a2i p(aix) − 1 a2i q(aix) ∥ ∥ ∥ ∥ ≤ 1 a2i kβ(aix), ∀x ∈ w ⇒ ‖tp(x) − tq(x)‖ ≤ lkβ(x), ∀x ∈ w ⇒ d(tp, tq) ≤ lk. this implies d(tp, tq) ≤ ld(p, q) for all p, q ∈ x. that is, t is a strictly contractive mapping on x with lipschitz constant l. replacing (x1, x2, x3) by (x, 0, 0) in (9.1) and using the evenness of g, we get ‖4g(ax) − 4a2g(x)‖ ≤ α(x, 0, 0), (8.6) ∥ ∥ ∥ ∥ g(x) − g(ax) n2 ∥ ∥ ∥ ∥ ≤ 1 4a2 α(x, 0, 0) (8.7) cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 251 for all x ∈ w . by using (8.4), for this case i = 0, it reduces to ∥ ∥ ∥ ∥ g(x) − g(ax) a2 ∥ ∥ ∥ ∥ ≤ 1 2a2 β(x) (8.8) for all x ∈ w . that is, d(g, t g) ≤ 1 a2 ⇒ d(g, t g) ≤ 1 a2 = l = l1 < ∞. again replacing x by x a in (8.6), we get ∥ ∥ ∥ g(x) − a2g ( x a ) ∥ ∥ ∥ ≤ 1 4 α ( x a , 0, 0 ) (8.9) for all x ∈ w . that is, d(g, t g) ≤ 1 2 < 1 ⇒ d(g, t g) ≤ 1 = l0 < ∞. in above case, we get d(g, t g) ≤ l1−i. the rest of the proof is similar to that of the previous theorem. this completes the proof of the theorem. corollary 5. let g : w → b be an even mapping and assume that there exist real numbers λ and s such that ‖dg(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); s 6= 2 λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s}; s 6= 1 3 (8.10) for all x1, x2, x3 ∈ x. then there exists a unique quadratic mapping q : w → b such that ‖gq(x) − q(x)‖ ≤        λ 4|a2−1| ; λ‖x‖s 4|a2−as| λ‖x‖3s 4|a2−a3s| (8.11) for all x ∈ x. 9 fixed point stability of (1.3): mixed mapping case in this method, we present the hyers-ulam stability of the functional equation (1.3) for a mixed mapping case by using fixed point method. theorem 10. let g : w → b be a mapping for which there exists a function α : w 3 → [0, ∞) with the condition (7.1) and (8.1) for ai =      a i = 0 1 a i = 1, such that the functional inequality ‖dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (9.1) 252 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) for all x1, x2, x3 ∈ w. if there exists l = l(i) such that the function x → β(x) = 1 2 α ( x a , 0, 0 ) satisfies (7.3) and (8.3) for all x ∈ w, then there exist a unique additive mapping a : w → b and a quadratic mapping q : w → b satisfying the functional equation (1.3) and ‖g(x) − a(x) − q(x)‖ ≤ l1−i 1 − l [β(x) + β(−x)] holds for all x ∈ w. proof. it follows from (6.2) and theorem 8 that ‖go(x) − a(x)‖ ≤ 1 2 l1−i 1 − l [β(x) + β(−x)]. (9.2) similarly, it follows from (7.5) and theorem 9 that ‖ge(x) − q(x)‖ ≤ 1 2 l1−i 1 − l [β(x) + β(−x)] (9.3) for all x ∈ w . then g(x) = go(x) + ge(x) for all x ∈ w . from (8.11), (9.2) and (9.3), we have ‖g(x) − a(x) − q(x)‖ = ‖ge(x) + go(x) − a(x) − q(x)‖ ≤ ‖go(x) − a(x)‖ + ‖ge(x) − q(x)‖ = l1−i 1 − l [β(x) + β(−x)] for all x ∈ w . hence the theorem is proved. corollary 6. let g : w → b be a mapping and assume that there exist real numbers λ and s such that ‖dg(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); s 6= 1, 2 λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s}; s 6= 1 3 , 2 3 for all x1, x2, x3 ∈ x. then there exist a unique additive mapping a : w → b and a unique quadratic mapping q : w → b such that ‖g(x) − a(x) − q(x)‖ ≤        λ 2|a−1| + λ 4|a2−1| λ‖x‖s 2|a−as| + λ‖x‖s 4|a2−as| λ‖x‖3s 2|a−a3s| + λ‖x‖3s 4|a2−a3s| for all x ∈ x. cubo 22, 2 (2020) hyers-ulam stability of an additive-quadratic functional equation 253 declarations availablity of data and materials not applicable. competing interests the authors declare that they have no competing interests. fundings this work was supported by basic science research program through the national research foundation of korea funded by the ministry of education, science and technology (nrf-2017r1d1a1b04032937). authors’ contributions the authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript. 254 v. govindan, c. park, s. pinelas & t. m. rassias cubo 22, 2 (2020) references [1] j. aczél and j. dhombres, functional equations in several variables, encyclopedia of mathematics and its applications, 31, cambridge university press, cambridge, 1989. 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[28] s. m. ulam, problems in modern mathematics, science editions john wiley & sons, inc., new york, 1964. introduction solution of the functional equation (1.3): when g is odd solution of the functional equation (1.3): when g is even stability results for (1.3): odd case and direct method stability results for (1.3): even case and direct method stability results of (1.3): mixed case fixed point stability of (1.3): odd mapping case fixed point stability of (1.3): even mapping case fixed point stability of (1.3): mixed mapping case cubo 6, 13-18 ( 1990) r11cilbldo14qo•lo 1989 on nuclear bernsteih algebras by rodolfo baeza v. 12 o. -abstract. in [ j] p. holgate preved that the core of any orthogonal bernstein algebra is a special train algebra and consequently a genetic algebra . let a be a bernstein algebra. then a is a nuclear algebra if and only if the ocre of a is a. the present paper preves that the core of a bernstein algebra is a special train algebra. 1.-prelimlnaries. in the following let k be an infinita commutati ve f ield whose characteristic is nei ther 2 nor 3. let a be a commutative nonassociative k algebra. for every sequence principal 8 1 •ª2, • • • 'ªk product a 1 of k elemente of a define the of all finita sume of products with a 1= a 1 . g'k is the set ak of k elemente in bu. g'k is called principal power of b. we say that aea is nilpotent if there existe ne~ such that a"""º, a is nilpotent if there existe teli euch that at• o , if all elements of a are 1 j fomotcyt 227-89; 01\ffll 710-881 ccint-usp-brazil ;ll l m apppa !ji l, • .a . a . pac.14 c u b o nilpotents we say that a is a nilalgebra. a la called a jordan algebra if xy=yx and x 2 (xy)=x(x2y) vx,yea. a.a.albert preved that all finite dimensional jordan nilalgebram or characteristic ~ 2 are nilpotent. lemma l. let a be a commutative algebra ,char(a)•2 and x3•o 't'xea. then a is a jordan algebra. proof. the identity x 3 "" (x+y) 3 • (x-y) 3 • o impliea that o ""' (x+y) 3 -(x-y) 3 = 2(2x(xy)+x2y). , but char(a) • 2, hence x 2y=-2x(xy). replacing y by xy we obtain x 2 (xy)•-2x(x(xy)) • x(-2x(xy)) "" x(x2y), le. a is a jordan algebra. let (a,w) be an (n+l) dimensional commutative non· associative baric k-algebra where w:a ---+ x is a waight function. (a,w) is called bernstein algebra iff (x 2 ) 2 ""w(x) 2 x 2 , vxea. in any bernstein (a,w) algebra th• nontrivial homomorphism w is uni(¡uely determined, and a possesses at least one non tri vial idempotent element ec.i, ( see { 5] ) • the e-canonical decomposi tion of a is xe•u•v whan u• { yeker { w) : ey• 'iy) and v-{ yeker ( w) : ey-o ) . the subspacea u and v satisfy the fundamental relations u2~, uv~, v2~, vv2•o, uy-o and the fundamental identities u~-o, u1(u 1v 1 )•o, u 1 (u 2 u 3 )+u 2 (u 3 u 1 }+u 3 (u 1 u 2 ) • o (jacobi's identity) 1 vu 1tu, v 1ev, and (xy)(zt)+(xz)(yt)+(xt)(yz) •o vx,y,z,ten • ker(w) • u• v. lemma 2. let (a,w) be a bernstein algebra. then n•xer(w) and its princ ipa l powers are ideals of a. proof . n is an ideal of a, since the kernel! of a h omomo rph i sm is an ideal. p.holgate preved [3,p 615), that all ye.h ea t isf y e( y 1 y2 ) ... and d is a n ideal of c . that ie , (c/d , w') h 1 berna te i n algebra, aince it is a homomorphic image ot c . naturall y w' {c+d) • w(c) defines the weight function in c/d. iat ua no te that x• c +d • ker(w') if f w(c) • o iff c • ker(w) i thua ke .r (1..1 ' ) • {c+d:cen ) • ii • ü ev. pr opo.t tion 2 . let a • keeu• v be a berna te i n a lgebra, and c• ic tmu• tf•jc be i t s core . then e is a apec i a l tra in algebra . proo f . pi rst , shall prove tha t c/ d i s a s pec i al train a lgebra . by lemma 3 we kn o w that c/ d i s a bernstein algebra . it only rema in s to prove tha t x 3 • o vx eke r (w ' ) and then uh propoai t lon 1. let x • n+d be an el e ment ot ke r {w') , thua n • u-tu , u;t . uaing the f undamental relat i o na and i dentitie• in oh hucl.ean .. e u b o pag.17 the bernstein algebra e, 2(u 1 u 2 )[u(u 1 u 2 ))eu. v 2s;:u, v2u=o and jacobi's identity imply v 2 s;:o and, consequently u 2 (u1 u 2 )ed . furthermore the second and the last fundamental identities imply that 2u3 ((u 1 u 2 )[u(u 1 u 2 ))) ""-2(u(u 1 u 2 )][u 3 (u 1 u 2 )] = (uu 3 )((u 1 u 2 )(u 1 u 2 ))e(uu 3 )v2-o vu3eu, because v 2g>. then we have (u 1 u 2 )(u{u 1 u 2 )) e uf\ann(u), and by jacobi's identity it is in d. '.1'hat is x 3=n3+d=-d and c/d is a special train algebra. this shows that r e n existe such that q. (ker(w 1 ) )r nr+1=nnrs;: nd .. o. by lemma 2 we know that n 1 is an ideal of e far all integer i>o. but n is nilpotent, hence e is a special train algebra. remark l. it was preved in (2} that the orthogonality is not a necessary condition to be a special train algebra. the present work provee that the orthogonal hypothesis can be removed from proposi tion 4 of [3) • remark 2. it is preved in (5] that the core e of a bernstein algebra a satisfy e = a2 • let a = keeu•v be a bernstein algebra, the.n a 2 ma implies v=if ,and the meaning of our proposition 2 is that "every nuclear bernstein algebra is a special train algebra". references . (1] m.t.a.lcalde, r.baeza, e.burgueño, aucour des algebres de bernstein, arch . math., vol. 53,134-140 (1989). ( 2] baeza r. ,a non orchogonal bernscein algebra vhich 1.~ r snj':r:1r1 trr1n rlrtt":hrr. a.tris dri x f.scnlri da alaabra cubo, a mathematical journal vol.22, n◦02, (177–201). august 2020 http://dx.doi.org/10.4067/s0719-06462020000200177 received: 28 february, 2020 | accepted: 15 june, 2020 mathematical modeling of chikungunya dynamics: stability and simulation ruchi arora, dharmendra kumar, ishita jhamb and avina kaur narang department of mathematics, sgtb khalsa college, university of delhi, delhi-110007, india ruchi@sgtbkhalsa.du.ac.in, dharmendrakumar@sgtbkhalsa.du.ac.in, ishita.jhamb@gmail.com, kaur.avina45@gmail.com abstract infection due to chikungunya virus (chikv) has a substantially prolonged recuperation period that is a long period between the stage of infection and recovery. however, so far in the existing models (sir and seir), this period has not been given due attention. hence for this disease, we have modified the existing seir model by introducing a new section of human population which is in the recuperation stage or in other words the human population that is no more showing acute symptoms but is yet to attain complete recovery. a mathematical model is formulated and studied by means of existence and stability of its disease free equilibrium (dfe) and endemic equilibrium (ee) points in terms of the associated basic reproduction number (r0). resumen la infección debida al virus chikungunya (chikv) tiene un peŕıodo de recuperación sustancialmente prolongado, que es un peŕıodo largo entre la etapa de infección y recuperación. sin embargo, hasta ahora en los modelos existentes (sir y seir), este peŕıodo no ha recibido suficiente atención. por tanto, para esta enfermedad, hemos modificado el modelo seir existente introduciendo una nueva sección de población humana que está en la etapa de recuperación o, en otras palabras, la población humana que ya no muestra śıntomas agudos pero todav́ıa no se recupera completamente. se formula y estudia un modelo matemático a través de la existencia y estabilidad de su equilibrio libre de enfermedad (dfe) y puntos de equilibrio endémico (ee) en términos del número de reproducción básico asociado (r0). keywords and phrases: equilibrium point, disease free equilibrium, endemic equilibrium, reproduction number, local stability, global stability. 2020 ams mathematics subject classification: 92b05, 93a30, 93c15 c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000200177 178 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) 1 introduction in recent past, the study of vector borne diseases has gained considerable attention and mathematics have become a useful tool for such studies. several temporal deterministic models have been proposed for diseases like dengue, malaria, chikungunya etc. chikungunya is a disease caused by the chikungunya virus, an rna genome which is a member of the alphavirus genus in the family of togaviridae. it is a mosquito borne viral disease which is transmitted to humans through aedes aegypti mosquito bite [1]. in 1952, chikungunya was first confirmed as the cause of an epidemic of dengue like illness on the comoros islands located on the eastern coast of northern mozambique [2]. since its discovery, numerous chikv outbreaks with irregular intervals of 2-20 years have affected asian, african, european and american countries. in thailand, the first report of chikungunya infection occurred in bangkok in 1958 [3]. in india, the virus emerged in parts of vellore, calcutta and maharashtra in the early 1960’s [4]. the virus continued to spread in sri lanka in 1969 and many countries of southeast asia such as myanmar, indonesia and vietnam [4]. later, some irregular cases of chikungunya fever were also seen in many provinces of thailand in the period from 1976 to 1995 [3]. from 1999 to 2000, the reemergence of chikungunya occurred in democratic republic of congo [2], 13,500 cases were reported in lamu, kenya in 2004 [5]. in the years 2005-2007, there occurred an outbreak in reunion islands in the indian ocean. in 2007, 197 cases were reported in europe due to chikungunya [1]. the outbreak mutated to facilitate the disease transmission by aedes albopictus from the tiger mosquito family. it was a mutation in one of the viral envelope genes which allowed the virus to be present in the mosquito saliva only two days after the infection and seven days in aedes aegypti mosquitoes. the results indicated that the areas where the tiger mosquitoes are present could have a greater risk of outbreak. after an effective bite from a mosquito infected with chikv, the incubation period (i.e., the time elapsed between exposure to pathogenic organism and when symptoms and signs are first apparent) usually lasts for 3-7 days with fever as the most prominent symptom. the symptoms of chikungunya fever differ from the normal fever as they are accompanied with acute joint pains. other common symptoms are nausea, rashes, headache and fatigue. some cases may result in neurological, retinal and carpological complications as well, which makes it difficult for older people to recover as against young people. in some instances, people live with joint pains for years which indicates that the recuperation period can last for a long time. the symptoms of chikungunya are generally mild and the disease may sometimes be misdiagnosed with zika and dengue due to similarity in symptoms. there have been very few cases where chikungunya resulted in death and mostly infected individuals are expected to make full recovery with lifelong immunity. as such, there is no preventive vaccine or cure for chikungunya. one can only manage the symptoms by taking medications for temporary relief. to prevent the spread of disease, breeding sites for the mosquitoes should be checked. using mosquito repellents and wearing long sleeve clothes and full cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 179 pants can help in preventing mosquito bite. for more such information one may refer to [1]. increasing globalization and factors contributing to climate change brought about a sudden expansion of mosquito breeding sites. this makes it necessary to improve the vector control techniques and to identify the indexes that monitor thresholds for such programs. through the 20th century, mathematical modeling has been extensively used to study epidemic diseases. futhermore, this branch of mathematics is also being used to devise optimal control strategies for various infectious diseases. like m. barro et al. [6] introduced an optimal control for a sir model governed by an ode system with time delay. and, o. k. oare [7] considered and analyzed a deterministic multipatch hepatitis c virus model for it. in context of infection due to chikungunya virus, y. dumont et al. [8] proposed a model associated with the time course of the first epidemic of chikungunya in several cities of reunion island. a model describing the mosquito population dynamics and the virus transmission to human population was discussed by d. moulay et al. [9]. although simplistic, l. yacob et al. [10] gave a model which provided a close approximation of the peak incidence of the outbreak and the final epidemic size. s. naowarat and i. m. tang [11] studied the model taking into consideration the presence of two species of aedes mosquito (aedes aegypti and aedes albopictus). d. h. palacio and j. ospina [12] derived measures of disease control, by means of three scenarios, namely a single vector, two vectors, and two vectors and human and non-human reservoirs. it also showed the need to periodically evaluate the effectiveness of vector control measures. f. b. agusto et al. [13] described the chikungunya model of three age structured transmission dynamics by considering juvenile, adult and senior population, where the dynamics of shift in individuals from one stage to another was studied. in this paper, we introduce a deterministic model to study the dynamics and transmission of chikungunya virus by considering a very significant section from the class of infected individuals. usually, the existing models focus on the sir or the seir human population model and sei mosquito population model. since the period from the infected stage to the complete recovery stage is quite long for this disease, so it becomes significant to study that particular class of human population which has recovered from acute symptoms of the disease but is yet to attain full recovery. though the class no longer shows the immediate symptoms like fever, rashes, nausea etc. but at the same time they are bearing the latent and the passive effects of the disease like joint pains, fatigue, headache etc. generally such ailments continue for a prolonged period which may vary from individual to individual. but as long as the patient is suffering from these ailments, he or she cannot be declared as fully recovered [14]. focussing on this category of patients, we introduce a new compartment between compartments of the infected and the recovered human population within the existing seir model. we refer to it as the recuperation compartment and denote it by r′. so, in this paper our aim is to study, analyse and investigate in detail the model showing the interaction between the human population divided into five compartments resulting 180 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) into a seir′r model and the mosquito population into the traditional three compartments which we denote by xyz model. the paper is divided as follows: section 2 deals with the formulation of the model, section 3 analyses its feasibility, section 4 determines the disease free equilibrium (dfe) and establishes its local and global stability , section 5 deals with the existence of endemic equilibrium (ee) and its local stability. also by means of simulation of the formulated model, we provide a visualization to the dynamics of this disease, in section 6. finally related to our model, some conclusions are stated. 2 model formulation in this section, an epidemic model is formulated for chikungunya disease. let nh represent the total human population which is further subdivided into five categories; susceptibles (s), humans exposed to infection (e), infected humans (i), population in recuperation phase (r′) and finally the population that has attained complete recovery (r). so, the traditional seir epidemic model has been modified to a more relevant and practically applicable seir′r model. hence in this case, at any time t nh(t) = s(t) + e(t) + i(t) + r ′(t) + r(t). (2.1) let nm represent the total mosquito population which is further subdivided into 3 parts; susceptible mosquitoes (x), mosquitoes exposed to infection (y) and infectious mosquitoes (z). so the total mosquito population is nm (t) = x(t) + y (t) + z(t). for human population, let µ be the constant birth rate and ζ be the natural death rate. then the rate of change of susceptible human population is given by ds dt = µ − λhs − ζs, (2.2) where λh = βbhz nh . bh is the transmission probability per contact for susceptible humans (s) and β is the mosquito biting rate for transfer of infection from infectious mosquito class (z) to susceptible human population (s). as only the susceptible human population out of the whole population is prone to get infection, thereby we divide the expression by nh. the rate of change of exposed human population is given by de dt = λhs − αe − ζe, (2.3) where α is the rate of progression from exposed (e) to infected (i) human population. here the inflow rate is λh and outflow rate is α + ζ. similarly, the rate of change of infected human population is cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 181 di dt = αe − γi − (ζ + ζ1)i, (2.4) where ζ1 is death rate due to infection and γ is progression rate of infected (i) to recuperated (r′) human population. now, rate of change of human population in recuperation phase is dr′ dt = γi − λr′ − (ζ + ζ2)r ′, (2.5) where ζ2 is the death rate of humans in recuperated phase due to virus and λ is the rate of progression from recuperation (r′) to the recovery phase (r). finally, rate of change of recovered human population is, dr dt = λr′ − ζr. (2.6) again for the mosquito population, let ρ be the constant birth rate and κ be the natural death rate, then the rate of susceptible mosquito population is given by dx dt = ρ − λmx − κx, (2.7) where λm = νbm (i + r ′) nh . bm is the transmission probability per contact for susceptible mosquito population (x) and ν is the mosquito biting rate for transfer of infection from infected (i) or recuperated (r′) human population to susceptible mosquito population (x). again there occurs division by nh because infection can be transfered to mosquitoes only by a certain fraction of human population. now, the rate of change of exposed mosquito population is given by dy dt = λmx − ψy − κy, (2.8) where ψ is the progression rate from exposed (y) to infectious (z) mosquito population. here the inflow rate is λm and outflow rate is ψ + κ. similarly, the rate of change of mosquito population carrying infection is dz dt = ψy − κz. (2.9) compiling the above discussion, we get the eight dimensional system of nonlinear ordinary differential equations that forms our chikungunya model (cm). the parameters and the variables used in the model (cm) are described in table 1. to get a clear view of the inter relationships between various compartments in discussion, one may refer to figure 1 which shows the schematic flow 182 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) diagram of the model. the model (cm) is as follows: (cm) ds dt = µ − βbhzs nh − ζs, de dt = βbhzs nh − αe − ζe, di dt = αe − γi − (ζ + ζ1)i, dr′ dt = γi − λr′ − (ζ + ζ2)r ′, dr dt = λr′ − ζr, dx dt = ρ − νbm (i + r ′)x nh − κx, dy dt = νbm (i + r ′)x nh − ψy − κy, dz dt = ψy − κz. figure 1: schematic diagram of chikungunya model (cm) cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 183 table 1: description of variables and parameters used in model (cm) variables description s susceptible human population. e exposed human population. (population that is infected but yet to show symptoms). i infected human population showing symptoms. r′ human population in recuperation phase. r fully recovered human population. x susceptible mosquito population. y exposed mosquito population. (carrying infection but not yet capable to spread it). z infectious mosquito population spreading the disease. parameters description µ human birth rate. β mosquito biting rate for transfer of infection from infectious mosquito class (z) to susceptible human population (s). α progression rate of exposed to infected human population. γ progression rate of infected to recuperated human population. λ progression rate of recuperated to fully recovered human population. ρ mosquito birth rate. ν mosquito biting rate for transfer of infection from infected human population(i) or population under recuperation phase (r′) to susceptible mosquito population (x). ψ progression rate from exposed to infectious mosquito population. ζ natural death rate for human population. ζ1 human death rate in infected stage due to viral infection. ζ2 human death rate due to infection under recovery phase. κ natural death rate for mosquito population. bh transmission probability per contact in susceptible humans. bm transmission probability per contact in susceptible mosquitoes. nh total human population, i.e. s+e+i+r ′+r. 184 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) table 2: range of parameters for the model (cm) parameters range references µ 400 × 1 15 × 365 400 × 1 12 × 365 [15, 16] β 0.19 0.39 [15, 17] α 1 4 1 2 [4, 15, 18, 19, 20, 21] γ 1 4 1 2 estimated [14] λ 1 8 1 4 estimated [14] ρ 500 × 0.015 500 × 0.33 [15, 16, 22, 23] ν 0.19 0.39 [15, 17] ψ 1 6 1 2 [9, 18, 20, 24] ζ 1 60 × 365 1 18 × 365 [13] ζ1 1 105 1 104 [25] ζ2 1 106 1 105 [25] κ 1 42 1 14 [9, 18, 19, 20, 21] bh 0.001 0.54 [8, 15, 26, 18, 27] bm 0.005 0.35 [8, 26, 27, 28, 29] cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 185 table 3: values of parameters for simulation parameters r0 < 1 r0 > 1 µ 400 × 1 15 × 365 400 × 1 15 × 365 β 0.25 0.30 α 1 3 1 4 γ 1 3 1 4 λ 1 7 1 8 ρ 500 × 0.1675 500 × 0.2 ν 0.25 0.30 ψ 1 3.5 1 4 ζ 1 40 × 365 1 30 × 365 ζ1 1 104 1 105 ζ2 1 105 1 106 κ 1 14 1 30 bh 0.24 0.30 bm 0.24 0.30 186 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) 3 preliminary results 3.1 positivity of solutions in order to establish the epidemiological meaningfullness [13], we prove the non negativity of the state variables for the formulated model at all t > 0. theorem 3.1: the solution m(t) = (s,e,i,r′,r,x,y,z) of model (cm) with m(0) ≥ 0, is non negative for all t > 0. moreover, lim t→∞ sup nh(t) = µ ζ and lim t→∞ sup nm(t) = ρ κ where nh(t) = s(t) + e(t) + i(t) + r ′(t) + r(t) and nm (t) = x(t) + y (t) + z(t). proof: let t1 = sup {t > 0 : m(t) > 0}. clearly t1 > 0. consider the first equation of the model (cm), ds dt = µ − βbhsz nh − ζs. solving the differential equation we have, d dt { s(t) exp [( ∫ t1 0 βbhz(τ) nh(τ) dτ + ζt )]} = µexp [( ∫ t1 0 βbhz(τ) nh(τ) dτ + ζt )] =⇒ s(t1) exp [( ∫ t1 0 βbhz(τ) nh(τ) dτ + ζt1 )] − s(0) = ∫ t1 0 µexp [( ∫ u 0 βbhz(τ) nh(τ) dτ + ζu )] du. furthermore, s(t1) =s(0) exp [( − ∫ t1 0 βbhz(τ) nh(τ) dτ + ζt1 )] + exp [( − ∫ t1 0 βbhz(τ) nh(τ) dτ + ζt1 )] ∫ t1 0 µexp [( ∫ u 0 βbhz(τ) nh(τ) dτ + ζu )] du > 0. similarly, the non negativity can be shown for all the state variables, i.e., m(t1) > 0 and therefore m(t) > 0 for all t > 0. in fact, we now have, 0 < s(t) ≤ nh(t), 0 < e(t) ≤ nh(t), 0 < i(t) ≤ nh(t), 0 < r ′(t) ≤ nh(t), 0 < r(t) ≤ nh(t); 0 < x(t) ≤ nm(t), 0 < y (t) ≤ nm(t), 0 < z(t) ≤ nm (t). as the total human population is given by nh(t) = s(t) + e(t) + i(t) + r ′(t) + r(t), the rate of change of human population with respect to time is given by dnh dt = µ − ζ(s + e + i + r′ + r) − ζ1i − ζ2r ′ = µ − ζnh − ζ1i − ζ2r ′ ≤ µ − ζnh. (3.1) cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 187 now for nm(t) = x(t) + y (t) + z(t), dnm dt ≤ ρ − κnm. let n = µ ζ . as t → ∞, the disease will disappear. therefore, lim t→∞ sup i(t) = 0 and lim t→∞ sup r′(t) = 0. now, dnh dt = µ − ζnh this implies nh(t) = µ ζ + ( nh(0) − µ ζ ) e−ζt, which further implies lim t→∞ nh(t) = µ ζ = n. this follows that 0 < lim t→∞ supnh(t) ≤ n = µ ζ if lim t→∞ sup i(t) = 0 and lim t→∞ sup r′(t) = 0. and if nh > n = µ ζ then from (3.1), dnh dt < 0. similarly, it can be seen that 0 < lim t→∞ supnm(t) ≤ ρ κ . 3.2 invariant region consider ℜ = ℜh × ℜm ⊂ r 5 + × r 3 +, where ℜh = { s,e,i,r′,r : nh(t) ≤ µ ζ } , ℜm = { x,y,z : nm(t) ≤ ρ κ } . now, we establish the positive invariance [13], of the region ℜ associated to the model (cm). that is, we show that solutions in ℜ remain in ℜ for all t > 0 . theorem 3.2: the region ℜ ⊂ r8+ is positively invariant for the model (cm), with non-negative initial conditions in r8+. proof : as seen in theorem 3.1, dnh dt ≤ µ − ζnh and dnm dt ≤ ρ − κnm. by using standard comparison theorem [30], it can be seen that, nh(t) ≤ µ ζ = n. so, clearly every solution in ℜh remains in ℜh for all t > 0. similar is the case for every solution of ℜm . hence, the region ℜ is positively invariant and contains all solutions of r8+ for model (cm). in the following sections, we show the existence and stability of the disease free equilibrium (dfe) and endemic equilibrium (ee) for the model (cm). 4 disease free equilibrium (dfe) in this section, we find a unique disease free equilibrium (dfe) for the model (cm) and then analyse its stability. 188 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) 4.1 existence of equilibrium to determine the disease free equilibrium (dfe) of the model, we consider the sections of populations that are free from disease and put their time derivatives equal to zero. let dfe be denoted by ed = (s ∗,e∗,i∗,r ′ ∗,r∗,x∗,y ∗,z∗). as sections of susceptible and recovered humans as well as susceptible mosquitoes are the only sections free from disease therefore ed = (s∗,0,0,0,r∗,x∗,0,0). solving the differential equations of the model (cm), dfe is obtained as ed = ( µ ζ ,0,0,0,0, ρ κ ,0,0 ) . 4.2 reproduction number let the basic reproduction number be denoted by r0, which is defined as the expected number of secondary cases produced by a single (typical) infection in a population that is completely disease free. to find the threshold quantity r0 [31, 32], we consider the next generation matrix g, which comprises of two matrices f and v −1, where f = dfi(x0) dxj and v = dvi(x0) dxj for 1 ≤ i,j ≤ 5. here, fi represents the new infection, whereas vi corresponds to the transfers of infection from one compartment to another. let x0 be the disease free equilibrium state. hence, the reproduction number is the largest eigen value of the next generation matrix g (defined as the product of matrices f and v −1), that is the largest eigen value of the matrix, g = fv −1. corresponding to the model (cm), f =       βbhsz nh 0 0 νbm (i+r ′)x nh 0       and v =       αe + ζe −αe + γi + (ζ + ζ1)i −γi + λr′ + (ζ + ζ2)r ′ ψy + κy −ψy + κz       . next, we find the jacobian f and v of the matrices f and v respectively and the eigen values of the matrix g = fv −1, gives the reproduction number as r0 = √ ρνψζαβbhbm (λ + γ + ζ + ζ2) κ √ µ(ψ + κ)(ζ + α)(ζ + γ + ζ1)(ζ + λ + ζ2) . cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 189 4.3 local stability theorem 4.1 : the dfe of the chikungunya model (cm) is locally asymptotically stable, if r0 < 1 and unstable if r0 > 1, where r0 is the associated reproduction number. proof : we consider the system of non linear differential equations, corresponding to the model (cm) to evaluate its jacobian matrix. let jd denote the jacobian of the system at dfe that is, jd =             −ζ 0 0 0 0 0 0 −bhβ 0 −α − ζ 0 0 0 0 0 bhβ 0 α −γ − ζ − ζ1 0 0 0 0 0 0 0 γ −λ − ζ − ζ2 0 0 0 0 0 0 0 λ −ζ 0 0 0 0 0 −νbm ρζ κµ −νbm ρζ κµ 0 −κ 0 0 0 0 νbm ρζ κµ νbm ρζ κµ 0 0 −ψ − κ 0 0 0 0 0 0 0 ψ −κ             clearly, the trace of the matrix jd is negative and determinant of matrix jd [33, 34], is given by det(jd) = −ζ 2[κ2µ(ψ + κ)(ζ(ζ + α + γ) + αγ + ζζ1 + ζ1α)(−ζ − λ − ζ2)] + ρνζψαβbhbm (ζ + λ + γ + ζ2) µ . for r0 < 1, we have √ ρνψζαβbhbm (ζ + γ + λ + ζ2) < κ √ µ(ψ + κ)(ζ + α)(ζ + λ + ζ2)(ζ + γ + ζ1). therefore, κ 2 µ(ψ + κ)(ζ + λ + ζ2)(ζ(ζ + α + γ) + αγ + ζζ1 + ζ1α) − ψ[ρνζαβbhbm(ζ + λ + γ + ζ2)] > 0 or det(jd) > 0. hence, dfe is locally asymptotically stable if r0 < 1 . 4.4 global stability consider the feasible region ℜ1 = {d ∈ ℜ : s ≤ s ∗,x ≤ x∗} where d = (s,e,i,r′,r,x,y,z), s∗ and x∗ are the components of dfe (ed). lemma 4.1: the region ℜ1 is positively invariant for the model (cm). 190 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) proof: from the first equation of the model (cm), ds dt = µ − βbhzs nh − ζs ≤ µ − ζs ≤ ζ ( µ ζ − s ) ≤ ζ(s∗ − s) s ≤ s∗ + (s(0) − s∗)e−ζt thus, if s∗ = µ ζ for all t ≥ 0 and s(0) ≤ s∗ , then s ≤ s∗ for all t ≥ 0. similarly, for dx dt = ρ − νbm (i + r ′)x nh − κx ≤ ρ − κx ≤ κ(x∗ − x) x ≤ x∗ + (x(0) − x∗)e−κt thus, if x∗ = ρ κ for all t ≥ 0 and x(0) ≤ x∗, then x ≤ x∗ for all t ≥ 0. hence, it has been shown that the region ℜ1 is positively invariant and attracts all solutions in ℜ 8 + for the model (cm). now in order to establish the global asymptotic stability of dfe [35], we rewrite the model (cm) as [ dtu dt = f(tu,ti) dti dt = g(tu,ti), g(tu,0) = 0 ] (rm) where tu = (s,r,x) ∈ r 3 and ti = (e,i,r ′,y,z) ∈ r5. let e∗d = (t ∗ u,0) be dfe of (rm) where t ∗ u = ( µ ζ ,0, ρ κ ) . we now state the following two conditions which must be satisfied to guarantee global asymptotic stability: (h1) for dtu dt = f(tu,0), t ∗ u is globally asymptotically stable. (h2) g(tu,ti) = ati − ĝ(tu,ti), ĝ(tu,ti) ≥ 0, (tu,ti) ∈ ℜ where a = ∂g(t ∗u,0) ∂ti is an m-matrix which by definition has the off diagonal elements non-negative. theorem 4.2: the fixed point e∗d = (t ∗ u,0) is globally asymptotic stable (g.a.s) equilibrium of (rm) provided that r0 < 1 and that assumptions (h1) and (h2) are satisfied. proof: for the system (rm), dtu dt = f(tu,0) =    µ − ζs 0 ρ − κx    cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 191 we solve the above linear differential system to get the s(t) = µ ζ + s∗(0)e−µt, r(t) = 0 and x(t) = ρ κ + x∗(0)e−κt which implies s(t) → µ ζ , r(t) → 0 and x(t) → ρ κ as t → ∞. therefore, disease free point t ∗u is a globally asymptotic stable (g.a.s) equilibrium of dtu dt = f(tu,0). hence (h1) holds. clearly it can be seen that g(tu,ti) =       βbhzs nh − αe − ζe αe − γi − (ζ + ζ1)i γi − λr′ − (ζ + ζ2)r ′ νbm (i+r ′)x nh − ψy − κy ψy − κz       also from (h2) g(tu,ti) = ati − ĝ(tu,ti), where a = ∂g(t ∗u,0) ∂ti =       −α − ζ 0 0 0 βbh α −γ − ζ − ζ1 0 0 0 0 γ −λ − ζ − ζ2 0 0 0 νbm ρζ κµ νbm ρζ κµ −ψ − κ 0 0 0 0 ψ −κ       . therefore, ∂g(t ∗u,0) ∂ti ti =       −αe − ζe − βbhz αe − γi − (ζ + ζ1)i γi − λr′ − (ζ + ζ2)r ′ νbm aζ κµ (i + r′) − (ψ + κ)y ψy − κz       . in view of (h2), ĝ(tu,ti) = ∂g(t ∗ u ,0) ∂ti ti − g(tu,ti) which gives ĝ(tu,ti) =        βbhz(1 − s nh ) 0 0 νbm (i + r ′) [ ρζ µκ − x nh ] 0        . clearly, βbhz(1 − s nh ) ≥ 0 as s nh < 1. also, κx ρ ≤ ζnh µ or ρζ µκ ≥ x nh ⇒ x∗ s∗ ≥ x nh and from lemma 4.1 we know x∗ ≥ x and n∗h = s ∗ ≥ nh, which implies νbm (i + r ′) [ ρζ µκ − x nh ] ≥ 0. therefore, (h2) holds true. hence, e∗d = (t ∗ u,0) is globally asymptotically stable in the region ℜ whenever r0 ≤ 1. 5 endemic equilibrium in this section, we first determine the endemic equilibrium points for the model (cm), establish its existence and then analyse its stability. 192 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) 5.1 endemic equilibrium points let endemic equilibrium points be denoted by ee = (s ∗∗,e∗∗,i∗∗,r′∗∗,r∗∗,x∗∗,y ∗∗,z∗∗). the components of ee are obtained by imposing constant solutions in the model (cm) and solving the algebraic equations. by computations, we have s∗∗ = µnh ζnh + z∗∗βbh , e ∗∗ = βbhz ∗∗µ (α + ζ)(βbhz∗∗ + ζnh) , i∗∗ = αβbhz ∗∗µ (γ + ζ + ζ1)(α + ζ)(βbhz∗∗ + ζnh) , r′∗∗ = αγβbhz ∗∗µ (λ + ζ + ζ2)(γ + ζ + ζ1)(α + ζ)(βbhz∗∗ + ζnh) , r∗∗ = λαβbhz ∗∗µγ ζ(λ + ζ + ζ2)(γ + ζ + ζ1)(α + ζ)(βbhz∗∗ + ζnh) , x∗∗ = ρ λm + κ , y ∗∗ = ρλm (λm + κ)(ψ + κ) , z∗∗ = ρψλm κ(λm + κ)(ψ + κ) . 5.2 existence and uniqueness of endemic equilibrium(e e ) theorem 5.1 : chikungunya model (cm) has a unique endemic equilibrium if r0 > 1. as seen in section 2, λm = νbm (i ∗∗ + r ′ ∗∗) nh = νbmζαβµbhz ∗∗(ζ + ζ2 + λ + γ) µ(βbhz∗∗ + µ)(α + ζ)(ζ + ζ1 + γ)(ζ + ζ2 + λ) = r20µz ∗∗κ2(ψ + κ) ρψ(βbhz∗∗ + µ) also, λh = βbhz ∗∗ nh = βbhρψλm κnh(λm + κ)(ψ + κ) , or equivalently λm = λhµκ 2(ψ + κ) βbhρψζ − µκ(ψ + κ)λh . cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 193 equating both values of λm , we get the following linear equation in terms of λh: λh(ρψβbh + r 2 0µκ(ψ + κ)) = (r 2 0 − 1)βbhρψζ. the unique solution to this equation exists and is given by λh = (r20 − 1)βbhρψζ ρψβbh + r 2 0µκ(ψ + κ) , which is positive if r20 > 1. this implies z ∗∗ > 0, for r0 > 1. hence, unique endemic equilibrium exists for r0 > 1. 5.3 local stability theorem 5.2: the endemic equilibrium of the chikungunya model (cm) is locally asymptotically stable if r0 > 1. proof: we evaluate the jacobian matrix for the system of nonlinear differential equations corresponding to the model (cm). let je denote the jacobian of the system at ee (which exists for r0 > 1). clearly, je = (j1,j2,j3,j4,j5,j6,j7,j8) t where j1 = ( −βbhz ∗∗ nh + βbhz ∗∗ s ∗∗ (nh ) 2 − ζ, βbh z ∗∗ s ∗∗ (nh ) 2 , βbh z ∗∗ s ∗∗ (nh ) 2 , βbh z ∗∗ s ∗∗ (nh ) 2 , βbh z ∗∗ s ∗∗ (nh ) 2 ,0, 0, −βbh s ∗∗ nh ) , j2 = ( βbhz ∗∗ nh − βbhz ∗∗s∗∗ (nh ) 2 , −βbh z ∗∗s∗∗ (nh ) 2 − α − ζ, −βbh z ∗∗s∗∗ (nh ) 2 , −βbh z ∗∗s∗∗ (nh ) 2 , −βbh z ∗∗s∗∗ (nh ) 2 ,0, 0, βbh s ∗∗ nh ) , j3 = (0,α,−γ − ζ − ζ1,0, 0,0, 0,0) , j4 = (0,0,γ,−λ − ζ − ζ2,0, 0,0, 0) , j5 = (0,0, 0,λ,−ζ,0,0, 0) , j6 = ( νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 , νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 , νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 − νbm x ∗∗ nh , νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 − νbm x ∗∗ nh , 0, − νbm (i ∗∗+r ′ ∗∗) nh − κ,0, 0 ) , j7 = ( −νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 , −νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 , −νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 + νbm x ∗∗ nh , −νbm (i ∗∗+r ′ ∗∗)x∗∗ (nh ) 2 + νbm x ∗∗ nh , 0, νbm (i ∗∗+r ′ ∗∗) nh ,−κ − ψ,0 ) , j8 = (0,0, 0,0, 0,0,ψ,−κ) further, we reduce je to the following upper triangular matrix (ue). ue = (u1,u2,u3,u4,u5,u6,u7,u8) t where u1 = ( −βbhz ∗∗ nh + βbhz ∗∗ s ∗∗ (nh)2 − ζ, βbhz ∗∗ s ∗∗ (nh)2 , βbhz ∗∗ s ∗∗ (nh)2 , βbhz ∗∗ s ∗∗ (nh)2 , βbh z ∗∗ s ∗∗ (nh)2 ,0,0, −βbhs ∗∗ nh ) , u2 = ( 0, −βbhz ∗∗ s ∗∗ (nh)2 − α − ζ, −βbhz ∗∗ s ∗∗ (nh)2 , −βbhz ∗∗ s ∗∗ (nh)2 , −βbhz ∗∗ s ∗∗ (nh)2 ,0,0, βbhs ∗∗ nh ) , u3 = (0,0,−γ − ζ − ζ1,0,0,0,0,0), u4 = (0,0,0,−λ − ζ − ζ2,0,0,0,0) , u5 = (0,0,0,0,−ζ,0,0,0), u6 = ( 0,0,0,0,0,− νbm(i ∗∗+r ′ ∗∗) nh − κ,0,0 ) , u7 = (0,0,0,0,0,0,−κ − ψ,0) , u8 = (0,0,0,0,0,0,0,−κ) 194 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) attached are the eigen values of ue: ( −ζ,−κ,−ψ − κ,−γ − ζ − ζ1,−λ − ζ − ζ2,− νbm (i ∗∗+r ′ ∗∗) nh − κ, z ∗∗ βbh(s ∗∗ −nh) (nh)2 − ζ, −βbhz ∗∗ s ∗∗ (nh)2 − α − ζ ) each of which is negative and by the criterion given in [36], the endemic equilibrium point (ee) is locally asymptotically stable if r0 > 1. 6 numerical simulation the values of parameters that would be used for simulation of the model (cm) are listed in table 3. the values used for simulation are taken with reference to their ranges, as stated in table 2. fig. 2a and fig. 2b are visualizations of the existence and stability of equilibria for the cases, r0 < 1 and r0 > 1, respectively. also, it illustrates that for r0 < 1, the infection dies out over a period of time as it is the case of dfe. however, in the same time period, it can been seen that the infection continues to persist in the population when r0 > 1 as it is the case of ee. (a) (b) figure 2: total number of infected humans (i) with respect to time. in fig. 3a, it is clear that the recuperated population ultimately falls down to zero for the case when r0 < 1, where finally the disease dies out and ultimately the entire population will shift to the recovered section with no more inflow into the recuperated part. in contrast, for the same time period, if r0 > 1 (fig. 3b), the disease persists in the population. therefore, we can see a substantial proportion of population which is still in the recuperated phase. fig. 4 and fig. 5, both show the time duration around which the number of infected population comes to a fall which is actually the same for recuperated population to reach the peak. in fig. 6a, again for r0 < 1, as the disease dies out so it is evidently a situation when the population of the infectious mosquitoes dies out. in contrast to it, for r0 > 1 (fig. 6b), the number cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 195 (a) (b) figure 3: total number of recuperated humans (r′) with respect to time. figure 4: total number of infected (i) and recuperated (r′) humans when r0 < 1. of infectious mosquitoes continue to persist in population as it is the case of endemic equilibrium (ee). fig. 7 shows the change in the number of infected, recuperated and recovered population with respect to time in accordance with model (cm) whereas fig. 8 is a simulation of the model (cm) without recuperated section of population. the curve representing the recovered population in fig. 8, is an increasing curve showing a rapid increase in the number of people attaining full recovery. but this does not fit in accordance to the case of chikungunya infection. however, in fig. 7, we can see the convexity of the curve representing recovered population for a substantial period of time and this is because of the presence of recuperation factor which has been considered in our model. during this period, the recuperation curve is rising higher which is practically more relevant and well in consensus with the nature of this particular disease. 196 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) figure 5: total number of infected (i) and recuperated (r′) humans when r0 > 1. (a) (b) figure 6: total number of infectious mosquitoes (z) with respect to time. 7 conclusion in this paper, a new deterministic model is formulated to study the transmission dynamics of chikungunya virus (chikv). making a considerable refinement to the existing models present in the literature, a so far neglected section of human population is introduced, namely the population in the recuperation phase. the study shows that the disease free equilibrium (dfe) of the model is locally as well as globally asymptotically stable whenever existence of an associated reproduction number r0, is less than 1 and unstable otherwise. also, an endemic equilibrium (ee) exists whenever r0 is greater than 1 and is locally asymptotically stable too. simulations of the model make it evident that introduction of the said compartment is well justified, as this model provides a more realistic illustration for chikungunya infection wherein the quantitative behaviour of disease has given a better visualisation. moreover, the qualitative behaviour of the disease as studied by cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 197 figure 7: variation of infected, recuperated and recovered human population with time for model (cm). figure 8: variation of infected and recovered human population with time for model (cm) without recuperation section. various researchers in [14] is very well taken into consideration through our model. if we do not consider the recuperation section in model (cm), then the following model becomes a special case of our model. it is clearly seen that our model (cm) gives a better illustration to the dynamics of the chikungunya virus and hence, the proposed model is indeed 198 ruchi arora, dharmendra kumar, ishita jhamb and avina kaur cubo 22, 2 (2020) more realistic and practical. ds dt = µ − βbhzs nh − ζs, de dt = βbhzs nh − αe − ζe, di dt = αe − γi − (ζ + ζ1)i, dr dt = γi − ζr, dx dt = ρ − νbmix nh − κx, dy dt = νbmix nh − ψy − κy, dz dt = ψy − κz, where nh(t) = s(t) + e(t) + i(t) + r(t). comparison of the above model with our model (cm) is done in section 6 with the help of the graphs shown in fig. 7 and fig. 8. acknowledgement the authors are thankful to dr. sukhanta dutta, dr. harsha kharbanda and tanvi for their valuable suggestions and inputs. this research was supported and funded by the science centre, sgtb khalsa college (project code: sgtbkc/sc/sp/2017/08/518). cubo 22, 2 (2020) mathematical modeling of chikungunya dynamics . . . 199 references [1] who. chikungunya. http://www.who.int/denguecontrol/arbo-viral/other aborvial chikun gunya/en/, 2014. 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[36] ex. dejesus and c. kaufman. routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. physical review. a, 35:5288–5290, 07 1987. introduction model formulation preliminary results positivity of solutions invariant region disease free equilibrium (dfe) existence of equilibrium reproduction number local stability global stability endemic equilibrium endemic equilibrium points existence and uniqueness of endemic equilibrium(ee) local stability numerical simulation conclusion cubo 8 , 8s-8& (1992) sext& jornada de matem6tica de la zona sur . análisis armónico sobre sl(2, .ft), .ft cuerpo p-ádico (p f. 2). roberto riquelme sepúlveda resumen. sea fl una extensión fini to di mellsional de !qp. thabajaudo en le consllrucción de las representaciones de sl(2, o), donde o es el anillo de enteros den, lema que constituye m.i tiraba.jo de tesis de magister, fue necesario analizar la construción de represen t aciouesunitar ias de sl(2, ji). en esta comunicación expondré los resultadosbé.sicos en que se apoya esta const 11ueci611: · transformada aditiva de fourier y proj:>iedades. representiación multivaluadas de sl (2, n). · la representación uni tiaria d(. e c1 ). definición 2. 1 a) por una representación proyectiva de g entenderemos a un homomorfism o p de e en u/ es un ca1·ácler n o trivial de n+, entonces el límite h(tp) = .... ~~o:/p"' ) w(b) es pa.r y es igual a q~g(qj,1 ) si w(l/j) w(b) es impar. ( consitlem r el carácter ~(x) = b(x)). construcción de r epre !:!entnciones unitarios d e s l (2, fl). sea v un espacio vectorial topológico de dimensión finita sobre n provisto de una forma cuadrática no degenerada q. resultados análogos a los de f/ (ef;) se pueden oblener para h(4>, q ) y h(tjj, a , q ). teorema 2.5 sea v tm espacio vectorial sobren. pam 11 en h v(= l2 (v+)) y p a ra cada carácter no trivial q, de n + sea tw = h (4>, q)li y paro b en n m,,li = f •, (donde / 6(2;) = (bq(x ))). enton ces la.s aplicaciones ( -~ ~) tw y (01 bl) mb puetlen .~er extendidas a 1ma representación proyec.tiva de sl (2.n). análisis orm6nlco ... cubo 8 85 denotemos a esl.8 representación por d(4', v). supongamos que dimnv = 2m. para h en hu y a en n• escribamos (u11 h)(x) =i a lm h(o.x). eot.ooccs a -u11 es una represenlación unitaria de n·. de aquí se tiene que si (u;h)(x) = h(~,a,q)[(~,qj-i ja jm h(ax) entonces a u~ es una reprl!sentaci6n unitaria de n·. además se puede ver que a h(cp,a,q)[h(fl,qj1 ¡a i"' define un carácter den·. denotemos este carácter por sign. si ug denota el operador unitario sobre h. corresponde al elemento g = (: ! ) en sl(2, !l) bajo la representación d(q,1 v) 1 entonces se puede ver que: (ugh)(x) = f l<(g¡x, y)h(y)d~y donde para c en n· : j((g;x,y) = .«gn(-l)h(~,qjw ~¡•91"1+"9)'1-8(.,,1] y para e= o; j((g,x, y) =j • ["' sign(a)~(baq(x))c.(ax-y). c. denota la función delta de dirac. teorema 2.6 la representación d(tjj, v) es continua. teorema 2. 7 sea c el algebra (con la topología débil) de todos los operadores acolados sobre hu que conm:utcm con d(l/j, v). entonces ui3le u.n homomorfismo continuo del grupo l1 (a) ele a en c ( clonrle a es el grupo ortogonal de la.forma cuadrática q). dirección del autor: depa11tiamento de matemática facultad de ciencias universidad de concepción. revista de matemáticas_0099 revista de matemáticas_0100 revista de matemáticas_0101 a mathematical journal vol. 6, no 4, (133-165). december 2004. symétries en dimension trois: une approche quantique nafaa chbili département de mathématiques. faculté des sciences de monastir bd de l’environnement, monastir 5000 tunisie nafaa.chbili@esstt.rnu.tn abstract one of the most intriguing problems in topology is the question of whether a given manifold is symmetric, i.e. whether there is a finite cyclic group that acts on it. this question has its origins in a number of interrelated facts. the problem of group action has been subject of extensive literature, where different kinds of classical techniques have been used to shed some lights on this subject. the recent years have seen many new invariants introduced to low dimensional topology. the discovery of these invariants is considered as a revolution in the theory of knots and three-manifolds. the main goal of our research is to use this kind of invariants to study the problem of group action on 3-manifolds. in the case of knots and links we use the new invariants to find necessary conditions for a knot to be freely periodic. we apply our criteria successfully to the 84 knot with less than 9 crossings. let n be an integer and m a compact oriented 3-manifold, m is said to be nperiodic if the cyclic group of order n acts semi-freely on m with a circle as the set of fixed points. we study the quantum invariants (su(2) and su(3)) of n-periodic manifolds (n is an odd prime) and show how these invariants reflect the periodicity of the considered manifold. the criteria we established enabled us to prove that the poincaré sphere is not n-periodic for some values of n. 134 nafaa chbili 6, 4(2004) 1 introduction une question fondamentale en topologie est la suivante : etant donnée une variété m et un groupe g, est-ce que g agit non trivialement sur m ? cette question trouve ses origines dans d’autres domaines de la science et de la vie. en effet, cette question n’est autre que la formulation mathématique de la notion de symétrie. une notion qui fascine et joue un rôle important dans de divers domaines. en topologie ce problème a fait l’objet d’une littérature abondante. la richesse et la complexité de la topologie en dimension trois donne plus d’importance à ce sujet. des techniques classiques variées ont été utilisées pour explorer certains aspects du problème. cependant plusieurs questions concernant les actions de groupes, même dans les cas les plus simples, restent à nos jours sans réponses. au cours des deux dernières décennies sont apparues des techniques nouvelles en dimension trois. en 1984, jones [27] a introduit un invariant de nœuds et entrelacs de s3 qui a permis de résoudre plusieurs problèmes dont des questions ouvertes vieilles de plus d’un siécle. cet invariant a été immédiatement généralisé en d’autres invariants du même genre pour les nœuds et entrelacs de s3. ces découvertes ont été suivies quelques années plus tard d’un progrès semblable en théorie des variétés de dimension trois. a partir de la théorie quantique des champs topologiques, witten [61] a montré l’existence d’une famille d’invariants topologiques des variétés compactes connexes orientées de dimension trois. il revient ensuite à reshitikhin et turaev [51] de donner une construction mathématique de ces invariants en utilisant l’algèbre quantique uq(sl2), d’où le nom invariants quantiques. cette construction a été suivie d’une autre beaucoup plus élémentaire, due à lickorish [36]. en effet ce dernier définit ces invariants en utilisant seulement la théorie skein associée au crochet de kauffman, évalué en les racines 4pème de l’unité. dans ce même esprit, l’invariant θp est introduit par blanchet, habbeager, masbaum et vogel, l’avantage de l’approche suivie par ces derniers [5] est qu’elle a permis de montrer qu’on peut étendre la définition des invariants de lickorish en évaluant le crochet de kauffman en les racines de l’unité d’ordre 2p. la théorie skein a été utilisée ensuite par ohtsuki et yamada [45] pour définir l’invariant quantique su(3), puis par yokota [60] et blanchet [4] qui ont défini une théorie skein pour l’invariant su(n). la question qu’on se pose dans ce travail est la suivante : comment se comportent les invariants quantiques lorsque les variétés considérées (nœuds, entrelacs, tresses, 3variétés) sont symétriques ? autrement dit, jusqu’à quel point ces invariants sont-ils sensibles à la géométrie de la variété considérée ? une réponse à cette question permettra non seulement d’avoir une idée sur les symétries de la variété en question, mais aussi de mesurer la sensibilité et la fidélité des invariants quantiques. fautil rappeler que ces invariants, dont on dispose de plusieurs façons pour les définir, manquent d’interprétation géométrique ? la première partie de ce travail s’intéresse au cas des symétries libres en dimension un (tresses, entrelacs). la deuxième partie traite le cas des 3-variétés périodiques (sur lesquelles un groupe cyclique fini agit semi-librement avec un cercle comme l’ensemble des points fixes). 6, 4(2004) symétries en dimension trois: une approche quantique 135 2 préliminaires ce paragraphe rappelle quelques notions de base concernant la théorie des nœuds. pour plus de détails nous renvoyons le lecteur à [7] et [52]. nous nous contenterons ici d’énoncer les définitions et les résultats principaux qui nous serviront par la suite pour la présentation de notre travail. 2.1 entrelacs et isotopies dans le reste de ce papier, nous parlerons indifféremment d’entrelacs dans s3 ou dans r3. définition 2.1.1 (entrelacs) un entrelacs l de s3 est une sous variété différentiable compacte sans bord de dimension 1 de la sphère s3. un nœud est un entrelacs qui possède une seule composante connexe. l’entrelacs l est dit en bande si pour tout x ∈ l on donne un vecteur normal qui dépend de x d’une façon continue. définition 2.1.2 (isotopie) on dit que deux entrelacs l1 et l2 sont isotopes si et seulement si il existe une famille d’applications ft pour t ∈ [0, 1] de s1 ∪ s1 ∪ ... ∪ s1 dans s3 telles que : (i) l’application : [0, 1] × (s1 ∪ s1 ∪ ... ∪ s1) −→ s3 (t,x) 7−→ ft(x) est différentiable. (ii) ∀t ∈ [0, 1], ft est un plongement. (iii) imf0 = l1 et imf1 = l2. définition 2.1.3 (diagramme d’un entrelacs) on appelle diagramme d’un entrelacs l un graphe dl qui est une projection de l contenue dans le plan telle que : (i) tout sommet est d’ordre 4. (ii) toute arête est différentiablement plongée dans le plan. par chaque sommet passent quatre arêtes. deux arêtes opposées sont considérées comme étant au dessus des deux autres. exemple figure 1. 136 nafaa chbili 6, 4(2004) remarque 2.1.4 reidemeister a montré [50] que l’étude des classes d’isotopies des entrelacs orientés se ramène à l’étude des diagrammes des entrelacs modulo la relation d’équivalence engendrée par les mouvements suivants (appelés mouvements de reidemeister) : de la même façon on peut démontrer que les classes d’isotopies des entrelacs en bandes correspondent aux diagrammes des entrelacs non orientés modulo les mouvements de reidemeister de type ii et iii. il existe aussi une version orientée des mouvements de reidemeister, cette version correspond aux diagrammes orientés [34]. figure 2. dans le reste de ce papier nous parlerons indifféremment d’entrelacs et de diagramme d’entrelacs. définition 2.1.5 (image miroir) soit l un entrelacs orienté. on appelle image miroir de l et on note l! l’entrelacs image de l par une symétrie plane de r3. si dl est un diagramme représentant l alors le diagramme dl! associé à l! est obtenu à partir de dl en inversant ses croisements. exemple dl dl! figure 3. 6, 4(2004) symétries en dimension trois: une approche quantique 137 définition 2.1.6 (nœud singulier) on appelle nœud singulier, une application différentiable du cercle orienté s1 dans r3 n’ayant ni point singulier ni point triple mais possédant (éventuelle-ment) des points doubles simples transverses. figure 4. de la même façon que pour les nœuds classiques, on peut définir une relation d’équivalence sur l’ensemble des nœuds singuliers. cette relation est dite isotopie à sommets rigides. 2.2 tresses et tangles nous notons par r2 le plan euclidien et par i l’intervalle [0, 1]. pour tout entier i on désigne par pi le point de r 2 de coordonnées (i, 0) dans la base canonique. notons par pn l’ensemble des points pi pour i variant entre 1 et n. définition 2.2.1 (tangle) soit n un entier naturel non nul. un n-tangle t est une sous variété de la bande r2 ×i compacte de dimension 1 telle que le bord ∂t vérifie la condition suivante : ∂t = r2 × {0, 1} ∩ t = pn × {0, 1}. deux tangles t et t ′ sont isotopes s’il existe une isotopie de la bande r2 ×i fixe sur le bord transformant t en t ′. figure 5. définition 2.2.2 (tresse) une tresse b à n brins est un n-tangle tel que pour tout réel z le plan r2 × {z} coupe b exactement en n points distincts. 138 nafaa chbili 6, 4(2004) figure 6. définition 2.2.3 (produit de deux tangles) soient t et t ′ deux n-tangles, on définit le produit tt ′ comme étant le n-tangle obtenu en plaçant les deux bandes r2 × i l’une au dessus de l’autre, celle contenant t au dessus (figure ci-dessous). figure 7. remarques. 1) il existe une définition un peu plus générale des tangles et des tresses. dans cette définition les points pi sont quelconques et ne correspondent pas forcément aux points (i, 0) de r2. les deux définitions sont équivalentes. dans les prochains paragraphes nous utilisons l’une ou l’autre selon le cas. 2) soit t un n tangle. il est possible de fermer t en connectant chaque point pi ×{0} au point pi×{1} sans ajouter de croisements. on obtient de ce fait un entrelacs qu’on note t̂. 6, 4(2004) symétries en dimension trois: une approche quantique 139 figure 8. la multiplication des tresses donnée par la définition précédente permet de définir une structure de groupe sur l’ensemble des classes d’isotopies des tresses à n brins. notons bn ce groupe, l’élément neutre de bn est la tresse triviale 1bn (voir figure 9). nous notons dans la suite par σi la tresse qui consiste à croiser les brins i et i + 1 de la façon illustrée par la figure suivante : 1bn σi σ −1 i figure 9. théorème 2.2.4 ([7]) le groupe de tresses bn admet la présentation suivante : générateurs : σ1, σ2, ..., σn−1. relations : 1) σjσj+1σj = σj+1σjσj+1, (1 ≤ j ≤ n − 2). 2) σjσk = σkσj, (1 ≤ j < k − 1 ≤ n − 2). théorème 2.2.5 ([7]) soit n un entier ≥ 3. le centre du groupe bn est le sous groupe cyclique infini engendré par l’élément : ωn = (σ1σ2...σn−1) n. dans la suite on désigne par sn le groupe des bijections d’un ensemble de n éléments. soit b une tresse de bn, il est clair qu’on peut associer à b d’une façon canonique un élément de sn. cet élément est appelé permutation induite par b et on le note par i(b). la permutation induite par ωn est la permutation identité ; c’est à dire que : i(ωn) = 1sn . 140 nafaa chbili 6, 4(2004) 2.3 tresses et entrelacs on a dit dans le paragraphe précédent qu’on peut obtenir un entrelacs en refermant une tresse. alexander [1] montre que tout entrelacs peut être vu comme la fermeture d’une tresse. quelques années plus tard, markov considère la relation d’équivalence ≡ sur l’ensemble b = ∐ n≥0 bn engendrée par les deux opérations suivantes appelées mouvements de markov : mouvement de type i ∀σ et τ ∈ bn on a τστ−1 ≡ σ mouvement de type ii ∀σ ∈ bn on a σ ≡ σσn ≡ σσ−1n ∈ bn+1 et montre le théorème suivant : théorème 2.3.1 ([38]) si α et β sont deux tresses telles que α̂ = β̂ alors α ≡ β; c’est à dire qu’on peut passer de α à β par une série de mouvements de types i, ii et leurs inverses. remarque 2.3.2 soit b une tresse de bn, alors b̂ est un nœud si et seulement si la permutation i(b) est un cycle de longueur n. plus généralement le nombre de composantes de b̂ est égal au nombre de cycles dans la décomposition de i(b) en produit de cycles à supports disjoints [7]. définition 2.3.3 (indice de tresse) soit l un entrelacs, on appelle indice de tresse (braid index) de l le plus petit entier n tel qu’il existe une tresse α ∈ bn vérifiant l = α̂. 2.4 noeuds et entrelacs toriques l’entrelacs torique de type (n,m) qu’on note t (n,m) est celui qu’on peut dessiner sur le tore s1 × s1 en tournant n fois autour du méridien et m fois autour de la longitude [52]. cette classe d’entrelacs joue un rôle très important dans notre travail. nous rappelons dans ce paragraphe quelques propriétés de ces objets. remarque 2.4.1 soient n et m deux entiers tels que n ∈ n∗. l’entrelacs torique t (n,m) est la fermeture de la tresse (σ1σ2...σn−1)m. exemple. le nœud de trèfle est le nœud torique t (2, 3). le nœud 51 est le nœud torique t (2, 5). remarque 2.4.2 le nombre de composantes de t (n,m) est égal à pgcd(n,m), donc t (n,m) est un nœud si et seulement si n et m sont premiers entre eux. les nœuds toriques sont classifiés par le théorème suivant [7] : théorème 2.4.3 soient a, b, a′ et b′ quatre entiers non nuls alors : i) t (a,−b) = t (a,b)!. ii) les deux nœuds toriques t (a,b) et t (a′,b′) sont isotopes si et seulement si (a′,b′) est égale à l’une des paires suivantes : (a,b), (−a,−b), (b,a) ou (−b,−a). 6, 4(2004) symétries en dimension trois: une approche quantique 141 2.5 les invariants polynômiaux dans ce paragraphe, nous rappelons les principaux invariants des entrelacs de s3. sans donner de démonstrations nous énonçons quelques propriétés du polynôme de homfly. dans les prochains paragraphes nous introduisons d’autres invariants. définition 2.5.1 une application de l’ensemble des diagrammes d’entrelacs orientés dans un anneau a qui est invariante par les mouvements de reidemeister de type i, ii et iii est appelée invariant d’entrelacs orientés (on dit aussi invariant d’isotopie ambiante). une application de l’ensemble des diagrammes d’entrelacs qui est invariante par les mouvements de reidemeister de type ii et iii est appelée invariant d’entrelacs en bandes (on dit aussi invariant d’isotopie régulière). soit d un diagramme d’entrelacs orienté et x un croisement de d. on dit que x est un croisement mixte s’il appartient à deux composantes différentes de d. sinon, on dit que x est un auto-croisement. a chaque croisement x de d on associe un nombre ε(x) ∈ {−1, 1} de la façon suivante: ε( ) = 1 ε( ) = −1 définition 2.5.2 (enlacement de deux composantes, enlacement total) soit d le diagramme d’un entrelacs orienté à deux composantes l1 et l2, on appelle enlacement de l1 et l2 le nombre entier : λ(d) = 1 2 ∑ croisements x mixtes ε(x). soit l un entrelacs à n composantes l1, l2, ..., ln. on appelle enlacement total de l’entrelacs l l’entier : λ(l) = ∑ 1≤i le groupe cyclique engendré par h. l’espace quotient de s3 par < h > est homéomorphe à s3. soit ψ la surjection canonique de s3 dans s3 / . si k est un entrelacs invariant par h, nous notons par k l’entrelacs ψ(k). comme le cercle b est invariant point par point sous l’action de h alors ψ(b) est aussi un cercle non noué. en particulier, le nombre d’enlacement de k et b est le même que celui de k et b. proposition 3.1.2 ([37]) soient k un nœud de s3 et k le nombre d’enlacement de k avec l’axe de la rotation h. alors le nombre de composantes de ψ−1(k) est égale à pgcd(k,p). en particulier ψ−1(k) est un nœud si et seulement si p et k sont premiers entre eux. définition 3.1.3 soit p un entier ≥ 2, un entrelacs k de s3 est dit p-librement périodique s’il existe un homéomorphisme direct h de s3 dans lui-même vérifiant : 1pour tout 1 ≤ i ≤ p − 1, hi n’admet pas de points fixes. 146 nafaa chbili 6, 4(2004) 2hp = id. 3h(k) = k. exemple : l’action lenticulaire. soit p un entier non nul. la sphère s3 est considérée comme la sous variété réelle de l’espace complexe c × c définie par : s3 = {(z1,z2) ∈ c × c tel que |z1|2 + |z2|2 = 1}. considérons l’application ϕp,s définie pour tout entier s premier avec p par : ϕp,s : s3 −→ s3 (z1,z2) 7−→ (e 2iπ p z1,e 2isπ p z2). l’application ϕp,s vérifie les deux premières conditions données par la définition précédente. l’espace quotient de s3 par cette action est par définition l’espace lenticulaire l(p,s). comme cette action est libre alors la surjection canonique πp,s est un revêtement à p feuillets. si k est un entrelacs de l(p,s) alors π−1p,s (k) est un entrelacs librement périodique de période p. définition 3.1.4 soient p et s deux entiers premiers entre eux. un entrelacs de s3 qui est invariant par ϕp,s est dit entrelacs (p,s)-lenticulaire. l’importance de cette classe d’entrelacs vient de la conjecture suivante [53]: conjecture. soit p un nombre premier, alors pour toute action libre et directe du groupe z/pz sur la sphère s3, il existe un entier s premier avec p tel que cette action est conjuguée à l’action définie par l’application ϕp,s. par conséquent, tout entrelacs librement périodique est isotope à un entrelacs lenticulaire, ainsi on réduit l’étude des entrelacs p−librement périodiques à l’étude des entrelacs (p,s)−lenticulaires. 3.2 une description combinatoire des entrelacs lenticulaires la théorie skein propose de définir les invariants quantiques directement à partir des diagrammes des entrelacs dans le plan. nous cherchons dans ce paragraphe à comprendre comment se traduit l’action lenticulaire sur les diagrammes. autrement dit, étant donnée un entrelacs lenticulaire de s3, est-ce qu’on peut représenter cet entrelacs par un diagramme symétrique ? la caractérisation que nous obtenons représente l’étape clef dans notre étude des invariants quantiques des nœuds et entrelacs lenticulaires. le premier résultat de notre travail consiste à caractériser les diagrammes des entrelacs lenticulaires. 6, 4(2004) symétries en dimension trois: une approche quantique 147 théorème a. soient p et s deux entiers tels que pgcd(p,s) = 1. soit l un entrelacs de s3, alors l est (p,s)-lenticulaire si et seulement s’il existe un entier m > 0 et t un m-tangle tels que : l = ̂t p(σ1σ2...σm−1)ms. la condition donnée par le théorème a généralise un résultat bien connu et facile à démontrer dans le cas des entrelacs périodiques (qui correspond au cas s = 0). ce que nous obtenons dans le théorème a est beaucoup moins évident. la difficulté vient de ce que la variété quotient n’est plus la sphère s3, mais un espace lenticulaire. autrement dit, il n’existe pas en général une surface de seifert invariante par l’action lenticulaire. la démonstration de notre résultat repose essentiellement sur l’analyse de l’action lenticulaire sur le tore solide. le théorème a permet de construire des exemples d’entrelacs lenticulaires. cependant, il n’existe pas un algorithme pratique permettant de transformer un diagramme quelconque d’un entrelacs lenticulaire en un diagramme de la forme ̂t p(σ1σ2...σm−1)ms. exemples les entrelacs toriques. l’entrelacs torique t (n,ns+p) est la fermeture de la tresse (σ1σ2...σn−1)ns+p, il se met alors sous la forme ̂t p(σ1σ2...σn−1)ns où t est la tresse (σ1σ2...σn−1). par conséquent, cet entrelacs est (p,s)−lenticulaire. en particulier : le nœud 51 qui est le nœud torique (2, 5) est (3, 1)-lenticulaire. le nœud 71 qui est le nœud torique (2, 7) est (3, 2)-lenticulaire. le nœud 10155 est (2,1)-lenticulaire. ce nœud est d’après hartley [23] la fermeture de la tresse (σ1σ2σ1)2(σ −3 1 σ2) 2, or (σ1σ2σ1)2 est égale à la tresse (σ1σ2)3 et par la suite : 10155 = ̂(σ−31 σ2) 2(σ1σ2) 3 le nœud 948 est (3,1)-lenticulaire. soit t le 2-tangle donné par la figure suivante : figure 11. conway [18] représente le nœud 948 par le tangle t 3σ −1 1 . d’après [18] (paragraphe 3, page 333) ce dernier tangle est équivalent à (tσ−11 ) 3σ21 . on conclut alors que le nœud 948 est (3, 1)-lenticulaire. 148 nafaa chbili 6, 4(2004) les entrelacs de montesinos : l’entrelacs de montesinos de type m(e, α1 β1 , ..., αp βp ) est un entrelacs admettant une projection comme dans la figure 12. dans cette figure la boite : (α, β) représente le tangle rationnel définie par la fraction rationnelle β α (une étude détaillée se trouve dans [7]). supposons que e est un nombre pair premier avec p. le théorème a nous montre que l’entrelacs de montesinos avec p tangles rationnels identiques m(e, α1 β1 , ..., α1 β1 ) est un entrelacs (p, e 2 )-lenticulaire. (α2,β2) (α1,β1) (αp,βp) . . . h � �� h � ��h h ... figure 12: entrelacs de montesinos. 3.3 généralisation des critères de murasugi étant donnée un nœud périodique k, l’idée de murasugi est d’établir le lien entre l’invariant de k et l’invariant du nœud quotient k. rappelons les deux résultats suivants de murasugi. théorème 3.3.1 (murasugi 1971). soient p un nombre premier et k un nœud ppériodique. notons par k le nombre d’enlacement de k avec l’axe de la rotation qui laisse invariant k et notons par k le nœud quotient. il existe un entier n tel que: ∆k (t) ≡ (∆k (t)) p(1 + t + t2 + ..... + tk−1)p−1tn mod p. 6, 4(2004) symétries en dimension trois: une approche quantique 149 théorème 3.3.2 (murasugi 1988). soient p un nombre premier et l un entrelacs p-périodique. alors on a : vl(t) ≡ (vl(t)) p mod j , où j est l’idéal de z[t±1/2] engendré par p et ξp(t) = p−1∑ j=0 (−t)j − t(p−1)/2. le crochet de kauffman est un invariant d’isotopie régulière introduit par l. kauffman dans [29]. pour un entrelacs en bande l, le crochet de kauffman est un polynôme de laurent en une variable a. la découverte de cet invariant a permis à kauffman de donner une autre définition au polynôme de jones. théorème 3.3.3 (kauffman, [29]) il existe un unique invariant < > de l’ensemble de classes d’isotopie régulière des entrelacs non orientés dans l’anneau des polynômes à une variable z[a±1] tel que : < l > = a < l0 > +a−1 < l∞ >, < © > = 1, où © désigne le nœud trivial, l, l0 et l∞ sont trois entrelacs identiques sauf au voisinage d’un croisement, où ils sont comme dans la figure 13. figure 13. avec ces notations kauffman a démontré que le polynôme de jones s’obtient facilement à partir du crochet. en effet, si on note par w(l) le nombre algébrique de croisements de l’entrelacs l, on a la formule suivante : vl(t) = (−t3/4)w(l) < l > (t−1/4). pour les entrelacs de l’espace lenticulaire, le polynôme de jones n’est pas bien défini. pour cette raison nous discutons le critère de murasugi en termes de tangles. en effet, en étudiant la catégorie des tangles on établit la congruence suivante : théorème b. soient s un entier, p un nombre premier et t un n-tangle. alors on a : < t̂ pωsn > (a) ≡ (< t̂ωsn > (a)) p mod i, où i est l’idéal de z[a±1] engendré par p, δp − δ et les éléments de la forme : (< ω̂sn−2i >) p − (< ω̂sn−2i >), pour 0 ≤ i ≤ [ n−1 2 ]. 150 nafaa chbili 6, 4(2004) l’étude des traces des représentations du groupe de tresses dans l’algèbre de hecke permet d’établir des formules explicites pour les générateurs du module i. remarquons que si on restreint la congruence du théorème b au cas s = 0, nous retrouvons un résultat équivalent au théorème de murasugi dont la démonstration d’origine utilise des techniques différentes de celles que nous utilisons ici. le polynôme d’alexander des tresses lenticulaires. comme nous l’avons déjà signalé, le cas du polynôme d’alexander des nœuds librement périodique a été traité par hartley [23]. nous nous intéressons ici a une question plus précise. en effet, nous cherchons à établir une condition nécessaire pour qu’un entrelacs soit obtenu comme la fermeture d’une tresse lenticulaire. définition 3.3.4 soient p et s deux entiers. une tresse β est dite (p,s)-lenticulaire s’il existe une tresse à n brins α telle que β = αpωsn. il est facile de voir que la fermeture d’une tresse lenticulaire donne naissance à un entrelacs lenticulaire. cependant, on ne sait pas si un entrelacs (p,s)−lenticulaire quelconque peut être obtenu comme la fermeture d’une tresse lenticulaire. dans le cas des entrelacs périodiques, des conditions nécessaires pour qu’un entrelacs admette une représentation comme une tresse périodique sont obtenues dans [33]. le polynôme d’alexander à plusieurs variables est un invariant d’isotopie régulière d’entrelacs orientées. il a été défini à partir des présentations du groupe fondamental du complémentaire de l’entrelacs dans la sphère s3. si l est un entrelacs à k-composantes alors le polynôme d’alexander à plusieurs variables qu’on note ∆l(t1, . . . , tk) vit dans l’anneau z[t±11 , . . . , t ±1 k ]. remarquons que si on prend t1 = t2 = · · · = tk = t alors on retrouve le polynôme d’alexander classique (modulo une normalisation). dans un papier récent, morton [39] a introduit une méthode pour associer à chaque tresse de bn une matrice à coefficients dans z[t±11 , . . . , t ±1 n ]. ainsi, il a pu définir le polynôme d’alexander à plusieurs variables à partir du groupe de tresses. cette méthode est inspirée par la représentation de burau à une variable. une représentation (classique) qui est reliée au polynôme d’alexander d’une façon très simple. dans la suite de ce paragraphe, nous trouvons plus commode d’introduire l’invariant suivant :{ dl = ∆l si k > 1 et ∆l = (1 − t)dl si k = 1. théorème c. soient p un nombre premier, s ∈ n et α une tresse pure à n brins. on a la congruence suivante modulo p (1 − t1 . . . tn)dα̂pωsn (t1, . . . , tn) ≡ 1 + (t1 . . . tn) sa p 1(t1, . . . , tn) + · · · + +(t1 . . . tn)(n−1)sa p n−1(t1, . . . , tn) où a1, . . . ,an−1 sont des éléments de z[t ±1 1 , ..., t ±1 n ]. si on écrit la congruence donnée par le théorème c dans le cas s = 0, on obtient une conséquence du théorème de murasugi concernant le polynôme d’alexander des 6, 4(2004) symétries en dimension trois: une approche quantique 151 entrelacs périodiques. la condition obtenue dans notre théorème n’est pas triviale. en effet, les polynômes ai ne sont pas quelconques puisqu’ils sont reliés au polynôme de α̂p par une relation de congruence. 3.4 le cas du polynôme de homfly soit k un nœud de s3. d’après [34], le polynôme de homfly pk (v,z) s’écrit sous la forme pk (v,z) = ∑ i≥0 p2i,k (v)z 2i, où p2i,k (v) ∈ z[v±2i]. nous discutons dans ce paragraphe le comportement des polynômes p0 et p2 des nœuds librement périodiques. rappelons que les travaux de traczyk et yokota ont montré que ces polynômes sont des témoins très précis de la périodicité des nœuds puisque cette symétrie est reflétée d’une façon très nette par ces polynômes. dans une première étape nous nous intéressons au premier coefficient du polynôme de homfly. soit p un nombre premier, nous notons par ifp le corps cyclique à p éléments. soient λp,s le ifp[v±2p]-module engendré par les polynômes p0,k où k parcourt l’ensemble des nœuds (p,s)-lenticulaires. nous démontrons que ce module est engendré par les polynômes des nœuds toriques de type t (n,ns + p) pour p premier avec n. dans le cas s = ±1, nous montrons que le module λp,s est de type fini et nous déterminons une famille finie de générateurs. le calcul de ces générateurs est possible grâce à une formule de jones [28]. dans la suite nous notons par p0,k (v)p la réduite modulo p du polynôme p0,k (v) ; c’est à dire que les coefficients de ce polynôme sont considérés dans le corps ifp. théorème d. soient p un nombre premier 6= 2 et s = ±1. si k est un nœud (p,s)−lenticulaire alors p0,k (v)p ∈ λp,s, où λp,s est le ifp[v±2p]-module engendré par les polynômes p0,t (β,βs±p)(v)p pour 1 ≤ β ≤ p − 1. pour les petites valeurs de p, le critère donné par le théorème d prend une forme simple et explicite. en effet, il est facile dans ce cas de donner une formule explicite pour les générateurs en utilisant la formule de jones pour le polynôme de homfly des nœuds toriques. dans les cas p = 3 et p = 5, la régularité du polynôme p0 est surprenante comme l’illustre les deux corollaires suivants. corollaire d1. soit k un nœud 3-librement périodique. alors on a : p0,k (v)3 ∈ if3[v±6]. corollaire d2. soit k un nœud (5,±1)-lenticulaire, alors p0,k (v)5 = ∑ a2iv 2i avec: a10k+4 = 2a10k+2 et a10k+6 = 2a10k+8 pour tout entier k. applications. pour illustrer le corollaire d1, nous considérons le nœud de trèfle 31. un calcul simple montre que le premier coefficient du polynôme de homfly p0,31 (v) = 2v 2 − v4. ce polyôme, à coefficients considérés modulo 3, n’appartient pas à if3[v±6]. par 152 nafaa chbili 6, 4(2004) conséquent, le nœud de trèfle n’est pas 3-librement périodique. dans le tableau de [52], il y a 84 nœuds ayant un nombre de croisements inférieur à 9. en examinant les polynômes de homfly de ces nœuds (voir les dernières pages de [34]), nous remarquons que 72 parmi ces nœuds ne vérifient pas la condition donnée par le corollaire d1. ils ne sont pas donc 3-librement périodiques. seuls les 12 nœuds suivants échappent à notre critère, c’est à dire que le corollaire reste indécis pour les nœuds donnés par la liste suivante : 51, 71, 82, 810, 821, 93, 96, 926, 938, 941, 948, 949. dans un certain sens, la condition donnée par le corollaire d2 est un peu plus large que celle obtenue pour les nœuds 3-librement périodiques. en effet parmi les 84 nœuds ayant un nombre de croisements inférieur à 9, le corollaire d2 exclut 60 nœuds et il reste indécis pour les 24 autres (voir tableau). bien que la condition obtenue dans le théorème précédent a permis de prouver que certains nœuds ne sont pas lenticulaires. le problème de savoir si un nœud donné est lenticulaire est loin d’être résolu. d’où l’idée de considérer les autres coefficients du polynôme de homfly afin d’établir d’autres conditions permettant de renforcer la condition donnée par le polynôme p0. soit l un entrelacs à k composantes. d’après [34], on sait que le polynôme p1−k est relié aux polynômes p0 des différentes composantes de l par une formule assez simple. cette formule joue un rôle important dans la démonstration du théorème précédent puisque elle permet de traduire les relations skein sur les polynômes p0. dans un papier récent [32], kanenobu et miyazawa ont réussi à démontrer une relation analogue entre le polynôme p3−k de l’entrelacs l et les polynômes p2 des composantes de l. bien que cette relation est beaucoup plus compliquée que celle obtenue par lickorich et millet dans le cas p0, elle a motivé notre étude du polynôme p2 des entrelacs lenticulaires. ce qui nous a permis d’établir le critère donné par le théorème suivant: théorème e. soient p > 3 un nombre premier et s = ±1. si k est un nœud (p,s)−lenticulaire alors p2,k (v)p ∈ γp,s, où γp,s est le ifp[v±2p]-module engendré par les polynômes p2,t (β,βs±p)(v)p pour 1 ≤ β ≤ p − 1. dans le cas p = 5, les huit générateurs du module γp,s peuvent être calculés facilement à l’aide de la formule de jones. on obtient le corollaire suivant: corollaire e1. soient s = ±1 et k un nœud (5,s)-lenticulaire. alors p2,k (v)5 appartient au if5[v±10]-module engendré par vs8. applications. les conditions obtenues dans les deux théorèmes précédents sont indépendantes, puisque il y a des nœuds qui vérifient la première sans vérifier la deuxième et vice versa. nous avons expliqué qu’en utilisant le polynôme p0 on a pu montrer que 60 parmi les 84 nœuds avec moins de 9 croisements ne sont pas (5,1)lenticulaires. les 24 nœuds restants vérifient la condition donnée par le corollaire d2, donc on ne peut pas savoir s’ils admettent une symétrie lenticulaire. si on applique la 6, 4(2004) symétries en dimension trois: une approche quantique 153 condition donnée par le polynôme p2 aux 84 nœuds avec moins de 9 croisements on peut exclure la possibilité d’être (5, 1)-lenticulaires à 80 nœuds. ce qui montre que la condition e1 est plus efficace que la condition d2. seul le nœud 819 échappe aux deux critères en même temps. dans le tableau suivant on utilise la mention d pour dire que le critère en question décide si le nœud n’est pas (5,1)-lenticulaire et la mention nd pour dire que le critère n’exclut pas la possibilité que le nœud soit (5,1)-lenticulaire. nous référons à la condition donnée par corollaire d2 (respectivement corollaire e1) par critère p0 (respectivement critère p2). nœud critère p0 critère p2 nœud critère p0 critère p2 31 n d d 98 d d 41 d n d 99 n d d 51 d d 910 n d d 52 d d 911 d d 61 d d 912 d d 62 n d d 913 n d d 63 d d 914 d d 71 n d n d 915 d d 72 d d 916 n d d 73 d d 917 d d 74 d d 918 d d 75 d d 919 d d 76 d d 920 d d 77 n d d 921 d d 81 d d 922 d d 82 d d 923 d d 83 d d 924 d d 84 d d 925 n d d 85 d d 926 d d 86 d d 927 d d 87 d d 928 d d 88 d d 929 d d 89 d d 930 d d 810 d d 931 n d d 811 d d 932 d d 812 d d 933 n d d 813 n d d 934 d d 814 n d d 935 n d d 815 d d 936 d d 816 n d d 937 d d 817 d d 938 n d d 818 d d 939 d d 819 n d n d 940 n d d 820 n d d 941 d d 821 d d 942 d d 91 n d n d 943 n d d 92 d d 944 d d 93 n d d 945 d d 94 d d 946 d d 95 d d 947 d d 96 n d d 948 d d 97 d d 949 n d d 154 nafaa chbili 6, 4(2004) remarque. les techniques que nous avons développées pour étudier le polynôme de homfly des nœuds lenticulaires peuvent être adaptées aux invariants de vassiliev. autrement dit nous définissons ”une version singulière” de notre théorie skein périodique. ce qui nous a permis d’établir des relations de congruences entre les invariants de vassiliev des nœuds lenticulaires et ceux des nœuds toriques. si on restreint ces relations aux nœuds périodiques, nous obtenons des précisions sur le critère de yokota concernant le polynôme de homfly des nœuds périodiques [58]. 4 cas des 3-variétés toutes les variétés de dimension trois considérées dans ce rapport sont compactes connexes sans bords et orientées. 4.1 présentation par chirurgie des 3-variétés périodiques définition 4.1.1 soient r ≥ 2 un entier et m une variété de dimension trois. on dit que m est r-périodique si et seulement si le groupe g = z/rz agit sur m et l’ensemble des points fixes par l’action de g sur m est un cercle. on note par m l’espace quotient. l’exemple le plus simple est celui de la sphère s3 munie de l’action de g = z/rz définie comme suit: g : s3 −→ s3 (z1,z2) 7−→ (z1,e 2iπ r z2). ainsi la sphère s3 est r-périodique pour tout r ≥ 2. la sphère d’homologie de poincaré σ est aussi r-périodique pour r = 2, 3 et 5. on peut expliciter facilement cette périodicité en regardant σ comme la variété de brieskorn σ(2, 3, 5), qui peut être vue comme l’intersection de la surface complexe d’équation z21 + z 3 2 + z 5 3 = 0 avec une sphère de dimension 5. par exemple, σ est 5-périodique, l’action est la suivante: h : σ(2, 3, 5) −→ σ(2, 3, 5) (z1,z2,z3) 7−→ (z1,z2,e 2iπ 5 z3). l’ensemble des points fixes pour cette action est le nœud de trèfle 31. rappelons que si m est une sphère d’homologie rationnelle périodique, alors l’espace m est aussi une sphère d’homologie rationnelle. d’après les travaux de lickorich et wallace [35], toute 3-variété est obtenue à partir de la sphère s3 par une opération topologique appelée chirurgie le long d’un entrelacs de s3. un tel entrelacs sera appelé présentation par chirurgie de la variété m. les mouvements de kirby (ou fenn-rourke) permettent de relier deux présentations par chirurgie de la même variété. goldsmith [22] a étudié les présentations par chirurgie des revêtements ramifiés au-dessus de la sphère de dimension 3 pour montrer qu’un 6, 4(2004) symétries en dimension trois: une approche quantique 155 tel revêtement est obtenu par chirurgie le long d’un entrelacs périodique. ce résultat a été généralisé récemment par przytycki et sokolov [49], nous rappelons par la suite ce résultat qui établit le lien entre entrelacs et variétés périodiques et qui joue un rôle crucial dans les démonstrations des nos principaux résultats concernant les variétés périodiques. définition 4.1.2 soient r ≥ 2 un entier et l un entrelacs r-périodique de s3. on dit que l est fortement r-périodique si et seulement si le nombre d’enlacement de chaque composante de l avec l’axe de la rotation est nul modulo r. théorème 4.1.3 [49]. soient r un nombre premier et m une 3-variété. alors m est r-périodique si et seulement si m s’obtient à partir de s3 par chirurgie le long d’un entrelacs fortement r-périodique. exemple. l’entrelacs dans la figure suivante est fortement 5-périodique. une chirurgie le long de cet entrelacs permet d’obtenir la sphère d’homologie de poincaré. ainsi on retrouve le fait que la sphère d’homologie de poincaré est 5-périodique. figure 14. les critères obtenus dans le cas des invariants quantiques des nœuds périodiques représentent un élément essentiel qui a motivé ce travail. en effet, comme les 3variétés périodiques sont obtenues par chirurgie le long d’entrelacs périodiques, il est tout à fait légitime de se demander si les résultats obtenus pour les entrelacs périodiques s’étendent aux 3-variétés. p. gilmer [21] a étudié l’invariant so(3). il a établi une certaine relation de congruence dans les cas des revêtements et des 3variétés périodiques. pendant la même période, d’une façon totalement indépendante et en utilisant des techniques complètement différentes, nous avons démontré une condition de congruence pour l’invariant su(2). ainsi le résultat de gilmer s’obtient comme un corollaire de notre condition donnée par le théorème f. 4.2 l’invariant su(2) comme nous l’avons déjà signalé, il existe plusieurs façons pour définir cet invariant. la version que nous allons considérer dans ce rapport est celle introduite dans [5] et 156 nafaa chbili 6, 4(2004) dont nous rappelons brièvement la définition par la suite. soient p ≥ 3 un entier et λp l’anneau z[a±1]/φ2p(a); où φ2p(a) est le polynôme cyclotomique d’ordre 2p. soient k l’ensemble des entrelacs du tore solide s1 × b2 et λp[k] le module libre engendré par k. le skein module de kauffman de s1 × b2 est défini comme étant le quotient de λp[k] par les relations suivantes: (r1) : © ∪ l = δl (r2) : l = al0 + a−1l∞, où δ = −a2 − a−2, l, l0 et l∞ sont trois entrelacs qui sont identiques sauf au voisinage d’un croisement où ils sont comme dans la figure 13. il est facile de voir que bp peut être muni d’une structure d’algèbre isomorphe à l’algèbre polynômiale λp[z], où z correspond à une bande axiale standard dans le tore s1 ×i ×i. pour tout réel d, on note par [d] la partie entière de d. dans bp, on définit l’élément ωp = [(p−3)/2]∑ i=0 〈ei〉ei, où (ei)i est définie par e0 = 1, e1 = z et la relation de récurrence ei+1 = zei − ei−1. rappelons qu’on a: 〈ei〉 = (−1)i a2(i+1) − a−2(i+1) a2 − a−2 . soit l un entrelacs en bandes à m composantes. le multi-crochet 〈., . . . , .〉l est une forme multilinéaire de bp ×bp × ...×bp −→ λp définie sur les éléments de la base de façon que 〈zi1,zi2, . . . ,zim〉l est égale à li1,i2,...,im qui représente la classe d’isotopie dans bp de l’entrelacs obtenu à partir de l en remplaçant la jème composante de l par ij bandes parallèles à celle-ci situées dans un voisinage suffisamment petit. dans la suite nous notons par t l’automorphisme de bp induit par un twist positif du tore solide, par t−1 on notera l’automorphisme inverse. enfin, par b+(l) (respectivement b−(l)) nous désignons le nombre de valeurs propres positives (respectivement négatives) de la matrice d’enlacement de l’entrelacs en bandes l. soit m une variété compacte connexe orientée de dimension trois. on sait que m s’obtient à partir de la sphère s3 par chirurgie le long d’un entrelacs en bandes l. l’invariant θp(m) est défini par la formule suivante: θp(m) = 〈ωp, . . . , ωp〉l 〈t(ωp)〉b+(l)〈t−1(ωp)〉b−(l) . notons que l’invariant θp(m) prend ses valeurs dans λp[ 1 p ]. cependant, si r est premier avec p alors p est inversible dans z/rz et par la suite si on considère bp à coefficients dans z/rz[a±1]/φp(a), alors θp(m) peut être vu comme un polynôme en a à coefficients dans z/rz. théorème f. soient r un nombre premier impair et m une sphère d’homologie rationnelle de dimension trois. si m est r-périodique alors pour tout entier p ≥ 3 premier avec r on a: θp(m) ≡ (θp(m))r(−a−6−p(p−1)/2)α, mod r,δr − δ, 6, 4(2004) symétries en dimension trois: une approche quantique 157 où δ = −a2 − a−2 et α est un entier. corollaire f1. soient r un nombre premier impair et m une sphère d’homologie rationnelle de dimension trois. supposons que m est r-périodique, alors pour tout p ≥ 3 tel que r ≡ ±1 mod p, il existe un entier α tel que: θp(m) ≡ (θp(m))r(−a−6−p(p−1)/2)α, mod r. corollaire f2. soient r un nombre premier impair et m le revêtement cyclique régulier à r feuillets ramifié au-dessus de s3. alors pour tout p ≥ 3 tel que r ≡ ±1 mod p, il existe un entier α tel que: θp(m) ≡ (−a−6−p(p−1)/2)α, mod r. applications. l’exemple suivant explique comment appliquer le corollaire f2 pour étudier les symétries de la sphère de poincaré σ. le polynôme cyclotomique d’ordre 10 est φ10(a) = a4 − a3 + a2 − a + 1. l’invariant θ5, à coefficients considérés modulo r, prend alors ses valeurs dans l’anneau z/rz[a±1]/a4−a3+a2−a+1. l’élément ω5 est formé seulement de deux termes. en effet : ω5 = 1 + (−a2 − a−2)z. en utilisant les propriétés élémentaires des sommes de gauss, on peut montrer que θ5(σ) = 1 − 2a + a2 + a3. soit r un nombre premier tel que r ≡ ±1 mod 5. supposons que σ est le revêtement cyclique régulier à r feuillets ramifié au-dessus de s3. alors on aura d’après le corollaire f2, θ5(σ) = (−a−16)α mod r, pour un certain entier α. rappelons que a est une racine primitive de l’unité d’ordre 10. par conséquent les valeurs possibles des puissances de a sont ±a, ±a2, ±a3 et ±a4 = ±(a3 − a2 + a − 1). on peut voir facilement que θ5(σ) n’est pas une puissance de a . ce qui permet de conclure que σ n’est pas le revêtement cyclique régulier à r feuillets ramifié au-dessus de s3, pour tout r congru à 1 ou -1 modulo 5. en particulier pour r = 11, 19 et 29. remarques. 1-) la condition donnée par le théorème f est nécessaire. cependant, il est difficile de voir s’il s’agit d’une condition suffisante. la difficulté vient essentiellement du fait qu’on ne dispose pas, même dans le cas des sphères d’homologie les plus simples, de formules calculant les invariants θp pour tout p ≥ 3. il est à noter aussi que le théorème f introduit une infinité de formules (pour tout p premier avec r). tout ce qu’on peut affirmer est que si la condition du corollaire f2 est vérifiée pour une seule valeur de p, alors elle est insuffisante. un contre exemple est facile à construire. en effet, prenons p = 4 et r = 11. comme θ4(σ) = 1, alors la congruence donnée par le corollaire f2 est vérifiée. cependant, σ n’est pas le revêtement cyclique régulier à 11 feuillets ramifié au-dessus de s3, comme on vient de le prouver. 158 nafaa chbili 6, 4(2004) 2-) pour certaines valeurs de p la condition donnée par le théorème f devient triviale. des considérations élémentaires en théorie des nombres montrent que les seuls cas intéressants dans la formule donnée par ce théorème sont les cas où r ≡ ±1 mod p. puisque si on n’a pas cette relation entre p et r alors l’idéal engendré par r et δr − δ est égal à λp[ 1p ]. 4.3 l’invariant su(3) ohtsuki et yamada [45] ont montré que la théorie skein introduite par kuperberg [31] pour les entrelacs peut servir pour construire l’invariant quantique su(3). nous rappelons brièvement cette théorie skein ainsi que la définition de l’invariant su(3). soit f une surface orientée. tous les graphes considérés dans la suite sont trivalents et orientés. de plus, nous supposons qu’à chaque sommet les trois arêtes sont orientées de la même façon (voir la figure 15). un diagramme dans la surface f est localement le diagramme d’un entrelacs ou un graphe trivalent. les diagrammes sont considérés modulo isotopie. dans la suite et pour tout entier k, nous posons [k] = q k 2 − q −k 2 q 1 2 − q −1 2 . les relations skein suivantes ont été introduites par kuperberg. @@ @@i � � �� = q−1k1 : 6 6 −q− 3 2 ? ��@i @�� i � �� @ @ @i �� = qk2 : 6 6 −q 3 2 ? ��@i @�� i � ? 6 @i � @r � � =k3 : 6 ? + � n6?? 6 = [2]k4 : 6 n?d = [3]dk5 : pour tout diagramme d. ∅ = 1k6 : figure 15. a l’aide de ces relations, kuperberg a défini un invariant de nœuds et entrelacs de s3. cet invariant est en fait une spécialisation du polynôme de homfly et peut être défini uniquement par les relations suivantes: (i) j©(q) = 1, (ii) q 3 2 jl+ (q) − q− 3 2 jl− (q) = (q 1 2 − q− 1 2 )jl0 (q), 6, 4(2004) symétries en dimension trois: une approche quantique 159 où l+, l− et l0 sont comme dans la figure 10. le skein module s(f) est défini comme étant le λr-module libre engendré par les diagrammes de f, quotienté par les relations de kuperberg. ohtsuki et yamada ont montré que le skein module de s1 ×i admet une structure d’algèbre isomorphe à l’algèbre polynômiale λr[x,y], où x et y sont comme dans la figure suivante: x : i�� ? &% '$ y : i�� 6&% '$ figure 16. en étudiant la théorie skein de kuperberg, nous donnons une généralisation au critère de murasugi pour les nœuds périodiques. plus précisémment nous démontrons le résultat suivant: théorème g. soient r un nombre premier et l un entrelacs r-périodque. alors on a la congruence suivante: j(l) ≡ (j(l))r modulo r, [3]r − [3]. inspirés par la démonstration de ce théorème, przytycki et sikora [48] ont obtenu une généralisation de ce résultat, étendant le théorème au cas des invariants quantiques su(n), pour n impair. pour introduire l’invariant su(3) des 3-variètés, nous trouvons plus commode de faire le changement de variable q = a6, dans les relations skein de kuperberg. soit p ≥ 4, nous définissons l’élément ωp de s(s1 × i) par la formule suivante:∑ n+m≤p−3,n,m≥0 [n + 1][m + 1][n + m + 2]pn,m. soit l = l1∪l2∪....∪lm un entrelacs à m-composantes. on note par l(i1,i′1),...,(ik,i′k) l’entrelacs obtenu à partir de l en remplaçant la jème composante de l par la réunion de ij bandes (positives) et i′j bandes (négatives) parallèles à celle-ci situées dans un voisinage suffisamment petit. d’une manière analogue au cas de l’invariant su(2) on définit le multi-crochet dans le cas su(3) comme étant la forme multi-linéaire 〈., . . . , .〉l de s × s × ... × s −→ z[a±1] dont les valeurs sur les générateurs sont données par: 〈xi1yi ′ 1, . . . ,xikyi ′ k〉l = j(l(i1,i′1),...,(ik,i′k)). en reprenant les notations du paragraphe précédent on définit l’invariant suivant: ip(m) = 〈ωp, . . . ,ωp〉l 〈t(ωp)〉b+(l)〈t−1(ωp)〉b−(l) . 160 nafaa chbili 6, 4(2004) cet invariant est défini en les racines primitives de l’unité d’ordre 3p. en se basant sur le fait que les termes 〈t(ωp)〉 et 〈t−1(ωp)〉 sont conjugués comme nombres complexes, nous montrons que ip peut être considéré comme un élément de λp[ 1 3p ] où λp est l’anneau z[a±1]/φ3p(a); ici φ3p(a) désigne le polynôme cyclotomique d’ordre 3p. en étudiant le comportement du multi-crochet, nous avons pu montrer que le critère que nous avons établi pour les entrelacs périodique (théorème g) se généralise aux entrelacs fortement périodiques. cette congruence représente l’étape clef dans la démonstration de notre critère de périodicité pour les invariants ip. avant d’énoncer ce résultat nous considérons l’élément suivant: gp(a) = a −36 ( p−1∑ k=0 a6k 2 )2( 3p−1∑ k=0 a2k 2 )2 3p2 . théorème h. soient r un nombre premier impair et m une sphère d’homologie rationnelle de dimension trois. si m est r-périodique alors pour tout entier p ≥ 4 premier avec 3r on a: ip(m) ≡ (ip(m))r(gp(a))α, modulo r, [3]r − [3], où α est un entier. la congruence donnée par ce théorème prend lieu dans λp[ 1 3p ]. pour montrer que la condition donnée par le théorème h n’est pas triviale, nous écrivons la congruence proposée par ce théorème dans le cas des revêtements branchés au-dessus de la sphère de dimension trois. dans ce cas la formule en question devient simple et plus explicite. corollaire h. soient r un nombre premier impair et m le revêtement cyclique régulier à r feuillets ramifié au-dessus de s3. alors pour tout p ≥ 4 tel que r ≡ ±1 mod p, il existe un entier α tel que : ip(m) ≡ (gp(a))α mod r. applications. on peut voir d’après la définition que le calcul de l’invariant ip n’est pas facile même pour les variétés les plus simples. dans la suite nous illustrons la condition obtenue dans le corollaire h1 en considérant l’espace lenticulaire l(2, 1). remarquons qu’on a g5(a) = a−36 et que φ15(a) = 1 −a + a3 −a4 + a5 −a7 + a8. par conséquent, pour montrer que l’espace lenticulaire l(2, 1) n’est pas le revêtement cyclique régulier à r feuillets ramifié au-dessus de s3 il suffit de montrer que i5(l(2, 1)) n’est une puissance de a dans λ5[ 1 15 ]. en écrivant la formule générale obtenue dans [43] dans le cas particulier p = 5, on obtient: i5(l(2, 1)) = 1 − a − a2 + a3 − a4 + a5 − a7. 6, 4(2004) symétries en dimension trois: une approche quantique 161 les puissances de a dans l’anneau λ5[ 1 15 ] sont données par la liste suivante: 1, a, a2, a3, a4, a5, a6, a7, −1 + a − a3 + a4 − a5 + a7, a2 − a6 − 1 − a3 + a7, −1 − a5, −a − a6, −a2 − a7, 1 − a − a4 + a5 − a7, 1 − a2 + a3 − a4 + a6 − a7. il est clair que i5(l(2, 1)) n’est pas une puissance de a. par conséquent, l(2, 1) n’est pas le revêtement cyclique régulier à r feuillets ramifié au-dessus de s3, pour tout r congru à 1 ou -1 modulo 5. en faisant des calculs similaires pour d’autres valeurs de p, nous pouvons avoir d’autres précisions sur les symétries des espaces lenticulaires. il est à signaler que les techniques que nous utilisons pour établir les conditions de congruences pour les invariants su(2) et su(3) permettent d’obtenir des résultats similaires pour l’invariant de murakami-ohtsuki-okada [44] qui est en un certain sens l’invariant quantique le plus simple. en effet, on peut construire une théorie skein très simple qui permet de définir cet invariant qu’on note zn . la congruence qu’on obtient est donnée 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[61] e. witten. quantum field theory and the jones polynomial. comm. math. phys. 121 (1989), 351-399. a mathematical journal vol. 7, no 2, (237 260). august 2005. positive operators and maximum principles for ordinary differential equations paul w. eloe 1 department of mathematics, university of dayton dayton, ohio 45469-2316, usa paul.eloe@notes.udayton.edu abstract we show an equivalence between a classical maximum principle in differential equations and positive operators on banach spaces. then we shall exhibit many types of boundary value problems for which the maximum principle is valid. finally, we shall present extended applications of the maximum principle that have arisen with the continued study of the qualitative properties of green’s functions. resumen mostramos una equivalencia entre el clásico principio del máximo en ecuaciones diferenciales y operadores positivos en espacios de banach. exhibiremos distintos tipos de problemas con valores en la frontera para los cuales el principio del máximo es válido. finalmente, mostraremos aplicaciones generalizadas del principio del máximo que resultan del estudio de las propiedades cualitativas de las funciones de green. key words and phrases: maximum principle, positive operators, sign properties of green’s functions math. subj. class.: 34b 1i thank claudio cuevas for his kind invitation to write this article. i also thank johnny henderson, my good friend, for many years of fulfilling collaboration. 238 paul w. eloe 7, 2(2005) table of contents 0. introduction 1. the maximum principle and positive operators on banach spaces 2. classes of boundary value problems 2.1 ordinary differential equations 2.2 discrete, time scale equations 2.3 systems of ordinary differential equations 2.4 impulsive problems with nonlinear boundary conditions 2.5 delay equations 2.6 singular equations 3. applications 3.1 monotone methods 3.2 krein-rutman theory 3.3 comparison theorems and a hierarchy of boundary value problems 3.4 a generalization of concavity 3.5 unbounded domains and unique green’s functions 0 introduction the purpose of this article is three-fold. first, in section 1, we shall show a relationship between a classical maximum principle in differential equations and positive operators on banach spaces. in loose terms, we shall refer to the “equivalence” of the maximum principle and known sign properties of an associated green’s function. second, in section 2, we shall exhibit many types of boundary value problems (bvps) for which the maximum principle is valid. finally, in section 3, we shall present some extended applications of the maximum principle that have arisen with the continued study of the qualitative properties of green’s functions. to begin, we recall the monograph of protter and weinberger [63]. in this work, the authors exhibit the impact of the maximum principle in ordinary differential equations, and elliptic, parabolic and hyperbolic partial differential equations. they give applications of the maximum principle to unique solvability, approximation methods, harnack inequalities, etc. in section 2, we shall exhibit families of bvps, not discussed by protter and weinberger [63], which can be considered in the context of a maximum principle or a positive fixed point operator. these families include higher order ordinary differential equations with a wide variety of boundary conditions, finite difference equations, and hence, dynamic equations on time scales, impulsive equations, bvps with nonlinear boundary conditions, and even some functional equations with delay. in section 3, we present extended applications of maximum principles that have emerged since the work of protter and weinberger [63]. again, these applications are available through sign property analysis of appropriate green’s functions. applications include elementary monotone methods coupled with upper and lower solution 7, 2(2005) positive operators and maximum principles for ... 239 methods, rapid convergence methods, krein-rutman theory [53], new comparison theorems, generalizations of concavity through harnack type inequalities, and limiting behavior on unbounded domains. 1 the maximum principle and positive operators on banach spaces to introduce this section, we consider a specific boundary value problem of the form x′′(t) = f(t), 0 < t < 1, (1.1) x(0) = x1, x(1) = x2. (1.2) it is well-known and one can show directly, that the solution x has the form x(t) = l(t) + ∫ 1 0 g(t,s)f(s)ds, (1.3) where g(t,s) = { t(s− 1), s ≥ t, s(t− 1), s ≤ t, (1.4) and l is the solution of the homogeneous differential equation, x′′ = 0, that satisfies the boundary conditions, (1.2). theorem 1.1 the statement x′′(t) ≤ 0, 0 < t < 1, x(0) = 0, x(1) = 0 ⇒ x(t) ≥ 0, 0 ≤ t ≤ 1, is equivalent to the statement g(t,s) ≤ 0, (t,s) ∈ (0, 1) × (0, 1). remark 1.1 the statement x′′(t) ≤ 0, 0 < t < 1, x(0) = 0, x(1) = 0, ⇒ x(t) ≥ 0, 0 ≤ t ≤ 1, is one form of the maximum principle [63], and more precisely it is a minimum principle. many authors prefer that a green’s function be positive pointwise; hence many authors choose to work with the operator, −d2/dt2. in this setting, signs are reversed; in particular, a green’s function is pointwise positive and the equivalent principle is a maximum principle. we will be lax with the phrase, maximum principle, throughout the paper and we do not use the phrase minimum principle. proof. one implication is immediately due to the representation, (1.3). the other implication employs a straightforward proof by contradiction. if g(t0,s0) > 0, one constructs a continuous nonpositive function f such that∫ 1 0 g(t,s)f(s)ds < 0. 240 paul w. eloe 7, 2(2005) the above theorem is trivial; but the above example carries over immediately to abstract boundary value problems of the form lx = f, bx = c, (1.5) where l denotes a linear operator on a banach space and b denotes linear boundary conditions. assume that (1.5) can be inverted; that is assume there exists g(t,s) such that the solution of (1.5) has the form x(t) = l(t) + ∫ ω g(t,s)f(s)ds, where l satisfies, lx = 0,bx = c. then the equivalence theorem applies in this setting as well. 2 classes of boundary value problems in this section, we intend to show that the simple equivalence that is exposed in the preceding section applies to a wide variety of problems. for a given operator, there will be applications to various examples depending on the boundary conditions. moreover, the equivalence applies to a broad variety of operators as well. we shall exhibit as examples ordinary differential operators, discrete and time scale operators, delay operators, and impulse operators. 2.1 ordinary differential equations we will present six types of bvps in this subsection. these are conjugate, right focal, lidstone, periodic, nonlocal, and sturm-liouville type bvps. this list is not exhaustive. the obvious place to begin is with the nth order disconjugate ordinary differential operator and the conjugate type boundary conditions. let i denote a bounded interval of the reals and let vi ∈ cn−i(i), i = 0, . . . ,n, be positive. for x ∈ cn(i), define the nth order ordinary differential operator, ln, by lnx(t) = vn(vn−1(. . . (v0x) ′ . . . )′)′(t), t ∈ i. (2.6) this particular factored operator has a long history of study and we refer the reader to the monograph of coppel [15]. let it suffice to comment that the primary motivation for the study of the operator, ln, is that the operator is homotopic to the operator, dn/dtn (vi ≡ 1), and solutions of lnx = 0 exhibit the same qualitative behavior as the family of nth order polynomials. hence, the study of bvps for disconjugate operators reduces to a study in interpolation theory. let a,b ∈ i, a < b. let k ∈{2, . . . ,n} and let a = t1 < t2 < · · · < tk = b be given. assume n1, . . . ,nk are positive integers such that α = k∑ i=1 ni = n. 7, 2(2005) positive operators and maximum principles for ... 241 for each j = 1, . . .k − 1, set αj = k∑ l=j+1 nl. the conjugate bvp associated with (2.6) is the problem, solve lnx(t) = f, t ∈ [a,b], (2.7) x(j−1)(ti) = 0, j = 1, . . . ,ni, i = 1, . . . ,k. (2.8) the bvp, (2.7), (2.8), is invertible in the following sense: there is a function g(t,s) : [a,b]2 → r such that the solution x of the bvp, (2.7), (2.8), has the representation, x(t) = ∫ b a g(t,s)f(s)ds, t ∈ [a,b]. (2.9) before we give the sign properties of g, we make two observations, the first with respect to the construction of g and the second with respect to nonlinear mathematics. there are various constructions of g. g is uniquely characterized by four properties [13, page 192]; this characterization leads to solving a linear system of 2n equations for 2n unknowns. we will refer to this type of characterization throughout this paper. for a second construction, g, as a function of t, satisfies lnx = 0 on triangles t < s, t > s; as a function of s, g satisfies the adjoint equation, l∗nx = 0. this observation leads to an independent construction of g, one we do not employ in this paper. a third construction, and one we shall use is as follows: let χ(t,s) denote the cauchy function associated with ln; that is, let χ denote the solution of the initial value problem, lnx(t) = 0, t ∈ [a,b], x(i−1)(s) = δi,n, i = 1, . . . ,n. χ is essentially the impulse function for this nth order ordinary problem. assume tl < s < tl+1. construct g as g(t,s) = { u(t,s), s > t, u(t,s) + χ(t,s), s ≤ t, where is u is the solution of the bvp, lnx(t) = 0, t ∈ [a,b], x(j−1)(ti) = 0, j = 1, . . . ,ni, i = 1, . . . , l, x(j−1)(ti) = −χ(j−1)(ti), j = 1, . . . ,ni, i = l + 1, . . . ,k. the factored form of ln implies that u is uniquely determined; if one thinks in terms of qualitative properties of polynomials, this is a well-posed interpolation problem. in the case, ln = dn/dtn, it is easy to see from this construction that as a function of t, g has precisely the n roots, counting multiplicities, given by the boundary conditions 242 paul w. eloe 7, 2(2005) (2.8), and no more. g is a cn−2 function and if g has an additional root, repeated applications of rolle’s theorem imply g(n−2) has three roots in (a,b). thus, g(n−1) vanishes for t < s or t > s. the zeros of each g(j) have been located by rolle’s theorem and so one can show inductively that each g(j) ≡ 0 for t < s or t > s, j = 0, . . . ,n− 1. for t < s, u ≡ 0 implies χ ≡ 0 by disconjugacy. this contradicts the construction of χ. a similar contradiction is obtained for s ≤ t. the representation (1.4) implies the following observation with respect to nonlinear problems. theorem 2.1 assume f : [a,b] × r → r is continuous. then x is a solution of the nonlinear bvp, lnx(t) = f(t,x(t)), t ∈ (a,b), with boundary conditions, (2.8), if, and only if, x ∈ c[a,b] and x satisfies the integral equation, x(t) = ∫ b a g(t,s)f(s,x(s))ds, t ∈ [a,b]. (2.10) the following sign condition of g is well-known [15]. theorem 2.2 assume ln has the factored form (2.6). then for (t,s) ∈ (ti, ti+1) × (a,b), (−1)αig(t,s) > 0. thus, the parity of the sign of g agrees with the number of boundary conditions specified to the right of t. couple the sign conditions given in theorem 2.2 with (2.10) and it is easy to see that cone theoretic fixed point theorems on banach spaces are very useful in the study of boundary value problems for nonlinear nth order ordinary differential equations. we also point out here that the nonlinear term f in theorem 2.1 is carefully constructed. only the sign of g is known; thus, the nonlinear term depends only on x and not on higher order derivatives of x. if one considers nonlinear effects from higher order derivatives of x, then one appeals to nagumo conditions [51]. although the theory of disconjugacy is summarized in the authoritative account due to coppel [15], we also refer to hartman [48], and levin [55], as these authors played key roles in the development of the theory. the second bvp we consider is the related right focal problem. for simplicity of exposition, we now consider the special case of (2.6), ln = dn/dtn. let k ∈ {2, . . . ,n− 1}. two-point (k,n−k) right focal boundary conditions are of the form x(i−1)(a) = 0, i = 1, . . . ,k, x(i−1)(b) = 0, i = k + 1, . . . ,n. (2.11) if one employs the more general factored operator, ln, then one needs stronger hypotheses than just the sign conditions on the vis [1]; if one chooses to employ two term differential operators [58], [21], then commonly, one employs quasi-derivatives in the boundary conditions, (2.11). hence, we only consider ln = dn/dtn. 7, 2(2005) positive operators and maximum principles for ... 243 theorem 2.3 a unique green’s function, g(t,s), exists for the bvp, x(n)(t) = f, t ∈ (a,b), with two-point right focal boundary conditions, (2.11). let gi(t,s) = ∂i(g(t,s))/∂ti. then (−1)n−kgi−1(t,s) > 0, (t,s) ∈ (a,b] × (a,b) i = 1, . . .k, (−1)n−i+1gi−1(t,s) ≥ 0, (t,s) ∈ [a,b) × (a,b) i = k + 1, . . .n. this theorem is very believable; beginning with the n−1 order derivative, one can sign gn−1 as the kernel of a first order initial value problem. one then signs lower order derivatives inductively via definite integration. note that each derivative of g satisfies a known sign condition; in particular, under suitable hypotheses, each derivative of a solution satisfies a maximum principle. so, cone theoretic methods apply immediately to nonlinear problems of the form, x(n)(t) = f(t,x(t), . . . ,x(n−1)(t)), t ∈ (a,b), with boundary conditions (2.11). no nagumo conditions [51] are required in an analysis here. a third and related bvp is the lidstone bvp [56], [2], [37], for example. again we restrict the discussion to a simple even order operator, l2n = d2n/dt2n, and we consider two-point boundary conditions of the form, x(2(i−1))(a) = 0, x(2(i−1))(b) = 0, i = 1, . . . ,n. (2.12) theorem 2.4 a unique green’s function, g(t,s), exists for the bvp, x(2n)(t) = f, t ∈ (a,b), with two-point lidstone boundary conditions, (2.12) and (−1)n−i+1g2(i−1)(t,s) > 0, (t,s) ∈ (a,b)2, i = 1, . . .n. as with theorem 2.3, theorem 2.4 is believable. the function, g2(n−1) is the green’s function for a second order conjugate bvp, x′′(t) = 0, a < t < b, x(a) = 0, x(b) = 0. hence, g2(n−1) has the representation (1.4) (with a = 0 and b = 1) and g2(n−1)(t,s) < 0, (t,s) ∈ (a,b)2. next, note that g2(n−2) is the solution of a nonhomogeneous bvp, x′′(t) = g2(n−1)(t,s), a < t < b, x(a) = 0, x(b) = 0. thus, g2(n−2) is the convolution g2(n−2)(t,s) = ∫ b a g2(n−1)(t,r)g2(n−1)(r,s)dr. 244 paul w. eloe 7, 2(2005) proceed inductively and calculate the convolution g2(i−2)(t,s) = ∫ b a g2(n−1)(t,r)g2(i−1)(r,s)dr at each step. the illustration in the preceding paragraph indicates that the lidstone problem can be considered as a nested collection of second order conjugate problems. upon further reflection, and in the same context, note that right focal bvps (and focal bvps as well) are nested initial value problems. with the maximum principle given in theorem 2.4, the natural nonlinear problem to consider has the form, x(2n)(t) = f(t,x(t),x′′(t), . . . ,x(2j)(t), . . . ,x(2(n−1))(t)), t ∈ (a,b). if one wishes to address nonlinear dependence on odd order derivatives of x, one must again appeal to nagumo conditions [18]. considerable work has been done recently in relation to the maximum principle for periodic bvps of the form, x(n)(t) = f, t ∈ (a,b), x(i−1)(a) = x(i−1)(b), i = 1, . . . ,n. see [12] for example. there is a current flurry to study nonlocal boundary value problems. key works include lomtatidze and co-authors [57], and gupta [46]. for example, consider the bvp, (1.1), with boundary conditions given by x(0) = 0, x(1) = x(1/2). a green’s function can be constructed directly and has the form g(t,s) = { g1(t,s), 0 < s ≤ 1/2, g2(t,s), 1/2 ≤ s ≤ 1, where g1(t,s) = { −t, t < s, −s, s < t, and g2(t,s) = { 2(s− 1)t, t < s, −s + (2s− 1)t, s < t. hence, the maximum principle is valid for these nonlocal bvps as well. finally, for second order problems, conjugate, right focal and periodic boundary conditions are all special cases of the general sturm-liouville boundary conditions [45]. 7, 2(2005) positive operators and maximum principles for ... 245 2.2 discrete, time scale problems the disconjugacy theory for forward difference equations was developed by philip hartman [47] in a landmark paper which has generated so much activity in the study of difference equations. sturm theory for a second order finite difference equation goes back to fort [43]. hartman considers the nth order linear finite difference equation, pu(m) = n∑ j=0 αj (m)u(m + j) = 0, αnα0 �= 0, m ∈ i = {a,a + 1,a + 2, . . .}, with conjugate boundary conditions, u(mi) = 0, i = 1, . . . ,n, where a ≤ m1 < m2 < · · · < mn. completely analogous to the development of disconjugacy for ordinary differential equations, he shows the existence of a green’s function, g(m,s), for this problem and (−1)σ(m)g(m,s) > 0, m ∈ i, m �= mj , s ∈ i, σ(m) =card {j : m < mj}. much of the theory for ordinary differential equations carries over to the discrete problems and we refer the reader to a comprehensive bibliography in [3]. in an effort to unify the continuous and discrete calculus, hilger [50] invented the calculus on time scales. rather than discuss the material here, we refer the reader to the authoritative account in [10]. a green’s function for the second order scalar function is developed there. in chapter 8 of [11], a green’s function for an nth order disconjugate equation with conjugate boundary conditions is signed. many of the applications discussed in section 3 for ordinary differential equations have extensions to difference equations. most of the extensions to time scales have not been developed. see chapter 8 of [11]. 2.3 systems of ordinary equations werner [66] wrote an interesting paper in which he developed an abstract maximum principle for systems of ordinary differential equations. let i = [a,b] be a subset of the reals and assume that f : i → rn. let m,n denote n × n matrices with real entries and let c ∈ rn. consider a two-point bvp of the form, x′(t) = f(t), t ∈ (a,b), (2.13) mx(a) + nx(b) = c. (2.14) the bvp, (2.13), (2.14), is not invertible. so, one considers an equivalent equation, x′(t) −d(t)x(t) = f(t) −d(t)x(t), t ∈ (a,b), (2.15) 246 paul w. eloe 7, 2(2005) where d is an n × n matrix with entries in c(i). one constructs d so that the equivalent bvp, (2.13), (2.14), is invertible, and so that a maximum principle is valid on a partial order induced by d. the green’s function has the form g(t,s) = { u(t)amu(a)u−1(s), a ≤ s < t ≤ b, u(t)(amu(a) −e)u−1(s), a ≤ t < s ≤ b, where u denotes a fundamental matrix for the system, x′ −dx = 0, e denotes the n×n identity matrix and a = (mu(a) + nu(b))−1. let b denote the banach space cn(i) with ||x|| = max ||xk|| and ||xk|| denotes the usual supremum norm on the kth component of x. define a partial order on b by x ≤ z if, and only if, xk(t) ≤ zk(t), t ∈ i, k = 1, . . . ,n. if h : b → b is an invertible linear operator, define a relation, ≤h , by x ≤h z if, and only if, hx ≤ hz. then ≤h denotes a partial order on b and b is a partially ordered banach space with respect to ≤h . werner [66] defines a partial order ≤hu−1 and a partial order ≤ju−1 where h and j are chosen so that hamu(a)j−1 ≥ 0, h(amu(a) −e)j−1 ≥ 0 elementwise. consider a modification of g: ĝ(t,s) = { u(t)amu(a)j−1, a ≤ s < t ≤ b, u(t)(amu(a) −e)j−1, a ≤ t < s ≤ b. then ĝ ≥hu−1 0 and the maximum principle applies. werner makes a very nice application to a second order scalar problem with periodic boundary conditions. this development in systems has been extended to multipoint problems [27] and problems with impulse [34]. 2.4 impulsive problems with nonlinear boundary conditions in this subsection, we will briefly discuss a second order impulsive bvp with nonlinear boundary conditions. we consider a problem with impulses at 0 < t1 < · · · < tm < 1; define an impulse δx(t) = x(t+)−x(t−). first, consider an impulsive problem with conjugate boundary conditions and consider the problem as linear, nonhomogeneous. x′′(t) = f(t), t ∈ (0, 1) \{t1, . . . , tm}, (2.16) δx(ti) = ui, δx ′(ti) = vi, i = 1, . . . ,m, (2.17) x(0) = x1, x(1) = x2. (2.18) then, see [35], x(t) = p(t) + m∑ i=1 ii(t) + ∫ b a g(t,s)f(s)ds, (2.19) 7, 2(2005) positive operators and maximum principles for ... 247 where p(t) = x1(t− 1) + x2t, and i(ti) = { t(−ui − (1 − tiv1)), t ≤ ti, (1 − t)(ui − tiv1), ti ≤ t, i = 1, . . . ,m. g has the representation given by (1.4). if one analyzes the g in terms of the maximum principle, it is clear that the crux of the matter is that the terms t, (1 − t),s, and (1 −s) are of fixed sign on (0, 1). note that the analogous terms in p and i are precisely t and (1 − t). as one readily imposes a sign condition on f to invoke a maximum principle, one can readily impose conditions on ui,vi or x1,x2 to invoke a maximum principle. we shall briefly return to this development below when we discuss monotone methods. if the problem is nonlinear in any term, differential equation, impulse, or boundary condition, then the integral expression (2.19) readily becomes a fixed point equation as in theorem 2.1. 2.5 delay equations azbelev, and co-authors in the perm group, have studied functional differential equations (fdes) for many years [5], [6]. domoshnitsky [17] and domoshnitsky and bainov [7] have studied the sign properties of associated green’s functions for linear fdes. eloe and henderson [31] studied a second order linear operator of the form lx(t) = x′′(t) + q(t)x′(0) + m∑ i=1 pi(t)x(hi(t)), 0 < t, (2.20) where q,pi,hi ∈ c[0,∞), i = 1, . . . ,m. eloe and henderson make the additional assumption, not assumed by the azbelev and co-authors; 0 ≤ hi(t) ≤ t, i = 1, . . . ,m. for each b > 0, consider the conjugate boundary conditions x(0) = 0, x(b) = 0. one can employ the construction of a green’s function as outlined by coddington and levinson [13, page 192] to show that a green’s function, g(b; t,s), exists for the conjugate bvp associated with (2.20). in this setting eloe and henderson proved that if q(t) ≥ 0 and 0 < b1 < b2, then 0 > g(b1; t,s) > g(b2; t,s), (t,s) ∈ (0,b1)2, and 0 > gt(b1; 0,s) > gt(b2; 0,s), s ∈ (0,b1). 248 paul w. eloe 7, 2(2005) 2.6 singular equations bebernes and jackson [9] studied a singular bvp on an unbounded domain of the form, x′′(t) = f, 0 < t, x(0) = 0, x ∈ bc[0,∞), where bc[0,∞) denotes the bounded continuous functions on [0,∞) with a usual supremum norm. a green’s function for this problem exists in the following sense: if f satisfies suitable asymptotic properties, then x is a solution of the the singular bvp, if, and only if, x(t) = ∫ ∞ 0 g(t,s)fds. the green’s function has the form g(t,s) = { −s, 0 ≤ s < t, −t, 0 ≤ t < s. if one calculates the second order conjugate green’s function on an interval [0,b] and formally lets b diverge to ∞, the characterization of a conjugate green’s functions “converges” to the form given above. likewise, if one calculates a second order right focal green’s function on an interval [0,b] and formally lets b diverge to ∞, the characterization of a right focal green’s functions “converges” to the form given above. this interesting behavior is addressed in the book by coddington and levinson [13] when they discuss limit circle and limit point cases. this would illustrate an example of the limit point case. elias [19] studied an nth order ordinary differential operator with a family of related two-point bvps and obtained a unique limiting green’s function satisfying known sign conditions. eloe and kaufmann [39] were motivated by elias, changed many of elias’ assumed sign conditions, and obtained a unique limiting green’s function satisfying known sign conditions. eloe and kaufmann [38] have shown similar results to be valid for discrete problems. eloe and henderson [30] have constructed unique limiting green’s functions with known sign conditions for singular problems on bounded domains as well. 3 applications 3.1 monotone methods collatz [14] provides an elegant discussion of monotone methods for positive operators on partially ordered banach spaces. let b be a partially ordered banach space with partial order, ≤. suppose t : b → b is an isotone operator; i.e., x1 ≤ x2 ⇒ tx1 ≤ tx2. 7, 2(2005) positive operators and maximum principles for ... 249 suppose there exist α0,β0 ∈ b such that α0 ≤ β0, (3.21) α0 ≤ tα0, tβ0 ≤ β0. (3.22) then if t is a completely continuous map, t has a fixed point x0 ∈ b such that α0 ≤ x0 ≤ β0. this follows immediately by the schauder fixed point theorem. define the convex set d := {x ∈ b : α0 ≤ x0 ≤ β0}. the monotonicity assumptions on t coupled with the assumptions (3.21), (3.22) immediately imply that t : d → d. in our setting, tx(t) = ∫ ω g(t,s)f(s,x(s))ds. for the simple problem, (1.1), (1.2), it is the case that g(t,s) < 0 on (0, 1) × (0, 1). hence, if f is decreasing in x, t is an isotone operator. there are methods of forced monotonicity as well (see [26], [45]). the goal then would be to construct a forcing term that essentially forces the nonlinear term to be decreasing in x. applications abound; we only list a few citations. if α0,β0 satisfy (3.21), (3.22) then α0 and β0 are typically called lower solution and upper solution, respectively. (3.21) is a straight forward assumption. to obtain (3.22), let us assume that g(t,s) < 0 on (0, 1) × (0, 1), and consider the equation (1.1). then one assumes α′′0 (t) ≥ f(t,α0(t)), β′′0 (t) ≤ f(t,β0(t)), t ∈ (0, 1). the inequalities make sense if one considers mental images due to concavity. they are precisely the correct inequalities when one appeals to the representation in (2.10) and notes the representation, α0(t) = ∫ ω g(t,s)α′′0 (s)ds. the analogous representation is valid for any x including β0. thus, couple the differential inequalities with the sign of g and obtain (3.22). šeda [65] has written a very interesting paper developing and exploiting properties of upper and lower solutions. a different type of monotone method coupled with the method of upper and lower solutions is a method of rapid convergence. an authoritative account has recently been published by lakshmikantham and vatsala [54]. the methods, and related rapid convergence methods are simply numerical methods related to taylor series expansions. how far one expands a taylor series dictates the order of convergence of the monotone iterates. regardless, it is an interesting and very delicate balance of the monotone methods and the method of upper and lower solutions. in addition to being applications of the maximum principle, the methods require uniqueness of solutions as well. 250 paul w. eloe 7, 2(2005) one requirement of the method is that if α and β satisfy (3.22), then α ≤ β; that is, (3.22) implies (3.21). this property implies the uniqueness of solutions since solutions are simultaneously lower solution and upper solutions. because of the requirement, (3.22) implies (3.21), the rapid convergence methods were initially applied to initial value problems. the methods have proved suitable to second order problems of the form x′′(t) = f(t,x(t),x′(t)), t ∈ i, with conjugate conditions, sturm liouville conditions, nonlocal conditions, or singular problems on unbounded domains, for example. in this setting, if f is increasing with respect to the second component, then one has uniqueness of solutions for various families of boundary conditions. one does not find many applications of the rapid convergence methods to higher order scalar problems. the methods have worked very nicely for nth order problems with periodic boundary conditions [12]; the methods should work as well on lidstone problems or other nested type problems based on the initial or boundary value problems discussed in section 2. the methods have yet to be successfully applied to the nth order conjugate type problems addressed in [15], for example. reasonable conditions for uniqueness of solutions have not been formulated for such problems. 3.2 krein-rutman theory a theorem due to perron states that if t is a square matrix with only positive entries, then the eigenvalue of largest magnitude is positive and simple, and there exists an associated eigenfunction that has all positive entries. the krein-rutman theory carries this idea over to partially ordered banach spaces. applications of the maximum principle carry this idea over to many of the problems illustrated in section 2. we shall begin with some definitions and results from the theory of cones. please refer to krasnosel’skǐı [52], amann [4], deimling [16], krein and rutman [53], schmitt and smith [64], and zeidler [67] for accounts of the material stated here. we shall assume the reader has some familiarity with cones in banach spaces. let b denote a real banach space, and let p denote a reproducing cone in b. recall that the cone is reproducing if for each x ∈ b, there exist u,v ∈ p such that x = u − v. let ≤ be the partial order on b induced by p ; that is, x ≤ y if, and only if, x− y ∈ p . let n1,n2 : b → b be bounded, linear operators. we will say n1 ≤ n2 if n1u ≤ n2u for each u ∈ p and we will say n is positive with respect to p if n : p → p . let r(n) denote the spectral radius of n. nussbaum [59] is responsible for the following result. theorem 3.1 let nb, α ≤ b ≤ β, be a family of compact, linear operators on a banach space such that the mapping b → nb is continuous in the uniform operator topology. then the mapping b → r(nb) is continuous. refer to [4] or [52] for proofs of the following three theorems. assume the maps n,n1,n2 are compact, linear, and positive with respect to a reproducing cone, p . 7, 2(2005) positive operators and maximum principles for ... 251 theorem 3.2 assume r(n) > 0. then r(n) is an eigenvalue of n, and there is a corresponding eigenvector in p. theorem 3.3 n1 ≤ n2 implies r(n1) ≤ r(n2). theorem 3.4 suppose there exists μ > 0, u ∈ b, −u /∈ p such that nu ≥ μu. then n has an eigenvector in p which corresponds to an eigenvalue, λ ≥ μ. for the sake of exposition we will consider the second order problem with dirichlet boundary conditions on an arbitrary interval (a,b). that is, we consider x′′(t) = p(t)x(t), a < t < b, (3.23) x(a) = 0, x(b) = 0. (3.24) the green’s function has the form g(b; t,s) = { (t−a)(s− b)/(b−a), a ≤ t < s ≤ b, (s−a)(t− b)/(b−a), a ≤ s < t ≤ b. assume that p is a nonpositive continuous function defined on [a,∞) and assume p does not vanish identically on each compact subinterval of [a,∞). of course, if p = −1 the eigenfunctions are sine functions. set b = {x ∈ bc[a,∞) : x(a) = 0}, where b is equipped with the the usual supremum norm. let p = {x ∈ b : x(t) ≤ 0, a ≤ t}. define nb : b → b by nbx(t) = { ∫ b a g(b; t,s)p(s)x(s)ds, a ≤ t ≤ b, 0, b < t. theorem 3.5 the following are equivalent: 1. b0 = inf{b > a : (3.23), (3.24) has a nontrivial solution}; 2. there exists a nontrivial solution x0 of the bvp (for b = b0) (3.23), (3.24) such that x ∈ pb0 ; 3. r(nb0 ) = 1. it is common to call b0 the principal eigenvalue with principal eigenvector x0. the principal eigenvector can be useful in nonlinear problems as an upper solution. we don’t prove theorem 3.5; details can be found in [25]. we do discuss how theorems 3.2, 3.3, and 3.4 can be applied due to the maximum principle. first, if one defines bb = {x ∈ c[a,b] : x(a) = 0} and one defines the obvious cone, pb, on a compact domain, then the restriction of nb to bb maps pb\{0} into the interior of pb. this observation is useful in the application of the theory of μ0 positive operators, a closely related theory [52]. 252 paul w. eloe 7, 2(2005) that the restriction of nb to bb maps pb{0} into the interior of pb follows because (∂/∂t)g(b; a,s) < 0, (∂/∂t)g(b; b,s) > 0. the above observation generalizes to all green’s functions discussed in section 2. more pertinent to the krein-rutman theory, one will compare (nb2 −nb1 )x(t). if b2 > b1 and t ∈ [a,b1], one considers ∫ b1 a (g(b2; t,s) −g(b1; t,s))p(s)x(s)ds. hence, one becomes interested in sign analysis of h(b; t,s) = (∂/∂b)g(b; t,s). the authors that come to mind that analyze h are bates and gustafson [8] and henderson [49]. h is a cn solution of a bvp and it’s sign can be analyzed using methods that will be discussed in the next section. for our simple second order example in this section, h satisfies the bvp, x′′(t) = 0, a < t < b, x(a) = 0, x(b) = −(∂/∂t)g(b; b,s) < 0. the krein-rutman theory has been applied to nth order bvps with conjugate conditions [25], right focal and in between two point problems [28], impulse problems [41] and [37], and functional differential equations [31]. it is interesting to note that in the impulse citations, the corresponding impulse function (∂/∂b)i satisfies the same sign properties as h. schmitt and smith [64] have shown the krein-rutman theory applies to elliptic bvps with dirichlet boundary conditions as well. the shape of the domain is a carefully constructed rectangle so that a concept of disconjugacy can be defined. 3.3 comparison theorems and a hierarchy of boundary value problems for equations of one independent variable, the difference of two green’s functions is, in fact, sufficiently smooth. the jump in the appropriate derivative to give the delta impulse effect adds out and so the difference is sufficiently smooth. this observation applies above to intuitively see that h is sufficiently smooth as well. this observation also applies in the case of laplace’s elliptic equation. here the delta impulse is generated with a logarithm term that adds out. in particular, the theory developed by schmitt and smith [64] can carry over to more irregularly shaped domains. before discussing a general problem, consider, for the sake of motivation, the second order equation, x′′(t) = 0, 0 < t < 1. let g(1; t,s) denote the green’s function associated with the conjugate conditions, x(0) = 0, x(1) = 0, 7, 2(2005) positive operators and maximum principles for ... 253 and let g(2; t,s) denote the green’s function associated with the right focal conditions, x(0) = 0, x′(1) = 0. then, for s ∈ (0, 1), h(t) = g(2; t,s) −g(1; t,s) is the solution of bvp, x′′(t) = 0, 0 < t < 1, x(0) = 0, x(1) = g(2; 1,s) < 0. note that x′′(t) = 0, 0 < t < 1, x(0) = 0, x(1) < 0, is a form of the maximum principle and so x(t) < 0, 0 < t < 1. to discuss the application of the maximum principle in this section, we shall appeal to an nth order ordinary differential operator lx = x(n)(t) + n∑ i=1 ai(t)x (n−i)(t), a < t < b. (3.25) where each ai ∈ c[a,b]. let k ∈ {1, . . . ,n − 1} be fixed; let w denote the set of nonnegative integers and define ωn−k ⊂ w n−k by ωn−k = {α = (α1, . . . ,αn−k) : 0 ≤ α1 < · · · < αn−k ≤ n− 1}. (3.26) consider two-point boundary conditions of the form x(l)(a) = 0, l = 0, . . . ,k − 1, x(l)(b) = 0, l = α1, . . . ,αn−k. (3.27) note the boundary conditions (3.27) depend on α and on b. also note that there is a natural partial order on ωn−k: α ≤ β ⇔ αi ≤ βi, i = 1, . . . ,n−k. we have need to assume that the operator l in (3.25) is right disfocal on an interval [a,b0] where a < b ≤ b0; that is, the only solution of lx = 0 satisfying x(l)(tl) = 0, l = 0, . . . ,n − 1, where a ≤ t0 ≤ ··· ≤ tn−1 ≤ b0, is x ≡ 0. for a given k,α,b, let g(k,α,b; t,s) denote the green’s function corresponding to the bvp, (3.25), (3.27). eloe and ridenhour [40] proved the following theorems. theorem 3.6 let k ∈ {1, . . . ,n − 1}, α,β ∈ ωn−k, α < β, a (−1)n−kgl(k,α,b; t,s) > 0, (3.28) (t,s) ∈ (0,b)2, and (−1)n−kgk(k,β,b; a,s) > (−1)n−kgk(k,α,b; a,s) > 0, (3.29) s ∈ (0,b). 254 paul w. eloe 7, 2(2005) theorem 3.7 let k ∈ {1, . . . ,n− 1}, α,β ∈ ωn−k, αn−k < n− 1. assume α ≤ β, a < b1 ≤ b2 ≤ b0 and that one of the inequalities (α ≤ β or b1 ≤ b2) is strict. then, for l = 0, . . . ,α1, (−1)n−kgl(k,β,b2; t,s) > (−1)n−kgl(k,α,b1; t,s) > 0, (3.30) (t,s) ∈ (0,b1)2, and (−1)n−kgk(k,β,b2; a,s) > (−1)n−kgk(k,α,b1; a,s) > 0, (3.31) s ∈ (0,b1). in particular, g is monotone with respect to b; g is monotone with respect to α. peterson [60], [61], elias [19], and peterson and ridenhour [62] have developed related inequalities for the two term operator x(n)(t)+p(t)x(t). peterson considers the cases p(t) < 0 and p(t) > 0 independently and employs an adjoint equation argument. elias considers the case (−1)n−kp(t) < 0 and obtains a wealth of inequalities that contain theorems 3.6 and 3.7. elias appeals to special features of the two-term operator [21]. peterson and ridenhour address the case when p is independent of sign. again they appeal to an adjoint equation argument. to obtain theorems 3.6 and 3.7 for the general operator l, one first observes the difference of two green’s functions to be n times continuously differentiable. one then replaces the adjoint equation argument with a double induction on k and ∑n−k i=1 αi. 3.4 a generalization of concavity in recent applications of cone theoretic fixed point theorems to boundary value problems (bvps), hadamard type inequalities that provide lower bounds for positive functions as a function of the supremum norm have been applied. a particular inequality to which we refer is as follows: if y′′(t) ≤ 0, 0 ≤ t ≤ 1 and y(t) ≥ 0, 0 ≤ t ≤ 1, then for 1/4 ≤ t ≤ 3/4, y(t) ≥ ||y||/4, (3.32) where ||·|| = sup0≤t≤1 |y(t)|. analogous inequalities are valid for functions that satisfy the differential inequality piecewise. in particular, analogous inequalities are valid for associated green’s functions. to obtain the above inequality, assume y′′(t) < 0, 0 ≤ t ≤ 1, y(0) ≥ 0, y(1) ≥ 0. let t0 ∈ (0, 1) be such that ||y|| = y(t0). construct the tent function = { (||y||/t0)t, 0 ≤ t ≤ t0, (||y||/(t0 − 1))(t− 1), t0 ≤ t ≤ 1. y lies above p because of concavity and and one evaluates p to obtain the estimate given in (3.32). the estimate in (3.32) was employed in [44] to obtain existence of solutions for bvps for singular ordinary differential equations. erbe and wang [42] employed the estimate in (3.32) in conjunction with krasnosel’skǐı-guo cone theoretic fixed point theorem and obtained existence of solutions in a cone when the nonlinear 7, 2(2005) positive operators and maximum principles for ... 255 term satisfies superlinear or sublinear asymptotic behavior. this landmark paper, [42], has stimulated considerable research in the applications of fixed point methods. see an authoritative account of the recent activity in the area of time scales in chapter 7 of [11]. upon inspection, p satisfies the differential equation piecewise. so, (3.32) has been very conducive to generalizations. the first such generalization was produced in [32]. let n ≥ 2 be an integer, and assume k ∈{1, . . . ,n− 1}. assume (−1)(n−k)y(n) ≥ 0, 0 ≤ t ≤ 1, (3.33) y(j)(0) = 0, j = 0, . . . ,k − 1, y(j)(1) = 0, j = 0, . . . ,n−k − 1. (3.34) then for 1/4 ≤ t ≤ 3/4, y(t) ≥ ||y||/4m, (3.35) where m = max{k,n− k}. with the development of (3.35), the work of erbe and wang [42] with superlinear or sublinear growth readily carried over to the two-point conjugate type boundary value, (3.33), (3.34) [33]. the method to obtain (3.35) is merely a generalization of the method to obtain (3.32). first, assume strict differential inequality in (3.33). let t0 ∈ (0, 1) be such that ||y|| = y(t0). construct the tent function p(t) = { (||y||/(tk0 ))tk, 0 ≤ t ≤ t0, (||y||/(t0 − 1)n−k)(t− 1)n−k, t0 ≤ t ≤ 1. one cannot appeal to generalized concavity to argue that y lies above p; using contradiction and repeated applications of rolle’s theorem, one proves that y lies above p and then one evaluates p to obtain the estimate given in (3.35). following [32], (3.32) has been generalized in various ways with applications to multipoint bvps [36], discrete problems, [22], and kiguradze type inequalities [20]. 3.5 unbounded domains and unique green’s functions we will close the paper with a discussion on a singular bvp, x′′(t) + q(t)x(t) = f(t,x(t)), t ∈ r+, (3.36) x(0) = x0, x(t) bounded on r +, (3.37) where x0 is real, f : r+ × r → r is continuous, q : r+ → r− is continuous, and q(t) ≤ −c2 < 0, t ∈ r+, for some c2 > 0. we model the singular bvp based on the work of bebernes and jackson [9]. assume that x1 ≤ x2 ⇒ f(t,x1) ≤ f(t,x2), t ∈ r+. 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[67] e. zeidler, nonlinear functional analysis and its applications, vol. i, springer-verlag, new york, 1985. cubo a mathematical journal vol.20, nβo2, (67–93). june 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000200067 some remarks on the non-real roots of polynomials shuichi otake 1 and tony shaska 2 1department of applied mathematics, waseda university, japan. 2department of mathematics and statistics , oakland university, rochester, mi, 48309. shuichi.otake.8655@gmail.com, shaska@oakland.edu abstract let f ∈ r(t)[x] be given by f(t,x) = xn + t · g(x) and β1 < · · · < βm the distinct real roots of the discriminant ∆(f,x)(t) of f(t,x) with respect to x. let γ be the number of real roots of g(x) = ∑s k=0 ts−kx s−k. for any ξ > |βm|, if n−s is odd then the number of real roots of f(ξ,x) is γ + 1, and if n − s is even then the number of real roots of f(ξ,x) is γ, γ + 2 if ts > 0 or ts < 0 respectively. a special case of the above result is constructing a family of degree n ≥ 3 irreducible polynomials over q with many non-real roots and automorphism group sn. resumen sea f ∈ r(t)[x] dada por f(t,x) = xn + t · g(x) y β1 < · · · < βm las diferentes ráıces reales del discriminante ∆(f,x)(t) de f(t,x) con respecto de x. sea γ el número de ráıces reales de g(x) = ∑s k=0 ts−kx s−k. para todo ξ > |βm|, si n − s es impar entonces el número de ráıces reales de f(ξ,x) es γ + 1, y si n − s es par entonces el número de ráıces reales de f(ξ,x) es γ, γ+2 si ts > 0 o ts < 0, respectivamente. un caso especial del resultado anterior es construyendo una familia de polinomios irreducibles sobre q de grado n ≥ 3 con muchas ráıces no-reales y grupo de automorfismos sn http://dx.doi.org/10.4067/s0719-06462018000200067 68 shuichi otake and tony shaska cubo 20, 2 (2018) keywords and phrases: polynomials, non-real roots, discriminant, bezoutian, galois groups. 2010 ams mathematics subject classification: 12d10, 12f10, 26c10. cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 69 1 introduction let f(x) ∈ q[x] be an irreducible polynomial of degree n ≥ 2 and gal (f) its galois group over q. let us assume that over r, f(x) is factored as f(x) = a r∏ j=1 (x − αj) s∏ i=1 (x2 + aix + bi), where a2i < 4bi, for all i = 1, . . . ,s. the pair (r,s) is called the signature of f(x). obviously degf = 2s + r. if s = 0 then f(x) is called totally real and if r = 0 it is called totally complex. equivalently the above terminology can be defined for binary forms f(x,z). by a reordering of the roots we may assume that if f(x) has 2s non-real roots then α := (1,2)(3,4) · · · (2s − 1,2s) ∈ gal(f). in [4] it is proved that if degf = p, for a prime p, and s satisfies s(s logs + 2 logs + 3) ≤ p then gal(f) = ap,sp. moreover, a list of all possible groups for various values of r is given for p ≤ 29; see [4, thm. 2]. there are some follow up papers to [4]. in [1] the author proves that if p ≥ 4s + 1, then the galois group is either sp or ap. this improves the bound given in [4]. the author also studies when polynomials with non-real roots are solvable by radicals, which are consequences of table 2 and theorem 2 in [4]. in [13] the author uses bezoutians of a polynomial and its derivative to construct polynomials with real coefficients where the number of real roots can be counted explicitly. thereby, irreducible polynomials in q[x] of prime degree p are constructed for which the galois group is either sp or ap. in this paper we study a family of polynomials with non-real roots whose degree is not necessarily prime. given a polynomial g(x) = ∑s i=0 tix i and with γ number of non-real roots we construct a polynomial f(t,x) = xn + tg(x) which has γ,γ + 1,γ + 2 non-real roots for certain values of t ∈ r; see theorem 3.2. the values of t ∈ r are given in terms of the bezoutian matrix of polynomials or equivalently the discriminant of f(t,x) with respect to x. this is the focus of section 3 in the paper. while most of the efforts have been focusing on the case of irreducible polynomials over q which have real roots, the case of polynomials with no real roots is equally interesting. how should an irreducible polynomial over q with all non-real roots must look like? what can be said about the galois group of such totally complex polynomials? in [5] is developed a reduction theory for such polynomials via the hyperbolic center of mass. a special case of theorem 3.2 provides a class of totally complex polynomials. notation for any polynomial f(x) we denote by ∆(f,x) its discriminant with respect to x. if f is a univariate polynomial then ∆f is used and the leading coefficient is denoted by led(f). throughout this paper the ground field is a field of characteristic zero. 70 shuichi otake and tony shaska cubo 20, 2 (2018) 2 preliminaries let f1(x), f2(x) be polynomials over a field f of characteristic zero and, let n be an integer which is greater than or equal to max{degf1,degf2}. then, we put bn(f1,f2) : = f1(x)f2(y) − f1(y)f2(x) x − y = n∑ i,j=1 αijx n−iyn−j ∈ f[x,y], mn(f1,f2) : = (αij)1≤i,j≤n. the matrix mn(f1,f2) is called the bezoutian of f1 and f2. clearly, bn(f1,f1) = 0 and hence mn(f1,f1) is the zero matrix. the following properties hold true; see [6, theorem 8.25] for details. proposition 1. the following are true: (1) mn(f1,f2) is an n × n symmetric matrix over f. (2) bn(f1,f2) is linear in f1 and f2, separately. (3) bn(f1,f2) = −bn(f2,f1). when f2 = f ′ 1, the formal derivative of f1 (with respect to the indeterminate x), we often write bn(f1) := bn(f1,f ′ 1). from now on, for any degree n ≥ 2 polynomial f(x) ∈ r[x] we will denote by mn(f) := mn(f,f ′) as above. the matrix mn(f) is called the bezoutian matrix of f. remark 2.1. it is often the case that the matrix m′n(f1,f2) = (α ′ ij)1≤i,j≤n defined by the generating function b′n(f1,f2) : = f1(x)f2(y) − f1(y)f2(x) x − y = n∑ i,j=1 α′ijx i−1yj−1 ∈ f[x,y] is called the bezoutian of f1 and f2. but no difference can be seen between these two definitions as far as we consider the corresponding quadratic forms n∑ i,j=1 αijxixj and n∑ i,j=1 α′ijxixj. in fact, these two quadratic forms are equivalent over the prime field q (⊂ f) since we have m′n(f1,f2) = tjnmn(f1,f2)jn, where jn =       0 1 1 ... 1 0       is an n × n anti-identity matrix. this implies that above two quadratic forms are equivalent over q or more precisely, over the ring of rational integers z. cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 71 let f(x) ∈ r[x] be a degree n ≥ 2 polynomial which is given by f(x) = a0 + a1x + · · · + anxn then over r this polynomial is factored as f(x) = a r∏ j=1 (x − αj) s∏ i=1 (x2 + aix + bi) for some α1, . . . ,αr ∈ r and ai,bi,a ∈ r, where a2i < 4bi, for all i = 1, · · · ,s. throughout this paper, for a univariate polynomial f, its discriminant will be denoted by ∆f. for any two polynomials f1(x), f2(x) the resultant with respect to x will be denoted by res(f1,f2,x). we notice the following elementary fact, its proof is elementary and we skip the details. remark 2.2. for any polynomial f(x), the determinant of the bezoutian is the same as the discriminant up to a multiplication by a constant. more precisely, ∆f = 1 led(f)2 detmn(f), where led(f) is the leading coefficient of f(x). if f(x) ∈ q[x] is irreducible and its degree is a prime number, say degf = p, then there is enough known for the galois group of polynomials with some non-real roots; see [4], [1], [13] for details. if the number of non-real roots is ”small” enough with respect to the prime degree degf = p of the polynomial, then the galois group is ap or sp. furthermore, using the classification of finite simple groups one can provide a complete list of possible galois groups for every polynomial of prime degree p which has non-real roots; see [4] for details. on the other extreme are the polynomials which have all roots non-real. we called them above, totally complex polynomials. we have the following: lemma 2.1. the followings are equivalent: i) f(x) ∈ r[x] is totally complex ii) f(x) can be written as f(x) = a n∏ i=1 fi where fi = x 2 + aix + bi, for i = 1, . . . ,n and ai,bi,a ∈ r, where a2i < 4bi, for all i = 1, . . . ,n. moreover, the determinant of the bezoutian mn(f) is given by ∆f = 1 led(f)2 detmn(f) = n∏ i=1 ∆fi · n∏ i,j,i6=j (res(fi,fj,x)) 2 72 shuichi otake and tony shaska cubo 20, 2 (2018) where led(f) is the leading coefficient of f(x). ii) the index of inertia of bezoutian m(f) is 0 iii) if ∆f 6= 0 then the equivalence class of m(f) in the witt ring w(r) is 0. proof. the equivalence between i), ii), and iii) can be found in [6]. it is not clear when such polynomials are irreducible over q. if that’s the case, what is the galois group gal (f)? clearly the group generated by the involution (1,2)(3,4) · · · (2n − 1,2n) is embedded in gal (f). is gal (f) larger in general? 3 on the number of real roots of polynomials for any degree n ≥ 2 polynomial f(x) ∈ r[x] and any symmetric matrix m := mn(f) with real entries, let nf be the number of distinct real roots of f and σ(m) be the index of inertia of m, respectively. the next result plays a fundamental role throughout this section ([6, theorem 9.2]). proposition 2. for any real polynomial f ∈ r[x], the number nf of its distinct real roots is the index of inertia of the bezoutian matrix mn(f). in other words, nf = σ(mn(f)) . let us cite one more result which says that the roots of a polynomial depend continuously on its coefficients ([11, theorem 1.4], [16, theorem 1.3.1]). proposition 3. let be given a polynomial f(x) = n∑ l=0 alx l ∈ c[x], with distinct roots α1, . . . ,αk of multiplicities m1, . . . ,mk respectively. then, for any given a positive ε < min 1≤i 0 such that any monic polynomial g(x) = ∑n l=0 blx l ∈ c[x] whose coefficients satisfy |bl − al| < δ, for l = 0, · · · ,n − 1, has exactly mj roots in the disk d(αj;ε) = {z ∈ c | |z − αj| < ǫ} (j = 1, · · · ,k). cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 73 let n, s be positive integers such that n > s and let g(t0, · · · ,ts;x) = s∑ k=0 ts−kx s−k, f(n)(t0, · · · ,ts,t;x) = xn + t · g(t0, · · · ,ts;x) (3.1) be polynomials in x over e1 = r(t0, · · · ,ts), e2 = r(t0, · · · ,ts,t), respectively. here, e1 (resp., e2) is a rational function field with s+1 (resp.,(s+2)) variables t0, · · · ,ts (resp.,(t0, · · · ,ts,t)). to ease notation, let us put g(x) = g(t0, · · · ,ts;x), f(t;x) = f(n)(t0, · · · ,ts,t;x) and for any real vector v = (v0, · · · ,vs) ∈ rs+1, we put gv(x) = g(v0, · · · ,vs;x), fv(t;x) = f(n)(v0, · · · ,vs,t;x). (3.2) by using proposition 2, we can prove the next theorem ([13, main theorem 1.3]). theorem 3.1. let r = (r0, · · · ,rs) ∈ rs+1 be a vector such that ngr = s. let us consider fr(t;x) = f (n)(r0, · · · ,rs,t;x) as a polynomial over r(t) in x and put pr(t) = detmn(fr(t;x)) = detmn(fr(t;x),f ′ r(t;x)), where f′r(t;x) is a derivative of fr(t;x) with respect to x. then, for any real number ξ > αr = max{α ∈ r | pr(α) = 0}, we have nfr(ξ;x) =    s + 1 if n − s : odd s if n − s : even, rs > 0 s + 2 if n − s : even, rs < 0. by this theorem and a theorem of oz ben-shimol [1, theorem 2.6], we can obtain an algorithm to construct prime degree p polynomials with given number of real roots, and whose galois groups are isomorphic to the symmetric group sp or the alternating group ap ([13, corollary 1.6]). in this section, we extend this theorem as follows; theorem 3.2. let r = (r0, · · · ,rs) ∈ rs+1 be a vector such that gr(x) is a degree s separable polynomial satisfying ngr(x) = γ (0 ≤ γ ≤ s). let us consider fr(t;x) = f(n)(r0, · · · ,rs,t;x) as a polynomial over r(t) in x and put pr(t) = detmn(fr(t;x)) = detmn(fr(t;x),f ′ r(t;x)), where f′r(t;x) is a derivative of fr(t;x) with respect to x. then, for any real number ξ > αr = max{α ∈ r | pr(α) = 0}, we have nfr(ξ;x) =    γ + 1 if n − s : odd γ if n − s : even, rs > 0 γ + 2 if n − s : even, rs < 0. (3.3) 74 shuichi otake and tony shaska cubo 20, 2 (2018) the above theorem can be restated as follows: corolary 1. let f ∈ r(t)[x] be given by f(t,x) = xn + t · s∑ k=0 ts−kx s−k and β1 < · · · < βm the distinct real roots of the degree s polynomial p(t) := 1 tn−1 ∆(f,x)(t). for any ξ > |βm|, the number of real roots of f(ξ,x) is nf(ξ,x) =    γ + 1 if n − s : odd γ if n − s : even, ts > 0 γ + 2 if n − s : even, ts < 0. where γ is the number or real roots of g(x) = f(x)−xn t ∈ r[x]. the rest of the section is concerned with proving thm. 3.2. 3.1 the bezoutian of f(t;x) first, let us put a(t0, · · · ,ts,t) = (aij(t0, · · · ,ts,t))1≤i,j≤n = mn(f(t;x)) ∈ symn(e2), b(t0, · · · ,ts) = (bij(t0, · · · ,ts))1≤i,j≤s = ms(g(x)) ∈ syms(e1). for ease of notation, we also write a(t0, · · · ,ts,t) = a(t) = (aij(t))1≤i,j≤n, b(t0, · · · ,ts) = b = (bij)1≤i,j≤s and we put b(t) = (bij(t))1≤i,j≤s = t 2b. then, by proposition 1, we have a(t) = mn(x n + tg(x),nxn−1 + tg′(x)) = nmn(x n,xn−1) − ntmn(x n−1,g(x)) + tmn(x n,g′(x)) + t2mn(g(x),g ′(x)) = nmn(x n,xn−1) − nt s∑ k=0 ts−kmn(x n−1,xs−k) + t s−1∑ k=0 (s − k)ts−kmn(x n,xs−k−1) + t2mn(g(x),g ′(x)). lemma 3.1. let λ,µ,ν be integers such that λ ≥ µ > ν ≥ 0. then mλ(xµ,xν) = (mij)1≤i,j≤λ, where mij = { 1 i + j = 2λ − (µ + ν) + 1 (λ − µ + 1 ≤ i, j ≤ λ − ν), 0 otherwise. cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 75 proof. by definition, we have bλ(x µ,xν) = xµyν − xνyµ x − y = µ−ν∑ k=1 xµ−kyν+k−1 = µ−ν∑ k=1 xλ−(λ−µ+k)yλ−(λ−ν−k+1), which implies mij = { 1 (i, j) = (λ − µ + k,λ − ν − k + 1) (1 ≤ k ≤ µ − ν) 0 otherwise = { 1 i + j = 2λ − (µ + ν) + 1 (λ − µ + 1 ≤ i, j ≤ λ − ν), 0 otherwise. this completes the proof. here, let us divide a(t) into two parts â(t) and ã(t), where â(t) = (âij(t))1≤i,j≤n = nmn(x n,xn−1) − nt s∑ k=0 ts−kmn(x n−1,xs−k) + t s−1∑ k=0 (s − k)ts−kmn(x n,xs−k−1), ã(t) = (ãij(t))1≤i,j≤n = t 2mn(g(x),g ′(x)) and put lk = n − s + k + 2 (= 2n − (n + s − k − 1) + 1). then, by lemma 3.1, we have { â11(t) = n â1,lk−1(t) = âlk−1,1(t) = (s − k)ts−kt (0 ≤ k ≤ s − 1). moreover, when i + j = lk, we have âij(t) = −ntts−k + t(s − k)ts−k = −(lk − 2)ts−kt (2 ≤ i, j ≤ lk − 2, 0 ≤ k ≤ s). (3.4) remark 3.3. note that, if s = n − 1, we have −nt s∑ k=0 ts−kmn(x n−1,xs−k) = −nt s∑ k=1 ts−kmn(x n−1,xs−k), thus, when i + j = lk, equation (3.4) should be modified by âij(t) = −ntts−k + t(s − k)ts−k = −(lk − 2)ts−kt (2 ≤ i, j ≤ lk − 2, 1 ≤ k ≤ s). we avoid this minor defect by considering that there is no entries satisfying 2 ≤ i, j ≤ l0 −2 when s = n − 1 since l0 − 2 = n − s = 1. 76 shuichi otake and tony shaska cubo 20, 2 (2018) proposition 4. put lk = n − s + k + 2. then âij(t) =    n (i, j) = (1,1) (s − k)ts−kt (i, j) = (1,lk − 1) or (lk − 1,1) (0 ≤ k ≤ s − 1) −(lk − 2)ts−kt i + j = lk, 2 ≤ i, j ≤ lk − 2, (0 ≤ k ≤ s) 0 otherwise. ãij(t) = { bi−(n−s),j−(n−s)t 2 n − s + 1 ≤ i, j ≤ n 0 otherwise. proof. the statement for âij(t) has just been proved. for ãij(t), it is enough to see that we can denote ms(g(x)) = s∑ ℓ=0 s∑ m=1 mtℓtmms(x ℓ,xm−1), mn(g(x)) = s∑ ℓ=0 s∑ m=1 mtℓtmmn(x ℓ,xm−1), that is, we can obtain mn(g(x)) from ms(g(x)) by just replacing s with n for all ms(x ℓ,xm), which, by lemma 3.1, means that s × s matrix ms(g(x)) occupies the part {b†ij | n − s + 1 ≤ i, j ≤ n} of the matrix mn(g(x)) = (b † ij)1≤i,j≤n. by proposition 4, we can express the matrix a(t) as follows; a(t) =                      n 0 .. . 0 stst (s − 1)ts−1t . . . t1t 0 −(n − s)tst −(n − s + 1)ts−1t . . . −(n − 1)t1t −nt0t . . . ... ... ... ... 0 0 −(n − s)tst ... ... 0 0 stst −(n − s + 1)ts−1t (s − 1)ts−1t . . . ... ... c(t). . . −(n − 1)t1t ... 0 t1t −nt0t 0 0                      . (3.5) here, c(t) = (cij(t))1≤i,j≤s = c(t0, · · · ,ts,t) = (cij(t0, · · · ,ts,t))1≤i,j≤s is an s × s symmetric matrix whose entries are of the form cij(t0, · · · ,ts,t) = bijt2 + λijt = bij(t0, · · · ,ts)t2 + λij(t0, · · · ,ts)t (λij = λij(t0, · · · ,ts) ∈ e1). next, let a(t)1 = (aij(t)1)1≤i,j≤n = a(t0, · · · ,ts,t)1 = (aij(t0, · · · ,ts,t)1)1≤i,j≤n be the n × n symmetric matrix obtained from a(t) by multiplying the first row and the first column by 1/ √ n and then sweeping out the entries of the first row and the first column by the (1,1) entry 1. here, let qm(k;c) = (qij)1≤i,j≤m and rm(k,l;c) = (rij)1≤i,j≤m be m × m elementary matrices such that cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 77 qm(k;c)=                1 ... 1 c 1 ... 1                , rm(k,l;c)=                 1 ... 1 c ... 1 ... 1                 , where qkk = c and rkl = c. moreover, for any m×m matrices m1, m2, · · · , ml, put ∏l k=1 mk = m1m2 · · ·ml. then, we have a(t)1 = ts(t)1a(t)s(t)1, where s(t)1 = qn(1;1/ √ n) s−1∏ k=0 rn(1,lk − 1; −a1,lk−1(t)/ √ n). the matrix a(t)1 can be expressed as follows; a(t)1 =                      1 0 . . . 0 0 0 . . . 0 0 0 . . . −(n − s)tst −(n − s + 1)ts−1t . . . −(n − 1)t1t −nt0t . . . . . . ... ... ... ... 0 0 −(n − s)tst ... ... 0 0 0 −(n − s + 1)ts−1t 0 . . . ... ... c(t)1.. . −(n − 1)t1t ... 0 0 −nt0t 0 0                      . (3.6) here, c(t)1 = (cij(t)1)1≤i,j≤s = c(t0, · · · ,ts,t)1 = (cij(t0, · · · ,ts,t)1)1≤i,j≤s is an s×s symmetric matrix whose entries are of the form cij(t0, · · · ,ts,t)1 = b̄ij(t0, · · · ,ts)t2 + λij(t0, · · · ,ts)t (b̄ij(t0, · · · ,ts) ∈ e1), where b̄ij(t0, · · · ,ts) = bij(t0, · · · ,ts) − (s − i + 1)(s − j + 1) n ts−i+1ts−j+1 (3.7) for any i, j (1 ≤ i, j ≤ s). we put b̄ij(t0, · · · ,ts) = b̄ij and b̄ = (b̄ij)1≤i,j≤s. 3.2 some results for the bezoutian of f r (t;x) let r = (r0, · · · ,rs) ∈ rs+1 be a vector as in theorem 3.2. we put ar(t) = (a (r) ij (t))1≤i,j≤n = a(r0, · · · ,rs,t) ∈ symn(r(t)), br = (b (r) ij )1≤i,j≤s = b(r0, · · · ,rs) ∈ syms(r) 78 shuichi otake and tony shaska cubo 20, 2 (2018) and br(t) = t 2br. let us also put ar(t)1 = a(r0, · · · ,rs,t)1. by equation (3.6), the matrix ar(t)1 can be expressed as follows; ar(t)1 =                      1 0 . . . 0 0 0 . . . 0 0 0 . . . −(n − s)rst −(n − s + 1)rs−1t . . . −(n − 1)r1t −nr0t . . . . . . ... ... ... ... 0 0 −(n − s)rst ... ... 0 0 0 −(n − s + 1)rs−1t 0 . . . ... ... cr(t)1.. . −(n − 1)r1t ... 0 0 −nr0t 0 0                      . here, cr(t)1 = (c (r) ij (t)1)1≤i,j≤s = c(r0, · · · ,rs,t)1 and c (r) ij (t)1 = b̄ij(r0, · · · ,rs)t 2 + λij(r0, · · · ,rs)t (b̄ij(r0, · · · ,rs),λij(r0, · · · ,rs) ∈ r). note that, by equation (3.7), we have b̄ij(r0, · · · ,rs) = b (r) ij − (s − i + 1)(s − j + 1) n rs−i+1rs−j+1 (1 ≤ i, j ≤ s). to ease notation, we put b̄ij(r0, · · · ,rs) = b̄ (r) ij and b̄r = (b̄ (r) ij )1≤i,j≤s. in particular, since ms(gr) = ms ( rsx s, s−1∑ k=0 (s − k)rs−kx s−k−1 ) + ms ( s∑ k=1 rs−kx s−k,g′r ) = s−1∑ k=0 (s − k)rsrs−kms(x s,xs−k−1) + ms ( s∑ k=1 rs−kx s−k,g′r ) , we have b (r) 1,k+1 = b (r) k+1,1 = (s − k)rsrs−k (0 ≤ k ≤ s − 1) (3.8) by lemma 3.1 and hence b̄ (r) 1j = (s − j + 1)rsrs−j+1 − s(s − j + 1) n rsrs−j+1 (3.9) = (s − j + 1) ( 1 − s n ) rsrs−j+1 (1 ≤ j ≤ s). lemma 3.2. put b̄r(t) = t 2b̄r. then, br(ξ) and b̄r(ξ) are equivalent over r for any real number ξ and we have σ(b̄r(ξ)) = ngr for any non-zero real number ξ. proof. let us denote by b∗r = (b (r,∗) ij )1≤i,j≤s (b̄ ∗ r = (b̄ (r,∗) ij )1≤i,j≤s) the matrix obtained from br (b̄r) by multiplying the first row and the first column by 1 / ± √ b (r) 11 ( 1 / ± √ b̄ (r) 11 ) (the sign cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 79 before √ b (r) 11 ( √ b̄ (r) 11 ) are the same as the sign of rs; see the definition of d ( d̄ ) below) and then sweeping out the entries of the first row and the first column by the (1,1) entry 1. since b11 = sr 2 s (> 0) and b̄11 = s(1 − s/n)r 2 s (> 0) by (3.8) and (3.9), we have b∗r = ttbrt, b̄ ∗ r = tt̄b̄rt̄, (3.10) where t = qs(1;1/d) s∏ k=2 rs(1,k; −b (r) 1k /d) (d = √ s · rs), t̄ = qs(1;1/d̄) s∏ k=2 rs(1,k; −b̄ (r) 1k /d̄) (d̄ = √ s(1 − s/n) · rs). note that in [13, lemma 3.3], we have proved b (r,∗) ij = b̄ (r,∗) ij (1 ≤ i, j ≤ s) and hence t 2b∗r = t 2b̄∗r, which, by (3.10), implies that symmetric matrices br(ξ) and b̄r(ξ) are equivalent over r for any real number ξ. then, since ngr = σ(br) = σ(br(ξ)) for any ξ ∈ r \ {0}, the latter half of the statement have also been proved. 3.3 nonvanishingness of some coefficients in this subsection, we prove the next lemma. lemma 3.3. let φ(x) = φ(t0, · · · ,ts;x) = s∑ k=0 hs−k(t0, · · · ,ts)xs−k ∈ e1[x] (3.11) be the characteristic polynomial of b̄. then, hs−k(t0, · · · ,ts) is a non-zero polynomial in e1 for any k (1 ≤ k ≤ s). proof. lemma 3.3 is clear for s = 1, since we have b = m1(t1x + t0) = [ t21 ] and hence, by equation (3.7), b̄ = [ t21 − 1 n t21 ] = [ n − 1 n t21 ] . next, suppose s ≥ 2. then, by equation (3.7) and the definition of the bezoutian, we have hs−k(t0, · · · ,ts) ∈ r[t0, · · · ,ts] for any k (1 ≤ k ≤ s). thus, we have only to prove that hs−k(t0, · · · ,ts) 6= 0 for any k (1 ≤ k ≤ s), which is clear from the next lemma 3.4. lemma 3.4. suppose s ≥ 2 and put u0 = us = 1, u1 = t1 and uk = 0 (2 ≤ k ≤ s − 1). then, hs−k(u0, · · · ,us) is a non-constant polynomial in r(t1) for any k (1 ≤ k ≤ s), i.e., hs−k(u0, · · · ,us) ∈ r[t1] \ r (1 ≤ k ≤ s). 80 shuichi otake and tony shaska cubo 20, 2 (2018) to prove lemma 3.4, let us put u = (u0, · · · ,us) and gu(x) = g(u0, · · · ,us;x) = xs + t1x + 1 ∈ r(t1)[x], fu(t;x) = x n + tgu(x) ∈ r(t1,t)[x] (n > s), au(t) = (a (u) ij (t))1≤i,j≤n = a(u0, · · · ,us,t) ∈ symn(r(t1,t)), bu = (b (u) ij )1≤i,j≤s = b(t0, · · · ,us) ∈ syms(r(t1)), bu(t) = t 2bu. then, by equation (3.5), we have au(t) =                      n 0 .. . 0 st 0 . . . t1t 0 −(n − s)t 0 . . . −(n − 1)t1t −nt . . . ... ... ... ... 0 0 −(n − s)t ... ... 0 0 st 0 0 . . . ... ... cu(t). . . −(n − 1)t1t ... 0 t1t −nt 0 0                      , where cu(t) = (c (u) ij (t))1≤i,j≤s = c(u0, · · · ,us,t) and c (u) ij (t) = bij(u0, · · · ,us)t 2 + λij(u0, · · · ,us)t (λij(u0, · · · ,us) ∈ r(t1)). moreover, by equation (3.6), we also have au(t)1 =                      1 0 . . . 0 0 0 . . . 0 0 0 . . . −(n − s)t 0 . . . −(n − 1)t1t −nt . . . . . . ... ... ... ... 0 0 −(n − s)t ... ... 0 0 0 0 0 . . . ... ... cu(t)1.. . −(n − 1)t1t ... 0 0 −nt 0 0                      . here, cu(t)1 = (c (u) ij (t)1)1≤i,j≤s = c(u0, · · · ,us,t)1 and c (u) ij (t)1 = b̄ij(u0, · · · ,us)t 2 + λij(u0, · · · ,us)t (b̄ij(u0, · · · ,us) ∈ r). note that, by equation (3.7), we have b̄ (u) ij =    b (u) 11 − (s 2/n) (i, j) = (1,1) b (u) 1s − (s/n)t1 (i, j) = (1,s) or (s,1) b (u) ss − (1/n)t 2 1 (i, j) = (s,s) b (u) ij otherwise. (3.12) cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 81 let us put b̄u = (b̄ (u) ij )1≤i,j≤s and b̄u(t) = t 2b̄u. then, since ms(gu) = ms(x s + t1x + 1,sx s−1 + t1) = sms(x s,xs−1) + t1ms(x s,1) − st1ms(x s−1,x) − sms(x s−1,1) + t21ms(x,1) + t1ms(1,1), we have (a) if s = 2, bu = [ 2 t1 t1 t 2 1 − 2 ] , (b) if s ≥ 3, b (u) ij =    s (i, j) = (1,1) t1 (i, j) = (1,s) or (s,1) (1 − s)t1 i + j = s + 1, 2 ≤ i, j ≤ s − 1 −s i + j = s + 2 t21 (i, j) = (s,s), 0 otherwise, which, by equation (3.12), implies (a′) if s = 2, b̄u = [ 2(n − 2)/n (n − 2)t1/n (n − 2)t1/n (n − 1)t 2 1/n − 2 ] , (b′) if s ≥ 3, b̄ (u) ij =    s(n − s)/n (i, j) = (1,1) (n − s)t1/n (i, j) = (1,s) or (s,1) (1 − s)t1 i + j = s + 1, 2 ≤ i, j ≤ s − 1 −s i + j = s + 2 (n − 1)t21/n (i, j) = (s,s), 0 otherwise. therefore, if s ≥ 3, the matrix b̄u = (b̄ (u) ij )1≤i,j≤s has the expression of the form                 s(n − s)/n 0 0 0 · · · 0 (n − s)t1/n 0 0 0 · · · 0 (1 − s)t1 −s 0 0 ... (1 − s)t1 −s 0 0 ... ... ... ... ... ... ... 0 (1 − s)t1 ... ... 0 0 (1 − s)t1 −s ... 0 (n − s)t1/n −s 0 · · · 0 0 (n − 1)t21/n                 . 82 shuichi otake and tony shaska cubo 20, 2 (2018) here, let us denote by φu(x) = s∑ k=0 h (u) s−kx s−k = φ(u0, · · · ,us;x) ( = s∑ k=0 hs−k(u0, · · · ,us)xs−k ) the characteristic polynomial of b̄u. note that since we have h (u) s−k ∈ r[t1] by the proof of lemma 3.3, we have only to prove h (u) s−k is non-constant for any k (1 ≤ k ≤ s). by the above expression of b̄u, we have (a′′) if s = 2, φu(x) = x 2 − (n − 1)t21 − 4 n x + (n − 2)t21 − 4n + 8 n , (b′′) if s ≥ 3, φu(x) =                                                                                                    ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ x − s(n − s)/n −(n − s)t1/n x (s − 1)t1 s . . . ... s . . . ... ... x + (s − 1)t1 s ... s x ... ... . . . (s − 1)t1 s x −(n − s)t1/n s x − (n − 1)t 2 1 /n ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (s is odd), ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ x − s(n − s)/n −(n − s)t1/n x (s − 1)t1 s . . . ... s x (s − 1)t1 ... (s − 1)t1 x + s ... ... . . . (s − 1)t1 s x −(n − s)t1/n s x − (n − 1)t 2 1 /n ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (s is even). example 3.1. (1) put s = 7 and n = 10. then, we have gu(x) = x 7 + t1x + 1, fu(t;x) = x 10 + t(x7 + t1x + 1), cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 83 φu(x) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ x − 21/10 0 0 0 0 0 −3t1/10 0 x 0 0 0 6t1 7 0 0 x 0 6t1 7 0 0 0 0 x + 6t1 7 0 0 0 0 6t1 7 x 0 0 0 6t1 7 0 0 x 0 −3t1/10 7 0 0 0 0 x − 9t 2 1 /10 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ =x7 + ( − 9 10 t21 + 6t1 − 21 10 ) x6 + ( − 27 5 t31 − 351 5 t21 − 63 5 t1 − 147 ) x5 + ( 324 5 t41 − 2106 5 t31 + 1197 5 t21 − 588t1 + 3087 10 ) x4 + ( 1944 5 t51 + 5832 5 t41 + 5859 5 t31 + 16758 5 t21 + 6174 5 t1 + 7203 ) x3 + ( − 5832 5 t61 + 34992 5 t51 − 21546 5 t41 + 50274 5 t31 − 95697 10 t21 + 14406t1 − 151263 10 ) x2 + ( − 34992 5 t71 + 11664 5 t61 − 81648 5 t51 + 15876 5 t41 − 111132 5 t31 + 21609 5 t21 − 151263 5 t1 − 117649 ) x + 69984 5 t71 + 2470629 10 . (2) put s = 8 and n = 12. then, we have gu(x) = x 8 + t1x + 1, fu(t;x) = x 12 + t(x8 + t1x + 1) and φu(x) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ x − 8/3 0 0 0 0 0 0 −t1/3 0 x 0 0 0 0 7t1 8 0 0 x 0 0 7t1 8 0 0 0 0 x 7t1 8 0 0 0 0 0 7t1 x + 8 0 0 0 0 0 7t1 8 0 x 0 0 0 7t1 8 0 0 0 x 0 −t1/3 8 0 0 0 0 0 x − 11t 2 1 /12 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ =x8 + ( − 11 12 t21 + 16 3 ) x7 + ( −152t21 − 640 3 ) x6 + ( 539 4 t41 − 256t 2 1 − 1024 ) x5 + ( 22736 3 t41 + 45824 3 t21 + 16384 ) x4 + ( − 26411 4 t61 − 22736 3 t41 + 31744 3 t21 + 65536 ) x3 + ( − 355348 3 t61 − 213248t 4 1 − 1064960 3 t21 − 524288 ) x2 + ( 1294139 12 t81 + 1075648 3 t61 + 1404928 3 t41 + 1835008 3 t21 − 4194304 3 ) x − 823543 3 t81 + 16777216 3 . proof of lemma 3.4. to prove lemma 3.4, it is enough to prove degh (u) s−k ≥ 1 for any k (1 ≤ k ≤ s). this is clear for s = 2 by (a′′) and we suppose s ≥ 3 hereafter. to prove degh(u)s−k ≥ 1 (1 ≤ k ≤ s), 84 shuichi otake and tony shaska cubo 20, 2 (2018) let us compute the leading term of h (u) s−k (∈ r[t1]). then, since h (u) s−k is the coefficient of the term h (u) s−kx s−k of the characteristic polynomial φu(x), we need to maximize the degree in t1 when we take ‘s − k’ x and the remaining k elements from r[t1]. (a) suppose s is odd. let us divide the case into three other sub-cases. (a1) suppose k is odd and 1 ≤ k ≤ s − 2. in this case, the degree of the leading term of h (u) s−k is k + 1. in fact, it is obtained by taking (a11) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (a12) ‘k − 1’ (s − 1)t1 from entries of the form (i,s + 1 − i) (2 ≤ i ≤ s − 1). first, suppose we take the (s,s) entry x − (n − 1)t21/n from the s-th row. then we must take the (1,1) entry from the first row. next, let us proceed to the (s − 1)-th row. if we take the (s −1,s− 1) entry x from the (s−1)-th row, then we must also take x from the second row, while if we take (s−1)t1 from the (s−1)-th row, then we must also take (s−1)t1 from the second row. the situation is the same for the (s − 2)-th row, the (s − 3)-th row ... and so on, which implies that (s − 1)t1 must occur in pair. hence, the leading term of h (u) s−k is − n − 1 n t21 · ( (s − 3)/2 (k − 1)/2 ) {(−1) · (s − 1)2t21} (k−1)/2 (( n 0 ) = 1 (n ≥ 0) ) and the degree of this term is k + 1 (≥ 2). (a2) suppose k is odd and k = s. if k = s, h (u) s−k = h (u) 0 is the constant term of φu(x). in this case, the degree of the leading term of h (u) 0 is s. in fact, it is obtained by taking (a21) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (a22) if s ≥ 5 (⇔ (s,k) 6= (3,3)), ‘(s−3)/2’ pairs of (s−1)t1 from entries of the form (i,s+1−i) (2 ≤ i ≤ (s − 1)/2, (s + 3)/2 ≤ i ≤ s − 1), (a23) (s − 1)t1 from the ((s + 1)/2,(s + 1)/2) entry x + (s − 1)t1, (a24) −s(n − s)/n from the (1,1) entry x − s(n − s)/n or by taking (a25) all anti-diagonal entries. cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 85 therefore, the leading term of h (u) 0 is − n − 1 n t21 · {(−1) · (s − 1) 2t21} (s−3)/2 · (s − 1)t1 · ( − s(n − s) n ) + (−1) · ( − n − s n t1 )2 · {(−1) · (s − 1)2t21} (s−3)/2 · (s − 1)t1 = (n − s)(s − 1) n · (−1)(s−3)/2(s − 1)s−2ts1 = (−1)(s−3)/2 (n − s)(s − 1)s−1 n ts1 for any s (s ≥ 3) and the degree of this term is s. (a3) suppose k is even. in this case, we have 2 ≤ k ≤ s − 1 and the degree of the leading term of h(u)s−k is k + 1. in fact, it is obtained by taking (a31) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (a32) if s ≥ 5 (⇔ (s,k) 6= (3,2)), ‘(k−2)/2’ pairs of (s−1)t1 from entries of the form (i,s+1− i) (2 ≤ i ≤ (s − 1)/2, (s + 3)/2 ≤ i ≤ s − 1), (a33) (s − 1)t1 from the ((s + 1)/2,(s + 1)/2) entry x + (s − 1)t1. therefore, the leading term of h (u) s−k is − n − 1 n t21 · ( (s − 3)/2 (k − 2)/2 ) {(−1) · (s − 1)2t21} (k−2)/2 · (s − 1)t1 for any s (s ≥ 3) and the degree of this term is k + 1 (≥ 3). (b) suppose s is even (s ≥ 4). we also divide this case into three other sub-cases. (b1) suppose k is odd. in this case, we have 1 ≤ k ≤ s − 1 and the degree of the leading term of h(u)s−k is k + 1. in fact, it is obtained by taking (b11) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (b12) ‘(k − 1)/2’ pairs of (s − 1)t1 from entries of the form (i,s + 1 − i) (2 ≤ i ≤ s − 1). therefore, the leading term of h (u) s−k is − n − 1 n t21 · ( (s − 2)/2 (k − 1)/2 ) {(−1) · (s − 1)2t21} (k−1)/2 and the degree of this term is k + 1 (≥ 2). (b2) suppose k is even and 2 ≤ k ≤ s − 2. in this case, the degree of the leading term of h (u) s−k is k. in fact, it is obtained by taking 86 shuichi otake and tony shaska cubo 20, 2 (2018) (b21) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (b22) ‘(k − 2)/2’ pairs of (s − 1)t1 from entries of the form (i,s + 1 − i) (2 ≤ i ≤ s − 1), (b23) −s(n − s)/n from the (1,1) entry x − s(n − s)/n or by taking (b24) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (b25) i f s ≥ 6 (⇔ (s,k) 6= (4.2)), ‘(k−2)/2’ pairs of (s−1)t1 from entries of the form (i,s+1−i) (2 ≤ i ≤ (s − 2)/2, (s + 4)/2 ≤ i ≤ s − 1), (b26) s from the ((s + 2)/2,(s + 2)/2) entry x + s or by taking (b27) ‘k/2’ pairs of (s − 1)t1 from entries of the form (i,s + 1 − i) (2 ≤ i ≤ s − 1) or by taking (b28) one pair of −(n − s)t1/n from the (1,s) and the (s,1) entry, (b29) ‘(k − 2)/2’ pairs of (s − 1)t1 from entries of the form (i,s + 1 − i) (2 ≤ i ≤ s − 1). here, note that if we take the (s,1) entry −(n − s)t1/n from the s-th row, we must also take the (1,s) entry −(n − s)t1/n from the first row. therefore, the leading term of h (u) s−k is − n − 1 n t21 · ( (s − 2)/2 (k − 2)/2 ) {(−1) · (s − 1)2t21} (k−2)/2 · ( − s(n − s) n ) − n − 1 n t21 · ( (s − 4)/2 (k − 2)/2 ) {(−1) · (s − 1)2t21} (k−2)/2 · s + ( (s − 2)/2 k/2 ) {(−1) · (s − 1)2t21} k/2 + ( (−1) · {−(n − s)}2 n2 t21 ) · ((s − 2)/2 (k − 2)/2 ) {(−1) · (s − 1)2t21} (k−2)/2 = ( s(n − s)(n − 1) n2 ((s − 2)/2 (k − 2)/2 ) − s(n − 1) n ((s − 4)/2 (k − 2)/2 ) − (s − 1)2 ((s − 2)/2 k/2 ) − (n − s)2 n2 ((s − 2)/2 (k − 2)/2 ) ) {(−1) · (s − 1)2t21} (k−2)/2t21. for any s (s ≥ 4). then, since ((s − 4)/2 (k − 2)/2 ) = s − k s − 2 ((s − 2)/2 (k − 2)/2 ) , ((s − 2)/2 k/2 ) = s − k k ((s − 2)/2 (k − 2)/2 ) , we have cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 87 s(n − s)(n − 1) n2 ((s − 2)/2 (k − 2)/2 ) − s(n − 1) n ((s − 4)/2 (k − 2)/2 ) (3.13) − (s − 1)2 ((s − 2)/2 k/2 ) − (n − s)2 n2 ((s − 2)/2 (k − 2)/2 ) = ( s(n − s)(n − 1) n2 − s(s − k)(n − 1) n(s − 2) − (s − 1)2(s − k) k − (n − s)2 n2 ) ((s − 2)/2 (k − 2)/2 ) = s{ ( k(k + s2 − 4s + 2) − s3 + 4s2 − 5s + 2 ) n − k(k + s2 − 4s + 2)} nk(s − 2) ((s − 2)/2 (k − 2)/2 ) . hence, if the above value becomes zero, we have ( k(k + s2 − 4s + 2) − s3 + 4s2 − 5s + 2 ) n − k(k + s2 − 4s + 2) = 0, which implies k(k + s2 − 4s + 2) = 0, −s3 + 4s2 − 5s + 2 = 0 (3.14) or n = k(k + s2 − 4s + 2) k(k + s2 − 4s + 2) − s3 + 4s2 − 5s + 2 . (3.15) here, (3.14) is impossible since −s3 + 4s2 − 5s + 2 = −(s − 1)2(s − 2) and s ≥ 4. also, (3.15) is impossible since, for any s ≥ 4 and 2 ≤ k ≤ s − 2, we have k(k + s2 − 4s + 2) ≥ 2(2 + s2 − 4s + 2) ≥ 2(s − 2)2 > 0 and k(k + s2 − 4s + 2) − s3 + 4s2 − 5s + 2 ≤ (s − 2){(s − 2) + s2 − 4s + 2} − s3 + 4s2 − 5s + 2 = −s2 + s + 2 = −(s + 1)(s − 2) < 0, which implies n < 0, a contradiction. thus, the above value (3.13) is non-zero and the degree of the leading term of h (u) s−k is k. (b3) suppose k is even and k = s. if k = s, h (u) s−k = h (u) 0 is the constant term of φu(x). in this case, the degree of the leading term of h (u) 0 is s. in fact, it is obtained by taking (b31) −(n − 1)t21/n from the (s,s) entry x − (n − 1)t 2 1/n, (b32) ‘(s − 2)/2’ pairs of (s − 1)t1 from entries of the form (i,s + 1 − i) (2 ≤ i ≤ s − 1), (b33) −s(n − s)/n from the (1,1) entry x − s(n − s)/n or by taking 88 shuichi otake and tony shaska cubo 20, 2 (2018) (b34) all anti-diagonal entries. therefore, the leading term of h (u) 0 is − n − 1 n t21 · {(−1) · (s − 1) 2t21} (s−2)/2 · ( − s(n − s) n ) + (−1) · ( − n − s n t1 )2 · {(−1) · (s − 1)2t21} (s−2)/2 = (−1)(s−2)/2 (n − s)(s − 1)s−1 n ts1 and the degree of this term is s (s ≥ 4). lemma 3.5. let v = (v0, · · · ,vs) ∈ rs+1 be a real vector and n (> s) be an integer. put pv(t) = detmn(fv(t;x)) = detmn(f (n) (v0, · · · ,vs,t;x)) and αv = max{α ∈ r | pv(α) = 0}. if there exists a real number ρ0 (> αv) such that nfv(ξ;x) = γ0 for any ξ > ρ0, we have nfv(ξ;x) = γ0 for any ξ > αv. proof. put av(t) = mn(fv(t;x)). then, by proposition 2, we have γ0 = σ(av(ξ)) for any ξ > ρ0. let us also put r = {ρ ∈ r | ρ > αv, σ(av(ξ)) = γ0 for any ξ > ρ}. since r is a nonempty set (ρ0 ∈ r) having a lower bound αv, r has the infimum ρv; ρv = inf r. then, it is enough to prove ρv = αv. here, suppose to the contrary that ρv > αv and we denote by ωv(t;x) = n∑ k=0 ωk(t)x k ∈ r(t)[x] the characteristic polynomial of av(t). note that ωk(t) ∈ r[t] (0 ≤ k ≤ n) and for any ξ > αv, ωv(ξ;x) has n non-zero real roots (counted with multiplicity) since av(ξ) is symmetric and detav(ξ) 6= 0. then, by proposition 3, there exists a positive real number δ such that ρv −δ > αv and for any ξ ∈ [ρv − δ,ρv + δ], ωv(ξ;x) has the same number of positive and hence negative real roots with ωv(ρv;x). on the other hand, since ρv = inf r, there exist real numbers ξ+ (ρv < ξ+ < ρv + δ) and ξ− (ρv − δ < ξ− < ρv) such that σ(av(ξ+)) 6= σ(av(ξ−)), which implies ωv(ξ+;x) and ωv(ξ−;x) have different number of positive and hence negative real roots. this is a contradiction and we have ρv = αv. 3.4 proof of theorem 3.2 let r = (r0, · · · ,rs) ∈ rs+1 be the vector as in theorem 3.2 and put n0 = { (n − s + 1)/2, n − s − 1 : even (n − s + 2)/2, n − s − 1 : odd. cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 89 when n − s ≥ 2, we inductively define the matrix ar(t)k = (a (r) ij (t)k)1≤i,j≤n (2 ≤ k ≤ n − s) as the matrix obtained from ar(t)k−1 by sweeping out the entries of the k-th row (k-th column) by the (k,l0 − k) entry −(n − s)rst ((l0 − k,k) entry −(n − s)rst). that is, we define ar(t)k = tsr(t)kar(t)k−1sr(t)k, where sr(t)k =    n∏ m=l0−k+1 rn ( l0 − k,m; − a (r) km(t)k−1 −(n − s)rst ) (2 ≤ k ≤ n0) rn ( l0 − k,k; − a (r) kk (t)k−1 −2(n − s)rst ) n∏ m=k+1 rn ( l0 − k,m; − a (r) km(t)k−1 −(n − s)rst ) (n0 < k ≤ n − s). then, if n − s ≥ 1, we can express the matrix ar(t)n−s as follows; ar(t)n−s =                  1 0 . . . 0 0 0 . . . −(n − s)rst ... ... ... 0 o 0 −(n − s)rst 0 0 o cr(t)n−s                  . note that a (r) km(t)k−1 and a (r) kk (t)k−1 appearing in sr(t)k are degree 1 monomials in t and hence the numbers −a (r) km(t)k−1/(−(n− s)rst), −a (r) kk (t)k−1/(−2(n − s)rst) appearing in sr(t)k are just real numbers. therefore, the entries of the s×s symmetric matrix cr(t)n−s = (c (r) ij (t)n−s)1≤i,j≤s (n − s ≥ 1) are of the form c (r) ij (t)n−s = b̄ (r) ij t 2 + λ̄ (r) ij t (λ̄ (r) ij ∈ r). (3.16) moreover, since the matrix dr(t)n−s =        1 0 . . . 0 0 0 . . . −(n − s)rst ... ... ... 0 0 −(n − s)rst 0 0        90 shuichi otake and tony shaska cubo 20, 2 (2018) is equivalent to the matrix d̄r(t)n−s =                                                                               1 0 −(n − s)rst −(n − s)rst 0 .. . 0 −(n − s)rst −(n − s)rst 0            (n − s : odd)              1 −(n − s)rst 0 −(n − s)rst −(n − s)rst 0 .. . 0 −(n − s)rst −(n − s)rst 0              (n − s : even) over r, we have σ(dr(ξ)n−s) = σ(d̄r(ξ)n−s) =    1 n − s : odd 0 n − s : even, rs > 0 2 n − s : even, rs < 0 (3.17) for any real number ξ > αr (≥ 0). here, note that since pr(0) = 0, we have αr ≥ 0. next, let φr(t;x), ψr(t;x) be characteristic polynomials of b̄r(t), cr(t)n−s, respectively. then, by equations (3.11) and (3.16), we have φr(t;x) = x s + h (r) s−1t 2xs−1 + · · · + h(r)1 t 2s−2x + h (r) 0 t 2s ( h (r) s−k = hs−k(r0, · · · ,rs) ∈ r (1 ≤ k ≤ s) ) , ψr(t;x) = x s + ( h (r) s−1t 2 + ψs−1(t) ) xs−1 + · · · + ( h (r) 1 t 2s−2 + ψ1(t) ) x + ( h (r) 0 t 2s + ψ0(t) ) (ψ0(t), · · · ,ψs−1(t) ∈ r[t],degψs−k(t) < 2k (1 ≤ k ≤ s)) . here, let us divide the proof into next two cases. (i) the case h (r) 0 h (r) 1 · · ·h (r) s−1 6= 0. in this case, we have ψr(t;x) = x s + h (r) s−1t 2 ( 1 + ψs−1(t) h (r) s−1t 2 ) xs−1 + · · · + h (r) 1 t 2s−2 ( 1 + ψ1(t) h (r) 1 t 2s−2 ) x + h (r) 0 t 2s ( 1 + ψ0(t) h (r) 0 t 2s ) cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 91 and 1+ψs−k(t) / h (r) s−kt 2k → 1 (t → ∞) for any k (1 ≤ k ≤ s). moreover, since h(r)0 h (r) 1 · · ·h (r) s−1 6= 0, we have h (r) 0 6= 0, which implies that for any non-zero real number ξ, φr(ξ;x) have s non-zero real roots (counted with multiplicity). thus, there exists a real number ρ0 (> αr) such that for any real number ξ > ρ0, ψr(ξ;x) have the same number of positive (hence also negative) real roots with φr(ξ;x) by proposition 3, which implies σ(cr(ξ)n−s) = σ(b̄r(ξ)) and hence σ(cr(ξ)n−s) = ngr = γ (ξ > ρ0) by lemma 3.2. then, by the equation (3.17), we have σ(ar(ξ)n−s) =    γ + 1 n − s : odd γ n − s : even, rs > 0 γ + 2 n − s : even, rs < 0 for any ξ > ρ0, which implies nfr(ξ;x) = σ(ar(ξ)) =    γ + 1 n − s : odd γ n − s : even, rs > 0 γ + 2 n − s : even, rs < 0 for any ξ > ρ0 since ar(ξ) and ar(ξ)n−s are equivalent over r. hence, by lemma 3.5, we have nfr(ξ;x) =    γ + 1 n − s : odd γ n − s : even, rs > 0 γ + 2 n − s : even, rs < 0 for any ξ > αr. (ii) general case. let ε0 be a positive real number and for any vector v ∈ rs+1, set α′v = max{|α| | α ∈ c,pv(α) = 0}. clearly, we have α′v ≥ αv for any v ∈ rs+1. here, let us put ρ′0 = α′r + ε0. then, by lemma 3.5, it is enough to prove the next claim. claim 1. for any real number ξ > ρ′0, we have nfr(ξ;x) =    γ + 1 n − s : odd γ n − s : even, rs > 0 γ + 2 n − s : even, rs < 0. proof. by the assumption that gr(x) is a separable polynomial of degree s and the fact that the non-real roots must occur in pair with its complex conjugate, there exists a real number δ0 such that for any vector v = (v0, · · · ,vs) ∈ rs+1 satisfying |r − v|0 = max0≤k≤s{|rk − vk|} < δ0, gv(x) is also a degree s separable polynomial satisfying ngv = ngr = γ by proposition 3. (s1) if a vector v ∈ rs+1 satisfies |r − v|0 < δ0, then gv(x) is also a degree s separable polynomial satisfying ngv = ngr = γ. 92 shuichi otake and tony shaska cubo 20, 2 (2018) next, we put p(t) = ∑ k≥0 xk(t0, · · · ,ts)tk = deta(t) (a(t) = a(t0, · · · ,ts,t)) and let us consider p(t) as a polynomial over e1 = r(t0, · · · ,ts) in t. then, since xk(t0, · · · ,ts) ∈ r[t0, · · · ,ts] for any k ≥ 0, there exists a real number δ1 > 0 such that for any vector v ∈ rs+1 satisfying |r − v|0 < δ1, we have |α ′ r − α ′ v| < ε0 by proposition 3; (s2) if a vector v ∈ rs+1 satisfies |r − v|0 < δ1, we have |α′r − α ′ v| < ε0. here, let ξ be any real number such that ξ > ρ′0 = α ′ r + ε0 and let ω(t0, · · · ,ts,ξ;x) = n∑ k=0 yk(t0, · · · ,ts)xk ∈ e1[x] be the characteristic polynomial of the bezoutian a(t0, · · · ,ts,ξ;x) = mn(f(n)(t0, · · · ,ts,ξ;x),f(n)(t0, · · · ,ts,ξ;x)′). here, f(n)(t0, · · · ,ts,ξ;x)′ is the derivative of f(n)(t0, · · · ,ts,ξ;x) = n∑ k=0 zk(t0, · · · ,ts)xk ∈ e1[x] with respect to x. then, since zk(t0, · · · ,ts) ∈ r[t0, · · · ,ts] (0 ≤ k ≤ n), we also have yk(t0, · · · ,ts) ∈ r[t0, · · · ,ts] (0 ≤ k ≤ n). moreover, since ξ > ρ′0 > αr, we have detar(ξ) = deta(r0, · · · ,rs,ξ) 6= 0. by these arguments, we can also deduce that there exists a positive real number δ2 such that for any vector v ∈ rs+1 satisfying |r − v|0 < δ2, the characteristic polynomial ωv(ξ;x) have the same number of positive and hence negative real roots with ωr(ξ;x) (counted with multiplicity), which implies nfr(ξ;x) = σ(ar(ξ)) = σ(av(ξ)) = nfv(ξ;x). (s3) if a vector v ∈ rs+1 satisfies |r − v|0 < δ2, we have nfr(ξ;x) = nfv(ξ;x). put δ = min{δ0,δ1,δ2} > 0. then, there exists a vector w = (w0, · · · ,ws) ∈ rs+1 such that (a) |r − w|0 < δ, (b) h (w) 0 h (w) 1 · · ·h (w) s−1 6= 0. here, we put h (w) s−k = hs−k(w0, · · · ,ws) for any k (1 ≤ k ≤ s). in fact, since hs−k(t0, · · · ,ts) is a non-zero polynomial for any k (1 ≤ k ≤ s) by lemma 3.3, the product ∏s k=1 hs−k(t0, · · · ,ts) is also non-zero, which implies that there exists a vector w ∈ rs+1 satisfying (a) and (b). let w ∈ rs+1 be the vector as above. then, since |r − w|0 < δ ≤ δ0, gw(x) is a degree s separable polynomial satisfying ngw = γ by (s1) and also, by (s2), we have αw ≤ α′w < α′r +ε0 = ρ′0 < ξ. thus, by (b) and the case (i), we have nfw(ξ;x) =    γ + 1 n − s : odd γ n − s : even, rs > 0 γ + 2 n − s : even, rs < 0, cubo 20, 2 (2018) some remarks on the non-real roots of polynomials 93 which, by (s3), implies nfr(ξ;x) =    γ + 1 n − s : odd γ n − s : even, rs > 0 γ + 2 n − s : even, rs < 0. since ξ is any real number such that ξ > ρ′0, this completes the proof of claim and hence the proof of theorem 3.2. proposition 5. let g(x) = ∑s i=0 aix i be a polynomial in r[x] such that ∆g 6= 0 and f(t,x) = xn + t · g(x) (3.18) if g(x) is totally complex, (n − s) is even, and as > 0 then f(β,x) is totally complex for all β > max{α |∆(f,x)(α) = 0}. proof. we have to show that f(β,x) has no real roots. since g(x) is totally complex we have that γ = 0. nf(β,x) = γ as β > max{α |∆(f,x)(α) = 0} and as > 0, so nf(β,x) = γ = 0. hence, f(β,x) is totally complex. let k := q(t,a0, . . . ,as) be the field of transcendental degree s + 1 and g(x) = ∑s i=0 aix i. then we have the following. corolary 2. let k := q(t,a0, . . . ,as) be the field of transcendental degree s+1, g(x) = ∑s i=0 aix i and f(t,x) = xn + t · g(x) for any value of (λ0, . . . ,λs) ∈ zs+1, if g(λ0, . . . ,λs,x) ∈ z[x] is irreducible and satisfies the conditions of the eisenstein criteria, then f(x) is irreducible, over q. we also note: remark 3.4. it can be verified computationally by maple that if n ≤ 9 and 1 ≤ s < n then the galois group gal k(f,x) is isomorphic to sn. remark 3.5. polynomials in eq. (3.18) for s = 1 and t = 1 has been treated by y. zarhin in [18] while studying mori trinomials. it is shown there that the galois group of f(x) over q is isomorphic to sn; see [18, cor. 3.5] for details. in general, if we let k := q(t,a0, . . . ,as) be the field of transcendental degree s + 1, for 1 ≤ s < n, then we expect that gal k(f) ∼= sn for all n ≥ 1. if true, this would generalize zarhin’s result to a more general class of polynomials. references [1] oz ben-shimol, on galois groups of prime degree polynomials with complex roots, algebra discrete math. 2 (2009), 99–107. mr2589076 http://www.ams.org/mathscinet-getitem?mr=2589076 94 shuichi otake and tony shaska cubo 20, 2 (2018) [2] l. beshaj, r. hidalgo, s. kruk, a. malmendier, s. quispe, and t. shaska, rational points in the moduli space of genus two, higher genus curves in mathematical physics and arithmetic geometry, 2018, pp. 83–115. mr3782461 [3] lubjana beshaj, reduction theory of binary forms, advances on superelliptic curves and their applications, 2015, pp. 84–116. mr3525574 [4] a. bialostocki and t. shaska, galois groups of prime degree polynomials with nonreal roots, computational aspects of algebraic curves, 2005, pp. 243–255. mr2182043 [5] artur elezi and tony shaska, reduction of binary forms via the hyperbolic center of mass (2017), available at 1705.02618. [6] paul a. fuhrmann, a polynomial approach to linear algebra, second, universitext, springer, new york, 2012. mr2894784 [7] ruben hidalgo and tony shaska, on the field of moduli of superelliptic curves, higher genus curves in mathematical physics and arithmetic geometry, 2018, pp. 47–62. mr3782459 [8] david joyner and tony shaska, self-inversive polynomials, curves, and codes, higher genus curves in mathematical physics and arithmetic geometry, 2018, pp. 189–208. mr3782467 [9] a. malmendier and t. shaska, the satake sextic in f-theory, j. geom. phys. 120 (2017), 290–305. mr3712162 [10] andreas malmendier and tony shaska, a universal genus-two curve from siegel modular forms, sigma symmetry integrability geom. methods appl. 13 (2017), paper no. 089, 17. mr3731039 [11] morris marden, geometry of polynomials, second edition. mathematical surveys, no. 3, american mathematical society, providence, r.i., 1966. mr0225972 [12] thomas mattman and john mckay, computation of galois groups over function fields, math. comp. 66 (1997), no. 218, 823–831. mr1401943 [13] shuichi otake, counting the number of distinct real roots of certain polynomials by bezoutian and the galois groups over the rational number field, linear multilinear algebra 61 (2013), no. 4, 429–441. mr3005628 [14] , a bezoutian approach to orthogonal decompositions of trace forms or integral trace forms of some classical polynomials, linear algebra appl. 471 (2015), 291–319. mr3314338 [15] shuichi otake and tony shaska, bezoutians and the discriminant of a certain quadrinomials, algebraic curves and their applications, 2019, pp. 55–72. [16] q. i. rahman and g. schmeisser, analytic theory of polynomials, london mathematical society monographs. new series, vol. 26, the clarendon press, oxford university press, oxford, 2002. mr1954841 [17] gene ward smith, some polynomials over q(t) and their galois groups, math. comp. 69 (2000), no. 230, 775–796. mr1659835 [18] yuri g. zarhin, galois groups of mori trinomials and hyperelliptic curves with big monodromy, eur. j. math. 2 (2016), no. 1, 360–381. mr3454107 http://www.ams.org/mathscinet-getitem?mr=3782461 http://www.ams.org/mathscinet-getitem?mr=3525574 http://www.ams.org/mathscinet-getitem?mr=2182043 1705.02618 http://www.ams.org/mathscinet-getitem?mr=2894784 http://www.ams.org/mathscinet-getitem?mr=3782459 http://www.ams.org/mathscinet-getitem?mr=3782467 http://www.ams.org/mathscinet-getitem?mr=3712162 http://www.ams.org/mathscinet-getitem?mr=3731039 http://www.ams.org/mathscinet-getitem?mr=0225972 http://www.ams.org/mathscinet-getitem?mr=1401943 http://www.ams.org/mathscinet-getitem?mr=3005628 http://www.ams.org/mathscinet-getitem?mr=3314338 http://www.ams.org/mathscinet-getitem?mr=1954841 http://www.ams.org/mathscinet-getitem?mr=1659835 http://www.ams.org/mathscinet-getitem?mr=3454107 introduction preliminaries on the number of real roots of polynomials the bezoutian of f(t; x) some results for the bezoutian of fr(t ; x) nonvanishingness of some coefficients proof of theorem ?? cubo a mathematical journal vol.20, no¯ 3, (37–47). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300037 yamabe solitons with potential vector field as torse forming yadab chandra mandal and shyamal kumar hui department of mathematics, the university of burdwan, burdwan, 713104, west bengal, india myadab436@gmail.com, skhui@math.buruniv.ac.in abstract the riemannian manifolds whose metric is yamabe soliton with potential vector field as torse forming admitting riemannian connection, semisymmetric metric connection and projective semisymmetric connection have been studied. an example is constructed to verify the theorem concerning riemannian connection. resumen se estudian las variedades riemannianas cuya métrica es un solitón de yamabe con vector de potencial que forma un virol (superficie desarrollable) con respecto a conexiones riemanniana, semisimétrica métrica y proyectiva semisimétrica. se construye un ejemplo expĺıcito para verificar las hipótesis del teorema en el caso de la conexión riemanniana. keywords and phrases: yamabe soliton, torse forming vector field, torqued vector field, semisymmetric metric connection, projective semisymmetric connection. 2010 ams mathematics subject classification: 53c21, 53c25. http://dx.doi.org/10.4067/s0719-06462018000300037 38 yadab chandra mandal and shyamal kumar hui cubo 20, 2 (2018) 1 introduction the curvature tensor, ricci tensor and scalar curvature of a riemannian manifold m of dimension n equipped with riemannian metric g with respect to levi-civita connection ∇ are denoted by r, s and r respectively. hamilton ([5], [6]) introduced the notion of yamabe flow, which is an evolution equation for metrics on m as follows: ∂ ∂t g = −rg. when n = 2, the yamabe flow is equivalent to the ricci flow. however, for n > 2, they do not agree. a yamabe soliton on m is, a special solution of the yamabe flow, a triplet (g,v,σ) such that 1 2 £vg = (r − σ)g, (1.1) where £v is the lie derivative in the direction of v ∈ χ(m) and σ is a constant. the nature of such soliton depends on the behaviour of σ. the yamabe soliton is said to be shrinking, steady and expanding according as σ < 0, = 0 and > 0 respectively. if σ ∈ c∞(m) then the metric satisfying (1.1) is called almost yamabe soliton [1]. for n = 2 such soliton is equivalent with ricci soliton, but for n > 2, they do not. yamabe solitons have been studied by several authors such as [5], [6], [9], [10] and references there in. as a generalization of concircular, concurrent and parallel vector field, yano [14] introduced the torse-forming vector field. a nowhere vanishing vector field τ is said to be a torse-forming on m if ∇xτ = fx + γ(x)τ, (1.2) where f ∈ c∞(m) and γ is an 1-form. if the 1-form γ in (1.2) vanishes identically, then τ is concircular [13]. concircular vector fields also known as geodesis vector fields since integral curves of such vector fields are geodesis. recently, chen [2] studied ricci solitons with concircular vector field. if f = 1 and γ = 0 then τ is concurrent [16]. the vector field τ is recurrent if it satisfies (1.2) with f = 0. also if f = γ = 0, the vector field τ in (1.2) is parallel vector field. as a consequence of torse forming vector field, recently chen [3] introduced a new vector field, called torqued vector field. if the vector field τ satisfies (1.2) with γ(τ) = 0 then τ is called torqued vector field. here, f is known as the torqued function and the 1-form is the torqued form of τ. in this paper we have studied yamabe solitons, whose potential vector field is torse forming, on riemannian manifolds with respect to riemannian connection (rc), semisymmetric metric connection (ssmc) and projective semisymmetric connection (pssc) and prove the following: cubo 20, 2 (2018) yamabe solitons with potential vector field as torse forming 39 theorem 1.1. let (g,τ,σ) be a yamabe soliton on m with respect to rc ∇. then the following holds: τ condition of existence conditions of shrinking, steady and expanding torse-forming r − f − 1 n γ(τ)=constant r − f − 1 n γ(τ) s 0 concircular r − f= constant r − f s 0 concurrent r=constant r s 1 recurrent r − 1 n γ(τ)= constant r − 1 n γ(τ) s 0 parallel r = constant r s 0 torqued r − f =constant r − f s 0 theorem 1.2. let (g,τ,σ) be a yamabe soliton on m with respect to ssmc ∇̄. then the following holds: τ condition of existence conditions of shrinking, steady and expanding torse-forming r − f − 2(n − 1)a r − f − 2(n − 1)a − 1 n {(n − 1)π(τ) + γ(τ)}=constant − 1 n {(n − 1)π(τ) + γ(τ)} s 0 concircular r − f − (n − 1){2a + 1 n π(τ)}=constant r − f − (n − 1){2a + 1 n π(τ)} s 0 concurrent r − 1 − (n − 1){2a + 1 n π(τ)} = constant r − 1 − (n − 1){2a + 1 n π(τ)}s 0 recurrent r − 2(n − 1)a r − 2(n − 1)a − 1 n {(n − 1)π(τ) + γ(τ)}= constant − 1 n {(n − 1)π(τ) + γ(τ)} s 0 parallel r − (n − 1){2a + 1 n π(τ)} = constant r − (n − 1){2a + 1 n π(τ)} s 0 torqued r − f − (n − 1){2a + 1 n π(τ)} =constant r − f − (n − 1){2a + 1 n π(τ)}s 0 40 yadab chandra mandal and shyamal kumar hui cubo 20, 2 (2018) theorem 1.3. let (g,τ,σ) be a yamabe soliton on m with respect to pssc ∇̃. then the following holds: τ condition of existence conditions of shrinking, steady and expanding torse-forming r − f + tr · β − (n − 1)tr · α r − f + tr · β − (n − 1)tr · α − 1 n {(n − 1)π(τ) + γ(τ)}=constant − 1 n {(n − 1)π(τ) + γ(τ)} s 0 concircular r − f + tr · β − (n − 1){tr · α r − f + tr · β − (n − 1){tr · α + 1 n π(τ)} =constant + 1 n π(τ)} s 0 concurrent r + tr · β − (n − 1){tr · α r − 1 + tr · β + 1 n π(τ)} =constant −(n − 1){tr · α + 1 n π(τ)} s 0 recurrent r + tr · β − (n − 1)tr · α r + tr · β − (n − 1)tr · α − 1 n {(n − 1)π(τ) + γ(τ)}= constant − 1 n {(n − 1)π(τ) + γ(τ)} s 0 parallel r + tr · β − (n − 1){tr · α + 1 n π(τ)} r + tr · β − (n − 1){tr · α+ =constant 1 n π(τ)} s 0 torqued r − f + tr · β − (n − 1){tr · α+ r − f + tr · β − (n − 1){tr · α+ 1 n π(τ)}=constant 1 n π(τ)} s 0 section 2 consists with preliminaries. the proof of our theorems are given in section 3. in section 4, we have constructed an example to verify theorem 1.1. remark. the conditions of existence of theorem 1.1, theorem 1.2 and theorem 1.3 are only necessary. finding sufficient conditions for the existence of solitons is a much deeper problem and this is not addressed in the present manuscript. 2 preliminaries the relation between the semisymmetric metric connection (ssmc) ∇̄ and ∇ of m is given by ([4], [7], [15]) ∇̄xy = ∇xy + π(y)x − g(x,y)ρ, (2.1) where π(x) = g(x,ρ) for all x ∈ χ(m). if r̄ (resp. s̄ and r̄) are the curvature tensor (respectively ricci tensor and scalar curvature) of m with respect to ssmc, then [4] r̄(x,y)z = r(x,y)z − p(y,z)x + p(x,z)y − g(y,z)lx + g(x,z)ly, (2.2) s̄(y,z) = s(y,z) − (n − 2)p(y,z) − ag(y,z), (2.3) r̄ = r − 2(n − 1)a, (2.4) cubo 20, 2 (2018) yamabe solitons with potential vector field as torse forming 41 where p is a tensor field of type (0,2) given by p(x,y) = g(lx,y) = (∇xπ)(y) − π(x)π(y) + 1 2 π(ρ)g(x,y) and a = tr.p for any x,y ∈ χ(m). the relation between projective semisymmetric connection ∇̃ and ∇ is [17] ∇̃xy = ∇xy + ψ(y)x + ψ(x)y + φ(y)x − φ(x)y, (2.5) where the 1-forms φ and ψ are given by φ(x) = 1 2 π(x) and ψ(x) = n−1 2(n+1) π(x). if r̃ (resp. r̃ and s̃) are the curvature tensor, ricci tensor and scalar curvature of m with respect to ∇̃, then ([12], [17]) r̃(x,y)z = r(x,y)z + β(x,y)z + α(x,z)y − α(y,z)x, (2.6) s̃(y,z) = s(y,z) + β(y,z) − (n − 1)α(y,z), (2.7) r̃ = r + tr.β − (n − 1)tr.α, (2.8) for all x,y,z ∈ χ(m), where β(x,y) = 1 2 [(∇yπ)(x) − (∇xπ)(y)], α(x,y) = n − 1 2(n + 1) (∇xπ)(y) + 1 2 (∇yπ)(x) − n2 (n + 1)2 π(x)π(y). 3 proof of the theorems proof of the theorem 1.1. let (g,τ,σ) be a yamabe soliton on m. then from (1.1) we get 1 2 (£τg)(x,y) = (r − σ)g(x,y). (3.1) now from (1.2) we have (£τg)(x,y) = g(∇xτ,y) + g(x,∇yτ) (3.2) = 2fg(x,y) + γ(x)g(τ,y) + γ(y)g(τ,x) for all x,y ∈ χ(m). in view of (3.2), (3.1) yields (r − σ − f)g(x,y) = 1 2 {γ(x)g(τ,y) + γ(y)g(τ,x)}. (3.3) taking contraction of (3.3) over x and y we get n(r − σ − f) = γ(τ). (3.4) this leads to the following: 42 yadab chandra mandal and shyamal kumar hui cubo 20, 2 (2018) proposition 3.1. let (g,τ,σ) be a yamabe soliton on m with respect to rc ∇. if τ is torseforming then this soliton is shrinking, steady and expanding according as r − f − 1 n γ(τ) s 0, provided as r − f − 1 n γ(τ) is constant. from proposition 3.1, we obtain theorem 1.1. proof of the theorem 1.2. we now consider (g,τ,σ) is a yamabe soliton on m with respect to semisymmetric metric connection. then we have 1 2 (£̄τg)(x,y) = (r̄ − σ)g(x,y), (3.5) where £̄τ is the lie derivative along τ of ∇̄. from (2.1) we get (£̄τg)(x,y) = g(∇̄xτ,y) + g(x,∇̄yτ) (3.6) = g(∇xτ + π(τ)x − g(x,τ)ρ,y) + g(x,∇yτ + π(τ)y − g(y,τ)ρ) = (£τg)(x,y) + 2π(τ)g(x,y) − [g(x,τ)π(y) + g(y,τ)π(x)]. using (2.4) and (3.6) in (3.5), we get 1 2 (£τg)(x,y) = (r − σ)g(x,y) − {2(n − 1)a + π(τ)}g(x,y) (3.7) + 1 2 [g(x,τ)π(y) + g(y,τ)π(x)]. in view of (3.2), (3.7) yields {r − σ − f − 2(n − 1)a − π(τ)}g(x,y) (3.8) + 1 2 [{π(y) − γ(y)}g(τ,x) + {π(x) − γ(x)}g(τ,y)] = 0. contracting (3.8) over x and y, we get n{r − σ − f − 2(n − 1)a} − (n − 1)π(τ) − γ(τ) = 0. (3.9) this leads to the following: proposition 3.2. let (g,τ,σ) be a yamabe soliton on m with respect to ssmc ∇̄. if τ is torseforming then this soliton is shrinking, steady and expanding according as r − f − 2(n − 1)a − 1 n {(n − 1)π(τ) + γ(τ)} s 0, cubo 20, 2 (2018) yamabe solitons with potential vector field as torse forming 43 provided r − f − 2(n − 1)a − 1 n {(n − 1)π(τ) + γ(τ)} is constant. from proposition 3.2, we obtain theorem 1.2. proof of the theorem 1.3. we now consider (g,τ,σ) is a yamabe soliton on m with respect to ∇̃. then we have 1 2 (£̃τg)(x,y) = (̃r − σ)g(x,y), (3.10) where £̃τ is the lie derivative along τ of ∇̃. from (2.5) we get (£̃τg)(x,y) = g(∇̃xτ,y) + g(x,∇̃yτ) (3.11) = (£τ)(x,y) + 1 n + 1 {2nπ(τ)g(x,y) − π(x)g(τ,y) − π(y)g(x,τ)}. using (2.8) and (3.11) in (3.10), we get 1 2 (£τg)(x,y) = (r − σ)g(x,y) (3.12) + [tr · β − (n − 1)tr · α]g(x,y) − 1 2(n + 1) {2nπ(τ)g(x,y) − π(x)g(y,τ) − π(y)g(x,τ)}. in view of (3.2), (3.12) yields {r − σ − f + tr · β − (n − 1)tr · α − n n + 1 π(τ)}g(x,y) (3.13) + 1 2 [{ π(y) n + 1 − γ(y)}g(τ,x) + { π(x) n + 1 − γ(x)}g(τ,y)] = 0. contracting (3.13) over x and y, we get n{r − σ − f + tr · β − (n − 1)tr · α} − (n − 1)π(τ) − γ(τ) = 0. (3.14) this leads to the following: proposition 3.3. let (g,τ,σ) be a yamabe soliton on m with respect to ∇̃. if τ is torse-forming then this soliton is shrinking, steady and expanding according as r − f + tr · β − (n − 1)tr · α − 1 n {(n − 1)π(τ) + γ(τ)} s 0, provided r − f + tr · β − (n − 1)tr · α − 1 n {(n − 1)π(τ) + γ(τ)} is constant. from proposition 3.3, we obtain theorem 1.3. 44 yadab chandra mandal and shyamal kumar hui cubo 20, 2 (2018) 4 example here we construct an example to verify theorem 1.1. example: let us consider a 3-dimensional manifold m = {(x,y,z) ∈ r3 : z 6= 0}. let {e1,e2,e3} be a linearly independent global frame on m given by e1 = z 2 ∂ ∂x , e2 = z 2 ∂ ∂y , e3 = ∂ ∂z . let g be the riemannian metric defined by g(ei,ej) = { 1, i=j 0, i 6= j . these vector field and such metric is used in ([8], [11]). using koszul formula, we have [11] ∇e1e1 = 2 z e3, ∇e1e2 = 0, ∇e1e3 = − 2 z e1, ∇e2e1 = 0, ∇e2e2 = 2 z e3, ∇e2e3 = − 2 z e2, ∇e3e1 = 0, ∇e3e2 = 0, ∇e3e3 = 0. the scalar curvature of this manifold is also computed in [11] and it is r = −32 z2 . since {e1,e2,e3} forms a basis, any vector field x,y,u ∈ χ(m) can be written as x = a1e1 + b1e2 + c1e3, y = a2e1 + b2e2 + c2e3, u = a3e1 + b3e2 + c3e3, where ai,bi,ci ∈ r + for i = 1,2,3 such that a1a2 + b1b2 c1 + c1( b2 b1 − a2 a1 − 1) 6= 0. if we choose the 1-form γ by γ(w) = g(w, 2 z e3) for any w ∈ χ(m) and considering f ∈ c ∞(m) as f = 2 z { a1a2 + b1b2 c1 + c1 ( b2 b1 − a2 a1 − 1 )} . then the relation ∇xy = fx + γ(x)y (4.1) holds. consequently y is a torse-forming vector field. now from (4.1) we get (£yg)(x,u) = g(∇xy,u) + g(x,∇uy) (4.2) = 2fg(x,u) + γ(x)g(y,u) + γ(u)g(y,x). also we can calculate      g(x,u) = a1a3 + b1b3 + c1c3, g(y,u) = a2a3 + b2b3 + c2c3, g(y,x) = a1a2 + b1b2 + c1c2, (4.3) γ(x) = 2c1 z ,γ(y) = 2c2 z ,γ(u) = 2c3 z . (4.4) cubo 20, 2 (2018) yamabe solitons with potential vector field as torse forming 45 in view of (4.3) and (4.4), (4.2) yields 1 2 (£yg)(x,u) = 1 z [{ 2(a1a2 + b1b2) c1 + 2c1 ( b2 b1 − a2 a1 − 1 )} (a1a3 + b1b3 + c1c3) + c1(a2a3 + b2b3 + c2c3) + c3(a1a2 + b1b2 + c1c2) ] . (4.5) also (r − σ)g(x,u) = (− 32 z2 − σ)(a1a3 + b1b3 + c1c3). (4.6) assuming that a1a3 + b1b3 + c1c3 6= 0 and 3c1(a2a3 + b2b3 + c2c3) + 3c3(a1a2 + b1b2 + c1c2) − 2c2(a1a3 + b1b3 + c1c3) = 0, we get (g,y,σ) is an yamabe soliton, i.e 1 2 (£yg)(x,u) = (r − σ)g(x,u) holds, provided σ = − 32 z2 − 2 z { (a1a2 + b1b2) c1 + c1 ( b2 b1 − a2 a1 − 1 )} − c1(a2a3 + b2b3 + c2c3) + c3(a1a2 + b1b2 + c1c2) (a1a3 + b1b3 + c1c3)z = r − f − 1 3 γ(y) = constant. thus the condition of existence of yamabe soliton (g,y,σ) on a 3-dimensional riemannian manifold with potential vector field y as torse forming in theorem 1.1 is verified. acknowledgement: the authors are thankful to the referee for his/her valuable suggestions towards to the improvement of the paper. the first author (y. c. mandal) gratefully acknowledges the university grants commission, government of india, for the award of junior research fellowship. 46 yadab chandra mandal and shyamal kumar hui cubo 20, 2 (2018) references [1] barbosa, e. and ribeiro, e., on conformal solutions of the yamabe flow, arch. math., 101 (2013), 79-89. [2] chen, b. y., some results on concircular vector fields and their applications to ricci solitons, bull. korean math. soc., 52 (2015), 1535-1547. [3] chen, b. y., classification of torqued vector fields and its applications to ricci solitons, kragujevac j. of math., 41(2) (2017), 239-250. [4] friedmann, a. and schouten, j. a., über die geometric derhalbsymmetrischen übertragung, math. zeitschr., 21 (1924), 211-223. [5] hamilton, r. s., the ricci flow on surfaces, mathematics and general relativity, contemp. math., 71 (1988), 237-262. [6] hamilton, r. s., lectures on geometric flows, unpublished manuscript, 1989. [7] hayden, h. a., subspaces of space with torsion, proc. london math. soc. 34 (1932), 27-50. [8] hui, s. k. and chakraborty, d., ricci almost solitons on concircular ricci pseudosymmetric β-kenmotsu manifolds, hacettepe j. of math. and stat., 47(3) (2018), 579-587. [9] hui, s. k. and mandal, y. c., yamabe solitons on kenmotsu manifolds, communications in korean math. soc., (2018). [10] mandal, y. c. and hui, s. k., on the existence of yamabe gradient solitons, int. j. math. eng. manag. sci., 3(4) (2018), 491-497. [11] shaikh, a. a. and hui, s. k., on extended generalized φ-recurrent β-kenmotsu manifolds, publ. de l’ inst. math., 89(103) (2011), 77-88. [12] shaikh, a. a. and hui, s. k., on pseudo cyclic ricci symmetric manifolds admitting semisymmetric metric connection, scientia series a: math. sci., 20 (2010), 73-80. [13] yano, k., concircular geometry i, concircular transformations, proc. imp. acad. tokyo, 16 (1940), 195-200. [14] yano, k., on torse forming direction in a riemannian space, proc. imp. acad. tokyo, 20 (1944), 340-345. [15] yano, k., on semi-symmetric metric connection, rev. roum. math. pures et appl. (bucharest), xv, 9, (1970), 1579-1586. [16] yano, k. and chen, b. y., on the concurrent vector fields of immersed manifolds, kodai math. sem. rep., 23 (1971), 343-350. cubo 20, 2 (2018) yamabe solitons with potential vector field as torse forming 47 [17] zhao, p., some properties of projective semisymmetric connections, int. math. forum, 3(7) (2008), 341-347. introduction preliminaries proof of the theorems example cubo a mathematical journal vol.21, no¯ 02, (15–40). august 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000200015 zk-magic labeling of path union of graphs p. jeyanthi 1 k. jeya daisy 2 and andrea semaničová-feňovč́ıková 3 1research centre, department of mathematics, govindammal aditanar college for women, tiruchendur 628215, tamilnadu, india jeyajeyanthi@rediffmail.com 2department of mathematics, holy cross college, nagercoil, tamilnadu, india jeyadaisy@yahoo.com 3department of applied mathematics and informatics, technical university, košice, slovak republic andrea.fenovcikova@tuke.sk abstract for any non-trivial abelian group a under addition a graph g is said to be a-magic if there exists a labeling f : e(g) → a − {0} such that, the vertex labeling f+ defined as f+(v) = ∑ f(uv) taken over all edges uv incident at v is a constant. an a-magic graph g is said to be zk-magic graph if the group a is zk, the group of integers modulo k and these graphs are referred as k-magic graphs. in this paper we prove that the graphs such as path union of cycle, generalized petersen graph, shell, wheel, closed helm, double wheel, flower, cylinder, total graph of a path, lotus inside a circle and n-pan graph are zk-magic graphs. http://dx.doi.org/10.4067/s0719-06462019000200015 16 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) resumen para cualquier grupo abeliano no-trivial a bajo adición, un grafo g se dice a-mágico si existe un etiquetado f : e(g) → a − {0} tal que el etiquetado de un vértice f+ definido como f+(v) = ∑ f(uv), tomado sobre todos los ejes uv incidentes en v, es constante. un grafo a-mágico g se dice zk-mágico si el grupo a es zk, el grupo de enteros módulo k y estos se llaman grafos k-mágicos. en este paper demostramos que los grafos tales como la unión por caminos de ciclos, grafos de petersen generalizados, concha, rueda, casco cerrado, rueda doble, flor, cilindro, el grafo total de un camino, lotos dentro de un ćırculo y n-sartenes son todos grafos zk-mágicos. keywords and phrases: a-magic labeling, zk-magic labeling, zk-magic graph, generalized petersen graph, shell, wheel, closed helm, double wheel, flower, cylinder, total graph of a path, lotus inside a circle, n-pan graph. 2010 ams mathematics subject classification: 05c78. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 17 1 introduction graph labeling is currently an emerging area in the research of graph theory. a graph labeling is an assignment of integers to vertices or edges or both subject to certain conditions. a detailed survey was done by gallian in [1]. if the labels of edges are distinct positive integers and for each vertex v the sum of the labels of all edges incident with v is the same for every vertex v in the given graph then the labeling is called a magic labeling. sedláček [10] introduced the concept of a-magic graphs. a graph with real-valued edge labeling such that distinct edges have distinct non-negative labels and the sum of the labels of the edges incident to a particular vertex is same for all vertices. low and lee [9] examined the a-magic property of the resulting graph obtained from the product of two a-magic graphs. shiu, lam and sun [12] proved that the product and composition of a-magic graphs were also a-magic. for any non-trivial abelian group a under addition a graph g is said to be a-magic if there exists a labeling f : e(g) → a−{0} such that, the vertex labeling f+ defined as f+(v) = ∑ f(uv) taken over all edges uv incident at v is a constant. an a-magic graph g is said to be zk-magic graph if the group a is zk, the group of integers modulo k. these zk-magic graphs are referred to as k-magic graphs. shiu and low [13] determined all positive integers k for which fans and wheels have a zk-magic labeling with a magic constant 0. kavitha and thirusangu [8] obtained a zk-magic labeling of two cycles with a common vertex. motivated by the concept of a-magic graph in [10] and the results in [9, 12, 13] jeyanthi and jeya daisy [2, 3, 4, 5, 6, 7] proved that some standard graphs admit zk-magic labeling. we use the following definitions in the subsequent section. definition 1.1. let g1, g2, . . . , gn, n ≥ 2, be copies of a graph g. let vi ∈ v (gi), i = 1, 2, . . . , n, be the vertex corresponding to the vertex v ∈ v (g) in the ith copy of gi. we denoted by p(n.g v) the graph obtained by adding the edge vivi+1, to gi and gi+1, 1 ≤ i ≤ n − 1, and we call p(n.g v) the path union of n copies of the graph g. note, that up to isomorphism, we obtain |v (g)| graphs p(n.gv). this operation was defined in [11]. definition 1.2. a generalized petersen graph p(n, m), n ≥ 3, 1 ≤ m < n 2 is a 3-regular graph with the vertex set {ui, vi : i = 1, 2, . . . , n} and the edge set {uivi, uiui+1, vivi+m : i = 1, 2, . . . , n}, where the indices are taken over modulo n. definition 1.3. a shell graph sn, n ≥ 4, is obtained by taking n − 3 concurrent chords in a cycle cn. the vertex at which all the chords are concurrent is called an apex. definition 1.4. a wheel graph wn, n ≥ 3, is obtained by joining the vertices of a cycle cn to an extra vertex called the centre. the vertices of degree three are called rim vertices. 18 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) definition 1.5. a helm graph hn, n ≥ 3, is obtained from a wheel wn by adjoining a pendant edge at each vertex of the wheel except the center. definition 1.6. a closed helm graph chn, n ≥ 3, is obtained from a helm hn by joining each pendent vertex to form a cycle. definition 1.7. a double wheel graph dwn, n ≥ 3, is obtained by joining the vertices of two cycles cn to an extra vertex called the hub. definition 1.8. a flower graph fln, n ≥ 3, is obtained from a helm hn by joining each pendent vertex to the central vertex of the helm. definition 1.9. a cartesian product of a cycle cn, n ≥ 3, and a path on two vertices is called a cylinder graph cn�p2. definition 1.10. a total graph t (g) is a graph with the vertex set v (g) ∪ e(g) in which two vertices are adjacent whenever they are either adjacent or incident in g. definition 1.11. a lotus inside a circle lcn, n ≥ 3, is a graph obtained from a wheel wn by subdividing every edge forming the outer cycle and joining these new vertices to form a cycle. definition 1.12. an n-pan graph, n ≥ 3, is obtained by attaching a pendent edge to a vertex of a cycle cn. 2 zk-magic labeling of path union of graphs in this section we prove that the graphs such as path union of cycle, generalized petersen graph, shell, wheel, closed helm, double wheel, flower, cylinder, total graph of a path, lotus inside a circle and n-pan graph are zk-magic graphs. let v be a vertex of a cycle cr, r ≥ 3. according to the symmetry all p(n.c v r ) are isomorphic. thus we use the notation p(n.cr). theorem 2.1. let r ≥ 3 and n ≥ 2 be integers. the path union of a cycle p(n.cr) is zk-magic for k ≥ 3 when r is odd. proof. let the vertex set and the edge set of p(n.cr) be v (p(n.cr)) = {v j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.cr)) = {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j 1v j+1 1 : 1 ≤ j ≤ n − 1}, where the index i is taken over modulo r. let a, k be positive integers, k > 2a. thus k ≥ 3. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 19 for r is odd, we define an edge labeling f : e(p(n.cr)) → zk − {0} as follows: f(v1i v 1 i+1) = f(v n i v n i+1) = { k − a, for i = 1, 3, . . . , r, a, for i = 2, 4, . . . , r − 1, f(v j i v j i+1) = { k − 2a, for i = 1, 3, . . . , r, j = 2, 3, . . . , n − 1, 2a, for i = 2, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(v j 1v j+1 1 ) = 2a, for j = 1, 2, . . . , n − 1. then the induced vertex labeling f+ : v (p(n.cr)) → zk is f +(v) ≡ 0 (mod k) for every vertex v in v (p(n.cr)). an example of a z10-magic labeling of p(4.c5) is shown in figure 1. b b b b 2 2 8 88 b b b b b b b 4 4 6 66 b b b b b b 4 4 6 66 b b b b b b 2 2 8 8 8 bb bbb 4 44 b figure 1: a z10-magic labeling of p(4.c5). up to isomorphism there are two graphs obtained by attaching n copies of a generalized petersen graph p(r, m), r ≥ 3, 1 ≤ m ≤ r−1 2 to a path pn to get a graph p(n.p(r, m) v). we deal with the case when v is a vertex in the outer polygon of p(r, m). theorem 2.2. let r ≥ 3, m ≤ r−1 2 and n ≥ 2 be positive integers. the path union of a generalized petersen graph p(n.p(r, m)v), where v is a vertex in the outer polygon of p(r, m), is zk-magic for k ≥ 5 when r is odd. proof. let the vertex set and the edge set of p(n.p(r, m)v) be v (p(n.p(r, m)v)) = {u j i, v j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.p(r, m)v)) = {u j iv j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j iu j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j 1u j+1 1 : 1 ≤ j ≤ n − 1} ∪ {v j i v j i+m : 1 ≤ i ≤ r, 1 ≤ j ≤ n}, where the index i is taken over modulo r. let a, k be positive integers, k > 4a. thus k ≥ 5. 20 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) define an edge labeling f : e(p(n.p(r, m)v)) → zk − {0} as follows: f(v j i v j i+m) = a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(u j iv j i ) = k − 2a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(u1i u 1 i+1) = { k − a, for i = 1, 3, . . . , r, 3a, for i = 2, 4, . . . , r − 1, f(u j iu j i+1) = a, for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(vni v n i+m) = { k − a, for n is odd, a, for n is even, f(uni v n i ) = { 2a, for n is odd, k − 2a, for n is even, f(uni u n i+1) =          a, for i = 1, 3, . . . , r and n is odd, k − 3a, for i = 2, 4, . . . , r − 1 and n is odd, k − a, for i = 1, 3, . . . , r and n is even, 3a, for i = 2, 4, . . . , r − 1 and n is even, f(u j 1u j+1 1 ) = { 4a, for j = 1, 3, . . . and j ≤ n − 1, k − 4a, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.p(r, m)v)) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.p(r, m)v)). thus v (p(n.p(r, m)v)) is a zk-magic graph. an example of a z15-magic labeling of p(5.p(5, 2) v) is shown in figure 2. b b b b b b b b b b 2 2 22 2 11 13 11 11 11 11 13 13 6 6 b b b b b b b b b b 2 2 22 2 11 11 11 11 11 b b b b b b b b b b 2 8 2 4 13 9 13 b b b b b b b b b b 2 2 22 2 11 11 11 11 11 b b b b b b b b b b 2 2 22 2 11 11 11 11 11 b b b b b 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 13 13 13 4 4 4 4 2 9 8 77 figure 2: a z15-magic labeling of p(5.p(5, 2) v). theorem 2.3. let r ≥ 4 and n ≥ 2 be positive integers. the path union of a shell graph p(n.svr ), where v ∈ v (sr) is the vertex of degree r − 1, is zk-magic for k ≥ 2r − 3 when r is odd and for k ≥ r − 1 when k is even. proof. let the vertex set and the edge set of p(n.svr ) be v (p(n.s v r )) = {v j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.svr )) = {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j 1v j i : 3 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j 1v j+1 1 : 1 ≤ j ≤ n − 1} with the index i taken over modulo r. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 21 we consider the following two cases according to the parity of r. case (i): when r is odd. let a, k be positive integers, k > 2(r − 2)a. thus k ≥ 2r − 3. define an edge labeling f : e(p(n.svr )) → zk − {0} as follows: f(v11v 1 i ) = 2a, for i = 3, 4, . . . , r − 1, f(v11v 1 2) = f(v 1 rv 1 1) = a, f(v1i v 1 i+1) = k − a, for i = 2, 3, . . . , r − 1, f(v j 1v j+1 1 ) = { k − 2a(r − 2), for j = 1, 3, . . . and j ≤ n − 1, 2a(r − 2), for j = 2, 4, . . . and j ≤ n − 1, f(v j 1v j i ) = a, for i = 3, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(v j i v j i+1) = { (r−3)a 2 , for i = 2, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, k − (r−1)a 2 , for i = 3, 5, . . . , r − 2, j = 2, 3, . . . , n − 1, f(v j 1v j 2) = f(v j rv j 1) = k − (r−3)a 2 , for j = 2, 3, . . . , n − 1, f(vn1 v n i ) = { k − 2a, for i = 3, 4, . . . , r − 1 and n is odd, 2a, for i = 3, 4, . . . , r − 1 and n is even, f(vn1 v n 2 ) = f(v n r v n 1 ) = { k − a, for n is odd, a, for n is even, f(vni v n i+1) = { a, for i = 2, 3, . . . , r − 1 and n is odd, k − a, for i = 2, 3, . . . , r − 1 and n is even. then the induced vertex labeling f+ : v (p(n.svr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.svr )). case (ii): when r is even. let a, k be positive integers, k > (r − 2)a. thus k ≥ r − 1. 22 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) define an edge labeling f : e(p(n.svr )) → zk − {0} in the following way. f(v11v 1 i ) = a, for i = 3, 4, . . . , r − 1, f(v11v 1 2) = k − a, f(v1rv 1 1) = 2a, f(v1i v 1 i+1) = { a, for i = 2, 4, . . . , r − 2, k − 2a, for i = 3, 5, . . . , r − 1, f(v j 1v j+1 1 ) = { k − a(r − 2), for j = 1, 3, . . . and j ≤ n − 1, a(r − 2), for j = 2, 4, . . . and j ≤ n − 1, f(v j 1v j i ) = k 2 , for i = 3, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(v j i v j i+1) =      3k 4 , for i = 2, 3, . . . , r − 1, j = 2, 3, . . . , n − 1 and k ≡ 0 (mod 4), 3k+2 4 , for i = 2, 4, . . . , r − 2, j = 2, 3, . . . , n − 1 and k ≡ 2 (mod 4), 3k−2 4 , for i = 3, 5, . . . , r − 1, j = 2, 3, . . . , n − 1 and k ≡ 2 (mod 4), f(v j 1v j 2) = { k 4 , for j = 2, 3, . . . , n − 1 and k ≡ 0 (mod 4), k−2 4 , for j = 2, 3, . . . , n − 1 and k ≡ 2 (mod 4), f(vjrv j 1) = { k 4 , for j = 2, 3, . . . , n − 1 and k ≡ 0 (mod 4), k+2 4 , for j = 2, 3, . . . , n − 1 and k ≡ 2 (mod 4), f(vn1 v n i ) = { k − a, for i = 3, 4, . . . , r − 1 and n is odd, a, for i = 3, 4, . . . , r − 1 and n is even, f(vn1 v n 2 ) = { a, for n is odd, k − a, for n is even, f(vnr v n 1 ) = { k − 2a, for n is odd, 2a, for n is even, f(vni v n i+1) =          k − a, for i = 2, 4, . . . , r − 2 and n is odd 2a, for i = 3, 5, . . . , r − 1 and n is odd, a, for i = 2, 4, . . . , r − 2 and n is even, k − 2a, for i = 3, 5, . . . , r − 1 and n is even. then the induced vertex labeling f+ : v (p(n.svr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.svr )). thus p(n.s v r ) is a zk-magic graph for r is even. an example of a z11-magic labeling of p(3.s v 7 ) is shown in figure 3. according to the symmetry of wheels there exist two non isomorphic graphs p(n.w vr ). we deal with the case when v is a rim vertex, that is a vertex of degree three in wr. theorem 2.4. let r ≥ 4 and n ≥ 2 be integers. the path union of a wheel graph p(n.w vr ), where v ∈ v (wr) is a vertex of degree 3, is zk-magic for k ≥ r when r is odd and for k ≥ 2r − 1 when r cubo 21, 2 (2019) zk-magic labeling of path union of graphs 23 b b b b b b 1 1 1 1 1 1010 b b b b b b 1 1 1 1 2 2 2 9 8 9 8 b b b b b b 2 2 2 2 1 1 10 10 10 10 10 b b b 9 9 9 9 1 10 figure 3: a z11-magic labeling of p(3.s v 7 ). is even. proof. let the vertex set and the edge set of p(n.w vr ) be v (p(n.w v r )) = {wj, v j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.w vr )) = {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {wjv j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j 1u j+1 1 : 1 ≤ j ≤ n − 1}, where the index i is taken over modulo r. we consider the following two cases according to the parity of r. case (i): when r is odd. let a, k be positive integers, k > (r − 1)a. this implies k ≥ r. define an edge labeling f : e(p(n.w vr )) → zk − {0} as follows: f(wjv j i ) = a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(wjv j 1) = k − (r − 1)a, for j = 1, 2, . . . , n − 1, f(v1i v 1 i+1) = { a, for i = 1, 3, . . . , r, k − 2a, for i = 2, 4, . . . , r − 1, f(v j i v j i+1) = { (r−1)a 2 , for i = 1, 3, . . . , r, j = 2, 3, . . . , n − 1, k − (r+1)a 2 , for i = 2, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(wnv n 1 ) = { (r − 1)a, for n is odd, k − (r − 1)a, for n is even, f(wnv n i ) = { k − a, for i = 2, 3, . . . , r and n is odd, a, for i = 2, 3, . . . , r and n is even, f(vni v n i+1) =          k − a, for i = 1, 3, . . . , r and n is odd, 2a, for i = 2, 4, . . . , r − 1 and n is odd, a, for i = 1, 3, . . . , r and n is even, k − 2a, for i = 2, 4, . . . , r − 1 and n is even, f(v j 1v j+1 1 ) = { a(r − 3), for j = 1, 3, . . . and j ≤ n − 1, k − a(r − 3), for j = 2, 4, . . . and j ≤ n − 1. this means that for the induced vertex labeling f+ : v (p(n.w vr )) → zk is f +(u) ≡ 0 (mod k) for 24 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) all u ∈ v (p(n.w vr )). case (ii): when r is even. let a, k be positive integers, k > 2(r − 1)a. define an edge labeling f : e(p(n.w vr )) → zk − {0} in the following way. f(w1v j 1) = f(wnv n 1 ) = k − (r − 1)a, f(w1v 1 i ) = f(wnv n i ) = a, for i = 2, 3, . . . , r, f(v1i v 1 i+1) = f(v n i v n i+1) = { a, for i = 1, 3, . . . , r − 1, k − 2a, for i = 2, 4, . . . , r, f(wjv j 1) = k − 2(r − 1)a, for j = 2, 3, . . . , n − 1, f(wjv j i ) = 2a, for i = 2, 3, . . . , r, j = 2, 3, . . . , n − 1, f(v j i v j i+1) = k − a, for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(v j 1v j+1 1 ) = ra, for j = 1, 2, . . . , n − 1. then the induced vertex labeling f+ : v (p(n.w vr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.w vr )). hence f + is constant that means p(n.w vr ) admits a zk-magic labeling. an example of a z12-magic labeling of p(3.w v 6 ) is shown in figure 4. b b b b b b b 2 2 2 2 2 2 11 11 11 11 1111 b b b b b b b 1 7 1 1 1 1 1 1 1 10 10 10 b b b b b b b 1 7 1 1 1 1 1 1 1 10 10 10 b b b 66 figure 4: a z12-magic labeling of p(3.w v 6 ). in the next theorem we deal with the path union of a closed helm graph p(n.chvr ), where v is a vertex of degree three in chr. theorem 2.5. let r ≥ 4 and n ≥ 2 be integers. the path union of a closed helm graph p(n.chvr ), where v is a vertex of degree 3 in chr, is zk-magic for k ≥ r when r is odd and for even k ≥ r when r is even. proof. let the vertex set and the edge set of p(n.chvr ) be v (p(n.ch v r )) = {wj, v j i , u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.chvr )) = {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j iu j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {wjv j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j 1u j+1 1 : 1 ≤ j ≤ n − 1}, where the index i is taken over modulo r. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 25 case (i): when r is odd. let a, k be positive integers, k > (r − 1)a. thus k ≥ r. define an edge labeling f : e(p(n.chvr )) → zk − {0} as follows: f(wjv j 1) = k − (r − 1)a, for j = 1, 2, . . . , n − 1, f(wjv j i ) = a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(v j i v j i+1) = { (r − 1)a, for i = 1, 3, . . . , r, j = 1, 2, . . . , n − 1, k − (r − 1)a, for i = 2, 4, . . . , r − 1, j = 1, 2, . . . , n − 1, f(u1i u 1 i+1) = { (r − 1)a, for i = 1, 3, . . . , r, k − (r − 2)a, for i = 2, 4, . . . , r − 1, f(v j i u j i ) = k − a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(v j 1u j 1) = k − (r − 1)a, for j = 1, 2, . . . , n − 1, f(u j i u j i+1) = { (r−1)a 2 , for i = 1, 3, . . . , r, j = 2, 3, . . . , n − 1, k − (r−3)a 2 , for i = 2, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(wnv n 1 ) = { (r − 1)a, for n is odd, k − (r − 1)a, for n is even, f(wnv n i ) =, { k − a, for i = 2, 3, . . . , r and n is odd, a, for i = 2, 3, . . . , r and n is even, f(vn1 u n 1 ) = { (r − 1), for n is odd, k − (r − 1)a, for n is even, f(vni u n i ) = { a, for i = 2, 3, . . . , r and n is odd, k − a, for i = 2, 3, . . . , r and n is even, f(vni v n i+1) =          k − (r − 1)a, for i = 1, 3, . . . , r and n is odd, (r − 1)a, for i = 2, 4, . . . , r − 1 and n is odd, (r − 1)a, for i = 1, 3, . . . , r and n is even, k − (r − 1)a, for i = 2, 4, . . . , r − 1 and n is even, f(uni u n i+1) =          k − (r − 1)a, for i = 1, 3, . . . , r and n is odd, (r − 2)a, for i = 2, 4, . . . , r − 1 and n is odd, (r − 1)a, for i = 1, 3, . . . , r and n is even, k − (r − 2)a, for i = 2, 4, . . . , r − 1 and n is even, f(u j 1u j+1 1 ) = { k − (r − 1)a, for j = 1, 3, . . . and j ≤ n − 1, (r − 1)a, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.chvr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.chvr )). 26 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) case (ii): when r is even. let a be a positive integer and k > (r − 2)a be an even integer. thus k ≥ r. define an edge labeling f : e(p(n.chvr )) → zk − {0} such that f(w1v 1 1) = k − (r − 1)a, f(w1v 1 i ) = a, for i = 2, 3, . . . , r, f(v1i v 1 i+1) = { (r − 1)a, for i = 1, 3, . . . , r − 1, k − (r − 1)a, for i = 2, 4, . . . , r, f(u1i u 1 i+1) = { (r − 1)a, for i = 1, 3, . . . , r − 1, k − (r − 2)a, for i = 2, 4, . . . , r − 1, f(v11u 1 1) = (r − 1)a, f(v1i u 1 i ) = k − a, for i = 2, 3, . . . , r, f(wjv j i ) = f(v j i u j i ) = f(v j i v j i+1) = k 2 , for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(u j i u j i+1) =      k 4 , for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1 and k ≡ 0 (mod 4), k−2 4 , for i = 1, 3, . . . , r − 1, j = 2, 3, . . . , n − 1 and k ≡ 2 (mod 4), k+2 4 , for i = 2, 4, . . . , r, j = 2, 3, . . . , n − 1 and k ≡ 2 (mod 4), f(wnv n 1 ) = { (r − 1)a, for n is odd, k − (r − 1)a, for n is even, f(wnv n i ) = { k − a, for i = 2, 3, . . . , r and n is odd, a, for i = 2, 3, . . . , r and n is even, f(vn1 u n 1 ) = { k − (r − 1)a, for n is odd, (r − 1)a, for n is even, f(vni u n i ) = { a, for i = 2, 3, . . . , r and n is odd, k − a, for i = 2, 3, . . . , r and n is even, f(vni v n i+1) =          k − (r − 1)a, for i = 1, 3, . . . , r − 1 and n is odd, (r − 1)a, for i = 2, 4, . . . , r and n is odd, (r − 1)a, for i = 1, 3, . . . , r − 1 and n is even, k − (r − 1)a, for i = 2, 4, . . . , r and n is even, f(uni u n i+1) =          k − (r − 1)a, for i = 1, 3, . . . , r − 1 and n is odd, (r − 2)a, for i = 2, 4, . . . , r and n is odd, (r − 1)a, for i = 1, 3, . . . , r − 1 and n is even, k − (r − 2)a, for i = 2, 4, . . . , r and n is even, f(u j 1u j+1 1 ) = { k − ra, for j = 1, 3, . . . and j ≤ n − 1, ra, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.chvr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ cubo 21, 2 (2019) zk-magic labeling of path union of graphs 27 v (p(n.chvr )). hence f + is constant equal to 0 (mod k). therefore p(n.chvr ) is a zk-magic graph. an example of a z6-magic labeling of p(3.ch v 6 ) is shown in figure 5. b b b b b b bb b b b b b b b b b b b bb b b b b b b b b b b b bb b b b 1 11 7 1 1 1 1 5 5 5 5 5 5 11 11 11 7 7 7 8 8 8 b b 11 5 b b 1 7 5 8 11 6 3 1 1 1 1 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6 5 5 5 5 5 8 8 11 11 11 11 7 7 7 5 3 3 3 3 3 6 6 figure 5: a z12-magic labeling of p(3.ch v 6 ). theorem 2.6. let r ≥ 3 and n ≥ 2 be integers. the path union of a double wheel graph p(n.dw vr ), where v ∈ v (dwr) is a vertex of degree 3, is zk-magic for k ≥ 5 when r is odd. proof. let the vertex set and the edge set of c(n.dw vr ) be v (p(n.dw v r )) = {vj, v j i , u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.dw vr )) = {vjv j i , vju j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪{u j iu j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪{u j 1u j+1 1 : 1 ≤ j ≤ n− 1} with index i taken over modulo r. let a, k be positive integers, k > 4a. thus k ≥ 5. 28 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) for r is odd we define an edge labeling f : e(p(n.dw vr )) → zk − {0} as follows: f(vjv j i ) = 2a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(vju j i ) = k − 2a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(v j i v j i+1) = k − a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(u1i u 1 i+1) = { k − a, for i = 1, 3, . . . , r, 3a, for i = 2, 4, . . . , r − 1, f(u j i u j i+1) = a, for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(vnv n i ) = { k − 2a, for i = 1, 2, . . . , r and n is odd, 2a, for i = 1, 2, . . . , r and n is even, f(vnu n i ) = { 2a, for i = 1, 2, . . . , r and n is odd, k − 2a, for i = 1, 2, . . . , r and n is even, f(vni v n i+1) = { a, for i = 1, 2, . . . , r − 1 and n is odd, k − a, for i = 1, 2, . . . , r − 1 and n is even, f(vnr v n 1 ) = { a, for n is odd, k − a, for n is even, f(uni u n i+1) =          a, for i = 1, 3, . . . , r and n is odd, k − 3a, for i = 2, 4, . . . , r − 1 and n is odd, k − a, for i = 1, 3, . . . , r and n is even, 3a, for i = 2, 4, . . . , r − 1 and n is even, f(u j 1u j+1 1 ) = { 4a, for j = 1, 3, . . . and j ≤ n − 1, k − 4a, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.dw vr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.dw vr )). an example of a z7-magic labeling of p(3.dw v 7 ) is shown in figure 6. theorem 2.7. let r ≥ 3 and n ≥ 2 be positive integers. the path union of a flower graph p(n.flvr), where v ∈ v (flr) is the vertex of degree 4, is zk-magic for k ≥ 5 when r is odd and for k ≥ 3 when k is even. proof. let the vertex set and the edge set of p(n.flvr) be v (p(n.fl v r )) = {wj, v j i , u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.flvr)) = {wjv j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {wju j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j 1v j+1 1 : 1 ≤ j ≤ n − 1}, with index i taken over modulo r. case (i): when r is odd. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 29 b 2 3 5 6 2 2 2 2 2 2 6 6 6 6 6 6 5 5 5 5 5 5 6 6 6 6 3 3 b 4 b 2 5 6 2 2 2 2 2 2 6 6 6 6 6 6 5 5 5 5 5 5 1 11 1 1 1 1 b 2 52 2 2 2 2 2 1 5 5 5 5 5 5 b 1 1 1 1 1 1 1 1 1 1 4 4 4 3b b b bb b b b b b b b b b b b bb b b b b b b b b b b b b bb b b b b b b b b figure 6: a z7-magic labeling of p(3.dw v 7 ). let a, k be positive integers, k > 4a. this means k ≥ 5. define an edge labeling f : e(p(n.flvr)) → zk − {0} as follows: f(wjv j i ) = f(v j i u j i ) = a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(u j i wj) = k − a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(v1i v 1 i+1) = { a, for i = 1, 3, . . . , r, k − 3a, for i = 2, 4, . . . , r − 1, f(v j i v j i+1) = k − a, for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(wnv n i ) = f(v n i u n i ) = { k − a, for i = 1, 2, . . . , r and n is odd, a, for i = 1, 2, . . . , r and n is even, f(uni wn) = { a, for i = 1, 2, . . . , r and n is odd, k − a, for i = 1, 2, . . . , r and n is even, f(vni v n i+1) =          k − a, for i = 1, 3, . . . , r and n is odd, 3a, for i = 2, 4, . . . , r − 1 and n is odd, a, for i = 1, 3, . . . , r and n is even, k − 3a, for i = 2, 4, . . . , r − 1 and n is even, f(v j 1v j+1 1 ) = { k − 4a, for j = 1, 3, . . . and j ≤ n − 1, 4a, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.flvr)) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.flvr)). case (ii): when r is even. let a, k be positive integers, k > 2a. thus k ≥ 3. 30 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) define an edge labeling f : e(p(n.flvr)) → zk − {0} as follows: f(w1v 1 1) = f(v 1 1u 1 1) = 2a, f(u11w1) = k − 2a, f(v j i v j i+1) = k − a, for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(wjv j i ) = f(v j i u j i ) = a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(wju j i) = k − a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(wnv n 1 ) = f(v n 1 u n 1 ) = { k − 2a, for n is odd, 2a, for n is even, f(wnu n 1 ) = { 2a, for n is odd, k − 2a, for n is even, f(wnv n i ) = f(v n i u n i ) = { k − a, for i = 2, 3, . . . , r and n is odd, a, for i = 2, 3, . . . , r and n is even, f(wnu n i ) = { a, for i = 2, 3, . . . , r and n is odd, k − a, for i = 2, 3, . . . , r and n is even, f(vni v n i+1) = { a, for i = 1, 2, . . . , r and n is odd, k − a, for i = 1, 2, . . . , r and n is even, f(v j 1v j+1 1 ) = { k − 2a, for j = 1, 3, . . . and j ≤ n − 1, 2a, for j = 2, 4, . . . and j ≤ n − 1. the induced vertex labeling f+ : v (p(n.flvr)) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.flvr)). an example of a z10-magic labeling of p(4.fl v 3) is shown in figure 7. b b 2 2 2 2 8 8 b b b b 2 8 8 8 8 2 b b b b 2 2 2 2 8 8 b b b b 2 8 8 8 2 b b 8 b b 2 2 2 2 4 8 8 2 b b b 2 8 2 b 2 b b b b 2 2 2 2 4 8 82 b b b b 2 8 2 b b 2 2 8 2 figure 7: a z10-magic labeling of p(4.fl v 3). let v be a vertex of a cylinder graph cr�p2, r ≥ 3. according to the symmetry all p(n.(cr�p2) v) are isomorphic. thus we use the notation p(n.(cr�p2)). theorem 2.8. let r ≥ 3, n ≥ 2 be integers. the path union of a cylinder graph p(n.(cr�p2)) is zk-magic for k ≥ 5 when r is odd. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 31 proof. let the vertex set and the edge set of p(n.(cr�p2)) be v (p(n.(cr�p2))) = {v j i , u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.(cr�p2))) = {u j iv j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j iu j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j 1u j+1 1 : 1 ≤ j ≤ n − 1}, with index i taken over modulo r. let a, k be positive integers, k > 4a. thus k ≥ 5. for r odd we define an edge labeling f : e(p(n.(cr�p2))) → zk − {0} as follows: f(v j i u j i) = k − 2a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(v j i v j i+1) = a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(u1i u 1 i+1) = { k − a, for i = 1, 3, . . . , r, 3a, for i = 2, 4, . . . , r − 1, f(u j iu j i+1) = a, for i = 1, 2, . . . , r, j = 2, 3, . . . , n − 1, f(vni v n i+1) = { k − a, for n is odd, a, for n is even, f(vni u n i ) = { 2a, for n is odd, k − 2a, for n is even, f(uni u n i+1) =          a, for i = 1, 3, . . . , r and n is odd, k − 3a, for i = 2, 4, . . . , r − 1 and n is odd, k − a, for i = 1, 3, . . . , r and n is even, 3a, for i = 2, 4, . . . , r − 1 and n is even, f(u j 1u j+1 1 ) = { 4a, for j = 1, 3, . . . , j ≤ n − 1, k − 4a, for j = 2, 4, . . . , j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.(cr�p2))) → zk is f +(v) ≡ 0 ≡ k for all v ∈ v (p(n.(cr�p2))). hence f + is constant and is equal to 0 ≡ k. an example of a z9-magic labeling of p(3.(c7�p2)) is shown in figure 8. 1 2 3 4 8 7 2 2 2 2 2 2 2 5 5 5 5 5 5 5 7 77 6 6 7 2 2 2 2 2 2 2 5 5 5 5 5 5 5 b b b 2 2 2 2 2 2 2 22 7 7 7 7 7 7 4 4 44 4 4 3 3 2 b b bb b b b b b bb b b b b bb b b b b b b b b b b b b bb b b b b b b b 6 b figure 8: a z9-magic labeling of p(3.(c7�p2) v). 32 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) theorem 2.9. let r ≥ 5 and n ≥ 2 be positive integers. the path union of a total graph of a path p(n.t (pr) v), where v ∈ v (t (pr)) is a vertex of degree two, is zk-magic for k ≥ 3. proof. let the vertex set and the edge set of p(n.t (pr) v) be v (p(n.t (pr) v)) = {u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i : 1 ≤ i ≤ r − 1, 1 ≤ j ≤ n} and e(p(n.t (pr) v)) = {u j iu j i+1 : 1 ≤ i ≤ r − 1, 1 ≤ j ≤ n} ∪ {v j i v j i+1 : 1 ≤ i ≤ r − 2, 1 ≤ j ≤ n} ∪ {u j i+1v j i : 1 ≤ i ≤ r − 1, 1 ≤ j ≤ n} ∪ {u j iv j i : 1 ≤ i ≤ r − 1, 1 ≤ j ≤ n} ∪ {u j 1u j+1 1 : 1 ≤ j ≤ n − 1}. we consider the following two cases according to the parity of r. case (i): when r is odd. let a, k be positive integers, k > 2a. thus k ≥ 3. define an edge labeling f : e(p(n.t (pr) v)) → zk − {0} as follows: f(u1i u 1 i+1) = { a, for i = 1, 3, . . . , r, 2a, for i = 2, 4, . . . , r − 3, f(u1r−1u 1 r) = f(v 1 1v 1 2) = a, f(v1i v 1 i+1) = { 2a, for i = 3, 5, . . . , r, a, for i = 2, 4, . . . , r − 1, f(u11v 1 1) = a, f(u12v 1 2) = k − a, f(u1i v 1 i ) = k − 2a, for i = 3, 4, . . . , r − 2, f(u1r−1v 1 r−1) = k − a, f(v11u 1 2) = k − 2a, f(v1i u 1 i+1) = k − a, for i = 2, 3, . . . , r − 1, f(u j 1v j 1) = f(u j 2v j 1) = a, for j = 2, 3, . . . , n − 1, f(u j 1u j 2) = f(u j r−1u j r) = k − a, for j = 2, 3, . . . , n − 1, f(u j iu j i+1) = k − 2a, for i = 2, 3, . . . , r − 2, j = 2, 3, . . . , n − 1, f(v j i v j i+1) = k − 2a, for i = 1, 2, . . . , r − 2, j = 2, 3, . . . , n − 1, f(u j iv j i ) = f(u j i+1v j i ) = 2a, for i = 2, 3, . . . , r − 2, j = 2, 3, . . . , n − 1, f(ujrv j r−1) = f(u j r−1v j r−1) = a, for j = 2, 3, . . . , n − 1, cubo 21, 2 (2019) zk-magic labeling of path union of graphs 33 f(unr−1u n r ) = { k − a, for n is odd, a, for n is even, f(uni u n i+1) =          k − a, for i = 1, 3, . . . , r and n is odd, k − 2a, for i = 2, 4, . . . , r − 3 and n is odd, a, for i = 1, 3, . . . , r and n is even, 2a, for i = 2, 4, . . . , r − 3 and n is odd, f(vn1 v n 2 ) = { k − a, for n is odd, a, for n is even, f(vni v n i+1) =          k − 2a, for i = 3, 5, . . . , r and n is odd, k − a, for i = 2, 4, . . . , r − 1 and n is odd, 2a, for i = 3, 5, . . . , r and n is even, a, for i = 2, 4, . . . , r − 1 and n is even, f(un1 v n 1 ) = { k − a, for n is odd, a, for n is even, f(un2 v n 2 ) = { a, for n is odd, k − a, for n is even, f(uni v n i ) = { 2a, for i = 3, 4, . . . , r − 2 and n is odd, k − 2a, for i = 3, 4, . . . , r − 2 and n is even, f(unr−1v n r−1) = { a, for n is odd, k − a, for n is even, f(vn1 u n 2 ) = { 2a, for n is odd, k − 2a, for n is even, f(vni u n i+1) = { a, for i = 2, 3, . . . , r − 1 and n is odd, k − a, for i = 2, 3, . . . , r − 1 and n is even, f(u j 1u j+1 1 ) = { k − 2a, for j = 1, 3, . . . and j ≤ n − 1, 2a, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.t (pr) v)) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.t (pr) v)). 34 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) case (ii): when r is even. let a, k be positive integers, k > 2a. thus k ≥ 3. define an edge labeling f : e(p(n.t (pr) v)) → zk − {0} as follows: f(u1i u 1 i+1) = f(v 1 i v 1 i+1) = { k − a, for i = 1, 3, . . . , r − 1, k − 2a, for i = 2, 4, . . . , r, f(v11u 1 1) = k − a, f(v1i u 1 i ) = a, for i = 2, 3, . . . , r − 1, f(v1i u 1 i+1) = 2a, for i = 1, 2, . . . , r − 2, f(v1r−1u 1 r) = a, f(u j 1v j 1) = f(u j 2v j 1) = a, for j = 2, 3, . . . , n − 1, f(u j 1u j 2) = f(u j r−1u j r) = k − a, for j = 2, 3, . . . , n − 1, f(u j i u j i+1) = k − 2a, for i = 2, 3, . . . , r − 2, j = 2, 3, . . . , n − 1, f(v j i v j i+1) = k − 2a, for i = 1, 2, . . . , r − 2, j = 2, 3, . . . , n − 1, f(u j i v j i ) = f(u j i+1v j i ) = 2a, for i = 2, 3, . . . , r − 2, j = 2, 3, . . . , n − 1, f(ujrv j r−1) = f(u j r−1v j r−1) = a, for j = 2, 3, . . . , n − 1, f(uni u n i+1) = f(v n i v n i+1) =          a, for i = 1, 3, . . . , r − 1 and n is odd, 2a, for i = 2, 4, . . . , r and n is odd, k − a, for i = 1, 3, . . . , r − 1 and n is even, k − 2a, for i = 2, 4, . . . , r and n is even, f(un1 v n 1 ) = { a, for n is odd, k − a, for n is even, f(uni v n i ) = { k − a, for i = 2, 3, . . . , r − 1 and n is odd, a, for i = 2, 3, . . . , r − 1 and n is even, f(vni u n i+1) = { k − 2a, for i = 1, 2, . . . , r − 2 and n is odd, 2a, for i = 1, 2, . . . , r − 2 and n is even, f(vnr−1u n r ) = { k − a, for n is odd, a, for n is even, f(u j 1u j+1 1 ) = { 2a, for j = 1, 3, . . . and j ≤ n − 1, k − 2a, for j = 2, 4, . . . and j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.t (pr) v)) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.t (pr) v)). hence p(n.t (pr) v) is a zk-magic graph. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 35 an example of a z5-magic labeling of p(5.t (p6) v) is shown in figure 9. b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 1 2 3 4 3 3 3 3 4 4 4 4 1 1 1 2 2 2 2 1 11 1 4 4 3 3 3 3 4 4 3 3 2 2 2 2 2 2 1 2 2 2 2 2 2 2 4 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 4 4 3 3 3 3 33 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 11 1 1 1 1 3 3 3 33 figure 9: a z5-magic labeling of p(5.t (p6) v). theorem 2.10. let r ≥ 3 and n ≥ 2 be integers. let v is a vertex of degree 2 in lcr. the path union of a lotus inside a circle graph p(n.lcvr ), is zk-magic for k ≥ r. proof. let the vertex set and the edge set of p(n.lcvr ) be v (p(n.lc v r )) = {wj, v j i , u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.lcvr )) = {wjv j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i u j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j i u j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j iu j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {u j 1u j+1 1 : 1 ≤ j ≤ n − 1}, where the index i is taken over modulo r. we consider the following two cases according to the parity of r. case (i): when r is odd. let a, k be positive integers, k > (r − 1)a. thus k ≥ r. 36 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) define an edge labeling f : e(p(n.lcvr )) → zk − {0} in the following way. f(wjv j 1) = k − (r − 1)a, for j = 1, 2, . . . , n − 1, f(wjv j i ) = a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(v j 1u j 1) = (r − 2)a, for j = 1, 2, . . . , n − 1, f(v j i u j i ) = k − 2a, for i = 2, 3, . . . , r, j = 1, 2, . . . , n − 1, f(v j i u j i+1) = a, for i = 1, 2, . . . , r, j = 1, 2, . . . , n − 1, f(u1i u 1 i+1) = { k − a, for i = 1, 3, . . . , r, 2a, for i = 2, 4, . . . , r − 1, f(u j iu j i+1) = { k − (r−1)a 2 , for i = 1, 3, . . . , r, j = 2, 3, . . . , n − 1, (r+1)a 2 , for i = 2, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(u j 1u j+1 1 ) = { k − (r − 3)a, for j = 1, 3, . . . , j ≤ n − 1, (r − 3)a, for j = 2, 4, . . . , j ≤ n − 1, f(wnv n 1 ) = { (r − 1)a, for n is odd, k − (r − 1)a, for n is even, f(wnv n i ) = { k − a, for i = 2, 3, . . . , r and n is odd, a, for i = 2, 3, . . . , r and n is even, f(vn1 u n 1 ) = { k − (r − 2)a, for n is odd, (r − 2)a, for n is even, f(vni u n i ) = { 2a, for i = 2, 3, . . . , r and n is odd, k − 2a, for i = 2, 3, . . . , r and n is even, f(vni u n i+1) = { k − a, for i = 1, 2, . . . , r and n is odd, a, for i = 1, 2, . . . , r and n is even, f(uni u n i+1) =          a, for i = 1, 3, . . . , r and n is odd, k − 2a, for i = 2, 4, . . . , r − 1 and n is odd, k − a, for i = 1, 3, . . . , r and n is even, 2a, for i = 2, 4, . . . , r − 1 and n is even. then the induced vertex labeling f+ : v (p(n.lcvr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.lcvr )). case (ii): when r is even. let a, k be positive integers, k > (r − 1)a. thus k ≥ r. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 37 define an edge labeling f : e(p(n.lcr)) → zk − {0} as follows: f(w1v 1 1) = k − (r − 1)a, f(w1v 1 i ) = a, for i = 2, 3, . . . , r, f(v11u 1 1) = (r − 2)a, f(v1i u 1 i ) = k − 2, for i = 2, 3, . . . , r, f(v1i u 1 i+1) = a, for i = 1, 2, . . . , r, f(u1i u 1 i+1) = { k − a, for i = 1, 3, . . . , r − 1, 2a, for i = 2, 4, . . . , r, f(wjv j i ) = { a, for i = 1, 3, . . . , r − 1, j = 2, 3, . . . , n − 1, k − a, for i = 2, 4, . . . , r, j = 2, 3, . . . , n − 1, f(v j i u j i) = { k − 2a, for i = 1, 3, . . . , r − 1, j = 2, 3, . . . , n − 1, k − a, for i = 2, 4, . . . , r, j = 2, 3, . . . , n − 1, f(v j i u j i+1) = { a, for i = 1, 3, . . . , r − 1, j = 1, 2, . . . , n − 1, 2a, for i = 2, 4, . . . , r, j = 1, 2, . . . , n − 1, f(u j i u j i+1) = { k − a, for i = 1, 3, . . . , r − 1, j = 2, 3, . . . , n − 1, a, for i = 2, 4, . . . , r, j = 2, 3, . . . , n − 1, f(wnv n 1 ) = { (r − 1)a, for n is odd, k − (r − 1)a, for n is even, f(wnv n i ) = { k − a, for i = 2, 3, . . . , r and n is odd, a, for i = 2, 3, . . . , r and n is even, f(vn1 u n 1 ) = { k − (r − 2)a, for n is odd, (r − 2)a, for n is even, f(vni u n i ) = { 2a, for i = 2, 3, . . . , r and n is odd, k − 2a, for i = 2, 3, . . . , r and n is even, f(vni u n i+1) = { k − a, for i = 1, 2, . . . , r and n is odd, a, for i = 1, 2, . . . , r and n is even, f(uni u n i+1) =          a, for i = 1, 3, . . . , r − 1 and n is odd, k − 2a, for i = 2, 4, . . . , r and n is odd, k − a, for i = 1, 3, . . . , r − 1 and n is even, 2a, for i = 2, 4, . . . , r and n is even, f(u j 1u j+1 1 ) = { k − ra, for j = 1, 3, . . . , j ≤ n − 1, ra, for j = 2, 4, . . . , j ≤ n − 1. then the induced vertex labeling f+ : v (p(n.lcvr )) → zk is f +(u) ≡ 0 (mod k) for all u ∈ 38 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) v (p(n.lcvr )). hence f + is constant and is equal to ≡ 0 (mod k). an example of a z10-magic labeling of p(3.lc v 6 ) is shown in figure 10. b b b bb b b b b b b b b b b b bb b b b b b b b b b b b bb b b b b b b b b b b b 1 2 4 5 6 8 9 1 1 1 1 1 1 1 1 1 8 8 8 8 1 1 1 9 9 9 2 2 4 1 1 1 9 9 2 8 9 2 2 8 8 9 1 9 9 9 9 9 5 6 9 9 9 9 9 9 2 2 2 2 2 1 8 1 1 8 8 9 9 9 1 1 1 9 figure 10: a z10-magic labeling of p(3.lc v 6 ). in the last theorem we deal with the path union of an r-pan graph p(n.(r-pan)v), where v is a vertex of degree two in an r-pan graph. theorem 2.11. let r ≥ 3, n ≥ 2 be integers. the path union of an r-pan graph p(n.(r-pan)v), where v is a vertex of degree two in an r-pan graph, is zk-magic for k ≥ 5 when r is odd. proof. let v be a vertex of degree two in an r-pan graph. let the vertex set and the edge set of p(n.(r-pan)v) be v (p(n.(r-pan)v)) = {wj, v j i : 1 ≤ i ≤ r, 1 ≤ j ≤ n} and e(p(n.(r-pan) v)) = {v j i v j i+1 : 1 ≤ i ≤ r, 1 ≤ j ≤ n} ∪ {v j 1wj : 1 ≤ j ≤ n} ∪ {w j 1w j+1 1 : 1 ≤ j ≤ n − 1}, where the index i is taken over modulo r. let a, k be positive integers, k > 2a. thus k ≥ 5. for r odd we define an edge labeling f : e(p(n.(r-pan)v)) → zk − {0} as follows: f(v1i v 1 i+1) = f(v n i v n i+1) = { k − a, for i = 1, 3, . . . , r, a, for i = 2, 4, . . . , r − 1, f(v j i v j i+1) = { k − 2a, for i = 1, 3, . . . , r, j = 2, 3, . . . , n − 1, 2a, for i = 2, 4, . . . , r − 1, j = 2, 3, . . . , n − 1, f(v11w1) = f(v n 1 wn) = 2a, f(v j 1wj) = 4a, for j = 2, 3, . . . , n − 1, f(w j 1w j+1 1 ) = k − 2a, for j = 1, 2, . . . , n − 1. then the induced vertex labeling f+ : v (p(n.(r-pan)v)) → zk is f +(u) ≡ 0 (mod k) for all u ∈ v (p(n.(r-pan)v)). this means that p(n.(r-pan)v) is a zk-magic graph. an example of a z9-magic labeling of p(4.(5-pan) v) is illustrated in figure 11. cubo 21, 2 (2019) zk-magic labeling of path union of graphs 39 b b b b b b 2 7 7 7 2 5 b b b b b b 8 b b b b b b 8 b b b b b b 2 7 7 7 2 b bbb 5 5 5 5 4 4 4 5 5 55 4 4 4 figure 11: a z9-magic labeling of p(4.(5-pan) v). acknowledgment this work was supported by the slovak research and development agency under the contract no. apvv-15-0116 and by vega 1/0233/18. 40 yp. jeyanthi, k. jeya daisy and andrea semaničová-feňovč́ıková cubo 21, 2 (2019) references [1] j.a. gallian, a dynamic survey of graph labeling, electron. j. comb., 2018, # ds6. [2] p. jeyanthi and k. jeya daisy, zk-magic labeling of subdivision graphs, discrete math. algorithm. appl., 8(3) (2016), 19 pages, doi: 10.1142/ s1793830916500464. [3] p. jeyanthi and k. jeya daisy, zk-magic labeling of open star of graphs, bull. inter. math. virtual inst., 7 (2017), 243–255. [4] p. jeyanthi and k. jeya daisy, certain classes of zk-magic graphs, j. graph labeling, 4(1) (2018), 38–47. [5] p. jeyanthi and k. jeya daisy, zk-magic labeling of some families of graphs, j. algorithm comput., 50(2) (2018), 1–12. [6] p. jeyanthi and k. jeya daisy, zk-magic labeling of cycle of graphs, int. j. math. combin., 1 (2019), 88–102. [7] p. jeyanthi and k. jeya daisy, some results on zk-magic labeling, palestine j. math., 8(2) (2019), 400–412. [8] k. kavitha and k. thirusangu, group magic labeling of cycles with a common vertex, int. j. comput. algorithm, 2 (2013), 239–242. [9] r.m. low and s.m. lee, on the products of group-magic graphs, australas. j. combin., 34 (2006), 41–48. [10] j. sedláček, on magic graphs, math. slov., 26 (1976), 329–335. [11] s.c. shee and y.s. ho, the cordiality of the path-union of n copies of a graph, discrete math., 151(1-3) (1996), 221–229. [12] w.c. shiu, p.c.b. lam and p.k. sun, construction of magic graphs and some a-magic graphs with a of even order, congr. numer., 167 (2004), 97–107. [13] w.c. shiu and r.m. low, zk-magic labeling of fans and wheels with magic-value zero, australas. j. combin., 45 (2009), 309–316. introduction zk-magic labeling of path union of graphs a mathematical journal vol. 7, no 2, (171 199). august 2005. spectral shift function for schrödinger operators in constant magnetic fields georgi raikov 1 depto matemática. facultad de ciencias. universidad de chile las palmeras 3425, santiago, chile graykov@uchile.cl abstract we consider the three-dimensional schrödinger operator with constant magnetic field, perturbed by an appropriate short-range electric potential, and investigate various asymptotic properties of the corresponding spectral shift function (ssf). first, we analyse the singularities of the ssf at the landau levels. further, we study the strong magnetic field asymptotic behaviour of the ssf; here we distinguish between the asymptotics far from the landau levels, and near a given landau level. finally, we obtain a weyl-type formula describing the high energy behaviour of the ssf. this is a survey article on recent published results obtained by the author jointly with vincent bruneau, claudio fernández, and alexander pushnitski. a shorter version will appear in the proceedings of the conference qmath9, giens, france, september 2004. resumen se considera el operador de schrödinger en tres dimensiones con campo magnético constante, perturbado por un potencial eléctrico de corto alcance apropiado, e investigamos una variedad de propiedades asintóticas de la función de corrimiento espectral (ssf). analizamos primero las singularidades de la ssf 1i would like to thank my co-authors vincent bruneau, claudio fernández, and alexander pushnitski for having let me present our joint results in this survey article. the major part of this work has been done during my visit to the institute of mathematics of the czech academy of sciences in december 2004. i am sincerely grateful to miroslav englǐs for his warm hospitality. the partial support by the chilean science foundation fondecyt under grant 1050716 is acknowledged. 172 georgi raikov 7, 2(2005) en los niveles de landau. además, estudiamos el comportamiento asintótico del campo magnético fuerte de la ssf; distinguimos aqúı entre propiedades asintóticas lejos de los niveles de landau y cerca de uno determinado. finalmente, obtenemos una fórmula de tipo weyl para describir el comportamiento de altas enerǵıas de la ssf. este es un art́ıculo prospectivo en torno a varios resultados obtenidos por el autor en conjunto con vincent bruneau, claudio fernández, y alexander pushnitski. una versión más compacta aparecerá en los proceedings of the conference qmath9, giens, france, september 2004. key words and phrases: spectral shift function, scattering phase, schrödinger operators, constant magnetic fields math. subj. class.: 81q10, 35j10, 47n50, 35p05 1 introduction in this survey article based on the papers [7], [10], and [8], we consider the 3d schrödinger operator with constant magnetic field of scalar intensity b > 0, perturbed by an electric potential v which decays fast enough at infinity, and discuss various asymptotic properties of the corresponding spectral shift function. more precisely, let h0 = h0(b) := (i∇ + a)2 − b be the unperturbed operator, essentially self-adjoint on c∞0 (r 3). here a = ( −bx2 2 , bx1 2 , 0 ) is the magnetic potential which generates the constant magnetic field b = curl a = (0, 0,b), b > 0. it is wellknown that σ(h0) = σac(h0) = [0,∞) (see [1]), where σ(h0) stands for the spectrum of h0, and σac(h0) for its absolutely continuous spectrum. moreover, the so-called landau levels 2bq, q ∈ z+ := {0, 1, . . .}, play the role of thresholds in σ(h0). for x = (x1,x2,x3) ∈ r3 we denote by x⊥ = (x1,x2) the variables on the plane perpendicular to the magnetic field. throughout the paper assume that v satisfies v �≡ 0, v ∈ c(r3), |v (x)| ≤ c0〈x⊥〉−m⊥〈x3〉−m3, x = (x⊥,x3) ∈ r3, (1.1) with c0 > 0, m⊥ > 2, m3 > 1, and 〈x〉 := (1 + |x|2)1/2, x ∈ rd, d ≥ 1. some of our results hold under a more restrictive assumption than (1.1), namely v �≡ 0, v ∈ c(r3), |v (x)| ≤ c0〈x〉−m0, m0 > 3, x ∈ r3. (1.2) note that (1.2) implies (1.1) with any m3 ∈ (0,m0) and m⊥ = m0−m3. in particular, we can choose m3 ∈ (1,m0 − 2) so that m⊥ > 2. on the domain of h0 define the operator h = h(b) := h0 +v . obviously, inf σ(h) ≤ inf σ(h0) = 0. moreover, if (1.1) holds, then for every e < inf σ(h) we have (h − e)−1 − (h0 − e)−1 ∈ s1 where s1 denotes the trace class. hence, there exists a unique function ξ = ξ(·; h,h0) ∈ l1(r; (1 + e2)−1de) which vanishes identically on 7, 2(2005) spectral shift function for magnetic schrödinger operators 173 (−∞, inf σ(h)) such that the lifshits-krein trace formula tr (f(h) −f(h0)) = ∫ r ξ(e; h,h0)f ′(e)de holds for each f ∈ c∞0 (r) (see the original works [22], [20], the survey article [5], or chapter 8 of the monograph [45]). the function ξ(·; h,h0) is called the spectral shift function (ssf) for the operator pair (h,h0). if e < 0 = inf σ(h0), then the spectrum of h below e could be at most discrete, and for almost every e < 0 we have ξ(e; h,h0) = −n(e; h) (1.3) where n(e; h) denotes the number of eigenvalues of h lying in the interval (−∞,e), and counted with their multiplicities. on the other hand, for almost every e ∈ [0,∞), the ssf ξ(e; h,h0) is related to the scattering determinant det s(e; h,h0) for the pair (h,h0) by the birman-krein formula det s(e; h,h0) = e −2πiξ(e;h,h0) (see [4] or [45, section 8.4]). a survey of various asymptotic results concerning the ssf for numerous quantum hamiltonians is contained in [40]. a priori, the ssf ξ(e; h,h0) is defined for almost every e ∈ r. in this article we will identify this ssf with a representative of its equivalence class which is welldefined on r \ 2bz+, bounded on every compact subset of r \ 2bz+, and continuous on r \ (2bz+ ∪ σpp(h)) where σpp(h) denotes the set of the eigenvalues of h. in the case of perturbations v of definite sign this representative is described explicitly in subsection 3.1 below; in the case of general non-sign-definite perturbations its description can be found in [7, section 3]. in the present article we investigate the behaviour of the ssf in several asymptotic regimes: • first, we analyse the singularities of the ssf at the landau levels. in other words, we fix q ∈ z+, and investigate the behaviour of ξ(2bq + λ; h,h0) as λ → 0. • further, we study the strong-magnetic-field asymptotics of the ssf, i.e. the behaviour of the ssf as b → ∞. here we distinguish between the asymptotics far from the landau levels, and the asymptotics near a given landau level. • finally, we obtain a weyl-type formula describing the high-energy asymptotics of the ssf. the paper is organised as follows. in section 2 we formulate our main results, and discuss briefly on them. more precisely, in subsection 2.1 we introduce some basic notations used throughout the paper, subsection 2.2 contains the results on the singularities of the ssf at the landau levels, subsection 2.3 is devoted to the strongmagnetic-field asymptotics of the ssf, and subsection 2.4 to its high-energy behaviour. section 3 contains some auxiliary results. in subsection 3.1 we describe the 174 georgi raikov 7, 2(2005) representation of the ssf in the case of perturbations of fixed sign, due to a. pushnitski (see [29]), while in subsection 3.2 we establish estimates of some auxiliary operators of birman-schwinger type which are used systematically in the proofs of the main results. some of these proofs could be found in section 4: in subsection 4.1 we prove the results of subsection 2.2, and in subsection 4.2 some of the results of subsection 2.3. since the detailed proofs have already been published in [10] and [7], the proofs presented here are somewhat sketchy, preference being given to the main ideas rather than to the technical details. 2 main results 2.1 notations and preliminaries in this subsection we introduce our basic notations used throughout the paper. we denote by s∞ the class of linear compact operators acting in a given hilbert space. let t = t ∗ ∈ s∞. denote by pi (t ) the spectral projection of t associated with the interval i ⊂ r. for s > 0 set n±(s; t ) := rank p(s,∞)(±t ). for an arbitrary (not necessarily self-adjoint) operator t ∈ s∞ put n∗(s; t ) := n+(s 2; t ∗t ), s > 0. (2.1) if t = t ∗, then evidently n∗(s; t ) = n+(s,t ) + n−(s; t ), s > 0. (2.2) moreover, if tj = t ∗j ∈ s∞, j = 1, 2, then the weyl inequalities n±(s1 + s2,t1 + t2) ≤ n±(s1,t1) + n±(s2,t2) (2.3) hold for each s1, s2 > 0. further, we denote by sp, p ∈ (0,∞), the schatten-von neumann class of compact operators for which the functional ‖t‖p : = ( p ∫ ∞ 0 sp−1n∗(s; t ) ds )1/p is finite. if t ∈ sp, p ∈ (0,∞), then the following elementary inequality of chebyshev type n∗(s; t ) ≤ s−p‖t‖pp (2.4) holds for every s > 0. if t = t ∗ ∈ sp, p ∈ (0,∞), then (2.2) and (2.4) imply n±(s; t ) ≤ s−p‖t‖pp, s > 0. (2.5) 2.2 singularities of the ssf at the landau levels introduce the landau hamiltonian h(b) := ( i ∂ ∂x1 − bx2 2 )2 + ( i ∂ ∂x2 + bx1 2 )2 − b, (2.6) 7, 2(2005) spectral shift function for magnetic schrödinger operators 175 i.e. the 2d schrödinger operator with constant scalar magnetic field b > 0, essentially self-adjoint on c∞0 (r 2). it is well-known that σ(h(b)) = ∪∞q=0 {2bq}, and each eigenvalue 2bq, q ∈ z+, has infinite multiplicity (see e.g. [1]). denote by pq = pq(b) the orthogonal projection onto the eigenspace ker (h(b) − 2bq), q ∈ z+. the estimates of the ssf for energies near the landau level 2bq, q ∈ z+, will be given in the terms of traces of certain functions of toeplitz-type operators pqupq where u : r2 → r decays in a certain sense at infinity. lemma 2.1 [31, lemma 5.1], [10, lemma 2.1] let u ∈ lr(r2), r ≥ 1, and q ∈ z+. then pqupq ∈ sr. assume that (1.1) holds. set w(x⊥) := ∫ r |v (x⊥,x3)|dx3, x⊥ ∈ r2. since v satisfies (1.1), we have w ∈ l1(r2), and lemma 2.1 with u = w implies pqwpq ∈ s1, q ∈ z+. evidently, pqwpq ≥ 0, and it follows from v �≡ 0 and v ∈ c(r2), that rankpqwpq = ∞ for all q ∈ z+ (see below lemma 2.4). if, moreover, v satisfies (1.2), then 0 ≤ w(x⊥) ≤ c′0〈x⊥〉−m0+1, x⊥ ∈ r2, with c′0 = c0 ∫ r 〈x〉−m0dx. in the following two theorems we assume that v has a definite sign, i.e. that either v ≤ 0 (then we will write h− instead of h), or v ≥ 0 (then we will write h+ instead of h). theorem 2.1 (cf. [10, theorem 3.1]) assume that (1.2) is valid, and ±v ≥ 0. let q ∈ z+, b > 0. then the asymptotic estimates ξ(2bq −λ; h+,h0) = o(1), (2.7) −n+((1−ε)2 √ λ; pqwpq)+o(1) ≤ ξ(2bq−λ; h−,h0) ≤−n+((1+ε)2 √ λ; pqwpq)+o(1), (2.8) hold as λ ↓ 0 for each ε ∈ (0, 1). suppose that the potential v satisfies (1.1). for λ ≥ 0 define the matrix-valued function wλ = wλ(x⊥) := ( w11 w12 w21 w22 ) , x⊥ ∈ r2, (2.9) where w11 := ∫ r |v (x⊥,x3)|cos2 ( √ λx3)dx3, w12 = w21 := ∫ r |v (x⊥,x3)|cos ( √ λx3) sin ( √ λx3)dx3, w22 := ∫ r |v (x⊥,x3)|sin2 ( √ λx3)dx3. it is easy to check that for λ ≥ 0 and q ∈ z+ the operator pqwλpq : l2(r2)2 → l2(r2)2 satisfies 0 ≤ pqwλpq ∈ s1, and rank pqwλpq = ∞. 176 georgi raikov 7, 2(2005) theorem 2.2 (cf. [10, theorem 3.2]) assume that (1.2) is valid, and ±v ≥ 0. let q ∈ z+, b > 0. then the asymptotic estimates ±1 π tr arctan (((1 ±ε)2 √ λ)−1pqwλpq) + o(1) ≤ ξ(2bq + λ; h±,h0) ≤ ±1 π tr arctan (((1 ∓ε)2 √ λ)−1pqwλpq) + o(1) (2.10) hold as λ ↓ 0 for each ε ∈ (0, 1). relations (2.8) and (2.10) allow us to reduce the analysis of the behaviour as λ → 0 of ξ(2bq + λ; h±,h0), to the study of the asymptotic distribution of the eigenvalues of toeplitz-type operators pqupq. the following three lemmas concern the spectral asymptotics of such operators. lemma 2.2 [31, theorem 2.6] let the function 0 ≤ u ∈ c1(r2) satisfy the estimates u(x⊥) = u0(x⊥/|x⊥|)|x⊥|−α(1 + o(1)), |x⊥|→∞, |∇u(x⊥)| ≤ c1〈x⊥〉−α−1, x⊥ ∈ r2, where α > 0, and u0 is a continuous function on s1 which is non-negative and does not vanish identically. then for each q ∈ z+ we have n+(s; pqupq) = b 2π ∣∣{x⊥ ∈ r2|u(x⊥) > s}∣∣ (1 + o(1)) = ψα(s) (1 + o(1)), s ↓ 0, where |.| denotes the lebesgue measure, and ψα(s) := s −2/α b 4π ∫ s1 u0(t) 2/αdt, s > 0. (2.11) lemma 2.3 [38, theorem 2.1, proposition 4.1] let 0 ≤ u ∈ l∞(r2). assume that ln u(x⊥) = −µ|x⊥|2β (1 + o(1)), |x⊥|→∞, for some β ∈ (0,∞), µ ∈ (0,∞). then for each q ∈ z+ we have n+(s; pqupq) = ϕβ (s)(1 + o(1)), s ↓ 0, where ϕβ (s) := ⎧⎪⎨ ⎪⎩ b 2µ1/β | ln s|1/β if 0 < β < 1, 1 ln (1+2µ/b) | ln s| if β = 1, β β−1 (ln | ln s|)−1| ln s| if 1 < β < ∞. s ∈ (0,e−1). (2.12) lemma 2.4 [38, theorem 2.2, proposition 4.1] let 0 ≤ u ∈ l∞(r2). assume that the support of u is compact, and that there exists a constant c > 0 such that u ≥ c on an open non-empty subset of r2. then for each q ∈ z+ we have n+(s; pqupq) = ϕ∞(s) (1 + o(1)), s ↓ 0, where ϕ∞(s) := (ln | ln s|)−1| ln s|, s ∈ (0,e−1). (2.13) 7, 2(2005) spectral shift function for magnetic schrödinger operators 177 employing lemmas 2.2, 2.3, 2.4, we easily find that asymptotic estimates (2.8) and (2.10) entail the following corollary 2.1 [10, corollaries 3.1 – 3.2] let (1.2) hold with m0 > 3. i) assume that the hypotheses of lemma 2.2 hold with u = w and α > 2. then we have ξ(2bq −λ; h−,h0) = − b 2π ∣∣∣ { x⊥ ∈ r2|w(x⊥) > 2 √ λ }∣∣∣ (1 + o(1)) = −ψα(2 √ λ) (1 + o(1)), λ ↓ 0, (2.14) ξ(2bq + λ; h±,h0) = ± b 2π2 ∫ r2 arctan ((2 √ λ)−1w(x⊥))dx⊥ (1 + o(1)) = ± 1 2 cos (π/α) ψα(2 √ λ) (1 + o(1)), λ ↓ 0, the function ψα being defined in (2.11). ii) assume that the hypotheses of lemma 2.3 hold with u = w. then we have ξ(2bq −λ; h−,h0) = −ϕβ (2 √ λ) (1 + o(1)), λ ↓ 0, β ∈ (0,∞), the functions ϕβ being defined in (2.12). if, in addition, v satisfies (1.1) for some m⊥ > 2 and m3 > 2, we have ξ(2bq + λ; h±,h0) = ± 1 2 ϕβ (2 √ λ) (1 + o(1)), λ ↓ 0, β ∈ (0,∞). iii) assume that the hypotheses of lemma 2.4 hold with u = w. then we have ξ(2bq −λ; h−,h0) = −ϕ∞(2 √ λ) (1 + o(1)), λ ↓ 0, the function ϕ∞ being defined in (2.13). if, in addition, v satisfies (1.1) for some m⊥ > 2 and m3 > 2, we have ξ(2bq + λ; h±,h0) = ± 1 2 ϕ∞(2 √ λ) (1 + o(1)), λ ↓ 0, the function ϕ∞ being defined in (2.13). in particular, we find that lim λ↓0 ξ(2bq −λ; h−,h0) ξ(2bq + λ; h−,h0) = 1 2 cos π α (2.15) if w has a power-like decay at infinity (i.e. if the assumptions of corollary 2.1 i) hold), or lim λ↓0 ξ(2bq −λ; h−,h0) ξ(2bq + λ; h−,h0) = 1 2 (2.16) 178 georgi raikov 7, 2(2005) if w decays exponentially or has a compact support (i.e. if the assumptions of corollary 2.1 ii) iii) are fulfilled). relations (2.15) and (2.16) could be interpreted as analogues of the classical levinson formulae (see e.g. [40]). remarks: i) since the ranks of pqwpq and pqwλpq are infinite, the quantities n+(s2 √ λ; pqwpq) and tr arctan ((s2 √ λ)−1pqwλpq) tend to infinity as λ ↓ 0 for every s > 0. therefore, theorems 2.1 and 2.2 imply that the ssf ξ(·; h±,h0) has a singularity at each landau level. the existence of singularities of the ssf at strictly positive energies is in sharp contrast with the non-magnetic case b = 0 where the ssf ξ(e;−∆ + v,−∆) is continuous for e > 0 (see e.g. [40]). the main reason for this phenomenon is the fact that the landau levels play the role of thresholds in σ(h0) while the free laplacian −∆ has no strictly positive thresholds in its spectrum. it is conjectured that the singularity of the ssf ξ(·; h±(b),h0(b)), b > 0, at a given landau level 2bq, q ∈ z+, could be related to a possible accumulation of resonances and/or eigenvalues of h at 2bq. here it should be recalled that in the case b = 0 the high energy asymptotics (see [27]) and the semi-classical asymptotics (see [28]) of the derivative of the ssf for appropriate compactly supported perturbations of the laplacian, are related by the breit-wigner formula to the asymptotic distribution near the real axis of the resonances defined as poles of the meromorphic continuation of the resolvent of the perturbed operator. ii) in the case q = 0, when by (1.3) we have ξ(−λ; h−,h0) = −n(−λ; h−) for λ > 0, asymptotic relations of the type of (2.14) have been known since long ago (see [43], [42], [44], [31], [17]). an important characteristic feature of the methods used in [31], and later in [38], is the systematic use, explicit or implicit, of the connection between the spectral theory of the schrödinger operator with constant magnetic field, and the theory of toeplitz operators acting in holomorphic spaces of fock-segal-bargmann type, and the related pseudodifferential operators with generalised anti-wick symbols (see [12], [3], [41], [15]). various important aspects of the interaction between these two theories have been discussed in [37] and [7, section 9]). the toeplitzoperator approach turned out to be especially fruitful in [38] where electric potentials decaying rapidly at infinity (i.e. decaying exponentially, or having compact support) were considered (see lemmas 2.3 2.4). it is shown in [11] that the precise spectral asymptotics for the landau hamiltonian perturbed by a compactly supported electric potential u of fixed sign recovers the logarithmic capacity of the support of u. iii) let us mention several other existing extensions of lemmas 2.2 – 2.4. lemmas 2.2 and 2.4 have been generalised to the multidimensional case where pq is the orthogonal projection onto a given eigenspace of the schrödinger operator with constant magnetic field of full rank, acting in l2(r2d), d > 1 (see [31] and [25] respectively). moreover, lemma 2.4 has been generalised in [25] to a relativistic setting where pq is an eigenprojection of the dirac operator. finally, in [36] lemmas 2.2 – 2.4 have been extended to the case of the 2d pauli operator with variable magnetic field from a certain class including the almost periodic fields with non-zero mean value (in this case the role of the landau levels is played by the origin), and electric potentials u satisfying the assumptions of lemmas 2.2 – 2.4. in the case of compactly supported u of definite sign, [11] contains a more precise version of the corresponding result of 7, 2(2005) spectral shift function for magnetic schrödinger operators 179 [36], involving again the logarithmic capacity of the support of u. iv) to the author’s best knowledge, the singularities at the landau levels of the ssf for the 3d schrödinger operator in constant magnetic field has been investigated for the first time in [10]. however, it is appropriate to mention here the article [19] where an axisymmetric potential v = v (|x⊥|,x3) has been considered. it is well-known (see e.g. [1]) that in this case the operators h0 and h are unitarily equivalent to the orthogonal sums ∑ m∈z ⊕h(m) and ∑ m∈z ⊕h (m) 0 respectively, where the operators h (m) 0 := − 1 � ∂ ∂� � ∂ ∂� − ∂ 2 ∂x23 + ( b� 2 + m � )2 − b, h(m) := h(m)0 + v (�,x3), m ∈ z, (2.17) are self-adjoint in l2(r+ ×r; �d�dx3). for a fixed magnetic quantum number m ∈ z the authors of [19] studied the behaviour of the ssf ξ(e; h(m),h(m)0 ) for energies e near the landau level 2m if m > 0, and near the origin if m ≤ 0, and deduced analogues of the classical levinson formulae for the operator pair ( h(m),h (m) 0 ) . later, the methods in [19] were developed in [23] and [24]. however, it is not possible to recover the results of our theorem 2.1, theorem 2.2 and/or corollary 2.1 from the results of [19], [23], and [24] even in the case of axisymmetric v . v) finally, [16] contains general bounds on the ssf for appropriate pairs of magnetic schrödinger operators. these bounds are applied in order to deduce wegner estimates of the integrated density of states for some random alloy-type models. 2.3 strong magnetic field asymptotics of the ssf our first theorem in this subsection treats the asymptotics as b →∞ of ξ(·; h(b),h0(b)) far from the landau levels. theorem 2.3 (cf. [7, theorem 2.1]) let (1.1) hold. assume that e ∈ (0,∞) \ 2z+, and λ ∈ r. then ξ(eb + λ; h(b),h0(b)) = b1/2 4π2 [e/2]∑ l=0 (e − 2l)−1/2 ∫ r3 v (x)dx + o(1), b →∞, (2.18) where [e/2] denotes the integer part of the real number e/2. the following two theorems concern the asymptotics of the ssf near a given landau level. in order to formulate our next theorem, we introduce the following self-adjoint operators χ0 := −d2/dx23, χ = χ(x⊥) := χ0 + v (x⊥, .), x⊥ ∈ r2, which are defined on the sobolev space h2(r), and depend on the parameter x⊥ ∈ r2. if (1.1) holds, then (χ(x⊥) − λ0)−1 − (χ0 − λ0)−1 ∈ s1 for each x⊥ ∈ r2 and λ0 < inf σ(χ(x⊥)). hence, the ssf ξ(.; χ(x⊥),χ0) is well-defined. set λ: = minx⊥∈r2 inf σ(χ(x⊥)). evidently, λ ∈ [−c0, 0]. moreover, λ = lim b→∞ inf σ(h(b)) (2.19) 180 georgi raikov 7, 2(2005) (see [1, theorem 5.8]). proposition 2.1 (cf. [7, proposition 2.2]) assume that (1.1) holds. i) for each λ ∈ r \{0} we have ξ(λ; χ(.),χ0) ∈ l1(r2). ii) the function (0,∞) � λ �→ ∫ r2 ξ(λ; χ(x⊥),χ0)dx⊥ is continuous, while the nonincreasing function (−∞, 0) � λ �→ ∫ r2 ξ(λ; χ(x⊥),χ0)dx⊥ = − ∫ r2 n(λ; χ(x⊥))dx⊥ (see (1.3)), is continuous at the point λ < 0 if and only if |{x⊥ ∈ r2|λ ∈ σ(χ(x⊥))}| = 0. (2.20) iii) assume ±v ≥ 0. if λ > λ, λ �= 0, then ± ∫ r2 ξ(λ; χ(x⊥),χ0)dx⊥ > 0. remark: the third part of proposition 2.1 was not included in [7, proposition 2.2]. however, it follows easily from the representation of the ssf described in subsection 3.1 below, and the hypotheses v �≡ 0 and v ∈ c(r3). theorem 2.4 (cf. [7, theorem 2.3]) assume that (1.1) holds. let q ∈ z+, λ ∈ r \{0}. if λ < 0, suppose also that (2.20) holds. then we have lim b→∞ b−1ξ(2bq + λ; h(b),h0(b)) = 1 2π ∫ r2 ξ(λ; χ(x⊥),χ0) dx⊥. (2.21) the proofs of theorems 2.3 and 2.4 are contained in subsection 4.2. we present these proofs under the additional assumption that v has a definite sign, and refer the reader to the original paper [7] for the proofs in the general case. by proposition 2.1 iii), if ±v ≥ 0, then the r.h.s. of (2.21) is different from zero if λ > λ, λ �= 0. unfortunately, we cannot prove that the same is true for general nonsign-definite electric potentials v . on the other hand, it is obvious that for arbitrary v we have ∫ r2 ξ(λ; χ(x⊥),χ0)dx⊥ = 0 if λ < λ. the last theorem of this subsection contains a more precise version of (2.21) for the case λ < λ. theorem 2.5 (cf. [7, theorem 2.4]) let (1.1) hold. i) let λ < λ. then for sufficiently large b > 0 we have ξ(λ; h(b),h0(b)) = 0. ii) let q ∈ z+, q ≥ 1, λ < λ. assume that the partial derivatives of 〈x3〉m3v with respect to the variables x⊥ ∈ r2 exist, and are uniformly bounded on r3. then we have lim b→∞ b−1/2ξ(2bq + λ; h(b),h0(b)) = 1 4π2 q−1∑ l=0 (2(q − l))−1/2 ∫ r3 v (x)dx. (2.22) the first part of the theorem is trivial, and follows immediately from (2.19). we omit the proof of theorem 2.5 ii) and refer the reader to the original work [7]. remarks: i) relations (2.18), (2.21), and (2.22) can be unified into a single asymptotic 7, 2(2005) spectral shift function for magnetic schrödinger operators 181 formula. in order to see this, notice that a general result on the high-energy asymptotics of the ssf for 1d schrödinger operators (see e.g. [40]) implies, in particular, that lim e→∞ e1/2ξ(e; χ(x⊥),χ0) = 1 2π ∫ r v (x⊥,x3) dx3, x⊥ ∈ r2. then relation (2.18) with 0 < e/2 �∈ z+, or relations (2.21) and (2.22) with e = 2q, q ∈ z+, entail ξ(eb+λ; h(b),h0(b)) = b 2π [e/2]∑ l=0 ∫ r2 ξ(b(e−2l)+λ; χ(x⊥),χ0)dx⊥ (1+o(1)), b →∞. (2.23) on its turn, (2.23) can be re-written as ξ(eb+λ; h(b),h0(b)) = ∫ r ∫ r2 ξ(eb+λ−s; χ(x⊥),χ0)dx⊥dνb(s) (1 +o(1)), b →∞, where νb(s) := b2π ∑∞ l=0 θ(s − 2bl), s ∈ r, and θ(s) := { 0 if s ≤ 0, 1 if s > 0, is the heaviside function. it is well-known that ν is the integrated density of states for the 2d landau hamiltonian (see (2.6)). ii) by (1.3) for λ < 0 we have ξ(λ; h(b),h0) = −n(λ; h(b)). the asymptotics as b → ∞ of the counting function n(λ; h0(b)) with λ < 0 fixed, has been investigated in [32] under considerably less restrictive assumptions on v than in theorems 2.3 – 2.5. the asymptotic properties as λ ↑ 0, and as λ ↓ λ if λ < 0, of the asymptotic coefficient − 1 2π ∫ r2 n(λ; χ(x⊥)dx⊥ which appears at the r.h.s. of (2.21) in the case of a negative perturbation, have been studied in [33]. the asymptotic distribution of the discrete spectrum for the 3d magnetic pauli and dirac operators in strong magnetic fields has been considered in [35] and [34] respectively. the main purpose in [32], [34], and [35] was to obtain the main asymptotic term (without any remainder estimates) of the corresponding counting function of the discrete spectrum under assumptions close to the minimal ones which guarantee that the hamiltonians are self-adjoint, and the asymptotic coefficient is well-defined. other results which again describe the asymptotic distribution of the discrete spectrum of the schrödinger and dirac operator in strong magnetic fields, but contain also sharp remainder estimates, have been obtained [17], [9], and [18] under assumptions on v which, naturally, are considerably more restrictive than those in [32], [34], and [35]. iii) generalisations of asymptotic relation (2.18) in several directions can be found in [26]. in particular, [26, theorem 4] implies that if v ∈ s(r3), then the ssf ξ(eb + λ; h(b),h0(b)), e ∈ (0,∞) \ 2z+, λ ∈ r, admits an asymptotic expansion of the form ξ(eb + λ; h(b),h0(b)) ∼ ∞∑ j=0 cjb 1−2j 2 , b →∞. iv) together with the pointwise asymptotics as b → ∞ of the ssf for the pair (h0(b),h(b)) (see (2.18), (2.21), or (2.22)), it also is possible to consider its weak 182 georgi raikov 7, 2(2005) asymptotics, i.e. the asymptotics of the convolution of the ssf with an arbitrary ϕ ∈ c∞0 (r). results of this type are contained in [6]. 2.4 high energy asymptotics of the ssf theorem 2.6 [8, theorem 2.1] assume that v satisfies (1.1). then we have lim e→∞,e∈or e−1/2ξ(e; h,h0) = 1 4π2 ∫ r3 v (x)dx, r ∈ (0,b), (2.24) where or := {e ∈ (0,∞)|dist(e, 2bz+)}. we omit the proof of theorem 2.6 which is quite similar to that of theorem 2.3, and refer the reader to the original paper [8]. remarks: i) it is essential to avoid the landau levels in (2.24), i.e. to suppose that e ∈or, r ∈ (0,b), as e →∞, since by theorems 2.1 2.2, the ssf has singularities at the landau levels, at least in the case ±v ≥ 0. ii) for e ∈ r set ξcl(e) := ∫ t ∗r3 ( θ(e −|p + a(x)|2) − θ(e −|p + a(x)|2 −v (x)) ) dxdp = 4π 3 ∫ r3 ( e 3/2 + − (e −v (x))3/2+ ) dx where θ, as above, is the heaviside function. note that ξcl(e) is independent of the magnetic field b ≥ 0. evidently, under the assumptions of theorem 2.6 we have lime→∞ e−1/2ξcl(e) = 2π ∫ r3 v (x)dx. hence, if ∫ r3 v (x)dx �= 0, then (2.24) is equivalent to ξ(e; h,h0) = (2π) −3ξcl(e)(1 + o(1)), e →∞, e ∈or, r ∈ (0,b). iii) as far as the author is informed, the high-energy asymptotics of the ssf for 3d schrödinger operators in constant magnetic fields was investigated for the first time in [8]. nonetheless, in [19] the asymptotic behaviour as e →∞, e ∈or, of the ssf ξ(e; h(m),h(m)0 ) for the operator pair (h (m),h (m) 0 ) (see (2.17)) with fixed m ∈ z has been been investigated. it does not seem possible to deduce (2.24) from the results of [19] even in the case of axial symmetry of v . 3 auxiliary results 3.1 a. pushnitski’s representation of the ssf in the first part of this subsection we summarise several results due to a. pushnitski on the representation of the ssf for a pair of lower-bounded self-adjoint operators (see [29]). let i ∈ r be a lebesgue measurable set. set µ(i) := 1 π ∫ i dt 1+t2 . note that µ(r) = 1. 7, 2(2005) spectral shift function for magnetic schrödinger operators 183 lemma 3.1 [29, lemma 2.1] let t1 = t ∗1 ∈ s∞ and t2 = t ∗2 ∈ s1. then∫ r n±(s1 + s2; t1 + tt2) dµ(t) ≤ n±(s1; t1) + 1 πs2 ‖t2‖1, s1,s2 > 0. (3.1) let h± and h0 be two lower-bounded self-adjoint operators. assume that v := ±(h± −h0) ≥ 0. (3.2) let λ0 < inf σ(h±) ∪σ(h0). suppose that (h0 −λ0)−γ − (h0 −λ0)−γ ∈ s2, γ > 0, (3.3) v1/2(h0 −λ0)−1/2 ∈ s∞, (3.4) v1/2(h0 −λ0)−γ ′ ∈ s2, γ′ > 0. (3.5) for z ∈ c with im z > 0 set t (z): = v1/2(h0 −z)−1v1/2. lemma 3.2 [29, lemma 4.1] let (3.3) – (3.5) hold. then for almost every e ∈ r the operator-norm limit t (e + i0) := n − limδ↓0 t (e + iδ) exists, and by (3.4) we have t (e + i0) ∈ s∞. moreover, 0 ≤ imt (e + i0) ∈ s1. theorem 3.1 [29, theorem 1.2] let (3.2) – (3.5) hold. then the ssf ξ(·;h±,h0) for the operator pair (h±,h0) is well-defined, and for almost every e ∈ r we have ξ(e;h±,h0) = ± ∫ r n∓(1; ret (e + i0) + t imt (e + i0)) dµ(t). remark: the representation of the ssf described in the above theorem was generalised to non-sign-definite perturbations in [14] in the case of trace-class perturbations, and in [30] in the case of relatively trace-class perturbations. these generalisations are based on the concept of index of orthogonal projections (see [2]). suppose now that v satisfies (1.1), and ±v ≥ 0. then relations (3.2) – (3.5) hold with v = |v |, h0 = h0, and γ = γ′ = 1. for z ∈ c, im z > 0, set t (z) := |v |1/2(h0−z)−1|v |1/2. by lemma 3.2, for almost every e ∈ r the operatornorm limit t (e + i0) := n − lim δ↓0 t (e + iδ) (3.6) exists, and 0 ≤ im t (e + i0) ∈ s1. (3.7) the following proposition contains a more precise version of the above statement, and provides estimates of the norm of t (e +i0), and the trace-class norm of im t (e +i0). proposition 3.1 [7, lemma 4.2] assume that (1.1) holds, and e ∈ r \2bz+. then the operator limit (3.6) exists, and we have ‖t (e + i0)‖≤ c1 (dist (e, 2bz+))−1/2 (3.8) 184 georgi raikov 7, 2(2005) with c1 independent of e and b. moreover, (3.7) holds, and if e < 0 then im t (e+i0) = 0, while for e ∈ (0,∞)\2bz+ we have ‖im t (e + i0)‖1 = tr im t (e + i0) = b 4π [ e2b ]∑ l=0 (e − 2bl)−1/2 ∫ r3 |v (x)|dx. (3.9) by lemma 3.1 and proposition 3.1, the quantity ξ̃(e; h±,h0) = ± ∫ r n∓(1; re t (e + i0) + t im t (e + i0)) dµ(t), e ∈ r \ 2bz+, (3.10) is well-defined for every e ∈ r \ 2bz+, and bounded on every compact subset of r \ 2bz+. moreover, by [7, proposition 2.5], ξ̃(·; h±,h0) is continuous on r \ {2bz+ ∪σpp(h±)}. on the other hand, by theorem 3.1 we have ξ̃(e; h±,h0) = ξ(e; h±,h0) (3.11) for almost every e ∈ r. as explained in the introduction, in the case of sign-definite perturbations we will identify the ssf ξ(e; h±,h0) with ξ̃(e; h±,h0), while in the case of non-sign-definite perturbations, we will identify it with the generalisation of ξ̃(e; h±,h0) described in [7, section 3] on the basis of the general results of [14] and [30]. here it should be underlined that in contrast to the case b = 0, we cannot rule out the possibility that the operator h has infinite discrete spectrum, or eigenvalues embedded in the continuous spectrum by imposing conditions about the fast decay of the potential v at infinity. first, it is well-known that if v satisfies v (x) ≤−cχ(x), x ∈ r3, (3.12) where c > 0, and χ is the characteristic function of a non-empty open subset of r3, then the discrete spectrum of h is infinite (see [1, theorem 5.1], [38, theorem 2.4]). further, if v is axisymmetric and satisfies (3.12), then the operator h(q) defined in (2.17) with q ≥ 0 has at least one eigenvalue in the interval (2bq −‖v‖l∞(r3), 2bq), and hence the operator h has infinitely many eigenvalues embedded in its continuous spectrum (see [1, theorem 5.1]). assume now that v is axisymmetric and satisfies the estimate v (x⊥,x3) ≤−cχ⊥(x⊥)〈x3〉−m3, (x⊥,x3) ∈ r3, (3.13) where c > 0, χ⊥ is the characteristic function of a non-empty open subset of r2, and m3 ∈ (0, 2) which is compatible with (1.1) if m3 ∈ (1, 2). then, using the argument of the proof of [1, theorem 5.1] and the variational principle, we can easily check that for each q ≥ 0 the operator h(q) has infinitely many discrete eigenvalues which accumulate to the infimum 2bq of its essential spectrum. hence, if v is axisymmetric and satisfies (3.13), then below each landau level 2bq, q ∈ z+, there exists an infinite 7, 2(2005) spectral shift function for magnetic schrödinger operators 185 sequence of finite-multiplicity eigenvalues of h, which converges to 2bq. note however that the claims in [10, p. 385] and [8, p. 3457] that [1, theorem 5.1]) implies the same phenomenon for axisymmetric non-positive potentials compactly supported in r3, are not justified. the challenging and interesting problem about the accumulation at a given landau level of embedded eigenvalues and/or resonances of h will be considered in a future work. finally, we note that generically the only possible accumulation points of the eigenvalues of h are the landau levels (see [1, theorem 4.7], [13, theorem 3.5.3 (iii)]). further information on the location of the eigenvalues of h can be found in [7, proposition 2.6]. 3.2 estimates for birman-schwinger operators for x, x′ ∈ r2 denote by pq,b(x, x′) the integral kernel of the orthogonal projection pq(b) onto the subspace ker (h(b)−2bq), q ∈ z+, the landau hamiltonian h(b) being defined in (2.6). it is well-known that pq,b(x, x′) = b 2π lq ( b|x − x′|2 2 ) exp ( −b 4 (|x − x′|2 + 2i(x1x′2 −x′1x2)) ) (3.14) (see [21] or [37, subsection 2.3.2]) where lq(t) := 1q!e t d q(tq e−t) dtq = ∑q k=0 ( q k ) (−t)k k! , t ∈ r, q ∈ z+, are the laguerre polynomials. note that pq,b(x, x) = b 2π , q ∈ z+, x ∈ r2. (3.15) define the orthogonal projections pq : l2(r3) → l2(r3), q ∈ z+, by pq := pq ⊗ i3 where i3 is the identity operator in l2(rx3 ). for z ∈ c with im z > 0, define the operator r(z) := ( − d2 dx23 −z )−1 bounded in l2(r). note that the operator r(z) admits the integral kernel rz (x3 − x′3) where rz(x) = iei √ z|x|/(2 √ z), x ∈ r, the branch of √z being chosen so that im √z > 0. define that the operators tq(z) := |v |1/2pq(h0 −z)−1|v |1/2, q ∈ z+, bounded in l2(r3). we have tq(z) = |v |1/2 ( pq(b) ⊗r(z − 2bq) ) |v |1/2. for λ ∈ r, λ �= 0, define r(λ) as the operator with integral kernel rλ(x3 −x′3) where rλ(x) := lim δ↓0 rλ+iδ (x) = ⎧⎨ ⎩ e− √−λ|x| 2 √ −λ if λ < 0, iei √ λ|x| 2 √ λ if λ > 0, x ∈ r. (3.16) evidently, if w1,w2 ∈ l2(r) and λ �= 0, then w1r(λ)w2 ∈ s2. for e ∈ r, e �= 2bq, q ∈ z+, set tq(e) := |v |1/2 ( pq(b) ⊗r(e − 2bq) ) |v |1/2. then limδ↓0 ‖tq(e + iδ) −tq(e)‖2 = 0 (see [10, proposition 4.1]). 186 georgi raikov 7, 2(2005) proposition 3.2 let e ∈ r, q ∈ z+, e �= 2bq. let (1.1) hold. then ‖tq(e)‖≤ c2|e − 2bq|−1/2, (3.17) ‖tq(e)‖22 ≤ c2b|e − 2bq|−1, (3.18) with c2 independent of e, b, and q. proof. we have tq(e) = mgq,m⊥ ⊗ t(e − 2bq)m (3.19) where m is the multiplier by the bounded function |v (x⊥,x3)|1/2〈x⊥〉m⊥/2〈x3〉m3/2, (x⊥,x3) ∈ r3, gq,m⊥ : l2(r2) → l2(r2) is the operator with integral kernel 〈x⊥〉−m⊥/2pb,q(x⊥,x′⊥)〈x′⊥〉−m⊥/2, x⊥,x′⊥ ∈ r2, and t(λ) : l2(r) → l2(r), λ ∈ r \{0}, is the operator with integral kernel 〈x3〉−m3/2rλ(x3 −x′3)〈x′3〉−m3/2, x3,x′3 ∈ r. then we have ‖tq(e)‖≤‖m‖2∞‖gq,m⊥‖‖t(e − 2bq)‖≤‖m‖2∞‖gq,m⊥‖‖t(e − 2bq)‖2, (3.20) ‖tq(e)‖2 ≤‖m‖2∞‖gq,m⊥‖2 ‖t(e − 2bq)‖2, (3.21) where ‖m‖∞ := ‖m‖l∞(r2). evidently, ‖gq,m⊥‖≤ 1, (3.22) ‖gq,m⊥‖22 ≤ tr pq〈x⊥〉−m⊥pq = b 2π ∫ r2 〈x⊥〉−m⊥dx⊥ (3.23) (see (3.15)), and ‖t(e − 2bq)‖22 ≤ 1 4|e − 2bq| ∫ r 〈x3〉−m3dx3 (3.24) (see (3.16)). now the combination of (3.20), (3.22), and (3.24) yields (3.17), while the combination of (3.21), (3.23), and (3.24) yields (3.18). remark: using more sophisticated tools than those of the proof of proposition 3.2, it is shown in [7] that for e �= 2bq we have not only tq(e) ∈ s2, but also tq(e) ∈ s1. we will not use this fact here. proposition 3.3 assume that v satisfies (1.1). let e ∈ r \ 2bz+, q ∈ z+. then we have 0 ≤ im tq(e) ∈ sp with any p > 2/m⊥. if e < 2bq, then im tq(e) = 0. if e > 2bq, then the estimate n+(s; im tq(e)) ≤ c3 ( 1 + b (e − 2bq)−1/m⊥s−2/m⊥ ) (3.25) holds for each s > 0 with c3 independent of s, b, and e. moreover, if e > 2bq, then we have ‖im tq(e)‖1 = tr im tq(e) = b 4π (e − 2bq)−1/2 ∫ r3 |v (x)|dx. (3.26) 7, 2(2005) spectral shift function for magnetic schrödinger operators 187 proof. by (3.19), we have im tq(e) = mgp,m⊥ ⊗ im t(e − 2bq)m. if e < 2bq, then im t(e−2bq) = 0. if e > 2bq, then im t(e−2bq) admits the integral kernel 1 2 √ e − 2bq〈x3〉 −m3/2 cos ( √ e − 2bq (x3 −x′3))〈x′3〉−m3/2, x3,x′3 ∈ r. since the function 〈x⊥〉−m⊥/2 is radially symmetric, the eigenvalues νk, k ∈ n, of the operator gp,m⊥ ≥ 0 can be computed explicitly, and for k ≥ k0 we have νk ≤ c′3bm⊥/2k−m⊥/2 with k0 ∈ n and c′3 independent of b and e (see the proof of [7, lemma 9.4]). further, if e > 2bq, we have rank im t(e − 2bq) = 2, and the eigenvalues of im t(e−2bq) are upper-bounded by 1 2 √ e−2bq ∫ r 〈x3〉−m3dx3. therefore, n+(s; im tq(e)) ≤ k0 + 2 ( c′3‖m‖2∞bm⊥/2s−1 2 √ e − 2bq ∫ r 〈x3〉−m3dx3 )2/m⊥ , s > 0, which entails immediately (3.25). finally, if we write the trace of the operator im tq(e) as the integral of the diagonal value of its kernel, and take into account (3.15) and (3.16), we get (3.26). proposition 3.4 [10, proposition 4.2] let q ∈ z+, λ ∈ r, |λ| ∈ (0,b), and δ > 0. assume that v satisfies (1.1). then the operator series t +q (2bq + λ + iδ) :=∑∞ l=q+1 tl(2bq + λ + iδ), and t +q (2bq + λ) := ∞∑ l=q+1 tl(2bq + λ) (3.27) converge in s2. moreover, ‖t +q (2bq + λ)‖22 ≤ c0b 8π ∞∑ l=q+1 (2b(l−q) −λ)−3/2 ∫ r3 v (x)dx. (3.28) finally, limδ↓0 ‖t +q (2bq + λ + iδ) −t +q (2bq + λ)‖2 = 0. 4 proofs of the main results 4.1 proofs of the results on the singularities of the ssf at the landau levels the first step in the proofs of both theorems 2.1 and 2.2 is to show that we can replace the operator t (e + i0) by tq(e) in the r.h.s of (3.10) when we deal with the first asymptotic term of ξ̃(e; h±,h0) as the energy e approaches a given landau 188 georgi raikov 7, 2(2005) level 2bq, q ∈ z+. more precisely, we pick q ∈ z+, λ ∈ r with |λ| ∈ (0,b), and set t −q (2bq + λ) := ∑q−1 l=0 tl(2bq + λ); if q = 0 the sum should be set equal to zero. evidently, t (2bq + λ + i0) = t −q (2bq + λ) + tq(2bq + λ) + t + q (2bq + λ), re t (2bq + λ + i0) = re t −q (2bq + λ) + re tq(2bq + λ) + t + q (2bq + λ), im t (2bq + λ + i0) = im t −q (2bq + λ) + im tq(2bq + λ), the operator t +q (2bq + λ) being defined in (3.27). combining the weyl inequalities (2.3), lemma 3.1, (3.26), the chebyshev-type estimates (2.5) with p = 2, (3.18), and (3.28), we easily find that the asymptotic estimates ∫ r n±(1 + ε; re tq(2bq + λ) + t im tq(2bq + λ)) dµ(t) + o(1) ≤ ∫ r n±(1; re t (2bq + λ + i0) + t im t (e + i0)) dµ(t) ≤ ∫ r n±(1 −ε; re tq(2bq + λ) + t im tq(2bq + λ)) dµ(t) + o(1) (4.1) hold as λ → 0 for each ε ∈ (0, 1) (see [10, proposition 5.1] for details). if λ > 0, then tq(2bq −λ) is a self-adjoint operator with integral kernel 1 2π √ |v (x⊥,x3)| pq,b(x⊥,x′⊥) ∫ r eip(x3−x ′ 3) p2 + λ dp √ |v (x′⊥,x′3)| = 1 2 √ λ √ |v (x⊥,x3)| pq,b(x⊥,x′⊥)e− √ λ|x3−x′3| √ |v (x′⊥,x′3)|, (x⊥,x3), (x′⊥,x′3) ∈ r3. in particular, im tq(2bq −λ) = 0, and re tq(2bq−λ) = tq(2bq −λ) ≥ 0. therefore,∫ r n±(s; re tq(2bq−λ) + t im tq(2bq−λ)) dµ(t) = n±(s; tq(2bq−λ)), s > 0, λ > 0. (4.2) since tq(2bq −λ) ≥ 0, we have n−(s; tq(2bq −λ)) = 0 for all s > 0 and λ > 0, which combined with (3.10), (4.1), and (4.2), implies (2.7). in order to prove (2.8), we write tq(2bq −λ) = oq(λ) + t̃q(λ) where oq(λ) is an operator with integral kernel 1 2 √ λ √ |v (x⊥,x3)| pq,b(x⊥,x′⊥) √ |v (x′⊥,x′3)|, (x⊥,x3), (x′⊥,x′3) ∈ r3, and t̃q(λ) := tq(2bq −λ) −oq(λ). by (1.2) we have n − limλ↓0 t̃q(λ) = t̃q(0) where t̃q(0) is a compact operator with integral kernel −1 2 √ |v (x⊥,x3)| pq,b(x⊥,x′⊥)|x3 −x′3| √ |v (x′⊥,x′3)|, (x⊥,x3), (x′⊥,x′3) ∈ r3. 7, 2(2005) spectral shift function for magnetic schrödinger operators 189 hence, the weyl inequalities easily imply that the asymptotic estimates n+(s ′;oq(λ)) + o(1) ≤ n+(s; tq(2bq −λ)) ≤ n+(s′′;oq(λ)) + o(1) (4.3) hold for every 0 < s′ < s < s′′ as λ ↓ 0. further, define the operator k : l2(r3) → l2(r2) by (ku)(x⊥) := ∫ r2 ∫ r pq,b(x⊥,x′⊥) √ |v (x′⊥,x′3)|u(x′⊥,x′3) dx′3 dx′⊥, x⊥ ∈ r2, where u ∈ l2(r3). the adjoint operator k∗ : l2(r2) → l2(r3) is given by (k∗v)(x⊥,x3) := √ |v (x⊥,x3)| ∫ r2 pq,b(x⊥,x′⊥)v(x′⊥) dx′⊥, (x⊥,x3) ∈ r3, where v ∈ l2(r2). obviously, oq(λ) = 1 2 √ λ k∗k, pqwpq = k k∗. therefore, n+(s;oq(λ)) = n+(s2 √ λ; pqwpq), s > 0, λ > 0. (4.4) now the combination of (3.10) with (4.1) – (4.4) entails (2.8). thus, we are done with the proof of theorem 2.1. in order to complete the proof of theorem 2.2, we recall that if λ > 0, then the operator re tq(2bq + λ) admits the integral kernel − 1 2 √ λ √ |v (x⊥,x3)|sin ( √ λ|x3 −x′3|)pq,b(x⊥,x′⊥) √ |v (x′⊥,x′3)|, (x⊥,x3), (x′⊥,x ′ 3) ∈ r3, and hence n − limλ↓0 re tq(2bq + λ) = t̃q(0). applying the weyl inequalities and the evident identities ∫ r n±(s; tt )dµ(t) = 1 π tr arctan (s−1t ), s > 0, where t = t ∗ ≥ 0, t ∈ s1, we find that asymptotic estimates 1 π tr arctan (((1 + ε)s)−1im tq(2bq + λ)) + o(1) ≤ ∫ r n±(s; re tq(2bq + λ) + t im tq(2bq + λ))dµ(t) ≤ 1 π tr arctan (((1 −ε)s)−1im tq(2bq + λ)) + o(1) (4.5) are valid as λ ↓ 0 for each s > 0 and ε ∈ (0, 1). define the operator k : l2(r3) → l2(r2)2 by ku := v = (v1,v2) ∈ l2(r2)2, u ∈ l2(r3), where v1(x⊥) := ∫ r2 ∫ r pq,b(x⊥,x′⊥) cos( √ λx′3) √ |v (x′⊥,x′3)|u(x′⊥,x′3) dx′3 dx′⊥, v2(x⊥) := ∫ r2 ∫ r pq,b(x⊥,x′⊥) sin( √ λx′3) √ |v (x′⊥,x′3)|u(x′⊥,x′3) dx′3 dx′⊥, x⊥ ∈ r2. 190 georgi raikov 7, 2(2005) evidently, the adjoint operator k∗ : l2(r2)2 → l2(r3) is given by (k∗v)(x⊥,x3) := cos( √ λx3) √ |v (x⊥,x3)| ∫ r2 pq,b(x⊥,x′⊥)v1(x′⊥) dx′⊥+ sin( √ λx3) √ |v (x⊥,x3)| ∫ r2 pq,b(x⊥,x′⊥)v2(x′⊥) dx′⊥, (x⊥,x3) ∈ r3, where v = (v1,v2) ∈ l2(r2)2. obviously, im tq(2bq + λ) = 1 2 √ λ k∗k, pqwλpq = kk∗, n+(s; im tq(2bq + λ)) = n+(s2 √ λ; pqwλpq), s > 0, λ > 0, and, therefore, tr arctan (s−1im tq(2bq + λ)) = tr arctan ((s2 √ λ)−1pqwλpq), s > 0, λ > 0. (4.6) now the combination of (3.10), (4.1), (4.5), and (4.6) yields (2.10). 4.2 proofs of the results on the strong-magnetic-field asymptotics of the ssf in this subsection we prove theorems 2.3 and 2.4 under the additional assumption that ±v ≥ 0. as before if v ≥ 0 (or if v ≤ 0), we will write h+ and χ+(x⊥), x⊥ ∈ r2, (or h− and χ−(x⊥)) instead of h and χ(x⊥) respectively. first, we prove theorem 2.3. for brevity set a = a(b) = re t (eb + λ), b = b(b) = im t (eb + λ). note that if e ∈ (0,∞) \ 2z+, and λ ∈ r, then (3.8) and (3.9) imply ‖a(b)‖ = o(b−1/2), ‖b(b)‖ = o(b−1/2), ‖b(b)‖1 = o(b1/2), b →∞. (4.7) assume that b is so large that ‖a(b)‖ < 1. then the operator i − a is boundedly invertible, and limb→∞ ‖(i − a(b))−1‖ = 1. by the birman-schwinger principle we have ∫ r n±(1; a + tb)dµ(t) = ∫ r n±(1; tb 1/2(i ∓a)−1b1/2)dµ(t) = ∫ ∞ 0 n+(s; b 1/2(i ∓a)−1b1/2)dµ(s) = 1 π tr arctan ( b1/2(i ∓a)−1b1/2 ) . (4.8) further, tr arctan ( b1/2(i ±a)−1b1/2 ) ≤ tr ( b1/2(i ±a)−1b1/2 ) = tr b ∓ tr ( (i ±a)−1ab ) , (4.9) 7, 2(2005) spectral shift function for magnetic schrödinger operators 191 tr arctan ( b1/2(i ±a)−1b1/2 ) ≥ tr ( b1/2(i ±a)−1b1/2 ) − 1 3 ‖b1/2(i ±a)−1b1/2‖33 = tr b ∓ tr ( (i ±a)−1ab ) − 1 3 ‖b1/2(i ±a)−1b1/2‖33. (4.10) by (4.7) we have |tr ( (i ±a)−1ab ) | ≤ ‖(i ±a)−1a‖‖b‖1 = o(1), b →∞, (4.11) ‖b1/2(i ±a)−1b1/2‖33 ≤ ≤‖b1/2(i ±a)−1b1/2‖2 ‖b1/2(i ±a)−1b1/2‖1 = o(b−1/2), b →∞. (4.12) putting together (4.8) – (4.12), and bearing in mind (3.10), we get ξ(eb + λ; h±(b),h0(b)) = ± 1 π tr b(b) + o(1), b →∞. (4.13) recalling (3.9), we find that the asymptotic estimate tr b(b) = b1/2 4π [e/2]∑ l=0 (e − 2l)−1/2 ∫ r3 |v (x)|dx + o(b−1/2) (4.14) holds as b →∞. now the combination of (4.13) and (4.14) yields (2.18). next, we pass to the proof of theorem 2.4 under the additional assumption that ±v ≥ 0. to this end we establish some auxiliary results. introduce the operator τ(x⊥; z) := |v (x⊥, .)|1/2(χ0 −z)−1|v (x⊥, .)|1/2, defined on l2(r), and depending on the parameters x⊥ ∈ r2 and z ∈ c with im z > 0. the operator τ(x⊥; z) admits the integral kernel |v (x⊥,x3)|1/2rz(x3 −x′3)|v (x⊥,x′3)|1/2, x3,x′3 ∈ r. evidently, τ(x⊥; z) ∈ s2. for x⊥ ∈ r2, λ ∈ r\{0}, define the operator τ(x⊥; λ+i0) : l2(r) → l2(r) as the operator with integral kernel |v (x⊥,x3)|1/2rλ(x3 −x′3)|v (x⊥,x′3)|1/2, x3,x′3 ∈ r, the function rλ(x), x ∈ r, being defined in (3.16). some explicit simple calculations with the kernel of the operator τ(x⊥; λ + i0) yield the following proposition 4.1 let x⊥ ∈ r2, λ ∈ r \{0}. assume that (1.1) holds. i) we have τ(x⊥; λ + i0) ∈ s2, ‖τ(x⊥; λ + i0)‖22 ≤ 1 4|λ| (∫ r |v (x⊥,x3)|dx3 )2 , (4.15) 192 georgi raikov 7, 2(2005) and τ(x⊥; λ+iδ) → τ(x⊥; λ+i0) in s2 as δ ↓ 0, uniformly with respect to x⊥ ∈ r2. ii) we have im τ(x⊥; λ + i0) ≥ 0, and im τ(x⊥; λ + i0) = 0 if λ < 0. if λ > 0, then rank im τ(x⊥; λ + i0) = 2, and n+(s; im τ(x⊥; λ + i0)) ≤ 2θ ( 1 2 √ λ ∫ r |v (x⊥,x3)|dx3 −s ) , s > 0. (4.16) for x⊥ ∈ r2, λ ∈ r \{0}, s > 0, set ξ±λ,s(x⊥) := ∫ r n±(s; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0) dµ(t). (4.17) corollary 4.1 let (1.1) hold. fix x⊥ ∈ r2, λ ∈ r \{0}, s > 0. then we have ξ(λ; χ±(x⊥),χ0) = ± ξ∓λ,1(x⊥) (4.18) where ξ(·; χ±(x⊥),χ0) is the representative of ssf for the operator pair (χ±(x⊥),χ0) which is monotonous and left-continuous for λ < 0, and continuous for λ > 0. proof. it suffices to apply theorem 3.1 with h± = χ±(x⊥) and h0 = χ0. corollary 4.2 under the assumptions of corollary 4.1 we have ξ±λ,s(·) ∈ l1(r2). (4.19) proof. combine lemma 3.1 for t1 = re τ(x⊥; λ + i0) and t2 = im τ(x⊥; λ + i0), with proposition 4.1. proposition 4.2 let λ > 0. assume that (1.1) holds. then the function∫ r2 ξ±λ,s(x⊥)dx⊥ is continuous with respect to s > 0. proof. fix s > 0. first of all we will show that for almost every (x⊥, t) ∈ r2 × r the functions s′ �→ n±(s′; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) are continuous at the point s′ = s. evidently, this is equivalent to ±s �∈ σ(re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)). (4.20) in order to prove (4.20), we will use an argument quite close to the one of the proof of [29, lemma 4.1]. note that the compact operator re τ(x⊥; λ+i0)+t im τ(x⊥; λ+i0) depends linearly on t. by the fredholm alternative the sets ω±(s,x⊥,λ) := {z ∈ c |±s ∈ σ(re τ(x⊥; λ + i0) + z im τ(x⊥; λ + i0))} either coincide with c, or are discrete. however, i ∈ ω±(s,x⊥,λ) is equivalent to dim ker (χ0 ∓ s−1|v (x⊥, .)| − λ) ≥ 1. on the other hand, it is well-known that the operators χ0 ∓s−1|v (x⊥, .)| have no positive eigenvalues (see e.g. [39, theorem 7, 2(2005) spectral shift function for magnetic schrödinger operators 193 xiii.58]) since (1.1) implies lim|x3|→∞ |x3|v (x⊥,x3) = 0. therefore, dim ker (χ0 ∓ s−1|v (x⊥, .)| − λ) = 0, i �∈ ω±(s,x⊥,λ), and the sets ω±(s,λ,x⊥) are discrete. in particular, |r ∩ ω±(s,x⊥,λ)| = 0. put ω̃±(s,λ) := {(x⊥, t) ∈ r2 × r|±s ∈ σ(re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0))}. the eigenvalues of the compact operator re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0) are continuous, and hence measurable with respect to (x⊥, t) ∈ r2 × r. therefore, the sets ω̃±(s,λ) are measurable, and by the fubini-tonelli theorem |ω̃±(s,λ)| = ∫ r2 ∫ r 1ω̃±(s,λ)(x⊥, t)dtdx⊥ = ∫ r2 |r ∩ ω±(s,x⊥,λ)|dx⊥ = 0 where 1ω̃±(s,λ) denotes the characteristic function of ω ±(s,λ). on the other hand, lim s′→s n±(s ′; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) = = n±(s; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) (4.21) if (x⊥, t) �∈ ω̃±(s,λ). the weyl inequalities (2.3) and estimates (4.15) – (4.16) imply n±(s ′; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) ≤ 1 s′2λ (∫ r |v (x⊥,x3)|dx3 )2 + 2θ ( |t|√ λ ∫ r |v (x⊥,x3)|dx3 −s′ ) . (4.22) note that the r.h.s. is in l1(r2 × r; dx⊥ dµ(t)) for each s′ > 0, and is a sum of two monotonous functions of s′ > 0. bearing in mind (4.21) – (4.22), we apply the dominated convergence theorem, and get lims′→s ∫ r2 ξ±λ,s′ (x⊥)dx⊥ =∫ r2 ξ±λ,s(x⊥)dx⊥. set φ±λ,s(t) := ∫ r2 n±(s; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) dx⊥, t ∈ r. corollary 4.3 assume that (1.1) holds. let λ > 0, s > 0. then lims′→s φ ± λ,s′ (t) = φ±λ,s(t) for almost every t ∈ r. proof. since the functions φ±λ,s(t) are non-increasing with respect to s > 0, the one-sided limits φ±λ,s−0(t) ≥ φ±λ,s+0(t) exist. next, proposition 4.2 implies∫ r2 ξ±λ,s−0(x⊥)dx⊥ = ∫ r2 ξ±λ,s+0(x⊥)dx⊥. by the fubini theorem ∫ r2 ξ±λ,s(x⊥)dx⊥ = ∫ r φ±λ,s(t)dµ(t). hence, ∫ r ( φ±λ,s−0(t) − φ±λ,s+0(t) ) dµ(t) = 0. since the functions φ±λ,s−0(t) − φ±λ,s+0(t) are non-negative, we conclude that ∣∣∣ { t ∈ r ∣∣∣φ±λ,s−0(t) > φ±λ,s+0(t) }∣∣∣ = 0. the following proposition contains key limiting relations used in the proof of theorem 2.4. 194 georgi raikov 7, 2(2005) proposition 4.3 (cf. [7, proposition 7.1]) let (1.1) hold. then we have lim b→∞ b−1tr (re tq(2bq + λ) + t im tq(2bq + λ)) p = 1 2π ∫ r2 tr (re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) p dx⊥ (4.23) for every t ∈ r and each integer p ≥ 2. proof. let t ∈ r, x ∈ r. if λ > 0, set r̃λ,t(x) := −sin( √ λ|x|) 2 √ λ + t cos( √ λ x) 2 √ λ . if λ < 0, then r̃λ,t(x) = rλ(x) = e− √ −λ|x| 2 √ −λ . we have tr (re tq(2bq + λ) + t im tq(2bq + λ)) p = � r2p � rp π p j=1|v (x⊥,j , x3,j )|π′ p j=1pq,b(x⊥,j , x⊥,j+1)r̃λ,t(x3,j − x3,j+1)πpj=1dx⊥,j dx3,j where the notation π′pj=1 means that in the product of p factors the variables x⊥,p+1 and x3,p+1 should be set equal respectively to x⊥,1 and x3,1. change the variables x⊥,1 = x ′ ⊥,1, x⊥,j = x ′ ⊥,1 + b −1/2x′⊥,j, j = 2, . . . ,p. (4.24) thus we obtain tr (re tq(2bq + λ) + t im tq(2bq + λ)) p = b ∫ r2p ∫ rp |v (x′⊥,1,x3,1)|πpj=2|v (x′⊥,1 + b−1/2x′⊥,j,x3,j )|pq,1(0,x′⊥,2)× πp−1j=2pq,1(x⊥′,j,x⊥′,j+1)pq,1(x′⊥,p, 0) π′ p j=1r̃λ,t(x3,j −x3,j+1)πpj=1dx′⊥,jdx3,j. (4.25) here and in the sequel, if p = 2, then the product πp−1j=2pq,b(x⊥′,j,x⊥′,j+1) should be set equal to one. bearing in mind (1.1) and (3.14), and applying the dominated convergence theorem, we easily find that (4.25) entails lim b→∞ b−1tr (re tq(2bq + λ + i0) + t im tq(2bq + λ + i0)) p = ∫ r2 ∫ rp πpj=1|v (x⊥,1,x3,j )|π′ p j=1r̃λ,t(x3,j −x3,j+1)dx⊥,1πpj=1dx3,j × ∫ r2(p−1) pq,1(0,x⊥,2)πp−1j=2pq,1(x⊥,j,x⊥,j+1)pq,1(x⊥,p, 0)πpj=2dx⊥,j = ∫ r2 tr ( re τ(x⊥,1; λ + i0) + t im τ(x⊥,1; λ + i0) )p dx⊥,1 × ∫ r2(p−1) pq,1(0,x⊥,2)πp−1j=2pq,1(x⊥,j,x⊥,j+1)pq,1(x⊥,p, 0)πpj=2dx⊥,j. 7, 2(2005) spectral shift function for magnetic schrödinger operators 195 in order to conclude that the above limiting relation is equivalent to (4.23), it remains to recall (3.15), and note that ∫ r2(p−1) pq,1(0,x⊥,2)πp−1j=2pq,1(x⊥,j,x⊥,j+1)pq,1(x⊥,p, 0)πpj=2dx⊥,j = pq,1(0, 0) = 1 2π . corollary 4.4 assume that the assumptions of theorem 2.4 hold. then we have lim b→∞ b−1n±(s; re tq(2bq + λ) + t im tq(2bq + λ)) = 1 2π ∫ r2 n±(s; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0))dx⊥, for each t ∈ r, provided that s > 0 is a continuity point of the r.h.s. proof. it suffices to notice that the norm of the operator tq(2bq + λ) is uniformly bounded with respect to b, and to apply a suitable version of the kac-murdockszegö theorem (see e.g. [32, lemma 3.1]) which tells us that under appropriate hypotheses the convergence of the moments of a given measure implies the convergence of the measure itself, and to take into account proposition 4.3. now we are in position to prove theorem 2.4. by a. pushnitski’s representation of the ssf (see (3.10) and (4.18)), in order to check the validity of (2.21), it suffices to show that lim b→∞ b−1 ∫ r n±(1; re t (2bq + λ + i0) + tim t (2bq + λ + i0))dµ(t) = 1 2π ∫ r ∫ r2 n±(1; re τ(x⊥; λ + i0) + tim τ(x⊥; λ + i0))dµ(t)dx⊥. (4.26) arguing as in the derivation of (4.1), we easily find that the asymptotic estimates ∫ r n±(1 + ε; re tq(2bq + λ) + t im tq(2bq + λ)) dµ(t) + o(b) ≤ ∫ r n±(1; re t (2bq + λ + i0) + t im t (2bq + λ + i0)) dµ(t) ≤ ∫ r n±(1 −ε; re tq(2bq + λ) + tim tq(2bq + λ)) dµ(t) + o(b), (4.27) hold as b → ∞ for each ε ∈ (0, 1). assume λ > 0. corollary 4.3 and corollary 4.4 imply lim b→∞ b−1n±(s; re tq(2bq + λ) + t im tq(2bq + λ)) = 1 2π ∫ r2 n±(s; re τ(x⊥; λ + i0) + tim τ(x⊥; λ + i0)) dx⊥ (4.28) 196 georgi raikov 7, 2(2005) for any fixed s > 0, and almost every t ∈ r. further, by (2.3), (2.5) with p = 2, proposition 3.2, and proposition 3.3 we have b−1n±(s; re tq(2bq+λ+i0)+t im tq(2bq+λ+i0)) ≤ c4(1+|t|2/m⊥ ), t ∈ r, (4.29) with c4 which may depend on s > 0, λ ∈ r \{0}, q and m⊥ but is independent of b ≥ 1 and t. note that the function on the r.h.s of (4.29) is in l1(r; dµ). by (4.28) – (4.29), the dominated convergence theorem and the fubini theorem imply lim b→∞ b−1 ∫ r n±(s; re tq(2bq + λ + i0) + t im tq(2bq + λ + i0) dµ(t) = 1 2π ∫ r2 ∫ r n±(s; re τ(x⊥; λ + i0) + t im τ(x⊥; λ + i0)) dµ(t) dx⊥, s > 0. 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[45] d. r. yafaev, mathematical scattering theory. general theory. translations of mathematical monographs, 105 ams, providence, ri, 1992. cubo a mathematical journal vol.21, no¯ 01, (37–48). april 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000100037 commutator criteria for strong mixing ii. more general and simpler s. richard 1 graduate school of mathematics, nagoya university, chikusa-ku, nagoya 464-8602, japan richard@math.nagoya-u.ac.jp r. tiedra de aldecoa 2 facultad de matemáticas, pontificia universidad católica de chile, av. vicuña mackenna 4860, santiago, chile rtiedra@mat.puc.cl abstract we present a new criterion, based on commutator methods, for the strong mixing property of unitary representations of topological groups equipped with a proper length function. our result generalises and unifies recent results on the strong mixing property of discrete flows {un}n∈z and continuous flows {e −ith}t∈r induced by unitary operators u and self-adjoint operators h in a hilbert space. as an application, we present a short alternative proof (not using convolutions) of the strong mixing property of the left regular representation of σ-compact locally compact groups. 1supported by the granttopological invariants through scattering theory and noncommutative geometry from nagoya university, and by jsps grant-in-aid for scientific research (c) no 18k03328, and on leave of absence from univ. lyon, université claude bernard lyon 1, cnrs umr 5208, institut camille jordan, 43 blvd. du 11 novembre 1918, f-69622 villeurbanne cedex, france. 2supported by the chilean fondecyt grant 1170008. http://dx.doi.org/10.4067/s0719-06462019000100037 38 s. richard and r. tiedra de aldecoa cubo 21, 1 (2019) resumen presentamos un nuevo criterio, basado en métodos de conmutadores, para la propiedad de mezcla fuerte de representaciones unitarias de grupos topológicos dotados de una función de longitud propia. nuestro resultado generaliza y unifica resultados recientes acerca de la propiedad de mezcla fuerte de flujos discretos {un}n∈z y flujos continuos {e−ith}t∈r inducidos por operadores unitarios u y operadores autoadjuntos h en un espacio de hilbert. como aplicación, presentamos una demostración corta alternativa (sin usar convoluciones) de la propiedad de mezcla fuerte de la representación regular de grupos localmente compactos σ-compactos. keywords and phrases: strong mixing, unitary representations, commutator methods. 2010 ams mathematics subject classification: 22d10, 37a25, 58j51, 81q10. cubo 21, 1 (2019) commutator criteria for strong mixing ii. . . . 39 1 introduction in the recent paper [14], itself motivated by the previous papers [8, 12, 13, 15], it has been shown that commutator methods for unitary and self-adjoint operators can be used to establish strong mixing. the main results of [14] are the following two commutator criteria for strong mixing. first, given a unitary operator u in a hilbert space h, assume there exists an auxiliary selfadjoint operator a in h such that the commutators [a,un] exist and are bounded in some precise sense, and such that the strong limit d1 := s-lim n→∞ 1 n [a,un]u−n (1.1) exists. then, the discrete flow {un}n∈z is strongly mixing in ker(d1) ⊥. second, given a selfadjoint operator h in h, assume there exists an auxiliary self-adjoint operator a in h such that the commutators [a,e−ith] exist and are bounded in some precise sense, and such that the strong limit d2 := s-lim t→∞ 1 t [a,e−ith] eith (1.2) exists. then, the continuous flow {e−ith}t∈r is strongly mixing in ker(d2) ⊥. these criteria were then applied to skew products of compact lie groups, furstenberg-type transformations, time changes of horocycle flows and adjacency operators on graphs. the purpose of this note is to unify these two commutator criteria into a single, more general, commutator criterion for strong mixing of unitary representations of topological groups, and also to remove an unnecessary invariance assumption made in [14]. our main result is the following. we consider a topological group x equipped with a proper length function ℓ : x → r+, a unitary representation u : x → u (h), and a net {xj}j∈j in x with xj → ∞ (see section 2 for precise definitions). also, we assume there exists an auxiliary self-adjoint operator a in h such that the commutators [a,u(xj)] exist and are bounded in some precise sense, and such that the strong limit d := s-lim j 1 ℓ(xj) [a,u(xj)]u(xj) −1 (1.3) exists. then, under these assumptions we show that the unitary representation u is strongly mixing in ker(d)⊥ along the net {xj}j∈j (theorem 2.3). as a corollary, we obtain criteria for strong mixing in the cases of unitary representations of compactly generated locally compact hausdorff groups (corollary 2.5) and the euclidean group rd (corollary 2.7). these results generalise the commutator criteria of [14] for the strong mixing of discrete and continuous flows, as well as the strong limit (1.3) generalises the strong limits (1.1) and (1.2) (see remarks 2.6 and 2.8). to conclude, we present in example 2.9 an application which was not possible to cover with the results of [14]: a short alternative proof (not using convolutions) of the strong mixing property of the left regular representation of σ-compact locally compact hausdorff groups. 40 s. richard and r. tiedra de aldecoa cubo 21, 1 (2019) we refer the reader to [4, 6, 9, 10, 11, 16] for references on strong mixing properties of unitary representations of groups. 2 commutator criteria for strong mixing we start with a short review of basic facts on commutators of operators and regularity classes associated with them. we refer to [1, chap. 5-6] for more details. let h be an arbitrary hilbert space with scalar product 〈·, ·〉 antilinear in the first argument, denote by b(h) the set of bounded linear operators on h, and write ‖ · ‖ both for the norm on h and the norm on b(h). let a be a self-adjoint operator in h with domain d(a), and take s ∈ b(h). for any k ∈ n, we say that s belongs to ck(a), with notation s ∈ ck(a), if the map r ∋ t 7→ e−ita seita ∈ b(h) (2.1) is strongly of class ck. in the case k = 1, one has s ∈ c1(a) if and only if the quadratic form d(a) ∋ ϕ 7→ 〈 ϕ,isaϕ 〉 − 〈 aϕ,isϕ 〉 ∈ c is continuous for the topology induced by h on d(a). we denote by [is,a] the bounded operator associated with the continuous extension of this form, or equivalently the strong derivative of the map (2.1) at t = 0. moreover, if we set aε := (iε) −1(eiεa −1) for ε ∈ r \ {0}, we have (see [1, lemma 6.2.3(a)]): s-lim εց0 [is,aε] = [is,a]. (2.2) now, if h is a self-adjoint operator in h with domain d(h) and spectrum σ(h), we say that h is of class ck(a) if (h −z)−1 ∈ ck(a) for some z ∈ c \σ(h). in particular, h is of class c1(a) if and only if the quadratic form d(a) ∋ ϕ 7→ 〈 ϕ,(h − z)−1aϕ 〉 − 〈 aϕ,(h − z)−1ϕ 〉 ∈ c extends continuously to a bounded form with corresponding operator denoted by [(h − z)−1,a] ∈ b(h). in such a case, the set d(h) ∩ d(a) is a core for h and the quadratic form d(h) ∩ d(a) ∋ ϕ 7→ 〈 hϕ,aϕ 〉 − 〈 aϕ,hϕ 〉 ∈ c is continuous in the topology of d(h) (see [1, thm. 6.2.10(b)]). this form then extends uniquely to a continuous quadratic form on d(h) which can be identified with a continuous operator [h,a] from d(h) to the adjoint space d(h)∗. in addition, the following relation holds in b(h) (see [1, thm. 6.2.10(b)]): [(h − z)−1,a] = −(h − z)−1[h,a](h − z)−1. (2.3) cubo 21, 1 (2019) commutator criteria for strong mixing ii. . . . 41 with this, we can now present our first result, which is at the root of the new commutator criterion for strong mixing. for it, we recall that a net {xj}j∈j in a topological space x diverges to infinity, with notation xj → ∞, if {xj}j∈j has no limit point in x. this implies that for each compact set k ⊂ x, there exists jk ∈ j such that xj /∈ k for j ≥ jk. in particular, x is not compact. we also fix the notations u (h) for the set of unitary operators on h and r+ := [0,∞). proposition 2.1. let {uj}j∈j be a net in u (h), let {ℓj}j∈j ⊂ r+ satisfy ℓj → ∞, assume there exists a self-adjoint operator a in h such that uj ∈ c 1(a) for each j ∈ j, and suppose that the strong limit d := s-lim j 1 ℓj [a,uj]u −1 j exists. then, limj 〈 ϕ,ujψ 〉 = 0 for all ϕ ∈ ker(d)⊥ and ψ ∈ h. before the proof, we note that for j ∈ j large enough (so that ℓj 6= 0) the operators 1 ℓj [a,uj]u −1 j are well-defined, bounded and self-adjoint. therefore, their strong limit d is also bounded and self-adjoint. proof. let ϕ = dϕ̃ ∈ dd(a) and ψ ∈ d(a), take j ∈ j large enough, and set dj := 1 ℓj [a,uj]u −1 j . since uj and u −1 j belong to c 1(a) (see [1, prop. 5.1.6(a)]), both ujψ and u −1 j ϕ̃ belong to d(a). thus, ∣∣〈ϕ,ujψ 〉∣∣ = ∣∣〈(d − dj)ϕ̃,ujψ 〉 + 〈 djϕ̃,ujψ 〉∣∣ ≤ ∥∥(d − dj)ϕ̃ ∥∥‖ψ‖ + 1 ℓj ∣∣〈[a,uj ] u−1j ϕ̃,ujψ 〉∣∣ ≤ ∥∥(d − dj)ϕ̃ ∥∥‖ψ‖ + 1 ℓj ∣∣〈aϕ̃,ujψ 〉∣∣ + 1 ℓj ∣∣〈ujau−1j ϕ̃,ujψ 〉∣∣ ≤ ∥∥(d − dj)ϕ̃ ∥∥‖ψ‖ + 1 ℓj ∥∥aϕ̃ ∥∥‖ψ‖ + 1 ℓj ∥∥ϕ̃ ∥∥‖aψ‖. since d = s-limj dj and ℓj → ∞, we infer that limj 〈 ϕ,ujψ 〉 = 0, and thus the claim follows by the density of dd(a) in dh = ker(d)⊥ and the density of d(a) in h. in the sequel, we assume that the unitary operators uj are given by a unitary representation of a topological group x. we also assume that the scalars ℓj are given by a proper length function on x, that is, a function ℓ : x → r+ satisfying the following properties (e denotes the identity of x): (l1) ℓ(e) = 0, 42 s. richard and r. tiedra de aldecoa cubo 21, 1 (2019) (l2) ℓ(x−1) = ℓ(x) for all x ∈ x, (l3) ℓ(xy) ≤ ℓ(x) + ℓ(y) for all x,y ∈ x, (l4) if k ⊂ r+ is compact, then ℓ −1(k) ⊂ x is relatively compact. remark 2.2 (topological groups with a proper left-invariant pseudo-metric). let x be a hausdorff topological group equipped with a proper left-invariant pseudo-metric d : x × x → r+ (see [7, def. 2.a.5 & 2.a.7]). then, simple calculations show that the associated length function ℓ : x → r+ given by ℓ(x) := d(e,x) satisfies the properties (l1)-(l4) above. examples of groups admitting a proper left-invariant pseudo-metric are σ-compact locally compact hausdorff groups [7, prop. 4.a.2], as for instance compactly generated locally compact hausdorff groups with the word metric [7, prop. 4.b.4(2)]. the next theorem provides a general commutator criterion for the strong mixing property of a unitary representation of a topological group. before stating it, we recall that if a topological group x is equipped with a proper length function ℓ, and if {xj}j∈j is a net in x with xj → ∞, then ℓ(xj) → ∞ (this can be shown by absurd using the property (l4) above). theorem 2.3 (topological groups). let x be a topological group equipped with a proper length function ℓ, let u : x → u (h) be a unitary representation of x, let {xj}j∈j be a net in x with xj → ∞, assume there exists a self-adjoint operator a in h such that u(xj) ∈ c 1(a) for each j ∈ j, and suppose that the strong limit d := s-lim j 1 ℓ(xj) [a,u(xj)]u(xj) −1 (2.4) exists. then, (a) limj 〈 ϕ,u(xj)ψ 〉 = 0 for all ϕ ∈ ker(d)⊥ and ψ ∈ h, (b) u has no nontrivial finite-dimensional unitary subrepresentation in ker(d)⊥. proof. the claim (a) follows from proposition 2.1 and the fact that ℓ(xj) → ∞. the claim (b) follows from (a) and the fact that matrix coefficients of finite-dimensional unitary representations of a group do not vanish at infinity (see for instance [3, rem. 2.15(iii)]). remark 2.4. (i) the result of theorem 2.3(a) amounts to the strong mixing property of the unitary representation u in ker(d)⊥ along the net {xj}j∈j, as mentioned in the introduction. if the strong limit (2.4) exists for all nets {xj}j∈j with xj → ∞, then theorem 2.3(a) implies the usual strong mixing property of the unitary representation u in ker(d)⊥. (ii) one can easily see that theorem 2.3 remains true if the scalars ℓ(xj) in (2.4) are replaced by (f ◦ ℓ)(xj), with f : r+ → r+ any proper function. for simplicity, we decided to present only the case f = idr+, but we note this additional freedom might be useful in applications. cubo 21, 1 (2019) commutator criteria for strong mixing ii. . . . 43 theorem 2.3 and remark 2.2 imply the following result in the particular case of a compactly generated locally compact group x : corollary 2.5 (compactly generated locally compact groups). let x be a compactly generated locally compact hausdorff group with generating set y and word length function ℓ, let u : x → u (h) be a unitary representation of x, let {xj}j∈j be a net in x with xj → ∞, assume there exists a selfadjoint operator a in h such that u(y) ∈ c1(a) for each y ∈ y, and suppose that the strong limit d := s-lim j 1 ℓ(xj) [a,u(xj)]u(xj) −1 (2.5) exists. then, (a) limj 〈 ϕ,u(xj)ψ 〉 = 0 for all ϕ ∈ ker(d)⊥ and ψ ∈ h, (b) u has no nontrivial finite-dimensional unitary subrepresentation in ker(d)⊥. proof. in order to apply theorem 2.3, we first note from remark 2.2 that the word length function ℓ is a proper length function. second, we note that x = ⋃ n≥1 (y ∪ y−1)n. therefore, for each x ∈ x there exist n ≥ 1, y1, . . . ,yn ∈ y and m1, . . . ,mn ∈ {±1} such that x = y m1 1 · · ·y mn n . thus, u(x) = u ( y m1 1 · · ·ymnn ) = u(y1) m1 · · ·u(yn) mn, and it follows from the inclusions u(y1), . . . ,u(yn) ∈ c 1(a) and standard results on commutator methods [1, prop. 5.1.5 & 5.1.6(a)] that u(x) ∈ c1(a). thus, we have u(xj) ∈ c 1(a) for each j ∈ j, and the commutators [a,u(xj)] appearing in (2.5) make sense. so, we can apply theorem 2.3 to conclude. remark 2.6. corollary 2.5 is a generalisation of [14, thm. 3.1] to the case of unitary representations of compactly generated locally compact hausdorff groups. indeed, if we let x be the additive group z with generating element 1, take the trivial net {xj = j}j∈n∗ = {n | n ∈ n ∗}, and set u := u(1) in corollary 2.5, then the strong limit (2.5) reduces to d = s-lim n→∞ 1 n [ a,un ] u−n = s-lim n→∞ 1 n n−1∑ n=0 un ( [a,u]u−1 ) u−n, which is the strong limit appearing in [14, thm. 3.1]. in corollary 2.5 we also removed the unnecessary invariance assumption η(d)d(a) ⊂ d(a) for each η ∈ c∞ c (r \ {0}). so, the strong mixing properties for skew products and furstenberg-type transformations established in [14, sec. 3] and [5, sec. 3] can be obtained more directly using corollary 2.5. in the next corollary we consider the case of a strongly continuous unitary representation u : rd → u (h) of the euclidean group rd, d ≥ 1. in such a case stone’s theorem implies the 44 s. richard and r. tiedra de aldecoa cubo 21, 1 (2019) existence of a family of mutually commuting self-adjoint operators h1, . . . ,hd such that u(x) = e−i ∑ d k=1 xkhk for each x = (x1, . . . ,xd) ∈ r d. therefore, we give a criterion for strong mixing in terms of the operators h1, . . . ,hd. we use the shorthand notations h := (h1, . . . ,hd), π(h) := (h1 + i) −1 · · · (hd + i) −1 and x · h := d∑ k=1 xkhk. corollary 2.7 (euclidean group rd). let rd, d ≥ 1, be the euclidean group with euclidean length function ℓ, let u : rd → u (h) be a strongly continuous unitary representation of rd, let {xj}j∈j be a net in r d with xj → ∞, assume there exists a self-adjoint operator a in h such that (hk − i) −1 ∈ c1(a) for each k ∈ {1, . . . ,d}, and suppose that the strong limit d := s-lim j 1 ℓ(xj) ∫1 0 ds e−is(xj·h) π(h) [ i(xj · h),a ] π(h)∗ eis(xj·h) (2.6) exists. then, (a) limj 〈 ϕ,u(xj)ψ 〉 = 0 for all ϕ ∈ ker(d)⊥ and ψ ∈ h, (b) u has no nontrivial finite-dimensional unitary subrepresentation in ker(d)⊥. proof. the proof consists in applying theorem 2.3 with a replaced by a new operator ã that we now define. the inclusions (h1 − i) −1, . . . ,(hd − i) −1 ∈ c1(a) and the standard result on commutator methods [1, prop. 5.1.5] imply that π(h)∗ ∈ c1(a). so, we have π(h)∗ d(a) ⊂ d(a), and the operator ãϕ := π(h)aπ(h)∗ϕ, ϕ ∈ d(a), is essentially self-adjoint (see [1, lemma 7.2.15]). take ϕ ∈ d(a) and j0 ∈ j such that ℓ(xj) > 0 for all j ≥ j0, and define for ε ∈ r \ {0} the operator aε := (iε) −1(eiεa −1). then, we have 〈 ãϕ,u(xj)ϕ 〉 − 〈 ϕ,u(xj)ãϕ 〉 = lim εց0 (〈 ϕ,π(h)aεπ(h) ∗ e−i(xj·h) ϕ 〉 − 〈 ϕ,e−i(xj·h) π(h)aεπ(h) ∗ϕ 〉) = lim εց0 ∫ℓ(xj) 0 dq d dq 〈 ϕ,ei(q−ℓ(xj))(xj·h)/ℓ(xj) π(h)aεπ(h) ∗ e−iq(xj·h)/ℓ(xj) ϕ 〉 = 1 ℓ(xj) lim εց0 ∫ℓ(xj) 0 dq 〈 ϕ,ei(q−ℓ(xj))(xj·h)/ℓ(xj) π(h) [ i(xj · h),aε ] π(h)∗ e−iq(xj·h)/ℓ(xj) ϕ 〉 . (2.7) but, (h1 − i) −1, . . . ,(hd − i) −1 ∈ c1(a). therefore, (2.2) and (2.3) imply that s-lim εց0 π(h) [ i(xj · h),aε ] π(h)∗ = π(h) [ i(xj · h),a ] π(h)∗, cubo 21, 1 (2019) commutator criteria for strong mixing ii. . . . 45 and we can exchange the limit and the integral in (2.7) to obtain 〈 ãϕ,u(xj)ϕ 〉 − 〈 ϕ,u(xj)ãϕ 〉 = 1 ℓ(xj) ∫ℓ(xj) 0 dq 〈 ϕ,ei(q−ℓ(xj))(xj·h)/ℓ(xj) π(h) [ i(xj · h),a ] π(h)∗ e−iq(xj·h)/ℓ(xj) ϕ 〉 = 1 ℓ(xj) ∫ℓ(xj) 0 dr 〈 ϕ,e−ir(xj·h)/ℓ(xj) π(h) [ i(xj · h),a ] π(h)∗ ei(r−ℓ(xj))(xj·h)/ℓ(xj) ϕ 〉 = ∫1 0 ds 〈 ϕ,e−is(xj·h) π(h) [ i(xj · h),a ] π(h)∗ eis(xj·h) u(xj)ϕ 〉 = 〈 ϕ,ℓ(xj)dju(xj)ϕ 〉 with dj := 1 ℓ(xj) ∫1 0 ds e−is(xj·h) π(h) [ i(xj · h),a ] π(h)∗ eis(xj·h) . since d(a) is a core for ã, this implies that u(xj) ∈ c 1(ã) with [ ã,u(xj) ] = ℓ(xj)dju(xj). therefore, we have dj = 1 ℓ(xj) [ ã,u(xj) ] u(xj) −1, and all the assumptions of theorem 2.3 are satisfied with a replaced by ã. remark 2.8. corollary 2.7 is a generalisation of [14, thm. 4.1] to the case of strongly continuous unitary representations of rd for an arbitrary d ≥ 1. indeed, if we set d = 1, write h for h1, and take the trivial net {xj = j}j∈(0,∞) = {t | t > 0} in corollary 2.7, then the strong limit (2.6) reduces to d = s-lim t→∞ 1 t ∫1 0 ds e−is(t·h)(h + i)−1 [ ith,a ] (h − i)−1 eis(t·h) = s-lim t→∞ 1 t ∫t 0 ds e−ish(h + i)−1 [ ih,a ] (h − i)−1 eish, which is (up to a sign) the strong limit appearing in [14, thm. 4.1]. in corollary 2.7, we also removed the unnecessary invariance assumption η(d)d(a) ⊂ d(a) for each η ∈ c∞ c (r \ {0}). so, the strong mixing properties for adjacency operators, time changes of horocycle flows, etc., established in [14, sec. 4] can be obtained more directly using corollary 2.7. to conclude, we add to the list of examples presented in [14] an application which was not possible to cover with the results of [14]. it is a short alternative proof, not using convolutions, of the strong mixing property of the left regular representation of σ-compact locally compact hausdorff groups (see for instance [2, sec. c.4] for the proof using convolutions): 46 s. richard and r. tiedra de aldecoa cubo 21, 1 (2019) example 2.9 (left regular representation). let x be a σ-compact locally compact hausdorff group with left haar measure µ and proper length function ℓ (see remark 2.2). let d ⊂ h be the set of functions x → c with compact support, and let u : x → u (h) be the left regular representation of x on h := l2(x,µ) given by u(x)ϕ := ϕ(x−1 ·), x ∈ x, ϕ ∈ h. let finally a be the maximal multiplication operator in h given by aϕ := ℓϕ ≡ ℓ(·)ϕ, ϕ ∈ d(a) := { ϕ ∈ h | ‖ℓϕ‖ < ∞ } . for ϕ ∈ d and x ∈ x, one has au(x)ϕ − u(x)aϕ = ( ℓ(·) − ℓ(x−1 ·) ) u(x)ϕ. furthermore, the properties (l2)-(l3) of a length function imply that ∣∣(ℓ(·) − ℓ(x−1 ·) )∣∣ ≤ ℓ(x). (2.8) therefore, since d is dense in d(a), it follows that u(x) ∈ c1(a) with [a,u(x)]u(x)−1 = ℓ(·) − ℓ(x−1 ·). now, we take {xj}j∈j a net in x with xj → ∞, and show that d := s-lim j 1 ℓ(xj) [a,u(xj)]u(xj) −1 = −1. (2.9) for this, we first note that for ϕ ∈ h we have ( 1 ℓ(xj) [a,u(xj)]u(xj) −1 + 1 ) ϕ = ℓ(·) − ℓ(x−1 j ·) + ℓ(xj) ℓ(xj) ϕ. next, we note that (2.8) implies that ∣∣∣∣∣ ℓ(·) − ℓ(x−1 j ·) + ℓ(xj) ℓ(xj) ϕ ∣∣∣∣∣ 2 ≤ 4|ϕ|2 ∈ l1(x,µ), and that the properties (l2)-(l3) imply that lim j ∣∣∣∣∣ ℓ(·) − ℓ(x−1 j ·) + ℓ(xj) ℓ(xj) ϕ ∣∣∣∣∣ 2 ≤ lim j ∣∣∣∣ 2ℓ(·) ℓ(xj) ϕ ∣∣∣∣ 2 = 0 µ-almost everywhere. therefore, we can apply lebesgue dominated convergence theorem to get the equality s-lim j ( 1 ℓ(xj) [a,u(xj)]u(xj) −1 + 1 ) ϕ = 0, which proves (2.9). so, theorem 2.3 applies with d = −1, and thus limj 〈 ϕ,u(xj)ψ 〉 = 0 for all ϕ,ψ ∈ h. cubo 21, 1 (2019) commutator criteria for strong mixing ii. . . . 47 references [1] w. o. amrein, a. boutet de monvel, and v. georgescu. c0-groups, commutator methods and spectral theory of n-body hamiltonians, volume 135 of progress in mathematics. birkhäuser verlag, basel, 1996. 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[16] r. j. zimmer. ergodic theory and semisimple groups, volume 81 of monographs in mathematics. birkhäuser verlag, basel, 1984. introduction commutator criteria for strong mixing a mathematical journal vol. 7, no 3, (75 85). december 2005. relations of al functions over subvarieties in a hyperelliptic jacobian shigeki matsutani 8-21-1 higashi-linkan, sagamihara, 228-0811, japan rxb01142@nifty.com abstract the sine-gordon equation has hyperelliptic al function solutions over a hyperelliptic jacobian for y2 = f (x) of arbitrary genus g. this article gives an extension of the sine-gordon equation to that over subvarieties of the hyperelliptic jacobian. we also obtain the condition that the sine-gordon equation in a proper subvariety of the jacobian is defined. resumen la ecuación de sine-gordon tiene soluciones funciones hipereĺıpticas sobre un jacobiano hipereĺıptico para y2 = f (x) de género arbitrario g. en este art́ıculo damos una extensión de la ecuación de sine-gordon sobre subvariedades de jacobiano hipereĺıptico. también obtenemos la condición para que la ecuación de sine-gordon esté definida en una subvariedad propia del jacobiano. key words and phrases: sine-gordon equation, nonlinear integrable differential equation, hyperelliptic functions, a subvariety in a jacobian math. subj. class.: primary 14h05, 14k12; secondary 14h51, 14h70 76 shigeki matsutani 7, 3(2005) 1 introduction for a hyperelliptic curve cg given by an affine curve y2 = ∏2g+1 i=1 (x − bi), where bi’s are complex numbers, we have a jacobian jg as a complex torus cg/λ by the abel map ω [mu]. due to the abelian theorem, we have a natural morphism from the symmetrical product symg(cg) to the jacobian jg ≈ ω[symg(cg)]/λ. as zeros of an appropriate shifted riemann theta function over jg, the theta divisor is defined as θ := ω[symg−1(cg)]/λ which is a subvariety of jg. similarly, it is natural to introduce a subvariety θk := ω[sym k(cg)]/λ and a sequence, θ0 ⊂ θ1 ⊂ θ2 ⊂ ··· ⊂ θg−1 ⊂ θg ≡ jg vanhaecke studied the structure of these subvarieties as stratifications of the jacobian jg using the strategies developed in the studies of the infinite dimensional integrable system [v1]. he showed that these stratifications of the jacobian are connected with stratifications of the sato grassmannian. further vanhaecke investigated lie-poisson structures in the jacobian in [v2]. he showed that invariant manifolds associated with poisson brackets can be identified with these strata; it implies that the strata are characterized by the lie-poisson structures. he also showed that the poisson brackets are connected with a finite-dimensional integrable system, henon-heiles system. following the study, abenda and fedorov [af] investigated these strata and their relations to henon-heiles system and neumann systems. on the other hand, functions over the embedded hyperelliptic curve θ1 in a hyperelliptic jacobian jg were also studied from viewpoint of number theory in [c, g, ô]. in [ô], ônishi also investigated the sequence of the subvarieties, and explicitly studied behaviors of functions over them in order to obtain higher genus analog of the frobenius-stickelberger relations for genus one case. though vanhaecke, abenda and fedorov found some relations of functions over these subvarieties explicitly using the infinite universal grassmannians and so-called mumford’s u v w expressions [mu], ônishi gave more explicit relations on some functions over the subvarieties using the theory of hyperelliptic functions in the nineteenth century fashion [ba1, ba2, ba3]. in this article, we will also investigate some relations of functions over the subvarieties based upon the studies of the hyperelliptic function theory developed in the nineteenth century [ba2, ba3, w]. especially this article deals with the “sine-gordon equation” over there. modern expressions of the sine-gordon equation in terms of riemann theta functions were given in [[mu] 3.241], ∂ ∂tp ∂ ∂tq log([2p − 2q]) = a([2p − 2q] − [2q − 2p]), (1.1) where p and q are ramified points of cg, a is a constant number, [d] is a meromorphic function over symg(cg) with a divisor d for each cg and tp′ is a coordinate in the 7, 3(2005) relations of al functions over subvarieties in a hyperelliptic jacobian 77 jacobi variety such that it is identified with a local parameter at a ramified point p′ up to constant. in the previous work [ma], we also studied (1.1) using the fashion of the nineteenth century. in [w] weierstrass defined al function by alr := γr √ fg(br) and fg(z) := (x1 − z) · · · (xg − z) over jg with a constant factor γr. let γr = 1 in this article. as weierstrass implicitly seemed to deal with it, (1.1) is naturally described by alfunctions as [ma], ∂2 ∂v (g) 1 ∂v (g) 2 log alr als = 1 (br − bs) ( f′(bs) ( alr als )2 + f′(br) ( als alr )2) . (1.2) here f′(x) := df (x)/dx and v(g)’s are defined in (2.4). ((1.2) was obtained in the previous article [ma] by more direct computations and will be shown as corollary 3.3 in this article). we call (1.2) weierstrass relation in this article. in this article, we will introduce an “al” function over the subvariety in the jacobian, al(m)r := √ fm(br) and fm(z) := (x1 − z) · · · (xm − z) for a point ((x1, y1), · · · , (xm, ym)) in the symmetric product of the m curves sym mcg (m = 1, · · · , g − 1). in [mu], mumford dealt with fm function (he denoted it by u ) for 1 ≤ m < g and studied the properties. further abenda and fedorov also studied some properties of the al(m)r and fm functions in [af] though they did not mention about weierstrass’s paper nor the relation (1.2). we will consider a variant of the weierstrass relation (1.2) to al(m)r over subvariety in non-degenerated and degenerated hyperelliptic jacobian. as in our main theorem 3.1, even on the subvarieties, we have a similar relation to (1.1), ∂ ∂v (m) r ∂ ∂v (m) s log al(m)r al(m)s = 1 (br − bs)   f′(br) (q(2)m (br))2 ( al(m)s al(m)r )2 + f′(bs) (q(2)m (bs))2 ( al(m)r al(m)s )2 + reminder terms. (1.3) here q(2)m is defined in (2.2). we regard (1.3) or (3.1) as a subvariety version of the weierstrass relation (1.2). in fact, (1.3) contains the same form as (1.1) up to the factors (q(2)m (bt))2 (t = r, s) and the reminder terms. thus (1.3) or (3.1) should be regarded as an extension of the sine-gordon equation (1.2) over the jacobian to that over the subvariety of the jacobian. further a certain degenerate curve, the remainders in (1.3) vanishes. then we have a relations over subvarieties in the jacobian, which is formally the same as the weierstrass relations (1.2) up to the factors (q(2)m (bt))2 (t = r, s), which means that we can find solutions of sine-gordon equation over subvarieties in hyperelliptic jacobian. we expect that our results shed a light on the new field of a relation between “integrability” and a subvariety in the jacobian, which was brought off by [v1, v2, af]. 78 shigeki matsutani 7, 3(2005) the author is grateful to the referee for directing his attensions to the references [af] and [v2]. 2 differentials of a hyperelliptic curve in this section, we will give our conventions of hyperelliptic functions of a hyperelliptic curve cg of genus g (g > 0) given by an affine equation, y2 = f (x) = (x − b1)(x − b2) · · · (x − b2g)(x − b2g+1) = q(x)p (x), (2.1) where bj ’s are complex numbers. here we use the expressions q(x) := q (1) m (x)q (2) m (x), q(1)m (x) := (x − a1)(x − a2) · · · (x − am), q(2)m (x) := (x − am+1)(x − am+2) · · · (x − ag), p (x) := (x − c1)(x − c2) · · · (x − cg)(x − c), (2.2) where ak ≡ bk, ck ≡ bg+k, (k = 1, · · · , g) c ≡ b2g+1. definition 2.1 [ba1, ba2] for a point (xi, yi) ∈ cg, we define the following quantities. 1. the unnormalized differentials of the first kind are defined by, dv (g,i) k := q(xi)dxi 2(xi − ak)q′(ak)yi , (k = 1, · · · , g) (2.3) 2. the abel map for g-th symmetric product of the curve cg is defined by, v(g) ≡ (v(g)1 , · · · , v (g) g ) : sym g(cg) −→ cg, ( v (g) k ((x1, y1), · · · , (xg, yg)) := g∑ i=1 ∫ (xi,yi) ∞ dv (g,i) k ) . (2.4) 3. for v(g) ∈ cg, we define the subspace, ξm := v (g)(symm(cg) × (am+1, 0) ×···× (ag, 0))/λλλ. (2.5) here c is a complex field and λλλ is a g-dimensional lattice generated by the related periods or the hyperelliptic integrals of the first kind. 7, 3(2005) relations of al functions over subvarieties in a hyperelliptic jacobian 79 the jacobi variety jg are defined as complex torus as jg := ξg. as ξm (m < g) is embedded in jg whose complex dimension as subvariety is m, the differential forms (dv(g)k )k=1,··· ,g are not linearly independent. we select linearly independent bases such as v(m)k := v (g) k ((x1, y1), · · · , (xm, ym), (am+1, 0), · · · , (ag, 0)), (k = 1, · · · , m) at ξm. ξ0 ⊂ ξ1 ⊂ ξ2 ⊂ ··· ⊂ ξg−1 ⊂ ξg ≡ jg for (x1, · · · , xm) ∈ symm(cg), we introduce fm(x) := (x − x1) · · · (x − xm), (2.6) and in terms of fm(x), a hyperelliptic al-function over (v(m)) ∈ ξm, [ba2 p.340, w], al(m)r (v (m)) = √ fm(br). (2.7) further we introduce m × m-matrices, mm :=   1 x1 − a1 1 x2 − a1 · · · 1 xm − a1 1 x1 − a2 1 x2 − a2 · · · 1 xm − a2 ... ... . . . ... 1 x1 − am 1 x2 − am · · · 1 xm − am   , qm =   √ q(x1) p (x1) √ q(x2) p (x2) . . . √ q(xm) p (xm)   , am =   q′(a1) q′(a2) . . . q′(am)   . lemma 2.2 1. det mm = (−1)m(m−1)/2p (x1, · · · , xm)p (a1, · · · , am)∏ k,l(xk − al) , 80 shigeki matsutani 7, 3(2005) where p (z1, · · · , zm) := ∏ i m), we have res(ak,0) f (x) (x − a1)2(x − a2)2fm(x)2(q (2) m (x))2 dx = 2f′(ak) (ak − a1)2(ak − a2)2fm(a2)2(q (2)′ m (ak))2 . by arranging them, we obtain (1). (2) is obvious. as a corollary, we have weierstrass relation (1.2) which was proved in [ma]: corollary 3.3 for m = g case, we have the weierstrass relation for a general curve cg, ∂ ∂v (g) r ∂ ∂v (g) s log al(g)r al(g)s = 1 (ar − as)  f′(ar) ( al(m)s al(m)r )2 + f′(as) ( al(m)r al(m)s )2 . (3.5) now we will give our final proposition as corollary. corollary 3.4 for a curve satisfying the relations cj = aj for j = m + 1, · · · , g, al(m)r and al (m) s (r, s ∈ {1, 2, · · · , m}) over ξm in (2.5) obey the relation, ∂ ∂v (m) r ∂ ∂v (m) s log al(m)r al(m)s = 1 (ar − as)   f′(ar) (q(2)m (ar))2 ( al(m)s al(m)r )2 + f′(as) (q(2)m (as))2 ( al(m)r al(m)s )2 . (3.6) 84 shigeki matsutani 7, 3(2005) proof. since the condition cj = aj for j = m + 1, · · · , g means f′(aj ) = 0 for j = m + 1, · · · , g, theorem 3.1 reduces to this one. under the same assumption of corollary 3.4, letting a = 2 √ f′(ar)f′(as) (ar − as)qm(ar)qm(as) , and φ(r,s)m (u) := 1 √ −1 log √ f′(ar) f′(as qm(ar) qm(as) fm(ar) fm(as) , defined over ξm, φ (r,s) m obeys the sin-gordon equation, ∂ ∂v (m) r ∂ ∂v (m) s φ(r,s)m = a cos(φ (r,s) m ). (3.7) received: september 2004. revised: november 2004. references [af] s. abenda, yu. fedorov, on the weak kowalevski-painleve property for hyperelliptically separable systems, acta appl. math., 60 (2000) 137-178. [ba1] h. f. baker, abelian functions – abel’s theorem and the allied theory including the theory of the theta functions –, cambridge univ. press, (1897) republication 1995. [ba2] h. f. baker, on the hyperelliptic sigma functions, amer. j. of math., xx (1898) 301-384. [ba3] h. f. baker, on a system of differential equations leading to periodic functions, acta math., 27 (1903) 135-156. [c] d.g. cantor, on the analogue of the division polynomials for hyperelliptic curves, j. reine angew. math., 447 (1994) 91-145. [g] d. grant, a generalization of a formula of eisenstein, 62 (1991) 121–132 proc. london math. soc., . [ma] s. matsutani, on relations of hyperelliptic weierstrass al-functions, int. j. appl. math., (2002) 11 295-307. [mu] d. mumford, tata lectures on theta, vol ii, birkhäuser, (1984) boston, . 7, 3(2005) relations of al functions over subvarieties in a hyperelliptic jacobian 85 [ô] y. ônishi, determinant expressions for hyperelliptic functions (with an appendix by shigeki matsutani), preprint math.nt/0105189, to appear in proc. edinburgh math. soc., , (2004) . [v1] p. vanhaecke, stratifications of hyperelliptic jacobians and the sato grassmannian, acta. appl. math., 40 (1995) 143-172. [v2] pol vanhaecke, integrable systems and symmetric products of curves, math. z., 227 (1998) 93-127. [t] t. takagi, daisuu-gaku-kougi (lecture of algebra), kyouritsu, tokyo, (1930) japanese. [w] k. weierstrass, zur theorie der abel’schen functionen, aus dem crelle’schen journal, 47 (1854) in mathematische werke i, mayer und müller, berlin, (1894) . cubo a mathematical journal vol.21, no¯ 03, (75–91). december 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000300075 weak solutions to neumann discrete nonlinear system of kirchhoff type rodrigue sanou1, idrissa ibrango2, blaise koné1, aboudramane guiro2 1 laboratoire d’analyse mathématiques et d’informatique (lami), institut des sciences exactes et appliquées, université joseph ki-zerbo, ouagadougou, burkinafaso. 2 laboratoire d’analyse mathématiques et d’informatique (lami), ufr, sciences et technique, université nazi boni, 01 bp 1091 bobo 01, bobo dioulasso, burkina-faso. drigoaime@gmail.com, ibrango2006@yahoo.fr, leizon71@yahoo.fr, abouguiro@yahoo.fr abstract we prove the existence of weak solutions for discrete nonlinear system of kirchhoff type. we build some hilbert spaces with suitable norms. we define the notion of weak solution corresponding to the problem (1.1). the proof of the main result is based on a minimization method of an energy functional j. resumen probamos la existencia de soluciones débiles para sistemas discretos no-lineales de tipo kirchhoff. construimos algunos espacios de hilbert con normas apropiadas. definimos la noción de solución débil correspondiente al problema (1.1). la demostración del resultado principal se basa en un método de minimización de un funcional de enerǵıa j. keywords and phrases: nonlinear difference equations, anisotropic nonlinear discrete systems, minimization methods, weak solutions. 2010 ams mathematics subject classification: 47a75; 35b38; 35p30; 34l05; 34l30. http://dx.doi.org/10.4067/s0719-06462019000300075 76 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) 1 introduction in this paper, we are going to investigate the existence of weak solutions for the following anisotropic nonlinear discrete system. for i = 1, · · · , n    −m (a(k − 1, ∆ui(k − 1))) ∆(a(k − 1, ∆ui(k − 1)))=fi(k, u(k)), k ∈ z[1, t] ∆ui(0) = ∆ui(t) = 0 (1.1) where ∆ui(k) = ui(k + 1) − ui(k) is the forward difference operator for any i = 1, · · · , n; z[1, t] = {1, . . . , t} for t ≥ 2 and a, fi are functions to be defined later. in the last few years, great attention has been paid to the study of fourth-order nonlinear difference equations. these equations have been widely used to study discrete models in many fields such as computer science, economics, neural network, ecology, cybernetics, etc. for background and recent results, we refer the reader to [2]-[12], [14] and the references therein. note that in recent years, much attention has been paid to problems not local since they appear in physical phenomena like the theory of nonlinear elasticity, heat diffusion, etc. among this problems, we find kirchhoff type problems, which are known by the presence of the term m( ∫ ω |∇u|2)∆u in the continuous case. as far as we know, the first study which deals with anisotropic discrete boundary value problems of p(.)-kirchhoff type difference equation was done by yucedag (see [11]). the function m(a(k − 1, ∆u(k − 1))) which appear in the left-hand side of problem (1.1) is more general. the main operator ∆(a(k − 1, ∆u(k − 1))) in problem (1.1) can be seen as a discrete counterpart of the anisotropic operator n∑ i=1 ∂ ∂xi a ( x, ∂ ∂xi u ) . the functional a derives from a potential with a(k, ξ) = ∂ ∂ξ a(k, ξ). our goal is to use a minimization method in order to establish some existence results of solutions of (1.1). the idea of the proof is to transfer the problem of the existence of solutions for (1.1) into the problem of existence of a minimizer for some associated energy functional. this method was successfully used by bonanno et al. [1] for the study of an eigenvalue nonhomogeneous neumann problem, where, under an appropriate oscillating behaviour of the nonlinear term, they proved the existence of a determined open interval of positive parameters for which the problem considered admits infinitely many weak solutions that strongly converge to zero, in an appropriate orlicz sobolev space. motivated by the work of [13] where j. zhao proved the existence of positive solutions, the approach presented in this article is different than the one given in the papers mentioned above. to the best of cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 77 our knowledge , results on existence of weak solutions of system (1.1), using minimization method, have not been found in the literature. the remaining part of this paper is organized as follows. section 2 is devoted to mathematical preliminaries. the main existence result is proved in section 3. in the section 4, we give an extension of our system. 2 mathematical background in the t-dimensional hilbert space h = { u : z[0, t + 1] −→ rn such that ∆u(0) = ∆u(t) = 0 } , with the inner product 〈u, v〉 = n∑ i=1 t+1∑ k=1 ∆ui(k − 1)∆vi(k − 1), ∀ u, v ∈ h, we consider the norm ‖u‖ = ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| 2 )1 2 . (2.1) we denote hi = { ui : z[0, t + 1] −→ r such that ∆ui(0) = ∆ui(t) = 0 } , for i = 1, · · · , n with the norm |ui|h = ( t+1∑ k=1 |∆ui(k − 1)| 2 )1 2 ∀ ui ∈ hi for i = 1, · · · , n. (2.2) moreover, we may consider hi with the following norm |ui|m = ( t∑ k=1 |ui(k)| m ) 1 m ∀ ui ∈ hi, m ≥ 2 for i = 1, · · · , n. (2.3) we have the following inequalities (see [2]) t(2−m)/(2m)|ui|2 ≤ |ui|m ≤ t 1/m|ui|2, ∀ ui ∈ hi, m ≥ 2 for i = 1, · · · , n. (2.4) let the function p : z[0, t] −→ (2, +∞) (2.5) denoted by p− = min k∈z[0,t] p(k) and p+ = max k∈z[0,t] p(k). 78 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) for the data a and fi, we assume the following. (h1). { a(k, .) : r → r, k ∈ z[0, t] and there exists a(., .) : z[0, t] × r → r which satisfies a(k, ξ) = ∂ ∂ξ a(k, ξ) and a(k, 0) = 0, for all k ∈ z[0, t]. (h2). for all k ∈ z[0, t] and ξ 6= η (a(k, ξ) − a(k, η)) .(ξ − η) > 0. (2.6) (h3). for any k ∈ z[0, t], ξ ∈ r, we have a(k, ξ) ≥ 1 p(k) |ξ|p(k). (2.7) (h4). for each k ∈ z[0, t], the function fi(k, .) : r n −→ r is jointly continuous and there exists (αi(.))1≤i≤n : z[0, t] −→ (0, +∞) and a function (ri(.))1≤i≤n : z[0, t] −→ [2, +∞) such that |fi(k, u)| ≤ αi(k) ( 1 + |ui(k)| ri(k)−1 ) (2.8) where 2 ≤ ri (k) < p − for i = 1, · · · , n. in what follows, we denote by : r− = min {(k,i)∈z[0,t]×z[1,n]} ri(k) and r + = max {(k,i)∈z[0,t]×z[1,n]} ri(k). for each i = 1, · · · , n, there exists hi ∈ r n such that ∇fi(k, u)(hi) = fi(k, u) ∀u ∈ h for i = 1, · · · , n. (2.9) by (2.8) there exists (βi(.))1≤i≤n : z[0, t] −→ (0, +∞) such that |fi(k, u)| ≤ βi(k) ( 1 + |ui(k)| ri(k) ) for i = 1, · · · , n (2.10) where 0 < β = inf {(k,i)∈ z[0,t]×z[1,n]} βi(k) ≤ sup {(k,i)∈ z[0,t]×z[1,n]} βi(k) = β < +∞. (2.11) (h5). we also assume that the function m : (0, +∞) −→ (0, +∞) is continuous and non-decreasing and there exist positive numbers b1, b2 with b1 ≤ b2 and α > 1 such that b1t α−1 ≤ m(t) ≤ b2t α−1 for t > t∗ > 0. (2.12) example 2.1. there are many functions satisfying both (h1) − (h4). let us mention the following. cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 79 • a(k, ξ) = 1 p(k) (( 1 + |ξ|2 )p(k)/2 − 1 ) , where a(k, ξ) = ( 1 + |ξ|2 )(p(k)−2)/2 ξ, ∀ k ∈ z[0, t], ξ ∈ r, • fi(k, ξ) = 1 + ∣∣ξi ∣∣p(k)−1, ∀ (k, i) ∈ z[0, t] × z[1, n] and ξ = (ξ1, · · · , ξn) , • m(t) = 1, ∀ t ∈ (0, +∞). moreover, we may consider h with the following norm ‖u‖m = n∑ i=1 ( t∑ k=1 |ui(k)| m ) 1 m , ∀ u ∈ h and m ≥ 2. (2.13) using the relation (2.4) we can prove the following lemma. lemma 2.2. we have the following inequalities t(2−m)/(2m)‖u‖2 ≤ ‖u‖m ≤ t 1/m‖u‖2, ∀ u ∈ h and m ≥ 2. (2.14) we need the following auxiliary results throughout our paper. lemma 2.3. (1) there exist two positive constant c1, c2 such that n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) ≥ c1 ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| 2 )p− 2 − c2, (2.15) for all u ∈ h with |ui|h > 1. (2) for any m ≥ 2 there exists a positive constant cm such that n∑ i=1 t∑ k=1 |ui(k)| m ≤ cm n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| m, ∀ u ∈ h. (2.16) indeed, (1) by [6], there exists the positive constants λi and µi for i = 1, · · · n t+1∑ k=1 |∆ui(k − 1)| p(k−1) ≥ λi ( t+1∑ k=1 |∆ui(k − 1)| 2 )p− 2 − µi ∀ ui ∈ hi and |ui|h > 1. n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) ≥ min 1≤i≤n (λi) n∑ i=1 ( t+1∑ k=1 |∆ui(k − 1)| 2 )p− 2 − max 1≤i≤n (µi) n. 80 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) since the function x 7−→ x p− 2 is convex because p− > 2, then we have n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) ≥ min 1≤i≤n (λi) ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| 2 )p− 2 − max 1≤i≤n (µi) n. we deduce that n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) ≥ c1 ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| 2 )p− 2 − c2. (2) by [8], for any m ≥ 2 there exists a positive constant cm such that for i = 1, · · · , n t∑ k=1 |ui(k)| m ≤ cm t+1∑ k=1 |∆ui(k − 1)| m ∀ ui ∈ hi. therefore n∑ i=1 t∑ k=1 |ui(k)| m ≤ cm n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| m ∀ u ∈ h. 3 existence of weak solutions in this section, we study the existence of weak solution of problem (1.1). definition 3.1. a weak solutions of problem (1.1) is u ∈ h such that n∑ i=1 [ m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) t+1∑ k=1 a(k − 1, ∆ui(k − 1))∆vi(k − 1) ] = n∑ i=1 t∑ k=1 fi(k, u(k))vi(k) (3.1) for all v ∈ h. note that, since h is a finite dimensional space, the weak solutions coincide with the classical solution the problem (1.1). theorem 3.2. assume that (h1)−(h5) holds. then, there exists a weak solution of the problem (1.1). cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 81 to prove this, we define the energy functional j : h −→ r by j(u) = n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) − n∑ i=1 t∑ k=1 fi ( k, u(k) ) (3.2) where m̂(t) = ∫t 0 m(s)ds. lemma 3.3. the functional j is well defined on h and is of class c1 ( h, r ) with the derivative given by 〈j′(u), v〉 = n∑ i=1 [ m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1) ) t+1∑ k=1 a(k − 1, ∆ui(k − 1))∆vi(k − 1) ] − n∑ i=1 t∑ k=1 fi(k, u(k))vi(k), (3.3) for all u, v ∈ h. indeed, let’s i(u) = n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) and λ(u) = n∑ i=1 t∑ k=1 fi ( k, u(k) ) . since m̂(.), a(k, .) and f(k, .) are continuous for all k ∈ z[0, t], then |i(u)| = ∣∣∣∣ n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) )∣∣∣∣ < +∞, |λ(u)| = ∣∣∣∣ n∑ i=1 t∑ k=1 fi ( k, u(k) )∣∣∣∣ < +∞. the energy functional j is well defined on h. it is not difficult to see that the functional i derivative are give by 〈i′(u), v〉= n∑ i=1 [ m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1) ) t+1∑ k=1 a(k − 1, ∆ui(k − 1))∆vi(k − 1) ] (3.4) 82 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) on the other hand, for all u, v ∈ h, there exists hi ∈ r n such that 〈λ′(u), v〉 = lim t→0+ λ(u + tv) − λ(u) t = lim t→0+ n∑ i=1 t∑ k=1 fi(k, u(k) + tv(k)) − fi(k, u(k)) t = n∑ i=1 t∑ k=1 lim t→0+ fi(k, u(k) + tv(k)) − fi(k, u(k)) t = n∑ i=1 t∑ k=1 ∇fi(k, u(k))(hi)vi(k) = n∑ i=1 t∑ k=1 fi(k, u(k))vi(k). the functional j is clearly of class c1 � lemma 3.4. the functional j is lower semi-continuous. indeed since the functional λ is completely continuous and weakly lower semi-continuous, we have to prove the semi-continuity of i. a is convex with respect to the second variable according (h1) and (h2). with the assumption (h5) we conclude that i is convex. thus, it is enough to show that i is lower semi-continuous. for this, we fix u ∈ h and ε > 0. since i is convex, we deduce that, for any v ∈ h. i(v) ≥ i(u) + 〈i′(u), v − u〉 ≥ i(u) − n∑ i=1 [ m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) × t+1∑ k=1 |a(k − 1, ∆ui(k − 1))||∆vi(k − 1) − ∆ui(k − 1)| ] ≥ i(u) − cm ( n∑ i=1 t+1∑ k=1 |a(k − 1, ∆ui(k − 1))||∆vi(k − 1) − ∆ui(k − 1)| ) , where cm = ( n∑ i=1 m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1) )) cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 83 by using schwartz inequality, we get : i(v) ≥ i(u) − cm n∑ i=1 [(t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2 × ( t+1∑ k=1 |∆vi(k − 1) − ∆ui(k − 1)| 2 )1 2 ] ≥ i(u) − cm   n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2   ×   n∑ i=1 ( t+1∑ k=1 |∆vi(k − 1) − ∆ui(k − 1)| 2 )1 2   by (2.2) i(v) ≥ i(u) − cm   n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2   [ n∑ i=1 |vi − ui|h ] . since hi is finite dimensional, there exist the positive constants θi for i = 1, · · · , n such that |vi|h ≤ θi|vi|2 ∀ vi ∈ hi. (3.5) then, i(v) ≥ i(u) − cm   n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2   [ n∑ i=1 θi|vi − ui|2 ] ≥ i(u) − max 1≤i≤n (θi) cm   n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2   [ n∑ i=1 |vi − ui|2 ] . also, the space h is finite dimensional, there exists a positive constant γ such that: ‖u‖2 ≤ γ‖u‖ ∀ u ∈ h. from this, we have i(v) ≥ i(u) − γ max 1≤i≤n (θi) cm   n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2  ‖v − u‖ ≥ i(u) −  1 + γ max 1≤i≤n (θi) cm n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2  ‖v − u‖ 84 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) finally i(v) ≥ i(u) − s(t, u)‖v − u‖ ≥ i(u) − ε, (3.6) for all v ∈ h with ‖v − u‖ < δ = ε s(t,u) , where s(t, u) = 1 + γ max 1≤i≤n (θi) cm n∑ i=1 ( t+1∑ k=1 |a(k − 1, ∆ui(k − 1))| 2 )1 2 . we conclude that j is weakly lower semi-continuous. proposition 3.5. the functional j is coercive and bounded from below. indeed, according to (2.7), (2.10)-(2.12) we have j(u) = n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) − n∑ i=1 t∑ k=1 fi ( k, u(k) ) ≥ b1 α(p+)α [ n∑ i=1 ( t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α] − n∑ i=1 t∑ k=1 fi ( k, u(k) ) ≥ b1 α(p+)α [ n∑ i=1 ( t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α] − n∑ i=1 t∑ k=1 βi(k) ( 1 + |ui(k)| ri(k) ) ≥ b1 α(p+)α [ n∑ i=1 ( t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α] − β n∑ i=1 t∑ k=1 ( 1 + |ui(k)| ri(k) ) ≥ b1 α(p+)α [ n∑ i=1 ( t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α] − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − βnt. there exist ηi and νi such that j(u) ≥ b1 α(p+)α [ min 1≤i≤n (ηi) ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α − max 1≤i≤n (νi) ] − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − βnt. (3.7) to prove the coerciveness of the functional j, we may assume that ||u|| > 1 and we deduce from the above inequality (2.15) that cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 85 j(u) ≥ b1 α(p+)α   min 1≤i≤n (ηi)  c1 ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| 2 )p− 2 − c2   α − max 1≤i≤n (νi)   −β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − βnt. there exist a function k(α, c) such that j(u) ≥ b1 α(p+)α ( min 1≤i≤n (ηi)c α 1 ||u|| αp− − min 1≤i≤n (ηi)k(α, c)c α 2 − max 1≤i≤n (νi) ) − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − βnt. namely j(u) ≥ a1||u|| αp− − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − a2, where a1 = b1 α(p+)α min 1≤i≤n (ηi)c α 1 and a2 = b1 α(p+)α ( min 1≤i≤n (ηi)k(α, c)c α 2 + max 1≤i≤n (νi) ) + βnt. so j(u) ≥ a1||u|| αp− − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − a2 ≥ a1||u|| αp− − β n∑ i=1 t∑ k=1 |ui(k)| r+ − β n∑ i=1 t∑ k=1 |ui(k)| r− − a2. using (2.16) j(u) ≥ a1||u|| αp− − (cr−)β n∑ i=1 t∑ k=1 |∆ui(k)| r− − (cr+)β n∑ i=1 t∑ k=1 |∆ui(k)| r+ − a2 by using (2.4) there exists the positive constants k1 and k2 such that j(u) ≥ a1||u|| αp− − k1 n∑ i=1 ( t∑ k=1 |∆ui(k)| 2 )r− 2 − k2 n∑ i=1 ( t∑ k=1 |∆ui(k)| 2 )r+ 2 − a2. 86 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) there exist the positive constants a3, a4, a5 and a6 such that j(u) ≥ a1||u|| αp− − k1a3 ( n∑ i=1 t∑ k=1 |∆ui(k)| 2 )r− 2 − k1 a4 − a5 k2 ( n∑ i=1 t∑ k=1 |∆ui(k)| 2 )r+ 2 − k2 a6 − a2. consequently, there exist the positive constants a7 , a8 and a9 such that j(u) ≥ a1||u|| αp− − a7||u|| r− − a8||u|| r+ − a9. (3.8) recall that p− > r+ α ≥ r− α . then j is coercive. besides, for ||u|| ≤ 1, we have with (3.7) j(u) ≥ b1 α(p+)α [ min 1≤i≤n (ηi) ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α − max 1≤i≤n (νi) ] − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − βnt ≥ − b1 α(p+)α max 1≤i≤n (νi) − β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − βnt ≥ − b1 α(p+)α max 1≤i≤n (νi) − β n∑ i=1 t∑ k=1 |ui(k)| r− − β n∑ i=1 t∑ k=1 |ui(k)| r+ − βnt. using (2.16) j(u) ≥ − b1 α(p+)α max 1≤i≤n (νi) − (kr−)β n∑ i=1 t∑ k=1 |∆ui(k)| r− − (kr+)β n∑ i=1 t∑ k=1 |∆ui(k)| r+ − βnt. by using (2.14) there exists the positives constants k′1 and k ′ 2 such that j(u) ≥ − b1 α(p+)α max 1≤i≤n (νi) − k ′ 1 n∑ i=1 ( t∑ k=1 |∆ui(k)| 2 )r− 2 − k′2 n∑ i=1 ( t∑ k=1 |∆ui(k)| 2 )r+ 2 − βnt. cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 87 there exist the positive constants c′3, c ′ 4, c ′ 5 and c ′ 6 such that j(u) ≥ − b1 α(p+)α max 1≤i≤n (νi) − k ′ 1c ′ 3 ( n∑ i=1 t∑ k=1 |∆ui(k)| 2 )r− 2 − k′1 c ′ 4 − c ′ 5 k ′ 2 ( n∑ i=1 t∑ k=1 |∆ui(k)| 2 )r+ 2 − k′2 c ′ 6 − βnt. consequently, there exist the positive constants c′7 and c ′ 8 such that j(u) ≥ − b1 α(p+)α max 1≤i≤n (νi) − c ′ 7||u|| r− − k′1 c ′ 4 − c ′ 8||u|| r+ − k′2 c ′ 6 − βnt ≥ − b1 α(p+)α max 1≤i≤n (νi) − c ′ 7 − k ′ 1 c ′ 4 − c ′ 8 − k ′ 2 c ′ 6 − βnt. thus, j is bounded from below � since j is weakly lower semi-continuous, bounded from below and coercive on h, using the relation between critical points of j and problem (1.1), we deduce that j has a minimizer which is a weak solution to problem (1.1). 4 an extension in this section we are going to show that the existence result obtained for system (1.1) can be extended. let’s consider the following system. for i = 1, · · · , n    −m (a(k − 1, ∆ui(k − 1))) ∆(a(k − 1, ∆ui(k − 1))) + σi(k)φ(k, ui(k)) = δi(k)fi(k, u(k)), ∀ k ∈ z[1, t] ∆ui(0) = ∆ui(t) = 0, (4.1) where t ≥ 2 is a fixed integer, and we shall use the following assumption. (h6). σi : z[1, t] −→ r and δi : z[1, t] −→ r are such that σi(k) ≥ σ0 > 0 for (k, i) ∈ z[1, t] × z[1, n] and 0 < δi(k) ≤ sup {(k,i)∈z[1,t]×z[1,n]} |δi(k)| = δ0. (h7). φ(k, t) = |t| p(k)−2t for (k, t) ∈ z[0, t] × r. 88 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) in the t−dimensional hilbert space h with the inner product 〈u, v〉 = n∑ i=1 t+1∑ k=1 ∆ui(k − 1)∆vi(k − 1) + n∑ i=1 t+1∑ k=1 ui(k)vi(k), we consider the norm ‖u‖ = √ 〈u, u〉 = ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| 2 + n∑ i=1 t∑ k=1 |ui(k)| 2 )1 2 . definition 4.1. a weak solution of problem (4.1) is a function u ∈ h such that n∑ i=1 [ m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1) ) t+1∑ k=1 a(k − 1, ∆ui(k − 1))∆vi(k − 1) ] + n∑ i=1 t∑ k=1 σi(k)|ui(k)| p(k)−2ui(k)vi(k) = n∑ i=1 t∑ k=1 δi(k)fi(k, u(k))vi(k). for all v ∈ h. theorem 4.2. under the assumptions (h1)(h6) the problem (4.1) has a least weak solution in h. indeed, for u ∈ h we define the energy functional corresponding to system (4.1) by j(u) = n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) + n∑ i=1 t∑ k=1 σi(k) p(k) |ui(k)| p(k) − n∑ i=1 t∑ k=1 δi(k)fi ( k, u(k) ) . obviously, j is class c1 (h, r) and is weakly lower semicontinuous, and we show that 〈j′(u), v〉 = n∑ i=1 [ m ( t+1∑ k=1 a(k − 1, ∆ui(k − 1) ) t+1∑ k=1 a(k − 1, ∆ui(k − 1))∆vi(k − 1) ] + n∑ i=1 t∑ k=1 σi(k)|ui(k)| p(k)−2ui(k)vi(k) − n∑ i=1 t∑ k=1 δi(k)fi ( k, u(k) ) vi(k). for all u, v ∈ h. this implies that the weak solution of system(4.1) coincides with the critical points of the functional j. it suffices to prove that j is bounded below and coercive in order to complete the proof. cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 89 j(u) = n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) + n∑ i=1 t∑ k=1 σi(k) p(k) |ui(k)| p(k) − n∑ i=1 t∑ k=1 δi(k)fi ( k, u(k) ) ≥ n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) − n∑ i=1 t∑ k=1 δi(k)fi ( k, u(k) ) ≥ n∑ i=1 m̂ ( t+1∑ k=1 a(k − 1, ∆ui(k − 1)) ) − δ0 n∑ i=1 t∑ k=1 fi ( k, u(k) ) . we obtain j(u) ≥ b1 α(p+)α [ min 1≤i≤n (ηi) ( n∑ i=1 t+1∑ k=1 |∆ui(k − 1)| p(k−1) )α − max 1≤i≤n (νi) ] − δ0β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − δ0βnt. (4.2) for ‖u‖ > 1, by the same procedure, we prove that j(u) ≥ a′1‖u‖ αp− − a′7‖u‖ r− − a′8‖u‖ r+ − a′9, where a′1, a ′ 7, a ′ 8 and a ′ 9 are the positive constants. hence p− > r+ α ≥ r− α , j is coercive. if ||u|| ≤ 1 by (4.2) we have j(u) ≥ − b1 α(p+)α max 1≤i≤n (νi) − δ0β n∑ i=1 t∑ k=1 |ui(k)| ri(k) − δ0βnt. by the same reasoning j(u) ≥ −d1 − δ0βnt where d1 > 0. thus, j is bounded from below � since j is weakly lower semi-continuous, bounded from below and coercive on h, using the relation between critical points of j and problem (4.1), we deduce that j has a minimizer which is a weak solution to problem (4.1). 90 r. sanou, i. ibrango, b. koné, a. guiro cubo 21, 3 (2019) competing interests the authors declare that there is no conflict of interest regarding the publication of the paper. acknowledgment the authors express their deepest thanks to the editor and anonymous referee for their comments and suggestions on the article. cubo 21, 3 (2019) weak solutions to neumann discrete nonlinear system of kirchhoff . . . 91 references [1] g. bonanno, g. molica bisci and v. radulescu; arbitrarity small weak solutions for nonlinear eigenvalue problem in orlicz-sobolev spaces, monatshefte fur mathematik, vol. 165, no. 3-4, pp. 305-318, 2012. 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[14] v. zhikov; averaging of functionals in the calculus of variations and elasticity, mathematics of the ussr-izvestiya, vol.29 (1987), pp. 33-66. introduction mathematical background existence of weak solutions an extension a mathematical journal vol. 7, no 2, (111 138). august 2005. symmetric spaces of noncompact type martin moskowitz 1 department of mathematics, cuny graduate center 365 fifth ave. new york ny 10016. usa mmoskowi@mat.uniroma2.it abstract this article gives a detailed introduction to symmetric spaces of non-compact type and their relation to corresponding semisimple lie groups. this is done more or less from scratch and explicitely without the reader having to know large parts of modern differential geometry. resumen este art́ıculo entrega una detallada introducción a espacios simétricos de tipo no compacto y su relación con los correspondientes grupos de lie semisimples. esto es hecho mas o menos en términos generales y expĺıcitamente sin que el lector tenga gran conocimiento de la geometŕıa diferencial moderna. key words and phrases: semisimple lie group and lie algebra of non-compact type, maximal compact subgroup and maximal subalgebra of compact type and their conjugacy. cartan and polar decomposition, rank, irreducible symmetric space,isometry group, killing form, exponential map, cartan involution, law of cosines and two point homogeneous space. math. subj. class.: 22e, 57s, 57t 1this article was written while the author was supported by a psc-cuny grant from the research foundation of cuny. he would like to take this opportunity to thank the rf for its generous support. 112 martin moskowitz 7, 2(2005) 1 introduction in this article we shall give an introduction to symmetric spaces of noncompact type. this subject, largely the creation of élie cartan (1869-1951), is of fundamental importance both to geometry and lie theory. indeed, one of the great achievements of the mathematics of the first half of the twentieth century was e. cartan’s discovery of the fact that these two categories correspond exactly. namely, given a connected, centerless, real semisimple lie group g without compact factors there is associated to it a unique symmetric space of noncompact type. this is g/k, where k is a maximal compact subgroup of g and g/k takes the riemannian metric induced from the killing form of g. conversely, if one starts with an arbitrary symmetric space, x, all of whose irreducible constituents are neither compact nor irn, then x = g/k, where g is the identity component of the isometry group of x. here g is a centerless, real semisimple lie group without compact factors. thus, we have a bijective correspondence between the two categories and this fact underlies an important reason why differential geometry and lie theory are so closely bound. as one might expect, this close relationship between the two will show up in some of the proofs. for the details of all this, see s. helgason [4] and g.d. mostow [9]. also, [4] has a particularly convenient and useful early chapter on differential geometry. concerning this correspondence, the same may be said of euclidean space and its group of isometries, or of compact semisimple groups and symmetric spaces of compact type, which were also studied by e. cartan. however, we shall not deal with these here. taken as a whole cartan’s work on symmetric spaces can be considered as the completion of the well known “erlangen program” first formulated by f. klein in 1872. in particular, it ties together euclidean, elliptic and hyperbolic geometry in any dimension. before turning to our subject proper it might be helpful to consider a most important example, namely that of g = sl(2, ir) and x the hyperbolic plane, which we view here as the poincaré upper half plane, h+, consisting of all complex numbers z = x + iy, where y > 0. we let g act on h+ by fractional linear transformations, g ·z = az+b cz+d , g = ( a b c d ) where a, b, c and d are real and det g = 1. since i( az+b cz+d ) = i(z)|cz+d|2 > 0, we see that g ·z ∈ h+. that this is an action is easy to verify. now this action is transitive. let c = 0, then a �= 0 and d = 1 a . then g · i = a2i + ab. evidently, by varying a > 0 and b ∈ ir this gives all of h+. a moment’s reflection tells us that the isotropy group, stabg(i), is given by a = d and c = −b. since det g = a2 + b2 = 1, we see stabg(i) = {g : g = ( cos t sin t −sin t cos t ) : t ∈ ir}. on h+ we place the riemannian metric ds2 = dx 2+dy2 y2 (meaning the hyperbolic metric ds = dseuc iz ) and check that g acts by isometries on h+ (for this see, for 7, 2(2005) symmetric spaces of noncompact type 113 example, p. 118 of [8]). since g is connected, its image, psl(2, ir), is contained in isom0(h+). (actually it is isom0(h+), but that will not matter. also, there are 2 connected components, the other is the anti-holomorphic automorphisms, but that will not matter either. from the point of view of the symmetric space it does not even matter whether we take sl(2, ir) or psl(2, ir). however, we note that psl(2, ir), the group that is really acting, is the centerless version.) another model for this symmetric space is the unit disk, d ⊆ c, called the disk model. it takes the metric ds2 = 4 dx 2+dy2 (1−r2)2 and has the advantage of radial symmetry about the origin, 0. here r is the usual radial distance from 0. the point of the 4 is, as we shall see, to make d isometric with h+, or put another way, to normalize the curvature on d to be −1. now the cayley transform c(z) = z−i z+i maps h+ diffeomorphically onto d. its derivative c′(z) = zi (z+i)2 . a direct calculation shows that for z ∈ h+ 2|c′(z)| 1 −|c(z)|2 = 1 i(z) . using this we see that if w = c(z), then |dw| = |c′(z)||dz| and so 2|dw| 1 −|w|2 = 2|c′(z)| 1 −|c(z)|2 |dz| = |dz| |i(z)|. thus c is an isometry. of course in the form of the disk, the group of isometries and its connected component will superficially look different. 2 the polar decomposition explaining the reasons for the relationship mentioned above will take some time and we shall begin by studying the exponential map on certain specific manifolds. the n × n complex matrices will be denoted by gl(n, c) and the real ones by gl(n, ir)). denote by h the set of all hermitian matrices in gl(n, c) and by h the positive definite ones. it is easy to see that h is a real (but not a complex!) vector space of dim n2. similarly, we denote by p the symmetric matrices in gl(n, ir) and by p those that are positive definite. p is a real vector space of dim n(n+1) 2 . as we shall see, h and p and certain of their subspaces will actually comprise all symmetric spaces of noncompact type. proposition 2.1 p and h are open in p and h, respectively. as open sets in a real vector space each is, in a natural way, a real analytic manifold of the appropriate dimension. proof. let p(z) = ∑ i piz i and q(z) = ∑ i qiz i be polynomials of degree n with complex coefficients, let z1 . . .zn and w1 . . .wn denote their respective roots counted according to multiplicity and let � > 0. it follows from rouché’s theorem (see [8]) that there exists a sufficiently small δ > 0 so that if for all i = 0, . . . ,n if |pi −qi| < δ, then after a possible reordering of the wi’s, |zi−wi| < � for all i. suppose h were not open 114 martin moskowitz 7, 2(2005) in h. then there would be an h ∈ h and a sequence xj ∈h−h converging to h in gl(n, c). since h is positive definite, all its eigenvalues are positive. choose � so small that the union of the � balls about the eigenvalues of h lies in the right half plane. since the coefficients of the characteristic polynomial of an operator are polynomials and therefore continuous functions of the matrix coefficients and xj converges to h, for j sufficiently large, the coefficients of the characteristic polynomial of xj are within δ of the corresponding coefficient of the characteristic polynomial of h. hence all the eigenvalues of such an xj are positive. this contradicts the fact that none of the xj are in h, proving h is open in h. intersecting everything in sight with gl(n, ir) shows p is also open in p. proposition 2.2 upon restriction, the exponential map of gl(n, c) = gl(v ) is a real analytic diffeomorphism between h and h. its inverse, is given by log h = log(tr h)i − ∞∑ i=1 (i − h tr h )i/i, which is an analytic function on h. as a consequence we see that the restriction of exp to any real subspace of h gives a real analytic diffeomorphism of the subspace with its image. in particular, exp is a real analytic diffeomorphism between between p and p . in particular, in all these cases exp is a bijection. proof. we shall do this for h, the real case being completely analogous. suppose h ∈ h is diagonal with eigenvalues hi > 0. then tr(h) > 0 and 0 < hitr(h) so log(tr(h)) is well defined and log( hi tr(h) ) is defined for all i. but since 0 < hi tr(h) < 1, we see that 0 < (1 − hi tr(h) )k < 1 for all positive integers k. hence log( i−h tr(h) ) is given by an absolutely convergent power series −∑∞i=1(i − htr h )i/i. if u is a unitary operator so that uhu−1 is diagonal, then tr(uhu−1) = tr(h) and since conjugation by u commutes with any convergent power series, this series actually converges for all h ∈ h and is a real analytic function log on h. because on the diagonal part of h this function inverts exp, and both exp and this power series commutes with conjugation, it inverts exp everywhere on h. finally, log(tr(h))i and log( h tr h ) commute and exp of a sum of commuting matrices is the product of the exp’s. since log inverts exp on the diagonal part of h it follows that log(h) = log(tr(h))i + log( h tr h ) = log(tr(h))i − ∞∑ i=1 (i − h tr h )i/i. we shall need the following elementary fact whose proof is left to the reader. lemma 2.3 for any g ∈ gl(n, c), g∗g ∈ h. 7, 2(2005) symmetric spaces of noncompact type 115 it follows that for all g ∈ gl(n, c), log(g∗g) ∈ h and since this is a real linear space also 1 2 log(g∗g) ∈h. this means we can apply exp and conclude the following: corollary 2.4 h(g) = exp( 1 2 log(g∗g)) ∈ h is a real analytic function from gl(n, c) → h. hence h(g)n = exp( n 2 log(g∗g)) ∈ h for every n ∈ zz. in particular, h(g)−2 = exp( 2 2 log(g∗g)) = g∗g. so that gh(g)−1(gh(g)−1)∗ = gh(g)−1h(g)−1∗g∗ = gh(g)−2g∗ and, since h(g)−1 ∈ h, g(g∗g)−1g∗ = i. thus, gh(g)−1 = u(g) is unitary for each g ∈ gl(n, c). since group multiplication and inversion are analytic, u(g) is also a real analytic function on gl(n, c) (as is h(g)). now this decomposition g = uh, where u ∈ u and h ∈ h is actually unique. to see this, suppose u1h1 = g = u2h2. then u−12 u1 = h2h −1 1 so that h2h −1 1 is unitary. this means (h2h −1 1 ) ∗ = (h2h −1 1 ) −1 and hence h21 = h 2 2. but since h1 and h2 ∈ h, each is an exponential of something in h; hi = exp xi. but then h2i = exp 2xi and since exp is 1 : 1 on h, we get 2x1 = 2x2 so x1 = x2 and therefore h1 = h2 and u1 = u2. the upshot of all this is that we have a real analytic map gl(n, c) → u(n, c) × h given by g �→ u(g)h(g). since g = u(g)h(g) for every g (multiplication in the lie group gl(n, c)), this map is surjective and has a real analytic inverse. we summarize these facts as the following polar decomposition theorem. theorem 2.5 the map g �→ u(g)h(g) gives a real analytic diffeomorphism gl(n, c) → u(n, c) × h. identical reasoning also shows that as a real analytic manifold gl(n, ir) is, in the same way, diffeomorphic to o(n, ir) ×p. from this it follows that, since h and p are each diffeomorphic with a euclidean space, and therefore are topologically trivial, in each case the topology of the noncompact group is completely determined by that of the compact one. in this situation, one calls the compact group a deformation retract of the noncompact group. since p and h are diffeomorphic images under exp of some euclidean space, one calls them exponential submanifolds. for example, connectedness, the number of components, simple connectedness and the fundamental group of the noncompact group are each the same as that of the compact one. thus for all n ≥ 1, gl(n, c) is connected and its fundamental group is zz, while for all n, gl(n, ir) has 2 components and the fundamental group of its identity component is zz2 for n ≥ 3 and zz for n = 2. these facts follow from the long exact homotopy sequence for a fibration and are explained in c. chevalley [2]. 3 the cartan decomposition of a real semi-simple lie group of noncompact type what we have done so far may seem rather special. we now turn to more general groups g and also streamline our notation. instead of h, we shall consider certain 116 martin moskowitz 7, 2(2005) real subspaces of h denoted by p whose exponential image will be p and make the following definition. definition 3.1 let g be a lie subgroup of gl(n, ir) with lie algebra g. we denote by k = o(n, ir) ∩ g, by p the positive definite symmetric matrices in g, by p the symmetric matrices of g, and by k the skew symmetric matrices in g. in the case that g be a lie subgroup of gl(n, c) we again denote the lie algebra by g, but now k = u(n, c) ∩g, p is the positive definite hermitian matrices of g, p is hermitian matrices of g and k the the skew hermitian matrices in g. lemma 3.2 let q(t) = ∑n j=1 cj exp(bjt) be a trigonometric polynomial, where cj ∈ c, and bj and t ∈ ir. if q vanishes for an unbounded set of real t’s, then q ≡ 0. an immediate consequence is that for a polynomial p ∈ c[z1, . . . ,zn] in n complex variables with complex coefficients and (x1, . . . ,xn) ∈ irn, if p(exp(tx1), . . . , exp(txn)) vanishes for an unbounded set of real t’s, then it vanishes identically in t. proof. first we can assume that the t’s for which q vanishes tend to +∞. otherwise, they would have to tend to −∞ and in this case we just let p(t) = q(−t). then p is also a trigonometric polynomial and if p = 0, then so is q. reorder the bj’s, if necessary, so that they are strictly increasing by combining terms by adding the corresponding cj ’s. of course, we can now assume that all the cj ’s are nonzero. let tk be a sequence tending to +∞ on which q vanishes. suppose there are two or more bj ’s. since q(t) cn exp(bnt) = n−1∑ j=1 cj cn exp((bj − bn)t) + 1, it follows that q(t) cn exp(bnt) → 1 as k →∞. but since q is identically 0 in k so is this quotient, a contradiction. this means that all the bj ’s are equal and so q(t) = c exp(bt) for some c ∈ c and b ∈ ir. this function cannot have an infinite number of zeros unless c = 0, that is q = 0. definition 3.3 one calls a subgroup g ⊆ gl(n, c) an algebraic group if it is the simultaneous zero set within gl(n, c) of a family of polynomials with complex coefficients in the xi,j coordinates of the matrices in gl(n, c). clearly, such a group is a closed subgroup of gl(n, c) in the usual euclidean topology and hence by a theorem of e. cartan is a lie group. further we shall call gir = g∩ gl(n, ir) its ir-points. similarly, the ir-points gir of an algebraic group g is also a lie group. if the family of polynomials defining g happens to have all its coefficients lying in some subfield f of c, we then say g is defined over f. typical examples of algebraic groups are gl(n, c) itself (the empty set of polynomials) and sl(n, c) itself (the single polynomial det−1 = 0). the respective real points are gl(n, ir) and sl(n, ir). the reader should check that each of the complex classical groups (see [4], or [2]) is an algebraic group. 7, 2(2005) symmetric spaces of noncompact type 117 proposition 3.4 suppose m is an algebraic subgroup of gl(n, c) and g be a lie subgroup of gl(n, ir) (alternatively gl(n, c)) with lie algebra g. let g have finite index in mir (alternatively m). if x ∈ p and exp x ∈ g, then exp tx ∈ p for all real t. in particular, x ∈ g and hence x ∈ p. proof. to avoid circumlocutions we shall prove the complex case, the real case being completely analogous. choose u ∈ u(n, c) so that uxu−1 is diagonal with real eigenvalues λj . replace g by ugu−1, a lie subgroup of gl(n, c) which is contained in umu−1 with finite index. now umu−1 is an algebraic subgroup of gl(n, c) (and in the real case umiru−1 = (umu−1)ir). hence we can assume x is diagonal. let p(zij ) be one of the complex polynomials defining m. since exp x ∈ g and g is a group, exp kx ∈ g ⊆ m for all k ∈ zz. but exp kx is diagonal with diagonal entries exp(kλj ). applying p to exp kx, we get p(exp kx) = 0 for all k. by the corollary, p(exp tx) = 0 for all t. because p was an arbitrary polynomial defining m, it follows that exp tx ∈ m for all real t. since g has finite index in m and the 1parameter group exp tx is connected, it must lie entirely in g and therefore in p . hence x ∈ g. definition 3.5 a subgroup g of gl(n, ir) (gl(n, c)) is called self-adjoint if it is stable under taking transpose (∗). here transpose and ∗ refer to any linear involution (conjugate linear involution) on irn (cn). for example, sl(n, ir) (sl(n, c)) are self-adjoint since det gt = det g (det g∗ = det(g). the routine calculations showing o(n,c), so(n, c), o(p,q) and so(p,q) are also self-adjoint are left to the reader. in fact, the reader can check that any classical noncompact simple group in e. cartan’s list (see [4]) is self-adjoint. clearly by their very definition these groups are either algebraic or have finite index in the real points of an algebraic group (essentially algebraic). now it is an important insight of mostow [10] that any linear real semisimple lie group is self-adjoint under an appropriate involution. moreover, by the root space decomposition the adjoint group of any semisimple group without compact factors is algebraic (actually over q). thus here we are really talking about all the semisimple groups without compact factors and, of course, this means our construction actually gives all symmetric spaces of noncompact type. but even if we did not know this, since any classical noncompact simple group is easily seen to be self-adjoint as well as essentially algebraic, we already get a plethora of symmetric spaces from them. particular cases of theorem 3.6 below are the following. we shall leave their routine verification to the reader. sl(n, ir) is real analytically diffeomorphic with so(n)×p1, where the latter is the positive definite symmetric matrices of det 1, which in turn is diffeomorphic under exp with the linear space of real symmetric matrices of trace 0. similarly, sl(n, c) is real analytically diffeomorphic with su(n) × h1, where the latter is the positive definite hermitian matrices of det 1, which in turn is diffeomorphic with the linear space of hermitian matrices trace 0. as deformation retracts, similar conclusions can be drawn about the topology of these, as well as the other groups mentioned earlier. 118 martin moskowitz 7, 2(2005) the following result is a special case of the iwasawa decomposition theorem which holds for an arbitrary lie group with a finite number of components, but with a somewhat more elaborate formulation (see g.p. hochschild [5]). here, we content ourselves with the matter at hand. namely, self-adjoint algebraic groups, or their real points. in this context, it is called the cartan decomposition. by a maximal compact subgroup of g we mean one not properly contained in a larger compact subgroup of g. our next result is the cartan decomposition. theorem 3.6 let g be a self-adjoint subgroup of gl(n, c) (gl(n, ir)) with lie algebra g. suppose that g has finite index in an algebraic subgroup m of gl(n, c) (g has finite index in mir, its real points). then 1. g = k ×p as real analytic manifolds. 2. g = k ⊕p as a direct sum of ir-vector spaces. 3. exp : p → p is a real analytic manifold diffeomorphism whose inverse is given by the global power series of proposition 2.2. 4. k is a maximal subgroup of g. in particular, p is simply connected and g is a deformation retract of k. proof. here again we deal with the complex case, the real case being similar. first we show each g ∈ g can be written uniquely as g = k exp x, where k ∈ k and x ∈ p. by theorem 2.5, g = up, where u ∈ u(n, c) and p ∈h. now g∗ = (up)∗ = p∗u∗ = pu−1, so g∗g = pu−1up = p2. since g is self-adjoint, p2 ∈ g, and therefore so is p2k for every k ∈ zz. now p = exp x for some hermitian x. hence p2k = exp 2kx = exp k2x. since 2x is hermitian, exp 2x ∈ g and exp k2x ∈ p for all k. by proposition 3.4, exp t2x ∈ p for all real t. now, just as above, taking t = 1 2 we get exp x = p ∈ p ⊆ g. but then gp−1 = u ∈ g. therefore u ∈ k. also, since exp tx ∈ p for all real t, x ∈ p. thus g = kp where k ∈ k and p = exp x for x ∈ p. thus we have a map g �→ (k,p) from g to k × p . as above, if we can show uniqueness of the representation g = kp, then the map is onto. but since k ⊆ u(n, c) and p ⊆ the positive definite hermitian matrices, this follows from the uniqueness result proven earlier. since multiplication inverts this map it is 1:1 and has a smooth inverse. the formula, p(g) = exp( 1 2 log(g∗g)) ∈ p derived in the case of gl(n, c) is still valid, if suitably interpreted, and gives a real analytic map g → p . arguing exactly as in the case of gl(n, c) we see that part 1 is true. part 3 follows immediately from the case of gl(n, c) treated earlier. for part 2, write x = x−x ∗ 2 + x+x ∗ 2 . since the first term is skew hermitian, the second is hermitian and each is an ir-linear function of x ∈ gl(n, c), this proves part 2 for the case gl(n, c). to prove it in general we need only show that x−x ∗ 2 ∈ k and x+x∗ 2 ∈ p. now for x and y ∈ gl(n, c), [x∗,y∗] = −[x,y]∗. hence the map θ sending x �→ −x∗ is an involutive automorphism of the lie algebra gl(n, c) called a cartan involution. if we show g is stable under this map, then x �→ x∗ also leaves g stable since it is an ir subspace of gl(n, c). hence x−x ∗ 2 ∈ k and x+x∗ 2 ∈ p. now for x ∈ g, 7, 2(2005) symmetric spaces of noncompact type 119 exp tx ∈ g for all t. since g is self-adjoint and (exp tx)∗ = exp t(x)∗, it follows that x∗ ∈ g. to prove part 4, we first consider the basic cases, gl(n, ir) and gl(n, c). proposition 3.7 let l be a compact subgroup of gl(n, c) (gl(n, ir)). then some conjugate glg−1, g ∈ gl(n, c) (gl(n, ir)) is contained in u(n, c) (o(n, ir)). in particular, u(n, c) is a maximal compact subgroup of gl(n, c) and o(n, ir) a maximal compact subgroup of gl(n, ir). in gl(n, c) and gl(n, ir) any two maximal compact subgroups are conjugate. proof. we deal with the complex case, the other being completely analogous. for an exposition of the existence of haar measure see, for example, [5]. if (, ) is a hermitian inner product on cn, using (finite) haar measure dl on l we can form an l-invariant hermitian inner product on cn given by 〈v,w〉 = ∫ l (lv, lw)dl. thus for some g ∈ gl(n, c), glg−1 is contained in u(n, c). if l ⊃ u(n, c), then it would have to have larger dimension, or if not, u(n, c) would be an open subgroup. as such, it would be the identity component of l since it is connected. because l is compact it would consist of a finite number of open components of u(n, c). but some conjugate, glg−1 is contained in u(n, c) so l can not have larger dimension. similarly, by continuity, there can only be one component. this is a contradiction, so l = u(n, c). in the real case we just work with the compact connected group so(n, ir) instead of u(n, c). thus u(n, c) and o(n, ir) are maximal compact subgroups of gl(n, c) and gl(n, ir), respectively. that any other maximal compact subgroup is conjugate to one of these now follows from the first statement of the proposition. in particular, if l is any compact subgroup of gl(n, c), all its elements have their eigenvalues on the unit circle. from this we see that if an element l ∈ l has all its eigenvalues equal to 1, then l = i. this is because glg−1 is unitary. hence for some u we know uglg−1u−1 is diagonal and also has all eigenvalues equal to 1. thus uglg−1u−1 = i and hence l itself equals i. finally, we turn to the proof of part 4 itself. proof. first suppose l is any compact subgroup of g. then l ∩ p = (1). to see this just observe that, by the theorem below, since l is compact, all its elements have all their eigenvalues on the unit circle. but the eigenvalues of elements of p are all positive. hence all the elements of l have all their eigenvalues equal to 1 and so, as above, each l = i. now let l ⊇ k. then each l ∈ l can be written l = kp, where k ∈ k and p ∈ p . but since k ∈ l, so is p. hence by the above p = i and l = k. hence l ⊆ k, so that actually l = k. thus k is a maximal compact subgroup of g. we have essentially used the conjugacy of maximal compact subgroups in gl(n, c) and gl(n, ir) to show that k is a maximal compact subgroup of g, in general. however to prove, in general, that any two maximal compact subgroups of g are conjugate will require something more. for this we will rely on the important differential geometric fact, called cartan’s fixed point theorem, that a compact group of isometries 120 martin moskowitz 7, 2(2005) acting on a complete simply connected riemannian manifold of nonpositive sectional curvature at every point (hadamard manifold) always has a unique fixed point and, for the reader’s convenience, we will prove cartan’s result as well in the next section. however, we will only prove it for symmetric spaces of noncompact type. this will also establish the fact that for each p ∈ p , stabg(p) is a maximal compact subgroup of g. we note that the cartan involution of g is given by k + p �→ k − p. it is an automorphism of g whose fixed point set is k. we also mention the cartan relations, which were also proved earlier. if the cartan decomposition of g is g = k ⊕ p, since k is a subalgebra and [x∗,y∗] = −[x,y]∗ and [xt,yt] = −[x,y]t it follows that 1. [k, k] ⊆ k, 2. [k, p] ⊆ p, 3. [p, p] ⊆ k. we conclude this section by observing that for all the g we are dealing with there is a natural smooth action of g on p given by (g,p) �→ gtpg. now this action is transitive. to see this, consider the g orbit of i ∈ p , og(i) = {gtg : g ∈ g}. as we saw earlier, this is {p2; p ∈ p}. but since everything in p is exp of a unique element x of p, it follows that everything in p has a unique square root in p , namely exp 1 2 x. this means the action is transitive. what is the isotropy group of stabg(i) of i? this is {g ∈ g : gtg = i} = g∩o(n, ir) = k. hence, by general principles, (g,p) is g-equivariantly diffeomorphic with g acting by left translation on g/k. as we shall see, this transitive action will be of great importance in what follows. observe that this action does not have the two-point homogeneity property. that is, given p,q and p′,q′, all in p , there may not be a g ∈ g so that g(p) = p′ and g(q) = q′, even when dim p = 1. note also that gt(exp x)g is not equal to exp(gtxg) so this is not equivariant with the ir-linear representation of g acting on p by (g,x) �→ gtxg, x ∈ p. concommitantly, the latter is not a transitive action because it is linear, so 0 is a single orbit. in fact, here the orbit space can be parametrized by the number of positive, negative and zero eigenvalues of a representative. 4 the case of hyperbolic space and the lorentz group we now make explicit the cartan decomposition in an important special case and give the lorentz model for hyperbolic n space, hn. we consider o(n, 1) the subgroup of gl(n + 1, ir) leaving invariant the nondegenerate quadratic form q(v,t) = v21 + . . . + v2n − t2, where v ∈ irn and t ∈ ir. equivalently, by polarization, this means leaving invariant the nondegenerate symmetric bilinear form 〈(v,t), (w,s)〉 = (v,w) − ts, where (v,w) is the usual (positive definite) inner product in irn. thus g is defined by the condition g−1 = gt (with respect to 〈,〉). it is easy to check that g is the set of ir-points of a self-adjoint algebraic group and, in particular, is a lie group. now 7, 2(2005) symmetric spaces of noncompact type 121 g is not compact. for example, so(1, 1) ⊆ o(1, 1), which sits inside o(n, 1), is given as follows. g = ( a b c d ) one checks easily that g ∈ so(1, 1) if and only if a2 − c2 = 1, ab − cd = 0 and b2 − d2 = −1. in particular, taking an arbitrary a and c = (a2 − 1) 12 , where a2 −1 = c2 > 0 and letting b and d be determined by the remaining two equations we see that b = (a2−1) 12 = c and d = a. now consider the identity component so0(1, 1). since the locus a2 − c2 = 1 has two connected components, if g ∈ so0(1, 1), then a > 0 and so there is a unique t ∈ ir for which a = cosh t and b = sinh t. thus g(t) = ( cosh t sinh t sinh t cosh t ) because these hyperbolic functions are unbounded, we see even so0(1, 1) is not compact. the identities satisfied by the hyperbolic functions show that this is an abelian subgroup. however, we shall see this without these identities; in fact, we will derive the identities. let x = ( 0 1 1 0 ) a direct calculation using the fact that x2 = i shows that exp tx = i cosh t + x sinh t = g(t), from which it follows that g(s + t) = g(s)g(t). this equation gives all the identities satisfied by the hyperbolic functions sinh and cosh and g is a smooth isomorphism of so0(1, 1) with ir. the geometric importance of such 1-parameter subgroups will be seen in a moment. by theorem 3.6 a maximal compact subgroup of g is given by o(n+1, ir)∩o(n, 1). because subgroups of gl(n, ir) can be regarded as subgroups of gl(n + 1, ir) via the imbedding g �→ diag(g, 1), we may regard o(n, ir) as a subgroup of gl(n + 1, ir) and, in fact, of o(n, 1). thus o(n, ir) ⊆ o(n + 1, ir) ∩ o(n, 1). clearly these are equal. since o(n, ir) has two components, so does o(n, 1) which equals o(n, ir)× an exponential submanifold, p . therefore, o(n, 1)0 = so(n, ir) × p . to identify this connected group, we note that g−1 = gt, ggt = i and so (det g)2 = 1. thus det g = ±1, a discrete set. it follows that so(n, 1) is open in o(n, 1) and hence has the same p . the same is true of so0(n, 1) because we are dealing with lie groups. thus so0(n, 1) = so(n, ir) ×p = g and we now work with this connected group. 2 as with a lie group defined by any nondegenerate bilinear form, the lie algebra g of g = so0(n, 1) is {x ∈ gl(n + 1, ir) : xt = −x}. this lie algebra evidently has dim = (n+1)n 2 . now consider the subspace of gl(n + 1, ir) consisting of 2actually, so(n, 1) is connected if n is even, and has two components if n is odd. 122 martin moskowitz 7, 2(2005) ( k v v 0 ) , where k is the lie algebra of so(n, ir) and v ∈ irn. it is clearly a subspace and has dimension (n−1)n 2 + n = (n+1)n 2 . now a direct calculation, which we leave to the reader, shows that this is a subspace of g, i.e., it consists of skew symmetric matrices with respect to 〈,〉. hence it must coincide with g. here the cartan decomposition is perfectly clear. the k part is ( k 0 0 0 ) , while the p part is ( 0 v v 0 ) . consider the locus of points, h = {(v,t) ∈ irn+1 : q(v,t) = −1}. for g ∈ o(n, 1), q(g(v,t)) = q(v,t). in particular, if q(v,t) = −1, then q(g(v,t)) = −1. thus h is invariant under o(n, 1). now h is a hyperboloid of two sheets: 1+ ‖ v ‖2= t2. so t = ±(1+ ‖ v ‖2) 12 . write h = h+ ∪ h−, a disjoint union of the upper and lower sheet. both sheets are open subsets of h since they are the intersection of the sheet with a half space. each is diffeomorphic with irn. in particular, each is connected and simply connected. we show that g = so0(n, 1) leaves both h+ and h− invariant. if g ∈ g, then g(h+) and g(h+) ⊆ h. so g(h+) = (g(h+) ∩h+) ∪ (g(h+) ∩h−). but h+ is connected and g is continuous. hence g(h+) is connected. therefore g(h+) ⊆ h+ or g(h+) ⊆ h−. since g itself is a diffeomorphism, g(h+) = h+ or g(h+) = h−. we show the former must hold. since g is itself connected and therefore arcwise connected, let gt be a smooth path in g joining g = g1 to i = g0 and let t + = {t ∈ [0, 1] : gt(h+) = h+} and t − = {t ∈ [0, 1] : gt(h+) = h−}. suppose g(h+) = h−. since i(h+) = h+, we have [0, 1] = t + ∪ t −, a disjoint union of nonempty sets. each of these is closed. for if tk → t and say gtk (h+) = h+, for all k, but gt(h+) = h−, then for x ∈ h+, gtk (x) → gt(x). this is impossible as the distance between h+ and h− is 2. we now know g operates on h+ which we shall call hn, the lorentz model of hyperbolic n-space. consider the lowest point, p0 = (0, . . . , 0, 1) ∈ hn. what is stabg(p0)? this is clearly a subgroup which does not change the t coordinate and is arbitrary in the other coordinates since it is linear and so always fixes 0. hence, stabg(p0) = so(n, ir), a maximal compact subgroup of g. next we look at g-orbit o(p0) and show g acts transitively on hn. let p = (v,t), where t = (1+ ‖ v ‖2) 12 , be 7, 2(2005) symmetric spaces of noncompact type 123 any point in hn and apply so(n, ir) to bring it to (‖ v ‖, 0, ...0, t). since we are now essentially in a two-dimensional situation, let us consider (x,y), where y2 −x2 = 1. we want to transform (0, 1) to p by something on the 1-parameter group g(s) = ( cosh s sinh s sinh s cosh s ) . but this is just the fundamental property of the right hand branch of the hyperbola mentioned earlier. therefore, g acts transitively and hn is equivariantly equivalent to so0(n, 1)/ so(n, ir). now consider the hyperplane t = 1 in irn+1. this is the tangent space t (p0) to hn at p0. thus there is a positive definite metric, namely (, ) on t (p0). if p is another point of hn, choose g ∈ g such that g(p) = p0. then dg(p) maps t (p) to t (p0). since g comes from a group, it is invertable. by the chain rule so is its derivative dg(p), so it maps t (p) to t (p0) bijectively. use this to transfer the inner product from t (p0) to t (p). now if h(p) also equals p0, then gh−1 ∈ stabg(p0) = so(n, ir). therefore dg(p)dh(p)−1 is a linear isometry in t (p0). this shows the inner product on t (p) is independent of g and is well defined. hence we get a riemannian metric on hn because g is a lie group acting smoothly on hn. evidently, g acts by isometries, the action is transitive and hn can be identified with g/ stabg(p0) = so0(n, 1)/ so(n, ir). notice that so(n, ir) = stabg(p0) acts transitively on k-dimensional subspaces for all 1 ≤ k ≤ n. in particular, this is so for 2-planes in irn = t (p0)(hn). since it acts by isometries, this means the sectional curvature is constant as both the point and the plane section vary. 5 the g-invariant metric geometry of p here we introduce a riemannian metric on any p and study its most basic differential geometric properties. from now on we will write exp and log instead of exp and log. lemma 5.1 if a and b are n ×n complex matrices, then tr(ab) = tr(ba). also tr(b∗b) ≥ 0 and equals 0 if and only if b = 0. evidently, tr(b)− = tr(b∗). proof. suppose a = (ai,j ) and b = (bk,l). then (ab)i,l = ∑ j ai,jbj,l. therefore tr(ab) = ∑ i,j ai,jbj,i. but then tr(ba) = ∑ i,j bi,jaj,i = ∑ i,j aj,ibi,j = ∑ j,i ai,jbj,i = tr(ab). taking b∗ for a we get tr(b∗b) = ∑ i,j b − j,ibj,i ≥ 0 and equals 0 if and only if b = 0. this enables us to put a hermitian inner product on gl(n, c) called the hilbert schmidt inner product and a symmetric inner product on gl(n, ir) by defining < y,x >= tr(y ∗x). for x hermitian (symmetric), we now study the linear operator adx on gl(n, c) (gl(n, ir)). as we saw from the cartan relations for t ∈ gl(n, c) and x hermitian, [x,t ]∗ = [t ∗,x] = −[x,t ∗]. 124 martin moskowitz 7, 2(2005) lemma 5.2 if x is hermitian, < adx (t ),s >=< t, adx (s) > for all s and t; that is, adx is self-adjoint. in particular, the eigenvalues of such an adx are all real. (this gives a direct proof of the fact that d(exp)x is invertible for all x ∈ p.) proof. we calculate tr([x,t ]∗s) = tr(−[x,t ∗]s) = −tr((xt ∗ − t ∗x)s) = tr(t ∗xs) − tr(xt ∗s). on the other hand, tr(t ∗[x,s]) = tr(t ∗xs) − tr(t ∗sx). thus we must show that tr(xt ∗s) = tr(t ∗sx). but this follows from the lemma above. a formal calculation, which we leave to the reader, proves the following: lemma 5.3 for each u ∈ gl(n, c), lexp(u) = exp(lu ) and rexp(u) = exp(ru ). definition 5.4 for x and y ∈ gl(n, c) let dx (y ) = d dt exp(−x/2) exp(x + ty ) exp(−x/2)|t=0. proposition 5.5 for x ∈ p, the operator dx is self-adjoint on gl(n, c). using functional calculus, this operator is given by the formula dx = sinh( adx 2 )/ adx 2 . proof. let t ∈ ir, x, y ∈ gl(n, c) and x(t) = x + ty . then dx (y ) = exp(−x/2) d dt exp(x(t))|t=0 exp(−x/2). now for all t, x(t) · exp(x(t)) = exp(x(t) ·x(t). differentiating we get x′(t) · exp(x(t)) + x(t) · d dt exp(x(t)) = d dt exp(x(t)) ·x(t) + exp(x(t) ·x′(t). evaluating at t = 0 and subtracting gives x · d dt exp(x(t))|t=0 − ddt exp(x(t))|t=0 · x = exp(x)y −y exp(x). multiplying on both the left and right by exp(−x/2) and taking into account the fact that exp(−x/2) and x commute, we get x · exp(−x/2) d dt exp(x(t))|t=0 exp(−x/2) − exp(−x/2) d dt exp(x(t))|t=0 exp(−x/2)x = exp(x/2)y exp(−x/2) − exp(−x/2)y exp(x/2). substituting for dx (y ), the left hand side becomes xdx (y ) −dx (y )x = adx dx (y ), while the right hand side is lexp(x/2)rexp(−x/2)(y ) −lexp(−x/2)rexp(x/2)(y ). but by the lemma above lexp(u) = exp(lu ) and rexp(u) = exp(ru ). 7, 2(2005) symmetric spaces of noncompact type 125 substituting we get adx dx (y ) = exp(lx/2) exp(r−x/2)(y ) − exp(l−x/2) exp(rx/2)(y ). since lu and ru′ commute for all u and u′, we see that exp(lx/2) exp(r−x/2) = exp(lx/2 + r−x/2) = exp(lx/2 −rx/2) = exp(adx /2). similarly, exp(l−x/2) exp(rx/2) = exp(l−x/2 + rx/2) = exp(−adx /2). so for all y , adx ·dx (y ) = (exp(adx /2) − exp(−adx /2))(y ). now let f(z) = ez/2 −e−z/2 = z + 2(z/2)3/3! + 2(z/2)5/5! + . . . . then f is an entire function and 0 is a removable singularity with f(0) = 0. in terms of f, the equation above says adx dx = f(adx ). this means if we let g(z) = f(z)/z = 1 + (z/2)2/3! + (z/2)4/5! + . . . , with g(0) = 1, then g is also entire and dx = g(adx ). now sinh z = z + z3/3! + z5/5! + . . . so g(z) = sinh(z/2)/(z/2) and hence the conclusion. finally, because dx = g(adx ), adx is self-adjoint and the taylor coefficients of g are real, dx is also self-adjoint. corollary 5.6 for x ∈ p, spec( sinh(adx ) adx ) consists of real numbers greater than or equal to 1. the same is so for the operator dx . proof. since for t ∈ ir, sinh t t = 1 + t2/3! + t4/5! + . . ., we see that sinh t t > 1 unless t = 0. now spec( sinh(adx ) adx ) = {sinh(λ) λ : λ ∈ spec adx}⊆{ sinh(λi −λj ) λi −λj } : λi,λj ∈ spec x}. if λ = λi − λj for distinct eigenvalues of x, then sinh(λ)λ > 1. if λi and λj are equal, then λ = 0 and sinh(λ) λ = 1. we now work exclusively over ir. the same type of arguments also work just as well over c. corollary 5.7 for x ∈ p and y ∈ gl(n, ir), tr(y 2) ≤ tr(dx (y ))2). equality occurs if and only if [x,y ] = 0. 126 martin moskowitz 7, 2(2005) proof. because adx is self-adjoint, we can choose an orthonormal basis of real eigenvectors of adx , y1, . . .yj ∈ gl(n, ir) which, since dx = g(adx ) are also eigenvectors for dx with corresponding real eigenvalues μ1, . . .μj . then dx (yk) = μkyk for all k. if y = ∑ k ak(y )yk, then dx (y ) = ∑ k ak(y )dx (yk) = ∑ k ak(y )μkyk. since the yk form an orthonormal basis, we see tr(dx (y )2) = ∑ k ak(y ) 2μ2k, while tr(y 2) = ∑ k ak(y ) 2. thus we are asking whether ∑ k ak(y ) 2 ≤ ∑k ak(y )2μ2k. since each μk ≥ 1, this is clearly so and equality occurs only if μk = 1 whenever ak(y ) �= 0. rearrange the eigenvectors so that the μk = 1 come first and for k ≥ k0, μk > 1. hence gl(n, ir) = w1 ⊕w∞ is the orthogonal direct sum of two adx -invariant subspaces. here w1 is the 1-eigenspace, and w∞ the sum of all the others. but since ak(y ) = 0 for k ≥ k0, y ∈ w1. but on w1 all eigenvalues of g(adx ) = dx are 1, and the eigenvalues of adx are 0 so adx = 0 on w1 and hence [x,y ] = 0. conversely, if [x,y ] = 0, then adx (y ) = 0. therefore dx = g(adx ) = i. hence w1 = gl(n, ir) and w∞ = (0). therefore all μk = 1 and equality holds. theorem 5.8 along any smooth path p(t) in p we have tr(( d dt log p(t))2) ≤ tr(p−1p′(t))2). with equality if and only if p(t) and p′(t) commute for that t. in particular, tr(p−1p′(t))2) ≥ 0 since if x(t) = log p(t) ∈ p, then x′(t) ∈ p· = p. hence tr(x′(t)tx′(t)) ≥ 0. proof. for each t, it is easy to see that p 1 2 p−1p′p− 1 2 = (p− 1 2 p′p−1p− 1 2 )2. it follows that tr(p−1p′(t))2) = tr(p− 1 2 p′p−1p− 1 2 )2). set x(t) = log p(t). then x(t) is a smooth path in p and p− 1 2 (t) = exp(−x(t)/2). let t be fixed and y = x′(t). since dx (y ) = exp(−x/2) dds exp(x + sy ))|s=0 exp(−x/2), this is p− 1 2 p′p− 1 2 , where p′ = d ds exp(x + sy ))|s=0 (the tangent vector to curve p(t) at p = exp x). hence tr(p− 1 2 p′p− 1 2 2 = tr(dx (x′))2. also tr( ddt log p(t)) 2 = tr x′(t)2. now by the corollary, for each t, tr x′(t)2 ≤ tr(dx(t)(x′(t))2 with equality if and only if x(t) and x′(t) commute for that t. finally we show that if exp x(t) = p(t), then for fixed t, x(t) and x′(t) commute if and only if p(t) and p′(t) commute. for by the chain rule and the formula for d(exp) (see l.s. varadarajan [11]), p′(t) = d(exp)x(t)x ′(t) = φ(−ad x(t))x′(t), where φ is the entire function given by φ(z) = ∑∞ n=0 zn (n+1)! . if x′ commutes with x for fixed t, then since φ(0) = 1, we see that φ(−ad(x))x′ = x′ so that p′ = x′. in particular, p′ commutes with x and therefore with exp x = p. on the other hand, if φ(−ad(x))x′ commutes with exp x = p, then since log : p → p is given by a convergent power series in p (see theorem 3.6, part 3), it must also commute with log p = x. looking at the specific form of the function φ, it follows that [x,x′ − ad(x)(x′)/2! + ad2(x)(x′)/3! . . .] = 0. 7, 2(2005) symmetric spaces of noncompact type 127 that is, ad(x)(x′) − ad2(x)(x′)/2! + ad3(x)(x′)/3! . . . = 0. hence exp(−ad(x)(x′)) = x′. taking exp(adx ) of both sides tells us exp(adx )(x′) = x′. therefore adexp x (x′) = x′ so x′ commutes with exp x. but then, reasoning as above, x′ must commute with log(exp x) = x. since what is inside the square root is real and positive, we make the following definition. definition 5.9 let p(t) be a smooth path in p, where a ≤ t ≤ b. then its length l(p) = ∫ b a tr(p−1p′(t))2) 1 2 dt. the riemannian metric is given by ds2 = tr((p−1p′)2)dt2. we call this metric d. proposition 5.10 g acts isometrically on p. proof. we calculate that (gtpg)−1(gtpg)′ = g−1p−1(gt)−1gtp′g = g−1p−1p′g. hence ((gtpg)−1(gtpg)′)2 = g−1(p−1p′)2g. taking traces we get tr((gtpg)−1(gtpg)′)2) = tr(p−1p′)2). on p we place the metric given infinitesimally by ds2 = tr(( d dt log p(t))2)dt2, that is, if x(t) is a smooth path in p, then ds2 = tr(x′(t)2)dt2. we call this metric dp. earlier we defined an inner product on gl(n, ir) by < y,x >= tr(y tx). hence the linear subspace p has an inner product on it by restriction, namely < y,x >= tr(y x). the associated norm is ‖ y ‖2= tr(y 2). this, together with the formula above, shows dp is the euclidean metric. if we transfer dp to p , then dp(p,q) =‖ log p − log q ‖. this will give us the opportunity to compare dp and d on p . since along any smooth path p(t) in p we have tr( d dt log p(t))2) ≤ tr(p(t)−1p′(t))2, we see that infinitesimally and hence globally dp ≤ d. now for x ∈ p, dx = sinh(adx /2)adx /2 . it follows that spec dx = { sinh(λ/2) λ/2 : λ ∈ spec adx}. as sinh t t is analytic, by continuity sinh t t → 1 as t → 0. this tells us that from the formulas for tr(dx (y ))2 and tr(y 2), if x → 0, then independently of y , tr(dx (y ))2 can be made as near as we want to tr(y 2). this last statement implies that for p and q in a sufficiently small neighborhood of a point p0, which by transitivity of g we may assume to be i, the nonpositively curved symmetric space and euclidean distances approach one another. lim p,q→p0 d(p,q) dp(p,q) = 1. 128 martin moskowitz 7, 2(2005) this has the interesting philosophical consequence that in the nearby part of the universe that man inhabits, because of experimental error in making measurements, nonpositively curved symmetric space distances and euclidean ones are indistinguishable. as we shall show below, angles at i are in any case identical. this means no experiment can tell us if we “really” live in a hyperbolic or euclidean world. corollary 5.11 if p = log x ∈ p, the 1-parameter subgroup exp tx is the unique geodesic in (p,d) joining i with p. moreover, any two points of p can be joined by a unique geodesic. (we shall see rather explicitly and directly how this geodesic is determined by its initial and terminal points.) this corollary also follows from more general facts in differential geometry. this is because as a 1-parameter subgroup every geodesic emanating from i has infinite length. since g acts transitively by isometries, this is true at every point. hence by the hopf-rinow theorem (see j. milnor [7]) p is complete. in particular, any two points can be joined by a shortest geodesic (also hopf-rinow). being diffeomorphic to euclidean space, p is simply connected. if p had nonpositive sectional curvature in every section and at every point, then this geodesic would be unique. this last fact is actually valid for any hadamard manifold and is called the cartan-hadamard theorem. we will give a direct proof of completeness of p shortly. proof. consider a path p(t) in p which happens to be a 1-parameter subgroup. since p(t) = exp tx, log p(t) = tx and its derivative is x. thus for each t, log p(t) and its derivative commute. hence, as we showed, p(t) and p′(t) also commute. this tells us that all along p(t), dp and d coincide. but the 1-dimensional subspaces of p are geodesics for dp. hence if p = log x ∈ p , the 1-parameter subgroup exp tx is the unique geodesic in (p,d) joining i with p. let p and q be distinct points of p . since g acts transitively on p , we can choose g so that g(q) = i. connect i with g(p) by its unique geodesic γ. since g acts isometrically, g−1(i) = q, g−1(g(p)) = p and g−1(γ) is the unique geodesic joining them. corollary 5.12 a curve p(t) in p is a geodesic through p0 ∈ p if and only if p(t) = g(exp tx)gt, where x ∈ p and g ∈ g. proof. since g acts transitively by isometries on p , choose g ∈ g so that gigt = p0. the result follows from the above since the 1-parameter subgroup exp tx is the unique geodesic in (p,d) beginning at i in the direction x. corollary 5.13 at i the angles in the two metrics coincide. proof. let x and y be two vectors in p and p(t) and q(t) be curves in p passing through i with tangent vectors x and y , respectively, and let p0(t) = exp tx and q0(t) = exp ty be two 1-parameter groups in p . then since x and y are also the tangent vectors of p0 and q0, respectively, the angle between p and q equals that between p0 and q0. we may therefore replace p and q by p0 and q0. now p−1p′q−1q′(0) is just xy so that tr(p−1p′q−1q′(0)) = tr(xy ). 7, 2(2005) symmetric spaces of noncompact type 129 corollary 5.14 for x ∈ p, d(i, exp x) = tr 12 (x2). proof. the 1-parameter group exp tx is a geodesic in p passing through i at t = 0. hence, infinitesimally along this curve, d = dp. this implies the same is true globally along it. put another way, at each point of exp tx, for 0 ≤ t ≤ 1, the theorem tells us the metric is tr( d dt (tx)2) = tr(x2). since this is independent of t, integrating from 0 to 1 gives tr(x2). corollary 5.15 for x and y ∈ p, d(exp x, exp y ) ≥ tr 12 ((x −y )2). corollary 5.16 p is complete. proof. let pk be a cauchy sequence in (p,d). by the inequality above, xk = log pk is a cauchy sequence in (p,dp) which must converge to x since euclidean space is complete. by continuity, pk converges to exp x = p. corollary 5.17 (law of cosines). let a, b and c be the lengths of the sides of a geodesic triangle in p and a, b and c be the corresponding vertices. then c2 ≥ a2 + b2 − 2ab cos c and the sum of the angles a + b + c ≤ π. moreover, if the vertex c is at i then equality holds if and only if 1. the triangle lies in a connected abelian subgroup of p, or equivalently, 2. a + b + c = π. proof. put c at the identity via an isometry from g. then the euclidean angle at c equals the angle in the metric d. also, lp(c) ≤ c and lp(a) = a and lp(b) = b. the inequality now follows from the euclidean law of cosines. equality holds if and only if lp(c) = c. this occurs if and only if log takes side c to a geodesic in p (i.e. a straight line) of the same length. this is also equivalent to tr(( d dt log p(t))2) = tr(p−1p′(t)2), for all t, where p(t) denotes the geodesic side of length c. this occurs if and only if p(t) satisfies the condition that p(t) and p′(t) commute for all t which, as we showed, is equivalent [x,y] = 0, where x and y are the infinitesimal generators of the sides a and b. thus equality in the law of cosines holds if and only if the euclidean triangle lies in a two-dimensional abelian subalgebra of g contained in p. equivalently, the geodesic triangle lies in a two-dimensional abelian subgroup of g contained in p . such a subgroup is called a flat of p . next we show that in general the sum of the angles ≤ π. since d is a metric and c = d(a,b), etc., it follows that each length a, b, or c is less than the sum of the other two. therefore there is an ordinary plane triangle with sides a, b and c. denote its angles by a′, b′ and c′. then a ≤ a′, b ≤ b′ and c ≤ c′. for by the law of cosines c2 ≥ a2 + b2 − 2ab cos c and c2 = a2 + b2 − 2ab cos c. this means cos c′ ≤ cos c. but then because c and c′ are between 0 and π and cos is 130 martin moskowitz 7, 2(2005) monotone decreasing there, we see c ≤ c′. similarly, this holds for the others. since a′ + b′ + c′ = π, it follows that a + b + c ≤ π. if c2 > a2 + b2 − 2ab cos c, then, as above, construct an ordinary plane triangle with sides a, b and c and angles a′, b′ and c′. then since here we have strict inequality, it follows as above that c < c′. but it is always the case that a ≤ a′ and b ≤ b′. hence a + b + c < a′ + b′ + c′ = π. conversely, if a + b + c = π, then c2 = a2 + b2 −2ab cosc and [x,y ] = 0. therefore x and y generate an abelian subalgebra, and the triangle lies in a flat. our next result is of fundamental importance. nonpositive and positive sectional curvature distinguish the symmetric spaces of noncompact type from those of compact type. corollary 5.18 the sectional curvature of p is ≤ 0 and strictly < 0 off flats. in particular, p is a hadamard manifold. before turning to the proof we remark that when x,y ∈ p and are orthonormal with respect to the killing form, one actually has k(x,y ) = − ‖ [x,y ] ‖2 (see j. cheeger and d. ebin [1]). however, we shall not need this formula. proof. each geodesic triangle lies in a plane section. we have just shown that each geodesic triangle in each such section has the sum of the angles ≤ π and the sum of the angles < π if we are off a flat. it is a standard result of two-dimensional riemannian geometry (gauss-bonnet theorem) that these conditions are equivalent to k ≤ 0 and k < 0, respectively, where k denotes the gaussian curvature of the section, that is, the sectional curvature. definition 5.19 a submanifold n of a riemannian manifold m is called totally geodesic if given any two points of n and a geodesic γ in m joining them, γ lies entirely in n. corollary 5.20 p is a totally geodesic submanifold in the set of all positive definite symmetric matrices. proof. let p and q ∈ p . since p12 and p− 12 are self-adjoint, p− 12 qp− 12 is positive definite and symmetric. but as we showed earlier, p− 1 2 ∈ g. hence p− 12 qp− 12 ∈ g. because p− 1 2 is self-adjoint we see that p− 1 2 qp− 1 2 ∈ p . let x ∈ p be its log. then exp tx lies in p , for all real t. therefore γ(t) = p 1 2 (exp tx)p 1 2 is a geodesic in p . clearly, γ(0) = p and γ(1) = q. since there is a unique geodesic joining these points in both p and in the positive definite matrices, this completes the proof. we conclude this section with the standard definition of a symmetric space. definition 5.21 a riemmanian manifold m is called a symmetric space if for each point p ∈ m there is an isometry σp of m satisfying the following conditions. 1. σ2p = i, but σp �= i 7, 2(2005) symmetric spaces of noncompact type 131 2. σp has only isolated fixed points among which is p. (in our case, p is actually the only fixed point.) 3. d(σp) on tp(m) = −i. thus the main feature of the definition is that for each point p there is an isometry which leaves p fixed and reverses geodesics through p. corollary 5.22 p is a symmetric space. proof. since g acts transitively and by isometries, we may restrict ourselves to the case p = i. take σi = σ(p) = p−1, for each p ∈ p . this map is clearly of order 2. if p is σ fixed, then p2 = i. hence every conjugate of p also has order 2. this means p ∈ k ∩p , which as we saw earlier = (i), thus i is the only fixed point. let p = exp x, then σ(p) = exp(−x) so that d(σp)p = −i, where here we identify ti (p) with p. it remains to see that σ is an isometry. for a curve p(t) in p , since p(t)p(t)−1 = i, differentiating tells us d dt (p(t)−1) = −p(t)−1 dp dt p(t)−1. substituting we get tr(p dp−1 dt )2 = tr(p−p(t)−1 dp dt p(t)−1p−p(t)−1 dp dt p(t)−1p). cancelling the minus signs and pp−1, we have tr( dp dt p(t)−1 dp dt p(t)−1). since tr(ab) = tr(ba), this is tr(p(t)−1 dp dt p(t)−1 dp dt ) = tr((p(t)−1 dp dt )2. thus for every t, tr(p(t) dp−1 dt )2 = tr((p(t)−1 dp dt )2. hence σ is an isometry of p and the latter is a symmetric space. 6 the conjugacy of maximal compact subgroups the theorem on the conjugacy of maximal compact subgroups of g in the present context is due to e. cartan. actually, the result is true for an arbitrary connected lie group and is due to k. iwasawa and the case of a finite number of components to g.d. mostow. in this more general context see [5]. we shall deal with this problem in the present context by means of cartan’s fixed point theorem which states that a compact group of isometries acting on a complete, simply connected riemannian manifold of nonpositive sectional curvature (hadamard manifold) has a unique fixed point. however, here we will prove the fixed point theorem where we need it, namely, in the special case when the manifold is a symmetric space of noncompact type. 132 martin moskowitz 7, 2(2005) theorem 6.1 let f : c → (p,d) be a continuous map where d denotes the distance on a symmetric space p of noncompact type and c is a compact space with a positive finite regular measure, μ, on it. then the functional j(p) = ∫ c d2(p,f(c))dμ(c),p ∈ p attains its minimum value at a unique point of p called the center of gravity of f(c) with respect to μ. proof. without loss of generality we may normalize the measure so that μ(c) = 1. we first prove the result for euclidean space irn. then f(c) = (f1(c), . . . ,fn(c)), p = (p1, . . . ,pn) and j(p) = ∫ c ∑ i (pi −fi(c))2dμ(c) = ∫ c ( ∑ i p2i − 2 ∑ i pifi(c) + ∑ i fi(c) 2)dμ(c). by our normalization, this is |p|2 − 2(a,p) + ∑i bi, where for i = 1, . . . ,n, ai =∫ c fi(c)dμ(c) and bi = ∫ c fi(c)2dμ(c). completing the square we get j(p) = |p|2 − 2(a,p) + |a|2 −|a|2 + ∑ i bi = |p−a|2 + ∑ i bi −|a|2. this is clearly minimized exactly at the point p = a. now let p be a symmetric space as above. we first show j is continuous on p . if pn → p in p , then since d is continuous, d2(pn,f(c)) → d2(p,f(c)) for each c ∈ c. hence for fixed c ∈ c, d2(pn,f(c)) ≤ a constant independent of n. since the measure on c is finite this constant function is integrable and so by the dominated convergence theorem j(pn) → j(p). we will now find a compact set k in m and an r > 0 so that j(p) > r2 on m − k and j(p0) ≤ r2 at some point p0 ∈ k. then the minimum value of j, if any, would have to be on k and there would be one since k is compact and j continuous. this would prove existence of a fixed point. to do this, choose p0 arbitrarily in p − f(c) (since f(c) is compact and p is not, the complement of f(c) is nonempty) and let infc∈c d(p0,f(c)) = r. then r is finite and positive. let k = {p ∈ p : d(p,f(c)) ≤ r}. if b is a compact set in p , then since exp is a global diffeomorphism, log b is compact in p. the formula d(exp x, exp y) ≥ dp(x,y) tells us that log{p ∈ p : d(p,b) ≤ r}⊆{x ∈ p : dp(x, log b) ≤ r}. the right hand side is closed and bounded so since p is a euclidean space, this set is compact. similarly, by elementary properties of the metric d, {p ∈ p : d(p,b) ≤ r} is closed in p and, since we have a diffeomorphism, the left side is closed and therefore is compact in p. but then exp of this set, i.e., k is compact in p . evidently p0 ∈ k and integrating and making use of the normalization of μ tells us j(p0) ≤ r2. if p ∈ m −k, then by compactness of f(c) there is a δ > 0 such that d(p,f(c)) > r + δ for all c ∈ c. upon integration we get j(p) > r2. 7, 2(2005) symmetric spaces of noncompact type 133 turning to the uniqueness, let log ·f = fp. this is a continuous map c → p and we can construct the functional jp on p defined by jp(x) = ∫ c d2p(x,fp(c))dμ(c),x ∈ p. for a point p ∈ p and x = log p, since dp(x,y) ≤ d(p,q) for all q = exp y ∈ p , we see that dp(x,fp(c)) ≤ d(p,f(c)) for all c. squaring and integrating tells us that jp(x) ≤ j(p),p ∈ p. let p0 be a point of p where a minimum value of j is attained and let p = log x ∈ p approach p0. then jp(log p0) ≤ jp(x). for if for some x → log p0, jp(x) < jp(log p0), which is, in turn, ≤ j(p0). then since j and jp involve integrating d and dp, respectively, over a compacta and lim p,q→p0 d(p,q)/dp(log p, log q) = 1, there is some sufficiently nearby point y to x for which j(log y) is near enough to jp(y) so that jp(y) ≤ j(log y) and so is < j(p0), contradicting the minimality of p0. thus for all p → p0, jp(log p0) ≤ jp(log p). this means log p0 is also a minimum value for jp. therefore if log p1 is another point of p where the minimum value of j is attained, then log p1 is also a minimum value for jp. by uniqueness in the euclidean case log p0 = log p1 and so p0 = p1. as usual, g is a self-adjoint essentially algebraic subgroup of gl(n, ir), or gl(n, c) acting on p by (g,p) �→ gtpg. corollary 6.2 if c is a compact subgroup of g, then c has a simultaneous fixed point acting on p. proof. let μ = dc be normalized haar measure on c, p0 a point of p and f : c → p be the continuous function given by f(c) = c · p0. then j(p) = ∫ c d2(p,c · p0)dc. now for c′ ∈ c, j(c′p) = ∫ c d2(c′p,c · p0)dc. since c acts by isometries this is∫ c d2(p, (c′)−1c·p0)dc. by left invariance of dc we get ∫ c d2(p,c·p0)dc. thus j(p0) = j(c ·p0) for all c ∈ c and p0 ∈ p . but by the fixed point theorem, j has a unique minimum value at some p0 ∈ p . this means c(p0) = p0 for all c ∈ c. we now prove the conjugacy theorem for maximal compact subgroups of g. the proof in [5] is similar to the one given here, but rather than involve differential geometry itself, it uses a convexity argument and a function which mimics the metric. theorem 6.3 let g be a self-adjoint essentially algebraic subgroup of gl(n, ir), or gl(n, c). then all maximal compact subgroups of g are conjugate. any compact subgroup of g is contained in a maximal one. proof. let c be a compact subgroup of g. by corollary 6.2 there is a point p0 ∈ p fixed under the action. thus c ⊆ stabg(p0). but this action is transitive so stabg(p0) = gkg−1 for some g ∈ g. since k is a maximal compact subgroup by theorem 3.6, so is the conjugate gkg−1. this proves the second statement. if c were itself maximal then c = gkg−1. 134 martin moskowitz 7, 2(2005) 7 the rank and two-point homogeneous spaces let g be the real (or complex) linear lie algebra of g, as above, and g = k ⊕ p be a cartan decomposition. by abuse of notation we shall call a subalgebra of g contained in p a subalgebra of p. when abelian, such subalgebras will play an important role in what follows. by finite dimensionality, maximal abelian subalgebras of p clearly exist. in fact, any abelian subset of p is contained in a maximal abelian subalgebra of p. consider the adjoint representation of k on g. then the subspace p is invariant under this action. since adk(p) ⊆ g, to see this we need only check that adk(p) is symmetric (hermitian). we shall always deal with the symmetric case, unless the hermitian one is harder. so for p ∈ p and k ∈ k we have adk(p) = kpk−1 = kpkt. hence the transpose is (kpkt)t = kpkt = adk(p). theorem 7.1 in g any two maximal abelian subalgebras a and a′ of p are conjugate by some element of k. in particular, their common dimension is an invariant of g called r = rank(g). this theorem was origianally proved by e. cartan; however, here we will use the following argument which is essentially due to g.a. hunt [6]. proof. let (, ) be the killing form on g. this is positive definite on p and negative definite on k. since k is compact and acts on g, by averaging with respect to haar measure on k we can, in addition, assume this form to be k-invariant. that is, each adk preserves the form. let a ∈ a and a′ ∈ a′ and consider the smooth numerical function on k given by f(k) = (adk a,a′). by compactness of k, this continuous function has a minimum value at k0 and by calculus, at this point the derivative is zero. thus for each x ∈ k, d dt (adexp tx·k0 a,a ′)|t=0 = 0. but (adexp tx·k0 a,a ′) = (adexp tx adk0 a,a ′) = (exp t ad(x) adk0 a,a ′). hence differentiating with respect to t at t = 0 gives (ad(x) adk0 a,a ′) = 0 for all x ∈ k. a calculation similar to the one just given shows that the k-invariance of the form on k has an infinitesimal version, ([x,y],z) + (y, [x,z]) = 0, valid for all x ∈ k and y,z ∈ p. hence, also for all x ∈ k, we get (x, [adk0 a,a ′]) = 0. now adk0 a and a ′ ∈ p and [p, p] ⊆ k. hence [adk0 a,a′] ∈ k and because (x, [adk0 a,a ′]) = 0 for all x ∈ k and (.) is nondegenerate on k, it follows that [adk0 a,a ′] = 0. finally, since a and a′ are arbitrary, [adk0 (a), a ′] = 0. now hold a ∈ a fixed. because [adk0 a, a′] = 0 we see by maximality of a′ that adk0 a ∈ a′ and since a is arbitrary adk0 a ⊆ a′. thus a ⊆ adk−10 (a ′). the latter is an abelian subalgebra of p and by maximality of a they coincide. thus adk0 (a) = a ′. it might be helpful to mention the significance of this theorem in the most elementary situation, namely, when g = gl(n, ir), or gl(n, c). as usual, we restrict our remarks to the real case. here p is the set of all symmetric matrices of order n. let d denote the diagonal matrices. these evidently form an abelian subalgebra of p. now d is actually maximal abelian. to see this, suppose there were a possibly larger abelian subalgebra a. each element of a is diagonalizable being symmetric. since all these elements commute they are simultaneously diagonalizable. this means, in effect, that a = d. thus d is a maximal abelian subalgebra of p. now since any 7, 2(2005) symmetric spaces of noncompact type 135 commuting family of symmetric matrices can be imbedded in a maximal abelian subalgebra of p, theorem 7.1 tells us that this commuting family can be simultaneously diagonalized by an orthonormal change of coordianates. similarly, over c it says any commuting family of hermitian matrices is simultaneously conjugate by a unitary matrix to the diagonal matrices. this is exactly the content of the theorem in these two cases. thus theorem 7.1 is a generalization of the classic result on simultaneous diagonalization of commuting families of quadratic or hermitian forms. we also note that the statement of theorem 7.1 without the stipulation that the subalgebras are in p is false. that is, in general, maximal abelian subalgebras of g are not conjugate. for example, in g = sl(2, ir), the diagonal elements, the skew symmetric elements and the unitriangular elements are each maximal abelian subalgebras of g, but no two of them are conjugate (by an element of k or anything else). we leave the verification of these facts to the reader. corollary 7.2 in g let a be a maximal abelian subalgebra of p. then the conjugates of a by k fill out p, that is, ∪k∈k adk(a) = p. of course, exponentiating and taking into account that exp commutes with conjugation, this translates on the group level to p = ∪k∈kkak−1, where a is the abelian analytic subgroup of g with lie algebra a. proof. let p ∈ p and choose a maximal abelian subalgebra a′ containing it. by our theorem there is some k ∈ k conjugating a′ to a. in particular, adk(p) ∈ a for some k ∈ k and so p ∈ adk−1 (a). our next corollary, also called the cartan decomposition, follows from this last fact together with the usual cartan decomposition, theorem 3.6. corollary 7.3 under the same hypothesis g = kak. an important use of this form of the cartan decomposition is that it reduces the study of the asymptotics at ∞ on g to a. that is, suppose gi is a sequence in g tending to ∞. now gi = kiaili, where ki and li ∈ k and ai ∈ a. since both ki and li have convergent subsequences, again denoted by ki and li, which converge to k and l, respectively, the sequence ai must also tend to ∞. thus in certain situations we can assume the original sequence started out in a. proof. g = kp ⊆ kkak = kak ⊆ g we now make explicit the notions of a homogeneous space and two-fold transitivity from differential geometry mentioned earlier. if x is a connected riemannian manifold, we shall say x is a homogeneous space if the isometry group isom(x) acts transitively on x. now even when the action may not be transitive it is a theorem of myers and steenrod (see [4]) that isom(x) is a lie group and the stabilizer kp of any point p is a compact subgroup. in the case of a transitive action it follows from general facts about actions that x is equivariantly equivalent as a riemannian manifold to isom(x)/kp with the quotient structure. of course, if some subgroup of the isometry group acted transitively then these same conclusions could be drawn replacing the isometry group by the subgroup. clearly, by its very construction, every symmetric space of noncompact type is a homogeneous space. 136 martin moskowitz 7, 2(2005) now suppose in our symmetric space p we are given points p and q and p′ and q′ of p with d(p,q) = d(p′,q′). we shall say a subgroup of the isometry group acts two-fold transitively if there is always an isometry g in the subgroup taking p to p′ and q to q′ for any choices of such points. when this occurs we shall say p is a two-point homogeneous space. clearly, every two-point homogeneous space is a homogeneous space. as we shall see the converse is not true and learn which of our symmetric spaces is actually a two-point homogeneous space. before doing so, we make a simple observation which follows immediately from transitivity. proposition 7.4 let g be as above and k be a maximal compact subgroup. then g/k = p is a two-point homogeneous space if and only if k acts transitively on the unit geodesic sphere u of p. for example, when g = so0(n, 1) and k = so(n), then g/k = hn, hyperbolic n-space. here k acts transitively on u. hence so0(n, 1) acts two fold-transitively on hn. as we shall see in theorem 7.5, this fact is a special case of a more general result. we also remark that this definition can be given for any connected riemannian manifold and indeed such a manifold is of necessity a symmetric space (see [4]). our last result tells us the significance of the rank in this connection. before proving it we observe that for all semisimple or reductive groups under consideration dim p ≥ 2. the lowest dimension arising is the case of the upper half plane introduced at the very beginning of this article. indeed, suppose dim p = 1. then since p is abelian and exp is a global diffeomorphism p → p , it follows easily from exp x + y = exp x exp y, where x,y ∈ p, that p is a connected one-dimensional abelian lie group. now since k acts on p by conjugation and in this case these form a connected group of automorphisms of p we see that this action is trivial because aut(p)0 = (1). thus k centralizes p and we have a direct product of groups. such a group is not semisimple. it is clearly also not gl(n, ir) or gl(n, c) for n ≥ 2. we now characterize two-point homogeneous symmetric spaces. theorem 7.5 let g be as above, g be its lie algebra and k be a maximal compact subgroup. then g/k is a two-point homogeneous space if and only if rank(g) = 1. proof. we first assume rank(g) = 1. by proposition 7.4, to see that g/k is a twopoint homogeneous space, it is sufficient to show int(k) acts transitively on geodesic spheres of p . of course, we know ad(k) acts linearly and isometrically on p. now by corollary 7.2 ∪k∈k adk(a) = p. hence each point p ∈ u is a conjugate by something in k of a point on the unit sphere of a. since the dimension of this sphere is zero, it consists of two points, ±a0. hence u = ad(k)(a0) ∪ ad(k)(−a0). in any case, u is a union of a finite number of orbits all of which are compact and therefore closed since k itself is compact. since these are closed, so is the union of all but one of them. hence u is the disjoint union of two nonempty closed sets. this is impossible since u is connected because dim p ≥ 2. thus there is only one orbit and therefore k acts transitively on u. before proving the converse, the following generic example will be instructive. let g = sl(n, ir), n ≥ 2. we shall see sl(n, ir)/ so(n) is a two-point homogeneous 7, 2(2005) symmetric spaces of noncompact type 137 space if and only if n = 2. this suggests that unless the rank = 1, one can never have a two-point homogeneous symmetric space. to see this, observe that since g/k = p is the set of positive definite n × n symmetric matrices of det 1, it follows that dim p = n(n+1) 2 −1. also dim k = n(n−1) 2 . hence if u denotes the geodesic unit sphere in p , its dimension is n(n+1) 2 −2. let k act on p and u by (k,p) �→ kpk−1 = kpkt. for p ∈ u the dimension of ok (p), the k-orbit of p, is dimok (p) = n(n− 1) 2 − (n− 1)(n− 2) 2 = n− 1. now if k were to act transitively on u, then dimok (p) = dim u. that is, n− 1 = n(n+1) 2 − 2. alternatively, (n− 2)(n + 1) = 0. since n ≥ 2, this holds if and only if n = 2. we conclude by proving the converse. proof. suppose (p,g) is a two-point homogeneous space and hence k acts transitively (by conjugation) on the unit geodesic sphere u in p. then u = ok (a0), where a0 ∈ p and ‖ a0 ‖= 1. since a0 is conjugate to something in a, we may assume a0 ∈ a. in particular, everything in u ∩ a is k-conjugate to everything else. because these matrices commute, they can be simultaneously diagonalized by some u0 (which may not be in k). by replacing these a’s by their u0 conjugates we may assume they are all diagonal. being conjugate under k these matrices have the same spectrum s. since s is finite and k is connected, k can not permute this finite set. thus the action of k leaves each of these matrices fixed. but k acts transitively on u ∩ a so u ∩ a must be a point. hence it has dim 0 and dim a = 1. received: june 2003. revised: august 2003. references [1] j. cheeger and d. ebin, comparison theorems in riemannian geometry, north holland, amsterdam, 1975. [2] c. chevalley, the theory of lie groups, princeton university press, princeton, 1946. [3] e. cartan, groupes simples clos et ouverts et geometrie riemannienne, journal de math. pures et appliqués 8, (1929) 1-33. [4] s. helgason, differential geometry, lie groups and symmetric spaces, academic press, new york, 1978. 138 martin moskowitz 7, 2(2005) [5] g. p. hochschild, the structure of lie groups, holden day, san francisco, 1965. [6] g. a. hunt, a theorem of e. cartan, proc. amer. math. soc., 7 (1956) 307-308. [7] j. milnor, morse theory, annals of math. studies 51, princeton university press, princeton, 1963. [8] m. moskowitz, a course in complex analysis in one variable, world scientific publishing co., singapore, 2002. [9] g. d. mostow, strong rigidity of locally symmetric spaces, annals of math. studies 78, princeton university press, princeton, 1973. [10] g. d. mostow, self adjoint groups, annals of math., 62 (1955), 44-55. [11] v. s. varadarajan, lie groups, lie algebras and their representations, prentice-hall series in modern analysis, englewood cliffs new jersey, 1974. r.eclbjdo· octybre 1992. on a pencil of k3 surfaces • víctor gonzález aguilera resumen. una superficie k3 es una superficie analítica compleja s(dimcs = 2) que es simplemente conexa y cuyo fibrado canónico k 5 es triv ia l. para las superficies k3 que son algebraicas (es decir, inmersas en pn), existe un espacio de módulos m(k3) que es 19 dimensional. en esta nota se construye una subvariedad !-dimensional de m (k3) con un grupo de simetrías fijo (z/4'71..) 2 x g.0 {y · µ(d ∩ {f > y})} . (3) the shilkret integral takes values in [0,∞]. the shilkret integral ([5]) has the following properties: (n∗) ∫ ω χedµ = µ(e) , (4) where χe is the indicator function on e ∈ f, (n∗) ∫ d cfdµ = c(n∗) ∫ d fdµ, c ≥ 0, (5) (n∗) ∫ d sup n∈n fndµ = sup n∈n (n∗) ∫ d fndµ, (6) where fn, n ∈ n, is an increasing sequence of elementary (countably valued) functions converging uniformly to f. furthermore we have (n∗) ∫ d fdµ ≥ 0, (7) f ≥ g implies (n∗) ∫ d fdµ ≥ (n∗) ∫ d gdµ, (8) where f,g : ω → [0,∞] are measurable. let a ≤ f(ω) ≤ b for almost every ω ∈ e, then aµ(e) ≤ (n∗) ∫ e fdµ ≤ bµ(e) ; (9) (n∗) ∫ e 1dµ = µ(e) ; (10) f > 0 almost everywhere and (n∗) ∫ e fdµ = 0 imply µ(e) = 0; (n∗) ∫ ω fdµ = 0 if and only f = 0 almost everywhere; (n∗) ∫ ω fdµ < ∞ implies that n(f) := {ω ∈ ω|f(ω) 6= 0} has σ-finite measure; 4 george a. anastassiou cubo 20, 1 (2018) (n∗) ∫ d (f + g)dµ ≤ (n∗) ∫ d fdµ + (n∗) ∫ d gdµ; (11) and ∣ ∣ ∣ ∣ (n∗) ∫ d fdµ − (n∗) ∫ d gdµ ∣ ∣ ∣ ∣ ≤ (n∗) ∫ d |f − g|dµ. (12) from now on in this article we assume that µ : f → [0,+∞). 3 univariate theory this section is motivated and inspired by [3] and [4]. let l be the lebesgue σ− algebra on r, and the set function µ : l → [0,+∞], which is assumed to be maxitive. let cu (r,r+) be the space of uniformly continuous functions from r into r+, and c(r,r+) the space of continuous functions from r into r+. for any f ∈ cu (r,r+) we have ω1 (f,δ) < +∞, δ > 0, where ω1 (f,δ) := sup x,y∈r: |x−y|≤δ |f(x) − f(y)| , δ > 0, is the first modulus of continuity. let {tk}k∈z be a sequence of positive sublinear operators that map cu (r,r+) into c(r,r+) with the property (tk (f)) (x) := l0 ( f ( 2−k· )) (x) , ∀ x ∈ r, ∀ f ∈ cu (r,r+) . (13) for a fixed a > 0 we assume that sup u,y∈r: |u−y|≤a |t0 (f,u) − f(y)| ≤ ω1 ( f, ma + n 2r ) , ∀ f ∈ cu (r,r+) , (14) where m ∈ n, n ∈ z+, r ∈ z. let ψ : r → r+ which is lebesgue measurable, such that (n∗) ∫a −a ψ(u)dµ(u) = 1. (15) we define the positive sublinear-shilkret operators (t0 (f)) (x) := (n ∗) ∫a −a (t0f) (x − u)ψ(u)dµ(u) , (16) and (tk (f)) (x) := ( t0 ( f ( 2−k· )))( 2kx ) , ∀ k ∈ z, ∀ x ∈ r. (17) cubo 20, 1 (2018) approximation by shift invariant univariate . . . 5 therefore it holds (tk (f)) (x) = (n ∗ ) ∫a −a ( t0 ( f ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) = (18) (n∗) ∫a −a (tk (f)) ( 2kx − u ) ψ(u)dµ(u) , ∀ x ∈ r, ∀ k ∈ z. indeed here we have (tk (f)) (x) (8) ≤ (n∗) ∫a −a ∥ ∥tk (f) ( 2kx − · )∥ ∥ ∞,[−a,a] ψ(u)dµ(u) (5) = ∥ ∥tk (f) ( 2kx − · )∥ ∥ ∞,[−a,a] ( (n∗) ∫a −a ψ(u)dµ(u) ) = (19) ∥ ∥tk (f) ( 2kx − · ) ∥ ∥ ∞,[−a,a] < +∞. hence (tk (f)) (x) ∈ r+ is well-defined. let f,g ∈ m (r,r+) (lebesgue measurable functions) where x ∈ a, a ⊂ r is a lebesgue measurable set. we derive that ∣ ∣ ∣ ∣ (n∗) ∫ a f(x)dµ(x) − n∗ ∫ a g(x)dµ(x) ∣ ∣ ∣ ∣ (12) ≤ (n∗) ∫ a |f(x) − g(x)|dµ(x) . (20) we need definition 3.1. let fα (·) := f(· + α), α ∈ r, and φ be an operator. if φ(fα) = (φf)α, then φ is called a shift invariant operator. we give theorem 3.2. assume that ( t0 ( f ( 2−k · +α )))( 2ku ) = ( t0 ( f ( 2−k· )))( 2k (u + α) ) , (21) for all k ∈ z, α ∈ r fixed, all u ∈ r and any f ∈ cu (r,r+). then tk is a shift invariant operator for all k ∈ z. proof. we have that (tk (f(· + α))) (x) = (tk (fα)) (x) (18) = (n∗) ∫a −a ( t0 ( fα ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) = (n∗) ∫a −a ( t0 ( f ( 2−k · +α )))( 2kx − u ) ψ(u)dµ(u) = 6 george a. anastassiou cubo 20, 1 (2018) (n∗) ∫a −a ( t0 ( f ( 2−k · +α )))( 2k ( x − 2−ku )) ψ(u)dµ(u) (21) = (22) (n∗) ∫a −a ( t0 ( f ( 2−k· )))( 2k ( x − 2−ku + α )) ψ(u)dµ(u) = (n∗) ∫a −a ( t0 ( f ( 2−k· )))( 2k (x + α) − u ) ψ(u)dµ(u) (18) = (tk (f)) (x + α) , that is tk (fα) = (tk (f))α , (23) proving the claim. it follows the global smoothness of the operators tk. theorem 3.3. for any f ∈ cu (r,r+) assume that, for all u ∈ r, |(t0 (f)) (x − u) − (t0 (f)) (y − u)| ≤ ω1 (f, |x − y|) , (24) for any x,y ∈ r. then ω1 (tkf,δ) ≤ ω1 (f,δ) , ∀ δ > 0. (25) proof. we observe that |(t0 (f)) (x) − (t0 (f)) (y)| = ∣ ∣ ∣ ∣ (n∗) ∫a −a (t0f) (x − u)ψ(u)dµ(u) − (n ∗ ) ∫a −a (t0f) (y − u)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (26) (n∗) ∫a −a |(t0f) (x − u) − (t0f) (y − u)|ψ(u)dµ(u) (by (24), (5)) ≤ ω1 (f, |x − y|) ( (n∗) ∫a −a ψ(u)dµ(u) ) (15) = ω1 (f, |x − y|) . so that |(t0 (f)) (x) − (t0 (f)) (y)| ≤ ω1 (f, |x − y|) . (27) from (17), (27) we get |(tk (f)) (x) − (tk (f)) (y)| (17) = ∣ ∣ ( t0 ( f ( 2−k· )))( 2kx ) − ( t0 ( f ( 2−k· )))( 2ky )∣ ∣ ≤ (28) ω1 ( f ( 2−k· ) ,2k |x − y| ) = ω1 (f, |x − y|) , i.e. global smoothness for tk has been proved. the convergence of tk to the unit operator, as k → +∞, k with rates follows: cubo 20, 1 (2018) approximation by shift invariant univariate . . . 7 theorem 3.4. for f ∈ cu (r,r+), under the assumption (14), we have |(tk (f)) (x) − f(x)| ≤ ω1 ( f, ma + n 2k+r ) , (29) where m ∈ n, n ∈ z+, k,r ∈ z. proof. we notice that |(tk (f)) (x) − f(x)| (17) = ∣ ∣ ( t0 ( f ( 2−k· )))( 2kx ) − f(x) ∣ ∣ (18) = ∣ ∣ ∣ ∣ (n∗) ∫a −a ( t0 ( f ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) − f(x) ∣ ∣ ∣ ∣ (15) = ∣ ∣ ∣ ∣ (n∗) ∫a −a ( t0 ( f ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) − (n∗) ∫a −a f(x)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (n∗) ∫a −a ∣ ∣ ( t0 ( f ( 2−k· )))( 2kx − u ) − f(x) ∣ ∣ψ(u)dµ(u) = (30) (n∗) ∫a −a ∣ ∣ ( t0 ( f ( 2−k· )))( 2kx − u ) − f ( 2−k· )( 2kx )∣ ∣ψ(u)dµ(u) (14) ≤ (here ∣ ∣ ( 2kx − u ) − 2kx ∣ ∣ = |u| ≤ a) ω1 ( f ( 2−k· ) , ma + n 2r )( (n∗) ∫a −a ψ(u)dµ(u) ) (15) = ω1 ( f ( 2−k· ) , ma + n 2r ) · 1 = ω1 ( f, ma + n 2k+r ) , (31) proving the claim. we give some applications. for each k ∈ z, we define (i) (bkf) (x) := (n ∗) ∫a −a f ( x − u 2k ) ψ(u)dµ(u) , (32) i.e., here (tk (f)) (u) = f ( u 2k ) , and (t0 (f)) (u) = f(u) , (33) are continuous in u ∈ r. also for k ∈ z, we define (ii) (γk (f)) (x) := (n ∗ ) ∫a −a γfk ( 2kx − u ) ψ(u)dµ(u) , (34) 8 george a. anastassiou cubo 20, 1 (2018) where (tk (f)) (u) = γ f k (u) := n∑ j=0 wjf ( u 2k + j 2kn ) , (35) n ∈ n, wj ≥ 0, n∑ j=0 wj = 1, is continuous in u ∈ r. notice here that (t0 (f)) (u) = γ f 0 (u) = n∑ j=0 wjf ( u + j n ) (36) is also continuous in u ∈ r. indeed we have (γk (f)) (x) = (n ∗) ∫a −a   n∑ j=0 wjf ( ( x − u 2k ) + j 2kn )  ψ(u)dµ(u) . (37) clealry here we have (bk (f)) (x) = ( b0 ( f ( 2−k· )))( 2kx ) , and (γk (f)) (x) = ( γ0 ( f ( 2−k· )))( 2kx ) , (38) ∀ k ∈ z, ∀ x ∈ r. we give proposition 3.5. bk,γk are shift invariant operators. proof. (i) for bk operators: here t0f = f. hence ( t0 ( f ( 2−k · +α )))( 2ku ) = f ( 2−k2ku + α ) = f(u + α) = (39) ( t0 ( f ( 2−k· )))( 2k (u + α) ) . (ii) for γk operators: (t0 (f)) (u) = n∑ j=0 wjf ( u + j n ) . hence ( t0 ( f ( 2−k · +α )))( 2ku ) = n∑ j=0 wjf ( 2−k ( 2ku + j n ) + α ) = n∑ j=0 wjf ( 2−k ( 2k (u + α) + j n )) = ( t0 ( f ( 2−k· )))( 2k (u + α) ) , (40) proving the claim. cubo 20, 1 (2018) approximation by shift invariant univariate . . . 9 next we show that the operators bk,γk possess the property of global smoothness preservation. theorem 3.6. for all f ∈ cu (r,r+) and all δ > 0 we have ω1 (bkf,δ) ≤ ω1 (f,δ) , and ω1 (γkf,δ) ≤ ω1 (f,δ) . (41) proof. (i) for bk operators: here t0f = f, therefore |(t0 (f)) (x − u) − (t0 (f)) (y − u)| = |f(x − u) − f(y − u)| ≤ ω1 (f, |x − y|) . (42) (ii) for γk operators: we observe that |(t0 (f)) (x − u) − (t0 (f)) (y − u)| = ∣ ∣γf0 (x − u) − γ f 0 (y − u) ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wj ( f ( x − u + j n ) − f ( y − u + j n )) ∣ ∣ ∣ ∣ ∣ ∣ ≤ n∑ j=0 wj ∣ ∣ ∣ ∣ f ( x − u + j n ) − f ( y − u + j n )∣ ∣ ∣ ∣ ≤ ω1 (f, |x − y|)   n∑ j=0 wj   = ω1 (f, |x − y|) , (43) proving the claim. the operators bk,γk, k ∈ z, converge to the unit operator with rates presented next. theorem 3.7. for k ∈ z, |(bk (f)) (x) − f(x)| ≤ ω1 ( f, a 2k ) , and |(γk (f)) (x) − f(x)| ≤ ω1 ( f, a+1 2k ) . (44) proof. (i) for bk operators: here (t0 (f)) (u) = f(u) and sup u,y∈r |u−y|≤a |(t0 (f)) (u) − f(y)| = sup u,y∈r |u−y|≤a |f(u) − f(y)| = ω1 (f,a) , (45) and we use theorem 3.4. (ii) for γk operators: here we see that sup u,y∈r |u−y|≤a |(t0 (f)) (u) − f(y)| = sup u,y∈r |u−y|≤a ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wjf ( u + j n ) − f(y) ∣ ∣ ∣ ∣ ∣ ∣ ≤ 10 george a. anastassiou cubo 20, 1 (2018) sup u,y∈r |u−y|≤a n∑ j=0 wj ∣ ∣ ∣ ∣ f ( u + j n ) − f(y) ∣ ∣ ∣ ∣ ≤ sup u,y∈r |u−y|≤a n∑ j=0 wjω1 ( f, ∣ ∣ ∣ ∣ u + j n − y ∣ ∣ ∣ ∣ ) ≤ (46) sup u,y∈r |u−y|≤a n∑ j=0 wjω1 ( f, j n + |u − y| ) ≤   n∑ j=0 wj  ω1 (f,1 + α) = ω1 (f,α + 1) . by (29) we are done. 4 higher order of approximation here all are as in section 3. see also earlier our work [1], and [2], chapter 16. we give theorem 4.1. let f ∈ cn (r,r+), n ≥ 1. consider the shilkret-sublinear operators (bkf) (x) = (n ∗ ) ∫a −a f ( x − u 2k ) ψ(u)dµ(u) , ∀ k ∈ z, ∀ x ∈ r. then |(bkf) (x) − f(x)| ≤ n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + an 2knn! ω1 ( f(n), a 2k ) . (47) if f(n) is uniformly continuous or bounded and continuous, then as k → +∞ we obtain that (bkf) (x) → f(x) pointwise with rates. proof. since f ∈ cn (r,r+), n ≥ 1, by taylor’s formula we have f ( x − u 2k ) − f(x) = n∑ i=1 f(i) (x) i! ( − u 2k )i + (48) ∫x− u 2k x ( f(n) (t) − f(n) (x) ) ( x − u 2k − t )n−1 (n − 1) ! dt. call γu (x) := ∣ ∣ ∣ ∣ ∣ ∫x− u 2k x ( f(n) (t) − f(n) (x) ) ( x − u 2k − t )n−1 (n − 1) ! dt ∣ ∣ ∣ ∣ ∣ . (49) next we estimate γu (x), where u ∈ [−a,a] . i) case of −a ≤ u ≤ 0, then x ≤ x − u 2k . then γu (x) ≤ ∫x− u 2k x ∣ ∣ ∣ f(n) (t) − f(n) (x) ∣ ∣ ∣ ( x − u 2k − t )n−1 (n − 1) ! dt ≤ cubo 20, 1 (2018) approximation by shift invariant univariate . . . 11 ∫x− u 2k x ω1 ( f(n), |t − x| ) ( x − u 2k − t )n−1 (n − 1) ! dt ≤ ω1 ( f(n), |u| 2k )∫x− u 2k x ( x − u 2k − t )n−1 (n − 1) ! dt ≤ (50) ω1 ( f(n), a 2k ) ( − u 2k )n n! ≤ ω1 ( f(n), a 2k ) an 2knn! . that is, when −a ≤ u ≤ 0, then γu (x) ≤ ω1 ( f(n), a 2k ) an 2knn! . (51) ii) case of 0 ≤ u ≤ a, then x ≥ x − u 2k . then γu (x) = ∣ ∣ ∣ ∣ ∣ ∫x x− u 2k ( f(n) (t) − f(n) (x) ) ( t − x + u 2k )n−1 (n − 1) ! dt ∣ ∣ ∣ ∣ ∣ ≤ ∫x x− u 2k ∣ ∣ ∣ f(n) (t) − f(n) (x) ∣ ∣ ∣ ( t − x + u 2k )n−1 (n − 1) ! dt ≤ (52) ∫x x− u 2k ω1 ( f(n), |t − x| ) ( t − x + u 2k )n−1 (n − 1) ! dt ≤ ω1 ( f(n), |u| 2k )∫x x− u 2k ( t − x + u 2k )n−1 (n − 1) ! dt ≤ ω1 ( f(n), a 2k ) ( u 2k )n n! ≤ ω1 ( f(n), a 2k ) an 2knn! . (53) that is, when 0 ≤ u ≤ a, then γu (x) ≤ ω1 ( f(n), a 2k ) an 2knn! . (54) we proved that γu (x) ≤ ω1 ( f(n), a 2k ) an 2knn! := ρ ≥ 0, (55) ∀ k ∈ z, ∀ x ∈ r, |u| ≤ a. by (48) we get that (|u| ≤ a) ∣ ∣ ∣ f ( x − u 2k ) − f(x) ∣ ∣ ∣ ≤ n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + ρ. (56) we observe that |(bkf) (x) − f(x)| = 12 george a. anastassiou cubo 20, 1 (2018) ∣ ∣ ∣ ∣ (n∗) ∫a −a f ( x − u 2k ) ψ(u)dµ(u) − (n∗) ∫a −a f(x)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (57) (n∗) ∫a −a ∣ ∣ ∣ f ( x − u 2k ) − f(x) ∣ ∣ ∣ ψ(u)dµ(u) ≤ ( n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + ρ ) ( (n∗) ∫a −a ψ(u)dµ(u) ) (15) = ( n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + ρ ) · 1 = (58) n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + an 2knn! ω1 ( f(n), a 2k ) , proving the claim. corollary 4.2. let f ∈ c1 (r,r+). then |(bkf) (x) − f(x)| ≤ a 2k ( |f′ (x)| + ω1 ( f′, a 2k )) , (59) ∀ k ∈ z, ∀ x ∈ r. proof. by (47) for n = 1. we also present theorem 4.3. let f ∈ cn (r,r+), n ≥ 1. consider the shilkret-sublinear operators (γk (f)) (x) = (n ∗) ∫a −a   n∑ j=0 wjf ( ( x − u 2k ) + j 2kn )  ψ(u)dµ(u) , (60) ∀ k ∈ z, ∀ x ∈ r. then |(γkf) (x) − f(x)| ≤ n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + (a + 1) n n!2kn ω1 ( f(n), a + 1 2k ) . (61) if f(n) is uniformly continuous or bounded and continuous, then as k → +∞ we obtain that (γkf) (x) → f(x) , pointwise with rates. corollary 4.4. let f ∈ c1 (r,r+). then |(γkf) (x) − f(x)| ≤ (a + 1) 2k [ |f′ (x)| + ω1 ( f′, a + 1 2k )] , (62) ∀ k ∈ z, ∀ x ∈ r. proof. by (61) for n = 1. cubo 20, 1 (2018) approximation by shift invariant univariate . . . 13 proof. of theorem 4.3. since f ∈ cn (r), n ≥ 1, by taylor’s formula we get n∑ j=0 wjf ( ( x − u 2k ) + j 2kn ) − f(x) = n∑ i=1 f(i) (x) i! n∑ j=0 wj ( − u 2k + j 2kn )i + (63) n∑ j=0 wj ∫(x− u 2k )+ j 2kn x ( f(n) (t) − f(n) (x) ) ( ( x − u 2k ) + j 2kn − t )n−1 (n − 1) ! dt. call ε(x,u,j) := ∫(x− u 2k )+ j 2kn x ( f(n) (t) − f(n) (x) ) ( ( x − u 2k ) + j 2kn − t )n−1 (n − 1) ! dt. (64) we estimate ε(x,u,j). here |u| ≤ a. i) case of u ≤ j n , iff u 2k ≤ j 2kn , iff x ≤ x − u 2k + j 2kn . hence |ε(x,u,j)| ≤ ∫(x− u 2k )+ j 2kn x ∣ ∣ ∣ f(n) (t) − f(n) (x) ∣ ∣ ∣ ( ( x − u 2k ) + j 2kn − t )n−1 (n − 1) ! dt ≤ (65) ∫(x− u 2k )+ j 2kn x ω1 ( f(n), |t − x| ) ( ( x − u 2k ) + j 2kn − t )n−1 (n − 1) ! dt ≤ ω1 ( f(n), [ j 2kn − u 2k ])∫(x− u 2k )+ j 2kn x ( ( x − u 2k ) + j 2kn − t )n−1 (n − 1) ! dt ≤ ω1 ( f(n), a + 1 2k ) ( j 2kn − u 2k )n n! ≤ ω1 ( f(n), a + 1 2k ) (a + 1) n 2knn! . (66) for u ≤ j n , we hve proved that |ε(x,u,j)| ≤ ω1 ( f(n), a + 1 2k ) (a + 1) n 2knn! . (67) ii) case of u ≥ j n , iff u 2k ≥ j 2kn , iff x ≥ x − u 2k + j 2kn . we observe that |ε(x,u,j)| = 14 george a. anastassiou cubo 20, 1 (2018) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫x (x− u 2k )+ j 2kn ( f(n) (t) − f(n) (x) ) ( t − [ ( x − u 2k ) + j 2kn ])n−1 (n − 1) ! dt ∣ ∣ ∣ ∣ ∣ ∣ ∣ ≤ (68) ∫x (x− u 2k )+ j 2kn ∣ ∣ ∣ f(n) (t) − f(n) (x) ∣ ∣ ∣ ( t − [ ( x − u 2k ) + j 2kn ])n−1 (n − 1) ! dt ≤ ∫x (x− u 2k )+ j 2kn ω1 ( f(n), |t − x| ) ( t − [ ( x − u 2k ) + j 2kn ])n−1 (n − 1) ! dt ≤ ω1 ( f(n), u 2k − j 2kn )∫x (x− u 2k )+ j 2kn ( t − [ ( x − u 2k ) + j 2kn ])n−1 (n − 1) ! dt ≤ ω1 ( f(n), a + 1 2k ) ( u 2k − j 2kn )n n! ≤ ω1 ( f(n), a + 1 2k ) (a + 1) n 2knn! . (69) so when u ≥ j n , we proved that |ε(x,u,j)| ≤ ω1 ( f(n), a + 1 2k ) (a + 1) n 2knn! . (70) therefore it always holds |ε(x,u,j)| ≤ ω1 ( f(n), a + 1 2k ) (a + 1) n 2knn! . (71) consequently we derive n∑ j=0 wj |ε(x,u,j)| ≤ ω1 ( f(n), a + 1 2k ) (a + 1) n 2knn! := ψ. (72) by (63) we find ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wjf ( ( x − u 2k ) + j 2kn ) − f(x) ∣ ∣ ∣ ∣ ∣ ∣ ≤ n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + ψ. (73) therefore we get |(γk (f)) (x) − f(x)| = ∣ ∣ ∣ ∣ ∣ ∣ (n∗) ∫a −a   n∑ j=0 wjf ( ( x − u 2k ) + j 2kn )  ψ(u)dµ(u) − (n∗) ∫a −a f(x)ψ(u)dµ(u) ∣ ∣ ∣ ∣ ∣ ∣ (20) ≤ (74) (n∗) ∫a −a ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wjf ( ( x − u 2k ) + j 2kn ) − f(x) ∣ ∣ ∣ ∣ ∣ ∣ ψ(u)dµ(u) (73) ≤ cubo 20, 1 (2018) approximation by shift invariant univariate . . . 15 [ n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + ψ ] (n∗) ∫a −a ψ(u)dµ(u) (15) = [ n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + ψ ] · 1 = n∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + (a + 1) n 2knn! ω1 ( f(n), a + 1 2k ) , (75) proving the claim. we finish with corollary 4.5. let f ∈ cn (r,r+), n ≥ 1, f (i) (x) = 0, i = 1, ...,n. then i) |(bk (f)) (x) − f(x)| ≤ an 2knn! ω1 ( f(n), a 2k ) , (76) and ii) |(γk (f)) (x) − f(x)| ≤ (a + 1) n n!2kn ω1 ( f(n), a + 1 2k ) , (77) ∀ k ∈ z, ∀ x ∈ r. proof. by (47) and (61). corollary 4.6. let f ∈ c1 (r,r+), f ′ (x) = 0. then i) |(bk (f)) (x) − f(x)| ≤ a 2k ω1 ( f′, a 2k ) , (78) and ii) |(γk (f)) (x) − f(x)| ≤ ( a + 1 2k ) ω1 ( f′, a + 1 2k ) , (79) ∀ k ∈ z, ∀ x ∈ r. proof. by (59) and (62). in inequalities (76)-(79) observe the high speed of convergence and approximation. 16 george a. anastassiou cubo 20, 1 (2018) 5 appendix let f ∈ cu (r,r+), and the positive sublinear shilkret operator (m(f)) (x) := (n∗) ∫a −a f(x + u)ψ(u)dµ(u) , ∀ x ∈ r. (80) we observe the following (for any x,y ∈ r): |(m(f)) (x) − (m(f)) (y)| = ∣ ∣ ∣ ∣ (n∗) ∫a −a f(x + u)ψ(u)dµ(u) − (n∗) ∫a −a f(y + u)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (n∗) ∫a −a |f(x + u) − f(y + u)|ψ(u)dµ(u) ≤ ω1 (f, |x − y|) ( (n∗) ∫a −a ψ(u)dµ(u) ) (15) = ω1 (f, |x − y|) · 1 = ω1 (f, |x − y|) . (81) therefore it holds the global smoothness preservation property: ω1 (m(f) ,δ) ≤ ω1 (f,δ) , ∀ δ > 0. (82) references [1] g.a. anastassiou, high order approximation by univariate shift-invariant integral operators, in: r. agarwal, d. o’regan (eds.), nonlinear analysis and applications, 2 volumes, vol. i, pp. 141-164, kluwer, dordrecht, (2003). [2] g.a. anastassiou, intelligent mathematics: computational analysis, springer, heidelberg, new york, 2011. [3] g.a. anastassiou, s. gal, approximation theory, birkhauser, boston, basel, berlin, 2000. [4] g.a. anastassiou, h.h. gonska, on some shift invariant integral operators, univariate case, ann. polon. math., lxi, 3, (1995), 225-243. [5] niel shilkret, maxitive measure and integration, indagationes mathematicae, 33 (1971), 109116. introduction background univariate theory higher order of approximation appendix cubo a mathematical journal vol.19, no¯ 02, (49–71). june 2017 on topological symplectic dynamical systems s. tchuiaga1, m. koivogui2, f. balibuno3 and v. mbazumutima3 1department of mathematics the university of buea, south west region, cameroon. tchuiagas@gmail.com 2ecole supérieure africaine des technologies de l’information et de communication, côte d’ ivoire. moussa.koivogui@esatic.ci 3institut de mathématiques et des sciences physiques bénin. balibuno.lugando@imsp-uac.org, mbazumutima.vianney@aims-cameroon.org abstract this paper contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. using the positivity result of symplectic displacement energy [4], we prove that any generator of a strong symplectic isotopy uniquely determine the latter. this yields a symplectic analogue of a result proved by oh [12], and the converse of the main theorem found in [6]. also, tools for defining and for studying the topological symplectic dynamical systems are provided: we construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all hamiltonian homeomorphism is equivalent to oh’s metric [12], define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. several open conjectures are elaborated. 50 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) resumen este art́ıculo contribuye al estudio de los sistemas dinámicos simplécticos topológicos, y por tanto a la extensión de los sistemas dinámicos simplécticos suaves. usando el resultado de la positividad de la enerǵıa de desplazamiento simpléctico [4], demostramos que cualquier generador de una isotoṕıa simpléctica determina esta última. esto entrega un análogo simpléctico de un resultado demostrado por oh [12], y el inverso del teorema principal encontrado en [6]. también entregamos herramientas para definir y estudiar los sistemas dinámicos simplécticos topológicos: construimos una métrica invariante por derecha en el grupo de homeomorfismos fuertemente simplécticos cuya restricción al grupo de homeomorfismos hamiltonianos es equivalente a la métrica de oh [12], definimos los análogos topológicos de la enerǵıa de desplazamiento simpléctico usual para conjuntos no-vaćıos, y demostramos que esta última es positiva. planteamos varios problemas abiertos. keywords and phrases: isotopies, diffeomorphisms, homeomorphisms, displacement energy, hofer-like norms, mass flow, riemannian metric, lefschetz type manifolds, flux geometry. 2010 ams mathematics subject classification: 53d05, 53d35, 57r52, 53c21. cubo 19, 2 (2017) on topological symplectic dynamical systems 51 1 introduction gromov [10] showed that on a symplectic manifold, the c0−closure of the group of symplectic diffeomorphisms in the group of diffeomorphisms is either the group of symplectic diffeomorphisms itself, or the group of volume preserving diffeomorphisms. eliashberg [9] proved that the symplectic nature of a sequence of symplectic diffeomorphisms survives topological limits. this result is known as the ”celebrated rigidity” result of eliashberg, which motivated various remarkable studies of continuum phenomena in the field of symplectic geometry. especially, based on this rigidity result, oh-müller [13] defined the group of symplectic homeomorphisms as the c0−closure of the group of symplectomorphisms in the group of homeomorphisms. they also defined both versions of c0−hamiltonian topologies on the space of hamiltonians paths and used them to define the group of hamiltonian homeomorphisms. this group is at the center of the study of topological hamiltonian dynamical systems (see viterbo [19], buhovsky and seyfaddini [8, 12]). more recently, motivated again by the celebrated rigidity result of eliashberg, banyaga [2, 3] defined two contexts of symplectic topologies on the space of symplectic isotopies that generalize the c0−hamiltonian topologies. these topology had been used to define a new class of symplectic homeomorphisms named the ”group strong symplectic homeomorphisms” (see banyaga [3]), which had been studied in banyaga-tchuiaga [6, 5, 14]. this group could be the right topological analogue of the identity component in the group of symplectomorphisms. however, for that to be possible, we have to define, and then study what will be the equivalents (or analogues) of some well known smooth symplectic objects in the world (or context) of strong symplectic homeomorphisms. therefore, one purpose of this paper is to point out further studies of generators for strong symplectic isotopies [6], and then use them to construct a framework in which the flux homomorphism and the hofer-like geometry can be extended to some category of continuous maps (see [18]). we organize the present paper as follows: in sections 2 and section 3, we recall some fundamental tools needed in the definition of strong symplectic homeomorphisms: the description of symplectic isotopies that was introduced in [5], and the displacement energy. section 4 deals with the definitions of the c0−compact open topology, the origin of strong symplectic homeomorphisms, and the definition of strong symplectic isotopies with their generators. these tools are used to show a bijective correspondence between the group of strong symplectic isotopies and the group of their generators (theorem 4.4). the hamiltonian version of this result is well known. this section also includes lemma 4.6 which shows that any strong symplectic isotopy which is a 1−parameter group decomposes as composition of a smooth harmonic flow and a continuous hamiltonian flow in the sense of oh-müller. in section 5, we use the results of section 4 to introduce a topological version of hofer-like geometry: we construct a topological counterpart of the hofer-like metric for strong symplectic homeomorphisms, and we prove that its restriction to the group of hamiltonian homeomorphisms is equivalent to oh’s metric [12]. therefore, the definition of a topological analogue of the usual symplectic displacement for non-empty sets is given, and we prove that it is positive. finally, in 52 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) section 6, we elaborate some conjectures and some examples are given. 2 preliminaries let m be an 2n−dimensional manifold of class c∞. a differential 2−form ω on m is called a symplectic form if it is closed and non-degenerate. the nondegeneracy of ω, implies that ωn is a volume form on m. a symplectic manifold is an even dimensional smooth manifold m that admits a symplectic form ω. from now on, we assume that m is an 2n−dimensional closed symplectic manifold with a symplectic form ω. we also equip m with a riemannian metric g. note that for technical reasons, we will sometimes assume that the symplectic manifold (m,ω) is of lefschetz type. that is, the mapping ωn−1 : h1(m, r) → h2n−1(m, r),α !→ α ∧ ωn−1, is an isomorphism. the category of lefschetz manifolds includes all kähler manifolds, such has oriented surfaces and even dimensional tori. 2.1 harmonics 1−forms let h1(m, r) denote the first de rham cohomology group (with real coefficients) of m, and let h1(m, g) denote the space of harmonic 1−forms on m with respect to the riemannian metric g. the set h1(m, g) forms a finite dimensional vector space over r which is isomorphic to h1(m, r), and whose dimension is denoted b1(m), and called the first betti number of the manifold m [20]. taking (hi)1≤i≤b1(m) as a basis of the vector space h 1(m, g), we equip h1(m, g) with the euclidean norm |.| defined as follows: for all h ∈ h1(m, g) with h = b1(m)∑ i=1 λihi, its norm is defined as |h| := b1(m)∑ i=1 |λi|. (2.1) we denote by ph1(m, g) the space of all smooth mappings h : [0, 1] → h1(m, g). 3 on the classical symplectic dynamical systems 3.1 symplectic diffeomorphisms and symplectic isotopies a diffeomorphism φ : m → m, is called symplectic if it preserves the symplectic form ω, i.e. φ∗(ω) = ω. we denote by symp(m,ω), the group of all symplectic diffeomorphisms of (m,ω). cubo 19, 2 (2017) on topological symplectic dynamical systems 53 an isotopy {φt} of a symplectic manifold (m,ω) is said to be symplectic if φt ∈ symp(m,ω) for each t, or equivalently, the vector field φ̇t := dφt dt ◦ φ−1 t is symplectic for each t. in particular, a symplectic isotopy {ψt} is a hamiltonian isotopy if for each t, the vector field ψ̇t := dψt dt ◦ ψ−1 t is hamiltonian, i.e. there exists a smooth function f : [0, 1] × m → r, called generating hamiltonian such that ι(ψ̇t)ω = dft, for each t. any hamiltonian isotopy determines its generating hamiltonian up to an additive constant. throughout this paper we assume that every generating hamiltonian f : [0, 1] × m → r is normalized, i.e. we require that ∫ m ftω n = 0 for all t. let n([0, 1] × m , r) denote the vector space of all smooth normalized hamiltonians. we denote by iso(m,ω) the group of all symplectic isotopies of (m,ω), and by symp0(m,ω), the group of time−1 maps of all symplectic isotopies. 3.2 description of the classical symplectic isotopies we now recall the description of symplectic isotopies introduced in [5]. given any symplectic isotopy φ = {φt}, one derives from hodge’s theory that the closed 1−form ι(φ̇t)ω decomposes in a unique way as the sum of an exact 1−form duφt and a harmonic 1−form h φ t [20]. denote by u the normalized hamiltonian of uφ = (uφ t ), and by h the smooth family of harmonic 1−forms hφ = (hφ t ). in [5], the cartesian product n([0, 1] × m, r) × ph1(m, g) is denoted t(m,ω, g), and equipped with a group structure which makes the bijection a : iso(m,ω) → t(m,ω, g),φ !→ (u, h) (3.1) a group isomorphism. under this identification, any symplectic isotopy φ is denoted by φ(u,h) to mean that a maps φ onto (u, h), and the pair (u, h) is called the “generator” of the symplectic isotopy φ. for instance, a symplectic isotopy φ(0,h), is a harmonic isotopy, and a symplectic isotopy φ(u,0), is a hamiltonian isotopy. 3.3 group structure on t(m,ω, g) the product rule in t(m,ω, g) is given by, (u, h) ✶ (v, k) = (u + v ◦ φ−1 (u,h) + !∆(k,φ−1 (u,h) ), h + k). (3.2) the inverse of (u, h), say (u, h) is given by (u, h) = (−u ◦ φ(u,h) − !∆(h,φ(u,h)), −h). (3.3) in (3.2) and (3.3) the quantity !∆ is defined as follows: for any symplectic isotopy ψ = {ψt}, and for any smooth family of closed 1−forms α = (αt), we have !∆t(α,ψ) = ∆t(α,ψ) − ∫ m ∆t(α,ψ)ω n ∫ m ωn , 54 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) where ∆t(α,ψ) := ∫t 0 αt(ψ̇ s) ◦ ψsds, for all t (see [5]). also, it is proved in [16, 18] that ∆(α,ψ) is a 1−cocycle. 3.4 metric structures on t(m,ω, g) for all (u, h), (v, k) ∈ t(m,ω, g), set d (1,∞) 0 ((u, h), (v, k)) = ∫ 1 0 [|ht − kt| + osc(ut − vt)] dt, (3.4) and d∞ 0 ((u, h), (v, k)) = max t [|ht − kt| + osc(ut − vt)] , (3.5) where osc(f) = max x f(x) − min x f(x), for all f ∈ c∞(m, r). therefore, the l∞−hofer-like metric and the l(1,∞)−hofer-like on t(m,ω, g) are defined respectively as follows: d(1,∞)((u, h), (v, k)) = d (1,∞) 0 ((u, h), (v, k)) + d(1,∞) 0 ((u, h), (v, k)) 2 , (3.6) and d∞((u, h), (v, k)) = d∞ 0 ((u, h), (v, k)) + d∞ 0 ((u, h), (v, k)) 2 . (3.7) (see [2, 5]). 3.5 displacement energy definition 3.1. ([4]) the symplectic displacement energy es(a) of a non empty set a ⊂ m is: es(a) = inf{∥φ∥hl|φ ∈ symp(m,ω)0,φ(a) ∩ a = ∅}. theorem 3.2. ([4]) for any non empty open set a ⊂ m, es(a) is a strict positive number. note that in the definition of the displacement energy, the quantity ∥.∥hl stands for the usual hofer-like norm defined in [2]. 4 on topological symplectic dynamical systems 4.1 the c0−topology let homeo(m) be the group of all homeomorphisms of m equipped with the c0− compact-open topology. this is the metric topology induced by the following distance d0(f, h) = max(dc0(f, h), dc0(f −1, h−1)), cubo 19, 2 (2017) on topological symplectic dynamical systems 55 where dc0(f, h) = supx∈m d(h(x), f(x)). on the space of all continuous paths λ : [0, 1] → homeo(m) such that λ(0) = idm, we consider the c 0−topology as the metric topology induced by the metric d̄(λ, µ) = max t∈[0,1] d0(λ(t), µ(t)). 4.2 the origin of strong symplectic isotopies a result found in [15] (corollary 3.7-[15], or more generally a result found in [16]) states that: let φi = {φ t i } be a sequence of symplectic isotopies, ψ = {ψt} be another symplectic isotopy, and η : t !→ ηt be a family of maps ηt : m → m, such that the sequence φi converges uniformly to η and l∞(ψ−1 ◦ φi) → 0, i → ∞, then η = ψ. note that if, ψ generated by (u, h) and φi generated by (ui, hi), then replacing the condition l∞(ψ−1 ◦ φi) → 0, i → ∞, by the condition d∞((u, h), (ui, hi)) → 0, i → ∞, does not break the result of corollary 3.7-[15, 16]. therefore, we can then ask the following question: if in corollary 3.7-[15, 16], the convergence d∞((u, h), (ui, hi)) → 0, i → ∞, is replaced by the condition d∞((ui+1, hi+1), (ui, hi)) → 0, i → ∞, then what can we say about the geometries and the structures of the space of all such paths η? the seek of a possible answer to the above question motivated the following definition: definition 4.1. ([6]) a continuous map ξ : [0, 1] → homeo(m) with ξ(0) = idm, is called strong symplectic isotopy if there exists a d∞−cauchy sequence {(fi,λi)} ⊂ t(m,ω, g) such that d̄(φ(fi,λi),ξ) → 0, i → ∞. we denote by pssympeo(m,ω) the space of all strong symplectic isotopies. it is proved in [6, 14] that pssympeo(m,ω) is a group. if the manifold is simply connected, then the group pssympeo(m,ω) reduces to the group of continuous hamiltonian flows [12]. the set of time−1 maps of all strong symplectic isotopies coincides with the group of all strong symplectic homeomorphisms, denoted here by ssympeo(m,ω) (see [5, 3]). 56 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) 4.3 generators of ssympeotopies let n 0([0, 1]×m , r) denotes the completion of the metric space n([0, 1]×m , r) with respect to the l∞−hofer norm, and let ph1(m, g)0 denotes the completion of the metric space ph1(m, g) with respect to the uniform sup norm. consider the map j0(m,ω, g) := n 0([0, 1] × m , r) × ph1(m, g)0, and the inclusion map i0 : t(m,ω, g) → j 0(m,ω, g). this map is uniformly continuous with respect of the topology induced by the metric d∞ on the space t(m,ω, g), and the natural topology of the complete metric space j0(m,ω, g). now, let l(m,ω, g) denotes the image of t(m,ω, g) under i0, and t(m,ω, g)0 be the closure of l(m,ω, g) inside the complete metric space j0(m,ω, g). that is, t(m,ω, g)0 consists of pairs (u, h) where the mappings (t, x) !→ ut(x) and t !→ ht are continuous, and for each t, ht lies in h1(m, g) such that there exists a d∞−cauchy sequence (ui, hi) ⊂ t(m,ω, g) that converges to (u, h) ∈ j0(m,ω, g). note that the sequence (fj,λj) in definition (4.1) converges necessarily in the complete metric space t(m,ω, g)0. the latter limit is called the ”generator” of strong symplectic isotopy (see [6]). we will often write (fi,λi) l ∞ −−→ (f,λ) to mean that the sequence (fi,λi) converges to (f,λ) in the space j0(m,ω, g). definition 4.2. ([6]) the set gssympeo(m,ω, g) is defined as the space of all the pairs (ξ, (u, h)) where ξ is a strong symplectic isotopy generated (u, h). group structure on the space gssympeo(m,ω, g) for all (ξ, (f,λ)), (µ, (v,θ)) ∈ gssympeo(m,ω, g), their product is given by, (ξ, (f,λ)) ∗ (µ, (v,θ)) = (ξ ◦ µ, (f + v ◦ ξ−1 + ∆0(θ,ξ−1),λ + θ)), and the inverse of the element (ξ, (f,λ)) is given by, (ξ, (f,λ)) = (ξ−1, (−f ◦ ξ − ∆0(λ,ξ), −λ)), with ∆0(θ,ξ−1) := lim l∞ (!∆(θi,φ−1 (fi,λi) ), (4.1) ∆0(λ,ξ) := lim l∞ (!∆(λi,φ(fi,λi)), (4.2) where (fi,λi), and (vi,θi) are two arbitrary sequences in t(m,ω, g) such that (fi,λi) l ∞ −−→ (f,λ), φ(fi,λi) d̄ −→ ξ, and (vi,θi) l ∞ −−→ (v,θ), φ(vi,θi) d̄ −→ µ, cubo 19, 2 (2017) on topological symplectic dynamical systems 57 with !∆t(λi,φ−1(fi,λi)) the normalized function of ∆t(λ i,φ−1 (fi,λi) ). this set is known as a topological group with respect to the symplectic topology [6]: the symplectic topology on the space gssympeo(m,ω, g) is defined to be the subspace topology induced by the inclusion of the latter in the complete topological space p(homeo(m), id) × t(m,ω, g)0. question (a) let (mi,ωi) be two closed symplectic manifolds equipped with two riemannian metrics gi, for i = 1, 2. if the group t(m1,ω1, g1)0 is isomorphic to the group t(m2,ω2, g2)0, then what can we say about: the manifolds m1 and m2? the symplectic structures ω1 and ω2? the riemannian structures g1 and g2? the following uniqueness results show that there is a bijective correspondence between the group of strong symplectic isotopies and that of their generators. theorem 4.3. ([6]) let (m,ω) be a lefschetz closed symplectic manifold. any strong symplectic isotopy determines a unique generator. in the presence of a positive symplectic displacement energy from banyaga-hurtubise-spaeth [4], we point out the following converse of theorem 4.3, which in the same time gives the symplectic analogue of a result prove by oh [12]. theorem 4.4. any generator corresponds to a unique strong symplectic isotopy, i.e. if (γ, (u, h)), (ξ, (u, h)) ∈ gssympeo(m,ω, g), then we must have γ = ξ. proof. let (γ, (u, h)) and (ξ, (u, h)) be two elements of gssympeo(m,ω, g). by definition of the group gssympeo(m,ω, g), there exist two sequences of symplectic isotopies φ(ui,hi) and φ(vi,ki) such that: φ(ui,hi) d̄ −→ ξ, (ui, hi) l ∞ −−→ (u, h), and φ(vi,ki) d̄ −→ γ, (vi, ki) l ∞ −−→ (u, h). assume that γ ≠ ξ, i.e. there exists s0 ∈]0, 1] such that γ(s0) ≠ ξ(s0). since the map γ −1(s0)◦ξ(s0) belongs to homeo(m), then we derive from the identity γ−1(s0) ◦ ξ(s0) ≠ id, that there exists of a closed ball b which is entirely moved by γ−1(s0) ◦ ξ(s0). from the compactness of b, and the uniform convergence of the sequence φ−1 (ui,hi) ◦ φ(vi,ki) to γ −1 ◦ ξ, we derive that (φ−s0 (ui,hi) ◦ φs0 (vi,ki) )(b) ∩ (b) = ∅, (4.3) 58 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) for all sufficiently large i. relation (4.3) implies that, 00, there exists a strong symplectic isotopy η with the same extremities that β ∗r γ, such that l∞(η)0 (we refer to [4] for the definition of symplectic displacement energy). for such a ball b, we set δ = es(b). the characterization of the infimum tells us that one can find a strong symplectic isotopy γδ (u,h) with γδ (u,h) (1) = h, such that ∥h∥shl>l∞(γ δ (u,h)) − δ 4 . on the other hand, it follows from the definition of strong symplectic isotopies that there exists a sequence (φ(fi,λi)) that converges to γ(u,h) with respect to the (c 0 + l∞)−topology. so, we can find a larger integer i0 for which the path φ(fi 0 ,λi 0 ) is sufficiently close to γ(u,h) [resp. φ −1 (fi 0 ,λi 0 ) sufficiently close to γ−1 (u,h) ] with respect to the (c0 + l∞)−topology, and so that l∞(γ δ (u,h))>l∞(φ(fi 0 ,λi 0 )) − δ 4 , where φi0 = φ 1 (fi 0 ,λi 0 ) displaces b. it follows from the definition of banyaga’s hofer-like norm ∥, ∥hl (see [2]) that l∞(φ(fi 0 ,λi 0 )) ≥ ∥φi0∥hl, i.e. l∞(φ(fi 0 ,λi 0 )) − 2δ 4 ≥ ∥φi0∥hl − δ 2 . then, we derive from the definition of symplectic displacement energy es (see [4]) that ∥φi0∥hl − δ 2 ≥ es(b) − δ 2 = δ − δ 2 = δ 2 >0. summarizing the above statements together gives, ∥h∥shl>l∞(γ δ (u,h)) − δ 4 >l∞(φ(fi 0 ,λi 0 )) − 2δ 4 ≥ ∥φi0∥hl − δ 2 = δ 2 >0. for (3), let h and f be two strong symplectic homeomorphisms, pick γ ∈ "(h) and β ∈ "(f), and derive from remark 5.1 that for all ϵ>0, there exists a strong symplectic isotopy η with the same extremities as the right concatenation γ ∗r β such that ē(h ◦ f) ≤ l∞(η)0, such that ∥φ∥shl ≤ ∥φ∥oh ≤ κ∥φ∥shl, for all hamiltonian homeomorphism φ. this result is motivated in party by a conjecture which can be found in [2]. the latter conjecture first was proved by buss-leclercq [7], and an alternate proof of the same conjecture is given 66 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) in [17]. this conjecture stated that the restriction of banyaga’s hofer-like norm to the group of hamiltonian diffeomorphisms is equivalent to hofer’s norm. proof of theorem 5.5. by construction, we always have ∥.∥shl ≤ ∥.∥oh. to complete the proof, we need to show that there exists a positive finite constant κ such that ∥.∥oh ≤ κ∥.∥shl, (5.15) or equivalently, via the sequential criterion, it suffices to prove that any sequence of hamiltonian homeomorphisms converging to the constant map identity with respect to the norm ∥.∥shl, converges to the constant map identity with respect to oh’s norm. the proof that we give here heavily relies the ideas that oh [12] and banyaga [2] used in the proof of the nondegeneracy of their norms. let ψi be a sequence of hamiltonian homeomorphisms that converges to the identity with respect to the norm ∥.∥shl. for each i, and any ϵ>0 there exists a strong symplectic isotopy ψ(ui,ϵ,hi,ϵ) such that ψ1 (ui,ϵ,hi,ϵ) = ψi, and l∞(ψ(ui,ϵ,hi,ϵ))<∥ψ i ∥shl + ϵ. (5.16) on the other hand, for a fixed i, there exists a sequence of symplectic isotopies φ(vi,j,ki,j) such that ψ(ui,ϵ,hi,ϵ) = lim c0+l∞ (φ(vi,j,ki,j)). (5.17) in particular, one can find a sufficiently large integer j0 such that φ(vi,j 0 ,ki,j 0 ) is sufficiently close to ψ(ui,ϵ,hi,ϵ) with respect to the (c 0 + l∞)−topology, and so that l∞(ψ(ui,ϵ,hi,ϵ))>l∞(φ(vi,j 0 ,ki,j 0 )) − ϵ 4 . (5.18) since ∥ψi∥shl → 0, i → ∞, then the hofer-like length of the isotopy φ(vi,j 0 ,ki,j 0 ) can be considered as being sufficiently small for i sufficiently large. this implies that the flux of the path φ(vi,j 0 ,ki,j 0 ) can be considered as arbitrarily small for all i sufficiently large. hence, it follows from banyaga [2] that, for all i sufficiently large, the time−1 map of φ(vi,j 0 ,ki,j 0 ) is a hamiltonian diffeomorphism. so, we can assume (without breaking the generality) that φ1 (vi,j 0 ,ki,j 0 ) is hamiltonian for i ≤ j0, and i sufficiently large. therefore, the above statements together with formula (5.18) imply that l∞(ψ(ui,ϵ,hi,ϵ))>∥φ 1 (vi,j 0 ,ki,j 0 )∥hl − ϵ 4 . (5.19) for i ≤ j0, and i sufficiently large, since the diffeomorphism φ 1 (vi,j 0 ,ki,j 0 ) is hamiltonian, we derive from a result found in [7, 16] that there exists a positive finite constant d which does not depend on neither i, nor j0, such that 1 d ∥φ1(vi,j 0 ,ki,j 0 )∥h ≤ ∥φ 1 (vi,j 0 ,ki,j 0 )∥shl, (5.20) cubo 19, 2 (2017) on topological symplectic dynamical systems 67 where ∥.∥h represents the hofer norm of hamiltonian diffeomorphisms. this implies that ∥ψi∥shl + ϵ> 1 d ∥φ1(vi,j 0 ,ki,j 0 )∥h − ϵ 4 , (5.21) for i ≤ j0, and i sufficiently large. at this level, we use the fact that oh’s norm restricted to the group of hamiltonian diffeomorphisms is bounded from above by hofer’s norm to get ∥ψi∥shl + ϵ> 1 d ∥φ1(vi,j 0 ,ki,j 0 )∥oh − ϵ 4 , (5.22) for i ≤ j0, and i sufficiently large. passing to the limit in the latter estimate, yields lim i→∞ ∥ψi∥shl + 5ϵ 4 ≥ 1 d lim j0≥i,i→∞ ∥φ1(vi,j 0 ,ki,j 0 )∥oh = 1 d lim i→∞ ∥ψi∥oh, (5.23) for all ϵ. finally, we have proved that for all positive real number δ (replacing ϵ by 4δ 5d ), we have δ ≥ lim i→∞ ∥ψi∥oh. (5.24) this completes the proof.! 5.3 topological symplectic displacement energy in this section, we extend the symplectic displacement energy to the world of strong symplectic homeomorphisms. this is motivated by the uniqueness result from [5] and the uniqueness of banyaga’s hofer-like geometry [15]. definition 5.6. the strong symplectic displacement energy e0,∞ s (b) of a non empty compact subset b ⊂ m is : e0,∞s (b) = inf{∥h∥shl|h ∈ ssympeo(m,ω), h(b) ∩ b = ∅}. lemma 5.7. for any non empty compact subset b ⊂ m, e0,∞ s (b) is a strict positive number. proof. let ϵ>0, by definition of e0,∞ s (b), there exists a strong symplectic isotopy ψ(fϵ,λϵ) such that ψ1 (fϵ,λϵ) = h, and e0,∞s (b) + ϵ 2 >l∞(ψ(fϵ,λϵ)). (5.25) on the other hand, there exists a sequence of symplectic isotopies (φ(fi,λi)) that converges to ψ(fϵ,λϵ) with respect to the (c 0 + l∞)−topology. so, one can choose integer j0 large enough such that φ1 (fi,λi) displaces b, and l∞(ψ(fϵ,λϵ))>l∞(φ(fi,λi)) − ϵ 4 . (5.26) 68 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) for all i>j0. it follows from the definition of symplectic displacement energy that ∥φ1(fi,λi)∥hl ≥ es(b)>0, (5.27) for all i>j0. relations (5.25), (5.26) and (5.27) implies that e0,∞ s (b) + ϵ 2 + ϵ 4 >l∞(φ(fi,λi)) ≥ ∥φ 1 (fi,λi) ∥hl ≥ es(b)>0, (5.28) for i sufficiently large. therefore, e0,∞s (b) + ϵ>es(b)>0, for all positive ϵ. this completes the proof. ✷ 6 conjectures and examples 6.1 conjectures: conjecture (a): for any h ∈ ssympeo(m,ω), we have ∥h∥shl = ∥h∥ (1,∞) shl . ⋆ conjecture−(a) is supported by the uniqueness result of hofer-like geometry from [15] or more generally in [16]. conjecture (b): if h ∈ ssympeo(m,ω), then the norm φ !→ ∥h ◦φ◦ h−1∥shl is equivalent to the norm φ !→ ∥φ∥shl.♣ conjecture−(b) is supported by a result found in [4] (theorem 7). conjecture (c): let σ be the canonical volume form on the unit circle s1 given by the orientation of the circle. if λ is a strong symplectic isotopy generated by (u, h), then the fathi’s mass flow of λ is exactly one of the following quantities: ± 1 (n − 1)! ∫ m ""∫1 0 htdt # ∧ ωn−1 ∧ f∗(σ) # , for any mapping f : m → s1.♥ in particular, any continuous hamiltonian flow has a trivial mass flow. therefore, is any strong symplectic isotopy with trivial fathi’s mass flow homotopic (relatively to fixed endpoints) to a continuous hamiltonian flow? cubo 19, 2 (2017) on topological symplectic dynamical systems 69 conjecture (d): if λ is a strong symplectic isotopy generated by (u, h), then the mapping, λ !→ $∫1 0 [ht]dt % ∈ h1(m, r), is a well defined group homomorphism which only depend on the homotopic class of λ relatively to fixed ends, where [, ] stands for the de rham cohomology class.$ 6.2 examples 6.2.1 example a harmonic 1−parameter group is an isotopy β = {βt} generated by the vector field x defined by ι(x)ω = k, where k is a harmonic 1−form, and let φ be a non-smooth hamiltonian homeomorphism [13]. by definition of φ, there exists a sequence of hamiltonian isotopies φj which is cauchy in (c0 + l∞), and φj := φj(1) → φ, uniformly. the isotopy ψj : t !→ φ −1 j ◦ βt ◦ φj, has time−1 map φ−1 j ◦β1 ◦φj, and it is generated by ( ∫ 1 0 k(φ̇j(s)) ◦φj(s)ds, k). in fact, the smooth function x !→ ∫ 1 0 k(φ̇j(s)) ◦ φj(s)ds(x), does not depend on the choice of any isotopy with time−1 map φj := φj(1) [5, 16, 18]. as it can be checked, the sequence of generators ( ∫ 1 0 k(φ̇j(s))◦φj(s)ds, k) is cauchy in d∞ if and only if the sequence of functions x !→ ( ∫1 0 k(φ̇j(s)) ◦ φj(s)ds)(x) is cauchy in the l∞−hofer norm. but, lemma 3.9 from [15, 16] shows that the sequence of functions x !→ ( ∫ 1 0 k(φ̇j(s)) ◦ φj(s)ds)(x), is cauchy in the l∞−hofer norm provided the sequence φj is cauchy in the metric d̄; which is the case. thus, the latter converges in the complete metric space n 0([0, 1] × m , r) to a time-independent continuous function that we denote f0. the strong symplectic isotopy φ : t !→ φ ◦ βt ◦ φ −1, is generated by (f0, k), and its time−1 map x !→ (φ◦β1 ◦φ −1)(x) is not necessary c1, but continuous. hence, we have constructed separately an example of strong symplectic isotopy which is a 1−parameter subgroup and whose generator is time independent. the hofer-like length of φ are given by l∞(φ) = osc(f0) + |k| = l(1,∞)(φ), (6.1) and we also have, ē0(φ ◦ β1 ◦ φ −1) ≤ ē(φ ◦ β1 ◦ φ −1) ≤ osc(f0) + |k|<∞. (6.2) △ 6.2.2 example consider the torus t 2l with coordinates (θ1, . . . ,θ2l) and equip it with the flat riemannian metric g0. note that all the 1−forms dθi, i = 1, . . . , 2l are harmonic. take the 1-forms dθi for i = 1, . . . , 2l as basis for the space of harmonic 1-forms and consider the symplectic form ω = ∑l i=1 dθi ∧dθi+l. given v = (a1, . . . , al, b1, . . . , bl) ∈ r 2l , the translation x !→ x+v on r2l induces a rotation rv on t 2l , which is a symplectic diffeomorphism. therefore, the smooth mapping {rt v } : t !→ rtv defines a symplectic isotopy generated by (0, h) with h = ∑ l i=1 (aidθi+l − bidθi). now, consider the 70 s. tchuiaga, m. koivogui, f. balibuno & v. mbazumutima cubo 19, 2 (2017) torus t 2 as the square: ✷ := {(p, q) | 0 ≤ p ≤ 1, 0 ≤ q ≤ 1} ⊂ r2, with opposite sides identified. then, the action of the unit circle s 1 on t 2 : ρ : s1 × t2 → t2, (α, (θ1,θ2)) !→ (θ1 + α,θ2 + α), induces a non-hamiltonian diffeomorphism ρα : t 2 → t2, because the latter has no fixed point for α small and non-trivial. assume this done. let d2 ⊂ r2 be the 2−disk of radius τ ∈]0, 1/8[ centered at a = (a, 0) with 7/8 ≤ a<1, and let λ2(τ) be the corresponding subset in t2. for any ν<1/4, consider the nonempty open subset b(ν) = {(x, y) | 0 0 there is n0 such that for all n > n0, ‖ d(xn, x) ‖≤ ǫ. we denote it by lim n→∞ xn = x or xn → x(n → ∞). (ii) (xn) is cauchy with respect to a if for any ǫ > 0 there is n0 such that for all n, m > n0, ‖ d(xn, xm) ‖≤ ǫ. (iii) (x, a, d) is a complete c∗-algebra valued b-metric space if every cauchy sequence with respect to a is convergent. example 2.7. if x is a banach space, then (x, a, d) is a complete c∗-algebra valued b-metric space with a = 2p−1i if we set d(x, y) =‖ x − y ‖p i where p > 1 is a real number. but (x, a, d) is not a c∗-algebra valued metric space because if x = r, then | x − y |p≤| x − z |p + | z − y |p is impossible for all x > z > y. definition 2.8. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i. we call a mapping f : x → x a c∗-algebra valued contraction mapping on x if there exists b ∈ a with ‖ b ‖2< 1 ‖a‖ such that d(fx, fy) � b∗ d(x, y)b for all x, y ∈ x. definition 2.9. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i. a mapping f : x → x is called a c∗-algebra valued fisher contraction if there exists b ∈ a ′ + with ‖ ba ‖< 1 ‖a‖+1 such that d(fx, fy) � b [d(fx, y) + d(fy, x)] for all x, y ∈ x. cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 45 definition 2.10. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i. a mapping f : x → x is called a c∗-algebra valued kannan operator if there exists b ∈ a ′ + with ‖ b ‖< 1 ‖a‖+1 such that d(fx, fy) � b [d(fx, x) + d(fy, y)] for all x, y ∈ x. definition 2.11. [2] let t and s be self mappings of a set x. if y = tx = sx for some x in x, then x is called a coincidence point of t and s and y is called a point of coincidence of t and s. definition 2.12. [19] the mappings t, s : x → x are weakly compatible, if for every x ∈ x, the following holds: t(sx) = s(tx) whenever sx = tx. proposition 2.13. [2] let s and t be weakly compatible selfmaps of a nonempty set x. if s and t have a unique point of coincidence y = sx = tx, then y is the unique common fixed point of s and t. definition 2.14. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i. a mapping f : x → x is called c∗-algebra valued expansive if there exists b ∈ a with 0 <‖ b ‖2< 1 ‖a‖ such that b∗d(fx, fy)b � d(x, y) for all x, y ∈ x. we next review some basic notions in graph theory. let (x, a, d) be a c∗-algebra valued b-metric space. let g be a directed graph (digraph) with a set of vertices v(g) = x and a set of edges e(g) contains all the loops, i.e., e(g) ⊇ ∆, where ∆ = {(x, x) : x ∈ x}. we also assume that g has no parallel edges and so we can identify g with the pair (v(g), e(g)). g may be considered as a weighted graph by assigning to each edge the distance between its vertices. by g−1 we denote the graph obtained from g by reversing the direction of edges i.e., e(g−1) = {(x, y) ∈ x × x : (y, x) ∈ e(g)}. let g̃ denote the undirected graph obtained from g by ignoring the direction of edges. actually, it will be more convenient for us to treat g̃ as a directed graph for which the set of its edges is symmetric. under this convention, e(g̃) = e(g) ∪ e(g−1). our graph theory notations and terminology are standard and can be found in all graph theory books, like [7, 12, 17]. if x, y are vertices of the digraph g, then a path in g from x to y of length n (n ∈ n) is a sequence (xi) n i=0 of n + 1 vertices such that x0 = x, xn = y and (xi−1, xi) ∈ e(g) for i = 1, 2, · · · , n. a graph g is connected if there is a path between any two vertices of g. g is weakly connected if g̃ is connected. 46 sushanta kumar mohanta cubo 20, 1 (2018) definition 2.15. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i and let g = (v(g), e(g)) be a graph. a mapping f : x → x is called a c∗-algebra valued gcontraction if there exists a b ∈ a with ‖ b ‖2< 1 ‖a‖ such that d(fx, fy) � b∗d(x, y)b, for all x, y ∈ x with (x, y) ∈ e(g). any c∗-algebra valued contraction mapping on x is a g0-contraction, where g0 is the complete graph defined by (x, x × x). but it is worth mentioning that a c∗-algebra valued g-contraction need not be a c∗-algebra valued contraction (see remark 3.23). definition 2.16. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i and let g = (v(g), e(g)) be a graph. a mapping f : x → x is called c∗-algebra valued fisher g-contraction if there exists b ∈ a ′ + with ‖ ba ‖< 1 ‖a‖+1 such that d(fx, fy) � b [d(fx, y) + d(fy, x)] for all x, y ∈ x with (x, y) ∈ e(g). it is easy to observe that a c∗-algebra valued fisher contraction is a c∗-algebra valued fisher g0-contraction. but it is important to note that a c ∗-algebra valued fisher g-contraction need not be a c∗-algebra valued fisher contraction. the following example supports the above remark. example 2.17. let x = [0, ∞) and b(h) be the set of all bounded linear operators on a hilbert space h. define d : x × x → b(h) by d(x, y) =| x − y |2 i for all x, y ∈ x. then (x, b(h), d) is a c∗-algebra valued b-metric space with the coefficient a = 2i. let g be a digraph such that v(g) = x and e(g) = ∆ ∪ {(3tx, 3t(x + 1)) : x ∈ x with x ≥ 2, t = 0, 1, 2, · · · }. let f : x → x be defined by fx = 3x for all x ∈ x. for x = 3tz, y = 3t(z + 1), z ≥ 2, we have d(fx, fy) = d ( 3t+1z, 3t+1(z + 1) ) = 32t+2i � 9 58 32t(8z2 + 8z + 10)i = b [ d ( 3t+1z, 3t(z + 1) ) + d ( 3t+1(z + 1), 3tz )] = b [d(fx, y) + d(fy, x)], where b = 9 58 i ∈ b(h) ′ + with ‖ ba ‖< 1 ‖a‖+1 . thus, f is a c∗-algebra valued fisher g-contraction. we now verify that f is not a c∗-algebra valued fisher contraction. in fact, if x = 3, y = 0, cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 47 then for any arbitrary b ∈ b(h) ′ + with ‖ ba ‖< 1 ‖a‖+1 = 1 3 (which implies 3ba ≺ i), we have b [d(fx, y) + d(fy, x)] = b [d(f3, 0) + d(f0, 3)] = 90bi = 45ba = 5 27 (3ba)(81i) ≺ 81i = d(fx, fy). definition 2.18. let (x, a, d) be a c∗-algebra valued b-metric space with the coefficient a � i and let g = (v(g), e(g)) be a graph. a mapping f : x → x is called c∗-algebra valued g-kannan if there exists b ∈ a ′ + with ‖ b ‖< 1 ‖a‖+1 such that d(fx, fy) � b [d(fx, x) + d(fy, y)] for all x, y ∈ x with (x, y) ∈ e(g). note that any c∗-algebra valued kannan operator is c∗-algebra valued g0-kannan. however, a c∗-algebra valued g-kannan operator need not be a c∗-algebra valued kannan operator (see remark 3.28). remark 2.19. if f is a c∗-algebra valued g-contraction(resp., g-kannan or fisher g-contraction), then f is both a c∗-algebra valued g−1-contraction(resp., g−1-kannan or fisher g−1-contraction) and a c∗-algebra valued g̃-contraction(resp., g̃-kannan or fisher g̃-contraction). 3 main results in this section we always assume that (x, a, d) is a c∗-algebra valued b-metric space with the coefficient a � i and g is a directed graph such that v(g) = x and e(g) ⊇ ∆. let f, g : x → x be such that f(x) ⊆ g(x). if x0 ∈ x is arbitrary, then there exists an element x1 ∈ x such that fx0 = gx1, since f(x) ⊆ g(x). proceeding in this way, we can construct a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · ·. definition 3.1. let (x, a, d) be a c∗-algebra valued b-metric space endowed with a graph g and f, g : x → x be such that f(x) ⊆ g(x). we define cgf the set of all elements x0 of x such that (gxn, gxm) ∈ e(g̃) for m, n = 0, 1, 2, · · · and for every sequence (gxn) such that gxn = fxn−1. if g = i, the identity map on x, then obviously cgf becomes cf which is the collection of all elements x of x such that (fnx, fmx) ∈ e(g̃) for m, n = 0, 1, 2, · · · . 48 sushanta kumar mohanta cubo 20, 1 (2018) theorem 3.2. let (x, a, d) be a c∗-algebra valued b-metric space endowed with a graph g and the mappings f, g : x → x be such that d(fx, fy) � b∗ d(gx, gy) b (3.1) for all x, y ∈ x with (gx, gy) ∈ e(g̃), where b ∈ a and ‖ b ‖2< 1 ‖a‖ . suppose f(x) ⊆ g(x) and g(x) is a complete subspace of x with the following property: (∗) if (gxn) is a sequence in x such that gxn → x and (gxn, gxn+1) ∈ e(g̃) for all n ≥ 1, then there exists a subsequence (gxni) of (gxn) such that (gxni, x) ∈ e(g̃) for all i ≥ 1. then f and g have a point of coincidence in x if cgf 6= ∅. moreover, f and g have a unique point of coincidence in x if the graph g has the following property: (∗∗) if x, y are points of coincidence of f and g in x, then (x, y) ∈ e(g̃). furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in x. proof. suppose that cgf 6= ∅. we choose an x0 ∈ cgf and keep it fixed. since f(x) ⊆ g(x), there exists a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · · and (gxn, gxm) ∈ e(g̃) for m, n = 0, 1, 2, · · · . it is a well known fact that in a c∗-algebra a, if a, b ∈ a+ and a � b, then for any x ∈ a both x∗ax and x∗bx are positive elements and x∗ax � x∗bx[23]. for any n ∈ n, we have by using condition (3.1) that d(gxn, gxn+1) = d(fxn−1, fxn) � b ∗d(gxn−1, gxn)b. (3.2) by repeated use of condition (3.2), we get d(gxn, gxn+1) � (b ∗ ) nd(gx0, gx1)b n = (bn)∗b0b n, (3.3) for all n ∈ n, where b0 = d(gx0, gx1) ∈ a+. cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 49 for any m, n ∈ n with m > n, we have by using condition (3.3) that d(gxn, gxm) � a[d(gxn, gxn+1) + d(gxn+1, gxm)] � ad(gxn, gxn+1) + a 2d(gxn+1, gxn+2) + · · · +am−n−1d(gxm−2, gxm−1) + a m−n−1d(gxm−1, gxm) � a(b∗)nb0b n + a2(b∗)n+1b0b n+1 + a3(b∗)n+2b0b n+2 + · · · +am−n−1(b∗)m−2b0b m−2 + am−n−1(b∗)m−1b0b m−1 � m−n−1∑ k=1 ak(b∗)n+k−1b0b n+k−1 + am−n(b∗)m−1b0b m−1 = m−n∑ k=1 ak(b∗)n+k−1b0b n+k−1 � m−n∑ k=1 ‖ ak(b∗)n+k−1b0b n+k−1 ‖ i � ‖ b0 ‖ m−n∑ k=1 ‖ a ‖k ‖ b ‖2(n+k−1) i = ‖ b0 ‖ ‖ b ‖ 2n ‖ a ‖ m−n∑ k=1 ( ‖ a ‖ ‖ b ‖2 )k−1 i � ‖ b0 ‖ ‖ b ‖ 2n ‖ a ‖ 1− ‖ a ‖ ‖ b ‖2 i, since ‖ b ‖2< 1 ‖ a ‖ → θ as n → ∞. therefore, (gxn) is a cauchy sequence with respect to a. since g(x) is complete, there exists an u ∈ g(x) such that lim n→∞ gxn = u = gv for some v ∈ x. as x0 ∈ cgf, it follows that (gxn, gxn+1) ∈ e(g̃) for all n ≥ 0, and so by property (∗), there exists a subsequence (gxni) of (gxn) such that (gxni, gv) ∈ e(g̃) for all i ≥ 1. using condition (3.1), we have d(fv, gv) � a[d(fv, fxni) + d(fxni, gv)] � ab∗d(gv, gxni)b + ad(gxni+1, gv) → θ as i → ∞. this implies that d(fv, gv) = θ and hence fv = gv = u. therefore, u is a point of coincidence of f and g. the next is to show that the point of coincidence is unique. assume that there is another point of coincidence u∗ in x such that fx = gx = u∗ for some x ∈ x. by property (∗∗), we have 50 sushanta kumar mohanta cubo 20, 1 (2018) (u, u∗) ∈ e(g̃). then, d(u, u∗) = d(fv, fx) � b∗d(gv, gx)b = b∗d(u, u∗)b, which implies that, ‖ d(u, u∗) ‖ ≤ ‖ b∗d(u, u∗)b ‖ ≤ ‖ b∗ ‖‖ d(u, u∗) ‖‖ b ‖ = ‖ b ‖2‖ d(u, u∗) ‖ . since ‖ b ‖2< 1 ‖a‖ ≤ 1, it follows that d(u, u∗) = θ i.e., u = u∗. therefore, f and g have a unique point of coincidence in x. if f and g are weakly compatible, then by proposition 2.13, f and g have a unique common fixed point in x. the following corollary gives fixed point of banach g-contraction in c∗-algebra valued bmetric spaces. corollary 3.3. let (x, a, d) be a complete c∗-algebra valued b-metric space endowed with a graph g and the mapping f : x → x be such that d(fx, fy) � b∗d(x, y)b (3.4) for all x, y ∈ x with (x, y) ∈ e(g̃), where b ∈ a with ‖ b ‖2< 1 ‖a‖ . suppose (x, a, d, g) has the following property: (∗)́ if (xn) is a sequence in x such that xn → x and (xn, xn+1) ∈ e(g̃) for all n ≥ 1, then there exists a subsequence (xni) of (xn) such that (xni, x) ∈ e(g̃) for all i ≥ 1. then f has a fixed point in x if cf 6= ∅. moreover, f has a unique fixed point in x if the graph g has the following property: (∗ ∗ )́ if x, y are fixed points of f in x, then (x, y) ∈ e(g̃). proof. the proof can be obtained from theorem 3.2 by considering g = i, the identity map on x. corollary 3.4. let (x, a, d) be a c∗-algebra valued b-metric space and the mappings f, g : x → x be such that (3.1) holds for all x, y ∈ x, where b ∈ a with ‖ b ‖2< 1 ‖a‖ . if f(x) ⊆ g(x) and cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 51 g(x) is a complete subspace of x, then f and g have a unique point of coincidence in x. moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in x. proof. the proof follows from theorem 3.2 by taking g = g0, where g0 is the complete graph (x, x × x). the following corollary is analogue of banach contraction principle. corollary 3.5. let (x, a, d) be a complete c∗-algebra valued b-metric space and the mapping f : x → x be such that (3.4) holds for all x, y ∈ x, where b ∈ a with ‖ b ‖2< 1 ‖a‖ . then f has a unique fixed point u in x and fnx → u for all x ∈ x. proof. it follows from theorem 3.2 by putting g = g0 and g = i. remark 3.6. we observe that banach contraction theorem in a complete metric space can be obtained from corollary 3.5 by taking a = c, a = 1. thus, theorem 3.2 is a generalization of banach contraction theorem in metric spaces to c∗-algebra valued b-metric spaces. from theorem 3.2, we obtain the following corollary concerning the fixed point of expansive mapping in c∗-algebra valued b-metric spaces. corollary 3.7. let (x, a, d) be a complete c∗-algebra valued b-metric space and let g : x → x be an onto mapping satisfying b∗d(gx, gy)b � d(x, y) for all x, y ∈ x, where b ∈ a with ‖ b ‖2< 1 ‖a‖ . then g has a unique fixed point in x. proof. the conclusion of the corollary follows from theorem 3.2 by taking g = g0 and f = i. corollary 3.8. let (x, a, d) be a complete c∗-algebra valued b-metric space endowed with a partial ordering ⊑ and the mapping f : x → x be such that (3.4) holds for all x, y ∈ x with x ⊑ y or, y ⊑ x, where b ∈ a and ‖ b ‖2< 1 ‖a‖ . suppose (x, a, d, ⊑) has the following property: (†) if (xn) is a sequence in x such that xn → x and xn, xn+1 are comparable for all n ≥ 1, then there exists a subsequence (xni) of (xn) such that xni, x are comparable for all i ≥ 1. if there exists x0 ∈ x such that f nx0, f mx0 are comparable for m, n = 0, 1, 2, · · · , then f has a fixed point in x. moreover, f has a unique fixed point in x if the following property holds: (††) if x, y are fixed points of f in x, then x, y are comparable. proof. the proof can be obtained from theorem 3.2 by taking g = i and g = g2, where the graph g2 is defined by e(g2) = {(x, y) ∈ x × x : x ⊑ y or y ⊑ x}. 52 sushanta kumar mohanta cubo 20, 1 (2018) theorem 3.9. let (x, a, d) be a c∗-algebra valued b-metric space endowed with a graph g and the mappings f, g : x → x be such that d(fx, fy) � b [d(fx, gy) + d(fy, gx)] (3.5) for all x, y ∈ x with (gx, gy) ∈ e(g̃), where b ∈ a ′ + and ‖ ba ‖< 1 ‖a‖+1 . suppose f(x) ⊆ g(x) and g(x) is a complete subspace of x with the property (∗). then f and g have a point of coincidence in x if cgf 6= ∅. moreover, f and g have a unique point of coincidence in x if the graph g has the property (∗∗). furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in x. proof. it follows from condition (3.5) that b(d(fx, gy) + d(fy, gx)) is a positive element. suppose that cgf 6= ∅. we choose an x0 ∈ cgf and keep it fixed. we can construct a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · · . evidently, (gxn, gxm) ∈ e(g̃) for m, n = 0, 1, 2, · · · . for any n ∈ n, we have by using condition (3.5) and lemma 2.1(iii) that d(gxn, gxn+1) = d(fxn−1, fxn) � b[d(fxn−1, gxn) + d(fxn, gxn−1)] = b[d(fxn−1, fxn−1) + d(fxn, fxn−2)] � ba[d(fxn, fxn−1) + d(fxn−1, fxn−2)] = ba d(gxn+1, gxn) + ba d(gxn, gxn−1)] which implies that, (i − ba)d(gxn, gxn+1) � bad(gxn, gxn−1). (3.6) now, a, b ∈ a ′ + implies that ba ∈ a ′ +. since ‖ ba ‖< 1 2 , by lemma 2.1, it follows that (i − ba) is invertible and ‖ ba(i − ba)−1 ‖=‖ (i − ba)−1ba ‖< 1. moreover, by lemma 2.1, ba � i i.e., i − ba � θ. since ba ∈ a ′ +, we have (i − ba) ∈ a ′ +. furthermore, (i − ba) −1 ∈ a ′ +. by using lemma 2.1(iv), it follows from (3.6) that d(gxn, gxn+1) � (i − ba) −1ba d(gxn, gxn−1) = td(gxn−1, gxn), (3.7) where t = (i − ba)−1ba ∈ a ′ +. by repeated use of condition (3.7), we get d(gxn, gxn+1) � t nd(gx0, gx1) = t nb0, (3.8) for all n ∈ n, where b0 = d(gx0, gx1) ∈ a+. cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 53 we now prove that if ‖ ba ‖< 1 ‖a‖+1 , then ‖ t ‖< 1 ‖a‖ . we have, ‖ t ‖ = ‖ (i − ba)−1ba ‖ ≤ ‖ (i − ba)−1 ‖‖ ba ‖ ≤ 1 1− ‖ ba ‖ ‖ ba ‖ < 1 ‖ a ‖ , since ‖ ba ‖< 1 ‖ a ‖ +1 . for any m, n ∈ n with m > n, we have by using condition (3.8) that d(gxn, gxm) � a[d(gxn, gxn+1) + d(gxn+1, gxm)] � ad(gxn, gxn+1) + a 2d(gxn+1, gxn+2) + · · · +am−n−1d(gxm−2, gxm−1) + a m−n−1d(gxm−1, gxm) � atnb0 + a 2tn+1b0 + a 3tn+2b0 + · · · +am−n−1tm−2b0 + a m−n−1tm−1b0 � m−n∑ k=1 aktn+k−1b0, since a � i and a ∈ a ′ + � m−n∑ k=1 ‖ aktn+k−1b0 ‖ i � ‖ b0 ‖ ‖ a ‖ ‖ t ‖ n m−n∑ k=1 (‖ a ‖ ‖ t ‖) k−1 i � ‖ b0 ‖ ‖ a ‖ ‖ t ‖ n 1 1− ‖ a ‖ ‖ t ‖ i → θ as n → ∞. therefore, (gxn) is a cauchy sequence with respect to a. as g(x) is complete, there exists an u ∈ g(x) such that lim n→∞ gxn = u = gv for some v ∈ x. by property (∗), there exists a subsequence (gxni) of (gxn) such that (gxni, gv) ∈ e(g̃) for all i ≥ 1. using condition (3.5), we have d(fv, gv) � a[d(fv, fxni) + d(fxni, gv)] � ab[d(fv, gxni) + d(fxni, gv)] + ad(gxni+1, gv) � aba[d(fv, gv) + d(gv, gxni)] + abd(gxni+1, gv) + ad(gxni+1, gv) which implies that, (i − ba2)d(fv, gv) � ba2d(gv, gxni) + abd(gxni+1, gv) + ad(gxni+1, gv). 54 sushanta kumar mohanta cubo 20, 1 (2018) since ‖ ba2 ‖< ‖a‖ ‖a‖+1 < 1, we have (i − ba2)−1 exists. by using lemma 2.1, it follows that d(fv, gv) � (i − ba2)−1ba2d(gv, gxni) + (i − ba 2)−1abd(gxni+1, gv) +(i − ba2)−1ad(gxni+1, gv) → θ as i → ∞. this implies that d(fv, gv) = θ i.e., fv = gv = u and hence u is a point of coincidence of f and g. finally, to prove the uniqueness of point of coincidence, suppose that there is another point of coincidence u∗ in x such that fx = gx = u∗ for some x ∈ x. by property (∗∗), we have (u, u∗) ∈ e(g̃). then, d(u, u∗) = d(fv, fx) � b[d(fv, gx) + d(fx, gv)] = b[d(u, u∗) + d(u, u∗)] � ab[d(u, u∗) + d(u, u∗)] which implies that, d(u, u∗) � (i − ab)−1 ab d(u, u∗). so, it must be the case that ‖ d(u, u∗) ‖ ≤ ‖ (i − ab)−1ab d(u, u∗) ‖ ≤ ‖ (i − ab)−1ab ‖ ‖ d(u, u∗) ‖ . since ‖ (i − ab)−1ab ‖< 1, we have ‖ d(u, u∗) ‖= 0 i.e., u = u∗. therefore, f and g have a unique point of coincidence in x. if f and g are weakly compatible, then by proposition 2.13, f and g have a unique common fixed point in x. corollary 3.10. let (x, a, d) be a complete c∗-algebra valued b-metric space endowed with a graph g and the mapping f : x → x be such that d(fx, fy) � b [d(fx, y) + d(fy, x)] (3.9) for all x, y ∈ x with (x, y) ∈ e(g̃), where b ∈ a ′ + and ‖ ba ‖< 1 ‖a‖+1 . suppose (x, a, d, g) has the property (∗)́. then f has a fixed point in x if cf 6= ∅. moreover, f has a unique fixed point in x if the graph g has the property (∗ ∗ )́. proof. the proof can be obtained from theorem 3.9 by putting g = i. cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 55 corollary 3.11. let (x, a, d) be a c∗-algebra valued b-metric space and the mappings f, g : x → x be such that (3.5) holds for all x, y ∈ x, where b ∈ a ′ + and ‖ ba ‖< 1 ‖a‖+1 . if f(x) ⊆ g(x) and g(x) is a complete subspace of x, then f and g have a unique point of coincidence in x. moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in x. proof. the proof can be obtained from theorem 3.9 by taking g = g0. corollary 3.12. let (x, a, d) be a complete c∗-algebra valued b-metric space and the mapping f : x → x be such that (3.9) holds for all x, y ∈ x, where b ∈ a ′ + with ‖ ba ‖< 1 ‖a‖+1 . then f has a unique fixed point in x. proof. the proof follows from theorem 3.9 by taking g = g0 and g = i. remark 3.13. we observe that brian fisher’s theorem in a complete metric space can be obtained from corollary 3.12 by taking a = c, a = 1. thus, theorem 3.9 is a generalization of brian fisher’s theorem in metric spaces to c∗-algebra valued b-metric spaces. corollary 3.14. let (x, a, d) be a complete c∗-algebra valued b-metric space endowed with a partial ordering ⊑ and the mapping f : x → x be such that (3.9) holds for all x, y ∈ x with x ⊑ y or, y ⊑ x, where b ∈ a ′ + with ‖ ba ‖< 1 ‖a‖+1 . suppose (x, a, d, ⊑) has the property (†). if there exists x0 ∈ x such that f nx0, f mx0 are comparable for m, n = 0, 1, 2, · · · , then f has a fixed point in x. moreover, f has a unique fixed point in x if the property (††) holds. proof. the proof can be obtained from theorem 3.9 by taking g = g2 and g = i. theorem 3.15. let (x, a, d) be a c∗-algebra valued b-metric space endowed with a graph g and the mappings f, g : x → x be such that d(fx, fy) � b [d(fx, gx) + d(fy, gy)] (3.10) for all x, y ∈ x with (gx, gy) ∈ e(g̃), where b ∈ a ′ + and ‖ b ‖< 1 ‖a‖+1 . suppose f(x) ⊆ g(x) and g(x) is a complete subspace of x with the property (∗). then f and g have a point of coincidence in x if cgf 6= ∅. moreover, f and g have a unique point of coincidence in x if the graph g has the property (∗∗). furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in x. proof. we observe that b(d(fx, gx) + d(fy, gy)) is a positive element. suppose that cgf 6= ∅. we choose an x0 ∈ cgf and keep it fixed. we can construct a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · · . evidently, (gxn, gxm) ∈ e(g̃) for m, n = 0, 1, 2, · · · . 56 sushanta kumar mohanta cubo 20, 1 (2018) for any n ∈ n, we have by using condition (3.10) that d(gxn, gxn+1) = d(fxn−1, fxn) � b[d(fxn−1, gxn−1) + d(fxn, gxn)] = b d(gxn, gxn−1) + b d(gxn, gxn+1) which implies that, (i − b)d(gxn, gxn+1) � bd(gxn, gxn−1). (3.11) since b ∈ a ′ + and ‖ b ‖< 1 2 , by lemma 2.1, it follows that b � i and (i − b) is invertible with ‖ b(i−b)−1 ‖=‖ (i−b)−1b ‖< 1. furthermore, (i−b), (i−b)−1 ∈ a ′ + and so, (i−b) −1b ∈ a ′ +. again, by using lemma 2.1(iv), it follows from condition (3.11) that d(gxn, gxn+1) � (i − b) −1b d(gxn, gxn−1) = td(gxn−1, gxn), (3.12) where t = (i − b)−1b ∈ a ′ +. by repeated use of condition (3.12), we get d(gxn, gxn+1) � t nd(gx0, gx1) = t nb0, (3.13) for all n ∈ n, where b0 = d(gx0, gx1) ∈ a+. we now prove that if ‖ b ‖< 1 ‖a‖+1 , then ‖ t ‖< 1 ‖a‖ . we have, ‖ t ‖ = ‖ (i − b)−1b ‖ ≤ ‖ (i − b)−1 ‖‖ b ‖ ≤ 1 1− ‖ b ‖ ‖ b ‖ < 1 ‖ a ‖ , since ‖ b ‖< 1 ‖ a ‖ +1 . cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 57 for any m, n ∈ n with m > n, we have by using condition (3.13) that d(gxn, gxm) � a[d(gxn, gxn+1) + d(gxn+1, gxm)] � ad(gxn, gxn+1) + a 2d(gxn+1, gxn+2) + · · · +am−n−1d(gxm−2, gxm−1) + a m−n−1d(gxm−1, gxm) � atnb0 + a 2tn+1b0 + a 3tn+2b0 + · · · +am−n−1tm−2b0 + a m−n−1tm−1b0 � m−n∑ k=1 aktn+k−1b0, since a � i and a ∈ a ′ + � m−n∑ k=1 ‖ aktn+k−1b0 ‖ i � ‖ b0 ‖‖ a ‖‖ t ‖ n m−n∑ k=1 (‖ a ‖‖ t ‖)k−1i � ‖ b0 ‖‖ a ‖‖ t ‖ n 1 1− ‖ a ‖‖ t ‖ i → θ as n → ∞. therefore, (gxn) is a cauchy sequence with respect to a. by completeness of g(x), there exists an u ∈ g(x) such that lim n→∞ gxn = u = gv for some v ∈ x . by property (∗), there exists a subsequence (gxni) of (gxn) such that (gxni, gv) ∈ e(g̃) for all i ≥ 1. using condition (3.10), we have d(fv, gv) � a[d(fv, fxni) + d(fxni, gv)] � ab[d(fv, gv) + d(fxni, gxni)] + ad(gxni+1, gv) which implies that, (i − ab)d(fv, gv) � abd(gxni+1, gxni) + ad(gxni+1, gv). since ‖ ab ‖< ‖a‖ ‖a‖+1 < 1, we have (i − ab)−1 exists and (i − ab) ∈ a ′ +. by using lemma 2.1, it follows that d(fv, gv) � (i − ab)−1abd(gxni+1, gxni) + (i − ab) −1ad(gxni+1, gv). then, ‖ d(fv, gv) ‖ ≤ ‖ (i − ab)−1ab ‖ ‖ d(gxni+1, gxni) ‖ + ‖ (i − ab)−1a ‖ ‖ d(gxni+1, gv) ‖ ≤ ‖ (i − ab)−1ab ‖ ‖ t ‖ni ‖ b0 ‖ + ‖ (i − ab)−1a ‖ ‖ d(gxni+1, gv) ‖ → 0 as i → ∞. 58 sushanta kumar mohanta cubo 20, 1 (2018) this implies that d(fv, gv) = θ i.e., fv = gv = u and hence u is a point of coincidence of f and g. finally, to prove the uniqueness of point of coincidence, suppose that there is another point of coincidence u∗ in x such that fx = gx = u∗ for some x ∈ x. by property (∗∗), we have (u, u∗) ∈ e(g̃). then, d(u, u∗) = d(fv, fx) � b[d(fv, gv) + d(fx, gx)] = θ which implies that, d(u, u∗) = θ i.e., u = u∗. therefore, f and g have a unique point of coincidence in x. if f and g are weakly compatible, then by proposition 2.13, f and g have a unique common fixed point in x. corollary 3.16. let (x, a, d) be a complete c∗-algebra valued b-metric space endowed with a graph g and the mapping f : x → x be such that d(fx, fy) � b [d(fx, x) + d(fy, y)] (3.14) for all x, y ∈ x with (x, y) ∈ e(g̃), where b ∈ a ′ + and ‖ b ‖< 1 ‖a‖+1 . suppose (x, a, d, g) has the property (∗)́. then f has a fixed point in x if cf 6= ∅. moreover, f has a unique fixed point in x if the graph g has the property (∗ ∗ )́. proof. the proof can be obtained from theorem 3.15 by putting g = i. corollary 3.17. let (x, a, d) be a c∗-algebra valued b-metric space and the mappings f, g : x → x be such that (3.10) holds for all x, y ∈ x, where b ∈ a ′ + and ‖ b ‖< 1 ‖a‖+1 . if f(x) ⊆ g(x) and g(x) is a complete subspace of x, then f and g have a unique point of coincidence in x. moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in x. proof. the proof can be obtained from theorem 3.15 by taking g = g0. corollary 3.18. let (x, a, d) be a complete c∗-algebra valued b-metric space and the mapping f : x → x be such that (3.14) holds for all x, y ∈ x, where b ∈ a ′ + with ‖ b ‖< 1 ‖a‖+1 . then f has a unique fixed point in x. proof. the proof follows from theorem 3.15 by taking g = g0 and g = i. cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 59 remark 3.19. we observe that kannan’s fixed point theorem in a complete metric space can be obtained from corollary 3.18 by taking a = c, a = 1. thus, theorem 3.15 is a generalization of kannan’s fixed point theorem in metric spaces to c∗-algebra valued b-metric spaces. corollary 3.20. let (x, a, d) be a complete c∗-algebra valued b-metric space endowed with a partial ordering ⊑ and the mapping f : x → x be such that (3.14) holds for all x, y ∈ x with x ⊑ y or, y ⊑ x, where b ∈ a ′ + with ‖ b ‖< 1 ‖a‖+1 . suppose (x, a, d, ⊑) has the property (†). if there exists x0 ∈ x such that f nx0, f mx0 are comparable for m, n = 0, 1, 2, · · · , then f has a fixed point in x. moreover, f has a unique fixed point in x if the property (††) holds. proof. the proof can be obtained from theorem 3.15 by taking g = g2 and g = i. we furnish some examples in favour of our results. example 3.21. let x = r and b(h) be the set of all bounded linear operators on a hilbert space h. define d : x × x → b(h) by d(x, y) =| x − y |3 i for all x, y ∈ x, where i is the identity operator on h. then (x, b(h), d) is a complete c∗-algebra valued b-metric space with the coefficient a = 4i. let g be a digraph such that v(g) = x and e(g) = ∆∪{( 1 n , 0) : n = 1, 2, 3 · · · }. let f, g : x → x be defined by fx = x 5 , if x 6= 4 5 = 1, if x = 4 5 and gx = 2x for all x ∈ x. obviously, f(x) ⊆ g(x) = x. if x = 0, y = 1 2n , n = 1, 2, 3, · · · , then gx = 0, gy = 1 n and so (gx, gy) ∈ e(g̃). for x = 0, y = 1 2n , we have d(fx, fy) = d ( 0, 1 10n ) = 1 103.n3 i ≺ 1 25n3 i = 1 25 d(gx, gy) = b∗ d(gx, gy)b, where b = 1 5 i ∈ b(h). 60 sushanta kumar mohanta cubo 20, 1 (2018) therefore, d(fx, fy) � b∗ d(gx, gy)b for all x, y ∈ x with (gx, gy) ∈ e(g̃), where b ∈ b(h) and ‖ b ‖2< 1 ‖a‖ . we can verify that 0 ∈ cgf. in fact, gxn = fxn−1, n = 1, 2, 3, · · · gives that gx1 = f0 = 0 ⇒ x1 = 0 and so gx2 = fx1 = 0 ⇒ x2 = 0. proceeding in this way, we get gxn = 0 for n = 0, 1, 2, · · · and hence (gxn, gxm) = (0, 0) ∈ e(g̃) for m, n = 0, 1, 2, · · · . also, any sequence (gxn) with the property (gxn, gxn+1) ∈ e(g̃) must be either a constant sequence or a sequence of the following form gxn = 0, if n is odd = 1 n , if n is even where the words ’odd’ and ’even’ are interchangeable. consequently it follows that property (∗) holds. furthermore, f and g are weakly compatible. thus, we have all the conditions of theorem 3.2 and 0 is the unique common fixed point of f and g in x. remark 3.22. it is worth mentioning that weak compatibility condition in theorem 3.2 cannot be relaxed. in example 3.21, if we take gx = 2x − 9 for all x ∈ x instead of gx = 2x, then 5 ∈ cgf and f(5) = g(5) = 1 but g(f(5)) 6= f(g(5)) i.e., f and g are not weakly compatible. however, all other conditions of theorem 3.2 are satisfied. we observe that 1 is the unique point of coincidence of f and g without being any common fixed point. remark 3.23. in example 3.21, f is a c∗-algebra valued g-contraction but it is not a c∗algebra valued contraction. in fact, for x = 4 5 , y = 0, we have d(fx, fy) = d(1, 0) = i = 125 64 . 64 125 i = 125 64 d(x, y) ≻ b∗ d(x, y) b, for any b ∈ b(h) with ‖ b ‖2< 1 ‖a‖ . this implies that f is not a c∗-algebra valued contraction. the following example shows that property (∗) is necessary in theorem 3.2. example 3.24. let x = [0, ∞) and b(h) be the set of all bounded linear operators on a hilbert space h. define d : x×x → b(h) by d(x, y) =| x−y |3 i for all x, y ∈ x, where i is the identity operator on h. then (x, b(h), d) is a complete c∗-algebra valued b-metric space with the coefficient cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 61 a = 4i. let g be a digraph such that v(g) = x and e(g) = ∆∪{(x, y) : (x, y) ∈ (0, 1]×(0, 1], x ≥ y}. let f, g : x → x be defined by fx = x 6 , if x 6= 0 = 1, if x = 0 and gx = x 2 for all x ∈ x. obviously, f(x) ⊆ g(x) = x. for x, y ∈ x with (gx, gy) ∈ e(g̃), we have d (fx, fy) = 1 27 d (gx, gy) � 1 9 d (gx, gy) = b∗ d(gx, gy) b, where b = 1 3 i ∈ b(h) with ‖ b ‖2< 1 ‖a‖ . we see that f and g have no point of coincidence in x. we now verify that the property (∗) does not hold. in fact, (gxn) is a sequence in x with gxn → 0 and (gxn, gxn+1) ∈ e(g̃) for all n ∈ n where xn = 2 n . but there exists no subsequence (gxni) of (gxn) such that (gxni, 0) ∈ e(g̃). example 3.25. let x = r and b(h) be the set of all bounded linear operators on a hilbert space h. choose a positive operator t ∈ b(h). define d : x×x → b(h) by d(x, y) =| x−y |5 t for all x, y ∈ x. then (x, b(h), d) is a complete c∗-algebra valued b-metric space with the coefficient a = 16i. let f, g : x → x be defined by fx = 2, if x 6= 5 = 3, if x = 5 and gx = 3x − 4 for all x ∈ x. obviously, f(x) ⊆ g(x) = x. let g be a digraph such that v(g) = x and e(g) = ∆ ∪ {(2, 3), (3, 5)}. if x = 2, y = 7 3 , then gx = 2, gy = 3 and so (gx, gy) ∈ e(g̃). again, if x = 7 3 , y = 3, then gx = 3, gy = 5 and so (gx, gy) ∈ e(g̃). it is easy to verify that condition (3.5) of theorem 3.9 holds for all x, y ∈ x with (gx, gy) ∈ e(g̃). furthermore, 2 ∈ cgf i.e., cgf 6= ∅, f and g are weakly compatible, and (x, b(h), d, g) has the property (∗). thus, all the conditions of theorem 3.9 are satisfied and 2 is the unique common fixed point of f and g in x. remark 3.26. it is observed that in example 3.25, f is not a fisher g-contraction. in fact, 62 sushanta kumar mohanta cubo 20, 1 (2018) for x = 3, y = 5, we have b [d(fx, y) + d(fy, x)] = b [d(2, 5) + d(3, 3)] = 243bt = 243 16 bat = 243 16 × 17 17bat ≺ t = d(fx, fy), for any b ∈ b(h) ′ + with ‖ ba ‖< 1 ‖a‖+1 . this implies that f is not a fisher g-contraction. the following example supports our theorem 3.15. example 3.27. let x = [0, ∞) and b(h) be the set of all bounded linear operators on a hilbert space h. choose a positive operator t ∈ b(h). define d : x×x → b(h) by d(x, y) =| x−y |2 t for all x, y ∈ x. then (x, b(h), d) is a complete c∗-algebra valued b-metric space with the coefficient a = 2i. let g be a digraph such that v(g) = x and e(g) = ∆ ∪ {(4tx, 4t(x + 1)) : x ∈ x with x ≥ 2, t = 0, 1, 2, · · · }. let f, g : x → x be defined by fx = 4x and gx = 16x for all x ∈ x. clearly, f(x) = g(x) = x. if x = 4t−2z, y = 4t−2(z + 1), then gx = 4tz, gy = 4t(z + 1) and so (gx, gy) ∈ e(g̃) for all z ≥ 2. for x = 4t−2z, y = 4t−2(z + 1), z ≥ 2 with b = 1 117 i, we have d(fx, fy) = d ( 4t−1z, 4t−1(z + 1) ) = 42t−2t � 1 117 42t−2(18z2 + 18z + 9)t = 1 117 [ d ( 4t−1z, 4tz ) + d ( 4t−1(z + 1), 4t(z + 1) )] = b [d(fx, gx) + d(fy, gy)]. thus, condition (3.10) is satisfied for all x, y ∈ x with (gx, gy) ∈ e(g̃). it is easy to verify that 0 ∈ cgf. also, any sequence (gxn) with gxn → x and (gxn, gxn+1) ∈ e(g̃) must be a constant sequence and hence property (∗) holds. furthermore, f and g are weakly compatible. thus, we have all the conditions of theorem 3.15 and 0 is the unique common fixed point of f and g in x. remark 3.28. it is easy to observe that in example 3.27, f is a c∗-algebra valued g-kannan operator with b = 16 117 i. but f is not a c∗-algebra valued kannan operator because, if x = 4, y = 0, cubo 20, 1 (2018) common fixed point results in c∗-algebra . . . 63 then for any arbitrary b ∈ b(h) ′ + with ‖ b ‖< 1 ‖a‖+1 = 1 3 (which implies 3b ≺ i), we have b [d(fx, x) + d(fy, y)] = b [d(f4, 4) + d(f0, 0)] = 144bt = 144 3 × 256 (3b)(256t) ≺ 256t = d(fx, fy). references [1] a. aghajani, m. abbas and j. r. roshan, common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, math. slovaca, 64, 2014, 941-960. 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[27] d. reem, s. reich, a. j. zaslavski, two results in metric fixed point theory, j. fixed point theory appl., 1, 2007, 149-157. introduction some basic concepts main results cubo a mathematical journal vol.21, no¯ 02, (77–98). august 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000200077 certain results for η-ricci solitons and yamabe solitons on quasi-sasakian 3-manifolds sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain3 1department of mathematics, poornima college of engineering, isi-6,riico institutional area, sitapura, jaipur302022, rajasthan,india prof sky16@yahoo.com 2,3department of mathematics & statistics d.d.u.gorakhpur university, gorakhpur-273009, uttar pradesh, india ab11sk1991@gmail.com, dhruwanarain.dubey@gmail.com abstract we classify quasi-sasakian 3-manifold with proper η-ricci soliton and investigate its geometrical properties. certain results of yamabe soliton on such manifold are also presented. finally, we construct an example of non-existence of proper η-ricci soliton on 3-dimensional quasi-sasakian manifold to illustrate the results obtained in previous section of the paper. resumen clasificamos 3-variedades cuasi-sasakianas con solitones η-ricci propios e investigamos sus propiedades geométricas. ciertos resultados sobre el solitón de yamabe en dichas variedades también se presentan. finalmente, construimos un ejemplo de la no existencia de solitones η-ricci propios en una 3-variedad cuasi-sasakiana para ilustrar los resultados contenidos en el art́ıculo. keywords and phrases: quasi-sasakian 3-manifold, infinitesimal contact transformation, ηricci soliton, yamabe soliton. 2010 ams mathematics subject classification: 53c15, 53c40, 53d10. http://dx.doi.org/10.4067/s0719-06462019000200077 78 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) 1 introduction in 1982, hamilton [17] introduced the notion of ricci flow to find a canonical metric on a smooth manifold. the ricci flow is an evolution equation for metric gij on a riemannian manifold defined as follows: ∂gij ∂t = −2rij, (1.1) where rij denotes the ricci tensor of a riemannian manifold and t is the time. ricci soliton are special solution of the ricci flow equation (1.1) of the form gij = σ(t)ψtgij with the initial condition gij(0) = gij, where ψt is the diffeomorphisms of m and σ(t) is the scaling function. a ricci soliton is a natural generalization of an einstein metric. we recall the notion of ricci soliton according to [9]. on a riemannian manifold m, a ricci soliton is a triple (g, v, µ) with the riemannian metric g, a vector field v, called potential vector field, µ a real scalar and s is the ricci tensor such that (lvg)(x, y) + 2s(x, y) + 2µg(x, y) = 0, (1.2) where lv is the lie-derivative along the vector field v on m. it is clear that a ricci soliton with v zero or a killing vector field reduces to an einstein metric. a ricci soliton is said to be shrinking, steady and expanding according as µ is negative, zero and positive, respectively. the ricci soliton have been studied by several authors such as ([12],[18],[20],[28],[36]). as a generalization of a ricci soliton, the notion of η-ricci soliton was introduced by cho and kimura [10]. this notion has also been studied in [10] for hopf hypersurfaces in complex-spaceforms. an η-ricci soliton is a 4-tuple (g, v, µ, α), where v is a vector field on m, µ and α are real constants and g is a riemannian (or pseudo-riemannian) metric satisfying the equation (lvg)(x, y) + 2s(x, y) + 2µg(x, y) + 2αη(x)η(y) = 0, (1.3) where s is the ricci tensor associated to g. in particular, if α = 0 then the notion of an η-ricci soliton (g, v, µ, α) reduces to the notion of a ricci soliton (g, v, µ). if α 6= 0, then the η-ricci soliton are known as the proper η-ricci soliton.thus the notion of η-ricci soliton have been studied by many authors like ([7],[8],[31],[32],[33]). the notion of yamabe flow was introduced by richard hamilton at the same time as the ricci flow [17], as a tool for constructing metrics of constant scalar curvature in a given conformal class of riemannian metrics on (mn, g)(n ≥ 3). a timedependent metric g(·, t) on a riemannian or, pseudo riemannian manifold m is said to evolve by the yamabe flow if the metric g satisfies ∂g(t) ∂t = −κg(t), g(0) = g0, (1.4) on m, where κ is the scalar curvature correspond to g. ye [35] has found that a point-wise elliptic gradient estimate for the yamabe flow on a locally conformally flat compact riemannian manifold. in case of ricci flow, yamabe soliton or the singularities of the yamabe flow appear naturally. cubo 21, 2 (2019) certain results for η-ricci solitons . . . 79 the significance of yamabe flow lies in the fact that it is a natural geometric deformation to metric of constant scalar curvature. one notes that yamabe flow corresponds to the fast diffusion case of the porous medium equation (the plasma equation) in mathematical physics. in dimension n = 2, the yamabe flow is equivalent to the ricci flow (defined by ∂ ∂t g(t) = −2α(t), where α stands for the ricci tensor). however in dimension n > 2, the yamabe and ricci flow do not agree, since the first one preserves the conformal class of metric but the ricci flow does not in general. just as ricci soliton is a special solution of the ricci flow, a yamabe soliton is a special solution of the yamabe flow that moves by one parameter family of diffeomorphism φt generated by a fixed vector field v on m , and homotheries, that is, g(., t) = σ(t)φ∗(t)g0. a yamabe soliton is defined on a riemannian or, pseudo-riemannian manifold (m, g) by a vector field v satisfying the equation [6]: 1 2 (lvg)(x, y) = (κ − λ)g(x, y), (1.5) where lv denotes the lie-derivative of the metric g along the vector field v, κ stands for the scalar curvature, while λ is a soliton constant. a yamabe soliton is said to be expanding, steady, or shrinking, respectively, if λ < 0, λ = 0 or λ > 0. otherwise, it will be called indefinite. given a yamabe soliton, if v = df holds for a smooth function f : m → ℜ on m, the equation (1.5) becomes hess f = (r − λ)g, where hess f denotes the hessian of f and d denotes the gradient operator of g on mn. in this case f is called the potential function of the yamabe soliton and g is said to be a gradient yamabe soliton. the notion of quasi-sasakian structure was introduced by blair [4] to unify sasakian and cosymplectic structures. tanno [30] also added some remarks on quasi-sasakian structure.the properties of such manifold have been studied by several authors,viz., gonzalez and chinea [16], kanemaki [21] and oubina [26]. kim [22] studied quasi-sasakian manifold and proved that fibred riemannian spaces with invariant fibers normal to the structure vector field do not admit nearly sasakian or contact structure but a sasakian or cosymplectic structure. recently, quasi-sasakian manifold have been the subject of growing interest in view of finding the significant of applications to physics, in particular to supargravity and magmatic theory ([3],[1]). quasi-sasakian structure have wide application in the mathematical analysis of string theory ([2],[14]). on a 3-dimensional quasi-sasakian manifold, the structure function β was defined by olszak [27] and with the help of this function he has obtained necessary and sufficient conditions for the manifold to be conformally flat [25]. next he has proved that if the manifold is additionally conformally flat with β=constant, then (a) it is locally a product of r and a two-dimensional kaehlerian space of constant gauss curvature ( the cosymplectic case), or, (b) it is constant positive curvature ( the non-cosymplectic case, here the quasi-sasakian structure is homothetic to a sasakian structure). now, we give some necessary definition and proposition that are uses in latter section. definition1.1[6] a vector field v is said to be conformal for yamabe soliton if it satisfying the 80 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) equation lvg = 2ωg, (1.6) where ω is called the conformal coefficient, that is, ω = (κ − λ). moreover, if ω = 0, is equivalent to v being killing. definition1.2[8] a vector field x on an almost contact riemannian manifold m is said to be infinitesimal transformation if there exists a smooth function υ on m such that (lxη)(y) = υη(y), (1.7) for every smooth vector field x and y. if υ = 0 then x is called a strict infinitesimal transformation. proposition1.1[34]on an n-dimensional riemannian or, pseudo riemannian manifold (mn, g) endowed with a conformal vector field v, we have (lvs)(x, y) = −(n − 2)g(∇xdω, y) + (∆ω)g(x, y), (lvκ) = −2ωκ + 2(n − 1)∆ω, for any vector fields x and y, where d denotes the gradient operator and ∆ = −divd denotes the laplacan operator of g. the outline of this paper is to consider 3-dimensional quasi-sasakian manifold with the structure function β is constant. in section 2, we recall the basic results of η-ricci soliton on quasi-sasakian 3-manifold. in section 3 and section 4, we examine the η-ricci soliton on quasi-sasakian 3-manifold admitting codazzi type and cyclic parallel ricci tensor, respectively. further, the section 5, section 6 and section 7, deals with an almost pseudo ricci symmetric, ϕ-ricci symmetric and conformally flat with η-ricci soliton on quasi-ssakian 3manifold respectively. the geometrical properties of a special weakly ricci symmetric and η-recurrent on quasi-sasakian 3-manifold are studied in section 8 and section 9, respectively. in section 10, we deals quasi-sasakian 3-manifolds with q · r = 0 and obtain new results for η-ricci soliton on such manifold. in section 11, we deduce some results related to yamabe soliton on quasi-sasakian 3-manifold. at last, we construct an example of non-existence of proper η-ricci soliton on quasi-sasakian 3-manifold. 2 preliminaries let m be a (2n + 1)-dimensional an almost contact metric manifold equipped with an almost contact metric structure (ϕ, ζ, η, g) consisting of a (1, 1) tensor field ϕ, a vector field ζ, a 1-form η and a riemannian metric g, which satisfies ϕ2 = −i + η ⊗ ζ, (2.1) cubo 21, 2 (2019) certain results for η-ricci solitons . . . 81 η(ζ) = 1, η ◦ ζ = 0, ϕζ = 0, (2.2) g(ϕu, ϕv) = g(u, v) − η(u)η(v), η(u) = g(u, ζ), (2.3) for all u, v ∈ χ(m), whereχ(m) is the lie-algebra of the vector fields of m2n+1. let φ be the fundamental 2-form of m2n+1 defined by φ(u, v) = g(u, ϕv), (2.4) for all u, v ∈ χ(m). then φ(u, ζ) = 0, u ∈ χ(m). m2n+1 is said to be quasi-sasakian if the almost contact structure (ϕ, ζ, η, g) is normal and the fundamental 2-form φ is closed, that is, every u, v ∈ ℑ2n+1, where ℑ2n+1 denotes the modulus of vector fields on m2n+1. (i) [ϕ, ϕ](u, v) + dη(u, v)ζ = 0, (ii) dφ = 0. (2.5) there are many types of quasi-sasakian structures ranging from the cosymplectic case, dη = 0(rank η = 1), to the sasakian case, η ∧ (dη)n 6= 0 (rank η = 2n + 1, φ = dη). the 1-form η has rank r = 2p if (dη)p 6= 0 and η∧(dη)p=0, and has rank r=2p+1 if (dη)p+1=0 and η∧(dη)p 6= 0. we also say that r is the rank of the quasi-sasakian structure. blair [7], proved that there are no quasi-sasakian structure of even rank, some theorems regarding kaehlerian manifolds and existence of quasi-sasakian manifold. an almost contact metric manifold m2n+1 is a 3-dimensional quasi-sasakian manifold if and only if [27] ∇uζ = −βϕu, u ∈ χ(m), (2.6) for a certain function β on m, such that ζβ = 0, ∇ being the operator of the covariant differentiation with respect to the levi-civita connection on m. clearly, such a quasi-sasakian manifold is cosymplectic if and only if β = 0. here we have shown that the assumption ζβ = 0 is not necessary. as per consequence (2.6), we find that [27] (∇uϕ)(v) = β[g(u, v)ζ − η(v)u]. (2.7) in view of (2.6) and (2.7), we obtain ∇u(∇vζ) = −(uβ)ϕv − β 2 [g(u, v)ζ − η(v)u] − βϕ∇uv. (2.8) this implies that r(u, v)ζ = −(uβ)ϕv + (vβ)ϕu + β2[η(v)u − η(u)v]. (2.9) thus from (2.9), we get 82 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) r(u, v, w, ζ) = (uβ)g(ϕv, w) − (vβ)g(ϕu, w) − β2[η(v)g(u, w) − η(u)g(v, w)]. (2.10) substituting u = ζ in (2.10), we have r(ζ, v, w, ζ) = β2[g(v, w) − η(v)η(w)] + g(ϕv, w)ζβ. (2.11) interchanging v and w of (2.11) it yields r(ζ, w, v, ζ) = β2[g(w, v) − η(w)η(v)] + g(ϕw, v)ζβ. (2.12) since r(ζ, v, w, ζ) = r(w, ζ, ζ, v) = r(ζ, w, v, ζ). then from (2.11) and(2.12), we obtain [g(ϕv, w) − g(ϕw, v)]ζβ = 0. (2.13) therefore, we can easily verify that ζβ = 0. in a 3-dimensional riemannian manifold we have r(u, v)w = {s(v, w)u − s(u, w)v + g(v, w)qu − g(u, w)qv} − κ 2 {g(v, w)u − g(u, w)v}, (2.14) where s and κ are the ricci tensor and the scalar curvature, respectively, and q denotes the ricci operator defined by g(qu, v) = s(u, v). it is well known that the ricci tensor s of a quasi-sasakian 3-manifold is given by [28] s(u, v) = [ κ 2 − β2 ] g(u, v) + [ 3β2 − κ 2 ] η(u)η(v). (2.15) as a consequence of (2.15), we find the ricci operator q qu = [ κ 2 − β2 ] u + [ 3β2 − κ 2 ] η(u)ζ. (2.16) from (2.15), we obtain s(u, ζ) = 2β2η(u). (2.17) keeping in mind the equ.(2.12),(2.13),(2.14) and (2.15), we have r(u, v)ζ = β2[η(v)u − η(u)v], (2.18) for all u, v, ∈ χ(m). also from (2.6), we have (∇uη)v = g(∇uζ, v) = −βg(ϕu, v). (2.19) again from (2.15), it follows that s(ϕu, ϕv) = s(u, v) − 2β2η(u)η(v). (2.20) cubo 21, 2 (2019) certain results for η-ricci solitons . . . 83 proposition 2.1. a 3-dimensional non-cosymplectic quasi-sasakian manifold with η-ricci soliton is an η-einstein manifold. proof. assume that the quasi-sasakian 3-manifold admits a proper η-ricci soliton (g, ζ, µ, α). then from (1.3), we have 2s(u, v) = −(lζg)(u, v) − 2µg(u, v) − 2αη(u)η(v), (2.21) for all smooth vector fields u, v ∈ χ(m). of the two natural situations regarding the vector field v : v ∈ span{ζ} and v⊥ζ, we investigate only the case v = ζ. our interest is in the expression for lζg + 2s + 2µg + 2αη ⊗ η. a straight forward computations give (lζg)(u, v) = g (∇uζ, v) + g (u, ∇v ζ) , = −β [g(ϕ u, v) + g(u, ϕ v) ] = 0. (2.22) using (2.22) in (2.21), we get s(u, v) = −µ g(u, v) − α η(u) η(v). (2.23) from the last equation,the proof ends. proposition 2.2. if a 3-dimensional non-cosymplectic quasi-sasakian manifold admits η-ricci soliton, then µ + α = −2β2. proof. from (2.15), we have s(u, v) = [ κ 2 − β2 ] g(u, v) + [ 3β2 − κ 2 ] η(u)η(v). (2.24) comparing (2.24) with (2.23), we get µ = 1 2 [2β2 − κ] and α = 1 2 [κ − 6β2]. in continuation we get µ + α = −2β2. from the last equation, the proof ends. 3 ricci tensor of codazzi type gray [15] introduced the notion of cyclic parallel and codazzi type ricci tensors. a riemannian manifold is said to possesses a cyclic parallel ricci tensor if its non-vanishing ricci tensor s of type (0, 2) satisfies the condition (∇us)(v, w) + (∇vs)(w, u) + (∇ws)(u, v) = 0, (3.1) 84 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) for arbitrary vector fields u, v, w on m. again a riemannian manifold is said to have a ricci tensor of codazzi type if s is non-zero and satisfies (∇us)(v, w) = (∇vs)(u, w), (3.2) for all the vector fields u, v, w on m. we consider proper η-ricci soliton on quasi-sasakian 3-manifold with ricci tensor of codazzi type. taking covariant derivative of (2.23) along w and using (2.19), we have (∇ws)(u, v) = −α [(∇w η)(u )η(v) + η(u)(∇wη)(v)] = αβ[g(u, ϕw)η(v) + g(v, ϕw)η(u)]. (3.3) since the ricci tensor s of m is of codazzi type. then (∇ws)(u, v) = (∇us)(w, v). (3.4) making use of (3.3) in (3.4), we yields αβ[g(u, ϕw)η(v) + g(v, ϕw)η(u)] = αβ[g(w, ϕu)η(v) + g(v, ϕu)η(w)]. (3.5) setting w = ζ in (3.5), we theorize β 6= 0, α = 0, which is a refutation. thus a non-cosymlectic quasi-sasakian 3-manifold with a ricci tensor of codazzi type does not admits a proper η-ricci soliton. in this way we terminate the following result: theorem 3.1. a non-cosymplectic quasi-sasakian 3-manifold accompanied by ricci tensor of codazzi type does not possess a proper η-ricci soliton. corollary 3.2. for a proper η-ricci soliton on a non-cosymplectic quasi-sasakian 3-manifold, the scalar curvature is constant if and only if the vector field ζ is harmonic. corollary 3.3. there exists no constant scalar curvature for a proper η-ricci soliton of noncosymplectic quasi-sasakian 3-manifold, provided the vector field ζ is non-harmonic. 4 cyclic parallel ricci tensor this section is affectionate to the study of proper η-ricci soliton on quasi-sasakian 3-manifold bearing cyclic parallel ricci tensor. on that account (∇us)(v, w) + (∇vs)(w, u) + (∇ws)(u, v) = 0. (4.1) on the other hand, we have (3.3) and left hand side of (4.1), we have (∇us)(v, w) + (∇vs)(w, u) + (∇ws)(u, v) = αβ [g(v, ϕu)η(w) + g(w, ϕu)η(v) + g(w, ϕv)η(u) +g(u, ϕv)η(w) + g(u, ϕw)η(v) + g(v, ϕw)η(u)]. (4.2) cubo 21, 2 (2019) certain results for η-ricci solitons . . . 85 taking in hand (2.3) and (4.2), we reached (∇us)(v, w) + (∇vs)(w, u) + (∇ws)(u, v) = 0. (4.3) thus we are in condition to plight the following result: theorem 4.1. a quasi-sasakian 3-manifold bearing proper η-ricci soliton always satisfies cyclic parallel ricci tensor. 5 almost pseudo ricci symmetric chaki and kawaguchi [11] introduced the concept of almost pseudo ricci symmetric manifolds as an extended class of pseudo symmetric manifolds. a riemannian manifold (m, g) is called an almost pseudo ricci symmetric manifold (aprs)n, if its ricci tensor s of type (0, 2) is not identically zero and satisfying the following condition: (∇us)(v, w) = [a(u) + b(u)]s(v, w) + a(v)s(u, w) + a(w)s(u, v), (5.1) where a and b are two non-zero 1-forms defined by a(u) = g(u, ρ1), b(u) = g(u, ρ2). (5.2) by taking cyclic sum of (5.1), we see that (∇us)(v, w) + (∇vs)(w, u) + (∇ws)(u, v) = [3a(u) + b(u)]s(v, w) +[3a(v) + b(v)]s(u, w) + [3a(w) + b(w)]s(u, v). (5.3) let m admits a cyclic ricci tensor, then (5.3) reduces [3a(u) + b(u)]s(v, w) + [3a(v) + b(v)]s(u, w) +[3a(w) + b(w)]s(u, v) = 0. (5.4) replacing w by ζ in (5.4) and using (2.23) and (5.2), we get −(µ + α)[3a(u) + b(u)]η(v) − (µ + α)[3a(v) + b(v)]η(u) +[3η(ρ1) + η(ρ2)]s(u, v) = 0. (5.5) in (5.5), substituting v = ζ and using (2.2), (2.23) and (5.2), we yield − (µ + α)[3a(u) + b(u)] − 2(µ + α)[3η(ρ1) + η(ρ2)]η(u) = 0. (5.6) again replacing u by ζ and using (5.2) in (5.6), we obtain − (µ + α) [3η (ρ1) + η (ρ2)] = 0, (5.7) 86 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) which implies 3η (ρ1) + η (ρ2) = 0, (5.8) since (µ + α) 6= 0. in view of (5.6) and (5.8), it follows that 3a(u) + b (u) = 0. thus we can state the following result. theorem 5.1. there is no almost pseudo ricci symmetric proper η-ricci soliton on non-cosymplectic quasi-sasakian 3-manifold admitting cyclic ricci tensor, unless 3a+b vanishes everywhere on m. consequently, if we keep in mind from (5.7) that 3η (ρ1) + η (ρ2) 6= 0, in this case µ + α = 0, but for η-ricci soliton on non-cosymplectic quasi-sasakian 3-manifold α + µ = −2β2. therefore for this condition α = −β2 and µ = −β2. thus we state the following result. corollary 5.2. a proper η-ricci soliton on almost pseudo ricci symmetric non-cosymplectic quasi-sasakian 3-manifold with cyclic ricci tensor is of type (g, v, −β2, − β2). 6 ϕ-ricci symmetric this segment is affectionate to the study of ϕ-ricci symmetric proper η-ricci soliton on a quasisasakian 3-manifold and deduce some result. a quasi-sasakian 3-manifold is said to be ϕ-ricci symmetric if the ricci operator q satisfies ϕ2(∇uq)v = 0, (6.1) for all smooth vector fields u, v ∈ χ(m). if x, y are orthogonal to ζ, then the manifold is said to be locally ϕ-ricci symmetric. it is well-known that ϕ-symmetric implies ϕ-ricci symmetric, but the converse, is not, in general true. ϕ-ricci symmetric sasakian manifolds have been studied by de and sarkar [13]. from (2.23), it follows that qu = −µ u − α η(u) ζ, (6.2) for all smooth vector fields u. proceeding covariant derivative of (6.2), we acquire (∇uq)v = −α β [g(u, ϕv)ζ − η(v) ϕu] = αβ[g(ϕ u, v)ζ + η(v)ϕu]. (6.3) applying ϕ2 on both sides of (6.3), we get ϕ2(∇uq)v = αβ η(v) ϕ 3u. (6.4) cubo 21, 2 (2019) certain results for η-ricci solitons . . . 87 making use of (6.1), from (6.4), it walk behind that β 6= 0, α = 0, which is a counter statement. thus we are in a condition to plight the following result: theorem 6.1. a ϕ-ricci symmetric non-cosymplectic quasi-sasakian 3-manifold does not admits a proper η-ricci soliton. in [24], authors prove that in a 3-dimensional non-cosymplectic quasi-sasakian mnaifold ϕricci symmetric and ϕ-symmetric are equivalent provided β is a constant. thus using this facts we state the following result. corollary 6.2. a ϕ-symmetric non-cosymplectic quasi-sasakian 3-manifold does not possess a proper η-ricci soliton. differentiating (2.16) covariantly along w and applying ϕ2 both side, we get ϕ2(∇wq)v = 1 2 [dκ(w)(−v + η(v)ζ) + (6β2 − κ)η(v)ϕ2(∇wζ)]. (6.5) if v is orthogonal to ζ, then from (6.4) and (6.5), we have 1 2 dκ(w)v = 0. (6.6) it implies that dκ = 0. hence the scalar curvature κ is constant.thus we state the following result. corollary 6.3. a non-cosymplectic quasi-sasakian 3-manifold bearing proper η-ricci soliton is locally ϕ-ricci symmetric if and only if the scalar curvature κ is constant. 7 conformally flat in this constituent we review conformally flat quasi-sasakian 3-manifolds with a proper η-ricci soliton. then we have [25]. (∇us)(v, w) − (∇vs)(u, w) = 1 4 [g(v, w)dκ(u) − g(u, w)dκ(v)]. (7.1) making use of (2.23) in (7.1), we have αβ[g(v, ϕu)η(w) + g(w, ϕu)η(v) − g(u, ϕv)η(w) − g(w, ϕv)η(u)] = 1 4 [g(v, w)dκ(u) − g(u, w)dκ(v)]. (7.2) substituting u = ζ in (7.2), we obtain 4αβg(ϕ w, v) = −η(w)dκ(v). (7.3) 88 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) that restricted to 4αβ ϕ v = −dκ(v)ζ. from the above equation it walk behind that 4αβ ϕ2 v = 0 and hence β 6= 0, α = 0, which is a counter statement. therefore we state the following result: theorem 7.1. a conformally flat non-cosymplectic quasi-sasakian 3-manifold does not possess a proper η-ricci soliton. 8 special weakly ricci symmetric the notion of a special weakly ricci symmetric manifold was introduced and studied by singh and quddus [29]. an n-dimensional riemannian manifold (m, g) is called a special weakly ricci symmetric manifold (swrs)n if (∇xs)(y, z) = 2ε(x)s(y, z) + ε(y)s(x, z) + ε(z)s(y, x), (8.1) where ε is a 1-form and is defined by ε(x) = g(x, ρ), (8.2) where ρ is the associated vector field. let the eq.(8.1) and (8.2) hold on quasi-sasakian 3-manifold. taking cyclic sum of (8.1), we get (∇xs)(y, z) + (∇ys)(z, x) + (∇zs)(x, y) = 4[ε(x)s(y, z) + ε(y)s(z, x) + ε(z)s(x, y)]. (8.3) let m admits a cyclic parallel ricci tensor. then (8.3) reduces to ε(x)s(y, z) + ε(y)s(z, x) + ε(z)s(x, y) = 0. (8.4) taking z = ζ in (8.4) and using (2.23) and (8.2), we have − (α + µ)[ε(x)η(y) + ε(y)η(x)] + η(ρ)s(x, y) = 0. (8.5) again, taking y = ζ in (8.5) and then using (2.23) and (8.2), we get − (α + µ)[ε(x) + 2η(ρ)η(x)] = 0. (8.6) taking x = ζ in (8.6) and using (8.2), we obtain − 3(α + µ)η (ρ) = 0. (8.7) in this case if η (ρ) = 0 and α + µ 6= 0, then from (8.6) we have ε(x) = 0, ∀ x ∈ χ(m). again if η (ρ) 6= 0, α + µ = 0 , in this case α = −β2, µ = −β2. it leads to the following result: cubo 21, 2 (2019) certain results for η-ricci solitons . . . 89 theorem 8.1. if a special weakly ricci symmetric non-cosymplectic quasi-sasakian 3-manifold with a proper η-ricci soliton admits a cyclic parallel ricci tensor, then the 1-form ε is vanish identically on m. corollary 8.2. a proper η-ricci soliton on a special weakly ricci symmetric non-cosymplectic quasi-sasakian 3-manifold admits cyclic ricci tensor is of type (g, v, −β2, −β2) if the 1-form ε 6= 0. again, if a complete einstein quasi-sasakian 3-manifold is compact. then we have [19]. s(x, y) = ϑg(x, y), ϑ = 2β2. it is well-known that for complete einstein quasi-sasakian 3-manifold, (∇xs)(y, z) = 0 and s(x, y) = ϑg(x, y). then (8.1) gives 2ε(x)g(y, z) + ε(y)g(x, z) + ε(z)g(y, x) = 0. (8.8) taking z = ζ in (8.8) and then using (8.2), we get 2ε(x)η(y) + ε(y)η(x) + η(ρ)g(y, x) = 0. (8.9) again taking x = ζ in (8.9) and then using (8.2), we get 3η(ρ)η(y) + ε(y) = 0. (8.10) taking y = ζ in (8.10) and using (8.2), we obtain η(ρ) = 0. (8.11) making use of (8.11) in (8.10), we get ε(y) = 0, ∀ y ∈ χ(m). finally we have the following result: theorem 8.3. a special weakly ricci symmetric non-cosymplectic quasi-sasakian 3-manifold can not be compact if the 1 -form ε 6= 0. 9 η-recurrent a quasi-sasakian manifold is said to be η-recurrent if its non-vanishing ricci tensor s satisfies the following condition (∇us)(ϕv, ϕw) = a(u)s(ϕv, ϕw), (9.1) for all u, v, w ∈ χ(m), where a(u) = g(u, ρ), ρ is the associated vector field of the 1-form a. in particular, if the 1-form a vanishes identically on m, then it is said to be η-parallel. this notion for sasakian manifold was first introduced by kon [21]. in view of (2.3), (2.19) and (2.23), we have (∇us)(ϕv, ϕw) = µβ[g(u, ϕv)η(w) + g(u, ϕw)η(v) −g(u, ϕw)η(v) − g(ϕv, u)η(w)] = 0. (9.2) 90 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) making use of (9.2) in (9.1), we get a(u)s(ϕv, ϕw) = 0. (9.3) again using (2.23) in (9.3), we obtain − µa(u)g(ϕv, ϕw) = 0. (9.4) this implies that a(u) 6= 0, g(ϕv, ϕw) 6= 0. therefore, we conclude that µ = 0, that is, the ricci soliton is always steady. so we have the following result. theorem 9.1. if a non-cosymplectic quasi-sasakian 3-manifold with proper η-ricci soliton satisfying η-recurrent, then the ricci soliton is always steady. corollary 9.2. the necessary condition for a non-cosymplectic quasi-sasakian 3-manifold with proper η-ricci soliton to be η-parallel, the ricci soliton is always shrinking. 10 the curvature condition q · r = 0. in this section we are going to study, a proper η-ricci soliton on a quasi-sasakian 3-manifold that satisfying the curvature condition q · r = 0. then (q · r)(u, v)w = 0, (10.1) for all smooth vector fields u, v, w ∈ χ(m). from (10.1), it is obvious that q(r(u, v)w) − r(qu, v)w − r(u, qv)w − r(u, v)qw = 0. (10.2) making use of (2.14) and (2.23), eq. (10.2) reduces to 4αµη(u)η(w)v + 2µ2η(u)η(w)v − 4αµη(v)η(w)u + ακη(u)η(w)v −ακη(v)η(w)u + 2µ[{−µg(v, w) − αη(v)η(w)}u −{−µg(u, w) − αη(u)η(w)}v + g(v, w){(−µu − αη(u)ζ} −g(u, w){(−µv − αη(v)ζ} − κ 2 {g(v, w)u − g(u, w)v}] −2α2η(v)η(w)u + αµg(v, w)η(u)ζ + α2g(v, w)η(u)ζ = 0. (10.3) putting u = w = ζ in (10.3), we get 4αµ v + 2µ2v − 4αµη (v)ζ + ακv − ακη (v)ζ +2µ[−µη (v)ζ − αη (v)ζ + µv + αv − µη(v)ζ −αη(v)ζ + µv + αη (v)ζ − κ 2 η(v)ζ + κ 2 (v)] −2α2η (v)ζ + µαη (v)ζ + α2η (v)ζ = 0. cubo 21, 2 (2019) certain results for η-ricci solitons . . . 91 applying the inner product of the above equation with ζ, we obtain α(µ + α)η(v) = 0. (10.4) it walk behind that α 6= 0, which is a counter statement. thus α + µ = 0. on other hand for a ηricci soliton on a quasi-sasakian 3-manifold, α + µ = −2β2. therefore for this condition α = −β2 and µ = −β2. thus we sate the following result: theorem 10.1. a proper η-ricci soliton on a non-cosymplectic quasi-sasakian 3-manifold satisfying the curvature condition q · r = 0 is of type (g, v, −β2, −β2). as the dissertation of our work, we keep in mind the corollary 5.2, corollary 8.2 and theorem 10.1, we state the following result. theorem 10.2. if a proper η-ricci soliton on a non-cosymplectic quasi-sasakian 3-manifold m is of type (g, v, −β2, −β2), then the following conditions are equivalent: i) mn is almost pseudo ricci symmetric with cyclic ricci tensor, ii) mn is special weakly ricci symmetric and its ricci tensor is cyclic parallel, iii) q · r = 0 holds on mn. 11 yamabe solitons in this section we find some results related to yamabe soliton on quasi-sasakian 3-manifolds. we consider a yamabe soliton (g, ζ). from (1.5) we have 1 2 (lvg)(x, y) = (κ − λ)g(x, y), (11.1) which implies that g(∇xζ, y) + g(x, ∇yζ) = 2(κ − λ)g(x, y). (11.2) keeping in mind (2.6), equ.(11.2) reduces to 2(κ − λ)g(x, y) = 0 (11.3) taking x = ζ in (11.3), we get λ = κ. then equation (1.5) reduces to lvg = 0, that is, v is killing vector field. moreover, λ is constant then the scalar curvature κ is also constant. thus we state the following result. theorem 11.1. if the metric of a 3-dimensional non-cosymplectic quasi-sasakian 3-manifold is a yamabe soliton then the manifold is space of constant curvature. 92 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) besides it, from (1.5), we have lvg = 0, thus v is killing. differentiating covariently along an arbitrary vector field x, we have ∇xlvg = 0. the identity (∇xlvg)(u, w) = g((lv ∇)(x, u), w) + g((lv ∇)(x, w), u), (11.4) can be reduced from the formula [34]. (lv ∇x g − ∇xlvg − ∇[v,x]g)(u, w) = −g((lv ∇)(x, u), w) − g((lv∇)(x, w), u). this implies that g((lv ∇)(w, x), u) + g((lv ∇)(w, u), x) = 0. (11.5) according to equation (11.4) and (11.5), the skew-symmetric property of φ, we get (lv∇) (u, w) = 0, which implies that (lv∇) (ζ, ζ) = 0. also, using geodesic properties of ζ, we have (lv∇) (x, u) = −∇x∇uv − ∇∇xuv + r(v, x)u, which yields ∇ζ∇ζ v + r(v, ζ)ζ = 0. this means that v is jacobi along the direction of ζ. so we have the following result. theorem 11.2. if the metric of a non-cosymplectic quasi-sasakian 3-manifold is a yamabe soliton, then the flow vector field v is killing and is jacobi along the direction of ζ. it is well-known that the reeb vector field ζ is a unit vector field, that is, g(ζ, ζ) = 1.taking lie-derivative of it along the vector fled v and using (1.5), we get η(lvζ) = −(lvη)(ζ) = (λ − κ). (11.6) moreover, in view of ω = (κ − λ), (n = 3) and proposition 1.1, we obtain (i) (lvs)(x, y) = −g(∇xdκ, y) + ∆κg(x, y). (ii) (lvκ) = −2κ(κ − λ) + 4∆κ. since g is a yamabe soliton, then taking the lie-derivative of (2.15), and using the above equation, we get −g(∆xdκ, y) = 1 2 (lvκ)[g(x, y) − η(x)η(y)] + [2( κ 2 − β2)(κ − λ)]g(x, y) + (3β2 − κ 2 )[(lvη)(x)η(y) + (lv η)(y)η(x)]. cubo 21, 2 (2019) certain results for η-ricci solitons . . . 93 since ζ is killing, therefore ζκ = 0. differentiating covariantly along the direction of an arbitrary vector field x, we have g(∆xdκ, ζ)=(βϕx)κ. substituting y = ζ in above equation, we have − 2β(ϕx)κ = [2(κ − 2β2)(κ − λ) − 2∆κ]η(x) + (6β2 − κ)[(lv η)x + (κ − λ)η(x)]. (11.7) taking x = ζ in (11.9), using (11.6) and proposition 1.1, we obtain ∆κ = 4β2(κ − λ) (11.8) in view of (11.9) and(11.10), we yields (6β2 − κ)(lvη)x = −2β(ϕx)κ − [(κ − λ)(κ − 6β 2]η(x) (11.9) since κ is constant then from (11.11) one can say that either κ = 6β2 or κ 6= 6β2. in particular if κ = 6β2 then from (2.15) we have s = 2β2g, that is, m is an einstein manifold of constant curvature β2. thus as per above consequences, we state the following result. corollary 11.3. if the metric of a 3-dimensional non-cosymplectic quasi-sasakian manifold admits a yamabe soliton and κ = 6β2 then the manifold is an einstein. corollary 11.4. for a 3-dimensional cosymplectic manifold which admits a yamabe soliton always has constant harmonic scalar curvature, that is ∆κ = 0. corollary 11.5. if a 3-dimensional non-cosymplectic quasi-sasakian manifold with constant harmonic scalar curvature admitting yamabe soliton then the manifold is space of constant curvature. on the other hand, if κ 6= 6β2 then from (11.8), we get lvη = 0. then the equation (1.7) implies that υ = 0. thus we state the following result. theorem 11.6. if the metric of a 3-dimensional non-cosymplectic quasi-sasakian manifold is a yamabe soliton, then the conformal contact transformation of the conformal vector field is strict. 12 an example we consider a 3-dimensional manifold m3 = {(u, v, w) ∈ ℜ3, (u, v, w) 6= (0, 0, 0)}, where (u, v, w) is the standard coordinate in ℜ3. let (e1, e2, e3) be linearly independent vector fields at each point of m, identify by e1 = ∂ ∂v , e2 = ∂ ∂w , e3 = β ( ∂ ∂u + v ∂ ∂w − w ∂ ∂v ) and [e1, e2] = 0, [e1, e3] = βe2, [e2, e3] = −βe1. 94 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) let the riemannian metric g on m3 is defined as g(e1, e2) = g(e2, e3) = g(e1, e3) = 0, g(e1, e1) = g(e2, e2) = g(e3, e3) = 1. and given by g = 1 β2 [ (1 − β2v2 − β2w2)du ⊗ du + β2dv ⊗ dv + β2dw ⊗ dw ] . let η be the 1-form has the significance η(u) = g(u, e3) for any u ∈ χ(m3) and ϕ be the (1, 1) tensor field defined by ϕe1 = −e2, ϕe2 = e1, ϕe3 = 0. making use of the linearity of ϕ and g, we have η(e3) = 1, ϕ2(u) = −u + η(u)e3 and g(ϕu, ϕv) = g(u, v) − η(u)η(v), for any u, w ∈ χ(m3). thus for e3 = ζ, the structure (m 3, η, ζ, ϕ, g) leads to a contact metric structure on m3. the riemannian connection ∇ of metric tensor g is given by the beauty of koszul’s formula 2g(∇uv, w) = u(g(v, w)) + v(g(w, x)) − w(g(u, v)) −g(u, [v, w]) − g(v, [u, w]) + g(w, [u, v]). making use of the koszul’s formula, we get    ∇e2e3 = −βe1, ∇e2e2 = 0, ∇e2e1 = −βe3, ∇e3e3 = 0, ∇e3e2 = 0, ∇e3e1 = −βe3, ∇e1e3 = βe2, ∇e1e2 = −βe3, ∇e1e1 = 0. consequently (m3, η, ζ, ϕ, g) is an quasi-sasakian structure that satisfies, (∇uϕ)v = β(g(u, v)ζ − η(v)u), ∇uζ = −βϕu, where β 6= 0. hence (m3, η, ζ, ϕ, g) define non-cosymplectic quasi-sasakian 3-manifold. therefore, we find the components of curvature tensor as follows:    r(e2, e3)e3 = β 2e2, r(e2, e3)e1 = −β 2e3, r(e3, e2)e2 = β 2e3, r(e1, e3)e3 = β 2e1, r(e3, e1)e1 = β 2e3, r(e2, e1)e1 = β 2e2 − β 2e1, r(e1, e2)e2 = β 2e1, r(e1, e2)e3 = β 2e3, r(e3, e1)e2 = 0. cubo 21, 2 (2019) certain results for η-ricci solitons . . . 95 from the above we can easily evaluate the value of the ricci tensor as follows: { s(e1, e1) = 2β 2, s(e2, e2) = 2β 2, s(e3, e3) = 2β 2 s(e1, e2) = 0 s(e2, e3) = 0, s(e2, e3) = 0. also, the scalar curvature κ is given by: κ = 3∑ i=1 g(eiei)s(ei, ei) = 6β 2 also from the equ.(2.23), we get s(e1, e1) = s(e2, e2) = −µ, s(e3, e3) = −µ − α. it is clear that µ = −2β2 and α = 0. thus the manifold does not admits proper η-ricci soliton. hence the theorem 3.1, theorem 4.1, theorem 6.1 and theorem 7.1 are verified. let {e1, e2, e3} be a basis of the tangent space at any point. for any vector x, y ∈ χ(m 2n+1), we have x = a1e1 + b1e2 + c1e3 , y = a2e1 + b2e2 + c2e3, where ai, bi, ci ∈ ℜ\{0}, for all i = 1, 2, 3. thus g(x, y) = a1a2 + b1b2 + c1c2, and s(x, y) = 2β 2{a1a2 + b1b2 + c1c2 }. then we obtain s(x, y) = 2β2g(x, y), that is, the manifold m is an einstein manifold. hence corollary 11.3 are hold. remark 12.1. in this example β 6= 0 and µ < 0. thus the ricci soliton in a 3-dimensional non-cosymplectic quasi-sasakian manifold is always shrinking. 96 sunil kumar yadav1, abhishek kushwaha2 and dhruwa narain cubo 21, 2 (2019) references [1] agricola, i. and friedrich, t., killing spinors in supergravity with 4-fluxes ,class.quant.grav.20(2003),4707-4717. 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[36] yadav, s. k., ricci solitons on para-kähler manifolds, extracta mathematicae, available online april 29, 2, 34 (2019). introduction preliminaries ricci tensor of codazzi type cyclic parallel ricci tensor almost pseudo ricci symmetric -ricci symmetric conformally flat special weakly ricci symmetric -recurrent the curvature condition qr=0. yamabe solitons an example cubo, a mathematical journal vol.22, n◦03, (379–393). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300379 received: 21 may, 2020 | accepted: 24 november, 2020 curves in low dimensional projective spaces with the lowest ranks edoardo ballico department of mathematics, university of trento, 38123 povo (tn), italy. ballico@science.unitn.it abstract let x ⊂ pr be an integral and non-degenerate curve. for each q ∈ pr the x-rank rx(q) of q is the minimal number of points of x spanning q. a general point of p r has x-rank ⌈(r + 1)/2⌉. for r = 3 (resp. r = 4) we construct many smooth curves such that rx(q) ≤ 2 (resp. rx(q) ≤ 3) for all q ∈ p r (the best possible upper bound). we also construct nodal curves with the same properties and almost all geometric genera allowed by castelnuovo’s upper bound for the arithmetic genus. resumen sea x ⊂ pr una curva integral y no-degenerada. para cada q ∈ pr el x-rango rx(q) de q es el mı́nimo número de puntos de x que generan q. un punto general de pr tiene x-rango ⌈(r + 1)/2⌉. para r = 3 (resp. r = 4) construimos muchas curvas suaves tales que rx(q) ≤ 2 (resp. rx(q) ≤ 3) para todo q ∈ p r (la mejor cota superior posible). también construimos curvas nodales con las mismas propiedades y casi todos los géneros geométricos permitidos por la cota superior de castelnuovo para el género aritmético. keywords and phrases: x-rank, projective curve, space curve, curve in projective spaces. 2020 ams mathematics subject classification: 14h51, 14n05. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300379 380 edoardo ballico cubo 22, 3 (2020) 1 introduction let x ⊂ pr be an integral and non-degenerate variety defined over an algebraically closed field with characteristic 0. for each q ∈ pr the x-rank rx(q) of q is the minimal cardinality of a finite set s ⊂ x such that q ∈ 〈s〉, where 〈〉 denotes the linear span. an interesting problem is the maximum of all integers rx(q), q ∈ p r ([2, 8]). an obvious lower bound for this integer is the generic x-rank rgen(x), i.e. the only integer such there is a non-empty zariski open subset u ⊂ p r such that rx(q) = rgen(x) for all q ∈ u. for each positive integer t set w 0 t (x) := {q ∈ p r | rx(q) = t}. let wt(x) denote the closure of w 0 t (x) in p r. if t ≤ rgen(x) the algebraic set wt(x) is the t-secant variety σt(x) of x. hence if 1 ≤ t ≤ rgen(x) the algebraic set wt(x) is non-empty, irreducible and dim wt(x) ≤ min{r, t(dim x + 1) − 1} with equality if dim x = 1 ([1, remark 1.6]). thus rgen(x) = ⌈(r + 1)/2⌉ if dim x = 1. for t > rgen(x) the geometry of wt(x) is described in [3, theorem 3.1], assuming of course wt(x) 6= ∅, i.e. w 0 t (x) 6= ∅. we prove the following results. theorem 1.1. fix integers b ≥ a > 0 such that a + b ≥ 5. set d := a + b and γ := ab − a − b + 1. then there exists an integral and non-degenerate nodal curve x ⊂ p3 with geometric genus g, deg(x) = d, exactly γ − g ordinary nodes and w 03 (x) = ∅. theorem 1.2. fix integers a, b such that a ≥ 2 and b ≥ 2a + 3. set d := a + b and γ := 1 + ab− a(a+ 1)/2 − b. fix an integer g such that 0 ≤ g ≤ γ. then there is an integral nodal curve x ⊂ p4 with degree d, geometric genus g, exactly γ − g ordinary nodes and with w 04 (x) = ∅. question 1.1. is there an integral and non-degenerate curve x ⊂ p5 with w 04 (x) = ∅ ? take an odd integer r > 5. is there an integral and non-degenerate curve x ⊂ pr with w 0 (r+3)/2 (x) = ∅ ? by [9, theorem 1] w 03 (x) 6= ∅ for x as in theorem 1.1, but with (a, b) ∈ {(1, 2), (1, 3), (2, 2)}. the case (a, b) = (3, 3) of theorem 1.1 is [9, theorem 2]. when a ≤ b ≤ a + 1 the integer γ appearing in theorem 1.1 is the maximal arithmetic genus of all non-degenerate space curves ([6, ch. iii]). many thanks are due to the referees for useful remarks. 2 preliminaries notation 2.1. for any q ∈ pr let ℓq : p r \ {q} −→ pr−1 denote the linear projection from q. let m be a projective scheme. let d ⊂ m be an effective cartier divisor of m. for any zero-dimensional scheme z ⊂ m the residual scheme resd(z) of z with respect to d is the closed subscheme of m with iz : id as its ideal sheaf. we have resd(z) ⊆ z and hence resd(z) is a cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 381 zero-dimensional scheme. we have deg(z) = deg(z ∩ d) + deg(resd(z)) and for any line bundle l on m we have an exact sequence of coherent sheaves on m: 0 −→ iresd(z) ⊗ l(−d) −→ iz ⊗ l −→ iz∩d,d ⊗ l|d −→ 0 (2.1) we will call (2.1) the residual exact sequence of d or the residual exact sequence of d in m. remark 2.1. let m be a smooth, projective and rational surface. thus h1(om ) = 0. assume that ω∨m is ample. this will be true in the cases in which we apply this remark, i.e. the case in which m is the smooth quadric surface and the case in which m is the hirzebruch surface f1. fix an integer e ≥ 2, a very ample line bundle l on m and a nodal curve d = d1 ∪ · · · ∪ de ∈ |l| with each di a smooth and connected curve. note that pa(d) = ∑e i=1 pa(di) + ♯(sing(d)) + 1 − e. since l is very ample, d is connected. since ω∨m is ample, we have di · ωm < 0 (intersection number) for all i. a subset a ⊆ sing(d) is said to be a disconnecting set of nodes if d \ a is not connected. fix a set a ⊂ sing(d) which is not disconnecting and set g := pa(d) − ♯(a). with the terminology of [10] we will say that a is the set of assigned nodes, while the set sing(d) \ a is the set of unassigned nodes. by [10, corollary 2.14] there are an affine smooth and connected curve ∆, o ∈ ∆, and a flat family {yt}t∈∆ of elements of |l| such that yo = d and yt is integral, nodal and with geometric genus g for all t ∈ ∆ \ {o}. moreover, the sets {sing(yt)}t∈∆\{o} have a as a limit. thus pa(d) = ♯(sing(d)) + 1 − e. we do not impose (or claim) that all yt are singular at the points of a, because it would require very strong restrictions on the integer ♯(a), only that the nodes of the curves yt near d are near a and that yt has only ♯(a) nodes. the quoted result [10, corollary 2.14] with movable assigned nodes is optimal, as shown by following particular case, the only one we will use. assume that each di is rational. in this case for each integer g with 0 ≤ g ≤ pa(d) there is a set of assigned nodes a ⊂ sing(d) such that the corresponding family of nodal curves has as a general member an integral nodal curve with geometric genus g. remark 2.2. let x be a smooth projective curve, l a line bundle on x and v ⊆ h0(l) a linear subspace. set g := pa(x), d := deg(l) and n := dim v − 1. assume n ≥ 1. for each p ∈ x and each integer t > 0 set v (−tp) := v ∩ h0(itp ⊗ l). we get n + 1 integers dim v (−tp), 1 ≤ t ≤ n + 1 ([5, pp. 264–277]). this is also done in details in [9]. the point p is said to be an osculating point of the pair (l, v ) (or of the linear system pv ) if dim(v (−(n + 1)p)) > 0. since we are in characteristic zero, there are only finitely many osculating points of (l, v ), say p1, . . . , ps, and at each point pi one can associate a positive integer w(pi) (the weight of pi), only depending on the n + 1 integers dim v (−tp), 1 ≤ t ≤ n + 1. moreover, there is an integer δ only depending on g, d and n such that w(p1) + · · · + w(ps) = δ. we have w(pi) = 1 if and only if dim v (−npi) = dim v (−(n + 1)pi) = 1. suppose for instance that pv induces an embedding of x into pn and see x has a curve of pn. since v ⊆ h0(l), x is non-degenerate. the point p ∈ x is an osculating point if and only if there is a hyperplane h ⊂ pn such that the connected component z of the scheme h ∩ x with p has its reduction has degree ≥ n + 1, i.e. h contains the divisor 382 edoardo ballico cubo 22, 3 (2020) (n + 1)p. the integer deg(z) is the order of contact of the osculating hyperplane h with x at p. the integer deg(z) − n is a lower bound for the weight of p. all non-osculating points have weight 0. 3 proof of theorem 1.1 in this section we fix a smooth quadric surface q ⊂ p3. for any irreducible curve y ⊂ p3, y not a line, let τ(y ) denote the tangential surface of y , i.e. the closure in p3 of the union of all tangent lines of y at its smooth points. τ(y ) is a plane if and only if 〈y 〉 is a plane. notation 3.1. for any reduced curve x ⊂ p3 with no irreducible component contained in a plane let t (x) be the set of all pairs (h, p), where h ⊂ p3 is a plane, p ∈ h ∩ x and the connected component of the scheme h ∩ x with p as its reduction has degree at least 5. remark 3.1. let ∆ a quasi-projective variety and x ⊂ p3 × ∆ a closed algebraic set such that the restriction u : x −→ ∆ to x of the projection p3 × ∆ −→ ∆ is proper and flat. assume that all fibers of u are reduced curves with no irreducible component contained in a plane. let t (x) or t (u) denote the set of all triples (s, h, p), where s ∈ ∆ and (h, p) ∈ t (u−1(s)). the map ut (x ) : t (x) −→ ∆ is proper. thus if ∆ is irreducible and if t (u −1(s0)) = ∅ for some s0 ∈ ∆, then t (u−1(s)) = ∅ for a general s ∈ ∆. let x ⊂ p3 be an integral and non-degenerate curve. fix p ∈ xreg. we say that p is a flex point of x or a flex of x or that the tangent line tpx is a flex tangent of x if the connected component of the zero-dimensional scheme tpx ∩ x with p as its reduction has degree at least 3. we say that p is a stall point of x or that tpx is a stall of x if tpx is not a flex tangent, but the osculating plane op(x) of x at p has order of contact at least 4 with x at p. thus a stall point is an osculating point which is not a flex point. remark 3.2. fix a smooth element y either of |oq(1, 1)| or of |oq(2, 1)| or of |oq(1, 2)|. since y is a rational normal curve in its linear span, it is easy to check that t (y ) = ∅ and that each q ∈ τ(y ) \ y is contained in at most 2 tangent lines of y . we collect in the next remark some standard tools and ideas which are used in the proofs of lemmas 3.1, 3.2 and 3.3 and which may be used in several other cases. in section 4 we will use this set-up for the hirzebruch surface f1 and the line bundle of1(ah + bf). remark 3.3. fix positive integers a, b and an integral quasi-projective family f of zero-dimensional subschemes of the smooth quadric q. suppose you want to compute the dimension of the family ψ of all c ∈ |oq(a, b)| containing at least one z ∈ f or of the family φ of all smooth c ∈ |oq(a, b)| containing at least one z ∈ f. in most lemmas we will need to check that cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 383 dim φ < dim |oq(a, b)|, i.e. that a general c ∈ |oq(a, b)| contains no z ∈ f. consider the incidence variety i := {(z, c) ∈ f × |oq(a, b)| : z ∈ c}. let π1 : i −→ f and π2 : i −→ |oq(a, b)| denote the restriction to i of the projections of f ×|oq(a, b)| onto its factors. note that ψ = π2(i). the algebraic set i is a closed subset of f × |oq(a, b)|. thus by chevalley’s theorem ψ is a constructible set ([7, ex. ii.3.18 and ii.3.19]). if i is irreducible, then ψ is irreducible. obviously φ = ∅, unless at least some z ∈ f is curvilinear. call u the set of all smooth c ∈ |oq(a, b)|. assume that at least some z ∈ f is curvilinear and let g denote the set of all curvilinear z ∈ f. the set g is an open subset of f. since f is assumed to be irreducible, g is irreducible. set j := i ∩ g × u. usually, if we are only interested in smooth curves c ∈ |oq(a, b)| it is better to start with g, i.e. take an irreducible family of curvilinear schemes. thus from now on we assume f = g, but we use i, i.e. we also consider singular curves, to quote below [7, iii.9.3, iii.9.6, iii.9.7]. suppose there is an integer z > 0 such that h0(q, iz(a, b)) = z for all z ∈ g. with this assumption all fibers of π1 are projective spaces of dimension z − 1. hence π1 is a proper flat map. since g is assumed to be irreducible, i is irreducible and dim i = dim g + z ([7, iii.9.3, iii.9.6, iii.9.7]). since j is a non-empty open subset of i, j is irreducible and dim j = dim i = dim g + z. thus φ is irreducible and dim φ ≤ dim g + z. if this inequality is not sufficient to conclude, one should look at a general c ∈ φ and try to compute dim(j∩π−12 (c)). suppose dim(j∩π −1 2 (c)) = x for a general c ∈ |oq(a, b)|. then dim φ = dim g + z − x. since c is smooth and dim c = 1, dim(j ∩ π−12 (c)) ≤ x if ♯(zred) ≤ x for all z ∈ j ∩ π −1 2 (c). moreover, dim(j ∩ π −1 2 (c)) = x if varying z ∈ j ∩ π−12 (c) the sets zred form an x-dimensional family of x distinct points of c. this set-up is classically summarized by the words “ a dimensional count shows that φ has dimension dim g + z − x ”. if our family g is not irreducible, we try to study separately each of its irreducible components. now we drop the assumption that all integers h0(q, iz(a, b)) are the same. there are a non-empty open subset g′ of g and an integer z such that h0(q, iz(a, b)) = z for all z ∈ g ′. moreover, there are a positive integer s and integers zi ≥ z, 1 ≤ i ≤ s, such that g \ g ′ is the union of finitely many irreducible quasi-projective varieties, say g \ g′ = g1 ∪ · · · ∪ gs, such that h0(q, iz(a, b)) = zi for all z ∈ gi. then we use the irreducible families g ′, g1, . . . , gs of curvilinear schemes. we will need only the case a = 1 of the next lemma, but its proof when a ≥ 2 requires no modification. lemma 3.1. fix integers a > 0, b > 0 such that a + b ≥ 4. let d be a general element of |oq(a, b)|. then d has no flex and t (d) = ∅. proof. we follow the classical approach outlined in remark 3.3. the key step in the proof of the lemma is the computation of the integer h0(q, iz(a, b)) for two types of zero-dimensional schemes z. 384 edoardo ballico cubo 22, 3 (2020) with no loss of generality we may assume b ≥ a and hence b ≥ 2. by bertini’s theorem d is smooth. since d ⊂ q, bezout theorem implies that each flex tangent line of d is contained in q and hence it is either an element of |oq(1, 0)| or an element of |oq(0, 1)|. (a) take l ∈ |oq(1, 0)| and any connected zero-dimensional scheme f ⊂ l such that deg(f) = 3. since deg(ol(a, b)) = b ≥ 2, we have h 1(l, if,l(a, b)) = 0. since h 1(oq(0, b)) = 0, the residual exact sequence of l gives h1(if (a, b)) = 0, i.e. h 0(if (a, b)) = h 0(oq(a, b)) − 3. since dim |oq(1, 0)| = 1 and each l ∈ |oq(1, 0)| contains ∞ 1 connected degree 3 subschemes, a general d ∈ |oq(a, b)| contains no f (for any l), i.e. no l ∈ |oq(1, 0)| is a flex tangent of d. (b) if a ≥ 2 step (a) shows that no r ∈ |oq(0, 1)| is a flex tangent of d. now assume a = 1. since d ∈ |oq(a, b)|, we have deg(r ∩ d) = 1 for all r ∈ |oq(0, 1)|. thus no element of |oq(0, 1)| is a flex tangent line of d. by steps (a) and (b) d has no flex. thus it is sufficient to prove that each osculating plane of d has order of contact 4 with d at the osculating point. fix a smooth element a ∈ |oq(1, 1)| and p ∈ a. let e be the connected zero-dimensional subscheme of a such that ered = {p} and deg(e) = 5. claim 1: we have h1(q, ie(a, b)) = 0. proof of claim 1: we have h1(a, ie,a(a, b)) = 0, because a ∼= p 1 and deg(oa(1, b)) = b + 1 ≥ 4. since e ⊂ a, resa(e) = ∅. thus it is sufficient to use the residual exact sequence of a in q and that h1(oq(0, b − 1)) = 0. by claim 1 we have h0(ie(a, b)) = (a + 1)(b + 1) − 5 for all e. since dim |oq(1, 1)| = 3 and each smooth a ∈ |oq(1, 1)| has ∞ 1 points and hence ∞1 schemes e’s. use claim 1. notation 3.2. let d ⊂ q be a reduced curve with no irreducible component of d being an element of |oq(1, 0)| or |oq(1, 0)| or |oq(1, 1)|. let f(d) be denote the set of all c ∈ |oq(1, 1)| such that the scheme c ∩ d contains at least two connected components, both of them of degree at least 4. lemma 3.2. fix integers a > 0 and b > 0 such that a + b ≥ 3. take a general d ∈ |oq(a, b)|. then f(d) = ∅. proof. the curve d is smooth and for each line l ⊂ q every connected component of the scheme l ∩ d has connected components of degree 1 or 2, with at most one having degree 2. thus it is sufficient to test the smooth c ∈ |oq(1, 1)|. since deg(d ∩ c) = a + b, we may assume a + b ≥ 8. call g the set of all zero-dimensional schemes z with 2 connected components, both of degree 4 and with z contained in some smooth c ∈ |oq(1, 1)|. each c contains ∞ 2 elements of g. fix z ∈ g and take c containing it. as in the proof of lemma 3.1 it is sufficient to observe that h1(iz(a, b)) = 0, because c ∼= p 1 and deg(oc(a, b)) = a + b ≥ deg(z) − 1 and hence cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 385 h1(c, oc(a, b)(−z)) = 0. since resc(z) = ∅ and h 1(q, oq(a − 1, b − 1)) = 0, the residual exact sequence of c gives h1(q, iz,q(a, b)) = 0. lemma 3.3. fix positive integers a, b, a′, b′ such that (a, b) 6= (1, 1) and (a′, b′) 6= (1, 1). take a general (d, d′) ∈ |oq(a, b)| × |oq(a ′, b′)|. set y := d ∪ d′. then there is no c ∈ |oq(1, 1)| such that the scheme c ∩ y has two connected components of degree at least 4. proof. by bertini’s theorem d and d′ are smooth and y is nodal. for a general pair (d, d′) for each line l ⊂ q the scheme l ∩ d has connected components of degree 1 or 2, with at most one being of degree 2. thus it is sufficient to test all smooth c ∈ |oq(1, 1)|. since (d, d ′) is general, each c contains at most 2 points of d ∩ d′. thus every smooth c ∈ |oq(1, 1)| containing some p ∈ d ∩ d′ satisfies the property that the connected component of c ∩ y with p as its reduction has degree ≤ 3. thus we only need to consider the schemes c ∩ (y \ d ∩ d′) with c smooth. by lemma 3.2 it is sufficient to exclude the smooth c such that c ∩ d has a connected component z1 of degree at least 4 and c ∩ (d ′ \ d ∩ d′) has a connected component z2 of degree at least 4. we may assume a + b ≥ 4 and a′ + b′ ≥ 4. as in the proof of lemma 3.1 we find only finitely many smooth ci ∈ |oq(1, 1)|, say ci, 1 ≤ i ≤ t, such that c ∩ d has a connected component of degree at least 4. for a general d′, the curve d′ is transversal to all ci, 1 ≤ i ≤ t. lemma 3.4. fix positive integers e ≥ 2, ai, bi, 1 ≤ i ≤ e, such that for each i ∈ {1, . . . , e} exactly one among ai and bi is 1. let d = d1 ∪ · · · ∪ ds ⊂ q be a general union with each di general in |oq(ai, bi)|. then d is nodal, no two of the nodes of d are contained in the same line of q, each line of q passing through a singular point of d is transversal to each di, t (d) = ∅ and there is no line j ⊂ q such that j ∩ d has a connected component of degree at least 3. proof. d is nodal by bertini’s theorem. lemma 3.1 gives t (d) ⊆ sing(d). fix p ∈ sing(d). call di and dj the irreducible components of d containing p. since d is general, neither di nor dj have a osculating plane at p with weight ≥ 2 and the tangent plane to one component, does not contain the tangent line to the other component. thus p /∈ t (d). for a general (d1, . . . , de) no two of the nodes of d are on the same line of q, because aibi 6= 0 for all i. we also see by induction on e that each line of q passing through a singular point of d is transversal to each di. fix any line j ⊂ p3. since d ⊂ q, we have deg(d ∩ j) ≤ 2 if j * q. now assume l ∈ |oq(1, 0)| (resp. r ∈ |oq(1, 0)|). we have deg(l ∩ d) = b (resp. deg(r ∩ d) = a). by lemma 3.1 each connected component of the zero-dimensional schemes l ∩ d and r ∩ d has degree ≤ 2. lemma 3.5. fix positive integers a, b and q ∈ p3 \q. then q /∈ τ(y ) for a general y ∈ |oq(a, b)|. 386 edoardo ballico cubo 22, 3 (2020) proof. the polar surface of q with respect to q is a plane, h, intersecting transversally q and q ∈ tpq if and only if p ∈ h ∩ q. take y intersecting transversally h ∩ q and not containing the degree 2 subscheme of 〈{p, q}〉 with p as its reduction at all p ∈ h ∩ q ∩ y . lemma 3.6. fix positive integers s ≥ 4, ai, bi, 1 ≤ i ≤ s. take a general (d1, . . . , ds) ∈ ∏s i=1 |oq(ai, bi)|. then for every q ∈ p 3 \ q there is sq ⊂ {1, . . . , s} such that ♯(sq) ≤ 3 and q /∈ τ(di) for all i ∈ {1, . . . , s} \ sq. proof. by lemma 3.5 and the generality of (d1, . . . , ds) we have dim((p 3 \ q) ∩ τ(d1)) = 2, dim((p3 \ q) ∩ τ(d1) ∩ τ(d2)) ≤ 1, dim((p 3 \ q) ∩ τ(d1) ∩ τ(d2) ∩ τ(d3)) ≤ 0 and (p 3 \ q) ∩ τ(d1) ∩ τ(d2) ∩ τ(d3) ∩ τ(d4) = ∅. using all subsets of {1, . . . , s} with cardinality 4 we get the lemma. lemma 3.7. fix positive integers a, b. take a general y ∈ |oq(a, b)|. then for every q ∈ p 3 \ q there are at most 3 points p ∈ y such that q ∈ tpy . proof. with no loss of generality we may assume b ≥ a. y is smooth. if a + b ≤ 3, then y is a rational normal curve in its linear span and the lemma is trivial in this case. thus we may assume a + b ≥ 4. the lemma is also easy to check using the linear projection ℓq and the genus formula for plane curves if (a, b) ∈ {(2, 2), (1, 3), (2, 3)} (all these cases are discussed in [9]). for any q ∈ p3 \ q the polar plane hq of q with respect to q has the following properties. the curve cq := hq ∩ q is a smooth conic and q ∈ t pq, p ∈ q, if and only if p ∈ cq. for any p ∈ cq let zp denote the degree 2 connected zero-dimensional subscheme of the line 〈{p, q}〉 with p as its reduction. for any curve e ⊂ q such that p ∈ ereg we have q ∈ tpe if and only if zp ⊂ e. let u denote the set of quadruples (z1, z2, z3, z4) with each zi a connected degree 2 zero-dimensional subscheme of q such that there is q ∈ p3 \ q and (p1, p2, p3, p4) ∈ c 4 q such that pi 6= pj for all i 6= j and zi = zpi. the lemma is equivalent to proving that a general y contains no scheme z1 ∪ z2 ∪ z3 ∪ z4 with (z1, z2, z3, z4) ∈ u. for each smooth c ∈ |oq(1, 1)| there is a unique q ∈ p3 \ q such that c = cq. each smooth c ∈ |oq(1, 1)| has ∞ 4 quadruples of distinct points. since dim |oq(1, 1)| = 3, we get dim u = 7. thus to prove the lemma it is sufficient to prove that dim |iz1∪z2∪z3∪z4(a, b)| = dim |oq(a, b)| − 8. fix (z1, z2, z3, z4) ∈ u, say zi = zpi with p1, p2, p3, p4 distinct points of a smooth c ∈ |oq(1, 1)|. set z := z1 ∪ z2 ∪ z3 ∪ z4. since deg(z) = 8, it is sufficient to prove that h1(iz(a, b)) = 0. we have c∩z = {p1, p2, p3, p4} (schemetheoretically), because each tangent line of c is contained in the plane 〈c〉 and if c = cq, then q /∈ 〈c〉. hence resc(z) = {p1, p2, p3, p4}. we have h 1(c, iz∩c(a, b)) = 0, because c ∼= p 1 and deg(oc(a, b)) = a + b. we have h 1(c, iresc(z)(a − 1, b − 1)) = 0, because deg(oc(a − 1, b − 1)) = a + b − 2 ≥ 3. we have h1(oq(a − 2, b − 2)) = 0. use twice the residual exact sequence of c, first with iz(a, b) as its middle term and then with iresc(z)(a − 1, b − 1) as its middle term. cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 387 lemma 3.8. fix positive integers a, b such that (a, b) 6= (1, 1). take a general y ∈ |oq(a, b)|. the set of all q ∈ p3 \ q such that there are 2 (resp. 3) points p ∈ y with q ∈ tpy has dimension ≤ 1 (resp. ≤ 0). proof. adapt the proof of lemma 3.7 using z1 ∪ z2 ∪ z3 (resp. z1 ∪ z2) instead of z1 ∪ z2 ∪ z3 ∪ z4. lemma 3.9. fix positive integers a1, b1, a2, b2. take a general pair (d1, d2) ∈ |oq(a1, b1)| × |oq(a2, b2)|. for each q ∈ p 3 \ q the following properties are true:. (a) there is no (p1, p2, p3, p4) ∈ d1 × d1 × d2 × d2 such that p1 6= p2, p3 6= p4 and q ∈ tp1d1 ∩ tp2d1 ∩ tp3d2 ∩ tp4d2; (b) there is no (p1, p2, p3, p4) ∈ d1 × d1 × d1 × d2 such that ♯({p1, p2, p3}) = 3 and q ∈ tp1d1 ∩ tp2d1 ∩ tp3d1 ∩ tp4d2. proof. part (b) follows from lemmas 3.5 and 3.8. now we prove part (a). this is trivial if (a2, b2) ∈ {(2, 1), (1, 2)}, i.e. if d2 is a rational normal curve. thus we may assume a2 + b2 ≥ 4. as in the proof of lemma 3.7 let hq be the polar hyperplane of q with respect to q and cq := hq ∩ q. for any p ∈ cq let zp denote the degree 2 connected zero-dimensional subscheme of the line 〈{p, q}〉 with p as its reduction. let u denote the set of all quadruples z1, z2, z3, z4 such that there is a smooth c ∈ |oq(1, 1)| and 4 distinct points pi ∈ c, 1 ≤ i ≤ 4, such that zpi = zi for all i. for a fixed d1 lemma 3.8 shows that we have at most ∞1 pairs (p1, p2) which may be prolonged to be the reduction of some (z1, z2, z3, z4). for a fixed p1, p2 we have h 0(q, ip1,p2(1, 1)) = 2 and hence there are only ∞ 1 c ∈ |oq(1, 1)| containing {p1, p2}. for a fixed c we have ∞ 2 pairs (p3, p4) ∈ c × c. we fix the general d1. to prove that a general d2 satisfies part (a) of the lemma it is sufficient to prove that h 1(iz1∪z2∪z3∪z4(a2, b2)) ≤ 2. we prove this inequality in the following way. recall that c ∩ (z∪z2 ∪ z3 ∪ z4) = {p1, p2, p3, p4} (scheme-theoretically), because each tangent line of c is contained in the plane 〈c〉 and if c = cq, then q /∈ 〈c〉. thus resc(z1 ∪ z2 ∪ z3 ∪ z4) = {p1, p2, p3, p4}. since a2 + b2 ≥ 4, we have h1(c, i{p1,p2,p3,p4}(a2, b2)) = 0 and h 1(c, i{p1,p2,p3,p4}(a2 − 1, b2 − 1)) ≤ 1. use twice the residual exact sequence of c, first with iz1∪z2∪z3∪z4(a2, b2) as its middle term and then with i{p1,p2,p3,p4}(a2 − 1, b2 − 1) as its middle term. lemma 3.10. fix positive integers a1, b1, a2, b2, a3, b3 such that (ai, bi) 6= (1, 1), 1 ≤ i ≤ 3. take a general (d1, d2, d3) ∈ |oq(a1, b1)|× |oq(a2, b2)|× |oq(a3, b3)|. take any q ∈ p 3 \ q. there are no (p1, p2, p3, p4) ∈ d1 × d1 × d2 × d3 such that p1 6= p2 and q ∈ tp1d1 ∩tp2d1 ∩tp3d2 ∩tp4d3. proof. the proof of part (a) of lemma 3.9 shows that there are only finitely many triples (p1, p2, p3) ∈ d1 × d1 × d2 such that p1 6= p2 and tp1d1 ∩ tp2d1 ∩ tp3d2 is a point of p 3 \ q. apply lemma 388 edoardo ballico cubo 22, 3 (2020) 3.5 to d3. proof of theorem 1.1: any y ∈ |oq(a, b)| has arithmetic genus γ. claim 1: there are integers e ≥ 2, ai, bi, 1 ≤ i ≤ e, such that for each i ∈ {1, . . . , e} exactly one among ai and bi is 1, a1 + · · · + ae = a and b1 + · · · + be = b. proof of claim 1: if d ≡ 0 (mod 6) we take e = d/3, (ai, bi) = (1, 2) for odd i and (ai, bi) = (2, 1) for even i. if d ≡ i (mod 6), 1 ≤ i ≤ 5, we take e = (d − i)/3, (a1, b1) = (1, 2 + i), (ai, bi) = (1, 2) for odd i ≥ 3 and (ai, bi) = (2, 1) for even i. take a nodal curve d = d1 ∪ · · · ∪ de ⊂ q satisfying the thesis of lemma 3.4. since each di is smooth and rational and pa(d) = γ, we have ♯(sing(d)) = γ + e − 1. since 0 ≤ g ≤ γ and each di is irreducible, there is a set a ⊂ sing(d) such that ♯(a) = γ − g and d \ a is connected. we fix one such set a and call it the set of all assigned nodes. the set sing(d) is called the set of all unassigned nodes (we are using the terminology of a. tannenbaum ([10]) who extended to other rational surfaces the classical theory of nodal plane curves due to severi). since d \ a is connected, [10, lemma 2.2 and theorem 2.13] gives the existence of a flat family {dt}t∈∆, ∆ an integral affine curve, and o ∈ ∆ such that dt ∈ |oq(a, b)| for all t ∈ ∆, do = d, each dt, t ∈ ∆ \ {o}, is integral, nodal and with geometric genus g, and the nodes of dt, t ∈ ∆ \ {o}, go to the set of assigned nodes. by remark 3.1 we have t (dt) = ∅ for a general t ∈ ∆. fix c ∈ ∆ \ {o} such that t (dc) = ∅ and set x := dc. x is an integral and nodal curve with geometric genus g. to conclude the proof of the theorem it is sufficient to prove that rx(q) = 2 for all q ∈ p 3 \ x. (a) fix q ∈ q. let l be the element of |oq(1, 0)| containing q. we have deg(l ∩ x) = b. by lemma 3.1 each connected component of l ∩ x has degree ≤ 2. thus ♯((l ∩ x)red) ≥ ⌈ba/2⌉. since b ≥ 3, we get rx(q) = 2. (b) fix q ∈ p3 \ q. assume rx(q) > 2, i.e. assume ℓq|x is injective. since ℓq(x) has degree d = a + b, it has arithmetic genus (a + b − 1)(a + b − 2)/2, while x has arithmetic genus γ = ab − a − b + 1. we silently use a small modification of remark 3.1 to get f(x) = ∅ (for a general partial smoothing x) knowing that f(d) = ∅. we use lemmas 3.1, 3.2, 3.3 to get t (d) = ∅ and hence (remark 3.1) we get t (x) = ∅. (b1) assume for the moment that q is not in the tangent space of one of the nodes of x. call oi, 1 ≤ i ≤ s, the points of xreg such that q ∈ toix. the following observation summarize lemmas 3.1, 3.2, 3.3, 3.4, 3.6, 3.7, 3.9, 3.10 first on d and then on x. observation 1: x has no flex, its osculating planes have weight 1 and each point of p3 \ q is contained in at most 3 tangent lines to smooth points of x. a dimensional count similar to the one needed to prove lemmas 3.2 and 3.7 gives the following cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 389 observation. observation 2: at each q ∈ p3 \ q such that there are 3 different smooth points p1, p2, p3 of xreg with q ∈ tpix, no tpi(x) is a stall. at each point of x at which there are 2 different smooth points p1, p2 of xreg with q ∈ tpix at most one among tp1x and tp2x is a stall. by observations 1 and 2 we have pa(ℓq(x)) ≤ pa(x) + 3. since ℓq(x) is a plane curve of degree a + b and pa(x) = ab − a − b + 1, we get (a + b − 1)(a + b − 2)/2 ≤ ab − a − b + 4, i.e. a2 + b2 ≤ a + b + 6, which is false if a = 1 and b ≥ 4 or a ≥ 2 and b ≥ 3. (b2) assume g < γ and that q is contained in at least one tangent plane at x at one of its points. first assume that q is contained in the tangent cone at one of the nodes, o, of x. for a general d (and hence a general partial smoothing x) no line in the tangent cones of x at its singular points are stalls and tangent cones at different singular points are disjoints. at most another singular point o′ of x has tangent plane containing q. now assume that q is not contained in any tangent cone at singular points. it is contained in at most 3 tangent spaces of x at its singular points and if at 3 it is not contained in any tangent line at a smooth point of x. we get a contradiction if (a + b − 1)(a + b − 2)/2 ≥ γ + 4, i.e. if a2 + b2 ≥ a + b + 8, which is true (for positive a, b) if and only if a + b ≥ 5. 4 curves in p4 let f1 ⊂ p 4 be a smooth and non-degenerate surface such that deg(f1) = 3. all such surfaces are projectively equivalent. the smooth or nodal curves we use to prove theorem 1.2 are contained in f1. the surface f1 is an embedding of the hirzebruch surface with the same name ([7, §v.2]). we have pic(f1) ∼= z 2 and we take as free generators of it the class f, of a fiber of the ruling of f1 and the section h of its ruling with negative self-intersection. we have h2 = −1, f2 = 0 and h · f = 1. we have of1(1) ∼= of1(h+2f) and h and the elements of the ruling |f| are the only lines contained in f1. each of1(ah + bf), b ≥ a ≥ 0, is globally generated; it is ample (and very ample, too) if and only if b > a > 0. fix d ∈ |ah + bf|, b ≥ a > 0. since ωf1 ∼= of1(−2h − 3f), the adjunction formula gives ωd ∼= od((a−2)h+(b−3)f). thus pa(d) = 1+ab−a(a+1)/2−b. for all b ≥ a−1 we have h1(of1(ah + bf)) = 0 and h 0(of1(ah + bf)) = ∑a i=0(b + 1 − i) = (2b + 2 − a)(a + 1)/2. remark 4.1. take any curve d ⊂ f1 and any line l ⊂ p 4 such that deg(d ∩ l) ≥ 3. since f1 is scheme-theoretically cut out by quadrics and d ⊂ f1, bezout theorem gives l ⊂ f1. lemma 4.1. fix an integer q ∈ p4 \ f1. then there is c ∈ |h + f| such that q ∈ 〈c〉. proof. since 3 is a prime integer and q /∈ f1, ℓq(f1) is an irreducible degree 3 ruled surface. this 390 edoardo ballico cubo 22, 3 (2020) surface has a double line l meeting all lines of the ruling of ℓq(f1) ([4, theorem 9.2.1]). thus there is a plane conic c ⊂ f1 (a priori even a double line) mapped by ℓq onto l. all conics c ⊂ f1 are elements of |h + f|. up to projective transformations there are exactly two degree 3 surfaces ℓq(f1), q ∈ p 4 \ f1, distinguished by the fact that the unique conic c ∈ |h + f| given by lemma 4.1 is smooth or not ([4, theorem 9.2.1]). proposition 4.1. let x ⊂ f1 ⊂ p 4 be a reduced and non-degenerate curve whose irreducible component have degrees at least 3. assume the following conditions: (1) ♯((l ∩ x)red) ≥ 2 for all l ∈ |f|; (2) ♯((h ∩ x)red) ≥ 2; (3) ♯((c ∩ x)red) ≥ 3 for all smooth c ∈ |h + f|. then rx(q) ≤ 3 for all q ∈ p 4. proof. the assumptions on the irreducible components of x is equivalent to assuming that x ∩ c contains no curve for all c ∈ |h + f|. first assume q ∈ f1. let l be the only element of |f| containing q. since l is a line and ♯((l ∩ x)red) ≥ 2, we have rx(q) ≤ 2. now assume q /∈ f1. take c ∈ |h + f| such that q ∈ 〈c〉. note that 〈c〉 is a plane. if c is smooth (and hence it is a smooth conic), we have rx(q) ≤ 3, because ♯((c ∩ x)red) ≥ 3 and hence (c ∩x)red spans 〈c〉. now assume that c is singular, i.e. c = h+l for some l ∈ |f|. both h and l are lines and h ∩ l is a single point. by assumption there are p1, p2 ∈ (l ∩ x)red with p1 6= p2 and hence l = 〈{p1, p2}〉. since ♯((h ∩ x)red) ≥ 2, there is p3 ∈ (h ∩ x)red such that p3 6= h ∩ l. since h = 〈{p3, h ∩ l}〉, we have 〈c〉 = 〈{p1, p2, p3}〉 and hence rx(q) ≤ 3. lemma 4.2. let ∆ a quasi-projective variety and x ⊂ f1 × ∆ a closed algebraic set such that the restriction u : x −→ ∆ to x of the projection f1 × ∆ −→ ∆ is proper and flat. for each t ∈ ∆ set xt := u −1(t). assume that all fibers of u are reduced curves with no irreducible component of degree ≤ 2. fix o ∈ ∆ and assume ♯((c ∩ xo)red) ≥ 3 for all c ∈ |h + f|. then for a general t ∈ ∆ we have ♯((c ∩ xt)red) ≥ 3 for all c ∈ |h + f|. proof. assume that the lemma is false. taking a neighborhood ω of o in ∆ and then a branch covering of ω we may assume that for each t ∈ ω\{o} there is ct ∈ |h+f| with ♯((ct ∩xt)red) ≤ 2. since |h + f| is a projective set, the family {ct}t∈ω\{o} has at least one limit point, c ′, and ♯((c′ ∩ xo)red) ≤ 2. cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 391 lemma 4.3. fix integers a, b such that either a = 1 and b ≥ 5 or a ≥ 2 and b ≥ max{4, a}. let x be a general element of |ah + bf|. then ♯((c ∩ x)red) ≥ 3 for all smooth c ∈ |h + f|. proof. for each c ∈ |h + f| we have deg(x ∩ c) = b. for e ∈ {1, 2} let u(e) denote the set of all degree zero-dimensional schemes z ⊂ f1 such that deg(z) = b, z has exactly e connected components and there is a smooth c ∈ |h+f| containing z. since each smooth c ∈ |h+f| has ∞e elements of u(e), we have dim u(e) = 2 + e. thus (since e ≤ 2) to prove the lemma it is sufficient to prove that dim |iz(ah + bf)| = dim |ah + bf| − 5 for all z ∈ u(e), i = 1, 2. fix z ∈ u(e) and take a smooth c ∈ |h + f| containing it. since deg(z) = b ≥ 5, it is sufficient to prove that h1(iz(ah + bf)) = 0. since h 1(of1((a − 1)h + (b − 1)f)) = 0, the residual exact sequence of c shows that it is sufficient to prove that h1(c, iz,c(ah + bf)) = 0. this is true, because c ∼= p 1 and deg(oc(ah + bf)) = b. lemma 4.4. fix q ∈ f1. there is a smooth c ∈ |h + f| such that q ∈ c if and only if q ∈ f1 \ h. proof. since h · (h + f) = 0, no irreducible c ∈ |h + f| (i.e. no smooth c ∈ |h + f|) meets h. now assume q ∈ |h + f|. since dim |iq(h + f)| = dim |h + f| − 1 = 1 and there is a unique singular element of |h + f| containing q, there is a smooth c ∈ |h + f| such that q ∈ c. proposition 4.2. fix integer a, b such that a ≥ 1 and b ≥ 2a + 3. (1) there is a nodal d ∈ |ah+bf| with exactly a smooth irreducible components, all of them rational and neither lines nor conics, such that ♯((d ∩ c)red) ≥ 3 for all c ∈ |h + f|. (2) if a ≥ 2 we have rd(q) ≤ 3 for all q ∈ p 4. proof. set bi := 2 for 2 ≤ i ≤ a and b1 := b − 2a + 2. take a general (d1, . . . , da) ∈ ∏a i=1 |h + bif| and set d := d1 ∪ · · · ∪ da. by bertini’s theorem each di is smooth and connected and d is nodal. set s := sing(d). each di is rational and pa(d) = 1 + ab − a(a + 1)/2 − b. thus ♯(s) = pa(d) + a − 1 = ab − a(a − 1)/2 − b. in the case a = 1 we have d = d1 with d1 a general element of |h + bf|. for a general (d1, . . . , da) the nodal curve d is transversal to h and hence ♯((h ∩ d)red) = b − a ≥ 4. hence part (1) is true for all singular c ∈ |h + f|. now we check part (1) for all smooth c ∈ |h + f|. if a = 1 it is sufficient to quote lemma 4.3. now assume a ≥ 2. since ♯((d1 ∩ c)red) ≥ 3 by lemma 4.3, we get part (1) for all smooth c ∈ |h + f|. now we prove part (2). by lemma 4.4 we have rd(q) ≤ 3 for all q ∈ f1\h. since ♯((h∩d)red) = b − a ≥ 2, we have rd(q) ≤ 2 for all q ∈ h. 392 edoardo ballico cubo 22, 3 (2020) take q ∈ p4 \ f1. take c ∈ |h + f| such that q ∈ 〈c〉 (lemma 4.1). if c is smooth we get rd(q) ≤ 3 by proposition 4.1. now assume c singular, say c = h ∪ l with l ∈ |f|. since d contains b − a points of d, it is sufficient to prove that l contains a point of d \ d ∩ h. this is true, because a ≥ 2 and d is transversal to h. proof of theorem 1.2: take the curve d given by proposition 4.2. use remark 2.1 to get x as in the proof of theorem 1.1. apply part (1) of proposition 4.2 and lemma 4.2. cubo 22, 3 (2020) curves in low dimensional projective spaces with the lowest ranks 393 references [1] b. ådlandsvik, “joins and higher secant varieties”, math. scand., vol. 62, pp. 213–222, 1987. [2] g. blekherman, z. teitler, “on maximum, typical and generic ranks”, math. ann., vol. 362, no. 3-4, pp. 1231–1031, 2015. [3] j. buczyński, k. han, m. mella, z. teitler, “on the locus of points of high rank”, eur. j. math., vol. 4, pp. 113–136, 2018. [4] i. v. dolgachev, classical algebraic geometry. a modern view, cambridge university press, cambridge, 2012. [5] j. p. griffiths, j. harris, principles of algebraic geometry, john wiley & sons, new york, 1978. [6] j. harris (with d. eisenbud): curves in projective space, les presses de l’université de montréal, montréal, 1982. [7] r. hartshorne, algebraic geometry, springer-verlag, berlin–heidelberg–new york, 1977. [8] j. m. landsberg, z. teitler, “on the ranks and border ranks of symmetric tensors”, found. comput. math., vol. 10, pp. 339–366, 2010. [9] p. piene, “cuspidal projections of space curves”, math. ann., vol. 256, no. 1, pp. 95–119, 1981. [10] a. tannenbaum, “families of algebraic curves with nodes”, composition math., vol. 41, no. 1, pp. 107–126, 1980. introduction preliminaries proof of theorem 1.1 curves in p4 cubo a mathematical journal vol.20, no¯ 02, (41–52). june 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000200041 on new types of sets via γ-open sets in (a)topological spaces b. k. tyagi1, sheetal luthra2 and harsh v. s. chauhan2 1department of mathematics, atmaram sanatan dharma college, university of delhi, new delhi-110021, india. 2department of mathematics, university of delhi, new delhi-110007, india. brijkishore.tyagi@gmail.com,premarora550@gmail.com, harsh.chauhan111@gmail.com abstract in this paper, we introduced the notion of γ-semi-open sets and γ-p-semi-open sets in (a)topological spaces which is a set equipped with countable number of topologies. several properties of these notions are discussed. resumen en este art́ıculo, introducimos la noción de conjuntos γ-semi-abiertos y conjuntos γ-psemi-abiertos en espacios (a)topológicos, el cual es un conjunto dotado con una cantidad numerable de topoloǵıas. discutimos diversas propiedades de estas nociones. keywords and phrases: (a)topological spaces, (a)-γ-semi-open sets, (a)-γ-p-semi-open sets. 2010 ams mathematics subject classification: 54a05, 54e55, 54a10. http://dx.doi.org/10.4067/s0719-06462018000200041 42 b. k. tyagi, sheetal luthra and harsh v. s. chauhan cubo 20, 2 (2018) 1 introduction the notion of bitopological space (x, τ1, τ2) (a non empty set x endowed with two topologies τ1 and τ2) is introduced by kelly [5]. kovár [7, 8] also studied the properties of a non empty set equipped with three topologies. many authors studied a countable number of topologies in (ω)topological spaces and (ℵ0)topological spaces in [1, 2, 3, 4]. ogata [9] defined an operation γ on a topological space (x, τ) as a mapping from τ into the power set p(x) of x such that u ⊆ γ(u) for each u ∈ τ, where γ(u) denotes the value of γ at u. a susbet a of x is said to be γ-open if for each x ∈ a, there exists an open set u containing x such that γ(u) ⊆ a. in topological spaces, γ-p-open set are defined by khalaf and ibrahim [6]. the main purpose of this paper is to introduce the concept of γ-p-semi-open sets and γ-semi-open sets in (a)topological spaces. we give some properties related to these sets and introduce some separation axioms in (a)topological spaces. further we define new types of functions in (a)topological spaces, namely (a)-γ-semicontinuous and (a)-γ-p-semi-continuous. an operation γ on (a)topological space (x, {τn}) is a mapping γ : ⋃ τn → p(x) such that u ⊆ γ(u) for each u ∈ ⋃ τn. throughout the paper, n denotes the set of natural numbers. the elements of n are denoted by i, m, n etc. µ stands for the discrete topology. the (τn)-closure (resp. (τn)-interior) of a set a is denoted by τn-cl(a) (resp. τn-int(a)). by τmγ-int(a) and τmγ-cl(a), we denote the τmγ-interior of a and τmγ-closure of a in (x, {τn}), respectively. if there is no scope of confusion, we denote the (a)topological space (x, {τn}) by x. 2 (a)topological spaces definition 2.1. [10] if {τn} is a sequence of topologies on a set x, then the pair (x, {τn}) is called an (a)topological space. definition 2.2. [9] a susbet a of x is said to be γ-open if for each x ∈ a, there exists an open set u containing x such that γ(u) ⊆ a. definition 2.3. let x be an (a)topological space. a subset s of x is said to be: (i). (m, n)-semi-open if s ⊆ τm-cl(τn-int(s)). (ii). (m, n)-γ-semi-open if s ⊆ τmγ-cl(τnγ-int(s)). (iii). (m, n)-γ-p-semi-open if s ⊆ τm-cl(τnγ-int(s)). the complements of (m, n)-semi-open set, (m, n)-γ-semi-open set and (m, n)-γ-p-semi-open set are (m, n)-semi-closed, (m, n)-γ-semi-closed and (m, n)-γ-p-semi-closed, respectively. definition 2.4. let x be an (a)topological space. a subset s of x is said to be: (i). (a)-semi-open if s is (m, n)-semi-open for all m 6= n. cubo 20, 2 (2018) on new types of sets via γ-open sets in (a)topological spaces. 43 (ii). (a)-γ-semi-open if s is (m, n)-γ-semi-open for all m 6= n. (iii). (a)-γ-p-semi-open if s is (m, n)-γ-p-semi-open for all m 6= n. the complements of (a)-semi-open set, (a)-γ-semi-open set and (a)-γ-p-semi-open set are (a)-semi-closed, (a)-γ-semi-closed and (a)-γ-p-semi-closed, respectively. by so(x), γso(x) and γpso(x), we denote the family of all (a)-semi-open sets, (a)-γ-semiopen sets and (a)-γ-p-semi-open sets in x, respectively. theorem 2.1. every (a)-γ-p-semi-open set is (a)-γ-semi-open. proof. let s be an (a)-γ-p-semi-open set. then s is (m, n)-γ-p-semi-open for all m 6= n. so s ⊆ τm-cl(τnγ-int(s)) ⊆ τmγ-cl(τnγ-int(s)) for all m 6= n. this implies that s is (m, n)-γ-semiopen for all m 6= n. thus, s is (a)-γ-semi-open. the following example shows that the converse of the above theorem is not true generally. example 2.5. consider x = {a, b, c, d} with topologies τ1 = {x, ∅, {b}, {d}, {b, d}, {a, b, c}}, τ2 = {x, ∅, {a}, {d}, {a, d}, {a, b}, {a, b, d}} and τi = µ for i 6= 1, 2. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u = {d} x, if u 6= {d} then {b, c, d} is (a)-γ-semi-open but it is not (a)-γ-p-semi-open. theorem 2.2. every (a)-γ-p-semi-open set is (a)-semi-open. proof. let s be an (a)-γ-p-semi-open set. then s is (m, n)-γ-p-semi-open for all m 6= n. so s ⊆ τm-cl(τnγ-int(s)) ⊆ τm-cl(τn-int(s)) for all m 6= n. this implies that s is (m, n)-semi-open for all m 6= n. thus, s is (a)-semi-open. the following example shows that the converse of the above theorem is not true generally. example 2.6. let x, τ1 and γ be as in example 2.6. and let τi = τ2 for all i 6= 1. then {a, b, c} is (a)-semi-open but not (a)-γ-p-semi-open. following example shows that there is no relation between (a)-semi-open sets and (a)-γ-semiopen sets. example 2.7. let (x, {τn}) and γ be as in example 2.8. then {a, b, c} is (a)-semi-open but not (a)-γ-semi-open and {b, d} is (a)-γ-semi-open but not (a)semi-open. following example shows that (a)-γ-p-semi-open set need not be τi-open set. 44 b. k. tyagi, sheetal luthra and harsh v. s. chauhan cubo 20, 2 (2018) example 2.8. consider x = {a, b, c, d} with topologies τ1 = {x, ∅, {a}, {d}, {a, d}, {a, b}, {a, b, d}}, τi = {x, ∅, {b}, {d}, {b, d}} for all i 6= 1. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u = {d} x, if u 6= {d} then {c, d} is (a)-γ-p-semi-open but is not τi-open. following example shows that (a)-γ-p-semi-open set need not be γi-open set. example 2.9. let (x, {τn}) and γ be as in example 2.10. then {c, d} is (a)-γ-p-semi-open but not γi-open . theorem 2.3. let {sα : α ∈ λ} be a class of (a)-γ-p-semi-open sets. then ⋃ α∈λ sα is also an (a)-γ-p-semi-open set. proof. since each sα is an (a)-γ-p-semi-open set, sα is (m, n)-γ-p-semi-open for all α ∈ λ and for all m 6= n. we have sα ⊆ τm-cl(τnγ-int(sα)) for all α ∈ λ and for all m 6= n. hence, it is obtained ⋃ α∈λ sα ⊆ ⋃ α∈λ τm-cl(τnγ-int(sα)) ⊆ τm-cl( ⋃ α∈λ τnγ-int(sα)) ⊆ τm-cl(τnγ-int( ⋃ α∈λ sα)). therefore, ⋃ α∈λ sα is also an (a)-γ-p-semi-open set. following example shows that the intersection of two (a)-γ-p-semi-open sets need not be again (a)-γ-p-semi-open. example 2.10. consider x = {a, b, c, d} with topologies τ1 = {x, ∅, {c}, {d}, {c, d}}, τi = {x, ∅, {c}, {d}, {c, d}, {b, c, d}} for all i 6= 1. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u ∈ {{c}, {d}} x, if u 6∈ {{c}, {d}} then {b, c} and {b, d} are (a)-γ-p-semi-open but their intersection {b} is not (a)-γ-p-semi-open. theorem 2.4. a subset f is (a)-γ-p-semi-closed in (a)topological space (x, {τn}) if and only if τm-int(τnγ-cl(f)) ⊆ f for all m 6= n. proof. let f be an (a)-γ-p-semi-closed set in x. then x\f is (a)-γ-p-semi-open, so x\f ⊆ τmcl(τnγ-int(x\f)) for all m 6= n. cubo 20, 2 (2018) on new types of sets via γ-open sets in (a)topological spaces. 45 it follows that f ⊇ x\τm-cl(τnγ-int(x\f)) = τm-int(x\τnγ-int(x\f)) = τm-int(τnγ-cl(f)). conversely, for all m 6= n, we obtain x\f ⊆ x\τm-int(τnγ-cl(f)) = τm-cl(x\τnγ-cl(f)) = τm-cl(τnγ-int(x\f)). which completes the proof. theorem 2.5. let {fα : α ∈ λ} be a class of (a)-γ-p-semi-closed sets. then ⋂ α∈λ fα is also an (a)-γ-p-semi-closed. proof. for each α ∈ λ, fα is an (a)-γ-p-semi-closed set. this implies that x\fα is an (a)-γ-psemi open set. by theorem 2.12., ⋃ α∈λ x\fα is an (a)-γ-p-semi open set. by de morgan’s law, x\ ⋂ α∈λ fα is an (a)-γ-p-semi open set. thus, ⋂ α∈λ fα is an (a)-γ-p-semi-closed set. following example shows that the union of two (a)-γ-p-semi-closed sets need not be (a)-γ-psemi-closed. example 2.11. let (x, {τn}) and γ be as in example 2.13. then {a, c} and {a, d} are (a)-γ-p-semi-closed but their union {a, c, d} is not (a)-γ-p-semi-closed. definition 2.12. in an (a)topological space x, a point x of x is said to be (a)-γ-p-semi interior ((a)-γ-semi interior) point of s if there exists an (a)-γ-p-semi-open ((a)-γ-semi-open) set v such that x ∈ v ⊆ s. by (a)-γ-ps-int(a) (resp.(a)-γ-s-int(a)), we denote the (a)-γ-ps-interior (resp.(a)-γ-sinterior) of a consisting of all (a)-γ-p-semi interior ((a)-γ-semi interior) points of a. theorem 2.6. the following properties hold for any subset a of (a)topological space x : (i). (a)-γ-ps-int(a) is the union of all (a)-γ-p-semi-open sets ( the largest (a)-γ-p-semi-open set) contained in a. (ii). (a)-γ-ps-int(a) is an (a)-γ-p-semi-open set. (iii). a is (a)-γ-p-semi-open if and only if a = (a)-γ-ps-int(a). proof. the proof follows from definitions. 46 b. k. tyagi, sheetal luthra and harsh v. s. chauhan cubo 20, 2 (2018) theorem 2.7. the following properties hold for any subsets a1, a2 and any class of subsets {aα : α ∈ λ} of (a)topological space x : (i). if a1 ⊆ a2, then (a)-γ-ps-int(a1) ⊆ (a)-γ-ps-int(a2). (ii). ⋃ α∈λ (a)-γ-ps-int(aα) ⊆ (a)-γ-ps-int( ⋃ α∈λ aα). (iii). (a)-γ-ps-int( ⋂ α∈λ aα) ⊆ ⋂ α∈λ (a)-γ-ps-int(aα). proof. (i). since a1 ⊆ a2, (a)-γ-ps-int(a1) is an (a)-γ-p-semi-open set contained in a2. but (a)-γ-ps-int(a2) is the largest (a)-γ-p-semi-open set contained in a2. so (a)-γ-psint(a1) ⊆ (a)-γ-ps-int(a2). (ii). from (i), we have (a)-γ-ps-int(aα) ⊆ (a)-γ-ps-int( ⋃ α∈λ aα) for all α ∈ λ. hence,⋃ α∈λ(a)-γ-ps-int(aα) ⊆ (a)-γ-ps-int( ⋃ α∈λ aα). (iii). from (i), (a)-γ-ps-int( ⋂ α∈λ aα) ⊆ (a)-γ-ps-int(aα) for all α ∈ λ. hence, (a)-γ-psint( ⋂ α∈λ aα) ⊆ ⋂ α∈λ (a)-γ-ps-int(aα). the reverse inclusion in (ii) and (iii) of theorem 2.19. may not be applicable as shown in the following examples. example 2.13. consider x = {a, b, c} with topologies τ1 = {x, ∅, {a}, {b, c}}, τi = {x, ∅, {b}} for all i 6= 1. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u = {c} x, if u 6= {c} {a, b, c} = (a)-γ-ps-int{a, b, c} * (a)-γ-ps-int{a} ∪ (a)-γ-ps-int{b, c} = ∅. example 2.14. let (x, {τn}) and γ be as in example 2.13. {b} = (a)-γ-ps-int{b, c} ∩ (a)-γ-ps-int{b, d} * (a)-γ-ps-int{b} = ∅. definition 2.15. in an (a)topological space x, a point x of x is said to be (a)-γ-p-semi cluster ((a)-γ-semi cluster) point of a subset a ⊂ x if a ∩ v 6= ∅ for every (a)-γ-p-semi-open ((a)-γsemi-open set) containing x. by (a)-γ-ps-cl(a) (resp.(a)-γ-s-cl(a)), we denote the (a)-γ-ps-closure (resp.(a)-γ-s-closure) of a consisting of all (a)-γ-p-semi cluster ((a)-γ-semi cluster) points of a. theorem 2.8. the following properties hold for any subset a of an (a)topological space x : (i). (a)-γ-ps-cl(a) is the intersection of all (a)-γ-p-semi-closed sets ( the smallest (a)-γ-psemi-closed set) containing a. cubo 20, 2 (2018) on new types of sets via γ-open sets in (a)topological spaces. 47 (ii). (a)-γ-ps-cl(a) is an (a)-γ-p-semi-closed set. (iii). a is (a)-γ-p-semi-closed if and only if a = (a)-γ-ps-cl(a). proof. the proof follows from definitions. theorem 2.9. the following properties hold for any subsets a1, a2 and any class of subsets {aα : α ∈ λ} of an (a)topological space x: (i). if a1 ⊆ a2, then (a)-γ-ps-cl(a1) ⊆ (a)-γ-ps-cl(a2). (ii). ⋃ α∈λ (a)-γ-ps-cl(aα) ⊆ (a)-γ-ps-cl( ⋃ α∈λ aα). (iii). (a)-γ-ps-cl( ⋂ α∈λ aα) ⊆ ⋂ α∈λ (a)-γ-ps-cl(aα). proof. (i). since a1 ⊆ a2, (a)-γ-ps-cl(a2) is an (a)-γ-p-semi-closed set containing a1. but (a)γ-ps-cl(a1) is the smallest (a)-γ-p-semi-closed set containing a1. so (a)-γ-ps-cl(a1) ⊆ (a)γ-ps-cl(a2). (ii). from (i), (a)-γ-ps-cl(aα) ⊆ (a)-γ-ps-cl( ⋃ α∈λ aα) for all α ∈ λ. hence, ⋃ α∈λ (a)-γ-pscl(aα) ⊆ (a)-γ-ps-cl( ⋃ α∈λ aα). (iii). from (i), (a)-γ-ps-cl( ⋂ α∈λ aα) ⊆ (a)-γ-ps-cl(aα) for all α ∈ λ. hence, (a)-γ-pscl( ⋂ α∈λ aα) ⊆ ⋂ α∈λ (a)-γ-ps-cl(aα). the reverse inclusion in (ii) and (iii) of theorem 2.24 may not be applicable as shown in the following examples. example 2.16. let (x, {τn}) and γ be as in example 2.13. {a, b, c, d} = (a)-γ-ps-cl{a, c, d} * (a)-γ-ps-cl{a, c} ∪ (a)-γ-ps-cl{a, d} = {a}. example 2.17. consider x = {a, b, c} with topologies τ1 = {x, ∅, {a}, {a, b}} and τi = {x, ∅, {b}, {a, b}} for all i 6= 1. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u = {a, b} x, if u 6= {a, b} {a, b, c} = (a)-γ-ps-cl{a, c} ∩ (a)-γ-ps-cl{b, c} * (a)-γ-ps-cl{c} = {c}. theorem 2.10. the following properties hold for a subset a of an (a)topological space x: (i). (a)-γ-ps-int(x\a) = x\(a)-γ-ps-cl(a). (ii). (a)-γ-ps-cl(x\a) = x\(a)-γ-ps-int(a). 48 b. k. tyagi, sheetal luthra and harsh v. s. chauhan cubo 20, 2 (2018) proof. 1. by part (i). of theorem 2.18., we have (a)-γ-ps-int(x\a) = ⋃ {s ⊂ x: s is (a)-γ-p-semi-open and s ⊂ x\a} = ⋃ {x\(x\s) ⊂ x: x\s is (a)-γ-p-semi-closed and a ⊂ x\s} = x\ ⋂ {x\s ⊂ x: x\s is (a)-γ-p-semi-closed and a ⊂ x\s} = x\ ⋂ {f ⊂ x: f is (a)-γ-p-semi-closed and a ⊂ f} = x\(a)-γ-ps-cl(a). 2. by part (i). of theorem 2.23., we have (a)-γ-ps-cl(x\a) = ⋂ {s ⊂ x: s is (a)-γ-p-semi-closed and x\a ⊂ s} = ⋂ {x\(x\s) ⊂ x: x\s is (a)-γ-p-semi-open and x\s ⊂ a} = x\ ⋃ {x\s ⊂ x: x\s is (a)-γ-p-semi-open and x\s ⊂ a} = x\ ⋃ {f ⊂ x: x\f is (a)-γ-p-semi-open and f ⊂ a} = x\(a)-γ-ps-int(a). definition 2.18. a set a is said to be (a)-γ-p-semi neighborhood of a point x in an (a)topological space x if there exists an (a)-γ-p-semi-open set u such that x ∈ u ⊆ a. theorem 2.11. a subset of an (a)topological space x is (a)-γ-p-semi-open if and only if it is (a)-γ-p-semi neighborhood of each of its points. proof. the proof follows from definition 2.28. definition 2.19. an (a)topological space x is said to be (a)-γ-ps-t0 if for every distinct points x and y of x, there exists an (a)-γ-p-semi-open set u such that x ∈ u but y 6∈ u or vice versa. theorem 2.12. an (a)topological space x is (a)-γ-ps-t0 if and only if for each distinct points x and y of x (a)-γ-ps-cl{x} 6= (a)-γ-ps-cl{y}. proof. let x and y be any two distinct points of x. then there exists an (a)-γ-p-semi-open set u such that x ∈ u but y 6∈ u or vice versa. without loss of generality, assume that u containing x but not y. then we have {y} ∩ u = ∅ which implies x 6∈ (a)-γ-ps-cl{y}. hence, (a)-γ-pscl{x} 6= (a)-γ-ps-cl{y}. conversely, let x and y be any two distinct points of x. then we have (a)-γ-ps-cl{x} 6= (a)-γps-cl{y}. without loss of generality let z ∈ (a)-γ-ps-cl{y} but z 6∈ (a)-γ-ps-cl{x}. then {y}∩u 6= ∅ for every (a)-γ-p-semi-open set u containing z and {x} ∩ u = ∅ for atleast one (a)-γ-p-semi-open set u containing z. thus, y ∈ u and x 6∈ u. hence, x is (a)-γ-ps-t0. cubo 20, 2 (2018) on new types of sets via γ-open sets in (a)topological spaces. 49 definition 2.20. an (a)topological space (x, {τn}) is said to be (a)-γ-ps-t1 if for every distinct points x and y of x, there exist two (a)-γ-p-semi-open sets which one of them contains x but not y and the other one contains y but not x. theorem 2.13. an (a)topological space x is (a)-γ-ps-t1 if and only if for each point x of x (a)-γ-ps-cl{x} = {x}. proof. since {x} ⊆ (a)-γ-ps-cl{x}, let y ∈ (a)-γ-ps-cl{x} be arbitrary. on contrary suppose that y 6∈ {x}. then there exists an (a)-γ-p-semi-open set u such that y ∈ u but x 6∈ u. then we have {x} ∩ u = ∅ which implies y 6∈ (a)-γ-ps-cl{x}. hence, contradiction. conversely, let x 6= y for x, y ∈ x. since x 6∈ (a)-γ-ps-cl{y} and y 6∈ (a)-γ-ps-cl{x}, there exist (a)γ-p-semi-open sets u and v containing x and y, respectively such that {y}∩u = ∅ and {x}∩v = ∅. thus, we have x ∈ u, y 6∈ u and y ∈ v, x 6∈ v. hence, x is (a)-γ-ps-t1. definition 2.21. an (a)topological space x is said to be (a)-γ-ps-t2 if for every distinct points x and y of x, there exist two disjoint (a)-γ-p-semi-open sets u and v containing x and y, respectively. theorem 2.14. an (a)topological space x is (a)-γ-ps-t2 if and only if for each distinct points x and y of x there exists an (a)-γ-p-semi-open set u containing x such that y 6∈ (a)-γ-ps-cl(u). proof. let x be an (a)-γ-ps-t2 space. on contrary suppose that y ∈ (a)-γ-ps-cl(u) for all (a)γ-p-semi-open set u containing x. then u ∩ v 6= ∅ for every (a)-γ-p-semi-open set v containing y and (a)-γ-p-semi-open set u containing x. thus, contradiction. conversely, let x and y be any two distinct point of x. then there exist two disjoint (a)-γ-psemi-open sets u and v containing x and y, respectively. this implies that {y} ∩ u = ∅. hence, y 6∈ (a)-γ-ps-cl(u). theorem 2.15. an (a)topological space x is (a)-γ-ps-t2 if and only if the intersection of all (a)-γ-ps-closed neighborhood of each point of x consists of only that point. proof. let x ∈ x be arbitrary and y ∈ x such that y 6= x. then there exist disjoint (a)-γp-semi-open sets uy and vy containing x and y, respectively. since uy ⊆ x\vy, x\vy is an (a)-γ-ps-closed neighborhood of x which does not contain y. hence, ∩{x\vy : y ∈ x, y 6= x} = {x}. conversely, let x and y be any two distinct points of x. since {x} = ∩{s ⊂ x: s is (a)-γ-ps-closed neighborhood of x}. this implies that there exists an (a)-γ-ps-closed neighborhood u of x not containing y. then, y ∈ x\u and x\u is (a)-γ-p-semi-open. since, u is an (a)-γ-ps-neighborhood of x, then there exists an (a)-γ-p-semi-open set v containing x such that v ⊆ u. clearly, v and x\u are disjoint. hence, (x, {τn}) is (a)-γ-ps-t2. remark 2.22. (i). every (a)-γ-ps-t2 (a)topological space is (a)-γ-ps-t1. (ii). every (a)-γ-ps-t1 (a)topological space is (a)-γ-ps-t0. following examples shows that converse of above remark need not be true. 50 b. k. tyagi, sheetal luthra and harsh v. s. chauhan cubo 20, 2 (2018) example 2.23. let x = {a, b, c, d} with topologies τ1 = {∅, x, {c}, {d}, {c, d} {a, b, c}} and τi = {∅, x, {c}, {d}, {c, d}} for all i 6= 1. γ(u) = { u, if u ∈ {{c}, {d}, {a, b, c}} x, if u 6∈ {{c}, {d}, {a, b, c}} then τnγ = τn for all n ∈ n and (a)-γ-pso = {∅, x, {c}, {d}, {c, d} {a, b, c}, {a, c}, {b, c}}. clearly, (x, {τn}) is (a)-γ-ps-t0 but not (a)-γ-ps-t1. example 2.24. let x = {a, b, c} with topologies τn = µ for all n. γ(u) = { u, if u ∈ {{a, b}, {a, c}, {b, c}} x, if u 6∈ {{a, b}, {a, c}, {b, c}} then τnγ = {∅, x, {a, b}, {a, c}, {b, c}} for all n ∈ n and (a)-γ-pso = {∅, x, {a, b}, {a, c}, {b, c}}. clearly (x, {τn}) is (a)-γ-ps-t1 but not (a)-γ-ps-t2. example 2.25. let x = {a, b, c} with topologies τn = µ for all n. γ(u) = { u, if u ∈ {{a}, {b}, {c}} x, if u 6∈ {{a}, {b}, {c}} then τnγ = µ for all n ∈ n and (a)-γ-pso = µ clearly x is (a)-γ-ps-t2 space. definition 2.26. let f : (x, {τn}) → (y, {ζn}) be a function and x be any point of x. f is said to be (a)-γ-p-semi continuous (resp.(a)-γ-semi continuous) at x if for every ζn open subset o of y containing f(x) there exists an (a)-γ-p-semi-open (resp. (a)-γ-semi-open) set g of x containing x such that f(g) ⊆ o. theorem 2.16. for a function f : (x, {τn}) → (y, {ζn}), the followings statements are equivalent : (i). f is (a)-γ-p-semi continuous (resp.(a)-γ-semi continuous). (ii). for every ζn open subset o of y, f −1(o) is an (a)-γ-p-semi-open (resp.(a)-γ-semi-open) set in x. (iii). for every ζn closed subset f of y, f −1(f) is an (a)-γ-p-semi-closed (resp.(a)-γ-semi-closed) set in x. (iv). for every subset t of x, f((a)-γ-ps-cl(t)) ⊆ ζn-cl(f(t)) (resp. f((a)-γ-s-cl(t)) ⊆ ζncl(f(t)). (v). for every subset f of y, (a)-γ-ps-cl(f−1f) ⊆ f−1(ζn-cl(f))(resp. (a)-γ-s-cl(f −1f) ⊆ f−1(ζncl(f)). cubo 20, 2 (2018) on new types of sets via γ-open sets in (a)topological spaces. 51 proof. (1). =⇒ (ii). let o be ζn open in y and x ∈ f−1(o) be arbitrary. since f is (a)-γ-p-semi continuous on x, there exists an (a)-γ-p-semi-open set g of x containing x such that f(g) ⊆ o. thus, we have g ⊆ f−1(o). hence, f−1(o) is an (a)-γ-p-semi-open set in x. (ii). =⇒ (i). let x be any point of x and h be a ζn open set containing f(x). we get f−1(h) is (a)-γ-p-semi-open and x ∈ f−1(h). take g = f−1(h), we have f(g) ⊆ h. hence, f is (a)-γ-p-semi continuous. (ii) ⇐⇒ (iii). obviously. (i). =⇒ (iv). let t be a subset of x and f(x) ∈ f((a)-γ-ps-cl(t)), for x ∈ (a)-γ-ps-cl(t). let h be any ζn open set of y containing f(x). by hypothesis there exists an (a)-γ-p-semi-open set g of x containing x such that f(g) ⊆ h. since g ∩ t 6= ∅, h ∩ f(t) 6= ∅. this implies that f(x) ∈ ζn-cl(f(t)). hence, (a)-γ-ps-cl(f −1f) ⊆ f−1(ζn-cl(f)). (iv). =⇒ (v). let f be a subset of y. by hypothesis, we have f((a)-γ-ps-cl(f)) ⊆ ζn-cl(f(f)). taking the pre-image on both sides, we get (a)-γ-ps-cl(f−1f) ⊆ f−1(ζn-cl(f)). (v). =⇒ (iii). let f be ζn-closed in y. by hypothesis, we have (a)-γ-ps-cl(f−1f) ⊆ f−1(f). hence, f−1(f) is (a)-γ-p-semi-closed in x. corolary 1. (i). every (a)-γ-p-semi continuous function is (a)-γ-semi continuous. (ii). every (a)-γ-p-semi continuous function is (a)-semi continuous. following example shows that (a)-γ-semi continuous function need not be (a)-γ-p-semi continuous. example 2.27. consider x = {a, b, c, d} with topologies τ1 = {x, ∅, {b}, {d}, {b, d}}, τi = {x, ∅, {a}, {d}, {a, d}, {a, b}, {a, b, d}} for all i 6= 1. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u = {d} x, if u 6= {d} define f : (x, {τn}) → (x, {τn}) as f{a, b, d} = d, f(c) = c. then f is (a)-γ-semi continuous function but not (a)-γ-p-semi continuous as {a, b, d} is not (a)-γ-p-semi-open. example 2.28. consider x = {a, b, c, d} with topologies τ1 = {x, ∅, {b}, {d}, {b, d}, {a, b, c}}, τi = {x, ∅, {a}, {d}, {a, d}, {a, b}, {a, b, d}} for all i 6= 1. let γ be an operation on ⋃ τn defined as follows : γ(u) = { u, if u = {a}, {b} x, if u 6= {a}, {b} define f : (x, {τn}) → (x, {τn}) as f{a, b, c} = d, f(d) = c. then f is (a)-semi continuous function but not (a)-γ-p-semi continuous as {d} is not (a)-γ-p-semi-open. references [1] bose, m. k. and tiwari, r. (ω)topological connectedness and hyperconnectedness, note mat. 31, 93-101, 2011. 52 b. k. tyagi, sheetal luthra and harsh v. s. chauhan cubo 20, 2 (2018) [2] bose, m. k. and mukharjee, a. on countable families of topologies on a set, novi sad j. math. 40 (2), 7-16, 2010. [3] bose, m. k. and tiwari, r. on (ω)topological spaces, riv. mat. univ. parma.(7) 9, 125-132, 2008. [4] bose, m. k. and tiwari, r. on increasing sequences of topologies on a set, riv. mat. univ. parma. (7) 7, 173-183, 2007. [5] kelly, j. c., bitopological spaces, j. proc. london math. soc. (1963), 13, 71-89. [6] khalaf, a. b. and ibrahim, h. z., some applications of γ-p-open sets in topological spaces, int. j. pure appl. math. sci. (2011), 5(1-2), 81-96. [7] kovár, m. m. on 3-topological version of θ-regularity, int. j. math. math. sci. 23, 393-398, 2000. [8] kovár, m. m. a note on the comparison of topologies, int. j. math. math. sci. 26, 233-237, 2001. [9] ogata, h., operation on topological spaces and associated topology, math. japonica (1991), 36, 175-184. [10] roy choudhuri, a., mukharjee, a. and bose, m. k., hyperconnectedness and extremal disconnectedness in (a)topological spaces, hacettepe journal of mathematics and statistics 44(2), 289-294, 2015. introduction (a)topological spaces cubo a mathematical journal vol.19, no¯ 02, (11–31). june 2017 on the hypercontractive property of the dunkl-ornstein-uhlenbeck semigroup iris a. lópez 1 departamento de matemáticas puras y aplicadas, universidad simón bolivar, aptdo 89000. caracas 1080-a. venezuela. iathamaica@usb.ve abstract the aim of this paper is to prove the hypercontractive propertie of the dunkl-ornsteinuhlenbeck semigroup, {e(tlk)}t≥0. to this end, we prove that the dunkl-ornsteinuhlenbeck differential operator lk with k ≥ 0 and associated to the zd2 group, satisfies a curvature-dimension inequality, to be precise, a c(ρ,∞)-inequality, with 0 ≤ ρ ≤ 1. as an application of this fact, we get a version of meyer’s multipliers theorem and by means of this theorem and fractional derivatives, we obtain a characterization of dunkl-potential spaces. resumen el objetivo de este art́ıculo es demostrar la propiedad hipercontractiva del semigrupo de dunkl-ornstein-uhlenbeck, {e(tlk)}t≥0. para lograr esto, probamos que el operador diferencial de dunkl-ornstein-uhlenbeck lk con k ≥ 0 y asociado al grupo zd2, satisface una desigualdad de curvatura-dimensión, para ser precisos, una c(ρ,∞)-desigualdad, con 0 ≤ ρ ≤ 1. como una aplicación de este hecho, obtenemos una versión del teorema de multiplicadores de meyer y a través de este teorema y derivadas fraccionales, obtenemos una caracterización de espacios dunkl-potenciales. keywords and phrases: dunkl-ornstein-uhlenbeck operator, generalized hermite polynomial, squared field operator, meyer’s multiplier theorem, dunkl-potential space, fractional integral, fractional derivative. 2010 ams mathematics subject classification: 33c45, 6a33, 33c52. 1the author’s research was partially supported by did-usb-cb-004-17 12 iris a. lópez cubo 19, 2 (2017) 1 preliminaries in this section we collect some notations and results in the dunkl theory (see [5]), but particularly for the zd2 group. let ν = (ν1, . . . ,νd) ∈ zd+ be a multi-index, where z+ = {0,1,2, . . .}, so ν! = ∏d j=1 vj! and |ν| = ∑d j=1 νj. for x = (x1, . . . ,xd) ∈ rd, we set xν = x ν1 1 . . .x νd d and |x| 2 2 = ∑d j=1 x 2 j . in what follows, we denote ∂j = ∂/∂xj, for each 1 ≤ j ≤ d, and ∂ν = ∂ν11 . . .∂ νd d . also, 〈., .〉 denotes the euclidean inner product in rd and finally, △ and ∇ denote the usual laplacean and the usual gradient, respectively. let us consider the finite reflection group generated by σj(x) = x − 2 〈x,ej〉 |ej| 2 2 ej, where (ej) d j=1 are the standard unit vectors of r d. so, for each j = 1, . . . ,d, σj(x1, . . . ,xj, . . . ,xd) = (x1, . . . ,−xj, . . . ,xd) and isomorphic to zd2 = {0,1} d. the reflection σj is in the hyperplane orthogonal to ej. then, we consider the root system r and the positive root system r+, respectively, as r = {± √ 2ej : j = 1, . . . ,d}, r+ = { √ 2ej : j = 1, . . . ,d} and let k be a nonnegative multiplicity function k : r+ → [0,∞), which is z d 2-invariant. then, we set k = (k1, . . . ,kd), where kj = αj + (1/2) and αj ≥ −1/2, for each j = 1, . . . ,d. thus, in this particular case, the dunkl differential difference operators, tkj , are given by tkj f(x) = ∂jf(x) + kj ( f(x) − f(σjx) xj ) , j = 1, . . . ,d with f ∈ c1(rd) and in the following, the operator △k = d∑ j=1 (tkj ) 2, given explicitty by △kf(x) = d∑ j=1 { ∂2jf(x) + 2kj xj ∂jf(x) − kj ( f(x) − f(σjx) x2j )} is called ”the generalized laplacian” or ”dunkl-laplacian” associated to zd2 and k. then the dunkl-ornstein-uhlenbeck differential operator is defined as, lk = △k 2 − 〈x,∇x〉 (1.1) cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 13 and therefore, from (1.1), the dunkl-ornstein-uhlenbeck differential operator can be written as lkf(x) = d∑ j=1 1 2 { ∂2jf(x) + 2kj xj ∂jf(x) − kj ( f(x) − f(σjx) x2j )} − xj∂jf(x). (1.2) here, the corresponding weight function is defined by wk(x) = ∏d j=1 |xj| 2kj and we consider the hilbert space l2(mk), where the probability measure, mk, is defined by mk(dx) = ck exp(−|x| 2 2)wk(x)dx, with x ∈ rd and ck = (∫ rd exp(−|x|22)wk(x)dx )−1 . now, we consider a complete system of orthogonal polynomials, with respect to the measure mk, which is known as generalized hermite polynomials. in dimension one, for the reflection group z2, the corresponding generalized hermite polynomials are defined as { hk2n(x) = (−1) n22nn!l α−(1/2) n (x 2), hk2n+1(x) = (−1) n22n+1n!xl α+(1/2) n (x 2), where lαn are the laguerre polynomials of degree n and order α, (see [11] and [4, pages 156,157]). in the multidimensional case the generalized hermite polynomials are defined by taking tensor products of the one-dimensional hkn; that is, h k ν(x) = ∏d j=1 h kj νj(xj), x ∈ rd, ν ∈ zd+. this way, we will denote hkν = 2 −|ν|/2hkν, ν ∈ zd+, from now on. hkν is a polynomial of degree |ν| and {h k ν}ν∈zd + forms an orthonormal basis of l2(mk). the generalized hermite polynomials satisfy the following important identity which is known as mehler’s formula. for r ∈ c with |r| < 1, ∑ ν∈zd + hkν(x)h k ν(y)r |ν| = 1 (1 − r2)|k|+d/2 exp ( − r2(|x|22 + |y| 2 2) 1 − r2 ) ek ( 2rx 1 − r2 ,y ) , where the sum is absolutely convergent and the dunkl kernel, ek(x,y), replaces the usual exponential function, exp〈x,y〉. then, from [11] the generalized hermite polynomials are eigenfunctions of lk; lk(h k ν) = −|ν|h k ν, ∀ν ∈ zd+. (1.3) also, let ckn be the closed subspace of l 2(mk) generate by linear combination of {h k ν : |ν| = n} and we denote by jkn the orthogonal projection of l 2(mk) onto c k n. if f is a polynomial, then jknf = ∑ |ν|=n ckν(f)h k ν, where given a function f ∈ l2(mk), its dunkl-fourier coefficient is defined by ckν(f) = ∫ rd f(x)hkν(x)mk(dx) and therefore, if f ∈ l2(mk), its dunkl-hermite expansion is given by f = ∑ ∞ n=0 j k nf. thus, the operator lkf = ∞∑ n=0 −njknf, 14 iris a. lópez cubo 19, 2 (2017) defined on the domain d2(lk) = { f ∈ l2(mk) : ∑ ∞ n=0 ∑ |ν|=n |c k ν(f)| 2 < ∞ } , is a self-adjoint extension of lk considered on c ∞ c (r d). more precisely, lk has a clousure which also will be denoted by lk. now, following [11, 13], the generalized heat kernel, γk(t,x,y), is given by γk(t,x,y) = ck exp{−(|x| 2 2 + |y| 2 2)/4t} (4t)|k|+d/2 ek ( x√ 2t , y√ 2t ) , (1.4) where x,y ∈ rd and t > 0. therefore, from [12] the dunkl-ornstein-uhlenbeck integral operator is defined as oktf(x) = ∫ rd γk ( (1 − e−2t) 4 ,e−tx,y ) f(y)wk(y)dy. but, (1.4) allows us to express exp(|y|22)γk ( (1 − e−2t) 4 ,e−tx,y ) = ck exp ( − e−2t(|x|22+|y| 2 2) 1−e−2t ) (1 − e−2t)|k|+d/2 ek ( √ 2e−tx√ 1 − e−2t , √ 2e−ty√ 1 − e−2t ) , and since, ek ( √ 2e−tx√ 1−e−2t , √ 2e−ty√ 1−e−2t ) = ek ( 2e−tx 1−e−2t ,y ) , by using mehler formula, we get explicity: oktf(x) = ∫ rd f(y)okt(x,y)mk(dy), where, okt(x,y) = ∑ ν∈zd + e−|ν|thkν(x)h k ν(y). besides, {o k t}t≥0 is a positive, strongly continuous contraction semigroup on c0(r d) with generator lk. thus, formally, we write o k t = e (tlk) and following [12, 14], if we consider mkt(x,dy) = γk ( (1 − e−2t) 4 ,e−tx,y ) wk(y)dy, (1.5) form, (together with the trivial kernel mk0), a semigroup of markov kernels. also, the corresponding dunkl-poisson semigroup {pkt }t≥0 is defined, by means of subordination principle, as pktf(x) = 1√ π ∫ ∞ 0 exp(−u)√ u okt2/4uf(x)du = ∫ rd f(y)pkt (x,y)mk(dy), where the kernel pkt (x,y) is defined as pkt (x,y) = 1√ π ∫ ∞ 0 exp(−u)√ u okt2/4u(x,y)du. again, {pkt }t≥0 is a positive, strongly continuous semigroup with infinitesimal generator (−lk) 1/2 and it is a markov process (see [12, 16]). in particular, by (1.3) we obtain that okt(h k ν) = e −|ν|thkν and p k t (h k ν) = e −t √ |ν|hkν cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 15 also, if f is a polynomial oktf = ∑ n≥0 e−ntjknf and p k tf = ∑ n≥0 e−t √ njknf. following [2, 3], let us consider the squared field operator, γ, associated to lk, as γ(f,g) = 1 2 [lk(fg) − flk(g) − glk(f)], ∀f,g ∈ a, (1.6) where we choose a as the set of all polynomials on rd, which is a dense subspace in l2(mk). besides, let us consider the operator γ2 defined as γ2(f,g) = 1 2 [lkγ(f,g) − γ(f,lkg) − γ(g,lkf)], ∀f,g ∈ a × a (1.7) and throughout this paper we denote γ(f) = γ(f,f) and γ2(f) = γ2(f,f). again, motivated by [2, 3], we say that the differential operator lk satisfies a cd(ρ,n)inequality, (a curvature-dimension inequality with curvature ρ and dimension n), if and only if γ2(f) ≥ ργ(f) + 1 n (lkf) 2, ∀f ∈ a, where ρ ∈ r and n ∈ [1,∞]. particularly, lk satisfies a cd(ρ,∞)-inequality, if and only if γ2(f) ≥ ργ(f) ∀f ∈ a. finally, we denote a dirichley form associated to the measure mk by e(f) = ∫ γ(f)(x)mk(dx) and the entropy of a positive function f as ent(f) = ∫ f(x) log(f)(x)mk(dx) − ∫ f(x)mk(dx) log (∫ f(x)mk(dx) ) . in this case, a logarithmic sobolev inequality, ls(a,c), has the form ent(f2) ≤ a ∫ f2(x)mk(dx) + ce(f), ∀f ∈ a. particularly, if a = 0 we say that the logarithmic sobolev inequality is tight. the logarithmic sobolev inequalities relate entropy to the dirichlet norm (the energy) and these type of inequalities were introduced by l. gross to study the hypercontractive propertie of the diffusion semigroups and the markov semigroups, (see [7, 8]). 2 the results now, we are ready to present the results of this paper. 16 iris a. lópez cubo 19, 2 (2017) 2.1 hypercontractivity of dunkl-ornstein-uhlenbeck semigroup following d. bakry [2, 3], we turn now to the study of the local structure of the dunkl-ornsteinuhlenbeck differential operator lk. then, we started recalling the operators, γ and γ2, defined in (1.6) and (1.7) respectively. that is, γ(f) = 1 2 [ lk(f 2) − 2flk(f) ] and γ2(f) = 1 2 [lkγ(f) − 2γ(f,lkf)] , (2.1) where, γ(f,lkf) = 1 2 [ lk(flkf) − flk(lkf) − (lkf) 2 ] , (2.2) ∀f ∈ a. again, we consider a as the space of all polynomials on rd. now, from (1.2), let us denote lkf = d∑ j=1 l j kf, ∀f ∈ a, (2.3) where, for each j = 1, . . . ,d, l j kf(x) = 1 2 { ∂2jf(x) + 2kj xj ∂jf(x) − kj ( f(x) − f(σjx) x2j )} − xj∂jf(x). (2.4) thus, in order to obtain our results, we prove the following technical lemmas. lemma 2.1. let lk be the dunkl-ornstein-uhlenbeck differential operator defined as in (1.2). then γ(f)(x) = |∇f(x)|2 2 + d∑ j=1 kj ( f(x) − f(σjx) 2xj )2 , ∀f ∈ a. proof. from (2.1) and (2.3), it is obvious that we can write γ(f)(x) = 1 2 d∑ j=1 {l j k(f 2)(x) − 2f(x)l j k(f)(x)}, where, considering (2.4), we denote l j kf(x) = l jf(x) + ω j kf(x), j = 1, . . . ,d with ljf(x) = 1 2 ∂2jf(x) − xj∂f(x) and ω j kf(x) = 1 2 [ 2kj xj ∂jf(x) − kj x2j (f(x) − f(σjx)) ] , cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 17 for each j = 1, . . . ,d. thus, first we have to compute lj(f2) and 2flj(f). we can see that { lj(f2)(x) = (∂jf) 2(x) + f(x)∂2jf(x) − 2xjf(x)∂jf(x), 2f(x)lj(f)(x) = f(x)∂2jf(x) − 2xjf(x)∂jf(x), since { ∂j(f 2)(x) = 2f(x)∂jf(x), ∂2j (f 2)(x) = 2(∂jf) 2(x) + 2f(x)∂2jf(x) and therefore lj(f2)(x) − 2f(x)ljf(x) = (∂jf) 2 (x). (2.5) on the other hand, ω j k(f 2)(x) = 1 2 [ 4kj xj f(x)∂jf(x) − kj x2j (f2(x) − f2(σjx)) ] and 2f(x)ω j kf(x) = 1 2 [ 4kj xj f(x)∂jf(x) − kj x2j (2f2(x) − 2f(x)f(σjx)) ] . then, we obtain that ω j k(f 2 )(x) − 2f(x)ω j k(f 2 )(x) = kj 2x2j (f(x) − f(σjx)) 2 (2.6) and consequently, the result of the lemma follows from (2.5) and (2.6), since γ(f)(x) = 1 2 d∑ j=1 {lj(f2)(x) − 2f(x)ljf(x)} + {ω j k(f 2)(x) − 2f(x)ω j kf(x)} = 1 2 { d∑ j=1 (∂jf) 2(x) + kj 2x2j (f(x) − f(σjx)) 2 } . lemma 2.2. let lk be the dunkl-ornstein-uhlenbeck differential operator defined as in (1.2). then, ∀f ∈ a, we have lk(flkf)(x) = (lkf) 2(x) + f(x)lk(lkf)(x) + 〈∇f(x),∇lkf(x)〉 + 1 2 d∑ i=1 d∑ j=1 ki x2i (f(x) − f(σix)) ( l j kf(x) − l j kf(σix) ) . proof. from (2.3) we obtain that lk(flkf)(x) = d∑ i=1 d∑ j=1 lik(fl j kf)(x) 18 iris a. lópez cubo 19, 2 (2017) and by using (2.4) with fl j kf instead of f, we can express lik(fl j kf)(x) = 1 2 [ ∂2i(fl j kf)(x) + 2ki xi ∂i(fl j kf)(x) − ki x2i ( f(x)l j kf(x) − f(σix)l j kf(σix) ) ] − xi∂i(fl j kf)(x). (2.7) now, for each j = 1, . . . ,d and i = 1, . . . ,d, we have { ∂i(fl j kf)(x) = ∂if(x)l j kf(x) + f(x)∂i(l j kf)(x), ∂2i(fl j kf)(x) = ∂ 2 if(x)l j kf(x) + 2∂if(x)∂i(l j kf)(x) + f(x)∂ 2 i(l j kf)(x) (2.8) and since, − ki x2i [f(x)l j kf(x) − f(σix)l j kf(σix)] = − ki x2i (f(x) − f(σix))l j kf(x) − ki x2i f(x) ( l j kf(x) − l j kf(σix) ) + ki x2i (f(x) − f(σix)) ( l j kf(x) − l j kf(σix) ) , (2.9) then substituing (2.8) and (2.9) in (2.7) we obtain explicitly lik(fl j kf)(x) = ( 1 2 [ ∂2if(x) + 2ki xi ∂if(x) − ki x2i (f(x) − f(σix)) ] − xi∂if(x) ) l j kf(x)+ ( 1 2 [ ∂2i(l j kf)(x) + 2ki xi ∂i(l j kf)(x) − ki x2i (l j kf(x) − l j kf(σix)) ] − xi∂i(l j kf)(x) ) f(x)+ ∂if(x)∂i(l j kf)(x) + ki 2x2i (f(x) − f(σix)) ( l j kf(x) − l j kf(σix) ) and we can express lik(fl j kf)(x) = likf(x)l j kf(x) + f(x)l i k(l j kf)(x) + ∂if(x)∂i(l j kf)(x)+ ki 2x2i (f(x) − f(σix))(l j kf(x) − l j kf(σix)), for each j = 1, . . . ,d and i = 1, . . . ,d. therefore, taking the sum with respect to i and j, we obtain the result of the lemma. cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 19 consequently, the identity (2.2) and the lemma 2.2 allows us to write γ2(f)(x) = 1 2 [ lk(γ(f))(x) − 〈∇f(x),∇lkf(x)〉 − d∑ i=1 d∑ j=1 ki 2x2i (f(x) − f(σix)) ( l j kf(x) − l j kf(σix) ) ] (2.10) and in the following result we obtain an explicit expression for the operator γ2(f). more precisely, proposition 2.3. let lk be the dunkl-ornstein-uhlenbeck differential operator defined as in (1.2). then, ∀f ∈ a, the operator γ2(f) can be rewritten as γ2(f)(x) = d∑ i=1 { (∂2if) 2(x) 4 + (∂if) 2(x) 2 + ki 4 [ (∂if(x) + ∂if(σix)) xi − (f(x) − f(σix)) x2i ]2 + ki 2 [ ∂if(x) xi − (f(x) − f(σix)) 2x2i ]2 + ki 4x2i (f(x) − f(σix)) 2 } + d∑ i=1 d∑ j=1,j6=i { (∂2ijf) 2(x) 4 + ki 8x2i (∂jf(x) − ∂jf(σix)) 2 + kj 8x2j (∂if(x) − ∂if(σjx)) 2 + kikj 16x2ix 2 j [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 } . proof. we have to compute each term in equation (2.10). as first step in this argument, observe that by using (2.3) and the lemma 2.1 we get lk(γf) = d∑ i=1 d∑ j=1 l j k(γif) = d∑ i=1 lik(γif) + d∑ i=1 d∑ j=1,j6=i l j k(γif), where we denote γif(x) = (∂if) 2(x) 2 + ki ( f(x) − f(σix) 2xi )2 . (2.11) using the identity (2.4) with γif instead of f, we can express l j k(γif)(x) = 1 2 { ∂2j (γif)(x) + 2kj xj ∂j(γif)(x) − kj ( γif(x) − γif(σjx) x2j )} − xj∂jγif(x). (2.12) then, let us consider two cases: i = j and i 6= j, with 1 ≤ i ≤ d and 1 ≤ j ≤ d. if i = j, differentiating (2.11) with respect to xi we obtain ∂i(γif)(x) = ∂if(x)∂ 2 if(x) + ki 2x2i (f(x) − f(σix))(∂if(x) + ∂if(σix)) − ki 2x3i (f(x) − f(σix)) 2 (2.13) 20 iris a. lópez cubo 19, 2 (2017) and ∂2i(γif)(x) = ∂if(x)∂ 3 if(x) + (∂ 2 if) 2 (x) + ki 2x2i (∂if(x) + ∂if(σix)) 2 + ki 2x2i (f(x) − f(σix))(∂ 2 if(x) − ∂ 2 if(σix)) − 2ki x3i (f(x) − f(σix))(∂if(x) + ∂if(σix)) + 3ki 2x4i (f(x) − f(σix)) 2, (2.14) (note that ∂i(σix) = −1). otherwise, considering (2.11) with σix instead of x, we can write γif(σix) = (∂if) 2(σix) 2 + ki ( f(σix) − f(x) 2xi )2 , 1 ≤ i ≤ d, since σi(σix) = x and we obtain that ki 2x2i (γif(x) − γif(σix)) = ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)). (2.15) therefore, if i = j, replacing (2.13), (2.14) and (2.15) in (2.12), we get that lik(γif)(x) = (∂2if) 2(x) 2 + ∂if(x)∂ 3 if(x) 2 − xi∂ 2 if(x)∂if(x) + ki 4x2i (∂if(x) + ∂if(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 if(x) − ∂ 2 if(σix)) − ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)) + [ k2i 2x3i − ki x3i − ki 2xi ] (f(x) − f(σix))(∂if(x) + ∂if(σix)) + ki xi ∂2if(x)∂if(x) + [ 3ki 4x4i − k2i 2x4i + ki 2x2i ] (f(x) − f(σix)) 2. (2.16) now, if i 6= j, again differentiating (2.11) with respect to xj we obtain ∂j(γif)(x) = ∂if(x)∂ 2 ijf(x) + ki 2x2i (f(x) − f(σix))(∂jf(x) − ∂jf(σix)) (2.17) and ∂2j (γif)(x) = (∂ 2 ijf) 2 (x) + ∂if(x)∂ 3 ijjf(x) + ki 2x2i (∂jf(x) − ∂jf(σix)) 2 + ki 2x2i (f(x) − f(σix))(∂ 2 jf(x) − ∂ 2 jf(σix)). (2.18) cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 21 again, considering (2.11) with σjx instead of x we have that kj 2x2j (γif(x) − γif(σjx)) = kj 4x2j ((∂if) 2(x) − (∂if) 2(σjx)) + kjki 8x2jx 2 i [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] . (2.19) therefore, if i 6= j, replacing the equations (2.17), (2.18) and (2.19) in (2.12) we can see that l j k(γif) can be expressed as l j k(γif)(x) = (∂2ijf) 2(x) 2 + ∂if(x)∂ 3 ijjf(x) 2 − xj∂jf(x)∂ 2 ijf(x) + ki 4x2i (∂jf(x) − ∂jf(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 jf(x) − ∂ 2 jf(σix)) + kj xj ∂if(x)∂ 2 ijf(x) + [ kikj 2x2ixj − kixj 2x2i ] (f(x) − f(σix))(∂jf(x) − ∂jf(σix)) − kj 4x2j ((∂if) 2 (x) − (∂if) 2 (σjx)) − kikj 8x2ix 2 j [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] . (2.20) thus, taking the sum with respect to i and j in (2.16) and (2.20) we obtain explicitly that lk(γf)(x) = d∑ i=1 { (∂2if) 2(x) 2 + ∂if(x)∂ 3 if(x) 2 − xi∂ 2 if(x)∂if(x) + ki 4x2i (∂if(x) + ∂if(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 if(x) − ∂ 2 if(σix)) − ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)) + [ k2i 2x3i − ki x3i − ki 2xi ] (f(x) − f(σix))(∂if(x) + ∂if(σix)) + ki xi ∂2if(x)∂if(x) + [ 3ki 4x4i − k2i 2x4i + ki 2x2i ] (f(x) − f(σix)) 2 } + d∑ i=1 d∑ j=1,j6=i { (∂2ijf) 2(x) 2 + ∂if(x)∂ 3 ijjf(x) 2 − xj∂jf(x)∂ 2 ijf(x) + ki 4x2i (∂jf(x) − ∂jf(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 jf(x) − ∂ 2 jf(σix)) + kj xj ∂if(x)∂ 2 ijf(x) + [ kikj 2x2ixj − kixj 2x2i ] (f(x) − f(σix))(∂jf(x) − ∂jf(σix)) − kj 4x2j ((∂if) 2(x) − (∂if) 2(σjx)) − kikj 8x2ix 2 j [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] } . (2.21) now, we will develop the second part of the proof of this proposition calculating the operator 22 iris a. lópez cubo 19, 2 (2017) 〈∇f,∇lkf〉, such can be written as 〈∇f(x),∇lkf(x)〉 = d∑ i=1 ∂if(x)∂i(lkf)(x) = d∑ i=1 d∑ j=1 ∂if(x)∂i(l j kf)(x) = d∑ i=1 ∂if(x)∂i(l i kf)(x) + d∑ i=1 d∑ j=1,j6=i ∂if(x)∂i(l j kf)(x). again, we consider i = j and i 6= j. if i = j, from (2.4) we obtain that ∂i(l i kf)(x) = ∂3if(x) 2 − ∂if(x) − xi∂ 2 if(x) + ki xi ∂2if(x) − ki x2i ∂if(x) − ki 2x2i (∂if(x) + ∂if(σix)) + ki x3i (f(x) − f(σix)). (2.22) but, if i 6= j, we get ∂i(l j kf)(x) = ∂3jjif(x) 2 + kj xj ∂2jif(x) − kj 2x2j (∂if(x) − ∂if(σjx)) − xj∂ 2 jif(x). (2.23) thus, taking the sum with respect to i and j in (2.22) and (2.23) we express 〈∇f(x),∇lkf(x)〉 = d∑ i=1 { ∂3if(x)∂if(x) 2 − (∂if) 2(x) − xi∂ 2 if(x)∂if(x) + ki xi ∂2if(x)∂if(x) − ki x2i (∂if) 2(x) − ki 2x2i (∂if(x) + ∂if(σix))∂if(x) + ki x3i (f(x) − f(σix))∂if(x) } + d∑ i=1 d∑ j=1,j6=i { ∂3jjif(x)∂if(x) 2 + kj xj ∂2jif(x)∂if(x) − kj 2x2j (∂if(x) − ∂if(σjx))∂if(x) − xj∂ 2 jif(x)∂if(x) } . (2.24) finally as third step, we turn now to compute explicitly the terms of the expression d∑ i=1 d∑ j=1 ki 2x2i (f(x) − f(σix)) ( l j kf(x) − l j kf(σix) ) . then, if i = j, by using (2.4) with σix instead of x, we can write likf(σix) = ∂2if(σix) 2 − ki xi ∂if(σix) − ki 2x2i (f(σix) − f(x)) + xi∂if(σix), cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 23 (remind that σix = (x1, . . . ,−xi, . . . ,xd) and σi(σix) = x), therefore, likf(x) − l i kf(σix) = (∂2if(x) − ∂ 2 if(σix)) 2 + ki xi (∂if(x) + ∂if(σix)) − ki x2i (f(x) − f(σix)) − xi(∂if(x) + ∂if(σix)). (2.25) but, if i 6= j, we get l j kf(x) − l j kf(σix) = (∂2jf(x) − ∂ 2 jf(σix)) 2 + kj xj (∂jf(x) − ∂jf(σix)) + kj 2x2j (f(σjx) − f(x) + f(σix) − f(σjσix)) − xj(∂jf(x) − ∂jf(σix)). (2.26) so, from (2.25) and (2.26) we can conclude that d∑ i=1 d∑ j=1 ki 2x2i (f(x) − f(σix)) ( l j kf(x) − l j kf(σix) ) = d∑ i=1 { ki 4x2i (∂2if(x) − ∂ 2 if(σix))(f(x) − f(σix)) + k2i 2x3i (∂if(x) + ∂if(σix))(f(x) − f(σix)) − k2i 2x4i (f(x) − f(σix)) 2 − ki 2xi (∂if(x) + ∂if(σix))(f(x) − f(σix)) } + d∑ i=1 d∑ j=1,j6=i { ki 4x2i (∂2jf(x) − ∂ 2 jf(σix))(f(x) − f(σix)) + kikj 2x2ixj (∂jf(x) − ∂jf(σix))(f(x) − f(σix)) + kikj 4x2ix 2 j (f(σjx) − f(x) + f(σix) − f(σjσix))(f(x) − f(σix)) − kixj 2x2i (∂jf(x) − ∂jf(σix))(f(x) − f(σix)) } . (2.27) then, at this point in our argument, replacing the identities (2.21), (2.24) and (2.27) in (2.10) and simplifying the terms that are equal, we can write γ2(f)(x) = e1(x) + e2(x), 24 iris a. lópez cubo 19, 2 (2017) where we denote by e1(x) = 1 2 { d∑ i=1 (∂2if) 2(x) 2 + (∂if) 2(x) + ki 4x2i (∂if(x) + ∂if(σix)) 2 − ki x3i (∂if(x) + ∂if(σix))(f(x) − f(σix)) + [ 3ki 4x4i + ki 2x2i ] (f(x) − f(σix)) 2 − ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)) + ki x2i (∂if) 2(x) + ki 2x2i (∂if(x) + ∂if(σix))∂if(x) − ki x3i (f(x) − f(σix))∂if(x) } and e2(x) = 1 2 { d∑ i=1 d∑ j=1,j6=i (∂2ijf) 2(x) 2 + ki 4x2i (∂jf(x) − ∂jf(σix)) 2 − kj 4x2j [(∂if) 2(x) − (∂if) 2(σjx)] − kjki 8x2jx 2 i [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] + kj 2x2j (∂if(x) − ∂if(σjx))∂if(x) − kikj 4x2ix 2 j [(f(σjx) − f(x) + f(σix) − f(σjσix))(f(x) − f(σix))] } . therefore, we only need to express e1(x) and e2(x) more easily. first, we consider e1(x) and associating the terms, we see that ki 4x2i (∂if(x) + ∂if(σix)) 2 + ki 2x2i (∂if(x) + ∂if(σix))∂if(x) − ki 4x2i ((∂if) 2 (x) − (∂if) 2 (σix)) = ki 2x2i (∂if(x) + ∂if(σix)) 2. (2.28) now, taking the identity (2.28) and completing squares in e1(x) we obtain that ki 2x2i (∂if(x) + ∂if(σix)) 2 − ki x3i (∂if(x) + ∂if(σix))(f(x) − f(σix)) = ki 2 [ (∂if(x) + ∂if(σix)) 2 x2i − 2 (∂if(x) + ∂if(σix)) xi (f(x) − f(σix)) x2i ± (f(x) − f(σix)) 2 x4i ] = ki 2 [ ( ∂if(x) + ∂if(σix) xi ) − ( f(x) − f(σix) x2i ) ]2 − ki 2x4i (f(x) − f(σix)) 2. (2.29) this way, from (2.29) we can write e1(x) = 1 2 { d∑ i=1 (∂2if) 2(x) 2 + (∂if) 2(x) + ki 2 [ ( ∂if(x) + ∂if(σix) xi ) − ( f(x) − f(σix) x2i ) ]2 + [ ki 4x4i + ki 2x2i ] (f(x) − f(σix)) 2 + ki x2i (∂if) 2(x) − ki x3i (f(x) − f(σix))∂if(x) } , cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 25 since, (3ki/4x 4 i) − (ki/2x 4 i) = ki/4x 4 i . then, associating the terms in the above expression, we have ki x2i (∂if) 2(x) − ki x3i (f(x) − f(σix))∂if(x) + ki 4x4i (f(x) − f(σix)) 2 = ki [ ∂if(x) xi − ( f(x) − f(σix) 2x2i )]2 and therefore, we can conclude that e1(x) = 1 2 { d∑ i=1 (∂2if) 2(x) 2 + (∂if) 2(x) + ki 2 [ ( ∂if(x) + ∂if(σix) xi ) − ( f(x) − f(σix) x2i ) ]2 + ki [ ∂if(x) xi − ( f(x) − f(σix) 2x2i )]2 + ki 2x2i (f(x) − f(σix)) 2 } . (2.30) now, we consider e2(x). once more, we observe that − kj 4x2j [(∂if) 2(x) − (∂if) 2(σjx)] + kj 2x2j (∂if(x) − ∂if(σjx))∂if(x) = kj 4x2j [∂if(x) − ∂if(σjx)] 2. (2.31) moreover, associating the terms − kjki 4x2jx 2 i (f(σjx) − f(x) + f(σix) − f(σjσix))(f(x) − f(σix)) = kjki 8x2jx 2 i [ 2(f(x) − f(σix)) 2 − 2(f(σjx) − f(σjσix))(f(x) − f(σix)) ] , then − kjki 8x2jx 2 i [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] + kjki 8x2jx 2 i [ 2(f(x) − f(σix)) 2 − 2(f(σjx) − f(σjσix))(f(x) − f(σix)) ] = kjki 8x2jx 2 i [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 . (2.32) therefore, replacing (2.31) and (2.32) in e2(x), we express e2(x) = 1 2 { d∑ i=1 d∑ j=1,j6=i (∂2ijf) 2(x) 2 + ki 4x2i [∂jf(x) − ∂jf(σix)] 2+ kj 4x2j [∂if(x) − ∂if(σjx)] 2 + kjki 8x2jx 2 i [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 } (2.33) and finally, the sum of (2.30) and (2.33) allows us to obtain the result of the proposition. in consecuense, we are able to prove that the dunkl-ornstein-uhlenbeck differential operator, lk, defined as in (1.2) and associated with the z d 2 group, satisfies a cd(ρ,∞)-inequality, if 0 ≤ ρ ≤ 1. 26 iris a. lópez cubo 19, 2 (2017) theorem 2.4. let lk be the dunkl-ornstein-uhlenbeck differential operator defined as in (1.2). then, if 0 ≤ ρ ≤ 1, the cd(ρ,∞)-inequality is satisfied. proof. from lemma 2.1 and the proposition 2.3, we have that γ2(f)(x) ≥ ργ(f)(x) is true, if and only if, d∑ i=1 { (∂2if) 2(x) 4 + (1 − ρ) (∂if) 2(x) 2 + ki 4 [ (∂if(x) + ∂if(σix)) xi − (f(x) − f(σix)) x2i ]2 + ki 2 [ ∂if(x) xi − (f(x) − f(σix)) 2x2i ]2 + (1 − ρ) ki 4x2i (f(x) − f(σix)) 2 } + d∑ i=1 d∑ j=1,j6=i { (∂2ijf) 2(x) 4 + ki 8x2i (∂jf(x) − ∂jf(σix)) 2 + kj 8x2j (∂if(x) − ∂if(σjx)) 2 + kikj 16x2ix 2 j [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 } ≥ 0. then, we only need to choose 0 ≤ ρ ≤ 1 to obtain the result. now, again we consider the family of measures mkt(x,dy) defined in (1.5). if the measures mk are replaced by mkt(x,dy), then the logarithmic sobolev inequalities ls(a,c) can be rewritten as okt(f 2 logf2) − okt(f 2) logokt(f 2) ≤ a(t)okt(f2) + c(t)okt(γf) and if a = 0, okt(f 2 logf2) − okt(f 2) logokt(f 2) ≤ c(t)okt(γf), (2.34) which are known as local log-sobolev inequalities and local tight-log-sobolev inequalities respectively, (see [2]). therefore, from the general criterion of d. bakry and m. emery cf. [1] we have that the curvature inequality c(ρ,∞) is equivalent to the local tight-log-sobolev inequality with c(t) = 1−e −2ρt ρ , (for details, we refer the reader to [2, proposition 2.6]). thus, from corolario 2.4 we obtain that inequality (2.34) is true and therefore, ∫ rd okt(f 2 logf2)(x)mk(dx) − ∫ rd okt(f 2)(x) logokt(f 2)(x)mk(dx) ≤ c(t) ∫ rd okt(γf)(x)mk(dx), where γ is defined as in the lemma 2.1. then, by using the propertie ∫ rd oktf(x)mk(dx) = ∫ rd f(x)mk(dx), we can conclude that ent(f2) ≤ ce(f), ∀f ∈ a. cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 27 in a general context, l. gross [8, 7] proved that logarithmic sobolev inequality is equivalent to the fact that for any t > 0 and p ∈ (1,∞), ‖htf‖q(t) ≤ em(t)‖f‖p, where (ht)t is a diffusion semigroup or a symmetric markov semigroup and the functions q(t) and m(t) are defined by q(t) − 1 p − 1 = exp(4t/c) and m(t) = a 16 ( 1 p − 1 q(t) ) , (see [8, theorems 1 and 2]). particularly, tight-logarithmic-sobolev inequality is equivalent to the hypercontractivity property. in consecuense, we can conclude that the dunkl-ornstein-uhlenbeck semigroup, {okt}t≥0, is hypercontractive, ∀t > 0 and 1 < p < ∞. this means, ‖oktf‖q(t),mk ≤ ‖f‖p,mk, ∀f ∈ l p(mk), where, exp(4t/c) = (q(t) − 1)/(p − 1) for some positive constant c. by using subordination formula we obtain the same result for the dunkl-poisson semigroup {pkt }t≥0. 2.2 applications as a consequence of the hypercontractivity propertie of {okt}t≥0 semigroup, we obtain the l p(mk)continuity of jkn operators for every 1 < p < ∞ and n = 0,1,2, ... the reasoning is similar as in the case of classical ornstein-uhlenbeck semigroup and we refer the reader to [17, lemma 1.1], where the identity okt(j k nf) = e −ntjknf, if f ∈ a, is a key condition in the argument. moreover, for 1 0, such that, ‖okt(i − · · · − jkn−1)f‖p,mk ≤ cp,ne−nt‖f‖p,mk, (2.35) and since the development is similar to the classical ornstein-uhlenbeck semigroup, we omit the details and refer to [17, lemma 1.2]. then we extend the celebrated p.a meyer’s multiplier theorem to dunkl-ornstein-uhlenbeck semigroup and the zd2 group. a first version of this theorem, associated with hermite expansions, has be obtained in [17, theorem 1.1] (see also, [18]). afterwards, similar versions to laguerre and jacobi setting have been obtained in [6, theorem 3.4] and [10, theorem 4.1], respectively. theorem 2.5 (meyer’s multiplier theorem). let {okt}t≥0 be the dunkl-onstein-uhlenbeck semigroup. assume that h is a function, which is analytic in a neighborhood of the origin. let {ψ(n)}n∈n 28 iris a. lópez cubo 19, 2 (2017) be a sequence of real numbers, such that ψ(n) = h(n−β), ∀n ≥ n0 and some β ∈ (0,1]. then, the operator tψf = ∑ n≥0 ψ(n)jknf, f = ∑ n≥0 jknf, defined initially in l2(mk), has a unique continuous linear extension to each of the spaces l p(mk), for 1 < p < ∞. next, we consider the fractional integrals, the fractional derivatives and the bessel potentials associated to the differential operator lk and the z d 2 group. since, lk is symmetric and has a selfadjoint extension, these can be defined by standarts ways, by example by spectral representation of bochner subordination. however, the use of these fractional operators together with meyer’s multipliers theorem allows us to obtain a characterization of dunkl-potential spaces, similar to the classic case (see [15]). then, dunkl-fractional integral of order s > 0, associated to dunkl-ornstein-uhlenbeck differential operator and the zd2 group, is defined by isk = (−lk) −s/2π0, where π0 denotes the orthogonal projection onto the orthogonal complement of the subspace spanned by the constant functions. immediately, from (1.3) we have isk(h k ν) = |ν| −s/2hkν, |ν| > 0, f ∈ a, and an integral representation of iskf can be obtained iskf = 1 γ(s) ∫ ∞ 0 ts−1pkt (i − j k 0)fdt, (2.36) which makes sense, for all f ∈ lp(mk), by means of (2.35) and subordination formula, because ‖iskf‖p,mk ≤ ap‖f‖p,mk for s > 0, and 1 < p < ∞, (we refer the reader to [6, 9] and [10]). also, we introduce the fractional derivative in the zd2-dunkl setting which is given formally by dsk = (−lk) s/2. for the generalized hermite polynomials we have dsk(h k ν) = |ν| s/2hkν, ∀s > 0 and therefore, by using the density of polynomials in lp(mk), the derivative d s k can be extended to lp(mk). particularly, if 0 < s < 1, we can write dskf = 1 cs ∫ ∞ 0 ts−1(pktf − f)dt, where, cs = ∫ ∞ 0 u−s−1(e−u − 1)du. (2.37) cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 29 the identity (2.37) may be regarded as the definiton of dsk, with 0 < s < 1, for all f ∈ c2b(rd), or for all f for which the corresponding integral is absolutely convergent. moreover, if f is a polynomial, we get dsk(i s kf) = i s k(d s kf) = π0f. now, the dunkl-bessel potential operator, associated to the dunkl-ornstein-uhlenbeck differential operator and the zd2 group, is defined as (i − lk) −s/2f = ∞∑ n=0 (1 + n)−s/2jknf, f ∈ a and we defined the dunkl-potential spaces l p,s k (mk), associated with generalized hermite expansions, as the completion of the space of all polynomials with respect to the norm ‖f‖p,s = ∥ ∥ ∥ (i − lk) s/2f ∥ ∥ ∥ p,mk . by means of meyer’s multiplier theorem, we can observe that the dunkl-bessel potential operator extends to a continuous linear operator on lp(mk), (for a similar argument see e.g lemma 6.1 in [6]). also, the potential spaces have the following properties: i) if 1 ≤ p ≤ q, then lq,sk (mk) ⊂ l p,s k (mk), for each s ≥ 0. ii) if 0 ≤ s ≤ r, then lp,rk (mk) ⊂ l p,s k (mk), for each 1 < p < ∞. moreover, the embeddings in i) and ii) are continuous. again, we omit the proofs of these two facts, but we refer the reader to the proposition 2.2 in [9] and the proposition 6.3 in [6]. finally, the following theorem allows us to extend the dunkl-fractional derivative, dsk, to the potential spaces l p,s k (mk), for 1 0 and associated to generalized hermite expansions, where we consider the zd2 group. thus, the union of these spaces; lsk(mk) = ⋃ p>1 l p,s k (mk) make up a natural domain of dsk. similar versions of this theorem has been obtained in [9, theorem 2.2], where we consider classical hermite expansions, in [6, theorem 6.4] related to laguerrre expansions and afterwards, in [10, theorem 5.1] in the jacobi context. theorem 2.6. let s ≥ 0 and 1 < p < ∞. i) if {pn}n is a sequence of polynomials such that limn→∞ pn = f in l p s(mk), then limn d s kpn exists in l p,s k (mk) and does not depend on the choice of a sequence {pn}n. if f ∈ lp,sk (mk) ∩ l p,r k (mk), then the limit does not depend on the choice of p or r. thus, dskf = limn→∞ d s kpn in l s,p k (mk), limn→∞ pn = f in l p,s k (mk), f ∈ lsk(mk), is well defined. 30 iris a. lópez cubo 19, 2 (2017) ii) f ∈ lp,sk (mk) if and only if dskf ∈ lp(mk). moreover, bp,s‖f‖p,s ≤ ‖dskf‖p,mk ≤ ap,s‖f‖p,s. references [1] d. bakry, m. emery, hypercontractivité de semi-groupes de diffusion, c.r acad. sci. paris sér, 1299, (15), 775-778, (1984). [2] d. bakry, on sobolev and logarithmic sobolev inequalities for markov semigroups, in: new trends in stochastic analysis (charingworth), 43-75, (1994). river edge n.j 1997. taniguchisymposium world. sci. publishing. [3] d. bakry, functional inequalities for markov semigroups, probability measures on groups: recent directions and trends, tata institute of fundamental research, mumbai, 91-147, (2006). [4] t.s chihara, an introduction to orthogonal polynomials, gordon and breach, new york, (1978). [5] c. f. dunkl, differential-difference operators associated to reflection groups, trans. amer. math. soc, 311, 167-183, (1989). [6] p. graczky, j. loeb, i. lópez, a. nowak, w. urbina, higher order riesz transforms, fractional derivatives and sobolev spaces for laguerre expansions, j. math. pures et appl, 84, 375-405, (2005). [7] l. gross, logarithmic sobolev inequalities and contractivity properties of semigroups, in: dirichlet forms (varenna, 1992) springer, verling, 54-88, (1993). [8] l. gross, logarithmic sobolev inequalities, amer. j. math. 97, 1061-1083, (1976). [9] i. lópez, w. urbina, fractional differentiation for the gaussian measure and applications, bull. sci. math, 83, vol 128, issue 7, 587-603, (2004). [10] i. lópez, operators associated with the jacobi semigroup, j. approx. theory, 161, 385-410, (2009). [11] m. rösler, generalize hermite polynomials and the heat equation for dunkl operators, commun. math. phys. 192, 519-542, (1998). [12] m. rösler, m. voit markov processes related with dunkl operators, adv. in appl. math, 21, 575-643, (1998). [13] m. rösler dunkl operators. theory and applications, orthogonal polynomials and special functions, (leuven 2002), 93-135. lecture notes in math. 1817, springer-berlin, (2003). cubo 19, 2 (2017) on the hypercontractive property of the dunkl-ornstein-uhlenbeck... 31 [14] m. rösler, m. voit, dunkl theory, convolution algebras and related markov processes, in: harmonic and stochastic analysis of dunkl processes, eds. p. graczyk, m. rösler, m. yor, travaux en cours 71, 1-112, hermann, paris, (2008). [15] e. stein, the characterization of functions arising as potentials i, bull. amer. math. soc. 97, 102-104, (1961). ii (ibid) 68, 577-582, (1962). [16] e. stein, topics in harmonic analysis related to the littlewood-paley theory, princenton univ. press. princenton (1971). [17] h. sugita, sobolev spaces of wiener functionals and malliavin’s calculus, j. math. kyoto univ, 25-1, 31-48, (1985). [18] s. watanabe, m. gopalan nair, b. rajeev lectures on stochastic differential equations and malliavin calculus, tata institute of fundamental research, vol. 73, springer verlag, berlin/heidelberg/new york/tokyo, (1984). preliminaries the results hypercontractivity of dunkl-ornstein-uhlenbeck semigroup applications clulioo a math ematical journal vol. 6, ~ 1, {11 6 -1 2 1). march 2004. on brouwer's fixed point theor em' n . tarkhanov un ivcrsitii.t potsdam ln stitu t für mathematik postfach 60 15 53 1441 5 potsdam ge rm any tark ha11 ov@math .uni -potsdam .de abstract thc pa ¡>cr presc nts a u cx pli cit fo rmul a for t hc numb er oí fixcd points of a c"" nrn p oí a segmc11t [a , b] c r. whi! c t hc formula can be derived frorn thc lcíschct:1. flx«i point thoo rcm for gc ncrnl cw -comp lcxes , t hc nc w p roof is ins truct ivc and hi ghli ght.s th c co 11 t ribu tiom1 of dcgcncratc fi xcd points . 1 introduction 111 19 10 orouwe r (cf. ib ro l 2]} provc h º(o '[a,bj) , ( h f ), h 1 (o' [a, bj)--> h ' (o' [a,bj) are cndomorphis ms of t hc cohomology of th c de rh a m co mp lex o n lc1, b], indu1,;cd by t hc pull-back o pera t or ¡n on differe nti a l fo rm s. sincc lh e cohomology is fi 11 it(' d im (' nsio nal al c\'e ry stcp , t hc t races of th ese endo mo rphi sms ar e well dcfi ncd . o. note th at in fact. l ( f) = 1 in our specia l case , fo r hº( l [a, bj) ~ c a nd h 1 (l [a , b]) ~ le mma 4 . 1 sup11os e all jixcd¡minl s of ll1 e ma¡j f on [a , bj are iso fot cci . tlw11 l b b " ) l(f ) = p·' · d( 0(j (y) y) b = . " a (4 . 1) proo f. appl ying t hc pu \1 -back opcrato r r t o both sid cs o f equalitics (3.2 ) wc g 1' i (f'p ) d f ' f' sº. d(f1p ) = f ' f's' . for ¡: and d rninmule. si ncc ¡ : p maps e 1 [a , b] to ljri , b] wc ded uce t ha t / 1 a nd /' s are liomo to pi c· cndomo r phi :-; ms of t h€' de rl mm com pl cx . hencc t hcy induce th e samc endomo rp hisms of t hc co homology, i.c ., fl f = fl (jds). !t. fol lows t hat l(/ ) = l( p s). \\'(' no w ob:;c rvc t.hat j ds is a trac e dass end omorp hism of t. he de jumm co mplcx . dy thc nltcrn at.i ng su 111 fo rmul a (cf. for ins t.a.ncc theo rcm 19 .1.1 5 of [hor85 ]) wc obtai n l(f) tr j' sº tr j' s' [ ll.'(u x l )'kso (! x l )' k s • ) wh(' rr ~ stands for t. hc d iagonal m ap [a , bj -+ [a . bl x lo ,bj . an cl k s is t hr schw<1rlz ke rn el of by as.-. ump1io n, th c se t fix(f, [a, b]) is d isc rc tc. since thc intcg ran cl is of cla."'~ l 1 j11. bj. w(' gf'l l (f) = li"' 1. ll.'(11 x l )'k so (/ x i )'ks •) ' 'º ¡ .. .1,¡\ u, wlu •rt• ( ' i!' l lw m't o f al l ¡wi1 1t s y e [a , bl wh osc d ist a ncc to fix( / . [o, b]) is l t~ss t hai1 on drouwer's flyed poillt theorem 119 wc no~ make use of cqualities (3.2) to evaluate lhe integrand in the latter integnll. nnmcly, tlhey imply that d~kp = -kso, d~kp = -ks• mvny from lhc diagonal oí [a, b] x [a, b]. il follows that holds on la, bj \ uc, whcnct: l(f) t hii; provt.'s thc formula. t-•(-d~(f x 1)'kp+d.,(fx 1)' i'u x 1)'kp) lim { d(t:::..~ (f x l)~kp) €-~u í¡n,bj\u, pv[ d(8(f(y)-y)-:=~) 5 fixed point theorem o '1'111• sccm1d term in t hc in lngral (4. l) is eas ily evaluated , he11ce t.his for1nula t r m1sf111·n1s '" l(f) = l + p.v. [ d8(f(y) y) . t heorem 5.1 let f be 11 c 00 mav of tite sr.9m e11t !n , b] with isofolerl jix1~1i ¡min/ .. '1. 'l'lie11 l(f) = 1 + 8(/(71) 11) r-+ l µ(p) n+ pefix(/,(n,b) ) wlum~ ¡1(1' ) i.s thc loco/ dcgree of 1 f at p. proof. writ.r r1 < p 1 < ... < pn < ¡, for 1.111' fixccl points oí f lha1. lit• in l.lu~ 01w11 i11t1•rv11i (u ,b}. siurr 1.ht! f1111ct.\011 0(f(y) y) is c:onstant away from tlw set. o f fixt~d poin!s uf f w1· ~1·t r1 • n 1•~+1 -• b -t l(f ) = 1 + i d8(f(71) -y) + l: i d8(f(y)-y ) + i d8(f(y)y) o+c k = l l'~ +c p ,, +r fur 1111 ( > o rnmll r.nourh. lfoncc it follows t hat n +r n 1•1+ t l(f ) = 1 (-l(/(71) y) i,_, l: 8(/(y) y ) i,.. _,. k = l 120 n. th.rkhanov a passage to the limit when € ~ o+ g ives the desired formula, for the local dcgrcc of 1 f at. pis opposite to that of f l . o lf a is a simple fixed poini of f then t he sign of (1 / )(a+) already unic¡ucly determines the local degree of a ny smooth extens ion of l f t.o a ncighbo urhoocl o f a . amely t he local degree of 1 f at. a just. amounts to s ign (1 /)(a+), or s ign {i /'(a )} . the s ame rcasoning applies to the case where bis a si mple fixcd point off. howcver these arguments no longer work if a or b is not s imple, for f can be extended to a smooth function in a ne ighbo urhood of a respecti vely b in di vcrse manners. fa r this reason we need another s pecification of fixed points of f 0 11 thl.' boundary. supposc / (a) = a. t hcn a is said to be an at.tracti ng fixed point off if ( 1 /)(a+) > o, a ncl repul!iing if ( j f)(a+) < o. lf f(b ) == b thcn t he fixccl point bis called a t tracti ng if ( 1 j)(b) < o, and repu\sing if ( 1 j)(b) > o. f'or thc atlracting fi.xcd points on the boundary we define t he local degree o r 1 f to be 1, and for thc repuls ing fixed points wc defi ne the loca l degree o f l f to be 1. thcn thcorcm 5. 1 can be reformulatcd in t hc following way. corollary 5.2 let f be a c 00 map of the segment [a, b] with isolated fixed 11oi11h. th en l(f) = µ (p), 11efix( / ,(n,b))u fix!• i( / ,o(o .b)) f'ix1ª 1(/,8(o, b)) being the set aj attracting fixed points o/f on tli e bo1mdary. for thc s mooth mapg or [a, b] the fi xed point theorcm of bro uwer [bro 12j is a n obvious consequcnce of corollary 5.2 because l(/) == 1. references iab67j m . f'. atiyah a nd r. bott, a lefschetz fixed point formula for ellipt.ic complezes. 1, ann. mat h. 86 (1967), no. 2, 374-407. ibd~hij l . boutet de m onve l, boundary probfem.t for pseudo-different.ial 011cr· aton, act a math. 126 (197 1) , no. 12, 11 5 1. id 91j a . v . bllennen a nd m. a. s tt un1n, t l1 e atiyah-bott-le/schetzform1jlajor ell1pltc complexes 011 manifolds with botmdary, c ur rent. problcms of mathe mat.ics. f\mdamental directions. vol. 38, vin it i, moscow, 1991 , pp. 11 9 183. jbro 12j l. 8 . j . b nouwen, über abbildungen uon mannig/alt1gke1 tcn, math. ann. 7 1 ( 1912) , 3053 14 . ido172) j\ . d olo , l ccturcs 011 algebrair topology, spring crverlag, bt•rlin c:l al.. 1972. on brouwer's fixed point theorem 121 [h0r85] l . h6rmander, 11~e analysis o/ linear partia/ diflerentfol operators. vol. 3: pseudo-dif!erential operators, springer-verlag, berlin et al., 1985. [hop29j h. hopf, úber die algebraische arizalll von fi.xpu11kten, math. z, .. 29 ( 1929~, 493524. [lcf26] s. lefschetz, jntersectio11s and transformation.s o/ complexes and manifoldj, tra.ns. amcr. matlh. soc. 28 (1926), 1-49. [tar95] n. n. tar.khanov, comvlexes o/ dif!erential operotors, kluwer academic publis hcrs, dordrccht., nl, 1995. cubo matemática educacional vol., 4. n" 2, junio 2002 un truco de naipes en la búsqueda de puntos fijos cristián mallo! departamento de matemática universidad de la frontera casilla 54-d, temuco, chile. tome el mazo de m.aipes con las figuras hacia aibajo. disponga 21 cartas repart.ié ndolas con las figuras hacia arri ba, de dered1a a izquierda y de a rriba a abajo en tres columnas, c uidando de colocar cada carta por columna un poco encima de la anterior. pida que ailgu1.ó.em. el·ij a una carta y le indique ea qt11.é coli11. rnna se encuentra. recoja las cartas dejando la columna señalada entre las otras dos. repita esta rnarm. ipuilación dos veces más (repartir, pregu'!iltarr, recoger). entonces, después crl.e la t'11l1t ima recogida, d eshágase de las diez primeras cartas; la que s igue, la onceava cart.a, es la escogida por el cá.rm.dirlo d e turno y usted se creerá un mago ta·m. bueno como el gran houdini. ei;i lo que s igue analizaremos y geaern1lizarernos esbe truco. dada la r;narnera de d isponer y recoger las cartas en este jiuego, desp ués de una manipul1ación es claro que las cartas de la cohuilil:id.a elegida ocuparán las posiciones seri.aladas con la letra e de este rnodo 1 si e1111meramos jas posiciones de la signiefj.t.e mane r a.: 16 un truco de naipes por lo que, dependieficlo ctle la col11unna en donde se em.cuentre la carta elegida, tenemos las sig;tiiefj.tes jdosi1idi1j1idades: esto nos lleva a modelizar el jilroblema con la función m: {o, 1, ... , 1§,20} ~{o, 1,. , rn, 20} q ue oper a sobr e las posiciones como sig l!l.e: o, 1, 2 7 3, 4, 5 8 0, 7, 8 9 0 , 10, 11 10 12, 13, 14 11 15, ~6. 17 12 18, 10, 20 13 ( cristián mallo/ 17 y que t iene por definición genérica: m (3q + r) ;,, 7 + q con o ,;;; q ,;;; 6 y o ,;;; r ,;;; 2. vemos que esta función tiene un y sólo un punto fijo, es d ecir una posición f tal que m (!) = /: se trata de la correspondiente a la posición enumerada con el 10. más aún , cons tatamos que para toda posición x se t iene: s i x < 10 ento nces x < 11/(x) ,;;; 10, s i x > 10 entonces x > al(x) ~ 10. todo esto quiere decir que por aplicación reiterada de la función /\1/ 1 siempre se llega a la posición 10; en este caso, m 3 (x) = 10 paratodox< {0, 1, .. . , 19,20} cuest ión que hubiéramos pod id o también ded ucir de la d efini ción ya que: max(lf-m (x)i) = 3. ahora bien, este p roblema se generaliza con un juego en que se d ispongan n columnas de p naipes cada una y escogiendo de antemano un lugar fij o, que denotamos por k, para colocar la columna de la carta elegida. 1~11~11~ 1 teniendo en claro q ue las columnas son enumeradas del o a l nl y que las filas lo son del o a l p 11 sabemos que la posición ocupada por el crnre d e la línea q con la columna r está dada por qn + r. si la r.art.a elegid a oc'npa esta posición, a l ubicar su columna en el lugar k, esta carta pasa a ocupar in posirión qn + k, s iendo ella la q-ésimn rartn de la co lumna. todo esto significa que d espués d e efect ua r una mnni p nlació n (es d eci r: reroger las rart.as seg1fo modo indicado, colocar la col11 mna elegida en el lugar k y d isponer las cartas seg1ín modo indicado) como se habrán repa rt.ido 18 un truco de nai pes prim ero k colwnnas de p cartas, la elegida se encont ra rá en la posición pk + q. es decir , es tamos estableciendo que la manipul ación se modeliza con la fun ción m : ln,p ~ ln,p dond e ln ,p = {o , ! , ... , np 1} , m(qn+r)= kp+q con o(q(p 1 y o (r(n1. ind agar si est e juego tiene solución pasa por estudiar si !vl tiene puntos fijos y adq ui ere ca lidad de "truco infalible" si encontramos sólo uno. lm conclusiones salen de estudiar la igualdad en aras de ver si existen reescribir como: m(qn+r)=qn+r y r que la verifiquen. esta igualdad se puede kp= q(n-1) +r expresión que nos recuerda la división euclidiana de kp por n l. sin embargo, hay dos situaciones posibles: l. si n 1 no divide a kp. obligadamente r# o, n 1 y, en este caso, existe un sólo punto fijo f=q¡n+r¡ donde q¡ y r¡ son el cociente y el resto de la división de kp por n -1. 2. si n 1 divide a kp. aquí hay dos casos con el mismo tipo de soluciones: dos puntos fijos contiguos • ya sea r = o; en este caso se tiene qu e: kp = q(n1) y podemos inferir que existen dos pun tos fij os, f = q¡n y j' = f 1 = (q¡ 1) n + (n 1) donde q¡ es el cociente de la di visión exacta de k p por n l. cristiá.n mallo! 19 • ya sea r = n 1, caso en el que reescribiendo se logra: kp= (q+ l )(n -1) obteniendo la misma situación anteriror. dada la escritura de f y f', estos puntos fijos siempre apa recerán distribuidos de la manera siguiente: no es mala idea que el lector comprue be que en t;odos los casos esthd iados ( 1 y 2) , los puntos firl os erncontrados lo son efectivamente. en los dos casos el t ruco funciona sin problemas, salvo ql1e en el segundo el 11mago 11 tiene dos posibles soluciones. sin embargo 1 debemos antes establecer que reiterafj.gio la manipulación un ciert.o número de veces, obligadamente llegaremos a i.m punto fijo. en lo que sigue, trartairemos el primer caso; el segtmdo se deja al lector. r ecordemos qt'le er.j. esta s ituación se tiene j = q¡n + r ¡ con r ¡ i' o, n l. observemos t.ambión que como m (!) = f, se tiene j = q¡n+r¡ = kp+q¡. sea x = qn + r. vamos a demostrar que si x < f entonces x < m(x) ~ f (') para la segnnda parte de la desig11aldad 1 veamos qt·1e si qn + r = x < f = q¡n + r¡ (º) 20 un truco de naipes obligadamente q ( q1 , de donde m(x) = kp +q ( kp+qt =f. en cuanto a la primera parte de la desigualdad (•): • si q = ql el resultado es inmediato ya que en tal caso m(x) = f , • si q < q¡ t enemos: de donde: j x = (qt q) n + (r1 r), m(x) f = q qf, m(x) x = (q¡ q) (n 1) + (r¡ r). ahora bien , como q¡ q ;, 1 ya que q < q¡, r¡ r < n 1 ya que tj < n 1, se deduce de lo anterior que m(x) x >o, que es lo que nos proponíamos demostrar. en definitiva, inductivamente, hemos establecido que: x < m(x) ( m 2 (x) ( m 3 (x) ( ... ( f . de la misma manera se demuestra que si f < x entonces : f ( ... ( m 3 (x) ( m 2 (x) ( m(x) < x. esto quiere decir que iterando la función m sobre un elemento x se llega en un cierto número de pasos al punto fijo f. es inmediato que ese número está acotado por max {lf m( x)i , x < ln .p ) invito al lector a estudiar casos concretos. más aún 1 lo invito a modelizar este truco con la variante de repartir las cartas torn ando el mazo con las fignras hacia arriba. espero que la magia le abra caminos insospechados .. cubo 10, 15-21 (1994) octava jornada de matemática de la zona su r. soluciones analíticas de un problema de valor inicial y de contorno.* eric paredes u. resumen en este trabajo se cons idera un problema de valor lnicial y de contorno (p.v.l.c .), en el cual la ecuación diferencial parcial es de segundo orde n , lineal y de tipo hiperbólico. el p.v.i.c. corresponde a una generahzación de p roblemas ya tratados en forma parcial e n s piegel [9] y en myint [6j. el objetivo del autor es estudiar el p. v. lc., d eterminando soluciones analíticas. en particular, se obtienen condiciones suficientes para obtener dichas soluciones. 1 introducción consideremos el problem a d e valor inicial y de contorno siguiente: { u:cy = k (x )u., + l(x),x > 0 ,y > o (p ) , u(o, y ) ~ f(y) u,(x , o) ~ g(x) (p i) (p2) donde k, l, f y c son funciones conocidas. supongamos que el problema (p) ad mi t e un a sol u ción u(x,y) t1d que " "y= " y.· determin a r emos a continuación d ich a soluci6n. ' proyecto p 4--02mf-94, univcr11idnd de magallmcll \ ó 16 cubo 1 0 e. paredes u. sean p = u,,, y p,,_ = u~. entonces, la edp del problema (p) se puede escr ibi r en la forma : py k(x)p = l(x) la cual es una edol, cu ya solución es: p(x , y)= u,(x, y)= e\b1(x,y)b1 (x , o)jdx , entonces, (7) se puede escribir de la forma: u(x, y)~ b,(x , y) + b3 (x, y)+ a( y ). aplicando (pl), obtenemos: a(y) ~ f(y)-b.(o , y)-83(0 , y ) luego , utilizando (10) y ( l l) se tiene: u(x, y)~ jb,(x,11) b,.(o, y)j + [b3 (x, y ) 8 3 (0. y)] + f(y), utilizando (8) y (9) se obtiene: ~ 1' ek(t)> \b1(t , y)b1(t,o)]dt + 1' ek(tl'g(t)dj. + f(y) , utilizan do (3) se obtiene: = 1"' ek(i)y{¡y l(t)ek(l)•dz}dt + 1" ek(l)yc(t )dl + f(y). así hemos obtenido como solución del problema ( p ) la función: pode mos o bservar q ue : 17 (8) (9 ) (16) (11 ) p ara q ue ( 12) sea una so lución del problema ( p ) de b en s atisfacerse algunas condicio nes de integrabilidad en la.o:; fondones k , l y c: , por ejemplo: k, l y c: co ntinu as. propo s ición 1.1 la. s olución d el probl e m a (p) es 1ín.ica. 18 cubo 10 e. paredes u. demostración sc&n u 1 y u'l soluciones de (p) tales que 111 -::!u:2 .y cons ideremos la función : v(x, y ) = u 1(x,y)t.12 (x,y) entonces: { v.,, k (x )v, =o (q)' v(o, y ) =o v,(x , o) =o sean q = v,.. y q, = v~, . entonces ( 13) se puede escr ibi r en ja forma: q, k (x)q =o que es un a edol, c uya soluci6n es: q(x, y ) = v, (x, y) = c(x) e k«» , donde c(x) es a r bitraria. al integnu· ( 14) con respecto ax: v(x , y)= / c(x)e kl•l>dx + a (y), dond e a (y) es a r bit ra ri a. aphcando(qi) y (q2) a ( 14) y ( 15), obtcncmo5' c(x) = a(y) = o. luego v(x,y)= o es !w>luci6n del problema (q) y en consecuencia: 111 (x, y) = u2(x, y ) de ma nera que h1. solución del proble ma ( p} es linica. ( 13) (q i ) (q2) ( 14) ( 15) • proposición 1.2 sean 11,(x, y ) 110/u c i ó n d el problema de valo r inicial y de r:nnl.n r.,1n: so luciones anaut icas .. cub o 10 { u, , (p)' u(o, y) u. (x, o) k (x) u, + l, (x) f; (y) g;(x) dondei= 1,2,3,···, n entonces: u (x, y) = l u; (x , y), 1= \ es .soluc i ó n d el proble ma d e v alor i nicial y d e conto m .o: ¡-· k (x) u, + l l , (x ) (p• ) o u (o, y) l f; (y) '";;"\ u:r:(x, o} ¿ c;(x) d e m ostración el res ultado es consecuencia di recta de le. lineal idad de los opere.dores : p[u] t l zy k (x)u., p,[uj u (o, y) p2juj u,(x, o) en efecto: pjt u,j = t p ju;] = t j( u, ),, k(x )( u,), j = t l;(x ) •=1 •=l p,[t u;] = t p,ju;j = t u, (o, y)= t f,(y) ... 1 , .. , •= l p,jt •4j = t p2j u;] = t (u; ),(x , o) = t g,(x) ·1 1111: 1 ,,,. 1 por lo tanto, si u(x, y)= l u;(x, y) : 19 20 c ubo 10 p[uj = l d;(x) •=j p,[uj = l f; (y ) •o;;l p2 [uj = l g;(x) y así u(x,y) es solución de (p'"). • ahora q ue hemos obtenido un a solució n ge neral del prob lem a {p ) .r además hemos clcmostrado su u nicid ad, pode mos estudi ar algunos casos partic ula res. por eje mplo, s i k (:r: ) = k 0 , do nde ko e ir, el problemn el e valor inicial y de co nto rno : (qj , { ::(o. u) u,.(x, o) t ie ne .!!ol ución dada p or : kou,. + l(x) f (y) g(x ) u (x , y ) = { (y,jj; l(t.)~t + ck" f,' g(t )dl + f(y) , , ; y fo l (t )dj +fo gbt,) + f(y ) , , ; k 0 ;"o, ko = o ( 16) . de in expresión ( 16), podemos obt ener condiciones s uficientes para que la 90lución d e (q) se pueda obtener d e ma nera exacta. en efect o si la.s integrales ind.cfin idm: j l (t )dt y j g(t.)dl., ~n elemen tales (en el sent ido d e abellan a.s y c61indo l!j), e nto nces la solució n d fl.cla e n {16) se p uede calc ulrr de mane ra exactr.. en base al, res ultado ant.er io r, es p osible obtene r unr. ín mil ia infini ta no n umcr1:\blc de soluciones exacl a.s d e (q), e ligiend o a.decundnm ente lr.s fu nc iones l y g pa r a f' y ko arbitrarios. r efer e n c ias [l j calindo a. ,\fétodo8 de cálc ri lo. me. c: ru.whi ll ( 1990 ). /2] ducluneau p., zachmann d.w. e c u ncionc.!i oi/crr. ri c iafc.., p a rciales. to.le c: rawhill ( 1988). cubo 10 2 1 [3) edwards p. ecuacioncb diferenciales e lementales co n a plicaciones. &l . p rent ice hall ( 1985) . [4 j fa r low s.j. pa,rtial d i.fferential equ atione.s for· s cienlist and engin eers. ed . john w il ey-sons (1 9 82) . [5] j iménez f. e cu acione.s dif erenci ales en derivadas parciales. uni versidad de concepción. [6) l ieberstein h.m. theory of partial differenlial equa t i on.s. ed. aca.dernic r ,ess (19 72). [7} myin ty ty n, lokenat h o. partial differe.ntial equation.s jor scienti3t and engineers . ed nor.thholila.nd ( 1987) . [8j pe trovski i. g. lecci ones sobre ecuaciones en derivada..~ parciale!l. ciencia téc nica (~969) . [9] sneddon l. e lemenls of partial differentin./ e qun.t.ions . ed . me c raw-hill ( 1957). [1 0) spiegel murray r. applied differe.ntail equ ation.s. e. p. u: h. (1 962}. [11 ] tijonov a., sam arsky a. ecuaciones de la fís i ca matemática. ed . mir. moscli [12j weinberger h. e. e cuaciones dif eren c iales e n d er ivadas parciales. ed . revert é, s. a. [13] willia ms w .e. p rwtial dif]'er'en tlal equatio n.s . cla rcndorc p ress-oxford (1 980 ) . d ir ección d e l a u t o r: departa ment o de ma tem át ica y física un ive rs id ad d e magallancs cmil!a 113-d. p u nta arenas cubo a mathematicai joumal vol.05/~0s october 2003 control of dynamic oligopsonies with production factors laszlo kapolyi systems consulting rt, 1535 budapest, 114, pf. 709, hv.ngary and ferenc szidarovszky systems and industrial engineering department, university of a rízona, tucson, arízona 85721-0020,u.s.a. abst ract. oynamic o!igopsony is examined with discrete time scales. the controllability of t he dynamic system is investigated where the competitive price is select ed as the control variable. for the single product factor case complete characterization is given. a simple example illustrates that for the more general case no general solution can be given. we al so elaborated the case when the fixed price of the production factors is the control variable. we show that the system is uncontrol!able in this case. introduction oligopoly and related models have been examined intensively during the last 4-5 decades . this research area goes back to cournot (1838), who is considered the founder of this field. a comprehensive summary of results up to the mid seventies is presented in okuguchi (1976) . variants of the classical cournot model with applications to natural resources rnanagement are discussed in okuguchi and szidarovszky ( 1999). most studies considered the cornpetition among the firm in the product market only. however there is a competition among th.e firms in the factor market as well. oligopsonies include this kind of competition in the oligopo\y models, as it is shown in szidarovszky and okuguchi (2001). recently a d y namic <132 control of dynamic o/igopsonies with production fac tors ollgopsony model was introduced by kapolyi and szidarovszky {2001) , in which the cxist ence , uniqueness and the global asymptotical stability of the nash-cournot equilibrium wcre prove o a nd c1r > o for ali k, and d1 and d2 are both positive, the system is control\able if and only if d1 :f:. d2, or equivalently c 1 '# c2. if n ~ 3, then we will prove that the system is not 434 control of dynamic oligopsonies with production factors controllab le. notice first that h~ b(l-l) where 1 is the n x n identity matrix , and ali e lements of l are equal to l. then !i' ~ 1!_2 ((n 2)1 + l) whi ch is a linear combination of 1 and ji, and similarly, ll.3 ,li.4 , .• are ali linear combination s of l and lf.... therefo re lf..2r;., h.3f , .. are linear comb inations off and fu, so the rank of k is nt most two. hence the system is not controllabl e. next we show that for m > 1 we cannot give a general answer. to illustrate the problem assume that n = 2, and matrix 11.. is diagonal. so let then further more and fu~(~ ff)(~)~(¡~;) . !i'o ~ ( ~ ~ ) ( ~~'. ) ~ ( ~:~; ) ' therefore the kalman-matrix has the special form subtract t he a 2 -multiple of th e first column from the third column, and the a: 2 -multiple of t he seco nd co lumn from t he last column t o have matrix ( d, ad, dz {jd4 dj adi d4 fj d2 (13' a 2 )d2 p(ij' ~ a 2 )d, ) o o ((32 a2)d4 /3(/32 a2) d2 1 laszlo kapolyi & ferenc szidarovszky with determinant (expanded with respect to its last column) -/3({fl a 2)ch{,b2 a 2)d2(adi ad~) +/3(/32 a 2)d4(/32 a 2)d4 (adi ad~) = a/3(.82 a2)2(di d~)(d~ 4 ) 435 which is zero if o = /3, or d1 = d3 , o r d2 = d4 , otherwise nonzero. ln t he first case the system is not contro\lable, and in the second case it is. for nond iagonal f1 and n > 2 matrix k is more comp\icated , a nd therefore no general couditions can be given. 4 control by fixed labor cost consider again t he d ynamic system (3) and assume that t he fixed price g of t he production factors is controlled. lf al! factors are cont ro lled in the same way, then g has to be replaced by a· u(t ), where 11(t ) shows the control. if the prod uction factors are controlled differently, t hen !! is replaced by d iag (a1 , . . , um ):!!(t), where !!(t) is a n m -dimensiona l control vector . in these cases the coefficient mat rix 11. is the same as in the previous section, however d,i; (k = l , 2, ... , n ) has t o be replaced by either notice t hat for n 2: 2 matrix k has rank at most m , since its first m rows are identica l to the second m rows, which are the same as the third m rows, and so on. therefore the system is a\ways uncontrol\able. references (1] c ournot, a. , recherchessur les principies mathém a.tiq ues de la. théorie des richcsses, hachett e, paris . (english trans lation ( 1960) resear ches into t he mat hema tical principies of the theory of wealth, kelley, new yor k), 1838. [2j ükuguchi , k. , expectations a.nd stability in oligopoly models, springer-verlag , berlin/ heidelberg/new york, 1976. [3] ü kuguchi , k. ano szjoarovszky , f ., the tlieory of oligopoly witli mu/ti-product f'irms (2"d edition), springer-verlag, berlin/ he idelberg/ new york, 1999. 436 c0ntr0l ef dynamic oligopsonies with production fa.ctors [4] kapolyt, l. ano sz.rn>arovsz.ky, f., dynamic qligopolies with producbien factors, southwest j. of pur.e and appj.ied mat:h., vol. 2001, issue 2 (20011), pp. 73-76. [5] szidarovszky, f. ano bahill, a.t., linear systems theory. crc press, boca raton/londen (2"d edition), 1998. [6] szidarovszky, f. ani!l ükuguchi, k., dynamic analysis ofojigepsony under adaptive expeetat;ions, seuthwest j. of pure and applied math., vel. 20011, issue 2 (2001), pp. 53-60. cu eo , vo l. 2 ma,. 1 0 l9b6 pú g 75-82 ' 75 solu cio n a pr obl emas pr opues tos pro blema 1: ---------s i a> 1 , x > o , demu est r' e q u e , ~ §_~~!!!~~ .!. ~~ s ea y = 1_, e e n to n ces 0 < y < l ; tone mo s : -log [ 1( 1 -e'i a ¡= -log ( 1 yªi n a a y _y _ _ n:>o n + 1 a ( ")" < y ~ n > o ( n+ l) a ' l..___ n :>o n+l 7 5 lu::1ón a probl 1-log (l-, pero log ( 1-y) • ic yo qui~: 1-e log 1( 1-o-x l d j < xª demu~a re que o • ( -1 l n t ----º ( n-+ 1 ) n 1 l . " o • • j e" logxdx • o t(~,,dx o o o n 1 o n ! pe r o : r o n• o en •fc c t o: sra x •e . dx •-e dt , log n i (x log•) dx • o sc.i {n •l l • z -t e (-11" i e(n•l) ctt o i • o n t dt cubo, voll/l'etl 2 1 logx) n ( -ll n l ueg o : i (• dx ( n+l )n+l o .!. 1 n r ( -1 ) n t f (' log ,¡ dx o n! o o (n+l) n+ln ! así tenemos finalmente qu e : o ~¡ ( 1) n ~~-n-+-1 r (n+l) , ya qu e (n+l) n ! o !'.!!~~!:~~~-~ ' s ea f(x) e c 2 (0 , cd ) tal que f(o ) oy f"(x) >o dem uestr e que: al existe lim '•o b) _!l!.!. ü , o o , f co n tinua en f (t) dx es est r ic tam e nt e co n vexa . a) apl1ca mos l ag r a ng e e n ( o.x 1 ,o s ea ] e x e ( 0 , x) tal que f (x) r (ol • x f'(c•) . pero f (o) • o 77 78 solucl.ón a problemas propuestos luego f(d = • r· 1c , 1 , o 'º ª .!:_k_!_ . r· o => r • j llm f ' {cx) x .• o •-o mo n óton a en onc~$ : b) pu esto que f" (x)>o . te n e mo s que f ' (x ) t!s crecie nt e y ove! to q ue cx f ' (t) ~ => f " (tl t f . ( t 1f ( t 1 t2 pero t es pos1 t.ivo y de (b) sa be mo s qu e f' ( xl ~ ,.q div1d1en co por t > o te n e m os ~ ......!..!..!.. >o t 2 y la fun c1ón es co n vexa . f " ( t i > ~~~!!!;~~~-~.:. dada& tre s recta s paral e las , c o n s tru y a un tr 1 ángulo eau1látero co n vértices e n cada re c ta . se a n l 1 • l 2 , l 3 r e ct a s tal e• oue l¡ 11 l 2 11 l3 co n o (l 1 , l 2 l • a . d ~ l 2 , l 3 l = b . co n sideremos t:j abc ped lcjo c on 8 e l l, a cl3 y c e l2 • al fijar dos vé r tices se r esuelve el prob lema . inicia lmen t e fije mo s 8 c l 1 de modo que 85 l l 3 c o n ocl 3 85 () l 2 =1el sea i a bg: gd p o r t ha l es , l u e g o t:jb f g ;t:j bad 6 b he [ rec t á~ g ul os] para fi jar a se co n si d e r a la co n st ru cció n d e fg solo e n f u n e lón de los va lore s a y b. ab () l 2 fh ge s1e n do = { h} p o r thales fb g b gb a + b ge ~=~ s i f b !:. !:. b a como y y => fh b+a collo aabc es e qui láte r o => cf € .e sien do ~'=c h = ~ g bf (de igua l naturaleza y lados re s pe c ti v a !tiente perpend icu l ares) l cf h lbeh (recto s ) 80 solución a pl"obloous f>roque$ ento n ces po,.. a . a . de semeja nza, se ll t!ga a qut?: iicfh .. abgf donde cf hf bg fg => fg • lo que co ndu ce a que ad ~ ~-=-~ j3 o a 2 . {3 püra f¡jar c :i.e procede co n 0 (a , a6j {) t 2 de modo que c r 85 /a . 2~~2~.!~~~!2~: (kna l i t i e a l p, el . q . en 6 abc e a8 a bc = ac . sea l 3 el l!ji! de absc1&cts , ol cjt? dl! ordenadas o r igen . o el lu ego a ( !!.::!! . oj ; 13 sea n ~ b • ta cf &e repre&cnta dor : 9 ( 0 , ai·bl ; f {~:! 2 ./3 ~=~--de modo que l a ( a +bj · /3 r e~ le l y = b repre &cnta a la rec a l 2 . ento~ ces : b n+b e 1~j!!:~ 2 b-n + ---2 . .p l > 3(a+b> ~ .~ --22 r ~:!__ . b ) . .¡3 e u e o, voll.l't'en 2 lo e go : a s : j_i!!.=;}._~ + ( a+b) z ~' / a 2 + a b + b 2 v3 v se = j ( -;-(a+ b l b a 2 2 ) + a 1 13 ( a + 2 b) 2 + 3a 2 ac = j(!?.=; + -~-i!!;_:!?_}._ + b-a ) +b2 ~ v •2+ ab+ bz' ••• ó.a bc eq uilá tero co n a e l 3 , b el 1 , ce l 2 . dados dos trazo s de med ida s a y b / prime ro s e co n s truy e ad . se t raza l co n st ruy e n do e n ella : ba l' /' .. / .y .., / ~. e¿ / ·~ ' " \ t sean as = b ; ts = a => rt=b -a en t h a ga qt l l qt : rt r(+> ) q y v (r, ro) (\ l = 1 p l) en p 1 hag a ~ l l 0 1 p 1 ; rt fl.t " ' i "---' r (c->j0 1 y g) (r , r0 1 1() l = ¡p2 ¡ como rp 1 i/z ( b a l => rp 2 = /3 ( bal ~) gl, ¡ / i / "" 81 2 , / 2 2 (5v ª +ab+ b p, por r se t r az d l ' l , ubicando en l ' l os pu n tos a1 , r2 . r3 siendo a1 a2 r3 · 82 1 ión lp 2 y cona1oerando ~ 11 r3 p2 con •.i e •'> am de donde ao • am a "o n inuac ión •• > ¡ º" mooo oue lz l b genera ei lado ab y co n (a, a8 n 0 (8, i>.8) • ( c ) el problema lempr• tiene oluc1ón . de~car ando la pos1b1l1oao de aue la, tres rectas co1nc1oan . al 51 a • b •> ce l 2 paralela mud1a o a.b • 80 oues fo •ad = o ~t~noo la con1 rucc1ón del aabc tnmcdia a . b) s1 a oo b •o•> l 2 co1nc1dc con l 1 o b1en co nl 3 con lo c u~l 80 e ~ran9forma n la al ura del ~abc oed1do y com o bo • §_ · ab entonces tambié n la cona rucc 1ón es inmeoiata. 48 cubo matemática educacional vol. !. junio 1999 acerca de una definición débil de números normales d. m. pelleg rino universidade federal da parafba departamento de matemática e estatísti ca caixa postal 10044 cep 58.109.970 campina grande-pe-brasil e-mail: dmp@dme.ufpb.br abstract. un número real, representado en la base g(número natural diferente de 1) por b1 b2 ... bm 1 a 1 a2 ... an.. se dice que es casi g-normal, si la sucesión (a;) contiene, como subsucesión, cada posible sucesión de dígitos o, 1, ... , g l. esta es una definición primaria acer ca del concepto de número g-normal , introducido por e. borel en 1909. damos un método simple para obtener números casi y-no rmales. l. introduccion un número rea l w está representado en la base g (g natural, diferente de 1) si m oo w, = ¿ b,g'+ l:a, gi,a, , b, e {0 , ... , g-1} vj e in ,vi e {l, .. . ,m}. i=-1 j=l esta representación es única si establecemos que ª" < g 1 para una cantidad infini ta de ín di ces n. los números a; se llaman dígi t os . a lo largo del texto siempre consideraremos que esta representación es única. el número w, t ambién se representa como b1 ... bm 1 a 1 a2 •• usaremos la notación acerca de un a definición débil de números normales 49 m [w9 ] :::; l:bigi donde denom ina remos [w9 ] como la parte entera de w, represen· j = l tada en la base g. m un número real w 9 = [w 9 j + l: aig i se llama g-s imple normal si j = l li m s(w,,a,n) =~ \lae{o, .. , g -1} , n-oo n g donde s(w 9 , a , n) denota ~a cantidad de índices i , 1 ::;; i ::;; n, tales que a; :::; a. decimos que w9 = [w 9 ] + l:aig i es g-normal si w9 es g-simple normal y si j=l lim s(w,, b., n ) = _!_ , n ...... oo n gk para cada bloque de k dígitos b"= c1 c2 ... c.i. y para cada entero positivo k , donde s(w,.b.,n )=#{i; 1$i$n-k+l, ª <+;-.=e, \lj, 1$j$k). en otras palabras s(w 9 , b", n) denota el número de ocurrencias del bloque b"en los n primeros dígitos de w9 . intuitivamente, un número gnormal es un número cuyos dígitos y sucesión finita de dígitos están , en algún sentido, bien distribuidos. es conocido que casi todos los números en el sentido de la medida de lebesgue son números g-normales (ver f7]). desafortunadamente , cada número g-simple normal está siempre construido ad ho c. no sabemos si números como j2, j3 son números g-simples normales. en [l] podemos encontrar un estudio estadístico de la normalidad de tales números. no sabemos si existe algún número a lgebraico que sea normal respecto a alguna base g. un ejemplo canónico de un número 10-normal, debido a d. g . champernowne [2] es a= o, !234567891011 1213141516171 8192021.. m un número real w 9 :::; [w 9 ] + l:aig-i, representado en la base g 1 tal que j=l (a1 )~ 1 contiene , como una subsucesión finita , to da p osible sucesión finit a de dígitos, será ll amado un número casi g-normal. formalmente , w, es casi g-normal si (\l k e in ), (\lc, ... c, ; e, e {o, ... , g-1}\lj e {l , ... , k} ), 3i e jjv ; a,., _, =e,, \lj e {l , .. , k) 50 d. m. pellegrino est a es una condi ción débil ante la condición requerida de números g-normales. es cl a ro que cad a número g-normal es casi g-normal 1 mientras que lo recíproco no es ciert o. por supues to , el número f3 = o, 01002000300004 .. ~ 12 .. 12 veces es cas i lonormal1 pero no es 10-normal. existen números g-simples normales que no son casi y-normales. un ejemplo de un número 10-simple normal que no es casi 10normal es 1 = o, 012345678901234567890123456789 .. luego , el concepto de número casi g-normal no es tan débil comparado con el concepto de número g-simple normal. p uesto que casi todos los números son g-normales , entonces casi todos los números son casi g-normales. en todo caso, este resultado puede ser fácilmen te probado en forma independiente . en (5] hay una demostración simple y directa de este hecho. nuestro propósito es mostrar un método simple de construir números casi g normales. 2. resultados debido al art ículo de h. davenport y paul erdéis [3], sabemos que si f es un polinomio cuyos valores , paran = 1, 2, 3, ... son enteros , entonces oj(l) 10 /(2) 10 .. . /(n)10 es 10-normal. estos números son , a priori, números casi 10-normales. demostraremos que para ciertas funciones, el número o,[f(l ), ][! (2), ] .. . [f(n),] es casi g-normal . proposición l. sea g -::j:. 1 un número natural. sea f :]01 oo[---¡.jo, oo [ una función es trictame nte crecient e1 a partir de algún punto, continua, con primera derivada co ntinua1 tal que lim f( x) = oo. además, supongamos que dado m > 01 exis te xm e]o, oo[ tal que ~00 f( x) x > x" = f'(x + b) > m , v 8 e [o, l]. entonces w, = o, [f( l ),][/(2),].. es un número casi g-normal. de most ración . sea x 1 e]o , oo[ , t al que f es est rictamente creciente para cada z ?: z 1 . sea c1 • •. e,. una sucesión arbitraria finita de dígitos . podemos suponer c1 # o. si c1 = o, consideramos 1 c1 • .. c.,. sea m = gk+l . a cerca de una definición d ébil de n úm eros 'orma/es 51 existe x,.., > x 1 tal que f(x) x > x., ==> f'(x + 8) > m , v 8 e ¡o, l ]. entonces f(x) > g'+1 ·!'(x+8), vx > x,,, v 8 e ¡o, !]. (1 ) sea c1 c2 . .. c" oo .. o una sucesión con r ceros tales que el número c1 c2 .. c" oo .. o es mayor qu e f (xm). sea x3 > xm tal que j (xj) = c¡ c2 ... c"oo .. . o = c¡ g r+ k1 + c2 . gr+k-2 + ... +e" . gr . (2 ) observe que esto es posible puesto que j~~ j(x ) = oo. si x 3 es un entero posit ivo, la sucesión arbitraria c1 c2 ••• c" oo .. o aparece en [f(x 3 ) 11 ] y, a priori1 la sucesión c1 c2 ... c" ocurre en [f( x 3 )j si x3 no es un entero 1 por el teorema del valor medio , existe () e [01 1] tal que f(x , + 1) = f(x, ) + j'( x, + 8). luego, por ( ! ), (2 ) y (3) concluimos que f(x ,) f( x,+ 1)< f (x,) + g;;+l (3 ) =(el. gr +k1 + c2 . gr+k2 + ... + ck . gr )+ (el. gr-2 + ... +e" . gr-k 1) sea l el entero tal que x, < l < x, + l. entonces f (x,) < f( l) < f(x, + 1). así, tenemos k c¡gr+k1 + e2 gr+k -2 + .. + ekgt < f(l) < clgr+k 1 + e2gr+k-2 + .. + ekgr + ¿: c,gri-1 i= l y entonces f(l ) =e,. gr+k1 +e, . gr+k-2 + .. + c,. g' + i :>i. gr-j, con di e {o , 1, ... , g 1}. j=2 así , la sucesión de dígitos c1 • • e" ocurre en [f(l )11 l con lo cual conclui mos la demostración . • es fácil observar que cada polinomio satisface las bjpótesis de la proposición. pero esto no es sorpresa debido a l resultado de davenport y erdos. si f es un polinomio entero que es posit ivo para x 2: 11 entonces o,j{1)¡ 0 f(2)i 0 f(3 )10 .. es un 52 d. m . pellegrino número trascendental (ver [4]) . ¿deberían los otros números creados mediante esta proposición ser también números t rascendentales? funciones tales como f (x) = xº + lf=1 {3, · xcr; 1 a > a, > o 1 donde o , o, y {3, son números reales, f (x) = a(x,.) con a y ¡ números rea les tales que o < ¡ < 1 y a > 11 satisfacen las hipótesis de la proposición (si ¡ = 1 entonces g(x ) = 1ox1 no satisface nuestras hipótesis y aún más, o,g( l),,g(2),, .. no es casi jo-normal). es posible que en alguno (o todos ) los casos, el número o,[f( l ),jl/ (2), ) .. . [f(n ),j sea g-normal, pero no lo sabemos (excepto para los polinomios). trivial , pero importante como ejemplo, es considerar la función h(x) = log 0 x , (a> 1). es obvio que <;, = o, [h( l ),jlh(2 ),) es un número casi g -normal (aún sin necesidad de la proposición), pero podemos, con un pequeño esfuerzo, observar que este número tampoco es g-simple norm al. por cierto , si g = 10 y a = 10, <;,,= o, t;jl .l;j 9 veces 90 veces y p odemos ver que lim s (~1 º 1 11 n) # ~ y1 además , este lími t e no existe. por tanto1 n -.oo n 10 concluimos que alguno de nu estros ejemplos no son números g normales. es posible también prob ar que si f y g sati sfacen las hipótesis de la proposición , entonces f +9 y f · g t ambién la satis facen. refer e ncias jl j b eye r , w . a. , metropo lis, n., a nd neerdgaard , j. r. , statistical study of digits of some square roots of int egers in various bases, math. comp. 24 1 455-4 73 (1970). 12] c h am p e rnown e, d. g. 1 the construction of decimals in the scale of ten, j. london lath. soc., 8, 254-260 (1933). 13] davenport , h. , a nd e rdos, p. , note on normal decimals, canad. j. math. 4, 5 -63 (1960). [4] ge lfo nd , a. o ., transcendental & algebraic numbe rs , dover publications1 loe. new york (1960). is) hard y, g. h. , a nd wright , e . m. , a n i ntroduction to the theory of num· bers, fou rth ed it ion 1 ox ford university press1 london (1975). (6 j p e ll eg rino , d.m., produto de medidas e tópicos em representaroes g-ádic.as de números reais, dissertat;ao de mestrado, imecc-unicamp (1998). [7) r é ny i, a ., foundations of probability, holden day, !ne (1970). cubo, a mathematical journal vol.22, n◦03, (325–350). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300325 received: 20 may, 2020 | accepted: 22 october, 2020 existence and attractivity theorems for nonlinear hybrid fractional integrodifferential equations with anticipation and retardation bapurao c. dhage kasubai, gurukul colony, thodga road, ahmedpur-413 515, dist. latur, maharashtra, india. bcdhage@gmail.com abstract in this paper, we establish the existence and a global attractivity results for a nonlinear mixed quadratic and linearly perturbed hybrid fractional integrodifferential equation of second type involving the caputo fractional derivative on unbounded intervals of real line with the mixed arguments of anticipations and retardation. the hybrid fixed point theorem of dhage is used in the analysis of our nonlinear fractional integrodifferential problem. a positivity result is also obtained under certain usual natural conditions. our hypotheses and claims have also been explained with the help of a natural realization. resumen en este art́ıculo, se establecen resultados de existencia y de atractividad global para una ecuación no lineal cuadrática mixta e h́ıbrida fraccionaria integrodiferencial linealmente perturbada de segundo tipo involucrando la derivada fraccional de caputo en intervalos no acotados de la recta real con argumentos mixtos de anticipación y retardo. el teorema de punto fijo h́ıbrido de dhage es usado en el análisis de nuestro problema no lineal fraccionario integrodiferencial. también se obtiene un resultado de positividad bajo ciertas condiciones naturales usuales. nuestras hipótesis y afirmaciones también se explican con la ayuda de una realización natural. keywords and phrases: hybrid fractional integrodifferential equation, dhage fixed point theorem, existence theorem, attractivity of solutions, asymptotic stability. 2020 ams mathematics subject classification: 34k10, 47h10. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300325 326 bapurao c. dhage cubo 22, 3 (2020) 1 introduction let t0 ∈ r be a fixed real number and let j∞ = [t0,∞) be a closed but unbounded interval in r. let crb(j∞) denote the class of pulling functions a : j∞ → (0,∞) satisfying the following properties: (i) a is continuous, and (ii) lim t→∞ a(t) = ∞. the notion of the pulling function is introduced in dhage [15, 17] and dhage et al. [21]. there do exist functions a : j∞ → (0,∞) satisfying the above two conditions. in fact, if a1(t) = |t| + 1, a2(t) = e |t|, then a1,a2 ∈ crb(j∞). again, the class of continuous and strictly monotone functions a : j∞ → (0,∞) going to ∞ satisfy the above criteria. note that if a ∈ crb(j∞), then the reciprocal function a : j∞ → r+ defined by a(t) = 1 a(t) is continuous and lim t→∞ a(t) = 0. it has been shown in dhage [16, 18, 19, 20] and dhage et. al [21] that the pulling functions are useful in proving different asymptotic characterizations of the solutions of nonlinear differential and integral equations. in this paper we employ the pulling functions for characterizing the solutions of a nonlinear hybrid fractional differential equation when the value of independent variable is large. it is now well-known that several anomalous real world problems in sciences and engineering are adequately modelled on fractional differential equations (see hilfer [25] and kilbas et. al [27]). sometimes one may be interested in the behaviour of the anomalous dynamic system in the long duration of time which depend upon both past history as well as the future data of the process in question. in such cases, we take help of fractional differential equations with retardatory and anticipatory arguments on the unbounded intervals of real line. motivated by this reason, in this paper we discuss asymptotic behaviour of a nonlinear hybrid fractional integrodifferential equation with retardation and anticipation on the unbounded intervals via hybrid fixed point theory of dhage [8, 9, 16]. we need the following fundamental definitions from fractional calculus (see podlubny [28], kilbas et al. [27] and references therein) in what follows. definition 1.1. if j∞ = [t0,∞) be an interval of the real line r for some t0 ∈ r with t0 ≥ 0, then for any x ∈ l1(j∞,r), the riemann-liouville fractional integral of fractional order q > 0 is defined as i q t0 x(t) = 1 γ(q) ∫ t t0 x(s) (t − s)1−q ds, t ∈ j∞, provided the right hand side is pointwise defined on (t0,∞), where γ is the euler’s gamma function defined by γ(q) = ∫ ∞ 0 e−ttq−1 dt. cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 327 definition 1.2. if x ∈ cn(j∞,r), then the caputo fractional derivative cd q t0 x of x of fractional order q is defined as cd q t0 x(t) = 1 γ(n − q) ∫ t t0 (t − s)n−q−1x(n)(s)ds, t ∈ j∞, where n − 1 < q ≤ n, n = [q] + 1, [q] denotes the integer part of the real number q, and γ is the euler’s gamma function. here cn(j∞,r) denotes the space of real valued functions x(t) which are n times continuously differentiable on j∞. given a pulling function a ∈ crb(j∞) ⋂ c1(j∞,r), we consider the following nonlinear hybrid fractional integrofractional differential equation (in short hfrigde) involving the caputo fractional derivative, cd q t0       a(t)x(t) − m ∑ j=1 iαjhj(t,x(t),x(η(t))) f(t,x(t),x(θ(t)))       = g(t,x(t),x(γ(t))), t ∈ j∞, x(t0) = x0,                (1.1) where cd q t0 is the caputo fractional derivative of fractional order 0 < q ≤ 1, iαj are the riemannlouville fractional integration of fractional order αj ≥ 0 for j = 1, . . . ,m, f : j∞×r×r → r\{0}, hj : j∞ ×r×r → r are continuous, g : j∞ ×r×r → r is carathéodory and η,θ,γ : j∞ → j∞ are the continuous functions such that η and θ are anticipatory and γ is retardatory, that is, η(t) ≥ t, θ(t) ≥ t and γ(t) ≤ t for all t ∈ j∞ with η(t0) = t0 = θ(t0). definition 1.3. by a solution for the hybrid fractional differential equation (1.1) we mean a function x ∈ c1(j∞,r) such that (i) the map (x,y,z) 7→ a(t)x − σmj=1i αjhj(t,x,z) f(t,x,y) is well defined for each t ∈ j∞, (ii) the map t 7→ a(t)x(t) − σmj=1i αjhj(t,x(t),x(θ(t))) f(t,x(t),x(θ(t))) = z(t) is differentiable on j∞ and z ′ ∈ c(j∞,r), and (iii) x satisfies the equations in (1.1) on j∞, where c1(j∞,r) is the space of continuous real-valued functions defined on j∞ whose first derivative x′ exists and x′ ∈ c(j∞,r). as the functions θ and γ in the hfrigde (1.1) are respectively anticipatory and retardatory, the arguments in the problem (1.1) are deviating over the unbounded interval j∞. therefore, the behaviour of the dynamic system modelled on the hfrigde (1.1) depends upon both back 328 bapurao c. dhage cubo 22, 3 (2020) history as well as future data. as a result the existence analysis of the hfrigde (1.1) involves both anticipation and retardation information of the state variable. in a nutshell, the hfrigde (1.1) is a nonlinear problem with anticipation and retardation. the hfrigde (1.1) is a mixed linear and quadratic perturbation of second type obtained by multiplying the unknown function under caputo derivative with a scalar function a together with a subtraction of the term containing unknown function and dividing by a nonlinearity f. the classification of the different types of perturbations of a differential equation is given in dhage [6]. when hj ≡ h on j∞ × r × r, the hfrigde (1.1) reduces to the nonlinear ordinary quadratic caputo fractional differential equation, cd q t0 [ a(t)x(t) − h(t,x(t),x(η(t))) f(t,x(t),x(θ(t))) ] = g(t,x(t),x(γ(t))), t ∈ j∞, x(t0) = x0,      (1.2) which again, when hj ≡ 0, includes the class of the nonlinear quadratic caputo fractional differential equations cd q t0 [ a(t)x(t) f(t,x(t),x(θ(t))) ] = g(t,x(t),x(γ(t))), t ∈ j∞, x(t0) = x0,      (1.3) as a special case. the hfrigde (1.2) is new to the literature whereas the hfrigde (1.3) is studied in dhage [18] for existence and attractivity of the solutions on unbounded interval j∞. when f(t,x,y) = 1 and g(t,x,y) = g(t,x) for all (t,x,y) ∈ j∞ × r × r, we obtain the following caputo fractional differential equation, cd q t0 [a(t)x(t)] = g(t,x(t)), t ∈ j∞, x(t0) = x0 ∈ r. } (1.4) the equation (1.4) is studied in dhage [17] for existence, uniqueness and asymptotic attractivity and stability of solutions via classical fixed point theory. we note that when q = 1, the hybrid fractional differential equations (in short hfrdes) (1.2), (1.3) and (1.4) reduce to the ordinary nonlinear hybrid differential equations, d dt [ a(t)x(t) − h(t,x(t),x(η(t))) f(t,x(t),x(θ(t))) ] = g(t,x(t),x(γ(t))), t ∈ j∞, x(t0) = x0,      , (1.5) d dt [ a(t)x(t) f(t,x(t),x(θ(t))) ] = g(t,x(t),x(γ(t))), t ∈ j∞, x(t0) = x0,      (1.6) cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 329 and d dt [a(t)x(t)] = g(t,x(t)), t ∈ j∞, x(t0) = x0 ∈ r,    (1.7) which are discussed in dhage [17], dhage [18] and [15] respectively. the hybrid differential equation (1.7) also includes the nonlinear differential equation treated in burton and furumochi [4] as the special case. therefore the existence and attractivity results of this paper include the similar results for the ordinary nonlinear hybrid classical and fractional differential equations (1.2) through (1.7) as special cases. now we state a couple of well-known results fractional calculus which are helpful in transforming the caputo fractional differential equations into riemann-louville fractional integral equations and vice versa. lemma 1.1 (kilbas et al. [27]). suppose that x ∈ cn(j,r) and q ∈ (n − 1,n), n ∈ n. then, the general solution of the fractional differential equation cd q t0 x(t) = 0 is given by x(t) = c0 + c1(t − t0) + c2(t − t0) 2 + · · · + cn−1(t − t0) n−1 for all t ∈ j, where ci, i = 0,1, . . . ,n − 1 are constants and c n(j,r) is the space of n times continuously differentiable real-valued functions defined on j = [a,b]. lemma 1.2. (kilbas et al. [27, page 96]) let x ∈ cn(j,r) and q > 0. then, we have i q t0 ( cd q t0 x(t) ) = x(t) − n−1 ∑ k=0 x(k)(t0) k ! (t − t0) k = x(t) + n−1 ∑ k=0 ck(t − t0) k for all t ∈ j = [a,b], where n − 1 < q ≤ n, n = [q] + 1 and c0, . . . ,cn−1 are constants. the converse of the above lemma is not true. it is mentioned in kilbas et al. [27, page 95] that if q > 0 and x ∈ c(j,r), then cd q t0 ( i q t0 x(t) ) = x(t) for all t ∈ j = [a,b], however it has been proved recently in cohen and salem [1, 2] that it is not true for any continuous function on j. remark 1.1. the conclusion of the above lemmas 1.1 and 1.2 also remains true if we replace the function spaces cn([a,b],r) and c([a,b],r) with the function spaces bcn(j∞,r) and bc(j∞,r) respectively. 2 auxiliary results let x be a non-empty set and let t : x → x. an invariant point under t in x is called a fixed point of t , that is, the fixed points are the solutions of the functional equation t x = x. any 330 bapurao c. dhage cubo 22, 3 (2020) statement asserting the existence of fixed point of the mapping t is called a fixed point theorem for the mapping t in x. the fixed point theorems are obtained by imposing the conditions on t or on x or on both t and x. by experience, better the mapping t or x, we have better fixed point principles. as we go on adding richer structure to the non-empty set x, we derive richer fixed point theorems useful for applications to different areas of mathematics and particularly to nonlinear differential and integral equations. below we give some fixed point theorems useful in establishing the attractivity and ultimate positivity of the solutions for hfrigde (1.1) on unbounded intervals. before stating these results we give some preliminaries. definition 2.1 (dhage [8, 9, 10]). an upper semi-continuous and nondecreasing function ψ : r+ → r+ satisfying ψ(0) = 0 is called a d-function on r+. let x be an infinite dimensional banach space with the norm ‖ · ‖. a mapping t : x → x is called d-lipschitz if there is a d-function ψt : r+ → r+ satisfying ‖t x − t y‖ ≤ ψt (‖x − y‖) (2.1) for all x,y ∈ x. if ψt (r) = k r, k > 0, then t is called lipschitz with the lipschitz constant k. in particular, if k < 1, then t is called a contraction on x with the contraction constant k. further, if ψt (r) < r for r > 0, then t is called nonlinear d-contraction and the function ψt is called d-function of t on x. there do exist d-functions and the commonly used d-functions are ψt (r) = k r and φ(r) = r 1 + r , etc. (see banas and dhage [3] and the references therein). definition 2.2. an operator t on a banach space x into itself is called totally bounded if for any bounded subset s of x, t (s) is a relatively compact subset of x. if t is continuous and totally bounded, then it is called completely continuous on x. the operator theoretic technique is a powerful method often times used in the analysis of different types of nonlinear equations. our essential tool used in the chapter is the following hybrid fixed point theorem of dhage [9, 16] for a quadratic operator equation involving three operators in a banach algebra x which uses arguments from analysis and topology. see also dhage [6, 7, 9, 16] and dhage and o’regan [22] for some related results and applications. theorem 2.1 (dhage fixed point theorem [9, 16]). let s be a non-empty, closed convex and bounded subset of the banach algebra x and let a,c : x → x and b : s → x be three operators such that (a) a and c are d-lipschitz with d-functions ψa and ψc respectively, (b) b is completely continuous, cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 331 (c) mb ψa(r) + ψc(r) < r, where mb = ‖b(s)‖ = sup{‖bx‖ : x ∈ s}, and (d) x = axby + cx =⇒ x ∈ s for all y ∈ s. then the operator equation axbx + cx = x has a solution in s. the above hybrid fixed point theorem of dhage is a fifth important operator theoretic technique or tool that used in the subject of nonlinear analysis in line with banach, schauder, krasnoselskii and dhage (see [23],[5]). the nonlinear alternatives related to dhage fixed point theorem, theorem 2.1 on the lines of leray-schauder and schafer are also available in the literature (see dhage [7, 8, 9, 10] and references therein), however the present version is more convenient to apply in the theory of nonlinear hybrid differential equations. a collection of a good number of applicable fixed point theorems may be found in the monographs of granas and dugundji [23], deimling [5], dhage [16] and the references therein. in the following section we give different types of characterizations of the solutions for nonlinear fractional integrodifferential equations on unbounded intervals of the real line. 3 characterizations of solutions we seek solutions of the hfrigde (1.1) in the function space bc(j∞,r) of continuous and bounded real-valued functions defined on j∞. define a standard supremum norm ‖ · ‖ and a multiplication “ · ” in bc(j∞,r) by ‖x‖ = sup t∈j∞ |x(t)| and (x · y)(t) = (xy)(t) = x(t)y(t), t ∈ j∞. clearly, bc(j∞,r) becomes a banach algebra w.r.t. the above norm and the multiplication. let a,b,c : bc(j∞,r) → bc(j∞,r) be three continuous operators and consider the following operator equation in the banach algebra bc(j∞,r), ax(t) bx(t) + cx(t) = x(t) (3.1) for all t ∈ j∞. below we give different characterizations of the solutions for the operator equation (3.1) in the function space bc(j∞,r). definition 3.1. we say that solutions of the operator equation (3.1) are locally attractive if there exists a closed ball br(x0) in the space bc(j∞,r) for some x0 ∈ bc(j∞,r) such that for arbitrary solutions x = x(t) and y = y(t) of equation (3.1) belonging to br(x0) we have that lim t→∞ (x(t) − y(t)) = 0. (3.2) 332 bapurao c. dhage cubo 22, 3 (2020) in the case when the limit (3.2) is uniform with respect to the set br(x0), i.e., when for each ε > 0 there exists t > 0 such that |x(t) − y(t)| ≤ ǫ (3.3) for all x,y ∈ br(x0) being solutions of ((3.1) and for t ≥ t, we will say that solutions of equation (3.1) are uniformly locally attractive on j∞. definition 3.2. a solution x = x(t) of equation (3.1) is said to be globally attractive if (3.2) holds for each solution y = y(t) of (3.1) in bc(j∞,r). in other words, we may say that solutions of the equation (3.1) are globally attractive if for arbitrary solutions x(t) and y(t) of (3.1) in bc(j∞,r), the condition (3.2) is satisfied. in the case when the condition (3.2) is satisfied uniformly with respect to the space bc(j∞,r), i.e., if for every ǫ > 0 there exists t > 0 such that the inequality (3.2) is satisfied for all x,y ∈ bc(j∞,r) being the solutions of (3.1) and for t ≥ t, we will say that solutions of the equation (3.1) are uniformly globally attractive on j∞. remark 3.1. let us mention that the details of the global attractivity of solutions may be found in a recent paper of hu and yan [26] while the concepts of uniform local and global attractivity (in the above sense) may be found in banas and dhage [3], dhage [10, 12, 13] and references therein. now we introduce the new concept of local and global ultimate positivity of the solutions for the operator equation (3.1) in the space bc(j∞,r). definition 3.3 (dhage [11]). a solution x of the equation (3.1) is called locally ultimately positive if there exists a closed ball br(x0) in the space bc(j∞,r) for some x0 ∈ bc(j∞,r) such that x ∈ br(0) and lim t→∞ [ |x(t)| − x(t) ] = 0. (3.4) in the case when the limit (3.4) is uniform with respect to the solution set of the operator equation (3.1) in bc(j∞,r), i.e., when for each ε > 0 there exists t > 0 such that | |x(t)| − x(t)| ≤ ǫ (3.5) for all x being solutions of (3.1) in bc(j∞,r) and for t ≥ t, we will say that solutions of equation (3.1) are uniformly locally ultimately positive on j∞. definition 3.4 (dhage [13]). a solution x ∈ bc(j∞,r) of the equation (3.1) is called globally ultimately positive if (3.4) is satisfied. in the case when the limit (3.5) is uniform with respect to the solution set of the operator equation (3.1) in bc(j∞,r), i.e., when for each ε > 0 there exists t > 0 such that (3.5) is satisfied for all x being solutions of (3.1) in in bc(j∞,r) and for t ≥ t, we will say that solutions of equation (3.1) are uniformly globally ultimately positive on j∞. cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 333 finally, we have the the following characterization of the asymptotic stability of the solution of the equation (3.1) on j∞. definition 3.5. a solution of the equation (3.1) is called asymptotically stable to t-axis or zero if limt→ x(t) = 0. again, x is called uniformly asymptotically stable to zero if for ǫ > 0 there exists a real number t ≥ t0 such that |x(t)| ≤ ǫ for all t ≥ t. remark 3.2. we note that global attractivity implies the local attractivity and uniform global attractivity implies the uniform local attractivity of the solutions for the operator equation (3.1) on j∞. similarly, global ultimate positivity implies local ultimate positivity of the solutions for the operator equation (3.1) on an unbounded interval j∞. however, the converse of the above two statements may not be true. 4 attractivity and positivity results now, in this section, we discuss the attractivity results for the ordinary hybrid functional fractional integrodifferential equation (1.1) on j∞. we need the following definition in the sequel. definition 4.1. a function β : j∞ × r × r → r is called carathéodory if (i) the map t 7→ β(t,x,y) is measurable for each x,y ∈ r, and (ii) the map (x,y) 7→ β(t,x,y) is jointly continuous for each t ∈ j∞. the following lemma is often used in the study of nonlinear differential equations (see granas et al. [24] and references therein). lemma 4.1 (carathéodory). let β : j∞ × r× r → r be a carathéodory function. then the map (t,x,y) 7→ β(t,x,y) is jointly measurable. in particular the map t 7→ β(t,x(t),y(t)) is measurable on j∞ for all x,y ∈ c(j∞,r). we need the following hypotheses in the sequel. (a1) the function f is continuous and there exists a function ℓ ∈ bc(j∞,r+) and a constant k > 0 such that ∣ ∣f(t,x1,x2) − f(t,x1,x2) ∣ ∣ ≤ ℓ(t) max{|x1 − x2|, |x2 − y2|} k + max{|x1 − x2|, |x2 − y2|} for all t ∈ j∞ and x1,x2,y1,y2 ∈ r. moreover, sup t∈j∞ ℓ(t) = l. (a2) the function t 7→ |f(t,0,0)| is bounded with bound f . 334 bapurao c. dhage cubo 22, 3 (2020) (b1) the function g is carathéodory and bounded on j∞ × r × r with bound mg. (c1) the functions hj are continuous and there exist a functions ℓj ∈ bc(j∞,r+) and a constants kj > 0 such that ∣ ∣hj(t,x1,x2) − hj(t,x1,x2) ∣ ∣ ≤ ℓj(t) max{|x1 − x2|, |x2 − y2|} kj + max{|x1 − x2|, |x2 − y2|} for all t ∈ j∞ and x1,x2,y1,y2 ∈ r, where j = 1, . . . ,m. moreover, sup t∈j∞ ℓj(t) = lj. (c2) the function t 7→ |hj(t,0,0)| is bounded with bound hj. (d1) the pulling function a satisfies lim t→∞ a(t)tq = 0 = lim t→∞ a(t)tαj for each j = 1, . . . ,m. remark 4.1. if a ∈ crb(j∞), then a ∈ bc(j∞,r+) and so the number ‖a‖ = supt∈j∞ a(t) exists. again, since the hypothesis (d1) holds, the function w : r+ → r+ defined by the expression w(t) = a(t)tq is continuous on j∞ and satisfies the relation lim t→∞ w(t) = 0. so the number w = supt≥t0 w(t) exists. similarly, the function wj : r+ → r+ defined by the expression wj(t) = a(t)t αj is continuous on j∞ and satisfies the relation lim t→∞ wj(t) = 0 for each j = 1, . . . ,m. hence, the number wj = supt≥t0 wj(t) exists for each j = 1, . . . ,m. the following lemma is useful in the sequel. lemma 4.2. if for any function h ∈ l1(j∞,r), the function x ∈ bc(j∞,r) is a solution of the hfrigde cd q t0 [ a(t)x(t) − ∑m j=1 i αj hj(t,x(t),x(η(t)) f(t,x(t),x(θ(t))) ] = h(t), t ∈ j∞, (4.1) and x(0) = x0, (4.2) then x satisfies the hybrid fractional integral equation (in short hfrie) x(t) = [ f(t,x(t),x(θ(t))) ] ( c0 a(t) + a(t) γq ∫ t t0 (t − s)q−1h(s)ds ) + a(t) m ∑ j=1 iαjhj(t,x(t),x(η(t)) (4.3) for all t ∈ j∞, where c0 = a(t0)x0 f(t0,x0,x0) and x0 6= 0. proof. let h ∈ l1(j∞,r). assume first that x is a solution of the hfrigde (4.1)-(4.2) defined on j∞ and x0 6= 0. we apply the riemann-liouville fractional integration i q t0 of fractional order q from t0 to t on both sides of the hfrigde (4.1). then, by an application lemma 1.2, the hfrigde (4.1)-(4.2) is transformed into the hfrie (4.3) on j∞. cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 335 definition 4.2. a solution x ∈ bc(j∞,r) of the frie (4.3) is called a mild solution of the hfrigde (4.1)-(4.2) defined on j∞. in the following we shall deal with the mild solution of the hfrigde (1.1) on unbounded interval j∞ of the real line r. our main existence and global attractivity result is as follows. theorem 4.1. assume that the hypotheses (a1) (a2), (b1), (c1) (c2) and (d1) hold. further, assume that (m + 1)· max { l ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) , l1 w1 γ(α1) , . . . , lm wm γ(αm) } ≤ min{k,k1, . . . ,km}. (4.4) then the hfrigde (1.1) has a mild solution and mild solutions are uniformly globally attractive defined on j∞. proof. now, using lemma 4.2, it can be shown that the mild solution x of the hfrigde (1.1) is equivalent to the nonlinear hybrid fractional integral equation (in short hfrie) x(t) = [ f(t,x(t),x(θ(t))) ] ( c0 a(t) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ) + a(t) m ∑ j=1 iαjhj(t,x(t),x(η(t)) (4.5) for all t ∈ j∞, where c0 = a(t0)x0 f(t0,x0,x0) . set x = bc(j∞,r) and define a closed ball br(0) in x centered at origin of radius r given by r = ( l + f ) ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) + m ∑ j=1 lj + hj γ(αj) wj. define three operators a and c on x and b on br(0) by ax(t) = f(t,x(t),x(θ(t))), t ∈ j∞, (4.6) bx(t) = c0a(t) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds, t ∈ j∞ (4.7) and cx(t) = a(t) m ∑ j=1 iαj hj(t,x(t),x(η(t)), t ∈ j∞. (4.8) then the hfrie (4.5) is transformed into the operator equation as ax(t) bx(t) + cx(t) = x(t), t ∈ j∞. (4.9) 336 bapurao c. dhage cubo 22, 3 (2020) we show that the operators a, b and c satisfy all the conditions of theorem 2.1 on bc(j∞,r). first we we show that the operators a, b and c define the mappings a,c : x → x and b : br(0) → x. let x ∈ x be arbitrary. obviously, ax is a continuous function on j∞. we show that ax is bounded on j∞. thus, if t ∈ j∞, then we obtain: |ax(t)| = |f(t,x(t),x(θ(t)))| ≤ |f(t,x(t),x(θ(t))) − f(t,0,0)| + |f(t,0,0)| ≤ ℓ(t) max{|x(t)|, |x(θ(t))|} k + max{|x(t)|, |x(θ(t))|} + f ≤ l + f. therefore, taking the supremum over t, ‖ax‖ ≤ l + f = n. thus ax is continuous and bounded on j∞. as a result ax ∈ x. again, we have ∣ ∣cx(t) ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ ∣ a(t) m ∑ j=1 iαjhj(t,x(t),x(η(t))) − a(t) m ∑ j=1 iαj hj(t,0,0) ∣ ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∣ a(t) m ∑ j=1 iαj hj(t,0,0) ∣ ∣ ∣ ∣ ∣ ∣ ≤ a(t) m ∑ j=1 iαj ∣ ∣h(t,x(t),x(η(t))) − h(t,0,0) ∣ ∣ + a(t) m ∑ j=1 iαj ∣ ∣hj(t,0,0) ∣ ∣ ≤ a(t) m ∑ j=1 iαj ℓj(t) max{|x(t)| , |x(η(t))|} kj + max{|x(t)| , |x(η(t))|} + a(t) m ∑ j=1 iαjhj ≤ a(t) m ∑ j=1 iαj ℓj(t) ‖x‖ kj + ‖x‖ + a(t) m ∑ j=1 iαjhj ≤ a(t) m ∑ j=1 iαjlj + a(t) m ∑ j=1 iαjhj ≤ m ∑ j=1 lj γ(αj) wj + m ∑ j=1 hj γ(αj) wj ≤ m ∑ j=1 lj + hj γ(αj) wj for all t ∈ t∞. taking the supremum over t as t → ∞, we obtain ‖cx‖ ≤ m ∑ j=1 lj + hj γ(αj) wj. as a result (cx) is continuous and bounded on j∞. hence, cx ∈ x. similarly, it can be shown that bx ∈ x and in particular, a,c : x → x and b : br(0) → x. we show that a is a lipschitz cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 337 on x. let x,y ∈ x be arbitrary. then, by hypothesis (a1), ‖ax − ay‖ = sup t∈j∞ |ax(t) − ay(t)| ≤ sup t∈j∞ ℓ(t) max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|} k + max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|} ≤ l‖x − y‖ k + ‖x − y‖ = ψa(‖x − y‖) for all x,y ∈ x. this shows that a is a d-lipschitz on x with d-function ψa(r) = lr k + r . similarly, by hypothesis (c1). we have ‖cx − cy‖ = sup t∈j∞ |cx(t) − cy(t)| ≤ sup t∈j∞ a(t) m ∑ j=1 iαj ∣ ∣hj(t,x(t),x(η(t))) − hj(t,y(t),y(η(t)) ∣ ∣ ≤ sup t∈j∞ a(t) m ∑ j=1 iαj ℓj(t) max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|} kj + max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|} ≤ sup t∈j∞ a(t) m ∑ j=1 iαj lj ‖x − y‖ kj + ‖x − y‖ ≤ m ∑ j=1 lj wj γ(αj) ‖x − y‖ kj + ‖x − y‖ ≤ m ∑ j=1 wj γ(αj) · lj ‖x − y‖ kj + ‖x − y‖ ≤ m · max { l1 w1 γ(α1) , . . . , lm wm γ(αm) } ‖x − y‖ min{k1, . . . ,km} + ‖x − y‖ this shows that c is a d-lipschitz on x with d-function ψc(r) given by ψc(r) = m · max { l1 w1 γ(α1) , . . . , lm wm γ(αm) } r min{k1, . . . ,km} + r . next, we shows that b is a completely continuous operator on br(0). first, we show that b is continuous on br(0). to do this, let us fix arbitrarily ǫ > 0 and let {xn} be a sequence of points 338 bapurao c. dhage cubo 22, 3 (2020) in br(0) converging to a point x ∈ br(0). then we get: |(bxn)(t) − (bx)(t)| ≤ a(t) γq ∫ t t0 (t − s)q−1|g(s,xn(s),xn(γ(s))) − g(s,x(s),x(γ(s)))|ds ≤ a(t) γq ∫ t t0 (t − s)q−1[|g(s,xn(s),xn(γ(s)))| + |g(s,x(s),x(γ(s)))|]ds ≤ 2mg a(t) γq ∫ t t0 (t − s)q−1 ds = 2mg γq · w(t), where, w(t) = a(t)tq. hence, by virtue of hypothesis (d1), we infer that there exists a t > 0 such that w(t) ≤ ǫ for t ≥ t . thus, for t ≥ t , from the estimate (3.3) we derive that |(bxn)(t) − (bx)(t)| ≤ 2mg γq ǫ as n → ∞. furthermore, let us assume that t ∈ [t0,t ]. then, by dominated convergence theorem, we obtain the estimate: lim n→∞ bxn(t) = lim n→∞ [ c0a(t) + a(t) γq ∫ t t0 (t − s)q−1g(s,xn(s),xn(γ(s)))ds ] = c0a(t) + a(t) γq ∫ t t0 (t − s)q−1 [ lim n→∞ g(s,xn(s),xn(γ(s))) ] ds = c0a(t) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds = bx(t) for all t ∈ [t0,t ]. moreover, it can be shown as below that {bxn} is an equicontinuous sequence of functions in x. now, following the arguments similar to that given in granas et al. [23], it is proved that b is a a continuous operator on br(0). next, we show that b is a compact operator on br(0). to finish, it is enough to show that every sequence {bxn} in b(br(0)) has a cauchy subsequence. now, proceeding with the earlier arguments it is proved that ‖bxn‖ ≤ |c0| ‖a‖ + mfw γq = r for all n ∈ n. this shows that {bxn} is a uniformly bounded sequence in b(br(0)). next, we show that {bxn} is also a equicontinuous sequence in b(br(0)). let ǫ > 0 be given. since limt→∞ w(t) = 0, there is a real number t1 > t0 ≥ 0 such that |w(t)| < ǫ 8mf/γ(q) for all cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 339 t ≥ t1. similarly, since lim t→∞ a(t) = 0, for above ǫ > 0, there is a real number t2 > t0 ≥ 0 such that |a(t)| < ǫ 8|c0| for all t ≥ t2. thus, if t = max{t1,t2}, then |w(t)| < ǫ 8mf/γ(q) and |a(t)| < ǫ 8|c0| (4.10) for all t ≥ t . let t,τ ∈ j∞ be arbitrary. if t,τ ∈ [t0,t ], then we have |bxn(t) − bxn(τ)| ≤ |c0| |a(t) − a(τ)| + ∣ ∣ ∣ ∣ a(t) γq ∫ t t0 (t − s)q−1f(s,x(s))ds − a(τ) γq ∫ τ t0 (τ − s)q−1f(s,x(s))ds ∣ ∣ ∣ ∣ ≤ |c0| |a(t) − a(τ)| + ∣ ∣ ∣ ∣ a(t) γq ∫ t t0 (t − s)q−1f(s,x(s))ds − a(τ) γq ∫ t t0 (τ − s)q−1f(s,x(s))ds ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ a(τ) γq ∫ t t0 (τ − s)q−1f(s,x(s))ds − a(τ) γq ∫ τ t0 (τ − s)q−1f(s,x(s))ds ∣ ∣ ∣ ∣ ≤ |c0| |a(t) − a(τ)| + mf γq ∫ t t0 ∣ ∣a(t)(t − s)q−1 − a(τ)(τ − s)q−1 ∣ ∣ ds + mf γq ∣ ∣ ∣ ∣ ∫ t τ ∣ ∣ ∣ a(τ)(τ − s)q−1 ∣ ∣ ∣ ds ∣ ∣ ∣ ∣ ≤ |c0| |a(t) − a(τ)| + mf γq ∫ t t0 ∣ ∣a(t)(t − s)q−1 − a(τ)(τ − s)q−1 ∣ ∣ ds + mf ‖a‖ γq |(τ − t)q| . since the functions t 7→ a(t) and t 7→ a(t)(t − s)q−1 are continuous on compact [t0,t ], they are uniformly continuous there. therefore, by the uniform continuity, for above ǫ we have the real numbers δ1 > 0 and δ2 > 0 depending only on ǫ such that |t − τ| < δ1 =⇒ |a(t) − a(τ)| < ǫ 9|c0| and |t − τ| < δ2 =⇒ ∣ ∣a(t)(t − s)q−1 − a(τ)(τ − s)q−1 ∣ ∣ < ǫ 9mft / γq . similarly, choose the real number δ3 = ( ǫ 9mf ‖a‖ / γ(q) )1/q > 0 so that |t − τ| < δ3 =⇒ |(t − τ) q| < ǫ 9mf‖a‖ / γ(q) . 340 bapurao c. dhage cubo 22, 3 (2020) let δ4 = min{δ1,δ2,δ3}. then |t − τ| < δ4 =⇒ |bxn(t) − bxn(τ)| < ǫ 3 for all n ∈ n. again, if t,τ > t , then we have a δ5 > 0 depending only on ǫ such that |bxn(t) − bxn(τ)| ≤ |c0| |a(t) − a(τ)| + a(t) γq ∣ ∣ ∣ ∣ ∫ t t0 (t − s)q−1f(s,xn(s))ds ∣ ∣ ∣ ∣ + a(τ) γq ∣ ∣ ∣ ∣ ∫ τ t0 (τ − s)q−1f(s,xn(s))ds ∣ ∣ ∣ ∣ ≤ ∣ ∣c0 [ |a(t)| + |a(τ)| ] + mf γ(q) [ w(t) + w(τ) ] < ǫ 2 < ǫ for all n ∈ n whenever |t − τ| < δ5. similarly, if t,τ ∈ r+ with t < t < τ, then we have |bxn(t) − bxn(τ)| ≤ |bxn(t) − bxn(t)| + |bxn(t) − bxn(τ)|. take δ = min{δ4,δ5} > 0 depending only on ǫ. therefore, from the above obtained estimates, it follows that |bxn(t) − bxn(t)| < ǫ 2 and |bxn(t) − bxn(τ)| < ǫ 2 for all n ∈ n whenever |t − τ| < δ. as a result, |bxn(t) − bxn(τ)| < ǫ for all t,τ ∈ j∞ and for all n ∈ n whenever |t − τ| < δ. this shows that {bxn} is a equicontinuous sequence in x. now an application of arzelà-ascoli theorem yields that {bxn} has a uniformly convergent subsequence on the compact subset [t0,t ] of j∞. without loss of generality, call the subsequence to be the sequence itself. we show that {bxn} is cauchy in x. now |bxn(t) − bx(t)| → 0 as n → ∞ for all t ∈ [t0,t ]. then for given ǫ > 0 there exists an n0 ∈ n such that sup t0≤t≤t a(t) γq ∫ t t0 (t − s)q−1 ∣ ∣f(s,xm(s)) − f(s,xn(s)) ∣ ∣ ds < ǫ 2 for all m,n ≥ n0. therefore, if m,n ≥ n0, then we have ‖bxm − bxn‖ = sup t0≤t<∞ ∣ ∣ ∣ ∣ a(t) γq ∫ t t0 (t − s)q−1 ∣ ∣f(s,xm(s)) − f(s,xn(s)) ∣ ∣ ds ∣ ∣ ∣ ∣ ≤ sup t0≤t≤t ∣ ∣ ∣ ∣ a(t) γq ∫ t t0 (t − s)q−1 ∣ ∣f(s,xm(s)) − f(s,xn(s)) ∣ ∣ ds ∣ ∣ ∣ ∣ + sup t≥t ∣ ∣ ∣ ∣ a(t) γq ∫ t t0 (t − s)q−1 [ ∣ ∣f(s,xm(s)) ∣ ∣ + ∣ ∣f(s,xn(s)) ∣ ∣ ] ds ∣ ∣ ∣ ∣ < ǫ. cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 341 this shows that {bxn} ⊂ b(br(0)) ⊂ x is cauchy. since x is complete, {bxn} converges to a point in x. as b(br(0)) is closed, we have that {bxn} converges to a point in b(br(0)). hence b(br(0)) is relatively compact and consequently b is a continuous and compact operator on br(0) into itself. next, we estimate the value of the constant mb of the hypothesis (c) of the theorem 2.1. by definition of mb, one has ‖b(br(0))‖ = sup{‖bx‖ : x ∈ br(0)} = sup { sup t∈j∞ |bx(t)| : x ∈ br(0) } ≤ sup x∈br(0) { sup t∈j∞ ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ |a(t)| + 1 γq · sup t∈j∞ |a(t)| ∫ t t0 (t − s)q−1|g(s,x(s),x(γ(s)))|ds } ≤ ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mg γq · sup t∈j∞ a(t) ∫ t t0 (t − s)q−1 ds ≤ ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mg γq · sup t∈j∞ a(t)tq ≤ ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq = mb. thus, ‖bx‖ ≤ ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq = mb for all x ∈ br(0). hence, we have mbψa(r) + ψc(r) ≤ l ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) r k + r + m · max { l1 w1 γ(α1) , . . . , lm wm γ(αm) } r min{k1, . . . ,km} + r ≤ (m + 1) · max { l ( ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ‖a‖ + mgw γq ) , l1 w1 γ(α1) , . . . , lm wm γ(αm) } r min{k,k1, . . . ,km} + r < r for r > 0, because (m + 1)· max { l ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) , l1 w1 γ(α1) , . . . , lm wm γ(αm) } ≤ min{k,k1, . . . ,km} therefore, hypothesis (c) of theorem 2.1 is satisfied. 342 bapurao c. dhage cubo 22, 3 (2020) next, let y ∈ br(0) be arbitrary and let x = axby + cx. then, |x(t)| ≤ |ax(t)| |by(t)| + |cx(t)| ≤ ‖ax‖ ‖by‖ + ‖cx‖ ≤ ‖a(x)‖ ‖b(br(0))‖ + ‖c(x)‖ ≤ ( l + f ) mb + m ∑ j=1 lj + hj γ(αj) wj ≤ ( l + f ) ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) + m ∑ j=1 lj + hj γ(αj) wj for all t ∈ j∞. therefore, we have: ‖x‖ ≤ ( l + f ) ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) + m ∑ j=1 lj + hj γ(αj) wj = r. this shows that x ∈ br(0) and hypothesis (c) of theorem 2.1 is satisfied. now we apply theorem 2.1 to the operator equation axbx + cx = x to yield that the hfrigde (1.1 ) has a mild solution on j∞. moreover, the mild solutions of the hfrigde (1.1) are in br(0). hence, mild solutions are global in nature. finally, let x,y ∈ br(0) be any two mild solutions of the hfrigde (1.1) on j∞. then, from cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 343 (4.5) we obtain |x(t) − y(t)| ≤ ∣ ∣ ∣ ∣ ∣ [ f(t,x(t),x(θ(t))) ] × × ( a(t0)x0a(t) f(t0,x0,x0) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ) − [ f(t,y(t),y(θ(t))) ] × × ( a(t0)x0a(t) f(t0,x0,x0) + a(t) γq ∫ t t0 (t − s)q−1g(s,y(s),y(γ(s)))ds ) ∣ ∣ ∣ ∣ ∣ + sup t∈j∞ a(t) m ∑ j=1 iαj ∣ ∣hj(t,x(t),x(η(t))) − hj(t,y(t),y(η(t)) ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ [ f(t,x(t),x(θ(t))) − f(t,y(t),y(θ(t))) ] × ( a(t0)x0a(t) f(t0,x0,x0) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ) ∣ ∣ ∣ ∣ ∣ + ∣ ∣ f(t,y(t),y(θ(t))) ∣ ∣ × × ∣ ∣ ∣ ∣ ( a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds − g(s,y(s),y(γ(s)))ds ) ∣ ∣ ∣ ∣ + sup t∈j∞ a(t) m ∑ j=1 iαj lj ‖x − y‖ kj + ‖x − y‖ ≤ ∣ ∣f(t,x(t),x(θ(t))) − f(t,y(t),y(θ(t))) ∣ ∣ × × ∣ ∣ ∣ ∣ ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ |a(t)| + mgw γq w(t) ) ∣ ∣ ∣ ∣ + 2 [ |f(t,x(t),x(θ(t))) − f(t,0,0)| + |f(t,0,0)| ]mgw γq w(t) + m ∑ j=1 wj(t) γ(αj) · lj ‖x − y‖ kj + ‖x − y‖ ≤ ℓ(t) |x(t) − y(t)| k + |x(t) − y(t)| ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) + 2 mgw γq [ ℓ(t) max { |x(t)| , |x(θ(t))| } k + max { |x(t)| , |x(θ(t))| } + f ] w(t) + m ∑ j=1 lj wj(t) γ(αj) ≤ l ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) |x(t) − y(t)| k + |x(t) − y(t)| + 2 mgw γq (l + f)w(t) + m ∑ j=1 lj wj(t) γ(αj) . (4.11) 344 bapurao c. dhage cubo 22, 3 (2020) taking the limit superior as t → ∞ in the above inequality (4.11) yields, limt→∞ |x(t)−y(t)| = 0. therefore, there is a real number t > 0 such that |x(t) − y(t)| < ǫ for all t ≥ t . consequently, the mild solutions of hfrigde (1.1) are uniformly globally attractive on j∞. this completes the proof. remark 4.2. the conclusion of theorem 4.1 also remains true under if we replace the hypotheses (a1), (a2), (c1) and (c2) with the following modified conditions: (a′1) the function f is continuous and there exists a d-function ψf ∈ d such that ∣ ∣f(t,x1,x2) − f(t,y1,y2) ∣ ∣ ≤ ψf ( max{|x1 − y1|, |x2 − y2|} ) for all t ∈ j∞ and x1,x2,y1,y2 ∈ r. (a′2) the function f is bounded on j∞ × r × r with bound mf. (c′1) the functions h ′ js are continuous and there exist d-functions ψhj ∈ d such that ∣ ∣hj(t,x1,x2) − hj(t,y1,y2) ∣ ∣ ≤ ψhj ( max{|x1 − y1|, |x2 − y2|} ) for all t ∈ j∞ and x1,x2,y1,y2 ∈ r, where j = 1, . . . ,m. (c′2) the functions hj are bounded on j∞ × r × r with bound mhj . theorem 4.2. assume that the hypotheses (a′1) (a ′ 2), (b1), (c ′ 1) (c ′ 2) and (d1) hold. furthermore, assume that ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ‖a‖ + mgw γq ) ψf (r) + m ∑ j=1 wj γ(αj) ψhj (r) < r, r > 0. (4.12) then the hfrigde (1.1) has a mild solution and mild solutions are uniformly globally attractive defined on j∞. proof. the proof is similar to theorem 4.1 and hence we omit the details. theorem 4.3. assume that the hypotheses (a1) (a2), (b1), (c1) (c2) and (d1) hold. then the hfrigde (1.1) has a mild solution and mild solutions are uniformly globally attractive and ultimately positive defined on j∞. proof. by theorem 4.1, the hfrigde (1.1) has a global mild solution in the closed ball br(0), where the radius r is given as in the proof of theorem 4.1, and the mild solutions are uniformly globally attractive on j∞. we know that for any x,y ∈ r, one has the inequality, |x| |y| = |xy| ≥ xy, cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 345 and therefore, ∣ ∣|xy| − (xy) ∣ ∣ ≤ |x| ∣ ∣|y| − y ∣ ∣ + ∣ ∣|x| − x ∣ ∣ |y| (4.13) for all x,y ∈ r. now, for any mild solution x of the hfrigde (1.1) in br(0), one has ∣ ∣|x(t)| − x(t) ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ f(t,x(t),x(θ(t))) ∣ ∣ ∣ × × ∣ ∣ ∣ ( a(t0)x0a(t) f(t0,x0,x0) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ) ∣ ∣ ∣ − [ f(t,x(t),x(θ(t))) ] × × ( a(t0)x0a(t) f(t0,x0,x0) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ) ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ m ∑ j=1 iαj ∣ ∣hj(t,x(t),x(η(t))) ∣ ∣ − m ∑ j=1 iαj hj(t,x(t),x(η(t))) ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ f(t,x(t),x(θ(t))) ∣ ∣ ∣ ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ − a(t0)x0 f(t0,x0,x0) ) a(t) ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ f(t,x(t),x(θ(t))) ∣ ∣ ∣ × × ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ∣ ∣ ∣ ∣ − a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ |f(t,x(t),x(θ(t))) ∣ ∣ − f(t,x(t),x(θ(t))) ∣ ∣ ∣ × ∣ ∣ ∣ ∣ a(t0)x0a(t) f(t0,x0,x0) + a(t) γq ∫ t t0 (t − s)q−1g(s,x(s),x(γ(s)))ds ∣ ∣ ∣ ∣ + m ∑ j=1 iαj ∣ ∣ ∣ ∣ ∣hj(t,x(t),x(η(t))) ∣ ∣ − hj(t,x(t),x(η(t))) ∣ ∣ ∣ ≤ [ 4(l + f) ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ ] a(t) + [ 4(l + f) mg γq ] w(t) + 2 m ∑ j=1 hj γ(αj) wj(t) (4.14) for all t ∈ j∞. taking the limit superior as t → ∞ in the above inequality (4.14), we obtain the estimate that lim t→∞ ∣ ∣|x(t)| − x(t) ∣ ∣ = 0. therefore, there is a real number t > 0 such that ∣ ∣ |x(t)| − x(t) ∣ ∣ ≤ ǫ for all t ≥ t . hence, mild solutions of the hfrigde (1.1) are uniformly globally attractive as well as ultimately positive defined on j∞. this completes the proof. 346 bapurao c. dhage cubo 22, 3 (2020) theorem 4.4. assume that the hypotheses (a1) (a2) and (b1) hold. then the hfrde (1.1) has a mild solution and mild solutions are uniformly globally attractive, uniformly ultimately positive and uniformly asymptotically stable to zero defined on j∞. proof. by theorems 4.1 and 4.2, the hfrigde (1.1) has a global mild solution in the closed ball br(0), where the radius r is given as in the proof of theorem 4.1, and the mild solutions are uniformly globally attractive and uniformly ultimately positive on j∞. now, for any mild solution x ∈ br(0), we have from (4.10), |x(t)| ≤ (l + f) ( ∣ ∣ ∣ a(t0)x0 f(t0,x0,x0) ∣ ∣ ∣ a(t) + mg γq w(t) ) + m ∑ j=1 lj + hj γ(αj) wj(t). taking the limit superior as t → ∞ in the above inequality yields that limt→∞ |x(t)| = 0. therefore, for ǫ > 0 there exists a real number t ≥ t0 such that |x(t)| < ǫ whenever t ≥ t . consequently, the mild solution x is a uniformly asymptotically stable to zero defined on j∞. this completes the proof. example 4.1 let j∞ = r+ = [0,∞) ⊂ r. given a pulling function a(t) = e t ∈ crb(r+), consider the following nonlinear hybrid fractional caputo differential equation with the mixed arguments of anticipation and retardation, cd q 0     etx(t) − t t2 + 1 i3/2 ( |x(t)| + |x(3t)| 4 + |x(t)| + |x(3t)| ) 1 + 1 t2 + 1 ( |x(t)| + |x(2t)| 2 + |x(t)| + |x(2t)| )     = e−t log ( 1 + |x(t)| + |x(t/2)| ) 2 + |x(t)| + |x(t/2)| , t ∈ r+, x(0) = 0,              (4.15) for all t ∈ r+, where cd q 0 is the caputo fractional derivative of fractional order 0 < q ≤ 1. here, a(t) = et, θ(t) = 2t, η(t) = 3t, γ(t) = t 2 for t ∈ r+ and hence θ(0) = 0 = η(0). next, α = 3/2 and the functions f : r+ × r × r → r+ \ {0} and g,h : r+ × r × r → r are defined by f(t,x,y) = 1 + 1 t2 + 1 [ |x| + |y| 2 + |x| + |y| ] , h(t,x,y) = t t2 + 1 [ |x| + |x| 4 + |x| + |x| ] and g(t,x,y) = e−t log(|x| + |y|) 1 + |x| + |y| . clearly, the function f is continuous and bounded real function on r+ × r × r with bound mf = 2 and in particular, f = 1. now, it can be shown as in banas and dhage [3] that the function f satisfies the hypothesis (a1) with ℓ(t) = 1 t2 + 1 and k = 1. so we have l = 1. furthermore, cubo 22, 3 (2020) existence and attractivity theorems for nonlinear hybrid . . . 347 the function h is also continuous and bounded on j∞ × r × r with bound mh = 1. next, the function h satisfies the hypothesis (c1) with the function ℓh(t) = t t2 + 1 so that we have lh = 1 2 and kh = 4. again, the function g is continuous and bounded on j∞ × r × r and therefore, satisfies the hypotheses (b1) with mg = 1. next, we have lim t→∞ w(t) = lim t→∞ e−ttq = 0 = lim t→∞ e−tt3/2 = lim t→∞ wh(t) and so the hypothesis (d1) is satisfied. now, ‖a‖ = supt∈r+ e −t = 1,w = supt∈r+ e −t tq = 1 and wh = 1. finally, it is verified that the the functions a, f, g and h satisfy the condition (4.4) of theorem 4.1. consequently, the hfrigde (4.15) has a mild solution and mild solutions are globally uniformly attractive, uniformly ultimately positive and uniformly asymptotically stable to zero defined on r+. in particular, the hfrigde cd 2/3 0     etx(t) − t t2 + 1 i3/2 ( |x(t)| + |x(3t)| 4 + |x(t)| + |x(3t)| ) 1 + 1 t2 + 1 ( |x(t)| + |x(2t)| 2 + |x(t)| + |x(2t)| )     = e−t log ( 1 + |x(t)| + |x(t/2)| ) 2 + |x(t)| + |x(t/2)| , t ∈ r+, x(0) = 0,              has a mild solution and mild solutions are globally uniformly attractive, uniformly ultimately positive and uniformly asymptotically stable to zero defined on r+. remark 4.3. finally, we remark that the ideas of this paper may be extended with appropriate modifications to a more general hybrid fractional integrdifferential equation with caputo fractional derivative, cd q t0       a(t)x(t) − m ∑ j=1 iαjhj(t,x(t),x(η1(t)), . . . ,x(ηn)) f(t,x(θ1(t)), · · · ,x(θn(t)))       = g(t,x(γ1(t)), · · · ,x(γn(t))), t ∈ j∞, x(t0) = x0 ∈ r,                        (4.16) where cd q t0 is the caputo fractional derivative of fractional order 0 < q ≤ 1, γ is a euler’s gamma function, f : j∞ × r × ...(n times) × r → r \ {0}, g,hj : j∞ × r × ...(n times) × r → r are continuous and θi,γi : j∞ → j∞ are continuous functions which are respectively anticipatory and retardatory, that is, θi(t) ≥ t and γi(t) ≤ t for all t ∈ j∞ with θi(t0) = t0 = ηi(t0) for i = 1, . . . ,n. remark 4.4. if g is assumed to be continuous function on j∞ × r × r, then the attractivity and existence results for the hfrigde (1.1) may be obtained via another approach of using measure of noncompactness. in that case we need to construct a handy tool for the measure of noncompactness which is not the case with the present approach in the qualitative study of such nonlinear fractional integrodifferential equations. see the details of this procedure that appears in banas and dhage [3], hu and yan [26], dhage [11, 14] and the references therein. 348 bapurao c. dhage cubo 22, 3 (2020) 5 the conclusion from the foregoing discussion, it is clear that the pulling functions and the hybrid fixed point theorems are very much useful for proving the existence theorems as well as characterizing the mild solutions of different types of nonlinear fractional integrodifferential equations on unbounded intervals of the real line when the nonlinearity is not necessarily continuous. the choices of the pulling function and the fixed point theorem depends upon the situations and the circumstances of the nonlinearities involved in the nonlinear problem. the clever selection of the fixed point theorems yields very powerful existence results as well as different characterizations of the nonlinear fractional differential equations. in this article, we have been able to prove in theorems 4.1, 4.2, 4.3 and 4.4 the existence as well as global attractivity, ultimate positivity and asymptotic stability of the mild solutions for a quadratic type of nonlinear hybrid fractional differential equation (1.1) on the unbounded interval j∞ = [t0,∞) of right half of the real line r+, however, other nonlinear fractional integrodifferential equations can be treated in the similar way for these and some other characterizations such as monotonic global attractivity, monotonic asymptotic attractivity and monotonic ultimate positivity etc. of the mild solutions on unbounded intervals of the real line. it is known that several real world phenomena in physics and chemistry such as growth and decay of the radioactive elements continue for a very long period of time and the existence results of the type proved in this paper may be applicable for the situation to understand the behavior of the process 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[28] i. podlubny, fractional differential equations, academic press, san diego, calif, usa, 1999. introduction auxiliary results characterizations of solutions attractivity and positivity results the conclusion c ubo 1 1, 2 0-~6 ( 190•) neyp111 j esnmi" dp mals;m 4ljo clr¡ i• 1.oua s ur , cá lcu lo s ubdife rencial y conjunto pola res . manuel bustos v_ a b wt r o c l. 111 lhi~ not.c wc 60l 1\ li nk bctwcun t-~ubdiffcrcnliili calculo~ n.nd ru1d 1><>lnr ool::j. t ltl., n.llow:t us to o\>(.-i n calc ulus ru le;:, o n 1><>llu~t..'i in ll nicc w fty md lt b rinp w: t.o bclíc vc llrnt t his ¡>0 inl o fv it.lw could gi vc risc to ncw n:::rnh:.1. ln íl\.cl by t hi:i b im wc oblnin t\ll im p rovcm e nl o f 11. f'c1uh 1ven in litcrf\l,urc .. 1 introdución. e n c~tc l~jo pre;,cnt.nmoi:i una nplic11ció n del ouculo :i:ubdilcrcncinl ni c ajc u lo de co nj11nc0ndicnlt.~ 1\1 c:ák:ulo de conjunl~ pol o.rl~. loci. conjunto!'i po tnrt.':'!j 11 cg1\ 11 11 11 ro l import l\ntc en""'"~ com •exo y este n uevo un foquc podri11 pc.rmi l.irnos obtu11 or mlb in fo r ml\ci6 n ~brc: e llos. eelu 1'upol'!lci6n no es infundfl.d& y n q ue por ctttc mét o do o bten r~ un l"c!nltado {propo:ojlción ti ) q ue mc:;ionrr. uhll eotlim n.c i6 n dnd f\ 0 11 11} ll t cr nt url\ c1s1. teottma 14 .7). el cont>c:clo gc ncrn.i e n lo quo i!-ig u e ~el do d~ o~ \•cq.o ria.10! l opo16gico:i lo ·nlmc.n1c rnn,cxort (c. v.t. l.c.) e y f , c.n dul\lidnd ~pararltc por un11 form 1;1. bilincl\i (-. -). ej c"jcm¡>lo ~16.~\co ~ d de 1111 c.\•. u .c. e y !fu dun tiopológico e " . 30 c ubo 11 m . bu.! {~ mo e:1 muft.! n 1m ».h!i l ~ convexo , r. dc no t11. el c:o njun 10 (, + j. si f : e r. 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'tz e e , . e b, / (xo) <=> / (.,..) + f'(x ' )(zo. z ") s c. oc 1 • d .-frn i io n~ j) r 1· ct1 d o.n lt.'i' !$! ~ i guc q ue s i (' e e ~ c:o n "'t" j< o, cc rrnd o y n o 'ow.do. cnlodca1 ,..,. e r o(j:.: ) y p11 rn lodo :ro e : r : e /j, 1jir(.r.o) ~ (.r .j:o, .r ") :s • 1-iz e e ~ 1.1;.(i")(ro.z ' ) ~' cu bo 11 31 la l"'t'-im:ión c ntrt el cálc ulo 6-!mbd ifcrcnci l\ly e l dlculo de conjunt.~ poh1.res l'ic <."!1ll\bl ri mcdlmlc lo noció n de co•v•rnlo normal •prox:imltdo: dado~ e e e convexo ~rn\tlo y no v1\cío, :r.o e e y e. ~ o, e l coajunao dt": dirca:ioncs 5-11ormnlcs n cen :roes c(c;"o) = {x' e p 1 {x"n'a todo oorij uni.o 8 , a e 8 e r.miv(au{o}) ~ li cnc 111. lgunldnd a·= aº, de modo que h1. hip6t.esl!f siguiente: " c f.: c'f un co1ij11nto con vexo currndo <1m· oont icn<' a (/' no c'i rc'i lric t.ivu. a~f. :ti c c e c:t un co njunto co nvexo cerrndo qu contiene o, de lo pn:c<:licl\n ln.'j r ugln .. 1 del calculo <-!lulxlifm"cncln.j rd cá.lc ulo de co njuntol'i poh1.f"c1 la.• rd rc ncl n.'i 11tllii.t1d11.:1 t ' ll ota "t"cxión m>ll ([2j y \5$, p11m 111 .. '1 r eg lm clmlcm dd c:::iüculo d e co 11j u11t06 1>0\n.res .. w n:-glm de calc ulo p nrt1 o l t:-:mbdi fc.rcncin pueden t:ncontrn.r~ on (l\j). 1,.,., híp61eó~ que l'c c nuncln n co ntinu n.ci6n no~ r~tricth-a en m odo n.l g uno, co mo lo hemos t'ic'ñmado c 11 in introducción : ('h) e, e,. 'c .. ' son :mbco1¡ju11tos coll\"c.\:0.-cr.t!ttadc...r de e (/llt: contfo11u11 11 o. bi lema que !sijtltc :-= dcmuc:it. rn dirc:ctnme ntc. 31 c ubo 11 lo1un 2. 1 poro roda ' > o .•r l.itm r: ejov,c(o) = n (c; o) . recordemos l.1t.mbié n q uo n (c ;o) e, tgunl a o .. ~. el cono 11.:11int 6 1ico de cº (vén.1c !sj, tcorc_mg l·l6). rig11c. cnlonc~ q ue ijot;.oc(o) c n ° pul\ todo a> o. nuc:strl\ primer" n.plicn ció n co rrejpon d c ni c i(jc ulo del co nj u nto 1><>lll r de in. s uml\ de un número fi nito de :rnbco nju nlo.'! d e b. p r opo lcióo 2.2 :ii g'1,c'7, . .. ,c11 , ,•ja ti...•/ n ccn in /j 1pólc...~i.~ ( h ) . cnt o n.ce.1 d c mos t rnción: $igu e por d nrc mo, li\ dcmo~lrnci611 ptl.rn u = 2; el r~ul ll\do gc nr: rnl ~ ind ucció n e n n . un dlculo ~imple mut..~lrn q ue: y '-"1· tllntbié.n o= o+ o e c1 n c2, d e mo d o q ue tpc, 'v tpc, ~ e:o.:m:l" e n o. l..o llntuior ~ugfo_rt: 11 p lic 1lt ill rcg ln el e calc ulo pnrn el i!·:rnbdifcrc n i1d dr h1. lnf· com-ol u c:ión de do~ funciom.~ co nvcxm p ro p i1t..'! (111, tcor mn 3.1): ( , +e,)• = 8,(v,c, 'i7 v•c.)(o) = u 10.,vc,(o) n a.,,.c, (o)j , • .,,,.¿:, lj la1 7íl o, ,j (o , .o,)< l:, • 0ntl08" oon1ln101ci6n 111h1 n11lic11d6 11 "' clilc ulo d d rl'lnjnntn pol" r rlc 111 inlc~ión dc' un número rnitoo de ~uhco1lju11u11>1 d e. s . p r op lc'd n 2.3 s t1¡hmr1 ri111n.• '1 uf' c, , c,,. ,, c.,, rrn/k,on in l 11 p 6 t ,.., (1<) 11 qu< rn aid.-caort. l(/('11/0 :tubdífntncila.i ... c ubo 11 oe ;,,t( íl c,)n c •. l ,s1,s n i ( n c. ). = con o ( u c7) l 5 1:: n \q ::111 oo mo•lrnclón: como en el cmo pr-cccdcnlc 3c dflri la dcmoslrllcíón parn n = 2. se llcnc:i./1c 1 íl ~ = 1/lc, + wc,, rehu:ión que 5ugicrt in. tlpllcuc ión de in. rcttl" de ca lc ulo d el t·:mbdifonmclnl de in :turnl\ de d~ funciond (l!j, tcoro m11 2. 1). 'omo ~. \' wr, ~n íu n ionl.':i co nvcxm pro pim y t..tc, es oontin11 11. n. lo mcnoo u11 n.li;(t'111 .r e in t ( 1) íl c,, ~e ,,iguc: (c, íl co)' = 8,(,¡,c, + .pc,)(o) = u 181 ,"<;,{0) + 8c, v•o,(o) j . (t:, .l1>e' l1 adum m 8,, v•c-, (o) = ,,c1c.. si e, > o • o·• '¡, ~¡ '· = o (v f\..~ lcm ... 2.1) lm.11:0 .ncw • u c<• i+c,c;) (c,.<, )llcftción co n!lid cn\lnoj d °" ¡>n~dc "·'"' j..c. en dun.li dnd (e1, f1 ), (j::.i. f 1). "" a : b1 /:).¡ un oj~nulor li11~1tl cn11cin110 y a ' : f'1 f'1 s u oj>t'rntlor iu(junio. si e e e., l!:'i 1111 co11j11nlo com·cxo cxn-ado que co nt iene 1\ 01 :ltlll /1 • uf' o a uj>ongn. moél q ue /1 :jl\li:tfncc in. hi p6tcd!l jiguicn lc: v7." e p1, /1'(!1: ' ) = iní{ ~(-(1/) 1 a ' v ' '""z' } lo cu 1 ocurn!v, po r ejemp lo, cx i!!lc x e e, ll\j qu(' a..i: e inl e 3'1 c ubo l l m. 81l•l (j!is ia 1(c)f' • b,(.¡,c o a)(o) = jt "lli ( a(o)) (( lj,tcorcm• 2) =1\' (c'' ) • n d fin de l'cillit. rum h\ t'11linm npll cncl ó n ncccsit.11mo.'i. m odificar im h l¡>6t.~l" d e ll\ propcdld6n 3. l~n lugn.r d o 1111 opcriidor lintttl .-1 : e1 /~ co n.1 id <: r1u110~ f\h r11. un11. funci ó n co 1wex 11 y propiil f : e fl, d conju ni o oo n,•cxo cc rrf\d o e =1 . af, ne r .v :-i h ir'm ft:c ll invc~n ¡ 1(c). p r op os ició n 2.s ,.,·cll f e l~o(b) rm.11 /tm c 1ó11 t nl qm. o e do m/ . s i r.i > / (o) . • •t:n o• {z e b 1 / (x) so). f 11t rm rr.• :r· e dº :o' 11 m5lo .•1 r..r1..1lc (c:,.t:.,) e ¿:,, 1 e r 1 /,nl qu e: 1 e [o.<,/(o / (o)) [ ,.,,,¡ (1/ )'(z ' ) s '' 1/(0). ; ldcmn .... ""' j 1~• 1mn /m1 c; i1i 1t po.~ it.iu11 u / (o) = o .. • tumc in tt.1 limn c1d11 i)' e " ' i• ' e e:' 1 /' (r ' ) so) e 20' dcm o:irtrnc.i ó n : si e =1. rr j clll oll c<::j: vd="''' ·•(c) = \.'c o i . adcmm 1.ye r o( fi.) e, 111111 fun ci6n crc<:ic nte y / (0) e / (dom/) ílj ,oj, luego dl' (( 11, tc"orcm11 !u ) ~ :'llgw·: .r. ' e d1(1j'c o / )(o) !ii ' ~lo :i:i cxil'l\jc n nl1111 cro:t rea.lc:!i poeiü'~ c1,!7, i , l nk· i1 q11 tl: ( t) (1 • ': = i , (2) 1 e 0.,-'r(/(0)) ,\', . e 1j, , (1/) (0). l'n lado dirtt'ld dn : oc, v•d / (o)) • !o« ./(n / (0))1 po r º "ª jl6m.r, r ' e oc,( 1 / )(0) i1i y ~l o .1ti (1/) (0) + (1/) ' (z ' )(o, .r ' ) ~ '' 1/(0) .. (11)" (.r' ) s ,, cubo 11 :m co n c:i fin de obtener la ~llm1u;l611 de d º o~rvcrnos que d c-·subdifore nc in.i ~ mm multifu nc i n c;:rockntc del j)l\r tlme tro ( positivo) t.. lo 1rnu:rior u nido a ll\.'! hip6tc::iil! :k)brc j noe1. dft: o s >. s 1• = 8<(>.l)(o) e 11tv•/)(o) . o· oqul,. deduce' 8,, (1./ ) (0) c 81(0 1 /)(o) , de modo que' • • e 8,, (1./)(0) = ("-'/)' (•') s 1 = j'( o-.<' ) s " ' y d e l\qul dcd ucirno." i" prlmern ejti m/\ción: 0 ° c o 1 { .:: ' e g • j /' (::. ' ) .$ cr ) . finn.lrncnlc, de in d~igun..ldnd : deducim~: ,¡.0(7') = • up(( (z,z' ) o ) + o 1 (-<") s 2o ) 2o (z ' e ,, 1 • •j,(z' ) s l} = 2o0'' lo que compl ti\ in dcm o5lrn.c lón . • eal c rcsuhmlo, n.1111n incl o e n ln introd ucció n de ~le trnhai. )o, bft sid o obte nid o madi· n.nl c d c4jculo c-~ubdiícrcnclnl. es mm cl\rl\cteduw:.ión c:om1>lc:l8 del conjunto p o lnr f 1(c) en tb-min~ d e lu ínncióu co njug&d" de j ,. pcmitc obt e ner h l eitl lmf\clón dndf\ por rodc.1t/cl lnr on (15). ti:orcnm 1•1.7), impl k:ita en (j.11). refer e 1lci as [ij l-líriajt·urru ty .j ., ~-sublliffot'onti'nl c nlculw, con\·cx ankly:tltj 11..ncl oplimizn· tíon. rocj11rch not.c!j 111 mnlhc l!ll\l íc'i scri~. pit me.n pul li~hur , 5 7, 4:j..02 ( 1932 ) . 121 lotl<" a .o., tlk ho rnlro v v.m ., dunlil11 o/ "on~·u fi:mciu>n., nntl sxt n::mum profllmliell~ 11. coll g de f'ttrncc ( 1966l967). ¡ 1¡ mo n:11.u j .. j., sur in / on c tio rrndlc p0fo1~ d"trnc. /oncj 1o n .,amioontinuc ~upcn 0}, and z+ the nonnegative integers. also, let bc denote the normed vector space of bounded functions φ : z → rk, with the norm ||φ|| = ∑k j=1 max n∈[0,ω−1] |φj(n)|, where φ = (φ1,φ2, ...,φk) t and [0,ω − 1] = {0,1, ...,ω − 1}. particularly for each x = (x1,x2, ...,xk) t ∈ rk, we define the norm |x|0 = ∑k j=1 |xj|. also, denote by bck+ = {φ ∈ bc : φ(n) ∈ r k + for n ∈ z}. in [12], raffoul used a krasnoselskii’s fixed point theorem in cones to prove the existence of positive periodic solutions of the scaler difference equation with parameter x(n + 1) = a(n)x(n) + λh(n)f(x(n − τ(n))). also, in [10], zhu and li generalized the work in [12] by proving that the system of difference equations with parameter x(n + 1) = a(n)x(n) + λh(n)f(x(n − τ(n))) where a(n) = diag[a1(n),a2(n), ...,am(n)] and h(n) = diag[h1(n),h2(n), ...,hm(n)] has positive periodic solutions. motivated by the above considerations we investigate the existence of multiple positive periodic solutions of the nonautonomous system of difference equations x(n + 1) = a(n,x(n))x(n) + λf(n,xn), (1.1) where, λ > 0 is a parameter, a(n,x(n)) = diag[a1(n,x(n)), ...,ak(n,x(n))], aj(n+ω,.) = aj(n,.), f(n,x) : z × bc → rk is continuous in x and f(n,x) is ω-periodic in n and x, whenever x is ωperiodic, ω ≥ 1 is an integer. if x ∈ bc, then xn ∈ bc for any n ∈ z is defined by xn(θ) = x(n+θ) for θ ∈ z. throughout this paper, we denote the product of y(n) from n = a to n = b by ∏b n=a y(n) with the understanding that ∏b n=a y(n) = 1 for all a > b. also, for two m×n matrices a and b, a ≥ b (a < b) means that the inequality is satisfied entrywisely. in particular, a is said to be a nonnegative matrix if a ≥ 0. definition 3.1. [4] let x be a banach space and p a closed, nonempty subset of x. p is a (convex) cone if (i) x,y ∈ p and α,β ∈ r+ imply αx + βy ∈ p. (ii) x ∈ p and −x ∈ p imply x = 0. definition 3.2. [4] let x be a banach space and d ⊂ x, 0 ∈ d. the operator l : d → x is such that l0 = 0. xλ 6= 0 is said to be an eigenvector of the eigenvalue λ of l if lxλ = λxλ. 82 youssef n. raffoul and ernest yankson cubo 21, 1 (2019) lemma 3.1. [4] suppose d is an open subset of an infinite-dimensional real banach space x, 0 ∈ d, and p is a cone of x. if the operator γ : p ∩ d → p is completely continuous with γ0 = 0 and satisfies infx∈p∩∂d ||γx|| > 0, then γ has an eigenvector on p ∩ ∂d associated with a positive eigenvalue. that is, there exist x0 ∈ p ∩ ∂d and µ0 > 0 such that γx0 = µ0x0. in this paper we make the following assumptions. (h1) 0 < aj(n) < 1, j = 1,2, ...k, and n ∈ [0, ω − 1]. (h2) there exist b(n) = diag[b1(n),b2(n), ...,bk(n)] and c(n) = diag[c1(n),c2(n), ...,ck(n)] where bj,cj : z → r+ are ω-periodic with 0 < bj,cj < 1, such that b(n) ≤ a(n,ϕ(n)) ≤ c(n) for all (n,ϕ) ∈ z × bck+. (h3) f(n,0) = 0 for all n ∈ z. (h4) f(n,ϕn) ≤ 0 for all (n,ϕ) ∈ z × bc k +. (h5) for any l > 0 and ǫ > 0, there exists δ > 0 such that [φ,ψ ∈ bck+, ||φ|| ≤ l, ||ψ|| ≤ l, ||φ − ψ|| < δ, 0 ≤ s ≤ ω] imply |f(s,φs) − f(s,ψs)| < ǫ. to study system (1.1) we let x = {x : z → rk, x(n + ω) = x(n)}, then it is clear that x ⊂ bc, endowed with the norm ||x|| = ∑k j=1 |xj|0, where |xj|0 = maxn∈[0,ω−1] |xj(n)|. for the next lemma we consider xj(n + 1) = aj(n,x(n))xj(n) + fj(n,xn), j = 1,2, ...,k. (1.2) the proof of the next lemma can be easily deduced from [12] and hence we omit it. lemma 3.2. suppose that (h1) hold. if x(n) ∈ x then xj(n) is a solution of equation (1.2) if and only if xj(n) = n+t−1∑ u=n gxj (n,u)fj(n,xn), j = 1,2, ...,k, (1.3) where gxj (n,u) = ∏n+t−1 s=u+1 aj(s,x(s)) 1 − ∏n+t−1 s=n aj(s,x(s)) , u ∈ [n,n + t − 1], j = 1,2, ...,k. (1.4) cubo 21, 1 (2019) positive periodic solutions of functional discrete systems . . . 83 let σ = min 1≤j≤k ( ∏ω−1 s=0 bj(s) )[ 1 − ∏ω−1 s=0 cj(s) ] ( ∏ω−1 s=0 cj(s) )[ 1 − ∏ω−1 s=0 bj(s) ] (1.5) it can easily be obtained from (h2) that σ < 1. we next define two cones in x as follows: p1 = { y ∈ x : yj(n) ≥ σ|yj|0,n ∈ z and j = 1, ...,k } , and p2 = { y ∈ x : y(n) ≥ 0, n ∈ z } . define an operator t on x by t : x → x by (tx) = (t1x,t2x,...,tkx) t. (1.6) where (tjx)(n) = n+ω−1∑ u=n gxj (n,u)fj(u,xu), j = 1, ...,k. it is not very difficult to see that gxj (n+ω,u+ω) = g x j (n,u). also, it can easily be verified that x∗(n) = (x∗1(n), ...,x ∗ k(n)) ≥ 0 is a positive ω-periodic solution of system (1.1) associated with λ ∗ if and only if x∗ ∈ p2 is an eigenvector of the operator t associated with the eigenvalue 1 λ∗ > 0, that is tx∗ = 1 λ∗ x∗. lemma 3.2. suppose that (h1) and (h2) hold. then the mapping t maps p1 into p1, i.e., tp1 ⊂ p1. proof. in view of (h1) and (h2), we have that, for j = 1,2, ...,k, and 0 ≤ u ≤ ω − 1, ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) ≤ gxj (n,u) ≤ ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) (1.7) 84 youssef n. raffoul and ernest yankson cubo 21, 1 (2019) |(tjx)(n)| ≤ n+ω−1∑ u=n ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) |fj(u,xu)| ≤ ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) ω−1∑ u=0 |fj(u,xu)| it follows that |(tjx)|0 ≤ ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) ω−1∑ u=0 |fj(u,xu)| or ω−1∑ u=0 |fj(u,xu)| ≥ 1 − ∏ω−1 s=0 cj(s) ∏ω−1 s=0 cj(s) |(tjx)|0. therefore, (tjx)(n) ≥ ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) ω−1∑ u=0 |fj(u,xu)| ≥ ( ∏ω−1 s=0 bj(s) )[ 1 − ∏ω−1 s=0 cj(s) ] ( ∏ω−1 s=0 cj(s) )[ 1 − ∏ω−1 s=0 bj(s) ] |(tjx)|o ≥ σ|(tjx)|o, which means that tx ∈ p1. this completes the proof. lemma 3.3. suppose (h5) hold. then the operator t : p2 → x is completely continuous. proof. in view of (h5) and the assumption that f(n,x) is continuous in x, we have that the operator t is continuous. we will show that t is compact. let u ⊆ p2 be any bounded set. then, by the (h5), there exists a constant m > 0 such that |fj(n,xn)| ≤ m, for (n,x) ∈ [0,ω − 1] × u, j = 1,2, ...,k. thus we have, |(tjx)| ≤ ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) mω. it follows that, cubo 21, 1 (2019) positive periodic solutions of functional discrete systems . . . 85 ||(tx)|| = k∑ j=1 |tjx|0 ≤ mω k∑ j=1 ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) ≤ mkωγ, where γ = max 1≤j≤k ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) . next, we show that t maps bounded subsets into compact sets. let j > 0 be given, and define ρ = {ϕ ∈ p2 :‖ ϕ ‖≤ j} and q = {(tϕ)(n) : ϕ ∈ ρ}, then ρ is a subset of r ωk which is closed and bounded thus compact. as t is continuous in ϕ it maps compact sets into compact sets. therefore q = t(ρ) is compact. this completes the proof of lemma 3.3. 2 main results in this section we state and prove our main results. for our main results we let f0 = lim φ∈p1, ||φ||→0 ∑ω−1 u=0 |f(u,xu)| ||φ|| , and f∞ = lim φ∈p1, ||φ||→∞ ∑ω−1 u=0 |f(u,xu)| ||φ|| . also, define, for r a positive number, ωr, by ωr = {x ∈ x : ||x|| < r }. theorem 4.1 suppose that (h1)-(h5) hold and 0 < f∞ < ∞. then there exist positive constants r0, λ1, and λ2 with λ1 < λ2 such that, for any r > r0, system (1.1) has a positive ω-periodic solution xr(n) associated with some λr ∈ [λ1,λ2] and ||x r|| = r. proof. since 0 < f∞ < +∞, there exist ǫ2 > ǫ1 > 0 and r0 > 0 such that ǫ1||φ|| < ω−1∑ u=0 |f(u,φu)| < ǫ2||φ|| for ||φ|| ≥ r0, φ ∈ p1. 86 youssef n. raffoul and ernest yankson cubo 21, 1 (2019) suppose r > r0, then ωr is a bounded open subset of x and 0 ∈ ωr. for x ∈ p1 ∩ ∂ωr, we have ||tx|| = k∑ j=1 max n∈[0,ω−1] |(tjx)(n)| ≥ k∑ j=1 |(tjx)(n)| = k∑ j=1 ω−1∑ u=0 gxj (n,u)fj(u,xu) ≥ k∑ j=1 ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) ω−1∑ u=0 fj(u,xu) ≥ min 1≤j≤k ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) ω−1∑ u=0 k∑ j=1 |fj(u,xu)| ≥ min 1≤j≤k ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) ǫ1r > 0. it follows that inf x∈p1∩∂ωr ||tx|| ≥ min 1≤j≤k { ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) } ǫ1r > 0. since, t is completely continuous with t(0) = 0, it follows from lemma 3.1 that the operator t has an eigenvector xr ∈ p1 associated with the eigenvalue µr > 0 such that ||x r|| = r. set λr = 1 µr . then xr is a positive ω-periodic solution of system (1.1). we next determine λ1 and λ2 as follows. from (xr)j(n) = λr n+ω−1∑ u=n gx r j (n,u)fj(u,x r u) ≤ λr ω−1∑ u=0 ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) |fj(u,x r u)| ≤ λr ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) ω−1∑ u=0 |fj(u,x r u)| ≤ λr ∏ω−1 s=0 cj(s) 1 − ∏ω−1 s=0 cj(s) ǫ2r, j = 1,2, ...,k, and ||xr|| = r we can get λr ≥ 1 ǫ2 ∑k j=1 ∏ ω−1 s=0 cj(s) 1− ∏ ω−1 s=0 cj(s) =: λ1 cubo 21, 1 (2019) positive periodic solutions of functional discrete systems . . . 87 on the other hand, (xr)j(n) ≥ λr ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) ω−1∑ u=0 |fj(u,x r u)|, j = 1, ...,k. it follows from ||xr|| = r ≥ λr min 1≤j≤k { ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) } ω−1∑ u=0 |f(u,xru)| ≥ λr min 1≤j≤k { ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) } ǫ1r that λr ≤ λr max 1≤j≤k {1 − ∏ω−1 s=0 bj(s) ǫ1 ∏ω−1 s=0 bj(s) } := λ2. therefore, λr ∈ [λ1,λ2] and this completes the proof. theorem 4.2. suppose that (h1)-(h5) hold and 0 < f0 < ∞. then there exist positive constants r0 > 0, λ̃1 and λ̃2 with λ̃1 < λ̃2 such that, for any 0 < r < r0, system (1.1) has a positive ω-periodic solution x̃r(n) associated with some λ̃r ∈ [λ̃1, λ̃2] and ||x̃r|| = r. proof. since 0 < f0 < ∞, there exist 0 < l1 < l2 and r0 > 0 such that l1||φ|| < ω−1∑ u=0 |f(u,φu)| < l2||φ|| for 0 < ||φ|| < r0, φ ∈ p1. for r ∈ (0,r0), ωr is a bounded subset of x and 0 ∈ ωr. moreover, for x ∈ p1 ∩ ∂ωr, ||tx|| ≥ k∑ j=1 |(tjx)(n)| = k∑ j=1 n+ω−1∑ u=n gxj (n,u)fj(u,xu) ≥ min 1≤j≤k { ∏ω−1 s=0 bj(s) 1 − ∏ω−1 s=0 bj(s) } l1r > 0. this implies that infx∈p1∩∂ωr ||tx|| > 0. the remaining part of the proof is similar to that of theorem 4.1 and so we omit it. this completes the proof. 88 youssef n. raffoul and ernest yankson cubo 21, 1 (2019) using arguments similar to that of theorem 4.1 and theorem 4.2, the following results can be established respectively. theorem 4.3. suppose that (h1)-(h5) hold and f∞ = ∞. then there exist positive constants r̆0 and λ̆ such that, for any r > r̆0, system (1.1) has a positive ω-periodic solution x̆ r(n) associated with some λ̆r ≤ λ̆ and ||x̆ r|| = r. theorem 4.4. suppose that (h1)-(h5) hold and f0 = ∞. then there exist positive constants r̄0 and λ̄ such that, for any 0 < r < r̄0, system (1.1) has a positive ω-periodic solution x̄ r(n) associated with some λ̄r ≤ λ̄ and ||x̄ r|| = r. 3 an application in this section, we apply our results from the previous section to the volterra discrete system xj(n + 1) = xj(n) [ aj(n) − λ k∑ i=1 ( bji(n)xi(n) + n∑ s=−∞ cji(n,s)gji(xi(s)) )] , j = 1,2, ...,k, (3.1) where xj(n) is the population of the jth species, aj,bji : z → r+ are ω-periodic and cji(n,s) ≥ 0 and cji(n + ω,s + ω) = cji(n,s) for all (n,s) ∈ z 2; gji : r+ → r+, i, j = 1, ...,k. theorem 5.1. suppose that maxn∈z ∑n s=−∞ |cji(n,s)| < +∞. then there exist positive constants r0 and λ0 such that, for any r > r0, system (3.1) has a positive ω-periodic solution x r(n) associated with λr ≤ λ0 and ||x r|| = r. proof. note that a(n,x(n)) = diag[a1(n),a2(n), ...,ak(n)] and f = (f1,f2, ...,fk) where fj(n,xn) = −xj(n) k∑ i=1 ( bji(n)xi(n) + n∑ s=−∞ cji(n,s)gji(xi(s)) ) for j = 1,2, ...,k and (h1)-(h5) are satisfied. cubo 21, 1 (2019) positive periodic solutions of functional discrete systems . . . 89 for x ∈ p1 and j = 1, ...,k we have ω−1∑ u=0 |fj(u,xu)| = k∑ i=1 ω−1∑ u=o xj(u) ( xi(u)bji(u) + u∑ s=−∞ cji(u,s)gji(xi(s)) ) ≥ k∑ i=1 ω−1∑ u=o xj(u)xi(u)bji(u) ≥ ω−1∑ u=o x2j (u)bjj(u) ≥ σ2|xj| 2 0 ω−1∑ u=o bjj(u). thus, ω−1∑ u=0 |f(u,xu)| = k∑ j=1 ω−1∑ u=0 |fj(u,xu)| ≥ k∑ j=1 σ2|xj| 2 0 ω−1∑ u=o bjj(u) ≥ σ2 min 1≤j≤k ω−1∑ u=o bjj(u) k∑ j=1 |xj| 2 0 ≥ σ2 k ||x||2 min 1≤j≤k ω−1∑ u=o bjj(u). it follows that ∑ω−1 u=0 |f(u,xu)| ||x|| → as ||x|| → ∞. the conclusion follows directly from theorem 4.3 and this completes the proof. 90 youssef n. raffoul and ernest yankson cubo 21, 1 (2019) references [1] a. datta and j. henderson, differences and smoothness of solutions for functional difference equations, proceedings difference equations, 1 (1995), 133-142. 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[8] j. henderson and w. n. hudson, eigenvalue problems for nonlinear differential equations, communications on applied nonlinear analysis, 3 (1996), 51-58. [9] m. a. krasnosel’skii, positive solutions of operator equations, noordhoff, groningen, (1964). [10] y. li and l. zhu, positive periodic solutions of higher-dimensional functional difference equations with a parameter, j. math. anal. appl. 290 (2004) 654-664. [11] f. merdivenci, two positive solutions of a boundary value problem for difference equations, journal of difference equations and application, 1 (1995), 263-270. [12] y.n. raffoul, positive periodic solutions of nonlinear functional difference equations, electron. j. differential equations, 55 (2002) 1-8. [13] y.n. raffoul, periodic solutions for scalar and vector nonlinear difference equations, panamerican journal of mathematics, 9 (1999), 97-111. [14] w. yin, eigenvalue problems for functional differential equations, journal of nonlinear differential equations, 3 (1997), 74-82. introduction main results an application cubo a mathematical journal vol.20, no¯ 02, (01–12). june 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000200001 an approach to f. riesz representation theorem rafael del rio, asaf l. franco and jose a. lara departamento de f́ısica matemática, instituto de investigaciones en matemáticas aplicadas y en sistemas, universidad nacional autónoma de méxico c.p. 04510, cdmx, méxico. delrio@iimas.unam.mx, asaflevif@hotmail.com, nekrotzar.ligeti@gmail.com abstract in this note we give a direct proof of the f. riesz representation theorem which characterizes the linear functionals acting on the vector space of continuous functions defined on a set k. our start point is the original formulation of riesz where k is a closed interval. using elementary measure theory, we give a proof for the case k is an arbitrary compact set of real numbers. our proof avoids complicated arguments commonly used in the description of such functionals. resumen en esta nota, damos una demostración directa del teorema de representación de f. riesz que caracteriza los funcionales lineales actuando en el espacio vectorial de funciones continuas definidas en un conjunto k. nuestro punto de partida es la formulación original de riesz, donde k es un intervalo cerrado. usando teoŕıa elemental de la medida, damos una demostración para el caso en que k es un conjunto arbitrario compacto de números reales. nuestra demostración evita argumentos complicados comúnmente usados en la descripción de dichos funcionales. keywords and phrases: riesz representation theorem, positive linear functionals, riemannstieltjes integral. 2010 ams mathematics subject classification: 46e11,46e15,28a25. http://dx.doi.org/10.4067/s0719-06462018000200001 2 rafael del rio, asaf l. franco and jose a. lara cubo 20, 2 (2018) 1 introduction the riesz representation theorem is a remarkable result which describes the continuous linear functionals acting on the space of continuous functions defined on a set k. it is very surprising that all these functionals are just integrals and vice versa. in case k is a closed interval of real numbers, any such functional is represented by riemann-stieltjes integral, which is a generalization of the usual riemann integral. this was first announced by f. riesz in 1909 [14]. in case k is compact set (not necessarily a closed interval), then a more general concept of integral is needed, because the riemann-stieltjes integral used by riesz is defined only for functions on intervals. in this work, we prove that there is a short path between the two cases. besides its aesthetic appeal, the above mentioned theorem has far-reaching applications. it allows a short proof of the kolmogoroff consistency theorem, see [3] thm 10.6.2., and can be used to give an elegant proof of the spectral theorem for selfadjoint bounded operators, see section vii.2 of [13]. both these theorems are main results in probability and functional analysis respectively. moreover, the entire theory of integration for general spaces can be recovered using the theorem of riesz. see for example [19], where the lebesgue measure on rn is constructed. more generally it can also be used to show the existence of the haar measure on a group, see [3] chap. 9. in this note we give a short proof of the riesz representation theorem for the case k is an arbitrary compact set of real numbers, see theorem 3.1 below. this is interesting because in many situations we have a compact set which is not a closed interval. to prove the spectral theorem, for example, one considers the set of continuous functions defined on the spectrum of selfadjoint bounded operator, which is a compact set of r, but not necessarily a closed interval. we get our result starting from the nondecreasing function that appears in the riemann-stieltjes integral representation of riesz original formulation. to this function we associate a measure which is used to integrate over general compact sets. then we show how this lebesgue integral representation can be seen as a riemann-stieltjes integral. our proof is new, avoids technical arguments which appear frequently in proofs of riesz theorem, it is elementary, direct and quite simple. 2 preliminaries let us introduce first some definitions and notations we shall use. 2.1 definitions and notation. let c(k) := {f : k → r : f continuous} where k is a compact subset of r, the real numbers. a functional is an assignment l : c(k) → r. the functional is linear if l(c1f+c2g) = c1l(f)+c2l(g) for all f, g ∈ c(k), c1, c2 ∈ r. it is continuous if there exists a fixed m > 0 such that |lf| ≤ m‖f‖∞ cubo 20, 2 (2018) an approach to f. riesz representation theorem 3 for all f ∈ c(k), where ‖ · ‖ ∞ denotes the uniform norm, that is, ‖f‖ ∞ = sup{|f(x)| : x ∈ k}. we define the norm of such functional as ‖l‖c(k) = ‖l‖ = sup{|l(f)| : f ∈ c(k) and ‖f‖∞ ≤ 1}, we denote the set of the linear continuous functionals on c(k) by c(k)∗. it is called the dual space. in general, the dual of normed linear space x is denoted by x∗. a functional l on c(k) is said to be a positive if l(f) ≥ 0 whenever f(x) ≥ 0 for every x ∈ r. we use the notation c(k)∗+ for the set of positive linear functionals on c(k). the function α : [a, b] −→ r is said to be normalized, if α(a) = 0 and α(t) = α(t+), a < t < b, that is, α is continuous from the right inside the interval (not at a! if it were right continuous at a, theorem (2.1) would not hold for the functional l(f) = f(a)). the total variation of a monotone increasing function α is defined as v(α) = α(b)−α(a). we denote the characteristic function of a set a ⊂ [a, b] by 1a where 1a(x) = 1 if x ∈ a and 0 if x ∈ [a, b] \ a. 2.2 representation theorem for functionals on c[a, b]. we formulate the above-mentioned result by f. riesz as follows: theorem 2.1. let l : c[a, b] −→ r be a positive linear functional. there exists a unique normalized monotone function α : [a, b] −→ r such that lf = ∫b a f(x)dα(x). (2.1) the integral is understood in the sense of riemann-stieltjes. moreover ‖l‖ = v(α). the riemann-stieltjes integral is a generalization of the riemann integral, where instead of taking the length of the intervals, a α-weighted length is taken. for an interval i the α-length is given by α(i) = α(y) − α(x), where x, y are the end points of i and α is a function of finite variation. the integral of a continuous function f on [a, b] is defined as the limit, when it exists, of the sum ∑ i f(ci)α(ii) where {ii} is a finite collection of subintervals whose endpoints form a partition of [a, b] and ci ∈ ii. see [17] p.122. there are different proofs of the above theorem, see for example [22]. here we will give a sketch of the proof which uses the following result about extensions of functionals known as the hahn-banach theorem: let x a normed linear space, y a subspace of x, and λ an element of y∗. then there exists a λ ∈ x∗ extending λ with the same norm. see [13] for a proof. proof of theorem 2.1. . we may assume that [a, b] = [0, 1]. since l ∈ c[0, 1]∗ we use hahn-banach theorem to conclude the existence of λ ∈ b[0, 1]∗ such that ‖λ‖ = ‖l‖ and l = λ on c[0, 1] and 4 rafael del rio, asaf l. franco and jose a. lara cubo 20, 2 (2018) where b[0, 1] is the set of bounded functions on [0, 1]. let us define the functions 1x := 1[0,x], that is 1x(t) = 1 when t ∈ [0, x] and zero otherwise. set α(x) = λ(1x) for all x ∈ [0, 1]. now for f ∈ c[0, 1], define fn = n∑ j=1 f(j/n)(1j/n − 1(j−1)/n). since f is continuous, it is uniformly continuous on [0, 1] and so ‖fn − f‖∞ → 0. thus lim n λ(fn) = λ(f) = l(f). using the definition of α we get λ(fn) = n∑ j=1 f(j/n)(α(j/n) − α((j − 1)/n)). this in turn implies λ(f) = lim n λ(fn) = ∫1 0 f dα. now to see that ‖l‖ = v(α) : let ε > 0 and choose f ∈ c[0, 1] such that ‖f‖ ∞ ≤ 1 and ‖l‖ ≤ |l(f)| + ε, we apply (2.1) and we get ‖l‖ ≤ |l(f)| + ε = ∣ ∣ ∣ ∣ ∣ ∫1 0 f(x)dα(x) ∣ ∣ ∣ ∣ ∣ + ε ≤ α(1) − α(0) + ε = v(α) + ε. it is possible to normalize α and in this case we easily have the other inequality, that is, v(α) = α(1) − α(0) = α(1) = λ(11) ≤ ‖λ‖ = ‖l‖. remarks. (1) the standard textbook’s proof uses hahn-banach’s theorem ([10],[22]), but the original proof of f. riesz does not use it. see [17] section 50 and [15],[16]. cubo 20, 2 (2018) an approach to f. riesz representation theorem 5 (2) e. helly [8] should have similar results. j. radon extended theorem 2.1 to compact subsets k ⊂ rn [12]. s. banach and s. saks extended the result to compact metric spaces, see appendix of [21] and [20]. the proof by s. saks is particularly elegant and clean. for compact hausdorff spaces the theorem was proven by s. kakutani [9] and for normal spaces by a. markoff [11]. nowadays this theorem is also known as riesz-markoff or riesz-markoffkakutani theorem. there is a great variety of proofs of f. riesz theorem using different methods and even category theory see [7]. our proof only uses basic knowledge of measure theory. more information on the history of this theorem can be found in [5] p. 231, the references therein, [23] p.238 and [6]. (3) positivity of a linear functional l implies continuity of l. to see it, we take the function 1(x) = 1 for all x ∈ k, then 1 ∈ c(k) and |f(x)| ≤ ‖f‖ ∞ 1(x), therefore ‖f‖ ∞ 1(x) ± f(x) ≥ 0 implies ‖f‖ ∞ l(1) ± l(f) ≥ 0 so |l(f)| ≤ l(1)‖f‖ ∞ . see [5] prop. 7.1. 3 main result next theorem is our main result. it is a generalization of theorem 2.1 to continuous functions defined on arbitrary compact sets k ⊂ r. since an ordinary riemann-stieltjes integral is not defined for functions on general compact k, we shall introduce the lebesgue integral which makes sense for such functions. in the appendix, we collect the basic facts and definitions of measure theory we need. theorem 3.1. let k a compact subset of r and let ℓ : c(k) → r be a positive linear functional. then, there is a unique finite borel measure µ such that µ(k) = ‖ℓ‖ c(k)∗ and ℓf = ∫ k fdµ. (3.1) proof. the proof proceeds in stages. i) integral representation. let [a, b] be a closed and bounded interval containing k. note that the technique used in what follows is independent of this interval. let r : c[a, b] −→ c(k) be the restriction operator, that is, for every f ∈ c[a, b], r(f)(x) = f(x) for x ∈ k. it is clear that r is a bounded linear operator, so we can define its transpose operator, see [23] p.11, also known as adjoint, see [22]. recall rt is defined as follows rt : c(k)∗ → c[a, b]∗, rt(ℓ)(f) = ℓ(r(f)) for f ∈ c[a, b]; the expression ℓ(r(f)) assigns a scalar to each function f ∈ c[a, b]. 6 rafael del rio, asaf l. franco and jose a. lara cubo 20, 2 (2018) let ℓ be a positive linear functional in c(k) and we define lf = rt(ℓ)(f) = ℓ(rf). since ℓ and r are positive linear functionals, so is l and we can apply theorem 2.1 and (c) in the appendix to find a monotone increasing function α and an associated borel measure µ such that lf = rt(ℓ)(f) = ∫b a fdα = ∫b a fdµ (3.2) for every f ∈ c[a, b]. denote kc := [a, b] \ k. we will show that µ(kc) = 0. let ε > 0 and choose fε as a closed subset of kc such that µ(kc \ fε) < ε, (3.3) see (a) in the appendix. let f̃ ∈ c[a, b] be a continuous function such that f̃(x) = 1 if x ∈ k, f̃(x) = 0 if x ∈ fε and ‖f̃‖ ∞ ≤ 1. one can take for instance f̃(x) = d(x, fε) d(x, fε) + d(x, k) , where d(x, a) = infy∈a |x − y|. note that since |d(x, a) − d(y, a)| ≤ |x − y| the function d(x, a) is even uniformly continuous, (cf. urysohn’s lemma. [5], 4.15.). therefore l(f̃) = ∫b a f̃dµ = ∫ k dµ + ∫ kc\fε f̃dµ + ∫ fε f̃dµ the third integral on the right is equal zero, by definition of f̃. we can estimate the second integral as follows, 0 ≤ ∫ kc\fǫ f̃dµ ≤ ∫ kc\fε dµ = µ(kc\fε) < ε, since f̃ ≤ 1 and using (3.3). then l(f̃) < ∫ k dµ + ε = µ(k) + ε. we have that µ(k) + µ(kc) = ∫b a dµ = l(1[a,b]) = l(f̃) < µ(k) + ε, the third equality follows from r(f̃) = r(1[a,b]). thus 0 ≤ µ(k c) < ε, since µ(k) < ∞. to conclude, let f ∈ c(k) and f∗ a continuous extension of f to the closed interval [a, b]. we can do this extension taking, for example, straight lines as follows: since kc is an open subset of [a, b], it is at most a countable union of pairwise disjoint open intervals (αi, βi) cubo 20, 2 (2018) an approach to f. riesz representation theorem 7 intersected with the interval [a, b], (see lindeloef’s thm., [18] prop.9. p.40). for x ∈ (αi, βi) we define f∗(x) = (1 − t)f(αi) + tf(βi) if x = αi(1−t)+tβi for t ∈ (0, 1). the function f ∗ is continuous on the interval [a, b] since on k coincides with the continuous function f and on kc consists of straight lines, (cf. tietze’s theorem [5], 4.16). then we have ℓ(f) = ℓ(r(f∗)) = lf∗ = ∫b a f∗dα = ∫b a f∗dµ = ∫ k f∗dµ = ∫ k fdµ. (3.4) as was to be shown. ii) conservation of norm. take f ∈ c(k) such that ‖f‖ ∞ ≤ 1. since (3.1) holds we have, |ℓ(f)| = ∣ ∣ ∣ ∣ ∫ k fdµ ∣ ∣ ∣ ∣ ≤ ‖f‖ ∞ µ(k) ≤ µ(k). for the reverse inequality, let 1(x) = 1 for all x, as defined in remark (3), so ‖ℓ‖ ≥ |ℓ(1)| = ∣ ∣ ∣ ∣ ∫ k 1 dµ ∣ ∣ ∣ ∣ = µ(k), we can conclude that µ(k) = ‖ℓ‖. iii) uniqueness. suppose µ and ν are finite measures that satisfy (3.1). since µ and ν are regular measures, from (a) in the appendix, it is enough to show that µ(c) = ν(c) for any closed set c of k. let c a nonempty closed set of k and set fk(x) := max{0, 1 − kd(x, c)} for all k and x ∈ k, where d(x, c) = infy∈c |x − y|. these functions are bounded, by 0 and 1, and continuous. thus fk belongs to c(k) for all k. notice that they form a sequence that decreases to the indicator of c, i.e., fk ↓ 1c, where 1c(x) = 1 if x ∈ c and 1c(x) = 0 if x /∈ c. thus, for all k we must have that ∫ k fkdµ = ∫ k fkdν, and so we can use the dominated convergence theorem, see (b) in the appendix, to conclude that µ(c) = lim k ∫ k fkdµ = lim k ∫ k fkdν = ν(c). remarks (a) it is possible to represent the linear positive functionals acting on c(k) as riemann-stieltjes integrals, similar to the original work of f. riesz. this follows immediately from the chain of 8 rafael del rio, asaf l. franco and jose a. lara cubo 20, 2 (2018) equalities (3.4). the caveat is that we cannot use f directly in order to define the riemannstieltjes integral, but any continuous extension of f works, cf. theorem 3.2 below. this integral is independent of the extension of f. (b) as just seen, the use of compact set k above allows us to extend the continuous functions to the entire interval [a, b], using an elementary version of the tietze’s theorem. this construction is in general not possible if k is an arbitrary subset of the real line. 3.1 isomorphic spaces as a consequence of the previous results, we shall see that two spaces of functionals are practically the same. one of the spaces consists of lebesgue integrals on compact subsets of [a, b] and the other of riemann-stieltjes integrals over the whole interval [a, b]. in this way we show how the lebesgue integral representation, that was introduced to represent functionals in the case of general compact sets, can be seen as a riemann-stieltjes integral. to state this precisely we need to introduce the terms isomorphic and constant in kc. a transformation t which preserves the norm, that is ‖tx‖ = ‖x‖, is called an isometry. two normed vector spaces x and y are said to be isomorphic if there is a linear, bijective, isometry t : x → y. such functions are called isomorphisms. since an isomorphism preserves the linear as well as the metric structure of the spaces, two isomorphic spaces can be considered identical, the isomorphism corresponding just to a labeling of the elements. we say that the monotone function α is constant in kc if it is constant in each interval of kc. recall that c(x)∗+ denotes the set of positive linear functionals on c(x). let lα denote the functional with corresponding monotone function α as introduced in (2.1). the result mentioned above can be then stated as follows: theorem 3.2. the normed spaces { lα ∈ c[a, b] ∗ + : α is constant in k c } and c(k)∗+ are isomorphic. before we prove this theorem we need two preparatory results. proposition 3.3. rt : c(k)∗+ → c[a, b]∗+ is an isometry. proof. ‖rtℓ‖c[a,b]∗ + = v(α) = α(b) − α(a) = µ([a, b]) = µ(k) + µ([a, b] \ k) = µ(k) = ‖ℓ‖c(k)∗ + . the first equality follows from theorem 2.1. the function α depends on ℓ. the second is the definition of the total variation of α and the third is the definition of µ. the last two equalities follow from the construction of theorem 3.1. cubo 20, 2 (2018) an approach to f. riesz representation theorem 9 we denote the range of rt by rang rt = { l ∈ c[a, b]∗+ : ∃l ∈ c(k) ∗ + s.t. l = r tl } proposition 3.4. rang rt = { lα ∈ c[a, b] ∗ + : α is constant in k c } proof. ”⊂” let l ∈ rang rt ⊂ c[a, b]∗+. then there exists ℓ ∈ c(k) ∗ + such that as in (3.2) rt(ℓ)(f) = lf = lαf = ∫b a fdα = ∫b a fdµ as was shown in the proof of theorem 3.1 i), µ(kc) = 0. since kc is a countable union of intervals, these have µ measure zero. by the relation which is given in (4.1) below, between the measure µ and the monotone function α we conclude that α is constant in each one of the intervals of kc. ” ⊃ ” let lα ∈ c[a, b] ∗ + with α constant in each interval of k c and µ be the measure associated with this α, as in appendix (c). define ℓ ∈ c(k)∗+ as ℓh = ∫ k hdµ. we shall show that rt(ℓ)(f) = lαf for every f ∈ c[a, b]. since α constant in each interval of kc this implies, using again (4.1), that µ(kc) = 0. then we have lαf = ∫b a fdα = ∫b a fdµ = ∫ k fdµ = ∫ k r(f)dµ = ℓ(r(f)) = rt(ℓ)(f) where r(f) denotes, as in theorem (3.1) i) above, the restriction of f to k. proof of theorem 3.2. . from proposition 3.3 and proposition 3.4 it follows that rt is a bijective isometry. since rt is linear as follows from its definition, then it is an isomorphism. acknowledgments we thank c. bosch and ma. c. arrillaga for useful comments. we are grateful to ma. r. sanchez for her help in the search of bibliographical information. 4 appendix a collection of subsets a of x is called an σ-algebra if it is closed under finite (countable) union, complements and x ∈ a. if our space is r, the borel σ-algebra, br, is the smallest σ-algebra containing all the open intervals. a function µ : a → [0, ∞], where a is a σ-algebra, it is called a measure if it is countable additive, that is µ( ⋃ an) = ∑ µ(an) whenever {an} is a disjoint sequence of elements in a, and µ(∅) = 0. a borel measure is a measure defined on br. we say 10 rafael del rio, asaf l. franco and jose a. lara cubo 20, 2 (2018) that a measure is regular if every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. a function f from (x, a, µ) to (r, br) is a-measurable if {x : f(x) ≤ t} ∈ a for all t ∈ r. the following results are used in the proof of theorem 3.1. (a) every borel measure in a metric space is regular. we will only use inner regularity, that is, for every borel set a and every ε > 0 there exist a compact set fε such that fε ⊂ a and µ(a \ fǫ) < ǫ . [2] thm 7.1.7. or [3] lemma 1.5.7. (b) (dominated convergence theorem) let (x, a, µ) a measure spaces. let g be a [0, ∞]-valued integrable function on x, that is, ∫ gdµ < ∞, and let f, f1, f2, . . . real-valued a-measurable functions on x such that f(x) = limn fn(x) and |fn(x)| ≤ g(x). then f and {fn} are integrable and ∫ fdµ = limn ∫ fndµ. (c) given a normalized monotone function α in the closed interval [a, b], there is a unique borel measure µ associated with it. this can be seen as follows (see for example [4]): for a ≤ s ≤ t ≤ b let define 〈s, t] where let f0 = { ⋃ finite 〈sk, tk] : 〈sk, tk] ⊂ [a, b] pairwise disjoint } then f0 is an algebra of subsets of [a, b] and therefore we can define a set function as µ0 ( ⋃ finite 〈sk, tk] ) = ∑ finite α(tk) − α(sk). (4.1) moreover, µ0 has a unique extension to a measure in the smallest σ-algebra containing f0 (caratheodory’s theorem). see [1] . moreover, for any continuous function f it happens that ∫b a fdα = ∫b a fdµ (4.2) where the integral on the left is a riemann-stieltjes integral, whereas the integral on the right is an integral in the sense of lebesgue. references [1] bartle, robert g. the elements of integration. john wiley & sons, inc., new york-londonsydney 1966 x+129 pp. [2] v. i. bogachev, measure theory. vol. i, ii, springer-verlag, berlin, 2007. cubo 20, 2 (2018) an approach to f. riesz representation theorem 11 [3] d. cohn, measure theory, secon ed., birkhäuser, boston, mass. 2013. [4] doob, j. l. measure theory. graduate texts in mathematics, 143. springer-verlag, new york, 1994. xii+210 pp. isbn: 0-387-94055-3 [5] gerald b. folland, real analysis, second ed., pure and applied mathematics (new york), john wiley & sons, inc., new york, 1999, modern techniques and their applications, a wiley-interscience publication. [6] gray, j. d. the shaping of the riesz representation theorem: a chapter in the history of analysis. arch. hist. exact sci. 31 (1984), no. 2, 127–187. [7] d. g. hartig, the riesz representation theorem revisited, amer. math. monthly 90 no. 4 (1983), pp. 277–280 [8] e. helly, über lineare funktionaloperationen, wien ber. 121 (1912), 265–297. [9] shizuo kakutani, concrete representation of abstract (m)-spaces. (a characterization of the space of continuous functions.), ann. of math. (2) 42 (1941), 994–1024. [10] erwin kreyszig, introductory functional analysis with applications, john wiley & sons, new york-london-sydney, 1978. [11] a. markoff, on mean values and exterior densities, mat. sbornik 4 (46) (1938), no. 1, 165–191. [12] johann radon, gesammelte abhandlungen. band 1, verlag der österreichischen akademie der wissenschaften, vienna; birkhäuser verlag, basel, 1987, with a foreword by otto hittmair, edited and with a preface by peter manfred gruber, edmund hlawka, wilfried nöbauer and leopold schmetterer. [13] michael reed and barry simon, methods of modern mathematical physics. i, second ed., academic press inc. [harcourt brace jovanovich publishers], new york, 1980, functional analysis. [14] f. riesz, sur les opérations fonctionnelles linéares, comptes rendus acad. sci. paris 149 (1909), 974–977. [15] demonstration nouvelle d’un théorème concernant les opérations, annales ecole norm. sup. 31 (1914), 9–14. [16] frédéric riesz, sur la représentation des opérations fonctionnelles linéaires par des intégrales de stieltjes, comm. sém. math. univ. lund [medd. lunds univ. mat. sem.] 1952 (1952), no. tome supplementaire, 181–185. [17] frigyes riesz and béla sz.-nagy, functional analysis, dover books on advanced mathematics, dover publications, inc., new york, 1990, translated from the second french edition by leo f. boron, reprint of the 1955 original. 12 rafael del rio, asaf l. franco and jose a. lara cubo 20, 2 (2018) [18] h. l. royden, real analysis, the macmillan co., new york; collier-macmillan ltd., london, 1963. [19] walter rudin, real and complex analysis 3rd. ed., mcgraw-hill, inc., 1987, new york, ny, usa. [20] s. saks, integration in abstract metric spaces, duke math. j. 4 (1938), no. 2, 408–411. [21] stanis law saks, theory of the integral, second revised edition. english translation by l. c. young. with two additional notes by stefan banach, dover publications, inc., new york, 1964. [22] martin schechter, principles of functional analysis, second ed., graduate studies in mathematics, vol. 36, american mathematical society, providence, ri, 2002. [23] barry simon, real analysis, a comprehensive course in analysis, part 1, american mathematical society, providence, ri, 2015, with a 68 page companion booklet. introduction preliminaries definitions and notation. representation theorem for functionals on c[a,b]. main result isomorphic spaces appendix cubo, a mathematical journal vol. 23, no. 02, pp. 239–244, august 2021 doi: 10.4067/s0719-06462021000200239 free dihedral actions on abelian varieties b. aguiló vidal 1 1 departamento de matemáticas, facultad de ciencias, universidad de chile, las palmeras 3425, ñuñoa, santiago, chile. bruno.aguilo@ug.uchile.cl abstract we give a simple construction for hyperelliptic varieties, defined as the quotient of a complex torus by the action of a finite group g that contains no translations and acts freely, with g any dihedral group. this generalizes a construction given by catanese and demleitner for d4 in dimension three. resumen damos una construcción simple de variedades hiperelípticas, definidas como el cociente de un toro complejo por la acción de un grupo finito g que no contiene traslaciones y actúa libremente, con g cualquier grupo diedral. esto generaliza la construcción de catanese y demleitner para d4 en dimensión tres. keywords and phrases: abelian varieties, dihedral group, free action. 2020 ams mathematics subject classification: 14k99, 14l30. accepted: 18 may, 2021 received: 27 august, 2020 c©2021 b. aguiló vidal. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200239 http://orcid.org/0000-0002-6826-1050 240 bruno aguiló vidal cubo 23, 2 (2021) 1 introduction a generalized hyperelliptic manifold x is defined as a quotient x = t/g of a complex torus t by the free action of a finite group g which contains no translations. the manifold x is called a generalized hyperelliptic variety if the torus t is also projective, i.e., it is an abelian variety. these have kodaira dimension zero, as mistretta showed it to be the case for any étale finite quotient of a torus [5]. furthermore, these type of manifolds are kähler and their fundamental group is given by the group of complex affine transformations of t which are lifts of transformations of the group g, as stated by catanese and corvaja in [2]. uchida and yoshihara showed that the only non abelian group that gives such an action in dimension three is the dihedral group d4 of order 8 [6]. later, catanese and demleitner gave a simple and explicit construction for that action [4] and completed the characterization of three-dimensional hyperelliptic manifolds [3]. some authors have worked with these objects in higher dimension as well. for example, auffarth and lucchini arteche showed that they can be constructed using any finite abelian group, and that there are simple ways to construct some varieties using non abelian groups [1], all of these formed as products of manifolds of lower dimension. however, not much is known about hyperelliptic manifolds of dimension greater than 3, and a characterization of these manifolds is far from completed. the purpose of this note is to help with the understanding of generalized hyperelliptic varieties in higher dimension, showing that catanese and demleitner’s construction is actually generalizable (in quite a natural way) to every odd dimension. specifically, for every n ∈ n we give a generalized hyperelliptic variety of dimension 2n + 1 defined by the action of the dihedral group d4n of order 8n acting on a family of abelian varieties, from which the construction by catanese and demleitner remains as the particular case for n = 1, and we end with a simple corollary that explains how this allows us to create this type of varieties using any dihedral group. 2 the construction let e, e′ be any two elliptic curves, e = c/ (z + zτ), e′ = c/ (z + zτ′) . now, for n ∈ n set a′ := e2n × e′ and a := a′/〈w〉, where w := (1/2, 1/2, ..., 1/2, 0). theorem 2.1. the abelian variety a admits a free action with no translations of the dihedral group d4n of order 8n. cubo 23, 2 (2021) free dihedral actions on abelian varieties 241 proof. first, let us recall that for k ∈ n, the dihedral group of order 2k is defined as dk := 〈 r, s | rk = 1, s2 = 1, (rs)2 = 1 〉 . so, in order to prove the result, we need to find automorphisms of a with the required characteristics, that satisfy the relations described above. and for that, we will use automorphisms of a′ that descends to those of a in a useful way. now, set, for z := (z1, z2, ..., z2n, z2n+1) ∈ a ′, the following linear automorphisms: r(z) := (−z2n, z1, ..., z2n−1, z2n+1) , s(z) := (−z2n, −z2n−1, ..., −z2, −z1, −z2n+1) , and with these, we define the following automorphisms of a′: r(z) := r(z) + ( 0, ..., 0, 1 4n ) , s(z) := s(z) + (b1, b2, ..., b2n, 0), where, for i = 1, ..., n, b2i−1 := 1/2 + τ/2 and b2i := τ/2. step 1. it is easy to verify that r and r have order exactly 4n on a′. also, since r is linear and r(w) = w, then r(z + w) = r(z) + w, and so r descends to an automorphism of a of order exactly 4n. moreover, any power rj , for 0 < j < 4n, acts freely on a since the (2n + 1)-th coordinate of rj(z) equals z2n+1 + j 4n (and so is distinct from z2n+1). also, clearly none of these powers are translations, due to the fact that their linear part modifies at least one dimension. step 2. s2(z) = z+w, since for i = 1, ..., 2n, bi −b2n+1−i = 1/2, and so s does not have order 2 on a′. neverthless, since the linearity of s and the fact that s(w) = w imply that s(z+w) = s(z)+w, s descends to an automorphism of a of order exactly 2. step 3. we have rs(z) = zm + b′ where m =      1 0 . . . 0 0 ... i ... 0 0 . . . 0 −1      , i =        0 . . . 0 −1 0 −1 0 ... −1 ... −1 . . . 0 0        and b′ = ( −b2n, b1, ..., b2n−1, 1 4n ) . 242 bruno aguiló vidal cubo 23, 2 (2021) hence, by simple computations, we have that (rs)2(z) = zm2 + b′m + b′ = z, also, because it was already shown for r and s, it is also true that rs(z + w) = rs(z) + w, and so rs descends to an automorphism of a of order 2. thus, we have an action of d4n on a, since the orders of r, s and rs are precisely 4n, 2 and 2, respectively. step 4. we claim that also the reflections in d4n are not translations, noticing that, since dihedral groups representing even polygons have two conjugacy classes of reflections, those of s and rs, it suffices to observe that these two transformations are not translations. in the next step we show that they both act freely on a. step 5. it is rather immediate that rs acts freely in a, since rs(z) = z in a is equivalent to the difference rs(z) − z = (−b2n, −z2n − z2 + b1, ..., −z2 − z2n + b2n−1, −2z2n+1 + 14n ) being a multiple of w in a′, but this is absurd since the only multiples of w are zero and w itself, while −b2n = τ/2 6= 0, 1/2. on the other hand, s acts freely on a because s(z) = z in a is equivalent to the difference s(z) − z = (−z2n − z1 + b1, −z2n−1 − z2 + b2, ..., −z1 − z2n + b2n, −2z2n+1) being a multiple of w in a′, but the first and 2n-th coordinate of multiples of w are equal, while here the difference between them is b1 − b2n = 1/2 6= 0. 3 using any dihedral group notice that, although the previous construction is somewhat restrictive because it works with very specific dihedral groups, since it is true that dn ⊆ dkn for all n, k ∈ n, we can always make a bigger dihedral group act on some variety using the method above, and then restrict ourselves to a smaller one. so we have the following corollary: corollary 3.1. for all n ∈ n, there exists a free action of the dihedral group dn of order 2n on some abelian variety of dimension lcm(4,n) 2 + 1 that contains no translations. it is interesting to observe that, although the relation is far away from being one-to-one, we have shown that for every odd dimension there is an abelian variety and a dihedral group acting on it, cubo 23, 2 (2021) free dihedral actions on abelian varieties 243 and for every dihedral group there is an abelian variety of odd dimension on which it acts, in a way that the quotient forms a generalized hyperelliptic variety. acknowledgement i would like to thank professors robert auffarth and giancarlo lucchini arteche for introducing me to this topic. 244 bruno aguiló vidal cubo 23, 2 (2021) references [1] r. auffarth and g. lucchini arteche, “smooth quotients of complex tori by finite groups”, preprint (2021). arxiv:1912.05327. [2] f. catanese and p. corvaja, “teichmüller spaces of generalized hyperelliptic manifolds”, complex and symplectic geometry, springer indam ser. 21, springer, cham, 2017, pp. 39-49. [3] f. catanese and a. demleitner, “the classification of hyperelliptic threefolds”, groups geom. dyn, vol. 14, no. 4, pp. 1447–1454, 2020. [4] f. catanese and a. demleitner, “hyperelliptic threefolds with group d4, the dihedral group of order 8”, preprint (2018), arxiv:1805.01835. [5] e. c. mistretta, “holomorphic symmetric differentials and parallelizable compact complex manifolds”, riv. math. univ. parma (n.s.), vol. 10, no. 1, pp. 187–197, 2019. [6] k. uchida and h. yoshihara, “discontinuous groups of affine transformations of c3 ”. tohoku math. j. (2), vol. 28, no. 1, pp. 89–94, 1976. introduction the construction using any dihedral group cubo a mathematical journal vol.21, no¯ 03, (29–38). december 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000300029 ostrowski-sugeno fuzzy inequalities george a. anastassiou department of mathematical sciences university of memphis memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract we present ostrowski-sugeno fuzzy type inequalities. these are ostrowski-like inequalities in the context of sugeno fuzzy integral and its special properties are investigated. tight upper bounds to the deviation of a function from its sugeno-fuzzy averages are given. this work is greatly inspired by [3] and [1]. resumen presentamos desigualdades de ostrowski-sugeno de tipo fuzzy. estas son desigualdades de tipo ostrowski en el contexto de integrales fuzzy de sugeno y se investigan sus propiedades especiales. se entregan cotas superiores ajustadas para la desviación de una función de sus promedios fuzzy de sugeno. este trabajo está inspirado principalmente por [3] y [1]. keywords and phrases: sugeno fuzzy, integral, function fuzzy average, deviation from fuzzy mean, fuzzy ostrowski inequality. 2010 ams mathematics subject classification: primary: 26d07, 26d10, 26d15, 41a44, secondary: 26a24, 26d20, 28a25. http://dx.doi.org/10.4067/s0719-06462019000300029 30 george a. anastassiou cubo 21, 3 (2019) 1 introduction the famous ostrowski ([3]) inequality motivates this work and has as follows: ∣ ∣ ∣ ∣ ∣ 1 b − a ∫b a f (y) dy − f (x) ∣ ∣ ∣ ∣ ∣ ≤ ( 1 4 + ( x − a+b 2 )2 (b − a) 2 ) (b − a) ‖f′‖ ∞ , where f ∈ c′ ([a, b]), x ∈ [a, b], and it is a sharp inequality. one can easily notice that ( 1 4 + ( x − a+b 2 )2 (b − a) 2 ) (b − a) = (x − a) 2 + (b − x) 2 2 (b − a) . another motivation is author’s article [1]. first we give a survey about sugeno fuzzy integral and its basic properties. then we derive a series of ostrowski-like inequalities to all directions in the context of sugeno integral and its basic important particular properties. we also give applications to special cases of our problem we deal with. 2 background in this section, some definitions and basic important properties of the sugeno integral which will be used in the next section are presented. definition 2.1. (fuzzy measure [5, 7]) let σ be a σ-algebra of subsets of x, and let µ : σ → [0, +∞] be a non-negative extended real-valued set function. we say that µ is a fuzzy measure iff: (1) µ (∅) = 0, (2) e, f ∈ σ : e ⊆ f imply µ (e) ≤ µ (f) (monotonicity), (3) en ∈ σ (n ∈ n), e1 ⊂ e2 ⊂ ..., imply lim n→∞ µ (en) = µ (∪ ∞ n=1en) (continuity from below); (4) en ∈ σ (n ∈ n), e1 ⊃ e2 ⊃ ..., µ (e1) < ∞, imply lim n→∞ µ (en) = µ (∩ ∞ n=1en) (continuity from above). let (x, σ, µ) be a fuzzy measure space and f be a non-negative real-valued function on x. we denote by f+ the set of all non-negative real valued measurable functions, and by lαf the set: lαf := {x ∈ x : f (x) ≥ α}, the α-level of f for α ≥ 0. definition 2.2. let (x, σ, µ) be a fuzzy measure space. if f ∈ f+ and a ∈ σ, then the sugeno integral (fuzzy integral) [6] of f on a with respect to the fuzzy measure µ is defined by (s) ∫ a fdµ := ∨α≥0 (α ∧ µ (a ∩ lαf)) , (1) where ∨ and ∧ denote the sup and inf on [0, ∞], respectively. cubo 21, 3 (2019) ostrowski-sugeno fuzzy inequalities 31 the basic properties of sugeno integral follow: theorem 2.3. ([4, 7]) let (x, σ, µ) be a fuzzy measure space with a, b ∈ σ and f, g ∈ f+. then 1) (s) ∫ a fdµ ≤ µ (a) ; 2) (s) ∫ a kdµ = k ∧ µ (a) for a non-negative constant k; 3) if f ≤ g on a, then (s) ∫ a fdµ ≤ (s) ∫ a gdµ; 4) if a ⊂ b, then (s) ∫ a fdµ ≤ (s) ∫ b fdµ; 5) µ (a ∩ lαf) ≤ α ⇒ (s) ∫ a fdµ ≤ α; 6) if µ (a) < ∞, then µ (a ∩ lαf) ≥ α ⇔ (s) ∫ a fdµ ≥ α; 7) when a = x, (s) ∫ a fdµ = ∨α≥0 (α ∧ µ (lαf)) ; 8) if α ≤ β, then lβf ⊆ lαf; 9) (s) ∫ a fdµ ≥ 0. theorem 2.4. ([7, p. 135]) let f ∈ f+, the class of all finite nonnegative measurable functions on (x, σ, µ). then 1) if µ (a) = 0, then (s) ∫ a fdµ = 0, for any f ∈ f+; 2) if (s) ∫ a fdµ = 0, then µ (a ∩ {x|f (x) > 0}) = 0; 3) (s) ∫ a fdµ = (s) ∫ a f · χadµ, where χa is the characteristic function of a; 4) (s) ∫ a (f + a) dµ ≤ (s) ∫ a fdµ + (s) ∫ a adµ, for any constant a ∈ [0, ∞). corollary 2.5. ([7, p. 136]) let f, f1, f2 ∈ f+. then 1) (s) ∫ a (f1 ∨ f2) dµ ≥ (s) ∫ a f1dµ ∨ (s) ∫ a f2dµ; 2) (s) ∫ a (f1 ∧ f2) dµ ≤ (s) ∫ a f1dµ ∧ (s) ∫ a f2dµ; 3) (s) ∫ a∪b fdµ ≥ (s) ∫ a fdµ ∨ (s) ∫ b fdµ; 4) (s) ∫ a∩b fdµ ≤ (s) ∫ a fdµ ∧ (s) ∫ b fdµ. in general we have (s) ∫ a (f1 + f2) dµ 6= (s) ∫ a f1dµ + (s) ∫ a f2dµ, and (s) ∫ a afdµ 6= a (s) ∫ a fdµ, where a ∈ r, see [7, p. 137]. lemma 2.6. ([7, p. 138]) (s) ∫ a fdµ = ∞ if and only if µ (a ∩ lαf) = ∞ for any α ∈ [0, ∞). we need 32 george a. anastassiou cubo 21, 3 (2019) definition 2.7. ([2]) a fuzzy measure µ is subadditive iff µ (a ∪ b) ≤ µ (a) + µ (b), for all a, b ∈ σ. we mention the following result theorem 2.8. ([2]) if µ is subadditive, then (s) ∫ x (f + g) dµ ≤ (s) ∫ x fdµ + (s) ∫ x gdµ, (2) for all measurable functions f, g : x → [0, ∞). moreover, if (2) holds for all measurable functions f, g : x → [0, ∞) and µ (x) < ∞, then µ is subadditive. notice here in (1) we have that α ∈ [0, ∞). we have the following corollary. corollary 2.9. if µ is aubadditive, n ∈ n, and f : x → [0, ∞) is a measurable function, then (s) ∫ x nfdµ ≤ n (s) ∫ x fdµ, (3) in particular it holds (s) ∫ a nfdµ ≤ n (s) ∫ a fdµ, (4) for any a ∈ σ. proof. by inequality (2). a very important property of sugeno integral follows. theorem 2.10. if µ is subadditive measure, and f : x → [0, ∞) is a measurable function, and c > 0, then (s) ∫ a cfdµ ≤ (c + 1) (s) ∫ a fdµ, (5) for any a ∈ σ. proof. let the ceiling ⌈c⌉ = m ∈ n, then by theorem 2.3 (3) and (4) we get (s) ∫ a cfdµ ≤ (s) ∫ a mfdµ ≤ m (s) ∫ a fdµ ≤ (c + 1) (s) ∫ a fdµ, proving (5). cubo 21, 3 (2019) ostrowski-sugeno fuzzy inequalities 33 3 main results from now on in this article we work on the fuzzy measure space ([a, b] , b, µ), where [a, b] ⊂ r, b is the borel σ-algebra on [a, b], and µ is a finite fuzzy measure on b. typically we take it to be subadditive. the functions f we deal with here are continuous from [a, b] into r+. we make the following remark remark 3.1. let f ∈ c1 ([a, b] , r+), and µ is a subadditive fuzzy measure such that µ ([a, b]) > 0, x ∈ [a, b]. we will estimate e := ∣ ∣ ∣ ∣ ∣ (s) ∫ [a,b] f (x) dµ (t) − µ ([a, b]) ∧ f (x) ∣ ∣ ∣ ∣ ∣ (6) (by theorem 2.3 (2)) = ∣ ∣ ∣ ∣ ∣ (s) ∫ [a,b] f (t) dµ (t) − (s) ∫ [a,b] f (x) dµ (t) ∣ ∣ ∣ ∣ ∣ . we notice that f (t) = f (t) − f (x) + f (x) ≤ |f (t) − f (x)| + f (x) , then (by theorem 2.3 (3) and theorem 2.4 (4)) (s) ∫ [a,b] f (t) dµ (t) ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) + (s) ∫ [a,b] f (x) dµ (t) , (7) that is (s) ∫ [a,b] f (t) dµ (t) − (s) ∫ [a,b] f (x) dµ (t) ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) . (8) similarly, we have f (x) = f (x) − f (t) + f (t) ≤ |f (t) − f (x)| + f (t) , then (by theorem 2.3 (3) and theorem 2.8) (s) ∫ [a,b] f (x) dµ (t) ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) + (s) ∫ [a,b] f (t) dµ (t) , that is (s) ∫ [a,b] f (x) dµ (t) − (s) ∫ [a,b] f (t) dµ (t) ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) . (9) by (8) and (9) we derive that ∣ ∣ ∣ ∣ ∣ (s) ∫ [a,b] f (t) dµ (t) − (s) ∫ [a,b] f (x) dµ (t) ∣ ∣ ∣ ∣ ∣ ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) . (10) 34 george a. anastassiou cubo 21, 3 (2019) consequently it holds e (by (6), (10)) ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) (and by |f (t) − f (x)| ≤ ‖f′‖ ∞ |t − x|) ≤ (s) ∫ [a,b] ‖f′‖ ∞ |t − x| dµ (t) (by (5)) ≤ (‖f′‖ ∞ + 1) (s) ∫ [a,b] |t − x| dµ (t) . (11) we have proved the following ostrowski-like inequality ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − µ ([a, b] ∧ f (x)) µ ([a, b]) ∣ ∣ ∣ ∣ ∣ ≤ (12) (‖f′‖ ∞ + 1) µ ([a, b]) (s) ∫ [a,b] |t − x| dµ (t) . the last inequality can be better written as follows: ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (‖f′‖ ∞ + 1) µ ([a, b]) (s) ∫ [a,b] |t − x| dµ (t) . (13) notice here that ( 1 ∧ f(x) µ([a,b]) ) ≤ 1, and 1 µ([a,b]) (s) ∫ [a,b] f (t) dµ (t) ≤ µ([a,b]) µ([a,b]) = 1, where (s) ∫ [a,b] f (t) dµ (t) ≥ 0. i.e. if f : [a, b] → r+ is a lipschitz function of order 0 < α ≤ 1, i.e. |f (x) − f (y)| ≤ k |x − y| α , ∀ x, y ∈ [a, b], where k > 0, denoted by f ∈ lipα,k ([a, b] , r+), then we get similarly the following ostrowski-like inequality: ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (k + 1) µ ([a, b]) (s) ∫ [a,b] |t − x| α dµ (t) . (14) we have proved the following ostrowski-sugeno inequalities: theorem 3.2. suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b] . 1) let f ∈ c1 ([a, b] , r+), then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (‖f′‖ ∞ + 1) µ ([a, b]) (s) ∫ [a,b] |t − x| dµ (t) . (15) cubo 21, 3 (2019) ostrowski-sugeno fuzzy inequalities 35 2) let f ∈ lipα,k ([a, b] , r+), 0 < α ≤ 1, then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (k + 1) µ ([a, b]) (s) ∫ [a,b] |t − x| α dµ (t) . (16) we make the following remark remark 3.3. let f ∈ c1 ([a, b] , r+) and g ∈ c 1 ([a, b]), by cauchy’s mean value theorem we get that (f (t) − f (x)) g′ (c) = (g (t) − g (x)) f′ (c) , for some c between t and x; for any t, x ∈ [a, b]. if g′ (c) 6= 0, we have (f (t) − f (x)) = ( f′ (c) g′ (c) ) (g (t) − g (x)) . here we assume that g′ (t) 6= 0, ∀ t ∈ [a, b]. hence it holds |f (t) − f (x)| ≤ ∥ ∥ ∥ ∥ f′ g′ ∥ ∥ ∥ ∥ ∞ |g (t) − g (x)| , (17) for all t, x ∈ [a, b] . we have again as before (see (11)) e ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) (by (17)) ≤ (s) ∫ [a,b] ∥ ∥ ∥ ∥ f′ g′ ∥ ∥ ∥ ∥ ∞ |g (t) − g (x)| dµ (t) (by (5)) ≤ ( ∥ ∥ ∥ ∥ f′ g′ ∥ ∥ ∥ ∥ ∞ + 1 ) (s) ∫ [a,b] |g (t) − g (x)| dµ (t) . (18) we have established the following general ostrowski-sugeno inequality: theorem 3.4. suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b]. let f ∈ c1 ([a, b] , r+) and g ∈ c 1 ([a, b]) with g′ (t) 6= 0, ∀ t ∈ [a, b] . then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ∥ ∥ ∥ f ′ g′ ∥ ∥ ∥ ∞ + 1 ) µ ([a, b]) (s) ∫ [a,b] |g (t) − g (x)| dµ (t) . (19) 36 george a. anastassiou cubo 21, 3 (2019) we give for g (t) = et the next result corollary 3.5. suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b]. let f ∈ c1 ([a, b] , r+), then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ∥ ∥ ∥ f ′ et ∥ ∥ ∥ ∞ + 1 ) µ ([a, b]) (s) ∫ [a,b] ∣ ∣et − ex ∣ ∣dµ (t) . (20) when g (t) = ln t we get the following corollary. corollary 3.6. suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b] and a > 0. let f ∈ c1 ([a, b] , r+) . then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (‖tf′ (t)‖ ∞ + 1) µ ([a, b]) (s) ∫ [a,b] ∣ ∣ ∣ ∣ ln t x ∣ ∣ ∣ ∣ dµ (t) . (21) many other applications of theorem 3.4 could follow but we stop it here. we make the following remark. remark 3.7. let f ∈ [ c ([a, b] , r+) ∩ c n+1 ([a, b]) ] , n ∈ n, x ∈ [a, b]. then by taylor’s theorem we get f (y) − f (x) = n∑ k=1 f(k) (x) k! (y − x) k + rn (x, y) , (22) where the remainder rn (x, y) := ∫y x ( f(n) (t) − f(n) (x) ) (y − t) n−1 (n − 1) ! dt; (23) here y can be ≥ x or ≤ x. by [1] we get that |rn (x, y)| ≤ ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! |y − x| n+1 , for all x, y ∈ [a, b] . (24) here we assume f(k) (x) = 0, for all k = 1, ..., n. therefore it holds |f (t) − f (x)| ≤ ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! |t − x| n+1 , for all t, x ∈ [a, b] . (25) cubo 21, 3 (2019) ostrowski-sugeno fuzzy inequalities 37 here we have again e ≤ (s) ∫ [a,b] |f (t) − f (x)| dµ (t) (by theorem 2.3 (3) and (25)) ≤ (s) ∫ [a,b] ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! |t − x| n+1 dµ (t) (by (5)) ≤ ( ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! + 1 ) (s) ∫ [a,b] |t − x| n+1 dµ (t) . (26) we have derived the following high order ostrowski-sugeno inequality: theorem 3.8. let f ∈ [ c ([a, b] , r+) ∩ c n+1 ([a, b]) ] , n ∈ n, x ∈ [a, b]. we assume that f(k) (x) = 0, all k = 1, ..., n. here µ is subadditive with µ ([a, b]) > 0. then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ‖f(n+1)‖ ∞ (n+1)! + 1 ) µ ([a, b]) (s) ∫ [a,b] |t − x| n+1 dµ (t) , (27) which generalizes (15). when x = a+b 2 we get the following corollary corollary 3.9. let f ∈ [ c ([a, b] , r+) ∩ c n+1 ([a, b]) ] , n ∈ n. assume that f(k) ( a+b 2 ) = 0, k = 1, ..., n. here µ is subadditive with µ ([a, b]) > 0. then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (s) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f ( a+b 2 ) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ‖f(n+1)‖ ∞ (n+1)! + 1 ) µ ([a, b]) (s) ∫ [a,b] ∣ ∣ ∣ ∣ t − a + b 2 ∣ ∣ ∣ ∣ n+1 dµ (t) . (28) 38 george a. anastassiou cubo 21, 3 (2019) references [1] g.a. anastassiou, ostrowski type inequalities, proc. amer. math. soc. 123(1995), 3775-3781. [2] m. boczek, m. kaluszka, on the minkowaki-hölder type inequalities for generalized sugeno integrals with an application, kybernetica, 52(3) (2016), 329-347. [3] a. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmittelwert, comment. math. helv., 10 (1938), 226-227. [4] e. pap, null-additive set functions, kluwer academic, dordrecht, 1995. [5] d. ralescu, g. adams, the fuzzy integral, j. math. anal. appl., 75 (1980), 562-570. [6] m. sugeno, theory of fuzzy integrals and its applications, phd thesis, tokyo institute of technology (1974). [7] z. wang, g.j. klir, fuzzy measure theory, plenum, new york, 1992. introduction background main results cubo, a mathematical journal vol. 23, no. 02, pp. 333–341, august 2021 doi: 10.4067/s0719-06462021000200333 on the conformally k-th gauduchon condition and the conformally semi-kähler condition on almost complex manifolds masaya kawamura 1 1 department of general education, national institute of technology, kagawa college, 355, chokushi-cho, takamatsu, kagawa, japan 761-8058. kawamura-m@t.kagawa-nct.ac.jp abstract we introduce the k-th gauduchon condition on almost complex manifolds. we show that if both the conformally k-th gauduchon condition and the conformally semi-kähler condition are satisfied, then it becomes conformally quasikähler. resumen introducimos la k-ésima condición de gauduchon en variedades casi complejas. mostramos que si la k-ésima condición de gauduchon conforme y la condición semikähler conforme se satisfacen ambas, entonces la variedad es cuasi-kähler conforme. keywords and phrases: almost hermitian manifold, k-th gauduchon metric, semi-kähler metric. 2020 ams mathematics subject classification: 32q60, 53c15, 53c55. accepted: 26 june, 2021 received: 17 november, 2020 c©2021 m. kawamura. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200333 https://orcid.org/0000-0003-1303-4237 334 masaya kawamura cubo 23, 2 (2021) 1 introduction s. ivanov and g. papadopoulous introduced the conditions on the hermitian form such that ωl ∧ ∂∂̄ωk = 0 for 1 ≤ k + l ≤ n − 1, which is called the (l|k)-skt condition. they have proven that every compact conformally balanced (l|k)-skt manifold, k < n − 1, n > 2, is kähler (cf. [5]). j. fu, z. wang and d. wu introduced and investigated the generalization of gauduchon metrics, which is called k-th gauduchon. the k-th gauduchon condition is the case l = n − k − 1, 1 ≤ k ≤ n−1 of the (l|k)-skt condition. by definition, (n−1)-th gauduchon metrics are the usual gauduchon metrics, astheno-kähler metrics are examples of (n − 2)-th gauduchon metrics, and pluriclosed metrics are in particular 1-st gauduchon. they proved that there exists a non-kähler 3-fold which can support a 1-gauduchon metric and a balanced metric simultaneously (cf. [2]). since k. liu and x. yang have shown that if a compact complex manifold is k-th gauduchon for 1 ≤ k ≤ n − 2 and also balanced, then it must be kähler, a 1-gauduchon metric and a balanced metric on a non-kähler 3-fold which fu, wang and wu discovered must be different hermitian metrics. liu and yang also have shown that the conformally kählerianity is equivalent to that both the conformally k-th gauduchon for 1 ≤ k ≤ n − 2, and the conformally balancedness are satisfied (cf. [7]). our aim in this paper is to generalize the liu-yang’s equivalence [7, corollary 1.17] to almost hermitian geometry. let (m2n,j) be an almost complex manifold with n ≥ 3 and let g be an almost hermitian metric on m. let {zr} be an arbitrary local (1,0)-frame around a fixed point p ∈ m and let {ζr} be the associated coframe. then the associated real (1,1)-form ω with respect to g takes the local expression ω = √ −1grk̄ζr ∧ ζk̄. we will also refer to ω as to an almost hermitian metric. we introduce the definition of a gauduchon metric and we define a k-th gauduchon metric as follows. definition 1.1. let (m2n,j) be an almost complex manifold. a metric g is called a gauduchon metric on m if g is an almost hermitian metric whose associated real (1,1)-form ω = √ −1gij̄ζi∧ζj̄ satisfies d∗(jd∗ω) = 0, where d∗ is the adjoint of d with respect to g, which is equivalent to d(jd(ωn−1)) = 0, or ∂∂̄(ωn−1) = 0. when an almost hermitian metric g is gauduchon, the triple (m2n,j,g) will be called a gauduchon manifold. for 1 ≤ k ≤ n − 1, an almost hermitian metric ω is called k-th gauduchon if it satisfies that ∂∂̄ωk ∧ ωn−k−1 = 0. notice that the condition ∂∂̄ωk ∧ ωn−k−1 = 0 for 1 ≤ k ≤ n − 2 is not equivalent to d(jd(ωk)) ∧ ωn−k−1 = 0 for 1 ≤ k ≤ n−2 since there exist a and ā parts of the exterior differential operator d in the almost complex setting (note that these conditions are equivalent in the case of k = n−1 as we confirmed in definition 1.1 since then we have a(ωn−1) = ā(ωn−1) = 0.). hence the condition ∂∂̄ωk ∧ ωn−k−1 = 0 for 1 ≤ k ≤ n − 1 can be regarded as a natural extension of the gauduchon condition on almost complex manifolds. we next introduce the definition of a semi-kähler metric. cubo 23, 2 (2021) on the conformally k-th gauduchon condition and the conformally ... 335 definition 1.2. let (m2n,j) be an almost complex manifold. a metric g is called a semi-kähler metric on m if g is an almost hermitian metric whose associated real (1,1)-form ω = √ −1gij̄ζi∧ζj̄ satisfies d(ωn−1) = 0. when an almost hermitian metric g is semi-kähler, the triple (m2n,j,ω) will be called a semi-kähler manifold. recall that on an almost hermitian manifold (m,j,g), a quasi-kähler structure is an almost hermitian structure whose real (1,1)-form ω satisfies (dω)(1,2) = ∂̄ω = 0, which is equivalent to the original definition of quasi-kählerianity: dxj(y ) + djxj(jy ) = 0 for all vector fields x,y (cf. [4]), where d is the levi-civita connection associated to g. it is important for us to study quasi-kähler manifolds since they include the classes of almost kähler manifolds and nearly kähler manifolds. an almost kähler or quasi-kähler manifold with j integrable is a kähler manifold. we define some conformally conditions. definition 1.3. let (m,j,ω) be an almost hermitian manifold. we say ω is conformally k-th gauduchon (resp. semi-kähler, quasi-kähler) if there exist a k-th gauduchon (resp. semi-kähler, quasi-kähler) metric ω̃ and a smooth function f ∈ c∞(m,r) such that ω = ef ω̃. our main result is as follows. theorem 1.4. on a compact almost hermitian manifold (m,j,ω), the following are equivalent: (1) (m,j,ω) is conformally quasi-kähler. (2) (m,j,ω) is conformally k-th gauduchon for 1 ≤ k ≤ n − 2, and conformally semi-kähler. in particular, the following are also equivalent: (a) (m,j,ω) is quasi-kähler. (b) (m,j,ω) is k-th gauduchon for 1 ≤ k ≤ n − 2, and conformally semi-kähler. this paper is organized as follows: in the second section, we recall some basic definitions and computations. in the last section, we will give a proof of the main result. notice that we assume the einstein convention omitting the symbol of sum over repeated indexes in all this paper. 2 preliminaries 2.1 the nijenhuis tensor of the almost complex structure let m be a 2n-dimensional smooth differentiable manifold. an almost complex structure on m is an endomorphism j of tm, j ∈ γ(end(tm)), satisfying j2 = −idt m . the pair (m,j) is called 336 masaya kawamura cubo 23, 2 (2021) an almost complex manifold. let (m,j) be an almost complex manifold. we define a bilinear map on c∞(m) for x,y ∈ γ(tm) by 4n(x,y ) := [jx,jy ] − j[jx,y ] − j[x,jy ] − [x,y ], (2.1) which is the nijenhuis tensor of j. the nijenhuis tensor n satisfies n(x,y ) = −n(y,x), n(jx,y ) = −jn(x,y ), n(x,jy ) = −jn(x,y ), n(jx,jy ) = −n(x,y ). for any (1,0)vector fields w and v , n(v,w) = −[v,w ](0,1), n(v,w̄) = n(v̄ ,w) = 0 and n(v̄ ,w̄) = −[v̄ ,w̄ ](1,0) since we have 4n(v,w) = −2([v,w ] + √ −1j[v,w ]), 4n(v̄ ,w̄) = −2([v̄ ,w̄] − √ −1j[v̄ ,w̄ ]). an almost complex structure j is called integrable if n = 0 everywhere on m. giving a complex structure on a differentiable manifold m is equivalent to giving an integrable almost complex structure on m. let (m,j) be an almost complex manifold. a riemannian metric g on m is called j-invariant if j is compatible with g, i.e., for any x,y ∈ γ(tm), g(x,y ) = g(jx,jy ). in this case, the pair (j,g) is called an almost hermitian structure. the fundamental 2-form ω associated to a j-invariant riemannian metric g, i.e., an almost hermitian metric, is determined by, for x,y ∈ γ(tm), ω(x,y ) = g(jx,y ). indeed we have, for any x,y ∈ γ(tm), ω(y,x) = g(jy,x) = g(j2y,jx) = −g(jx,y ) = −ω(x,y ) (2.2) and ω ∈ γ( ∧2 t ∗m). we will also refer to the associated real fundamental (1,1)-form ω as an almost hermitian metric. the form ω is related to the volume form dvg by n!dvg = ω n. let a local (1,0)-frame {zr} on (m,j) with an almost hermitian metric g and let {ζr} be a local associated coframe with respect to {zr}, i.e., ζi(zj) = δij for i,j = 1, . . . ,n. since g is almost hermitian, its components satsfy gij = gīj̄ = 0 and gij̄ = gj̄i = ḡīj. we write t rm for the real tangent space of m. then its complexified tangent space is given by t cm = t rm ⊗r c. by extending j c-linearly and g, ω, c-bilinearly to t cm, they are also defined on t cm and we observe that the complexified tangent space t cm can be decomposed as t cm = t 1,0m⊕t 0,1m, where t 1,0m, t 0,1m are the eigenspaces of j corresponding to eigenvalues √ −1 and − √ −1, respectively: t 1,0m = {x − √ −1jx ∣∣x ∈ tm}, t 0,1m = {x + √ −1jx ∣∣x ∈ tm}. (2.3) let λrm = ⊕ p+q=r λ p,qm for 0 ≤ r ≤ 2n denote the decomposition of complex differential r-forms into (p,q)-forms, where λp,qm = λp(λ1,0m) ⊗ λq(λ0,1m), λ1,0m = {α + √ −1jα ∣∣α ∈ λ1m}, λ0,1m = {α − √ −1jα ∣∣α ∈ λ1m} (2.4) and λ1m denotes the dual of tm. for any α ∈ λ1m, we define jα(x) = −α(jx) for x ∈ tm. let (m2n,j,g) be an almost hermitian manifold. an affine connection d on tm is called almost hermitian connection if dg = dj = 0. for the almost hermitian connection, we have the following lemma (cf. [3, 9, 11]). cubo 23, 2 (2021) on the conformally k-th gauduchon condition and the conformally ... 337 lemma 2.1. let (m,j,g) be an almost hermitian manifold with dimr m = 2n. then for any given vector valued (1,1)-form θ = (θi)1≤i≤n, there exists a unique almost hermitian connection d on (m,j,g) such that the (1,1)-part of the torsion is equal to the given θ. if the (1,1)-part of the torsion of an almost hermitian connection vanishes everywhere, then the connction is called the second canonical connection or the chern connection. we will refer the connection as the chern connection and denote it by ∇. note that for any p-form ψ, there holds that dψ(x1, . . . ,xp+1) = p+1∑ i=1 (−1)i+1xi(ψ(x1, . . . ,x̂i, . . . ,xp+1)) + ∑ i 0. then, ū(x∗,γ) stands for the closure of u(x∗,γ). we base the local convergence on this notation and the conditions (c). (c1) f : d −→ b2 is differentiable according to fréchet, and x∗ ∈ d with f(x∗) = 0 is a simple solution. (c2) there exists an increasing and continuous real function ω0 on i satisfying ω0(0) = 0 and such that for all x ∈ d ‖f ′(x∗)−1(f ′(x) −f ′(x∗))‖≤ ω0(‖x−x∗‖). set u0 = d ∩u(x∗,ρ0). (c3) there exists a function ω on i0 continuous and increasing satisfying ω(0) = 0 such that for all x,y ∈ u0 ‖f ′(x∗)−1(f ′(y) −f ′(x))‖≤ ω(‖y −x‖). set u1 = d ∩u(x∗,ρ4). (c4) there exists a function v on i2 continuous and increasing, such that for all x ∈ u1 ‖f ′(x∗)−1f ′(x)‖≤ v(‖x−x∗‖). (c5) ū(x∗,r) ⊆ d. (c6) there exists r1 ≥ r such that ∫ 1 0 ω0(θr1)dθ < 1. set u2 = d ∩ ū(x∗,r1). cubo 23, 1 (2021) extended domain for fifth convergence order schemes 101 theorem 2.1. assume hypotheses (c) hold and starting point x0 ∈ u(x∗,r) −{x∗}. then the following assertions are valid, sequence {xn} belongs in u(x∗,r) −{x∗} and converges to x∗ ∈ u(x∗,r) so that this limit point uniquely solves equation f(x) = 0 in the set u2. proof. let z ∈ u(x∗,r) −{x∗} and utilize (c2), (2.4) and (2.5) to obtain ‖f ′(x∗)−1(f ′(z) −f ′(x∗))‖≤ ω0(‖z −x∗‖) ≤ ω0(r) < 1, which together with a result by banach [12] for linear operators whose inverse exists imply ‖f ′(z)−1f ′(x∗)‖≤ 1 1 −ω0(‖z −x∗‖) . (2.10) in particular, by scheme (1.2) y0,z0 are well defined since if we set z = x0 ∈ u(x∗,r) −{x∗}, and f ′(x0) is invertible. then, by (2.4), (2.8) (for k = 1), (c1), (c3) and (2.10) (for z = x0), we have ‖y0 −x∗‖ = ‖x0 −x∗ −f ′(x0)−1f(x0)‖ ≤ ‖f ′(x0)−1f ′(x∗)‖ [∫ 1 0 ‖f ′(x∗)−1[f ′(x0 + θ(x0 −x∗)) −f ′(x0)](x0 −x∗)dθ‖ ] ≤ ∫ 1 0 ω((1 −θ)‖x0 −x∗‖)dθ 1 −ω0(‖x0 −x∗‖) ‖x0 −x∗‖ ≤ λ1(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖ < r. (2.11) hence, y0 ∈ u(x∗,r). using the second substep of method (1.2) and replacing x0,y0, by y0,z0, respectively as in (2.10) and (2.11), we get ‖z0 −x∗‖ ≤ ∫ 1 0 ω((1 −θ)‖y0 −x∗‖)dθ 1 −ω0(‖y0 −x∗‖) ‖y0 −x∗‖ ≤ ∫ 1 0 ω((1 −θ)λ1(‖x0 −x∗‖)‖x0 −x∗‖)dθλ1(‖x0 −x∗‖)‖x0 −x∗‖ 1 −ω0(λ1(‖x0 −x∗‖)‖x0 −x∗‖ ≤ λ2(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖. (2.12) that is z0 ∈ u(x∗,r) and also x1 exists (for y0 = z, in (2.10)). notice that (c1), (c4), (2.12) and f(z0) = f(z0) −f(x∗) = ∫ 1 0 f ′(x∗ + θ(z0 −x∗))dθ(z0 −x∗), we obtain that ‖f ′(x∗)−1f ′(z0)‖ ≤ ∫ 1 0 v(θ‖z0 −x∗‖)dθ‖z0 −x∗‖ ≤ ∫ 1 0 v(θλ2(‖x0 −x∗‖)‖x0 −x∗‖dθλ2(‖x0 −x∗‖)‖x0 −x∗‖. (2.13) moreover, by the last substep of method (1.2), (2.4), (2.5), (2.8) (for k = 3), (2.10), (2.13) (for 102 i. k. argyros & s. george cubo 23, 1 (2021) z = x0,y0), (2.11) and (2.12), we have in turn that ‖x1 −x∗‖ ≤ ‖z0 −x∗ −f ′(z0)−1f(z0)‖ (2.14) +‖f ′(z0)−1[(f ′(y0) −f ′(x∗)) + (f ′(x∗) −f ′(z0))]f ′(y0)−1f(z0)‖ ≤ [∫ 1 0 ω((1 −θ)‖z0 −x∗‖)dθ 1 −ω0(‖z0 −x∗‖) + (ω0(‖z0 −x∗‖) + ω0(‖y0 −x∗‖)) ∫ 1 0 v(θ‖z0 −x∗‖)dθ (1 −ω0(‖z0 −x∗‖))(1 −ω0(‖y0 −x∗‖)) ] ‖z0 −x∗‖ ≤ λ3(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖, (2.15) so x1 ∈ u(x∗,r). replacing x0,y0,z0,x1 by xk,yk,zk,xk+1, in the previous computations we obtain ‖yk −x∗‖≤ λ1(‖xk −x∗‖)‖xk −x∗‖≤‖xk −x∗‖ < r, (2.16) ‖zk −x∗‖≤ λ2(‖xk −x∗‖)‖xk −x∗‖≤‖xk −x∗‖ (2.17) and ‖xk+1 −x∗‖≤ λ3(‖xk −x∗‖)‖xk −x∗‖≤‖xk −x∗‖, (2.18) so yk,zk,xk+1 stay in u(x∗,r) and limk−→∞xk = x∗. furthermore, let x 1 ∗ ∈ u2 with f(x1∗) = 0. in view of (c2) and (c6) we obtain∣∣∣∣ ∣∣∣∣f ′(x∗)−1 (∫ 1 0 f ′(x∗ + θ(x 1 ∗ −x∗))dθ −f ′(x∗) )∣∣∣∣ ∣∣∣∣ ≤ ∫ 1 0 ω0(θ‖x1∗ −x∗‖)dθ ≤ ∫ 1 0 ω0(θr1)dθ < 1, so x1∗ = x∗, since t = ∫ 1 0 f ′(x∗ + θ(x 1 ∗ −x∗))dθ is invertible and 0 = f(x1∗) −f(x∗) = t(x 1 ∗ −x∗). remark 2.2. 1. in view of (2.10) and the estimate ‖f ′(x∗)−1f ′(x)‖ = ‖f ′(x∗)−1(f ′(x) −f ′(x∗)) + i‖ ≤ 1 + ‖f ′(x∗)−1(f ′(x) −f ′(x∗))‖≤ 1 + l0‖x−x∗‖ condition (2.13) can be dropped and m can be replaced by m(t) = 1 + l0t or m(t) = m = 2, since t ∈ [0, 1 l0 ). cubo 23, 1 (2021) extended domain for fifth convergence order schemes 103 2. the results obtained here can be used for operators f satisfying autonomous differential equations [2] of the form f ′(x) = p(f(x)) where p is a continuous operator. then, since f ′(x∗) = p(f(x∗)) = p(0), we can apply the results without actually knowing x∗. for example, let f(x) = ex − 1. then, we can choose: p(x) = x + 1. 3. let ω0(t) = l0t, and ω(t) = lt. in [2, 3] we showed that ra = 2 2l0+l is the convergence radius of newton’s method: xn+1 = xn −f ′(xn)−1f(xn) for each n = 0, 1, 2, · · · (2.19) under the conditions (2.11) and (2.12). it follows from the definition of r in (2.4) that the convergence radius r of the method (1.2) cannot be larger than the convergence radius ra of the second order newton’s method (2.19). as already noted in [2, 3] ra is at least as large as the convergence radius given by rheinboldt [12] rr = 2 3l , (2.20) where l1 is the lipschitz constant on d. the same value for rr was given by traub [15]. in particular, for l0 < l1 we have that rr < ra and rr ra → 1 3 as l0 l1 → 0. that is the radius of convergence ra is at most three times larger than rheinboldt’s. 4. it is worth noticing that method (1.2) is not changing when we use the conditions of theorem 2.1 instead of the stronger conditions used in [13, 14]. moreover, we can compute the computational order of convergence (coc) defined by ξ = ln ( ‖xn+1 −x∗‖ ‖xn −x∗‖ ) / ln ( ‖xn −x∗‖ ‖xn−1 −x∗‖ ) or the approximate computational order of convergence ξ1 = ln ( ‖xn+1 −xn‖ ‖xn −xn−1‖ ) / ln ( ‖xn −xn−1‖ ‖xn−1 −xn−2‖ ) . this way we obtain in practice the order of convergence in a way that avoids the bounds involving estimates using estimates higher than the first fréchet derivative of operator f. note also that the computation of ξ1 does not require the usage of the solution x ∗. 104 i. k. argyros & s. george cubo 23, 1 (2021) 3 semi-local convergence analysis let γ0 = f ′(x0) −1 ∈ l(b2,b1) exists at x0 ∈ d, where l(b2,b1) denotes the set of bounded linear operators from b2,b1 and the following conditions hold. (1) ‖γ0‖≤ β0. (2) ‖γ0f(x0)‖≤ η0. (3)’ ‖f ′(x) −f ′(x0)‖≤ m0‖x−x0‖ for all x ∈ d. set d0 = d ∩u ( x0, 1 β0m0 ) . (3) ‖f ′′(x)‖≤ m for all x ∈ d0. (4) ‖f ′′(x) − f ′′(y)‖ ≤ ω(‖x − y‖) for all x,y ∈ d0 for a continuous nondecreasing function ω, ω(0) ≥ 0 such that ω(tx) ≤ tpω(x) for t ∈ [0, 1],x ∈ (0,∞) and p ∈ [0, 1]. then, as in [13, 14], let r0 = mβ0η0, s0 = β0η0ω(η0) and define sequences {rk},{sk} and {ηk} for k = 0, 1, 2, . . . , by rk+1 = rkϕ(rk) 2ψ(rk,sk), (3.1) sk+1 = skϕ(rk) 2+pψ(rk,sk) 1+p, (3.2) ηk+1 = ηkϕ(rk)ψ(rk,sk), (3.3) where ϕ(t) = 1 1 − tg(t) (3.4) g(t) = ( 1 + t 2 + t2 2(1 − t) ( 1 + t 4 )) (3.5) and ψ(t,s) = t2 2(1 − t) (1 + t 4 ) [ s 1 + p ( t1+p 21+p + 1 2 + p ( t2 2(1 − t) ( 1 + t 4 ))1+p) + t 2 ( t + t2 2(1 − t) ( 1 + t 4 ))] . (3.6) remark 3.1. in [14] the following conditions were used instead of (3), (4), respectively (3)’ ‖f ′′(x)‖≤ m1 for all x ∈ d (4)’ ‖f ′′(x) −f ′′(y)‖≤ ω1(‖x−y‖) for all x,y ∈ d and ω1 as ω. but, we have d0 ⊆ d, so m0 ≤ m1 cubo 23, 1 (2021) extended domain for fifth convergence order schemes 105 m ≤ m1 and ω(θ) ≤ ω1(θ). examples where the preceding items are strict can be found in [1, 2, 3, 4, 5, 6]. notice that (3)’ is used to determine d0 leading to m = m(d0,x)). hence, the results in [13, 14] can be rewritten with m replacing m1. so, if m < m1 the new semi-local convergence analysis is finer. this is also done under the same computational effort because in practice finding ω1,m1 requires finding ω,m0,m as special cases. this technique can be used to extend the applicability of other schemes involving inverses in an analogous fashion. hence, the proof of the following semi-local convergence result for scheme (1.2) is omitted. theorem 3.2. let r0 = mβ0η0 < ν,s0 = β0η0ω(η0) and assumptions (1)-(4) hold. then, for ū(x0,rη0) ⊆ d, where r = g(r0) 1−δγ , the sequence {xk} generated by (1.2) converges to the solution x∗ of f(x) = 0. moreover, yk,zk,xk+1,x∗ ∈ ū(x0,rη0) and x∗ is the unique solution in u ( x0, 2 m0β0 −rη0 ) ∩d. furthermore, we have ‖xk −x∗‖≤ g(r0)δk γ (4+q)k−1 3+q 1 − δγ(4+q)k η0. 4 numerical examples example 4.1. let us consider a system of differential equations governing the motion of an object and given by f ′1(x) = e x, f ′2(y) = (e− 1)y + 1, f ′ 3(z) = 1 with initial conditions f1(0) = f2(0) = f3(0) = 0. let f = (f1,f2,f3). let b1 = b2 = r3,d = ū(0, 1),p = (0, 0, 0)t . define function f on d for w = (x,y,z)t by f(w) = ( ex − 1, e− 1 2 y2 + y,z )t . the fréchet-derivative is defined by f ′(v) =   ex 0 0 0 (e− 1)y + 1 0 0 0 1   . notice that using the (a) conditions, we get for α = 1, w0(t) = (e−1)t,w(t) = e 1 e−1 t,v(t) = e 1 e−1 . the radii are r1 = 0.38269191223238574472986783803208,r2 = 0.33841523581069998805048726353562, r3 = 0.32249343047238987480795913143083 and r = r3. 106 i. k. argyros & s. george cubo 23, 1 (2021) example 4.2. let b1 = b2 = c[0, 1], the space of continuous functions defined on [0, 1] be equipped with the max norm. let d = u(0, 1). define function f on d by f(ϕ)(x) = ϕ(x) − 5 ∫ 1 0 xθϕ(θ)3dθ. (4.1) we have that f ′(ϕ(ξ))(x) = ξ(x) − 15 ∫ 1 0 xθϕ(θ)2ξ(θ)dθ, for each ξ ∈ d. then, we get that x∗ = 0, so w0(t) = 7.5t,w(t) = 15t and v(t) = 2. then the radii are r1 = 0.066666666666666666666666666666667,r2 = 0.059338915721683857529278327547217, r3 = 0.047722035514509826559237382070933 and r = r3. example 4.3. returning back to the motivational example at the introduction of this study, we have w0(t) = w(t) = 96.6629073t and v1(t) = 2. the parameters for method (1.2) are r1 = 0.0068968199414654552878434223828208,r2 = 0.0061008926455964288676492301988219, r3 = 0.004463243021326804456372361329386 and r = r3. 5 conclusion in general, the convergence domain of iterative schemes is small limiting their applications. hence, any attempt to increase it is very important. this is achieved here by finding smaller ω− functions than before which are also specialization of the previous ones. hence, the extensions are obtained under the same computational cost. our idea can be used to extend the usage of other schemes in a similar way. numerical experiments further demonstrate the superiority of our findings. cubo 23, 1 (2021) extended domain for fifth convergence order schemes 107 references [1] i. k. argyros, “a new convergence theorem for the jarratt method in banach space”, comput. math. appl., vol. 36, pp. 13–18, 1998. 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[13] s. singh, d. k. gupta, e. mart́ınez, and j. l. hueso, “semilocal convergence analysis of an iteration of order five using recurrence relations in banach spaces”, mediterr. j. math., vol. 13, pp. 4219–4235, 2016. 108 i. k. argyros & s. george cubo 23, 1 (2021) [14] s. singh, e. mart́ınez, a. kumar, and d. k. gupta, “domain of existence and uniqueness for nonlinear hammerstein integral equations”, mathematics, vol. 8, no. 3, 2020. [15] j. f. traub, iterative methods for the solution of equations, ams chelsea publishing, 1982. [16] x. wang, j. kou, and c. gu, “semi-local convergence of a class of modified super halley method in banach space”, j. optim. theory. appl., vol. 153, pp. 779–793, 2012. [17] q. wu, and y. zhao, “newton-kantorovich type convergence theorem for a family of new deformed chebyshev method”, appl. math. comput., vol. 192, pp. 405–412, 2008. [18] y. zhao, and q. wu, “newton-kantorovich theorem for a family of modified halley’s method under hölder continuity conditions in banach space”, appl. math. comput., vol. 202, pp. 243–251, 2008. [19] a. emad, m. o. al-amr, a. yıldırım, w. a. alzoubi, “revised reduced differential transform method using adomian’s polynomials with convergence analysis”, mathematics in engineering, science & aerospace (mesa), vol. 11, no. 4, pp. 827–840, 2020. introduction local convergence semi-local convergence analysis numerical examples conclusion cubo, a mathematical journal vol.22, no¯ 01, (23–37). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100023 η-ricci solitons on 3-dimensional trans-sasakian manifolds sampa pahan department of mathematics, mrinalini datta mahavidyapith kolkata-700051, india. sampapahan25@gmail.com abstract in this paper, we study η-ricci solitons on 3-dimensional trans-sasakian manifolds. firstly we give conditions for the existence of these geometric structures and then observe that they provide examples of η-einstein manifolds. in the case of φ-ricci symmetric trans-sasakian manifolds, the η-ricci soliton condition turns them to einstein manifolds. afterward, we study the implications in this geometric context of the important tensorial conditions r · s = 0, s · r = 0, w2 · s = 0 and s · w2 = 0. resumen en este art́ıculo estudiamos solitones η-ricci en variedades trans-sasakianas tridimensionales. en primer lugar damos condiciones para la existencia de estas estructuras geométricas y luego observamos que ellas dan ejemplos de variedades η-einstein. en el caso de variedades trans-sasakianas φ-ricci simétricas, la condición de solitón η-ricci las convierte en variedades einstein. a continuación estudiamos las implicancias en este contexto geométrico de las importantes condiciones tensoriales r · s = 0, s · r = 0, w2 · s = 0 y s · w2 = 0. keywords and phrases: trans-sasakian manifold, η-ricci solitons. 2010 ams mathematics subject classification: 53c21, 53c25, 53c44. http://doi.org/10.4067/s0719-06462020000100023 24 sampa pahan cubo 22, 1 (2020) 1 introduction in 1982, the notion of the ricci flow was introduced by hamilton [10] to find a canonical metric on a smooth manifold.the ricci flow is an evolution equation for riemannian metric g(t) on a smooth manifold m given by ∂ ∂t g(t) = −2s. a solution to this equation (or a ricci flow) is a one-parameter family of metrics g(t), parameterized by t in a non-degenerate interval i, on a smooth manifold m satisfying the ricci flow equation. if i has an initial point t0, then (m, g(t0)) is called the initial condition of or the initial metric for the ricci flow (or of the solution) [14]. ricci solitons and η-ricci solitons are natural generalizations of einstein metrics. a ricci soliton on a riemannian manifold (m, g) is defined by s + 1 2 lxg = λg where lxg is the lie derivative along the vector field x, s is the ricci tensor of the metric and λ is a real constant. if x = ∇f for some function f on m, the ricci soliton becomes gradient ricci soliton. ricci solitons appear as self-similar solutions to hamiltons’s ricci flow and often arise as limits of dilations of singularities in the ricci flow [11]. a soliton is called shrinking, steady and expanding according as λ > 0, λ = 0 and λ < 0 respectively. in 2009, the notion of η-ricci soliton was introduced by j.c. cho and m. kimura [6]. j.c. cho and m. kimura proved that a real hypersurface admitting an η-ricci soliton in a non-flat complex space form is a hopf-hypersurface [6]. an η-ricci soliton on a riemannian manifold (m, g) is defined by the following equation 2s + lξg + 2λg + 2µη ⊗ η = 0, (1.1) where lξ is the lie derivative operator along the vector field ξ, s is the ricci tensor of the metric and λ, µ are real constants. if µ = 0, then η-ricci soliton becomes ricci soliton. in the last few years, many authors have worked on ricci solitons and their generalizations in different contact metric manfolds in [1], [7], [8], [9], [12] etc. in 2014, b. y. chen and s. deshmukh have established the characterizations of compact shrinking trivial ricci solitons in [5]. also, in [2], a. bhattacharyya, t. dutta, and s. pahan studied the torqued vector field and established some applications of torqued vector field on ricci soliton and conformal ricci soliton. a.m. blaga [3], d. g. prakasha and b. s. hadimani [17] observed η-ricci solitons on different contact metric manifolds satisfying some certain curvature conditions. cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 25 in this paper we study the existence of η-ricci soliton on 3-dimensional trans-sasakian manifold. next we show that η-ricci soliton on 3-dimensional trans-sasakian manifolds becomes η-einstein manifold under some conditions. next we prove that φ-ricci symmetric trans-sasakian manifold (m, g) manifold satisfying an η-ricci soliton becomes an einstein manifold. next we give an example of an η-ricci soliton on 3-dimensional trans-sasaian manifold with λ = −2 and µ = 6. later we obtain some different types of curvature tensors and their properties under certain conditions. 2 preliminaries the product m̄ = m×r has a natural almost complex structure j with the product metric g being hermitian metric. the geometry of the almost hermitian manifold (m̄, j, g) gives the geometry of the almost contact metric manifold (m, φ, ξ, η, g). sixteen different types of structures on m like sasakian manifold, kenmotsu manifold etc are given by the almost hermitian manifold (m̄, j, g) . the notion of trans-sasakian manifolds was introduced by oubina [15] in 1985. then j. c. marrero [13] have studied the local structure of trans-sasakian manifolds. in general a trans-sasakian manifold (m, φ, ξ, η, g, α, β) is called a trans-sasakian manifold of type (α, β). an n (= 2m + 1) dimensional riemannian manifold (m, g) is called an almost contact manifold if there exists a (1,1) tensor field φ, a vector field ξ and a 1-form η on m such that φ2(x) = −x + η(x)ξ, (2.1) η(ξ) = 1, η(φx) = 0, (2.2) φξ = 0, (2.3) η(x) = g(x, ξ), (2.4) g(φx, φy) = g(x, y) − η(x)η(y), (2.5) g(x, φy) + g(y, φx) = 0, (2.6) for any vector fields x, y on m. a 3-dimensional almost contact metric manifold m is called a trans-sasakian manifold if it satisfies the following condition (∇xφ)(y) = α{g(x, y)ξ − η(y)x} + β{g(φx, y)ξ − η(y)φx}, (2.7) for some smooth functions α, β on m and we say that the trans-sasakian structure is of type (α, β). for 3-dimensional trans-sasakian manifold, from (2.7) we have, ∇xξ = −αφx + β(x − η(x)ξ), (2.8) 26 sampa pahan cubo 22, 1 (2020) (∇xη)(y) = −αg(φx, y) + βg(φx, φy). (2.9) in a 3-dimensional trans-sasakian manifold, we have r(x, y)z = [ r 2 − 2(α2 − β2 − ξβ)][g(y, z)x − g(x, z)y] − [ r 2 − 3(α2 − β2) + ξβ][g(y, z)η(x) − g(x, z)η(y)]ξ + [g(y, z)η(x) − g(x, z)η(y)][φ grad α − grad β] − [ r 2 − 3(α2 − β2) + ξβ]η(z)[η(y)x − η(x)y] − [zβ + (φz)α]η(z)[η(y)x − η(x)y] − [xβ + (φx)α][g(y, z)ξ − η(z)y] − [yβ + (φy)α][g(x, z)ξ − η(z)x], s(x, y) = [ r 2 − (α2 − β2 − ξβ)]g(x, y) − [ r 2 − 3(α2 − β2) + ξβ]η(x)η(y) − [yβ + (φy)α]η(x) − [xβ + (φx)α]η(y). when α and β are constants the above equations reduce to, r(ξ, x)ξ = (α2 − β2)(η(x)ξ − x), (2.10) s(x, ξ) = 2(α2 − β2)η(x), (2.11) r(ξ, x)y = (α2 − β2)(g(x, y)ξ − η(y)x). (2.12) r(x, y)ξ = (α2 − β2)(η(y)x − η(x)y). (2.13) definition 2.1. a trans-sasakian manifold m3 is said to be η-einstein manifold if its ricci tensor s is of the form s(x, y) = ag(x, y) + bη(x)η(y), where a, b are smooth functions. cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 27 3 η-ricci solitons on trans-sasakian manifolds to study the existence conditions of η-ricci solitons on 3-dimensional trans-sasakian manifolds, we prove the following theorem. theorem 3.1: let (m, g, φ, η, ξ, α, β) be a 3-dimensional trans-sasakian manifold with α, β constants (β 6= 0). if the symmetric (0, 2) tensor field h satisfying the condition βh(x, y) − α 2 [h(φx, y) + h(x, φy)] = lξg(x, y) + 2s(x, y) + 2µη(x)η(y) is parallel with respect to the levicivita connection associated to g. then (g, ξ, µ) becomes an η-ricci soliton. proof: we consider a symmetric (0, 2)-tensor field h which is parallel with respect to the levicivita connection (∇h = 0). then it follows that h(r(x, y)z, w) + h(r(x, y)z, w) = 0, (3.1) for an arbitary vector field w, x, y, z on m. put x = z = w = ξ we get h(r(x, y)ξ, ξ) = 0, (3.2) for any x, y ∈ χ(m) by using the equation (2.13) h(y, ξ) = g(y, ξ)h(ξ, ξ), (3.3) for any y ∈ χ(m). differentiating the equation (3.3) covariantly with respect to the vector field x ∈ χ(m) we have h(∇xy, ξ) + h(y, ∇xξ) = g(∇xy, ξ)h(ξ, ξ) + g(y, ∇xξ)h(ξ, ξ), (3.4) using the equation (2.8) we have βh(x, y) − αh(φx, y) = −αg(φx, y)h(ξ, ξ) + βh(ξ, ξ)g(x, y). (3.5) interchanging x by y we have βh(x, y) − αh(x, φy) = −αg(x, φy)h(ξ, ξ) + βh(ξ, ξ)g(x, y). (3.6) then adding the above two equations we get βh(x, y) − α 2 [h(φx, y) + h(x, φy)] = βh(ξ, ξ)g(x, y). (3.7) we see that βh(x, y)− α 2 [h(φx, y)+h(x, φy)] is a symmetric tensor of type (0, 2). since lξg(x, y), s(x, y), η(x) = g(x, ξ) and η(y) = g(y, ξ) are symmetric tensors of type (0, 2) and λ, µ are real constants, the sum lξg(x, y) + 2s(x, y) + 2µη(x)η(y) is a symmetric tensor of type (0, 2). 28 sampa pahan cubo 22, 1 (2020) therefore, we can take the sum as an another symmetric tensor field of type (0, 2). hence for we can assume that βh(x, y) − α 2 [h(φx, y) + h(x, φy)] = lξg(x, y) + 2s(x, y) + 2µη(x)η(y). then we compute βh(ξ, ξ)g(x, y) = lξg(x, y) + 2λg(x, y) + 2µη(x)η(y). as h is parallel so, h(ξ, ξ) is constant. hence, we can write h(ξ, ξ) = − 2 β λ where β is constant and β 6= 0. so, from the equation (3.7) we have βh(x, y) − α 2 [h(φx, y) + h(x, φy)] = −2λg(x, y), (3.8) for any x, y ∈ χ(m). therefore lξg(x, y) + 2s(x, y) + 2µη(x)η(y) = −2λg(x, y) and so (g, ξ, µ) becomes an η-ricci soliton. corollary 3.2: let (m, g, φ, η, ξ, α, β) be a 3-dimensional trans-sasakian manifold with α, β constants (β 6= 0). if the symmetric (0, 2) tensor field h admitting the condition βh(x, y) − α 2 [h(φx, y) + h(x, φy)] = lξg(x, y) + 2s(x, y) is parallel with respect to the levi-civita connection associated to g with λ = 2n. then (g, ξ) becomes a ricci soliton. next theorem shows the necessary condition for the existence of η-ricci soliton on 3-dimensional trans-sasakian manifolds. theorem 3.3: if 3-dimensional trans-sasakian manifold satisfies an η-ricci soliton then the manifold becomes η-einstein manifold with α and β constants. proof: from the equation (1.1) we get 2s(x, y) = −g(∇xξ, y) − g(x, ∇yξ) − 2λg(x, y) − 2µη(x)η(y). (3.9) by using the equation (2.8) we get s(x, y) = −(β + λ)g(x, y) + (β − µ)η(x)η(y) (3.10) and s(x, ξ) = −(λ + µ)η(x). (3.11) also from (2.11) we have λ + µ = 2(β2 − α2). (3.12) the ricci operator q is defined by g(qx, y) = s(x, y). then we get qx = (µ − β + 2(α2 − β2))x + (β − µ)η(x)ξ. (3.13) cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 29 then we can easily see that the manifold is an η-einstein manifold. we know a manifold is φ-ricci symmetric if φ2 ◦ ∇q = 0. now we prove the next theorem. theorem 3.4: if a φ-ricci symmetric trans-sasakian manifold (m, g) satisfies an η-ricci soliton then µ = β, λ = 2(β2 − α2) − β and (m, g) is an einstein manifold. proof: from the equation (3.13) we have (∇xq)y = ∇xqy − q(∇xy) = −α(β − µ)η(y)φx + β(β − µ)η(y)x − (β − µ)η(y)η(x)ξ +(β − µ)[−αg(φx, y) + βg(φx, φy)]ξ. now applying φ2 both sides we have µ = β, λ = 2(β2 −α2)−β and (m, g) is an einstein manifold. we construct an example of η-ricci soliton on 3-dimensional trans-sasakian manifolds in the the next section. 4 example of η-ricci solitons on 3-dimensional trans-sasakian manifolds we consider the three dimensional manifold m = {(x, y, z) ∈ r3 : y 6= 0} where (x, y, z) are the standard coordinates in r3. the vector fields e1 = e 2z ∂ ∂x , e2 = e 2z ∂ ∂y , e3 = ∂ ∂z are linearly independent at each point of m. let g be the riemannian metric defined by gij = { 1 for i = j, 0 for i 6= j. let η be the 1-form defined by η(z) = g(z, e3) for any z ∈ χ(m 3). let φ be the (1, 1) tensor field defined by φ(e1) = e2, φ(e2) = −e1, φ(e3) = 0. then using the linearity property of φ and g we have η(e2) = 1, φ 2(z) = −z + η(z)e2, g(φz, φw) = g(z, w) − η(z)η(w), for any z, w ∈ χ(m3). thus for e2 = ξ, (φ, ξ, η, g) defines an almost contact metric structure on m. now, after some calculation we have, 30 sampa pahan cubo 22, 1 (2020) [e1, e3] = −2e1, [e2, e3] = −2e2, [e1, e2] = 0. the riemannian connection ∇ of the metric is given by the koszul’s formula which is 2g(∇xy, z) = xg(y, z) + yg(z, x) − zg(x, y) − g(x, [y, z]) − g(y, [x, z]) + g(z, [x, y]). by koszul’s formula we get, ∇e1e1 = 2e3, ∇e2e1 = 0, ∇e3 e1 = 0, ∇e1e2 = 0, ∇e2e2 = 2e3, ∇e3e2 = 0, ∇e1e3 = −2e1, ∇e2e3 = −2e2, ∇e3e3 = 0. from the above it can be easily shown that m3(φ, ξ, η, g) is a trans-sasakian manifold of type (0, −2). here r(e1, e2)e2 = −4e1, r(e3, e2)e2 = 4e2, r(e1, e3)e3 = −4e1, r(e2, e3)e3 = −4e2, r(e3, e1)e1 = −4e2, r(e2, e1)e1 = 4e3. so, we have s(e1, e1) = 0, s(e2, e2) = 0, , s(e3, e3) = −8. (4.1) from the equation (1.1) we get λ = −2 and µ = 6. therefore, (g, ξ, λ, µ) is an η-ricci soliton on m3(φ, ξ, η, g). in the next sections we consider η-ricci solitons on 3-dimensional trans-sasakian manifolds satisfying some curvature conditions. 5 η-ricci solitons on 3-dimensional trans-sasakian manifolds satisfying r(ξ, x) · s = 0 first we suppose that 3-dimensional trans-sasakian manifolds with η-ricci solitons satisfy the condition r(ξ, x) · s = 0. then we have s(r(ξ, x)y, z) + s(y, r(ξ, x)z) = 0 cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 31 for any x, y, z ∈ χ(m). using the equations (2.12), (3.10), (3.11) we get (β − µ)g(x, y)η(z) + (β − µ)g(x, z)η(y) − 2(β − µ)η(x)η(y)η(z) = 0. put z = ξ we have (β − µ)g(x, y) − (β − µ)η(x)η(y) = 0. setting x = φx and y = φy in the above equation we get (β − µ)g(φx, φy) = 0. again using the equation (3.12) we have µ = β, λ = 2(β2 − α2) − β. also we can easily see that m is an einstein manifold. so we have the following theorem. theorem 5.1: if a 3-dimensional trans-sasakian manifold (m, g, φ, η, ξ, α, β) with α, β constants admitting an η-ricci soliton satisfies the condition r(ξ, x) · s = 0 then µ = β, λ = 2(β2 − α2) − β and m is an einstein manifold. corollary 5.2: a 3-dimensional trans-sasakian manifold with α, β constants satisfies the condition r(ξ, x) · s = 0, there is no ricci soliton with the potential vector field ξ. 6 η-ricci solitons on 3-dimensional trans-sasakian manifolds satisfying s(ξ, x) · r = 0 we consider 3-dimensional trans-sasakian manifolds with η-ricci solitons satisfying the condition s(ξ, x) · r = 0. 32 sampa pahan cubo 22, 1 (2020) so we have s(x, r(y, z)w)ξ − s(ξ, r(y, z)w)x + s(x, y)r(ξ, z)w − s(ξ, y)r(x, z)w +s(x, z)r(y, ξ)w − s(ξ, z)r(y, x)w + s(x, w)r(y, z)ξ − s(ξ, w)r(y, z)x = 0. taking inner product with ξ then the above equation becomes s(x, r(y, z)w) − s(ξ, r(y, z)w)η(x) + s(x, y)η(r(ξ, z)w) −s(ξ, y)η(r(x, z)w) + s(x, z)η(r(y, ξ)w) − s(ξ, z)η(r(y, x)w) + s(x, w)η(r(y, z)ξ) − s(ξ, w)η(r(y, z)x) = 0. (6.1) put w = ξ and using the equations (2.10), (2.12), (3.10), (3.11) we get − (β + λ)g(x, r(y, z)ξ) + (λ + µ)η(r(y, z)x) = 0. (6.2) also we have η(r(y, z)x) = −g(x, r(y, z)ξ). so from the equation (6.2) we get (β + 2λ + µ)g(x, r(y, z)ξ) = 0. again using the equation (3.12) we have µ = β + 4(β2 − α2), λ = −[2(β2 − α2) + β]. so we have the following theorem. theorem 6.1: if a 3-dimensional trans-sasakian manifold (m, g, φ, η, ξ, α, β) with α, β constants admitting an η-ricci soliton satisfies the condition s(ξ, x) · r = 0 then µ = β + 4(β2 − α2), λ = −[2(β2 − α2) + β]. cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 33 corollary 6.2: a 3-dimensional trans-sasakian manifold with α, β constants satisfies the condition s(ξ, x) · r = 0, there is no ricci soliton with the potential vector field ξ. 7 η-ricci solitons on 3-dimensional trans-sasakian manifolds satisfying w2(ξ, x) · s = 0 definition 7.1. let m be 3-dimensional trans-sasakian manifold with respect to semi-symmetric metric connection. the w2-curvature tensor of m is defined by [16] w2(x, y)z = r(x, y)z + 1 2 (g(x, z)qy − g(y, z)qx). (7.1) we assume 3-dimensional trans-sasakian manifolds with η-ricci solitons satisfying the condition w2(ξ, x) · s = 0. then we have s(w2(ξ, x)y, z) + s(y, w2(ξ, x)z) = 0 for any x, y, z ∈ χ(m). using the equations (2.12), (3.10), (3.11), (7.1) we get [− (β + λ) 2 (λ + µ) + (β + λ)2 2 + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 ]g(x, y)η(z) +[ (β + λ)2 2 − (β + λ) 2 (λ + µ) + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 ]g(x, z)η(y) +[−(β + λ)(β − µ) − 2(β − µ)(α2 − β2) − (β − µ)(λ + µ)]η(x)η(y)η(z) = 0. put z = ξ in the above equation we get [− (β + λ) 2 (λ + µ) + (β + λ)2 2 + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 ]g(x, y) +[ (β + λ)2 2 − (β + λ) 2 (λ + µ) + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 34 sampa pahan cubo 22, 1 (2020) −(β + λ)(β − µ) − 2(β − µ)(α2 − β2) − (β − µ)(λ + µ)]η(x)η(y) = 0. setting x = φx and y = φy in the above equation we get (β − µ)( (β + 2λ + µ + 2(α2 − β2)) 2 )g(φx, φy) = 0. again using the equation (3.12) we have µ = β, λ = 2(β2 − α2) − β or µ = 2(β2 − α2) + β, λ = −β. so we have the following theorem. theorem 7.1: if a 3-dimensional trans-sasakian manifold (m, g, φ, η, ξ, α, β) with α, β constants admitting an η-ricci soliton satisfies the condition w2(ξ, x)·s = 0 then µ = β, λ = 2(β 2 −α2)−β or µ = 2(β2 − α2) + β, λ = −β. corollary 7.2: a 3-dimensional trans-sasakian manifold with α, β constants satisfies the condition w2(ξ, x) · s = 0, there is no ricci soliton with the potential vector field ξ. 8 η-ricci solitons on 3-dimensional trans-sasakian manifolds satisfying s(ξ, x) · w2 = 0 suppose that 3-dimensional trans-sasakian manifolds with η-ricci solitons satisfy the condition s(ξ, x) · w2 = 0. so we have s(x, w2(y, z)v)ξ − s(ξ, w2(y, z)v)x + s(x, y)w2(ξ, z)v − s(ξ, y)w2(x, z)v +s(x, z)w2(y, ξ)v − s(ξ, z)w2(y, x)v + s(x, v)w2(y, z)ξ − s(ξ, v)w2(y, z)x = 0. cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 35 taking inner product with ξ then the above equation becomes s(x, w2(y, z)v) − s(ξ, w2(y, z)v)η(x) + s(x, y)η(w2(ξ, z)v) −s(ξ, y)η(w2(x, z)v) + s(x, z)η(w2(y, ξ)v) − s(ξ, z)η(w2(y, x)v) + s(x, v)η(w2(y, z)ξ) − s(ξ, v)η(w2(y, z)x) = 0. (8.1) put v = ξ and using the equations (2.10), (2.12), (3.10), (3.11), (7.1) we get − (β + λ)g(x, w2(y, z)ξ) + (λ + µ)η(w2(y, z)x) = 0. (8.2) using the equations (3.10), (3.11), (7.1) then the equation (8.2) becomes [(β + λ)2 + (λ + µ)2 + 2(α2 − β2)(β + 2λ + µ)]g(x, r(y, z)ξ) = 0. using the equation (3.12) we have µ = β, λ = 2(β2 − α2) − β or µ = 2(β2 − α2) + β, λ = −β. so we have the following theorem. theorem 8.1: if let a 3-dimensional trans-sasakian manifold (m, g, φ, η, ξ, α, β) with α, β constants admitting an η-ricci soliton satisfies the condition s(ξ, x) · w2 = 0 then µ = β, λ = 2(β2 − α2) − β or µ = 2(β2 − α2) + β, λ = −β. corollary 8.2: a 3-dimensional trans-sasakian manifold with α, β constants satisfies the condition s(ξ, x) · w2 = 0, there is no ricci soliton with the potential vector field ξ. acknowledgement: the author wish to express her sincere thanks and gratitude to the referee for valuable suggestions towards the improvement of the paper. 36 sampa pahan cubo 22, 1 (2020) references [1] c. s. bagewadi, g. ingalahalli, s. r. ashoka, a stuy on ricci solitons in kenmotsu manifolds, isrn geometry, (2013), article id 412593, 6 pages. [2] a. bhattacharyya, t. dutta, and s. pahan, ricci soliton, conformal ricci soliton and torqued vector fields, bulletin of the transilvania university of brasov series iii: mathematics, informatics, physics,, vol 10(59), no. 1 (2017), 39-52. [3] a. m. blaga, eta-ricci solitons on para-kenmotsu manifolds, balkan journal of geometry and its applications, vol.20, no.1, 2015, pp. 1-13. [4] c. călin, m. crasmareanu, eta-ricci solitons on hopf hypersurfaces in complex forms, revue roumaine de math. pures et app., 57 (1), (2012), 53-63. [5] b. y. chen, s. deshmukh, geometry of compact shrinking ricci solitons, balkan journal of geometry and its applications, vol.19, no.1, 2014, pp. 13-21 [6] j.c. cho, m. kimura ricci solitons and real hypersurfaces in a complex space form, tohoku math. j. 61 (2), (2009), 205-2012. [7] o. chodosh, f. t.-h fong, rational symmetry of conical kähler-ricci solitons, math. ann., 364(2016), 777-792. [8] a. futaki, h. ono, g. wang, transverse kähler geometry of sasaki manifolds and toric sasaki-einstein manifolds, j. diff. geom. 83 (3), (2009), 585-636. [9] s. golab, on semi-symmetric and quarter-symmetric linear connection, tensor. n. s., 29(1975), 249-254. [10] r. s. hamilton, the formation of singularities in the ricci flow, surveys in differential geometry (cambridge, ma, 1993), 2, 7-136, international press, combridge, ma, 1995. [11] r. s. hamilton, the ricci flow on surfaces, mathematical and general relativity, contemp. math, 71(1988), 237-261. [12] g. ingalahalli, c. s. bagewadi, ricci solitons on α-sasakian manifolds, isrn geometry, (2012), article id 421384, 13 pages. [13] j. c. marrero, ihe local structure of trans-sasakian manifolds, ann. mat. pura. appl., (4), 162(1992), 77-86. [14] j. morgan, g. tian, ricci flow and the poincaŕe conjecture, american mathematical society clay mathematics institute, (2007). cubo 22, 1 (2020) η-ricci solitons on 3-dimensional trans-sasakian manifolds 37 [15] j. a. oubina, new classes of almost contact metric structures, pub. math. debrecen, 20 (1), (2015), 1-13. [16] g. p. pokhariyal, r. s. mishra, the curvature tensors and their relativistic significance, yokohama math. j., 18(1970), 105-106. [17] d. g. prakasha, b. s. hadimani, η-ricci solitons on para-sasakian manifolds, journal of geometry, (2016), doi: 10.1007/s00022-016-0345-z, pp 1-10. introduction preliminaries -ricci solitons on trans-sasakian manifolds example of -ricci solitons on 3-dimensional trans-sasakian manifolds -ricci solitons on 3-dimensional trans-sasakian manifolds satisfying r(, x)s=0 -ricci solitons on 3-dimensional trans-sasakian manifolds satisfying s(, x)r=0 -ricci solitons on 3-dimensional trans-sasakian manifolds satisfying w2(, x)s=0 -ricci solitons on 3-dimensional trans-sasakian manifolds satisfying s(, x)w2=0 cubo a mathematical journal vol.19, no¯ 03, (31–42). october 2017 on the solution set of a fractional integro-differential inclusion involving caputo-katugampola derivative aurelian cernea faculty of mathematics and computer science university of bucharest academiei 14, 010014 bucharest, romania academy of romanian scientists splaiul independenţei 54, 050094 bucharest, romania acernea@fmi.unibuc.ro abstract we study an initial value problem associated to a fractional integro-differential inclusion defined by caputo-katugampola derivative and by a set-valued map with nonconvex values. we prove the arcwise connectedness of the solution set and that the set of selections corresponding to the solutions of the problem considered is a retract of the space of integrable functions on a given interval. resumen estudiamos un problema de valor inicial asociado a la inclusión ı́ntegro-diferencial fraccionaria definida por la derivada de caputo-katugampola y por una aplicación multivaluada con valores no-convexos. demostramos la arco-conexidad del conjunto solución y que el conjunto de selecciones correspondientes a las soluciones del problema considerado es un retracto del espacio de funciones integrables en un intervalo dado. keywords and phrases: differential inclusion, fractional derivative, initial value problem. 2010 ams mathematics subject classification: 34a60, 26a33, 34b15. 32 aurelian cernea cubo 19, 3 (2017) 1 introduction in the last years one may see a strong development of the theory of differential equations and inclusions of fractional order ([2,5,7-9] etc.). the main reason is that fractional differential equations are very useful tools in order to model many physical phenomena. recently, a generalized caputo-katugampola fractional derivative was proposed in [6] by katugampola and afterwards he provided the existence of solutions for fractional differential equations defined by this derivative. this caputo-katugampola fractional derivative extends the well known caputo and caputo-hadamard fractional derivatives. also, in some recent papers [1,12], several qualitative properties of solutions of fractional differential equations defined by caputokatugampola derivative were obtained. differential inclusion is a generalization of the notion of ordinary differential equation that provides powerful tools for various fields of mathematical analysis. at the same time there are dynamics of economical, social and biological systems that are set-valued. therefore, differential inclusions serve as natural models in such systems. an outstanding example comes from control theory. namely, the equivalence between a control system and the corresponding differential inclusion, established by filippov, allowed to obtained necessary and sufficient conditions of optimality using set-valued techniques. in the present paper we study the following cauchy problem dα,ρc x(t) ∈ f(t, x(t), v(x)(t)) a.e. ([0, t]), x(0) = x0, (1.1) where α ∈ (0, 1], ρ > 0, d α,ρ c is the caputo-katugampola fractional derivative, f : [0, t]×r×r → p(r) is a set-valued map, v : c([0, t], r) → c([0, t], r) is a nonlinear volterra integral operator defined by v(x)(t) = ∫t 0 k(t, s, x(s))ds with k(., ., .) : [0, t] × r × r → r a given function and x0 ∈ r. our goal is twofold. on one hand, we prove the arcwise connectedness of the solution set of problem (1.1) when the set-valued map is lipschitz in the second and third variable. on the other hand, under such type of hypotheses on the set-valued map we establish a more general topological property of the solution set of problem (1.1). namely, we prove that the set of selections of the set-valued map f that correspond to the solutions of problem (1.1) is a retract of l1([0, t], r). both results are essentially based on bressan and colombo results ([3]) concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values. in the theory of ordinary differential equations kneser’s theorem states that the solution set of an ordinary differential equation is connected, i.e., it cannot be represented as a union of two closed sets without common points. in the case of differential inclusions, although the solution set multifunction is not, in general, convex valued we are able to prove its arcwise connectedness and therefore, our result may be regarded as an extension of the classical theorem of kneser. we note that similar results for fractional differential inclusions defined by classical caputo fractional derivative are obtained in our previous paper [4]. the results in [4] extend to the case of cubo 19, 3 (2017) on the solution set of a fractional integro-differential inclusion . . . 33 fractional differential inclusions the results in [10,11] obtained for ordinary differential inclusions. the present paper generalizes and unifies all these results in the case of the more general problem (1.1). the paper is organized as follows: in section 2 we present the notations, definitions and the preliminary results to be used in the sequel and in section 3 we prove our main results. 2 preliminaries let t > 0, i := [0, t] and denote by l(i) the σ-algebra of all lebesgue measurable subsets of i. let x be a real separable banach space with the norm |.|. denote by p(x) the family of all nonempty subsets of x and by b(x) the family of all borel subsets of x. if a ⊂ i then χa(.) : i → {0, 1} denotes the characteristic function of a. for any subset a ⊂ x we denote by cl(a) the closure of a. the distance between a point x ∈ x and a subset a ⊂ x is defined as usual by d(x, a) = inf{|x−a|; a ∈ a}. we recall that pompeiu-hausdorff distance between the closed subsets a, b ⊂ x is defined by dh(a, b) = max{d ∗ (a, b), d ∗ (b, a)}, d ∗ (a, b) = sup{d(a, b); a ∈ a}. as usual, we denote by c(i, x) the banach space of all continuous functions x : i → x endowed with the norm |x|c = supt∈i|x(t)| and by l 1(i, x) the banach space of all (bochner) integrable functions x : i → x endowed with the norm |x|1 = ∫t 0 |x(t)|dt. we recall first several preliminary results we shall use in the sequel. a subset d ⊂ l1(i, x) is said to be decomposable if for any u, v ∈ d and any subset a ∈ l(i) one has uχa + vχb ∈ d, where b = i\a. we denote by d(i, x) the family of all decomposable closed subsets of l1(i, x). next (s, d) is a separable metric space; we recall that a multifunction g : s → p(x) is said to be lower semicontinuous (l.s.c.) if for any closed subset c ⊂ x, the subset {s ∈ s; g(s) ⊂ c} is closed. the next lemmas may be found in [3]. lemma 2.1. if f : i → d(i, x) is a lower semicontinuous multifunction with closed nonempty and decomposable values then there exists f : i → l1(i, x) a continuous selection from f. lemma 2.2. let g(., .) : i×s → p(x) be a closed-valued l(i)⊗b(s)-measurable multifunction such that g(t, .) is l.s.c. for any t ∈ i. then the multifunction g∗(.) : s → d(i, x) defined by g∗(s) = {f ∈ l1(i, x); f(t) ∈ g(t, s) a.e. (i)} is l.s.c. with nonempty closed values if and only if there exists a continuous mapping q(.) : s → l1(i, x) such that d(0, g(t, s)) ≤ q(s)(t) a.e. (i), ∀s ∈ s. 34 aurelian cernea cubo 19, 3 (2017) lemma 2.3. let h(.) : s → d(i, x) be a l.s.c. multifunction with closed decomposable values and let a(.) : s → l1(i, x), b(.) : s → l1(i, r) be continuous such that the multifunction f(.) : s → d(i, x) defined by f(s) = cl{f ∈ h(s); |f(t) − a(s)(t)| < b(s)(t) a.e. (i)} has nonempty values. then f(.) has a continuous selection. let ρ > 0. definition 2.4. ([6]) a) the generalized left-sided fractional integral of order α > 0 of a lebesgue integrable function f : [0, ∞) → r is defined by iα,ρf(t) = ρ1−α γ(α) ∫t 0 (tρ − sρ)α−1sρ−1f(s)ds, (2.1) provided the right-hand side is pointwise defined on (0, ∞) and γ(.) is the (euler’s) gamma function defined by γ(α) = ∫ ∞ 0 tα−1e−tdt. b) the generalized fractional derivative, corresponding to the generalized left-sided fractional integral in (2.1) of a function f : [0, ∞) → r is defined by dα,ρf(t) = (t1−ρ d dt ) n (in−α,ρ)(t) = ρα−n+1 γ(n − α) (t1−ρ d dt ) n ∫t 0 sρ−1f(s) (tρ − sρ)α−n+1 ds if the integral exists and n = [α]. c) the caputo-katugampola generalized fractional derivative is defined by dα,ρc f(t) = (d α,ρ[f(s) − n−1∑ k=0 f(k)(0) k! sk])(t) we note that if ρ = 1, the caputo-katugampola fractional derivative becames the well known caputo fractional derivative. on the other hand, passing to the limit with ρ → 0+, the above definition yields the hadamard fractional derivative. in what follows ρ > 0 and α ∈ [0, 1] lemma 2.5. for a given integrable function h(.) : [0, t] → r, the unique solution of the initial value problem dα,ρc x(t) = h(t) a.e. ([0, t]), x(0) = x0, is given by x(t) = x0 + ρ1−α γ(α) ∫t 0 (tρ − sρ)α−1sρ−1h(s)ds cubo 19, 3 (2017) on the solution set of a fractional integro-differential inclusion . . . 35 for the proof of lemma 2.2, see [6]; namely, lemma 4.2. a function x ∈ c(i, r) is called a solution of problem (1.1) if there exists a function f ∈ l1(i, r) with f(t) ∈ f(t, x(t), v(x)(t)) a.e. (i) such that d α,ρ c x(t) = f(t) a.e. (i) and x(0) = x0. in this case (x(.), f(.)) is called a trajectory-selection pair of problem (1.1). we shall use the following notations for the solution sets and for the selection sets of problem (1.1). s(x0) = {x ∈ c(i, r); x is a solution of(1.1)}, f̃(t) = x0 + ρ 1−α γ(α) ∫t 0 (tρ − sρ)α−1sρ−1f(s)ds, t (x0) = {f ∈ l 1(i, r); f(t) ∈ f(t, f̃(t), v(f̃)(t)) a.e. i}. 3 the main results in order to prove our topological properties of the solution set of problem (1.1) we need the following hypotheses. hypothesis 3.1. i) f(., .) : i × r × r → p(r) has nonempty closed values and is l(i) ⊗ b(r × r) measurable. ii) there exists l(.) ∈ l1(i, (0, ∞)) such that, for almost all t ∈ i, f(t, ., .) is l(t)-lipschitz in the sense that dh(f(t, x1, y1), f(t, x2, y2)) ≤ l(t)(|x1 − x2| + |y1 − y2|) ∀ x1, x2, y1, y2 ∈ r. iii) there exists p ∈ l1(i, r) such that dh({0}, f(t, 0, v(0)(t))) ≤ p(t) a.e. i. iv) k(., ., .) : i × r × r → r is a function such that ∀x ∈ r, (t, s) → k(t, s, x) is measurable. v) |k(t, s, x) − k(t, s, y)| ≤ l(t)|x − y| a.e. (t, s) ∈ i × i, ∀ x, y ∈ r. we use next the following notations m(t) := l(t)(1 + ∫t 0 l(u)du), t ∈ i, iα,ρm := sup t∈i |iα,ρm(t)|. theorem 3.2. assume that hypothesis 3.1 is satisfied and iα,ρm < 1. then for any ξ0 ∈ r the solution set s(ξ0) is arcwise connected in the space c(i, r). proof. let ξ0 ∈ r and x0, x1 ∈ s(ξ0). therefore there exist f0, f1 ∈ l 1(i, r) such that x0(t) = ξ0 + ρ 1−α γ(α) ∫t 0 (tρ −uρ)α−1uρ−1f0(u)du and x1(t) = ξ0 + ρ 1−α γ(α) ∫t 0 (tρ −uρ)α−1uρ−1f1(u)du, t ∈ i. 36 aurelian cernea cubo 19, 3 (2017) for λ ∈ [0, 1] define x0(λ) = (1 − λ)x0 + λx1 and g 0(λ) = (1 − λ)f0 + λf1 obviously, the mapping λ 7→ x0(λ) is continuous from [0, 1] into c(i, r) and since |g0(λ) − g0(λ0)|1 = |λ − λ0|.|f0 − f1|1 it follows that λ 7→ g 0(λ) is continuous from [0, 1] into l1(i, r). define the set-valued maps ψ1(λ) = {v ∈ l1(i, r); v(t) ∈ f(t, x0(λ)(t), v(x0(λ))(t)) a.e. i}, φ1(λ) =    {f0} if λ = 0, ψ1(λ) if 0 < λ < 1, {f1} if λ = 1 and note that φ1 : [0, 1] → d(i, r) is lower semicontinuous. indeed, let c ⊂ l1(i, r) be a closed subset, let {λm}m∈n converges to some λ0 and φ 1(λm) ⊂ c for any m ∈ n. let v0 ∈ φ 1(λ0). since the multifunction t 7→ f(t, x0(λm)(t), v(x 0(λm))(t)) is measurable, it admits a measurable selection vm(.) such that |vm(t) − v0(t)| = d(v0(t), f(t, x 0(λm)(t), v(x 0(λm))(t)) a.e. i. taking into account hypothesis 3.1 one may write |vm(t) − v0(t)| ≤ dh(f(t, x 0(λm)(t), v(x 0(λm))(t)), f(t, x 0(λ0)(t), v(x0(λ0))(t)) ≤ l(t)[|x 0(λm)(t) − x 0(λ0)(t)| + ∫t 0 l(s)|x0(λm)(s)− x0(λ0)(s)|ds] = l(t)|λm − λ0|[|x0(t) − x1(t)| + ∫t 0 l(s)|x0(s) − x1(s)|ds] hence |vm − v0|1 ≤ |λm − λ0| ∫t 0 l(t)[|x0(t) − x1(t)| + ∫t 0 l(s)|x0(s) − x1(s)|ds]dt which implies that the sequence vm converges to v0 in l 1(i, r). since c is closed we infer that v0 ∈ c; hence φ 1(λ0) ⊂ c and φ 1(.) is lower semicontinuous. next we use the following notation p0(λ)(t) = |g 0 (λ)(t)| + p(t) + l(t)(|x0(λ)(t)| + ∫t 0 l(s)|x0(λ)(s)|ds), t ∈ i, λ ∈ [0, 1]. since |p0(λ)(t) − p0(λ0)(t)| ≤ |λ − λ0|[|f1(t) − f0(t)|+ l(t)(|x0(t) − x1(t)| + ∫t 0 l(s)|x0(s) − x1(s)|ds)] cubo 19, 3 (2017) on the solution set of a fractional integro-differential inclusion . . . 37 we deduce that p0(.) is continuous from [0, 1] to l 1(i, r). at the same time, from hypothesis 3.1 it follows d(g0(λ)(t), f(t, x0(λ)(t), v(x0(λ))(t)) ≤ p0(λ)(t) a.e. i. (3.1) fix δ > 0 and for m ∈ n we set δm = m+1 m+2 δ. we shall prove next that there exists a continuous mapping g1 : [0, 1] → l1(i, r) with the following properties a) g1(λ)(t) ∈ f(t, x0(λ)(t), v(x0(λ))(t)) a.e. i, b) g1(0) = f0, g 1(1) = f1, c) |g1(λ)(t) − g0(λ)(t)| ≤ p0(λ)(t) + δ0 ρ α γ(α+1) tρα a.e. i. define g1(λ) = cl{v ∈ φ1(λ); |v(t) − g0(λ)(t)| < p0(λ)(t) + δ0 ραγ(α + 1) tρα , a.e. i} and, by (3.1), we find that g1(λ) is nonempty for any λ ∈ [0, 1]. moreover, since the mapping λ 7→ p0(λ) is continuous, we apply lemma 2.3 and we obtain the existence of a continuous mapping g1 : [0, 1] → l1(i, r) such that g1(λ) ∈ g1(λ) ∀λ ∈ [0, 1], hence with properties a)-c). define now x1(λ)(t) = ξ0 + ρ1−α γ(α) ∫t 0 (tρ − uρ)α−1uρ−1g1(λ)(u)du, t ∈ i and note that, since |x1(λ) − x1(λ0)|c ≤ t ρα ραγ(α+1) |g1(λ) − g1(λ0)|1, x 1(.) is continuous from [0, 1] into c(i, r). set pm(λ) := (i α,ρm)m−1( t ρα ραγ(α+1) |p0(λ)|1 + δm). we shall prove that for all m ≥ 1 and λ ∈ [0, 1] there exist xm(λ) ∈ c(i, r) and gm(λ) ∈ l1(i, r) with the following properties i) gm(0) = f0, g m(1) = f1, ii) gm(λ)(t) ∈ f(t, xm−1(λ)(t), v(xm−1(λ))(t)) a.e. i, iii) gm : [0, 1] → l1(i, r)is continuous, iv) |g1(λ)(t) − g0(λ)(t)| ≤ p0(λ)(t) + δ0 ρ α γ(α+1) tρα , v) |gm(λ)(t) − gm−1(λ)(t)| ≤ m(t)pm(λ), m ≥ 2, vi) xm(λ)(t) = ξ0 + ρ 1−α γ(α) ∫t 0 (tρ − uρ)α−1uρ−1gm(λ)(u)du, t ∈ i. assume that we have already constructed gm(.) and xm(.) with i)-vi) and define ψm+1(λ) = {v ∈ l1(i, r); v(t) ∈ f(t, xm(λ)(t), v(xm(λ))(t)) a.e. i}, 38 aurelian cernea cubo 19, 3 (2017) φm+1(λ) =    {f0} if λ = 0, ψm+1(λ) if 0 < λ < 1, {f1} if λ = 1. as in the case m = 1 we obtain that φm+1 : [0, 1] → d(i, r) is lower semicontinuous. from ii), v) and hypothesis 3.1, for almost all t ∈ i, we have |xm(λ)(t) − xm−1(λ)(t)| ≤ ρ1−α γ(α) ∫t 0 (tρ − uρ)α−1uρ−1|gm(λ)(u) − gm−1(λ)(u)|du ≤ ρ1−α γ(α) ∫t 0 (tρ − uρ)α−1uρ−1m(u)pm(λ)du = i α,ρm(t)pm(λ) ≤ i α,ρmpm(λ) < pm+1(λ). for λ ∈ [0, 1] consider the set gm+1(λ) = cl{v ∈ φm+1(λ); |v(t) − gm(λ)(t)| < m(t)pm+1(λ) a.e. i}. to prove that gm+1(λ) is not empty we note first that rm := (i α,ρm)m(δm+1 − δm) > 0 and by hypothesis 3.1 and ii) one has d(gm(t), f(t, xm(λ)(t), v(xm(λ))(t)) ≤ l(t)(|xm(λ)(t) − xm−1(λ)(t)|+ ∫t 0 l(s)|xm(λ)(s) − xm−1(λ)(s)|ds) ≤ l(t)(1 + ∫t 0 l(s)ds)|iα,ρm(t)|pm(λ) = m(t)(pm+1(λ) − rm) < m(t)pm+1(λ). moreover, since φm+1 : [0, 1] → d(i, r) is lower semicontinuous and the maps λ → pm+1(λ), λ → hm(λ) are continuous we apply lemma 2.3 and we obtain the existence of a continuous selection gm+1 of gm+1. therefore, |xm(λ) − xm−1(λ)|c ≤ i α,ρmpm(λ) ≤ (i α,ρm)m( tρα ραγ(α + 1) |p0(λ)|1 + δ) and thus {xm(λ)}m∈n is a cauchy sequence in the banach space c(i, r), hence it converges to some function x(λ) ∈ c(i, r). let g(λ) ∈ l1(i, r) be such that x(λ)(t) = ξ0 + ρ1−α γ(α) ∫t 0 (tρ − uρ)α−1uρ−1g(λ)(u)du, t ∈ i. the function λ 7→ t ρα ραγ(α+1) |p0(λ)|1 + δ is continuous, so it is locally bounded. therefore the cauchy condition is satisfied by {xm(λ)}m∈n locally uniformly with respect to λ and this implies that the mapping λ → x(λ) is continuous from [0, 1] into c(i, r). obviously, the convergence of the sequence {xm(λ)} to x(λ) in c(i, r) implies that gm(λ) converges to g(λ) in l1(i, r). cubo 19, 3 (2017) on the solution set of a fractional integro-differential inclusion . . . 39 finally, from ii), hypothesis 3.1 and from the fact that the values of f are closed we obtain that x(λ) ∈ s(ξ0). from i) and v) we have x(0) = x0, x(1) = x1 and the proof is complete. in what follows we use the notations ũ(t) = x0 + ρ1−α γ(α) ∫t 0 (tρ − sρ)α−1sρ−1u(s)ds, u ∈ l1(i, r) (3.2) and p0(u)(t) = |u(t)| + p(t) + l(t)(|ũ(t)| + ∫t 0 l(s)|ũ(s)|ds), t ∈ i (3.3) let us note that d(u(t), f(t, ũ(t), v(ũ)(t)) ≤ p0(u)(t) a.e. i (3.4) and, since for any u1, u2 ∈ l 1(i, r) |p0(u1) − p0(u2)|1 ≤ (1 + |i α,ρm(t)|)|u1 − u2|1 the mapping p0 : l 1(i, r) → l1(i, r) is continuous. proposition 3.3. assume that hypothesis 3.1 is satisfied and let φ : l1(i, r) → l1(i, r) be a continuous map such that φ(u) = u for all u ∈ t (x0). for u ∈ l 1(i, r), we define ψ(u) = {u ∈ l1(i, r); u(t) ∈ f(t, φ̃(u)(t), v(φ̃(u))(t)) a.e. i}, φ(u) = { {u} if u ∈ t (x0), ψ(u) otherwise. then the multifunction φ : l1(i, r) → p(l1(i, r)) is lower semicontinuous with closed decomposable and nonempty values. the proof of proposition 3.3 is similar to the proof of proposition 3.2 in [4]. theorem 3.4. assume that hypothesis 3.1 is satisfied, consider x0 ∈ r and assume i α,ρm < 1. then there exists a continuous mapping g : l1(i, r) → l1(i, r) such that i) g(u) ∈ t (x0), ∀u ∈ l 1(i, r), ii) g(u) = u, ∀u ∈ t (x0). proof. fix δ > 0 and for m ≥ 0 set δm = m+1 m+2 δ and define pm(u) := (i α,ρm)m−1( t ρα ραγ(α+1) |p0(u)|1+ δm), where ũ and p0(.) are defined in (3.2) and (3.3). by the continuity of the map p0(.), already proved, we obtain that pm : l 1(i, r) → l1(i, r) is continuous. we define g0(u) = u and we shall prove that for any m ≥ 1 there exists a continuous map gm : l 1(i, r) → l1(i, r) that satisfies a) gm(u) = u, ∀u ∈ t (x0), 40 aurelian cernea cubo 19, 3 (2017) b) gm(u)(t) ∈ f(t, ˜gm−1(u)(t), v( ˜gm−1(u))(t)) a.e. i, c) |g1(u)(t) − g0(u)(t)| ≤ p0(u)(t) + δ0 ραγ(α + 1) tρα a.e. i, d) |gm(u)(t) − gm−1(t)| ≤ m(t)pm(u) a.e. i, m ≥ 2. for u ∈ l1(i, r), we define ψ1(u) = {v ∈ l 1(i, r); v(t) ∈ f(t, ũ(t), v(ũ)(t)) a.e. i}, φ1(u) = { {u} if u ∈ t (x0), ψ1(u) otherwise and by proposition 3.3 (with φ(u) = u) we obtain that φ1 : l 1(i, r) → d(i, r) is lower semicontinuous. moreover, due to (3.4), the set g1(u) = cl{v ∈ φ1(u); |v(t) − u(t)| < p0(u)(t) + δ0 ραγ(α + 1) tρα a.e. i} is not empty for any u ∈ l1(i, r). so applying lemma 2.3, we find a continuous selection g1(.) of g1(.) that satisfies a)-c). suppose we have already constructed gi(.), i = 1, . . . , m satisfying a)-d). for u ∈ l 1(i, r) we define ψm+1(u) = {v ∈ l 1(i, r); v(t) ∈ f(t, g̃m(u)(t), v(g̃m(u))(t)) a.e. i}, φm+1(u) = { {u} if u ∈ t (x0), ψm+1(u) otherwise. we apply proposition 3.3 (with φ(u) = gm(u)) and obtain that φm+1(.) is a lower semicontinuous multifunction with closed decomposable and nonempty values. define the set gm+1(u) = cl{v ∈ φm+1(u); |v(t) − gm+1(u)(t)| < m(t)pm+1(u) a.e. i}. to prove that gm+1(u) is not empty we note first that rm := (i α,ρm)m(δm+1 − δm) > 0 and by hypothesis 3.1 and b) one has d(gm(t), f(t, g̃m(u)(t), v(g̃m(u))(t)) ≤ l(t)(|g̃m(u)(t) − ˜gm−1(u)(t)|+ ∫t 0 l(s)|g̃m(u)(s) − ˜gm−1(u)(s)|ds ≤ m(t)(i α,ρm)pm(u) = m(t)(pm+1(u) − rm) < m(t)pm+1(u). thus gm+1(u) is not empty for any u ∈ l 1(i, r). with lemma 2.3, we find a continuous selection gm+1 of gm+1, satisfying a)-d). cubo 19, 3 (2017) on the solution set of a fractional integro-differential inclusion . . . 41 therefore, we obtain that |gm+1(u) − gm(u)|1 ≤ (i α,ρm)m( tρα−1 ραγ(α + 1) |p0(u)|1 + δ) and this implies that the sequence {gm(u)}m∈n is a cauchy sequence in the banach space l 1(i, r). let g(u) ∈ l1(i, r) be its limit. the function u → |p0(u)|1 is continuous, hence it is locally bounded and the cauchy condition is satisfied by {gm(u)}m∈n locally uniformly with respect to u. hence the mapping g(.) : l1(i, r) → l1(i, r) is continuous. from a) it follows that g(u) = u, ∀u ∈ t (x0) and from b) and the fact that f has closed values we obtain that g(u)(t) ∈ f(t, g̃(u)(t), v(g̃(u))(t)) a.e. i ∀u ∈ l1(i, r). and the proof is complete. remark 3.5. we recall that if y is a hausdorff topological space, a subspace x of y is called retract of y if there is a continuous map h : y → x such that h(x) = x, ∀x ∈ x. therefore, by theorem 3.4, for any x0 ∈ r, the set t (x0) of selections of solutions of (1.1) is a retract of the banach space l1(i, r). example 3.6. consider α = 1 2 , ρ = 1, t = 1, x0 = 1 and c < min{1, 3 6+2γ( 1 2 ) }. define f(., .) : i × r × r → p(r) by f(t, x, y) = [−a |x| 1 + |x| , 0] ∪ [0, a |y| 1 + |y| ], a = cγ( 1 2 ) and k(., ., .) : i × r × r → r by k(t, s, x) = ax. since sup{|u|; u ∈ f(t, x, y)} ≤ a ∀ t ∈ [0, 1], x, y ∈ r, dh(f(t, x1, y1), f(t, x2, y2)) ≤ a|x1 − x2| + a|y1 − y2| ∀ x1, x2, y1, y2 ∈ r, in this case p(t) ≡ a, l(t) ≡ a, m(t) = a(1 + at) and, taking into account the choice of c, i 1 2 ,1m(t) = 2ct1/2[1 + 2cγ(1 2 ) 3 t] ≤ 2c[1 + 2cγ(1 2 ) 3 ] < 1 ∀t ∈ [0, 1]. therefore, applying theorems 3.2 and 3.4 to the problem d 1 2 ,1 c x(t) ∈ [−a |x(t)| 1 + |x(t)| , 0] ∪ [0, a2 | ∫t 0 x(s)ds| 1 + a| ∫t 0 x(s)ds| ], x(0) = 1, we deduce that its solution set s(1) is arcwise connected in the space c([0, 1], r) and its set of selections of solutions t (1) is a retract of the banach space l1([0, 1], r). 42 aurelian cernea cubo 19, 3 (2017) references [1] r. almeida, a. b. malinowski and t. odzijewicz, fractional differential equations with dependence on the caputo-katugampola derivative, j. comput. nonlin. dyn., 11 (2016), id 061017, 11 pp. [2] d. băleanu, k. diethelm, e. scalas and j. j. trujillo, fractional calculus models and numerical methods, world scientific, singapore, 2012. [3] a. bressan and g. colombo, extensions and selections of maps with decomposable values, studia math., 90 (1988), 69-86. [4] a. cernea, on a fractional integrodifferential inclusion, electronic j. qual. theory differ. equ., 2014 (2014), no. 25, 1-11. [5] k. diethelm, the analysis of fractional differential equations, springer, berlin, 2010. [6] u. n. katugampola, a new approach to generalized fractional derivative, bull. math. anal. appl., 6 (2014) 1-15. [7] a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [8] k. miller and b. ross, an introduction to the fractional calculus and differential equations, john wiley, new york, 1993. [9] i. podlubny, fractional differential equations, academic press, san diego,1999. [10] v. staicu, on the solution sets to differential inclusions on unbounded interval, proc. edinburgh math. soc., 43 (2000), 475-484. [11] v. staicu, arcwise conectedness of solution sets to differential inclusions, j. math. sciences 120 (2004), 1006-1015. [12] s. zeng, d. băleanu, y. bai, g. wu, fractional differential equations of caputo-katugampola type and numerical solutions, appl. math. comput., 315 (2017), 549-554. introduction preliminaries the main results cubo, a mathematical journal vol. 23, no. 01, pp. 145–159, april 2021 doi: 10.4067/s0719-06462021000100145 existence and attractivity results for ψ-hilfer hybrid fractional differential equations fatima si bachir1 säıd abbas2 maamar benbachir3 mouffak benchohra4 gaston m. n’guérékata5 1 laboratory of mathematics and applied sciences, university of ghardaia, 47000, algeria. sibachir.fatima@univ-ghardaia.dz.com 2 department of mathematics, university of säıda–dr. moulay tahar, p.o. box 138, en-nasr, 20000 säıda, algeria. said.abbas@univ-saida.dz 3 department of mathematics, saad dahlab blida1, university of blida, algeria. mbenbachir2001@gmail.com 4 laboratory of mathematics, djillali liabes university of sidi bel-abbès, p.o. box 89, sidi bel-abbès 22000, algeria. benchohra@yahoo.com 5 neerlab, department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore m.d. 21252, usa. gaston.nguerekata@morgan.edu abstract in this work, we present some results on the existence of attractive solutions of fractional differential equations of the ψ-hilfer hybrid type. the results on the existence of solutions are a consequence of the schauder fixed point theorem. next, we prove that all solutions are uniformly locally attractive. resumen en este trabajo, presentamos algunos resultados sobre la existencia de soluciones atractivas de ecuaciones diferenciales fraccionarias de tipo ψ-hilfer h́ıbridas. los resultados de existencia de soluciones son consecuencia del teorema de punto fijo de schauder. a continuación, probamos que todas las soluciones son uniformemente localmente atractivas. keywords and phrases: ψ-hilfer fractional derivative; schauder fixed-point theorem; uniformly locally attractive. 2020 ams mathematics subject classification: 26a33, 34a08. accepted: 01 march, 2021 received: 12 november, 2020 ©2021 f. si bachir et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100145 https://orcid.org/0000-0002-1947-9213 https://orcid.org/0000-0002-2518-8658 https://orcid.org/0000-0003-3519-1153 https://orcid.org/0000-0003-3063-9449 https://orcid.org/0000-0001-5765-7175 146 f. si bachir, s. abbas, m. benbachir, m. benchohra & g. m. n’guérékata cubo 23, 1 (2021) 1 introduction the theory of derivatives and integrals to a real or complex order is none other than the fractional theory which began in 1695 between g.a. de l’hospital and g.w. leibniz. the fractional integration and differentiation go back to leibniz, riemann, liouville, abel, weyl, and riesz [27]. many monographs to which the reader can refer such as abbas et al. [1, 5, 6], diethelm [13], kilbas et al. [17], oldham et al. [22], podlubny [23], samko et al. [24], zhou [32, 33], zhou et al. [34] and the works by abbas and benchohra [2], lakshmikantham et al. [19, 20, 21]. recently several works have been done concerning hybrid fractional differential equations see [9, 12, 14, 26, 31], and the references therein. functional ψ− fractional differential equations received a great importance in applied mathematics and other sciences, see [8, 16, 18, 25, 28, 29, 30], and the references therein. some interesting results on existence and attractivity have been obtained in [3, 4, 7]. in this work, we are interested in the existence and attractivity of solutions for the following problem  d λ,σ;ψ 0+ u(t) v(t,u(t)) = w (t,u(t)) ; a.e. t ∈ r+, (ψ(t) −ψ(0))1−ςu(t) |t=0= u0; u0 ∈ r, (1.1) where r+ := [0, +∞), 0 < λ < 1, 0 ≤ σ ≤ 1, ς = λ + σ(1−λ), hd λ,σ;ψ 0+ is the ψ -hilfer fractional derivative of order λ and type σ, v : r+ ×r → r∗ and w : r+ ×r → r, are given functions. special cases: • for σ = 0,ψ(t) = t,u0 = 0, we will get nonlinear hybrid fdes of the form  rldλ 0+ [ u(t) v(t,u(t)) ] = w(t,u(t)), a.e. t ∈ r+, u(0) = 0. • for λ = 1,σ = 1,ψ(t) = t, we will get nonlinear integer order hybrid differential equations of the form   d dt [ u(t) v(t,u(t)) ] = w(t,y(t)), a.e. t ∈ r+, u(0) = u0 ∈ r. for v = 1, we will get nonlinear ψ-hilfer fdes of the form  hd λ,σ;ψ 0+ u(t) = w(t,y(t)), a.e. t ∈ r+, (ψ(t) −ψ(0))1−ςu(t) ∣∣ t=0 = u0 ∈ r. • for v = 1,σ = 0 (in this case ς = λ),ψ(t) = t, we will get nonlinear fdes involving riemannliouville fractional derivative rldλ0+u(t) = w(t,y(t)), a.e. t ∈ r+. cubo 23, 1 (2021) existence and attractivity results for ψ-hilfer hybrid fractional. . . 147 2 preliminaries let ψ : [a,b] → r be an increasing differentiable function such that ψ′(t) 6= 0, for all t ∈ [a,b], (−∞≤ a < b ≤ +∞). define on [a,b], (0 < a < b < ∞) the weighted space cςψ[a,b] = {τ : (a,b] → r : (ψ(t) −ψ(a))ςτ(t) ∈ c[a,b]}, 0 ≤ ς < 1, with the norm ‖τ‖cς;ψ[a,b] = ‖(ψ(t) −ψ(a)) ςτ(t)‖c[a,b] = max{|(ψ(t) −ψ(a)) ςτ(t)| : t ∈ [a,b]} , where c([a,b]) denotes the banach space of all real continuous functions on [a,b]. let bc := bc(r+) be the banach space of all bounded and continuous functions from r+ into r. by bcς := bcς(r+), we denote the weighted space of all bounded and continuous functions defined by bcς = {φ : r+ → r : (ψ(t) −ψ(0))1−ςφ(t) ∈ bc}, with the norm ‖φ‖bcς := sup t∈r+ ∣∣(ψ(t) −ψ(0))1−ςφ(t)∣∣ . let us recall some definitions and properties of fractional calculus. definition 2.1. [17] the left-sided ψ-riemann-liouville fractional integral and fractional derivative of order λ, (n − 1 < λ < n) for an integrable function φ : [a,b] → r with respect to another function ψ : [a,b] → r, that is an increasing differentiable function such that ψ′(t) 6= 0, for all t ∈ [a,b], (−∞≤ a < b ≤ +∞), are respectively defined as follows: i λ;ψ a+ φ(t) = 1 γ(λ) ∫ t a ψ′(s)(ψ(t) −ψ(s))λ−1φ(s)ds, and d λ;ψ a+ φ(t) = ( 1 ψ′(t) d dt )n i n−α;ψ a+ φ(t) = 1 γ(n−λ) ( 1 ψ′(t) d dt )n ∫ t a ψ′(s)(ψ(t) −ψ(s))n−λ−1φ(s)ds, where γ(·) is the euler gamma function defined by γ(δ) = ∫ ∞ 0 e−ttδ−1dt, δ > 0. definition 2.2. [10] the left-sided ψ -caputo fractional derivative of function χ ∈ cn[a,b], (n− 1 < λ < n) n = [α] + 1 with respect to another function ψ is defined by cd λ;ψ a+ φ(t) = i n−λ;ψ a+ ( 1 ψ′(t) d dt )n φ(t) = 1 γ(n−λ) ∫ t a ψ′(s)(ψ(t) −ψ(s))n−λ−1φ[n]ψ (s)ds, where φ [n] ψ (t) = ( 1 ψ′(t) d dt )n φ(t) and ψ defined as in definition q. moreover, the ψ− caputo fractional derivative of function φ ∈ acn[a,b] is determined as cd λ;ψ a+ φ(t) = d λ;ψ a+  φ(t) −n−1∑ k=0 [ 1 ψ′(t) d dt ]k φ(a) k! (ψ(t) −ψ(a))k   . 148 f. si bachir, s. abbas, m. benbachir, m. benchohra & g. m. n’guérékata cubo 23, 1 (2021) definition 2.3. [29] let n − 1 < λ < n,n ∈ n, with [a,b],−∞ ≤ a < b ≤ +∞, and ψ ∈ cn([a,b],r) a function such that ψ(t) is increasing and ψ′(t) 6= 0, for all t ∈ [a,b]. the ψ -hilfer fractional derivative (left-sided) of function φ ∈ cn([a,b],r) of order λ and type σ ∈ [0, 1] is determined as d λ,σ;ψ a+ φ(t) = i σ(n−λ);ψ a+ [ 1 ψ′(t) d dt ]n i (1−σ)(n−λ);ψ a+ φ(t), t > a. in other way d λ,σ;ψ a+ φ(t) = i σ(n−λ);ψ a+ d γ;ψ a+ φ(t), t > a, where d γ;ψ a+ φ(t) = [ 1 ψ′(t) d dt ]n i (1−σ)(n−λ);ψ a+ φ(t). in particular, the ψ -hilfer fractional derivative of order λ ∈ (0, 1) and type σ ∈ [0, 1], can be written in the following form d λ,σ;ψ a+ φ(t) = 1 γ(ς −λ) ∫ t a (ψ(t) −ψ(s))ς−λ−1dγ;ψ a+ φ(s)ds = i ς−λ;ψ a+ d ς;ψ a+ φ(t), where ς = λ + σ −λσ, and dς;ψ a+ φ(t) = [ 1 ψ′(t) d dt ] i 1−ς;ψ a+ φ(t). lemma 2.4. [29] let λ > 0, 0 ≤ ς < 1 and φ ∈ l1(a,b). then i λ;ψ a+ i σ;ψ a+ φ(t) = i λ+σ;ψ a+ φ(t), a.e. t ∈ [a,b]. in particular (i) if φ ∈ cς;ψ[a,b], then i λ;ψ a+ i σ;ψ a+ φ(t) = i λ+σ;ψ a+ φ(t), t ∈ (a,b]. (ii) if φ ∈ c[a,b], then iλ;ψ a+ i σ;ψ a+ φ(t) = i λ+σ;ψ a+ φ(t), t ∈ [a,b]. lemma 2.5. [29] let λ > 0, 0 ≤ σ ≤ 1 and 0 ≤ ς < 1. if h ∈ cς;ψ[a,b] then d λ,σ;ψ a+ i λ;ψ a+ φ(t) = φ(t), t ∈ (a,b]. if φ ∈ c1[a,b] then d λ,σ;ψ a+ i α;ψ a+ φ(t) = φ(t), t ∈ [a,b]. lemma 2.6. let λ > 0, 0 ≤ ς < 1 and φ ∈ cς;ψ[a,b]. if λ > ς, then i λ;ψ a+ φ ∈ c[a,b] and i λ;ψ a+ φ(a) = lim t→a+ i λ;ψ a+ φ(t) = 0. lemma 2.7. [29] let φ ∈ cn[a,b],n− 1 < λ < n, 0 ≤ σ ≤ 1, and ς = λ + σ −λσ. then for all t ∈ (a,b] i λ;ψ a+ d λ,σ;ψ a+ φ(t) = φ(t) − n∑ k=1 [ψ(t) −ψ(a)]ς−k γ(ς −k + 1) φ (n−k) ψ i (1−σ)(n−λ);ψ a+ φ(a). in particular, if 0 < λ < 1, we have i λ;ψ a+ d λ,σ;ψ a+ φ(t) = φ(t) − [ψ(t) −ψ(a)]ς−1 γ(ς) i (1−σ)(1−λ);ψ a+ φ(a). cubo 23, 1 (2021) existence and attractivity results for ψ-hilfer hybrid fractional. . . 149 moreover, if φ ∈ c1−ς;ψ[a,b] and i 1−ς;ψ a+ φ ∈ c11−ς;ψ[a,b] such that 0 < ς < 1. then for all t ∈ (a,b] i ς;ψ a+ d ς;ψ a+ φ(t) = φ(t) − [ψ(t) −ψ(a)]γ−1 γ(ς) i 1−ς;ψ a+ φ(a). we deduce from the above lemma the following lemmas: lemma 2.8. [18] let v ∈ c(υ ×r,r∗); υ := [0,d], d > 0, κ ∈ c1−ζ,ψ(υ). then the problem  d λ,σ;ψ 0+ u(t) v(t,u(t)) = κ(t),a.e. t ∈ υ. (ψ(t) −ψ(0))1−ςu(t) |t=0= u0, u0 ∈ r, has a unique solution given by u(t) = v(t,u(t)) { u0 v(0,u(0)) (ψ(t) −ψ(0))ς−1 + iλ;ψ 0+ κ(t) } . lemma 2.9. let v ∈ c(υ × r,r∗), w : υ × r → r be a function such that w(·,u(·)) ∈ bcς for any u ∈ bcς. then the problem (1.1) is equivalent to the problem of obtaining the solutions of the integral equation u(t) = v(t,u(t)) { u0 v(0,u(0)) (ψ(t) −ψ(0))ς−1 + iλ;ψ 0+ w(·,u(·))(t) } . let ∅ 6= λ ⊂ bc and let k : λ → λ. we consider the solution of the equation (ku)(t) = u(t). (2.1) we introduce the concept of attractivity of solutions for equation (2.1). definition 2.10. solutions of equation (2.1) are locally attractive if there exists a ball b (u0,µ) in the space bc such that, for any solutions τ = τ(t) and ξ = ξ(t) of equations (2.1) that belong to b (u0,µ) ∩ λ, we can write lim t→∞ (τ(t) − ξ(t)) = 0. (2.2) if the limit (2.2) is uniform with respect to b (u0,µ) ∩ λ, then the solutions of equation (2.1) are said to be uniformly locally attractive (or, equivalently, that the solutions of (2.1) are locally asymptotically stable). lemma 2.11. [11] let m ⊂ bc. then m is relatively compact in bc if the following conditions are satisfied: (a) m is uniformly bounded in bc; (b) the functions belonging to m are almost equicontinuous in r+, i.e., equicontinuous on every compact set in r+; 150 f. si bachir, s. abbas, m. benbachir, m. benchohra & g. m. n’guérékata cubo 23, 1 (2021) (c) the functions from m are equiconvergent, i.e., given ε > 0, there exists l(ε) > 0 such that ∣∣∣u(t) − lim t→∞ u(t) ∣∣∣ < ε, for any t ≥ l(ε) and u ∈ m. theorem 2.12. (schauder fixed-point theorem [15]). let f be a banach space, let u be a nonempty bounded convex and closed subset of f, and let k : u → u be a compact and continuous map. then k has at least one fixed point in u. 3 existence and attractivity results definition 3.1. a measurable function u ∈ bcς is a solution of problem (1.1) if it verifies the initial condition (ψ(t)−ψ(0))1−ςu(t) |t=0= u0 and the equation d λ,σ;ψ 0+ u(t) v(t,u(t)) = w (t,u(t)) on r+. we will give the following hypotheses: (h1) the function t 7→ w(t,u) is measurable on r+ for each u ∈ bcς, the function u 7→ w(t,u) is continuous on bcς for a.e. t ∈ r+, and the function v is bounded such that u 7→ v(t,u) is continuous. (h2) there exists a continuous function t : r+ → r+ such that for a.e. t ∈ r+ and each u ∈ r, |w(t,u)| ≤ t(t) 1 + |u| , and lim t→∞ (ψ(t) −ψ(0))1−ς ( i λ;ψ 0+ t ) (t) = 0. set t∗ = sup t∈r+ (ψ(t) −ψ(0))1−ς ( i λ;ψ 0+ t ) (t) < ∞. now we present a theorem on the existence and attractivity of solutions of the problem (1.1). theorem 3.2. assume that the hypotheses (h1) and (h2) hold. then the problem (1.1) has at least one solution defined on r+ and the solutions of problem (1.1) are uniformly locally attractive. proof. consider the operator k such that, for any u ∈ bcς, (ku)(t) = v(t,u(t)) { u0 v(0,u(0)) (ψ(t) −ψ(0))ς−1 + 1 γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds } . cubo 23, 1 (2021) existence and attractivity results for ψ-hilfer hybrid fractional. . . 151 let l be a bound of the function v. for any u ∈ bcς, and for each t ∈ r+, we have∣∣∣∣(ψ(t) −ψ(0))1−ς(ku)(t) ∣∣∣∣ ≤|v(t,u(t))| {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ςγ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds } ≤|v(t,u(t))| {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ςγ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1t(s)ds } ≤l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + t∗ } :=r∗. so |k(u)‖bc ≤ r∗. (3.1) therefore, k(u) ∈ bcς. since, the map k(u) is continuous on r+; for any u ∈ bcς, and k(bcς) ⊂ bcς, then the operator k maps bcς into itself. furthermore, equation (3.1) implies that the operator k transforms the ball br∗ := b(0,r∗) = {v ∈ bcς : ‖v‖bcς ≤ r∗} into itself. from lemma 2.9 the solution of problem (1.1) is reduced to finding the solutions of the operator equation k(u) = u. we show that the operator k : bcς → bcς satisfies all assumptions of theorem 2.12. the proof is divided into several steps: step 1. k is continuous. let {un}n∈n be a sequence such that un → u in br∗. then, for each t ∈ r+, we have∣∣((ψ(t) −ψ(0))1−ς (kun) (t) − ((ψ(t) −ψ(0))1−ς(ku)(t)∣∣ ≤ ∣∣∣∣v(t,un(t)) { u0 v(0,u(0)) + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds } −v(t,u(t)) { u0 v(0,u(0)) + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds }∣∣∣∣ ≤ ∣∣∣∣v(t,un(t)) { u0 v(0,u(0)) + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds } −v(t,u(t)) { u0 v(0,u(0)) + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds } + v(t,u(t)) { u0 v(0,u(0)) + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,un(s))ds } −v(t,u(t)) { u0 v(0,u(0)) + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds }∣∣∣∣ 152 f. si bachir, s. abbas, m. benbachir, m. benchohra & g. m. n’guérékata cubo 23, 1 (2021) ≤ ∣∣∣∣v(t,un(t)) −v(t,u(t)) ∣∣∣∣ ∣∣∣∣ u0v(0,u(0)) + (ψ(t) −ψ(0)) 1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1 ×w(s,un(s))ds ∣∣∣∣ + |v(t,u(t))|(ψ(t) −ψ(0))1−ςγ(λ) × ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,un(s)) −w(s,u(s))|ds. hence ∣∣(ψ(t) −ψ(0))1−ς (kun) (t) − (ψ(t) −ψ(0))1−ς(ku)(t)∣∣ ≤ ∣∣∣∣v(t,un(t)) −v(t,u(t)) ∣∣∣∣ {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,un(s))|ds } + l (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,un(s)) −w(s,u(s))|ds. (3.2) case 1. if t ∈ [0,d], then, in view of the facts that un → u as n →∞, v and w are continuous, by the lebesgue dominated convergence theorem, from the equation (3.2), we have ‖k (un) −k (u)‖bcς → 0 as n →∞. case 2. if t ∈ (d,∞), then, from the hypotheses and (3.2), we have∣∣(ψ(t) −ψ(0))1−ς (kun) (t) − (ψ(t) −ψ(0))1−ς(ku)(t)∣∣ ≤ ∣∣∣∣v(t,un(t)) −v(t,u(t)) ∣∣∣∣ {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1t(s)ds } + 2l (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1t(s)ds. then ∣∣(ψ(t) −ψ(0))1−ς (kun) (t) − (ψ(t) −ψ(0))1−ς(ku)(t)∣∣ ≤ ∣∣∣∣v(t,un(t)) −v(t,u(t)) ∣∣∣∣ {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + ((ψ(t) −ψ(0))1−ς (iλ;ψ0+ t) (t) } + 2l((ψ(t) −ψ(0))1−ς ( i λ;ψ 0+ t ) (t). (3.3) since un → u as n →∞, v is continuous and (ψ(t)−ψ(0))1−ς ( i λ;ψ 0+ t ) (t) → 0 as t →∞, it follows from (3.3) that ‖k (un) −k(u)‖bcς → 0 as n →∞. step 2. l (br∗) is uniformly bounded, and equicontinuous on every compact subset [0,d] of r+, d > 0. cubo 23, 1 (2021) existence and attractivity results for ψ-hilfer hybrid fractional. . . 153 we have l (br∗) ⊂ br∗ and br∗ is bounded, so l (br∗) is uniformly bounded. next, for each t1, t2 ∈ [0,d], t1 < t2, and u ∈ br∗, we have ∣∣(ψ(t2) −ψ(0))1−ς (ku) (t2) − (ψ(t1) −ψ(0))1−ς(ku)(t1)∣∣ ≤ ∣∣∣∣v(t2,u(t2)) { u0 v(0,u(0)) + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds } −v(t1,u(t1)) { u0 v(0,u(0)) + (ψ(t1) −ψ(0))1−ς γ(λ) ∫ t1 0 ψ′(s)(ψ(t1) −ψ(s))λ−1w(s,u(s))ds }∣∣∣∣ ≤ ∣∣∣∣v(t2,u(t2)) { u0 v(0,u(0)) + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds } −v(t1,u(t1)) { u0 v(0,u(0)) + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds } + v(t1,u(t1)) { u0 v(0,u(0)) + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds } −v(t1,u(t1)) { u0 v(0,u(0)) + (ψ(t1) −ψ(0))1−ς γ(λ) ∫ t1 0 ψ′(s)(ψ(t1) −ψ(s))λ−1w(s,u(s))ds }∣∣∣∣. thus ∣∣(ψ(t2) −ψ(0))1−ς (ku) (t2) − (ψ(t1) −ψ(0))1−ς(ku)(t1)∣∣ ≤ |v(t2,u(t2)) −v(t1,u(t1))| ∣∣∣∣ u0v(0,u(0)) + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds ∣∣∣∣ + |v(t1,u(t1))| ∣∣∣∣(ψ(t2) −ψ(0))1−ςγ(λ) ∫ t1 0 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 t1 ψ′(s)(ψ(t2) −ψ(s))λ−1w(s,u(s))ds − (ψ(t1) −ψ(0))1−ς γ(λ) ∫ t1 0 ψ′(s)(ψ(t1) −ψ(s))λ−1w(s,u(s))ds ∣∣∣∣. hence ∣∣(ψ(t2) −ψ(0))1−ς (ku) (t2) − (ψ(t1) −ψ(0))1−ς(ku)(t1)∣∣ ≤ |v(t2,u(t2)) −v(t1,u(t1))| (∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1|w(s,u(s))|ds ) + l (∫ t1 0 ∣∣∣∣(ψ(t2) −ψ(0))1−ςγ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1 − (ψ(t1) −ψ(0))1−ς γ(λ) ψ′(s)(ψ(t1) −ψ(s))λ−1 ∣∣∣∣ |w(s,u(s))|ds + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 t1 ψ′(s)(ψ(t2) −ψ(s))λ−1|w(s,u(s))|ds ) 154 f. si bachir, s. abbas, m. benbachir, m. benchohra & g. m. n’guérékata cubo 23, 1 (2021) ≤ |v(t2,u(t2)) −v(t1,u(t1))| (∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1t(s)ds ) + l (∫ t1 0 ∣∣∣∣(ψ(t2) −ψ(0))1−ςγ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1 − (ψ(t1) −ψ(0))1−ς γ(λ) ψ′(s)(ψ(t1) −ψ(s))λ−1 ∣∣∣∣ t(s)ds + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 t1 ψ′(s)(ψ(t2) −ψ(s))λ−1t(s)ds ) . from the continuity of the functions t and v, by setting t∗ = supt∈[0,d] t(t), we obtain∣∣(ψ(t2) −ψ(0))1−ς (ku) (t2) − (ψ(t1) −ψ(0))1−ς(ku)(t1)∣∣ ≤ |v(t2,u(t2)) −v(t1,u(t1))| (∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + t∗(ψ(t2) −ψ(0))1−ςγ(λ) ∫ t2 0 ψ′(s)(ψ(t2) −ψ(s))λ−1ds ) + lt∗ (∫ t1 0 ∣∣∣∣(ψ(t2) −ψ(0))1−ςγ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1 − (ψ(t1) −ψ(0))1−ς γ(λ) ψ′(s)(ψ(t1) −ψ(s))λ−1 ∣∣∣∣ds + (ψ(t2) −ψ(0))1−ς γ(λ) ∫ t2 t1 ψ′(s)(ψ(t2) −ψ(s))λ−1ds ) ≤ |v(t2,u(t2)) −v(t1,u(t1))| (∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + t∗(ψ(t2) −ψ(0))1−ς+λγ(λ + 1) ) + lt∗ (∫ t1 0 ∣∣∣∣(ψ(t2) −ψ(0))1−ςγ(λ) ψ′(s)(ψ(t2) −ψ(s))λ−1 − (ψ(t1) −ψ(0))1−ς γ(λ) ψ′(s)(ψ(t1) −ψ(s))λ−1 ∣∣∣∣ds + (ψ(t2) −ψ(0))1−ςγ(λ + 1) (ψ(t2) −ψ(t1))λ ) . as t1 → t2, the right-hand side of the inequality tends to zero. step 3. l (br) is equiconvergent. let u ∈ br∗. then, for each t ∈ r+ we have∣∣∣∣(ψ(t) −ψ(0))1−ς(ku)(t)| ∣∣∣∣ ≤ |v(t,u(t))| {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + ∣∣∣∣(ψ(t) −ψ(0))1−ςγ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1w(s,u(s))ds ∣∣∣∣ } ≤ |v(t,u(t))| {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + ∣∣∣∣(ψ(t) −ψ(0))1−ςγ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1t(s)ds ∣∣∣∣ } ≤ l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ς (iλ;ψ0+ t) (t) } . since (ψ(t) −ψ(0))1−ς ( i λ;ψ 0+ t ) (t) → 0 as t → +∞, cubo 23, 1 (2021) existence and attractivity results for ψ-hilfer hybrid fractional. . . 155 we find |(ku)(t)| ≤ l {∣∣∣∣ u0(ψ(t) −ψ(0))1−ςv(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0)) 1−ς ( i λ;ψ 0+ t ) (t) (ψ(t) −ψ(0))1−ς } . hence, |(lu)(t) − (lu)(+∞)|→ 0 as t → +∞, in view of lemma 2.11 as a consequence of steps 1 − 4, we conclude that k : br∗ → br∗ is compact and continuous. applying the theorem 2.12, we have that k has a fixed point u, which is a solution of problem (1.1) on r+. step 4. the uniform local attractivity of solutions. we assume that u∗ is a solution of problem (1.1) under the conditions of this theorem. set u ∈ b ( u∗, 2l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2t∗ }) , we have ∣∣(ψ(t) −ψ(0))1−ς (ku) (t) − (ψ(t) −ψ(0))1−ς(u∗)(t)∣∣ ≤ ∣∣(ψ(t) −ψ(0))1−ς (ku) (t) − (ψ(t) −ψ(0))1−ς(ku∗)(t)∣∣ ≤ |v(t,u(t)) −v(t,u∗(t))| {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds } + l (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s)) −w(s,u∗(s))|ds ≤ 2l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + (ψ(t) −ψ(0))1−ςγ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1t(s)ds } + 2l (ψ(t) −ψ(0))1−ς γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1t(s)ds ≤ 2l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2t∗ } . thus, we get ‖k(u) −u∗‖bcς ≤ 2l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2t∗ } . so, we conclude that k is a continuous function such that k ( b ( u∗, 2l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2t∗ })) ⊂ b ( u∗, 2l {∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2t∗ }) . 156 f. si bachir, s. abbas, m. benbachir, m. benchohra & g. m. n’guérékata cubo 23, 1 (2021) moreover, if u is a solution of problem (1.1), then |u(t) −u∗(t)| = |(ku)(t) − (ku∗) (t)| ≤ |v(t,u(t)) −v(t,u∗(t))| { (ψ(t) −ψ(0))ς−1 ∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 1 γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds } + l γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s)) −w(s,u∗(s))|ds ≤ 2l { (ψ(t) −ψ(0))ς−1 ∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 1 γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s))|ds } + l γ(λ) ∫ t 0 ψ′(s)(ψ(t) −ψ(s))λ−1|w(s,u(s)) −w(s,u∗(s))|ds ≤ 2l { (ψ(t) −ψ(0))ς−1 ∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2(iλ;ψ0+ t)(t) } . therefore, |u(t) −u∗(t)| ≤ 2l { (ψ(t) −ψ(0))ς−1 ∣∣∣∣ u0v(0,u(0)) ∣∣∣∣ + 2 (ψ(t) −ψ(0))1−ς(iλ;ψ0+ t)(t)(ψ(t) −ψ(0))1−ς } . (3.4) by using (3.4) and the fact that lim t→∞ (ψ(t) −ψ(0))1−ς(iλ;ψ 0+ t)(t) = 0, we conclude lim t→∞ |u(t) −u∗(t)| = 0. consequently, all solutions of problem (1.1) are uniformly locally attractive. 4 an example as an application of our results, we consider the following problem for a ψ-hilfer fractional differential equation   d 1 2 , 1 2 ;ψ 0+ u(t) v(t,u(t)) = w (t,u(t)) ,a.e. t ∈ r+, (ψ(t) −ψ(0)) 1 4 u(t) |t=0= 1, (4.1) where ψ : [0, 1] → r with ψ(t) = √ t + 3, v(t,u) = 1 (1 + t)(1 + |u|) ,   w(t,u) = β(ψ(t) −ψ(0)) −1 4 sin t 64(1 + √ t)(1 + |u|) , t ∈ (0,∞), u ∈ r, w(0,u) = 0, u ∈ r, cubo 23, 1 (2021) existence and attractivity results for ψ-hilfer hybrid fractional. . . 157 and β = 9 √ π 16 . clearly, the function w is continuous. the hypothesis (h2) is satisfied with  t(t) = β(ψ(t) −ψ(0)) −1 4 |sin t| 64(1 + √ t) , t ∈ (0,∞), t(0) = 0. in addition, we have (ψ(t) −ψ(0)) 1 4 ( i 1 2 ;ψ 0+ t ) (t) = (ψ(t) −ψ(0)) 1 4 γ ( 1 2 ) ∫ t 0 ψ′(τ)(ψ(t) −ψ(τ)) −1 2 t(τ)dτ ≤ 1 4 (ψ(t) −ψ(0)) −1 4 → 0 as t →∞. simple computations show that all conditions of theorem 3.2 are satisfied. consequently, our problem (4.1) has at least one solution defined on r+, and all solutions of this problem are uniformly locally attractive. 5 conclusion in this paper, we provided some sufficient conditions ensuring the existence and the uniform locally attractivity of solutions of some ψ-hilfer fractional differential equations. the technique used is based on schauder’s fixed point theory theorem. references [1] s. abbas, and m. benchohra, advanced functional evolution equations and inclusions, dev. math., vol. 39, springer, cham, 2015. 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[34] y. zhou, j. wang, and l. zhang, basic theory of fractional differential equations, second edition, world scientific, singapore, 2017. introduction preliminaries existence and attractivity results an example conclusion caracterizacicn !!! alge8ra.s ~ berlls'iein ~tmjj[ c. l blll.\y. c. llllwb~ u::: ando la cara.cter12ac 1ón de l as al.j3eb.ras re berns't'ein que son de jordan, conseguuoos 1.ida nueva caracter l'zac ión que requ ier e sólo de d os cond 1ciones en subespaci os del álgebra. a.d1 c1onalment e obtuv l..l'oos otr a deroostrac1 ón para caract er i zar l as alge8ra.s r::e bertstein que s on njfmaj...es, tr.m,jo ¡wci.allbtt! f inn:ia:kl ¡ula du'&tifl de lrl'l'plt~illl de la u.f.t.o . .lallle.c:m ptma •eta en el ~de j.gtailk2 y lltafütu:a de 1~ u.no. 76 preliminares sea [ lill1 cuerpo de caracter!.stica distü1ta file 2 y a un álgebra rnmutativa sot>re k. decilms que (a,w) es un algebra foni.'erada ss1 existe un hcm::rmrfism:> <.:le álgebras w : a____. k n© trivial. cc;xoo w es no tr i vial, existe a e a tal ~e w(a)=i, luege teneiros que a= ka @ ker (w) . sea (.a¡ w) una k-.álgetira p0n<:ierada. dec;:imes (!jl!le (.a¡ w) es un alcíebra oc bernstein ss i (x2 )2=w(x )2x2 't/xea. s i a es un álgetira eie bernste in, es p@sihlle p;irebar ~e jos únicos idem¡:>0tentes son ker(w). el endoo)'ilrf ism0 le (x) = ex vxea cleja ifjvariante el subespaci0 ker(w) ( ya <:¡ue w(ex) = w(e)w(x) = 1.0 =o vxeker(w ) ), l1.:.1ego ker(wl=le (ker (w ) )g ker (be). sean u:, l,, (ker(w))' lex/w(x) , üj y v:' ker(l,,i, = {xeker(w)/ex=ol=!xe.a/w(x)=o y ex=ol. en•t0nces a=ke@ u@ v. si n: =dim(ker(w) ), r; :d im(u) y s; =dim(v), deci.jnc¡is que a es de tipo (1+r,s). en (2) est.á. demostrad© qw.e para cualquier desc0lllp0sic:::iór.i de n en la forma r+s, existe un álgebra de bernsteion de tij:i0 (1+r, s ) . sea {a, w) un ájgetira p0m:l.erada s0tire k . i!lecim::ls (;fl.:le (a, w) es un .algebra }[)ñma!... ssi x2y = w(x)xy 'v'x, yu... observemos cque si a es un álgebra normal y xea entonces (x2 ¡2 = x2 •x2=w(x}x•x2=w(x) (x2 •x ) =w ( x) (w(x )x•x )=w(x ¡2x2 es decir, a es un algetira c:l.e bernstein. 77 ( sea a un algegra sobre k. ~cimas que a es un a1..gelffia oc j oeid\n ssi x2(yx) = (x2y )x vx., yea. notemos que toda á l l¡!ebra nonnal es de jarcian, porque x2 (yx) ' (w(x )x ) (xy i ' w (x )(xy)x ' (x2y¡x \fx, yea eil [ 1) se denuestra la si guiente proposición : profos!c!oo _!: sea a = ke itj u lil v \..u'la kálgebra de bemste m. entonces: a es un á lgebra de j ordan ssi vuiru, vif\/ se cunple 1.1 vi vz=ü 1. 2 (u1 v1 )vz + (u1v2 )v1 ' o 2 . 3 (u1uz)v1 + 2((u 1v1 )uz)u1 ' o i .~ ( (u 1v1 )v2 jv1 ' o 2 1 . s (u 1u2 )u1 ' o 1.6 ( (u1v1 )v2 )u1 ca.~jzn:.icnes para me.jor·ar la propos1ci6n 1 usaremos los dos lemas s i guientes: sea a=ke iti u el v uru kálgebra de bemstein, entonces \t\.lie u, v 1e v: 2 a ) u 1u2 + 2(u1uzju1 : o 7 8 rec ordems que { a). ( b) y (e) son parte de 1 lena 2 que aparece en [1). para las dos ú l tuna.s af1rmac1ones notems que: d ) el1 ( b) obten i endo (u 1v1 )(u1v2) sust1tuldx>s v 1 u1 ( (u1 v1 )vz ) + o, por (e ) , por v2 y uz por u1v1 (u¡ v¡) (u¡ v2 1 ' o ' pero l uego ui(v2(u1v1)): o. 2 e ) en ( b ) , ahora sust 1tuijoos v por u 1, obteniendo 2 2 u 1 (uzu1 ) + u 2 (u 1u 1 j : o. pero en toda .{,,lgebra de bemstem 2 t eneroos que u3 : o para cualqu ier uru, juego u1 (u2u1 ) =ú, es decir vale (e ) . sea a : ke @ u @ v u n álgebra de bernstein sobre k, en que v(vu ):o vueu , vev . entonces son v.illdas las s1gu1entes igu aldad es vu ieu, v 1€ v l ) ( u 1v 1 )v 2+(u 1v 2 ) v 1 : 2 ll ) ( u1u2) v1-+ 2 ( ( u 1v1 ) u2 )u 1 demootracjon: l ) haciendo v:v 1+v 2 , obtenemos o:( u (v 1+vz ))( v 1+vz ) = :(uv 1 )v 2+(uv z)v1+ ( uv 2 )v 2 , pero ( uv1 )v 1 = o para cualquier v 1ev, luego v ale (1) 2 11) observemos q ue ( u 1u 2 jv 1 ... 2((u 1v 1¡u 2 )u 1 = 2 2 u 1 ( u 2v1)+ 2((u 1v 1 )u 2 )u 1 = u 1 ( u 2v 1 j-2 (( u 2v 1 ) u 1 )u 1 : o. estas igualdades están justificadas por (l), (b ) y ( a ) anteri ores. 111 } por ( 1 ) t en emos que ( u 1v 1 )vz ( u 1vzjv 1 , lu ego ((u1 v1 jv2jv 1 = -((u 1vz)v t)v 1 : o 79 ( evi!'l'a cu nq'i phopos!c!on ~: sea a = ke ~ u li) v un álgebra de bernstein sobre k, entonces a es un álgebra de jordan ss1 v2=o y v ( vu),0 'tvev. phoposic!on ~: sea a ::: ke i'! u g v un álgebra de bernstern sobre k, en ton ces a es normal ss1 v2:o y uv:o demcs!'racion: sean x = ae + u 1 + v1e.a. y = ¡:le + u2 + v2ea. recordando que ev¡ = o, eui = yµ 1 , u¡v2 j = o obtenams que 2 2 2 2 2 2 2 x y : a · 13 · e + r yp u •u u +u v +yiav + v v + ytoeu + au v + l.. 2 1 2 1 2 112 1 12 + ~ v + 2 (u v )v ] • [ au u + 2 (u v )u ] 1 1 1 1 2 1 2 1 1 2 y adenis w (x)xy:a2 .13 •• elyp2u +y.aau +au v +av u +av vj+ íau uj l 2 1 12 12 1 l 1 luego a es normal s1 y sólo s1 2 2 2 2 u1u2+u1 v2•vet'jv1•v1v2+yjlu1 v1•2{u1v 1 )v2 = av 1u2+av1v2 si y sólo s1 u1 u2 + u1 v2 + v 1v2 + 2(u 1v 1 )v2 =o 2 kv° t + yil.j1vt : 0 (l! ) en l a tercera igua ldad d e ( :;t ) po dem os hacer u 2 : o, obteniendo v 1vz = o y con v! = o se ob t iene v 1u2 = o. es decir (#) implic a que v : o y que u v =o. el recip r oco 60 revista cul!g .. nq'i rnmecliato usan<:1 0 las pr0 pi eclades u2 ~ v, v 2 u y uv !:. u váli<:las en t oda álgebra de bernstein. colorar id: t @da k.:i !getira n•©rma l es wn•a k-áj.getira cle j 0 r c:í an . bibliografla [1) a.worz-busbkr0s, berns:rein algeeras. aroro. matro. v o l.'18,388398(1987 ). (2) a.worz-busekros, 0 for all n ≥ 0 therefore, from contraction condition (3.1), we have d(t xn+1, t xn) ≤ α d(fxn+1, t xn+1) d(fxn, t xn) d(fxn+1, fxn) + β [d(fxn+1, t xn+1) + d(fxn, t xn)] +γd(fxn+1, fxn) which intern implies that d(t xn+1, t xn) ≤ αd(t xn, t xn+1) + β [d(t xn, t xn+1) + d(t xn−1, t xn)] + γd(t xn, t xn−1) finally, we arrive at d(t xn+1, t xn) ≤ ( β + γ 1 − α − β ) d(t xn, t xn−1) continuing the same process up to (n − 1) times, we get d(t xn+1, t xn) ≤ ( β + γ 1 − α − β )n d(t x1, t x0) let k = β+γ 1−α−β ∈ [0, 1), then from triangular inequality for m ≥ n, we have d(t xm, t xn) ≤ d(t xm, t xm−1) + d(t xm−1, t xm−2) + ........ + d(t xn+1, t xn) ≤ ( km−1 + km−2 + .......... + kn ) d(t x1, t x0) ≤ kn 1 − k d(t x1, t x0) as m, n → +∞, d(t xm, t xn) → 0, which shows that the sequence {t xn} is a cauchy sequence in x. so, by the completeness of x, there exists a point µ ∈ x such that t xn → µ as n → +∞. again, by the continuity of t , we have lim n→+∞ t (t xn) = t ( lim n→+∞ t xn ) = t µ. but fxn+1 = t xn, then fxn+1 → µ as n → +∞ and from the compatibility for t and f, we have lim n→+∞ d(t (fxn), f(t xn)) = 0. further by triangular inequality, we have d(t µ, fµ) = d(t µ, t (fxn)) + d(t (fxn), f(t xn)) + d(f(t xn), fµ) cubo 22, 2 (2020) contractive mapping theorems in partially ordered metric spaces 207 on taking limit as n → +∞ in both sides of the above equation and using the fact that t and f are continuous then, we get d(t µ, fµ) = 0. thus, t µ = fµ. hence, µ is a coincidence point of t and f in x. corollary 1. let (x, d, �) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying the following condition d(t x, t y) ≤ α d(fx, t x) d(fy, t y) d(fx, fy) + β [d(fx, t x) + d(fy, t y)] for all x, y in x with f(x) 6= f(y) are comparable and for some α, β ∈ [0, 1) with 0 ≤ α + 2β < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and the mappings t and f are compatible, then t and f have a coincidence point in x. proof. set γ = 0 in theorem 1. corollary 2. let (x, d, �) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying the following condition d(t x, t y) ≤ β [d(fx, t x) + d(fy, t y)] + γd(fx, fy) for all x, y in x with f(x) 6= f(y) are comparable and for some β, γ ∈ [0, 1) with 0 ≤ 2β + γ < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and the mappings t and f are compatible, then t and f have a coincidence point in x. proof. the proof can be obtained by setting α = 0 in theorem 1. we may remove the continuity criteria of t in theorem 1 is still valid by assuming the following hypothesis in x: if {xn} is a nondecreasing sequence in x such that xn → x, then xn � x for all n ∈ n. theorem 2. let (x, d, �) be a complete partially ordered metric space. suppose that f and t are self-mappings on x, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying d(t x, t y) ≤ α d(fx, t x) d(fy, t y) d(fx, fy) + β [d(fx, t x) + d(fy, t y)] + γd(fx, fy) (3.2) for all x, y in x with f(x) 6= f(y) are comparable and for some α, β, γ ∈ [0, 1) with 0 ≤ α+2β+γ < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and {xn} is a nondecreasing sequence in x such that xn → x, then xn � x for all n ∈ n. if f(x) is a complete subset of x, then t and f have a coincidence point in x. further, if t and f are weakly compatible, then t and f have a common fixed point in x. moreover, the set 208 n.seshagiri rao, k.kalyani & kejal khatri cubo 22, 2 (2020) of common fixed points of t and f is well ordered if and only if t and f have one and only one common fixed point in x. proof. suppose f(x) is a complete subset of x. as we know from the proof of theorem 1, the sequence {t xn} is a cauchy sequence and hence {fxn} is also a cauchy sequence in (f(x), d) as fxn+1 = t xn and t (x) ⊆ f(x). since f(x) is complete then there exists some fu ∈ f(x) such that lim n→+∞ t (xn) = lim n→+∞ f(xn) = f(u). also note that the sequences {t xn} and {fxn} are nondecreasing and from hypotheses, we have t (xn) � f(u) and f(xn) � f(u) for all n ∈ n. but t is a monotone f-nondecreasing then, we get t (xn) � t (µ) for all n. letting n → +∞, we obtain that f(u) � t (u). suppose that f(u) ≺ t (u) then define a sequence {un} by u0 = u and fun+1 = t un for all n ∈ n. an argument similar to that in the proof of theorem 1 yields that {fun} is a nondecreasing sequence and lim n→+∞ f(un) = lim n→+∞ t (un) = f(v) for some v ∈ x. so from hypotheses, it is clear that sup f(un) � f(v) and sup t (un) � f(v), for all n ∈ n. notice that f(xn) � f(u) � f(u1) � ........ � f(un) � .... � f(v). case:1 suppose if there exists some n0 ≥ 1 such that f(xn0) = f(un0) then, we have f(xn0) = f(u) = f(un0) = f(u1) = t (u). hence, u is a coincidence point of t and f in x. case:2 suppose that f(xn0) 6= f(un0) for all n then, from (3.2), we have d(fxn+1, fun+1) =d(t xn, t un) ≤α d(fxn, t xn) d(fun, t un) d(fxn, fun) + β [d(fxn, t xn) + d(fun, t un)] +γd(fxn, fun) taking limit as n → +∞ on both sides of the above inequality, we get d(fu, fv) ≤γ d(fu, fv) < d(fu, fv), since γ < 1. thus, we have f(u) = f(v) = f(u1) = t (u). hence, we conclude that u is a coincidence point of t and f in x. cubo 22, 2 (2020) contractive mapping theorems in partially ordered metric spaces 209 now, suppose that t and f are weakly compatible. let w be a coincidence point then, t (w) = t (f(z)) = f(t (z)) = f(w), since w = t (z) = f(z), for some z ∈ x. now by contraction condition, we have d(t (z), t (w)) ≤α d(fz, t z) d(fw, t w) d(fz, fw) + β [d(fz, t z) + d(fw, t w)] + γd(fz, fw) ≤γ d(t (z), t (w)) as γ < 1, then d(t (z), t (w)) = 0. therefore, t (z) = t (w) = f(w) = w. hence, w is a common fixed point of t and f in x. now suppose that the set of common fixed points of t and f is well ordered, we have to show that the common fixed point of t and f is unique. let u and v be two common fixed points of t and f such that u 6= v then from (3.2), we have d(u, v) ≤α d(fu, t u) d(fv, t v) d(fu, fv) + β [d(fu, t u) + d(fv, t v)] + γd(fu, fv) ≤γ d(u, v) < d(u, v), since γ < 1, which is a contradiction. thus, u = v. conversely, suppose t and f have only one common fixed point then the set of common fixed points of t and f being a singleton is well ordered. this completes the proof. corollary 3. let (x, d, �) be a complete partially ordered metric space. suppose that f and t are self-mappings on x, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying d(t x, t y) ≤ α d(fx, t x) d(fy, t y) d(fx, fy) + β [d(fx, t x) + d(fy, t y)] for all x, y in x with f(x) 6= f(y) are comparable and for some α, β ∈ [0, 1) with 0 ≤ α + 2β < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and {xn} is a nondecreasing sequence in x such that xn → x, then xn � x for all n ∈ n. if f(x) is a complete subset of x, then t and f have a coincidence point in x. further, if t and f are weakly compatible, then t and f have a common fixed point in x. moreover, the set of common fixed points of t and f is well ordered if and only if t and f have one and only one common fixed point in x. proof. set γ = 0 in theorem 2. corollary 4. let (x, d, �) be a complete partially ordered metric space. suppose that f and t are self-mappings on x, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying d(t x, t y) ≤β [d(fx, t x) + d(fy, t y)] + γd(fx, fy) 210 n.seshagiri rao, k.kalyani & kejal khatri cubo 22, 2 (2020) for all x, y in x for which f(x) 6= f(y) are comparable and for some β, γ ∈ [0, 1) with 0 ≤ 2β +γ < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and {xn} is a nondecreasing sequence in x such that xn → x, then xn � x for all n ∈ n. if f(x) is a complete subset of x, then t and f have a coincidence point in x. further, if t and f are weakly compatible, then t and f have a common fixed point in x. moreover, the set of common fixed points of t and f is well ordered if and only if t and f have one and only one common fixed point in x. proof. set α = 0 in theorem 2. remark 1. (i). if β = 0, in theorem 1 and theorem 2, we obtain theorem 2.1 and theorem 2.3 of chandok [28]. (ii). if f = i and β = 0, in theorem 1 and theorem 2, then we get theorem 2.1 and theorem 2.3 of harjani et al. [19]. 4 applications in this section, we state some applications of the main result for a self mapping involving the integral type contractions. let us denote τ, a set of all functions ϕ defined on [0, +∞) satisfying the following conditions: (1) each ϕ is lebesgue integrable mapping on each compact subset of [0, +∞) and (2) for any ǫ > 0, we have ∫ ǫ 0 ϕ(t)dt > 0. theorem 3. let (x, d, �) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying the following condition ∫ d(t x,t y) 0 ϕ(t)dt ≤ α ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + β ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + γ ∫ d(fx,fy) 0 ϕ(t)dt (4.1) for all x, y in x with f(x) 6= f(y) are comparable, ϕ(t) ∈ τ and for some α, β, γ ∈ [0, 1) such that 0 ≤ α + 2β + γ < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and the mappings t and f are compatible, then t and f have a coincidence point in x. similarly, we can obtain the following results in complete partially ordered metric space, by putting γ = 0 and α = 0 in an integral contraction of theorem 3. cubo 22, 2 (2020) contractive mapping theorems in partially ordered metric spaces 211 theorem 4. let (x, d, �) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying the following condition ∫ d(t x,t y) 0 ϕ(t)dt ≤ α ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + β ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt (4.2) for all x, y in x with f(x) 6= f(y) are comparable, ϕ(t) ∈ τ and where α, β ∈ [0, 1) such that 0 ≤ α + 2β < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and the mappings t and f are compatible, then t and f have a coincidence point in x. theorem 5. let (x, d, �) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t (x) ⊆ f(x) and satisfying the following condition ∫ d(t x,t y) 0 ϕ(t)dt ≤ β ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + γ ∫ d(fx,fy) 0 ϕ(t)dt (4.3) for all x, y in x with f(x) 6= f(y) are comparable, ϕ(t) ∈ τ and for some β, γ ∈ [0, 1) such that 0 ≤ 2β + γ < 1. if there exists a point x0 ∈ x such that f(x0) � t (x0) and the mappings t and f are compatible, then t and f have a coincidence point in x. corollary 5. by replacing β = 0 in theorem 3, we obtain the corollary 2.5 of chandok [28]. we illustrate the usefulness of the obtained results for the existence of the coincidence point in the space. example 1. define a metric d : x × x → [0, +∞) by d(x, y) = |x − y|, where x = [0, 1] with usual order ≤. suppose that t and f be two self mappings on x such that t x = x 2 2 and fx = 2x 2 1+x , then t and f have a coincidence in point x. proof. by definition of a metric d, it is clear that (x, d) is a complete metric space. obviously, (x, d, ≤) is a partially ordered complete metric space with usual order. let x0 = 0 ∈ x then f(x0) ≤ t (x0) and also by definition; t , f are continuous, t is a monotone fnondecreasing and t (x) ⊆ f(x). now for any distinct x, y in x, we have d(t x, t y) = 1 2 |x2 − y2| = 1 2 (x + y)|x − y| < α 4 x2y2 (x + y + xy) |x − 3||y − 3| |x − y| + β 2 x2(1 + y)|x − 3| + y2(1 + x)|y − 3| (1 + x)(1 + y) + γ 2(x + y + xy) (1 + x)(1 + y) |x − y| 212 n.seshagiri rao, k.kalyani & kejal khatri cubo 22, 2 (2020) < α x 2|x−3| 2(1+x) . y 2|y−3| 2(1+y) 2|x − y| x+y+xy (1+x)(1+y) + β [ x2|x − 3| 2(1 + x) + y2|y − 3| 2(1 + y) ] + γ 2(x + y + xy) (1 + x)(1 + y) |x − y| < α d(fx, t x) d(fy, t y) d(fx, fy) + β [d(fx, t x) + d(fy, t y)] + γd(fx, fy) then, the contraction condition in theorem 1 holds by selecting proper values of α, β, γ in [0, 1) such that 0 ≤ α + 2β + γ < 1. therefore t , f have a coincidence point 0 ∈ x. similarly the following is one more example of main theorem 1. example 2. a distance function d : x × x → [0, +∞) by d(x, y) = |x − y|, where x = [0, 1] with usual order ≤. define two self mappings t and f on x by t x = x2 and fx = x3, then t and f have two coincidence points 0, 1 in x with x0 = 1 2 . acknowledgments the authors would like to thank the editor and referees for their precise remarks which improve the presentation of the paper. cubo 22, 2 (2020) contractive mapping theorems in partially ordered metric spaces 213 references [1] s. banach, sur les operations dans les ensembles abstraits et leur application aux equations untegrales, fund. math. 3 (1922), 133–181. 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[30] x. zhang, fixed point theorems of multivalued monotone mappings in ordered metric spaces, appl. math. lett. 23 (2010), no. 3, 235–240. introduction preliminaries main results applications cubo a mathematical journal vol.20, no¯ 3, (81–95). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300081 the basic ergodic theorems, yet again jairo bochi facultad de matemáticas, pontificia universidad católica de chile jairo.bochi@mat.uc.cl abstract a generalization of rokhlin’s tower lemma is presented. the maximal ergodic theorem is then obtained as a corollary. we also use the generalized rokhlin lemma, this time combined with a subadditive version of kac’s formula, to deduce a subadditive version of the maximal ergodic theorem due to silva and thieullen. in both the additive and subadditive cases, these maximal theorems immediately imply that “heavy” points have positive probability. we use heaviness to prove the pointwise ergodic theorems of birkhoff and kingman. resumen se presenta una generalización del lema de la torre de rokhlin. el teorema ergódico maximal se obtiene como corolario. también usamos el lema de rokhlin generalizado, esta vez combinado con una versión subaditiva de la fórmula de kac, para deducir una versión subaditiva del teorema ergódico maximal obtenida por silva y thieullen. tanto en el caso aditivo como en el subaditivo, estos teoremas maximales inmediatamente implican que puntos “pesados” tienen probabilidad positiva. usamos esta pesadez para probar los teoremas ergódicos puntuales de birkhoff y kingman. keywords and phrases: maximal ergodic theorem, birkhoff’s ergodic theorem, rokhlin lemma, kingman’s subadditive ergodic theorem. 2010 ams mathematics subject classification: 37a05; 37a30. http://dx.doi.org/10.4067/s0719-06462018000300081 82 jairo bochi cubo 20, 3 (2018) 1 introduction ergodic theory is the subfield of dynamical systems concerned with measure-preserving dynamics, and it has applications throughout mathematics. its most fundamental result is the pointwise ergodic theorem proved by birkhoff [2] in 1931. an important extension was obtained by kingman [10] in 1968, and is known as the subadditive ergodic theorem. a multitude of other proofs of these basic results were obtained by many authors. one of the most popular methods of proof (especially in the case of birkhoff’s theorem) involves maximal inequalities, which have intrinsic interested by their own. in this note, we provide self-contained proofs of birkhoff’s and kingman’s theorems by means of maximal inequalities; the only prerequisite is basic measure theory. our approach has one novelty: it is based on a seemingly new extension of rokhlin tower lemma, which is another basic tool used in many constructions in ergodic theory. let us proceed directly with the precise statements and proofs. we will provide further connections with the literature in the final section 5. standing hypothesis: let (x,a,µ) be a lebesgue probability space. let t : x → x be an automorphism; this means that t and t−1 are measurable and preserve the measure µ. we assume that t is aperiodic, that is, the set of periodic points has zero measure. 2 a generalized rokhlin lemma a measurable set b ⊆ x is called sweeping if µ ( ⋃ n≥0 t n(b) ) = 1. theorem 2.1 (generalized rokhlin lemma). for any ε > 0 and any measurable function n : x → z+, there exists a sweeping set b ⊆ x such that (1) if x ∈ b and 1 ≤ i < n(x) then tix 6∈ b; (2) ∫ b ndµ > 1 − ε. the case of constant n corresponds to the classical rokhlin lemma [19]. let us introduce some useful terminology. if b ⊆ x is any set of positive measure, poincaré recurrence theorem says that a.e. x ∈ b returns to b under iteration of t. so return time function rb(x) := min{k ≥ 1 ; t kx ∈ b} is finite for a.e. x ∈ b. (2.1) if this function admits a lower bound n then the union b ∪ t(b) ∪ · · · ∪ tn−1(b) is disjoint; such a set is called a tower of height n with base b and levels b, t(b), . . . , tn−1(b). a skyscraper is a countable union of disjoint towers; its base is defined as the union of the bases of the towers. cubo 20, 3 (2018) the basic ergodic theorems, yet again 83 if b ⊆ x is a set of positive measure, the kakutani skyscraper with base b is the union of the towers ci (i = 1,2, . . .) with respective bases bi := {x ∈ b ; rb(x) = i}. as a set, it equals ⋃ n≥0 t n(b). so the set b is sweeping if and only if the kakutani skyscraper has full measure. in that case, ∫ b rb dµ = ∞∑ i=1 iµ(bi) = ∞∑ i=1 µ(ci) = 1; (2.2) this is kac’s lemma. a set b has property (1) in theorem 2.1 if and only if rb ≥ n on b. if this property is satisfied and moreover b is a sweeping set, then the following error set: e := { tix ; x ∈ b and n(x) ≤ i < rb(x) } . (2.3) has measure ∫ b (rb − n) dµ, which by kac’s lemma equals 1 − ∫ b ndµ. so we can restate theorem 2.1 in an equivalent form replacing conclusions (1) and (2) by the following ones: (i’) rb ≥ n on b; (ii’) the error set (2.3) has measure µ(e) < ε. before proving theorem 2.1, we need a preliminary result which is also used in the proof of the usual rokhlin lemma: lemma 2.2. for any integer m ≥ 2, there exists a sweeping set a with ra ≥ m. proof. since we are working on a non-atomic lebesgue probability space, we can assume x is the unit interval and µ is lebesgue measure. since t is aperiodic, the function ϕ(x) := inf{|tj(x) − x| ; 1 ≤ j < m} is positive a.e. as a consequence, any set e of positive measure contains another set f of positive measure such that rf ≥ m; indeed, it suffices to take a positive measure set g ⊆ e where ϕ is bigger than some δ > 0, and the take a positive measure subset f ⊆ g with diameter less than this δ. now consider the family f formed by the sets f ⊆ x of positive measure that satisfy rf ≥ m, partially ordered as follows: f1 ≺ f2 if f1 ⊆ f2 and µ(f2 \ f1) = 0. increasing chains are at most countable, and so by zorn’s lemma f contains a maximal element a. we claim that a is sweeping. indeed, the kakutani skyscraper s with base a is a mod 0 invariant set, and if its complement sc had positive measure then, using the fact established at the beginning, we could find a positive measure set f ⊆ sc such that rf ≥ m. then a ≺ a ∪ f, contradicting the maximality of a.1 1incidentally, this construction yields a set a whose kakutani skyscraper has all towers with heights between m and 2m − 1. 84 jairo bochi cubo 20, 3 (2018) the following proof is due to anthony quas [17]: proof of theorem 2.1. fix an integer n > 1 such that the set {x ∈ x ; n(x) ≥ n}, which we will call bad set, has measure less than ε/2. fix another integer m > 2n/ε. by lemma 2.2, there exists a sweeping set a such that ra ≥ m. consider the kakutani skyscraper with base a. the set ⊔n i=1 t −i(a), which we will call penthouse, has measure nµ(a) < n/m < ε/2, and consists on the n topmost levels of the skyscraper. let us define the set b. for each point x ∈ a, we follow the steps: 1. if x is in the penthouse, then we stop. 2. if x is in the bad set, then we replace x with t(x), and go back to step 1. 3. otherwise (i.e. x is neither in the penthouse nor in the bad set), then we put x inside b, replace x with tn(x)(x), and go back to step 1. the set b constructed in this way is clearly measurable and satisfies rb ≥ n. the associated error set (2.3) is contained in the union of the bad set and the penthouse, and therefore has measure less than ε. 3 the maximal and birkhoff’s ergodic theorems 3.1 maximal ergodic theorem as a first application of the generalized rokhlin lemma, we will give a short proof of the maximal ergodic theorem. the birkhoff sums of f are denoted as: f(n) := f + f ◦ t + · · · + f ◦ tn−1 . theorem 3.1 (wiener, yosida, and kakutani’s maximal ergodic theorem). let f ∈ l1(µ). let p be the set of x ∈ x such that f(n)(x) > 0 for some n ≥ 1. then ∫ p fdµ ≥ 0. proof. let l := x \ p be the set where all birkhoff sums are non-positive. define a function n : x → z+ as follows: if x ∈ p, let n(x) be the least n ≥ 1 such that f(n)(x) > 0, while if x ∈ l, let n(x) := 1. apply theorem 2.1 to the function n and a small ε > 0, obtaining a measurable set b whose return time function is ≥ n, and such that the error set (2.3) has measure µ(e) < ε. then: ∫ x f = ∫ b [ f(n(x))(x) + rb(x)−1∑ i=n(x) f(tix) ] dµ(x) = ∫ b f(n(x))(x) dµ(x) + ∫ e f, cubo 20, 3 (2018) the basic ergodic theorems, yet again 85 by invariance of the measure. the integrand f(n(x))(x) equals f(x) if x ∈ l and is positive otherwise, and so ∫ x f ≥ ∫ b∩l f+ ∫ e f. but ∫ b∩l f ≥ ∫ l f, since f ≤ 0 on l. so ∫ p f = ∫ x f− ∫ p f ≥ ∫ e f. since f is integrable and e can be made of arbitrarily small measure, we conclude that ∫ p f ≥ 0. 3.2 heaviness we present a corollary of the maximal ergodic theorem 3.1 that is sufficient for some applications. let a, b ∈ r. we say that a point x ∈ x is a-heavy (resp. b-light with respect to a function f if for all n ≥ 1, we have f(n)(x) ≥ an (resp. f(n)(x) ≤ bn). a measurable function f : x → [−∞,+∞] is called quasi-integrable if at least one of the functions f+ or f− is integrable (where, as usual, f+ := max(f,0) and f− := max(−f,0)), and so ∫ f = ∫ f+ − ∫ f− is defined. lemma 3.2 (heaviness). let f be a quasi-integrable function, and let a, b ∈ r. (1) if a < ∫ fdµ then the set of a-heavy points has positive measure. (2) if b > ∫ fdµ then the set of b-light points has positive measure. proof. by symmetry, it is sufficient to prove part (2). adding a constant to f, we can assume that b = 0, so ∫ fdµ is strictly negative. let l be the set of points that are 0-light w.r.t. f. first consider the case of integrable f. the set p in the statement of the maximal ergodic theorem 3.1 is the complement of l and therefore ∫ l f = ∫ x f− ∫ p f is strictly negative. in particular, µ(l) > 0, as we wanted to show. now consider the case that f is not integrable, so ∫ f = −∞. for sufficiently large k > 0, the integrable function f∗ := max(f,−k) has ∫ f∗ < 0. by the previous case, its set l∗ of 0-light points has positive measure. but l ⊇ l∗, so l has positive measure as well. 3.3 birkhoff’s pointwise ergodic theorem the conditional expectation of a quasi-integrable function f with respect to a sub-σ-algebra b ⊆ a is the b-measurable quasi-integrable function denoted e(f | b) such that ∫ b e(f | b) dµ = ∫ b fdµ for every b ∈ b. existence and essential uniqueness are immediate consequences of the radon–nikodym theorem. let i ⊆ a denote the σ-algebra of t-invariant sets. theorem 3.3 (birkhoff’s ergodic theorem). if f is a quasi-integrable function then f(n) n → e(f | i) a.e. proof. define functions g ≤ h respectively as the the lim inf and the lim sup of the sequence f(n)/n. it follows from the equality f(n) = f1 + f (n−1) ◦ t that the functions g and h are invariant. let 86 jairo bochi cubo 20, 3 (2018) ϕ := e(f | i), which by definition is also invariant. we want to prove that g = ϕ = h a.e. our plan is to prove the following inequalities: g ≥ ϕ ≥ h a.e. (3.1) in order to prove the first inequality, it is sufficient to show that for all real numbers α < β, the set eα,β := {x ∈ x ; g(x) < α < β < ϕ(x)} has zero measure; indeed in that case the functions g and ϕ coincide over the full-measure set ⋂ α,β∈q,α<β e c α,β. so let us assume by contradiction that µ(eα,β) > 0 for certain numbers α < β. let c := 1 µ(eα,β) ∫ eα,β fdµ = 1 µ(eα,β) ∫ eα,β ϕdµ ≥ β, where the equality between the integrals is due to the fact that the set eα,β is invariant. applying lemma 3.2.(1) to the measurable dynamical system (t |eα,β, µ|eα,β µ(eα,β) ), we conclude that for any real a < c, there is a positive measure set of points x ∈ eα,β that are a-heavy. such points clearly satisfy g(x) ≥ a. therefore α > a. since this holds for every a < c, we conclude that α ≥ c ≥ β. this is a contradiction, and the first inequality in (3.1) is therefore proved. the second inequality in (3.1) follows from the first one applied to −f. 4 subadditive ergodic theorems 4.1 subadditivity a sequence (an)n≥1 taking values in r ∪ {−∞} is called subadditive if an+k ≤ an + ak for all n, k ≥ 1. by a well-known exercise (sometimes called fekete lemma), the limit limn→∞ an n exists in r∪{−∞} and equals infn an n . we will denote it by: linf an n . a sequence (fn)n≥1 of measurable functions is called subadditive with respect to t if, for all n, k ≥ 1, fn+k ≤ fn + fk ◦ t n for all n, k ≥ 1. suppose that the positive part f+1 is µ-integrable. then we define the asymptotic average of the subadditive sequence by: λ := linf ∫ fn n dµ. cubo 20, 3 (2018) the basic ergodic theorems, yet again 87 4.2 subadditive kac’s formula given a sweeping set b ⊆ x, let t̂ : b → b be the first-return map. it preserves µ̂ := µ|b µ(b) , the normalized restriction of µ. kac’s formula (2.2) becomes ∫ b rb dµ̂ = 1 µ(b) . now suppose (fn)n≥1 is a subadditive sequence with respect to t : (x,µ) ←֓, and f + 1 ∈ l 1(µ). we define a sequence (f̂n)n≥1 of functions on b by: f̂n(y) := frb(y)+rb(t̂y)+···+rb(t̂n−1y)(y) . then (f̂n)n≥1 is a subadditive sequence with respect to t̂; we call it the induced subadditive sequence. note that f̂+1 ∈ l 1(µ̂). indeed, considering the kakutani skyscraper and using invariance of µ, we see that in fact ∫ b f̂+1 dµ̂ ≤ 1 µ(b) ∫ x f+1 dµ. integrability allows us to define the asymptotic average of the induced subadditive sequence, that is, λ̂ := linf ∫ b f̂n n dµ̂ . the next result relates the asymptotic averages (f̂n): of the two subadditive sequences: theorem 4.1 (subadditive kac’s formula). λ̂ = λ µ(b) . in the case of the additive sequence fn := n, the result coincides with the usual kac’s formula. in the case where the asymptotic average λ is a lyapunov exponent, theorem 4.1 appeared in [23, lemma 2.2]; see also [11, lemma 2.2]. actually theorem 4.1 is a easy consequence of kingman’s subadditive ergodic theorem 4.5; this is the argument used in [23, 11]. here we go in the opposite direction; our ultimate aim is to prove kingman’s theorem. so, to avoid circular reasoning, we should provide an independent proof of theorem 4.1. though this is possible, we won’t do it, because the following weaker version is sufficient for our purposes: lemma 4.2 (subadditive kac inequality). λ ≤ ∫ b f̂1 dµ. proof. for each positive integer k, let bk := {x ∈ b ; rb(x) = k}, and ck := ⊔k−1 j=0 t j(bk). so ck is the tower of the kakutani skyscraper with height k, and bk is its base. moreover, the bk’s form a mod 0 partition of b, and the ck’s form a mod 0 partition of x. we claim that for every m ≥ 1, the following inequality holds (the integrals being w.r.t. µ): ∫ x fm ≤ m∑ k=1 (m + 1 − k) ∫ bk f̂1 + ∞∑ ℓ=2 min(ℓ − 1,m) ∫ cℓ f+1 . (4.1) in order to prove this, fix m and, for each point x ∈ x, consider all times n1 < n2 < · · · < np in the interval {0,1, . . . ,m} such that tnjx ∈ b. if p ≥ 1 (i.e., the segment of orbit x, . . . , tmx hits 88 jairo bochi cubo 20, 3 (2018) b at least once) then we use subadditivity to bound fm(x) by the following sum: n1−1∑ i=1 f+1 (t ix) + p−1∑ j=1 f̂1(t njx) + m−1∑ i=np f+1 (t ix) . (4.2) in the case that p = 0 (i.e., there are no hits), we bound fm(x) simply by m−1∑ i=0 f+1 (t ix) . (4.3) now we analyze the terms that appear in these sums: • given k ≥ 1 and a non-periodic point y ∈ bk, can a term f̂1(y) appear in a sum (4.2)? the answer is clearly “no” if k > m. on the other hand, if k ≤ m then the term f̂1(y) does appear in a sum (4.2): namely it appears once for each x in the set {y,t−1y,. . . ,t−(m−k)y}, and there is a total of m − k + 1 appearances. • similarly we ask: given ℓ ≥ 1 and a non-periodic point z ∈ cℓ, how many times does a term f+1 (z) appear in a sum (4.2) or in a sum (4.3)? the answer is min(ℓ − 1,m). indeed, if ℓ ≥ m + 1 then the term f+1 (z) appears once for each x in the set {z,t −1z, . . . ,t−(m−1)z}, while if ℓ ≤ m then m + 1 − ℓ of these points x do not contribute with a term of the form f+1 (z) and generate a term of the previous type f̂1(z) instead. using these counts and the t-invariance of the measure µ, we obtain the claimed inequality (4.1). next, note the following two facts about series, which follow from fatou’s lemma and dominated convergence theorem, respectively: bk ∈ [−∞,+∞), ∞∑ k=1 b+k < +∞ ⇒ lim sup m→∞ 1 m m∑ k=1 (m − k + 1)bk ≤ ∞∑ k=1 bk ; cℓ ∈ [0,+∞), ∞∑ ℓ=1 cℓ < +∞ ⇒ lim m→∞ 1 m ∞∑ ℓ=2 min(ℓ − 1,m)cℓ = 0. it follows that the lim sup of the right hand side of (4.1) divided by m is at most ∫ b f̂1. but the left hand side of (4.1) divided by m tends to λ. so λ ≤ ∫ b f̂1, as we wanted to show. 4.3 subadditive maximal ergodic theorem the following result extends the maximal ergodic theorem 3.1 to the subadditive context: theorem 4.3 (silva and thieullen’s maximal subadditive ergodic theorem). let (fn) be a subadditive sequence of functions satisfying the integrability condition f+1 ∈ l 1(µ), and let λ be its asymptotic average. let h be the set of x ∈ x such that fn(x) ≥ 0 for all n ≥ 1. then λ ≤ ∫ h f1 dµ. cubo 20, 3 (2018) the basic ergodic theorems, yet again 89 actually, the result is not stated (nor named) exactly in this form by silva and thieullen, but it is a corollary of [20, lemma 2.4(a)]. using the generalized rokhlin theorem 2.1 and the subadditive kac inequality (lemma 4.2), theorem 4.3 becomes almost obvious; its proof is of course similar to the proof of theorem 3.1: proof of theorem 4.3. define a function n : x → z+ as follows: if x 6∈ h then n(x) is the least n ≥ 1 such that fn(x) < 0, while if x ∈ h then n(x) := 1. apply theorem 2.1 to the function n and a small ε > 0, obtaining a measurable set b whose return time satisfies rb ≥ n on b, and such that the error set (2.3) has measure µ(e) < ε. then (all integrals are w.r.t. µ): λ ≤ ∫ b f̂1 (by lemma 4.2) ≤ ∫ b [ fn(x)(x) + rb(x)−1∑ i=n(x) f+1 (t ix) ] dµ(x) (by subadditivity) = ∫ b fn(x)(x) dµ(x) + ∫ e f+1 (by invariance) ≤ ∫ b∩h f1 + ∫ e f+1 (by definition of n) ≤ ∫ h f1 + ∫ e f+1 (since f1 ≤ 0 on h). since µ(e) can be made arbitrarily small, so can ∫ e f+1 . therefore λ ≤ ∫ h f1. 4.4 subadditive heaviness let (fn) be a subadditive sequence of functions, and let a, b ∈ r. we say that a point x ∈ x is a-heavy, or respectively eventually b-light, with respect to the sequence (fn) if fn(x) ≥ an for all n ≥ 1, or respectively fn(x) ≤ bn for all sufficiently large n ≥ 1. the following is the subadditive version of lemma 3.2.2 lemma 4.4 (subadditive heaviness). let (fn) be a subadditive sequence of functions satisfying the integrability condition f+1 ∈ l 1(µ), and let λ be its asymptotic average. let a, b ∈ r. (1) if a < λ then the set of a-heavy points has positive measure. (2) if b > λ then the set of eventually b-light points has positive measure. 2see the discussion in the comments (section 5) about the occurrence of lemma 4.4 in the literature. 90 jairo bochi cubo 20, 3 (2018) proof. replacing fn by fn − an, we can assume that a = 0 and so λ > 0. let h be the set of 0-heavy points. by theorem 4.3, ∫ h f1 ≥ λ > 0; in particular, µ(h) > 0, proving part (1). now let us prove part (2). suppose b > λ, and take ε > 0 such that b − ε > λ. fix m ≥ 1 such that ∫ fm m < b − ε. let ψ := max(f+1 ,f + 2 , . . . ,f + m−1). by subadditivity, fn(x) ≤ ⌊n/m⌋−1∑ i=0 fm(t mix) + ψ(tm⌊n/m⌋x) −: 1 + 2 we deal with these two terms as follows: • let l be the set of points that that are (b − ε)m-light with respect to the function fm and the dynamics tm. so µ(l) > 0 by lemma 3.2.(2). note that for all x ∈ l and all n ≥ 1 we have 1 ≤ (b − ε)n. • since ψ ∈ l1(µ), by birkhoff limk→∞ 1 k ϕ ◦ tk = 0 a.e. 3 in particular, for almost every x and all sufficiently large n, we have 2 ≤ εn. it follows that for almost every x ∈ l and all sufficiently large n, we have fn(x) ≤ 1 + 2 ≤ bn. this proves part (2). 4.5 kingman’s subadditive ergodic theorem finally, let us use lemma 4.4 to prove the following fundamental result: theorem 4.5 (kingman’s subadditive ergodic theorem). let (fn) be a subadditive sequence of functions satisfying the integrability condition f+1 ∈ l 1(µ). let ϕ := linf e ( fn n ∣ ∣ ∣ ∣ i ) . then fn n → ϕ a.e. proof. if e is an invariant (or mod 0 invariant) set of positive measure, let λ(e) denote the asymptotic average of the subadditive sequence restricted to e, with respect to the restricted system (t |e, µ|e µ(e) ), that is, λ(e) := 1 µ(e) linf ∫ e fn n dµ. we claim that λ(e) = 1 µ(e) ∫ e ϕdµ. (4.4) indeed, on one hand, for every n we have e(fn n | i) ≥ ϕ and in particular ∫ fn n ≥ ∫ ϕ; taking limits we obtain the ≥ inequality in (4.4). on the other hand, by subadditivity, each e(fn n | i) can be bounded from above by e(f+1 | i), which is a integrable function. using fatou’s lemma we obtain the ≤ inequality in (4.4). 3for a simple proof of this fact that does not rely on birkhoff, see [1, lemma 2]. cubo 20, 3 (2018) the basic ergodic theorems, yet again 91 the rest of the proof of kingman’s theorem 4.5 is analogous to our proof of birkhoff’s theorem 3.3, using lemma 4.4 instead of lemma 3.2. define functions g ≤ h respectively as the the lim inf and the lim sup of the sequence fn/n. it follows from the inequality fn ≤ f1 +fn−1 ◦t that the functions g and h are sub-invariant, that is, g ≤ g ◦ t and h ≤ h ◦ t. by invariance and finiteness of the measure µ, every sub-invariant function is a.e. invariant. we must prove that g = ϕ = h a.e., and our plan is to show that: g ≥ ϕ ≥ h a.e. (4.5) assume by contradiction that the inequality g ≥ ϕ fails on a positive measure set; then there exist real numbers α < β such that the (mod 0 invariant) set eα,β := {x ∈ x ; g(x) < α < β < ϕ(x)} has positive measure. applying lemma 4.4.(1) to the system restricted to eα,β, we conclude that for any real a < λ(eα,β), there is a positive measure set of points x ∈ eα,β that are a-heavy. such points satisfy g(x) ≥ a. therefore α > a. since this holds for every a < λ(eα,β), we conclude that α is at least λ(eα,β), which by (4.4) is at least β. so α ≥ β, a contradiction. the first inequality in (4.5) is therefore proved. the second inequality is proved similarly: assume by contradiction that there are real numbers α < β such that the (mod 0 invariant) set fα,β := {x ∈ x ; ϕ(x) < α < β < h(x)} has positive measure. applying lemma 4.4.(2) to the system restricted to fα,β, we conclude that for any real b > λ(fα,β), there is a positive measure set of points x ∈ fα,β that are eventually b-light. such points satisfy h(x) ≤ a. therefore β < b. since this holds for every b > λ(fα,β), we conclude that β is at most λ(fα,β), which by (4.4) is at most β. so β ≤ β, a contradiction. this proves the second inequality in (4.5). 5 comments to summarize, our approach to prove birkhoff’s and kingman’s ergodic theorems was: generalized rokhlin lemma ⇒ maximal inequality ⇒ heaviness lemma ⇒ ergodic theorem in the subadditive case, the first arrow also relies on a generalization of kac’s lemma. these intermediate results are also interesting by themselves. this path to the ergodic theorems is not the shortest one4, but we hope that it has a gentle slope. 4the proof in [7, p. 136] is unbeatable. 92 jairo bochi cubo 20, 3 (2018) there are many extensions and variations of rokhlin lemma (see [22, 12]), but nevertheless theorem 2.1 appears to be new. there are other proofs of the maximal ergodic theorem 3.1 using towers: see [16, p.27ff]. garsia’s celebrated short proof uses a different idea; see e.g. [16, p.75ff].5 quoting steele [21], the proof has become a textbook standard, but the inequality and its proof are widely regarded as mysterious. it is our hope that the generalized rokhlin lemma makes the maximal ergodic theorem more plain to see6. silva and thieullen deduce their subadditive ergodic theorem [20, lemma 2.4(a)] (which implies theorem 3.1 in this note) from a pointwise inequality. this type of proof is probably the shortest in this case, and appears in other proofs of the ergodic theorems [8, 4, 9, 1].7 while kac’s formula and rokhlin lemma also hold for non-invertible t (see [13, 3]), it turns out that the generalized rokhlin lemma introduced here is false for non-invertible t: see proposition 5.1 below. on the other hand, we can easily drop the invertibility assumption in the heaviness lemmas 3.2 and 4.4, by considering the natural extension of t [16, p.13]. the “heaviness” terminology comes from ralston [18] (who used it in a slightly different context). lessa [14] also uses heaviness (without this terminology) to prove birkhoff’s and kingman’s theorems. lessa’s work is perhaps the first place where the statement of lemma 4.4 appears explicitly. lemma 4.4.(1), which as we have seen follows immediately from silva–thieullen’s result, is also contained in a deeper result by karlsson and margulis, namely [6, lemma 4.1]. lemma 4.4.(2) is [14, teorema 3.10]. neither karlsson–margulis [6] nor lessa [14] use maximal inequalities to obtain heaviness; instead they use riesz’ combinatorial lemma about leaders. see also karlsson [5] for a related approach. for another version of heaviness in a subadditive context, see [15, p. 144ff]. we conclude with the following example, also due to quas, which shows that invertibility of t is necessary for the validity of theorem 2.1: proposition 5.1 (quas). let t the shift on the space x := {1,2}n, and let µ be the (1 2 , 1 2 )-bernoulli measure. consider the function n(x) := x0. then for any measurable set b ⊆ x such that rb ≥ n on b, we have ∫ b (rb − n) dµ ≥ 1 9 . proof. if µ(b) < 4 9 , then by kac’s formula ∫ b (rb − n)dµ = 1 − ∫ b ndµ ≥ 1 − 2µ(b) > 1 9 . so let us assume that µ(b) ≥ 4 9 . 5as made clear by steele, garsia’s proof boils down to a pointwise inequality involving a coboundary: see inequality (3) in [21]. 6in the case of finite measure, at least. 7incidentally, as remarked by karlsson [5], garsia’s argument has a minor subadditive extension which is unfortunately insufficient to prove kingman’s theorem. cubo 20, 3 (2018) the basic ergodic theorems, yet again 93 consider s := [1] ∩ t−1([2]) ∩ t−2(b), so that µ(s) = 1 4 µ(b) ≥ 1 9 . note that t−1(s) ⊆ bc, as a consequence of the hypothesis rb ≥ n. now let f := 1b · (rb − n). we claim that for any x ∈ s that returns to s in finite time n := rs(x), the birkhoff sum f (n)(t(x)) = f(tx) + · · · + f(tnx) is at least 1. indeed, consider the biggest k ∈ {2,3, . . . ,n} such that tk(x) ∈ b; such k exists because n ≥ 2 and t2(x) ∈ b. since tn+1(x) 6∈ b and tn+2(x) ∈ b, we have rb(t k(x)) = n + 2 − k. now, if k = n then n(tk(x)) = 1, while if k < n then rb(t k(x)) ≥ 3. in either case, f(tk(x)) ≥ 1, proving the claim. it follows the asymptotic average of f along almost every orbit is at least the frequency that the set s is visited. since µ is ergodic, this means that ∫ fdµ ≥ µ(b) ≥ 1 9 , as we wanted to prove. acknowledgements. i initially proved theorem 2.1 under the assumption that n is bounded; this weaker result is sufficient for the proof of maximal inequalities, but an extra step is required. i thank anthony quas for removing this assumption, for telling me that invertibility of t cannot be relaxed (proposition 5.1), and for several comments and corrections. i thank godofredo iommi for stimulating conversations. finally, i thank the referee for suggestions that improved the presentation. 94 jairo bochi cubo 20, 3 (2018) references [1] avila, a.; bochi, j. – on the subadditive ergodic theorem. manuscript, 2009. www.mat.uc.cl/˜jairo.bochi/docs/kingbirk.pdf [2] birkhoff, g.d. – proof of the ergodic theorem. proc. nat. acad. sci. usa, 17 (1931), 656–660. 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[14] lessa, p. – teoremas ergódicos en espacios hiperbólicos. master’s thesis, universidad de la república, montevideo (2009). www.cmat.edu.uy/˜lessa/masters.html http://www.mat.uc.cl/~jairo.bochi/docs/kingbirk.pdf http://www.ams.org/mathscinet-getitem?mr=1843737 http://www.ams.org/mathscinet-getitem?mr=0687641 http://www.ams.org/mathscinet-getitem?mr=3644014 http://www.ams.org/mathscinet-getitem?mr=1729880 http://www.ams.org/mathscinet-getitem?mr=1326374 http://www.ams.org/mathscinet-getitem?mr=0682312 http://www.ams.org/mathscinet-getitem?mr=2306205 http://www.ams.org/mathscinet-getitem?mr=0254907 http://www.ams.org/mathscinet-getitem?mr=1178949 http://www.ams.org/mathscinet-getitem?mr=2087594 http://www.ams.org/mathscinet-getitem?mr=0797411 http://www.cmat.edu.uy/~lessa/masters.html cubo 20, 3 (2018) the basic ergodic theorems, yet again 95 [15] morris, i.d. – mather sets for sequences of matrices and applications to the study of joint spectral radii. proc. lond. math. soc. 107 (2013), no. 1, 121–150. [16] petersen, k. – ergodic theory. corrected reprint of the 1983 original. cambridge studies in advanced mathematics, 2. cambridge univ. press, cambridge, 1989. [17] quas, a. – mathoverflow.net/q/279635 [18] ralston, d. – heaviness: an extension of a lemma of y. peres. houston j. math. 35 (2009), no. 4, 1131–1141. [19] rokhlin, v. – a “general” measure-preserving transformation is not mixing. (russian) doklady akad. nauk sssr 60, (1948), 349–351. [20] silva, c.e.; thieullen, p. – the subadditive ergodic theorem and recurrence properties of markovian transformations. j. math. anal. appl. 154 (1991), no. 1, 83–99. [21] steele, j.m. – explaining a mysterious maximal inequality – and a path to the law of large numbers. amer. math. monthly 122 (2015), no. 5, 490–494. [22] weiss, b. – on the work of v. a. rokhlin in ergodic theory. ergodic theory dynam. systems 9 (1989), no. 4, 619–627. [23] wojtkowski, m. – invariant families of cones and lyapunov exponents. ergodic theory dynam. systems 5 (1985), no. 1, 145–161. http://www.ams.org/mathscinet-getitem?mr=3083190 http://www.ams.org/mathscinet-getitem?mr=1073173 https://mathoverflow.net/q/279635 http://www.ams.org/mathscinet-getitem?mr=2577147 http://www.ams.org/mathscinet-getitem?mr=0024503 http://www.ams.org/mathscinet-getitem?mr=1087960 http://www.ams.org/mathscinet-getitem?mr=3352814 http://www.ams.org/mathscinet-getitem?mr=1036900 http://www.ams.org/mathscinet-getitem?mr=0782793 introduction a generalized rokhlin lemma the maximal and birkhoff's ergodic theorems maximal ergodic theorem heaviness birkhoff's pointwise ergodic theorem subadditive ergodic theorems subadditivity subadditive kac's formula subadditive maximal ergodic theorem subadditive heaviness kingman's subadditive ergodic theorem comments ((jullm jl j\foth r:rno hool jool'1w i fo/ h, /\"' 2, ('1 · 51} . . ·tu91u t ijq(j{i. global solutions of yang-mili equat ion qike n g lu 1 l11 stit.nlt' oí ~ l uthl'lllhtk:-;, a<·1hll'll\y uf t<.lill.il1 •1m1ti1'.s 1utcl )"'-lt•m sril'llrt· hirn .. 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(5) whc rc a. b are 2 x 2 co111plcx urnlric~ 11ml satis fy the couditio n a 1 a + 131 ij = 1, a 1 ij 8 1 a = o (g) with a' . b' d enotcd the co r11 ple x conj ugute n nd transpose: matrices of a,8 respcc· 1iw ly associnled lo the t.rn nsfo ri nnlion (5) thcrc is l.\ l(2. c ) matrix 'll.,.(.") = dcl( a + h,8)1(a + 11, 8 )1 • (7) inl't' ki .._,. crnmsltlvc undcr lhc group {i formcd from the 1r1\1ts formntions (5). lh•' rom':'ponding {ijlr(.r) }re.q urc thc trnn~ition func tions o f thc ll{lt11ral principnl b11ndl1 ri l . l(2, )). \ve opply t)i(' following thcorcm(c r ijj throrcm 2..t.2) clobnl sol11t.io11 s o í ynngmills ec1111u.io11 •19 thoor e m a. lf 9)1 l.'l o .{-rlm11•11sfo1"'l f~orr:u t.z ·"11¡,, mmufold. the n (fl~ = '72if,.17v i~ a !11(2. c )-t:o11ncc t1on 011 thc principal bmulle p(~ul , sl(2 , c )), whcrc fl.111l d s 2 = .'ji k1j:r.i tl:é = 1¡111,w" wb i.~ lh c !~ore.ni;. rnctrft· wit11 w" = e.y' )rl!1;i tmd "fu) 1ml:isfyi,1g ~ .. )e~b) = ág. s i11cc u¿ = oo n11d o~ = 0 0 (0· = l , 2, 3), tl1c !1l(2, c )-conncction in thcornm a cnri b u wril,tcn into (8) w! wru t.hu grcck iud iccii run fr0t 11 1t.o31.111d {a 0 ,'i.a 0 } 0 ::::i1 , 2 ,:i is l\ bnsiil oí t he lic ulgubrn lil(2, c ) oí s j..,(2, c ) nud {'ia0 }c,.=1,2,:j is t.lmt. o í t.he lie nlgcbrn s u(2) oí su(2). siuce {uauu1 } 0 .1,2.3 íor nny u e su(2) is swl n bns is oí lhc vcclor spnce geuern t.ed by {u0 } 0 = 1•2 ,3 , nccordin g l,o lhe red uct.io 11 t.heore m o í con neclio ns, (y) is/\ o·n(2 )-co1111cc io n on t hc rcd 11ced p rit icitnil bumlle p 1 (9jt. su(2)) or p (9jt, s /_,(2 , c}) . in cnsc t !lnt. 9jl = af nnd d.~2 is dofincd by (4 ), aj is cxnctly cxprcssed by ( 1). !!~ ru111 uit1s l:o provc t hot. such a 1 sllt.isflcs tfo,: y11 11g-ivlilh; cq\ullion (2). in fnct., lll:corcli11t1. t.o ( ! ) t ite •lc mcnt.s o í t.hc nial.rices aj(j = o, l , 2, 3) m e llll odd íunct.ions o f ~,) . ohvio 11s ly [aj(.t))r=o =o. hc11 ec itll olmucuts oí f j.1: nrc cvcu fonclions oí :jj. t hcrcforu 11 11 d 1::mcuts o í f, .1::/ nrc odd fu11ct io11s o f :r,j und con$1.-quc11lly jf (r)jk:il.r.=o = o. s i11ce mis lntns.ili,·efl l uuder 1.lie g ro up q, for uuy poi nl :r0 of :1 t here is t\i. jcosl n lrnnsformnt.ion (5) which cmries t lie poi11 t. :t = xo to t.he poi nl y= o. sincc bot.h !}jl· 011d f ja·:i nre c0\'8.til\nl ullder t.he lhe lro.ns fo rm otio n (5), o= [ f}.r,p {j:r,q fj:r.rl (y),. ,,j •• o = ~lr(")f('"),,.,,..'llr{r)1 8yo 8y' 8y' z~z,' whi uh im pl k-s thnt jf (.:r)1111,r].r.m:r.u = o. siucu :ro c1m be 1111 l\rbitrnry point o f m, wc lnwu f(:i.:)~1 =o nnd obvious ly il s1.1l. ii! lics t.hc yung-~ l ills cquotion gt·l p jl·:i = o. since m is lhe compncted minkowski spucc m nnd lhc yru1g·~lills eq uut.ion is confo r111 nl inv8.tinnt, the i:.11 (2)-co1111ect.io 11 aj dcfüicd by{! ) nlso satis fles 1.hc yn 11gj\i!il ls cqu ntion "( o f f ) '1 f).r.1 ' 11;+ a 1 · jkf ,.1,. a 1 = o i11 l1hu mi ukow"ski s pucc m . q ikcug: lu ano thc r :.0lu1io n o f 11bc y1111g-t-.11i lls eq1 mt.io11 on m cnu be d<.'(lucc in cuse thut. ds2 is d e f111ccl by {11 ). this is n tat ic solut ion bec:lu1se a ; tu·e nol dep e nd 011 x 0 . o ur mc thod cn n nlso be appl ierl to cons truct. ::;olutions of t hc yungmills l'flllb· tion on 1hc d c-sit.t.c r s p11ccs d eli11orl by oirnc15j. thc d c-sittc r spnc of cosmology con.:;tntll ;\ is demote t-1idc'e el ( a) b, iuvmiu nl 11wlor lhu trnnsforumtion ' = a{u)~~/y. 11 l a111,,,fjl'a'' k ' (15) bnou.o¡jy .. whcn a = o, {15) is 11 1,orcntz trnnsformntio n. the me1ric ds~ is in\lnrinnt u ndrr tll<' 1mn. .. for111ulio11 {15) 11ud ds( a) is trnm1lti\'c und r thc g·rou p of 1111 s11ch trul."fonnntmll'l. in fncl t his is thc group 50(2, 3)(in ca.«e a < o) o r t.hc group ~ot l. -l)(m c".(l.'<' a > o) 1lm1, uc1s · o11 r/s(a) . ~loroovcr 1hc mc lril' el.¡~ undc.r llw coordillftt c 1nuu.fo r111nt io11 (16) .~pl . lolml 'e>lu tio us o í y1 11 1g-~ l il\~ e:q untion 51 i ~ d1u11 gcd to bl• ( 17) wlii cl 1 is n co u~ rmal flnt. me t ric. th nt th e d e-sitt r s pnc nre s pi n urn nifolds is imp lied in tht> dirnc' oonstru c ll on or th c '¡ji,,4 wnve l"ql1ation15j. app ly ing t heorcm a nncl the theorem of recl uct io11 of coiincc ti o ns, wc obl ai n t. hc !lll(2)-co n ncc ti on a ,= -~1\ ( i t\~111u1 11 1'11.'1 ) 1 (&~ 11 ti 6:u°')6!}'1o1 , ( 18) wh k h im ti s fi th '{nng-1\ lills cqu nt io n bcc nu se t. hc el mcnts oí , 8ít' odd fun ct.i oua of " ' · w(• co 11 cl11 cil' in gc nr rnl 1 l111t. t li coro m b. /f 9)1 1.~ fl 4·tfimtmsimwl .~pi 1 l maiujoltl anti pda"l('. 0, . . . , xn > 0} and c(u, v) = min 1≤i≤n viui, ([14]), cubo 22, 1 (2020) level sets regularization with application to optimization problems 139 (2) (u, d) a metric space, α > 0, v = u and c(u, v) = −αd(u, v), ([6]), (3) (u, d) a metric space, v = u×]0, +∞[ and c(u, (v, α)) = −αd(u, v), ([6]), (4) u = v = rn, 0 < α ≤ 1, β > 0 and c(u, v) = −β‖u − v‖α, ([7]), (5) u a topological space, v = c (u,r), space of countinuous real functions on u and c(u, v) = v(u), ([5],[10]). given a function h : u → r, the following notation and definitions will be needed: dom h = {u ∈ u | h(u) < +∞}, the effective domain of h, [h ≤ t] := {u ∈ u | h(u) ≤ t}, the t-sub-level set of h(or level set of h in short). given a subset a of u, we define its indicator function ia by ia(u) = 0 if u ∈ a and ia(u) = +∞ if u ∈ u \ a. following the terminology introduced in [9] we will also use the valley function va of a defined by va(u) = −∞ if u ∈ a and va(u) = +∞ if u ∈ u \ a. 3 γ−regularization of functions and hull of sets 3.1 γ−regularization of functions the notion of continuous affine functions can be generalized by those of c−elementary functions. in this work, we call c−elementary function on u (resp. on v ), the function of the form c(., v) − r (resp. c(u, .) − r) with v ∈ v (resp. u ∈ u) and r ∈ r. the upper hull (i.e., the supremum) of a family of c−elementary functions is called c-regular. we denote by γc(u), the set of c−regular functions defined on u. we call c-hull or γc−regularization of h : u → r, the greatest c−regular minorant of h. this function is denoted by hγc. it is well known ([8]) that hcc = hγc, for each h : u → r. (3.1) remark 3.1. the equality (3.1) is still valid if the coupling function is an extended real-valued function. in this case, one must interpret the conjugate hc as follows hc(v) = − inf u∈u {h(u) − c(u, v)}, with the usual conventions (+∞) − (+∞) = (−∞) − (−∞) = +∞. there exists an equivalent approach to generalized convex duality in terms of φ -convexity [2], which consists of working with a set u and a class of functions φ ⊂ r u . 140 moussa barro & sado traore cubo 22, 1 (2020) 3.2 hull of sets let p be a nonvoid subset of r. the following definition generalizes the notion of half space. definition 1. we call (c,p)−elementary subset of u any subset of uof the form {u ∈ u | r − c(u, v) ∈ p}, where (v, r) ∈ v × r. we note it by epv,r. note that, if p = r, then epv,r = u for any (v, r) ∈ v × r. in this case, the only (c,p)−elementary subset of u is u itself. the (c,p)−elementary subsets of u allow us to define a notion of hull of a subset a of u. definition 2. the (c,p)−hull of a ⊂ u is the intersection of all (c,p)−elementary subsets of u containing a. the (c,p)−hull of a is denoted by 〈a〉c,p. remark 3.2. if there is not (c,p)−elementary subset of u containing a, then 〈a〉c,p = u by convention. proposition 1. if p 6= r, then 〈∅〉c,p = ∅. proof let s ∈ r \ p. assume 〈∅〉c,p 6= ∅. let a ∈ 〈∅〉c,p. then r − c(a, v) ∈ p for any (v, r) ∈ v × r. in particular s = (s + c(a, v)) − c(a, v) ∈ p, absurd. it follows from the definition of 〈.〉c,p that, for each a ⊂ u, and for each u ∈ u, one has u /∈ 〈a〉c,p ⇐⇒ ∃(v, r) ∈ v × r : a ⊂ e p v,r and r − c(u, v) /∈ p. (3.2) by definition 1, one has a ⊂ 〈a〉c,p , for any a ⊂ u. moreover, if a ⊂ b then 〈a〉c,p ⊂ 〈b〉c,p. therefore, 〈〈a〉c,p〉c,p = 〈a〉c,p, ∀a ⊂ u. we deduce that 〈.〉c,p is an algebraic closure operator. definition 3. a subset a of u is said to be (c,p)−regular if a = 〈a〉c,p. we denote rc,p(u), the set of all (c,p)-regular subsets of u. observe that (c,p)−elementary sets are (c,p)−regular. more generally, any intersection of (c,p)−regular subsets is (c,p)−regular and the (c,p)−regular hull of a ⊂ u coincides with the intersection of all (c,p)−regular subsets of u containing a. in what follows, we will use the following values for p : p1 = r+ := [0, +∞[, p2 = r ∗ + := ]0, +∞[, p3 = r ∗ := r \ {0} and p4 = {0}. for i = 1, 2, 3, 4, 〈.〉c,i := 〈.〉c,pi for short. for i = 1, 2, 3, the set 〈a〉c,i can be explained as follows: proposition 2. for any a ⊂ u, one has: 〈a〉c,1 = {u ∈ u | c(u, v) ≤ sup a∈a c(a, v), ∀v ∈ v }, (3.3) 〈a〉c,2 = {u ∈ u | ∀v ∈ v, ∃a ∈ a | c(u, v) ≤ c(a, v)}, (3.4) 〈a〉c,3 = {u ∈ u | ∀v ∈ v, ∃a ∈ a | c(u, v) = c(a, v)}. (3.5) cubo 22, 1 (2020) level sets regularization with application to optimization problems 141 proof by (3.2), one has: a /∈ 〈a〉c,1 ⇐⇒ ∃(v, r) ∈ v × r : a ⊂ [c(., v) ≤ r] and r < c(a, v) ⇐⇒ ∃(v, r) ∈ v × r : sup u∈a c(u, v) ≤ r < c(a, v) ⇐⇒ ∃v ∈ v : sup u∈a c(u, v) < c(a, v). thus, a ∈ 〈a〉c,1 ⇐⇒ ∀v ∈ v, c(a, v) ≤ sup u∈a c(u, v), and (3.3) holds. a /∈ 〈a〉c,2 ⇐⇒ ∃(v, r) ∈ v × r : a ⊂ [c(., v) < r] and r ≤ c(a, v) ⇐⇒ ∃(v, r) ∈ v × r : c(u, v) < r ≤ c(a, v), ∀u ∈ a ⇐⇒ ∃v ∈ v : c(u, v) < c(a, v), ∀u ∈ a. thus, a ∈ 〈a〉c,2 ⇐⇒ ∀v ∈ v, ∃u ∈ a : c(a, v) ≤ c(u, v), and (3.4) holds. a /∈ 〈a〉c,3 ⇐⇒ ∃(v, r) ∈ v × r : a ⊂ [c(., v) 6= r] and r = c(a, v) ⇐⇒ ∃v ∈ v : c(u, v) 6= c(a, v), ∀u ∈ a. thus a ∈ 〈a〉c,3 ⇐⇒ ∀v ∈ v, ∃u ∈ a : c(a, v) = c(u, v), and (3.5) holds. remark 3.3. observe that one cannot remove the real parameter r in the definition of 〈a〉c,4. example 3.4. we observe the situation in topological vector case. assume u is a topological vector space with topological dual v and c the standard coupling function. the c−elementary functions are affine continuous functions, and we have: 1. (c,p1)−elementary sets are ∅, u and closed half spaces. moreover, if u is locally convex, then by hahn-banach separation theorem and (3.3), 〈a〉c,1 = conva, the closed convex hull of a. 2. (c,p2)−elementary sets are ∅, u and half open spaces. the (c,p2)−hull of a subset of u is its evenly convex hull ([4],[7],. . . ). 3. (c,p3)−elementary sets are ∅, u and complementary set of closed hyperplane. the (c,p3)−hull of a subset of u is its evenly co-affine hull ([15]). observe that ([15], corollary 2.2) a is evenly convex if and only if a is evenly co-affine and convex. 4. (c,p4)−elementary sets are ∅, u and closed hyperplane. moreover, if u is locally convex, then by the hahn-banach separation theorem and (3.2), the (c,p4)−hull of a non empty subset of u is its closed affine hull. proposition 3. let p and q be two nonvoid subsets of r. assume that any (c, p)-elementary set is (c, q)-regular. then, 〈a〉c,q ⊂ 〈a〉c,p , ∀a ⊂ u. 142 moussa barro & sado traore cubo 22, 1 (2020) proof let a /∈ 〈a〉c,p . by definition, there exists an (c, p)-elementary set e such that a ⊂ e and a /∈ e. since e is also (c, q)-regular, it follows from (3.2) that a /∈ 〈a〉c,q, and we are done. corollary 3.5. for any a ⊂ u, one has: 〈a〉c,1 ⊃ 〈a〉c,2 ⊃ 〈a〉c,3 and 〈a〉c,4 ⊃ 〈a〉c,3. proof let v ∈ v and r ∈ r, it is obvious that {u ∈ u | c(u, v) ≤ r} = ⋂ s>r {u ∈ u | c(u, v) < s}. consequently, any (c,p1)−elementary subset is (c,p2)−regular, and by proposition 3, one has 〈a〉c,1 ⊃ 〈a〉c,2 for any a ⊂ u. it is easy to verify that: {u ∈ u | c(u, v) < r} = ⋂ s≥r {u ∈ u | c(u, v) 6= s}, {u ∈ u | c(u, v) = r} = ⋂ s6=r {u ∈ u | c(u, v) 6= s}, therefore, 〈a〉c,2 ⊃ 〈a〉c,3 ⊂ 〈a〉c,4. we derive from corollary 3.5, the following comparison between the sets rc,pi(u): rc,p1(u) ⊂ rc,p2(u) ⊂ rc,p3(u) and rc,p4(u) ⊂ rc,p3(u). (3.6) remark 3.6. observe that 1. p1 ⊃ p2 and rc,p1(u) ⊂ rc,p2(u). 2. p3 ⊃ p2 and rc,p3(u) ⊃ rc,p2(u). 3. rc,p1(u) ⊂ rc,p3(u) whereas p1 and p3 are not comparable in the sense of inclusion. 4. in particular, in the case of locally convex vector space, we recover the fact that every closed convex subset is evenly convex. proposition 4. assume that the coupling function c satisfies the property: ∀v ∈ v, ∃w ∈ v | − c(., v) = c(., w). (3.7) then, one has: rc,p4(u) ⊂ rc,p1(u) ⊂ rc,p2(u) ⊂ rc,p3(u). (3.8) cubo 22, 1 (2020) level sets regularization with application to optimization problems 143 proof by (3.6), we only need to show that rc,p4(u) ⊂ rc,p1(u). let (v, r) ∈ v × r. we have [c(., v) = r] = [c(., v) ≤ r] ⋂ [−c(., v) ≤ −r]. by assumption on the coupling function, there exists w ∈ v such that −c(., v) = c(., w). consequently, [c(., v) = r] = [c(., v) ≤ r] ⋂ [c(., w) ≤ −r]. we conclude with proposition 3. example 3.7. assume that u = v = rn, and coupling function c is defined by c(u, v) = ‖u − v‖, where ‖.‖ is the euclidean norm. the non trivial (c,p4)−elementary sets are spheres (not convex) whereas the non trivial (c,p1)−elementary sets are closed balls (closed convex). in this case, rc,p1(u) and rc,p4(u) are not comparable. observe that in this case assumption (3.7) does not hold. proposition 5. for any a ⊂ u, one has 〈a〉c,1 = [i γc a ≤ 0]. proof by (3.2), one has a /∈ 〈a〉c,1 ⇐⇒∃v ∈ v : i c a(v) < c(a, v) ⇐⇒0 < sup v∈v {c(a, v) − ica(v)} ⇐⇒0 < iγca (a) ⇐⇒0 /∈ [ iγca ≤ 0 ] . thus 〈a〉c,1 = [ iγca ≤ 0 ] . the following result makes the link between hull of set and γ-regularization of function by means of indicator function. theorem 3.8. assume that the coupling function c satisfies the condition: ∀ (v, β) ∈ v × r∗+, ∃v̄ ∈ v | βc(., v) = c(., v̄). (3.9) then for each a ⊂ u such that dom ica 6= ∅, one has: i γc a = i〈a〉c,1. proof let b ∈ u. (1) assume that b /∈ 〈a〉c,1. by (3.3), there exists (v, ǫ) ∈ v ×r ∗ + such that c(b, v)−sup a∈a c(a, v) ≥ ǫ. from (3.9), one has: ∀n ≥ 1, ∃vn ∈ v : nc(., v) = c(., vn). 144 moussa barro & sado traore cubo 22, 1 (2020) consequently, nǫ ≤ c(b, vn) − sup a∈a c(a, vn) = c(b, vn) − i c a(vn) ≤ i γc a (b), ∀n ≥ 1. therefore iγca (b) = +∞. (2) assume that b ∈ 〈a〉c,1. by (3.3), one has c(b, w) − sup a∈a c(a, w) ≤ 0, ∀v ∈ v. thus iγca (b) = sup v∈v { c(b, v) − sup a∈a c(a, v) } ≤ 0. let v ∈ dom ica. by (3.9), one gets ∀n ≥ 1, ∃ vn ∈ v : 1 n c(., v) = c(., vn). consequently, 1 n ( c(b, v) − sup a∈a c(a, v) ) = c(b, vn) − sup a∈a c(a, vn) ≤ i γc a (b), ∀n ≥ 1. therefore, 0 = lim n→+∞ 1 n ( c(b, v) − sup a∈a c(a, v) ) ≤ iγca (b), and finally, iγca (b) = 0. remark 3.9. assumption (3.9) is satisfied by coupling functions (1), (3) and (5) of example 2.2. coupling functions (2) and (4) of the same example do not satisfy assumption (3.9). 4 level set regularization of functions in this section, we introduce a notion of (c,p)−level set regularization of extended real-valued functions. we show that this level set regularization can be interpreted as bi-conjugacy relative to a couple of dual polarities by decomposition of the closure operator. we then give some other expressions of these regularizations. 4.1 definitions and properties definition 4. a function h : u → r is said to be (c,p)−level regular if all of its sub-level sets are (c,p)−regular, i.e 〈[h ≤ r]〉c,p = [h ≤ r], ∀r ∈ r. cubo 22, 1 (2020) level sets regularization with application to optimization problems 145 we denote nc,p(u), the set of (c,p)−level regular functions defined on u to r. observe that this set contains the constant function −∞. proposition 6. the set nc,p(u) is closed under pointwise suprema, i.e given (hi)i∈i a family of (c,p)−level regular functions,then h := sup i∈i hi is (c,p)−level regular. proof let r ∈ r. since [h ≤ r] = ∩i∈i[hi ≤ r], the conclusion follows from the fact that any intersection (c,p)−regular sets is (c,p)−regular. we define the (c,p)−level set regularization of an extended real-valued function as follows. definition 5. the (c,p)−level set regularization of a function h : u → r is the greatest (c,p)−level regular minorant of h. this function is denoted by h〈〉c,p. example 4.1. assume u is topological vector space with topological dual v and c the standard coupling function. (c,p2)−level regular functions are evenly quasi-convex functions. moreover, if u is locally convex then (c,p1)−level regular functions are lower semi-continuous quasi-convex functions. example 4.2. assume that u is a metric space, v = c (u,r) a space of continuous functions from u to r, and c : u × v → r defined by c(u, v) = v(u). a function h : u → r ∪ {+∞} is (c,p1)−level regular if and only if h is lower semi-continuous ([3], corollary 11). proposition 7. any c−elementary function is (c,pi)−level regular for i = 1, 2, 3. more precisely, one has γc(u) ⊂ nc,p1(u) ⊂ nc,p2(u) ⊂ nc,p3(u) and nc,p4(u) ⊂ nc,p3(u). proof let h := c(., v) − r an c−elementary function. for any t ∈ r, we have [h ≤ t] = {u ∈ u | t + r − c(u, v) ≥ 0}, which is obviously (c,p1)−elementary set. therefore γc(u) ⊂ nc,p1(u). the other inclusions follow from (3.6). remark 4.3. c−elementary functions are not necessary (c,p4)−level set regular functions. for example, in the topological case, one cannot write a half space as an intersection of affine hyperplanes. example 4.4. let n ≥ 1, an integer number. assume that u = v = rn and c a standard scalar product of rn. let h1, h2 : r n → j0; nk two functions defined by h1(x) = { 0 if x = 0 max{i ∈ j1, nk | xi 6= 0} if x 6= 0, 146 moussa barro & sado traore cubo 22, 1 (2020) h2(x) = { 0 if xi 6= 0, ∀i ∈ j1, nk max{i ∈ j1, nk | xi = 0} else . for any r ∈ r, we have [h1 ≤ r] =      ∅ if r < 0 {x ∈ rn | xi+1 = . . . = xn = 0} if i ≤ r < i + 1, i = 0, 1, . . . , n − 1 r n if n ≤ r, [h2 ≤ r] =      ∅ if r < 0 {x ∈ rn | xi+1 6= 0, . . . , xn 6= 0} if i ≤ r < i + 1, i = 0, 1, . . . , n − 1 r n if n ≤ r. it is clear that: (1) h1 is (c,p4)−level regular. in particular, h1 ∈ nc,pi(u), for i = 1, 2, 3, 4. (2) h2 is (c,p3)−level regular but not (c,p2)−level regular since [h2 ≤ n−1] = {x ∈ r n | xn 6= 0} is not convex. example 4.5. let u = v = r, c the standard product of r. the indicator function of r∗,i r ∗ is (c,p3)−level regular but not quasi-convex. 4.2 decomposition of 〈〉c,p let us consider a map ∆c,p : 2 u → 2v ×r defined by: ∆c,p(a) := {(v, r) ∈ v × r | a ⊂ e p v,r}, (4.1) which, we simply denote ∆ in the sequel. given (ai)i∈i a family of subsets of u, we have ∆ ⋃ i∈i ai := { (v, r) ∈ v × r | ⋃ i∈i ai ⊂ e p v,r } = { (v, r) ∈ v × r | ai ⊂ e p v,r, ∀i ∈ i } = ⋂ i∈i { (v, r) ∈ v × r | ai ⊂ e p v,r } = ⋂ i∈i ∆ai. therefore ∆ is said to be a polarity ([16]). we associate to ∆, its dual polarity ∆∗ : 2v ×r → 2u defined by ∆∗(b) = ⋃ {a ∈ 2u | b ⊂ ∆(a)}. (4.2) observe that for each (v, r) ∈ v × r and for each u ∈ u, one has u ∈ ∆∗(v, r) ⇐⇒ (v, r) ∈ ∆(u) ⇐⇒ u ∈ epv,r, (4.3) cubo 22, 1 (2020) level sets regularization with application to optimization problems 147 therefore ∆∗(v, r) = epv,r. since ∆ ∗ is a polarity, then we have for each b ⊂ v × r, ∆∗(b) = ∆∗ ( ⋃ (v,r)∈b {(v, r)} ) = ⋂ (v,r)∈b ∆∗(v, r) = ⋂ (v,r)∈b epv,r. (4.4) the operator 〈〉c,p can be decomposed as follows. proposition 8. for any a ⊂ u, we have (∆∗ ◦ ∆)(a) = 〈a〉c,p. proof given a ⊂ u, one has (∆∗ ◦ ∆)(a) = ∆∗({(v, r) ∈ v × r | a ⊂ epv,r}) = ⋂ a⊂epv,r epv,r = 〈a〉c,p. 4.3 conjugacy associated to polarities ∆ and ∆∗ ([16, 17]) the conjugate of a function h : u → r relative to the polarity ∆ is the function h∆ : v × r → r given by h∆(v, r) := sup u/∈∆∗(v,r) −h(u) = sup u/∈ep v,r −h(u). (4.5) analogously, the conjugate of a function k : v × r → r relative to the polarity ∆∗ is defined by k∆ ∗ (u) := sup (v,r)/∈∆(u) −k(v, r) = sup u/∈epv,r −k(v, r). (4.6) thus, the bi-conjugacy relative to polarities ∆, ∆∗ of a function h : u → r is the function h∆∆ ∗ : u → r given by h∆∆ ∗ (a) := sup (v,r)/∈∆(a) inf u/∈∆∗(v,r) h(u) = sup a/∈ep v,r inf u/∈epv,r h(u). (4.7) it is well known ([16]) that this conjugacy can be interpreted by means of coupling function δ : u × (v × r) → r defined by δ(u, (v, r)) = { 0 si u /∈ epv,r −∞ si u ∈ epv,r. more precisely, given a function h : u → r, we have h∆ = hδ and h∆∆ ∗ = hδδ. theorem 4.6 ([16]). the (c,p)−level regularization of a function h : u → r coincides with bi-conjugacy relative to polarities ∆, ∆∗: h〈〉c,p = h∆∆ ∗ . corollary 4.7. for any subset a of u, we have i 〈〉c,p a = i〈a〉c,p and v 〈〉c,p a = v〈a〉c,p. 148 moussa barro & sado traore cubo 22, 1 (2020) proof let a ⊂ u. let a ∈ a. it follows from theorem 4.6 that i 〈〉c,p a (a) = sup a/∈ep v,r inf u/∈ep v,r ia(u). we distinguish two cases: • we first assume that a /∈ 〈a〉c,p. there exists (v, r) ∈ v ×r such that a /∈ e p v,r and a ⊂ e p v,r. consequently, inf u/∈ep v,r ia(u) = +∞. • secondly, assume that a ∈ 〈a〉c,p. for all (v, r) ∈ v × r such that a /∈ e p v,r, there exists u ∈ a such that u /∈ epv,r. consequently, i 〈〉c,p a (a) = sup a/∈ep v,r inf u/∈ep v,r ia(u) = { 0 if a ∈ 〈a〉c,p +∞ if a /∈ 〈a〉c,p =i〈a〉c,p(a). analogously, we have v 〈〉c,p a (a) = sup a/∈epv,r inf u/∈ep v,r va(u) = { −∞ if a ∈ 〈a〉c,p +∞ if a /∈ 〈a〉c,p =v〈a〉c,p(a). proposition 9. for any function h : u → r and for any real number t, one has [h〈〉c,p ≤ t] = ⋂ s>t 〈[h ≤ s]〉c,p. proof let s > t and a /∈ 〈[h ≤ s]〉c,p. there exists (v̄, r̄) ∈ v × r such that a /∈ e p v̄,r̄ and [h ≤ s] ⊂ epv̄,r̄. we deduce that h〈〉c,p(a) = sup a/∈ep v,r inf u/∈ep v,r h(u) ≥ inf u/∈epv̄,r̄ h(u) ≥ s > t. let a /∈ 〈 [h〈〉c,p ≤ s] 〉 c,p . there exists s ∈ r such that h〈〉c,p(a) > s > t. by theorem 4.6, there exists (v, r) ∈ v × r such that a /∈ epv,r and inf u/∈ep v,r h(u) > s, thus [h ≤ s] ⊂ epv,r, and finally a /∈ 〈[h ≤ s]〉c,p. cubo 22, 1 (2020) level sets regularization with application to optimization problems 149 4.4 other expressions of (c,p)−level regularizations we now give another expression of the (c,p)−level regularization of an extended real-valued function h. these expressions give the value of the (c,p)−level regularization of h at a given point. given h : u → r and a ∈ u, we define sets ih(a) and jh(a) by: ih(a) := {t ∈ r | a /∈ 〈[h ≤ t]〉c,p} and jh(a) = {t ∈ r | a ∈ 〈[h ≤ t]〉c,p}. (4.8) sets ih(a) and jh(a) are two intervals of r such that ih(a) ∩ jh(a) = ∅ and ih(a) ∪ jh(a) = r. moreover, for any (r, s) ∈ ih(a) × jh(a), we have r < s. we deduce that sup ih(a) = inf jh(a). proposition 10. h〈〉c,p(a) = sup { t ∈ r : a /∈ 〈[h ≤ t]〉c,p } = inf { t ∈ r : a ∈ 〈[h ≤ t]〉c,p } . proof let t ∈ ih(a). there exists (v, r) ∈ v × r such that a /∈ e p v,r and [h ≤ t] ⊂ e p v,r. therefore inf u/∈ep v,r h(u) ≥ t and so sup a/∈ep v,r inf u/∈ep v,r h(u) ≥ t. by theorem 4.6, one gets h〈〉c,p(a) ≥ t. hence h〈〉c,p(a) ≥ sup ih(a). conversely, let t < h 〈〉c,p(a), then a /∈ [h〈〉c,p ≤ t] and by proposition 9, there exists s > t such that a /∈ 〈[h ≤ s]〉c,p. consequently, sup ih(a) ≥ s > t. hence sup ih(a) ≥ h 〈〉c,p(a). 5 applications to an optimization problem: sub-level set duality let us consider the following minimization problem: min x f(x), s.t x ∈ x, (p) where x is a nonempty set and f : x → r ∪ {+∞} is an extended real-valued function. 5.1 level set perturbational duality we consider a perturbation function f : x × u → r satisfying ∃a ∈ u : f(., a) = f(.). (5.1) we associate to f a valued function h : u → r defined by h(u) := inf x∈x f(x, u). (5.2) 150 moussa barro & sado traore cubo 22, 1 (2020) we denote by α the optimal value of (p). it is obvious that α = h(a). the perturbational dual of (p) ([16]) is given by max (v,r) −h∆(v, r) s.t a /∈ epv,r. (d) we denote by β the optimal value of (d). by definition, one has − ∞ ≤ β := sup (d) = h∆∆ ∗ (a) ≤ h(a) =: α = inf (p) ≤ +∞. (5.3) thus, the weak duality holds. the following theorem gives a necessary and sufficient condition to assure the strong duality. theorem 5.1. the following statements are equivalent: (1) the strong duality holds for (p) i.e inf (p) = max (d), (2) a /∈ 〈[h < α]〉c,p. proof. assume that (1) holds. there exists (v̄, r̄) ∈ v × r such that a /∈ epv̄,r̄ and α = h(a) = −h ∆(v̄, r̄) := inf u/∈epv̄,r̄ h(u). we deduce that [h < α] ⊂ epv̄,r̄. thus a /∈ 〈[h < α]〉c,p. conversely, assume that (2) holds. there exists (v̄, r̄) ∈ v × r such that a /∈ epv̄,r̄ and [h < α] ⊂ e p v̄,r̄ . therefore inf u/∈epv̄,r̄ h(u) ≥ α ≥ β. remember that β := sup a/∈ep v,r −h∆(v, r) = sup a/∈ep v,r inf u/∈ep v,r h(u). thus β ≥ inf u/∈epv̄,r̄ h(u) = −h∆(v̄, r̄) ≥ α ≥ β. hence β = −h∆(v̄, r̄) = α. theorem 5.1 is interesting in evenly convex case which is used in economic theory. 5.2 evenly quasi-convex duality ([1],[7],[11],[12],[17]) we assume x and u are topological vector spaces, v = u∗ the topological dual of u, c = 〈, 〉 the standard coupling function between u and u∗. cubo 22, 1 (2020) level sets regularization with application to optimization problems 151 corollary 5.2. assume that function f : x × u → r is quasi-convex and for each x ∈ x, f(x, .) : u → r is upper semi-continuous. one has: inf (p) = max u∗∈u∗ inf (x,u)∈x×u 〈u−a,u∗〉≥0 f(x, u). proof since f is quasi-convex and for each x ∈ x, f(x, .) is upper semi-continuous then h is quasi-convex and upper semi-continuous. consequently, [h < α] is open convex set and so it is evenly convex. as a /∈ [h < α], it results from theorem 5.1 that inf (p) = max (d) = max r−〈a,u∗〉≤0 inf r−〈u,u∗〉≤0 h(u) = max u∗∈u∗ inf 〈u,u∗〉≥〈a,u∗〉 h(u), where the last equality follows from the fact that for each u∗ ∈ u∗, function ku∗ : r → r defined by ku∗(r) = inf r−〈u,u∗〉≤0 h(u) is not decreasing. corollary 5.3. assume that function f : x × u → r is quasi-convex and for each x ∈ x, f(x, .) : u → r is upper semi-continuous. one has: inf (p) = max u∗∈u∗ inf (x,u)∈x×u 〈u−a,u∗〉=0 f(x, u). proof we know that under these assumptions on f , [h < α] is convex open set, therefore it is (〈, 〉,r∗)−regular. since a /∈ [h < α], it results from theorem 5.1 that inf (p) = max (d) = max (u∗,r)∈u∗×r 〈a,u∗〉 = r inf u∈u 〈u,u∗〉=r h(u) = max u∗∈u∗ max r∈r 〈a,u∗〉=r inf u∈u 〈u,u∗〉=r h(u) = max u∗∈u∗ inf u∈u 〈u−a,u∗〉=0 h(u) = max u∗∈u∗ inf (x,u)∈x×u 〈u−a,u∗〉=0 f(x, u) by definition of h. 6 conclusion in this work, we introduced a closure operator by means of coupling function and a subset of r. this operator allowed us to define a hull of sets and level set regularization of extended real-valued functions. by decomposition of closure operator, we showed that a level set regularization of any 152 moussa barro & sado traore cubo 22, 1 (2020) extended real-valued function coincides with its bi-conjugacy relative to a couple of dual polarities. we derive an analytic expression of a level set regularization of extended real-valued function. our results are applied to derive a strong duality for a minimization problem. acknowledgements the authors are grateful to the anonymous referees and the editor for their constructive comments which have contributed to the final presentation of the paper. cubo 22, 1 (2020) level sets regularization with application to optimization problems 153 references [1] crouzeix,j-p.: contributions à l’étude des fonctions quasiconvexes. thesis. university of clermont-ferrand, france (1977) [2] dolecki, s. and kurcyusz, s.: on φ-convexity in extremal problems. siam j. control optim. 16, 277–300 (1978) [3] elias, l.m. and mart́ınez-legaz, j.e.: a simplified conjugation scheme for lower semicontinuous functions. optimization, 65(4):751–763 (2016) [4] fenchel,w.: a remark on convex sets and polarity. comm. sém. math. univ. lund[medd. lunds univ. mat. sem.], 82–89 (1952) [5] flores-bazán, f.: on a notion of subdifferentiability for non-convex functions. optimization, 33(1):1–8 (1995) [6] guillaume,s. and volle,m.: level set relaxation, epigraphical relaxation and conditioning in optimization. positivity, 19:769–795 (2015) [7] mart́ınez-legaz, j.: generalized convex duality and its economic applicatons. nonconvex optimization and its application, handbook of generalized convexity and generalized monotonicity. springer, new york, (2005) [8] moreau, j.j.: inf-convolution, sous-additivité, convexité des fonctions numériques. j. math. pures appl., 49:109–154 (1970) [9] penot,j.p.: what is quasiconvex analysis? 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[17] volle,m.: conjugaison par tranche et dualité de toland. optimization, 18(5):633–642 (1987) introduction preliminaries -regularization of functions and hull of sets -regularization of functions hull of sets level set regularization of functions definitions and properties decomposition of "426830a "526930b c,p conjugacy associated to polarities and * (volle-1985,volle-1987) other expressions of (c,p)-level regularizations applications to an optimization problem: sub-level set duality level set perturbational duality evenly quasi-convex duality (crouzeix-1977,legaz-book,penot-volle-1990,penot-volle-2004,volle-1987) conclusion cubo a mathematical journal vol.21, no¯ 01, (49–60). april 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000100049 certain integral transforms of the generalized lommel-wright function s. haq department of applied mathematics, faculty of engineering and technology, aligarh muslim university, aligarh-202002, up, india sirajulhaq007@gmail.com k.s. nisar department of mathematics, college of arts and science, prince sattam bin abdulaziz university, wadi aldawaser, riyadh region 11991, saudi arabia ksnisar1@gmail.com a.h. khan department of applied mathematics, faculty of engineering and technology, aligarh muslim university, aligarh-202002, up, india ahkhan.amu@gmail.com d.l. suthar department of mathematics, wollo university, ethiopia dlsuthar@gmail.com abstract the aim of this article is to establish some integral transforms of the generalized lommel-wright functions, which are expressed in terms of wright hypergeometric function. some integrals involving trigonometric, generalized bessel and struve functions are also indicated as special cases of our main results. http://dx.doi.org/10.4067/s0719-06462019000100049 50 s. haq, k.s. nisar, a.h. khan and d.l. suthar cubo 21, 1 (2019) resumen el objetivo de este art́ıculo es establecer algunas transformadas integrales de las funciones generalizadas de lommel-wright, que se expresan en términos de la función hipergeométrica de wright. algunas integrales que involucran funciones trigonométricas, de bessel generalizadas y de struve también se obtienen como casos especiales de nuestros resultados principales. keywords and phrases: gamma function, generalized wright hypergeometric function pψq , generalized lommel-wright functions j µ m ν,λ (z), integral transforms. 2010 ams mathematics subject classification: 33b20, 33b15, 65r10, 33c20. cubo 21, 1 (2019) certain integral transforms of the generalized lommel-wright . . . 51 1 introduction the k-pochhammer symbol (λ)ν,k is defined (for ν,λ ∈ c;k ∈ r) by [4] (λ)ν,k = γk(λ + ν k) γk(λ) (λ ∈ c/0) (1.1) and the k-gamma function has the relation γk(z) = k z/k−1γ(z/k), (1.2) is such that γk(z) → γ(z) if k → 1 . the wright hypergeometric function defined by the series [21] pψq     (α1,a1), ...,(αp,ap); (β1,b1), ...,(βq,bq) z     = ∞∑ k=0 p∏ j=1 γ(αj + ajk)z k q∏ j=1 γ(βj + bjk)k! , (1.3) where the coefficients a1, ...,ap and b1, ...,bq are positive real numbers such that 1 + q∑ j=1 bj − p∑ j=1 aj ≥ 0. (1.4) can be slightly generalized (1.3) as given below. pψq     (α1,1), ...,(αp,1); (β1,1), ...,(βq,1); z     = p∏ j=1 γ(αj) q∏ j=1 γ(βj) pfq     α1, ..,αp; β1, ...,βq; z     , (1.5) where pfq is the generalized hypergeometric function defined by [19, 21] pfq     α1, ...,αp; β1, ...,βq z     = ∞∑ k=0 (α1)n, ...,(αp)nz n (β1)n, ...,(βq)nn! = pfq(α1, ...,αp;β1, ...,βq;z), (1.6) where (λ)n is the well known pochhammer symbol [21]. the generalization of (λ)n is given as (λ)n = λ(λ + 1)(λ + 2), ...,(λ + n − 1)) ,n > 0 (1.7) (λ)n = n∏ m=1 (λ + m − 1), (λ)0 = 1, λ 6= 0 52 s. haq, k.s. nisar, a.h. khan and d.l. suthar cubo 21, 1 (2019) (λ)n = γ(λ + n) γ(λ) generalized bessel, lommel, struve and lommel-wright function have originated from concrete problems in mechanics, physics, engineering and astronomy. the series representation of the generalized lommel wright function as [8]; j µ,m ν,λ (z) = ∞∑ k=0 (−1)kγ(k + 1)( z 2 )2k+ν+2λ γ(λ + k + 1)mγ(ν + kµ + λ + 1)k! , (1.8) (z ∈ n/(−∞,0] m ∈ n, ν,λ ∈ c,µ > 0). also, we have the following relations of generalized lommel wright functions with trigonometric functions and the generalized bessel function and struve function: j1,11/2,0(z) = √ ( 2 πz ) sin(z) (1.9) j1,1−1/2,0(z) = √ ( 2 πz ) cos(z) (1.10) j µ,1 ν,λ(z) = j µ ν,λ(z) (1.11) j1,1ν,1/2(z) = hν(z) (1.12) further, we recall the following results [5]. ∫ ∞ 0 tu−1 exp(−t/2)wλ,µ(t)dt = γ(1/2 + µ + u)γ(1/2 − µ + u) γ(1 − λ + u) , (1.13) (re(u ± µ) > −1/2), where the whittaker function wλ,µ(t) is given in[5, 11]. wλ,µ(t) = γ(−2µ) γ(1/2 − µ − λ) mλ,µ(t) + γ(2µ) γ(1/2 + µ − λ) mλ,−µ(t) where mλ,µ(t) is defined as mλ,µ(t) = z 1/2+µ exp(−t/2) 1f1 ( 1/2 + µ + u;2µ + 1;t ) definition 1.1. euler transform: let ρ,σ ∈ c and re(ρ),re(σ) > 0, then the euler transform of the function f(z) is defined by b(f(z);ρ,σ) = ∫1 0 zρ−1(1 − z)σ−1f(z)dz (1.14) cubo 21, 1 (2019) certain integral transforms of the generalized lommel-wright . . . 53 definition 1.2. laplace transform: the laplace transform of the function f(t) is defined as f(δ) = l(f(t);δ) = ∫ ∞ 0 exp(−tδ)f(t)dt, re(δ) > 0 (1.15) definition 1.3. fourier transform: the following integral gives the fourier transform u = im[u](w) = ∫ r u(t) exp(iwt)dt, (1.16) where u = u(t) be a function of the space s(r) shwartzian space of the function that decay rapidly at ∞ together with all derivatives. definition 1.4. the fractional fourier transform (fft): let u be the function belonging to φ(r), the lizorkin space of function, where φ(r) = {φ ∈ s(r)} : im[φ] ∈ v(r) and v(r) is the set of functions defined by v(r) = {v ∈ s(r)} : vu0 = 0,n = 0,1,2, ... then fft of order α, 0 ≤ α ≤ 1 is given by uα(w) = imα(w) = ∫ r exp(i wα t)u(t)dt (1.17) particularly, if α = 1 (1.17) reduces to ft and for w > 0 (1.17) reduces to fft given by luchko et al [10]. the aim of this paper is to obtain the euler, laplace, whittaker and fractional fourier transforms of lommel-wright function. various generalizations, integrals, transforms and fractional calculus of special functions have been investigated by many researchers (see, for details, [1, 2, 6, 7, 9, 12, 13, 14, 15, 16, 17, 18, 20]). in this sequel, here, we aim at establishing certain new generalized integral formula involving the generalized lommel-wright function j µ,m ν,λ (z) interesting integral formulas which are derived as special cases. 2 main results this section deals with some integral formulas involving lommel-wright function. 54 s. haq, k.s. nisar, a.h. khan and d.l. suthar cubo 21, 1 (2019) theorem 2.1. for t ∈ n/(−∞,0] m ∈ n, ν,λ ∈ c and µ > 0 , the following integral formula holds true ∫1 0 tα−1(1 − t)β−1j µ,m ν,λ (x t σ)dt = ( x 2 )ν+2λ γ(β) × 2ψm+2 [ (1,1),(α + νσ + 2λσ,2σ); (λ + 1,1), ...,(λ + 1,1),(ν + λ + 1,µ),(α + β + νσ + 2λσ,2σ); − x2 4 ] . (2.1) proof. on using (1.8) in the integrand of (2.1) and then interchanging the order of integral sign and summation which is verified by uniform convergence of the involved series under the given conditions we get ∫1 0 tα−1(1 − t)β−1j µ,m ν,λ (x t σ)dt = ( x 2 )ν+2λ ∞∑ k=0 γ(k + 1)(−x2/4)k γ(λ + k + 1)mγ(ν + kµ + λ + 1)k! × ∫1 0 tα+σ(2k+ν+2λ)−1(1 − t)β−1dt. (2.2) now using (1.14) in the above equation we get ∫1 0 tα−1(1 − t)β−1j µ,m ν,λ (x t σ)dt = γ(β) ( x 2 )ν+2λ × ∞∑ k=0 γ(k + 1)γ(α + νσ + 2λσ + 2kσ)(−x 2 4 )k γ(λ + k + 1)mγ(α + β + νσ + 2λσ + 2kσ)γ(ν + kµ + λ + 1)k! . (2.3) finally, using (1.3) in the above equation, we get our assertion (2.1). this completes the proof of theorem 2.1. theorem 2.2. for t ∈ n/(−∞,0] m ∈ n, ν,λ ∈ c and µ > 0 , the following integral formula holds true ∫ ∞ 0 tα−1 exp(−tδ)j µ,m ν,λ (x t σ )dt = ( x 2 δ−α )ν+2λ (δ)−α × 2ψm+1 [ (1,1),(α + νσ + 2λσ,2σ); (λ + 1,1), ...,(λ + 1,1),(ν + λ + 1,µ); − x2 4 δ2σ ] . (2.4) proof. on using (1.8) in the integrand of (2.4) and then interchanging the order of integral sign and summation which is verified by uniform convergence of the involved series under the given cubo 21, 1 (2019) certain integral transforms of the generalized lommel-wright . . . 55 conditions we get ∫ ∞ 0 tα−1 exp(−δ t)j µ,m ν,λ (x t σ )dt = ( x 2 )ν+2λ ∞∑ k=0 γ(k + 1)(−x2/4)k γ(λ + k + 1)mγ(ν + kµ + λ + 1)k! × ∫ ∞ 0 tα+σ(2k+ν+2λ)−1 exp(−δ t)dt. (2.5) now using (1.15) in the above equation we get ∫ ∞ 0 tα−1 exp(−δ t)j µ,m ν,λ (x t σ)dt = (δ)−α ( x 2δσ )ν+2λ × ∞∑ k=0 γ(k + 1)γ(α + νσ + 2λσ + 2kσ)( −x 2 4 δ2σ )k γ(λ + k + 1)mγ(ν + kµ + λ + 1)k! . (2.6) finally, using (1.3) in the above equation, we get our assertion (2.6). this completes the proof of theorem 2.2. theorem 2.3. for t ∈ n/(−∞,0] m ∈ n, ν,λ ∈ c and µ > 0 , the following integral formula holds true ∫ ∞ 0 tη−1 exp(−p t)/2 wλ,µ(p t)j µ,m ν,λ (w t δ)dt = ( w pδ )ν+2λ × 3ψm+2 [ (1,1),(1/2 + µ + η + δ ν + 2δλ,2δ),(1/2 − µ + η + δ ν + 2δλ,2δ); (λ + 1,1), ...,(λ + 1,1),(ν + λ + 1,µ),(1 − λ + η + νδ + 2δλ,2δ); − w2 4 p2δ ] . (2.7) proof. on using (1.8) in the integrand of (2.7) and then interchanging the order of integral sign and summation which is verified by uniform convergence of the involved series under the given conditions we get ∫ ∞ 0 (u/p)η−1 exp(−u/2)wλ,µ(u)j µ,m ν,λ (w (u/p) δ)du = ( w pδ )ν+2λ ∞∑ k=0 γ(k + 1)(−w2/4 p2δ)k γ(λ + k + 1)mγ(ν + kµ + λ + 1)k! × ∫ ∞ 0 uη+δ(2k+ν+2λ)−1 exp(−u/2)wλ,µ(u)du. (2.8) now using (1.13) in the above equation we get ∫ ∞ 0 tη−1 exp(−p t)/2 wλ,µ(p t)j µ,m ν,λ (w t δ)dt = ( w pδ )ν+2λ × ∞∑ k=0 γ(k + 1)γ(1/2 + µ + η + 2kδ + δν + 2δλ)γ(1/2 − µ + η + 2kδ + δν + 2δλ)( −w 2 4 p2δ )k γ(λ + k + 1)mγ(ν + kµ + λ + 1)γ(1 − λ + η + 2kδ + δν + 2δλ)k! . (2.9) 56 s. haq, k.s. nisar, a.h. khan and d.l. suthar cubo 21, 1 (2019) finally, using (1.3) in the above equation, we get our assertion (2.9). this completes the proof of theorem 2.3. 3 special cases in this section, we get some integral formulas involving trigonometric function and generalized lommel-wright function as follows: corollary 3.1. if we take m = 1,µ = 1,λ = 0 and ν = 1/2 in (2.1) and then by using (1.9), we derive the following integral formula: ∫1 0 tα−σ/2−1(1 − t)(β−1) sin(x tσ)dt = √ π ( x 2 ) γ(β) 1ψ2     (α + σ/2,2σ); (3/2,1),(α + β + σ/2,2σ); − x2 4     (3.1) corollary 3.2. if we take m = 1,µ = 1,λ = 0 and ν = 1/2 in (2.4) and then by using (1.9), we derive the following integral formula: ∫ ∞ 0 tα−σ/2−1 exp(−δ t) sin(x tσ)dt = δ−α √ π δσ ( x 2 ) γ(β) 1ψ1     (α + σ/2,2σ); (3/2,1); − x2 4 δ2σ     (3.2) corollary 3.3. further if we take m = 1,µ = 1,λ = 0 and ν = 1/2 in (2.7) and then by using (1.9), we derive the following integral formula: ∫ ∞ 0 tη−δ/2−1 exp(−pt/2)wλ,µ(p t) sin(w t δ)dt = w √ π 2 pδ 2ψ2     (η + δ/2 + 3/2,2δ)(η + δ/2 − 1/2,2δ), ; (3/2,1),(η + δ/2 + 1,2δ); − w2 4 p2δ     (3.3) corollary 3.4. if we take m = 1,µ = 1,λ = 0 and ν = −1/2 in (2.1) and then by using (1.10), we derive the following integral formula: ∫1 0 tα−σ/2−1(1 − t)(β−1) cos(x tσ)dt = √ πγ(β) 1ψ2     (α − σ/2,2σ); (1/2,1),(α + β − σ/2,2σ); − x2 4     (3.4) cubo 21, 1 (2019) certain integral transforms of the generalized lommel-wright . . . 57 corollary 3.5. if we take m = 1,µ = 1,λ = 0 and ν = −1/2 in (2.4) and then by using (1.10), we derive the following integral formula: ∫ ∞ 0 tα−σ/2−1 exp(−δ t) cos(x tσ)dt = δ(σ−α) √ π 1ψ1     (α − σ/2,2σ); (1/2,1); − x2 4 δ2σ     (3.5) corollary 3.6. further if we take m = 1,µ = 1,λ = 0 and ν = −1/2 in (2.7) and then by using (1.10), we derive the following integral formula: ∫ ∞ 0 tη−δ/2−1 exp(−pt/2)wλ,µ(p t) cos(w t δ)dt = w √ π 2 2ψ2     (η − δ/2 + 3/2,2δ)(η − δ/2 − 1/2,2δ), ; (1/2,1),(η − δ/2 + 1,2δ); − w2 4 p2δ     (3.6) corollary 3.7. if we take m = 1 in (2.1) and then by using (1.11), we derive the following integral formula: ∫1 0 tα−1(1 − t)(β−1)j µ ν,λ(x t σ )dt = ( x 2 )ν+2λ γ(β) ×2ψ3     (1,1),(α + νσ + 2λσ,2σ); (λ + 1,1),(ν + λ + 1,µ),(α + β + νσ + 2λσ,2σ); − x2 4     (3.7) corollary 3.8. if we take m = 1 in (2.4) and then by using (1.11), we derive the following integral formula: ∫ ∞ 0 tα−1 exp(−δ t)j µ ν,λ(x t σ)dt = ( x 2 )ν+2λ δ−α 2ψ2     (1,1),(α + νσ + 2λσ,2σ); (λ + 1,1),(ν + λ + 1,µ); − x2 4 δ2σ     (3.8) corollary 3.9. further if we take m = 1 in (2.7) and then by using (1.11), we derive the following integral formula: ∫ ∞ 0 tη−1 exp(−pt/2)wλ,µ(p t)j µ ν,λ(w t δ )dt = ( w pδ )ν+2λ ×3ψ3     (1,1),(1/2 + µ + η + νδ + 2λδ,2δ),(1/2 − µ + η + νδ + 2λδ,2δ); (λ + 1,1),(ν + λ + 1,µ),(1 − λ + η + δν + 2δλ,2δ); − w2 4 p2δ     (3.9) 58 s. haq, k.s. nisar, a.h. khan and d.l. suthar cubo 21, 1 (2019) corollary 3.10. if we take µ = 1,m = 1 and λ = 1/2 in (2.1) and then by using (1.12), we derive the following integral formula: ∫1 0 tα−1(1 − t)(β−1)hν(x t σ)dt = ( x 2 )ν+1 γ(β) ×2ψ3     (1,1),(α + νσ + σ,2σ); (3/2,1),(ν + 3/2,1),(α + β + νσ + σ,2σ); − x2 4     (3.10) corollary 3.11. if we take µ = 1,m = 1 and λ = 1/2 in (2.4) and then by using (1.12), we derive the following integral formula: ∫ ∞ 0 tα−1 exp(−δ t)hν(x t σ )dt = ( x 2 δσ )ν+1 δ−α ×2ψ2     (1,1),(α + νσ + σ,2σ); (3/2,1),(ν + 3/2,1); − x2 4 δ2σ     (3.11) corollary 3.12. further if we take µ = 1,m = 1 and λ = 1/2 in (2.7) and then by using (1.12), we derive the following integral formula: ∫ ∞ 0 tη−1 exp(−pt/2)wλ,µ(p t)hν(w t δ)dt = ( w pδ )ν+1 ×3ψ3     (1,1),(η + νδ + δ + 3/2,2δ),(η + νδ + δ − 1/2,2δ); (3/2,1),(ν + 3/2,1),(η + δν + δ + 1/2,2δ); − w2 4 p2δ     (3.12) cubo 21, 1 (2019) certain integral transforms of the generalized lommel-wright . . . 59 references [1] j. choi and p. agarwal, certain unified integrals associated with bessel functions, bound. value probl., 95, (2013), pages 9. 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[21] h.m. srivastava, and h.l. manocha, a treatise on generating functions, john wily and sons (halsted press, new york,ellis horwood, chichester), 1984. introduction main results special cases cubo, a mathematical journal vol. 24, no. 01, pp. 105–114, april 2022 doi: 10.4067/s0719-06462022000100105 some results on the geometry of warped product cr-submanifolds in quasi-sasakian manifold shamsur rahman department of mathematics, maulana azad national urdu university polytechnic satellite campus darbhanga bihar846002, india. shamsur@rediffmail.com abstract the present paper deals with a study of warped product submanifolds of quasi-sasakian manifolds and warped product cr-submanifolds of quasi-sasakian manifolds. we have shown that the warped product of the type m = d⊥×ydt does not exist, where d⊥ and dt are invariant and antiinvariant submanifolds of a quasi-sasakian manifold m̄, respectively. moreover we have obtained characterization results for cr-submanifolds to be locally cr-warped products. resumen el presente art́ıculo trata de un estudio de subvariedades producto alabeadas de variedades cuasi-sasakianas y crsubvariedades producto alabeadas de variedades cuasisasakianas. hemos mostrado que el producto alabeado de tipo m = d⊥×ydt no existe, donde d⊥ y dt son subvariedades invariantes y anti-invariantes de una variedad cuasisasakiana m̄, respectivamente. más aún, hemos obtenido resultados de caracterización para que cr-subvariedades sean localmente cr-productos alabeados. keywords and phrases: warped product, cr-submanifolds, quasi sasakian manifold, canonical structure. 2020 ams mathematics subject classification: 53c25, 53c40. accepted: 18 january, 2022 received: 02 may, 2021 c©2022 s. rahman. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100105 https://orcid.org/0000-0003-0995-2860 mailto:shamsur@rediffmail.com 106 s. rahman cubo 24, 1 (2022) 1 introduction if (d, gd) and (e, ge) are two semi-riemannian manifolds with metrics gd and ge respectively and y a positive differentiable function on d, then the warped product of d and e is the manifold d×ye = (d×e, g), where g = gd +y 2ge. further, let t be tangent to m = d×e at (p, q). then we have ‖t ‖2 = ‖dπ1t ‖ 2 + y2‖dπ2t ‖ 2 where πi(i = 1, 2) are the canonical projections of d×e onto d and e. a warped product manifold d×ye is said to be trivial if the warping function y is constant. in a warped product manifold, we have ∇u v = ∇v u = (u ln y)v (1.1) for any vector fields u tangent to d and v tangent to e [5]. the idea of a warped product manifold was introduced by bishop and o’neill [5] in 1969. chen [2] has studied the geometry of warped product submanifolds in kaehler manifolds and showed that the warped product submanifold of the type d⊥×ydt is trivial where dt and d⊥ are φ-invariant and anti-invariant submanifolds of a sasakian manifold, respectively. many research articles appeared exploring the existence or nonexistence of warped product submanifolds in different spaces [1, 10, 6]. the idea of cr-submanifolds of a kaehlerian manifold was introduced by a. bejancu [9]. later, a. bejancu and n. papaghiue [11], introduced and studied the notion of semi-invariant submanifolds of a sasakian manifold. these submanifolds are closely related to cr-submanifolds in a kaehlerian manifold. however the existence of the structure vector field implies some important changes. later on, binh and de [4] studied cr-warped product submanifolds of a quasi-saskian manifold. the purpose of this paper is to study the notion of a warped product submanifold of quasi-sasakian manifolds. in the second section we recall some results and formulae for later use. in the third section, we prove that the warped product in the form m = d⊥×ydt does not exist except for the trivial case, where dt and d⊥ are invariant and anti-invariant submanifolds of a quasisasakian manifold m̄, respectively. also, we obtain a characterization result of the warped product cr-submanifolds of the type m = d⊥×ydt . 2 preliminaries if m̄ is a real (2n+ 1) dimensional differentiable manifold, endowed with an almost contact metric structure (f, ξ, η, g), then f2u = −u + η(u)ξ, η(ξ) = 1, f(ξ) = 0, η(fu) = 0, (2.1) cubo 24, 1 (2022) some results on the geometry of warped product ... 107 η(u) = g(u, ξ), g(fu, fv ) = g(u, v ) − η(u)η(v ), (2.2) for any vector fields u, v tangent to m̄, where i is the identity on the tangent bundle γm̄ of m̄. throughout the paper, all manifolds and maps are differentiable of class c∞. we denote by ̥m̄ the algebra of the differentiable functions on m̄ and by γ(e) the ̥m̄ module of the sections of a vector bundle e over m̄. the nijenhuis tensor field, denoted by nf , with respect to the tensor field f, is given by nf(u, v ) = [fu, fv ] + f 2[u, v ] − f[fu, v ] + f[u, fv ], and the fundamental 2-form λ is given by λ(u, v ) = g(u, fv ), ∀u, v ∈ γ(t m̄). the curvature tensor field of m̄, denoted by r̄ with respect to the levi-civita connection ∇̄, is defined by r̄(u, v )w = ∇̄u ∇̄v w − ∇̄v ∇̄uw − ∇̄[u,v ]w, ∀u, v ∈ γ(t m̄), definition 2.1. (a) an almost contact metric manifold m̄(f, ξ, η, g) is called normal if nf (u, v ) + 2dη(u, v )ξ = 0, ∀u, v ∈ γ(t m̄), or equivalently (∇̄fu f)v = f(∇̄uf)v − g(∇̄uξ, v )ξ, ∀u, v ∈ γ(t m̄). (b) the normal almost contact metric manifold m̄ is called cosympletic if dλ = dη = 0. if m̄ is an almost contact metric manifold, then m̄ is a quasi-sasakian manifold if and only if ξ is a killing vector field [7] and (∇̄u f)v = g(∇̄fuξ, v )ξ − η(v )∇̄fu ξ, ∀u, v ∈ γ(t m̄). (2.3) next we define a tensor field f of type (1, 1) by fu = −∇̄uξ, ∀u ∈ γ(t m̄). (2.4) 108 s. rahman cubo 24, 1 (2022) lemma 2.1. for a quasi-sasakian manifold m̄, we have (i) (∇̄ξf)u = 0, ∀u ∈ γ(t m̄), (ii) f ◦ f = f ◦ f, (iii) fξ = 0, (iv) g(fu, v ) + g(u, fv ) = 0, (v) η ◦ f = 0, (vi) (∇̄u f)v = r̄(ξ, u)v , for all u, v ∈ γ(t m̄). the tensor field f defines on m̄ an f-structure in sense of k. yano [12], that is f3 + f = 0. if m is a submanifold of a quasi-sasakian manifold m̄ and denote by n the unit vector field normal to m. denote by the same symbol g the induced tensor metric on m, by ∇ the induced levicivita connection on m and by t m⊥ the normal vector bundle to m. the gauss and weingarten methods are ∇̄u v = ∇u v + σ(u, v ), (2.5) ∇̄u λ = −aλu + ∇ ⊥ u λ, ∀u, v ∈ γ(t m), (2.6) where ∇⊥ is the induced connection in the normal bundle, σ is the second fundamental form of m and aλ is the weingarten endomorphism associated with λ. the second fundamental form σ and the shape operator a are related by g(aλu, v ) = g(h(u, v ), λ), (2.7) where g denotes the metric on m̄ as well as the induced metric on m [7]. for any u ∈ t m, we write fu = ru + su, (2.8) where ru is the tangential component of fu and su is the normal component of fu, respectively. similarly, for any vector field λ normal to m, we put fλ = jλ + kλ (2.9) where jλ and kλ are the tangential and normal components of fλ, respectively. for all u, v ∈ γ(t m) the covariant derivatives of the tensor fields r and s are defined as (∇̄u r)v = ∇urv − r∇u v, (2.10) (∇̄u s)v = ∇ ⊥ u sv − s∇u v. (2.11) cubo 24, 1 (2022) some results on the geometry of warped product ... 109 3 warped product submanifolds if dt and d⊥ are invariant and anti-invariant submanifolds of a quasi-sasakian manifold m̄, then their warped product cr-submanifolds are one of the following forms: (i) m = d⊥×ydt , (ii) m = dt ×yd⊥. for case (i), when ξ ∈ t dt , we have the following theorem. theorem 3.1. there do not exist warped product cr-submanifolds m = d⊥×ydt in a quasisasakian manifold m̄ such that dt is an invariant submanifold, d⊥ is an anti-invariant submanifold of m̄ and ξ is tangent to m. proof. if m = d⊥×ydt is a warped product cr-submanifold of a quasi-sasakian manifold m̄ such that dt is an invariant submanifold tangent to ξ and d⊥ is an anti-invariant submanifold of m̄, then from (1.1), we have ∇u w = ∇w u = (w ln y)u, for any vector fields w and u tangent to d⊥ and dt , respectively. in particular, ∇w ξ = (w ln y)ξ, (3.1) using (2.4), (2.5) and ξ is tangent to d⊥, we have ∇w ξ = −fw, h(w, ξ) = 0. (3.2) it follows from (3.1) and (3.2) that w ln y = 0, for all w ∈ t d⊥, i. e., y is constant for all w ∈ t d⊥. now, the other case, when ξ tangent to d⊥ is dealt in the following two results. lemma 3.1. let m = d⊥×ydt be a warped product cr-submanifold of a quasi-sasakian manifold such that ξ is tangent to d⊥, where d⊥ and dt are any riemannian submanifolds of m̄. then (i) ξ ln y = −f, (ii) g(σ(u, fu), sw) = −{η(w)f + (w ln y)}‖u‖2, for any u ∈ t dt and w ∈ t d⊥. 110 s. rahman cubo 24, 1 (2022) proof. let ξ ∈ t d⊥ then for any u ∈ t dt , we have ∇u ξ = (ξ ln y)u, (3.3) from (2.4) and the fact that ξ is tangent to d⊥, we have ∇̄u ξ = −fu. with the help of (2.5), we have ∇w ξ = −fw, h(w, ξ) = 0. (3.4) from (3.3) and (3.4), we have ξ ln y = −f . now, for any u ∈ t dt and w ∈ t d⊥, we have ∇̄ufw = (∇̄u f)w + f(∇̄uw). using (2.3), (2.6), (2.8), (2.9) and by the orthogonality of the two distributions, we derive −η(w)∇̄fu ξ = −asw u + ∇ ⊥ u sw − r∇u w − s∇u w − jh(u, w) − kh(u, w). equating the tangential components, we get −η(w)ffu = asw u + r∇u w + jh(u, w). taking the product with fu and using (2.2) and (2.3), we derive −η(w)fg(fu, fu) = g(asw u, fu) + (w ln y)g(ru, fu) + g(jh(u, w), fu) = g(h(fu, fu), sw) + (w ln y)g(fu, fu) + g(fh(u, w), fu). using (2.2), we obtain g(σ(u, fu), sw) = −{η(w)f + (w ln y)}‖u‖2. (3.5) theorem 3.2. if m = d⊥×ydt is a warped product cr-submanifold of a quasi-sasakian manifold m̄ such that ξ is tangent to d⊥ and if σ(u, fu) ∈ µ the invariant normal subbundle of m, then w ln y = −η(w)f, for all u ∈ t dt and z ∈ t n⊥. proof. the affirmation follows from formula (3.5) by means of the known truth. the warped product m = dt ×yd⊥, we have the following theorem. theorem 3.3. there do not exist warped product cr-submanifolds m = dt ×yd⊥ in a quasisasakian manifold m̄ such that ξ is tangent to d⊥. proof. if ξ ∈ t n⊥, then from (1.1), we have ∇u ξ = (u ln y)ξ, (3.6) for any u ∈ t dt . while using (2.4), (2.5) and ξ ∈ t d⊥, we have ∇uξ = −fu, h(u, ξ) = 0. (3.7) from (3.6) and (3.7), it follows that u ln y = 0, for all u ∈ t dt , and this means that y is constant on nt . cubo 24, 1 (2022) some results on the geometry of warped product ... 111 the remaining case, when ξ ∈ t dt is dealt with the following two theorems. theorem 3.4. let m = dt ×yd⊥ be a warped product cr-submanifold of a quasi-sasakian manifold m̄ such that ξ is tangent to dt . then (∇̄u f)w ∈ µ, for each u ∈ t dt and w ∈ t d⊥, where µ is an invariant normal subbundle of t m. proof. for any u ∈ t dt and w ∈ t d⊥, we have g(f∇̄uw, fw) = g(∇̄uw, w) = g(∇u w, w). using (1.1), we get g(f∇̄uw, fw) = (u ln y)‖w‖ 2 . (3.8) on the other hand, we have ∇̄u fw = (∇̄u f)w + f(∇̄uw), for any u ∈ t dt and w ∈ t d⊥. using (2.3) and the fact that ξ is tangent to dt , the left-hand side of the above equation is identically zero, that is ∇̄ufw = f(∇̄uw). (3.9) taking the product with fw in (3.9) and making use of formula (2.6), we obtain g(f∇̄uw, fw) = g(∇ ⊥ u sw, sw). then from (2.10), we derive g(f∇̄u w, fw) = g((∇̄u s)w, sw) + g(s∇uw, sw). from (1.1) we have g(f∇̄uw, fw) = (u ln y)g(sw, sw) + g((∇̄u s)w, sw) = (u ln y)g(fw, fw) + g((∇̄u s)w, sw). therefore by (2.2), we obtain g(f∇̄uw, fw) = (u ln y)‖w‖ 2 + g((∇̄u s)w, sw). (3.10) thus (3.8) and (3.9) imply g((∇̄u s)w, sw) = 0. (3.11) also, as dt is an invariant submanifold then fq ∈ t dt , for any q ∈ t dt , thus on using (2.11) and the fact that the product of tangential components with normal is zero, we obtain g((∇̄us)w, fq) = 0. (3.12) hence from (3.11) and (3.12), it follows that (∇̄u s)w ∈ µ, for all u ∈ t dt and w ∈ t d⊥. 112 s. rahman cubo 24, 1 (2022) theorem 3.5. a cr-submanifold m of a quasi-sasakian manifold (m̄, f, ξ, g) is a cr-warped product if and only if the shape operator of m satisfies afw u = (fuµ)w, u ∈ b ⊕ 〈ξ〉, w ∈ b ⊥ , (3.13) for some function µ on m, fulfilling c(µ) = 0, for each c ∈ b⊥. proof. if m = dt ×yd⊥ is a cr-warped product submanifold of a quasi-sasakian manifold m̄, with ξ ∈ t dt , then for any u ∈ t dt and w, q ∈ t d⊥, we have g(afw u, q) = g(σ(u, q), fw) = g(∇̄qu, fw) = g(f∇̄qu, w) = g(∇̄qfu, w) − g((∇̄qf)u, w). by equations (1.1), (2.3) and the fact that ξ is tangent to dt , we derive g(afw u, q) = (fu ln y)g(w, q). (3.14) on the other hand, we have g(σ(u, v ), sw) = g(f∇̄u v, w) = −g(fv, ∇̄uw), for each u, v ∈ t dt and w ∈ t n⊥. using (1.1), we obtain g(σ(u, v ), sw) = 0. taking into account this fact in (3.14), we obtain (3.13). conversely, suppose that m is a proper cr-submanifold of a quasi-sasakian manifold m satisfying (3.13), then for any u, v ∈ b ⊕ 〈ξ〉, g(σ(u, v ), fw) = g(afw u, v ) = 0. this implies that g(∇̄u fv, w) = 0, that is, g(∇u v, w) = 0. this means b ⊕ 〈ξ〉 is integrable and its leaves are totally geodesic in m. now, for any w, q ∈ b⊥ and u ∈ b ⊕ 〈ξ〉, we have g(∇w q, fu) = g(∇̄w q, fu) = g(f∇̄w q, u) = g(∇̄w fq, u) − g((f∇̄w f)q, u). by equations (2.3) and (2.6), it follows that g(∇w q, fu) = −g(afqw, u). thus from (2.6), we arrive at g(∇w q, fu) = −g(σ(w, u), fq). again using (2.7) and (3.13), we obtain g(∇w q, fu) = −g(afqu, w) = −(fuµ)g(w, q). (3.15) if n⊥ is a leaf of b ⊥ and σ⊥ is the second fundamental form of the immersion of d⊥ into m, then for any w, q ∈ b⊥, we have g(σ⊥(w, q), fu) = g(∇w q, fu). (3.16) hence, from (3.15) and (3.16), we find that g(σ⊥(w, q), fu) = −(fuµ)g(w, q). cubo 24, 1 (2022) some results on the geometry of warped product ... 113 this means that the integral manifold d⊥ of b ⊥ is totally umbilical in m. since c(µ) = 0 for each c ∈ b⊥, which implies that the integral manifold of b⊥ is an extrinsic sphere in m, this means that the curvature vector field is nonzero and parallel along n⊥. hence by virtue of a result in [7], m is locally a warped product dt ×yd⊥, where dt and n⊥ denote the integral manifolds of the distributions b ⊕ 〈ξ〉 and b⊥, respectively and y is the warping function. acknowledgements the authors grateful the referee(s) for the corrections and comments in the revision of this paper. 114 s. rahman cubo 24, 1 (2022) references [1] k. arslan, r. ezentas, i. mihai and c. murathan, “contact cr-warped product submanifolds in kenmotsu space forms”, j. korean math. soc., vol. 42, no. 5, pp. 1101–1110, 2005. [2] a. bejancu, “cr-submanifold of a kaehler manifold. i”, proc. amer. math. soc., vol. 69, no. 1, 135–142, 1978. [3] a. bejancu and n. papaghiuc, “semi-invariant submanifolds of a sasakian manifold.”, an. ştiinţ. univ. “al. i. cuza” iaşi secţ. i a mat. (n.s.), vol 27, no. 1, pp. 163–170, 1981. [4] t.-q. binh and a. de, “on contact cr-warped product submanifolds of a quasi-sasakian manifold”, publ. math. debrecen, vol. 84, no. 1-2, pp. 123–137, 2014. [5] r. l. bishop and b. o’neill, “manifolds of negative curvature”, trans. amer. math. soc., vol. 145, pp. 1–49, 1969. [6] d. e. blair, contact manifolds in riemannian geometry, lecture notes in math., vol. 509, berlin-new york: springer-verlag, 1976. [7] c. calin, “contributions to geometry of cr-submanifold”, phd thesis, university of ias,i, ias,i, romania, 1998. [8] b.-y. chen, “geometry of warped product cr-submanifolds in kaehler manifolds”, monatsh. math., vol. 133, no. 3, pp. 177–195, 2001. [9] i. hasegawa and i. mihai, “contact cr-warped product submanifolds in sasakian manifolds”, geom. dedicata, vol. 102, pp. 143–150, 2003. [10] s. hiepko, “eine innere kennzeichnung der verzerrten produkte”, math. ann., vol. 241, no. 3, pp. 209–215, 1979. [11] m.-i. munteanu, “a note on doubly warped product contact cr-submanifolds in transsasakian manifolds”, acta math. hungar., vol 116, no. 1-2, pp. 121–126, 2007. [12] k. yano, “on structure defined by a tensor field f of type (1, 1) satisfying f3 +f = 0”, tensor (n.s.), vol. 14, pp. 99–109, 1963. [13] k. yano and m. kon, structures on manifolds, series in pure mathematics, vol. 3, singapore: world scientific publishing co., 1984. introduction preliminaries warped product submanifolds cubo a mathematical journal vol.20, no¯ 3, (01–11). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300001 quantitative approximation by a kantorovich-shilkret quasi-interpolation neural network operator george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract in this article we present multivariate basic approximation by a kantorovich-shilkret type quasi-interpolation neural network operator with respect to supremum norm. this is done with rates using the multivariate modulus of continuity. we approximate continuous and bounded functions on rn, n ∈ n. when they are additionally uniformly continuous we derive pointwise and uniform convergences. resumen en este art́ıculo presentamos un resultado de aproximación básico multivariado a través de un operador de cuasi-interpolación en red neuronal de tipo kantorovich-shilkret con respecto a la norma del supremo. esto se realiza con tasas usando el módulo de continuidad multivariado. aproximamos funciones continuas y acotadas en rn, n ∈ n. cuando ellas son adicionalmente uniformemente continuas, derivamos convergencias puntuales y uniformes. keywords and phrases: error function based activation function, multivariate quasi-interpolation neural network approximation, kantorovich-shilkret type operator. 2010 ams mathematics subject classification: 41a17, 41a25, 41a30, 41a35. http://dx.doi.org/10.4067/s0719-06462018000300001 2 george a. anastassiou cubo 20, 3 (2018) 1 introduction the author here performs multivariate error function based neural network approximation to continuous functions over rn, n ∈ n, and then he extends his results to complex valued functions. the convergences here are with rates expressed via the multivariate modulus of continuity of the involved function and give by very tight jackson type inequalities. the author comes up with the ”right” precisely defined flexible quasi-interpolation baskakovshilkret type integral coefficient neural network operator associated to the error function. feed-forward neural network (fnns) with one hidden layer with deal with, are expressed mathematicaly as nn (x) = n∑ j=0 cjσ(〈aj · x〉 + bj) , x ∈ rs, s ∈ n, where for 0 ≤ j ≤ n, bj ∈ r are the thresholds, aj ∈ rs are the connection weights, cj ∈ r are the coefficients, 〈aj · x〉 is the inner product of aj and x, and σ is the activation function of the network. in many fundamental neural network models the activation function is error function generated. about neural networks in general you may read [4], [5], [6]. in recent years non-additive integrals, like the n. shilkret one [7], have become fashionable and more useful in economic theory, etc. 2 background here we follow [7]. let f be a σ-field of subsets of an arbitrary set ω. an extended non-negative real valued function µ on f is called maxitive if µ(∅) = 0 and µ(∪i∈iei) = sup i∈i µ(ei) , (1) where the set i is of cardinality at most countable. we also call µ a maxitive measure. here f stands for a non-negative measurable function on ω. in [7], niel shilkret developed his non-additive integral defined as follows: (n∗) ∫ d fdµ := sup y∈y {y · µ(d ∩ {f ≥ y})} , (2) where y = [0,m] or y = [0,m) with 0 < m ≤ ∞, and d ∈ f. here we take y = [0,∞). it is easily proved that (n∗) ∫ d fdµ = sup y>0 {y · µ(d ∩ {f > y})} . (3) cubo 20, 3 (2018) quantitative approximation by a kantorovich-shilkret . . . 3 the shilkret integral takes values in [0,∞]. the shilkret integral ([7]) has the following properties: (n∗) ∫ ω χedµ = µ(e) , (4) where χe is the indicator function on e ∈ f, (n∗) ∫ d cfdµ = c(n∗) ∫ d fdµ, c ≥ 0, (5) (n∗) ∫ d sup n∈n fndµ = sup n∈n (n∗) ∫ d fndµ, (6) where fn, n ∈ n, is an increasing sequence of elementary (countably valued) functions converging uniformly to f. furthermore we have (n∗) ∫ d fdµ ≥ 0, (7) f ≥ g implies (n∗) ∫ d fdµ ≥ (n∗) ∫ d gdµ, (8) where f,g : ω → [0,∞] are measurable. let a ≤ f(ω) ≤ b for almost every ω ∈ e, then aµ(e) ≤ (n∗) ∫ e fdµ ≤ bµ(e) ; (n∗) ∫ e 1dµ = µ(e) ; f > 0 almost everywhere and (n∗) ∫ e fdµ = 0 imply µ(e) = 0; (n∗) ∫ ω fdµ = 0 if and only f = 0 almost everywhere; (n∗) ∫ ω fdµ < ∞ implies that n(f) := {ω ∈ ω|f(ω) 6= 0} has σ-finite measure; (9) (n∗) ∫ d (f + g)dµ ≤ (n∗) ∫ d fdµ + (n∗) ∫ d gdµ; and ∣ ∣ ∣ ∣ (n∗) ∫ d fdµ − (n∗) ∫ d gdµ ∣ ∣ ∣ ∣ ≤ (n∗) ∫ d |f − g|dµ. (10) from now on in this article we assume that µ : f → [0,+∞). 4 george a. anastassiou cubo 20, 3 (2018) 3 main results we consider here the (gauss) error special function ([1], [3]) erf(x) = 2√ π ∫x 0 e−t 2 dt, x ∈ r, (11) which is a sigmoidal type function and a strictly increasing function. it has the properties erf(0) = 0, erf(−x) = erf(x) , erf(+∞) = 1, erf(−∞) = −1, and (erf(x)) ′ = 2√ π e−x 2 , x ∈ r, ∫ erf(x)dx = xerf(x) + e−x 2 √ π + c, where c is a constant. the error function is related to the cumulative probability distribution function of the standard normal distribution φ(x) = 1 2 + 1 2 erf ( x√ 2 ) . we consider the activation function χ(x) = 1 4 (erf(x + 1) − erf(x − 1)) , x ∈ r, (12) and we notice that χ(−x) = χ(x) , (13) and even function. clearly χ(x) > 0, all x ∈ r. we see that χ(0) = erf(1) 2 ≃ 0.4215. (14) let x > 0, we have that χ′ (x) < 0, for x > 0. (15) that is χ is strictly decreasing on [0,∞) and is strictly increasing on (−∞,0], and χ′ (0) = 0. clearly the x-axis is the horizontal asymptote on χ. conclusion, χ is a bell symmetric function with maximum χ(0) ≃ 0.4215. we further need cubo 20, 3 (2018) quantitative approximation by a kantorovich-shilkret . . . 5 theorem 3.1. ([2]) we have that ∞∑ i=−∞ χ(x − i) = 1, all x ∈ r. (16) theorem 3.2. ([2]) it holds ∫∞ −∞ χ(x)dx = 1. (17) so χ(x) is a density function on r. theorem 3.3. ([2]) let 0 < α < 1, and n ∈ n with n1−α ≥ 3. it holds ∞∑    k = −∞ : |nx − k| ≥ n1−α χ(nx − k) < 1 2 √ π(n1−α − 2)e(n 1−α−2) 2 . (18) remark 3.4. we introduce z(x1, ...,xn) := z(x) := n∏ i=1 χ(xi) , (19) x = (x1, ...,xn) ∈ rn, n ∈ n. it has the properties: (i) z(x) > 0, ∀ x ∈ rn, (20) (ii) ∞∑ k=−∞ z(x − k) := ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞ z(x1 − k1, ...,xn − kn) = 1, (21) where k := (k1, ...,kn) ∈ zn, ∀ x ∈ rn, hence (iii) ∞∑ k=−∞ z(nx − k) = 1, ∀ x ∈ rn, n ∈ n, (22) and (iv) ∫ rn z(x)dx = 1, (23) that is z is a multivariate density function. 6 george a. anastassiou cubo 20, 3 (2018) here ‖x‖∞ := max {|x1| , ..., |xn|}, x ∈ rn, also set ∞ := (∞, ...,∞), −∞ = (−∞, ...,−∞) upon the multivariate context. it is also clear that (see (18)) (v) ∞∑    k = −∞ ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ z(nx − k) ≤ 1 2 √ π(n1−β − 2)e(n 1−β−2) 2 , (24) 0 < β < 1, n ∈ n : n1−β ≥ 3, x ∈ rn. for f ∈ c+ b ( r n ) (continuous and bounded functions from rn into r+), we define the first modulus of continuity ω1 (f,h) := sup x,y∈rn ‖x−y‖∞≤h |f(x) − f(y)| , h > 0. (25) given that f ∈ c+ u ( r n ) (uniformly continuous from rn into r+), we have that lim h→0 ω1 (f,h) = 0. (26) we make definition 3.5. let l be the lebesgue σ-algebra on rn, n ∈ n, and the maxitive measure µ : l → [0,+∞), such that for any a ∈ l with a 6= ∅, we get µ(a) > 0. for f ∈ c+ b ( r n ) , we define the multivariate kantorovich-shilkret type neural network operator for any x ∈ rn : tµn (f,x) = t µ n (f,x1, ...,xn) := ∞∑ k=−∞   (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) µ ( [ 0, 1 n ]n )  z(nx − k) = ∞∑ k1=−∞ ∞∑ k2=−∞ ... ∞∑ kn=−∞   (n∗) ∫ 1 n 0 ... ∫ 1 n 0 f ( t1 + k1 n ,t2 + k2 n , ...,tn + kn n ) dµ(t1, ...,tn) µ ( [ 0, 1 n ]n )   (27) · ( n∏ i=1 z(nxi − ki) ) , where x = (x1, ...,xn) ∈ rn, k = (k1, ...,kn), t = (t1, ...,tn), n ∈ n. clearly here µ ( [ 0, 1 n ]n ) > 0, ∀ n ∈ n. above we notice that ‖tµn (f)‖∞ ≤ ‖f‖∞ , (28) so that t µ n (f,x) is well-defined. cubo 20, 3 (2018) quantitative approximation by a kantorovich-shilkret . . . 7 remark 3.6. let t ∈ [ 0, 1 n ]n and x ∈ rn, then f ( t + k n ) = f ( t + k n ) − f(x) + f(x) ≤ ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ + f(x) , (29) hence (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) ≤ (n∗) ∫ [0, 1n ] n ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ dµ(t) + f(x)µ ( [ 0, 1 n ]n ) . (30) that is (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) − f(x)µ ( [ 0, 1 n ]n ) ≤ (31) (n∗) ∫ [0, 1n ] n ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ dµ(t) . similarly we have f(x) = f(x) − f ( t + k n ) + f ( t + k n ) ≤ ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ + f ( t + k n ) , hence (n∗) ∫ [0, 1n ] n f(x)dµ(t) ≤ (n∗) ∫ [0, 1n ] n ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ dµ(t) + (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) . that is f(x)µ ( [ 0, 1 n ]n ) − (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) ≤ (32) (n∗) ∫ [0, 1n ] n ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ dµ(t) . by (31) and (32) we derive ∣ ∣ ∣ ∣ ∣ (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) − f(x)µ ( [ 0, 1 n ]n ) ∣ ∣ ∣ ∣ ∣ ≤ (n∗) ∫ [0, 1n ] n ∣ ∣ ∣ ∣ f ( t + k n ) − f(x) ∣ ∣ ∣ ∣ dµ(t) . (33) in particular it holds ∣ ∣ ∣ ∣ ∣ ∣ (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) µ ( [ 0, 1 n ]n ) − f(x) ∣ ∣ ∣ ∣ ∣ ∣ ≤ (n∗) ∫ [0, 1n ] n ∣ ∣f ( t + k n ) − f(x) ∣ ∣dµ(t) µ ( [ 0, 1 n ]n ) . (34) 8 george a. anastassiou cubo 20, 3 (2018) we present theorem 3.7. let f ∈ c+ b ( r n ) , 0 < β < 1, x ∈ rn; n,n ∈ n with n1−β ≥ 3. then i) sup µ |tµn (f,x) − f(x)| ≤ ω1 ( f, 1 n + 1 nβ ) + ‖f‖∞√ π(n1−β − 2)e(n 1−β−2) 2 =: λn, (35) ii) sup µ ‖tµn (f) − f‖∞ ≤ λn. (36) given that f ∈ ( c+ u ( r n ) ∩ c+ b ( r n )) , we obtain lim n→∞ t µ n (f) = f, uniformly. proof. we observe that |tµn (f,x) − f(x)| = ∣ ∣ ∣ ∣ ∣ ∣ ∞∑ k=−∞   (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) µ ( [ 0, 1 n ]n )  z(nx − k) − ∞∑ k=−∞ f(x)z(nx − k) ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ ∞∑ k=−∞     (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) µ ( [ 0, 1 n ]n )   − f(x)  z(nx − k) ∣ ∣ ∣ ∣ ∣ ∣ ≤ (37) ∞∑ k=−∞ ∣ ∣ ∣ ∣ ∣ ∣   (n∗) ∫ [0, 1n ] n f ( t + k n ) dµ(t) µ ( [ 0, 1 n ]n )   − f(x) ∣ ∣ ∣ ∣ ∣ ∣ z(nx − k) (34) ≤ ∞∑ k=−∞   (n∗) ∫ [0, 1n ] n ∣ ∣f ( t + k n ) − f(x) ∣ ∣dµ(t) µ ( [ 0, 1 n ]n )  z(nx − k) = (38) ∞∑    k = −∞ : ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ   (n∗) ∫ [0, 1n ] n ∣ ∣f ( t + k n ) − f(x) ∣ ∣dµ(t) µ ( [ 0, 1 n ]n )  z(nx − k) + ∞∑    k = −∞ : ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ   (n∗) ∫ [0, 1n ] n ∣ ∣f ( t + k n ) − f(x) ∣ ∣dµ(t) µ ( [ 0, 1 n ]n )  z(nx − k) ≤ ∞∑    k = −∞ : ∥ ∥ k n − x ∥ ∥ ∞ ≤ 1 nβ   (n∗) ∫ [0, 1n ] n ω1 ( f,‖t‖∞ + ∥ ∥ k n − x ∥ ∥ ∞ ) dµ(t) µ ( [ 0, 1 n ]n )  z(nx − k) (39) cubo 20, 3 (2018) quantitative approximation by a kantorovich-shilkret . . . 9 +2‖f‖∞           ∞∑    k = −∞ : ∥ ∥ k n − x ∥ ∥ ∞ > 1 nβ z(nx − k)           (24) ≤ ω1 ( f, 1 n + 1 nβ ) + ‖f‖∞√ π(n1−β − 2)e(n 1−β−2) 2 , (40) proving the claim. additionally we give definition 3.8. denote by c+ b ( r n,c ) = {f : rn → c|f = f1 + if2, where f1,f2 ∈ c+b ( r n ) , n ∈ n}. we set for f ∈ c+ b ( r n,c ) that tµn (f,x) := t µ n (f1,x) + it µ n (f2,x) , (41) ∀ n ∈ n, x ∈ rn, i = √ −1. theorem 3.9. let f ∈ c+ b ( r n,c ) , f = f1 + if2, n ∈ n, 0 < β < 1, x ∈ rn; n ∈ n with n1−β ≥ 3. then i) sup µ |tµn (f,x) − f(x)| ≤ [ ω1 ( f1, 1 n + 1 nβ ) + ω1 ( f2, 1 n + 1 nβ )] + (‖f1‖∞ + ‖f2‖∞)√ π(n1−β − 2)e(n 1−β−2) 2 =: ψn, (42) and ii) sup µ ‖tµn (f) − f‖ ≤ ψn. (43) proof. |tµn (f,x) − f(x)| = |t µ n (f1,x) + it µ n (f2,x) − f1 (x) − if2 (x)| = |(tµn (f1,x) − f1 (x)) + i(t µ n (f2,x) − f2 (x))| ≤ |tµn (f1,x) − f1 (x)| + |t µ n (f2,x) − f2 (x)| (35) ≤ (44) ( ω1 ( f1, 1 n + 1 nβ ) + ‖f1‖∞√ π(n1−β − 2)e(n 1−β−2) 2 ) + ( ω1 ( f2, 1 n + 1 nβ ) + ‖f2‖∞√ π(n1−β − 2)e(n 1−β−2)2 ) = 10 george a. anastassiou cubo 20, 3 (2018) [ ω1 ( f1, 1 n + 1 nβ ) + ω1 ( f2, 1 n + 1 nβ )] + (‖f1‖∞ + ‖f2‖∞)√ π(n1−β − 2)e(n 1−β−2) 2 , (45) proving the claim. cubo 20, 3 (2018) quantitative approximation by a kantorovich-shilkret . . . 11 references [1] m. abramowitz, i.a. stegun, eds, handbook of mathematical functions with formulas, graphs, and mathematical tables, new york, dover publication, 1972. [2] g.a. anastassiou, univariate error function based neural network approximation, indian j. of math., vol. 57, no. 2 (2015), 243-291. [3] l.c. andrews, special functions of mathematics for engineers, second edition, mc graw-hill, new york, 1992. [4] i.s. haykin, neural networks: a comprehensive foundation (2 ed.), prentice hall, new york, 1998. [5] w. mcculloch and w. pitts, a logical calculus of the ideas immanent in nervous activity, bulletin of mathematical biophysics, 7 (1943), 115-133. [6] t.m. mitchell, machine learning, wcb-mcgraw-hill, new york, 1997. [7] niel shilkret, maxitive measure and integration, indagationes mathematicae, 33 (1971), 109116. introduction background main results cubo, a mathematical journal vol. 23, no. 02, pp. 191–206, august 2021 doi: 10.4067/s0719-06462021000200191 the zamkovoy canonical paracontact connection on a para-kenmotsu manifold d. g. prakasha 1 h. harish 2,3 p. veeresha 4 venkatesha 5 1 department of mathematics, davangere university, davangere 577 007, india. prakashadg@gmail.com 2 department of mathematics, mahaveera college, mudbidri, india. 3 department of mathematics, karnatak university, dharwad 580 003, india. harishjonty@gmail.com 4 department of mathematics, christ (deemed to be university), bengaluru 560029, india. pundikala.veeresha@christuniversity.in; viru0913@gmail.com 5 department of mathematics, kuvempu university, shankaraghatta 577 4511, shimoga district, india. vensmath@gmail.com abstract the object of the paper is to study a type of canonical linear connection, called the zamkovoy canonical paracontact connection on a para-kenmotsu manifold. resumen el objetivo de este art́ıculo es estudiar un tipo de conexión lineal canónica, llamada la conexión canónica paracontacto de zamkovoy en una variedad para-kenmotsu. keywords and phrases: para-kenmotsu manifold; zamkovoy canonical paracontact connection; local φ-symmetry; local φ-ricci symmetry; recurrent; η-einstein manifold. 2020 ams mathematics subject classification: 53c21, 53c25, 53c44. accepted: 01 december, 2020 received: 27 may, 2020 c©2021 d. g. prakasha et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200191 https://orcid.org/0000-0001-6453-0308 https://orcid.org/0000-0002-2175-187x https://orcid.org/0000-0002-4468-3048 https://orcid.org/0000-0002-2799-2535 192 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) 1 introduction in recent years, many authors started to the study of paracontact geometry due to its unexpected relation with the most activated contact geometry. as a result of this, in 1985, s. kaneyuki and f. l. williams [9] introduced the notion of paracontact metric manifold as a natural counter part of the well known contact metric manifold. since then, several authors studied these manifolds by focusing on various special cases. a systematic study of paracontact metric manifolds and their subclasses were carried out by s. zamkovoy [25] by emphasizing similarities and differences with respect to the contact case. further, the notion of para-kenmotsu manifold was introduced by j. welyczko [23] for 3-dimensional normal almost paracontact metric structures. this structure is an analogy of kenmotsu manifold [10] in paracontact geometry. again the similar notion called p-kenmotsu manifold was studied by b. b. sinha and k. l. sai prasad [20] and they obtained many results. at this point, we refer the papers [1, 4, 14, 15, 16, 26] and the references therein to reader for a wide and detailed overview of the results on para-kenmotsu manifolds. in the context of para-kenmotsu geometry, author a. m. blaga [2] studied certain canonical linear connections (levi-civita, schouten-van kampen, golab and zamkovoy canonical paracontact connections) with a special view towards φ-conjugation. some properties of generalized dual connections of the above said canonical linear connections on a para-kenmotsu manifold was also studied in [3]. as a continuation of this, we are considering one of such canonical linear connection on a para-kenmotsu manifold. so we undertake the study of zamkovoy canonical paracontact connection on a para-kenmotsu manifold. this connection on a paracontact manifolds was adapted and studied rigorously by s. zamkovoy [25]. this connection plays the role of the (generalized) tanaka-webster connection [22] in paracontact geometry. the main feature of this connection is that, it is metrical but not symmetrical. throughout the paper, we refer the canonical linear connection as zamkovoy canonical paracontact connection. on the other hand, the notion of locally symmetric manifolds have been weakened by many authors in several ways to a different extent. in 1977, t. takahashi [21] introduced the notion of local φ-symmetry on a sasakian manifold as a weaker version of local symmetry of such a manifold. since then, several authors studied this notion on various structures and their generalizations or extension in [6, 7, 12, 13, 17, 18, 20]. a para-kenmotsu manifold is said to be locally φ-symmetric if its curvature tensor r satisfies the condition φ2((∇w r)(x, y )u) = 0 (1.1) for any vector fields x, y, u, w orthogonal to ξ on m, where ∇ denotes the operator of covariant differentiation with respect to the metric tensor g. recently, u. c. de and a. sarkar [5] introduced the notion of local φ-ricci symmetry on a sasakian manifold. further, this notion was studied by s. ghosh and u. c. de [8] in the context of (κ, µ)cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 193 contact metric manifolds and obtained interesting results. a para-kenmotsu manifold m is said to be locally φ-ricci symmetric if the ricci operator q satisfies φ 2((∇w q)x) = 0, for any vector fields x, w orthogonal to ξ on m and s(x, w) = g(qx, w). the object of the present paper is to study the zamkovoy canonical paracontact connection on a para-kenmotsu manifold. this paper is organized as follows: section 2 is devoted to preliminaries on para-kenmotsu manifolds. in section 3, we give a brief account of information regarding the zamkovoy canonical paracontact connection ∇z on a para-kenmotsu manifold and obtain a relationship between the levi-civita connection ∇ and the zamkovoy canonical paracontact connection ∇z. in section 4, we characterize locally φ-symmetric and locally concircular φ-symmetric para-kenmotsu manifolds with respect to the connection ∇z. it is prove that the notion of local φ-symmetry (also, locally concircular φ-symmetry) with respect to the connections ∇z and ∇ are equivalent. section 5, covers the study of locally φ-ricci symmetric para-kenmotsu manifold with respect to the connection ∇z and prove that a para-kenmotsu manifold is locally φ-symmetric with respect to the connection ∇z, then the manifold is ricci symmetric and hence it is an einstein manifold. a para-kenmotsu manifold whose curvature tensor is covariant constant with respect to the connection ∇z and the manifold is recurrent with respect to the connection ∇ is studied in section 6 and shown that in this situation the manifold is η-einstein manifold. finally, we construct an example of a 3-dimensional para-kenmotsu manifold admitting the connection ∇z to illustrate some results. 2 preliminaries let m be an n-dimensional differentiable manifold, n is odd, with an almost paracontact structure (φ, ξ, η), that is, φ is a (1, 1)-tensor field, ξ is a vector field, and η is a 1-form such that φ2 = i − η ⊗ ξ, η(ξ) = 1, (2.1) φξ = 0, η · φ = 0, rank(φ) = n − 1. (2.2) let g be a pseudo-riemannian metric compatible with (φ, ξ, η), that is, g(φx, φy ) = −g(x, y ) + η(x)η(y ) (2.3) for any vector fields x, y ∈ χ(m), where χ(m) is the set of all differentiable vector fields on m, then the manifold is said to be an almost paracontact metric manifold. from (2.3) it can be easily deduce that g(x, φy ) = −g(φx, y ) and g(x, ξ) = η(x), (2.4) 194 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) for any vector fields x, y ∈ χ(m). an almost paracontact metric manifold becomes a paracontact metric manifold [25] if g(x, φy ) = dη(x, y ) with the associated metric g and is denoted by (m, g). if moreover, (∇xφ)y = g(x, φy )ξ − η(y )φx, (2.5) where ∇ denotes the pseudo-riemannian connection of g holds, then (m, g) is called an parakenmotsu manifold. from (2.5), it follows that ∇xξ = x − η(x)ξ, (2.6) (∇xη)y = g(x, y ) − η(x)η(y ), (2.7) moreover, in a para-kenmotsu manifold (m, g) of dimension n, the curvature tensor r and the ricci tensor s satisfy [23]: r(x, y )ξ = η(x)y − η(y )x, (2.8) η(r(x, y )z) = g(x, z)η(y ) − g(y, z)η(x), (2.9) r(ξ, x)y = η(y )x − g(x, y )ξ, (2.10) s(x, ξ) = −(n − 1)η(x), (2.11) s(φx, φy ) = s(x, y ) + (n − 1)η(x)η(y ), (2.12) for any vector fields x, y, z ∈ χ(m). a para-kenmotsu manifold m is said to be an η-einstein manifold if its ricci tensor s of the levi-civita connection is of the form s(x, w) = ag(x, w) + bη(x)η(w), where a and b are smooth functions on the manifold. in particular, if b = 0, then m reduces to an einstein manifold with some constant a. 3 zamkovoy canonical paracontact connection on a parakenmotsu manifold in the following, we consider a connection ∇z on an almost paracontact metric manifold using the levi-civita connection ∇ of the structure [25]: ∇zxy = ∇xy + (∇xη)y.ξ − η(y )∇xξ + η(x)φy. (3.1) if we use (2.6) and (2.7) in (3.1), we obtain ∇zxy = ∇xy + g(x, y )ξ − η(y )x + η(x)φy, (3.2) cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 195 for any vector fields x, y ∈ χ(m). we call the connection ∇z defined by (3.2) on a para-kenmotsu manifold, the zamkovoy canonical paracontact connection on a para-kenmotsu manifold. the expression for the curvature tensor r∇z with respect to the connection ∇ z is defined by r∇z (x, y )u = ∇ z x∇ z y u − ∇ z y ∇ z xu − ∇ z [x,y ]u. then, in a para-kenmotsu manifold, we have r∇z (x, y )u = r(x, y )u + g(y, u)x − g(x, u)y, (3.3) where r(x, y )u = ∇x∇y u − ∇y ∇xu − ∇[x,y ]u, is the curvature tensor of m with respect to the connection ∇. the expression (3.3) is treated as the curvature tensor of a para-kenmotsu manifold with respect to the connection ∇z. proposition 3.1. a para-kenmotsu manifold is ricci-flat with respect to the zamkovoy canonical paracontact connection if and only if it is an einstein manifold of the form s(y, u) = −(n − 1)g(y, u). proof. in a para-kenmotsu manifold m, the ricci tensor s∇z and scalar curvature r∇z of the zamkovoy canonical paracontact connection ∇z are defined by s∇z (y, u) = s(y, u) + (n − 1)g(y, u), (3.4) r∇z = r + n(n − 1), (3.5) where s and r denote the ricci tensor and scalar curvature of levi-civita connection ∇, respectively. remark 3.2. for a para-kenmotsu manifold m with respect to the zamkovoy canonical paracontact connection ∇z: (a) the curvature tensor r∇z is given by (3.3), (b) the ricci tensor s∇z is given by (3.4), (c)r∇z (x, ξ)u = r∇z (ξ, y )u = r∇z (x, y )ξ = 0, (d) r′ ∇z (x, y, u, v ) + r′ ∇z (x, y, v, u) = 0, (e) r′ ∇z (x, y, u, v ) + r′ ∇z (y, x, v, u) = 0, (f) r′ ∇z (x, y, u, v ) − r′ ∇z (u, v, x, y ) = 0, (g) r∇z (x, ξ)u = r∇z (ξ, y )u = r∇z (x, y )ξ = 0, (h) s∇z (y, ξ) = 0, (i) the ricci tensor s∇z is symmetric, (j) the scalar curvature r∇z is given by (3.5). next, suppose that a para-kenmotsu manifold is ricci flat with respect to the zamkovoy canonical paracontact connection. then from (3.4) we get s(y, u) = −(n − 1)g(y, u). (3.6) 196 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) conversely, if the manifold is an einstein manifold of the form s(y, u) = −(n − 1)g(y, u), then from (3.4) it follows that s∇(y, u) = 0: proposition 3.3. if in a para-kenmotsu manifold the curvature tensor of the zamkovoy canonical paracontact connection vanishes, then the sectional curvature of the plane determined by two vectors x, y ∈ ξ⊥ is −1. proof. let ξ⊥ denote the (n − 1)-dimensional distribution orthogonal to ξ in a para-kenmotsu manifold with respect to the zamkovoy canonical paracontact connection whose curvature tensor vanishes. then for any x ∈ ξ⊥, g(x, ξ) = 0 or, η(x) = 0. now we shall determine the sectional curvature ′r of the plane determine by the vectors x, y ∈ ξ⊥. taking inner product on both sides of (3.3) with x and then for u = y , we have r∇z (x, y, y, x) = r(x, y, y, x) + g(y, y )g(x, x) − g(x, y )g(x, y ). (3.7) putting r∇z = 0 in (3.7) we get ′r(x, y ) = r(x, y, y, x) g(x, x)g(y, y ) − g(x, y )2 = −1. this proves the require result. 4 local φ-symmetry and local concircular φ-symmetry with respect to the connections ∇z and ∇ definition 4.1. a para-kenmotsu manifold is said to be locally φ-symmetric with respect to the zamkovoy canonical paracontact connection ∇z if its curvature tensor r∇z with respect to the connection ∇z satisfies the condition φ2((∇zw r∇z )(x, y )u) = 0, (4.1) for any vector fields x, y, u, w orthogonal to ξ. proposition 4.2. a para-kenmotsu manifold is locally φ-symmetric with respect to the zamkovoy canonical paracontact connection ∇z if and only if it is so with respect to the levi-civita connection ∇. proof. let us suppose that a para-kenmotsu manifold m is locally φ-symmetric with respect to the zamkovoy canonical paracontact connection ∇z. then, by the help of (3.2), (4.1) simplifies as follow (∇zw r∇z )(x, y )u = (∇w r∇z )(x, y )u + g(w, r∇z (x, y )u)ξ − η(r∇z (x, y )u)w + η(w)φr∇z (x, y )u. (4.2) cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 197 by virtue of η(r∇z (x, y )u) = 0, (4.2) reduces to (∇zw r∇z )(x, y )u = (∇w r∇z )(x, y )u + g(w, r∇z (x, y )u)ξ + η(w)φr∇z (x, y )u. (4.3) now covariant differentiation of (3.3) with respect to w , we obtain (∇w r∇z )(x, y )u = (∇w r)(x, y )u. (4.4) using (4.4) in (4.3), we get (∇zw r∇z )(x, y )u = (∇w r)(x, y )u + {r ′(x, y, u, w) + g(y, u)g(x, w) − g(x, u)g(y, w)}ξ + η(w){φr(x, y )u + g(y, u)φx − g(x, u)φy }. (4.5) applying φ2 on both sides of (4.5); then using (2.1) and (2.2), we obtain φ2(∇zw r∇z )(x, y )u = φ 2(∇w r)(x, y )u + η(w){φr(x, y )u + g(y, u)φx − g(x, u)φy }. (4.6) if we consider x, y, u, w orthogonal to ξ, (4.6) gives to φ2((∇zw r∇z )(x, y )u) = φ 2((∇w r)(x, y )u). it completes the proof. definition 4.3. for an n-dimensional (n > 1) para-kenmotsu manifold the concircular curvature tensor c∇z with respect to the zamkovoy canonical paracontact connection is defined by c∇z (x, y )u = r∇z (x, y )u − r∇z n(n − 1) [g(y, u)x − g(x, u)y ]. (4.7) where r∇z and r∇z are the riemannian curvature tensor and scalar curvature with respect to the connection ∇z, respectively. using (3.3) and (3.5) in (4.7), we get c∇z (x, y )u = c(x, y )u, (4.8) where c(x, y )u = r(x, y )u − r n(n − 1) [g(y, u)x − g(x, u)y ] (4.9) is the concircular curvature tensor [24] with respect to the levi-civita connection ∇. thus, the concircular curvature tensor with respect to the connections ∇z and ∇ are equal. 198 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) definition 4.4. a para-kenmotsu manifold is said to be locally concircular φ-symmetric with respect to the zamkovoy canonical paracontact connection ∇z if its concircular curvature tensor c∇z with respect to the connection ∇ z satisfies the condition φ2((∇zw c∇z )(x, y )u) = 0, (4.10) for any vector fields x, y, u, w orthogonal to ξ. proposition 4.5. a para-kenmotsu manifold is locally concircular φ-symmetric with respect to the zamkovoy canonical paracontact connection ∇z if and only if it is so with respect to the levicivita connection ∇. proof. if a para-kenmotsu manifold m is locally concircular φ-symmetric with respect to the zamkovoy canonical paracontact connection ∇z, then using (3.2), (4.10) simplifies to (∇zw c∇z )(x, y )u = (∇w c∇z )(x, y )u + g(w, c∇z (x, y )u)ξ − η(c∇z (x, y )u)w + η(w)(φc∇z )(x, y )u. (4.11) now covariant differentiation of (4.8) with respect to w , yields (∇w c∇z )(x, y )u = (∇w c)(x, y )u. (4.12) making use of (4.8) and (4.12) in (4.11) we obtain (∇zw c∇z )(x, y )u = (∇w c)(x, y )u + g(w, c(x, y )u)ξ − η(c(x, y )u)w + η(w)(φc)(x, y )u. (4.13) taking account of (4.9), we write (4.13) as (∇zw c∇z )(x, y )u = (∇w c)(x, y )u + r ′(x, y, u, w)ξ + η(w)φr(x, y )u − r n(n − 1) {g(y, u)(g(x, w)ξ + η(w)φx) − g(x, u)(g(y, w)ξ + η(w)φy )} − [ r n(n − 1) + 1 ] {g(x, u)η(y )w − g(y, u)η(x)w}. (4.14) applying φ2 on both sides of above equation; then using (2.1) and (2.2) in (4.14) we have φ2(∇zw c∇z )(x, y )u = φ2(∇w c)(x, y )u + η(w)φr(x, y )u − r n(n − 1) {g(y, u)φx − g(x, u)φy )}η(w) − [ r n(n − 1) + 1 ] {g(x, u)η(y ) − g(y, u)η(x)}(w − η(w)ξ). (4.15) cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 199 if we consider x, y, u, w orthogonal to ξ, (4.15) reduces to φ2((∇zw c∇z )(x, y )u) = φ 2((∇w c)(x, y )u). (4.16) this ends the proof of the required result. proposition 4.6. let m be an n-dimensional (n > 1) locally concircular φ-symmetric parakenmotsu manifold with respect to the zamkovoy canonical paracontact connection ∇z. if the scalar curvature r with respect to the levi-civita connection ∇ is constant, then m is locally φ-symmetric. proof. now, from (4.9) we have (∇w c)(x, y )u = (∇w r∇)(x, y )u − (∇w r) n(n − 1) [g(y, u)x − g(x, u)y ]. (4.17) from (4.17) in (4.16) we obtain φ2((∇zw c∇z )(x, y )u) = φ 2((∇w r)(x, y )u) − (∇w r) n(n − 1) [g(y, u)φ2x − g(x, u)φ2y ]. (4.18) by virtue of (2.1) in (4.18) and then taking x, y, u, w orthogonal to ξ, we get φ2(∇zw c∇z )(x, y )u = φ 2((∇w r)(x, y )u) − (∇w r) n(n − 1) [g(y, u)x − g(x, u)y ]. (4.19) if r is constant, then ∇w r is zero. therefore, (4.19) gives φ2(c∇z )(x, y )u = φ 2((∇w r)(x, y )u). hence, it completes the proof of the required result. 5 local φ-ricci symmetry with respect to the connections ∇z and ∇ definition 5.1. a para-kenmotsu manifold m is said to be locally φ-ricci symmetric with respect to the zamkovoy canonical paracontact connection ∇z if its ricci operator q∇z satisfies φ2((∇zw q∇z )x) = 0, (5.1) for any vector fields x, w orthogonal to ξ, and s∇z (x, w) = g(q∇z x, w). proposition 5.2. if a para-kenmotsu manifold is locally φ-ricci symmetric with respect to the zamkovoy canonical paracontact connection, then the manifold is ricci symmetric. 200 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) proof. let us consider a para-kenmotsu manifold, which is locally φ-ricci symmetric with respect to the connection ∇z. then by virtue of (2.1) it follows from (5.1) that (∇zw q∇z )x − η((∇ z w q∇z )x)ξ = 0. (5.2) from (3.4) we can write q∇z x = qx + (n − 1)x. (5.3) again we have (∇zw q∇z )x = ∇ z w q∇z x − q∇z (∇ z w x), (5.4) using (5.3) in (5.4) we get (∇zw q∇z )x = (∇ z w q)x. (5.5) taking account of (5.5), (5.2) reduces to (∇zw q)x − η((∇ z w q)x)ξ = 0. (5.6) from (3.2) it follows that, (∇zw q)x = ∇ z w qx − q(∇ z w x), = (∇w q)x + s(w, x)ξ + (n − 1)(η(x)w + g(w, x)ξ) + η(x)qw + η(w)(φqx − qφx) (5.7) and η((∇zw q)x) = η((∇w q)x) + s(w, x) + (n − 1)g(w, x). (5.8) using (5.7) and (5.8) we get from (5.6) that (∇w q)x + (n − 1)η(x)w + η(w)(φqx − qφx) + η(x)qw − η((∇w q)x)ξ = 0. (5.9) taking inner product with u of (5.9) and considering x, w, u orthogonal to ξ, we get (∇w s)(x, u) = 0, (5.10) which implies that the manifold is ricci symmetric with respect to the levi-civita connection ∇. hence the proof. proposition 5.3. a locally φ-ricci symmetric para-kenmotsu manifold with respect to the zamkovoy canonical paracontact connection is an einstein manifold. proof. putting x = ξ in (5.10) and using (2.11), we get s(w, u) = −(n − 1)g(w, u), (5.11) for any vector fields w, u ∈ χ(m). this ends the required proof. cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 201 6 a para-kenmotsu manifold m whose curvature tensor is covariant constant with respect to the connection ∇z and m is recurrent with respect to the connection ∇ definition 6.1. a para-kenmotsu manifold m with respect to the levi-civita connection is said to be recurrent [11] if its curvature tensor r satisfies the condition. (∇w r)(x, y )u = a(w)r(x, y )u, (6.1) where a is a non-zero 1-form and x, y, u, w ∈ χ(m). proposition 6.2. if in a para-kenmotsu manifold the curvature tensor is covariant constant with respect to the zamkovoy canonical paracontact connection and the manifold is recurrent with respect to the levi-civita connection, then the manifold is an η-einstein manifold. proof. from (3.2), we can write (6.1) as (∇zw r)(x, y )u = ∇ z w r(x, y )u − r(∇ z w x, y )u − r(x, ∇ z w y )u − r(x, y )∇ z w u, = (∇w r)(x, y )u + g(w, r(x, y )u)ξ − η(r(x, y )u)w + η(x)r(w, y )u + η(y )r(x, w)u + η(u)r(x, y )w + η(w){φr(x, y )u − r(φx, y )u − r(x, φy )u − r(x, y )φu} − g(x, w)r(ξ, y )u − g(y, w)r(x, ξ)u − g(u, w)r(x, y )ξ. (6.2) using (2.7)-(2.9) in (6.2), we obtain (∇zw r)(x, y )u = (∇w r)(x, y )u + g(w, r(x, y )u)ξ + η(x)r(w, y )u + η(y )r(x, w)u + η(u)r(x, y )w + η(w){φr(x, y )u − r(φx, y )u − r(x, φy )u − r(x, y )φu} − g(x, u)η(y )w + g(y, u)η(x)w − g(x, w){η(u)y − g(y, u)ξ} − g(y, w){g(x, u)ξ − η(u)x} − g(w, u){η(x)y − η(y )x}. (6.3) let (∇zw r)(x, y )u = 0, then from (6.3), it follows that (∇w r)(x, y )u + g(w, r(x, y )u)ξ + η(x)r(w, y )u + η(y )r(x, w)u + η(u)r(x, y )w + η(w){φr(x, y )u − r(φx, y )u − r(x, φy )u − r(x, y )φu} − g(x, u)η(y )w + g(y, u)η(x)w − g(x, w){η(u)y − g(y, u)ξ} − g(y, w){g(x, u)ξ − η(u)x} − g(w, u){η(x)y − η(y )x} = 0. (6.4) 202 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) now using (6.1) in (6.4), we have a(w)r(x, y )u + g(w, r(x, y )u)ξ + η(x)r(w, y )u + η(y )r(x, w)u + η(u)r(x, y )w + η(w){φr(x, y )u) − r(φx, y )u − r(x, φy )u − r(x, y )φu} − g(x, u)η(y )w + g(y, u)η(x)w − g(x, w){η(u)y − g(y, u)ξ} − g(y, w){g(x, u)ξ − η(u)x} − g(u, w){η(x)y − η(y )x} = 0. (6.5) taking the inner product of (6.5) with ξ and using (2.2) and (2.9), it follows that a(w){g(x, u)η(y ) − g(y, u)η(x)} + g(w, r(x, y )u) + η(w){g(φy, u)η(x) − g(φx, u)η(y ) − g(x, φu)η(y ) + g(y, φu)η(x)} + g(x, w)g(y, u) − g(y, w)g(x, u) = 0. (6.6) contracrting (6.6) over x and w , we obtain s(y, u) = {a(ξ) − (n − 1)}g(y, u) − a(u)η(y ). (6.7) since the ricci tensor s with respect to the connection ∇ is symmetric; then from (6.7), we get a(u)η(y ) = a(y )η(u). (6.8) putting y = ξ in (6.8) and using (2.2) we have a(u) = a(ξ)η(u). (6.9) combining (6.7) and (6.9), it follows that s(y, u) = {a(ξ) − (n − 1)}g(y, u) − a(ξ)η(y )η(u). (6.10) this results shows that the manifold is an η-einstein manifold. hence the proof. 7 example we consider the 3-dimensional manifold m3 = {(x, y, z) ∈ r3, z 6= 0} where (x, y, z) are the standard coordinates in r3. the vector fields (see [27], example of section 7) x = ∂ ∂x , φx = ∂ ∂y , ξ = (x + 2y) ∂ ∂x + (2x + y) ∂ ∂y + ∂ ∂z are linearly independent at each point of m3. cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 203 the 1-form η = dz defines an almost paracontact structure on m3 with characteristic vector field ξ. let g, φ be the semi-riemannian metric and the (1, 1) tensor field given by g =     1 0 −(x + 2y) 0 −1 (2x + y) −(x + 2y) (2x + y) 1 − (2x + y)2 + (x + 2y)2     ϕ =     0 1 −(2x + y) 1 0 −(x + 2y) 0 0 0     with respect to the basis ∂ ∂x , ∂ ∂y , ∂ ∂z . clearly, (φ, ξ, η, g) defines an almost paracontact metric structure on m3. let ∇ be the levi-civita connection with metric g, then we have [x, φx] = 0, [x, ξ] = x + 2φx, [φx, ξ] = 2x + φx. next, by using the well-known koszul’s formula, we obtain ∇xx = −ξ, ∇φxx = 0, ∇ξx = −2φx, ∇xφx = 0, ∇φxφx = ξ, ∇ξφx = −2x, ∇xξ = x, ∇φxξ = φx, ∇ξξ = 0. hence, from the above it can be easily shown that m3(φ, ξ, η, g) is a para-kenmotsu manifold. by the above results, one can easily compute r(x, φx)ξ = 0, r(φx, ξ)ξ = −φx, r(x, ξ)ξ = −x, r(x, φx)φx = x, r(φx, ξ)φx = −ξ, r(x, ξ)φx = 0, (7.1) r(x, φx)x = φx, r(φx, ξ)x = 0, r(x, ξ)x = ξ. using (7.1), we have constant scalar curvature as follows: r = s(x, x) − s(φx, φx) + s(ξ, ξ) = −6. now consider the zamkovoy canonical paracontact connection ∇z defined by (3.2) such that ∇zxx = 0, ∇ z φxx = 0, ∇ z ξ x = −φx, ∇zxφx = 0, ∇ z φxφx = 0, ∇ z ξ φx = −x, ∇zxξ = 0, ∇ z φxξ = 0, ∇ξξ = 0. again, by the above results we can compute the components of curvature tensors with respect to the connection ∇z as follows: r∇z (x, φx)ξ = 0, r∇z (φx, ξ)ξ = 0, r∇z (x, ξ)ξ = 0, r∇z (x, φx)φx = 0, r∇z (φx, ξ)φx = 0, r∇z (x, ξ)φx = 0, (7.2) r∇z (x, φx)x = 0, r∇z (φx, ξ)x = 0, r∇z (x, ξ)x = 0. 204 d. g. prakasha, h. harish, p. veeresha & venkatesha cubo 23, 2 (2021) using (7.2), we have constant scalar curvature r∇z as follows: r∇z = s∇z (x, x) − s∇z (φx, φx) + s∇z (ξ, ξ) = 0. the above arguments easily verifies all the properties of remark 3.2 and proposition 3.1. acknowledgement authors are grateful to the referees for their valuable suggestions in improvement of the paper. cubo 23, 2 (2021) the zamkovoy canonical paracontact connection on a para-kenmotsu manifold 205 references [1] a. m. blaga, “η-ricci solitons on para-kenmotsu manifolds”, balkan j. geom. appl., vol. 20, no. 1, pp. 1-13, 2015. 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[27] s. zamkovoy, g. nakova, “the decomposition of almost paracontact metric manifolds in eleven classes revisited”, j. geom., vol. 109, no. 18, 23 pages, 2018. introduction preliminaries zamkovoy canonical paracontact connection on a para-kenmotsu manifold local -symmetry and local concircular -symmetry with respect to the connections z and local -ricci symmetry with respect to the connections z and a para-kenmotsu manifold m whose curvature tensor is covariant constant with respect to the connection z and m is recurrent with respect to the connection example cubo, a mathematical journal vol.22, n◦03, (299–314). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300299 received: 28 december, 2019 | accepted: 01 september, 2020 odd harmonious labeling of some classes of graphs p. jeyanthi 1 and s. philo 2 1research centre, department of mathematics, govindammal aditanar college for women, tiruchendur 628 215, tamil nadu, india. 2department of mathematics, st.xavier’s college, palayamkottai, tirunelveli -627002, tamilnadu, india. jeyajeyanthi@rediffmail.com, lavernejudia@gmail.com abstract a graph g(p, q) is said to be odd harmonious if there exists an injection f : v (g) → {0, 1, 2, · · · , 2q − 1} such that the induced function f∗ : e(g) → {1, 3, · · · , 2q − 1} defined by f∗(uv) = f(u) + f(v) is a bijection. in this paper we prove that tptree, t ◦̂pm, t ◦̂2pm, regular bamboo tree, cn◦̂pm, cn◦̂2pm and subdivided grid graphs are odd harmonious. resumen un grafo g(p, q) se dice impar armonioso si existe una inyección f : v (g) → {0, 1, 2, · · · , 2q − 1} tal que la función inducida f∗ : e(g) → {1, 3, · · · , 2q − 1} definida por f∗(uv) = f(u) + f(v) es una biyección. en este art́ıculo probamos que los grafos tp-árboles, t ◦̂pm, t ◦̂2pm, árboles bambú regulares, cn◦̂pm, cn◦̂2pm y cuadŕıculas subdivididas son impar armoniosos. keywords and phrases: harmonious labeling, odd harmonious labeling, transformed tree, subdivided grid graph, regular bamboo tree. 2020 ams mathematics subject classification: 05c78. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300299 300 p. jeyanthi and s. philo cubo 22, 3 (2020) 1 introduction throughout this paper by a graph is implied as a finite, simple and undirected. for standard terminology and notation we follow harary [3]. a graph g(v, e) with p vertices and q edges is called a (p, q) – graph. the graph labeling is an assignment of integers to the set of vertices or edges or both, subject to certain conditions. an extensive survey of various graph labeling problems is available in [1]. graham and sloane [2] introduced harmonious labeling during their study of modular versions of additive bases problems stemming from error correcting codes. a graph g is said to be harmonious if there exists an injection f : v (g) → zq such that the induced function f∗ : e(g) → zq defined by f∗(uv) = (f(u) + f(v)) (mod q) is a bijection and f is called harmonious labeling of g. the concept of an odd harmonious labeling was due to liang and bai [14]. a graph g is said to be odd harmonious if there exists an injection f : v (g) → {0, 1, 2, · · · , 2q − 1} such that the induced function f∗ : e(g) → {1, 3, · · · , 2q − 1} defined by f∗(uv) = f(u) + f(v) is a bijection. if f : v (g) → {0, 1, 2, · · · , q} then f is called as strongly odd harmonious labeling and g is called a strongly odd harmonious graph. the odd harmoniousness of a graph is useful for the solution of undetermined equations. the following results have been proved in [14]: 1. if g is an odd harmonious graph, then g is a bipartite graph. hence any graph that contains an odd cycle is not an odd harmonious. 2. if a (p, q) – graph g is odd harmonious, then 2 √ q ≤ p ≤ (2q − 1). 3. if g is an odd harmonious eulerian graph with q edges, then q ≡ 0, 2(mod 4). followed by this, vaidya and shah [18], [19] showed that shadow and splitting graphs are odd harmonious. selvaraju et al. [17] established that some path related graphs are odd harmonious. jeyanthi et al. proved that the following graphs are odd harmonious: double quadrilateral snake and banana tree [5], cycle related graphs [6], plus graphs [7], super subdivision graphs [8], subdivided shell graphs [9], spider and necklace graphs [10], m-shadow, m-splitting and m-mirror graphs [11] and [12], grid graphs [13]. we use the following definitions in the subsequent section. definition 1.1. let g = (v, e) be a graph. g is called a path pn if v = {v1, v2, · · · , vn} such that 1 ≤ i ≤ n, (vi, vi+1) ∈ e. definition 1.2. the cartesian product of graphs g and h denoted as g�h, is the graph with vertex set v (g) × v (h) = {(u, v)|u ∈ v (g) and v ∈ v (h)} and (u, v) is adjacent to (ú, v́) if and only if either u = ú and (v, v́) ∈ e(h) or v = v́ and (u, ú) ∈ e(g). the cartesian product of two paths pm and pn denoted by pm × pn is known as a grid graph on mn vertices and 2mn − (m + n) edges. cubo 22, 3 (2020) odd harmonious labeling of some classes of graphs 301 definition 1.3. let g be a graph with p vertices and h be any graph and x be a vertex of h. a graph g◦̂h is obtained from g and p copies of h by identifying vertex x of ith copy of h with ith vertex of g. definition 1.4. [4] let t be a tree and u0 and v0 be the two adjacent vertices in t . let u and v be the two pendant vertices of t such that the length of the path u0 − u is equal to the length of the path v0 − v. if the edge u0v0 is deleted from t and u and v are joined by an edge uv, then such a transformation of t is called an elementary parallel transformation (or an ept) and the edge u0v0 is called transformable edge. if by some sequence of ept’s, t can be reduced to a path, then t is called a tptree (transformed tree) and such sequence regarded as a composition of mappings (ept’s) denoted by p is called a parallel transformation of t . the path, the image of t under p is denoted as p(t ). a tptree and the sequence of two ept’s reducing it to a path are illustrated in figure 1. a) a tptree t b) an ept p1(t ) c) second ept p2(t ) r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r u0 v0 x0 y0 u v x0 u v x y y0 figure 1: transformed tree definition 1.5. [15] let t be a tp-tree with n vertices v1, v2, · · · , vn. the graph t ◦̂pm is obtained from t and n copies of pm by identifying a pendant vertex of i th copy of pm with vertex vi of t . definition 1.6. [16] consider k copies of paths pn of length n − 1 and stars sm with m pendant vertices. identify one of the two pendant vertices of the jth path with the centre of the jth star. identify the other pendant vertex of each path with a single vertex u0 (u0 is not in any of the star and path). the graph obtained is a regular bamboo tree. 302 p. jeyanthi and s. philo cubo 22, 3 (2020) 2 main results in this section, we prove that tptree, t ◦̂pm, t ◦̂2pm, regular bamboo tree, cn◦̂pm, cn◦̂2pm and subdivided grid graphs are odd harmonious. theorem 2.1. every tptree is strongly odd harmonious. proof. let t be a tp-tree with n vertices. by definition, there exists a parallel transformation p of t , we have v (p(t )) = v (t ) and e(p(t )) = (e(t ) − ed) ∪ ea, where ed is the set of deleted edges and ea is the set of newly added edges through the sequence p = (p1, p2, · · · , pl) of the ept’s used to obtain p(t ). hence ed and ea have the same number of edges. let u1, u2, · · · , un be the vertices of p(t ) successively, from one pendant vertex of p(t ) right up to the other. this tp-tree has n vertices and n − 1 edges. we define a labeling f : v (g) → {0, 1, 2, · · · , q = n − 1} as follows: f(ui) = i − 1, 1 ≤ i ≤ n. let (uiuj) be an edge of t , 1 ≤ i ω̂, the following inequality | (λi − a)−n | ≤ m (λ − ω̂)n , where ρ(a) is the resolvent set of a. the history function yt : [−r,0] → e defined for each θ ∈ [−r,0] by yt(θ) = y(t + θ), belongs to c([−r,0],e) the space of continuous functions equipped with the supremum norm. l : c → e is a linear bounded operator and h is a continuous function from r to e. almost periodic and periodic solutions remain the most interesting subject in the qualitative analysis of pde in view of their important applications in many real phenomena and fields. recall that the concept of almost periodic is more general than the one of periodicity. it was introduced by bochner and studied by many authors. for more details on almost periodic function we refer to [9, 16, 17, 18]. for the periodicity, there is an extensive literature related to this topic, see for example [10, 11, 25] for more details. moreover, the choice of a suitable fixed point theorem is a fundamental tool to establish the periodicity of solutions for different classes of differential equations, in fact, to find a fixed point of the well known poincaré map is equivalent to find the initial data of the periodic solution of the equation. after a long period of research and development, massera’s theorem [24] is the first result explaining the relation between the existence of bounded and periodic solutions for periodic differential equations. in finite dimensional spaces, several works have been developed on this subject. the authors in [4, 12] proved the periodicity of solutions when the solutions of periodic system are just bounded and ultimately bounded by the use of the horn’s fixed point theorem. especially, in infinite dimensional spaces, the authors in [8], used the poincaré map approach to get the periodicity of solutions for a class of retarded differential equation, they supposed that the infinitesimal generator satisfies the hille-yosida condition and generates a compact semigroup (t (t))t≥0 by applying an appropriate fixed point theorem. in [22], the authors proved the periodicity for a nonhomogeneous linear differential equation when the linear part generates a c0-semigroup on e and they obtained the existence and uniqueness of periodic solutions for this class of equations. cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 471 the present work would be a continuation and extension of the work [8] for inhomogeneous linear retarded pde, we establish the periodicity of solution for equation (1.1) by using the perturbation theory of semi-fredholm operators and without considering the compactness condition of (t (t))t≥0. to achieve this goal, we suppose that equation (1.1) admits a bounded solution on the positive real line and under suitable estimations on the norm of the operator l, we derive periodic solution of equation (1.1) from bounded ones on the positive real line by using the perturbation theory of semi-fredholm operators and the chow and hale’s fixed point theorem. this work is treated as follows, in section 2, we give some definitions and results about integral solutions of equation (1.1). moreover, we give some definitions and properties concerning the semifredholm operators. section 3 is devoted to prove and introduce some useful estimations on the integral solutions of equation (1.1). in section 4, we discuss the problem of existence of periodic solutions of equation (1.1). finally, we apply our theoretical results to an equation appearing in physical systems. 2 preliminary results in this article, we assume that: (h0) a satisfies the hille-yosida condition. we consider the following definition and results. definition 2.1 ([1]). a continuous function y : [0,+∞) → e is said to be an integral solution of equation (1.1) if the following conditions hold: (i) y : [0,+∞) → e is continuous, such that y0 = ψ, (ii) ∫ t 0 y(s)ds ∈ d(a) for t ≥ 0, (iii) y(t) = ψ(0) + a ∫ t 0 y(s)ds + ∫ t 0 (l(ys) + h(s)) ds for t ≥ 0. we can deduce from the continuity of the integral solution y that y(t) ∈ d(a), for all t ≥ 0. moreover ψ(0) ∈ d(a). in the next we define the part a0 of the operator a in d(a) as follows d(a0) = {y ∈ d(a) : ay ∈ d(a)}, and a0y = ay for y ∈ d(a0). lemma 2.2 ([2]). the operator a0 is the infinitesimal generator of a strongly continuous semigroup denoted by (t0(t))t≥0 on d(a). 472 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) theorem 2.3 ([1]). under the assumption (h0). for all ψ ∈ c such that ψ(0) ∈ d(a), equation (1.1) has a unique integral solution y(.) on [−r,+∞). furthermore, y(.) is given by y(t) = t0(t)ψ(0) + lim λ→+∞ ∫ t 0 t0(t − s)λr(λ,a) (l(ys) + h(s)) ds for t ≥ 0. through this work, the integral solutions of equation (1.1) are called plainly solutions. let y(.,ψ,l,h) be the solution of equation (1.1). we define c0 the phase space of equation (1.1) as c0 = {ψ ∈ c : ψ(0) ∈ d(a)}. let x(t) be the linear operator defined on c0 for each t ≥ 0, by x(t)ψ = yt(.,ψ,0,0), where yt(.,ψ,0,0) is the solution of the following equation  d dt y(t) = ay(t) for t ≥ 0, y0 = ψ. theorem 2.4 ([1]). (x(t))t≥0 is a linear strongly continuous semigroup on c0: (i) for all t ≥ 0, x(t) is a bounded linear operator on c0 such that x(0) = i and x(t + s) = x(t)x(s) for all t,s ≥ 0, (ii) for t ≥ 0 and θ ∈ [−r,0], (x(t))t≥0 satisfies: [x(t)ϕ](θ) =   [x(t + θ)ϕ](0), if t + θ ≥ 0,ϕ(t + θ), if t + θ ≤ 0. (iii) for all ψ ∈ c0 and t ≥ 0, x(t)ψ is a continuous function with values in c0. theorem 2.5 ([1]). under the assumption (h0). the solution y(t)ψ = yt(.,ψ,l,0) of equation (1.1) with h = 0 can be decomposed as follows: y(t)ψ = x(t)ψ + z(t)ψ for t ≥ 0, where z(t), is a bounded linear operator defined on c0, by [z(t)ψ](θ) =   lim λ→+∞ ∫ t+θ 0 t0(t + θ − s)λr(λ,a)l(ys(ψ))ds t + θ ≥ 0, 0 t + θ ≤ 0. (2.1) for each t ≥ 0. to discuss the existence of periodic solutions, we use the theory of semi-fredholm operators. we consider some definitions and propositions which are taken from [21]. cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 473 definition 2.6. let e, f be two banach spaces. a bounded linear operator f from e to f, denoted by f ∈ l(e,f), is said to be semi-fredholm from e to f if (i) dim ker(f) < ∞, where ker(f) is the null space. (ii) im(f) the range of f is closed in f. we designate by φ+(e,f) the set of all semi-fredholm operators and φ+(e) = φ+(e,e). now we recall some well known theorems for the closed range operators. let f ∈ b(e,f). then the quotient space e/ker(f) is a banach space equipped with the norm |[x]| = inf{|x + m| : m ∈ ker(f)}, where [x] = x + ker(f) := {x + m : m ∈ ker(f)}. furthermore, if dim ker(f) < ∞, then there exists a closed subspace m of e such that e = ker(f) ⊕ m. theorem 2.7 ([21]). let f be a bounded linear operator in e. then, im(f) is closed if and only if there exists a constant c̃ such that |[x]| ≤ c̃∥fx∥ for all x ∈ e. theorem 2.8 ([21]). let f be a bounded linear operator in e such that dim ker(f) < ∞. then, the following assertions are equivalent. (i) f ∈ φ+(e). (ii) there exists a constant c̃ such that |[x]| ≤ c̃∥fx∥ for all x ∈ e. (iii) there exists a constant b such that ∥(i − p)x∥ ≤ b∥fx∥ for all x ∈ e, where p is the projection operator onto ker(f) along m. we present now a result for bounded perturbation of semi-fredholm operators. theorem 2.9 ([21]). let f be an operator in φ+(e,f). if s ∈ l(e) satisfying ∥s∥ < 1 2b , where b is the constant given in theorem 2.8. then, f + s ∈ φ+(e,f) with dim ker(f + s) ≤ dim ker(f). 474 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) now, we need to introduce some well known definitions and results about the spectral theory. let j be a linear bounded operator on f, we define the measure of kuratowski of noncompactness of the operator j as follows |j |α = inf{ϵ > 0 : α(j (b)) ≤ ϵα(b), for every bounded set b ⊂ f}, where α(.) is the measure of kuratowski of noncompactness of bounded sets b ⊂ f defined by α(j) = inf{ϵ > 0 : b has a finite cover of ball of diameter < ϵ}. the essential radius ress(j ) is characterized by the following nussbaum formula introduced in [19]: ress(j ) = lim n→+∞ [|j n|α]1/n. moreover, if j is bounded and ress(j ) < 1, then i − j ∈ φ+(f). let us define the essential growth bound of a strongly continuous semigroup s := (s(t))t≥0 on a banach space f as ωess(s) := lim t→+∞ 1 t log |s(t)|α. it is well know that ress(s(t)) = exp (tωess(s)) , t > 0. 3 several estimates before discussing the periodicity of solution of equation (1.1), we need some preparatory estimates. proposition 3.1. suppose that |t0(t)| ≤ m0 eω0t for t ≥ 0. then ∥z(t)∥c ≤ m0 eω + 0 t(em0|l|m t − 1) for t ≥ 0, where ω+0 = max{ω0,0}. to prove the above proposition, we need to introduce the following lemma. lemma 3.2. let |t0(t)| ≤ m0eω0t for t ≥ 0. then, the solution of equation (1.1) in the case where h = 0 is estimated as |y(t)| ≤ m0 e(ω + 0 +m0|l|m)t. proof. for t ≥ 0, θ ∈ [−r,0], one has ∥y(t)ψ∥c = sup θ∈[−r,0] |y(t + θ,ψ)| = sup ξ∈[t−r,t] |y(ξ,ψ)|. cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 475 then for t ≥ r, sup ξ∈[t−r,t] |y(ξ,ψ)| ≤ sup ξ∈[t−r,t] |t0(ξ)ψ(0)| + sup ξ∈[t−r,t] ∣∣∣∣∣ limλ→+∞ λ ∫ ξ 0 t0(ξ − s)r(λ,a)l(y(s)ψ)ds ∣∣∣∣∣ ≤ m0eω + 0 t∥ψ∥c + m0|l|m (∫ t 0 eω + 0 (t−s) ∥y(s)ψ∥c ds ) . for t < r, sup ξ∈[t−r,t] |y(ξ,ψ)| = max { sup ξ∈[t−r,0] |y(ξ,ψ)|; sup ξ∈[0,t] |y(ξ,ψ)| } ≤ max { ∥ψ∥c; sup ξ∈[0,t] |y(ξ,ψ)| } , and sup ξ∈[0,t] |y(ξ,ψ)| ≤ m0eω + 0 t∥ψ∥c + m0|l|m (∫ t 0 eω + 0 (t−s) ∥y(s)ψ∥cds ) . finally, we obtain that ∥y(t)ψ∥c ≤ m0eω + 0 t |ψ|c + m0 |l|m ∫ t 0 eω + 0 (t−s) ∥y(s)ψ∥c ds. hence, ∥e−ω + 0 t y(t)ψ∥c ≤ m0 ∥ψ∥c + m0 |l|m ∫ t 0 e−ω + 0 s ∥y(s)ψ∥c ds. gronwall’s inequality implies that ∥e−ω + 0 t y(t)ψ∥c ≤ m0 em0 |l| m t ∥ψ∥c, and then ∥y(t)ψ∥c ≤ m0 e(ω + 0 +m0 |l| m) t ∥ψ∥c proof of proposition 3.1. let t ≥ 0, t + θ ≥ 0. then ∥z(t)ψ∥c = sup θ∈[−r,0] |(z(t)ψ)(θ)| ≤ m0|l|m (∫ t 0 eω + 0 (t−s)∥y(s)ψ∥c ds ) . from lemma 3.2, we have ∥z(t)ψ∥c ≤ m20 |l|m e ω + 0 t (∫ t 0 em0|l| m s ds ) ∥ψ∥c ≤ m0 eω + 0 t(em0|l| m t − 1) ∥ψ∥c. this implies our inequality. proposition 3.3. a function ϕ ∈ ker(i − x(ω)) if and only if ϕ(0) ∈ ker(i − t0(ω)), furthermore dim ker(i − x(ω)) = dim ker(i − t0(ω)). 476 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) proof. let ϕ ∈ ker(i − x(ω)). then, x(ω)ϕ = ϕ and (x(ω)ϕ)(θ) = ϕ(θ) for θ ∈ [−r,0]. since (x(ω)ϕ)(0) = t0(ω)ϕ(0), then ϕ(0) = t0(ω)ϕ(0), and hence ϕ(0) ∈ ker(i − t0(ω)). conversely, let x ∈ ker(i − t0(ω)) and ϕn(θ) = t0(nω + θ)x for n ≥ [ rω ] + 1, where [.] denotes the integer part. then, t0(t + ω)x = t0(t)t0(ω)x = t0(t)x for t ≥ 0, and ϕn(θ) is independent of the integer n and then ϕn(θ) = t0(ω + θ)x = ϕ(θ) and ϕ(0) = x. if ω + θ ≥ 0, then (x(ω)ϕ)(θ) = t0(ω + θ)ϕ(0) = t0(ω + θ)x = ϕ(θ). if ω + θ ≤ 0, then (x(ω)ϕ)(θ) = ϕ(ω + θ) = ϕn(ω + θ) = t0(θ + ω + nω)x = t0(nω + θ)x = ϕn(θ) = ϕ(θ), hence, x(ω)ϕ = ϕ, which implies that ϕ ∈ ker(i − x(ω)). moreover, ker(i − t0(ω)) is mapped bijectively onto the space ker(i − x(ω)). therefore, dim ker(i − x(ω)) = dim ker(i − t0(ω)). let us define the constant mω by mω = sup 0≤t≤ω |t0(t)|. as it is shown in [22], the proof is omitted here, if i − t0(ω) is semi-fredholm on d(a), then, the operator defined by sm := i − t0(ω) : m → im(i − t0(ω)), cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 477 is bijective, such that m is a subset of e and d(a) is decomposed as d(a) = ker(i − t0(ω)) ⊕ m. let s−1m be the inverse operator of sm and let k ∈ n ∗ such that (k − 1)ω < r ≤ kω. we set ik = [−r,−(k − 1)ω) and ip = [−pω,−(p − 1)ω) for p = 1,2, . . . ,k − 1 with k ≥ 2. let g be a linear operator defined from d(g) to c0 by (gϕ)(θ) = p−1∑ j=0 ϕ(θ + jω) + t0(θ + pω)s−1m ϕ(0) for θ ∈ ip, with d(g) = {ϕ ∈ c0 : ϕ(0) ∈ im(i − t0(ω))}. clearly, for θ ∈ ip,p = 1,2, . . . ,k, ∥gϕ∥c = sup θ∈[−r,0] |(gϕ)(θ)| ≤ p−1∑ j=0 ∥ϕ∥c + sup s∈[0,ω] |t0(s)||s−1m ||ϕ(0)|. then ∥gϕ∥c ≤ ( k + mω|s−1m | ) ∥ϕ∥c. (3.1) consequently, we have the following theorem. theorem 3.4. i − t0(ω) is semi-fredholm on d(a) if and only if i − x(ω) is semi-fredholm on c0 . to prove theorem 3.4, we need the following lemma lemma 3.5 ([14]). im(i − x(ω)) = d(g). proof of theorem 3.4. suppose that im(i − t0(ω)) is closed, let ϕn ∈ d(g) such that ϕn → ϕ as n → ∞. then ϕn(0) ∈ im(i − t0(ω)) and ϕn(0) → ϕ(0) ∈ im(i − t0(ω)), which implies that ϕ ∈ d(g) and d(g) is closed. lemma 3.5 implies that im(i − x(ω)) is closed. now, we suppose that im(i − x(ω)) is closed and xn ∈ im(i − t0(ω)) such that xn → x as n → ∞. let ϕn,ϕ ∈ c0 be such that ϕn(θ) = xn and ϕ(θ) = x. it is clear that ϕn → ϕ as n → ∞ and by lemma 3.5 we have that (ϕn) ∈ im(i − x(ω)). then ϕ ∈ im(i − x(ω)) and ϕ(0) = x ∈ im(i − t0(ω)). consequently, im(i − t0(ω)) is closed. therefore, by the use of proposition 3.3 we obtain the desired result. 478 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) in the nondensely defined case, we can prove the following result as in [22], the proof is omitted here. theorem 3.6. suppose that i−t0(ω) is semi-fredholm on d(a) such that dim ker(i−t0(ω)) = n. if |z(ω)| < 1 2c̃(1 + √ n) . then, i − y(ω) ∈ φ+(c0) and dim ker(i − y(ω)) ≤ n. proposition 3.7. suppose that i − t0(ω) is semi-fredholm on d(a). if |l| satisfies |l| < log ( e−ω + 0 ω 2m0c̃(1 + √ n) + 1 ) m0mω . (3.2) then, i − y(ω) ∈ φ+(c0) and dim ker(i − y(ω)) ≤ n. proof. by the inequality (3.2), it follows that m0 e ω + 0 ω(em0m|l| ω − 1) < 1 2c̃(1 + √ n) , and by proposition 3.1, one has |z(ω)| < 1 2c̃(1 + √ n) . theorem 3.6 gives that i − y(ω) ∈ φ+(c0) and dim ker(i − y(ω)) ≤ n. 4 periodic solutions for equation (1.1) to discuss the existence of periodic solutions of equation (1.1), we introduce the following fixed point theorem for a linear affine map t defined from e to e by tu = tu + v for u ∈ e, where t ∈ b(e) and v ∈ e is fixed. let ft be the set of all fixed points of the map t . theorem 4.1 ([5]). let t be a linear affine map on a banach space e such that the range im(i−t) is closed. if there is an u0 ∈ e such that {tku0,k ∈ n} is bounded in e, then ft ̸= ∅. if there exists some v ∈ ft , then ft = v + ker(i − t). cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 479 dim ft is defined as dim ft = dim ker(i − t). if i − t ∈ φ+(x). then, theorem 4.1 is refined as follows theorem 4.2 ([22]). let t be a linear affine map on a banach space e. if i − t ∈ φ+(e) and if there exists an u0 ∈ e such that {tku0,k ∈ n} is bounded, then ft ̸= ∅ and dim ft is finite. through the rest of this work, we suppose that (h1) h is an ω-periodic function. furthermore, by property (r) we mean the following equivalence: equation (1.1) has an ω−periodic solution if and only if it has a bounded one on the positive real line. then, we have the following result. theorem 4.3. under assumptions (h0) and (h1). if i − t0(ω) is semi-fredholm on d(a) and if the operator l satisfies the following estimate |l| < log ( e−ω + 0 ω 2m0c̃(1 + √ n) + 1 ) m0mω , where c̃ and n are the constants given in theorem 3.6. then, equation (1.1) satisfies the property (r). proof. let y(.,ψ,h) be the solution of equation (1.1). we introduce the poincaré map p defined from c0 to c0 as follows pω(ψ) = yω(.,ψ,h), then, pωψ = yω(.,ψ,0) + yω(.,0,h), and hence pω is an affine map such that pωψ = pψ + φ, with pψ = yω(.,ψ,0) and φ = yω(.,0,h). according to the second section, p is decomposed as p = x(ω) + z(ω). moreover, proposition 3.7 gives that i − p ∈ φ+(c0). 480 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) now, let y(.,ψ,h) denote the bounded solution of equation (1.1) on r+. thus, for each n ∈ n, we have pnωψ = ynω(.,ψ,h), and then (pnωψ)n≥0 is a bounded sequence in c0. all conditions of theorem 4.2 are satisfied and then fpω ̸= ∅, which yields an ω-periodic solution of equation (1.1). corollary 4.4. under assumptions (h0) and (h1). if i − t0(ω) is semi-fredholm in d(a) and if |l| satisfies the following inequality |l| < log ( e−ω + 0 ω 2m0 ( k + mω|s−1m | ) (1 + √ n) + 1 ) m0mω . then, equation (1.1) satisfies the property (r). to establish the proof, we need the following lemma. lemma 4.5 ([14]). suppose that i − t0(ω) is semi-fredholm on d(a). if there exists a constant c̃ > 0 such that ∥gψ∥c ≤ c̃∥ψ∥c for all ψ ∈ d(g). then, |[φ]| ≤ c̃∥(i − x(ω))φ∥c for all φ ∈ c0. proof of corollary 4.4: since, |l| < log ( e−ω + 0 ω 2m0 ( k + mω|s−1m | ) (1 + √ n) + 1 ) m0mω . it follows that, (k + mω|s−1m |)(e m0mω|l| − 1) < e−ω + 0 ω 2m0 (1 + √ n) . lemma 4.5 and estimation (3.1) implies that c̃ ≤ k + mω|s−1m |, and then c̃(em0mω|l| − 1) < e−ω + 0 ω 2m0 (1 + √ n) . finally |l| < log ( e−ω + 0 ω 2m0c̃(1 + √ n) + 1 ) m0mω . now, theorem 4.3 shows that equation (1.1) satisfies the property (r). cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 481 in the particular case where the semigroup (t0(t))t≥0 is exponentially stable, we have the following theorem. theorem 4.6. under assumptions (h0) and (h1). if the semigroup (t0(t))t≥0 is exponentially stable and if the operator l satisfies the following inequality |l| < log ( 1 2m0 ( k + mω|s−1m | ) + 1 ) m0mω . then, equation (1.1) satisfies the property (r). proof. from the exponential stability of (t0(t))t≥0, we have ωess(t0) = lim t→+∞ 1 t log |t0(t)|α ≤ lim t→+∞ 1 t log |t0(t)| = −ω0 < 0. consequently, ress(t0(ω)) = exp (ωωess(t0)) < 1. which implies that im(i − t0(ω)) is closed. on the other hand, one has |t0(ω)n| = |t0(nω)| ≤ m0e−ω0nω and |t0(nω)| 1 n ≤ m 1 n 0 e −ω0ω, which implies that the spectral radius is estimated as r(t0(ω)) = lim n→+∞ |t0(ω)n| 1 n ≤ e−ω0ω lim n→+∞ m 1 n 0 < lim n→+∞ m 1 n 0 < 1. consequently 1 /∈ σ(t0(ω)) and n = dim ker(i − t0(ω)) = 0. all conditions of corollary 4.4 are satisfied with n = 0. then, equation (1.1) satisfies the property (r). 5 application in order to apply our theoretical results, we consider the following delayed partial differential equation:  ∂ ∂t y(t,ζ) = ∂2 ∂ζ2 y(t,ζ) − ay(t,ζ) + by(t − r,ζ) + g(t,ζ) for t ∈ r+ and ζ ∈ r, y(θ,ζ) = ψ0(θ,ζ) for θ ∈ [−r,0] and ζ ∈ r, (5.1) where a and b are positive constants, g : r × r → r and ϕ : [−r,0] × r → r are continuous functions where ϕ(θ,ζ) has a finite limit at ±∞. 482 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) note that equation (5.1) can be written in the form of equation (1.1). in fact: we set r := [−∞,+∞] and we say that z ∈ ck(r) if z ∈ ck(r) and all derivatives of z up to the order k have finite limits at ±∞. then, the space of continuous functions on r, denoted by e = c(r), endowed with the norm ∥z∥∞ = sup −∞<ζ<+∞ |z(ζ)| becomes a banach space. we consider the linear operator ∆ defined from d(∆) ⊂ e to e by  d(∆) = { z ∈ c2 ( r ) : lim ζ→±∞ z(ζ) = 0 } , ∆z = z′′. then, we have lemma 5.1 ([7]). (0,+∞) ⊂ ρ(∆) and for each λ > 0 ∣∣∣(λi − ∆)−1∣∣∣ ≤ 1 λ . clearly d(∆) = { z ∈ c ( r ) : lim ζ→±∞ z(ζ) = 0 } . we write the part ∆0 of ∆ in d(∆) as  d(∆0) = { z ∈ c2 ( r ) : lim ζ→±∞ z(ζ) = lim ζ→±∞ z′′(ζ) = 0 } , ∆0z = z ′′. lemma 5.2 ([7]). ∆0 is the infinitesimal generator of a strongly continuous semigroup (t∆0(t))t≥0 on d(∆). furthermore, |t∆0(t)| ≤ 1 for t ≥ 0. let a : d(a) ⊂ e → e defined by:  d(a) = { z ∈ c2 ( r ) : lim ζ→±∞ z(ζ) = 0 } , az = z′′ − az. by lemma 5.1, it is clear that lemma 5.3. (−a,+∞) ⊂ ρ(a) and for each λ > −a ∣∣∣(λi − a)−1∣∣∣ ≤ 1 λ + a . cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 483 lemma 5.3 guarantees that the assumption (h0) is satisfied with ω̂ = −a and m = 1. moreover, d(a) = { z ∈ c ( r ) : lim ζ→±∞ z(ζ) = 0 } ⊊ e. moreover, we write the part a0 of the linear operator a in d(a) as:  d(a0) = { z ∈ c2 ( r ) : lim ζ→±∞ z(ζ) = 0 = lim ζ→±∞ z′′(ζ) = 0 } , a0z = z′′ − az. lemma 5.4. a0 is the infinitesimal generator of an exponentially stable continuous semigroup (t0(t))t≥0 on d(a). moreover, for t ≥ 0, we have |t0(t)| ≤ e−at. consider the following notations:  y(t)(ζ) = y(t,ζ) for t ∈ r +, ζ ∈ r, ψ(θ)(ζ) = ψ0(θ,ζ) for θ ∈ [−r,0], ζ ∈ r, and define the function l : c → e as follows l(ϕ)(ζ) = bϕ(−r)(ζ) for ζ ∈ r and ϕ ∈ c. h : r −→ e is defined by h(t)(ζ) = g(t,ζ) for t ∈ r and ζ ∈ r. clearly, l is a linear bounded operator from c to e. then, equation (5.1) can be written in e as follows   d dt y(t) = ay(t) + l(yt) + h(t) for t ≥ 0, y0 = ψ ∈ c. (5.2) we suppose that lim ζ→±∞ ψ0(0,ζ) = 0, then equation (5.2) has a unique integral solution y on [−r,+∞). to get the periodicity of solutions of equation (5.2), we suppose that (h2) b < a. let ρ = 1 + |h|∞ a − b where |h|∞ = sup 0≤t≤ω |h(t)|. then, we have lemma 5.5. under assumption (h2). for every ψ ∈ c such that ∥ψ∥c < ρ, the solution of equation (5.2) is bounded by ρ on r+. 484 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) proof. we proceed by contradiction. let t∗ = inf{t > 0 : |y(t,ψ)| > ρ}. from the continuity of y, one has |y(t∗,ψ)| = ρ, and there is α > 0, with |y(t,ψ)| > ρ for each t ∈ (t∗, t∗ + α). applying the variation-of-constants formula for equation (5.2) with the initial value ψ, y(t) = t0(t)ψ(0) + lim λ→+∞ λ ∫ t 0 t0(t − s)r(λ,a) (l(ys) + h(s)) ds. then, for t ≥ 0 |y(t∗,ψ)| ≤ |t0(t∗)| |ψ(0)| + ∫ t∗ 0 |t0(t∗ − s)| (|l(ys)| + |h(s)|) ds. since for 0 < s < t∗, it follows that −r ≤ s − r ≤ t∗ − r < t∗ and then |l(ys)| = b|y(s − r)| ≤ bρ, hence |y(t∗,ψ)| ≤ ρe−at∗ + (bρ + |h|∞) ∫ t∗ 0 e−a(t∗−s) ds ≤ ρe−at∗ + (1 − e−at∗) a (bρ + |h|∞) . consequently, |y(t∗,ψ)| ≤ ρe−at∗ + (bρ + (a − b)(ρ − 1)) (1 − e−at∗) a ≤ ρe−at∗ + ( ρ − 1 + b a ) (1 − e−at∗) ≤ ρe−at∗ + ρ(1 − e−at∗) ≤ ρ, which contradicts the definition of t∗, and we deduce that |y(t,ψ)| ≤ ρ for t ≥ 0. to discuss the periodicity of solutions of equation (5.2), we assume that: (h3) h is an ω-periodic function in t. theorem 5.6. suppose that (h2) and (h3) hold true. if |l| < ω−1 log ( (1 − e−aω)(1 + 2k) + 2 2 + 2k(1 − e−aω) ) , then, equation (5.2) has an ω-periodic solution. cubo 23, 3 (2021) on the periodic solutions for some retarded partial differential... 485 proof. let mω be the constant defined by mω = sup 0≤t≤ω |t0(t)|. then mω ≤ sup 0≤t≤ω e−at = 1. moreover, since |t0(ω)| < 1, one has |s−1m | = |(i − t0(ω)) −1| ≤ 1 i − |t0(ω)| ≤ 1 1 − e−aω . thus, k + mω|s−1m | ≤ k + 1 1 − e−aω , and |l| < ω−1 log ( (1 − e−aω)(1 + 2k) + 2 2 + 2k(1 − e−aω) ) < ω−1 log ( 1 2 ( k + mω|s−1m | ) + 1 ) . all condition of theorem 4.6 are satisfied. then, lemma 5.5 implies that equation (5.2) has an ω−periodic solution. acknowledgement the authors would like to thank the anonymous referees for their constructive comments and valuable suggestions, which are helpful to improve the quality of this paper. 486 a. elazzouzi, k. ezzinbi & m. kriche cubo 23, 3 (2021) references [1] m. adimy and k. ezzinbi, “local existence and linearized stability for partial functional differential equations”, dyn. syst. appl., vol. 7, no. 3, pp. 389–404, 1998. 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[26] t. yoshizawa, stability theory by liapunov’s second method, publications of the mathematical society of japan, vol. 9, tokyo: math. soc. japan, 1966. introduction preliminary results several estimates periodic solutions for equation (1.1) application cubo a mathematical journal vol.21, no¯ 03, (01–08). december 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000300001 the k-theory ranks for crossed products of c∗-algebras by the group of integers takahiro sudo department of mathematical sciences, faculty of science, university of the ryukyus, senbaru 1, nishihara, okinawa 903-0213, japan. sudo@math.u-ryukyu.ac.jp abstract we study the k-theory ranks for crossed products of c∗-algebras by the group of integers. as an application, we obtain certain estimates for the k-theory ranks of the group c∗-algebras of torsion free, finitely generated, nilpotent or solvable discrete groups, written as successive semi-direct products. resumen estudiamos los rangos de k-teoŕıa para productos cruzados de c∗-álgebras por el grupo de los enteros. como aplicación, obtenemos ciertas estimaciones para los rangos de kteoŕıa de las c∗-álgebras de grupos libres de torsión, finitamente generados, nilpotentes o solubles, escritos como productos semidirectos sucesivos. keywords and phrases: k-theory, c*-algebra, crossed product, betti number, discrete group. 2010 ams mathematics subject classification: 46l05, 46l55, 46l80 http://dx.doi.org/10.4067/s0719-06462019000300001 2 takahiro sudo cubo 21, 3 (2019) 1 introduction in this paper we study the (free or z) ranks of the k-theory groups for crossed products of c∗-algebras by z the group of integers. such c∗-algebras and their k-theory play fundamental roles in the theory of c∗-algebras and k-theory (cf. blackadar [1], pedersen [2], tomiyama [10], wegge-olsen [11]). by using the pimsner-voiculescu six-term exact sequence (pv) of the ktheory groups of the crossed product c∗-algebra a ⋊α z of a c ∗-algebra a by an action α of z by automorphisms (pimsner and voiculescu [3], cf. [1]), in section 2 we estimate the k-theory group ranks of a ⋊α z in terms of those of a. this simple result should be new in some insight and interesting in some sense, as another introductory step in this developed research area. as an easy, direct application of pv, in section 3 we obtain certain estimates for the k-theory ranks of the group c∗-algebras of torsion free, finitely generated, nilpotent or solvable discrete groups, written as successive semi-direct products by torsion free, abelian groups. there may be more other applications left to be considered, but not so many probably. may as well refer to [5], [6], [7], [8], [9] for some related details. in particular, in [5], [7], and [9], the k-theory groups of the c∗-algebras of the generalized heisenberg discrete nilpotent groups as typical examples of non-type i discrete amenable groups are computed by some methods of determining k-theory class generators as projections or unitaries, of the k-theory groups, but it seems that still, the k-theory groups of the c∗-algebras of general (torsion free, finitely generated) nilpotent (or solvable) discrete groups are not yet done completely, because of some difficulties involving successive unknown group actions. however, this time, without determining their k-theory groups as groups, the k-theory group rank estimates are obtained by us in such a way mentioned above, as the motivated examples, as given in section 3. 2 the k-theory ranks for crossed product c∗-algebras by z let a be a c∗-algebra. we denote by a ⋊α z the crossed product c ∗-algebra of a by an action α of z on a by automorphisms, where αn = α n = α ◦ · · · ◦ α as the n-fold composition of α = α1 : a → a for n ∈ z (cf. blackadar [1], pedersen [2], tomiyama [10]). there is the following pimsner-voiculescu six-term exact sequence of the k-theory abelian groups (k0 additive and k1 multiplicative) (pimsener and voiculescu [3], cf. [1]): k0(a) (id−α)∗−−−−−→ k0(a) i∗−−−−→ k0(a ⋊α z) ∂ x   ind exp   y ∂ k1(a ⋊α z) i∗ ←−−−− k1(a) (id−α)∗ ←−−−−− k1(a), where id : a → a is the identity map and i : a → a ⋊α z is the canonical inclusion map and the k-theory group maps (id − α)∗ and i∗ are induced by id − α and i, respectively, and the upward cubo 21, 3 (2019) the k-theory ranks for crossed products of c∗-algebras . . . 3 and downward arrows as the boundary maps ∂ are the index map as ind and the exponential map as exp, respectively. it follows from exactness of the pv diagram above that lemma 2.1. for any c∗-algebra a and any a ⋊α z, we have the following short exact sequences: for j = 0, 1, 0 −→ kj(a)/(id − α)∗kj(a) = kj(a)/ker(i∗) i∗−→ kj(a ⋊α z) ∂−→ im(∂) = ker(id − α)∗ → 0 with (id−α)∗kj(a) = ker(i∗) ⊂ kj(a), where (id−α)∗kj(a) is the image of kj(a) under (id −α)∗ and ker(id − α)∗ is the kernel of (id − α)∗ on k0 or k1, and im(∂) is the image of the boundary map ∂ equal to exp or ind. let g be an abelian group. we denote by rankz g the z-rank (or free rank) of g, which is also called the betti number of g, denoted as b(g). for a c∗-algebra a, set bj(a) = b(kj(a)) for j = 0, 1, each of which we call the j-th betti number of a (cf. [6]). we denote by t(g) the torsion rank of g, which is defined to be the number of direct sum components of indecomposable, finite cyclic groups in g. set tj(a) = t(kj(a)) for j = 0, 1, each of which we may call the j-th torsion rank of a. recall as a fundament fact in group theory that a finitely generated abelian group h has the following direct product decomposition: h ∼= z b(h) × z p n1 1 × · · · z p nt(h) t(h) , where p1, · · · pt(h) are primes and n1, · · · , nt(h) are some positive integers and each zpnj j = z/p nj j z for 1 ≤ j ≤ t(h) is the finite cyclic group of order pnj j , that is indecomposable, and these powers of primes are distinct. lemma 2.2. for a short exact sequence 1 → h → g → g/h → 1 of finitely generated, abelian groups, we have b(h) ≤ b(g) and b(g/h) ≤ b(g) and b(g) = b(h) + b(g/h). proof. note that there is no homomorphism from a finite torsion group to a torsion free group. hence b(h) ≤ b(g), and b(g/h) = b(g) − b(h) ≤ b(g). proposition 1. for any a ⋊α z, we have that for j = 0, 1, bj(a ⋊α z) ≤ b0(a) + b1(a) and b(kj(a)/(id − α)∗kj(a)) ≤ bj(a ⋊α z). 4 takahiro sudo cubo 21, 3 (2019) proof. by using the lemmas 2.1 and 2.2 above, we obtain bj(a ⋊α z) = bj(kj(a)/ker(i∗)) + bj+1(ker(id − α)∗) ≤ bj(kj(a)) + bj+1(kj+1(a)) for j = 0, 1 and j + 1 (mod 2), and bj(a ⋊α z) ≥ bj(kj(a)/ker(i∗)). let g be an abelian group. let gf and gt denote the free and torsion parts of g respectively, so that g ∼= gf × gt with b(g) = b(gf) and t(g) = t(gt). lemma 2.3. let g be a finitely generated, abelian group and h a subgroup. then there is the following short exact sequence of groups, preserving the free and torsion parts of h and g : 0 → h = hf × ht → g = gf × gt → g/h = (gf/hf) × (gt/ht) → 0 with gt ∼= ht × (gt/ht) and (gf/hf)t × (gt/ht) ∼= (g/h)t and (gf/hf)f = (g/h)f. it then follows that t(h) ≤ t(g) ≤ t(h) + t(g/h). proof. note that there are injective maps from z to z and from zk to zl with k ≤ l, but there is no injective map from z to a finite cyclic group. it follows that an injective map from h to g preserves their free and torsion parts. note also that gt/ht is a torsion group, but gf/hf may have its free part (gf/hf)f and torsion part (gf/hf)t. remark. the inequality t(g/h) ≤ t(g) does not hold in general. for instance, there is a quotient map from g = z to z2 = z/2z, with h = 2z, so that t(h) = t(g) = 0 < 1 = t(g/h) = t(h) + t(g/h). proposition 2. it then follows that for j = 0, 1 ∈ z2, tj(a ⋊α z) ≤ t(kj(a)/(id − α)∗kj(a)) + t(ker(id − α)∗) with ker(id − α)∗ ⊂ kj+1(a) as a subgroup, and t(kj(a)/(id − α)∗kj(a)) ≤ tj(a ⋊α z). remark. let a be a c∗-algebra. set χ(a) = b0(a) − b1(a), which is called the euler characteristic of a, where we assume that it is defined to be an integer or ±∞ (or formally ∞ − ∞). if χ(a) and χ(a ⋊α z) are finite, then it holds that χ(a ⋊α z) = 0 by using the pv diagram (see [6] or [8]). let a be a c∗-algebra. we denote by a ⋊α(1) z · · · ⋊α(n) z the n-fold successive crossed product c∗-algebra of a by successive actions α(j) of z (1 ≤ j ≤ n). it then follows that cubo 21, 3 (2019) the k-theory ranks for crossed products of c∗-algebras . . . 5 theorem 2.1. for such an n-fold successive crossed product c∗-algebra of a c∗-algebra a by n successive actions of z as above or below, we have bj(a ⋊α(1) z · · · ⋊α(n) z) ≤ 2n−1(b0(a) + b1(a)) for j = 0, 1. proof. when n = 2, we have bj(a ⋊α(1) z ⋊α(2) z) ≤ b0(a ⋊α(1) z) + b1(a ⋊α(1) z) ≤ 2(b0(a) + b1(a)). when n = 3, we have bj(a ⋊α(1) z ⋊α(2) z ⋊α(3) z) ≤ b0(a ⋊α(1) z ⋊α(2) z) + b1(a ⋊α(1) z ⋊α(2) z) ≤ 2[b0(a ⋊α(1) z) + b1(a ⋊α(1) z)] ≤ 22(b0(a) + b1(a)). the general case follows by induction with respect to n. 3 examples and more example 1. let c(tn) be the c∗-algebra of all continuous, complex-valued functions on the ndimensional torus tn, which is also the univesal c∗-algebra generated by mutually commuting n unitaries. the c∗-algebra is regarded as the successive crossed product c∗-algebra of c by trivial actions id of z: c(tn) ∼= c ∗ (z n ) ∼= c ⋊α(1) z · · · ⋊α(n) z with α(j) = id for 1 ≤ j ≤ n, via the fourier transform from c∗(zn) to c(tn), with tn as the dual group of zn. it then follows that bj(c(t n )) ≤ 2n−1(b0(c) + b1(c)) = 2n−1(1 + 0) = 2n−1 for j = 0, 1. moreover, the estimate equality holds. because kj(c(t n)) ∼= z2 n−1 (cf. [11]), which is also deduced by using the pimsner-voiculescu six-term exact sequence repeatedly. example 2. let tnθ denote the n-dimensional noncommutative torus, which is the c ∗-algebra generated by n unitaries uj such that ujuk = e 2πiθj,kukuj for 1 ≤ j, k ≤ n, where i = √ −1 and θ = (θj,k) is a n × n skew adjoint matrix over r of reals so that −θ = θt the transpose of θ (cf. [1], [11]). the c∗-algebra is regarded as the successive crossed product c∗-algebra of c by id of z: t n θ ∼= c ⋊id z ⋊α(2) z · · · ⋊α(n) z 6 takahiro sudo cubo 21, 3 (2019) and by successive actions α(j) for 2 ≤ j ≤ n given by α(j)uk = ad(uj)uk = ujuku ∗ j = e 2πiθj,kuk for 1 ≤ k ≤ j − 1. it then follows that bj(t n θ) ≤ 2 n−1(b0(c) + b1(c)) = 2 n−1(1 + 0) = 2n−1 for j = 0, 1. moreover, the estimate equality holds. b because kj(t n θ) ∼= z2 n−1 , which is deduced by using the pimsner-voiculescu six-term exact sequence repeatedly. note that example 3.1 is just the case where θ is the zero matrix. example 3. let h2n+1 be the discrete heisenberg nilpotent group of rank 2n + 1, consisting of the following (n + 2) × (n + 2) invertible matrices: h2n+1 =        1 a c 0n,1 1n b t 0 01,n 1     ∈ gln+2(r) | a, b ∈ zn, c ∈ z    where 1n is the n × n identity matrix and 0j,k is the j × k zero matrix, and with a, b ∈ zn as row vectors and bt the transpose of b. the group h2n+1 is viewed as the semi-direct product z n+1 ⋊α z n of tuples (c, b, a) identified with the matrices above, where the action α is defined by matrix multiplication as αa(c, b) = a(c, b)a −1 = (c + n∑ j=1 ajbj, b) ∈ zn+1, where a = (a1, · · · , an) = (0, 0n, a) and (c, b) = (c, b1, · · · , bn) = (c, b, 0n), with 0n = (0, · · · , 0) the zero of zn. then the group c∗-algebra c∗(h2n+1) = c ∗(zn+1 ⋊α z n) is regarded as the crossed product c∗-algebra c∗(zn+1) ⋊α z n, where the action α of the semi-direct product group is extended and identified with that of the crossed product c∗-algebra, by the same symbol as α (also in what follows). note that each element of an amenable (such as nilpotent or solvable) discrete group γ is identified with the corresponding unitary under the left regular representation λ on l2(γ) the hilbert space of all square summable, complex-valued functions on γ (cf. [2]). let ej (1 ≤ j ≤ 2n + 1) be the canonical basis for zn+1 and zn in zn+1 ⋊α zn and let uj = λej (1 ≤ j ≤ 2n + 1) be the corresponding unitaries in c∗(zn+1 ⋊α zn). then we have that αa(u1) = λαa(e1) = λe1 = u1, αa(uj) = λαa(ej) = λaj−1e1+ej = u aj−1 1 uj for 2 ≤ j ≤ n + 1. it then follows that bj(c ∗(h2n+1)) ≤ 2n−1(b0(c(tn+1)) + b1(c(tn+1)) = 2n−1(2n + 2n) = 22n for j = 0, 1. in fact, it is computed in [9, theorem 4.7] that kj(c ∗(h2n+1)) ∼= z 2 n (2 n −1)+1 for j = 0, 1, with 2n(2n − 1) + 1 ≤ 22n for n ≥ 1 (cf. [5], [7]). cubo 21, 3 (2019) the k-theory ranks for crossed products of c∗-algebras . . . 7 theorem 3.1. let g be a successive semi-direct product of torsion free, finitely generated discrete group, written as g = zn0 ⋊α(1) z n1 · · · ⋊α(k) znk for some n0, · · · , nk ≥ 1, k ≥ 1. let c∗(g) be the group c∗-algebra of g. then bj(c ∗(g)) ≤ 2n0+n1+···+nk−1 for j = 0, 1. proof. note that c∗(g) ∼= c ∗(zn0) ⋊α(1) z n1 · · · ⋊α(k) znk with c∗(zn0) ∼= c(tn0), where the right hand side above is viewed as an n1 +· · ·+nk fold, crossed product c∗-algebra by the successive actions of z. theorem 3.2. let g be a torsion free, finitely generated nilpotent discrete group, with b(g) = n. then bj(c ∗(g)) ≤ 2n−1 for j = 0, 1. proof. it is well known that such a nilpotent discrete group can be written as such a successive semi-direct product as in the theorem above. remark. these theorems above partially answer to a question as given in the remark of [9, theorem 4.7]. note that any torsion free, finitely generated solvable discrete group may be not be written as such a successive semi-direct product as above, in the sense as neither always being split nor being supper-solvable with such a normal series (cf. [4]). acknowledgement. the author would like to thank the referee for several critical comments and suggestions for some improvement as in the introduction. 8 takahiro sudo cubo 21, 3 (2019) references [1] b. blackadar, k-theory for operator algebras, second edition, cambridge, (1998). [2] g. k. pedersen, c∗-algebras and their automorphism groups, academic press (1979). [3] m. pimsner and d. voiculescu, exact sequences for k-groups and ext-groups of certain cross-product c∗-algebras, j. operator theory 4 (1) (1980), 93-118. [4] derek j. s. robinson, a course in the theory of groups, second edition, graduate texts in math., 80, springer (1996). [5] t. sudo, k-theory of continuous fields of quantum tori, nihonkai math. j. 15 (2004), 141-152. [6] t. sudo, k-theory ranks and index for c∗-algebras, ryukyu math. j. 20 (2007), 43-123. [7] t. sudo, k-theory for the group c∗-algebras of nilpotent discrete groups, cubo a math. j. 15 (2013), no. 3, 123-132. [8] t. sudo, the euler characteristic and the euler-poincaré formula for c∗-algebras, sci. math. japon. 77, no. 1 (2014), 119-138 :e-2013, 621-640. [9] t. sudo, the k-theory for the c∗-algebras of nilpotent discrete groups by examples, bulletin of the faculty of science, university of the ryukyus, no. 104, (september 2017), 1-40. [10] j. tomiyama, invitation to c∗-algebras and topological dynamics, world scientific (1987). [11] n. e. wegge-olsen, k-theory and c∗-algebras, oxford univ. press (1993). introduction the k-theory ranks for crossed product c*-algebras by z examples and more cubo a mathematical journal vol.21, no¯ 02, (41–49). august 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000200041 totally umbilical proper slant submanifolds of para-kenmotsu manifold m.s. siddesha1, c.s. bagewadi2 and d. nirmala3 1department of mathematics, new horizon college of engineering, bangalore, india 2,3department of mathematics, kuvempu university, shankaraghatta, shimoga, karnataka, india mssiddesha@gmail.com, prof bagewadi@yahoo.co.in, nirmaladraj14@gmail.com abstract in this paper, we study slant submanifolds of a para-kenmotsu manifold. we prove that totally umbilical slant submanifold of a para-kenmotsu manifold is either invariant or anti-invariant or dimension of submanifold is 1 or the mean curvature vector h of the submanifold lies in the invariant normal subbundle. resumen en este paper estudiamos subvariedades inclinadas en variedades para-kenmotsu. demostramos que una subvariedad inclinada en una variedad para-kenmotsu totalmente umbilical es invariante, o anti-invariante, o una subvariedad de dimensión 1, o el vector de curvatura media h de la subvariedad vive en el fibrado normal invariante. keywords and phrases: para-kenmotsu manifold; totally umbilical; slant submanifold. 2010 ams mathematics subject classification: 53c25, 53c40, 53d15. http://dx.doi.org/10.4067/s0719-06462019000200041 42 m.s. siddesha, c.s. bagewadi and d. nirmala cubo 21, 2 (2019) 1 introduction the study of submanifolds of an almost contact manifold is one of the utmost interesting topics in differential geometry. according to the behaviour of the tangent bundle of a submanifold with respect to action of the almost contact structure φ of the ambient manifold, there are two well known classes of submanifolds, namely, invariant and anti-invariant submanifolds. chen [4], introduced the notion of slant submanifolds of the almost hermitian manifolds. the contact version of slant submanifolds were given by lotta [12]. since then many research articles have been appeared on the existence of different contact and lorentzian manifolds (see. [1, 3, 7, 14, 15]). motivated by the above studies, in the present paper we study slant submanifolds of almost para-kenmotsu manifold and give a classification of results. also we prove that totally umbilical slant submanifolds of para-kenmotsu manifolds are totally geodesic. the paper is organized as follows: in section 2, we review some basic concepts of para-kenmotsu manifold and submanifold theory. section 3 is the main section of this paper. in this section we give the classification result of totally umbilical slant submanifolds of para-kenmotsu manifold. further, we prove that totally umbilical slant submanifolds of a para-kenmotsu manifold are totally geodesic. 2 preliminaries let m̃ be a (2m + 1)-dimensional smooth manifold, φ a tensor field of type (1, 1), ξ a vector field and η a 1-form. we say that (φ, ξ, η) is an almost para contact structure on m̃ if [18] φξ = 0, η · φ = 0, rank(φ) = 2m, (2.1) φ2 = i − η ⊗ ξ, η(ξ) = 1. (2.2) if an almost paracontact manifold admits a pseudo riemannian metric g of signature (m + 1, m) satisfying g(φ·, φ·) = −g + η ⊗ η (2.3) called almost para contact metric manifold. examples of almost para contact metric structure are given in [6] and [9]. analogous to the definition of kenmotsu manifold [10], welyczko [17] introduced para-kenmotsu structure for three dimensional normal almost para contact metric structures. the similar notion called p-kenmotsu structure appears in the sinha and sai prasad [16]. definition 2.1. an almost para contact metric manifold m(φ, ξ, η, g) is para-kenmotsu manifold if the levi-civita connection ∇̃ of g satisfies (∇̃xφ)y = g(φx, y)ξ − η(y)φx, (2.4) cubo 21, 2 (2019) totally umbilical proper slant submanifolds . . . 43 for any x, y ∈ χ(m), (where χ(m) is the set of all differential vector fields on m). from (2.4), we have ∇̃xξ = x − η(x)ξ. (2.5) assume m is a submanifold of a para-kenmotsu manifold m̃. let g and ∇ be the induced riemannian metric and connections of m, respectively. then the gauss and weingarten formulae are given respectively, by ∇̃xy = ∇xy + σ(x, y), (2.6) ∇̃xn = −anx + ∇ ⊥ xn, (2.7) for all x, y on tm and n ∈ t⊥m, where ∇⊥ is the normal connection and a is the shape operator of m with respect to the unit normal vector n. the second fundamental form σ and the shape operator a are related by: g(σ(x, y), n) = g(anx, y). (2.8) now for any x ∈ γ(tm) and v ∈ γ(t⊥m), we write φx = px + fx, (2.9) φv = pv + fv. (2.10) for x, y ∈ γ(tm), it is easy to observe from (2.1) and (2.9) that g(px, y) = −g(x, py). (2.11) the covariant derivatives of the endomorphisms φ, p and f are defined respectively as (∇̃xφ)y = ∇̃xφy − φ∇̃xy, ∀x, y ∈ γ(tm̃), (2.12) (∇̃xp)y = ∇xpy − p∇xy, ∀x, y ∈ γ(tm), (2.13) (∇̃xf)y = ∇xfy − f∇xy, ∀x, y ∈ γ(tm). (2.14) the structure vector field ξ has been considered to be tangential to m throughout this paper, else m is simply anti-invariant [12]. since ξ ∈ tm, for any x ∈ γ(tm) by virtue of (2.5) and (2.6), we have ∇xξ = x − η(x)ξ and σ(x, ξ) = 0. (2.15) making use of (2.4), (2.6), (2.7), (2.9), (2.10) and (2.12)-(2.14), we obtain (∇̃xp)y = pσ(x, y) + afyx + g(px, y)ξ − η(y)px, (2.16) (∇̃xf)y = fσ(x, y) − σ(x, py) − η(y)fx. (2.17) a submanifold m of an almost para contact metric manifold m̃ is said to be totally umbilical if σ(x, y) = g(x, y)h, (2.18) where h is the mean curvature vector of m. further m is totally geodesic if σ(x, y) = 0 and minimal if h = 0. 44 m.s. siddesha, c.s. bagewadi and d. nirmala cubo 21, 2 (2019) 3 slant submanifolds of an almost contact metric manifold for any x ∈ m and x ∈ txm such that x, ξ are linearly independent, the angle θ(x) ∈ [0, π 2 ] between φx and txm is a constant θ, that is θ does not depend on the choice of x and x ∈ m. θ is called the slant angle of m in m̃. invariant and anti-invariant submanifolds are slant submanifolds with slant angle θ equal to 0 and π 2 , respectively [5]. a slant submanifold which is neither invariant nor anti-invariant is called a proper slant submanifold. we mention the following results for later use. theorem 3.1. [1] let m be a submanifold of an almost contact metric manifold m̃ such that ξ ∈ tm. then, m is slant if and only if there exists a constant λ ∈ [0, 1] such that p2 = −λ(i − η ⊗ ξ). (3.1) further more, if θ is the slant angle of m, then λ = cos2θ. corolary 1. [1] let m be a slant submanifold of an almost contact metric manifold m̃ with slant angle θ. then, for any x, y ∈ tm, we have g(px, py) = −cos2θ(g(x, y) − η(x)η(y)), (3.2) g(fx, fy) = −sin2θ(g(x, y) − η(x)η(y)). (3.3) theorem 3.2. let m be a totally umbilical slant submanifold of a para-kenmotsu manifold m̃. then either one of the following statements is true: (i) m is invariant; (ii) m is anti-invariant; (iii) m is totally geodesic; (iv) dimm= 1; (v) if m is proper slant, then h ∈ γ(µ); where h is the mean curvature vector of m. proof. suppose m is totally umbilical slant submanifold, then we have σ(px, px) = g(px, px)h = cos2θ{−‖x‖2 + η2(x)}h. by virtue of (2.6), one can get cos2θ{−‖x‖2 + η2(x)}h = ∇̃pxpx − ∇pxpx. from (2.9), we have cos2θ{−‖x‖2 + η2(x)}h = ∇̃pxφx − ∇̃pxfx − ∇pxpx. cubo 21, 2 (2019) totally umbilical proper slant submanifolds . . . 45 applying (2.7) and (2.12), we get cos2θ{−‖x‖2 + η2(x)}h = (∇̃pxφ)x + φ∇̃pxx + afxpx − ∇ ⊥ pxfx − ∇pxpx. using (2.4) and (2.6), we obtain cos2θ{−‖x‖2 + η2(x)}h = g(φpx, x)ξ − η(x)φpx + φ(∇pxx + σ(x, px)) +afxpx − ∇ ⊥ pxfx − ∇pxpx. from (2.9), (2.11), (2.18) and the fact that x and px are orthogonal vector fields on m, we arrive at cos2θ{−‖x‖2 + η2(x)}h = −g(px, px)ξ − η(x)p2x − η(x)fpx + p∇pxx + f∇pxx +afxpx − ∇ ⊥ pxfx − ∇pxpx. then applying (3.1) and (3.2), we obtain cos2θ{−‖x‖2 + η2(x)}h = cos2θ{‖x‖2 − η2(x)}ξ + cos2θη(x){x − η(x)}ξ − η(x)fpx +p∇pxx + f∇pxx + afxpx − ∇ ⊥ pxfx − ∇pxpx. (3.4) taking inner product with px in (3.4), we get 0 = g(p∇pxx, px) + g(afxpx, px) − g(∇pxpx, px). (3.5) by virtue of (3.2), the first term of (3.5) can be written as g(p∇pxx, px) = −cos 2θ{g(∇pxx, x) − η(x)g(∇pxx, ξ)}. (3.6) we simplify the third term of (3.5) as follows g(∇pxpx, px) = g(∇̃pxpx, px) = 1 2 pxg(px, px). = 1 2 px[−cos2θ{(g(x, x) − η2(x))}] = − 1 2 cos2θ[pxg(x, x) − p(x)(g(x, ξ)g(x, ξ))] = − 1 2 cos2θ[pxg(x, x) − 2η(x)p(x)g(x, ξ)] = − 1 2 cos2θ[2g(∇̃pxx, x) − 2η(x){g(∇̃pxx, ξ) + g(x, ∇̃pxξ)}]. using (2.5), (2.6), (3.6) and the fact that x and px are orthogonal vector fields on m, we derive g(∇pxpx, px) = −cos 2θ[g(∇pxx, x) − η(x)g(∇pxx, ξ) −η(x)g(x, px − η(px)ξ)] = −cos2θ[g(∇pxx, x) − η(x)g(∇pxx, ξ)] → g(∇pxpx, px) = g(p∇pxx, px). 46 m.s. siddesha, c.s. bagewadi and d. nirmala cubo 21, 2 (2019) using this fact in (3.5), we obtain 0 = g(afxpx, px) = g(σ(px, px), fx). as m is totally umbilical slant, then from (2.18) and (3.2), we obtain 0 = −cos2θ{‖x‖2 − η2(x)}g(h, fx). (3.7) thus from (3.7), we conclude that either θ = π 2 that is m is anti-invariant which part (ii) or the vector field x is parallel to the structure vector field ξ, i.e., m is 1-dimensional submanifold which is fourth part of the theorem or h ⊥ fx, for all x ∈ γ(tm), i.e., h ∈ γ(µ) which is the last part of the theorem or h = 0, i.e., m is totally geodesic which is (iii) or fx = 0, i.e., m is invariant which is part (i). this completes the proof of the theorem. theorem 3.3. every totally umbilical proper slant submanifold of a para-kenmotsu manifold is totally geodesic. proof. let m be a totally umbilical proper slant submanifold of a para-kenmotsu manifold m̃, then for any x, y ∈ γ(tm), we have ∇̃xφy − φ∇̃xy = g(φx, y)ξ − η(y)φx. using equations (2.6) and (2.9), we get ∇̃xpy + ∇̃xfy − φ(∇xy + σ(x, y)) = g(px, y)ξ − η(y)px − η(y)fx. again from (2.6), (2.7) and (2.9), we obtain g(px, y)ξ − η(y)px − η(y)fx = ∇xpy + σ(x, py) − afyx +∇⊥xfy − p∇xy − f∇xy − φσ(x, y). as m is totally umbilical, then g(px, y)ξ − η(y)px − η(y)fx = ∇xpy + g(x, py)h − afyx + ∇ ⊥ xfy −p∇xy − f∇xy − g(x, y)φh. (3.8) taking the inner product with φh in (3.8) and from the fact that h ∈ γ(µ), we obtain g(∇⊥xfy, φh) = −g(x, y)‖h‖ 2. applying (2.7) and the property of riemannian connection the above equation takes the form g(fy, ∇⊥x φh) = g(x, y)‖h‖ 2. (3.9) cubo 21, 2 (2019) totally umbilical proper slant submanifolds . . . 47 now for any x ∈ γ(tm), we have ∇̃xφh = (∇̃xφ)h + φ∇̃xh. using the fact h ∈ γ(µ) and by virtue of (2.4), (2.7) and (2.9), we obtain − aφhx + ∇ ⊥ xφh = −pahx − fahx + φ∇ ⊥ xh. (3.10) also for any x ∈ γ(tm), we have g(∇⊥xh, fx) = g(∇̃xh, fx) = −g(h, ∇̃xfx). using (2.9), we obtain g(∇⊥xh, fx) = −g(h, ∇̃xφx) + g(h, ∇̃xpx). further from (2.6) and (2.12), we derive g(∇⊥xh, fx) = −g(h, (∇̃xφ)x) − g(h, φ∇̃xx) + g(h, σ(x, px)). using (2.4) and (2.18), the first and last term of right hand side of the above equation are identically zero and hence by (2.3), the second term gives g(∇⊥xh, fx) = g(φh, ∇̃xx). again by using (2.6) and (2.18), we obtain g(∇⊥xh, fx) = g(φh, h)‖x‖ 2 = 0. this means that ∇⊥xh ∈ γ(µ). (3.11) now, taking the inner product in (3.10) with fy for any y ∈ γ(tm), we get g(∇⊥xφh, fy) = −g(fahx, fy) + g(φ∇ ⊥ xh, fy). using (3.11) and from (3.3) and (3.9), we obtain g(x, y)‖h‖2 = sin2θ{g(ahx, y) − η(y)g(ahx, ξ)}. (3.12) hence by (2.8) and (2.18), the above equation reduces to g(x, y)‖h‖2 = sin2θ{g(x, y)‖h‖2 − η(y)g(σ(x, y), h)}. (3.13) since for a para-kenmotsu manifold m̃, σ(x, ξ) = 0 for any x tangent to m̃, thus we obtain g(x, y)‖h‖2 = sin2θ{g(x, y)‖h‖2. therefore, the above equation can be written as cos2θg(x, y)‖h‖2 = 0. (3.14) since m is proper slant submanifold, thus from (3.14) we conclude that h = 0, i.e., m is totally geodesic in m̃. this completes the proof. 48 m.s. siddesha, c.s. bagewadi and d. nirmala cubo 21, 2 (2019) references [1] a.m. blaga, invariant, anti-invariant and slant submanifolds of a para-kenmotsu manifold, bsg proceedings, 24 (2017), 19-28. [2] a.m. blaga, eta-ricci solitons on para-kenmotsu manifolds, balkan journal of geometry and its applications, 20(1) (2015), 1-13. [3] j.l. cabrerizo, a. carriazo and l.m. fernandez, slant submanifolds in sasakian manifolds, glasgow math. j., 42 (2000), 125-138. [4] b.y. chen, slant immersions, bull. aust. math. soc., 41 (1990), 135-147. [5] b.y. chen, geometry of slant submanifolds, katholieke universiteit leuven, (1990). 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[18] s. zamkovoy, canonical connections on paracontact manifolds, ann. global anal. geom., 36(1) (2008), 37-60. introduction preliminaries slant submanifolds of an almost contact metric manifold cubo, a mathematical journal vol. 23, no. 02, pp. 287–298, august 2021 doi: 10.4067/s0719-06462021000200287 weakly strongly star-menger spaces gaurav kumar 1 brij k. tyagi 2 1 department of mathematics, university of delhi, new delhi-110007, india. gaurav.maths.du@gmail.com 2 atma ram sanatan dharma college, university of delhi, new delhi-110021, india. brijkishore.tyagi@gmail.com abstract a space x is called weakly strongly star-menger space if for each sequence (un : n ∈ ω) of open covers of x, there is a sequence (fn : n ∈ ω) of finite subsets of x such that ⋃ n∈ω st(fn, un) is x. in this paper, we investigate the relationship of weakly strongly star-menger spaces with other related spaces. it is shown that a hausdorff paracompact weakly star menger p-space is star-menger. we also study the images and preimages of weakly strongly star-menger spaces under various type of maps. resumen un espacio x se llama débilmente fuertemente estrellamenger si para cada sucesión (un : n ∈ ω) de cubrimientos abiertos de x, existe una sucesión (fn : n ∈ ω) de subconjuntos finitos de x tales que ⋃ n∈ω st(fn, un) es x. en este art́ıculo, investigamos la relación entre espacios débilmente fuertemente estrella-menger con otros espacios relacionados. se muestra que un p-espacio hausdorff paracompacto débilmente estrella menger es estrella-menger. también estudiamos las imágenes y preimágenes de espacios débilmente fuertemente estrella-menger bajo diversos tipos de aplicaciones. keywords and phrases: stronlgy star-menger, star-menger, almost star-menger, weakly star-menger, covering topological spaces. 2020 ams mathematics subject classification: 54c10, 54d20, 54g10. accepted: 03 june, 2021 received: 28 january, 2021 c©2021 g. kumar et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200287 https://orcid.org/0000-0002-4010-7879 https://orcid.org/0000-0003-2660-2432 288 gaurav kumar & brij k. tyagi cubo 23, 2 (2021) 1 introduction the study of selection principles in topology and their relations to game theory and ramsey theory was started by scheepers [24] (see also [12]). in the last two decades, these have gained enough importance to become one of the most active areas of set theoretic topology. several covering properties are defined based on these selection principles ([17, 18]). a number of results in the literature show that many topological properties can be described and characterized in terms of star covering properties ([7, 21, 22]). the method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions. let us recall that a space x is countably compact (cc) if every countable open cover of x has a finite subcover. fleischman [10] defined a space x to be starcompact if for every open cover u of x, there exists a finite subset f of x such that st(f,u) = x, where st(f,u) = ⋃ {u ∈ u : u∩f 6= φ}. he proved that every countably compact space is starcompact. van douwen in [7] showed that every t2 starcompact space is countably compact, but this does not hold for t1-spaces (see [26, example 2.5]). matveev [20] defined a space x to be absolutely countably compact (acc) if for each open cover u of x and each dense subset d of x, there exists a finite subset f of d such that st(f,u) = x. it is clear that every t2-absolutely countably compact space is countably compact. kočinac et al. ([1, 2, 15, 16]), defined a space x to be strongly star-menger (ssm) if for each sequence (un : n ∈ ω) of open covers of x, there exists a sequence (fn : n ∈ ω) of finite subsets of x such that {st(fn,un) : n ∈ ω} is an open cover of x. the ssm property is stronger than the star-menger (sm) property. pansera [23], defined a space x to be weakly star-menger (wsm) if for each sequence (un : n ∈ ω) of open covers of x there is a sequence (vn : n ∈ ω) with vn a finite subset of un for each n ∈ ω, and ⋃ n∈ω st(∪vn,un) = x. wsm is weaker than the ssm property. in this paper we introduce a star property which lies between ssm and wsm called weakly strongly star-menger (wssm). the paper is organized as follows. section 2 contains some preliminaries used in the paper. in section 3 we investigate the relationship of wssm spaces with other related spaces. section 4 contains the information on subspaces and product spaces of wssm and in the last section 5 we study the image and preimage of wssm spaces under continuous maps. cubo 23, 2 (2021) weakly strongly star-menger spaces 289 2 preliminaries throughout this paper a space means topological space. the cardinality of a set a is denoted by |a| . let ω be the first infinite cardinal and ω1 the first uncountable cardinal, c the cardinality of the set of all real numbers. as usual, a cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. every cardinal is often viewed as a space with the usual order topology. other terms and symbols that we define follow [9]. we make use of two of the cardinals defined in [8]. define ωω as the set of all functions from ω to itself. for all f,g ∈ ωω, we say f ≤∗ g if and only if f(n) ≤ g(n) for all but finitely many n. the unbounding number, denoted by b, is the smallest cardinality of an unbounded subset of (ωω,≤∗). the dominating number, denoted by d, is the smallest cardinality of a cofinal subset of (ωω,≤∗). it is not difficult to show that ω1 ≤ b ≤ d ≤ c and it is known that ω1 < b = c, ω1 < d = c and ω1 ≤ b < d = c are all consistent with the axioms of zfc (see [8] for details). a space x is said to be absolutely strongly star-menger (assm) [6], if for each sequence (un : n ∈ ω) of open covers of x and each dense subset d of x, there exists a sequence (fn : n ∈ ω) of finite subsets of d such that {st(fn,un) : n ∈ ω} is an open cover of x. a space x is called star-menger (sm) [15], if for each sequence (un : n ∈ ω) of open covers of x there is a sequence (vn : n ∈ ω) with vn a finite subset of un for each n ∈ ω, and {st(∪vn,un) : n ∈ ω} is a cover of x. a space x is called almost star-menger (asm) [14], if for each sequence (un : n ∈ ω) of open covers of x there is a sequence (vn : n ∈ ω) with vn a finite subset of un for each n ∈ ω, and {st(∪vn,un) : n ∈ ω} is a cover of x. definition 2.1. a space x is called weakly strongly star-menger (wssm) if for each sequence (un : n ∈ ω) of open covers of x, there is a sequence (fn : n ∈ ω) of finite subsets of x such that ⋃ n∈ω st(fn,un) = x. from the above definitions we have the following diagram of implications: acc assm asm cc starcompact ssm wssm wsm sm t2−space t2−space p −space t2+p aracompact+p −space the purpose of this paper is to investigate the relationships of weakly strongly star-menger spaces with other spaces. in example 3.3, we have shown that the wssm property need not be ssm 290 gaurav kumar & brij k. tyagi cubo 23, 2 (2021) property. presently, we do not know that the wsm property implies wssm and asm property. on the other hand, there are several examples in the literature on star-selection principles showing that other reverse implications need not be true. 3 weakly strongly star-menger spaces and related spaces in this section, we give some results and examples showing relationships of weakly strongly starmenger with other properties. a subspace (subset) y of a space x is wssm if y is wssm as a subspace. theorem 3.1. if x has a dense wssm subspace, then x is wssm. proof. if d = x then we are done. let d be a non-trivial dense wssm subspace of x and (un : n ∈ ω) be a sequence of open covers of x. then (u ′ n : n ∈ ω) is a sequence of open covers of d, where u ′ n = {u ∩ d : u ∈ un}. therefore there exists a sequence (f ′ n : n ∈ ω) of finite subsets of d with ⋃ n∈ω st(f ′ n,u ′ n) = d. hence ⋃ n∈ω st(f ′ n,un) = x as d is dense in x. corollary 3.2. every separable topological space is wssm. given an almost disjoint family a of infinite subsets of ω (that is, the intersection of every two distinct elements of a is finite) the ψ-space or the isbell-mrówka space associated to a (denoted by ψ(a) has ω ∪ a as the underlying set, the points of ω being isolated, while the basic open neighborhoods of a ∈ a are of the form {a} ∪ (a\ f), where f ranges over all finite subsets of ω. for more details (see [3, 11]). example 3.3. there exists a tychonoff wssm space x which is not ssm. proof. let x = ω ∪ a be the isbell-mrówka space, where a is the maximal almost disjoint family of infinite subsets of ω with |a| = c. then x is not strongly star-menger ([25, example 2.3]). but x is wssm, ω being a countable dense subset of x. recall that a topological space x is a p-space [13] if every intersection of countably many open subsets of x is open. proposition 3.4. a wssm p-space x is almost star-menger. proof. let (un : n ∈ ω) be sequence of open covers of x. then there exists a sequence (fn : n ∈ ω) of finite subsets of x with ⋃ n∈ω st(fn,un) = x as x is wssm. since x is p-space, ⋃ n∈ω st(fn,un) is a closed subset of x which contains ⋃ n∈ω st(fn,un). hence ⋃ n∈ω st(fn,un) = ⋃ n∈ω st(fn,un). then we can find a sequence vn of finite subsets of un containing fn such that ⋃ n∈ω st(∪vn,un) = x. cubo 23, 2 (2021) weakly strongly star-menger spaces 291 theorem 3.5. a hausdorff paracompact weakly star-menger p-space x is star-menger. proof. let (un : n ∈ ω) be a sequence of open covers of x. since a hausdorff paracompact space is regular, for each x ∈ x there exists an open neighborhood say vn,x of x with vn,x ⊆ u for some u ∈ un. let vn be a locally finite open refinement of the open cover {vn,x : x ∈ x}. since x is wsm there exists a sequence (v ′ n : n ∈ ω) such that v ′ n is a finite subset of vn for each n ∈ ω with ⋃ n∈ω st(∪v ′ n,vn) = x. now for each v ∈ vn there is a uv ∈ un with v ⊆ uv . then for each fixed n ∈ ω, st(∪v ′ n,vn) ⊆ st(∪u ′ n,un), where u ′ n is finite subset of un such that for every v ∈ v ′ n there is u ∈ u ′ n contaning v . therefore, x = ⋃ n∈ω st(∪v ′ n,vn) = ⋃ n∈ω st(∪v ′ n,vn) = ⋃ n∈ω st(∪u ′ n,un), because x is a p-space. in [15], kočinac has shown that the property strongly star menger is equivalent to the property star-menger in hausdorff paracompact space x. then we have the following corollary: corollary 3.6. for a hausdorff paracompact p-space x, the following statements are equivalent: (1) x is strongly star-menger; (2) x is weak strongly star-menger; (3) x is almost star-menger; (4) x is weakly star-menger; (5) x is star-menger. in [13], kocev defined d-paracompact space. a space x is said to be d-paracompact if every dense family of subsets of x has a locally finite refinement. theorem 3.7. a wsm and d-paracompact space x is almost star-menger. proof. let (un : n ∈ ω) be a sequence of open covers of x. as x is wsm, there exists a sequence (vn : n ∈ ω), where vn is a finite subset of un with ⋃ {st(∪vn,un) : n ∈ ω} dense in x. by the assumption {st(∪vn,un) : n ∈ ω} has a locally finite refinement say, w. then ∪w = ⋃ n∈ω st(∪vn,un) and therefore ∪w = ⋃ n∈ω st(∪vn,un). as w is a locally finite family, we have ∪w = ⋃ w∈w w. since each w ∈ w is contained in st(∪vn,un) for some n ∈ ω, ⋃ n∈ω st(∪vn,un) = x. corollary 3.8. for a hausdorff paracompact d-paracompact space x, the following statements are equivalent: (1) x is strongly star-menger; (2) x is weak strongly star-menger; 292 gaurav kumar & brij k. tyagi cubo 23, 2 (2021) (3) x is weakly star-menger; (4) x is almost star-menger; (5) x is star-menger. at the end of this section, we study the relation of wssm property to lindelöf covering properties. recall, a space x is called lindelöf if for each open cover u of x there is countable subset v of u such that x = ⋃ v. let x be a space of the example 4.1, then x is wssm space but it is not lindelöf, because x has a uncountable discrete closed subset. that means wssm property does not imply lindelöf property. theorem 3.9. every t2-paracompact wssm space is lindelöf. proof. let u be an open cover of x. for each x ∈ x there exists an open neighborhood say vx of x such that vx ⊆ u for some u ∈ u, because a t2-paracompact space is regular. let v be a locally finite open refinement of the cover {vx : x ∈ x}. then (vn : n ∈ ω) be a sequence of open covers of x, where vn = v for each n ∈ ω. since x is wssm, there exists a sequence (fn : n ∈ ω) of finite subsets of x such that ⋃ n∈ω st(fn,vn) = x. since vn is locally finite family, there exist finite subset v ′ n of vn such that st(fn,vn) ⊂ ∪v ′ n, so x = ⋃ n∈ω st(fn,vn) ⊂ ⋃ n∈ω ∪{v ′ : v ′ ∈ v′n} = ⋃ n∈ω ∪{v ′ : v ′ ∈ v′n} = ⋃ n∈ω ∪{v ′ : v ′ ∈ v′n}. for each v ∈ vn there is a uv ∈ u with v ⊆ uv . hence we can constuct a countable subset u′ of u such that ⋃ n∈ω ∪{v ′ : v ′ ∈ v′n} ⊂ ⋃ u′. definition 3.10. [7] a space x is called strongly star-lindelöf (in short, ssl) if for each open cover u of x there is a countable subset f of x such that st(f,u) = x. clearly, ssm property implies ssl property. but next we will show that wssm property need not be ssl property. a space x is almost star countable [28], if for each open cover u of x there exists a countable subset f of x such that ⋃ x∈f st(x,u) = x. evidently, strongly star-lindelöf ⇒ almost star-countable. example 3.11. a wssm space need not be ssl. proof. let d be a discrete space of cardinality ω1, x = (βd × (ω + 1)) \ ((βd \ d) × {ω}) is a subspace of the product space βd × (ω + 1). then x is wssm by lemma 4.4., because βd × ω is a dense σ-compact (hence, σ-countably compact) subset of x. but x is not ssl, because x is not almost star countable (see [28, example 2.5]). cubo 23, 2 (2021) weakly strongly star-menger spaces 293 theorem 3.12. every t2-paracompact wssm space is ssl. proof. the proof follows the same constructions of theorem 3.9, thus omitted. 4 subspaces and product spaces in this section we study subspaces of a wssm space and also show that product of two wssm spaces need not be wssm. for some relative version of star selection principles see ([4, 5, 19]). example 4.1. a closed subset of wssm space need not be wssm. proof. let r be the set of real numbers, i the set of irrational numbers and q the set of rational numbers. for each irrational x we choose a sequence {xi : i ∈ ω} of rational numbers converging to x in the euclidean topology. the rational sequence topology τ (see [29, example 65]) is then defined by declaring each rational open and selecting the sets un(x) = {xn,i : i ∈ ω} ∪ {x} as a basis for the irrational point x. then the set of irrational points i is a closed subset of (r,τ) and i as a subspace of the space (r,τ) is not wssm, because it is uncountable discrete subspace. on the other hand, (r,τ) is wssm, because q is dense in (r,τ). proposition 4.2. every clopen subset of a wssm space is wssm. proof. let y be a clopen subset of a wssm space x and let (un : n ∈ ω) be a sequence of open covers of y. then (vn : n ∈ ω), where vn = un ∪ {x \ y } is a sequence of open covers of x. since x is wssm, there exists a sequence of finite subsets fn of x with ⋃ n∈ω st(fn,vn) = x. put f ′ n = y ∩ fn. then clearly, ⋃ n∈ω st(f ′ n,un) = y. song [25] gave an example showing that the product of two countably compact spaces is not strongly star compact. this example also shows that the product of two wssm spaces need not be wssm. we sketch it below. example 4.3. there exist two countably compact (and hence wssm) spaces x and y such that x × y is not wssm. proof. let d be the discrete space of the cardinality c. we define x = ⋃ α<ω1 eα, y = ⋃ α<ω1 fα, where eα and fα are the subsets of β(d) which are defined inductively so as to satisfy the following three conditions: (1) eα ∩ fβ = d if α 6= β. (2) |eα| ≤ c and |fα| ≤ c. (3) every infinite subset of eα (resp., fα) has an accumulation point in eα+1 (resp, fα+1). 294 gaurav kumar & brij k. tyagi cubo 23, 2 (2021) these sets eα and fα are well-defined since every infinite closed set in β(d) has the cardinality 2c, for detail see [30]. then, x ×y is not wssm since the diagonal {< d,d >: d ∈ d} is a discrete open and closed subset of x × y with the cardinality c and the property wssm is preserved by open and closed subsets. hence product of wssm spaces need not be wssm. now we give some positive results. recall that a subset a of a space x is said to be σ-countably compact if it is union of countably many countably compact subset of x. lemma 4.4. if a space x has a σ-countably compact dense subset, then x is wssm. proof. let d = ⋃ n∈ω dn be a dense subset of x, where each dn is countably compact subset of x. let (un : n ∈ ω) be a sequence of open covers of x. then for each n ∈ ω there exists a finite subset say fn of dn such that dn ⊂ st(fn,un). so (fn : n ∈ ω) is a sequence of finite subsets of d such that d = ⋃ n∈ω st(fn,un). theorem 4.5. let {xn : n ∈ ω} be a countable collection of countably compact subsets of space x such that x = ⋃ n∈ω xn. then x is a wssm space. corollary 4.6. if {xn : n ∈ ω} is a countable collection of mutually disjoint countably compact spaces, then the topological sum ⊕ n∈ω xn is wssm if and only if each xn is wssm. corollary 4.7. if x is countably compact space, then x × ω, where ω has discrete topology is wssm. proof. since x × ω is homeomorphic to ⊕ n∈ω x × {n} and x × {n} is homeomorphic to x for each n ∈ ω. then, by corollary 4.6, ⊕ n∈ω x × {n} is wssm. 5 images and preimages in this section we study the images and preimages of wssm spaces under continuous maps. theorem 5.1. a continuous image of a wssm space is wssm. proof. let f : x → y be a continuous surjection and x be a wssm space. let (un : n ∈ ω) be sequence of open covers of y . then (u ′ n : n ∈ ω), where u ′ n = {f −1(u) : u ∈ un} is a sequence of open covers of x. thus there exists a sequence (f ′ n : n ∈ ω) of finite subsets of x such that ⋃ n∈ω st(f ′ n,u ′ n) = x. let fn = f(f ′ n). then (fn : n ∈ ω) is a sequence of finite subsets of y. hence result follows from the fact that for an arbitrary y ∈ y and each neighbourhood u of y, u ⋂ ⋃ n∈ω st(fn,un) 6= φ. cubo 23, 2 (2021) weakly strongly star-menger spaces 295 next we turn to consider preimages. we show that the preimage of a wssm space under a closed 2-to-1 continuous map need not be wssm. first we discuss examples. recall the alexandroff duplicate a(x) of a space x. the underlying set a(x) is x × {0,1}; each point of x × {1} is isolated and a basic neighborhood of < x,0 > ∈ x ×{0} is a set of the form (u × {0}) ∪ ((u × {1}) \ {< x,1 >}), where u is a neighborhood of x in x. example 5.2. assuming d = c, there exists a wssm space x such that a(x) is not wssm. proof. assume that d = c. let x = ω ∪ a be the isbell–mrówka space with |a| = ω1. then x is absolutely strongly star-menger ([27, example 3.5]), and hence wssm. however a(x) is not wssm. since the set a × {1} is an open and closed subset of a(x) with |a × {1}| = ω1, and for each a ∈ a, the point < a,1 > is isolated in a(x) . hence a(x) is not wssm, since every open and closed subset of a wssm space is wssm, and a × {1} is not wssm . example 5.3. assuming d = c, there exists a closed 2-to-1 continuous map f : x → y such that y is a wssm space, but x is not a wssm. proof. let y be the space x of example 5.2. then y is wssm. let x be the space a(y ). then x is not wssm. let f : x → y be the projection. then f is a closed 2-to-1 continuous map, which completes the proof. theorem 5.4. let f be an open and closed, finite-to-one continuous map from a space x onto a wssm space y. then x is wssm. proof. let (un : n ∈ ω) be a sequence of open covers of x and let y ∈ y. since f −1(y) is finite, for each n ∈ ω there exists a finite sub-collection uny of un such that f −1(y) ⊂ ∪uny and u ∩ f−1(y) 6= φ for each u ∈ uny . since f is closed, there exists an open neighbourhood vny of y in y such that f −1(vny ) ⊆ ∪{u : u ∈ uny }. since f is open, we can assume that vny ⊆ ∩{f(u) : u ∈ uny }. for each n ∈ ω, take such open set vny for each y ∈ y, and put vn = {vny : y ∈ y } of y . thus (vn : n ∈ ω) is a sequence of open covers of y. since y is wssm, there exists a sequence (fn : n ∈ ω) of finite subsets of y such that ⋃ n∈ω st(fn,vn) = y . since f is finite to one, the sequence (f−1(fn) : n ∈ ω) is a sequence of finite subsets of x. we show that ⋃ n∈ω st(f−1(fn),un) = x. let x ∈ x and v be an arbitrary neighbourhood of x in x, then f(v ) is a neighbourhood of y = f(x) as f is an open map. then there exist n ∈ ω and y′ ∈ y such that y ∈ f(v ) ∩vn y′ with vn y′ ∩fn 6= φ. choose u ∈ un y′ . then vn y′ ⊆ f(u). hence u ∩f−1(fn) 6= φ as vn y′ ∩ fn 6= φ. therefore, x ∈ ⋃ n∈ω st(f−1(fn),un). this shows that x is wssm. 296 gaurav kumar & brij k. tyagi cubo 23, 2 (2021) acknowledgements (1) the first author acknowledges the fellowship grant of university grant commission, india. 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[30] r. c. walker, the stone-čech compactification, ergebnisse der mathematik und ihrer grenzgebiete, band 83, new york-berlin: springer, 1974. introduction preliminaries weakly strongly star-menger spaces and related spaces subspaces and product spaces images and preimages cubo, a mathematical journal vol.22, no¯ 01, (71–84). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100071 certain results on the conharmonic curvature tensor of (κ, µ)-contact metric manifolds divyashree g. 1 and venkatesha 2 1 department of mathematics, govt., science college, chitradurga-577501, karnataka, india. 2 department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india. gdivyashree9@gmail.com, vensmath@gmail.com abstract the paper presents a study of (κ, µ)-contact metric manifolds satisfying certain conditions on the conharmonic curvature tensor. resumen el art́ıculo presenta un estudio de variedades (κ, µ)-contacto métricas satisfaciendo ciertas condiciones sobre el tensor de curvatura conharmónico. keywords and phrases: (κ, µ)-contact metric manifold, conharmonically flat, conharmonically locally φ-symmetric, φ-conharmonically semisymmetric, h-conharmonically semisymmetric. 2010 ams mathematics subject classification: 53c25, 53c50, 53d10 http://doi.org/10.4067/s0719-06462020000100071 72 divyashree g. & venkatesha cubo 22, 1 (2020) 1 introduction in 1995, blair et al.[3] introduced the idea of a class of contact metric manifolds for which the characteristic vector field ξ belongs to the (κ, µ)-nullity distribution for some real numbers κ and µ and such type of manifolds are called (κ, µ)-contact metric manifold. the non-sasakian (κ, µ)contact metric manifolds have two classes, namely, the class consists of the unit tangent sphere bundles of spaces of constant curvature, equipped with the natural contact metric structure and the class contains all the three-dimensional unimodular lie groups, except the commutative one admitting the structure of a left invariant (κ, µ)-contact metric manifold [3, 4, 9]. boeckx [4] given a full classification of (κ, µ)-contact metric manifolds. (κ, µ)-contact metric manifolds have been studied by several authors in [5, 6, 13, 11] and others. a rank-four tensor n that remains invariant under conharmonic transformation for a (2n+1)dimensional riemannian manifold m is given by n(x, y)z = r(x, y)z − 1 2n − 1 [s(y, z)x − s(x, z)y (1.1) +g(y, z)qx − g(x, z)qy], which is also of the form n(x, y, z, t) = r(x, y, z, t) − 1 2n − 1 [s(y, z)g(x, t) − s(x, z)g(y, t) (1.2) +g(y, z)g(qx, t) − g(x, z)g(qy, t)], where r, s and q represents the riemannian curvature tensor, ricci tensor and ricci operator respectively. a manifold whose conharmonic curvature vanishes at every point of the manifold is called conharmonically flat manifold. such a curvature tensor have been extensively studied by siddiqui and ahsan [12], ozgur [8], avijit sarkar et al. [10], asghari and taleshian [7] and many others. our present work is organised in the following way: after introduction, section 2 includes basics related to (κ, µ)-contact metric manifold which will be used later. section 3 deals with conharmonically flat (κ, µ)-contact metric manifolds. we proved that conharmonically locally φsymmetric (κ, µ)-contact metric manifold is locally isometric to the riemannian product en+1(0)× sn(4) in section 4. section 5 and 6 are devoted to the study of h-conharmonically semisymmetric and φ-conharmonically semisymmetric non-sasakian (κ, µ)-contact metric manifolds respectively. finally, we have shown that if the conharmonic curvature tensor on a (κ, µ)-contact metric manifold is divergent free then the ricci tensor s is a codazzi tensor. cubo 22, 1 (2020) certain results on the conharmonic curvature tensor of (κ, µ) . . . 73 2 preliminaries a (2n+1)-dimensional differentiable manifold m2n+1 is called a contact manifold [1] if it carries a global 1-form η such that η ∧(dη)2n+1 6= 0 everywhere on m2n+1. it is well known that a contact metric manifold admits an almost contact metric structure (φ, ξ, η, g), where φ is a (1, 1)-tensor field, ξ is the characteristic vector field, and a riemannian metric g such that φ2 = −i + η ⊗ ξ, g(x, ξ) = η(x), (2.1) η(ξ) = 1, g(x, y) = g(φx, φy) + η(x)η(y). (2.2) dη(x, y) = g(x, φy), g(x, φy) = −g(y, φx), (2.3) for all vector fields x, y ∈ tm2n+1 and then we call a structure as contact metric structure. a manifold m2n+1 with such a structure is said to be contact metric manifold and it is denoted by (φ, ξ, η, g). φξ = 0, η ◦ φ = 0, dη(ξ, x) = 0. (2.4) we define a (1, 1)-tensor field h by h = 1 2 £ξφ, where £ξ is the lie differentiation in the direction of ξ. since the tensor field h is self-adjoint and anticommutes with φ, we have hξ = 0, φh + hφ = 0, trh = trφh = 0, (2.5) ∇xξ = −φx − φhx, (2.6) (∇xφ)y = g(x, y)ξ − η(y)x, (2.7) where ∇ is the levi-civita connection and if x 6= 0 is an eigenvector of h corresponding to the eigenvalue λ, then φx is an eigenvector of h corresponding to the eigenvalue −λ. blair et al. [3] studied the (κ, µ)-nullity condition and the (κ, µ)-nullity distribution n(κ, µ) of a contact metric manifold m is defined by [3] n(κ, µ) : p −→ np(κ, µ) (2.8) = [z ∈ tpm : r(x, y)z = (κi + µh){g(y, z)x − g(x, z)y}], for all x, y ∈ tm2n+1. a contact metric manifold m2n+1 with ξ ∈ n(κ, µ) is called a (κ, µ)-contact metric manifold. in a (κ, µ)-contact metric manifold, we have r(x, y)ξ = κ{η(y)x − η(x)y} + µ{η(y)hx − η(x)hy}, (2.9) for all x, y ∈ tm2n+1. in a (κ, µ)-contact metric manifold, the following relations hold [3, 11]: 74 divyashree g. & venkatesha cubo 22, 1 (2020) h2 = (κ − 1)φ2, (2.10) (∇xφ)y = g(x + hx, y)ξ − η(y)(x + hx), (2.11) r(ξ, x)y = κ[g(x, y)ξ − η(y)x] + µ[g(hx, y)ξ − η(y)hx], (2.12) s(x, ξ) = 2nκη(x), (2.13) s(x, y) = [2(n − 1) − nµ]g(x, y) + [2(n − 1) + µ]g(hx, y) (2.14) +[2(1 − n) + n(2κ + µ)]η(x)η(y), qx = [2(n − 1) − nµ]x + [2(n − 1) + µ]hx (2.15) +[2(n − 1) + n(2κ + µ)], s(φx, φy) = s(x, y) − 2nkη(x)η(y) − 2(2n − 2 + µ)g(hx, y), (2.16) g(qx, y) = s(x, y). (2.17) from (2.6), we have (∇xη)y = g(x + hx, φy), (2.18) (∇xh)y = {(1 − κ)g(x, φy) + g(x, hφy)}ξ + η(y){h(φx + φhx)} (2.19) −µη(x)φhy, where s is the ricci tensor of type (0, 2), q is the ricci operator and r is the scalar curvature of the manifold. it is well known that in a sasakian manifold, the ricci operator q commutes with φ. but in a (κ, µ)-contact metric manifold q does not commute with φ. in general, in a (κ, µ)-contact metric manifold blair et al.[3] proved the following: proposition 1. let mn be a (κ, µ)-contact metric manifold, then the relation qφ − φq = 2[2(n − 1) + µ]hφ, holds. from the definition of η-einstein manifold, it follows easily that qφ = φq. hence from proposition 2.1 we obtain either µ = −2(n−1), or the manifold is sasakian. using µ = −2(n−1), from (2.14) we obtain that the manifold is an η-einstein manifold. therefore yildiz and de [13] proved the following: proposition 2. in a non-sasakian (κ, µ)-contact metric manifold, the following conditions are equivalent: (i) η-einstein manifold, (ii) qφ = φq. cubo 22, 1 (2020) certain results on the conharmonic curvature tensor of (κ, µ) . . . 75 for n = 1, from proposition 2.1 and proposition 2.2, yildiz and de [13] obtained the following: corolary 1. a 3-dimensional non-sasakian (κ, µ)-contact η-einstein manifold is an n(k)-contact metric manifold. lemma 2.1. [2]:let m2n+1 (φ, ξ, η, g) be a contact metric manifold with r(x, y)ξ = 0 for all vector fields x, y tangent to m2n+1. then m2n+1 is locally isometric to the riemannian product en+1(0) × sn(4). 3 conharmonically flat (κ, µ)-contact metric manifolds from (1.2), for a (2n+1)-dimensional conharmonically flat (κ, µ)-contact metric manifold, we have r(x, y, z, t) = 1 2n − 1 [s(y, z)g(x, t) − s(x, z)g(y, t) + g(y, z)g(qx, t) (3.1) −g(x, z)g(qy, t)]. substituting z = ξ in (3.1) and using (2.1), (2.9) and (2.13), we obtain κ[η(y)g(x, t) − η(x)g(y, t)] + µ[η(y)g(hx, t) − η(x)g(hy, t)] (3.2) = 1 2n − 1 [2nκη(y)g(x, t) − 2nκη(x)g(y, t) + η(y)g(qx, t) − η(x)g(qy, t)]. again, by taking y = ξ and using (2.1), (2.2), (2.5) and (2.13), (3.2) becomes s(x, t) = −κg(x, t) + (2n + 1)κη(x)η(t) + (2n − 1)µg(hx, t). (3.3) from the equation (3.3), it follows that if µ = 0, then the manifold is an η-einstein manifold. conversely, if the manifold is η-einstein, then we can write s(x, t) = a1g(x, t) + b1η(x)η(t). (3.4) on equating (3.3) and (3.4), we find a1g(x, t) + b1η(x)η(t) = −κg(x, t) + (2n + 1)κη(x)η(t) + (2n − 1)µg(hx, t).(3.5) now, in (3.5) replacing t by φx and using (2.3), we get (2n − 1)µg(hx, φx) = 0, (3.6) for all x. consequently, µ = 0. hence, an n-dimensional conharmonically flat (κ, µ)-contact metric manifold is an η-einstein manifold if and only if µ = 0. but from (2.14), it follows that a (κ, µ)-contact metric manifold is 76 divyashree g. & venkatesha cubo 22, 1 (2020) η-einstein if and only if {2(n − 1) + µ} = 0. if we consider a (2n + 1)-dimensional (n > 1) conharmonically flat η-einstein (κ, µ)-contact metric manifold, then n = 1, which contradicts the fact that n > 1. hence, the theorem can be stated as follows: theorem 3.1. an (2n + 1)-dimensional (n > 1) conharmonically flat (κ, µ)-contact metric manifold cannot be an η-einstein manifold. 4 conharmonically locally φ-symmetric (κ, µ)-contact metric manifolds definition 4.1. an (2n + 1)-dimensional (n > 1) (κ, µ)-contact metric manifold m2n+1 is said to be conharmonically locally φ-symmetric if it satisfies φ2((∇wn)(x, y)z) = 0, (4.1) for all x, y, z, w orthogonal to ξ. taking covariant differentiation of (1.1), we have (∇wn)(x, y)z = (∇wr)(x, y)z − 1 2n − 1 [(∇ws)(y, z)x − (∇ws)(x, z)y (4.2) +g(y, z)(∇wq)(x) − g(x, z)(∇wq)(y)], where ∇ denotes the levi-civita connection on the manifold. differentiating equations (2.8), (2.14) and (2.15) covariantly with respect to w, we obtain (∇wr)(x, y)z = wκ{g(y, z)x − g(x, z)y} + wµ{g(y, z)hx − g(x, z)hy} (4.3) +µ[g(y, z)({(1 − κ)g(w, φx) + g(w, hφx)}ξ +η(x){h(φw + φhw)} − µη(w)φhx) −g(x, z)({(1 − κ)g(w, φy) + g(w, hφy)}ξ +η(y){h(φw + φhw)} − µη(w)φhy)], (∇ws)(y, z)x = {2(1 − n) + n(2κ + µ)}[g(w, φy)η(z)x (4.4) +g(hw, φy)η(z)x + g(w, φz)η(y)x + g(hw, φz)η(y)x] +(2(n − 1) + µ)[{(1 − κ)g(w, φy)η(z)x + g(w, hφy)η(z)x +g(h(φw + φhw), z)η(y)x} − µg(φhy, z)η(w)x] cubo 22, 1 (2020) certain results on the conharmonic curvature tensor of (κ, µ) . . . 77 and (∇wq)(x) = {2(n − 1) + µ}[{(1 − κ)g(w, φx) + g(w, hφx)}ξ (4.5) +η(x){h(φw + φhx)} − µη(w)φhx] +{2(n − 1) + n(2κ + µ)}g(w, φx)ξ +{2(n − 1) + n(2κ + µ)}g(hw, φx)ξ −{2(n − 1) + n(2κ + µ)}η(x)φw −{2(n − 1) + n(2κ + µ)}η(x)φhw. now, considering equations (4.3), (4.4) and (4.5) in (4.2) and also taking x, y, z, w orthogonal to ξ, we get (∇wn)(x, y)z = wκ[g(y, z)x − g(x, z)y] + wκ[g(y, z)hx − g(x, z)hy] (4.6) +µ[(1 − κ)g(y, z)g(w, φx)ξ + (1 − κ)g(y, z)g(w, hφx)ξ −(1 − κ)g(x, z)g(w, φy)ξ − (1 − κ)g(x, z)g(w, hφy)ξ] − 1 2n − 1 [{2(n − 1) + µ}{(1 − κ)[g(y, z)g(w, φx)ξ −g(x, z)g(w, φy)ξ] + g(y, z)g(w, hφx)ξ −g(x, z)g(w, hφy)ξ} +{2(n − 1) + n(2κ + µ)}[g(y, z)g(w, φx)ξ +g(y, z)g(hw, φx)ξ − g(x, z)g(w, φy)ξ −g(x, z)g(hw, φy)ξ]]. applying φ2 on both sides of (4.6), one can obtain φ2((∇wn)(x, y)z) = φ 2 {wκ[g(y, z)x − g(x, z)y] + wκ[g(y, z)hx (4.7) −g(x, z)hy] + µ[(1 − κ)g(y, z)g(w, φx)ξ +(1 − κ)g(y, z)g(w, hφx)ξ − (1 − κ)g(x, z)g(w, φy)ξ −(1 − κ)g(x, z)g(w, hφy)ξ] − 1 2n − 1 [{2(n − 1) + µ}{(1 − κ)[g(y, z)g(w, φx)ξ −g(x, z)g(w, φy)ξ] + g(y, z)g(w, hφx)ξ −g(x, z)g(w, hφy)ξ} +{2(n − 1) + n(2κ + µ)}{g(y, z)g(w, φx)ξ +g(y, z)g(hw, φx)ξ − g(x, z)g(w, φy)ξ −g(x, z)g(hw, φy)ξ}]}. 78 divyashree g. & venkatesha cubo 22, 1 (2020) from (4.1) and using (2.1), (4.7) becomes (wκ)[g(x, z)y − g(y, z)x] + (wκ)[g(y, z)η(x) − g(x, z)η(y)]ξ (4.8) +(wµ)[g(x, z)hy − g(y, z)hx] = 0. again, considering x, y orthogonal to ξ, one can get (wκ)[g(x, z)y − g(y, z)x] + (wµ)[g(x, z)hy − g(y, z)hx] = 0. (4.9) by taking inner product of (4.9) with v, we have (wκ)[g(x, z)g(y, v) − g(y, z)g(x, v)] + (wµ)[g(x, z)g(hy, v) (4.10) −g(y, z)g(hx, v)] = 0. on contraction, the above equation yields −2n(wκ)g(y, z) + (wµ)g(z, hy) = 0. (4.11) setting y = ξ in (4.11) and using (2.5), we get 2n(wκ)η(z) = 0. (4.12) if we assume that κ = 0 in (4.11) then either µ = 0 or g(z, hy) = 0. further, if κ = 0 = µ in (2.9), then we get r(x, y)ξ = 0 for all x, y and in the light of lemma 2.1, the manifold under consideration is locally isometric to the riemannian product en+1 × sn(4). so from lemma 2.1, we can state the theorem as follows: theorem 4.2. let m2n+1 (φ, ξ, η, g) be a conharmonically locally φ-symmetric (κ, µ)-contact metric manifold. then the manifold is locally isometric to the riemannian product en+1(0)×sn(4). 5 h-conharmonically semisymmetric non-sasakian (κ, µ)-contact metric manifolds definition 5.1. a riemannian manifold (m2n+1, g) is said to be h-conharmonically semisymmetric if it satisfies n(x, y) · h = 0. (5.1) the following lemma which was proved in [3] is helpful to state our theorem. cubo 22, 1 (2020) certain results on the conharmonic curvature tensor of (κ, µ) . . . 79 lemma 5.1. [3]: let m2n+1(φ, ξ, η, g) be a contact metric manifold with ξ belonging to the (κ, µ)-nullity distribution. then for any vector fields x, y, z, r(x, y)hz − hr(x, y)z = {κ[g(hy, z)η(x) − g(hx, z)η(y)] (5.2) +µ(κ − 1)[g(x, z)η(y) − g(y, z)η(x)]}ξ +κ{g(y, φz)φhx − g(x, φz)φhy + g(z, φhy)φx −g(z, φhx)φy + η(z)[η(x)hy − η(y)hx]} −µ{η(y)[(1 − κ)η(z)x + µη(x)hz] −η(x)[(1 − κ)η(z)y + µη(y)hz] + 2g(x, φy)φhz}. let m2n+1 be h-conharmonically semisymmetric non-sasakian (κ, µ)-contact metric manifold. the condition n(x, y) · h = 0 can be expressed as follows, (n(x, y) · h)z = n(x, y)hz − hn(x, y)z = 0, (5.3) for any vector fields x, y, z. with the help of (1.1) and (5.2), (5.3) can be written as [κ{g(hy, z)η(x) − g(hx, z)η(y)} + µ(κ − 1){g(x, z)η(y) − g(y, z)η(x)}]ξ (5.4) +κ{g(y, φz)φhx − g(x, φz)φhy + g(z, φhy)φx − g(z, φhx)φy + η(z)[η(x)hy −η(y)hx]} − µ{η(y)[(1 − κ)η(z)x + µη(x)hz] − η(x)[(1 − κ)η(z)y + µη(y)hz] +2g(x, φy)φhz} − 1 2n − 1 [s(y, hz)x − s(x, hz)y + g(y, hz)qx − g(x, hz)qy −s(y, z)hx + s(x, z)hy − g(y, z)qhx + g(x, z)qhy] = 0. by taking inner product of (5.4) with t, we get [κ{g(hy, z)η(x) − g(hx, z)η(y)} + µ(κ − 1){g(x, z)η(y) − g(y, z)η(x)}]η(t) (5.5) +κ{g(y, φz)g(φhx, t) − g(x, φz)g(φhy, w) + g(z, φhy)g(φx, t) −g(z, φhx)g(φy, w) + η(z)[η(x)g(hy, w) − η(y)g(hx, t)]} −µ{η(y)[(1 − κ)η(z)g(x, t) + µη(x)g(hz, t)] − η(x)[(1 − κ)η(z)g(y, t) +µη(y)g(hz, t)] + 2g(x, φy)g(φhz, t)} − 1 2n − 1 [s(y, hz)g(x, t) −s(x, hz)g(y, t) + g(y, hz)s(x, t) − g(x, hz)s(y, t) − s(y, z)g(hx, t) +s(x, z)g(hy, t) − g(y, z)s(hx, t) + g(x, z)s(hy, t)] = 0. setting y = t = ξ in (5.5) and using (2.2) and (2.5), we get 1 2n − 1 s(x, hz) = −µ(1 − κ)g(x, z) + [2(1 − µ) + (1 − κ)]η(x)η(z) (5.6) +[κ − 2(2n + 1)κ 2(n − 1) g(x, hz)]. 80 divyashree g. & venkatesha cubo 22, 1 (2020) replacing x by hx in the above equation and using (2.10), we have s(x, z) = −κg(x, z) + κη(x)η(z) − 2µ(n − 1)g(hx, z). (5.7) if we consider µ = 0 in (5.7) then it is an η-einstein manifold. using (2.14) in (5.7) and simplifying, we finally obtain s(x, z) = n1g(x, z) + n2η(x)η(z), (5.8) where n1 = −κ[2(n−1)+µ]+µ(2n−1)[2(n−1)+nµ] [2(n−1)+µ]+µ(2n−1) and n2 = κ[2(n−1)+µ]+µ(2n−1)[2(1−n)+n(2κ+µ)] [2(n−1)+µ]+µ(2n−1) . thus from (5.8), we can conclude the following theorem: theorem 5.2. let m2n+1(φ, ξ, η, g) be a non-sasakian (κ, µ)-contact metric manifold. if m is h-conharmonically semisymmetric, then the manifold is an η-einstein manifold with constant coefficients. from proposition 2.2 and theorem 5.5 we can state the following: corolary 2. if m2n+1 is a h-conharmonically semisymmetric (κ, µ)-contact metric manifold then the ricci operator q commutes with φ i.e., qφ = φq. 6 φ-conharmonically semisymmetric non-sasakian (κ, µ)-contact metric manifolds definition 6.1. a riemannian manifold (m2n+1, g) is said to be φ-conharmonically semisymmetric if n(x, y) · φ = 0. (6.1) now we need the following lemma: lemma 6.1. [3]: let m2n+1(φ, ξ, η, g) be a contact metric manifold with ξ belonging to the (κ, µ)-nullity distribution. then for any vector fields x, y, z, r(x, y)φz − φr(x, y)z = {(1 − κ)[g(φy, z)η(x) − g(φx, z)η(y)] (6.2) +(1 − µ)[g(φhy, z)η(x) − g(φhx, z)η(y)]}ξ −g(y + hy, z)(φx + φhx) + g(x + hx, z)(φy +φhy) − g(φy + φhy, z)(x + hx) + g(φx +φhx, z)(y + hy) − η(z){(1 − κ)[η(x)φy −η(y)φx] + (1 − µ)[η(x)φhy − η(y)φhx)]}. cubo 22, 1 (2020) certain results on the conharmonic curvature tensor of (κ, µ) . . . 81 let m2n+1 be a (2n+1)-dimensional φ-conharmonically semisymmetric non-sasakian (κ, µ)contact metric manifold. the condition n(x, y) · φ = 0 turns into, (n(x, y) · φ)z = n(x, y)φz − φn(x, y)z = 0, (6.3) for any vector fields x, y, z. in view of (1.1) and (6.2), (6.3) becomes {(1 − κ)[g(φy, z)η(x) − g(φx, z)η(y)] + (1 − µ)[g(φhy, z)η(x) − g(φhx, z)η(y)]}ξ (6.4) −g(y + hy, z)(φx + φhx) + g(x + hx, z)(φy + φhy) − g(φy + φhy, z)(x + hx) +g(φx + φhx, z)(y + hy) − η(z){(1 − κ)[η(x)φy − η(y)φx] + (1 − µ)[η(x)φhy −η(y)φhx)]} − 1 2n − 1 [s(y, φz)x − s(x, φz)y + g(y, φz)qx − g(x, φz)qy −s(y, z)φx + s(x, z)φy − g(y, z)qφx + g(x, z)qφy] = 0. taking inner product of (6.4) with t, we get {(1 − κ)[g(φy, z)η(x) − g(φx, z)η(y)] + (1 − µ)[g(φhy, z)η(x) (6.5) −g(φhx, z)η(y)]}η(t) − g(y, z)g(φx, t) − g(hy, z)g(φx, t) − g(y, z)g(φhx, t) −g(hy, z)g(φhx, t) + g(x, z)g(φy, t) + g(hx, z)g(φy, t) + g(x, z)g(φhy, t) +g(hx, z)g(φhy, t) − g(φy, z)g(x, t) − g(φy, z)g(hx, t) − g(φhy, z)g(x, t) −g(φhy, z)g(hx, t) + g(φx, z)g(y, t) + g(φhx, z)g(y, t) + g(φx, z)g(hy, t) +g(φhx, z)g(hy, t) − η(z){(1 − κ)[η(x)g(φy, t) − η(y)g(φx, t)] +(1 − µ)[η(x)g(φhy, t) − η(y)g(φhx, t)]} − 1 2n − 1 [s(y, φz)g(x, t) −s(x, φz)g(y, t) + g(y, φz)g(qx, t) − g(x, φz)g(qy, t) − s(y, z)g(φx, t) +s(x, z)g(φy, t) − g(y, z)g(qφx, t) + g(x, z)g(qφy, t)] = 0. treating y = t = ξ in (6.5) and using (2.1), (2.2), (2.4), (2.5) and (2.13), we have 1 2n − 1 s(x, φz) = {(κ − 2) + 2(2n + 1)κ 2n − 1 }g(x, φz) − µg(φx, hz). (6.6) substituting x by φx in (6.6) and using (2.1), (2.2) and (2.16), one can get s(x, z) = [(κ − 2)(2n − 1) + 2nκ]g(x, z) − [(κ − 2)(2n − 1)]η(x)η(z) (6.7) +[µ(κ − 1)(2n − 1) + 2{2(n − 1) + µ}]g(hx, z). making use of (2.14), (6.7) yields s(x, z) = n3g(x, z) + n4η(x)η(z), (6.8) 82 divyashree g. & venkatesha cubo 22, 1 (2020) where n3 = {(κ−2)(2n−1)+2nκ}{2(n−1)+µ}−{µ(κ−1)(2n−1)+2[2(n−1)+µ]}{2(n−1)−nµ} [2(n−1)+µ]−[µ(κ−1)(2n−1)+2{2(n−1)+µ}] and n4 = [(2−κ)(2n−1)][2(n−1)+µ]−{µ(κ−1)(2n−1)+2[2(n−1)+µ]}[2(1−n)+2n(2κ+µ)] [2(n−1)+µ]−[µ(κ−1)(2n−1)+2{2(n−1)+µ}] . hence from (6.8), the theorem can be stated as follows: theorem 6.2. if a (2n + 1)-dimensional non-sasakian (κ, µ)-contact metric manifold m2n+1 is φ-conharmonically semisymmetric then the manifold is an η-einstein manifold with constant coefficients. similarly, from proposition 2.2 and theorem 6.6, we get the following statement: corolary 3. if m2n+1 is a φ-conharmonically semisymmetric (κ, µ)-contact metric manifold then the ricci operator q commutes with φ i.e., qφ = φq. 7 (κ, µ)-contact metric manifold with divergent free conharmonic curvature tensor in this section, we study divergent free conharmonic curvature tensor on (κ, µ)-contact metric manifold. let m2n+1(φ, ξ, η, g) (n > 1) be a (κ, µ)-contact metric manifold satisfying the following condition (divn)(x, y)z = 0. (7.1) in view of (7.1), (1.1) leads to (divr)(x, y)z = 1 2n − 1 [(∇xs)(y, z) − (∇ys)(x, z) + g(y, z)dr(x) (7.2) −g(x, z)dr(y)]. the above equation simplifies to, 2(n − 1) (2n − 1) [(∇xs)(y, z) − (∇ys)(x, z)] − 1 (2n − 1) [g(y, z)dr(x) (7.3) −g(x, z)dr(y)] = 0. on contracting and taking summation over i, 1 ≤ i ≤ n in (7.3), we get 2(3n − 1)dr(y) = 0, (7.4) which implies dr(y) = 0, (7.5) cubo 22, 1 (2020) certain results on the conharmonic curvature tensor of (κ, µ) . . . 83 since 2(3n − 1) 6= 0. further, considering (7.5) in (7.3), we obtain (∇xs)(y, z) − (∇ys)(x, z) = 0, (7.6) which gives (∇xq)y = (∇yq)x. (7.7) thus, we can state: theorem 7.1. let m2n+1(φ, ξ, η, g) (n > 1) be a (κ, µ)-contact metric manifold. if the manifold has divergent free conharmonic curvature tensor then the ricci tensor s is a codazzi tensor. 84 divyashree g. & venkatesha cubo 22, 1 (2020) references [1] blair d. e, contact manifolds in riemannian geometry, lecture notes in math. 509, springerverlag, 1976. [2] d. e. blair, two remarks on contact metric structures, tohoku math. j., 29, 319-324, 1977. [3] d.e. blair, t. koufogiorgos and b.j. papantoniou, contact metric manifolds satisfying a nullity condition, israel j. math., 19, 189-214, 1995. [4] e. boeckx, a full classification of contact metric (κ, µ)-spaces, illinois j. of math. 44, 212-219, 2000. [5] a. ghosh, r. sharma and j.t. cho, contact metric manifolds with η-parallel torsion tensor, ann. glob. anal. geom. 34, 287-299, 2008. [6] jun j. b. yildiz a. and de u. c, on φ-recurrent (κ, µ)contact metric manifolds, bull. korean math. soc. 45, 689-700, 2008. [7] nader asghari and abolfazl taleshian, on the conharmonic curvature tensor of kenmotsu manifolds, thai journal of mathematics, 12(3), 525-536, 2014. [8] c. özgür, on φ-conformally flat lorentzian para-sasakian manifolds, radovi matematicki, 12(1), 99-106, 2003. [9] d. perrone, homogeneous contact riemannian three-manifolds, illinois j. math. 42, 243-258, 1998. [10] a. sarkar, matilal sen and ali akbar, generalized sasakian space forms with conharmonic curvature tensor, palestine journal of mathematics, 4(1), 84-90, 2015. [11] a. a. shaikh and k. kanti baishya, on (κ, µ)-contact metric manifolds, differential geometry dynamical systems, 8, 253-261, 2006. [12] s. a. siddiqui and z. ahsan, conharmonic curvature tensor and the space-time of general relativity, differential geometry-dynamical systems, 12, 213-220, 2010. [13] a. yildiz and u. c. de, a classification of (κ, µ)-contact metric manifolds, commun. korean math. soc., 2, 327-339, 2012. introduction preliminaries conharmonically flat (,)-contact metric manifolds conharmonically locally -symmetric (,)-contact metric manifolds h-conharmonically semisymmetric non-sasakian (,)-contact metric manifolds -conharmonically semisymmetric non-sasakian (,)-contact metric manifolds (,)-contact metric manifold with divergent free conharmonic curvature tensor cubo, a mathematical journal vol. 23, no. 03, pp. 369–384, december 2021 doi: 10.4067/s0719-06462021000300369 the structure of extended function groups r. a. hidalgo1 1 departamento de matemática y estad́ıstica, universidad de la frontera, temuco, chile. ruben.hidalgo@ufrontera.cl abstract conformal (respectively, anticonformal) automorphisms of the riemann sphere are provided by the möbius (respectively, extended möbius) transformations. a kleinian group (respectively, an extended kleinian group) is a discrete group of möbius transformations (respectively, a discrete group of möbius and extended möbius transformations, necessarily containing extended ones). a function group (respectively, an extended function group) is a finitely generated kleinian group (respectively, a finitely generated extended kleinian group) with an invariant connected component of its region of discontinuity. a structural decomposition of function groups, in terms of the kleinmaskit combination theorems, was provided by maskit in the middle of the 70’s. one should expect a similar decomposition structure for extended function groups, but it seems not to be stated in the existing literature. the aim of this paper is to state and provide a proof of such a decomposition structural picture. resumen los automorfismos conformes (respectivamente, anticonformes) de la esfera de riemann son dados por las transformaciones de möbius (respectivamente, möbius extendidas). un grupo kleiniano (respectivamente, un grupo kleiniano extendido) es un grupo discreto de transformaciones de möbius (respectivamente, un grupo discreto de transformaciones de möbius y transformaciones de möbius extendidas, necesariamente conteniendo extendidas). un grupo función (respectivamente, un grupo función extendido) es un grupo kleiniano finitamente generado (respectivamente, un grupo kleiniano extendido finitamente generado) con una componente conexa invariante de su región de discontinuidad. una descomposición estructural de los grupos función, en términos de los teoremas de combinación de klein-maskit, fue dado por maskit a mediados de los 70’s. se debiera esperar una estructura de descomposición similar para los grupos función extendidos, pero no parece estar enunciado en la literatura existente. el objetivo de este art́ıculo es enunciar y dar una demostración de una tal descomposición estructural. keywords and phrases: kleinian groups, equivariant loop theorem. 2020 ams mathematics subject classification: 30f10, 30f40. accepted: 20 july, 2021 received: 24 march, 2021 ©2021 r. a. hidalgo. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300369 https://orcid.org/0000-0003-4070-2819 370 rubén a. hidalgo cubo 23, 3 (2021) 1 introduction the conformal (respectively, anticonformal) automorphisms of the riemann sphere ĉ = c ∪ {∞} are provided by the möbius (respectively, extended möbius) transformations, that is, transformations of the form t(z) = (az+b)/(cz+d) (respectively, l(z) = (az+b)/(cz+d)) where a, b, c, d ∈ c are such that ad − bc = 1. the group of möbius transformations m is isomorphic to the special projective linear group psl2(c) and the group of möbius and extended möbius transformations is m̂ = ⟨m, j(z) = z⟩. a kleinian group (respectively, an extended kleinian group) is a discrete subgroup of m (respectively, a discrete subgroup of m̂ necessarily containing extended möbius transformations). the region of discontinuity of a (extended) kleinian group k is the locus of points p ∈ ĉ admitting an open neighborhood p ∈ u ⊂ ĉ such that k(u) ∩ u ̸= ∅ only for finitely many elements k ∈ k. by definition, the region of discontinuity is an open set (it might be empty). the complement of the region of discontinuity is called the limit set and it is the place where the dynamics of the group action is chaotic. the history of kleinian groups can be traced back to poincaré [17] and a classical source is the book [13]. a function group is a finitely generated kleinian group (with a non-empty region of discontinuity) admitting an invariant connected component of its region of discontinuity. basic examples of function groups are provided by elementary groups (kleinian groups with finite limit set), quasifuchsian groups (function groups whose limit set is a jordan curve) and totally degenerate groups (non-elementary finitely generated kleinian groups whose region of discontinuity is both connected and simply-connected). in a serie of papers, maskit provided the following decomposition structure of function groups, in terms of the klein-maskit combination theorems [7, 8, 13]. theorem 1.1 (maskit’s decomposition of function groups [6, 9, 10, 11]). every function group is constructed from elementary groups, quasifuchsian groups and totally degenerate groups by a finite number of applications of the klein-maskit combination theorems. moreover, in the construction, the amalgamated free products and the hnn-extensions are realized along either (i) a finite cyclic group (including the trivial group) or (ii) a cyclic group generated by an accidental parabolic element. an extended function group is a finitely generated extended kleinian group with an invariant connected component of its region of discontinuity. basic examples of extended function groups are the extended elementary groups (extended kleinian groups with finite limit set), extended quasifuchsian groups (finitely generated extended function groups whose limit set is a jordan curve) and extended totally degenerate groups (non-elementary extended finitely generated kleinian groups with connected and simply-connected region of discontinuity). note that the term “extended quasifuchsian group” used in this paper is different from the given cubo 23, 3 (2021) the structure of extended function groups 371 by other authors in the sense that they refer it to kleinian groups whose limit set is a jordan curve and contains elements permuting the two discs bounded by it. as it is for the case of function groups, one should expect a similar decomposition result for the extended function groups (see theorem 1.2). it seems that such a result is missing in the literature. the aim of this paper is to provide such an structural decomposition of extended function groups. theorem 1.2 (decomposition of extended function groups). every extended function group is constructed from (extended) elementary groups, (extended) quasifuchsian groups and (extended) totally degenerate groups by a finite number of applications of the klein-maskit combination theorems. moreover, in the construction, the amalgamated free products and the hnn-extensions are realized along either (i) a finite cyclic group (including the trivial group) or (ii) an infinite dihedral group generated by two reflections or (iii) a cyclic group generated by either an accidental parabolic element or by an accidental pseudo-parabolic element (i.e., its square is accidental parabolic). the above structure description is a consequence of theorems 3.1 and 3.2 (which are new results); their statements are at the beginning of section 3. their proofs build upon a sequence of lemmas 3.5, 3.7, 3.8, 3.9, which also are not found in the literature. the idea of the proof is the following. let k be an extended function group, with invariant connected component ∆. its index two orientation preserving half k+ = k ∩ m is a function group with the same invariant component. as k+ is finitely generated, selberg’s lemma [19] asserts the existence of a torsion free finite index normal subgroup g1 of k + (which is again a function group). since k = ⟨k+, τ⟩, where τ2 ∈ k+, the group g = g1 ∩τg1τ−1 is a finite index torsion free normal subgroup of k. by ahlfors finiteness theorem [1], the quotient space s = ∆/g is an analytically finite riemann surface, that is, s = ŝ − c, where ŝ is a closed riemann surface and c ⊂ ŝ is a finite set of points (it might be empty). the finite group h = k/g is a group of conformal and anticonformal automorphisms of s. maskit’s decomposition of function groups may be applied to g. there are many possible decompositions, but in order to get one which can be used to obtain a decomposition of k, we must find one which is in some sense equivariant with respect to g. this is solved by theorem 2.2 (equivariant theorem for function groups) obtained by maskit and the author in [4]. this result permits us to obtain a first decomposition structural picture (see theorem 3.1). in such a picture, there may appear (extended) b-groups as factors. a b-group (respectively, an extended b-group) is a function group (respectively, an extended function group) with a simply-connected invariant component in its region of discontinuity. a subtle modification to maskit’s arguments for the case of b-groups, to deal with these extended b-groups, is provided (see theorem 3.2). a. haas’s thesis [3] concerns with uniformizing groups of conformal and anticonformal automorphisms acting on plane domains. it leads naturally to extended function groups, but it seems that the above decomposition does not follow immediately from it. 372 rubén a. hidalgo cubo 23, 3 (2021) 2 preliminaries 2.1 riemann orbifolds a riemann orbifold o consists of a (possible non-connected) riemann surface s (called the underlying riemann surface of the orbifold), an isolated collection of points of s (called the cone points of the orbifold) and associated to each cone point an integer at least 2 (called the cone order). a connected riemann orbifold is analytically finite if its underlying (connected) riemann surface is the complement of a finite number of points of a closed riemann surface and the number of cone points is also finite. we may think of a riemann surface as a riemann orbifold without cone points. a conformal automorphism (respectively, anticonformal automorphism) of the riemann orbifold o is a conformal automorphism (respectively, anticonformal) of the underlying riemann surface s which preserves both its set of cone points together with their cone orders (cone points can be permuted but preserving their cone orders). we denote by aut(o) (respectively, aut(s)) the group of conformal/anticonformal automorphisms of o (respectively, s) and by aut+(o) (respectively, aut+(s)) its subgroup of conformal automorphisms. 2.2 kleinian and extended kleinian groups in the following, we recall some facts on (extended) kleinian groups. a good source on the topic are the classical books [13, 14]. let us start by observing that, if k1 < k2 < m̂ and k1 has finite index in k2, then both are discrete if one of them is and, in the discreteness case, both have the same region of discontinuity. let k < m̂ and set k+ := k ∩ m. if k ̸= k+, then k+ is called the orientation-preserving half of k and, in this case, k is an extended kleinian group if and only if k+ is a kleinian group; in which case both have the same region of discontinuity. if moreover, k is an extended kleinian group and k+ is a function group, then either: (i) k is an extended function group or (ii) k+ is a quasifuchsian group and there is an element of k − k+ permuting both discs bounded by the limits set jordan curve (so k is not an extended function group). 2.3 accidental parabolic elements a b-group is a function group k with a simply-connected invariant component ∆. let us assume k is non-elementary (i.e., its limit set is not finite). by the klein-poincaré uniformization theorem [18], there is a bi-holomorphism f : h2 → ∆, where h2 denotes the hyperbolic upper-half plane. the group γ = f−1kf is a group of conformal automorphisms of h2, i.e., a fuchsian group of the first kind, in particular, a b-group with h2 as an invariant connected component of its region of discontinuity. in this case, h2/γ has finite hyperbolic area. it is known that f sends parabolic cubo 23, 3 (2021) the structure of extended function groups 373 transformations to parabolic transformations, but it may send a hyperbolic transformation to a parabolic one. a parabolic element p ∈ k is called accidental if f−1pf is a hyperbolic transformation. in this case, the image under f of the axis of the hyperbolic transformation f−1pf is called the axis of p (in maskit’s notation this is the true axis of p). if k is an extended b-group, that is, an extended function group with a simply-connected invariant component, then we say that an element of k is accidental pseudo-parabolic if its square is an accidental parabolic element of k+. 2.4 klein-maskit’s decomposition theorems let k be a kleinian group with region of discontinuity ω and let h be a subgroup of k with limit set λ(h). a set x ⊂ ĉ is called precisely invariant under h in k if e(x) = x, for every e ∈ h, and t(x) ∩ x = ∅, for every t ∈ k \ h. we will assume h to be either (i) the trivial group, (ii) a finite cyclic group or (iii) an infinite cyclic group generated by a parabolic transformation. if h is a cyclic subgroup, a precisely invariant disc b is the interior of a closed topological disc b, where b − λ(h) ⊂ ω is precisely invariant under h in k. theorem 2.1 (klein-maskit’s combination theorems [7, 8]). (1) (amalgamated free products). for j = 1, 2, let kj be a kleinian group, let h ≤ k1 ∩ k2 be a cyclic subgroup (either trivial, finite or generated by a parabolic transformation), h ̸= kj, and let bj be a precisely invariant disc under h in kj. assume that b1 and b2 have as a common boundary the simple loop σ and that b1 ∩ b2 = ∅. then k = ⟨k1, k2⟩ is a kleinian group isomorphic to the free product of k1 and k2 amalgamated over h, that is, k = k1 ∗h k2, and every elliptic or parabolic element of k is conjugated in k to an element of either k1 or k2. moreover, if k1 and k2 are both geometrically finite, then k is also geometrically finite. (2) (hnn extensions). let k be a kleinian group. for j = 1, 2, let bj be a precisely invariant disc under the cyclic subgroup hj (either trivial, finite or generated by a parabolic) in k, let σj be the boundary loop of bj and assume that t(b1)∩b2 = ∅, for every t ∈ k. let a be a loxodromic transformation such that a(σ1) = σ2, a(b1) ∩ b2 = ∅, and a−1h2a = h1. then ka = ⟨k, a⟩ is a kleinian group, isomorphic to the hnn-extension k∗⟨a⟩ (that is, every relation in ka is consequence of the realtions in k and the relations a−1h2a = h1). if each hj, for j = 1, 2, is its own normalization in k, then every elliptic or parabolic element of ka is conjugated to some element of k. moreover, if k is geometrically finite, then ka is also geometrically finite. 374 rubén a. hidalgo cubo 23, 3 (2021) 2.5 an equivariant loop theorem for function groups let k be a function group and ∆ be a k-invariant connected component of its region of discontinuity. by the alhfors’ finiteness theorem [1, 2], the quotient o = ∆/k turns out to be an analytically finite riemann orbifold. let b ⊂ o be the (finite) collection of the cone points and let g ⊂ o − b be the collection of loops which lift to loops under the natural regular holomorphic covering π : ∆0 → o − b, where ∆0 is the open dense subset of ∆ consisting of those points with trivial k-stabilizer. in [5], maskit proved the existence of a finite subcollection f ⊂ g of pairwise disjoint loops inside o − b, each one being a finite power of a simple loop, such that the cover π is determined as a highest regular planar cover for which the loops in f lift to loops (such a collection of loops is not unique). the collection f is called a fundamental system of loops of the above regular planar covering. assume that there is a finite group h < aut(o) whose elements lift to automorphisms of ∆ under π. then, in [4], maskit and the author proved that there is a fundamental system of loops f being equivariant under h. theorem 2.2 (equivariant loop theorem for function groups [4]). let k be a function group, with invariant connected component ∆ in its region of discontinuity, o = ∆/k (which is an analytically finite riemann orbifold) and let b be the finite set of cone points of o. let π : ∆ → o be the natural regular branched regular covering induced by k. let g be the collection of loops in o − b which lift to loops in ∆ under π. if h < aut(o) lifts to a group of automorphisms of ∆, then there is a finite sub-collection f ⊂ g such that: (1) f consists of pairwise disjoint powers of simple loops; (2) f is h-invariant; and (3) every loop in g is homotopic to the product of finite powers of a finite loop in f. the collection f is called a fundamental set of loops for the pair (k, h). remark 2.3. the condition (3) above is equivalent to say that f is a fundamental system of loops for π. also, if the function group k is torsion-free, then o is an analytically finite riemann surface and each of the loops in the finite collection f turns out to be a simple loop. as a consequence of the above, one may write the following equivariant result for kleinian groups. theorem 2.4 (equivariant loop theorem for kleinian groups). let k be a kleinian group with region of discontinuity ω ̸= ∅, let ∆ be a (non-empty) collection of connected components of ω which is invariant under the action of k, let o = ∆/k, let b be the cone points of o and let h < aut(o) be a finite group of automorphisms of o. let us assume that o consists of (may be infinitely many) analytically finite riemann orbifolds. fix some regular (branched) covering map π : ∆ → o with k as its deck group. let g be the collection of loops in o−b which lift, with respect cubo 23, 3 (2021) the structure of extended function groups 375 to π, to loops in ∆. if h lifts to a group of automorphisms of ∆, then there is a sub-collection f ⊂ g such that: (1) f consists of pairwise disjoint powers of simple loops; (2) f is h-invariant; and (3) every loop in g is homotopic to the product of finite powers of a finite sub-collection of loops in f. proof. let us consider a maximal subcollection of non-equivalent components of ∆ under the action of k, say ∆j for j ∈ j. let kj be the k-stabilizer of ∆j under the action of k. by theorem 2.2, on oj = ∆j/kj there is a collection of loops, say fj, satisfying the properties on that theorem. clearly the collection of fundamental loops f = ∪j∈jfj is the required one. remark 2.5. the condition for o = ∆/k to consist of analytically finite riemann orbifolds is equivalent, by the ahlfors finiteness theorem, for the k-stabilizer of each connected component in ∆ to be finitely generated. in particular, if k is finitely generated, then o is a finite collection of analytically finite riemann surfaces and f turns out to be a finite collection. if, in theorem 2.4, we assume k to be torsion-free, then the loops in f will be simple loops. 2.6 a connection to kleinian 3-manifolds let k be a kleinian group, with region of discontinuity ω ⊂ ĉ. there is a natural discrete action (by poincaré extension) of k on the upper half-space h3 = {(z, t) : z ∈ c, t ∈ (0, +∞)}, which is given by isometries in the hyperbolic metric ds2 = (|dz|2+dt2)/t2. the quotient mk = (h3∪ω)/k carries the structure of a 3-orbifold, its interior h3/k has a structure of a complete hyperbolic 3-orbifold and ω/k the structure of a riemann orbifold. in the case that k is torsion free, all the above are manifolds and we say that mk is a kleinian 3-manifold. a direct consequence of theorem 2.4 is the equivariant theorem for kleinian 3-manifolds in the case that the conformal boundary is non-empty and it consists of analytically finite riemann surfaces. corollary 2.6. let k be a torsion free kleinian group, with non-empty region of discontinuity ω, such that sk = ω/k is a collection (it might be infinitely many of them) of analytically finite riemann surfaces. let h be a finite group of automorphismsm of the kleinian 3-manifold mk = (h3 ∪ ω)/k. if g is the collection of loops on sk that are homotopically nontrivial in sk but homotopically trivial in mk, then there exists a collection of pairwise disjoint simple loops f ⊂ g, equivariant under the action of h, so that g is the smallest normal subgroup of π1(sk) generated by f. 376 rubén a. hidalgo cubo 23, 3 (2021) remark 2.7. let k be a torsion free kleinian group and let h be as in corollary 2.6. then the following hold. (1) if π1(m) is finitely generated, then the collection f is finite. (2) by lifting h to the universal cover space, one obtains a (extended) kleinian group k̂ containing k as a finite index normal subgroup so that h = k̂/k. corollary 2.6 may be used to obtain a geometric structure picture of k̂, in the sense of the klein-maskit combination theorems, in terms of the algebraic structure of h. (3) if mk is compact, then the result follows from meeks-yau’s equivariant loop theorem [15, 16], whose arguments are based on minimal surfaces theory. if k is not a purely loxodromic geometrically finite kleinian group, then mk is non-compact and the result is no longer a consequence of meek’s-yau’s equivariant theorem. 3 proof of theorem 1.2 the proof of theorem 1.2 is a direct consequence of theorem 3.1, which is the main step, and theorem 3.2 as described below. if the word “extended” is removed, the statements of these theorems are simply maskit’s original theorems (see [6, 9, 10, 11]). theorem 3.1 (first step in maskit-type decomposition of an extended function group). every extended function group is constructed, using the klein-maskit combination theorems, as amalgamated free products and hnn-extensions using a finite collection of (extended) b-groups. moreover, the amalgamations and hnn-extensions are realized along either trivial or a finite cyclic group or a dihedral group generated by two reflections (this last one only in the amalgamated free product operation). the above result asserts that every extended function group is constructed from (extended) bgroups by applying the klein-maskit combination theorems. maskit’s results provide a geometrical decomposition of b-groups (see theorem 3.2 below and delete the word “extended”). we now need to take care of the extended b-groups, which is exactly what the next result is about. theorem 3.2 (decomposition of extended b-groups). let k be an extended b-group with a simply-connected invariant component ∆. then either (i) k is an elementary extended kleinian group or (ii) k is an extended quasifuchsian group or (iii) k is an extended degenerate group or (iv) ∆ is the only invariant component and k is constructed as amalgamated free products and hnnextensions, by use of the klein-maskit combination theorems, using (extended) elementary groups, (extended) quasifuchsian groups and (extended) totally degenerate groups. the amalgamated free products and hnn-extensions are given along axes of accidental parabolic transformations. remark 3.3. we note for the reader that the proof of theorem 3.1 includes remarks 3.4 and 3.6 and lemmas 3.5 and 3.7 and that the proof of theorem 3.2 includes lemmas 3.8 and 3.9. cubo 23, 3 (2021) the structure of extended function groups 377 3.1 proof of theorem 3.1 let k be an extended function group and let ∆ be a k-invariant connected component of its region of discontinuity (we may assume k to be non-elementary). if there is another different invariant connected component of its region of discontinuity, then k+ = k ∩ m ̸= k is known to be a quasifuchsian group [12]; so k is an extended quasifuchsian group. let us assume, from now on, that ∆ is the unique invariant connected component. by selberg’s lemma [19], there is a torsion free finite index normal subgroup g1 of k +. as k = ⟨k+, τ⟩, where τ2 ∈ k+, one has that g = g1 ∩ τg1τ−1 is a torsion free finite index normal subgroup of k. it follows that g is a function group with ∆ as an invariant connected component of its region of discontinuity (the same as for k). also, ∆ is the only invariant connected component of g; otherwise g is a quasifuchsian group and k will have two different invariant connected components, which is a contradiction to our assumption on k. let s = ∆/g (an analytically finite riemann surface by ahlfors finiteness theorem) and consider a regular planar unbranched cover p : ∆ → s with g as its deck group. set h = k/g < aut(s), which is a non-trivial finite group (since g ̸= k). theorem 2.2 asserts the existence of a fundamental set of loops f ⊂ s for the pair (g, h). such a collection of loops cuts s into some finite number of connected components and such a collection of components is invariant under h. the h-stabilizer of each of these connected components and each of the loops in f is a finite group. remark 3.4 (decomposition structure of h). the h-equivariant fundamental system of loops f permits to obtain a structure of h as a finite iteration of amalgamated free products and hnnextensions of certain subgroups of h as follows. let us consider a maximal collection of components of s −f, say s1. . . , sn, so that any two different components are not h-equivalent. let us denote by hj the h-stabilizer of sj. it is possible to choose these surfaces so that, by adding some on the boundary loops, we obtain a planar surface s∗ (containing each sj in its interior). if two surfaces si and sj have a common boundary in s ∗, then hi ∩ hj is either trivial or a cyclic group (this being exactly the h-stabilizer of the common boundary loop). we perform the amalgamated free product of hi and hj along the trivial or cyclic group hi ∩ hj. set sij be the union of si, sj with the common boundary loop in s∗ and set hij the constructed group. now, if sk is another of the surfaces which has a common boundary loop in s∗ with sij, then we again perform the amalgamated free product of hij and hk along the trivial or cyclic group hij ∩ hk. continuing with this procedure, we end with a group h∗ obtained as amalgamated free product along finite cyclic groups or trivial groups. for each boundary of s∗ we add a boundary loop, in order to stay with a planar compact surface (we are out of s in this part). if α is any of the boundary loops of s∗, there should be another boundary loop β of s∗ and an element h ∈ h so that h(α) = β. by the choice of the surfaces sj, we must have that h(s ∗) ∩ s∗ = ∅. in particular, β ̸= α. if 378 rubén a. hidalgo cubo 23, 3 (2021) there is another element k ∈ h − {h} so that k(α) = β, then k−1h is a non-trivial element that stabilizes α and k−1h(s∗) ∩ s∗ ̸= ∅, a contradiction. also, if there is another boundary loop γ of s∗ (different from β) and an element u ∈ h so that u(α) = γ, then uh−1 ∈ h − {i} satisfies that uh−1(s∗) ∩ s∗ ̸= ∅, which is again a contradiction. we may now perform the hhn-extension of h∗ by the finite cyclic group generated by h. if α1 = α,. . . , αm are the boundary loops of s ∗, which are not h-equivalent, then we perform the hhn-extension with each of them. at the end, we obtain an isomorphic copy of h. we may assume the fundamental set of loops f to be minimal, that is, by deleting any non-empty subcollection of loops from it, we obtain a collection which fails to be a fundamental set of loops for (g, h). the minimality condition asserts that each connected component of s − f is different from either a disc or an annulus. by lifting f to ∆, under p , one obtains a collection f̂ ⊂ ∆ of pairwise disjoint simple loops, so that f̂ is invariant under the group k. each of the loops in f̂ is called a structure loop and each of the connected components of ∆ − f̂ a structure region. these structure loops and regions are permuted by the action of k. the k-stabilizer (respectively, the g-stabilizer) of each structure loop and each structure region is called a structure subgroup of k (respectively, a structure subgroup of g). if r is a structure region, then its k-stabilizer, denoted by kr, is a finite extension of its gstabilizer, denoted by gr. similarly, if α is a structure loop, then its k-stabilizer is a finite extension of its g-stabilizer. lemma 3.5. let α be a structure loop and let r be a structure region containing α on its border. then the kr-stabilizer of α is either trivial or a finite cyclic group or a dihedral group generated by two reflections (both circles of fixed points intersecting at two points, one inside of one of the two discs bounded by α and the other point contained inside the other disc). moreover, the k-stabilizer of α is either equal to its kr-stabilizer or it is generated by its kr-stabilizer and an involution (conformal or anticonformal) that sends r to the other structure region containing α in its border. proof. let α ∈ f̂ be a structure loop. as α is contained in the region of discontinuity of k, the k-stabilizer of α is a finite group; so also its g-stabilizer is finite. note that the k+-stabilizer of α is either trivial, a finite cyclic group or a dihedral group. moreover, in the dihedral case, one of the involutions interchanges both discs bounded by α. let r be a structure region containing α as a boundary loop. then the k+r -stabilizer of α is either trivial or a finite cyclic group. it follows that the kr-stabilizer of α is either trivial or a finite cyclic group or a dihedral group generated by two reflections (both circles of fixed points intersecting at two points, one inside of one of the two discs bounded by α and the other point contained inside the other disc). the k-stabilizer of α is generated by the kr-stabilizer and probably an extra involution (conformal or anticonformal) that interchanges both discs bounded by α. cubo 23, 3 (2021) the structure of extended function groups 379 remark 3.6. we do not need this extra information for the rest of the proof, but it may help with a clarification of the gluing process at the klein-maskit combination theorems. it follows, from lemma 3.5, that the k-stabilizer of α ∈ f̂ must be one of the following: (1) the trivial group, (2) a cyclic group generated by a reflection with α as its circle of fixed points (so it permutes both discs bounded by α), (3) a cyclic group generated by a reflection that keeps invariant each of the two discs bounded by α (the reflection has exactly two fixed points over α), (4) a cyclic group generated by an imaginary reflection (it permutes both discs bounded by α), (5) a cyclic group generated by an elliptic transformation of order two (permuting the two discs bounded by α), (6) a cyclic group generated by an elliptic transformation (preserving each of the two discs bounded by α), (7) a group generated by an elliptic transformation (preserving each of the two discs bounded by α) and a reflection whose circle of fixed points is α, (8) a group generated by an elliptic transformation (preserving each of the two discs bounded by α) and an imaginary reflection (permuting both discs bounded by α), (9) a group generated by an elliptic transformation of order two (permuting the two discs bounded by α) and an imaginary reflection that keeps α invariant (it permutes both discs bounded by α), (10) a dihedral group generated by two reflections (both circles of fixed points intersecting at two points, one inside of one of the two discs bounded by α and the other point contained inside the other disc), (11) a group generated by an elliptic transformation (preserving each of the two discs bounded by α) and an imaginary reflection that keeps α invariant (it permutes both discs bounded by α), (12) a group generated by a dihedral group of möbius transformations and a reflection with α as circle of fixed points, (13) a group generated by a dihedral group generated by two reflections (both circles of fixed points intersecting at two points, one inside of one of the two discs bounded by α and the other point contained inside the other disc) and an elliptic transformation of order two that permutes both discs bounded by α, to obtain the above, we use the following fact. let α be a loop which is invariant under (i) an elliptic transformation e, of order two that interchanges both discs bounded by it, and (ii) also invariant under an imaginary reflection τ. then eτ is necessarily a reflection whose circle of fixed points is transversal to α. let r be a structure region and let α ∈ f̂ be on the boundary of r. by lemma 3.5, the krstabilizer of α is some finite group; either trivial or a finite cyclic group or a dihedral group generated by two reflections (both circles of fixed points intersecting at two points, one inside of one of the two discs bounded by α and the other point contained inside the other disc). let dα be the topological disc bounded by α and disjoint from r. clearly, the kr-stabilizer of such a disc is contained in the kr-stabilizer of α (each element of kr that stabilizes dα also stabilizes α), so dα is contained in the region of discontinuity of kr. it follows that kr is a (extended) function group with an invariant connected component ∆r of its region of discontinuity containing r and all the discs dα, for every structure loop α on its boundary. 380 rubén a. hidalgo cubo 23, 3 (2021) lemma 3.7. ∆r is simply-connected. proof. if ∆r is not simply-connected, then there is a simple loop β ⊂ r bounding two topological discs, each one containing limit points of kr (so limit points of k). the projection on s of β produces a loop β̃ ⊂ s which lifts to a loop under p . but, we know that β̃ is homotopic to the product of finite powers of the simple loops on the boundary of the finite domain p(r) ⊂ s. it follows that β must be homotopic to the product of finite powers of a finite collection of structure loops on the boundary of r. as each of these boundary loops bounds a disc containing no limit points, we get a contradiction for β to bound two discs, each one containing limit points. we may follow the same lines as described in remark 3.4 to obtain that k is constructed, using the klein-maskit combination theorems [13, 7], as amalgamated free products and hnn-extensions using a finite collection of the structure subgroups of kr (which, by lemma 3.7, are extended b-groups with invariant simply-connected component ∆r). by lemma 3.5, the amalgamations and hnn-extensions are realized along either trivial or a finite cyclic group or a dihedral group generated by two reflections. this ends the proof of theorem 3.1. □ 3.2 proof of theorem 3.2 we proceed to describe the subtle modifications in maskit’s arguments in the decomposition of b-groups [10, 11] adapted to the case of extended b-groups (see also chapter ix.h. in [13]). let us assume that k is an extended b-group and that it is neither an (extended) elementary group or a (extended) quasifuchsian group or a (extended) degenerate group. let ∆ be the simply-connected invariant component of the region of discontinuity of k. every other connected component of the region of discontinuity of k is simply-connected (see proposition ix.d.2. in [13]). by our assumptions on k, we have that k+ is neither elementary nor degenerate kleinian group. it may be, even if k is not an extended quasifuchsian, that k+ is a quasifuchsian. but in this case, we have that k is just a hnn-extension of a quasifuchsian group along a cyclic group. so, from now on, we assume that k+ is neither a quasifuchsian group. as k is non-elementary, we may consider a bi-holomorphism f : h2 → ∆ and consider the fuchsian group f−1kf. as it is well known that no rank two parabolic subgroup can preserve a disc in the riemann sphere, it follows that f−1k+f does not contain rank two parabolic subgroup, in particular, k+ neither does contain a rank two parabolic subgroup. theorem ix.d.21 in [13] states that k+ is either quasifuchsian or totally degenerate or it contains accidental parabolics. by our assumptions on k and k+, we note that k+ necessarily must have accidental parabolic transformations. moreover, there is a finite number of conjugacy classes of primitive accidental parabolic transformations in k+. let us consider a collection of accidental parabolic transformations in k+, say p1,..., pm, so that pj is not k +-conjugate to p ±1r if j ̸= r, and pj is primitive, cubo 23, 3 (2021) the structure of extended function groups 381 that is it is not of the form qa for some q ∈ k and a ≥ 2. let us denote by lj ⊂ ∆ the axis of pj (note that lj is a geodesic for the hyperbolic metric of ∆ and that pj keeps it invariant acting by a translation on it). lemma 3.8. (1) if j ̸= r, then the k+-translates of lj do not intersect the k+-translates of lr. (2) for each fixed j, any k+-translates of lj is either disjoint from lj or it coincides with it. proof. let us consider a riemann map f : h2 → ∆, where h2 is the upper half-plane with the hyperbolic metric ds2 = |dz|2/im(z)2. it is well known that any two different geodesics in h2 are either disjoint of they intersect at exactly one point. the push-forward of the hyperbolic metric in h2 provides the hyperbolic metric of ∆. it follows that any k+-translate of lj and any k+ translate of lr (for j not necessarily different from r) are either disjoint or they intersect exactly at one point or they are the same. let us first prove (1), that is, we assume j ̸= r. if there are k+-translates of lj and lr which are the same, as pj and pr are primitive parabolic, share the same fixed point and k+ is discrete, then pj is conjugate to either p ±1 r , a contradiction. if there are k+-translates of lj and lr which intersect at a point, then the planarity of ∆ asserts that the non-empty intersection only may happen if a k+ conjugate of pj and a k +-conjugate of pr share their unique fixed point. the discreteness of k+ asserts that k+ must contain a rank two parabolic subgroup, a contradiction. let us now prove (2), that is, we assume j = r. this follows the same lines a the previous case to see that either the translates are either disjoint or equal. lemma 3.9. if t ∈ k − k+, then t preserves the collection of k+-translates of {l1, ...., lm}. proof. t acts as an isometry on ∆ and must permute the accidental parabolic transformations. as the axis is unique for each accidental parabolic, we are done. let l̂j be equal to lj together the corresponding fixed point of pj. then the collection f given by the k+-translates of {l̂1, ..., l̂m} consists of pairwise disjoint simple loops; each one is called a structure loop for the group k. such a collection of structure loops is still invariant for any t ∈ k − k+ by lemma 3.9. the structure loops cut ω (the region of discontinuity of k) and ∆ into regions; called structure regions for k. these are different from our previous definitions of structure loops and regions as these ones are not completely contained in the region of discontinuity. let α ∈ f be a structure loop and let r1 and r2 be the two structure regions containing α in their common boundary. let kj < k be the k-stabilizer of rj, let kα be the k-stabilizer of α and let p ∈ k be the primitive accidental parabolic transformation whose axis is α (which is then k-conjugated to some of the pj’s). clearly, ⟨p⟩ is contained in kj, ⟨p⟩ < kα and either (i) ⟨p⟩ = kα or (ii) ⟨p⟩ has index two in kα or (iii) ⟨p⟩ has index four in kα (this last case means that ⟨p⟩ has index two inside the kj-stabilizer of α). the region r3−j is contained in a disc d3−j, whose kj-stabilizer is equal to the kj-stabilizer of the loop α; this is either the 382 rubén a. hidalgo cubo 23, 3 (2021) cyclic group generated by p or it contains it as an index two subgroup. it follows that d3−j is contained in the region of discontinuity ωj of kj and that there is an invariant connected component ∆j ⊂ ωj containing ∆. lemma ix.h.10 in [13] states that k+j is a b-group, with ∆j as invariant simply-connected component, without accidental parabolic transformations. it follows that k+j is either elementary or quasifuchsian or totally degenerate, in particular, that kj is either (extended) elementary or (extended) quasifuchsian or (extended) totally degenerate. one possibility is that kα is an extension of degree two of the kj-stabilizer of α. in this case, there is an element q ∈ kα that permutes r1 with r2 (q is either a pseudo-parabolic whose square is p or an involution). in this case, ⟨k1, k2⟩ is the hnn-extension of k1 by q (in the sense of the second klein-maskit combination theorem). the other possibility is that kα is equal to k1 ∩ k2 (either the cyclic group generated by the parabolic p or a group generated by two reflections sharing as a common fixed point the fixed point of p). in this case, ⟨k1, k2⟩ is the free product of k1 and k2 amalgamated over k1 ∩ k2 (in the sense of the first klein-maskit combination theorem). now, following the same ideas in [10, 11], one obtains a decomposition of k as an amalgamated free products and hnn-extensions, by use of the klein-maskit combination theorems, using (extended) elementary groups, (extended) quasifuchsian groups and (extended) totally degenerate groups. □ acknowledgment the author would like to thank the referees for their valuable comments, suggestions and corrections to the previous versions. cubo 23, 3 (2021) the structure of extended function groups 383 references [1] l. v. ahlfors, “finitely generated kleinian groups”, amer. j. of math., vol. 86, pp. 413–429, 1964. [2] l. v. ahlfors, “correction to “finitely generated kleinian groups”. amer. j. math., vol. 87, p. 759, 1965. [3] a. haas, “linearization and mappings onto pseudocircle domains”, trans. amer. math. soc., vol. 282, no. 1, pp. 415–429, 1984. [4] r. a. hidalgo and b. maskit, “a note on the lifting of automorphisms” in geometry of riemann surfaces, lecture notes of the london mathematics society, vol. 368, new york: cambridge university press, 2009, pp. 260–267. [5] b. maskit, “a theorem on planar covering surfaces with applications to 3-manifolds”, ann. of math. vol. 81, no. 2, pp. 341–355, 1965. [6] b. maskit, “construction of kleinian groups,” in proceedings of the conference on complex analysis, minneapolis: springer-verlag, 1965, pp. 281–296. 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[14] k. matsuzaki and m. taniguchi, hyperbolic manifolds and kleinian groups, oxford mathematical monographs, new york: the clarendon press, oxford university press, 1998. 384 rubén a. hidalgo cubo 23, 3 (2021) [15] w. meeks and s.-t. yau, “the equivariant dehn’s lemma and loop theorem”, comment. math. helvetici, vol. 56, no. 2, pp. 225–239, 1981. [16] w. meeks and s.-t.yau, “the equivariant loop theorem for three-dimensional manifolds and a review of the existence theorems for minimal surfaces”, in the smith conjecture (new york, 1979), pure appl. math., vol. 112, orlando, fl: academic press, 1984, pp.153–163. [17] h. poincaré, “mémoire: les groupes kleinéens”, acta math., vol. 3, pp. 193–294, 1982. [18] h. poincaré, “sur l’uniformisation des fonctions analytiques”, acta math., vol. 31, no. 1, pp. 1–63, 1908. [19] a. selberg, “on discontinuous groups in higher dimensional symmetric spaces” in contributions to function theory, internat. colloq. function theory, bombay: tata institute of fundamental research, 1960, pp. 147–164. introduction preliminaries riemann orbifolds kleinian and extended kleinian groups accidental parabolic elements klein-maskit's decomposition theorems an equivariant loop theorem for function groups a connection to kleinian 3-manifolds proof of theorem 1.2 proof of theorem 3.1 proof of theorem 3.2 a mathematical journal vol. 6, no 4, (73 94). december 2004. a restricted additive schwarz preconditioner with harmonic overlap for symmetric positive definite linear systems xiao-chuan cai 1 department of computer science, university of colorado boulder, co 80309 cai@cs.colorado.edu maksymilian dryja 2 faculty of mathematics, informatics and mechanics warsaw university, warsaw dryja@mimuw.edu.pl marcus sarkis 3 mathematical sciences department, worcester polytechnic institute worcester, ma 01609 msarkis@wpi.edu abstract a restricted additive schwarz (ras) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. in this paper, we provide an extension of ras for symmetric positive definite problems using the so-called harmonic overlaps (rasho). both ras and rasho outperform their counterparts of the classical additive schwarz variants (as). the design of rasho is based on a much deeper understanding of the behavior of schwarz type methods in overlapping subregions, and in the construction of 1supported in part by the nsf grants asc-9457534, ecs-9725504, and aci-0072089. 2supported in part by the nsf grant ccr-9732208 and in part by the polish science foundation grant 2 p03a 021 16. 3supported in part by the nsf grant ccr-9984404. 74 6, 4(2004) the overlap. in rasho, the overlap is obtained by extending the nonoverlapping subdomains only in the directions that do not cut the boundaries of other subdomains, and all functions are made harmonic in the overlapping regions. as a result, the subdomain problems in rasho are smaller than that of as, and the communication cost is also smaller when implemented on distributed memory computers, since the right-hand sides of discrete harmonic systems are always zero that do not need to be communicated. we also show numerically that rasho preconditioned cg takes fewer number of iterations than the corresponding as preconditioned cg. a nearly optimal theory is included for the convergence of rasho/cg for solving elliptic problems discretized with a finite element method. key words and phrases: restricted additive schwarz preconditioner, domain decomposition, harmonic overlap, elliptic equations, finite elements math. subj. class.: 65n30, 65f10 1 introduction a restricted additive schwarz (ras) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems [1, 5, 7, 13, 15, 16, 17]. ras outperforms the classical additive schwarz preconditioner (as) [8, 20] in the sense that it requires fewer number of iterations, as well as smaller communication and cpu time costs when implemented on distributed memory computers, [1]. unfortunately, ras in its original form is nonsymmetric and therefore the conjugate gradient method (cg) cannot be used [14]. although a symmetrized version was constructed in [7], our numerical experiments show that it often takes more iterations than the corresponding as/cg. in this paper we propose another modification of ras and show in both theory and numerical experiments that this new variant works well for symmetric positive definite sparse linear systems and is superior to as. recall that the basic building blocks of classical schwarz type algorithms are realized by solving the linear systems of the form aδi w = r δ i v (1) on each extended subdomain, where aδi is the extended subdomain stiffness matrix and rδi is the restriction operator for the extended subdomain (formal definitions will be given later in the paper). the key idea of ras is that equation (1) is replaced by aδi w = { v inside the un-extended subdomain 0 in the overlapping part of the subdomain. (2) 6, 4(2004) a restricted additive schwarz preconditioner with ... 75 note that the solution of (2) is discrete harmonic in the overlapping part of the subdomain, and therefore carries minimum energy in some sense. setting part of the right-hand side vector to zero reduces the energy of the solution, and also destroys the symmetry of the additive schwarz operator. in this paper, we further explore the idea of “harmonic overlap” and at the same time keep the symmetry of the schwarz preconditioner. we mention that other approaches can also be taken to modify the schwarz algorithm in the overlapping regions, such as allowing the functions to be discontinuous [4]. the algorithm to be discussed below is applicable for general symmetric positive definite problems. however, in order to provide a complete mathematical analysis, we restrict our discussion to a finite element problem, [3]. we consider a simple variational problem: find u ∈ h10 (ω), such that a(u, v) = f (v), ∀ v ∈ h10 (ω), (3) where a(u, v) = ∫ ω ∇u · ∇v dx and f (v) = ∫ ω f v dx for f ∈ l2(ω). for simplicity, let ω be a bounded polygonal region in �2 with a diameter of size o(1). the extension of the results to �3 can be carried out easily by using the theory developed here in this paper and the well-known three-dimensional additive schwarz techniques; [9, 10, 12]. let t h(ω) be a shape regular, quasi-uniform triangulation, of size o(h), of ω and v ⊂ h10 (ω) the finite element space consisting of continuous piecewise linear functions associated with the triangulation. we are interested in solving the following discrete problem associated with (3): find u∗ ∈ v such that a(u∗, v) = f (v), ∀ v ∈ v. (4) using the standard basis functions, (4) can be rewritten as a linear system of equations au∗ = f. (5) for simplicity, we understand u∗ and f both as functions and vectors depending on the situation. the paper is organized as follows. in section 2, we introduce notations. the new algorithm is described in section 3. section 4 is devoted to the mathematical analysis of the new algorithm. we conclude the paper in section 5 by providing some numerical results and final remarks. through out this paper, c and c0, are positive generic constants that are independent of any of the mesh parameters and the number of subdomains. all the domains and subdomains are assumed to be open; i.e., boundaries are not included in their definitions. 2 notations let n be the total number of interior nodes of t h(ω) and w the set containing all the interior nodes. we assume that a node-based partitioning has been applied and 76 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) resulted in n nonoverlapping subsets w 0i , i = 1, . . . , n , whose union is w . for each w 0i , we define a subregion ω r i as the union of all elements of t h(ω) that have all three vertices in w 0i ∪∂ω. note that ∪ω̄ri is not equal to ω̄; see fig. 1(b). we denote by h as the representative size (diameter) of the subregion ωri . we define the overlapping partition of w as follows. let {w 1i } be the one-overlap partition of w , where w 1i ⊃ w 0i is obtained by including all the immediate neighboring vertices of all vertices in w 0i ; see fig. 1(c). using the idea recursively, we can define a δ-overlap partition of w , w = n⋃ i=1 w δi . here the integer δ indicates the level of overlap with its neighboring subdomains and δh is approximately the extend of the extension. the definition of w δi , as well as many other subsets, can be found in an illustrative picture, fig. 1. we next define a subregion of ω induced by a subset of nodes of t h(ω) as follows. let z be a subset of w . the induced subregion, denoted as ω(z), is defined as the union of: (1) the set z itself; (2) the union all the open elements (triangles) of t h(ω) that have at least one vertex in z; and (3) the union of the open edges of these triangles that have at least one endpoint as a vertex of z. note that ω(z) is always an open region. the extended subregion ωδi is defined as ω(w δ i ), and the corresponding subspace as vδi ≡ v ∩ h10 (ωδi ) extended by zero to ω\ωδi . it is easy to verify that v = vδ1 + vδ2 + · · · + vδn . this decomposition is used in defining the classical one-level additive schwarz algorithm [8]. note that for δ = 0 this decomposition is a direct sum. let us define p δi : v → vδi by: for any u ∈ v, a(p δi u, v) = a(u, v), ∀v ∈ vδi . (6) then, the classical one-level additive schwarz operator has the form p δ = p δ1 + · · · + p δn . in the classical as as defined above, all the nodes of w δi are treated equally even through some subsets of the nodes play different roles in determining the convergence rate of the as preconditioned cg. to further understand the issue, we classify the nodes as follows. let γδi = ∂ω δ i \∂ω; i.e., the part of the boundary of ωδi that does not belong to the dirichlet part of the physical boundary ∂ω. we define the interface overlapping boundary γδ as the union of all γδi ; i.e., γ δ = ∪ni=1γδi . we also need to define the following subsets of w , see, for examples, fig. 1, where δ = 1 6, 4(2004) a restricted additive schwarz preconditioner with ... 77 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) figure 1: the partition of a finite element mesh into 9 subdomains with the overlapping factor δ = 1. (a) the finite element mesh and nodal points; (b) a node-based partition of the mesh into 9 nonoverlapping subsets, and the collection of “•” forms the set w 0i ; (c) w δi ; (d) w γ δ ; (e) w γ δ i ; (f) w γδ i,in; (g) w γδ i,cut; (h) w δ i,ovl; (i) w δ i,non; (j) w δ i,in; (k) w̃ δ i ; (l) the shadowed area is ωδi . 78 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) • w γδ ≡ w ⋂γδ (interface nodes) • w γδi ≡ w γ δ ⋂ w δi (local interface nodes) • w γδi,in ≡ w γ δ ⋂ w 0i (local internal interface nodes) • w γδi,cut ≡ w γ δ i \w γ δ i,in (local cut interface nodes) • w δi,ovl ≡ (w δi \w γ δ i ) ⋂ ( ⋃ j �=i w δ j ) (local overlapping nodes) • w δi,non ≡ w δi \(w γ δ i ⋃ w δi,ovl) (local nonoverlapping nodes) • w δi,in ≡ w δi,non ⋃ w γ δ i,in (internal nodes) we note that the most northwest and the southeast nodes in (c) were added to γδi in order to make ω δ i a rectangle. this just to simplify the presentation and it is not required in the implementation of the algorithms. we frequently use functions that are discrete harmonic at certain nodes. let xk ∈ w be a mesh point and φxk (x) ∈ v the finite element basis function associated with xk; i.e., φxk (xk) = 1, and φxk (xj ) = 0, j �= k. we say u ∈ v is discrete harmonic at xk if a(u, φxk ) = 0. if u is discrete harmonic at a set of nodal points z, we say u is discrete harmonic in ω(z). our new algorithm will be built on the subspace ṽδi defined as a subspace of vδi . ṽδi consists of all functions that vanish on the cuting nodes w γ δ i,cut and discrete harmonic at the nodes of w δi,ovl. note that the support of the subspace ṽδi is w̃ δi ≡ w δi \w γ δ i,cut and, since the values at the harmonic nodes are not independent, they can not be counted toward the degree of freedoms. the dimension of ṽδi is dim ( ṽδi ) = |w δi,in|. let ω(w̃ δi ) be the induced domain. it is easy to see that ω(w̃ δ i ) is the same as ω δ i but with cuts. we denote ω(w̃ δi ) by ω̃ δ i . we have then ṽδi = v ∩ h10 (ω̃δi ) and discrete harmonic on ω(w δi,ovl). we denote ω(w δ i,ovl) by ω δ i,ovl. we define ṽδ ⊂ vδ as ṽδ = ṽδ1 ⊕ · · · ⊕ ṽδn , 6, 4(2004) a restricted additive schwarz preconditioner with ... 79 which is a direct sum. we remark that functions in ṽδ are, by definition, the sum of functions ui ∈ ṽδi , i = 1, · · · , n . functions in ṽδ can, in fact, be characterized easily as in the following lemma. lemma 2.1 if u ∈ v and u is discrete harmonic at all the overlapping nodes, i.e., on ∪ni=1w δi,ovl, then u ∈ ṽδ. proof. to prove that u ∈ ṽδ, all we need is to find a decomposition u = n∑ i=1 ui, with ui ∈ ṽδi , i = 1, · · · , n. for the given u, we define ui piece by piece as follows. on the nodes in w δi,in we let ui = u. on the nodes in w δi,cut we let ui be zero. on the nodes outside w δ i we set ui to zero. we now need only to define ui on the nodes belong to w δi,ovl. there, we extend ui as a discrete harmonic function with boundary data given by ui just defined. 3 restricted additive schwarz with harmonic overlap (rasho) using notations introduced in the previous section, we now describe a new method, namely a restricted additive schwarz with harmonic overlap. we first define p̃ δi : ṽδ → ṽδi as a projection operator, such that, for any u ∈ ṽδ a(p̃ δi u, v) = a(u, v), ∀v ∈ ṽδi . (7) the rasho operator can then be defined as p̃ δ = p̃ δ1 + · · · + p̃ δn . (8) note, however, that the solution u∗ of (4), see also (5), is not, generally speaking, in the subspace ṽδ, therefore, the operator p̃ δ cannot be used to solve the linear system (5) directly. we will need to modify the right-hand side of the system (5). a reformulated (5) will be presented in lemma 3.1 below. we will show that the elimination of the variables associated with the overlapping nodes is not needed in order to apply p̃ δ to any given vector v ∈ p̃ δ. we now introduce a matrix form of (8). we define the restriction operator, or a matrix, r̃δi as follows. let v = (v1, . . . , vn) t be a vector corresponding to the nodal values of a function u ∈ v; namely for any node xi ∈ w , vi = u(xi). for convenience, we say “v is defined on w ”. its restriction on w̃ δi , r̃ δ i v, is defined as ( r̃δi v ) (xi) = ⎧⎨ ⎩ vi if xi ∈ w̃ δi 0 otherwise. (9) 80 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) the matrix representation of r̃δi is given by a diagonal matrix with 1 for nodal points in w̃ δi and zero for the remaining nodal points. we remark that, by way of definition, the operator r̃δi is symmetric; i.e., (r̃ δ i ) t = r̃δi . use this restriction operator, we define the subdomain stiffness matrix as ãδi = r̃ δ i a (r̃ δ i ) t , which can also be obtained by the discretization of the original finite element problem on w̃ δi with zero dirichlet data on nodes w \ w̃ δi . the matrix ãδi is block diagonal with blocks corresponding to the structure of r̃δi and its inverse is understood as an inverse of the nonzero block. a matrix representation of p̃ δi denoted also by p̃ δ i is equals to p̃ δi = ( ãδi )−1 a and p̃ δ = ( (ãδ1) −1 + · · · + (ãδn )−1 ) a. (10) using the matrix notations, the next lemma shows how to modify the system (5) so that its solution belongs to ṽδ. lemma 3.1 let u∗ and f be the exact solution and the right-hand side of (5), and w = n∑ i=1 (ãδi ) −1r̃0i f, (11) then, we have ũ∗ = u∗ − w ∈ ṽδ, which is the solution of the modified linear system of equations aũ∗ = f − aw = f̃ . proof. if we can show that a(w, φk) = f (φk), for a regular basis function associated with an arbitrary overlapping node xk ∈ w δi,ovl, for some i, then we will have a(u∗ − w, φk) = f (φk) − f (φk) = 0, (12) which says that ũ∗ = u∗ − w is discrete harmonic at the overlapping node xk. we can then use lemma 2.1 to conclude the proof. let us now consider wi = (ã δ i ) −1r̃0i f, which, by definition, is the same as a(wi, φj ) = (φj , r̃ 0 i f ), ∀xj ∈ w̃ δi . 6, 4(2004) a restricted additive schwarz preconditioner with ... 81 here and in the rest of the proof, φj is the basis function associated with the node xj ∈ w̃ δi . using that r̃0i is symmetric and (φj , r̃ 0 i f ) = (f, r̃ 0 i φj ) = a(u ∗, r̃0i φj ), we get a(wi, φj ) = a(u ∗, r̃0i φj ). (13) let us compute a(wi, φk). since xk is an overlapping node, it cannot be on the boundary of ω̃δi . this leaves us with the following two cases. case (1): the support of φk(x) belongs to the exterior of ω̃δi . since the supports of wi and φk do not overlap, we have a(wi, φk) = 0. case (2): the support of φk(x) belongs to the interior of ω̃δi . in this case, we have a(wi, φk) = a(u ∗, r̃0i φk). taking the sum of the above equality for i = 1, · · · , n , a(w, φk) = a ( n∑ i=1 wi, φk ) = a ( u∗, n∑ i=1 r̃0i φk ) = a(u∗, φk), which proves (12). here the fact ∑n i=1 r̃ 0 i = i is used. there are basically two ways to compute w in practice. suppose that subdomain problems are solved using some lu factorization based method. one can use the same factorization of ãδi to modify the right-hand side of the system and to solve subdomain problems in the preconditioning steps, as what was suggested in lemma 3.1. or, one can obtain w by solving several small poisson problems on each subdomain with zero dirichlet boundary conditions in the overlapping regions ωδi,ovl. in both strategies, the computation can be done in parallel and no communication is needed in a distributed memory implementation. let f̃ = f − aw, then ũ∗ is the solution of the following linear system of equations aũ∗ = f̃ . (14) since ũ∗ ∈ ṽδ, g ≡ p̃ δũ∗ is well defined, and can be computed without knowing ũ∗ by using the following relations: a(p̃ δi ũ ∗, v) = a(ũ∗, v) = (f̃ , v), ∀v ∈ ṽδi and i = 1, · · · , n. 82 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) more precisely speaking, we can obtain g by solving the subdomain problems a(gi, v) = (f̃ , v), ∀v ∈ ṽδi , for i = 1, · · · , n , and taking g = g1 + · · · + gn . with such a right-hand side, we introduce a new linear system p̃ δũ∗ = g, (15) which is equivalent to the linear system (14). the system (15) is a symmetric positive definite system under the usual energy inner product, and therefore, can be solved using the conjugate gradient method. rasho has a few advantages over the classical as preconditioner. let us recall as briefly. let ( rδi v ) (xi) = ⎧⎨ ⎩ vi if xi ∈ w δi 0 otherwise. (16) then the as operator takes the following matrix form p δ = ( (aδ1) −1 + · · · + (aδn )−1 ) a (17) where aδi = r δ i a(r δ i ) t . because of the inclusion of the cut interface nodes, the size of the matrix aδi is |w δi |, which is slightly larger than the size of the matrix ãδi , which is |w̃ δi |. in a distributed memory implementation, the operation rδi v involves moving data from one processor to another, but the operation r̃δi v does not involve any communication. more precisely speaking, in rasho, if u ∈ ṽδ, then it is easy to see that r̃δi au = r̃ δ i,inau, (18) where r̃δi,in is defined as ( r̃δi,inv ) (xi) = ⎧⎨ ⎩ vi if xi ∈ w δi,in 0 otherwise. (19) therefore, for functions in ṽδ, we can rewrite p̃ δ, as in (10), in the following form p̃ δ = ( (ãδ1) −1r̃δ1,in + · · · + (ãδn )−1r̃δn,in ) a. (20) although the operator (20) does not look like a symmetric operator, but it is indeed symmetric when applying to functions in the subspace ṽδ. the form (18) takes the advantage of the fact that the operator r̃δi,in is communication-free in the sense that it needs only the residual associated with nodes in w γ δ i,in ⊂ ω0i . we make some further comments on how the residual au can be calculated in a distributed memory environment, for a given vector u ∈ ṽδ. in a typical implementation, the matrix a is constructed and stored in the form of {ãδi }, each processor has 6, 4(2004) a restricted additive schwarz preconditioner with ... 83 one or several of the subdomain matrix ãδi . similarly u is stored in the form of {ui}, where ui ∈ ṽδi . we note, however, that to compute the residual at nodes w γ δ i,in some communications are required. the processor associated with subdomain ωδi needs to obtain the local solution from the neighboring subdomains at nodes connected to w γ δ i,in. it is important to note that the amount of communications does not depend on the size of the overlap since only one layer of nodes is required. this shows that in terms of communications, the rasho is superior to as and ras. 4 theoretical analysis the algorithm presented in the previous section is applicable for general sparse, symmetric positive definite linear systems. the notions of subdomains, harmonic overlaps, the classification of the nodal points, etc, can all be defined in terms of the graph of the sparse matrix. in this section we provide a nearly optimal estimate for a poisson equation discretized with a piecewise linear finite element method. we estimate the condition number of the rasho operator p̃ δ in terms of the fine mesh size h, the subdomain size h, and the overlapping factor δ. we note that because we do not include a coarse space, the constant will depend on the subdomain size h. we shall follow the abstract additive schwarz theory [20]: lemma 4.1 suppose the following assumptions hold: i) there exists a constant c0 such that for any u ∈ ṽδ there exists a decomposition u = n∑ i=1 ui, where ui ∈ ṽδi , and n∑ i=1 |ui|2h1(ωδ i ) ≤ c20 |u|2h1(ω). ii) there exist constants �ij , i, j = 1, . . . , n such that a(ui, uj ) ≤ �ij a(ui, ui)1/2a(uj, uj )1/2, ∀ui ∈ ṽδi , ∀uj ∈ ṽδj . then, p̃ δ is invertible, symmetric; i.e., a(p̃ δu, v) = a(u, p̃ δv), ∀u, v ∈ ṽδ, and c−20 a(u, u) ≤ a(p̃ δu, u) ≤ ρ(e)a(u, u), ∀u ∈ ṽδ. (21) here ρ(e) is the spectral radius of e, which is a (n ) × (n ) matrix made of {�ij}. it is trivial to see that ρ(e) ≤ c. so our focus in the rest of the paper is in bounding c0. 84 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) 4.1 a partition of unity and a comparison function the construction of a partition of unity is one of the key steps in an additive schwarz analysis. we construct φi(x) as follows: φi(xk) = ⎧⎪⎨ ⎪⎩ 1 if xk ∈ w γ δ i,in discrete harmonic if xk ∈ w δi,ovl ∪ w δi,non 0 if xk ∈ w \w̃ δi note that φi(x) = 0 if x �∈ ω̃δi . let us denote ω(w δi,non) by ωδi,non, then φi(xk) = 1 at xk ∈ w δi,non for the case ωδi,non ∩ ∂ω = ∅ since all the boundary nodes of ωδi,non belong to w γ δ i,in. also, it is easy to see that {φi(x), i = 1, . . . , n} restricted to w γ δ form a partition of unit. �� h �� type (2) type (3) type (4) type (1) figure 2: the partition of ωδi into the union of four types of subregions. this is a ‘floating’ subdomain with δ = 2. the collection of “•” forms the set w 0i . in addition to φi(x), we also need to construct a comparison function θi(x) for each subdomain ωδi . comparison functions, or barrier functions, are very useful for many schwarz algorithms, such as these on non-matching grids [6]. we will show 6, 4(2004) a restricted additive schwarz preconditioner with ... 85 that, even though θi(x) ∈ vδi , not in ṽδi as we wished, it can still be used to bound functions in ṽδi . both θi(x) and φi(x) depend on the overlapping factor δ. because φi(x) is discrete harmonic at w δi,ovl ∪ w δi,non, we have a(φi, φi) ≤ a(θi, θi). to construct the function θi(x), we first consider the case when ω0i is a floating square subdomain. “floating” refers to the fact that the subdomain doesn’t touch the boundary ∂ω. the extension to cases when ωδi touches the boundary is simple and we will comment on it later. to further simplify our arguments, we assume that ωδi and its neighboring extended subdomains ωδj are squares of the same size; i.e. sides length equals to h + 2(δ + 1)h. this assumption is equivalent to that ωr has size h and δ levels of overlap is applied; see fig. 2. and we also assume the overlap is not too large; for the analysis given below δh no larger than h/4 is enough. our techniques can be modified to consider larger overlaps and more complex subdomains, although too large of an overlap has little practical value. roughly speaking, θi(x) equals to φi(x) on w \w δi,ovl. on the overlapping region w δi,ovl we need to define θi(x) carefully so that we can control its energy in the semi h1 norm. for this purpose, we decompose ωδi into subregions of four types: ω δ i,non, ωδδi , ω δh i , and ω δδ̃ i and define θi(x) on each piece of the subregion separately. type (1): the first subregion is ωδi,non, which a square with sides of size h − 2δh. type (2): the second subregion ωδδi is the area where ω δ i overlaps simulatneously with three neighbors ωδj . ω δδ i therefore represents the union of the four corner pieces of ωδi ; i.e. four squares with sides of size (2δ + 1)h. type (3) and (4): the area where ωδi overlaps only one neighbor are four rectangles of size h − 2δh by (2δ + 1)h. we further partition each of the four rectangles into three smaller rectangles; i.e. two of them are of ωδδ̃i type and one of them of ωδhi type. for instance, without lost of generality, let us consider the intersection of ωδi with its right neighbor ω δ j , excluding the corner parts. in this case, the subregion to be partitioned is a rectangle of size (2δ + 1)h in the x direction and h − 2δh in the y direction. the partition of this rectangles gives two smaller rectangles of ωδδ̃i type with dimensions 2(δ + 1)h × δh and each one has an edge in common with a square of ωδδi type. we denote them as transition subregions because they are placed between a corner type subregion ωδδi and a face type subregion ω δh i . the ω δh i face type subregions are the smaller rectangles that are placed between the two smaller rectangles of ωδδ̃i type. ω δh i face type regions are of size (2δ + 1)h by h − 4δh. for any node x belonging to a type (1) region ωδi,non, we define θi(x) to be equal to one; i.e., equals to φi(x). therefore |φi(x)|2 h1(ωδ i,non ) = |θi(x)|2h1(ωδ i,non ) = 0. we next define θi(x), node by node, in ωδi,ovl, which is the union of corner, transition and face type regions defined above. 86 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) for a type (2) region ωδδi . let q be such a square with vertices v1 = (a, b), v2 = (a + (2δ + 1)h, b), v3 = (a, b + (2δ + 1)h), and v4 = (a + (2δ + 1)h, b + (2δ + 1)h). we assume that v1, v2, and v4 belong to ∂ωδi . in other words, q is located on the southeast corner of ωδi . let use also introduce another square region q̃, with vertices v3 = (a, b + (2δ + 1)h), ṽ1 = (a, b + δh), ṽ2 = (a + (δ + 1)h, b + δh), and ṽ4 = (a + (δ + 1)h, b + (2δ + 1)h). note that q̃ is contained in q, with v3 as the common vertex. to define θi(x) on q, we set θi(v3) = 1, θi(ṽ1) = 0, θi(ṽ2) = 0, θi(ṽ4) = 0. at the remaining nodes x on the edges ṽ1ṽ2 and ṽ2ṽ4 we set θi(x) = 0, and on the edges v3ṽ1 and v3ṽ4 we set θi(x) = 1. for nodes on q\q̃ we set θi(x) = 0. it remains only to define θi(x) for nodes x in the interior of q̃. to define θi(x) there we use a well-known cutoff function technique, such as the one introduced in lemma 4.4 of [10] but for two-dimensional square regions. an illustrative picture of θi(x) in a typical region ωδδi is shown in fig 3. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 figure 3: an illustrative picture of θi(x) in a typical region ωδδi . for the completeness of this paper, we include the construction below. let c be the center of the square q̃. the construction of θi(x) is defined by the following steps: (1) define θi(v3) = 1, θi(ṽ2) = 0, θi(ṽ1) = 0 and θi(ṽ4) = 0. (2) for a point p that belongs to the segments v3ṽ4 or v3ṽ1, define θi(p ) = 1. for a point p that belongs to the segments ṽ4ṽ2 or ṽ1ṽ2, define θi(p ) = 0. (3) for a point y that belongs to the line segment connecting c to v3, define θi(y ) by linear interpolation between values θi(c) = 1/2 and θi(v3) = 1. for a point 6, 4(2004) a restricted additive schwarz preconditioner with ... 87 y that belongs to the line segment connecting c to ṽ2, define θi(y ) by linear interpolation between values θi(c) = 1/2 and θi(ṽ2) = 0. (4) for a point s that belongs to a line segment connecting a point y to a vertex ṽ1 or ṽ4, define θi(s) = θi(y ). (5) note that the θi is defined everywhere on q̃ ∪ ∂q̃. θi is continuous everywhere except at the points ṽ1 and ṽ4. we redefine θi as the continuous piecewise linear finite element function given by the standard pointwise interpolation. the most important observation of the construction of θi(x) inside q̃ is that |∇θi(x)| ≤ c/r near ṽ1 or ṽ4. here r is the distance of x to ṽ1 or ṽ4. therefore, we obtain (see [10] and [19]) |θi(x)|2h1(q) = |θi(x)|2h1(q̃) ≤ c ( 1 + log ( (δ + 1)h h )) = c(1 + log(δ + 1)). since inside of ωδi there are four of those squares we obtain |θi(x)|2h1(ωδδ i ) ≤ c (1 + log(δ + 1)) . type (3) regions consist of transition type rectangles. let us consider one of them and denote it by t , which we assume has vertices at v3 = (a, b + (2δ + 1)h), v4 = (a+(2δ+1)h, b+(2δ+1)h), v5 = (a, b+(3δ+1)h), and v6 = (a+(2δ+1)h, b+(3δ+1)h). note that t stands on the top of the square q introduced above and has the common edge v3v4. we define θi(x) over the edge v3v4 to be equal to φi(x). over the edge v3v5, we set θi(x) = 1. over the edge v4v6, we set θi(x) = 0. and over the edge v5v6 we let θi(x) decrease linearly from the value 1 to 0. what remains is to define θi(x) inside t . let us define the nodes vl = (a + δh, b + (2δ + 1)h) and vr = (a + (δ + 1)h, b + (2δ + 1)h), which is the same as the node ṽ4 used in the description of type (2) regions. the nodes vl and vr are exactly the places on the edge v3v4 where φi(x) jumps from 1 to 0. on the triangle v3vlv5 we set θi(x) = 1. on the triangle vrv4v6 we set θi(x) = 0. on the region vlvrv6v5, we let θi(x) decrease linearly in the x direction from the value 1 to 0. we note that next to the nodes vlvr , θi(x) has a singular behavior similar to |∇θi(x)| ≤ c/r where r is the distance from x to the line vl vr. similarly, we have |θi(x)|2h1(t ) ≤ c (1 + log(δ + 1))) . since there are eight rectangles of type (3) inside ωδδ̃i , we obtain |θi(x)|2h1(ωδδ̃ i ) ≤ c (1 + log(δ + 1)) . type (4) regions are rectangles of face type. let r be one of them, and we assume that the vertices are given by v5 = (a, b+(3δ +1)h), v6 = (a+(2δ +1)h, b+(3δ +1)h), v7 = (a, b + h − (δ − 1)h), and v8 = (a + (2δ + 1)h, b + h − (δ − 1)h). note that r is 88 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) on the top of the rectangle t defined above and its height is h − 4δh. the vertices v6 and v8 are the vertices that belong to ∂ωδi . we define θi(x) = 1 if x is on the edge v5v7, and equals zero if x is on the edge v6v8, and linear in the x direction for the remaining points. we obtain then |θi(x)|2h1 (r) ≤ h − 4δh (2δ + 1)h . since there are four of those rectangles inside ωδhi , we obtain |θi(x)|h1(ωδh i ) ≤ c h − 4δh (2δ + 1)h ≤ c h (2δ + 1)h . for the cases in which ω0i touches the boundary ∂ω, the analysis needs to be modified slightly. the first modification is because the shape of the overlapping region changes slightly, i.e. the longer side is shorter. it is easy to see that we get similar bounds as before. the other modification is because φi on ωδi,non is not identically equal to one and therefore the corresponding energy is not necessarily zero. for this case we can design θi similarly and obtain |θi(x)|2h1(ωδi,non) ≤ c ( 1 + log ( h h )) . putting all pieces of θi(x) together, we see that θi(x) ∈ vδi and it equals to φi(x) on w γ δ . adding all the estimates on subregions of four types, we arrive at the following lemma. lemma 4.2 for i = 1, · · · , n. θi(x) ∈ vδi , and φi(x) ∈ ṽδi , and also (1) |φi|2 h1(ωδ i ) ≤ |θi|2h1(ωδ i ) . (2) |θi|2h1(ωδ i \ωδ i,non ) ≤ c ( 1 + log(δ + 1) + h (2δ + 1)h ) . (3) if ωδi,non ∩ ∂ω = ∅ then |θi|2h1(ωδ i,non ) = 0. (4) if ωδi,non ∩ ∂ω �= ∅ then |θi|2h1(ωδ i,non ) ≤ c ( 1 + log ( h h )) . here c > 0 is independent of the parameters h, h and δ. 4.2 a bounded partition lemma to obtain the parameter c0 of assumption i) of the abstract additive schwarz theory, see lemma 4.1, we construct a decomposition of ṽδ and prove its boundedness below. 6, 4(2004) a restricted additive schwarz preconditioner with ... 89 lemma 4.3 there exists a constant c > 0, independent of h, h, and δ, such that for any u ∈ ṽδ, there exist ui ∈ ṽδi , such that u = n∑ i=1 ui, (22) and n∑ i=1 |ui|2h1(ω) ≤ c (1 + log(δ + 1)) ( 1 + log ( h h )) |u|2 h1(ω) + c 1 h2 ( 1 + log(δ + 1) + h (2δ + 1)h ) |u|2h1(ω). (23) proof. we first construct the decomposition (22). for any given u ∈ ṽδ, we define ui ∈ ṽδi as ui(xk) = ⎧⎨ ⎩ u(xk) if xk ∈ w δi,in discrete harmonic if xk ∈ w δi,ovl 0 if xk ∈ w \w̃ δi . it is easy to see (22) holds. for i = 1, . . . , n , let us define vi ∈ ṽδi by vi = ui − ūiφi ∈ ṽδi , where ūi = 1 |ωδi | ∫ ωδ i udx is the average of u on the extended region ωδi . here |ωδi | is the area of the region ωδi . the next step is to bound the sums ∑n i=1 |vi|2h1(ω) and ∑n i=1 |ūiφi|2h1(ω). for the second sum, we use lemma 4.2 to obtain n∑ i=1 |ūiφi|2h1(ω) ≤ c ( 1 + log ( h h )) ∑ i∈∂ω |ūi|2+ c ( 1 + log(δ + 1) + h (2δ + 1)h )∑ i |ūi|2. here we use i ∈ ∂ω to denote the subdomains ω0i that touch the boundary ∂ω with a face. by cauchy-schwarz and friedrichs inequalities we have n∑ i=1 |ūi|2 = n∑ i=1 ( 1 |ωδi | ∫ ωδ i udx )2 ≤ c n∑ i=1 1 h2 ‖u‖2 l2(ωδ i ) ≤ c 1 h2 ‖u‖2l2(ω) ≤ c 1 h2 |u|2h1(ω). 90 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) and for the cases i ∈ ∂ω, we can use a poincaré inequality to obtain ∑ i∈∂ω |ūi|2 ≤ c ∑ i∈∂ω 1 h2 ‖u‖2 l2(ωδ i ) ≤ c ∑ i∈∂ω |u|2 h1(ωδ i ) ≤ c|u|2h1(ω). to bound the other terms |vi|2h1(ω), i = 1, . . . , n , we use θi(x), i = 1, . . . , n , introduced before. consider ṽi ∈ vδi defined as follows ṽi(x) = ih(θi(x)(u(x) − ūi)). note that ṽi(x) is equal to vi(x) on w γ δ i and on ∂ω δ i . on ω δ i,ovl and ω δ i,non, vi is discrete harmonic. therefore, we have |vi|2h1(ωδ i ) ≤ |ṽi|2h1(ωδ i ) . the rest of the proof will be devoted to the estimate of |ṽi|2h1(ωδ i ) in terms of |u|2 h1(ωδ i ) . let k be an element of ωδi and let us denote wi = u − ūi then |ṽi|2h1(k) = |ih(θiwi)|2h1(k) ≤ 2|θ̄iwi|2h1(k) + 2|ih((θ̄i − θi)wi)|2h1(k). (24) here, θ̄i is the average of θi on k, and ih is the standard pointwise interpolation. to estimate the first part of (24) we use the fact that |θ̄i| ≤ 1, to obtain |θ̄iwi|2h1(k) = |θ̄i(u − ūi)|2h1(k) ≤ |u − ūi|2h1(k) = |u|2h1(k). the last equality comes from the fact that ūi is a constant. for the second part of (24), according to an inverse inequality we have |ih((θ̄i − θi)wi)|2h1(k) ≤ c 1 h2 ‖ih((θ̄i − θi)wi)‖2l2(k). (25) to obtain the bound for the right-hand side of (25), we consider the element k in four different situations corresponding to the four types of subregions into which the the subregion ωδi is split i.e., ω δ i,non, ω δh i , ω δδ̃ i and ω δδ i . the proof for the cases k ⊂ ωδhi and k ⊂ ωδδ̃i are nearly the same, so we only consider one of them here. for k ⊂ ωδhi , since ‖θ̄i − θi‖2l∞(k) ≤ c ( h (2δ + 1)h )2 , we obtain 1 h2 ‖ih((θ̄i − θi)wi)‖2l2(k) ≤ c 1 ((2δ + 1)h)2 ‖wi‖2l2(k). applying a technique developed in dryja and widlund [11], we obtain 1 ((2δ + 1)h)2 ‖wi‖2l2(ωδh i ) ≤ c ( h (2δ + 1)h |wi|2h1(ωδ i ) + 1 h((2δ + 1)h) ‖wi‖2l2(ωδ i ) ) . (26) 6, 4(2004) a restricted additive schwarz preconditioner with ... 91 using the fact |wi|2h1(ωδ i ) = |u|2 h1(ωδ i ) and a friedrichs inequality ‖wi‖2l2(ωδ i ) ≤ ch2|u|2 h1(ωδ i ) . (27) combining the estimates (26) and (27), we obtain 1 ((2δ + 1)h)2 ‖wi‖2l2(ωδh i ) ≤ c h (2δ + 1)h |u|2 h1(ωδ i ) . for the case when k ⊂ ωδδi , we use similar arguments as in dryja, smith and widlund [10] to obtain ∑ k∈ωδδ i 1 h2 ‖ih((θ̄i − θi)wi‖2l2(k) ≤ ∑ k∈ωδδ i c 1 r2 ‖wi‖2l2(k), (28) where ch ≤ r ≤ c((δ + 1)h) is the distance to those “cut pieces”. we have used here that θi(x) has the singular behavior c/r on ωδδi . we have then ∑ k∈ωδδ i 1 r2 ‖wi‖2l2(k) ≤ c ∫ c(δ+1)h ch ∫ α r−2r‖wi‖2l∞(ωδδ i ) dαdr (29) and ‖wi‖2l∞(ωδδ i ) ≤ c ( 1 + log ( h h )) |u|2 h1(ωδ i ) . (30) for the inequality (30), we have used a well-known result (see bramble [2]) ‖u − ūi‖2l∞(ωδδ i ) ≤ ‖u − ūi‖l∞(ωδ i ) ≤ c ( 1 + log ( h h )) ‖u − ūi‖2h1(ωδ i ) and that ūi is the average of u on ωδi ; i.e., a friedrichs inequality ‖u − ūi‖2h1(ωδ i ) ≤ c|u|2 h1(ωδ i ) . putting (29) and (30) together, we obtain ∑ k∈ωδδ i 1 r2 ‖wi‖2l2(k) ≤ c ( (1 + log(δ + 1)) ( 1 + log ( h h ))) |u|2 h1ωδi . (31) for the case k ⊂ ωδi,non. if ω0i is a floating subdomain, which is to say that ωδi,non does not touch ∂ω, then θ̄i − θi is zero. if ωδi,non touches the boundary ∂ω, then the estimate becomes |vi|2h1(ωδ i,non ) ≤ c ( |u|2 h1(ωδ i,non ) + |ūi|2|φi|2h1(ωδ i,non ) ) ≤ c ( 1 + log ( h h )) |u|2 h1(ωδ i ) . (32) 92 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) 4.3 the main theorem we state the main theorem of this paper here, the proof follows directly from the abstract as theory and the lemmas just proved. theorem 4.1 the rasho operator p̃ δ is symmetric in the inner product a(·, ·), nonsingular, and bounded in the following sense c−20 a(u, u) ≤ a(p̃ δu, u) ≤ c1a(u, u) ∀u ∈ ṽδ. (33) here c20 = c ( (1 + log(δ + 1)) ( 1 + log ( h h )) + 1 h2 ( 1 + log(δ + 1) + h (2δ + 1)h )) . the constants c, c1 > 0 are independent of h, h, and δ. we remark that the corresponding convergence rate estimate for the regular onelevel as [11], in terms of the constant c0, is c20 = c ( 1 + 1 h(2δ + 1)h ) . the lower bound c20 of rasho is theoretically slightly worse than the lower bound of as in case of large overlap, but roughly the same for small overlap. on the other hand, the upper bound c1 of rasho is smaller than the upper bound of as. we can see this since ṽδk ⊂ vδk , ∀k, implies that the positive numbers �ij defined in lemma 4.1 are smaller for rasho than the correspondent �ij for as. consequently, the spectral radius of e in rasho is smaller. because c1 of rasho is smaller, the numerical performance of rasho presented in the next section is better than that of as. we also remark that the results of the paper is for one-level schwarz algorithms. because of the “harmonic overlap“ requirement, the extension of the algorithm to multiply levels is not as trivial as the multilevel as. 5 numerical experiments in this section, we present some numerical results for solving the poisson’s equation on the unit square with zero dirichlet boundary conditions. we compare the performance of rasho and as preconditioned conjugate gradient methods in terms of the number of iterations and the condition numbers. we pay particular attention to the dependence on the number of subdomains and the size of overlap. we first discuss a few implementation issues related to the new preconditioner. in order to apply the rasho/cg method, it is necessary to force the solution to belong to ṽδ. to do so, a pre-cg-computation is needed, and it is done through the formula (11). we note that u = u∗ − w ∈ ṽδ, see lemma 3.1, and therefore, we can apply the regular pcg to the rasho preconditioned system (15). the as/cg is 6, 4(2004) a restricted additive schwarz preconditioner with ... 93 table 1: rasho and as preconditioned cg for solving the poisson’s equation on a 128 × 128 mesh decomposed into 2 × 2 = 4 subdomains with overlap = ovlp. the as/cg results are shown in ( ). the “+1” is for the preprocessing step needed for rasho. ovlp iter cond max min h 42 (42) 129.(129.) 1.98 (1.98) 0.0154 (0.0154) 3h 24+1 (28) 48.4 (86.3) 1.94 (4.00) 0.0402 (0.0464) 5h 20+1 (23) 33.3 (51.8) 1.91 (4.00) 0.0574 (0.0773) 7h 18+1 (20) 27.2 (37.0) 1.89 (4.00) 0.0694 (0.1081) table 2: rasho and as preconditioned cg for solving the poisson’s equation on a 32 ∗ dom × 32 ∗ dom mesh decomposed into dom × dom subdomains with overlap = 3h, i.e. δ = 1. dom × dom iter cond max min 2 × 2 19+1 (20) 26.8 (43.7) 1.89 (4.00) 0.0708 (0.0916) 4 × 4 39+1 (42) 86.9 (145.) 1.95 (4.00) 0.0225 (0.0276) 8 × 8 75+1 (78) 328. (550.) 1.97 (4.00) 0.0060 (0.0073) 16 × 16 147+1 (156) 1295 (2168.) 1.98 (4.00) 0.0015 (0.0018) the classical additive schwarz preconditioned cg as described in [8]. we note that in the case δ = 0, i.e. ovlp = h, rasho and as are the same. the stopping condition for cg is to reduce the initial residual by a factor of 10−6. the exact solution of the equation is u(x, y) = e5(x+y) sin(πx) sin(πy). all subdomain problems are solved exactly. the iteration counts (iter), condition numbers (cond), maximum (max) and minimum (min) eigenvalues of the preconditioned matrix are summerized in table 1, and table 2. from table 1 and table 2, it is clear that rasho/cg is always better than the classical as/cg in terms of the iteration counts and condition numbers. note that there is a practical suggestion for as that the overlap should be 3h − 5h width. in this case the condition number of rasho is almost twice smaller than as. this is an important result since it is easy to modify a (parallel) as/cg code to obtain a rasho/cg implementation. although we do not have any parallel results to report here, we are confident to predict that rasho/cg would be even better than as/cg on a parallel computer with distributed memory since much less communications are required. also the local solvers in rasho are slightly cheaper since the local solvers have slightly smaller numbers of unknowns than for the regular as. 94 xiao-chuan cai, maksymilian dryja and marcus sarkis 6, 4(2004) references [1] s. balay, w. gropp, l. mcinnes, and b. smith, the portable extensible toolkit for scientific computing (petsc), www.mcs.anl.gov/petsc, 2001. 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[20] b. smith, p. bjørstad, and w. gropp, domain decomposition: parallel multilevel methods for elliptic partial differential equations, cambridge university press, 1996. cubo, a mathematical journal vol.22, n◦02, (257–271). august 2020 http://dx.doi.org/10.4067/s0719-06462020000200257 received: 07 november, 2019 | accepted: 20 july, 2020 results on para-sasakian manifold admitting a quarter symmetric metric connection vishnuvardhana. s.v. 1 and venkatesha 2 1 department of mathematics, gitam school of science, gitam (deemed to be university) bengaluru, karnataka-561 203, india. 2 department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india. svvishnuvardhana@gmail.com, vensmath@gmail.com abstract in this paper we have studied pseudosymmetric, ricci-pseudosymmetric and projectively pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection and constructed examples of 3-dimensional and 5-dimensional para-sasakian manifold admitting a quarter-symmetric metric connection to verify our results. resumen en este art́ıculo hemos estudiado variedades para-sasakianas seudosimétricas, ricciseudosimétricas y proyectivamente seudosimétricas que admiten una conexión métrica cuarto-simétrica, y construimos ejemplos de variedades para-sasakianas 3-dimensional y 5-dimensional que admiten una conexión métrica cuarto-simétrica para verificar nuestros resultados. keywords and phrases: para-sasakian manifold, pseudosymmetric, ricci-pseudosymmetric, projectively pseudosymmetric, quarter-symmetric metric connection. 2020 ams mathematics subject classification: 53c35, 53d40. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000200257 258 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) 1 introduction one of the most important geometric property of a space is symmetry. spaces admitting some sense of symmetry play an important role in differential geometry and general relativity. cartan [5] introduced locally symmetric spaces, i.e., the riemannian manifold (m, g) for which ∇r = 0, where ∇ denotes the levi-civita connection of the metric. the integrability condition of ∇r = 0 is r · r = 0. thus, every locally symmetric space satisfies r · r = 0, whereby the first r stands for the curvature operator of (m, g), i.e., for tangent vector fields x and y one has r(x, y ) = ∇x∇y −∇y ∇x −∇[x,y ], which acts as a derivation on the second r which stands for the riemannchristoffel curvature tensor. the converse however does not hold in general. the spaces for which r · r = 0 holds at every point were called semi-symmetric spaces and which were classified by szabo [19]. semisymmetric manifolds form a subclass of the class of pseudosymmetric manifolds. in some spaces r · r is not identically zero, these turn out to be the pseudo-symmetric spaces of deszcz [9, 10, 11], which were characterized by the condition r · r = l q(g, r), where l is a real function on m and q(g, r) is the tachibana tensor of m. if at every point of m the curvature tensor satisfies the condition r(x, y ) · j = lj [(x ∧g y ) · j ], (1.1) then a riemannian manifold m is called pseudosymmetric (resp., ricci-pseudosymmetric, projectively pseudosymmetric) when j = r(resp., s, p) . here (x ∧g y ) is an endomorphism and is defined by (x ∧g y )z = g(y, z)x − g(x, z)y and lj is some function on uj = {x ∈ m : j 6= 0} at x. a geometric interpretation of the notion of pseudosymmetry is given in [13]. it is also easy to see that every pseudosymmetric manifold is ricci-pseudosymmetric, but the converse is not true. an analogue to the almost contact structure, the notion of almost paracontact structure was introduced by sato [18]. an almost contact manifold is always odd-dimensional but an almost paracontact manifold could be of even dimension as well. kaneyuki and williams [14] studied the almost paracontact structure on a pseudo-riemannian manifold. recently, almost paracontact geometry in particular, para-sasakian geometry has taking interest, because of its interplay with the theory of para-kahler manifolds and its role in pseudo-riemannian geometry and mathematical physics ([4, 7, 8], etc.,). as a generalization of semi-symmetric connection, quarter-symmetric connection was introduced. quarter-symmetric connection on a differentiable manifold with affine connection was defined and studied by golab [12]. from thereafter many geometers studied this connection on different manifolds. para-sasakian manifold with respect to quarter-symmetric metric connection was studied by cubo 22, 2 (2020) results on para-sasakian manifold admitting a quarter symmetric . . . 259 de et.al., [16, 1], pradeep kumar et.al., [17] and bisht and shanker [15]. motivated by the above studies in this article we study properties of projective curvature tensor on para-sasakian manifold admitting a quarter-symmetric metric connection. the organization of the paper is as follows: in section 2, we present some basic notions of para-sasakian manifold and quarter-symmetric metric connection on it. section 3 and 4 are respectively devoted to study the pseudosymmetric and ricci-pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection. here we prove that if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is pseudosymmetric (resp., ricci pseudosymmetric) then mn is an einstein manifold with respect to quarter-symmetric metric connection or it satisfies l r̃ = −2 (resp., l s̃ = −2). section 5 and 6 are concerned with projectively flat and projectively pseudosymmetric para-sasakian manifold mn admitting a quarter-symmetric metric connection. finally, we construct examples of 3-dimensional and 5-dimensional para-sasakian manifold admitting a quarter-symmetric metric connection and we find some of its geometric characteristics. 2 preliminaries a differential manifold mn is said to admit an almost paracontact riemannian structure (φ, ξ, η, g), where φ is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form and g is a riemannian metric on mn such that φ2x = x − η(x)ξ, η(ξ) = 1, φ(ξ) = 0, η(φx) = 0, (2.1) g(x, ξ) = η(x), g(φx, φy ) = g(x, y ) − η(x)η(y ), (2.2) for all vector fields x, y ∈ χ(mn). if (φ, ξ, η, g) on mn satisfies the following equations (∇xφ)y = −g(x, y )ξ − η(y )x + 2η(x)η(y )ξ, (2.3) dη = 0 and ∇xξ = φx, (2.4) then mn is called para-sasakian manifold [3]. in a para-sasakian manifold, the following relations hold [6]: (∇xη)y = −g(x, y ) + η(x)η(y ), (2.5) η(r(x, y )z) = g(x, z)η(y ) − g(y, z)η(x), (2.6) r(x, y )ξ = η(x)y − η(y )x, r(ξ, x)y = η(y )x − g(x, y )ξ, (2.7) s(x, ξ) = −(n − 1)η(x), (2.8) s(φx, φy ) = s(x, y ) + (n − 1)η(x)η(y ), (2.9) for every vector fields x, y, z on mn. here ∇ denotes the levi-civita connection, r denotes the riemannian curvature tensor and s denotes the ricci curvature tensor. 260 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) here we consider a quarter-symmetric metric connection ∇̃ on a para-sasakian manifold [16] given by ∇̃xy = ∇xy + η(y )φx − g(φx, y )ξ. (2.10) the relation between curvature tensor r̃(x, y )z of mn with respect to quarter-symmetric metric connection ∇̃ and the curvature tensor r(x, y )z with respect to the levi-civita connection ∇ is given by r̃(x, y )z = r(x, y )z + 3g(φx, z)φy − 3g(φy, z)φx +{η(x)y − η(y )x}η(z) − [g(y, z)η(x) − η(y )g(x, z)]ξ. (2.11) also from (2.11) we obtain s̃(y, z) = s(y, z) + 2g(y, z) − (n + 1)η(y )η(z) − 3traceφ g(φy, z), (2.12) where s̃ and s are ricci tensors of connections ∇̃ and ∇ respectively. 3 pseudosymmetric para-sasakian manifold admitting a quartersymmetric metric connection a para-sasakian manifold mn admitting a quarter-symmetric metric connection is said to be pseudosymmetric if r̃(x, y ) · r̃ = l r̃ [(x ∧g y ) · r̃], (3.1) holds on the set u r̃ = {x ∈ mn : r̃ 6= 0 at x}, where l r̃ is some function on u r̃ . suppose that mn be pseudosymmetric, then in view of (3.1) we have r̃(ξ, y )r̃(u, v )w − r̃(r̃(ξ, y )u, v )w − r̃(u, r̃(ξ, y )v )w −r̃(u, v )r̃(ξ, y )w = l r̃ [(ξ ∧g y )r̃(u, v )w − r̃((ξ ∧g y )u, v )w −r̃(u, (ξ ∧g y )v )w − r̃(u, v )(ξ ∧g y )w ]. (3.2) by virtue of (2.7) and (2.11), (3.2) takes the form (l r̃ + 2)[η(r̃(u, v )w)y − g(y, r̃(u, v )w)ξ − η(u)r̃(y, v )w + g(y, u)r̃(ξ, v )w −η(v )r̃(u, y )w + g(y, v )r̃(u, ξ)w − η(w)r̃(u, v )y + g(y, w)r̃(u, v )ξ] = 0. (3.3) taking inner product of (3.3) with ξ and using (2.6) and (2.11), we get (l r̃ + 2)[g(y, r(u, v )w) + 3g(φu, w)g(φv, y ) − 3g(φv, w)g(φu, y ) +η(w){η(u)g(v, y ) − η(v )g(u, y )} − {g(v, w)η(u) − η(v )g(u, w)}η(y ) +2{g(v, w)g(y, u) − g(v, y )g(u, w)}] = 0. (3.4) cubo 22, 2 (2020) results on para-sasakian manifold admitting a quarter symmetric . . . 261 assuming that l r̃ + 2 6= 0, the above equation becomes g(y, r(u, v )w) + 3g(φu, w)g(φv, y ) − 3g(φv, w)g(φu, y ) +η(w){η(u)g(v, y ) − η(v )g(u, y )} − [g(v, w)η(u) − η(v )g(u, w)]η(y ) +2[g(v, w)g(y, u) − g((v, y )g(u, w)] = 0. (3.5) putting v = w = ei, where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, i = 1, 2, 3, · · · , n, we get s̃(y, u) = −2(n − 1)g(y, u). (3.6) hence, we can state the following: theorem 1. if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is pseudosymmetric then mn is an einstein manifold with respect to quarter-symmetric metric connection or it satisfies l r̃ = −2. 4 ricci-pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection a para-sasakian manifold mn admitting a quarter-symmetric metric connection is said to be ricci-pseudosymmetric if the following condition is satisfied r̃(x, y ) · s̃ = l s̃ [(x ∧g y ) · s̃], (4.1) on u s̃ . let para-sasakian manifold mn admitting a quarter-symmetric metric connection be riccipseudosymmetric. then we have s̃(r̃(x, y )z, w) + s̃(z, r̃(x, y )w) = l s̃ [s̃((x ∧g y )z, w) + s̃(z, (x ∧g y )w)]. (4.2) by taking y = w = ξ and making use of (2.7), (2.8) and (2.11), the above equation turns into (l s̃ + 2)[s̃(x, z) + 2(n − 1)g(x, z)] = 0 (4.3) thus, we have the following assertion: theorem 2. if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is ricci-pseudosymmetric then mn is an einstein manifold with respect to quarter-symmetric metric connection or it satisfies l s̃ = −2. 262 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) 5 projectively flat para-sasakian manifold admitting a quartersymmetric metric connection the projective curvature tensor on a riemannian manifold is defined by [2] p(x, y )z = r(x, y )z − 1 (n − 1) [s(y, z)x − s(x, z)y ]. (5.1) for an n-dimensional para-sasakian manifold mn admitting a quarter-symmetric metric connection, the projective curvature tensor is given by p̃(x, y )z = r̃(x, y )z − 1 (n − 1) [s̃(y, z)x − s̃(x, z)y ]. (5.2) theorem 3. a projectively flat para-sasakian manifold mn admitting a quarter-symmetric metric connection is an einstein manifold with respect to quarter-symmetric metric connection. proof. consider a projectively flat para-sasakian manifold admitting a quarter-symmetric metric connection. then from (5.2) we have g(r̃(x, y )z, w) = 1 (n − 1) [s̃(y, z)g(x, w) − s̃(x, z)g(y, w)]. (5.3) setting x = w = ξ in (5.3) and using (2.7), (2.8), (2.11) and (2.12), we get s̃(x, z) = −2(n − 1)g(x, z). (5.4) hence, the proof is completed. 6 projectively pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection a para-sasakian manifold admitting a quarter-symmetric metric connection is said to be projectively pseudosymmetric if r̃(x, y ) · p̃ = l p̃ [(x ∧g y ) · p̃], (6.1) holds on the set u p̃ = {x ∈ mn : p̃ 6= 0 at x}, where l p̃ is some function on u p̃ . let mn be projectively pseudosymmetric, then we have r̃(x, ξ)p̃(u, v )ξ − p̃(r̃(x, ξ)u, v )ξ − p̃(u, r̃(x, ξ)v )ξ −p̃(u, v )r̃(x, ξ)ξ = l p̃ [(x ∧g ξ)p̃(u, v )ξ − p̃((x ∧g ξ)u, v )ξ −p̃(u, (x ∧g ξ)v )ξ − p̃(u, v )(x ∧g ξ)ξ]. (6.2) cubo 22, 2 (2020) results on para-sasakian manifold admitting a quarter symmetric . . . 263 by virtue of (2.11), (2.12) and (5.2), (6.2) becomes (l p̃ + 2)p̃(u, v )x = 0. (6.3) so, one can state that: theorem 4. if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is projectively pseudosymmetric then mn is projectively flat with respect to quarter-symmetric metric connection or l p̃ = −2. in view of theorem 3, one can state the above theorem as theorem 5. if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is projectively pseudosymmetric then mn is an einstein manifold with respect to quarter-symmetric metric connection or l p̃ = −2. 7 examples 7.1 example we consider a 3-dimensional manifold m = {(x, y, z) ∈ r3 : z 6= 0}, where (x, y, z) are standard coordinates in r3. let {e1, e2, e3} be a linearly independent global frame field on m given by e1 = e z ∂ ∂y , e2 = e z( ∂ ∂y − ∂ ∂x ), e3 = ∂ ∂z , if g is a riemannian metric defined by g(ei, ej) =      1, i = j 0, i 6= j for 1 ≤ i, j ≤ 3, and if η is the 1-form defined by η(z) = g(z, e3) for any vector field z ∈ χ(m). we define the (1, 1)-tensor field φ as φ(e1) = e1, φ(e2) = −e2, φ(e3) = 0. the linearity property of φ and g yields that η(e3) = 1, φ2u = u − η(u)e3, g(φu, φv ) = g(u, v ) − η(u)η(v ), 264 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) for any u, v ∈ χ(m). now we have [e1, e2] = 0, [e1, e3] = e1, [e2, e3] = e2. the riemannian connection ∇ of the metric g known as koszul’s formula and is given by 2g(∇xy, z) = xg(y, z) + y g(z, x) − zg(x, y ) − g(x, [y, z]) −g(y, [x, z]) + g(z, [x, y ]). using koszul’s formula we get the followings in matrix form     ∇e1e1 ∇e1e2 ∇e1e3 ∇e2e1 ∇e2e2 ∇e2e3 ∇e3e1 ∇e3e2 ∇e3e3     =     −e3 0 e1 0 −e3 e2 0 0 0     . clearly (φ, ξ, η, g) is a para-sasakian structure on m. thus m(φ, ξ, η, g) is a 3-dimensional para-sasakian manifold. using (2.10) and the above equation, one can easily obtain the following:     ∇̃e1e1 ∇̃e1e2 ∇̃e1e3 ∇̃e2e1 ∇̃e2e2 ∇̃e2e3 ∇̃e3e1 ∇̃e3e2 ∇̃e3e3     =     −2e3 0 2e1 0 −2e3 2e2 0 0 0     . with the help of the above results it can be easily verified that r(e1, e2)e3 = 0, r(e2, e3)e3 = −e2, r(e1, e3)e3 = −e1, r(e1, e2)e2 = −e1, r(e2, e3)e2 = e3, r(e1, e3)e2 = 0, r(e1, e2)e1 = e2, r(e2, e3)e1 = 0, r(e1, e3)e1 = e3. and r̃(e1, e2)e3 = 0, r̃(e2, e3)e3 = −2e2, r̃(e1, e3)e3 = −2e1, r̃(e1, e2)e2 = −4e1, r̃(e2, e3)e2 = 2e3, r̃(e1, e3)e2 = 0, r̃(e1, e2)e1 = 4e2, r̃(e2, e3)e1 = 0, r̃(e1, e3)e1 = 2e3. (7.1) since e1, e2, e3 forms a basis, any vector field x, y, z ∈ χ(m) can be written as x = a1e1 + b1e2 + c1e3, y = a2e1 + b2e2 + c2e3, z = a3e1 + b3e2 + c3e3, where ai, bi, ci ∈ r, i = 1, 2, 3. using the expressions of the curvature tensors, we find values of riemannian curvature cubo 22, 2 (2020) results on para-sasakian manifold admitting a quarter symmetric . . . 265 and ricci curvature with respect to quarter-symmetric metric connection as; r̃(x, y )z = [−4{a1b2 − b1a2}b3 + 2{c1a2 − a1c2}c3]e1 + [−4{b1a2 − a1b2}a3 + 2{c1b2 − b1c2}c3]e2 + [−2{c1a2 − a1c2}a3 − 2{c1b2 − b1c2}b3]e3, (7.2) s̃(e1, e1) = s̃(e2, e2) = −6, s̃(e3, e3) = −4. (7.3) using (7.1), (7.3) and the expression of the endomorphism (x ∧g y )z = g(y, z)x −g(x, z)y , one can easily verify that s̃(r̃(x, e3)y, e3) + s̃(y, r̃(x, e3)e3) = −2[s̃((x ∧g e3)y, e3) + s̃(y, (x ∧g e3)e3)], (7.4) here l s̃ = −2. thus, the above equation verify one part of the theorem 2 of section 4. moreover, the manifold under consideration satisfies r̃(x, y )z = −r̃(y, x)z, r̃(x, y )z + r̃(y, z)x + r̃(z, x)y = 0. hence, from the above equations one can say that this example verifies the condition (c) of theorem 3.1 in [1] and first bianchi identity. 7.2 example we consider a 5-dimensional manifold m = {(x1, x2, x3, x4, x5) ∈ r 5}, where (x1, x2, x3, x4, x5) are standard coordinates in r5. we choose the vector fields e1 = ∂ ∂x1 , e2 = ∂ ∂x2 , e3 = ∂ ∂x3 , e4 = ∂ ∂x4 , e5 = x1 ∂ ∂x1 + x2 ∂ ∂x2 + x3 ∂ ∂x3 + x4 ∂ ∂x4 + ∂ ∂x5 , which are linearly independent at each point of m. let g be a riemannian metric defined by g(ei, ej) =      1, i = j 0, i 6= j for 1 ≤ i, j ≤ 5, and if η is the 1-form defined by η(z) = g(z, e5) for any vector field z ∈ χ(m). let φ be the (1, 1)-tensor field defined by φ(e1) = e1, φ(e2) = e2, φ(e3) = e3, φ(e4) = e4, φ(e5) = 0. 266 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) the linearity property of φ and g yields that η(e5) = 1, φ 2 u = u − η(u)e5, g(φu, φv ) = g(u, v ) − η(u)η(v ), for any u, v ∈ χ(m). now we have [e1, e2] = 0, [e1, e3] = 0, [e1, e4] = 0, [e1, e5] = e1, [e2, e3] = 0, [e2, e4] = 0, [e2, e5] = e2, [e3, e4] = 0, [e3, e5] = e3, [e4, e5] = e4. by virtue of koszul’s formula we get the followings in matrix form           ∇e1e1 ∇e1e2 ∇e1e3 ∇e1e4 ∇e1e5 ∇e2e1 ∇e2e2 ∇e2e3 ∇e2e4 ∇e2e5 ∇e3e1 ∇e3e2 ∇e3e3 ∇e3e4 ∇e3e5 ∇e4e1 ∇e4e2 ∇e4e3 ∇e4e4 ∇e4e5 ∇e5e1 ∇e5e2 ∇e5e3 ∇e5e4 ∇e5e5           =           −e5 0 0 0 e1 0 −e5 0 0 e2 0 0 −e5 0 e3 0 0 0 −e5 e4 0 0 0 0 0           . above expressions satisfies all the properties of para-sasakian manifold. thus m(φ, ξ, η, g) is a 5-dimensional para-sasakian manifold. from the above expressions and the relation of quarter symmetric metric connection and riemannian connection, one can easily obtain the following:           ∇̃e1e1 ∇̃e1e2 ∇̃e1e3 ∇̃e1e4 ∇̃e1e5 ∇̃e2e1 ∇̃e2e2 ∇̃e2e3 ∇̃e2e4 ∇̃e2e5 ∇̃e3e1 ∇̃e3e2 ∇̃e3e3 ∇̃e3e4 ∇̃e3e5 ∇̃e4e1 ∇̃e4e2 ∇̃e4e3 ∇̃e4e4 ∇̃e4e5 ∇̃e5e1 ∇̃e5e2 ∇̃e5e3 ∇̃e5e4 ∇̃e5e5           =           −2e5 0 0 0 2e1 0 −2e5 0 0 2e2 0 0 −2e5 0 2e3 0 0 0 −2e5 2e4 0 0 0 0 0           . with the help of the above results it can be easily obtain the non-zero components of curvature tensors as r(e1, e2)e1 = e2, r(e1, e2)e2 = −e1, r(e1, e3)e1 = e3, r(e1, e3)e3 = −e1, r(e1, e4)e1 = e4, r(e1, e4)e4 = −e1, r(e1, e5)e1 = e5, r(e1, e5)e5 = −e1, r(e2, e3)e2 = e3, r(e2, e3)e3 = −e2, r(e2, e4)e2 = e4, r(e2, e4)e4 = −e2, r(e2, e5)e2 = e5, r(e2, e5)e5 = −e2, r(e3, e4)e3 = e4, r(e3, e4)e4 = −e3, r(e3, e5)e3 = e5, r(e3, e5)e5 = −e3, r(e4, e5)e4 = e5, r(e4, e5)e5 = −e4, cubo 22, 2 (2020) results on para-sasakian manifold admitting a quarter symmetric . . . 267 and r̃(e1, e2)e1 = 4e2, r̃(e1, e2)e2 = −4e1, r̃(e1, e3)e1 = 4e3, r̃(e1, e3)e3 = −4e1, r̃(e1, e4)e1 = 4e4, r̃(e1, e4)e4 = −4e1, r̃(e1, e5)e1 = 2e5, r̃(e1, e5)e5 = −2e1, r̃(e2, e3)e2 = 4e3, r̃(e2, e3)e3 = −4e2, r̃(e2, e4)e2 = 4e4, r̃(e2, e4)e4 = −4e2, r̃(e2, e5)e2 = 2e5, r̃(e2, e5)e5 = −2e2, r̃(e3, e4)e3 = 4e4, r̃(e3, e4)e4 = −4e3, r̃(e3, e5)e3 = 2e5, r̃(e3, e5)e5 = −2e3, r̃(e4, e5)e4 = 2e5, r̃(e4, e5)e5 = −2e4. (7.5) since e1, e2, e3, e4, e5 forms a basis, any vector field x, y, z ∈ χ(m) can be written as x = a1e1 + b1e2 + c1e3 + d1e4 + f1e5, y = a2e1 + b2e2 + c2e3 + d2e4 + f2e5, z = a3e1 + b3e2 + c3e3 + d3e4 + f3e5, where ai, bi, ci, di, fi ∈ r, i = 1, 2, 3, 4, 5. using the expressions of the curvature tensors, we find values of riemannian curvature and ricci curvature with respect to quarter-symmetric metric connection as; r̃(x, y )z = [−4{a1(b2b3 + c2c3 + d2d3) − a2(b1b3 + c1c3 + d1d3)} − 2(a1f2 − f1a2)f3]e1 + [−4{b1(a2a3 + c2c3 + d2d3) − b2(a1a3 + c1c3 + d1d3)} − 2(b1f2 − f1b2)f3]e2 + [−4{c1(a2a3 + b2b3 + d2d3) − c2(a1a3 + b1b3 + d1d3)} − 2(c1f2 − f1c2)f3]e3 + [−4{d1(a2a3 + b2b3 + c2c3) − d2(a1a3 + b1b3 + c1c3)} − 2(d1f2 − f1d2)f3]e4 + [2{(a1f2 − f1a2)a3 + (b1f2 − f1b2)b3 + (c1f2 − f1c2)c3 + (d1f2 − f1d2)d3}]e5, s̃(e1, e1) = s̃(e2, e2) = s̃(e3, e3) = s̃(e4, e4) = −14, s̃(e5, e5) = −8. (7.6) in view of (7.5), (7.6) and the expression of the endomorphism one can easily verify the equation (7.4) and hence the theorem 2 of section 4 is verified. this example also verifies the condition (c) of theorem 3.1 in [1] and first bianchi identity. above two examples verifies the one part of the theorem 2, that is, if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is ricci pseudosymmetric then mn satisfies l s̃ = −2 (mn is not einstein manifold with respect to quarter-symmetric metric connection). another part of the theorem is that, if a para-sasakian manifold mn admitting a quarter-symmetric metric connection is ricci pseudosymmetric then mn is an einstein manifold with respect to quarter-symmetric metric connection (l s̃ 6= −2). now, the second part of the theorem 2 can be verified by using the proper example. 7.3 example we consider a 5-dimensional manifold m = {(x, y, z, u, v) ∈ r5}, where (x, y, z, u, v) are standard coordinates in r5. let {e1, e2, e3, e4, e5} be a linearly independent global frame field on m given 268 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) by e1 = ∂ ∂x , e2 = e −x ∂ ∂y , e3 = e −x ∂ ∂z , e4 = e −x ∂ ∂u , e5 = e −x ∂ ∂v . let g be a riemannian metric defined by g(ei, ej) =      1, i = j 0, i 6= j for 1 ≤ i, j ≤ 5, and if η is the 1-form defined by η(z) = g(z, e1) for any vector field z ∈ χ(m). let the (1, 1)-tensor field φ be defined by φ(e1) = 0, φ(e2) = e2, φ(e3) = e3, φ(e4) = e4, φ(e5) = e5. with the help of linearity property of φ and g, we have η(e1) = 1, φ2v = v − η(v )e1, g(φx, φy ) = g(x, y ) − η(x)η(y ), for any x, y ∈ χ(m). now we have [e1, e2] = −e2, [e1, e3] = −e3, [e1, e4] = −e4, [e1, e5] = −e5, [e2, e3] = [e2, e4] = [e2, e5] = [e3, e4] = [e3, e5] = e4, e5] = 0. with the help of koszul’s formula we get the followings in matrix form           ∇e1e1 ∇e1e2 ∇e1e3 ∇e1e4 ∇e1e5 ∇e2e1 ∇e2e2 ∇e2e3 ∇e2e4 ∇e2e5 ∇e3e1 ∇e3e2 ∇e3e3 ∇e3e4 ∇e3e5 ∇e4e1 ∇e4e2 ∇e4e3 ∇e4e4 ∇e4e5 ∇e5e1 ∇e5e2 ∇e5e3 ∇e5e4 ∇e5e5           =           0 0 0 0 0 e2 −e1 0 0 0 e3 0 −e1 0 0 e4 0 0 −e1 0 e5 0 0 0 −e1           . in this case, (φ, ξ, η, g) is a para-sasakian structure on m and hence m(φ, ξ, η, g) is a 5dimensional para-sasakian manifold. using (2.10) and the above equation, one can easily obtain the following:           ∇̃e1e1 ∇̃e1e2 ∇̃e1e3 ∇̃e1e4 ∇̃e1e5 ∇̃e2e1 ∇̃e2e2 ∇̃e2e3 ∇̃e2e4 ∇̃e2e5 ∇̃e3e1 ∇̃e3e2 ∇̃e3e3 ∇̃e3e4 ∇̃e3e5 ∇̃e4e1 ∇̃e4e2 ∇̃e4e3 ∇̃e4e4 ∇̃e4e5 ∇̃e5e1 ∇̃e5e2 ∇̃e5e3 ∇̃e5e4 ∇̃e5e5           =           0 0 0 0 0 2e2 −2e1 0 0 0 2e3 0 −2e1 0 0 2e4 0 0 −2e1 0 2e5 0 0 0 −2e1           . cubo 22, 2 (2020) results on para-sasakian manifold admitting a quarter symmetric . . . 269 from above results it can be easily obtain the non-zero components of riemannian curvature and ricci curvature tensors as r(e1, e2)e1 = e2, r(e1, e2)e2 = −e1, r(e1, e3)e1 = e3, r(e1, e3)e3 = −e1, r(e1, e4)e1 = e4, r(e1, e4)e4 = −e1, r(e1, e5)e1 = e5, r(e1, e5)e5 = −e1, r(e2, e3)e2 = e3, r(e2, e3)e3 = −e2, r(e2, e4)e2 = e4, r(e2, e4)e4 = −e2, r(e2, e5)e2 = e5, r(e2, e5)e5 = −e2, r(e3, e4)e3 = e4, r(e3, e4)e4 = −e3, r(e3, e5)e3 = e5, r(e3, e5)e5 = −e3, r(e4, e5)e4 = e5, r(e4, e5)e5 = −e4, and r̃(e1, e2)e1 = 2e2, r̃(e1, e2)e2 = −2e1, r̃(e1, e3)e1 = 2e3, r̃(e1, e3)e3 = −2e1, r̃(e1, e4)e1 = 2e4, r̃(e1, e4)e4 = −2e1, r̃(e1, e5)e1 = 2e5, r̃(e1, e5)e5 = −2e1, r̃(e2, e3)e2 = 2e3, r̃(e2, e3)e3 = −2e2, r̃(e2, e4)e2 = 2e4, r̃(e2, e4)e4 = −2e2, r̃(e2, e5)e2 = 2e5, r̃(e2, e5)e5 = −2e2, r̃(e3, e4)e3 = 2e4, r̃(e3, e4)e4 = −2e3, r̃(e3, e5)e3 = 2e5, r̃(e3, e5)e5 = −2e3, r̃(e4, e5)e4 = 2e5, r̃(e4, e5)e5 = −2e4, (7.7) s̃(e1, e1) = s̃(e2, e2) = s̃(e3, e3) = s̃(e4, e4) = s̃(e5, e5) = −8, (7.8) s̃(x, y ) = −2(5 − 1)g(x, y ) = −8g(x, y ), where x = a1e1 + b1e2 + c1e3 + d1e4 + f1e5 and y = a2e1 + b2e2 + c2e3 + d2e4 + f2e5. from (7.7), (7.8) and the expression of the endomorphism one can easily substantiate, the equation (7.4) and hence second part of the theorem 2 (for l s̃ 6= −2). 270 vishnuvardhana. s.v. & venkatesha cubo 22, 2 (2020) references [1] abul kalam mondal and u.c. de, quarter-symmetric nonmetric connection on p-sasakian manifolds, isrn geometry, (2012), 1–14. 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[19] z. i. szabó, structure theorems on riemannian spaces satisfying r(x, y )·r = 0. i. the local version, j. differential geometry 17 (1982), 4, 531–582. introduction preliminaries pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection ricci-pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection projectively flat para-sasakian manifold admitting a quarter-symmetric metric connection projectively pseudosymmetric para-sasakian manifold admitting a quarter-symmetric metric connection examples example example example cubo, a mathematical journal vol.22, n◦03, (395–410). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300395 received: 31 july, 2020 | accepted: 04 december, 2020 toric, u(2), and lebrun metrics brian weber department of mathematics, shanghaitech university, 319 yueyang road, xuhui district, shanghai, china, cn 201210. bjweber@shanghaitech.edu.cn abstract the lebrun ansatz was designed for scalar-flat kähler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. we state the conditions for a metric to be (locally) expressible in lebrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is kähler, scalar-flat, or extremal kähler. second, toric kähler metrics (such as the generalized taub-nuts) and u(2)-invariant metrics (such as the fubini-study or page metrics) are certainly expressible in the lebrun ansatz. we give general formulas for such transitions. we close the paper with examples, and find expressions for two examples—the exceptional half-plane metric and the page metric—in terms of the lebrun ansatz. resumen el ansatz de lebrun fue diseñado para métricas kähler escalares-planas con una simetŕıa continua; acá mostramos que es generalizable a clases mucho más amplias de métricas con una simetŕıa. establecemos las condiciones para que una métrica sea (localmente) expresable con la forma de ansatz de lebrun, las condiciones bajo las cuales su estructura compleja natural es integrable, y las condiciones que producen una métrica que es kähler, escalar-plana, o kähler extremal. en segundo lugar, métricas tóricas kähler (tales como las taub-nut generalizadas) y métricas u(2)-invariantes (tales como la métrica de fubini-study o la de page) son ciertamente expresables en el ansatz de lebrun. damos fórmulas generales para tales transiciones. concluimos el art́ıculo con ejemplos, y encontramos expresiones para dos ejemplos—la métrica excepcional del semiplano y la métrica de page—en términos del ansatz de lebrun. keywords and phrases: differential geometry, kähler geometry, canonical metrics, ansatz. 2020 ams mathematics subject classification: 53b21, 53b35. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300395 396 brian weber cubo 22, 3 (2020) 1 introduction lebrun [19] created an ansatz for scalar-flat kähler metrics with a continuous symmetry. this was an expansion of the gibbons-hawking ansatz for ricci-flat kähler metrics with a symmetry, itself a version of the kaluza ansatz [18] [6]. in the original construction kaluza showed that if a lorentzian 5-metric is endowed with a spacelike continuous symmetry, the einstein equations will partially linearize, with the linear part being the maxwell equations. the gibbons-hawking construction utilized this idea except in euclidean signature and a dimension lower, where the maxwell equations reduce to just the laplace equation on a potential, and the “gravity” equations (the ricci-flat equations) fully linearize. lebrun’s ansatz, which also works for 4-dimensional riemannian metrics with a circle symmetry, partially linearizes the scalar-flat kähler (sfk) equations. these sfk equations, normally exceedingly complicated and nonlinear, were shown to reduce to a pair of second order equations, one linear and the other quasilinear. we show that lebrun’s ansatz is much more general than this original use, and is suitable for expressing interesting 4-metrics that are not scalar-flat, kähler, or even have an integrable complex structure. we show the conditions under which a metric is expressible in terms of the lebrun ansatz, and give the explicit transformations into the lebrun ansatz from two toric kähler ansätze, and from the u(2)-invariant ansatz. in the last section we use these translations to express several common metrics in the lebrun ansatz. finally we indicate how the lebrun ansatz can be used, at least in principle, to create new metrics of special kinds, a subject we shall take up elsewhere. 2 the lebrun ansatz we lay out the basic definitions in the lebrun ansatz and determine when the ansatz possesses an integrable complex structure and when it possesses a closed kähler 2-form. we end with some expressions for curvature quantities of such metrics, and state when such a metric is extremal kähler. the reference for this section is [19]. 2.1 the ansatz the lebrun ansatz is an s1-fibration π : m4 → n3 along with the metric g = weu ( dx2 + dy2 ) + w dz2 + w−1 (dτ + π∗a) 2 (2.1) cubo 22, 3 (2020) toric, u(2), and lebrun metrics 397 where (x,y,z) are local coordinates on n3, w = w(x,y,z) and u = u(x,y,z) are functions, and a is a 1-form a = ax(x,y,z)dx + ay(x,y,z)dy + az(x,y,z)dz on n 3.1 the coordinate τ is defined after a choice of a transversal: after setting τ = 0 on this transversal, τ is pushed forward via the s 1-action. the field d dτ is invariant under rechoosing the transversal so it is globally defined, and it is killing. the exterior derivative of a will be important. because dπ∗a = π∗da, it is immaterial whether we compute on m4 or n3. letting b = da we have b = bx dy ∧ dz − by dx ∧ dz + bz dx ∧ dy, where bx = ay,x − ax,y, by = ax,z − az,x, bz = az,y − ay,z. (2.2) in the spirit of kaluza’s work, we may interpret a as a vector potential over 3-space and b = da as the corresponding maxwell field strength. it so closely resembles a magnetostatic field that we will sometimes call it the metric’s magnetic field. in all curvature computations a never appears; only its field b appears. a g-compatible almost-complex structure on (m4,g) is j(dx) = −dy, j(dz) = −w−1(dτ + π∗a), (2.3) which dualizes to j(∇x) = ∇y, j(∇z) = ∂ ∂τ (2.4) where the duality convention is j(η) , η ◦j for η ∈ ∧1 . the corresponding antisymmetric form is ω = g(j·, ·) = weudx ∧ dy + dz ∧ (dτ + π∗a) . (2.5) 2.2 the complex and symplectic structures as usual, the almost complex structure splits ∧1 c = ∧1 (m4) ⊗ c into holomorphic and antiholomorphic bundles, where ∧1 c = ∧1,0 ⊕ ∧0,1 are the respective ± √ −1 eigenspaces of j. in bases, ∧ 1,0 = spanc { dx + √ −1dy, dz + √ −1w−1(dτ + π∗a) } , ∧ 0,1 = spanc { dx − √ −1dy, dz − √ −1w−1(dτ + π∗a) } . (2.6) of the many ways to check the integrability of an almost-complex structure, the most convenient will be verifying that d : ∧0,1 → ∧1 c ∧ ∧0,1 . lemma 2.1. the complex structure (2.3) is integrable if and only if wx = bx and wy = by. (2.7) 1lebrun denotes ω = dτ + π∗a, and interprets this as a connection. following a different but very standard convention, we shall prefer using the symbol ω for the 2-form ω = g(j·, ·). 398 brian weber cubo 22, 3 (2020) proof. this comes from out of the proof of proposition 1 of [19]. we compute on bases. certainly d(dx − √ −1dy) = 0. then d ( dz − √ −1w−1(dτ + π∗a) ) = w−1 ( dw ∧ ( dz − √ −1w−1(dτ + π∗a) ) − dw ∧ dz − √ −1b ) . (2.8) from (2.6), the first term is in ∧1 c ∧ ∧0,1 . the second and third terms become − dw ∧ dz − √ −1b = −(wx − √ −1by)dx ∧ dz − (wy + √ −1bx)dy ∧ dz − √ −1bzdx ∧ dy = 1 2 ( (wx − bx) − √ −1(wy − by) ) dz ∧ (dx + √ −1dy) + 1 2 ( (wx + bx)dz − √ −1(by + wy)dz − √ −1bz(dx + √ −1dy) ) ∧ (dx − √ −1dy). (2.9) because dx − √ −1dy ∈ ∧0,1 the second term on the right is in ∧ 1 c ∧ ∧ 0,1. but the first term is in ∧1 c ∧ ∧1,0 . we conclude j is integrable if and only if this term is zero, which is the same as (wx − bx) − √ −1(wy − by) = 0. lemma 2.2. we have dω = (−bz + (weu)z) dz ∧ dx ∧ dy. in particular, the antisymmetric form ω of (2.5) is closed if and only if bz = (we u)z. proof. using ω = dz ∧ (dτ + π∗a) + weudx ∧ dy and dπ∗a = π∗da = π∗b, dω = −dz ∧ dπ∗a + (weu)zdz ∧ dx ∧ dy = (−bz + (weu)z)dz ∧ dx ∧ dy, (2.10) from which the assertion follows. theorem 2.1. the triple (g,j,ω) always has g(j·,j·) = g(·, ·). it is i) hermitian if and only if bx = wx and by = wy, ii) symplectic if and only if bz = (we u)z, and iii) kähler if and only if bx = wx, by = wy, and bz = (we u)z. condition (iii) implies wxx + wyy + (we u)zz = 0. (2.11) cubo 22, 3 (2020) toric, u(2), and lebrun metrics 399 proof. after lemmas 2.2 and 2.1, we must only verify equation (2.11). but with b = da, after assuming the relations in (iii) then equation (2.11) is just db = 0. remark. the metric is almost kähler if (ii) holds but (i) does not. remark. the original approach of lebrun [19] was essentially the reverse of this. lebrun solves (2.11) for w first, and then finds a 1-form a (which will have dirac string singularities) whose field b satisfies (iii). this contrasts with our method which starts with a metric of the form (2.1), finds conditions on a and w that give it special traits, and from such traits derives equation (2.11). we have the following characterization of the lebrun ansatz. theorem 2.2. let g be a metric on m4. then g can be expressed locally via the lebrun ansatz if and only if the following three conditions hold: i) m4 has a vector field v and an almost-complex structure j compatible with g so that, letting ω = g(j·, ·) be the associated antisymmetric form, then ω, g, and j are all v-invariant, ii) given any simply connected domain ω ⊂ m4, there is a function z : ω → r with ivω = dz, and iii) the action of ∇z on j, when restricted to the rank-2 distribution p ⊂ ∧1 m4 that is null on span{v,jv}, is zero. remark. regarding condition (iii), p is specifically the distribution p = {η ∈ ∧1 m4 such that η(v) = 0 and η(jv) = 0}. remark. condition (iii) is certainly the most technical; it exists so that the first two terms in the ansatz can be written in the form f(x,y,z)(dx2 +dy2), instead of f1dx 2 +f2(dxdy+dydx)+ f3dy 2. condition (iii) could also be written l∇z(j ∣ ∣ p ) = 0 where l is the lie derivative. proof. supposing g can be expressed via the lebrun ansatz, we simply set v = ∂ ∂t and let j be as in (2.3) or equivalently (2.4). the work above shows j and ω are v-invariant and ivω = dz. we compute l∇zj ∣ ∣ p by (l∇zj)(dx) = l∇z(jdx) − jl∇zdx = l∇z(dy) − jl∇zdx. (2.12) the cartan formula gives l∇zdx = di∇zdx = d〈dz,dx〉. but this inner product is zero, as is easily verified after computing the inverse matrix gij. similarly l∇zdy = 0, so we have shown l∇zj(dx) = 0. the same argument works for l∇zj(dy), so we have shown that l∇z(j ∣ ∣ p ) = 0. for the converse we assume g, j, ω are v-invariant, and that ivω = dz for some function z. this allows us to perform a version of the kähler reduction. because z is itself v-invariant (due to the fact that lvz = ivivω = 0), the function z passes to the quotient manifold n3 = m4/v 400 brian weber cubo 22, 3 (2020) where the quotient is by the action of the killing field v—this works if the orbits of v are closed; if not then a second killing field must exist, and we can take an appropriate linear combination to find a killing field with closed orbits. pick a level-set σ2z = {z = const} on which to place isothermal coordinates (x,y), and then extend (x,y) along trajectories of ∇z so the functions x, y are now defined on some region of n3. we show that (x,y) remains isothermal on all other nearby level-sets of z; this is a consequence of j|p being invariant under trajectories of ∇z. to see this, note that j|p restricts to the hodge-star ∗2 on any level-set of z, and x, y are isothermal if and only if d ∗2 dx = d ∗2 dy = 0 and dx ∧ ∗dy = 0. by construction, d ∗2 dx = d ∗2 dy = 0 and dx ∧ ∗dy = 0 holds on one level-set of z; to see it is true on all nearby level-sets we compute l∇zd ∗2 dx = dl∇zj|p dx = dj|p l∇zdx = dj|p dl∇zx = 0. (2.13) where we used the facts that d always commutes with l∇z, that by hypothesis l∇zj|p = 0, and that by construction l∇zx = 0. therefore d ∗2 dx remains zero on all level-sets. similarly we compute l∇z (dx ∧ ∗2dy) = (l∇zdx) ∧ ∗2dy + dx ∧ (l∇z ∗2 dy) = dx ∧ ∗2 (l∇zdy) = 0 (2.14) where again we used l∇zdx = l∇zdy = 0 and l∇z∗2 = l∇zj|p = 0. now, because the functions x, y remain an isothermal system on any level-set of z, we may express the metric g3 on the quotient manifold n 3 in the form g3 = f1(x,y,z)dz 2 +f2(x,y,z) ( dx2 + dy2 ) . we define the functions w, eu by w , |dz|−2g3 = f1 weu , |dx|−2g3 = |dy| −2 g3 = f2. (2.15) the functions x and y pull back from n3 to m4, where we now have three coordinate functions x, y, and z. for the fourth coordinate τ, after choosing a transversal to v, we may set τ = 0 along this transversal, and push τ along trajectories of v—incidentally, this establishes ∂ ∂τ = v and j∇z = ∂ ∂τ . we now have coordinates (x,y,z,τ) on m4. from (2.15) we have w−1 = |dz|2 = |∇z|2 = |j∇z|2 = |∂/∂τ|2. we define functions c, ax, ay, and az in terms of the complex structure j by −c (dτ + axdx + aydy + azdz) = jdz. (2.16) we can compute the value of c. transvecting both sides of (2.16) with ∂ ∂τ gives −c = jdz ( ∂ ∂τ ) = 〈 ∇z, j ∂ ∂τ 〉 = −|∇z|2 = −|dz|2 = −w−1. (2.17) therefore c = w−1. finally because the distribution {∇x,∇y} is perpendicular to the distribution {∇z, ∂/∂τ}, we arrive at the expression cubo 22, 3 (2020) toric, u(2), and lebrun metrics 401 g = weu ( dx2 + dy2 ) + wdz2 + w−1 (dτ + axdz + aydy + azdz) 2 . (2.18) 2.3 curvature quantities proposition 2.1. assume the metric (2.1) is kähler, meaning (iii) of theorem 2.1 holds. then the ricci curvature of g is ric = −1 2 ( hess u (·, ·) + hess u (j·, j·) ) (2.19) proof. the proof of proposition 1 of [19] gives ricci form and ricci curvature ρ = − √ −1∂∂̄u, and ric = ρ(·, j·) = −1 2 ( hessu (·, ·) + hessu (j·, j·) ) . (2.20) proposition 2.2. assume the metric (2.1) is kähler, meaning (iii) of theorem 2.1 holds. then the scalar curvature s of g is s = − 1 weu (uxx + uyy + (e u)zz) . (2.21) proof. this is computed in the proof of proposition 1 of [19]. proposition 2.3 (the extremal condition). assume the metric (2.1) is kähler. then it is an extremal kähler metric if constants m,b ∈ r exist so − 1 weu (uxx + uyy + (e u)zz) = mz + b. (2.22) proof. if (2.22) holds then s = mz + b and so ∇s = m∇z and j∇s = m ∂ ∂τ ; thus j∇s is a killing field. the proposition is established after recalling that a kähler metric is extremal if and only if j∇s is killing [7] [8]. remark. whether g is kähler or not, its scalar curvature is s = − 1 weu ( ( uxx + uyy + (e u)zz ) + 1 w ( wxx + wyy + (we u)zz ) + 1 2w2 (b2x − (wx)2) + 1 2w2 (b2y − (wy)2) + e−u 2w2 (b2z − ( (weu)z )2 ) . (2.23) 402 brian weber cubo 22, 3 (2020) 3 expressing toric kähler metrics using the lebrun ansatz the lebrun ansatz operates on 4-manifolds with one symmetry. on kähler 4-manifolds with two holomorphic symmetries, there are more specialized ansätze. letting x 1, x 2 be commuting holomorphic killing fields (recall that “holomorphic” means lx ij = 0, just as killing means lx ig = 0), then (m4,g,j,x 1,x 2) can be considered a toric kähler 4-manifold. this situation has been studied in [17] [1] [13] [14] [2] [9] and many other works. certainly a toric kähler metric can be translated into the lebrun ansatz once a distinguished killing field is chosen. we do this here. 3.1 the two toric ansätze there are two standard presentations for toric kähler 4-manifolds. these were originally explored by guillemin [17], who also discovered that they are equivalent via a legendre transform. the lebrun ansatz is a mixture of the two. the first of the two presentations is the symplectic ansatz. if {x 1,x 2} are independent commuting holomorphic killing fields, we can use the arnold-liouville construction [3] to produce the so-called action-angle coordinates on m4. to execute this construction, one defines action variables (up to a constant) by ∇ϕi = −jx i or equivalently by dϕi = ix iω, and defines angle variables, denoted θ1, θ2, by choosing a transversal and then pushing forward the action of the fields x 1, x 2. in these coordinates, the ansatz demands the metric be expressed g = uijdϕ i ⊗ dϕj + uijdθi ⊗ dθj (3.1) where u = u(ϕ1,ϕ2) is a convex function of the action variables. the matrix (uij) is defined by uij , ∂ 2 u ∂ϕiϕj , and we define (uij) , (uij) −1. the map m4 → r2 given by p 7→ (ϕ1(p),ϕ2(p)) sends m4 to a region σ2 ⊂ r2; this is sometimes called the arnold-liouville reduction or, by abuse of terminology, the moment map. if m4 is compact then its image σ2 is a compact polygon in r2. this polygon encodes the topology of m4, via the delzant gluing rules [11]. if m4 is non-compact, then σ2 need not be a polygon nor even be topologically closed. the second ansatz, the holomorphic ansatz, also begins with the fields {x 1,x 2}. again we may produce corresponding coordinates θ1, θ2 after choosing a transversal. because x 1, x 2 are not only symplectomorphic but holomorphic, the variables θi are actually pluriharmonic, meaning d(jdθi) = 0. the poincaré lemma then guarantees functions ξ1, ξ2 exist (at least locally) so that dξi = jdθi, and we have two holomorphic functions fi = ξi+ √ −1θi which constitute a holomorphic chart (f1,f2) : ω → c2 on some subdomain ω ⊆ m4. the kähler form on this chart, as usual, can be expressed ω = √ −1∂∂̄v for some pseudoconvex function v . because v is θ1-θ2 invariant, it is cubo 22, 3 (2020) toric, u(2), and lebrun metrics 403 convex instead of just pseudoconvex. the metric is then g = v ijdξi ⊗ dξi + v ijdθi ⊗ dθj (3.2) where (v ij) is the matrix with components v ij , ∂ 2 v ∂ξi∂ξj . we might consider the map p 7→ (ξ1(p), ξ2(p)) for p ∈ m4, just as we considered the moment map p 7→ (ϕ1(p),ϕ2(p)). but it is much less interesting than the moment map. if m4 is compact then its image is all of r2. in particular there is no way to read off the topology of m4 from its image. a duality relationship exists between the symplectic system (ϕ1,θ1,ϕ 2,θ2) with its symplectic potential u and the holomorphic system (ξ1,θ1,ξ2,θ2) with its kähler potential v . as shown in [17], they are legendre transforms of each other: ξi = ∂u ∂ϕi , ϕi = ∂v ∂ξi , and u(ϕi) + v (ξi) = ∑ i ϕiξi. (3.3) 3.2 translation to the lebrun ansatz it is now possible to relate these two systems to the lebrun ansatz, which is a mixed symplecticholomorphic system. we define the lebrun variable τ to be the angle variable θ1 corresponding to x 1, and y the angle variable θ2 corresponding to x 2. let z be the symplectic variable corresponding to the angle τ, meaning z = ϕ1, and x the holomorphic variable corresponding the angle variable y, meaning x = ξ2. then we create the lebrun functions w and u, and determine the 1-form a. we record the change of frame from the symplectic frame { ∂ ∂ϕ1 , ∂ ∂θ1 , ∂ ∂ϕ2 , ∂ ∂θ2 } to the lebrun frame { ∂ ∂z , ∂ ∂τ , ∂ ∂x , ∂ ∂y } . one easily computes ∂ ∂ϕ1 = ∂ ∂z + u21 ∂ ∂x dϕ1 = dz ∂ ∂θ1 = ∂ ∂τ dθ1 = dτ ∂ ∂ϕ2 = u22 ∂ ∂x dϕ2 = −u21 u22 dz + 1 u22 dx ∂ ∂θ1 = ∂ ∂y dθ2 = dy. (3.4) upon substituting the symplectic frame components into the lebrun metric (2.1), we find the functions w, u and the components ax, ay, and az to be w = 1/u11, u = log ( u11u22 − (u12)2 ) ax = 0, ay = u12 u11 , az = 0. (3.5) we express this in the form of a proposition. 404 brian weber cubo 22, 3 (2020) proposition 3.1. assume (m4,j,g,x 1,x 2) is a toric kähler manifold. let (ϕ1,θ1,ϕ2,θ2) be symplectic coordinates and (ξ1,θ1,ξ2,θ2) holomorphic coordinates on m 4. there exists a convex function u(ϕ1,ϕ2) on σ2, where σ2 is the image of the moment map (ϕ1,ϕ2) : m4 → r2, so that g = uijdϕ i ⊗ dϕj + uijdθi ⊗ dθj (3.6) where uij = ∂ 2 u ∂ϕiϕj and (uij) = (uij) −1. there also exists a convex function v = v (ξ1,ξ2) on r 2 so that g = v ijdξi ⊗ dξj + v ijdθi ⊗ dθj (3.7) where v ij = ∂ 2 v ∂ξiξj . these systems are related via the legendre transform: ϕi = ∂v ∂ξi , ξi = ∂u ∂ϕi , u(ϕ1,ϕ2) + v (ξ1,ξ2) = ϕ 1ξ1 + ϕ 2ξ2. (3.8) the metric (m4,g,j,x 1,x 2) can be expressed in the lebrun ansatz after setting ( z, τ, x, y ) = ( ϕ1, θ1, ξ2, θ2 ) . (3.9) a lebrun ansatz expression of g is obtained by setting u = log detuij = log ( u11u22 − (u12)2 ) , w = 1 u11 , and a = aydy = u12 u11 dy (3.10) (the components ax and az are zero). the components of the magnetic 2-form are bx = −ay,z, by = 0, and bz = ay,x. 3.3 variation of lebrun structures in our construction of section 3.2 we began by setting τ = θ1, but we could have chosen τ = θ2 or indeed any linear combination of the cyclic variables. up to scale a toric metric automatically has a 1-parameter family of distinct lebrun structures. if α ∈ [0,π/2] is a constant and x 1, x 2 are symplectomorphic killing fields, then for each α we may select the field x = cos(α)x 1 + sin(α)x 2. (3.11) then, referring to the construction of section 3.2, the corresponding angle variable is τ = cos(α)θ1 +sin(α)θ2 with conjugate momentum variable z = cos(α)ϕ 1 +sin(α)ϕ2. the holomorphic variables are then x = − sin(α)ξ1 + cos(α)ξ2 and y = − sin(α)θ1 + cos(α)θ2. this allows for a “tuning” or selection of a distinguished 1-parameter symmetry field form which the lebrun ansatz metric can be constructed. the variable y remains cyclic (that is, its field cubo 22, 3 (2020) toric, u(2), and lebrun metrics 405 remains a symmetry direction), and u, w will remain functions of x and z. these functions will change with α, so we may write u = uα(x,z) and w = wα(x,z). we remark that a third auxiliary function u̇α , d dα uα exists. if the uα solve the lebrun equation (uα)xx + (e uα)zz = 0 then u̇α will solve the linearized equation (u̇α)xx + (u̇αe uα)zz = 0. under some conditions uα will be positive, and setting w = u̇α we have an entirely new lebrun metric. 4 expressing u(2)-invariant metrics in the lebrun ansatz the usual ansatz for u(2)-invariant metrics is g = adr2 + b (η1) 2 + c ( (η2) 2 + (η3) 2 ) (4.1) where {η1,η2,η3} is a standard left-invariant coframe on s3, and a, b, c are functions of the radial variable r. if (ψ,ϕ,θ) are euler coordinates on on s3, the usual frame transitions are η1 = 1 2 (dψ + cos(θ)dϕ) η2 = 1 2 (sin(θ) cos(ψ)dϕ − sin(ψ)dθ) η3 = 1 2 (sin(θ) sin(ψ)dϕ + cos(ψ)dθ) . (4.2) from this we deduce (η2) 2 + (η3) 2 = 1 4 ( dθ2 + sin2(θ)dϕ2 ) , so in euler coordinates g = adr2 + b 4 (dψ + cos(θ)dϕ) 2 + c 4 ( dθ2 + sin2(θ)dϕ2 ) (4.3) this is already close to lebrun ansatz form. to place it precisely in lebrun ansatz form we make the change of variables x = log cot θ 2 , y = ϕ, z = 1 2 ∫ √ ab dr, τ = ψ. (4.4) this gives dθ2 + sin2(θ)dϕ2 = sech2(x)(dx2 + dy2), and the metric now reads g = 4 b dz2 + b 4 (dτ + tanh(x)dy) 2 + c 4 sech2(x) ( dx2 + dy2 ) . (4.5) reading off the lebrun ansatz quantities from (2.1), we have w = 4 b , u = log ( bc 16 sech2(x) ) ax = 0, ay = tanh(x), az = 0 (4.6) where b and c are now functions of the new variable z, via the transition from r to z given in (4.4). because u(2) has a rank 2 toral subgroup, any u(2)-invariant metric is also t2-invariant— if the metric is kähler then it is toric. one can see directly that the metric (4.5) has no τor y-dependency so has t2 symmetry. 406 brian weber cubo 22, 3 (2020) 5 examples we give two examples of our method. the exceptional half-plane metric from [21] was originally written in a toric ansatz, and the page metric on cp2♯cp2 was originally written in the u(2) ansatz. we use our methods to express both in the lebrun ansatz. in the last section we outline methods for creating new metrics that are einstein, half-conformally flat, or bach-flat. 5.1 the exceptional half-plane metric on c2. this toric sfk metric on c2 appears in [21]. it has one translational and one rotational field. in rectangular coordinates (x1,y1,x2,y2) on c 2, these fields are x 1 = ∂ ∂y1 and x 2 = −y2 ∂∂x2 +x2 ∂ ∂y2 , which are clearly translational and rotational, respectively. let u = u(ϕ1,ϕ2) be the symplectic potential u = 1 2 ( (ϕ2)2 1 + 2mϕ1 + ϕ1 log(ϕ1) + m(ϕ1)2 ) (5.1) where m ≥ 0 is a constant. the case m = 0 produces the flat metric. when m > 0, the resulting metric is the exceptional half-plane metric; the fact that (5.1) is the correct symplectic potential for the exceptional half-plane metric can be verified directly from equations (6-1) and (6-3) of [21]. the kähler potential v is difficult to write explicitly, as it involves inverting a function with transcendental and algebraic parts. however it is possible to find lebrun coordinates, which in terms of the symplectic coordinates are x = ϕ2 1 + 2mϕ1 , y = θ2, z = ϕ 1, τ = θ1. (5.2) the lebrun functions w and u are w = m + 1 2z , u = log (2z) (5.3) and the vector potential and field strength are a = 2mxdy, which is ax = 0, ay = 2mx, az = 0, b = 2mdx ∧ dy, which is bx = 0, by = 0, bz = 2m. (5.4) we notice that u = log(2z) gives what lebrun called the hyperbolic ansatz in section 4 of [19]. if m = 0 this is the flat metric, which lebrun wrote down on p. 233 of [19] (unfortunately lebrun’s equations are mostly unnumbered). the exceptional half-plane metric in lebrun ansatz form is g = (1 + 2mz)(dx2 + dy2) + 1 + 2mz 2z dz2 + 2z 1 + 2mz (dτ + 2mxdy)2. (5.5) cubo 22, 3 (2020) toric, u(2), and lebrun metrics 407 5.2 the page metric the page metric was originally developed in [20], and can be found explicitly in (3.25) of [16] (unfortunately its expression in the appendix of [15] has a typo). methods for building ricci-flat metrics, including the page metric, can be found [4]; see also 9.125 of [5]. this metric exists on cp2♯cp 2; it is einstein, hermitian, and bach-flat, but not half-conformally flat. it is conformal to an extremal kähler metric, which calabi [7] [8] independently wrote down; see [10] for the specific conformal transformation, or [12] for a more general theory of conformal transformations between extremal kähler and einstein metrics on 4-manifolds. from [16] the page metric is g = 3(1 + ν2) λ [ 1 − ν cos2(r) 3 − ν2 − ν2(1 + ν2) cos2(r) dr2+ + 3 − ν2 − ν2(1 + ν2) cos2(r) (3 + ν2)2(1 − ν cos2(r)) sin2(r)η21 + 4 1 − ν2 cos2(r) 3 + 6ν2 − ν4 ( η22 + η 2 3 ) ] . (5.6) the method of section 4 gives its expression in the lebrun ansatz: g = weu ( dx2 + dy2 ) + wdz2 + 1 w (dτ + tanh(x)dy) 2 , where w = f(z) g(z) and weu = 1 3λ(1 + ν2)(3 + 6ν2 − ν4) h(z) sech2(x) (5.7) and f , g, h are the polynomials f(z) = 27(1 + ν2 − ν4 − ν6) + 36(4ν2 + 4ν4 + ν6)λz − 12(9ν2 + 6ν4 + ν6)λ2z2 g(z) = 27(1 + ν2 − ν4 − ν6) + 3(−9 + 9ν2 + 11ν4 + 15ν6)λz − 24(3ν2 + 3ν4 − ν6)λ2z2 + 4(9ν2 + 6ν3 + ν6)λ3z3 h(z) = 9(1 + ν2 − ν4 − ν6) + 12(3ν2 + 16ν4 + ν6)λz − 4(9ν2 + 6ν4 + ν6)λ2z2. (5.8) the domain for (x,z) is x ∈ r and z ∈ [ 0, 3(1+ν2) λ(3+ν2) ] . 5.3 new metrics creation of special metrics, namely einstein, half-conformally flat, or bach-flat metrics are of considerable importance in differential geometry. one may regard the metric g, if expressed in the lebrun ansatz, as a dynamic variable with five unknowns w, u, bx, by, bz which are each functions of the coordinates (x,y,z). these values can be specified independently, subject to the single requirement that bx,x +by,y +bz,z = 0 which is equivalent to the definition of b from (2.2), 408 brian weber cubo 22, 3 (2020) which is that b = da for a 1-form a. in a sense, there are four completely independent variables that may be chosen, with the choice of a fifth being partially constrained. letting w+ be the self-dual part of the weyl tensor, one might consider the condition w+ = 0. because the operator w+ : ∧+ → ∧+ has three eigenvalues which are subject to the condition that they sum to zero, the condition w+ = 0 imposes two differential identities on our five variables. with the fifth constraint discussed above, we arrive at an underdetermined system, which surely has a large solutions space. there remain many obstacles, both technical and theoretical, to fully understanding this system. similar comments hold for systems like rı ◦ c = 0 and b = 0 where rı ◦ c is the trace-free ricci tensor and b is the bach tensor. this subject will be taken up elsewhere. cubo 22, 3 (2020) toric, u(2), and lebrun metrics 409 references [1] m. abreu, “kähler geometry of toric varieties and extremal metrics”, international journal of mathematics, vol. 9, pp. 641–651, 1998. [2] m. abreu and r. sena-dias, “scalar-flat kähler metrics on non-compact symplectic toric 4manifolds”, annals of global analysis and geometry, vol. 41, no. 2, pp. 209–239, 2012. [3] v. arnold, mathematical methods of classical mechanics, springer science & business media, vol. 60, 2013. [4] l. bérard-bergery: “sur de nouvelles variétés riemanniennes d’einstein”, publications de l’institut élie cartan, vol. 6, 1982. [5] a. besse: einstein manifolds. springer science & business media, 2007. [6] j. bourguignon, “a mathematician’s visit to kaluza-klein theory”, presented at conference on differential geometry and partial differential equations, torino, italy, rend. semin. mat. torino fasc., pp. 143-163, 1989. [7] e. calabi: extremal kähler metrics. in seminar on differential geometry, princeton university press, vol. 102, pp. 259–290, 1982. [8] e. calabi, extremal kähler metrics ii. in differential geometry and complex analysis, springer, berlin, heidelberg, pp. 95-114, 1985. [9] d. calderbank, l. david, and p. gauduchon, “the guillemin formula and kähler metrics on toric symplectic manifolds”, journal of symplectic geometry, vol. 4, no. 1, pp. 767–784, 2002. [10] t. chave and g. valent, “compact extremal versus compact einstein metrics”, classical and quantum gravity, vol. 13, no. 8, pp. 2097–2108, 1996. [11] t. delzant, “hamiltoniens périodiques et images convexes de l’application moment”, bulletin de la société mathématique de france, vol. 116, pp. 315–339, 1988. [12] a. derdzinski, “self-dual kähler manifolds and einstein manifolds of dimension four”, compositio mathematica, vol. 49, no. 3, pp. 405–433, 1983. [13] s. donaldson, “a generalized joyce construction for a family of nonlinear partial differential equations”, journal of gökova geometry/topology conferences, vol. 3, 2009. [14] s. donaldson, “constant scalar curvature metrics on toric surfaces”, geometric and functional analysis, vol. 19, no. 1, pp. 83–136, 2009. 410 brian weber cubo 22, 3 (2020) [15] t. eguchi, p. gilkey, and a. hanson. “gravitation, gauge theories and differential geometry”, physics reports, vol. 66, no. 6, pp. 213–393, 1980. [16] g. gibbons and s hawking, “classification of gravitational instanton symmetries”, communications in mathematical physics, vol. 66, no. 3, pp. 291–310, 1979. [17] v. guillemin, “kähler structures on toric varieties”, journal of differential geometry, vol. 40, pp. 285–309, 1994. [18] t. kaluza, “zum unitatsproblem der physik”, sitzungsber. d. berl. akad., pp. 966–972, 1921. [19] c. lebrun, “explicit self-dual metrics on cp2# · · · #cp2”, journal of differential geometry, vol. 34, no. 1, pp. 223–253, 1991. [20] d. page. “a compact rotating gravitational instanton”, physics letters b, vol. 79, no. 3, pp. 235–238, 1978. [21] b. weber, “generalized kähler taub-nut metrics and two exceptional instantons”, arxiv:1602.06178 (to appear in communications in analysis and geometry). introduction the lebrun ansatz the ansatz the complex and symplectic structures curvature quantities expressing toric kähler metrics using the lebrun ansatz the two toric ansätze translation to the lebrun ansatz variation of lebrun structures expressing u(2)-invariant metrics in the lebrun ansatz examples the exceptional half-plane metric on c2. the page metric new metrics cubo, a mathematical journal vol.22, n◦03, (351–359). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300351 received: 17 april, 2020 | accepted: 05 october, 2020 the multivariable aleph-function involving the generalized mellin-barnes contour integrals abdi oli1, kelelaw tilahun2, g. v. reddy3 1,2department of mathematics, wollo university, p.o. box: 1145, dessie, south wollo, amhara region, ethiopia. abdioli30@gmail.com, kta3151@gmail.com 3department of mathematics, jigjiga university, p.o. box: 1020, jigjiga, ethiopia gvreddy16673@gmail.com abstract in this paper, we have evaluated three definite integrals involving the product of two hypergeometric functions and multivariable aleph-function. certain special cases of the main results are also pointed out. resumen en este art́ıculo, hemos evaluado tres integrales definidas que involucran el producto de dos funciones hipergeométricas y la función aleph multivariada. también se señalan ciertos casos especiales del resultado principal. keywords and phrases: hypergeometric function, multivariable aleph function. 2020 ams mathematics subject classification: 33c20, 33c05. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300351 352 a. oli, k. tilahun, g. v. reddy cubo 22, 3 (2020) 1 introduction the aleph-function is among very significant special functions and its closely related ones are widely used in physics and engineering. therefore they are of high interest to physicists and engineers as well as mathematicians. in recent years, many integral formulas involving a diversity of special functions have been presented by many authors (see e.g., [3, 9, 12, 13, 14, 15, 16]). motivated by these recent papers, three generalized integral formulae involving product of two hypergeometric functions and multivariable aleph-function are established in the form of three theorems: for our study, we recall the following three integral formulas (see [5], p. 77, equations (3.1), (3.2) and (3.3)): ∫ ∞ 0 [ ( αx + β x )2 + γ ]−ρ−1 dx = √ π γ ( ρ + 1 2 ) 2α(4αβ + γ) ρ+ 1 2 γ (ρ + 1) (1.1) ( α > 0; β ≥ 0; γ + 4αβ > 0; ℜ(ρ) + 1 2 > 0 ) . ∫ ∞ 0 1 x2 [ ( αx + β x )2 + γ ]−ρ−1 dx = √ π γ ( ρ + 1 2 ) 2β (4αβ + γ) ρ+ 1 2 γ (ρ + 1) (1.2) ( α ≥ 0; β > 0; γ + 4αβ > 0; ℜ(ρ) + 1 2 > 0 ) . ∫ ∞ 0 [ ( α + β x2 ) ( αx + β x )2 + γ ]−ρ−1 dx = √ π γ ( ρ + 1 2 ) (4αβ + γ) ρ+ 1 2 γ (ρ + 1) (1.3) ( α > 0; β ≥ 0; γ + 4αβ > 0; ℜ(ρ) + 1 2 > 0 ) . we also recall the following identity involving the hypergeometric series 2f1(.) ([8] p. 75, theorem 1): if (1 − y)α+β−γ 2f1 (2α,2β; 2γ;y) = ∞ ∑ k=1 aky k, (1.4) then 2f1 ( a, b; c + 1 2 ; y ) 2f1 ( c − a, c − b; c + 1 2 ; y ) = ∞ ∑ k=0 (c) k ( c + 1 2 ) k aky k. (1.5) the multivariable aleph-function defined by sharma and ahmad [6] as: ℵ (z1, z2, ..., zr) = ℵ0, n: m1n1; m2n2; ....;mrnr pi, qi, τi; r; p i(1) , q i(1) , τ i(1) ; r(1) , ..., p i(r) , q i(r) , τ i(r) ; r(r)      z1 ... zr ∣ ∣ ∣ ∣ ∣ ∣ ∣ b1 : b2 b3 : b4      cubo 22, 3 (2020) the multivariable aleph-function involving the generalized . . . 353 = 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 (φi (ςi) (zi) ςi)dς1 · · ·dςr (1.6) where, ω = √ −1, b1 = ( ( aj;α (1) j , · · · , α (r) j ) 1, n ) , ( τi ( aji;α (1) ji , · · · , α (r) ji ) n+1, pi ) b2 = ( ( c (1) j ,γ (r) j ) 1, n1 ) , ( τi(1) ( c (1) ji(1) ,γ (1) ji(1) ) n1+1, p i(1) ) ; · · · ; ( ( c (r) j ,γ (r) j ) 1, nr ) , ( τi(r) ( c (r) ji(r) ,γ (r) ji(r) ) nr+1, p i(r) ) b3 = ( τi ( bji;β (1) ji , · · · , β (r) ji ) m+1,qi ) b4 = ( ( d (1) j ,δ (1) j ) 1, m1 ) , ( τi(1) ( d (1) ji(1) ,δ (1) ji(1) ) m1+1, q i(1) ) ; · · · ; ( ( d (r) j ,δ (r) j ) 1, mr ) , ( τi(r) ( d (r) ji(r) ,δ (r) ji(r) ) mr+1, q i(r) ) and ψ (ς1, · · · , ςr) = ∏n j=1 γ ( 1 − aj + ∑r k=1 α (k) j ςk ) ∑r i=1 [ τi ∏pi j=n+1 γ ( aji − ∑r k=1 α (k) ji ςk ) ∏qi j=1 γ ( 1 − bji + ∑r k=1 β (k) ji ςk )], (1.7) φk (ςk) = ∏mk j=1 γ ( d (k) j − δ (k) j ςk ) ∏nk j=1 γ ( 1 − c(k)j + γ (k) j ςk ) ∑r(k) i(k)=1 [ τi(k) ∏q i(k) j=mk+1 γ ( 1 − d(k) ji(k) + δ (k) ji(k) ςk ) ∏p i(k) j=nk+1 γ ( c (k) ji(k) − γ(k) ji(k) ςk )], (1.8) the parameters d (k) ji(k) (j = mk + 1, · · · , qi(k) ), (k = 1, · · · , r; i = 1, · · · , r & i(k) = 1, · · · ,r(k) are complex numbers. also positive real numbers α’s, β’s, γ’ sand δ’s for standardization purpose such that u (k) i = n ∑ j=1 α (k) j + τi pi ∑ j=n+1 α (k) ji + nk ∑ j=1 γ (k) j + τi(k) p i(k) ∑ j=nk+1 γ (k) ji(k) − τi qi ∑ j=1 β (k) ji − mk ∑ j=1 δ (k) j − τi(k) q i(k) ∑ j=mk+1 δ (k) ji(k) ≤ 0 (1.9) 354 a. oli, k. tilahun, g. v. reddy cubo 22, 3 (2020) the real numbers τi > 0 (i = 1, ...,r) and τi(k) > 0 (i = 1, · · · , r). the contour is in the sk−plane and run from σ − ω ∞toσ + ω ∞, where σ is real number with loop, if necessary, ensure that the poles of γ ( d (k) j − δ (k) j ςk ) with j = 1, . . . , mk are separated from those of γ ( 1 − aj + ∑r k=1 α (k) j ςk ) with j = 1, . . . , n and γ ( 1 − c(k)j + γ (k) j ςk ) with j = 1, . . . , nk to the left of the contour lk. the condition for absolute convergence of multiple mellin-barnes type contours (1.6) can be obtained by extension of corresponding conditions for multi variable h-function as: |arg zk| < 12a (k) i π where. a (k) i = n ∑ j=1 α (k) j − τi pi ∑ j=n+1 α (k) ji − τi qi ∑ j=1 β (k) ji + nk ∑ j=1 γ (k) j − τi(k) p i(k) ∑ j=nk+1 γ (k) ji(k) + mk ∑ j=1 δ (k) j − τi(k) q i(k) ∑ j=mk+1 δ (k) ji(k) > 0 (1.10) with k = 1, · · · , r; i = 1, · · · , r and i(k) = 1, · · · , r(k). remark 1: by setting τi = τi (k) = 1, the multivariable aleph function reduces to multivariable i-function (see [4, 7]). remark 2: by setting τi = τi (k) = 1 (k = 1, ...,r) and r = r (1) =, ...,r(r) = 1, the multivariable aleph-function reduces to multivariable h-function defined by srivastava and panda [10]. remark 3: when we set r = 1, the multivariable aleph function reduces to aleph-function of one variable defined by sudland [11]. 2 main results theorem 2.1. let α > 0, β ≥ 0, γ + 4αβ > 0, µi > 0, η ≥ 0, ℜ(ρ) + 12 > 0; − 1 2 < α − β − γ < 1 2 ; ℜ ( λ + µi min 1≤j≤mi { re ( d (i) j ) δ (i) j }) > 0 (i = 1, · · · , r) , and σ = [ ( αx + β x )2 + γ ] then the following formula holds: ∫ ∞ 0 σ−ρ−12f1 ( α, β; γ + 1 2 ; σ ) 2f1 ( γ − α, γ − β; γ + 1 2 ; σ ) ×ℵ ( z1σ −η1, ..., zrσ −ηr ) dx = √ π 2α(4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ) h ( γ + 1 2 ) h ah × ℵ0, n+1: m1n1; m2n2; ....;mrnr pi+1, qi+1, τi; r; p i(1) , q i(1) , τ i(1) ; r(1) , ..., p i(r) , q i(r) , τ i(r) ; r(r) cubo 22, 3 (2020) the multivariable aleph-function involving the generalized . . . 355      z1 (4αβ+γ)η1 ... zr (4αβ+γ)ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( −1 2 − ρ + h;η1, · · · ,ηr ) ,b1 : b2 (−ρ + h;η1, · · · ,ηr) ,b3 : b4      . (2.1) proof. assume that ω in l.h.s. of (2.1), then by virtue of equation (1.5) and (1.6), we have the following ω = ∫ ∞ 0 σ−ρ−1 ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ahσ h ℵ0, n: m1n1, m2n2, ...., mrnr pi, qi, τi; r; p i(1) , q i(1) , τ i(1) ; r(1) , ..., p i(r) , q i(r) , τ i(r) ; r(r)      z1σ −η1 ... zrσ −ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ b1 : b2 b3 : b4      dx = ∫ ∞ 0 σ −ρ−1 ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ahσ h × { 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 ( φi (ςi) ( ziσ −ηi )ςi ) dς1 · · ·dςr } dx = ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ah { 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 (φi (ςi) (zi) ςi)dς1 · · ·dςr } × ∫ ∞ 0 σ −ρ−1+h− ∑ s k=1 ηkςkdx by using equation (1.1), we can obtain the following equation ω = ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ah { 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 (ϕi (ςi) (zi) ςi)dς1 · · ·dςr } × √ π γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) 2α(4αβ + γ) ρ+ 1 2 −h+ ∑ s k=1 ηkςk γ (ρ − h + ∑s k=1 ηkςk + 1) = ∞ ∑ h=0 (γ) h ah ( γ + 1 2 ) h √ π γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) 2α(4αβ + γ) ρ+ 1 2 −h+ ∑ s k=1 ηkςk γ (ρ − h + ∑s k=1 ηkςk + 1) × 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 (ϕi (ςi) (zi) ςi)dς1 · · ·dςr = √ π 2α(4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ah (4αβ + γ) −h 356 a. oli, k. tilahun, g. v. reddy cubo 22, 3 (2020) × γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) (4αβ + γ) ∑ s k=1 ηkςk γ (ρ − h + ∑s k=1 ηkςk + 1) × 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 (ϕi (ςi) (zi) ςi)dς1 · · ·dςr = √ π 2α(4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ) h ( γ + 1 2 ) h ahx { γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) γ (ρ − h + ∑s k=1 ηkςk + 1) } × 1 (2πω)r ∫ l1 · · · ∫ lr ψ (ς1, · · · , ςr) r ∏ i=1 ( φi (ςi) [ zi (4αβ + γ) ηi ]ςi ) dς1 · · ·dςr we readily arrive at the right hand side of (2.1) in view of the presentation of aleph function in mellin barnes contour integral. theorem 2.2. let α ≥ 0, β > 0, γ + 4αβ > 0, µi > 0, η ≥ 0, ℜ(ρ)+ 12 > 0; − 1 2 < α−β −γ < 1 2 ℜ ( λ + µi min 1≤j≤mi { re ( d (i) j ) δ (i) j }) > 0 (i = 1, · · · , r) , and σ = [ ( αx + β x )2 + γ ] then the following formula holds: ∫ ∞ 0 1 x2 σ−ρ−12f1 ( α, β; γ + 1 2 ; σ ) 2f1 ( γ − α, γ − β; γ + 1 2 ; σ ) ×ℵ ( z1σ −η1, ..., zrσ −ηr ) dx = √ π 2β (4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ)h ( γ + 1 2 ) h ah × ℵ0, n+1: m1n1; m2n2; ....;mrnr pi+1, qi+1, τi; r; p i(1) , q i(1) , τ i(1) ; r(1) , ..., p i(r) , q i(r) , τ i(r) ; r(r) ×      z1 (4αβ+γ)η1 ... zr (4αβ+γ)ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( −1 2 − ρ + h;η1, · · · ,ηr ) ,b1 : b2 (−ρ + h;η1, · · · ,ηr) ,b3 : b4      . (2.2) proof. in the similar manner of theorem 2.1 and using (1.2) we easily arrive at the result (2.2). theorem 2.3. let α > 0, β > 0, γ +4αβ > 0, µi > 0, η ≥ 0, ℜ(ρ)+ 12 > 0; − 1 2 < α−β−γ < 1 2 ; ℜ ( λ + µi min 1≤j≤mi { re ( d (i) j ) δ (i) j }) > 0 (i = 1, · · · , r) , and σ = [ ( αx + β x )2 + γ ] then the following formula holds: ∫ ∞ 0 ( α + β x2 ) σ−ρ−12f1 ( α, β; γ + 1 2 ; σ ) 2f1 ( γ − α, γ − β; γ + 1 2 ; σ ) ×ℵ ( z1σ −η1, ..., zrσ −ηr ) dx cubo 22, 3 (2020) the multivariable aleph-function involving the generalized . . . 357 = √ π (4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ) h ( γ + 1 2 ) h ah × ℵ0, n+1: m1n1; m2n2; ....;mrnr pi+1, qi+1, τi; r; p i(1) , q i(1) , τ i(1) ; r(1) , ..., p i(r) , q i(r) , τ i(r) ; r(r)      z1 (4αβ+γ)η1 ... zr (4αβ+γ)ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( −1 2 − ρ + h;η1, · · · ,ηr ) ,b1 : b2 (−ρ + h;η1, · · · ,ηr) ,b3 : b4      . (2.3) proof. in the similar way of theorem 2.1 and using (1.3) we easily arrive at the result (2.3). 3 special cases (1) if we put τi = 1, in (2.1), (2.2) and (2.3), we get the results in terms of multivariable i-function [4, 7]. (2) some suitable parametric changes in (1.1), we obtain single variable i-function, then we arrive at the results due to chand [1]. (3) also, multivariable aleph function reduces to multivariable h-function with some suitable parameters;we get the known result due to daiya et al. [2]. 4 conclusion in this article, we analyze the generalized fractional calculus involving definite integrals of gradshteynryzhik of the multivariable aleph-function. as the special cases of our main results, which are related to i-function, h-function and g-function, we can also get the number of special functions. 358 a. oli, k. tilahun, g. v. reddy cubo 22, 3 (2020) references [1] m. chand, “new theorems involving the i-function and general class of polynomials”, global journal of science frontier research, vol. 12, no 4-f, pp. 56–68, 2012. [2] j. daiya, j. ram and d. kumar, “the multivariable h-function and the general class of srivastava polynomials involving the generalized mellin-barnes contour integrals”, filomat, vol. 30, no. 6, pp. 1457–1464, 2016. [3] d. kumar, s.d. purohit and j. choi, “generalized fractional integrals involving product of multivariable h-function and a general class of polynomials”, j. nonlinear sci. appl., vol. 9, pp. 8–21, 2016. [4] y. n. prasad, “multivariable i-function, vijñāna parishad anusandhan patrika”, vol. 29, no. 4, pp. 231–235, 1986. [5] m. i. qureshi, k.a. quraishi and r. pal, “some definite integrals of gradshteynryzhil and other integrals”, global journal of science frontier research, vol. 11, no. 4, pp. 75–80, 2011. [6] c. k. sharma and s.s. ahmad, “on the multivariable i-function”. acta ciencia indica math, vol. 20, no. 2, pp. 113–116, 1994. [7] c. k. sharma and g.k. mishra, “exponential fourier series for the multivariable i-function”, acta ciencia indica mathematics, vol. 21, no. 4, pp. 495–501, 1995. [8] l. j. slater, “generalized hypergeometric functions”, cambridge univ. press, cambridge, london and new york, (1996). [9] r. k. saxena, j. ram and d.l. suthar, “unified fractional derivative formulas for the multivariable h-function”, vijnana parishad anusandhan patrika, vol. 49, no. 2, (2006), pp. 159–175, 2006. [10] h. m. srivastava, r. panda, “some expansion theorems and generating relationsfor the hfunction of several complex variables”, comment. math. univ. st. paul. vol. 24, pp. 119–137, 1975. [11] n. sudland, b. baumann and t. f. nonnenmacher, “open problem: who knows about the aleph-function?”, fract. calc. appl. anal., vol. 1, no. 4, pp. 401–402, 1998. [12] d. l. suthar and p. agarwal, “generalized mittag-leffler function and the multivariable h-function involving the generalized mellin-barnes contour integrals”, communications in numerical analysis, vol. 1, pp. 25–33, 2017. cubo 22, 3 (2020) the multivariable aleph-function involving the generalized . . . 359 [13] d. l. suthar, b. debalkie, and m. andualem, “modified saigo fractional integral operators involving multivariable h-function and general class of multivariable polynomials”, advances in difference equation, vol. 213, 2019. doi: 10.1186/s13662-019-2150-0 [14] d. l. suthar, h. habenom and h. tadesse, “certain integrals involving aleph function and wright’s generalized hypergeometric function”, acta universitatis apulensis, vol. 52, pp. 1–10, 2017. [15] d. l. suthar, s. agarwal and d. kumar, “certain integrals involving the product of gaussian hypergeometric function and aleph function”. honam mathematical j., vol. 41, no. 1, pp. 1–17, 2019. [16] d. l. suthar and b. debalkie, “a new class of integral relation involving aleph-functions”, surveys in mathematics and its applications, vol. 12, pp. 193–201, 2017. introduction main results special cases conclusion cubo a mathematical journal vol.21, no¯ 03, (93–105). december 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000300093 wave propagation through a gap in a thin vertical wall in deep water b. c. das1, soumen de 1 and b. n. mandal2 1department of applied mathematics, university of calcutta, 92, a.p.c.road, kolkata-700009, india. 2 physics and applied mathematics unit, indian statistical institute, 203, b.t.road kolkata-700108, india. findbablu10@gmail.com, sdeappmath@caluniv.ac.in, bnm2006@rediffmail.com abstract the problem of oblique scattering of surface water waves by a vertical wall with a gap submerged in infinitely deep water is re-investigated in this paper. it is formulated in terms of two first kind integral equations, one involving the difference of potential across the wetted part of the wall and the other involving the horizontal component of velocity across the gap. the integral equations are solved approximately using oneterm galerkin approximations involving constants multiplied by appropriate weight functions whose forms are dictated by the physics of the problem. this is in contrast with somewhat complicated but known solutions of corresponding deep water integral equations for the case of normal incidence, used earlier in the literature as one-term galerkin approximation. ultimately this leads to very closed (numerically) upper and lower bounds of the reflection and transmission coefficients so that their averages produce fairly accurate numerical estimates for these coefficients. known numerical results for normal incidence and for a narrow gap obtained by other methods in the literature are recovered, thereby confirming the correctness of the method employed here. http://dx.doi.org/10.4067/s0719-06462019000300093 94 b. c. das, soumen de, b. n. mandal cubo 21, 3 (2019) resumen en este art́ıculo re-investigamos el problema de dispersión oblicua de ondas superficiales de agua por una pared vertical con una abertura sumergida en agua infinitamente profunda. se formula en términos de dos ecuaciones integrales de primera especie, una involucrando la diferencia de potencial a través de la parte mojada de la pared y la otra involucrando la componente horizontal de la velocidad a través de la apertura. las ecuaciones integrales son resueltas aproximadamente usando aproximaciones de galerkin de un término involucrando constantes multiplicadas por funciones peso apropiadas, cuyas formas son dictadas por la f́ısica del problema. esto se contrapone con lo complicado de soluciones conocidas para las correspondientes ecuaciones integrales de agua profunda para el caso de incidencia normal, usadas anteriormente en la literatura como aproximaciones de galerkin de un término. últimamente esto lleva a cotas superiores e inferiores muy cercanas (numéricamente) para los coeficientes de reflexión y transmisión de tal suerte que sus promedios producen estimaciones numéricas razonablemente precisas para estos coeficientes. se recuperan resultados numéricos conocidos en la literatura para la incidencia normal y para una apertura delgada, confirmando que los métodos empleados son correctos. keywords and phrases: thin vertical wall, submerged gap, integral equations, one-term galerkin approximations, constant as basis, reflection and transmission coefficients. 2010 ams mathematics subject classification: 76b07, 76b15. cubo 21, 3 (2019) wave propagation through a gap in a thin vertical wall in deep water 95 1 introduction the problem of oblique scattering of surface water waves by a thin vertical wall with a gap of arbitrary width submerged in infinitely deep water is re investigated here within the framework of linearized theory of water waves. porter [10] investigated this problem for normal incidence of a surface wave train employing a reduction procedure and also an integral equation formulation, both leading to the same riemann-hilbert problem in the theory of complex variable, and the reflection and transmission coefficients are obtained in closed forms in terms of some definite integrals which could be computed numerically. when the gap is narrow, tuck [12] earlier employed the method of matched asymptotic expansion to obtained the transmission coefficient approximately in terms of an analytical expression. packham and williams [9] employed an integral equation formulation based on green’s integral theorem to reduce the problem of narrow gap in uniform finite depth water to a first kind integral equation in horizontal component of velocity across the gap. they solved the integral equation approximately exploiting the concept of narrowness of the gap, and obtained an approximate analytical expression for the transmission coefficient. mandal [7] employed an integral equation formulation based on havelock’s [6] expansion of water wave potential to solve the narrow gap problem in deep water for normal incidence, and obtained the transmission coefficients approximately by exploiting the concept of narrowness of the gap as has been done by packham and williams [9]. chakrabarti et al [1] re-investigated porter’s problem by reducing it to a special logarithmic singular integral equation involving two unknown constants, one involving the unknown reflection coefficient, which were ultimately determined by two solvability criteria. das et al [2] investigated the oblique scattering problem by formulating it in terms of two first kind integral equations after employing havelock’s [6] expansion of water wave potential, one involving the horizontal component of velocity across the gap and the other involving the difference of potential across the wetted parts of the wall. these were then solved approximately employing one-term galerkin approximations involving somewhat complicated but exact solutions of the corresponding integral equations for the case of normal incidence as could be found from porter [10]. also, one-term galerkin technique was employed recently by roy et al [11] while studying the problem of water wave scattering by a pair of thin vertical barriers with unequal gaps submerged in deep water. however, it involves somewhat complicated but known exact solutions of the corresponding integral equations for a single barrier partially immersed in deep water and for normal incidence, as basis functions. in the present paper, this problem is re-investigated employing one-term galerkin approximation technique wherein the one-term approximations are taken to be simply constants multiplied by appropriate weight functions whose forms are dictated by the physics of the problem. this technique leads to very accurate close bounds(numerical) for the reflection and transmission coef96 b. c. das, soumen de, b. n. mandal cubo 21, 3 (2019) ficients so that their averages produce accurate numerical estimates for these coefficients. known numerical results for normal incidence and also for a narrow gap obtained by other methods in the literature are recovered from the results obtained by the present method as special cases, thereby confirming the correctness of the method. numerical results obtained by the present method are displayed graphically in a number of figures. it may be noted that this type of one-term galerkin method to solve integral equations has not been employed in the literature on water waves earlier. (0,a) re -ky-iµx e -ky+iµx 0 x free surface te -ky+iµx (0,b) y (a) barrier configuration (b) angle of incident wave figure 1: sketch of the problem. 2 mathematical formulation and solution a cartesian co-ordinate system is taken in which y-axis is chosen vertically downwards in the fluid region and the x, z-plane is taken as the rest position of the free surface. for a thin vertical wall with a gap submerged in deep water, its wetted parts are represented by x = 0, y ∈ l = (0, a) ⋃ (b, ∞), wherein the gap is represented by x = 0, y ∈ l̄ = (a, b). the problem is described in figure 1 wherein r and |t | denote the reflection and transmission coefficient respectively. full details of the problem is given in das et al.[2] . for the problem of oblique scattering of surface water waves by the wall with a gap, let f(y)(y ∈ l̄) denote the horizontal component of velocity across the gap, g(y)(y ∈ l̄) denote the difference of potential function across the wetted parts of the wall, r and t denote the reflection and transmission coefficients respectively. then the behaviors of f(y) and g(y) at the end points y = a, y = b are given by f(y) =    o ( (y − a) − 1 2 ) as y → a + 0, o ( (b − y) − 1 2 ) as y → b − 0, (2.1a) and cubo 21, 3 (2019) wave propagation through a gap in a thin vertical wall in deep water 97 g(y) =    o ( (a − y) 1 2 ) as y → a − 0, o ( (y − b) 1 2 ) as y → b + 0. (2.1b) the relation between r, t and f(y), g(y) are given by t = 1 − r = −2i sec α ∫ l̄ f(y)e−kydy, (2.2a) r = −k ∫ l g(y)e−kydy, (2.2b) where α is the angle of incidence of train of surface water waves on the thin wall, k = σ 2 g , σ being the angular frequency and g is the gravity. let f(y) = − 2 πr f(y), y ∈ l̄, (2.3a) g(y) = 1 πik cosα(1 − r) g(y), y ∈ l, (2.3b) then it is easy to see that g(y) and f(y) satisfy the first kind integral equations (cf. das et al [2], mandal and chakrabarti [8]) (mg)(y) ≡ ∫ l g(u)m(y, u)du = e−ky, y ∈ l (2.4a) and (nf)(y) ≡ ∫ l f(u)n(y, u)du = e−ky, y ∈ l̄ (2.4b) where m(y, u) = lim ǫ→+0 ∫ ∞ 0 k1s(k, y)s(k, u) k2 + k2 e−ǫkdk, (2.5a) and n(y, u) = ∫ ∞ 0 s(k, y)s(k, u) k1(k 2 + k2) dk, (2.5b) where k1 = ( k2 + ν2 ) 1 2 , ν = k sin α, s(k, y) = k cos ky − k sin ky while (2.2a) and (2.2b) produce ∫ l f(y)e−kydy = c, (2.6a) ∫ l g(y)e−kydy = 1 π2k2c (2.6b) where c = 1 − r iπr cos α. (2.7) 98 b. c. das, soumen de, b. n. mandal cubo 21, 3 (2019) (2.5a) and (2.5b) show that m(y, u) and n(y, u) are real and symmetric so that g(u), f(u) satisfying (2.4a) and (2.4b) respectively are real and hence, c satisfying (2.6a) as well as (2.6b), is an unknown real quantity. once c is found r and t(= 1 − r) can be calculated using (2.7). if g(y) and f(y) are chosen as one-term galerkin approximations given by g(y) ≈ c0g0(y), y ∈ l; f(y) ≈ d0f0(y), y ∈ l̄, (2.8) then exploiting the properties of symmetry, self-adjointness and positive semi-definiteness of the integral operators (mg)(y) and (nf)(y) defined by (2.4) proceeding as in evans and morris [4] and das et al [2], it can be shown that c has the bounds a, b b ≤ c ≤ a (2.9) where a and b can be expressed in terms of integrals involving g0(y) and f0(y) respectively as given by a = 1 π2k2 ∫ l g0(y)(mg0)(y)dy ( ∫ l g0(y)e −kydy)2 , (2.10) b = ( ∫ l̄ f0(y)e −kydy)2 ∫ l̄ f0(y)(nf)(y)dy . (2.11) it may be noted that a, b are independent of c0, d0 so that these can be chosen to be unity. the upper and lower bounds for |r| and |t | are now obtained as r1 ≤ |r| ≤ r2, t1 ≤ |t | ≤ t2 (2.12) where r1 = 1 (1 + π2a2 sec2 α) 1 2 , r2 = 1 (1 + π2b2 sec2 α) 1 2 , (2.13a) t1 = πb sec α (1 + π2a2 sec2 α) 1 2 , t2 = πa sec α (1 + π2b2 sec2 α) 1 2 . (2.13b) das et al [2] chose g0(y) and f0(y) as the exact solutions of the integral equations (2.4a) and (2.4b) for the case of normal incidence(α = 00) and these involve quite complicated expressions (cf. mandal and chakrabarti [8]). here we choose g0(y), f0(y) as g0(y) =    ( 1 − y a ) 1 2 , 0 < y < a, e−ky ( y b − 1 ) 1 2 , b < y < ∞ (2.14a) and f0(y) = a {(y − a)(b − y)} 1 2 , a < y < b. (2.14b) cubo 21, 3 (2019) wave propagation through a gap in a thin vertical wall in deep water 99 this choice of f0(y) and g0(y) is dictated by the behaviors of f(y) and g(y) at the end points y = a and y = b. then, after substituting (2.14a) in (2.10), a is obtained as a = ∫ ∞ 0 k1 k2+k2 [ku(a, b, k, k) − kv(a, b, k, k)] 2 dk π2k2w2(a, b, k, k) (2.15) where u(a, b, k, k) = ∫a 0 ( 1 − y a ) 1 2 cos kydy + ∫ ∞ b e−ky ( y b − 1 ) 1 2 cos kydy, v(a, b, k, k) = ∫a 0 ( 1 − y a ) 1 2 sin kydy + ∫ ∞ b e−ky ( y b − 1 ) 1 2 sin kydy, w(a, b, k) = ∫a 0 e−ky ( 1 − y a ) 1 2 dy + ∫ ∞ b e−2ky ( y b − 1 ) 1 2 dy. u(a, b, k, k), v(a, b, k, k) and w(a, b, k) can be expressed analytically in terms of young’s and lower incomplete gamma functions(cf. gradshteyn and ryzhik [5]). similarly, after substituting (2.14b) in (2.11), b is obtained as b = m20,0(k(b − a))e −k(a+b) k(b − a) ∫ ∞ 0 j2 0 ( k(b−a) 2 ) k1(k 2+k2) [ k cos k(a+b 2 ) − k sin k(a+b 2 ) ]2 dk (2.16) where m0,0 is the whittaker function and j0 is the bessel function. 3 numerical results the lower and upper bounds of the reflection and transmission coefficients |r| and |t | respectively are evaluated numerically for various values of different parameters such as wavenumber kb, angle of incidence α and a b = 0.5. only the lower and upper bounds r1 and r2 of |r| are displayed in table 1. here we put α = 00 in the expressions for r1 and r2 for obtaining numerical estimates for |r| for the case of normal incidence and the bounds are also compared with exact values derived from porter’s [10] exact analytical results. numerical values of upper and lower bounds of |r| coincide within 3 to 4 decimal places and hence their averages provide very accurate estimates for the reflection coefficients. similar computations have been carried out for the upper and lower 100 b. c. das, soumen de, b. n. mandal cubo 21, 3 (2019) bounds of |t |. however, these results are not displayed here. it has also been checked that these numerical estimates satisfy the energy identity |r|2 + |t |2 = 1, which provides a partial check on the correctness of the method. there are also other checks as described below. also the numerical results presented in table 1 are compared with those in table 3 of das et al [2]. almost the same results are obtained. it may be noted that for the present method, the basis function g0(y) given by (2.14a) decays exponentially as y → ∞ while for the method employed in das et al [2] the basis function f1(y) given by (5.2) (and (5.3)) of das et al [2] decays algebraically as y → ∞. because of this, the one-term galerkin method with simplified basis functions employed here provides high accuracy in the numerical results. α = 00 α = 300 α = 600 α = 850 kb r1 r2 |r| porter[1] r1 r2 r1 r2 r1 r2 0.05 0.7251 0.7257 0.7251 0.6582 0.6587 0.4106 0.4109 0.0831 0.0831 0.4 0.4343 0.4344 0.4343 0.3605 0.3625 0.1823 0.1875 0.0306 0.0307 1.2 0.6500 0.6504 0.6502 0.5872 0.5877 0.3752 0.3755 0.0733 0.0772 2.0 0.9448 0.9472 0.9466 0.9236 0.9238 0.7950 0.7954 0.2092 0.2099 3.0 0.9960 0.9987 0.9960 0.9936 0.9937 0.9725 0.9771 0.6100 0.6107 4.0 0.9996 0.9999 0.9996 0.9993 0.9994 0.9967 0.9969 0.9206 0.9206 table 1. lower and upper bounds for the reflection coefficient of |r| for various values of the parameters kb, α and a b = 0.5 as in porter [10] and tuck [12], let a = h(1 − µ 2 ), b = h(1 + µ 2 ), λ = 2π k where h is the depth of the center of the gap below the free surface , µ is the ratio of the width of the gap to its mean depth and it lies between 0 to 2 and λ is the wavelength of the incident wave. 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kh (=σ2h/g) µ=0.5 µ=1.0 µ=1.5 µ=0.1 µ=0.1 µ=0.5 µ=1.0 µ=1.5 x x x porter(1972) |t|(present method) −−−− |r| (present mehod) figure 2: |r|(...) and |t |(−) against kh for different values of µ, and α = 00. in figure 2, |r| and |t | are depicted against kh(= k(a+b) 2 ) for different values of µ(= 2(b−a) b+a ) cubo 21, 3 (2019) wave propagation through a gap in a thin vertical wall in deep water 101 and for normal incidence(α = 00). also |r| and |t | calculated from porter’s [1] exact expressions obtained by a completely different method are indicated in figure 2 by cross marks (x). from this figure it is observed that the curves of |r| and |t | plotted on the basis of the numerical results obtained by the present method and plotted on the basis of porter’s [10] exact results coincide. this gives another check on the correctness of the method. in figure 3, |t |2 is depicted against h λ (= k(a+b) 4π ) for different small values of µ = 0.05, 0.15, 0.4 and for normal incidence(α = 00) so that the gap is narrow. also |t2| calculated from tuck’s [12] result (expression given in (6.2) there) are indicated in figure 3 by cross marks (x). from this figure it is observed that the curves of |t2| plotted on the basis of the numerical results obtained by the present method and plotted on the basis of tuck’s [12] approximate result obtained by the method of matched asymptotic expansion coincide. this provides yet another check for the correctness of the results obtained by the present method. 0 0.05 0.1 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h/λ |t |2 µ=0.15 µ=0.4 µ=0.05 present method x x x tuck(1971) figure 3: |t |2 against h λ for different values of µ, and α = 00. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kb |r|(present method) |t|(present method) |r|(dean(1945) |t|(dean (1945) 102 b. c. das, soumen de, b. n. mandal cubo 21, 3 (2019) figure 4: |r|(...) and |t |(−) against kb for α = 00. in figure 4, |r| and |t | are depicted graphically against the wavenumber kb for a b = 0 so that the upper part of the wall is absent and the wall becomes a submerged barrier considered by dean [3]. the curves of |r| and |t | almost coincide with the corresponding curves given by dean [12] (indicated here by cross (x) marks). this produces a final check for the correctness of the results obtained by the present method. 0 0.05 0.1 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h/λ |t |2 α=600 α=750 α=300 α=00 figure 5: |t |2 against h λ for different values of α, and µ = 0.05 in figure 5, |t |2 is depicted against h λ for different values of α with fixed µ = 0.05(narrow gap). this is in fact an extension of tuck’s figure for a narrow gap and normal incidence to oblique incidence. all the conclusion drawn by tuck [12] for normal incidence about the transmission of energy through a narrow gap can be extended for oblique incidence. for example, considerable transmission of energy occurs for long waves. from the figure 5 it is observed that transmission increases with the increase in the angle of incidence which is plausible. also for a fixed angle of incidence, transmission first increases as the wavenumber increases and then it decreases steadily as the wavenumber further increases. this is due to the fact that for large wavenumber the waves are confined near the free surface so that most of these are reflected by the upper part of the thin wall. cubo 21, 3 (2019) wave propagation through a gap in a thin vertical wall in deep water 103 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kh |r |( .. .) |t |( − ) α=00 α=600 α=300 α=600 α=00 α=750 α=750 α=300 figure 6: |r|(...) and |t |(−) against kh for different values of α, and µ = 0.5 in figure 6, |r| and |t | are depicted against kh for different values of α with fixed µ = 0.5(moderate gap). this is again an extension of porter’s [10] curves for oblique incidence. this figure shows that for a wall with a moderate gap, as the angle of incidence increases, reflection coefficient decreases while transmission increases for fixed wavenumber. incident waves are reflected by two parts of the wall. obviously this reflection is maximum when waves are incident normally (α = 00) on the wall and then reflection decreases gradually as α increases. this is plausible from physical considerations. here however results for values of α from 00 to 750 are presented. again for fixed angle of incidence the reflection coefficient first decreases with increase of wavenumber and then increases asymptotically to unity as the wavenumber further increases. this is also plausible since for large wavenumber, the waves are confined near the free surface as mentioned earlier, so that most of the incident waves are reflected back. reverse of this happens for the transmission coefficient i.e, transmission increases with the increase in the angle of incidence and for a fixed angle of incidence, transmission increases first with the increase of wavenumber and then decreases steadily to zero as the wavenumber further increases. it is interesting to note that for fixed angle of incidence, µ(0 < µ < 2) is a crucial parameter in determining the transmission of wave energy through the gap at certain wavelengths. for µ = 1.0, |t | attains maximum near kh = 0.5 corresponding to about more than 90 percent of wave energy transmission. for fixed α, as µ decreases i.e, as gap becomes smaller, |t | decreases for all finite kh which is shown in the figure 2. the curves in figures 5 and 6 may be regarded as new results. 4 conclusion the problem of water wave scattering by a thin vertical wall with a gap submerged in infinitely deep water is re-investigated by using integral equation formulations based on havelock’s expan104 b. c. das, soumen de, b. n. mandal cubo 21, 3 (2019) sion of water wave potential. two first kind integral equations involving horizontal component of velocity across the gap and difference of velocity potential across the upper and lower parts of the wall are obtained. these are solved here approximately by using one-term galerkin approximations involving constants multiplied by appropriate weight functions whose forms are dictated by the behaviour at the end points of the gap and at infinite depth. exploitation of the symmetry and positive semi-definiteness of the operators of the integral equations lead to expressions for upper and lower bounds for the reflection and transmission coefficients. these bounds, when computed numerically, coincide upto 3-4 decimal places so that their averages produce very accurate numerical estimates for the reflection and transmission coefficients. known numerical results(in the form of graphs) for the problem of water wave scattering by a thin wall with a gap, available in the literature by employing different methods, are recovered from the results obtained by the present method as special cases. the method employed here appears to be quite simple in comparison to other known methods employed for this problem. it is felt that this type of one-term galerkin technique involving simple basis functions can be employed to study wave scattering by other types of obstacles with submerged edges such as multiple thin vertical barriers, thick rectangular barriers, wave scattering by step-type bottom topography etc. 5 acknowledgments the authors thank the referee for his comments and suggestions to improve the paper in the present form. b c das thanks the ugc, india, for providing financial support (file no: 22/12/2013(ii)euv), as a research scholar of the university of calcutta, india. this work is also supported by serb through the research project no. emr/2016/005315 cubo 21, 3 (2019) wave propagation through a gap in a thin vertical wall in deep water 105 references [1] a. chakrabrti, s.r. manam, s. banerjea, scattering of surface water waves involving a vertical barrier with a gap, j. eng. math., 45 (2003), 183-194. [2] p. das, s. banerjea, b. n. mandal, scattering of oblique waves by a thin vertical wall with a submerged gap, arch. mech.,, 48 (1996), 959-972. [3] w. r. dean, on the reflection of surface waves by a submerged plane barrier, proc. camb. phil. soc., 41 (1945), 231-238. [4] d. v. evans, c.a.n. morris, the effect of a fixed vertical barrier on oblique incidence surface waves in deep water, j. inst. math. applic., 9 (1972), 198-204. [5] i. s. gradshteyn, i. m. ryzhik, table of integrals, series and products, london, academic press, 1980. [6] t. h. havelock, forced surface waves on water, phil. mag., 8 (1929), 569-576. [7] b. n. mandal, a note on the diffraction of water waves by a vertical wall with a narrow gap, arch. mech., 39 (1987), 269-273. [8] b. n. mandal, a. chakrabarti, water wave scattering by barrier, wit press, southampton, uk, 2000. [9] b. a. packham, w. e. williams, a note on the transmission of water waves through small apertures, j. math. anal. appl., 10 (1972), 176-184. [10] d. porter, the transmission of surface waves through a gap in a vertical barrier, proc. camb. phil. soc., 71 (1972), 411-421. [11] r. roy, u. basu, b. n. mandal, water wave scattering by a pair of thin vertical barriers with submerged gaps, j. eng. math., 105 (2017), 85-97. [12] e. o. tuck, transmission of water waves through small apertures, j. fluid mech., 49 (1971), 65-74. introduction mathematical formulation and solution numerical results conclusion acknowledgments cubo a mathematical journal vol.20, no¯ 02, (53–66). june 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000200053 optimal control of a sir epidemic model with general incidence function and a time delays moussa barro, aboudramane guiro and dramane ouedraogo département de mathématiques, ufr des sciences et techniques, université nazi boni, laboratoire d’analyse mathématique et d’informatique (lami) b.p. 1091 bobo-dioulasso, burkina faso. mousbarro@yahoo.fr, abouguiro@yahoo.fr, dramaneouedraogo268@yahoo.ca. abstract in this paper, we introduce an optimal control for a sir model governed by an ode system with time delay. we extend the stability studies of model (2.2) in section 2, by incorporating suitable controls. we consider two control strategies in the optimal control model, namely: the vaccination and treatment strategies. the model has a time delays that represent the incubation period. we derive the first-order necessary conditions for the optimal control and perform numerical simulations to show the effectiveness as well as the applicability of the model for different values of the time delays. these numerical simulations show that the model is sensitive to the delays representing the incubation period. resumen en este art́ıculo, introducimos un control óptimo para un modelo sir gobernado por un sistema de edos con retardo temporal. extendemos los estudios de estabilidad del modelo (2) en la sección 2, incorporando controles apropiados. consideramos dos estrategias de control en el modelo de control óptimo, llámense: las estrategias de vacunación y tratamiento. el modelo tiene un retardo en el tiempo que representa el peŕıodo de incubación. derivamos las condiciones necesarias de primer orden para http://dx.doi.org/10.4067/s0719-06462018000200053 54 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) el control óptimo y realizamos simulaciones numéricas para mostrar la efectividad y también la aplicabilidad del modelo para diferentes valores de los retardos temporales. estas simulaciones numéricas muestran que el modelo es sensible a los retardos que representan el peŕıodo de incubación. keywords and phrases: sir, general incidence, delays, optimal control, epidemic models, hamiltonian. 2010 ams mathematics subject classification: 34k19, 34k20, 49k25, 49k30, 65n06, 90c90. cubo 20, 2 (2018) optimal control of a sir epidemic model with general incidence . . . 55 1 introduction mathematical modeling of population are often used to describe the dynamics of epidemic diseases. this is a fast growing research area and has been paying important roles in discovering relations between species and their interactions. there have been many variations such as classical epidemiological models [11]. these models are based on the standard susceptible-infectious-susceptible (sis), susceptible-infectious-recovered (sir) and susceptible-exposed-infectious-recovered (seir) models, which are determined according to the difference on the method of transmission, nature of the disease, those with short/long incubation period, killer/curable diseases, etc, and the response of the individuals to it, for instance, gaining transient/permanent immunity, dying from the disease, etc.[6, 16]. the main purpose of formulating a such epidemiological model is to understand the long-term behavior of the epidemic disease and to determine the possible strategies to control it. differential equations, whether there are ordinary, delay, partial or stochastic are one of the main mathematical tools being used to formulate many epidemiological models. the focus in such epidemiological models has been on the general incidence at which people move from the class of susceptible individuals to the class of infective individuals.these general incidence have been modeled mostly by using bilinear and holling type of functional responses [10, 12]. on the other hand, optimal control has extensively been used a strategy to control the epidemic outbreaks [8]. the main idea behind using the optimal control in epidemics is to search for, among the available strategies, the most effective strategy that reduces the infection rate to a minimum level while optimizing the cost of deploying a therapy or preventive vaccine that is used for controlling the disease progression [18]. in terms of epidemic diseases, such strategies can include therapies, vaccines, isolation and educational campaigns [3, 5]. mathematical models have become important tools in analyzing the spread and control of infectious diseases. the model formulation process clarifies assumptions, variables parameters. there have been many studies that have mathematically analyzed infectious diseases [4, 7, 15]. recently, many control optimal models pertaining to epidemic disease to epidemic diseases have appeared in the literature. they include, but not limited to, delayed sirs epidemic model [13], delayed sir model [1], tuberculosis model [17], hiv model [9] and dengue fever [2]. in this paper, we consider an optimal control problem governed by a system of delay differential equations with general incidence function and time delays. the governing state equations of the optimal control are described in a sir framework with a general incidence function and a time delays representing the incubation period. then we derive first-order necessary conditions for existence of the optimal control and develop a numerical method to solve them. the rest of this paper is organized as follows. in section 2, we give the statement of the optimal control problem. we derive the necessary conditions for existence of the optimal control in section 3. in section 4, we describe the numerical method and present the resulting numerical simulations. finally, we discuss these results in section 5 along with some concluding remarks. 56 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) 2 statement of the optimal control problem compute the optimal pair of vaccination and treatment strategies (u1, u2) that would maximize the recovered population and minimize both the infected and susceptible population, and at the same time minimize the costs of applying the vaccination and treatment strategies. so we consider the optimal control problem of the form (see eihab b. m. et al): min (u1,u2)∈u j(u1(t), u2(t)) =    s(t) + i(t) − r(t) + ∫t 0 (c1u 2 1(t) + c2u 2 2(t) + s(t) + i(t) − r(t))dt, (2.1) subject to the quation    ṡ = b − µ1s − f(s, iτ) − u1s, i̇ = f(s, iτ) − (µ2 + γ)i − u2i, ṙ = γi − µ3r. (2.2) the two functions u1(t) and u2(t) represent vaccination and treatment strategies. these control functions are assumed to be l∞(0, t) functions belonging to a set of admissible controls u = {(u1, u2) ∈ (l ∞ (0, t))2 : u1min ≤ u1(t) ≤ u1max, u2min ≤ u2(t) ≤ u2max}, where 0 ≤ u1min < u1max ≤ 1 and 0 ≤ u2min < u2max ≤ 1. the two constants c1 and c2 are weighted cost associated with the use of the controls u1(t) and u2(t), respectively. the state equations are formulated from an sir model with general incidence model, where s(t), i(t), r(t) are the numbers of susceptible, infected and recovered individuals at time t, respectively. the parameters b is the recruitment rate, the death rates for the classes are µ1, µ2 and µ3, respectively. the average time spent in class i before recovery is 1/γ. for biological reasons, we assume that µ1 ≤ µ2 + γ; that is, removal of infectives is at least as fast as removal of susceptibles. the time delays τ represents the incubation period. that is to say, only susceptible individuals who got infected a time t − τ are able to communicate the disease at time t. as general as possible, the incidence function f must satisfy technical conditions. thus, we assume that h1 f is non-negative c1 functions on the non-negative quadrant, h2 for all (s, i) ∈ r2+, f(s, 0) = f(0, i) = 0. let us denote by f1 and f2 the partial derivatives of f with respect to the first and to the second variable the differential equation model described by (2.2) without controls (u1 = u2 = 0) has two equilibrium points: a disease-free equilibrium e0 given by e0 = ( b µ1 , 0, 0 ) cubo 20, 2 (2018) optimal control of a sir epidemic model with general incidence . . . 57 and an endemic equilibrium e∗ = (s∗, i∗, r∗) where, s∗ = b − (µ2 + γ)i ∗ µ1 i∗ = i∗ r∗ = γ µ3 i∗ the basic reproduction number of (2.2) without controls is given by r0 = f2(s 0, 0) µ2 + γ it was proven that if r0 < 1, then the disease-free equilibrium is asymptotically stable and if r0 > 1 then it is unstable. on the other-hand, when the controls is not null (u1 6= 0 and or u2 6= 0), we have the sir model (2.2). the disease-free equilibrium for system (2.2) is given by ec0 = ( b µ1 + u1 , 0, 0 ) (2.3) whereas the endemic equilibrium e∗c is given by e∗c = ( b − (µ2 + γ + u2)i ∗ µ1 + u1 , i∗, γ µ3 i∗ ) the basic reproduction number rc of system (2.2) is given by rc = f2(s 0, 0) µ2 + γ + u2 and it is clear that when u1 → 0 and u2 → 0 then rc → r0 3 existence and characterization of the optimal control in this section, we discuss the existence of the optimal control and then construct the hamiltonian of the optimal control problem to derive the first order necessary conditions for the optimal control. 3.1 existence of optimal control to show the existence of the optimal control for the problem under consideration, we notice that the set of admissible controls u is, by definition, closed and bounded. it is also convex because [u1min, u1max] × [u2min, u2max] is convex in r 2. it is obvious that there is an admissible pair ((u1(t), u2(t))) for the problem. hence, the existence of the optimal control comes as a direct result from the filippove-cesari theorem [14]. we therefore, have the following result: 58 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) theorem 3.1. consider the optimal control problem (2.1) subject to (2.2). then there exists an optimal pair of controls (u∗ 1 , u∗ 2 ) and a corresponding optimal states (s∗, i∗, r∗) that minimizes the objective function j(u1, u2) over set of admissible controls u. proof. to prove the existence of an optimal control pair, it is important to verify the following assertion. (1) the set of controls and corresponding state variables is nonempty. (2) the admissible set u is convex and closed. (3) the right-hand side of the state system (2.2) is bounded by a linear function in the state and control variables. (4) the integrand of the objective functional ls,i,r(u1, u2) is convex on the set u. the hessian matrix of ls,i,r(u1, u2) on u is done by : m = ( 2c1 0 0 2c2 ) , sp(m) = {2c1, 2c2} ⊂ r ∗ +, then, ls,i,r(u1, u2) is strictly convex in u. (5) there exists constants ω1 > 0, ω2 and ρ > 1 such that the integrand ls,i,r(u1, u2) of the objective functional satisfies ls,i,r(u1, u2) ≥ ω1|(u1, u2)| ρ − ω2. ls,i,r(u1, u2) = c1u 2 1(t) + c2u 2 2(t) + s(t) + i(t) − r(t) ≥ min(c1, c2)(u 2 1(t) + u 2 2(t)) − r(t) r(t) is bounded because n = s + i + r i.e ∃α, β, α < r(t) < β, ∀t let ω1 = min(c1, c2) and ω2 = β. we have, ls,i,r(u1, u2) ≥ ω1‖(u1; u2)‖ 2 − ω2. cubo 20, 2 (2018) optimal control of a sir epidemic model with general incidence . . . 59 3.2 characterization of optimal control in this subsection, we derive the first order necessary conditions for the existence of optimal control, by constructing the hamiltonian h and applying the pontryagin’s maximum principle. to simplify the notations, we write x(t) = [s(t), i(t), r(t)] t , u(t) = [u1(t), u2(t)] t and λ(t) = [λ1(t), λ2(t), λ3(t)]. we denote by g(u(t), x(t)) the integrand part of the objective function (2.1).with these notations and terminologies, the hamiltonian is given by h = h(u(t), x(t), λ(t)) h = g(u(t), x(t)) + λt (t).ẋ(t) = c1u 2 1 + c2u 2 2 + s + i − r + λ1 ( b − µ1s − f(s, iτ) − u1s ) (3.1) +λ2 ( f(s, iτ) − (µ2 + γ)i − u2i ) + λ3 ( γi − µ3r ) . let χ[a,b](t) be the characteristic function defined by χ[a,b](t) =    1, if t ∈ [a, b], 0, otherwise. (3.2) let u∗ = [u∗1, u ∗ 2] t be the optimal control and x∗(t) = [s∗(t), i∗(t), r∗(t)]t be the corresponding optimal trajectory. then there exists λ(t) ∈ r3 such that the first order necessary conditions for the existence of optimal control are given by the equations ∂h ∂u (t) = 0, (3.3) dx dt (t) = ∂h ∂λ , (3.4) dλ dt (t) = − ∂h ∂x . (3.5) the optimality conditions: [ ∂h ∂u1 (t) ] u(t)=u∗(t) = 0, (3.6) [ ∂h ∂u2 (t) ] u(t)=u∗(t) = 0. (3.7) simplifying (3.5) and (3.6), we obtain 2c1u ∗ 1 − sλ1 = 0, (3.8) 2c2u ∗ 2 − λ2i = 0. (3.9) 60 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) further simplification of (3.8) and (3.9) yields u∗1(t) = min { u1max; max { 0; s(t)λ1(t) 2c1 }} (3.10) and u∗2(t) = min { u1max; max { 0; i(t)λ2(t) 2c2 }} . (3.11) the state equations: given by the forms (2.2) the co-state equations: dλ1 dt (t) = − ∂h ∂s (t), dλ2 dt (t) = − [ ∂h ∂i (t) + χ[0,t−τ] ∂h ∂iτ (t + τ) ] , dλ3 dt (t) = − ∂h ∂r (t), which when simplified, lead to dλ1 dt = −1 + (λ1(t) − λ2(t))f1(s, i) + (µ1 + u1)λ1(t), dλ2 dt = −1 + (λ1(t + τ) − λ2(t + τ))χ[0,t−τ](t)f2(s, i) + (µ2 + γ + u2)λ2(t) − γλ3(t), dλ3 dt = 1 + µ3λ3(t). the transversality conditions: λ1(t) = 1, λ2(t) = 1, λ3(t) = −1. remark 3.2. it is noting that 1. the hamiltonian function h is strongly convex in the control variables. 2. the right-hand sides of the state and co-state equations are lipschitz continuous. 3. the set of the admissible controls u is convex 4 numerical simulations in this section, we apply the above optimal control theory with consideration of its applicability. we discuss the discretization of the optimal control problem described and present the numerical cubo 20, 2 (2018) optimal control of a sir epidemic model with general incidence . . . 61 results obtained through our simulations. the algorithm describing the approximation method to obtain the optimal control is the following algorithm inspired from [13]. the algorithm used here is a numerical variation of forward euler method with a step size h. we explicitly write the forward euler method for the state and the adjoint. step1: for i = −m, ..., 0, do : si = s0; ii = i0; ri = r0; u i 1 = 0; ui 2 = 0 end for for i = n, ..., n + m λi1 = 1; λ i 2 = 1; λ i 3 = −1 end for step2 :for i = 0, ..., n − 1 si+1 = si + h(b − µ1si − βsiii − u i 1si) ii+1 = ii + h(βsiii − (µ2 + γ)ii − u i 2 ii) ri+1 = ri + h(γii − µ3ri) λn−i−1 1 = λn−i 1 − h(−1 + (λn−i 1 − λn−i 2 )βii+1 + (µ1 + u i 1 )λn−i 1 λn−i−1 2 = λn−i 2 − h(−1 + (λn+m−i 1 − λn+m−i 2 )χ[0,t−τ](tn−i)βsi+1 + (µ2 + γ + u i 2 )λn−i 2 − γλn−i 3 ) λn−i−1 3 = λn−i 3 − h(1 + µ3λ n−i 3 ) ui+1 1 = si+1λ n−i 1 /2c1 ui+1 2 = ii+1λ n−i 2 /2c2 end for step3 :for i = 1, ..., n, write s∗(ti) = si, i ∗(ti) = ii r ∗(ti) = ri u ∗ 1 (ti) = u i 1 and u∗ 2 (ti) = u i 2 . 62 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) comments fig 1. represent the different dynamics of the susceptible population for different aspect of control. the orange color represent the population when there is treatment but not vaccination (u1 = 0 and u2 6= 0). the blue curve represent the population when there are vaccination and treatment (u1 6= 0 and u2 6= 0). the green curve show the evolution of the susceptible population when there is just treatment but not vaccination (u1 = 0 and u2 6= 0). this show that, without vaccination so many people are exposed to disease. fig 1. dynamic of susceptible population with different aspect of control. cubo 20, 2 (2018) optimal control of a sir epidemic model with general incidence . . . 63 fig 2. represent the different dynamics of the infected population for different aspect of control. the orange color present the evolution of the infected population when there is treatment but not vaccination (u1 = 0 and u2 6= 0). the blue curve represent the population when there are vaccination and treatment (u1 6= 0 and u2 6= 0). the green curve show the evolution of the infected population when there is just treatment but not vaccination (u1 = 0 and u2 6= 0). naturally, as many people are exposed to the disease without vaccination, we see the growth of the infected population. fig 2. dynamic of infected population with different aspect of control. 64 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) fig 3. represent the different dynamics of the infected population for different aspect of control. the orange color present the evolution of the recovered population when there is treatment but not vaccination (u1 = 0 and u2 6= 0). the blue curve represent the population when there are vaccination and treatment (u1 6= 0 and u2 6= 0). the green curve show the evolution of the recovered population when there is just treatment but not vaccination (u1 = 0 and u2 6= 0). as many people are exposed to the disease without vaccination, indeed we see the growth of the recovered population. fig 3. dynamic of recovered population with different aspect of control. in conclusion, observing the figures, we can deduce that the strategy leading to the vaccination alone (u1 6= 0 and u2 = 0) should be preferable to the joint use of vaccination (u1 6= 0) and treatment (u2 6= 0). the optimal control strategy here shows that prevention is more effective for the eradication of the disease. cubo 20, 2 (2018) optimal control of a sir epidemic model with general incidence . . . 65 5 conclusion in this paper, we considered an optimal control problem for a sir model with time delay (representing the incubation period ) and general incidence function. the main idea developed here is the optimal control in epidemics in order to search among the available strategies, the most effiscience one that reduce the infection rate to a minimum level while optimizing the cost deploying a therapy and preventive vaccine that is used to control the disease progression. the two control functions u1(t) and u2(t), which represent the vaccination and the treatment strategies are subject to time delays before being effective. then we formulated the objective function of the optimal control problem. we discussed the existence of the optimal control and then derived the first order necessary conditions for the optimal control through constructing the hamiltonian and using the pontryagin’s maximum principle to achieve our aim. finally, to end our study, we do a numerical simulation to corroborate the theoretical results obtained. acknowledgments the authors want to thank the anonymous referee for his valuable comments on the paper. competing interests the authors declare that they have no competing interests. author’s contribution aboudramane guiro provide the subject, wrote the introduction and the conclusion and verified some calculation. moussa barro conceived the study and computed the equilibria and their local stabilities. dramane ouedraogo wrote mathematical formula, he bring up the control strategy and did all the calculus with the second author. all the authors read and approved the final manuscript. references [1] a. abta, h. laarabi, h. t. alaoui , the hopf bifurcation analysis and optimal control of a delayed sir epidemic model, int. j. anal. (2014), 1-10. [2] d. aldila, t. gotz, e. soewono, an optimal control problem arising from a dengue disease transmission model, math. biosci. 242 (1)(2013), 9-16. [3] f. g. ball, e. s. knock, p.d. o’neil control of emerging infectious diseases using responsive imperfect vaccination and isolation , math. biosci. 216 (1) (2008), 100-113. [4] n. becker, the use of epidemic models, biometrics 35(1978) 295-305. 66 moussa barro, aboudramane guiro and dramane ouedraogo cubo 20, 2 (2018) [5] c. castilho , optimal control of an epidemic through educational campaigns , electron. j. differ. equ. 2006 (2006), 1-11. [6] c. chiyaka, w. garira, s. dube, transmission model of endemic human malaria in a partially immune population. math. comput. model. 46 (2007), 806-822. [7] k. dietz, the first epidemic model: a historical note on p.d. en’ko. aust. j. stat. 30a (1988), 56-65. [8] h. gaff, e.schaefer, optimal control applied to vaccination and treatment strategies for various epidemiological models, math. biosci. eng. 6 (3) (2009), 469-492. [9] k. hattaf, n. yousfi, optimal control of a delayed hiv infection model with immune response using an efficient numerical method , int. sch. res. netw. (2012) (2012), 1-7. [10] h. w. hethcote, p. van den driessche; some epidemiological models with nonlinear incidence. j.math. biol. 29 (1991), 271-287. [11] h. w. hethcote, the mathematics of infectious, siam rev.42 (2000), 599-653. [12] a. kaddar; on the dynamics of a delayed sir epidemic model with a modified saturated incidence rate, electron. j. differ. equ. 13 (2009), 1-7. [13] h. laarabi, a. abta, k. hattaf , optimal control of a delayed sirs epidemic model with vaccination and treatment , acta biotheor. 63 (15) (2015), 87-97. [14] s. nababan ; a filippov-type lemma for functions involving delays and its application to timedelayed optimal control problems,optim. theory appl. 27 3 (1979), 357-376. [15] p. ogren, c. f. martin, vaccination strategies for epidemics in highly mobile populations. appl. math. comput. 127 (2002), 261-276. [16] s. ruan, d. xiao, j. c. beier; on the delayed ross-macdonald model for malaria transmission, bull. math. biol. 70 (2008), 1007-1025. [17] c. j. silva, d. f. torres, optimal control strategies for tuberculosis treatment: a case study in angola , numer. algebra control optim. 2 (3) (2012), 601-617. [18] g. zaman, y.h. kang, j.h. jung, optimal treatment of an sir epidemic model with time delay , biosystems 98 (1) (2009), 43-50. introduction statement of the optimal control problem existence and characterization of the optimal control existence of optimal control characterization of optimal control numerical simulations conclusion cubo a mathematical journal vol.20, no¯ 01, (65–78). march 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000100065 on rigid hermitean lattices, ii ana cecilia de la maza departamento de matemática y estad́ıstica, universidad de la frontera, temuco, chile. remo moresi cerfim, cp 1132, 6601 locarno, switzerland, anace.delamaza@ufrontera.cl, romicatj@yahoo.it abstract we study the indexed hermitean lattice of type 0 generated by a single element a subjected to the relation a ≤ b⊥ ∧ bb⊥ = 0. we prove that it is finite, provided that two crucial indices are finite. we show that index relations imply algebraic relations and describe the lattice by means of its subdirectly irreducible factors. we finally use the results to confirm a conjecture appeared in 2000. resumen estudiamos el reticulado hermitiano finito indexado de tipo 0 generado por un solo elemento a sujeto a la relación a ≤ b⊥ ∧bb⊥ = 0. probamos que es finito, suponiendo que dos ı́ndices cruciales son finitos. mostramos que las relaciones de ı́ndices implican relaciones algebraicas y describimos el reticulado a travs de sus factores subdirectamente irreductibles. finalmente, usamos nuestros resultados para confirmar una conjetura aparecida el ao 2000. keywords and phrases: lattices, semilattices, modular lattices, hermitean lattices, orthogonal geometry. 2010 ams mathematics subject classification: 03g10,06a12, 06c05, 06b25. http://dx.doi.org/10.4067/s0719-06462018000100065 ignacio castillo ignacio castillo ignacio castillo ignacio castillo 66 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) 1 introduction the importance of lattices in infinite-dimensional orthogonal geometry was brought to attention by the pioneering work of herbert gross (1936-1989): see in particular [g1] and [g2]. all examples treated in origin are sublattices of some l(e), the subspace-lattice of an ℵ0-dimensional vector space e over an appropriate division ring k, together with the orthogonal operation induced by a hermitean form φ (i.e. x 7→ x⊥ := {y ∈ e | φ(y, x) = 0 ∀x ∈ x}) and were used to study geometric invariants, for instance dimension of quotient spaces or intersections with the subspace e∗ of trace-valued vectors in e. the fact that e∗ 6= e only if char(k)= 2 was also playing some role. after some time of concrete investigations with subspace lattices (see e.g. [m1]), the natural idea to insert all considerations into an abstract setting gave rise to the following definitions (cf. [kkw], ch. iv): a hermitean lattice (hl for short) is an algebra (l, 0, 1, · , + , ⊥, b) of type 〈0, 0, 2, 2, 1, 0〉 such that i) (l, 0, 1, · , +) is a modular lattice with universal bounds 0 , 1; ii) ⊥ : l → l is a unary operation with 1⊥ = 0 and x ≤ (x⊥y)⊥ ∀ x, y ∈ l (1.1) iii) b ∈ l is a nullary operation with xx⊥ ≤ b ∀ x ∈ l. in case b is explicitly not trivial (i.e. b 6= 1), the modular law in i) is sometimes replaced by the stronger fano identity (w + v)(y + z) ≤ (w + y)(v + z) + (w + z)(v + y). if we drop the operation “+”, then we obtain the notion of hermitean semilattice (hsl for short). in the present paper we will endow hl l with a so-called index function of type 0 (if for short), i.e a function δ from the set of quotients of l into the set of cardinals ≤ ℵo, with the following properties: δ(x/y) ≥ δ(xz/yz), (1.2) δ(x/y) ≥ δ(x + z/y + z), (1.3) δ(x/y) ≥ δ(y⊥/x⊥), (1.4) δ(x/y) + δ(y/z) = δ(x/z), (1.5) δ(x/y) = 0 ⇐⇒ x = y. (1.6) cubo 20, 1 (2018) on rigid hermitean lattices, ii 67 we will speak about indexed hermitean lattices (ihl). by dropping (1.3), we obtain the notion of indexed hermitean semilattices (ihsl). a major task of the theory of h(s)l consists in describing the free objects s[a] and f[a], generated by a single element a in the varieties of hsl and hl, respectively. since such objects are infinite, a more realistic project consists in studying appropriate presentations under (index) relations suggested by geometrical choice (see [g1], [m1], [m2] and also the bibliography in [kkw] for many known examples). one of these options is given by the relation a ≤ b⊥, which was introduced in [dm3] and gave rise to the concept of rigid h (s)l. here we continue such investigation and consider rigid hl with the (somewhat complementary) property bb⊥ = 0. in the above work the hsl s := s[a; a ≤ b⊥ ∧ bb⊥ = 0] was already computed, but here we briefly reproduce its description, without proofs, to make this paper more self-contained. since the corresponding hl is most probably infinite, we work with an if δ and start our research with the following hypothesis: δ(a⊥/d⊥1 ) < ℵ0 ∧ δ(b ⊥/c⊥1 ) < ℵ0, (1.7) where c1 := d ⊥e⊥, d1 := c ⊥e⊥, and c := a⊥e⊥, d := b⊥e⊥, e := a⊥b⊥. (1.8) the algebraic relations forced by the index condition (1.7) are given below in (4.4), theorem 4.1, and have the following important consequence: f := f[a; (1) ∧ (4.4)] is finite and has 23 subdirectly irreducible factors. the factors are listed in tables ii, iii and iv, section 7, together with the associated critical quotients. we will finally use these results to confirm conjecture 2 in [m2] and to suggest an application in orthogonal geometry. we conclude this introduction with two more remarks: without (1.7), f would be most probably infinite (cf. also the arguments given in [m2]). thus we can recognize the importance of the intervals [d⊥1 , a ⊥] and [c⊥1 , b ⊥] in the above hl. moreover, it is easy to prove that (1.7) is a weakening of the condition δ(1/b) < ℵ0, which has a natural interpretation in orthogonal geometry (cf. section 6) and was used as hypothesis in many precedent investigations. s appeared naturally as substructure in other works (see [m2] and [dm2]). this important fact was an additional motivation for the present study. 2 preliminaries lemma 2.1. any countable hl is indexable. 68 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) proof. each hl admits the trivial if, defined to have value ℵ0 on each nontrivial quotient. lemma 2.2. the class of ihl is closed with respect to subalgebras, homomorphic images and countable products. proof. this is just a slight generalization of proposition 21 in [kkw], ch. iv. clearly, the existence of a nontrivial if on some hl is controlled by prime quotients. our lattices do not present difficulties such as described in [s] because the subdirectly irreducible factors are finite and known. the next result represents the key to obtain algebraic relations from index relations (cf. proof of theorem 4.1): lemma 2.3. let u/v be any finite quotient of an ihl. if v = v⊥⊥ then u = u⊥⊥. proof. δ(u/v) ≥ δ(v⊥/u⊥) ≥ δ(u⊥⊥/v⊥⊥) = δ(u⊥⊥/v) ≥ δ(u/v). for the sake of precision we give also the following definition 2.4. s[a : a ≤ b⊥] is the initial object of the class of rigid hsl. similarly, f[a : a ≤ b⊥] is the initial object of the class of rigid hl. thus any rigid h(s)l is a homomorphic image of the initial object. we could have been even more precise by saying that this is in fact the concept of a 1-generated rigid h(s)l, a special case of n-generated rigid h(s)l, but of course, for the moment, all this is not necessary. we conclude this section by remarking that the axiom (1.1) is equivalent with the following conditions: (i) x ≤ x⊥⊥; (ii) x ≤ y ⇒ y⊥ ≤ x⊥. this may facilitate some computations. 3 description of s theorem 3.1. the hsl s has 18 elements and its structure is given by the diagram depicted in figure 1 (see section 7). proof. see [dm3]. cubo 20, 1 (2018) on rigid hermitean lattices, ii 69 since we are interested in indices, we consider an if δ on s and put β1 := δ(a/0), β2 := δ(b/0), β3 := δ(e/0), β4 := δ(c/c1), β5 := δ(a ⊥⊥/a), β6 := δ(b ⊥⊥/b), β7 := δ(c1/b ⊥⊥), β8 := δ(d1/a ⊥⊥). (3.1) theorem 3.2. (relations among indices in s) (i) all other indices of s are determined by β1, · · · , β8 as is shown in figure 2, section 7. (ii) in particular, the following relations hold: a) β4 6= 0 implies β1 = β2 = ℵ0; b) β5 6= 0 implies β1 = ℵ0; c) β6 6= 0 implies β2 = ℵ0; d) β7 6= 0 or β8 6= 0 implies β1 = β2 = β3 = ℵ0. proof. see [dm3]. remark 3.3. using the above theorem, we find 8 subdirectly irreducible factors of s. they are reproduced in tables i and ii, section 7. 4 description of f remembering (1.8), let us consider the two descending chains {a1, a2, a3} := {a ⊥, d⊥1 , d ⊥} and {b1, b2, b3} := {b ⊥, c⊥1 , c ⊥}. for 1 ≤ i, j ≤ 3 we define aij := ai(bj + e ⊥ ), bij := bj(ai + e ⊥ ), eij := e ⊥ (ai + bj). (4.1) let i1, i2 and i3 be the modular sublattices of f generated by {a⊥, d⊥1 , d ⊥, c, b⊥31, b ⊥ 21, b ⊥ 11, b, e}∪{aij}, {b ⊥, c⊥1 , c ⊥, d, a⊥13, a ⊥ 12, a ⊥ 11, a, e}∪{bij} and {e ⊥, c, c1, b, b ⊥ 31, b ⊥ 21, b ⊥ 11, d, d1, a ⊥ 13, a {eij}, respectively. by the main result in [dm1], i1, i2 and i3 coincide with the principal ideals of f0 :=< i1 ∪ i2 ∪ i3 > generated by a ⊥, b⊥ and e⊥ respectively. moreover, they are distributive and additively generate f0. we want to show that f0 = f. to this end it will be useful to define the following indices: 70 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) αi := βi for i = 1, 2, 3, 4, 5, 6 and further α7 := δ(e ⊥/e11), α8 := δ(c ⊥/b13), α9 := δ(d ⊥/a31), α10 := δ(1/a ⊥ + b⊥ + e⊥), α11 := δ(b ⊥ 11/b ⊥⊥ ), α12 := δ(a ⊥ 11/a ⊥⊥ ), α13 := δ(d ⊥ 1 /a21 + d ⊥ ), α14 := δ(c ⊥ 1 /b12 + c ⊥ ), α15 := δ(b33/d1 + e), α16 := δ(d1/a ⊥ 13), α17 := δ(c1/b ⊥ 31), α18 := δ(a22/a23 + a32), α19 := δ(a ⊥ 12/a ⊥ 11), α20 := δ(a ⊥ 13/a ⊥ 12), α21 := δ(b ⊥ 31/b ⊥ 21), α22 := δ(b23/b33), α23 := δ(a32/a33). (4.2) theorem 4.1. (description of i1, i2 and i3 in f): 1) the plain structure of i1, i2 and i3 is represented by the diagrams depicted in fig 3, fig 4 and fig 5 of section 7. 2) the ideals are connected by the following relations between indices: α4 = δ(d/d1) = δ(c/c1), α11 = δ(b ⊥ 11/b ⊥⊥ ) = δ(b⊥/b⊥⊥11), α12 = δ(a ⊥ 11/a ⊥⊥ ) = δ(a⊥/a⊥⊥11), α15 = δ(b33/d1 + e) = δ(a33/c1 + e) = δ(e33/c1 + d1), α16 = δ(d1/a ⊥ 13) = δ(a13/a23) = δ(b13/b23) = δ(e13/e23), α17 = δ(c1/b ⊥ 31) = δ(a31/a32) = δ(b31/b32) = δ(e31/e32), α18 = δ(a22/a23 + a32) = δ(e22/e23 + e32), (4.3) α19 = δ(a ⊥ 12/a ⊥ 11) = δ(b ⊥ 21/b ⊥ 11) = δ(e11/e12 + e21), α20 = δ(a ⊥ 13/a ⊥ 12) = δ(a12/a13 + a22) = δ(b12/b13 + b22) = δ(e12/e13 + e22), α21 = δ(b ⊥ 31/b ⊥ 21) = δ(a21/a22 + a31) = δ(b21/b22 + b31) = δ(e21/e22 + e31), α22 = δ(b23/b33) = δ(a23/c + a33), α23 = δ(a32/a33) = δ(b32/b33 + d) = δ(e32/d + e33). 3) i1 ∪ i2 ∪ i3 is orthogonally closed in force of the following relations: a11 + d ⊥ 1 = a ⊥⊥ 11, a12 + d ⊥ 1 = a ⊥⊥ 12, a13 + d ⊥ 1 = a ⊥⊥ 13, b11 + c ⊥ 1 = b ⊥⊥ 11, b21 + c ⊥ 1 = b ⊥⊥ 21, b31 + c ⊥ 1 = b ⊥⊥ 31. (4.4) proof. 1) this is routine verification. 2) δ(d/d1) ≥ δ(d ⊥ 1 /d ⊥) ≥ δ(d⊥1 e ⊥/d⊥e⊥) ≥ δ(c/c1) ≥ δ(c ⊥ 1 /c ⊥) ≥ δ(c⊥1 e ⊥/c⊥e⊥) ≥ δ(d/d1). this shows the first equality. the second and third ones are evident. as to the fourth, just consider the free modular lattice generated by the triple (d⊥, c⊥, e⊥). the other equalities are proved analogously. cubo 20, 1 (2018) on rigid hermitean lattices, ii 71 3) we just show the first equality (the others follow in the same manner): δ(a11 + d ⊥ 1 /d ⊥ 1 ) = δ(a11/a11d ⊥ 1 ) = δ(a11/a21) ≤ δ(a ⊥/d⊥1 ) < ℵ0 (by (1.7)). thus a11 + d ⊥ 1 = (a11 + d ⊥ 1 ) ⊥⊥ by lemma 2.3, because d⊥1 = (d ⊥ 1 ) ⊥⊥. since d⊥1 ≤ a ⊥⊥ 11 (because c + e ≤ a11), we obtain the desired equality. the rest is easy and it follows f = f0. theorem 4.2. (forced relations among indices): i) if α7 6= 0 then α1 = α2 = ℵ0; ii) if α8 6= 0 then α1 = α3 = ℵ0; iii) if α9 6= 0 then α2 = α3 = ℵ0; iv) for i ∈ {10, 11, 12, 15, 16, 17, 19}, if αi 6= 0 then α1 = α2 = α3 = ℵ0; v) for i ∈ {13, 14, 18, 20, 21, 22, 23}, if αi 6= 0 then α1 = α2 = α3 = α4 = ℵ0; vi) α11 + α12 + α16 + α17 + α19 + α20 + α21 < ℵ0 proof. each implication follows in a way as was shown in the proof of theorem 3.2, possibly in conjunction with lemma 2.3. the well known rule (x + y)⊥ = x⊥y⊥ may also be useful for computations. the last assertion is just the translation of (1.7) in terms of the indices αi. 5 the subdirectly irreducible factors of f. in order to discover the factors of f it is sufficient to work out i1, i2 and i3 at the same time, using the relations given in theorem 4.1 and theorem 4.2. the essence of the procedure consists in collecting all prime quotients that are connected with a given one via the algebraic operations: this will produce automatically the corresponding subdirectly irreducible factor, together with the associated relation. observe how useful are indices in this procedure: on the one hand they are associated in natural way to congruences, on the other hand the forced relations among them give directly the non minimal congruences in the subdirectly irreducible factors. a little final caution is needed: there is a quotient which does not appear in the ideals, namely 1/(a⊥+b⊥+e⊥) (see the factor corresponding to α9 in table iii). since (a ⊥+b⊥)⊥⊥ = (a⊥+e⊥)⊥⊥ = (e⊥ + b⊥)⊥⊥ = 1 this is the only exception. the factors are labelled from 1 to 23 in tables ii, iii and iv. the last table contains all non distributive members. 72 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) remark 5.1. from all the above results we deduce in particular that conjecture 2 in [m2] is true: in fact, the finite codimensions indicated in the conjecture correspond to the ones given by (1.7). remark 5.2. there are plain lattice isomorphisms between different factors. nevertheless we chose to give explicitly all diagrams, in order to facilitate visualization. it is worth noticing that the majority of this plain isomorphisms are induced by the map a 7→ b and b 7→ a, which defines an involution of s that extends naturally to f. more precisely, there are eight pairs of symmetric factors, namely (1,2), (5,6), (8,9), (11,12), (13,14), (16,17), (20,21) and (2,23), all other factors being self symmetric. 6 remarks concerning applications to hermitean spaces it is possible to prove that all factors of f are implemented by hermitean models. hence they can be used to describe the congruence class of a subspace a in a hermitean space (e, φ) of denumerable dimension under the starting assumptions, where a, e, e∗ correspond to a, 1, b, respectively. in general, these ihl will not suffice to build a complete set of geometric invariants, but they constitute a very important part. details on these aspects cannot be discussed in the present work. 7 diagrams q q q q q q q q q q q q q q q q q q ✟ ✟ ✟ ✟ ✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟✟ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟✟ c d⊥ 1 b⊥⊥ b c1 d⊥ a⊥ 0 e e⊥ 1 a a⊥⊥ b⊥ d1 c⊥ d c⊥ 1 s = s[a; a ≤ b⊥ ∧ bb⊥ = 0] figure 1 q q q q q q q q q q q q q q q q q q ✟ ✟ ✟ ✟ ✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟✟ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟✟β1 β1 β1 β1 β2 β2 β2 β2 β3 β3 β3 β3 β6 β5 β7 β7 β8 β8 β4 β4 β4 β4 s = s[a; a ≤ b⊥ ∧ bb⊥ = 0] figure 2 cubo 20, 1 (2018) on rigid hermitean lattices, ii 73 q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q � � � � � � � � ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ � � � � � ❍ ❍ ❍ ❍❍ � � �� ❍ ❍ ❍❍ � � ❍ ❍ ❍ ❍❍ � � � � � � � � � � � � � � � �� ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ c + e c a13 a12 a11 c1 a23 a⊥⊥ 11 a⊥ b⊥ 31 a22 a⊥⊥ 12 b⊥ 21 a33 a⊥⊥ 13 b⊥ 11 a32 d⊥ 1 = (c + e)⊥⊥ b b⊥⊥ 0 a31 d⊥ = (b + e)⊥⊥ e a21 b + e α6 α2 α3 α4 α9 α11 α12 α13 α15 α16 α17 α17 α18 α19 α19 α20 α21 α21 α22 α23 figure 3: the ideal i1 in f q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ❅ ❅ ✟ ✟✟ ✟ ✟✟ ✟ ✟✟ ❅ ❅ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟✟❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✟ ✟ ✟✟✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟✟ ❅ ❅ e a + e 0 (a + e)⊥⊥ = c⊥ b13 b23 b33 (d + e)⊥⊥ = c⊥ 1 b12 b22 b32 b⊥ b⊥⊥ 11 b⊥⊥ 31 b⊥⊥ 21 b11 b21 b31 d a⊥⊥ a a⊥ 11 a⊥ 12 a⊥ 13 d1 d + e α5 α1α3 α4 α8 α12 α11 α14 α15 α16 α16 α17 α18 α19 α19 α22 α23 α20 α20 α21 figure 4: the ideal i2 in f 74 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r rr r ✟✟ ✟ ✟ ✟ ✟ ✟✟ ❍❍ ❍ ❍ ❍ ❍ ❍❍ ✟✟ ❍❍ ❍ ❍ ❍❍ ✟ ✟ ✟✟ ✟✟ ❍❍ ❍❍ ✟✟ ✟ ✟ ✟✟ ❍ ❍ ❍❍ ✟ ✟ ✟✟ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍❍ ✟ ✟ ✟ ✟ ✟✟ ✟✟❍❍ ❍❍✟✟ ❍ ❍ ❍❍ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ c c1 b⊥ 31 b⊥ 21 b⊥ 11 b⊥⊥ b 0 a⊥⊥ a a⊥ 11 a⊥ 12 a⊥ 13 d1 d e13 e23 e12 e33 e22 e11 e⊥ = (a + b)⊥⊥ e32 e21 e31 α1 α5α6 α2 α7 α4 α4 α12α11 α15 α16 α16 α17 α17 α18 α19α19 α19 α22 α23 α20 α20α21 α21 figure 5: the ideal i3 in f table i r r r r r r b a b⊥a ⊥ � � �� ❅ ❅ ❅❅ ❅ ❅ ❅❅ β4 = δ ( c/c1 ) β4 β2 β1 β1 β4 β2 r r r r r r r r r r b c c⊥ e a b⊥a⊥ e⊥� � � � ❅ ❅ ❅❅ ❅ ❅ ❅❅ � � � � ❅ ❅ ❅❅ ❅ ❅ ❅❅ β7 = δ ( c1/b ⊥⊥ ) β3 β7 β1 β1 β2 β1 β3 β3 β2 β2 β1 β1 β3 β7 r r r r rr r r r r b d⊥ d e a⊥ a e⊥ b⊥ � � �� ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ β8 = δ ( d1/a ⊥⊥ ) β3 β2 β2 β8 β1 β3 β1 β1 β1 β3 β8 β2 β3 β2 table ii cubo 20, 1 (2018) on rigid hermitean lattices, ii 75 r r β1 = α1 = δ (a/0) a b α1 r r β2 = α2 = δ (b/0) b a α2 r r β3 = α3 = δ (e/0) e ba α3 r r r β5 = α5 = δ ( a ⊥⊥ /a ) a⊥ b a⊥⊥ a α1 α5 α1 = ℵ0 r r r β6 = α6 = δ ( b ⊥⊥ /b ) b⊥ a b⊥⊥ b α2 α6 α2 = ℵ0 76 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) table iii r r r r r r r r r α4 = δ ( c/c1 ) e b = c1 a = d1 b⊥a⊥ � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α4 α4 α2 α1 = α2 = ℵ0 r r r r r α7 = δ ( e ⊥ /e11 ) a = b⊥b = a⊥�� ❅❅ ❅❅ α7 α1 α2 α1 = α2 = ℵ0 r r r r r α8 = δ ( c ⊥ /b13 ) b⊥ b a = a⊥⊥a⊥ �� ❅❅ ❅❅ α8 α1 α3 α1 = α3 = ℵ0 r r r r r α9 = δ ( d ⊥ /a31 ) a a⊥ b⊥b⊥⊥ = b�� ❅❅ ❅❅ α9 α3 α2 α2 = α3 = ℵ0 r r r r r r r r r r α10 = δ ( 1 a⊥ + b⊥ + e⊥ ) b⊥e ⊥a⊥ e ab �� ❅❅ ❅❅ �� ❅❅ ❅❅ α1 α2 α3 α10 α1 = α2 = α3 = ℵ0 r r r r r r r r r r r r r r r α11 = δ ( b ⊥ 11/b ⊥⊥ ) b⊥⊥ = b c c ⊥ e a = d b⊥a ⊥ e⊥� � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ �� ❅ ❅ ❅ ❅ ❅ ❅ α1 α11 α11 α2 α3 α1 = α2 = α3 = ℵ0 r r r r r r r r r r r r r r r α12 = δ ( a ⊥ 11/a ) c = b d⊥ = a11 d = a ⊥ 11 e a = a⊥⊥ e⊥ b⊥a⊥ � � � � � � � � � � � � � � � ❅❅ ❅❅ ❅❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α12 α12 α2 α3 α1 = α2 = α3 = ℵ0 r r r r r r r r r r r r r r r r r r r r r α13 = δ   d⊥ 1 a21 + d ⊥   b = c1 d⊥ e a = d1 c⊥ b⊥ a⊥ e⊥ c d � � � � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α4 α13 α4 α2 α3 α1 = α2 = α3 = α4 = ℵ0 r r r r r r r r r r r r r r r r r r r r r α14 = δ   c⊥ 1 b12 + c ⊥   b = c1 d⊥ e a = d1 c⊥ b⊥e⊥ a⊥ c d � � � � � � � � � � � �❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α4 α14 α4 α2 α3 α1 = α2 = α3 = α4 = ℵ0 cubo 20, 1 (2018) on rigid hermitean lattices, ii 77 table iv q q q q q qq q q q q q qq q α15 = δ ( b33 d1 + e ) c = b e a = d = a⊥ 11 b⊥ = b33a ⊥ e⊥ �� ❅❅ ❅❅❅ ❅❅ ❅❅ ❅ ❅❅❅❅❅ ❅❅ α1 = α2 = α3 = ℵ0 α1 α15 α15 α2 α3 q q q q q qq q q q q q qq q q qq q q α16 = δ ( d1/a ⊥ 13 ) c = b e a = a⊥ 13 d = d1 b33 d⊥ b⊥ = b13 a⊥ e⊥ ❅❅ ❅❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅❅ � �� α1 = α2 = α3 = ℵ0 α1 α16 α16 α16 α2 α3 q q q q q q q q q q q q q q q q q q q q α17 = δ ( c1/b ⊥ 31 ) b c⊥ b⊥ a⊥ c e a = d e⊥ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ � �� α1 = α2 = α3 = ℵ0 α1 α17 α17 α17 α2 α3 q q q q q q q q q q q q q q q q q q q q q q q q q q q α18 = δ ( a22 a23 + a32 ) b = c1 d⊥ e a = d1 c⊥ b⊥a⊥ c d e⊥ � � �� ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ � � �� ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ α1 = α2 = α3 = α4 = ℵ0 α1 α4 α18 α18 α4 α2 α3 q q q q q q q q q q q q q q q q q q q q q q q q q q q α19 = δ ( a ⊥⊥ 11/a ⊥⊥ 12 ) b e a b⊥a⊥ d⊥ c1 = c d = d1 e⊥ c⊥ � � �� ❅ ❅ ❅❅❅ ❅❅ ❅ ❅ ❅❅❅❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ α1 = α2 = α3 = ℵ0 α1 α19 α19 α19 α19 α3 α2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q α20 = δ ( a ⊥ 13/a ⊥ 12 ) c1 = b d⊥ e a b⊥ a⊥ c d1 c⊥ d e⊥ d⊥ 1 � � � �� ❅ ❅ ❅❅ ❅ ❅ ❅❅❅ ❅❅ ❅ ❅ ❅❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ α1 = α2 = α3 = α4 = ℵ0 α4 α20 α20 α3 α1 α4 α2α20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q α21 = δ ( b ⊥ 31/b ⊥ 21 ) c1 d⊥ b e a = d1 c⊥ b⊥ a⊥ c d e⊥ c⊥ 1 � � �� ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅❅ ❅❅ ❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅❅ α1 = α2 = α3 = α4 = ℵ0 α4 α21α21 α4 α3 α21 α2 α1 q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q α22 = δ ( b23/b33 ) a = d1 b⊥a⊥ c⊥ d⊥ ec1 = b dc e⊥ ✑ ✑ ✑✑❅ ❅❅ ❅ ❅❅ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ✑✑ ✑✑❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ✑✑ ✑✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑✑ ✑ ✑✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑✑ ✑ ✑✑ ✑✑ ✑✑ α1 = α2 = α3 = α4 = ℵ0 α1 α4 α22 α4 α2 α3 α22 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q α23 = δ ( a32/a33 ) a = d1 b⊥ a⊥ c⊥ d⊥ ec1 = b dc e⊥✪ ✪✪ ✪ ✪✪ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍❍ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑✑✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑✑ α1 = α2 = α3 = α4 = ℵ0 α1 α4 α23 α4 α2 α23 α3 references [dm1] a.c. de la maza, r.moresi, on modular lattices generated by chains, algebra universalis 54 (2005), 475-488. [dm2] a.c. de la maza, r.moresi, hermitean (semi) lattices and rolf’s lattice, algebra universalis 66 (2011), 49-62. [dm3] a.c. de la maza, r.moresi, on rigid hermitean lattices, i, preprint. [g1] h. gross, quadratic forms in infinite dimensional vector spaces, birkäuser, boston, 1979. [g2] h. gross, lattices and infinite-dimensional forms. “the lattice method”, order 4 (1987), 233-256. 78 ana cecilia de la maza and remo moresi cubo 20, 1 (2018) [kkw] h. a. keller, u.-m. künzi, m. wild (eds), orthogonal geometry in infinite dimensional vector spaces, heft 53, bayreuther mathematische schriften, bayreuth, 1998. [m1] r. moresi, modular lattices and hermitean forms, algebra universalis 22 (1986), 279-297. [m2] r. moresi, a test-example of a quadratic lattice, order 17 (2000), 215-226. [r] h. l. rolf, the free lattice generated by a set of chains, pacific j. math. 8 (1958), 585-595. [s] e. t. schmidt, on finitely generated simple modular lattice, periodica mathematica hungarica 6(3) (1975), 213-216. introduction preliminaries description of s description of f the subdirectly irreducible factors of f. remarks concerning applications to hermitean spaces diagrams cubo, a mathematical journal vol. 23, no. 03, pp. 423–440, december 2021 doi: 10.4067/s0719-06462021000300423 foundations of generalized prabhakar-hilfer fractional calculus with applications george a. anastassiou1 1 department of mathematical sciences university of memphis, memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract here we introduce the generalized prabhakar fractional calculus and we also combine it with the generalized hilfer calculus. we prove that the generalized left and right side prabhakar fractional integrals preserve continuity and we find tight upper bounds for them. we present several left and right side generalized prabhakar fractional inequalities of hardy, opial and hilbert-pachpatte types. we apply these in the setting of generalized hilfer calculus. resumen introducimos el cálculo fraccionario generalizado de prabhakar y también lo combinamos con el cálculo generalizado de hilfer. demostramos que las integrales fraccionarias generalizadas de prabhakar izquierda y derecha preservan la continuidad y encontramos cotas superiores ajustadas para ellas. presentamos diversas desigualdades fraccionarias generalizadas de prabhakar izquierda y derecha de tipos hardy, opial y hilbert-pachpatte. aplicamos estos resultados en el contexto del cálculo generalizado de hilfer. keywords and phrases: prabhakar fractional calculus, hilfer fractional calculus, fractional hardy, opial and hilbert-pachpatte inequalities. 2020 ams mathematics subject classification: 26a33, 26d10, 26d15. accepted: 13 september, 2021 received: 08 april, 2021 ©2021 george a. anastassiou. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300423 https://orcid.org/0000-0002-3781-9824 424 george a. anastassiou cubo 23, 3 (2021) 1 background during the last 50 years fractional calculus due to its wide applications to many applied sciences has become a main trend in mathematics. its predominant kinds are the old riemann-liouville fractional calculus and the newer one of caputo type. around these two versions have been built a plethora of other variants and all of these involve singular kernels. more recently researchers presented also new fractional calculi of non singular kernels. the recent hilfer fractional calculus unifies the riemann-liouville and caputo fractional calculi and the prabhakar fractional calculus unifies both singular and non-singular kernel fractional calculi. finally the newer hilfer-prabhakar fractional calculus is the most general one unifying all trends and for different values of its parameters we get the particular fractional calculi. in this article we present and employ unifying advanced and generalized versions of prabhakar and hilfer-prabhakar fractional calculi and we establish related unifying fractional integral inequalities of the following types: hardy, opial and hilbert-pachpatte. the advantage of this unification is the uniform action taken in describing the various natural phenomena. we are inspired by [7], [6] and [1]. we start by introducing our own generalized ψ-prabhakar type of fractional calculus, then mixing it with the ψ-hilfer fractional calculus. then, we prove a variety of generalized hardy, opial and hilbert-pachpatte type left and right fractional integral inequalities related to ψ-hilfer ([8]) and ψ-prabhakar fractional calculi. we involve several functions. we consider the prabhakar function (also known as the three parameter mittag-leffler function), (see [4, p. 97]; [3]) e γ α,β (z) = ∞∑ k=0 (γ)k k!γ (αk + β) zk, (1.1) where γ is the gamma function; α,β,γ ∈ r : α,β > 0, z ∈ r, and (γ)k = γ (γ + 1) · · · (γ + k − 1). it is e0α,β (z) = 1 γ (β) . let a,b ∈ r, a < b and x ∈ [a,b]; f ∈ c ([a,b]) . let also ψ ∈ c1 ([a,b]) which is increasing. the left and right prabhakar fractional integrals with respect to ψ are defined as follows:( e γ;ψ ρ,µ,ω,a+f ) (x) = ∫ x a ψ′ (t) (ψ (x) − ψ (t))µ−1 eγρ,µ [ω (ψ (x) − ψ (t)) ρ ]f (t)dt, (1.2) and ( e γ;ψ ρ,µ,ω,b−f ) (x) = ∫ b x ψ′ (t) (ψ (t) − ψ (x))µ−1 eγρ,µ [ω (ψ (t) − ψ (x)) ρ ]f (t)dt, (1.3) where ρ,µ > 0; γ,ω ∈ r. functions (1.2) and (1.3) are continuous, see theorem 3.1. next, additionally, assume that ψ′ (x) ̸= 0 over [a,b] . let ψ,f ∈ cn ([a,b]), where n = ⌈µ⌉, (⌈·⌉ is the ceiling of the number), 0 < µ /∈ n. we define the cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 425 ψ-prabhakar-caputo left and right fractional derivatives of order µ as follows (x ∈ [a,b]):( cd γ;ψ ρ,µ,ω,a+f ) (x) = ∫ x a ψ′ (t) (ψ (x) − ψ (t))n−µ−1 e −γ ρ,n−µ [ω (ψ (x) − ψ (t)) ρ ] ( 1 ψ′ (t) d dt )n f (t)dt, (1.4) and ( cd γ;ψ ρ,µ,ω,b−f ) (x) = (−1)n ∫ b x ψ′ (t) (ψ (t) − ψ (x))n−µ−1 e −γ ρ,n−µ [ω (ψ (t) − ψ (x)) ρ ] ( 1 ψ′ (t) d dt )n f (t)dt. (1.5) one can write (see (1.4), (1.5))( cd γ;ψ ρ,µ,ω,a+f ) (x) = ( e −γ;ψ ρ,n−µ,ω,a+f [n] ψ ) (x) , (1.6) and ( cd γ;ψ ρ,µ,ω,b−f ) (x) = (−1)n ( e −γ;ψ ρ,n−µ,ω,b−f [n] ψ ) (x) , (1.7) where f [n] ψ (x) = f (n) ψ f (x) := ( 1 ψ′ (x) d dx )n f (x) , (1.8) ∀ x ∈ [a,b]. functions (1.6) and (1.7) are continuous on [a,b]. next we define the ψ-prabhakar-riemann-liouville left and right fractional derivatives of order µ as follows (x ∈ [a,b]): ( rld γ;ψ ρ,µ,ω,a+f ) (x) = ( 1 ψ′ (x) d dx )n ∫ x a ψ′ (t) (ψ (x) − ψ (t))n−µ−1 e −γ ρ,n−µ [ω (ψ (x) − ψ (t)) ρ ]f (t)dt, (1.9) and ( rld γ;ψ ρ,µ,ω,b−f ) (x) = ( − 1 ψ′ (x) d dx )n ∫ b x ψ′ (t) (ψ (t) − ψ (x))n−µ−1 e −γ ρ,n−µ [ω (ψ (t) − ψ (x)) ρ ]f (t)dt. (1.10) that is we have ( rld γ;ψ ρ,µ,ω,a+f ) (x) = ( 1 ψ′ (x) d dx )n ( e −γ;ψ ρ,n−µ,ω,a+f ) (x) , (1.11) and ( rld γ;ψ ρ,µ,ω,b−f ) (x) = ( − 1 ψ′ (x) d dx )n ( e −γ;ψ ρ,n−µ,ω,b−f ) (x) , (1.12) ∀ x ∈ [a,b]. 426 george a. anastassiou cubo 23, 3 (2021) we define also the ψ-hilfer-prabhakar left and right fractional derivatives of order µ and type 0 ≤ β ≤ 1, as follows( hdγ,β;ψρ,µ,ω,a+f ) (x) = e −γβ;ψ ρ,β(n−µ),ω,a+ ( 1 ψ′ (x) d dx )n e −γ(1−β);ψ ρ,(1−β)(n−µ),ω,a+f (x) , (1.13) and ( hdγ,β;ψρ,µ,ω,b−f ) (x) = e −γβ;ψ ρ,β(n−µ),ω,b− ( − 1 ψ′ (x) d dx )n e −γ(1−β);ψ ρ,(1−β)(n−µ),ω,b−f (x) , (1.14) ∀ x ∈ [a,b]. when β = 0, we get the riemann-liouville version, and when β = 1, we get the caputo version. we call ξ = µ + β (n − µ), we have that n − 1 < µ ≤ µ + β (n − µ) ≤ µ + n − µ = n, hence ⌈ξ⌉ = n. we can easily write that( hdγ,β;ψρ,µ,ω,a+f ) (x) = e −γβ;ψ ρ,ξ−µ,ω,a+ rld γ(1−β);ψ ρ,ξ,ω,a+ f (x) , (1.15) and ( hdγ,β;ψρ,µ,ω,b−f ) (x) = e −γβ;ψ ρ,ξ−µ,ω,b− rld γ(1−β);ψ ρ,ξ,ω,b− f (x) , (1.16) ∀ x ∈ [a,b]. 2 main results we start with a left ψ-prabhakar fractional hardy type integral inequality involving several functions. theorem 2.1. here i = 1, . . . ,m; fi ∈ c ([a,b]), ψ ∈ c1 ([a,b]) and ψ is increasing. let ρi,µi > 0, γi,ωi ∈ r. also let r1,r2,r3 > 1 : 1r1 + 1 r2 + 1 r3 = 1, and assume that µi > 1r2 + 1 r3 , for all i = 1, . . . ,m. then ∥∥∥∥∥ m∏ i=1 e γi;ψ ρi,µi,ωi,a+ fi ∥∥∥∥∥ lr1 ([a,b],ψ) ≤ (ψ (b) − ψ (a)) [ m∑ i=1 µi−m+ mr1 + 1 r1 − 1 r1r2 ] ( r1r3 ( m∑ i=1 µi − m ) + mr3 + 1 ) 1 r1r3 ( m∏ i=1 (r1 (µi − 1) + 1) ) 1 r1 {∫ b a [ m∏ i=1 (∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) )]r1 dψ (x) } 1 r1r2 ( m∏ i=1 ∥fi∥lr3 ([a,b],ψ) ) . (2.1) cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 427 proof. by (1.2) we have( e γi;ψ ρi,µi,ωi,a+ fi ) (x) = ∫ x a ψ′ (t) (ψ (x) − ψ (t))µi−1 eγiρi,µi [ωi (ψ (x) − ψ (t)) ρi]fi (t)dt, (2.2) i = 1, . . . ,m; ∀ x ∈ [a,b]. by hölder’s inequality and (2.2) we obtain∣∣∣(eγi;ψρi,µi,ωi,a+fi)(x)∣∣∣ ≤∫ x a ψ′ (t) (ψ (x) − ψ (t))µi−1 ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣ |fi (t)|dt ≤(∫ x a (ψ (x) − ψ (t))r1(µi−1) dψ (t) ) 1 r1 (∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) ) 1 r2 (∫ x a |fi (t)| r3 dψ (t) ) 1 r3 ≤ (2.3) (ψ (x) − ψ (a))µi−1+ 1 r1 (r1 (µi − 1) + 1) 1 r1(∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) ) 1 r2 ∥fi∥lr3 ([a,b],ψ) . so far we have ∣∣∣(eγi;ψρi,µi,ωi,a+fi)(x)∣∣∣ ≤ (ψ (x) − ψ (a))µi−1+ 1 r1 (r1 (µi − 1) + 1) 1 r1(∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) ) 1 r2 ∥fi∥lr3 ([a,b],ψ) , (2.4) ∀ x ∈ [a,b], with µi > 1r2 + 1 r3 , for any i = 1, . . . ,m. hence it holds ( m∏ i=1 ∣∣∣(eγi;ψρi,µi,ωi,a+fi)(x)∣∣∣ )r1 ≤ (ψ (x) − ψ (a)) r1 m∑ i=1 µi−mr1+m( m∏ i=1 (r1 (µi − 1) + 1) ) [ m∏ i=1 (∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) )]r1r2 ( m∏ i=1 ∥fi∥lr3 ([a,b],ψ) )r1 , (2.5) ∀ x ∈ [a,b] . therefore we obtain ∫ b a ( m∏ i=1 ∣∣∣(eγi;ψρi,µi,ωi,a+fi)(x)∣∣∣ )r1 dψ (x) ≤ ( m∏ i=1 ∥fi∥lr3 ([a,b],ψ) )r1 ( m∏ i=1 (r1 (µi − 1) + 1) ) (2.6) [∫ b a (ψ (x) − ψ (a)) r1 m∑ i=1 µi−mr1+m 428 george a. anastassiou cubo 23, 3 (2021) [ m∏ i=1 (∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) )]r1r2 dψ (x)   (again by hölder’s inequality) ≤ ( m∏ i=1 ∥fi∥lr3 ([a,b],ψ) )r1 ( m∏ i=1 (r1 (µi − 1) + 1) ) (∫ b a (ψ (x) − ψ (a)) r1r3 m∑ i=1 µi−mr1r3+mr3 dψ (x) ) 1 r3 {∫ b a [ m∏ i=1 (∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) )]r1 dψ (x) } 1 r2 (ψ (b) − ψ (a)) 1 r1 = ( m∏ i=1 ∥fi∥lr3 ([a,b],ψ) )r1 (ψ (b) − ψ (a)) r1 m∑ i=1 µi−mr1+m+1− 1r2 ( m∏ i=1 (r1 (µi − 1) + 1) )( r1r3 m∑ i=1 µi − mr1r3 + mr3 + 1 ) 1 r3 (2.7) {∫ b a [ m∏ i=1 (∫ x a ∣∣eγiρi,µi [ωi (ψ (x) − ψ (t))ρi]∣∣r2 dψ (t) )]r1 dψ (x) } 1 r2 , where µi > 1r2 + 1 r3 , i = 1, . . . ,m. the claim is proved. we continue with a right ψ-prabhakar fractional hardy type integral inequality involving several functions. theorem 2.2. all as in theorem 2.1. it holds∥∥∥∥∥ m∏ i=1 e γi;ψ ρi,µi,ωi,b−fi ∥∥∥∥∥ lr1 ([a,b],ψ) ≤ (ψ (b) − ψ (a)) [ m∑ i=1 µi−m+ mr1 + 1 r1 − 1 r1r2 ] ( r1r3 ( m∑ i=1 µi − m ) + mr3 + 1 ) 1 r1r3 ( m∏ i=1 (r1 (µi − 1) + 1) ) 1 r1 {∫ b a [ m∏ i=1 (∫ b x ∣∣eγiρi,µi [ωi (ψ (t) − ψ (x))ρi]∣∣r2 dψ (t) )]r1 dψ (x) } 1 r1r2 ( m∏ i=1 ∥fi∥lr3 ([a,b],ψ) ) . (2.8) proof. similar to the proof of theorem 2.1 and omitted. next we apply theorems 2.1, 2.2. we give the related hardy type inequalities: cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 429 theorem 2.3. here i = 1, . . . ,m; fi ∈ cni ([a,b]), where ni = ⌈µi⌉, 0 < µi /∈ n; θ := max {n1, . . . ,nm} , ψ ∈ cθ ([a,b]) with ψ′ ̸= 0 and increasing. let ρi > 0, γi,ωi ∈ r. also let r1,r2,r3 > 1 : 1r1 + 1 r2 + 1 r3 = 1, and assume that ni − µi > 1r2 + 1 r3 , for all i = 1, . . . ,m. then i) ∥∥∥∥∥ m∏ i=1 cd γi;ψ ρi,µi,ωi,a+ fi ∥∥∥∥∥ lr1 ([a,b],ψ) ≤ (ψ (b) − ψ (a)) [ m∑ i=1 (ni−µi)−m+ mr1 + 1 r1 − 1 r1r2 ] ( r1r3 ( m∑ i=1 (ni − µi) − m ) + mr3 + 1 ) 1 r1r3 ( m∏ i=1 (r1 (ni − µi − 1) + 1) ) 1 r1 {∫ b a [ m∏ i=1 (∫ x a ∣∣∣e−γiρi,ni−µi [ωi (ψ (x) − ψ (t))ρi]∣∣∣r2 dψ (t) )]r1 dψ (x) } 1 r1r2 ( m∏ i=1 ∥∥∥f[ni]iψ ∥∥∥ lr3 ([a,b],ψ) ) , (2.9) and ii) ∥∥∥∥∥ m∏ i=1 cd γi;ψ ρi,µi,ωi,b−fi ∥∥∥∥∥ lr1 ([a,b],ψ) ≤ (ψ (b) − ψ (a)) [ m∑ i=1 (ni−µi)−m+ mr1 + 1 r1 − 1 r1r2 ] ( r1r3 ( m∑ i=1 (ni − µi) − m ) + mr3 + 1 ) 1 r1r3 ( m∏ i=1 (r1 (ni − µi − 1) + 1) ) 1 r1 {∫ b a [ m∏ i=1 (∫ b x ∣∣∣e−γiρi,ni−µi [ωi (ψ (t) − ψ (x))ρi]∣∣∣r2 dψ (t) )]r1 dψ (x) } 1 r1r2 ( m∏ i=1 ∥∥∥f[ni]iψ ∥∥∥ lr3 ([a,b],ψ) ) . (2.10) proof. by (1.6), (1.7) and theorems 2.1, 2.2. we also present other hardy type related inequalities: theorem 2.4. here i = 1, . . . ,m; fi ∈ cni ([a,b]), where ni = ⌈µi⌉, 0 < µi /∈ n; θ := max {n1, . . . ,nm} , ψ ∈ cθ ([a,b]) , ψ′ ̸= 0, and ψ is increasing. let ρi > 0, γi,ωi ∈ r, 0 ≤ βi ≤ 1, ξi = µi +βi (ni − µi) . also let r1,r2,r3 > 1 : 1r1 + 1 r2 + 1 r3 = 1, and assume that ξi −µi > 1r2 + 1 r3 , for all i = 1, . . . ,m. also assume that rldγi(1−βi);ψρi,ξi,ωi,a+fi, rld γi(1−βi);ψ ρi,ξi,ωi,b−fi ∈ c ([a,b]), i = 1, . . . ,m. then 430 george a. anastassiou cubo 23, 3 (2021) i) ∥∥∥∥∥ m∏ i=1 hdγi,βi;ψρi,µi,ωi,a+fi ∥∥∥∥∥ lr1 ([a,b],ψ) ≤ (ψ (b) − ψ (a)) [ m∑ i=1 (ξi−µi)−m+ mr1 + 1 r1 − 1 r1r2 ] ( r1r3 ( m∑ i=1 (ξi − µi) − m ) + mr3 + 1 ) 1 r1r3 ( m∏ i=1 (r1 (ξi − µi − 1) + 1) ) 1 r1 (2.11) {∫ b a [ m∏ i=1 (∫ x a ∣∣∣e−γiβiρi,ξi−µi [ωi (ψ (x) − ψ (t))ρi]∣∣∣r2 dψ (t) )]r1 dψ (x) } 1 r1r2 ( m∏ i=1 ∥∥∥rldγi(1−βi);ψρi,ξi,ωi,a+fi∥∥∥lr3 ([a,b],ψ) ) , and ii) ∥∥∥∥∥ m∏ i=1 hdγi,βi;ψρi,µi,ωi,b−fi ∥∥∥∥∥ lr1 ([a,b],ψ) ≤ (ψ (b) − ψ (a)) [ m∑ i=1 (ξi−µi)−m+ mr1 + 1 r1 − 1 r1r2 ] ( r1r3 ( m∑ i=1 (ξi − µi) − m ) + mr3 + 1 ) 1 r1r3 ( m∏ i=1 (r1 (ξi − µi − 1) + 1) ) 1 r1 (2.12) {∫ b a [ m∏ i=1 (∫ b x ∣∣∣e−γiβiρi,ξi−µi [ωi (ψ (t) − ψ (x))ρi]∣∣∣r2 dψ (t) )]r1 dψ (x) } 1 r1r2 ( m∏ i=1 ∥∥∥rldγi(1−βi);ψρi,ξi,ωi,b−fi∥∥∥lr3 ([a,b],ψ) ) . proof. by (1.15), (1.16) and theorems 2.1, 2.2. from now on all entities are according and respectively to section 1. background. next we give opial type inequalities related to prabhakar fractional calculus. a left side one follows: theorem 2.5. let p,q > 1 : 1 p + 1 q = 1. then∫ x a ∣∣∣(eγ;ψρ,µ,ω,a+f)(w)∣∣∣ |f (w)|ψ′ (w)dw ≤ 2− 1q [∫ x a {∫ w a (ψ (w) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (w) − ψ (t))ρ]∣∣p dt}dw] 1 p (∫ x a |f (w)|q (ψ′ (w))q dw )2 q , (2.13) ∀ x ∈ [a,b] . cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 431 proof. by (1.2), using hölder’s inequality, we have∣∣∣(eγ;ψρ,µ,ω,a+f)(x)∣∣∣ ≤ ∫ x a ψ′ (t) (ψ (x) − ψ (t))µ−1 ∣∣eγρ,µ [ω (ψ (x) − ψ (t))ρ]∣∣ |f (t)|dt ≤ (∫ x a (ψ (x) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (x) − ψ (t))ρ]∣∣p dt) 1 p (∫ x a (ψ′ (t) |f (t)|)q dt )1 q . (2.14) call ϕ(x) = ∫ x a (ψ′ (t) |f (t)|)q dt, ϕ(a) = 0. (2.15) thus ϕ′ (x) = (ψ′ (x) |f (x)|)q ≥ 0, (2.16) and (ϕ′ (x)) 1 q = ψ′ (x) |f (x)| ≥ 0, ∀ x ∈ [a,b] . consequently, we get ∣∣∣(eγ;ψρ,µ,ω,a+f)(w)∣∣∣ψ′ (w) |f (w)| ≤(∫ w a (ψ (w) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (w) − ψ (t))ρ]∣∣p dt) 1 p (ϕ(w)ϕ′ (w)) 1 q , ∀ w ∈ [a,b] . (2.17) then, by applying again hölder’s inequality:∫ x a ∣∣∣(eγ;ψρ,µ,ω,a+f)(w)∣∣∣ |f (w)|ψ′ (w)dw ≤ (2.18) ∫ x a {∫ w a (ψ (w) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (w) − ψ (t))ρ]∣∣p dt} 1 p (ϕ(w)ϕ′ (w)) 1 q dw ≤[∫ x a {∫ w a (ψ (w) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (w) − ψ (t))ρ]∣∣p dt}dw] 1 p (∫ x a ϕ(w)dϕ(w) )1 q = [∫ x a {∫ w a (ψ (w) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (w) − ψ (t))ρ]∣∣p dt}dw] 1 p ( ϕ2 (x) 2 )1 q = 2− 1 q [∫ x a {∫ w a (ψ (w) − ψ (t))p(µ−1) ∣∣eγρ,µ [ω (ψ (w) − ψ (t))ρ]∣∣p dt}dw] 1 p (∫ x a (ψ′ (w) |f (w)|)q dw )2 q . (2.19) the theorem is proved. 432 george a. anastassiou cubo 23, 3 (2021) the right side opial inequality follows: theorem 2.6. let p,q > 1 : 1 p + 1 q = 1. then ∫ b x ∣∣∣(eγ;ψρ,µ,ω,b−f)(w)∣∣∣ |f (w)|ψ′ (w)dw ≤ 2− 1q [∫ b x {∫ b w (ψ (t) − ψ (w))p(µ−1) ∣∣eγρ,µ [ω (ψ (t) − ψ (w))ρ]∣∣p dt } dw ]1 p (∫ b x |f (w)|q (ψ′ (w))q dw )2 q , (2.20) ∀ x ∈ [a,b] . proof. as it is similar to the proof of theorem 2.5, is omitted. we continue with more interesting opial type prabhakar-caputo fractional inequalities: theorem 2.7. let p,q > 1 : 1 p + 1 q = 1. then i) ∫ x a ∣∣∣(cdγ;ψρ,µ,ω,a+f)(w)∣∣∣ ∣∣∣f[n]ψ (w)∣∣∣ψ′ (w)dw ≤ 2− 1q [∫ x a {∫ w a (ψ (w) − ψ (t))p(n−µ−1) ∣∣∣e−γρ,n−µ [ω (ψ (w) − ψ (t))ρ]∣∣∣p dt } dw ]1 p (∫ x a ∣∣∣f[n]ψ (w)∣∣∣q (ψ′ (w))q dw )2 q , (2.21) and ii) ∫ b x ∣∣∣(cdγ;ψρ,µ,ω,b−f)(w)∣∣∣ ∣∣∣f[n]ψ (w)∣∣∣ψ′ (w)dw ≤ 2− 1q [∫ b x {∫ b w (ψ (t) − ψ (w))p(n−µ−1) ∣∣∣e−γρ,n−µ [ω (ψ (t) − ψ (w))ρ]∣∣∣p dt } dw ]1 p (∫ b x ∣∣∣f[n]ψ (w)∣∣∣q (ψ′ (w))q dw )2 q , (2.22) ∀ x ∈ [a,b] . proof. by theorems 2.5, 2.6 and (1.6)-(1.8). next come ψ-hilfer-prabhakar left and right opial type fractional inequalities: cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 433 theorem 2.8. let p,q > 1 : 1 p + 1 q = 1. additionally here assume that rld γ(1−β);ψ ρ,ξ,ω,a+ f, rl d γ(1−β);ψ ρ,ξ,ω,b− f ∈ c ([a,b]) . then i) ∫ x a ∣∣∣(hdγ,β;ψρ,µ,ω,a+f)(w)∣∣∣ ∣∣∣(rldγ(1−β);ψρ,ξ,ω,a+ f)(w)∣∣∣ψ′ (w)dw ≤ 2− 1q[∫ x a {∫ w a (ψ (w) − ψ (t))p(ξ−µ−1) ∣∣∣e−γβρ,ξ−µ [ω (ψ (w) − ψ (t))ρ]∣∣∣p dt } dw ]1 p (∫ x a ∣∣∣(rldγ(1−β);ψρ,ξ,ω,a+ f)(w)∣∣∣q (ψ′ (w))q dw )2 q , (2.23) and ii) ∫ b x ∣∣∣(hdγ,β;ψρ,µ,ω,b−f)(w)∣∣∣ ∣∣∣(rldγ(1−β);ψρ,ξ,ω,b− f)(w)∣∣∣ψ′ (w)dw ≤ 2− 1q[∫ b x {∫ b w (ψ (t) − ψ (w))p(ξ−µ−1) ∣∣∣e−γβρ,ξ−µ [ω (ψ (t) − ψ (w))ρ]∣∣∣p dt } dw ]1 p (∫ b x ∣∣∣(rldγ(1−β);ψρ,ξ,ω,b− f)(w)∣∣∣q (ψ′ (w))q dw )2 q , (2.24) ∀ x ∈ [a,b] . proof. by theorems 2.5, 2.6 and (1.15), (1.16). next we give several prabhakar hilbert-pachpatte fractional inequalities. we start with a left side one. theorem 2.9. let p,q > 1 : 1 p + 1 q = 1; i = 1,2. let [ai,bi] ⊂ r, ψi ∈ c1 ([ai,bi]) and strictly increasing, fi ∈ c ([ai,bi]); ρi,µi > 0, γi,ωi ∈ r. then ∫ b1 a1 ∫ b2 a2 ∣∣∣(eγ1;ψ1ρ1,µ1,ω1,a1+f1)(x1)∣∣∣ ∣∣∣(eγ2;ψ2ρ2,µ2,ω2,a2+f2)(x2)∣∣∣dx1dx2  [∫ x1 a1 {(ψ1(x1)−ψ1(t1))µ1−1|eγ1ρ1,µ1 [ω1(ψ1(x1)−ψ1(t1)) ρ1 ]|}pdt1 ] p +[∫ x2 a2 {(ψ2(x2)−ψ2(t2))µ2−1|eγ2ρ2,µ2 [ω2(ψ2(x2)−ψ2(t2)) ρ2 ]|}qdt2 ] q   ≤ (b1 − a1) (b2 − a2) ∥ψ′1f1∥q ∥ψ ′ 2f2∥p . (2.25) proof. we have that (i = 1,2) ( e γi;ψi ρi,µi,ωi,ai+ fi ) (xi) (1.2) =∫ xi ai ψ′i (ti) (ψi (xi) − ψi (ti)) µi−1 eγiρi,µi [ωi (ψi (xi) − ψi (ti)) ρi]fi (ti)dti, (2.26) 434 george a. anastassiou cubo 23, 3 (2021) ∀ xi ∈ [ai,bi], where ρi,µi > 0; γi,ωi ∈ r. then ∣∣∣(eγi;ψiρi,µi,ωi,ai+fi)(xi)∣∣∣ ≤∫ xi ai ψ′i (ti) (ψi (xi) − ψi (ti)) µi−1 ∣∣eγiρi,µi [ωi (ψi (xi) − ψi (ti))ρi]∣∣ |fi (ti)|dti, (2.27) i = 1,2, ∀ xi ∈ [ai,bi] . by appying hölder’s inequality twice we get:∣∣∣(eγ1;ψ1ρ1,µ1,ω1,a1+f1)(x1)∣∣∣ ≤ [∫ x1 a1 { (ψ1 (x1) − ψ1 (t1)) µ1−1 ∣∣eγ1ρ1,µ1 [ω1 (ψ1 (x1) − ψ1 (t1))ρ1]∣∣}p dt1 ]1 p (∫ x1 a1 (ψ′1 (t1) |f1 (t1)|) q dt1 )1 q , (2.28) ∀ x1 ∈ [a1,b1] , and ∣∣∣(eγ2;ψ2ρ2,µ2,ω2,a2+f2)(x2)∣∣∣ ≤[∫ x2 a2 { (ψ2 (x2) − ψ2 (t2)) µ2−1 ∣∣eγ2ρ2,µ2 [ω2 (ψ2 (x2) − ψ2 (t2))ρ2]∣∣}q dt2 ]1 q (∫ x2 a2 (ψ′2 (t2) |f2 (t2)|) p dt2 )1 p , (2.29) ∀ x2 ∈ [a2,b2] . hence we have (by (2.28), (2.29))∣∣∣(eγ1;ψ1ρ1,µ1,ω1,a1+f1)(x1)∣∣∣ ∣∣∣(eγ2;ψ2ρ2,µ2,ω2,a2+f2)(x2)∣∣∣ ≤ [∫ x1 a1 { (ψ1 (x1) − ψ1 (t1)) µ1−1 ∣∣eγ1ρ1,µ1 [ω1 (ψ1 (x1) − ψ1 (t1))ρ1]∣∣}p dt1 ]1 p [∫ x2 a2 { (ψ2 (x2) − ψ2 (t2)) µ2−1 ∣∣eγ2ρ2,µ2 [ω2 (ψ2 (x2) − ψ2 (t2))ρ2]∣∣}q dt2 ]1 q ∥ψ′1f1∥q ∥ψ ′ 2f2∥p ≤ (2.30) (using young’s inequality for a,b ≥ 0, a 1 p b 1 q ≤ a p + b q )   [∫x1 a1 { (ψ1 (x1) − ψ1 (t1)) µ1−1 ∣∣eγ1ρ1,µ1 [ω1 (ψ1 (x1) − ψ1 (t1))ρ1]∣∣}p dt1] p + [∫x2 a2 { (ψ2 (x2) − ψ2 (t2)) µ2−1 ∣∣eγ2ρ2,µ2 [ω2 (ψ2 (x2) − ψ2 (t2))ρ2]∣∣}q dt2] q   ∥ψ′1f1∥q ∥ψ ′ 2f2∥p , cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 435 ∀ xi ∈ [ai,bi] , i = 1,2. so far we have∣∣∣(eγ1;ψ1ρ1,µ1,ω1,a1+f1)(x1)∣∣∣ ∣∣∣(eγ2;ψ2ρ2,µ2,ω2,a2+f2)(x2)∣∣∣  [∫ x1 a1 {(ψ1(x1)−ψ1(t1))µ1−1|eγ1ρ1,µ1 [ω1(ψ1(x1)−ψ1(t1)) ρ1 ]|}pdt1 ] p +[∫ x2 a2 {(ψ2(x2)−ψ2(t2))µ2−1|eγ2ρ2,µ2 [ω2(ψ2(x2)−ψ2(t2)) ρ2 ]|}qdt2 ] q   ≤ ∥ψ′1f1∥q ∥ψ ′ 2f2∥p , (2.31) ∀ xi ∈ [ai,bi] , i = 1,2. the denominator in (2.31) can be zero only when x1 = a1 and x2 = a2. therefore we obtain (2.25) by integrating (2.31) over [a1,b1] × [a2,b2] . it follows the corresponding to (2.25) right side inequality. theorem 2.10. all as in theorem 2.9. then ∫ b1 a1 ∫ b2 a2 ∣∣∣(eγ1;ψ1ρ1,µ1,ω1,b1−f1)(x1)∣∣∣ ∣∣∣(eγ2;ψ2ρ2,µ2,ω2,b2−f2)(x2)∣∣∣dx1dx2  [∫ b1 x1 {(ψ1(t1)−ψ1(x1))µ1−1|eγ1ρ1,µ1 [ω1(ψ1(t1)−ψ1(x1)) ρ1 ]|}pdt1 ] p +[∫ b2 x2 {(ψ2(t2)−ψ2(x2))µ2−1|eγ2ρ2,µ2 [ω2(ψ2(t2)−ψ2(x2)) ρ2 ]|}qdt2 ] q   ≤ (b1 − a1) (b2 − a2) ∥ψ′1f1∥q ∥ψ ′ 2f2∥p . (2.32) proof. as similar to the proof of theorem 2.9 is omitted. we continue with applications of theorems 2.9, 2.10. theorem 2.11. let p,q > 1 : 1 p + 1 q = 1; i = 1,2. let [ai,bi] ⊂ r, ψi ∈ cmax(n1,n2) ([ai,bi]) , ψ′i ̸= 0, and strictly increasing; fi ∈ c ni ([ai,bi]), where ni = ⌈µi⌉, 0 < µi /∈ n. here ρi > 0; γi,ωi ∈ r. then ∫ b1 a1 ∫ b2 a2 ∣∣∣(cdγ1;ψ1ρ1,µ1,ω1,a1+f1)(x1)∣∣∣ ∣∣∣(cdγ2;ψ2ρ2,µ2,ω2,a2+f2)(x2)∣∣∣dx1dx2  [∫ x1 a1 { (ψ1(x1)−ψ1(t1))n1−µ1−1 ∣∣∣e−γ1ρ1,n1−µ1 [ω1(ψ1(x1)−ψ1(t1))ρ1 ] ∣∣∣}pdt1] p +[∫ x2 a2 { (ψ2(x2)−ψ2(t2))n2−µ2−1 ∣∣∣e−γ2ρ2,n2−µ2 [ω2(ψ2(x2)−ψ2(t2))ρ2 ] ∣∣∣}qdt2] q   ≤ (b1 − a1) (b2 − a2) ∥∥∥ψ′1f[n1]1ψ1 ∥∥∥q ∥∥∥ψ′2f[n2]2ψ2 ∥∥∥p . (2.33) proof. by theorem 2.9 and (1.2), (1.6). we also give 436 george a. anastassiou cubo 23, 3 (2021) theorem 2.12. all as in theorem 2.11. then ∫ b1 a1 ∫ b2 a2 ∣∣∣(cdγ1;ψ1ρ1,µ1,ω1,b1−f1)(x1)∣∣∣ ∣∣∣(cdγ2;ψ2ρ2,µ2,ω2,b2−f2)(x2)∣∣∣dx1dx2  [∫ b1 x1 { (ψ1(t1)−ψ1(x1))n1−µ1−1 ∣∣∣e−γ1ρ1,n1−µ1 [ω1(ψ1(t1)−ψ1(x1))ρ1 ] ∣∣∣}pdt1] p +[∫ b2 x2 { (ψ2(t2)−ψ2(x2))n2−µ2−1 ∣∣∣e−γ2ρ2,n2−µ2 [ω2(ψ2(t2)−ψ2(x2))ρ2 ] ∣∣∣}qdt2] q   ≤ (b1 − a1) (b2 − a2) ∥∥∥ψ′1f[n1]1ψ1 ∥∥∥q ∥∥∥ψ′2f[n2]2ψ2 ∥∥∥p . (2.34) proof. by theorem 2.10 and (1.3), (1.7). we present theorem 2.13. let p,q > 1 : 1 p + 1 q = 1; i = 1,2. let [ai,bi] ⊂ r, ψi ∈ cmax(n1,n2) ([ai,bi]) , ψ′i ̸= 0, and strictly increasing; fi ∈ c ni ([ai,bi]), where ni = ⌈µi⌉, 0 < µi /∈ n. here ρi > 0; γi,ωi ∈ r and ξi = µi + βi (ni − µi), i = 1,2, where 0 ≤ βi ≤ 1. then ∫ b1 a1 ∫ b2 a2 ∣∣∣(hdγ1,β1;ψ1ρ1,µ1,ω1,a1+f1)(x1)∣∣∣ ∣∣∣(hdγ2,β2;ψ2ρ2,µ2,ω2,a2+f2)(x2)∣∣∣dx1dx2  [∫ x1 a1 { (ψ1(x1)−ψ1(t1))ξ1−µ1−1 ∣∣∣e−γ1β1ρ1,ξ1−µ1 [ω1(ψ1(x1)−ψ1(t1))ρ1 ] ∣∣∣}pdt1] p +[∫ x2 a2 { (ψ2(x2)−ψ2(t2))ξ2−µ2−1 ∣∣∣e−γ2β2ρ2,ξ2−µ2 [ω2(ψ2(x2)−ψ2(t2))ρ2 ] ∣∣∣}qdt2] q   ≤ (b1 − a1) (b2 − a2) ∥∥∥ψ′1 rldγ1(1−β1);ψ1ρ1,ξ1,ω1,a1+f1∥∥∥q ∥∥∥ψ′2 rldγ2(1−β2);ψ2ρ2,ξ2,ω2,a2+f2∥∥∥p . (2.35) proof. by theorem 2.9 and (1.15). we also give theorem 2.14. all as in theorem 2.13. then ∫ b1 a1 ∫ b2 a2 ∣∣∣(hdγ1,β1;ψ1ρ1,µ1,ω1,b1−f1)(x1)∣∣∣ ∣∣∣(hdγ2,β2;ψ2ρ2,µ2,ω2,b2−f2)(x2)∣∣∣dx1dx2  [∫ b1 x1 { (ψ1(t1)−ψ1(x1))ξ1−µ1−1 ∣∣∣e−γ1β1ρ1,ξ1−µ1 [ω1(ψ1(t1)−ψ1(x1))ρ1 ] ∣∣∣}pdt1] p +[∫ b2 x2 { (ψ2(t2)−ψ2(x2))ξ2−µ2−1 ∣∣∣e−γ2β2ρ2,ξ2−µ2 [ω2(ψ2(t2)−ψ2(x2))ρ2 ] ∣∣∣}qdt2] q   ≤ (b1 − a1) (b2 − a2) ∥∥∥ψ′1 rldγ1(1−β1);ψ1ρ1,ξ1,ω1,b1−f1∥∥∥q ∥∥∥ψ′2 rldγ2(1−β2);ψ2ρ2,ξ2,ω2,b2−f2∥∥∥p . (2.36) proof. by theorem 2.10 and (1.16). cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 437 3 appendix we give the following important fundamental results: theorem 3.1. let ρ,µ > 0; γ,ω ∈ r; and ψ ∈ c1 ([a,b]) increasing, f ∈ c ([a,b]). then( e γ;ψ ρ,µ,ω,a+f ) , ( e γ;ψ ρ,µ,ω,b−f ) ∈ c ([a,b]) . proof. we only prove ( e γ;ψ ρ,µ,ω,a+f ) ∈ c ([a,b]). we skip the proof for the other is similar. we consider the power series e γ ρ,µ (z) = ∞∑ k=0 |(γ)k| k!γ (ρk + µ) (ρk + µ) zk, z ∈ r. (3.1) we form r −1 := lim k→∞ |(γ)k+1| (k+1)!γ(ρ(k+1)+µ)(ρ(k+1)+µ) |(γ)k| k!γ(ρk+µ)(ρk+µ) = lim k→∞ |γ+k| (k+1)γ(ρ(k+1)+µ)(ρ(k+1)+µ) 1 γ(ρk+µ)(ρk+µ) = (3.2) lim k→∞ |γ + k| γ (ρk + µ) (ρk + µ) (k + 1) γ (ρ(k + 1) + µ) (ρ(k + 1) + µ) = lim k→∞ ( |γ + k| γ (ρk + µ) (k + 1) γ (ρ(k + 1) + µ) ) lim k→∞ ( ρk + µ (ρk + µ) + ρ ) =: (ξ) . (3.3) notice that lim k→∞ ( ρk + µ (ρk + µ) + ρ ) = 1. (3.4) from (1.1) we have that its radius of convergence is r = lim k→∞ |(γ)k| k!γ(ρk+µ) |(γ)k+1| (k+1)!γ(ρ(k+1)+µ) = lim k→∞ 1 γ(ρk+µ) |γ+k| (k+1)γ(ρ(k+1)+µ) = lim k→∞ (k + 1) γ (ρ(k + 1) + µ) |γ + k| γ (ρk + µ) = ∞, because (1.1) is an entire function. therefore, we have that lim k→∞ |γ + k| γ (ρk + µ) (k + 1) γ (ρ(k + 1) + µ) = 0. consequently by (3.3), (3.4), we get that (ξ) = 0. thus r −1 = 0 and the radius of convergence of e γ ρ,µ (z), see (3.1), is r = ∞, hence (3.1) is convergent everywhere. consequently it holds ∞∑ k=0 |(γ)k| (|ω| (ψ (x) − ψ (a)) ρ ) k k!γ (ρk + µ) (ρk + µ) < ∞, (3.5) ∀ x ∈ [a,b] . we notice that ∞∑ k=0 |(γ)k| |ω| k k!γ (ρk + µ) ∫ x a ψ′ (t) (ψ (x) − ψ (t))(ρk+µ)−1 |f (t)|dt ≤ 438 george a. anastassiou cubo 23, 3 (2021) ∥f∥∞ ∞∑ k=0 |(γ)k| |ω| k k!γ (ρk + µ) (ψ (x) − ψ (a))ρk+µ ρk + µ ≤ (3.6) ∥f∥∞ (ψ (b) − ψ (a)) µ ∞∑ k=0 |(γ)k| (|ω| (ψ (x) − ψ (a)) ρ ) k k!γ (ρk + µ) (ρk + µ) (3.5) < ∞. consequently, by [5, p. 175], we derive ( e γ;ψ ρ,µ,ω,a+f ) (x) (1.2) = ∫ x a ψ′ (t) (ψ (x) − ψ (t))µ−1 ( ∞∑ k=0 (γ)k k!γ (ρk + µ) (ω (ψ (x) − ψ (t))ρ)k ) f (t)dt = ∞∑ k=0 (γ)k ω k k!γ (ρk + µ) ∫ x a ψ′ (t) (ψ (x) − ψ (t))(ρk+µ)−1 f (t)dt, (3.7) ∀ x ∈ [a,b] . by [2, p. 98], we obtain that the function λ(k)ρ,µ (f,x) = ∫ x a ψ′ (t) (ψ (x) − ψ (t))(ρk+µ)−1 f (t)dt, x ∈ [a,b], is absolutely continuous for ρk +µ ≥ 1 and continuous for ρk +µ ∈ (0,1); ψ ∈ c1 ([a,b]) and increasing. that is always λ(k)ρ,µ (|f| ,x) ∈ c ([a,b]), for all k = 0,1, . . . by (3.5), one can derive that ∞∑ k=0 |(γ)k| |ω| k k!γ (ρk + µ) λ(k)ρ,µ (|f| ,x) ≤ ∥f∥∞ (ψ (b) − ψ (a)) µ ∞∑ k=0 |(γ)k| (|ω| (ψ (b) − ψ (a)) ρ ) k k!γ (ρk + µ) (ρk + µ) < ∞. (3.8) notice that ∣∣∣λ(k)ρ,µ (f,x)∣∣∣ ≤ λ(k)ρ,µ (|f| ,x) = ∫ x a ψ′ (t) (ψ (x) − ψ (t))(ρk+µ)−1 |f (t)|dt ≤ ∥f∥∞ (ψ (b) − ψ (a))(ρk+µ) (ρk + µ) , k = 0,1, . . . (3.9) and even more we get: |(γ)k| |ω| k k!γ (ρk + µ) ∣∣∣λ(k)ρ,µ (f,x)∣∣∣ ≤ |(γ)k| |ω|kk!γ (ρk + µ)λ(k)ρ,µ (|f| ,x) ≤( |(γ)k| |ω| k k!γ (ρk + µ) ) ∥f∥∞ (ψ (b) − ψ (a)) (ρk+µ) (ρk + µ) =: mk, k = 0,1, . . . ; (3.10) and by (3.8) that ∞∑ k=0 mk < ∞, converges. by weierstrass m-test we get that ∞∑ k=0 (γ) k ωk k!γ(ρk+µ) λ (k) ρ,µ (f,x) is uniformly and absolutely convergent for x ∈ [a,b]. consequently by (3.7) we derive that ( e γ;ψ ρ,µ,ω,a+f ) ∈ c ([a,b]) . the proof is completed. cubo 23, 3 (2021) foundations of generalized prabhakar-hilfer fractional calculus... 439 we finish with corollary 3.2. all as in theorem 3.1. we have that ∥∥∥eγ;ψρ,µ,ω,a+(b−)f∥∥∥∞ ≤ ( ∞∑ k=0 |(γ)k| |ω| k (ψ (b) − ψ (a))ρk+µ k!γ (ρk + µ + 1) ) ∥f∥∞ < +∞. (3.11) that is eγ;ψ ρ,µ,ω,a+(b−) are bounded linear operators and positive operators if γ,ω > 0. proof. by (3.7), (3.8). 440 george a. anastassiou cubo 23, 3 (2021) references [1] g. a. anastassiou, fractional differentiation inequalities, new york: springer-verlag, 2009. [2] g. a. anastassiou, intelligent computations: abstract fractional calculus inequalities, approximations, cham: springer, 2018. [3] a. giusti, i. colombaro, r. garra, r. garrappa, f. polito, m. popolizio and f. mainardi, “a practical guide to prabhakar fractional calculus”, fract. calc. appl. anal., vol. 23, no. 1, pp. 9–54, 2020. [4] r. gorenflo, a. kilbas, f. mainardi and s. rogosin, mittag-leffler functions, related topics and applications, heidelberg: springer, 2014. [5] e. hewith and k. stromberg, real and abstract analysis. a modern treatment of the theory of functions of a real variable, new york: springer, 1965. [6] f. polito and ž. tomovski, “some properties of prabhakar-type fractional calculus operators”, fract. differ. calc., vol. 6, no. 1, pp. 73–94, 2016. [7] t. r. prabhakar, “a singular integral equation with a generalized mittag-leffler function in the kernel”, yokohama math. j., vol. 19, pp. 7–15, 1971. [8] j. vanterler da c. sousa, e. capelas de oliveira, “on the ψ-hilfer fractional derivative”, commun. nonlinear sci. numer. simul., vol. 60, pp. 72–91, 2018. background main results appendix cubo, a mathematical journal vol. 24, no. 01, pp. 21–35, april 2022 doi: 10.4067/s0719-06462022000100021 infinitely many positive solutions for an iterative system of singular bvp on time scales k. rajendra prasad 1 mahammad khuddush 2 k. v. vidyasagar 3 1department of applied mathematics, college of science and technology, andhra university, visakhapatnam, 530003, india. rajendra92@rediffmail.com 2department of mathematics, dr. lankapalli bullayya college, resapuvanipalem, visakhapatnam, 530013, india. khuddush89@gmail.com 3department of mathematics, s. v. l. n. s. government degree college, bheemunipatnam, bheemili, 531163, india. vidyavijaya08@gmail.com abstract in this paper, we consider an iterative system of singular twopoint boundary value problems on time scales. by applying hölder’s inequality and krasnoselskii’s cone fixed point theorem in a banach space, we derive sufficient conditions for the existence of infinitely many positive solutions. finally, we provide an example to check the validity of our obtained results. resumen en este art́ıculo, consideramos un sistema iterativo de problemas de valor en la frontera singulares de dos puntos en escalas de tiempo. aplicando la desigualdad de hölder y el teorema de punto fijo cónico de krasnoselskii en un espacio de banach, derivamos condiciones suficientes para la existencia de una cantidad infinita de soluciones positivas. finalmente, entregamos un ejemplo para verificar la validez de nuestros resultados. keywords and phrases: iterative system, time scales, singularity, cone, krasnoselskii’s fixed point theorem, positive solutions. 2020 ams mathematics subject classification: 34b18, 34n05. accepted: 15 october, 2021 received: 18 january, 2021 c©2022 k. r. prasad et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://dx.doi.org/10.4067/s0719-06462022000100021 https://orcid.org/0000-0001-8162-1391 https://orcid.org/0000-0002-1236-8334 https://orcid.org/0000-0003-4532-8176 mailto:rajendra92@rediffmail.com mailto:khuddush89@gmail.com mailto:vidyavijaya08@gmail.com 22 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) 1 introduction the theory of time scales was created to unify continuous and discrete analysis. difference and differential equations can be studied simultaneously by studying dynamic equations on time scales. a time scale is any closed and nonempty subset of the real numbers. so, by this theory, we can extend known results from continuous and discrete analysis to a more general setting. as a matter of fact, this theory allows us to consider time scales which possess hybrid behaviours (both continuous and discrete). these types of time scales play an important role for applications, since most of the phenomena in the environment are neither only discrete nor only continuous, but they possess both behaviours. moreover, basic results on this issue have been well documented in the articles [1, 2] and the monographs of bohner and peterson [6, 7]. there is a great deal of research activity devoted to existence of solutions to the dynamic equations on time scales, see for example [8,9,13,16–19] and references therein. in [14], liang and zhang studied countably many positive solutions for nonlinear singular m–point boundary value problems on time scales, ( ϕ(υ∆(t)) )∇ + a(t)f ( υ(t) ) = 0, t ∈ [0, t]t, υ(0) = m−2 ∑ i=1 aiυ(ξi), υ ∆(t) = 0, by using the fixed-point index theory and a new fixed-point theorem in cones. in [12], khuddush, prasad and vidyasagar considered second order n-point boundary value problem on time scales, υ ∆∇ i (t) + λ(t)gℓ ( υi+1(t) ) = 0, 1 ≤ i ≤ n, t ∈ (0, σ(a)]t, υn+1(t) = υ1(t), t ∈ (0, σ(a)]t, υ ∆ i (0) = 0, υi(σ(a)) = n−2 ∑ k=1 ckυi(ζk), 1 ≤ i ≤ n, and established existence of positive solutions by applying krasnoselskii’s fixed point theorem. inspired by the aforementioned works, in this paper by applying hölder’s inequality and krasnoselskii’s cone fixed point theorem in a banach space, we establish the existence of infinitely many positive solutions for the iterative system of two-point boundary value problems with n– singularities on time scales, υ ∆∆ ℓ (t) + λ(t)gℓ ( υℓ+1(t) ) = 0, 1 ≤ ℓ ≤ m, t ∈ (0, t)t, υm+1(t) = υ1(t), t ∈ (0, t)t,    (1.1) υℓ(0) = υ ∆ ℓ (0), 1 ≤ ℓ ≤ m, υℓ(t) = −υ∆ℓ (t), 1 ≤ ℓ ≤ m,    (1.2) cubo 24, 1 (2022) infinitely many positive solutions for an iterative system of... 23 where m ∈ n, λ(t) = ∏k i=1 λi(t) and each λi(t) ∈ l pi ∆([0, t]t) (pi ≥ 1) has n–singularities in the interval (0, t) t . we assume the following conditions are true throughout the paper: (h1) gℓ : [0, +∞) → [0, +∞) is continuous. (h2) lim t→ti λi(t) = ∞, where 0 < tn < tn−1 < · · · < t1 < t. 2 preliminaries in this section, we introduce some basic definitions and lemmas which are useful for our later discussions. definition 2.1 ( [6]). a time scale t is a nonempty closed subset of the real numbers r. t has the topology that it inherits from the real numbers with the standard topology. it follows that the jump operators σ, ρ : t → t, and the graininess µ : t → [0, +∞) are defined by σ(t) = inf{τ ∈ t : τ > t}, ρ(t) = sup{τ ∈ t : τ < t}, and µ(t) = σ(t) − t, respectively. • the point t ∈ t is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) > t, respectively. • if t has a right-scattered minimum m, then tκ = t\{m}; otherwise tκ = t. • if t has a left-scattered maximum m, then tκ = t\{m}; otherwise tκ = t. • a function f : t → r is called rd-continuous provided it is continuous at right-dense points in t and its left-sided limits exist (finite) at left-dense points in t. the set of all rd-continuous functions f : t → r is denoted by crd = crd(t) = crd(t, r). • a function f : t → r is called ld-continuous provided it is continuous at left-dense points in t and its right-sided limits exist (finite) at right-dense points in t. the set of all ld-continuous functions f : t → r is denoted by cld = cld(t) = cld(t, r). • by an interval time scale, we mean the intersection of a real interval with a given time scale, i.e., [a, b]t = [a, b] ∩ t. other intervals can be defined similarly. 24 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) definition 2.2 ([5,11]). let µ∆ and µ∇ be the lebesgue ∆−measure and the lebesgue ∇−measure on t, respectively. if a ⊂ t satisfies µ∆(a) = µ∇(a), then we call a measurable on t, denoted µ(a) and this value is called the lebesgue measure of a. let p denote a proposition with respect to t ∈ t. (i) if there exists γ1 ⊂ a with µ∆(γ1) = 0 such that p holds on a\γ1, then p is said to hold ∆–a.e. on a. (ii) if there exists γ2 ⊂ a with µ∇(γ2) = 0 such that p holds on a\γ2, then p is said to hold ∇–a.e. on a. definition 2.3 ( [4,5]). let e ⊂ t be a ∆−measurable set and p ∈ r̄ ≡ r ∪ {−∞, +∞} be such that p ≥ 1 and let f : e → r̄ be a ∆−measurable function. we say that f belongs to lp∆(e) provided that either ∫ e |f|p(s)∆s < ∞ if p ∈ [1, +∞), or there exists a constant m ∈ r such that |f| ≤ m, ∆ − a.e. on e if p = +∞. lemma 2.4 ( [20]). let e ⊂ t be a ∆−measurable set. if f : t → r is ∆−integrable on e, then ∫ e f(s)∆s = ∫ e f(s)ds + ∑ i∈ie ( σ(ti) − ti ) f(ti) + r(f, e), where r(f, e) =        µn(e)f(m), if n ∈ t, 0, if n /∈ t, ie := {i ∈ i : ti ∈ e} and {ti}i∈i, i ⊂ n, is the set of all right-scattered points of t. lemma 2.5. for any y(t) ∈ crd([0, t]t), the boundary value problem, υ ∆∆ 1 (t) + y(t) = 0, t ∈ (0, t)t, (2.1) υ1(0) = υ ∆ 1 (0), υ1(t) = −υ∆1 (t), (2.2) has a unique solution υ1(t) = ∫ t 0 ℵ(t, τ)y(τ)∆τ, (2.3) where ℵ(t, τ) = 1 2 + t    (t − t + 1)(σ(τ) + 1), if σ(τ) < t, (t − σ(τ) + 1)(t + 1), if t < τ. (2.4) cubo 24, 1 (2022) infinitely many positive solutions for an iterative system of... 25 proof. suppose υ1 is a solution of (2.1), then υ1(t) = − ∫ t 0 ∫ τ 0 y(τ1)∆τ1∆τ + a1t + a2 = − ∫ t 0 (t − σ(τ))y(τ)∆τ + a1t + a2, where a1 = υ ∆ 1 (0) and a2 = υ1(0). by the conditions (2.2), we get a1 = a2 = 1 2 + t ∫ t 0 (t − σ(τ) + 1)y(τ)∆τ. so, we have υ1(t) = ∫ t 0 (t − σ(τ))y(τ)∆τ + 1 2 + t ∫ t 0 (t − σ(τ) + 1)(1 + t)y(τ)∆τ = ∫ t 0 ℵ(t, τ)y(τ)∆τ. this completes the proof. lemma 2.6. suppose (h1)–(h2) hold. for ε ∈ (0, t2 )t, let g(ε) = ε + 1 t + 1 < 1. then ℵ(t, τ) has the following properties: (i) 0 ≤ ℵ(t, τ) ≤ ℵ(τ, τ) for all t, τ ∈ [0, 1]t, (ii) g(ε)ℵ(τ, τ) ≤ ℵ(t, τ) for all t ∈ [ε, t − ε]t and τ ∈ [0, 1]t. proof. (i) is evident. to prove (ii), let t ∈ [ε, t − ε]t and t ≤ τ. then ℵ(t, τ) ℵ(τ, τ) = t + 1 τ + 1 ≥ ε + 1 t + 1 = g(ε). for τ ≤ t, ℵ(t, τ) ℵ(τ, τ) = t − t + 1 t − τ + 1 ≥ ε + 1 t + 1 = g(ε). this completes the proof. notice that an m−tuple (υ1(t), υ2(t), υ3(t), . . . , υm(t)) is a solution of the iterative boundary value problem (1.1)–(1.2) if and only if υℓ(t) = ∫ 1 0 ℵ(t, τ)λ(τ)gℓ(υℓ+1(τ))∆τ, t ∈ (0, t)t, 1 ≤ ℓ ≤ m, υm+1(t) = υ1(t), t ∈ (0, t)t, i.e., υ1(t) = ∫ 1 0 ℵ(t, τ1)λ(τ1)g1 ( ∫ 1 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ 1 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ 1 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1. 26 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) let b be the banach space crd((0, t)t, r) with the norm ‖υ‖ = max t∈(0,t)t |υ(t)|. for ε ∈ ( 0, t 2 ) t , we define the cone kε ⊂ b as kε = { υ ∈ b : υ(t) is nonnegative and min t∈[ε, t−ε]t υ(t) ≥ g(ε)‖υ(t)‖ } . for any υ1 ∈ kε, define an operator ω : kε → b by (ωυ1)(t) = ∫ 1 0 ℵ(t, τ1)λ(τ1)g1 ( ∫ 1 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ 1 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ 1 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1. lemma 2.7. assume that (h1)–(h2) hold. then for each ε ∈ ( 0, t 2 ) t , ω(kε) ⊂ kε and ω : kε → kε are completely continuous. proof. from lemma 2.6, ℵ(t, τ) ≥ 0 for all t, τ ∈ (0, t)t. so, (ωυ1)(t) ≥ 0. also, for υ1 ∈ kε, we have ‖ωυ1‖ = max t∈(0,t)t ∫ 1 0 ℵ(t, τ1)λ(τ1)g1 ( ∫ 1 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ 1 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ 1 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1 ≤ ∫ 1 0 ℵ(τ1, τ1)λ(τ1)g1 ( ∫ 1 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ 1 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ 1 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1. again from lemma 2.6, we get min t∈[ε,t−ε]t { (ωυ1)(t) } ≥ g(ε) ∫ 1 0 ℵ(τ1, τ1)λ(τ1)g1 ( ∫ 1 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ 1 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ 1 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1. it follows from the above two inequalities that min t∈[ε,t−ε]t { (ωυ1)(t) } ≥ g(ε)‖ωυ1‖. so, ωυ1 ∈ kε and thus ω(kε) ⊂ kε. next, by standard methods and the arzela-ascoli theorem, it can be proved easily that the operator ω is completely continuous. the proof is complete. 3 infinitely many positive solutions for the existence of infinitely many positive solutions for iterative system of boundary value problem (1.1)–(1.2), we apply following theorems. theorem 3.1 ( [10]). let e be a cone in a banach space x and let m1, m2 be open sets with 0 ∈ m1, m1 ⊂ m2. let a : e ∩ (m2\m1) → e be a completely continuous operator such that cubo 24, 1 (2022) infinitely many positive solutions for an iterative system of... 27 (a) ‖av‖ ≤ ‖v‖, v ∈ e ∩ ∂m1, and ‖av‖ ≥ ‖v‖, v ∈ e ∩ ∂m2, or (b) ‖av‖ ≥ ‖v‖, v ∈ e ∩ ∂m1, and ‖av‖ ≤ ‖v‖, v ∈ e ∩ ∂m2. then a has a fixed point in e ∩ (m2\m1). theorem 3.2 ( [7,15]). let f ∈ lp∇(j) with p > 1, g ∈ l q ∆(j) with q > 1, and 1 p + 1 q = 1. then fg ∈ l1∆(j) and ‖fg‖l1∆ ≤ ‖f‖lp∆‖g‖lq∆, where ‖f‖lp ∆ :=      [ ∫ j |f|p(s)∆s ] 1 p , p ∈ r, inf { m ∈ r / |f| ≤ m ∆ − a.e. on j } , p = ∞, and j = [a, b)t. theorem 3.3 (hölder’s inequality [3,4,15]). let f ∈ lpi∆(j) with pi > 1, for i = 1, 2, . . . , n and ∑n i=1 1 pi = 1. then ∏k i=1 gi ∈ l1∆(j) and ∥ ∥ ∥ ∏k i=1 gi ∥ ∥ ∥ 1 ≤ ∏k i=1 ‖gi‖pi. further, if f ∈ l1∆(j) and g ∈ l∞∆ (j), then fg ∈ l1∆(j) and ‖fg‖1 ≤ ‖f‖1‖g‖∞. we need the following condition in the sequel: (h3) there exists δi > 0 such that λi(t) > δi (i = 1, 2, . . . , n) for t ∈ [0, t]t. consider the following three possible cases for λi ∈ lpi∆(0, t)t : n ∑ i=1 1 pi < 1, n ∑ i=1 1 pi = 1, n ∑ i=1 1 pi > 1. firstly, we seek infinitely many positive solutions for the case n ∑ i=1 1 pi < 1. theorem 3.4. suppose (h1)–(h3) hold, let {εr}∞r=1 be such that 0 < ε1 < t/2, ε ↓ t∗ and 0 < t∗ < tn. let {γr}∞r=1 and {λr}∞r=1 be such that γr+1 < g(εr)λr < λr < θλr < γr, r ∈ n, where θ = max { [ g(ε1) k ∏ i=1 δi ∫ t−ε1 ε1 ℵ(τ, τ)∆τ ]−1 , 1 } . assume that gℓ satisfies (c1) gℓ(υ) ≤ n1γr ∀ t ∈ (0, t)t, 0 ≤ υ ≤ γr, where n1 < [ ‖ℵ‖lq ∆ k ∏ i=1 ‖λi‖lpi ∆ ]−1 , (c2) gℓ(υ) ≥ θλr ∀ t ∈ [εr, t − εr]t, g(εr)λr ≤ υ ≤ λr. 28 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) then the iterative boundary value problem (1.1)–(1.2) has infinitely many solutions {(υ[r]1 , υ [r] 2 , . . . , υ [r] m )}∞r=1 such that υ [r] ℓ (t) ≥ 0 on (0, t)t, ℓ = 1, 2, . . . , m and r ∈ n. proof. let m1,r = {υ ∈ b : ‖υ‖ < γr}, m2,r = {υ ∈ b : ‖υ‖ < λr}, be open subsets of b. let {εr}∞r=1 be given in the hypothesis and we note that t∗ < tr+1 < εr < tr < t 2 , for all r ∈ n. for each r ∈ n, we define the cone kεr by kεr = { υ ∈ b : υ(t) ≥ 0, min t∈[εr, t−εr]t υ(t) ≥ g(εr)‖υ(t)‖ } . let υ1 ∈ kεr ∩ ∂m1,r. then, υ1(τ) ≤ γr = ‖υ1‖ for all τ ∈ (0, t)t. by (c1) and for τm−1 ∈ (0, t)t, we have ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ≤ ∫ t 0 ℵ(τm, τm)λ(τm)gm(υ1(τm))∆τm ≤ n1γr ∫ t 0 ℵ(τm, τm) k ∏ i=1 λi(τm)∆τm. there exists a q > 1 such that 1 q + n ∑ i=1 1 pi = 1. so, ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ≤ n1γr ∥ ∥ℵ ∥ ∥ l q ∆ ∥ ∥ ∥ ∥ ∥ k ∏ i=1 λi ∥ ∥ ∥ ∥ ∥ l pi ∆ ≤ n1γr‖ℵ‖lq ∆ k ∏ i=1 ‖λi‖lpi ∆ ≤ γr. it follows in similar manner (for τm−2 ∈ (0, t)t), that ∫ t 0 ℵ(τm−2, τm−1)λ(τm−1)gm−1 ( ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) ∆τm−1 ≤ ∫ t 0 ℵ(τm−2, τm−1)λ(τm−1)gm−1(γr)∆τm−1 ≤ ∫ t 0 ℵ(τm−1, τm−1)λ(τm−1)gm−1(γr)∆τm−1 ≤ n1γr ∫ t 0 ℵ(τm−1, τm−1) k ∏ i=1 λi(τm−1)∆τm−1 ≤ n1γr‖ℵ‖lq ∆ k ∏ i=1 ‖λi‖lpi ∆ ≤ γr. cubo 24, 1 (2022) infinitely many positive solutions for an iterative system of... 29 continuing with this bootstrapping argument, we get (ωυ1)(t) = ∫ t 0 ℵ(t, τ1)λ(τ1)g1 ( ∫ t 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ t 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1 ≤ γr. since γr = ‖υ1‖ for υ1 ∈ kεr ∩ ∂m1,r, we get ‖ωυ1‖ ≤ ‖υ1‖. (3.1) let t ∈ [εr, t − εr]t. then, λr = ‖υ1‖ ≥ υ1(t) ≥ min t∈[εr,t−εr]t υ1(t) ≥ g(εr) ‖υ1‖ ≥ g(εr)λr. by (c2) and for τm−1 ∈ [εr, t − εr]t, we have ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ≥ ∫ t−εr εr ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ≥ g(εr)θλr ∫ t−εr εr ℵ(τm, τm)λ(τm)∆τm ≥ g(εr)θλr ∫ t−εr εr ℵ(τm, τm) k ∏ i=1 λi(τm)∆τm ≥ g(ε1)θλr k ∏ i=1 δi ∫ t−ε1 ε1 ℵ(τm, τm)∆τm ≥ λr. continuing with the bootstrapping argument, we get (ωυ1)(t) = ∫ t 0 ℵ(t, τ1)λ(τ1)g1 ( ∫ t 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ t 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1 ≥ λr. thus, if υ1 ∈ kεr ∩ ∂k2,r, then ‖ωυ1‖ ≥ ‖υ1‖. (3.2) it is evident that 0 ∈ m2,k ⊂ m2,k ⊂ m1,k. from (3.1)–(3.2), it follows from theorem 3.1 that the operator ω has a fixed point υ [r] 1 ∈ kεr ∩ ( m1,r\m2,r ) such that υ [r] 1 (t) ≥ 0 on (0, t)t, and r ∈ n. next setting υm+1 = υ1, we obtain infinitely many positive solutions {(υ[r]1 , υ [r] 2 , . . . , υ [r] m )}∞r=1 of (1.1)–(1.2) given iteratively by υℓ(t) = ∫ t 0 ℵ(t, τ)λ(τ)gℓ(υℓ+1(τ))∆τ, t ∈ (0, t)t, ℓ = m, m − 1, . . . , 1. the proof is completed. 30 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) for n ∑ i=1 1 pi = 1, we have the following theorem. theorem 3.5. suppose (h1)–(h3) hold, let {εr}∞r=1 be such that 0 < ε1 < t/2, ε ↓ t∗ and 0 < t∗ < tn. let {γr}∞r=1 and {λr}∞r=1 be such that γr+1 < g(εr)λr < λr < θλr < γr, r ∈ n, where θ = max { [ g(ε1) k ∏ i=1 δi ∫ t−ε1 ε1 ℵ(τ, τ)∆τ ]−1 , 1 } . assume that gℓ satisfies (c2) and (c3) gj(υ) ≤ n2γr ∀ t ∈ (0, t)t, 0 ≤ υ ≤ γr, where n2 < min    [ ‖ℵ‖l∞ ∆ k ∏ i=1 ‖λi‖lpi ∆ ]−1 , θ    . then the iterative boundary value problem (1.1)–(1.2) has infinitely many solutions {(υ[r]1 , υ [r] 2 , . . . , υ [r] m )}∞r=1 such that υ [r] ℓ (t) ≥ 0 on (0, t)t, ℓ = 1, 2, . . . , m and r ∈ n. proof. for a fixed r, let m1,r be as in the proof of theorem 3.4 and let υ1 ∈ kεr ∩ ∂m2,r. again υ1(τ) ≤ γr = ‖υ1‖, for all τ ∈ (0, t)t. by (c3) and for τℓ−1 ∈ (0, t)t, we have ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ≤ ∫ t 0 ℵ(τm, τm)λ(τm)gm(υ1(τm))∆τm ≤ n1γr ∫ t 0 ℵ(τm, τm) k ∏ i=1 λi(τm)∆τm ≤ n1γr ∥ ∥ℵ ∥ ∥ l∞ ∆ ∥ ∥ ∥ ∥ ∥ k ∏ i=1 λi ∥ ∥ ∥ ∥ ∥ l pi ∆ ≤ n1γr‖ℵ‖l∞ ∆ k ∏ i=1 ‖λi‖lpi ∆ ≤ γr. cubo 24, 1 (2022) infinitely many positive solutions for an iterative system of... 31 it follows in similar manner (for τm−2 ∈ [0, 1]t), that ∫ t 0 ℵ(τm−2, τm−1)λ(τm−1)gm−1 ( ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) ∆τm−1 ≤ ∫ t 0 ℵ(τm−2, τm−1)λ(τm−1)gm−1(γr)∆τm−1 ≤ ∫ t 0 ℵ(τm−1, τm−1)λ(τm−1)gm−1(γr)∆τm−1 ≤ n1γr ∫ t 0 ℵ(τm−1, τm−1) k ∏ i=1 λi(τm−1)∆τm−1 ≤ n1γr‖ℵ‖l∞ ∆ k ∏ i=1 ‖λi‖lpi ∆ ≤ γr. continuing with this bootstrapping argument, we get (ωυ1)(t) = ∫ t 0 ℵ(t, τ1)λ(τ1)g1 ( ∫ t 0 ℵ(τ1, τ2)λ(τ2)g2 ( ∫ t 0 ℵ(τ2, τ3) · · · × gm−1 ( ∫ t 0 ℵ(τm−1, τm)λ(τm)gm(υ1(τm))∆τm ) · · · ∆τ3 ) ∆τ2 ) ∆τ1 ≤ γr. since γr = ‖υ1‖ for υ1 ∈ kεr ∩ ∂m1,r, we get ‖ωυ1‖ ≤ ‖υ1‖. (3.3) now define m2,r = {υ1 ∈ b : ‖υ1‖ < λr}. let υ1 ∈ kεr ∩ ∂m2,r and let τ ∈ [εr, t − εr]t. then, the argument leading to (3.2) can be done to the present case. hence, the theorem. lastly, the case n ∑ i=1 1 pi > 1. theorem 3.6. suppose (h1)–(h3) hold, let {εr}∞r=1 be such that 0 < ε1 < t/2, ε ↓ t∗ and 0 < t∗ < tn. let {γr}∞r=1 and {λr}∞r=1 be such that γr+1 < g(εr)λr < λr < θλr < γr, r ∈ n, where θ = max { [ g(ε1) k ∏ i=1 δi ∫ t−ε1 ε1 ℵ(τ, τ)∆τ ]−1 , 1 } . assume that gℓ satisfies (c2) and (c4) gj(υ) ≤ n3γr ∀ t ∈ (0, t)t, 0 ≤ υ ≤ γr, where n3 < min { [ ‖ℵ‖l∞ ∆ ∏k i=1 ‖λi‖l1 ∆ ]−1 , θ } . then the iterative boundary value problem (1.1)–(1.2) has infinitely many solutions {(υ[r]1 , υ [r] 2 , . . . , υ [r] m )}∞r=1 such that υ [r] ℓ (t) ≥ 0 on (0, t)t, ℓ = 1, 2, . . . , m and r ∈ n. proof. the proof is similar to the proof of theorem 3.4. so, we omit the details here. 32 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) 4 example in this section, we present an example to check validity of our main results. example 4.1. consider the following boundary value problem on t = r. υ ′′ ℓ (t) + λ(t)gℓ(υℓ+1(t)) = 0, ℓ = 1, 2, υ3(t) = υ1(t),    (4.1) υℓ(0) = υ ′ ℓ(0), υℓ(1) = −υ′ℓ(1),    (4.2) where λ(t) = λ1(t)λ2(t) in which λ1(t) = 1 |t − 1 4 | 12 and λ2(t) = 1 |t − 3 4 | 12 , g1(υ) = g2(υ) =                                      1 5 × 10−4, υ ∈ (10−4, +∞), 25×10−(4r+3)− 1 5 ×10−4r 10−(4r+3)−10−4r (υ − 10−4r)+ 1 5 × 10−8r, υ ∈ [ 10−(4r+3), 10−4r ] , 25 × 10−(4r+3), υ ∈ ( 1 5 × 10−(4r+3), 10−(4r+3) ) , 25×10−(4r+3)− 1 5 ×10−8r 1 5 ×10−(4r+3)−10−(4r+4) (υ − 10−(4r+4))+ 1 5 × 10−8r, υ ∈ ( 10−(4r+4), 1 5 × 10−(4r+3) ] , 0, υ = 0. let tr = 31 64 − r ∑ k=1 1 4(k + 1)4 , εr = 1 2 (tr + tr+1), r = 1, 2, 3, . . ., then ε1 = 15 32 − 1 648 < 15 32 , and tr+1 < εr < tr, εr > 1 5 . therefore, g(εr) = εr + 1 t + 1 = εr + 1 2 > 1 5 , r = 1, 2, 3, . . . it is clear that t1 = 15 32 < 1 2 , tr − tr+1 = 1 4(r + 2)4 , r = 1, 2, 3, . . . since ∞ ∑ x=1 1 x4 = π4 90 and ∞ ∑ x=1 1 x2 = π2 6 , it follows that t∗ = lim r→∞ tr = 31 64 − ∞ ∑ k=1 1 4(r + 1)4 = 47 64 − π 4 360 = 0.4637941914, cubo 24, 1 (2022) infinitely many positive solutions for an iterative system of... 33 λ1, λ2 ∈ lp[0, 1] for all 0 < p < 2, and δ1 = δ2 = (4/3)1/4 , g(ε1) = 0.7336033951. ∫ t−ε1 ε1 ℵ(τ, τ)∆τ = ∫ 1− 15 32 + 1 648 15 32 − 1 648 (2 − τ)(1 + τ) 3 dτ = 0.04918197801. thus, we get θ = max { [ g(ε1) k ∏ i=1 δi ∫ t−ε1 ε1 ℵ(τ, τ)∇τ ]−1 , 1 } = max { 1 0.04166167167 , 1 } = 24.00287746. next, let 0 < a < 1 be fixed. then λ1, λ2 ∈ l1+a[0, 1] and 21+a > 1 for 0 < a < 1. it follows that k ∏ i=1 ‖λi‖lpi ∆ ≈ π − ln(7 − 4 √ 3), and also ‖ℵ‖∞ = 23. so, for 0 < a < 1, we have n1 < [ ‖ℵ‖∞ k ∏ i=1 ‖λi‖lpi ∆ ]−1 ≈ 0.2597173925. taking n1 = 1 4 . in addition if we take γr = 10 −4r, λr = 10 −(4r+3), then γr+1 = 10 −(4r+4) < 1 5 × 10−(4r+3) < g(εr)λr < λr = 10−(4r+3) < γr = 10−4r, θλr = 24.00287746×10−(4r+3) < 14 ×10 −4r = n1γr, r = 1, 2, 3, . . . , and g1, g2 satisfy the following growth conditions: g1(υ) = g2(υ) ≤ n1γr = 1 4 × 10−4r, υ ∈ [ 0, 10−4r ] , g1(υ) = g2(υ) ≥ θλr = 24.00287746 × 10−(4r+3), υ ∈ [ 1 5 × 10−(4r+3), 10−(4r+3) ] . then all the conditions of theorem 3.4 are satisfied. therefore, by theorem 3.4, the iterative boundary value problem (1.1) has infinitely many solutions {(υ[r]1 , υ [r] 2 )}∞r=1 such that υ [r] ℓ (t) ≥ 0 on [0, 1], ℓ = 1, 2 and r ∈ n. acknowledgements the authors would like to thank the referees for their valuable suggestions and comments for the improvement of the paper. 34 k. r. prasad, m. khuddush & k. v. vidyasagar cubo 24, 1 (2022) references [1] r. p. agarwal and m. bohner, “basic calculus on time scales and some of its applications”, results math., vol. 35, no. 1–2, pp. 3–22, 1999. 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[20] p. a. williams, “unifying fractional calculus with time scales”, ph.d. thesis, university of melbourne, melbourne, australia, 2012. introduction preliminaries infinitely many positive solutions example art n5.dvi cubo a mathematical journal vol.19, no¯ 02, (73–85). june 2017 a trigonometrical approach to morley’s observation ioannis gasteratos1, spiridon kuruklis2 and thedore kuruklis3 1department of mathematics and statistics, boston university, boston, ma 02215, usa. igaster@bu.edu 2eurobank, group information and it security, 14234 athens, greece. skuruklis@gmail.com 3theoklitos, 17672 kallithea, greece. tkuruklis@gmail.com abstract simple trigonometrical arguments verify that in a triangle the trisectors, proximal to sides respectively, meet at the vertices of an equilateral triangle by showing that the length of each side is 8r times the sines of the angles between the sides of the triangle and the trisectors that determine it, where r is the radius of the circumcircle of the triangle. the 27 meeting points of the trisectors, proximal to a side, determine 18 such equilaterals, which in pairs share a vertex having two collinear sides and the third parallel. hence these points are located 6 by 6 on three triples of parallel lines. resumen argumentos trigonométricos simples verifican que en un triángulo los trisectores, próximos a los lados respectivamente, se encuentran en los vértices de un triángulo equilátero mostrando que la longitud de cada lado es 8r veces los senos de los ángulos entre los lados del triángulo y los trisectores que lo determinan, donde r es el radio del circunćırculo del triángulo. los 27 puntos de encuentro de los trisectores, próximos a un lado, determinan 18 tales equiláteros, que a pares comparten un vértice teniendo dos lados colineales y el tercero paralelo. luego estos puntos están ubicados 6 por 6 en tres triples de ĺıneas paralelas. keywords and phrases: angle trisection, proximal trisector, triangle trisectors, morley’s theorem, morley triangle, morley’s magic, morley’s miracle, morley’s mystery. 2010 ams mathematics subject classification: 00a05, 00a08 74 ioannis gasteratos, spiridon kuruklis & thedore kuruklis cubo 19, 2 (2017) 1. introduction one of the infamous contemporary problems in mathematics refers to the angle trisection in a triangle. the problem appeared suddenly around 1900, when frank morley made a shocking observation, known since then as morley’s theorem, usually expressed with the statement: in a triangle, the trisectors of its angles, proximal to sides respectively, meet at the vertices of an equilateral triangle. the above theorem is considered among the most unexpected discoveries in mathematics, as strangely went unnoticed during the ages, even though it expresses a property for trisectors analog to bisectors. ancient greeks studied the triangle in depth and they could have discovered it, but simply ignored it. more than one hundred years since its discovery and a very respectable number of publications, several authored by distinguished mathematicians, we are still struggling to fully comprehend it. words like magic, miracle, mystery, or paradox have appeared in titles of several articles. notably, morley’s theorem has been included in the list of the hundred greatest theorems.[1] morley inferred it while studying the behavior of cardioids from an observation that plainly asserts: in a triangle, the trisectors proximal to a side intersect on three sets of three parallel lines forming equilateral triangles... thus, if we take the interior trisectors of the angles of a triangle, the points where those proximal to a side meet form an equilateral triangle. [13, p.469] in fig.1 appear 27 equilaterals. their placement reveals a structure with startling symmetry, where an impressive number of overlapping and interconnected equilaterals are arranged with common vertices and parallel or collinear sides. their existence, in fact with arrangement, is interpreted as evidence of regularity in the behavior of angle trisectors in a triangle, like the incenter and the excenters of a triangle express regularity in the behavior of its angle bisectors.[11] visual inspection easily verifies that only 18 from the 27 triangles determined by the meeting points of trisectors of all three angles, proximal to sides respectively, are equilateral and they are called morley triangles. fig.2 illustrates the trisectors of ∠abc = 3β, 0 < β < 60◦. the proximal to side bc trisector is the one (bt) which coincides with side ba after two rotations around b towards ba by ∠cbt. note that the inclinations of the proximal trisectors to the corresponding proximal sides are β, 60◦ − β and 60◦ + β. the used formulation of morley’s theorem does not specify the type of trisectors, intersecting in pairs at the vertices of an equilateral. this becomes crucial in the presence of many triangles formed by the intersections of trisectors, proximal to sides respectively. however, several from these triangles but not all are equilateral. as the theorem can be valid for other types of trisectors this ambiguity may have been deliberately used, although most publications have focused only on the interior trisectors. in contrast to the elementary, sharp and clean statement of the theorem, approaches dealing with the observation remain abstract or in higher mathematics, where it was cubo 19, 2 (2017) a trigonometrical approach to morley’s observation 75 figure 1: fig.1: the 27 intersections of trisectors, proximal to sides, determine 18 equilaterals discovered. [8],[12], [16] in this presentation, we prove several instances of morley’s theorem yielding the 18 morley triangles. in fact, it is shown: the side length of a morley triangle is 8r times the sin of the angles between the sides of the triangle and the trisectors that determine it, where r is the radius of the circumcircle of the triangle. then, utilizing the arrangement of these triangles, we establish morley’s observation about the alignment of the intersections of trisectors proximal to a side. the approach relies on the following trigonometrical property that combines the sine and cosine laws. proposition: in a ∆stv with ∠stv = φ, if st = p sin θ and tv = p sin ω, where φ + θ + ω = 180◦, then sv = p sin φ while ∠svt = θ and ∠tsv = ω. proof. in ∆stv by applying the law of cosines we get (sv)2 = (ts)2+(tv)2−2(ts)(tv) cos φ = p2 sin2 φ since sin2 θ + sin2 ω − 2 sin θ sin ω cos φ = sin2 φ by the law of cosines in the triangle with sides sin θ, sin ω and sin φ. thus sv = p sin φ. now from the law of sines st/ sin(∠svt) = tv/ sin(∠tsv) = sv/ sin φ. as sv = p sin φ, st = p sin θ and tv = p sin ω, we get sin θ = sin(∠svt), sin ω = sin(∠tsv). therefore, ∠svt = θ or ∠svt = 180◦−θ and ∠tsv = ω or∠tsv = 180◦ − ω. since φ + θ + ω = 180◦ and φ + ∠svt + ∠tsv = 180◦, only the case ∠svt = θ and ∠tsv = ω may hold. 76 ioannis gasteratos, spiridon kuruklis & thedore kuruklis cubo 19, 2 (2017) figure 2: fig.2: the three types of trisectors given a ∆abc, with ∠a = 3α, ∠b = 3β and ∠c = 3γ, where α + β + γ = 60◦, we establish that ∆a′b′c′ is equilateral, where a′, b′ and c′ are the intersections of trisectors proximal to sides of ∆abc respectively, by showing a′b′ = b′c′ = c′a′. to prove this we apply the above proposition to the adjacent triangles ∆a′cb′, ∆b′ac′ and ∆c′ba′, formed by a side of ∆a′b′c′ and the trisectors that determine it. the lengths of the trisectors are found from the surrounding triangles ∆ba′c, ∆cb′a and ∆ac′b after the sides are expressed by the formulas ab = 2r sin 3γ, bc = 2r sin 3α, ac = 2r sin 3β, obtained from the law of sines. in the appearing expressions, it is convenient to represent angles θ + 60◦ and 60◦ − θ as θ+ and θ− respectively. hence θ+ = 60◦ + θ and θ− = 60◦ − θ, while θ++ = 120◦ + θ and θ−+ = 120◦ − θ. also, the formula sin 3θ = 4 sin(60◦ − θ) sin θ sin(60◦ + θ) = 4 sin θ− sin θ sin θ+ will be used to simplify expressions. as sin 3θ = 3 sin θ−4 sin3 θ the above follows after factoring out the right hand side. in the article are these the most beautiful? hofstadter is quoted that would have given a very high score to morley’s theorem as it follows from this trigonometrical identity. [15] in the sequel, morley triangles are grouped according to the type of trisectors that determine them and only representatives of groups are examined. the groups are: the primitive triangles formed by the same type of trisectors, the mix triangles formed properly by one type of trisectors of an angle and another type of trisectors of the other two angles, and the complete triangles formed by trisectors of one distinct type for each angle. to avoid degenerate cases in which some of the considered triangles are not formed, in the sequel we assume that the angles of the given triangle are different and not multiples of 30◦. cubo 19, 2 (2017) a trigonometrical approach to morley’s observation 77 2. the 18 morley triangles in this section, we will prove that the 18 morley triangles are indeed equilaterals by producing formulas for the lengths of their sides. i. primitive morley triangles (3) the primitive triangles are formed exclusively by the intersections of same type trisectors. hence the two pairs of trisectors determining one of its sides form the same three angles with the corresponding sides of ∆abc. in the sequel, we find an expression of the length for one side. as this is independent of the side then all sides have the same length and so the specific morley triangle is equilateral. figure 3: fig.3: the primitive morley triangles theorem 0.1. in a triangle, the same type trisectors of its angles, proximal to sides respectively, meet at the vertices of a corresponding equilateral. proof. assume that the intersections of the interior, exterior and explementary trisectors, proximal to the sides, bc, ca and ab, meet at a′, b′ and c′, a′′, b′′ and c′′, a′′′, b′′′ and c′′′, defining ∆a′b′c′, ∆a′′b′′c′′ and ∆a′′′b′′′c′′′, called inner triangle, central triangle and peripheral triangle respectively. the proximal trisectors that determine one of their sides form angles with the corresponding 78 ioannis gasteratos, spiridon kuruklis & thedore kuruklis cubo 19, 2 (2017) sides of ∆abc equal to α, β and γ, for the inner, or α−, β− and γ−, for the central, or α+, β+ and γ+, for the peripheral triangle, regardless of the side. a. the lengths of the interior trisectors determining side b′c′ are found from the law of sines in ∆ab′c and ∆bc′a : ab′/ sin γ = ac/ sin(180◦ − α − γ) = 2r sin 3β/ sin β− = 8r sin β sin β+; so ab′ = 8r sin β sin β+ sin γ. similarly, ac′ = 8r sin γ sin γ+ sin β. then in ∆b′ac′, ∠b′ac′ = α, ab′ = p sin β+, ac′ = p sin γ+, where p = 8r sin β sin γ. as α+β++γ+ = 180◦, b′c′ = p sin α = 8r sin β sin γ sin α while ∠ab′c′ = γ+ and ∠ac′b′ = β+. we conclude that ∆a′b′c′ is equilateral. b. the lengths of the exterior trisectors determining side b′′c′′ are found from the law of sines in ∆ab′′c and ∆bc′′a: ab′′/ sin γ− = ac/ sin(180◦ − α− − γ−) = 2r sin 3β/ sin β+ = 8r sin β sin β− and so ab′′ = 8r sin β sin β− sin γ−. similarly, ac′′ = 8r sin γ sin γ− sin β−. then in ∆b′′ac′′, ∠b′′ac′′ = 2α−+3α = α++, ab′′ = p sin β, ac′′ = p sin γ, where p = 8r sin β− sin γ−. as α++ +β+γ = 180◦, b′′c′′ = p sin α++ = 8r sin α− sin β− sin γ− while ∠ab′′c′′ = γ and ∠ac′′b′′ = β. we conclude that ∆a′′b′′c′′ is equilateral. c. the lengths of the explementary trisectors determining side b′′′c′′′ are found from the law of sines in ∆ab′′′c and ∆bc′′′a: ab′′′/ sin γ+ = ac/ sin(180◦ − α+ − γ+) = 2r sin 3β/ sin β = 8r sin β+ sin β−; so ab′′′ = 8r sin β+ sin β− sin γ+. similarly, ac′′′ = 8r sin γ+ sin γ− sin β+. then in ∆b′′′ac′′′, ∠b′′′ac′′′ = 2α+ − 3α = α−+, ab′′′ = p sin β−, ac′′′ = p sin γ−, where p = 8r sin β+ sin γ+. as α−+ + β− + γ− = 180◦, b′′′c′′′ = p sin α−+ = 8r sin α+ sin β+ sin γ+ while ∠ab′′′c′′′ = γ− and ∠ac′′′b′′′ = β−. we conclude that ∆a′′′b′′′c′′′ is equilateral. ii. mix morley triangles (9) the mix triangles are formed by the intersections of same type trisectors of an angle combined with a different type trisectors of the other two. hence, they share a vertex with a primitive triangle. for a short statement of the next theorem we define the mixable type of explementary, interior and exterior trisectors to be the interior, exterior and explementary trisectors respectively. theorem 0.2. in a triangle, the same type trisectors of an angle and the corresponding mixable type trisectors of the other two, proximal to sides respectively, meet at the vertices of an equilateral. proof. a. the intersections of the explementary trisectors of an angle and the interior trisectors of the other two proximal to sides respectively, form three triangles, referred as mix inner triangles since each of them shares a vertex with the inner triangle. we will study only one representative from them. consider the explementary trisectors of ∠b and the interior trisectors of the other two. cubo 19, 2 (2017) a trigonometrical approach to morley’s observation 79 assume the interior trisectors proximal to ac meet at b′ while the explementary and the interior proximal to bc and ab meet at bc and ba defining ∆b ′babc, called b-inner triangle as it shares vertex b′ with the inner triangle. the angles between the trisectors that determine it and the corresponding sides are equal to β+ (explementary) and α, γ (interior). find the lengths of the trisectors defining ∆b′babc: in ∆bbca, as ∠bbca = 180 ◦ − α − β+ = γ+, bbc/ sin α = abc/ sin β + = ab/ sin γ+ = 2r sin 3γ/ sin γ+ = 8r sin γ sin γ− and so bbc = 8r sin γ sin γ − sin α and abc = 8r sin γ sin γ − sin β+. similarly, in ∆cbab, bba = 8r sin α sin α − sin γ and cba = 8r sin α sin α − sin β+. in ∆ab′c, as ∠ab′c = 180◦ − α − γ = β++, ab′/ sin γ = cb′/ sin α = ac/ sin β− = 2r sin 3β/ sin β− = 8r sin β sin β+ and so ab′ = 8r sin β sin β+ sin γ and cb′ = 8r sin β sin β+ sin α. now in ∆bcab ′, ab′ = 8r sin β sin β+ sin γ = p sin β and abc = 8r sin γ sin γ ++ sin β+ = p sin γ++ where p = 8r sin β+ sin γ. as α + β + γ++ = 180◦, b′bc = 8r sin β+ sin γ sin α while ∠bcb ′a = γ++. similarly, in ∆bacb ′, b′bc = 8r sin β + sin γ sin α while ∠bab ′c = α++. also, in ∆bbabc, ∠bcbba = 2β + − 3β = β−+, bba = p sin α −, bbc = p sin γ −, where p = 8r sin α sin γ. as β−+ + α− + γ− = 180◦, babc = p sin β −+ = 8r sin α sin γ sin β−+ while ∠bbcba = α −, ∠bbabc = γ −. since sin β−+ = sin β+, we conclude that ∆b′babc is equilateral. corollary 2a. a mix inner and the inner triangle have two collinear sides and the third parallel. proof. consider for instance the b-inner ∆b′babc. it was seen ∠ab ′bc = γ ++. as it is equilateral ∠ab′ba = γ +. also, it was shown in the inner ∆a′b′c′ that ∠ab′c′ = γ+. thus, b′ba and b ′c′ are collinear. b. the intersections of the interior trisectors of an angle and the exterior trisectors of the other two, proximal to sides respectively, form three triangles, referred as mix central triangles since each of them shares a vertex with the central triangle. we will study only a representative from them. consider the interior trisectors of ∠c and the exterior trisectors of the other two. assume the exterior trisectors proximal to ab meet at c′′ while the interior and the exterior proximal to bc and ac meet at c′′a and c ′′ b defining ∆c ′′c′′ac ′′ b, called c-central triangle as it shares vertex c ′′ with the central triangle. the angles between the trisectors that determine it and the corresponding sides are equal to γ (interior) and α−, β− (exterior). find the lengths of the trisectors defining ∆c′′c′′ac ′′ b: in ∆cac′′b, as ∠c ′′ bac = 2α − + 3α = α++ and so ∠cc′′ba = β. hence ac′′b/ sin γ = cb ′′ a/ sin α ++ = ac/ sin β = 2r sin 3β/ sin β = 8r sin β− sin β+ and so ac′′b = 8r sin β − sin β+ sin γ and cc′′b = 8r sin β − sin β+ sin α++. similarly, in ∆cbc′′a, bc ′′ a = 8r sin α − sin α+ sin γ and cc′′a = 8r sin α − sin α+ sin β++. also in ∆ac′′b, 80 ioannis gasteratos, spiridon kuruklis & thedore kuruklis cubo 19, 2 (2017) figure 4: fig.4: mix morley triangles as ∠ac′′b = 180◦ − β− − α− = γ−+, ac′′/ sin β− = bc′′/ sin α− = ab/ sin γ−+ = 8r sin 3γ/ sin γ = 8r sin γ sin γ− and so ac′′ = 8r sin γ sin γ− sin β− and bc′′ = 8r sin γ sin γ− sin α−. then in ∆c′′bac ′′, ∠c′′bac ′′ = α−, ac′′b = p sin β −+, ac′′ = p sin γ− where p = 8r sin γ sin β−. as α− + β−+ + γ− = 180◦, c′′bc ′′ = p sin α− = 8r sin γ sin β− sin α−, ∠ac′′bc ′′ = γ−, ∠ac′′c′′b = β −+. similarly in ∆c′′abc ′′, c′′ac ′′ = 8r sin γ sin β− sin α− and ∠bc′′ac ′′ = γ−, ∠bc′′c′′a = α −+. also in ∆c′′acc ′′ b, ∠c′′acc ′′ b = γ, cc ′′ b = p sin β +, cc′′a = p sin α +, where p = 8r sin α− sin β−. as γ + β+ + α+ = 180◦, c′′ac ′′ b = p sin γ = 8r sin α − sin β− sin γ and ∠cc′′ac ′′ b = β +, ∠cc′′bc ′′ a = α +. we conclude that ∆cc′′ac ′′ b is equilateral. � corollary 2b. a mix central and the central triangle have two collinear sides and the third parallel. proof. consider for instance the mix central ∆c′′c′′ac ′′ b. it was seen ∠ac ′′c′′b = β −+. also, it was shown that in the central ∆a′′b′′c′′, ∠ac′′b′′ = β. as it is equilateral, ∠ac′′c′′b + ∠ac ′′b′′ + ∠b′′c′′a′′ = 180◦. so c′′bc ′′ and c′′a′′ are collinear. c. the intersections of the exterior trisectors of an angle and the explementary trisectors of the other two, proximal to sides respectively, define three triangles, referred as mix peripherals since each of them shares a vertex with the peripheral triangle. we will study only one representative from them. consider the exterior trisectors of ∠b and the explementary trisectors of ∠c and ∠a. cubo 19, 2 (2017) a trigonometrical approach to morley’s observation 81 assume that the explementary trisectors proximal to ac meet at b′′′ while the explementary and the exterior proximal to ab and bc meet at c′a and a ′ c defining ∆b ′′′c′aa ′ c, called b-peripheral triangle as it shares vertex b′′′ with the peripheral. the angles between the trisectors that determine it and the corresponding sides are equal to β− (exterior) and α+, γ+ (explementary). the lengths of trisectors that determine ∆b′′′c′aa ′ c are: in ∆bc′aa, ∠abc ′ a = β −, ∠bac′ = 180◦ − α+ = α−+ and so ∠bc′aa = γ −, ac′a/ sin β − = bc′a/ sin α −+ = ab/ sin γ− = 2r sin 3γ/ sin γ− = 8r sin γ sin γ+; thus ac′a = 8r sin γ sin γ + sin β− and bc′a = 8r sin γ sin γ + sin α−+. similarly, in ∆ca′cb, ca ′ c = 8r sin α sin α + sin β− and ba′c = 8r sin α sin α + sin γ−+. also in ∆ab′′′c, as ∠ab′′′c = 180◦ − α+ − γ+ = β, ab′′′/ sin γ+ = cb′′′/ sin α+ = ac/ sin β = 2r sin 3β/ sin β = 8r sin β+ sin β−; thus ab′′′ = 8r sin β+ sin β− sin γ+ and cb′′′ = 8r sin β+ sin β− sin α+. then in ∆c′aab ′′′, ∠c′aab ′′′ = 180◦ − 2α+ + 3α = α+, ac′a = p sin γ, ab′′′ = p sin β+ where p = 8r sin β− sin γ+. as α+ + γ + β+ = 180◦, b′′′c′a = p sin α + = 8r sin β− sin γ+ sin α+,∠ab′′′c′a = γ, ∠ac ′ ab ′′′ = β+. similarly, in ∆cb′′′a′c, b′′′a′c = 8r sin β − sin α+ sin γ+, ∠cb′′′a′c = α, ∠ca ′ cb ′′′ = β+. also, in ∆a′cbc ′ a, as ∠a ′ cbc ′ a = 2β − + 3β = β++, bc′a = p sin γ, ba ′ c = p sin α, where p = 8r sin γ+ sin α+. as β++ + α + γ = 180◦, c′aa ′ c = p sin β ++ = 8r sin γ+ sin α+ sin β++ while ∠ba′cc ′ a = γ, ∠bc ′ aa ′ c = α. since sin β ++ = sin β−, we conclude that ∆b′′′c′aa ′ c is equilateral. corollary 2c. a mix peripheral and the peripheral triangle have two collinear sides and the third parallel. proof. consider for instance the mix peripheral ∆b′′′c′aa ′ c. it was seen ∠ab ′′′c′a = γ. also in the peripheral equilateral, it was shown ∠ab′′′c′′′ = γ−. hence ∠ab′′′a′′′ = γ. thus, b′′′c′a and b′′′c′′′ are collinear. iii. complete morley triangles (6) the complete morley triangles are formed by the intersections of the interior, exterior and explementary trisectors from each angle, proximal to sides respectively. apparently, there are 3x2 such triangles. for example, the interior trisectors of ∠c combined with the explementary trisectors proximal to cb or ca form two different complete triangles. theorem 0.3. in a triangle, the trisectors of a distinct type from each angle, proximal to sides respectively, meet at the vertices of an equilateral. proof. from the 6 triangles formed by the intersections of the interior, exterior and explementary trisectors from each angle, proximal to sides, we will study only a representative as the rest are 82 ioannis gasteratos, spiridon kuruklis & thedore kuruklis cubo 19, 2 (2017) figure 5: fig.5: the ba-complete morley triangle similar. consider for instance the interior trisectors of ∠c, the exterior trisectors of ∠a and the explementary trisectors of ∠b, proximal to sides respectively. assume that the interior and the explementary proximal to cb meet at ba, the interior with the exterior proximal to ca meet at c ′′ b and the exterior with the explementary meet at c′b defining ∆bac ′′ bc ′ b called ba-complete triangle as it shares vertex ba with the b-inner triangle. it also shares vertex c ′′ b with the c-central, and vertex c′b with the a-peripheral triangle. the angles between the trisectors that determine it and the corresponding sides are equal to α−(exterior), β+ (explementary) and γ (interior). the lengths of the trisectors that determine it are: in ∆ac′bb, ∠bac ′ b = α −, ∠c′bba = 180 ◦ − β+ = β−+ and so ∠ac′bb = γ −. hence ac′b/ sin β −+ = c′b/ sin α − = ab/ sin γ− = 2r sin 3γ/ sin γ− = 8r sin γ sin γ+; thus ac′b = 8r sin γ sin γ + sin β−+ and bc′b = 8r sin γ sin γ + sin α−. similarly, in ∆bbac, as ∠babc = β +, ∠bbac = α +, cba/ sin β + = bba/ sin γ = bc/ sin α + = 2r sin 3α/ sin α+ = 8r sin α sin α−; thus cba = 8r sin α sin α − sin β+ and bba = 8r sin α sin α − sin γ. also in ∆cc′′ba, as ∠acc ′′ b = γ and ∠c ′′ bac = 2α − + 3α = α++, ∠ac′′bc = β, cc′′b/ sin α ++ = ac′′b/ sin γ = ac/ sin β = 2r sin 3β/ sin β = 8r sin β − sin β+; thus cc′′b = 8r sin β − sin β+ sin α++ and ac′′b = 8r sin β − sin β+ sin γ. then, in ∆c′′bac ′ b, ∠c ′′ bac ′ b = α −, ac′′b = p sin β −, ac′b = p sin γ + = p sin γ−+ where p = 8r sin γ sin β−+. as α− + β− + γ−+ = 180◦, c′′bc ′ b = p sin α − = 8r sin γ sin β−+ sin α− while ∠ac′bc ′′ b = β −, ∠ac′′bc ′ b = γ −+. cubo 19, 2 (2017) a trigonometrical approach to morley’s observation 83 also in ∆c′bbba, ∠c ′ bbba = 180 ◦ − 2β+ + 3β = β+, bc′b = p sin γ +, bba = p sin α where p = 8r sin γ sin α−. as β+ + γ+ + α = 180◦, c′bba = p sin β + = 8r sin γ sin α− sin β+ while ∠bbac ′ b = γ + and ∠bc′bba = α. furthermore, in ∆bacc ′′ b, ∠bacc ′′ b = γ, cc ′′ b = p sin β − = p sin β++, cba = p sin α, where p = 8r sin α++ sin β+. as γ + β++ + α = 180◦, c′′bac = p sin γ = 8r sin α ++ sin β+ sin γ while ∠cacc ′′ b = 120 ◦ + β = β++ and ∠cc′′bba = α. we conclude that ∆bac ′′ bc ′ b is equilateral. corollary 3. a complete and a mix inner, or a mix central or a mix peripheral triangle, sharing a vertex, have two collinear sides and the third parallel. proof. consider for instance the complete ∆bac ′′ bc ′ b. this shares a vertex with the mix inner ∆b′babc, the mix central ∆c ′′c′′ac ′′ b, and the mix peripheral ∆a ′′′b′cc ′ b. it was shown ∠bbac ′ b = γ +. however, in the b-inner ∆b′babc, ∠bbabc = γ −. since it is equilateral, ∠bcbab ′ = 60◦. then ∠c′bbab ′ = ∠c′bbab+∠bbabc +∠bcbab ′ = γ+ +γ− +60◦ = 180◦ and so c′bba and bab ′ are collinear. hence, ∆bac ′′ bc ′ b and ∆b ′babc have two collinear sides with the third ones parallel. it was shown ∠ac′′bc ′ b = γ −+. as it is equilateral ∠ac′′bba = γ −. however, in the c-central, ∆c′′c′′ac ′′ b ∠ac ′′ bc ′′ = γ−. thus, c′′bc ′′ and c′′bba are collinear. hence, ∆bac ′′ bc ′ b and ∆c ′′c′′ac ′′ b have two collinear sides with the third parallel. it was shown ∠ac′bc ′′ b = β −. since is ∆bac ′′ bc ′ b equilateral, ∠ac ′ bba = β. however, in the b-peripheral, ∠bc′aa ′ c = α. symmetrically, in the a-peripheral ∆a ′′′b′cc ′ b, ∠ac ′ bb ′ c = β. hence ∠ac′bba = ∠ac ′ bb ′ c and so c ′ bba and c ′ bb ′ c are collinear. thus ∆bac ′′ bc ′ b and ∆a ′′′b′cc ′ b have two collinear sides with the third parallel. from the above we infer that a precise formulation of the general morley’s theorem yielding 18 equilaterals is: in a triangle, the same type trisectors of the three angles, the same type trisectors of an angle with its mixable type trisectors of the other two, and the trisectors of a distinct type from each angle, proximal to sides respectively, meet at the vertices of an equilateral. 3. arrangement of morley triangles and alignment of intersections of proximal trisectors the trisectors of a triangle, proximal to one of its sides, meet at 27 points. each of them is a common vertex of two morley triangles which are arranged with two sides collinear and the third parallel. from corollary 3, sides c′′bba and c ′′ bc ′ b of the complete ∆bac ′′ bc ′ b and the mix central ∆c′′c′′bc ′′ a are collinear. hence, c ′′ b and ba lie on the line determined by the side c ′′a′′ of the 84 ioannis gasteratos, spiridon kuruklis & thedore kuruklis cubo 19, 2 (2017) central triangle ∆a′′b′′c′′. thus, on the line determined by a side of the central triangle lie 6 intersections of trisectors proximal to a side, two of the interior with the exterior trisectors, two of the interior with the explementary trisectors and two between the exterior trisectors of the central triangle. also, sides bac ′ b and bab of the complete ∆bac ′′ bc ′ b and the mix inner are collinear. hence, c′′b and ba lie on the line which is determined by a side of the inner triangle. thus, on the line determined by a side of the inner triangle lie 6 intersections of trisectors proximal to a side, two of the exterior with the explementary trisectors, two of interior with the explementary trisectors and two intersections between interior trisectors of the inner triangle. finally, sides c′bc ′′ b and c ′ ba ′′′ of the complete ∆bac ′′ bc ′ b and the mix peripheral triangle ∆a′′′b′cc ′ b are collinear. as the a-peripheral and the peripheral have sides a ′′′c′b and a ′′′b′′′ collinear, c′′b and c ′ b lie on side a ′′′b′′′. symmetrically c′′b and c ′ b lie on side a ′′′b′′′ as well. thus, on a side of the peripheral triangle lie 6 intersections of trisectors proximal to a side, two of the interior with the exterior trisectors, two of the explementary with the exterior trisectors and two between the explementary trisectors of the peripheral triangle. conclude that the intersections of trisectors proximal to a side lie 6 by 6 on three triples of parallel lines intersecting with 60◦ angles. 4. open problems next there are three basic questions stemming out from this work inviting further exploration. 1. how many equilaterals do the intersections of trisectors in a triangle determine? 2. are there lines or circles, beyond morley’s 3 triples of parallel lines, on which the intersections of trisectors lie? 3. do theorems exist, analog to morley’s theorem regarding angle trisectors, for the side or perpendicular trisectors? very special thanks to gerry ladas and thanasis fokas for their unceasing encouragement to our long efforts in demystifying morley’s mystery. references [1] paul and jack abad, the hundred greatest theorems, in actes de la confrence jfla2008. [2] b. bollobás, the art of mathematics, cambridge university press, (2006), 126-127. [3] alain connes, a new proof of morley’s theorem, publications mathmatiques de l’ ihés, (1998), 43-46. cubo 19, 2 (2017) a trigonometrical approach to morley’s observation 85 [4] john conway, the power of mathematics, in power, cambridge university press, (2006), 36-50. [5] john conway, on morley’s trisector theorem, the mathematical intelligencer, 36, no. 3(2014), 3-3 [6] edsger w. dijkstra, a collection of beautiful proofs, selected writings on computing: a personal perspective, springer, (1982), 174-183. [7] edsger w. dijkstra, on the design of a simple proof for morley’s theorem, programming and mathematical method, volume 88 of the nato asi series, springer (1992) 3-9. [8] w. j. dobbs, morley’s triangle, the mathematical gazette, 1938 jstor. [9] richard k. guy, the lighthouse theorem a budget of paradoxes, amer. math. monthly, 114 (2007), 97-141 [10] clark kimberling, central points and central lines in the plane of a triangle, mathematics magazine, 67 (1994), 163-187. [11] spiridon a. kuruklis, trisectors like bisectors with equilaterals instead of points, cubo 16, 2 (2014), 71-110. [12] henri lebesgue, sur les n-sectrices d’un triangle [en memoire de frank morley (1860-1930)], enseingnement math., 38 (1940), 39-58. [13] frank morley, extensions of clifford’s chain-theorem, american j. of math., 51 (1929), 465472. [14] roger penrose, morley’s trisector theorem, eureka the archimedeans journal, cambridge, 16 (1953), 6-7. [15] david wells, are these the most beautiful? the mathematical intelligencer, 12 (1990), 37-41. [16] glanville f. taylor and l.w. marr, the six trisectors of each of the angles of a triangle, proceedings of the edinburgh mathematical society 33 (1913-14), 119-131. [17] alexander bogomolny, morley’s miracle, interactive mathematics miscellany and puzzles, http://www.cut-the-knot.org/triangle/morley/ cubo, a mathematical journal vol. 24, no. 03, pp. 393–412, december 2022 doi: 10.56754/0719-0646.2403.0393 two nonnegative solutions for two-dimensional nonlinear wave equations svetlin georgiev1, b mohamed majdoub2,3 1 department of differential equations, faculty of mathematics and informatics, university of sofia, sofia, bulgaria. svetlingeorgiev1@gmail.com b 2 department of mathematics, college of science, imam abdulrahman bin faisal university, p. o. box 1982, dammam, saudi arabia. 3 basic and applied scientific research center, imam abdulrahman bin faisal university, p.o. box 1982, 31441, dammam, saudi arabia. mmajdoub@iau.edu.sa abstract we study a class of initial value problems for two-dimensional nonlinear wave equations. a new topological approach is applied to prove the existence of at least two nonnegative classical solutions. the arguments are based upon a recent theoretical result. resumen estudiamos una clase de problemas de valor inicial para ecuaciones de onda no lineales en dimensión dos. se aplica un nuevo enfoque topológico para probar la existencia de al menos dos soluciones clásicas no negativas. los argumentos se basan en un resultado teórico reciente. keywords and phrases: hyperbolic equations, positive solution, fixed point, cone, sum of operators. 2020 ams mathematics subject classification: 47h10, 58j20, 35l15. accepted: 26 september, 2022 received: 21 september, 2021 ©2022 s. georgiev et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0393 https://orcid.org/0000-0001-8015-4226 https://orcid.org/0000-0001-6038-1069 mailto:svetlingeorgiev1@gmail.com mailto:mmajdoub@iau.edu.sa 394 s. georgiev & m. majdoub cubo 24, 3 (2022) 1 introduction global existence for nonlinear wave equations is an important mathematical topic. mathematicians, including f. john, s. kleinerman, l. hörmander, etc., have made investigations to this subject. the first non-trivial long-time existence result was established by f. john and s. kleinerman in [19], where it is proved the almost global existence for a class 3d quasilinear scalar wave equations. global existence for 3d quasilinear wave equations was established firstly by s. kleinerman in [20] and by d. christodoulou, independently by s. kleinerman, in [5]. the problem in 2d case is quite delicate. introducing the ghost weight, in [1] was proved the global well-posedness for a class 2d nonlinear wave equations. using a class hardy-type inequality depending on the compact support of the initial data, in [21] was proved almost global existence for 2d case. here we propose a new approach for investigations for classical solutions of a class 2d nonlinear wave equations. we investigate for existence of at least two positive solutions for the following ivp utt − ∆u = f(t, x, u, ut, ux), t > 0, x = (x1, x2) ∈ r2, u(0, x) = u0(x), x = (x1, x2) ∈ r2, (1.1) ut(0, x) = u1(x), x = (x1, x2) ∈ r2, where ∆u = ux1x1 + ux2x2, ux = (ux1, ux2). the initial value problem (1.1) has attracted considerable attention in the mathematical community and the well-posedness theory in the sobolev spaces for polynomial type nonlinearities has been extensively studied. the case of exponential nonlinearity was recently investigated (see [18] and references therein). in particular, if the nonlinearity f and the initial data u0, u1 are smooth then the cauchy problem (1.1) has a classical local (in time) solution. this follows from duhamel’s formula via the usual fixed point argument in the space hsloc × h s−1 loc , s > 2. such an s guarantee that u, ut, ∇u are in l∞. note that u ∈ hsloc means that the h s norm over a ball centered at x0 and with radius 1 is uniformly bounded by a constant independent of x0. we refer the reader to [23] and references therein for more properties and information on nonlinear wave equations. in [17] is proved existence and uniqueness of generalized solutions of the first initial boundary value problem for strongly hyperbolic systems in bounded domains. in the case when f(t, x, u, ut, ux) = f(u(x)), t > 0, x ∈ r2, and u0(x) = u1(x) = 0, x ∈ r2, the problem (1.1) is investigated in [14] where the authors prove existence of at least one nontrivial classical solution of the problem (1.1). cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 395 we make the following assumptions on the non-linearity and initial data trough the paper. (h1) u0, u1 ∈ c2(r2), 0 ≤ u0, |u0x1|, |u0x1x1|, |u0x2|, |u0x2x2| ≤ r, 0 ≤ u1, |u1x1|, |u1x1x1|, |u1x2|, |u1x2x2| ≤ r on r 2, where r > 0 is a given constant. (h2) f ∈ c([0, ∞) × r6), 0 ≤ f(t, x, w1, w2, w3, w4) ≤ l∑ j=1 (aj(t, x)|w1|pj + bj(t, x)|w2|pj + cj(t, x)|w3|pj + dj(t, x)|w4|pj ) , (t, x) ∈ [0, ∞) × r2, where aj, bj, cj, dj ∈ c([0, ∞) × r2), 0 ≤ aj, bj, cj, dj ≤ a, pj > 0, j ∈ {1, . . . , l}, where a > 0 and l ∈ n are given constants. our main result reads as follows. theorem 1.1. suppose (h1) and (h2). then the ivp (1.1) has at least two nonnegative classical solutions. to prove our main result we use a new topological approach. this approach can be used for investigations for existence of at least one and at least two classical solutions for initial value problems, boundary value problems and initial boundary value problems for some classes ordinary differential equations, partial differential equations and fractional differential equations (see [2, 3, 4, 7, 10, 12, 13, 15, 16] and references therein). so far, for the authors they are not known investigations for existence of multiple solutions for the ivp (1.1). the paper is organized as follows. in the next section, we give some auxiliary results. in section 3, we prove our main result. in section 4, we give an example. 396 s. georgiev & m. majdoub cubo 24, 3 (2022) 2 auxiliary results let x be a real banach space. definition 2.1. a mapping k : x → x is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets. the concept for k-set contraction is related to that of the kuratowski measure of noncompactness which we recall for completeness. definition 2.2. let ωx be the class of all bounded sets of x. the kuratowski measure of noncompactness α : ωx → [0, ∞) is defined by α(y ) = inf  δ > 0 : y = m⋃ j=1 yj and diam(yj) ≤ δ, j ∈ {1, . . . , m}   , where diam(yj) = sup{∥x − y∥x : x, y ∈ yj} is the diameter of yj, j ∈ {1, . . . , m}. for the main properties of measure of noncompactness we refer the reader to [6]. definition 2.3. for a given number k ≥ 0, a map k : x → x is said to be k-set contraction if it is continuous, bounded and α(k(y )) ≤ kα(y ) for any bounded set y ⊂ x. obviously, if k : x → x is a completely continuous mapping, then k is 0-set contraction. definition 2.4. let x and y be real banach spaces. a mapping k : x → y is said to be expansive if there exists a constant h > 1 such that ∥kx − ky∥y ≥ h∥x − y∥x for any x, y ∈ x. definition 2.5. a closed, convex set p in x is said to be a cone if (1) αx ∈ p for any α ≥ 0 and for any x ∈ p, (2) x, −x ∈ p implies x = 0. cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 397 let p ⊂ x be a cone and define p∗ = p\{0}, pr1 = { u ∈ p : ∥u∥ ≤ r1 } , pr1,r2 = { u ∈ p : r1 ≤ ∥u∥ ≤ r2 } for positive constants r1, r2 such that 0 < r1 ≤ r2. the following result will be used to prove theorem 1.1 . we refer the reader to [8] and [11] for more details. theorem 2.6. let p be a cone of a banach space e; ω a subset of p and u1, u2 and u3 three open bounded subsets of p such that u1 ⊂ u2 ⊂ u3 and 0 ∈ u1. assume that t : ω → p is an expansive mapping with constant h > 1, s : u3 → e is a k-set contraction with 0 ⩽ k < h − 1 and s(u3) ⊂ (i − t)(ω). suppose that (u2 \ u1) ∩ ω ̸= ∅, (u3 \ u2) ∩ ω ̸= ∅, and there exists u0 ∈ p∗ such that the following conditions hold: (i) sx ̸= (i − t)(x − λu0), for all λ > 0 and x ∈ ∂u1 ∩ (ω + λu0), (ii) there exists ϵ > 0 such that sx ̸= (i − t)(λx), for all λ ≥ 1 + ϵ, x ∈ ∂u2 and λx ∈ ω, (iii) sx ̸= (i − t)(x − λu0), for all λ > 0 and x ∈ ∂u3 ∩ (ω + λu0). then t + s has at least two non-zero fixed points x1, x2 ∈ p such that x1 ∈ ∂u2 ∩ ω and x2 ∈ (u3 \ u2) ∩ ω or x1 ∈ (u2 \ u1) ∩ ω and x2 ∈ (u3 \ u2) ∩ ω. note that (see [9]) the function g(t, x, τ, ξ) = − 1 2π h(t − τ − |x − ξ|)√ (t − τ)2 − |x − ξ|2 , t, τ > 0, x, ξ ∈ r2, where |x − ξ| = √ (x1 − ξ1)2 + (x2 − ξ2)2, is the green function for the two-dimensional wave equation utt − ∆u = h(t, x), t > 0, x = (x1, x2) ∈ r2, u(0, x) = ut(0, x) = 0, x = (x1, x2) ∈ r2, where h(·) denotes the heaviside function. observe that g(t, x, τ, ξ) ≤ 0, t, τ > 0, x, ξ ∈ r2. 398 s. georgiev & m. majdoub cubo 24, 3 (2022) a key lemma in our proof is the following. lemma 2.7. for h1, h2, p > 0, we have∣∣∣∣ ∫ r2 ∫ ∞ 0 (h1 + h2τ) pg(t, x, τ, ξ)dτdξ ∣∣∣∣ ≤ (h1 + h2t)pi(t), (t, x) ∈ (0, ∞) × r2, (2.1) where i(t) = t3 + t2(1 + | log t|). proof. let h1, h2, p > 0 and t > 0. one has ∣∣∣∣ ∫ r2 ∫ ∞ 0 (h1 + h2τ) pg(t, x, τ, ξ)dτdξ ∣∣∣∣ ≤ 12π ∫ |x−ξ|≤t ∫ t−|x−ξ| 0 (h1 + h2τ) p√ (t − τ)2 − |x − ξ|2 dτdξ ≤ (h1 + h2t) p 2π ∫ |x−ξ|≤t ( log(t + √ t2 − |x − ξ|2) − log |x − ξ| ) dξ = (h1 + h2t) p 2π (∫ |x−ξ|≤t log(t + √ t2 − |x − ξ|2)dξ − ∫ |x−ξ|≤t log |x − ξ|dξ ) ≤ (h1 + h2t) p 2π ( log(2t) ∫ |x−ξ|≤t dξ − 2π ∫ t 0 r1 log r1dr1 ) = (h1 + h2t) p 2π ( πt2 log(2t) − π ( t2 log t − t2 2 )) ≤ (h1 + h2t) p 2 ( t2 log(1 + 2t) + t2| log t| + t2 2 ) ≤ (h1 + h2t) p 2 ( 2t3 + t2| log t| + t2 2 ) ≤ (h1 + h2t)p ( t3 + t2(1 + | log t|) ) . this gives (2.1) as desired. we make the change u = v + u0 + tu1. then, we get the ivp vtt − ∆v = f(t, x, v + u0 + tu1, vt + u1, vx + u0x + tu1x) + ∆u0 + t∆u1 = f1(t, x, v, vt, vx), t > 0, x ∈ r2, (2.2) v(0, x) = vt(0, x) = 0, x ∈ r2. lemma 2.8. suppose (h2). if wk ∈ r, |wk| ≤ b, k ∈ {1, . . . , 4}, for some positive b, then f(t, x, w1, w2, w3, w4) ≤ 4a l∑ j=1 bpj . cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 399 proof. we have 0 ≤ f(t, x, w1, w2, w3, w4) ≤ l∑ j=1 (aj(t, x)|w1|pj + bj(t, x)|w2|pj + cj(t, x)|w3|pj + dj(t, x)|w4|pj ) ≤ l∑ j=1 (abpj + abpj + abpj + abpj ) = 4a l∑ j=1 bpj , (t, x, w1, w2, w3, w4) ∈ [0, ∞) × r6. this completes the proof. let e = c2([0, ∞) × r2) and for any u ∈ e, denote ∥u∥ = max { ∥u∥∞, ∥ut∥∞, ∥utt∥∞∥uxj ∥∞, ∥uxjxj ∥∞, j ∈ {1, 2} } , provided that it is finite, where ∥v∥∞ = sup (t,x)∈[0,∞)×r2 |v(t, x)|. lemma 2.9. suppose (h1) and (h2). let v ∈ e, ∥v∥ ≤ b, for some positive b. then f(t, x, v + u0 + tu1, vt + u1, vx + u0x + tu1x) ≤ 4a l∑ j=1 (b + r(1 + t))pj , (t, x) ∈ [0, ∞) × r2. proof. let w1 = v + u0 + tu1, w2 = vt + u0 + tu1, w3 = ux1 + u0x1 + tu1x1, w4 = vx2 + u0x2 + tu1x2. then |wj| ≤ b + r(1 + t), j ∈ {1, . . . , 4}, t ≥ 0. hence and lemma 2.8, we get the desired result. this completes the proof. 400 s. georgiev & m. majdoub cubo 24, 3 (2022) lemma 2.10. suppose (h1) and (h2). let v ∈ e, ∥v∥ ≤ b, for some positive b. then |f1(t, x, v, vt, vx)| ≤ 4a l∑ j=1 (b + r(1 + t))pj + 2r(1 + t), (t, x) ∈ [0, ∞) × r2. proof. by (h1), we get |∆u0| ≤ 2r, |∆u1| ≤ 2r on r2. using lemma 2.9, we obtain |f1(t, x, v, vt, vx)| ≤ f(t, x, v + u0 + tu1, vt + u1, vx + u0x + tu1x) + |∆u0| + t|∆u1| ≤ 4a l∑ j=1 (b + r(1 + t))pj + 2r(1 + t), (t, x) ∈ [0, ∞) × r2. this completes the proof. now, applying lemma 2.10 and (2.1), we obtain the following result. lemma 2.11. suppose (h1) and (h2). then ∣∣∣∣ ∫ r2 ∫ ∞ 0 g(t, x, τ, ξ)f1(τ, ξ, v(τ, ξ), vt(τ, ξ), vx(τ, ξ))dτdξ ∣∣∣∣ ≤  4a l∑ j=1 (b + r(1 + t))pj + 2r(1 + t)  i(t) ≤  4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj tpj + 2r(1 + t)  i(t), (t, x) ∈ [0, ∞) × r2. take a nonnegative function g ∈ c([0, ∞) × r2). suppose that v ∈ e is a solution to the integral equation. 0 = 1 8 ∫ t 0 ∫ x1 0 ∫ x2 0 (x1 − s1)2(x2 − s2)2(t − t1)2g(t1, s1, s2)v(t1, s1, s2)ds2ds1dt1 − 1 16π ∫ t 0 ∫ x1 0 ∫ x2 0 (x1 − s1)2(x2 − s2)2(t − t1)2g(t1, s1, s2) ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1ds2ds1dt1, (2.3) t ≥ 0, (x1, x2) ∈ r2. we differentiate three times in t, three times in x1 and three times in x2 the equation (2.3) and we obtain 0 = g(t, x)v(t, x) − 1 2π g(t, x) ∫ r2 ∫ ∞ 0 g(t, x, τ, ξ)f1(τ, ξ, v(τ, ξ), vt(τ, ξ), vx(τ, ξ))dτdξ, cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 401 t ≥ 0, x ∈ r2, whereupon 0 = v(t, x) − 1 2π ∫ r2 ∫ ∞ 0 g(t, x, τ, ξ)f1(τ, ξ, v(τ, ξ), vt(τ, ξ), vx(τ, ξ))dτdξ, t ≥ 0, x ∈ r2. hence, using the green function, we conclude that v is a solution of the ivp (2.2). thus, any solution v ∈ e of the integral equation (2.3) is a solution to the ivp (2.2). (h3) let m > 0 be large enough and a, r1, l1, r1 be positive constants that satisfy the following conditions r1 < l1 < r1, r1 < r, r1 > ( 2 5m + 1 ) l1, a  r1 + 4a l∑ j=1 (2(r1 + r)) pj + 4a l∑ j=1 (2r)pj + 2r   < l1 20 . (h4) there exists a nonnegative function g ∈ c([0, ∞) × r2) such that q(t, x1, x2) = ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)g(t1, s1, s2) × ( 1 + |x1 − s1| + (x1 − s1)2 )( 1 + |x2 − s2| + (x2 − s2)2 ) × ( 1 + (t − t1) + (t − t1)2 )1 +  1 + t1 + l∑ j=1 t pj 1  i(t1)  ds2ds1dt1 ≤ a, (t, x1, x2) ∈ [0, ∞) × r2. in the last section we will give an example for the constants m, a, r, l1, r1 and r and for a function g that satisfy (h3) and (h4). for v ∈ e, define the operator fv(t, x1, x2) = 1 8 ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2g(t1, s1, s2) × v(t1, s1, s2)ds2ds1dt1 − 1 16π ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2g(t1, s1, s2) × ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1ds2ds1dt1, (t, x1, x2) ∈ [0, ∞) × r2. lemma 2.12. suppose (h1)–(h3). then, for v ∈ e, ∥v∥ ≤ b, for some positive b, we have ∥fv∥ ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   . 402 s. georgiev & m. majdoub cubo 24, 3 (2022) proof. using lemma 2.11 and (h3), we get |fv(t, x1, x2)| ≤ 1 8 ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2g(t1, s1, s2) × |v(t1, s1, s2)|ds2ds1dt1 + 1 16π ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2g(t1, s1, s2) × ∣∣∣∣ ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1 ∣∣∣∣ds2ds1dt1 ≤ ba + 4a l∑ j=1 (2(b + r))pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2 × g(t1, s1, s2)i(t1)ds2ds1dt1 + 4a l∑ j=1 (2r)pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2 × g(t1, s1, s2)t pj 1 i(t1)ds2ds1dt1 + 2r ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)2 × g(t1, s1, s2)(1 + t1)i(t1)ds2ds1dt1 ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2, and ∣∣∣∣ ∂∂tfv(t, x1, x2) ∣∣∣∣ ≤ 14 ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)g(t1, s1, s2) × |v(t1, s1, s2)|ds2ds1dt1 + 1 8π ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1)g(t1, s1, s2) × ∣∣∣∣ ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1 ∣∣∣∣ds2ds1dt1 ≤ ba + 4a l∑ j=1 (2(b + r))pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1) × g(t1, s1, s2)i(t1)ds2ds1dt1 cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 403 + 4a l∑ j=1 (2r)pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1) × g(t1, s1, s2)t pj 1 i(t1)ds2ds1dt1 + 2r ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2(t − t1) × g(t1, s1, s2)(1 + t1)i(t1)ds2ds1dt1 ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2, and ∣∣∣∣ ∂2∂t2 fv(t, x1, x2) ∣∣∣∣ ≤ 14 ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2g(t1, s1, s2) × |v(t1, s1, s2)|ds2ds1dt1 + 1 8π ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2g(t1, s1, s2) × ∣∣∣∣ ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1 ∣∣∣∣ds2ds1dt1 ≤ ba + 4a l∑ j=1 (2(b + r))pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2 × g(t1, s1, s2)i(t1)ds2ds1dt1 + 4a l∑ j=1 (2r)pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2 × g(t1, s1, s2)t pj 1 i(t1)ds2ds1dt1 + 2r ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x1 − s1)2(x2 − s2)2 × g(t1, s1, s2)(1 + t1)i(t1)ds2ds1dt1 ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2, and ∣∣∣∣ ∂∂x1 fv(t, x1, x2) ∣∣∣∣ ≤ 14 ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)|x1 − s1|(x2 − s2)2(t − t1)2g(t1, s1, s2) × |v(t1, s1, s2)|ds2ds1dt1 + 1 8π ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)|x1 − s1|(x2 − s2)2(t − t1)2g(t1, s1, s2) 404 s. georgiev & m. majdoub cubo 24, 3 (2022) × ∣∣∣∣ ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1 ∣∣∣∣ds2ds1dt1 ≤ ba + 4a l∑ j=1 (2(b + r))pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)|x1 − s1|(x2 − s2)2(t − t1)2 × g(t1, s1, s2)i(t1)ds2ds1dt1 + 4a l∑ j=1 (2r)pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)|x1 − s1|(x2 − s2)2(t − t1)2 × g(t1, s1, s2)t pj 1 i(t1)ds2ds1dt1 + 2r ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)|x1 − s1|(x2 − s2)2(t − t1)2 × g(t1, s1, s2)(1 + t1)i(t1)ds2ds1dt1 ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2, and ∣∣∣∣ ∂2∂x21 fv(t, x1, x2) ∣∣∣∣ ≤ 14 ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x2 − s2)2(t − t1)2g(t1, s1, s2) × |v(t1, s1, s2)|ds2ds1dt1 + 1 8π ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x2 − s2)2(t − t1)2g(t1, s1, s2) × ∣∣∣∣ ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ 0 g(t1, s1, s2, t2, ξ1, ξ2) × f1(t2, ξ1, ξ2, v(t2, ξ1, ξ2), vt(t2, ξ1, ξ2), vx(t2, ξ1, ξ2))dt2dξ2dξ1 ∣∣∣∣ds2ds1dt1 ≤ ba + 4a l∑ j=1 (2(b + r))pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x2 − s2)2(t − t1)2 × g(t1, s1, s2)i(t1)ds2ds1dt1 + 4a l∑ j=1 (2r)pj ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x2 − s2)2(t − t1)2 × g(t1, s1, s2)t pj 1 i(t1)ds2ds1dt1 + 2r ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)(x2 − s2)2(t − t1)2 × g(t1, s1, s2)(1 + t1)i(t1)ds2ds1dt1 cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 405 ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2. as above, one can obtain ∣∣∣∣ ∂∂x2 fv(t, x1, x2) ∣∣∣∣ ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2, and ∣∣∣∣ ∂2∂x22 fv(t, x1, x2)| ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   , (t, x1, x2) ∈ [0, ∞) × r2. consequently ∥fv∥ ≤ a  b + 4a l∑ j=1 (2(b + r))pj + 4a l∑ j=1 (2r)pj + 2r   . this completes the proof. 3 proof of the main result let p̃ = {u ∈ e : u ≥ 0 on [0, ∞) × r2}. with p we will denote the set of all equi-continuous families in p̃. note that fv ≥ 0 for any v ∈ p. let ϵ > 0. for v ∈ e, define the operators tv(t, x) = (1 + mϵ)v(t, x) − ϵ l1 10 , sv(t, x) = −ϵfv(t, x) − mϵv(t, x) − ϵ l1 10 , (t, x) ∈ [0, ∞) × r2. note that any fixed point v ∈ e of the operator t + s is a solution to the ivp (2.2). define u1 = pr1 = {v ∈ p : ∥v∥ < r1}, u2 = pl1 = {v ∈ p : ∥v∥ < l1}, u3 = pr1 = {v ∈ p : ∥v∥ < r1}, 406 s. georgiev & m. majdoub cubo 24, 3 (2022) r2 = r1 + a m  r1 + 4a l∑ j=1 (2(r1 + r)) pj + 4a l∑ j=1 (2r)pj + 2r   + l1 5 , ω = pr2 = {v ∈ p : ∥v∥ ≤ r2}. 1. for v1, v2 ∈ ω, we have ∥tv1 − tv2∥ = (1 + mϵ)∥v1 − v2∥, whereupon t : ω → e is an expansive operator with a constant 1 + mϵ > 1. 2. for v ∈ pr1, we get ∥sv∥ ≤ ϵ∥fv∥ + mϵ∥v∥ + l1 10 ≤ ϵ ( a  r1 + 4a l∑ j=1 (2(r1 + r)) pj + 4a l∑ j=1 (2r)pj + 2r   + mr1 + l1 10 ) . therefore s(pr1) is uniformly bounded. since s : pr1 → e is continuous, we have that s(pr1) is equi-continuous. consequently s : pr1 → e is a 0-set contraction. 3. let v1 ∈ pr1. set v2 = v1 + 1 m fv1 + l1 5m . note that by the second inequality of (h3) and by lemma 2.12, it follows that ϵfv+ϵl1 5 ≥ 0 on [0, ∞) × r2. we have v2 ≥ 0 on [0, ∞) × r2 and ∥v2∥ ≤ ∥v1∥ + 1 m ∥fv1∥ + l1 5m ≤ r1 + a m ( r1 + 4a l∑ j=1 (2(r1 + r)) pj + 4a l∑ j=1 (2r)pj + 2r ) + l1 5 = r2. therefore v2 ∈ ω and −ϵmv2 = −ϵmv1 − ϵfv1 − ϵ l1 10 − ϵ l1 10 or (i − t)v2 = −ϵmv2 + ϵ l1 10 = sv1. consequently s(pr1) ⊂ (i − t)(ω). 4. suppose that there exists an v0 ∈ p∗ such that t(v−λv0) ∈ p, v ∈ ∂pr1, v ∈ ∂pr1 ⋂ (ω+λu0) cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 407 and sv = v − λv0 for some λ ≥ 0. then r1 = ∥v − λv0∥ = ∥sv∥ ≥ −sv(t, x) = ϵfv(t, x) + ϵmv(t, x) + ϵ l1 10 ≥ ϵ l1 20 , (t, x) ∈ [0, ∞) × r2, because by the second inequality of (h3) and by lemma 2.12, it follows that ϵfv + ϵl1 20 ≥ 0 on [0, ∞) × r2. 5. suppose that for any ϵ1 > 0 small enough there exist a u ∈ ∂pl and λ1 ≥ 1 + ϵ1 such that λ1u ∈ pr1 and su = (i − t)(λ1u). (3.1) in particular, for ϵ1 > 2 5m , we have u ∈ ∂pl, λ1u ∈ pr1, λ1 ≥ 1 + ϵ1 and (3.1) holds. since u ∈ ∂pl and λ1u ∈ pr1, it follows that( 2 5m + 1 ) l < λ1l = λ1∥u∥ ≤ r1. moreover, −ϵfu − mϵu − ϵ l 10 = −λ1mϵu + ϵ l 10 , or fu + l 5 = (λ1 − 1)mu. from here, 2 l 5 ≥ ∥∥∥∥fu + l5 ∥∥∥∥ = (λ1 − 1)m∥u∥ = (λ1 − 1)ml and 2 5m + 1 ≥ λ1, which is a contradiction. therefore all conditions of theorem 2.6 hold. hence, the ivp (2.2) has at least two solutions v1 and v2 so that r1 < ∥v1∥ < l1 < ∥v2∥ < r1, and u = v1 + u0 + tu1, w = v2 + u0 + tu1 408 s. georgiev & m. majdoub cubo 24, 3 (2022) are two different positive solutions of the ivp (1.1). this completes the proof. 4 an example let l = 1, p1 = 3 5 , r1 = r = 1, a = 200, l1 = 1 2 , r1 = 1 100 , m = 1050, ϵ = 50, a = 1 1010 , r = 100. then r1 > ( 2 5m + 1q ) l1, r1 < l1 < r1, r1 < l1 20 . also, a  r1 + 4a l∑ j=1 (2(r1 + r)) pj + 4a l∑ j=1 (2r)pj + 2r   = 1 1010 ( 1 + 800 · (4)2 + 800 · 4 + 2 ) < 1 40 = l1 20 . consequently (h3) holds. now, we will construction the function g in (h4). let h(x) = log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 , l(s) = arctan s11 √ 2 1 − s22 , s ∈ r. then h′(s) = 22 √ 2s10(1 − s22) (1 − s11 √ 2 + s22)(1 + s11 √ 2 + s22) , l′(s) = 11 √ 2s10(1 + s20) 1 + s40 , s ∈ r. therefore −∞ < lim s→±∞ (1 + s + s2)h(s) < ∞, −∞ < lim s→±∞ (1 + s + s2)l(s) < ∞. hence, there exists a positive constant c1 so that (1 + s + s2) ( 1 44 √ 2 log 1 + s11 √ 2 + s22 1 − s11 √ 2 + s22 + 1 22 √ 2 arctan s11 √ 2 1 − s22 ) ≤ c1, s ∈ r. cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 409 note that by [22, p. 707, integral 79], we have ∫ dz 1 + z4 = 1 4 √ 2 log 1 + z √ 2 + z2 1 − z √ 2 + z2 + 1 2 √ 2 arctan z √ 2 1 − z2 . let q(s) = s10 (1 + s44)(1 + s + s2)2(1 + ((1 + s + s2)i(s))2) , s ∈ r, and g1(t, x1, x2) = q(t)q(x1)q(x2), t ∈ [0, ∞), x1, x2 ∈ r. then there exists a constant c2 > 0 so that c2 ≥ ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)g1(t1, s1, s2) × ( 1 + |x1 − s1| + (x1 − s1)2 )( 1 + |x2 − s2| + (x2 − s2)2 ) × ( 1 + (t − t1) + (t − t1)2 )( 1 + (1 + t1 + t 2 1)i(t1) ) ds2ds1dt1, (t, x1, x2) ∈ [0, ∞) × r2. now, we take g(t, x1, x2) = 1 1020c2 g1(t, x1, x2), (t, x1, x2) ∈ [0, ∞) × r2. then a = 1 1010 ≥ ∫ t 0 ∫ x1 0 ∫ x2 0 sign(x1)sign(x2)g(t1, s1, s2) × ( 1 + |x1 − s1| + (x1 − s1)2 )( 1 + |x2 − s2| + (x2 − s2)2 ) × ( 1 + (t − t1) + (t − t1)2 )( 1 + (1 + t1 + t 2 1)i(t1) ) ds2ds1dt1, (t, x1, x2) ∈ [0, ∞) × r2. now, consider the ivp utt − ux1x1 − ux2x2 = w(t)u 3 5 , (t, x1, x2) ∈ (0, ∞) × r2, u(0, x) = ut(0, x) = 0, (x1, x2) ∈ r2, (4.1) where w(t) =   10(9t2 − 9t + 2) t ∈ [0, 1] 20 t > 1. 410 s. georgiev & m. majdoub cubo 24, 3 (2022) here l = 1, a1(t, x1, x2) = |w(t)| ≤ a = 200, b1(t, x1, x2) = c1(t, x1, x2) = d1(t, x1, x2) = 0, (t, x1, x2) ∈ [0, ∞) × r2, and u0(x) = u1(x) = 0 ≤ 1 = r, (x1, x2) ∈ r2. we have that (h1) and (h2) hold. the ivp (4.1) has two nonnegative solutions u1(t, x) = 0, (t, x) ∈ [0, ∞) × r2, and u2(t, x) =   (t(1 − t))5 (t, x) ∈ [0, 1] × r2 0 (t, x) ∈ (1, ∞) × r2. acknowledgements the authors thank the reviewers for the careful reading of the manuscript and helpful comments. cubo 24, 3 (2022) two nonnegative solutions for two-dimensional nonlinear... 411 references [1] s. alinhac, “the null condition for quasilinear wave equations in two space dimensions i”, invent. math., vol. 145, no. 3, pp. 597–618, 2001. 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[23] c. d. sogge, lectures on nonlinear wave equations, 2nd edition, boston: international press, inc., 2013. introduction auxiliary results proof of the main result an example cubo a mathematical journal vol.20, no¯ 02, (13–21). june 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000200013 odd vertex equitable even labeling of cycle related graphs p. jeyanthi 1 and a. maheswari 2 1 research centre, department of mathematics, govindammal aditanar college for women, tiruchendur-628215, tamilnadu, india. 2 department of mathematics, kamaraj college of engineering and technology, virudhunagar, tamil nadu, india. jeyajeyanthi@rediffmail.com, bala nithin@yahoo.co.in abstract let g be a graph with p vertices and q edges and a = {1, 3, ..., q} if q is odd or a = {1, 3, ..., q + 1} if q is even. a graph g is said to admit an odd vertex equitable even labeling if there exists a vertex labeling f : v(g) → a that induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv such that for all a and b in a, |vf(a) − vf(b)| ≤ 1 and the induced edge labels are 2, 4, ..., 2q where vf(a) be the number of vertices v with f(v) = a for a ∈ a. a graph that admits an odd vertex equitable even labeling is called an odd vertex equitable even graph. here, we prove that the subdivision of double triangular snake (s(d(tn))), subdivision of double quadrilateral snake (s(d(qn))), da(qm) ⊙ nk1 and da(tm) ⊙ nk1 are odd vertex equitable even graphs. http://dx.doi.org/10.4067/s0719-06462018000200013 14 p. jeyanthi and a. maheswari cubo 20, 2 (2018) resumen sea g un grafo con p vértices y q aristas, y a = {1, 3, ..., q} si q es impar o a = {1, 3, ..., q + 1} si q es par. se dice que un grafo g admite un etiquetado par equitativo de vértices impares si existe un etiquetado de vértices f : v(g) → a que induce un etiquetado de ejes f∗ definido por f∗(uv) = f(u) + f(v) para todos los ejes uv tales que para todo a y b en a, |vf(a) − vf(b)| ≤ 1 y las etiquetas de ejes inducidas son 2, 4, ..., 2q donde vf(a) es el número de vértices v con f(v) = a para a ∈ a. un grafo que admite un etiquetado par equitativo de vértices impares se dice grafo par equitativo de vértices impares. aqúı demostramos que la subdivisión de serpientes triangulares dobles (s(d(tn))), la subdivisión de serpientes cuadriláteras dobles (s(d(qn))), da(qm) ⊙ nk1 y da(tm) ⊙ nk1 son grafos pares equitativos de vértices impares. keywords and phrases: odd vertex equitable even labeling, odd vertex equitable even graph, double triangular snake, subdivision of double quadrilateral snake, double alternate triangular snake, double alternate quadrilateral snake, subdivision graph. 2010 ams mathematics subject classification: 05c78. cubo 20, 2 (2018) odd vertex equitable even labeling of cycle related graphs 15 1 introduction: all graphs considered here are simple, finite, connected and undirected. let g(v, e) be a graph with p vertices and q edges. we follow the basic notations and terminology of graph theory as in [2]. the vertex set and the edge set of a graph are denoted by v(g) and e(g) respectively. a graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions and a detailed survey of graph labeling can be found in [1]. the concept of vertex equitable labeling was due to lourdusamy and seenivasan [6]. let g be a graph with p vertices and q edges and a = {0, 1, 2, ..., ⌈ q 2 ⌉ }. a graph g is said to be vertex equitable if there exists a vertex labeling f : v(g) → a that induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv such that for all a and b in a, |vf(a) − vf(b)| ≤ 1 and the induced edge labels are 1, 2, 3, ..., q, where vf(a) be the number of vertices v with f(v) = a for a ∈ a. the vertex labeling f is known as vertex equitable labeling. a graph g is said to be a vertex equitable if it admits vertex equitable labeling. motivated by the concept of vertex equitable labeling [6], jeyanthi, maheswari and vijayalakshmi extend this concept and introduced a new labeling namely odd vertex equitable even (ovee) labeling in [3]. a graph g with p vertices and q edges and a = {1, 3, ..., q} if q is odd or a = {1, 3, ..., q + 1} if q is even. a graph g is said to admit an odd vertex equitable even labeling if there exists a vertex labeling f : v(g) → a that induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv such that for all a and b in a, vf(a) − vf(b) ≤ 1 and the induced edge labels are 2, 4, ..., 2q where vf(a) be the number of vertices v with f(v) = a for a ∈ a. a graph that admits an odd vertex equitable even (ovee) labeling then g is called an odd vertex equitable even (ovee) graph. in [3], [4] and [5] the same authors proved that nc4-snake, cs(n1, n2, ..., nk, ni ≡ 0(mod4), ni ≥ 4, be a generalized kcn -snake, tôqsn and tõqsn are odd vertex equitable even graphs. they also proved that the graphs path, pn ⊙ pm(n, m ≥ 1), k1,n ∪ k1,n−2(n ≥ 3), k2,n, tp-tree, cycle cn (n≡ 0 or 1 (mod4)), quadrilateral snake qn, ladder ln, ln ⊙ k1, arbitrary super subdivision of any path pn, s(ln), lmôpn, ln ⊙ km and〈 lnôk1,m 〉 are odd vertex equitable even graphs. also they proved that the graphs k1,n is an odd vertex equitable even graph iff n ≤ 2 and the graph g = k1,n+k ∪ k1,n is an odd vertex equitable even graph if and only if k = 1, 2 and cycle cn is an odd vertex equitable even graph if and only if n≡ 0 or 1(mod4). let g be a graph with p vertices and q edges and p ≤ ⌈ q 2 ⌉ + 1, then g is not an odd vertex equitable even graph. in addition they proved that if every edge of a graph g is an edge of a triangle, then g is not an odd vertex equitable even graph. we use the following definitions in the subsequent section. definition 1.1. the double triangular snake d(tn) is a graph obtained from a path pn with vertices v1, v2, ..., vn by joining vi and vi+1 to the new vertices wi and ui for i = 1, 2, ..., n − 1. definition 1.2. the double quadrilateral snake d(qn) is a graph obtained from a path pn with vertices u1, u2, ..., un by joining ui and ui+1 to the new vertices vi, xi and wi, yi respectively and then joining vi, wi and xi, yi for i = 1, 2, ..., n − 1. definition 1.3. a double alternate triangular snake da(tn) consists of two alternate triangular snakes that have a common path. that is, a double alternate triangular snake is obtained from 16 p. jeyanthi and a. maheswari cubo 20, 2 (2018) a path u1, u2, ..., un by joining ui and ui+1(alternatively) to the two new vertices vi and wi for i = 1, 2, ..., n − 1. definition 1.4. a double alternate quadrilateral snake da(qn) consists of two alternate quadrilateral snakes that have a common path. that is, a double alternate quadrilateral snake is obtained from a path u1, u2, ..., un by joining ui and ui+1 (alternatively) to the two new vertices vi, xi and wi, yi respectively and adding the edges viwi and xiyi for i = 1, 2, ..., n − 1. definition 1.5. let g be a graph. the subdivision graph s(g) is obtained from g by subdividing each edge of g with a vertex. definition 1.6. the corona g1 ⊙ g2 of the graphs g1 and g2 is defined as the graph obtained by taking one copy of g1 (with p vertices) and p copies of g2 and then joining the i th vertex of g1 to every vertex of the ith copy of g2. 2 main results in this section, we prove that s(d(tn)), s(d(qn)), da(qm) ⊙ nk1 and da(tm) ⊙ nk1 are odd vertex equitable even graphs. theorem 2.1. let g1(p1, q1), g2(p2, q2),...,gm(pm, qm) be an odd vertex equitable even graphs with each qi is even for i = 1, 2, ..., m − 1, qm is even or odd and let ui, vi be the vertices of gi(1 ≤ i ≤ m) labeled by 1, qi if qi is odd or qi + 1 if qi is even. then the graph g obtained by identifying v1 with u2 and v2 with u3 and v3 with u4 and so on until we identify vm−1 with um is also an odd vertex equitable even graph. proof. the graph g has p1 + p2 + ... + pm − (m − 1) vertices and ∑m i=1 qi edges and fi be an odd vertex equitable even labeling of gi(1 ≤ i ≤ m). let a = { 1, 3, 5, ..., ∑m i=1 qi, if ∑m i=1 qi is odd 1, 3, 5, ..., ∑m i=1 qi + 1, if ∑m i=1 qi is even } . define a vertex labeling f : v(g) → a as follows: f(x) = f1(x) if x ∈ v(g1), f(x) = fi(x)+ ∑i−1 k=1 qk if x ∈ v(gi) for 2 ≤ i ≤ m. the edge labels of the graph g1 will remain fixed, the edge labels of the graph gi(2 ≤ i ≤ m) are 2q1 +2, 2q1 +4, ..., 2(q1 +q2);2(q1 +q2)+2, 2(q1 +q2)+4, ..., 2(q1 + q2 + q3);...,2 ∑m−1 i=1 qi + 2, 2 ∑m−1 i=1 qi + 4, ..., 2 ∑m i=1 qi. hence the edge labels of g are distinct and is { 2, 4, 6, ..., 2 ∑m i=1 qi } . also |vf(a) − vf(b)| ≤ 1 for all a, b ∈ a. hence g is an odd vertex equitable even graph. theorem 2.2. the graph s(d(tn)) is an odd vertex equitable even graph. proof. let gi = s(d(t2)) 1 ≤ i ≤ n − 1 and ui, vi be the vertices with labels 1 and q + 1 respectively. by theorem 2.1, s(d(t2)) admits an odd vertex equitable even labeling. an odd vertex equitable even labeling of gi = s(d(t2)) is given in figure 1. cubo 20, 2 (2018) odd vertex equitable even labeling of cycle related graphs 17 r rr ❅ ❅ ❅ ❅ ❅ r r r r r r 1 11 3 1 7 9 5 5 7 figure 1. theorem 2.3. the graph s(d(qn)) is an odd vertex equitable even graph. proof. let gi = s(d(q2)) 1 ≤ i ≤ n − 1 and ui, vi be the vertices with labels 1 and q + 1 respectively. by theorem 2.1, s(d(q2)) admits an odd vertex equitable even labeling. an odd vertex equitable even labeling of gi = s(d(q2)) is given in figure 2. r rr r r r rr r r r r r 1 1 15 3 9 9 13 11 13 773 5 figure 2. theorem 2.4. the double quadrilateral graph d(q2n) is an odd vertex equitable even graph. proof. let gi = d(q4) 1 ≤ i ≤ n−1 and ui, vi be the vertices with labels 1 and q+1 respectively. by theorem 2.1, d(q4) admits an odd vertex equitable even labeling. an odd vertex equitable even labeling of gi = d(q4) is given in figure 3. 18 p. jeyanthi and a. maheswari cubo 20, 2 (2018) r r r r r r r r r r r❇ ❇ ❇ ❇❇ 1 7 15 1 5 9 11 3 7 11 13 figure 3. theorem 2.5. let g1(p1, q), g2(p2, q), ..., gm(pm, q) be an odd vertex equitable even graphs with q odd and ui,vi be vertices of gi(1 ≤ i ≤ m) labeled by 1 and q. then the graph g obtained by joining v1 with u2 and v2 with u3 and v3 with u4 and so on until joining vm−1 with um by an edge is also an odd vertex equitable even graph. proof. the graph g has p1 + p2 + ... + pm vertices and mq + (m − 1) edges. let fi be the odd vertex equitable even labeling of gi(1 ≤ i ≤ m) and let a = {1, 3, ..., mq + (m − 1)}. define a vertex labeling f : v(g) → a as f(x) = fi(x) + (i − 1)(q + 1) if x ∈ gi for 1 ≤ i ≤ m. the edge labels of gi are incresed by 2(i − 1)(q + 1) for i = 1, 2, ..., m under the new labeling f. the bridge between the two graphs gi, gi+1 will get the label 2i(q + 1), 1 ≤ i ≤ m − 1. hence the edge labels of g are distinct and is {2, 4, ..., 2(mq + m − 1)}. also |vf(a) − vf(b)| ≤ 1 for all a, b ∈ a. then the graph g is an odd vertex equitable even graph. theorem 2.6. the graph da(t2) ⊙ nk1 is an odd vertex equitable even graph for n ≥ 1. proof. let g = da(t2) ⊙ nk1. let v(g) = {u1, u2, u, w} ∪ {uij : 1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vi, wi : 1 ≤ i ≤ n} and e(g) = {u1u2, u1v, vu2, u1w, wu2} ∪ {uiuij : 1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vvi, wwi : 1 ≤ i ≤ n}. here |v(g)| = 4(n + 1) and |e(g)| = 4n + 5. let a = {1, 3, ..., 4n + 5}. define a vertex labeling f : v(g) → a as follows: for 1 ≤ i ≤ n f(u1) = 1, f(u2) = 4n + 5, f(v) = 2n + 1, f(w) = 2n + 5, f(u1i) = 2i − 1, f(u2i) = 4n + 5 − 2(i − 1), f(vi) = { 3 if i=1 2i + 3 if 2 ≤ i ≤ n, f(wi) = { 2(n + i) + 1 if 1 ≤ i ≤ n − 1 4n + 3 if i=n. cubo 20, 2 (2018) odd vertex equitable even labeling of cycle related graphs 19 it can be verified that the induced edge labels of da(t2)⊙nk1 are 2, 4, ..., 8n+10 and |vf(a) − vf(b)| ≤ 1 for all a, b ∈ a. hence f is an odd vertex equitable even labeling da(t2) ⊙ nk1. an odd vertex equitable even labeling of da(t2) ⊙ 3k1 is shown in figure 4. s s s s s s s s s s s s s s s s 1 11 17 7 15 11 9 17 15 13 9 7 3 5 3 1 figure 4. theorem 2.7. the graph da(q2) ⊙ nk1 is an odd vertex equitable even graph for n ≥ 1. proof. let g = da(q2) ⊙ nk1. let v(g) = {u1, u2, v, w, x, y} ∪ {vi, wi, xi, yi : 1 ≤ i ≤ n} ∪ {uij : 1 ≤ i ≤ 2, 1 ≤ j ≤ n} and e(g) = {u1u2, u1v, vw, wu2, u1x, xy, yu2} ∪ {vvi, wwi, xxi, yyi : 1 ≤ i ≤ n} ∪ {uiuij : 1 ≤ i ≤ 2, 1 ≤ j ≤ n}. here |v(g)| = 6(n + 1) and |e(g)| = 6n + 7. let a = {1, 3, ..., 6n + 7}. define a vertex labeling f : v(g) → a as follows: for 1 ≤ i ≤ n f(u1) = 1, f(u2) = 6n + 7, f(u1i) = 2i − 1, f(u2i) = 6n − 2i + 9, f(v) = 2n + 1, f(w) = 2n + 3, f(x) = 4n + 5, f(y) = 4n + 7, f(vi) = 2i + 1, f(wi) = f(xi) = 2n + 2i + 3, f(yi) = 4n + 2i + 5. it can be verified that the induced edge labels of da(q2)⊙nk1 are 2, 4, ..., 12n+14 and |vf(a) − vf(b)| ≤ 1 for all a, b ∈ a. hence f is an odd vertex equitable even labeling of da(q2) ⊙ nk1. 20 p. jeyanthi and a. maheswari cubo 20, 2 (2018) an odd vertex equitable even labeling of da(q2) ⊙ 4k1 is shown in figure 5. s s s s s s s s s s s s s s s s s s s s sss s s s s s s s 1 21 23 31 119 13 15 17 19 23 25 27 29 25 27 29 31 19 17 15139 7 5 3 1 3 5 7 figure 5. theorem 2.8. the graph da(qm) ⊙ nk1 is an odd vertex equitable even graph for m, n ≥ 1. proof. by theorem 2.7, da(q2)⊙nk1 is an odd vertex equitable even graph. let gi = da(q2)⊙ nk1 for 1 ≤ i ≤ m − 1. since each gi has 6n+7 edges, by theorem 2.5, da(qm) ⊙ nk1 admits odd vertex equitable even labeling. an odd vertex equitable even labeling of da(q4) ⊙ 4k1 is shown in figure 6. s s s s s s s s s s s s✑ ✑ ✑✑ ✡ ✡ ✡✡ ❚ ❚ ❚ ✭✭✭✭ ❜ ❜ ❜ ✔ ✔ ✔✔ s s s s s s s s s s s s s s s s s s s ss s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 21 23 31 119 33 53 55 63 4341 1 3 5 7 13 15 17 19 23 25 27 29 31 29 27 25 19 17 15 139 7 5 3 39 37 35 33 35 37 39 41 45 47 49 51 63 61 59 57 61 59 57 5551 49 47 45 figure 6. cubo 20, 2 (2018) odd vertex equitable even labeling of cycle related graphs 21 theorem 2.9. the graph da(tm) ⊙ nk1 is an odd vertex equitable even graph for m, n ≥ 1. proof. by theorem 2.6, da(t2)⊙ nk1 is an odd vertex equitable even graph. let gi = da(t2)⊙ nk1 for 1 ≤ i ≤ m − 1. since each gi has 4n+5 edges, by theorem 2.5, da(tm) ⊙ nk1 admits odd vertex equitable even labeling. an odd vertex equitable even labeling of da(t4) ⊙ 3k1 is shown in figure 7. r r r r r r r r❏ ❏ ❏ ❏ ❏ ◗ ◗ ◗◗ ✑ ✑ ✑r r r r r r r r r r r r r r r r r r r r r r r r 1 11 17 7 15 11 9 17 15 13 973 5 3 1 19 21 23 33 29 27 35 33 31 272511 19 29 35 25 figure 7. references [1] a. gallian, a dynamic survey of graph labeling, the electronic journal of combinatorics, 19(2017), #ds6. [2] f. harary, graph theory, addison wesley, massachusetts, 1972. [3] p. jeyanthi, a. maheswari and m. vijaya lakshmi, odd vertex equitable even labeling, proyecciones journal of mathematics, vol.36(1)(2017), 1-11. [4] p. jeyanthi, a. maheswari and m. vijaya lakshmi, odd vertex equitable even labeling of cyclic snake related graphs, proyecciones journal of mathematics, vol.37(4)(2018), 613-625. [5] p. jeyanthi, a. maheswari and m. vijaya lakshmi, odd vertex equitable even labeling of ladder graphs, jordon journal of mathematics and statistics, to appear. [6] a. lourdusamy and m. seenivasan, vertex equitable labeling of graphs, journal of discrete mathematical sciences and cryptography, vol.11,(6)(2008), 727-735. introduction: main results cubo a mathematical journal vol.20, no¯ 01, (17–29). march 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000100017 w2-curvature tensor on generalized sasakian space forms venkatesha and shanmukha b. department of mathematics, kuvempu university shankaraghatta 577 451, shimoga, karnataka, india. vensmath@gmail.com, meshanmukha@gmail.com abstract in this paper, we study w2-pseudosymmetric, w2-locally symmetric, w2-locally φsymmetric and w2-φ-recurrent generalized sasakian space form. further, illustrative examples are given. resumen en este art́ıculo, estudiamos formas espaciales sasakianas generalizadas w2-seudosimétricas, w2-localmente φ-simétricas y w2-φ-recurrentes. ejemplos ilustrativos son dados. keywords and phrases: generalized sasakian space form, w2-curvature tensor, pseudosymmetric, φ-recurrent, einstein manifold. 2010 ams mathematics subject classification: 53c15, 53c25, 53c50. http://dx.doi.org/10.4067/s0719-06462018000100017 ignacio castillo ignacio castillo ignacio castillo ignacio castillo 18 venkatesha and shanmukha b. cubo 20, 1 (2018) 1 introduction the nature of a riemannian manifold depends on the curvature tensor r of the manifold. it is well known that the sectional curvatures of a manifold determine its curvature tensor completely. a riemannian manifold with constant sectional curvature c is known as a real space form and its curvature tensor is given by r(x, y)z = c{g(y, z)x − g(x, z)y}. representation for these spaces are hyperbolic spaces (c < 0), spheres (c > 0) and euclidean spaces (c = 0). the φ-sectional curvature of a sasakian space form is defined by sasakian manifold and it has a specific form of its curvature tensor. same notion also holds for kenmotsu and cosymplectic space forms. in order to generalize such space forms in a common frame alegre, blair and carriazo [1] introduced and studied generalized sasakian space forms. a generalized sasakian space form is an almost contact metric manifold (m2n+1, φ, ξ, η, g), whose curvature tensor is given by r(x, y)z = f1{g(y, z)x − g(x, z)y} + f2{g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz} + f3{η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ}, (1.1) the riemanian curvature tensor of a generalized sasakian space form m2n+1(f1, f2, f3) is simply given by r = f1r1 + f2r2 + f3r3, where f1, f2, f3 are differential functions on m 2n+1(f1, f2, f3) and r1(x, y)z = g(y, z)x − g(x, z)y, r2(x, y)z = g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz, and r3(x, y)z = η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ, where f1 = c+3 4 , f2 = f3 = c−1 4 . here c denotes the constant φ-sectional curvature. the properties of generalized sasakian space form was studied by many geometers such has [2, 9, 10, 14, 17, 18, 19, 21, 26]. the concept of local symmetry of a riemanian manifold has been studied by many authors in several ways to a different extent. the locally φ-symmetry of sasakian manifold was introduce by takahashi in [28]. de and et al generalize this to the notion of φ-symmetry and then introduced the notion of φ-recurrent sasakian manifold in [11]. further φ-recurrent condition was studied on kenmotsu manifold [8], lp-sasakian manifold [29] and (lcs)n-manifold [20]. in[16], pokhariyal and mishra have defined the w2-curvature tensor, given by w2(x, y)z = r(x, y)z + 1 2n {g(x, z)qy − g(y, z)qx}, (1.2) cubo 20, 1 (2018) w2-curvature tensor on generalized sasakian space forms 19 here r and q are the riemanian curvature tensor and ricci operator of riemanian manifold respectively. in a generalized sasakian space forms, the w2-curvature tensor satisfies the condition η(w2(x, y)z) = 0. (1.3) many geometers studied the w2 curvature tensor studied on different manifolds such has generalized sasakian space forms [13], lorentzian para sasakian manifolds [30] and kenmotsu manifolds [25] motivated by these ideas, we made an attempt to study the properties of generalized sasakian space form. the present paper is organized as follows: in section 2, we review some preliminary results. in section 3, we study w2-pseudosymmetric generalized sasakian space form. section 4, deals with the w2-locally symmetric generalized sasakian space forms and it is shown that a generalized sasakian space form of dimension greater than three is w2-locally symmetric if and only if it is conformally flat. section 5, is devoted to the study of w2-locally φ-symmetric generalized sasakian space forms. finally in last section, we discus the w2-φ-recurrent generalized sasakian space form and found to be einstein manifold. 2 generalized sasakian space-forms the riemannian manifold m2n+1 is called an almost contact metric manifold if the following result holds [5, 6]: φ2x = −x + η(x)ξ, (2.1) η(ξ) = 1, φξ = 0, η(φx) = 0, g(x, ξ) = η(x), (2.2) g(φx, φy) = g(x, y) − η(x)η(y), (2.3) g(φx, y) = −g(x, φy), g(φx, x) = 0 (2.4) (∇xη)(y) = g(∇xξ, y), ∀ x, y ∈ (tpm). (2.5) a almost contact metric manifold is said to be sasakian if and only if [5, 23] (∇xφ)y = g(x, y)ξ − η(y)x, (2.6) ∇xξ = −φx. (2.7) 20 venkatesha and shanmukha b. cubo 20, 1 (2018) again we know that [1] in (2n + 1)-dimensional generalized sasakian space form: s(x, y) = (2nf1 + 3f2 − f3)g(x, y) − (3f2 + (2n − 1)f3)η(x)η(y), (2.8) s(φx, φy) = s(x, y) + 2n(f1 − f3)η(x)η(y), (2.9) qx = (2nf1 + 3f2 − f3)x − (3f2 + (2n − 1)f3)η(x)ξ, (2.10) r = 2n(2n + 1)f1 + 6nf2 − 4nf3, (2.11) r(x, y)ξ = (f1 − f3){η(y)x − η(x)y}, (2.12) r(ξ, x)y = (f1 − f3){g(x, y)ξ − η(y)x}, (2.13) η(r(x, y)z) = (f1 − f3){g(y, z)η(x) − g(x, z)η(y)}, (2.14) s(x, ξ) = 2n(f1 − f3)η(x). (2.15) here r, s, q and r are the riemannian curvature tensor, ricci tensor, ricci operator and scalar curvature tensor of generalized sasakian space forms in that order. 3 w2-pseudosymmetric generalized sasakian space forms the concept of a pseudosymmetric manifold was introduced by chaki [7] and deszcz [12]. in this article we shall study properties of pseudosymmetric manifold according to deszcz. semisymmetric manifolds satisfies the condition r·r = 0 and were categorized by szabo in [27]. every pseudosymmetric manifold is semisymmetric but semisymmetric manifold need not be pseudosymmetric. an (2n + 1)-dimensional riemannian manifold m2n+1 is said to be pseudosymmetric, if (r(x, y) · r)(u, v)w = lr{((x ∧ y) · r)(u, v)w)}. (3.1) where lr is some smooth function on ur = {x ∈ m 2n+1|r − r n(n−1) g 6= 0 at x}, where g is the (0, 4)-tensor defined by g(x1, x2, x3, x4) = g((x1 ∧x2)x3, x4) and (x∧y)z is the endomorphism and it is defined as, (x ∧ y)z = g(y, z)x − g(x, z)y (3.2) an (2n+1)-dimensional generalized sasakian space form m2n+1 is said to be w2-pseudosymmetric, if (r(x, y) · w2)(u, v)z = lw2{(x ∧ y) · w2)(u, v)z}, (3.3) holds on the set uw2 = {x ∈ m 2n+1|w2 6= 0 at x}, where lw2 is some function on uw2. suppose that generalized sasakian space form is w2-pseudosymmetric. now the lefthand side of (3.3) is r(ξ, y)w2(u, v)z − w2(r(ξ, y)u, v)z − w2(u, r(ξ, y)v)z − w2(u, v)r(ξ, y)z = 0. (3.4) cubo 20, 1 (2018) w2-curvature tensor on generalized sasakian space forms 21 in the view of (2.12) the above expression becomes (f1 − f3){g(y, w2(u, v)z)ξ − η(w2(u, v)z)y − g(y, u)w2(ξ, v)z + η(u)w2(y, v)z − g(y, v)w2(u, ξ)z + η(v)w2(u, y)z − g(y, z)w2(u, v)ξ + η(z)w2(u, v)y} = 0. (3.5) next the right hand side of (3.3) is lw2{(ξ ∧ y)w2(u, v)z − w2((ξ ∧ y)u, v)z − w2(u, (ξ ∧ y)v)z − w2(u, v)(ξ ∧ y)z} = 0. (3.6) by virtue of (3.2), (3.6) becomes lw2{g(y, w2(u, v)z)ξ − η(w2(u, v)z)y − g(y, u)w2(ξ, v)z + η(u)w2(y, v)z − g(y, v)w2(u, ξ)z + η(v)w2(u, y)z − g(y, z)w2(u, v)ξ + η(z)w2(u, v)y} = 0. (3.7) using the expressions (3.5) and (3.7) in (3.3) and taking inner product with ξ, we obtain {lw2 − (f1 − f3)}{w2(u, v, z, y) − η(w2(u, v)z)η(y) − g(y, u)η(w2(ξ, v)z) + η(u)η(w2(y, v)z) − g(y, v)η(w2(u, ξ)z) + η(v)η(w2(u, v)z) − g(y, z)η(w2(u, v)ξ) + η(z)η(w2(u, v)z)} = 0, (3.8) where w2(u, v, z, y) = g(y, w2(u, v)z) and using(1.3) we get either lw2 = (f1 − f3) or w2(u, v, z, y) = 0. (3.9) thus we have following: theorem 3.1. if m2n+1(f1, f2, f3) is w2-pseudosymmetric generalized sasakian space form, then m2n+1(f1, f2, f3) is either w2-flat, or lw2 = (f1 − f3) if (f1 6= f3). also in a generalized sasakian space form, singh and pandey [24] proved the following, theorem 3.2. a (2n+1)-dimensional (n > 1) generalized sasakian space form satisfying w2 = 0 is an η-einstein manifolds. in view of theorem (3.1) and theorem (3.2) we can state the following corollary. corolary 1. if m2n+1(f1, f2, f3) is a w2-pseudosymmetric generalized sasakian space forms then m2n+1 is either η-einstein manifold or lw2 = (f1 − f3) if (f1 6= f3). 22 venkatesha and shanmukha b. cubo 20, 1 (2018) 4 w2-locally symmetric generalized sasakian space forms definition 1. a (2n+1) dimensional (n > 1) generalized sasakian space form is called projectively locally symmetric if it satisfies [18]. (∇wp)(x, y)z = 0. for all vector fields x, y, z orthogonal to ξ and an arbitrary vector field w. analogous to this definition, we define a (2n + 1) dimensional (n > 1) w2-locally symmetric generalized sasakian space form if (∇ww2)(x, y)z = 0, for all vector fields x, y, z orthogonal to ξ and an arbitrary vector field w. from (1.1) and (1.2), we have w2(x, y)z = f1{g(y, z)x − g(x, z)y} + f2{g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz} + f3{η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ} + 1 2n {g(x, z)qy − g(y, z)qx}. (4.1) taking covariant differentiation of (4.1) with respect to an arbitrary vector field w, we get (∇ww2)(x, y)z = df1(w){g(y, z)x − g(x, z)y} + df2(w){g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz} + f2{g(x, φz)(∇wφ)y + g(x, (∇wφ)z)φy − g(y, φz)(∇wφ)x − g(y, (∇wφ)z)φx + 2g(x, φy)(∇wφ)z + 2g(x, (∇wφ)y)φz} + df3(w){η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ} + f3{(∇wη)(x)η(z)y + η(x)(∇wη)(z)y − (∇wη)(y)η(z)x − η(y)(∇wη)η(z)x + g(x, z)(∇wη)(y)ξ + g(x, z)η(y)∇wξ − g(y, z)(∇wη)(x)ξ − g(y, z)η(x)∇wξ} + 1 2n {g(x, z)(∇wq)(y) − g(y, z)(∇wq)(x)}. (4.2) where ∇ denotes the riemannian connection on the manifold. differentiating (2.10) covariantly with respect to a w, one can get (∇wq)(y) = d(2nf1 + 3f2 − f3)(w)y − d(3f2 + (2n − 1)f3)(w)η(y)ξ − (3f2 + (2n − 1)f3)[(∇wη)(y)ξ + η(y)(∇wξ)]. (4.3) cubo 20, 1 (2018) w2-curvature tensor on generalized sasakian space forms 23 in view of (4.3) and (4.2), it follows that (∇ww2)(x, y)z = df1(w){g(y, z)x − g(x, z)y} + df2(w){g(x, φz)φy − g(y, φz)φx + 2g(x, φy)φz} + f2{g(x, φz)(∇wφ)y + g(x, (∇wφ)z)φy − g(y, φz)(∇w φ)x − g(y, (∇wφ)z)φx + 2g(x, φy)(∇wφ)z + 2g(x, (∇wφ)y)φz} + df3(w){η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ} + f3{(∇wη)(x)η(z)y + η(x)(∇wη)(z)y − (∇wη)(y)η(z)x − η(y)(∇wη)η(z)x + g(x, z)(∇wη)(y)ξ + g(x, z)η(y)∇wξ − g(y, z)(∇wη)(x)ξ − g(y, z)η(x)∇wξ} + 1 2n [g(x, z){d(2nf1 + 3f2 − f3)(w)y − d(3f2 + (2n − 1)f3)(w)η(y)ξ − (3f2 + (2n − 1)f3)[(∇wη)(y)ξ + η(y)(∇wξ)]} − g(y, z){d(2nf1 + 3f2 − f3)(w)x − d(3f2 + (2n − 1)f3)(w)η(x)ξ − (3f2 + (2n − 1)f3)[(∇wη)(x)ξ + η(x)(∇wξ)]}]. (4.4) taking x, y, z orthogonal to ξ in (4.4) and then taking the inner product of the resultant equation with v, followed by setting v = z = ei in the above equation, where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, i = 1, 2, ......, 2n + 1, we get f2{−g(φx, (∇wφ)y) + n∑ i=1 g(x, (∇wφ)ei)g(φy, ei) + g(φy, (∇wφ)x) − n∑ i=1 g(y, (∇wφ)ei)g(φx, ei) + 2 n∑ i=1 g(x, φy)g((∇wφ)ei, ei)} = 0. (4.5) for levi civita connection ∇, (∇wg)(x, y) = 0, which gives (∇wg)(x, y) − g(∇wx, y) − g(x, ∇wy) = 0. putting x = ei and y = φei in the above equation, we obtain − g(∇wei, φei) − g(ei, (∇wφ)ei) = 0, 24 venkatesha and shanmukha b. cubo 20, 1 (2018) which can be written as g(ei, φ(∇wei)) − g(ei, (∇wφ)ei) = 0. thus we have g(ei, (∇wφ)ei) = 0. (4.6) by the virtue of (4.5) and (4.6) takes the form f2{−g(φx, (∇wφ)y) + ∑ i=1 g(x, (∇wφ)ei)g(φy, ei) + g(φy, (∇wφ)x) − ∑ i=1 g(y, (∇wφ)ei)g(φx, ei)} = 0. (4.7) the above equation yields f2 = 0. it is known that a generalized sasakian space form of dimension greater than three is conformally flat if and only if f2 = 0 [14]. hence the manifold under consideration is conformally flat. conversely, suppose that the manifold is conformally flat. then f2 = 0. in addition, if we consider x, y, z orthogonal to ξ then (1.1) yields r(x, y)z = f1{g(y, z)x − g(x, z)y}. the above equation gives, r = 2n(2n + 1)f1. (4.8) in view of (2.11) and (4.8), we obtain f3 = 0. hence from (4.4), we get (∇ww2)(x, y)z = 0. therefore, the manifold is w2-locally symmetric. thus we have the following assertion. theorem 4.1. a (2n + 1) dimensional (n > 1) generalized sasakian space form is w2-locally symmetric if and only if it is conformally flat. or theorem 4.2. a (2n + 1) dimensional (n > 1) generalized sasakian space form is w2-locally symmetric if and only if f1 is constant. 5 w2-locally φ-symmetric generalized sasakian space forms definition 2. a generalized sasakian space form m2n+1(f1, f2, f3) of dimension greater than three is called w2-locally φ-symmetric if it satisfies φ2((∇ww2)(x, y)z) = 0, (5.1) for all vector fields x, y, z orthogonal to ξ on m2n+1. let us consider a w2-locally φ-symmetric generalized sasakian space form of dimension greater than three. then from the definition and (2.1), we have cubo 20, 1 (2018) w2-curvature tensor on generalized sasakian space forms 25 − ((∇ww2)(x, y)z) + η(∇ww2)(x, y)z)ξ = 0, (5.2) taking the inner product g in both sides of the above equation with respect to w, we get − g((∇ww2)(x, y)z, w) + η(∇ww2)(x, y)z)η(w) = 0, (5.3) if we take orthogonal to w, then the above equation yields, g((∇ww2)(x, y)z, w) = 0, (5.4) the above equation is true for all w orthogonal to ξ. if we choose w 6= 0 and not orthogonal to (∇ww2)(x, y)z, then it follows that (∇ww2)(x, y)z = 0 (5.5) hence, the manifold is w2-locally symmetric and hence by theorem 4.3, it is conformally flat. conversely, let the manifold is conformally flat and hence f2 6= 0. again, for x, y, z orthogonal to ξ, we have applying φ2 on both side to equation (4.4), one can get φ2(∇ww2)(x, y)z = −df2(w){g(x, φz)φx − g(y, φz) + 2g(x, φy)φz} − 1 2n {d(3f2 − f3)(w)[g(x, z)y − g(y, z)x]}. (5.6) if f2 = f3 = 0, the above equation yields φ2(∇ww2)(x, y)z = 0 for all x, y, z are orthogonal to ξ, therefore the manifold is w2-locally φ-symmetric. now we are in a position to state the following statement, theorem 5.1. a (2n + 1)-dimensional (n > 1) generalized sasakian space form m2n+1 is w2locally φ-symmetric if and only if it is conformally flat. 6 w2-φ-recurrent generalized sasakian space form definition 3. a generalized sasakian space form is said to be φ-recurrent if there exists a non-zero 1-form a such that,(see[11]) φ2((∇wr)(x, y)z) = a(w)r(x, y)z, for arbitrary vector fields x, y, z, w. if the 1-form a vanishes, then the manifold reduces to a φ-symmetric manifold. 26 venkatesha and shanmukha b. cubo 20, 1 (2018) according to the definition of φ-recurrent generalized sasakian space form, we define w2-φrecurrent generalized sasakian space form by φ2((∇ww2)(x, y)z) = a(w)w2(x, y)z. (6.1) then by (2.1) and (6.1), we have − (∇ww2)(x, y)z + η((∇ww2)(x, y)z)ξ = a(w)w2(x, y)z, (6.2) for arbitrary vector fields x, y, z, w. from the above equation it follows that − g((∇ww2)(x, y)z, u) + η((∇ww2)(x, y)z)η(u) = a(w)g(w2(x, y)z, u). (6.3) let {ei}, i = 1, 2, ......2n + 1, be an orthogonal basis of the tangent space at any point of the manifold. then putting x = u = ei in (6.3) and taking summation over i, 1 ≤ i ≤ 2n + 1, we get − (∇ws)(y, z) − 1 2n [(∇ws(y, z)) − g(y, z)dr(w)] + 2n+1∑ i=1 η((∇ww2)(ei, y)z)η(ei) = a(w){(∇ws)(y, z) − 1 2n [(∇ws)(y, z) − g(y, z)dr(w)]}. (6.4) setting z = ξ in (6.4) then using (2.5), (2.13) and (2.7) and then replace y by φy in (6.4), we get s(y, w) = 2n(f1 − f3)g(y, w). (6.5) hence we can state following theorem: theorem 6.1. let generalized sasakian space forms m2n+1is w2-φ-recurrent, then it is an einstein manifold, provided (f1 − f3) 6= 0. 7 example in [1], generalized complex space-form of dimension two is n(a, b) and the warped product m = r×n endowed with the almost contact metric structure is a three dimensional generalized sasakianspace-form whose smooth functions f1 = a−(f ′ ) 2 f2 , f2 = b f2 and f3 = a−(f ′ ) 2 f2 + f ′′ f . here f = f(t), t ∈ r and f ′ indicates the derivative of f with respect to t. suppose we set a = 2, b = 0 and f(t) = t with t 6= 0, then f1 = 1 t2 , f2 = 0 and f3 = 1 t2 , we have from (1.2) w2(x, y)z = 1 t2 {g(y, z)x − g(x, z)y + η(x)η(z)y − η(y)η(z)x + g(x, z)η(y)ξ − g(y, z)η(x)ξ} + 1 2t2 {g(x, z)y − g(y, z)x − g(x, z)η(y)ξ + g(y, z)η(x)ξ}. 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[31] venkatesha and b. sumangala, on m-projective curvature tensor of generalised sasakian space form. acta math. univ. comenianae, 2 (2013), 209–217. introduction generalized sasakian space-forms w2-pseudosymmetric generalized sasakian space forms w2-locally symmetric generalized sasakian space forms w2-locally -symmetric generalized sasakian space forms w2–recurrent generalized sasakian space form example cubo a mathematical journal vol.21, no¯ 02, (01–13). august 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000200001 caputo fractional iyengar type inequalities george a. anastassiou department of mathematical sciences, university of memphis, memphis, tn 38152, u.s.a. ganastss@memphis.edu abstract here we present caputo fractional iyengar type inequalities with respect to lp norms, with 1 ≤ p ≤ ∞. the method is based on the right and left caputo fractional taylor’s formulae. resumen aqúı presentamos desigualdades de tipo caputo fraccional iyengar con respecto a las normas lp, con 1 ≤ p ≤ ∞. el método se basa en las fórmulas de taylor fraccionales de caputo derecha e izquierda. keywords and phrases: iyengar inequality, right and left caputo fractional, taylor formulae, caputo fractional derivative. 2010 ams mathematics subject classification: 26a33, 26d10, 26d15. http://dx.doi.org/10.4067/s0719-06462019000200001 2 george a. anastassiou cubo 21, 2 (2019) 1 introduction we are motivated by the following famous iyengar inequality (1938), [4]. theorem 1. let f be a differentiable function on [a, b] and |f′ (x)| ≤ m. then ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − 1 2 (b − a) (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ m (b − a) 4 2 − (f (b) − f (a)) 2 4m . (1) we need definition 2. ([1], p. 394) let ν > 0, n = ⌈ν⌉ (⌈·⌉ the ceiling of the number), f ∈ acn ([a, b]) (i.e. f(n−1) is absolutely continuous on [a, b]). the left caputo fractional derivative of order ν is defined as dν ∗af (x) = 1 γ (n − ν) ∫x a (x − t) n−ν−1 f(n) (t) dt, (2) ∀ x ∈ [a, b], and it exists almost everywhere over [a, b] . we need definition 3. ([2], p. 336-337) let ν > 0, n = ⌈ν⌉, f ∈ acn ([a, b]). the right caputo fractional derivative of order ν is defined as dνb−f (x) = (−1) n γ (n − ν) ∫b x (z − x) n−ν−1 f(n) (z) dz, (3) ∀ x ∈ [a, b], and exists almost everywhere over [a, b] . 2 main results we present the following caputo fractional iyengar type inequalities: theorem 4. let ν > 0, n = ⌈ν⌉ (⌈·⌉ is the ceiling of the number), and f ∈ acn ([a, b]) (i.e. f(n−1) is absolutely continuous on [a, b]). we assume that dν ∗af, d ν b−f ∈ l∞ ([a, b]). then i) ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 + (−1)kf(k) (b) (b − t) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l∞([a,b]) , ∥ ∥dνb−f ∥ ∥ l∞([a,b]) } γ (ν + 2) [ (t − a) ν+1 + (b − t) ν+1 ] , (4) ∀ t ∈ [a, b] , cubo 21, 2 (2019) caputo fractional iyengar type inequalities 3 ii) at t = a+b 2 , the right hand side of (4) is minimized, and we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! (b − a) k+1 2k+1 [ f(k) (a) + (−1)kf(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l∞([a,b]) , ∥ ∥dνb−f ∥ ∥ l∞([a,b]) } γ (ν + 2) (b − a) ν+1 2ν , (5) iii) if f(k) (a) = f(k) (b) = 0, for all k = 0, 1, ..., n − 1, we obtain ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l∞([a,b]) , ∥ ∥dνb−f ∥ ∥ l∞([a,b]) } γ (ν + 2) (b − a) ν+1 2ν , (6) which is a sharp inequality, iv) more generally, for j = 0, 1, 2, ..., n ∈ n, it holds ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! ( b − a n )k+1 [ jk+1f(k) (a) + (−1)k (n − j) k+1 f(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l∞([a,b]) , ∥ ∥dνb−f ∥ ∥ l∞([a,b]) } γ (ν + 2) ( b − a n )ν+1 [ jν+1 + (n − j) ν+1 ] , (7) v) if f(k) (a) = f(k) (b) = 0, k = 1, ..., n − 1, from (7) we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a n ) [jf (a) + (n − j) f (b)] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l∞([a,b]) , ∥ ∥dνb−f ∥ ∥ l∞([a,b]) } γ (ν + 2) ( b − a n )ν+1 [ jν+1 + (n − j) ν+1 ] , (8) j = 0, 1, 2, ..., n, vi) when n = 2 and j = 1, (8) turns to ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a 2 ) (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l∞([a,b]) , ∥ ∥dνb−f ∥ ∥ l∞([a,b]) } γ (ν + 2) (b − a) ν+1 2ν , (9) vii) when 0 < ν ≤ 1, inequality (9) is again valid without any boundary conditions. 4 george a. anastassiou cubo 21, 2 (2019) proof. let ν > 0, n = ⌈ν⌉, and f ∈ acn ([a, b]). then by ([3], p. 54) left caputo fractional taylor’s formula we have f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k = 1 γ (ν) ∫x a (x − t) ν−1 dν ∗af (t) dt, (10) ∀ x ∈ [a, b] . also by ([2], p. 341) right caputo fractional taylor’s formula we get: f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k = 1 γ (ν) ∫b x (z − x) ν−1 dνb−f (z) dz, (11) ∀ x ∈ [a, b] . by (10) we derive ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ ‖dν ∗af‖l∞([a,b]) γ (ν + 1) (x − a) ν , (12) and by (11) we obtain ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ ∥ ∥dνb−f ∥ ∥ l∞([a,b]) γ (ν + 1) (b − x) ν , (13) ∀ x ∈ [a, b] . call γ1 := ‖dν ∗af‖l∞([a,b]) γ (ν + 1) , (14) and γ2 := ∥ ∥dνb−f ∥ ∥ l∞([a,b]) γ (ν + 1) . (15) set γ := max (γ1, γ2) . (16) that is ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ γ (x − a) ν , (17) and ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ γ (b − x) ν , (18) ∀ x ∈ [a, b] . cubo 21, 2 (2019) caputo fractional iyengar type inequalities 5 hence it holds n−1∑ k=0 f(k) (a) k! (x − a) k − γ (x − a) ν ≤ f (x) ≤ n−1∑ k=0 f(k) (a) k! (x − a) k + γ (x − a) ν (19) and n−1∑ k=0 f(k) (b) k! (x − b) k − γ (b − x) ν ≤ f (x) ≤ n−1∑ k=0 f(k) (b) k! (x − b) k + γ (b − x) ν , (20) ∀ x ∈ [a, b] . let any t ∈ [a, b], then by integration over [a, t] and [t, b], respectively, we obtain n−1∑ k=0 f(k) (a) (k + 1) ! (t − a) k+1 − γ (ν + 1) (t − a) ν+1 ≤ ∫t a f (x) dx ≤ n−1∑ k=0 f(k) (a) (k + 1) ! (t − a) k+1 + γ (ν + 1) (t − a) ν+1 , (21) and − n−1∑ k=0 f(k) (b) (k + 1) ! (t − b) k+1 − γ (ν + 1) (b − t) ν+1 ≤ ∫b t f (x) dx ≤ − n−1∑ k=0 f(k) (b) (k + 1) ! (t − b) k+1 + γ (ν + 1) (b − t) ν+1 . (22) adding (21) and (22), we obtain { n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 − f(k) (b) (t − b) k+1 ] } − γ (ν + 1) [ (t − a) ν+1 + (b − t) ν+1 ] ≤ ∫b a f (x) dx ≤ { n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 − f(k) (b) (t − b) k+1 ] } + γ (ν + 1) [ (t − a) ν+1 + (b − t) ν+1 ] , (23) ∀ t ∈ [a, b] . consequently we derive: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 + (−1)kf(k) (b) (b − t) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) [ (t − a) ν+1 + (b − t) ν+1 ] , (24) 6 george a. anastassiou cubo 21, 2 (2019) ∀ t ∈ [a, b] . let us consider g (t) := (t − a) ν+1 + (b − t) ν+1 , ∀ t ∈ [a, b] . hence g′ (t) = (ν + 1) [ (t − a) ν − (b − t) ν ] = 0, giving (t − a) ν = (b − t) ν and t − a = b − t, that is t = a+b 2 the only critical number here. we have g (a) = g (b) = (b − a) ν+1 , and g ( a+b 2 ) = (b−a) ν+1 2ν , which the minimum of g over [a, b]. consequently the right hand side of (24) is minimized when t = a+b 2 , with value γ (ν+1) (b−a) ν+1 2ν . assuming f(k) (a) = f(k) (b) = 0, for k = 0, 1, ..., n − 1, then we obtain that ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) (b − a) ν+1 2ν , (25) which is a sharp inequality. when t = a+b 2 , then (24) becomes ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! (b − a) k+1 2k+1 [ f(k) (a) + (−1)kf(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) (b − a) ν+1 2ν . (26) next let n ∈ n, j = 0, 1, 2, ..., n and tj = a + j ( b−a n ) , that is t0 = a, t1 = a + b−a n , ..., tn = b. hence it holds tj − a = j ( b − a n ) , (b − tj) = (n − j) ( b − a n ) , j = 0, 1, 2, ..., n. (27) we notice that (tj − a) ν+1 + (b − tj) ν+1 = ( b − a n )ν+1 [ jν+1 + (n − j) ν+1 ] , (28) j = 0, 1, 2, ..., n, and (for k = 0, 1, ..., n − 1) [ f(k) (a) (tj − a) k+1 + (−1)kf(k) (b) (b − tj) k+1 ] = [ f(k) (a) jk+1 ( b − a n )k+1 + (−1)kf(k) (b) (n − j) k+1 ( b − a n )k+1 ] = cubo 21, 2 (2019) caputo fractional iyengar type inequalities 7 ( b − a n )k+1 [ f(k) (a) jk+1 + (−1)kf(k) (b) (n − j) k+1 ] , (29) j = 0, 1, 2, ..., n. by (24) we get ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! ( b − a n )k+1 [ f(k) (a) jk+1 + (−1)kf(k) (b) (n − j) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) ( b − a n )ν+1 [ jν+1 + (n − j) ν+1 ] , (30) j = 0, 1, 2, ..., n. if f(k) (a) = f(k) (b) = 0, k = 1, ..., n − 1, then (30) becomes ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a n ) [jf (a) + (n − j) f (b)] ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) ( b − a n )ν+1 [ jν+1 + (n − j) ν+1 ] , (31) j = 0, 1, 2, ..., n. when n = 2 and j = 1, then (31) becomes ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a 2 ) (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) 2 ( b − a 2 )ν+1 = γ (ν + 1) (b − a) ν+1 2ν . (32) let 0 < ν ≤ 1, then n = ⌈ν⌉ = 1. in that case, without any boundary conditions, we derive from (32) again that ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a 2 ) (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ γ (ν + 1) (b − a) ν+1 2ν . (33) the theorem is proved in all cases. we give theorem 5. let ν ≥ 1, n = ⌈ν⌉, and f ∈ acn ([a, b]). we assume that dν ∗af, d ν b−f ∈ l1 ([a, b]). then i) ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 + (−1)kf(k) (b) (b − t) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ 8 george a. anastassiou cubo 21, 2 (2019) max { ‖dν ∗af‖l1([a,b]) , ∥ ∥dνb−f ∥ ∥ l1([a,b]) } γ (ν + 1) [ (t − a) ν + (b − t) ν ] , (34) ∀ t ∈ [a, b] , ii) when ν = 1, from (34), we have ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − [f (a) (t − a) + f (b) (b − t)] ∣ ∣ ∣ ∣ ∣ ≤ ‖f′‖ l1([a,b]) (b − a) , ∀ t ∈ [a, b] , (35) iii) from (35), we obtain (ν = 1 case) ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a 2 ) (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ ‖f′‖ l1([a,b]) (b − a) , (36) iv) at t = a+b 2 , ν > 1, the right hand side of (34) is minimized, and we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! (b − a) k+1 2k+1 [ f(k) (a) + (−1)kf(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l1([a,b]) , ∥ ∥dνb−f ∥ ∥ l1([a,b]) } γ (ν + 1) (b − a) ν 2ν−1 , (37) v) if f(k) (a) = f(k) (b) = 0, for all k = 0, 1, ..., n − 1; ν > 1, from (37), we obtain ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l1([a,b]) , ∥ ∥dνb−f ∥ ∥ l1([a,b]) } γ (ν + 1) (b − a) ν 2ν−1 , (38) which is a sharp inequality, vi) more generally, for j = 0, 1, 2, ..., n ∈ n, it holds ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! ( b − a n )k+1 [ jk+1f(k) (a) + (−1)k (n − j) k+1 f(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l1([a,b]) , ∥ ∥dνb−f ∥ ∥ l1([a,b]) } γ (ν + 1) ( b − a n )ν [ jν + (n − j) ν ] , (39) vii) if f(k) (a) = f(k) (b) = 0, k = 1, ..., n − 1, from (39) we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a n ) [jf (a) + (n − j) f (b)] ∣ ∣ ∣ ∣ ∣ ≤ cubo 21, 2 (2019) caputo fractional iyengar type inequalities 9 max { ‖dν ∗af‖l1([a,b]) , ∥ ∥dνb−f ∥ ∥ l1([a,b]) } γ (ν + 1) ( b − a n )ν [ jν + (n − j) ν ] , (40) j = 0, 1, 2, ..., n, viii) when n = 2 and j = 1, (40) turns to ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − (b − a) 2 (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖l1([a,b]) , ∥ ∥dνb−f ∥ ∥ l1([a,b]) } γ (ν + 1) (b − a) ν 2ν−1 . (41) proof. here ν ≥ 1 and dν ∗af, d ν b−f ∈ l1 ([a, b]). by (10) we get ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ 1 γ (ν) (x − a) ν−1 ∫x a |dν ∗af (t)| dt ≤ (x − a) ν−1 γ (ν) ‖dν ∗af‖l1([a,b]) , (42) ∀ x ∈ [a, b] . that is ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ ‖dν ∗af‖l1([a,b]) γ (ν) (x − a) ν−1 , (43) ∀ x ∈ [a, b] . by (11) we get ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ 1 γ (ν) (b − x) ν−1 ∫b x ∣ ∣dνb−f (z) ∣ ∣dz ≤ (b − x) ν−1 γ (ν) ∥ ∥dνb−f ∥ ∥ l1([a,b]) , (44) ∀ x ∈ [a, b] . that is ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ ∥ ∥dνb−f ∥ ∥ l1([a,b]) γ (ν) (b − x) ν−1 , (45) ∀ x ∈ [a, b] . call δ1 := ‖dν ∗af‖l1([a,b]) γ (ν) , (46) 10 george a. anastassiou cubo 21, 2 (2019) and δ2 := ∥ ∥dνb−f ∥ ∥ l1([a,b]) γ (ν) . (47) set δ := max (δ1, δ2) . (48) that is ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ δ (x − a) ν−1 , (49) and ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ δ (b − x) ν−1 , (50) ∀ x ∈ [a, b] . as in the proof of theorem 4, we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 + (−1)kf(k) (b) (b − t) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ δ ν [ (t − a) ν + (b − t) ν ] , (51) ∀ t ∈ [a, b] . the rest of the proof is similar to the proof of theorem 4. we continue with theorem 6. let p, q > 1 : 1 p + 1 q = 1, ν > 1 q , n = ⌈ν⌉ ; f ∈ acn ([a, b]), with dν ∗af, d ν b−f ∈ lq ([a, b]). then i) ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 + (−1)kf(k) (b) (b − t) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) ( ν + 1 p ) (p (ν − 1) + 1) 1 p [ (t − a) ν+ 1 p + (b − t) ν+ 1 p ] , (52) ∀ t ∈ [a, b] , ii) at t = a+b 2 , the right hand side of (52) is minimized, and we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! (b − a) k+1 2k+1 [ f(k) (a) + (−1)kf(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ cubo 21, 2 (2019) caputo fractional iyengar type inequalities 11 max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) ( ν + 1 p ) (p (ν − 1) + 1) 1 p (b − a) ν+ 1 p 2 ν− 1 q , (53) iii) if f(k) (a) = f(k) (b) = 0, for all k = 0, 1, ..., n − 1, we obtain ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) ( ν + 1 p ) (p (ν − 1) + 1) 1 p (b − a) ν+ 1 p 2 ν− 1 q , (54) which is a sharp inequality, iv) more generally, for j = 0, 1, 2, ..., n ∈ n, it holds ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! ( b − a n )k+1 [ jk+1f(k) (a) + (−1)k (n − j) k+1 f(k) (b) ] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) ( ν + 1 p ) (p (ν − 1) + 1) 1 p ( b − a n )ν+ 1 p [ j ν+ 1 p + (n − j) ν+ 1 p ] , (55) v) if f(k) (a) = f(k) (b) = 0, k = 1, ..., n − 1, from (55) we get: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a n ) [jf (a) + (n − j) f (b)] ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) ( ν + 1 p ) (p (ν − 1) + 1) 1 p ( b − a n )ν+ 1 p [ j ν+ 1 p + (n − j) ν+ 1 p ] , (56) for j = 0, 1, 2, ..., n, vi) when n = 2 and j = 1, (56) turns to ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − ( b − a 2 ) (f (a) + f (b)) ∣ ∣ ∣ ∣ ∣ ≤ max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) ( ν + 1 p ) (p (ν − 1) + 1) 1 p (b − a) ν+ 1 p 2 ν− 1 q , (57) vii) when 1/q < ν ≤ 1, inequality (57) is again valid but without any boundary conditions. proof. here ν > 0, n = ⌈ν⌉, f ∈ acn ([a, b]) ; p, q > 1 : 1 p + 1 q = 1, with dν ∗af, d ν b−f ∈ lq ([a, b]). by (10) we have ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ 1 γ (ν) ∫x a (x − t) ν−1 |dν ∗af (t)| dt ≤ 12 george a. anastassiou cubo 21, 2 (2019) 1 γ (ν) (∫x a (x − t) p(ν−1) dt ) 1 p (∫x a |dν ∗af (t)| q dt ) 1 q ≤ 1 γ (ν) (x − a) p(ν−1)+1 p (p (ν − 1) + 1) 1 p ‖dν ∗af‖lq([a,b]) . (58) here we assume that ν > 1 q ⇔ p (ν − 1) + 1 > 0. so, we get ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ ‖dν ∗af‖lq([a,b]) γ (ν) (p (ν − 1) + 1) 1 p (x − a) ν− 1 q , (59) ∀ x ∈ [a, b] . by (11) we have ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ 1 γ (ν) ∫b x (z − x) ν−1 ∣ ∣dνb−f (z) ∣ ∣dz ≤ 1 γ (ν) (∫b x (z − x) p(ν−1) dz ) 1 p (∫b x ∣ ∣dνb−f (z) ∣ ∣ q dz ) 1 q ≤ 1 γ (ν) (b − x) p(ν−1)+1 p (p (ν − 1) + 1) 1 p ∥ ∥dνb−f ∥ ∥ lq([a,b]) . (60) so, we get ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ ∥ ∥dνb−f ∥ ∥ lq([a,b]) γ (ν) (p (ν − 1) + 1) 1 p (b − x) ν− 1 q , (61) ∀ x ∈ [a, b] . call ρ1 := ‖dν ∗af‖lq([a,b]) γ (ν) (p (ν − 1) + 1) 1 p , (62) and ρ2 := ∥ ∥dνb−f ∥ ∥ lq([a,b]) γ (ν) (p (ν − 1) + 1) 1 p . (63) set ρ := max (ρ1, ρ2) , m := ν − 1 q > 0. (64) that is ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (a) k! (x − a) k ∣ ∣ ∣ ∣ ∣ ≤ ρ (x − a) m , (65) and ∣ ∣ ∣ ∣ ∣ f (x) − n−1∑ k=0 f(k) (b) k! (x − b) k ∣ ∣ ∣ ∣ ∣ ≤ ρ (b − x) m , (66) cubo 21, 2 (2019) caputo fractional iyengar type inequalities 13 ∀ x ∈ [a, b] . as in the proof of theorem 4, we obtain: ∣ ∣ ∣ ∣ ∣ ∫b a f (x) dx − n−1∑ k=0 1 (k + 1) ! [ f(k) (a) (t − a) k+1 + (−1)kf(k) (b) (b − t) k+1 ] ∣ ∣ ∣ ∣ ∣ ≤ ρ (m + 1) [ (t − a) m+1 + (b − t) m+1 ] = max { ‖dν ∗af‖lq([a,b]) , ∥ ∥dνb−f ∥ ∥ lq([a,b]) } γ (ν) (p (ν − 1) + 1) 1 p ( ν + 1 p ) [ (t − a) ν+ 1 p + (b − t) ν+ 1 p ] , (67) ∀ t ∈ [a, b] . the rest of the proof is similar to the proof of theorem 4. references [1] george a. anastassiou, fractional differentiation inequalities, springer, heidelberg, new york, 2009. [2] george a. anastassiou, intelligent mathematical computational analysis, springer, heidelberg, new york, 2011. [3] k. diethelm, the analysis of fractional differential equations, springer, heidelberg, new york, 2010. [4] k.s.k. iyengar, note on an inequality, math. student, 6 (1938), 75-76. introduction main results cubo, a mathematical journal vol. 23, no. 03, pp. 343–355, december 2021 doi: 10.4067/s0719-06462021000300343 non-algebraic limit cycles in holling type iii zooplankton-phytoplankton models homero g. d́ıaz-maŕın1 osvaldo osuna2 1facultad de ciencias f́ısico-matemáticas, universidad michoacana, edif. alfa, ciudad universitaria, c.p. 58040, morelia, michoacán, méxico. homero.diaz@umich.mx 2instituto de f́ısica y matemáticas, universidad michoacana, edif. c-3, ciudad universitaria, c.p. 58040, morelia, michoacán, méxico. osvaldo.osuna@umich.mx abstract we prove that for certain polynomial differential equations in the plane arising from predator-prey type iii models with generalized rational functional response, any algebraic solution should be a rational function. as a consequence, limit cycles, which are unique for these dynamical systems, are necessarily trascendental ovals. we exemplify these findings by showing a numerical simulation within a system arising from zooplankton-phytoplankton dynamics. resumen probamos que para ciertas ecuaciones diferenciales polinomiales en el plano que aparecen a partir de modelos predador-presa de tipo iii con respuesta funcional racional generalizada, toda solución algebraica debe ser una función racional. como consecuencia, los ciclos ĺımite, que son únicos para estos sistemas dinámicos, son necesariamente óvalos trascendentes. ejemplificamos estos resultados mostrando una simulación numérica para un sistema que aparece en la dinámica de zooplancton-fitoplancton. keywords and phrases: predator-prey models, functional-response, puiseux series, newton polygon, limit cycles, invariant algebraic curve. 2020 ams mathematics subject classification: 34c25, 34m25, 34m35, 37n25, 92d25. accepted: 07 july, 2021 received: 08 january, 2021 ©2021 h. g. d́ıaz-maŕın et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300343 https://orcid.org/0000-0002-2453-9049 344 h. g. dı́az-maŕın & o. osuna cubo 23, 3 (2021) 1 introduction we consider predator-prey model u̇ = ru ( 1 − u k ) − vp(u), v̇ = v (−d + γp(u)) , (1.1) with functional response proposed in [13] p(u) = mun/(a + un). where the parameters are: the maximal feeding rate m; an affinity constant a, related to handling times, capture efficiencies, etc.; and the number of encounters n ≥ 1 a predator must have with a prey item before becoming maximally efficient at utilizing the prey item as a resource. according to [13], this last parameter is derived from an analogy with michaelis-menten equation for enzymatic kinetics. here n measures the amount of ‘learning’ exhibited by the predator. for n > 1, this functional response has holling type iii, while for n = 1 it has holling type ii, that is why this functional is also called generalized functional response. increasing the attack exponent 1 < n < 2 introduces the stability of simple consumer-resource population models, theoretical findings reveal that this increases biodiversity, see [14] and references therein. by fitting parameters, it is shown that n ≥ 2 appear in certain models in ecology, where predator free-space is a component of the habitat structure, see [1]. other theoretical models of biological relevance consider the specific attack exponent n = 2, see [15, 18]. for 1 < n < 2 existence and uniqueness of limit cycles for predator-prey system (1.1) is proved in [16]. existence and uniqueness for 0 ≤ n ≤ 1, n ≥ 2 also holds true under certain conditions, see [17]. along this work we consider only integer values n ≥ 2. existence of non-algebraic limit cycles for the lotka-volterra model were first exhibited by [12]. since then, existence of trascendental ovals as limit cycles in system generalizing lotka-voltera models have been proved, see for instance [9, 5, 6]. motivated by these results we explore this question for generalized functional responses. our main result is contained in theorem 2.1 which asserts that limit cycles can not be algebraic ovals in the case of holing type iii predator-prey models. the proof uses puiseux series at infinity in the variable x. we estimate the number of branches of solutions given by the puiseux series. we perform calculation of the leading term and prove that there exists at most one determination or branch of such series. to see this we show how each coefficient cn is completely specified by the parameters of the system. thus, we conclude that any invariant algebraic curve must have at most degree one in y. thus any algebraic invariant curve y(x), should be a rational function. for related works which also apply formal and puiseux series to planar polynomial systems see [4, 7, 8]. cubo 23, 3 (2021) non-algebraic limit cycles in holling type iii ... 345 2 rational functions as invariant algebraic curves under a suitable change of variables, x = u, y = −v/m, and time reparametrization, ds dt = 1/(a + u2), system (1.1) becomes ẋ = x ( (r − 1/k)a + (r − 1/k)xn + xn−1y ) , ẏ = y (−d + (γm − d)xn) . (2.1) thus we study the algebraic system: ẋ = x ( a0 + anx n + xn−1y ) , ẏ = y (b0 + bnx n) , an ̸= bn. (2.2) notice that the axes x = 0, y = 0 are algebraic solutions of (2.1). take the ode defined by system (2.1) in the complex domain dy dx = y (b0 + bnx n) x (a0 + anxn + xn−1y) . (2.3) solutions are riemann surfaces immersed in cx × cy, where cx ≃ c and poles of solutions correspond to values y = ∞ in the compactification cy = cy ∪ {∞} ≃ cp1. if we ask for the existence of algebraic solutions f(x, y) = 0 for f ∈ c[x, y], of the dynamical system (2.1). then, such algebraic curve should be rational. theorem 2.1. suppose that the following conditions hold, a0 ̸= b0, an ̸= bn. (2.4) if there exists an invariant algebraic curve f(x, y) = 0 of equation (2.3) with x, y ∤ f(x, y), then degy f = 1. therefore, any algebraic (possibly multivalued) solution should also be a rational (univalued) solution, y = ϕ(x), provided we exclude the trivial solution, y(x) ≡ 0. the following claim becomes of interest. corollary 2.2. there can not exist algebraic limit cycles of the dynamical system (2.1) as a real vector field in r2, whenever conditions (2.4) hold true. for the proof of theorem 2.1 we consider the newton-puiseux algorithm to describe explicitly the nature of solutions at the infinites x = ∞ and y = ∞. for further explanation of the newtonpuiseux method for ode, see [2, 10, 11]. the crucial step of the proof is to apply the following result. theorem 2.3 (theorem 1.4 in [3]). let g(z, w) = 0 be an invariant algebraic curve, ∂wg ̸= 0 of the polynomial ode p(z, w) dw dz − q(z, w) = 0. (2.5) 346 h. g. dı́az-maŕın & o. osuna cubo 23, 3 (2021) then degw g is at most the number of puiseux series w(z) = c0z µ0 + ∞∑ l=1 clz l m0 +µ0, (2.6) solving (2.5), whenever the number of these series is finite. here µ0 = l0/m0 with m0, l0 relatively prime integers m0 ≥ 0. proof of theorem 2.1. we proceed analyzing poles and algebraic branch points according to painlevé methodology, see [10, 11]. notice that under the blow-up change of coordinates ξ = 1 x , equation (2.1) yields an equation at x = ∞ corresponding to ξ = 0 dy dξ = − y(a0ξ n + an) ξ(b0ξn + bn + ξy) , (2.7) at infinity the trivial solution y ≡ 0 yields a trivial solution which tends to ξ = 0. to find an expansion of non-trivial solutions along ξ = 0, with ξ = 1/x, in equation (2.7), we adopt the following puiseux series expansion: y(ξ) = c0ξ µ0 + ∞∑ l=1 clξ l m0 +µ0, (2.8) where µ0 = l0/m0 and −1/µ0 is one of many possible slopes of the corresponding newton polygon, and l0, m0 are relatively prime integers. for equation (2.7) the newton polygon is a right-angled triangle whose only oblique side is the hypothenuse, see fig. 1. y l 6 . . .pp pp pp ppp� � � 2 1 2 3 4 5 1 n ξ l figure 1: newton polygon associated to the ode (2.7) and used to calculate µ0. circled vertices correspond to monomials appearing in by′ within the expression a(ξ, y) + b(ξ, y)dy dξ = 0. therefore, the only slope to consider is −1/µ0 = 1. accordingly, µ0 = −1 and c0(bn −an)−c20 = 0, with two possible roots: c0 = 0, bn −an ∈ c. if we make a direct substitution c0 = 0 of the laurent expansion, ∑∞ l=0 clξ −1+l. this yields the trivial solution, y ≡ 0. we claim that the remaining value, c0 = bn − an ̸= 0 (2.9) cubo 23, 3 (2021) non-algebraic limit cycles in holling type iii ... 347 gives rise to a unique laurent series of a simple pole at x = ∞, and therefore to just one branch of the puiseux series. indeed, under substitution ξ1 = ξ, y = c0ξ −1 1 + y1, we obtain a newton polygon for (a0ξ n 1 + bn + ξ1y1)ξ1 dy1 dξ1 + (b0ξ n 1 + an)y1 + (an − bn)(a0 − b0)ξ n−1 1 = 0 (2.10) which has two possible slopes and corresponding values µ1 = 1, −1/(n − 1). see fig. 2. n−1 l l 6 . . . ..... ppppppppp� � � pp pp pp ppp� � � 2 1 2 3 4 5 1 n ξ y l figure 2: newton polygon associated to the ode (2.10) and used to calculate µ1. according to the algorithm given in [2], for positive slope −1/µ1, we choose as principal side, µ1 = n − 1. we have cn = (an − bn)(a0 − b0) (1 − n)bn − an . (2.11) in the following step, we have a principal side with µ2 = 2n − 1. see the corresponding newton polygon used to calculate µ2 in fig. 3. this determines c2n. therefore, there is a unique determination for the puiseux-laurent series: y = c0ξ −1 + ∞∑ k=1 cknξ kn−1 + . . . . (2.12) by direct substitution of the puiseux-laurent series in eq. (2.7) we can also verify that the middle coefficients vanish, i.e. for each k = 0, 1, 2, . . . , we have cl = 0, ∀l = kn + 1, . . . , (k + 1)n − 1. theorem 2.3 implies that degy f ≤ 1. thus, under the hypothesis of theorem 2.1 we conclude that y = ϕ(x) is a rational function which cannot contain an algebraic limit cycle because of the uniqueness of its determination with respect to x. 348 h. g. dı́az-maŕın & o. osuna cubo 23, 3 (2021) 1 l l 6 . . . . � � aaaaaaaaaahhh hhh hhh hh1 y 2 3 4 n 2n − 1 l figure 3: newton polygon used to calculate µ2. remark 2.4. we have chosen puiseux-laurent series of the form y = y(x) because there is a recognizable pattern in the successive newton polygons, namely triangles with a moving low vertex. this yields a unique side with a unique slope. therefore a unique µk yields a unique linear relation that allows us to compute all the coefficients ck. 3 on the degree with respect to x we may ask whether the degree in x for an invariant curve can be estimated with the same methods. notice that expression (2.12) suggests that degx f = nk for some k ∈ n. we illustrate the difficulties to calculate an upper bound for degx f using the same techniques by considering n = 3. if we take, x = x(y) at y = ∞, then we may take the coordinate change y = 1 η . thus system (2.1) becomes dx dη = a0xη + anx n+1η − xn−1 η2(b0 + bnxnη2) the corresponding newton polygon is shown in fig. 4. there are three posible cases for puiseux-laurent series x(η) = c0η µ0 + ∞∑ l=1 clη l m0 +µ0, corresponding to slopes −1/µ0 equal to 1, ∞, −(n − 1) which yield µ0 equal to −1, 0, 1n−1. no infinite values c0 ∈ r arise in each case. this can be verified as follows: (1) case µ0 = −1. under substitution η1 = η, x = c0η µ0 1 + x1, cubo 23, 3 (2021) non-algebraic limit cycles in holling type iii ... 349 x i 6 b b b b b b b bb � � � n n+1 1 1 η i figure 4: newton polygon for x(y) at infinity used to calculate µ0 = 1 n−1. we regard the least degree coefficient. there exists a unique puiseux series where c0 is determined by a linear relation cn0 (c0(bn − an) − 1) = 0, therefore there exists just one branch. (2) case µ0 = 0. corresponds to the trivial solution x ≡ 0 with c0 = 0. (3) case µ0 = 1 n−1. puiseux-laurent series arise as c0 solve a relation: a0c0 + b0c0 n − 1 + cn0 = 0, therefore there exists n − 1 posible branches. each branch corresponds to a (n − 1)−th root c0 = ( −a0 − b0 n − 1 )1/n−1 . if we choose µ0 = 1 n−1, then we get the ode, a(η1, x1) + b(η1, x1) dx1 dη1 , with extended expression, a(3,0)η 3 1 +a(1,1)η1x1 + a(5/2,1)η 5/2 1 x1 + a(1/2,2)η 1/2 1 x 2 1 + a(2,2)η 2 1x 2 1 +a(0,3)x 3 1 + a(3/2,1)η 3/2 1 x1 + a(1,4)η1x 4 1 + dx1 dη1 × [b(1,1)η21 + b(5/2,1)η 7/2 +b(2,2)η 3 1x1 + b(3/2,3)η 5/2 1 x 2 1 + b(1,4)η 2 1x 4 1] = 0. 350 h. g. dı́az-maŕın & o. osuna cubo 23, 3 (2021) whose newton polygon is shown in fig 5. circled vertices correspond to monomials appearing in bx′1 within the ode. under the same assumptions of theorem 2.1, if there exists an invariant algebraic curve f(x, y) = 0 of (2.3) with x, y ∤ f(x, y), then degx f has upper bound at least n − 1, provided we exclude the trivial solution, y(x) ≡ 0. this would require a0 + b0 n − 1 ̸= 0. (3.1) we still can not conclude that degx f ≤ n − 1, since the proof of this fact would require a suitable description of successive newton polygons, as well as an effective calculation of the number of branches of the corresponding puiseux-laurent series. two main difficulties arise: on one hand these newton polygons may follow a complex pattern. on the other hand, we may have several different relations defining general coefficients ck, k > 0 requiring enough conditions so that there is a finite number of branches rather than a continuum. 4 l l ll 6 . . . . . . . . a a a a h h hha a a a a a aa� � x 1 2 3 η 1 2 3 l figure 5: newton polygon that determines µ1 = 2 with n = 3. in the second step we have the possibility to choose either µ1 = 2 or µ1 = 1/2. if we choose µ1 = 2. then, the corresponding relations arising from the least degree terms in the substitution η1 = η2, x1 = c1η µ1 2 + x2 become, 4c1(4a0 − b0) = 8a3a20 + 4a 2 0b3 + 8a3a0b0 + 4a0b3b0 + 2a3b 2 0 + b3b 2 0, therefore, we would require the additional condition 4a0 ̸= b0 (3.2) in order to be able to calculate c1 and thus have a finite number of branches. cubo 23, 3 (2021) non-algebraic limit cycles in holling type iii ... 351 according to [2, lemma 2], in order to have a finite number of branches, i.e. a good side of the newton polygon in its terminology, it is sufficient that the following conditions hold for the high and low vertices of such a side, (a, b), (a′, b′), respectively: (1) b(a,b) ̸= 0 and a(a,b) b(a,b) /∈ q≥µ1 = {q ∈ q : q ≥ µ}, (2) a(a′,b′) + µb(a′,b′) ̸= 0, where a(a,b), b(a,b) refer to the coefficients for the monomials in the equation a + b dx1 dη1 = 0 associated to the vertex (a, b). in our concrete example, in fig. 5 we have chosen µ1 = 2 because it corresponds to the slope −1/µ1 of the unique good side which has vertices (a, b) = (1, 1) and (a′, b′) = (3, 0). calculations yield a(1,1) = 1 8 (16a0 + 12b0), b(1,1) = −b0, a(3,0) = −c40 ( a0 + b0 2 ) , b(3,0) = 0. recall that under our conventions, b(3,0) = 0 is implied by the fact that the vertex (3, 0) is not circled. conditions for a good side which are sufficient to have a finite number of branches read as follows: (1) 16a0+12b0 8b0 /∈ q≥µ1 = {q ∈ q : q ≥ µ1}. that is, either a0 b0 < 1 4 or a0 b0 ≥ 1 4 but a0 b0 /∈ q. (3.3) (2) a0 + b0 2 ̸= 0. we recover condition (3.1). notice that condition (3.3) is stronger than (3.2). in the following step we choose µ2 = 7/2 by considering the slopes of the newton polygon shown in fig. 6 with the unique good side which has vertex (a′, b′) on the η1−axis. remark the increasing complexity of the polygon. thus in the step k ≥ 2, we can always choose the good side largest negative slope −1/µk with vertex (a′, b′) = (a′, 0) on the ηk−axis and vertex (a, b) = (1, 1). but even if in each step k ≥ 2 we achieve linear relations to determine coefficients ck, we still can not conclude that there is a finite number of determinations. an additional calculation needs to be done, namely to verify that no other side in the newton polygon, yield a continuous indetermination ck ∈ c. those additional sides are not good. to illustrate this difficulty suppose that we do not choose the unique good side in fig. 5. suppose that on the contrary we choose the side with vertices (a, b) = (0, 3) and (a′, b′) = (1, 1) which is not good. further calculation yields (a0 + 2b0)c0 = 0. 352 h. g. dı́az-maŕın & o. osuna cubo 23, 3 (2021) therefore, either c0 = 0 or an indetermination c0 ∈ c arises whenever the following relation does or does not hold: a0 + 2b0 ̸= 0. (3.4) x l ll l l ll l l l. . . . . . . . . . . . . . . . . 6 @ @xxxxxxx h h h h h h h h h h h h h hh� � 1 2 3 1 2 3 4 η4 5 6 7 8 9 l figure 6: newton polygon to determine µ2 = 7/2. notice its increasing complexity with respect to figs. 5 and 4. summarizing, if we follow the same strategy, to find an effective upper bound for degx f requires further calculations and a detailed and complete description of the conditions that allow a finite number of branches. we leave it for a future work. 4 zooplankton-phytoplankton dynamics we consider the dependence of a predator’s (zooplankton) grazing rate on prey (phytoplankton) is taken as that of holling type iii response as in [15], instead of type ii as in (2.67), (2.68) in [18]. suppose that phytoplankton grows in logistic form whereas the zooplankton predation by fish is neglected. we then get the following system u̇ = u (1 − u) − vu2 h + u2 , v̇ = γvu2 h + u2 − δv, (4.1) which yields a system similar to (2.1): ẋ = x (xy) , ẏ = y ( −δ + (γ − δ)x2 ) . (4.2) cubo 23, 3 (2021) non-algebraic limit cycles in holling type iii ... 353 the criterion exposed in [17] for existence and uniqueness of a limit cycle adapted to system (1.1) with m = 1 and n ≥ 2 states that (nd − (n − 2)γ) · n √ ad γ − d < (pd − (p − 1)γ)k, which for (4.1) yields 2δ √ hδ γ − δ < 2δ − γ. (4.3) for the specific choice of parameters: δ = 0.25, γ = 0.35, h = 0.01, condition (4.3) holds true. therefore there exists a unique limit cycle. indeed, numerical evidence for the existence of a limit cycle is given in fig. 7. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 figure 7: solutions of system (4.1) with initial conditions (0.1, 0.05) and (0.2, 0.2) converge to a limit cycle. on the other hand, due to corollary 2.2 this limit cycle can not be an algebraic curve. indeed, for an algebraic invariant curve f(x, y) = 0, there is only one branch of a simple pole at x = ∞. the corresponding puiseux-laurent series of this branch is y = ∞∑ k=0 c2kξ 2k−1 = c0ξ −1 + c2ξ + c4ξ 3 + . . . , where ξ = 1/x. a straightforward calculation of the coefficients yields c0 = γ − δ = 0.1, c2 = δ = 0.25, c4 = 0 = c2k, k ≥ 2. therefore, y = c0ξ −1 + c2ξ. hence for an invariant algebraic curve, degy f = 1, and y = ϕ(x), should be rational with f(x, ϕ(x)) = 0. in this example degx f = 2. 354 h. g. dı́az-maŕın & o. osuna cubo 23, 3 (2021) references [1] d. barrios-o’neill, j. t. a. dick, m. c. emmerson, a. ricciardi and h. j. macisaac, “predator-free space, functional responses and biological invasions”, functional ecology, vol. 29, no. 3, pp. 377–384, 2015. 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[8] j. giné and j. llibre, “formal weierstrass nonintegrability criterion for some classes of polynomial differential systems in c2”, internat. j. bifur. chaos appl. sci. engrg., vol. 30, no. 4, 7 pages, 2020. [9] m. hayashi, “on polynomial liénard systems which have invariant algebraic curves”, funkcial. ekvac., vol. 39, no. 3, pp. 403–408, 1996. [10] e. hille, ordinary differential equations in the complex domain, dover publications, inc., mineola, ny, 1976. [11] e. l. ince, ordinary differential equations, dover publications, new york, 1944. [12] k. odani, “the limit cycle of the van der pol equation is not algebraic”, j. differential equations, vol. 115, no. 1, pp. 146–152, 1995. [13] l. a. real, “the kinetics of functional response”, the american naturalist, vol. 111, no. 978, pp. 289–300, 1977. [14] b. rosenbaum and b. c. rall, “fitting functional responses: direct parameter estimation by simulating differential equations”, methods in ecology and evolution, vol. 9, no. 10, pp. 2076–2090, 2018. cubo 23, 3 (2021) non-algebraic limit cycles in holling type iii ... 355 [15] v. a. ryabchenko, m. j. r. fasham, b. a. kagan and e. e. popova, “what causes short-term oscillations in ecosystem models of the ocean mixed layer?”, journal of marine systems, vol. 13, no. 1, pp. 33–50, 1997. [16] j. sugie, “uniqueness of limit cycles in a predator-prey system with holling-type functional response”, quart. appl. math., vol. 58, no. 3, pp. 577–590, 2000. [17] j. sugie, r. kohno, and r. miyazaki, “on a predator-prey system of holling type”, proc. amer. math. soc., vol. 125, no. 7, pp. 2041–2050, 1997. [18] r. k. upadhyay and s. r. k. iyengar, introduction to mathematical modeling and chaotic dynamics, crc press, 2013. introduction rational functions as invariant algebraic curves on the degree with respect to x zooplankton-phytoplankton dynamics cubo, a mathematical journal vol. 23, no. 01, pp. 109–119, april 2021 doi: 10.4067/s0719-06462021000100109 inequalities and sufficient conditions for exponential stability and instability for nonlinear volterra difference equations with variable delay ernest yankson department of mathematics, university of cape coast, ghana. ernestoyank@gmail.com abstract inequalities and sufficient conditions that lead to exponential stability of the zero solution of the variable delay nonlinear volterra difference equation x(n + 1) = a(n)h(x(n)) + n−1∑ s=n−g(n) b(n, s)h(x(s)) are obtained. lyapunov functionals are constructed and employed in obtaining the main results. a criterion for the instability of the zero solution is also provided. the results generalizes some results in the literature. resumen se obtienen desigualdades y condiciones suficientes que implican la estabilidad exponencial de la solución cero de la ecuación en diferencias no lineal de volterra con retardo variable x(n + 1) = a(n)h(x(n)) + n−1∑ s=n−g(n) b(n, s)h(x(s)). se construyen funcionales de lyapunov y se utilizan para obtener los resultados principales. se entrega también un criterio para la inestabilidad de la solución cero. los resultados generalizan algunos resultados en la literatura. keywords and phrases: exponential stability, lyapunov functional, instability. 2020 ams mathematics subject classification: 34d20, 34d40, 34k20. accepted: 01 february, 2021 received: 02 june, 2020 ©2021 e. yankson. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100109 https://orcid.org/0000-0002-5621-1048 110 ernest yankson cubo 23, 1 (2021) 1 introduction let r and z+ denote the set of real numbers and the set of positive integers respectively. in recent times, research into the stability properties of solutions of difference equations have gained the attention of many mathematicians, see [1], [2], [4], [6], [7], [8] and the references cited therein. we are mainly motivated by the work of kublik and raffoul in [6] in which the authors obtained inequalities that lead to the exponential stability of the zero solution of the linear volterra difference equation with finite delay x(n + 1) = a(n)x(n) + n−1∑ s=n−r b(n,s)x(s), (1.1) for some positive constant r. in this paper we consider the scalar nonlinear volterra difference equation with variable delay x(n + 1) = a(n)h(x(n)) + n−1∑ s=n−g(n) b(n,s)h(x(s)), (1.2) where a : z+ → r, b : z+ × [−g0,∞) → r, h : r → r and 0 < g(n) ≤ g0, for all n ∈ z+ for some positive constant g0. we will obtain some inequalities regarding the solutions of (1.2) by employing lyapunov functionals. these inequalities can be used to deduce exponential stability of the zero solution. also, by means of a lyapunov functional an instability criterion of the zero solution of equation (1.2) will be provided. let ψ : [−g0, 0] → (−∞,∞) be a given bounded initial function with ||ψ|| = max −g0≤s≤0 |ψ(s)|. we further denote the norm of a function ϕ : [−g0,∞) → (−∞,∞) by ||ϕ|| = sup −g0≤s≤∞ |ϕ(s)|. throughout this paper we let h(x) = xh1(x). the notation xn means that xn(τ) = x(n + τ),τ ∈ [−g0, 0] as long as x(n + τ) is defined. thus, xn is a function mapping an interval [−g0, 0] into r. we say that x(n) ≡ x(n,n0,ψ) is a solution of (1.2) if x(n) satisfies (1.2) for n ≥ n0 and xn0 = x(n0 + s) = ψ(s), s ∈ [−g0, 0]. in this paper we use the convention that ∑b s=a h(s) = 0 if a > b. the following notation is introduced. let a(n,s) = γ∑ u=n−s b(u + s,s), where 0 < γ ≤ g(n− 1) for all n ∈ z+. (1.3) cubo 23, 1 (2021) inequalities and sufficient conditions for exponential stability . . . 111 it follows from (1.3) that a(n,n−g(n− 1) − 1) = 0. (1.4) we assume throughout the paper that ∆na 2(n,z) ≤ 0, for all n + s + 1 ≤ z ≤ n− 1. (1.5) due to (1.3) we can express (1.2) in the equivalent form ∆x(n) = ( a(n)h1(x(n)) + a(n + 1,n)h1(x(n)) − 1 ) x(n) − ∆n n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s)). (1.6) definition 1.1. the zero solution of (1.2) is said to be exponentially stable if any solution x(n,n0,ψ) of (1.2) satisfies |x(n,n0,ψ)| ≤ c(||ψ||,n0)ζγ(n−n0), for all n ≥ n0, where ζ is a constant with 0 < ζ < 1,c : r+ × z+ → r+, and γ is a positive constant. the zero solution of (1.2) is said to be uniformly exponentially stable if c is independent of n0. we end this section by stating a fact which will be used in the proof of lemma 2.1, that is, if u(n) is a sequence, then ∆u2(n) = u(n + 1)∆u(n) + u(n)∆u(n). for more on the calculus of difference equations we refer to [3] and [5]. 2 exponential stability in this section we obtain inequalities that can be used to deduce the exponential stability of (1.2). to simplify notation we let q(n,x) = ( a(n) + a(n + 1,n) ) h1(x(n)) − 1, and q1(n) = ( a(n) + a(n + 1,n) ) − 1. lemma 2.1. suppose that (1.3), (1.5) and for δ > 0, − δ δg0 + g(n) ≤ q(n,x) ≤−δg0a2(n + 1,n)h21(x(n)) −q 2(n,x), (2.1) 112 ernest yankson cubo 23, 1 (2021) holds. if 1 ≤ h1(x), and v (n) =  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))  2 + δ −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)), (2.2) then based on the solutions of (1.2) we have ∆v (n) ≤ q1(n)v (n). (2.3) proof. let x(n,n0,ψ) be a solution of (1.2) and let v (n) be defined by (2.2). it must also be noted that in view of condition (2.1), q(n,x) < 0 for all n ≥ 0. this together with the fact that 1 ≤ h1(x) also implies that q(n,x) ≤ q1(n) < 0. then based on the solutions of (1.2) we have ∆v (n) =  x(n + 1) + n∑ s=n−g(n) a(n + 1,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))   +  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))   + δ∆n −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)). (2.4) but x(n + 1) + n∑ s=n−g(n) a(n + 1,s)h(x(s)) = ( q(n,x) + 1 ) x(n) − ∆n n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s)) + n∑ s=n−g(n) a(n + 1,s)h(x(s)) = ( q(n,x) + 1 ) x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s)) = ( q(n,x) + 1 ) x(n) + n−1∑ s=n−g(n) a(n,s)h(x(s)) (2.5) cubo 23, 1 (2021) inequalities and sufficient conditions for exponential stability . . . 113 where we have used the fact that a(n,n−g(n− 1) − 1) = 0. using (2.5) in (2.4) we obtain ∆v (n) =  (q(n,x) + 1) x(n) + n−1∑ s=n−g(n) a(n,s)h(x(s))  q(n,x)x(n) +  x(n) + n−1∑ s=n−g(n) a(n,s)h(x(s))  q(n,x)x(n) + δ∆n −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) = q(n,x)v (n) + (q2(n,x) + q(n,x))x2(n) + δ∆n −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) − q(n,x)   n−1∑ s=n−g(n) a(n,s)h(x(s))  2 − δq(n,x) −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) (2.6) considering the third term on the right hand side of (2.6) we obtain ∆n −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) = −1∑ s=−g0 n∑ z=n+s+1 a2(n + 1,z)h2(x(z)) − −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) = −1∑ s=−g0 [ a2(n + 1,n)h(x2(n)) + n−1∑ z=n+s+1 a2(n + 1,z)h2(x(z)) − n−1∑ z=n+s+1 a2(n,z)h2(x(z)) −a2(n,n + s)h2(x(n + s)) ] = −1∑ s=−g0 ( a2(n + 1,n)h21(x(n))x 2(n) −a2(n,n + s)h2(x(n + s)) ) + −2∑ s=−g0 n−1∑ z=n+s+1 ∆na 2(n,z)h2(x(z)) = g0a 2(n + 1,n)h21(x(n))x 2(n) − −1∑ s=−g0 a2(n,n + s)h2(x(n + s)) + −2∑ s=−g0 n−1∑ z=n+s+1 ∆na 2(n,z)h2(x(z)) ≤ g0a2(n + 1,n)h21(x(n))x 2(n) − −1∑ s=−g0 a2(n,n + s)h2(x(n + s)). = g0a 2(n + 1,n)h21(x(n))x 2(n) − n−1∑ z=n−g0 a2(n,z)h2(x(z)) (2.7) 114 ernest yankson cubo 23, 1 (2021) applying the holder’s inequality to the squared term in the fourth term on the right hand side of (2.6) gives   n−1∑ s=n−g(n) a(n,s)h(x(s))  2 ≤ g(n) n−1∑ s=n−g(n) a2(n,s)h2(x(s)) ≤ g(n) n−1∑ s=n−g0 a2(n,s)h2(x(s)). (2.8) considering the last term on the right hand side of (2.6) we obtain −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) ≤ g0 n−1∑ s=n−g0 a2(n,s)h2(x(s)) (2.9) substituting (2.7), (2.8) and (2.9) in (2.6) we obtain ∆v (n) ≤ q(n,x)v (n) + (q2(n,x) + q(n,x) + δg0a2(n + 1,n)h21(x(n)))x 2(n) + [−(g(n) + δg0)q(n,x) − δ] n−1∑ s=n−g0 a2(n,s)h2(x(s)) ≤ q(n,x)v (n) + (q2(n,x) + q(n,x) + δg0a2(n + 1,n))x2(n) + [−(g(n) + δg0)q(n,x) −δ] n−1∑ s=n−g0 a2(n,s)h2(x(s)) ≤ q(n,x)v (n) ≤ q1(n)v (n). theorem 2.2. suppose the hypothesis of lemma 2.1 hold. then any solution x(n) = x(n,n0,ψ) of (1.2) satisfies the exponential inequality |x(n)| ≤ √√√√g0 + δ δ v (n0) n−1∏ s=n0 ( a(n) + a(n + 1,n) ) (2.10) for n ≥ n0. proof. let v (n) be defined by (2.2). changing the order of summation in the second term on the right hand side of (2.2) we obtain δ −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) = δ n−1∑ z=n−g0 z−n∑ s=−g0 a2(n,z)h2(x(z)) = δ n−1∑ z=n−g0 a2(n,z)h2(x(z))(z −n + g0 + 1) ≥ δ n−1∑ z=n−g0 a2(n,z)h2(x(z)) ≥ δ n−1∑ z=n−g(n) a2(n,z)h2(x(z)), cubo 23, 1 (2021) inequalities and sufficient conditions for exponential stability . . . 115 where we have used the fact that if n − g0 ≤ z ≤ n − 1 then 1 ≤ z − n + g0 + 1 ≤ g0 and n−g0 ≤ n−g(n). also, we note that   n−1∑ z=n−g(n) a(n,z)h(x(z))  2 ≤ g0 n−1∑ z=n−g(n) a2(n,z)h2(x(z)). hence, δ −1∑ s=−g0 n−1∑ z=n+s a2(n,z)h2(x(z)) ≥ δ g0   n−1∑ z=n−g(n) a(n,z)h(x(z))  2 thus, v (n) ≥  x(n) + n−1∑ s=n−g(n) a2(n,z)h2(x(z))  2 + δ g0   n−1∑ z=n−g(n) a(n,z)h(x(z))  2 = δ g0 + δ x2(n) +  √ g0 g0 + δ x(n) + √ g0 + δ g0 n−1∑ z=n−g(n) a(n,z)h(x(z))  2 ≥ δ g0 + δ x2(n). but v (n) ≤ v (n0) n−1∏ s=n0 ( (a(n) + a(n + 1,n) ) this implies that δ g0 + δ x2(n) ≤ v (n0) n−1∏ s=n0 ( (a(n) + a(n + 1,n) ) hence, |x(n)| ≤ √√√√g0 + δ δ v (n0) n−1∏ s=n0 ( a(n) + a(n + 1,n) ) . (2.11) this completes the proof. corollary 2.3. suppose that the hypotheses of theorem 3.2 hold. suppose that there exists a positive number α < 1 such that 0 < a(n) + a(n + 1,n) ≤ α. then the zero solution of (1.2) is exponentially stable. 116 ernest yankson cubo 23, 1 (2021) proof. it follows from (2.10) that |x(n)| ≤ √√√√g0 + δ δ v (n0) n−1∏ s=n0 (a(n) + a(n + 1,n)) ≤ √ g0 + δ δ v (n0)αn−n0 for n ≥ n0. since α ∈ (0, 1) the proof is complete. 3 instability criteria in this section we consider the problem of finding a criteria for instability of the zero solution of (1.2). a suitable lyapunov functional will be used to obtain the instability criteria. theorem 3.1. assume that (1.3), (1.5) hold and let ρ > g0 be a constant. assume that q1(n) > 0 and q(n,x) > 0 such that q2(n,x) + q(n,x) −ρa2(n + 1,n)h21(x(n)) ≥ 0. (3.1) if 1 ≤ h1(x) and v (n) =  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))  2 −ρ n−1∑ s=n−g(n−1)−1 a2(n,s)h2(x(s)) (3.2) then, based on the solutions of (1.2) we have ∆v (n) ≥ q1(n)v (n). proof. let x(n,n0,ψ) be a solution of (1.2) and let v (n) be defined by (3.2). then based on the solutions of (1.2) we have ∆v (n) =  x(n + 1) + n−1∑ s=n−g(n) a(n,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))   +  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))   − ρ  a2(n + 1,n)h2(x(n)) + n−1∑ s=n−g(n) ∆na 2(n,s)h2(x(s))   cubo 23, 1 (2021) inequalities and sufficient conditions for exponential stability . . . 117 ≥  (q(n,x) + 1)x(n) + n−1∑ s=n−g(n) a(n,s)h(x(s))  q(n,x)x(n) +  x(n) + n−1∑ s=n−g(n) a(n,s)h(x(s))  q(n,x)x(n) − ρa2(n + 1,n)h2(x(n)) = q(n,x)v (n) + (q2(n,x) + q(n,x) −ρa2(n + 1,n)h21(x(n)))x 2(n) − q(n,x)   n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))  2 + q(n,x)ρ n−1∑ s=n−g(n−1)−1 a2(n,s)h2(x(s)) ≥ q(n,x)v (n) + (q2(n,x) + q(n,x) −ρa2(n + 1,n)h21(x(n)))x 2(n) + q(n,x)(ρ−g0) n−1∑ s=n−g(n−1)−1 a2(n,s)h2(x(s)) ≥ q(n,x)v (n) ≥ q1(n)v (n). this completes the proof. theorem 3.2. suppose the hypothesis of theorem 3.1 hold. then the zero solution of (1.2) is unstable, provided that ∞∏ s=0 (a(n) + a(n + 1,n)) = ∞. proof. we have from theorem 3.1 that ∆v (n) ≥ q1(n)v (n), which implies that v (n) ≥ v (n0) ∞∏ s=n0 (a(s) + a(s + 1,s)). (3.3) using the definition of v (n) in (3.2) we have that v (n) = x2(n) + 2x(n) n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s)) +   n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))  2 −ρ n−1∑ s=n−g(n−1)−1 a2(n,s)h2(x(s)) (3.4) now let β = ρ−g0, then from (√g0√ β a− √ β √ g0 b )2 ≥ 0, 118 ernest yankson cubo 23, 1 (2021) we have 2ab ≤ g0 β a2 + β g0 b2. it follows from this inequality that 2x(n) n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s)) ≤ 2|x(n)| ∣∣∣∣∣∣ n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s)) ∣∣∣∣∣∣ ≤ g0 β x2(n) + β g0   n−1∑ s=n−g(n−1)−1 a(n,s)h(x(s))  2 ≤ g0 β x2(n) + β g0 g0 n−1∑ s=n−g(n−1)−1 a2(n,s)h2(x(s)). (3.5) substituting (3.5) into (3.4) we obtain v (n) ≤ x2(n) + g0 β x2(n) + (β + g0 −ρ) n−1∑ s=n−g(n−1)−1 a2(n,s)h2(x(s)) = β + g0 β x2(n) ≤ ρ ρ−g0 x2(n). using the last inequality and (3.3) we obtain |x(n)|2 ≥ ρ−g0 ρ v (n) = ρ−g0 ρ v (n0) ∞∏ s=n0 [a(n) + a(n + 1,n)]. this completes the proof. cubo 23, 1 (2021) inequalities and sufficient conditions for exponential stability . . . 119 references [1] i. berezansky, and e. braverman, “exponential stability of difference equations with several delays: recursive approach”, adv. difference edu., article id 104310, pp. 13, 2009. [2] el-morshedy, “new explicit global asymptotic stability criteria for higher order difference equations”, vol. 336, no. 1, pp. 262–276, 2007. [3] s. elaydi, an introduction to difference equations, springer verlage, new york, 3rd edition, 2005. [4] m. islam, and e. yankson, “boundedness and stability in nonlinear delay difference equations employing fixed point theory”, electron. j. qual. theory differ. equ., vol. 26, 2005. [5] w. kelley, and a. peterson, difference equations: an introduction with applications, second edition, academic press, new york, 2001. [6] c. kublik, and y. raffoul, “lyapunov functionals that lead to exponential stability and instability in finite delay volterra difference equations”, acta mathematica vietnamica, vol. 41, pp. 77–89, 2016. [7] y. raffoul, “stability and periodicity in discrete delay equations”, j. math. anal. appl., vol. 324, pp. 1356–1362, 2006. [8] y. raffoul, “inequalities that lead to exponential stability and instability in delay difference equations”, j. inequal. pure appl. math, vol. 10, no. 3, article 70, pp. 9, 2009. introduction exponential stability instability criteria cubo a mathematical journal vol.20, no¯ 3, (49–63). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300049 study of global asymptotic stability in nonlinear neutral dynamic equations on time scales abdelouaheb ardjouni 1,2 and ahcene djoudi 2 1department of mathematics and informatics, university of souk ahras, p.o. box 1553, souk ahras, 41000, algeria. 2applied mathematics lab, faculty of sciences, department of mathematics, university of annaba, p.o. box 12, annaba 23000, algeria. abd ardjouni@yahoo.fr, adjoudi@yahoo.com abstract this paper is mainly concerned the global asymptotic stability of the zero solution of a class of nonlinear neutral dynamic equations in c1rd. by converting the nonlinear neutral dynamic equation into an equivalent integral equation, our main results are obtained via the banach contraction mapping principle. the results obtained here extend the work of yazgan, tunc and atan [17]. resumen este art́ıculo está mayormente interesado en la estabilidad global asintótica de la solución cero de una clase de ecuaciones nolineales neutrales dinámicas en c1rd. transformando la ecuación nolineal neutral dinámica en una ecuación integral equivalente, nuestros resultados principales son obtenidos a través del principio de la aplicación contractiva de banach. los resultados obtenidos aqúı son una extensión del trabajo de yazgan, tunc y atan [17]. keywords and phrases: fixed points, neutral dynamic equations, asymptotic stability, time scales. http://dx.doi.org/10.4067/s0719-06462018000300049 50 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) 2010 ams mathematics subject classification: 34k20, 34k30, 34k40. cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 51 the concept of time scales analysis is a fairly new idea. in 1988, it was introduced by the german mathematician stefan hilger in his ph.d. thesis [13]. it combines the traditional areas of continuous and discrete analysis into one theory. after the publication of two textbooks in this area by bohner and peterson [7] and [8], more and more researchers were getting involved in this fast-growing field of mathematics. the study of dynamic equations brings together the traditional research areas of differential and difference equations. it allows one to handle these two research areas at the same time, hence shedding light on the reasons for their seeming discrepancies. in fact, many new results for the continuous and discrete cases have been obtained by studying the more general time scales case (see [1, 3, 4, 6, 14] and the references therein). there is no doubt that the lyapunov method have been used successfully to investigate stability properties of wide variety of ordinary, functional and partial equations. nevertheless, the application of this method to problem of stability in differential equations with delay has encountered serious difficulties if the delay is unbounded or if the equation has unbounded term. it has been noticed that some of theses difficulties vanish by using the fixed point technic. other advantages of fixed point theory over lyapunov’s method is that the conditions of the former are average while those of the latter are pointwise (see [2, 5, 9, 10, 11, 12, 15, 17] and references therein). in paper, we consider the following neutral nonlinear dynamic equations with variable delays given by x△ (t) = −a (t) xσ (t) + b (t) g (x (t)) + c (t) f ( x△̃ (t − τ1 (t)) ) + q (t, x (t) , x (t − τ2 (t))) , (0.1) with the initial condition x (t) = ϕ (t) , t ∈ [dt0, t0] ∩ t, where dt0 = inf t∈[t0,∞)∩t {t − τ1 (t) , t − τ2 (t)} , for each t0 ∈ [0, ∞) ∩ t and t is an unbounded above and below time scale and such that t0 ∈ t. our results are obtained with no need of further assumptions on the delta-differentiable of the neutral coefficient c and the twice delta-differentiable of τi with τ △ i (t) 6= 1 for t ∈ [0, ∞) ∩ t, so that for a given initial function ϕ ∈ φt0 a mapping p for (0.1) is constructed in such a way to map a, carefully chosen, closed convex nonempty subset d of a banach space x into itself on which p is a contraction mapping possessing a fixed point. this procedure will enable us to establish and prove by means of the contraction mapping theorem ([16], p. 2) the global asymptotic stability in c1rd for the zero solution of (0.1) with a less restrictive conditions. in the special case t = r, yazgan, tunc and atan in [17] show that the zero solution of (0.1) is globally asymptotically stable in c1rd by using the contraction mapping theorem. then, the results obtained here extend the work of yazgan, tunc and atan [17]. 52 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) 1 preliminaries in this section, we consider some advanced topics in the theory of dynamic equations on a time scales. again, we remind that for a review of this topic we direct the reader to the monographs of bohner and peterson [7] and [8]. a time scale t is a closed nonempty subset of r. for t ∈ t the forward jump operator σ, and the backward jump operator ρ, respectively, are defined as σ (t) = inf {s ∈ t : s > t} and ρ (t) = sup {t ∈ t : s < t}. these operators allow elements in the time scale to be classified as follows. we say t is right scattered if σ (t) > t and right dense if σ (t) = t. we say t is left scattered if ρ (t) < t and left dense if ρ (t) = t. the graininess function µ : t → [0, ∞), is defined by µ (t) = σ (t) − t and gives the distance between an element and its successor. we set inf ∅ = sup t and sup ∅ = inf t. if t has a left scattered maximum m, we define tk = t� {m}. otherwise, we define tk = t. if t has a right scattered minimum m, we define tk = t� {m}. otherwise, we define tk = t. let t ∈ tk and let f : t → r. the delta derivative of f (t), denoted f△ (t), is defined to be the number (when it exists), with the property that, for each ǫ > 0, there is a neighborhood u of t such that ∣∣f (σ (t)) − f (s) − f△ (t) [σ (t) − s] ∣∣ ≤ ǫ |σ (t) − s| , for all s ∈ u. if t = r then f△ (t) = f′ (t) is the usual derivative. if t = z then f△ (t) = △f (t) = f (t + 1) − f (t) is the forward difference of f at t. a function f is right dense continuous (rd-continuous), f ∈ crd = crd (t, r), if it is continuous at every right dense point t ∈ t and its left-hand limits exist at each left dense point t ∈ t. the function f : t → r is differentiable on tk provided f△ (t) exists for all t ∈ tk. f ∈ c1rd = c 1 rd (t, r) if f△ ∈ crd (t, r). we are now ready to state some properties of the delta-derivative of f. note fσ (t) = f (σ (t)). theorem 1.1 ([7, theorem 1.20]). assume f, g : t → r are differentiable at t ∈ tk and let α be a scalar. (i) (f + g) △ (t) = g△ (t) + f△ (t). (ii) (αf) △ (t) = αf△ (t). (iii) the product rules (fg) △ (t) = f△ (t) g (t) + fσ (t) g△ (t) , (fg) △ (t) = f (t) g△ (t) + f△ (t) gσ (t) . (iv) if g (t) gσ (t) 6= 0 then ( f g )△ (t) = f△ (t) g (t) − f (t) g△ (t) g (t) gσ (t) . cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 53 the next theorem is the chain rule on time scales ([7, theorem 1.93]). theorem 1.2 (chain rule). assume ν : t → r is strictly increasing and t̃ := ν (t) is a time scale. let ω : t̃ → r. if ν△ (t) and ω△̃ (ν (t)) exist for t ∈ tk, then (ω ◦ ν) △ = ( ω△̃ ◦ ν ) ν△. in the sequel we will need to differentiate and integrate functions of the form f (t − τ (t)) = f (ν (t)) where, ν (t) := t − τ (t). our next theorem is the substitution rule ([7, theorem 1.98]). theorem 1.3 (substitution). assume ν : t → r is strictly increasing and t̃ := ν (t) is a time scale. if f : t → r is rd-continuous function and ν is differentiable with rd-continuous derivative, then for a, b ∈ t, ∫b a f (t) ν△ (t) △t = ∫ν(b) ν(a) ( f ◦ ν−1 ) (s) △̃s. a function p : t → r is said to be regressive provided 1 + µ (t) p (t) 6= 0 for all t ∈ tk. the set of all regressive rd-continuous function f : t → r is denoted by r. the set of all positively regressive functions r+, is given by r+ = {f ∈ r : 1 + µ (t) f (t) > 0 for all t ∈ t}. let p ∈ r and µ (t) 6= 0 for all t ∈ t. the exponential function on t is defined by ep (t, s) = exp (∫t s 1 µ (z) log (1 + µ (z) p (z)) ∆z ) . it is well known that if p ∈ r+, then ep (t, s) > 0 for all t ∈ t. also, the exponential function y (t) = ep (t, s) is the solution to the initial value problem y △ = p (t) y, y (s) = 1. other properties of the exponential function are given by the following lemma. lemma 1.4 ([7, theorem 2.36]). let p, q ∈ r. then (i) e0 (t, s) = 1 and ep (t, t) = 1, (ii) ep (σ (t) , s) = (1 + µ (t) p (t)) ep (t, s), (iii) 1 ep(t,s) = e⊖p (t, s), where ⊖p (t) = − p(t) 1+µ(t)p(t) , (iv) ep (t, s) = 1 ep(s,t) = e⊖p (s, t), (v) ep (t, s) ep (s, r) = ep (t, r), (vi) e △ p (., s) = pep (., s) and ( 1 ep(.,s) )△ = − p(t) eσp(.,s) . lemma 1.5 ([1]). if p ∈ r+, then 0 < ep (t, s) ≤ exp (∫t s p (u) △u ) , ∀t ∈ t. 2 global asymptotic stability in this section, we shall study the global asymptotic stability in c1rd of the zero solution to (0.1). we introduce the following hypotheses. 54 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) (h1) a, b, c ∈ crd ([0, ∞) ∩ t, r), g, f ∈ c (r, r), q ∈ crd ([0, ∞) ∩ t × r×r, r), τi ∈ crd([0, ∞) ∩ t, (0, ∞) ∩ t) and (id − τi) ([0, ∞) ∩ t) is closed with t − τi (t) → ∞ as t → ∞, i = 1, 2. (h2) for t ∈ [0, ∞) ∩ t, g (0) = f (0) = q (t, 0, 0) = 0, and there exist lg, lf > 0, l1, l2 ∈ crd ([0, ∞) ∩ t, (0, ∞)) such that |g (x1) − g (x2)| ≤ lg |x1 − x2| , |f (x1) − f (x2)| ≤ lf |x1 − x2| , |q (t, x1, y1) − q (t, x2, y2)| ≤ l1 (t) |x1 − x2| + l2 (t) |y1 − y2| , for any xi, yi ∈ r, i = 1, 2. (h3) a ∈ r + is bounded on [0, ∞) ∩ t and lim t→∞ inf ∫t 0 1 µ(s) log (1 + µ (s) a (s)) ∆s > −∞. (h4) there exists α ∈ (0, 1) such that for t ∈ [0, ∞) ∩ t, ∫t 0 e⊖a (t, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ α, and |a (t)| ∫σ(t) 0 e⊖a (σ (t) , u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u + lg |b (t)| + lf |c (t)| + l1 (t) + l2 (t) ≤ α. for each t0 ∈ [0, ∞) ∩ t denote c 1 rd (t0) = c 1 rd ([dt0, t0] ∩ t, r) with the norm defined by |x| t0 = max t∈[dt0 ,t0]∩t { |x (t)| , ∣∣x△ (t) ∣∣} for x ∈ c1rd (t0). in addition, denote φt0 = {ϕ ∈ c 1 rd (t0) : ϕ △ (t0) = −a (t0) ϕ σ (t0) + b (t0) g (ϕ (t0)) + c (t0) f ( ϕ△̃ (t0 − τ1 (t0)) ) + q (t0, ϕ (t0) , ϕ (t0 − τ2 (t0)))}. for each t0 ∈ [0, ∞)∩t, we always assume that the initial function for (0.1) is of the type ϕ ∈ φt0. for convenience of stating our main result, we shall give the following definitions. definition 2.1. for each (t0, ϕ) ∈ [0, ∞) ∩ t×φt0, x is said to be a solution of (0.1) through (t0, ϕ) if x ∈ c 1 rd ([dt0, ∞) ∩ t) satisfies (0.1) on [t0, ∞)∩t and x (t) = ϕ (t) for t ∈ [dt0, t0]∩t. we denote such a solution by x (t) = x (t, t0, ϕ). definition 2.2. (i) the zero solution of (0.1) is said to be stable in c1rd if, for any t0 ∈ [0, ∞)∩t, ε > 0 there is a δ = δ (ε, t0) such that ϕ ∈ φt0 and |ϕ|t0 < δ implies max s∈[dt0 ,t]∩t { |x (s)| , ∣∣x△ (s) ∣∣} < ε, cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 55 for t ∈ [t0, ∞) ∩ t. (ii) the zero solution of (0.1) is said to be globally asymptotically stable in c1rd if it is stable in c1rd, and for any t0 ∈ [0, ∞) ∩ t, ϕ ∈ φt0 implies lim t→∞ x (t, t0, ϕ) = lim t→∞ x△ (t, t0, ϕ) = 0. in view of the definition of solution of (0.1), it is clear that the conditions imposed on the initial functions are very natural. from the above assumptions, it is easy to see that for each (t0, ϕ) ∈ [0, ∞) ∩ t×φt0, there exists a unique solution x (t) = x (t, t0, ϕ) of (0.1) defined on [dt0, ∞) ∩ t. by (h2), (0.1) has the zero solution. theorem 2.3. assume that (h1) − (h4) hold. then the zero solution of (0.1) is globally asymptotically stable in c1rd if and only if ∫t 0 1 µ (s) log (1 + µ (s) a (s)) ∆s → ∞ as t → ∞. (2.1) proof. (i) suppose that (2.1) holds. for any t0 ∈ [0, ∞) ∩ t, let x = { x ∈ c1rd ([dt0, ∞) ∩ t) : lim t→∞ x (t) = lim t→∞ x△ (t) = 0 } , with the norm defined by ‖x‖ t0 = max t∈[dt0 ,∞)∩t { |x (t)| , ∣∣x△ (t) ∣∣} , for x ∈ x. since x is a closed vectorial subspace of c1rd ([dt0, ∞) ∩ t) and c 1 rd ( [dt0, ∞) ∩ t, ‖.‖t0 ) is a banach space, then ( x, ‖.‖ t0 ) is also a banach space. for any ϕ ∈ φt0, let d = {x ∈ x : x (t) = ϕ (t) for t ∈ [dt0, t0] ∩ t} . it is easy to see that d is a nonempty, closed subset of x. multiply both sides of (0.1) by ea (t, t0) and then integrate from t0 to t to obtain ∫t t0 [x (u) ea (u, t0)] ∆ ∆u = ∫t t0 ea (u, t0) [ b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))] ∆u. as a consequence, we arrive at x (t) ea (t, t0) − x (t0) = ∫t t0 ea (u, t0) [ b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))] ∆u. 56 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) by dividing both sides of the above equation by ea (t, t0), we obtain x (t) = ϕ (t0) e⊖a (t, t0) + ∫t t0 e⊖a (t, u) [ b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))] ∆u. (2.2) use (2.2) to define the operator p : d → crd ([dt0, ∞) ∩ t) by (px) (t) = ϕ (t) for t ∈ [dt0, t0] ∩ t and (px) (t) = ϕ (t0) e⊖a (t, t0) + ∫t t0 e⊖a (t, u) [ b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))] ∆u, (2.3) for t ∈ [t0, ∞) ∩ t. firstly, we prove px ∈ d for any x ∈ d. from (2.3), for t > t0, (px) △ (t) = −ϕ (t0) a (t) e⊖a (σ (t) , t0) + e⊖a (σ (t) , t) [ b (t) g (x (t)) + c (t) f ( x△̃ (t − τ1 (t)) ) + q (t, x (t) , x (t − τ2 (t))) ] − a (t) ∫t t0 e⊖a (σ (t) , u) [ b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))] ∆u = −a (t) ϕ (t0) e⊖a (σ (t) , t0) − a (t) ∫σ(t) t0 e⊖a (σ (t) , u) [ b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))] ∆u + b (t) g (x (t)) + c (t) f ( x△̃ (t − τ1 (t)) ) + q (t, x (t) , x (t − τ2 (t))) = −a (t) (px) σ (t) + b (t) g (x (t)) + c (t) f ( x△̃ (t − τ1 (t)) ) + q (t, x (t) , x (t − τ2 (t))) . (2.4) by the definition of φt0, (2.4) yields on a time scale (px) △ + (t0) = −a (t0) ϕ σ (t0) + b (t0) g (ϕ (t0)) + c (t0) f ( ϕ△̃ (t0 − τ1 (t0)) ) + q (t0, ϕ (t0) , ϕ (t0 − τ2 (t0))) = ϕ △ − (t0) . hence, px ∈ c1rd ([dt0, ∞) ∩ t) for x ∈ d. for x ∈ d, limt→∞ x (t) = limt→∞ x △ (t) = 0. note that limt→∞ t − τi (t) = ∞, i = 1, 2. therefore, for any ε > 0, there exists t > 0 such that for t ≥ t, max { |x (t)| , |x (t − τ2 (t))| , ∣∣∣x△̃ (t − τ1 (t)) ∣∣∣ } < ε. (2.5) cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 57 it follows from (2.3), (2.5) and (h2) and (h4) that for t > t and x ∈ d, |(px) (t)| ≤ |ϕ (t0)| e⊖a (t, t0) + ∫t t0 e⊖a (t, u) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))| ∆u + ∫t t e⊖a (t, u) ∣∣∣b (u) (g (x (u)) − g (0)) + c (u) ( f ( x△̃ (u − τ1 (u)) ) − f (0) ) +q (u, x (u) , x (u − τ2 (u))) − q (u, 0, 0)| ∆u ≤ e⊖a (t, t0) [ |ϕ (t0)| + ∫t t0 ea (u, t0) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))| ∆u] + ∫t t e⊖a (t, u) [ lg |b (u)| |x (u)| + lf |c (u)| ∣∣∣x△̃ (u − τ1 (u)) ∣∣∣ +l1 (u) |x (u)| + l2 (u) |x (u − τ2 (u))|] ∆u ≤ e⊖a (t, t0) [ |ϕ (t0)| + ∫t t0 ea (u, t0) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))| ∆u] + ε ∫t t e⊖a (t, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ e⊖a (t, t0) [ |ϕ (t0)| + ∫t t0 ea (u, t0) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))| ∆u] + αε. from (2.1), there exists t1 > t such that for t > t1, e⊖a (t, t0) [ |ϕ (t0)| + ∫t t0 ea (u, t0) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) +q (u, x (u) , x (u − τ2 (u)))| ∆u] < ε. hence, limt→∞ (px) (t) = 0 for x ∈ d. in addition, it follows from (2.4) and (h2) that ∣∣∣(px)△ (t) ∣∣∣ ≤ ∣∣a (t) (px)σ (t) ∣∣ + |b (t) (g (x (t)) − g (0))| + ∣∣∣c (t) ( f ( x△̃ (t − τ1 (t)) ) − f (0) )∣∣∣ + |q (t, x (t) , x (t − τ2 (t))) − q (t, 0, 0)| ≤ ∣∣a (t) (px)σ (t) ∣∣ + lg |b (t)| |x (t)| + lf |c (t)| ∣∣∣x△̃ (t − τ1 (t)) ∣∣∣ + l1 (t) |x (t)| + l2 (t) |x (t − τ2 (t))| . 58 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) this, together with (h3) and (h4), yields limt→∞ (px) △ (t) = 0 for x ∈ d. therefore, px ∈ d for x ∈ d, i.e. p : d → d. secondly, we show that p : d → d is a contraction mapping. for any x, y ∈ d, it follows from (2.3), (h2) and (h4) that for t ∈ [t0, ∞) ∩ t, |(px) (t) − (py) (t)| ≤ ∫t t0 e⊖a (t, u) [ lg |b (u)| |x (u) − y (u)| + lf |c (u)| ∣∣∣x△̃ (u − τ1 (u)) − y△̃ (u − τ1 (u)) ∣∣∣ + |q (u, x (u) , x (u − τ2 (u))) − q (u, y (u) , y (u − τ2 (u)))|] ∆u ≤ ‖x − y‖ t0 ∫t t0 e⊖a (t, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ α ‖x − y‖ t0 . (2.6) in addition, it follows from (2.4), (2.6), (h2) and (h4) that for t ∈ [t0, ∞) ∩ t, ∣∣∣(px)△ (t) − (py)△ (t) ∣∣∣ ≤ |a (t)| ∣∣(px)σ (t) − (py)σ (t) ∣∣ + lg |b (t)| |x (t) − y (t)| + lf |c (t)| ∣∣∣x△̃ (t − τ1 (t)) − y△̃ (t − τ1 (t)) ∣∣∣ + |q (t, x (t) , x (t − τ2 (t))) − q (t, y (t) , y (t − τ2 (t)))| ≤ ‖x − y‖ t0 [ |a (t)| ∫σ(t) t0 e⊖a (σ (t) , u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u +lg |b (t)| + lf |c (t)| + l1 (t) + l2 (t)] ≤ α ‖x − y‖ t0 . (2.7) from (2.6) and (2.7), p : d → d is a contraction mapping. by the contraction mapping principle, p has a unique fixed point x in d, which is a unique solution of (0.1) through (t0, ϕ) and satisfies lim t→∞ x (t) = lim t→∞ x△ (t) = 0. (2.8) finally, we show that the zero solution of (0.1) is stable in c1rd. let k = sup t∈[t0,∞)∩t {e⊖a (t, t0)} and a = sup t∈[t0,∞)∩t {|a (t)|} . from (2.1) and (h3), k, a ∈ (0, ∞). for any ε > 0, let δ > 0 such that δ < ε min { 1, 1 − α k , 1 − α ka } . if x (t) = x (t, t0,ϕ) is a solution of (0.1) with |ϕ|t0 < δ, then x (t) = (px) (t) on [t0, ∞) ∩ t. we claim that ‖x‖ t0 < ε. otherwise, there exists t1 > t0 such that max { |x (t1)| , ∣∣x△ (t1) ∣∣} = ε, cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 59 and max { |x (t)| , ∣∣x△ (t) ∣∣} < ε, for t ∈ [dt0, t1) ∩ t. if |x (t1)| = ε, then it follows from (2.3), (h2) and (h4) that |x (t1)| ≤ |ϕ (t0)| e⊖a (t1, t0) + ∫t1 t0 e⊖a (t1, u) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) + q (u, x (u) , x (u − τ2 (u))) ∣∣∣ ∆u ≤ kδ + ε ∫t1 t0 e⊖a (t1, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ kδ + αε < ε. this is a contradiction. if ∣∣x△ (t1) ∣∣ = ε, then it follows from (2.4) and (h2) and (h4) that ∣∣x△ (t1) ∣∣ ≤ |ϕ (t0) a (t1)| e⊖a (σ (t1) , t0) + |b (t1)| |g (x (t1))| + |c (t1)| ∣∣∣f ( x△̃ (t1 − τ1 (t1)) )∣∣∣ + |q (t1, x (t1) , x (t1 − τ2 (t1)))| + |a (t1)| ∫σ(t1) t0 e⊖a (σ (t1) , u) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) + q (u, x (u) , x (u − τ2 (u))) ∣∣∣ ∆u ≤ kaδ + ε { |a (t1)| ∫σ(t1) t0 e⊖a (σ (t1) , u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u +lg |b (t1)| + lf |c (t1)| + l1 (t1) + l2 (t1)} ≤ kaδ + αε < ε. this is also a contradiction. hence, the zero solution of (0.1) is stable in c1rd. this, together with (2.8), implies that the zero solution of (0.1) is globally asymptotically stable in c1rd. (ii) assume that the zero solution of (0.1) is globally asymptotically stable in c1rd. now we prove that (2.1) holds. otherwise, set l = lim t 7→∞ inf ∫t 0 1 µ (s) log (1 + µ (s) a (s)) ∆s, k0 = sup t∈[0,∞)∩t {e⊖a (t, 0)} , a0 = sup t∈[0,∞)∩t {|a (t)|} , thus it follows from (h3) that l ∈ (−∞, ∞), k0 ∈ (0, ∞), a0 ∈ [0, ∞). hence, there exists an increasing sequence {tn} ⊂ [0, ∞) ∩ t such that limt→∞ tn = ∞ and lim n7→∞ ∫tn 0 1 µ (s) log (1 + µ (s) a (s)) ∆s = l. (2.9) denote in = ∫tn 0 ea (u, 0) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u, n = 1, 2, .... 60 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) from (h4), it follows that in = ea (tn, 0) ∫tn 0 e⊖a (tn, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ αea (tn, 0) . this, together with (2.9), implies that the sequence {in} is bounded. further, there exists a convergent subsequence. for brevity of notation, we may assume that {in} is convergent. therefore, there exists a positive integer m such that for any integer n > m, ∫tn tm ea (u, 0) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u < 1 − α 8b (e−l + 1) , (2.10) and e⊖a (tn, tm) > 1 2 , e⊖a (tn, 0) < e −l + 1, ea (tm, 0) < e l + 1, (2.11) where b = max { k0 ( el + 1 ) , k0a0 ( el + 1 ) , 1 } . for any δ > 0, consider the solution x (t) = x (t, tm, ϕ) of (0.1) with |ϕ|tm < δ and |ϕ (tm)| > δ/2. it follows from (2.3), (2.4), (2.11), (h2) and (h4) that for t ∈ [tm, ∞) ∩ t, |x (t)| ≤ |ϕ (tm)| e⊖a (t, tm) + ∫t tm e⊖a (t, u) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) + q (u, x (u) , x (u − τ2 (u))) ∣∣∣ ∆u ≤ |ϕ (tm)| e⊖a (t, 0) ea (tm, 0) + ‖x‖tm ∫t tm e⊖a (t, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ k0 ( el + 1 ) δ + ‖x‖ tm ∫t 0 e⊖a (t, u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u ≤ bδ + α ‖x‖tm , and ∣∣x△ (t) ∣∣ ≤ |ϕ (tm)| |a (t)| e⊖a (σ (t) , tm) + |b (t)| |g (x (t))| + |c (t)| ∣∣∣f ( x△̃ (t − τ1 (t)) )∣∣∣ + |q (t, x (t) , x (t − τ2 (t)))| + |a (t)| ∫σ(t) tm e⊖a (σ (t) , u) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) + q (u, x (u) , x (u − τ2 (u))) ∣∣∣ ∆u ≤ k0a0 ( el + 1 ) δ + ‖x‖ tm { |a (t)| ∫σ(t) tm e⊖a (σ (t) , u) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u +lg |b (t)| + lf |c (t)| + l1 (t) + l2 (t)} ≤ bδ + α ‖x‖ tm . cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 61 hence, ‖x‖ tm ≤ bδ + α ‖x‖ tm , i.e. ‖x‖ tm ≤ b 1 − α δ. (2.12) it follows from (2.3),(2.10)-(2.12) and (h2) that for any n > m, |x (tn)| ≥ |ϕ (tm)| e⊖a (tn, tm) − e⊖a (tn, 0) ∫tn tm ea (u, 0) ∣∣∣b (u) g (x (u)) + c (u) f ( x△̃ (u − τ1 (u)) ) + q (u, x (u) , x (u − τ2 (u))) ∣∣∣ ∆u ≥ |ϕ (tm)| e⊖a (tn, tm) − ‖x‖tm e⊖a (tn, 0) ∫tn tm ea (u, 0) [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] ∆u > 1 4 δ − b 1 − α δ ( e−l + 1 ) 1 − α 8b (e−l + 1) = 1 8 δ. this contradicts the fact that limn→∞ tn = ∞ and the zero solution of (0.1) is globally asymptotically stable in c1rd. the proof is complete. example 2.4. let t = r. consider the following neutral differential equation x′ (t) = −a (t) x (t) + b (t) g (x (t)) + c (t) f (x′ (t − τ1 (t))) + q (t, x (t) , x (t − τ2 (t))) , (2.13) where τ1 (t) = t/2 + 1, τ2 (t) = t/3 + 2, a (t) = 1 t + 1 , b (t) = 1 15 (t + 1) , c (t) = 1 12 (t + 1) , g (x) = 1 − cos (x) , f (x) = sin (x) , q (t, x, y) = 1 16 (t + 1) sin (x + y) . obviously a, b, c ∈ c ([0, ∞) , r), g, f ∈ c (r, r), q ∈ c ([0, ∞) × r × r, r), τ1, τ2 ∈ c ([0, ∞) , (0, ∞)) with t − τi (t) → ∞ as t → ∞, i = 1, 2. a simple calculation shows that g (0) = f (0) = q (t, 0, 0) = 0, ∫∞ 0 a (s) ds = ∞, l1 (t) = l2 (t) = 1 16 (t + 1) , lg = 1, lf = 1, ∫t 0 e− ∫ t u a(s)ds [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] du ≤ 11 40 , and a (t) ∫t 0 e− ∫ t u a(s)ds [lg |b (u)| + lf |c (u)| + l1 (u) + l2 (u)] du +lg |b (t)| + lf |c (t)| + l1 (t) + l2 (t) ≤ 22 40 . it is easy to see that all the conditions of theorem 2.3 hold for α = 22 40 < 1. thus, theorem 2.3 implies that the zero solution of (2.13) is globally asymptotically stable in c1. 62 abdelouaheb ardjouni and ahcene djoudi cubo 20, 3 (2018) acknowledgments. the authors would like to thank the anonymous referee for his/her valuable remarks. cubo 20, 3 (2018) study of global asymptotic stability in nonlinear neutral dynamic . . . 63 references [1] m. adıvar, y. n. raffoul, existence of periodic solutions in totally nonlinear delay dynamic equations. electronic journal of qualitative theory of differential equations 2009, 1 (2009), 1–20. 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[17] r. yazgan, c. tunc and o. atan, on the global asymptotic stability of solutions to neutral equations of first order, palestine journal of mathematics, vol. 6(2) (2017), 542–550. preliminaries global asymptotic stability cubo, a mathematical journal vol. 24, no. 01, pp. 63–82, april 2022 doi: 10.4067/s0719-06462022000100063 the topological degree methods for the fractional p(·)-laplacian problems with discontinuous nonlinearities hasnae el hammar 1 chakir allalou 1 adil abbassi 1 abderrazak kassidi 1 1laboratory lmacs, fst of beni-mellal, sultan moulay slimane university, morocco. hasnaeelhammar11@gmail.com chakir.allalou@yahoo.fr. abbassi91@yahoo.fr abderrazakassidi@gmail.com abstract in this article, we use the topological degree based on the abstract hammerstein equation to investigate the existence of weak solutions for a class of elliptic dirichlet boundary value problems involving the fractional p(x)-laplacian operator with discontinuous nonlinearities. the appropriate functional framework for this problems is the fractional sobolev space with variable exponent. resumen en este art́ıculo, usamos el grado topológico basado en la ecuación abstracta de hammerstein para investigar la existencia de soluciones débiles para una clase de problemas eĺıpticos de valor en la frontera de dirichlet que involucran el operador p(x)-laplaciano fraccional con no linealidades discontinuas. el marco funcional apropiado para estos problemas es el espacio de sobolev fraccional con exponente variable. keywords and phrases: fractional p(x)-laplacian, weak solution, discontinuous nonlinearity, topological degree theory. 2020 ams mathematics subject classification: 35r11, 35j60, 47h11, 35a16. accepted: 05 november, 2021 received: 12 may, 2021 c©2022 h. el hammar et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100063 https://orcid.org/0000-0003-3532-4673 https://orcid.org/0000-0002-4885-9397 https://orcid.org/0000-0003-0284-8390 https://orcid.org/0000-0002-9105-1123 mailto:hasnaeelhammar11@gmail.com mailto:chakir.allalou@yahoo.fr mailto:abbassi91@yahoo.fr mailto:abderrazakassidi@gmail.com 64 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) 1 introduction and main result the study of fractional sobolev spaces and the corresponding nonlocal equations has received a tremendous popularity in the last two decades considering their intriguing structure and great application in many fields, such as social sciences, fractional quantum mechanics, materials science, continuum mechanics, phase transition phenomena, image process, game theory, and levy process, see [34, 35] and references therein for more details. on the other hand, in recent years, a great deal of attention has been paid to the study of differential equations and variational problems involving p(x)-growth conditions since they can be used to model a variety of physical phenomena that occur in the fields of elastic mechanics, electro-rheological fluids (”smart fluids”), and image processing, etc. the readers are guided to [19, 20, 27] and its references. it is only normal to wonder what results can be obtained when the fractional p(·)-laplacian is used instead of the p(·)-laplacian. the fractional p(·)-laplacian has also recently been investigated in elliptic problems; see [8, 10, 25, 26]. u. kaufmann et al. [26] presented a new class of fractional sobolev spaces with variable exponents in a recent paper. the authors in [8, 9] showed some additional basic properties on this function space as well as the associated nonlocal operator. they used the critical point theory in [4] to prove the existence of solutions for fractional p(·)laplacian equations. k. ho and y.-h. kim [25] managed to obtain fundamental imbeddings for a new fractional sobolev space with variable exponents, which is a generalization of previously defined fractional sobolev spaces. let ω ⊂ rn (n ≥ 1) be a bounded open set with lipschitz boundary and let p : ω×ω → (1, +∞) be a continuous bounded function. the purpose of this paper is to establish the existence of nontrivial weak solutions for the following fractional p(x)-laplacian problems with discontinuous nonlinearities.    (−△p(x)) su(x) + |u(x)|q(x)−2u(x) + λh(x,u) ∈ − [ ψ(x,u),ψ(x,u) ] in ω, u = 0 on rn\ω, (1.1) where ps < n with 0 < s < 1 and (−△p(x)) s is the fractional p(x)-laplacian operator defined by (−∆)sp(x)u(x) = p.v. ∫ rn \bε(x) |u(x) − u(y)|p(x,y)−2(u(x) − u(y)) |x − y|n+sp(x,y) dy, x ∈ rn (1.2) ∀x ∈ ω, where p.v. is a commonly used abbreviation in the principal value sense and let p ∈ c(rn × rn) satisfying 1 < p− = min (x,y)∈ω×ω p(x, y) ≤ p(x, y) ≤ p+ = max (x,y)∈ω×ω p(x, y) < +∞, (1.3) p is symmetric i. e. p(x, y) = p(y, x), ∀(x, y) ∈ ω × ω; (1.4) cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 65 and bε(x) := { y ∈ rn : |x − y| < ε } . let as denote by: p̃(x) = p(x, x), ∀x ∈ ω. furthermore, the carathéodory’s functions h satisfy only the growth condition, for all s ∈ r and a. e. x ∈ ω. (h0) |h(x,s)| ≤ ̺(e(x) + |s| q(x)−1), where ̺ is a positive constant, e(x) is a positive function in lp ′(x)(ω). in the simplest case p = 2, we have the well-known fractional laplacian, a large amount of papers were written on this direction see [6, 15]. moreover, if s = 1, we get the classic laplacian. some related results can be found in [21, 39, 40, 41, 42]. notice that when s = 1, the problems like (1.1) have been studied in many papers, we refer the reader to [1, 5, 24], in which the authors have used various methods to get the existence of solutions for (1.1). in the case when p = p(x) is a continuous function, problem (1.1) has also been studied by many authors. for more information, see [11, 23]. in order to prove the existence of nontrivial weak solutions, the main difficulties are reflected in the following aspect, we cannot directly use the topological degree methods in a natural way because the nonlinear term ψ is discontinuous. in order to overcome the discontinuous difficulty, we will transform this dirichlet boundary value problem involving the fractional p-laplacian operator with discontinuous nonlinearities into a new one governed by a hammerstein equation. then, we shall employ the topological degree theory developed by kim in [29, 28] for a class of weakly upper semi-continuous locally bounded set-valued operators of (s+) type in the framework of real reflexive separable banach spaces, based on the berkovits-tienari degree [12]. the topological degree theory was constructed for the first time by leray-schauder [31] in their study of the nonlinear equations for compact perturbations of the identity in infinite-dimensional banach spaces. furthermore, browder [14] has developed a topological degree for operators of class (s+) in reflexive banach spaces, see also [37, 38]. among many examples, we refer the reader to the classical works [2, 3, 18, 45] for more details. to this end, we always assume that ψ : ω × r → r is a possibly discontinuous function, we “fill the discontinuity gaps” of ψ, replacing ψ by an interval [ ψ(x,u),ψ(x,u) ] , where ψ(x,s) = lim inf η→s ψ(x,η) = lim δ→0+ inf |η−s|<δ ψ(x,η), ψ(x,s) = lim sup η→s ψ(x,η) = lim δ→0+ sup |η−s|<δ ψ(x,η). such that 66 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) (h1) ψ and ψ are super-positionally measurable (i. e., ψ(·,u(·)) and ψ(·,u(·)) are measurable on ω for every measurable function u : ω → r). (h2) ψ satisfies the growth condition: |ψ(x,s)| ≤ b(x) + c(x)|s|γ(x)−1, for almost all x ∈ ω and all s ∈ r, where b ∈ lγ ′(x)(ω), c ∈ l∞(ω), where 1 < γ(x) < p(x) for all x ∈ ω. first of all, we define the operator n acting from w s,p(x,y) 0 (ω) into 2 ( w s,p(x,y) 0 (ω) ) ∗ by nu = { ϕ ∈ ( w s,p(x,y) 0 (ω) )∗ \ ∃ h ∈ lp ′(x)(ω); ψ(x,u(x)) ≤ h(x) ≤ ψ(x,u(x)) a. e. x ∈ ω and 〈ϕ,v〉 = ∫ ω hvdx ∀v ∈ w s,p(x,y) 0 (ω) } . in this spirit, we consider f : w s,p(x,y) 0 (ω) −→ ( w s,p(x,y) 0 (ω) )∗ such that 〈fu,v〉 = ∫ ω×ω |u(x) − u(y)|p(x,y)−2(u(x) − u(y))(v(x) − v(y)) |x − y|n+sp(x,y) dxdy, (1.5) for all v ∈ w s,p(x,y) 0 (ω) and the operator a : w0 → w ∗ 0 setting by 〈au,v〉 = ∫ ω |u(x)|q(x)−2(u(x)v(x) + λh(x,u))v(x)dx, ∀u,v ∈ w0, where the spaces w s,p(x,y) 0 (ω) := w0 will be introduced in section 2. next, we give the definition of weak solutions for problem (1.1). definition 1.1. a function u ∈ w s,p(x,y) 0 (ω) is called a weak solution to problem (1.1), if there exists an element ϕ ∈ nu verifying 〈fu,v〉 + 〈au,v〉 + 〈ϕ,v〉 = 0, for all v ∈ w s,p(x,y) 0 (ω). now we are in a position to present our main result. theorem 1.2. assume that ψ satisfies (h1),(h2) and h satisfies (h0). then, the problem (1.1) has a weak solution u in w s,p(x,y) 0 (ω). 2 preliminaries 2.1 lebesgue and fractional sobolev spaces with variable exponent in this subsection, we first recall some useful properties of the variable exponent lebesgue spaces lp(x)(ω) . for more details we refer the reader to [22, 30, 44]. cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 67 denote c+(ω) = {h ∈ c(ω)| inf x∈ω h(x) > 1}. for any h ∈ c+(ω) , we define h+ := max{h(x), x ∈ ω}, h− := min{h(x), x ∈ ω}. for any p ∈ c+(ω) we define the variable exponent lebesgue spaces l p(x)(ω) = { u; u : ω → r is measurable and ∫ ω |u(x)|p(x)dx < +∞ } . endowed with luxemburg norm ‖u‖p(x) = inf { λ > 0 |ρp(·) ( u λ ) ≤ 1 } where ρp(·) (u) = ∫ ω |u(x)|p(x)dx, ∀u ∈ lp(x) (lp(x)(ω), ||.||p(x)) is a banach space, separable and reflexive. its conjugate space is l p ′(x)(ω) where 1 p(x) + 1 p′(x) = 1 for all x ∈ ω. we have also the following result proposition 2.1. ([22]) for any u ∈ lp(x)(ω) we have (i) ‖u‖p(x) < 1(= 1;> 1) ⇔ ρp(·)(u) < 1(= 1;> 1), (ii) ‖u‖p(x) ≥ 1 ⇒ ‖u‖ p − p(x) ≤ ρp(·)(u) ≤ ‖u‖ p + p(x) , (iii) ‖u‖p(x) ≤ 1 ⇒ ‖u‖ p + p(x) ≤ ρp(·)(u) ≤ ‖u‖ p − p(x) . from this proposition, we can deduce the inequalities ‖u‖p(x) ≤ ρp(·)(u) + 1, (2.1) ρp(·)(u) ≤ ‖u‖ p − p(x) + ‖u‖ p + p(x) . (2.2) if p,q ∈ c+(ω) such that p(x) ≤ q(x) for any x ∈ ω, then there exists the continuous embedding lq(x)(ω) → lp(x)(ω) . next, we present the definition and some results on fractional sobolev spaces with variable exponent that was introduced in [8, 26]. let s be a fixed real number such that 0 < s < 1, and let q : ω → (0,∞) and p : ω×ω → (0,∞) be two continuous functions. furthermore, we suppose that the assumptions (1.3) and (1.4) be satisfied, we define the fractional sobolev space with variable exponent via the gagliardo approach as follows: w = ws,,q(x),p(x,y)(ω) = { u ∈ lq(x)(ω) : ∫ ω×ω |u(x) − u(y)|p(x,y) λp(x,y)|x − y|n+sp(x,y) dxdy < +∞, ∫ for some λ > 0 } . 68 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) we equip the space w with the norm ‖u‖w = ‖u‖q(x) + [u]s,p(x,y), where [·]s,p(x,y) is a gagliardo seminorm with variable exponent, which is defined by [u]s,p(x,y) = inf { λ > 0 : ∫ ω×ω |u(x) − u(y)|p(x,y) λp(x,y)|x − y|n+sp(x,y) dxdy ≤ 1 } . the space (w, ‖ · ‖w ) is a banach space (see [17]), separable and reflexive (see [8, lemma 3.1]). we also define w0 as the subspace of w which is the closure of c ∞ 0 (ω) with respect to the norm || · ||w . from [7, theorem 2.1 and remark 2.1] ‖ · ‖w0 := [·]s,p(x,y) is a norm on w0 which is equivalent to the norm ‖ · ‖w , and we have the compact embedding w0 →֒→֒ l q(x). so the space (w0, ‖ · ‖w0) is a banach space separable and reflexive. we defne the modular ρp(·,·) : w0 → r by ρp(·,·)(u) = ∫ ω×ω |u(x) − u(y)|p(x,y) |x − y|n+sp(x,y) dxdy. the modular ρp checks the following results, which is similar to proposition 2.1 (see [43, lemma 2.1]) proposition 2.2. ([30]) for any u ∈ w0 we have (i) ‖u‖w0 ≥ 1 ⇒ ‖u‖ p − w0 ≤ ρp(·,·)(u) ≤ ‖u‖ p + w0 , (ii) ‖u‖w0 ≤ 1 ⇒ ‖u‖ p + w0 ≤ ρp(·,·)(u) ≤ ‖u‖ p − w0 . 2.2 some classes of operators and an outline of berkovits degree now, we introduce the theory of topological degree which is the major tool for our results. we start by defining some classes of mappings. let x be a real separable reflexive banach space with dual x∗ and with continuous dual pairing 〈 · , · 〉 between x∗ and x in this order. the symbol ⇀ stands for weak convergence. let y be another real banach space. definition 2.3. (1) we say that the set-valued operator f : ω ⊂ x → 2y is bounded, if f maps bounded sets into bounded sets; (2) we say that the set-valued operator f : ω ⊂ x → 2y is locally bounded at the point u ∈ ω, if there is a neighborhood v of u such that the set f(v) = ⋃ u∈v fu is bounded. cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 69 definition 2.4. the set-valued operator f : ω ⊂ x → 2y is called (1) upper semicontinuous (u.s.c.) at the point u, if, for any open neighborhood v of the set fu, there is a neighbhorhood u of the point u such that f(u) ⊆ v . we say that f is upper semicontinuous (u.s.c) if it is u.s.c at every u ∈ x; (2) weakly upper semicontinuous (w.u.s.c.), if f−1(u) is closed in x for all weakly closed set u in y. definition 2.5. let ω be a nonempty subset of x, (un)n≥1 ⊆ ω and f : ω ⊂ x → 2 x ∗ \ ∅. then, the set-valued operator f is (1) of type (s+), if un ⇀ u in x and for each sequence (hn) in x ∗ with hn ∈ fun such that lim sup n→∞ 〈hn,un − u〉 ≤ 0, we get un → u in x; (2) quasi-monotone, if un ⇀ u in x and for each sequence (wn) in x ∗ such that wn ∈ fun yield lim inf n→∞ 〈wn,un − u〉 ≥ 0. definition 2.6. let ω be a nonempty subset of x such that ω ⊂ ω1, (un)n≥1 ⊆ ω and t : ω1 ⊂ x → x ∗ be a bounded operator. then, the set-valued operator f : ω ⊂ x → 2x \ ∅ is of type (s+)t , if    un ⇀ u in x, tun ⇀ y in x ∗, and for any sequence (hn) in x with hn ∈ fun such that lim sup n→∞ 〈hn,tun − y〉 ≤ 0, we have un → u in x. next, we consider the following sets : f1(ω) := {f : ω → x ∗|f is bounded, demicontinuous and of type (s+)}, ft (ω) := {f : ω → 2 x|f is locally bounded, w.u.s.c. and of type (s+)t }, for any ω ⊂ df and each bounded operator t : ω → x ∗, where df denotes the domain of f . remark 2.7. we say that the operator t is an essential inner map of f, if t ∈ f1(g). lemma 2.8. ([29, lemma 1.4]) let x be a real reflexive banach space and g ⊂ x is a bounded open set. assume that t ∈ f1(g) is continuous and s : ds ⊂ x ∗ → 2x weakly upper semicontinuous and locally bounded with t(g) ⊂ ds. then the following alternative holds: 70 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) (1) if s is quasi-monotone, yield i + s ◦ t ∈ ft (g), where i denotes the identity operator. (2) if s is of type (s+), yield s ◦ t ∈ ft (g). definition 2.9. ([29]) let t : g ⊂ x → x∗ be a bounded operator, a homotopy h : [0,1]×g → 2x is called of type (s+)t , if for every sequence (tk,uk) in [0,1]×g and each sequence (ak) in x with ak ∈ h(tk,uk) such that uk ⇀ u ∈ x, tk → t ∈ [0,1], tuk ⇀ y in x ∗ and lim sup k→∞ 〈ak,tuk − y〉 ≤ 0, we get uk → u in x. lemma 2.10. ([29]) let x be a real reflexive banach space and g ⊂ x is a bounded open set, t : g → x∗ is continuous and bounded. if f, s are bounded and of class (s+)t , then an affine homotopy h : [0,1] × g → 2x given by h(t,u) := (1 − t)fu + tsu, for (t,u) ∈ [0,1] × g, is of type (s+)t . now, we introduce the topological degree for a class of locally bounded, w.u.s.c. and satisfies condition (s+)t for more details see [29]. theorem 2.11. let l = { (f,g,g)|g ∈ o, t ∈ f1(g), f ∈ ft (g), g 6∈ f(∂g) } , where o denotes the collection of all bounded open sets in x. there exists a unique (hammerstein type) degree function d : l −→ z such that the following alternative holds: (1) ( normalization) for each g ∈ g, we have d(i,g,g) = 1. (2) ( domain additivity) let f ∈ ft (g). we have d(f,g,g) = d(f,g1,g) + d(f,g2,g), with g1, g2 ⊆ g disjoint open such that g 6∈ f(g\(g1 ∪ g2)). (3) ( homotopy invariance) if h : [0,1] × g → x is a bounded admissible affine homotopy with a common continuous essential inner map and g: [0,1] → x is a continuous path in x such that g(t) 6∈ h(t,∂g) for all t ∈ [0,1], then the value of d(h(t, ·),g,g(t)) is constant for any t ∈ [0,1]. (4) ( solution property) if d(f,g,g) 6= 0, then the equation g ∈ fu has a solution in g. cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 71 3 proof of theorem 1.2 in the present section, following compactness methods (see [18, 32]), we prove the existence of weak solutions for the problem (1.1) in fractional sobolev spaces. in doing so, we transform this elliptic dirichlet boundary value problem involving the fractional p-laplacian operator with discontinuous nonlinearities into a new problem governed by a hammerstein equation. more precisely, by means of the topological degree theory introduced in section 2, we establish the existence of weak solutions to the state problem, which holds under appropriate assumptions. first, we give several lemmas. lemma 3.1. let 0 < s < 1 and 1 < p(x,y) < +∞, (or sp+ < n) the operator f defined in (1.5) is (i) bounded and strictly monotone operator. (ii) of type (s+). proof. (i) it is clear that f is a bounded operator. for all ξ,η ∈ rn, we have the simon inequality (see [36]) from which we can obtain the strictly monotonicity of f :    |ξ − η|p ≤ cp ( |ξ|p−2ξ − |η|p−2η ) (ξ − η); p ≥ 2 |ξ − η|p ≤ cp [( |ξ|p−2ξ − |η|p−2η ) (ξ − η) ] p 2 (|ξ|p + |η|p) 2−p 2 ; 1 < p < 2, where cp = (1 2 )−p and cp = 1 p − 1 . (ii) let (un) ∈ w s,p(x,y) 0 (ω) be a sequence such that un ⇀ u and lim sup n→∞ 〈fun − fu,un −u〉 ≤ 0. in view of (i), we get lim n→∞ 〈fun − fu,un − u〉 = 0. thanks to proposition 2.1, we obtain un(x) → u(x), a.e. x ∈ ω. (3.1) in the sequel, we denote by l(x,y) = |x − y|−n−sp(x,y). by fatou’s lemma and (3.1), we get lim inf n→+∞ ∫ ω×ω |un(x) − un(y)| p(x,y)l(x,y)dxdy ≥ ∫ ω×ω |u(x) − u(y)|p(x,y)l(x,y)dxdy. (3.2) on the other hand, from un ⇀ u we have lim n→+∞ 〈fun, un − u〉 = lim n→+∞ 〈fun − fu,un − u〉 = 0. (3.3) 72 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) now, by using young’s inequality, there exists a positive constant c such that 〈fun,un − u〉 = ∫ ω×ω |un(x) − un(y)| p(x,y)l(x,y)dxdy − ∫ ω×ω |un(x) − un(y)| p(x,y)−2(un(x) − un(y))(u(x) − u(y))l(x,y)dxdy ≥ ∫ ω×ω |un(x) − un(y)| p(x,y) l(x,y)dxdy (3.4) − ∫ ω×ω |un(x) − un(y)| p(x,y)−1|u(x) − u(y)|l(x,y)dxdy ≥ c ∫ ω×ω |un(x) − un(y)| p(x,y)l(x,y)dxdy − c ∫ ω×ω |u(x) − u(y)|p(x,y)l(x,y)dxdy, combining (3.2), (3.3) and (3.4), we obtain lim n→+∞ ∫ ω×ω |un(x) − un(y)| p(x,y)l(x,y)dxdy = ∫ ω×ω |u(x) − u(y)|p(x,y)l(x,y)dxdy. (3.5) according to (3.1), (3.5) and the brezis-lieb lemma [13], our result is proved. proposition 3.2. ([16, proposition 1]) for any fixed x ∈ ω, the functions ψ(x,s) and ψ(x,s) are upper semicontinuous (u.s.c.) functions on rn. lemma 3.3. let ω ⊂ rn (n ≥ 1) be a bounded open set with smooth boundary. the operator a : w s,p(x,y) 0 (ω) → ( w s,p(x,y) 0 (ω) )∗ defined by 〈au,v〉 = ∫ ω (|u(x)|q(x)−2u(x) + λh(x,u))vdx, ∀u,v ∈ w0 is compact. proof. the proof is broken down into three sections. step 1. let φ : w0 → l q ′ (x)(ω) be the operator defined by φu(x) := −|u(x)|q(x)−2u(x) for u ∈ w0 and x ∈ ω. it is obvious that φ is continuous. next we show that φ is bounded. for every u ∈ w0, we have by the inequalities (2.1) and (2.2) that ‖φu‖q′(x) ≤ ρq′(·)(φu) + 1 = ∫ ω ∣∣∣|u|q(x)−1 ∣∣∣ q ′(x) dx + 1 = ρq(·)(u) ≤ ‖u‖ q − q(x) + ‖u‖ q + q(x) + 1. by the compact embedding w0 →֒→֒ l q(x)(ω) we have ‖φu‖q′(x) ≤ const ( ‖u‖ q − w0 + ‖u‖ q + w0 ) + 1. this implies that φ is bounded on w0. cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 73 step 2. we show that the operator ψ defined from w0 into l p ′(x)(ω) by ψu(x) := −λh(x,u) for u ∈ w0 and x ∈ ω is bounded and continuous. let u ∈ w0, by using the growth condition (h0) we obtain ‖ψu‖ p ′(x) p′(x) ≤ ∫ ω |λh(x,u)|p ′(x)dx ≤ (̺λ)p ′(x) ∫ ω ( |e(x)|p ′(x) + |u|(q(x)−1)p ′(x) ) dx ≤ (̺λ)p ′(x) ∫ ω ( |e(x)|p ′(x) + |u|(p(x)−1)p ′(x) ) dx (3.6) ≤ (̺λ)p ′(x) ∫ ω |e(x)|p ′(x) dx + (̺λ)p ′(x) ∫ ω |u|p(x) dx ≤ (̺λ)p ′(x)(‖e‖ p ′+ p′(x) + ‖e‖ p ′− p′(x) ) + (̺λ)p ′(x)(‖u‖ p+ p(x) + ‖u‖ p− p(x) ) ≤ cm(‖u‖ p+ w0 + ‖u‖ p− w0 + 1), where cm = max ( (̺λ)p ′(x)(‖e‖ p ′+ p′(x) +‖e‖ p ′− p′(x) ),(̺λ)p ′(x) ) . (due to e(x) is a positive function in lp ′(x)(ω)). therefore ψ is bounded on ws,q(x),p(x,y)(ω). next, we show that ψ is continuous, let un → u in w s,q(x),p(x,y)(ω), then un → u in l p(x)(ω). thus there exists a subsequence still denoted by (un) and measurable function ϕ in l p(x)(ω) such that un(x) → u(x), |un(x)| ≤ ϕ(x), for a.e. x ∈ ω and all n ∈ n. since h satisfies the carathéodory condition, we obtain h(x,un(x)) → h(x,u(x)) a.e. x ∈ ω. (3.7) thanks to (h0) we obtain |h(x,un(x))| ≤ ̺ ( e(x) + |ϕ(x)|q(x)−1 ) for a.e. x ∈ ω and for all k ∈ n. since e(x) + |ϕ(x)|p(x)−1 ∈ lp ′(x)(ω), and from (3.7), we get ∫ ω |h(x,uk(x)) − h(x,u(x))| p ′(x)dx −→ 0, by using the dominated convergence theorem we have ψuk → ψu in l p ′(x)(ω). thus the entire sequence (ψun) converges to ψu in l p ′(x)(ω) and then ψ is continuous. 74 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) step 3. since the embedding i : w0 → l q(x)(ω) is compact, it is known that the adjoint operator i∗ : lq ′(x)(ω) → w∗0 is also compact. therefore, the compositions i ∗◦φ and i∗◦ψ : w0 → w ∗ 0 are compact. we conclude that s = i∗ ◦ φ + i∗ ◦ ψ is compact. lemma 3.4. let ω ⊂ rn (n ≥ 1) be a bounded open set with smooth boundary. if the hypotheses (h1) and (h2) hold, then the set-valued operator n defined above is bounded, upper semicontinuous (u.s.c.) and compact. proof. let λ : lp(x)(ω) → 2l p′(x)(ω) be a set-valued operator defined as follows λu = { h ∈ lp ′(x)(ω)| ψ(x,u(x)) ≤ h(x) ≤ ψ(x,u(x)) a. e. x ∈ ω } . let u ∈ w0, by the assumption (h2) we obtain max { |ψ(x,s)| ; |ψ(x,s)| } ≤ b(x) + c(x)|s|γ(x)−1. for all (x,t) ∈ ω × r where 1 < γ(x) < p(x) for all x ∈ r . as a result ∫ ω |ψ(x,u(x))|p ′(x) dx ≤ 2p ′++1 ( ∫ ω |b(x)|p ′(x) dx + ∫ ω |c|p ′(x)|u(x)|p(x)dx ) . a same inequality is shown for ψ(x,s), it follows that the set-valued operator λ is bounded on w0(ω). it remains to prove that λ is upper semi-continuous (u.s.c.), i. e., ∀ε > 0, ∃δ > 0, ‖u − u0‖p < δ ⇒ λu ⊂ λu0 + bε, where bε is the ε-ball in l p ′(x)(ω). to come to an end, given u0 ∈ l p(x)(ω), let us consider the sets gm,ε = ⋂ t∈rn kt, where kt = { x ∈ ω, if |t − u0(x)| < 1 m , then [ψ(x,t),ψ(x,t)] ⊂ ] ψ(x,u0(x)) − ε r ,ψ(x,u0(x)) + ε r [} , m being an integer, |t| = max 1≤i≤n |ti| and r is a constant to be determined in the following pages. in view of proposition 3.2, we define the sets of points as follows gm,ε = ⋂ r∈rn a kr, where rna denotes the set of all rational grids in r n. for any r = (r1, . . . ,rn ) ∈ r n a , kr = { x ∈ ω | u0(x) ∈ c n∏ i=1 ] ri − 1 m ,ri + 1 m [} ∪ { x ∈ ω | u0(x) ∈ n∏ i=1 ] ri − 1 m ,ri + 1 m [} ∩ { x ∈ ω | ψ(x,r) < ψ(x,u0(x)) + ε r and ψ(x,r) > ψ(x,u0(x)) − ε r } , cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 75 so that kr and therefore gm,ε are measurable. it is obvious that g1,ε ⊂ g2,ε ⊂ · · · in light of proposition 3.2, we have ∞⋃ m=1 gm,ε = ω, therefore there exists m0 ∈ n such that m(gm0,ε) > m(ω) − ε r . (3.8) but for each ε > 0, there is η = η(ε) > 0, such that m(t) < η yields 2p ′+−1 ∫ t 2|b(x)|p ′(x) + cp ′(x)(x)(2p ′+−1 + 1)|u0(x)| p(x)dx < ( ε 3 )p′+ , (3.9) because of b ∈ lp ′(x)(ω) and u0 ∈ l p(x)(ω). let now 0 < δ < min { 1 m0 ( η 2 ) 1 p− , 1 2p +−2 ( ε 6c ) p′+ θ } , (3.10) r > max { 2ε η ,3 ( m(ω) ) 1 p′− } , (3.11) where θ =    p + if ‖u − u0‖p(x) ≤ 1 p− if ‖u − u0‖p(x) ≥ 1. suppose that ‖u − u0‖p(x) < δ and define the set g = { x ∈ ω \ |u(x) − u0(x)| ≥ 1 m0 } , we get m(g) < (m0δ) p(x) < η 2 . (3.12) if x ∈ gm0,ε\g, then, for any h ∈ λu, |u(x) − u0(x)| < 1 m0 and h(x) ∈ ] ψ(x,u0(x)) − ε r , ψ(x,u0(x)) + ε r [ . let k0 = { x ∈ ω; h(x) ∈ [ ψ(x,u0(x)),ψ(x,u0(x)) ]} , k− = { x ∈ ω; h(x) < ψ(x,u0(x)) } , k + = { x ∈ ω; h(x) > ψ(x,u0(x)) } , 76 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) and w(x) =    ψ(x,u0(x)), for x ∈ k +; h(x) , for x ∈ k0; ψ(x,u0(x)), for x ∈ k −. hence w ∈ λu0 and |w(x) − h(x)| < ε r for all x ∈ gm0,ε \ g. (3.13) from (3.11) and (3.13), we have ∫ gm0,ε \ g |w(x) − h(x)|p ′(x)dx < ( ε r )p′+ m(ω) < ( ε 3 )p′+ . (3.14) assume that v is a coset in ω of gm0,ε \ g, then v = (ω \ gm0,ε) ∪ (gm0,ε ∩ g) and m(v ) ≤ m(ω \ gm0,ε) + m(gm0,ε ∩ g) < ε r + m(g) < η. according to (3.8), (3.11) and (3.12). from (h2), (3.9) and (3.10), we obtain ∫ v |w(x) − h(x)|p ′(x)dx ≤ ∫ v |w(x)|p ′(x) + |h(x)|p ′(x)dx ≤ 2p ′+−1 (∫ v |b(x)|p ′(x) + cp ′(x)(x)|u0(x)| p(x) + |b(x)|p ′(x) + cp ′ (x)|u(x)|p(x)dx ) ≤ 2p ′+−1 ( ∫ v 2|b(x)|p ′(x) + cp ′(x)(x)(2p +−1 + 1)|u0(x)| p(x)dx ) + 2p ′+−1 ( ∫ v 2p +−1cp ′(x)(x)|u(x) − u0(x)| p(x)dx ) (3.15) ≤ 2p ′+−1 ∫ v 2|b(x)|p ′(x) + cp ′(x)(x)(2p +−1 + 1)|u0(x)| p(x)dx + 2p ++p ′+−2 ‖cp ′+ ‖l∞(ω) ∫ v |u(x) − u0(x)| p(x) dx ≤ ( ε 3 )p′+ + 2p ++p ′+−2 ‖cp ′+ ‖l∞(ω)δ θ ≤ 2 ( ε 3 )p′+ ≤ εp ′+ . thanks to (3.14), (3.15) and (2.1), we get ‖w − h‖p′(x) ≤ ∫ ω |w(x) − h(x)|p ′(x)dx + 1 < ε. hence λ is upper semicontinuous (u.s.c.). hence n = i∗ ◦ λ ◦ i is clearly bounded, upper semicontinuous (u.s.c.) and compact. next, we give the proof of theorem 1.2. let s := a + n : w s,p(x,y) 0 (ω) → 2 ( w s,p(x,y) 0 (ω) ) ∗ , where a and n were defined in lemma 3.3 and in section 2 respectively. this means that a point u ∈ w s,p(x,y) 0 (ω) is a weak solution of (1.1) if and only if fu ∈ −su, (3.16) with f defined in (1.5). by the properties of the operator f given in lemma 3.1 and the mintybrowder’s theorem on monotone operators in [45, theorem 26 a], we guarantee that the inverse cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 77 operator t := f−1 : ( w s,p(x,y) 0 (ω) )∗ → w s,p(x,y) 0 (ω) is continuous, of type (s+) and bounded. moreover, thanks to lemma 3.3 the operator s is quasi-monotone, upper semicontinuous (u.s.c.) and bounded. as a result, the equation (3.16) is equivalent to the abstract hammerstein equation u = tv and v ∈ −s ◦ tv. (3.17) we will apply the theory of degrees introduced in section 3 to solve the equations (3.17). for this, we first show the following lemma. lemma 3.5. the set b := { v ∈ ( w0 )∗ such that v ∈ −ts ◦ tv for some t ∈ [0,1] } is bounded. proof. let v ∈ b , so, v + ta = 0 for every t ∈ [0,1], with a ∈ s ◦ tv. setting u := tv, we can write a = au + ϕ ∈ su, where ϕ ∈ nu, namely, 〈ϕ,u〉 = ∫ ω h(x)u(x)dx, for each h ∈ lp ′(x)(ω) with ψ(x,u(x)) ≤ h(x) ≤ ψ(x,u(x)) for almost all x ∈ ω. if ‖u‖w0 ≤ 1, then ‖tv‖w0 is bounded. if ‖u‖w0 > 1, then we get by the implication (i) in proposition 2.1 and the inequality (2.2) and using (h0), the young inequality, the compact embedding w0 →֒→֒ l q(x)(ω), the estimate ‖tv‖ p − w0 = ‖u‖ p − w0 ≤ ρp(·,·)(u) ≤ t|〈a,tv〉| ≤ t ∫ ω |u|q(x) dx + t ∫ ω λ|h(x,u)|udx + t ∫ ω |hu|dx ≤ t ∫ ω |u|q(x) + tcp′ ∫ ω |λh(x,u)|q ′(x) dx + tcp ∫ ω |u|q(x) dx + cγt (∫ ω |u|γ(x)dx ) + cγ′t (∫ ω |h|γ ′(x)dx ) ≤ const ( ‖u‖ q − q(x) + ‖u‖ q + q(x) + ‖u‖ γ − γ(x) + ‖u‖ γ+ γ(x) + 1 ) ≤ const ( ‖u‖ q − w0 + ‖u‖ q + w0 + ‖u‖ γ − w0 + ‖u‖ γ+ w0 + 1 ) ≤ const ( ‖tv‖ q + w0 + ‖tv‖ γ + w0 + 1 ) . hence it is obvious that { tv | v ∈ b } is bounded. as the operator s is bounded and from (3.17), we deduce the set b is bounded in ( w0 )∗ . 78 h. el hammar, c. allalou, a. abbassi & a. kassidi cubo 24, 1 (2022) thanks to lemma 3.5, we can find a positive constant r such that ‖v‖( w0 ) ∗ < r for any v ∈ b. this says that v ∈ −ts ◦ tv for each v ∈ ∂br(0) and each t ∈ [0,1]. under the lemma 2.8, we get i + s ◦ t ∈ ft (br(0)) and i = f ◦ t ∈ ft (br(0)). now, we are in a position to consider the affine homotopy h : [0,1] × br(0) → 2 ( w0 ) ∗ defined by h(t,v) := (1 − t)iv + t(i + s ◦ t)v for (t,v) ∈ [0,1] × br(0). by applying the normalization and homotopy invariance property of the degree d fixed in theorem 2.11, we have d(i + s ◦ t,br(0),0) = d(i,br(0),0) = 1. it follows that, we can get a function v ∈ br(0) such that v ∈ −s ◦ tv. which implies that u = tv is a weak solution of (1.1). this completes the proof. cubo 24, 1 (2022) the topological degree methods for the fractional p(·)-laplacian... 79 references [1] a. abbassi, c. allalou and a. kassidi, “existence of weak solutions for nonlinear p-elliptic problem by topological degree”, nonlinear dyn. syst. theory, vol. 20, no. 3, pp. 229–241, 2020. 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[45] e. zeidler, nonlinear functional analysis and its applications ii/b, nonlinear monotone operators, new york: springer, 1990. introduction and main result preliminaries lebesgue and fractional sobolev spaces with variable exponent some classes of operators and an outline of berkovits degree proof of theorem 1.2 cubo, a mathematical journal vol. 24, no. 03, pp. 457-466, december 2022 doi: 10.56754/0719-0646.2403.0457 on the minimum ergodic average and minimal systems manuel saavedra 1 helmuth villavicencio 2, b 1 instituto de matemática, universidade federal do rio de janeiro, rio de janeiro, brazil. saavmath@pg.im.ufrj.br 2 instituto de matemática y ciencias afines, universidad nacional de ingenieŕıa, lima, peru. hvillavicencio@imca.edu.pe b abstract we prove some equivalences associated with the case when the average lower time is minimal. in addition, we characterize the minimal systems by means of the positivity of invariant measures on open sets and also the minimum ergodic averages. finally, we show that a minimal system admits an open set whose measure is minimal with respect to a set of ergodic measures and its value can be chosen in [0, 1]. resumen demostramos algunas equivalencias asociadas con el caso cuando el tiempo inferior promedio es mı́nimo. además, caracterizamos los sistemas minimales a través de la positividad de medidas invariantes en conjuntos abiertos y también los promedios ergódicos mı́nimos. finalmente, mostramos que un sistema minimal admite un conjunto abierto cuya medida es mı́nima con respecto a un conjunto de medidas ergódicas y su valor puede ser elegido en [0, 1]. keywords and phrases: time average, minimum ergodic average, minimal systems. 2020 ams mathematics subject classification: 37c35, 37b05. accepted: 08 november, 2022 received: 25 july, 2022 c©2022 m. saavedra et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0457 https://orcid.org/0000-0002-3045-7648 https://orcid.org/0000-0002-5812-9021 mailto:saavmath@pg.im.ufrj.br mailto:hvillavicencio@imca.edu.pe 458 m. saavedra & h. villavicencio cubo 24, 3 (2022) 1 introduction the main motivation of this paper is the result obtained by jenkinson in [3] which states that given an invariant measure there exists a continuous function that achieves the maximum ergodic average using such measure. this result has been used in several recent works [1, 6, 7]. a minimizing version of this result is possible to obtain in a straightforward way. in this sense, given the behavior of uniquely ergodic systems, it is natural to ask whether this version admits any relation to minimal systems and time averages. the present paper addresses both problems in the following way. firstly, we prove some equivalences associated with the case when the average lower time is minimal. we also characterize the minimal systems by means of the positivity of invariant measures on open sets and also the minimum ergodic averages (this result was inspired by theorem 6.17 in [9]). finally, we show that given a finite set of ergodic measures in a minimal system it is possible to find an open set whose measure is minimal and its value can be chosen in [0, 1]. let us state our results in a precise way. throughout this paper, the pair (x, d) denotes a compact metric space and c(x) denotes the space of all continuous real-valued functions on x. we denote by m(x) the set of all borel probability measures of x, provided with the weak* topology. let t : x → x be a continuous transformation. given µ an element of m(x), we say that µ is t -invariant if µ(t −1(a)) = µ(a) for every borel subset a of x. we denote by mt (x) the set of t -invariant probability measures. a probability measure µ is called ergodic if µ(a) ∈ {0, 1} for each t -invariant set a. denote by et (x) the set of ergodic measures. for x ∈ x, let δx be denote the dirac point measure of x defined by δx(a) = 1 when x ∈ a and δx(a) = 0 otherwise. let f : x → r be a continuous function, we say that an invariant measure µ is f-minimizing if the minimum ergodic average [4] defined by α(f) = min {∫ x fdm : m ∈ mt (x) } , satisfies α(f) = ∫ x fdµ. given x ∈ x, recall that the lower time average is τ(x, f) = lim inf n→∞ 1 n n−1∑ i=0 f ◦ t i(x). we consider the number e(x, f) = τ(x, f) − α(f). this number quantifies the non-minimal time average. note that e(x, f) ≥ 0. next we state our first result that characterizes the cases where the non-minimal time average is equal to zero totally and partially uniform. for this purpose, we must recall that (x, t ) is said to be uniquely ergodic if there is a unique invariant probability measure on x. cubo 24, 3 (2022) on the minimum ergodic average and minimal systems 459 theorem 1.1. let t : x → x be a continuous transformation of a compact metric space. for every x ∈ x and f ∈ c(x), we have the following equivalences (1) e ≡ 0 if and only if (x, t ) is uniquely ergodic. (2) e(·, f) = 0 if and only if α(f) = ∫ x fdµ, for all µ ∈ mt (x). (3) e(x, ·) = 0 if and only if every ergodic measure is a limit point of the sequence { 1 n n−1∑ i=0 δt i(x) } . our next result shows a characterization of the minimal systems through the open sets and minimum ergodic averages. recall that a dynamical system (x, t ) is called minimal if x does not contain any non-empty, proper, closed t -invariant subset. theorem 1.2. let t : x → x be a continuous transformation of a compact metric space. the following statements are equivalents: (1) (x, t ) is a minimal system. (2) for each non empty open set a ⊂ x and each µ ∈ mt (x), we have µ(a) > 0. (3) every non-zero f ∈ c(x) with f ≥ 0 satisfies α(f) > 0. finally, in the case of non-discrete minimal systems, it is satisfied that the minimum ergodic average reaches all values of [0, 1] for continuous functions of norm one. (see lemma 2.8). motivated by this, we found a condition on the ergodic measures [2] to obtain a version of this result through open sets. theorem 1.3. let t : x → x be a continuous transformation of a non-discrete compact metric space. if (x, t ) is minimal and f is a finite subset of et (x), then for every r ∈ [0, 1] there is an open set a such that r is the minimun value of µ(a) whenever µ ∈ f. the paper is organized as follows. in section 2, we will prove several results necessary for the proof of the main theorems. finally, in section 3, we will prove theorems 1.1, 1.2 and 1.3. 2 preliminary lemmas let x be a compact metric space and t : x → x be a continuous transformation. we denote the applications α : c(x) −→ r f 7−→ min µ∈mt (x) ∫ x fdµ, 460 m. saavedra & h. villavicencio cubo 24, 3 (2022) and e : x × c(x) −→ [0, +∞) (x, f) 7−→ τ(x, f) − α(f). below are some properties of these applications that are straightforward from the definition. let z be a convex set of a vector space v . a subset f of z is called face of z if whenever x, y ∈ z and λx + (1 − λ)y ∈ f with 0 < λ < 1, then {x, y} ⊂ f . proposition 2.1. we have the following properties (1) α is continuous and t -invariant. (2) α(1) = 1. (3) e(x, f) = e(x, f ◦ t ). (4) α(f) ≤ α(g) whenever f ≤ g. (5) the set { µ ∈ mt (x) : α(f) = ∫ x fdµ } is a non-empty closed face of mt (x). we will prove some additional properties of e lemma 2.2. let t : x → x be a continuous transformation of a compact metric space. it holds that e(x, f) = 0 for every x ∈ x and f ∈ c(x) if and only if the system (x, t ) is uniquely ergodic. proof. it is sufficient to prove that if e ≡ 0 then the system is uniquely ergodic. by theorem 1 in [3] for every ergodic measure ν there exists an f ∈ c(x) such that ν is the unique f-minimizing measure, that is, ν is the unique satisfying ∫ x fdν = α(f). since e(x, f) = 0 for each x ∈ x, we obtain lim inf n→∞ 1 n n−1∑ i=0 f ◦ t i(x) = ∫ x fdν. (2.1) if the system is not uniquely ergodic, there exists ω ∈ et (x) such that ω 6= ν. let p be a generic point for ω. using (2.1), we have ∫ x fdω = lim n→∞ 1 n n−1∑ i=0 f ◦ t i(p) = ∫ x fdν < ∫ x fdω. it is a contradiction. so (x, t ) is uniquely ergodic. cubo 24, 3 (2022) on the minimum ergodic average and minimal systems 461 lemma 2.3. let t : x → x be a continuous transformation of a compact metric space. given f ∈ c(x). then, e(x, f) = 0 for every x ∈ x if and only if α(f) = ∫ x fdµ, for all µ ∈ mt (x). proof. by proposition 2.1, we know that the set h = { ν ∈ mt (x) : α(f) = ∫ x fdν } , is a non-empty closed face of mt (x). if h 6= mt (x), then there is µ ∈ et (x) \ h. let p be a generic point for µ, so ∫ x fdµ = lim n→∞ 1 n n−1∑ i=0 f ◦ t i(p) = α(f), (2.2) the last equality in (2.2) is a consequence of the hypothesis e(p, f) = 0. thus µ ∈ h, which is absurd. conversely, given x ∈ x we can find a sequence {nk} such that the inferior mean sojourn time is written as τ(x, f) = lim k→∞ 1 nk nk−1∑ i=0 f ◦ t i(x), (2.3) and also { n−1 k nk−1∑ i=0 δt i(x) } is convergent to ν ∈ mt (x). therefore e(x, f) = 0 since τ(x, f) = ∫ x fdν = α(f). a consequence of the above result is the following corollary 2.4. the set {f ∈ c(x) : e(x, f) = 0 for every x ∈ x} is a closed linear subspace of c(x). lemma 2.5. let t : x → x be a continuous transformation of a compact metric space. given x ∈ x. then, e(x, f) = 0 for each f ∈ c(x) if and only if every ergodic measure is a limit point of the sequence { 1 n n−1∑ i=0 δt i(x) } . proof. denote by λ the set of the limit points of the sequence { 1 n n−1∑ i=0 δt i(x) } . suppose there is µ ∈ et (x) \ λ. by theorem 1 in [3], for every ergodic measure µ there exists an f ∈ c(x) such that µ is the unique with the property ∫ x fdµ = α(f). since e(x, f) = 0, we have ∫ x fdµ = α(f) = τ(x, f). 462 m. saavedra & h. villavicencio cubo 24, 3 (2022) moreover, using (2.3), there is a sequence {mk} in m(x) such that τ(x, f) = lim k→∞ ∫ x fdmk. we can assume that {mk} converges to ν ∈ mt (x). then ∫ x fdµ = ∫ x fdν with ν 6= µ. it is a contradiction. conversely, given f ∈ c(x) there exists an ergodic measure µ such that α(f) = ∫ x fdµ. on the other hand, there is a sequence {nk} satisfying µ = lim k→∞ n−1 k nk−1∑ i=0 δt i(x). therefore α(f) ≤ τ(x, f) ≤ lim k→∞ 1 nk nk−1∑ i=0 f ◦ t i(x) = ∫ x fdµ = α(f), so e(x, f) = 0. now, we introduce the following auxiliary application ̺ : t × c −→ [0, 1] (a, f) 7−→ ̺(a, f) = min µ∈f µ(a), where t denotes the topology associated with x and c denotes the space of all closed subsets of mt (x). we write ̺(a) = ̺(a, mt (x)). note that ̺(a) can be interpreted as the capacity of an open set (see lemma 4.1 in [5]). lemma 2.6. let t : x → x be a continuous transformation of a compact metric space. it holds that ̺(a) > 0 for every non-empty open set a if and only if α(f) > 0 for each non-zero f ∈ c(x) with f ≥ 0. proof. given a non-zero f ∈ c(x) with f ≥ 0. we can find a non-empty open a and a constant c > 0 that verify f(x) ≥ c for all x ∈ a. it follows that α(f) ≥ min µ∈mt (x) ∫ a fdµ ≥ c̺(a) > 0. hence α(f) > 0. conversely, let a be a non-empty open set in x. by urysohn’s lemma choose f ∈ c(x) with 0 ≤ f ≤ 1, f(p) = 1 and f = 0 on ac for some p ∈ a. if ̺(a) = 0, there exists some ν ∈ mt (x) such that ν(a) = 0, therefore 0 < α(f) ≤ ∫ a fdν = 0. it is a contradiction. cubo 24, 3 (2022) on the minimum ergodic average and minimal systems 463 lemma 2.7. let t : x → x be a continuous transformation of a compact metric space. if (x, t ) is a minimal system, then ̺(a) > 0 for every non-empty open set a. proof. since (x, t ) is minimal, for every non-empty open set a we have x = n⋃ i=0 t −i(a), for some n ∈ n, therefore ̺(a) ≥ 1 n+1 , that is, ̺(a) > 0. lemma 2.8. let t : x → x be a continuous transformation of a non-discrete compact metric space. if (x, t ) is minimal, then given r ∈ [0, 1], there is an f ∈ c(x) with ‖f‖ ∞ = 1 such that α(f) = r. proof. note that the set b = {f ∈ c(x) : ‖f‖ ∞ = 1} is connected in (c(x), ‖.‖ ∞ ). given r ∈ (0, 1), by lemma 2.7 and since (x, t ) is non-discrete, we obtain a non-empty open set a with the following property 0 < ̺(a) < r/2. by urysohn’s lemma choose g ∈ c(x) with 0 ≤ g ≤ 1, g(p) = 1 and g = 0 on ac for some p ∈ a. therefore α(g) = min µ∈mt (x) ∫ a gdµ ≤ ̺(a) < r/2. by proposition 2.1, α is continuous on b and α(1) = 1. so, there exists f ∈ c(x) with ‖f‖ ∞ = 1 such that α(f) = r. now for the remaining cases it is sufficient to consider the constant functions f ≡ 1 and g ≡ −1. then α(f) = 1 and α(g) = −1, it follows that there is h ∈ b such that α(h) = 0. if we denote ẽ(x, a) = τ(x, χa) − ̺a, this value represents the non-minimal mean sojourn time on a. also we can obtain that ẽ(x, a) ∈ [0, 1]. recall that a point x ∈ x is periodic for t : x → x if t n(x) = x for some n ∈ n and the minimal such n is called the period of t . a point x ∈ x is called pre-periodic if some iterate of x is periodic. we denote by o(x) the orbit of x. lemma 2.9. let t : x → x be a continuous transformation of a compact metric space. it holds that ẽ(x, a) ≡ 0 for every x ∈ x and a ∈ t if and only if every point in x is pre-periodic and there is only one periodic orbit. proof. it is enough to prove the sufficiency. first, we claim that ẽ ≡ 0 implies that each measure in mt (x) is atomic. suppose there is a non-atomic µ invariant measure. given z ∈ x, we can find open sets {v zn }n∈n such that t n(z) ∈ v zn and µ(v z n ) < 1/2 n+1. therefore, the open 464 m. saavedra & h. villavicencio cubo 24, 3 (2022) set az = ⋃ n v zn contains the orbit of z, so τ(z, az) = 1. thus ẽ(z, az) > 1/2 since ̺(az) ≤ µ(az) < 1/2. this proves our claim. let ν be an ergodic measure. there is p ∈ x with ν(p) > 0. by the poincaré’s recurrence (theorem 1.2.4 in [8]), the point p is periodic. given x ∈ x, if x = o(p), then there is nothing to prove. otherwise, the open set b = x \ o(p) satisfies ̺b = 0, so τ(x, b) = 0. this implies that o(x) 6⊂ b. hence, there exists a periodic point p such that for each x ∈ x there exists k ∈ n satisfying t k(x) ∈ o(p). 3 proof of the theorems proof of theorem 1.1. the proof of this result is actually contained in the lemmas 2.2, 2.3 and 2.5. proof of theorem 1.2. to prove that item (1) implies item (2), we use lemma 2.7. to prove that item (2) implies item (1), assume that (x, t ) is not a minimal system. there exists some point x ∈ x whose orbit is not dense in x. we consider the non-empty open set a = x \ o(x), so ̺(a) > 0. on the other hand, there are a sequence {nk} and a measure µ ∈ mt (x) satisfying µ = lim k→∞ n−1 k nk−1∑ i=0 δt i(x), therefore ̺(a) ≤ µ(a) ≤ lim inf k→∞ 1 nk |{0 ≤ i ≤ nk − 1 : t i(x) ∈ a}| = 0. it is a contradiction. finally, the lemma 2.6 proves the equivalence between item (2) and item (3). proof of theorem 1.3. suppose that there exists r ∈ (0, 1) such that ̺(a, f) 6= r for every open set a . we consider the set z = {b : b is open in x and 0 < ̺(b, f) < r}. by lemma 2.7, we obtain that z is non-empty since (x, t ) is non-discrete. we partially order z by inclusion. assume {bi}i∈i ⊂ z is a totally ordered subset of z where i is infinite. an upper bound for the bi’s in z is the open set b = ⋃ i∈i bi. since i is infinite, we can suppose that n ⊂ i. we choose an increasing sequence {aj} such that b = ⋃ j∈n aj. if f = {µℓ} n ℓ=1, then there exists ℓ such that the set kℓ = {j ∈ n : ̺(aj, f) = µℓ(aj)} is infinite. thus, given ν ∈ f we have ν(b) = lim j∈kℓ ν(aj) ≥ lim j∈kℓ µℓ(aj) = µℓ(b), cubo 24, 3 (2022) on the minimum ergodic average and minimal systems 465 then ̺(b, f) = µℓ(b). on the other hand, since b = ⋃ j∈n aj using the regularity of the measure we have that there are j ∈ n and a compact k such that k ⊂ aj and µℓ(k) ≤ µℓ(aj) < r. so, µℓ(b) ≤ r but for the hypothesis ̺(a, f) 6= r. hence ̺(b, f) = µℓ(b) < r, therefore b ∈ z. zorn’s lemma now tells us that z contains a maximal element a. let µ ∈ f such that µ(a) = ̺(a, f) < r. given x ∈ x, there is an open set ux with µ(ux) < r − µ(a). hence ̺(a ∪ ux, f) ≤ µ(a ∪ ux) ≤ µ(a) + µ(ux) < r. using the maximality of a it is concluded that ux ⊂ a for every x ∈ x, so a = x. it implies ̺(a, f) = 1 > r, which is absurd. acknowledgements ms was partially supported by capes and cnpq-brazil. hv was partially supported by fondecytconcytec contract 100-2018 and universidad nacional de ingenieŕıa, peru, projects fc-pf-33-2021 and p-cc-2022-000956. 466 m. saavedra & h. villavicencio cubo 24, 3 (2022) references [1] s. addas-zanata and f. a. tal, “support of maximizing measures for typical c0 dynamics on compact manifolds”, discrete contin. dyn. syst., vol. 26, no. 3, pp. 795–804, 2010. 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[9] p. walters, an introduction to ergodic theory, graduate texts in mathematics 79, new york: springer new york, 1982. introduction preliminary lemmas proof of the theorems cubo, a mathematical journal vol. 23, no. 01, pp. 161–170, april 2021 doi: 10.4067/s0719-06462021000100161 idempotents in an ultrametric banach algebra alain escassut université clermont auvergne, umr cnrs 6620, lmbp, f-63000 clermont-ferrand, france. alain.escassut@uca.fr abstract let ik be a complete ultrametric field and let a be a unital commutative ultrametric banach ik-algebra. suppose that the multiplicative spectrum admits a partition in two open closed subsets. then there exist unique idempotents u, v ∈ a such that φ(u) = 1, φ(v) = 0 ∀φ ∈ u, φ(u) = 0 φ(v) = 1 ∀φ ∈ v . suppose that ik is algebraically closed. if an element x ∈ a has an empty annulus r < |ξ − a| < s in its spectrum sp(x), then there exist unique idempotents u, v such that φ(u) = 1, φ(v) = 0 whenever φ(x − a) ≤ r and φ(u) = 0, φ(v) = 1 whenever φ(x−a) ≥ s. resumen sea ik un cuerpo ultramétrico completo y sea a una ik-algebra de banach ultramétrica unital conmutativa. suponga que el espectro multiplicativo admite una partición en dos conjuntos abiertos y cerrados. luego, existen idempotentes únicos u, v ∈ a tales que φ(u) = 1, φ(v) = 0 ∀φ ∈ u, φ(u) = 0 φ(v) = 1 ∀φ ∈ v . suponga que ik es algebraicamente cerrado. si un elemento x ∈ a tiene un anillo vaćıo r < |ξ−a| < s en su espectro sp(x), entonces existen idempotentes únicos u, v tales que φ(u) = 1, φ(v) = 0 cada vez que φ(x−a) ≤ r y φ(u) = 0, φ(v) = 1 cada vez que φ(x−a) ≥ s. keywords and phrases: ultrametric banach algebras, multiplicative semi-norms, idempotents, affinoid algebras. 2020 ams mathematics subject classification: 12j25, 30d35, 30g06. accepted: 17 march, 2021 received: 05 november, 2020 ©2021 a. escassut. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100161 https://orcid.org/0000-0003-0002-0194 162 a. escassut cubo 23, 1 (2021) 1 introduction and main theorem ultrametric banach algebras have been a topic of many resarch along the last years [1], [3], [4], [5],[6], [10], [11], [12]. the following theorem 1.1 (stated in [14]) corresponds in ultrametric banach algebras to a well known theorem in complex banach algebra: if the spectrum of maximal ideals admits a partition in two open closed subsets u and v with respect to the gelfand topology, there exist idempotents u and v such that χ(u) = 1, χ(v) = 0 ∀χ ∈ u and χ(u) = 0, χ(v) = 1 ∀χ ∈ v . in an ultrametric banach algebra, it is impossible to have a similar result because a partition in two open closed subsets for the gelfand topology on the spectrum of maximal ideals then makes no sense, due to the total disconnection of the spectrum. b. guennebaud first had the idea to consider the set of continuous multiplicative semi-norms of an ultrametric banach algebra, denoted by mult(a,‖ . ‖) instead of the spectrum of maximal ideals [14], an idea that later suggested berkovich theory [2]. recall that mult(a,‖ . ‖) is compact with respect to the topology of pointwise convergence (theorem 1.11 in [7]). the proof of theorem 1.1, stated in [14], was heavy and involved many particular notions in a chapter of over 40 pages that was never published. we will use propositions 2.10, 2.11, 2.12 in order to assure the unicity. finally, we will show that if the theorem is proven for affinoid algebras, that may be generalised to all ultrametric banach algebras (proposition 2.12). notations: we denote by ik a complete ultrametric field. given a ik-algebra a, we denote by mult(a) the set of multplicative semi-norms of a and if a is a normed ik-algebra, we denote by mult(a,‖ . ‖) the set of continuous multplicative semi-norms of a provided with the topology of pointwise convergence. next, we denote by multm(a,‖ . ‖) the set of continuous multplicative semi-norms of a whose kernel is a maximal ideal of a. given φ ∈ mult(a,‖ . ‖), we denote by ker(φ) the closed prime ideal of the x ∈ a such that φ(x) = 0. it is well known that every maximal ideal is the kernel of at least one multiplicative semi-norm on a (see for example [9]). the algebra a is said to be multbijective if for every maximal ideal m, a m admits only one absolute value that is an expansion of this of ik. it is easily seen that if every maximal ideal is of finite codimension, then the algebra a is multbijective. consider then a multbijective unital commutative ultrametric ik-banach algebra a. we denote by x(a) the set of algebra homomorphisms from a onto a field extension of ik of the form a m where m is a maximal ideal of a. so, for every χ ∈ x(a), the mapping |χ| defined on a by |χ|(x) = |χ(x)| belongs to multm(a,‖ . ‖) and this is the unique φ ∈ multm(a,‖ .‖) such that ker(φ) = ker(χ). theorem 1.1. let a be a unital commutative ultrametric ik-banach algebra such that mult(a,‖ . ‖) admits a partition in two compact subsets u, v . there exist unique idempotents u,v ∈ a such that φ(u) = 1, φ(v) = 0, ∀φ ∈ u and φ(u) = 0, φ(v) = 1, ∀φ ∈ v . cubo 23, 1 (2021) idempotents in an ultrametric banach algebra 163 corollary 1.2. let a be a unital commutative ultrametric ik-banach algebra such that mult(a,‖ . ‖) admits a partition in two compact subsets u, v . then a is isomorphic to a direct product of two ik-banach algebras au ×av such that mult(au,‖ . ‖) = u and mult(av ,‖ . ‖) = v . given the idempotent u ∈ a such that φ(u) = 1 ∀φ ∈ u, φ(u) = 0 ∀φ ∈ v , then au = ua, av = (1 −u)a. as an easy consequence, we have theorem 1.3. a few definitions are necessary: definitions and notations: suppose that ik is algebraically closed. let a ∈ ik and r, s ∈ ir+ with 0 < r < s. we denote by γ(a,r,s) the set {x ∈ ik| r < |x−a| < s}. let d be a subset of ik, let a ∈ d be such that d ∩ γ(a,r,s) = ∅ and that r = sup{|a − x|, x ∈ d, |a − x| ≤ r} and s = inf{|a−x|, x ∈ d, |a−x| ≥ s}. the annulus γ(a,r,s) is called an empty-annulus of d. let a be a unital commutative ik-algebra and let x ∈ a. we denote by sp(x) the set of all λ ∈ ik such that x−λ is not invertible. theorem 1.3. suppose that ik is algebraically closed. let a be a unital commutative ultrametric ik-banach algebra such that multm(a,‖ . ‖) is dense in mult(a,‖ . ‖) and let x ∈ a be such that sp(x) admits an empty-annulus γ(a,r,s). then there exist a unique idempotent u ∈ a and a unique idempotent v ∈ a such that χ(u) = 1, χ(v) = 0 ∀χ ∈ x(a) satisfying |χ(x) −a| ≤ r and χ(u) = 0, χ(v) = 1 ∀χ ∈x(a) satisfying |χ(x) −a| ≥ s. 2 the proofs proving theorem 1.1 requires some preparation. we will use propositions 2.10, 2.11 and 2.12 and mainly theorem 2.7. definitions and notations: let a be a unital commutative ultrametric ik-banach algebra whose norm is ‖ . ‖. we define the spectral semi-norm ‖ . ‖sp as ‖f‖sp = limn→+∞‖fn‖ 1 n . by [13] we have theorem 2.1 (see also [9], theorem 6.19). theorem 2.1. ‖f‖sp = sup{φ(f) | φ ∈ mult(a,‖ . ‖)}. affinoid algebras were introduced by john tate in [17] who called them algebras topologically of finite type and are now usually called affinoid algebras. as this first name suggests, such an algebra is the completion of an algebra of finite type for a certain norm. definitions and notation: the ik-algebra of polynomials in n variables ik[x1, . . . ,xn] is equipped with the gauss norm ‖ . ‖ defined as∣∣∣∣∣∣ ∣∣∣∣∣∣ ∑ i1,...,in ai1,...,inx i1 1 · · ·x in n ∣∣∣∣∣∣ ∣∣∣∣∣∣ = supi1,...,in |ai1,...,in|. 164 a. escassut cubo 23, 1 (2021) we denote by ik{x1, . . . ,xn} the set of power series in n variables∑ i1,...,in ai1,...,inx i1 1 · · ·x in n such that lim i1+...+in→∞ ai1,...,in = 0. the elements of such an algebra are called the restricted power series in n variables, with coefficients in ik. hence, by definition, ik[x1, . . . ,xn] is dense in ik{x1, . . . ,xn}. then ik{x1, . . . ,xn} is a ik-banach algebra which is just the completion of ik[x1, . . . ,xn] and is denoted by tn. by [16] (see also [9]): we have theorem 2.2: theorem 2.2. every algebra ik{x1, . . . ,xn} is factorial and all ideals are closed. a ikaffinoid algebra corresponds to a quotient of any algebra of the form ik{x1, . . . ,xn} by one of its ideals equipped with its quotient norm of banach ik-algebra. by theorems 31.1 and 32.7 of [9] (see also [17] and [14]): theorem 2.3. let a be a ik-affinoid algebra. then a is noetherian and all its ideals are closed. each maximal ideal is of finite codimension. moreover the nilradical of a is equal to its jacobson radical. further, a has finitely many minimal prime ideals. by theorems 35.4 in [9] or proposition 2.8 of iii in [14], we have theorem 2.4: theorem 2.4. let a be a ik-affinoid algebra. then multm(a,‖ . ‖) is dense in mult(a,‖ . ‖) for the topology of pointwise convergence. by theorems 35.4 in [9] we can state theorem 2.5: theorem 2.5. let a be a reduced ik-affinoid algebra. then the spectral norm ‖ . ‖ of a is a norm and is equivalent to the norm of affinoid algebra. remark 2.6. the proofs given in [9] for theorems 2.2, 2.3, 2.4, 2.5 are given for algebraically closed complete ultrametric field but they hold on any complete ultrametric field. by corollary 2.2.7 in [2] we have theorem 2.7: theorem 2.7. let a be a reduced ik-affinoid algebra such that mult(a,‖ . ‖) admits a partition in two compact subsets u1 and u2. then a is isomorphic to a direct product a1 ×a2 where aj is a ik-affinoid algebra such that mult(aj,‖ . ‖) is homeomorphic to uj, j = 1, 2. proposition 2.8. let a be a ik-affinoid algebra of jacobson radical r and let w ∈ r. the equation x2 −x + w = 0 has a solution in r. proof. since a is affinoid, by theorem 2.3, w is nilpotent, hence we can consider the element u = − 1 2 +∞∑ n=1 ( 1 2 n ) (−4w)n. cubo 23, 1 (2021) idempotents in an ultrametric banach algebra 165 now we can check that (2u− 1)2 = 1 − 4w and then u2 −u−w = 0. proposition 2.9. let a be a ik-affinoid algebra of jacobson radical r and let w ∈ a be such that w2 −w ∈r. there exists an idempotent u ∈ a such that w −u ∈r. proof. we will roughly follow the proof known in complex algebra [15]. let r = w2 −w. we first notice that 1 + 4r = (2w − 1)2. next, r 1 + 4r belongs to r hence by proposition 2.8, there exists x ∈r such that x2 −x + r 1 + 4r = 0, and hence ((2w − 1)x)2 − (2w − 1)2x + r = 0. now set s = (2w − 1)x. then s belongs to r, as x. then we obtain s2 − (2w − 1)s + r = 0. let us now put u = w −s and compute u2: (w −s)2 = w2 − 2ws + s2 = w + r − 2ws + s2. but s2 = −r + (2w − 1)s, hence finally: (w + s)2 = w −r + 2ws + r − (2w − 1)s = w + s. thus u is an idempotent such that u−w ∈r. proposition 2.10. [14] let a be a commutative unital ultrametric ik-banach algebra and assume that mult(a,‖ . ‖) admits a partition in two compact subsets u, v . suppose that there exist two idempotents u and e such that ∀φ ∈ u, φ(u) = φ(e) = 1 and ∀φ ∈ v, φ(u) = φ(e) = 0 . then u = e. proof. put e = u + r. since e2 = e, we have (u + r)2 = u + 2ur + r2 hence u + r = u + 2ur + r2 and hence r = 2ur + r2, therefore r(2u + r − 1) = 0. suppose r 6= 0. then 2u + r−1 is a divisor of zero. now, when φ ∈ u, we have φ(1−u) = 0, hence φ(−1 + 2u + r) = φ(u + r) = φ(e) = 1, and when φ ∈ v , we have φ(u) = φ(e) = 0, hence φ(1−2u−r) = φ(1−u−r) = φ(1−e) = 1. hence, ∀φ ∈ mult(a,‖ . ‖), we have φ(1−2u−r) = 1. consequently, 1 − 2u− r does not belong to any maximal ideal of a and hence is invertible. but then 1 − 2u−r is not a divisor of zero, which proves that r = 0 and hence e = u. proposition 2.11. [14] let a be a ik-affinoid algebra such that mult(a,‖ . ‖) admits a partition in two compact subsets u1, u2. there exist unique idempotents e1,e2 ∈ a such that φ(e1) = 1,φ(e2) = 0 ∀φ ∈ u1 and φ(e1) = 0, φ(e2) = 1∀φ ∈ u2. 166 a. escassut cubo 23, 1 (2021) proof. suppose first that a is reduced. by theorem 2.7, a is isomorphic to the direct product a1 × a2 where aj is a ik-affinoid algebra such that mult(aj,‖ . ‖) = uj, j = 1, 2. let φ be the isomorphism from a1 × a2 onto a, let uj be the unity of aj, j = 1, 2 and let e1 = φ(u1, 0), e2 = φ(0,u2). so e1, e2 are idempotents of a. let a ′ 1 = {φ(x, 0) |x ∈ a1} and let a′2 = {φ(0,x) |x ∈ a2}. then, given ϕ ∈ uj, it factorizes in the form ψ ◦ φ−1 with ψ ∈ mult(aj, ‖ . ‖), (j = 1, 2) and for ϕ ∈ u1, we have ϕ(e1) = 1, ϕ(e2) = 0, and given ϕ ∈ u2, we have ϕ(e1) = 0, ϕ(e2) = 1. by proposition 2.10, the idempotents e1, e2 are unique. we can easily greneralize when a is no longer supposed to be reduced. let r be the jacobson radical of a and let b = a r . let θ be the canonical surjection from a onto b. every φ ∈ mult(a,‖ . ‖) is of the form ϕ ◦ θ with ϕ ∈ mult(b,‖ . ‖). let u′1 = {ϕ ∈ mult(b,‖ . ‖)} be such that ϕ◦θ ∈ u1 and let u′2 = {ϕ ∈ mult(b,‖ . ‖)} be such that ϕ◦θ ∈ u2. then u′1 and u′2 are two compact subsets making a partition of mult(b,‖ . ‖). therefore, b has an idempotent u1 such that ϕ(u1) = 1 ∀ϕ ∈ u′1 and ϕ(u1) = 0 ∀ϕ ∈ u′2. let w ∈ a be such that θ(w) = u1. then we can check that φ(w) = 1 ∀φ ∈ u1 and φ(w) = 0 ∀φ ∈ u2. but by proposition 2.9, there exists an idempotent e1 ∈ a such that e1 − w ∈ r. then χ(e1) = χ(w) ∀χ ∈ x(a) and hence φ(e1) = φ(w) ∀φ ∈ mult(a,‖ . ‖) because, by theorem 2.4 multm(a,‖ . ‖) is dense in mult(a,‖ . ‖). the unicity of e1 follows from proposition 2.10. similarly, there exists a unique idempotent e2 ∈ a such that φ(e2) = 1 ∀φ ∈ u2 and φ(e2) = 0 ∀φ ∈ u1. definition and notations: we will denote by | . |∞ the archimedean absolute value of ir. given a unital commutative ultrametric ik-normed algebra a and φ ∈ mult(a,‖ . ‖), y1, . . .yq ∈ a and � > 0, we will denote by w(φ,y1, . . . ,yq,�) the set of θ ∈ mult(a,‖ .‖) such that |φ(yj)−θ(yj)|∞ ≤ � ∀j = 1, . . . ,q. given a unital commutative ultrametric ik-normed algebra a and a subalgebra b, we call canonical mapping from mult(a,‖ . ‖) to mult(b,‖ . ‖) the mapping φ defined by φ(ϕ)(x) = ϕ(x) ∀x ∈ b, ϕ ∈ mult(a,‖ . ‖). proposition 2.12. [14] let a be a unital commutative ultrametric ik-banach algebra and assume that mult(a,‖ . ‖) admits a partition in two compact subsets u, v . there exists a ik-affinoid algebra b included in a, admitting for norm this of a, such that mult(b,‖ . ‖) admits a partition in two open subsets u′, v ′ where the canonical mapping φ from mult(a,‖ . ‖) to mult(b,‖ . ‖) satisfies φ(u) ⊂ u′, φ(v ) ⊂ v ′. proof. since u and v are compact sets, we can easily define a covering of open sets (oj)j∈j such that oj ∩v = ∅ ∀j ∈ j. from this, we can extract a finite covering (ui)1≤i≤n of u where the ui are of the form w(fi,xi,1, . . . ,xi,mi,�i) with xi,j ∈ a, such that ui ∩ v = ∅ ∀i = 1, . . . ,n. let ã be the finite type ik-subalgebra generated by all the xi,j, 1 ≤ j ≤ mi, 1 ≤ i ≤ n. consider the image of mult(a,‖ . ‖) in mult(ã,‖ . ‖) through the mapping φ that associates to each cubo 23, 1 (2021) idempotents in an ultrametric banach algebra 167 φ ∈ mult(a,‖ . ‖) its restriction to ã and let ũ = φ(u), ṽ = φ(v ). then both ũ, ṽ are compact with respect to the topology of mult(ã,‖ . ‖) and hence there exist open neighborhoods u′ of ũ and v ′ of ṽ in mult(ã,‖ . ‖) such that u′ ∩v ′ = ∅. let y = u′ ∪v ′. by construction we have φ(u) ⊂ u′, φ(v ) ⊂ v ′. let φ ∈ mult(ã,‖ . ‖) \ y . there exists a finite type algebra ãφ containing ã, such that the canonical image hϕ of mult(ãφ,‖ . ‖) in mult(ã,‖ . ‖) does not contain φ. since this image hφ is compact, there exists a neighborhood g(φ) of φ such that g(φ) ∩ hφ = ∅. next, we notice that mult(ã,‖ . ‖) \y is compact, hence we can find φ1, . . . ,φn ∈ mult(ã,‖ . ‖) \y and neighborhoods z(φ1), . . . ,z(φn) making a covering of mult(ã,‖ . ‖) \y . let e be the finite type algebra generated by the ãφi, 1 ≤ i ≤ n. then e is a ik-subalgebra of a of finite type which contains ã and hence is equipped with the ik-algebra norm ‖ . ‖ of a. moreover, by construction, mult(e,‖ . ‖) is equal to y = u′ ∪v ′. let {x1, . . . ,xn} be a finite subset of the unit ball of e such that ik[x1, . . . ,xn ] = e. let t be the topologically pure extension ik{x1, . . . ,xn} and consider the canonical morphism θ from ik[x1, . . . ,xn ] equipped with the gauss norm, into e, equipped with the norm ‖ . ‖ of a, defined as θ(f(x1, . . . ,xn )) = f(x1, . . . ,xn ). since by hypotheses, ‖xj‖≤ 1 ∀j = 1, . . . ,n, θ is continuous and has expansion to a continuous morphism θ from t into a. let i be the closed ideal of the elements f ∈ t such that θ(f) = 0. then θ(t) is the ik-affinoid algebra b = t i containing e and included in a. by construction, the ik-affinoid norm of b is the restriction of the norm ‖ . ‖ of a. since by construction e is dense in b, we have mult(b,‖ . ‖) = mult(e,‖ . ‖) = u′ ∪v ′. consequently, φ(u) ⊂ u′, φ(v ) ⊂ v ′, which ends the proof. remark 2.13. proposition 2.12 was roughly stated in [14]. however, its proof was confusing about subsets containing u and v and norms defined on an affinoid subalgebra b, which then puts in doubt the conclusion. we can now conclude. proof of theorem 1.1. by proposition 2.12, there exists a ik-affinoid algebra b included in a such that mult(b,‖ . ‖) admits a partition in two open disjoint subsets u′, v ′ and such that the canonical mapping φ from mult(a,‖ .‖) to mult(b,‖ . ‖) satisfies φ(u) ⊂ u′, φ(v ) ⊂ v ′. now, by proposition 2.11, there exist idempotents u′, v′ ∈ b such that φ(u′) = 1 ∀φ ∈ u′ and φ(u′) = 0 ∀φ ∈ v ′. consequently, we have φ(u) = 1 ∀φ ∈ u, φ(u) = 0 ∀φ ∈ v and φ(v) = 0 ∀φ ∈ u, φ(v) = 1 ∀φ ∈ v . the unicity follows from proposition 2.11. that ends the proof. proof of theorem 1.3. without loss of generality, we can suppose a = 0. let u = {φ ∈ mult(a,‖ . ‖)} such that φ(x) ≤ r, and let v = {φ ∈ mult(a,‖ . ‖)} such that φ(x) ≥ s. since multm(a,‖ . ‖) is dense in mult(a,‖ . ‖), it is clear that no φ ∈ mult(a,‖ . ‖) can satisfy r < φ(x) < s. consequently, u,v make a partition of mult(a,‖ . ‖). next, one can easily 168 a. escassut cubo 23, 1 (2021) check that u and v are open and closed with respect to the pointwise convergence. indeed, given φ ∈ mult(a,‖ . ‖), g1, . . . ,gt ∈ a and � > 0, we denote by w(φ,g1, . . . ,gt,�) the neighborhood of φ defined as {θ ∈ mult(a,‖ . ‖) |φ(gj) − θ(gj)|∞ ≤ � ∀j = 1, . . . , t}. so, let � ∈ ] 0, s−r 2 [ and consider the families of neighborhoods of u and v of the form w(φ,x,f1, . . . ,fm,�)φ∈u and w(ψ,x,g1, . . . ,gn,�)ψ∈v respectively. then given any w(φ,x,f1, . . . ,fm,�), φ ∈ u and w(ψ,x,g1, . . . ,gn,�), ψ ∈ v we have w(φ,x,f1, . . . ,fm,�) ∩ w(ψ,x,g1, . . . ,gn,�) = ∅ hence u and v are two open subsets such that u ∩v = ∅. by construction mult(a,‖ . ‖) = u ∪v . consequently, u and v are two open subsets making a partition of mult(a,‖ . ‖), which by theorem 1.1, ends the proof. remark 2.14. the proof of theorem 1.3 consists of injecting the krasner-tae algebra [8] h(γ(a,r,s)) into a. acknowledgement: i am grateful to the referee for useful remarks. cubo 23, 1 (2021) idempotents in an 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[17] j. tate, “rigid analytic spaces”, inventiones mathematicae, vol. 12, pp. 257–289, 1971. introduction and main theorem the proofs a mathematical journal vol. 7, no 2, (201 221). august 2005. an introduction to the fractional fourier transform and friends a. bultheel 1 department of computer science k.u.leuven, belgium adhemar.bultheel@cs.kuleuven.ac.be h. mart́ınez national experimental univ. of guayana, port ordaz, state boĺıvar, venezuela hmartine@uneg.edu.ve abstract in this survey paper we introduce the reader to the notion of the fractional fourier transform, which may be considered as a fractional power of the classical fourier transform. it has been intensely studied during the last decade, an attention it may have partially gained because of the vivid interest in timefrequency analysis methods of signal processing, like wavelets. like the complex exponentials are the basic functions in fourier analysis, the chirps (signals sweeping through all frequencies in a certain interval) are the building blocks in the fractional fourier analysis. part of its roots can be found in optics and mechanics. we give an introduction to the definition, the properties and approaches to the continuous fractional fourier transform. 1the work of the first author is partially supported by the belgian programme on interuniversity poles of attraction, initiated by the belgian federal science policy office. the scientific responsibility rests with the authors. 202 a. bultheel and h. mart́ınez 7, 2(2005) resumen en este art́ıculo de prospección introducimos al lector en la noción de la transformada de fourier fraccional, que puede ser considerada como una potencia fraccional de la transformada de fourier clásica. ha sido objeto de intensos estudios durante la última década, que puede deberse parcialmente al interés respecto de los métodos de análisis tiempo-frecuencia en el proceso de señales, como es el caso de los wavelets. tal como las exponenciales complejas son las funciones básicas del análisis de fourier, los llamados chirps (señales que barren todas las frecuencias en un intervalo dado) son los elementos básicos del análisis de fourier fraccional. parte de sus oŕıgenes se pueden encontrar en la óptica y la mecánica. damos una introducción a la definición, las propiedades y acercamientos a la transformada de fourier fraccional. key words and phrases: fourier transform, fractional transforms, signal processing, chirp, phase space math. subj. class.: 42a38, 65t20 1 introduction the idea of fractional powers of the fourier operator appears in the mathematical literature as early as 1929 [32, 8, 11]. it has been rediscovered in quantum mechanics [19, 16], optics [17, 21, 2] and signal processing [3]. the boom in publications started in the early years of the 1990’s and it is still going on. a recent state of the art can be found in [22]. the outline of the paper is as follows. section 2 gives a motivation for our definition of the fractional fourier transform (frft) given in the next section. whereas in the classical fourier transform, the harmonics and the delta functions play a prominent role, these are for the frft replaced by a more general class of chirp functions introduced in section 4. the wigner distribution is a function that essentially gives the distribution of the energy of the signal in a time-frequency or phase plane. the effect of a frft can be effectively visualized with the help of this function. this is described in section 5. relations with the windowed or short time fourier transform, with wavelets and chirplets can be found in section 6. the frft may be seen as a special case of a more general linear canonical transform (lct). whereas the frft corresponds to a rotation of the wigner distribution in the time-frequency plane, the lct will correspond to any linear transform that can be represented by a unimodular 2 × 2 matrix. this is the subject of section 7. thus everything that is explained in this section will also hold for the fractional fourier transform. this includes computational aspects, filtering in the transform domain, generalization to higher dimension, etc. to define the lct in higher dimensions, we give a brief introduction to a group theoretic approach in section 8. we conclude by a section giving a quick review of some closely related transforms. because of page limitations, we shall have to refer 7, 2(2005) an introduction to the fractional fourier transform and friends 203 the reader for all the details to the literature. 2 the classical fourier transform we recall some of the definitions and properties that are related to the classical continuous fourier transform (ft) so that we can motivate our definition of the fractional fourier transform (frft) later. on an appropriate function space l like e.g., l2(ir), the classical ft operator f : f → f and its inverse are defined as f(ξ) = 1√ 2π ∫ ∞ −∞ f(x)e−iξxdx, f(x) = 1√ 2π ∫ ∞ −∞ f(ξ)eiξxdξ. (1) in signal processing applications, f is often a time depending signal so that x denotes time and ξ frequency. therefore f(x) is a time domain description of the signal and f(ξ) a frequency domain description. furthermore, it is immediately verified that (f2f)(x) = f(−x), (f3f)(ξ) = f(−ξ), and (f4f)(x) = f(x). this means that for a ∈ z we may identify fa with a rotation in the (x,ξ)-plane over an angle α = aπ/2. the idea of the frft is to define fa for any a ∈ ir. it will be useful to introduce some notation. let ra denote the rotation matrix ra = [ cos α sin α −sin α cos α ] = ejα, j = [ 0 1 −1 0 ] and suppose that (xa,ξa)t = ra(x,ξ)t , or switching to complex variables z = x−iξ, then za = eiαz. note that with this notation ξ = x1, and in general ξa = xa+1. the notation ra will also be used as an operator working on a function of two variables to mean raf(x,ξ) = f(xa,ξa) and to indicate that ra(x,ξ) = (xa,ξa). 3 the fractional fourier transform in [22] the authors give 6 different possible definitions of the frft and others can be found elsewhere. we prefer to follow an intuitive approach and define it as an extension of fa for a ∈ z to a ∈ ir. 3.1 eigenfunctions how to define fa for a ∈ ir? the key is the eigenvalue decomposition of f. it is known that f has a complete set of eigenvectors that span l2(ir). since f4 = i, the different eigenvalues are {1,−i,−1, i} each with an infinite dimensional eigenspace. the eigenvectors are thus not unique, but a possible choice of orthonormal eigenfunctions is given by the set of normalized hermite-gauss functions: φn(x) = 21/4√ 2nn! e−x 2/2hn(x), where hn(x) = (−i)nex 2dne−x2, d = −i d dx , 204 a. bultheel and h. mart́ınez 7, 2(2005) is an hermite polynomial of degree n. we have fφn = λnφn with λn = e−inπ/2. so, provided we properly define λa for a ∈ ir, we may set faφn = λanφn, and since {φn} is a complete set, this defines fa on l. if we define the analysis operator tφ, the synthesis operator t ∗φ and the scaling operator sλ as tφ : f �→ {cn = 〈f,φn〉2}, sλ : {cn} �→{λncn}, t ∗φ : {dn} �→ ∞∑ n=0 dnφn, (〈·, ·〉2 is the inner product in l2(ir)) then it is clear that we may write f = t ∗φ sλtφ and fa = t ∗φ saλtφ. (2) note that the operator tφ is unitary on l2(ir) and that t ∗φ is its adjoint. the formula (2) gives a general procedure to define the fractional power of any operator that has a complete set of eigenfunctions. this definition implies that fa can be written as a operator exponential fa = e−iαh = e−iaπh/2 where the hamiltonian operator h is given by h = 1 2 (d2 +u2−i) with d = −id/dx and u the shift operator of l2(ir) defined as (uf)(x) = xf(x) or u = fdf−1 (see [16, 19, 22]). the form of the operator h can be readily checked by differentiating the relation e−iαh ( e−x 2/2hn(x) ) = e−inα ( e−x 2/2hn(x) ) with respect to α, setting α = 0 and then using the differential equation (d + 2iu)dhn = 2nhn satisfied by the hermite polynomials. note that this form identifies fa as a unitary operator, and hence the parseval equality holds in l2(ir). several simple properties can now be derived, the most glamorous one being fafb = fa+b, which reflects the group structure of the rotations. 3.2 integral representation any function f ∈ l2(ir) can be expanded as f = ∑n 〈f,φn〉2 φn, so that after application of fa we have (faf)(ξ) = 〈f(x),∑n φn(x)λanφn(ξ)〉2, which identifies fa as an integral transform with kernel ka(ξ,x) = ∑ n φn(x)λ a nφn(ξ)/ √ 2π. for a = ±1 this reduces to the ft kernel k±1(ξ,x) = e∓ixξ/ √ 2π. for a �= ±1, this is not so simple. using the eigenvalues and eigenfunctions for the transform fa, we obtain ka(ξ,x) = ∞∑ n=0 e−inaπ/2hn(ξ)hn(x) 2nn! √ π e−(x 2+ξ2)/2 = 1√ π √ 1 −e−2iα exp { 2xξe−iα −e−2iα(ξ2 + x2) 1 −e−2iα } exp { −ξ 2 + x2 2 } 7, 2(2005) an introduction to the fractional fourier transform and friends 205 where in the last step we used mehler’s formula ([19, p. 244] or [4, eq. (6.1.13)]) ∞∑ n=0 e−inαhn(ξ)hn(x) 2nn! √ π = exp { 2xξe−iα−e−2iα(ξ2+x2) 1−e−2iα } √ π(1 −e−2iα) . to rewrite this expression, we observe that the following identities hold (they are easily checked) 2xξe−iα 1 −e−2iα = −ixξ csc α 1√ π √ 1 −e−2iα = e− i 2 ( π 2 α̂−α)√ 2π|sin α| e−2iα 1 −e−2iα + 1 2 = − i 2 cot α where α̂ = sgn (sin α). obviously, such relations only make sense if sin α �= 0, i.e., if α �∈ πz or equivalently a �∈ 2z. the branch of (sin α)1/2 we are using for sin α < 0 is the one with 0 < |α| < π. with these expressions, we obtain a more tractable integral representation of fa for a �∈ 2z viz. fa(ξ) := (faf)(ξ) = e− i 2 ( π 2 α̂−α)e i 2 ξ 2 cot α√ 2π|sin α| ∫ ∞ −∞ exp { −i xξ sinα + i 2 x2 cot α } f(x)dx, (3) where α̂ = sgn (sin α) and 0 < |α| < π. previously we defined (faf)(ξ) = f(ξ), if α = 0, and (faf)(ξ) = f(−ξ), if α = ±π. that is consistent with this integral representation because for these special values, it holds that lim�→0 fa+� = fa. thus, with this limiting property, we can assume that the integral representation holds on the whole interval |α| ≤ π. when |α| > π, the definition is taken modulo 2π and reduced to the interval [−π,π]. defining the frft via this integral transform, we can say that the frft exists for f ∈ l1(ir) (and hence in l2(ir)) or when it is a generalized function. indeed, in that case, the integrand in (3) is also in l1(ir) (or l2(ir)) or is a generalized function. thus the frft exists under exactly the same conditions as under which the ft exists. thus we have proved theorem 3.1 assume α = aπ/2 then the frft has an integral representation fa(ξ) := (faf)(ξ) = ∫ ∞ −∞ ka(ξ,x)f(x)dx. the kernel is defined as follows: for a �∈ 2z, then with α̂ = sgn (sin α), ka(ξ,x) = cα exp { −i xξ sin α + i 2 (x2 + ξ2) cot α } 206 a. bultheel and h. mart́ınez 7, 2(2005) with cα = e− i 2 ( π 2 α̂−α)√ 2π|sin α| = √ 1 − i cot α 2π . for a ∈ 4z the frft becomes the identity, hence k4n(ξ,x) = δ(ξ −x), n ∈ z and for a ∈ 2 + 4z, it is the parity operator: k2+4n(ξ,x) = δ(ξ + x), n ∈ z. if we restrict a to the range 0 < |a| < 2, then fa is a homeomorphism of l2(ir) (with inverse f−a). the last statement is proved in [16, p. 162]. it is directly verified that the kernel ka has the following properties. theorem 3.2 if ka(x,t) is the kernel of the frft as in theorem 3.1, then 1. ka(ξ,x) = ka(x,ξ) (diagonal symmetry) 2. k−a(ξ,x) = ka(ξ,x) (complex conjugate) 3. ka(−ξ,x) = ka(ξ,−x) (point symmetry) 4. ∫ ∞ −∞ ka(ξ,t)kb(t,x)dt = ka+b(ξ,x) (additivity) 5. ∫ ∞ −∞ ka(t,ξ)ka(t,x)dt = δ(ξ −x) (orthogonality) 4 the chirp function a chirp function (or chirp for short) is a signal that contains all frequencies in a certain interval and sweeps through it while it progresses in time. the interval can be swept in several ways (linear, quadratic, logarithmic,. . . ), but we shall restrict us here to the case where the sweep is linear. the complex exponential eiωt contains just one frequency: ω. this type of functions is essential in fourier analysis. in fact, they form a basis for the space of functions treated by the ft. indeed, the relation f(t) = 1√ 2π ∫ ∞ −∞ f(ω)e iωtdω can be seen as a decomposition of f into a (continuous) combination of the basis functions {eω(t) = eiωt}ω∈ir. on the other hand, if the frequencies of the signal sweeps linearly through the frequency interval [ω0,ω1] in the time interval [t0, t1], then we should have ω = ω0 + ω1−ω0 t1−t0 (t − t0). thus, a chirp will have the form exp{i(χt + γ)t}. the parameter χ is called the sweep rate. now consider the frft kernel ka(ξ,x), then, seen as a function of x and taking ξ as a parameter, this is a chirp with sweep rate 1 2 cot α. so, by rearranging the kernel (see also section 7), it can be seen that one way of describing a frft is 1. multiply by a chirp 2. do an ordinary ft 3. do some scaling 4. multiply by a chirp. 7, 2(2005) an introduction to the fractional fourier transform and friends 207 the inverse frft can be written as f(x) = ∫ ∞ −∞ fa(ξ)ψξ(x)dξ where ψξ(x) = k−a(ξ,x) is a chirp parameterized in ξ with sweep rate −1 2 cot α. thus we see that the role played by the harmonics in classical ft, is now taken by chirps, and the latter relation is a decomposition of f(x) into a linear combination of chirps with a fixed sweep rate determined by α. note also that in this expansion in chirp series, the basis functions are orthogonal by property 5 of the previous theorem. however, there is more. the chirps are in between harmonics and delta functions. indeed, up to a rotation in the time-frequency plane, the chirps are delta functions and harmonics. to see this, take the frft of a delta function δ(x−γ). that is (faδ(·−γ))(ξ) = ka(ξ,γ), which is a chirp with sweep rate 1 2 cot α. thus, given a (linear) chirp with sweep rate 1 2 cot α, we can transform it by a frft f−a into a delta function and hence by taking the ft of the delta function, we can take the chirp by a frft f1−a into an harmonic function. 5 the wigner distribution and the frft the relation between the multiplication operator u and the complex differentiation operator d, in the case of the classical fourier transform is uf = fd, which can be generalized as follows fa [ u d ] = [ ua da ] fa where [ ua da ] = [ cos α sin α −sin α cos α ][ u d ] . thus ua and da correspond to multiplication and complex differentiation in the variable of the frft domain. it are rotations of the usual u and d: (ua,da) = ra(u,d). this property is intuitively clear: by first applying u or d (i.e., multiplication in the x, respectively ξ direction) followed by a rotation in the (x,ξ)-plane must be the same as the rotation followed by the same operations applied to the rotated variables. because the rotation is an orthogonal transformation, we also have d2a +u2a = d2 +u2, so that the hamiltonian is rotation invariant: ha = 12 (d2a + u2a −i) = 12 (d2 +u2 −i) = h. the rotation property of the frft that has been mentioned several times now, can be visualised by the wigner distribution which is what will be defined next. let f be in l2(ir), then its wigner distribution or wigner transform wf is defined as (wf)(x,ξ) = 1√ 2π ∫ ∞ −∞ f(x + u/2)f(x−u/2)e−iξudu. its meaning is roughly speaking one of energy distribution of the signal in the timefrequency plane. indeed, setting f1 = ff, we have∫ ∞ −∞ (wf)(x,ξ)dξ = |f(x)|2 and ∫ ∞ −∞ (wf)(x,ξ)dx = |f1(ξ)|2, so that 1√ 2π ∫ ∞ −∞ ∫ ∞ −∞ (wf)(x,ξ)dξdx = ‖f‖2 = ‖f1‖2, which is the energy of the signal f. an important property of the frft is the following. 208 a. bultheel and h. mart́ınez 7, 2(2005) theorem 5.1 the wigner distribution of a signal and its frft are related by a rotation over an angle −α: (wfa)(x,ξ) = r−a(wf)(x,ξ) where α = aπ/2, fa = faf. equivalently ra(wfa)(x,ξ) = (wfa)(xa,ξa) = (wf)(x,ξ) with (xa,ξa) = ra(x,ξ). this theorem says that if we have the wigner distribution of f, then the wigner distribution of fa is obtained by rotating it clockwise over an angle α in the (x,ξ)plane. the proof is tedious but straightforward. for details see [3, p. 3087]. looking at figure 1, the result is in fact obvious since it just states that before and after a rotation of the coordinate axes, the wigner distribution is computed in two different ways taking the new variables into account, and that should of course give the same result. figure 1: wigner distribution of a signal f and the wigner distribution of its frft are related by a rotation. this implies for example∫ ∞ −∞ (wfa)(x,ξ)dξ = |fa(x)|2 and 1√ 2π ∫ ∞ −∞ ∫ ∞ −∞ (wfa)(x,ξ)dxdξ = ‖f‖2. the ambiguity function is closely related to the wigner distribution. its definition is (af)(x,ξ) = 1√ 2π ∫ ∞ −∞ f(u + x/2)f(u−x/2)e−iuξdu. thus it is like the wigner distribution, but now the integral is over the other variable. the ambiguity function and the wigner distribution are related by what is essentially a 2-dimensional fourier transform. whereas the wigner distribution gives an idea about how the energy of the signal is distributed in the (x,ξ)-plane, the ambiguity function will have a correlative interpretation. indeed (af)(x, 0) is the autocorrelation function of f and (af)(0,ξ) is the autocorrelation function of f1 = ff. 7, 2(2005) an introduction to the fractional fourier transform and friends 209 6 windowed transform, wavelets and chirplets the short time fourier transform or windowed fourier transform (wft) is defined as (fwf)(x,ξ) = 1√ 2π ∫ ∞ −∞ f(t)w(t−x)e−iξtdt where w is a window function. it is a local transform in the sense that the window function more or less selects an interval, centered at x to cut out some filtered information of the signal. so it gives information that is local in the time-frequency plane in the sense that we can find out which frequencies appear in the time intervals that are parameterized by their centers x. it can be shown that (fwf)(x,ξ) = e−ixξ(fw1f1)(ξ,−x) = e−ixξ(fw1f1)(x1,ξ1) where w1 = fw and f1 = ff. because of the asymmetric factor e−ixξ, it is more convenient to introduce a modified wft defined by (f̃wf)(x,ξ) = eixξ/2(fwf)(x,ξ). then it can be shown that (f̃wf)(x,ξ) = (f̃wafa)(xa,ξa). for more information on windows applied in the frft domain see [27]. from its definition fa(ξ) = ∫ ka(ξ,u)f(u)du, we get by setting x = ξ sec α and g(x) = fa(x/ sec α) g(x) = c(α)e−i4x 2 sin(2α) ∫ ∞ −∞ exp [ i 2 ( x−u tan1/2 α )2] f(u)du. c(α) is a constant that depends on α only. although, there are some characteristics of a wavelet transform, this can not exactly be interpreted as a genuine wavelet transform. we do have a scaling parameter tan1/2 α and a translation by u of the basic function ψ(t) = eit 2 but since ∫ ∞ −∞ ψ(x)dx �= 0 and it has no compact support, this is not really a wavelet. multiscale chirp functions were introduced in [5, 15]. a. bultan [6] has developed a so called chirplet decomposition which is related to wavelet package techniques. it is especially suited for the decomposition of signals that are chirps, i.e., whose wigner distribution corresponds to straight lines in the (x,ξ)-plane. the idea is that a dictionary of chirplets is obtained by scaling and translating an atom whose wigner distribution is that of a gaussian that has been stretched and rotated. so, we take a gaussian g̃(t) = π−1/4e−x 2/2 with wigner distribution (wg̃)(x,ξ) = (2/π)1/2 exp{−(x2 + ξ2)}. next we stretch it as g(x) = s−1/2g̃(x/s) giving (wg)(x,ξ) = (wg̃)(x/s,sξ). finally we rotate (wg) to give (wc)(x,ξ) with c = fag. the chirplet c depends on two parameters s and a and its main support 210 a. bultheel and h. mart́ınez 7, 2(2005) in the (x,ξ)-plane can be thought of as a stretched (by s) and rotated (by a) ellipse centered at the origin. to cover the whole (x,ξ)-plane, we have to tile it with shifted versions of this ellipse, i.e., we need the shifted versions (wg)(x − u,ξ − ν) corresponding to the functions c(x−u)eiνx. with these four-parameters ρ = (s,a,u,ν) we have a redundant dictionary {cρ}. the next step is to find a discretization of these 4 parameters such that the dictionary is complete when restricted to that lattice. it has been shown [31] that such a system can be found for a = 0 that is indeed complete, and the rotation does not alter this fact. if the discrete dictionary is {cn} with cn = cρn , then a chirplet representation of the signal f has to be found of the form f(x) = ∑ n ancn(x). such a discrete dictionary for a signal with n samples has a discrete chirplet dictionary with o(n2) elements. therefore a matching pursuit algorithm [14] can be adapted from wavelet analysis. the main idea is that among all the atoms in the dictionary the one that matches best the data is retained. this gives a first term in the chirplet expansion. the approximation residual is then again approximated by the best chirplet from the dictionary, which gives a second term in the expansion etc. this algorithm has a complexity of the order o(mn2 log n) to find m terms in the expansion. this is far too much to be practical. a faster o(mn) algorithm based on local optimization has been published [10]. this approach somehow neglects the nice logarithmic and dyadic tiling of the plane that made more classical wavelets so attractive. so this kind of decomposition will be most appropriate when the signal is a composition of a number of chirplets. such signals do exist like the example of a signal emitted by a bat which consists of 3 nearly parallel chirps in the (x,ξ)-plane. other examples are found in seismic analysis. for more details we refer to [6]. an example in acoustic analysis was given in [10]. 7 the linear canonical transform as we have seen, the frft is essentially a rotation in the (x,ξ)-plane. so, it can be characterized by a 2 × 2 rotation matrix which depends on one parameter, namely the rotation angle. it is a subgroup so(2) of the group gl(2) of 2 × 2 real invertible matrices. most of what has been said can be generalized to a more general linear transform, which is characterized by a general matrix m in the subgroup sl(2) = {m ∈ ir2×2 : det(m) = 1}. these generalizations are called linear canonical transforms (lct). 7.1 definition consider a 2×2 unimodular matrix (i.e., whose determinant is 1). such a matrix has 3 free parameters u,v,w which we shall arrange as follows m = [ a b c d ] = [ w v 1 v −v + uw v u v ] = [ u v −1 v v − uw v w v ]−1 = [ d −b −c a ]−1 . 7, 2(2005) an introduction to the fractional fourier transform and friends 211 the parameters can be recovered from the matrix by u = d b = 1 a ( 1 b + c ) , v = 1 b , w = a b = 1 d ( 1 b + c ) a typical example is the rotation matrix associated with rα where a = d = cos α and b = −c = sin α. let us call this matrix rα. although m ∈ ir2×2 is a matrix, we shall for typographical reasons often write m = (a,b,c,d). the linear canonical transform fm of a function f is an integral transform with kernel km (ξ,x) defined by km (ξ,x) = √ v 2πi e i 2 (uξ 2−2vξx+wx2) = 1√ 2πib e i 2b (dξ 2−2ξx+ax2). note that, just like in the case of the frft, there is some ambiguity since we have to choose the branch of the square root in the definition of the kernel. 7.2 effect on wigner distribution and ambiguity function note that if m is the rotation matrix rα, then the kernel km reduces almost to the frft kernel because m = rα implies u = w = cot α while v = csc α. hence frα = e−iα/2fa. if f denotes a signal, and fm its linear canonical transform, then the wigner transform gives (wfm )(ax + bξ,cx + dξ) = (wf)(x,ξ). (4) the latter equation can be directly obtained from the definition of linear canonical transform and the definition of wigner distribution. thus if rm is the operator defined by rmf(x) = f(mx), then w = rmwfm . note that this generalizes theorem 5.1, since (up to a unimodular constant factor which does not influence the wigner distribution) frα and fa are the same. similarly for the ambiguity function: a = rmafm . the group structure can be used to show that fm is unitary in l2(ir) and it holds that fafb = fc if and only if c = ab. 7.3 special cases when we restrict ourselves to real matrices m, there are several interesting special cases, the frft being one of them. others are • the fresnel transform: this is defined by gz(ξ) = eiπz/l√ ilz ∫ ∞ −∞ ei(π/lz)(ξ−z) 2 f(x)dx. this corresponds to the choice m = (1,b, 0, 1) with b = zl 2π , because, with this m we have gz(ξ) = eiπz/l(fmf)(ξ). 212 a. bultheel and h. mart́ınez 7, 2(2005) • dilation: the operation f(x) �→ gs(ξ) = √ sf(sξ), can be also obtained as a lct because with m = (1/s, 0, 0,s) we have gs(ξ) = √ sgn (s)(fmf)(ξ). • gauss-weierstrass transform or chirp convolution: this is obtained by the choice m = (1,b, 0, 1): (fmf)(ξ) = 1√ 2πib ∫ ∞ −∞ exp{i(x− ξ)2/2b}f(x)dx. • chirp (or gaussian) multiplication: here we take m = (1, 0,c, 1) and get (fmf)(ξ) = exp{icξ2/2}f(ξ). 7.4 on the computation of the lct to compute the lct, it is only in exceptional cases that the integral can be evaluated exactly. so in most practical cases, the integral will have to be approximated numerically. two forms depending on different factorizations of the m matrix are interesting for the fast computation or the lct and thus also for the frft. the first one reflects the decomposition [ a b c d ] = [ 1 0 (d− 1)/b 1 ][ 1 b 0 1 ][ 1 0 (a− 1)/b 1 ] (5) which means (see section 7.3) that the computation can be reduced to a chirp multiplication, followed by a chirp convolution, followed by a chirp multiplication. taking into account that the convolution can be computed in o(n log n) operations using the fast fourier transform (fft), the resulting algorithm is a fast algorithm. another interesting decomposition is given by [ a b c d ] = [ 1 0 db−1 1 ][ b 0 0 b−1 ][ 0 1 −1 0 ][ 1 0 b−1a 1 ] (6) and this is to be interpreted as a chirp multiplication, followed by an ordinary fourier transform (which can be obtained using fft), followed by a dilation, followed eventually by another chirp multiplication. again it is clear that this gives a fast way of computing the frft or lct. in figure 2, the effect of the lct on a unit square is illustrated showing the different steps when the matrix m, which is for this example m = (2, 0.5, 0, 1, 0.525), is decomposed as in (5) or as in (6). as we can see the two methods compute quite different intermediate results. in the example given there, it is clear that the second decomposition on the right stretches the initial unit square much more and shifts it over larger distances compared to first decomposition on the left. this is an indication that more severe numerical rounding errors are to be expected with the second way of computing than with the first one. the straightforward implementation of these steps may be a bit naive because for example in the frft case, the kernel may be highly oscillating for certain values of 7, 2(2005) an introduction to the fractional fourier transform and friends 213 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 figure 2: the effect of a lct on a square. left when the matrix m is decomposed as in (5) and right when it is decomposed as in (6). a. it is clear that those values should be avoided. therefore it is best to evaluate the frft only for a in the interval [0.5, 1.5] and to use the relation fa = ff1−a for a ∈ [0, 0.5) ∪ (1.5, 2]. a discussion in [20] follows the approach given by the first decomposition (5). 7.5 filtering in the lct domain one may now set up a program of generalizing all the properties that were given in the case of the frft to the lct. usually this does not pose a big problem and the generalization is smoothly obtained. we just pick one general approach to what could be called canonical filtering operations. for its definition, we go back to the classical fourier transform. if we want to filter a signal, then we have to compute a convolution of the signal and the filter. however, as we know, the convolution in the x-domain corresponds to a multiplication in the ξ-domain. thus the filtering operation is characterized by f∗g = f−1[(ff)(fg)]. the natural fractional generalization would then be to define a fractional convolution f ∗a g = (fa)−1[(faf)(fag)] and the canonical convolution would be f ∗m g = (fm )−1[(fmf)(fmg)]. clearly, if m = i or a = 1, the classical convolution is recovered. this definition has been used in many papers. see for example [18] and [22, p. 420]. similar definitions can be given in connection with correlation instead of convolution operations. the essential difference between convolution and correlation is a complex conjugate, so that a canonical correlation can be defined as f �m g = (fm )−1[(fmf)(fmg)∗]. one could generalize even more and define for example an operation like fm3 [(fm1f)(fm2g)] (see [24]). if we consider the convolution in the x-domain and the multiplication in the ξ214 a. bultheel and h. mart́ınez 7, 2(2005) domain as being dual operations, then we can ask for the notion of dual operations in the fractional or the canonical situation. a systematic study of dual operations has been undertaken in [12], but we shall not go into details here. the windowed fourier transform can be seen as a special case. indeed, as we have seen, applying a window in the x-domain corresponds to applying a transformed window in the xa-domain. so it may well be that in some fractional domain, it may be easier to design a window that will separate different components of the signal, or that can better catch some desired property of the signal because its spread is smaller in the transform domain [27]. also the hilbert transform which is defined as 1 π ∫ ∞ −∞ f(x) x−x′ dx ′ (7) (integral in the sense of principal value) corresponds to filtering in the x domain with a filter g(x) = 1/x. a somewhat different approach to the definition of a canonical convolution is taken in [1]. it is based on the fact that a classical convolution f ∗ g = f ∗0 g is an inner product of f with a time-inverted and shifted version of g: (f ∗0 g)(x) = 1√ 2π ∫ ∞ −∞ f(x′)g(x−x′)dx′ = 〈f(·),g∗(x−·)〉2 . if we denote a shift in the x-domain as (t0(x′)f)(x) = f(x− x′) and recalling that time-inversion is obtained by the parity operator f2, it is clear that g∗(x − x′) = (f2t0(x)g∗)(x′). so f ∗0 g = 〈 f,f2t0(x)g∗ 〉 2 . if we now define a canonical shift as tm (xm ) = (fm )−1t0(xm )fm that is, we transform the signal, shift it in the transform domain, and then transform back, then another definition of a canonical convolution could be (f ∗m g)(x) = 〈 f,f2tm (x)g∗ 〉 2 . it still has the classical convolution as a special case when m = i, but it is different from the previous definition. 8 groups and generalization to higher dimensions there is a nice interpretation of the lct as a group representation. the purpose of [28] is to find a unitary operator v on l2(irn) such that it has the effect that the wigner transform of vf is the wigner transform of f subject to a general linear transformation. the n-dimensional wigner transform is defined as (wf)(x, ξ) = 1 (2π)n/2 ∫ irn f(x + u/2)f(x − u/2)e−iξ·udu. the dot represents the standard inner product in irn. thus we want to find the unitary operator v = fm on l2(irn) for which a matrix m ∈ gl(2n) = {m ∈ ir2n×2n : det m �= 0} can be found such that w = rmwv, where as before (rmf)(x) = f(mx). gl(2n) is a lie group and some subgroups are sl(2n) = {m ∈ gl(2n) : det m = 1}, o(2n) = {m ∈ gl(2n) : mt m = 1} (1 is the identity 7, 2(2005) an introduction to the fractional fourier transform and friends 215 in irn×n) and so(2n) = sl(2n) ∩o(2n). a symplectic form on ir2n can be defined as a lie bracket: [f, g] = f t jg with f, g ∈ ir2n and j = [ 0 1 −1 0 ] . the symplectic group sp(n) is the group of real 2n× 2n matrices that leave a symplectic form invariant, i.e., that satisfy mt jm = j. this implies that a symplectic matrix has determinant ±1. we have in fact sp(n) ⊂ sl(2n) with equality if n = 1. the heisenberg group hn is identified with ir n × irn × ir with group law (x1, ξ1, t1)(x2, ξ2, t2) = (x1 + x2, ξ1 + ξ2, t1 + t2 + (ξ1 · x2 − x1 · ξ2)/2). a representation of a topological group g on a hilbert space h is a mapping μ from g to the space b(h) of bounded operators on h such that μ(x)μ(y) = μ(xy), μ(e) = i with e the identity in g and i the identity operator in b(h) and x → μ(x)f is a continuous mapping for all f ∈ h. the representation μ is unitary if b(h) can be replaced by u(h), the unitary operators on h. and μ is called irreducible if {0} and h are the only invariant subspaces of h under the group action μ(x) for all x ∈ g. it can be shown that a unitary irreducible representation of hn in the space l2(ir n) is the schrödinger representation defined as (μ(x, ξ, t)f)(u) = ex·uei(t+x·ξ/2)f(u + ξ). it takes a couple of lines to show that the relation with the wigner distribution is that we can write (wf)(x, ξ) = [f〈μ(·, ·, 0)f,f〉2](x, ξ) where 〈·, ·〉2 is the inner product in l2(irn) and f the 2n-dimensional ft acting on the variables indicated by a dot. given a unitary v ∈ u(l2(irn)), another equivalent unitary representation would be given by ρ(h) = v∗μ(h)v for all h ∈ hn so that (wvf)(x, ξ) = [f〈ρ(·, ·, 0)f,f〉2](x, ξ) = 1 (2π)n/2 ∫ irn ∫ irn 〈ρ(u, v, 0)f,f〉2 e−iu·xe−iv·ξdu dv. this implies that if there is a matrix m ∈ ir2n×2n such that μ(g, 0) = ρ(mg, 0) for all g ∈ h′n = {g ∈ ir2n : (g, t) ∈ hn}, then by a change of variables in the last expression, we get (wvf)(g) = |det m|(wf)(mg). now consider the subgroup of u(l2(irn)) g = {v ∈ u(l2(irn)) : ∀(g, t) ∈ ir2n+1,∃g′ ∈ ir2n s.t. v∗μ(g, t)v = μ(g′, t)} the g′ is uniquely defined so that there is a homomorphism ν(v) : ir2n → ir2n given by ν(v)g = g′. this ν is a continuous mapping from g onto sp(n) in the subspace 216 a. bultheel and h. mart́ınez 7, 2(2005) topology of g ⊂ u(l2(irn)) with kernel {ci : |c| = 1}. this means that ν−1 is only defined up to a unimodular factor. we obtain the metaplectic group which is a twofold covering of the symplectic group. this shows up in the formulas in the form of a square root for which the sign has to be chosen. with these tools, our original problem of finding a unitary v that causes an arbitrary linear transform of the wigner distribution, can be solved. it requires some more lines to show that w = rmwv, if and only if v ∈ g and m = ν(v)−1 ∈ sp(n). several simple examples from the group g can be found. • fourier transform for example, the n-dimensional ft f satisfies all the properties and ν(f) = jt . • dilation a second example is the dilation operator: d∗bμ(x, ξ, t)db = μ(bt x, b−1ξ, t) with b ∈ gl(n). we have now ν(db) = [ b−1 0 0 bt ] . note that if b is symmetric then (dbf)(x) = (det b)−1/2f(b−1x). • chirp multiplication a third example is a chirp multiplication c∗s μ(x, ξ, t)cs = μ(x + sξ, ξ, t). ν(cs) = [ 1 0 s 1 ] . with an n-dimensional chirp defined as cs(x) = exp{ i2 xt sx}, and the effect is that (csf)(x) = cs(x)f(x). it is clearly no restriction if we assume that s is symmetric. in view of the decomposition of the lct in the scalar case, it is natural to define the n-dimensional lct as fm = ccdb−1dbfcb−1a with |c| = 1. it is represented by a matrix ν(cdb−1dbfcb−1a) = [ a b c d ] . the special case of the separable n-dimensional frft corresponds to a = d = diag(cos α1, . . . , cos αn) and b = −c = diag(sin α1, . . . , sin αn). then m ∈ sp(n) ∩ so(2n), the orthogonal symplectic group. for more details on this approach see e.g. [9, 28] and for the integral representation [29]. 9 other transforms probably motivated by the success of the frft and the lct, quite some effort has been put in the design of fractional versions of related classical transforms. 7, 2(2005) an introduction to the fractional fourier transform and friends 217 9.1 radial canonical transforms it should be clear that for problems with circular symmetry, this symmetry should be taken into account when defining the transforms. take for example the 2-dimensional case. instead of cartesian (x,y) coordinates, one should switch to polar coordinates so that, because of the symmetry, the transform will only depend on the radial distance. for example, it is well known that the hankel transform appears naturally as a radial form of the (2-sided) laplace transform [34, sec. 8.4]. giving directly the n-dimensional formulation, we shall switch from the n-dimensional variables x and ξ to the scalar variables x = ‖x‖ and ξ = ‖ξ‖, and the n-dimensional lct will become canonical hankel transforms [33, 30]. it is a one-sided integral transform∫ ∞ 0 km (ξ,x)f(x)dx with kernel km (ξ,x) = x n−1 e − π2 ( n2 +ν) b (xξ)1−n/2 exp { i 2b (ax2 + dξ2) } jn/2+ν−1 ( xξ b ) , where jν is the bessel function of the first kind of order ν. the fractional hankel transform is a special case of the canonical hankel transform when the matrix m is replaced by a rotation matrix. 9.2 fractional hilbert transform the definition of the hilbert transform has been given before in (7). note that the convolution defining the transform can be characterized by a multiplication with −isgn (ξ) in the fourier domain. since −isgn (ξ) = e−iπ/2h(ξ) + eiπ/2h(−ξ) with h the heaviside step function: h(ξ) = 1 for ξ ≥ 0 and h(ξ) = 0 for ξ < 0, we can now define a fractional hilbert transform as (fm )−1[(e−iφh(ξ) + eiφh(−ξ))(fmf)(ξ)], with m the rotation matrix m = ra. for further reading see [35, 13, 23, 7]. 9.3 cosine, sine and hartley transform while in the classical fourier transform, the integral is taken of f(x)e−iξx, one shall in the cosine, sine, and hartley transform replace the complex exponential by cos(ξx), sin(ξx) or cas(ξx) = cos(ξx) + sin(ξx) respectively. since cos and sin are the real and imaginary part of the complex exponential, one might think of defining the fractional cosine and sine transforms by replacing the kernel in the frft by its real or imaginary part. however, this will not lead to index additivity for the transforms. we could however use the general fractionalization procedure given in (2). we just have to note that the hermite-gauss eigenfunctions are also eigenfunctions of the cosine and sine transform, except that for the cosine transform, the odd eigenfunctions will correspond to eigenvalues zero and for the sine transform, the even eigenfunctions will correspond to eigenvalue zero. this implies that the odd part of f will be killed by the cosine transform. so, the cosine transform will not be invertible unless we restrict ourselves 218 a. bultheel and h. mart́ınez 7, 2(2005) to the set of even functions. a similar observation holds for the sine transform: it can only be invertible when we restrict the transform to the set of odd functions. this motivates the habit to define sine and cosine transforms by one sided integrals over ir+. see [26]. the bottom line of the whole fractionalization process is that to obtain the good fractional forms of these operators we essentially have to replace in the definition of the frft the factor eiξx in the kernel of the transform by cos(ξx), sin(ξx) or cas(ξx) to obtain the kernel for the fractional cosine, sine or hartley transforms respectively. in the case of the cosine or sine transform, the restriction to even or odd functions implies that we need only to transform half of the function, which means that the integral over ir can be replaced by two times the integral over ir+. besides the fractional forms of these operators there are also canonical forms for which we refer to [26]. also here simplified forms exist [25, 26]. 9.4 other transforms the list of transforms that have been fractionalized is too long to be completed here. some examples are: laplace, mellin, hadamard, haar, gabor, radon, derivative, integral, bragman, barut-girardello,. . . the fractionalization procedure of (2) can be used in general. this means the following. if we have a linear operator t in a complex separable hilbert space with inner product 〈·, ·〉2 and if there is a complete set of orthonormal eigenvectors φn with corresponding eigenvalues λn, then any element in the space can be represented as f = ∑∞ n=0 anφn, an = 〈f,φn〉2, so that (t f) =∑∞ n=0 anλnφn. the fractional transform can the be defined as (t af)(ξ) = ∞∑ n=0 anλ a nφn(ξ) = ∞∑ n=0 λan 〈f,φn〉2 φn(ξ) = 〈f,ka(ξ, ·)〉2 , where ka(ξ,x) = ∞∑ n=0 λ a nφn(ξ)φn(x). of course a careful analysis will require some conditions like for example if it concerns the hilbert space l2μ(i) of square integrable functions on an interval i with respect to a measure μ, then we need ka(ξ, ·) to be in this space, which means that∑∞ n=0 |λn|2a|φn(ξ)|2 < ∞ for all ξ. in view of the general development for the construction of fractional transforms, it is clear that the main objective is to find a set on orthonormal eigenfunctions for the transform that one wants to “fractionalize”. there were several papers that give eigenvalues and eigenvectors for miscellaneous transforms. zayed [36] has given an alternative that uses instead of the kernel ka(ξ,x) =∑ n λ a nφ ∗ n(x)φn(ξ) the kernel ka(ξ,x) = lim |λ|→1− ∑ n |λ|einαφn(ξ)∗φn(x). 7, 2(2005) an introduction to the fractional fourier transform and friends 219 thus λan is replaced by |λ|neinα and the φn can be any (orthonormal) set of basis functions. in this way he obtains fractional forms of the mellin, and hankel transforms, but also of the riemann-liouville derivative and integral, and he defines a fractional transform for the space of functions defined on the interval [−1, 1] based on jacobi-functions which play the role of the eigenfunctions. to the best of our knowledge, a further generalization by taking a biorthogonal system spanning the hilbert space, which is very common in wavelet analysis, has not yet been explored in this context. received: june 2003. revised: march 2004. references [1] o. akay and g. f. boudreaux-bartels, fractional convolution and correlation via operator methods and an application to detection of linear fm signals. ieee trans. sig. proc., 49(5):979 –993, 2001. 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[9] g. b. folland, harmonic analysis in phase space. annals of mathematical studies. princeton university press, new jersey, 1989. 220 a. bultheel and h. mart́ınez 7, 2(2005) [10] r. gribonval, fast matching pursuit with a multiscale dictionary of gaussian chirps. ieee trans. sig. proc., 49(5):994 –1001, 2001. [11] h. kober, wurzeln aus der hankelund fourier und anderen stetigen transformationen. quart. j. math. oxford ser., 10:45–49, 1939. [12] p. kraniauskas, g. cariolaro, and t. erseghe, method for defining a class of fractional operations. ieee trans. sig. proc., 46(10):2804–2807, 1998. [13] a. w. lohmann, d. mendlovic, and z. zalevsky, fractional hilbert transform. opt. lett., 21(4):281–283, 1996. [14] s. mallat and z. zhang, matching pursuit with time-frequency dictionaries. technical report, courant institute of mathematical sciences, 1993. [15] s. mann and s. haykin, the chirplet transform: physical considerations. ieee trans. sig. proc., 43:2745–2761, 1995. [16] a. c. mcbride and f. h. kerr, on namias’s fractional fourier transforms. ima j. appl. math., 39:159–175, 1987. [17] d. mendlovic and h. m. ozaktas, fractional fourier transforms and their optical implementation: i. j. opt. soc. amer. a, 10:1875–1881, 1993. [18] d. mustard, fractional convolution. j. australian math. soc. b, 40:257– 265, 1998. [19] v. namias, the fractional order fourier transform and its application in quantum mechanics. j. inst. math. appl., 25:241–265, 1980. [20] h. m. ozaktas, m. a. kutay, and g. bozdaği, digital computation of the fractional fourier transform. ieee trans. sig. proc., 44:2141–2150, 1996. [21] h. m. ozaktas and d. mendlovic, fractional fourier transforms and their optical implementation: ii. j. opt. soc. amer. a, 10:2522–2531, 1993. [22] h. m. ozaktas, z. zalevsky, and m. a. kutay, the fractional fourier transform. wiley, chichester, 2001. [23] a. c. pei and p. h. wang, analytical design of maximally flat fir fractional hilbert transformers. signal processing, 81:643–661, 2001. [24] s. c. pei and j. j. ding, simplified fractional fourier transforms. j. opt. soc. amer. a, 17:2355–2367, 2000. 7, 2(2005) an introduction to the fractional fourier transform and friends 221 [25] s. c. pei and j. j. ding, fractional, canonical, and simplified fractional cosine transforms. in proc. int. conference on acoustics, speech and signal processing. ieee, 2001. [26] s. c. pei and j. j. ding, eigenfunctions of linear canonical transform. ieee trans. sig. proc., 50(1):11–26, 2002. [27] l. stankovic, t. alieva, and m. j. bastiaans, time-frequency signal analysis based on the windowed fractional fourier transform. signal processing, 83(11):2459–2468, 2003. [28] h. g. ter morsche and p. j. oonincx, integral representations of affine transformations in phase space with an application to energy localization problems. technical report pna-r9919, cwi, amsterdam, 1999. [29] h. g. ter morsche and p. j. oonincx, on the integral representations for metaplectic operators. j. fourier anal. appl., 8(3):245–257, 2002. [30] a. torre, linear and radial canonical transforms of fractional order. j. comput. appl. math., 153:477–486, 2003. [31] b. torresani, wavelets associated with representations of the affine weyl-heisenberg group. j. math. phys., 32:1273–1279, 1991. [32] n. wiener, hermitian polynomials and fourier analysis. j. math. phys., 8:70–73, 1929. [33] k. b. wolf, canonical transforms. ii. complex radial transforms. j. math. phys. a, 15(12):2102–2111, 1974. [34] k. b. wolf, integral transforms in science and engineering. plenum, new york, 1979. [35] a. i. zayed, hilbert transform associated with the fractional fourier transform. ieee sig. proc. letters, 5:206–209, 1998. [36] a. i. zayed, a class of fractional integral transforms: a generalization of the fractional fourier transform. ieee trans. sig. proc., 50(3):619 –627, 2002. cubo, a mathematical journal vol.22, no¯ 01, (39–53). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100039 a sufficiently complicated noded schottky group of rank three rubén a. hidalgo 1 departamento de matemática y estad́ıstica, universidad de la frontera, temuco, chile. ruben.hidalgo@ufrontera.cl abstract in 1974, marden proved the existence of non-classical schottky groups by a theoretical and non-constructive argument. explicit examples are only known in rank two; the first one by yamamoto in 1991 and later by williams in 2009. in 2006, maskit and the author provided a theoretical method to construct non-classical schottky groups in any rank. the method assumes the knowledge of certain algebraic limits of schottky groups, called sufficiently complicated noded schottky groups. the aim of this paper is to provide explicitly a sufficiently complicated noded schottky group of rank three and explain how to use it to construct explicit non-classical schottky groups. resumen en 1974, marden demostró la existencia de grupos de schottky no-clásicos con un argumento teórico y no-constructivo. se conocen ejemplos expĺıcitos solo en rango dos; el primero por yamamoto en 1991 y después por williams en 2009. en 2006, maskit y el autor entregaron un método teórico para construir grupos de schottky no-clásicos en cualquier rango. el método asume el conocimiento de ciertos ĺımites algebraicos de grupos de schottky, llamados grupos de schottky anodados suficientemente complicados. el objetivo de este paper es dar un grupo de schottky anodado suficientemente complicado expĺıcitamente de rango tres y explicar cómo usarlo para construir grupos de schottky no-clásicos expĺıcitos. keywords and phrases: riemann surfaces; schottky groups. 2010 ams mathematics subject classification: 30f40, 30f10. 1partially supported by projects fondecyt 1190001. http://doi.org/10.4067/s0719-06462020000100039 40 rubén a. hidalgo cubo 22, 1 (2020) 1 introduction a kleinian group g is called a schottky group of rank g ≥ 2 if it is generated by loxodromic transformations a1, . . . ,ag ∈ psl2(c) such that there is a collection of 2g pairwise disjoint simple loops α1,α ′ 1,α2,α ′ 2, . . . ,αg,α ′ g on the riemann sphere ĉ, all of them bounding a common domain d of connectivity 2g, with aj(αj) = α ′ j and aj(d) ∩ d = ∅. the above set of generators is called geometrical and the above loops a fundamental set of loops for g. it is well known that g is a free group of rank g and that d is a fundamental domain for it. in [3] chukrow proved that every set of g generators of g is geometrical. we say that g is a classical schottky group if it has a set of geometrical generators, called a classical set of generators, for which we may find a fundamental set of loops being circles. classical schottky groups were firstly considered by schottky around 1882. in general, a classical schottky group may have non-classical set of generators. examples of classical schottky groups are given by the finitely generated purely hyperbolic fuchsian groups representing a closed surface with holes [2]. moreover, if such a fuchsian group is a two generator group representing a torus with one hole, then every pair of generators is a classical set of generators [21]. in 1974, marden [14] provided the existence of non-classical schottky groups (his proof is non-constructive). in 1975, zarrow [27] claimed to have constructed an explicit example of a nonclassical schottky group of rank two, but it was lately noted by sato in [22] to be incorrect. the first explicit (correct) construction was provided by yamamoto [26] in 1991 and in 2009 another example was provided by williams in his ph.d. thesis [24], both for rank two. it seems that, for rank at least three, there is not explicit example in the literature. in [7], maskit and the author described a theoretical method to construct non-classical schottky groups in any rank g ≥ 2. the idea is to consider certain kleinian groups, obtained as geometrically finite algebraic limits of schottky groups of rank g, called sufficiently complicated noded schottky groups of rank g (see section 2). in this paper (see section 3) we provide an explicit construction of a sufficiently complicated noded schottky group of rank three and we used it to describe how to obtain a infinite family (one-dimensional) non-classical schottky group of rank three. to finish this introduction, and as a matter of completeness, let us mention another conjecture related to classical schottky groups. if ω is the region of discontinuity of a schottky group g of rank g, then it is a connected set and ω/g is a closed riemann surface of genus g. conversely, if s is a closed riemann surface, then there is a schottky group g such that s and ω/g are isomorphic (koebe’s uniformization theorem). as we have the existence of non-classical schottky group, it might be that g is non-classical. a conjecture (due to bers) asserts that we may chose g to be classical. some positive answers were obtained by bobenko [1], koebe [11], maskit [17], seppälä [23] (for the case in which the surface admits antiholomorphic involutions with fixed points) and cubo 22, 1 (2020) a sufficiently complicated noded schottky . . . 41 mcmullen [13] (for the case in which the surface has sufficiently many shorts geodesics). recently, hou [8, 9] have announced a proof of this conjecture (by using haussdorf dimension of the limit set of kleinian groups) and another approach in [6] (by using belyi curves). 2 sufficiently complicated noded schottky groups 2.1 noded schottky groups a noded schottky group of rank g ≥ 2 is geometrically defined as follows. consider a collection of pairwise disjoint open topological discs d1, d ′ 1, . . . ,dg and d ′ g on the riemann sphere ĉ so that the corresponding boundaries α̂1 = ∂d1, α̂ ′ 1 = ∂d ′ 1, . . . , α̂g = ∂dg, α̂ ′ g = ∂d ′ g are simple loops and they only intersect in at most finitely many points. let â1, . . . , âg be möbius transformations such that âj(α̂j) = α̂ ′ j and âj(dj)∩d ′ j = ∅, for each j = 1, . . . ,g. observe that the transformation âj may only be loxodromic or parabolic. the group ĝ, generated by these transformations, is a kleinian group isomorphic to a free group of rank g. if p is a point of intersection of two of the above loops, then either it is a fixed point of a parabolic transformation of ĝ or it has trivial ĝ-stabilizer. in the second situation, one may deform in a suitable manner these loops in order to avoid the intersection at p and not adding extra intersections. in this way, two possibilities appear (up to performing the above deformation), either: (i) g is a schottky group of rank g or (ii) there are intersection points, each of them being a fixed point of a parabolic transformation of ĝ. in case (ii) we say that ĝ is a noded schottky group of rank g; we call the above set of loops a fundamental set of loops and the generators a set of geometrical generators. remark 1. in [18], as an application of the klein-maskit’s combination theorems, it was noted that a noded schottky group ĝ is geometrically finite, that each of its parabolic elements is a conjugate of a power of one of the transformations fixing a common point of two of the fundamental loops, and that the complement d̂ of the union of the closures of d1,d ′ 1, . . . ,dg, d ′ g is a fundamental domain for ĝ. different as for the case of schottky groups, not every set of free generators of a noded schottky group is necessarily geometrical. 2.2 the extended region of discontinuity if ω is the region of discontinuity of a noded schottky group ĝ of rank g, then by adding to it the parabolic fixed points of ĝ, and with the appropriate cusped topology (see [5, 12]), we obtain its extended region of discontinuity ω+; it happens that s+ = ω+/ĝ is a stable riemann surface of genus g. in the case that the number of nodes of s+ is 3g − 3, we say that ĝ is a maximal noded schottky group (in this case, there are exactly 2g − 2 connected components of the complement of the nodes of s+, each one being a triple-punctured sphere). in [4] it was observed that every stable riemann surface of genus g is obtained as above; so every point of the deligne-mumford 42 rubén a. hidalgo cubo 22, 1 (2020) figure 1—a stable riemann surface of genus 3 with 3 nodes figure 2—a stable riemann surface of genus 3 with 6 nodes compactification of the moduli space of genus g can be realized by a suitable noded schottky group of rank g. 2.3 neoclassical noded schottky groups a noded schottky group for which there is a set of geometrical generators admitting a fundamental set of loops all of which are euclidean circles is called neoclassical; the corresponding set of generators is called a neoclassical set of generators. in [7] it was proved that if g is a noded schottky group such that ω+/g is a stable riemann surface as in figure 1, then it cannot be neoclassical (this should be still true for every noded schottky group of rank g ≥ 4 whose corresponding stable riemann surface has g +1 components, one being of genus zero and the others being of genus one). 2.4 sufficiently complicated noded schottky groups the space of deformations of a schottky group of rank g, denoted by salg, is a subset of the representation space of the free group of rank g in psl2(c), modulo conjugation. regard h 3 as being the set {(z,t) : z ∈ c,t > 0 ∈ r}. we likewise identify c with the boundary of h3, except for the point at infinity; that is, we identify c with {(z,t) : t = 0}. 2.4.1 the relative conical neighbourhood of a noded schottky group let ĝ be a noded schottky group of rank g ≥ 2, with a set of geometrical generators â1, . . . , âg, and corresponding fundamental set of loops α̂1, . . . , α̂ ′ g, these being the corresponding boundary loops of a collection of pairwise disjoint open discs d1, . . . ,d ′ g. the complement of the closures of these discs is a fundamental domain d̂ for ĝ. let p̂1, . . . , p̂q be a maximal set of primitive parabolic elements of ĝ generating non-conjugate cyclic subgroups, where q ≥ 1 (we may assume the fix point of these parabolic transformations to be contained in the intersection of two fundamental loops). we denote by ω(ĝ) its region of discontinuity and by ω+(ĝ) its extended region of discontinuity. cubo 22, 1 (2020) a sufficiently complicated noded schottky . . . 43 next, we proceed to recall a construction done in [7] of a one-real family of schottky groups gτ whose geometric limit is ĝ. (i) the infinite shoebox construction for each i = 1, . . . ,q, choose a particular möbius transformation hi conjugating p̂i to the transformation p(z) = z + 1 and consider the renormalized group hiĝh −1 i . for this group, there is a number τ0 > 1 so that the set {| im(z)| ≥ τ0} is precisely invariant under the stabilizer stab(∞) of ∞ in the group hiĝh −1 i . in this normalization, for each parameter τ, with τ > τ0, we define the infinite shoebox to be the set b0,τ = {(z,t) : | im(z)| ≤ τ,t ≤ τ}. since τ0 > 1, we easily observe that for every τ > τ0, the complement of b0,τ in h 3 ∪ c is precisely invariant under stab(∞) ⊂ hiĝh −1 i , where we are now regarding möbius transformations as hyperbolic isometries, which act on the closure of h3. then for ĝ, the infinite shoebox with parameter τ at zi, the fixed point of p̂i, is bi,τ = h −1 i (b0,τ). if p̂ is any parabolic element of ĝ, conjugate to some power of p̂i, then the corresponding infinite shoebox at the fixed point of p̂, is given by t(bi,τ), where p̂ = tp̂it −1. it was observed in [19] that, for each fixed τ > τ0, ĝ acts as a group of conformal homeomorphisms on the expanded regular set bτ = ⋂ â(bi,τ), where the intersection is taken over all â ∈ ĝ and all i = 1, . . . ,q. further, ĝ acts as a (topological) schottky group (in the sense of our geometrical definition) on the boundary of bτ. each parabolic p̂ ∈ ĝ appears to have two fixed points on the boundary of bτ; that is, p̂, as it acts on the boundary of bτ, appears to be loxodromic. the flat part of bτ is the intersection of bτ with the extended complex plane ĉ. the complement of the flat part (on the boundary of bτ) is the disjoint union of 3-sided boxes, where each box has two vertical sides (translates of the sets {im(z) = ±τ,0 < t < τ}) and one horoball side (a translate of the set {| im(z)| ≤ τ,t = τ}). for each i = 1. . . ,q and for each integer n ≥ 1, we set bi,τ,n = h −1 i (b0,τ ∩ {| re(z)| ≤ n}) and b τ,n = ⋂ â∈ĝ â(bi,τ,n). the truncated flat part of bτ,n is the intersection bτ,n ∩ ĉ. the boundary of the truncated flat part near a parabolic fixed point, renormalized so as to lie at ∞, is a euclidean rectangle. let us renormalize ĝ so that ∞ ∈ ω(ĝ). then, for each τ > τ0, there is a conformal map f τ, mapping the boundary of bτ to ĉ, and conjugating ĝ onto a schottky group gτ, where fτ is defined by the requirement that, near ∞, fτ(z) = z+o(|z|−1). the group gτ depends on the choice of the möbius transformations h1, . . . ,hq as well as on the choice of τ. the elements a τ 1 = f τâ1(f τ)−1, . . . ,aτg = f τâg(f τ)−1 provide a set of free generators for the schottky group gτ. (ii) vertical projection loops next, we proceed to construct a fundamental set of loops for gτ for the above generators in terms of the fundamental set of loops for ĝ. in [19] it was shown that, with the above normalization, fτ converges to the identity i, uniformly on compact subsets of ω(ĝ), and, for each j ∈ {1, . . . ,g}, aτj converges to âj, as τ → ∞. in particular, if we fix τ0, and fix n, then f τ → i uniformly on 44 rubén a. hidalgo cubo 22, 1 (2020) compact subsets of the truncated flat part of bτ0,n. the boundary of bτ0,n consists of a disjoint union of quadrilaterals with circular sides. after renormalization, the part of the boundary of bτ0,n corresponding to {| im(z)| = τ0} is the horizontal part of the boundary, while the part of the boundary corresponding to {| re(z)| = n} is the vertical part of the boundary. we make a fixed choice of the conjugating maps, hi, i = 1, . . . ,q, and we fix a choice of the parameter τ > τ0 in the above construction. we may deform all the loops α̂i and α̂ ′ i, within ω +(ĝ) to an equivalent fundamental set of loops, with the same geometric generators â1, . . . , âg, so that, after appropriate renormalization, each connected component of each of the deformed loops appears, in each component of the complement of the flat part of bτ, as a pair of half-infinite euclidean vertical lines, one in {im(z) ≥ τ}, the other in {im(z) ≤ −τ}, both with the same real part (the technical details of such a deformation can be found in [7]). we call such a deformed loops the vertical projection loops. these vertical projection loops, which we still denoting as α̂1, . . . , α̂ ′ g, yields a fundamental set of loops, ατ1, . . . ,α ′ g τ for the generators aτ1, . . . ,a τ g of the schottky group gτ. (iii) the relative conical neighborhood the relative conical neighborhood of ĝ is to be defined as the set of all marked schottky groups gτ = 〈aτ1, . . . ,a τ g〉, with the fundamental set of loops α τ 1, . . . ,α ′ g τ , as constructed above. remark 2. recall that we are assuming that ∞ is an interior point of the flat part corresponding to τ0, and f τ(z) = z+o(|z|−1) near ∞. as, with these normalizations, fτ → i uniformly on compact subsets of ω(ĝ), we obtain that gτ → ĝ algebraically. it now follows, from the jørgensen-marden criterion [10], that gτ → ĝ geometrically and that each relative conical neighborhood contains infinitely many distinct marked schottky groups. it is also easy to see, as in [19], that, for each primitive parabolic element p̂ ∈ ĝ, as τ → ∞, there is a corresponding geodesic on sτ = ω(gτ)/gτ whose length tends to zero. it follows that each relative conical neighborhood of a noded schottky group contains schottky groups representing infinitely many distinct riemann surfaces. 2.4.2 pinchable loops of schottky groups let g be a schottky group of rank g ≥ 2, with generators a1, . . . ,ag, and let π : ω(g) → s be a regular covering with deck group g. (iv) pinchable loops let γ1, . . . ,γq be a set of simple disjoint geodesics loops on s. each γj corresponds to a conjugacy class of a cyclic subgroup of g (including the trivial subgroup) by the lifting under π; let 〈wj〉 be a representative of such a class. if these q cyclic subgroups are non-trivial, they are pairwise non-conjugated in g and the generators wj are non-trivial powers in g (i.e., there is no tj ∈ g so cubo 22, 1 (2020) a sufficiently complicated noded schottky . . . 45 that wj = t mj j for some mj ≥ 2), then we say that this set of geodesics is pinchable in g. remark 3. (1) it was shown in [20] (see also yamamoto [25]) that if γ1, . . . ,γq is a set of pinchable simple disjoint geodesics loops on s in g, defined by the words w1, . . . ,wq, as above, then there is a noded schottky group ĝ, and there is an isomorphism ψ : g → ĝ, where ψ(w1), . . . ,ψ(wq), and their powers and conjugates, are exactly the parabolic elements of ĝ. more precisely, it was shown in [20] that there is a path in schottky space, salg, which converges to a set of generators for ĝ, along which the lengths of the geodesics, γ1, . . . ,γq, all tend to zero. (2) on the other direction, let us consider a noded schottky group ĝ of rank g ≥ 2, with a set of geometrical generators â1, . . . , âg and corresponding fundamental set of loops α̂1, . . . , α̂ ′ g. let g τ be a schottky group of rank g with fundamental set of loops ατ1, . . . ,α ′ g τ and generators aτ1, . . . ,a τ g, in a relative conical neighborhoodof ĝ as previously described in section 2.4.1. let s = ω(gτ)/gτ be the closed riemann surface of genus g represented by gτ, and let vi ⊂ s be the projection of α τ i , i = 1, . . . ,g. then v1, . . . ,vg is a set of g homologically independent simple disjoint loops on s. let ψ : gτ → ĝ be the isomorphism defined by aτi 7→ âi, i = 1, . . . ,g. the elements of g τ which are sent to parabolic elements of ĝ are called the pinched elements of gτ. there are simple disjoint geodesics γ1, . . . ,γq on s, defined by pinched elements of g τ, given by the words w1, . . . ,wq in the generators aτ1, . . . ,a τ g, so that every parabolic element of ĝ is a power of a conjugate of one of their ψ-image. it happens that this collection of loops γ1, . . . ,γq is a set of pinchable geodesics of gτ. the construction in [19] shows that we can choose the above parameter τ so that the γi are all arbitrarily short. proposition 1 ([7]). every non-empty set of k < 3g − 3 pinchable geodesics is contained in a maximal set of 3g − 3 pinchable geodesics. (v) valid sets of fundamental loops and their complexity let γ1, . . . ,γq ⊂ s be a pinchable set of geodesics in g. set ŝ + the stable riemann surface obtained from s by pinching these q geodesics; it consists of a finite number of compact riemann surfaces, called parts, which are joined together at a finite number of nodes. also, let ĝ be the noded schottky group obtained from g by pinching these q geodesics. let v1, . . . ,vg, be a fundamental set of loops for g on s (that is, the components of the lifting of these loops under π are simple loops and such a lifting set of loops contains a fundamental set of loops for g) and let v̂1, . . . , v̂g be the corresponding loops on ŝ + obtained by pinching γ1, . . . ,γq. we observe that the lifts of the v̂i to ω +(ĝ) are all loops, but they are generally not disjoint and they need not to be simple. there are certainly some number of these lifts passing through each parabolic fixed point, and some of them might pass more than once through the same parabolic fixed point. the set of loops, v1, . . . ,vg, is called a valid set of fundamental loops for γ1, . . . ,γq, if every lift of every v̂i to ω +(ĝ) is a simple loop; that is, it passes at most once through each parabolic fixed point (i.e., the set of loops, v̂1, . . . , v̂g, forms a fundamental set of loops for ĝ on 46 rubén a. hidalgo cubo 22, 1 (2020) ŝ+). we note that there are exactly q equivalence classes of parabolic fixed points in ĝ, one for each of the loops γi. proposition 2 ([4, 7]). there is at least one valid set of fundamental loops v1, . . . ,vg, for every set of pinchable geodesics, γ1, . . . ,γq. (vi) the complexity let us now consider a valid set of fundamental loops, v1, . . . ,vg, for a set of pinchable geodesics γ1, . . . ,γq. we can deform the vi on s so that they are all geodesics. then the geometric intersection number, vi •γj, of vi with γj is well defined; it is the number of points of intersection of these two geodesics. looking on the corresponding noded surface ŝ+, vi • γj is the number of times the curve v̂i obtained from vi by contracting γj to a point, passes through that point (node). the complexity of v1, . . . ,vg, with respect to γ1, . . . ,γq, is given by ξ(γ1, . . . ,γq;v1, . . . ,vg) = max 1≤j≤q g∑ i=1 vi • γj, and the complexity ξ(γ1, . . . ,γq) is the minimum of ξ(γ1, . . . ,γq;v1, . . . ,vg), where the minimum is taken over all valid sets of fundamental loops. if ξ(γ1, . . . ,γq) ≥ n, then, for every valid fundamental set v1, . . . ,vg, there is a node p on s + so that the total number of crossings of p by v̂1, . . . , v̂g is at least n. proposition 3 ([7]). let g ≥ 2 and g be a schottky group of rank g. for each positive integers n there are only finitely many topologically distinct maximal (i.e. q = 3g − 3) pinchable set of geodesics in g and complexity n. in particular, there are infinitely many topologically distinct maximal noded schottky groups of rank g and there are only finitely many topologically distinct maximal neoclassical noded schottky groups in each rank g. (vii) sufficiently complicated pinchable sets of geodesics now, we consider a maximal set of pinchable geodesics in g, say γ1, . . . ,γ3g−3; so ĝ is a maximal noded schottky group. observe that ĝ is rigid, and that every part of s+ is a sphere with three distinct nodes. also, every connected component ∆ ⊂ ω(ĝ) is a euclidean disc ∆, where ∆/stab(∆) is a sphere with three punctures; the three punctures correspond to the three nodes of the corresponding part of ŝ+. let v1, . . . ,vg be a valid set of fundamental loops on s for the given set of pinchable geodesics (the existence is given by proposition 2), and let v̂1, . . . , v̂g be the corresponding loops on ŝ +. for each i = 1, . . . ,g, the intersection of a lifting of v̂i with a component of ĝ (i.e., a connected component of ω(ĝ)) is called a strand of that lifting v̂i. similarly, the loops v̂1, . . . , v̂g appear on the corresponding parts of ŝ+ as collections of strands connecting the nodes on the boundary cubo 22, 1 (2020) a sufficiently complicated noded schottky . . . 47 of each part. there are two possibilities for these strands; either a strand connects two distinct nodes on some part, or it starts and ends at the same node. since the loops v1, . . . ,vg are simple and disjoint, there are at most three sets of parallel strands of the v̂i in each part; that is, there are at most three sets of strands, where any two strands in the same set are homotopic arcs with fixed endpoints at the nodes. we regard each of these sets of strands on a single part as being a superstrand, so that there are at most 3 superstrands on any one part. we next look in some component ∆ of ω(ĝ), and look at a parabolic fixed point x on its boundary, where x corresponds to the node n on the part si of ŝ +. in general, there will be infinitely many liftings of superstrands emanating from x in ∆, but, modulo stab(∆) there are only finitely many. in fact, there are at most 4 such liftings of superstrands emanating from x. if there is exactly one superstrand on si with one endpoint at n, and the other endpoint at a different node, then modulo stab(∆) there will be exactly the one lifting of this superstrand emanating from x. if there is only one superstrand on si with both endpoints at the same node n, then this superstrand has two liftings starting at x, one in each direction; so, in this case, we see two lifts of superstrands modulo stab(∆) emanating from x. it follows that, modulo stab(∆), we can have 0, 1, 2, 3 or 4 liftings of superstrands starting at x. we note that these liftings of superstrands all end at distinct parabolic fixed points on the boundary of ∆. we say that the fundamental set of loops, v̂1, . . . , v̂p is sufficiently complicated if there are two (different) lifts α̂i and α̂j, of some v̂i and some not necessarily distinct v̂j, respectively, so that α̂i and α̂j both pass through the parabolic fixed point z1, into a component ∆1 of ĝ, then both travel through ∆1 to the same parabolic fixed point on its boundary, z2, and into another component ∆2, which they again traverse together to the same boundary point, z3, necessarily a parabolic fixed point, where they enter ∆3, and they leave ∆3 at different parabolic fixed points. 2.4.3 sufficiently complicated noded schottky groups a maximal noded schottky group ĝ is sufficiently complicated if every set of valid fundamental loops on ŝ+ is sufficiently complicated. we note that (keeping the notation of last section), inside ∆1, α̂i and α̂j are disjoint; they both enter ∆1 at the same point, and they both leave ∆1 at the same point; hence they cannot both be circles. in [7] the following result, stating a sufficient condition in terms of the complexity for a maximal noded schottky group to be sufficiently complicated, was obtained. theorem 1 ([7]). if a maximal noded schottky group ĝ has complexity at least 11, then it is sufficiently complicated. the previous theorem, together proposition 3, asserts the existence of infinitely many topologically different sufficiently complicated maximal noded schottky groups in every rank g ≥ 2. the following result states sufficient conditions for a schottky group to be non-classical. theorem 2 ([7]). let ĝ be a maximal noded schottky group. 48 rubén a. hidalgo cubo 22, 1 (2020) (1) if ĝ is sufficiently complicated, then, for τ sufficiently large, the schottky group gτ in the relative conical neighborhood of ĝ is non-classical. (2) if s+ = ω+(ĝ)/ĝ is the stable riemann surface as shown in figure 2, then ĝ is sufficiently complicated. 3 explicit construction of a sufficiently complicated noded schottky group in this section we construct explicitly a noded schottky group as in part (2) of theorem 2, so part (1) of the same theorem asserts that any schottky group sufficiently near to ĝ is necessarily non-classical. 3.1 a family schottky groups of rank three let l0 be the unit circle, l1 be the real line, l2 be the line through the points 0 and w0 = e πi/3, and set (see figure 7) f = { (p,r) : 1/2 < p < 1, 0 < r < r ∗ (p) := √ 1 + p2 + p4 + p2 − 1 √ 3p } . for each (p,r) ∈ f we set l3 to be the circle with center at 0 and radius r and l4 to be the circle orthogonal to l0, intersecting l1 at the points p and 1/p with angle π/3 (see figure 3). the circle l4 has its center at c = ( 1+p2 2p ) + i√ 3 ( 1−p2 2p ) and it has radius r = 2√ 3 ( 1−p2 2p ) . the condition p > 1/2 asserts that l2 and l4 are disjoint (tangency occurs when p = 1/2) and the condition r < r∗(p) asserts that l3 and l4 are disjoint (tangency occurs when r = r ∗(p)). let τj be the reflection on lj, for j = 0,1,2,3,4, so τ0(z) = 1/z, τ1(z) = z, τ2(z) = w 2z, τ3(z) = r 2/z, τ4(z) = cz − 1 z − c , and let kr,p = 〈τ0,τ1,τ2,τ3,τ4〉. it turns out that kr,p is an extended kleinian group with connected region of discontinuity ωr,p and so that ωr,p/kr,p is a closed disc with 5 branch values, of orders 2, 2, 2, 2 and 3, on its border. as a consequence of the klein-maskit combination theorems [18], the group kr,p has no parabolic transformations, its limit set is a cantor set and it is geometrically finite. if we set w = τ2τ1, j = τ0τ1 and l = τ1τ4, then w 3 = l3 = j2 = (wj)2 = (lj)2 = 1. now, if a1 = l −1w−1 = τ4τ2, a2 = τ1a1τ1, and a3 = τ0τ3, so a1(z) = cw0z − 1 w0z − c , a2(z) = cw20z − 1 w20z − c , a3(z) = r 2z, cubo 22, 1 (2020) a sufficiently complicated noded schottky . . . 49 τ 1 2 3 4 0 ττ τ τ figure 3—a set of lines and circles 1a a a lw 3 2 figure 4—the schottky group gr,p of rank 3: the six darkest loops are a fundamental set of loops then we set gr,p = 〈a1,a2,a3〉. in figure 4 we show the situation for values of p near to 1 and r near to 0; in which case gr,p turns out to be a classical schottky group of rank three. lemma 1. if (p,r) ∈ f, then gr,p is a schottky group of rank three. proof. it can be seen that gr,p is a finite index normal subgroup of kr,p and kr,p/gr,p = 〈τ0 : τ 2 0 = 1〉 × 〈τ1,τ2 : τ 2 1 = τ 2 2 = (τ2τ1) 3 = 1〉 ∼= z2 ⊕ d3. in particular, gr,p has the same region of discontinuity as for kr,p (so a function group), it is geometrically finite and does not have parabolic transformations. as any of the elliptic transformations of kr,p goes into an element of the same order in the quotient kr,p/gr,p, we also have that gr,p is torsion free. now, as a consequence of the classification of function groups [15, 16], the group gr,p is a schottky group of rank three (in figure 4 there is shown a fundamental set of loops). the closed riemann surface sr,p = ωr,p/gr,p of genus 3 admits the group z2 ⊕ d3 as a group of conformal/anticonformal automorphisms. on sr,p we have simple closed curves γ1,..., γ6 which are pinchable (see figures 5 and 6) with respect to the schottky group gr,p; these pinchable curves correspond to the conjugacy classes of cyclic groups of gr,p as follows: γ1 corresponds to a −1 2 ; γ2 corresponds to a1; γ3 corresponds to a −1 1 a2; γ4 corresponds to a −1 1 a2a −1 3 a −1 2 a1a3; γ5 corresponds to a −1 2 a −1 3 a2a3; γ6 corresponds to a1a −1 3 a −1 1 a3. 50 rubén a. hidalgo cubo 22, 1 (2020) 3 γ γ γ γ γ γ γ γ γ γ 1 2 3 γ 4 4 4 γ 4 5 5 5 5 6 6 6 γ γ γ 6 γ figure 5—a set of pinchable curves seen at the schottky uniformization 5 α α α γ γ γ γ γ 2 1 3 3 1 2 6 4 γ figure 6—a set of pinchable curves seen on the riemann surface sr,p 3.2 a sufficiently complicated noded schottky groups of rank three to obtain the desired noded schottky group, we need to move the pair (p,r) ∈ f to some point in the boundary in order to have that the loxodromic transformations a−12 , a1, a −1 1 a2, a−11 a2a −1 3 a −1 2 a1a3, a −1 2 a −1 3 a2a3 and a1a −1 3 a −1 1 a3 are transformed into parabolic transformations. as the order three automorphism w permutes cyclically γ1,γ2,γ3 and also γ4,γ5,γ6, we only need to take care of a1 and a1a −1 3 a −1 1 a3. first, in order to transform a1 into a parabolic transformation we only need to have τ4(w0) = w0, equivalently, that the circle l4 is tangent to the line l2 at w0. this happens exactly when p = 1/2. now, assuming p = 1/2, in order for a1a −1 3 a −1 1 a3 to be a parabolic transformation we only need to have tangency of the circle l3 with l4, that is, r = r ∗(1/2) = √ 7− √ 3 2 . the group gr∗(1/2),1/2 turns out to be a noded schottky group that uniformizes a stable riemann surface as shown in figure 2 and, by (2) in theorem 2, it is a sufficiently complicated noded schottky group. non−classical schottky groups r=r *(p) p r p=1/2 p=1p0 (1/2)*r r 0 figure 7—the region f and the filled part for the non-classical schottky groups 3.3 non-classical schottky groups of rank three by (1) in theorem 2, there exist (p0,r0) ∈ f with the property that if (p,r) ∈ f, 1/2 < p < p0 and r0 < r < r ∗(1/2), then gr,p is a non-classical schottky group of rank three (see filled part region in figure 7). moreover, each of these schottky groups is contained in a kleinian group kr,p cubo 22, 1 (2020) a sufficiently complicated noded schottky . . . 51 as a finite index normal subgroup with kr,p/gr,p ∼= z2 ⊕ d3, in other words, the closed riemann surfaces sr,p = ω(gr,p)/gr,p admit a group of conformal automorphisms isomorphic to z2 ⊕d3. the family of these riemann surfaces degenerates to a stable riemann surface sr∗(1/2),1/2 as figure 2 keeping the above group of automorphisms invariant. 52 rubén a. hidalgo cubo 22, 1 (2020) references [1] a. i. bobenko. schottky uniformization and finite-gap integration, soviet math. dokl. 36 no. 1 (1988), 38–42 (transl. from russian: dokl. akad. nauk sssr, 295, no.2 (1987)). 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[27] r. zarrow. classical and non-classical schottky groups. duke math. j. 42 (1975), 717–724. introduction sufficiently complicated noded schottky groups noded schottky groups the extended region of discontinuity neoclassical noded schottky groups sufficiently complicated noded schottky groups the relative conical neighbourhood of a noded schottky group pinchable loops of schottky groups sufficiently complicated noded schottky groups explicit construction of a sufficiently complicated noded schottky group a family schottky groups of rank three a sufficiently complicated noded schottky groups of rank three non-classical schottky groups of rank three cubo, a mathematical journal vol. 24, no. 03, pp. 501–519, december 2022 doi: 10.56754/0719-0646.2403.0501 infinitely many solutions for a nonlinear navier problem involving the p-biharmonic operator filippo cammaroto1, b 1 department of mathematical and computer sciences, physical sciences and earth sciences, university of messina, viale f. stagno d’alcontres, 31, 98166 messina, italy. fdcammaroto@unime.it b abstract in this paper we establish some results of existence of infinitely many solutions for an elliptic equation involving the p-biharmonic and the p-laplacian operators coupled with navier boundary conditions where the nonlinearities depend on two real parameters and do not satisfy any symmetric condition. the nature of the approach is variational and the main tool is an abstract result of ricceri. the novelty in the application of this abstract tool is the use of a class of test functions which makes the assumptions on the data easier to verify. resumen en este art́ıculo establecemos algunos resultados sobre la existencia de infinitas soluciones para una ecuación eĺıptica que involucra los operadores p-biarmónico y p-laplaciano acoplados con condiciones de borde de navier, donde las nolinealidades dependen de dos parámetros reales y no satisfacen ninguna condición simétrica. la naturaleza del enfoque es variacional y la herramienta principal es un resultado abstracto de ricceri. la novedad de la aplicación de esta herramienta abstracta es el uso de una clase de funciones test que hacen que las hipótesis sobre la data sean más fáciles de verificar. keywords and phrases: p-biharmonic operator, p-laplacian operator, navier problem, multiplicity. 2020 ams mathematics subject classification: 35j35, 35j60. accepted: 28 november, 2022 received: 21 october, 2022 ©2022 f. cammaroto. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0501 https://orcid.org/0000-0001-8229-8848 mailto:fdcammaroto@unime.it 502 f. cammaroto cubo 24, 3 (2022) 1 introduction in this paper we investigate the existence of infinitely many solutions to the following p-biharmonic elliptic equation with navier conditions,   ∆ 2 pu − ∆pu + v (x)|u|p−2u = λf(x, u) + µg(x, u) in ω u = ∆u = 0 on ω (pλ,µ) where ω ⊂ rn (n ⩾ 1) is a bounded domain with smooth boundary ∂ω, p > max { 1, n 2 } , ∆2pu = ∆(|∆u|p−2∆u) is the p-biharmonic operator, ∆pu = ∇(|∇u|p−2∇u) is the p-laplacian operator, v ∈ c(ω) satisfying infω v > 0, f, g : ω × r → r are two carathéodory functions with suitable behaviors, λ ∈ r and µ > 0. in the last years several authors have showed their interest in fourth-order differential problems involving biharmonic and p-biharmonic operators, motivated by the fact that this type of equations finds applications in fields such as the elasticity theory, or more in general, in continuous mechanics. in particular, the fourth-order elliptic equations can describe the static form change of beam or the motion of rigid body, so they are widely applied in physics and engineering. in 1990 lazer and mckenna, in a large paper in which they investigated the oscillatory phenomena that led to the collapse of the tacoma narrows bridge, considered fourth-order problems with the nonlinearity (u + 1)+ − 1; this nonlinearity is useful to study traveling waves in suspension bridges. anyway the same authors observed that this kind of problems are interesting also when this particular nonlinearity is replaced by a somewhat more general function f(·, u) (see [24, 31, 32]). as regards fourth-order differential problems involving biharmonic and p-biharmonic operators, a non-negligible part of the literature is devoted to the study of the existence of infinitely many solutions to problems involving only the biharmonic or p-biharmonic operator (see, for instance, [2, 4, 5, 6, 9, 10, 17, 18, 19, 29, 30, 40]) or considering also the presence of laplacian or p-laplacian operator ([22, 26, 38, 42, 43]) and/or a term with a potential function ([11, 12, 13, 25, 28]); some authors have also recently considered the case in which a nonlocal term is present ([16, 41]). unlike some papers concerning problems set in an unbounded domain (see [2, 4, 11, 12, 13, 18, 19, 30] and above all [25] which inspired us in the choice of this type of problem), most of the literature is devoted to the bounded case. in this case, different approaches have been adopted for obtaining infinitely many solutions. in a lot of papers symmetry conditions on the nonlinearities are assumed together with the use of the symmetric mountain pass theorem of ambrosetti rabinowitz (see [26, 40]) or with the use of the fountain theorem ([38, 42, 43]). in our investigation the approach is variational. more precisely we will apply the following critical point theorem that ricceri established in 2000 ([34, theorem 2.5]), recalled below for the reader’s convenience. cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 503 theorem 1.1. let x be a reflexive real banach space, and let φ, ψ : x → r be two sequentially weakly lower semicontinuous and gâteaux differentiable functionals. assume also that ψ is (strongly) continuous and coercive. for each r > infx ψ, we put φ(r) := inf x∈ψ−1(]−∞,r[) φ(x) − inf ψ−1(]−∞,r[) ω φ r − ψ(x) where ψ−1(] − ∞, r[)w is the closure of ψ −1(] − ∞, r[) in the weak topology. fixed λ ∈ r, then a) if {rk} is a real sequence such that lim k→∞ rk = +∞ and φ(rk) < λ, for each k ∈ n, the following alternative holds: either φ + λψ has a global minimum or there exists a sequence {xk} of critical points of φ + λψ such that lim k→∞ ψ(xk) = +∞; b) if {sk} is a real sequence such that lim k→∞ sk = (inf x ψ)+ and φ(sk) < λ for each k ∈ n, the following alternative holds: either there exists a global minimum of ψ which is a local minimum of φ + λψ or there exists a sequence {xk} of pairwise distinct critical points of φ + λψ with lim k→∞ ψ(xk) = inf x ψ, which weakly converges to a global minimum of ψ. since its appearance in 2000 until our days, it has been a powerful tool to get multiplicity results for different kinds of problems. in particular, it has been widely applied to obtain theorems of existence of infinitely many solutions to problems associated with a vast range of differential equations. in each of these applications, in order to guarantee that φ(rk) < λ (or φ(sk) < λ), for each k ∈ n, and that the functional φ + λψ has no global minimum, it is necessary to use some sequences of functions defined ad hoc. generally, in these functions the norm of the variable is raised to a suitable power which depends on the nature of the problem and that gives them the requested regularity properties: in some applications the norm is used without power (see, for instance, [3, 7, 14, 15, 23, 27, 39]), in some others it is raised to the second ([9, 10, 29, 33, 35, 36]) or to the third ([22, 28]) or to the fourth power ([1]); in [20, 21] the authors combined the norm with trigonometric functions. the choice of a particular sequence of functions inside the proof reflects heavily on the assumptions and while there are some cases in which probably the choice is optimal, in some other cases it could happen that a different choice of the sequence would make the result applicable in a greater number of cases. this is the reason we have introduced an abstract class of test functions serving our purpose. we will clarify this fact in section 3, showing some examples. a similar line of reasoning can be found in [8] and above all in [37] where the author does not choose the test functions arbitrarily during the proof but he uses two generic functions whose properties are described in the statement of his result. 504 f. cammaroto cubo 24, 3 (2022) 2 preliminaries in this section we describe the variational framework in which we will work in our investigations. to begin with, we denote by ω := π n 2 /γ ( n 2 + 1 ) the measure of the unit ball in rn. if x is a banach space, the symbol b(x, r) stands for the open ball centered at x ∈ x and of radius r > 0. let ω be a bounded smooth domain of rn, n ≥ 1, p > max { 1, n 2 } and let v ∈ c(ω) satisfy infω v > 0. put e = w 2,p(ω) ∩ w 1,p0 (ω); it is a reflexive banach space when endowed with the standard norm ∥u∥ = (∫ ω |∆u|pdx )1 p . moreover, the assumptions on v assure that the position ∥u∥v = (∫ ω (|∆u|p + |∇u|p + v (x)|u|p) dx )1 p for any u ∈ e, defines a norm equivalent to the standard one. being p > n 2 , the rellich-kondrachov theorem assures that e is compactly embedded in c0(ω); in particular, there exists a constant c∞ > 0 such that ∥u∥∞ ≤ c∞ ∥u∥ ≤ c∞ ∥u∥v (2.1) for every u ∈ e. now, motivated by the reasons that we have illustrated in the introduction, let us introduce the following class of functions. if {ak}, {bk}, {σk} are three real sequences with 0 < ak < bk and σk > 0, for each k ∈ n, let us denote by h({ak} , {bk} , {σk}) the space of all sequences {χk} ⊂ w 2,p(]ak, bk[) satisfying i) 0 ≤ χk(x) ≤ σk for a.e. x ∈]ak, bk[; ii) lim x→a+ k χk(x) = σk, lim x→b− k χk(x) = 0; iii) lim x→a+ k χ′k(x) = lim x→b− k χ′k(x) = 0; iv) for all j ∈ {1, 2} there exists cj > 0, independent of k, such that |χ(j)k (x)| ≤ cj σk (bk − ak)j (2.2) for a.e. x ∈]ak, bk[ and for all k ∈ n. now, we show how the space h({ak} , {bk} , {σk}) help us to build some sequences in e that play a crucial role in the proof of the main result. if x0 ∈ ω, {bk} ⊂]0, +∞[ such that b(x0, bk) ⊂ ω, for each k ∈ n, and {χk} ∈ h({ak} , {bk} , {σk}), consider the function uk : ω → r defined by setting cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 505 uk(x) =   0 in ω \ b(x0, bk), σk in b(x0, ak), χk(|x − x0|) in b(x0, bk) \ b(x0, ak) for each k ∈ n. simple computations show that, fixed k ∈ n, for each i ∈ {1, . . . , n}, we have ∂uk ∂xi (x) =   0 in ω \ b(x0, bk), 0 in b(x0, ak), χ′k(|x − x0|) xi − x0i |x − x0| in b(x0, bk) \ b(xo, ak) and ∂2uk ∂x2i (x) =   0 in ω \ b(x0, bk), 0 in b(x0, ak), χ′′k(|x − x0|) (xi − x0i ) 2 |x − x0|2 + χ′k(|x − x0|) |x − x0|2 − (xi − x0i ) 2 |x − x0|3 in b(x0, bk) \ b(x0, ak) using these computations together with (2.2), we get the following inequalities |∇uk(x)| ⩽ |χ′k(|x − x0|)| ≤ c1 σk (bk − ak) , and |∆uk(x)| ⩽ |χ′′k(|x − x0|)| + |χ ′ k(|x − x0|)| (n − 1) |x − x0| ≤ c2 σk (bk − ak)2 + c1 σk (bk − ak) (n − 1) ak . these inequalities allow us to estimate the norm of the functions uk as follows ∥uk∥ p v = ∫ ω (|∆uk|p + |∇uk|p + v (x)|uk(x)|p) dx = ∫ b(x0,bk)\b(x0,ak) |∆uk(x)|pdx + ∫ b(x0,bk)\b(x0,ak) |∇uk(x)|pdx + ∫ b(x0,bk) v (x)|uk(x)|pdx ≤ ωσpk {[ c2 (bk − ak)2 + c1(n − 1) ak(bk − ak) ]p (bnk − a n k) + [ c1 (bk − ak) ]p (bnk − a n k) + b n k max b(x0,bk) v } . let us denote by c the class of all carathéodory functions η : ω×r → r satisfying sup|t|≤ξ |η(·, t)| ∈ l1(ω) for all ξ > 0 and let f, g ∈ c. 506 f. cammaroto cubo 24, 3 (2022) we say that a function u ∈ e is a weak solution to (pλ,µ) if∫ ω ( |∆u|p−2∆u∆v + |∇u|p−2∇u∇v + v (x)|u|p−2uv ) dx = λ ∫ ω f(x, u(x))v(x)dx + µ ∫ ω g(x, u(x))v(x)dx for each v ∈ e. obviously the weak solutions to (pλ,µ) are exactly the critical points in e of the energy functional defined, for each u ∈ e, by e(u) := 1 p ψ(u) + λφf (u) + µφg(u), where ψ(u) := ∥u∥pv , φf (u) := − ∫ ω f(x, u(x))dx, φg(u) := − ∫ ω g(x, u(x))dx, where, for each (x, t) ∈ ω × r, f(x, t) := ∫ t 0 f(x, s)ds, g(x, t) := ∫ t 0 g(x, s)ds. 3 results the first multiplicity result deals with the case in which f has a global (m − 1)-sublinear growth, with m < p, while different cases are considered for the behaviour of function g. theorem 3.1. let v ∈ c(ω) satisfy infω v > 0 and let f, g ∈ c such that: (i1) there exist 1 < m 0, p1, p2 > 1 such that b(x0, ρ) ⊆ ω and lim inf t→+∞ ∫ ω max|ξ|≤t g(x, ξ)dx tp1 := a < +∞, lim sup t→+∞ ∫ b(x0,ρ) g(x, t)dx tp2 := b > 0. then the following facts hold: (r1) if p1 0, the problem (pλ,µ) admits a sequence of non-zero weak solutions; (r2) if p1 0 such that for all λ ∈ r and for all µ > µ1, the problem (pλ,µ) admits a sequence of non-zero weak solutions; cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 507 (r3) if p1 = p < p2, there exists µ2 > 0 such that for all λ ∈ r and for all µ ∈]0, µ2[, the problem (pλ,µ) admits a sequence of non-zero weak solutions; (r4) if p1 = p2 = p, there exists γ > 1 and cv,γ,ρ > 0 such that, if cv,γ,ρ < b ωc p ∞a , (3.1) (the previous inequality always being satisfied whether a = 0 or b = +∞) then µ1 < µ2 and for all λ ∈ r and for all µ ∈]µ1, µ2[, the problem (pλ,µ) admits a sequence of non-zero weak solutions. proof. to prove (r1), let us apply part a) of theorem 1.1 choosing x = e, ψ defined as in the preliminaries and φ = λφf + µφg. as we have already observed the critical points of the functional φ + 1 p ψ are precisely the weak solution of problem (pλ,µ). the functionals φ and ψ are sequentially weak lower semicontinuous and moreover ψ is strongly continuous and coercive. in our case the function φ is defined by setting φ(r) = inf ∥u∥p v 0. now, we wish to find a sequence {rk}k∈n such that lim k→∞ rk = +∞ and φ(rk) < 1p for each k ∈ n. to this aim it suffices to prove that for each k ∈ n there exists a function uk ∈ x, with ∥uk∥ p v < rk, such that sup ∥w∥p v ≤rk { λ ∫ ω f(x, w(x))dx + µ ∫ ω g(x, w(x))dx } − λ ∫ ω f(x, uk(x))dx+ −µ ∫ ω g(x, uk(x))dx < 1 p (rk − ∥uk∥ p v ) . thanks to (i3), fixed a > a, for each k ∈ n there exists αk ≥ k such that∫ ω max |ξ|≤αk g(x, ξ)dx ≤ aαp1k . now we choose uk = θe and rk = 1 c p ∞ α p k. obviously we have lim k→∞ rk = +∞. before proving (3), observe that, for each w ∈ x with ∥w∥ p v ≤ rk, one has ∥w∥∞ ≤ c∞ ∥w∥v ≤ c∞r 1 p k = αk for each k ∈ n. therefore, we obtain 508 f. cammaroto cubo 24, 3 (2022) λ ∫ ω f(x, w(x))dx + µ ∫ ω g(x, w(x))dx ≤ |λ| ∫ ω |h(x)| ( |w(x)| + |w(x)|m m ) dx + µ ∫ ω max |ξ|≤αk g(x, ξ)dx ≤ |λ|∥h∥1 ( αk + αmk m ) + µaα p1 k ≤ |λ|∥h∥1c∞r 1 p k + |λ| m ∥h∥1cm∞r m p k + µac p1 ∞r p1 p k < 1 p rk for k large enough, being 1 < m < p and p1 < p. so, thanks to part a) of theorem 1.1, the functional φ + 1 p ψ has a global minimum, or there exists a sequence of weak solutions {uk} ⊂ e such that lim k→∞ ∥uk∥ = +∞. this part of the proof will end if we show that the functional φ + 1pψ has no global minimum. to this aim, using (i3), fixed 0 1 such that b(x0, γρ) ⊆ ω and a sequence {χk} ∈ h(ρ, γρ, {αk}), we consider wk(x) =   0, in ω \ b(x0, γρ), βk, in b(x0, ρ), χk(|x − x0|) in b(x0, γρ) \ b(x0, ρ). using the estimation of the norm made in the previous section, we get ∥wk∥ p v ≤ ωβ p k [ 2p−1(γn − 1) ρ2p−n(γ − 1)2p c p 2 + (2p−1(n − 1)p + ρp)(γn − 1) ρ2p−n(γ − 1)p c p 1 + γ nρn max b(x0,γρ) v ] . if we put cv,γ,ρ = 2p−1(γn − 1) ρ2p−n(γ − 1)2p c p 2 + (2p−1(n − 1)p + ρp)(γn − 1) ρ2p−n(γ − 1)p c p 1 + γ nρn max b(x0,γρ) v we have φ(wk) + 1 p ψ(wk) = −λ ∫ ω f(x, wk(x))dx − µ ∫ ω g(x, wk(x))dx + 1 p ∥wk∥ p v ≤ |λ| ∫ ω |h(x)| ( |wk(x)| + |wk(x)|m m ) dx − µ ∫ b(x0,ρ) g(x, βk)dx + ωcv,γ,ρ p β p k ≤ |λ|∥h∥1βk + |λ|∥h∥1 βmk m − µbβp2k + ωcv,γ,ρ p β p k and, since 1 < m < p < d2 and lim k→∞ βk = +∞, the functional φ + 1pψ has no global minimum, being lim k→∞ φ(wk) + 1 p ψ(wk) = −∞. this concludes the proof of (r1). cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 509 the proof of (r2) is similar. if p1 µ1, choosing b such that ωcv,γ,ρ pµ < b < b, in a similar way we have φ(wk) + 1 p ψ(wk) ≤ |λ|∥h∥1βk + |λ|∥h∥1 βmk m − ( µb − ωcv,ρ,γ p ) β p k and, thanks to the choice of b, also in this case the functional φ + 1 p ψ has no global minimum. this concludes (r2). as for the proof of (r3), if p1 = p and p2 > p, we choose µ2 = 1 pc p ∞a (obviously if a = 0 we choose µ2 = +∞). then, fixing λ ∈ r and 0 < µ < µ2, we can choose a such that a < a < 1pcp∞µ. similar computations give λ ∫ ω f(x, w(x))dx + µ ∫ ω g(x, w(x))dx ≤ |λ|∥h∥1c∞r 1 p k + |λ| m ∥h∥1cm∞r m p k + µac p ∞rk < 1 p rk for k large enough, being 1 < m < p and µacp∞ < 1 p . finally, the proof of (r4) relies on the considerations made in the previous two cases. we have only to prove that µ1 < µ2, but this is guaranteed by the assumption (3.1). now, we are interested in the existence of infinitely many weak solutions in the case that the nonlinearities f and g have a particular form. theorem 3.2. let v ∈ c(ω) satisfy infω v > 0, m < p, h ∈ l1(ω), and r ∈ l1(ω) \ {0} with r ≥ 0 a.e. in ω. let s : r → r be a continuous function with ∫ t 0 s(ξ)dξ ≥ 0, for all t ≥ 0. moreover assume that there exists p1, p2 > 1, α, β > 0 and {αk}, {βk} satisfying lim k→∞ αk = lim k→∞ βk = +∞, such that max |ξ|≤αk ∫ ξ 0 s(t)dt ≤ ααp1k , ∫ βk 0 s(t)dt ≥ ββp2k for each k ∈ n. then, for the problem   ∆ 2 pu − ∆pu + v (x)|u|p−2u = λh(x)|u|m−2u + µr(x)s(u) in ω u = ∆u = 0 on ω (p λ,µ) the following facts hold: (r1) if p1 0, the problem (p λ,µ) admits a sequence of non-zero weak solutions; (r2) if p1 0 such that for all λ ∈ r and for all µ > µ1, the problem (p λ,µ) admits a sequence of non-zero weak solutions; 510 f. cammaroto cubo 24, 3 (2022) (r3) if p1 = p < p2, there exists µ2 > 0 such that for all λ ∈ r and for all µ ∈]0, µ2[, the problem (p λ,µ) admits a sequence of non-zero weak solutions; (r4) if p1 = p2 = p, there exist x0 ∈ ω, ρ > 0, γ > 1 and cv,γ,ρ > 0, such that, if cv,γ,ρ < β∥r∥l1(b(x0,ρ)) αωc p ∞∥r∥l1(ω) , (3.2) then µ1 < µ2 and for all λ ∈ r and for all µ ∈]µ1, µ2[, the problem (p λ,µ) admits a sequence of non-zero weak solutions. proof. we want to apply theorem 3.1 choosing f(x, t) = h(x)|t|m−2t and g(x, t) = r(x)s(t) for all (x, t) ∈ ω × r. the hypotheses (i1), (i2) are obviously verified. since r ̸≡ 0 we can choose x0 ∈ ω and ρ > 0 such that b(x0, ρ) ⊂ ω and r > 0 in b(x0, ρ). then we have: ∫ ω max |ξ|≤αk g(x, ξ)dx = ∫ ω max |ξ|≤αk (∫ ξ 0 r(x)s(t)dt ) dx = ∥r∥l1(ω) max |ξ|≤αk ∫ ξ 0 s(t)dt ≤ ∥r∥l1(ω)αα p1 k and ∫ b(x0,ρ) g(x, βk)dx = ∫ b(x0,ρ) (∫ βk 0 r(x)s(t)dt ) dx = ∥r∥l1(b(x0,ρ)) ∫ βk 0 s(t)dt ≥ ∥r∥l1(b(x0,ρ))ββ p2 k . therefore lim inf t→+∞ ∫ ω max|ξ|≤t g(x, ξ)dx tp1 ≤ ∥r∥l1(ω)α < +∞ and lim sup t→+∞ ∫ b(x0,ρ) g(x, t)dx tp2 ≥ ∥r∥l1(b(x0,ρ))β > 0. so, (i3) is also verified with a = α∥r∥l1(ω) and b = β∥r∥l1(b(x0,ρ)). therefore we can apply the theorem 3.1 and obtain the conclusions (r1)–(r4). now, we want to exhibit two examples. in the first one we present a function s verifying the hypotheses of theorem 3.2. example 3.3. let p > 1, δ > 1 and let s : r → r be the function such that s(t) = ∫ t 0 s(ξ)dξ =   0, in ] − ∞, 0], −2δt3 + 3δt2, in ]0, 1], 2p(k−1)δk in ] 2k−1δ k−1 p , 2k−1δ k p ] k ≥ 1, akt 3 + bkt 2 + ckt + dk in ] 2k−1δ k p , 2kδ k p ] k ≥ 1 cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 511 where ak := −2(p−3)k+4δ (p−3)k p ( δ − 2−p ) , bk := 9 · 2(p−2)k+2δ (p−2)k p ( δ − 2−p ) , ck := −3 · 2(p−1)k+3δ (p−1)k p ( δ − 2−p ) , dk := 2 pkδk ( 5δ − 22−p ) . using matlab by mathworks, we have plotted the graph of the function s (for δ = 2 and p = 2), showed in the following image. -2 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 140 160 180 200 the function s satisfies all the assumption of theorem 3.2 with α = 1, β = δ, αk = 2 k−1δ k p and βk = 2 kδ k p , for each k ∈ n. in particular max |ξ|≤αk ∫ ξ 0 s(t)dt = ∫ 2k−1δ kp 0 s(t)dt = 2p(k−1)δk = α p k and ∫ βk 0 s(t)dt = 2pkδk+1 = δβ p k for all k ∈ n. in theorems 3.1 and 3.2, inequalities (3.1) and (3.2) serve to assure that µ1 < µ2; moreover the value of cv,γ,ρ depends heavily also on constants cj and then on the choice of the sequence {χk}. obviously, fixed the nonlinearity, the smaller the constant cv,γ,ρ the easier the inequalities (3.1) and (3.2) will be verified. the next example is in this direction. example 3.4. let p > 1, ω = b(0, 1) in rn, x0 = 0, r ∈ l1(ω) \ 0, with r ≥ 0, v (x) = |x|2r2 + 1, for all x ∈ b(0, 1), ρ = 1 2 , γ = 2 and {σk} ⊂]0, +∞[ with limk→∞ σk = +∞. let { χ1k } , { χ2k } ∈ h(1 2 , 1, {σk}) the sequences defined by χ1k(x) = 4σk(4x 3 − 9x2 + 6x − 1) 512 f. cammaroto cubo 24, 3 (2022) and χ2k(x) = σk 2 cos(π(2x − 1) + 1) for all x ∈]1 2 , 1[ and for each k ∈ n. we observe that, for each x ∈]1 2 , 1[, |χ1k ′ (x)| ≤ 3σk, |χ1k ′′ (x)| ≤ 24σk and then the constants cj( { χ1k } ), defined in (2.2), are respectively c1( { χ1k } ) = 3 2 and c2( { χ1k } ) = 6. in a similar way, for each x ∈]1 2 , 1[, we have |χ2k ′ (x)| ≤ πσk, |χ2k ′′ (x)| ≤ 2π2σk and, in this case, the constants cj( { χ2k } ) are respectively c1( { χ2k } ) = π 2 and c2( { χ2k } ) = π 2 2 . now let us consider a sequence of functions that, in combination with the norm, raises it to the second power; namely χ3k(x) =   σk ( −8x2 + 8x − 1 ) in ]1 2 , 3 4 [ αk(8x 2 − 16x + 8) in ]3 4 , 1[ (3.3) for each k ∈ n. in this case |χ3k ′ (x)| ≤ 4σk, |χ3k ′′ (x)| ≤ 16σk and then c1( { χ3k } ) = 2 and c2( { χ3k } ) = 4. with respect to these three sequences of test functions the smallest cv,γ,ρ (among the three) depends on the values of n and p. for instance, for n = 3 and p = 2 the smallest cv,γ,ρ is the one in correspondence with the sequence {χ3k}; in fact, using matlab again to compute these constants, one has cv,γ,ρ({χ1k}) ≈ 1270, cv,γ,ρ({χ 2 k}) ≈ 969, cv,γ,ρ({χ 3 k}) = 912. but, for instance, for n = 4 and p = 3, the smallest cv,γ,ρ is the one in correspondence with the sequence {χ2k} being cv,γ,ρ({χ1k}) ≈ 73737, cv,γ,ρ({χ 2 k}) ≈ 53988, cv,γ,ρ({χ 3 k}) = 67262. obviously if we consider the function s of example 3.3, taking a posteriori δ > ωcp∞∥r∥l1(ω)cv,γ,ρ ∥r∥ l1(b(0, 1 2 ) the corresponding problem admits a sequence of non-zero weak solutions; but if δ is fixed a priori, theorems 3.1 and 3.2 could be always applied as long as one manages to find an appropriate sequence {χk} while it is not sure that a generic application of theorem 1.1 can be applied because the assumptions depends heavily by the particular sequence of test functions fixed during the proof. cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 513 the last theorem concerns the case in which the growth exponent of nonlinearity f(x, t) is exactly p − 1. in this situation the existence of infinite weak solutions will be obtained not for each λ ∈ r but in an appropriate interval. theorem 3.5. let v ∈ c(ω) satisfy infω v > 0 and let f, g ∈ c such that (i2) and (i3) are verified. moreover, suppose that: (̃i1) there exist h ∈ l1(ω) such that |f(x, t)| = h(x) ( 1 + |t|p−1 ) for a.e. x ∈ ω and for all t ∈ r. then the following facts hold: (r̃1) if p1 0, the problem (pλ,µ) admits a sequence of non-zero weak solutions; (r̃2) if p1 0 such that, for all µ > µ1, there exists λµ > 0 such that, for all |λ| < λµ, the problem (pλ,µ) admits a sequence of non-zero weak solutions; (r̃3) if p1 = p < p2, there exists µ2 > 0 such that, for all µ ∈]0, µ2[, there exists λµ > 0 such that, for all |λ| < λµ, the problem (pλ,µ) admits a sequence of non-zero weak solutions; (r̃4) if p1 = p2 = p, there exists γ > 1 and cv,γ,ρ > 0 such that, if cv,γ,ρ 0 such that, for all |λ| < λµ the problem (pλ,µ) admits a sequence of non-zero weak solutions. proof. the proof is similar to that of theorem 3.1. in fact, computing the two main evaluations for m = p, we get: λ ∫ ω f(x, w(x))dx + µ ∫ ω g(x, w(x))dx ≤ |λ|∥h∥1c∞r 1 p k + |λ| p ∥h∥1cp∞rk + µac p1 ∞r p1 p k (3.5) and φ(wk) + 1 p ψ(wk) ≤ |λ|∥h∥1βk + |λ|∥h∥1 β p k p − µbβp2k + ωcv,γ,ρ p β p k. (3.6) to prove (r̃1), fix λ such that |λ| ≤ 1∥h∥1cp∞ and µ > 0. thanks to the choice of λ and to the fact that p1 < p then, from (3.5) we get λ ∫ ω f(x, w(x))dx + µ ∫ ω g(x, w(x))dx < 1 p rk (3.7) for k large enough (remember that lim k→∞ rk = +∞); moreover, from (3.6) we obtain lim k→∞ φ(wk) + 1 p ψ(wk) = −∞ (3.8) because p < p2. 514 f. cammaroto cubo 24, 3 (2022) to prove (r̃2), it is sufficient to choose µ1 = ωcv,γ,ρ pb . fixed µ > µ1 and b in a similar way as done in theorem 3.1, we define λµ = min { 1 ∥h∥1c p ∞ , µpb−ωcv,γ,ρ ∥h∥1 } . fixed λ such that |λ| < λµ, obviously, from (3.5), we get (3.7) (for k large enough) because p1 < p and thanks to the choice of λ. moreover, using (3.6), the choice of λ and µ guarantees that (3.8) holds. to prove (r̃3), it is sufficient to choose µ2 = 1 pc p ∞a . fixed µ ∈]0, µ2[ and a in a similar way as done in theorem 3.1, we choose λµ = 1−µpcp∞a ∥h∥1c p ∞ . fixed λ such that |λ| < λµ, obviously, from (3.6), we get (3.8) because p < p2. moreover, using (3.5), the choice of λ and µ guarantees that (3.7) holds. in the last case, to prove (r̃4), we observe that, thanks to (3.4), we have µ1 < µ2. so, fixed µ ∈]µ1, µ2[, and choosing a and b in a similar way as done in theorem 3.1, we define λµ = min { 1−µcp∞a ∥h∥1 , µpb−ωcv,γ,ρ ∥h∥1 } . fixed λ such that |λ| < λµ, obviously, from (3.5), we get (3.7) (for k large enough) because of the choice of λ and µ. moreover, using (3.6), the choice of λ and µ guarantees that (3.8) holds. we conclude with an example related to case (r̃4) of theorem 3.5. in this case we consider the one-dimensional setting, providing an explicit estimate of the constant c∞ in (3.4). example 3.6. let n = 1, ω =] − 1, 1[, p1 = p2 = p = 2, v (x) = x2 + 1 for all x ∈] − 1, 1[, h ∈ l1(]−1, 1[), r ∈ l1(]−1, 1[)\{0} with r ≥ 0 in ]−1, 1[ and ∫ 1/2 −1/2 r(x)dx > 0. it is well-known that, for all u ∈ w 2,2(] − 1, 1[) ∩ w 1,20 (] − 1, 1[), one has max x∈]−1,1[ |u(x)| ≤ √ 2 2 ∥u′∥l2(]−1,1[) and ∥u′∥l2(]−1,1[) ≤ 2 π ∥u′′∥l2(]−1,1[) , so max x∈]−1,1[ |u(x)| ≤ √ 2 π ∥u′′∥l2(]−1,1[) ≤ √ 2 π ∥u∥v and then c∞ = √ 2 π . now choosing x0 = 0, ρ = 1 2 , γ = 2, δ > 1064∥r∥ l1(]−1,1[) π2∥r∥ l1(]− 12 , 1 2 [) , and g(t, x) = r(x)s(t) (where the function s is that of example 3.3), assumptions (i2) and (i3) are satisfied with a = ∥r∥l1(]−1,1[) and b = δ∥r∥l1(]− 12 , 12 [). using the sequence {χ 3 k} of example 3.4 as test function, we compute cv,γ,ρ = 266 (lower than those associated with the other two sequences). it is easy to see that b ωc p ∞a = δπ2∥r∥ l1(]− 12 , 1 2 [) 8∥r∥l1(]−1,1[) > 266 then (3.4) is satisfied and then the fact (r̃4) holds. in particular, for all µ ∈ ] 266 δ , π 2 4∥r∥ l1(]−1,1[) [ , there exists λµ > 0 (defined inside the proof of theorem 3.5) such that, for all |λ| < λµ the problem (pλ,µ) admits a sequence of non-zero weak solutions. cubo 24, 3 (2022) infinitely many solutions for a nonlinear navier problem involving... 515 acknowledgment the author is member of the gruppo nazionale per l’analisi matematica, la probabilità e le loro applicazioni (gnampa) of the istituto nazionale di alta matematica (indam). 516 f. cammaroto cubo 24, 3 (2022) references [1] g. a. afrouzi and s. shokooh, “existence of infinitely many solutions for quasilinear problems with a p(x)-biharmonic operator”, electron. j. differ. equations, paper no. 317, 14 pages, 2015. 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[43] j. zhang and z. wei, “infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems”, nonlinear anal., vol. 74, no. 18, pp. 7474–7485, 2011. introduction preliminaries results cubo, a mathematical journal vol.22, n◦03, (315–324). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300315 received: 17 march, 2020 | accepted: 05 october, 2020 characterization of upper detour monophonic domination number m. mohammed abdul khayyoom department of mathematics ptm govt. college, perintalmanna, malappuram, kerala, india. khayyoom.m@gmail.com abstract this paper introduces the concept of upper detour monophonic domination number of a graph. for a connected graph g with vertex set v (g), a set m ⊆ v (g) is called minimal detour monophonic dominating set, if no proper subset of m is a detour monophonic dominating set. the maximum cardinality among all minimal monophonic dominating sets is called upper detour monophonic domination number and is denoted by γ+ dm (g). for any two positive integers p and q with 2 ≤ p ≤ q there is a connected graph g with γm(g) = γdm(g) = p and γ + dm (g) = q. for any three positive integers p, q, r with 2 < p < q < r, there is a connected graph g with m(g) = p, γdm(g) = q and γ+ dm (g) = r. let p and q be two positive integers with 2 < p < q such that γdm(g) = p and γ+ dm (g) = q. then there is a minimal dmd set whose cardinality lies between p and q. let p, q and r be any three positive integers with 2 ≤ p ≤ q ≤ r. then, there exist a connected graph g such that γdm(g) = p, γ + dm (g) = q and |v (g)| = r. resumen este artículo introduce el concepto de número de dominación de desvío monofónico superior de un grafo. para un grafo conexo g con conjunto de vértices v (g), un conjunto m ⊆ v (g) se llama conjunto dominante de desvío monofónico minimal, si ningún subconjunto propio de m es un conjunto dominante de desvío monofónico. la cardinalidad máxima entre todos los conjuntos dominantes de desvío monofónico minimales se llama número de dominación de desvío monofónico superior y se denota por γ+ dm (g). para cualquier par de enteros positivos p y q con 2 ≤ p ≤ q existe un grafo conexo g con γm(g) = γdm(g) = p y γ + dm (g) = q. para cualquiera tres enteros positivos p, q, r con 2 < p < q < r, existe un grafo conexo g con m(g) = p, γdm(g) = q y γ+ dm (g) = r. sean p y q dos enteros positivos con 2 < p < q tales que γdm(g) = p y c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300315 316 m. mohammed abdul khayyoom cubo 22, 3 (2020) γ+ dm (g) = q. entonces existe un conjunto dmd mínimo cuya cardinalidad se encuentra entre p y q. sean p, q y r tres enteros positivos cualquiera con 2 ≤ p ≤ q ≤ r. entonces existe un grafo conexo g tal que γdm(g) = p, γ + dm (g) = q y |v (g)| = r. keywords and phrases: monophonic number, domination number, detour monophonic number, detour monophonic domination number, upper detour monophonic domination number. 2020 ams mathematics subject classification: 05c69, 05c12. 1 introduction consider an undirected connected graph g(v, e) without loops or multiple edges. let p : u1, u2, ...un be a path of g. an edge e is said to be a chord of p if it is the join of two non adjacent vertices of p . a path is said to be monophonic path if there is no chord. if s is a set of vertices of g such that each vertex of g lies on an u − v monophonic path in g for some u, v ∈ s, then s is called monophonic set. monophonic number is the minimum cardinality among all the monophonic sets of g. it is denoted by m(g) [1,2]. a vertex v in a graph g dominates itself and all its neighbours. a set t of vertices in a graph g is a dominating set if n[t ] = v (g). the minimum cardinality among all the dominating sets of g is called domination number and is dented by γ(g)[4]. a set t ⊂ v (g) is a monophonic dominating set of g if t is both monophonic set and dominating set. the monophonic domination number is the minimum cardinality among all the monophonic dominating sets of g and is denoted by γm(g)[5,6]. a monophonic set m in a connected graph g is minimal monophonic set if no proper subset of m is a monophonic set. the upper monophonic number is the maximum cardinality among all minimal monophonic sets and is denoted by m+(g)[9]. the shortest x − y path is called geodetic path and longest x − y monophonic path is called detour monophonic path. if every vertex of g lies on a x−y detour monophonic path in g for some x, y ∈ m ⊆ v (g), m could be identified as a detour monophonic set. the minimum cardinality among all the detour monophonic set is the detour monophonic number and is denoted by dm(g). a minimal detour monophonic set d of a connected graph g is a subset of v (g) whose any proper subset is not a detour monophonic set of g. the maximum cardinality among all minimal detour monophonic sets is called upper detour monophonic set, denoted by dm+(g) [10]. if d is both a detour monophonic set and a dominating set, it could be a detour monophonic dominating set. the minimum cardinality among all detour monophonic dominating sets of g is the detour monophonic dominating number ( dmd number) and is denoted by γdm(g)[7,8]. a vertex v is an extreme vertex if the sub graph induced by its neighbourhood is complete. a vertex u in a connected graph g is a cut-vertex of g, if g − u is disconnected. in this article, we consider cubo 22, 3 (2020) characterization of upper detour monophonic domination number 317 g as a connected graph of order n ≥ 2 if otherwise not stated. for basic notations and terminology refer [3]. theorem 1.1 (8). each extreme vertex of a connected graph g belongs to every detour monophonic dominating set of g. example 1.1. consider the graph g given in figure 1. here m1 = {v1, v4} is a monophonic set. therefore m(g) = 2. m1 also dominate g. hence γ(g) = 2. the set m2 = {v1, v2, v3} is a minimum detour monophonic set. thus dm(g) = 3. m2 does not dominate g. m2 ∪ {v4} is a minimum dmd set. therefore γdm(g) = 4. 2 udmd number of a graph definition 2.1. a detour monophonic dominating set m in a connected graph g is called minimal detour monophonic dominating set if no proper subset of m is a detour monophonic dominating set. the maximum cardinality among all minimal detour monophonic dominating sets is called upper detour monophonic domination number and is denoted by γ+ dm (g). figure 1: graph g with udmd number 5 example 2.1. consider the graph g given in figure 1. the set m = {v1, v5, v6, v7, v8} is a minimal dmd set with maximum cardinality. therefore γ+ dm (g) = 5. theorem 2.1. let g be a connected graph and v an extreme vertex of g. then v belongs to every minimal detour monophonic dominating set of g. proof. every minimal detour monophonic dominating set is a minimum detour monophonic set. since each extreme vertex belongs to every minimum detour monophonic dominating set, the result follows. 318 m. mohammed abdul khayyoom cubo 22, 3 (2020) theorem 2.2. let v be a cutvertex of a connected graph g. if m is a minimal dmd set of g, then each component of g − v have an element of m. proof. suppose let a is a component of g − v having no vertices of m. let u be any one of the vertex in a. since m is a minimal dmd set, there exist two vertices p, q in m such that u lies on a p − q detour monophonic path p : p, u0, u1, ..., u, ..., um = q in g. consider two sub-paths p1 : p − u and p2 : u − q of p . given v is a cut-vertex of g. therefore both p1 and p2 contain v. hence p is not a path. this is a contradiction. that is, each component of g − v have an element of every minimal dmd set. theorem 2.3. for a connected graph g of order n, γdm(g) = n if and only if γ + dm (g) = n. proof. first, suppose γ+ dm (g) = n. that is m = v (g) is the unique minimal dmd set of g, so that no proper subset of m is a dmd set. hence m is the unique dmd set. therefore γdm(g) = n. conversely, let γdm(g) = n. since every dmd set is a minimal dmd set, γdm(g) ≤ γ + dm (g). therefore γ+ dm (g) ≥ n. since v (g) is the maximum dmd set, γ+ dm (g) = n. 3 udmd number of some standard graphs example 3.1. complete bipartite graph km,n for complete bipartite graph g = km,n, γ+ dm (g) =          2, if m = n = 1; n, if n ≥ 2, m = 1; 4, if m = n = 3 max{m, n} if m, n ≥ 2, m, n 6= 3 proof. case (i): let m = n = 1. then km,n = k2. therefore γ + dm (g) = 2. case (ii): let n ≥ 2, m = 1. this graph is a rooted tree. there are n end vertices. all these are extreme vertices. therefore they belong to every dmd set and consequently every minimal dmd set. case (iii): if m = n = 3, then exactly two vertices from both the particians form a minimal dmd set. case (iv): let m, n ≥ 2, m, n 6= 3. assume that m ≤ n. let a = {a1, a2, ...am} and b = {b1, b2, ...bn} be the partitions of g. first, prove m = b is a minimal dmd set. take a vertex aj, 1 ≤ j ≤ m, which lies in a detour monophonic path biajbk for k 6= j so that m is a detour monophonic set. they also dominate g. hence m is a dmd set. cubo 22, 3 (2020) characterization of upper detour monophonic domination number 319 next, let s be any minimal dmd set such that |s| > n. then s contains vertices from both the sets a and b. since a and b are themselves minimal dmd sets, they do not completely belongs to s. note that if s contains exactly two vertices from a and b, then it is a minimum dmd set. thus γ+ dm (g) = n = max{m, n}. example 3.2. complete graph kn for complete graph g = kn, γ + dm (g) = n. proof. for a complete graph g, every vertex in g is an extreme vertex. by theorem 2.1 they belong to every minimal dmd set. example 3.3. cycle graph cn for cycle graph g = cn with n vertices , γ+ dm (g) =        3, if n ≤ 7, n 6= 4 2, if n = 4 4 + n − 7 − r 3 , if n ≥ 8, n − 7 ≡ r mod(3) proof. for n ≤ 7 the results are trivial. for n ≥ 8, let cn : v1, v2, v3, ..., vn, v1 be the cycle with n vertices. then the set of vertices {v1, v3, vn−1} is a minimal detour monophonic set but not dominating. this set dominates only seven vertices. there are n − 7 remaining vertices. if r is the reminder when n − 7 is divided by 3, then n − 7 − r 3 + 1 vertices dominate the remaining vertices. therefore every minimal dmd set contains 4 + n − 7 − r 3 vertices. 4 characterization of γ+dm(g) theorem 4.1. for any two positive integers p and q with 2 ≤ p ≤ q there is a connected graph g with γm(g) = γdm(g) = p and γ + dm (g) = q. proof. construct a graph g as follows. let c6 : u1, u2, u3, u4, u5, u6, u1 be the cycle of order 6. join p−1 disjoint vertices m1 = {x1, x2, ..., xp−1} with the vertex u1. let m2 = {y1, y2, ..., yq−p−1} be a set of q − p− 1 disjoint vertices. add each vertex in m2 with u4 and u6. let xp−1 be adjacent with u2 and u6. this is the graph g given in figure 2. since all vertices except xp−1 in m1 are extreme, they belong to every minimum monophonic dominating set and dmd set. the set m = m1 ∪ {u4} is a minimum monophonic dominating set. therefore γm(g) = p. moreover, the set of all vertices in m form a dmd set and is minimum. that is γdm(g) = p. 320 m. mohammed abdul khayyoom cubo 22, 3 (2020) next, we prove that γ+ dm (g) = q. clearly n = m1 ∪ m2 ∪ {u5, u6} is a dmd set. n is also a minimal dmd set of g. for the proof, let n′ be any proper subset of n. then there exists at least one vertex u ∈ n and u /∈ n′. if u = yi, for 1 ≤ i ≤ q − p − 1, then yi does not lie on any x − y detour monophonic path for some x, y ∈ n′. similarly if u ∈ {u5, u6, xp−1}, then that vertex does not lie on any detour monophonic path in n′. thus n is a minimal dmd set. therefore γ+ dm (g) ≥ q. figure 2: γm(g) = γdm(g) = p and γ + dm (g) = q. note that n is a minimal dmd set with maximum cardinality. on the contrary, suppose there exists a minimal dmd set, say t , whose cardinality is strictly greater than q. then there is a vertex u ∈ t, u /∈ n. therefore u ∈ {u2, u3, u4}. if u = u4, then m1 ∪ {u4} is a dmd set properly contained in t which is a contradiction. if u = u3, then the set m1 ∪ {u3, u5} is a dmd set which is a proper subset of t and is a contradiction. if u = u2, then the set (n − {u6}) ∪ {u2} is a dmd set properly contained in t and is a contradiction. thus γ+ dm (g) = q. theorem 4.2. for any three positive integers p, q, r with 2 < p < q < r, there is a connected graph g with m(g) = p, γdm(g) = q and γ + dm (g) = r. proof. let g be the graph constructed as follows. take q − p copies of a cycle of order 5 with each cycle ci has a vertex set {di, ei, fi, gi, hi}, for 1 ≤ i ≤ q − p. join each ei with all other vertices in ci. also join the vertex fi−1 of ci−1 with the vertex di of ci. let {u, v} and {b1, b2, ..., br−q+1} be two sets of mutually non adjacent vertices. join each bi with u and v, for 1 ≤ i ≤ r − q + 1. join another p − 2 pendent vertices with u and one pendent vertex with d1. this is the graph g given in figure 3. the set m1 = {a0, a1, a2..., ap−2} is the set of all extreme vertices and belongs to every monophonic dominating set and dmd set (theorem 1.1). clearly m1 is not monophonic. but m1 ∪ {v} is a monophonic set and is minimum. therefore m(g) = p. take m2 = {e1, e2, ..., eq−p}. then m1 ∪ m2 ∪ {v} is a dmd set and is minimum. therefore γdm(g) = p − 1 + q − p + 1 = q. cubo 22, 3 (2020) characterization of upper detour monophonic domination number 321 figure 3: graph g with m(g) = p, γdm(g) = q and γ + dm (g) = r. let m3 = {b1, b2, ..., br−q+1}. then m = m1 ∪ m2 ∪ m3 is a dmd set. now m is a minimal dmd set. on the contrary, suppose n is any proper dmd subset of m so that there exists at least one vertex in m which does not belong to n. let u ∈ m and u /∈ n. clearly u /∈ m1 since m1 is the set of all extreme vertices. if u = ei for some i, then the vertex ei does not belong to any detour monophonic path induced by n. therefore u /∈ m2. similarly u /∈ m3. this is a contradiction. hence m is a minimal dmd set with maximum cardinality. therefore γ+ dm (g) = |m1| + |m2| + |m3| = (p − 1) + (q − p) + (r − q + 1) = r. theorem 4.3. let p and q be two positive integers with 2 < p < q such that γdm(g) = p and γ+ dm (g) = q. then there is a minimal dmd set whose cardinality lies between p and q. proof. consider three sets of mutually disjoint vertices m1 = {a1, a2, ..., aq−n+1}, m2 = {b1, b2, ..., bn−p+1} and m3 = {x, y, z}. join each vertex ai with x and z and each vertex bj with y and z. add p − 2 pendent vertices m4 = {c1, c2, ..., cp−2} with the vertex y. this is the graph g given in figure 4. since m4 is the set of all extreme vertices, it belongs to every dmd set. but m4 is not a dmd set. the set m = m4 ∪ {x, z} is a minimum dmd set. therefore γdm(g) = p. consider the set n = m1 ∪ m2 ∪ m4. we claim n is a minimal dmd set with maximum cardinality. on the contrary, suppose there is a set n′ ⊂ n which is a dmd set of g. then there exists at least one vertex, say u in n which does not belong to n′. clearly u /∈ m4 since it is the set of all extreme vertices. if u ∈ m1, then u = ai for some i. then the vertex ai does not lie on any detour monophonic path, which is a contradiction. similarly, if u ∈ m2, we get a contradiction. thus n is a minimal dmd set. therefore γ+ dm (g) ≥ q. 322 m. mohammed abdul khayyoom cubo 22, 3 (2020) figure 4: graph g with γdm(g) = p and γ + dm (g) = q next, we claim that n has the maximum cardinality of any minimal dmd set. if γ+ dm (g) > q, there is at least one vertex v ∈ v (g), v /∈ n and belongs to a minimal dmd set. therefore v ∈ m3. if v = x, then the set m2 ∪ m4 ∪ {v} is a minimal dmd set having less than q vertices. similarly if v = z, then the set m1 ∪ m4 ∪ {v} is a minimal dmd set. for v = y, the set n ∪ {y} is not a minimal dmd set. therefore γ+ dm (g) ≤ q. let n be any number which lies between p and q. then there is a minimal dmd set of cardinality n. for the proof, consider the set t = m2 ∪ m4 ∪ {x}. t is a minimal dmd set. if t is not a minimal dmd set, there is a proper subset t ′ of t such that t ′ is a minimal dmd set. let u ∈ t and u /∈ t ′. since each vertex in m4 is an extreme vertex, v /∈ m4. if u = x, then the vertex u is not an internal vertex of any detour monophonic path in t ′. a similar argument may be made if u ∈ m2. this leads to a contradiction. therefore t is a minimal dmd set with cardinality (n − p + 1) + (p − 2) + 1 = n. theorem 4.4. let p , q and r be any three positive integers with 2 ≤ p ≤ q ≤ r. then, there exists a connected graph g such that γdm(g) = p, γ + dm (g) = q and |v (g)| = r. proof. let k1,p is a star graph with leaves set m1 = {u1, u2, ..., up} and let u be the support vertex of k1,p. insert r − q − 1 vertices m2 = {v1, v2, ..., vr−q−1} in the edges uui respectively for 1 ≤ i ≤ r − q − 1. add q − p vertices m3 = {x1, x2, ..., xq−p} with this graph and join each xi with u and u1. this is the graph g as shown in figure 5. here |v (g)| = (q −p)+p+(r −q −1)+1 = r. the length of a detour monophonic path is 4. cubo 22, 3 (2020) characterization of upper detour monophonic domination number 323 figure 5: graph g with γdm(g) = p and γ + dm (g) = q let t = m1 −{u1}. all the vertices in t are extreme vertices and belong to all dmd sets and minimal dmd sets. clearly m1 is a dmd set with minimum cardinality. therefore γdm(g) = p. let n = t ∪ m3 ∪ {v1}. then |n| = (p − 1) + (q − p) + 1 = q. we claim that n is a minimal dmd set with maximum cardinality. on the contrary, suppose there is a proper subset n′ of n which is a minimal dmd set of g. then there exists at least one vertex x ∈ n, x /∈ n′. clearly x /∈ t . if x ∈ m3, then x = xi for some i, 1 ≤ i ≤ q − p. then the vertex xi does not lie on any u − v detour monophonic path for u, v ∈ n′. if x = v1 then v1 does not lies on any detour monophonic path in n ′. thus no such vertex x exists. this is a contradiction. therefore γ+ dm (g) ≥ q. to prove maximum cardinality of n, suppose there exists a minimal dmd set s with |s| > q. since s contains t , the set of all extreme vertices, the vertex x lies on some u−v detour monophonic path for all x ∈ {u, v2, v3, .., vr−q−1}. now s is a minimal dmd set having more than q vertices and u, v2, v3, ..., vr−q−1 /∈ s. therefore s = {v1} ∪ m3 ∪ {u1} ∪ t . then n is properly contained in s. this is a contradiction. therefore γ+ dm (g) = q. hence the proof. 324 m. mohammed abdul khayyoom cubo 22, 3 (2020) references [1] p. a. p. sudhahar, m. m. a. khayyoom and a. sadiquali, “edge monophonic domination number of graphs”. j.adv.in mathematics, vol. 11, no. 10, pp. 5781–5785, 2016. 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[10] p. titus, a. p. santhakumaran, k. ganesamoorthy, “upper detour monophonic number of a graph”, electronic note in discrete mathematics, vol. 53, pp. 331–342, 2016. introduction udmd number of a graph udmd number of some standard graphs characterization of dm+(g) cubo, a mathematical journal vol. 23, no. 02, pp. 245–264, august 2021 doi: 10.4067/s0719-06462021000200245 approximate solution of abel integral equation in daubechies wavelet basis jyotirmoy mouley 1 m. m. panja 2 b. n. mandal 3 1 department of applied mathematics, university of calcutta, 92, a.p.c. road, kolkata-700 009, india. jyoti87.cu.wavelet@gmail.com 2 department of mathematics,visva-bharati, santiniketan, west bengal, 731235, india madanpanja2005@yahoo.co.in 3 physics and applied mathematics unit, indian statistical institute, 203, b.t. road, kolkata-700108, india. bnm2006@rediffmail.com abstract this paper presents a new computational method for solving abel integral equation (both first kind and second kind). the numerical scheme is based on approximations in daubechies wavelet basis. the properties of daubechies scale functions are employed to reduce an integral equation to the solution of a system of algebraic equations. the error analysis associated with the method is given. the method is illustrated with some examples and the present method works nicely for low resolution. resumen este art́ıculo presenta un nuevo método computacional para resolver la ecuación integral de abel (tanto de primer como de segundo tipo). el esquema numérico está basado en aproximaciones en la base de ondeletas de daubechies. se emplean las propiedades de las funciones de escala de daubechies para reducir una ecuación integral a la solución de un sistema algebraico de ecuaciones. se entrega el análisis de error asociado con el método. el método es ilustrado con algunos ejemplos donde el método presentado funciona bien en baja resolución. keywords and phrases: abel integral equation, daubechies scale function, daubechies wavelet, gauss-daubechies quadrature rule. 2020 ams mathematics subject classification: 45d05. accepted: 20 may, 2021 received: 02 may, 2020 c©2021 j. mouley et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200245 https://orcid.org/0000-0001-8353-0813 https://orcid.org/0000-0002-3690-6395 https://orcid.org/0000-0001-9771-1287 246 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) 1 introduction the theory of integral equations is a very important tool to deal with problems arising in mathematical physics. abel integral equation appears in many physical problems of water waves, astrophysics, solid mechanics and in many applied sciences (see [1, 2, 3, 4]). in the year 1823, abel integral equation was derived directly from the tautochorone problem in physics. in fact this gave birth to the topic known as integral equation. before 1930, the branch of mathematics which is related to wavelet began with joseph fourier with his theories of frequency analysis, now often referred to fourier synthesis (see [5]). the concept of wavelet was first mentioned in an appendix of the thesis of a. haar (see [6]), but the formulation of problems involving wavelets has been developed mostly over last 30 years. grossman and morelet [7] developed the continuous wavelet transform and the orthogonal one was developed by lamarie and meyer [8]. daubechies (see [9, 10]) constructed a compactly supported orthogonal wavelet basis that can be generated from a single function with the aim to serve the multiresolution analysis (mra of l2 (r)). wavelets allow to represent variety of functions and operators very accurately. furthermore, wavelets setup a connections with fast numerical algorithms [11]. hence wavelets are used as an efficient tool to solve integral equations. in this paper we consider the abel integral equations in the form first kind : ∫ x 0 y (t) dt (x − t)µ = f (x) , (1.1) second kind : y(x) + λ ∫ x 0 y (t) dt (x − t)µ = f (x) . (1.2) here 0<µ<1, 0 ≤ x ≤ 1 and the forcing term f(x) ∈ c[0,1] in order to confirm the existence and uniqueness of the solution y(x) ∈ c[0,1], the space of all continuous function defined on [0,1]. the abel integral equation has been solved earlier analytically and numerically by various methods in the literature. for instance, yousefi [12] constructed a numerical scheme based on legendre multiwavelets to solve abel integral equation. a system of generalized abel integral equations was solved using fractional calculus by mandal et al [13]. liu and tao [14] applied mechanical quadrature methods for solving first kind abel integral equation. numerical solution of abel integral equation is obtained using orthogonal functions by derili and sohrabi [15]. alipour and rostamy [16] used bernstein polynomials to solve abel integral equations. shahsavaram [17] used haar wavelet as the basis function in the collocation method to solve volterra integral equation with weakly singular kernel. in this paper, the unknown function in the integral equation is expanded by employing daubechies wavelet basis with unknown coefficients. the integral equation is converted into a system algebraic equations utilizing the properties of daubechies scale functions. after evaluating the unknown coefficients, the values of the unknown function in the integral equations can be determined at any cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 247 dyadic point in [0,1]. 2 preliminary concept of daubechies scale function here some important properties of daubechies scale function with a compact support are presented in a finite interval [a,b] ⊂ r , where a and b(> a) are integers. 2.1 two-scale relations daubechies constructed a whole new class of orthogonal wavelets that can be generated from a single function φ(x), known as daubechies scale function. this scale function has some interesting features like compact support, fractal nature, and unknown structure at all resolutions. daubechies -k (dau-k) scale function(k ∈ n) has 2k filter coefficients and compact support [0,2k −1]. the two-scale relation of scale function is given by φ(·) = √ 2ht φ(·), (2.1) where h = [h0,h1,h2, ...,h2k−1] t 2k×1 (2.2) and φ(·) = [φ(2·),φ(2 · −1),φ(2 · −2), ...,φ(2 · −2k + 1)]t2k×1 (2.3) with the normalization condition ∫ r φ(x)dx = 1. (2.4) the elements hl (l = 0,1,2, ...,2k − 1) are known as filter coefficients or low pass filters. these filter coefficients satisfy the following algebraic relations 2k−1 ∑ l=0 hl = √ 2 ; 2k−1 ∑ l=0 hlhl−2m = δm0. (2.5) here we define two operators, one is the translation operator t and other is the scale transformation operator d as t kφ(x) = φk(x) = φ(x − k) (2.6) and djφ(x) = 2 j 2 φ(2jx). (2.7) for a specific value of resolution j, the translate of scaling functions are orthonormal to each other viz. ∫ r φjk1 (x)φjk2 (x)dx = δk1k2 (2.8) 248 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) where φjk(x) = 2 j 2 φ(2jx − k). (2.9) it is evident that all the properties of scaling functions are applicable on r. but in the finite interval [a,b] the translation property (2.6) does not hold good for all k ∈ z as well as the orthogonalization condition (2.8) cannot be applied for φjk(x). so in order to apply the machinery of dau-k scale function on a finite interval [a,b], we divide the translate of φ(x) for a specific resolution j into three classes (cf. mouley et al. [18] and panja et al. [19]) φljk(·) = φjk(·)χk(x) if k ∈ { a2j − 2k + 2, ...,a2j − 1 } , φi jk (·) = φjk(·)χk(x) if k ∈ { a2j, ...,b2j − 2k + 1 } , φrjk(·) = φjk(·)χk(x) if k ∈ { b2j − 2k + 2, ...,b2j − 1 } . (2.10) here χk(x) is the characteristic function assuming the value 1 or 0 according as x ∈ [a,b] or x 6∈ [a,b]. 2.2 scale function at dyadic points a number of the form m 2n is known as a dyadic fraction or dyadic rational (m is an integer and n is a natural number). it has extensive application in measurement, the inch is normally subdivided in dyadic rather than decimal fraction. the ancient egyptians also used dyadic fractions in measurement, with denominators up to 64 [20]. after knowing the value of scale function at integer points within support, it is possible to determine the scale function at any dyadic point with in the support [21] . using the two-scale relation (2.1) the value of dau-k scale function φ(x) at x = m 2n is calculated as φ (m 2n ) = 2k−1 ∑ l1=0 √ 2hl1φ ( m − 2n−1l1 2n−1 ) . (2.11) again using the two-scale relation (2.1) we get φ (m 2n ) = 2k−1 ∑ l1=0 2k−1 ∑ l2=0 2hl1hl2φ ( m − 2n−1l1 − 2n−2l2 2n−2 ) . (2.12) repeating the two-scale relation (2.1) n times, we get φ (m 2n ) = 2k−1 ∑ l1=0 2k−1 ∑ l2=0 ... 2k−1 ∑ ln=0 2 m 2 hl1hl2...hlnφ(m − 2n−1l1 − 2n−2l2...2ln−1 − ln). (2.13) 3 multiresolution analysis (mra) and daubechies wavelet basic concepts of mra and daubechies wavelet are discussed in most of the texts on wavelets (see [9, 10, 18, 19, 21]). why wavelet has started to dominate in different applications such as technology, engineering and applied mathematics, one serious reason behind it is mra. a mra cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 249 on r is defined as a sequence of nested subspaces vj of function l 2 on r with scaling function φ(x) if the following properties hold, ∀j ∈ z, vj ⊆ vj+1, (3.1) closl2 (∪j∈zvj) = l2 (r) , (3.2) ∩j∈z vj = {0} , (3.3) φ(x) ∈ vj ⇔ φ(2x) ∈ vj+1, ∀j ∈ z. (3.4) here vj’s are called approximation spaces. the scale function φ(x) belongs to v0 and the set {φ(x − k) : k ∈ z} is a riesz basis of v0. the scale function φ(x) satisfies the two-scale relation (2.1). also the set {φjk(x) : k ∈ z} given by (2.9) is a riesz basis of vj. from the property (3.1), it is evident that each element of vj+1 can be uniquely written as the orthogonal sum of an element in vj and an element in wj that contains the complementary details i.e. vj+1 = vj ⊕ wj = v0 ⊕ w0 ⊕ w1 ⊕ w2 ⊕ ... ⊕ wj. (3.5) let wj be the span of ψjk(x) = 2 j 2 ψ(2jx − k), which is called wavelet function. the wavelet function ψ(x) satisfies the relation ψ(·) = √ 2gt φ(·) (3.6) where g = [g0,g1,g2, ...,g2k−1] t 2k×1. (3.7) here φ(·) is given by (2.3) and gl (l = 0,1,2, ...,2k − 1) are known as high pass filter coefficients and are given by gl = (−1)lh2k−1−l. (3.8) 4 method of approximation we approximate the unknown function of the integral equations (1.1) and (1.2) in the approximation space vj as y(x) ≈ ymsj (x) = 2j−1 ∑ k=0 cjkφjk(x) = 2j−2k+1 ∑ k=0 cijkφ i jk(x) + 2j−1 ∑ k=2j −2k+2 crjkφ r jk(x) = ct ~φ(x). (4.1) 250 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) as the support of φ(x) is [0,2k − 1], so ymsj (x) always vanishes at x = 0. the value of y(x) at x = 0 for second kind abel integral equation is obviously f(0) but for the first kind abel integral equation y(x) cannot be evaluated at x = 0 but as y(x) can be evaluated at any dyadic point in (0,1], it can be evaluated very close to x = 0 by making the resolution fairly large. here c and ~φ(x) both are 2j × 1 vectors, given by c = [ cij0,c i j1, ...,c i j2j −2k+1,c r j2j−2k+2, ...,c r j2j −1 ]t (4.2) and ~φ(x) = [ φij0(x),φ i j1(x), ...,φ i j2j −2k+1(x),φ r j2j −2k+2(x), ...,φ r j2j −1(x) ]t . (4.3) using the approximate form of y(x) in (4.1) in both the first and second kind integral equations (1.1) and (1.2) we get, c t ∫ x 0 ~φ(t)dt (x − t)µ = f (x) (4.4) and c t [ ~φ(x) + λ ∫ x 0 ~φ(t)dt (x − t)µ ] = f (x) . (4.5) we choose total 2j number of points by xjk′ = k ′ 2j (k ′ = 1,2,3, ...,2j) and substituting these points in both the equations (4.4) and (4.5) we get, c t b (k ′ ) = f ( k ′ 2j ) (4.6) and c t [ a (k ′ ) + λb(k ′ ) ] = f ( k ′ 2j ) (4.7) where a (k ′ ) = ~φ ( k ′ 2j ) = [ φij0 ( k ′ 2j ) ,φij1 ( k ′ 2j ) , ...,φij2j −2k+1 ( k ′ 2j ) ,φrj2j −2k+2 ( k ′ 2j ) , ...,φrj2j −1 ( k ′ 2j )]t (4.8) and b (k ′ ) =   ∫ k ′ 2j 0 φij0 (t) dt (k ′ 2j − t)µ , ..., ∫ k ′ 2j 0 φi j2j −2k+1 (t) dt (k ′ 2j − t)µ , ∫ k ′ 2j 0 φr j2j −2k+2 (t) dt (k ′ 2j − t)µ , ..., ∫ k ′ 2j 0 φr j2j −1 (t) dt (k ′ 2j − t)µ   t . (4.9) as k = 0,1,2, ...,2j − 1 and k′ = 1,2,3, ....,2j, each of the equation (4.6) and (4.7) represents a system of 2j equations in 2j variables ci jk and cr jk . solving these systems the unknown coefficients cijk and c r jk are obtained. cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 251 in the last part of this section, we explain the procedure for calculating the matrix elements of the matrix b(k ′ ). we use the notation iµ j(k ′ ,k) = ∫ k ′ 2j 0 φjk (t) dt (k ′ 2j − t)µ . (4.10) in the relation (4.10), for 0 ≤ k ≤ 2j −2k+1, φjk (t) means φijk (t) and for 2j −2k+2 ≤ k ≤ 2j −1, φjk (t) means φ r jk (t) . using (2.9) we find iµ j(k ′ ,k) = 2 ( µ− 1 2 ) j lµ(k ′ − k), (4.11) where lµ(k) = ∫ k 0 φ(t) dt (k − t)µ . (4.12) as the support of dau-k scale function φ(t) is [0,2k−1], so if k ≤ 0 the range of the integration in (4.12) is completely outside of the support. in this case lµ(k) vanishes. again if k ≥ 2k,lµ(k) has no singularity within the support [0,2k − 1]. using gauss-daubechies quadrature rule involving daubechies scale function [22], lµ(k) is evaluated as lµ(k) = m ∑ i=1 wi (k − ti)µ , (k ≥ 2k). (4.13) here wi , ti are weights are nodes of gauss-daubechies quadrature rule involving daubechies scale function [22]. for 0 < k ≤ 2k − 1, lµ(k) has integrable singularity at the upper limit so that evaluation of such integrals by using the quadrature rule may not provide their approximate value with desired order of accuracy within less computational time. the two-scale relation (2.1) for φ(t), may be used to obtain a recurrence relation for lµ(k) as lµ(k) = 2µ− 1 2 2k−1 ∑ l=0 hllµ(2k − l). (4.14) using the symbols hk =           h1 h0 0 0 · · · 0 0 h3 h2 h1 h0 · · · 0 0 ... ... ... ... · · · ... ... 0 0 0 0 · · · h2k−2 h2k−3 0 0 0 0 · · · 0 h2k−1           (4.15) and bµ k =           0 0 ... ∑2k−4 l=0 hllµ(4k − 4 − l) ∑2k−2 l=0 hllµ(4k − 2 − l)           (4.16) 252 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) the relation (4.14) can be put in the form ( i − 2µ− 12 hk ) lµ = bµ k. (4.17) so, the singular integrals in lµ are found as lµ = ( i − 2µ− 12 hk )−1 bµ k. (4.18) thus, evaluation of lµ(k) is summarized as l(k) =            0 k ≤ 0, solution obtained by (4.18) 1 ≤ k ≤ 2k − 1, ∑m i=0 wi (k − ti)µ k ≥ 2k. (4.19) table 1: values of l(k) k µ = 1 4 µ = 1 3 µ = 1 2 1 0.925995 1.098666 1.643812 2 1.064183 1.042183 0.954199 3 0.808341 0.759600 0.682604 4 0.748236 0.679445 0.560703 5 0.699178 0.620553 0.488824 in table 1 the values of l(k) for k = 1,2, ...,5 are given taking dau-3 scale function for µ = 1 4 , 1 3 , 1 2 . for other values of µ (0 < µ < 1) these can be easily calculated. table 2: accuracy of l(2k) for dau-3 scale function µ detemined by (4.13) detemined by (4.18) 1/4 0.662722 0.662722 1/3 0.577792 0.577792 1/2 0.439182 0.439182 in table 2 the values of l(2k) for dau-3 scale function are presented for µ = 1 4 , 1 3 , 1 2 using the relations (4.13) and (4.18) separately. for the two methods the values of l(2k) are found to be same. the values of l(2k) establish the efficiency of the relation (4.18) in the determination of l(k) (k = 0,1,2, ...,2k − 1). cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 253 5 error estimation in this section, the error of the proposed method is estimated in detail. for this we need the following definitions and theorems. definition 5.1 ([23]). in a σ-finite measure space (x,f,µ∗) (x denotes underlying space, f is the σ-algebra of measurable sets and µ∗ is the measure) the lp-norm (1 ≤ p < ∞) of a function f is defined by ‖f‖lp(x, f, µ∗) = ( ∫ x |f(x)|pdµ∗(x) ) 1 p . the abbreviations ‖f‖lp(x) , ‖f‖lp , ‖f‖p are also used to mean lpnorm. definition 5.2 ([24]). the inner product of two functions f and g on a measure space x is defined by < f,g >= ∫ x fḡdµ. theorem 5.3 (minkowski [23]). if 1 ≤ p < ∞ and f,g ∈ lp then f + g ∈ lp and ‖f + g‖lp ≤ ‖f‖lp + ‖g‖lp. theorem 5.4. let {φjk(x) : k ∈ z} and {ψjk(x) : k ∈ z} be the riesz bases of approximation space vj and detail space wj. if n b j:k,k′ = ∫ b a φbjk(x)φ b jk′ (x)dx and t b j:k,k′ = ∫ b a ψbjk(x)ψ b jk′ (x)dx (b stands for l or r) then t bj:k,k′ = 2k−1 ∑ l1=0 2k−1 ∑ l2=0 gl1gl2n b j+1:2k+l1,2k′+l2 . proof. here nbj:k,k′ = ∫ b a φbjk(x)φ b jk′ (x)dx. now t bj:k,k′ = ∫ b a ψbjk(x)ψ b jk′ (x)dx = 2j ∫ b a ψb(2jx − k)ψb(2jx − k′)dx (using expression of ψj,k(x)) = ∫ b2j a2j ψb(z − k)ψb(z − k′)dz = 2k−1 ∑ l1=0 2k−1 ∑ l2=0 gl1gl2 ∫ b2j+1 a2j+1 φb(z − 2k − l1)φb(z − 2k′ − l2)dz (using equation (3.6)) = 2k−1 ∑ l1=0 2k−1 ∑ l2=0 gl1gl2n b j+1:2k+l1,2k′+l2 . this completes the proof. 254 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) so to evaluate t bj:k,k′ , we need to evaluate n b j+1:2k+l1,2k′+l2 (l1, l2 = 0,1,2, ...,2k − 1). the values of nb j:k,k′ are tabulated in table 3 and table 4 in [25]. in section 3 to find the approximate solution, the projection of the unknown function ymsj (x) is used in the approximation space (the linear span of φjk(x),k = 0,1,2, ....2 j − 1). to estimate the error of the unknown function y(x) ∈ l2([0,1]) satisfying both the integral equations (1.1) and (1.2), we employ the fact that the multiscale expansion of y(x) (the projection of y(x) into the approximation space vj and detail space wj) is y(x) = 2j−1 ∑ k=0 cjkφjk(x) + ∞ ∑ j′=j 2j ′ −1 ∑ k=0 dj′kψj′k(x) (5.1) where cjk ≈ ∫ 1 0 φjk(x)y(x)dx, (5.2) and djk ≈ ∫ 1 0 ψjk(x)y(x)dx. (5.3) using the two-scale relation (2.1) and the equation (3.6), (5.2) and (5.3) are reduced to cjk = 2k−1 ∑ l=0 hlcj+1,2k+l , (5.4) djk = 2k−1 ∑ l=0 glcj+1,2k+l. (5.5) to evaluate cjk and djk,(k = 0,1,2, ...,2 j − 1) at level j, we need the values of cj+1,2k+l and dj+1,2k+l at level j +1. if 0 ≤ k ≤ 2j −2k +1, cjk and djk are denoted by cijk and dijk respectively. again if 2j − 2k + 2 ≤ k ≤ 2j − 1, cjk and djk are denoted by crjk and drjk respectively. now using the expression for ymsj (x) given by (4.1), (5.1) is reduced to y(x) = ymsj (x) + ∞ ∑ j′=j δyj′ (5.6) where δyj′ is given by δyj′ = 2j ′ −1 ∑ k=0 dj′kψj′k(x) = 2j ′ −2k+1 ∑ k=0 dij′kψ i j′k(x) + 2j ′ −1 ∑ k=2j ′ −2k+2 drj′kψ r j′k(x). (5.7) the error in the multiscale approximation is given by e(x) = y(x) − ymsj (x) = ∞ ∑ j′=j δyj′. (5.8) cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 255 now ‖e(x)‖2l2[0,1] = ∥ ∥ ∥ ∥ ∥ ∥ ∞ ∑ j′=j δyj′ ∥ ∥ ∥ ∥ ∥ ∥ 2 l2[0,1] ≤ ∞ ∑ j′=j ‖δyj′‖2l2[0,1] = ‖δyj‖2l2[0,1] [ 1 + ‖δyj+1‖2l2[0,1] ‖δyj‖2l2[0,1] + ‖δyj+2‖2l2[0,1] ‖δyj‖2l2[0,1] + .... ] (5.9) we choose max η ‖δyj+η‖2l2[0,1] ‖δyj+η−1‖2l2[0,1] = τ for η = 1,2,3, ... and τ is found to satisfy the condition 0 < τ < 1, which is verified by taking a few examples of abel first kind and second kind integral equations. the values of τ are different for different examples. then the expression in (5.9) becomes ‖δyj‖2l2[0,1] [ 1 + ‖δyj+1‖2l2[0,1] ‖δyj‖2l2[0,1] + ‖δyj+2‖2l2[0,1] ‖δyj‖2l2[0,1] + .... ] ≤ ‖δyj‖2l2[0,1] [ 1 + τ + τ2 + τ3 + ... ] = ‖δyj‖2l2[0,1] 1 1 − τ . (5.10) the expression for ‖δyj‖2l2[0,1] is obtained by using orthonormality property of ψjk(x) within its support and theorem 5.4 for the partial support of ψjk(x). this is given by ‖δyj‖2l2[0,1] = 〈 2j−1 ∑ k=0 djkψjk(x), 2j −1 ∑ k=0 djkψjk(x) 〉 = 2j−2k+1 ∑ k=0 2j−2k+1 ∑ k′=0 dijkd i jk′δkk′ + 2j−1 ∑ k=2j −2k+2 2j−1 ∑ k′=2j−2k+2 drjkd r jk′t r j:kk′. (5.11) as ∫ 1 0 ψr jk (x)ψi jk′ (x)dx and ∫ 1 0 ψi jk (x)ψr jk′ (x)dx vanish, so we neglect those terms in the expression (5.11) which contain these specific integrals. the bound of l2norm of error ‖e(x)‖l2[0,1] can be estimated from the inequality (5.10). 6 illustrative examples example 1 consider the first kind abel integral equation ∫ x 0 y(t)dt (x − t)µ = b (1 − µ,1 + ν)x1+ν−µ, 0 < µ < 1, ν > 0 which has the exact solution y(x) = xν. here b(m,n) is the beta function and defined by b(m,n) = ∫ 1 0 xm−1(1 − x)n−1dx, m > 0, n > 0. 256 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) table 3 shows the exact and approximate solutions of the example 1 at the points x = i 8 for i = 1,2, ...,7 taking dau-3 scale function and m = 5. in this table, four sets of values of µ and ν are considered taking both fraction and integer values of ν. table 3: comparison of exact and approximate solutions of example 1 x exact solution approximate solution j = 4 j = 6 j = 8 µ = 1 4 ,ν = 1 2 1/8 0.353553 0.309319 0.352867 0.353554 2/8 0.500000 0.486212 0.499995 0.500000 3/8 0.612372 0.608044 0.612374 0.612373 4/8 0.707107 0.705733 0.707108 0.707107 5/8 0.790569 0.790135 0.790570 0.790569 6/8 0.866025 0.865890 0.866026 0.866025 7/8 0.935414 0.935375 0.935415 0.935414 µ = 1 4 ,ν = 3 1/8 0.001953 0.001805 0.001951 0.001953 2/8 0.015625 0.015478 0.015623 0.015625 3/8 0.052734 0.052588 0.052732 0.052734 4/8 0.125000 0.124854 0.124998 0.125000 5/8 0.244141 0.243995 0.244138 0.244141 6/8 0.421875 0.421730 0.421873 0.421875 7/8 0.669922 0.669776 0.669920 0.669922 µ = 3 4 ,ν = 1 2 1/8 0.353553 0.358049 0.353775 0.353575 2/8 0.500000 0.500476 0.500099 0.500009 3/8 0.612372 0.613000 0.612433 0.612378 4/8 0.707107 0.707550 0.707149 0.707111 5/8 0.790569 0.790915 0.790602 0.790572 6/8 0.866025 0.866305 0.866051 0.866028 7/8 0.935414 0.935647 0.935436 0.935416 µ = 3 4 ,ν = 3 1/8 0.001953 0.001870 0.001951 0.001953 2/8 0.015625 0.015525 0.015623 0.015625 3/8 0.052734 0.052630 0.052732 0.052734 4/8 0.125000 0.124892 0.124998 0.125000 5/8 0.244141 0.244030 0.244139 0.244141 6/8 0.421875 0.421763 0.421873 0.421875 7/8 0.669922 0.669809 0.669920 0.669922 cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 257 table 4: values of ‖δyj‖2l2[0,1] for different resolution j j for di jk for both di jk and dr jk 4 5.75007 × 10−8 1.14422 × 10−3 5 1.43777 × 10−8 5.71189 × 10−4 µ = 1 4 ,ν = 1 2 6 3.59444 × 10−9 2.85378 × 10−4 7 8.9861 × 10−10 1.42637 × 10−4 8 2.24653 × 10−10 7.13055 × 10−5 9 5.61631 × 10−11 3.56496 × 10−5 4 3.92808 × 10−9 1.15222 × 10−3 5 7.16154 × 10−11 5.74173 × 10−4 µ = 1 4 ,ν = 3 6 1.19898 × 10−12 2.86245 × 10−4 7 1.93591 × 10−14 1.42869 × 10−4 8 3.07368 × 10−16 7.13653 × 10−5 9 4.84076 × 10−18 3.56647 × 10−5 4 1.28416 × 10−7 1.14442 × 10−3 5 3.21065 × 10−8 5.71226 × 10−4 µ = 3 4 ,ν = 1 2 6 2.85385 × 10−9 2.86245 × 10−4 7 2.00666 × 10−9 1.42638 × 10−4 8 5.01665 × 10−10 7.13058 × 10−5 9 1.25416 × 10−10 3.56496 × 10−5 4 3.92901 × 10−9 1.15223 × 10−3 5 7.16226 × 10−11 5.74174 × 10−4 µ = 3 4 ,ν = 3 6 1.9904 × 10−12 2.86245 × 10−4 7 1.93595 × 10−14 1.42869 × 10−4 8 3.07371 × 10−16 7.13653 × 10−5 9 4.84079 × 10−18 3.56648 × 10−5 table 5: comparison of sup error and bound of l2-norm of error ‖e(x)‖l2[0,1] j sup error bound of ‖e(x)‖l2[0,1] taking dijk taking d i jk and d r jk µ = 1 4 ,ν = 1 2 4 4.423400 × 10−2 2.768973 × 10−4 4.783764 × 10−2 6 6.867130 × 10−4 6.923054 × 10−5 3.389050 × 10−2 8 7.100990 × 10−7 1.730766 × 10−5 1.194198 × 10−2 258 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) µ = 1 4 ,ν = 3 4 1.48274 × 10−4 6.324620 × 10−5 4.800458 × 10−2 6 2.27344 × 10−6 1.104969 × 10−6 1.691878 × 10−2 8 3.55446 × 10−8 1.769186 × 10−8 1.194201 × 10−2 µ = 3 4 ,ν = 1 2 4 4.49596 × 10−3 4.165755 × 10−4 4.784182 × 10−2 6 2.21534 × 10−4 1.041480 × 10−4 2.389079 × 10−2 8 2.13190 × 10−5 2.603701 × 10−5 1.194201 × 10−2 µ = 3 4 ,ν = 3 4 8.32861 × 10−5 6.316013 × 10−5 4.800479 × 10−2 6 1.68850 × 10−6 1.105110 × 10−6 2.023927 × 10−2 8 2.90795 × 10−8 1.769375 × 10−8 1.194699 × 10−2 example 2 consider the second kind abel integral equation [12] y (x) = x2 + 16 5 x 5 2 − ∫ x 0 y(t)dt√ x − t which has the exact solution y(x) = x2. table 6 shows the exact and approximate solutions of the example 2 at the points x = i 8 for i = 0,1,2, ...,7 taking dau-3 scale function and m = 5. table 6: comparison of exact and approximate solutions of example 2 x exact solution approximate solution j = 4 j = 6 j = 8 0 0 0 0 0 1/8 0.015625 0.015508 0.015624 0.015625 2/8 0.062500 0.062463 0.062499 0.062500 3/8 0.140625 0.140603 0.140625 0.140625 4/8 0.250000 0.249984 0.250000 0.250000 5/8 0.390625 0.390613 0.390625 0.390625 6/8 0.562500 0.562490 0.562500 0.562500 7/8 0.765625 0.765617 0.765625 0.765625 cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 259 table 7: values of ‖δyj‖2l2[0,1] for different resolution j j for di jk for both di jk and dr jk 4 1.45784 × 10−12 1.15094 × 10−3 5 4.66357 × 10−14 5.73210 × 10−4 6 1.46128 × 10−15 2.85926 × 10−4 7 4.53812 × 10−17 1.42779 × 10−4 8 1.40555 × 10−18 7.13417 × 10−5 9 4.35409 × 10−20 3.56587 × 10−5 table 8: comparison of sup error and bound of l2norm of error ‖e(x)‖l2[0,1] j sup error bound of ‖e(x)‖l2[0,1] taking dijk taking d i jk and d r jk 4 1.16723 × 10−4 1.22720 × 10−6 4.79779 × 10−2 6 9.56852 × 10−7 3.88534 × 10−8 2.39134 × 10−2 8 1.34341 × 10−8 1.20500 × 10−9 1.19450 × 10−2 example 3 consider the second kind abel integral equation [17] y (x) = 1√ x + 1 + π 8 − 1 4 sin−1 ( 1 − x 1 + x ) − 1 4 ∫ x 0 y(t)dt√ x − t which has the exact solution y(x) = 1√ x + 1 . table 9 shows the exact and approximate solutions of the example 3 at the points x = i 8 for i = 0,1,2, ...,7 taking dau-3 scale function and m = 5. 260 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) table 9: comparison of exact and approximate solutions of example 3 x exact solution approximate solution j = 4 j = 6 j = 8 0 1 1 1 1 1/8 0.942809 0.964541 0.947179 0.943883 2/8 0.894427 0.905166 0.897201 0.895110 3/8 0.852803 0.861371 0.854894 0.853318 4/8 0.816497 0.823468 0.818192 0.816914 5/8 0.784465 0.790355 0.785898 0.784818 6/8 0.755929 0.761042 0.757173 0.756236 7/8 0.730297 0.734819 0.731398 0.730568 table 10: values of ‖δyj‖2l2[0,1] for different resolution j j for dijk for both d i jk and d r jk 4 4.60915 × 10−6 5.80526 × 10−4 5 2.39761 × 10−6 2.88967 × 10−4 6 1.22773 × 10−6 1.44416 × 10−4 7 6.23369 × 10−7 7.20037 × 10−5 8 3.14886 × 10−7 3.59832 × 10−5 9 1.58536 × 10−7 1.79872 × 10−5 table 11: comparison of sup error and bound of l2norm of error ‖e(x)‖l2[0,1] j sup error bound of ‖e(x)‖l2[0,1] taking di jk taking di jk and dr jk 4 2.17315 × 10−2 3.09877 × 10−3 3.40742 × 10−2 6 4.37017 × 10−3 1.59930 × 10−3 1.69951 × 10−2 8 1.07361 × 10−3 8.09946 × 10−4 8.48330 × 10−3 we present in tables 4, 7 and 10 the values of ‖δyj‖2l2[0,1] (j = 4,5,6, .....,9) given by equation (5.11) for the examples 1, 2 and 3 respectively. second column of all tables present the values ‖δyj‖2l2[0,1] taking only d i jk i.e. taking only first term of (5.11), whereas third column presents the values ‖δyj‖2l2[0,1] taking both dijk and drjk. from these tables it appears that the values of ‖δyj‖2l2[0,1] gradually decrease if the resolution j increases. the presence of a few d r jk in (5.11) makes a lot of difference in calculating ‖δyj‖2l2[0,1] taking only dijk and taking both dijk and drjk. cubo 23, 2 (2021) approximate solution of abel integral equation in daubechies.... 261 in tables 5, 8 and 11, the sup errors are compared with the bound of l2-norm of error ‖e(x)‖l2[0,1] taking di jk and taking both di jk and dr jk for examples 1, 2 and 3 respectively. to evaluate bound of l2norm of error ‖e(x)‖l2[0,1], τ = 0.250044,τ = 0.50; τ = 0.250044,τ = 0.50; τ = 0.250044,τ = 0.50 and τ = 0.250044,τ = 0.50 are used for the four sets of values of µ and ν taking only di jk and taking both dijk and d r jk for example 1. also to evaluate bound of l 2 norm of error ‖e(x)‖l2[0,1], τ = 0.032, τ = 0.50 and τ = 0.52, τ = 0.50 are used for examples 2 and 3 respectively. sup errors are calculated taking maximum absolute difference of exact and approximate solutions from tables 3, 6 and 9. figures 1 to 6 display the exact and approximate solutions of examples 1, 2 and 3 for different resolutions (j = 4,6,8). we observe from these figures that as j increases, an approximate solution becomes closer to exact solution. this demonstrates efficiency of the proposed method. j=8 j=6 j=4 exact 0.0 0.2 0.4 0.6 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x y h x l figure 1: example 1 (µ = 1 4 ,ν = 1 2 ) j=8 j=6 j=4 exact 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x y h x l figure 2: example 1 (µ = 1 4 ,ν = 3) j=8 j=6 j=4 exact 0.0 0.2 0.4 0.6 0.8 0.4 0.5 0.6 0.7 0.8 0.9 x y h x l figure 3: example 1 (µ = 3 4 ,ν = 1 2 ) j=8 j=6 j=4 exact 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x y h x l figure 4: example 1 (µ = 3 4 ,ν = 3) 262 j. mouley, m. m. panja & b. n. mandal cubo 23, 2 (2021) j=8 j=6 j=4 exact 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x y h x l figure 5: example 2 j=8 j=6 j=4 exact 0.0 0.2 0.4 0.6 0.8 0.75 0.80 0.85 0.90 0.95 1.00 x y h x l figure 6: example 3 7 conclusion the purpose of the present work is to develop an efficient and accurate numerical scheme based on daubechies wavelet basis to solve abel integral equation. as wavelets are orthogonal systems, they have different resolution capabilities. the detail error estimation shows that the bound of l2-norm of error ‖e(x)‖l2[0,1] depends on resolution j. from tables 3, 6 and 9 it appears that the present numerical scheme works nicely for low resolution (j = 4,6,8). the results can be further improved by taking larger resolution j. acknowledgement jm acknowledges financial support from university grants commission, new delhi, for the award of research fellowship (file no. 16-9(june2017/2018(net/csir))). cubo 23, 2 (2021) approximate solution of 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[25] m. m. panja and b. n. mandal, “evaluation of singular integrals using daubechies scale function”, adv. comput. math. appl., vol. 1, pp. 64-75, 2012. introduction preliminary concept of daubechies scale function two-scale relations scale function at dyadic points multiresolution analysis (mra) and daubechies wavelet method of approximation error estimation illustrative examples conclusion cubo, a mathematical journal vol. 24, no. 02, pp. 291–305, august 2022 doi: 10.56754/0719-0646.2402.0291 graded weakly 1-absorbing prime ideals ünsal tekir1, b suat koç2 rashid abu-dawwas3 eda yıldız4 1department of mathematics, marmara university, istanbul, turkey. utekir@marmara.edu.tr b 2department of mathematics, istanbul medeniyet university, istanbul, turkey. suat.koc@medeniyet.edu.tr 3department of mathematics, yarmouk university, jordan. rrashid@yu.edu.jo 4department of mathematics, yildiz technical university, istanbul, turkey. edyildiz@yildiz.edu.tr abstract in this paper, we introduce and study graded weakly 1absorbing prime ideals in graded commutative rings. let g be a group and r be a g-graded commutative ring with a nonzero identity 1 6= 0. a proper graded ideal p of r is called a graded weakly 1-absorbing prime ideal if for each nonunits x, y, z ∈ h(r) with 0 6= xyz ∈ p , then either xy ∈ p or z ∈ p . we give many properties and characterizations of graded weakly 1-absorbing prime ideals. moreover, we investigate weakly 1-absorbing prime ideals under homomorphism, in factor ring, in rings of fractions, in idealization. resumen en este art́ıculo, introducimos y estudiamos ideales primos débilmente 1-absorbentes en anillos conmutativos gradados. sea g un grupo y r un anillo conmutativo g-gradado con identidad no cero 1 6= 0. un ideal gradado propio p de r se llama ideal primo gradado débilmente 1-absorbente si para cualquiera x, y, z ∈ h(r) no-unidades con 0 6= xyz ∈ p , entonces o bien xy ∈ p o z ∈ p . entregamos muchas propiedades y caracterizaciones de ideales primos gradados débilmente 1-absorbentes. más aún, investigamos ideales primos débilmente 1-absorbentes bajo homomorfismo, en anillos cociente, en anillos de fracciones, en idealización. keywords and phrases: graded ideal, 1-absorbing prime ideal, weakly 1-absorbing prime ideal, graded weakly 1-absorbing prime ideal. 2020 ams mathematics subject classification: 13a02, 13a15, 16w50. accepted: 6 june, 2022 received: 24 november, 2021 c©2022 ü. tekir et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2402.0291 mailto:utekir@marmara.edu.tr https://orcid.org/0000-0003-0739-1449 https://orcid.org/0000-0003-1622-786x https://orcid.org/0000-0001-8998-7590 https://orcid.org/0000-0002-6469-6698 mailto:utekir@marmara.edu.tr mailto:suat.koc@medeniyet.edu.tr mailto:rrashid@yu.edu.jo mailto:edyildiz@yildiz.edu.tr 292 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) 1 introduction throughout the paper, we focus only on graded commutative rings with a nonzero identity. r will always denote such a ring and g denotes a group with identity e. u(r), n(r) and reg(r) denote the set of all unit elements, all nilpotent elements and all regular elements of r, respectively. over the years, several types of ideals have been developed such as prime, maximal, primary, etc. the concept of prime ideals and its generalizations have a significant place in commutative algebra since they are used in understanding the structure of rings [6, 19, 11, 4]. r is said to be g-graded if r = ⊕ g∈g rg with rgrh ⊆ rgh for all g, h ∈ g where rg is an additive subgroup of r for all g ∈ g. sometimes we denote the g-graded ring r by g(r). the elements of rg are called homogeneous of degree g. if x ∈ r, then x can be written as ∑ g∈g xg, where xg is the component of x in rg. also, we set h(r) = ⋃ g∈g rg. the support of g(r) is defined as supp(g(r)) = {g ∈ g : rg 6= {0}}. moreover, as shown for example in [13] that re is a subring of r and 1 ∈ re. let p be an ideal of a graded ring r. then p is said to be graded ideal if p = ⊕ g∈g (p ∩ rg), i.e., for x ∈ p , x = ∑ g∈g xg where xg ∈ p for all g ∈ g. it is known that an ideal of a graded ring need not be graded. let r be a g-graded ring and p be a graded ideal of r. then r/p is g-graded by (r/p) g = (rg +p)/p for all g ∈ g. if r and s are g-graded rings, then r×s is a g-graded ring by (r × s) g = rg ×sg for all g ∈ g. lemma 1.1 ([9, lemma 2.1]). let r be a g-graded ring. (1) if p and q are graded ideals of r, then p + q, pq and p ⋂ q are graded ideals of r. (2) if x ∈ h(r), then rx = (x) is a graded ideal of r. let p be a proper graded ideal of r. then the graded radical of p is denoted by grad(p) and it is defined as follows: grad(p) =    x = ∑ g∈g xg ∈ r : for all g ∈ g, there exists ng ∈ n such that x ng g ∈ p    . note that grad(p) is always a graded ideal of r (see [15]). in [15], refai et al. defined and studied graded prime ideals. a proper graded ideal p of a graded ring r is called graded prime ideal if whenever xy ∈ p for some x, y ∈ h(r) then either x ∈ p or y ∈ p . clearly, if p is a prime ideal of r and it is a graded ideal of r, then p is a graded prime ideal of r. on the other hand, the importance of graded prime ideals comes from the fact that graded prime ideals are not necessarily prime ideals, as we see in the next example. example 1.2. consider r = z[i] and g = z2. then r is g-graded by r0 = z and r1 = iz. cubo 24, 2 (2022) graded weakly 1-absorbing prime ideals 293 consider the graded ideal i = 17r of r. we show that i is a graded prime ideal of r. let xy ∈ i for some x, y ∈ h(r). case (1): assume that x, y ∈ r0. in this case, x, y ∈ z such that 17 divides xy, and then either 17 divides x or 17 divides y as 17 is a prime number, which implies that x ∈ i or y ∈ i. case (2): assume that x, y ∈ r1. in this case, x = ia and y = ib for some a, b ∈ z such that 17 divides xy = −ab, and then 17 divides a or 17 divides b in z, which implies that 17 divides x = ia or 17 divides y = ib in r. then we have that x ∈ i or y ∈ i. case (3): assume that x ∈ r0 and y ∈ r1. in this case, x ∈ z and y = ib for some b ∈ z such that 17 divides xy = ixb in r, that is ixb = 17(α + iβ) for some α, β ∈ z. then we obtain xb = 17β, that is 17 divides xb in z, and again 17 divides x or 17 divides b, which implies that 17 divides x or 17 divides y = ib in r. thus, x ∈ i or y ∈ i. one can similarly show that x ∈ i or y ∈ i in other cases. so, i is a graded prime ideal of r. on the other hand, i is not a prime ideal of r since (4 − i)(4 + i) ∈ i, (4 − i) /∈ i and (4 + i) /∈ i. several generalizations of graded prime ideals attracted attention by many authors. in [14], refai and al-zoubi introduced graded primary ideals which is a generalization of graded prime ideals. a proper graded ideal p of a graded ring r is called graded primary ideal if xy ∈ p for some x, y ∈ h(r) implies that either x ∈ p or y ∈ grad(p). they also studied graded primary gdecomposition related to graded primary ideals. atani defined a generalization of graded prime ideals as graded weakly prime ideals in [5]. a proper graded ideal p of a graded ring r is said to be graded weakly prime ideal if whenever x, y ∈ h(r) such that 0 6= xy ∈ p then either x ∈ p or y ∈ p . they gave some characterizations of graded weakly prime ideals and their homogeneous components. in [12], naghani and moghimi introduced 2-absorbing version of graded prime ideals and graded weakly prime ideals. a proper graded ideal p of a graded ring r is called graded 2absorbing (graded weakly 2-absorbing) if whenever x, y, z ∈ h(r) such that xyz ∈ p (0 6= xyz ∈ p) then xy ∈ p or yz ∈ p or xz ∈ p . they investigated some properties of this new class of graded ideals. yassine et al. studied 1-absorbing prime ideals which is a generalization of prime ideals in [19]. a proper ideal p of r is said to be 1-absorbing prime ideal if for some nonunit elements x, y, z ∈ r such that xyz ∈ p implies that either xy ∈ p or z ∈ p . authors determined 1-absorbing prime ideals in some special rings such as principal ideal domains, valuation domains and dedekind domains. currently, koç et al. defined weakly 1-absorbing prime ideals which is a generalization of 1-absorbing prime ideals in [11]. a proper ideal p of r is called weakly 1-absorbing prime ideal if 0 6= xyz ∈ p for some nonunits x, y, z ∈ r implies xy ∈ p or z ∈ p . they gave certain properties of this new concept and characterized rings that every proper ideal is weakly 1-absorbing ideal. more recently, in [1], dawwas et al. defined graded version of 1-absorbing prime ideals which is a 294 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) generalization of both graded prime ideals and 1-absorbing prime ideals. a proper graded ideal p of a graded ring r is called graded 1-absorbing prime ideal if whenever for some nonunits x, y, z in h(r) such that xyz ∈ p then either xy ∈ p or z ∈ p . moreover, many studies have been made by researchers related to graded versions of known structures [3, 7, 8, 10, 16]. in this paper, we define graded weakly 1-absorbing prime ideal which is a generalization of graded 1-absorbing prime ideals. a proper graded ideal p of a graded ring r is said to be graded weakly 1-absorbing prime ideal if whenever for some nonunits x, y, z in h(r) such that 0 6= xyz ∈ p then either xy ∈ p or z ∈ p . every graded 1-absorbing prime ideal is a graded weakly 1absorbing prime ideals but the converse is not true in general (see, example 3.2). in addition to many properties of this new class of graded ideals, we also investigate behavior of graded weakly 1absorbing ideals under homomorphism, in factor ring, in rings of fractions, in idealization (see, theorem 3.15, proposition 3.14, theorem 3.16, theorem 3.18 and theorem 3.23). 2 motivation graded prime ideals play an essential role in graded commutative ring theory. indeed, graded prime ideals are interesting because they correspond to irreducible varieties and schemes in the graded case and because of their connection to factorization. also, graded prime ideals are important because they have applications to combinatorics and they have structural significance in graded ring theory. thus, this concept has been generalized and studied in several directions. the significance of some of these generalizations is same as the graded prime ideals. in a feeling of animate being, they determine how far an ideal is from being graded prime. several generalizations of graded prime ideals attracted attention by many authors. for instance, graded weakly prime ideals, graded primary ideals, graded almost prime ideals, graded 2-absorbing ideals, graded 2-absorbing primary ideals and graded 1-absorbing prime ideals. in continuation of these generalizations, we present the concept of graded weakly 1-absorbing prime ideals, as a new generalization to graded prime ideals, in order to benefit from this new concept in more applications, and to make the study of graded prime ideals more flexible. 3 graded weakly 1-absorbing prime ideals definition 3.1. let r be a g-graded ring and p be a proper graded ideal of r. then, p is called graded weakly 1-absorbing prime ideal of r if whenever 0 6= xyz ∈ p for some nonunit elements x, y, z in h(r) then xy ∈ p or z ∈ p. example 3.2. every graded 1-absorbing prime ideal is a graded weakly 1-absorbing prime ideal. the converse may not be true. let r = z21 and consider the trivial grading on r. p = (0̄) is graded cubo 24, 2 (2022) graded weakly 1-absorbing prime ideals 295 weakly 1-absorbing prime ideal. but it is not graded 1-absorbing prime ideal since 3̄.3̄.7̄ = 0̄ ∈ p, 3̄.3̄ 6∈ p and 7̄ 6∈ p. example 3.3. let r = z8[i] = z8 ⊕ iz8. then note that r is a z2-graded ring and h(r) = z8 ∪iz8. now, put p = (4). since 2(1+i)(1−i) = 4 ∈ p but 2(1+i) /∈ p and (1−i) /∈ p, it follows that p is not a weakly 1-absorbing prime ideal of r. however, the set of nonunit homogeneous elements of r is {0, 2, 4, 6, 2i, 4i, 6i}. let x, y, z ∈ h(r) be nonunit elements. then note that xyz = 0 ∈ p, which implies that p is a graded weakly 1-absorbing prime ideal of r. n(r) denotes the set of all nilpotent elements of r. recall that a ring r is said to be reduced if n(r) = 0. theorem 3.4. let r be a g-graded reduced ring and p be a graded weakly 1-absorbing prime ideal of r. then, grad(p) is a graded weakly prime ideal of r. proof. suppose that 0 6= xy ∈ grad(p) where x, y ∈ h(r). then there exists n ∈ n such that (xy)n ∈ p . we have 0 6= (xy)n = xkxn−kyn ∈ p for some positive integer k < n. if x or y is unit in h(r), we are done. so, assume that x and y are nonunit elements in h(r). as p is graded weakly 1-absorbing prime ideal, xn ∈ p or yn ∈ p showing that x ∈ grad(p) or y ∈ grad(p). theorem 3.5. let r be a g-graded ring and p be a graded weakly 1-absorbing prime ideal of r. then, (p : a) is a graded weakly prime ideal of r where a is a regular nonunit element in h(r)−p. proof. from [1, lemma 2.4], (p : a) is a graded ideal of r. suppose 0 6= xy ∈ (p : a) for some x, y ∈ h(r). then 0 6= (xa)y ∈ p where xa, y ∈ h(r). if x or y is unit, there is nothing to prove. so, we can assume that x and y are nonunit elements in h(r). since p is graded weakly 1-absorbing prime ideal of r, we get either xa ∈ p or y ∈ p . it gives x ∈ (p : a) or y ∈ (p : a), as needed. definition 3.6. let r be a g-graded ring and p be a graded ideal of r. then, p is called g-weakly 1-absorbing prime ideal of r for g ∈ g with pg 6= rg if 0 6= xyz ∈ p for some nonunit elements x, y, z in rg implies that xy ∈ p or z ∈ p. we say that a proper graded ideal p of a g-graded ring r is said to be a g-weakly prime for g ∈ g if pg 6= rg and whenever 0 6= xy ∈ p for some x, y ∈ rg implies x ∈ p or y ∈ p . proposition 3.7. let r be a g-graded reduced ring and p be a gn-weakly 1-absorbing prime ideal of r for each n ∈ n. then, grad(p) is a g-weakly prime ideal of r. proof. it immediately follows from theorem 3.4. 296 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) recall from [1] that a proper graded ideal p of a g-graded ring r is said to be a g-1-absorbing prime for g ∈ g if pg 6= rg and whenever xyz ∈ p for some nonunits x, y, z ∈ rg implies xy ∈ p or z ∈ p . proposition 3.8. let r be a g-graded ring. if r has a g-weakly-1-absorbing prime ideal that is not a g-weakly prime ideal of r and (0) is a g-1-absorbing prime ideal of r, then, for each unit element u in rg and for each nonunit element v in rg the sum u + v is a unit element in rg. proof. assume that p is a g-weakly 1-absorbing prime ideal of r that is not a g-weakly prime ideal of r. then, there exist nonunit elements x, y ∈ rg such that xy ∈ p but x 6∈ p and y 6∈ p . then we have vxy ∈ p where v is a nonunit element in rg. if vxy = 0 ∈ (0), then vx ∈ p since (0) is a g-1 absorbing prime ideal and y 6∈ p . if 0 6= vxy ∈ p , we have vx ∈ p since p is a g-weakly 1-absorbing prime ideal of r. now we will show that u + v is a unit element in rg where u is a unit element in rg. suppose to the contrary. if (u + v)xy = 0 ∈ (0), we get (u + v)x ∈ p . this implies ux ∈ p giving that x ∈ p which is a contadiction. if we assume 0 6= (u + v)xy ∈ p , then again we get a contradiction by using the fact that p is a g-weakly 1-absorbing prime ideal and so it completes the proof. theorem 3.9. let r be a g-graded ring and p be a proper graded ideal of r. consider the following statements. (i) p is a graded weakly 1-absorbing prime ideal of r. (ii) if xy 6∈ p for some nonunits x, y ∈ h(r), then (p : xy) = p ∪ (0 : xy). (iii) if xy 6∈ p for some nonunits x, y ∈ h(r), then either (p : xy) = p or (p : xy) = (0 : xy). (iv) if 0 6= xyk ⊆ p for some nonunits x, y ∈ h(r) and proper graded ideal k of r, then either xy ∈ p or k ⊆ p. (v) if 0 6= xjk ⊆ p for some nonunit x ∈ h(r) and proper graded ideals j, k of r, then either xj ⊆ p or k ⊆ p. (vi) if 0 6= ijk ⊆ p for proper graded ideals i, j, k of r, then either ij ⊆ p or k ⊆ p. then, (vi) ⇒ (v) ⇒ (iv) ⇒ (iii) ⇒ (ii) ⇒ (i). proof. (vi) ⇒ (v) : suppose that 0 6= xjk ⊆ p for some nonunit x ∈ h(r) and proper graded ideals j, k of r. now, put i = (x). then i is a proper graded ideal of r and 0 6= ijk ⊆ p. by (vi), we have xj ⊆ ij ⊆ p or k ⊆ p, which completes the proof. (v) ⇒ (iv) : suppose that 0 6= xyk ⊆ p for some nonunits x, y ∈ h(r) and proper graded ideal k of r. now, consider the proper graded ideal j = (y) of r and note that 0 6= xjk ⊆ p. so by (v), we get xy ∈ xj ⊆ p or k ⊆ p. cubo 24, 2 (2022) graded weakly 1-absorbing prime ideals 297 (iv) ⇒ (iii) : let x, y ∈ h(r) be nonunit elements such that xy /∈ p. it is easy to see that xy(p : xy) ⊆ p. case 1: assume that xy(p : xy) = 0. this gives (p : xy) ⊆ (0 : xy) ⊆ (p : xy), that is, (p : xy) = (0 : xy). case 2: assume that xy(p : xy) 6= 0. then by (iv), we have (p : xy) ⊆ p which implies that (p : xy) = p. (iii) ⇒ (ii) : it is straightforward. (ii) ⇒ (i) : let x, y, z ∈ h(r) be nonunits such that 0 6= xyz ∈ p. if xy ∈ p, then we are done. so assume that xy /∈ p. since z ∈ (p : xy) − (0 : xy) and (p : xy) = (0 : xy) ∪ p, we have z ∈ p which completes the proof. in the following example, we show that the condition “p is a graded weakly 1-absorbing prime ideal” does not ensure that the conditions (ii)-(vi) in theorem 3.9 hold. in fact, we will show that (i) ; (ii). example 3.10. let r = z12[x], where x is an indeterminate over z12. then r = ⊕ n∈z rn is a z-graded ring, where r0 = z12 and rn = z12x n if n > 0, otherwise rn = 0. then note that h(r) = ⋃ n≥0 z12x n and the set of nonunits homogeneous elements of r is nh(r) = {2k, 3k, axn : k, a ∈ z and n ∈ n}. consider the graded ideal p = (x, 4) of r. let f, g, h ∈ nh(r) such that 0 6= fgh ∈ p. if at least one of the f, g, h is of the form axn, then we are done. so assume that f, g, h ∈ {2k, 3k : k ∈ z}. since 0 6= fgh ∈ p = (x, 4), we have 0 6= fgh = 4k for some k ∈ z with gcd(k, 3) = 1. since 4|fgh and 3 ∤ fgh, we conclude that f, g, h ∈ {2, 4, 8, 10}. this implies that fg ∈ p, that is, p is a graded weakly 1-absorbing prime ideal of r. now, we will show that p does not satisfy the condition (ii) in theorem 3.9. take nonunits homogeneous elements c = 2, d = 3 of r. then note that cd /∈ p. on the other hand, it is clear that 2 ∈ (0 : cd) − p and x ∈ p − (0 : cd). this gives z = 2 + x ∈ (p : cd) − ((0 : cd) ∪ p) . thus, we have (p : cd) ) (0 : cd) ∪ p, i.e., p does not satisfy the condition (ii) in theorem 3.9. definition 3.11. let p be a graded weakly 1-absorbing prime ideal of r and xg1, yg2, zg3 be nonunits in h(r). then, (xg1 , yg2, zg3) is called graded 1-triple zero if xg1 yg2zg3 = 0, xg1 yg2 6∈ p and zg3 6∈ p, where g1, g2, g3 ∈ g. theorem 3.12. let p = ⊕ g∈g pg be a graded weakly 1-absorbing prime ideal that is not graded 1-absorbing prime ideal and (xg1 , yg2, zg3) be a graded 1-triple zero of p, where g1, g2, g3 ∈ g. then, (i) xg1yg2pg3 = 0. (ii) xg1zg3 6∈ pg1g3 and yg2zg3 6∈ pg2g3 imply that xg1 zg3pg2 = yg2zg3pg1 = xg1 pg2pg3 = yg2pg1 pg3 = zg3pg1pg2 = 0. in particular, pg1pg2pg3 = 0. 298 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) proof. (i) : let p = ⊕ g∈g pg be a graded weakly 1-absorbing prime ideal that is not graded 1-absorbing prime ideal and (xg1 , yg2, zg3) be a graded 1-triple zero of p . assume that xg1yg2pg3 6= 0. then, there exists a ∈ pg3 = p ∩ rg3 such that 0 6= xg1 yg2a. so, we have 0 6= xg1 yg2a = xg1yg2(zg3 + a) ∈ p . if zg3 + a is unit, then xg1yg2 ∈ p which gives a contradiction. since p is graded weakly 1-absorbing prime ideal and xg1yg2 6∈ p , zg3 +a ∈ p . this shows zg3 ∈ p , a contradiction. (ii) : let xg1zg3 6∈ pg1g3 and yg2zg3 6∈ pg2g3. then, xg1zg3, yg2zg3 6∈ p . now choose a ∈ pg2. so, we have xg1(yg2 + a)zg3 = xg1azg3 ∈ p since xg1yg2zg3 = 0. if yg2 + a is unit, then we obtain xg1zg3 ∈ p , which is a contradiction. thus, yg2 + a is not unit. if xg1azg3 6= 0, then 0 6= xg1(yg2 + a)zg3 ∈ p . thus, xg1(yg2 + a) ∈ p or zg3 ∈ p implying that xg1 yg2 ∈ p or zg3 ∈ p , a contradiction. this shows xg1azg3 = 0 and so xg1 zg3pg2 = 0. similarly, yg2zg3pg1 = 0. now assume that xg1 pg2pg3 6= 0. then there exist ag2 ∈ pg2 , bg3 ∈ pg3 such that xg1ag2bg3 6= 0. this gives 0 6= xg1 (yg2 + ag2)(zg3 + bg3) = xg1yg2zg3 + xg1yg2bg3 + xg1ag2zg3 + xg1ag2bg3 = xg1ag2bg3 ∈ p . if (yg2 + ag2) is unit, xg1(zg3 + bg3) ∈ p . it means that xg1 zg3 ∈ p , which is a contradiction. hence, (yg2 + ag2) is nonunit. similar argument shows that (zg3 + bg3) is nonunit. since p is graded weakly 1-absorbing prime ideal, xg1(yg2 +ag2) ∈ p or zg3 +bg3 ∈ p . this proves xg1yg2 ∈ p or zg3 ∈ p which is a contradiction. so, xg1pg2pg3 = 0. similarly we have yg2pg1pg3 = zg3pg1pg2 = 0. suppose pg1pg2 pg3 6= 0. then there exist ag1 ∈ pg1 , bg2 ∈ pg2, cg3 ∈ pg3 such that ag1bg2cg3 6= 0. so, we have 0 6= (ag1 + xg1 )(bg2 + yg2)(cg3 + zg3) = ag1bg2cg3 ∈ p since xg1zg3pg2 = yg2zg3pg1 = xg1 pg2pg3 = yg2pg1pg3 = zg3pg1 pg2 = 0 and xg1yg2zg3 = 0. if ag1 + xg1 is unit, (bg2 + yg2)(cg3 + zg3) ∈ p and it implies yg2zg3 ∈ p , a contradiction. so, ag1 + xg1 is not unit. similar argument shows that bg2 + yg2, cg3 + zg3 are nonunits. since p is graded weakly 1-absorbing prime ideal, we have either (ag1 + xg1)(bg2 + yg2) ∈ p or cg3 + zg3 ∈ p . thus, we conclude that xg1yg2 ∈ p or zg3 ∈ p giving a contradiction. therefore, pg1pg2pg3 = 0. let r be a g-graded ring. it is clear that for each g ∈ g, rg is an re-module and pg is an re-submodule of rg. theorem 3.13. let p = ⊕ g∈g pg be a graded 1-absorbing prime ideal of g(r) and g ∈ g. if x, y ∈ rg are nonunits such that xy 6∈ p, then (pg2 :re xy) = pe. proof. let z ∈ (pg2 :re xy), where x, y ∈ rg are nonunits. then, xyz ∈ pg2 ⊆ p . if z is a unit, xy ∈ p which gives a contradiction. so, z is not unit. as p is graded 1-absorbing prime ideal and xy 6∈ p we get z ∈ p . thus, z ∈ p ∩ re = pe. this shows (pg2 :re xy) ⊆ pe. cubo 24, 2 (2022) graded weakly 1-absorbing prime ideals 299 on the other hand, suppose z ∈ pe ⊆ p . then, xyz ∈ p ∩ rg2 = pg2 proving z ∈ (pg2 :re xy), as desired. proposition 3.14. let r be a g-graded ring and j ⊆ i be proper graded ideals of r. then the followings statements are satisfied. (i) if i is graded weakly 1-absorbing prime ideal, then i/j is graded weakly 1-absorbing prime ideal of r/j. (ii) suppose that j consists of all nilpotent elements of r. if j is a graded weakly 1-absorbing prime ideal of r and i/j is a graded weakly 1-absorbing prime ideal of r/j, then i is a graded weakly 1-absorbing prime ideal of r. (iii) if (0) is graded 1-absorbing prime ideal of r and i is graded weakly 1-absorbing prime ideal of r, then i is graded 1-absorbing prime ideal of r. proof. (i) : let 0+j 6= (x+j)(y +j)(z +j) ∈ i/j for some nonunits x+j, y +j, z +j ∈ h(r/j). then, 0 6= xyz + j ∈ i/j and so 0 6= xyz ∈ i where x, y, z are nonunits in h(r). as i is a graded weakly 1-absorbing prime ideal, either xy ∈ i or z ∈ i. hence, xy + j ∈ i/j or z + j ∈ i/j, as desired. (ii) : suppose 0 6= xyz ∈ i for some nonunits x, y, z ∈ h(r). then, xyz+j = (x+j)(y+j)(z+j) ∈ i/j. if xyz ∈ j, then xy ∈ j ⊆ i or z ∈ j since j ⊆ i is graded weakly 1-absorbing prime ideal. so we can assume xyz 6∈ j. then we have 0 + j 6= (x + j)(y + j)(z + j) ∈ i/j. as i/j is graded weakly 1-absorbing prime ideal of r/j, (x + j)(y + j) ∈ i/j or z + j ∈ i/j. it implies either xy ∈ i or z ∈ i. (iii) : suppose that xyz ∈ i for some nonunits x, y, z ∈ h(r). if xyz 6= 0, then we are done. so, we can assume xyz = 0 ∈ (0). then, we get either xy = 0 ∈ i or z = 0 ∈ i since (0) is graded 1-absorbing prime ideal. therefore, we conclude that xy ∈ i or z ∈ i. let r and s be two g-graded rings. a ring homomorphism f : r → s is said to be graded homomorphism if f(rg) ⊆ sg for all g ∈ g. theorem 3.15. let r1 and r2 be two g-graded rings and f : r1 −→ r2 be a graded homomorphism such that f(1r1) = 1r2. the following statements are satisfied. (i) if f is injective, j is a graded weakly 1-absorbing prime ideal of r2 and f(x) is a nonunit element of r2 for all nonunit elements x ∈ h(r1), then f −1(j) is a graded weakly 1-absorbing prime ideal of r1. (ii) if f is surjective and i is a graded weakly 1-absorbing prime ideal of r1 with ker(f) ⊆ i, then f(i) is a graded weakly 1-absorbing prime ideal of r2. 300 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) proof. (i) : it is clear that f−1(j) is a graded ideal of r1. let 0 6= xyz ∈ f −1(j) for some nonunits x, y, z in h(r1). so, f(x), f(y) and f(z) are nonunits in h(r2) by the assumption. since f is injective and xyz 6= 0, we have f(xyz) 6= 0. then we get 0 6= f(x)f(y)f(z) = f(xyz) ∈ j. as j is a graded weakly 1-absorbing prime ideal of r2, f(x)f(y) ∈ j or f(z) ∈ j. it implies that we have either xy ∈ f−1(j) or z ∈ f−1(j). (ii) : suppose that 0 6= abc ∈ f(i) for some nonunits a, b, c ∈ h(r2). then, there exist nonunits x, y, z ∈ h(r1) such that f(x) = a, f(y) = b and f(z) = c. it gives that 0 6= f(x)f(y)f(z) = abc ∈ f(i). so, there exists i ∈ i such that f(xyz) = f(i). this means xyz − i ∈ ker(f) ⊆ i giving xyz ∈ i. since i is a graded weakly 1-absorbing prime ideal and 0 6= xyz ∈ i, we conclude that xy ∈ i or z ∈ i. it shows f(x)f(y) = ab ∈ f(i) or f(z) = c ∈ f(i), as needed. let s ⊆ h(r) be a multiplicative set and r be a g-graded ring. then s−1r is a g-graded ring with (s−1r)g = { a s : a ∈ rh, s ∈ s ∩ rhg−1 } . let i be a graded ideal of r. then we denote the set {a ∈ r : ab ∈ i for some b ∈ r − i} by zi(r). theorem 3.16. let r be a g-graded ring and s ⊆ h(r) be a multiplicatively closed subset. the following statements are satisfied. (i) if i is a graded weakly 1-absorbing prime ideal of r with i ∩ s = ∅, then s−1i is a graded weakly 1-absorbing prime ideal of s−1r. (ii) if s−1i is a graded weakly 1-absorbing prime ideal of s−1r, u(s−1r) = {x s : x ∈ u(r), s ∈ s}, s ⊆ reg(r) and s ∩ zi(r) = ∅, then i is a graded weakly 1-absorbing prime ideal of r. proof. (i) : suppose that 0 6= x s y t z u ∈ s−1i for some nonunits x s , y t , z u ∈ h(s−1r). then 0 6= a(xyz) = (ax)yz ∈ i for some a ∈ s. here, ax, y, z are nonunits in h(r). otherwise, we would have x s , y t , z u are units in s−1r, a contradiction. as i is a graded weakly 1-absorbing prime ideal of r, we have either axy ∈ i or z ∈ i. this implies that x s y t = axy ast ∈ s−1i or z u ∈ s−1i. thus, s−1i is a graded weakly 1-absorbing prime ideal of s−1r. (ii) : let 0 6= xyz ∈ i for some nonunits x, y, z ∈ h(r). since s ⊆ reg(r), we conclude that 0 6= x 1 y 1 z 1 ∈ s−1i. here, x 1 , y 1 , z 1 are nonunits in h(s−1r). since s−1i is a graded weakly 1-absorbing prime ideal of s−1r, we conclude either x 1 y 1 = xy 1 ∈ s−1i or z 1 ∈ s−1i. then there exists s ∈ s such that sxy ∈ i or sz ∈ i. we can assume that sxy ∈ i. if xy /∈ i, then we have s ∈ zi(r)∩s which is a contradiction. thus we have xy ∈ i. in other case, similarly, we get z ∈ i. therefore, i is a graded weakly 1-absorbing prime ideal of r. theorem 3.17. let p = ⊕ g∈g pg be a graded weakly 1-absorbing prime ideal of r and g ∈ g. then, (pg2 :re xy) = pe ∪ (0 :re xy) where x, y ∈ rg are nonunits such that xy 6∈ p. cubo 24, 2 (2022) graded weakly 1-absorbing prime ideals 301 proof. clearly (0 :re xy) ⊆ (pg2 :re xy). let z ∈ pe ⊆ p . this implies that xyz ∈ p ∩ rg2 = pg2 and so z ∈ (pg2 :re xy). hence, pe ∪ (0 :re xy) ⊆ (pg2 :re xy). now, we will show that (pg2 :re xy) ⊆ (0 :re xy) ∪ pe. let z ∈ (pg2 :re xy). then, we have xyz ∈ pg2 ⊆ p . if z is a unit, then we have xy ∈ p, a contradiction. suppose that z is a nonunit of r. if xyz 6= 0, then z ∈ p ∩ re = pe. so assume that xyz = 0. it gives z ∈ (0 :re xy). thus we have z ∈ pe ∪ (0 :re xy). therefore, (pg2 :re xy) = pe ∪ (0 :re xy). let r = ⊕ g∈g rg be a graded ring. recall from [18] that r is said to be a graded field if every nonzero homogenous element is a unit in r. theorem 3.18. suppose that r1, r2 be two g-graded commutative rings that are not graded fields and r = r1 × r2. let p be a nonzero proper graded ideal of r. the following statements are equivalent. (i) p is a graded weakly 1-absorbing prime ideal of r. (ii) p = p1 × r2 for some graded prime ideal p1 of r1 or p = r1 × p2 for some graded prime ideal p2 of r2. (iii) p is a graded prime ideal of r. (iv) p is a graded weakly prime ideal of r. (v) p is a graded 1-absorbing prime ideal of r. proof. (i) ⇒ (ii) : let p be a nonzero proper graded ideal of r. then we can write p = p1×p2 for some graded ideals p1 of r1 and p2 of r2. since p is nonzero, p1 6= 0 or p2 6= 0. without loss of generality, we may assume that p1 6= 0. then there exists a homogeneous element 0 6= x ∈ p1. since p is a graded weakly 1-absorbing prime ideal and (0, 0) 6= (1, 0)(1, 0)(x, 1) ∈ p, we conclude either (1, 0) ∈ p or (x, 1) ∈ p. then we have either p1 = r1 or p2 = r2. assume that p1 = r1. now we will show that p2 is a graded prime ideal of r2. let yz ∈ p2 for some y, z ∈ h(r2). if y or z is a unit, then we have either y ∈ p2 or z ∈ p2. so assume that y, z are nonunits in h(r2). since r1 is not a graded field, there exists a nonzero nonunit t ∈ h(r1). this implies that (0, 0) 6= (t, 1)(1, y)(1, z) = (t, yz) ∈ p. as p is a graded weakly 1-absorbing prime ideal of r, we conclude either (t, 1)(1, y) = (t, y) ∈ p or (1, z) ∈ p. thus we get y ∈ p2 or z ∈ p2 and so p2 is a graded prime ideal of r2. in other case, one can similarly show that p = p1 × r2 and p1 is a graded prime ideal of r1. (ii) ⇒ (iii) ⇒ (iv) ⇒ (i) : it is obvious. (iii) ⇒ (v) : it is clear. (v) ⇒ (i) : it is straightforward. 302 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) definition 3.19. let r be a ring and m be an r-module. a proper submodule n of m is called a 1-absorbing r-submodule if whenever xym ∈ n where x, y ∈ r are nonunits, m ∈ m, then either xy ∈ (n :r m) or m ∈ n. theorem 3.20. let p = ⊕ g∈g pg be a graded 1-absorbing prime ideal of g(r). if pg 6= rg, then pg is a 1-absorbing re-submodule of rg. proof. let xyr ∈ pg ⊆ p for some nonunits x, y ∈ re and r ∈ rg. as p is graded 1-absorbing prime ideal, xy ∈ p or r ∈ p . this implies that xy ∈ (pg :re rg) since xyrg ⊆ prg ⊆ p ∩rg = pg or r ∈ p ∩ rg = pg . definition 3.21. let p = ⊕ g∈g pg be a graded ideal of g(r). a graded component pg of p is called 1-absorbing prime subgroup of rg if xyz ∈ pg for some nonunits x, y, x ∈ h(r) implies either xy ∈ pg or z ∈ pg. proposition 3.22. let p = ⊕ g∈g pg be a graded ideal of g(r). if pg is a 1-absorbing prime subgroup of rg for all g ∈ g, then p is a graded 1-absorbing prime ideal of r. proof. suppose xyz ∈ p for some nonunits x, y, z ∈ h(r). then, xyz ∈ pg for some g ∈ g. since pg is 1-absorbing prime subgroup of rg, xy ∈ pg or z ∈ pg. this gives xy ∈ p or z ∈ p , as needed. let m be an r-module. the idealization r⋉m = {(r, m) : r ∈ r and m ∈ m} of m is a commutative ring with componentwise addition and multiplication: (x, m1)+(y, m2) = (x+y, m1 +m2) and (x, m1)(y, m2) = (xy, xm2 + ym1) for each x, y ∈ r and m1, m2 ∈ m. let g be an abelian group and m be a g-graded r-module. then x = r⋉m is a g-graded ring by xg = rg ⋉mg = rg ⊕mg for all g ∈ g. note that, xg is an additive subgroup of x for all g ∈ g. also, for g, h ∈ g, xgxh = (rg ⋉mg)(rh⋉mh) = rgrh⋉(rgmh+rhmg) ⊆ rgh⋉(mgh+mhg) ⊆ rgh⋉mgh = xgh as g is abelian (see [2, 17]). theorem 3.23. let g be an abelian group, m be a g-graded r-module and p be an ideal of r. then, the following statements are equivalent. (i) p ⋉ m is a graded weakly 1-absorbing prime ideal of r ⋉ m. (ii) p is a graded weakly 1-absorbing prime ideal of r and if xg1yg2zg3 = 0 such that xg1 yg2 6∈ p and zg3 /∈ p for some nonunit elements xg1, yg2, zg3 in h(r), where g1, g2, g3 ∈ g, then xg1yg2mg3 = xg1 zg3mg2 = yg2zg3mg1 = 0. proof. (i) ⇒ (ii) : by [17, theorem 3.3], p is a graded ideal of r. suppose that 0 6= abc ∈ p where a, b, c are nonunits in h(r). since (0, 0) 6= (a, 0)(b, 0)(c, 0) ∈ p ⋉ m and (a, 0), (b, 0), (c, 0) cubo 24, 2 (2022) graded weakly 1-absorbing prime ideals 303 are nonunits in h(r ⋉ m), we get (a, 0)(b, 0) ∈ p ⋉ m or (c, 0) ∈ p ⋉ m. thus, we conclude that ab ∈ p or c ∈ p , as needed. now suppose xg1 yg2zg3 = 0 such that xg1 yg2 6∈ p and zg3 /∈ p for some nonunit elements xg1 , yg2, zg3 in h(r), where g1, g2, g3 ∈ g. let xg1yg2mg3 6= 0. then there exists mg3 ∈ mg3 such that xg1 yg2mg3 6= 0. this gives (0, 0) 6= (xg1 , 0)(yg2, 0)(zg3, mg3) = (0, xg1yg2mg3) ∈ p ⋉ m for some nonunits (xg1 , 0), (yg2, 0), (zg3, mg3) ∈ h(r ⋉ m) and p ⋉ m is a graded weakly 1-absorbing prime ideal, we have (xg1 , 0)(yg2, 0) = (xg1yg2, 0) ∈ p ⋉ m or (zg3, mg3) ∈ p ⋉ m. this gives xg1yg2 ∈ p or zg3 ∈ p , a contradiction. hence, xg1yg2mg3 = 0. similar argument shows that xg1zg3mg2 = yg2zg3mg1 = 0. (ii) ⇒ (i) : by [17, theorem 3.3], p ⋉ m is a graded ideal of r ⋉ m. assume that (0, 0) 6= (xg1 , mg1)(yg2 , mg2)(zg3 , mg3) = (xg1 yg2zg3, xg1yg2mg3 +xg1zg3mg2 +yg2zg3mg1) ∈ p ⋉m for some nonunits (xg1 , mg1), (yg2, mg2), (zg3, mg3) in h(r ⋉ m). then we get xg1 yg2zg3 ∈ p for some nonunits xg1, yg2, zg3 ∈ h(r). case 1: assume that xg1yg2zg3 = 0. if xg1 yg2 6∈ p and zg3 6∈ p , we have xg1 yg2mg3 = xg1zg3mg2 = yg2zg3mg1 = 0. this implies that xg1yg2mg3 + xg1zg3mg2 + yg2zg3mg1 = 0 and so (xg1 , mg1)(yg2, mg2)(zg3, mg3) = (0, 0) giving a contradiction. hence, we must have xg1yg2 ∈ p or zg3 ∈ p . this gives (xg1 , mg1)(yg2 , mg2) ∈ p ⋉ m or (zg3, mg3) ∈ p ⋉ m. case 2: now, assume that xg1 yg2zg3 6= 0. this gives xg1yg2 ∈ p or zg3 ∈ p since p is graded weakly 1-absorbing prime ideal. then we conclude that (xg1 , mg1)(yg2 , mg2) ∈ p ⋉m or (zg3, mg3) ∈ p ⋉ m which completes the proof. acknowledgements the authors would like to thank the referee for his/her valuable comments that improved the paper. 304 ü. tekir, s. koç, r. abu-dawwas & e. yıldız cubo 24, 2 (2022) references [1] r. abu-dawwas, e. yıldız, ü. tekir and s. koç, “on graded 1-absorbing prime ideals‘”, são paulo j. math. sci., vol. 15, no. 1, pp. 450–462, 2021. 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[19] a. yassine, m. j. nikmehr and r. nikandish, “on 1-absorbing prime ideals of commutative rings”, j. algebra appl., vol. 20, no. 10, paper no. 2150175, 12 pages, 2021. introduction motivation graded weakly 1-absorbing prime ideals cubo a mathematical journal vol.21, no¯ 03, (39–61). december 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000300039 stability and boundedness in nonlinear and neutral difference equations using new variation of parameters formula and fixed point theory youssef n raffoul department of mathematics, university of dayton dayton, oh 45469-2316 usa yraffoul1@udayton.edu abstract in the case of nonlinear problems, whether in differential or difference equations, it is difficult and in some cases impossible to invert the problem and obtain a suitable mapping that can be effectively used in fixed point theory to qualitatively analyze its solutions. in this paper we consider the existence of a positive sequence and utilize it in the capacity of integrating factor to obtain a new variation of parameters formula. then, we will use the obtained new variation of parameters formula and revert to the contraction principle to arrive at results concerning, boundedness, periodicity and stability. the author is working on parallel results for the continuous case. resumen en el caso de problemas no-lineales, ya sea en ecuaciones diferenciales o en diferencias, es dif́ıcil y en algunos casos imposible invertir el problema y obtener una aplicación apropiada que pueda ser efectivamente usada en teoŕıa de punto fijo para analizar quantitativamente sus soluciones. en este paper consideramos la existencia de una sucesión positiva y la usamos en la capacidad del factor de integración para obtener una nueva fórmula de variación de parámetros. luego, usaremos esta nueva fórmula de variación de parámetros y volver al principio de contracción para obtener resultados que involucran acotamiento, periodicidad y estabilidad. el autor se encuentra trabajando en resultados paralelos para el caso continuo keywords and phrases: new variation of parameters, difference, neutral, sability, boundedness, fixed point theorems, contraction mapping, equi-boundedness. 2010 ams mathematics subject classification: 39a11. http://dx.doi.org/10.4067/s0719-06462019000300039 40 youssef n raffoul cubo 21, 3 (2019) 1 introduction for motivational purpose we consider the linear difference equation x(t + 1) = a(t)x(t), x(t0) = x0, t ≥ t0 ≥ 0. (1.1) it is clear that the solution of (1.1) is given by x(t) = x0 t−1 ∏ s=t0 a(s), (1.2) provided that a(t) 6= 0 for all t ∈ z+. throughout this paper we adopt the convention that for any sequence x(k) b ∑ k=a x(k) = 0 and b ∏ k=a x(k) = 1 whenever a > b. for more on the calculus of difference equations we refer to [6][8] and [13]. let v(t) be a sequence such that v : z+ → r with v(t) 6= 0 for all t ∈ z+. multiply both sides of (1.1) by t ∏ s=t0 v−1(s) to obtain x(t + 1) t ∏ s=t0 v−1(s) = a(t)x(t) t ∏ s=t0 v−1(s), thus the above expression can be written in the compact form ∆ [ x(t) t−1 ∏ s=t0 v−1(s) ] = [ ( a(t) − v(t) ) x(t) ] t ∏ s=t0 a−1(s). (1.3) summing equation (1.3) from t0 to t-1 gives x(t) = x0 t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 ( a(r) − v(r) ) x(r) t−1 ∏ s=r+1 v(s). (1.4) note that (1.4) reduces to (1.2) if we set v(t) = a(t) in (1.4). to obtain asymptotic stability of the zero solution of (1.1) using (1.2) one would have to assume that t ∏ s=t0 a(s) → 0, as t → ∞. on the hand, if we use (1.4) instead, then such requirement is not necessary. but instead, we would have to ask that t ∏ s=t0 v(s) → 0, as t → ∞. cubo 21, 3 (2019) new variation of parameters 41 such technique of inversion is of more importance when the right hand of (1.1) is either totally nonlinear or totally delayed. to see this, we consider the nonlinear difference equation x(t + 1) = f(t,x(t)), x(t0) = x0, (1.5) where the function f : z+ × r → r is continuous. the subject of stability and boundedness in difference equations is vast and we refer to [16] and [17]. we begin by stating some definitions . definition 1.1. we say x(t) := x(t,t0,x0) is a solution of (1.5) if x(t0) = x0 and satisfies (1.5) for t ≥ t0 ≥ 0. definition 1.2. the zero solution of (1.5) is stable if for any ǫ > 0 and any integer t0 ≥ 0 there exists a δ > 0 such that |x0| ≤ δ implies |x(t,t0,x0)| ≤ ǫ for t ≥ t0. definition 1.3. the zero solution of (1.5) is asymptotically stable if it is stable and |x(t,t0,x0)| → 0 as t → ∞. for more on stability we refer to [9] and [11]. we begin with the following lemma. its proof follows along the lines of the derivation of (1.4). lemma 1.4. if x(t) is a solution of (1.5) on an interval z+ ∩ [0,t ] and satisfies the initial condition x(t0) = x0, t0 ≥ 0, then x(t) is a solution of the summation equation if and only if x(t) = x0 t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 ( f(r,x(r)) − v(r)x(r) ) t−1 ∏ s=r+1 v(s), (1.6) where v : z+ → r with v(t) 6= 0 for all t ∈ z+. next, we will use (1.6) to define a mapping on the proper space and show the zero solution is (as). let c be the set of all real-valued bounded sequences. define the space s = {φ : [0,∞) → r/φ ∈ c, |φ(t)| ≤ l,φ(t) → 0, as t → ∞}. then (s, || · ||) is a complete metric space under the uniform metric ρ(φ1,φ2) = ||φ1 − φ2||, where ||φ|| = sup t∈z+ {|φ(t)|}. assume f(t,0) = 0. (1.7) 42 youssef n raffoul cubo 21, 3 (2019) we assume the function f is locally lipschitz on the set s. that is, for any φ1 and φ2 ∈ s, we have |f(t,φ1) − f(t,φ2)| ≤ λ(t)||φ1 − φ2||, (1.8) for λ : [0,∞) → (0,∞). assume for φ ∈ s and positve constant l, we have that ∣ ∣ ∣ x0 t−1 ∏ s=t0 v(s) ∣ ∣ ∣ + l t−1 ∑ r=t0 ( |v(r)| + λ(r) ) ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ l. (1.9) note that (1.9) implies that t−1 ∑ r=t0 ( |v(r)| + λ(r) ) ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ α < 1. the next theorem offers results about stability and boundedness. for more results on the stability and boundedness using fixed point theory, we refer the interest reader to the book [18] and to the paper [19]. theorem 1.5. assume (1.7)-(1.9). suppose there exists a positive constant k such that ∣ ∣ ∣ t−1 ∏ s=t0 v(s) ∣ ∣ ∣ ≤ k, (1.10) then the unique solution of (1.5) is bounded and its zero solution is stable. if, in addition, t−1 ∏ s=t0 v(s) → 0, (1.11) then the zero solution of (1.5) is asymptotically stable. proof. for φ ∈ s, define the mapping p : s → s by (pφ)(t) = x0 t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 ( f(r,φ(r)) − φ(r)v(r) t ∏ s=r+1 v(s) (1.12) it is clear that (pφ)(t0) = x0. now for φ ∈ s, we have that ∣ ∣(pφ)(t) ∣ ∣ ≤ |x0|k + t−1 ∑ r=t0 ( λ(r)|φ(r)| + |φ(r)|v(r) ) ∣ ∣ ∣ t ∏ s=r+1 v(s) ∣ ∣ ∣ . consequently, ‖pφ‖ ≤ |x0|k + t−1 ∑ r=t0 ( |v(r)| + λ(r) ) ∣ ∣ ∣ t ∏ s=r+1 v(s) ∣ ∣ ∣ ||φ|| cubo 21, 3 (2019) new variation of parameters 43 or, ‖pφ‖ ≤ |x0|k + α‖φ‖ ≤ l. (1.13) since p is continuous we have that p : s → s. next we show that p is a contraction. for φ1,φ2 ∈ s, we have from (1.12) that ∣ ∣(pφ1)(t) − (pφ2)(t) ∣ ∣ ≤ t−1 ∑ r=t0 ( |v(r)| + λ(r) ) ∣ ∣ ∣ t ∏ s=r+1 v(s) ∣ ∣ ∣ ||φ1 − φ2|| ≤ α||φ1 − φ2||. this shows that p is a contraction. by banach’s contraction mapping principle,p has a unique fixed point x ∈ s which is bounded. moreover, the unique fixed point is a solution of (1.5) on [0,∞). next we show the zero solution is stable. let x be the unique solution. let ε > 0 be given and chose δ = ε1−α k . thus for |x0| < δ, we have by (1.13) that (1 − α)||x|| ≤ |x0|k < δk. or ||x|| ≤ ε. left to prove that (pϕ)(t) → 0, as t → ∞. we have already proved that the zero solution of (1.5) is stable. let δ be the one from stability such that |x0| < δ and define s∗ = { ϕ : z+ → r| ϕ(t0) = x0 , ||ϕ|| ≤ ǫ and ϕ(t) → 0 as t → ∞ } . (1.14) let p be given by (1.12) and define p : s∗ → s∗. the map p is contraction and it maps from s∗ into itself. we next show that (pϕ)(t) goes to zero as t goes to infinity. the first term on the right of (1.12) goes to zero due to condition (1.11). left to show that | t−1 ∑ r=t0 ( f(r,x(r)) − v(r)φ(r) ) t−1 ∏ s=r+1 v(s)| → 0, as t → ∞. let ϕ ∈ s∗ then |ϕ(t)| ≤ ǫ. also, since ϕ(t) → 0 as t → ∞, there exists a t1 > 0 such that for t > t1, |ϕ(t)| < ǫ1 for ǫ1 > 0. due to condition (1.11) there exists a t2 > t1 such that for t > t2 implies that ∣ ∣ ∣ t ∏ s=t1 v(s) ∣ ∣ ∣ < ǫ1 αǫ . 44 youssef n raffoul cubo 21, 3 (2019) thus for t > t2, we have ∣ ∣ ∣ t−1 ∑ r=t0 ( f(r,x(r)) − v(r)φ(r) ) t−1 ∏ s=r+1 v(s)| ∣ ∣ ∣ ≤ t−1 ∑ r=t0 ( λ(r) + v(r) ) |φ(r)| ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ t1−1 ∑ r=t0 ( λ(r) + v(r) ) |φ(r)| t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + t−1 ∑ r=t1 ( λ(r) + v(r) ) |φ(r)| ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ ǫ t1−1 ∑ r=t0 ( λ(r) + v(r) ) |φ(r)| ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + ǫ1α ≤ ǫ t1−1 ∑ r=t0 ( λ(r) + v(r) ) | t1−1 ∏ s=r+1 v(s) ∣ ∣ ∣ t−1 ∏ s=t1 v(s) ∣ ∣ ∣ + ǫ1α ≤ ǫ ∣ ∣ ∣ t−1 ∏ s=t1 v(s) ∣ ∣ ∣ t1−1 ∑ r=t0 ( λ(r) + v(r) ) ∣ ∣ ∣ t1−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + ǫ1α ≤ ǫα| t−1 ∏ s=t1 v(s)| + ǫ1α ≤ ǫ1 + ǫ1α. since ǫ1 is arbitrary small, this shows that (pϕ)(t) → 0 as t → ∞. as p has a unique fixed point, say x it implies the asymptotic stability of the zero solution of (1.11). this completes the proof. 2 contraction versus large contraction now we consider particular nonlinear equation and rewrite so we can invert the usual way. consequently, contraction mapping principle can no longer be useful. let f(t,x) = −a(t)x3 + l(t,x), where l(t,x) continuous and satisfies a smallness condition. thus, we consider x(t + 1) = −a(t)x3 + l(t,x). (2.1) mentioned paper as an example for illustrating the need for large contraction. in [4], the author put (1.12) in the form x(t + 1) = −a(t)x + a(t)(x − x3) + l(t,x). (2.2) cubo 21, 3 (2019) new variation of parameters 45 then by the variation of parameters formula we have x(t) = x0 t−1 ∏ s=t0 a(s) + t−1 ∑ r=t0 ( a(r) ( x(r) − x3(r) ) + l(t,x(r)) ) t−1 ∏ s=r+1 a(s). (2.3) it is naive to believe that every map can be defined so that it is a contraction, even with the strictest conditions. to see this, we consider g(x) = x − x3 then for x,y ∈ r with |x|, |y| ≤ √ 3 3 we have that |g(x) − g(y)| = |x − x3 − y + y3| ≤ |x − y| ( 1 − x 2 + y2 2 ) and the contraction constant tends to one as x2+y2 → 0. as a consequence, the regular contraction mapping principle failed to produce any results. for more on this and large contraction, we refer to [18], p: 52. to get around it, we let v(t) be a sequence such that v : z+ → r with v(t) 6= 0 for all t ∈ z+. by similar steps as in the development of (1.4) we arrive at the variation of parameters formula x(t) = x0 t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 ( v(r)x(r) − a(t)x3(r) + l(t,x(r)) ) t−1 ∏ s=r+1 v(s) (2.4) thus, one can show that the map given by f(x) = v(r)x(r) − a(t)x3(r), is a contraction on some bounded and small set provided a and v have small magnitudes. to better illustrate our intention we set l(t,x) = 0, and consider (2.1). then from the above variation of parameters formula, we have that x(t) = x0 t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 ( v(r)x(r) − a(t)x3(r)) ) t−1 ∏ s=r+1 v(s). (2.5) assume for φ ∈ s and positve constant l, we have that ∣ ∣ ∣ x0 t−1 ∏ s=t0 v(s) ∣ ∣ ∣ + t−1 ∑ r=t0 ( l|v(r)| + l3|a(r)| ) ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ l, (2.6) and t−1 ∑ r=t0 ( |v(r)| + 3l2|a(r)| ) ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ α < 1. (2.7) the next theorem offers results about stability and boundedness. for more results on the stability and boundedness using fixed point theory, we refer the interest reader to the book [18] and to the paper [19]. 46 youssef n raffoul cubo 21, 3 (2019) theorem 2.1. assume (1.7), (1.10), (2.6) and (2.7). then the unique solution of (2.1) is bounded and its zero solution is stable. if, in addition, (1.11) holds, then the zero solution of (2.1) is asymptotically stable. proof. for φ ∈ s, define the mapping p : s → s, by (pφ)(t) = x0 t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 ( v(r)φ(r) − φ3(r)a(r) ) t ∏ s=r+1 v(s) (2.8) it is clear that (pφ)(t0) = x0. now for φ ∈ s, we have that ∣ ∣(pφ)(t) ∣ ∣ ≤ |x0|k + t−1 ∑ r=t0 ( |v(r)||φ(r)| + |φ3(r)||a(r)| ) ∣ ∣ ∣ t ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ ∣ ∣ ∣ x0 t−1 ∏ s=t0 v(s) ∣ ∣ ∣ + t−1 ∑ r=t0 ( l|v(r)| + l3|a(r)| ) ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ . thus, ‖pφ‖ ≤ l. since p is continuous we have that p : s → s. next we show that p is a contraction. for φ1,φ2 ∈ s, we have from (1.12) that ∣ ∣(pφ1)(t) − (pφ2)(t) ∣ ∣ ≤ t−1 ∑ r=t0 ( |v(r)||φ1(r) − φ2(r)| + t−1 ∑ r=t0 |a(r)||φ1(r) − φ2(r)|(φ21(r) + |φ1(r)φ2(r)| + φ21(r) ) ∣ ∣ ∣ t ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ t−1 ∑ r=t0 ( |v(r)| + 3l2|a(r)| ) ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ||φ1 − φ2|| ≤ α||φ1 − φ2||. this shows that p is a contraction. by banach’s contraction mapping principle,p has a unique fixed point x ∈ s which is bounded. the proof for stability and asymptotic stability follow along the lines of the proof of theorem 1.5. for the rest of this section we set l(t,x) = 0 in (2.3) and use large contraction and prove parallel theorem to theorem 2.1. we saw before that the function or map, g(x) = x − x3 does not define a contraction. to get around it we use the notion of large contraction that was introduced by burton in [5] . we will restate the contraction mapping principle and krasnoselskii’s fixed point theorems in which the regular contraction is replaced with large contraction. then based on the notion of large contraction, we introduce a theorem to obtain boundedness results in which large contraction is substituted for regular contraction. cubo 21, 3 (2019) new variation of parameters 47 definition 2.2. let (m,d) be a metric space and b: m → m. the map b is said to be large contraction if φ,ϕ ∈ m, with φ 6= ϕ then d(bφ,bϕ) ≤ d(φ,ϕ) and if for all ε > 0, there exists a δ ∈ (0,1) such that [φ,ϕ ∈ m,d(φ,ϕ) ≥ ε] ⇒ d(bφ,bϕ) ≤ δd(φ,ϕ). the next theorems are alternative to the regular contraction mapping principle and krasnoselskii’s fixed point theorem in which we substitute large contraction for regular contraction. the proofs of the two theorems and the statement of definition 2.2 can be found in [5]. theorem 2.3. let (m,ρ) be a complete metric space and b be a large contraction. suppose there are an x ∈ m and an l > 0 such that ρ(x,bnx) ≤ l for all n ≥ 1. then b has a unique fixed point in m. next we state and prove a remarkable theorem by adivar, raffoul and islam that generalizes the concept of large contraction. its proof can be found in [18]. the theorem provides easily checked sufficient conditions under which a mapping is a large contraction. several authors have published it in their work without the proper citations. consider the mapping h defined by h(x(u)) = x(u) − h(x(u)). (2.9) let α ∈ (0,1] be a fixed real number and define the set mα by mα = {φ : φ ∈ c(r,r) and ‖φ‖ ≤ α} . (2.10) h.1. h : r → r is continuous on [−α,α] and differentiable on (−α,α), h.2. the function h is strictly increasing on [−α,α], h.3. sup t∈(−α,α) h′(t) ≤ 1. theorem 2.4. ([1] )[adivar-raffoul-islam] (classifications of large contraction theorem) let h : r → r be a function satisfying (h.1-h.3). then the mapping h in (2.9) is a large contraction on the set mα. example 2.5. let α ∈ (0,1) and k ∈ n be fixed elements and u ∈ (−1,1). 1. the condition (h.2) is not satisfied for the function h1(u) = 1 2k u2k. 2. the function h2(u) = 1 2k+1 u2k+1 satisfies (h.1-h.3). proof. since h′1(u) = u 2k−1 < 0 for −1 < u < 0, the condition (h.2) is not satisfied for h1. evidently, (h.1-h.2) hold for h2. (h.3) follows from the fact that h ′ 2(u) ≤ α2k and α ∈ (0,1). 48 youssef n raffoul cubo 21, 3 (2019) we have the following lemma. define the mapping h(x) = x − x3. (2.11) lemma 2.6. let ‖ · ‖ denote the supremum norm. if m = { φ : z → r | φ(0) = φ0, and ‖φ‖ ≤ √ 3 3 } , then the mapping h defined by (2.11) is a large contraction on the set m. proof. let α = √ 3 3 and h(x) = x3. then, clearly h satisfies (h.1-h.2). moreover, sup x∈(−α,α) h′(x) = 1, which satisfies h.3. hence by theorem 2.4 defines a large contraction. for ψ ∈ m, we define the map b : m → m by (bψ)(t) = ψ0 t−1 ∏ s=0 a(s) + t−1 ∑ s=0 ( a(s)h(ψ(s)) t−1 ∏ u=s+1 a(u) ) (2.12) lemma 2.7. assume for all t ∈ z |ψ0| ∣ ∣ ∣ t−1 ∏ s=0 a(s) ∣ ∣ ∣ + 2 √ 3 9 t−1 ∑ s=0 ∣ ∣ ∣ t−1 ∏ u=s a(u) ∣ ∣ ∣ ≤ √ 3 3 . (2.13) if h is a large contraction on m, then so is the mapping b. proof. it is easy to see that |h(x(t))| = |x(t) − x(t)3| ≤ 2 √ 3 9 for all x ∈ m. by lemma 2.6, h is a large contraction on m. hence, for x,y ∈ m with x 6= y, we have ‖hx−hy‖ ≤ ‖x − y‖. hence, |bx(t) − by(t)| ≤ t−1 ∑ s=0 |h(x(s)) − h(y(s))| ∣ ∣ ∣ t−1 ∏ u=s a(u) ∣ ∣ ∣ ≤ 2 √ 3 9 t−1 ∑ s=0 ∣ ∣ ∣ t−1 ∏ u=s a(u) ∣ ∣ ∣ ‖x − y‖ = ‖x − y‖. taking supremum norm over the set [0,∞), we get that ‖bx − by‖ ≤ ‖x − y‖. for a given ε ∈ (0,1), suppose x,y ∈ m with ‖x − y‖ ≥ ε. then for δ = min { 1 − ε2/16,1/2 } , which implies that 0 < δ < 1. hence, for all such ε > 0 we know that [x,y ∈ m,‖x − y‖ ≥ ε] ⇒ ‖hx − hy‖ ≤ δ‖x − y‖. cubo 21, 3 (2019) new variation of parameters 49 therefore, using (2.13), one easily verify that ‖bx − by‖ ≤ δ‖x − y‖. the proof is complete. we arrive at the following theorem in which we prove boundedness. theorem 2.8. assume (2.13). then (2.1) has a unique solution in m which is bounded. proof. (m,‖ · ‖) is a complete metric space of bounded sequences. for ψ ∈ m we must show that (bψ)(t) ∈ m. from (2.12) and the fact that |h(x(t))| = |x(t) − x(t)3| ≤ 2 √ 3 9 for all x ∈ m, we have |(bψ)(t)| ≤ |ψ0| ∣ ∣ ∣ t−1 ∏ s=0 a(s) ∣ ∣ ∣ + 2 √ 3 9 t−1 ∑ s=0 ∣ ∣ ∣ t−1 ∏ u=s a(u) ∣ ∣ ∣ ≤ √ 3 3 . this shows that (bψ)(t) ∈ m. lemma 2.6 implies the map b is a large contraction and hence by theorem 2.3, the map b has a unique fixed point in m which is a solution of (2.1). this completes the proof. 3 periodic solutions in this section we apply our new method to linear or nonlinear difference equations to show the existence of periodic solutions without the requirement of some classic conditions. to better illustrate our approach, we consider the nonlinear difference equation x(t + 1) = a(t)x(t) + f(t,x(t)) (3.1) where f is continuous in x. let t be an integer such that t ≥ 1. we assume the periodicity condition a(t + t) = a(t), and f(t + t, ·) = f(t, ·). (3.2) let bc is the space of bounded sequences φ : z → r with the maximum norm || · ||. define pt = {φ ∈ bc,φ(t + t) = φ(t)}. then pt is a banach space when it is endowed with the maximum norm ‖x‖ = max t∈[0,t −1] |x(t)|. 50 youssef n raffoul cubo 21, 3 (2019) also, we assume that t−1 ∏ s=t−t a(s) 6= 1. (3.3) throughout this section we assume that a(t) 6= 0 for all t ∈ [0,t − 1]. let x ∈ pt . then eqn. (3.1) is equivalent to ∆ [ x(t) t−1 ∏ s=t0 a−1(s) ] = f(t,x(t)) t ∏ s=t0 a−1(s). (3.4) summing equation (3.4) from t − t to t − 1 and using the fact that x(t − t) = x(t), gives x(t) = ( 1 − t−1 ∏ s=t−t a(s) )−1 t−1∑ r=t−t f(r,x(r)) t−1 ∏ s=r+1 a(s). (3.5) define the mapping p on pt by (pφ)(t) = ( 1 − t−1 ∏ s=t−t a(s) )−1 t−1∑ r=t−t f(r,φ(r)) t−1 ∏ s=r+1 a(s). (3.6) one can easily verify that (pφ)(t + t) = (pφ)(t), and hence p : pt → pt . theorem 3.1. suppose a(t) 6= 0 for all t ∈ [0,t − 1] and assume (3.3). suppose the function f is lipschitz continuous with lipschitz constant k. if k ∣ ∣ ∣ ( 1 − t−1 ∏ s=t−t a(s) )−1∣ ∣ ∣ t−1 ∑ r=t−t ∣ ∣ ∣ t−1 ∏ s=r+1 a(s) ∣ ∣ ∣ ≤ α, for α ∈ (0,1), then eqn. (3.1) has a unique periodic solution. proof. the proof is easily obtained by direct application of contraction mapping principle on the set pt . next, we use our new technique to avoid the requirement that a(t) 6= 0 for all t ∈ [0,t − 1] along with condition (3.3). let v(t) be a sequence such that v : z+ → r with v(t) 6= 0 for all t ∈ [0,t − 1. assume (3.2) and for v ∈ pt , multiply both sides of (3.1) by t ∏ s=t0 v−1(s) to obtain ∆ [ x(t) t−1 ∏ s=t0 v−1(s) ] = [ ( a(t)x(t) − v(t)x(t) + f(t,x(t)) ] t ∏ s=t0 v−1(s). (3.7) summing equation (3.7) from t − t to t-1 gives and using the fact that x(t − t) = x(t), gives x(t) = ( 1 − t−1 ∏ s=t−t v(s) )−1 t−1∑ r=t−t [a(r)x(r) − v(r)x(r) + f(r,x(r))] t−1 ∏ s=r+1 v(s). (3.8) cubo 21, 3 (2019) new variation of parameters 51 define the mapping p on pt by (pφ)(t) = ( 1 − t−1 ∏ s=t−t v(s) )−1 t−1∑ r=t−t [a(r)x(r) − v(r)x(r) + f(r,φ(r)) t−1 ∏ s=r+1 v(s). (3.9) one can easily verify that (pφ)(t + t) = (pφ)(t), and hence p : pt → pt . theorem 3.2. suppose v(t) 6= 0 for all t ∈ [0,t − 1] and assume t−1 ∏ s=t−t v(s) 6= 1. (3.10) suppose the function f is lipschitz continuous with lipschitz constant k,. if ∣ ∣ ∣ ( 1 − t−1 ∏ s=t−t v(s) )−1∣ ∣ ∣ t−1 ∑ r=t−t [|a(r)| + |v(r)| + k] ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ α, for α ∈ (0,1), then eqn. (3.1) has a unique periodic solution. proof. the proof is easily obtained by direct application of the contraction mapping principle on the set pt . next we display an example. example 3.3. for positive constant k, we consider the difference equation x(t + 1) = ( 1 − (−1)t ) x(t) + kx 1 + x2 (3.11) it is clear that a(t) = ( 1 − (−1)t ) is periodic of period t = 2 and a(0) = 0. hence theorem 3.1 can not be applied. on the other hand we may apply theorem 3.2 by taking v(t) = (−1)t 2 , for sufficiently small k. 4 neutral difference equations we extend the results of the previous sections to the neutral difference equation with functional delay x(t + 1) = a(t)x(t) + b(t)x(t − g(t)) + c(t)∆x(t − g(t)) (4.1) where where a,b,c : z → r, and g : z → z+. moreover, we will discuss the concept of equiboundedness. if for some positive constant k, |g| ≤ k then for any integer t0 ≥ 0, we define z0 to be the set of integers in [t0 − k,t0]. if g is unbounded then z0 will be the set of integers in (−∞, t0]. we assume 52 youssef n raffoul cubo 21, 3 (2019) the existence of a given bounded initial sequence ψ(t) : z0 → r. we will use the summation by parts formula ∑ ( ex(t)∆z(t) ) = x(t)z(t) − ∑ z(t)∆x(t) where e is defined as ex(t) = x(t + 1). definition 4.1. we say x(t) := x(t,t0,ψ) is a solution of (4.1) if x(t) = ψ(t) on z0 and satisfies (4.1) for t ≥ t0. definition 4.2. the zero solution of (4.1) is stable if for any ǫ > 0 and any integer t0 ≥ 0 there exists a δ > 0 such that |ψ(t)| ≤ δ on z0 implies |x(t,t0,ψ)| ≤ ǫ for t ≥ t0. definition 4.3. the zero solution of (4.1) is asymptotically stable if it is stable and if for any integer t0 ≥ 0 there exists r(t0) > 0 such that |ψ(t)| ≤ r(t0) on z0 implies |x(t,t0,ψ)| → 0 as t → ∞. definition 4.4. a solution x(t,t0,ψ) of (4.1) is said to be bounded if there exist a b(t0,ψ) > 0 such that |x(t,t0,ψ)| ≤ b(t0,ψ) for t ≥ t0. definition 4.5. the solutions of (4.1) are said to be equi-bounded if for any t0 and any b1 > 0, there exists a b2 = b2(t0,b1) > 0 such that |ψ(t)| ≤ b1 on z0 implies |x(t,t0,ψ)| ≤ b2 for t ≥ t0. for the remaining of the section we assume that there is a positive constant k, |g| ≤ k. lemma 4.6. if x(t) is a solution of (4.1) and satisfies the initial condition x(t) = ψ(t) for t ∈ z0, then x(t) is a solution of the summation equation if and only if x(t) = [ x(t0) − c(t0 − 1)x(t0 − g(t0)) ] t−1 ∏ s=t0 v(s) + c(t − 1)x(t − g(t)) + t−1 ∑ r=t0 [ (a(r) − v(r))x(r) t−1 ∏ s=r+1 v(s) ] + t−1 ∑ r=t0 ( [b(r) − φ(r)]x(r − g(r)) t−1 ∏ s=r+1 v(s) ) , t ≥ t0 (4.2) where φ(r) = c(r) − c(r − 1)v(r). [0,t ] where v : z ∩ [−k,∞) → r with v(t) 6= 0. multiply both sides of (4.1) by t ∏ s=t0 v−1(s) and then notice the resulting expression is equivalent cubo 21, 3 (2019) new variation of parameters 53 to ∆ [ x(t) t−1 ∏ s=t0 v−1(s) ] = [ (a(t) − v(t))x(t) + b(t)x(t − g(t)) + c(t)∆x(t − g(t)) ] t ∏ s=t0 v−1(s). summing the above expression from t0 to t-1 gives x(t) t−1 ∏ s=t0 v−1(s) − x(t0) = t−1 ∑ r=t0 [ (a(r) − v(r))x(r) + b(r)x(r − g(r)) + c(r)∆x(r − g(r)) ] r ∏ s=t0 v−1(s) dividing both sides by t−1 ∏ s=t0 v−1(s), gives x(t) = x(t0) t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 [ (a(r) − v(r))x(r) t−1 ∏ s=r+1 v(s) ] + t−1 ∑ r=t0 [ b(r)x(r − g(r)) +c(r)∆x(r − g(r)) ] r ∏ s=t0 v−1(s) t−1 ∏ s=t0 v(s) = x(t0) t−1 ∏ s=t0 v(s) + t−1 ∑ r=t0 [ (a(r) − v(r))x(r) t−1 ∏ s=r+1 v(s) ] + t−1 ∑ r=t0 [ b(r)x(r − g(r)) ] t−1 ∏ s=r+1 v(s) + t−1 ∑ r=t0 [ c(r)∆x(r − g(r)) ] t−1 ∏ s=r+1 v(s) using summation by parts and after some calculations and simplification we arrive at (4.2). theorem 4.7. suppose v(t) 6= 0 for t ≥ t0 and v(t) satisfies ∣ ∣ ∣ t−1 ∏ s=t0 v(s) ∣ ∣ ∣ ≤ m 54 youssef n raffoul cubo 21, 3 (2019) for m > 0. also, suppose that there is an α ∈ (0,1) such that |c(t − 1)| + t−1 ∑ r=t0 ∣ ∣a(r) − v(r ∣ ∣ ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + t−1 ∑ r=t0 [ |b(r) − φ(r)| ] ∣ ∣ ∣ t−1 ∏ s=r+1 a(s) ∣ ∣ ∣ ≤ α, t ≥ t0. (4.3) then solutions of (4.1) are equi-bounded. proof. let b1 and b2 be two positive constants to be defined later in the proof and let ψ(t) be a bounded initial function satisfying |ψ(t)| ≤ b1 on z0. define s = { ϕ : z → r| ϕ(t) = ψ(t) on z0 and ||ϕ|| ≤ b2 } , where ||ϕ|| = sup |ϕ(t)|. t ∈ z then ( s, || · || ) is a complete metric space. define mapping p : s → s by ( pϕ ) (t) = ψ(t) on z0 and ( pϕ ) (t) = [ ψ(t0) − c(t0 − 1)ψ(t0 − g(t0)) ] t−1 ∏ s=t0 v(s) + c(t − 1)ϕ(t − g(t)) + t−1 ∑ r=t0 [ (a(r) − v(r))ϕ(r) t−1 ∏ s=r+1 v(s) ] + t−1 ∑ r=t0 [ ( b(r) − φ(r) ) ϕ(r − g(r)) t−1 ∏ s=r+1 v(s) ] , t ≥ t0. (4.4) let b1 > 0 be given. choose b2 such that |1 − c(t0 − 1)|mb1 + αb2 ≤ b2 (4.5) we first show that p maps from s to s. by (4.5) |(pϕ)(t)| ≤ |1 − c(t0 − 1)|mb1 + αb2 ≤ b2 for t ≥ t0 cubo 21, 3 (2019) new variation of parameters 55 thus p maps from s into itself. we next show that p is a contraction under the supremum norm. let ζ,η ∈ s. then |(pζ)(t) − (pη)(t)| ≤ ( |c(t − 1)| + t−1 ∑ r=t0 [ |b(r) − φ(r)| ] ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ) ||ζ − η|| + t−1 ∑ r=t0 ∣ ∣a(r) − v(r ∣ ∣ ∣ ∣ ∣ t−1 ∏ s=r+1 v(s)||ζ − η|| ≤ α||ζ − η||. this shows that p is a contraction. thus, by the contraction mapping principle, p has a unique fixed point in s which solves (4.1). hence solutions of (4.1) are equi-bounded. theorem 4.8. assume that the hypotheses of theorem 4.7 hold. then the zero solution of (4.1) is stable. proof. let ǫ > 0 be given. choose δ > 0 such that |1 − c(t0 − 1)|mδ + αǫ ≤ ǫ. (4.6) let ψ(t) be a bounded initial function satisfying |ψ(t)| ≤ δ. define the complete metric space s by s = { ϕ : z → r| ϕ(t) = ψ(t) on z0 and ||ϕ|| ≤ ǫ } . let p : s → s be defined by (4.4). then, from the proof of theorem 4.8 we have that p is a contraction map and for any ϕ ∈ s, ||pϕ|| ≤ ǫ. hence the zero solution of (4.1) is stable. theorem 4.9. assume that the hypotheses of theorem 4.7 hold. also assume that t−1 ∏ s=t0 v(s) → 0 as t → ∞, (4.7) then the zero solution of (4.1) is asymptotically stable. proof. we have already shown that the zero solution of (4.1) is stable. let r(t0) be the δ of stability of the zero solution. let ψ(t) be any initial discrete function satisfying |ψ(t)| ≤ r(t0). define s∗ = { ϕ : z → r| ϕ(t) = ψ(t) on z0, ||ϕ|| ≤ ǫ and ϕ(t) → 0 as t → ∞ } . 56 youssef n raffoul cubo 21, 3 (2019) define p : s∗ → s∗ by (4.4). the from theorem 4.7, the map p is a contraction and it maps from s∗ into itself. left to show that (pϕ)(t) → 0 as t → ∞. let ϕ ∈ s∗. then the first first term on the right of (4.4) goes to zero. the second term on the right side of (4.4) goes to zero due condition (4.7) and the fact that ϕ ∈ s∗. now we show that the second term on the right side of (4.7) goes to zero as t → ∞. let ϕ ∈ s∗ then |ϕ(t)| ≤ ǫ. also, since ϕ(t) → 0 as t → ∞, there exists a t1 > 0 such that for t > t1, |ϕ(t)| < ǫ1 for ǫ1 > 0. due to condition (4.7) there exists a t2 > t1 such that for t > t2 implies that ∣ ∣ ∣ t ∏ s=t1 v(s) ∣ ∣ ∣ < ǫ1 αǫ . thus for t > t2, we have ∣ ∣ ∣ t−1 ∑ r=t0 [ a(r) − v(r) ] ϕ(r) t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ t−1 ∑ r=t0 ∣ ∣ ∣ (a(r) − v(r))ϕ(r) t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ t1−1 ∑ r=t0 ∣ ∣ ∣ (a(r) − v(r))ϕ(r) t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + t−1 ∑ r=t1 ∣ ∣ ∣ (a(r) − v(r))ϕ(r) t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ ǫ t1−1 ∑ r=t0 ∣ ∣ ∣ (a(r) − v(r)) t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + ǫ1α ≤ ǫ t1−1 ∑ r=t0 ∣ ∣ ∣ [a(r) − v(r)] t1−1 ∏ s=r+1 v(s) t−1 ∏ s=t1 v(s) ∣ ∣ ∣ + ǫ1α ≤ ǫ ∣ ∣ ∣ t−1 ∏ s=t1 v(s) ∣ ∣ ∣ t1−1 ∑ r=t0 ∣ ∣ ∣ [a(r) − v(r)] t1−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + ǫ1α ≤ ǫα| t−1 ∏ s=t1 v(s)| + ǫ1α ≤ ǫ1 + ǫ1α. this shows that the second term of (4.4) goes to zero as t goes to infinity. showing that the last term on the right side of (4.7) goes to zero as t → ∞ is similar, and hence we omit. this implies that (pϕ)(t) → 0 as t → ∞. by the contraction mapping principle, p has a unique fixed point that solves (4.1) and goes to zero as t goes to infinity. this concludes that the zero solution of (4.1) is asymptotically stable. cubo 21, 3 (2019) new variation of parameters 57 remark 4.10. if the delay function g(t) is unbounded, then we may prove a similar theorem to theorem 4.9 by making the additional requirement that t − g(t) → 0, as t → ∞. 5 example example 5.1. solutions of the linear neutral difference equation x((t + 1) = 2t+1 8(1 + t)! x(t − 2) + 2t+1 8(1 + t)! ∆x(t − 2), t ≥ 0 (5.1) are equi-bounded and the zero solution is asymptotically stable. proof. let v(t) = 1 3(1 + t) . comparing terms, we see that a(t) = 0, b(t) = c(t) = 2t+1 8(1 + t)! . set t0 = 0. then (4.3) is equivalent to |c(t − 1)| + t−1 ∑ r=0 ∣ ∣v(r) ∣ ∣ ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + t−1 ∑ r=0 [ |b(r) − φ(r)| ] ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ 2 t 8(t)! + t−1 ∑ r=0 t−1 ∏ s=r 1 3(1 + s) + t−1 ∑ r=0 2r 8(1 + r)! t−1 ∏ s=r+1 1 3(1 + s) . now, t−1 ∑ r=0 2r 8(1 + r)! t−1 ∏ s=r+1 1 3(1 + s) ≤ 1/3 t−1 ∑ r=0 2r 8(1 + r)! 1 (r + 2)(r + 3)...(t) ≤ 1/3 t−1 ∑ r=0 2r 8t! ≤ 1 24t! (2t − 1) ≤ 2 t 24t! . similarly, by estimating 1 1+s ≤ 1, for s ≥ 0, we have that t−1 ∑ r=0 t−1 ∏ s=r 1 3(1 + s) ≤ t−1 ∑ r=0 ( 1 3 )t−r ≤ (1 3 )t t−1 ∑ r=0 3r = ( 1 3 )t[ 3r 2 ]|t−10 ≤ 1 6 [1 − 21−t] ≤ 1/6 58 youssef n raffoul cubo 21, 3 (2019) combining the two inequalities we end up with |c(t − 1)| + t−1 ∑ r=0 ∣ ∣v(r) ∣ ∣ ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ + t−1 ∑ r=0 [ |b(r) − φ(r)| ] ∣ ∣ ∣ t−1 ∏ s=r+1 v(s) ∣ ∣ ∣ ≤ 2t 8(t)! + 1 3 + 2t 24t! ≤ 1 4 + 1 6 + 1 12 = 1 2 < 1. hence (4.3) is satisfied. it is clear that condition (4.7) is satisfied for the specified value of v. this implies the zero solution is asymptotically stable, by theorem 4.9. left to show solutions are equi-bounded. since t0 = 0, we have that z0 = [−2,0]. let b1 > 0 be given and ψ(t) : z0 → r be a given initial function with |ψ(t)| ≤ b1. we need to choose b2 so that (4.5) is satisfied. it is clear that c(t0 − 1) = c(−1) = 1 8 , and hence |1 − c(t0 − 1)| = 1 − 1 8 = 7 8 . in addition ∣ ∣ ∣ t−1 ∏ s=0 v(s) ∣ ∣ ∣ ≤ m is satisfied for m = 1 3 . from the above calculation for asymptotic stability, we see that α = 1 2 . now we choose b2 such that 7 24 b1 ≤ b2 2 . then, in our case, inequality (4.5) corresponds to |1 − c(t0 − 1)|mb1 + αb2 ≤ b2. or equivalently, 7 24 b1 + b2 2 ≤ b2, is satisfied. remark 5.2. we mention that the work of islam-yankson in [12] can not be applied to our example due to the absence of the linear term a(t)x(t). it is worth mentioning that the results of section 4 can be easily extended to the nonlinear neutral difference equation x(t + 1) = a(t)x(t) + c(t)∆x(t − g(t)) + q(t,x(t),x(t − g(t))) (5.2) cubo 21, 3 (2019) new variation of parameters 59 where a(t),c(t) and g(t) are defined as before. we assume that, q(t,0,0) = 0 for the stability and q is locally lipschitz in x and y. that is, there is a k > 0 so that if |x|, |y|, |z| and |w| ≤ k then |q(x,y) − q(z,w)| ≤ l|x − z| + e|y − w| for some positive constants l and e. note that |q(x,y)| = |q(x,y) − q(0,0) + q(0,0)| ≤ |q(x,y) − q(0,0)| + |q(0,0)| ≤ l|x| + e|y|. remark 5.3. the method of section 4 can be easily used to extend the existence of periodic solutions to systems of the form of (4.1) and (5.2), see [15]. 60 youssef n raffoul cubo 21, 3 (2019) references [1] adivar, m,. islam, m. and raffoul, y., separate contraction and existence of periodic solutions in totally nonlinear delay differential equations, hacettepe journal of mathematics and statistics, 41 (1) (2012), 1-13. 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[14] j. liu, a first course in the qualitative theory of differential equations,pearson education, inc., upper saddle river, new jersey 07458, 2003. cubo 21, 3 (2019) new variation of parameters 61 [15] m. maroun and y. raffoul , periodic solutions in nonlinear neutral difference equations with functional delay, j. korean math. soc. 42 (2005), no. 2, 255-268. [16] r. medina, asymptotic behavior of volterra difference equations, computers math. appl. 41(2001), 679-687. [17] r. medina, the asymptotic behavior of the solutions of a volterra difference equation, computers math. appl. 181(1994), 19-26. [18] y. raffoul, qualitative theory of volterra difference equations, springer, new york, 2018. [19] y. raffoul, stability and periodicity in discrete delay equations, j. math. anal. appl. 324 (2006) 1356-1362. [20] y. raffoul, stability in neutral nonlinear differential equations with functional delays using fixed point theory, mathematical and computer modelling, 40(2004), 691-700. [21] y. raffoul, general theorems for stability and boundedness for nonlinear functional discrete systems, j. math. anal. appl. ,279(2003), 639-650. introduction contraction versus large contraction periodic solutions neutral difference equations example cubo, a mathematical journal vol. 24, no. 02, pp. 343–368, august 2022 doi: 10.56754/0719-0646.2402.0343 fixed point results of (φ,ψ)-weak contractions in ordered b-metric spaces n. seshagiri rao 1, b k. kalyani 1 1department of mathematics, school of applied science, vignan’s foundation for science, technology & research, vadlamudi-522213, andhra pradesh, india. seshu.namana@gmail.com b kalyani.namana@gmail.com abstract the purpose of this paper is to prove some results on fixed point, coincidence point, coupled coincidence point and coupled common fixed point for the mappings satisfying generalized (φ,ψ)-contraction conditions in complete partially ordered b-metric spaces. our results generalize, extend and unify most of the fundamental metrical fixed point theorems in the existing literature. a few examples are illustrated to support our findings. resumen el propósito de este artículo es demostrar algunos resultados sobre puntos fijos, puntos de coincidencia, puntos de coincidencia acoplados y puntos de coincidencia acoplados comunes para aplicaciones que satisfacen condiciones de (φ,ψ)contracción generalizadas en b-espacios métricos completos parcialmente ordenados. nuestros resultados generalizan, extienden y unifican la mayoría de los teoremas de punto fijo métricos fundamentales en la literatura existente. se ilustran algunos ejemplos para apoyar nuestros resultados. keywords and phrases: fixed point, coupled coincidence point, coupled common fixed point, partially ordered b-metric space, compatible, mixed f-monotone. 2020 ams mathematics subject classification: 47h10, 54h25. accepted: 03 august, 2022 received: 02 june, 2021 c©2022 n. seshagiri rao et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2402.0343 https://orcid.org/0000-0003-2409-6513 https://orcid.org/0000-0002-4531-5976 mailto:seshu.namana@gmail.com mailto:kalyani.namana@gmail.com 344 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) 1 introduction the usual metric space has been generalized and enhanced in many different directions, one of such generalizations is a b-metric space which was first coined by czerwik in [16] and is also known as metric type space (khamsi and hussain [35] used recently the term “metric type space”)‘. indeed, in some papers it is considered that this concept has been introduced by bourbaki [14] in 1974, or that it has been introduced by bakhtin [12] in 1989, or by czerwik [16] in 1993 or even by czerwik [17] in 1998. after extensive searches in zbmath and mathematical reviews, it appears that the first fixed point theorem in a quasimetric space (b-metric spaces) has been established in 1981 by vulpe et al. [55], who transposed the picard-banach contraction mapping principle from metric spaces to the framework of a quasimetric space. some important information on the introduction of a b-metric spaces can be found from the article “the early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects” by berinde and pacurar [13]. later, a series of papers have been dedicated to the improvement of fixed point results for single valued and multi-valued operators on b-metric spaces by following various topological properties, some of such are from [1, 3, 6, 5, 9, 20, 22, 28, 29, 30, 32, 34, 36, 39, 40, 41, 43, 53]. the concept of coupled fixed points for certain mappings in ordered spaces was first introduced by bhaskar et al. [23] and applied their results to study the existence and uniqueness of the solutions for boundary valued problems. while the concept of coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings with monotone property in complete partially ordered metric spaces was first introduced by lakshmikantham et al. [37]. since then, several authors have carried out further generalizations and improvements in various spaces (see [10, 18, 21, 24, 44, 48]). aghajani et al. [2] proved some coupled coincidence and coupled fixed point results for mappings satisfying generalized (ψ,φ,θ)-contractive conditions in partially ordered complete b-metric spaces. later, the results of [2] have been improved and generalized by huaping huang et al. [27] in the same space. more works on coupled coincidence and coupled fixed point results for generalized contraction mappings in ordered spaces can be seen from [4, 7, 8, 11, 15, 19, 25, 26, 31, 38, 42, 45, 46, 47, 49, 50, 51, 52]. recently, some results on fixed point, coincidence point and coupled coincidence points for the mappings satisfying generalized weak contraction contractions in partially ordered b-metric spaces have been discussed by belay mituku et al. [39], seshagiri rao et al. [53, 54] and kalyani et al. [33]. the aim of this work is to provide some results on fixed point and coincidence point, coupled coincidence point for the mappings satisfying generalized (φ,ψ)-contractive conditions in an ordered b-metric space. our results are the variations and the generalizations of the results of [25, 26, 31, 38, 42, 45, 52] and several comparable results in the existing literature. a few numerical examples are illustrated to support the findings. cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 345 2 mathematical preliminaries the following definitions and results will be needed in what follows. definition 2.1 ([39, 53]). a mapping d : p ×p → [0,+∞), where p is a non-empty set is said to be a b-metric, if it satisfies the properties given below for any υ,ξ,µ ∈ p and for some real number s ≥ 1, (a) d(υ,ξ) = 0 if and only if υ = ξ, (b) d(υ,ξ) = d(ξ,υ), (c) d(υ,ξ) ≤ s(d(υ,µ) + d(µ,ξ)). then (p,d,s) is known as a b-metric space. if (p,�) is still a partially ordered set, then (p,d,s,�) is called a partially ordered b-metric space. definition 2.2 ([39, 53]). let (p,d,s) be a b-metric space. then (1) a sequence {υn} is said to converge to υ, if lim n→+∞ d(υn,υ) = 0 and written as lim n→+∞ υn = υ. (2) {υn} is said to be a cauchy sequence in p, if lim n,m→+∞ d(υn,υm) = 0. (3) (p,d) is said to be complete, if every cauchy sequence in it is convergent. definition 2.3. if the metric d is complete then (p,d,s,�) is called complete partially ordered b-metric space. definition 2.4 ([39]). let (p,�) be a partially ordered set and let f,g : p → p be two mappings. then (1) g is called monotone non-decreasing, if gυ � gξ for all υ,ξ ∈ p with υ � ξ. (2) an element υ ∈ p is called a coincidence (common fixed) point of f and g, if fυ = gυ (fυ = gυ = υ). (3) f and g are called commuting, if fgυ = gfυ, for all υ ∈ p. (4) f and g are called compatible, if any sequence {υn} with lim n→+∞ fυn = lim n→+∞ gυn = µ, for µ ∈ p then lim n→+∞ d(gfυn,fgυn) = 0. (5) a pair of self maps (f,g) is called weakly compatible, if fgυ = gfυ, when gυ = fυ for some υ ∈ p. (6) g is called monotone f-non-decreasing, if fυ � fξ implies gυ � gξ, for any υ,ξ ∈ p. 346 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) (7) a non empty set p is called well ordered set, if every two elements of it are comparable i.e., υ � ξ or ξ � υ, for υ,ξ ∈ p. definition 2.5 ([2, 37]). let (p,�) be a partially ordered set and, let h : p × p → p and f : p → p be two mappings. then (1) h has the mixed f-monotone property, if h is non-decreasing f-monotone in its first argument and is non-increasing f-monotone in its second argument, that is for any υ,ξ ∈ p υ1,υ2 ∈ p, fυ1 � fυ2 implies h(υ1,ξ) � h(υ2,ξ) and ξ1,ξ2 ∈ p, fξ1 � fξ2 implies h(υ,ξ1) � h(υ,ξ2). suppose, if f is the identity mapping then h is said to have the mixed monotone property. (2) an element (υ,ξ) ∈ p × p is called a coupled coincidence point of h and f, if h(υ,ξ) = fυ and h(ξ,υ) = fξ. note that, if f is the identity mapping then (υ,ξ) is said to be a coupled fixed point of h. (3) an element υ ∈ p is called a common fixed point of h and f, if h(υ,υ) = fυ = υ. (4) h and f are commutative, if for all υ,ξ ∈ p, h(fυ,fξ) = f(hυ,hξ). (5) h and f are said to be compatible, if lim n→+∞ d(f(h(υn,ξn)),h(fυn,fξn)) = 0 and lim n→+∞ d(f(h(ξn,υn)),h(fξn,fυn)) = 0, whenever {υn} and {ξn} are any two sequences in p such that lim n→+∞ h(υn,ξn) = lim n→+∞ fυn = υ and lim n→+∞ h(ξn,υn) = lim n→+∞ fξn = ξ, for any υ,ξ ∈ p. we know that a b-metric is not continuous and then we use frequently the following lemma in the proof of our results for the convergence of sequences in b-metric spaces. lemma 2.6 ([2]). let (p,d,s,�) be a b-metric space with s > 1 and suppose that {υn} and {ξn} are b-convergent to υ and ξ respectively. then we have 1 s2 d(υ,ξ) ≤ lim n→+∞ inf d(υn,ξn) ≤ lim n→+∞ supd(υn,ξn) ≤ s 2d(υ,ξ). in particular, if υ = ξ, then lim n→+∞ d(υn,ξn) = 0. moreover, for each τ ∈ p, we have 1 s d(υ,τ) ≤ lim n→+∞ inf d(υn,τ) ≤ lim n→+∞ supd(υn,τ) ≤ sd(υ,τ). cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 347 3 main results the following distance functions are used throughout the paper. a self mapping φ defined on [0,+∞) is said to be an altering distance function, if it satisfies the following conditions: (i) φ is non-decreasing and continuous function, (iii) φ(t) = 0 if and only if t = 0. let us denote the set of all altering distance functions on [0,+∞) by φ. similarly, ψ denotes the set of all functions ψ : [0,+∞) → [0,+∞) satisfying the following conditions: (i) ψ is lower semi-continuous, (ii) ψ(t) = 0 if and only if t = 0. let (p,d,s,�) be a partially ordered b-metric space with parameter s > 1 and, let g : p → p be a mapping. set m(υ,ξ) = max { d(ξ,gξ) [1 + d(υ,gυ)] 1 + d(υ,ξ) , d(υ,gυ) d(υ,gξ) 1 + d(υ,gξ) + d(ξ,gυ) ,d(υ,ξ) } , (3.1) and n(υ,ξ) = max { d(ξ,gξ) [1 + d(υ,gυ)] 1 + d(υ,ξ) ,d(υ,ξ) } . (3.2) let φ ∈ φ and ψ ∈ ψ. the mapping g is a generalized (φ,ψ)-contraction mapping if it satisfies the following condition φ(sd(gυ,gξ)) ≤ φ(m(υ,ξ)) − ψ(n(υ,ξ)), (3.3) for any υ,ξ ∈ p with υ � ξ and m,n are same as above. now, we prove some results for the existence of fixed point, coincidence point, coupled coincidence point and coupled common fixed point of the mappings satisfying a generalized (φ,ψ)-contraction condition in the context of partially ordered b-metric space. we begin with the following fixed point theorem in this paper. theorem 3.1. suppose that (p,d,s,�) is a complete partially ordered b-metric space with parameter s > 1. let g : p → p be a generalized (φ,ψ)-contractive mapping, and be continuous, non-decreasing mapping with respect to �. if there exists υ0 ∈ p with υ0 � gυ0, then g has a fixed point in p. 348 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) proof. for some υ0 ∈ p such that gυ0 = υ0, then we have the result. assume that υ0 ≺ gυ0, then construct a sequence {υn} ⊂ p by υn+1 = gυn, for n ≥ 0. since g is non-decreasing, then by induction we obtain that υ0 ≺ gυ0 = υ1 � · · · � υn � gυn = υn+1 � · · · . (3.4) if for some n0 ∈ n such that υn0 = υn0+1 then from (3.4), υn0 is a fixed point of g and we have nothing to prove. suppose that υn 6= υn+1, for all n ≥ 1. since υn > υn−1 for all n ≥ 1 and then by condition (3.3), we have φ(d(υn,υn+1)) = φ(d(gυn−1,gυn)) ≤ φ(sd(gυn−1,gυn)) ≤ φ(m(υn−1,υn)) − ψ(n(υn−1,υn)). (3.5) from (3.5), we get d(υn,υn+1) = d(gυn−1,gυn) ≤ 1 s m(υn−1,υn), (3.6) where m(υn−1,υn) = max { d(υn,gυn) [1 + d(υn−1,gυn−1)] 1 + d(υn−1,υn) , d(υn−1,gυn−1) d(υn−1,gυn) 1 + d(υn−1,gυn) + d(υn,gυn−1) , 1 11 d(υn−1,υn) } = max { d(υn,υn+1), d(υn−1,υn) d(υn−1,υn+1) 1 + d(υn−1,υn+1) ,d(υn−1,υn) } ≤ max{d(υn,υn+1),d(υn−1,υn)}. (3.7) if max{d(υn,υn+1),d(υn−1,υn)} = d(υn,υn+1) for some n ≥ 1, then from (3.6) follows d(υn,υn+1) ≤ 1 s d(υn,υn+1), (3.8) which is a contradiction. this means that max{d(υn,υn+1),d(υn−1,υn)} = d(υn−1,υn) for n ≥ 1. hence, we obtain from (3.6) that d(υn,υn+1) ≤ 1 s d(υn−1,υn). (3.9) since, 1 s ∈ (0,1) then the sequence {υn} is a cauchy sequence by [1, 6, 41, 22]. but p is complete, then there exists µ ∈ p such that υn → µ. also, the continuity of g implies that gµ = g( lim n→+∞ υn) = lim n→+∞ gυn = lim n→+∞ υn+1 = µ. (3.10) cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 349 therefore, µ is a fixed point of g in p . last result is still valid for g not necessarily continuous, assuming an additional hypothesis on p . theorem 3.2. in theorem 3.1 assume that p satisfies, if a non-decreasing sequence {υn} → µ in p, then υn � µ for all n ∈ n, i.e., µ = supυn. then a non-decreasing mapping g has a fixed point in p. proof. from theorem 3.1, we take the same sequence {υn} in p such that υ0 � υ1 � · · · � υn � υn+1 � · · · , that is, {υn} is non-decreasing and converges to some µ ∈ p . thus from the hypotheses, we have υn � µ, for any n ∈ n, implies that µ = supυn. next, we prove that µ is a fixed point of g in p , that is gµ = µ. suppose that gµ 6= µ. let m(υn,µ) = max { d(µ,gµ) [1 + d(υn,gυn)] 1 + d(υn,µ) , d(υn,gυn) d(υn,gµ) 1 + d(υn,gµ) + d(µ,gυn) ,d(υn,µ) } , (3.11) and n(υn,µ) = max { d(µ,gµ) [1 + d(υn,gυn)] 1 + d(υn,µ) ,d(υn,µ) } . (3.12) letting n → +∞ and from the fact that lim n→+∞ υn = µ, we get lim n→+∞ m(υn,µ) = max{d(µ,gµ),0,0} = d(µ,gµ), (3.13) and lim n→+∞ n(υn,µ) = max{d(µ,gµ),0} = d(µ,gµ). (3.14) we know that υn � µ for all n, then from contraction condition (3.3), we get φ(d(υn+1,gµ)) = φ(d(gυn,gµ) ≤ φ(sd(gυn,gµ) ≤ φ(m(υn,µ)) − ψ(n(υn,µ)). (3.15) letting n → +∞ and use of (3.13) and (3.14), we get φ(d(µ,gµ)) ≤ φ(d(µ,gµ)) − ψ(d(µ,gµ)) < φ(d(µ,gµ)), (3.16) which is a contradiction under (3.16). thus, gµ = µ, that is g has a fixed point µ in p . now we give a sufficient condition for the uniqueness of the fixed point that exists in theorem 3.1 and theorem 3.2. every pair of elements has a lower bound or an upper bound. (3.17) 350 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) this condition is equivalent to, for every υ,ξ ∈ p, there exists w ∈ p which is comparable to υ and ξ. theorem 3.3. in addition to the hypotheses of theorem 3.1 (or theorem 3.2), condition (3.17) provides the uniqueness of a fixed point of g in p. proof. from theorem 3.1 (or theorem 3.2), we conclude that g has a nonempty set of fixed points. suppose that υ∗ and ξ∗ be two fixed points of g then, we claim that υ∗ = ξ∗. suppose that υ∗ 6= ξ∗, then from the hypotheses we have φ(d(gυ∗,gξ∗)) ≤ φ(sd(gυ∗,gξ∗)) ≤ φ(m(υ∗,ξ∗)) − ψ(n(υ∗,ξ∗)). (3.18) consequently, we get d(υ∗,ξ∗) = d(gυ∗,gξ∗) ≤ 1 s m(υ∗,ξ∗), (3.19) where m(υ∗,ξ∗) = max { d(ξ∗,gξ∗) [1 + d(υ∗,gυ∗)] 1 + d(υ∗,ξ∗) , d(υ∗,gυ∗) d(υ∗,gξ∗) 1 + d(υ∗,gξ∗) + d(ξ∗,gυ∗) ,d(gυ∗,gξ∗) } = max { d(ξ∗,ξ∗) [1 + d(υ∗,υ∗)] 1 + d(υ∗,ξ∗) , d(υ∗,υ∗) d(υ∗,ξ∗) 1 + d(υ∗,ξ∗) + d(ξ∗,υ∗) ,d(υ∗,ξ∗) } = max{0,0,d(υ∗,ξ∗)} = d(υ∗,ξ∗). (3.20) from (3.19), we obtain that d(υ∗,ξ∗) ≤ 1 s d(υ∗,ξ∗) < d(υ∗,ξ∗), (3.21) which is a contradiction. hence, υ∗ = ξ∗. this completes the proof. let (p,d,s,�) be a partially ordered b-metric space with parameter s > 1, and let g,f : p → p be two mappings. set mf(υ,ξ) = max { d(fξ,gξ) [1 + d(fυ,gυ)] 1 + d(fυ,fξ) , d(fυ,gυ) d(fυ,gξ) 1 + d(fυ,gξ) + d(fξ,gυ) ,d(fυ,fξ) } , (3.22) and nf(υ,ξ) = max { d(fξ,gξ) [1 + d(fυ,gυ)] 1 + d(fυ,fξ) ,d(fυ,fξ) } . (3.23) cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 351 now, we introduce the following definition. definition 3.4. let (p,d,s,�) be a partially ordered b-metric space with s > 1. the mapping g : p → p is called a generalized (φ,ψ)-contraction mapping with respect to f : p → p for some φ ∈ φ and ψ ∈ ψ, if φ(sd(gυ,gξ)) ≤ φ(mf(υ,ξ)) − ψ(nf(υ,ξ)), (3.24) for any υ,ξ ∈ p with fυ � fξ, where mf(υ,ξ) and nf(υ,ξ) are given by (3.22) and (3.23) respectively. theorem 3.5. suppose that (p,d,s,�) is a complete partially ordered b-metric space with s > 1. let g : p → p be a generalized (φ,ψ)-contractive mapping with respect to f : p → p and, g and f are continuous such that g is a monotone f-non-decreasing mapping, compatible with f and gp ⊆ fp. if for some υ0 ∈ p such that fυ0 � gυ0, then g and f have a coincidence point in p. proof. by following the proof of theorem 2.2 in [8], we construct two sequences {υn} and {ξn} in p such that ξn = gυn = fυn+1 for all n ≥ 0, (3.25) for which fυ0 � fυ1 � · · · � fυn � fυn+1 � · · · . (3.26) again from [8], we have to show that d(ξn,ξn+1) ≤ λd(ξn−1,ξn), (3.27) for all n ≥ 1 and where λ ∈ [0, 1 s ). now from (3.24) and using (3.25) and (3.26), we get φ(sd(ξn,ξn+1)) = φ(sd(gυn,gυn+1)) ≤ φ(mf(υn,υn+1)) − ψ(nf(υn,υn+1)), (3.28) where mf(υn,υn+1) = max { d(fυn+1,gυn+1) [1 + d(fυn,gυn)] 1 + d(fυn,fυn+1) , d(fυn,gυn) d(fυn,gυn+1) 1 + d(fυn,gυn+1) + d(fυn+1,gυn) , 1 11 d(fυn,fυn+1) } = max { d(ξn,ξn+1) [1 + d(ξn−1,ξn)] 1 + d(ξn−1,ξn) , d(ξn−1,ξn) d(ξn−1,ξn+1) 1 + d(ξn−1,ξn+1) + d(ξn,ξn) ,d(ξn−1,ξn) } = max{d(ξn−1,ξn),d(ξn,ξn+1)} 352 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) and nf(υn,υn+1) = max { d(fυn+1,gυn+1) [1 + d(fυn,gυn)] 1 + d(fυn,fυn+1) ,d(fυn,fυn+1) } = max { d(ξn,ξn+1) [1 + d(ξn−1,ξn)] 1 + d(ξn−1,ξn) ,d(ξn−1,ξn) } = max{d(ξn−1,ξn),d(ξn,ξn+1)}. therefore from equation (3.28), we get φ(sd(ξn,ξn+1)) ≤ φ(max{d(ξn−1,ξn),d(ξn,ξn+1)}) − ψ(max{d(ξn−1,ξn),d(ξn,ξn+1)}). (3.29) if 0 < d(ξn−1,ξn) ≤ d(ξn,ξn+1) for some n ∈ n, then from (3.29) we get φ(sd(ξn,ξn+1)) ≤ φ(d(ξn,ξn+1)) − ψ(d(ξn,ξn+1)) < φ(d(ξn,ξn+1)), (3.30) or equivalently sd(ξn,ξn+1) ≤ d(ξn,ξn+1). (3.31) this is a contradiction. hence from (3.29) we obtain that sd(ξn,ξn+1) ≤ d(ξn−1,ξn). (3.32) thus equation (3.27) holds, where λ ∈ [0, 1 s ). therefore from (3.27) and lemma 3.1 of [32], we conclude that {ξn} = {gυn} = {fυn+1} is a cauchy sequence in p and then converges to some µ ∈ p as p is complete such that lim n→+∞ gυn = lim n→+∞ fυn+1 = µ. thus by the compatibility of g and f, we obtain that lim n→+∞ d(f(gυn),g(fυn)) = 0, (3.33) and from the continuity of g and f, we have lim n→+∞ f(gυn) = fµ, lim n→+∞ g(fυn) = gµ. (3.34) further, from the triangular inequality of a b-metric and, from equations (3.33) and (3.34) , we get 1 s d(gµ,fµ) ≤ d(gµ,g(fυn)) + sd(g(fυn),f(gυn)) + sd(f(gυn),fµ). (3.35) finally, we arrive at d(gv,fv) = 0 as n → +∞ in (3.35). therefore, v is a coincidence point of g cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 353 and f in p . relaxing the continuity of the mappings f and g in theorem 3.5, we obtain the following result. theorem 3.6. in theorem 3.5, assume that p satisfies for any non-decreasing sequence {fυn} ⊂ p with lim n→+∞ fυn = fυ in fp, where fp is a closed subset of p implies that fυn � fυ,fυ � f(fυ) for n ∈ n. if there exists υ0 ∈ p such that fυ0 � gυ0, then the weakly compatible mappings g and f have a coincidence point in p. furthermore, g and f have a common fixed point, if g and f commute at their coincidence points. proof. the sequence, {ξn} = {gυn} = {fυn+1} is a cauchy sequence from the proof of theorem 3.5. since fp is closed, then there is some µ ∈ p such that lim n→+∞ gυn = lim n→+∞ fυn+1 = fµ. thus from the hypotheses, we have fυn � fµ for all n ∈ n. now, we have to prove that µ is a coincidence point of g and f. from equation (3.24), we have φ(sd(gυn,gυ)) ≤ φ(mf(υn,υ)) − ψ(nf(υn,υ)), (3.36) where mf(υn,µ) = max { d(fµ,gµ) [1 + d(fυn,gυn)] 1 + d(fυn,fµ) , d(fυn,gυn) d(fυn,gµ) 1 + d(fυn,gµ) + d(fµ,gυn) ,d(fυn,fµ) } → max{d(fµ,gµ),0,0} = d(fµ,gµ) as n → +∞, and nf(υn,µ) = max { d(fµ,gµ) [1 + d(fυn,gυn)] 1 + d(fυn,fµ) ,d(fυn,fµ) } → max{d(fµ,gµ),0} = d(fµ,gµ) as n → +∞. therefore equation (3.36) becomes φ(s lim n→+∞ d(gυn,gυ)) ≤ φ(d(fµ,gµ)) − ψ(d(fµ,gµ)) < φ(d(fµ,gµ)). (3.37) 354 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) consequently, we get lim n→+∞ d(gυn,gυ) < 1 s d(fµ,gµ). (3.38) further by triangular inequality, we have 1 s d(fµ,gµ) ≤ d(fµ,gυn) + d(gυn,gµ), (3.39) then (3.38) and (3.39) lead to contradiction, if fµ 6= gµ. hence, fµ = gµ. let fµ = gµ = ρ, that is g and f commute at ρ, then gρ = g(fµ) = f(gµ) = fρ. since fµ = f(fµ) = fρ, then by equation (3.36) with fµ = gµ and fρ = gρ, we get φ(sd(gµ,gρ)) ≤ φ(mf(µ,ρ)) − ψ(nf(µ,ρ)) < φ(d(gµ,gρ)), (3.40) or equivalently, sd(gµ,gρ) ≤ d(gµ,gρ), which is a contradiction, if gµ 6= gρ. thus, gµ = gρ = ρ. hence, gµ = fρ = ρ, that is ρ is a common fixed point of g and f. definition 3.7. let (p,d,s,�) be a complete partially ordered b-metric space with s > 1, φ ∈ φ and ψ ∈ ψ. a mapping h : p × p → p is said to be a generalized (φ,ψ)-contractive mapping with respect to f : p → p such that φ(skd(h(υ,ξ),h(ρ,τ))) ≤ φ(mf(υ,ξ,ρ,τ)) − ψ(nf(υ,ξ,ρ,τ)), (3.41) for all υ,ξ,ρ,τ ∈ p with fυ � fρ and fξ � fτ, k > 2 where mf(υ,ξ,ρ,τ) = max { d(fρ,h(ρ,τ)) [1 + d(fυ,h(υ,ξ))] 1 + d(fυ,fρ) , d(fυ,h(υ,ξ)) d(fυ,h(ρ,τ)) 1 + d(fυ,h(ρ,τ)) + d(fρ,h(υ,ξ)) , 1 11 d(fυ,fρ) } , and nf(υ,ξ,ρ,τ) = max { d(fρ,h(ρ,τ)) [1 + d(fυ,h(υ,ξ))] 1 + d(fυ,fρ) ,d(fυ,fρ) } . theorem 3.8. let (p,d,s,�) be a complete partially ordered b-metric space with s > 1. suppose that h : p × p → p be a generalized (φ,ψ)contractive mapping with respect to f : p → p and, h and f are continuous functions such that h has the mixed f-monotone property and commutes with f. also assume that h(p × p) ⊆ f(p). then h and f have a coupled coincidence point in p, if there exists (υ0,ξ0) ∈ p × p such that fυ0 � h(υ0,ξ0) and fξ0 � h(ξ0,υ0). cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 355 proof. from the hypotheses and following the proof of theorem 2.2 of [8], we construct two sequences {υn} and {ξn} in p such that fυn+1 = h(υn,ξn), fξn+1 = h(ξn,υn), for all n ≥ 0. in particular, {fυn} is non-decreasing and {fξn} is non-increasing sequences in p . now from (3.41) by replacing υ = υn,ξ = ξn,ρ = υn+1,τ = ξn+1, we get φ(skd(fυn+1,fυn+2)) = φ(s kd(h(υn,ξn),h(υn+1,ξn+1))) ≤ φ(mf(υn,ξn,υn+1,ξn+1)) − ψ(nf(υn,ξn,υn+1,ξn+1)), (3.42) where mf(υn,ξn,υn+1,ξn+1) ≤ max{d(fυn,fυn+1),d(fυn+1,fυn+2)} (3.43) and nf(υn,ξn,υn+1,ξn+1) = max{d(fυn,fυn+1),d(fυn+1,fυn+2)}. (3.44) therefore from (3.42), we have φ(skd(fυn+1,fυn+2)) ≤ φ(max{d(fυn,fυn+1),d(fυn+1,fυn+2)}) − ψ(max{d(fυn,fυn+1),d(fυn+1,fυn+2)}). (3.45) similarly by taking υ = ξn+1,ξ = υn+1,ρ = υn,τ = υn in (3.41), we get φ(skd(fξn+1,fξn+2)) ≤ φ(max{d(fξn,fξn+1),d(fξn+1,fξn+2)}) − ψ(max{d(fξn,fξn+1),d(fξn+1,fξn+2)}). (3.46) from the fact that max{φ(c),φ(d)} = φ{max{c,d}} for all c,d ∈ [0,+∞). then combining (3.45) and (3.46), we get φ(skδn) ≤ φ(max{d(fυn,fυn+1),d(fυn+1,fυn+2),d(fξn,fξn+1),d(fξn+1,fξn+2)}) − ψ(max{d(fυn,fυn+1),d(fυn+1,fυn+2),d(fξn,fξn+1),d(fξn+1,fξn+2)}), (3.47) where δn = max{d(fυn+1,fυn+2),d(fξn+1,fξn+2)}. (3.48) let us denote, ∆n = max{d(fυn,fυn+1),d(fυn+1,fυn+2),d(fξn,fξn+1),d(fξn+1,fξn+2)}. (3.49) 356 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) hence from equations (3.45)-(3.48), we obtain s k δn ≤ ∆n. (3.50) next, we prove that δn ≤ λδn−1, (3.51) for all n ≥ 1 and where λ = 1 sk ∈ [0,1). suppose that if ∆n = δn then from (3.50), we get s kδn ≤ δn which leads to δn = 0 as s > 1 and hence (3.51) holds. if ∆n = max{d(fυn,fυn+1),d(fξn,fξn+1)}, i.e., ∆n = δn−1 then (3.50) follows (3.51). now from (3.50), we obtain that δn ≤ λ nδ0 and hence, d(fυn+1,fυn+2) ≤ λ n δ0 and d(fξn+1,fξn+2) ≤ λ n δ0. (3.52) therefore from lemma 3.1 of [32], the sequences {fυn} and {fξn} are cauchy sequences in p . hence, by following the remaining proof of theorem 2.2 of [2], we can show that h and f have a coincidence point in p . corollary 3.9. let (p,d,s,�) be a complete partially ordered b-metric space with s > 1, and h : p × p → p be a continuous mapping such that h has a mixed monotone property. suppose there exists φ ∈ φ and ψ ∈ ψ such that φ(skd(h(υ,ξ),h(ρ,τ))) ≤ φ(mf(υ,ξ,ρ,τ)) − ψ(nf(υ,ξ,ρ,τ)), for all υ,ξ,ρ,τ ∈ p with υ � ρ and ξ � τ, k > 2 where mf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) , d(υ,h(υ,ξ)) d(υ,h(ρ,τ)) 1 + d(υ,h(ρ,τ)) + d(ρ,h(υ,ξ)) ,d(υ,ρ) } , and nf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) ,d(υ,ρ) } . then h has a coupled fixed point in p, if there exists (υ0,ξ0) ∈ p × p such that υ0 � h(υ0,ξ0) and ξ0 � h(ξ0,υ0). proof. set f = ip in theorem 3.8. corollary 3.10. let (p,d,s,�) be a complete partially ordered b-metric space with s > 1, and h : p × p → p be a continuous mapping such that h has a mixed monotone property. suppose cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 357 there exists ψ ∈ ψ such that d(h(υ,ξ),h(ρ,τ)) ≤ 1 sk mf(υ,ξ,ρ,τ) − 1 sk ψ(nf(υ,ξ,ρ,τ)), for all υ,ξ,ρ,τ ∈ p with υ � ρ and ξ � τ, k > 2 where mf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) , d(υ,h(υ,ξ)) d(υ,h(ρ,τ)) 1 + d(υ,h(ρ,τ)) + d(ρ,h(υ,ξ)) ,d(υ,ρ) } , and nf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) ,d(υ,ρ) } . if there exists (υ0,ξ0) ∈ p × p such that υ0 � h(υ0,ξ0) and ξ0 � h(ξ0,υ0), then h has a coupled fixed point in p. theorem 3.11. in addition to theorem 3.8, if for all (υ,ξ),(r,s) ∈ p × p, there exists (c∗,d∗) ∈ p × p such that (h(c∗,d∗),h(d∗,c∗)) is comparable to (h(υ,ξ),h(ξ,υ)) and to (h(r,s),h(s,r)), then h and f have a unique coupled common fixed point in p × p. proof. from theorem 3.8, we know that there exists at least one coupled coincidence point in p for h and f. assume that (υ,ξ) and (r,s) are two coupled coincidence points of h and f, i.e., h(υ,ξ) = fυ, h(ξ,υ,) = fξ and h(r,s) = fr, h(s,r) = fs. now, we have to prove that fυ = fr and fξ = fs. from the hypotheses, there exists (c∗,d∗) ∈ p × p such that (h(c∗,d∗),h(d∗,c∗)) is comparable to (h(υ,ξ),h(ξ,υ)) and to (h(r,s),h(s,r)). suppose that (h(υ,ξ),h(ξ,υ)) ≤ (h(c∗,d∗),h(d∗,c∗)) and (h(r,s),h(s,r)) ≤ (h(c∗,d∗),h(d∗,c∗)). let c∗0 = c ∗ and d∗0 = d ∗ and then choose (c∗1,d ∗ 1) ∈ p × p as fc∗1 = h(c ∗ 0,d ∗ 0), fd ∗ 1 = h(d ∗ 0,c ∗ 0) (n ≥ 1). by repeating the same procedure above, we can obtain two sequences {fc∗n} and {fd ∗ n} in p such that fc ∗ n+1 = h(c ∗ n,d ∗ n), fd ∗ n+1 = h(d ∗ n,c ∗ n) (n ≥ 0). similarly, define the sequences {fυn}, {fξn} and {frn}, {fsn} as above in p by setting υ0 = υ, ξ0 = ξ and r0 = r, s0 = s. further, we have that fυn → h(υ,ξ), fξn → h(ξ,υ), frn → h(r,s), fsn → h(s,r) (n ≥ 1). (3.53) since, (h(υ,ξ),h(ξ,υ)) = (fυ,fξ) = (fυ1,fξ1) is comparable to (h(c ∗,d∗),h(d∗,c∗)) = (fc∗,fd∗) = 358 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) (fc∗1,fd ∗ 1) and hence we get (fυ1,fξ1) ≤ (fc ∗ 1,fd ∗ 1). thus, by induction we obtain that (fυn,fξn) ≤ (fc ∗ n,fd ∗ n) (n ≥ 0). (3.54) therefore from (3.41), we have φ(d(fυ,fc∗n+1)) ≤ φ(s kd(fυ,fc∗n+1)) = φ(s kd(h(υ,ξ),h(c∗n,d ∗ n))) ≤ φ(mf(υ,ξ,c ∗ n,d ∗ n)) − ψ(nf(υ,ξ,c ∗ n,d ∗ n)), (3.55) where mf(υ,ξ,c ∗ n,d ∗ n) = max { d(fc∗n,h(c ∗ n,d ∗ n)) [1 + d(fυ,h(υ,ξ))] 1 + d(fυ,fc∗n) , d(fυ,h(υ,ξ)) d(fυ,h(c∗n,d ∗ n)) 1 + d(fυ,h(c∗n,d ∗ n)) + d(fc ∗ n,h(υ,ξ)) ,d(fυ,fc∗n) } = max{0,0,d(fυ,fc∗n)} = d(fυ,fc∗n) and nf(υ,ξ,c ∗ n,d ∗ n) = max { d(fc∗n,h(c ∗ n,d ∗ n)) [1 + d(hυ,h(υ,ξ))] 1 + d(fυ,fc∗n) ,d(fυ,fc∗n) } = d(fυ,fc∗n). thus from (3.55), φ(d(fυ,fc∗n+1)) ≤ φ(d(fυ,fc ∗ n)) − ψ(d(fυ,fc ∗ n)). (3.56) as by the similar process, we can prove that φ(d(fξ,fd∗n+1)) ≤ φ(d(fξ,fd ∗ n)) − ψ(d(fξ,fd ∗ n)). (3.57) from (3.56) and (3.57), we have φ(max{d(fυ,fc∗n+1),d(fξ,fd ∗ n+1)}) ≤ φ(max{d(fυ,fc ∗ n),d(fξ,fd ∗ n)}) − ψ(max{d(fυ,fc∗n),d(fξ,fd ∗ n)}) < φ(max{d(fυ,fc∗n),d(fξ,fd ∗ n)}). (3.58) hence by the property of φ, we get max{d(fυ,fc∗n+1),d(fξ,fd ∗ n+1)} < max{d(fυ,fc ∗ n),d(fξ,fd ∗ n)}, which shows that max{d(fυ,fc∗n),d(fξ,fd ∗ n)} is a decreasing sequence and by a result there exists cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 359 γ ≥ 0 such that lim n→+∞ max{d(fυ,fc∗n),d(fξ,fd ∗ n)} = γ. from (3.58) taking upper limit as n → +∞, we get φ(γ) ≤ φ(γ) − ψ(γ), (3.59) from which we get ψ(γ) = 0, implies that γ = 0. thus, lim n→+∞ max{d(fυ,fc∗n),d(fξ,fd ∗ n)} = 0. consequently, we get lim n→+∞ d(fυ,fc∗n) = 0 and lim n→+∞ d(fξ,fd∗n) = 0. (3.60) by similar argument, we get lim n→+∞ d(fr,fc∗n) = 0 and lim n→+∞ d(fs,fd∗n) = 0. (3.61) therefore from (3.60) and (3.61), we get fυ = fr and fξ = fs. since fυ = h(υ,ξ) and fξ = h(ξ,υ), then by the commutativity of h and f, we have f(fυ) = f(h(υ,ξ)) = h(fυ,fξ) and f(fξ) = f(h(ξ,υ)) = h(fξ,fυ). (3.62) let fυ = a∗ and fξ = b∗ then (3.62) becomes f(a∗) = h(a∗,b∗) and f(b∗) = h(b∗,a∗), (3.63) which shows that (a∗,b∗) is a coupled coincidence point of h and f. it follows that f(a∗) = fr and f(b∗) = fs that is f(a∗) = a∗ and f(b∗) = b∗. thus from (3.63), we get a∗ = f(a∗) = h(a∗,b∗) and b∗ = f(b∗) = h(b∗,a∗). therefore, (a∗,b∗) is a coupled common fixed point of h and f. for the uniqueness, let (u∗,v∗) be another coupled common fixed point of h and f, then we have u∗ = fu∗ = h(u∗,v∗) and v∗ = fv∗ = h(v∗,u∗). since (u∗,v∗) is a coupled common fixed point of h and f, then we get fu∗ = fυ = a∗ and fv∗ = fξ = b∗. thus, u∗ = fu∗ = fa∗ = a∗ and v∗ = fv∗ = fb∗ = b∗. hence the result. theorem 3.12. in addition to the hypotheses of theorem 3.11, if fυ0 and fξ0 are comparable, then h and f have a unique common fixed point in p. 360 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) proof. from theorem 3.11, h and f have a unique coupled common fixed point (υ,ξ) ∈ p . now, it is enough to prove that υ = ξ. from the hypotheses, we have fυ0 and fξ0 are comparable then we assume that fυ0 � fξ0. hence by induction we get fυn � fξn for all n ≥ 0, where {fυn} and {fξn} are from theorem 3.8. now by use of lemma 2.6, we get φ(sk−2d(υ,ξ)) = φ ( sk 1 s2 d(υ,ξ) ) ≤ lim n→+∞ supφ(skd(υn+1,ξn+1)) = lim n→+∞ supφ(skd(h(υn,ξn),h(ξn,υn))) ≤ lim n→+∞ supφ(mf(υn,ξn,ξn,υn)) − lim n→+∞ inf ψ(nf(υn,ξn,ξn,υn)) ≤ φ(d(υ,ξ)) − lim n→+∞ inf ψ(nf(υn,ξn,ξn,υn)) < φ(d(υ,ξ)), which is a contradiction. thus, υ = ξ, i.e., h and f have a common fixed point in p . remark 3.13. it is well known that b-metric space is a metric space when s = 1. so, from the result of jachymski [31], the condition φ(d(h(υ,ξ),h(ρ,τ))) ≤ φ(max{d(fυ,fρ),d(fξ,fτ)}) − ψ(max{d(fυ,fρ),d(fξ,fτ)}) is equivalent to, d(h(υ,ξ),h(ρ,τ)) ≤ ϕ(max{d(fυ,fρ),d(fξ,fτ)}), where φ ∈ φ, ψ ∈ ψ and ϕ : [0,+∞) → [0,+∞) is continuous, ϕ(t) < t for all t > 0 and ϕ(t) = 0 if and only if t = 0. so, in view of above our results generalize and extend the results of [15, 23, 25, 31, 37, 38] and several other comparable results. corollary 3.14. suppose (p,d,s,�) be a complete partially ordered b-metric space with parameter s > 1. let g : p → p be a continuous, non-decreasing mapping with regards to � such that there exists υ0 ∈ p with υ0 � gυ0. suppose that φ(sd(gυ,gξ)) ≤ φ(m(υ,ξ)) − ψ(m(υ,ξ)), (3.64) where m(υ,ξ) and the conditions upon φ,ψ are same as in theorem 3.1. then g has a fixed point in p. proof. set n(υ,ξ) = m(υ,ξ) in a contraction condition (3.3) and apply theorem 3.1, we have the required proof. cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 361 note 1. similarly by removing the continuity of a non-decreasing mapping g and taking a nondecreasing sequence {υn} as above in theorem 3.2, we can obtain a fixed point for g in p. also one can obtain the uniqueness of a fixed point of g by using condition (3.17) in p as by following the proof of theorem 3.3. note 2. by following the proofs of theorems 3.5 3.6, we can find the coincidence point for the mappings g and f in p. similarly, from theorem 3.8, theorem 3.11 and theorem 3.12, one can obtain a coupled coincidence point and its uniqueness, and a unique common fixed point for the mappings h and f in p × p and on p satisfying an almost generalized contraction condition (3.64), where m(υ,ξ), mf(υ,ξ), mf(υ,ξ,ρ,τ) and the conditions upon φ,ψ are same as above defined. corollary 3.15. suppose that (p,d,s,�) be a complete partially ordered b-metric space with s > 1. let g : p → p be a continuous, non-decreasing mapping with regards to �. if there exists k ∈ [0,1) and for any υ,ξ ∈ p with υ � ξ such that d(gυ,gξ) ≤ k s max { d(ξ,gξ) [1 + d(υ,gυ)] 1 + d(υ,ξ) , d(υ,gυ) d(υ,gξ) 1 + d(υ,gξ) + d(ξ,gυ) ,d(υ,ξ) } . (3.65) if there exists υ0 ∈ p with υ0 � gυ0, then g has a fixed point in p. proof. set φ(t) = t and ψ(t) = (1 − k)t, for all t ∈ (0,+∞) in corollary 3.14. note 3. relaxing the continuity of a map g in corollary 3.15, one can obtains a fixed point for g on taking a non-decreasing sequence {υn} in p by following the proof of theorem 3.2. example 3.16. define a metric d : p × p → p as below and ≤ is an usual order on p, where p = {1,2,3,4,5,6} d(υ,ξ) = d(ξ,υ) = 0, if υ,ξ = 1,2,3,4,5,6 and υ = ξ, d(υ,ξ) = d(ξ,υ) = 3, if υ,ξ = 1,2,3,4,5 and υ 6= ξ, d(υ,ξ) = d(ξ,υ) = 12, if υ = 1,2,3,4 and ξ = 6, d(υ,ξ) = d(ξ,υ) = 20, if υ = 5 and ξ = 6. define a map g : p → p by g1 = g2 = g3 = g4 = g5 = 1,g6 = 2 and let φ(t) = t 2 , ψ(t) = t 4 for t ∈ [0,+∞). then g has a fixed point in p. proof. it is apparent that, (p,d,s,�) is a complete partially ordered b-metric space for s = 2. consider the possible cases for υ, ξ in p : case 1 suppose υ,ξ ∈ {1,2,3,4,5}, υ < ξ then d(gυ,gξ) = d(1,1) = 0. hence, φ(2d(gυ,gξ)) = 0 ≤ φ(m(υ,ξ)) − ψ(m(υ,ξ)). 362 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) case 2 suppose that υ ∈ {1,2,3,4,5} and ξ = 6, then d(gυ,gξ) = d(1,2) = 3, m(6,5) = 20 and m(υ,6) = 12, for υ ∈ {1,2,3,4}. therefore, we have the following inequality, φ(2d(gυ,gξ)) ≤ m(υ,ξ) 4 = φ(m(υ,ξ)) − ψ(m(υ,ξ)). thus, condition (3.64) of corollary 3.14 holds. furthermore, the remaining assumptions in corollary 3.14 are fulfilled. hence, g has a fixed point in p as corollary 3.14 is appropriate to g,φ,ψ and (p,d,s,�). example 3.17. a metric d : p × p → p, where p = {0,1, 1 2 , 1 3 , 1 4 , . . . 1 n , . . .} with usual order ≤ is defined as follows d(υ,ξ) =                  0, if υ = ξ 1, if υ 6= ξ ∈ {0,1} |υ − ξ|, if υ,ξ ∈ { 0, 1 2n , 1 2m : n 6= m ≥ 1 } 3, otherwise. a map g : p → p be such that g0 = 0,g 1 n = 1 12n for all n ≥ 1 and let φ(t) = t, ψ(t) = 4t 5 for t ∈ [0,+∞). then, g has a fixed point in p. proof. it is obvious that for s = 12 5 , (p,d,s,�) is a complete partially ordered b-metric space and also by definition, d is discontinuous b-metric space. now for υ,ξ ∈ p with υ < ξ, we have the following cases: case 1 if υ = 0 and ξ = 1 n , n ≥ 1, then d(gυ,gξ) = d(0, 1 12n ) = 1 12n and m(υ,ξ) = 1 n or m(υ,ξ) = {1,3}. therefore, we have φ ( 12 5 d(gυ,gξ) ) ≤ m(υ,ξ) 5 = φ(m(υ,ξ)) − ψ(m(υ,ξ)). case 2 if υ = 1 m and ξ = 1 n with m > n ≥ 1, then d(gυ,gξ) = d ( 1 12m , 1 12n ) and m(υ,ξ) ≥ 1 n − 1 m or m(υ,ξ) = 3. therefore, φ ( 12 5 d(gυ,gξ) ) ≤ m(υ,ξ) 5 = φ(m(υ,ξ)) − ψ(m(υ,ξ)). hence, condition (3.64) of corollary 3.14 and remaining assumptions are satisfied. thus, g has a fixed point in p . cubo 24, 2 (2022) fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 363 example 3.18. let p = c[a,b] be the set of all continuous functions. let us define a b-metric d on p by d(θ1,θ2) = sup t∈c[a,b] {|θ1(t) − θ2(t)| 2} for all θ1,θ2 ∈ p with partial order � defined by θ1 � θ2 if a ≤ θ1(t) ≤ θ2(t) ≤ b, for all t ∈ [a,b], 0 ≤ a < b. let g : p → p be a mapping defined by gθ = θ 5 ,θ ∈ p and the two altering distance functions by φ(t) = t, ψ(t) = t 3 , for any t ∈ [0,+∞]. then g has a unique fixed point in p. proof. from the hypotheses, it is clear that (p,d,s,�) is a complete partially ordered b-metric space with parameter s = 2 and fulfill all the conditions of corollary 3.14 and note 1. furthermore for any θ1,θ2 ∈ p , the function min(θ1,θ2)(t) = min{θ1(t),θ2(t)} is also continuous and the conditions of corollary 3.14 and note 1 are satisfied. hence, g has a unique fixed point θ = 0 in p . acknowledgements the authors thankful to the editor and anonymous referees for their valuable suggestions and comments which improved the contents of the paper. 364 n. seshagiri rao & k. kalyani cubo 24, 2 (2022) references [1] m. abbas, b. ali, b. bin-mohsin, n. dedović, t. nazir and s. radenović, “solutions and ulam-hyers stability of differential inclusions involving suzuki type multivalued mappings on b-metric spaces”, vojnotehnički glasnik/military technical courier, vol. 68, no. 3, pp. 438–487, 2020. 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[55] i. m. vul’pe, d. ostrăıh and f. hŏıman, “the topological structure of a quasimetric space”, (russian) in investigations in functional analysis and differential equations, math. sci., interuniv. work collect., shtiintsa, pp. 14–19, 1981. introduction mathematical preliminaries main results cubo, a mathematical journal vol. 24, no. 02, pp. 187–209, august 2022 doi: 10.56754/0719-0646.2402.0187 numerical analysis of nonlinear parabolic problems with variable exponent and l1 data stanislas ouaro 1, b noufou rabo 1 urbain traoré 1 1laboratoire de mathématiques et informatique (lami), unité de formation et de recherche en sciences exactes et appliquées, université joseph ki-zerbo, 03 bp. 7021 ouagadougou 03, burkina faso. ouaro@yahoo.fr b rabonouf@gmail.com urbain.traore@yahoo.fr abstract in this paper, we make the numerical analysis of the mild solution which is also an entropy solution of parabolic problem involving the p(x)−laplacian operator with l1− data. resumen en este art́ıculo, realizamos el análisis numérico de la solución mild que también es una solución de entroṕıa del problema parabólico involucrando el operador p(x)−laplaciano con datos en l1. keywords and phrases: elliptic-parabolic, numerical iterative method, variable exponent, mild solution, renormalized solution. 2020 ams mathematics subject classification: 65m12, 65n22, 35k55, 35k65, 46e35. accepted: 17 december, 2021 received: 06 june, 2021 c©2022 s. ouaro et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://doi.org/10.56754/0719-0646.2402.0187 mailto:ouaro@yahoo.fr https://orcid.org/0000-0003-0671-2378 https://orcid.org/0000-0003-3659-0906 https://orcid.org/0000-0002-9729-4724 mailto:ouaro@yahoo.fr mailto:rabonouf@gmail.com mailto:urbain.traore@yahoo.fr 188 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) 1 introduction we consider a bounded open domain ω ⊂ rd (d ≥ 2) with a lipschitz boundary denoted by ∂ω. let t > 0 and p : ω → (1, ∞) be a continuous function. in this paper, one of our main goals is the numerical approximation of the mild solution of the following nonlinear parabolic problem involving the p(x)−laplacian operator            ∂u ∂t − div(|∇u|p(x)−2∇u) = f in q ≡ ω × (0, t ), u = 0 on ∂ω × (0, t ), u(x, 0) = u0 in ω, (1.1) where u0 ∈ l 1(ω), f ∈ l1(q). the assumptions on the variable exponent p(x) will be specified later. partial differential equations with nonlinearities involving non-constant exponents have attracted an increasing amount of attention on recent years. their study is an interesting topic which raises many mathematical difficulties (see [1, 2, 14, 16, 27, 30]). there are many results devoted to questions on existence and uniqueness of solutions to problems like (1.1), we refer for example the reader to the bibliography [3, 4, 5, 9, 24, 29] and references therein. many of these models have already been analyzed for constant exponents of nonlinearity (see the references therein), but it seems to be more realistic to assume the exponent to be variable. from numerical point of view, in the classical evolution problem case where p(x) ≡ p, the numerical analysis was firstly considered in [7, 22]. afterward, jäger and kačur [18] and kačur [20] studied the numerical approximation. inspired by these works, maitre [23] proposed a numerical scheme to approximate the mild solutions. on the other side, for problems with variable exponent, in recent years, there are some papers devoted to their numerical analysis (see for example [8, 10, 12, 13, 17, 19, 26]). thus, in [13] the authors used a quasi-newton minimization method to approach the solution of the p(x)−lapacian problems; in [12], they present an inverse power method to compute the first homogeneous eigenpair. in [26], an interior penalty discontinuous galerkin method has been used by the authors to approximate the minimizer of a variational problem related to the p(x)−laplacian. other authors use finite elements to approximate the solution (see [10]). nevertheless, there are scarcely papers about the numerical analysis of nonlinear parabolic problems with variable exponent (see for example [11]). the importance of investigating the problem (1.1) lies in their occurrence in modeling various physical problems involving strong anisotropic phenomena related to electrorheological fluids (an important class of non-newtonian fluids, see [27]) which are characterized by their ability to change the mechanical properties under the influence of the exterior electromagnetic field. other important applications are related to image processing, elasticity [30], the processes of filtration in complex media, stratigraphy problems and also mathematical biology. the study of problem (1.1) involves using of generalized lebesgue and sobolev spaces i.e., lp(.) and w 1,p(.) respectively (see [15]). cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 189 throughout this paper we assume that the exponent p(.) appearing in (1.1) is a continuous function p : ω → (1, ∞) such that:            ∃c > 0 : |p(x) − p(y)| ≤ c − log |x − y| for every x, y ∈ ω with |x − y| ≤ 1 2 2d d + 2 < p− := min x∈ω p(x) ≤ p + := max x∈ω p(x) < ∞. (1.2) the first condition says that p(.) belongs to the class of log-hölder continuous functions. these assumptions are used to obtain several regularity results for sobolev spaces with variable exponents; in particular, c∞(ω) is dense in w 1,p(.)(ω) and w 1,p(.) 0 (ω) = w 1,p(.)(ω) ∩ w 1,1 0 (ω). our paper was inspired by the work of maitre (see [23]) where the author studied the numerical analysis of an elliptic-parabolic problem in the context of constant exponent setting. the rest of this paper is organized as follows: in section 2, we give some results for the study of (1.1). in section 3, we recall the notion of mild solution. in section 4, we proceed to the numerical study, where we show the existence and uniqueness of solution of numerical scheme for the approximation of mild solution and the study of the convergence of this numerical scheme. we conclude this section by numerical tests. 2 preliminaries we first recall in what follows some definitions and basic properties of generalized lebesgue-sobolev spaces with variable exponent. we define the lebesgue space with a variable exponent p(.) by lp(.)(ω) = { u : ω → r; u is measurable with ρp(.)(u) < ∞ } , where ρp(.)(u) = ∫ ω |u(x)|p(x)dx, is called a modular. we define a norm, the so-called luxemburg norm, on this space by the formula |u|p(.) = inf { µ > 0 : ρp(.) ( u µ ) ≤ 1 } . the space (lp(.)(ω), |.|p(.)) is a separable banach space. moreover, if 1 < p − ≤ p+ < +∞, then lp(.)(ω) is uniformly convex, hence reflexive, and its dual space is isomorphic to lp ′(.)(ω), where 1 p(x) + 1 p′(x) = 1. finally, we have the hölder type inequality: ∣ ∣ ∣ ∣ ∫ ω uv dx ∣ ∣ ∣ ∣ ≤ ( 1 p− + 1 p′− ) |u|p(.)|v|p′(.) for all u ∈ lp(.)(ω) and v ∈ lp ′(.)(ω). 190 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) we define also the variable sobolev space w 1,p(.)(ω) = { u ∈ lp(.)(ω) : |∇u| ∈ lp(.)(ω) } . on w 1,p(.)(ω) we may consider the following norm ‖u‖1,p(.) = |u|p(.) + |∇u|p(.). the space (w 1,p(.)(ω), ‖u‖1,p(.)) is a separable and reflexive banach space. next, we define w 1,p(.) 0 (ω) as the closure of c ∞ 0 (ω) in w 1,p(.)(ω) under the norm ‖u‖ := |∇u|p(.). the space (w 1,p(.) 0 (ω), ‖u‖) is a separable and reflexive banach space. for the interested reader, more details about lebesgue and sobolev spaces with variable exponent can be found in [15] (see also [21]). since ω is bounded and p : ω → (1, ∞) is log-hölder continuous, the poincaré inequality holds (see [28]) |u|p(.) ≤ c|∇u|p(.), ∀ u ∈ w 1,p(.) 0 (ω), where c is a constant which depends on ω and on the function p. an important role in manipulating the generalized lebesgue and sobolev spaces is played by modular ρp(.) of the space l p(.). we have the following result (see [28]). lemma 2.1. if un, u ∈ l p(.) and p+ < ∞, then the following relations hold: (1) |u|p(.) > 1 ⇒ |u| p − p(.) ≤ ρp(.)(u) ≤ |u| p + p(.) ; (2) |u|p(.) < 1 ⇒ |u| p + p(.) ≤ ρp(.)(u) ≤ |u| p − p(.) ; (3) |u|p(.) < 1 (respectively = 1; > 1) ⇐⇒ ρp(.)(u) < 1 (respectively = 1; > 1); (4) |u|p(.) → 0 (respectively → ∞) ⇐⇒ ρp(.)(u) → 0 (respectively → ∞); (5) ρp(.) ( u/|u|p(.) ) = 1. following [4], we extend a variable exponent p : ω → [1, +∞) to q = [0, t ] × ω by setting p(t, x) := p(x) for all (t, x) ∈ q. we also consider the generalized lebesgue space lp(.)(q) = { u : q → r measurable such that ∫∫ q |u(x, t)|p(x) d(x, t) < ∞ } endowed with the norm ‖u‖lp(.) := inf { µ > 0 : ∫∫ q ∣ ∣ ∣ ∣ u(x, t) µ ∣ ∣ ∣ ∣ p(x) d(x, t) < 1 } cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 191 which shares the same properties as lp(.)(ω). now, we recall the main results for the study of (1.1). in order to approximate the mild solution of (1.1), let us recall that ouaro and traoré have studied in [25] the existence and uniqueness of weak energy and entropy solutions of the following stationary problem associated to the problem (1.1)        u − div a(x, ∇u) = f in ω, u = 0 on ∂ω, (2.1) where ω ⊂ rd is a bounded domain with smooth boundary and f ∈ l1(ω). for the vector field a(x, ξ) : ω × rd → rd, in addition to be carathéodory, is the continuous derivative with respect to ξ of the mapping a : ω × rd → rd, i.e. a(x, ξ) = ∇ξa(x, ξ) such that: a(x, 0) = 0 for almost every x ∈ ω. (2.2) there exists a positive constant c1 such that |a(x, ξ)| ≤ c1(j(x) + |ξ| p(x)−1), (2.3) for almost every x ∈ ω and for every ξ ∈ rd where j is a non-negative function in lp ′(.)(ω), with 1 p(x) + 1 p′(x) = 1. the following inequalities hold (a(x, ξ) − a(x, η)).(ξ − η) > 0, (2.4) for almost every x ∈ ω and for every ξ, η ∈ rd, with ξ 6= η and 1 c |ξ|p(x) ≤ a(x, ξ).ξ ≤ cp(x)a(x, ξ), (2.5) for almost every x ∈ ω, c > 0 and for every ξ ∈ rd. the exponent appearing in (2.3) and (2.5) is defined as follows.        p(.) : ω → r is a measurable function such that 1 < p− := ess infx∈ω p(x) ≤ p + := ess supx∈ω p(x) < ∞. (2.6) for more details, see [24, 25]. as example of models with respect to above assumptions, we can give the following. set a(x, ξ) = 1 p(x) |ξ|p(x), a(x, ξ) = |ξ|p(x)−2ξ. then, we get the p(x)−laplace operator div (|∇u|p(x)−2∇u). note that the weak solution of (2.1) is defined as follows. 192 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) definition 2.2. a weak solution of (2.1) is a function u ∈ w 1,1 0 (ω) such that a(., ∇u) ∈ ( l1loc(ω) )d and ∫ ω a(., ∇u).∇ϕ dx + ∫ ω uϕ dx = ∫ ω fϕ dx, (2.7) for all ϕ ∈ c∞0 (ω). a weak energy solution is a weak solution such that u ∈ w 1,p(.) 0 (ω). now, we recall one of main results. theorem 2.3. assume that (2.2)–(2.6) hold and f ∈ l∞(ω). then there exists a unique weak energy solution of (2.1). we also recall a useful result needed in this paper (see [23]). lemma 2.4 ([23]). let x be a banach space and c a convex subset of x, containing 0. let t̄ be a non-expansive map on c such that t̄ (c) ⊂ c, admitting a unique fixed point x∗ in c. let λk be a sequence of (0, 1) verifying lim k→∞ λk = 1, ∏ k≥0 λk = 0, ∑ k≥0 |λk+1 − λk| < ∞. then the sequence (xk) generated by the iterative scheme x0 ∈ c, xk+1 = λk+1t̄(x k) (2.8) verifies limk→∞ x k − t̄(xk) = 0. consequently, if all subsequences of (xk) have in turn a subsequence converging to a point of c, then the whole sequence (xk) converges toward x∗. recall that a self-mapping t̄ of c is non-expansive if ‖t̄(x) − t̄(y)‖ ≤ ‖x − y‖ for all x, y ∈ c. in the next section, we give the definition of mild solution. 3 notion of mild solution let f ∈ l1(0, t ; l1(ω)), u0 ∈ l 1(ω) and ε > 0 be given. we consider the time discretization of problem (1.1) by an implicit euler scheme          uεn+1 − u ε n tn+1 − tn − div(|∇uεn+1| p(x)−2∇uεn+1) = f ε n+1 in d ′(ω) for n = 0, . . . , n − 1, uεn+1 ∈ w 1,p(.) 0 (ω) ∩ l ∞(ω); (3.1) cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 193 where                                                          n ∈ n∗, 0 = t0 < t1 < · · · < tn ≤ t is a partition of [0, t ]. fεn ∈ l ∞(ω) for n = 1, . . . , n such that n ∑ n=1 ∫ tn tn−1 ‖f(t) − fεn‖l1(ω)dt → 0 as ε → 0, maxn=1,...,n(tn − tn−1) → 0, t − tn → 0 as ε → 0, u ε 0 ∈ l ∞(ω) such that ‖u0 − u ε 0‖l1(ω) → 0 as ε → 0, with uε the piecewise constant function defined by uε(t) = uεn on (tn−1, tn] with n = 1, . . . , n; u ε(0) = uε0. (3.2) definition 3.1. a mild solution of (1.1) is a function u ∈ c([0, t ]; l1(ω)) with u(0) = u0 ∈ l 1(ω) such that, for all ε > 0, there exists (t0, t1, . . . , tn ; f ε 1 , f ε 2 , . . . , f ε n) and u ε 0 verifying (3.2); and for which there exists (uε1, . . . , u ε n) verifying (3.1) such that ‖u(t) − u ε n‖l1(ω) ≤ ε for all t ∈ (tn−1, tn], n = 1, . . . , n. remark 3.2. in this paper, for the sake of simplicity and readability, we chose to present the constant step subdivision algorithm, i.e. that we set tn+1 − tn = h = t n for all n = 0, . . . , n − 1. however, the techniques developed thereafter can be adapted to a varying step subdivision without difficulty. note that using the nonlinear semigroups theory [6], ouaro and ouédraogo have proved in [24] the existence and uniqueness of mild solutions of the following parabolic problem            ∂u ∂t − div a(x, ∇u) = f in q ≡ ω × (0, t ), u = 0 on ∂ω × (0, t ), u(x, 0) = u0 in ω, where u0 ∈ l 1(ω) and f ∈ l1(q). the assumptions on the vector field are the same than those given in (2.2)–(2.5) and those on the variable exponent p(x) are the same as (2.6). thanks to their paper, one has the existence and uniqueness of the mild solution of problem (1.1). 194 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) 4 numerical study 4.1 numerical scheme we are now interested in the numerical resolution of (3.1). let f1, f2, . . . , fn, u0 be some functions satisfying (3.2), we use the following iterative scheme (proposed by maitre in [23]) to get uεn+1 from uεn.        let u ε,0 n+1 = u ε n ∈ l ∞(ω), solve for k = 0, 1, . . . , u ε,k+1 n+1 − ρ div(|∇u ε,k+1 n+1 | p(x)−2∇u ε,k+1 n+1 ) = λku ε,k n+1 − ρ h (λku ε,k n+1 − u ε n) + ρf ε n+1, (4.1) where ρ > 0 is a given parameter and (λk) is a sequence of (0, 1) such that lim k→∞ λk = 1, ∏ k≥0 λk = 0, ∑ k≥0 |λk+1 − λk| < ∞. (4.2) for example, we can take λk = 1 − 1 k + 1 . remark 4.1. for the sake of simplicity, we could take ρ = h, but in this paper our idea is to build a non-expansive map and use the halpern algorithm to approach the solution of (3.1). in the numerical simulation one will give examples where ρ = h. 4.2 existence and uniqueness of solution of (4.1) in this section, we state and prove the well-posedness of our scheme. definition 4.2. for any n = 0, . . . , n − 1, ε > 0 and uεn ∈ l ∞(ω), a weak solution of (4.1) is a sequence ( u ε,k+1 n+1 ) k≥0 such that u ε,k+1 n+1 ∈ w 1,p(.) 0 (ω) ∩ l ∞(ω) for all k = 0, 1, . . . , and ∫ ω u ε,k+1 n+1 ϕ dx + ρ ∫ ω |∇u ε,k+1 n+1 | p(x)−2∇u ε,k+1 n+1 .∇ϕ dx = ∫ ω gεn,kϕ dx, (4.3) for all ϕ ∈ w 1,p(.) 0 (ω), where gε,kn := λku ε,k n+1 − ρ h (λku ε,k n+1 − u ε n) + ρf ε n+1. theorem 4.3. let ε > 0. for any n = 0, . . . , n − 1 let u ε,0 n+1 = u ε n ∈ l ∞(ω) and fεn+1 ∈ l ∞(ω). then, problem (4.1) admits a unique weak solution u ε,k+1 n+1 ∈ w 1,p(.) 0 (ω) for all k = 0, 1, . . . furthermore, for k = 0, 1, . . . , u ε,k+1 n+1 ∈ l ∞(ω). proof. let ε > 0 and fix n. for k = 0 we rewrite problem (4.1) as        u ε,1 n+1 − ρ div(|∇u ε,1 n+1| p(x)−2∇u ε,1 n+1) = g ε,0 n in ω u ε,1 n+1 = 0 on ∂ω, (4.4) cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 195 where gε,0n = [ λ0 ( 1 − ρ h ) + 1 ] uεn + ρf ε n+1. consider the energy functional jρ on w 1,p(.) 0 (ω) associated to (4.4) given by jρ(u) = 1 2 ∫ ω u2dx + ρ ∫ ω |∇u|p(x) p(x) dx − ∫ ω gε,0n u dx. we will establish that jρ(u) has a minimizer u ε,1 n+1 in w 1,p(.) 0 (ω). note that jρ is well-defined and gateaux differentiable on w 1,p(.) 0 (ω), since w 1,p(.) 0 (ω) →֒ l 2(ω) thanks to (1.2). for ‖u‖ w 1,p(.) 0 (ω) ≥ 1 we have from the continuous embedding of w 1,p(.) 0 (ω) in l p − (ω) and gε,0n ∈ l∞(ω), jρ(u) = 1 2 ∫ ω u2dx + ρ ∫ ω |∇u|p(x) p(x) dx − ∫ ω gε,0n u dx ≥ ρ p+ ‖u‖ p − w 1,p(x) 0 (ω) − c‖u‖ w 1,p(x) 0 (ω) . as p− > 1, then jρ is coercive. jρ(u) is lower bounded and furthermore weakly lower semicontinuous; therefore, admits a global minimizer u ε,1 n+1 ∈ w 1,p(.) 0 (ω) which is a weak solution to (4.4). the global minimizer u ε,1 n+1 is also unique. it remains to show that u ε,1 n+1 ∈ l ∞(ω). to do this, let us show that ‖u ε,1 n+1‖∞ ≤ ‖g ε,0 n ‖∞. as u ε,1 n+1 is a weak solution of (4.4), we have ∫ ω u ε,1 n+1ϕ dx + ρ ∫ ω |∇u ε,1 n+1| p(x)−2∇u ε,1 n+1.∇ϕ dx = ∫ ω gε,0n ϕ dx, (4.5) for all ϕ ∈ w 1,p(.) 0 (ω). let τ ∈ r+. then, u ε,1 n+1 − τ ∈ w 1,p(.) 0 (ω) and ( u ε,1 n+1 − τ )+ ∈ w 1,p(.) 0 (ω). note that for r ∈ r, r+ := max(r, 0) and r− := min(r, 0). taking ( u ε,1 n+1 − τ )+ as a test function, it follows from (4.5) that ∫ ω u ε,1 n+1(u ε,1 n+1 − τ) + dx + ρ ∫ ω |∇u ε,1 n+1| p(x)−2∇u ε,1 n+1.∇(u ε,1 n+1 − τ) + dx = ∫ ω gε,0n (u ε,1 n+1 − τ) + dx. setting aτ = { x ∈ ω : u ε,1 n+1 ≥ τ } , we have ρ ∫ ω |∇u ε,1 n+1| p(x)−2∇u ε,1 n+1.∇(u ε,1 n+1 − τ) + dx = ρ ∫ aτ |∇u ε,1 n+1| p(x)−2∇u ε,1 n+1.∇(u ε,1 n+1 − τ) dx = ρ ∫ aτ |∇u ε,1 n+1| p(x)dx ≥ 0. therefore, ∫ ω u ε,1 n+1(u ε,1 n+1 − τ) + dx ≤ ∫ ω gε,0n (u ε,1 n+1 − τ) + dx. 196 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) as ω is a bounded open domain, we have ∫ ω [(u ε,1 n+1 − τ) +]2 dx ≤ ∫ ω (gε,0n − τ)(u ε,1 n+1 − τ) + dx. taking τ = ‖gε,0n ‖∞, then g ε,0 n − τ ≤ 0 a.e. in ω. therefore, we have (u ε,1 n+1 − τ) + = 0 a.e. in ω for all τ = ‖gε,0n ‖∞ which is equivalent to saying u ε,1 n+1 ≤ ‖g ε,0 n ‖∞ a.e. in ω. it remains to prove that u ε,1 n+1 ≥ −‖g ε,0 n ‖∞ a.e. in ω. to do this we take (u ε,1 n+1 + τ) − as test function in (4.5) and use the same argument as previously. thus, setting c = ‖gε,0n ‖∞ implies that u ε,1 n+1 ∈ l ∞(ω). in short u ε,1 n+1 ∈ w 1,p(.) 0 (ω) ∩ l ∞(ω). by induction, we deduce in the same manner that the problem (4.1) has a unique weak solution ( u ε,k+1 n+1 ) k≥0 such that u ε,k+1 n+1 ∈ w 1,p(.) 0 (ω) ∩ l ∞(ω) for all k ∈ n. 4.3 study of the convergence we begin with the following lemma which provides a crucial l∞ uniform bound for the sequence ( u ε,k n+1 ) k≥0 . lemma 4.4. let ε > 0 and fix n. if ρ ≤ h, there exists m > 0 independent of k such that ‖u ε,k n+1‖∞ ≤ m. proof. let m = max ( ‖u ε,0 n+1‖∞, ‖hf ε n+1 + u ε n‖∞ ) . now let us show by induction that ‖u ε,k n+1‖∞ ≤ m. we first note that ‖u ε,0 n+1‖∞ ≤ m. one assumes that ‖u ε,k n+1‖∞ ≤ m, and one shows that ‖u ε,k+1 n+1 ‖∞ ≤ m. as u ε,k+1 n+1 ∈ l ∞(ω) and verifies u ε,k+1 n+1 − div ( ρ|∇u ε,k+1 n+1 | p(x)−2∇u ε,k+1 n+1 ) = λku ε,k n+1 − ρ h (λku ε,k n+1 − u ε n) + ρf ε n+1, then, from the previous proof, it is established that for all k = 1, 2, . . . , ‖u ε,k+1 n+1 ‖∞ ≤ ∥ ∥ ∥ λku ε,k n+1 − ρ h (λku ε,k n+1 − u ε n) + ρf ε n+1 ∥ ∥ ∥ ∞ . since ρ ≤ h, we then obtain using the induction assumption ‖u ε,k+1 n+1 ‖∞ ≤ ( 1 − ρ h ) m + ρ h ‖hfεn+1 + u ε n‖∞ ≤ m. thanks to m defined in the above proof we have the following convergence result. cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 197 theorem 4.5. assume that conditions in theorem 4.3 are satisfied. then, for ρ ≤ h, the iterative scheme (4.1) converges, i.e. u ε,k n+1 −→ u ε n+1 strongly in l 1(ω) as k → +∞, where uεn+1 verifies (3.1). proof. thanks to lemma 4.4, we can write (4.1) as 1 λk+1 ū ε,k+1 n+1 − ρ div ( |∇ 1 λk+1 ū ε,k+1 n+1 | p(x)−2∇ 1 λk+1 ū ε,k+1 n+1 ) = ū ε,k n+1 − ρ h (ū ε,k n+1 − u ε n) + ρf ε n+1, (4.6) where we put ū ε,k n+1 = λku ε,k n+1 and ū ε,k+1 n+1 = λk+1u ε,k+1 n+1 . let a(u) = −div(|∇u|p(x)−2∇u). we identify the operator a : l1(ω) → l1(ω) associated with the p(x)−laplacian problem (1.1) with its graph i.e. g(a) = { (u, v) ∈ l1(ω) × l1(ω); v ∈ a(u) } . therefore, a is t −accretive as soon as u is an entropy solution of problem (2.1) where a(x, ∇u) = (|∇u|p(x)−2∇u). for more details, see [6] and [24, proposition 4.3]. a is called t −accretive if ‖(u − û)+‖1 ≤ ‖(u − û + ρ(v − v̂)) +)‖1, for any (u, v), (û, v̂) ∈ a, ρ > 0; equivalently, if ∫ {u>û} (v − v̂) + ∫ {u=û} (v − v̂)+ ≥ 0 for any (u, v), (û, v̂) ∈ a. hence, (4.6) yields (i + ρa) ( 1 λk+1 ū ε,k+1 n+1 ) = ū ε,k n+1 − ρ h (ū ε,k n+1 − u ε n) + ρf ε n+1. (4.7) to complete the proof of theorem 4.5, we use the following technical lemma. lemma 4.6. let ρ ≤ 2h and m defined in the above proof such that cm = { u ∈ l1(ω), ‖u‖∞ ≤ m } . the iteration operator t̃(ū) = (i + ρa)−1 ( ū − ρ h (ū − uεn) + ρf ε n+1 ) is an l1-non-expanding operator from cm to cm . proof. the fact that t̃ maps cm to cm is easily seen thanks to the proof of the lemma 4.4 and (4.7). now let (ū, v̄) ∈ c2m . one has from the t −accretiveness of a on l 1(ω) that (i + ρa)−1 is a t −contraction in l1(ω) (see [6]), thus, a contraction. therefore, ‖t̃(ū) − t̃(v̄)‖1 = ∥ ∥ ∥ (i + ρa)−1 ( ū − ρ h (ū − un) + ρfn+1 ) − (i + ρa)−1 ( v̄ − ρ h (v̄ − un) + ρfn+1 ) ∥ ∥ ∥ 1 ≤ ∥ ∥ ∥ ( 1 − ρ h ) ū − ( 1 − ρ h ) v̄ ∥ ∥ ∥ 1 . 198 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) since ρ ≤ 2h, we obtain ‖t̃(ū) − t̃(v̄)‖1 ≤ ‖ū − v̄‖1. consequently, from (4.7) one has the iteration ū ε,k+1 n+1 = λk+1t̃(ū ε,k n+1) where t̃ is a non-expansive operator in l1(ω) defined as in lemma 4.6. now, we are going to apply the lemma 2.4 with x = l1(ω) and c = cm which is clearly a convex subset of l 1(ω) containing 0. the uniqueness of a fixed point is verified thanks to theorem 2.3. indeed a fixed point u∗ of t̃ verifies u∗ − ρ div (|∇u∗|p(x)−2∇u∗) = u∗ − ρ h (u∗ − uεn) + ρf ε n+1. thus, u∗ − h div (|∇u∗|p(x)−2∇u∗) = uεn + hf ε n+1. from theorem 2.3 this equation has a unique solution and from the definition of mild solution it is uεn+1. to conclude the proof of convergence of (4.1), we point out that each subsequence of ū ε,k n+1 has a convergent subsequence to an element of cm , using the l ∞ bound of ū ε,k n+1 and the monotonicity of (|∇ū ε,k n+1| p(x)−2∇ū ε,k n+1), to the equation (4.6). applying lemma 2.4, we conclude that the sequence ū ε,k n+1 converges strongly in l 1(ω) toward uεn+1. the same occurs for u ε,k n+1 = 1 λk ū ε,k n+1. 4.4 convergence when ε → 0 toward a solution of (1.1) note that for a mild solution we do not need to show the convergence in time since it is included in its definition: once convergence in k is achieved for uεn+1, then, by the definition of mild solution, uεn+1 approaches u ε(t) on (tn, tn+1] up to ε. thus, our scheme converges to the mild solution when ε goes to zero. we can state also the following result. proposition 4.7. let u0 ∈ l ∞(ω), f ∈ l∞(q) and u the unique mild solution of (1.1). then u is a weak solution of (1.1). by a weak solution we understand a solution in the sense of distributions that belongs to the energy space, i.e., u ∈ v := { v ∈ lp − (0, t ; w 1,p(.) 0 (ω)); |∇v| ∈ l p(.)(q) } , ∂u ∂t − div(|∇u|p(x)−2∇u) = f in d ′ (q), u(., 0) = u0. (4.8) remark 4.8. note that a proof of the above proposition exists in [24]. here, we use l∞ uniform boundedness and the strong convergence in l1(ω) of the solution of our numerical scheme to prove proposition 4.7. moreover, these two results lead to the l∞ uniform boundedness of the weak solution. proof of proposition 4.7. let u be the mild solution of (1.1). for n = 0, . . . , n − 1, uεn+1 is the unique weak solution of (3.1). we have ∫ ω uεn+1 − u ε n h ϕ dx + ∫ ω |∇uεn+1| p(x)−2∇uεn+1.∇ϕ dx = ∫ ω fεn+1ϕ dx, (4.9) cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 199 ∀ϕ ∈ w 1,p(.) 0 (ω) ∩ l ∞(ω) and                        • 0 = t0 < · · · < tn = t such that tn − tn−1 = h ≤ ε for n = 1, . . . , n, • n ∑ n=1 ∫ tn tn−1 ‖f(t) − fεn‖l1(ω) dt ≤ ε ⇒ ‖f ε n‖l∞(ω) ≤ ‖f(t)‖l∞(ω), • n ∑ n=1 h‖fεn‖l∞(ω) ≤ ∫ t 0 ‖f(., t)‖l∞(ω) dt, • ‖u0 − u ε 0‖l1(ω) ≤ ε ⇒ ‖u ε 0‖l∞(ω) ≤ ‖u0‖l∞(ω). (4.10) note that relations in (4.10) are equivalent to relations in (3.2). let us set uε(t) = u ε n+1 ∀ t ∈ (tn, tn+1], uε(0) = u ε 0 and fε(t) = f ε n+1, ∀ t ∈ (tn, tn+1]. lemma 4.4, theorem 4.5 and the above relations in (4.10) imply that ‖uε‖l∞(q) ≤ c(‖u0‖l∞(ω); ‖f‖l∞(q)). (4.11) let ζ be the function defined by ζ(r) = r2 2 that satisfies ζ(r) − ζ(r̃) ≤ (r − r̃)r. taking ϕ = uεn+1 as test function in (4.9) and integrating over (tn, tn+1] and summing over n = 0, . . . , n − 1, we get ∫ ω ζ(uε(t)) dx + ∫ q |∇uε| p(x) dx dt ≤ ∫ q fεuε dx dt + ∫ ω ζ(uε0) dx. thanks to the uniform boundedness of uε in ε and as u ε 0 ∈ l ∞(ω), we have ∫ q |∇uε| p(x) dx dt ≤ c. moreover, ∫ t 0 ‖∇uε‖ p − lp(.)(ω) dt ≤ ∫ t 0 max   ∫ ω |∇uε| p(x); ( ∫ ω |∇uε| p(x) ) p− p+   dt. hence, ∫ t 0 ‖uε‖ p − w 1,p(.) 0 (ω) dt ≤ c. as a consequence, there exists a subsequence still denoted (uε)ε>0, such that uε ⇀ u, weakly-* in l ∞(q), uε ⇀ u, weakly in l p − (0, t ; w 1,p(.) 0 (ω)), |∇uε| p(.)−2∇uε ⇀ φ, weakly in ( lp ′(.)(q) )d . using the monotonicity method we show that φ = |∇u|p(.)−2∇u a.e. in q. now, let ũε be the piecewise linear function defined by ũε(t) = u ε n + t − tn h (uεn+1 − u ε n) for t ∈ [tn, tn+1], n = 0, . . . , n − 1. 200 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) the function ũε verifies (ũε)t (t) = uεn+1 − u ε n h and ũε → u in l ∞(0, t ; l1(ω)). hence, u ∈ c([0, t ]; l1(ω)). integrating (4.9) over (tn, tn+1) and summing over n = 0, . . . , n − 1, we find − ∫ t 0 ∫ ω ϕtũε dx dt − ∫ ω ϕ(0)uε0 dx + ∫ t 0 ∫ ω ( |∇uε| p(x)−2∇uε ) .∇ϕ dx dt = ∫ t 0 ∫ ω fεϕ dx dt. (4.12) using the convergence results and passing to the limit in (4.12) as ε → 0, we get the result. remark 4.9. for u0 ∈ l 1(ω), f ∈ l1(q) the unique mild solution u of (1.1) is also an entropy solution. indeed, since l∞ is dense in l1, we consider two sequences of functions (fm)m≥1 ⊂ l∞(q) and (u0m)m≥1 ⊂ l ∞(ω) satisfying        fm → f in l 1(q), u0m → u0 in l 1(ω), as m → ∞, ‖fm‖l1(q) ≤ ‖f‖l1(q), ‖u0m‖l1(ω) ≤ ‖u0‖l1(ω). (4.13) then, we get the following approximate problem of (1.1).            ∂um ∂t − div(|∇um| p(x)−2∇um) = fm in q, um = 0 on ∂ω × (0, t ), um(x, 0) = u0m in ω. (4.14) thanks to [24], for each m = 1, 2, . . . , we can find a unique mild solution um ∈ c([0, t ]; l 1(ω)) for problem (4.14) which verifies the l1−contraction principle, i.e. the following estimate holds for almost all t ∈ (0, t ), ‖um(., t)‖l1(ω) ≤ ‖u0m‖l1(ω) + ∫ t 0 ‖fm(., s)‖l1(ω) ds ≤ ‖u0‖l1(ω) + ∫ t 0 ‖f(., s)‖l1(ω) ds. by proposition 4.7, and following the proof of [24, theorem 5.1] we get the result. note that this entropy solution is equivalent to the renormalized solution of (1.1). indeed, in [29], zhang and zhou have proved thanks to the assumptions (1.2) the existence and uniqueness of renormalized and entropy solutions of (1.1). in their paper, they have showed the equivalence between entropy and renormalized solutions. cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 201 4.5 numerical tests 4.5.1 implementation we know that solving the equation (4.1) is equivalent to solve the following minimization problem for n = 0, 1, . . . , n − 1 and k = 0, 1, . . . u ε,k+1 n+1 = argminv ∈wj(v), (4.15) where, w := { v ∈ w 1,p(.) 0 (ω) ∩ l ∞(ω) } and the functional j is j(v) = 1 2 ∫ ω v2 dx + ρ ∫ ω 1 p(x) |∇v|p(x) dx − ( 1 − ρ h ) λk ∫ ω u ε,k n+1v − ρ h ∫ ω uεnv dx −ρ ∫ ω fεn+1v dx. (4.16) we formulate a basic procedure for solving problem (4.15) following the split bregman technique (see [17]). we solve the minimization problem by introducing an auxiliary variable b. we have min v { 1 2 ∫ ω v2 dx + ρ ∫ ω 1 p(x) |b|p(x) dx − ( 1 − ρ h ) λk ∫ ω u ε,k n+1v dx − ρ h ∫ ω uεnv dx − ρ ∫ ω fεn+1v dx subject to b = ∇v } . (4.17) by adding one quadratic penalty function term, we convert equation (4.17) to an unconstrained splitting formulation as follow. min v,b { 1 2 ∫ ω v2 dx + ρ ∫ ω 1 p(x) |b|p(x) dx + γ 2 ∫ ω |b − ∇v|2 dx − ( 1 − ρ h ) λk ∫ ω u ε,k n+1v dx − ρ h ∫ ω uεnv dx − ρ ∫ ω fεn+1v dx } , (4.18) where γ is a positive parameter which controls the weight of the penalty term. similar to the split bregman iteration, we propose the following scheme.                        (vl+1, bl+1) = argminv,b { 1 2 ∫ ω v2 dx + ρ ∫ ω 1 p(x) |b|p(x) dx + γ 2 ∫ ω |b − ∇v − δl|2 dx − ( 1 − ρ h ) λk ∫ ω u ε,k n+1v dx − ρ h ∫ ω uεnv dx − ρ ∫ ω fεn+1v dx } , δl+1 = δl + ∇vl+1 − bl+1. (4.19) alternatively, this joint minimization problem can be solved by decomposing into several subproblems. 202 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) 4.5.2 subproblem v with fixed b and δ given the fixed variable bl and δl, our aim is to find the solution of the following problem vl+1 = argminv { 1 2 ∫ ω v2 dx + γ 2 ∫ ω |bl − ∇v − δl|2 dx − ( 1 − ρ h ) λk ∫ ω u ε,k n+1v dx − ρ h ∫ ω uεnv dx − ρ ∫ ω fεn+1v dx } . (4.20) we know that solve (4.20) is equivalent to solve the following optimality condition. v − γ∆v = γ∇.(δl − bl) + ( 1 − ρ h ) λku ε,k n+1 + ρ h uεn + ρf ε n+1. (4.21) since the discrete system is strictly diagonally dominant with neumann boundary condition, the most natural choice is the gauss-seidel method. 4.5.3 subproblem b with fixed v and δ similarly, we solve bl+1 = argminb { ρ ∫ ω 1 p(x) |b|p(x) dx + γ 2 ∫ ω |b − ∇vl+1 − δl|2 dx } (4.22) in two dimensional space. here, setting b = (bx, by) and δ = (δx, δy). then, the resolution of (4.22) is equivalent to solve the following optimality condition.        ρ|b|p(x,y)−2bx + γ(bx − ∇xv l+1 − δlx) = 0 ρ|b|p(x,y)−2by + γ(by − ∇yv l+1 − δly) = 0, (4.23) where ∇v = (∇xv, ∇yv). if bx and by are not zero, then, bx = ∇xv l+1 + δlx ∇yvl+1 + δly by. (4.24) substituting (4.24) into (4.23), we obtain sign(by)t |by| p(x,y)−1 + γ(by − ∇yv l+1 − δly) = 0, (4.25) where t = ρ ( ( ∇xv l+1 + δlx ∇yvl+1 + δly )2 + 1 ) p(x,y)−2 2 . here, sign is defined as follows. sign(ω) :=        1 if ω > 0, 0 if ω = 0, −1 if ω < 0. cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 203 note that sign(bx) = sign(∇xv l+1 + δlx) (4.26) and sign(by) = sign(∇yv l+1 + δly). (4.27) so, (4.25) can be expressed as sign(∇yv l+1 + δly)t |by| p(x,y)−1 + γ(by − ∇yv l+1 − δly) = 0. (4.28) unfortunately, we cannot obtain the explicit solution of the equation (4.28). we can use newton method to get an approximate solution. if by is solved, bx can be easily determined using (4.24) and (4.26). 4.5.4 applications in the following numerical simulation the iteration process stops when the following condition is satisfied ‖uk+1n+1 − u k n+1‖2 ‖uk+1n+1‖2 ≤ stop := 10−5, (4.29) where ‖.‖2 is the euclidean norm and u k n+1 the vector approaching, at iteration k, the spacediscretization of un+1. after stopping the iterations at k = klast, we denote un+1 = u klast n+1 and switch to the next time step. note that for implementation, finite difference method is used to approximate the partial derivatives. moreover, for sake of simplicity, the domain ω will be a square. the domain ω will be subdivided into n2x uniform squares. for numerical simulation, we will use the following parameters nx = 80 and h = 0.02. let us recall that h is the time step. the space step is easily computed thanks to nx and ω. example 4.10. in this example, we take ω = (0, 1) × (0, 1), t = 1, p(x, y) = 2, and f = xy(1 − x)(1 − y) + 2t((1 − y)y + (1 − x)x). as initial condition, we set u0(x, y) = 0. let us note that with these data p, u0 and f, the exact solution is u(x, y, t) = txy(1 − x)(1 − y). 204 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) 0 1 0.01 0.02 1 0.03 u e x 0.04 0.8 exact solution at t=1.000 y 0.05 0.5 0.6 x 0.06 0.4 0.2 0 0 0 1 0.01 0.02 1 0.03u 0.04 0.8 numerical solution at t=1.000 y 0.05 0.5 0.6 x 0.06 0.4 0.2 0 0 figure 1: left: u(x, y, t) = txy(1 − x)(1 − y) right: for ρ = h and γ = 0.02 . 0 1 0.01 0.02 1 0.03 u e x 0.04 0.8 exact solution at t=1.000 y 0.05 0.5 0.6 x 0.06 0.4 0.2 0 0 0 1 0.01 0.02 1 0.03u 0.04 0.8 numerical solution at t=1.000 y 0.5 0.05 0.6 x 0.06 0.4 0.2 0 0 figure 2: left: u(x, y, t) = txy(1 − x)(1 − y) right: for ρ = h/2 and γ = 0.02 figure 1 shows the exact solution and the numerical solution for γ = 0.02 and ρ = h. while, figure 2 shows the exact solution and the numerical solution for γ = 0.02 and ρ = h/2. as we can see, we always get a good numerical approximation of the solution even if ρ varies. denoting uh the numerical solution and u the exact solution of example 4.10, with ρ = h and γ = 0.02, we get the following table of the error approximation. t 0.1 0.2 0.3 0.4 0.5 ‖uh − u‖1 2.5099.10 −5 5.6941.10−5 7.9789.10−5 1.0717.10−4 1.345.10−4 t 0.6 0.7 0.8 0.9 1 ‖uh − u‖1 1.6192.10 −4 1.8930.10−4 2.1668.10−4 2.4406.10−4 2.7144.10−4 cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 205 example 4.11. in this example, we set ω = (0, 1) × (0, 1), t = 5, p(x, y) = 2 + |x| 2 , and f = 1. as initial condition we set u0(x, y) = 0. as parameters we set ρ = h and γ = 0.02. 1 0 1 x 0.02 0.5 numerical solution at t=1.000 0.8 0.04 y 0.6 0.06u 0.4 0.08 0.2 0.1 0 0.12 0 1 0 1 x 0.02 0.5 numerical solution at t=5.000 0.8 0.04 y 0.6 0.06u 0.4 0.08 0.2 0.1 0 0.12 0 figure 3: numerical solution for p(x, y) = 2 + |x| 2 , ρ = h and γ = 0.02. figure 3 shows the numerical solution at t = 1 and at t = 5. one can see that both figures are the same. example 4.12. in this example, we take ω = (−1, 1) × (−1, 1), t = 5, p(x, y) = 9 5 − x2 2 and f =    1 if x ≥ 0 0 if x < 0. as the initial condition, we set u0(x, y) = e (1−x2)(1−y2) − 1. we use the same parameters ρ and γ as previously. figure 4 shows the numerical solution at t = 1 and t = 5. 206 s. ouaro, n. rabo & u. traoré cubo 24, 2 (2022) 0 1 0.02 0.04 0.5 1 u 0.06 numerical solution at t=1.000 0.5 y 0.08 0 x 0.1 0 -0.5 -0.5 -1 -1 0 1 0.02 0.04 0.5 1 u 0.06 numerical solution at t=5.000 0.5 y 0.08 0 x 0.1 0 -0.5 -0.5 -1 -1 figure 4: numerical solution for p(x, y) = 9 5 − x2 2 , ρ = h and γ = 0.02. we remark that the exponents p(x) considered in the three examples satisfy the condition 1.2. also, note that the choice of γ results from the knowledge of the explicit solution of the example 4.10. indeed, knowing the explicit solution, we choose γ so as to obtain a better approximation of this explicit solution. this leads to the choice of γ = 0.02. conclusion and discussion inspired by the work of maitre (see [23]), we have in this paper made a numerical analysis of the mild solution of parabolic problem involving the p(x)−laplacian operator. using the works of zhang and zhou (see [29]), and ouaro and ouédraogo (see [24]), we have shown that the mild solution is also an entropy solution which is equivalent to the renormalized solution. for the numerical tests, we have used the split bregman iteration. in a forthcoming paper, we will make a comparison of the solutions of our numerical scheme (4.1) to those of the classical backward euler scheme. cubo 24, 2 (2022) numerical analysis of nonlinear parabolic problems with variable... 207 references [1] s. n. antontsev and s. i. shmarev, “a model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of 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(pomi), vol. 310, pp. 67–81, 2004. introduction preliminaries notion of mild solution numerical study numerical scheme existence and uniqueness of solution of (4.1) study of the convergence convergence when 0 toward a solution of (1.1) numerical tests implementation subproblem v with fixed b and subproblem b with fixed v and applications cubo a mathematical journal vol.21, no¯ 02, (51–64). august 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000200051 the perimeter of a flattened ellipse can be estimated accurately even from maclaurin’s series vito lampret university of ljubljana, 386 slovenia vito.lampret@guest.arnes.si abstract for the perimeter p(a, b) of an ellipse with the semi-axes a ≥ b ≥ 0 a sequence qn(a, b) is constructed such that the relative error of the approximation p(a, b) ≈ qn(a, b) satisfies the following inequalities 0 ≤ − p(a, b) − qn(a, b) p(a, b) ≤ (1 − q2)n+1 (2n + 1)2 ≤ 1 (2n + 1)2 e−q 2(n+1), true for n ∈ n and q = b a ∈ [0, 1]. resumen para el peŕımetro p(a, b) de una elipse con semiejes a ≥ b ≥ 0, se construye una sucesión qn(a, b) tal que el error relativo de la aproximación p(a, b) ≈ qn(a, b) satisface las siguientes desigualdades 0 ≤ − p(a, b) − qn(a, b) p(a, b) ≤ (1 − q2)n+1 (2n + 1)2 ≤ 1 (2n + 1)2 e−q 2(n+1), válidas para n ∈ n y q = b a ∈ [0, 1]. keywords and phrases: approximation, elementary, ellipse, estimate, maclaurin series, mathematical validity, perimeter, simple. 2010 ams mathematics subject classification: 40a25, 65b10. http://dx.doi.org/10.4067/s0719-06462019000200051 52 vito lampret cubo 21, 2 (2019) 1 introduction injective parametric equations of the border of an ellipse having semi-axes of lengths a and b ≤ a are given as x = x(t) = a cos(t), y = y(t) = b sin(t), where t ∈ [0, 2π). its perimeter p(a, b) is determined as p(a, b) = ∫ 2π 0 √ ẋ2(t) + ẏ2(t) dt = 4 ∫ π 2 0 √ a2 sin2(t) + b2 cos2(t) dt = 4a ∫ π 2 0 √ 1 − ǫ2 cos2(t) dt = ︷︸︸︷ t = π/2 − τ 4a ∫ 0 π 2 √ 1 − ǫ2 sin2(τ)(− dτ). thus, the perimeter p(a, b) of an ellipse having semi-axes of lengths a and b ≤ a, is given as p(a, b) = 4a e(ǫ), (1.1) where e(ǫ) := ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ (1.2) is complete elliptic integral of the second kind and ǫ := √ 1 − ( b a )2 = √ a2−b2 a2 ∈ [0, 1), (1.3) is the eccentricity of an ellipse. for b ≈ 0 it is intuitively evident that p(a, b) > 2 × 2a = 4a. moreover, since the functions ǫ 7→ 1 − ǫ2 sin2(τ) are decreasing on the interval [0, 1] for any τ ∈ [0, π/2], the function e(ǫ) is decreasing too. therefore, we have 1 = ∫ π 2 0 cos(τ) dτ = e(1) ≤ e(ǫ) ≤ e(0) = π 2 , for 0 ≤ ǫ ≤ 1. consequently, due to (1.1), inf 0 n; consequently w0 = 1 54 vito lampret cubo 21, 2 (2019) we obtain ( 1 2 i ) = ∏i−1 j=0( 1 2 − j) i! = (−1)i−1 1 2i · ∏i−1 j=1(2j − 1) ∏i j=1 j = (−1)i−1 1 2i − 1 i ∏ j=1 2j − 1 2j = (−1)i−1 wi 2i − 1 . (2.3) thus, thanks to (2.1), replacing x by −x, we get (1 − x) 1 2 = 1 − n∑ i=1 wi 2i − 1 xi + rn(x), (2.4) with the remainder rn(x) = −x n+1 wn+1 2n + 1 (n + 1) ∫ 1 0 ( 1 − t 1 − tx )n dt (1 − tx) 1 2 , estimated, for x ∈ (0, 1), as 0 < −rn(x) = xn+1 (1 − x) 1 2 · wn+1 2n + 1 (n + 1) ∫ 1 0 ( 1 − t 1 − tx )n dt < wn+1 (1 − x) 3 2 (2n + 1) xn+1 . (2.5) indeed, using the substitution τ = 1−t 1−tx , i.e. t = 1−τ 1−τx we have (considering x ∈ (0, 1)) ∫ 1 0 ( 1 − t 1 − tx )n dt = ∫ 0 1 τ n ( − 1 − x (1 − τx)2 ) dτ = ∫ 1 0 τ n · 1 − x (1 − τx)2 dτ < ∫ 1 0 τ n · 1 − x (1 − x)2 dτ = 1 (1 − x)(n + 1) . 2.2 wallis ratios estimates the integrals in := ∫ π 2 0 sinn(x) dx (n ≥ 0), (2.6) satisfy the recurrence relation in = n − 1 n in−2, for n ≥ 2, where, obviously, we have i0 = π 2 and i1 = 1. consequently, by induction we find i2i = i ∏ j=1 2j − 1 2j · π 2 = wi · π 2 (2.7) cubo 21, 2 (2019) the perimeter of a flattened ellipse can be estimated accurately . . . 55 and i2i+1 = i∏ j=1 2j 2j + 1 = 1 (2i + 1)wi . (2.8) obviously, we estimate 0 < sin2i+2(x) < sin2i+1(x) < sin2i(x) < 1, for x ∈ ( 0, π 2 ) and i ∈ n. integrating, we obtain 0 < i2i+2 < i2i+1 < i2i < 1, for all i ∈ n. hence, thanks to (2.7)–(2.8), we get 2i + 1 2i + 2 wi · π 2 = wi+1 · π 2 < 1 (2i + 1)wi < wi · π 2 . consequently, 2 π · 1 2i + 1 < w2i < 2 π · 1 2i − 1 (i ∈ n). (2.9) we remark that there exists a huge literature on useful, more accurate estimates for wn, e.g. [4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 31]. however, for our needs, there suffice rather rough estimates (2.9). 2.3 some logarithmic formula expansion for p ≥ 1 and −1 < t < 1 we have 2 p−1 ∑ j=0 t2j = 2(p−1) ∑ k=0 ( tk + (−t)k ) = 2(p−1) ∑ k=0 tk + 2(p−1) ∑ k=0 (−t)k = 1 − t2p−1 1 − t + 1 − (−t)2p−1 1 + t . consequently, integrating from 0 to x ∈ (−1, 1), the first and the last part of these equalities, we obtain 2 p−1 ∑ j=0 x2j+1 2j + 1 = ∫ x 0 1 1 − t dt − ∫ x 0 t2p−1 1 − t dt + ∫ x 0 1 1 + t dt + ∫ x 0 t2p−1 1 + t dt = − ln(1 − x) + ln(1 + x) − ∫ x 0 ( 1 1 − t − 1 1 + t ) t2p−1 dt ︸︷︷︸ =r∗ p (x) . thus, ln ( 1 + x 1 − x ) = 2 p ∑ i=1 x2i−1 2i − 1 + r∗p(x), (2.10) 56 vito lampret cubo 21, 2 (2019) with the remainder r∗p(x), r∗p(x) := ∫ x 0 2t2p 1 − t2 dt ≥ ∫ x 0 2t2p dt. (0 < x < 1), estimated as 2x2p+1 2p + 1 < r∗p(x) < 2x2p+1 (1 − x2)(2p + 1) (p ∈ n, 0 < x < 1) (2.11) from (2.10)–(2.11) we end up with the expansion ln ( 1 + x 1 − x ) = 2 ∞ ∑ i=1 x2i−1 2i − 1 , (2.12) true for x ∈ (0, 1) and, consequently, also for x ∈ (−1, 0]. 3 the maclaurin series 3.1 derivation referring to (2.4)–(2.5) and (2.6)–(2.7), we have, for any n ∈ n, ∫ π 2 0 √ 1 − ǫ2 sin2(τ) ︸︷︷︸ dτ = π 2 − n ∑ i=1 wi ǫ 2i 2i − 1 ∫ π 2 0 sin2i(τ) dτ + r∗n(ǫ) = π 2 − n ∑ i=1 wi ǫ 2i 2i − 1 ( wi π 2 ) + r∗n(ǫ). hence ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ = π 2 ( 1 − n ∑ i=1 w2i 2i − 1 ǫ2i ) + r∗n(ǫ), (3.1) where wi is the i-th wallis’ ratio and the error term r ∗ n(ǫ) := ∫ π/2 0 rn ( ǫ2 sin2(τ) ) dτ is estimated, due to (2.5) and considering (2.6)–(2.7), as 0 ≤ −r∗n(ǫ) ≤ ǫ2n+2 1 − ǫ2 · wn+1 2n + 1 ∫ π 2 0 sin2n+2(τ) dτ = ǫ2n+2 wn+1 (1 − ǫ2)(2n + 1) · wn+1 π 2 . thus, according to (2.9), 0 ≤ −rn(ǫ) ≤ π 2 · 1 1 − ǫ2 · w2n+1 2n + 1 ǫ2n+2 ≤ 1 1 − ǫ2 · ǫ2n+2 (2n + 1)2 . (3.2) this estimate is not usable for ǫ ≈ 1, i.e. for b ≈ 0 (for a very flattened ellipse). cubo 21, 2 (2019) the perimeter of a flattened ellipse can be estimated accurately . . . 57 as w2n ≤ 1, we have lim n→∞ rn(ǫ) = 0 for any ǫ < 1, which is always true for ordinary ellipse, due to the equivalence ǫ = 1 ⇔ b = 0. hence, there holds the so-called maclaurin series expansion5 ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ = π 2 ( 1 − ∞ ∑ i=1 w2i 2i − 1 ǫ2i ) , (3.3) valid for 0 ≤ ǫ < 1. in addition, the series on the right is convergent also for ǫ = 1 due to (2.9). indeed, we have w2i 2i−1 < 1 i2 , which implies the convergence of the series ∑∞ i=1 w2i 2i−1 . remark 3.1. about fifty years after maclaurin’s book [24], including the series (3.3), ivory published article [13], where he presented the expansion ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ = π(a + b) 4a ( 1 + ∞∑ i=1 w2i (2i − 1)2 λ2i ) ( λ = a − b a + b ) , where the series on the right converges slightly faster than the series in (3.3). applying (2.9) for the partial sums µn(ǫ) := n ∑ i=1 w2i 2i − 1 ǫ2i (n ∈ n ∪ {∞}), (3.4) we shall estimate the series µ∞(ǫ) figuring in (3.3). 3.2 approximating µ ∞ (ǫ) using (2.9) we estimate, 2 π(2i − 1)(2i + 1) < w2i 2i − 1 < 2 π(2i − 1)2 (i ∈ n) . (3.5) therefore µ∞(ǫ) ≈ ∞ ∑ i=1 2ǫ2i π(2i − 1)(2i + 1) (0 ≤ ǫ < 1). this idea produces the next theorem. theorem 3.2. we have µ∞(ǫ) = mn(ǫ) + δn(ǫ), (3.6) where mn(ǫ) = a(ǫ) + bn(ǫ), (3.7) a(ǫ) := 1 2π [( ǫ − 1 ǫ ) ln (1 + ǫ 1 − ǫ ) + 2 ] ∈ ( 0, 1 π ) , (3.8) 5the coefficients of the original maclaurin series [24] have a visually more complicated form. 58 vito lampret cubo 21, 2 (2019) bn(ǫ) := n ∑ i=1 ( w2i − 2 π(2i + 1) ) ǫ2i 2i − 1 , (3.9) and 0 < δn(ǫ) < δ ∗ n(ǫ) := 2ǫ2n+2 π(2n + 1)2 , (3.10) valid for any integer n ≥ 1 and every 0 < ǫ < 1. the basic function a(ǫ) is strictly increasing having the range ( 0, 1 π ) , where lim ǫ↓0 a(ǫ) = 0 and lim ǫ↑1 a(ǫ) = 1 π . both sequences, n 7→ bn(ǫ) and n 7→ δn(ǫ), are strictly increasing, for every ǫ ∈ (0, 1). the sequence n 7→ mn(ǫ) converges strictly increasingly to µ∞(ǫ), for any ǫ ∈ (0, 1). additionally, for every n ∈ n, the functions ǫ 7→ mn(ǫ) and ǫ 7→ δn(ǫ) are strictly increasing on the interval (0, 1). figure 1 shows, on the left, the graph6 of the basic function a(ǫ), and, on the right, the graphs of the functions m∗1 (ǫ) and µ∞(ǫ). as an example, we present b ∗ 4(ǫ) and δ ∗ 4(ǫ) as follows: b∗4(ǫ) = ( 1 4 − 2 3π ) ǫ2 + 1 3 ( 9 64 − 2 5π ) ǫ4 + 1 5 ( 25 256 − 2 7π ) ǫ6 + 1 7 ( 1225 16384 − 2 9π ) ǫ8 ≈ 0.037 793 409 ǫ2 + 0.004 433 682 ǫ4 + 0.001 342 114 ǫ6 + 0.000 576 077 ǫ8, δ∗4(ǫ) ≤ 2ǫ10 81π ≤ 0.00786 ǫ10 ( ǫ = √ 1 − ( b a )2 ) . 0.2 0.4 0.6 0.8 1.0 0.05 0.10 0.15 0.20 0.25 0.30 ahεl 0.2 0.4 0.6 0.8 1.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ahεl m1 hεl» μ¥hεl figure 1: the graph of the basic function a(ǫ) (left) and the graphs of the functions m1(ǫ), µ∞(ǫ) and a(ǫ) (right). proof of theorem 3.2. we have, for 0 < ǫ < 1, ∞ ∑ i=1 w2i ǫ2i 2i − 1 = ∞ ∑ i=1 2 ǫ2i π(2i − 1)(2i + 1) (3.11) + n ∑ i=1 ( w2i 2i − 1 − 2 π(2i − 1)(2i + 1) ) ǫ2i + δn(ǫ), 6all the graphics and calculations in this paper are made using the mathematica [30] computer system. cubo 21, 2 (2019) the perimeter of a flattened ellipse can be estimated accurately . . . 59 where δn(ǫ) = ∞ ∑ i=n+1 ( w2i − 2 π(2i + 1) ) ǫ2i 2i − 1 . (3.12) moreover, using (2.12), we have ∞ ∑ i=1 2 π(2i − 1)(2i + 1) ǫ2i = 1 π ∞ ∑ i=1 ( 1 2i − 1 − 1 2i + 1 ) ǫ2i = 1 π ( ǫ 2 · 2 ∞ ∑ i=1 ǫ2i−1 2i − 1 − 1 2ǫ · 2 ∞ ∑ i=1 ǫ2i+1 2i + 1 ) = 1 π [ ǫ 2 ln (1 + ǫ 1 − ǫ ) − 1 2ǫ ( ln (1 + ǫ 1 − ǫ ) − 2ǫ )] = 1 2π [( ǫ − 1 ǫ ) ln (1 + ǫ 1 − ǫ ) + 2 ] = a(ǫ). concerning a(ǫ) = 1 2π ( f(ǫ) + 2 ) , the function f(ǫ) := ( ǫ − 1 ǫ ) ln ( 1+ǫ 1−ǫ ) (0 < ǫ < 1) has the derivative f′(ǫ) = g(ǫ)/ǫ2, where g(ǫ) = (1 + ǫ2) ln ( 1+ǫ 1−ǫ ) − 2ǫ, having the derivative g′(ǫ) = 2ǫ 1 − ǫ2 ( 2ǫ + (1 − ǫ2) ln ( 1 + ǫ 1 − ǫ )) > 0 (0 < ǫ < 1). thus, g is strictly increasing on [0, 1). consequently, g(ǫ) > g(0) = 0, i.e. f′(ǫ) > 0, for 0 < ǫ < 1. therefore, f is strictly increasing on (0, 1) too. moreover, using (2.10)–(2.11) with p = 1, we have f(ǫ) = ǫ 2 −1 ǫ · 2 ( ǫ + ϑ · 2ǫ 3 3(1−ǫ2) ) = 2(ǫ2 − 1) ( 1 + ϑ · 2 1−ǫ2 · ǫ 2 3 ) , for some ϑ = ϑ(ǫ) ∈ (0, 1). hence, lim ǫ↓0 f(ǫ) = −2, i.e. lim ǫ↓0 a(ǫ) = lim ǫ↑1 1 2π ( f(ǫ) + 2 ) = 0. in addition, lim ǫ↑1 f(ǫ) = lim ǫ↑1 [ ǫ2−1 ǫ · 2 ln(1 + ǫ) ] − 1 1 · lim h↓0 ( − h ln(h) ) = 0, where h = 1 − ǫ2. thus, lim ǫ↑1 a(ǫ) = lim ǫ↑1 1 2π ( f(ǫ) + 2 ) = 1 π . according to (3.5), all summands in bn(ǫ) and δn(ǫ) (see (3.12)) are positive, i.e. the sequences n 7→ bn(ǫ) and n 7→ δn(ǫ) are strictly increasing; consequently the sequence n 7→ mn(ǫ) is also strictly increasing, for every ǫ ∈ (0, 1). since all coefficients of the power series bn(ǫ) and δn(ǫ) (see (3.9) and (3.12)) are positive, due to (3.5), the functions ǫ 7→ mn(ǫ) and ǫ 7→ δn(ǫ) are strictly increasing on the interval (0, 1), for any n ∈ n. according to (3.12) and (3.5), we estimate, for ǫ ∈ (0, 1], 0 < δn(ǫ) < ∞ ∑ i=n+1 ( 2 π(2i − 1) − 2 π(2i + 1) ) ǫ2n+2 2n + 1 = 2ǫ2n+2 π(2n + 1)2 , 60 vito lampret cubo 21, 2 (2019) using the telescoping method of summation. example 3.3. theorem 3.2 is quite useful for an estimate of µ∞(ǫ), consequently for an estimate of the perimeter of an ellipse. for example, for a very flattened ellipse with q = 0.01 we have 0.99994 < ǫ(q) < 0.99995 where 0.36315 < m20(0.99995) < 0.36316 . . . and δ ∗ 20(0.99995) < 0.00038. therefore, 0.36315 < µ∞(0.99995) < 0.36316+0.00038 = 0.36354. thus, to three decimal places, we have µ∞(0.99995) = 0.363 . . .. consequently, the perimeter p(a, b) of the corresponding ellipse is given as p(a, b) = 4a · π 2 ( 1 − µ∞(0.99995) ) ≈ 4a · π 2 ( 1 − 0.363 ) ≈ 4.002 a (compare with relations (1.4)). remark 3.4. referring to abel’s theorem on the boundary behavior of a power series, if we continuously extend the domain of the function a(ǫ) to the closed interval [0, 1] by using limits, a(0) := 0 and a(1) := 1 π , then (3.6), (3.7), (3.9) and (3.10) are all valid also for ǫ = 0 and ǫ = 1. remark 3.5. alternatively, we can estimate the remainder r∗∗n (ǫ) := µ∞(ǫ) − mn(ǫ) as follows: r∗∗n (ǫ) ≤ ∞ ∑ i=n+1 w2i ǫ 2i 2i − 1 ≤ w2n+1ǫ 2n+2 2n + 1 ∞ ∑ j=0 ǫ2j = w2n+1ǫ 2n+2 (2n + 1)(1 − ǫ2) ≤ 1 1 − ǫ2 · 2ǫ2n+2 π(2n + 1)2 . this simple method works quite well for ǫ, which “differs enough from 1”, but it is useless for ǫ, which is close to 1. 4 the main result theorem 4.1. for every n ∈ n, the perimeter p(a, b) of an ellipse having semi-major and semiminor axes, a and b, the aspect ratio q = q(a, b) := b/a, and the eccentricity ǫ = ǫ(a, b) := √ 1 − q2, the n-th approximation qn(a, b) ≈ p(a, b), qn(a, b) := 2πa ( 1 − mn ( ǫ ) ) = 2πa ( 1 − a(ǫ) − bn ( ǫ ) ) , (4.1) has the relative error, p(a, b) − qn(a, b) p(a, b) =: ρn(q) ( q = q(a, b) = ( b a )2 ) , estimated as − 1 (2n + 1)2 e−q 2(n+1) ≤ − ( 1 − q2 )n+1 (2n + 1)2 =: ρ∗n(q) ≤ ρn(q) ≤ 0 . here, a(ǫ) and bn ( ǫ ) are defined in theorem 3.2 and we have bn+1 ( ǫ ) = bn ( ǫ ) + ( w2n+1 − 2 π(2n+3) ) ǫ2n+2 2n+1 , for n ∈ n and 0 ≤ ǫ ≤ 1. cubo 21, 2 (2019) the perimeter of a flattened ellipse can be estimated accurately . . . 61 proof. thanks to (1.1), (1.2), (1.4) and (3.3), we estimate − p(a, b) − qn(a, b) p(a, b) = − 2πa ( 1 − mn(ǫ) − δn(ǫ) ) − 2πa ( 1 − mn(ǫ) ) p(a, b) (1.4) < 2πa δn(ǫ) 4a ≤ π δn(ǫ) 2 < π 2 · 2ǫ2n+2 π(2n + 1)2 = ǫ2n+2 (2n + 1)2 , where, considering the convexity of the exponential function or, referring to [16, (6a)] with ε = q2 and h = −q2 , we have ǫ2n+2 = (1 − q2)n+1 ≤ e−q 2(n+1) (0 ≤ q < 1). figures 2–3 show, for several values of n, the graphs of actual relative errors −ρn(q) = [ µ∞ ( ǫ(q) ) − mn ( ǫ(q) )] / [ 1 − µ∞ ( ǫ(q) )] (left) together with their upper bounds −ρ∗n(q) (right). 0.2 0.4 0.6 0.8 1.0 0.002 0.004 0.006 0.008 0.010 -ρ1hql 0.2 0.4 0.6 0.8 1.0 0.02 0.04 0.06 0.08 0.10 -ρ1 * hql figure 2: the graphs of the functions q 7→ −ρ1(q) and q 7→ −ρ ∗ 1(q). 0.2 0.4 0.6 0.8 1.0 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 -ρ9hql 0.2 0.4 0.6 0.8 1.0 0.0005 0.0010 0.0015 0.0020 0.0025 -ρ9 * hql figure 3: the graphs of the functions q 7→ −ρ9(q) and q 7→ −ρ ∗ 9(q). table 1 additionally confirms the usefulness of the derived formula. conclusion. the article demonstrates that with the help of 277 years old maclaurin series the perimeter of an ellipse can be accurately estimated, even if an ellipse flattens into a line segment. this is done only by elementary means, not using complex analysis or elliptical integral theory, neither arithmetic-geometric means nor hypergeometric functions. 62 vito lampret cubo 21, 2 (2019) q 0.00001 0.1 0.2 0.3 0, 4 0, 5 −ρ20(q) < 8·10 −5 < 6·10−5 < 2·10−5 < 5·10−6 < 6·10−7 < 4·10−8 −ρ∗20(q) < 6·10 −4 < 5·10−4 < 3·10−4 < 9·10−5 < 2·10−5 < 2·10−6 table 1: the actual error ρ20(q) and the a priori estimated error ρ ∗ 20(q). references [1] s. adlaj, an eloquent formula for the perimeter of an ellipse, notices of the ams 59 (2012), no. 8, 1094–1099. 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[25] f. qi and c. mortici, some best approximation formulas and the inequalities for the wallis ratio, appl. math. comput. 253 (2015), 363–368. [26] , f. qi, an improper integral, the beta function, the wallis ratio, and the catalan numbers, problemy analiza–issues of analysis 7 (25) (2018), no. 1, 104–115. [27] j.-s. sun and c.-m. qu, alternative proof of the best bounds of wallis’ inequality, commun. math. anal. 2 (2007), 23–27. 64 vito lampret cubo 21, 2 (2019) [28] m. b. villarino, a direct proof of landen’s transformation, arxiv:math/0507108v1 [math.ca]. [29] m. b. villarino, ramanujan’s inverse elliptic arc approximation, ramanujan j., 34 (2014), no. 2, 157–161. [30] s. wolfram, mathematica, version 7.0, wolfram research, inc., 1988–2009. [31] x.-m. zhang, t. q. xu and l. b. situ geometric convexity of a function involving gamma function and application to inequality theory, j. inequal. pure appl. math. 8 (2007) 1, art. 17, 9 p. introduction background the binomial approximation wallis ratios estimates some logarithmic formula expansion the maclaurin series derivation approximating () the main result cubo, a mathematical journal vol. 24, no. 02, pp. 239–261, august 2022 doi: 10.56754/0719-0646.2402.0239 on an a priori l∞ estimate for a class of monge-ampère type equations on compact almost hermitian manifolds masaya kawamura 1, b 1department of general education, national institute of technology, kagawa college 355, chokushi-cho, takamatsu, kagawa, japan, 761-8058. kawamura-m@t.kagawa-nct.ac.jp b abstract we investigate monge-ampère type equations on almost hermitian manifolds and show an a priori l∞ estimate for a smooth solution of these equations. resumen investigamos ecuaciones de tipo monge-ampère en variedades casi hermitianas y mostramos una estimación l∞ a priori para una solución suave de estas ecuaciones. keywords and phrases: monge-ampère type equation, almost hermitian manifold, chern connection. 2020 ams mathematics subject classification: 32q60, 53c15, 53c55. accepted: 7 april, 2022 received: 19 july, 2021 c©2022 m. kawamura. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2402.0239 mailto:kawamura-m@t.kagawa-nct.ac.jp https://orcid.org/0000-0003-1303-4237 mailto:kawamura-m@t.kagawa-nct.ac.jp 240 m. kawamura cubo 24, 2 (2022) 1 introduction let (m2n,j,ω) be a compact almost hermitian manifold of real dimension 2n with n ≥ 2. let χ be a smooth real (1,1)-form on m. we define for a function u ∈ c2(m), χu := χ + √ −1∂∂̄u and [χ] := {χu|u ∈ c2(m)}, [χ]+ := {χ′ ∈ [χ]|χ′ > 0}, h(m,χ) := {u ∈ c2(m)|χu > 0} and cα(ψ) := {[χ]|∃χ′ ∈ [χ]+,nχ′n−1 > (n − α)ψχ′n−α−1 ∧ ωα}. we consider the following fully nonlinear monge-ampère type equations, which are called the (n,n − α)-quotient equations for 1 ≤ α ≤ n: χnu = ψχ n−α u ∧ ωα with χu > 0, (1.1) where ψ is a smooth positive function. we will call a function u ∈ c2(m) admissible if it satisfies that u ∈ h(m,χ). when solutions u are admissible, the equations (1.1) are elliptic. since the equation (1.1) is invariant under the addition of constants to u, we may assume that u satisfies the normalized condition such that sup m u = 0. (1.2) w. sun has studied a class of fully nonlinear elliptic equations on closed hermitian manifolds and derived some a priori estimates for these equations (cf. [5, 6]). in [5], w. sun has proven a uniform a priori c∞ estimates of a smooth solution of the equation (1.1) and shown the existence of a solution of (1.1) on a closed hermitian manifold. in [12], j. zhang has shown that on a compact almost hermitian manifold (m2n,j,ω), if there exists an admissible c-subsolution and an admissible supersolution for the equation (1.1) for χ = ω, there exists a pair of (u,b) with b ∈ r such that u ∈ h(m,ω), supm u = 0, ωnu = ebψωn−α ∧ ωα for 1 ≤ α ≤ n on m. l. chen has studied a hessian equation with its structure as a combination of elementary symmetric functions on a closed kähler manifold and chen has provided a sufficient and necessary condition for the solvability of this equation in [1]. q. tu and n. xiang have investigated the dirichlet problem for a class of hessian type equation with its structure as a combination of elementary symmetric functions on a closed hermitian manifold with smooth boundary and they have derived a priori estimates for the complex mixed hessian equation in [9]. in this paper, we show that we have the a priori l∞ estimate for a smooth solution of the equation (1.1) on general almost hermitian manifolds. cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 241 theorem 1.1. let (m,j,ω) be a compact almost hermitian manifold of real dimension 2n with n ≥ 2 and u be a smooth admissible solution to (1.1). suppose that χ ∈ cα(ψ). then there is a uniform a priori l∞ estimate for u depending only on (m,j,ω), χ, ψ. this paper is organized as follows: in section 2, we recall some basic definitions and computations on an almost hermitian manifold (m,j,ω). in section 3, for an arbitrary chosen smooth function ϕ on m, we show the result that ∂∂∂̄ϕ and ∂̄∂∂̄ϕ depend only on the first derivative of ϕ and some geometric quantities of (m,j,ω). in section 4, we give a proof for theorem 1.1. notice that we assume the einstein convention omitting the symbol of sum over repeated indexes in all this paper. 2 preliminaries 2.1 the nijenhuis tensor of the almost complex structure let m be a 2n-dimensional smooth differentiable manifold. an almost complex structure on m is an endomorphism j of tm, j ∈ γ(end(tm)), satisfying j2 = −idt m , where tm is the real tangent vector bundle of m. the pair (m,j) is called an almost complex manifold. let (m,j) be an almost complex manifold. we define a bilinear map on c∞(m) for x,y ∈ γ(tm) by 4n(x,y ) := [jx,jy ] − j[jx,y ] − j[x,jy ] − [x,y ], (2.1) which is the nijenhuis tensor of j. the nijenhuis tensor n satisfies n(x,y ) = −n(y,x), n(jx,y ) = −jn(x,y ), n(x,jy ) = −jn(x,y ), n(jx,jy ) = −n(x,y ). for any (1,0)vector fields w and v , n(v,w) = −[v,w ](0,1), n(v,w̄) = n(v̄ ,w) = 0 and n(v̄ ,w̄) = −[v̄ ,w̄ ](1,0) since we have 4n(v,w) = −2([v,w ] + √ −1j[v,w ]), 4n(v̄ ,w̄) = −2([v̄ ,w̄] − √ −1j[v̄ ,w̄ ]). an almost complex structure j is called integrable if n = 0 on m. giving a complex structure to a differentiable manifold m is equivalent to giving an integrable almost complex structure to m (cf. [4]). a riemannian metric g on m is called j-invariant if j is compatible with g, i.e., for any x,y ∈ γ(tm), g(x,y ) = g(jx,jy ). in this case, the pair (j,g) is called an almost hermitian structure. the complexified tangent vector bundle is given by t cm = tm ⊗r c for the real tangent vector bundle tm. by extending j c-linearly and g c-bilinearly to t cm, they are also defined on t cm and we observe that the complexified tangent vector bundle t cm can be decomposed as t cm = t 1,0m⊕t 0,1m, where t 1,0m, t 0,1m are the eigenspaces of j corresponding to eigenvalues √ −1 and − √ −1, respectively: t 1,0 m = {x − √ −1jx ∣∣x ∈ tm}, t 0,1m = {x + √ −1jx ∣∣x ∈ tm}. (2.2) 242 m. kawamura cubo 24, 2 (2022) let λrm = ⊕ p+q=r λ p,qm for 0 ≤ r ≤ 2n denote the decomposition of complex differential r-forms into (p,q)-forms, where λp,qm = λp(λ1,0m) ⊗ λq(λ0,1m), λ1,0m = {η + √ −1jη ∣∣η ∈ λ1m}, λ0,1m = {η − √ −1jη ∣∣η ∈ λ1m} (2.3) and λ1m denotes the dual of t cm. let {zr} be a local (1,0)-frame on (m,j) with an almost hermitian metric g and let {ζr} be a local associated coframe with respect to {zr}, i.e., ζi(zj) = δij for i,j = 1, . . . ,n. since g is almost hermitian, its components satisfy gij = gīj̄ = 0 and gij̄ = gj̄i = ḡīj. using these local frame {zr} and coframe {ζr}, we have n(zī,zj̄) = −[zī,zj̄](1,0) =: nkīj̄zk, n(zi,zj) = −[zi,zj] (0,1) = nk īj̄ zk̄, and n = 1 2 nk īj̄ zk̄ ⊗ (ζi ∧ ζj) + 1 2 nk īj̄ zk ⊗ (ζī ∧ ζj̄). (2.4) let (m,j,g) be an almost hermitian manifold with dimr m = 2n. an affine connection d on t cm is called almost hermitian connection if dg = dj = 0. for the almost hermitian connection, we have the following lemma (cf. [10, 13]). lemma 2.1. let (m,j,g) be an almost hermitian manifold with dimr m = 2n. then for any given vector valued (1,1)-form θ = (θi)1≤i≤n, there exists a unique almost hermitian connection ∇ on (m,j,g) such that the (1,1)-part of the torsion is equal to the given θ. if the (1,1)-part of the torsion of an almost hermitian connection vanishes everywhere, then the connection is called the second canonical connection or the chern connection. we will refer the connection as the chern connection and denote it by ∇. now let ∇ be the chern connection on m. we denote the structure coefficients of lie bracket by [zi,zj] = b r ijzr + b r̄ ijzr̄, [zi,zj̄] = b r ij̄ zr + b r̄ ij̄ zr̄, [zī,zj̄] = b r īj̄ zr + b r̄ īj̄ zr̄. we have bkij = −bkji since [zi,zj] = −[zj,zi]. notice that j is integrable if and only if the br̄ij’s vanish. for any p-form ψ, there holds that dψ(x1, . . . ,xp+1) = p+1∑ i=1 (−1)i+1xi(ψ(x1, . . . ,x̂i, . . . ,xp+1)) + ∑ i 0, ∀ 1 ≤ i ≤ k} and 0 ≤ l < k ≤ n, 0 ≤ s < r, r ≤ k, s ≤ l, we have [ sk(λ) ck n sl(λ) cln ] 1 k−l ≤ [ sr(λ) cr n ss(λ) csn ] 1 r−s . (4.2) 250 m. kawamura cubo 24, 2 (2022) in this section, the positive constant c may be changed from line to line, but it depends on the allowed data. proof of theorem 1.1. it suffices to show the following key inequality: ∫ m |∂e− p 2 u|2gωn ≤ cp ∫ m e−puωn (4.3) for p large enough. lemma 4.2. let u be a smooth admissible solution to the monge-ampère type equation (1,1). then, there are uniform constants c, p0 such that for any p ≥ p0, we have the inequality (4.3). proof. without loss of generality, we may assume that nχn−1 > (n − α)ψχn−α−1 ∧ ωα, (4.4) and there exist uniform positive constants λ,λ > 0 such that λω ≤ χ ≤ λω. (4.5) as the local expression (4.1): χnu χ n−α u ∧ ωα = cαn sn(χu) sn−α(χu) = ψ, we locally have that c α n−1 sn−1(χu) sn−α−1(χu) = χn−1u χ n−α−1 u ∧ ωα and which implies that the following inequality nχn−1u > (n − α)ψχn−α−1u ∧ ωα (4.6) is equivalent to sn−1(χu) sn−α−1(χu) > sn(χu) sn−α(χu) (4.7) since we have locally that n − α n · ψ = n − α n · cαn sn(χu) sn−α(χu) = cαn−1 sn(χu) sn−α(χu) . note that we may apply lemma 4.1 to χu since χu > 0. applying the inequality (4.2), we have [ sn(χu) cn n sn−α(χu) c n−α n ] 1 α ≤ [ sn−1(χu) c n−1 n sn−α−1(χu) c n−α−1 n ] 1 α , which can be written by sn(χu) sn−α(χu) ≤ c n−α−1 n c n−1 n c n−α n sn−1(χu) sn−α−1(χu) = n − α n(α + 1) sn−1(χu) sn−α−1(χu) < sn−1(χu) sn−α−1(χu) , cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 251 where we used that n−α n(α+1) < 1. therefore, the inequality (4.7) holds and as a consequence, we have the inequality (4.6). we estimate that i := ∫ m e −pu((χnu − χn) − ψ(χn−αu ∧ ωα − χn−α ∧ ωα)) = ∫ m e−pu ( χnu χ n−α u ∧ ωα − χ n χn−α ∧ ωα ) χn−α ∧ ωα ≤ c ∫ m e−puωn. (4.8) on the other hand, we have that by stokes’ theorem, i = ∫ 1 0 ∫ m e −pu d dt (χntu − ψχn−αtu ∧ ωα)dt = ∫ 1 0 ∫ m e −pu √ −1∂∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt = ∫ 1 0 ∫ m d(e−pu √ −1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα))dt − ∫ 1 0 ∫ m √ −1∂e−pu ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt + ∫ 1 0 ∫ m √ −1e−pu∂̄u ∧ ∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt = p ∫ 1 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt −1 p ∫ 1 0 ∫ m √ −1∂̄e−pu ∧ ∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt = p ∫ 1 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt −1 p ∫ 1 0 ∫ m d( √ −1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα))dt + 1 p ∫ 1 0 ∫ m e −pu √ −1∂̄∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα))dt = p ∫ 1 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt −1 p ∫ 1 0 ∫ m e−pu √ −1∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt, (4.9) where we have used that d = a + ∂ + ∂̄ + ā, ∂̄(e−pu √ −1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0, a(e−pu √ −1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0, ā(e−pu √ −1∂̄u ∧ (nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0, ∂( √ −1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0, 252 m. kawamura cubo 24, 2 (2022) a( √ −1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0, ā( √ −1e−pu∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)) = 0 and from (2.9), ∂̄∂(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = −(∂∂̄ + aā + āa)(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = −∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) since we have a(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = ā(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = 0. we compute that for 0 ≤ t ≤ 1, −1 p ∫ m e−pu √ −1∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα) = −1 p ∫ m e−pu √ −1∂ ( n(n − 1)χn−2tu ∧ (∂̄χ + √ −1t∂̄∂∂̄u) − (n − α)∂̄ψ ∧ χn−α−1tu ∧ ωα −(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂̄χ + √ −1t∂̄∂∂̄u) ∧ ωα − α(n − α)ψχn−α−1tu ∧ ωα−1 ∧ ∂̄ω ) = −1 p ∫ m √ −1e−pu { n(n − 1)(n − 2)χn−3tu ∧ (∂χ + t √ −1∂∂∂̄u) ∧ (∂̄χ + t √ −1∂̄∂∂̄u) +n(n − 1)χn−2tu ∧ (∂∂̄χ + t √ −1∂∂̄∂∂̄u) − (n − α)∂∂̄ψ ∧ χn−α−1tu ∧ ωα +(n − α)(n − α − 1)∂̄ψ ∧ χn−α−2tu ∧ (∂χ + t √ −1∂∂∂̄u) ∧ ωα +α(n − α)∂̄ψ ∧ χn−α−1tu ∧ ωα−1 ∧ ∂ω −(n − α)(n − α − 1)∂ψ ∧ χn−α−2tu ∧ (∂̄χ + t √ −1∂̄∂∂̄u) ∧ ωα −(n − α)(n − α − 1)(n − α − 2)ψχn−α−3tu ∧ (∂χ + t √ −1∂∂∂̄u) ∧ (∂̄χ + t √ −1∂̄∂∂̄u) ∧ ωα −(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂∂̄χ + t √ −1∂∂̄∂∂̄u) ∧ ωα −α(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂̄χ + t √ −1∂̄∂∂̄u) ∧ ωα−1 ∧ ∂ω −α(n − α)∂ψ ∧ χn−α−1tu ∧ ωα−1 ∧ ∂̄ω −α(n − α)(n − α − 1)ψχn−α−2tu ∧ (∂χ + t √ −1∂∂∂̄u) ∧ ωα−1 ∧ ∂̄ω −α(n − α)(α − 1)ψχn−α−1tu ∧ ωα−2 ∧ ∂ω ∧ ∂̄ω −α(n − α)ψχn−α−1tu ∧ ωα−1 ∧ ∂∂̄ω } ≥ −c p ∫ m e−puχ n−3 tu ∧ ω3 − c p ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−3tu ∧ ω2 − c ∫ m e−puχ n−2 tu ∧ ω2 −c p ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2tu ∧ ω −c p ∫ m e−puχ n−α−1 tu ∧ ωα+1 − c p ∫ m e−puχ n−α−2 tu ∧ ωα+2 −c p ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−α−2tu ∧ ωα+1 − c p ∫ m e −pu χ n−α−3 tu ∧ ωα+3 − c p ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−α−3tu ∧ ωα+2, (4.10) cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 253 where we have used that for instance, by applying (3.12), ∫ m √ −1e−puχn−2tu ∧ t √ −1∂∂̄∂∂̄u = ∫ m √ −1e−puχn−2tu ∧ t √ −1(t1 ∗ ∂u + t2 ∗ ∂̄u) ≤ c ∫ m e −pu χ n−2 tu ∧ √ −1∂u ∧ ∂̄u ∧ ω + c ∫ m e −pu χ n−2 tu ∧ ω2, (4.11) ∫ m ∂̄ψ ∧ χn−α−2tu ∧ t √ −1∂∂∂̄u ∧ ωα = ∫ m ∂̄ψ ∧ χn−α−2tu ∧ t √ −1t3 ∗ ∂̄u ∧ ωα ≤ c ∫ m χ n−α−2 tu ∧ √ −1∂u ∧ ∂̄u ∧ ωα+1 + c ∫ m χ n−α−2 tu ∧ ωα+2, (4.12) ∫ m √ −1e−puχn−3tu ∧ ∂χ ∧ t √ −1∂̄∂∂̄u ∧ ω = ∫ m √ −1e−puχn−3tu ∧ ∂χ ∧ t √ −1t4 ∗ ∂u ∧ ω ≤ c ∫ m e−puχ n−3 tu ∧ √ −1∂u ∧ ∂̄u ∧ ω + c ∫ m e−puχ n−3 tu ∧ ω3. (4.13) since we have assumed that χ,χu > 0, then we have that χtu > 0 for any 0 ≤ t ≤ 1. now we introduce the following crucial inequalities (cf. [6]): lemma 4.3. for any 0 < t ≤ 1, 1 < l ≤ n, one has that l l − 1 ∫ t 0 ∫ m e−pu √ −1∂u∧∂̄u∧χl−1su ∧ωn−lds ≥ λ ∫ t 0 ∫ m e−pu √ −1∂u∧∂̄u∧χl−2su ∧ωn−l+1ds, (4.14) and for any 0 < t ≤ 1, 1 ≤ k ≤ n, one has that k + 1 k ∫ t 0 χksu ∧ ωn−kds ≥ λ ∫ t 0 χk−1su ∧ ωn−k+1ds, (4.15) where λ > 0 is the uniform constant in (4.5). proof. by using integration by parts and g̊arding’s inequality as in [6, (2.22)], we have that by using χ ≥ λω, ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds = ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−2su ∧ (χ + s √ −1∂∂̄u) ∧ ωn−lds ≥ λ ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−2su ∧ ωn−l+1ds + 1 l − 1 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ s d ds χl−1su ∧ ωn−lds ≥ λ ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−2su ∧ ωn−l+1ds − 1 l − 1 ∫ t 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds, (4.16) 254 m. kawamura cubo 24, 2 (2022) where we used that ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ s d ds χl−1su ∧ ωn−lds = t ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−1tu ∧ ωn−l − ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds ≥ − ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χl−1su ∧ ωn−lds. the inequality (4.16) gives the desired one (4.14). next we compute that by using integration by parts and g̊arding’s inequality as in [6, (3.7)], for 1 ≤ k ≤ n, using χ ≥ λω, ∫ t 0 χ k su ∧ ωn−kds = ∫ t 0 χ k−1 su ∧ (χ + s √ −1∂∂̄u) ∧ ωn−kds ≥ λ ∫ t 0 χ k−1 su ∧ ωn−k+1ds + 1 k ∫ t 0 s d ds (χksu ∧ ωn−k)ds = λ ∫ t 0 χ k−1 su ∧ ωn−k+1ds + t k χ k tu ∧ ωn−k − 1 k ∫ t 0 χ k su ∧ ωn−kds ≥ λ ∫ t 0 χ k−1 su ∧ ωn−k+1ds − 1 k ∫ t 0 χ k su ∧ ωn−kds, which implies the inequality (4.15). by applying these inequalities (4.14) and (4.15) for t = 1 to the estimate (4.10), we obtain that −1 p ∫ 1 0 ∫ m e −pu √ −1∂∂̄(nχn−1tu − (n − α)ψχn−α−1tu ∧ ωα)dt ≥ − c p ∫ 1 0 ∫ m e −pu χ n−1 tu ∧ ωdt − c p ∫ 1 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−1tu dt. (4.17) combining (4.17) with (4.9), we have that i ≥ p ∫ 1 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ {( n − c p2 ) χ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα } dt −c p ∫ 1 0 ∫ m e −pu χ n−1 tu ∧ ωdt. (4.18) by the concavity of hyperbolic polynomials, for 0 < τ < 1, 1 ≤ k ≤ n, we have (cf. [6, (2.13)]) 1 τ s 1 k k (χτtu) + ( 1 − 1 τ ) s 1 k k (χ) ≥ s 1 k k (χtu), which gives sk(χτtu) ≥ τksk(χtu). for τ = 1 2 , k = n − 1, we obtain that ∫ 1 0 ∫ m e−puχ n−1 tu ∧ ωdt ≤ 2n−1 ∫ 1 0 ∫ m e−puχ n−1 tu 2 ∧ ωdt = 2n ∫ 1 2 0 ∫ m e −pu χ n−1 tu ∧ ωdt. (4.19) cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 255 by combining (4.8), (4.18) and (4.19), we have that p ∫ 1 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ {( n − c p2 ) χ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα } dt ≤ c ∫ m e −pu ω n + c p ∫ 1 2 0 ∫ m e −pu χ n−1 tu ∧ ωdt. (4.20) since we have χtu > 0 and nχ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα > 0 for any 0 ≤ t ≤ 1, we can choose a sufficiently large p so that nχ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα − c p2 χ n−1 tu > 0. then we have that by the concavity of the quotient equation, for some 0 < δ < 1, we have (cf. [6, (3.10)]) nχ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα > n { 1 − 1 (1 + δ − tδ)α } χ n−1 tu , hence for sufficiently large p, ∫ 1 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ {( n − c p2 ) χ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα } dt ≥ ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ {( n − c p2 ) χ n−1 tu − (n − α)ψχn−α−1tu ∧ ωα } dt ≥ ∫ 1 2 0 n { 1 − c np2 − 1 (1 + δ − tδ)α } ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−1tu dt ≥ n { 1 − c np2 − 1 (1 + δ 2 )α } ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−1tu dt. (4.21) on the other hand, we compute by stokes’ theorem, 1 p ∫ 1 2 0 ∫ m e −pu χ n−1 tu ∧ ωdt = 1 p ∫ 1 2 0 ∫ t 0 d ds ( ∫ m e−puχn−1su ∧ ω ) dsdt + 1 2p ∫ m e−puχn−1 ∧ ω = n − 1 p ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂∂̄u ∧ χn−2su ∧ ωdsdt + 1 2p ∫ m e−puχn−1 ∧ ω = n − 1 p ∫ 1 2 0 ∫ t 0 ∫ m d(e−pu √ −1∂̄u ∧ χn−2su ∧ ω)dsdt −n − 1 p ∫ 1 2 0 ∫ t 0 ∫ m √ −1∂e−pu ∧ ∂̄u ∧ χn−2su ∧ ωdsdt + n − 1 p ∫ 1 2 0 ∫ t 0 ∫ m e −pu √ −1∂̄u ∧ ∂(χn−2su ∧ ω)dsdt + 1 2p ∫ m e −pu χ n−1 ∧ ω 256 m. kawamura cubo 24, 2 (2022) = (n − 1) ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt −n − 1 p2 ∫ 1 2 0 ∫ t 0 ∫ m √ −1∂̄e−pu ∧ ∂(χn−2su ∧ ω)dsdt + 1 2p ∫ m e −pu χ n−1 ∧ ω = (n − 1) ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt −n − 1 p2 ∫ 1 2 0 ∫ t 0 ∫ m d( √ −1e−pu∂(χn−2su ∧ ω))dsdt + n − 1 p2 ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂̄∂(χn−2su ∧ ω)dsdt + 1 2p ∫ m e−puχn−1 ∧ ω = (n − 1) ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt −n − 1 p2 ∫ 1 2 0 ∫ t 0 ∫ m e −pu √ −1∂∂̄(χn−2su ∧ ω)dsdt + 1 2p ∫ m e −pu χ n−1 ∧ ω, (4.22) where we used that as in the computation in (4.9), d(e−pu √ −1∂̄u∧χn−2su ∧ω) = (∂+∂̄+a+ā)(e−pu √ −1∂̄u∧χn−2su ∧ω) = ∂(e−pu √ −1∂̄u∧χn−2su ∧ω), d( √ −1e−pu ∧∂(χn−2su ∧ω)) = (∂+∂̄+a+ā)( √ −1e−pu∧∂(χn−2su ∧ω)) = ∂̄( √ −1e−pu ∧∂(χn−2su ∧ω)), and ∂̄∂(χn−2su ∧ ω) = −(∂∂̄ + aā + āa)(χn−2su ∧ ω) = −∂∂̄(χn−2su ∧ ω). applying (3.12), we estimate that as in (4.11)-(4.13) such as ∫ m √ −1e−puχn−3su ∧ s √ −1∂∂̄∂∂̄u ∧ ω ≤ c ∫ m e−puχn−3su ∧ √ −1∂u ∧ ∂̄u ∧ ω2 + c ∫ m e−puχn−3su ∧ ω3, (4.23) ∫ m e−puχn−4su ∧ s √ −1∂∂∂̄u ∧ ∂̄χ ∧ ω ≤ c ∫ m e −pu χ n−4 su ∧ √ −1∂u ∧ ∂̄u ∧ ω3 + c ∫ m e −pu χ n−4 su ∧ ω4, (4.24) ∫ m e−puχn−4su ∧ ∂χ ∧ s √ −1∂̄∂∂̄u ∧ ω2 ≤ c ∫ m e−puχn−4su ∧ √ −1∂u ∧ ∂̄u ∧ ω3 + c ∫ m e−puχn−4su ∧ ω4. (4.25) then we estimate that by applying these estimates (4.23)-(4.25) and the inequalities (4.14)-(4.15), n − 1 p2 ∫ t 0 ∫ m e−pu √ −1∂∂̄(χn−2su ∧ ω)ds = n − 1 p2 ∫ t 0 ∫ m e−pu √ −1∂((n − 2)χn−3su ∧ (∂̄χ + s √ −1∂̄∂∂̄u) ∧ ω)ds = n − 1 p2 ∫ t 0 ∫ m e−pu √ −1 { (n − 2)(n − 3)χn−4su ∧ (∂χ + s √ −1∂∂∂̄u) ∧ (∂̄χ + s √ −1∂̄∂∂̄u) ∧ ω +(n − 2)χn−3su ∧ (∂∂̄χ + s √ −1∂∂̄∂∂̄u) ∧ ω + (n − 2)χn−3su ∧ (∂̄χ + s √ −1∂̄∂∂̄u) ∧ ∂ω cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 257 +(n − 2)χn−3su ∧ (∂χ + s √ −1∂∂∂̄u) ∧ ∂̄ω + χn−2su ∧ ∂∂̄ω } ds ≤ c p2 ∫ t 0 ∫ m e −pu χ n−4 su ∧ ω4ds + c p2 ∫ t 0 ∫ m e −pu χ n−4 su ∧ √ −1∂u ∧ ∂̄u ∧ ω3ds + c p2 ∫ t 0 ∫ m e −pu χ n−3 su ∧ ω3ds + c p2 ∫ t 0 ∫ m e −pu χ n−3 su ∧ √ −1∂u ∧ ∂̄u ∧ ω2ds + c p2 ∫ t 0 ∫ m e−puχn−2su ∧ ω2ds ≤ c1 p2 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2su ∧ ωds + c2 p2 ∫ t 0 ∫ m e−puχn−2su ∧ ω2ds. (4.26) by choosing p sufficiently large such that c1 p2 < n − 1, c2 p < λ · n−1 n , by combining (4.22) with (4.26), and applying (4.15) for t = 1 2 , k = n − 1 such that ∫ 1 2 0 ∫ m e −pu χ n−2 tu ∧ ω2dt ≤ 1 λ · n n − 1 ∫ 1 2 0 ∫ m e −pu χ n−1 tu ∧ ωdt, we obtain that for 0 ≤ t ≤ 1 2 , 1 p ∫ 1 2 0 ∫ m e −pu χ n−1 tu ∧ ωdt ≤ (n − 1) ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt + 1 2p ∫ m e−puχn−1 ∧ ω + c1 p2 ∫ 1 2 0 ∫ t 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2su ∧ ωdsdt + c2 p2 ∫ 1 2 0 ∫ t 0 ∫ m e−puχn−2su ∧ ω2dsdt ≤ (n − 1) ∫ 1 2 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt + 1 2p · λ(n − 1) n ∫ 1 2 0 ∫ m e−puχ n−2 tu ∧ ω2dt + 1 2p ∫ m e−puχn−1 ∧ ω ≤ (n − 1) ∫ 1 2 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt + 1 2p ∫ 1 2 0 ∫ m e−puχ n−1 tu ∧ ωdt + 1 2p ∫ m e−puχn−1 ∧ ω (4.27) which implies that we have by applying (4.14) for t = 1 2 , l = n, 1 2p ∫ 1 2 0 ∫ m e −pu χ n−1 tu ∧ ωdt ≤ (n − 1) ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt + 1 2p ∫ m e−puχn−1 ∧ ω ≤ n λ ∫ 1 2 0 ∫ m e −pu √ −1∂u ∧ ∂̄u ∧ χn−1tu dt + 1 2p ∫ m e −pu χ n−1 ∧ ω. (4.28) therefore, by combining (4.28) with (4.20), (4.21), we obtain that [ np { 1 − c np2 − 1 (1 + δ 2 )α } − c 2n λ ] ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−1tu dt ≤ c ∫ m e −pu ω n + c p ∫ m e −pu χ n−1 ∧ ω ≤ c ∫ m e −pu ω n . (4.29) 258 m. kawamura cubo 24, 2 (2022) we choose p sufficiently large such that [ n { 1 − c np2 − 1 (1 + δ 2 )α } − c 2n λp ] > 0. by applying (4.14) for t = 1 2 repeatedly, we obtain ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−1tu dt ≥ λ n − 1 n ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−2tu ∧ ωdt ≥ λ2 n − 1 n n − 2 n − 1 ∫ 1 2 0 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ χn−3tu ∧ ω2dt · · · ≥ λ n−1 n 1 2 ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ ωn−1. (4.30) by combining (4.29) with (4.30), we finally obtain that for sufficiently large p, p ∫ m e−pu √ −1∂u ∧ ∂̄u ∧ ωn−1 ≤ c ∫ m e−puωn, which tells us that there exists a sufficiently large p0 such that for all p ≥ p0, the desired inequality (4.3) holds. the rest of the proof is similar to the ones in [7, 8]. in the following, we give a brief proof for reader’s convenience. we introduce the definition of gauduchon metrics on almost complex manifolds. definition 4.4. let (m2n,j) be an almost complex manifold. a metric g is called a gauduchon metric on m if g is an almost hermitian metric whose associated real (1,1)-form ω = √ −1gij̄ζi∧ζj̄ satisfies d∗(jd∗ω) = 0, where d∗ is the adjoint of d with respect to g, which is equivalent to d(jd(ωn−1)) = 0, or ∂∂̄(ωn−1) = 0. one has the following well-known result. proposition 4.5 (cf. [2, theorem 2.1], [3]). let (m2n,j,ω) be a compact almost hermitian manifold with n ≥ 2. then there exists a smooth function σ, unique up to addition of a constant, such that the conformal almost hermitian metric eσω is gauduchon. thanks to proposition 4.5, there exists a smooth function σ : m → r with supm σ = 0 such that ωg := e σω is gauduchon on m. lemma 4.6 (cf. [8, lemma 2.3]). let m be a compact almost complex manifold of real dimension 2n (n ≥ 2) with a gauduchon metric ωg. if φ is a smooth nonnegative function on m with ∆gφ ≥ −c0, where ∆g is the laplacian operator with respect to ωg, then there exists a positive constant c1, c2 depending only on (m,ωg) and c0 such that ∫ m |∂φ p+1 2 |2ωgω n g ≤ c1p ∫ m φpωng (4.31) for any p ≥ 1, and sup m φ ≤ c2 max { ∫ m φωng,1 } . (4.32) cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 259 proof. we compute for p ≥ 1, by stokes’ theorem, ∫ m |∂φ p+1 2 |2ωgω n g = n ∫ m √ −1∂φ p+1 2 ∧ ∂̄φ p+1 2 ∧ ωn−1g = n(p + 1)2 4 ∫ m √ −1φp−1∂φ ∧ ∂̄φ ∧ ωn−1 g = n(p + 1)2 4p ∫ m √ −1∂(φp) ∧ ∂̄φ ∧ ωn−1g = n(p + 1)2 4p ∫ m √ −1(∂ + ∂̄ + a + ā)(φp∂̄φ ∧ ωn−1g ) −n(p + 1) 2 4p ∫ m φp √ −1∂∂̄φ ∧ ωn−1 g + n(p + 1) 4p ∫ m √ −1∂̄(φp+1) ∧ ∂ωn−1 g = −(p + 1) 2 4p ∫ m φpn √ −1∂∂̄φ ∧ ωn−1g ωng ωng + n(p + 1) 4p ∫ m √ −1(∂ + ∂̄ + a + ā)(φp+1∂ωn−1g ) −n(p + 1) 4p ∫ m φp+1 √ −1∂̄∂ωn−1 g = (p + 1)2 4p ∫ m φ p(−∆gφ)ωng ≤ c1p ∫ m φ p ω n g, (4.33) where we used that (∂̄ + a + ā)(φp∂̄φ ∧ ωn−1g ) = 0, (∂ + a + ā)(φp+1∂ω n−1 g ) = 0, and that ∂̄∂ω n−1 g = −(∂∂̄ + aā + āa)ωn−1 g = −∂∂̄ωn−1 g = 0 since we have aωn−1 g = āωn−1 g = 0. we apply the sobolev inequality: for β := n n−1 > 1, and for any smooth function f, ( ∫ m f 2β ω n ) 1 β ≤ c ( ∫ m |∂f|2gωn + ∫ m f 2 ω n ) . (4.34) taking ω = ωg and f = φ q 2 , where we put q := p + 1, then for q ≥ 2, we have that ( ∫ m φqβωng ) 1 β ≤ cq max { ∫ m φqωng,1 } . by repeatedly replacing q by qβ and iterating, after setting q = 2, then we obtain that sup m φ ≤ c max {( ∫ m φ2ωng ) 1 2 ,1 } ≤ c max {( sup m φ ) 1 2 ( ∫ m φωng ) 1 2 ,1 } , which gives us the desired estimate (4.32). by applying the inequality (4.3) and the sobolev inequality (4.34), for any p ≥ p0, we obtain that ‖e−u‖lpβ ≤ c 1 p p 1 p ‖e−u‖lp, 260 m. kawamura cubo 24, 2 (2022) and by the standard iteration, we have that e−p0 infm u ≤ c ∫ m e−p0uωn. (4.35) we need the following lemma, whose proof goes in the same way as in the hermitian case. lemma 4.7 (cf. [7, lemma 3.2], [8, lemma 2.2]). let f be a smooth function on a compact almost hermitian manifold (m,j,ω). write dµ := ω n ∫ m ωn . if there exists a constant c1 such that e− infm f ≤ ec1 ∫ m e−fdµ, (4.36) then |{f ≤ inf m f + c1 + 1}| ≥ e−c1 4 , (4.37) where | · | denotes the volume of the set with respect to dµ. we apply lemma 4.6 to f = p0u, and then since we have the inequality (4.35), there exist uniform constants c, δ > 0 such that |{u ≤ inf m u + c}| ≥ δ. (4.38) now, we define φ := u − infm u. since it satisfies that ∆gφ = e−σ∆φ > −c, where ∆ is the laplacian operator with respect to ω, we may apply lemma 4.3 to the function φ. from the poincaré inequality and the estimate (4.31) with p = 1, we obtain that ‖φ − φ‖l2 ≤ c ( ∫ m |∂φ|2ωgω n g ) 1 2 ≤ c‖φ‖ 1 2 l1 , (4.39) where we put φ := 1∫ m ωn g ∫ m φωng. by making use of (4.38), the set s := {φ ≤ c} satisfies that |s|g ≥ δ, where | · |g denotes the volume of a set with respect to ωng. therefore, we obtain that δφ ≤ ∫ s φωng ≤ ∫ s (|φ − φ| + c)ωng ≤ ∫ m |φ − φ|ωng + c, which gives that by applying (4.39), ‖φ‖l1 ≤ c(‖φ − φ‖l1 + 1) ≤ c(‖φ − φ‖l2 + 1) ≤ c(‖φ‖ 1 2 l1 + 1). hence, φ is uniformly bounded in l1, and from (4.32) and (1.2), we obtain a uniform bound of u in the l∞ norm. acknowledgments this work was supported by jsps kakenhi grant number jp21k13798. cubo 24, 2 (2022) on an a priori l∞ estimate for a class of monge-ampère type... 261 references [1] l. chen, “hessian equations of krylov type on kähler manifolds”, preprint, arxiv:2107.12035v3, 2021. [2] j. chu, v. tosatti and b. weinkove, “the monge-ampère equation for non-integrable almost complex structures”, j. eur. math. soc., vol. 21, no. 7, pp. 1949–1984, 2019. [3] p. gauduchon, “le théorème de l’excentricité nulle”, c. r. acad. sci. paris sér. a-b, vol. 285, no. 5, pp. a387–a390, 1977. [4] a. newlander and l. nirenberg, “complex analytic coordinates in almost complex manifolds”, ann. of math. (2), vol. 65, pp. 391–404, 1957. [5] w. sun, “on a class of fully nonlinear elliptic equations on closed hermitian manifolds”, j. geom. anal., vol. 26, no. 3, pp. 2459–2473, 2016. [6] w. sun, “on a class of fully nonlinear elliptic equations on closed hermitian manifolds ii: l∞ estimate”, comm. pure appl. math., vol. 70, no. 1, pp. 172–199, 2017. [7] v. tosatti and b. weinkove, “estimates for the complex monge-ampère‘ equation on hermitian and balanced manifolds”, asian j. math., vol. 14, pp. 19–40, 2010. [8] v. tosatti and b. weinkove, “the complex monge-ampère equation on compact hermitian manifolds”, j. amer. math. soc., vol. 23, pp. 1187–1195, 2010. [9] q. tu and n. xiang, “the dirichlet problem for mixed hessian equations on hermitian manifolds”, preprint, arxiv:2201.05030v1, 2022. [10] l. vezzoni, “on hermitian curvature flow on almost complex manifolds”, differential geom. appl., vol. 29, no. 5, pp. 709–722, 2011. [11] c.-j. yu, “nonpositively curved almost hermitian metrics on products of compact almost complex manifolds”, acta math. sin., vol. 31, no. 1, pp. 61–70, 2015. [12] j. zhang, “monge-ampère type equations on almost hermitian manifolds”, preprint, arxiv:2101.00380, 2022. [13] t. zheng, “an almost complex chern-ricci flow”, j. geom. anal., vol. 28, no. 3, pp. 2129– 2165, 2018. https://arxiv.org/pdf/2107.12035.pdf https://arxiv.org/pdf/2201.05030.pdf https://arxiv.org/pdf/2101.00380.pdf introduction preliminaries the nijenhuis tensor of the almost complex structure the torsion and the curvature on almost complex manifolds some results for a smooth function on almost hermitian manifolds proof of theorem 1.1 cubo a mathematical journal vol.21, no¯ 01, (01–19). april 2019 http: // dx. doi. org/ 10. 4067/ s0719-06462019000100001 on algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds m.i.belishev and a.f.vakulenko saint-petersburg department of the steklov mathematical institute, st-petersburg state university, supported by the rfbr grant 18-01-00269. belishev@pdmi.ras.ru, vak@pdmi.ras.ru abstract let ω be a smooth compact oriented 3-dimensional riemannian manifold with boundary. a quaternion field is a pair q = {α, u} of a function α and a vector field u on ω. a field q is harmonic if α, u are continuous in ω and ∇α = rot u, div u = 0 holds into ω. the space q(ω) of harmonic fields is a subspace of the banach algebra c (ω) of continuous quaternion fields with the point-wise multiplication qq′ = {αα′ − u · u′, αu′ + α′u + u ∧ u′}. we prove a stone-weierstrass type theorem: the subalgebra ∨q(ω) generated by harmonic fields is dense in c (ω). some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided. comprehensive study of harmonic fields is motivated by possible applications to inverse problems of mathematical physics. resumen sea ω una variedad riemanniana 3-dimensional suave con borde, orientada y compacta. un campo cuaterniónico es un par q = {α, u} dado por una función α y un campo de vectores u en ω. un campo q es armónico si α, u son continuos en ω y ∇α = rot u, div u = 0 vale en todo ω. el espacio q(ω) de campos armónicos es un subespacio del álgebra de banach c (ω) de campos cuaterniónicos continuos con la multiplicación punto a punto qq′ = {αα′ − u · u′, αu′ + α′u + u ∧ u′}. probamos un teorema de tipo stone-weierstrass: la subálgebra ∨q(ω) generada por campos armónicos es densa en c (ω). se entregan también algunos resultados acerca de 2-jets de funciones armónicas y los conjuntos de unicidad campos armónicos. http://dx.doi.org/10.4067/s0719-06462019000100001 2 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) keywords and phrases: 3d quaternion harmonic fields, real uniform banach algebras, stoneweierstrass type theorem on density, uniqueness theorems. 2010 ams mathematics subject classification: 30f15, 35qxx, 46jxx. cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 3 1 introduction motivation there is an approach to inverse problems of mathematical physics (the so-called boundary control method), which was originally based on the relations between inverse problems and the boundary control theory [4, 7, 9]. the bc-method recovers riemannian manifolds via spectral and/or dynamical boundary data. later on, its version that makes use of connections with banach algebras, was proposed in [2, 5, 6]. the problem of recovering the manifold via its dn-map (the so-called impedance tomography problem) in dimensions > 3 isn’t yet properly solved. however, beginning from the papers [3, 10] it becomes clear that harmonic quaternion fields may play the key role in the 3d itp. it is the reason, which has stimulated the study of their properties [8, 11]. here we consider certain of algebraic and uniqueness properties of the harmonic quaternion fields with hope for their future application to itp [8]. in the mean time, our results may be of certain independent interest for functional analysis: namely, the real uniform banach algebras theory [1, 13, 15]. main result • let ω be a smooth compact oriented 3-dimensional riemannian manifold with boundary, tωx the tangent space at x ∈ ω, u · v and u ∧ v the inner and vector products in tωx. elements of the space hx := r ⊕ tωx (the pairs q = {α, u}) endowed with a multiplication qq′ = {αα′ − u · u′, αu′ +α′u+u∧u′} are said to be the geometric quaternions. as an algebra, hx is isometrically isomorphic to the quaternion algebra h. • a quaternion field is a pair q = {α, u} of a function α and vector field u on ω; in other words, q is an hx-valued function on the manifold. the space c(ω; h) of continuous quaternion fields endowed with the point-wise linear operations and multiplication, and the relevant sup-norm, is a real uniform banach algebra [1, 13, 15]. a field q = {α, u} ∈ c(ω; h) is harmonic if α, u are continuous in ω and ∇α = rot u, div u = 0 holds into ω. the space q(ω) of harmonic fields is a subspace of c(ω, h) (but not a subalgebra!). • let a be an algebra. for a set a ⊂ a by ∨a we denote the minimal subalgebra that contains a. the main result of the paper is a stone-weierstrass type theorem 1 which claims that ∨q(ω) is dense in c(ω; h). 4 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) more results and comments • in the course of proving theorem 1 we show that q(ω) (and, hence, ∨q(ω)) separates points of ω. it is quite evident for ω ⊂ r3 [11] but far from being evident for a 3d-manifold of arbitrary topology. the separation property is derived from the so-called h-controllability of ω from the boundary, which is much stronger than separability. the h-controllability is proved by the use of the results [18] on existence of the global green function and the landis type uniqueness theorems for the second order elliptic equations [16]. the key step in proving theorem 1 is to show that ∨q(ω) contains the algebra of scalar fields { {α, 0} | α ∈ cr(ω) } . the latter resembles the trick applied in [14]. • in sec 4 we prove that the 2-jets of harmonic functions are point-wise controllable from the boundary. the proof also makes use of the elliptic uniqueness theorems. then this result is applied to show that harmonic functions determine the riemannian structure of 3d manifold. as we hope, it is a step towards the main prospective goal: application to the 3d impedance tomography problem on riemannian manifolds. • one more result, which is of certain independent interest, is the following uniqueness property of harmonic quaternion fields (sec 5). if q ∈ q(ω) vanishes on a piece of a smooth surface then it vanishes in ω identically. • everywhere in the paper we deal with real functions, fields, spaces, etc. everywhere smooth means c∞-smooth. acknowledgements we’d like to thank dr c.shonkwiler for helpful remarks and useful references. 2 quaternion fields quaternions • let e be an oriented 3d euclidean space, u · v and u ∧ v the scalar (inner) and vector products, |u| = √ u · u. elements p = {α, u} of the space h := r⊕e endowed with the norm |p| = √ α2 + |u|2 and a (noncommutative) multiplication pp′ := {αα′ − u · u′, αu′ + α′u + u ∧ u′} , (2.1) are said to be geometric quaternions. the norm obeys |p2| = |p|2, cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 5 • let h be the algebra of (standard) quaternions. recall that it is the real algebra generated by 1, i, j, k with the unit 1 and multiplication defined by the table i 2 = j2 = k2 = −1, ij = k, jk = i, ki = j . • for an orthogonal normalized basis ε = {e1, e2, e3} in e, the correspondence e1 7→ i, e2 7→ j, e3 7→ k determines an isometric isomorphism µε : h → h, {α, ae1 + be2 + ce3} µε7→ α1 + ai + bj + ck , (2.2) (we write h ∼= h). any isometric isomorphism µ : h → h is of the form (2.2) by proper choice of the basis ε. vector analysis in the sequel, the following assumptions are accepted. convention 1. ω is a smooth compact oriented riemannian 3d-manifold with the smooth boundary ∂ω. it is endowed with the metric tensor g ∈ c2; dµ is the riemannian volume 3-form; ⋆ is the hodge operator. on such a manifold, the intrinsic operations of vector analysis are well defined on smooth functions and vector fields (sections of the tangent bundle tω). following [21], chapter 10, we recall their definitions. • for a vector field u, one defines the conjugate 1-form u♭ by u♭(v) = g(u, v), ∀v. for a 1-form f, the conjugate field f♭ is defined by g(f♭, u) = f(u), ∀u. • a scalar product: {fields} × {fields} ·→ {functions} is defined point-wise by u · v = g(u, v). a vector product: {fields} × {fields} ∧→ {fields} is defined point-wise by g(u ∧ v, w) = dµ (u, v, w), ∀w. • a gradient: {functions} ∇→ {fields} and a divergence: {fields} div→ {functions} are defined by ∇α = (dα)♭ and div u = ⋆ d⋆ u♭ respectively, where d is the exterior derivative. • a rotor: {fields} rot→ {fields} is defined by rot u = (⋆ d u♭)♭. recall the basic identities: div rot = 0 and rot ∇ = 0. the equalities ∇α = rot u and dα = ⋆ d u♭ are equivalent. • the laplacian {functions} ∆→ {functions} is ∆ = div ∇. the vector laplacian {fields} ~∆ → {fields} is ~∆ = ∇ div − rot rot . 6 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) remark 1. under the above accepted assumptions on the smoothness of ω and g, the (harmonic) functions and fields, which obey ∆α = 0 and ~∆u = 0 in the relevant weak sense, do belong to the class c2 loc : see, e.g, [12], part ii, chapter 1. fields let ω̇ := ω\∂ω be the set of the inner points, c(ω) and ~c(ω) the spaces of continuous functions and vector fields. let hx := r ⊕ tωx, x ∈ ω be the point-wise geometric quaternion algebras. • a quaternion field is a pair p = {α, u} with the components α ∈ c(ω) and u ∈ ~c(ω), the values p(x) = {α(x), u(x)} ∈ hx being regarded as geometric quaternions. by c(ω; h) we denote the space of continuous quaternion fields. one can regard them as sections of the bundle c(ω; h) = ∪x∈ωhx. • elements of the subspace q(ω) := { p ∈ c(ω; h) ∣ ∣ ∇α = rot u, div u = 0 in ω̇ } are called harmonic fields. to be rigorous, here the conditions on the components of p are understood in the relevant sense of distributions but imply ∆α = 0 and ~∆u = 0, so that α and u are automatically smooth enough by remark 1. 3 density theorem algebra c(ω; h) the space c(ω; h) with the point-wise multiplication (2.1) and the norm ‖p‖ = sup x∈ω |p(x)| = sup x∈ω √ |α(x)|2 + |u(x)|2 tωx satisfying ‖qp‖ 6 ‖q‖‖p‖, ‖p2‖ = ‖p‖2 is a real uniform noncommutative banach algebra. • the fields {α, 0} constitute a subalgebra c(ω; r) of c(ω; h), which is isometrically isomorphic to the real continuous function algebra on ω: c(ω; r) ∼= c r (ω) . (3.1) we say {α, 0} to be the scalar fields and often identify them with functions α via the map α 7→ {α, 0}, which embeds cr(ω) in c(ω; h). • the harmonic subspace q(ω) ⊂ c(ω; h) is not an algebra since, in general, p, q ∈ q(ω) does not imply pq ∈ q(ω). it is easy to see that q(ω) ∩ c(ω; r) = {{c, 0} | c is a constant function} , cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 7 whereas {1, 0} is the unit of c(ω; h). main result for an algebra a and a set s ⊂ a by ∨s we denote a minimal (sub)algebra in a , which contains s. our main results is the following. theorem 1. the algebra ∨q(ω) is dense in c(ω; h). the proof occupies the rest of sec 3. green function • a well-known in geometry fact is that the assumptions of convention 1, in particular, provide the existence of a compact 3-dimensional c∞manifold ω′ ⋑ ω endowed with the tensor g′ ∈ c2 such that g′|ω = g. this enables one to apply the results by m.mitrea and m.taylor [18] (existence of the fundamental solution, green function, poisson formula, etc) which are valid for much weaker smoothness restrictions on g and ∂ω. also, one can apply the results on the uniqueness of continuation of solutions to the elliptic pde [12, 16]. • the following results are mostly taken from [18]. also we use some well-known facts of the elliptic 2-nd order equations theory [17, 12, 16]. by wlp(ω) we denote the sobolev space of functions which possess the (generalized) derivatives of the order l = 1, 2, . . . belonging to lp(ω) (p > 1). recall that ω̇ = ω \ ∂ω. also we put d := {(x, y) ∈ ω × ω | x = y}. the distance in ω is denoted by rxy. let d(ω̇) be a space of the smooth compactly supported into ω functions (test functions) endowed with the standard topology, d ′(ω̇) the corresponding distributions. for an h ∈ l2(ω), the dirichlet problem ∆v = h in ω̇ v = 0 on ∂ω has a unique solution vh ∈ w22(ω) vanishing at the boundary. the solution is represented in the form vh(x) = ∫ ω g(x, y) h(y) dµ(y), x ∈ ω (3.2) via the green function g, which possesses the following properties. 1. g ∈ c2 loc ([ω × ω] \ d); g(x, y) = g(y, x), (x, y) 6∈ d; g(x, ·)|∂ω = 0, x ∈ ω̇ . (3.3) 8 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) for the closed sets k, k′ ⊂ ω provided k ∩ k′ = ∅ the map y 7→ g(·, y) is continuous from k to c2(k′). 2. the estimates g(x, y) 6 c rxy , |∇yg(x, y)| 6 c r2xy hold and imply g(x, ·) ∈ w1p(ω) for x ∈ ω, 1 6 p < 32 . 3. as a distribution of the class d ′(ω̇) on the test functions (of the variable y) of the class d(ω̇), the green function satisfies ∆yg(x, ·) = δx, (3.4) where δx is the dirac measure supported at x. note that in (3.4), and below in (3.8), (3.9), the variable x ∈ ω̇ plays the role of parameter. 4. for f ∈ c∞(∂ω), the inhomogeneous boundary value problem ∆w = 0 in ω̇ (3.5) w = f on ∂ω (3.6) has a unique classical solution w = wf(x), which is represented in the form wf(x) = ∫ ∂ω ∂νyg(x, y) f(y) dσ(y), x ∈ ω̇ , (3.7) where νy is the outward unit normal at the boundary, dσ is the boundary surface element. this is a poisson formula derived from (3.2) by integration by parts. function f in (3.6) is said to be a boundary control. • fix a point x ∈ ω̇ and a vector e ∈ tωx, |e| = 1. let γe be the geodesic that emanates from x in direction e. define a functional ∂xeδx ∈ d ′(ω̇) by 〈∂xeδx, ϕ〉 := lim γe∋ x′→x ϕ(x′) − ϕ(x) rxx′ = 〈 lim γe∋ x′→x δx′ − δx rxx′ , ϕ 〉 = e · ∇ϕ(x) . the relevant limit passage in (3.4) determines a derivative ∂xeg(x, ·) ∈ d ′(ω̇) which satisfies ∆y[∂ x eg(x, ·)] = ∂xeδx . (3.8) in the mean time, by the properties 1 and 2, ∂xeg(·, y) is a (classical) function belonging to lp(ω) for 1 6 p < 3 2 . moreover it is harmonic (and hence c2-smooth) in ω \ {x} and satisfies ∂xeg(x, ·)|∂ω = 0 , x ∈ ω̇. (3.9) • the relevant limit passage in the poisson formula (3.7) implies e · ∇wf(x) = ∫ ∂ω ∂νy [∂ x eg(x, y)] f(y) dσ(y), x ∈ ω̇ . (3.10) cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 9 h-controllability • the following result plays the key role in the proof of theorem 1. recall that hx = r⊕tωx ∼= h, and ω obeys convention 1. for a set of points a = {a1, . . . , an} ⊂ ω define a 4n-dimensional space ha := ⊕ ∑n i=1 hai and a map ma : c ∞(∂ω) → ha: f 7→ ⊕ n∑ i=1 {wf(ai), ∇wf(ai)} (each summand {wf(ai), ∇wf(ai)} belongs to the corresponding hai). we say ω to be hcontrollable from boundary if this map is surjective for any finite set a. lemma 1. the manifold ω is h-controllable from boundary. proof. the opposite means that ha⊖ran ma 6= {0}, i.e. there is a nonzero element ⊕ ∑n i=1 {αi, βiei} ∈ ha (αi, βi ∈ r, |ei| = 1) such that n∑ i=1 αiw f(ai) + βi ei · ∇wf(ai) = 0 (3.11) holds for all f ∈ c∞(∂ω). show that such an assumption leads to contradiction. 1. let a ⊂ ω̇, i.e., all ai are the interior points. a function φ(y) := n∑ i=1 αig(ai, y) + βi∂ x ei g(ai, y) (3.12) satisfies ∆φ = 0 in ω \ a (3.13) φ|∂ω = 0 (3.14) by (3.3), (3.4), (3.8), and (3.9). the relations (3.7), (3.10) and (3.11) easily follow to ∫ ∂ω ∂νφ(y) f(y) dσ(y) = 0 that implies ∂νφ|∂ω = 0 (3.15) by arbitrariness of f. 10 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) 2. so, φ is harmonic in ω \ a and has the zero cauchy data at the boundary: see (3.14) and (3.15). by the well-known uniqueness property of solutions to elliptic pde (see, e.g., [16], sec. 4.3, remark 4.17), we get φ = 0 in ω \ a, i.e., almost everywhere in ω. since g(ai, ·) ∈ w1p(ω) and ∂eig(ai, ·) ∈ lp(ω), we have φ ∈ lp(ω) for some p > 1. therefore, φ is a summable function equal zero a.e. in ω. thus, φ = 0 as a distribution of the class d ′(ω̇). in the mean time, by (3.4) and (3.8) one has ∆φ = n∑ i=1 αiδai + βi∂ x ei δai 6= 0 , i.e., φ is a nonzero element of d ′(ω̇). we arrive at the contradiction that proves the lemma for a ∈ ω̇. 3. let a contain the points of ∂ω. the smoothness assumptions on ω enable one to provide ω′, g′ obeying convention 1 and such that ω ⋐ ω′ and g′|ω = g holds. then one has a ⊂ ω̇′ that reduces this case to the previous one. note that relations between controllability and uniqueness theorems (like the one used in the proof) are widely exploited in control theory for pde (see, e.g., [9]). • recall that wf is a harmonic function that solves (3.5), (3.6). as immediate consequence of lemma 1 we have corollary 1. the algebra ∨ { |∇wf|2 | f ∈ c∞(ω) } is dense in cr(ω). indeed, by lemma 1, for any a, b ∈ ω there is a smooth f such that |∇wf(a)|2 6= |∇wf(b)|2, i.e., the functions |∇wf(·)|2 separate points of ω. in the mean time, by the same lemma, there is no x0 ∈ ω, at which all these functions vanish simultaneously. hence, by the classical stoneweierstrass theorem (see, e.g., [19]), the above mentioned density does hold. note that {0, ∇wf} ∈ q(ω) and {0, ∇wf}2 = −{|∇wf(·)|2, 0} ∈ ∨q(ω). hence, the algebra ∨ { {|∇wf|2, 0} | f ∈ c∞(ω) } is a subalgebra in ∨q(ω). by (3.1), corollary 1 implies that this algebra is dense in c(ω; r). as a result, denoting c := ∨q(ω) we arrive at the important relation c ⊃ c(ω; r) . (3.16) cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 11 strong separation we say that a family f ⊂ c(ω; h) strongly separates points (of ω) if for any a, b ∈ ω and ha ∈ ha, hb ∈ hb there is a p ∈ f such that p(a) = ha and p(b) = hb holds [13]. lemma 2. the space q(ω) strongly separates points. proof. • let ~l2(ω) be the space of square-integrable vector fields and h := {v ∈ ~l2(ω) | div v = 0, rot v = 0} its harmonic subspace. the well-known hodge-morrey-friedrichs decomposition claims that h = g ⊕ n = r ⊕ d , (3.17) where g := {v ∈ h | v = ∇α}, n := {v ∈ h | v · ν = 0} , r := {v ∈ h | v = rot u}, d := {v ∈ h | v ∧ ν = 0} . (see, e.g., [21], corollary 3.5.2). the subspaces n and d determined by the boundary conditions are called the neumann and dirichlet spaces respectively. their finite dimensions are equal to the betti numbers: dim n = β1, dim d = β2 [21]. note that n ∩ d = {0} [3, 21]. also note that dim g = dim r = ∞. • as a consequence of (3.17), a field v ∈ h is represented in the form v = ∇α = rot u if and only if v ∈ g ∩ r or, equivalently, v⊥[n +̇d]. if w = wf(x) solves (3.5), (3.6) then for any d ∈ d one has (∇wf, d) = ∫ ω ∇wf · d dµ = ∫ ∂ω f d·ν dσ . in the mean time, since ∇wf ∈ g , the representation ∇wf = rot u holds if and only if ∇wf⊥d, which is equivalent to ∫ ∂ω f d·ν dσ = 0 , d ∈ d. (3.18) in particular, taking f = 1 one has wf = 1 in ω and gets ∫ ∂ω d·ν dσ = 0 , d ∈ d. (3.19) • now, fix two distinct points a, b ∈ ω and elements ha = {ca, ka} ∈ ha, hb = {cb, kb} ∈ hb. to prove the lemma we need to show that there is a smooth f, which provides wf(a) = ca, w f (b) = cb; ∇wf = rot u; u(a) = ha, u(b) = hb . (3.20) 12 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) step 1. at first assume a, b ∈ ω̇. let px(y) := ∂νyg(x, y) be the poisson kernel. by (3.7) for f = 1 we have ∫ ∂ω px(y) dσ(y) = 1 , x ∈ ω . (3.21) in accordance with (3.7) and (3.18), to satisfy the relations wf(a) = ca, w f(b) = cb; ∇wf = rot u in (3.20) we need to find f provided ∫ ∂ω pa(y) f(y) dσ(y) = ca , ∫ ∂ω pb(y) f(y) dσ(y) = cb ; ∫ ∂ω f(y) d(y)·ν dσ(y) = 0 , d ∈ d, or, equivalently, (pa, f) = ca, (pb, f) = cb , f⊥ ν·d (3.22) (the inner products in l2(∂ω)), where ν · d := {ν · d | d ∈ d}. comparing (3.19) with (3.21), we conclude that neither pa nor pb belong to ν · d. in the mean time, pa 6= pb as elements of l2(∂ω). indeed, otherwise we’d have wf(a) = wf(b) for any f that is impossible by lemma 2. hence, span{pa, pb} ∩ ν · d may consist of {c(pa − pb) | c ∈ r} only. as a result, to proof the solvability of the linear system (3.22) (with respect to f) in the case of ca 6= cb we must show that pa − pb 6∈ ν · d. step 2. assume the opposite: there is a d ∈ d such that pa − pb = d · ν, and show that this assumption leads to a contradiction. compare the fields ∇[g(a, ·) − g(b, ·)] and d. since g(a, ·) = g(b, ·) = 0 on ∂ω both of them are normal on the boundary. hence, by the assumption, they are equal on ∂ω. in the mean time, the field ∇[g(a, ·) − g(b, ·)] is harmonic in ω̇ \ [{a} ∪ {b}], whereas d is harmonic in the whole ω̇. the coincidence at the boundary implies the coincidence in the domain of harmonicity. hence, ∇[g(a, ·) − g(b, ·)] can be extended by continuity to the whole ω and ∇[g(a, ·) − g(b, ·)] = d everywhere. however, the latter is impossible since div ∇[g(a, ·) − g(b, ·)] = ∆[g(a, ·) − g(b, ·)] = δa − δb , whereas div d = 0 everywhere in ω̇. this contradiction shows that pa − pb 6∈ ν · d. step 3. the case of a and/or b belonging to the boundary is reduced to the previous one by the collar theorem arguments, which were applied at the end of the proof of lemma 1. corollary 2. the algebra ∨q(ω) ⊂ c(ω; h) strongly separates points of ω. this property plays important role in proving density theorems [13]. cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 13 completing the proof of theorem 1 recall that c = ∨q(ω) and prove that c = c(ω; h). the fact, which will play the key role, is the embedding c ⊃ c(ω; r) ∼= cr(ω): see (3.16). • fix an x ∈ ω and choose the smooth boundary controls fx1, fx2, fx3 such that ∇wf x 1 (x), ∇wfx2 (x), ∇wfx3 (x) constitute a basis of tωx. it is possible owing to lemma 1. by continuity, there is a ball br(x)[x] ⊂ ω centered at x, of (small enough) radius r(x), such that ∇wf x 1 (y), ∇wfx2 (y), ∇wfx3 (y) is a basis of tωy for each y ∈ br(x)[x]. let such a choice be done for each x ∈ ω. • the balls provide an open cover ω = ∪x∈ωbr(x)[x]. by compactness there is a finite subcover ω = ∪nn=1brn[xn], where rn := r(xn). let η1, . . . , ηn be a partition of unit subordinated to the subcover, so that η1, . . . , ηn ∈ c∞(ω), supp ηn ⊂ brn[xn], n∑ n=1 ηn ≡ 1 in ω holds. • take p = {α, u} ∈ c(ω; h) and represent p = n∑ n=1 ηnp = { n∑ n=1 ηnα, n∑ n=1 ηnu} = n∑ n=1 {ηnα, 0} + n∑ n=1 {0, ηnu} with {ηnα, 0} ∈ c(ω; r) ⊂ c . in the mean time, one has ηnu = 3∑ k=1 κ n k ∇wf xn k with the certain κnk ∈ cr(ω) supported in brn[xn]. note that {κnk , 0} ∈ c(ω; r) ⊂ c . summarizing, we arrive at the representation p = n∑ n=1 {ηnα, 0} + n∑ n=1 3∑ k=1 {κnk , 0}{0, ∇wf xn k } , where all cofactors and summands do belong to c . thus p ∈ c and, hence, c(ω; h) = c . theorem 1 is proved. remark 2. analyzing the proof, it is easy to recognize that the family w := { {0, ∇wf} | f is smooth } , which is smaller than q(ω), also generates the whole of the continuous field algebra: ∨w = c(ω; h). 14 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) 4 controllability of 2-jets fix an a ∈ ω̇; let x1, x2, x3 be the local coordinates in a neighborhood ω ∋ a. with a smooth function φ one associates the row of its 0,1,2-order derivatives ja[φ] := {φ(a); φx1(a), φx2(a), φx3(a); φx1x1(a), φx1x2(a), φx1x3(a), φx2x2(a), φx2x3(a), φx3x3(a)} ∈ r10, which provides a coordinate representation of its second jet at the point a [20]. for short, we say ja[φ] to be a 2-jet of φ at a and consider r 10 with the (standard) inner product 〈j, j′〉 as a space of 2-jets. recall that in coordinates the laplacian acts by ∆φ = g− 1 2 [g 1 2 gikφxk]xi , where {gik} is the inverse to the metric tensor matrix {gik} and g = det{gik} (summation over repeating indexes is in the use). we say the row λa := = {0; g− 1 2 [g 1 2 gi1]xi, g − 1 2 [g 1 2 gi2]xi, g − 1 2 [g 1 2 gi3]xi; g 11, 2g12, 2g13, g22, 2g23, g33} ∣ ∣ x=a to be the laplace jet and represent (∆φ)(a) = 〈λa, ja[φ]〉. the harmonicity ∆w = 0 is equivalent to the orthogonality 〈ja[w], λa〉 = 0, a ∈ ω. therefore one has ja[w] ∈ r10 ⊖ span λa. let us show that the 2-jets of harmonic functions exhaust the subspace r10 ⊖ span λa. this result may be interpreted as a point-wise boundary controllability of 2-jets by harmonic functions. recall that wf is a solution to (3.5), (3.6). lemma 3. for any a ∈ ω and s ∈ r10 ⊖ span λa there is a smooth f such that ja[wf] = s. proof. taking into account the structure of the laplace jet, we may deal with s = {0; s1, s2, s3; s11, . . . , s33}, and let it be such that 0 6= s ∈ r10 ⊖ span λa but 〈s, ja[wf]〉 = 0 for any smooth f. show that such an assumption leads to contradiction. • for a differential operator l with smooth coefficients in ω, by l∗ we denote its adjoint by lagrange that is defined by (lη, ζ)l2(ω) = (η, l ∗ζ)l2(ω), η, ζ ∈ d(ω̇) . for a distribution h ∈ d ′(ω̇) one defines lh by (lh, η) := (h, l∗η)l2(ω), η ∈ d(ω̇). cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 15 let s be a differential operator, which acts by (sv)(x) = = [s1vx1 + s2vx2 + s3vx3 + s11vx1x1 + s12vx1x2 + · · · + s33vx3x3] (x) = = 〈s, jx[v]〉, x ∈ ω in a coordinate neighborhood ω of a ∈ ω̇, where the (constant) coefficients are the components of the above chosen jet s. • let δa ∈ d ′(ω̇) be the dirac measure supported at the point a ∈ ω̇. consider the problem ∆h = s∗δa (4.1) h ∣ ∣ ∂ω = 0 . (4.2) the equation is understood as a relation in d ′(ω̇); its r.h.s. is a distribution acting by (s∗δa, η)l2(ω) = (sη)(a). the boundary condition does make sense since h is harmonic outside supp s∗δa = {a}. also, the normal derivative ∂νh is a smooth function on ∂ω. formally by green, for a function v ∈ c2(ω) one has 〈s, ja[v]〉 = (sv)(a) = ∫ ω δa sv dµ = ∫ ω s∗δa v dµ (4.1) = ∫ ω ∆h v dµ = (4.2) = ∫ ω h ∆v dµ + ∫ ∂ω ∂νh v dσ . to justify the final equality 〈s, ja[v]〉 = ∫ ω h ∆v dµ + ∫ ∂ω ∂νh v dσ (4.3) one can use the standard regularization technique, approximating δa by δ ε a ∈ d(ω̇) supported near a. • by the choice of s, for v = wf the equality (4.3) provides ∫ ∂ω ∂νh w f dσ = ∫ ∂ω ∂νh f dσ = 0 . by arbitrariness of f we get ∂νh = 0 on ∂ω. so, h is harmonic in ω\{a} and has the zero cauchy data on the boundary. by the uniqueness theorem, h vanishes everywhere outside a. hence, the distribution h is supported at a. the well-known fact of the distribution theory is that such an h is a linear combination of δa and its derivatives. in the mean time, comparing the orders of singularities in the left and right hand sides of (4.1), one easily concludes that h = cδa 16 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) with c = const 6= 0. indeed, otherwise ∆h contains the derivatives of δa of the order > 3 that makes the equality (4.1) impossible. for an η ∈ d(ω̇) one has 〈s, ja[η]〉 = (δa, sη) = (s∗δa, η) (4.1) = (∆cδa, η) = (cδa, ∆η) = 〈cλa, ja[η]〉 . comparing the beginning with the end and referring to the evident {ja[η] | η ∈ d(ω̇)} = r10, we arrive at s = cλa that contradicts to the starting assumption s⊥λa. • the case a ∈ ∂ω is reduced to the previous one by means of the trick already used at the end of the proof of lemma 1: embedding ω ⋐ ω′. as is easy to recognize, lemma 3 implies the assertion of lemma 1 for the case of the single point a. however, lemma 3 may be generalized on the finite set a1, . . . , an so that the relevant boundary controllability of 2-jets of harmonic functions holds up to the natural defect in ⊕ ∑ i r 10 ai . determination of metric from harmonic functions the metric on ω determines the family of harmonic functions. the converse is also true in the following sense. • let c > 0 be a smooth function on ω and cg a conformal deformation of the metric g. by ∆cg and ∆g we denote the corresponding laplacians. a simple calculation leads to the relation ∆cgy = c −1∆gy − 2 −1 ∇c−1 · ∇y , (4.4) which is specific for the 3d case. taking y = wf, we see that the metrics cg and g have the same reserve of harmonic functions wf if and only if ∇c−1 · ∇wf = 0 holds for any smooth f. in the mean time, by lemma 1 the gradients ∇wf = 0 constitute the local bases in ω. hence, the latter equality implies ∇c−1 = 0, i.e., c = const. • fix a point a in a coordinate neighborhood ω ∋ a. by λga we denote the laplace jet of the given metric g. by lemma 3, the space of jets is r 10 a = {ja[φ] | φ is smooth} = {ja[w f] | f is smooth} ⊕ span λga . (4.5) therefore, writing (∆wf)(a) = 0 in the form 〈λga, ja[wf]〉 = 0, f is smooth and varying f = f1, f2, . . . , we get a linear homogeneous algebraic system with respect to the components of the jet λ g a, which determines them up to a factor, which may depend on a. along cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 17 with the components, we determine the tensor g up to a factor, possibly depending on a. however, by the above mentioned geometric reasons, this factor is a constant. thus, the family {wf | f is smooth} determines the metric g up to a constant positive factor. if g is known at least at a single point x0 ∈ ω, then it is uniquely determined everywhere. notice in addition that in two-dimensional case relation (4.4) is of the form ∆cgy = c −1∆gy, so that the metrics cg and g determine the same reserve of harmonic functions. it is the reason, because of which in 2d impedance tomography problem the metric is recovered up to conformal equivalence [2]. • here we describe a trick, which is used in dynamical/spectral inverse problems and 2d impedance tomography problem, for recovering the metric via boundary data[9]. the hope is that it may be useful in future investigation of 3d itp. assume that a topological space ω̃ is homeomorphic to ω via a homeomorphism β : ω → ω̃. also assume that the family of functions {w̃f = wf ◦ β−1 | f is smooth} is given. the following procedure enables one to determine the metric g̃ = β∗g in ω̃. 1. fix a point a ∈ ω̃ and choose its neighborhood ω̃ with the coordinates x1, x2, x3. by the way, lemma 1 enables one to use the images w̃f as local coordinates. 2. find span λ g̃ a by (4.5) (replacing functions w f on ω with w̃f on ω̃). as was shown above, the family of these subspaces given for a ∈ ω̃ determines the metric up to a constant factor. so, cg̃ is recovered. assuming g̃ to be known at least at a single point a0 ∈ ω̃, one recovers g̃ uniquely. 3. covering ω̃ by the coordinate neighborhoods and repeating the previous steps, we determine g̃ in ω̃. 5 uniqueness properties of harmonic fields roughly speaking, the following result means that the set of zeros of a harmonic quaternion field may be at most of dimension 1. lemma 4. let σ ∈ ω be a c2-smooth surface (2-dim submanifold). if p ∈ q(ω) obeys p|σ = 0 then p = 0 in the whole ω. proof. since the claimed result is of local character, we assume σ to be a both-side surface endowed with a smooth field of the unit normals ν. also, σ possesses the (induced) riemannian metric and 18 m.i.belishev and a.f.vakulenko cubo 21, 1 (2019) is provided with the corresponding operations on vector fields. in particular, a divergence, which is denoted by divς, is well defined. • for a point x ∈ σ and vector v ∈ tωx we represent v = vθ + vν : vν = v · ν ν, vθ = v − vν and, by default, identify vθ with the proper vector of tσx. by the latter, for a smooth vector field v given in a neighborhood of σ, the value [divς vθ](x) is of clear meaning. also, recall the well-known vectot analysis relation ν · rot v = divς ν ∧ vθ on σ (5.1) (see, e.g. [21]). • begin with the case σ ⊂ ω̇. let p = {α, u} ∈ q(ω), so that ∇α = rot u, div u = 0 in ω̇ (5.2) holds. let p|σ = 0. since α|σ = 0, we have (∇α)θ|σ = 0 that implies (rot u)θ ∣ ∣ σ = 0 by (5.2). in the mean time, u|σ = 0 is equivalent to uθ = uν = 0 on σ; hence (rot u)ν|σ = divς ν ∧ uθ = 0 by virtue of (5.1). thus we get (rot u)θ|σ = (rot u)ν|σ = 0, i.e. rot u|σ = 0. the latter equality and (5.2) lead to (∇α)|σ = 0 (along with α|σ = 0). so, α is a harmonic function with the zero cauchy data on σ. therefore α = 0 in ω by the elliptic uniqueness theorems [16]. as a result, rot u = ∇α = 0 everywhere in ω. since div u = 0, the vector field u is harmonic in ω and vanishes on σ. therefore, locally near the points x ∈ σ one represents u = ∇ϕ with a harmonic function ϕ provided ∇ϕ|σ = 0. such a function is a constant; hence u = 0 near σ. by its harmonicity, u vanishes globally in ω. so, we have p = 0 in ω. • the case σ ⊂ ∂ω is reduced to the previous one by means of the trick already used at the end of the proof of lemma 1: embedding ω ⋐ ω′. cubo 21, 1 (2019) on algebraic and uniqueness properties of harmonic . . . 19 references [1] m.abel and k.jarosz. noncommutative uniform algebras. studia mathematica, 162 (3) (2004), 213–218. 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[21] g.schwarz. hodge decomposition a method for solving boundary value problems. lecture notes in math., 1607. springer–verlag, berlin, 1995. introduction quaternion fields density theorem controllability of 2-jets uniqueness properties of harmonic fields cubo, a mathematical journal vol.22, n◦02, (215–231). august 2020 http://dx.doi.org/10.4067/s0719-06462020000200215 received: 12 march, 2019 | accepted: 14 july, 2020 d-metric spaces and composition operators between hyperbolic weighted family of function spaces a. kamal 1 and t.i.yassen 2 1 port said university, faculty of science, department of mathematics and computer science, port said, egypt. 2 the higher engineering institute in al-minya (est-minya) minya , egypt. alaa mohamed1@yahoo.com, taha hmour@yahoo.com abstract the aim of this paper is to introduce new hyperbolic classes of functions, which will be called b∗α, log and f ∗ log(p, q, s) classes. furthermore, we introduce d-metrics space in the hyperbolic type classes b∗α, log and f ∗ log(p, q, s). these classes are shown to be complete metric spaces with respect to the corresponding metrics. moreover, necessary and sufficient conditions are given for the composition operator cφ to be bounded and compact from b∗α, log to f ∗ log(p, q, s) spaces. resumen el objetivo de este art́ıculo es introducir nuevas clases hiperbólicas de funciones, que serán llamadas clases b∗α, log y f ∗ log(p, q, s). a continuación, introducimos d-espacios métricos en las clases de tipo hiperbólicas b∗α, log y f ∗ log(p, q, s). mostramos que estas clases son espacios métricos completos con respecto a las métricas correspondientes. más aún, damos condiciones necesarias y suficientes para que el operador composición cφ sea acotado y compacto desde el espacio b ∗ α, log a f ∗ log(p, q, s). keywords and phrases: d-metric spaces, logarithmic hyperbolic classes, composition operators. 2020 ams mathematics subject classification: 47b38, 46e15. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000200215 216 a. kamal & t.i.yassen cubo 22, 2 (2020) 1 introduction let φ be an analytic self-map of the open unit disk d = {z ∈ c : |z| < 1} in the complex plane c and let ∂d be its boundary. let h(d) denote the space of all analytic functions in d and let b(d) be the subset of h(d) consisting of those f ∈ h(d) for which |f(z)| < 1 for all z ∈ d. let the green’s function of d be defined as g(z, a) = log 1 |ϕa(z)| , where ϕa(z) = a−z 1−āz is the möbius transformation related to the point a ∈ d. a linear composition operator cφ is defined by cφ(f) = (f ◦ φ) for f in the set h(d) of analyticfunctions on d (see [9]). a function f ∈ b(d) belongs to α-bloch space bα, 0 < α < ∞, if ||f‖bα = sup z∈d (1 − |z|)α|f′(z)| < ∞. the little α-bloch space bα, 0 consisting of all f ∈ bα so that lim |z|→1− (1 − |z|2)|f′(z)| = 0. definition 1. [15] for an analytic function f on d and 0 < α < ∞, if ||f||bα log = sup z∈d (1 − |z|2)α|f′(z)| ( log 2 1 − |z|2 ) < ∞, then, f belongs to the weighted α-bloch spaces bαlog. if α = 1, the weighted bloch space blog is the set for all analytic functions f in d for which ||f||blog < ∞. the expression ||f||blog defines a seminorm while the norm is defined by ||f||blog = |f(0)| + ||f||blog. definition 2. [14] for 0 < p, s < ∞, −2 < q < ∞ and q + s > −1, a function f ∈ h(d) is in f(p, q, s), if sup a∈d ∫ d |f′(z)|p(1 − |z|2)qgs(z, a)da(z) < ∞. moreover, if lim |a|→1− ∫ d |f′(z)|p(1 − |z|2)qgs(z, a)da(z) = 0, then f ∈ f0(p, q, s). el-sayed and bakhit [5] gave the following definition: cubo 22, 2 (2020) d-metric spaces and composition operators 217 definition 3. for 0 < p, s < ∞, −2 < q < ∞ and q + s > −1, a function f ∈ h(d) is said to belong to flog(p, q, s), if sup i⊂∂d ( log 2 |i| )p |i|s ∫ s(i) |f′(z)|p(1 − |z|2)q ( log 1 |z| )s da(z) < ∞. where |i| denotes the arc length of i ⊂ ∂d and s(i) is the carleson box defined by (see [8, 6]) s(i) = {z ∈ d : 1 − |i| < |z| < 1, z |z| ∈ |i|}. the interest in the flog(p, q, s)-spaces rises from the fact that they cover some well known function spaces. it is immediate that flog(2, 0, 1) = bmoalog and flog(2, 0, p) = q p log, where 0 < p < ∞. 2 preliminaries definition 4. [11] the hyperbolic bloch space b∗α is defined as b∗α = {f : f ∈ b(d) and sup z∈d (1 − |z| 2 )αf∗(z) < ∞}. denoting f∗(z) = |f′(z)| 1−|f(z)|2 , the hyperbolic derivative of f ∈ b(d). [7] the little hyperbolic bloch space b∗α, 0 is a subspace of b ∗ α consisting of all f ∈ b ∗ α so that lim |z|→1− (1 − |z|2)αf∗(z) = 0. the space b∗α is banach space with the norm defined as ||f||b∗ α = |f(0)| + sup z∈d (1 − |z|)α|f∗(z)|. definition 5. for 0 < p, s < ∞, −2 < q < ∞, α = q+2 p and q + s > −1, a function f ∈ h(d) is said to belong to f ∗(p, q, s), if sup a∈d ∫ d (f∗(z))p(1 − |z|2)αp−2gs(z, a)da(z) < ∞. definition 6. for f ∈ b(d) and 0 < α < ∞, if ||f||b∗ α, log = sup z∈d (1 − |z|2)α(f∗(z)) ( log 2 1 − |z|2 ) < ∞, then f belongs to the b∗α, log. 218 a. kamal & t.i.yassen cubo 22, 2 (2020) we must consider the following lemmas in our study: lemma 2.1. [12] let 0 < r ≤ t ≤ 1, then log 1 t ≤ 1 r (1 − t2) lemma 2.2. [12] let 0 ≤ k1 < ∞, 0 ≤ k2 < ∞, and k1 − k2 > −1, then c(k1, k2) = ∫ d ( log 1 |z| )k1 (1 − |z|2)−k2da(z) < ∞. to study composition operators on b∗α, log and f ∗ log(p, q, s) spaces, we need to prove the following result: theorem 1. if 0 −1. then the following are equivalent: (a) f ∈ b∗α, log. (b) f ∈ f ∗log(p, q, s). (c) sup a∈d ( log 2 1−|a|2 )p ∫ d (f∗(z))p(1 − |z|2)αp−2(1 − |ϕ(z)|2)sda(z) < ∞, (d) sup a∈d ( log 2 1−|a|2 )p ∫ d (f∗(z))p(1 − |z|2|αp−2gs(z, a)da(z) < ∞. proof. let 0 < p < ∞, −2 < q < ∞, 1 < s < ∞ and 0 < r < 1. by subharmonicity we have for an analytic function g ∈ d that |g(0)|p ≤ 1 πr2 ∫ d(0,r) |g(w)|pda(w). for a ∈ d, the substitution z = ϕa(z) results in jacobian change in measure given by da(w) = |ϕ′a(z)| 2 da(z). for a lebesgue integrable or a non-negative lebesgue measurable function f on d, we thus have the following change of variable formula: ∫ d(0,r) f(ϕa(w))da(w) = ∫ d(a,r) f(z)|ϕ′a(z)| 2da(z). let g = f ′◦ϕa 1−|f◦ϕa|2 then we have ( |f′(a)| 1 − |f(a)|2 )p = (f∗(a))p ≤ 1 πr2 ∫ d(0,r) ( |f′(ϕa(w))| 1 − |f(ϕa(w))|2 )p da(w) = 1 πr2 ∫ d(a,r) (f∗(z))p|ϕ′a(z)| 2da(z). cubo 22, 2 (2020) d-metric spaces and composition operators 219 since |ϕ′a(z)| = 1 − |ϕa(z)| 2 1 − |z|2 , and 1 − |ϕa(z)| 2 1 − |z|2 ≤ 4 1 − |a|2 a, z ∈ d. so we obtain that (f∗(a))p ≤ 16 πr2(1 − |a|2)2 ∫ d(a,r) (f∗(z))pda(z). again f ∈ b∗α, log, and (1 − |z| 2)2 ≈ (1 − |a|2)2 ≈ d(a, r), for z ∈ d(a, r). thus, we have ( log 2 1−|a|2 )p (f∗(a))p(1 − |a|2)αp ≤ 16 πr2(1 − |a|2)2−αp × ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))pda(z) ≤ 16 πr2 × ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))p(1 − |z|2)αp−2da(z) ≤ 16 πr2 × ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))p(1 − |z|2)αp−2 × ( 1 − |ϕa(z)| 2 1 − |ϕa(z)|2 )s da(z) ≤ 16 πr2(1 − r2)s × ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ m(r) × ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))p(1 − |z|2)αp−2(1 − |ϕ′a(z)| 2)sda(z). where m(r) is a constant depending on r. thus, the quantity (a) is less than or equal to constant times the quantity (c). from the fact (1 − |ϕa(z)| 2) ≤ 2 log 1 |ϕa(z)| = 2g(z, a) for a, z ∈ d, we have ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ ( log 2 1 − |a|2 )p∫ d(a,r) (f∗(z))p(1 − |z|2)αp−2gs(z, a)da(z). hence, the quantity (c) is less than or equal to a constant times (d). by taking α = q+2 p , it follows f ∈ f ∗log(p, q, s). thus, the quantity (c) is less than or equal to a constant times the quantity (b). 220 a. kamal & t.i.yassen cubo 22, 2 (2020) finally, from the following inequality, let z = ϕa(w) then w = ϕa(z). hence, ( log 2 1 − |a|2 )p∫ d (f∗(ϕa(w))) p(1 − |ϕa(w)| 2)αp−2 ( log 1 |w| )s |ϕ′a(w)| 2 da(w) = ( log 2 1 − |a|2 )p∫ d (f∗(ϕa(w))) p(1 − |ϕa(w)| 2)αp ( log 1 |w| )s |ϕ′a(w)| 2 (1 − |ϕa(w)|2)2 da(w) = ( log 2 1 − |a|2 )p∫ d (f∗(ϕa(w))) p(1 − |ϕa(w)| 2)αp ( log 1 |w| )s 1 (1 − |w|2)2 da(w) ≤ ||f|| p b∗ α, log ( log 2 1 − |a|2 )p∫ d ( log 1 |w| )s (1 − |w|2)−2da(w) = c(s, 2)||f|| p b∗ α, log . by lemma 2.2, c(s, 2) = ∫ d ( log 1 |w| )s (1 − |w|2)−2da(w) < ∞, for 1 < s < ∞. thus, the quantity (d) is less than or equal to a constant times the quantity (a). hence, it is proved. let us we give the following equivalent definition for f ∗log(p, q, s). definition 7. for 0 < p, s < ∞, −2 < q < ∞, α = q+2 p and q + s > −1, a function f ∈ h(d) is said to belong to f ∗log(p, q, s), if sup a∈d ( log 2 1 − |a|2 )p∫ d (f∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ∞. definition 8. a composition operator cφ : b ∗ α, log → f ∗ log(p, q, s) is said to be bounded if there is a positive constant c so that ||cφf||f ∗ log (p, q, s) ≤ c||f||b∗ α, log for all f ∈ b∗p, α. definition 9. a composition operator cφ : b ∗ α, log → f ∗ log(p, q, s) is said to be compact if it maps any ball in b∗p, α onto a precompact set in f ∗(p, q, s). the following lemma follows by standard arguments similar to those outline in [13]. hence, we omit the proof. lemma 2.3. assume φ is a holomorphic mapping from d into itself. let 0 < p, s, α < ∞, −2 < q < ∞, then cφ : b ∗ α, log → f ∗ log(p, q, s) is compact if and only if for any bounded sequence {fn}n∈n ∈ b ∗ α, log which converges to zero uniformly on compact subsets of d as n → ∞ we have lim n→∞ ||cφfn||f ∗ log (p,q,s) = 0. 3 d-metric space topological properties of generalized metric space called dmetric space was introduced in [1], see for example, ([2] and [3]). this structure of d-metric space is quite different from a 2-metric space and natural generalization of an ordinary metric space in some sense. cubo 22, 2 (2020) d-metric spaces and composition operators 221 definition 10. [4] let x denote a nonempty set and r the set of real numbers. a function d : x×x×x → r is said to be a d-metric on x if it satisfies the following properties: (i) d(x, y, z) ≥ 0 for all x, y, z ∈ x and equality holds if and only if x = y = z (nonnegativity), (ii) d(x, y, z) = d(x, z, y) = · · ·· (symmetry), (iii) d(x, y, z) ≤ d(x, y, a) + d(x, a, z) + d(a, y, z) for all x, y, z, a ∈ x (tetrahedral inequality). a nonempty set x together with a d-metric d is called a d-metric space and is represented by (x, d). the generalization of a d-metric space with d-metric as a function of n variables is provided in dhage [2]. example1.1: [4] let (x, d) be an ordinary metric space and define a function d1 on x 3 by d1(x, y, z) = max{d(x, y), d(y, z), d(z, x)}, for all x, y, z ∈ x. then, the function d1 is a d-metric on x and (x, d1) is a d-metric space. example1.2: [4] let (x, d) be an ordinary metric space and define a function d2 on x 3 by d2(x, y, z) = d(x, y) + d(y, z) + d(z, x) for x, y, z ∈ x. then, d2 is a metric on x and (x, d2) is a d-metric space. remark 1. geometrically, the d-metric d1 represents the diameter of a set consisting of three points x, y and z in x and the d-metric d2(x, y, z) represents the perimeter of a triangle formed by three points x, y, z in x as its vertices. definition 11. (cauchy sequence , completeness)[10] for every m, n > n. a sequence (xn) in a metric space x = (x, d) is said to be-cauchy if for every ε > 0 there is an n = n(ε) such that d(xm, xn) < ε. the space x is said to be complete if every cauchy sequence in x converges (that is, has a limit which is an element of x ). the following theorem can be found in [4]: theorem 2. [4] let d be an ordinary metric on x and let d1 and d2 be corresponding associated d-metrics on x. then, (x, d1) and (x, d2) are complete if and only if (x, d) is complete. 222 a. kamal & t.i.yassen cubo 22, 2 (2020) 4 d-metrics in b∗α, log and f ∗ log(p, q, s) in this section, we introduce a d-metric on b∗α, log and f ∗ log(p, q, s). let 0 < p, s < ∞, −2 < q < ∞, and 0 < α < 1. first, we can find a d-metric in b∗α, log, for f, g, h ∈ b∗α, log by defining d(f, g, h; b∗α, log) := db∗α, log (f, g, h) + ||f − g||bα, log + ||g − h||bα, log + ||h − f||bα, log +|f(0) − g(0)| + |g(0) − h(0)| + |h(0) − f(0)|, where db∗ α, log (f, g, h) := db∗ α, log (f, g) + db∗ α, log (g, h) + db∗ α, log (h, f) and db∗ α, log (f, g, h) := ( sup z∈d |f∗(z) − g∗(z)| + sup z∈d |g∗(z) − h∗(z)| + sup z∈d |h∗(z) − f∗(z)| ) × ( (1 − |z|2)α ( log 2 1 − |z|2 )) . also, for f, g, h ∈ f ∗log(p, q, s) we introduce a d-metric on f ∗ log(p, q, s) by defining d(f, g, h; f ∗log(p, q, s)) := df ∗log(p,q,s)(f, g, h) + ||f − g||flog(p,q,s) + ||g − h||flog(p,q,s)+ ||h − f||flog(p,q,s) + |f(0) − g(0)| + |g(0) − h(0)| + |h(0) − f(0)|, where df ∗ log (p,q,s)(f, g, h) := df ∗ log (p,q,s)(f, g) + df ∗ log (p,q,s)(g, h) + df ∗ log (p,q,s)(h, f) and df ∗ log (p,q,s)(f, g) := ( sup z∈d ℓ p(a) ∫ d |f∗(z) − g∗(z)|p(1 − |z|2)q(1 − |ϕ(z)|2)sda(z) ) 1 p . proposition 1. the class b∗α, log equipped with the d-metric d(., .; b ∗ α, log) is a complete metric space. moreover, b∗α, log, 0 is a closed (and therefore complete) subspace of b ∗ α, log. proof. let f, g, h, a ∈ b∗α, log. then, clearly (i) d(f, g, h; b∗α, log) ≥ 0, for all f, g, h ∈ b ∗ α, log. cubo 22, 2 (2020) d-metric spaces and composition operators 223 (ii)d(f, g, h; b∗α, log) = d(f, h, g; b ∗ α, log) = d(g, h, f; b ∗ α, log). (iii)d(f, g, h; b∗α, log) ≤ d(f, g, a; b ∗ α, log) + d(f, a, h; b ∗ α, log) + d(a, g, h; b ∗ α, log) for all f, g, h, a ∈ b∗α, log. (iv)d(f, g, h; b∗α, log) = 0 implies f = g = h. hence, d is a d-metric on b∗α, log, and (b ∗ α, log, d) is d-metric space. to prove the completeness, we use theorem 2, let (fn) ∞ n=1 be a cauchy sequence in the metric space (b∗α, log, d), that is, for any ε > 0 there is an n = n(ε) ∈ n such that d(fn, fm; b ∗ α, log) < ε, for all n, m > n. since (fn) ⊂ b(d), the family (fn) is uniformly bounded and hence normal in d. therefore, there exists f ∈ b(d) and a subsequence (fnj ) ∞ j=1 such that fnj converges to f uniformly on compact subsets of d. it follows that fn also converges to f uniformly on compact subsets, and by the cauchy formula, the same also holds for the derivatives. now let m > n. then, the uniform convergence yields ∣ ∣ ∣ ∣ f∗(z) − f∗m(z) ∣ ∣ ∣ ∣ (1 − |z|2)α ( log 2 1 − |z|2 ) = lim n→∞ ∣ ∣ ∣ ∣ f ∗ n(z) − f ∗ m(z) ∣ ∣ ∣ ∣ (1 − |z|2)α ( log 2 1 − |z|2 ) ≤ lim n→∞ d(fn, fm; b ∗ α, log) ≤ ε for all z ∈ d, and it follows that ||f||b∗ α, log ≤ ||fm||b∗ α, log +ε. thus f ∈ b∗α, log as desired. moreover, the above inequality and the compactness of the usual b∗α, log space imply that (fn) ∞ n=1 converges to f with respect to the metric d, and (b∗α, log, d) is complete d-metric space. since lim n→∞ d(fn, fm; b ∗ α, log) ≤ ε, the second part of the assertion follows. next we give characterization of the complete d-metric space d(., .; f ∗log(p, q, s)). proposition 2. the class f ∗log(p, q, s) equipped with the d-metric d(., .; f ∗ log(p, q, s)) is a complete metric space. moreover, f ∗log, 0(p, q, s) is a closed (and therefore complete) subspace of f ∗ log(p, q, s). proof. let f, g, h, a ∈ f ∗log(p, q, s). then clearly (i) d(f, g, h; f ∗log(p, q, s)) ≥ 0, for all f, g, h ∈ f ∗ log(p, q, s). (ii)d(f, g, h; f ∗log(p, q, s)) = d(f, h, g; f ∗ log(p, q, s)) = d(g, h, f; f ∗ log(p, q, s)). 224 a. kamal & t.i.yassen cubo 22, 2 (2020) (iii)d(f, g, h; f ∗log(p, q, s)) ≤ d(f, g, a; f ∗ log(p, q, s)) + d(f, a, h; f ∗ log(p, q, s)) +d(a, g, h; f ∗log(p, q, s)) for all f, g, h, a ∈ f ∗log(p, q, s). (iv)d(f, g, h; f ∗log(p, q, s)) = 0 implies f = g = h. hence, d is a d-metric on f ∗log(p, q, s), and (f ∗ log(p, q, s), d) is d-metric space. for the complete proof, by using theorem 2, let (fn) ∞ n=1 be a cauchy sequence in the metric space (f ∗log(p, q, s), d), that is, for any ε > 0 there is an n = n(ε) ∈ n so that d(fn, fm; f ∗ log(p, q, s)) < ε, for all n, m > n. since (fn) ⊂ b(d), such that fnj converges to f uniformly on compact subsets of d. it follows that fn also converges to f uniformly on compact subsets, now let m > n, and 0 < r < 1. then, the fatou’s yields ∫ d(0,r) ∣ ∣ ∣ ∣ f∗(z) − f∗m(z) ∣ ∣ ∣ ∣ p (1 − |z|2)q(1 − |ϕa(z)| 2)sda(z) = ∫ d(0,r) lim n→∞ ∣ ∣ ∣ ∣ f ∗ n(z) − f ∗ m(z) ∣ ∣ ∣ ∣ p (1 − |z|2)q(1 − |ϕa(z)| 2)sda(z) ≤ lim n→∞ ∫ d(0,r) ∣ ∣ ∣ ∣ f∗(z) − f∗m(z) ∣ ∣ ∣ ∣ p (1 − |z|2)q(1 − |ϕa(z)| 2)sda(z) ≤ εp, and by taking r → 1 − , it follows that, ∫ d (f∗(z))p(1 − |z|2)q(1 − |ϕa(z)| 2)sda(z) ≤ 2pεp + 2p ∫ d (f∗m(z)) p(1 − |z|2)q(1 − |ϕa(z)| 2)sda(z). this yields ||f|| p f ∗ log (p,q,s) ≤ 2p||fm|| p f ∗ log (p,q,s) + 2pεp. and thus f ∈ f ∗log(p, q, s). we also find that fn → f with respect to the metric of (f ∗ log(p, q, s), d) and (f ∗log(p, q, s), d) is complete d-metric space. the second part of the assertion follows. 5 composition operators of cφ : b ∗ α, log → f ∗ log(p, q, s) in this section, we study boundedness and compactness of composition operators on b∗α, log and f ∗log(p, q, s) spaces. we need the following notation: φφ(α, p, s; a) = ℓ p(a) ∫ d |φ′(z)|p (1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)|2)αp ( log 2 (1−|φ(z)|2) )p da(z), cubo 22, 2 (2020) d-metric spaces and composition operators 225 where ℓp(a) = ( log 2 1−|a|2 )p . for 0 < α < 1, we suppose there exist two functions f, g ∈ b∗α, log such that for some constant c, (|f∗(z)| + |g∗(z)|) ≥ c (1 − |z|2)α ( log 2 1−|a|2 )p > 0, for each z ∈ d. now, we provide the following theorem: theorem 3. assume φ is a holomorphic mapping from d into itself and let 0 < p, 1 < s < ∞, 0 < α ≤ 1. then the induced composition operator cφ maps b ∗ α, log into f ∗ log(p, αp − 2, s) is bounded if and only if, sup z∈d φφ(α, p, s; a) < ∞. (5.1) proof. first assume that sup z∈d φφ(α, p, s; a) < ∞ is held, and f ∈ b ∗ α, log with ||f||bα, log ≤ 1, we can see that ||cφf|| p f ∗ log (p,αp−2,s) = sup a∈d ℓp(a) ∫ d ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) = sup a∈d ℓp(a) ∫ d (f∗(φ(z)))p|φ′(z)|αp−2(1 − |ϕa(z)| 2)sda(z) ≤ ||f|| p b∗ α, log sup a∈d ℓp(a) ∫ d |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)2|)pα(log 2 1−|z|2 ) da(z) = ||f|| p b∗ α, log φφ(α, p, s; a) < ∞. for the other direction, we use the fact that for each function f ∈ b∗α, log, the analytic function 226 a. kamal & t.i.yassen cubo 22, 2 (2020) cφ(f) ∈ f ∗ log(p, αp − 2, s). then, using the functions of lemma 1.2 2p { ||cφf1|| p f ∗ log (p,αp−2,s) + ||cφf2|| p f ∗ log (p,αp−2,s) } = 2p { sup a∈d ℓp(a) ∫ d [ ((f1 ◦ φ) ∗(z))p + ((f2 ◦ φ) ∗(z))p ] ×(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) } ≥ { sup a∈d ℓ p(a) ∫ d [ (f1 ◦ φ) ∗(z) + (f2 ◦ φ) ∗(z) ]p ×(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) } ≥ { sup a∈d ℓp(a) ∫ d [ (f∗1 (φ))(z) + (f ∗ 2 (φ))(z) ]p ×|φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) } ≥ c { sup a∈d ℓp(a) ∫ d |φ′(z)|p (1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)|2)αp ( log 2 (1−|φ(z)|2) )p da(z) } ≥ c sup a∈d φφ(α, p, s; a). hence cφ is bounded, the proof is completed. the composition operator cφ : b ∗ α, log → f ∗ log(p, αp − 2, s) is compact if and only if for every sequence fn ∈ n ⊂ f ∗ log(p, αp − 2, s) is bounded in f ∗ log(p, αp − 2, s) norm andfn → 0, n → ∞, uniformly on compact subset of the unit disk (where n be the set of all natural numbers), hence, ||cφ(fn)||f ∗ log (p,αp−2,s) → 0, n → ∞. now, we describe compactness in the following result: theorem 4. let 0 < p, 1 < s < ∞, α < ∞. if φ is an analytic self-map of the unit disk, then the induced composition operator cφ : b ∗ α, log → f ∗ log(p, αp − 2, s) is compact if and only if φ ∈ f ∗log(p, αp − 2, s), and lim r→1 sup a∈d φφ(α, p, s; a) → 0. (5.2) proof. let cφ : b ∗ α, log → f ∗ log(p, αp − 2, s) be compact. this means that φ ∈ f ∗log(p, αp − 2, s). let u1r = {z : |φ(z)| > r, r ∈ (0, 1)}, cubo 22, 2 (2020) d-metric spaces and composition operators 227 and u2r = {z : |φ(z)| ≤ r, r ∈ (0, 1)}. let fn(z) = z n n if α ∈ [0, ∞) or fn(z) = z n n1−α if α ∈ (0, 1). without loss of generality, we only consider α ∈ (0, 1). since ||fn||b∗ α, log ≤ m and fn(z) → 0 as n → ∞, locally uniformly on the unit disk, then ||cφ(fn)||f ∗ log (p,αp−2,s), n → ∞. this means that for each r ∈ (0, 1) and for all ε > 0, there exist n ∈ n so that if n ≥ n, then nαp rp(1−n) sup a∈d ℓ p(a) ∫ u1 r |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ε. if we choose r so that n αp rp(1−n) = 1, then sup a∈d ℓp(a) ∫ u1 r |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ε. (5.3) let now f be with ||f||b∗ α, log ≤ 1. we consider the functions ft(z) = f(tz), t ∈ (0, 1). ft → f uniformly on compact subset of the unit disk as t → 1 and the family (ft) is bounded on b ∗ α, log, thus ||(ft ◦ φ ) − (f ◦ φ )|| → 0. due to compactness of cφ, we get that for ε > 0 there is t ∈ (0, 1) so that sup a∈d ℓp(a) ∫ d |ft(φ(z))| p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ε, where ft(φ(z)) = [ (f ◦ φ )∗ − (ft ◦ φ ) ∗ ] . thus, if we fix t, then sup a∈d ℓp(a) ∫ u1 r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ 2p sup a∈d ℓp(a) ∫ u1 r |ft(φ(z))| p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) +2p sup a∈d ℓp(a) ∫ u1 r ((ft ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ 2pε + ||f∗t || p h∞ sup a∈d ℓ p(a) ∫ u1 r |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ 2pε + 2pε||f∗t || p h∞ . 228 a. kamal & t.i.yassen cubo 22, 2 (2020) i.e, sup a∈d ℓp(a) ∫ u1r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ 2pε(1 + ||f∗t || p h∞ ), (5.4) where we have used (4). on the other hand, for each ||f||b∗ α, log ≤ 1 and ε > 0, there exists a δ depending on f and ε, so that for r ∈ [δ, 1), sup a∈d ℓp(a) ∫ u1 r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ε. (5.5) since cφ is compact, then it maps the unit ball of b ∗ α, log to a relatively compact subset of f ∗log(p, q, s). thus, for each ε > 0, there exists a finite collection of functions f1, f2, ..., fn in the unit ball of b∗α, log so that for each ||f||b∗α, log, there is k ∈ {1, 2, 3, ..., n} so that sup a∈d ℓp(a) ∫ u1r |fk(φ(z))| p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ε, where fk(φ(z)) = [ (f ◦ φ )∗ − (fk ◦ φ ) ∗ ] . also, using (5), we get for δ = max1≤k≤nδ(fk, ε) and r ∈ [δ, 1), that sup a∈d ℓp(a) ∫ u1r ((fk ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) < ε. hence, for any f, ||f||b∗ α, log ≤ 1, combining the two relations as above, we get the following sup a∈d ℓp(a) ∫ u1r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ 2pε. therefore, we get that (2) holds. for the sufficiency, we use that φ ∈ f ∗log(p, αp − 2, s) and (2) holds. let {fn}n∈n be a sequence of functions in the unit ball of b ∗ α, log so that fn → 0 as n → ∞, uniformly on the compact subsets of the unit disk. let also r ∈ (0, 1). then, ||fn ◦ φ|| p f ∗ log (p,αp−2,s) ≤ 2p|fn(φ(0))| +2p sup a∈d ℓp(a) ∫ u2r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) +2p sup a∈d ℓp(a) ∫ u1 r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) = 2p(i1 + i2 + i3). cubo 22, 2 (2020) d-metric spaces and composition operators 229 since fn → 0 as n → ∞, locally uniformly on the unit disk, then i1 = |fn(φ(0))| goes to zero as n → ∞ and for each ε > 0, there is n ∈ n so that for each n > n, i2 = sup a∈d ℓp(a) ∫ u2r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ ε||φ|| p f ∗ log (p,αp−2,s) . we also observe that i3 = sup a∈d ℓp(a) ∫ u1r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sda(z) ≤ ||f|| p b∗ α, log × sup a∈d ℓp(a) ∫ u1r |φ′(z)|p (1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)|2)αp ( log 2 (1−|φ(z)|2) )p da(z). under the assumption that (2) holds, then for every n > n and for every ε > 0, there exists r1 so that for every r > r1, i3 < ε. thus, if φ(z) ∈ f ∗log(p, αp − 2, s), we get ||fn ◦ φ|| p f ∗ log (p,αp−2,s) ≤ 2p { 0 + ε||φ|| p f ∗ log (p,αp−2,s) + ε } ≤ cε. combining the above, we get ||cφ(fn)|| p f ∗ log (p,αp−2,s) → 0 as n → ∞ which proves compactness. thus, the theorem we presented is proved. 6 conclusions we have obtained some essential and important d-metric spaces. moreover, the important properties for d-metric on b∗α, log and f ∗ log(p, q, s) are investigated in section 4. finally, we introduced composition operators in hyperbolic weighted family of function spaces. 7 acknowledgements the authors would like to thank the referees for their useful comments, which improved the original manuscript. 230 a. kamal & t.i.yassen cubo 22, 2 (2020) references [1] b. c. dhage, a study of some fixed point theorem. ph.d. thesis, marathwada univ. aurangabad, india, 1984. [2] b. c. dhage, generalized metric spaces and mappings with fixed point, bull. cal. math. soc, 84(4)(1992), 329–336. [3] b. c. dhage, on generalized metric spaces and topological structure. ii, pure appl. math. sci. 40 (1994), no. 1-2, 37–41 [4] b. c. dhage, generalized metric spaces and topological structure. i, an. ştiinţ. univ. al. i. cuza iaşi. mat. (n.s.) 46 (2000), no. 1, 3–24 (2001). [5] a. el-sayed ahmed and m. a. bakhit, composition operators acting between some weighted möbius invariant spaces, ann. funct. anal. 2 (2011), no. 2, 138–152. [6] a. el-sayed ahmed, a. kamal and t. i. yassen, characterizations for certain analytic functions by series expansions with hadamard gaps, cubo 16 (2014), no. 1, 81–93. [7] t. hosokawa, differences of weighted composition operators on the bloch spaces, complex anal. oper. theory 3 (2009), no. 4, 847–866. 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[14] r. zhao, on a general family of function spaces, ann. acad. sci. fenn. math. diss. no. 105 (1996), 56 pp. cubo 22, 2 (2020) d-metric spaces and composition operators 231 [15] r. zhao, on logarithmic carleson measures, acta sci. math. (szeged) 69 (2003), no. 3-4, 605–618. introduction preliminaries d-metric space d-metrics in b*,log and f*log(p,q,s) composition operators of c:b* ,log f *log(p,q,s) conclusions acknowledgements cubo, a mathematical journal vol. 23, no. 01, pp. 21–62, april 2021 doi: 10.4067/s0719-06462021000100021 anisotropic problem with non-local boundary conditions and measure data a. kaboré s. ouaro laboratoire de mathematiques et informatiques (lami), ufr. sciences exactes et appliquées, université joseph ki-zerbo, 03 bp 7021 ouaga 03, ouagadougou, burkina faso. kaboreadama59@yahoo.fr; ouaro@yahoo.fr abstract we study a nonlinear anisotropic elliptic problem with nonlocal boundary conditions and measure data. we prove an existence and uniqueness result of entropy solution. resumen estudiamos un problema eĺıptico nolineal anisotrópico con condiciones de borde no-locales y data de medida. probamos un resultado de existencia y unicidad de la solución de entroṕıa. keywords and phrases: entropy solution, non-local boundary conditions, leray-lions operator, bounded radon diffuse measure, marcinkiewicz spaces. 2020 ams mathematics subject classification: 35j05, 35j25, 35j60, 35j66. accepted: 06 january, 2021 received: 13 november, 2019 ©2021 a. kaboré et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100021 https://orcid.org/0000-0003-0671-2378 22 a. kaboré & s. ouaro cubo 23, 1 (2021) 1 introduction and assumptions let ω be a bounded domain in rn (n ≥ 3) such that ∂ω is lipschitz and ∂ω = γd ∪ γne with γd ∩ γne = ∅. our aim is to study the following problem. p(ρ,µ,d)   − n∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pm (x)−2u = µ in ω u = 0 on γd ρ(u) + n∑ i=1 ∫ γne ai ( x, ∂ ∂xi u ) ηi = d u ≡ constant   on γne, (1.1) where the right-hand side µ is a bounded radon diffuse measure (that is µ does not charge the sets of zero pm(.)-capacity), ρ : r → r a surjective, continuous and non-decreasing function, with ρ(0) = 0, d ∈ r and ηi, i ∈{1, ...,n} are the components of the outer normal unit vector. for any ω ⊂ rn , we set c+(ω̄) = {h ∈ c(ω̄) : inf x∈ω h(x) > 1} (1.2) and we denote h+ = sup x∈ω h(x), h− = inf x∈ω h(x). (1.3) for the exponents, ~p(.) : ω̄ → rn , ~p(.) = (p1(.), ...,pn (.)) with pi ∈ c+(ω̄) for every i ∈{1, ...,n} and for all x ∈ ω̄. we put pm (x) = max{p1(x), ...,pn (x)} and pm(x) = min{p1(x), ...,pn (x)} . we assume that for i = 1, ...,n, the function ai : ω × r → r is carathéodory and satisfies the following conditions. • (h1): ai(x,ξ) is the continuous derivative with respect to ξ of the mapping ai = ai(x,ξ), that is, ai(x,ξ) = ∂ ∂ξ ai(x,ξ) such that the following equality holds. ai(x, 0) = 0, (1.4) for almost every x ∈ ω. • (h2) : there exists a positive constant c1 such that |ai(x,ξ)| ≤ c1(ji(x) + |ξ|pi(x)−1), (1.5) for almost every x ∈ ω and for every ξ ∈ r, where ji is a non-negative function in lp ′ i(.)(ω), with 1 pi(x) + 1 p′i(x) = 1. cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 23 • (h3) : there exists a positive constant c2 such that (ai(x,ξ) −ai(x,η)).(ξ −η) ≥   c2|ξ −η|pi(x) if |ξ −η| ≥ 1, c2|ξ −η|p − i if |ξ −η| < 1, (1.6) for almost every x ∈ ω and for every ξ, η ∈ r, with ξ 6= η. • (h4) : for almost every x ∈ ω and for every ξ ∈ r, |ξ|pi(x) ≤ ai(x,ξ).ξ ≤ pi(x)ai(x,ξ). (1.7) • (h5) : the variable exponents pi(.) : ω̄ → [2,n) are continuous functions for all i = 1, ...,n such that p̄(n − 1) n(p̄− 1) < p−i < p̄(n − 1) n − p̄ , n∑ i=1 1 p−i > 1 and p+i −p − i − 1 p−i < p̄−n p̄(n − 1) , (1.8) where 1 p̄ = 1 n n∑ i=1 1 p−i . as examples under assumptions (h1) -(h5), we can give the following. (1) set ai(x,ξ) = ( 1 pi(x) )|ξ|pi(x) and ai(x,ξ) = |ξ|pi(x)−2ξ , where 2 ≤ pi(x) < n. (2) ai(x,ξ) = ( 1 pi(x) )((1 + |ξ|2) pi(x) 2 − 1) and ai(x,ξ) = (1 + |ξ|2) pi(x)−2 2 ξ , where 2 ≤ pi(x) < n. we put for all x ∈ ∂ω, p∂(x) =   (n − 1)p(x) n −p(x) if p(x) < n, ∞ if p(x) ≥ n. we introduce the numbers q = n(p̄− 1) n − 1 , q∗ = nq n −q = n(p̄− 1) n − p̄ . (1.9) we denote by mb(ω) the space of bounded radon measure in ω, equipped with its standard norm ‖.‖mb(ω). note that, if u belongs to mb(ω), then |µ|(ω) (the total variation of µ) is a bounded positive measure on ω. given µ ∈ mb(ω), we say that µ is diffuse with respect to the capacity w 1,p(.) 0 (ω) (p(.)-capacity for short) if µ(a) = 0, for every set a such that capp(.)(a, ω) = 0. for every a ⊂ ω, we denote sp(.)(a) = {u ∈ w 1,p(.) 0 (ω) ∩c0(ω) : u = 1 on a,u ≥ 0 on ω}. 24 a. kaboré & s. ouaro cubo 23, 1 (2021) the p(.)-capacity of every subset a with respect to ω is defined by capp(.)(a, ω) = inf u∈sp(.)(a) { ∫ ω |∇u|p(x)dx}. in the case sp(.)(a) = ∅, we set capp(.)(a, ω) = ∞. the set of bounded radon diffuse measure in the variable exponent setting is denoted by mp(.)b (ω). we use the following result of decomposition of bounded radon diffuse measure proved by nyanquini et al. (see [31]). theorem 1.1. let p(.) : ω̄ → (1,∞) be a continuous function and µ ∈mb(ω). then µ ∈m p(.) b (ω) if and only if µ ∈ l1(ω) + w−1,p ′(.)(ω). remark 1.2. since µ ∈ mpm(.)b (ω), the theorem 1.1 implies that there exist f ∈ l 1(ω) and f ∈ (lp ′ m(.)(ω))n such that µ = f − divf, (1.10) where 1 pm(x) + 1 p′m(x) = 1, ∀x ∈ ω. the study of nonlinear elliptic equations involving the p-laplace operator is based on the theory of standard sobolev spaces wm,p(ω) in order to find weak solutions. for the nonhomogeneous p(.)-laplace operators, the natural setting for this approach is the use of the variable exponent lebesgue and sobolev spaces lp(.)(ω) and wm,p(.)(ω). variable exponent lebesgue spaces appeared in the literature for the first time in a article by orlicz in 1931. in the 1950’s, this study was carred on by nakano who made the first systematic study of spaces with variable exponent (called modular spaces). nakano explicitly mentioned variable exponent lebesgue spaces as an example of more general spaces he considered (see [30], p. 284). later, the polish mathematicians investigated the modular function spaces (see [29]). note also that h. hudzik [18] investigated the variable exponent sobolev spaces. variable exponent lebesgue spaces on the real line have been independently developed by russian researchers, notably sharapudinov [40] and tsenov [42]. the next major step in the investigation of variable exponent lebesgue and sobolev spaces was the comprehensive paper by o. kovacik and j. rakosnik in the early 90’s [23]. this paper established many of basic properties of lebesgue and sobolev spaces with variables exponent. variable sobolev spaces have been used in the last decades to model various phenomena. in [9], chen, levine and rao proposed a framework for image restoration based on a laplacian variable exponent. another application which uses nonhomogeneous laplace operators is related to the modelling of electrorheological fluids see [38]. the first major discovery in electrorheological fluids was due to winslow in 1949 (cf. [43]). these fluids have the interesting property that their viscosity depends on the electric field in the fluid. they can raise the viscosity by as much as five orders of magnitude. this phenomenon is known as the winslow effect. for some technical applications, we refer the readers to the work by pfeiffer et al [33]. electrorheological fluids have been used in robotics and space technology. the experimental research has been done mainly in cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 25 the usa, for instance in nasa laboratories. for more information on properties, modelling and the application of variable exponent spaces to these fluids, we refer to diening [11], rajagopal and ruzicka [35], and ruzicka [36]. in this paper, the operator involved in (1.1) is more general than the p(.)-laplace operator. thus, the variable exponent sobolev space w 1,p(.)(ω) is not adequate to study nonlinear problems of this type. this leads us to seek entropy solutions for problems (1.1) in a more general variable exponent sobolev space which was introduced for the first time by mihäılescu et al. [28], see also [34, 26, 27]. the need for such theory comes naturally every time we want to consider materials with inhomogeneities that have different behavior on different space directions. non-local boundary value problems of various kinds for partial differential equations are of great interest by now in several fields of application. in a typical non-local problem, the partial differential equation (resp. boundary conditions) for an unknown function u at any point in a domain ω involves not only the local behavior of u in a neighborhood of that point but also the non-local behavior of u elsewhere in ω. for example, at any point in ω the partial differential equation and/or the boundary conditions may contains integrals of the unknown u over parts of ω, values of u elsewhere in d or, generally speaking, some non-local operator on u. beside the mathematical interest of nonlocal conditions, it seems that this type of boundary condition appears in petroleum engineering model for well modeling in a 3d stratified petroleum reservoir with arbitrary geometry (see [12] and [15]). a lot of papers ( see [34], [24], [25], [2], [19], [1]) on problems like (1.1) considered cases of generally boundary value condition. in [6], bonzi et al. studied the following problems.   − n∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pm (x)−2u = f in ω n∑ i=1 ai ( x, ∂ ∂xi u ) ηi = −|u|r(x)−2u on ∂ω, (1.11) which correspond to the robin type boundary condition. the authors used minimization techniques used in [8] to prove the existence and uniqueness of entropy solution. by the same techniques, koné and al. proved the existence and uniqueness of entropy solution for the following problem.   − n∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pm (x)−2u = f in ω n∑ i=1 ai ( x, ∂ ∂xi u ) ηi + λu = g on ∂ω, (1.12) which correspond to the fourier type boundary condition. in a recent paper we studied a nonlinear elliptic anisotropic problem involving nonlocal conditions. we also considered variable exponent and general maximal monotone graph datum at the boundary 26 a. kaboré & s. ouaro cubo 23, 1 (2021) and proved existence and uniqueness of weak solution to the following problem. s(ρ,µ,d)   − n∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pm (x)−2u = f in ω u = 0 on γd ρ(u) + n∑ i=1 ∫ γne ai ( x, ∂ ∂xi u ) ηi 3 d u ≡ constant   on γne, where the right-hand side f ∈ l∞(ω) and ρ a maximal monotone graph on r such that d(ρ) = im(ρ) = r and 0 ∈ ρ(0), d ∈ r, by using the technique of monotone operators in banach spaces (see [21]) and approximation methods. there are two difficulties associated with the study of problem p(ρ,µ,d). the first is to give a sense to the partial derivative of u which appear in the term ai ( x, ∂ ∂xi u ) . as µ is a measure (even if µ is a integrable function), then we cannot take the partial derivative of u in the usual distribution sense. the idea consists in considering troncatures of the solution u (see [5]). the second difficulty appears with the question of uniqueness of solutons. we obtain existence and uniqueness of a special class of solutions of problem p(ρ,µ,d) that satisfy an extra condition that we call the entropy condition (see formula (2.9)). an alternative notion of solution which can leads to existence and uniqueness of solution to problem p(ρ,µ,d) is the notion of renormalized solution. but in this work, we consider the notion of entropy solution. the paper is organized as follows. section 2 is devoted to mathematical preliminaries including, among other things, a brief discussion on variable exponent lebesgue, sobolev, anisotropic and marcinkiewicz spaces. in section 3, we study an approximated problem and in section 4, we prove by using the results of the section 3, the existence and uniqueness of entropy solution of problem p(ρ,µ,d). 2 preliminary this part is related to anisotropic lebesgue and sobolev spaces with variable exponent and some of their properties. given a measurable function p(.) : ω → [1,∞). we define the lebesgue space with variable exponent lp(.)(ω) as the set of all measurable functions u : ω → r for which the convex modular ρp(.)(u) := ∫ ω |u|p(x)dx is finite. if the exponent is bounded, i.e, if p+ < ∞, then the expression |u|p(.) := inf { λ > 0 : ρp(.)( u λ ) ≤ 1 } cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 27 defines a norm in lp(.)(ω), called the luxembourg norm. the space (lp(.)(ω), |.|p(.)) is a separable banach space. then, lp(.)(ω) is uniformly convex, hence reflexive and its dual space is isomorphic to lp ′(.)(ω), where 1 p(x) + 1 p′(x) = 1, for all x ∈ ω. we have the following properties (see [13]) on the modular ρp(.). if u,un ∈ lp(.)(ω) and p+ < ∞, then |u|p(.) < 1 ⇒|u| p+ p(.) ≤ ρp(.)(u) ≤ |u| p− p(.) , (2.1) |u|p(.) > 1 ⇒|u| p− p(.) ≤ ρp(.)(u) ≤ |u| p+ p(.) , (2.2) |u|p(.) < 1(= 1; > 1) ⇒ ρp(.)(u) < 1(= 1; > 1), (2.3) and |un|p(.) → 0 (|un|p(.) →∞) ⇔ ρp(.)(un) → 0 (ρp(.)(un) →∞). (2.4) if in addition, (un)n∈n ⊂ lp(.)(ω), then limn→∞ |un −u|p(.) = 0 ⇔ limn→∞ρp(.)(un −u) = 0 ⇔ (un)n∈n converges to u in measure and limn→∞ρp(.)(un) = ρp(.)(u). we introduce the definition of the isotropic sobolev space with variable exponent, w 1,p(.)(ω) := { u ∈ lp(.)(ω) : |∇u| ∈ lp(.)(ω) } , which is a banach space equipped with the norm ‖u‖1,p(.) := |u|p(.) + |∇u|p(.). now, we present the anisotropic sobolev space with variable exponent which is used for the study of p(ρ,µ,d). the anisotropic variable exponent sobolev space w 1,~p(.)(ω) is defined as follow. w 1,~p(.)(ω) := { u ∈ lpm (.)(ω) : ∂u ∂xi ∈ lpi(.)(ω), for all i ∈{1, ...,n} } . endowed with the norm ‖u‖~p(.) := |u|pm (.) + n∑ i=1 ∣∣∣∣ ∂u∂xi ∣∣∣∣ pi(.) , the space ( w 1,~p(.)(ω),‖.‖~p(.) ) is a reflexive banach space (see [14], theorem 2.1 and theorem 2.2). as consequence, we have the following. theorem 2.1. (see [14]) let ω ⊂ rn (n ≥ 3) be a bounded open set and for all i ∈{1, ...,n}, pi ∈ l∞(ω), pi(x) ≥ 1 a.e. in ω. then, for any r ∈ l∞(ω) with r(x) ≥ 1 a.e. in ω such that ess inf x∈ω (pm (x) −r(x)) > 0, we have the compact embedding w 1,~p(.)(ω) ↪→ lr(.)(ω). 28 a. kaboré & s. ouaro cubo 23, 1 (2021) we also need the following trace theorem due to [7]. theorem 2.2. let ω ⊂ rn (n ≥ 2) be a bounded open set with smooth boundary and let ~p(.) ∈ c(ω̄) satisfy the condition 1 ≤ r(x) < min x∈∂ω {p∂1 (x), ...,p ∂ n (x)}, ∀x ∈ ∂ω. (2.5) then, there is a compact boundary trace embedding w 1,~p(.)(ω) ↪→ lr(.)(∂ω). let us introduce the following notation: ~p− = (p − 1 , ...,p − n ). we will use in this paper, the marcinkiewicz spaces mq(ω) (1 < q < ∞) with constant exponent. note that the marcinkiewicz spaces mq(.)(ω) in the variable exponent setting was introduced for the first time by sanchon and urbano (see [37]). marcinkiewicz spaces mq(ω) (1 < q < ∞) contain all measurable function h : ω → r for which the distribution function λh(γ) := meas({x ∈ ω : |h(x)| > γ}), γ ≥ 0, satisfies an estimate of the form λh(γ) ≤ cγ−q, for some finite constant c > 0. the space mq(ω) is a banach space under the norm ‖h‖∗mq(ω) = sup t>0 t 1 q ( 1 t ∫ t 0 h∗(s)ds ) , where h∗ denotes the nonincreasing rearrangement of h. h∗(t) := inf { c : λh(γ) ≤ cγ−q, ∀γ > 0 } , which is equivalent to the norm ‖h‖∗mq(ω) (see [3]). we need the following lemma (see [4], lemma a-2). lemma 2.3. let 1 ≤ q 0 {λpmeas[x ∈ ω : |u| > λ]}≤‖u‖pmp(ω). moreover, (ii) ∫ k |u|qdx ≤ p p−q ( p q ) q p‖u‖qmp(ω)(meas(k)) p−q p , for every measurable subset k ⊂ ω. in particular, mp(ω) ⊂ lqloc(ω), with continuous embedding and u ∈ m p(ω) implies |u|q ∈ m p q (ω). cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 29 the following result is due to troisi (see [39]). theorem 2.4. let p1, ...,pn ∈ [1,∞), ~p = (p1, ...,pn ); g ∈ w 1,~p(ω), and let  q = p̄∗ if p̄∗ < n, q ∈ [1,∞) if p̄∗ ≥ n; (2.6) where p∗ = n∑n i=1 1 pi − 1 , ∑n i=1 1 pi > 1 and p̄∗ = np̄ n − p̄ . then, there exists a constant c > 0 depending on n, p1, ...,pn if p̄ < n and also on q and meas(ω) if p̄ ≥ n such that ‖g‖lq(ω) ≤ c n∏ i=1 [ ‖g‖lpm (ω) + ‖ ∂g ∂xi ‖lpi(ω) ] 1 n , (2.7) where pm = max{p1, ...,pn} and 1p̄ = 1 n ∑n i=1 1 pi . in particular, if u ∈ w 1,~p0 (ω), we have ‖g‖lq(ω) ≤ c n∏ i=1 [∥∥∥∥ ∂g∂xi ∥∥∥∥ lpi(ω) ] 1 n . (2.8) in the sequel, we consider the following spaces. w 1,~p(.) d (ω) = {ξ ∈ w 1,~p(.)(ω) : ξ = 0 on γd} and w 1,~p(.) ne (ω) = {ξ ∈ w 1,~p(.) d (ω) : ξ ≡ constant on γne}. t 1,~p(.)d (ω) = {ξ measurable on ω such that ∀k > 0, tk(ξ) ∈ w 1,~p(.) d (ω)} and t 1,~p(.)ne (ω) = {ξ measurable on ω such that ∀k > 0, tk(ξ) ∈ w 1,~p(.) ne (ω)}, where tk is a truncation function defined by tk(s) =   k if s > k, s if |s| ≤ k, −k if s < −k. for any v ∈ w 1,~p(.)ne (ω), we set vn = vne := v|γne. definition 2.5. a measurable function u : ω → r is an entropy solution of p(ρ,µ,d) if u ∈ t 1,~p(.)ne (ω) and for every k > 0,  ∫ ω ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi tk(u− ξ) ) dx + ∫ ω |u|pm (x)−2utk(u− ξ)dx ≤∫ ω tk(u− ξ)dµ + (d−ρ(une))tk(une − ξ), (2.9) for all ξ ∈ w 1,~p(.)ne (ω) ∩l ∞(ω). 30 a. kaboré & s. ouaro cubo 23, 1 (2021) our main result in this paper is the following theorem. theorem 2.6. assume (h1)-(h5). then for any (µ,d) ∈m pm(.) b (ω) × r, the problem p(ρ,µ,d) admits a unique entropy solution u. 3 the approximated problem corresponding to p(ρ,µ,d) we define a new bounded domain ω̃ in rn as follow. we fix θ > 0 and we set ω̃ = ω ∪{x ∈ rn/dist(x, γne) < θ}. then, ∂ω̃ = γd ∪ γ̃ne is lipschitz with γd ∩ γ̃ne = ∅. figure 1: domains representation let us consider ãi(x,ξ) (to be defined later) carathéodory and satisfying (1.4), (1.5), (1.6) and (1.7), for all x ∈ ω̃. we also consider a function d̃ in l1(γ̃ne) such that∫ γ̃ne d̃dσ = d. (3.1) for any � > 0, we set µ� = f� − divf, where f� = t1 � (f) ∈ l∞(ω) . note that f� → f as � → 0 in l1(ω) and ‖f�‖1 ≤‖f‖1. we set µ̃� = f�χω − divfχω, d̃� = t1 � (d̃) and we consider the problem p(ρ̃, µ̃�, d̃�)   − n∑ i=1 ∂ ∂xi ãi(x, ∂ ∂xi u�) + |u�|pm (x)−2u�χω(x) = µ̃� in ω̃ u� = 0 on γd ρ̃(u�) + n∑ i=1 ãi(x, ∂ ∂xi u�)ηi = d̃� on γ̃ne, (3.2) where the function ρ̃ is defined as follow. cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 31 • ρ̃(s) = 1 |γ̃ne| ρ(s), where |γ̃ne| denotes the hausdorff measure of γ̃ne. we obviously have ∀� > 0, d̃� ∈ l∞(γ̃ne). the following definition gives the notion of solution for the problem p�(ρ̃, µ̃�, d̃�). definition 3.1. a measurable function u� : ω̃ → r is a solution to problem p�(ρ̃, µ̃�, d̃�) if u� ∈ w 1,~p(.) d (ω̃) and∫ ω̃ n∑ i=1 ãi(x, ∂ ∂xi u�) ∂ ∂xi ξ̃dx + ∫ ω |u�|pm (x)−2u�ξ̃dx = ∫ ω f�ξ̃dx + ∫ ω f.∇ξ̃ + ∫ γ̃ne (d̃� − ρ̃(u�))ξ̃dσ, (3.3) for any ξ̃ ∈ w 1,~p(.)d (ω̃) ∩l ∞(ω). theorem 3.2. the problem p�(ρ̃, µ̃�, d̃�) admits at least one solution in the sense of definition 3.1. step 1: approximated problem we study an existence result to the following problem. for any k > 0 we consider p�,k(ρ̃, µ̃�, d̃�)   − n∑ i=1 ∂ ∂xi ãi(x, ∂ ∂xi u�,k) + tk(b(u�,k))χω(x) = µ̃� in ω̃ u�,k = 0 on γd tk(ρ̃(u�,k)) + n∑ i=1 ãi(x, ∂ ∂xi u�,k)ηi = d̃� on γ̃ne, (3.4) where b(u) = |u|pm (x)−2u. we have to prove that p�,k(ρ̃, µ̃�, d̃�) admits at least one solution in the following sense.  u�,k ∈ w 1,~p(.) d (ω̃) and for all ξ̃ ∈ w 1,~p(.) d (ω̃),∫ ω̃ n∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi ξ̃dx + ∫ ω tk(b(u�,k))ξ̃dx = ∫ ω ξ̃dµ� + ∫ γ̃ne (d̃� −tk(ρ̃(u�,k)))ξ̃dσ. (3.5) for any k > 0, let us introduce the operator λk : w 1,~p(.) d (ω̃) → (w 1,~p(.) d (ω̃)) ′ such that for any (u,v) ∈ w 1,~p(.)d (ω̃) ×w 1,~p(.) d (ω̃), 〈λk(u),v〉 = ∫ ω̃ ( n∑ i=1 ãi(x, ∂ ∂xi u) ∂ ∂xi v ) dx + ∫ ω tk(b(u))vdx + ∫ γ̃ne tk(ρ̃(u))vdσ. (3.6) we need to prove that for any k > 0, the operator λk is bounded, coercive, of type m and therefore, surjective. (i) boundedness of λk. let (u,v) ∈ f ×w 1,~p(.) d (ω̃) with f a bounded subset of w 1,~p(.) d (ω̃) . 32 a. kaboré & s. ouaro cubo 23, 1 (2021) we have  |〈λk(u),v〉| ≤ n∑ i=1 (∫ ω̃ ∣∣∣∣ãi(x, ∂∂xiu) ∣∣∣∣ ∣∣∣∣ ∂∂xiv ∣∣∣∣dx ) + ∫ ω̃ |tk(b(u))||v|dx + ∫ γ̃ne |tk(ρ̃(u))||v|dσ = i1 + i2 + i3, where we denote by i1, i2 and i3 the three terms on the right hand side of the first inequality. by (h2) and the hölder type inequality, we have  i1 ≤ c1 n∑ i=1 (∫ ω̃ |ji(x)| ∣∣∣∣ ∂∂xiv ∣∣∣∣dx + ∫ ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣ ∂∂xiv ∣∣∣∣dx ) ≤ c1 n∑ i=1 ( 1 p′−i + 1 p−i ) |ji|p′ i (.) ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) + n∑ i=1 ( 1 p′−i + 1 p−i )∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.) ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) . as u ∈ f, ∀ i ∈{1, ...,n}, there exists a constant m > 0 such that n∑ i=1 ∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.) < m; so ∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.) < m, ∀ i ∈{1, ...,n}. let c4 = max i=1,...,n   ∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.)   . as ji ∈ lp ′ i(.)(ω̃), we have i1 ≤ c5(c1,p−i , (p ′ i) −,c3(ji)) n∑ i=1 ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) + c6(c1,p − i , (p ′ i) −,c4) n∑ i=1 ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) . it is easy to see that i2 ≤ k ∫ ω̃ |v|dx. using theorem 2.1, we have ‖v‖l1(ω̃) ≤ c7‖v‖w1,~p(.) d (ω̃) . so, i2 ≤ kc7‖v‖w1,~p(.) d (ω̃) . similarly, by using theorem 2.2, we have i3 ≤ kc8‖v‖w1,~p(.) d (ω̃) � therefore, λk maps bounded subsets of w 1,~p(.) d (ω̃) into bounded subsets of (w 1,~p(.) d (ω̃)) ′. thus, λk is bounded on w 1,~p(.) d (ω̃). cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 33 (ii) coerciveness of λk. we have to show that for any k > 0, 〈λk(u),u〉 ‖u‖ w 1,~p(.) d (ω̃) → ∞ as ‖u‖ w 1,~p(.) d (ω̃) →∞. for any u ∈ w 1,~p(.)d (ω̃), we have 〈λk(u),u〉 = 〈λ(u),u〉 + ∫ ω tk(b(u))udx + ∫ γ̃ne tk(ρ̃(u))udσ, (3.7) where 〈λ(u),u〉 = n∑ i=1 (∫ ω̃ ãi(x, ∂ ∂xi u) ∂ ∂xi udx ) . the last two terms on the right-hand side of (3.7) are non-negative by the monotonicity of tk, b and ρ̃. we can assert that  〈λk(u),u〉≥ 〈λ(u),u〉 ≥ 1 np − m−1 ‖u‖p − m w 1,~p(.) d (ω̃) −n. indeed, since ∫ ω̃ |tk(b(u))||u|dx + ∫ γ̃ne |tk(ρ̃(u))||u|dσ ≥ 0, for all u ∈ w 1,~p(.) d (ω̃), we have 〈λk(u),u〉≥ 〈λ(u),u〉. so, 〈λk(u),u〉 ≥ n∑ i=1 (∫ ω̃ ãi(x, ∂ ∂xi u) ∂ ∂xi udx ) ≥ n∑ i=1 (∫ ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) . we make the following notations: i = { i ∈{1, ...,n} : ∣∣∣∣ ∂∂xiu ∣∣∣∣ pi(.) ≤ 1 } and j = { i ∈{1, ...,n} : ∣∣∣∣ ∂∂xiu ∣∣∣∣ pi(.) > 1 } . we have 〈λk(u),u〉 ≥ ∑ i∈i (∫ ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) + ∑ i∈j (∫ ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) ≥ ∑ i∈i (∣∣∣∣ ∂∂xiu ∣∣∣∣p + i pi(.) ) + ∑ i∈j (∣∣∣∣ ∂∂xiu ∣∣∣∣p − i pi(.) ) ≥ ∑ i∈j (∣∣∣∣ ∂∂xiu ∣∣∣∣p − i pi(.) ) ≥ ∑ i∈j (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) ≥ n∑ i=1 (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) − ∑ i∈i (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) ≥ n∑ i=1 (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) −n. 34 a. kaboré & s. ouaro cubo 23, 1 (2021) we now use jensen’s inequality on the convex function z : r+ → r+, z(t) = tp − m, p−m > 1 to get   〈λk(u),u〉≥ 〈λ(u),u〉 ≥ 1 np − m−1 ‖u‖p − m w 1,~p(.) d (ω̃) −n. hence, λk is coercive (as p − m > 1). (iii) the operator λk is of type m. lemma 3.3. (cf [41]) let a and b be two operators. if a is of type m and b is monotone and weakly continuous, then a + b is of type m. now , we set 〈au,v〉 := 〈λ(u),v〉 and 〈bku,v〉 := ∫ ω tk(b(u))vdx + ∫ γ̃ne tk(ρ̃(u))vdσ. then, for every k > 0, we have λk = a + bk. we now have to show that for every k > 0, bk is monotone and weakly continuous, because it is well-known that a is of type m. for the monotonicity of bk, we have to show that 〈bku−bkv,u−v〉≥ 0 for all (u,v) ∈ w 1,~p(.) d (ω̃) ×w 1,~p(.) d (ω̃). we have 〈bku−bkv,u−v〉 = ∫ ω (tk(b(u)) −tk(b(v)))(u−v)dx + ∫ γ̃ne (tk(ρ̃(u)) −tk(ρ̃(v)))(u−v)dσ. from the monotonicity of b, ρ̃ and the map tk, we conclude that 〈bku−bkv,u−v〉≥ 0. (3.8) we need now to prove that for each k > 0 the operator bk is weakly continuous, that is, for all sequences (un)n∈n ⊂ w 1,~p(.) d (ω̃) such that un ⇀ u in w 1,~p(.) d (ω̃), we have bkun ⇀ bku as n →∞. for all φ ∈ w 1,~p(.)d (ω̃), we have 〈bkun,φ〉 := ∫ ω tk(b(un))φdx + ∫ γ̃ne tk(ρ̃(un))φdσ. (3.9) passing to the limit in (3.9) as n goes to ∞ and using the lebesgue dominated convergence theorem, since un ⇀ u in w 1,~p(.) d (ω̃); up to a subsequence, we have un → u in l 1(ω̃) and a.e. in ω̃. as |tk(b(un))φ| ≤ k|φ| and φ ∈ w 1,~p(.) d (ω̃) ↪→ l 1(ω̃), for the first term on the right-hand side of (3.9), we obtain lim n→∞ ∫ ω tk(b(un))φdx = ∫ ω tk(b(u))φdx. (3.10) furthermore, since un ⇀ u in w 1,~p(.) d (ω̃); up to a subsequence, we have un → u in l 1(∂ω̃) and a.e. on ∂ω̃ . as |tk(ρ̃(un))φ| ≤ k|φ| and φ ∈ w 1,~p(.) d (ω̃) ↪→ l 1(∂ω̃), we deduce by the lebesgue dominated convergence theorem that lim n→∞ ∫ γ̃ne tk(ρ̃(un))φdx = ∫ γ̃ne tk(ρ̃(u))φdx. (3.11) cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 35 from (3.10) and (3.11) we conclude that for every k > 0, bk(un) →bk(u) as n →∞. the operator a is type m and as bk is monotone and weakly continuous, thanks to lemma 3.3, we conclude that the operator λk is of type m. then for any l ∈ (w 1,~p(.) d (ω̃)) ′, there exists u�,k ∈ w 1,~p(.) d (ω̃), such that λk(u�,k) = l. we now consider l ∈ (w 1,~p(.)d (ω̃)) ′ defined by l(v) = ∫ ω vdµ� + ∫ γ̃ne d̃�vdσ, for v ∈ w 1,~p(.) d (ω̃) and we obtain (3.5)� step 2: a priori estimates lemma 3.4. let u�,k a solution of p�,k(ρ̃, µ̃�, d̃�). then  |ρ̃(u�,k)| ≤ k1 := max{‖d̃�‖∞, (ρ̃� ◦ b−1)(‖µ�‖∞)} a.e. on γ̃ne, |b(u�,k)| ≤ k2 := max{|µ�‖∞; (b◦ρ−10 )(|γ̃ne|‖d̃�‖∞)} a.e. in ω. (3.12) proof. for any τ > 0, let us introduce the function hτ : r → r by hτ (s) =   0 if s < 0, s τ if 0 ≤ s ≤ τ, 1 if s > τ. in (3.5) we set ξ̃ = hτ (u�,k −m), where m > 0 is to be fixed later. we get  ∫ ω̃ n∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi hτ (u�,k −m)dx + ∫ ω tk(b(u�,k))hτ (u�,k −m)dx =∫ ω hτ (u�,k −m)dµ� + ∫ γ̃ne (d̃� −tk(ρ̃(u�,k)))hτ (u�,k −m)dσ. (3.13) the first term in (3.13) is non-negative. indeed,∫ ω̃ n∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi hτ (u�,k −m)dx = 1 τ ∫ {0≤u�,k−m≤τ} n∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi u�,kdx ≥ 0. from (3.13) we obtain∫ ω tk(b(u�,k))hτ (u�,k −m)dx ≤ ∫ ω hτ (u�,k −m)dµ� + ∫ γ̃ne (d̃� −tk(ρ̃(u�,k)))hτ (u�,k −m)dσ. then, one has  ∫ ω (tkb(u�,k) −tk(b(m)))hτ (u�,k −m)dx + ∫ γ̃ne (tk(ρ̃(u�,k)) −tk(ρ̃(m)))hτ (u�,k −m)dx ≤∫ ω (µ� −tk(b(m)))hτ (u�,k −m)dx + ∫ γ̃ne (d̃� −tk(ρ̃(m)))hτ (u�,k −m)dσ. letting τ go to 0 in the inequality above, we get  ∫ ω (tk(b(u�,k)) −tk(b(m)))+dx + ∫ γ̃ne (tk(ρ̃(u�,k)) −tk(ρ̃(m)))+dσ ≤∫ ω (µ� −tk(b(m)))sign+0 (uk −m)dx + ∫ γ̃ne (d̃� −tk(ρ̃(m)))sign+0 (u�,k −m)dσ. 36 a. kaboré & s. ouaro cubo 23, 1 (2021) as im(b) = im(ρ) = r, we can fix m = m0 = max{b−1(‖µ�‖∞),ρ−10 (|γ̃ne|‖d̃�‖∞)}. from the above inequality we obtain  ∫ ω (tk(b(u�,k)) −tk(b(m0)))+dx + ∫ γ̃ne (tk(ρ̃(u�,k) −tk(ρ̃(m0)))+dσ ≤∫ ω (µ� −tk(‖µ�‖∞))sign+0 (u�,k −m0)dx + ∫ γ̃ne (d̃−tk(‖d̃�‖∞))sign+0 (u�,k −m0)dσ. for k > k0 := max{‖µ�‖,‖d̃�‖∞}, it follows that∫ ω (tk(b(u�,k)) −tk(b(m0)))+dx + ∫ γ̃ne (tk(ρ̃(u�,k)) −tk(ρ̃(m0)))+dσ ≤ 0. (3.14) from (3.14), we deduce that  tk(ρ̃(u�,k)) ≤ tk(ρ̃(m0)) a.e. on γ̃ne, tk(b(u�,k)) ≤ tk(b(m0)) a.e. in ω. (3.15) from (3.15), we deduce that for every k > k1 := max{‖d̃�‖∞,‖µ�‖∞, b(m0), ρ̃(m0)}, ρ̃(u�,k) ≤ ρ̃(m0) a.e. on γ̃ne and b(u�,k) ≤ b(m0) a.e. in ω. note that with the choice of m0 and the fact that d(ρ) = d(b) = r, for every k > k1 := max{‖d̃�‖∞,‖µ�‖∞, b(m0), ρ̃(m0)}, we have  b(u�,k) ≤ max{‖µ�‖∞,b◦ρ−10 (|γ̃ne|‖d̃�‖∞) } a.e. in ω, ρ̃(u�,k) ≤ max{‖d̃�‖∞, (ρ̃◦ b−1)(‖µ�‖∞)} a.e. on γ̃ne. (3.16) we need to show that for any k large enough,  b(u�,k) ≥−max{‖µ�‖∞,b◦ρ−10 (|γ̃ne|‖d̃�‖∞)} a.e. in ω, ρ̃(u�,k) ≥−max{‖d̃�‖∞, (ρ̃◦ b−1)(‖µ�‖∞)} a.e. on γ̃ne. (3.17) it is easy to see that if (u�,k) is a solution of p�,k(ρ̃, µ̃�, d̃�), then (−u�,k) is a solution of p�,k(ρ̂, µ̂�, d̂�)   − n∑ i=1 ∂ ∂xi âi(x, ∂ ∂xi u�,k) + tk(b̂(u�,k))χω(x) = µ̂� in ω̃ u�,k = 0 on γd tk(ρ̂(u�,k)) + n∑ i=1 âi(x, ∂ ∂xi u�,k)ηi = d̂� on γ̃ne, where âi(x,ξ) = −ãi(x,−ξ), ρ̂(s) = −ρ̃(−s), b̂(s) = −b(−s), µ̂� = −µ̃� and d̂ = −d̃�. then for every k > k2 := max{‖d̃�‖∞,‖µ�‖∞, −b(−m0), −ρ̃(−m0)}, we have  −b(u�,k) ≤ max{‖µ�‖∞,b◦ρ−10 (|γ̃ne|‖d̃�‖∞)} a.e. in ω, −ρ̃(u�,k) ≤ max{‖d̃�‖∞, (ρ̃◦ b−1)(‖µ�‖∞)} a.e. on γ̃ne, cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 37 which implies (3.17). from (3.16) and (3.17), we deduce (3.12). step 3. convergence since u�,k is a solution of p�,k(ρ̃, µ̃�, d̃�), thanks to lemma 3.4 and the fact that ω is bounded, we have ρ̃(u�,k) ∈ l1(γ̃ne) and b(u�,k) ∈ l1(ω). for k = 1 + max(k1,k2) fixed, by lemma 3.4, one sees that problem p�(ρ̃, µ̃�, d̃�) admits at least one solution u� � remark 3.5. using the relation (3.12) and the fact that the functions b and ρ are non-decreasing, it follows that for k large enough, the solution of the problem p(ρ̃, µ̃�, d̃�) belongs to l ∞(ω) ∩ l∞(γ̃ne) and |u�| ≤ c(b,k1) a.e. in ω and |u�| ≤ c(ρ,k2) a.e. on γ̃ne. now, we set ãi(x,ξ) = ai(x,ξ)χω(x) + 1 �pi(x) |ξ|pi(x)−2ξχω̃\ω(x) for all (x,ξ) ∈ ω̃ × r n and we consider the following problem. p�(ρ̃, µ̃�, d̃�)  − n∑ i=1 ∂ ∂xi ( ai ( x, ∂ ∂xi u� ) χω(x) + 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu�χω̃\ω(x) ) + |u�|pm (x)−2u�χω = µ̃� in ω̃ u� = 0 on γd ρ̃(u�) + n∑ i=1 ãi(x, ∂ ∂xi u�)ηi = d̃� on γ̃ne. (3.18) thanks to theorem 3.2, p�(ρ̃, µ̃�, d̃�) has at least one solution. so, there exists at least one measurable function u� : ω̃ → r such that  n∑ i=1 ∫ ω ai ( x, ∂ ∂xi u� ) ∂ ∂xi ξ̃dx + n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) | ∂ ∂xi u�|pi(x)−2 ∂ ∂xi u�. ∂ ∂xi ξ̃ ) dx + ∫ ω |u�|pm (x)−2u�ξ̃dx = ∫ ω ξ̃dµ� + ∫ γ̃ne (d̃� − ρ̃(u�)ξ̃dσ, (3.19) where u� ∈ w 1,~p(.) d (ω̃) and ξ̃ ∈ w 1,~p(.) d (ω̃) ∩l ∞(ω). moreover u� ∈ l∞(ω) ∩l∞(γ̃ne). our aim is to prove that these approximated solutions u� tend, as � goes to 0, to a measurable function u which is an entropy solution of the problem p(ρ̃, µ̃, d̃). to start with, we establish some a priori estimates. proposition 3.6. let u� be a solution of the problem p�(ρ̃, µ̃�, d̃�). then, the following statements hold. (i) ∀k > 0, n∑ i=1 ∫ ω ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) dx + n∑ i=1 ∫ ω̃\ω ( 1 � ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣ )pi(x) dx ≤ k(‖d̃‖l1(γ̃ne) + |µ|(ω)); 38 a. kaboré & s. ouaro cubo 23, 1 (2021) (ii) ∫ ω |u�|pm (x)−1dx + ∫ γ̃ne |ρ̃(u�)|dx ≤ (‖d̃‖l1(γ̃ne) + |µ|(ω)); (iii) ∀k > 0, n∑ i=1 ∫ ω̃ ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) dx ≤ k(‖d̃‖l1(γ̃ne) + |µ|(ω)). proof. for any k > 0, we set ξ̃ = tk(u�) in (3.19), to get  n∑ i=1 ∫ ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi tk(u�) ) dx + n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu� ∂∂xitk(u�) ) dx∫ ω |u�|pm (x)−2u�tk(u�)dx = ∫ ω tk(u�)dµ� + ∫ γ̃ne (d̃� − ρ̃(u�))tk(u�)dσ. (3.20) (i) obviously, we have n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu� ∂∂xitk(u�) ) dx = n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) ) dx ≥ 0,∫ γ̃ne ρ̃(u�)tk(u�)dσ ≥ 0 and ∫ ω |u�|pm (x)−2u�tk(u�)dx ≥ 0. moreover,   ∫ ω tk(u�)dµ� + ∫ γ̃ne d̃�tk(u�)dσ ≤ k ∫ ω dµ� + k ∫ γ̃ne |d̃�|dσ ≤ k ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) . (3.21) using the inequalities above and (1.7), it follows that n∑ i=1 ∫ ω ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣pi(x) dx ≤ k ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) . (3.22) as n∑ i=1 ∫ ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi tk(u�) ) dx ≥ 0, ∫ γ̃ne ρ̃(u�)tk(u�)dσ ≥ 0 and∫ ω |u�|pm (x)−2u�tk(u�)dx ≥ 0, therefore, we get from (3.20), n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) | ∂ ∂xi tk(u�)|pi(x) ) dx ≤ k ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) (3.23) adding (3.22) and (3.23), we obtain (i). (ii) the first two terms in (3.20) are non-negative and using (3.21), we have from (3.20) the following ∫ γ̃ne ρ̃(u�)tk(u�)dσ + ∫ ω |u�|pm (x)−2u�tk(u�)dx ≤ k ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) . we divide the above inequality by k > 0 and let k go to zero, to get∫ γ̃ne ρ̃(u�)sign(u�)dσ + ∫ ω |u�|pm (x)−2u�sign(u�)dx = ∫ γ̃ne |ρ̃(u�)|dσ + ∫ ω |u�|pm (x)−1dx ≤ ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) . cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 39 (iii) for all k > 0, we have n∑ i=1 ∫ ω̃ ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) dx ≤ n∑ i=1 ∫ ω ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) dx + n∑ i=1 ∫ ω̃\ω ∣∣∣∣1� ∂∂xitk(u�) ∣∣∣∣pi(x) dx, for any 0 < � < 1. according to (i ), we deduce that n∑ i=1 ∫ ω̃ ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) dx ≤ k ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) . lemma 3.7. there is a positive constant d such that meas{|u�| > k}≤ dp − m (1 + k) kp − m−1 , ∀k > 0. proof. let k > 0; by using proposition 3.6-(iii), we have n∑ i=1 ∫ ω̃ ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣p − m(x) dx ≤ n∑ i=1 ∫  ∣∣∣∣∣∣ ∂tk(u�) ∂xi ∣∣∣∣∣∣>1   ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣p − m(x) dx + nmeas(ω̃) ≤ n∑ i=1 ∫ ω̃ ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣pi(x) dx + nmeas(ω̃) ≤ k ( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) + nmeas(ω̃) ≤ c′(k + 1), with c′ = max (( |µ|(ω) + ∫ γ̃ne |d̃|dσ ) ; nmeas(ω̃) ) . we can write the above inequality as n∑ i=1 ∥∥∥∥∂tk(u�)∂xi ∥∥∥∥p − m p − m ≤ c′(1 + k) or ‖tk(u�)‖ w 1,p − m d (ω̃) ≤ [c′(1 + k)] 1 p − m . by the poincaré inequality in constant exponent, we obtain ‖tk(u�)‖lp−m(ω̃) ≤ d(1 + k) 1 p − m . the above inequality implies that∫ ω̃ |tk(u�)|p − mdx ≤ dp − m(1 + k), from which we obtain meas{|u�| > k}≤ dp − m (1 + k) kp − m , since ∫ ω̃ |tk(u�)|p − mdx = ∫ {|u�|>k} |tk(u�)|p − mdx + ∫ {|u�|≤k} |tk(u�)|p − mdx, 40 a. kaboré & s. ouaro cubo 23, 1 (2021) we get ∫ {|u�|>k} |tk(u�)|p − mdx ≤ ∫ ω̃ |tk(u�)|p − mdx and kp − mmeas{|u�| > k}≤ ∫ ω̃ |tk(u�)|p − mdx ≤ dp − m(1 + k) lemma 3.8. there is a positive constant c such that n∑ i=1 ∫ ω̃ (∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i ) dx ≤ c(k + 1), ∀k > 0. (3.24) proof. let k > 0, we set ω1 = { |u| ≤ k; ∣∣∣∣ ∂∂xiu� ∣∣∣∣ ≤ 1 } and ω2 = { |u| ≤ k; ∣∣∣∣ ∂∂xiu� ∣∣∣∣ > 1 } ; using proposition 3.6-(iii), we have n∑ i=1 ∫ ω̃ (∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i ) dx = n∑ i=1 ∫ ω1 (∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i ) dx + n∑ i=1 ∫ ω2 (∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i ) dx ≤ nmeas(ω̃) + n∑ i=1 ∫ ω̃ (∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) ) dx ≤ nmeas(ω̃) + k ( |µ|(ω) + ‖d̃‖l1(γ̃ne) ) ≤ c(k + 1), with c = max { nmeas(ω̃); ( |µ|(ω) + ‖d̃‖l1(γ̃ne) )} . lemma 3.9. for all k > 0, there is two constants c1 and c2 such that (i) ‖u�‖mq∗(ω̃) ≤ c1; (ii) ∣∣∣∣ ∣∣∣∣ ∂∂xiu� ∣∣∣∣ ∣∣∣∣ mp − i q/p (ω̃) ≤ c2. proof. (i) by lemma 3.8, we have n∑ i=1 ∫ ω̃ ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i dx ≤ c(1 + k), ∀k > 0 and i = 1, ...,n. • if k > 1, we have n∑ i=1 ∫ ω̃ ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i dx ≤ c′k, which means tk(u�) ∈ w 1,(p − 1 ,...,p − n )(ω̃). using relation (2.8), we deduce that ‖tk(u�)‖l(p̄)∗(ω̃ ≤ c1 n∏ i=1 ∥∥∥∥ ∂∂xitk(u�) ∥∥∥∥ 1 n l p − i (ω̃) . cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 41 so, ∫ ω̃ |tk(u�)| (p̄)∗ dx ≤ c   n∏ i=1 (∫ ω̃ ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i dx ) 1 np−i   (p̄)∗ ≤ c′′   n∏ i=1 (k) 1 np−i   (p̄)∗ ≤ c′′  k n∑ i=1 1 np−i   (p̄)∗ ≤ c′′k (p̄)∗ p̄ . thus, ∫ {|u�|>k} |tk(u�)| (p̄)∗ dx ≤ ∫ ω̃ |tk(u�)| (p̄)∗ dx ≤ c′k (p̄)∗ p̄ and so, (k)(p̄) ∗ meas{x ∈ ω̃ : |u�| > k} ≤ c′k (p̄)∗ p̄ ; which means that λu�(k) ≤ c ′k (p̄)∗( 1 p̄ −1) = c′k−q ∗ , ∀k ≥ 1. • if 0 < k < 1, we have λu�(k) = meas { x ∈ ω̃ : |u�| > k } ≤ meas(ω̃) ≤ meas(ω̃)k−q ∗ . so, λu�(k) ≤ (c ′ + meas(ω̃))k−q ∗ = c1k −q∗. therefore, ‖u�‖mq∗(ω̃) ≤ c1. 42 a. kaboré & s. ouaro cubo 23, 1 (2021) (ii) • let α ≥ 1. for all k ≥ 1, we have λ∂u� ∂xi (α) = meas ({∣∣∣∣∂u�∂xi ∣∣∣∣ > α }) = meas ({∣∣∣∣∂u�∂xi ∣∣∣∣ > α; |u�| ≤ k }) + meas ({∣∣∣∣∂u�∂xi ∣∣∣∣ > α; ; |u�| > k }) ≤ ∫  ∣∣∣∣∣∣ ∂u� ∂xi ∣∣∣∣∣∣>α;|u�|≤k   dx + λu�(k) ≤ ∫ {|u�|≤k} ( 1 α ∣∣∣∣∂u�∂xi ∣∣∣∣ )p− i dx + λu�(k) ≤ α−p − i c′k + ck−q ∗ ≤ b ( α−p − i k + k−q ∗ ) , with b = max(c′; c). let g : [1;∞) → r, x 7→ g(x) = x αp − i + x−q ∗ . we have g′(x) = 0 with x = ( q∗αp − ) 1 q∗ + 1 . we set k = ( q∗αp − i ) 1 q∗ + 1 ≥ 1 in the above inequality to get, λ∂u� ∂xi (α) ≤ b  α−p−i ×(q∗αp−i ) 1 q∗ + 1 + ( q∗αp − i ) −q∗ q∗ + 1   ≤ b  (q∗) 1 q∗ + 1 ×α −p− i ( 1− 1 q∗ + 1 ) + (q∗) −q∗ q∗ + 1 ×α −p−i q ∗ q∗ + 1   ≤ b  (q∗) 1 q∗ + 1 ×α −p− i   q∗ q∗ + 1   + (q∗) −q∗ q∗ + 1 ×α −p−i q ∗ q∗ + 1   ≤ mα −p− i q∗ q∗ + 1 ≤ mα −p− i q p̄ , where m = b × max  (q∗) 1 q∗ + 1 ; (q∗) −q∗ q∗ + 1   and as q∗ = n(p̄− 1) n − p̄ , q = n(p̄− 1) n − 1 . cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 43 so, q∗ q∗ + 1 = q∗(n − p̄) n(p̄− 1) + n − p̄ = q∗(n − p̄) np̄− p̄ = n(p̄− 1) (n − 1)p̄ = q p̄ . • if 0 ≤ α < 1, we have. λ∂u� ∂xi (α) = meas ({ x ∈ ω̃ : ∣∣∣∣∂u�∂xi ∣∣∣∣ > α }) ≤ meas(ω̃)α −p− i q p̄ . therefore, λ∂u� ∂xi (α) ≤ ( m + meas(ω̃) ) α −p− i q p̄ , ∀ α ≥ 0. so, ∥∥∥∥∂u�∂xi ∥∥∥∥ h ≤ c2, where h = m(ω̃) p−i q p̄ proposition 3.10. let u� be a solution of the problem p(ρ̃, µ̃�, d̃�). then, (i) u� → u in measure, a.e. in ω and a.e. on γ̃n ; (ii) for all i = 1, ...n, ∂tk(u�) ∂xi ⇀ ∂tk(u) ∂xi = 0 in lp − i (ω̃ \ ω). proof. (i) by proposition 3.6 (i), we deduce that (tk(u�))�>0 is bounded in w 1,~p(.) d (ω̃) ↪→ lpm(.)(ω̃) ↪→ lp − m(ω̃) (with compact embedding). therefore, up to a subsequence, we can assume that as � → 0, (tk(u�))�>0 converges strongly to some function σk in lp − m(ω̃), a.e. in ω̃ and a.e. on γ̃ne. let us see that the sequence (u�)�>0 is cauchy in measure. indeed, let s > 0 and define: e1 = [|u�1| > k], e2 = [|u�2| > k] and e3 = [|tk(u�1 ) −tk(u�2 )| > s], where k > 0 is fixed. we note that [|u�1 −u�2| > s] ⊂ e1 ∪e2 ∪e3; 44 a. kaboré & s. ouaro cubo 23, 1 (2021) hence, meas([|u�1 −u�2| > s]) ≤ 3∑ i=1 meas(ei). (3.25) let θ > 0, using lemma 3.7, we choose k = k(θ) such that meas(e1) ≤ θ 3 and meas(e2) ≤ θ 3 . (3.26) since (tk(u�))�>0 converges strongly in l p−m(ω̃), then, it is a cauchy sequence in lp − m(ω̃). thus, meas(e3) ≤ 1 sp − m ∫ ω |tk(u�1 ) −tk(u�2 )| p−mdx ≤ θ 3 , (3.27) for all �1,�2 ≥ n0(s,θ). finally, from (3.25), (3.26) and (3.27), we obtain meas([|u�1 −u�2| > s]) ≤ θ for all �1,�2 ≥ n0(s,θ); (3.28) which means that the sequence (u�)�>0 is cauchy in measure, so u� → u in measure and up to a subsequence, we have u� → u a.e. in ω̃. hence, σk = tk(u) a.e. in ω̃ and so, u ∈t 1,~p(.) d (ω). (ii) according to the proof of (i), we have tk(u�) ⇀ tk(u) in w 1,~p(.) d (ω̃) ↪→ w 1,~p− d (ω̃) which implies on one hand that for all i = 1, ...n, ∂tk(u�) ∂xi ⇀ ∂tk(u) ∂xi in lpi(.)(ω̃) and on the other hand that for all i = 1, ...n, ∂tk(u�) ∂xi ⇀ ∂tk(u) ∂xi in lpi(.)(ω̃) and then for all i = 1, ...n, ∂tk(u�) ∂xi ⇀ ∂tk(u) ∂xi in lp − i (ω̃ \ ω). let i = 1, ...,n, by proposition 3.6-(i), we can assert that ( 1 � ∂tk(u�) ∂xi ) �>0 is bounded in lp − i (ω̃ \ ω). indeed, let k > 0, we set ω1 = { x ∈ ω̃ \ ω; |u(x)| ≤ k; ∣∣∣∣ ∂∂xiu�(x) ∣∣∣∣ ≤ � } and ω2 = { x ∈ ω̃ \ ω; |u| ≤ k; ∣∣∣∣ ∂∂xiu�(x) ∣∣∣∣ > � } ; using proposition 3.6-(i), we have n∑ i=1 ∫ ω̃\ω ( 1 � ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣p − i ) dx = n∑ i=1 ∫ ω1 ( 1 � ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣p − i ) dx + n∑ i=1 ∫ ω2 ( 1 � ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣p − i ) dx ≤ nmeas(ω̃ \ ω) + n∑ i=1 ∫ ω̃\ω ( 1 � ∣∣∣∣ ∂∂xitk(u�) ∣∣∣∣pi(x) ) dx ≤ nmeas(ω̃ \ ω) + k ( |µ|(ω) + ‖d̃‖l1(γ̃ne) ) ≤ c′(k + 1), with c′ = max { nmeas(ω̃ \ ω); ( |µ|(ω) + ‖d̃‖l1(γ̃ne) )} . to end, we have ∫ ω̃\ω ( 1 � ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣p − i ) dx ≤ n∑ i=1 ∫ ω̃\ω ( 1 � ∣∣∣∣∂tk(u�)∂xi ∣∣∣∣p − i ) dx, for anyi = 1, . . . ,n. cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 45 therefore, there exists θk ∈ lp − i (ω̃ \ ω) such that 1 � ∂tk(u�) ∂xi ⇀ θk in l p − i (ω̃ \ ω) as � → 0. for any ψ ∈ l(p ′ i) − (ω̃ \ ω), we have∫ ω̃\ω ∂tk(u�) ∂xi ψdx = ∫ ω̃\ω ( 1 � ∂tk(u�) ∂xi − θk ) (�ψ)dx + � ∫ ω̃\ω θkψdx. (3.29) as (�ψ)�>0 converges strongly to zero in l (p′i) − (ω̃\ω), we pass to the limit as � → 0 in (3.29), to get ∂tk(u�) ∂xi ⇀ 0 in lp − i (ω̃ \ ω). hence, one has ∂tk(u�) ∂xi ⇀ ∂tk(u) ∂xi = 0 in lp − i (ω̃ \ ω), for any i = 1, ...,n. lemma 3.11. b(u) ∈ l1(ω) and ρ̃(u) ∈ l1(γ̃ne). proof. having in mind that by proposition 3.6-(ii),∫ ω |b(u�)|dx + ∫ γ̃ne |ρ̃(u�)|dσ ≤ (|µ|(ω) + ‖d̃‖l1(γ̃ne)), we deduce that ∫ ω |b(u�)|dx ≤ (|µ|(ω) + ‖d̃‖l1(γ̃ne)) (3.30) and ∫ γ̃ne |ρ̃(u�)|dσ ≤ (|µ|(ω) + ‖d̃‖l1(γ̃ne)). (3.31) by fatou’s lemma, the continuity of b, ρ̃ and using proposition 3.10, we have lim inf �→0 ∫ ω |b(u�)|dx ≥ ∫ ω |b(u)|dx (3.32) and lim inf �→0 ∫ γ̃ne |ρ̃(u�)|dσ ≥ ∫ γ̃ne |ρ̃(u)|dσ. (3.33) using (3.30)-(3.33), we deduce that∫ ω |b(u)|dx ≤ (|µ|(ω) + ‖d̃‖l1(γ̃ne)) and ∫ γ̃ne |ρ̃(u)|dσ ≤ (|µ|(ω) + ‖d̃‖l1(γ̃ne)). therefore, b(u) ∈ l1(ω) and ρ̃(u) ∈ l1(γ̃ne). 46 a. kaboré & s. ouaro cubo 23, 1 (2021) lemma 3.12. assume (1.4)-(1.8) hold and u� be a weak solution of the problem p(ρ,µ̃�, d̃�). then, (i) ∂ ∂xi u� converges in measure to ∂ ∂xi u . (ii) ai ( x, ∂tk(u�) ∂xi ) → ai(x, ∂tk(u) ∂xi ) strongly in l1(ω) and weakly in lp ′ i(.)(ω), for all i=1,...,n. in order to give the proof of lemma 3.12, we need the following lemmas. lemma 3.13 (cf [6]). let u ∈ t 1,~p(.)(ω). then, there exists a unique measurable function νi : ω → r such that νiχ{|u| 0 and i = 1, ...,n; where χa denotes the characteristic function of a measurable set a. the functions νi are denoted ∂ ∂xi u. moreover, if u belongs to w 1,~p(.)(ω), then νi ∈ lpi(.)(ω) and coincides with the standard distributional gradient of u i.e. νi = ∂ ∂xi u. lemma 3.14 (cf [37], lemma 5.4). let (vn)n∈n be a sequence of measurable functions. if vn converges in measure to v and is uniformly bounded in lp(.)(ω) for some 1 << p(.) ∈ l∞(ω), then vn → v strongly in l1(ω). the third technical lemma is a standard fact in measure theory (cf [16]). lemma 3.15. let (x,m,µ) be a measurable space such that µ(x) < ∞. consider a measurable function γ : x → [0;∞] such that µ({x ∈ x : γ(x) = 0}) = 0. then, for every � > 0, there exists δ such that µ(a) < �, for all a ∈m with ∫ a γdx < δ. proof of lemma 3.12. (i) we claim that ( ∂ ∂xi u� ) �∈n is cauchy in measure. indeed, let s > 0, consider an,m := {∣∣∣∣ ∂∂xiun ∣∣∣∣ > h } ∪ {∣∣∣∣ ∂∂xium ∣∣∣∣ > h } , bn,m := {|un −um| > k} and cn,m := {∣∣∣∣ ∂∂xiun ∣∣∣∣ ≤ h, ∣∣∣∣ ∂∂xium ∣∣∣∣ ≤ h, |un −um| ≤ k, ∣∣∣∣ ∂∂xiun − ∂∂xium > s ∣∣∣∣ } , where h and k will be chosen later. one has{∣∣∣∣ ∂∂xiun − ∂∂xium ∣∣∣∣ > s } ⊂ an,m ∪bn,m ∪cn,m. (3.34) let ϑ > 0. by lemma 3.9, we can choose h = h(ϑ) large enough such that meas(an,m) ≤ ϑ 3 for all n,m ≥ 0. on the other hand, by proposition 3.10, we have that meas(bn,m) ≤ ϑ 3 cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 47 for all n,m ≥ n0(k,ϑ). moreover, by assumption (h3), there exists a real valued function γ : ω → [0,∞] such that meas{x ∈ ω : γ(x) = 0} = 0 and (ai(x,ξ) −ai(x,ξ′)).(ξ − ξ′) ≥ γ(x), (3.35) for all i = 1, ...,n, |ξ|, |ξ′| ≤ h, |ξ − ξ′| ≥ s, for a.e. x ∈ ω. indeed, let’s set k = {(ξ,η) ∈ r×r : |ξ| ≤ h, |η| ≤ h, |ξ −η| ≥ s}. we have k ⊂ b(0,h) ×b(0,h) and so k is a compact set because it is closed in a compact set. for all x ∈ ω and for all i = 1, ...,n, let us define ψ : k → [0;∞] such that ψ(ξ,η) = (ai(x,ξ) −ai(x,η)).(ξ −η). as for a.e. x ∈ ω, ai(x,.) is continuous on r, ψ is continuous on the compact k, by weierstrass theorem, there exists (ξ0,η0) ∈ k such that ∀(ξ,η) ∈ k, ψ(ξ,η) ≥ ψ(ξ0,η0). now let us define γ on ω as follows. γ(x) = ψi(ξ0,η0) = (ai(x,ξ0) −ai(x,η)).(ξ −η0). since s > 0, the function γ is such that meas ({x ∈ ω : γ(x) = 0}) = 0. let δ = δ(�) be given by lemma 3.15, replacing � and a by � 3 and cn,m respectively. taking respectively ξ̃ = tk(un −um) and ξ̃ = tk(um −un) for the weak solutions un and um in (3.19) and after adding the two relations, we have  n∑ i=1 ∫ {|un−um| s }) ≤ ϑ, (3.36) for all n,m ≥ n0(s,ϑ), and then the claim is proved. as consequence, ( ∂ ∂xi u� ) �∈n converges in measure to some measurable function νi. in order to end the proof of lemma 3.12, we need the following lemma. lemma 3.16. (a) for a.e. k ∈ r, ∂ ∂xi tk(u�) converges in measure to νiχ{|u| 0, { |χ{|u�| δ } ⊂ { |χ{|u�| δ }) ≤ meas ({|u| = k}) + meas ({u� < k < u}) +meas ({u < k < u�}) +meas ({u� < −k < u}) +meas ({u < −k < u�}) . (3.37) note that meas ({|u| = k}) ≤ meas ({k −h 0, since u is fixed function. next, meas ({u� < k < u}) ≤ meas ({k h}) , for all cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 49 h > 0. due to proposition 3.10, we have for all fixed h > 0, meas ({|u� −u| > h}) → 0 as � → 0. since meas ({k 0, one can find n such that for all n > n, meas ({|u| = k}) < ϑ 2 + ϑ 2 = ϑ by choosing h and then n. each of the other terms on the right-hand side of (3.37) can be treated in the same way as for meas ({u� < k < u}) . thus, meas ({ |χ{|u�| δ} }) → 0 as � → 0. finally, since ∂ ∂xi tk(u�) = ∂ ∂xi u�χ{|u�| 0, i = 1, ...,n, ai ( x, ∂ ∂xi tk(u�) ) converges to ai ( x, ∂ ∂xi tk(u) ) in l1(ω) strongly. indeed, let s,k > 0, consider e4 = {∣∣∣∣∂un∂xi − ∂um∂xi ∣∣∣∣ > s, |un| ≤ k, |um| ≤ k } , e5 = {∣∣∣∣∂um∂xi ∣∣∣∣ > s, |un| > k, |um| ≤ k } , e6 ={∣∣∣∣∂un∂xi ∣∣∣∣ > s, |un| ≤ k, |um| > k } . we have {∣∣∣∣∂tk(un)∂xi − ∂tk(um)∂xi ∣∣∣∣ > s } ⊂ e4 ∪e5 ∪e6. (3.38) 50 a. kaboré & s. ouaro cubo 23, 1 (2021) ∀ϑ > 0 , by lemma 3.7, there exists k(ϑ) such that meas(e5) ≤ ϑ 3 and meas(e6) ≤ ϑ 3 . (3.39) using (3.36)-(3.39), we get meas ({∣∣∣∣ ∂∂xitk(un) − ∂∂xitk(um) ∣∣∣∣ > s }) ≤ ϑ, (3.40) for all n,m ≥ n1(s,ϑ). therefore, ∂tk(u�) ∂xi converges in measure to ∂tk(u) ∂xi . using lemmas 3.8 and 3.14, we deduce that ∂tk(u�) ∂xi converges to ∂tk(u) ∂xi in l1(ω). so, after passing to a suitable subsequence of ( ∂tk(u�) ∂xi ) �>0 , we can assume that ∂tk(u�) ∂xi converges to ∂tk(u) ∂xi a.e. in ω. by the continuity of ai(x,.), we deduce that ai ( x, ∂tk(u�) ∂xi ) converges to ai ( x, ∂tk(u) ∂xi ) a.e. in ω. as ω is bounded, this convergence is in measure. using lemmas 3.14 and 3.16, we deduce that for all k > 0, i = 1, ...,n, ai ( x, ∂ ∂xi tk(u�) ) converges to ai ( x, ∂ ∂xi tk(u) ) in l1(ω) strongly and ai ( x, ∂ ∂xi tk(u�) ) converges to χk ∈ lp ′ i(.)(ω) weakly in lp ′ i(.)(ω). since each of the convergences implies the weak l1-convergence, χk can be identified with ai ( x, ∂ ∂xi tk(u) ) ; thus, ai ( x, ∂ ∂xi tk(u) ) ∈ lp ′ i(.)(ω) by using lebesgue generalized convergence theorem and above results, we deduce the following result. proposition 3.17. for any k > 0 and any i = 1, ...,n , as � tends to 0, we have (i) ∂tk(u�) ∂xi → ∂tk(u) ∂xi a.e. in ω, (ii) ai ( x, ∂tk(u�) ∂xi ) ∂tk(u�) ∂xi → ai ( x, ∂tk(u) ∂xi ) ∂tk(u) ∂xi a.e. in ω and strongly in l1(ω), (iii) ∂tk(u�) ∂xi → ∂tk(u) ∂xi strongly in lpi(x)(ω). 4 existence and uniqueness of solution to p(ρ,µ,d) we are now able to prove theorem 2.6. proof of theorem 2.6 thanks to the proposition 3.10 and as ∀k > 0, ∀i = 1, ...,n, ∂tk(u) ∂xi = 0 in lp − i (ω̃ \ ω), then, ∀k > 0, tk(u) = constant a.e. on ω̃ \ ω. hence, we conclude that u ∈t 1,~p(.) ne (ω). cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 51 we already state that b(u) ∈ l1(ω). to show that u is an entropy solution of p(ρ,µ,d), we only have to prove the inequality in (2.9). let ϕ ∈ w 1,~p(.)d (ω) ∩l ∞(ω). we consider the function ϕ1 ∈ w 1,~p(.) d (ω̃) ∩l ∞(ω) such that ϕ1 = ϕχω + ϕnχω̃\ω. we set ξ̃ = tk(u� −ϕ1) in (3.19) to get  n∑ i=1 ∫ ω ( ai ( x, ∂ ∂xi u� ) . ∂ ∂xi tk(u� −ϕ) ) dx + n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu�. ∂∂xitk(u� −ϕn ) ) dx∫ ω b(u�)tk(u� −ϕ)dx = ∫ ω tk(u� −ϕ)dµ� + ∫ γ̃ne (d̃� − ρ̃(u�))tk(u� −ϕn )dσ. (4.1) the following convergence result hold true. lemma 4.1. for any k > 0, for all i = 1, ...,n, as � → 0, ∂ ∂xi tk(u� −ϕ) → ∂ ∂xi tk(u−ϕ) strongly in lpi(.)(ω). proof. let k > 0, i = 1, ...,n. we have∫ ω ∣∣∣∣ ∂∂xitk(u� −ϕ) − ∂∂xitk(u−ϕ) ∣∣∣∣pi(x) dx = ∫ ω∩[|u�−ϕ|≤k,|u−ϕ|≤k] ∣∣∣∣ ∂∂xiu� − ∂∂xiu ∣∣∣∣pi(x) dx ≤ ∫ ω∩[|u�|≤l,|u|≤l] ∣∣∣∣∂u�∂xi − ∂u∂xi ∣∣∣∣pi(x) dx, with l = k + ‖ϕ‖∞ = ∫ ω ∣∣∣∣ ∂∂xitl(u�) − ∂∂xitl(u) ∣∣∣∣pi(x) dx → 0 as � → 0 by proposition 3.17 − (iii). we need to pass to the limit in (4.1) as � → 0. we have n∑ i=1 ∫ ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi tk(u� −ϕ) ) dx = n∑ i=1 ∫ ω ( ai ( x, ∂tl(u�) ∂xi ) ∂ ∂xi tk(u� −ϕ) ) dx, with l = k + ‖ϕ‖∞, then, by lemma 3.12(ii) and lemma 4.1, we have lim �→0 n∑ i=1 ∫ ω ( ai ( x, ∂tl(u�) ∂xi ) ∂ ∂xi tk(u� −ϕ) ) dx = n∑ i=1 ∫ ω ( ai ( x, ∂tl(u) ∂xi ) ∂ ∂xi tk(u−ϕ) ) dx; that is lim �→0 n∑ i=1 ∫ ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi tk(u� −ϕ) ) dx = n∑ i=1 ∫ ω ( ai ( x, ∂tl(u) ∂xi ) ∂ ∂xi tk(u−ϕ) ) dx. (4.2) 52 a. kaboré & s. ouaro cubo 23, 1 (2021) for the second term in the left hand side of (4.1), we have lim sup �→0 n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) | ∂ ∂xi u�|pi(x)−2 ∂ ∂xi u� ∂ ∂xi tk(u� −ϕn ) ) dx ≥ 0. (4.3) indeed   n∑ i=1 ∫ ω̃\ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu� ∂∂xitk(u� −ϕn ) ) dx = n∑ i=1 ∫ ω̃\ω∩[|u�−ϕ|≤k] ( 1 �pi(x) | ∂ ∂xi u�|pi(x)−2 ∂ ∂xi u� ∂ ∂xi (u� −ϕn ) ) dx = n∑ i=1 ∫ ω̃\ω∩[|u�−ϕ|≤k] ( 1 �pi(x) | ∂ ∂xi u�|pi(x) ) dx ≥ 0. hence, we get (4.3). let us examine the last term in the left hand side of (4.1). we have ∫ ω b(u�)tk(u� −ϕ)dx = ∫ ω (b(u�) − b(ϕ))tk(u� −ϕ)dx + ∫ ω b(ϕ)tk(u� −ϕ)dx. as b is non-decreasing, (b(u�) − b(ϕ))tk(u� −ϕ) ≥ 0 a.e. in ω and we get by fatou’s lemma that lim inf �→0 ∫ ω (b(u�) − b(ϕ))tk(u� −ϕ)dx ≥ ∫ ω (b(u) − b(ϕ))tk(u−ϕ)dx. as ϕ ∈ l∞(ω), we obtain b(ϕ) ∈ l∞(ω) and so b(ϕ) ∈ l1(ω) (as ω is bounded) and by lebesgue dominated convergence theorem, we deduce that lim �→0 ∫ ω b(ϕ)tk(u� −ϕ)dx = ∫ ω b(ϕ)tk(u−ϕ)dx. consequently, lim sup �→0 ∫ ω b(u�)tk(u� −ϕ)dx ≥ ∫ ω b(u)tk(u−ϕ)dx. (4.4) as f� → f strongly in l1(ω) and tk(u�−v) ⇀∗ tk(u−v) in l∞(ω), using the lebesgue generalized convergence theorem we have  lim �→0 ∫ ω f�tk(u� −ϕ)dx = ∫ ω tk(u−ϕ)dx, lim �→0 ∫ γ̃ne d̃�tk(u� −ϕn )dσ = ∫ ω d̃tk(u−ϕn )dσ. (4.5) since ∇tk(u� −ϕ) ⇀ ∇tk(u−ϕ) in (lpm(.)(ω))n and f ∈ (lp ′ m(.)(ω))n, lim �→0 ∫ ω f.∇tk(u� −ϕ)dx = ∫ ω f.∇tk(u−ϕ)dx. (4.6) we know that ∀k > 0, tk(u) = constant on ω̃ \ ω, then, it yields that u = constant on ω̃ \ ω. so, one has lim �→0 ∫ γ̃ne d̃�tk(u� −ϕ)dx = dtk(un −ϕn ). (4.7) cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 53 at last, we have ∫ γ̃ne ρ̃(u�)tk(u� −ϕn )dσ = ∫ γ̃ne (ρ̃(u�) − ρ̃(ϕn ))tk(u� −ϕn )dσ + ∫ γ̃ne ρ̃(ϕn )tk(u� −ϕn )dσ. as ρ̃ is non-decreasing, (ρ̃(u�) − ρ̃(ϕn ))tk(u� −ϕn ) ≥ 0 a.e. in γ̃ne and we get by fatou’s lemma that lim inf �→0 ∫ γ̃ne (ρ̃(u�) − ρ̃(ϕn ))tk(u� −ϕn )dσ ≥ ∫ γ̃ne (ρ̃(un ) − ρ̃(ϕn ))tk(un −ϕn )dσ = (ρ(un ) −ρ(ϕn ))tk(un −ϕn ). as ϕn ∈ l∞(γ̃ne), we obtain ρ̃(ϕn ) ∈ l∞(γ̃ne) and so ρ̃(ϕn ) ∈ l1(γ̃ne) (as γ̃ne is bounded) and by the lebesgue dominated convergence theorem, we deduce that lim �→0 ∫ γ̃ne ρ̃(ϕn )tk(u� −ϕn )dσ = ∫ γ̃ne ρ̃(ϕn )tk(un −ϕn )dσ = ρ(ϕn )tk(un −ϕn ). hence, lim sup �→0 ∫ γ̃ne ρ̃(u�)tk(u� −ϕn )dσ ≥ ρ(ϕn )tk(un −ϕn ). (4.8) passing to the limit as � → 0 in (4.1) and using (4.2)-(4.8), we see that u is an entropy solution of p(ρ,µ,d). we now prove the uniqueness part of theorem 2.6. let u and v be two entropy solutions of p(ρ,µ,d). let h > 0. for u, we take ξ = th(v) as test function and for v, we take ξ = th(u) as test function in (2.9), to get for any k > 0 with k < h,   ∫ ω ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi tk(u−th(v)) ) dx + ∫ ω b(u)tk(u−th(v))dx ≤∫ ω ftk(u−th(v))dx + ∫ ω f.∇tk(u−th(v))dx + (d−ρ(une))tk(une −th(v)) (4.9) and   ∫ ω ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi tk(v −th(u)) ) dx + ∫ ω b(v)tk(v −th(u))dx ≤∫ ω ftk(v −th(u))dx + ∫ ω f.∇tk(v −th(u))dx + (d−ρ(vne))tk(vne −th(u)). (4.10) 54 a. kaboré & s. ouaro cubo 23, 1 (2021) by adding (4.9) and (4.10), we obtain  ∫ ω ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi tk(u−th(v)) ) dx + ∫ ω ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi tk(v −th(u)) ) dx := a(h,k) + ∫ ω b(u)tk(u−th(v))dx + ∫ ω b(v)tk(v −th(u))dx := b(h,k) +ρ(une)tk(une −th(v)) + ρ(vne)tk(vne −th(u)) := c(h,k) ≤ ∫ ω ftk(u−th(v))dx + ∫ ω ftk(v −th(u))dx := d(h,k) + ∫ ω f.∇tk(u−th(v))dx + ∫ ω f.∇tk(v −th(u))dx := t(h,k) +dtk(une −th(v)) + dtk(vne −th(u)) := e(h,k). (4.11) let us introduce the following subsets of ω. a0 := [|u−v| < k, |u| < h, |v| < h] a1 := [|u−th(v)| < k, |v| ≥ h] a′1 := [|v −th(u)| < k, |u| ≥ h] a2 := [|u−th(v)| < k, |u| ≥ h, |v| < h] a′2 := [|v −th(u)| < k, |v| ≥ h, |u| < h]. we have the following assertion (see [22] for the proof). assertion 4.2. if u is an entropy solution of p(ρ,µ,d), then a2 ⊂ fh,k and a1 ⊂ fh−k,2k, where fh,k = {h ≤ |u| < h + k,h > 0,k > 0}. assertion 4.3. let u be an entropy solution of p(ρ,µ,d). on a2 (and on a1) we have according to hölder inequality. (1) ∫ a2 f.∇udx ≤ (∫ a2 |f|(p ′ m) − dx ) 1 (p′m) − (∫ a2 |∇u|p − m ) 1 p − m dx, (4.12) with lim h→∞ (∫ a2 |f|(p ′ m) − dx ) 1 (p′m) − (∫ a2 |∇u|p − mdx ) 1 p − m = 0. (2) ∫ a1 f.∇udx ≤ (∫ a1 |f |(p ′ m) − dx ) 1 (p′m) − (∫ a1 |∇u|p − mdx ) 1 p − m , (4.13) with lim h→∞ (∫ a1 |f|(p ′ m) − dx ) 1 (p′m) − (∫ a1 |∇u|p − mdx ) 1 p − m = 0. cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 55 proof. (1) lim h→∞ (∫ a2 |f |(p ′ m) − dx ) 1 (p′m) − = 0 (see [22]). now, it remains to prove that (∫ a2 |∇u|p − mdx ) 1 p − m is bounded with respect to h. we make the following notations: i = { i ∈{1, ...,n} : {∣∣∣∣ ∂∂xiu ∣∣∣∣ } ≤ 1 } and j = { i ∈{1, ...,n} : {∣∣∣∣ ∂∂xiu ∣∣∣∣ } > 1 } . we have n∑ i=1 ∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx = ∑ i∈i (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) + ∑ i∈j (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) ≥ ∑ i∈j (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) ≥ ∑ i∈j (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m dx ) ≥ n∑ i=1 (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m dx ) − ∑ i∈i (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m dx ) ≥ n∑ i=1 (∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m ) −nmeas(ω) ≥ n∑ i=1 ∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣ ∣∣∣∣p − m (lp − m(fh,k))n −nmeas(ω) ≥ c‖∇u‖p − m (lp − m(fh,k))n −nmeas(ω). we deduce that n∑ i=1 ∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ≥ c ∫ fh,k |∇u|p − mdx−nmeas(ω). (4.14) choosing th(u) as test function in (2.9), we get  ∫ ω ( n∑ i=1 ai ( x, ∂ ∂xi u ) ) ∂ ∂xi tk(u−th(u)) ) dx + ∫ ω |u|pm (x)−2utk(u−th(u))dx ≤∫ ω ftk(u−th(u))dx + ∫ ω f.∇tk(u−th(u))dx + (d−ρ(une))tk(une −th(une)). (4.15) according to the fact that ∇tk(u−th(u)) = ∇u on {h ≤ |u| < h + k} and zero elsewhere,∫ ω |u|pm (x)−2utk(u−th(u))dx ≥ 0 and ρ(une)tk(une−th(une)) ≥ 0, we deduce from (4.15) that   ∫ fh,k ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi tk(u−th(u)) ) dx ≤ k ∫ |u|≥h |f|dx + ∫ fh,k ∣∣∣∣∣∣∣ ( 2 cp−m ) 1 p−m f ∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣ ( cp−m 2 ) 1 p−m ∇u ∣∣∣∣∣∣∣dx + k|d|. (4.16) 56 a. kaboré & s. ouaro cubo 23, 1 (2021) using (1.7) (in the left hand side of (4.16)), young inequality (in the right hand side of(4.16)) and setting c = ( 2 cp−m )(p′m)− p−m p − m − 1 p−m , we obtain   n∑ i=1 ∫ fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ≤ k ∫ |u|≥h |f|dx + c ∫ fh,k |f|(p ′)−mdx + c 2 ∫ fh,k |∇u|p − mdx + k|d|. (4.17) from (4.14) and (4.17), we deduce  c ∫ fh,k |∇u|p − mdx ≤ k ∫ |u|≥h |f|dx + c ∫ fh,k |f|(p ′)−mdx + c 2 ∫ fh,k |∇u|p − mdx + k|d| + nmeas(ω). therefore,   c 2 ∫ fh,k |∇u|p − mdx ≤ k ∫ {|u|≥h} |f|dx + c ∫ fh,k |f|(p ′)−mdx + k|d| + nmeas(ω). (4.18) since a2 ⊂ fh,k , we deduce from (4.18) that ∫ a2 |∇u|p − mdx is bounded. (2) lim h→∞ (∫ a1 |f|(p ′ m) − dx ) 1 (p′m) − = 0 (see [22]). now, it remains to prove that (∫ a1 |∇u|p − mdx ) 1 p − m is bounded with respect to h. since a1 ⊂ fh−k,2k , we deduce from (4.18) that ∫ a2 |∇u|p − mdx is bounded. remark 4.4. similarly, we prove that if v is an entropy solution of p(ρ,f,d), then lim h→∞ ∫ a′2 f.∇vdx ≤ 0 and lim h→∞ ∫ a′1 f.∇vdx ≤ 0. now, we have  a(h,k) = ∫ a0 ( n∑ i=1 ( ai ( x, ∂ ∂xi u ) −ai ( x, ∂ ∂xi v )) ∂ ∂xi (u−v) ) dx := i0(h,k) + ∫ a1 ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi u ) dx + ∫ a′1 ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi v ) dx := i1(h,k) + ∫ a2 ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi (u−v) ) dx + ∫ a′2 ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi (v −u) ) dx := i2(h,k). cubo 23, 1 (2021) anisotropic problem with non-local boundary conditions and measure data 57 the term i1(h,k) is non-negative since each term in i1(h,k) is non-negative. for the term i2(h,k), as i2(h,k) + ∫ a2 ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi v ) dx + ∫ a′2 ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi u ) dx = i1(h,k), so, i2(h,k) ≥− (∫ a2 ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi v ) dx + ∫ a′2 ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi u ) dx ) . let us show that − (∫ a2 ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi v ) dx ) goes to 0 as h →∞. we have   ∣∣∣∣∣ ∫ a2 ( n∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi (v) ) dx ∣∣∣∣∣ ≤ c n∑ i=1 ( |ji|p′ i (.) + ∣∣∣∣ ∂u∂xi ∣∣∣∣pi(x)−1 lpi(.)({h<|u|≤h+k}) )∣∣∣∣ ∂v∂xi ∣∣∣∣ lpi(.)({h−k<|v|≤h}) . for all i = 1, ...n, the quantity ( |ji|p′ i (.) + ∣∣∣∣ ∂u∂xi ∣∣∣∣pi(x)−1 lpi(.)({h<|u|≤h+k}) ) is finite since u = th+k(u) ∈t 1,~p(.) ne (ω) and ji ∈ l p′i(.)(ω); then by lemma 3.8, the last expression converges to zero as h tends to infinity. similarly we can show that − (∫ a2 ( n∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi (u) ) dx ) goes to 0 as h → ∞, hence, we obtain lim sup h→∞ a(h,k) ≥ ∫ [|u−v| 0, ∫ [|u−v| β. especially, we will use p instead of p(0). 302 h. özlem güney, g. murugusundaramoorthy & k. vijaya cubo 23, 2 (2021) theorem 1.2. [6] the function p̃(z) = 1 + τ2z2 1 − τz − τ2z2 belongs to the class p(β) with β = √ 5/10 ≈ 0.2236. now we give the following lemma which will use in proving. lemma 1.3. [10] let p ∈p with p(z) = 1 + c1z + c2z2 + · · · , then |cn| ≤ 2, for n ≥ 1. (1.6) 2 bi-univalent function class pslλs,σ(α,p̃(z)) in this section, we introduce a new subclass of σ associated with shell-like functions connected with fibonacci numbers and obtain the initial taylor coefficients |a2| and |a3| for the function class by subordination. firstly, let p(z) = 1 + p1z + p2z 2 + · · · , and p ≺ p̃. then there exists an analytic function u such that |u(z)| < 1 in u and p(z) = p̃(u(z)). therefore, the function h(z) = 1 + u(z) 1 −u(z) = 1 + c1z + c2z 2 + · · · (2.1) is in the class p. it follows that u(z) = c1z 2 + ( c2 − c21 2 ) z2 2 + ( c3 − c1c2 + c31 4 ) z3 2 + · · · (2.2) and p̃(u(z)) = 1 + p̃1c1z 2 + { 1 2 ( c2 − c21 2 ) p̃1 + c21 4 p̃2 } z2 + { 1 2 ( c3 − c1c2 + c31 4 ) p̃1 + 1 2 c1 ( c2 − c21 2 ) p̃2 + c31 8 p̃3 } z3 + · · · . (2.3) and similarly, there exists an analytic function v such that |v(w)| < 1 in u and p(w) = p̃(v(w)). therefore, the function k(w) = 1 + v(w) 1 −v(w) = 1 + d1w + d2w 2 + · · · (2.4) is in the class p(0). it follows that v(w) = d1w 2 + ( d2 − d21 2 ) w2 2 + ( d3 −d1d2 + d31 4 ) w3 2 + · · · (2.5) and p̃(v(w)) = 1 + p̃1d1w 2 + { 1 2 ( d2 − d21 2 ) p̃1 + d21 4 p̃2 } w2 + { 1 2 ( d3 −d1d2 + d31 4 ) p̃1 + 1 2 d1 ( d2 − d21 2 ) p̃2 + d31 8 p̃3 } w3 + · · · . (2.6) cubo 23, 2 (2021) subclasses of λ-bi-pseudo-starlike functions with respect to .... 303 the class lλ(α) of λ-pseudo-starlike functions of order α (0 ≤ α < 1) were introduced and investigated by babalola [1] whose geometric conditions satisfy < ( z(f′(z))λ f(z) ) > α, λ > 0. he showed that all pseudo-starlike functions are bazilevič of type ( 1 − 1 λ ) order α 1 λ and univalent in open unit disk u. if λ = 1, we have the class of starlike functions of order α, which in this context, are 1-pseudo-starlike functions of order α. a function f ∈ a is starlike with respect to symmetric points in u if for every r close to 1, r < 1 and every z0 on |z| = r the angular velocity of f(z) about f(−z0) is positive at z = z0 as z traverses the circle |z| = r in the positive direction. this class was introduced and studied by sakaguchi [13] presented the class s∗s of functions starlike with respect to symmetric points. this class consists of functions f(z) ∈s satisfying the condition < ( 2zf′(z) f(z) −f(−z) ) > 0, z ∈ u. motivated by s∗s , wang et al. [18] introduced the class ks of functions convex with respect to symmetric points, which consists of functions f(z) ∈s satisfying the condition < ( 2(zf′(z))′ (f(z) −f(−z))′ ) > 0, z ∈ u. it is clear that, if f(z) ∈ks , then zf′(z) ∈s∗s . for such a function φ, ravichandran [12] presented the following subclasses: a function f ∈ a is in the class s∗s (φ) if 2zf′(z) f(z) −f(−z) ≺ φ(z), z ∈ u, and in the class ks(φ) if 2(zf′(z))′ (f(z) −f(−z))′ ≺ φ(z) z ∈ u. motivated by aforementioned works [1, 13, 12, 18] and recent study of sokól [14] (also see [11]), in this paper we define the following new subclass f ∈ pslλs,σ(p̃(z)) of σ named as λ-bi-pseudostarlike functions with respect to symmetric points, related to shell-like curves connected with fibonacci numbers, and determine the initial taylor-maclaurin coefficients |a2| and |a3|. further we determine the fekete-szegö result for the function class pslλs,σ(p̃(z)) and the special cases are stated as corollaries which are new and have not been studied so far. definition 2.1. for 0 ≤ α ≤ 1; λ > 0; λ 6= 1 3 ,a function f ∈ σ of the form (1.1) is said to be in the class pslλs,σ(α,p̃(z)) if the following subordination hold:( 2z(f′(z))λ f(z) −f(−z) )α ( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ )1−α ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.7) and ( 2w(g′(w))λ g(w) −g(−w) )α ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ )1−α ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.8) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). 304 h. özlem güney, g. murugusundaramoorthy & k. vijaya cubo 23, 2 (2021) specializing the parameter λ = 1 we have the following definitions, respectively: definition 2.2. for 0 ≤ α ≤ 1, a function f ∈ σ of the form (1.1) is said to be in the class psl1s,σ(α,p̃(z)) ≡msls,σ(α,p̃(z)) if the following subordination hold:( 2zf′(z) f(z) −f(−z) )α ( 2(z(f′(z)))′ [f(z) −f(−z)]′ )1−α ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.9) and ( 2wg′(w) g(w) −g(−w) )α ( 2(w(g′(w)))′ [g(w) −g(−w)]′ )1−α ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.10) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). further by specializing the parameter α = 1 and α = 0 we state the following new classes sl∗s,σ(p̃(z)) and kls,σ(p̃(z)) respectively. definition 2.3. a function f ∈ σ of the form (1.1) is said to be in the class psl1s,σ(1, p̃(z)) ≡ sl∗s,σ(p̃(z)) if the following subordination hold: 2zf′(z) f(z) −f(−z) ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.11) and 2wg′(w) g(w) −g(−w) ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.12) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). definition 2.4. a function f ∈ σ of the form (1.1) is said to be in the class psl1s,σ(0, p̃(z)) ≡ kls,σ(p̃(z)) if the following subordination hold: 2(z(f′(z)))′ [f(z) −f(−z)]′ ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.13) and 2(w(g′(w)))′ [g(w) −g(−w)]′ ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.14) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). definition 2.5. for λ > 0; λ 6= 1 3 , a function f ∈ σ of the form (1.1) is said to be in the class pslλs,σ(p̃(z)) if the following subordination hold:( 2z(f′(z))λ f(z) −f(−z) ) ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.15) and ( 2w(g′(w))λ g(w) −g(−w) ) ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.16) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). cubo 23, 2 (2021) subclasses of λ-bi-pseudo-starlike functions with respect to .... 305 definition 2.6. for λ > 0; λ 6= 1 3 , a function f ∈ σ of the form (1.1) is said to be in the class gslλs,σ(p̃(z)) if the following subordination hold:( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ ) ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.17) and ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ ) ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.18) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). in the following theorem we determine the initial taylor coefficients |a2| and |a3| for the function class pslλs,σ(α,p̃(z)). later we will reduce these bounds to other classes for special cases. theorem 2.7. let f given by (1.1) be in the class pslλs,σ(α,p̃(z)). then |a2| ≤ |τ|√ 4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ . (2.19) and |a3| ≤ 2λ|τ| [ 2λ(α− 2)2 −{5λ(α− 2)2 + 4 − 3α}τ ] (3λ− 1)(3 − 2α) [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] (2.20) where 0 ≤ α ≤ 1; λ > 0 and λ 6= 1 3 . proof. let f ∈pslλs,σ(α,p̃(z)) and g = f−1. considering (2.7) and (2.8), we have ( 2z(f′(z))λ f(z) −f(−z) )α ( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ )1−α = p̃(u(z)) (2.21) and ( 2w(g′(w))λ g(w) −g(−w) )α ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ )1−α = p̃(v(w)) (2.22) for some schwarz functions u and v where τ = (1− √ 5)/2 ≈−0.618 where z,w ∈ u and g is given by (1.2). since ( 2z[f′(z)]λ f(z) −f(−z) )α ( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ )1−α = 1 − 2λ(α− 2)a2z + {[2λ2(α− 2)2 + 2λ(3α− 4)]a22 + (3λ− 1)(3 − 2α)a3}z 2 + · · · and ( 2w(g′(w))λ g(w) −g(−w) )α ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ )1−α = 1 + 2λ(α−2)a2w +{[2λ2(α−2)2 + 2λ(5−3α) + 2(2α−3)]a22 + (3λ−1)(2α−3)a3}w 2 + · · · . 306 h. özlem güney, g. murugusundaramoorthy & k. vijaya cubo 23, 2 (2021) thus we have 1 − 2λ(α− 2)a2z + {[2λ2(α− 2)2 + 2λ(3α− 4)]a22 + (3λ− 1)(3 − 2α)a3}z 2 + · · · = 1 + p̃1c1z 2 + [ 1 2 ( c2 − c21 2 ) p̃1 + c21 4 p̃2 ] z2 + [ 1 2 ( c3 − c1c2 + c31 4 ) p̃1 + 1 2 c1 ( c2 − c21 2 ) p̃2 + c31 8 p̃3 ] z3 + · · · . (2.23) and 1 + 2λ(α− 2)a2w + {[2λ2(α− 2)2 + 2λ(5 − 3α) + 2(2α− 3)]a22 + (3λ− 1)(2α− 3)a3}w 2 = 1 + p̃1d1w 2 + [ 1 2 ( d2 − d21 2 ) p̃1 + d21 4 p̃2 ] w2 + [ 1 2 ( d3 −d1d2 + d31 4 ) p̃1 + 1 2 d1 ( d2 − d21 2 ) p̃2 + d31 8 p̃3 ] w3 + · · · . (2.24) it follows from (1.5), (2.23) and (2.24) that − 2λ(α− 2)a2 = c1τ 2 , (2.25) [2λ2(α− 2)2 + 2λ(3α− 4)]a22 + (3λ− 1)(3 − 2α)a3 = 1 2 ( c2 − c21 2 ) τ + 3 4 c21τ 2, (2.26) and 2λ(α− 2)a2 = d1τ 2 , (2.27) [2λ2(α− 2)2 + 2λ(5 − 3α) + 2(2α− 3)]a22 + (3λ− 1)(2α− 3)a3 = 1 2 ( d2 − d21 2 ) τ + 3 4 d21τ 2. (2.28) from (2.25) and (2.27), we have c1 = −d1, (2.29) and a22 = (c21 + d 2 1) 32λ2(α− 2)2 τ2. (2.30) now, by summing (2.26) and (2.28), we obtain [ 4λ2(α− 2)2 + 2(λ + 2α− 3) ] a22 = 1 2 (c2 + d2)τ − 1 4 (c21 + d 2 1)τ + 3 4 (c21 + d 2 1)τ 2. (2.31) by putting (2.30) in (2.31), we have 2 [ 8λ2(α− 2)2 −{20λ2(α− 2)2 − 2(λ + 2α− 3)}τ ] a22 = (c2 + d2)τ 2. (2.32) therefore, using lemma 1.3 we obtain |a2| ≤ |τ|√ 4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ . (2.33) now, so as to find the bound on |a3|, let’s subtract from (2.26) and (2.28). so, we find 2(3λ− 1)(3 − 2α)a3 − 2(3λ− 1)(3 − 2α)a22 = 1 2 (c2 −d2) τ. (2.34) cubo 23, 2 (2021) subclasses of λ-bi-pseudo-starlike functions with respect to .... 307 hence, we get 2(3λ− 1)(3 − 2α)|a3| ≤ 2|τ| + 2(3λ− 1)(3 − 2α)|a2|2. (2.35) then, in view of (2.33), we obtain |a3| ≤ 2λ|τ| [ 2λ(α− 2)2 −{5λ(α− 2)2 + 4 − 3α}τ ] (3λ− 1)(3 − 2α) [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] . (2.36) if we can take the parameter λ = 1 in the above theorem, we have the following the initial taylor coefficients |a2| and |a3| for the function classes msls,σ(α,p̃(z)). corollary 2.8. let f given by (1.1) be in the class msls,σ(α,p̃(z)). then |a2| ≤ |τ|√ 4(α− 2)2 − 2(5α2 − 21α + 21)τ (2.37) and |a3| ≤ |τ| [ 2(α− 2)2 −{5α2 − 23α + 24}τ ] (3 − 2α) [4(α− 2)2 −{10α2 − 42α + 42}τ] . (2.38) further by taking α = 1 and α = 0 and τ = −0.618 in corollary 2.8, we have the following improved initial taylor coefficients |a2| and |a3| for the function classes sl∗s,σ(p̃(z)) and kls,σ(p̃(z)) respectively. corollary 2.9. let f given by (1.1) be in the class sl∗s,σ(p̃(z)). then |a2| ≤ |τ| √ 4 − 10τ ' 0.19369 (2.39) and |a3| ≤ |τ|(1 − 3τ) 2 − 5τ ' 0.3465. (2.40) corollary 2.10. let f given by (1.1) be in the class kls,σ(p̃(z)). then |a2| ≤ |τ| √ 16 − 42τ ' 0.0954 (2.41) and |a3| ≤ 4|τ|(1 − 3τ) 3(8 − 21τ) ' 0.17647. (2.42) corollary 2.11. let f given by (1.1) be in the class pslλs,σ(p̃(z)). then |a2| ≤ |τ|√ 4λ2 −{10λ2 −λ + 1}τ (2.43) and |a3| ≤ 2λ|τ| [2λ−{5λ + 1}τ] (3λ− 1) [4λ2 −{10λ2 −λ + 1}τ] (2.44) where λ > 0 and λ 6= 1 3 . 308 h. özlem güney, g. murugusundaramoorthy & k. vijaya cubo 23, 2 (2021) corollary 2.12. let f given by (1.1) be in the class gslλs,σ(p̃(z)). then |a2| ≤ |τ|√ 16λ2 −{40λ2 −λ + 3}τ (2.45) and |a3| ≤ 2λ|τ| [8λ−{20λ + 4}τ] 3(3λ− 1) [16λ2 −{40λ2 −λ + 3}τ] (2.46) where λ > 0 and λ 6= 1 3 . 3 fekete-szegö inequality for the function class pslλs,σ(α,p̃(z)) fekete and szegö [7] introduced the generalized functional |a3−µa22|, where µ is some real number. due to zaprawa [19], in the following theorem we determine the fekete-szegö functional for f ∈ pslλs,σ(α,p̃(z)). theorem 3.1. let λ ∈ r with λ > 1 3 and let f given by (1.1) be in the class pslλs,σ(α,p̃(z)) and µ ∈ r. then we have |a3 −µa22| ≤   |τ| 4(3λ− 1)(3 − 2α) , 0 ≤ |h(µ)| ≤ |τ| 4(3λ− 1)(3 − 2α) 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ− 1)(3 − 2α) where h(µ) = (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] . (3.1) proof. from (2.32) and (2.34) we obtain a3 −µa22 = (1 −µ)(c2 + d2)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] + τ(c2 −d2) 4(3λ− 1)(3 − 2α) = ( (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] + τ 4(3λ− 1)(3 − 2α) ) c2 + ( (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] − τ 4(3λ− 1)(3 − 2α) ) d2. so we have a3 −µa22 = ( h(µ) + τ 4(3λ− 1)(3 − 2α) ) c2 + ( h(µ) − τ 4(3λ− 1)(3 − 2α) ) d2 (3.2) where h(µ) = (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] . then, by taking modulus of (3.2), we conclude that cubo 23, 2 (2021) subclasses of λ-bi-pseudo-starlike functions with respect to .... 309 |a3 −µa22| ≤   |τ| 4(3λ− 1)(3 − 2α) , 0 ≤ |h(µ)| ≤ |τ| 4(3λ− 1)(3 − 2α) 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ− 1)(3 − 2α) . taking µ = 1, we have the following corollary. corollary 3.2. if f ∈pslλs,σ(α,p̃(z)), then |a3 −a22| ≤ |τ| 4(3λ− 1)(3 − 2α) . (3.3) if we can take the parameter λ = 1 in theorem 3.1, we can state the following: corollary 3.3. let f given by (1.1) be in the class msls,σ(α,p̃(z)) and µ ∈ r. then we have |a3 −µa22| ≤   |τ| 8(3 − 2α) , 0 ≤ |h(µ)| ≤ |τ| 8(3 − 2α) 4|h(µ)|, |h(µ)| ≥ |τ| 8(3 − 2α) where h(µ) = (1 −µ)τ2 4 [4(α− 2)2 −{10(α− 2)2 − 2α + 2}τ] . further by fixing λ = 1 taking α = 1 and α = 0 in the above corollary, we have the following the fekete-szegö inequalities for the function classes sl∗s,σ(p̃(z)) and kls,σ(p̃(z)), respectively. corollary 3.4. let f given by (1.1) be in the class sl∗s,σ(p̃(z)) and µ ∈ r. then we have |a3 −µa22| ≤   |τ| 24 , 0 ≤ |h(µ)| ≤ |τ| 24 4|h(µ)|, |h(µ)| ≥ |τ| 24 where h(µ) = (1 −µ)τ2 8 [2 − 5τ] . corollary 3.5. let f given by (1.1) be in the class kls,σ(p̃(z)) and µ ∈ r. then we have |a3 −µa22| ≤   |τ| 8 , 0 ≤ |h(µ)| ≤ |τ| 8 4|h(µ)|, |h(µ)| ≥ |τ| 8 where h(µ) = (1 −µ)τ2 8 [8 − 21τ] . by assuming λ ∈ r; λ > 1 3 and taking α = 1 and α = 0 we have the following the fekete-szegö inequalities for the function classes pslλs,σ(p̃(z)) and gsl λ s,σ(p̃(z)), respectively. 310 h. özlem güney, g. murugusundaramoorthy & k. vijaya cubo 23, 2 (2021) corollary 3.6. let λ ∈ r with λ > 1 3 and let f given by (1.1) be in the class pslλs,σ(p̃(z)) and µ ∈ r. then we have |a3 −µa22| ≤   |τ| 4(3λ− 1) , 0 ≤ |h(µ)| ≤ |τ| 4(3λ− 1) 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ− 1) where h(µ) = (1 −µ)τ2 4 [4λ2 −{10λ2 −λ + 1}τ] . corollary 3.7. let λ ∈ r with λ > 1 3 and let f given by (1.1) be in the class gslλs,σ(p̃(z)) and µ ∈ r. then we have |a3 −µa22| ≤   |τ| 12(3λ− 1) , 0 ≤ |h(µ)| ≤ |τ| 12(3λ− 1) 4|h(µ)|, |h(µ)| ≥ |τ| 12(3λ− 1) where h(µ) = (1 −µ)τ2 4 [16λ2 −{40λ2 −λ + 3}τ] . conclusions our motivation is to get many interesting and fruitful usages of a wide variety of fibonacci numbers in geometric function theory. by defining a subclass λ-bi-pseudo-starlike functions with respect to symmetric points of σ related to shell-like curves connected with fibonacci numbers we were able to unify and extend the various classes of analytic bi-univalent function, and new extensions were discussed in detail. further, by specializing α = 0 and α = 1 and τ = −0.618 we have attempted at the discretization of some of the new and well-known results. our main results are new and better improvement to initial taylor-maclaurin coefficients |a2| and |a3|. acknowledgements the authors thank the referees of this paper for their insightful suggestions and corrections to improve the paper in present form. cubo 23, 2 (2021) subclasses of λ-bi-pseudo-starlike functions with respect to .... 311 references [1] k. o. babalola, “on λ-pseudo-starlike functions”, j. class. anal., vol. 3, no. 2, pp. 137–147, 2013. 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[19] p. zaprawa, “on the fekete-szegö problem for classes of bi-univalent functions”, bull. belg. math. soc. simon stevin, vol. 21, no. 1, pp. 169-178, 2014. introduction bi-univalent function class psls,(,(z)) fekete-szegö inequality for the function class psls,(,(z)) cubo, a mathematical journal vol. 23, no. 03, pp. 489–501, december 2021 doi: 10.4067/s0719-06462021000300489 extension of exton’s hypergeometric function k16 ahmed ali atash1 maisoon ahmed kulib2 1 department of mathematics, faculty of education shabwah, aden university, aden, yemen. ah-a-atash@hotmail.com 2department of mathematics, faculty of engineering, aden university, aden, yemen. maisoonahmedkulib@gmail.com abstract the purpose of this article is to introduce an extension of exton’s hypergeometric function k16 by using the extended beta function given by özergin et al. [11]. some integral representations, generating functions, recurrence relations, transformation formulas, derivative formula and summation formulas are obtained for this extended function. some special cases of the main results of this paper are also considered. resumen el propósito de este artículo es introducir una extensión de la función hipergeométrica de exton k16 usando la función beta extendida dada por özergin et al. [11]. se obtienen algunas representaciones integrales, funciones generatrices, relaciones de recurrencia, fórmulas de transformación, fórmulas de derivadas y fórmulas de sumación para esta función extendida. se consideran también algunos casos especiales de los resultados principales de este artículo. keywords and phrases: extended beta function, extended exton’s function, integral representations, generating functions, recurrence relation, transformation formula, derivative formula, summation formula. 2020 ams mathematics subject classification: 33b15, 33c05, 33c15, 33c65. accepted: 12 october, 2021 received: 31 march, 2021 ©2021 a. a. atash et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300489 https://orcid.org/0000-0001-7762-6341 https://orcid.org/0000-0002-2503-3496 490 a. a. atash & m. a. kulib cubo 23, 3 (2021) 1 introduction in recent years, some extensions of beta function and gauss hypergeometric function have been considered by several authors (see [3, 5, 6, 11]). the following extended beta function and extended gauss hypergeometric function are introduced by özergin et al. [11]: b(α,β)p (x, y) = ∫ 1 0 tx−1(1 − t)y−11f1 ( α; β; − p t(1 − t) ) dt, (1.1) (ℜ(α) > 0, ℜ(β) > 0, ℜ(p) ≥ 0, ℜ(x) > 0, ℜ(y) > 0) and f (α,β)p (a, b; c; z) = ∞∑ n=0 (a)n b (α,β) p (b + n, c − b) b(b, c − b) zn n! , (1.2) (ℜ(c) > ℜ(b) > 0, |z| < 1). they [11] presented the following integral representation: f (α,β)p (a, b; c; z) = 1 b(b, c − b) ∫ 1 0 tb−1(1 − t)c−b−1(1 − zt)−a1f1 ( α; β; − p t(1 − t) ) dt, (1.3) ℜ(p) > 0; p = 0 and |arg(1 − z)| < π; ℜ(c) > ℜ(b) > 0. clearly, we have b (α,β) 0 (x, y) = b(x, y) and f (α,β) 0 (a, b; c; z) = 2f1(a, b; c; z), where b(x, y) and 2f1(z, b; c; z) are the classical beta function and gauss hypergeometric function defined by (see [13]) b(x, y) = ∫ 1 0 tx−1(1 − t)y−1dt, ℜ(x) > 0, ℜ(y) > 0 (1.4) and 2f1(a, b; c; z) = ∞∑ n=0 (a)n(b)n (c)n zn n! , c ̸= 0, −1, −2, . . . , (1.5) where (λ)n (n ∈ n0 = n ∪ {0}) denotes the pochhammer’s symbol defined by [13] (λ)n =   1, n = 0 λ(λ + 1)(λ + 2) . . . (λ + n − 1), n ∈ n. (1.6) many authors have considered certain interesting extensions of some hypergeometric functions of two and three variables (see [1, 2, 8, 10]). by using the extended beta function in (1.1), liu [8] defined the extended appell’s function f1 as follows: f (α,β) 1,p (a, b, c; d; x, y) = ∞∑ m,n=0 b (α,β) p (a + m + n, d − a)(b)m(c)n b(a, d − a) xm m! yn n! (1.7) cubo 23, 3 (2021) extension of exton’s hypergeometric function k16 491 and obtained the following integral representation: f (α,β) 1,p (a, b, c; d; x, y) = 1 b(a, d − a) × ∫ 1 0 ta−1(1 − t)d−a−1(1 − xt)−b(1 − yt)−c1f1 ( α; β; − p t(1 − t) ) dt. (1.8) observe that f (α,β) 1,0 (a, b, c; d; x, y) = f1(a, b, c; d; x, y), where f1(a, b, c; d; x, y) is appell’s hypergeometric function [13] f1(a, b, c; d; x, y) = ∞∑ m,n=0 (a)m+n(b)m(c)n (d)m+n xm m! yn n! . (1.9) the exton’s hypergeometric function k16 is defined by [7] as follows: k16(a1, a2, a3, a4; b; x, y, z, t) = ∞∑ m,n,p,q=0 (a1)m+n(a2)m+p(a3)n+q(a4)p+q x m yn zp tq (b)m+n+p+q m! n! p! q! . (1.10) in this paper, we use the extended beta function given in (1.1) to define extended exton’s hypergeometric function k(α,β)16,p (a, b, c, d; e; x, y, z, u) as follows: k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = ∞∑ m,n,r,s=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r(c)n+s(d)r+s xmynzrus b(a, e − a)(e − a)r+s m! n! r! s! . (1.11) the extended exton’s hypergeometric function k(α,β)16,p (a, b, c, d; e; x, y, z, u) given in (1.11) can be written as follows: k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = ∞∑ r,s=0 (d)r+s(b)r(c)s (e)r+s f (α,β) 1,p (a, b + r, c + s; e + r + s; x, y) zr us r! s! . (1.12) observe that: • the special case d = e − a of (1.11) yields the following extended exton’s hypergeometric function k(α,β)16,p : k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) = ∞∑ m,n,r,s=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r(c)n+s xm yn zr us b(a, e − a) m! n! r! s! . (1.13) • the special case p = 0 of (1.11) yields the exton’s hypergeometric function k16 k (α,β) 16,0 (a, b, c, d; e; x, y, z, u) = k16(a, b, c, d; e; x, y, z, u). (1.14) 492 a. a. atash & m. a. kulib cubo 23, 3 (2021) 2 integral representations in this section, we present some integral representations for the extended exton’s hypergeometric function k(α,β)16,p (a, b, c, d; e; x, y, z, u) in (1.11). theorem 2.1. the integral representations (2.1), (2.4), (2.5) of k(α,β)16,p (a, b, c, d; e; x, y, z, u) hold for ℜ(p) > 0, ℜ(e) > ℜ(a) > 0; |x| + |z| < 1, |y| + |u| < 1 and the others hold for ℜ(p) > 0, ℜ(e) > ℜ(a) > ℜ(d) > 0; |x| + |z| < 1, |y| + |u| < 1: k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = 1 b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a−1(1 − xt)−b(1 − yt)−c × f1 ( d, b, c; e − a; z(1 − t) 1 − xt , u(1 − t) 1 − yt ) 1f1 ( α; β; − p t(1 − t) ) dt (2.1) k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = 1 b(a, e − a) 1 b(d, e − a − d) ∫ 1 0 ∫ 1 0 ta−1sd−1(1 − t)e−a−1(1 − s)e−a−d−1 × (1 − xt − zs(1 − t))−b(1 − yt − us(1 − t))−c1f1 ( α; β; − p t(1 − t) ) dtds (2.2) k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = 1 b(a, e − a) b(d, e − a − d) × ∫ 1 0 ∫ 1 0 ta−1sd−1(1 − t)e−a−1(1 − s)e−a−d−1(1 − zs)−b(1 − us)−c × ( 1 − ( x − zs 1 − zs ) t )−b ( 1 − ( y − us 1 − us ) t )−c 1f1 ( α; β; − p t(1 − t) ) dtds (2.3) k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = 2 b(a, e − a) × ∫ π 2 0 sin2a−1 θ cos2e−2a−1 θ(1 − x sin2 θ)−b(1 − y sin2 θ)−c × f1 ( d, b, c; e − a; z cos2 θ 1 − x sin2 θ , u cos2 θ 1 − y sin2 θ ) 1f1 ( α; β; − p sin2 θ cos2 θ ) dθ (2.4) k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = 1 b(a, e − a) × ∫ ∞ 0 ξa−1(1 + ξ)c+b−e(1 + (1 − x)ξ)−b(1 + (1 − y)ξ)−c × f1 ( d, b, c; e − a; z 1 + (1 − x)ξ , u 1 + (1 − y)ξ ) 1f1 ( α; β; − p(1 + ξ)2 ξ ) dξ. (2.5) cubo 23, 3 (2021) extension of exton’s hypergeometric function k16 493 proof of (2.1). using (1.1) in (1.11) and interchanging the order of summation and integration, we have k (α,β) 16,p (a, b, c, d; e; x, y, z, u) = 1 b(a, e − a) × ∫ 1 0 ta−1(1 − t)e−a−1 ∞∑ r,s=0 (d)r+s(b)r(c)s(z(1 − t))r(u(1 − t))s (e − a)r+s r! s! × 1f1 ( α; β; − p t(1 − t) )( ∞∑ m=0 (b + r)m(xt) m m! )( ∞∑ n=0 (c + s)n(yt) n n! ) dt = 1 b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a−1(1 − xt)−b(1 − yt)−c × 1f1 ( α; β; − p t(1 − t) ){ ∞∑ r,s=0 (d)r+s(b)r(c)s (e − a)r+s r! s! ( z(1 − t) 1 − xt )r ( u(1 − t) 1 − yt )s} dt, which by applying the definition of appell hypergeometric function f1 (1.9), we have the desired result (2.1). the integral representation (2.2) can be obtained easily from (2.1) by using the following integral representation of f1 [12]: f1(a, b, c; d; x, y) = 1 b(a, d − a) ∫ 1 0 ta−1(1 − t)d−a−1(1 − xt)−b(1 − yt)−cdt. (2.6) also the integral representation (2.3) can be obtained directly from (2.2) if we use the following relation: (1 − xt − z(1 − t))−a = (1 − z)−a ( 1 − (x − z)t 1 − z )−a . (2.7) finally, the integral representations (2.4) and (2.5) can be easily obtained by taking the transformations t = sin2 θ and t = ξ 1+ξ in (2.1), respectively. this completes the proof of theorem 2.1. the special case d = e − a of (2.1), (2.4) and (2.5), yields the following results: corollary 2.2. k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) = 1 b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a−1(1 − xt − z(1 − t))−b(1 − yt − u(1 − t))−c × 1f1 ( α; β; − p t(1 − t) ) dt, (2.8) k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) = 2 b(a, e − a) × ∫ π 2 0 sin2a−1 θ cos2e−2a−1 θ(1 − x sin2 θ − z cos2 θ)−b(1 − y sin2 θ − u cos2 θ)−c × 1f1 ( α; β; − p sin2 θ cos2 θ ) dθ (2.9) 494 a. a. atash & m. a. kulib cubo 23, 3 (2021) and k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) = 1 b(a, e − a) × ∫ ∞ 0 ξa−1(1 + ξ)c+b−e(1 + (1 − x)ξ − z)−b(1 + (1 − y)ξ − u)−c × 1f1 ( α; β; − p(1 + ξ)2 ξ ) dξ. (2.10) 3 generating functions in this section, we derive certain generating functions for the extended exton’s hypergeometric function k(α,β)16,p (a, b, c, d; e; x, y, z, u) in (1.11). theorem 3.1. the following generating functions holds true: ∞∑ k=0 (b)kt k k! k (α,β) 16,p (a, b + k, c, d; e; x, y, z, u) = (1 − t) −bk (α,β) 16,p ( a, b, c, d; e; x 1 − t , y, z 1 − t , u ) (3.1) ∞∑ k=0 (c)kt k k! k (α,β) 16,p (a, b, c + k, d; e; x, y, z, u) = (1 − t) −ck (α,β) 16,p ( a, b, c, d; e; x, y 1 − t , z, u 1 − t ) (3.2) ∞∑ k=0 (d)kt k k! k (α,β) 16,p (a, b, c, d+k; e; x, y, z, u) = (1−t) −dk (α,β) 16,p ( a, b, c, d; e; x, y, z 1 − t , u 1 − t ) . (3.3) proof of (3.1). using (1.11) in the l.h.s. of equation (3.1), we get ∞∑ k=0 (b)kt k k! k (α,β) 16,p (a, b + k, c, d; e; x, y, z, u) = ∞∑ m,n,r,s,k=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r+k(c)n+s(d)r+s xmynzrustk b(a, e − a)(e − a)r+s m! n! r! s! k! = ∞∑ m,n,r,s=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r(c)n+s(d)r+s xmynzrus b(a, e − a)(e − a)r+s m! n! r! s! ∞∑ k=0 (b + m + r)kt k k! = (1 − t)−bk(α,β)16,p ( a, b, c, d; e; x 1 − t , y, z 1 − t , u ) . this completes the proof of (3.1). the generating functions (3.2) and (3.3) can be proved by a similar method as in the proof of (3.1). setting p = 0 in (3.1), (3.2) and (3.3), we get known results [4]. theorem 3.2. the following generating functions holds true: ∞∑ k=0 (λ)kt k k! k (α,β) 16,p (a, b, c, −k; e; x, y, z, u) = (1 − t) −λk (α,β) 16,p ( a, b, c, λ; e; x, y, −zt 1 − t , −ut 1 − t ) (3.4) cubo 23, 3 (2021) extension of exton’s hypergeometric function k16 495 ∞∑ k=0 (λ)kt k k! k (α,β) 16,p (a, b, −k, d; e; x, y, z, u) = (1 − t) −λk (α,β) 16,p ( a, b, λ, d; e; x, −yt 1 − t , z, −ut 1 − t ) (3.5) ∞∑ k=0 (λ)kt k k! k (α,β) 16,p (a, −k, c, d; e; x, y, z, u) = (1 − t) −λk (α,β) 16,p ( a, λ, c, d; e; −xt 1 − t , y, −zt 1 − t , u ) . (3.6) proof of (3.4). using (1.11) in the l.h.s. of equation (3.4) and using the result [13] (−k)r = (−1)rk! (k − r)! , 0 ≤ r ≤ k, (3.7) we have ∞∑ k=0 (λ)kt k k! k (α,β) 16,p (a, b, c, −k; e; x, y, z, u) = ∞∑ m,n,k=0 k∑ r=0 k−r∑ s=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r(c)n+s(λ)k xmyn(−z)r(−u)stk b(a, e − a)(e − a)r+s m! n! r! s! (k − r − s)! = ∞∑ m,n,r,s,k=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r(c)n+s(λ)k+r+s xmyn(−zt)r(−ut)stk b(a, e − a)(e − a)r+s m! n! r! s! k! = ∞∑ m,n,r,s=0 b (α,β) p (a + m + n, e − a + r + s)(b)m+r(c)n+s(λ)r+s xmyn(−zt)r(−ut)s b(a, e − a)(e − a)r+s m! n! r! s! ∞∑ k=0 (λ + r + s)kt k k! = (1 − t)−λk(α,β)16,p ( a, b, c, λ; e; x, y, −zt 1 − t , −ut 1 − t ) . this completes the proof of (3.4). the generating functions (3.5) and (3.6) can be proved by a similar method as in the proof of (3.4). 4 recurrence relations in this section, we deduce some recurrence relations for the extended exton’s hypergeometric function k(α,β)16,p (a, b, c, d; e; x, y, z, u) in (1.11) by using the recurrence relations of the confluent function 1f1 and appell’s function f1. theorem 4.1. the following recurrence relation holds true: k (α,β) 16,p (a, b, c, d + 1; e; x, y, z, u) − k (α,β) 16,p (a, b, c, d; e; x, y, z, u) − bz e k (α,β) 16,p (a, b + 1, c, d + 1; e + 1; x, y, z, u) − cu e k (α,β) 16,p (a, b, c + 1, d + 1; e + 1; x, y, z, u) = 0 (4.1) proof. to prove theorem 4.1, we consider the following recurrence relation of appell’s function f1 [14]: f1(α + 1, β1, β2; γ; x, y) − f1(α, β1, β2; γ; x, y) − xβ1 γ f1(α + 1, β1 + 1, β2; γ + 1; x, y) − yβ2 γ f1(α + 1, β1, β2 + 1; γ + 1; x, y) = 0 (4.2) 496 a. a. atash & m. a. kulib cubo 23, 3 (2021) in (4.2) replacing α, β1, β2, γ, x, y by d, b, c, e − a, z(1−t) 1−xt , u(1−t) 1−yt respectively, multiplying both sides by 1 b(a,e−a)t a−1(1 − t)e−a−1(1 − xt)−b(1 − yt)−c1f1 ( α; β; − p t(1−t) ) and integrating the resulting equation with respect to t between the limits 0 to 1, we get 1 b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a−1(1 − xt)−b(1 − yt)−c × f1 ( d + 1, b, c; e − a; z(1 − t) 1 − xt , u(1 − t) 1 − yt ) 1f1 ( α; β; − p t(1 − t) ) dt − 1 b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a−1(1 − xt)−b(1 − yt)−c × f1 ( d, b, c; e − a; z(1 − t) 1 − xt , u(1 − t) 1 − yt ) 1f1 ( α; β; − p t(1 − t) ) dt − bz (e − a)b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a(1 − xt)−b−1(1 − yt)−c × f1 ( d + 1, b + 1, c; e − a + 1; z(1 − t) 1 − xt , u(1 − t) 1 − yt ) 1f1 ( α; β; − p t(1 − t) ) dt − cu (e − a)b(a, e − a) ∫ 1 0 ta−1(1 − t)e−a(1 − xt)−b(1 − yt)−c−1 × f1 ( d + 1, b, c + 1; e − a + 1; z(1 − t) 1 − xt , u(1 − t) 1 − yt ) 1f1 ( α; β; − p t(1 − t) ) dt = 0. finally, using the integral representation (2.1), we get the desired result (4.1). theorem 4.2. the following recurrence relations hold true: (i) (β − α)k(α−1,β)16,p (a, b, c, e − a; e; x, y, z, u) − αk (α+1,β) 16,p (a, b, c, e − a; e; x, y, z, u) + (2α − β)k(α,β)16,p (a, b, c, e − a; e; x, y, z, u) + pb(a − 1, e − a − 1) b(a, e − a) k (α,β) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) = 0 (4.3) (ii) β(β − 1)k(α,β−1)16,p (a, b, c, e − a; e; x, y, z, u) − β(β − 1)k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) − βpb(a − 1, e − a − 1) b(a, e − a) k (α,β) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) + p(α − β)b(a − 1, e − a − 1) b(a, e − a) k (α,β+1) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) = 0 (4.4) (iii) αβk (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) − αβk (α+1,β) 16,p (a, b, c, e − a; e; x, y, z, u) + pβb(a − 1, e − a − 1) b(a, e − a) k (α,β) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) − p(β − α)b(a − 1, e − a − 1) b(a, e − a) k (α,β+1) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) = 0 (4.5) cubo 23, 3 (2021) extension of exton’s hypergeometric function k16 497 (iv) βk (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) − βk (α−1,β) 16,p (a, b, c, e − a; e; x, y, z, u) + pb(a − 1, e − a − 1) b(a, e − a) k (α,β+1) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) = 0 (4.6) (v) (β − α − 1)k(α,β)16,p (a, b, c, e − a; e; x, y, z, u) + αk (α+1,β) 16,p (a, b, c, e − a; e; x, y, z, u) − (β − 1)k(α,β−1)16,p (a, b, c, e − a; e; x, y, z, u) = 0 (4.7) (vi) (α − 1)k(α,β)16,p (a, b, c, e − a; e; x, y, z, u) + pb(a − 1, e − a − 1) b(a, e − a) k (α,β) 16,p (a − 1, b, c, e − a − 1; e − 2; x, y, z, u) + (β − α)k(α−1,β)16,p (a, b, c, e − a; e; x, y, z, u) − (β − 1)k(α,β−1)16,p (a, b, c, e − a; e; x, y, z, u) = 0. (4.8) proof. to prove our results in theorem 4.2, we require the following recurrence relations of the confluent function 1f1 [9]: (β − α)1f1(α − 1; β; z) − α 1f1(α + 1; β; z) + (2α − β + z)1f1(α; β; z) = 0 (4.9) β(β − 1)1f1(α; β − 1; z) − β(β − 1 + z)1f1(α; β; z) + (β − α)z1f1(α; β + 1; z) = 0 (4.10) β(α + z)1f1(α; β; z) − α β 1f1(α + 1; β; z) − (β − α)z 1f1(α; β + 1; z) = 0 (4.11) β 1f1(α; β; z) − β 1f1(α − 1; β; z) − z 1f1(α; β + 1; z) = 0 (4.12) (β − α − 1)1f1(α; β; z) + α 1f1(α + 1; β; z) − (β − 1)1f1(α; β − 1; z) = 0 (4.13) (α + z − 1)1f1(α; β; z) + (β − α)1f1(α − 1; β; z) − (β − 1)1f1(α; β − 1; z) = 0. (4.14) proof of (4.3). in (4.9) replacing z by − p t(1−t) , multiplying both sides by t a−1(1−t)e−a−1(1−xt− z(1 − t))−b(1 − yt − u(1 − t))−c/b(a, e − a) and integrating the resulting equation with respect to t between the limits 0 to 1, we get β − α b(a, e − a) ∫ 1 0 t a−1 (1 − t)e−a−1(1 − xt − z(1 − t))−b(1 − yt − u(1 − t))−c1f1 ( α − 1; β; − p t(1 − t) ) dt − α b(a, e − a) ∫ 1 0 t a−1 (1 − t)e−a−1(1 − xt − z(1 − t))−b(1 − yt − u(1 − t))−c1f1 ( α; β; − p t(1 − t) ) dt + 2α − β b(a, e − a) ∫ 1 0 t a−1 (1 − t)e−a−1(1 − xt − z(1 − t))−b(1 − yt − u(1 − t))−c1f1 ( α; β; − p t(1 − t) ) dt + p b(a, e − a) ∫ 1 0 t a−2 (1 − t)e−a−2(1 − xt − z(1 − t))−b(1 − yt − u(1 − t))−c1f1 ( α; β; − p t(1 − t) ) dt = 0 finally, using the integral representation (2.8), we get the desired result (4.3). 498 a. a. atash & m. a. kulib cubo 23, 3 (2021) the results (4.4)-(4.8) can be proved by a similar method as in the proof of (4.3) and we use here the recurrence relations (4.10)-(4.14). 5 transformation, differentiation and summation formulas in this section, we derive certain transformation, derivative and summation formulas for the extended exton’s hypergeometric function k(α,β)16,p (a, b, c, d; e; x, y, z, u) in (1.11). theorem 5.1. the following transformation formula of kα,β16,p(a, b, c, d; e; x, y, z, u) holds true: k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) = (1 − z) −b(1 − u)−cf (α,β)1,p ( a, b, c; e; x − z 1 − z , y − u 1 − u ) . (5.1) proof. using (2.7) in (2.8), we have k (α,β) 16,p (a, b, c, e − a; e; x, y, z, u) = (1 − z)−b(1 − u)−c b(a, e − a) × ∫ 1 0 t a−1 (1 − t)e−a−1 ( 1 − ( x − z 1 − z ) t )−b ( 1 − ( y − u 1 − u ) t )−c 1f1 ( α; β; − p t(1 − t) ) dt, which by using (1.8), we get the desired result (5.1). setting p = 0 in (5.1), we get a known result [7] k16(a, b, c, e − a; e; x, y, z, u) = (1 − z)−b(1 − u)−cf1 ( a, b, c; e; x − z 1 − z , y − u 1 − u ) . (5.2) theorem 5.2. the following derivative formula holds true: dm+n+r+s dxm dyn dzr dus { k (α,β) 16,p (a, b, c, d; e; x, y, z, u) } = (a)m+n(b)m+r(c)n+s(d)r+s (e)m+n+r+s × k(α,β)16,p (a + m + n, b + m + r, c + n + s, d + r + s; e + m + n + r + s; x, y, z, u). (5.3) proof. differentiating (1.11) with respect to x, y, z and u, we have d dx dy dz du { k (α,β) 16,p (a, b, c, d; e; x, y, z, u) } = ∞∑ m=1 ∞∑ n=1 ∞∑ r=1 ∞∑ s=1 bp(a + m + n, e − a + r + s)(b)m+r(c)n+s(d)r+sxm−1yn−1zr−1us−1 b(a, e − a)(e − a)r+s(m − 1)!(n − 1)!(r − 1)!(s − 1)! setting m → m + 1, n → n + 1, r → r + 1, s → s + 1 and using the following identities: b(a, e − a) = e a b(a + 1, e − a) = e(e + 1) a(a + 1) b(a + 2, e − a), (a)p+q+2 = a(a + 1)(a + 2)p+q, we obtain d dx dy dz du { k (α,β) 16,p (a, b, c, d; e; x, y, z, u) } = (a)2(b)2(c)2(d)2 (e)2(e − a)2 × ∞∑ m,n,r,s=0 b (α,β) p (a + m + n + 2, e − a + r + s + 2)(b + 2)m+r(c + 2)n+s(d + 2)r+sxmynzrus b(a + 2, e − a)(e − a + 2)r+s m! n! r! s! cubo 23, 3 (2021) extension of exton’s hypergeometric function k16 499 now using b(a + 2, e − a) = (e)4 (e)2(e − a)2 b(a + 2, e − a + 2), we have d dx dy dz du { k (α,β) 16,p (a, b, c, d; e; x, y, z, u) } = (a)2(b)2(c)2(d)2 (e)4 × k(α,β)16,p (a + 2, b + 2, c + 2, d + 2; e + 4; x, y, z, u). thus by repeatedly differentiating, we find that the result (5.3) can be derived by induction. theorem 5.3. the following summation formulas hold true: k (α,β) 16,p (a, b, c, d; e; 1, 1, 1, 1) = γ(e)γ(e − a − b − c − d) γ(a)γ(e − a − d)γ(e − a − b − c) b(α,β)p (a, e − a − b − c) (5.4) k (α,β) 16,p (a, b, c, d; 1 + a + b + d − c; 1, 1, 1, −1) = γ(1 − c)γ(1 + 1 2 d)γ(1 + a + b + d − c) γ(a)γ(1 + d)γ(1 + b − c)γ(1 + 1 2 d − c) b(α,β)p (a, d − 2c + 1). (5.5) proof. setting x = y = z = u = 1 in (2.1) and using the following formula: f1(a, b, c; d; 1, 1) = γ(d)γ(d − a − b − c) γ(d − a)γ(d − b − c) , (5.6) we get k (α,β) 16,p (a, b, c, d; e; 1, 1, 1, 1) = γ(e)γ(e − a − b − c − d) γ(a)γ(e − a − d)γ(e − a − b − c) × ∫ 1 0 ta−1(1 − t)e−a−b−c−11f1 ( α; β; − p t(1 − t) ) dt (5.7) now, by using (1.1) in (5.7), we obtain the desired result (5.4). the summation formula (5.5) can be obtained easily by putting e = 1 + a + b + d − c, x = y = z = 1, u = −1 in (2.1) and using the formula f1(a, b, c; 1 + a + b − c; 1, −1) = γ(1 − c)γ(1 + 1 2 a)γ(1 + a + b − c) γ(1 + a)γ(1 + b − c)γ(1 + 1 2 a − c) . (5.8) this completes the proof of the theorem (5.3). setting p = 0 in (5.4) and (5.5), we get respectively the following summation formulas of exton’s hypergeometric function k16: k16(a, b, c, d; e; 1, 1, 1, 1) = γ(e)γ(e − a − b − c − d) γ(e − a − d)γ(e − b − c) (5.9) and k16(a, b, c, d; 1 + a + b + d − c; 1, 1, 1, −1) = γ(1 − c)γ(1 + 1 2 d)γ(1 + a + b + d − c)γ(d − 2c + 1) γ(1 + d)γ(1 + b − c)γ(1 + 1 2 d − c)γ(a + d − 2c + 1) . (5.10) 500 a. a. atash & m. a. kulib cubo 23, 3 (2021) 6 conclusion in this paper, we have introduced the extended exton’s hypergeometric function kα,β16,p(a, b, c, d; e; x, y, z, u) by using the extended beta function bα,βp (x, y) given by özergin et al. [11]. for this function we have presented some integral representations, generating functions, recurrence relations, transformation formulas, derivative formula and summation formulas. we have also established some a known and new generating functions, transformation formulas, and summation formulas for the classical exton’s hypergeometric function k16(a, b, c, d; e; x, y, z, u). acknowledgements the authors are thankful to the referees for useful comments and suggestions towards the improvement of this paper. cubo 23, 3 (2021) extension of exton’s hypergeometric function k16 501 references [1] p. agarwal, j. choi and s. jain, “extended hypergeometric functions of two and three variables”, commun. korean math. soc., vol. 30, no. 4, pp. 403–414, 2015. [2] r. p. agarwal, m. j. luo and p. agarwal, “on the extended appell-lauricella hypergeometric functions and their applications”, filomat, vol. 31, no. 12, pp. 3693–3713, 2017. [3] a. çetinkaya, i. o. kıymaz, p. agarwal and r. agarwal, “a comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators”, adv. difference equ., vol. 2018, paper no. 156, pp. 1–11, 2018. [4] r. c. singh chandel and a. tiwari, “generating relations involving hypergeometric functions of four variables”, pure appl. math. sci., vol. 36, no. 1-2, pp. 15–25, 1991. [5] m. a. chaudhry, a. qadir, m. rafique and s. m. zubair, “extension of euler’s beta function”, j. comp. appl. math., vol. 78, no. 1, pp. 19–32, 1997. [6] m. a. chaudhry, a. qadir, h. m. srivastava and r. b. paris, “extended hypergeometric and confluent hypergeometric functions”, appl. math. comp., vol. 159, no. 2, pp. 589–602, 2004. [7] h. exton, multiple hypergeometric functions and applications, new york: halsted press, 1976. [8] h. liu, “some generating relations for extended appell’s and lauricella’s hypergeometric functions”, rocky mountain j. math., vol. 44, no. 6, pp. 1987–2007, 2014. [9] y. l. luke, the special functions and their approximations, new york: academic press, 1969. [10] m. a. özarslan and e. özergin, “some generating relations for extended hypergeometric functions via generalized fractional derivative operator”, math. comput. modelling, vol. 52, no. 9-10, pp. 1825–1833, 2010. [11] e. özergin, m. a. özarslan, and a. altin, “extension of gamma, beta and hypergeometric functions”, j. comp. appl. math., vol. 235, no. 16, pp. 4601–4610, 2011. [12] h. m. srivastava and p. w. karlsson, multiple gaussian hypergeometric series, new york: halsted press, 1985. [13] h. m. srivastava and h. l. manocha, a treatise on generating functions, new york: halsted press, 1984. [14] x. wang, “recursion formulas for appell functions”, integral transforms spec. funct., vol. 23, no. 6, pp. 421–433, 2012. introduction integral representations generating functions recurrence relations transformation, differentiation and summation formulas conclusion a mathematical journal vol. 6, no 4, (1-32). december 2004. spectral methods for partial differential equations bruno costa 1 abstract in this article we present the essential aspects of spectral methods and their applications to the numerical solution of partial differential equations. starting from the fundamental ideas, we go further on building auxilliary techniques, as the treating of boundary conditions, and presenting accessory tools like mapping and filtering, finishing with a complete algorithm to solve a classical problem of fluid dynamics: the flow through a circular obstacle. we also present a short comparison with finite differences, showing the superior efficiency of spectral methods in problems with smooth solutions. several equations like the wave equation in one spatial dimension, burgers in a 2d domain and a simple multidomain setting for the navier-stokes 2d are solved numerically and the results are presented at the applications sections. we end with a brief presentation on the software pseudopack2000 and a quick discussion on the relevant literature. 1 introduction spectral methods have been used extensively during the last decades for the numerical solution of partial differential equations (pde) due to their bigger accuracy when compared to finite differences (fd) and finite elements (fe) methods. the rate of convergence of spectral approximations depends only on the smoothness of the solution, yielding the ability to achieve high precision with a small number of data. this fact is known in literature as ”spectral accuracy”. in fd and fe, the order of 1this work was supported by cnpq grant # 300315/98–8. 2 bruno costa 6, 4(2004) accuracy depends on some fixed negative power of n , the number of gridpoints being used, even when only analytic functions are involved. the expression spectral methods has different meanings for several subareas of mathematics, like functional analysis and signal processing. in this article, spectral methods has the meaning of a high accuracy numerical method to solve partial differential equations. the numerical solution is expressed as a finite expansion of some set of basis functions. when the pde is written in terms of the coefficients of this expansion, the method is known as a galerkin spectral method. spectral collocation methods, also known as pseudospectral methods, is another subclass of spectral methods and are similar to finite differences methods due to direct use of a set of gridpoints, which are called ¨collocation points¨. a third class are the tau spectral methods. these methods are similar to the galerkin spectral methods, however the expanding basis is not obliged to satisfy boundary conditions, requiring extra equations. in this article, we will mainly concentrate on collocation methods. the reader interested on galerkin and tau methods must check the works of canuto et al [8] and gotllieb and orszag [20]. in the past several years, the activity on both theory and application of spectral methods have been concentrated on collocation spectral methods. one of the reasons is that collocation methods deals with nonlinear terms more easily than galerkin methods. the nonlinear terms are treated on a fd manner, via gridpoints values multiplication. the underlying idea of a collocation spectral method is to approximate the unknown solution in the entire computational domain by an interpolating high order polynomial at the collocation points. the spatial derivatives of the solution are approximated by the derivatives of the polynomial and the time derivative, if it exists, is solved through classical finite differences schemes. periodic and non-periodic problems are respectively treated with trigonometric and algebraic polynomials. some of the methods commonly used in the literature are the fourier collocation methods for periodic domains and the jacobi polynomials for non-periodic domains, with the chebyshev and legendre polynomials as special cases. pseudospectral methods have a wide range of applications going from 3-d seismic wave propagation to turbulence, combustion, non-linear optics and passing through aero-acoustics and electromagnetics. the material exposed in this article intends to enable the interested reader to solve the ubiquitous navier-stokes equations, whose solution is exemplified in the case of the two-dimensional flow through an obstacle. thus, the necessary main ingredients as mapping and boundary conditions are treated in some extent along with several basic examples. this article is divided as follows: in section 2, we state the fundamentals of spectral methods by describing the galerkin spectral methods for linear and nonlinear problems. the collocation version of spectral methods comes in section 3 where we introduce polinomial interpolating algorithms. in section 4, we discuss mapping techniques for spectral methods in one and two dimensions in space. the subjects of filtering and boundary conditions are dealt with in section 5, together with applications to basic equations of fluid dynamics. we finish the article with a quick description of the library pseudopack and a brief discussion on the relevant literature. 6, 4(2004) spectral methods for partial differential equations 3 2 preliminaries we start this section showing a direct comparison between spectral methods and finite diferences. the main idea to be passed is the following: spectral methods achieve a greater precision with a smaller number of points than finite difference methods. the table below, extracted from [8] shows the error decay of approximating the first derivative of esin x through a second and a fourth order fd method and a fourier spectral collocation method: n fd 2nd order fd 4th order fourier spectral 16 3.4020e-001 1.3844e-001 4.3179e-003 32 9.3589e-002 1.5750e-002 1.7619e-007 64 2.5833e-002 1.1318e-003 2.3870e-014 128 6.5118e-003 7.5900e-005 7.2054e-014 the graph below in figure (1) shows the decay of the error with varying number of points. figure 1: error comparison between spectral and finite differences schemes. in [35] one can find quick matlab codes to generate graphics like the one above. 4 bruno costa 6, 4(2004) 2.1 linear periodic problems in spectral methods the solution of a pde is approximated by a finite linear combination of a chosen set of orthogonal functions { ϕj (x) }∞ j=0 as fn (x) = n∑ k=0 f̂kϕk(x), (1a) where the f̂k’s are called the spectral coefficients of f with respect to the basis {ϕk}; f̂k is a measure of how much of f is in the ”direction ” of ϕk and can be evaluated through an integral like f̂k = ∫ f (x) ϕk(x)w (x) dx, (2) where w (x) is a weight function associated to the basis {ϕk}. for instance, if we consider periodic functions in the interval [0, 2π], one such a basis is the fourier basis formed by sines and cosines as {cos kx, sin kx}∞k=0 , which in its concise complex form becomes {eikx}∞k=−∞. if f satisfies some conditions, like continuity and bounded variation, one can prove that fn given by fn (x) = n∑ k=0 f̂ke ikx with f̂k = ∫ f (x) eikxdx, converges to f uniformly as n increases. the derivative of f is approximated by f′n which is given by f′n (x) = n∑ k=0 f̂kϕ ′ k(x). (3) f′n is named the galerkin spectral derivative of f . once we know the specifics about the derivatives ϕ′k (a recurrence formulae, in general), we can work out formula (3) and find out the coefficients f̂′k to write f ′ n (x) = ∑ f̂′kϕk(x). for instance, let f be an even periodic function in [0, 2π]. an appropriate basis to approximate f is the fourier basis {cos kx}∞k=0. if we write fn (x) = n∑ k=0 f̂k cos kx, we easily see that an approximation to the second derivative of f is given by: f′′n (x) = n∑ k=0 ( −k2f̂k ) cos kx, meaning that once we know the spectral coefficients f̂k of f we obtain an approximation to f′′ by multiplying each fk by (−k2). in this way, double differentiation in spectral space is equal to multiplication by (−k2). the case of the first derivative is a little bit more ”complex”, for we need to use the basis {eikx}, since the first derivative of an even function is odd. 6, 4(2004) spectral methods for partial differential equations 5 with the above technique to compute second derivatives, let us solve the heat equation below ut − νuxx = 0, x ∈ [−π, π], t > 0, (4) u(x, 0) = e−x 2 , x ∈ [−π, π], where u is the temperature and ν is the difusion coefficient. using the approximation un (x) = n∑ k=0 ûk cos kx (5) and substituting into the above equation one obtains n∑ k=0 ( ∂ûk ∂t + νk2ûk ) cos kx = 0. (6) since the functions {cos kx}nk=0 are orthogonal, which means that ∫ π −π cos kx cos lxdx = δkl, we can reduce equation (6) to the system of n + 1 ordinary diferential equations ∂ûk ∂t + νk2ûk = 0, for k = 0, .., n, (7) ûk (0) = (u0)k , for k = 0, .., n. the example above shows the essential steps to solve a pde with spectral methods. first, is necessary to choose an appropriate basis for the problem to be solved, for instance, periodic problems are solved with the fourier basis and nonperiodic ones with a polynomial basis. next, we need the formula relating the spectral coefficients of f with those of its derivative. in this way, a system of odes analogous to the one in (7) can be formed and solved. note that one does not work with the physical values of the solution, but with its set of spectral coefficients. in fact, the physical values will be necessary to approximate the integrals (2) through quadrature formulae, as we shall see later. we now show why the error |f − fn| decays so fast with the increase of number of points, as seen in figure 1 above. if f is such that we can substitute it by its expansion series ∞∑ k=0 f̂kϕk(x), the error in approximating f by fn can be measured by the size of the tail of the above series, given by εn = |f − fn| = ∣∣∣∣ ∞∑ k>n f̂kϕk(x) ∣∣∣∣. considering the fourier basis { eikx } , we may write f̂k = 1 2π ∫ 2π 0 f (x) e−ikxdx = 1 2πik [f (2π) − f (0)] + 1 2πik ∫ 2π 0 f′ (x) e−ikxdx. 6 bruno costa 6, 4(2004) therefore, we can see that ∣∣∣f̂k∣∣∣ = o (1k) and that εn = o ( 1n ) . however, if f̂ is periodic, the boundary term vanishes and we can integrate by parts once again to obtain ∣∣∣f̂k∣∣∣ = o ( 1k2 ) or εn = o ( 1n 2 ). further iteration of this argument shows that the decay of the spectral coefficients of f depend on the diferentiability of f , together with the periodicity of its derivatives. in particular, if f is c∞, we can iterate the integration by parts above indefinitely to affirm that the decay of the spectral coefficients is faster than any negative power of n, or that εn = o ( e−cn ) , for some c > 0. this is what is known as exponential or spectral convergence and it is this property that has given its name to spectral methods. 2.2 nonlinear problems and pseudospectral methods mounting the system of odes as in (7) gets an extra degree of complexity in the presence of nonlinear terms in the pde. for instance, consider the same heat equation above with a diffusion coefficient that now depends on the temperature itself, say, ν (u) = cu, where c is a constant. thus, the equation to be considered is ut −cuuxx = 0. this time, let us use the complex form of the fourier expansion and substitute the truncated series (5) into the equation to obtain: n∑ k=0 ∂ûk ∂t eikx + ( c n∑ l=0 ûle ilx ) n∑ k=0 k2ûke ikx = n∑ n=0 ∂ûk ∂t einx + c n∑ n=0 n∑ k+l=n ( k2ûlûk ) einx = n∑ n=0 ∂ûk ∂t einx + c n∑ n=0 ŵne inx = 0. the computation of the coefficients ŵn takes o ( n 2 ) operations, which is too expensive if compared to the o (n ) cost of a finite difference algorithm to treat the same nonlinear term. in 3 dimensions, this cost rises up to o ( n 4 ) in the spectral method and o ( n 3 ) for finite diferences. it was this single fact that historically kept spectral methods from being used on applications involving a large number of data. this situation changed when transform methods between the spectral coefficients and the physical values were developed independently by orszag() and eliasen, machenhauer and rasmussen (). these tranforms, known by the general name of fast fourier transforms (fft), allow the computation of the coefficients of the nonlinear term to be performed in o (n log n ) operations. the general idea of the transform technique to compute the spectral coefficients of a general nonlinear term as f (x) g (x) is as follows: start by computing the physical 6, 4(2004) spectral methods for partial differential equations 7 values of f and g through ffts (o (n log n ) operations), perform the pointwise product in physical space (o (n ) operations) and go back to spectral space, applying the inverse transformation to generate the coeficients ˆ(f g)k (o (n log n )). orszag (1971) termed the conjugation of the galerkin spectral method with the transform technique to compute nonlinear terms as a pseudospectral method. this term is also used in the literature to denote collocation methods, which we will introduce in the next section. this happens, because depending on the way the collocation method is designed, it becomes equivalent to a pseudospectral method. for the interested reader, we advance that if the collocation points are chosen as nodes of the quadrature formulae used in the pseudospectral method, than this equivalence occurs (see canuto and hesthaven). 2.3 non-periodic problems for non-periodic problems, the sets of functions choosen as bases are solutions of certain differential equations, the sturm-liouville problems. for instance, the set of chebyshev polynomials defined as tk (x) = cos (k arccos (x)) are solutions to the following differential equation in the interval [−1, 1]: − (√ 1 − x2t ′k (x) )′ = k2 √ 1 − x2 tk (x) . (8) it is easy to prove that these functions are indeed polynomials, since t0 ≡ 1, t1 (x) = x and by using trigonometric identities one can arrive to the recurrence relation tk+1 = 2xtk − tk−1. the expansion assumes the form fn (x) = n∑ k=0 f̂ktk (x) with f̂k = ∫ 1 −1 f (x) tk (x) w (x) dx and w (x) = ( 1 − x2 )− 12 . note that, after the change of variable x = cos (θ), the chebyshev polynomials become cosine functions, enabling many theoretical results to be transported from the fourier system, including spectral convergence. if a function f is expanded on a chebyshev series as f = ∞∑ k=0 f̂ktk, then its derivative can also be expressed on a chebyshev series as f′ = ∞∑ k=0 f̂ (1) k tk, whose coefficients f̂ (1)k are computed with the help of the recurrence relation 2tk (x) = 1 k + 1 t ′k+1 (x) − 1 k − 1 t ′k−1 (x) , k ≥ 1. (9) for instance, when considering an approximation fn as above, we have from (9) that 2kf̂ (1)k = ck−1f̂ (1) k−1 − f̂ (1) k+1, for k ≥ 1, with ck = 2, for j = 0, n and 1 otherwise. we 8 bruno costa 6, 4(2004) also have f̂ (1)k = 0, for k ≥ n , and the remaining coefficients can be computed in decreasing order by 2kf̂ (1)k = ck−1f̂ (1) k−1 − f̂ (1) k+1, for 0 ≤ k ≤ n − 1. this relation can be easily generalized to obtain derivatives of higher order (see [8]). another property inherited from the fourier system is the possibility of applying fast transforms between spectral and physical spaces. this makes the chebyshev polynomials the preferred system among the many possibilities of nonperiodic bases. much more information on the chebyshev polynomials can be found at the books of fox and parker (1968) and rivlin (1974) . another set of great use is the set of legendre polynomials, defined as the solutions of the sturm-liouville problem − (( 1 − x2 ) l′k (x) )′ = k (k + 1) lk (x) . (10) although there is no fast transform for these polynomials, the spectral coefficients of the expansion f = ∞∑ k=0 f̂klk are defined as f̂k = ∫ 1 −1 f (x) lk (x) dx, where the weight function w (x) = 1 makes much easier to prove theorems for this set than for the chebyshev polynomials. here, the book of reference is the one by szegö (1939) . note that both functions in the left hand side of equations (8) and (10) vanish at ±1. it is this property that allows the vanishing of the boundary terms on the integration by parts process, yielding spectral convergence to the chebyshev and legendre bases. sturm-liouville problems with this property are named singular. infinite domains can also be treated by polynomials. hermite and laguerre polynomials are the common bases used for infinite and doubly infinite domains respectively. it is also common to map infinite domains to finite ones and use the chebyshev polynomials (see [8]). 3 polinomial interpolation in the collocation version of spectral methods, one requires the numerical approximation to be exact on a set of pre-defined points, the collocation points. the general idea is to interpolate the solution on the collocation points and approximate its derivatives by the derivatives of the interpolating polinomial. as we shall see below, collocation spectral methods treat with equal cost problems with linear, variable coefficients or non-linear terms. all products are performed pointwise at the physical space and this makes collocation schemes very similar to a high order finite differences scheme (see [18]). 6, 4(2004) spectral methods for partial differential equations 9 for the sake of clarity, let us start with the step of computing the numerical approximation of the derivative fx of a real function f , from which we only know its values {f (xj )} n j=0 at a set of collocation points {xj} n j=0 . we need to find a polynomial of degree n that interpolates f at these n + 1 points. note that if the degree is less than n, such polynomial might not exist (not all sets of 3 points are colinear). it is also easy to show that we do not have uniqueness (monic uniqueness) if the degree is greater than n . a general formula for such a polynomial depending only on the values {f (xj )} n j=0 is given by (in f ) (x) = n∑ j=0 f (xj )gj (x), (11) where gj (x) is a nth-degree polynomial satisfying gj (xk) = δjk = { 1 if j = k, 0, otherwise. the polynomials {gj (x)} n j=0 are called cardinal functions. it is easily seen that in f (x) is a nth-degree polynomial interpolating f , since it is a linear combination of the nth-degree polynomials gj (x), and it satisfies (in f ) (xk) = f (xk), k = 0, ..., n. (12) in this way, given a set of collocation points, we only need to find the n + 1 nthdegree polynomials gj (x), which can be easily computed by means of lagrangian interpolation through the formula gj (x) = p (x) p ′(xj )(x − xj ) , where p (x) = n∏ j=0 (x − xj ). the derivative of the interpolating polynomial is simply given by the (n − 1)thdegree polynomial (in f ) ′ (x) = n∑ j=0 f (xj )g ′ j (x), (13) providing an easy way to obtain the approximated values (in f ) ′ (xi): we only need to multiply the vector composed of the function values {f (xj )} n j=0 by the differentiation matrix d, whose elements are given by di,j = g′j (xi).in this way, numerical differentiation through collocation spectral methods can be accomplished through matrix-vector multiplication. once the collocation points are provided and the differentiation matrix is obtained, the solution of a pde is very close. for instance, given an initial temperature distribution ~u0 = [ u00, u 0 1, ..., u 0 n ] , with u0j = u (xj , t = 0), one can obtain numerical approximations to future temperature profiles through a first order finite difference temporal 10 bruno costa 6, 4(2004) discretization of the heat equation ut − νuxx = 0, conjugated with the collocation spatial derivative: ~un+1 − ~un ∆t − νdxx~un = 0, or ~un+1 = ~un + ∆tdxx~un, (14) where dxx is the second derivative pseudospectral matrix that can be obtained by squaring the differentiation matrix d. if we suppose, as in section 3.2, that the diffusion coefficient is given by ν (u) = cu. thus, scheme (14) becomes ~un+1 = ~un + c∆t~un · dxx~un, where the product ′·′means pointwise product. note that at equation (14), all values of u, including boundary values, are computed simultaneously. in section 5, we will see that some of the boundary values must be discarded in order to respect the physics of the problem being solved. 3.1 the chebyshev collocation method we will now present the chebyshev collocation method starting by its most used set of collocation points, the chebyshev-gauss-lobatto points: xi = cos ( πi n ) i = 0, ..., n, (15) which take this name because they are the nodes of the gauss-lobatto quadrature formula for the chebyshev polynomials (see [8]). they are also the extrema of the n–th order chebyshev polynomial tn (x) and can be seen as the projection on the x−axis of the equispaced set of points θi = πin , in the semicircle, as in figure (2) below figure 2: the chebyshev points as projections on the x-axis of an equispaced set of points in the semicircle. note that points are concentrated close to the boundaries. 6, 4(2004) spectral methods for partial differential equations 11 the n–th order cardinal function gj (x) is given in this case by gj (x) = (−1)j+1(1 − x2)t ′n (xj ) cj n 2(x − xj ) cj = 1 + δj,0 + δj,n , (16) and the differentiation matrix in closed form is given by dkj = ck cj (−1)j+k xk − xj , k 6= j (17) dkk = −1 2 xk 1 − x2k , k 6= 0, n (18) d00 = −dn n = 2n 2 + 1 6 . (19) we solved the linear heat equation (4) in the interval [−1, 1], using the chebyshev collocation method, see figure (3). since we have imposed temperature zero at both boundaries, we see that the initial condition e−cx 2 converges to a uniform zero temperature on the whole interval, as it should be if we heat a metal bar in the center and immerse its ends on ice. figure 3: solution of the heat equation by the chebyshev pseudospectral method. we end these two introductory sections on spectral methods pointing out that the value of the derivative of a function f at a point xj is a linear combination of all values of f at the remaining points, as it can be seen from (13). this is why spectral methods are called global methods and also means that the differentiation matrices in collocation spectral methods are full, making more expensive in general the numerical differentiation when compared to finite differences or finite elements, where derivatives are computed at each point using only a small number of adjacent points, a characteristic of local methods. 12 bruno costa 6, 4(2004) 4 mapping the chebyshev and legendre bases of polynomials are the most used sets of functions for the spectral expansions of solutions of partial differential equations. however, being eigenfunctions of sturm-liouville problems, the number of domains which can be modelled with them is limited. in this section we will see how to make use of mappings between two-dimensional domains in order to overcome this barrier and still be able to use valuable numerical tools such as the fast fourier transform. mappings also have different applications other than only conforming physical to computational domains. for instance, concentration of points on a region of large gradients might improve resolution. on the other hand, increasing the minimum distance of a set of collocation points increases the stability of temporal integration schemes. below, we deal with such questions when developing a general framework for 1d mappings, aiming the integration of mappings to differentiation operators in order to solve equations on more complex domains without including the metrics terms into the equations. 4.1 grid transformation in 1d consider a set of collocation points {yj} j = 0, ..n, and a mapped set xj = g (yj ) . given a function u, with values u (xj ) at the new points {xj},the derivative of u can be evaluated as du dx = du dy dy dx , (20) where dy dx = 1 g′(y) . in matricial form, (20) can be rewritten as ~u′ = m d~u, (21) where m is the diagonal matrix with elements mjj = 1g′(yj ) and d is the differentiation matrix with respect to the set of points {yj}. note that we need only o(n ) more operations to apply m , after the differentiation process d~u is concluded. since fast transforms are only applicable to the chebyshev-gauss-lobato points (15), we see that the above algorithm keeps the possibility of using the fft for differentiation on sets of collocation points which are mapped from the chebyshev ones. if we take the example of the most simple and common 1d mapping, which is used to conform the chebyshev computational domain [−1, 1] to a physical domain [a, b] of interest: x = g (y) = a + (y + 1) 2 (b − a) . (22) then, the matrix m is the constant diagonal matrix mjj = b−a2 . in general, 2d or 3d grids are cross-products of 1d sets of points. for instance, one can simulate a periodic channel by considering a chebyshev grid in the vertical and a fourier grid in the horizontal direction. one can also define a grid on the circle by using this same arrangement in the radial and angular directions, respectively. 6, 4(2004) spectral methods for partial differential equations 13 this is a natural way to proceed since the derivative operators work in each direction separatly. thus, characteristics of the 1d distribution are transported to the 2d or 3d grid and some properties of the multidimensional method are determined by individual properties of each onedimensional grid component. a relevant practical issue when numerically solving evolutionary pdes is the size of the maximum timestep allowed by the numerical method. it is empirically shown that this size is in inverse proportion to the minimum separation between the points, which is directly proportional to the number of points. the chebyshev points are naturally agglomerated at the boundaries and this causes the timestep to be exceedingly small in comparison to other methods. for instance, finite differences schemes allow a o ( n−1 ) time step when solving the first order wave equation, while a o ( n−2 ) size must be respected to obtain stability with spectral methods. in [26], kosloff and tal-ezer proposed the following mapping in order to mitigate this problem: x = g (y, α) = arcsin (αy) arcsin (α) , (23) the mapping above stretches the grid spacing on the boundary pushing the points to the center of the domain, generating a quasi-uniform grid, where the strength of the endpoints separation depends on α ∈ (0, 1). kosloff and tal-ezer claimed that the mapping decreases the spectral radius of the derivative operator from o ( n 2 ) to o (n ) , increasing the allowed time step from o ( n−2 ) to o ( n−1 ) , in the case of the one-way wave equation. however, it can be easily seen that the mapping has singularities at y = ± 1 α , which introduce some approximation error and make necessary some expertise on choosing the correct value of α. see [26] and [17] for detailed theoretical and numerical information on the choice of α, which might depend on n. besides the increase of the timestep, the mapping (23) also increases the resolution of the high modes because the maximum grid spacing ∆xmax is diminished. resolution of a mode is attained when exponential decay is observed on the spectral coefficients. thus, the improvement of resolution can be measured by looking to the number of points necessary to achieve this order of decay for a fixed frequency. figure(4) shows the size of the spectral coefficients of cos (mx) for varying m and n , where we see that resolution is attained exactly at the point where n > mg′ (0, α) = mα arcsin α , where it should be noticed that ∆xmax = g′ (0, α) ∆ymax increases when α → 1. for a more up to date discussion on the tal-ezer mapping see [4],[9], [13],[30],[31] and [28]. 14 bruno costa 6, 4(2004) figure 4: decay of the spectral coefficients of cos(mx) for several values of m and number of gridpoints n showing the dependence of spatial resolution on the parameter α. 4.2 transfinite mapping in this section we define a simple and useful bidimensional mapping which consists of mapping the square [0, 1]2 onto any quadrilateral with curved boundaries. the transfinite mapping is defined by four functions from the segment [0, 1] onto the curves defining the sides of the quadrilateral. let ~πi (ξ) = (xi (ξ) , yi (ξ)) , ξ ∈ [0, 1] , i = 1, 3 and ~πj (η) = (xj (η) , yj (η)) , η ∈ [0, 1] , j = 2, 4 be the functions defining the sides of the quadrilateral, then the transfinite mapping is expressed by ~ω (ξ, η) = 1 − η 2 ~π1 (ξ) + 1 + η 2 ~π3 (ξ) (24) + 1 + ξ 2 ( ~π2 (η) − 1 + η 2 ~π2 (1) − 1 − η 2 ~π2 (−1) ) + 1 + ξ 2 ( ~π4 (η) − 1 + η 2 ~π4 (1) − 1 − η 2 ~π4 (−1) ) . this mapping was first presented in [19] and in the figures below we show some domains which can be generated. two very interesting cases are the ones of the circle and the ellipse. below, in figure (5), we show a circular grid which is obtained by dividing the boundary of the circle in four curves and mapping them to the interval [0, 1] as above. this is an easy way to avoid the singularities of polar coordinates which appears at the center of the circle due to the vanishing of the radius. 6, 4(2004) spectral methods for partial differential equations 15 figure 5: circular grid generated by the transfinite mapping. figure (6) below shows the solution of the second-order wave equation utt − c2∆u = 0, on a elliptical domain, generated in a similar way as the circular one above, simulating the travelling of sound waves between the foci of an elipse. figure 6: acoustic wave equation simulation on the elliptical grid generated by the transfinite mapping. 4.3 two-dimensional mappings consider the classical case of the polar coordinates transformation (x (r, θ) , y (r, θ)) 7→ (r cos θ, r sin θ) , (r, θ) ∈ [r1, r2] × [0, 2π] , 16 bruno costa 6, 4(2004) where the radial direction is discretized with the chebyshev collocation points and the angular, with the fourier points. if we need to solve a problem in the annular domain of figure (7), this is clearly the transformation to be used. figure 7: grid obtained from the polar mapping. for instance, the problem one wants to solve might be the case of a fluid between two concentric and rotating cylinders. suppose we are only interested in the dynamics on a cross section perpendicular to the axis and, to simplify even more, let us look only to the diffusion of heat on the space between the cylinders. thus, the equation to be discretized is the two-dimensional heat equation ut − ν (uxx + uyy) = 0. (25) we have two alternatives: the first one is to apply the change of variables to the equation and solve it in the new variables r and θ. for instance, the new equation would become ut − ν ( urr + ur r + uθθ r2 ) = 0. (26) this approach has the drawback of requiring some analytical work that sometimes can get cumbersome if the mapping is a little bit more involved. take a look at the transfinite mapping (24) above. the second approach is to express only the derivatives in (x, y) of the original equation in terms of the computational variables (r, θ) . for instance, differentiation with respect to x is carried out using the chain rule: ∂u ∂x = ∂u ∂r ∂r ∂x + ∂u ∂θ ∂θ ∂x . the derivatives with respect to r and θ are respectively the chebyshev and fourier derivatives, but we still need to spend some work on the computation of the quantities 6, 4(2004) spectral methods for partial differential equations 17 ∂r ∂x , ∂r ∂y ∂θ ∂x and ∂θ ∂y . no problem in the polar mapping case, however, look again to the transfinite mapping and see that it can get very troublesome. thus, it would be handy to have a formula for ∂u ∂x and ∂u ∂y not involving the metrics ∂r ∂x , ∂r ∂y , ∂θ ∂x and ∂θ ∂y , but the inverse metrics ∂x ∂r , ∂x ∂θ , ∂y ∂r and ∂y ∂θ , which can be computed on a straightforward way. we can resort to some very simple linear algebra and write ( ∂u ∂r ∂u ∂θ ) = [ ∂x ∂r ∂y ∂r ∂x ∂θ ∂y ∂θ ]( ∂u ∂x ∂u ∂y ) . we solve the system above for [ ∂u ∂x , ∂u ∂y ] by simply using the straightforward formula for the inverse of a 2 × 2 matrix and obtain:( ∂u ∂x ∂u ∂y ) = 1 j [ ∂y ∂θ −∂y ∂r −∂x ∂θ ∂x ∂r ]( ∂u ∂r ∂u ∂θ ) , (27) where j = ( ∂x ∂r ∂y ∂θ − ∂y ∂r ∂x ∂θ ) is the jacobian of the mapping. at this point, we would like to advocate a computational procedure which makes simpler the solution of pdes on mapped domains. looking at formulae, we see that after computing the inverse metrics and the jacobian, which can be done at the begining of the computations, we can define the operators ∂ ∂x and ∂ ∂y and code the equation in its original form, with no need to include the metrics. the library pseudopack contains routines for the computation of the metrics and the jacobian, which makes the computation of a term like uxx as simpler as call ps diff (d t, d r, u, uxx, 2, radx, rady, thx, thy, jacob) we will see below, in section 5, that this procedure is of great help on clarifying the coding and implementation of more involved equations as the navier-stokes and enables the clear use of more complex differential operators, like the ones of vector calculus. 5 applications 5.1 the wave equation the first-order wave equation:{ ut + cux = 0, t > 0, x ∈ [−1, 1], u (x, 0) = u0 (x) , x ∈ [−1, 1], (28) provides us with a typical situation on the study of boundary conditions: the one where some quantity is carried through the domain towards its boundary. it can be seen as the modelling of a travelling wave on a unidimensional medium as a material pulse on a rope or an acoustic wave on a piece of wire. the constant c is the speed of the wave. one can easily check that u (x, t) = u0 (x − ct) (29) 18 bruno costa 6, 4(2004) is the solution to equation (28), and if c > 0, it means that information is being propagated to the right, or that values of the solution at a point x depends on values of the function at points to the left of x and at previous times. if one is interested in the propagation of the initial pulse u0 (x) with no other influence than the pure advection phenomenon, no boundary condition must be set at the point x = 1, since the solution values are going to be ”brought” there by the numerical integration itself (remember that spectral differentiation also computes values for the boundary points, see section 3). the imposition of a boundary condition such as u (1) = 0, as it might be naively proposed, would simulate a numerical barrier on which the travelling pulse would hit and reflect backwards with inverse values. just think about swinging a rope with one end fixed at a wall. on the other hand, we necessarily have to impose a boundary condition at x = −1 and discard the value computed by the spectral differentiation, otherwise, we would be going against the restriction expressed by equation (29) that information must always come from the left. if we use the gauss-lobato set of points {xj}, then x0 = 1 and xn = −1 are boundary points and both are updated at the temporal integration process, since the differentiation matrix includes lines for these points. however, the value of u at −1 has to be thrown out and substituted by a convenient boundary condition, such as u (−1) = 0, (30) at the end of every step (at the end of every stage, when using a runge-kutta scheme, see [11]). remark 5.1 in the case above, the inclusion of the boundary point x = 1 is not essential for the simulation. in these situations one could use instead the gaussradau set of points xj = cos 2πj 2n + 1 , 1 ≤ j ≤ n. in the case of the second-order wave equation, however, both endpoints are needed, since we will have waves moving in both directions, as we shall see below. remark 5.2 statement (30) is equivalent to zeroing the last row of the differentiation matrix. this can be generalized to the case of a second order pde with homogeneous boundary conditions, where one would fill the first and last lines with zeroes. this does not apply if one has nonzero dirichlet boundary conditions. the typical situation provided by the one-way wave equation (28) will help us to design boundary conditions for problems of greater interest like the navier-stokes equations. equations whose solutions involve propagation of information at a finite speed are called hyperbolic equations and (28) is their very simplest example. in what follows, we will try to reduce more complicated hyperbolic equations to a form which is similar to the one-way wave equation: ut + aux = 0, (31) 6, 4(2004) spectral methods for partial differential equations 19 where u is a vector with the dependent variables and a is a matrix that does not depend on the derivatives of u , but may depend of u itself. for this reason, we denote (31) a quasilinear system of partial differential equations. another characteristic of hyperbolic equations in the form of the system (31), is that all the eigenvalues of a are nonzero real numbers, in this way we are able to diagonalize a = γλγ−1(for the sake of simplicity, let us disregard the case of defective eigenvalues), where γ and λ are the matrices of eigenvectors and eigenvalues of a, respectively. applying the variables transformation v = γ−1u to equation (31) yields the uncoupled system of equations vt + λvx = 0. (32) since each equation of the above system is similar to the one-way wave equation, we are able to construct boundary conditions for all the transformed variables. the eigenvalues of a are the velocities with which the new dependent variables in v are being propagated. for practical applications, the diagonalization of a is performed ahead of the temporal integration process and the system (32) is used only to impose the boundary conditions to the transformed variables. the process ends by the computation of the boundary values of the original variables by an inverse transformation. below, we illustrate this technique with the case of the second order wave equation utt − c2uxx = 0 by reducing it to the first order linear system ut = aux, where u = ( u v ) and a = [ 0 c c 0 ] . the diagonalization of the matrix a = γλγ−1 makes possible to write the system above in its characteristic form:( u + v u − v ) t = c ( u + v u − v ) x , (33) since γ = [ 1 1 1 −1 ] and λ = [ c 0 0 −c ] . defining the characteristic functions r1 = u + v and r2 = u − v, we easily obtain the solution of (33) as r1 (x, t) = r1 (x − ct, 0) and r2 = r2(x + ct, 0). thus, r1 has to have its right boundary condition imposed and r2, its left one. this reasoning is particularly useful when simulating infinite domains, since they impose no barrier for the travelling quantity to leave the computational domain. due to its derivation, these are called characteristic boundary conditions. another name for the characteristic functions above is ”riemann quantities ” and they are the main subject of study when designing numerical schemes for hyperbolic conservation laws. 20 bruno costa 6, 4(2004) for us, they will be of great use in section 6 when performing this same characteristic decomposition for the more complex (and more interesting) system of the navier stokes equations. they will also be useful for the setting of interface conditions for multi-domain implementation, in section 6. figure 8: second-order wave equation with characteristic boundary conditions. in figure (8) above, we show the solution of the second-order wave equation using characteristic boundary conditions with an initial pulse at the middle of the domain. this initial pulse is decomposed into two smaller ones travelling to opposite directions. note that when passing through the boundaries, little wiggles start to appear, but nothing is reflected inside the domain and once the waves are gone, nothing is ”sounding”. 5.2 nonlinearity: burgers equation the viscous burgers equation in 1d is given by: ut + uux − νuxx = 0, (34) and is the simplest equation where the phenomena of diffusion and nonlinear advection are present. it is therefore very used on testing numerical methods for fluid dynamics and their proposed improvements. another important feature of the burgers equation is the development of discontinuities, or shocks, in finite time. in figure (9) below, we used the fourier collocation method with 64 points in order to show the numerical solution of (34), with a gaussian function centered at the middle of the periodic domain as the initial condition. the advective speed is given by u itself, which means that the initial profile is going to move to the right, however, since the velocity start to decrease at center of the domain, particles departing from the center will colide with slower particles on their right, causing the discontinuity observed in the numerical solution. the value of the diffusion constant ν is the measure of how much the solution is going to behave as the heat equation. thus, smaller the value of ν, sharper is the discontinuity. in this example, a value of v = 0.001 was used. the occurrence of a discontinuity in the solution poses an extra difficulty for spectral methods. the osccilations emanating from the discontinuity are known as the gibbs phenomenon and are caused by the fact that a discontinuous solution is being approximated by an osccilatory set of smooth functions. 6, 4(2004) spectral methods for partial differential equations 21 filtering is an effective tool to get rid of the osccilations and recover exponential accuracy away from the discontinuity. the central idea of using filters is to increase the order of convergence away from the discontinuity by attenuating the high order coefficients, which decay only linearly. care must be taken in order to maintain the important information contained in these coefficients, otherwise we might obtain a strongly smeared function. the gibbs phenomenon is a relevant issue due to the contamination of the solution with the spurious osccilations. the use of filters is recomended in these situations and in figure (5.2) below, we see the attenuation of the osccilations after the use of a filter of the exponential type, also paying the price of an extra smoothing of the discontinuity. more information on the several types of filters and their properties can be found in [8]. figure 9: burgers equation in 1d. in the two-dimensional case, the viscous burgers equation assumes the form:   ut − ν∆u + (u · ∇u ) = f, in (0, 1)2, t in [0, t ] , u = g on ∂ω, u (x, y, 0) = u0(x, y), ∀ (x, y) ∈ ω, (35) where u = (u, v). the initial condition u0(x, y) = ( cos (πx) cos (πy) , ( x+y 4 )) and forcing function f (x, y) = (sin(3πx) sin(4πy + 3 sin(16πx) sin(8πy)), sin(4πx) sin(2πy + 2 sin(10πx) sin(16πy))) are chosen in such a way to promote enough dynamics in the fluid, forcing an interaction between the several frequencies involved, and generate some ¨turbulence¨. as it can be seen in figure (10), we have a discontinuity front close to x = 0.6. in this example, taken from [7], we used 64 points in each direction and a timestep of size ∆t = 10−4. without the use of the exponential filter, the code became unstable. 22 bruno costa 6, 4(2004) figure 10: 5.3 2d flow through an obstacle -the compressible navier stokes equation we now have all the necessary ingredients (mapping, filtering, boundary conditions) to simulate a more realistic example. we chose the classical case of the flow past a circular cylinder. we will use the two–dimensional compressible viscous navier– stokes equations in strong conservation form (see [2] and [32]). these equations describe the laws of conservation of mass, momentum and energy when the system is isolated from the influence of external forces. they are much more involved equations than the ones we have been dealing, nevertheless, their derivation can be found in many basic books on fluid dynamics, for instance, anderson [2] presents a clear and thorough construction of these equations in two and three dimensions. the non-dimensional form of these equations in the cartesian coordinates (x , y) is ∂q ∂t + ∂f ∂x + ∂g ∂y = 1 re ( ∂fν ∂x + ∂gν ∂y ) (36) where q =   ρ ρu ρv e   , 6, 4(2004) spectral methods for partial differential equations 23 and f =   ρu ρu2 + p ρuv (e + p ) u   , fν =   0 τ xx τ xy γκ pr−1 ∂t ∂x + uτ xx + vτ xy   , (37) g =   ρv ρuv ρv2 + p (e + p ) v   , gν =   0 τ xy τ yy γκ pr−1 ∂t ∂y + uτ xy + vτ yy   . the variables are the density ρ, the velocity field (u, v) and the total energy e. the parameter γ = cρ cv is the ratio of specific heats at constant density and volume, pr is the prandtl number and κ is the coefficient of thermal conductivity. the stress tensor elements are given by τ xx = 2 3 µ ( 2 ∂u ∂x − ∂v ∂y ) , τ xy = µ ( ∂u ∂y + ∂v ∂x ) , τ yy = 2 3 µ ( 2 ∂v ∂y − ∂u ∂x ) . and the viscosity µ is related to the temperature by the sutherland viscosity law µ = (1 + s)t 3 2 s + t , (38) where s is a thermodynamic non-dimensionalized constant. the navier stokes equations are coupled with the non-dimensional equations of state and internal energy, respectively: p = (γ − 1) ρ t, e = ρ [ t + 1 2 (u2 + v2) ] . the navier–stokes equations (36) and their necessary supplementary relations such as the sutherland viscosity law are characterized by four parameters [32]. these parameters are the free–stream mach number m∞, the free–stream reynolds number re, the diameter of the cylinder d, and the dimensional temperature t0. the free– stream reynolds number re∞ = ρ∞u∞d/µ∞ of the flow is based on the diameter of the cylinder d, the free–stream velocity u∞, the free–stream density ρ∞ and the dynamic viscosity µ∞, which are the values of these quantities away from the cylinder. the physical domain will be represented by the ring-shaped region shown in figure (7) below, which is the image of the computational domain (ξ , η) ∈ [0, 2π] × [−1, 1] through the polar mapping. the navier–stokes equations (36) written in the coordinates (ξ , η) assume the form ∂q̄ ∂t + ∂f̄ ∂ξ + ∂ḡ ∂η = 1 re ( ∂f̄ν ∂ξ + ∂ḡν ∂η ) , (39) 24 bruno costa 6, 4(2004) where q̄ = 1 j   ρ ρu ρv e   , f̄ = f j ∂ξ ∂x + g j ∂ξ ∂y , f̄ν = fν j ∂ξ ∂x + gν j ∂ξ ∂y , (40) ḡ = f j ∂η ∂x + g j ∂η ∂y , ḡν = fν j ∂η ∂x + gν j ∂η ∂y ; and j is the jacobian of the transformation (ξ, η) → (x, y), given by j = ξxηy −ξyηx. all the derivative terms in (37) have to be computed with respect to the coordinates (ξ, η). therefore, the elements of the stress tensor are given by τ xx = 2 3 µ [ 2 ( ∂u ∂ξ ∂ξ ∂x + ∂u ∂η ∂η ∂x ) − ( ∂v ∂ξ ∂ξ ∂y + ∂v ∂η ∂η ∂y )] , τ xy = µ [ ( ∂u ∂ξ ∂ξ ∂y + ∂u ∂η ∂η ∂y ) + ( ∂v ∂ξ ∂ξ ∂x + ∂v ∂η ∂η ∂x )] , τ yy = 2 3 µ [ 2 ( ∂v ∂ξ ∂ξ ∂y + ∂v ∂η ∂η ∂y ) − ( ∂u ∂ξ ∂ξ ∂x + ∂u ∂η ∂η ∂x )] and the gradient of the temperature t : ∂t ∂x = ∂t ∂ξ ∂ξ ∂x + ∂t ∂η ∂η ∂x , ∂t ∂y = ∂t ∂ξ ∂ξ ∂y + ∂t ∂η ∂η ∂y . in this problem, we take γ = 1.4. and pr = 0.72 figure (11) shows a contour plot of the density function ρ after the trail of vortices started to develop, as the streamlines of the velocity field show. go to www.labma. ufrj.br/ ˜bcosta to see the movie. we used 64 points in both directions and applied the kosloff-tal-ezer mapping in the radial direction. an exponential filter of order 16 was also applied. the boundary conditions were imposed through the characteristic decomposition as above, in order to simulate a free boundary. for more information on the characteristic variables decomposition and further results on the flow through an obstacle problem, the interested reader can check [22]. 6, 4(2004) spectral methods for partial differential equations 25 figure 11: contour density plot for the flow through an obstacle problem. equations (36) and (39) can be written in the general form ∂q ∂t + ∇x,y · (f, g) = 1 re ∇x,y · (fv, gv) (41) and one only has to note that ∇ξ,η · (f̄ , ḡ) is the conservative form of the divergence ∇x,y · (f, g) in the general cartesian coordinates (x, y)(see [2]). it is now worth pointing that the software pseudopack2000 has all the tools needed for the transformation from a classical set of computational variables like fourier, chebyshev or legendre to a general set of curvilinear coordinates like polar, cylindrical, spherical or user defined, including the computation of the metrics and the jacobian. below, we show the one-page code in fortran 90 necessary to implement the main part of the navier-stokes program to generate figure (11) when using the subroutines of pseudopack2000. 26 bruno costa 6, 4(2004) !********************************************************************** ! compute the velocities u and v and the temperature !********************************************************************** u = q(:,:,2)/q(:,:,1) v = q(:,:,3)/q(:,:,1) t = q(:,:,4)/q(:,:,1) t = t half*(u**2+v**2) !********************************************************************** ! compute the viscosity array mu !********************************************************************** mu = (one+s)*t*sqrt(t)/(s+t) !********************************************************************** ! compute the derivatives of u and v with respect to x and y !********************************************************************** call psgrad(dt, dr, u, ux, uy, radx, rady, thx, thy, jacob) call psgrad(dt, dr, v, vx, vy, radx, rady, thx, thy, jacob) !********************************************************************** ! compute the stress terms !********************************************************************** tauxx = rre*mu*(c43*ux-c23*vy) tauxy = rre*mu*( uy+ vx) tauyy = rre*mu*(c43*vy-c23*ux) !********************************************************************** ! compute tx and ty !********************************************************************** call psgrad(dt, dr, t, t x, t y, radx, rady, thx, thy,jacob) tx = cq*rre*mu*tx ty = cq*rre*mu*ty !********************************************************************** ! compute the pressure term !********************************************************************** p = q(:,:,1)*t*gm1 !********************************************************************** ! compute the divergence of (f-fv, g-gv) in the conservative form !********************************************************************** f(:,:,1) = q(:,:,2) f(:,:,2) = q(:,:,2) *u tauxx + p f(:,:,3) = q(:,:,3) *u tauxy f(:,:,4) = (q(:,:,4)+p)*u tauxx*u tauxy*v tx g(:,:,1) = q(:,:,3) g(:,:,2) = q(:,:,2) *v tauxy g(:,:,3) = q(:,:,3) *v tauyy + p g(:,:,4) = (q(:,:,4)+p)*v tauxy*u tauyy*v ty do i = 1,4 call psdiv(dt, dr, f (:, :, i), g(:, :, i), df lux(:, :, i), radx, rady, thx, thy,jacob, 1) enddo 6, 4(2004) spectral methods for partial differential equations 27 5.4 multidomain as we saw in section 4, the bases used in spectral methods are sets of eigenfunctions of sturm-liouville problems, having limitations with the types of domain that they can be used at. a common way to represent a solution in complex geometries is to use a multidomain setting, where each domain element is of a simple type, or can be mapped to a simple one. for instance, the figure below shows a multidomain representation of the circular obstacle problem with an inner tube through the obstacle. figure 12: multidomain grid for the cylinder with an inner tube. this setting came from the general problem of vortex induced vibrations (viv) common to submarine raisers in oil extraction. it was noticed experimentally that positioning the tube in order to allow passage of the fluid through the obstacle would eliminate the ressonance of the obstacle induced by the vortex generation. in this simple representation, we are only extending the problem of last section to a multidomain configuration, therefore, the equations are being solved separatly in each subdomain. for pasting information through the domain interfaces, we are simply averaging the solution, ensuring its continuity. while being a very simple setting, it shows the potentiallity of the transfinite mapping associated to a multidomain spectral scheme. it is also worth pointing that heavy filtering is necessary in order to sustain stability on the corners of the tube. in this experiment we applied a global filtering of the exponential type, however, this can be improved by also applying a heavier filter only to the solution at the corners. there is a whole new area of research for the multidomain setting, the spectral element method, where the idea is to use much smaller elements than the ones above ( see [25]). 28 bruno costa 6, 4(2004) 5.5 the fortran 90 library pseudopack2000 while several software tools for the solution of partial differential equations (pdes) exist in the commercial (e.g. diffpack) as well as the public domain (e.g. petsc), they are almost exclusively based on the use of low-order finite difference, finite element or finite volume methods. geometric flexibility is one of their main advantages. for most pde solvers employing pseudospectral (collocation) methods, one major component of the computational kernel is the differentiation operator. the differentiation must be done accurately and efficiently on a given computational platform for a successful numerical simulation. it is not an easy task given the number of choice of algorithms for each new and existing computational platform. issues involving the complexity of the coding, efficient implementation, geometric restriction and lack of high quality software library tended to discourage the general use of pseudospectral methods in scientific research and practical applications. in particular, the lack of standard high quality library for pseudospectral methods forced individual researchers to build codes that were not optimal in efficiency and accuracy. furthermore, while pseudospectral methods are at a fairly mature level, many critical issues regarding efficiency and accuracy have only recently been addressed and resolved. the knowledge of these solutions is not widely known and appears to restrict a more general usage of such schemes. when developing the software pseudopack2000, the authors aimed to develop a numerical software to make available to the user, in a high performance computing environment, a library of subroutines providing an accurate, versatile, optimal and efficient implementation of the basic components of global pseudospectral methods on which to address a variety of applications of interest to scientists. this library provides subroutines for computing the derivative of the fourier collocation methods for periodical domain and chebyshev and legendre collocation methods for the non-periodical domain. state-of-the-art numerical techniques such as even-odd decomposition [33], and specialized fast algorithms are employed to increase the efficiency of the library. kosloff-tal-ezer mapping is used for reducing roundoff error in the chebyshev and legendre collocation methods [17]. moreover, highly accurate lagrange polynomial interpolation is used when forming the differentiation matrices [12], decreasing solution contamination by roundoff error when dealing with a large number of grid points. other routines for filtering and grid mapping are also included. since the user is shielded from any coding errors in the main computational kernels, reliability of the solution is enhanced. pseudopack will speed up code development, increase productivity and enhance re-usability. the library contains several simple user callable subroutines that return the derivatives and/or filtering (smoothing) of, possibly multi-dimensional, data sets. in term of flexibility and user interaction, any aspect of the library can be modified by a simple change of a small set of input parameters. all source codes are written in fortran 90. using the macro and conditional capability of the c preprocessor, this software package can be compiled into several versions with several different computational platforms. several popular computational platforms (ibm rs6000, sgi cray, sgi, sun) are supported to take advantages of any existing optimized native library such 6, 4(2004) spectral methods for partial differential equations 29 as general matrix-matrix multiply (gemm) from basic linear algebra level 3 subroutine (blas 3), fast fourier transform (fft) and fast cosine/sine transform (cft/sft).the fortran 90 library pseudopack2000 is available by request at the webpage of the authors and it comes with a manual. 6 conclusions the main goal of this article was to provide a more than rough introduction to the subject of spectral methods and their applications in the study of pdes and the field of the dynamics of fluids. we followed the approach of starting from the very begining and only builting the necessary elements to arrive at a meaningful application, like the flow through an obstacle. thus, many points of relevant interest were ignored and, sometimes, only quickly mentioned through the text. among them we can cite time integration processes and their consequences towards numerical stability. nevertheless, the reader with some experience on fd or fe noticed that the mere substitution of the discrete differentiation operator by the spectral one will make most of the temporal integration formulae work with higher precision, with a much smaller time step, however. the interested reader might now check the cited literature and a very effective starting point is the book of trefethen [35], containing the essentials of spectral collocation methods and a large range of applications. the book is rich of practical examples using a very powerful tool for scientific computing, the matlab language. those interested on a more complete overview will be very pleased. for more extensive insights, the work of canuto et al [8] is a good encyclopedia in spectral methods directed towards fluid dynamics applications, as well as the more recent monography by fornberg [18], where a close relationship between collocation spectral methods and high order finite diference schemes is developed. the book by boyd [6] also contains plenty information on spectral methods and several areas of research, as well as a substantious discussion on spherical harmonics and other bases for problems in the sphere, with applications to meteorology. the reader interested in multidomain 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[2] anderson, j. d.; degrez, g.; dick, e.; grundmann, r. computational fluid dynamics. an introduction. edited and with a preface by john f. wendt. a von karman institute book. springer-verlag, berlin, 1992. 30 bruno costa 6, 4(2004) [3] j. augenbaum, an adaptive pseudospectral method for discontinuous problems, icase report no. 88-54. appl. numer. math. (1988) [4] m. r. abril-raimundo and b. garćia-archilla, approximation properties of a mapped chebyshev method, appl. numer. math., vol. 32, (2000) pp. 119-136. [5] c. m. bender and s. a. orszag, advanced mathematical methods for scientists and engineers, mcgraw hill book, new york (1978) [6] j.p. boyd, chebyshev and fourier spectral methods, 2nd ed., dover, new york, 2000. [7] calgaro, c.; laminie, j.; temam, r. dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization. appl. numer. math. 23 (1997), no. 4, 403–442. 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[36] l. n. trefethen and m. r. trummer, an instability phenomenon in spectral methods, siam j. numer. anal., vol. 24, no. 5, (1987) [37] j. a. c. weideman and l. n. trefethen, the eigenvalues of second-order spectral differentiation matrices, siam j. numer. anal., vol. 23, no. 6, (1988) cubo, a mathematical journal vol.22, n◦02, (273–288). august 2020 http://dx.doi.org/10.4067/s0719-06462020000200273 received: 11 september, 2019 | accepted: 27 july, 2020 fixed point theorems on cone s-metric spaces using implicit relation g. s. saluja department of mathematics, govt. kaktiya p. g., college jagdalpur, jagdalpur 494001 (c.g.), india. saluja1963@gmail.com abstract in this paper, we establish some fixed point theorems in the framework of cone s-metric spaces using implicit relation. our results extend, unify and generalize several results from the current existing literature. especially, they extend the corresponding results of sedghi and dung [24] to the setting of complete cone s-metric spaces. resumen en este art́ıculo, establecemos algunos teoremas de punto fijo en el marco de espacios s-métricos del cono usando una relación impĺıcita. nuestros resultados extienden, unifican y generalizan diversos resultados de la literatura actual existente. especialmente, extienden los resultados correspondientes de sedghi y dung [24] en el contexto de espacios s-métricos de cono completo. keywords and phrases: fixed point, implicit relation, cone s-metric space, cone. 2020 ams mathematics subject classification: 47h10, 54h25. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000200273 274 g. s. saluja cubo 22, 2 (2020) 1 introduction and preliminaries in 2007, huang and zhang [8] introduced the concept of cone metric spaces as a generalization of metric spaces by replacing the set of real numbers by a general banach space e which is partially ordered with respect to a cone p ⊂ e and establish some fixed point theorems for contractive mappings in normal cone metric spaces. in 2012, sedghi et al. [23] introduced the concept of s-metric space which is different from other space and proved fixed point theorems in s-metric space. they also give some examples of s-metric space which shows that s-metric space is different from other spaces. in 2016, rahman and sarwar [20] have discussed the fixed point results of altman integral type mappings in s-metric spaces and in the same year ozgur and tas [14] have studied new contractive conditions of integral type in complete s-spaces. recently, dhamodharan and krishnakumar [6] introduced the concept of cone s-metric space and proved some fixed point theorems using various contractive conditions in the above said space. due to great importance of the fixed point theory, it is immensely interesting to study fixed point theorems on different concepts. many authors studied the fixed points for mappings satisfying contractive conditions in complete s-metric spaces (see, e.g., [6, 11, 13, 14, 20, 23, 25, 26]) and others). popa [15] and [16], on the other hand, considered an implicit contraction type condition instead of the usual explicit condition. this direction of research produced a consistent literature on fixed point and common fixed point theorems in various ambient spaces. for more details see [1, 2, 3, 9, 17, 18, 19, 24]. motivated and inspired by popa [15, 16], sedghi and dung [24] and others, this paper is aimed to study and establish some fixed point theorems in the setting of complete cone s-metric spaces under implicit contractive condition which is used in [24]. following the current literature there is ample vicinity to explore and improve this new avenue of research area. here, we prove an important result of cone s-metric space and then obtain some classical fixed point theorems as corollaries, for example, banach’s contraction mapping principle, kannan’s fixed point theorem, chatterjae’s fixed point theorem, reich fixed point theorem and ćirić’s fixed point theorem in this setting. our results extend and generalize several results from the existing literature, especially, the results of sedghi and dung [24] from complete s-metric spaces to the setting of complete cone s-metric spaces. the present work is to encouraged by its possible application, especially in discrete models for numerical analysis, where iterative schemes are extensively used due to their versatility for computer simulation. these models play an important role in applied mathematics. cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 275 we need the following definitions and lemmas in the sequel. definition 1. ([8]) let e be a real banach space. a subset p of e is called a cone whenever the following conditions hold: (c1) p is closed, nonempty and p 6= {0}; (c2) a,b ∈ r, a,b ≥ 0 and x,y ∈ p imply ax + by ∈ p; (c3) p ∩ (−p) = {0}. given a cone p ⊂ e, we define a partial ordering ≤ in e with respect to p by x ≤ y if and only if y − x ∈ p. we shall write x < y to indicate that x ≤ y but x 6= y, while x ≪ y will stand for y − x ∈ p0, where p0 stands for the interior of p. if p0 6= ∅ then p is called a solid cone (see [28]). there exist two kinds of conesnormal (with the normal constant k) and non-normal ones ([7]). let e be a real banach space, p ⊂ e a cone and ≤ partial ordering defined by p . then p is called normal if there is a number k > 0 such that for all x,y ∈ p , 0 ≤ x ≤ y imply ‖x‖ ≤ k‖y‖, (1.1) or equivalently, if (∀n) xn ≤ yn ≤ zn and lim n→∞ xn = lim n→∞ zn = x imply lim n→∞ yn = x. (1.2) the least positive number k satisfying (1.1) is called the normal constant of p . the cone p is called regular if every increasing sequence which is bounded from above is convergent, that is, if {xn} is a sequence such that x1 ≤ x2 ≤ · · · ≤ xn ≤ · · · ≤ y for some y ∈ e, then there is x ∈ e such that ‖xn − x‖ → 0 as n → ∞. equivalently, the cone p is regular if and only if every decreasing sequence which is bounded from below is convergent. it is well known that a regular cone is a normal cone. suppose e is a banach space, p is a cone in e with int(p) 6= ∅ and ≤ is partial ordering in e with respect to p . example 1. ([12]) let k > 1 be given. consider the real vector space e = { ax + b : a,b ∈ r;x ∈ [ 1 − 1 k ,1 ]} with supremum norm and the cone p = { ax + b ∈ e : a ≥ 0,b ≥ 0 } in e. the cone p is regular and so normal. 276 g. s. saluja cubo 22, 2 (2020) definition 2. ([8, 29]) let x be a nonempty set. suppose that the mapping d: x × x → e satisfies: (cm1) 0 ≤ d(x,y) for all x,y ∈ x with x 6= y and d(x,y) = 0 ⇔ x = y; (cm2) d(x,y) = d(y,x) for all x,y ∈ x; (cm3) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z ∈ x. then d is called a cone metric [8] on x and (x,d) is called a cone metric space [8] or simply cms. the concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric space where e = r and p = [0,+∞). lemma 1. ([22]) every regular cone is normal. example 2. ([8]) let e = r2, p = {(x,y) ∈ r2 : x ≥ 0,y ≥ 0}, x = r and d: x × x → e defined by d(x,y) = (|x − y|,α|x − y|), where α ≥ 0 is a constant. then (x,d) is a cone metric space with normal cone p where k = 1. clearly, the above example shows that the class of cone metric spaces contains the class of metric spaces. definition 3. ([23, 14]) let x be a nonempty set and s : x3 → [0,∞) be a function satisfying the following conditions for all x, y, z, t ∈ x: (sm1) s(x,y,z) ≥ 0; (sm2) s(x,y,z) = 0 if and only if x = y = z; (sm3) s(x,y,z) ≤ s(x,x,t) + s(y,y,t) + s(z,z,t). then the function s is called an s-metric on x and the pair (x,s) is called an s-metric space or simply sms. example 3. ([27]) let x be a nonempty set and d be the ordinary metric on x. then s(x,y,z) = d(x,z) + d(y,z) is an s-metric on x. example 4. ([23]) let x = rn and ‖.‖ a norm on x, then s(x,y,z) = ‖y + z − 2x‖ + ‖y − z‖ is an s-metric on x. example 5. ([23]) let x = rn and ‖.‖ a norm on x, then s(x,y,z) = ‖x − z‖ + ‖y − z‖ is an s-metric on x. example 6. ([24]) let x = r be the real line. then s(x,y,z) = ‖x − z‖ + ‖y − z‖ for all x,y,z ∈ r is an s-metric on x. this s-metric on x is called the usual s-metric on x. cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 277 definition 4. ([6]) suppose that e is a real banach space, p is a cone in e with intp 6= ∅ and ≤ is partial ordering with respect to p. let x be a nonempty set and let the function s : x3 → e satisfy the following conditions: (csm1) s(x,y,z) ≥ 0; (csm2) s(x,y,z) = 0 if and only if x = y = z; (csm3) s(x,y,z) ≤ s(x,x,a) + s(y,y,a) + s(z,z,a), ∀x,y,z,a ∈ x. then the function s is called a cone s-metric on x and the pair (x,s) is called a cone s-metric space or simply csms. example 7. ([6]) let e = r2, p = {(x,y) ∈ r2 : x ≥ 0,y ≥ 0}, x = r and d be the ordinary metric on x. then the function s : x3 → e defined by s(x,y,z) = ( d(x,z) + d(y,z),α(d(x,z) + d(y,z)) ) , where α > 0 is a cone s-metric on x. lemma 2. ([6]) let (x,s) be a cone s-metric space. then we have s(x,x,y) = s(y,y,x). definition 5. ([6]) let (x,s) be a cone s-metric space. (i) a sequence {un} in x converges to u if and only if s(un,un,u) → 0 as n → ∞, that is, there exists n0 ∈ n such that for all n ≥ n0, s(un,un,u) ≪ c for each c ∈ e, 0 ≪ c. we denote this by limn→∞ un = u or limn→∞ s(un,un,u) = 0. (ii) a sequence {un} in x is called a cauchy sequence if s(un,un,um) → 0 as n,m → ∞, that is, there exists n0 ∈ n such that for all n,m ≥ n0, s(un,un,um) ≪ c for each c ∈ e, 0 ≪ c. (iii) the cone s-metric space (x,s) is called complete if every cauchy sequence is convergent. in the following lemma, we see the relationship between a cone metric and a cone s-metric. lemma 3. ([6]) let (x,d) be a cone metric space. then, the following properties are satisfied: (1) s(u,v,z) = d(u,z) + d(v,z) for all u,v,z ∈ x, is a cone s-metric on x. (2) un → u in (x,d) if and only if un → u in (x,sd). (3) {un} is cauchy in (x,d) if and only if {un} is cauchy in (x,sd). (4) (x,d) is complete if and only if (x,sd) is complete. lemma 4. ([24]) let f : x → y be a map from an s-metric space x to an s-metric space y . then f is continuous at x ∈ x if and only if f(xn) → f(x) whenever xn → x. now, we introduce an implicit relation to investigate some fixed point theorems on cone smetric spaces. let ψ be the family of all continuous functions of five variables φ: r5 + → r+. for some k ∈ [0,1), we consider the following conditions. 278 g. s. saluja cubo 22, 2 (2020) (a1) for all x,y,z ∈ r+, if y ≤ φ(x,x,y,z,0) with z ≤ 2x + y, then y ≤ kx. (a2) for all y ∈ r+, if y ≤ φ(y,0,0,y,y), then y = 0. (a3) if xi ≤ yi + zi for all xi,yi,zi ∈ r+, i ≤ 5, then φ(x1, . . . ,x5) ≤ φ(y1, . . . ,y5) + φ(z1, . . . ,z5). moreover, for all y ∈ x, φ(0,0,2y,y,0) ≤ ky. remark 1. note that the coefficient k in conditions (a1) and (a3) may be different, for example, k1 and k3 respectively. but we may assume that they are equal by taking k = max{k1,k3}. 2 main results in this section, we shall prove some fixed point theorems using implicit relation in the setting of cone s-metric spaces. theorem 1. let t be a self-map on a complete cone s-metric space (x,s), p be a normal cone with normal constant k and s(tx,tx,ty) ≤ φ ( s(x,x,y),s(x,x,tx),s(y,y,ty), s(x,x,ty),s(y,y,tx) ) (2.1) for all x,y ∈ x and some φ ∈ ψ. then we have (1) if φ satisfies the condition (a1), then t has a fixed point. moreover, for any x0 ∈ x and the fixed point x, we have s(txn,txn,x) ≤ ( 2kn 1 − k ) s(x0,x0,tx0). (2) if φ satisfies the condition (a2) and t has a fixed point, then the fixed point is unique. (3) if φ satisfies the condition (a3) and t has a fixed point x, then t is continuous at x. cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 279 proof. (1) for each x0 ∈ x and n ∈ n, put xn+1 = txn. it follows from (2.1) and lemma 2 that s(xn+1,xn+1,xn+2) = s(txn,txn,txn+1) ≤ φ ( s(xn,xn,xn+1),s(xn,xn,txn),s(xn+1,xn+1,txn+1), s(xn,xn,txn+1),s(xn+1,xn+1,txn) ) = φ ( s(xn,xn,xn+1),s(xn,xn,xn+1),s(xn+1,xn+1,xn+2), s(xn,xn,xn+2),s(xn+1,xn+1,xn+1) ) = φ ( s(xn,xn,xn+1),s(xn,xn,xn+1),s(xn+1,xn+1,xn+2), s(xn,xn,xn+2),0 ) . (2.2) by condition (csm3) and lemma 2, we have s(xn,xn,xn+2) ≤ 2s(xn,xn,xn+1) + s(xn+2,xn+2,xn+1) = 2s(xn,xn,xn+1) + s(xn+1,xn+1,xn+2). (2.3) since φ satisfies the condition (a1), there exists k ∈ [0,1) such that s(xn+1,xn+1,xn+2) ≤ ks(xn,xn,xn+1) ≤ k n+1s(x0,x0,x1). (2.4) thus for all n < m, by using (csm3), lemma 2 and equation (2.4), we have s(xn,xn,xm) ≤ 2s(xn,xn,xn+1) + s(xm,xm,xn+1) = 2s(xn,xn,xn+1) + s(xn+1,xn+1,xm) . . . ≤ 2[kn + · · · + km−1]s(x0,x0,x1) ≤ ( 2kn 1 − k ) s(x0,x0,x1). this implies that ‖s(xn,xn,xm)‖ ≤ (2knk 1 − k ) ‖s(x0,x0,x1)‖. taking the limit as n,m → ∞, we get ‖s(xn,xn,xm)‖ → 0, since 0 < k < 1. thus, we have s(xn,xn,xm) → 0 as n,m → ∞. this shows that the sequence {xn} is a cauchy sequence in the complete cone s-metric space (x,s). by the completeness of the space, we have limn→∞ xn = x ∈ x. moreover, taking the limit as m → ∞ we get s(xn,xn,x) ≤ (2kn+1 1 − k ) s(x0,x0,x1). 280 g. s. saluja cubo 22, 2 (2020) it implies that s(txn,txn,x) ≤ ( 2kn 1 − k ) s(x0,x0,tx0). now we prove that x is a fixed point of t . by using inequality (2.1) again we obtain s(xn+1,xn+1,tx) = s(txn,txn,tx) ≤ φ ( s(xn,xn,x),s(xn,xn,txn),s(x,x,tx), s(xn,xn,tx),s(x,x,txn) ) = φ ( s(xn,xn,x),s(xn,xn,xn+1),s(x,x,tx), s(xn,xn,tx),s(x,x,xn+1) ) . note that φ ∈ ψ, then using lemma 3 and taking the limit as n → ∞, we get s(x,x,tx) ≤ φ ( 0,0,s(x,x,tx),s(x,x,tx),0 ) . since φ satisfies the condition (a1), then s(x,x,tx) ≤ k.0 = 0. this shows that x = tx. thus x is a fixed point of t . (2) let x1,x2 be fixed points of t . we shall prove that x1 = x2. it follows from equation (2.1) and lemma 2 that s(x1,x1,x2) = s(tx1,tx1,tx2) ≤ φ ( s(x1,x1,x2),s(x1,x1,tx1),s(x2,x2,tx2), s(x1,x1,tx2),s(x2,x2,tx1) ) = φ ( s(x1,x1,x2),s(x1,x1,x1),s(x2,x2,x2), s(x1,x1,x2),s(x2,x2,x1) ) = φ ( s(x1,x1,x2),0,0,s(x1,x1,x2),s(x2,x2,x1) ) = φ ( s(x1,x1,x2),0,0,s(x1,x1,x2),s(x1,x1,x2) ) . since φ satisfies the condition (a2), then s(x1,x1,x2) = 0. this shows that x1 = x2. thus the fixed point of t is unique. (3) let x be the fixed point of t and yn → x ∈ x. by lemma 4, we need to prove that cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 281 tyn → tx. it follows from inequality (2.1) and lemma 2 that s(x,x,tyn) = s(tx,tx,tyn) ≤ φ ( s(x,x,yn),s(x,x,tx),s(yn,yn,tyn), s(x,x,tyn),s(yn,yn,tx) ) = φ ( s(x,x,yn),s(x,x,x),s(yn,yn,tyn), s(x,x,tyn),s(yn,yn,x) ) = φ ( s(x,x,yn),0,s(tyn,tyn,yn), s(tyn,tyn,x),s(x,x,yn) ) . since φ satisfies the condition (a3), by lemma 2 and (csm3), we have s(tyn,tyn,yn) ≤ 2s(tyn,tyn,x) + s(yn,yn,x) = 2s(tyn,tyn,x) + s(x,x,yn) then we have s(x,x,tyn) ≤ φ ( s(x,x,yn),0,0,0,s(x,x,yn) ) +φ ( 0,0,2s(tyn,tyn,x),s(tyn,tyn,x),0 ) ≤ φ ( s(x,x,yn),0,0,0,s(x,x,yn) ) +ks(tyn,tyn,x) = φ ( s(x,x,yn),0,0,0,s(x,x,yn) ) +ks(x,x,tyn). (by lemma 2) therefore s(x,x,tyn) ≤ ( 1 1 − k ) φ ( s(x,x,yn),0,0,0,s(x,x,yn) ) . note that φ ∈ ψ, hence taking the limit as n → ∞, we get s(x,x,tyn) → 0. this shows that tyn → x = tx. this completes the proof. next, we give some analogues of fixed point theorems in metric spaces for cone s-metric spaces by combining theorem 1 with φ ∈ ψ and φ satisfies the conditions (a1), (a2) and (a3). the following corollary is an analogue of banach’s contraction principle. corollary 1. let (x,s) be a complete cone s-metric space and p be a normal cone with normal constant k. suppose that the mapping t : x → x satisfies the following condition: s(tx,tx,ty) ≤ hs(x,x,y) 282 g. s. saluja cubo 22, 2 (2020) for all x,y ∈ x, where h ∈ [0,1) is a constant. then t has a unique fixed point in x. moreover, t is continuous at the fixed point. proof. the assertion follows using theorem 1 with φ(x,y,z,s,t) = hx for some h ∈ [0,1) and all x,y,z,s,t ∈ r+. the following corollary is an analogue of r. kannan’s result [10]. corollary 2. let (x,s) be a complete cone s-metric space and p be a normal cone with normal constant k. suppose that the mapping t : x → x satisfies the following condition: s(tx,tx,ty) ≤ q [s(x,x,tx) + s(y,y,ty)] for all x,y ∈ x, where q ∈ [0, 1 2 ) is a constant. then t has a unique fixed point in x. moreover, t is continuous at the fixed point. proof. the assertion follows using theorem 1 with φ(x,y,z,s,t) = q(y + z) for some q ∈ [0, 1 2 ) and all x,y,z,s,t ∈ r+. indeed, φ is continuous. first, we have φ(x,x,y,z,0) = q(x + y). so, if y ≤ φ(x,x,y,z,0) with z ≤ 2x + y, then y ≤ ( q 1−q ) x with ( q 1−q ) < 1. thus, t satisfies the condition (a1). next, if y ≤ φ(y,0,0,y,y), then y = 0. thus, t satisfies the condition (a2). finally, if xi ≤ yi + zi for i ≤ 5, then φ(x1, . . . ,x5) = q(x2 + x3) ≤ q[(y2 + z2) + (y3 + z3)] = q(y2 + y3) + q(z2 + z3) = φ(y1, . . . ,y5) + φ(z1, . . . ,z5). moreover φ(0,0,2y,y,0) = q(0 + 2y) = 2qy where 2q < 1. thus, t satisfies the condition (a3). the following corollary is an analogue of s. k. chatterjae’s result [4]. corollary 3. let (x,s) be a complete cone s-metric space and p be a normal cone with normal constant k. suppose that the mapping t : x → x satisfies the following condition: s(tx,tx,ty) ≤ p [s(x,x,ty) + s(y,y,tx)] for all x,y ∈ x, where p ∈ [0, 1 2 ) is a constant. then t has a unique fixed point in x. moreover, t is continuous at the fixed point. cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 283 proof. the assertion follows using theorem 1 with φ(x,y,z,s,t) = p(s + t) for some p ∈ [0, 1 2 ) and all x,y,z,s,t ∈ r+. indeed, φ is continuous. first, we have φ(x,x,y,z,0) = p(z + 0). so, if y ≤ φ(x,x,y,z,0) with z ≤ 2x + y, then y ≤ ( 2p 1−p ) x with ( 2p 1−p ) < 1. thus, t satisfies the condition (a1). next, if y ≤ φ(y,0,0,y,y) = 2py, then y = 0 since p < 1 2 . thus, t satisfies the condition (a2). finally, if xi ≤ yi + zi for i ≤ 5, then φ(x1, . . . ,x5) = p(x4 + x5) ≤ p[(y4 + z4) + (y5 + z5)] = p(y4 + y5) + p(z4 + z5) = φ(y1, . . . ,y5) + φ(z1, . . . ,z5). moreover φ(0,0,2y,y,0) = p(y + 0) = py where p < 1. thus, t satisfies the condition (a3). the following corollary is an analogue of s. reich’s result [21]. corollary 4. let (x,s) be a complete cone s-metric space and p be a normal cone with normal constant k. suppose that the mapping t : x → x satisfies the following condition: s(tx,tx,ty) ≤ as(x,x,y) + bs(x,x,tx) + cs(y,y,ty) for all x,y ∈ x, where a,b,c ≥ 0 are constants with a+b+c < 1. then t has a unique fixed point in x. moreover, if c < 1 2 , then t is continuous at the fixed point. proof. the assertion follows using theorem 1 with φ(x,y,z,s,t) = ax+by+cz for some a,b,c ≥ 0 are constants with a + b + c < 1 and all x,y,z,s,t ∈ r+. indeed, φ is continuous. first, we have φ(x,x,y,z,0) = ax + bx + cy. so, if y ≤ φ(x,x,y,z,0) with z ≤ 2x + y, then y ≤ ( a+b 1−c ) x with ( a+b 1−c ) < 1. thus, t satisfies the condition (a1). next, if y ≤ φ(y,0,0,y,y) = ay, then y = 0 since a < 1. thus, t satisfies the condition (a2). finally, if xi ≤ yi + zi for i ≤ 5, then φ(x1, . . . ,x5) = ax1 + bx2 + cx3 ≤ a(y1 + z1) + b(y2 + z2) + c(y3 + z3) = (ay1 + by2 + cy3) + (az1 + bz2 + cz3) = φ(y1, . . . ,y5) + φ(z1, . . . ,z5). 284 g. s. saluja cubo 22, 2 (2020) moreover φ(0,0,2y,y,0) = a.0 + b.0 + c.2y = 2cy where 2c < 1. thus, t satisfies the condition (a3). the following corollary is an analogue of l. b. ćirić’s result [5]. corollary 5. let (x,s) be a complete cone s-metric space and p be a normal cone with normal constant k. suppose that the mapping t : x → x satisfies the following condition: s(tx,tx,ty) ≤ h max { s(x,x,y),s(x,x,tx),s(y,y,ty), s(x,x,ty),s(y,y,tx) } for all x,y ∈ x, where h ∈ [0, 1 3 ) is a constant. then t has a unique fixed point in x. moreover, t is continuous at the fixed point. proof. the assertion follows using theorem 1 with φ(x,y,z,s,t) = h max{x, y,z,s,t} for some h ∈ [0, 1 3 ) and all x,y,z,s,t ∈ r+. indeed, φ is continuous. first, we have φ(x,x,y,z,0) = h max{x,x,y,z,0}. so, if y ≤ φ(x,x,y,z,0) with z ≤ 2x + y, then y ≤ hx or y ≤ hz ≤ h(2x + y). then y ≤ kx with k = max { h, 2h 1−h } < 1. thus, t satisfies the condition (a1). next, if y ≤ φ(y,0,0,y,y) = h max{y,0,0,y,y} = hy, then y = 0 since h < 1 3 . thus, t satisfies the condition (a2). finally, if xi ≤ yi + zi for i ≤ 5, then φ(x1, . . . ,x5) = h max{x1, . . . ,x5} ≤ h max{y1 + z1, . . . ,y5 + z5} ≤ h max{y1, . . . ,y5} + h max{z1, . . . ,z5} = φ(y1, . . . ,y5) + φ(z1, . . . ,z5). moreover φ(0,0,2y,y,0) = h max{0,0,2y,y,0} = 2hy where 2h < 1. thus, t satisfies the condition (a3). example 8. let e = r2, the euclidean plane, p = {(x,y) ∈ r2 : x ≥ 0,y ≥ 0} a normal cone in e and x = r. then the function s : x3 → e defined by s(x,y,z) = |x − z| + |y − z| for all x,y,z ∈ x. then (x,s) is a cone s-metric space. now, we consider the mapping t : x → x by t(x) = x 2 and {xn} = { 1 2n } for all n ∈ n is a sequence converging to zero. cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 285 result analysis (1) taking x = xn−1 and y = xn in inequality (2.1) and using (csm3), we have s(xn,xn,xn+1) = s(txn−1,txn−1,txn) ≤ φ ( s(xn−1,xn−1,xn),s(xn−1,xn−1,txn−1),s(xn,xn,txn), s(xn−1,xn−1,txn),s(xn,xn,txn−1) ) = φ ( s(xn−1,xn−1,xn),s(xn−1,xn−1,xn),s(xn,xn,xn+1), s(xn−1,xn−1,xn+1),s(xn,xn,xn) ) = φ ( s(xn−1,xn−1,xn),s(xn−1,xn−1,xn),s(xn,xn,xn+1), s(xn−1,xn−1,xn+1),0 ) ≤ φ ( s(xn−1,xn−1,xn),s(xn−1,xn−1,xn),s(xn,xn,xn+1), 2s(xn−1,xn−1,xn) + s(xn,xn,xn+1),0 ) . since φ satisfies the condition (a1), so there exists k ∈ [0,1) such that s(xn,xn,xn+1) ≤ ks(xn−1,xn−1,xn) or 2 ( xn − xn+1 ) ) ≤ k.2 ( xn−1 − xn ) or ( 1 2n − 1 2n+1 ) ) ≤ k ( 1 2n−1 − 1 2n ) or k ≥ 1 2 . if we take 0 < k < 1, then inequality (2.1) is satisfied. thus all the conditions of theorem 1 are satisfied. hence by theorem 1, t has a unique fixed point. here, note that ′0′ is the unique fixed point of t. (2) let {yn} = { 1 3n } be a sequence in x converging to the fixed point z = 0, then we have to show that tyn → z as n → ∞, that is, t is continuous at the fixed point of t, we have lim n→∞ tyn = t( lim n→∞ yn) = t(0) = 0 = z. 286 g. s. saluja cubo 22, 2 (2020) that is, tyn → z as n → ∞. thus, t is continuous at the fixed point of t. example 9. let e = r2, the euclidean plane, p = {(x,y) ∈ r2 : x ≥ 0,y ≥ 0} a normal cone in e and x = r. then the function s : x3 → e defined by s(x,y,z) = |x − z| + |y − z| for all x,y,z ∈ x. then (x,s) is a cone s-metric space. now, we consider the mapping t : x → x by t(x) = x 3 . then s(tx,tx,ty) = |tx − ty| + |tx − ty| = 2|tx − ty| = 2 ∣ ∣ ∣ ( x 3 ) − ( y 3 ) ∣ ∣ ∣ = 2 3 |x − y| = 1 3 ( 2|x − y| ) ≤ 1 2 ( 2|x − y| ) = hs(x,x,y) where h = 1 2 < 1. thus t satisfies all the conditions of corollary 1 and clearly 0 ∈ x is the unique fixed point of t. 3 conclusion in this paper, we establish some fixed point theorems using implicit relation in the framework of complete cone s-metric spaces. our results extend, unify and generalize several results from the existing literature. especially, they extend the corresponding results of sedghi and dung [24] from complete s-metric spaces to the setting of complete cone s-metric spaces. however, these results have vast potential in solving various nonlinear problems in functional analysis, differential and integral equations, computer science and engineering. 4 acknowledgement the author is grateful to the anonymous referees for their careful reading and valuable suggestions to improve the manuscript. cubo 22, 2 (2020) fixed point theorems on cone s-metric spaces using implicit relation 287 references [1] a. aliouche and v. popa, general common fixed point theorems for occasionally weakly compatible hybmappings and applications, novi sad j. math. 39(1) (2009), 89–109. 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[29] p. p. zabrejko, k-metric and k-normed linear spaces: survey, collect. math. 48 (1997), no. 4-6, 825–859. introduction and preliminaries main results conclusion acknowledgement cubo, a mathematical journal vol.22, n◦02, (155–175). august 2020 http://dx.doi.org/10.4067/s0719-06462020000200155 received: 09 december, 2019 | accepted: 28 may, 2020 a new iterative method based on the modified proximal-point algorithm for finding a common null point of an infinite family of accretive operators in banach spaces t.m.m. sow amadou mahtar mbow university, dakar senegal sowthierno89@gmail.com abstract in this paper, we introduce and study a new iterative method for finding a common null point of an infinite family of accretive operators with a strongly accretive and lipschitzian operator, by using the proximal-point algorithm. and also we prove that the common null point is a unique solution of variational inequality without imposing any compactness-type condition on either the operators or the space considered. finally, some applications of the main results to equilibrium problems and fixed point problems with an infinite family of pseudocontractive mappings are given. the main result is a generalization and improvement of numerous well-known results in the available literature. resumen en este art́ıculo, introducimos y estudiamos un nuevo método iterativo para encontrar un cero común de una familia infinita de operadores acretivos con un operador lischitziano fuertemente acretivo, usando el algoritmo punto-proximal. también demostramos que el cero común es la única solución de una desigualdad variacional sin imponer ninguna condición de tipo compacidad en ninguno de los operadores o los espacios considerados. finalmente, se entregan algunas aplicaciones de los resultados principales a problemas de equilibrio y problemas de punto fijo con una familia infinita de aplicaciones pseudo-contractivas. el resultado principal es una generalización y mejora de numerosos resultados bien conocidos en la literatura disponible. keywords and phrases: proximal-point algorithm; accretive operators; variational inequality; common zeros. 2020 ams mathematics subject classification: 46t05; 47h06; 47h09; 47h10. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000200155 156 t.m.m. sow cubo 22, 2 (2020) 1 introduction let h be a real hilbert space and k be a nonempty subset of h. for a set-valued map a : h → 2h, the domain of a, d(a), the image of a subset s of h, a(s) the range of a, r(a) and the graph of a, g(a) are defined as follows: d(a) := {x ∈ h : ax 6= ∅}, a(s) := ∪{ax : x ∈ s}, r(a) := a(h), g(a) := {(x, u) : x ∈ d(a), u ∈ ax}. a multi-valued map a : d(a) ⊂ h → 2h is called monotone if the inequality 〈u − v, x − y〉 ≥ 0 holds for each x, y ∈ d(a), u ∈ ax, v ∈ ay. a single-valued operator a : k → h is said to be strongly positive bounded linear if there exists a constant k > 0 such that 〈ax, x〉 ≥ k‖x‖2, ∀ x, y ∈ k. remark 1. it is immediate that if a is k-strongly positive bounded linear, then a is k-strongly monotone and ‖a‖-lipschitz continuous. a monotone operator a is called maximal monotone if its graph g(a) is not properly contained in the graph of any other monotone operator. it is well known that a is maximal monotone if and only if a is monotone and r(i + ra) = h for all r > 0 and a is said to satisfy the range condition if d(a) ⊂ r(i + ra). many problems arising in different areas of mathematics, such as optimization, variational analysis and differential equations, can be modeled by the equation 0 ∈ ax, (1.1) where a is a monotone mapping. the solution set of this equation coincide to a null points set of a. such operators have been studied extensively (see, e.g., bruck jr [5], chidume [9], rockafellar [29], xu [30] and the references therein). consider, for example, the following: let f : h → r ∪ {∞} be a proper lower semi continuous and convex function. the subdifferential, ∂f : h → 2h of f at x ∈ h is defined by ∂f(x) = { x∗ ∈ h : f(y) − f(x) ≥ 〈y − x, x∗〉 ∀ y ∈ h } . it is easy to check that ∂f : h → 2h is a monotone operator on h, and that 0 ∈ ∂f(x) if and only if x is a minimizer of f. setting ∂f ≡ a, it follows that solving the inclusion 0 ∈ au, in this case, is solving for a minimizer of f. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 157 in order to find a solution of problem (1.1), rockafellar [29] introduced a powerful and successful algorithm which is recognized as rockafellar proximalpoint algorithm: for any initial point x0 ∈ h, a sequence {xn} is generated by: xn+1 = jrn(xn + en), ∀ n ≥ 0, where jr = (i + ra) −1 for all r > 0, is the resolvent of a and {en} is an error sequence in a hilbert space. in the recent years, the problem of finding a common element of the set of solutions of convex minimization, variational inequality and the set of fixed point problems in real hilbert spaces, banach spaces and complete cat(0) (hadamard) spaces have been intensively studied by many authors; see, for example, [20, 21, 19, 29, 30] and the references therein. very recently, eslamian and vahidi [10] introduced a new iterative method base on proximal point algorithm with strongly positive bounded linear operator for solving a system of inclusion problem. they established a strong convergence theorem which extends the corresponding results in [30, 2, 32, 28, 13, 14, 15, 16, 16, 17, 18]. theorem 2 (eslamian and vahidi [10]). let h be a real hilbert space and k be a nonempty, closed and convex subset of h. let {bi}, i ∈ n ∗ := {1, 2, 3, ...} be an infinite family of operators of h such that ∞ ⋂ i=1 bi −1(0) 6= ∅ and ∞ ⋂ i=1 d(bi) ⊂ k ⊂ ∞ ⋂ i=1 r(i + rbi), for all r > 0. let a : h → h be a k-strongly bounded linear operator with a coefficient γ̄ and f be a b− contraction mapping of k into itself with a constant b ≥ 0. let {xn} be a sequence defined iteratively from arbitrary x0 ∈ k by:      yn = βn,0xn + ∞ ∑ i=1 βn,ij bi rn xn xn+1 = αnγf(xn) + (i − αna)yn. (1.2) let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ n. assume that 0 < γ < γ̄ b . then, the sequence {xn} generated by (1.2) converges strongly to x ∗ ∈ ∞ ⋂ i=1 bi −1(0). above discussion yields the following questions. question 1:can results of eslamian and vahidi [10], and so on be extended from hilbert spaces to banach spaces? 158 t.m.m. sow cubo 22, 2 (2020) question 2: we know that lipschitzian mapping is more general than contraction. what happens if the contraction is replaced by lipschitzian mapping ? question 3: we know that k-strongly accretive operators and l-lipchizian operators is more general than the strong positive bounded linear operators. what happens if the strongly positive bounded linear operators is replaced by kstrongly accretive operators and l-lipchizian operators ? the purpose of this paper is to give affirmative answers to these questions mentioned above. applications are also included to valide our new findings. 2 preliminairies let e be a real banach space and c be a nonempty, closed and convex subset of e. we denote by j the normalized duality map from e to 2e ∗ (e∗ is the dual space of e) defined by: j(x) := {x∗ ∈ e∗ : 〈x, x∗〉 = ||x||2 = ||x∗||2}, ∀ x ∈ e. let s := {x ∈ e : ‖x‖ = 1}. e is said to be smooth if lim t→0+ ‖x + ty‖ − ‖x‖ t exists for each x, y ∈ s. e is said to be uniformly smooth if it is smooth and the limit is attained uniformly for each x, y ∈ s. let e be a normed space with dime ≥ 2. the modulus of smoothness of e is the function ρe : [0, ∞) → [0, ∞) defined by ρe(τ) := sup { ‖x + y‖ + ‖x − y‖ 2 − 1 : ‖x‖ = 1, ‖y‖ = τ } ; τ > 0. it is known that a normed linear space e is uniformly smooth if lim τ→0 ρe(τ) τ = 0. if there exists a constant c > 0 and a real number q > 1 such that ρe(τ) ≤ cτ q, then e is said to be q-uniformly smooth. typical examples of such spaces are the lp, ℓp and w m p spaces for 1 < p < ∞ where, lp (or lp) or w m p is { 2 − uniformly smooth and p − uniformly convex if 2 ≤ p < ∞; 2 − uniformly convex and p − uniformly smooth if 1 < p < 2. (2.1) it is known that a normed linear space e is uniformly smooth if lim τ→0 ρe(τ) τ = 0. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 159 if there exists a constant c > 0 and a real number q > 1 such that ρe(τ) ≤ cτ q, then e is said to be q-uniformly smooth. typical examples of such spaces are the lp, ℓp and w m p spaces for 1 < p < ∞ where, lp (or lp) or w m p is { 2 − uniformly smooth and p − uniformly convex if 2 ≤ p < ∞; 2 − uniformly convex and p − uniformly smooth if 1 < p < 2. let jq denote the generalized duality mapping from e to 2 e ∗ defined by jq(x) := { f ∈ e∗ : 〈x, f〉 = ‖x‖q and ‖f‖ = ‖x‖q−1 } where 〈., .〉 denotes the generalized duality pairing. notice that for x 6= 0, jq(x) = ‖x‖ q−2j2(x), q > 1. following browder [3], we say that a banach space has a weakly continuous normalized duality map if j is a single-valued and is weak-to-weak∗ sequentially continous, i.e., if {xn} ⊂ e, xn ⇀ x, then j(xn) ⇀ j(x) in e ∗. weak continuity of duality map j plays an important role in the fixed point theory for nonlinear operators. finally recall that a banach space e satisfies opial property (see, e.g., [24]) if lim sup n→+∞ ‖xn − x‖ < lim sup n→+∞ ‖xn − y‖ whenever xn ⇀ x, x 6= y. a banach space e that has a weakly continuous normalized duality map satisfies opial’s property. remark 3. note also that a duality mapping exists in each banach space. we recall from [1] some of the examples of this mapping in lp, lp, w m,p-spaces, 1 < p < ∞. (i) lp : jx = ‖x‖ 2−p lp y ∈ lq, x = (x1, x2, · · · , xn, · · · ), y = (x1|x1| p−2, x2|x2| p−2, · · · , xn|xn| p−2, · · · ), (ii) lp : ju = ‖u‖ 2−p lp |u|p−2u ∈ lq, (iii) w m,p : ju = ‖u‖ 2−p w m,p ∑ |α≤m|(−1) |α|dα ( |dαu|p−2dαu ) ∈ w −m,q, where 1 < q < ∞ is such that 1/p + 1/q = 1. finally recall that a banach space e satisfies opial’s property (see, e.g., [24]) if lim sup n→+∞ ‖xn − x‖ < lim sup n→+∞ ‖xn − y‖ whenever xn w −→ x, x 6= y. recall that an operator a : k → e is said to be accretive if there exists j ∈ jq(x − y) such that 〈ax − ay, j〉 ≥ 0, ∀x, y ∈ k. 160 t.m.m. sow cubo 22, 2 (2020) it is said to be strongly accretive if there exists a positive constant k ∈ (0, 1) and such that for all x, y ∈ k, such that 〈ax − ay, j〉 ≥ k‖x − y‖q, ∀x, y ∈ k. in a hilbert space, the normalized duality map is the identity map. hence, in hilbert spaces, monotonicity and accretivity coincide. a multi-valued map a defined on a real banach space e is called m-accretive if it is accretive and r(i + ra) = e for some r > 0 and it is said to satisfy the range condition r(i + ra) = e for all r > 0. the operator a in the following example satisfies range condition. example 4. let a : r → 2r defined by ax = { sgn(x), x 6= 0, [−1, 1] , x = 0, (2.2) where a is the subdifferential of the absolute value function, ∂|.|, then a is m-accretive. it can be shown that if r(i + ra) = e for some r > 0, then this holds for all r > 0. hence, m-accretive condition implies range condition. the demiclosedness of a nonlinear operator t usually plays an important role in dealing with the convergence of fixed point iterative algorithms. definition 1. let e be a real banach space and t : d(t ) ⊂ e → e be a mapping. i − t is said to be demiclosed at 0 if for any sequence {xn} ⊂ d(t ) such that {xn} converges weakly to p and ‖xn − t xn‖ converges to zero, then p ∈ f(t ), where f(t ) denote the set of fixed points of the mapping t. lemma 5 (demiclosedness principle, [3]). let e be a real banach space satisfying opial’s property, k be a closed convex subset of e, and t : k → k be a nonexpansive mapping such that f(t ) 6= ∅. then i − t is demiclosed; that is, {xn} ⊂ k, xn ⇀ x ∈ k and (i − t )xn → y implies that (i − t )x = y. lemma 6 ([22]). let e be a smooth real banach space. then, we have ‖x + y‖2 ≤ ‖x‖2 + 2〈y, j(x + y)〉 ∀x, y ∈ e. lemma 7 ([31]). assume that {an} is a sequence of nonnegative real numbers such that an+1 ≤ (1 − αn)an + σn for all n ≥ 0, where {αn} is a sequence in (0, 1) and {σn} is a sequence in r such that (a) ∞ ∑ n=0 αn = ∞, (b) lim sup n→∞ σn αn ≤ 0 or ∞ ∑ n=0 |σn| < ∞. then lim n→∞ an = 0. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 161 theorem 8. [9] let q > 1 be a fixed real number and e be a smooth banach space. then the following statements are equivalent: (i) e is q-uniformly smooth. (ii) there is a constant dq > 0 such that for all x, y ∈ e ‖x + y‖q ≤ ‖x‖q + q〈y , jq(x)〉 + dq‖y‖ q. (iii) there is a constant c1 > 0 such that 〈x − y , jq(x) − jq(y)〉 ≤ c1‖x − y‖ q ∀ x, y ∈ e. lemma 9 ( [8]). let e be a uniformly convex real banach space. for arbitrary r > 0, let b(0)r := {x ∈ e : ||x|| ≤ r}, a closed ball with center 0 and radius r > 0. for any given sequence {u1, u2, ....., un, .....} ⊂ b(0)r and any positive real numbers {λ1, λ2, ...., λn, ....} with ∞ ∑ k=1 λk = 1, there exists a continuous, strictly increasing and convex function g : [0, 2r] → r+, g(0) = 0, such that for any integer i, j with i < j, ‖ ∞ ∑ k=1 λkuk‖ 2 ≤ ∞ ∑ k=1 λk‖uk‖ 2 − λiλjg(‖ui − uj‖). lemma 10. [33] let h be a real hilbert space and k a nonempty, closed convex subset of h. let a : k → h be a k-strongly monotone and l-lipschitzian operator with k > 0, l > 0. assume that 0 < η < 2k l2 and τ = η ( k − l2η 2 ) . then for each t ∈ ( 0, min{1, 1 τ } ) , we have ‖(i − tηa)x − (i − tηa)y‖ ≤ (1 − tτ)‖x − y‖ ∀x, y ∈ k. let c be a nonempty subsets of a real banach space e. a mapping qc : e → c is said to be sunny if qc(qcx + t(x − qcx)) = qcx for each x ∈ e and t ≥ 0. a mapping qc : e → c is said to be a retraction if qcx = x for each x ∈ c. lemma 11. [26] let c and d be nonempty subsets of a smooth real banach space e with d ⊂ c and qd : c → d a retraction from c into d. then qd is sunny and nonexpansive if and only if 〈z − qdz, j(y − qdz)〉 ≤ 0 (2.3) for all z ∈ c and y ∈ d. 162 t.m.m. sow cubo 22, 2 (2020) remark 12. if k is a nonempty closed convex subset of a hilbert space h, then the nearest point projection pk from h to k is the sunny nonexpansive retraction. the resolvent operator has the following properties: lemma 13. [12] for any r > 0. (i) a is accretive if and only if the resolvent jar of a is single-valued and nonexpansive; (ii) a is m-accretive if and only if jar of a is single-valued and nonexpansive and its domain is the entire e; (iii) 0 ∈ a(x∗) if and only if x∗ ∈ f(jar ), where f(j a r ) denotes the fixed-point set of j a r . lemma 14. ( [23]) for any r > 0 and µ > 0, the following holds: µ r x + (1 − µ r )jar x ∈ d(j a r ) and jar x = j a µ ( µ r x + (1 − µ r )jar x). lemma 15. [7] let a be a continuous accretive operator defined on a real banach space e with d(a) = e. then a is m-accretive. 3 main results for our main theorem, we shall need the following lemma. lemma 16. let q > 1 be a fixed real number and e be a q-uniformly smooth real banach space with constant dq. let a : e → e be a k-strongly accretive and l-lipschitzian operator with k > 0, l > 0. assume that η ∈ ( 0, min { 1, ( kq dqlq ) 1 q−1 }) and τ = η ( k − dql qηq−1 q ) . then for each t ∈ ( 0, min{1, 1 τ } ) , we have ‖(i − tηa)x − (i − tηa)y‖ ≤ (1 − tτ)‖x − y‖, ∀ x, y ∈ e. (3.1) proof. without loss of generality, assume k < 1 q . then, as η < ( kq dqlq ) 1 q−1 , we have 0 < qk − dql qηq−1. furthermore, from k < 1 q , we have qk − dql qηq−1 < 1 so that 0 < qk − dql qηq−1 < 1. by using (ii) of theorem 8 and properties of a, it follows that ‖(i − tηa)x − (i − tηa)y‖q ≤ ‖x − y‖q + q〈tηay − tηax , jq(x − y)〉 + dq‖tηax − tηay‖ q ≤ ‖x − y‖q − qtη〈ax − ay , jq(x − y)〉 + dq(tη) q‖ax − ay‖q ≤ ‖x − y‖q − qtkη‖x − y‖q + dq(ltη) q‖x − y‖q ≤ ( 1 − qtkη + dql qtqηq ) ‖x − y‖q. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 163 therefore ‖(i − tηa)x − (i − tηa)y‖ ≤ ( 1 − qtkη + dql qtηq ) 1 q ‖x − y‖. (3.2) using definition of τ, inequality (3.2) and inequality (1 + x)s ≤ 1 + sx, for x > −1 and 0 < s < 1, we have ‖(i − tηa)x − (i − tηa)y‖ ≤ ( 1 − tkη + dql qtηq q ) ‖x − y‖ ≤ ( 1 − tη(k − dql qηq−1 q ) ) ‖x − y‖ ≤ (1 − tτ)‖x − y‖, which gives us the required result (3.1). this completes the proof. remark 17. lemma 16 is one generalization of lemma 10 for a banach space. we are now in a position to state and prove our main result. theorem 18. let q > 1 be a fixed real number and e be a q-uniformly smooth and uniformly convex real banach space having a weakly continuous duality map. let k be a nonempty, closed and convex subset of e which is a nonexpansive retract of e with qk as the nonexpansive retraction. let {bi}, i ∈ n ∗ be an infinite family of accretive operators of e such that f := ∞ ⋂ i=1 bi −1(0) 6= ∅ and ∞ ⋂ i=1 d(bi) ⊂ k ⊂ ∞ ⋂ i=1 r(i + rbi), for all r > 0. let a : k → e be a k-strongly accretive and l-lipschitzian operator and f : k → e be a b-lipschitzian mapping with a constant b ≥ 0. let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ k by:        yn = βn,0xn + ∞ ∑ i=1 βn,ij bi rn xn, xn+1 = qk ( αnγf(xn) + (i − ηαna)yn ) . (3.3) let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ n. assume that 0 < η < ( kq dqlq ) 1 q−1 and 0 < bγ < τ, where τ = η ( k − dql qηq−1 q ) . then the sequence {xn} generated by (3.3) converges strongly to x ∗ ∈ f, which is a unique solution of variational inequality 〈ηax∗ − γf(x∗), j(x∗ − p)〉 ≤ 0, ∀p ∈ f. (3.4) 164 t.m.m. sow cubo 22, 2 (2020) proof. first of all, we show that the uniqueness of a solution of the variational inequality (3.4). suppose both x∗ ∈ f and x∗∗ ∈ f are solutions to (3.4). then 〈ηax∗ − γf(x∗), j(x∗ − x∗∗)〉 ≤ 0 (3.5) and 〈ηax∗∗ − γf(x∗∗), j(x∗∗ − x∗)〉 ≤ 0. (3.6) adding up (3.5) and (4.3) yields 〈ηax∗∗ − ηax∗ + γf(x∗) − γf(x∗∗), j(x∗∗ − x∗)〉 ≤ 0. (3.7) dql qηq−1 q > 0 ⇐⇒ k − dql qηq−1 q < k ⇐⇒ η ( k − dql qηq−1 q ) < kη ⇐⇒ τ < kη. it follows that 0 < bγ < τ < kη. noticing that 〈ηax∗∗ − ηax∗ + γf(x∗) − γf(x∗∗), jϕ(x ∗∗ − x∗)〉 ≥ (kη − bγ)‖x∗ − x∗∗‖2, which implies that x∗ = x∗∗ and the uniqueness is proved. below we use x∗ to denote the unique solution of (3.4). without loss of generality, we can assume αn ∈ ( 0, min{1 , 1 τ } ) . now, we prove that the sequences {xn} and {yn} are bounded. let p ∈ f. using (3.3) and the fact that jbirn are nonexpansive, we have ‖yn − p‖ = ‖βn,0xn + ∞ ∑ i=1 βn,ij bi rn xn − p‖ ≤ βn,0‖xn − p‖ + ∞ ∑ i=1 βn,i‖j bi rn xn − p‖ ≤ βn,0‖xn − p‖ + ∞ ∑ i=1 βn,i‖xn − p‖ ≤ ‖xn − p‖. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 165 using lemma 16, we have ‖xn+1 − p‖ = ‖qk ( αnγf(xn) + (i − ηαna)yn ) − p‖ ≤ ‖αnγf(xn) + (i − ηαna)yn − p‖ ≤ αnγ‖f(xn) − f(p)‖ + (1 − ταn)‖yn − p‖ + αn‖γf(p) − ηap‖ ≤ (1 − αn(τ − bγ))‖xn − p‖ + αn‖γf(p) − ηap‖ ≤ max {‖xn − p‖, ‖γf(p) − ηap‖ τ − bγ }. by induction, it is easy to see that ‖xn − p‖ ≤ max {‖x0 − p‖, ‖γf(p) − ηap‖ τ − bγ }, n ≥ 1. hence {xn} is bounded also are {f(xn)}, and {axn}. let k ∈ n∗, from lemma 9 and (3.3), we have ‖yn − p‖ 2 = ‖βn,0xn + ∞ ∑ i=1 βn,ij bi rn xn − p‖ 2 ≤ βn,0‖xn − p‖ 2 + ∞ ∑ i=1 βn,i‖j bi rn xn − p‖ 2 − βn,0βn,kg(‖j bk rn xn − xn‖) ≤ ‖xn − p‖ 2 − βn,0βn,kg(‖j bk rn xn − xn‖). consequently, we obtain ‖xn+1 − p‖ 2 = ‖qk ( αnγf(xn) + (i − ηαna)yn ) − p‖2 ≤ ‖αn(γf(xn) − ηap) + (i − ηαna)(yn − p)‖ 2 ≤ α2n‖γf(xn) − ηap‖ 2 + (1 − ταn) 2‖yn − p‖ 2 + 2αn(1 − ταn)‖γf(xn) −ηap‖‖yn − p‖ ≤ α2n‖γf(xn) − ηap‖ 2 + (1 − ταn) 2‖xn − p‖ 2 − (1 − ταn) 2βn,0βn,kg(‖j bk rn xn − xn‖) +2αn(1 − ταn)‖γf(xn) − ηap‖‖xn − p‖ thus, for every k ∈ n∗, we get (1 − ταn) 2βn,0βn,kg(‖j bk rn xn − xn‖) ≤ ‖xn − p‖ 2 − ‖xn+1 − p‖ 2 + α2n‖γf(xn) − ηap‖ 2 +2αn(1 − ταn)‖γf(xn) − ηap‖‖xn − p‖. (3.8) since {xn} and {f(xn)} are bounded, there exists a constant c > 0 such that (1 − ταn) 2βn,0βn,kg(‖j bk rn xn − xn‖) ≤ ‖xn − p‖ 2 − ‖xn+1 − p‖ 2 + αnc. (3.9) let v i(a, f) the solutions set of variational inequality (3.4). now, we prove v i(a, f) is nonempty. let t0 be a fixed real number such that t0 ∈ ( 0, min{1 , 1 τ } ) . we observe that qf (i+(t0γf−t0ηa)) 166 t.m.m. sow cubo 22, 2 (2020) is a contraction, where qf is the sunny nonexpansive retraction from e to f. indeed, for all x, y ∈ k, by lemma 16, we have ‖qf (i + (t0γf − t0ηa))x − qf (i + (t0γf − t0ηa))x‖ ≤ ‖(i + (t0γf − t0ηa))x −(i + (t0γf − t0ηa))x‖ ≤ t0γ‖f(x) − f(y)‖ +‖(i − t0ηa)x − (i − t0ηa)y‖ ≤ (1 − t0(τ − γ))‖x − y‖. banach’s contraction mapping principle guarantees that qf (i +(t0γf −t0ηa)) has a unique fixed point, say x1 ∈ e. that is, x1 = qf (i + (t0γf − t0ηa))x1. thus, in view of lemma 11, it is equivalent to the following variational inequality problem 〈ηax1 − γf(x1), j(x1 − p)〉 ≤ 0, ∀ p ∈ f. hence, x1 ∈ v i(a, f). by the uniqueness of the solution of (3.4), we have x1 = x ∗. next, we prove that {xn} converges strongly to x ∗. we divide the proof into two cases. case 1. assume that the sequence {‖xn − p‖} is monotonically decreasing. then {‖xn − p‖} is convergent. clearly, we have ‖xn − p‖ 2 − ‖xn+1 − p‖ 2 → 0. it then implies from (3.9) that lim n→∞ βn,0βn,kg(‖j bk rn xn − xn‖) = 0. (3.10) since limn→∞ inf βn,0βn,k > 0 and property of g, we have lim n→∞ ‖xn − j bk rn xn‖ = 0. (3.11) by using the resolvent identity (lemma 14), for any r > 0, we conclude that ‖xn − j bk r xn‖ ≤ ‖xn − j bk rn xn‖ + ‖j bk rn xn − j bk r xn‖ ≤ ‖xn − j bk rn xn‖ + ‖j bk r xn ( r rn xn + (1 − r rn )jbkrn xn ) − jbkr xn‖ ≤ ‖xn − j bk rn xn‖ + ‖ r rn xn + ( 1 − r rn ) jbkrn xn − xn‖ ≤ ‖xn − j bk rn xn‖+|1 − r rn | ‖jbkrn xn − xn‖ → 0, n → ∞, ∀k ∈ n ∗. hence, lim n→∞ ‖xn − j bk r xn‖ = 0. (3.12) we show that lim sup n→+∞ 〈ηax∗ − γf(x∗), j(x∗ − xn)〉 ≤ 0. since e is reflexive and {xn} is bounded, there exists a subsequence {xnj } of {xn} such that {xnj } converges weakly to a in k and lim sup n→+∞ 〈ηax∗ − γf(x∗), j(x∗ − xn)〉 = lim j→+∞ 〈ηax∗ − γf(x∗), j(x∗ − xnj )〉. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 167 from (3.12), the fact that jbkr , k ∈ n ∗ are nonexpansive and lemma 5, we obtain a ∈ f. on the other hand, the assumption that the duality mapping is weakly continuous and the fact that x∗ ∈ v i(a, f), we then have lim sup n→+∞ 〈ηax∗ − γf(x∗), j(x∗ − xn)〉 = lim j→+∞ 〈ηax∗ − γf(x∗), j(x∗ − xnj )〉 = 〈ηax∗ − γf(x∗), j(x∗ − a)〉 ≤ 0. finally, we show that xn → x ∗. applying lemma 6, we get that ‖xn+1 − x ∗‖2 = ‖qk(αnγf(xn) + (i − ηαna)yn) − x ∗‖2 ≤ 〈αnγf(xn) + (i − ηαna)yn − x ∗, j(xn+1 − x ∗)〉 = 〈αnγf(xn) + (i − ηαna)yn − x ∗ − αnγf(x ∗) + αnγf(x ∗) − αnηax ∗ +αnηax ∗, j(xn+1 − x ∗)〉 ≤ ( αnγ‖f(xn) − f(x ∗)‖ + ‖(i − αnηa)(yn − x ∗)‖ ) ‖xn+1 − x ∗‖ +αn〈ηax ∗ − γf(x∗), j(x∗ − xn+1)〉 ≤ (1 − αn(τ − bγ))‖xn − x ∗‖‖xn+1 − x ∗‖ + αn〈ηax ∗ − γf(x∗), j(x∗ − xn+1)〉 ≤ (1 − αn(τ − bγ))‖xn − x ∗‖2 + 2αn〈ηax ∗ − γf(x∗), j(x∗ − xn+1)〉. from lemma 7, its follows that xn → x ∗. case 2. assume that the sequence {‖xn−x ∗‖} is not monotonically decreasing. set bn = ‖xn−x ∗‖ and τ : n → n be a mapping for all n ≥ n0 (for some n0 large enough) by τ(n) = max{k ∈ n : k ≤ n, bk ≤ bk+1}. we have τ is a non-decreasing sequence such that τ(n) → ∞ as n → ∞ and bτ(n) ≤ bτ(n)+1 for n ≥ n0. let i ∈ n ∗, from (3.9), we have (1 − τατ(n)) 2βτ(n),0βτ(n),ig(‖j bi rτ(n) xτ(n) − xτ(n)‖) ≤ ατ(n)c → 0 as n → ∞. furthermore, we have βτ(n),0βτ(n),ig(‖j bi rτ(n) xτ(n) − xτ(n)‖) → 0 as n → ∞. hence, lim n→∞ ‖jbirτ(n)xτ(n) − xτ(n)‖ = 0. (3.13) by same argument as in case 1, we can show that xτ(n) and yτ(n) are bounded in k and lim sup τ(n)→+∞ 〈ηax∗ − γf(x∗), j(x∗ − xτ(n))〉 ≤ 0. we have for all n ≥ n0, 0 ≤ ‖xτ(n)+1−x ∗‖2−‖xτ(n)−x ∗‖2 ≤ ατ(n)[−(τ−bγ)‖xτ(n)−x ∗‖2+2〈ηax∗−γf(x∗), j(x∗−xτ(n)+1)〉], which implies that ‖xτ(n) − x ∗‖2 ≤ 2 τ − bγ 〈ηax∗ − γf(x∗), j(x∗ − xτ(n)+1)〉. 168 t.m.m. sow cubo 22, 2 (2020) then, we have lim n→∞ ‖xτ(n) − x ∗‖2 = 0. therefore, lim n→∞ bτ(n) = lim n→∞ bτ(n)+1 = 0. furthermore, for all n ≥ n0, we have bτ(n) ≤ bτ(n)+1 if n 6= τ(n) (that is, n > τ(n)); because bj > bj+1 for τ(n) + 1 ≤ j ≤ n. as a consequence, we have for all n ≥ n0, 0 ≤ bn ≤ max{bτ(n), bτ(n)+1} = bτ(n)+1. hence, lim n→∞ bn = 0, that is {xn} converges strongly to x ∗. this completes the proof. as a consequence of theorem 18, we have the following theorem. theorem 19. let q > 1 be a fixed real number and e be a q-uniformly smooth and uniformly convex real banach space having a weakly continuous duality map. let {bi}, i ∈ n ∗ be an infinite family of m-accretive operators of e such that f := ∞ ∩ i=1 bi −1(0) 6= ∅. let a : e → e be a kstrongly accretive and l-lipschitzian operator and and f : k → e be a b-lipschitzian mapping with a constant b ≥ 0. let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ e by:      yn = βn,0xn + ∞ ∑ i=1 βn,ij bi rn xn, xn+1 = αnγf(xn) + (i − ηαna)yn. (3.14) let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ n. assume that 0 < η < ( kq dqlq ) 1 q−1 and 0 < bγ < τ, where τ = η ( k − dql qηq−1 q ) . then the sequence {xn} generated by (3.14) converges strongly to x ∗ ∈ f, which is a unique solution of variational inequality (3.4) . proof. since bi are m-accretive operators, we conclude that bi are accretive and satisfy the condition r(i + rbi) = e for all r > 0. setting k = e in theorem 18, we obtain the desired result. corollary 1. let h be a real hilbert space. let k be a nonempty, closed and convex subset of h. let {bi}, i ∈ n ∗ be an infinite family of monotone operators of h such that f := ∞ ⋂ i=1 bi −1(0) 6= ∅ and ∞ ⋂ i=1 d(bi) ⊂ k ⊂ ∞ ⋂ i=1 r(i + rbi), for all r > 0. let a : k → h be a strongly bounded linear cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 169 operator and and f : k → e be a b-lipschitzian mapping with a constant b ≥ 0. let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ k by:        yn = βn,0xn + ∞ ∑ i=1 βn,ij bi rn xn, xn+1 = pk ( αnγf(xn) + (i − ηαna)yn ) . (3.15) let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ n. assume that 0 < η < 2k ‖a‖2 and 0 < bγ < τ, where τ = η ( k − ‖a‖2η 2 ) . then the sequence {xn} generated by (3.15) converges strongly to x ∗ ∈ f, which is the optimality condition for the minimization problem min x∈f η 2 〈ax, x〉 − h(x), (3.16) where h is a potential function for γf (i.e. h ′ (x) = γf(x) on k ). proof. from remark 1, we have a is strongly monotone and ‖a‖-lipschitz. the proof follows theorem 18. 4 applications in this section, as applications, we will utilize theorem 18 to deduced several results. as a direct consequence of theorem 18, we have the following results: 4.1 application to equilibrium problems let h be a real hilbert space and let c be a nonempty, closed and convex subset of h. let f be a bifunction of c × c into r, where r is the real numbers. the equilibrium problem for f is to find x ∈ c such that f(x, y) ≥ 0, ∀y ∈ c. (4.1) the set of solutions is denoted by ep(f). equilibrium problems which were introduced by fan [11] and blum and oettli [4] have had a great impact and influence on the development of several branches of pure and applied sciences. for solving the equilibrium problem for a bifunction f : c × c → r, let us assume that f satisfies the following conditions: (a1) f(x, x) = 0 for all x ∈ c; (a2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y ∈ c; 170 t.m.m. sow cubo 22, 2 (2020) (a3) for each x, y, z ∈ c, lim t→0 f(tz + (1 − t)x, y) ≤ f(x, y) (a4) for each x ∈ c, y → f(x, y) is convex and lower semicontinuous. lemma 20. [6] assume that f : c × c → r satisfying (a1)-(a4). for r > 0 and x ∈ h, define a mapping tr : h → c as follows tr(x) = {z ∈ c, f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0, ∀y ∈ c}, for all x ∈ h. then, the following hold: 1.tr is single-valued; 2.tr is firmly nonexpansive, i.e., ‖tr(x) − tr(y)‖ 2 ≤ 〈trx − try, x − y〉 for any x, y ∈ h; 3.f(tr) = ep(f); 4.ep(f) is closed and convex. the following lemma appears implicitly in [29]. lemma 21. [29] let h be a hilbert space and let c be a nonempty closed convex subset of h. let f : c × c → r satisfy (a1) − (a4). let af be a set-valued mapping of h into itself defined by: af x = { {z ∈ h, f(x, y) ≥ 〈y − x, z〉, ∀y ∈ c, } ∀x ∈ c ∅, x /∈ c. (4.2) then ep(f) = af −1(0) and af is a maximal monotone operator with d(af ) ⊂ c. furthermore, for any x ∈ h and r > 0, the map tr defined as lemma 20 coincides with the resolvent of af , i.e, trx = ( i + raf )−1 x. using theorem 18 , we prove a strong convergence theorem for an equilibrium problem in a hilbert space. theorem 22. let h be a real hilbert space and f : h × h → r satisfying (a1)-(a4) such that ep(f) 6= ∅. let a : h → h be a k-strongly monotone and l-lipschitzian operator and f : k → e be a b-lipschitzian mapping with a constant b ≥ 0. let {xn}, {un} and {yn} be a sequences defined iteratively from arbitrary x0 ∈ h by:      f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ h yn = βnxn + (1 − βn)un, xn+1 = αnγf(xn) + (i − ηαna)yn. (4.3) let {rn} ⊂]0, ∞[, {βn} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, βn ∈ [a, b] ⊂ (0, 1). cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 171 (iii) lim n→∞ inf rn > 0. assume that 0 < η < 2k l2 and 0 < bγ < τ, where τ = η ( k − l2η 2 ) . then the sequence {xn} generated by (4.3) converge strongly to x∗ ∈ ep(f), which is a unique solution of variational inequality 〈ηax∗ − γf(x∗), x∗ − p〉 ≤ 0, ∀p ∈ ep(f). (4.4) proof. since f : h × h → r satisfying (a1)-(a4), we have that the mapping af defined by lemma 21 is a maximal and monotone operator. put b = af in theorem 19 (with i=1). then, we obtain that un = trnxn = j b rn xn. therefore, we arrive at the desired results. 4.2 application to an infinite family of continuous pseudocontractive mappings. let k be a nonempty, closed convex subset of a real banach spacee. a mapping t : k → k is said to be pseudocontractive if there exists j(x − y) ∈ j(x − y) such that 〈t x − t y, j(x − y)〉 ≤ ‖x − y‖2, ∀x, y ∈ k. it is well known that the class of pseudocontractive mapping is more general than the class of nonexpansive mapping. moreover, there exists a relationship between the class of accretive mappings and the class of pseudocontractive mappings. a mapping a : k → e is said to be pseudocontractive if t := i − a is accretive. we can observe that x∗ is a zero of the accretive mapping a if and only if it is a fixed point of the pseudocontractive mapping t := i − a. hence, one has the following result. theorem 23. let q > 1 be a fixed real number and e be a q-uniformly smooth and uniformly convex real banach space having a weakly continuous duality map. let k be a nonempty, closed and convex subset of e which is a nonexpansive retract of e with qk as the nonexpansive retraction. let ti : k → e, i ∈ n ∗ be an infinite family of continuous pseudo-contractive mappings of such that ∞ ⋂ i=1 f(ti) 6= ∅. for each r > 0, let j i r := (i + r(i − ti)) −1, i ∈ n∗. let a : k → e be a k-strongly accretive and l-lipschitzian operator and f : k → e be an b-lipschitzian mapping with a constant b ≥ 0. let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ k by:        yn = βn,0xn + ∞ ∑ i=1 βn,ij i rn xn xn+1 = qk ( αnγf(xn) + (i − ηαna)yn ) . (4.5) let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, 172 t.m.m. sow cubo 22, 2 (2020) (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ n ∗. assume that 0 < η < ( kq dqlq ) 1 q−1 and 0 < bγ < τ, where τ = η ( k − dql qηq−1 q ) . then the sequence {xn} generated by (4.5) converges strongly to x ∗ ∈ ∞ ⋂ i=1 f(ti), which is a unique solution of variational inequality 〈ηax∗ − γf(x∗), j(x∗ − p)〉 ≤ 0, ∀p ∈ ∞ ⋂ i=1 f(ti). (4.6) proof. for each i ∈ n∗, we set bi = i − ti into theorem 18. then f(ti) = bi −1(0), for all i ∈ n∗ and hence ∞ ⋂ i=1 f(ti) = ∞ ⋂ i=1 bi −1(0). furthermore, each bi is m-accretive. therefore, the proof is complete from theorem 18. cubo 22, 2 (2020) a new iterative method based on the modified proximal-point . . . 173 references [1] ya. alber, metric and generalized projection operators in banach space:properties and applications in theory and applications of nonlinear operators of accretive and monotone type, (a. g kartsatos, ed.), marcel dekker, new york, (1996). pp. 15–50. [2] o. a. boikanyo and g. moroşanu, modified rockafellar’s algorithm, math. sci. res. j. 13 (2009), no. 5, 101–122. [3] f. e. browder, convergenge theorem for sequence of nonlinear operator in banach spaces, math.z.100–201–225, (1976). vol. eviii, part 2. [4] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, math. student 63 (1994), no. 1-4, 123–145. 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[33] s. wang, a general iterative method for obtaining an infinite family of strictly pseudocontractive mappings in hilbert spaces, appl. math. lett. 24 (2011), no. 6, 901–907. introduction preliminairies main results applications application to equilibrium problems application to an infinite family of continuous pseudocontractive mappings. cubo a mathematical journal vol.20, no¯ 3, (13–29). october 2018 http: // dx. doi. org/ 10. 4067/ s0719-06462018000300013 mean curvature flow of certain kind of isoparametric foliations on non-compact symmetric spaces naoyuki koike department of mathematics, faculty of science, tokyo university of science, 1-3 kagurazaka shinjuku-ku, tokyo 162-8601, japan koike@rs.kagu.tus.ac.jp abstract in this paper, we investigate the mean curvature flows starting from all leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. we prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal leaf of the foliation in c∞-topology. these results give the geometric information between the leaves. resumen en este art́ıculo, investigamos el flujo por curvatura media comenzando desde cualquier hoja de una foliación isoparamétrica dada por la acción de un cierto grupo soluble en un espacio simétrico de tipo no-compacto. demostramos que el flujo por curvatura media comenzando desde cualquier hoja no mı́nima de la foliación existe para tiempo infinito, si la foliación no admite hojas mı́nimas, entonces el flujo es asintótico al flujo autosemejante comenzando desde otra hoja; en cambio si el flujo admite una hoja mı́nima (en este caso, se muestra que la hoja mı́nima es única), entonces el flujo converge a dicha hoja mı́nima de la foliación en la topoloǵıa c∞. estos resultados entregan información geométrica entre las hojas. keywords and phrases: error function based activation function, multivariate quasi-interpolation neural network approximation, kantorovich-shilkret type operator. http://dx.doi.org/10.4067/s0719-06462018000300013 14 naoyuki koike cubo 20, 3 (2018) 2010 ams mathematics subject classification: 41a17, 41a25, 41a30, 41a35. cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 15 1 introduction in [6], we proved that the mean curvature flow starting from any non-minimal compact isoparametric (equivalently, equifocal) submanifold in a symmetric space of compact type collapses to one of its focal submanifolds in finite time. here we note that parallel submanifolds and focal ones of the isoparametric submanifold give an isoparametric foliation consisting of compact leaves on the symmetric space, where an isoparametric foliation means a singular riemannian foliation satisfying the following conditions: (i) the mean curvature form is basic, (ii) the regular leaves are submanifolds with section. a singular riemannian foliation satisfying only the first condition is called a generalized isoparametric foliation. recently, m. m. alexandrino and m. radeschi [1] investigated the mean curvature flow starting from a regular leaf of a generalized isoparametric foliation consisting of compact leaves on a compact riemannian manifold. in particular, they [1] generalized our result to the mean curvature flow starting from a regular leaf of the foliation in the case where the foliation is isoparametric and the ambient space curves non-negatively. on the other hand, we [7] proved that the mean curvature flow starting from a certain kind of non-minimal (not necessarily compact) isoparametric submanifold in a symmetric space of non-compact type (which curves non-positively) collapses to one of its focal submanifolds in finite time. here we note that the isoparametric foliation associated with this isoparametric submanifold consists of curvature-adapted leaves. see the next paragraph about the definition of the curvature-adaptedness. in this paper, we study the mean curvature flow starting from leaves of the isoparametiric foliation given by the action of a certain kind of solvable subgroup (see examples 1 and 2) of the (full) isometry group of a symmetric space of non-compact type. here we note that this isoparametric foliation consists of (not necessarily curvature-adapted) non-compact regular leaves. we shall explain the solvable group action which we treat in this paper. let g/k be a symmetric space of non-compact type, g = k + p (k := lie k) be the cartan decomposition associated with the symmetric pair (g, k), a be the maximal abelian subspace of p, ã be the cartan subalgebra of g containing a and g = k + a + n be the iwasawa’s decomposition. let a, ã and n be the connected lie subgroups of g having a, ã and n as their lie algebras, respectively. let π : g → g/k be the natural projection. given metric. in this paper, we give g/k the g-invariant metric induced from the restriction b|p×p of the killing form b of g to p × p. the symmetric space g/k is identified with the solvable group an with a left-invariant metric through π|an. fix a lexicographic ordering of a. let g = g0 + ∑ λ∈△ gλ, p = a + ∑ λ∈△+ pλ and 16 naoyuki koike cubo 20, 3 (2018) k = k0 + ∑ λ∈△+ kλ be the root space decompositions of g, p and k with respect to a, where we note that gλ = {x ∈ g | ad(a)x = λ(a)x for all a ∈ a} (λ ∈ △), pλ = {x ∈ p | ad(a)2x = λ(a)2x for all a ∈ a} (λ ∈ △+), kλ = {x ∈ k | ad(a)2x = λ(a)2x for all a ∈ a} (λ ∈ △+ ∪ {0}). note that n = ∑ λ∈△+ gλ. let g = kan be the iwasawa decomposition of g. now we shall give examples of a solvable group contained in an whose action on g/k(= an) is (complex) hyperpolar. since g/k is of non-compact type, π gives a diffeomorphism of an onto g/k. denote by 〈 , 〉 the left-invariant metric of an induced from the metric of g/k by π|an. also, denote by 〈 , 〉g the bi-invariant metric of g induced from the killing form b. note that 〈 , 〉 6= ι∗〈 , 〉g, where ι is the inclusion map of an into g. denote by exp the exponential map of the riemannian manifold an(= g/k) at e and by expg the exponential map of the lie group g. let l be a r-dimensional subspace of a + n and set s := (a + n) ⊖ l, where (a + n) ⊖ l denotes the orthogonal complement of l in a + n with respect to 〈 , 〉e (e : is the identity element of g). according to the result in [5], if s is a subalgebra of a + n and lp := prp(l) (prp : the orthogonal projection of g onto p) is abelian, then the s-action (s := expg(s)) gives an isoparametric foliation without singular leaf. we [5] gave examples of such a subalgebra s of a + n. example 1. let b be a r(≥ 1)-dimensional subspace of a and sb := (a + n) ⊖ b. it is clear that bp(= b) is abelian and that sb is a subalgebra of a + n. example 2. let {λ1, · · · , λk} be a subset of a simple root system π of △ such that hλ1, · · · , hλk are mutually orthogonal, b be a subspace of a ⊖ span{hλ1, · · · , hλk} (where b may be {0}) and li (i = 1, · · · , k) be a one-dimensional subspace of rhλi + gλi with li 6= rhλi, where hλi is the element of a defined by 〈hλi, ·〉 = λi(·) and rhλi is the subspace of a spanned by hλi. set l := b + k∑ i=1 li. then, it is shown that lp is abelian and that sb,l1,··· ,lk := (a + n) ⊖ l is a subalgebra of a + n. in example 2, a unit vector of li is described as 1 cosh(||λi||ti) ξi − 1 ||λi|| tanh(||λi||ti)hλi for a unit vector ξi of gλi and some ti ∈ r, where ||λi|| := ||hλi||. then we denote li by lξi,ti if necessary and set ξiti := 1 cosh(||λi||ti) ξi − 1 ||λi|| tanh(||λi||ti)hλi. set sb := expg(sb) and sb,,l1,··· ,lk := expg(sb,l1,··· ,lk). denote by fb and fb,l1,··· ,lk the isoparametric foliations given by the sb-action and the sb,l1,··· ,lk-one, respectively. a submanifold in a riemannian manifold is said to be curvature-adapted if, for each normal vector v of the submanifold, the normal jacobi operator r(v) := r(·, v)v preserves the tangent space of the submanifold invariantly and the restriction of r(v) to the tangent space commutes with the shape operator av, where r is the curvature tensor of the ambient riemannian manifold. according to the results in[5], the following facts hold for cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 17 isoparametric foliations fb and fb,l1,··· ,lk: (i) all leaves of fb are curvature-adapted. (ii) let λ1, · · · , λk (∈ △+) be as in example 2. if the root system △ of g/k is non-reduced and 2λi0 ∈ △+ for some i0 ∈ {1, · · · , k}, then all leaves of fb,l1,··· ,lk are not curvature-adapted. (iii) if b 6= {0}, then fb,l1,··· ,lk admits no minimal leaf. on the other hand, if b = {0}, then this action admits the only minimal leaf. (iv) let l1, · · · , lk be as in example 2 and li (i = 1, · · · , k) be the orthogonal projection of li onto gλi. then fb,l1,··· ,lk is congruent to fb,l1,··· ,lk. in more detail, we have lb·γ ξ1 (t1)· ··· ·γξk(tk) (sb,l1,··· ,lk · e) = sb,l1,··· ,lk · (b · γξ1(t1) · · · · · γξk(tk)), where γξi (i = 1, · · · , k) is the geodesic in an(= g/k) with γ′ξi(0) = ξ i, b is an element of exp(b) and lb·γ ξ1 (t1)· ··· ·γξk(tk) is the left translation by b · γξ1(t1) · · · · · γξk(tk). for example, in case of k = 1 and b = e, the positional relation among the leaves of these foliations is as in figure 1. sb,l1 · e s b,l1 · e sb,l1 · γξ1(t1) = lγξ1(t1)(sb,l1 · e) exp(b + l1) exp(b + l1) γξ1 γξ1(t1) γξ1 t1 e figure 1. according to the above facts (i) and (ii), the leaves of fb,l1,··· ,lk give examples of interesting isoparametric submanifolds in g/k. in this paper, we shall prove the following facts for the mean curvature flows starting from the non-minimal leaves of f b,l1,··· ,lk . theorem a. assume that b 6= {0}. let m be any leaf of f b,l1,··· ,lk . and mt (0 ≤ t < t) be the mean curvature flow starting from m. then the following statements (i) − (iii) hold. (i) t = ∞ holds. (ii) if m passes through exp(b), then the mean curvature flow mt is self-similar. (iii) if m does not pass through exp(b), then the mean curvature flow mt asymptotes the mean curvature flow starting from the leaf of f b,l1,··· ,lk passing through a point of exp(b). remark 1.1. the mean curvature flow starting from any leaf of fb is self-similar. 18 naoyuki koike cubo 20, 3 (2018) exp(l)exp(b) m1 m3m2 the mean curvature flows starting from leaves m1 and m3 of fb,̄l1,··· ,̄lk (b 6= {0}) asymptotes the mean curvature flow (which is self-similar) starting from a leaf m2 of fb,̄l1,··· ,̄lk. figure 2. also, in case of b = {0}, we obtain the following fact. theorem b. let m be a leaf of f {0},l1,··· ,lk -action other than s {0},l1,··· ,lk · e and mt (0 ≤ t < t) be the mean curvature flow starting from m. then the following statements (i) − (ii) hold. (i) t = ∞ holds. (ii) mt convergres to the only minimal leaf s{0},l1,··· ,lk · e (in c ∞-topology) as t → ∞. exp(l) m1 m3 m2 the mean curvature flows starting from leaves m1, m2 and m3 of e f{0},̄l1,··· ,̄lk converge to the only minimal leaf m 0 of f{0},̄l1,··· ,̄lk. m0 figure 3. the following question arises naturally. question. let f be an isoparametric foliation consisting of non-compact regular leaves on a nonpositively curved riemannian manifold. assume that the leaves of f are cohomogeneity compact (i.e., each leaf l is invariant under some subgroup action hl of the isometry group of the ambient space and the quotient space l/hl is compact). in what case, does the result similar to theorem a or b hold for f? cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 19 2 mean curvature flow. in this section, we shall recall the notion of the mean curvature flow. let ft’s (t ∈ [0, t)) be a oneparameter c∞-family of immersions of a manifold m into a riemannian manifold m̃, where t is a positive constant or t = ∞. define a map f : m×[0, t) → m̃ by f(x, t) = ft(x) ((x, t) ∈ m×[0, t)). denote by π the natural projection of m × [0, t) onto m. for a vector bundle e over m, denote by π∗e the induced bundle of e by π. also, denote by ht and gt the mean curvature vector field and the induced metric of ft, respectively. define a section g of π ∗(t(0,2)m) by g(x,t) := (gt)x ((x, t) ∈ m×[0, t)) and sections h of f∗tm̃ by h(x,t) := (ht)x ((x, t) ∈ m×[0, t)), where t(0,2)m is the tensor bundle of degree (0, 2) of m and tm̃ is the tangent bundle of m̃. the family ft’s (0 ≤ t < t) is called a mean curvature flow if it satisfies (1.1) f∗ ( ∂ ∂t ) = h. in particular, if ft’s are embeddings, then we call mt := ft(m)’s (0 ∈ [0, t)) rather than ft’s (0 ≤ t < t) a mean curvature flow. see [3], [4] and [2] and so on about the study of the mean curvature flow (treated as the evolution of an immersion). 3 the non-curvature-adaptedness of the leaves. in [5], we proved the following statement: (∗) if the root system △ of g/k is non-reduced and 2λi0 ∈ △+ for some i0 ∈ {1, · · · , k}, then all leaves of f b,l1,··· ,lk are not curvature-adapted. (see the statement (ii) of proposition 3.5 in [5]). however, there is a gap in the second-half part of the proof. in this section, we shall close the gap by recalculating the normal jacobi operators of the leaves (see proposition 3.5). we shall use the notations in introduction. according to the fact (iv) stated in introduction, we have lb·γ ξ1 (t1)· ··· ·γξk(tk) (sb,l1,··· ,lk · e) = sb,l1,··· ,lk · (b · γξ1(t1) · · · · · γξk(tk)). hence we suffice to show that the leaves sb,l1,··· ,lk · e’s are not curvature-adapted. as stated in example 2, we set ξiti := 1 cosh(||λi||ti) ξi − 1 ||λi|| tanh(||λi||ti)hλi. for the shape operator of sb,l1,··· ,lk · e, we showed the following facts (see lemma 3.2 of [5]). lemma 3.1[5]. let a be the shape tensor of sb,l1,··· ,lk · e (⊂ an). then, for aξ0 (ξ0 ∈ b) and aξiti (i = 1, · · · , k), the following statements (i) ∼ (vii) hold: (i) for x ∈ a ⊖ (b + k∑ i=1 rhλi), we have aξ0x = aξiti x = 0 (i = 1, · · · , k). 20 naoyuki koike cubo 20, 3 (2018) (ii) for x ∈ ker(ad(ξi)|gλi ) ⊖ rξ i, we have aξ0x = 0 and aξiti x = −||λi|| tanh(||λi||ti)x. (iii) assume that 2λi ∈ △+. for x ∈ g2λi, we have aξ0([θξi, x]) = 0 and aξiti x = −2||λi|| tanh(||λi||ti)x − 1 2 cosh(||λi||ti) [θξi, x], aξi ti ([θξi, x]) = − ||λi|| 2 cosh(||λi||ti) x − ||λi|| tanh(||λi||ti)[θξ i, x], where θ is the cartan involution of g with fix θ = k. (iv) for x ∈ (rξi + rhλi) ⊖ li, we have aξ0x = 0 and aξiti x = −||λi|| tanh(||λi||ti)x. (v) for x ∈ (gλj ⊖ rξj) + ((rξj + rhλj) ⊖ lj) + g2λj (j 6= i), we have aξ0x = aξiti x = 0. (vi) for x ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have aξ0x = µ(ξ0)x. (vii) let ki := exp ( π√ 2||λi|| (ξi + θξi) ) , where exp is the exponential map of g. then ad(ki)◦ aξiti = −aξiti ◦ ad(ki) holds over n⊖ k∑ i=1 (gλi +g2λi), where ad is the adjoint representation of g. remark 3.1. if λi ∈ △+, then we have ||λi|| = √ 2 from how to choose the metric of g/k (see introduction). according to (5.3) in page 310 of [8], we have the following fact. lemma 3.2[8]. let x and y be left-invariant vector fields on an and ∇ be the levi-civita connection of the left-invariant metric 〈 , 〉 of an. then we have (3.2) ∇xy = 1 2 ( [x, y] − ad(x)∗(y) − ad(y)∗(x) ) , where ad(x)∗ (resp. ad(y)∗) is the adjoint operator of ad(x) (resp. ad(y)) with respect to 〈 , 〉e and (•)a+n is the the (a + n)-component of (•). let pr1a+n (resp. pr 2 a+n) be the projection of g onto a + n with respect to the decomposition g = k + (a + n) (resp. g = (k0 + ∑ λ∈△+ pλ) + (a + n)). we [5] showed the following facts (see the proof of lemma 3.2 in [5]). lemma 3.3[5]. (i) for any h ∈ a, we have (3.3) ad(h)∗ = ad(h). cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 21 (ii) for any x ∈ gλ, we have (3.4) ad(x)∗ = −pra+n ◦ ad(θx) =    0 on a −〈x, ·〉e ⊗ hλ − prn ◦ pr1a+n ◦ ad(xk) +prn ◦ pr2a+n ◦ ad(xp) on n, where (•)k (resp. (·)p) denotes the k-component (resp. p-component) of (•). according to (3.4), we have (3.5) ad(x)∗(y) =    0 (λ − µ ∈ △+) −〈x, y〉hλ (λ = µ) −[θx, y] (µ − λ ∈ △+) 0 (λ − µ /∈ △ ∪ {0}) for any x ∈ gλ (λ ∈ △+) and any y ∈ gµ (µ ∈ △+). for each x ∈ a + n, we denote by x̃ the left-invariant vector field on an with (x̃)e = x. by using lemma 3.2, (3.3), (3.4) and (3.5), we can derive the facts directly. lemma 3.4. for any unit vector xλ, yλ of gλ (λ ∈ △+) and hλ (λ ∈ △+), we have ∇ h̃λ h̃µ = ∇h̃λx̃µ = 0, ∇x̃λh̃µ = −λ(hµ)x̃λ (λ, µ ∈ △+) and ∇ x̃λ ỹµ =    1 2 ( [x̃λ, ỹµ] + θ̃[yµ, θxλ] ) (λ − µ ∈ △+) 1 2 [x̃λ, ỹµ] + 〈x̃λ, ỹµ〉h̃λ (λ = µ) 1 2 ( [x̃λ, ỹµ] + θ̃[xλ, θyµ] ) (µ − λ ∈ △+) 1 2 [x̃λ, ỹµ] (λ − µ /∈ △ ∪ {0}) from lemma 3.4 and (3.5), we can derive the following facts for the normal jacobi operators by somewhat long calculations. proposition 3.5. let r be the curvature tensor of an(= g/k). then, for r(ξ0) (ξ 0 ∈ b) and r(ξiti) (i = 1, · · · , k), the following statements (i) ∼ (vi) hold: (i) for x ∈ a ⊖ (b + k∑ i=1 rhλi), we have r(ξ0)(x) = r(ξ i ti )(x) = 0 (i = 1, · · · , k). 22 naoyuki koike cubo 20, 3 (2018) (ii) for x ∈ ker(ad(ξi)|gλi ) ⊖ rξ i, we have r(ξ0)(x) = 0 and r(ξ i ti )(x) = ||λi|| 2 2 (1 − 3 tanh2(||λi||ti))x. (iii) assume that 2λi ∈ △+ (hence ||λi|| = √ 2). for x ∈ g2λi, we have r(ξ0)(x) = r(ξ0)([θξ i, x]) = 0 and r(ξiti)(x) = −||λi|| 2(1 + 3 tanh2(||λi||ti))x − 3||λi|| tanh(||λi||ti) 2 cosh(||λi||ti) [θξi, x] r(ξiti)([θξ i, x]) = − 6||λi|| tanh(||λi||ti) cosh(||λi||ti) x + √ 2||λi|| 4 (1 − 3 tanh2(||λi||ti))[θξ i, x]. (iv) for x ∈ (rξi + rhλi) ⊖ li, we have r(ξ0)(x) = 0 and r(ξiti)(x) = −||λi|| 2x. (v) for x ∈ (gλj ⊖ rξj)+((rξj +rhλj)⊖ lj)+g2λj (j 6= i), we have r(ξ0)(x) = r(ξiti)(x) = 0. (vi) for x ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have r(ξ0)(x) = −µ(ξ0)2x. from lemma 3.1 and proposition 3.5, we can derive the following facts directly. proposition 3.6. for [aξ0, r(ξ0)] (ξ0 ∈ b) and [aξiti , r(ξ i ti )] (i = 1, · · · , k), the following statements (i) ∼ (vi) hold: (i) for x ∈ a ⊖ (b + k∑ i=1 rhλi), we have [a, r(ξ0)](x) = [aξiti , r(ξiti)](x) = 0 (i = 1, · · · , k). (ii) for x ∈ ker(ad(ξi)|gλi ) ⊖ rξ i, we have [aξ0, r(ξ0)](x) = [aξiti , r(ξiti)](x) = 0. (iii) assume that 2λi ∈ △+ (hence ||λi|| = √ 2). for x ∈ g2λi, we have [aξ0, r(ξ0)](x) = [aξ0, r(ξ0)]([θξ i, x]) = 0 and [aξi ti , r(ξiti)](x) = − 3 2 cosh3( √ 2ti) [θξi, x] [aξiti , r(ξiti)]([θξ i, x]) = − 6 cosh3( √ 2ti) x. (iv) for x ∈ (rξi + rhλi) ⊖ li, we have [aξ0, r(ξ0)](x) = [aξiti , r(ξ i ti )](x) = 0. (v) for x ∈ (gλj ⊖ rξj) + ((rξj + rhλj) ⊖ lj) + g2λj (j 6= i), we have [aξ0, r(ξ0)](x) = [aξiti , r(ξiti)](x) = 0. (vi) for x ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have [aξ0, r(ξ0)](x) = [aξiti , r(ξ i ti )](x) = 0. from (iv) of proposition 3.6, we can derive the statement (∗). also, we [5] showed the following fact in terms of lemma 3.1. cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 23 proposition 3.7[5]. if b = {0}, then fb,l ξ1,t1 ,··· ,l ξk,tk admits the only minimal leaf. 4 proof of theorem a in this section, we shall prove theorem a. we use the notations in sections 1 and 3. note that exp|a = exp |a and exp|n 6= exp |n. set σ := exp(t⊥e sb,l1,··· ,lk · e)(= exp(b + r{ξ 1, · · · , ξk})), which is the flat section of the s b,l1,··· ,lk -action through e. each leaf of f b,l1,··· ,lk meets σ at the only one point. that is, σ is regarded as the leaf space of this foliation. for ξ0 ∈ b and ti ∈ r (i = 1, · · · , k), we set xξ0,t1,··· ,tk := expξ0 · γξ1(t1) · · · · · γξk(tk). also, denote by d ds (•) the covariant derivative of vector fields (•) along curves in an (with respect to the left-invariant metric). the following fact is well-known about the geodesics in rank one symmetric spaces of non-compact type but we shall give the proof. lemma 4.1. the velocity vector γ′ ξi (s) (i = 1, · · · , k) is described as (4.1) γ′ ξi (s) = 1 cosh(||λi||s) (ξ̃i)γ ξi (s) − tanh(||λi||s) ||λi|| (h̃λi)γξi (s) and γ′ξ0(s) is described as (4.2). γ′ξ0(s) = (ξ̃0)γξ0(s) proof. set y(s) := 1 cosh(||λi||s) (ξ̃i)γ ξi (s) − tanh(||λi||s) ||λi|| (h̃λi)γξi(s) . it is clear that y(0) = ξi. by using lemma 3.4, we can show d ds y = 0. hence we obtain y(s) = γ′ ξi (s). also, it is clear that (ξ̃0)γξ0 (0) = ξ0. by using lemma 3.4, we can show d ds (ξ̃0)γξ0 (s) = 0. hence we obtain (ξ̃0)γξ0 (s) = γ′ξ0(s). q.e.d. next we shall show the following fact. lemma 4.2. the point xξ0,t1,··· ,tk belongs to σ. proof. it is clear that exp(ξ0) belongs to σ. first we shall show that exp(ξ0) · γξ1(t1) belongs to σ. let γξ0 be the geodesic in an with γ ′ ξ0 (0) = ξ0. since γξ1 is a geodesic in an and lexp(ξ0) is an isometry of an, lexp(ξ0) ◦ γξ1 is a geodesic in an. hence we suffice to show that (lexp(ξ0) ◦γξ1)′(0) = (ξ̃1)exp(ξ0) is tangent to σ. denote by ξ̂1 the parallel vector field along γξ0. take orthonormal bases {eλ1, · · · , eλmλ} of gλ (λ ∈ △+). also, take an orthonormal base {e 0 1, · · · , e0r} 24 naoyuki koike cubo 20, 3 (2018) of a. we describe ξ̂1 as ξ̂1(s) = r∑ i=1 a0i (s)(ẽ 0 i )γξ0 (s) + ∑ λ∈△+ mλ∑ i=1 aλi (s)(ẽ λ i )γξ0 (s) (s ∈ r), where a0i and a λ i are functions over r. fix s0 ∈ r. by using lemma 3.4, we can show d ds ∣∣∣∣ s=s0 ξ̂1 = r∑ i=1 ( (a0i ) ′(s0)(ẽ 0 i )γξ0(s0) + (a0i )(s0) d ds ∣∣∣∣ s=s0 ((ẽ0 i )γξ0 (s) ) ) + ∑ λ∈△+ mλ∑ i=1 ( (aλi ) ′(s0)(ẽ λ i )γξ0(s0) + aλi (s0) d ds ∣∣∣∣ s=s0 ((ẽλ i )γξ0 (s) ) ) = r∑ i=1 ( (a0i ) ′(s0)(ẽ 0 i )γξ0 (s0) + (a0i )(s0)∇γ′ξ0 (s0)((ẽ 0 i )γξ0(s0) ) ) + ∑ λ∈△+ mλ∑ i=1 ( (aλi ) ′(s0)(ẽ λ i )γξ0 (s0) + aλi (s0)∇γ′ξ0 (s0)((ẽ λ i )γξ0(s0) ) ) = r∑ i=1 ( (a0i ) ′(s0)(ẽ 0 i )γξ0 (s0) + (a0i )(s0)(∇ξ̃0ẽ 0 i )γξ0 (s0) ) + ∑ λ∈△+ mλ∑ i=1 ( (aλi ) ′(s0)(ẽ λ i )γξ0 (s0) + aλi (s0)(∇ξ̃0ẽ λ i )γξ0(s0) ) ) = r∑ i=1 (a0i ) ′(s0)(ẽ 0 i )γξ0 (s0) + ∑ λ∈△+ mλ∑ i=1 (aλi ) ′(s0)(ẽ λ i )γξ0 (s0) = 0, that is, (a0i ) ′(s0) = (a λ i ) ′(s0) = 0, where we use γ ′ ξ0 (s0) = ξ̃0γξ0(s0) . from the arbitrariness of s0, we see that a 0 i and a λ i are constant. hence we obtain ξ̂ 1(s) = (ξ̃1)γξ0(s) . on the other hand, since ξ1 is tangent to σ and σ is totally geodesic, ξ̂1(1) also is tangent to σ. hence we see that (ξ̃1)exp(ξ0) is tangent to σ. therefore exp(ξ0) · γξ1(t1) belongs to σ. next we shall show that exp(ξ0) · γξ1(t1) · γξ2(t2) belongs to σ. since γξ2 is a geodesic in an and lexp(ξ0)·γξ1(t1) is an isometry of an, lexp(ξ0)·γξ1(t1) ◦ γξ2 is a geodesic in an. hence we suffice to show that (lexp(ξ0)·γξ1(t1) ◦ γξ2) ′(0) = (ξ̃2)exp(ξ0)·γξ1(t1) is tangent to σ. denote by ξ̂2 the parallel vector field along γξ1 := lexp(ξ0) ◦ γξ1 with ξ̂ 2(0) = (ξ̃2)exp(ξ0). we describe ξ̂2 as ξ̂2(s) = r∑ i=1 b0i (s)(ẽ 0 i )γ ξ1 (s) + ∑ λ∈△+ mλ∑ i=1 bλi (s)(ẽ λ i )γ ξ1 (s) (s ∈ r), cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 25 where b0i and b λ i are functions over r. fix s0 ∈ r. by using lemma 3.4, we can show (4.3) d ds ∣∣∣∣ s=s0 ξ̂2 = r∑ i=1 ( (b0i ) ′(s0)(ẽ 0 i )γ ξ1 (s0) + (b 0 i )(s0) d ds ∣∣∣∣ s=s0 ((ẽ0 i )γ ξ1 (s)) ) + ∑ λ∈△+ mλ∑ i=1 ( (bλi ) ′(s0)(ẽ λ i )γ ξ1 (s0) + b λ i (s0) d ds ∣∣∣∣ s=s0 ((ẽλ i )γ ξ1 (s)) ) = r∑ i=1 ( (b0i ) ′(s0)(ẽ 0 i )γ ξ1 (s0) + (b 0 i )(s0)∇γ′ ξ1 (s0)((ẽ 0 i )γ ξ1 (s)) ) + ∑ λ∈△+ mλ∑ i=1 ( (bλi ) ′(s0)(ẽ λ i )γ ξ1 (s0) + b λ i (s0)∇γ ′ ξ1 (s0)((ẽ λ i )γ ξ1 (s)) ) = 0. since γ′ ξ1 (s0) = 1 cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (h̃λ1)γξ1 (s0) by lemma 4.1, γ′ξ1(s0) is described as γ′ ξ1 (s0) = (lexp(ξ0))∗(γ ′ ξ1 (s0)) = 1 cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (h̃λ1)γξ1 (s0) . hence, by using lemma 3.4, we have (4.4) ∇γ′ ξ1 (s0)((ẽ 0 i )γ ξ1 = 1 cosh(||λ1||s0) (∇ ξ̃1 ẽ0 i )γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (∇ h̃λ1 ẽ0 i )γ ξ1 (s0) = − λ1(e 0 i ) cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) and (4.5) ∇γ′ ξ1 (s0)((ẽ λ i )γ ξ1 = 1 cosh(||λ1||s0) (∇ ξ̃1 ẽλ i )γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (∇ h̃λ1 ẽλ i )γ ξ1 (s0) =    1 2 cosh(||λ1||s0) ( [ξ̃1, ẽλi ] + θ̃[e λ i , θξ 1] ) (λ1 − λ ∈ △+) 1 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + 2〈ξ̃1, ẽλ i 〉h̃λ1 ) (λ1 = λ) 1 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + θ̃[ξ1, θeλi ] ) (λ − λ1 ∈ △+) 1 2 cosh(||λ1||s0) [ξ̃1, ẽλ i ] (λ1 − λ /∈ △ ∪ {0}). 26 naoyuki koike cubo 20, 3 (2018) by substituting (4.4) and (4.5) into (4.3), we obtain (4.6) d ds ∣∣∣∣ s=s0 ξ̂2 = r∑ i=1 ( (b0i ) ′(s0)(ẽ 0 i )γ ξ1 (s0) − λ1(e 0 i )(b 0 i )(s0) cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) ) + ∑ λ∈△+ mλ∑ i=1 (bλi ) ′(s0)(ẽ λ i )γ ξ1 (s0) + ∑ λ1−λ∈△+ mλ∑ i=1 bλi (s0) 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + θ̃[eλ i , θξ1] ) + ∑ λ−λ1∈△+ mλ∑ i=1 bλi (s0) 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + θ̃[ξ1, θeλi ] ) + ∑ λ−λ1 /∈△∪{0} mλ∑ i=1 bλi (s0) 2 cosh(||λ1||s0) [ξ̃1, ẽλ i ] + mλ1∑ i=1 b λ1 i (s0) 2 cosh(||λ1||s0) ( [ξ̃1, ẽ λ1 i ] + 2〈ξ̃1, ẽ λ1 i 〉h̃λ1 ) = 0. without loss of generality, we may assume that eλ2 1 = ξ2. hence we have bλ2 1 (0) = 1 and bλi (0) = 0 for any (λ, i) other than (λ2, 1). from (4.6) and these relations, we obtain b λ2 1 ≡ 1 and bλi ≡ 0 for any (λ, i) other than (λ2, 1), where we note that λ1 − λ2 /∈ △ ∪ {0}. therefore we obtain ξ̂2 = (ξ̃2)γ ξ1 (s). on the other hand, since (ξ̂ 2)(0) is tangent to σ and σ is totally geodesic, ξ̂2(t1) also is tangent to σ. hence we see that (ξ̃2)exp(ξ0)·γξ1(t1) is tangent to σ. therefore exp(ξ0) · γξ1(t1) · γξ2(t2) belongs to σ. in the sequel, by repeating the same discussion, we can derive that xξ0,t1,··· ,tk = exp(ξ0)·γξ1(t1)· · · · ·γξk(tk) belongs to σ. it is clear that any point of σ is described as xξ0,t1,··· ,tk for some ξ0 ∈ b and some t1, · · · , tk ∈ r. fix an orthonormal base {e01, · · · , e0m0} of b, where m0 := dim b. define vector fields e 0 i (i = 1, · · · , m0) and ej (j = 1, · · · , k) along σ by (e0i )xξ0,t1,··· ,tk := (lxξ0,t1,··· ,tk )∗(e 0 i )(= (ẽ 0 i )xξ0,t1,··· ,tk ) and (ej)xξ0,t1,··· ,tk := (lxξ0,t1,··· ,tk )∗(ξ j tj )(= (ξ̃ j tj )xξ0,t1,··· ,tk ). by imitating the discussions in the proofs of lemmas 4.1 and 4.2, we can show the following fact for these vector fields. lemma 4.3. the vector fields e0i (i = 1, · · · , m0) and ej (j = 1, · · · , k) are tangent to σ and they give a parallel orthonormal tangent frame field on σ. proof. let (ξ̂i)j (resp. (ξ̂i)0) be the parallel vector field along γξj (i 6= j) (resp. γξ0) with (ξ̂i) j 0 = ξi (resp. (ξ̂i)00 = ξ i) and (ξ̂0) j be the parallel vector field along γξj with (ξ̂0) j 0 = ξ0. according to lemma 4.1, we have (γξi) ′(t) = (lγ ξi (t))∗(ξ i t) and (γξ0) ′(t) = (lγξ0 (t) )∗(ξ0). also, we can cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 27 show (ξ̂i) j γ ξj (t) = (lγ ξj (t))∗(ξ i) (j 6= i), (ξ̂i)0 γξ0(t) = (lγξ0 (t) )∗(ξ i) and (ξ̂0) j γ ξj (t) = (lγ ξj (t))∗(ξ0) by imitating the discussion in the proof of lemma 4.2. on the basis of these facts, we can derive the statement of this lemma, where we note that σ is flat. ξ̃j ξ̃ j tj e e e σ σ σ hλj ej γξj(tj) γξj(tj) γξj(tj) hλj ξj ξj ξj hλj figure 4. by using these lemmas, we prove theorem a. proof of theorem a. in this proof, we use the notations as in example 2. set mxξ0,t1,··· ,tk := s b,l1,··· ,lk · xξ0,t1,··· ,tk. denote by hxξ0,t1,··· ,tk the mean curvature vector field of mxξ0,t1,··· ,tk . let {e01, · · · , e0m0} be an orthonormal base of b and (hλ)b = ∑m0 i=1 hiλe 0 i be the b-component of hλ. according to the fact (iv) stated in introduction, we have mxξ0,t1,··· ,tk = lxξ0,t1,··· ,tk (sb,l ξ1,t1 ,··· ,l ξk,tk · e). denote by ĥξ0,t1,··· ,tk the mean curvature vector field of sb,l ξ1,t1 ,··· ,l ξk,tk ·e. according to lemma 3.1, we have (ĥξ0,t1,··· ,tk)e = ∑ λ∈△+ mλ(hλ)b − k∑ i=1 ||λi|| tanh(||λi||ti)(mλi + 2m2λi)ξ i ti and hence (4.7) (hxξ0,t1,··· ,tk )xξ0,t1,··· ,tk = ∑ λ∈△+ m0∑ i=1 mλh i λ(e 0 i )xξ0,t1,··· ,tk − k∑ i=1 ||λi|| tanh(||λi||ti)(mλi + 2m2λi)(e i)xξ0,t1,··· ,tk . 28 naoyuki koike cubo 20, 3 (2018) define a tangent vector field z over σ by zx := (h x)x (x ∈ σ). according to (4.7), we have (4.8) zxξ0,t1,··· ,tk = ∑ λ∈△+ m0∑ i=1 mλh i λ(e 0 i )xξ0,t1,··· ,tk − k∑ i=1 ||λi|| tanh(||λi||ti)(mλi + 2m2λi)(e i)xξ0,t1,··· ,tk . define a coordinate φ = (u1, · · · , um0+k) : σ → rm0+k of σ by φ(x∑m0 i=1 sie 0 i ,t1,··· ,tk ) := (s1, · · · , sm0, t1, · · · , tk) (s1, · · · , sm0, t1, · · · , tk ∈ r). we can show ∂∂ui = e 0 i (i = 1, · · · , m0) and ∂∂um0+j = e j (j = 1, · · · , k). hence φ is a euclidean coordinate of σ. under the identification of σ and rm0+k by φ, we regard z as a tangent vector field on rm0+k. then z is described as (4.9) z(u1,··· ,um0+k) = ( ∑ λ∈△+ mλh 1 λ, · · · , ∑ λ∈△+ mλh m0 λ , −||λ1|| tanh(||λ1||um0+1)(mλ1 + 2m2λ1), · · · , −||λk|| tanh(||λk||um0+k)(mλk + 2m2λk)). fix (a1, · · · , am0, t1, · · · , tk) ∈ rm0+k. let c be the integral curve of z starting from (a1, · · · , am0, t1, · · · , tk) and let c = (c1, · · · , cm0+k). we suffice to investigate c to investigate the mean curvature flow starting from mx∑m0 i=1 aie 0 i ,t1,··· ,tk from c′(t) = zc(t), we have c ′ i(t) = ∑ λ∈△+ mλh i λ (i = 1, · · · , m0) and c′m0+j(t) = −(mλj + 2m2λj )||λj|| tanh (||λj||cm0+j(t)) (j = 1, · · · , k). by solving c′i(t) = ∑ λ∈△+ mλh i λ under the initial condition ci(0) = ai, we have (4.10) ci(t) = ai + t ∑ λ∈△+ mλh i λ. also, by solving c′m0+j(t) = −(mλj + 2m2λj )||λj|| tanh(||λj||cm0+j(t)) under the initial condition cm0+j(0) = tj, we have (4.11) cm0+j(t) = 1 ||λj|| arcsinh ( e −||λj|| 2 (mλj +2m2λj)t sinh(||λj||tj) ) . from (4.10) and (4.11), we can derive t = ∞, lim t→∞ ∑m0 i=1 ci(t) 2 = ∞ (i = 1, · · · , m0) and lim t→∞ cm0+j(t) = 0 (j = 1, · · · , k). if t1 = · · · = tk = 0, then we have cm0+j ≡ 0 (j = 1, · · · , m0). hence the mean curvature flow starting from mxξ0,0,··· ,0 (xξ0,0,··· ,0 ∈ exp(b)) consists of the leaves of f b,l1,··· ,lk through points of exp(b). also, according to the fact (iv) stated in introduction, the leaves of f b,l1,··· ,lk through points of exp(b) are congruent to s b,l1,··· ,lk · e. therefore, the mean curvature flow starting from mxξ0,0,··· ,0 is self-similar. from limt→∞ ∑m0 i=1 ci(t) 2 = ∞ (i = 1, · · · , m0) and lim t→∞ cm0+j(t) = 0 (j = 1, · · · , k), we see that the mean curvature flow starting cubo 20, 3 (2018) mean curvature flow of certain kind of isoparametric . . . 29 from any leaf of f b,l1,··· ,lk asymptotes the mean curvature flow starting from the leaf of f b,l1,··· ,lk passing through a point of exp(b). q.e.d. according to this proof, we obtain the following fact. corollary 4.1. (i) the mean curvature flow starting from mxξ0,0,··· ,0 is self-similar. (ii) the mean curvature flow starting from mxξ0,t1,··· ,tk ((t1, · · · , tk) 6= (0, · · · , 0)) asymptotes the flow starting from mxξ0,0,··· ,0. in more detail, the distance between mxξ0,t1,··· ,tk and mxξ0,0,··· ,0 is equal to √√√√ k∑ j=1 1 ||λj|| 2 arcsinh2 ( e −||λj|| 2(mλj+2m2λj )t sinh(||λj||tj) ) , which converges to zero as t → ∞. next we prove theorem b. proof of theorem b. in case of b = {0}, the relation (4.9) is as follows: (4.12) z(u1,··· ,uk) = (−||λ1|| tanh(||λ1||um0+1)(mλ1 + 2m2λ1 ), · · · , −||λk|| tanh(||λk||um0+k)(mλk + 2m2λk)). hence, according to the dicussion in the proof of theorem a, the mean curvature flow starting from any leaf of f b,l1,··· ,lk converges to the only minimal leaf s b,l,··· ,lk · e. furthermore, the flow converges to the minimal leaf in c∞-topology because the flow consists of s b,l1,··· ,lk -orbits and the limit submanifold also is a s b,l1,··· ,lk -orbit. q.e.d. 30 naoyuki koike cubo 20, 3 (2018) references [1] m. m. alexandrino and m. radeschi, mean curvature flow of singular riemannian foliations, j. geom. anal. 26 2204–2220 (2015). [2] b. andrews and c. baker, mean curvature flow of pinched submanifolds to spheres, j. differential geom. 85 (2010) 357-396. [3] g. huisken, flow by mean curvature of convex surfaces into spheres, j. differential geom. 20 (1984) 237-266. [4] g. huisken, contracting convex hypersurfaces in riemannian manifolds by their mean curvature, invent. math. 84 (1986) 463-480. [5] n. koike, examples of a complex hyperpolar action without singular orbit, cubo a math. j. 12 (2010) 131-147. [6] n. koike, collapse of the mean curvature flow for equifocal submanifolds, asian j. math. 15 (2011) 101-128. [7] n. koike, collapse of the mean curvature flow for isoparametric submanifolds in a symmetric space of non-compact type, kodai math. j. 37 (2014) 355-382. [8] j. milnor, curvatures of left invariant metrics on lie groups, adv. math. 21 (1976) 293–329. introduction mean curvature flow. the non-curvature-adaptedness of the leaves. proof of theorem a cubo, a mathematical journal vol.22, n◦03, (361–377). december 2020 http://dx.doi.org/10.4067/s0719-06462020000300361 received: 29 may, 2020 | accepted: 10 november, 2020 mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions abdeldjalil aouane1,3, smäıl djebali2,3 and mohamed aziz taoudi4 1département de sciences exactes et informatique, école normale supérieure, constantine, algeria. 2department of mathematics, faculty of sciences, imam mohammad ibn saud islamic university (imsiu), pb 90950. riyadh 11623, saudi arabia. 3laboratoire théorie du point fixe et applications ens, bp 92 kouba, algiers, 16006. algeria. 4 cadi ayyad university, national school of applied science marrakesh, morocco. abdeldjalilens@hotmail.com, djebali@hotmail.com, a.taoudi@uca.ma abstract in this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order β ∈ (1,2] subjected to noncompact initial nonlocal conditions. we assume that the linear part generates an arbitrarily strongly continuous β-order fractional cosine family, while the nonlinear forcing term is of carathéodory type and satisfies some fairly general growth conditions. our approach combines the monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. our conclusions improve and generalize many earlier related works. an example is provided to illustrate the main results. c©2020 by the author. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://dx.doi.org/10.4067/s0719-06462020000300361 362 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) resumen en este art́ıculo, probamos la existencia de soluciones leves de una clase de ecuaciones integro-diferenciales fraccionales semilineales de orden β ∈ (1,2] con condiciones nocompactas iniciales no-locales. asumimos que la parte lineal genera una familia coseno de orden fraccional β arbitrariamente fuertemente continua , mientras que el término no-lineal de forzamiento es de tipo carathéodory y satisface algunas condiciones de crecimiento bastante generales. nuestro enfoque combina el teorema de punto fijo de monch con algunos resultados recientes sobre la medida de no-compacidad de operadores integrales. nuestras conclusiones mejoran y generalizan muchos trabajos anteriores relacionados . se provee un ejemplo para ilustrar los resultados principales. keywords and phrases: cosine operator, fractional integro-differential operator, abstract differential equation, noncompact nonlocal condition. 2020 ams mathematics subject classification: 34a08, 34g20, 35f25, 47d09, 47d60, 47h08, 47h10, 47g20. 1 introduction in recent years, the investigation of fractional differential equations in banach spaces has attracted many research works due to its applications in various areas of engineering, physics, bio-engineering, and other applied sciences. notable contributions have been made to both theory and applications of fractional differential equations; we refer, e.g., to [1, 6, 13, 14, 15, 16, 18, 19, 25] and the references therein. actually, it has been found that differential equations involving fractional derivatives in time are more realistic to describe many phenomena in practical situations than those of integer order. the most significant advantage of fractional derivatives compared with integer derivatives is that it can be used to describe the property of memory and heredity of various materials and processes [5, 8, 22]. for more details about fractional calculus and fractional differential equations, we refer the reader to [2, 4, 10]. in this paper, we are concerned with the existence of mild solutions of the following class of fractional semilinear integro-differential equations:    cd β t u(t) = au(t) + f(t,u(t),gu(t),su(t)), t ∈ [0,a], u(0) = u0 + q(u), u′(0) = v0 + p(u), (1.1) where β ∈ (1,2] and cd β t is the standard caputo fractional derivative of order β. the operator a is the infinitesimal generator of a strongly continuous β-order fractional cosine family {cβ(t) : t ≥ 0} cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 363 in a banach space e, f, q,p are suitably defined functions satisfying certain conditions to be specified later, x0, y0 are given elements of e and g,s are two linear operators defined by gu(t) = ∫ t 0 k(t,s)u(s)ds and su(t) = ∫ a 0 h(t,s)u(s)ds, t ∈ [0,a], (1.2) where h ∈ c [[0,a] × [0,a],r+] , k ∈ c [u,r+] , and u = { (t,s) ∈ r2 : 0 ≤ s ≤ t ≤ a } . here r+ refers to the set of nonnegative real numbers. the problem of the existence of mild solutions to (1.1) has been addressed by many investigators in the case where β ∈ (0,1]. we quote for instance the contributions by shu and wang [21], qin et al. [20], and the pioneering works of travis and webb [23, 24]. however, only a few papers have been up to now devoted to the case β ∈ (1,2]. we quote the paper [25], where the authors proved the existence of mild solutions to (1.1) with β ∈ (1,2] when p and q are compact. in many applications, nonlocal conditions are not compact. specifically, periodic p(u) = u(a), anti-periodic p(u) = −u(a), or multipoint discrete nonlocal conditions p(u) = ∑m i=1 ciu(ti), 0 < t1 < · · · < tm are not compact. as a matter of fact, the first and major aim of this paper is to address the problem of existence of mild solutions to (1.1) in the case where p and q are not necessarily compact. moreover, we merely assume that the operator a generates an arbitrarily strongly continuous β-order fractional cosine family, which is an extra interesting feature. our approach combines the monch fixed point theorem with some recent results concerning the measure of noncompactness of integral operators. the outline of the paper is as follows: in section 2, we present the main technical tools which will be used in this work. in section 3, we investigate the existence of mild solution to problem (1.1) by means of a fixed point method. finally, in section 4, we include an example to illustrate our results. 2 preliminaries and auxiliary results in this section, we recall some background and collect several useful results which are crucial for our further work. to do this, let (e,‖ · ‖) be a banach space and c([0,a],e) be the space of all continuous functions defined on [0,a] with values in e, equipped with the standard sup-norm. let l(e) denote the space of all bounded linear operators on e endowed with the classical operator norm. we first list some basic definitions and properties of the fractional calculus theory which are used further in this paper. definition 2.1. [4] for 0 < γ < 1, consider the function of wright type defined by φγ(z) = ∞ ∑ n=0 (−z)n n!γ(−γn + 1 − γ) = 1 2πi ∫ γ µγ−1 exp (µ − zµγ) dµ, (2.1) 364 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) where γ is a contour which starts and ends at −∞ and encircles the origin once counterclockwise. φγ(t) is a probability density function: φγ(t) ≥ 0 for t > 0 and ∫ ∞ 0 φγ(t)dt = 1. (2.2) definition 2.2. [4] the riemann-liouville fractional integral of order β > 0 of a function f ∈ l1([0,a];e) is defined by i β t f(t) = 1 γ(β) ∫ t 0 (t − s)β−1f(s)ds, t > 0, (2.3) where γ(·) stands for the gamma function. definition 2.3. [4] the riemann-liouville fractional derivative of order 1 < β ≤ 2 is defined by d β t f(t) = d2 dt2 i 2−β t f(t), (2.4) where f ∈ l1([0,a];e) and d β t f ∈ l 1([0,a];e). definition 2.4. [4] the caputo fractional derivative of order β ∈ (1,2] is defined by cd β t f(t) = d β t (f(t) − f(0) − f ′(0)t) , (2.5) where f ∈ l1([0,a];e) ∩ c1([0,a];e) and d β t f ∈ l 1([0,a];e). consider the following problem cd β t x(t) = ax(t), x(0) = η, x ′(0) = 0, (2.6) where β ∈ (1,2], a : d(a) ⊂ e → e is a closed densely defined linear operator in banach space e. definition 2.5. [4] let β ∈ (1,2]. a family {cβ}β≥0 ⊂ l(e) is called a solution operator (or a strongly continuous β-order fractional cosine family) for the problem (2.6) if the following conditions are satisfied: (a) cβ(t) is strongly continuous for t ≥ 0 and cβ(0) = i, (b) cβ(t)d(a) ⊂ d(a) and acβ(t)η = cβ(t)aη for all η ∈ d(a), t ≥ 0, (c) cβ(t)η is a solution of x(t) = η + ∫ t 0 (t−s)β−1 γ(β) ax(s)ds for all η ∈ d(a), t ≥ 0. in this case, a is called the infinitesimal generator of cβ(t). definition 2.6. [15] the fractional sine family sβ : r + → l(e) associated with cβ is defined by sβ(t) = ∫ t 0 cβ(s)ds, t ≥ 0. (2.7) cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 365 definition 2.7. [15] the fractional riemann-liouville family pβ : r + → l(e) associated with cβ is defined by pβ(t) = i β−1 t cβ(t) = 1 γ(β − 1) ∫ t 0 (t − s)β−2cβ(s)ds, t ≥ 0. (2.8) definition 2.8. [4] the strongly continuous β-order fractional cosine family cβ(t) is called exponentially bounded if there are constants m ≥ 1 and ω ≥ 0 such that ‖cβ(t)‖ ≤ me ωt, t ≥ 0. (2.9) an operator a is said to belong to cβ(m,ω), if the problem (2.6) has a strongly continuous β-order fractional cosine family cβ(t) satisfying (2.9). denote c β(ω) = ⋃ {cβ(m,ω);m ≥ 1}. theorem 2.1. [4, theorem 3.1] let 0 < β′ < β ≤ 2, γ = β ′ β , ω ≥ 0. if a ∈ cβ(ω) then a ∈ cβ ′ (ω 1 γ ) and the following representation holds cβ′(t) = ∫ ∞ 0 ϕt,γ(s)cβ(s)ds, t > 0, (2.10) where ϕt,γ(s) := t −γφγ (st −γ) and φγ(z) is defined by (2.1). for more details regarding β-order fractional cosine families, we refer the reader to [4]. definition 2.9. a function ψ defined on the set of all bounded subsets of a banach space e with values in r+ is called a measure of noncompactness (mnc in short) on e if for any bounded subset m of e we have ψ(com) = ψ(m), where com stands for the closed convex hull of m. an mnc is said to be (i) full: ψ(m) = 0 if and only if m is a relatively compact set. (ii) monotone: for all bounded subsets m1 and m2 of e, we have m1 ⊂ m2 =⇒ ψ(m1) ≤ ψ(m2). (iii) nonsingular: ψ(m ∪ {x}) = ψ(m), for every bounded subset m of e and for all x ∈ e. one of most important measures of noncompactness is the hausdorff measure of noncompactness defined by χ(m) = inf{r > 0;m can be covered by finitely many balls with radii ≤ r}, for each bounded subset m of e. the hausdorff measure of noncompactness is full, monotone and nonsingular. moreover, it enjoys the following additional properties. 366 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) lemma 2.1. [3] (i) χ(m1 + m2) ≤ χ(m1) + χ(m2). (ii) χ(λm) = |λ|χ(m), for all λ ∈ r. (iii) χ(co(m)) = χ(m). (iv) χ(a + x) = χ(a), ∀x ∈ e. (v) if b : e −→ e is a lipschitz continuous map with constant k, then χ(b(m)) ≤ kχ(m) for all bounded subset m of e. lemma 2.2. [17, 9] if {un}n∈n ⊂ l 1([0,a];e) is uniformly integrable, then the function t 7→ χ({un(t)}n∈n) for t ∈ [0,a] is measurable and χ ({ ∫ t 0 un(s)ds }∞ n=1 ) ≤ ∫ t 0 χ ( {un(s)} ∞ n=1 ) ds. in the sequel, we use a measure of noncompactness in the space c(i;e) which was investigated in [11, 12]. in order to define this measure, let us fix a nonempty bounded subset ω of the space c(i;e). let modc(ω) = sup {modc(ω(t)) : t ∈ i} , where modc(ω(t)) = lim δ→0 sup x∈ω {sup {|x(t2) − x(t1)| : t1, t2 ∈ (t − δ,t + δ)}} , and χ∞(ω) = sup {χ(ω(t)) : t ∈ i} , where χ denotes the hausdorff measure of noncompactness in e. it is worth noticing that χ∞ and modc are monotone nonsingular mncs on c(i;e) (see [3, 12]). from an application view point, one of the main disadvantages of these mncs is the lack of fullness. to overcome this problem, we can define the function ψc on the family of bounded subsets in c(i;e) by taking ψc(ω) = χ∞(ω) + modc(ω) lemma 2.3. [11, lemma 3.1] ψc is a full monotone and nonsingular mnc on the space c(i;e). finally, we will make use of monch’s fixed point theorem. theorem 2.2. [17] let c be a closed, convex subset of a banach space e with x0 ∈ c. suppose there is a continuous map t : c → c with the following property: { d ⊆ c countable and d ⊆ co ({x0} ∪ t(d)) imply that d is relatively compact. then, t has at least one fixed point in c. cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 367 let f be a function from [0,+∞) into l(e). suppose that f is continuous for the strong operator topology, namely the mapping [0,+∞) ∋ t → f(t)x ∈ e is continuous for every x ∈ e. (2.11) notice that from the uniform boundedness principle, we know that f is uniformly bounded on any interval [0,a], i.e., ma := supt∈[0,a] ‖f(t)‖l(e) < +∞. for later use, let us define the quantity ω(f(t)) = lim δ→0 sup ‖x‖≤1 {‖f (t2) x − f (t1)x‖e : t1, t2 ∈ (t − δ,t + δ)} . recall that a family (f(t))t≥0 is said to be equicontinuous if {f(·)x : x ∈ ω} is equicontinuous at any t > 0 for any bounded subset ω ⊂ x. it is easily seen that a family (f(t))t≥0 is equicontinuous if and only if ω(f(t)) = 0 for any t > 0. theorem 2.3. [7] let f be a function from [0,+∞) into l(e). suppose that f is continuous for the strong operator topology. then, for any bounded set ω ⊂ e and for any t ≥ 0, we have modc(f(t)ω) ≤ ω(f(t))χ(ω). in particular, for any t ∈ [0,a] we have modc(f(t)ω) ≤ 2maχ(ω). now, we present two crucial results concerning the integral operator: (s0f) (t) = ∫ t 0 f(t − s)f(s)ds for t ∈ [0,a] where f ∈ l1([0,a];e) and f : [0,+∞) → l(e) verifies (2.11). theorem 2.4. [7] let {fn} ∞ n=1 ⊂ l 1([0,a];e) be integrably bounded, that is, ‖fn(t)‖ ≤ ν(t) for all n = 1,2, · · · and a.e. t ∈ [0,a], (2.12) where ν ∈ l1([0,a]). assume that χ({fn(t)} ∞ n=1) ≤ q(t) (2.13) for a.e. t ∈ [0,a] where q ∈ l1([0,a]). then, for every t ∈ [0,a] we have: mod c ({s0fn(t)} ∞ n=1) ≤ 4ma ∫ t 0 q(s)ds. (2.14) theorem 2.5. [7] let {fn} ∞ n=1 ⊂ l 1([0,a];e) be as in (2.12) assume that (2.13) holds. then χ({s0fn(t)} ∞ n=1) ≤ 2ma ∫ t 0 q(s)ds, for all t ∈ [0,a] 368 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) 3 existence results in this section, we discuss the existence of mild solutions to the semilinear fractional integrodifferential equation (1.1). before doing so, it is appropriate to clarify the definition of solution we will consider. definition 3.1. assume a ∈ cβ(m,ω) and cβ(t) is the solution operator. we say that u ∈ c[i,e] is a mild solution of (1.1) if u satisfies u(t) = cβ(t) (u0 + q(u)) + sβ(t) (v0 + p(u)) + ∫ t 0 pβ(t − s)f(s,u(s),gu(s),su(s))ds, t ∈ i. (3.1) to allow the abstract formulation of our problem, we define the operator t : c([0,a];e) → c([0,a];e) by tu(t) = cβ(t) (u0 + q(u)) + sβ(t) (v0 + p(u)) + ∫ t 0 pβ(t − s)f(s,u(s),gu(s),su(s))ds, t ∈ [0,a] (3.2) for all t ∈ [0,a]. it is clear that u is a mild solution of (1.1) if and only if it is a fixed point of t . our problem will be investigated under the following assumptions: (c1) p,q : c([0,a];e) → e are continuous functions and there exist nonnegative constants kp and kq, such that for all bounded subset d ⊂ c([0,a];e), we have maχ(q(d)) + amaχ(p(d)) ≤ (makq + amakp)χ∞(d), where ma = supt∈[0,a] ‖cβ(t)‖l(e). (c2) there exist nondecreasing continuous functions σ1,σ2 : r + → r+ such that { ‖q(u)‖e ≤ σ1 (‖u‖∞) , for all u ∈ c([0,a];e), ‖p(u)‖e ≤ σ2 (‖u‖∞) , for all u ∈ c([0,a];e). (c3)                f : [0,a] × e × e × e → e is a carathéodory function, i.e., (i) the map t 7→ f(t,u1,u2,u3) is measurable for all (u1,u2,u3) ∈ e × e × e, (ii) the functions u1 7→ f(t,u1,u2,u3), u2 7→ f(t,u1,u2,u3) and u3 7→ f(t,u1,u2,u3) are continuous for almost t ∈ [0,a], (c4) there exist functions ρ1,ρ2,ρ3 ∈ l 1((0,a); r+) and nondecreasing continuous functions ω1,ω2,ω3 : r + → r+ such that ‖f(t,u1,u2,u3)‖e ≤ 3 ∑ i=1 ρi(t)ωi(‖ui‖e), for all t ∈ [0,a] and ui ∈ e. cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 369 (c5) there exist functions m1,m2,m3 ∈ l 1([0,a]; r+) such that for all bounded subset d1,d2,d3 ⊂ e χ(f(t,d1,d2,d3)) ≤ 3 ∑ i=1 mi(t)χ(di), for almost every t ∈ [0,a]. (c6) makq + amakp + 2 maa β−1 γ(β) ‖m‖1 < 1, where m(s) = m1(s) + ak0m2(s) + ah0m3(s), k0 = sup{k(t,s); (t,s) ∈ u}, h0 = sup{h(t,s); (t,s) ∈ u}, and u = { (t,s) ∈ r2 : 0 ≤ s ≤ t ≤ a } . remark 3.1. it is easy to prove that for every t ≥ 0, we have sup t∈[0,a] ‖sβ(t)‖l(e) ≤ ama and sup t∈[0,a] ‖pβ(t)‖l(e) ≤ maa β−1 γ(β) . (3.3) in light of this, we shall show that operator t fulfills all conditions of theorem 2.2. this will be done in a series of lemmas. lemma 3.1. t : c([0,a];e) → c([0,a];e) is continuous. proof. let (un) ⊂ c([0,a];e) be a sequence which converges to u ∈ c([0,a];e). then ‖tun − tu‖∞ ≤ ma‖q(un) − q(u)‖e + ama‖p(un) − p(u)‖e +maa β−1 γ(β) ∫ a 0 ‖f(s,un(s),gun(s),sun(s)) −f(s,u(s),gu(s),su(s))‖eds. with assumptions (c1) and (c3) in mind, the continuity of g and s entails lim n→∞ f(s,un(s),gun(s),sun(s)) = f(s,u(s),gu(s),su(s)). since (un) is convergent then there exists r > 0 such that ‖un‖∞ ≤ r, for all n ∈ n and ‖u‖∞ ≤ r. so by (c4) we have ‖f(s,un(s),gun(s),sun(s)) − f(s,u(s),gu(s),su(s))‖∞ ≤ 2 (ρ1(s)ω1(r) + ρ2(s)ω2(ak0r) + ρ3(s)ω3(ah0r)) . using the dominated convergence theorem, we deduce that t is continuous. lemma 3.2. assume that ma lim inf r→∞ ( σ(r) r + aβ−1 γ(β) ω(r) r ) < 1, (3.4) where σ(r) = σ1(r) + aσ2(r) and ω(r) = ω1(r)‖ρ1‖l1 + ω2(ak0r)‖ρ2‖l1 + ω3(ah0r)‖ρ3‖l1. then, there is a r0 > 0 such that t selfmaps the closed ball br0 = {u ∈ c([0,a];e) : ‖u‖∞ ≤ r0} . 370 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) proof. for u ∈ br and t ∈ [0,a], we have ‖(tu)(t)‖e ≤ ‖cβ(t) (u0 + q(u))‖e + ‖sβ(t) (v0 + p(u))‖e + ∥ ∥ ∥ ∫ t 0 pβ(t − s)f(s,u(s),gu(s),su(s))ds ∥ ∥ ∥ e ≤ ma (‖u0‖e + σ1(r)) + ama (‖v0‖e + σ2(r)) +maa β−1 γ(β) ∫ a 0 ω1(r)ρ1(s) +ω2(ak0r)ρ2(s) + ω3(ah0r)ρ3(s)ds. we claim that there exists r0 > 0 such that tu ∈ br0 whenever u ∈ br0. if is not the case, then for each r > 0 there exists u ∈ br such that tu /∈ br, that is r < ‖tu‖∞ ≤ ma (‖u0‖e + σ1(r)) + ama (‖v0‖e + σ2(r)) + maa β−1 γ(β) ω(r), which implies when dividing by r that 1 < ma ‖u0‖e + ama‖v0‖e r + ma σ(r) r + maa β−1 γ(β) ω(r) r . taking the lim inf as r → ∞, we obtain 1 ≤ ma lim inf r→∞ ( σ(r) r + aβ−1 γ(β) ω(r) r ) , which contradicts the assumption (3.4) therefore, there exists r0 > 0 such that ‖tu‖∞ ≤ r0, for all ‖u‖ ≤ r0. thus, tu ∈ br0 for all u ∈ br0. lemma 3.3. let r0 be as in lemma 3.2 and let x0 ∈ br0. let d be a countable subset of br0. then d ⊆ co ({x0} ∪ t(d)) implies that d is relatively compact. proof. let d = {un} ∞ n=1 be any countable subset of br0 such that d ⊆ co ({x0} ∪ t(d)) . (3.5) we show that d is relatively compact. notice first that for each t ∈ [0,a], we have χ(t(d)(t)) ≤ χ(cβ(t)(u0 + q(d))) + χ(sβ(t)(v0 + p(d))) + χ ( { ∫ t 0 pβ(t − s)f(s,un(s),gun(s),sun(s))ds }∞ n=1 ) . since ‖f(s,un(s),gun(s),sun(s))‖e ≤ ω1(r0)ρ1(s) + ω2(ak0r0)ρ2(s) + ω3(ah0r0)ρ3(s) cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 371 and ρ1,ρ2,ρ3 ∈ l 1([0,a];r+), then, in view of theorem 2.5 and lemma 2.2, we obtain the following estimates: χ(t(d)(t)) ≤ maχ(q(d)) + amaχ(p(d)) +2maa β−1 γ(β) ∫ t 0 m1(s)χ(d(s)) + m2(s)χ(g(d(s))) + m3(s)χ(s(d(s)))ds ≤ (makq + amakp)χ∞(d) +2maa β−1 γ(β) ∫ t 0 m1(s)χ(d(s)) + ak0m2(s)χ(d(s)) + ah0m3(s)χ(d(s))ds ≤ ( makq + amakp + 2 maa β−1 γ(β) ‖m‖1 ) χ∞(d). thus, χ∞(t(d)) ≤ [ makq + amakp + 2 maa β−1 γ(β) ‖m‖1 ] χ∞(d). (3.6) on the other hand, referring to theorem 2.3, theorem 2.4, and lemma 2.2, we can see that modc(t(d)(t)) ≤ modc(cβ(t)q(d)) + modc(sβ(t)p(d)) +4maa β−1 γ(β) ∫ t 0 m(s)χ(d(s))ds ≤ 2(makq + amakp)χ∞(d) + 4 maa β−1 γ(β) ‖m‖1χ∞(d). thus, mod c(t(d)) ≤ [ 2(makq + amakp) + 4 maa β−1 γ(β) ‖m‖1 ] χ∞(d). (3.7) combining (3.5) and (3.6), we arrive at χ∞(td) = χ∞(d) = 0. by (3.7) we get mod c(t(d)) = 0 and therefore t(d) is equicontinuous. going back to (3.5) we deduce that d is equicontinuous and so relatively compact in c([0,a];e). this achieves the proof. theorem 3.1. assume that (c1) − −(c6) hold. then, the nonlocal problem (1.1) has at least one mild solution in c([0,a];e), provided that (3.4) holds. proof. invoking theorem 2.2 together with lemmas 3.1, 3.2, and 3.3, we infer that t has at least one fixed point in br0 which is, in turn, a mild solution of (1.1). 372 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) 4 application to illustrate the application of the theoretical results of this work, we consider the following integro-differential equation:                                  cd β t w(t,x) = ∂2w(t,x) ∂2x + ρ1(t)f1(w(t,x)) + ρ2(t)f2 ( ∫ t 0 ts 2 w(s,x)ds ) +ρ3(t)f3 ( ∫ 1 0 t 2 s 2 2 w(s,x)ds ) , t ∈ i = [0,1], x ∈ [0,π], w(t,0) = w(t,π) = 0, t ∈ i, w(0,x) = w0(x) + m ∑ i=1 ciw(si,x), x ∈ [0,π], s1 < s2 < ... < sm, ti ∈ i, ci ∈ r, ∂w(t,x) ∂t ∣ ∣ t=0 = y0(x) + n ∑ i=1 diw(ti,x), x ∈ [0,π], t1 < t2 < ... < tn, ti ∈ i, di ∈ r, (4.1) where β ∈ (1,2], the functions ρi : i → r and fi : e → e for i ∈ {1,2,3} satisfy appropriate conditions which are specified later. to allow the abstract formulation of (4.1), let e = l2([0,π]; r) be the banach space of square integrable functions from [0,π] into r. define the operator a : d(a) ⊂ e → e by aw = w′′ with domain d(a) = {w ∈ e : w,w′ are absolutely continuous ,w′′ ∈ e,w(0) = w(π) = 0}. it is well known that a is the generator of strongly continuous cosine functions {c(t) : t ∈ r} on e. moreover a has a discrete spectrum whose eigenvalues are −n2, n ∈ n with corresponding normalized eigenvectors zn(τ) = √ 2 π sin(nτ), and the following properties hold: (a) {zn : n ∈ n} is an orthonormal basis of e. (b) if z ∈ e, then az = − ∑∞ n=1 n 2 < z, zn > zn. (c) for z ∈ e, c(t)z = ∑∞ n=1 cos(nt) < z, zn > zn, and the associated sine family is s(t)z = ∑∞ n=1 sin(nt) n < z,zn > zn. s(t) is compact for every t ∈ i and ‖c(t)‖l(e) = ‖s(t)‖l(e) ≤ 1, for every t ∈ r. for β ∈ (1,2], since a is the infinitesimal generator of a strongly continuous cosine family c(t), from the subordinate principle (theorem 2.1), it follows that a is the infinitesimal generator of a strongly continuous exponentially bounded fractional cosine family cβ(t). cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 373 with u(t) = w(t, ·), equation (4.1) may be written in the abstract form:    cd β t u(t) = au(t) + f(t,u(t),gu(t),su(t)), t ∈ i, u(0) = u0 + q(u), u′(0) = v0 + p(u), (4.2) where the function f : i × e × e × e → e is given by f(t,x,y,z) = ρ1(t)f1(x) + ρ2(t)f2(y) + ρ3(t)f3(z). here ρi : i → r is integrable on i, fi : e → e is a lipschitz continuous function with a lipschitz constant li, the functions p,q : c(i,e) → e are given by q(u) = m ∑ i=1 ciu(si), 0 < s1 < s2 < · · · < sm ≤ 1, and p(u) = n ∑ i=1 diu(ti), 0 < t1 < t2 < · · · < tn ≤ 1, and the functions g,s : c(i,e) → c(i,e) are defined by gu(t) = ∫ t 0 ts 2 u(s)ds, su(t) = ∫ 1 0 t2s2 2 u(s)ds, where h0 = k0 = 1 2 . in order to obtain a mild solution, our strategy is to apply theorem 3.1. first, by (c) we have ‖c(t)‖l(e) ≤ 1, for every t ∈ r +. in view of theorem 2.1 and (2.2) we see that there exists a real number ma = 1 > 0 such that ‖cβ(t)‖l(e) ≤ ma for t ≥ 0. observe further that the function f : i × e × e × e → e is given by f(t,x,y,z) = ρ1(t)f1(x) + ρ2(t)f2(y) + ρ3(t)f3(z), where ρi : i → r is integrable on i and fi : e → e is a lipschitz continuous function with a lipschitz constant li (i = 1,2,3). this shows that (c3) is satisfied. on one hand, ‖q(u)‖e ≤ ( m ∑ i=1 |ci| ) ‖u‖∞ = σ1(‖u‖∞) (4.3) and ‖p(u)‖e ≤ ( n ∑ i=1 |di| ) ‖u‖∞ = σ2(‖u‖∞), (4.4) where σ1(r) = ( ∑m i=1 |ci|)r and σ2(r) = ( ∑n i=1 |di|) r. in addition, it is easily seen that for any bounded subset d of c([0,1],e) we have χ(q(d)) ≤ m ∑ i=1 |ci|χ(d (si)) ≤ ( m ∑ i=1 |ci| ) χ∞(d) = kqχ∞(d) (4.5) 374 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) and χ(p(d)) ≤ n ∑ i=1 |di|χ(d (ti)) ≤ ( n ∑ i=1 |di| ) χ∞(d) = kpχ∞(d). (4.6) thus maχ(q(d)) + amaχ(p(d)) ≤ (makq + amakp)χ∞(d), (4.7) for any bounded subset d of c([0,1];e). this shows that (c1) and (c2) are satisfied. moreover the function f satisfies ‖f(t,u1,u2,u3)‖e ≤ |ρ1(t)|‖f1(u1)‖e + |ρ2(t)|‖f2(u2)‖e + |ρ3(t)|‖f3(u3)‖e ≤ |ρ1(t)|(‖f1(0)‖e + l1‖u1‖e) + |ρ2(t)|(‖f2(0)‖e + l2‖u2‖e) + |ρ3(t)|(‖f3(0)‖e + l3‖u3‖e) ≤ |ρ1(t)|ω1(‖u1‖e) + |ρ2(t)|ω2(‖u2‖e) + |ρ3(t)|ω3(‖u3‖e) ≤ 3 ∑ i=1 |ρi(t)|ωi(‖ui‖e), where ωi(‖ui‖e) = ‖fi(0)‖e + li‖ui‖e. by virtue of lemma 2.1, (v) we have χ(f(t,d1,d2,d3)) ≤ |ρ1(t)|χ(f1(d1)) + |ρ2(t)|χ(f2(d2)) + |ρ3(t)|χ(f3(d3)) ≤ |ρ1(t)|l1χ(d1) + |ρ2(t)|l2χ(d2) + |ρ3(t)|l3χ(d3) ≤ 3 ∑ i=1 mi(t)χ(di), for any t ∈ [0,a] and for any bounded subsets d1,d2,d3 of e. thus, (c4) and (c5) are satisfied. now the condition (c6) is given by taking 2 aβ−1ma γ(β) ( l1‖ρ1‖l1 + 1 2 l2‖ρ2‖l1 + 1 2 l3‖ρ3‖l1 ) + (makq + amakp) < 1, because, we have m(s) = m1(s) + ak0m2(s) + ah0m3(s) = l1|ρ1(s)| + 1 2 l2|ρ2(s)| + 1 2 l3|ρ3(s)|. then ‖m‖1 = l1‖ρ1‖l1 + 1 2 l2‖ρ2‖l1 + 1 2 l3‖ρ3‖l1. finally, for ω(r) = ω1(r)‖ρ1‖l1 + ω2(ak0r)‖ρ2‖l1 + ω3(ah0r)‖ρ3‖l1 = ω1(r)‖ρ1‖l1 + ω2( 1 2 r)‖ρ2‖l1 + ω3( 1 2 r)‖ρ3‖l1, we have lim r→∞ ω(r) r = l1‖ρ1‖l1 + 1 2 l2‖ρ2‖l1 + 1 2 l3‖ρ3‖l1, cubo 22, 3 (2020) mild solutions of a class of semilinear fractional . . . 375 and for σ(r) = σ1(r) + aσ2(r) = (kq + akp)r, notice that lim r→∞ σ(r) r = kq + akp. then ma lim infr→∞ ( σ(r) r +a β−1 γ(β) ω(r) r ) = a β−1 ma γ(β) ( l1‖ρ1‖l1 + 1 2 l2‖ρ2‖l1 + 1 2 l3‖ρ3‖l1 ) + (makq + amakp) ≤ 2a β−1 ma γ(β) ( l1‖ρ1‖l1 + 1 2 l2‖ρ2‖l1 + 1 2 l3‖ρ3‖l1 ) + (makq + amakp) < 1. thus, all conditions of theorem 3.1 are fulfilled. therefore equation (4.1) has a mild solution. acknowledgment the authors thank the three referees for their helpful remarks. the first two authors are grateful to the direction générale de la recherche scientifique et de développement technologique in algeria for supporting this work. 376 abdeldjalil aouane, smäıl djebali, mohamed aziz taoudi cubo 22, 3 (2020) references [1] d. araya, c. lizama, “almost automorphic mild solutions to fractional differential equations”, nonlinear analysis. vol. 69, no. 11, pp. 3692–3705, 2008. 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[25] b. zhu, l. liu, and y. wu, “local and global existence of mild solutions for a class of semilinear fractional integro-differential equations”, fract. calc. appl. anal., vol. 20, no. 6, pp. 1338–1355, 2017. doi: 10.1515/fca-2017-0071. introduction preliminaries and auxiliary results existence results application cubo, a mathematical journal vol. 23, no. 02, pp. 207–224, august 2021 doi: 10.4067/s0719-06462021000200207 coincidence point results of nonlinear contractive mappings in partially ordered metric spaces k. kalyani 1 n. seshagiri rao 2 1 department of mathematics, vignan’s foundation for science, technology & research, vadlamudi-522213, andhra pradesh, india. kalyani.namana@gmail.com 2 department of applied mathematics, school of applied natural sciences, adama science and technology university, post box no.1888, adama, ethiopia. seshu.namana@gmail.com abstract in this paper, we proved some coincidence point results for fnondecreasing self-mapping satisfying certain rational type contractions in the context of a metric space endowed with a partial order. moreover, some consequences of the main result are given by involving integral type contractions in the space. some numerical examples are illustrated to support our results. as an application, we have discussed the existence of a unique solution of integral equation. resumen en este art́ıculo, probamos algunos resultados sobre puntos de coincidencia para un auto-mapeo no decreciente f satisfaciendo ciertas contracciones de tipo racional en el contexto de un espacio métrico dotado de un orden parcial. más aún, se entregan algunas consecuencias del resultado principal que involucran contracciones de tipo integral en el espacio. se ilustran algunos ejemplos numéricos en apoyo a nuestros resultados. como una aplicación, discutimos la existencia de una única solución de una ecuación integral. keywords and phrases: ordered metric spaces; rational contractions; compatible mappings; weakly compatible mappings; coupled fixed point; common fixed point. 2020 ams mathematics subject classification: 41a50, 47h10. accepted: 01 april, 2021 received: 25 nov, 2020 c©2021 k. kalyani et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200207 https://orcid.org/0000-0002-4531-5976 https://orcid.org/0000-0003-2409-6513 208 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) 1 introduction a remarkable fixed point theorem was first introduced by banach [4] in 1922, which is one of the most influential results in analysis. it is being used widely in many different areas of mathematics and its applications. it needs the structure of complete metric spaces together with a contractive condition on the self map which is easy to test in many circumstances. basically this principle gives a sequence of approximate solutions and also give a valuable information about the convergence rate of a fixed point. this kind of iteration process has been used both in mathematics and computer science. in particular, fixed point iterations together with monotone iterative techniques are the central methods when solving a large class of problems in theoretical and applied mathematics and play an important role in many algorithms. many authors have extended this theorem by introducing more generalized contractive conditions, which imply the existence of a fixed point [6, 7, 8, 9, 11, 12, 13, 14, 15, 16]. the existence of fixed point results for self-mappings in partially ordered sets have been considered first by ran and reurings [36] and presented some applications to matrix equations therein. these results were again generalized and extended by nieto et al. [32, 33] in partially ordered sets and applied their results to study the ordinary differential equations. prominent works on various existence and uniqueness theorems on fixed point and common fixed point for monotone mappings in cone metric spaces, partially ordered metric spaces and others spaces, refer the readers to [5, 10, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], which generate natural interest to establish usable fixed point theorems by weakening its hypothesis. various types of contraction conditions have been used to find a fixed point of a single and multivalued mappings on metric spaces by altun et al. [1], aslantas et al. [2, 3], and sahin et al. [37]. it is well known that a powerful technique for proving existence results for nonlinear problems is the method of upper and lower solutions. in many cases it is possible to find a minimal and a maximal solution between the lower and the upper solution by an iterative scheme: the monotone iterative technique. this method provides a constructive procedure for the solutions and it is also useful for the investigation of qualitative properties of solutions. this method has been used to acquire the unique solution of periodic boundary value problems of ordinary and partial differential equations, integro ordinary and partial differential equations by several authors, some of which are in [23, 32, 33]. the aim of this paper is to prove the coincidence point and common fixed point results for fnondecreasing self-mapping satisfying generalized contractive conditions of rational type in the context of partially ordered metric spaces. these results generalize and extend the result of [7, 12, 14, 25, 26] in partially ordered metric spaces. some consequences of the main results are given in terms of integral type contractions in the same space. further, some examples and an application for the existence of the unique solution for an integral equation are presented at the end. cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 209 2 preliminaries the following definitions are frequently used in our study. definition 2.1. [40] the triple (x,d,≤) is called a partially ordered metric space, if (x,≤) is a partially ordered set together with (x,d) is a metric space. definition 2.2. [40] if (x,d) is a complete metric space, then the triple (x,d,≤) is called a complete partially ordered metric space. definition 2.3. [38] let (x,≤) be a partially ordered set. a mapping f : x → x is said to be strictly increasing (strictly decreasing), if f(x) < f(y) (f(x) > f(y)) for all x,y ∈ x with x < y. definition 2.4. [40] a point x ∈ a, where a is a non-empty subset of a partially ordered set (x,≤) is called a common fixed (coincidence) point of two self-mappings f and t, if fx = tx = x (fx = tx). definition 2.5. [39] the two self-mappings f and t defined over a subset a of a partially ordered metric space (x,d,≤) are called commuting, if ftx = tfx for all x ∈ a. definition 2.6. [39] two self-mappings f and t defined over a ⊂ x are compatible, if for any sequence {xn} with lim n→+∞ fxn = lim n→+∞ txn = µ for some µ ∈ a, then lim n→+∞ d(tfxn,ftxn) = 0. definition 2.7. [40] two self-mappings f and t defined over a ⊂ x are said to be weakly compatible, if they commute only at their coincidence points (i.e., if fx = tx, then ftx = tfx). definition 2.8. [40] let f and t be two self-mappings defined over a partially ordered set (x,≤). a mapping t is called monotone f-nondecreasing, if fx ≤ fy implies tx ≤ ty, for all x,y ∈ x. definition 2.9. [38] let a be a non-empty subset of a partially ordered set (x,≤). if every two elements of a are comparable, then it is called a well ordered set. definition 2.10. [39] a partially ordered metric space (x,d,≤) is called an ordered complete, if for each convergent sequence {xn} ∞ n=0 ⊂ x, one of the following conditions holds: • if {xn} is a non-decreasing sequence in x such that xn → x implies xn ≤ x, for all n ∈ n that is, x = sup{xn} or, • if {xn} is a non-increasing sequence in x such that xn → x implies x ≤ xn, for all n ∈ n that is, x = inf{xn}. 210 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) 3 main results we start this section with the following coincidence point theorem in the context of a partially ordered metric space. theorem 3.1. let (x,d,≤) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t(x) ⊆ f(x) and satisfying the following condition d(tx,ty) ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + β d(fx,tx) d(fy,ty) d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.1) for all x, y in x for which fx 6= fy are comparable, and for some α,β,γ,δ,λ ∈ [0,1) with 0 ≤ α+β + 2(γ +δ)+λ < 1. if there exists a point x0 ∈ x such that fx0 ≤ tx0 and the mappings f and t are compatible, then f and t have a coincidence point in x. proof. suppose for some x0 ∈ x such that fx0 ≤ tx0. from the hypothesis, we have t(x) ⊆ f(x), then choose a point x1 ∈ x such that fx1 = tx0. but tx1 ∈ f(x), then there exists another point x2 ∈ x such that fx2 = tx1. as by a similar argument above, we obtain a sequence {xn} in x such that fxn+1 = txn for all n ≥ 0. since, fx0 ≤ tx0 = fx1 and t is monotone f-nondecreasing mapping, then we have that tx0 ≤ tx1. similarly, we get tx1 ≤ tx2 as fx1 ≤ fx2. continuing the same process, we obtain that tx0 ≤ tx1 ≤ ... ≤ txn ≤ txn+1 ≤ ... . now, we discuss the following two cases. case 1: if d(txn0,txn0+1) = 0 for some n0 ∈ n, then txn0+1 = txn0 and by the above argument, we have txn0+1 = txn0 = fxn0+1. therefore, xn0+1 is a coincidence point of t and f, and so we have the result. case 2: if d(txn,txn+1) > 0 for all n ∈ n, then from contraction condition (3.1), we have d(txn+1,txn) ≤ α d(fxn+1,txn+1) [1 + d(fxn,txn)] 1 + d(fxn+1,fxn) + β d(fxn+1,txn+1) d(fxn,txn) d(fxn+1,fxn) + γ [d(fxn+1,txn+1) + d(fxn,txn)] + δ [d(fxn+1,txn) + d(fxn,txn+1)] + λd(fxn+1,fxn), which implies that d(txn+1,txn) ≤ αd(txn,txn+1) + βd(txn,txn+1) + γ [d(txn,txn+1) + d(txn−1,txn)] + δ [d(txn,txn) + d(txn−1,txn+1)] + λd(txn,txn−1). cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 211 therefore, we arrive at d(txn+1,txn) ≤ ( γ + δ + λ 1 − α − β − γ − δ ) d(txn,txn−1). continuing the same process up to n times, we obtain that d(txn+1,txn) ≤ ( γ + δ + λ 1 − α − β − γ − δ )n d(tx1,tx0). let k = γ+δ+λ 1−α−β−γ−δ < 1. moreover, from the triangular inequality for m ≥ n, we have d(txm,txn) ≤ d(txm,txm−1) + d(txm−1,txm−2) + ... + d(txn+1,txn) ≤ ( km−1 + km−2 + ... + kn ) d(tx1,tx0) ≤ kn 1 − k d(tx1,tx0), as m,n → +∞, d(txm,txn) → 0, this shows that the sequences {txn} is a cauchy sequence in x. so, by the completeness of x, there exists a point µ ∈ x such that txn → µ as n → +∞. the continuity of t implies that lim n→+∞ t(txn) = t ( lim n→+∞ txn ) = tµ. since, fxn+1 = txn then fxn+1 → µ as n → +∞. further, the compatibility of t and f, we have lim n→+∞ d(tfxn,ftxn) = 0. from the triangular inequality of a metric d, we have d(tµ,fµ) = d(tµ,tfxn) + d(tfxn,ftxn) + d(ftxn,fµ), on taking limit as n → +∞ in the above inequality and using the fact that t and f are continuous, we obtain that d(tµ,fµ) = 0. thus, tµ = fµ. hence, µ is a coincidence point of t and f in x. we obtain the following consequences from theorem 3.1 on taking zero value to α,β,γ,δ and λ as special cases. corollary 3.2. let (x,d,≤) be a complete partially ordered metric space. suppose that the selfmappings f and t on x are continuous, t is a monotone f-nondecreasing, t(x) ⊆ f(x) and satisfying the following contraction conditions (a) d(tx,ty) ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.2) for some α,γ,δ,λ ∈ [0,1) with 0 ≤ α + 2(γ + δ) + λ < 1, 212 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) (b) d(tx,ty) ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] + λd(fx,fy), (3.3) where α,γ,λ ∈ [0,1) such that 0 ≤ α + 2γ + λ < 1, (c) d(tx,ty) ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.4) there exist α,δ,λ ∈ [0,1) such that 0 ≤ α + 2δ + λ < 1, (d) d(tx,ty) ≤ γ [d(fx,tx) + d(fy,ty)] + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.5) for some γ,δ,λ ∈ [0,1) with 0 ≤ 2(γ + δ) + λ < 1, for all x, y in x for which fx 6= fy are comparable. if there exists a point x0 ∈ x such that fx0 ≤ tx0 and the mappings t and f are compatible, then t and f have a coincidence point in x. corollary 3.3. let (x,d,≤) be a complete partially ordered metric space. suppose that the mappings f,t : x → x are continuous, t is a monotone f-nondecreasing, t(x) ⊆ f(x) and satisfying the following contraction conditions (i) d(tx,ty) ≤ β d(fx,tx) d(fy,ty) d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] +δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.6) where β,γ,δ,λ ∈ [0,1) such that 0 ≤ β + 2(γ + δ) + λ < 1, (ii) d(tx,ty) ≤ β d(fx,tx) d(fy,ty) d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] + λd(fx,fy), (3.7) for some β,γ,λ ∈ [0,1) with 0 ≤ β + 2γ + λ < 1, (iii) d(tx,ty) ≤ β d(fx,tx) d(fy,ty) d(fx,fy) + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.8) there exist β,δ,λ ∈ [0,1) such that 0 ≤ β + 2δ + λ < 1, cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 213 (iv) d(tx,ty) ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + β d(fx,tx) d(fy,ty) d(fx,fy) + λd(fx,fy), (3.9) where α,β,λ ∈ [0,1) such that 0 ≤ α + β + λ < 1, for all x, y in x for which fx 6= fy are comparable. if there exists a point x0 ∈ x such that fx0 ≤ tx0 and the mappings t and f are compatible, then t and f have a coincidence point in x. corollary 3.4. let (x,d,≤) be a complete partially ordered metric space. suppose that t : x → x is a mapping such that for all comparable x,y ∈ x, the contraction condition(s) in theorem 3.1 (or corollaries 3.2 and 3.3 ) is satisfied. assume that t satisfies the following hypotheses: (i). t is continuous, (ii). t(tx) ≤ tx for all x ∈ x. if there exists a point x0 ∈ x such that x0 ≤ tx0, then t has a fixed point in x. proof. follow from theorem 3.1 by taking f = ix (the identity map). we may remove the continuity criteria of t in theorem 3.1, is still valid by assuming the following hypothesis in x: if {xn} is a non-decreasing sequence in x such that xn → x, then xn ≤ x for all n ∈ n. theorem 3.5. let (x,d,≤) be a complete partially ordered metric space. suppose that t,f : x → x are two mappings such that t is a monotone f-nondecreasing, t(x) ⊆ f(x) and satisfying d(tx,ty) ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + β d(fx,tx) d(fy,ty) d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy), (3.10) for all x, y in x for which fx 6= fy are comparable and where α,β,γ,δ,λ ∈ [0,1) such that 0 ≤ α+ β + 2(γ + δ) + λ < 1. assume that there exists x0 ∈ x such that fx0 ≤ tx0 and {xn} is a non-decreasing sequence in x such that xn → x, then xn ≤ x for all n ∈ n. if f(x) is a complete subset of x, then t and f have a coincidence point in x. further, if t and f are weakly compatible then t and f have a common fixed point in x. moreover, the set of common fixed points of t and f are well ordered if and only if t and f have one and only one common fixed point in x. 214 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) proof. suppose f(x) is a complete subset of x. as we know from theorem 3.1, the sequence {txn} is a cauchy sequence and hence, {fxn} is also a cauchy sequence in (f(x),d) as fxn+1 = txn and t(x) ⊆ f(x). since f(x) is complete then there exists fu ∈ f(x) such that lim n→+∞ txn = lim n→+∞ fxn = fu. (3.11) also note that the sequences {txn} and {fxn} are nondecreasing and from the hypothesis, we have txn ≤ fu and fxn ≤ fu for all n ∈ n. since t is a monotone f-nondecreasing, we get txn ≤ tu for all n. letting n → +∞, we obtain fu ≤ tu. suppose that fu < tu, define a sequence {un} by u0 = u and fun+1 = tun for all n ∈ n. an argument similar to that in the proof of theorem 3.1 yields that {fun} is a nondecreasing sequence and lim n→+∞ fun = lim n→+∞ tun = fv for some v ∈ x. (3.12) so from the hypothesis, we have that sup n∈n fun ≤ fv and sup n∈n tun ≤ fv. notice that fxn ≤ fu ≤ fu1 ≤ fu2 ≤ ... ≤ fun ≤ ... ≤ fv. now, we discuss the following two cases: case 1: if there exists some n0 ≥ 1 with fxn0 = fun0, then we have fxn0 = fu = fun0 = fu1 = tu, this is a contradiction to fu < tu. thus, fu = tu, that is, u is a coincidence point of t and f in x. case 2: suppose fxn 6= fun+1 for all n. then from condition (3.10), we have d(fxn+1,fun+1) = d(txn,tun) ≤ α d(fxn,txn) [1 + d(fun,tun)] 1 + d(fxn,fun) + β d(fxn,txn) d(fun,tun) d(fxn,fun) + γ [d(fxn,txn) + d(fun,tun)] + δ [d(fxn,tun) + d(fun,txn)] + λd(fxn,fun). on taking limit as n → +∞ in the above inequality and from equations (3.11) and (3.12), we get d(fu,fv) ≤ (2δ + λ) d(fu,fv) < d(fu,fv), since 2δ + λ < 1. so, we have fu = fv = fu1 = tu, cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 215 this is again a contradiction to fu < tu. hence, we conclude that u is a coincidence point of t and f in x. now, we suppose that t and f are weakly compatible. let w be the coincidence point then tw = tfz = ftz = fw, since w = tz = fz, for some z ∈ x. now from (3.10), we have d(tz,tw) ≤ α d(fz,tz) [1 + d(fw,tw)] 1 + d(fz,fw) + β d(fz,tz) d(fw,tw) d(fz,fw) + γ [d(fz,tz) + d(fw,tw)] + δ [d(fz,tw) + d(fw,tz)] + λd(fz,fw) ≤ (2γ + 2δ + λ) d(tz,tw). as 2γ + 2δ + λ < 1, then d(tz,tw) = 0. therefore, tz = tw = fw = w. hence, w is a common fixed point of t and f in x. now, suppose that the set of common fixed points of t and f is well ordered, we have to show that the common fixed point of t and f is unique. let u and v be two common fixed points of t and f such that u 6= v, then from condition (3.10), we have d(u,v) ≤ α d(fu,tu) [1 + d(fv,tv)] 1 + d(fu,fv) + β d(fu,tu) d(fv,tv) d(fu,fv) + γ [d(fu,tu) + d(fv,tv)] + δ [d(fu,tv) + d(fv,tu)] + λd(fu,fv) ≤ (2γ + 2δ + λ) d(u,v) < d(u,v), since 2γ + 2δ + λ < 1, which is a contradiction and hence, u = v. conversely, suppose t and f have only one common fixed point, then the set of common fixed points of t and f being a singleton is well ordered. besides, in corollary 3.2 and corollary 3.3 by relaxing the continuity criteria on t and satisfying the hypotheses given in theorem 3.5, then t and f have a coincidence point, a common fixed point and its uniqueness in x. corollary 3.6. let (x,d,≤) be a complete partially ordered metric space. suppose that t : x → x is a mapping such that for all comparable x,y ∈ x, the contraction condition (3.10) is satisfied. suppose that the following hypotheses are satisfied (i). if {xn} is a non-decreasing sequence in x with respect to ≤ such that xn → x ∈ x as n → +∞, then xn ≤ x, for all n ∈ n and (ii). t(tx) ≤ tx for all x ∈ x. if there exists a point x0 ∈ x such that x0 ≤ tx0, then t has a fixed point in x. 216 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) proof. follow from theorem 3.5 by taking f = ix (the identity map). remark 3.7. (i). if α = γ = δ = 0 in theorem 3.1 and 3.5, we obtain theorem 2.1 and 2.3 of chandok [25]. (ii). if f = i and α = γ = δ = 0 in theorem 3.1 and 3.5, then we get theorem 2.1 and 2.3 of harjani et al. [26]. some other consequences of the main result for the self mappings involving the integral type contractions are as follows. let χ denote the set of all functions ϕ : [0,+∞) → [0,+∞) satisfying the following hypotheses: (a) each ϕ is lebesgue integrable function on every compact subset of [0,+∞) and (b) for any ǫ > 0, we have ∫ ǫ 0 ϕ(t)dt > 0, for t ∈ [0,+∞). corollary 3.8. let (x,d,≤) be a complete partially ordered metric space. suppose that the mappings t,f : x → x are continuous, t is a monotone f-nondecreasing, t(x) ⊆ f(x) and satisfying ∫ d(t x,t y) 0 ϕ(t)dt ≤ α ∫ d(fx,t x)[1+d(fy,t y)] 1+d(fx,fy) 0 ϕ(t)dt + β ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + δ ∫ d(fx,t y)+d(fy,t x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.13) for all x, y in x for which fx 6= fy are comparable, ϕ ∈ χ and where α,β,γ,δ,λ ∈ [0,1) such that 0 ≤ α+β + 2(γ +δ)+λ < 1. if there exists a point x0 ∈ x such that fx0 ≤ tx0 and the mappings t and f are compatible, then t and f have a coincidence point in x. similarly, we obtain the following results from corollaries 3.2 and 3.3 in a complete partially ordered metric space. corollary 3.9. let (x,d,≤) be a complete partially ordered metric space. suppose that the selfmappings f,t on x are continuous, t is a monotone f-nondecreasing, t(x) ⊆ f(x) satisfying the following contraction conditions (a) ∫ d(t x,t y) 0 ϕ(t)dt ≤ α ∫ d(fx,t x)[1+d(fy,t y)] 1+d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + δ ∫ d(fx,t y)+d(fy,t x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.14) for some α,γ,δ,λ ∈ [0,1) with 0 ≤ α + 2(γ + δ) + λ < 1, cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 217 (b) ∫ d(t x,t y) 0 ϕ(t)dt ≤ α ∫ d(fx,t x)[1+d(fy,t y)] 1+d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.15) where α,γ,λ ∈ [0,1) with 0 ≤ α + 2γ + λ < 1, (c) ∫ d(t x,t y) 0 ≤ α ∫ d(fx,t x)[1+d(fy,t y)] 1+d(fx,fy) 0 ϕ(t)dt + δ ∫ d(fx,t y)+d(fy,t x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.16) where α,δ,λ ∈ [0,1) such that 0 ≤ α + 2δ + λ < 1, (d) ∫ d(t x,t y) 0 ≤ γ ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + δ ∫ d(fx,t y)+d(fy,t x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.17) there exist γ,δ,λ ∈ [0,1) such that 0 ≤ 2(γ + δ) + λ < 1, for all x, y in x for which fx 6= fy are comparable, and where ϕ ∈ χ. if there exists a point x0 ∈ x such that fx0 ≤ tx0 and the mappings t and f are compatible, then t and f have a coincidence point in x. corollary 3.10. let (x,d,≤) be a complete partially ordered metric space. suppose that the mappings f,t : x → x are continuous, t is a monotone f-nondecreasing, t(x) ⊆ f(x) and satisfying the following integral type contraction conditions: (i) ∫ d(t x,t y) 0 ϕ(t)dt ≤ β ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + δ ∫ d(fx,t y)+d(fy,t x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.18) for some β,γ,δ,λ ∈ [0,1) with 0 ≤ β + 2(γ + δ) + λ < 1, (ii) ∫ d(t x,t y) 0 ϕ(t)dt ≤ β ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + γ ∫ d(fx,t x)+d(fy,t y) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.19) where β,γ,λ ∈ [0,1) such that 0 ≤ β + 2γ + λ < 1, 218 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) (iii) ∫ d(t x,t y) 0 ϕ(t)dt ≤ β ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + δ ∫ d(fx,t y)+d(fy,t x) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.20) there exist β,δ,λ ∈ [0,1) such that 0 ≤ β + 2δ + λ < 1, (iv) ∫ d(t x,t y) 0 ϕ(t)dt ≤ α ∫ d(fx,t x)[1+d(fy,t y)] 1+d(fx,fy) 0 ϕ(t)dt + β ∫ d(fx,t x) d(fy,t y) d(fx,fy) 0 ϕ(t)dt + λ ∫ d(fx,fy) 0 ϕ(t)dt, (3.21) where α,β,λ ∈ [0,1) with 0 ≤ α + β + λ < 1, for all x, y in x for which fx 6= fy are comparable, and where ϕ ∈ χ. if there exists a point x0 ∈ x such that fx0 ≤ tx0 and the mappings t and f are compatible, then t and f have a coincidence point in x. remark 3.11. if α = γ = δ = 0 in corollary 3.8, then we obtain the corollary 2.5 of chandok [25]. now, we give the examples for the main theorem 3.1. example 3.12. define a metric d : x × x → [0,+∞) by d(x,y) = |x − y|, where x = [0,1] with usual order ≤. let t and f be two self mappings on x such that tx = x 2 2 and fx = 2x 2 1+x , then t and f have a coincidence point in x. proof. note that (x,d) is a complete metric space and thus, (x,d,≤) be a complete partially ordered metric space with respect to usual order ≤. let x0 = 0 ∈ x then fx0 ≤ tx0 and also note that t and f are continuous, t is a monotone f-nondecreasing and t(x) ⊆ f(x). now consider the following for any x, y in x with x < y, d(tx,ty) = 1 2 |x2 − y2| = 1 2 (x + y)|x − y| ≤ 2(x + y + xy) (1 + x)(1 + y) |x − y| ≤ α 2x2|3 − x| [ (1 + y) + y2|3 − y| ] 4(1 + x)(1 + y) + 2|x − y|(x + y + xy) + β 4 x2y2 (x + y + xy) |x − 3||y − 3| |x − y| + γ 2 x2(1 + y)|x − 3| + y2(1 + x)|y − 3| (1 + x)(1 + y) + δ (1 + y)|4x2 − y2(1 + x)| + (1 + x)|4y2 − x2(1 + y)| 2(1 + x)(1 + y) + λ 2(x + y + xy) (1 + x)(1 + y) |x − y| cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 219 d(tx,ty) ≤ α x 2|x−3| 2(1+x) · 2(1+y)+y2|3−y| 2(1+y) 1 + 2|x−y|(x+y+xy) (1+x)(1+y) + β x 2|x−3| 2(1+x) · y 2|y−3| 2(1+y) 2|x − y| x+y+xy (1+x)(1+y) + γ [ x2|x − 3| 2(1 + x) + y2|y − 3| 2(1 + y) ] + δ [ ∣ ∣ ∣ ∣ x2 (1 + x) − y2 2 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ 2y2 (1 + y) − x2 2 ∣ ∣ ∣ ∣ ] + λ 2(x + y + xy) (1 + x)(1 + y) |x − y| ≤ α d(fx,tx) [1 + d(fy,ty)] 1 + d(fx,fy) + β d(fx,tx) d(fy,ty) d(fx,fy) + γ [d(fx,tx) + d(fy,ty)] + δ [d(fx,ty) + d(fy,tx)] + λd(fx,fy). then, the contraction condition in theorem 3.1 holds by selecting proper values of α,β,γ,δ,λ in [0,1) such that 0 ≤ α + β + 2(γ + δ) + λ < 1. therefore, t and f have a coincidence point 0 ∈ x. example 3.13. define a distance function d : x × x → [0,+∞) by d(x,y) = |x − y|, where x = [0,1] with usual order ≤. let t and f be two self mappings on x such that tx = x3 and fx = x4, then t and f have two coincidence points 0, 1 in x with x0 = 1 4 . 4 applications now our aim is to give an existence theorem for a solution of the following integral equation. h(x) = ∫ m 0 µ(x,y,h(y))dy + g(x), x ∈ [0,m], (4.1) where m > 0. let x = c[0,m] be the set of all continuous functions defined on [0,m]. now, define d : x × x → r+ by d(u,v) = sup x∈[0,m] {|u(x) − v(x)|} then, (x,≤) is a partially ordered set. now, we prove the following result. theorem 4.1. suppose the following hypotheses holds: (i) µ : [0,m] × [0,m] × r+ → r+ and g : r → r are continuous, (ii) for each x,y ∈ [0,m], we have µ ( x,y, ∫ m 0 µ(x,z,h(z))dz + g(x) ) ≤ µ(x,y,h(y)), (iii) there exists a continuous function n : [0,m] × [0,m] → [0,+∞] such that |µ(x,y,a) − µ(x,y,b)| ≤ n(x,y)|a − b| and 220 k. kalyani & n. seshagiri rao cubo 23, 2 (2021) (iv) sup x∈[0,m] ∫ m 0 n(x,y)dy ≤ γ for some γ < 1. then, the integral equation (4.1) has a solution a ∈ c[0,m]. proof. define t : c[0,m] → c[0,m] by tw(x) = ∫ m 0 µ(x,y,w(x))dx + g(x), x ∈ [0,m]. now, we have t(tw(x)) = ∫ m 0 µ(x,y,tw(x))dx + g(x) = ∫ m 0 µ ( x,y, ∫ m 0 µ(x,z,w(z))dz + g(x) ) dx + g(x) ≤ ∫ m 0 µ(x,y,w(z))dz + g(x) = tw(x) thus, we have t(tx) ≤ tx for all x ∈ c[0,m]. for any x∗,y∗ ∈ c[0,m] with x ≤ y, we have d(tx∗,ty∗) = sup x∈[0,m] |tx∗(x) − ty∗(x)| = sup x∈[0,m] ∣ ∣ ∣ ∣ ∣ ∫ m 0 µ(x,y,x∗(x)) − µ(x,y,y∗(x))dx ∣ ∣ ∣ ∣ ∣ ≤ sup x∈[0,m] ∫ m 0 |µ(x,y,x∗(x)) − µ(x,y,y∗(x))| dx ≤ sup x∈[0,m] ∫ m 0 n(x,y)|x∗(x) − y∗(x)|dx ≤ sup x∈[0,m] |x∗(x) − y∗(x)| sup x∈[0,m] ∫ m 0 n(x,y)dx = d(x∗,y∗) sup x∈[0,m] ∫ m 0 n(x,y)dx ≤ γd(x∗,y∗). moreover, {x∗n} is a nondecreasing sequence in c[0,m] such that x ∗ n→ x ∗, then x∗n ≤ x ∗ for all n ∈ n. thus all the required hypotheses of corollary 3.6 are satisfied. thus, there exists a solution a ∈ c[0,m] of the integral equation (4.1). cubo 23, 2 (2021) coincidence point results of nonlinear contractive mappings in ... 221 references [1] i. altun, m. aslantas and h. sahin, “best proximity point results for p-proximal contractions”, acta math. hungar. vol. 162, no. 2, pp. 393-402, 2020. 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leandrorenanf@gmail.com abstract in this paper, we introduce a new general class of trigonometric distributions based on the tangent function, called the tan-g class. a mathematical procedure of the tan-g class is carried out, including expansions for the probability density function, moments, central moments and rényi entropy. the estimates are acquired in a non-closed form by the maximum likelihood estimation method. then, an emphasis is put on a particular member of this class defined with the burr xii distribution as baseline, called the tan-bxii distribution. the inferential properties of the tan-bxii model are investigated. finally, the tan-bxii model is applied to a practical data set, illustrating the interest of the tan-g class for the practitioner. resumen en este art́ıculo, introducimos una nueva clase general de distribuciones trigonométricas basada en la función tangente, llamada la clase tan-g. se lleva a cabo un procedimiento matemático para la clase tan-g, incluyendo expansiones para la función de densidad de probabilidad, momentos, momentos centrales y entroṕıa de rényi. las estimaciones se obtienen en forma no-cerrada para el método de estimación de máxima verosimilitud. luego, se pone énfasis en un miembro particular de esta clase definido con la distribución burr xii como ĺınea de base, llamada la distribución tanbxii. se investigan las propiedades inferenciales del modelo tan-bxii. finalmente, el modelo tan-bxii es aplicado para un conjunto de datos prácticos, ilustrando el interés de la clase tan-g para el practicante. keywords and phrases: trigonometric class of distributions, tangent function, burr xii distribution, maximum likelihood estimation, entropy. 2020 ams mathematics subject classification: 60e05, 62e15, 62f10. accepted: 23 december, 2020 received: 12 february, 2020 ©2021 l. souza et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100001 https://orcid.org/0000-0001-9029-7714 http://orcid.org/0000-0002-3261-8265 http://orcid.org/0000-0002-3699-5156 https://orcid.org/0000-0002-1522-9292 https://orcid.org/0000-0001-9553-5515 https://orcid.org/0000-0002-2131-9825 2 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) 1 introduction the recent years of research on probabilistic distributions have been marked by the rise of general classes of trigonometric distributions, more or less sophisticated. modern statistical developments can be found in, e.g., [10], [16], [18], [19], [11], [4] and [8]. in particular, among the most fundamental of them, [18] introduced the sin-g class defined by the cumulative distribution function (cdf) given by h (1) g (x) = sin (π 2 g(x) ) , x ∈ r, where g(x) denotes a baseline cdf of a continuous distribution and [19] proposed the cos-g class defined by the cdf given by h (2) g (x) = 1 − cos (π 2 g(x) ) , x ∈ r. one can notice that the eventual parameter(s) of these classes is (are) (the one) (those) of g(x) only, and that the following elementary equation hold: [h (1) g (x)] 2 +[1−h(2)g (x)] 2 = 1, i.e., h (2) g (x) = 1−√ 1 − [h(1)g (x)]2 (showing that h (2) g (x) belongs to the so-called kum-g class with the parameters 1/2 and 2 and the baseline cdf h (1) g (x), see [5]). in addition to their simplicity, both of these two trigonometric classes benefit from the smooth periodic oscillations of the involved trigonometric functions to attain new levels of flexibility in statistical modeling. in [18] and [19], this fact is illustrated by means of several practical data sets, with winning results in comparison to useful model competitors. in this study, following the spirit of [18] and [19], we introduce a new and simple general class of trigonometric distributions having the feature to be centered around the tangent function. for the purpose of this paper, we call it the tan-g class. it is defined by the following cdf: hg(x) = tan (π 4 g(x) ) , x ∈ r. (1.1) several existing constructions give this cdf, beginning by the integral techniques developed by [2]; we have hg(x) = ∫ (π/4)g(x) 0 sec2(t)dt, where sec(t) = 1/ cos(t). after some algebra, one can also notice that hg(x) can be expressed in terms of the cdfs h (1) g (x) and h (2) g (x) as hg(x) = √ 1 − [1 −h(2)g (x)]2 2 −h(2)g (x) , hg(x) = h (1) g (x) 1 + √ 1 − [h(1)g (x)]2 . from these expressions, we immediately get the following stochastic ordering: hg(x) ≤ h (1) g (x), attesting that hg(x) can provide different statistical models to those of h (1) g (x). in full generality, the main qualities of the tan-g class are to be simple: there is no additional parameter and the related functions are very tractable, and its ability to create flexible statistical models, well-adapted to fit with precision several kinds of data sets, beyond those related to the sin-g or cos-g class. cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 3 all these aspects are developed in this paper according to the following plan. in section 2, the main theoretical features of the tan-g class are presented. section 3 is devoted to a special member of the class defined with the burr xii distribution as baseline. concluding remarks are given in section 4. 2 main theoretical features of the tan-g class a theoretical treatment of the tan-g class is performed in this section, investigating the related distributional functions, asymptotic and critical points, useful expansion, moments and central moments, expansion for the general coefficient, entropy and the mathematics of the maximum likelihood estimation. 2.1 distributional functions we recall that the tan-g class of distributions is defined by the cdf given by (1.1). upon differentiation, the corresponding pdf is given by hg(x) = π 4 g(x) sec2 (π 4 g(x) ) , x ∈ r, (2.1) where g(x) denotes the pdf corresponding to g(x). the hazard function (hf) of the tan-g class is given by rg(x) = π 4 g(x) sec2 (π 4 g(x) ) 1 − tan (π 4 g(x) ) , x ∈ r. (2.2) the curvatures properties of hg(x) and rg(x) are crucial to define an appropriate statistical model for a given data set. further elements on these curvature properties will be presented in the subsection below. another important function is the quantile function (qf) given by q(u) = h−1g (u) = g −1 [ 4 π arctan(u) ] , u ∈ (0, 1). that is, the median of the tan-g class is given by m = q(0.5) ≈ g−1 (0.5903345) . other properties of the tan-g class can be studied through this qf. for instance, the main steps to generate random numbers from the tan-g class via the qf are described in table 1. 4 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) table 1: generated numbers from the tan-g class by the use of the qf algorithm 1. generate n values from u ∼ u(0, 1) 2. specify g−1(x) 3. obtain an outcome of x with cdf (1.1) by x = q(u) 2.2 asymptotic and critical points let us now investigate the asymptotic and critical points for hg(x) and rg(x). owing to (2.1) and (2.2), when g(x) → 0, we have hg(x) ∼ π 4 g(x), hg(x) ∼ π 4 g(x), rg(x) ∼ π 4 g(x). also, when g(x) → 1, we have hg(x) ∼ 1 − π 2 (1 −g(x)), hg(x) ∼ π 2 g(x), rg(x) ∼ g(x) 1 −g(x) . if x∗ denotes a critical point for hg(x), then it satisfies the following equation: {ln[hg(x)]} ′ |x=x∗ = 0, i.e., g(x)′ |x=x∗ + π 2 g(x∗) 2 tan (π 4 g(x∗) ) = 0. with similar arguments, if x∗∗ denotes a critical point for rg(x), then it satisfies the following equation: {ln[rg(x)]} ′ |x=x∗∗ = 0, i.e.,[ g(x)′ |x=x∗∗ + π 2 g(x∗∗) 2 tan (π 4 g(x∗∗) )][ 1 − tan (π 4 g(x∗∗) )] + π 4 g(x∗∗) 2 sec2 (π 4 g(x∗∗) ) = 0. none of these non-linear equations has solution(s) with closed form. that is, for a specific g(x), we can determine x∗ and x∗∗ numerically by the use of any scientific software as r, matlab, mathematica. . . 2.3 useful expansion the following result presents an useful expansion of the pdf of the tan-g class involving functions of the exponentiated-g class (see [7]). theorem 2.1. the pdf of the tan-g class given by (2.1) can be expressed as a linear combination of pdfs of the exponentiated-g class as hg(x) = +∞∑ k=1 ωkg(2k−1)(x), cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 5 where ωk = (π 4 )2k−1 b2k(−4)k(1 − 4k) (2k)! , (2.3) b2k is the so-called 2kth bernoulli number and g(2k−1)(x) = (2k − 1)g(x)g2k−2(x) is the pdf of the exponentiated-g class with parameter 2k − 1. proof. using the taylor series for the tangent function, since (π/4)g(x) ∈ (0,π/2), we have tan (π 4 g(x) ) = +∞∑ k=1 b2k(−4)k(1 − 4k) (2k)! (π 4 g(x) )2k−1 . thus, we obtain the following expansion for hg(x): hg(x) = +∞∑ k=1 (π 4 )2k−1 b2k(−4)k(1 − 4k) (2k)! g2k−1(x)· the desired expansion for hg(x) is deduced by differentiation. this ends the proof of theorem 2.1. 2.4 moments and central moments an expansion for the moment of order m of the tan-g class is studied in the following result. theorem 2.2. let µm be the moment of order m of the tan-g class and µ (2k−1) m be the moment of order m of the exponentiated-g class with parameter 2k − 1. then, we have µm = +∞∑ k=1 ωkµ (2k−1) m , where ωk is given by (2.3). proof. the moment of order m of the tan-g class is defined by µm = ∫ +∞ −∞ xmdhg(x). it follows from theorem 2.1 that µm = ∫ +∞ −∞ xm +∞∑ k=1 ωkg(2k−1)(x)dx = +∞∑ k=1 ωk ∫ +∞ −∞ xmg(2k−1)(x)dx = +∞∑ k=1 ωkµ (2k−1) m . this ends the proof of theorem 2.2. the mean is given by µ = µ1. remark 2.3. by applying the change of variable u = g(x), we can express µ (2k−1) m as µ(2k−1)m = (2k − 1) ∫ +∞ −∞ xmg(x)g2k−2(x)dx = (2k − 1) ∫ 1 0 [ g−1(u) ]m u2k−2du. 6 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) similarly, we can obtain an expansion of the central moments of order m by using theorem 2.2. corollary 2.4. let µ′m be the central moment of order m of the tan-g class and µ (2k−1) m be the moment of order m of the exponentiated-g class with parameter 2k − 1. then, we have µ′m = +∞∑ k=1 m∑ r=0 γk,m,rµ (2k−1) m−r , where γk,m,r = ωk ( m r ) (−1)rµr and ωk is defined by (2.3). proof. the central moment of order m of the tan-g class is defined by µ′m = ∫ +∞ −∞ (x−µ)mdhg(x). by using the binomial theorem and theorem 2.2, we have µ′m = m∑ r=0 ( m r ) (−1)rµr ∫ +∞ −∞ xm−rdhg(x) = m∑ r=0 ( m r ) (−1)rµrµm−r = m∑ r=0 ( m r ) (−1)rµr +∞∑ k=1 ωkµ (2k−1) m−r = +∞∑ k=1 m∑ r=0 γk,m,rµ (2k−1) m−r . the proof of corollary 2.4 is ended. by considering m = 2, the variance is given by σ2 = µ′2 = +∞∑ k=1 2∑ r=0 γk,2,rµ (2k−1) 2−r . by using similar summation techniques, one can set expansions of the incomplete moments, the moment generating function and the characteristic function, among others. 2.5 expansion to the general coefficient the general coefficient of the tan-g class is defined by cm = µ′m σm . by applying corollary 2.4, it can be written as cm = ∑+∞ k=1 ∑m r=0 γk,m,rµ (2k−1) m−r[∑+∞ k=1 ∑2 r=0 γk,2,rµ (2k−1) 2−r ]m 2 . cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 7 so, the asymmetry and kurtosis of the tan-g class can be respectively expressed by c3 = ∑+∞ k=1 ∑3 r=0 γk,3,rµ (2k−1) 3−r[∑+∞ k=1 ∑2 r=0 γk,2,rµ (2k−1) 2−r ]3 2 , c4 = ∑+∞ k=1 ∑4 r=0 γk,4,rµ (2k−1) 4−r[∑+∞ k=1 ∑2 r=0 γk,2,rµ (2k−1) 2−r ]2 . 2.6 entropy entropy measures the uncertainty; the greater the entropy, the higher the disorder and the less likely it will be to observe a phenomenon; the lower the entropy, the lower its disorder and the higher the probability of observing a particular event. among the most useful entropy, there is the rényi entropy introduced by [13]. in the context of the tan-g class, it is defined by lg(γ) = 1 1 −γ ln [∫ +∞ −∞ h γ g(x) ] dx, where γ > 0 with γ 6= 1 and h γ g(x) = (π 4 )γ gγ(x) sec2γ (π 4 g(x) ) . let us now consider the function w(s) = sec2γ[(π/4)s], s ∈ (0, 1). by applying the taylor series formula to w(s) at a fixed point s0 ∈ (0, 1) (say s0 = 0.5), we get sec2γ [π 4 s ] = +∞∑ k=0 ak(s−s0)k = +∞∑ k=0 k∑ r=0 ( k r ) aks r(−1)k−rsk−r0 , where ak = w (k)(s) |s=s0 /k!. we are now able to derive an expansion of the rényi entropy of the tan-g class. after some algebra, we obtain lg(γ) = 1 1 −γ { γ ln (π 4 ) + ln [ +∞∑ k=0 k∑ r=0 aks r(−1)k−rsk−r0 ir ]} , (2.4) where ir = ∫ +∞ −∞ gr(x)gγ(x)dx. even if it has no closed form, the integral ir can be computed numerically. the shannon entropy, pioneered by [15], is given by sg = − ∫ +∞ −∞ ln[hg(x)]hg(x)dx. it can deduced from lg(γ) via the relation limγ→1 lg(γ) = sg. 8 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) 2.7 maximum likelihood estimation and scores here, we consider the estimation of the parameters of the tan-g class by the method of maximum likelihood. let x̃ = (x1, . . . ,xn) > be a random sample observations from the tan-g class with vector parameter θ̃ = (θ1, . . . ,θp) (thus, p is the number of parameters of the distribution). then, the log-likelihood (ll) function for the tan-g class is given by `(θ̃) = n ln (π 4 ) + n∑ i=1 ln ( g(xi|θ̃) ) + 2 n∑ i=1 ln [ sec (π 4 g(xi|θ̃) )] · the maximum likelihood estimators (mles) are obtained by maximizing this function according to θ̃. in this regards, if g(x|θ̃) is differentiable according to θ̃, one can consider the jth score given by u(θj) = ∂`(θ̃) ∂θj = n∑ i=1 1 g(xi|θ̃) ∂g(xi|θ̃) ∂θj + π 2 n∑ i=1 tan (π 4 g(xi|θ̃) ) ∂g(xi|θ̃) ∂θj and consider the following equations: u(θ1) = 0, . . . ,u(θp) = 0. thus, the mles are defined as the simultaneous solutions of these equations. 3 the tan-bxii distribution we now focus on a special distribution of the tan-g class, called the tan-bxii distribution. 3.1 definition tan-bxii distribution is defined by the cdf given by (1.1) with the cdf g(x) of the burr xii distribution, i.e., g(x) = 1 − [ 1 + (x s )c]−κ , x,s,c,κ > 0. hence, the cdf of the tan-bxii distribution is given by hg(x) = tan { π 4 ( 1 − [ 1 + (x s )c]−κ)} , x > 0. the corresponding pdf is given by hg(x) = π 4 { xc−1cκs−c [ 1 + (x s )c]−κ−1} sec2 { π 4 ( 1 − [ 1 + (x s )c]−κ)} , x > 0. finally, the corresponding hf is given by rg(x) = π 4 { xc−1cκs−c [ 1 + (x s )c]−κ−1} sec2 { π 4 ( 1 − [ 1 + (x s )c]−κ)} 1 − tan { π 4 ( 1 − [ 1 + (x s )c]−κ)} , x > 0. it is expected that the hf is unimodal or decreasing, as it can be seen in figures 3 and 4, respectively, but an analytic verification of this fact using all three parameters is an unnecessarily complicated computation. one can check for given parameters that it is indeed the case using computing software. cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 9 3.2 shape characteristics of probability density and hazard functions the asymptotic and critical points for hg(x) and rg(x) can be obtained in non-closed form by applying subsection 2.2. also, some possible shapes of hg(x) for some parameter values are displayed in figure 1. some plots of hg(x) are given in figure 2. 0 1 2 3 4 5 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 x d e n s it y c = 2.5, κ = 3, s = 3.5 c = 2, κ = 3, s = 3 c = 2.5, κ = 1, s = 1 c = 3, κ = 1, s = 1 c = 3.5, κ = 1, s = 1 c = 4, κ = 1.5, s = 2 figure 1: plots of the pdf of the tanbxii distribution 0 1 2 3 4 5 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 x c u m u la ti v e c = 1, k = 1, s = 1 c = 2, k = 1, s = 1 c = 2.5, k = 1, s = 1 c = 3, k = 1, s = 1 c = 3.5, k = 1, s = 1 c = 4, k = 1, s = 1 figure 2: plots of the cdf of the tanbxii distribution figures 3 and 4 present plots of rg(x) for some parameter values. we observe that the hf can be unimodal or only be decreasing. 0 2 4 6 8 10 0 .0 0 .5 1 .0 1 .5 x h a z a rd c = 0.5, k = 1.7, s = 1.1 c = 0.5, k = 2.3, s = 1.5 c = 0.5, k = 2.6, s = 1.9 c = 0.5, k = 3.3, s = 2.3 c = 0.5, k = 3.5, s = 3.7 c = 0.5, k = 3.9, s = 4.6 figure 3: plots of decreasing hf of the tan-bxii distribution. 0 1 2 3 4 5 0 .0 0 .5 1 .0 1 .5 2 .0 x h a z a rd c = 1.5, κ = 1.5, s = 0.8 c = 2.2, κ = 1.5, s = 1.2 c = 2.5, κ = 1.5, s = 1.3 c = 3, κ = 1.5, s = 1.9 c = 4.5, κ = 1.5, s = 2 c = 4.7, κ = 1.5, s = 3 figure 4: plots of unimodal hf of the tan-bxii distribution. 10 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) 3.3 expansion of the probability density function here, we use the general results proved for the tan-g class of distributions to reveal properties for the tan-bxii distribution. an useful expansion of the pdf is presented below. theorem 3.1. the pdf of the tan-g class can be expanded as a mixture of pdfs of the burr xii distribution, i.e., hg(x) = +∞∑ k=1 2k−2∑ j=0 ωj,kgburrxii(x; s,c,κ(j + 1)), where ωj,k = ωk(2k − 1) ( 2k − 2 j ) (−1)j 1 j + 1 , (3.1) ωk is given by (2.3) and gburrxii(x; s,c,κ(j + 1)) is the pdf of the burr xii distribution with parameters s, c and κ(j + 1), i.e., gburrxii(x; s,c,κ(j + 1)) = x c−1cκ(j + 1)s−c [1 + (x/s) c ] −κ(j+1)−1 , x > 0. proof. owing to theorem 2.1, we can write hg(x) = +∞∑ k=1 ωkg(2k−1)(x), where ωk is given by (2.3) and g(2k−1)(x) = (2k − 1)g(x)g2k−2(x) = (2k − 1)xc−1cκs−c [ 1 + (x s )c]−κ−1 { 1 − [ 1 + (x s )c]−κ}2k−2 . the standard binomial theorem gives g(2k−1)(x) = (2k − 1)xc−1cκs−c 2k−2∑ j=0 ( 2k − 2 j ) (−1)j [ 1 + (x s )c]−κ(j+1)−1 = (2k − 1) 2k−2∑ j=0 ( 2k − 2 j ) (−1)j 1 j + 1 gburrxii(x; s,c,κ(j + 1)). the proof ends by putting the above equalities together. 3.4 moments and central moments by using identical manipulations to those used in theorem 2.2, we introduce the moment expansion of the tan-bxii distribution in the following result. cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 11 theorem 3.2. first of all, the moment of order m of the tan-bxii distribution exists if and only if cκ > m. in this case, the moment of order m of the tan-bxii distribution is given by µm = +∞∑ k=1 2k−2∑ j=0 ωj,ks mκ(j + 1)b ( κ(j + 1) −mc−1, 1 + mc−1 ) , where ωj,k is given by (3.1) and b(a,b) = ∫ 1 0 ta−1(1−t)b−1dt, a,b > 0 (the standard beta function). proof. it follows from theorem 3.1 that µm = +∞∑ k=1 2k−2∑ j=0 ωj,kjj,k,m, where jj,k,m = ∫ +∞ 0 xmgburrxii(x; s,c,κ(j + 1))dx = ∫ +∞ 0 xmxc−1cκ(j + 1)s−c [ 1 + (x s )c]−κ(j+1)−1 dx. by applying the changes of variables u = (x s )c and ν = (1 + u)−1, in turn, we get jj,k,m = s mκ(j + 1) ∫ +∞ 0 u m c (1 + u)−κ(j+1)−1du = smκ(j + 1) ∫ 1 0 νκ(j+1)− m c −1(1 −ν) m c dν = smκ(j + 1)b ( κ(j + 1) −mc−1, 1 + mc−1 ) . by combining the above equalities together, we end the proof of theorem 3.2. the mean is given by µ = µ1. remark 3.3. by adopting the notations introduced in section 2, following the lines of the proof of theorem 3.2, one can show that µ(2k−1)m = (2k − 1)s mκ 2k−2∑ j=0 ( 2k − 2 j ) (−1)jb ( κ(j + 1) −mc−1, 1 + mc−1 ) . similarly to corollary 2.4, the central moment of order m of the tan-bxii distribution is given µ′m = m∑ r=0 ( m r ) (−1)rµrµm−r = +∞∑ k=1 2k−2∑ j=0 m∑ r=0 ρj,k,m,rb ( κ(j + 1) − (m−r)c−1, 1 + (m−r)c−1 ) , where ρj,k,m,r = ωj,ks m−rκ(j + 1) ( m r ) (−1)rµr. by considering m = 2, we get the following expansion for variance of the distribution: σ2 = µ′2 = +∞∑ k=1 2k−2∑ j=0 2∑ r=0 ρj,k,2,rb ( κ(j + 1) − (2 −r)c−1, 1 + (2 −r)c−1 ) . 12 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) 3.5 expansion to the general coefficient the general coefficient of the tan-bxii distribution can be expressed as cm = µ′m σm = ∑+∞ k=1 ∑2k−2 j=0 ∑m r=0 ρj,k,m,rb ( κ(j + 1) − (m−r)c−1, 1 + (m−r)c−1 ) {∑+∞ k=1 ∑2k−2 j=0 ∑2 r=0 ρj,k,2,rb (κ(j + 1) − (2 −r)c−1, 1 + (2 −r)c−1) }m/2 . thus, the asymmetry and kurtosis can be expressed by taking m = 3 and m = 4, respectively, which is the object of the next part. 3.6 figures of asymmetry and kurtosis in figures 5, 6 and 7, we present the asymmetry and kurtosis graphs for the tan-bxii distribution. it is possible to observe that this new distribution has a great flexibility on these aspects, showing varying values, small and large. 3.0 3.5 4.0 4.5 5.0 2 4 6 8 1 0 c s k e w n e s s κ = 1, s = 2 κ = 1.1, s = 2.1 κ = 1.2, s = 2.2 κ = 1.3, s = 2.3 κ = 1.4, s = 2.4 (a) 5.0 5.5 6.0 6.5 7.0 6 7 8 9 1 0 c k u r to s i κ = 1, s = 2.5 κ = 1.1, s = 2.6 κ = 1.2, s = 2.7 κ = 1.3, s = 2.8 κ = 1.4, s = 2.9 (b) figure 5: plots of the skewness and kurtosis coefficients of the tan-bxii distribution as a function of c for selected values of κ and s cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 13 3.0 3.5 4.0 4.5 5.0 2 .0 2 .1 2 .2 2 .3 2 .4 2 .5 κ s k e w n e s s c = 1.5, s = 2.5 c = 3.8, s = 1.8 c = 2.6, s = 2.7 c = 2.9, s = 2.9 c = 8.1, s = 3 (a) 3.0 3.5 4.0 4.5 5.0 5 1 0 1 5 2 0 κ k u r to s is c = 1.5, s = 2.5 c = 3.8, s = 1.8 c = 2.6, s = 2.7 c = 2.9, s = 2.9 c = 8.1, s = 3 (b) figure 6: plots of the skewness and kurtosis coefficients of the tan-bxii distribution as a function of κ for selected values of c and s 3.0 3.5 4.0 4.5 5.0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0 s s k e w n e s s c = 1, κ = 0.2 c = 1.1, κ = 0.2 c = 1.2, κ = 0.2 c = 1.3, κ = 0.2 c = 1.4, κ = 0.2 (a) 4.0 4.2 4.4 4.6 4.8 5.0 5 1 0 1 5 2 0 2 5 s k u r to s is c = 1, κ = 0.7 c = 1.1, κ = 0.7 c = 1.2, κ = 0.7 c = 1.3, κ = 0.7 c = 1.4, κ = 0.7 (b) figure 7: plots of the skewness and kurtosis coefficients of the tan-bxii distribution as a function of s for selected values of c and κ 14 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) 3.7 entropy by applying (2.4), the rényi entropy is given by lg(γ) = 1 1 −γ { γ ln (π 4 ) + ln [ +∞∑ k=0 k∑ r=0 aks r(−1)k−rsk−r0 ir ]} , where γ > 0 with γ 6= 1 and, after some algebra, ir = ∫ +∞ −∞ gr(x)gγ(x)dx = r∑ j=0 ( r j ) (−1)jκγs−(γ−1)cγ−1b(κ(j + γ) + (γ − 1)c−1, (γ − 1)(c− 1)c−1 + 1), assuming that κγ + (γ − 1)c−1 > 0 and (γ − 1)(c− 1)c−1 + 1 > 0. figure 8 displays this rényi entropy for some values of the parameters. 5 10 15 20 0 2 4 6 8 1 0 c e n tr o p y κ = 1, s = 2 κ = 1.1, s = 2.1 κ = 1.2, s = 2.2 κ = 1.3, s = 2.3 κ = 1.4, s = 2.4 figure 8: plots of the rényi entropy of the tan-bxii distribution as a function of c for selected values of κ and s 3.8 maximum likelihood estimation here, we provide the mathematical background related to the mles of the tan-bxii model parameters, i.e., c, κ and s. let x = {x1, . . . ,xn} > be n independent random variables from the tan-bxii distribution. then, the log-likelihood function is given by l = n ln (π 4 ) + n ln(c) + n ln(κ) −nc ln(s) + (c− 1) n∑ i=1 ln(xi) − (κ + 1) n∑ i=1 ln [ 1 + (xi s )c] + 2 n∑ i=1 ln [ sec { π 4 ( 1 − [ 1 + (xi s )c]−κ)}] . cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 15 the scores are presented below: uc = n c −n ln(s) + n∑ i=1 ln(xi) − (κ + 1) n∑ i=1 xci ln (xi s ) sc + xci + π 2 κ n∑ i=1 (xi s )c ln (xi s )[ 1 + (xi s )c]−κ−1 tan { π 4 ( 1 − [ 1 + (xi s )c]−κ)} , uκ = n κ − n∑ i=1 ln [ 1 + (xi s )c] + π 2 n∑ i=1 [ 1 + (xi s )c]−κ ln [ 1 + (xi s )c] tan { π 4 ( 1 − [ 1 + (xi s )c]−κ)} and us = − nc s + c(κ + 1)s−1 n∑ i=1 xci sc + xci − π 2 cκs−(c+1) n∑ i=1 xci [ 1 + (xi s )c]−κ−1 tan { π 4 ( 1 − [ 1 + (xi s )c]−κ)} . the mles of c, κ and s are defined by the simultaneous solutions of the following non-linear equations: uc = 0, uκ = 0 and us = 0 according to c, κ and s. under some standard regularity conditions, the well-known theory on mle can be applied, ensuring nice asymptotic properties (see [3]). 3.9 simulation using the tanb r package [17], we perform a simulation study using several random samples of the tan-bxii distribution. for each sample, we calculate the mles using native r language’s optim implementation. biases, and mean square errors (mses) are also calculated using the mles obtained. for this simulation, we use samples with sizes 10, 20, 30, . . . , 100 and 1000 replicas for the parameter’s configuration: c = 1, κ = 1.4 and s = 0.15. figures 9a, 9b and 9c show the bias for c, κ and s, respectively, in this simulation and we can see it decreasing over the sample sizes. figures 10a, 10b and 10c show the mse for the same parameters and also decreases over the sample sizes. table 2 summarizes the simulation, given the means of mles, biases and mses of the samples with sizes of 10, 20, 30, 50 and 100. we can see in the table that all the parameters are overestimated by the maximum likelihood method. the biases and mses decrease over the sample sizes as we see in figures 9a, 9b, 9c, 10a, 10b and 10c. 16 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) table 2: mles, biases and mses for c = 1, κ = 1.4, s = 0.15 using 1000 replicas sample size(n) parameters mles biases mses c 1.5102 0.5102 1.1065 10 κ 7.6587 6.2587 86.6797 s 2.5062 2.3562 15.5951 c 1.2998 0.2998 0.4181 20 κ 6.7327 5.3327 68.2502 s 2.3631 2.2131 12.9993 c 1.2444 0.2444 0.2478 30 κ 5.5806 4.1806 47.7063 s 1.8732 1.7232 8.7874 c 1.1787 0.1787 0.111 50 κ 4.7807 3.3807 32.0412 s 1.6109 1.4609 6.7689 c 1.1636 0.1636 0.066 100 κ 3.4506 2.0506 11.3414 s 0.9844 0.8344 2.0205 20 40 60 80 100 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 sample sizes (n) b ia s ( c ) (a) plots of bias(c) 20 40 60 80 100 0 1 2 3 4 5 6 sample sizes (n) b ia s ( k ) (b) plots of bias(κ) 20 40 60 80 100 0 .0 0 .5 1 .0 1 .5 2 .0 sample sizes (n) b ia s ( s ) (c) plots of bias(s) figure 9: plots of the biases for the simulated experiment related to the tan-burxii model parameters 20 40 60 80 100 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 sample sizes (n) m s e ( c ) (a) plots of mse(c) 20 40 60 80 100 0 2 0 4 0 6 0 8 0 sample sizes (n) m s e ( k ) (b) plots of mse(κ) 20 40 60 80 100 0 5 1 0 1 5 sample sizes (n) m s e ( s ) (c) plots of mse(s) figure 10: plots of the mses for the simulated experiment related to the tan-burxii model parameters cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 17 3.10 application now, we apply the tan-bxii model to fit a practical data set and compare it with three other models, namely kum-bxii, burrxii and kum-w models. these data are on the aircraft windshield failures (thousands of hours) reported in murthy [12] (see table 3). a brief statistical description of these data can be found in table 4. table 5 shows the mles of the parameters of the tan-bxii, kum-bxii, burrxii and kum-w models with error in parentheses, as well as the related akaike information criterion (aic), corrected akaike information criterion (caic), bayesian information criterion (bic), cramér-von mises (w∗) and anderson-darling (a∗) statistics. we refer to [1], [6] and the book of [9] for precise definitions and use of these fundamental statistical tools. table 3: data on aircraft windshield failures (thousands of hours) 0.040 1.866 2.385 3.443 0.301 1.876 2.481 3.467 0.309 1.899 2.610 3.478 0.557 1.911 2.625 3.578 0.943 1.912 2.632 3.595 1.070 1.914 2.646 3.699 1.124 1.981 2.661 3.779 1.248 2.010 2.688 3.924 1.281 2.038 2.823 4.035 1.281 2.085 2.890 4.121 1.303 2.089 2.902 4.167 1.432 2.097 2.934 4.240 1.480 2.135 2.962 4.255 1.505 2.154 2.964 4.278 1.506 2.190 3.000 4.305 1.568 2.194 3.103 4.376 1.615 2.223 3.114 4.449 1.619 2.224 3.117 4.485 1.652 2.229 3.166 4.570 1.652 2.300 3.344 4.602 1.757 2.324 3.376 4.663 table 4: descriptive statistics of the considered data min. q1 median mean q3 max. var. 0.040 1.839 2.354 2.557 3.393 4.663 1.252 table 5: mles of the parameters of the tan-bxii, kum-bxii, kum-w and burrxii models, with errors in parentheses, and aic, bic, caic, w∗ and a∗ statistics models estimates aic bic caic w∗ a∗ tan-bxii(c,κ,s) 2.27 186.02 26.00 — — 267.76 275.09 268.06 0.06 0.58 (0.20) (659.52) (41.42) — — kum-bxii(a,b,c,d,k) 0.28 1.96 7.17 4.54 5.82 267.95 280.17 268.71 0.08 0.64 (0.11) (1.36) (2.38) (5.07) (1.46) kum-w(a,b,c,β) 0.38 8.53 5.78 0.13 — 268.82 278.59 269.32 0.06 0.56 (0.04) (6.89) (0.06) (0.04) — bxii(a,c,k) 2.48 11.31 7.47 — — 270.24 277.57 270.54 0.06 0.63 (0.23) (8.05) (2.57) — — 18 l. souza, w. júnior, c. de brito, c. chesneau, r. fernandes & t. ferreira cubo 23, 1 (2021) it follows from table 5 that, when compared to other ones, the tan-bxii model is the best. we illustrate this claim by showing the fits of the estimated pdfs and cdfs in figures 11 and 12, respectively. thus, we conclude that the tan-bxii distribution is quite flexible in the modeling of the proposed data. x p d f 0 1 2 3 4 5 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 tan − b kum − b burr kumw figure 11: some fitted pdfs of the data 0 1 2 3 4 5 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 x c d f tan − b kum − b burr kumw figure 12: some fitted cdfs of the data 4 concluding remarks in this paper, we introduced and discussed a new class of trigonometric distributions, called the tan-g class, with a focus on a new lifetime trigonometric distribution of the class, called the tan-bxii distribution. we obtain probability density function, cumulative distribution function, hazard function and various moments. the entropy is also calculated. a complete part is devoted to the estimation of the model parameters via the maximum likelihood method. we put the light on the applicability of the new related models by considering a practical data set. even though our class of distributions does not optimally fit the data presented, it still proves to be a powerful tool for statistical analysis. we will apply this distribution to other data sets to show its full power and it will be reported elsewhere. acknowledgments we would like to thank the reviewer and the associated editor for constructive comments on the article, improving it on several important aspects. cubo 23, 1 (2021) tan-g class of trigonometric distributions and its applications 19 references [1] t. w. anderson and d. a. 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[17] l. souza, l. gallindo, and l. serafim-de-souza, (2016). tanb: the tanb distribution. r package version 0.2. available at https://cran.r-project.org/web/packages/tanb/ index.html or by running install.packages("tanb");library("tanb");help("rtanb") inside r([14]). [18] l. souza, w. r. o. junior, c. c. r. de brito, c. chesneau, t. a. e. ferreira, and l. soares, “on the sin-g class of distributions: theory, model and application”, journal of mathematical modeling, vol. 7, no. 3, pp. 357–379, 2019. [19] l. souza, w. r. o. junior, c. c. r. de brito, c. chesneau, t. a. e. ferreira, and l. soares, “general properties for the cos-g class of distributions with applications”, eurasian bulletin of mathematics, vol. 2, no. 2, pp. 63–79, 2019. https://cran.r-project.org/web/packages/tanb/index.html https://cran.r-project.org/web/packages/tanb/index.html introduction main theoretical features of the tan-g class distributional functions asymptotic and critical points useful expansion moments and central moments expansion to the general coefficient entropy maximum likelihood estimation and scores the tan-bxii distribution definition shape characteristics of probability density and hazard functions expansion of the probability density function moments and central moments expansion to the general coefficient figures of asymmetry and kurtosis entropy maximum likelihood estimation simulation application concluding remarks cubo, a mathematical journal vol. 24, no. 01, pp. 83–94, april 2022 doi: 10.4067/s0719-06462022000100083 existence, uniqueness, continuous dependence and ulam stability of mild solutions for an iterative fractional differential equation abderrahim guerfi 1 abdelouaheb ardjouni 1,2 1applied mathematics lab, faculty of sciences, department of mathematics, university of annaba, p.o. box 12, annaba 23000, algeria. abderrahimg21@gmail.com 2department of mathematics and informatics, university of souk ahras, p.o. box 1553, souk ahras, 41000, algeria. abd ardjouni@yahoo.fr abstract in this work, we study the existence, uniqueness, continuous dependence and ulam stability of mild solutions for an iterative caputo fractional differential equation by first inverting it as an integral equation. then we construct an appropriate mapping and employ the schauder fixed point theorem to prove our new results. at the end we give an example to illustrate our obtained results. resumen en este trabajo, estudiamos la existencia, unicidad, dependencia continua y estabilidad de ulam de soluciones mild para una ecuación diferencial fraccionaria de caputo iterativa, invirtiéndola primero como ecuación integral. luego construimos una aplicación apropiada y empleamos el teorema del punto fijo de schauder para demostrar nuestros nuevos resultados. finalmente damos un ejemplo para ilustrar los resultados obtenidos. keywords and phrases: iterative fractional differential equations, fixed point theorem, existence, uniqueness, continuous dependence, ulam stability. 2020 ams mathematics subject classification: 34k40, 34k14, 45g05, 47h09, 47h10. accepted: 20 december, 2021 received: 02 april, 2021 c©2022 a. guerfi et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100083 https://orcid.org/0000-0002-8221-9545 https://orcid.org/0000-0003-0216-1265 mailto:abderrahimg21@gmail.com mailto:abd_ardjouni@yahoo.fr 84 a. guerfi & a. ardjouni cubo 24, 1 (2022) 1 introduction fractional differential equations have gained considerable importance due to their applications in various sciences, such as physics, mechanics, chemistry, engineering, etc. in recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives, see the monographs of kilbas et al. [10], miller and ross [12], podlubny [14]. in particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1]–[4], [6]–[16], [18] and the references therein. recently, iterative functional differential equations of the form x′ (t) = h ( x[0] (t) ,x[1] (t) ,x[2] (t) , . . . ,x[n] (t) ) , have appeared in several papers, where x[0] (t) = t, x[1] (t) = x(t) , x[2] (t) = x(x(t)) , . . . , x[n] (t) = x[n−1] (x(t)) are the iterates of the state x(t). iterative differential equations often arise in the modeling of a wide range of natural phenomena such as disease transmission models in epidemiology, two-body problem of classical electrodynamics, population models, physical models, mechanical models and other numerous models. this kind of equations which relates an unknown function, its derivatives and its iterates, is a special type of the so-called differential equations with state-dependent delays, see [5, 9, 19] and the references therein. in this paper, inspired and motivated by the references [1]–[16], [18, 19], we concentrate on the existence, uniqueness, continuous dependence and ulam stability of mild solutions for the nonlinear iterative fractional differential equation    cdα 0+ x(t) = f ( x[0] (t) ,x[1] (t) ,x[2] (t) , . . . ,x[n] (t) ) , t ∈ j, x(0) = x′ (0) = 0, (1.1) where j = [0,t ], cdα 0+ is the standard caputo fractional derivative of order α ∈ (1,2) and f is a positive continuous function with respect to its arguments and satisfies some other conditions that will be specified later. to reach our desired end we have to transform (1.1) into an integral equation and then use the schauder fixed point theorem to show the existence and uniqueness of mild solutions. the organization of this paper is as follows. in section 2, we introduce some definitions and lemmas, and state some preliminary results needed in later sections. also, we present the inversion of (1.1) and state the schauder fixed point theorem. for details on the schauder theorem we refer the reader to [17]. in section 3, we present our main results on the existence, uniqueness, continuous cubo 24, 1 (2022) existence, uniqueness, continuous dependence and ulam stability... 85 dependence and ulam stability of mild solutions for the problem (1.1) and provide an example to illustrate our results. 2 preliminaries let c (j,r) be the banach space of all real-valued continuous functions defined on the compact interval j, endowed with the norm ‖x‖ = sup t∈j |x(t)| . for 0 < l ≤ t and m > 0, define the sets c (j,l) = {x ∈ c (j,r) : 0 ≤ x(t) ≤ l, ∀t ∈ j} , and cm (j,l) = {x ∈ c (j,l) : |x(t2) − x(t1)| ≤ m |t2 − t1| , ∀t1, t2 ∈ j}. then, cm (j,l) is a closed convex and bounded subset of c (j,r). furthermore, we suppose that the positive function f is globally lipschitz in xi, that is, there exist positive constants c1, c2, . . . , cn such that |f (t,x1,x2, . . . ,xn) − f (t,y1,y2, . . . ,yn)| ≤ n ∑ i=1 ci |xi − yi| . (2.1) we introduce the constants ρ = sup t∈j {f (t,0,0, . . . ,0)} , ζ = ρ + l n ∑ i=1 ci i−1 ∑ j=0 mj, where mj = m × mj−1. definition 2.1 ([10]). the fractional integral of order α > 0 of a function x : r+ −→ r is given by iα0+x(t) = 1 γ (α) ∫ t 0 (t − s) α−1 x(s)ds, provided the right side is pointwise defined on r+, where γ is the gamma function. for instance, iα 0+ x exists for all α > 0, when x ∈ c(r+) then iα 0+ x ∈ c (r+) and moreover iα 0+ x(0) = 0. definition 2.2 ([10]). the caputo fractional derivative of order α > 0 of a function x : r+ −→ r is given by cdα0+x(t) = i n−α 0+ x(n) (t) = 1 γ (n − α) ∫ t 0 (t − s) n−α−1 x(n) (s)ds, where n = [α] + 1, provided the right side is pointwise defined on r+. 86 a. guerfi & a. ardjouni cubo 24, 1 (2022) lemma 2.3 ([10]). suppose that x ∈ cn−1 ([0,+∞)) and x(n) exists almost everywhere on any bounded interval of r+. then ( iα c0+ d α 0+x ) (t) = x(t) − n−1 ∑ k=0 x(k) (0) k! tk. in particular, when α ∈ (1,2) , ( iα c 0+ dα 0+ x ) (t) = x(t) − x(0) − x′ (0)t. definition 2.4. a function x ∈ cm (j,l) is a mild solution of the problem (1.1) if x satisfies the corresponding integral equation of (1.1). from lemma 2.3, we deduce the following lemma. lemma 2.5. let x ∈ cm (j,l) is a mild solution of (1.1) if x satisfies x(t) = 1 γ (α) ∫ t 0 (t − s) α−1 f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ds, t ∈ j. (2.2) lemma 2.6 ([19]). if ϕ,ψ ∈ cm (j,l), then ∥ ∥ ∥ ϕ[m] − ψ[m] ∥ ∥ ∥ ≤ m−1 ∑ j=0 mj ‖ϕ − ψ‖ , m = 1,2, . . . theorem 2.7 (schauder fixed point theorem [17]). let m be a nonempty compact convex subset of a banach space (b,‖·‖) and a : m → m is a continuous mapping. then a has a fixed point. 3 main results in this section, we use theorem 2.7 to prove the existence of mild solutions for (1.1). moreover, we will introduce the sufficient conditions of the uniqueness of mild solutions of (1.1). to transform (2.2) to be applicable to the schauder fixed point, we define an operator a : cm (j,l) → c (j,r) by (aϕ)(t) = 1 γ (α) ∫ t 0 (t − s) α−1 f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ds, t ∈ j. (3.1) since cm (j,l) is a compact set as a uniformly bounded, equicontinuous and closed subset of the space c (j,r). to prove that operator a has at least one fixed point, we will prove that a is well defined, continuous and a(cm (j,l)) ⊂ cm (j,l), i. e. aϕ ∈ cm (j,l) for all ϕ ∈ cm (j,l) . lemma 3.1. suppose that (2.1) holds. then the operator a : cm (j,l) → c (j,r) given by (3.1) is well defined and continuous. cubo 24, 1 (2022) existence, uniqueness, continuous dependence and ulam stability... 87 proof. let a be defined by (3.1). clearly, a is well defined. to show the continuity of a. let ϕ,ψ ∈ cm (j,l), we have |(aϕ)(t) − (aψ)(t)| ≤ 1 γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) −f ( ψ[0] (s) ,ψ[1] (s) ,ψ[2] (s) , . . . ,ψ[n] (s) ) ∣ ∣ ∣ ds. by (2.1), we obtain |(aϕ)(t) − (aψ)(t)| ≤ 1 γ (α) ∫ t 0 (t − s) α−1 n ∑ i=1 ci ∥ ∥ ∥ ϕ[i] − ψ[i] ∥ ∥ ∥ ds. it follows from lemma 2.6 that |(aϕ)(t) − (aψ)(t)| ≤ 1 γ (α) ∫ t 0 (t − s) α−1 n ∑ i=1 ci i−1 ∑ j=0 mj ‖ϕ − ψ‖ds ≤ t α γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 mj ‖ϕ − ψ‖ , which proves that the operator a is continuous. lemma 3.2. suppose that (2.1) holds. if ζt α γ (α + 1) ≤ l, (3.2) and ζt α−1 γ (α) ≤ m, (3.3) then a(cm (j,l)) ⊂ cm (j,l). proof. for ϕ ∈ cm (j,l), we get |(aϕ)(t)| ≤ 1 γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ ds. but ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ = ∣ ∣ ∣ f ( s,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) − f (s,0,0, . . . ,0) + f (s,0,0, . . . ,0) ∣ ∣ ∣ ≤ ∣ ∣ ∣ f ( s,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) − f (s,0,0, . . . ,0) ∣ ∣ ∣ + |f (s,0,0, . . . ,0)| ≤ ρ + n ∑ i=1 ci i−1 ∑ j=0 mj ‖ϕ‖ ≤ ρ + l n ∑ i=1 ci i−1 ∑ j=0 mj = ζ, 88 a. guerfi & a. ardjouni cubo 24, 1 (2022) then |(aϕ)(t)| ≤ ζ γ (α) ∫ t 0 (t − s) α−1 ds ≤ ζt α γ (α + 1) ≤ l. from (3.2), we have 0 ≤ (aϕ)(t) ≤ |(aϕ)(t)| ≤ l. let t1, t2 ∈ j with t1 < t2, we have |(aϕ) (t2) − (aϕ) (t1)| ≤ 1 γ (α) ∫ t1 0 ∣ ∣ ∣ (t2 − s) α−1 − (t1 − s) α−1 ∣ ∣ ∣ ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ ds + 1 γ (α) ∫ t2 t1 (t2 − s) α−1 ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ ds ≤ ζ γ (α) ( ∫ t1 0 ( (t2 − s) α−1 − (t1 − s) α−1 ) ds + ∫ t2 t1 (t2 − s) α−1 ds ) ≤ ζ γ (α + 1) (tα2 − t α 1 ) ≤ ζt α−1 γ (α) |t2 − t1| . using (3.3), we obtain |(aϕ) (t2) − (aϕ) (t1)| ≤ m |t2 − t1| . therefore, aϕ ∈ cm (j,l) for all ϕ ∈ cm (j,l). so, we conclude that a(cm (j,l)) ⊂ cm (j,l). theorem 3.3. suppose that conditions (2.1), (3.2) and (3.3) hold. then (1.1) has at least one mild solution x in cm (j,l). proof. from lemma 2.5, the problem (1.1) has a mild solution x on cm (j,l) if and only if the operator a defined by (3.1) has a fixed point. from lemmas 3.1 and 3.2, all conditions of the schauder fixed point theorem are satisfied. consequently, a has at least one fixed point on cm (j,l) which is a mild solution of (1.1). theorem 3.4. in addition to the assumptions of theorem 3.3, if we suppose that t α γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 mj < 1, (3.4) then (1.1) has a unique mild solution in cm (j,l). proof. let ϕ and ψ be two distinct fixed points of the operator a. similarly as in the proof of lemma 3.1 we have |ϕ(t) − ψ (t)| = |(aϕ) (t) − (aψ) (t)| ≤ t α γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 mj ‖ϕ − ψ‖ . cubo 24, 1 (2022) existence, uniqueness, continuous dependence and ulam stability... 89 it follows from (3.4) that ‖ϕ − ψ‖ < ‖ϕ − ψ‖ . therefore, we arrive at a contradiction. we conclude that a has a unique fixed point which is the unique mild solution of (1.1). theorem 3.5. suppose that the conditions of theorem 3.4 hold. the unique mild solution of (1.1) depends continuously on the function f. proof. let f1,f2 : j × r n → [0,+∞) two continuous functions with respect to their arguments. from theorem 3.4, it follows that there exist two unique corresponding functions x1 and x2 in cm (j,l) such that x1 (t) = 1 γ (α) ∫ t 0 (t − s) α−1 f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ds, and x2 (t) = 1 γ (α) ∫ t 0 (t − s) α−1 f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) ds. we get |x2 (t) − x1 (t)| ≤ 1 γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ ds. but ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ = ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) − f2 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) + f2 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ . using (2.1) and lemma 2.6, we arrive at ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ ≤ ‖f2 − f1‖ + n ∑ i=1 ci i−1 ∑ j=0 mj ‖x2 − x1‖ . hence ‖x2 − x1‖ ≤ t α γ (α + 1) ‖f2 − f1‖ + t α γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 mj ‖x2 − x1‖ . therefore ‖x2 − x1‖ ≤ t α γ(α+1) 1 − t α γ(α+1) n ∑ i=1 ci i−1 ∑ j=0 mj ‖f2 − f1‖ . this completes the proof. 90 a. guerfi & a. ardjouni cubo 24, 1 (2022) now, we investigate the ulam-hyers stability and generalized ulam-hyers stability for the problem (1.1). definition 3.6 ([18]). the problem (1.1) is said to be ulam-hyers stable if there exists a real number kf > 0 such that for each ǫ > 0 and for each mild solution y ∈ cm (j,l) of the inequality ∣ ∣ ∣ cdα0+y (t) − f ( y[0] (t) ,y[1] (t) ,y[2] (t) , . . . ,y[n] (t) ) ∣ ∣ ∣ ≤ ǫ, t ∈ j, (3.5) with y (0) = y′ (0) = 0, there exists a mild solution x ∈ cm (j,l) of the problem (1.1) with |y (t) − x(t)| ≤ kfǫ, t ∈ j. definition 3.7 ([18]). the problem (1.1) is generalized ulam-hyers stable if there exists ψ ∈ c (j,r+) with ψ (0) = 0 such that for each ǫ > 0 and for each mild solution y ∈ cm (j,l) of the inequality (3.5) with y (0) = y′ (0) = 0, there exists a mild solution x ∈ cm (j,l) of the problem (1.1) with |y (t) − x(t)| ≤ ψ (ǫ) , t ∈ j. theorem 3.8. assume that the assumptions of theorem 3.4 hold. then the problem (1.1) is ulam-hyers stable. proof. let y ∈ cm (j,l) be a mild solution of the inequality (3.5) with y (0) = y ′ (0) = 0, i.e.    ∣ ∣ cdα 0+ y (t) − f ( y[0] (t) ,y[1] (t) ,y[2] (t) , . . . ,y[n] (t) ) ∣ ∣ ≤ ǫ, t ∈ j, y (0) = y′ (0) = 0. (3.6) let us denote by x ∈ cm (j,l) the unique mild solution of the problem (1.1). by using lemma 2.5, we get x(t) = 1 γ (α) ∫ t 0 (t − s) α−1 f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ds, t ∈ j. by integration of (3.6), we have ∣ ∣ ∣ ∣ y (t) − 1 γ (α) ∫ t 0 (t − s) α−1 f ( y[0] (s) ,y[1] (s) ,y[2] (s) , . . . ,y[n] (s) ) ds ∣ ∣ ∣ ∣ ≤ tα γ (α + 1) ǫ ≤ t α γ (α + 1) ǫ. on the other hand, we obtain, for each t ∈ j |y (t) − x(t)| = ∣ ∣ ∣ ∣ y (t) − 1 γ (α) ∫ t 0 (t − s) α−1 f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ds ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ y (t) − 1 γ (α) ∫ t 0 (t − s) α−1 f ( y[0] (s) ,y[1] (s) ,y[2] (s) , . . . ,y[n] (s) ) ds ∣ ∣ ∣ ∣ + 1 γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f ( y[0] (s) ,y[1] (s) ,y[2] (s) , . . . ,y[n] (s) ) −f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ∣ ∣ ∣ ds ≤ t α γ (α + 1) ǫ + t α γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 mj ‖y − x‖ . cubo 24, 1 (2022) existence, uniqueness, continuous dependence and ulam stability... 91 thus, in view of (3.4) ‖y − x‖ ≤ t α γ(α+1) 1 − t α γ(α+1) n ∑ i=1 ci i−1 ∑ j=0 mj ǫ. then, there exists a real number kf = t α/ ( γ (α + 1) − t α n ∑ i=1 ci i−1 ∑ j=0 mj ) > 0 such that |y (t) − x(t)| ≤ kfǫ, t ∈ j. (3.7) thus, the problem (1.1) is ulam-hyers stable, which completes the proof. corollary 3.9. suppose that all the assumptions of theorem 3.8 are satisfied. then the problem (1.1) is generalized ulam-hyers stable. proof. let ψ (ǫ) = kfǫ in (3.7) then ψ (0) = 0 and the problem (1.1) is generalized ulam-hyers stable. example 3.10. let us consider the following nonlinear fractional initial value problem    cd 3 2 0+ x(t) = 1 4 + 1 4 cost + 1 18 cos2 (t)x[1] (t) + 1 19 sin2 (t) x[2] (t) , t ∈ [0,1] , x(0) = x′ (0) = 0, (3.8) where t = 1, j = [0,1] and f (t,x,y) = 1 4 + 1 4 cost + 1 18 xcos2 (t) + 1 19 y sin2 (t) . we have |f (t,x1,x2) − f (t,y1,y2)| ≤ 1 18 |x1 − y1| + 1 19 |x2 − y2| , then |f (t,x1,x2) − f (t,y1,y2)| ≤ 2 ∑ i=1 ci ‖xi − yi‖ . with c1 = 1 18 , c2 = 1 19 . furthermore, if l = 1 and m = 4 in the definition of cm (j,l), then f is positive, ρ = sup t∈j {f (t,0,0)} = 1 2 and ζ = 0.5 + ( 1 18 + 4 19 ) ≃ 0.766. for α = 3 2 , we get ζt α γ (α + 1) = 0.766 γ ( 5 2 ) ≃ 0.576 ≤ l = 1, and ζt α−1 γ (α) = 0.766 γ ( 3 2 ) ≃ 0.864 ≤ m = 4. so, t α γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 mj = 1 γ ( 5 2 ) ( 1 18 + 4 19 ) ≃ 0.2 < 1. then, by theorems 3.4 and 3.5, (3.8) has a unique mild solution which depends continuously on the function f. also, from theorem 3.8, (3.8) is ulam-hyers stable, and from corollary 3.9, (3.8) is generalized ulam-hyers stable. 92 a. guerfi & a. ardjouni cubo 24, 1 (2022) 4 conclusion in the current paper, under some sufficient conditions on the nonlinearity, we established the existence, uniqueness, continuous dependence and ulam stability of a mild solution for an iterative caputo fractional differential equation. the main tool of this work is the schauder fixed point theorem. the obtained results have a contribution to the related literature. acknowledgement the authors would like to thank the anonymous referee for his/her valuable comments and good advice. cubo 24, 1 (2022) existence, uniqueness, continuous dependence and ulam stability... 93 references [1] s. abbas, “existence of solutions to fractional order ordinary and delay differential equations and applications”, electron. j. differential equations, no. 9, 11 pages, 2011. 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[19] h. y. zhao and j. liu, “periodic solutions of an iterative functional differential equation with variable coefficients”, math. methods appl. sci., vol. 40, no. 1, pp. 286–292, 2017. introduction preliminaries main results conclusion cubo, a mathematical journal vol. 23, no. 03, pp. 357–368, december 2021 doi: 10.4067/s0719-06462021000300357 basic asymptotic estimates for powers of wallis’ ratios vito lampret1 1 university of ljubljana, ljubljana, 386 slovenia, eu. vito.lampret@guest.arnes.si abstract for any a ∈ r, for every n ∈ n, and for n-th wallis’ ratio wn := ∏n k=1 2k−1 2k , the relative error r0(a, n) := ( v0(a, n) − wan ) /wan of the approximation w a n ≈ v0(a, n) := (πn) −a/2 is estimated as ∣∣r0(a, n)∣∣ < 14n . the improvement wan ≈ v(a, n) := (πn)−a/2 ( 1 − a 8n + a 2 128n2 ) is also studied. resumen para cualquier a ∈ r, para todo n ∈ n, y para el nésimo cociente de wallis wn := ∏n k=1 2k−1 2k , el error relativo r0(a, n) := ( v0(a, n) − w a n ) /wan de la aproximación wan ≈ v0(a, n) := (πn) −a/2 se estima como ∣∣r0(a, n)∣∣ < 1 4n . también se estima la mejora wan ≈ v(a, n) := (πn)−a/2 ( 1 − a 8n + a 2 128n2 ) . keywords and phrases: approximation, asymptotic, estimate, inequality, power, wallis’ ratio. 2020 ams mathematics subject classification: 41a60, 65b10, 11y60, 33e20, 33f05, 40a25. accepted: 09 july, 2021 received: 16 january, 2021 ©2021 v. lampret. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300357 https://orcid.org/0000-0002-2921-3451 358 vito lampret cubo 23, 3 (2021) 1 introduction the sequence of wallis1 ratios wn := n∏ k=1 2k − 1 2k = (2n − 1)!! (2n)!! = 4−n ( 2n n ) (1.1) is often encountered in pure and applied mathematics, in physics, and in several other exact sciences too. for example, the perimeter p(a, b) of an ellipse having semi-axes of length a and b ≤ a is given as p(a, b) = 4a ∫ π/2 0 √ 1 − ε2 sin2(τ) dτ = 2aπ ( 1 − ∞∑ k=1 w2k 2k − 1 ε2k ) (1.2) [20], where ε = √ 1 − b2 a2 , the eccentricity of an ellipse. similarly, the period t of a simple pendulum, located in the gravitational field with the acceleration g and having the length l and the amplitude of the oscillation α ∈ (0, π), is determined by the formula t = 4 √ l g ∫ π/2 0 dτ√ 1 − ε2 sin2(τ) = 2π √ l g ( 1 + ∞∑ k=1 w2kε 2k ) (1.3) [21, p. 26], where ε = sin(α/2). not only in mechanics, but also in other parts of physics, the wallis ratio has several interesting roles, see for example [9] and [12]. in mathematics, the sequence of the landau constants gn, important in the theory of analytic functions, see [1], is also defined by the wallis ratios as gn := n∑ k=1 w2k (n ∈ n). (1.4) the wallis ratio attracts mathematicians also because of its direct connections with catalan numbers cn := 1 n+1 ( 2n n ) , also important objects for pure and applied mathematics [15, 29]. in fact, the wallis ratio, i.e. the sequence n 7→ wn, was investigated by many researches, see for example the papers [2, 3, 4, 5, 6, 7, 8, 11, 14, 16, 22, 23, 26, 27, 28, 29, 31, 33]. in 2007 was presented [33] aesthetically pleasing double inequality 1 √ eπn ( 1 + 1 2n )n− 1 12n < wn ≤ 1 √ eπn ( 1 + 1 2n )n− 1 12n+16 , (1.5) true for all n ∈ n. in 2013 was demonstrated [10] the estimate√ e π ( 1 − 1 2n )n √ n − 1 n < wn ≤ 4 3 ( 1 − 1 2n )n √ n − 1 n , (1.6) 1john wallis, 1616 – 1703 cubo 23, 3 (2021) basic asymptotic estimates for powers of wallis’ ratios 359 true for n ≥ 2. in 2015 was derived [11] the inequalities( 2 3 )3/2 ( 1 − 1 2n )n+1/2 ( n − 3 2 )−1/2 ≤ wn < √ e π ( 1 − 1 2n )n+1/2 ( n − 3 2 )−1/2 , (1.7) valid for n ≥ 2. at the same time, in [28, theorems 4.2 and 5.2] were presented the estimates wn > √ e πn ( 1 − 1 2n )n exp ( 1 24n2 + 1 48n3 + 1 160n4 + 1 960n5 ) (1.8) wn > √ e πn ( 1 − 1 2(n + 1/3) )n+1/3 (1.9) and wn < √ e πn ( 1 − 1 2(n + 1/3) )n+1/3 exp ( 1 144n3 ) , (1.10) all true for n ≥ 1. in the mentioned formulas for the perimeter of an ellipse and the period of a simple pendulum, as well as for the landau sequence, see (1.2)–(1.4), we met the second powers of the wallis ratios. this fact initiated our desire to approximate any power of the wallis ratio. but, all the inequalities (1.5)–(1.10) are less suitable for estimating the power wan for a ∈ r. fortunately, the approximation formula for the wallis ratio, presented in [19], is more convenient for this task. in this contribution we shall show the first two steps how to approximate simply and accurately the powers of the wallis ratios having real exponents. 2 basic discussion the sequence of wallis’ ratios was estimated recently [19] as wn = 1 √ π n exp ( − s̃r(n) + δr(n) ) (n ∈ n), (2.1) where s̃r(n) = r∑ i=1 (1 − 4−i)b2i i(2i − 1)n2i−1 (n, r ∈ n) (2.2) and, for any n, r ∈ n, the error δr(n) is estimated as − ∣∣b2r+2∣∣ (r + 1)(2r + 1)n2r+1 < (−1)rδr(n) < ∣∣b2r+2∣∣ 2(r + 1)(2r + 1)(2n)2r+1 . (2.3) here b2 = 1 6 , b4 = − 130 , b6 = 1 42 , . . . are the bernoulli numbers, defined by the identity x ex−1 ≡ ∑∞ j=0 bj xj j! ( |x| < 2π ). we obtain the basic approximation by using r = 1, wan = (πn) −a/2 exp ( − as̃1(n) + aδ1(n) ) (a ∈ r, n ∈ n), (2.4) 360 vito lampret cubo 23, 3 (2021) with, for n ∈ n, s̃1(n) := 1 8n > 0 (2.5) and − 1 180 n3 < − 1 2880 n3 < δ1(n) < 1 180 n3 . (2.6) thus, due to (2.5), as̃1(n) = a 8n (a ∈ r, n ∈ n) . (2.7) moreover, thanks to (2.5)–(2.6), we estimate, for n ∈ n, −s̃1(n) ± ∣∣δ1(n)∣∣ ≥ −s̃1(n) − ∣∣δ1(n)∣∣ > − 1 8n − 1 180n3 > − 1 7n (2.8) and −s̃1(n) ± ∣∣δ1(n)∣∣ ≤ −s̃1(n) + ∣∣δ1(n)∣∣ < − 1 8n + 1 180n3 < − 1 9n . (2.9) therefore, − a 7n < a ( − s̃1(n) ± δ1(n) ) < − a 9n , for a > 0 and − a 7n > a ( − s̃1(n) ± δ1(n) ) > − a 9n , for a < 0. thus, min { − a 7n , − a 9n } < a ( − s̃1(n) ± ∣∣δ1(n)∣∣) < max{− a7n, − a9n} (a ̸= 0, n ∈ n). (2.10) hence, considering (2.4), together with the equality min(−s) = − max(s), for every s ⊆ r, we derive the following theorem. theorem 2.1. for a ∈ r ∖ {0} and n ∈ n, the following double inequality holds: (πn)−a/2 exp ( − max { a 7n , a 9n }) < wan < (πn) −a/2 exp ( − min { a 7n , a 9n }) . (2.11) figure 1 shows2 the graphs of the function a 7→ wa2 and its approximation (dashed line) a 7→ (π · 2)−a/2. -2 -1 1 2 1 2 3 4 5 6 7 n=2 -0.10 -0.05 0.05 0.10 0.95 1.00 1.05 1.10 n=2 figure 1: the graphs of the function a 7→ wa2 and its approximation (dashed line) a 7→ (π · 2)−a/2. 2all graphics and calculations in this paper are made using the mathematica [32] computer system. cubo 23, 3 (2021) basic asymptotic estimates for powers of wallis’ ratios 361 example 2.2. for any n ∈ n we have (πn)−50 exp ( − 100 7n ) < w100n < (πn) −50 exp ( − 100 9n ) , (πn)50 exp ( 100 9n ) < w−100n < (πn) 50 exp ( 100 7n ) . from theorem 2.1 there follows the next corollary. corollary 2.3. for every a ∈ r ∖ {0} and for any positive integer n ≥ a we have wan > 6 7 (πn)−a/2 . (2.12) proof. for k ≥ a > 0, using (2.11), we obtain3 wak > (πk) −a/2 exp ( − a 7k ) > (πk)−a/2(1 − a 7k ) ≥ ( πk)−a/2(1 − 1 7 ) = 6 7 (πk)−a/2. furthermore, for a < 0, due to (2.11), we estimate wak > (πk)−a/2 exp ( − a 9k ) = (πk)−a/2 exp ( |a| 9k ) > (πk)−a/2 · 1. lemma 2.4. let real numbers α, β, v and w satisfy the inequalities αβ ≥ 0, β ≤ 1 2 , v > 0 and eαv < w < eβv. then we have |v − w| < 3 2 v · max{|α|, |β|}. proof. supposing that all conditions of lemma 2.4 are satisfied, we have only two possibilities α < β ≤ 0 or 0 ≤ α < β, together with the estimate (eα − 1)v < w − v < (eβ − 1)v. therefore, in case α ≤ 0, we have (1 − eα)v > v − w > (1 − eβ)v ≥ 0. thus, see footnote 3, |v−w| < −α = |α|. additionally, using the first order taylor’s formula and the estimate 0 ≤ β ≤ 1 2 , in case α ≥ 0, we obtain, 0 ≤ (eα − 1)v < −(v − w) < (eβ − 1)v < β + 1 2 eββ2 ≤ β + 1 2 e1/2 1 2 β < 3 2 β. hence, in both cases we have |v − w| < v · max{|α|, 3 2 |β|}. corollary 2.5 (relative error). for every a ∈ r ∖ {0} and for any positive integer n ≥ a the relative error r0(a, n) := ( wan − v0(a, n) ) /wan of the approximation w a n ≈ v0(a, n) := (πn)−a/2 is roughly estimated as ∣∣r0(a, n)∣∣ < 1 4n . proof. thanks to theorem 2.1 and lemma 2.4, using the notations α = − max { a 7n , a 9n } , β = − min { a 7n , a 9n } , v = v0(a, n) = (πn) −a/2 and w = wan, we obtain∣∣v0(a, n) − wan∣∣ < 32(πn)−a/2 · max {∣∣max{ a 7n , a 9n }∣∣ , ∣∣min{ a 7n , a 9n }∣∣}. thus, according to the identity max { |max {x, y}| , |min {x, y}| } = max { |x|, |y| } , we get ∣∣v0(a, n) − wan∣∣ < 32(πn)−a/2 · |a|7n . 3considering the well-known estimate ex > 1 + x, true for x ∈ r ∖ {0}. 362 vito lampret cubo 23, 3 (2021) hence, using corollary 2.3,∣∣v0(a, n) − wan∣∣ wan < 3 2 (πn)−a/2 |a| 7n · 7 6 (πn)a/2 = |a| 4n . figure 2 shows, on the left – the graph of the actual relative error function a 7→ r0(a, n) and on the right – the graphs of the functions a 7→ r0(a, n) and a 7→ |a| 4×1000 (dashed line). -10 000 -5000 -2000 2000 4000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 a b b=r0ha,1000l -8 -6 -4 -2 2 4 6 8 0.0005 0.0010 0.0015 0.0020 a b b=r0ha,1000l b= a 4´1000 figure 2: left – the graph of the actual relative error function a 7→ r0(a, 1000); right – the graphs of the actual relative error a 7→ r0(a, 1000) and its approximation (dashed line) a 7→ |a| 4×1000 . 3 improvement the relations (2.4)–(2.6) can be exploited more accurately to derive the next theorem. theorem 3.1. for any a ∈ r and every integer n ≥ |a| 8 , we have wan = v(a, n) + ε(a, n), (3.1) where v(a, n) := (πn)−a/2 ( 1 − a 8n + a2 128n2 ) , (3.2) and the error ε(a, n) is estimated as ∣∣ε(a, n)∣∣ ≤ ε∗(a, n) := (πn)−a/2 [ a2 100 + 1 18 exp ( − min { a 7n , a 9n })] |a| 10n3 (3.3) ≤ (πn)−a/2 ( a2 100 + 1 18 exp (|a| 7n )) |a| 10n3 ≤ ε∗∗(a, n) := (πn)−a/2 ( a2 + 35 2 ) |a| (10n)3 . (3.4) proof. using the second order taylor’s formula, we have exp ( − as̃1(n) ) = exp ( − a 8n ) = 1 − a 8n + 1 2 ( − a 8n )2 + r2(a, n) (3.5) cubo 23, 3 (2021) basic asymptotic estimates for powers of wallis’ ratios 363 with r2(a, n) = 1 6 exp ( −ϑ · a 8n )( − a 8n )3 , for some ϑ = ϑ(a, n) ∈ (0, 1). therefore, for a ∈ r and n ≥ |a| 8 , ∣∣r2(a, n)∣∣ ≤ 1 6 exp ( |a| 8n ) · ( |a| 8n )3 ≤ e 6 · |a|3 512n3 ≤ |a|3 1000n3 . (3.6) similarly, exp ( aδ1(n) ) = 1 + exp ( ϑ · aδ1(n) ) · aδ1(n), (3.7) for some ϑ = ϑ(a, n) ∈ (0, 1). thanks to (3.7), (2.10) and (2.6), we estimate, using some θ = θ(a, n) ∈ (0, 1),∣∣∣exp( − as̃1(n) + aδ1(n)) − exp( − as̃1(n))︸︷︷︸ =∆(a,n) ∣∣∣ = exp( − as̃1(n)) · ∣∣exp(θ · aδ1(n)) · aδ1(n)∣∣ ≤ exp ( − as̃1(n) ) · exp ( |aδ1(n)| ) · ∣∣aδ1(n)∣∣ = exp ( a ( − s̃1(n) ± |δ1(n)| )) |a| ∣∣δ1(n)∣∣ (2.10) ≤ (2.6) exp ( max { − a 7n , − a 9n }) · |a| 180n3 . (3.8) consequently, according to (2.4) and (3.5), we obtain wan (2.4) = (πk)−a/2 ( exp ( −as̃1(n)+aδ1(n) )) (3.5) = (πn)−a/2 ( 1 − a 8n + a2 128n2 + r2(a, n)︸︷︷︸ =exp(−as̃1(n)) +∆(a, n) ) , where, considering (3.6) and (3.8), for a ∈ r and n ≥ |a| 8 , we estimate the error ε(a, n) := (πn)−a/2 ( r2(a, n) + ∆(a, n) ) as ∣∣ε(a, n)∣∣ ≤ (πn)−a/2 [ |a|3 1000n3 + exp ( − min { a 7n , a 9n }) |a| 180n3 ] = (πn)−a/2 [ a2 100 + 1 18 exp ( − min { a 7n , a 9n })] |a| 10n3 ≤ (πn)−a/2 [ a2 100 + 1 18 exp (|a| 8n · 8 7 )] |a| 10n3 ≤ (πn)−a/2 ( a2 100 + 1 18 exp ( 1 · 8 7 )) |a| 10n3 ≤ (πn)−a/2 ( a2 100 + 7 40 ) |a| 10n3 . remark 3.2. the sequence n 7→ wn := 12n+1 (∏n k=1 2k 2k−1 )2 , called the wallis sequence, is closely connected to the sequence of the wallis ratios wn by the identity wn = w −2 n /(2n + 1). so, wn can be estimated easily using theorem 3.1, e.g. its consequence (3.14). 364 vito lampret cubo 23, 3 (2021) remark 3.3. according to theorem 3.1, the constant π can be easily approximated using certain rational functions r∓(n). for example, from (3.14) we get, for any n ∈ n, 1 n ( w−2n − 1 5n2 )( 1 + 1 4n + 1 32n2 )−1 < π < 1 n ( w−2n + 1 5n2 )( 1 + 1 4n + 1 32n2 )−1 . directly from theorem 3.1 and corollary 2.3, from (3.4) and (2.12), we read the next corollary. corollary 3.4 (relative error). for every a ∈ r and for any positive integer n ≥ |a| the relative error of the approximation wan ≈ v(a, n), r(a, n) := wan − v(a, n) wan , (3.9) is a priori estimated as ∣∣r(a, n)∣∣ ≤ r∗(a, n) := (a2 + 13 2 ) 7|a| 6(10n)3 . (3.10) for any a ∈ r and all integers n ≥ |a| the rough estimate r∗(a, n) < 8.2 h holds true. figure 3 shows the graphs of the actual relative error functions a 7→ r(a, n), for n ∈ {10, 100}. -10 -5 5 10 0.0001 0.0002 0.0003 a b b = rha,10l -100 -50 50 100 0.0001 0.0002 0.0003 a b b = rha,100l figure 3: the graphs of the actual relative error functions a 7→ r(a, n) for n ∈ {10, 100} . figures 4–5 compare the actual relative error functions a 7→ r(a, n) and their approximations a 7→ r∗(a, n), for n ∈ {1, 3, 10, 100}. -1.0 -0.5 0.5 1.0 0.002 0.004 0.006 0.008 n=1 a b b = rha,1l b = r * ha,1l -3 -2 -1 1 2 3 0.0005 0.0010 0.0015 0.0020 n=3 a b b = rha,3l b = r * ha,3l figure 4: the graphs of the actual relative error functions a 7→ r(a, n) and their approximations a 7→ r∗(a, n), for n ∈ {1, 3} . cubo 23, 3 (2021) basic asymptotic estimates for powers of wallis’ ratios 365 -10 -5 5 10 0.0002 0.0004 0.0006 0.0008 n=10 a b = rha,10l b = r * ha,10l -100 -50 50 100 0.0002 0.0004 0.0006 0.0008 0.0010 n=100 a b b = rha,100l b = r * ha,100l figure 5: the graphs of the actual relative error functions a 7→ r(a, n) and their approximations a 7→ r∗(a, n), for n ∈ {10, 100} . using a ∈ {1, −1, 2, −2, 1 2 , π, −2π} in theorem 3.1, considering (3.1) and (3.4), we obtain several inequalities for wallis’ ratios, presented in the next corollary. corollary 3.5. for every4 positive integer n we have 1 √ π n ( 1 − 1 8n + 1 128n2 ) − 1 95 n7/2 < wn < 1 √ π n ( 1 − 1 8n + 1 128n2 ) + 1 95 n7/2 , (3.11) √ π n ( 1 + 1 8n + 1 128n2 ) − 1 30 n5/2 < 1 wn < √ π n ( 1 + 1 8n + 1 128n2 ) + 1 30 n5/2 , (3.12) 1 π n ( 1 − 1 4n + 1 32n2 ) − 1 73 n4 < w2n < 1 π n ( 1 − 1 4n + 1 32n2 ) + 1 73 n4 , (3.13) (π n) ( 1 + 1 4n + 1 32n2 ) − 1 7 n2 < 1 w2n < (π n) ( 1 + 1 4n + 1 32n2 ) + 1 7 n2 , (3.14) 1 4 √ π n ( 1 − 1 16n + 1 512n2 ) − 1 150n13/4 < √ wn < 1 4 √ π n ( 1 − 1 16n + 1 512n2 ) + 1 150n13/4 , (3.15) 1 (π n)π/2 ( 1 − π 8n + π 2 128n2 ) − 1 70n3+π/2 < wπn < 1 (π n)π/2 ( 1 − π 8n + π 2 128n2 ) + 1 70n3+π/2 , (3.16) (π n)π ( 1 + π 4n + π 2 32n2 ) − 14 nπ−3 < w−2πn < (π n) π ( 1 + π 4n + π 2 32n2 ) + 14 nπ−3 . 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[33] x.-m. zhang, t. q. xu and l. b. situ “geometric convexity of a function involving gamma function and application to inequality theory”, jipam. j. inequal. pure appl. math., vol. 8, no. 1, art. 17, 9 pages, 2007. introduction basic discussion improvement cubo, a mathematical journal vol. 23, no. 03, pp. 441–455, december 2021 doi: 10.4067/s0719-06462021000300441 existence and uniqueness of solutions to discrete, third-order three-point boundary value problems saleh s. almuthaybiri1,2 jagan mohan jonnalagadda3 christopher c. tisdell1 1 school of mathematics and statistics, the university of new south wales, unsw, sydney, nsw, 2052, australia. cct@unsw.edu.au 2 department of mathematics, college of science and arts in uglat asugour, qassim university, buraydah, kingdom of saudi arabia. s.almuthaybiri@qu.edu.sa 3 department of mathematics, birla institute of technology and science pilani, hyderabad, 500078, telangana, india. j.jaganmohan@hotmail.com abstract the purpose of this article is to move towards a more complete understanding of the qualitative properties of solutions to discrete boundary value problems. in particular, we introduce and develop sufficient conditions under which the existence of a unique solution for a third-order difference equation subject to three-point boundary conditions is guaranteed. our contributions are realized in the following ways. first, we construct the corresponding green’s function for the problem and formulate some new bounds on its summation. second, we apply these properties to the boundary value problem by drawing on banach’s fixed point theorem in conjunction with interesting metrics and appropriate inequalities. we discuss several examples to illustrate the nature of our advancements. resumen el propósito de este artículo es avanzar hacia un entendimiento más completo de las propiedades cualitativas de las soluciones a problemas discretos de valor en la frontera. en particular, introducimos y desarrollamos condiciones suficientes bajo las cuales se garantiza la existencia de una única solución para una ecuación en diferencias de tercer orden sujeta a condiciones de borde en tres puntos. nuestras contribuciones son de dos tipos. en primer lugar, construimos las funciones de green correspondientes para el problema y formulamos nuevas cotas para su suma. en segundo lugar, aplicamos estas propiedades al problema de valor en la frontera usando el teorema del punto fijo de banach junto con métricas interesantes y desigualdades apropiadas. discutimos varios ejemplos para ilustrar la naturaleza de nuestros avances. keywords and phrases: forward difference, boundary value problem, green’s function, contraction, fixed point, existence, uniqueness. 2020 ams mathematics subject classification: 39a12. accepted: 14 september, 2021 received: 09 january, 2021 ©saleh s. almuthaybiri et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300441 https://orcid.org/0000-0002-9399-3253 https://orcid.org/0000-0002-1310-8323 https://orcid.org/0000-0002-3387-2505 442 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) 1 introduction discrete boundary value problems are of significant interest to scientific and technical communities. for instance, their perceived utility is partly due to their ability to act as a mathematical framework to model purely discrete processes and phenomena that arise in various fields of science and engineering. in addition, developing a theory of discrete boundary value problems has the potential to inform our understanding of continuous boundary value problems. for example, discrete boundary value problems can arise as approximations to “continuous” boundary value problems that involve differential equations, where the numerical aspects of solutions are of importance. furthermore, it is also possible to construct a theory of differential equations by only using difference equations [7]. although discrete problems have enjoyed continued interest, the mathematics community is yet to reach a complete understanding of the qualitative and quantitative properties of their solutions. this includes, for example, discrete boundary value problems of the third order, which have not been advanced to the same degree as their “continuous cousins” or to the same extent as discrete problems of the second order. moreover, we are yet to achieve a total comprehension of the mathematical similarities and distinctions between such continuous and discrete problems. motivated by the above discussion, the purpose of the current paper is to make progress towards a more complete theory concerning the existence and uniqueness of solutions to discrete boundary value problems of the third order. “knowing an equation has a unique solution is important from both a modelling and theoretical point of view” [19, p. 794] as it informs our mathematical understanding from applied and pure perspectives. for example, by developing a deeper understanding of the existence and uniqueness of solutions to discrete boundary value problems we are simultaneously expanding capacity and knowledge of the associated models and the mathematical frameworks that attempt to describe them. for any a, b ∈ r such that (b − a) ∈ n, we will denote na = {a, a + 1, a + 2, . . . } and nba = {a, a + 1, a + 2, . . . , b}. let ∆ denote the usual forward difference operator defined by( ∆u ) (t) = u(t + 1) − u(t), t ∈ nt +20 , herein we will consider the following third-order, three-point discrete boundary value problem  ( ∆3u ) (t − 2) + f(t, u(t)) = 0, t ∈ nt +22 , u(0) = ( ∆u ) (0) = 0, u(t + 3) = ku(η) (1.1) where f is a continuous function from nt +30 × r to r which we denote via f ∈ c [ nt +30 × r, r ] . in addition, t ∈ n1, k ∈ r and η ∈ nt +21 . let us briefly outline recent and relevant literature to situate and contextualize our work. agarwal and henderson [2] initiated the study of positive solutions to the third-order three-point discrete cubo 23, 3 (2021) existence and uniqueness of solutions to discrete.... 443 boundary value problem  ( ∆3u ) (t − 2) + a(t)g(u(t)) = 0, t ∈ nt +22 , u(0) = u(1) = u(t + 3) = 0, (1.2) where a : nt +20 → r + and g ∈ c [r+, r+]. following this work, anderson [5] and anderson and avery [6] examined the existence of multiple solutions to third-order, three-point discrete focal boundary value problems. positive solutions to discrete, third-order problems have been shown to exist using fixed point theory in cones [13]. in addition, several authors have discussed various qualitative properties of different classes of third-order three-point discrete boundary value problems and a detailed discussion can be found in [11, 12, 23, 24, 25, 13] and the references therein. motivated by the recent work [4, 15], where the differential equation version of (1.1) was analyzed, in the present article we investigate the discrete boundary value problem (1.1). when compared with the ideas in [4, 15] our methods and results herein are different; and they reveal some thoughtprovoking distinctions and connections between the sets of works. for example, the present work develops alternative bounds on the green’s functions to those in [4, 15] and we employ purely discrete ways of working. in particular, we observe that some of our bounds for the discrete case are sharp, while others are rougher. the bounds are different from those developed for the continuous case [4]. this highlights some of the interesting distinctions between the discrete and the continuous in terms of results and methods within the domain of third order problems. our article is organized as follows: in section 2, we construct the green’s function corresponding to the boundary value problem (1.1) and establish new bounds on its summation. in section 3 we apply the properties of the green’s function to the boundary value problem (1.1) in conjunction with banach’s contraction mapping theorem to establish sufficient conditions for the existence of a unique solution. we provide a discussion of examples in section 4 to illustrate how our ideas can be put into practice and the relationships between them. finally, we conclude with some ideas for further work in section 5. for more on discrete problems, see the monographs [1, 8, 9, 10, 14]. 2 green’s function and its properties in order to develop the green’s function for the three-point case, we first analyze the two-point discrete boundary value problem  ( ∆3v ) (t − 2) + h(t) = 0, t ∈ nt +22 , v(0) = ( ∆v ) (0) = v(t + 3) = 0, (2.1) 444 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) where h ∈ c [ nt +22 , r ] . the boundary value problem (2.1) can be equivalently rewritten as  ( ∆3v ) (t − 2) + h(t) = 0, t ∈ nt +22 , v(0) = v(1) = v(t + 3) = 0. (2.2) yang and weng [25] derived a green’s function for the boundary value problem (2.2) and also investigated its sign. the following two results are found therein and will be helpful in our present analysis. lemma 2.1 ([25]). the unique solution of the boundary value problem (2.2) (or (2.1)) is given by v(t) = t +2∑ s=2 h(t, s)h(s), t ∈ nt +30 , (2.3) where h(t, s) =   t(t − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) − (t − s)(t − s + 1) 2 , s ∈ nt−10 , t(t − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) , s ∈ nt +3t . (2.4) lemma 2.2 ([25]). the green’s function h(t, s) in (2.4) satisfies h(t, s) ≥ 0 for all (t, s) ∈ nt +30 × n t +2 2 . now let us construct the green’s function for the boundary value problem  ( ∆3u ) (t − 2) + h(t) = 0, t ∈ nt +22 , u(0) = ( ∆u ) (0) = 0, u(t + 3) = ku(η) (2.5) to form the following new result. lemma 2.3. let h ∈ c [ nt +22 , r ] and assume (t + 2)(t + 3) ̸= kη(η − 1). the unique solution to the boundary value problem (2.5) is given by u(t) = t +2∑ s=2 g(t, s)h(s), t ∈ nt +30 , (2.6) where g(t, s) = h(t, s) + kt(t − 1) (t + 2)(t + 3) − kη(η − 1) h(η, s). (2.7) proof. assume the solution of the boundary value problem (2.5) can be expressed as u(t) = v(t) + [c0 + c1t + c2t(t − 1)] v(η), (2.8) cubo 23, 3 (2021) existence and uniqueness of solutions to discrete.... 445 where c0, c1 and c2 are constants to be determined and v be the unique solution of the boundary value problem (2.1). when v(η) = 0, the u(t) defined by (2.8) is the same as v(t) and it is actually the solution of (2.5). in what follows, we assume that v(η) ̸= 0. it follows from (2.8) that ( ∆u ) (t) = ( ∆v ) (t) + [c1 + 2c2t] v(η). (2.9) from (2.1), (2.5), (2.8) and (2.9), we have u(0) = 0 ⇒ v(0) + c0v(η) = 0 ⇒ c0 = 0, (2.10)( ∆u ) (0) = 0 ⇒ ( ∆v ) (0) + c1v(η) = 0 ⇒ c1 = 0, (2.11) and u(t + 3) = ku(η) ⇒ v(t + 3) + c2(t + 2)(t + 3)v(η) = k [v(η) + c2η(η − 1)v(η)] ⇒ c2 = k (t + 2)(t + 3) − kη(η − 1) . (2.12) using (2.3) and (2.10) – (2.12) in (2.8) and rearranging the terms, we obtain (2.6) and (2.7). the proof is complete. let us now establish new bounds on the summation of the green’s functions h(t, s) and g(t, s) via the following result, which is of interest in its own right, for example, the bounds may prove useful in areas beyond the scope of this paper, such as in the application of topological ways of working with fixed point theory. we will draw on it to establish the main existence and uniqueness results of section 3. the bounds will be formulated in terms of t, k, η. to assist with notation, we define the following constant λ (that depends on the form of t ) that we will use below. for n ∈ n1 we define λ =   n(n + 2)(2n + 1) 3 , if t = 3n, n(n + 1)(2n + 1) 3 , if t = 3n − 1, n(n + 1)(2n − 1) 3 , if t = 3n − 2. (2.13) lemma 2.4. the green’s function g(t, s) in (2.7) satisfies t +2∑ s=2 |g(t, s)| ≤ γ, where γ depends on t, k, η and is explicitly given by γ = λ + ∣∣∣∣ k(t + 2)(t + 3) − kη(η − 1) ∣∣∣∣(t + 2)(t + 3) [ η(η − 1)(t + 7) 6 + η ] (2.14) and λ is defined in (2.13). 446 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) proof. consider t +2∑ s=2 h(t, s) = t−1∑ s=2 [ t(t − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) − (t − s)(t − s + 1) 2 ] + t +2∑ s=t [ t(t − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) ] = t +2∑ s=2 [ t(t − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) ] − t−1∑ s=2 [ (t − s)(t − s + 1) 2 ] = t(t − 1) 2(t + 3)(t + 2) t +1∑ s=1 s(s + 1) − 1 2 t−2∑ s=1 s(s + 1) = t(t − 1) 2(t + 3)(t + 2) [ t +1∑ s=1 s2 + t +1∑ s=1 s ] − 1 2 [ t−2∑ s=1 s2 + t−2∑ s=1 s ] = t(t − 1) 2(t + 3)(t + 2) [ (t + 1)(t + 2)(2t + 3) 6 + (t + 1)(t + 2) 2 ] − 1 2 [ (t − 2)(t − 1)(2t − 3) 6 + (t − 2)(t − 1) 2 ] = t(t − 1)(t + 1) − t(t − 1)(t − 2) 6 = t(t − 1)(t + 3 − t) 6 . clearly, t +2∑ s=2 h(0, s) = t +2∑ s=2 h(1, s) = 0. now we wish to maximize t +2∑ s=2 h(t, s) for t ∈ nt +32 . denote by g(t) = t(t − 1)(t + 3 − t) 6 , t ∈ nt +32 . the first forward difference of g with respect to t is given by( ∆g ) (t) = t (2t − 3t + 5) 6 . in this expression, the term t/6 is positive for all t ∈ nt +32 . the equation 2t − 3t + 5 = 0 has the solution t = 2t +5 3 , so we consider t = ⌊2t +5 3 ⌋ ∈ nt +32 . if t ≤ ⌊ 2t +5 3 ⌋, the difference 2t − 3t + 5 is positive, and thus g is increasing. if t > ⌊2t +5 3 ⌋, the quantity 2t − 3t + 5 is negative, and thus g is decreasing. hence, the maximum value of g occurs at t = ⌊2t +5 3 ⌋. we observe that, for n ∈ n1, ⌊ 2t + 5 3 ⌋ =   2n + 1, if t = 3n, 2n + 1, if t = 3n − 1, 2n, if t = 3n − 2. therefore, max t∈nt +30 t +2∑ s=2 h(t, s) = max t∈nt +32 g(t) = g (⌊ 2t + 5 3 ⌋) = λ. (2.15) cubo 23, 3 (2021) existence and uniqueness of solutions to discrete.... 447 consider t +2∑ s=2 h(η, s) = t−1∑ s=2 [ η(η − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) − (η − s)(η − s + 1) 2 ] + t +2∑ s=t [ η(η − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) ] = t +2∑ s=2 [ η(η − 1)(t + 3 − s)(t + 4 − s) 2(t + 3)(t + 2) ] − t−1∑ s=2 [ (η − s)(η − s + 1) 2 ] = η(η − 1) 2(t + 3)(t + 2) t +1∑ s=1 s(s + 1) − 1 2 t−2∑ s=1 (η − s)(η − s − 1) = η(η − 1) 2(t + 3)(t + 2) [ t +1∑ s=1 s2 + t +1∑ s=1 s ] − 1 2 [ η(η − 1) t−2∑ s=1 1 − (2η − 1) t−2∑ s=1 s + t−2∑ s=1 s2 ] = η(η − 1) 2(t + 3)(t + 2) [ (t + 1)(t + 2)(2t + 3) 6 + (t + 1)(t + 2) 2 ] − 1 2 [ η(η − 1)(t − 2) − (2η − 1) (t − 2)(t − 1) 2 + (t − 2)(t − 1)(2t − 3) 6 ] = η(η − 1)(t − 3t + 7) + (3η − t)(t − 1)(t − 2) 6 . now we wish to maximize t +2∑ s=2 h(η, s) for t ∈ nt +30 . denote by h(t) = η(η − 1)(t − 3t + 7) + (3η − t)(t − 1)(t − 2) 6 , t ∈ nt +30 . the first forward difference of h with respect to t is given by ( ∆h ) (t) = − t2 − (2η + 1)t + η(η + 1) 2 . we observe that ( ∆h ) (t)   < 0, for t ∈ nη−10 , = 0, for t = η, = 0, for t = η + 1, < 0, for t ∈ nt +3η+2 , implying that max t∈nt +30 h(t) = h(0) = η(η − 1)(t + 7) 6 + η. that is, max t∈nt +30 t +2∑ s=2 h(η, s) = η(η − 1)(t + 7) 6 + η. 448 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) now, consider t +2∑ s=2 |g(t, s)| = t +2∑ s=2 ∣∣∣∣h(t, s) + kt(t − 1)(t + 2)(t + 3) − kη(η − 1)h(η, s) ∣∣∣∣ ≤ t +2∑ s=2 |h(t, s)| + ∣∣∣∣ k(t + 2)(t + 3) − kη(η − 1) ∣∣∣∣t(t − 1) t +2∑ s=2 |h(η, s)| = t +2∑ s=2 h(t, s) + ∣∣∣∣ k(t + 2)(t + 3) − kη(η − 1) ∣∣∣∣t(t − 1) t +2∑ s=2 h(η, s) ≤ λ + ∣∣∣∣ k(t + 2)(t + 3) − kη(η − 1) ∣∣∣∣(t + 2)(t + 3) [ η(η − 1)(t + 7) 6 + η ] = γ. the proof is complete. remark 2.5. from the proof of lemma 2.4 we see that (2.15) implies that the bound λ on t +2∑ s=2 h(t, s), t ∈ nt +30 is sharp therein. if we compare this sharp bound with the sharp bound for the integral of the corresponding green’s function in the continuous case of (t + 3)3/81 in [4] then we see the bounds between the discrete and continuous cases are different. this is partly due to the differing forms of the green’s function for the discrete and continuous problems. however, it is possible to establish a connection between the two theories by forming a new bound that is common to both problems simply by choosing the larger of the two bounds. the price to pay for this unity in this situation is that one of the bounds will no longer be sharp. thus we see a trade-off between unification and sharpness in this situation. 3 application of banach’s theorem in this section we establish sufficient conditions on the existence of a unique solution for the boundary value problem (1.1) using banach’s fixed point theorem. “the field of fixed point theory aims to establish conditions under which certain classes of problems will admit one, or more, fixed points [21, 20]” [16, c16]. first let us recall the statement of this theorem. theorem 3.1 ([3]). let (x, d) be a complete metric space and t : x → x be a contraction mapping, that is, there is an α, 0 ≤ α < 1, such that d(tx, ty) ≤ αd(x, y), for all x, y in x. then t has a unique fixed point z in x, that is, tz = z. cubo 23, 3 (2021) existence and uniqueness of solutions to discrete.... 449 every solution of the boundary value problem (1.1) can be treated as a (t + 4)-tuple real vector. denote the set x = rt +4 and consider the following metrics defined on x: d(u, v) = max t∈nt +30 |u(t) − v(t)| , δ(u, v) = ( t +3∑ t=0 |u(t) − v(t)|p )1 p , p > 1, for all u, v ∈ x. the pair (x, d) forms a complete metric space, and the pair (x, δ) also forms a complete metric space. define the operator t : x → x by ( tu ) (t) = t +2∑ s=2 g(t, s)f(s, u(s)), t ∈ nt +30 . note that u is a solution of the boundary value problem (1.1) if and only if u is a fixed point of t . we apply theorem 3.1 to show that t has a unique fixed point in x with the ideas manifested in the following two new theorems. theorem 3.2. let f ∈ c [ nt +30 × r, r ] , let f(t, 0) ̸= 0 for all t ∈ nt +30 , let (t + 2)(t + 3) ̸= kη(η − 1) and let γ be defined in (2.14). if f satisfies a lipschitz condition with respect to the second variable on nt +30 × r with lipschitz constant k, that is, there is a nonnegative constant k, such that |f(t, x) − f(t, y)| ≤ k|x − y|, for all t ∈ nt +30 and all x, y ∈ r and kγ < 1, (3.1) then the boundary value problem (1.1) has a unique nontrivial solution. proof. for u, v ∈ x and t ∈ nt +30 , consider ∣∣(tu)(t) − (tv)(t)∣∣ = ∣∣∣∣∣ t +2∑ s=2 g(t, s)f(s, u(s)) − t +2∑ s=2 g(t, s)f(s, v(s)) ∣∣∣∣∣ ≤ t +2∑ s=2 |g(t, s)| |f(s, u(s)) − f(s, v(s))| ≤ k t +2∑ s=2 |g(t, s)| |u(s) − v(s)| ≤ kd(u, v) t +2∑ s=2 |g(t, s)| ≤ kγd(u, v), implying that d(tu, tv) ≤ αd(u, v), 450 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) where α = kγ < 1. thus, t is a contraction mapping on x. hence, by theorem 3.1, our t has a unique fixed point in x. this is equivalent to the boundary value problem (1.1) admitting a unique nontrivial solution. the proof is complete. the following result sharpens the inequality (3.1) in theorem 3.2 through the strategic use of a different metric. theorem 3.3. let the conditions of theorem 3.2 hold, with the assumption (3.1) removed. if there are constants p > 1 and q > 1 such that 1/p + 1/q = 1 and k  t +3∑ t=0 ( t +2∑ s=2 |g(t, s)|q )p q   1 p < 1, (3.2) then the boundary value problem (1.1) has a unique nontrivial solution. proof. we apply theorem 3.1 to show that t has a unique fixed point in x where x is defined in the proof of theorem 3.2 but is now coupled with the metric δ(u, v) := ( t +3∑ t=0 |u(t) − v(t)|p )1 p . consider ∣∣(tu)(t) − (tv)(t)∣∣ = ∣∣∣∣∣ t +2∑ s=2 g(t, s)f(s, u(s)) − t +2∑ s=2 g(t, s)f(s, v(s)) ∣∣∣∣∣ ≤ t +2∑ s=2 |g(t, s)| |f(s, u(s)) − f(s, v(s))| ≤ k t +2∑ s=2 |g(t, s)| |u(s) − v(s)| . (3.3) by holder’s inequality, we have t +2∑ s=2 |g(t, s)| |u(s) − v(s)| ≤ ( t +2∑ s=2 |u(s) − v(s)|p )1 p ( t +2∑ s=2 |g(t, s)|q )1 q . (3.4) thus, ∣∣(tu)(t) − (tv)(t)∣∣ ≤ k ( t +2∑ s=2 |u(s) − v(s)|p )1 p ( t +2∑ s=2 |g(t, s)|q )1 q (3.5) ≤ k ( t +2∑ s=2 |g(t, s)|q )1 q δ(u, v) and so, we have ( t +3∑ t=0 ∣∣(tu)(t) − (tv)(t)∣∣p )1 p ≤ k  t +3∑ t=0 ( t +2∑ s=2 |g(t, s)|q )p q   1 p δ(u, v), cubo 23, 3 (2021) existence and uniqueness of solutions to discrete.... 451 implying that δ(tu, tv) ≤ γδ(u, v), where γ = k  t +3∑ t=0 ( t +2∑ s=2 |g(t, s)|q )p q   1 p < 1. thus, the conditions of theorem 3.1 hold. hence, by theorem 3.1, our t has a unique fixed point in x. this is equivalent to the boundary value problem (1.1) furnishing a unique nontrivial solution. the proof is complete. for the choices p = q = 2, theorem 3.3 takes the following form: theorem 3.4. let the conditions of theorem 3.2 hold, with the assumption (3.1) removed. if k ( t +3∑ t=0 ( t +2∑ s=2 |g(t, s)|2 ))1 2 < 1, (3.6) then the boundary value problem (1.1) has a unique nontrivial solution. 4 discussion of examples let us discuss two examples to illustrate the nature of our new theorems and the relationship between them. example 4.1. consider the following discrete boundary value problem  ( ∆3u ) (t − 2) + 1 150 cos(u(t)) = 0, t ∈ n112 , u(0) = ( ∆u ) (0) = 0, u(12) = u(6). (4.1) we claim that this problem admits a unique solution. proof. observe that (4.1) is a special case of (1.1) with t = 9, k = 1, η = 6 and f(t, u) = f(u) = (cos(u))/150. we show that the conditions of theorem 3.2 are satisfied. since t is a multiple of 3 we have n = 3 and so λ = 35. furthermore, appropriate calculations reveal γ ≈ 146.294 < 147. our f satisfies a lipschitz condition due to the property that its derivative with respect to u is uniformly bounded by 1/150 and we may choose this bound to be the lipschitz constant, that is, on r we have |∂f/∂u| = | − sin(u)|/150 ≤ 1/150 = k. 452 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) finally, we see that kγ < 147/150 < 1. thus, all of the conditions of theorem 3.2 hold and we conclude that the discrete boundary value problem (4.1) admits a unique solution. let us now discuss an example that illustrates theorem 3.3 and its distinction from theorem 3.2. example 4.2. consider the following discrete boundary value problem  ( ∆3u ) (t − 2) + 1 54 tan−1(u(t)) + t2 + 1 = 0, t ∈ n112 , u(0) = ( ∆u ) (0) = 0, u(12) = u(6). (4.2) we claim that this problem admits a unique solution. proof. observe that (4.2) is a special case of (1.1) with t = 9, k = 1, η = 6 and f(t, u) = (tan−1(u))/54 + t2 + 1. we show that the conditions of theorem 3.3 are satisfied with p = 2 = q, that is, theorem 3.4 will hold. appropriate calculations using maple reveal( t +3∑ t=0 ( t +2∑ s=2 |g(t, s)|2 ))1 2 ≈ 52.3839 < 53. our f satisfies a lipschitz condition due to the property that its derivative with respect to u is uniformly bounded by 1/54 and we may choose this bound to be the lipschitz constant, that is, on r we have |∂f/∂u| = |1/(54(u2 + 1))| ≤ 1/54 = k. finally, we see that (3.6) holds since k ( t +3∑ t=0 ( t +2∑ s=2 |g(t, s)|2 ))1 2 < 53/54 < 1. thus, all of the conditions of theorem 3.3 hold with p = 2 = q (that is, theorem 3.4 holds) and we conclude that the discrete boundary value problem (4.2) admits a unique solution. remark 4.3. we note that theorem 3.2 cannot be directly applied to the boundary value problem (4.2) in example 4.2. the reason is because the condition kγ < 1 is not satisfied in this situation. thus, we observe that theorem 3.4 is more general than theorem 3.2. 5 concluding remarks and further work this paper deepened our understanding of the existence and uniqueness of solutions to discrete boundary value problems. we showed that a larger class of problems admitted a unique solution cubo 23, 3 (2021) existence and uniqueness of solutions to discrete.... 453 and achieved this by drawing on fixed-point theory and the use of new bounds. our results add to the recent literature on discrete boundary value problems and difference equations [17, 18] and move us closer to a more complete understanding of the underlying theory and application. although our bound on the summation of h(t, s) herein is sharp, the corresponding bound involving g(t, s) remains rough and a natural question for further work is: can this bound be sharpened? one of the main limitations with many fixed point theorems is the very nature of their assumptions. because sufficient conditions are involved, it may be the case that the conditions of these theorems do not hold, yet the problem under consideration does actually admit a unique solution (or solutions). thus it is important to also look beyond these types of sufficient assumptions and the development of new methods and altenative perspectives in mathematics are needed [21, 22] to advance the associated existence and uniqueness theory. 454 s. s. almuthaybiri, j. m. jonnalagadda & c. c. tisdell cubo 23, 3 (2021) references [1] r. p. agarwal, difference equations and inequalities. theory, methods, and applications, second edition, monographs and textbooks in pure and applied mathematics, vol. 228. new york: marcel dekker, 2000. 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[25] ch. yang and p. weng, “green functions and positive solutions for boundary value problems of third-order difference equations”, comput. math. appl., vol. 54, no. 4, pp. 567–578, 2007. introduction green's function and its properties application of banach's theorem discussion of examples concluding remarks and further work cubo, a mathematical journal vol. 23, no. 03, pp. 385–409, december 2021 doi: 10.4067/s0719-06462021000300385 entropy solution for a nonlinear parabolic problem with homogeneous neumann boundary condition involving variable exponents u. traoré1 1 laboratoire de mathématiques et informatique (lami), université joseph ki-zerbo 03, bp 7021 ouaga 03, ouagadougou, burkina faso. urbain.traore@yahoo.fr abstract in this paper we prove the existence and uniqueness of an entropy solution for a non-linear parabolic equation with homogeneous neumann boundary condition and initial data in l1. by a time discretization technique we analyze the existence, uniqueness and stability questions. the functional setting involves lebesgue and sobolev spaces with variable exponents. resumen en este artículo probamos la existencia y unicidad de una solución de entropía para una ecuación parabólica no lineal con condiciones de borde neumann homogéneas y data inicial en l1. usando una técnica de discretización del tiempo, analizamos las preguntas de existencia, unicidad y estabilidad. el contexto funcional involucra espacios de lebesgue y sobolev con exponentes variables. keywords and phrases: nonlinear parabolic problem, variable exponents, entropy solution, neumann-type boundary conditions, semi-discretization. 2020 ams mathematics subject classification: 35k55, 35k61, 35j60, 35dxx. accepted: 12 august, 2021 received: 19 december, 2020 ©2021 u. traoré. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300385 https://orcid.org/0000-0002-9729-4724 386 u. traoré cubo 23, 3 (2021) 1 introduction and main result let ω be a smooth bounded open domain of rd, (d ≥ 3) with lipschitz boundary ∂ω, t is a fixed positive number, in this paper we study the existence and uniqueness of an entropy solution for the following nonlinear parabolic problem (p)   ∂u ∂t − diva(x,∇u) + b(u) = f in qt = ]0,t[×ω, a(x,∇u) .η = 0 on ∑ t = ]0,t[×∂ω, u(0, .) = u0 in ω, where f ∈ l1(qt ), b : r → r, a(x,ξ) : ω × rd → r is carathéodory function and verifying some assumptions which will be given later, η denotes the unit vector normal on ∂ω. the usual weak formulations of parabolic problems in the case where the initial data are in l1 do not ensure existence and uniqueness of solutions. for this reason, new formulations and types of solutions are given in order to obtain existence and uniqueness. for that, three notions of solution have been adopted: solutions named sola (solution obtained as the limit of approximations) defined by a. dall’aglio (see [10]); renormalized solutions defined by r. diperna and p.-l. lions (see [12]); and entropy solutions defined by ph. bénilan et al. in [8]. in this paper, we will be interested in the entropy formulation. the stationary version of the problem for the problem (p) has been already studied by bonzi et al. (cf. [9]), where they proved the existence and uniqueness of an entropy solution for the initial data in l1. the study of parabolic equations with variable exponents is a very active field (see [1, 2, 20, 21, 23, 27, 29]), in these papers, the authors consider the homogeneous dirichlet boundary conditions, which permit them to use many results in the generalized sobolev space w1,p(.)(ω) and the many results concerned the differential equation in the literature to achieve there works. in particular in the case of p(x)-laplace, where b ≡ 0, bendahmane et al. (see [6]) have proved the existence and uniqueness of renormalized solution. we can also point out that the well-posedness of triply nonlinear degenerate ellipticparabolic-hyperbolic problems: b(u)t − diva(x,∇ϕ(u)) +ψ(u) = f in a bounded domain with homogeneous dirichlet boundary conditions by k. h. karlsen et al. in [3]. unfortunately, in this paper, due to the neumann boundary condition, we cannot use the ideas developed in these papers and also some functional analysis results which play and important role in the a priori estimation, in particular the famous poincaré inequality. to overcome these difficulties we apply a time discretization of given continuous problem by the euler forward scheme. let’s recall that this method has been used in the literature for the study cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 387 of some nonlinear parabolic problems, we refer for example to [7, 13, 16, 17] for some details. this scheme is usually used to prove existence of solutions as well as to compute numerical approximations. in this paper, our assumptions are the following:  p(.) : ω → r is a continuous function such that1 < p− ≤ p+ < +∞, (1.1) where p− := ess inf x∈ω p(x) and p+ := ess sup x∈ω p(x) and b : ω → r is a continuous, nondecreasing function, surjective such that b(0) = 0. (1.2) also, we assume that a(x,ξ) : ω × rn → rn is carathéodory such that: • there exists a positive constant c1 with |a(x,ξ)| ≤ c1 ( j(x) + |ξ|p(x)−1 ) (1.3) for almost every x ∈ ω and for every ξ ∈ rn, where j is a nonnegative function in lp ′(.)(ω) with 1 p(x) + 1 p′(x) = 1; • there exists a positive constant c2 such that for every x ∈ ω and every ξ1, ξ2 ∈ rd with ξ1 ̸= ξ2, the following two inequalities hold (a(x,ξ1) − a(x,ξ2)) .(ξ1 − ξ2) > 0 (1.4) a(x,ξ) .ξ ≥ c2|ξ|p(x). (1.5) the rest of the paper is organized as follows: after some preliminary results in section 2, we introduce the euler forward scheme associated with the problem (p) in section 3. we analyze the stability of the discretized problem and we study the existence of an entropy solution to the parabolic problem (p) in the section 4. 2 preliminaries we define the lebesgue space with variable exponent lp(.) (ω) (see [11]) as the set of all measurable functions u : ω → r for which the convex modular ρp(.) (u) := ∫ ω |u|p(x) dx is finite. if the exponent is bounded, i.e., if p+ < +∞, then the expression ∥u∥p(.) := inf { λ > 0 : ρp(.) (u/λ) ≤ 1 } 388 u. traoré cubo 23, 3 (2021) defines a norm in lp(.) (ω) , called the luxembourg norm. the space ( lp(.) (ω) ,∥.∥p(.) ) is a separable banach space. moreover, if 1 < p− ≤ p+ < +∞, then lp(.) (ω) is uniformly convex, hence reflexive and its dual space is isomorphic to lp ′(.) (ω) , where 1 p(x) + 1 p′ (x) = 1. finally, we have the hölder type inequality∣∣∣∣ ∫ ω uvdx ∣∣∣∣ ≤ ( 1 p− + 1 (p−)′ ) ∥u∥p(.) ∥v∥p′(.) , (2.1) for all u ∈ lp(.) (ω) and v ∈ lp ′(.) (ω) . let w1,p(.) (ω) := { u ∈ lp(.) (ω) : |∇u| ∈ lp(.) (ω) } , which is banach space equipped with the following norm ∥u∥1,p(.) := ∥u∥p(.) + ∥∇u∥p(.) . the space ( w1,p(.) (ω) ,∥.∥1,p(.) ) is a separable and reflexive banach space. an important role in manipulating the generalized lebesgue and sobolev spaces is played by the modular ρp(.) of the space l p(.) (ω) . we have the following result. proposition 2.1 (see [14, 28]). if un,u ∈ lp(.) (ω) and p+ < ∞, the following properties hold true: (i) ∥u∥p(.) > 1 ⇒ ∥u∥ p− p(.) < ρp(.) (u) < ∥u∥ p+ p(.) ; (ii) ∥u∥p(.) < 1 ⇒ ∥u∥ p+ p(.) < ρp(.) (u) < ∥u∥ p− p(.) ; (iii) ∥u∥p(.) < 1 (respectively = 1;> 1) ⇔ ρp(.) (u) < 1 (respectively = 1;> 1) ; (iv) ∥un∥p(.) → 0 (respectively → +∞) ⇔ ρp(.) (un) < 1 (respectively → +∞) ; (v) ρp(.) ( u/∥u∥p(.) ) = 1. for a measurable function u : ω → r we introduce the following notation: ρ1,p(.) (u) = ∫ ω |u|p(x) dx + ∫ ω |∇u|p(x) dx. proposition 2.2 (see [25, 26]). if u ∈ w1,p(.) (ω) , the following properties hold true: (i) ∥u∥1,p(.) > 1 ⇒ ∥u∥ p− 1,p(.) < ρ1,p(.) (u) < ∥u∥ p+ 1,p(.) ; (ii) ∥u∥1,p(.) < 1 ⇒ ∥u∥ p+ 1,p(.) < ρp(.) (u) < ∥u∥ p− 1,p(.) ; (iii) ∥u∥1,p(.) < 1 (respectively = 1;> 1) ⇔ ρ1,p(.) (u) < 1 (respectively = 1;> 1) . cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 389 put p∂ (x) := (p(x)) ∂ =   (n − 1)p(x) n − p(x) , if p(x) < n ∞, if p(x) ≥ n. proposition 2.3 (see [26]). let p ∈ c ( ω̄ ) and p− > 1. if q ∈ c (∂ω) satisfies the condition 1 < q (x) < p∂ (x) ∀x ∈ ∂ω, then, there is a compact embedding w1,p(.) (ω) ↪→ lq(.) (∂ω) . in particular, there is a compact embedding w1,p(.) (ω) ↪→ lp(.) (∂ω) . following [29], we extend a variable exponent p : ω → [1,+∞) to qt = [0,t] × ω by setting p(t,x) = p(x) for all (t,x) ∈ qt . we may also consider the generalized lebesgue space lp(.) (q) = { u : q → r measurable such that ∫∫ q |u(t,x)|p(x) d(t,x) < ∞ } endowed with the norm ∥u∥lp(.)(qt ) := inf { λ > 0, ∫∫ qt ∣∣∣∣u(t,x)λ ∣∣∣∣p(x) d(t,x) < 1 } , which share the same properties as lp(.) (ω) . for a measurable set u in rd, meas(u) denotes its measure, ci and c will denote various positive constants. for a banach space x and a < b, lq(a,b;x) is the space of measurable functions u : [a,b] → x such that (∫ b a ∥u∥qx dt )1 q := ∥u∥lq(a,b;x) < ∞. (2.2) for a given constant k > 0 we define the cut-off function tk : r → r by tk(s) :=   s if |s| ≤ kk sign(s) if |s| > k with sign(s) :=   1 if s > 0 0 if s = 0 −1 if s < 0. let jk : r → r+ defined by jk(x) = ∫ x 0 tk(s)ds (jk is a primitive of tk). we have (see [15])〈 ∂v ∂t ,tk(s) 〉 = d dt (∫ ω jk(v)dx ) in l1(]0,t[) 390 u. traoré cubo 23, 3 (2021) which implies that ∫ t 0 〈 ∂v ∂t ,tk(s) 〉 = ∫ ω j(v(t))dx − ∫ ω j(v(0))dx for all u ∈ w1,p(.) (ω) we denote by τ (u) the trace of u on ∂ω in the usual sense. in the sequel, we will identify at the boundary, u and τ (u) . set t 1,p(.) (ω) = { u : ω → r, measurable such that tk (u) ∈ w1,p(.) (ω) , for any k > 0 } . proposition 2.4 (see [8]). let u ∈ t 1,p(.) (ω) . then there exists a unique measurable function v : ω → rn such that ∇tk (u) = vχ{|u| 0. the function v is denoted by ∇u. moreover, if u ∈ w1,p(.) (ω) then v ∈ ( lp(.) (ω) )n and v = ∇u in the usual sense. we denote by t 1,p(.)tr (ω) (cf. [4, 5, 18, 19]) the set of functions u ∈ t 1,p(.) (ω) such that there exists a sequence (un)n∈n ⊂ w 1,p(.) (ω) satisfying the following conditions: i) un → u a.e. in ω. ii) ∇tk (un) → ∇tk (u) in ( l1 (ω) )n for any k > 0. iii) there exists a measurable function v on ∂ω, such that un → v a.e. on ∂ω. the function v is the trace of u in the generalized sense introduced in [4, 5]. in the sequel, the trace of u ∈ t 1,p(.)tr (ω) on ∂ω will be denoted by tr (u) . if u ∈ w1,p(.) (ω) , tr (u) coincides with τ (u) in the usual sense. moreover u ∈ t 1,p(.)tr (ω) and for every k > 0, τ (tk (u)) = tk (tr (u)) and if φ ∈ w1,p(.) (ω) ∩ l∞ (ω) then (u − φ) ∈ t 1,p(.)tr (ω) and tr (u − φ) = tr (u) − tr (φ) . 3 the semi-discrete problem in this section, we study the euler forward scheme associated with the problem (p): (pn)   un − τdiv a(x,∇un) + τb(un) = τfn + un−1 in ω a(x,∇un) .η = 0 on ∂ω, u0 = u0 in ω where nτ = t, 0 < τ < 1, 1 ≤ n ≤ n and fn(.) = 1 τ ∫ nτ (n−1)τ f(s, .)ds in ω. cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 391 definition 3.1. an entropy solution to the discretized problems (pn) is a sequence (un)0≤n≤n such that u0 = u0 ∈ l1 (ω) and un is defined by induction as an entropy solution to the problem  un − τdiv a(x,∇un) + τb(un) = τfn + un−1 in ω a(x,∇un) .η = 0 on ∂ω i.e. un ∈ t 1,p(.)tr (ω), b(un) ∈ l1(ω), and for every k > 0 τ ∫ ω a(x,∇un).∇tk(un −φ)dx+ ∫ ω (τb(un) +un)tk(u n −φ)dx ≤ ∫ ω (τfn +u n−1)tk(u n −φ)dx (3.1) for all φ ∈ w1,p(.)(ω) ∩ l∞(ω). we have the following result lemma 3.2. let hypotheses (1.3) − (1.5) be satisfied. if (un)0≤n≤n is an entropy solution of problems (pn), then un ∈ l1(ω) for all n = 1, . . . ,n. proof. for n = 1, we take φ = 0 in (3.1), to get, τ ∫ ω a(x,∇u1).∇tk(u1)dx + ∫ ω (τb(u1) + u1)tk(u 1)dx ≤ ∫ ω (τf1 + u0)tk(u 1)dx, which is equivalent to τ ∫ ω a(x,∇tk(u1))∇tk(u1)dx + ∫ ω τb(u1)tk(u 1)dx + ∫ ω u1tk(u 1)dx ≤ ∫ ω (τf1 + u0)tk(u 1)dx, (3.2) by the assumption (1.5) and the properties of the function b, we have τ ∫ ω a(x,∇tk(u1))∇tk(u1)dx + ∫ ω τb(u1)tk(u 1)dx ≥ 0, then it follows that ∫ ω u1tk(u 1)dx ≤ kτ ∥f1∥1 + k ∥u0∥1 . since n∑ n=1 τ ∥fn∥1 ≤ ∥f∥1 . then, it follows that ∫ ω u1tk(u 1)dx ≤ k(∥f∥1 + ∥u0∥1). (3.3) since lim k→0 u1 tk(u 1) k = |u1|. then dividing (3.3) by k and letting k → 0; we deduce by fatou’s lemma that∥∥u1∥∥ 1 ≤ (∥f∥1 + ∥u0∥1) (3.4) 392 u. traoré cubo 23, 3 (2021) theorem 3.3. let hypotheses (1.3) − (1.5) be satisfied. then for all n ∈ n, the problems (pn) have unique entropy solution un ∈ t 1,p(.)tr (ω) ∩ l1(ω) for all n = 1, . . . ,n. proof. the problem (p1) can be rewritten in the following form −τdiva(x,∇u) + b(u) = f1 in ω a(x,∇u).η = 0 on ∂ω with b(s) := τb(s) + s, f1 := τf1 + u0. from the assumption (h2), we have f1 ∈ l1(ω), and using the properties of b, we obtain b is a continuous, nondecreasing function, surjective such that b(0) = 0. hence, using [9, theorem 4.3], we have the existence of unique entropy solution u1 ∈ t 1,p(.)tr (ω), b ( u1 ) ∈ l1(ω). thanks to lemma 3.2, by induction, we deduce that for n = 2, . . . ,n, the problem u − τdiva(x,∇u) + τα(u) = τfn + un−1 in ω a(x,∇u) .η = 0 on ∂ω, has an unique entropy solution un ∈ t 1,p(.)tr (ω) ∩ l1(ω), b(un) ∈ l1(ω). 4 stability this section is devoted to the a priori estimates for the discrete entropy solution (un)1≤n≤n. these result are essentials for the study of the convergence of the euler forward scheme. theorem 4.1. let hypotheses (1.3)−(1.5) be satisfied. then there exist positive constants c(u0,f), c(u0,f,ω) depending on the data but not on n such that for all n = 1, . . . ,n, we have 1. ∥un∥1 ≤ c(u0,f) 2. τ ∑n i=1 ∥∥b(ui)∥∥ 1 ≤ c(u0,f) 3. ∑n i=1 ∥∥ui − ui−1∥∥ 1 ≤ c(,u0,f) 4. τ ∑n i=1 ρp(.)(∇tk(u i)) ≤ kc(u0,f) 5. τ ∑n i=1 ∫ {|ui|≤k} |∇u i|p−dx ≤ kc(u0,f,ω) proof. 1 and 2. for φ = 0 as a test function in (3.1), we have τ k ∫ ω a(x,∇tk(ui))∇tk(ui)dx + ∫ ω ui tk(u i) k dx + ∫ ω τb(ui) tk(u i) k dx ≤ τ ∥fi∥1 + ∥∥ui−1∥∥ 1 dx. cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 393 since ∫ ω a(x,∇tk(ui))∇tk(ui)dx ≥ 0. then, it follows that∫ ω ui tk(u i) k dx + ∫ ω τb(ui) tk(u i) k dx ≤ τ ∥fi∥1 + ∥∥ui−1∥∥ 1 . then letting k → 0 and using fatou’s lemma, we deduce that ∥∥ui∥∥ 1 + τ ∥∥b(ui)∥∥ 1 ≤ τ ∥fi∥1 + ∥∥ui−1∥∥ 1 . (4.1) now, we sum (4.1) from i = 1 to n to obtain ∥un∥1 + τ n∑ i=1 ∥∥b(ui)∥∥ 1 ≤ ∥f∥1 + ∥u0∥1 (4.2) which give, the inequalities 1 and 2. 3. for k ≥ 1, we take φ = th(ui −sign(ui −ui−1)), (h > 1) as a test function in (3.1), then letting h → ∞, for k ≥ 1, we obtain, τ lim h→∞ i(k,h) + ∥∥ui − ui−1∥∥ 1 ≤ τ ( ∥fi∥1 + ∥∥b(ui)∥∥ 1 ) where i(k,h) := ∫ ω a(x,∇ui)∇tk(ui − th(ui − sign(ui − ui−1)))dx = ∫ ωk,h∩ω(k) a(x,∇ui)∇uidx and ωk,h := { |ui − th(ui − sign(ui − ui−1))| ≤ k } ω(k) = { |ui − sign(ui − ui−1)| > h } . then by the hypothesis (1.3) , we have lim h→∞ i(k,h) ≥ 0. then, it follows that ∥∥ui − ui−1∥∥ 1 ≤ kτ ( ∥fi∥1 + ∥∥b(ui)∥∥ 1 ) . (4.3) then, summing (4.3) from i = 1 to n and by the stability result 2, we obtain the stability result 3. 4. we take φ = 0 as a test function in 3.1 to get τ (∫ ω |a(x,∇tk(ui))∇tk(ui)dx ) ≤ kτ(∥fi∥1 + ∥∥b(ui)∥∥ 1 ) + k ∥∥ui − ui−1∥∥ 1 . 394 u. traoré cubo 23, 3 (2021) therefore, using the assumption (1.5) it follows that τρp(x)(∇tk(ui)) ≤ c3[kτ(∥fi∥1 + ∥∥b(ui)∥∥ 1 ) + k ∥∥ui − ui−1∥∥ 1 ]. (4.4) now, summing (4.4) from i = 1 to n and using the stability results 1, 2, 3, we get τ n∑ i=1 ρp(x)(∇tk(ui)) ≤ c3k [ ∥f∥1 + τ n∑ i=1 ∥∥b(ui)∥∥ 1 + n∑ i=1 ∥∥ui − ui−1∥∥ 1 ] ≤ kc(f,u0). (4.5) 5. according to (4.5), we get from the above estimate τ n∑ i=1 ∫ {|ui|≤k} |∇ui|p(x)dx ≤ kc(u0,f). (4.6) now, note that∫ {|ui|≤k} |∇ui|p−dx = ∫ {|ui|≤k, |∇ui|> 1n } |∇ui|p−dx + ∫ {|ui|≤k |∇ui|≤ 1n } |∇ui|p−dx ≤ ∫ {|ui|≤k, |∇ui|> 1n } |∇ui|p−dx + 1 n meas(ω) ≤ ∫ {|ui|≤k} |∇ui|p(x)dx + 1 n meas(ω). by the inequalities above, thanks to (4.6), we obtain τ n∑ i=1 ∫ {|ui|≤k} |∇ui|p−dx ≤ kc(u0,f) + n n meas(ω) ≤ kc(u0,f) + meas(ω) ≤ k(c(u0,f) + meas(ω)) (4.7) for all k ≥ 1. 5 convergence and existence result in this section, we prove the existence of an entropy solution of problem (p). first of all, we introduce the appropriate spaces for the entropy formulation of the nonlinear parabolic problem (p). we define the space: v = { v ∈ lp−(0,t;w1,p(·)(ω)) : ∇v ∈ (lp(·)(qt ))d } , and t 1,p(·)(qt ) = { u : ω × (0,t]; measurable | tk(u) ∈ lp−(0,t;w1,p(·)(ω)) with ∇tk(u) ∈ (lp(·)(qt ))d for every k > 0 } . cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 395 definition 5.1. an entropy solution to problem (p) is a function u ∈ t 1,p(·)(qt )∩c(0,t;l1(ω)) such that and for all k > 0 we have ∫ t 0 ∫ ω a(x,∇u)∇tk(u − φ) + ∫ t 0 ∫ ω b(u)tk(u − φ) ≤ − ∫ t 0 〈 ∂φ ∂s ,tk(u − φ) 〉 + ∫ ω jk(u(0) − φ(0)) − ∫ ω jk(u(t) − φ(t)) + ∫ t 0 ∫ ω ftk(u − φ) for all φ ∈ l∞(q) ∩ v ∩ w1,1(0,t;l1(ω)) and t ∈ [0,t]. our main result is theorem 5.2. let hypotheses (h1)−(h3) be satisfied. then the nonlinear parabolic problem (p) has an entropy solution. proof. the proof is divided into two steps step 1: the rothe function. we introduce a piecewise linear extension:   un(0) := u0, un(t) := un−1 + (un − un−1)t−t n−1 τ (5.1) for all t ∈]tn−1, tn], n = 1, · · · ,n, in ω and a piecewise constant function   un(0) := u0, un(t) := un, ∀t ∈]tn−1, tn], n = 1, · · · ,n, in ω, (5.2) where tn := nτ and (un)1≤n≤n an entropy solution of (pn). by theorem 3.3, for any n ∈ n; the solution (un)n∈n of problems (pn) is unique. thus, un and un are uniquely defined. consequently, by the theorem 4.1, we deduce the existence of a constant c(t,u0,f) not depending on n such that for all n ∈ n, we have ∥∥un − un∥∥ l1(qt ) ≤ 1 n c(t,u0,f)∥∥un∥∥ l1(qt ) ≤ c(t,u0,f)∥∥un∥∥ l1(qt ) ≤ c(t,u0,f) (5.3)∥∥∥∥∂un∂t ∥∥∥∥ l1(qt ) ≤ c(t,u0,f)∥∥b(un)∥∥ l1(qt ) ≤ c(t,u0,f) 396 u. traoré cubo 23, 3 (2021) moreover combining proposition 2.1 and young inequality, we get ∥∥∇tk(un)∥∥p−p(x) ≤ max { ρp(x)(∇tk(un)),ρ1,p(x)(∇tkun) p− p+ } ≤ ρp(x)(∇tk(un)) + ρ1,p(x)(∇tkun) p− p+ ≤ ρp(x)(∇tk(un)) + p− p+ ρp(x)(∇tk(un)) + 1 − p− p+ (5.4) ≤ 2ρp(x)(∇tk(un)) + 1. thanks to poincaré-wirtinger inequality, we have ∥∥tk(un)∥∥p(x) ≤ cmeas(ω)∥∥∇tk(un)∥∥p(x) + k ∥1∥p(x) , which implies that ∥∥tk(un)∥∥p−p(x) ≤ 2p−−1 ((cmeas(ω))p− ∥∥∇tk(un)∥∥p−p(x) + kp− ∥1∥p−p(x)) , (5.5) then it follows that, ∥∥tk(un)∥∥p−1,p(x) ≤ 2p−−1 [(cmeas(ω))p− (2ρp(x)(∇tk(un)) + 1) + kp− ∥1∥p−p(x)] (5.6) +2ρp(x)(∇tk(un)) + 1. therefore, ∫ t 0 ∥∥tk(un)∥∥p−1,p(.) dt ≤ 2p−−1 [ (cmeas(ω))p− ( 2 ∫ t 0 ρp(.)(∇tk(un))dt + t ) +tkp− ∥1∥p− p(x) n∑] + 2 ∫ t 0 ρp(.)(∇tk(un))dt + t ≤ 2p−−1 [ (cmeas(ω))p− ( 2 n∑ n=1 ∫ nτ (n−1)τ ρp(.)(∇tk(un))dt + t ) +tkp− ∥1∥p− p(.) n∑] + 2 n∑ n=1 ∫ nτ (n−1)τ ρp(.)(∇tk(un))dt + t (5.7) ≤ 2p−−1 [ (cmeas(ω))p− ( 2 n∑ n=1 τρp(.)(∇tk(un)) + t ) +tkp− ∥1∥p− p(.) n∑] + 2 n∑ n=1 τρ1,p(.)(tk(u n)) + t. consequently from stability result 4 it follows that ∥∥tk(un)∥∥lp− (0,t ;w 1,p(x)(ω)) ≤ c(t,k,u0,f,p−). (5.8) lemma 5.3. let hypotheses (1.3) − (1.5) be satisfied. then the sequence (un)n∈n converges in measure and a.e. in qt . cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 397 proof. let ε,r,k be positive numbers. for n,m ∈ n, we have the inclusion { |un − um| > r } ⊂ { |un| > k } ∪ { |um| > k } ∪ { |un| ≤ k, |um| ≤ k, |un − um| > r } . firstly, we have meas { |un| > k } ≤ 1 k ∥∥un∥∥ l1(qt ) ≤ 1 k c(t,u0,f). (5.9) similarly, we have meas { |um| > k } ≤ 1 k ∥∥un∥∥ l1(qt ) ≤ 1 k c(t,u0,f). (5.10) therefore, for k large enough, we have meas( { |um| > k } ∪ { |um| > k } ) ≤ ε 2 . (5.11) secondly, by the proposition 2.1 and young inequality, we have ∥∥∥∇tk(un )∥∥∥ lp(.)(qt ) ≤ max {(∫ t 0 ∫ ω |∇tk(un )|p(x)dxdt ) 1 p− ; (∫ t 0 ∫ ω |∇tk(un )|p(x)dxdt ) 1 p+ } ≤ (∫ t 0 ∫ ω |∇tk(un )|p(x)dxdt ) 1 p− + (∫ t 0 ∫ ω |∇tk(un )|p(x)dxdt ) 1 p+ and also, we have∫ t 0 ∫ ω |∇tk(un )|p(x)dxdt = ∫ t 0 ρp(.)(tk(∇u n )) = n∑ n=1 ∫ nτ (n−1)τ ρp(.)(∇tk(u n ))dt ≤ n∑ n=1 τρp(.)(∇tk(u n )). therefore, using the stability result 4 and proposition 2.1, it follows∥∥∥∇tk(un )∥∥∥ (lp(x)(qt )) d ≤ (kc(u0, f)) 1 p− + (kc(u0, f)) 1 p+ . (5.12) since by the poincaré-wirtinger inequality, we have∥∥∥tk(un )∥∥∥ lp(x)(qt ) ≤ cmeas(ω) ∥∥∥∇tk(un )∥∥∥ lp(x)(qt ) + k ∥1∥ lp(x)(qt ) , then by (5.12), we get∥∥∥tk(un )∥∥∥ lp(x)(qt ) ≤ cmeas(ω) ( (kc(u0, f)) 1 p− + (kc(u0, f)) 1 p+ ) + k ∥1∥ |lp(x)(qt ). (5.13) hence, the sequences (tk(un ))n∈n are bounded in lp(.)(qt ). then, there exists a subsequence, still denoted by (tk(un ))n∈n, that is a cauchy sequence in lp(.)(qt ) and in measure. thus, there exists n0 ∈ n such that for all n, m ≥ n0, we have meas ({ |un | ≤ k, |um | ≤ k, |un − um | > r }) < ε 2 . (5.14) then, by (5.11) and (5.14), (un )n∈n converges in measure. therefore there exists an element u ∈ m(qt ) such that u n → u a.e. in qt . 398 u. traoré cubo 23, 3 (2021) now, by (5.12) (∇tk(un))n∈n is uniformly bounded in, (lp(.)(qt ))d. (5.15) hence there exists a subsequence, still denoted by (∇tk(un))n∈n converges weakly to an element v in lp(.)(qt ). since tk(u n) converges weakly to tk(u) in lp(.)(qt ). then ∇tk(un) converges weakly to ∇tk(u) in (lp(.)(qt ))d. (5.16) and by (5.8) we conclude that tk(u) ∈ lp−(0,t;w1,p(.)(ω)) for all k > 0. in the sequel, we need the following lemma (see [22]). lemma 5.4. let (vn)n≥1 be a sequence of measurable functions in ω. if (vn)n≥1 converges in measure to v and is uniformly bounded in lp(.)(ω) for some 1 << p(.) ∈ l∞(ω), then (vn)n≥1 → v strongly in l1(ω). now, we have the following result lemma 5.5. let hypotheses (1.3) − (1.5) be satisfied. then (i) (∇tk(un))n∈n converges in measure to ∇tk(u); (ii) (a(x,tk(un)))n∈n converges strongly to a(x,∇tk(u)) in (l1(qt ))d and weakly in (lp ′(.) (qt )) d. proof. (i) let h ≥ 1, from the hölder type inequality, we have meas { |∇tk(un) − ∇tk(u)| > h } ≤ 1 h ∫ qt |∇tk(un) − ∇tk(u)|dxds ≤ 1 h ( 1 p− + 1 p+ )∥∥∇tk(un) − ∇tk(u)∥∥p(.) ∥1∥p′(.) (5.17) ≤ 1 h ( 1 p− + 1 (p−)′ )(∥∥∇tk(un)∥∥p(.) + ∥∇tk(u)∥p(.))∥1∥p′(.) . so by (5.15), meas { |∇tk(un) − ∇tk(u)| > h } → 0 as h → ∞ for any fixed k > 0 and the proof of (i) is complete. as a consequence of (i), up to a subsequence, we can assume that ∇tk(un) → ∇tk(u) a.e in qt . (ii) since a(x,ξ) is continuous with respect to ξ ∈ rn, then by (i) we deduce that (a(x,tk(u n)))n∈n converges in measure to a(x,∇tk(u)) and a.e. in qt . cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 399 moreover, using the hypotheses (1.3) and (5.12) one shows that (a(x,∇tk(un)))n∈n is uniformly bounded in (lp ′(.)(qt )) d. consequently, in the one part thanks to lemma 5.4 it follows that (a(x,tk(un)))n∈n → a(x,∇tk(u)) strongly in ( l1(qt ) )d . on the other part, we can extract a subsequence still denoted by (a(x,∇tk(un)))n∈n such that a(x,∇tk(un)) ⇀ ζk in (lp ′(.)(qt )) d. since each of the convergence implies the weak l1convergence, ζk can be identified with a(x,∇tk(u)), thus a(x,∇tk(u)) ∈ (lp ′(.)(qt )) d. this completes the proof. lemma 5.6. (un)n∈n converges a.e. in σt . proof. we know that the trace operator is compact from w1,1 (ω) into l1 (∂ω) , then there exists a constant c such that∫ t 0 ∥∥tk(un(t)) − tk(u(t))∥∥l1(∂ω) dt ≤ c ∫ t 0 ∥∥tk(un(t)) − tk(u(t))∥∥w 1,1(ω) dt. since w1,p(.) (ω) ↪→ w1,1 (ω) for all p(.) ≥ 1, then by the hölder type inequality, we deduce that tk(u n(t)) → tk(u) in l1 (σt ) and a.e. on σt . so, there exists a ⊂ σt such that tk(un(t)) converges to tk(u(t)) on σt \ a with meas(a) = 0. for every k > 0, we set ak = {(t,x) ∈ σt : |tk(u(t))| < k} , and b = σt \ ∞⋃ k=1 ak. we have, by hölder’s inequality meas (b) ≤ 1 k ∫ b |tk (u)|dσ ≤ 1 k ∫ t 0 ∥tk(u)∥l1(∂ω) dt ≤ 1 k ∫ t 0 ∥tk (u)∥w 1,1(ω) dt (5.18) ≤ 1 k ∫ t 0 ∫ ω (|tk (u) | + |∇tk (u) |) ≤ 1 k ( 1 p− + 1 (p−)′ ) ∥1∥lp′(x)(qt ) ( ∥tk (u)∥lp(x)(qt ) + ∥∇tk (u)∥(lp(x)(qt ))d ) . thanks to (5.12) and (5.13), for all k > 0, we have ∥∥tk (un)∥∥lp(x)(q) + ∥∥∇tk (un)∥∥(lp(x)(q))d ≤ 2(k 1p− + k 1p+ ) (5.19) ×max { c(u0,p+,f,g) 1 p+ ,c(u0,p+,f,g) 1 p+ } 400 u. traoré cubo 23, 3 (2021) we now use the fatou’s lemma in (5.19) to get ∥tk (u)∥lp(x)(q) + ∥∇tk (u)∥(lp(x)(q))d ≤ 2 ( k 1 p− + k 1 p+ ) ×max { c(u0,p+,f,g) 1 p+ ,c(u0,p+,f,g) 1 p+ } , and (5.18) becomes meas (b) ≤ 2 ( 1 k 1− 1 p− + 1 k 1− 1 p+ ) max { c(u0,p+,f,g) 1 p+ ,c(u0,p+,f,g) 1 p+ } . (5.20) therefore, we get by letting k → ∞ in (5.20) that meas (b) = 0. let us now define on ∂ω, the function v by v(t,x) = tk(u(t))(x) if (x,t) ∈ ak. we take (x,t) ∈ σt \ (a ∪ b); then there exists k > 0 such that (x,t) ∈ ak and we have un (t,x) − v (t,x) = (un (t,x) − tk(un(t))(x)) + (tk(un(t))(x) − tk(u(t))(x)). since (x,t) ∈ ak, we have ∣∣tk(un(t))(x)∣∣ < k from which we deduce that tk(un(t))(x) = un (t,x) . therefore, un (t,x) − v (t,x) = (tk(un(t))(x) − tk(u(t))(x)) → 0, as n → ∞. this means that ( un ) converges to v a.e. on σt . lemma 5.7. the sequence (un)n∈n converges to u in c(0,t;l1(ω)). proof. let (tn = nτn)nn=1 and (t m = mτm) m n=1 be two partitions of the interval [0,t] and let (un(t),un(t)), (um(t);um(t)); be the semi-discrete solutions defined by (5.1), (5.2) and corresponding to the respective partitions. let φ ∈ l∞(ω) ∩ v ∩ w1,1(0,t;l1(ω)). we rewrite (3.1) in the forms ∫ t 0 〈 ∂un ∂s ,tk(u n − φ) 〉 ds + ∫ t 0 ∫ ω a(x,∇un).∇tk(un − φ)dxds + ∫ t 0 ∫ ω b(un)tk(u n − φ)dxds ≤ ∫ t 0 ∫ ω fntk(u n − φ)dxds (5.21) and ∫ t 0 〈 ∂um ∂s ,tk(u m − φ) 〉 ds + ∫ t 0 ∫ ω a(x,∇um).∇tk(um − φ)dxds + ∫ t 0 ∫ ω b(um)tk(u m − φ)dxds ≤ ∫ t 0 ∫ ω fmtk(u m − φ)dxds (5.22) cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 401 where fn(t,x) = fn(x) ∀t ∈]tn−1, tn] fm(t,x) = fm(x) ∀t ∈]tm−1, t m ] let h > 1, in inequality (5.21) we take φ = th(um) and in inequality (5.22) we take φ = th(un). summing both inequalities, we get, for k = 1,∫ t 0 〈 ∂(un − um) ∂s ,t1(u n − um) 〉 ds + in,m(h) + ∫ t 0 ∫ ω b(un)t1(u n − th(um))dxds + ∫ t 0 ∫ ω b(um)t1(u m − th(un))dxds ≤ ∫ t 0 〈 ∂(un − um) ∂s ,t1(u n − um) 〉 − 〈 ∂un ∂s ,t1(u n − th(um)) 〉 ds (5.23) − ∫ t 0 〈 ∂um ∂s ,t1(u m − th(un)) 〉 ds + ∫ t 0 ∫ ω [fnt1(u n − th(um)) + fmt1(um − th(un))]dxds where in,m(h) = ∫ t 0 ∫ ω a(x,∇un).∇t1(un − th(um))dxds + ∫ t 0 ∫ ω a(x,∇um).∇t1(um − th(un))dxds. we have∣∣∣∣ ∫ t 0 〈 ∂(un − um) ∂s ,t1(u n − um) 〉 ds ∣∣∣∣ ≤ ∥∥∥∥∂(un − um)∂s ∥∥∥∥ l1(qt ) ∥∥t1(un − um)∥∥l∞(qt ) ≤ 2c(t,f,u0) ∥∥t1(un − um)∥∥l∞(qt ) . since lim n,m→∞ ∥∥t1(un − um)∥∥l∞(qt ) = 0. then it follows that lim h→∞ lim n,m→∞ ∫ t 0 〈 ∂(un − um) ∂s ,t1(u n − um) 〉 ds = 0. (5.24) similarly, we show that lim h→∞ lim n,m→∞ (∫ t 0 〈 ∂un ∂s ,t1(u n − th(um)) 〉 + 〈 ∂um ∂s ,t1(u m − th(un)) 〉 ds ) = 0 lim h→∞ lim n,m→∞ ∫ t 0 ∫ ω [fnt1(u n − th(um)) + fmt1(um − th(un))]dxds = 0 402 u. traoré cubo 23, 3 (2021) and lim h→∞ lim n,m→∞ ∫ t 0 ∫ ω b(un)t1(u n − th(um))dxds + ∫ t 0 ∫ ω b(um)t1(u m − th(un))dxds = 0. then, letting n, m → ∞ and h → ∞, in (5.23)we get lim h→∞ lim n,m→∞ ∫ t 0 〈 ∂(un − um) ∂s ,t1(u n − um) 〉 ds + lim h→∞ lim n,m→∞ in,m(h) ≤ 0. (5.25) since 〈 ∂v ∂t ,tk(v) 〉 = d dt ∫ ω jk(v) in l1(]0,t[), inequality (5.25) becomes lim n,m→∞ ∫ ω j1(u n(t) − um(t))dx + lim h→∞ lim n,m→∞ in,m(h) ≤ 0. (5.26) now, we show that lim h→∞ lim n,m→∞ in,m(h) ≥ 0. we consider the decomposition in,m(h) = 4∑ i=1 li(h), where li(h) = ∫ t 0 ∫ ωi(h) a(x,∇un).∇t1(un − th(um))dxds + ∫ t 0 ∫ ωi(h) a(x,∇um).∇t1(um − th(un))dxds and ω1(h) = { |un| ≤ h, |um| ≤ h } ω2(h) = { |un| ≤ h, |um| > h } ω3(h) = { |un| > h, |um| ≤ h } ω4(h) = { |un| > h, |um| > h } . on the one hand, thanks to assumption (1.4) we have l1(h) = ∫ t 0 ∫ ω11(h) [a(x,∇un) − a(x,∇um)].∇(un − um)dxds ≥ 0. therefore lim h→∞ lim n,m→∞ l1(h) ≥ 0. on the other hand, we have l2(h) = ∫ t 0 ∫ ω12(h) a(x,∇un).∇undxds + ∫ t 0 ∫ ω22(h) a(x,∇um).∇(um − un)dxds ≥ − ∫ t 0 ∫ ω22(h) a(x,∇um).∇undxds, cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 403 where ω12(h) = { |un| ≤ h, |um| > h, |un − hsign(um)| ≤ 1 } , ω22(h) = { |un| ≤ h, |um| > h, |un − um| ≤ 1 } . now, taking φ = th(un) in (5.21), we deduce that lim h→∞ lim n→∞ ∫ t 0 ∫ {h≤|un |≤h+k} a(x,∇un).∇un = 0. this implies lim h→∞ lim n→∞ ∫ t 0 ∫ {h≤|un |≤h+k} |∇un|p(x) = 0, k > 0. (5.27) by the young inequality, we have∣∣∣∣∣ ∫ t 0 ∫ ω22(h) a(x,∇um).∇undxds ∣∣∣∣∣ ≤ ∫ t 0 ∫ ω22(h) |∇um|p(x)−1|∇un|dxds ≤ ∫ t 0 ∫ {h≤|um |≤h+1} 1 p′(x) |∇um|p(x)dxds + ∫ t 0 ∫ {h−1≤|un |≤h} 1 p(x) |∇um|p(x)dxds ≤ ∫ t 0 ∫ {h≤|um |≤h+1} 1 p′− |∇um|p(x)dxds + ∫ t 0 ∫ {h−1≤|un |≤h} 1 p− |∇um|p(x)dxds. thus (5.27) gives lim n,m→∞ ∫ t 0 ∫ t 0 ∫ ω22(h) a(x,∇um).∇undxds = 0, which implies that lim h→∞ lim n,m→∞ l2(h) ≥ 0. similarly, we show that lim h→∞ lim n,m→∞ (l3(h) + l4(h)) ≥ 0. therefore lim h→∞ lim n,m→∞ in,m(h) ≥ 0. thus (5.26) becomes lim n,m→∞ ∫ ω j1(u n(t) − um(t))dx = 0. (5.28) since 1 2 ∫ {|un −um |≤1} |un(t) − um(t)|2dx + ∫ {|un −um |≥1} |un(t) − um(t)|dx ≤ ∫ ω j1(u n(t) − um(t)); 404 u. traoré cubo 23, 3 (2021) we have ∫ {|un −um |≥1} |un(t) − um(t)|dx = ∫ {|un −um |≤1} |un(t) − um(t)|dx + ∫ {|un −um |≥1} |un(t) − um(t)|dx ≤ cω (∫ {|un −um |≤1} |un(t) − um(t)|2dx )1 2 + ∫ {|un −um |≥1} |un(t) − um(t)|dx ≤ c2(ω) (∫ ω j1(u n(t) − um(t))dx )1 2 + ∫ ω j1(u n(t) − um(t))dx. by (5.26), we deduce that (un)n∈n is a cauchy sequence in c(0,t;l1(ω)). hence (un)n∈n converges to u in c(0,t;l1(ω)). step 2: existence of entropy solution. now, we prove that the limit function u is an entropy solution of the problem (p). since un(0) = u0 = u0 for all n ∈ n, we have u(0, .) = u0, and inequality (5.21) implies∫ t 0 〈 ∂un ∂s ,tk(u n − φ) − tk(un − φ) 〉 ds + ∫ t 0 ∫ ω a(x,∇un).∇tk(un − φ)dxds + ∫ t 0 ∫ ω b(un)tk(u n − φ)dxds (5.29) ≤ ∫ t 0 〈 φ ∂s ,tk(u n − φ) − tk(un − φ) 〉 ds + ∫ ω jk(u n(0) − φ(0))dx − ∫ ω jk(u n(t) − φ(t))dx + ∫ t 0 ∫ ω fntk(u n − φ)dxds. let k = k + ∥φ∥∞ . then∫ t 0 ∫ ω a(x,∇un).∇tk(un − φ)dxds = ∫ t 0 ∫ ω a(x,∇tk(u n)).∇tk(tk(u n) − φ)dxds = ∫ t 0 ∫ ω [a(x,∇tk(u n)).∇tk(u n) −a(x,∇tk(u n)).∇φ]1q(n,k)dxds, where q(n,k) = { |tk(u n) − φ| ≤ k } . thus, the inequality (5.29) becomes ∫ t 0 〈 ∂un ∂s ,tk(u n − φ) − tk(un − φ) 〉 ds − ∫ t 0 ∫ ω a(x,∇tk(u n)).∇φ1q(n,k) + ∫ t 0 ∫ ω [a(x,∇tk(u n)).∇tk(u n)]1q(n,k) + ∫ t 0 ∫ ω b(un)tk(u n − φ)dxds (5.30) ≤ − ∫ t 0 〈 ∂φ ∂s ,tk(u n − φ) 〉 ds + ∫ ω jk(u n(0) − φ(0))dx − ∫ ω jk(u n(t) − φ(t))dx + ∫ t 0 ∫ ω fntk(u n − φ)dxds. cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 405 on the one hand, thanks to lemma 5.5 a(x,∇tk(u n)) converges weakly to a(x,∇tk(u)) in( lp ′(.) (ω) )d . therefore, lim n→∞ ∫ t 0 ∫ ω a(x,∇tk(u n)).∇φ1q(n,k) = ∫ t 0 ∫ ω a(x,∇tk(u)).∇φ1q(k), (5.31) where q(k) = { |tk(u) − φ| ≤ k } . moreover, a(x,∇tk(u n)).∇tk(u n) is nonnegative and converges a.e. in qt to a(x,∇tk(u)).∇tk(u) (see lemma 5.5). therefore by fatou’s lemma, we obtain lim inf n→∞ ∫ t 0 ∫ t 0 ∫ ω [a(x,∇tk(u n)).∇tk(u n)]1q(n,k)dxds ≥ ∫ t 0 ∫ t 0 ∫ ω [a(x,∇tk(u)).∇tk(u)]1q(k)dxds. for the fourth term of (5.30), we have∫ t 0 ∫ ω b(un)tk(u n −φ)dxds = ∫ t 0 ∫ ω (b(un)−b(φ))tk(un −φ)dxds+ ∫ t 0 ∫ ω b(φ)tk(u n −φ)dxds. the quantity (b(un) − b(φ))tk(un − φ) is is nonnegative and since for all s ∈ r, s 7→ b(s) is continuous, we obtain (b(un) − b(φ))tk(un − φ) → (b(u) − b(φ))tk(un − φ) a.e. in ω. then, it follows by fatou’s lemma that lim inf n→∞ ∫ t 0 ∫ ω (b(un) − b(φ))tk(un − φ)dxds ≥ ∫ t 0 ∫ ω (b(u) − b(φ))tk(u − φ)dxds. we have b(φ) ∈ l1(qt ). since tk(un − φ) converges weakly−∗ to tk(u − φ) and b(φ) ∈ l1(qt ), it follows that lim inf n→∞ ∫ t 0 ∫ ω b(φ)tk(u n − φ)dxds ≥ ∫ t 0 ∫ ω b(φ)tk(u − φ)dxds. by lemma 5.7 , we deduce that un(t) → u(t) in l1(ω) for all t ∈ [0,t], which implies that∫ ω jk(u n(t) − φ(t))dx → ∫ ω jk(u(t) − φ(t))dx ∀t ∈ [0,t]. (5.32) we follow the method used in the proof of equality (5.24) to show that lim n→∞ ∫ t 0 〈 ∂un ∂s ,tk(u n − φ) − tk(un − φ) 〉 ds = 0. (5.33) finally, letting n → ∞ and using the above results, the continuity of b and the facts that fn → f in l1(qt ), tk(u n − φ) → tk(u − φ) in l ∞(qt ), we deduce that u is an entropy solution of the nonlinear parabolic problem (p). 406 u. traoré cubo 23, 3 (2021) 6 conclusion in this paper we prove the existence and uniqueness of an entropy solution for a nonlinear parabolic equation with homogeneous neumann boundary conditions and initial data in l1 by a time discretization technique. this method turns out to be better suited for the study of parabolic problems under neumanntype boundary conditions. however, this technique assumes that the associated elliptic problem is well posed. this study opens up new perspectives, we could always in the context of the sobolev space with variable exponents look at the problem with measure data or consider the function b as maximal monotone graph. cubo 23, 3 (2021) entropy solutions for nonlinear parabolic problems with... 407 references [1] m. abdellaoui, e. azroul, s. ouaro and u. traoré, “nonlinear parabolic capacity and renormalized solutions for pdes with diffuse measure data and variable exponent”, an. univ. craiova ser. mat. inform., vol. 46, no. 2, pp. 269–297, 2019. 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[29] a. zimmermann, “renormalized solutions for a nonlinear parabolic equation with variable exponents and l1 data”, ph. d. thesis, t. u. berlin, 2010. introduction and main result preliminaries the semi-discrete problem stability convergence and existence result conclusion cubo, a mathematical journal vol. 24, no. 01, pp. 37–51, april 2022 doi: 10.4067/s0719-06462022000100037 smooth quotients of abelian surfaces by finite groups that fix the origin robert auffarth1 giancarlo lucchini arteche1 pablo quezada2 1departamento de matemáticas, facultad de ciencias, universidad de chile, las palmeras 3425, ñuñoa, santiago, chile. rfauffar@uchile.cl luco@uchile.cl 2facultad de matemáticas, pontificia universidad católica, vicuña mackenna 4860, macul, santiago, chile. psquezada@uc.cl abstract let a be an abelian surface and let g be a finite group of automorphisms of a fixing the origin. assume that the analytic representation of g is irreducible. we give a classification of the pairs (a, g) such that the quotient a/g is smooth. in particular, we prove that a = e2 with e an elliptic curve and that a/g ≃ p2 in all cases. moreover, for fixed e, there are only finitely many pairs (e2, g) up to isomorphism. this fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origin started by the first two authors. resumen sea a una superficie abeliana y sea g un grupo finito de automorfismos de a fijando el origen. se asume que la representación anaĺıtica de g es irreducible. damos una clasificación de los pares (a, g) tales que el cociente a/g es suave. en particular, probamos que a = e2 con e una curva eĺıptica y que a/g ≃ p2 en todos los casos. más aún, para e fija, hay solo una cantidad finita de pares (e2, g), salvo isomorfismo. esto llena una pequeña brecha en la literatura y completa la clasificación de cocientes suaves de variedades abelianas por grupos finitos fijando el origen comenzado por los dos primeros autores. keywords and phrases: abelian surfaces, automorphisms. 2020 ams mathematics subject classification: 14l30, 14k99. accepted: 26 october, 2021 received: 21 may, 2021 c©2022 r. auffarth et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100037 https://orcid.org/0000-0001-7243-0315 https://orcid.org/0000-0003-3269-1814 mailto:rfauffar@uchile.cl mailto:luco@uchile.cl mailto:psquezada@uc.cl 38 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) 1 introduction the purpose of this paper is to give a complete classification of all smooth quotients of abelian surfaces by finite groups that fix the origin, and is to be seen as the completion of the classification given in [2] of smooth quotients of abelian varieties that fix the origin. this kind of quotients of abelian surfaces has already been studied by tokunaga and yoshida in [6], where infinite 2dimensional complex reflection groups, which are extensions of a finite complex reflection group g by a g-invariant lattice, are classified. however, these do not cover all possible g-invariant lattices and hence not all possible group actions on abelian surfaces. moreover, there seem to be some complex reflection groups that the authors missed, as can be seen by looking at popov’s classification of the same groups in [3]. the techniques used in this paper are similar, but not exactly the same, to the methods used in [2]. indeed, the ideas used in this last paper have been modified in order to apply them to the two-dimensional case. moreover our approach is far different from that used in [6]. our main theorem states the following: theorem 1.1. let a be an abelian surface and let g be a (non-trivial) finite group of automorphisms of a that fix the origin. then the following conditions are equivalent: (1) a/g is smooth and the analytic representation of g is irreducible. (2) a/g ≃ p2. (3) there exists an elliptic curve e such that a ≃ e2 and (a, g) satisfies exactly one of the following: (a) g ≃ c2 ⋊ s2 where c is a non-trivial (cyclic) subgroup of automorphisms of e that fix the origin; here the action of c2 is coordinate-wise and s2 permutes the coordinates. (b) g ≃ s3 and acts on a ≃ {(x1, x2, x3) ∈ e 3 : x1 + x2 + x3 = 0}, by permutations. (c) e = c/z[i] and g is the order 16 subgroup of gl2(z[i]) generated by:      −1 1 + i 0 1   ,   −i i − 1 0 i   ,   −1 0 i − 1 1      , acting on a ≃ e2 in the obvious way. the first two cases found in item (3) of the above theorem were studied in detail in [1] (in arbitrary dimension), where it was proven that both examples give the projective plane as a quotient. cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 39 throughout the paper we will refer to these two examples as example (a) and example (b), respectively. the equivalent assertion for example (c) is proposition 4.1 in this paper. note that, aside from examples (a) and (b) which belong to infinite families, example (c) is the only new case of an action of g on an abelian variety satisfying condition (1) from theorem 1.1, cf. [2, thm. 1.1]. remark 1.2. if a/g is smooth and the analytic representation of g is reducible, then the results in [2] imply that a is isogenous to a product of two elliptic curves. the quotient is then either p 1 × p1 (in which case a = e1 × e2) or a bielliptic surface. in [7], yoshihara introduces the notion of a galois embedding of a smooth projective variety. if x is a smooth projective variety of dimension n and d is a very ample divisor that induces an embedding x →֒ pn, then the embedding is said to be galois if there exists an (n − n − 1)dimensional linear subspace w of pn such that x ∩ w = ∅ and the restriction of the linear projection πw : p n 99k p n to x is galois. yoshihara specifically studies when abelian surfaces have a galois embedding. he gives a classification of abelian surfaces having a galois embedding, along with their galois groups, and proves that after taking the quotient of the original abelian variety by the translations of the galois group, the abelian variety must be isomorphic to the selfproduct of an elliptic curve. unfortunately, his results were incomplete since they depended on a classification of smooth quotients like the one given in this paper, which yoshihara attributed to tokunaga and yoshida in [6]. but as stated before, tokunaga and yoshida’s results do not imply such a classification. nevertheless, we can now safely say, thanks to theorem 1.1, that yoshihara’s results remain correct. the structure of this paper is as follows: in section 2 we fix notations and give some preliminary results that will be needed in the proofs of theorem 1.1. the implication (2) ⇒ (1) is obvious and (3) ⇒ (2) was already treated in [1] in the case of examples (a) and (b). thus, we are mainly concerned with (1) ⇒ (3), which we treat in section 3. finally, in section 4 we treat (3) ⇒ (2) for example (c), which is a construction of a different nature that only exists in the 2-dimensional case. 2 preliminaries on group actions on abelian varieties we recall here some elementary results that were proved in [2]. let a be an abelian surface and let g be a group of automorphisms of a that fix the origin, such that the quotient variety a/g is smooth. by the chevalley-shephard-todd theorem, the stabilizer in g of each point in a must be generated by pseudoreflections; that is, elements that fix pointwise a divisor (i.e. a curve) containing the point. in particular, g = stabg(0) is generated by pseudoreflections and g acts on the tangent space at the origin t0(a) (this is the analytic representation). in this context, a pseudoreflection is an element that fixes a line pointwise. we will often abuse notation and display 40 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) g as either acting on a or t0(a); it will be clear from the context which action we are considering. in what follows, let l be a fixed g-invariant polarization on a (take the pullback of an ample class on a/g, for example). for σ a pseudoreflection in g of order r, define dσ := im(1 + σ + · · · + σ r−1), eσ := im(1 − σ). these are both abelian subvarieties of a. the following result corresponds to [2, lem. 2.1]. lemma 2.1. we have the following: 1. dσ is the connected component of ker(1 − σ) that contains 0 and eσ is the complementary abelian subvariety of dσ with respect to l. in particular, dσ and eσ are elliptic curves. 2. fσ := dσ ∩ eσ consists of 2-torsion points for r = 2, 4, of 3-torsion points for r = 3 and dσ ∩ eσ = 0 for r = 6. we will consider now a new abelian surface b equipped with a g-equivariant isogeny to a, which we will call g-isogeny from now on. let λa denote the lattice in c 2 such that a = c2/λa. let λb ⊂ λa be a g-invariant sublattice, and let b := c 2/λb be the induced abelian surface, along with the g-isogeny π : b → a, whose analytic representation is the identity. note that this implies that σ ∈ g is a pseudoreflection of b if and only if it is a pseudoreflection of a. we may then consider the subvarieties eσ, dσ, fσ ⊂ a defined as above, which we will denote by eσ,a, dσ,a and fσ,a. we do similarly for b. note that, by definition, π sends eσ,b to eσ,a and dσ,b to dσ,a, hence fσ,b to fσ,a. the following result was proved in [2, prop. 2.4]. proposition 2.2. let σ ∈ g be a pseudoreflection and let l be the line defining both eσ,a and eσ,b. assume that the map fσ,b → fσ,a is surjective and that λa ∩ l = λb ∩ l. then ker(π) is contained in dτστ−1,b for every τ ∈ g. define ∆ := ker(π). since π is g-equivariant, g acts on ∆ and hence we may consider the group ∆⋊ g. this group acts on b in the obvious way: ∆ acts by translations and g by automorphisms that fix the origin. in particular, we see that the quotient b/(∆ ⋊ g) is isomorphic to a/g. we conclude this section by recalling a result on pseudoreflections in ∆ ⋊ g (cf. [2, lem. 2.5]). lemma 2.3. let σ ∈ ∆ ⋊ g be a pseudoreflection. then σ = (t, τ) with τ ∈ g a pseudoreflection and t ∈ ∆ ∩ eτ,b. cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 41 3 proof of (1) ⇒ (3) assume (1), that is, we have an abelian surface a with an action of a finite group g that fixes the origin and such that a/g is smooth and the analytic representation of g is irreducible. under these conditions, we see that g is an irreducible finite complex reflection group in the sense of shephard-todd [4]. these groups were completely classified by shephard and todd in [4]. in the particular case of dimension 2, we get that g is either one of 19 sporadic cases or it is isomorphic to a semidirect product g(m, p) := h(m, p) ⋊ s2, where p|m, m ≥ 2, and h = h(m, p) = {(ζa1m , ζ a2 m ) | a1 + a2 ≡ 0 (mod p)} ⊂ µ 2 m, with ζm denoting a primitive m-th root of unity. the action of s2 on h is the obvious one. the case g = g(2, 2) is excluded since g is then a klein group and thus the representation is not irreducible. the action of g on c2 is given as follows: h acts on c2 coordinate-wise while s2 permutes the coordinates. emulating the work done in [2], we wish to describe which of these actions actually appear on abelian surfaces and give smooth quotients. the sporadic cases were already treated in [2] and were proven not to give a smooth quotient (cf. [2, §3.3]), so we may and will assume henceforth that g = g(m, p) as above. this fixes a g-equivariant isomorphism of t0(a) with c 2. we denote by e1 and e2 the canonical basis of t0(a) thus obtained. lemma 3.1. assume that g acts on a as above. then m ∈ {2, 3, 4, 6}. proof. we have that (ζm, ζ −1 m ) acts on a, and so the characteristic polynomial of (ζm, ζ −1 m ) ⊕ (ζm, ζ −1 m ) must have integer coefficients, and so must be the k-th power of the m-th cyclotomic polynomial φm, where k = 2 if m ≥ 2 and k = 4 if m = 2. looking at the degrees, we get that 4 = kϕ(m), where ϕ is euler’s totient function. therefore, if m 6= 2 then ϕ(m) = 2 and so m ∈ {3, 4, 6}. having proved this result, we see that there is a finite list of cases to be analyzed, that is: (m, p) ∈ {(2, 1), (2, 2), (3, 1), (4, 1), (6, 1), (3, 3), (4, 2), (4, 4), (6, 2), (6, 3), (6, 6)}. recall that we have already eliminated the case (2, 2) since the analytic representation of g(2, 2) is not irreducible. moreover, it is well-known that there is an exceptional isomorphism of complex reflection groups between g(4, 4) and g(2, 1). we will prove then the following: • if g = g(m, 1) and a/g is smooth, then the pair (a, g) corresponds to example (a) (sections 3.1, 3.3, 3.4, 3.5); • if g = g(3, 3) and a/g is smooth, then the pair (a, g) corresponds to example (b) (section 3.8); 42 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) • if g = g(4, 2) and a/g is smooth, then the pair (a, g) corresponds to example (c) (section 3.2); • if g = g(6, p) with p ≥ 2, then a/g cannot be smooth (sections 3.6, 3.7, 3.9). in order to do this, we will construct a g-isogeny b → a such that the action of g on b is “well-known”. let us concentrate first on the cases where m 6= p. then we obtain b as follows: let ei be the image of cei in a via the exponential map. we claim that it corresponds to an elliptic curve. indeed, consider the non-trivial element τ = (ζpm, 1) ∈ h. then a direct computation shows that im(1 − τ) = ce1. this tells us that e1 = (1 − τ)(a) and hence it corresponds to an elliptic curve. the same proof works for e2. now, let λa be a lattice for a in c 2. then cei ∩ λa corresponds to the lattice of ei in c = cei. we can thus define the g-stable sublattice of λa λb := (ce1 ∩ λa) ⊕ (ce2 ∩ λa). as in section 2, this defines a g-isogeny π : b → a. moreover, we see that b ≃ e1 ×e2 ≃ e 2 and that π|ei is an injection. let ∆ be the kernel of π. we will study the different possible quotients a/g by studying the possible quotients b/(∆ ⋊ g) and thus by studying the possible ∆’s. our first result is the following: lemma 3.2. assume that m 6= p. then the coordinates of every element in ∆ are invariant by ζpm, so in particular these elements are • 2-torsion if (m, p) ∈ {(2, 1), (4, 1), (4, 2), (6, 3)}; • 3-torsion if (m, p) ∈ {(3, 1), (6, 2)}; • trivial if (m, p) = (6, 1). proof. let t̄ = (t1, t2) ∈ ∆. then, since ∆ is g-stable, we have that, for τ1 = (ζ p m, 1) ∈ h, (1 − τ1)(t̄) = ((1 − ζ p m)t1, 0) ∈ ∆. but, by construction, there are no elements of the form (x, 0) in ∆. we deduce then that t1 is ζpm-invariant. the same proof works for t2. the assertion on the torsion of t1 and t2 follows immediately. let us study now pseudoreflections in ∆ ⋊ g. define the elements ρ := (ζm, ζ −1 m ) ∈ h ⊂ g; σ := (1 2) ∈ s2 ⊂ g; τ := (ζ p m, 1) ∈ h ⊂ g. then there are two types of pseudoreflections in g: cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 43 • conjugates of ρaσ for 0 ≤ a < m p ; • conjugates of powers of τ; and the corresponding elliptic curves in b are respectively: eρaσ = {(x, −ζ a mx) | x ∈ e}; eτ = {(x, 0) | x ∈ e}. recall that elements of the form (x, 0) are not in ∆ by construction of the isogeny π : b → a. using lemmas 2.3 and 3.2, we obtain immediately the following result: lemma 3.3. every pseudoreflection in ∆ ⋊ g that is not in g is a conjugate of (t̄, ρaσ), where 0 ≤ a < m p , t̄ = (t, −ζamt) ∈ ∆ and t is ζ p m-invariant. with these considerations, we can start a case by case study of the non-trivial ∆’s. we recall that the main tool will be the chevalley-shephard-todd theorem, which states that a/g = b/(∆⋊g) is smooth if and only if the stabilizer in ∆⋊g of each point in b is generated by pseudoreflections. 3.1 the case g = g(2, 1) by lemma 3.2, we know that ∆ is 2-torsion. since we also know that there are no elements of the form (t, 0) for t ∈ e, we get the following possible options for ∆: (1) ∆ = {0}; (2) ∆ = 〈(t, t)〉 with t ∈ e[2]; (3) ∆ = {(t, t) | t ∈ e[2]}; (4) ∆ = {(0, 0), (t1, t2), (t2, t1), (t1 + t2, t1 + t2)} with t1, t2 ∈ e[2], t1 6= t2. case (1) clearly corresponds to example (a) (which gives a smooth quotient, cf. [2, prop. 3.4]). case (2) cannot give a smooth quotient and this follows directly from [2, prop. 3.7].1 in case (3), we claim that the pair (a, g) is isomorphic to the pair (b, g). this will reduce us to the case with trivial ∆, which was already dealt with. to prove the claim, consider the canonical basis of t0(a) = t0(b) = c 2. then the analytic representation of g is given by the following values in its generators: ρa((1, −1)) =   1 0 0 −1   , ρa((1 2)) =   0 1 1 0   1the proof of this proposition only uses two variables and thus it works in dimension 2 as well. 44 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) now, with this basis and this ∆, we can view the g-isogeny b → a as the morphism e2 → e2 given by the following matrix: m =   1 1 1 −1   , (*) for which one can check that its kernel is precisely the elements in ∆. in order to prove that the pairs (a, g) and (b, g) are isomorphic, it suffices thus to prove that the image of this representation of g under conjugation by m is g once again. direct computations give: mρa((1, −1))m −1 =   0 1 1 0   = ρa((1 2)), mρa((1 2))m −1 =   1 0 0 −1   = ρa((1, −1)). and these clearly generate the same group g. in case (4), consider the element t̄ = (t′ 1 , t′ 2 ) where 2t′i = ti. note that g cannot fix t̄ as t ′ 1 and t′ 2 lie in different orbits by the action of µ2. now, it is easy to see that there is no way the action of ∆ can compensate the action of g except in the case when we add the element (t1, t2). a direct computation tells us then that the only element fixing t̄ is ((t1, t2), (−1, −1)) ∈ ∆ ⋊ g and since this stabilizer is not generated by pseudoreflections by lemma 3.3, we see that a/g is not smooth. 3.2 the case g = g(4, 2) since g(4, 2) contains g(2, 1), we may start from the precedent list of possible non-trivial ∆’s. however, these must also be stable by the new element (i, i) ∈ h(4, 2) (where i = ζ4). note that such an element acts on each component e of b by multiplication by i, which implies in particular that e = c/z[i]. defining by t0 the only non-trivial i-invariant element in e, we get the following possibilities: (1) ∆ = {0}; (2) ∆ = 〈(t0, t0)〉; (3) ∆ = {(t, t) | t ∈ e[2]}; (4) ∆ = {(0, 0), (t, t + t0), (t + t0, t), (t0, t0)} with t ∈ e[2], t 6= t0. case (1) does not give a smooth quotient a/g, cf. [2, prop. 3.4]. case (2) corresponds to example (c) (and it actually gives a smooth quotient a/g as we prove in section 4). indeed, the g-isogeny b → a corresponds in this case to the morphism e2 → e2 with e = c/z[i] given by the matrix   1 −1 0 i − 1   , cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 45 and the generators given in example (c) correspond to the conjugates by this matrix of the following respective matrices:      −1 0 0 1   ,   −i 0 0 i   ,   0 1 1 0      . but these are clearly the matrix expressions of the generators (−1, 1), (−i, i) ∈ h and (1 2) ∈ s2 of g = h ⋊ s2. remark 3.4. since the first and third matrices above generate the subgroup g(2, 1) ⊂ g(4, 2), we see that if we take f to be the subgroup of g spanned by the pseudoreflections   −1 i + 1 0 1   and   −1 0 i − 1 1   , then f is isomorphic to g(2, 1) and a/f ≃ p2. in particular, the pair (a, f) is isomorphic to example (a) with c cyclic of order 2. in cases (3) and (4), we claim that the pair (a, g) is isomorphic to the pair (b, g). this will reduce us to the case with trivial ∆, which was already dealt with. to prove the claim, we consider as for g = g(2, 1) the canonical basis of t0(a) = t0(b) = c 2. then the analytic representation of g is given by the following values in its generators: ρa((i, −i)) =   i 0 0 −i   , ρa((−1, 1)) =   −1 0 0 1   , ρa((1 2)) =   0 1 1 0   now, with this basis and the ∆ from case (2), we already know that b → a looks like e2 → e2 with matrix m from (*). it suffices to check then that the new generator ρa((i, −i)) falls into ρa(g) after conjugation by m. and indeed we have that mρa((i, −i))m −1 = ρa((i, i))ρa((1 2)). with the ∆ from case (3), the corresponding matrix for b → a is: n =   1 i i 1   . and once again, direct computations give: nρa((i, −i))n −1 =   0 −1 1 0   = ρa((−1, 1))ρa((1 2)), nρa((−1, 1))n −1 =   0 −i i 0   = ρa((1 2))ρa((i, −i)), nρa((1 2))n −1 =   0 1 1 0   = ρa((1 2)). and these clearly generate the same group g. 46 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) 3.3 the case g = g(4, 1) since g(4, 1) contains g(4, 2), we may start from the precedent list of possible non-trivial ∆’s. now, by lemma 3.2, we know that the coordinates of the elements in ∆ are i-invariant. we get then that there are only two options for ∆, that is the trivial case and ∆ = 〈(t0, t0)〉. in the trivial case, we immediately see that (a, g) corresponds to example (a). assume then that ∆ is non-trivial and consider the element (s, t) ∈ b with s ∈ e[2], s not i-invariant and 2t = t0. since clearly these elements have different order, we see that the orbits of these elements by the action of 〈t0〉 × µ4 are different. thus no action of an element in ∆ × h ⊂ ∆ ⋊ g can compensate the action of (1 2) ∈ g in order to fix (s, t). in other words, the stabilizer of t̄ must be contained in ∆ × h. it is easy to see then that it corresponds to 〈((t0, t0), (i, −1))〉. by lemma 3.3, this stabilizer is not generated by pseudoreflections and hence a/g is not smooth in this case. 3.4 the case g = g(3, 1) by lemma 3.2, we know that the coordinates of the elements in ∆ are ζ3-invariant. now, there are only two such non-trivial elements that we will denote by s0 and −s0. since we also know that there are no elements of the form (t, 0) for t ∈ e, we get the following possible options for a non-trivial ∆: (1) ∆ = {0}; (2) ∆ = 〈(s0, s0)〉; (3) ∆ = 〈(s0, −s0)〉. we immediately see that the trivial case gives us example (a). in case (2), lemma 3.3 tells us that the only pseudoreflections in ∆ ⋊ g are those coming from g. in particular, in order to prove that a/g cannot be smooth, it suffices to exhibit an element in b such that its stabilizer in ∆ ⋊ g has elements that are not in g. let τ = (ζ3, ζ3) ∈ h ⊂ g, then 1 − τ is surjective. then there exists an element z̄ ∈ b such that z̄ − τ(z̄) = (s0, s0). this implies that ((s0, s0), τ) ∈ ∆ ⋊ g stabilizes z, proving thus that a/g is not smooth in this case. in case (3), consider the element s̄ = (s, −s) ∈ b with s ∈ e[3] and s not ζ3-invariant. note that 〈s0〉 × µ3 acts on e[3] and a direct computation tells us that the orbit of s is {s, s + s0, s − s0}. in particular, we see that s and −s lie in different orbits for this action. the same argument used in the case of g(4, 1) tells us then that the stabilizer of s̄ must be contained in ∆ × h. it is easy to see then that, up to changing s̄ by −s̄, it corresponds to 〈((s0, −s0), (ζ3, ζ3))〉. since this stabilizer is not generated by pseudoreflections by lemma 3.3, we see that a/g is not smooth in this case as well. cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 47 3.5 the case g = g(6, 1) by lemma 3.2, we know that the only possibility is a trivial ∆. this clearly corresponds to example (a). 3.6 the case g = g(6, 2) since g(6, 2) contains g(3, 1), we may start from the possible non-trivial ∆’s for that case. note that these are all 3-torsion subgroups. thus, if x̄ ∈ b denotes a 2-torsion element, we see that its stabilizer in ∆ ⋊ g can only contain elements in g. consider then the element t̄ = (t, 0) where t is a non-trivial 2-torsion element. as it is proven in [2, prop. 3.4], the stabilizer of this element in g is not generated by pseudoreflections. this implies that a/g = b/(∆ ⋊ g) cannot be smooth regardless of the choice of possible ∆. 3.7 the case g = g(6, 3) since g(6, 3) contains g(2, 1), we may start from the possible non-trivial ∆’s for that case. note that these are all 2-torsion subgroups. thus, like we noticed in the previous case, if x̄ ∈ b denotes a 3-torsion element, its stabilizer in ∆ ⋊ g only contains elements in g. consider then the element s̄ = (s0, 0) where s0 is a ζ3-invariant element (hence 3-torsion). once again, as proven in [2, prop. 3.4], the stabilizer of this element in g is not generated by pseudoreflections, which implies that a/g cannot be smooth in any case of ∆. this finishes the study of the cases where m 6= p. we are left thus with the cases g(3, 3) and g(6, 6). in these particular cases we forget all the constructions done before and start from scratch. 3.8 the case g = g(3, 3) the group g(3, 3) is easily seen to be isomorphic as a complex reflection group to s3 acting on c 2 via the standard representation. as such, it has already been treated by the first two authors in [2, §3.1] and we know that in that case we get a smooth quotient if and only if we are in example (b). 3.9 the case g = g(6, 6) note that g(6, 6) is isomorphic to the direct product g(3, 3) × {±1}. since the actions of s3 and µ2 = {±1} commute, we may follow the approach taken by [2] for s3 and we will prove the following: 48 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) proposition 3.5. let g(6, 6) = s3 × µ2 act on an abelian surface a in such a way that its action on t0(a) is the standard one for s3 and the obvious one for µ2. then a/g is not smooth. proof. let σ = (1 2) ∈ s3 and e = eσ be induced by a line lσ ⊂ t0(a), and define the lattice λb := ∑ τ∈s3 τ(lσ ∩ λa). since clearly all lattices are µ2-invariant, this gives us a g-invariant sublattice of λa. therefore, we get a g-equivariant isogeny π : b → a with kernel ∆. applying this construction to example (b), to which we can naturally add the action of µ2 in order to get an action of g, we see that it gives the whole lattice. we can thus see b as b = {(x1, x2, x3) ∈ e 3 | x1 + x2 + x3 = 0}, where s3 and µ2 act in their respective natural ways. using the notations from section 2, we see by inspection that fσ,b = eσ,b[2] ≃ e[2], hence the map π : fσ,b → fσ,a is surjective since by lemma 2.1, case 2., we have fσ,a ⊂ eσ,a[2] ≃ e[2]. by proposition 2.2, we have that ∆ is contained in the fixed locus of all the conjugates of σ, which clearly generate s3. thus, ∆ consists of elements of the form (x, x, x) ∈ e3 such that 3x = 0. in particular, ∆ is isomorphic to a subgroup of e[3] and hence of order 1, 3 or 9. assume that ∆ is trivial, that is, that a = b. then the action of g = s3 × µ2 on b ≃ e 2 induces an action of µ2 on b/s3 ≃ p 2 (recall that the action of s3 on b is that of example (b)). we only need to notice then that any quotient of p2 by a non trivial action of the group µ2 is not smooth. this is well-known. assume now that ∆ has order 3 and let t̄ = (t, t, t) ∈ ∆ be a non-trivial element (thus t ∈ e[3]). let x ∈ e[3] be a non-trivial element different from ±t and consider x̄ = (x, x + t, x − t). it is then easy to see that the element (t̄, (1 2 3)) ∈ ∆ ⋊ g fixes x̄ and that stabg(x̄) = {1}, so that every pseudoreflection fixing x̄ must lie outside g. let (s̄, σ) be such a pseudoreflection. using lemma 2.3, we see that σ ∈ {−(1 2), −(2 3), −(1 3)}, where −τ denotes (τ, −1) ∈ s3 × µ2 = g. now, for any such σ, direct computations tell us that (s̄, σ) fixes x̄ if and only s̄ = (s, s, s) with s = aσx+bσt for some aσ 6= 0. since x 6∈ 〈t〉 ⊂ e[3], we see that s̄ 6∈ ∆ and hence these pseudoreflections do not exist. we get then that stab∆⋊g(x̄) is not generated by pseudoreflections and hence a/g cannot be smooth. assume finally that ∆ has order 9. we claim that in this case the pair (a, g) is isomorphic to the pair (b, g). this will reduce us to the case with trivial ∆, which was already dealt with. to prove the claim, fix the basis {(1, 0, −1), (0, 1, −1)} of t0(b) = t0(a) ⊂ c 3. then the analytic representation of g is given by the following values in its generators: ρa((1 2)) =   0 1 1 0   , ρa(−1) =   −1 0 0 −1   , ρa((1 2 3)) =   −1 −1 1 0   cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 49 now, with this basis and this ∆, the analytic representation of b → a is given by the inverse of the following matrix: m =   −1 −2 2 1   . indeed, this corresponds to the morphism that sends (x, y, −x − y) ∈ b ⊂ e3 to (−x − 2y, 2x + y, −x+ y) ∈ a ⊂ e3 and thus its kernel is precisely the elements of the form (x, x, x) ∈ e[3]3 ⊂ b, that is, ∆. in order to prove that the pairs (a, g) and (b, g) are isomorphic, it suffices thus to prove that the image of this representation of g under conjugation by m is g once again. direct computations give: mρa(−1)m −1 = ρa(−1), mρa((1 2 3))m −1 = ρa((1 2 3)), mρa((1 2))m −1 =   0 −1 −1 0   = ρa((1 2))ρa(−1). and these clearly generate the same group g. 4 proof of (3) ⇒ (2) the only thing left to prove is that example (c) satisfies property (2) from theorem 1.1 (the other two are proved in [1]). let us then study this example in detail. recall that in section 3.2 we proved that the pair (a, g) from example (c) can be obtained as follows. let g = g(4, 2) and let b = e2 with e = c2/z[i]. denote by t0 the i-invariant element in e and denote by q0 the quotient morphism e → e/〈t0〉 ≃ e. then a = b/∆ with ∆ = 〈(t0, t0)〉 ∈ e 2 = b and the action of g on a is the one induced by b → a. note now that g has an index 2 subgroup g1 := g(2, 1) = h1 ⋊ s2, which is thus normal in g (here, h1 = {±1} 2). moreover, the pair (b, g1) corresponds to example (a), so that b/g1 ≃ p 2. finally, note that ∆ is an order 2 subgroup of b and thus g acts trivially on it. in particular, the actions of g and ∆ on b commute and hence we have a commutative diagram of galois covers b ∆ // g1 �� g �� a �� g �� p 2 // g/g1 �� a/g1 �� b/g // a/g, where parallel arrows have the same galois group. since ∆ and g/g1 have both order 2, we see then that a/g is a quotient of p2 by the action of a klein group. 50 r. auffarth, g. lucchini arteche & p. quezada cubo 24, 1 (2022) proposition 4.1. the quotient a/g is isomorphic to p2. this proposition finishes the proof of (3) ⇒ (2) in theorem 1.1. remark 4.2. this example was already known to tokunaga and yoshida (cf. [6, §5, table ii]). however, in order to prove that a/g ≃ p2, they cite an article by švarcman which contains no proofs (cf. [5]). proof. since a/g is a quotient of p2 by the action of a klein group k, the only thing we need to check is that this action gives p2 as a quotient. note first that the action is faithful since it comes from the faithful action of g × ∆ on b. consider then k as a subgroup of pgl3 = aut(p 2) and let k1 be its preimage in sl3. this is an order 12 group and hence any 2-sylow subgroup of k1 gives a lift of k to a subgroup of gl3. this implies that the action lifts to c 3 and it can thus be seen as a linear representation of k. since there are exactly four irreducible representations of k, all of dimension 1, a direct check tells us that any choice of three different representations gives the same faithful action on p2 up to conjugation, whereas any other choice gives a non-faithful action. we may assume then that the nontrivial elements xi ∈ k for i = 1, 2, 3 act on p 2 via the diagonal matrices with 1 on the i-th coordinate and −1 elsewhere. the quotient of p2 by such a group is the weighted projective space p(2, 2, 2), which is well-known to be isomorphic to p(1, 1, 1) = p2. this concludes the proof. acknowledgments the first and third authors were partially supported by conicyt pia act1415. the second author was partially supported by fondecyt grant 11170016 and pai grant 79170034. we would like to thank the anonymous referees for their comments and specially one of them for showing us a more elegant proof of lemma 3.1. cubo 24, 1 (2022) smooth quotients of abelian surfaces by finite groups... 51 references [1] r. auffarth, “a note on galois embeddings of abelian varieties”, manuscripta math., vol. 154, no. 3–4, pp. 279–284, 2017. [2] r. auffarth and g. lucchini arteche, “smooth quotients of abelian varieties by finite groups”, ann. sc. norm. sup. pisa cl. sci. (5), vol. 21, pp. 673–694, 2020. [3] v. popov. discrete complex reflection groups, communications of the mathematical institute, rijksuniversiteit utrecht, 15, netherland: rijksuniversiteit utrecht, 1982. [4] g. c. shephard and j. a. todd, “finite unitary reflection groups”. canad. j. math., vol. 6, pp. 274–304, 1954. [5] o. v. švarcman, “a chevalley theorem for complex crystallographic groups that are generated by mappings in the affine space c2” (russian), uspekhi mat. nauk, vol. 34, no.1(205), pp. 249–250, 1979. [6] s. tokunaga and m. yoshida.“complex crystallographic groups. i.”, j. math. soc. japan, vol. 34, no. 4, pp. 581–593, 1982. [7] h. yoshihara, “galois embedding of algebraic variety and its application to abelian surface”, rend. semin. mat. univ. padova, vol. 117, pp. 69–85, 2007. introduction preliminaries on group actions on abelian varieties proof of (1)(3) the case g=g(2,1) the case g=g(4,2) the case g=g(4,1) the case g=g(3,1) the case g=g(6,1) the case g=g(6,2) the case g=g(6,3) the case g=g(3,3) the case g=g(6,6) proof of (3)(2) cubo, a mathematical journal vol. 24, no. 03, pp. 413–437, december 2022 doi: 10.56754/0719-0646.2403.0413 existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale saroj panigrahi1, b sandip rout1 1 school of mathematics and statistics, university of hyderabad, hyderabad, 500 046, india. panigrahi2008@gmail.com b sandiprout7@gmail.com abstract in this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: (ψ(t)y ∆ (t)) ∇ +λ1g(t, y(t))+λ2h(t, y(t)) = 0, t ∈ [ρ(c), σ(d)]t, with mixed boundary conditions αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c)) = 0, γy(σ(d)) + δψ(d)y ∆ (d) = 0, where ψ : c[ρ(c), σ(d)]t, ψ(t) > 0 for all t ∈ [ρ(c), σ(d)]t; both g and h : [ρ(c), σ(d)]t × [0, ∞) → r are continuous and semipositone. we have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a banach space, when g and h are both superlinear or sublinear or one is superlinear and the other is sublinear for λi > 0; i = 1, 2 are sufficiently small. accepted: 28 september, 2022 received: 12 april, 2022 ©2022 s. panigrahi et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0413 https://orcid.org/0000-0003-4704-5102 https://orcid.org/0000-0002-6575-9910 mailto:panigrahi2008@gmail.com mailto:sandiprout7@gmail.com cubo, a mathematical journal vol. 24, no. 03, pp. 413–437, december 2022 doi: 10.56754/0719-0646.2403.0413 resumen en este art́ıculo estudiamos la existencia de soluciones positivas del siguiente problema de valor de frontera semipositón en escalas de tiempo: (ψ(t)y ∆ (t)) ∇ +λ1g(t, y(t))+λ2h(t, y(t)) = 0, t ∈ [ρ(c), σ(d)]t, con condiciones de frontera mixtas αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c)) = 0, γy(σ(d)) + δψ(d)y ∆ (d) = 0, donde ψ : c[ρ(c), σ(d)]t, ψ(t) > 0 para todo t ∈ [ρ(c), σ(d)]t; ambas g y h : [ρ(c), σ(d)]t × [0, ∞) → r son continuas y semipositón. hemos establecido la existencia de al menos una solución positiva o múltiples soluciones positivas del problema de valor en la frontera anterior usando un teorema de punto fijo en un cono en un espacio de banach, cuando g y h son ambas superlineales o sublineales o una es superlineal y la otra es sublineal para λi > 0; i = 1, 2 suficientemente pequeños. keywords and phrases: positive solutions, boundary value problems, fixed point theorem, cone, time scales. 2020 ams mathematics subject classification: 34b15, 34b16, 34b18, 34n05, 39a10, 39a13. accepted: 28 september, 2022 received: 12 april, 2022 ©2022 s. panigrahi et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0413 cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 415 1 introduction the study of dynamic equations on time scales goes to the seminal work of stefan hilger [11] and has received a lot of attention in recent years. time scales were created to unify the study of continuous and discrete mathematics and particularly used in differential and difference equations. we are interested to prove the results for a dynamic equation where the domain of the unknown function is a time scale t, which is a non-empty closed subset of real numbers r. we consider the second order semipositone boundary value problem on time scales: (ψ(t)y∆(t))∇ + λ1g(t, y(t)) + λ2h(t, y(t)) = 0, t ∈ [ρ(c), σ(d)]t, (1.1) with mixed boundary conditions αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c)) = 0, γy(σ(d)) + δψ(d)y∆(d) = 0, (1.2) where λ1 and λ2 are positive and (h1) ψ : c[ρ(c), σ(d)]t, ψ(t) > 0 for all t ∈ [ρ(c), σ(d)]t; (h2) α, β, γ, δ, ≥ 0 and αδ + βγ + αγ > 0; (h3) g and h : [ρ(c), σ(d)]t × [0, ∞) → r are continuous satisfying with both g and h are semipositone. d. r. anderson and p. y. wong [1], have established the existence result for the sl-bvp (1.1) and (1.2) where g is superlinear such that g(t, y) ≥ −m for some constant m > 0 and λ is in some interval of r with h(t, y) = 0. they did not establish any results concerning the existence of positive solutions for the boundary value problem (1.1) and (1.2), when g is sublinear. many findings have also been obtained for the existence of positive solution of the boundary value problem (1.1) and (1.2), when h(t, y) = 0, but only a few results have been established for the existence of positive solutions when h(t, y) ̸= 0. motivated by the work of [1] and the references cited therein, we would like to establish the sufficient conditions for the existence of positive solution of the boundary value problem (1.1) and (1.2), when g and h are both superlinear or sublinear or one is superlinear and the other is sublinear for λi > 0; i = 1, 2 are sufficiently small. it is worthy of mention that results of this paper not only apply to the set of real numbers or the set of integers but also to more general time scales such as t = n20 = {t2 : t ∈ n0}, t = { √ n : n ∈ n0}, etc. for basic notations and concepts on time scale calculus, we refer the readers to monographs [5, 6] and references cited therein. the study of nonlinear, semipositone boundary value problem has considerable importance even in differential equations. in recent years, several researchers studied 416 s. panigrahi & s. rout. cubo 24, 3 (2022) semipositone boundary value problem on time scales [1, 2, 4, 7, 10, 16, 17]. semipositone problems arise in many physical and chemical processes such as in chemical reactor theory, astrophysics, gas dynamics and fluidmechanics, relativistic mechanics, nuclear physics, design of suspension bridges, bulking of mechanical systems, combustion and management of natural resources (see [3, 9, 12, 15]). let a and b such that 0 ≤ ρ(a) ≤ a < b ≤ σ(b) < ∞ and (ρ(a), σ(b))t has at least two points. the plan of the paper is as follows. in section 2, we provided some preliminary results concerning the green’s function for the homogeneous boundary value problem and some important lemmas. these results allow us in section 3 to discuss the existence of at least one or multiple positive solutions. finally, in section 4, we illustrate few examples to justify the results obtained in the previous section. 2 preliminaries in this section, we have obtained some basic results related to green’s function for the homogeneous boundary value problem and some important lemmas. now let us consider the homogenoeous dynamic boundary value problem (ψ(t)y∆(t))∇ = 0, t ∈ [ρ(c), σ(d)]t, (2.1) with boundary conditions (1.2). green’s function g(t, s) (see [7]) for the boundary value problem (2.1) and with the boundary conditions (1.2) is given by g(t, s) = 1 φ   ( β + α ∫ t ρ(c) ∇τ ψ(τ) )( δ + γ ∫σ(d) s ∇τ ψ(τ) ) , ρ(c) ≤ t ≤ s ≤ σ(d), ( β + α ∫ s ρ(c) ∇τ ψ(τ) )( δ + γ ∫σ(d) t ∇τ ψ(τ) ) , ρ(a) ≤ s ≤ t ≤ σ(b), (2.2) where φ = αδ + βγ + αγ ∫ σ(d) ρ(c) ∇τ ψ(τ) > 0. lemma 2.1 ([17]). assume (h1) and (h2) hold. then the green function g(t, s) satisfies (ψ(t)y∆(t))∇ + q(t) = 0, t ∈ (ρ(c), σ(d))t, (2.3) with mixed boundary conditions (1.2), where q ∈ crd[ρ(c), σ(d)]t, q(t) ≥ 0; then y(t) ≥ q(t)∥y∥, t ∈ [ρ(c), σ(d)]t, s ∈ [a, b]t, (2.4) cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 417 where q(t) is given by q(t) = min   β + α ∫ t ρ(c) ∇τ ψ(τ) β + α ∫σ(d) ρ(c) ∇τ ψ(τ) , δ + γ ∫σ(d) t ∇τ ψ(τ) δ + γ ∫σ(d) ρ(c) ∇τ ψ(τ)   . lemma 2.2 ([1]). for all t ∈ [ρ(c), σ(d)]t and s ∈ [c, d]t, then q(t)g(s, s) ≤ g(t, s) ≤ g(s, s), (2.5) where g(t, s) is given in (2.2) and q(t) is defined as in lemma 2.1. lemma 2.3 ([1]). let (h1) and (h2) hold and let y1 be the solution of (ψ(t)y∆(t))∇ + 1 = 0, t ∈ (ρ(c), σ(d))t, (2.6) with mixed boundary conditions (1.2), then there exists a positive constant c such that y1(t) ≤ c q(t), t ∈ [ρ(c), σ(d)]t, (2.7) where c = 1 φ (σ(d) − ρ(c)) ( β + α ∫ σ(d) ρ(c) ∇τ ψ(τ) )( δ + γ ∫ σ(d) ρ(c) ∇τ ψ(τ) ) . lemma 2.4 ([8]). let lim y→∞ g(t, y) y = ∞ and define g : [0, ∞) → [0, ∞) by g = max ρ(c)≤t≤σ(d), 0≤y≤r g(t, y). (2.8) then (i) g is non-decreasing; (ii) lim r→∞ g(r) r = ∞; (iii) there exists r∗ > 0 such that g(r) > 0 for r ≥ r∗. lemma 2.5 ([8]). let limy→∞ g(t, y) y = 0 holds. then g defined by (2.8) is a nondecreasing function, such that lim r→∞ g(r) r = 0. define a function for y ∈ c[ρ(c), σ(d)]t, g(t, y) =   g(t, y), y ≥ 0, g(t, 0), y < 0. 418 s. panigrahi & s. rout. cubo 24, 3 (2022) and h(t, y) =   h(t, y), y ≥ 0, h(t, 0), y < 0. let us consider the nonlinear boundary value problem: (ψ(t)y∆)∇ = −[λ1g(t, y − x) + λ2h(t, y − x) + m], (2.9) with boundary conditions (1.2). lemma 2.6. assume that x(t) = my1(t), where y1(t) is a unique solution of the boundary value problem (2.6) and (1.2). then y(t) is a solution of the boundary value problem (1.1) and (1.2) if and only if y(t) = y(t) + x(t) is a positive solution of the boundary value problem (2.9) and (1.2) with y(t) > x(t) for t ∈ [ρ(c), σ(d)]t. proof. let us assume that y(t) is a solution of the boundary value problem (2.9) and (1.2) such that y(t) ≥ x(t) for any t ∈ [ρ(c), σ(d)]t. let y(t) = y(t) −x(t) > 0 on [ρ(c), σ(d)]t as y(t) ≥ x(t). now, for any t ∈ [ρ(c), σ(d)]t, we have (ψ(t)y∆(t))∇ + [λ1g(t, (y(t) − x(t)) + λ2h(t, (y(t) − x(t))) + m] = 0, that is, (ψ(t)y∆(t))∇ + (ψ(t)x∆(t))∇ + [λ1g(t, (y(t) − x(t)) + λ2h(t, (y(t) − x(t))) + m] = 0. by using the definition of y together with the definition of x, we have (ψ(t)y∆(t))∇ + [λ1g(t, y(t)) + λ2h(t, y(t)) + m] + m(ψ(t)y ∆ 1 (t)) ∇(t) = 0. thus, (ψ(t)y∆(t))∇ + λ1g(t, y(t)) + λ2h(t, y(t)) = 0. on the other hand, αy(ρ(c)) − βψ(ρ(c))y(ρ(c)) = (αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c))) − (αx(ρ(c)) − βψ(ρ(c))x∆(ρ(c))) = (αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c))) − m(αy1(ρ(c)) − βψ(ρ(c))y∆1 (ρ(c))) = 0, and γy(σ(d)) + δψ(d)y∆(d) = γy(σ(d)) + δψ(d)y∆(d) − (γx(σ(d)) + δψ(d)x∆(d)) cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 419 = γy(σ(d)) + δψ(d)y∆(d) − m(γy1(σ(d)) + δψ(d)y∆1 (d)) = 0. hence, y(t) is a solution of the boundary value problem (1.1) and (1.2). hence this completes the proof of the lemma. let us define a banach space e = {y : c[ρ(c), σ(d)]t → r} endowed with the norm ∥y∥ = max{|y(t)|, t ∈ [ρ(c), σ(d)]t}. define a cone k on e by k = {y ∈ c[ρ(c), σ(d)]t : y(t) ≥ q(t)∥y∥, t ∈ [ρ(c), σ(d)]t}, where q(t) is defined as in lemma 2.1. let us define an operator tλ on k by tλy(t) = ∫ d ρ(c) g(t, s)[λ1g(s, y(s) − x(s)) + λ2h(s, y(s) − x(s)) + m]∇s. (2.10) lemma 2.7. assume that (h1)–(h3) hold. then tλ(k) ⊂ k and tλ : k → k is a completely continuous operator. proof. first we show that tλ(k) ⊂ k. let y ∈ k and t ∈ [ρ(a), σ(b)]t. note that (tλy)(t) = ∫ d ρ(c) g(t, s)[λ1g(s, y(s) − x(s)) + λ2h(s, y(s) − x(s)) + m]∇s, that is, (tλy)(t) ≤ ∫ d ρ(c) g(s, s)[λ1g(s, y(s) − x(s)) + λ2h(s, y(s) − x(s)) + m]∇s. hence, ∥tλy∥ ≤ ∫ d ρ(c) g(s, s)[λ1g(s, y(s) − x(s)) + λ2h(s, y(s) − x(s)) + m]∇s. by use of the lemma (2.2), we obtain (tλy)(t) ≥ q(t) ∫ d ρ(c) g(s, s)[λ1g(s, y(s) − x(s)) + λ2h(s, y(s) − x(s)) + m]∇s, which implies (tλy)(t) ≥ q(t)∥tλy∥. thus, tλ(k) ⊂ k. since f and g are continuous, it shows that tλ is continuous and by the 420 s. panigrahi & s. rout. cubo 24, 3 (2022) arzelà-ascoli theorem [14], it is easy to verify that tλ is a completely continuous operator. hence this completes the proof of the lemma lemma 2.8 ([13]). let e be a real banach space, and let k ⊂ e be a cone. let ω1, ω2 be two bounded open subsets of e with 0 ∈ ω1, ω1 ⊂ ω2. assume that t : k ∩ (ω2 \ ω1) → k be a completely continuous operator such that either ∥ty∥ ≤ ∥y∥ for all y ∈ k ∩ ∂ω1 and and ∥ty∥ ≥ ∥y∥ for all y ∈ k ∩ ∂ω2, or ∥ty∥ ≥ ∥y∥ for all y ∈ k ∩ ∂ω1 and ∥ty∥ ≤ ∥y∥ for all y ∈ k ∩ ∂ω2, then t has at least one fixed point in k ∩ (ω2 \ ω1). let us define the following: (l1) lim y→∞ g(t, y) y = ∞; (l2) lim y→∞ g(t, y) y = 0; (l3) lim y→0 g(t, y) y = 0; (l4) lim y→0 g(t, y) y = ∞; (l5) lim y→∞ h(t, y) y = ∞; (l6) lim y→∞ h(t, y) y = 0; (l7) lim y→0 h(t, y) y = 0; (l8) lim y→0 h(t, y) y = ∞. note that the limits (li), i ∈ n81, are assumed to be inform with respect t. we would like to establish the existence of solutions for the boundary value problem (1.1) and (1.2) under the following cases: (i) l1 and l5; (ii) l1 and l6; (iii) l1 and l7; (iv ) l2 and l5; (v ) l2 and l6; (v i) l2 and l8; (v ii) l3 and l5; (v iii) l3 and l7; (ix) l3 and l8; (x) l4 and l6; (xi) l4 and l7; (xii) l4 and l8. remark 2.9. we fails to apply the lemma 2.8 for the pairs such as (xiii) l1 and l8, (xiv ) l2 and l7, (xv ) l3 and l6 & (xv i) l4 and l5. cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 421 3 main results theorem 3.1. let (h1)–(h3), (l1) and (l5) hold. then the boundary value problem (1.1) and (1.2) has a positive solution for λi, i = 1, 2 are sufficiently small. proof. let λ1 and λ2 satisfy 0 < λ1 + λ2 < 1 max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y) , (3.1) where r1 = max{(m + 1)∥y1∥, r∗, cm}, c and r∗ are defined as in lemma 2.3 and lemma 2.4, respectively and y1 be the solution of (1.2) and (2.6). define ωr1 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r1}. for y ∈ k ∩ ∂ωr1, we have (tλy)(t) = ∫ d ρ(c) g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≤ (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y)  ∫ d ρ(c) g(t, s)∇s + ∫ d ρ(c) g(t, s)m∇s =  (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y)   + m  y1(t) ≤ (1 + m)y1(t) ≤ r1 = ∥y∥. thus, ∥tλy∥ ≤ ∥y∥ for y ∈ k ∩ ∂ωr1. (3.2) let us choose a constant m > 0 such that 1 2 m(λ1 + λ2)µ ( min t1≤t≤t2 ∫ t2 t1 g(t, s)∇s ) ≥ 1, (3.3) where µ = min t1≤s≤t2 q(s). (3.4) from (l1) and (l5), we have for same m > 0 there exists a constant l > 0 such that g(t, y) ≥ my for y ∈ [l, ∞), h(t, y) ≥ my for y ∈ [l, ∞). now set r2 = max { 2r1, 2cm, 2l1 µ } . define ωr2 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r2}. for y ∈ 422 s. panigrahi & s. rout. cubo 24, 3 (2022) k ∩ ∂ωr2, we have y(s) − x(s) = y(s) − my1(s) ≥ y(s) − mcq(s) ≥ y(s) − cm ∥y∥ y(s) ≥ y(s) − cm r2 y(s) ≥ 1 2 y(s), and min t1≤s≤t2 (y(s) − x(s)) ≥ min t1≤s≤t2 y(s) 2 ≥ min t1≤s≤t2 ∥y∥ 2 q(s) = r2µ 2 ≥ l. for y ∈ k ∩ ∂ωr2, we have min t∈[t1, t2] (tλy)(t) = min t∈[t1, t2] ∫ d ρ(c) g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)(λ1 + λ2)m(y(s) − x(s))∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)(λ1 + λ2)m y(s) 2 ∇s ≥ 1 2 (λ1 + λ2)mµ min t∈[t1, t2] ∫ t2 t1 g(t, s)∥y∥∇s ≥ ∥y∥. thus, ∥tλy∥ ≥ ∥y∥ for y ∈ k ∩ ∂ωr2. (3.5) by lemma 2.8, tλ has a fixed point y with r1 ≤ ∥y∥ ≤ r2. by use of the lemma 2.3, it follows cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 423 that y(t) ≥ r1q(t) ≥ r1 y1(t) c ≥ my1(t) = x(t). hence, y = y − x is a positive solution of the boundary value problem (1.1) and (1.2). this completes the proof of the theorem. theorem 3.2. let (h1)–(h3), (l4) and (l8) hold. then the boundary value problem (1.1) and (1.2) has a positive solution for λi, i = 1, 2 are sufficiently small. proof. the proof of theorem 3.2 is similar to that of theorem 3.1, hence it is omitted. theorem 3.3. assume that (h1), (h2), (l2) and (l6) hold. let there exist two constant d > 0 and η > 0 such that g(t, y) ≥ η for t ∈ [ρ(c), σ(d)], y ∈ [d, ∞), h(t, y) ≥ η for t ∈ [ρ(c), σ(d)], y ∈ [d, ∞), then the boundary value problem (1.1) and (1.2) has a positive solution for λi, i = 1, 2 are sufficiently small. proof. set r1 = max { 2d µ , 2mc } , (3.6) and a = 2r1 ( min t1≤t≤t2 ∫ t2 t1 g(t, s)(λ1 + λ2)η∇s )−1 , (3.7) where µ = min t1≤s≤t2 q(s). our claim is that for λi ∈ [a, ∞), i = 1, 2, the boundary value problem (1.1) and (1.2) has a positive solution. define ωr1 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r1}. for y ∈ k ∩ ∂ωr1, we have y(s) − x(s) = y(s) − my1(s) ≥ y(s) − mcq(s) ≥ y(s) − cm r3 y(s) ≥ 1 2 y(s), 424 s. panigrahi & s. rout. cubo 24, 3 (2022) and min t1≤s≤t2 (y(s) − x(s)) ≥ min t1≤s≤t2 y(s) 2 ≥ min t1≤s≤t2 ∥y∥ 2 q(s) = r1µ 2 ≥ d. for y ∈ k ∩ ∂ωr1, we have min t∈[t1, t2] (tλy)(t) = min t∈[t1, t2] ∫ d ρ(c) g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)(λ1 + λ2)η∇s = r1 = ∥y∥. thus, ∥tλy∥ ≥ ∥y∥ for y ∈ k ∩ ∂ωr1. (3.8) from (l2) and (l6), we have g(t, y) ≤ ϵy for t ∈ [ρ(c), σ(d)], y ≥ l, h(t, y) ≤ ϵy for t ∈ [ρ(c), σ(d)], y ≥ l, on the other hand, by use of the lemma 2.4, there exists a r > 0 such that r > max { 2r1, max ρ(c)≤t≤σ(d) ∫ d ρ(c) [g(t, s)m + 1]∇s } . and ϵ satisfies max ρ(c)≤t≤σ(d) ∫ d ρ(c) g(t, s)[ϵλ1r + ϵλ2r + m]∇s ≤ r. let ωr = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r}. cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 425 for y ∈ k ∩ ∂ωr, we have tλy(t) = ∫ d ρ(c) g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≤ ∫ d ρ(c) g(t, s)[λ1ϵr + ϵλ2r + m]∇s ≤ r = ∥y∥. thus, ∥tλy∥ ≤ ∥y∥ for y ∈ k ∩ ∂ωr. (3.9) by lemma 2.8, tλ has a fixed point y with r1 ≤ ∥y∥ ≤ r. it follows that y(t) ≥ r1q(t) ≥ r1 y1(t) c ≥ 2my1(t) ≥ x(t). hence, y = y − x is a positive solution of the boundary value problem (1.1) and (1.2). this completes the proof of the theorem. theorem 3.4. assume that (h1)–(h3), (l3) and (l7) hold. let there exist two constant d > 0 and η > 0 such that g(t, y) ≥ η for t ∈ [ρ(c), σ(d)]y ∈ [d, l], h(t, y) ≥ η for t ∈ [ρ(c), σ(d)]y ∈ [d, l], then the boundary value problem (1.1) and (1.2) has a positive solution for λi, i = 1, 2 are sufficiently small. proof. the proof of the theorem 3.4 is similar to that of theorem 3.3, hence it is omitted. theorem 3.5. let (h1)–(h3), (l1) and (l6) hold. then the boundary value problem (1.1) and (1.2) has at least two positive solutions for λi, 1 = 1, 2 are sufficiently small. proof. if (l6) holds, then by the lemma 2.5, there exists a constant r1 > 0 such that g(r1) ≤ nr1. 426 s. panigrahi & s. rout. cubo 24, 3 (2022) since λ1 and λ2 are sufficiently small, we have λ1 max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + λ2 max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y) + m  ∫ d ρ(c) g(s, s)∇s ≤ r1. let ωr1 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r1}. for y ∈ ∂ωr1, we have (tλy)(t) = ∫ d ρ(c) g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≤  λ1 max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + λ2 max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y)  ∫ d ρ(c) g(s, s)∇s + ∫ d ρ(c) g(s, s)m∇s ≤ r1 = ∥y∥. thus, ∥tλy∥ ≤ ∥y∥ for all y ∈ k ∩ ∂ωr1. (3.10) from (l1), we have g(t, y) > n1y for all y ≤ l. let r2 = max { 2cm, 2l µ ,2r1 } and ωr2 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r2}. for y ∈ ∂k ∩ ωr2, we have y(s) − x(s) = y(s) − my1(s) ≥ y(s) − mcq(s) ≥ y(s) − cm r5 y(s) ≥ 1 2 y(s), and min t1≤s≤t2 (y(s) − x(s)) ≥ min t1≤s≤t2 y(s) 2 ≥ min t1≤s≤t2 ∥y∥ 2 q(s) = r2µ 2 ≥ l. cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 427 for y ∈ k ∩ ∂ωr2, we have min t∈[t1, t2] (tλy)(t) = min t∈[t1, t2] ∫ d ρ(c) g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)[λ1g(s, y − x) + λ2h(s, y − x) + m]∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)λ1n1(y(s) − x(s))∇s = r2 = ∥y∥. thus, ∥tλy∥ ≥ ∥y∥ for all y ∈ k ∩ ∂ωr2. (3.11) let r = max    λ1 max ρ(c)≤t≤σ(d) 0≤y≤r g(t, y) + λ2 max ρ(c)≤t≤σ(d) 0≤y≤r h(t, y) + m  (∫ d ρ(c) g(s, s)∇s ) ,2r2   , then r1 < r2 < r. let ωr = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r}. for y ∈ k ∩ ωr, t ∈ [ρ(c), σ(d)]t, we have (tλy)(t) = ∫ d ρ(c) g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≤  λ1 max ρ(c)≤t≤σ(d) 0≤y≤r g(t, y) + λ2 max ρ(c)≤t≤σ(d) 0≤y≤r h(t, y) + m  ∫ d ρ(c) g(s, s)∇s ≤ r = ∥y∥. thus, ∥tλy∥ ≤ ∥y∥ for all y ∈ k ∩ ∂ωr. (3.12) thus by the lemma 2.8, tλ has at least two fixed points. hence, the boundary value problem (1.1) and (1.2) has at least two positive solutions. theorem 3.6. let (h1)–(h3), (l2) and (l5) hold. then the boundary value problem (1.1) and (1.2) has at least two positive solutions for λi, i = 1, 2 are sufficiently small. proof. the proof of the theorem 3.6 is similar to that of theorem 3.5. theorem 3.7. let (h1)–(h3), (l4) and (l7) hold. then the boundary value problem (1.1) and (1.2) has at least two positive solutions for λi, i = 1, 2 are sufficiently small. 428 s. panigrahi & s. rout. cubo 24, 3 (2022) proof. from (l7), we have lim y→0 h(t, y) y = 0. for ϵ > 0, there exists a r1 > 0 such that h(t, y) ≤ ϵy for y ∈ [0, r1). since λ1 and λ2 are sufficiently small, we have (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y)   + m  ∫ d ρ(c) g(s, s)∇s ≤ r1. let ωr1 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r1}. for y ∈ ∂ωr1, we have (tλy)(t) = ∫ d ρ(c) g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≤ ∫ d ρ(c) g(s, s) ( λ1g(s, y − x) + λ2h(s, y − x) ) ∇s + ∫ d ρ(c) g(s, s)m∇s ≤  (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y)   + m  ∫ d ρ(c) g(s, s)∇s ≤ r1 = ∥y∥ thus, ∥tλy∥ ≤ ∥y∥ for all y ∈ k ∩ ∂ωr1. (3.13) from (l4), we have g(t, y) > n1y for all y ≤ l. let r2 = max { 2cm, 2l µ ,2r1 } and ωr2 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r2}. for y ∈ k ∩ ∂ωr2, we have y(s) − x(s) = y(s) − my1(s) ≥ y(s) − mcq(s) ≥ y(s) − cm r5 y(s) ≥ 1 2 y(s), cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 429 and min t1≤s≤t2 (y(s) − x(s)) ≥ min t1≤s≤t2 y(s) 2 ≥ min t1≤s≤t2 ∥y∥ 2 q(s) = r2µ 2 ≥ l. for y ∈ k ∩ ∂ωr2, we have min t∈[t1, t2] (tλy)(t) = min t∈[t1, t2] ∫ d ρ(c) g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)λ1n1(y(s) − x(s))∇s = r2 = ∥y∥. thus, ∥tλy∥ ≥ ∥y∥ for all y ∈ k ∩ ∂ωr2. (3.14) let r = max    λ1 max ρ(c)≤t≤σ(d) 0≤y≤r g(t, y) + λ2 max ρ(c)≤t≤σ(d) 0≤y≤r h(t, y) + m  (∫ d ρ(c) g(s, s)∇s ) ,2r2   , then r1 < r2 < r. let ωr = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r}. for y ∈ k ∩ ωr, t ∈ [ρ(c), σ(d)]t, we have (tλy)(t) = ∫ d ρ(c) g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≤  (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r h(t, y)   + m  ∫ d ρ(c) g(s, s)∇s ≤ r = ∥y∥ thus, ∥tλy∥ ≤ ∥y∥ for all y ∈ k ∩ ∂ωr. (3.15) thus by the lemma 2.8, tλ has at least two fixed points. hence, the boundary value problem (1.1) and (1.2) has at least two positive solutions. 430 s. panigrahi & s. rout. cubo 24, 3 (2022) theorem 3.8. let (h1)–(h3), (l3) and (l8) hold. then the boundary value problem (1.1) and (1.2) has at least two positive solutions for λi, i = 1, 2 are sufficiently small. proof. the proof of the theorem 3.8 is similar to that of theorem 3.5. theorem 3.9. let (h1)–(h3), (l1) and (l7) hold. then the boundary value problem (1.1) and (1.2) has at least one positive solution for λi, i = 1, 2 are sufficiently small. proof. from (l1), we have lim y→∞ g(t, y) y = ∞. for k > 0, there exists a r1 > 0 such that g(t, y) ≥ ky for y > r1. let ωr1 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r1} and let k satisfy kµ 2 λ1 min t∈[t1, t2] ∫ t2 t1 g(t, s)∇s ≥ 1. for y ∈ k ∩ ∂ωr1, we have min t∈[t1, t2] (tλy)(t) = min t∈[t1, t2] ∫ d ρ(c) g(t, s) [ λ1g(s, y − x) + λ2h(s, y − x) + m ] ∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)λ1k(y − x)∇s ≥ k 2 λ1 min t∈[t1, t2] ∫ t2 t1 g(t, s)∥y∥q(s)∇s ≥ ∥y∥. thus, ∥tλy∥ ≥ ∥y∥ for y ∈ k ∩ ∂ωr1. (3.16) from (l7), we have lim y→0 h(t, y) y = 0. for ϵ > 0, there exists a r2 > 0 such that h(t, y) ≤ ϵy for y ∈ [0, ∞). cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 431 since λ1 and λ2 are sufficiently small, let (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r2 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r2 h(t, y) + m    ∫ d ρ(c) g(s, s)∇s ≤ r2. let ωr2 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r2}. now for any y ∈ k ∩ ∂ωr2, we have (tλy)(t) = ∫ d ρ(c) g(t, s) [ λ1g(t, y − x) + λ2h(s, y − x) + m ] ∇s ≤ ∫ d ρ(c) g(s, s) [ λ1g(t, y − x) + λ2h(s, y − x) + m ] ∇s ≤  (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r2 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r2 h(t, y)   + m  ∫ d ρ(c) g(s, s)∇s ≤ r2 = ∥y∥. thus, ∥tλy∥ ≤ ∥y∥ for y ∈ k ∩ ∂ωr2. (3.17) hence, by the lemma 2.8, tλ has a fixed point y with r1 < ∥y∥ < r2. by the lemma 2.6, the boundary value problem (1.1) and (1.2) has at least one positive solution. theorem 3.10. let (h1)–(h3), (l3) and (l5) hold. then the boundary value problem (1.1) and (1.2) has at least one positive solution for λi, i = 1, 2 are sufficiently small. proof. the proof of the theorem 3.10 is similar to that of theorem 3.9. theorem 3.11. let (h1)–(h3), (l2) and (l8) hold. then the boundary value problem (1.1) and (1.2) has at least one positive solution for λi, i = 1, 2 are sufficiently small. proof. from (l2), we have lim y→∞ g(t, y) y = 0. by lemma 2.5, there exist r1 > 0 and k1 > 0 such that g(r1) ≤ k1r1. let ωr1 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r1}. since λ1 and λ2 are sufficiently small, we have (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r1 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r1 h(t, y)   + m  ∫ d ρ(c) g(s, s)∇s ≤ r1. 432 s. panigrahi & s. rout. cubo 24, 3 (2022) for any y ∈ k ∩ ∂ωr1, we obtain (tλy)(t) = ∫ d ρ(c) g(t, s) [ λ1g(t, y − x) + λ2h(s, y − x) + m ] ∇s ≤ ∫ d ρ(c) g(s, s) [ λ1g(t, y − x) + λ2h(s, y − x) + m ] ∇s ≤  (λ1 + λ2)   max ρ(c)≤t≤σ(d) 0≤y≤r2 g(t, y) + max ρ(c)≤t≤σ(d) 0≤y≤r2 h(t, y)   + m  ∫ d ρ(c) g(s, s)∇s ≤ r1 = ∥y∥. thus, ∥tλy∥ ≤ ∥y∥ for y ∈ k ∩ ∂ωr1. (3.18) from (l8), we have lim y→0 h(t, y) y = ∞. for k > 0, there exists a l > 0 such that h(t, y) ≥ ky for y ∈ [0, l]. let r2 = { 2cm, 2l µ ,2r1 } and ωr2 = {y ∈ c[ρ(c), σ(d)]t : ∥y∥ < r2}. for any y ∈ k ∩ ∂ωr2, we have min t∈[t1, t2] (tλy)(t) = min t∈[t1, t2] ∫ d ρ(c) g(t, s) [ λ1g(t, y − x) + λ2h(s, y − x) + m ] ∇s ≥ min t∈[t1, t2] ∫ t2 t1 g(t, s)λ2k(y(s) − x(s))∇s ≥ k 2 λ2µ min t∈[t1, t2] ∫ t2 t1 g(t, s)∥y∥∇s ≥ r2 = ∥y∥. thus, ∥tλy∥ ≥ ∥y∥ for y ∈ k ∩ ∂ωr2. (3.19) hence, by the lemma 2.8, tλ has a fixed point y with r1 < ∥y∥ < r2. by the lemma 2.6, the boundary value problem (1.1) and (1.2) has at least one positive solution. theorem 3.12. let (h1)–(h3), (l4) and (l6) hold. then the boundary value problem (1.1) and (1.2) has at least one positive solution for λi, , i = 1, 2 are sufficiently small. proof. the proof of the theorem 3.12 is similar to that of theorem 3.11. cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 433 4 examples we shall illustrate few examples in different time scales to justify the results obtained in the preceding section. example 4.1. let us consider the following boundary value problem on time scale t = r, ((1 + t2)y′)′ + 1 2 1 + y2 52 + 1 4 y2 sin2 y 35 = 0, t ∈ [0, 1], (4.1) with boundary conditions y(0) − y′(0) = 0, y(1) + 2y′(1) = 0, (4.2) where ψ(t) = 1+t2, m = 1, α, β, γ, δ ≥ 0, g(t, y) = 1+y 2 52 and h(t, y) = y 2 sin2 y 35 . green’s function for the boundary value problem (4.1) and (4.2) is given by g(t, s) = 1 2 + π 4   ( 1 + tan−1 t )( 1 + π 4 − tan−1 s ) , t ≤ s,( 1 + tan−1 s )( 1 + π 4 − tan−1 t ) , s ≤ t. all the conditions (h1)–(h3), (l1) and (l5) are satishfied for (t, y) ∈ [0, 1]×[0, 100]. by theorem 3.1, boundary value problem (4.1) and (4.2) has at least one positive solution for λ1 = 1 2 and λ2 = 1 4 . example 4.2. let us consider the following boundary value problem on time scale t = z, ∇((1 + t)−1y∆) + λ1 sin2 y + λ2 √ y cos y = 0, t ∈ [0, 3], (4.3) with boundary conditions y(0) − ∆y(0) = 0, y(3) + 1 3 ∆y(2) = 0, (4.4) where ψ(t) = (1 + t)−1, m = 1, α, β, γ, δ ≥ 0, g(t, y) = sin2 y and h(t, y) = √ y cos y. green’s function for the boundary value problem (4.3) and (4.4) is given by g(t, s) = 1 11   ( 1 + t 2+3t 2 )( 1 + (3−s)(s+6) 2 ) , t ≤ s,( 1 + s 2+3s 2 )( 1 + (3−t)(t+6) 2 ) , s ≤ t. all the conditions (h1)–(h3), (l2) and (l6) are satishfied for (t, y) ∈ [0, 3] × [0, 100]. let d = 1 434 s. panigrahi & s. rout. cubo 24, 3 (2022) and η = 1 2 such that g(t, y) ≥ 1 2 and h(t, y) ≥ 1 2 for t ∈ [0, 3], y ∈ [1, ∞). by theorem 3.3, boundary value problem (4.3) and (4.4) has at least one positive solution for λi; i = 1, 2 are sufficiently small. example 4.3. consider the boundary value problem on time scale t = qz = {2k : k ∈ z} ∪ {0}, where q = 2 > 1, dq ( (1 + t)−1dqy(t) ) + λ1 y2 sin y + λ2 ln(y) = 0, t ∈ [0, 2], (4.5) with boundary conditions y(0) − dqy(0) = 0, y(2) + 1 2 dqy(1) = 0, (4.6) where ψ(t) = (1+t)−1, m = 1, α, β, γ, δ ≥ 0, g(t, y) = y 2 sin y and h(t, y) = ln(y). green’s function for the boundary value problem (4.5) and (4.6) is given by g(t, s) = 3 20   ( 2t2+3t+3 3 )( 17−3s−2s2 3 ) , t ≤ s,( 2s2+3s+3 3 )( 17−3t−2t2 3 ) , s ≤ t. the conditions (h1)–(h3), (l1) and (l6) are satishfied for (t, y) ∈ [0, 2] × [0, 100]. by theorem 3.5, boundary value problem (4.5) and (4.6) has at least two positive solutions for λi; i = 1, 2 are sufficiently small. example 4.4. let us consider the time scale t = pa,b = ∞⋃ k=0 [k(a + b),k(a + b) + a] = p1, 1 = ∞⋃ k=0 [2k,2k + 1], where a = b = 1. consider the following boundary value problem:   y∆∇ + λ1 √ y + λ2 y ln(1 + y), t ∈ (0, 2), y(0) = 0, y(2) = 0, (4.7) where ψ(t) = 1, m = 1, α, β, γ, δ ≥ 0, g(t, y) = √ y and h(t, y) = y ln(y). green’s function for the boundary value problem (4.7) is given by g(t, s) = 1 2   t(1 − s), t ≤ s, (1 + s)(2 − t), s ≤ t. the conditions (h1)–(h3), (l4) and (l7) are satishfied for (t, y) ∈ [0, 2] × [0, 100]. by theorem 3.7, boundary value problem (4.7) has at least two positive solutions for λi; i = 1, 2 are sufficiently small. cubo 24, 3 (2022) existence of positive solutions for a nonlinear semipositone... 435 example 4.5. consider the boundary value problem on time scale t = {n 2 : t ∈ n0}: y∆∇(t) + λ1y ln(1 + y) + λ2 √ y siny 6 = 0, t ∈ [ 0, 3 2 ] , (4.8) with boundary conditions y(0) − y∆(0) = 0, y ( 3 2 ) + y∆(1) = 0, (4.9) where ψ(t) = 1, m = 1, α, β, γ, δ ≥ 0, g(t, y) = y ln(1+y) and h(t, y) = √ y sin y 6 . green’s function for the boundary value problem (4.8) and (4.9) is given by g(t, s) = 2 7   (1 + s) ( 5 2 − t ) , t ≤ s, (1 + t) ( 5 2 − s ) , s ≤ t. the conditions (h1)–(h3), (l1) and (l7) are satisfied for (t, y) ∈ [0, 32] × [0, 100]. by theorem 3.9, boundary value problem (4.8) and (4.9) has at least one positive solutions for λi; i = 1, 2 are sufficiently small. example 4.6. consider the following boundary value problem in time scale t = hz = {hk : k ∈ z}, where h = 1 2 > 0, ( (1 + t)−1y∆ )∇ + λ1 √ y siny + λ2 = 0 for t ∈ [0, 2], (4.10) with boundary conditions y(0) − y∆(0) = 0, y(2) + 2 5 y∆ ( 3 2 ) = 0, (4.11) where ψ(t) = (1+t)−1, m = 1, α, β, γ, δ ≥ 0, g(t, y) = √ y siny and h(t, y) = 1. green’s function for the boundary value problem (4.10) and (4.11) is given by g(t, s) = 2 13   ( 1 + s(2s+5) 4 )( 1 + (2−t)(2t+9) 4 ) , s ≤ t,( 1 + t(2t+5) 4 )( 1 + (2−s)(2s+9) 4 ) , t ≤ s. the conditions (h1)–(h3), (l2) and (l8) are satishfied for (t, y) ∈ [0, 2] × [0, 100]. by theorem 3.11, boundary value problem (4.10) and (4.11) has at least one positive solutions for λi; 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[17] y. yang and f. meng, “positive solutions of the singular semipositone boundary value problem on time scales”, math. comput. modelling, vol. 52, no. 3–4, pp. 481–489, introduction preliminaries main results examples cubo, a mathematical journal vol. 24, no. 03, pp. 467-484, december 2022 doi: 10.56754/0719-0646.2403.0467 positive solutions of nabla fractional boundary value problem n. s. gopal1 j. m. jonnalagadda1, b 1 department of mathematics, birla institute of technology and science pilani, hyderabad 500078, telangana, india. nsgopal94@gmail.com j.jaganmohan@hotmail.com b abstract in this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with dirichlet boundary conditions. − ( ∇νρ(a)u ) (t) + λu(t) = f(t, u(t)), t ∈ nba+2, u(a) = u(b) = 0, where 1 < ν < 2, a, b ∈ r with b−a ∈ n3, nba+2 = {a+2, a+ 3, . . . , b}, |λ| < 1, ∇νρ(a)u denotes the ν th-order riemann– liouville nabla difference of u based at ρ(a) = a − 1, and f : nba+2 × r → r+. we make use of guo–krasnosels’kǐı and leggett–williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. we establish sufficient requirements for at least one, at least two, and at least three positive solutions of the considered boundary value problem. we also provide an example to demonstrate the applicability of established results. accepted: 10 november, 2022 received: 25 april, 2022 ©2022 n. s. gopal et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0467 https://orcid.org/0000-0002-1166-3446 https://orcid.org/0000-0002-1310-8323 mailto:nsgopal94@gmail.com mailto:j.jaganmohan@hotmail.com cubo, a mathematical journal vol. 24, no. 03, pp. 467–485, december 2022 doi: 10.56754/0719-0646.2403.0467 resumen en este art́ıculo consideramos el siguiente problema de valor en la frontera de dos puntos discreto fraccional con coeficientes constantes asociado a condiciones de frontera de tipo dirichlet − ( ∇νρ(a)u ) (t) + λu(t) = f(t, u(t)), t ∈ nba+2, u(a) = u(b) = 0, donde 1 < ν < 2, a, b ∈ r con b − a ∈ n3, nba+2 = {a + 2, a + 3, . . . , b}, |λ| < 1, ∇νρ(a)u denota la nabla diferencia de riemann–liouville de u de orden ν basada en ρ(a) = a − 1, y f : nba+2 × r → r+. usamos los teoremas de punto fijo de guo–krasnosels’kĭı y leggett–williams en conos adecuados y bajo condiciones apropiadas en la parte nolineal de la ecuación en diferencias. establecemos requerimientos suficientes para al menos una, al menos dos, y al menos tres soluciones positivas del problema de valor en la frontera considerado. también entregamos un ejemplo para mostrar la aplicabilidad de los resultados. keywords and phrases: nabla fractional difference, boundary value problem, dirichlet boundary conditions, positive solution, existence, fixed-point. 2020 ams mathematics subject classification: 39a12. accepted: 10 november, 2022 received: 25 april, 2022 ©2022 n. s. gopal et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0467 cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 469 1 introduction nabla fractional calculus is a branch of mathematics that deals with arbitrary order differences and sums in the backward sense. the theory of nabla fractional calculus is still in its early stages, with the most important contributions appearing in the last two decades. gray & zhang [15] and miller & ross in [34] first introduced the concept of nabla fractional difference and sum. atici & eloe [2] developed the riemann–liouville type nabla fractional difference operator. they also studied the nabla fractional initial value problem, and established the exponential law, product rule, and nabla laplace transform in this line. several mathematicians [2, 3, 4, 5, 6, 7, 8, 16, 17, 21, 22] have contributed to the development of the theory of discrete fractional calculus in line with the theory of continuous fractional calculus. for historical references on continuous fractional calculus, see [28, 31, 32]. as a result of their works, today discrete fractional calculus has turned into a fruitful field of research in science and engineering. we refer here to recent monographs [9, 12, 29] and the references therein, which are important resources pertaining to this field of work. the study of boundary value problems (bvps) has a long past and can be followed back to the work of euler and taylor on vibrating strings. on the discrete fractional side, there is a sudden growth in interest for the development of nabla fractional bvps. many authors have studied nabla fractional bvps recently. to name a few, goar [11] and ikram [18] worked with self-adjoint caputo nabla bvps. gholami et al. [10] obtained the green’s function for a non-homogeneous riemann– liouville nabla bvp with dirichlet boundary conditions. jonnalagadda [19, 20, 23] analysed some qualitative properties of two-point non-linear riemann–liouville nabla fractional bvps associated with a variety of boundary conditions. as pointed out earlier, many authors have studied the discrete fractional two-point boundary value problem like in [4, 19] and recently authors in [23] have worked with general nabla fractional difference equation with constant coefficients coupled with dirichlet conditions, which resulted in for the first time green’s function in terms of discrete mittag–leffler function along with a few properties of the same. compared to discrete taylor monomial, discrete mittag–leffler function is an infinite series because of which it poses a challenge while proving positivity of green’s function. in the article, [23] the authors have overcome this challenge of proving positivity of green’s function. in the present article, we use the positivity of green’s function and prove an important lemma which helps us deal with conical mappings by proving that a ratio of infinite series is increasing or decreasing with respect to the ratio of its coefficient. to the best of our knowledge, no work has been done with leggett–williams fixed-point theorem in the nabla setting. we consider the following boundary value problem  − ( ∇ν ρ(a) u ) (t) + λu(t) = f(t, u(t)), t ∈ nba+2, u(a) = u(b) = 0, (1.1) 470 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) where 1 < ν < 2, a, b ∈ r with b−a ∈ n3, nba+2 = {a+2, a+3, . . . , b}, |λ| < 1, ∇νρ(a)u denotes the νth-order riemann–liouville nabla difference of u based at ρ(a) = a − 1, and f : nba+2 × r → r+. the present paper is organized as follows: section 2 contains preliminaries on nabla fractional calculus. in section 3, we establish some properties of the green’s function associated with the nabla fractional boundary value problem (1.1) and construct the existence of at least one, at least two and at least three positive solutions with the help of guo–krasnosel’skĭı and leggett–williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. finally, we conclude this article with an example to demonstrate the applicability of our results. 2 preliminaries denote the set of all real numbers and positive integers by r and z+, respectively. we use the following notations, definitions and known results of nabla fractional calculus [12]. assume empty sums and products are 0 and 1, respectively. definition 2.1. for a ∈ r, the sets na and nba, where b − a ∈ z+, are defined by na = {a, a + 1, a + 2, . . .}, nba = {a, a + 1, a + 2, . . . , b}. let u : na → r and n ∈ n1. the first order backward (nabla) difference of u is defined by( ∇u ) (t) = u(t) − u(t − 1), for t ∈ na+1, and the nth-order nabla difference of u is defined recursively by ( ∇nu ) (t) = ( ∇ ( ∇n−1u )) (t), for t ∈ na+n. definition 2.2 ([12]). for t ∈ r\{. . . , −2, −1, 0} and r ∈ r such that (t+r) ∈ r\{. . . , −2, −1, 0}, the generalized rising function (many authors employ the pochhammer symbol [33] to denote the same) is defined by tr = γ(t + r) γ(t) . here γ(·) denotes the euler gamma function. also, if t ∈ {. . . , −2, −1, 0} and r ∈ r such that (t + r) ∈ r \ {. . . , −2, −1, 0}, then we use the convention that tr = 0. definition 2.3 ([12]). let t, a ∈ r and µ ∈ r \ {. . . , −2, −1}. the µth-order nabla fractional taylor monomial is given by hµ(t, a) = (t − a)µ γ(µ + 1) , provided the right-hand side exists. we observe the following properties of the nabla fractional taylor monomials. cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 471 lemma 2.4 ([18, 19]). let µ > −1 and s ∈ na. then the following hold: (1) if t ∈ nρ(s), then hµ(t, ρ(s)) ≥ 0 and if t ∈ ns, then hµ(t, ρ(s)) > 0. (2) if t ∈ ns and −1 < µ < 0, then hµ(t, ρ(s)) is an increasing function of s. (3) if t ∈ ns+1 and −1 < µ < 0, then hµ(t, ρ(s)) is a decreasing function of t. (4) if t ∈ nρ(s) and µ > 0, then hµ(t, ρ(s)) is a decreasing function of s. (5) if t ∈ nρ(s) and µ ≥ 0, then hµ(t, ρ(s)) is a non-decreasing function of t. (6) if t ∈ ns and µ > 0, then hµ(t, ρ(s)) is an increasing function of t. (7) if 0 < v ≤ µ, then hv(t, a) ≤ hµ(t, a), for each fixed t ∈ na. definition 2.5 ([12]). let u : na+1 → r and ν > 0. the νth-order nabla sum of u is given by ( ∇−νa u ) (t) = t∑ s=a+1 hν−1(t, ρ(s))u(s), t ∈ na+1. definition 2.6 ([12]). let u : na+1 → r, ν > 0 and choose n ∈ n1 such that n − 1 < ν ≤ n. the νth-order riemann–liouville nabla difference of u is given by ( ∇νau ) (t) = ( ∇n ( ∇−(n−ν)a u )) (t), t ∈ na+n. lemma 2.7 ([13]). let a, b be two real numbers such that 0 < a ≤ b and 1 < α < 2. then (a−s)α−1 (b−s)α−1 is a decreasing function of s for s ∈ na−10 . lemma 2.8 ([12]). assume the successive fractional nabla taylor monomials are well defined. (1) let ν > 0 and α ∈ r. then, ∇−νa hα(t, a) = hα+ν(t, a), for t ∈ na. (2) let ν, α ∈ r and n ∈ n1 such that n − 1 < ν ≤ n. then, ∇νahα(t, a) = hα−ν(t, a), for t ∈ na+n. finally, we present the definition of the nabla mittag–leffler function which is the nabla analogue of classical mittag-leffler function [14, 30]. definition 2.9 ([12]). let α, β, λ ∈ r such that α > 0 and |λ| < 1. the nabla mittag–leffler function is defined by eλ,α,β(t, a) = ∞∑ n=0 λnhαn+β(t, a), for t ∈ na. 472 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) theorem 2.10 ([23]). assume 1 < ν < 2, −1 < λ < 1 and h : na+2 → r. the unique solution of the nabla fractional boundary value problem  − ( ∇ν ρ(a) u ) (t) + λu(t) = h(t), t ∈ nba+2, u(a) = u(b) = 0, (2.1) is given by u(t) = b∑ s=a+2 g(t, s)h(s), t ∈ nba, (2.2) where g(t, s) =   g1(t, s) = eλ,ν,ν−1(t, a) eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s)), s ∈ nbt+1, g2(t, s) = eλ,ν,ν−1(t, a) eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s)) − eλ,ν,ν−1(t, ρ(s)), s ∈ nta+2. (2.3) now, we state some positive properties of the green’s function (2.3). lemma 2.11 ([23]). assume 1 < ν < 2 and t ∈ na+2. for each 0 ≤ λ < 1, denote by g(λ) = ∞∑ n=0 λnhνn+ν−3(t, ρ(a)) (2.4) = ∞∑ n=0 λn γ(t − a + νn + ν − 2) γ(t − a + 1)γ(νn + ν − 2) . (2.5) then there exists a unique λ̄ = λ̄(t) ∈ (0, 1) such that g(λ̄) = 0. (2.6) take λ∗ = min t∈nb a+2 λ̄(t). then, 0 < λ∗ < 1. we observe the following properties of the nabla mittag-leffler function lemma 2.12 ([23]). assume 1 < ν < 2 and 0 ≤ λ < 1. then, (1) 0 < hν−1(t, ρ(a)) ≤ eλ,ν,ν−1(t, ρ(a)) for t ∈ na; (2) eλ,ν,ν−1(t, ρ(a)) is an increasing function with respect to t for t ∈ na; (3) 0 < hν−2(t, ρ(a)) ≤ ∇eλ,ν,ν−1(t, ρ(a)) for t ∈ na+1; (4) ∇eλ,ν,ν−1(t, ρ(a)) is a decreasing function with respect to t for t ∈ na+1 and λ ∈ (0, λ∗]; (5) eλ,ν,ν−1(t, ρ(s)) ≤ eλ,ν,ν−1(t, a) for t ∈ ns and s ∈ na+1; (6) ∇eλ,ν,ν−1(t, ρ(s)) ≥ ∇eλ,ν,ν−1(t, a) for t ∈ ns, s ∈ na+1 and λ ∈ (0, λ∗]. cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 473 lemma 2.13 ([27]). let (an) and (bn) (n = 0, 1, 2, . . . ) be real numbers and let the power series a(x) = ∞∑ n=0 anx n and b(x) = ∞∑ n=0 bnx n be convergent for |x| < r. if bn > 0, n = 0, 1, 2, . . . and the sequence ( an bn ) n≥0 is (strictly) increasing (decreasing), then the function a(x) b(x) is also (strictly) increasing (decreasing) on [0, r). theorem 2.14 ([23]). assume 1 < ν < 2 and 0 ≤ λ < 1 such that λ ∈ (0, λ∗]. the green’s function g(t, s) defined in (2.3) satisfies g(t, s) ≥ 0 for each (t, s) ∈ nba × nba+2. in particular, g(a, s) = g(b, s) = 0 and g(t, s) > 0 for each (t, s) ∈ nb−1a+1 × n b a+2. 3 multiple positive solutions in this section, we establish sufficient conditions on existence of at least one, at least two and at least three positive solutions of (1.1) using guo–krasnosel’skĭı and leggett–williams fixed-point theorems on conical shells. definition 3.1. let b be a banach space over r. a closed nonempty convex set k ⊂ b is called a cone provided, (i) λ1u ∈ k, for all u ∈ k and λ1 ≥ 0. (ii) u ∈ k and −u ∈ k implies u = 0. definition 3.2. a functional α2 is said to be a non-negative continuous concave functional on a cone k of a real banach space β, if α2 : k → [0, ∞) is continuous and α2(tx + (1 − t)y) ≥ tα2(x) + (1 − t)α2(y), for all x, y ∈ k and t ∈ [0, 1]. definition 3.3. an operator is called completely continuous, if it is continuous and maps bounded sets into precompact sets. theorem 3.4 (guo–krasnosel’skĭı fixed-point theorem, [24]). let b be a banach space and k ⊆ b be a cone. assume that ω1 and ω2 are open sets contained in b such that 0 ∈ ω1 and ω1 ⊆ ω2. assume further that t : k ∩ (ω2 \ ω1) −→ k is a completely continuous operator. if, either (1) ∥tu∥ ≤ ∥u∥ for u ∈ k ∩ ∂ω1 and ∥tu∥ ≥ ∥u∥ for u ∈ k ∩ ∂ω2; or (2) ∥tu∥ ≥ ∥u∥ for u ∈ k ∩ ∂ω1 and ∥tu∥ ≤ ∥u∥ for u ∈ k ∩ ∂ω2; holds, then t has at least one fixed-point in k ∩ (ω2 \ ω1). the following results are useful for the main results of this section. 474 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) lemma 3.5. let a, b be two real numbers such that 0 < a ≤ b and 1 < ν < 2. then eλ,ν,ν−1(a, ρ(s)) eλ,ν,ν−1(b, ρ(s)) is a decreasing function of s for s ∈ na−10 . proof. for each s ∈ na−10 , denote by an = hνn+ν−1(a, ρ(s)) and bn = hνn+ν−1(b, ρ(s)), n ∈ n0. clearly, an and bn for n ∈ n0 are real numbers. further, denote by a(λ) = eλ,ν,ν−1(a, ρ(s)) and b(λ) = eλ,ν,ν−1(b, ρ(s)). we know that the power series a(λ) and b(λ) are convergent for |λ| < 1. also, bn > 0, n ∈ n0 and the sequence ( an bn ) n≥0 = ( hνn+ν−1(a, ρ(s)) hνn+ν−1(b, ρ(s)) ) n≥0 is strictly decreasing, by lemma 2.7. then, by lemma 2.13, the function a(λ) b(λ) = eλ,ν,ν−1(a, ρ(s)) eλ,ν,ν−1(b, ρ(s)) is also strictly decreasing on [0, 1) for each s ∈ na−10 . the proof is complete. theorem 3.6. there exists a number γ ∈ (0, 1), such that min t∈ndc g(t, s) ≥ γ max t∈nba g(t, s) = γg(s − 1, s), (3.1) for λ ∈ (0, λ∗] and c, d ∈ nb−1a+1 such that c = a + ⌈ b − a + 1 4 ⌉ and d = a + 3 ⌊b − a + 1 4 ⌋ . proof. it follows from the proof of theorem 2.14 in [23] that for each λ ∈ (0, λ∗], g(t, s) is an increasing function of t for ∈ ns−1a and is a decreasing function of t for ∈ nbs. thus, we have max t∈nba g(t, s) = g(s − 1, s) for s ∈ nba+2. consider g(t, s) g(s − 1, s) =   eλ,ν,ν−1(t, a) eλ,ν,ν−1(s − 1, a) , s ∈ nbt+1, eλ,ν,ν−1(t, a) eλ,ν,ν−1(s − 1, a) − eλ,ν,ν−1(t, ρ(s))eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s))eλ,ν,ν−1(s − 1, a) , s ∈ nta+2. now, for s > t and c ≤ t ≤ d, g1(t, s) is an increasing function with respect to t. then, we have min t∈ndc g1(t, s) = g1(c, s) = eλ,ν,ν−1(c, a) eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s)), s ∈ nbt+1. for t > s and c ≤ t ≤ d, g2(t, s) is a decreasing function with respect to t. then, we have min t∈ndc g2(t, s) = g2(d, s) = eλ,ν,ν−1(d, a) eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s)) − eλ,ν,ν−1(d, ρ(s)), s ∈ nta+2. cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 475 thus, min t∈ndc g(t, s) =   g1(c, s), for s ∈ nbd, min{g2(d, s), g1(c, s)}, for s ∈ nd−1c+1, g2(d, s), for s ∈ nca+2, =   g2(d, s), for s ∈ nra+2, g1(c, s), for s ∈ nbr, where c < r < d. consider mint∈ndc g(t, s) g(s − 1, s) =   eλ,ν,ν−1(c, a) eλ,ν,ν−1(s − 1, a) , s ∈ nbr, eλ,ν,ν−1(d, a) eλ,ν,ν−1(s − 1, a) − eλ,ν,ν−1(d, ρ(s))eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s))eλ,ν,ν−1(s − 1, a) , s ∈ nra+2. hence, min t∈ndc g(t, s) ≥ γ(s) max t∈nba g(t, s), (3.2) where γ(s) = min [ eλ,ν,ν−1(c, a) eλ,ν,ν−1(s − 1, a) , eλ,ν,ν−1(d, a) eλ,ν,ν−1(s − 1, a) − eλ,ν,ν−1(d, ρ(s))eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s))eλ,ν,ν−1(s − 1, a) ] . for s ∈ nbr, denote by γ1(s) = eλ,ν,ν−1(c, a) eλ,ν,ν−1(s − 1, a) ≥ eλ,ν,ν−1(c, a) eλ,ν,ν−1(b − 1, a) . similarly, for s ∈ nra+2, we take γ2(s) = eλ,ν,ν−1(d, a) eλ,ν,ν−1(s − 1, a) − eλ,ν,ν−1(d, ρ(s))eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, ρ(s))eλ,ν,ν−1(s − 1, a) . by lemma 3.5, we see that eλ,ν,ν−1(d, ρ(s)) eλ,ν,ν−1(b, ρ(s)) is a decreasing function for s ∈ nra+2. then, γ2(s) ≥ 1 eλ,ν,ν−1(s − 1, a) [ eλ,ν,ν−1(d, a) − eλ,ν,ν−1(d, a + 1)eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, a + 1) ] > 1 eλ,ν,ν−1(d, a) [ eλ,ν,ν−1(d, a) − eλ,ν,ν−1(d, a + 1)eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, a + 1) ] . thus, min t∈ndc g(t, s) ≥ γ max t∈nba g(t, s), (3.3) where γ = min [ eλ,ν,ν−1(c, a) eλ,ν,ν−1(b − 1, a) , 1 − eλ,ν,ν−1(d, a + 1)eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, a + 1)eλ,ν,ν−1(d, a) ] . 476 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) since g1(c, s) > 0 and g2(d, s) > 0, we have γ(s) > 0 for all s ∈ nba+2, implying that γ > 0. it would be suffice to prove that one of the terms eλ,ν,ν−1(c, a) eλ,ν,ν−1(b − 1, a) , 1− eλ,ν,ν−1(d, a + 1)eλ,ν,ν−1(b, a) eλ,ν,ν−1(b, a + 1)eλ,ν,ν−1(d, a) is less than 1. it follows from lemma 2.12 that eλ,ν,ν−1(c, a) eλ,ν,ν−1(b − 1, a) < 1. therefore, we conclude that γ ∈ (0, 1). the proof is complete. by theorem 2.10, we observe that u is a solution of (1.1) if and only if u is a solution of the summation equation u(t) = b∑ s=a+2 g(t, s)f(s, u(s)), t ∈ nba. (3.4) note that any solution u : nba → r of (1.1) can be viewed as a real (b − a + 1)-tuple vector. consequently, u ∈ rb−a+1. define the operator t : rb−a+1 → rb−a+1 by ( tu ) (t) = b∑ s=a+2 g(t, s)f(s, u(s)), t ∈ nba. (3.5) clearly, u is a fixed-point of t if and only if u is a solution of (1.1). we use the fact that rb−a+1 is a banach space equipped with the maximum norm ∥u∥ = maxt∈nba |u(t)|, for any u ∈ r b−a+1. denote by b = {u : nba → r | u(a) = u(b) = 0} ⊆ r b−a+1. (3.6) clearly b is a banach space equipped with the maximum norm i.e. ∥u∥ = max t∈nba |u(t)|. since t is defined on a discrete finite domain, it is trivially completely continuous. define the cone k = {u ∈ b : u(t) ≥ 0 for t ∈ nba, and min t∈ndc u(t) ≥ γ∥u∥}. (3.7) lemma 3.7. for λ ∈ (0, λ∗] the operator t maps k into itself. proof. let u ∈ k. clearly, (tu) (t) ≥ 0, whenever u ∈ k. consider min t∈ndc (tu) (t) = min t∈ndc b∑ s=a+2 g(t, s)f(s, u(s)) ≥ b∑ s=a+2 min t∈ndc [g(t, s)] f(s, u(s)) ≥ b∑ s=a+2 γ max t∈nba [g(t, s)] f(s, u(s)) ≥ γ max t∈nba b∑ s=a+2 g(t, s)f(s, u(s)) = γ max t∈nba ∣∣∣∣∣ b∑ s=a+2 g(t, s)f(s, u(s)) ∣∣∣∣∣ = γ∥tu∥. thus, we have t : k → k and it is completely continuous. the proof is complete. cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 477 take η = 1 b∑ s=a+2 g(s − 1, s) . theorem 3.8. assume f(t, u(t)) satisfies the following conditions for 0 < r1 < r2 (i) there exists a number r1 > 0 such that f(t, u(t)) ≤ ηr1, whenever 0 ≤ u ≤ r1. (ii) there exists a number r2 > 0 such that f(t, u(t)) ≥ ηr2γ , whenever γr2 ≤ u ≤ r2. then, for λ ∈ (0, λ∗] the bvp (1.1) has at least one positive solution. proof. we know that t : k → k is completely continuous. define the set ω1 = {u ∈ k : ∥u∥ < r1}. clearly, ω1 ⊆ β is an open set with 0 ∈ ω1. since ∥u∥ = r1 for u ∈ ∂ω1, condition (i) holds for all u ∈ ∂ω1. so, it follows that ∥tu∥ = max t∈nba b∑ s=a+2 g(t, s)f(s, u(s)) ≤ b∑ s=a+2 max t∈nba g(t, s)f(s, u(s)) ≤ ηr1 b∑ s=a+2 g(s − 1, s) = r1 = ∥u∥. implying that ∥tu∥ ≤ ∥u∥ whenever u ∈ k ∩ ∂ω1. on the other hand, define the set ω2 = {u ∈ k : ∥u∥ < r2}. clearly, ω2 ⊆ β is an open set and ω1 ⊆ ω2. since ∥u∥ = r2 for u ∈ ∂ω2, condition (ii) holds for all u ∈ ∂ω2. thus, we have ∥tu∥ ≥ min t∈ndc b∑ s=a+2 g(t, s)f(s, u(s)) ≥ b∑ s=a+2 min t∈ndc g(t, s)f(s, u(s)) ≥ γ b∑ s=a+2 g(s − 1, s)f(s, u(s)) ≥ ηr2 b∑ s=a+2 g(s − 1, s) = r2 = ∥u∥ implying that ∥tu∥ ≥ ∥u∥ whenever u ∈ k ∩ ∂ω2. hence by part 1 of theorem 3.4, t has at least one fixed-point in k ∩ (ω1\ω1), say u0 satisfying r1 < ∥u0∥ < r2 478 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) theorem 3.9. assume f(t, u(t)) satisfies the following conditions (i) there exists a number r2 > 0 such that f(t, u(t)) ≤ ηr2, whenever 0 ≤ u ≤ r2. (ii) lim u→0+ min t∈nba f(t, u(t)) u = ∞, lim u→∞ min t∈nba f(t, u(t)) u = ∞. then, for λ ∈ (0, λ∗] the bvp (1.1) has at least two positive solution. proof. let us choose a number n > 0 such that nγ η > 1, by condition (ii) there exists a number r∗ > 0 such that r∗ < r1 < r2 and f(t, u(t)) ≥ nu for u ∈ [0, r∗] and t ∈ nba. define the set ωr∗ = {u ∈ k : ∥u∥ < r∗}. it can easily be shown that ∥tu∥ > ∥u∥, for u ∈ ∂ωr∗ ∩ k. next for the same n, we can find a number r1 > 0 such that f(t, u) ≥ nu for u ≥ r1 and t ∈ nba. choose r such that r = max { r2, r1 γ } . define the set ωr = {u ∈ k : ∥u∥ < r}. we can show that ∥tu∥ > ∥u∥, for u ∈ ∂ωr ∩ k. finally define the set ω2 = {u ∈ k : ∥u∥ < r2}. since ∥u∥ = r2 condition (i) holds for all u ∈ ∂ω2. then, we have ∥tu∥ = max t∈na b b∑ s=a+2 g(t, s)f(s, u(s)) ≤ b∑ s=a+2 max t∈na b [g(t, s)] f(s, u(s) ≤ r2η b∑ s=a+2 g(s − 1, s) = r2. implying ∥tu∥ ≤ ∥u∥, for u ∈ ∂ωr2 ∩ k. hence, we conclude that t has at least two fixed-points say u1 ∈ ω2\ω̂r∗ and u2 ∈ ωr\ω̂2, where ω̂ denoted the interior of the set ω. in particular (1.1) has at least two positive solutions, say u1 and u2 satisfying 0 < ∥u1∥ < r2 < ∥u2∥. the proof is complete. we state here the leggett–williams fixed-point theorem as follows. the proof can be found in [26] and also, we would like to refer here a paper by kwong [25] on the same. denote kc ={u ∈ k : ∥u∥ < c}, kα2(a, b) ={u ∈ k : a ≤ α2(u), ∥u∥ ≤ b}, where α2 is defined as in definition 3.2. cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 479 theorem 3.10 ([1]). let t : k̄c → k̄c be completely continuous and α2 be a non-negative continuous concave functional on k, such that α2(u) ≤ ∥u∥, for all u ∈ k̄c. suppose there exists 0 < d < a a} ≠ ∅ and α2(tu) > a, for u ∈ kα2(a, b); (2) ∥tu∥ < d, for ∥u∥ ≤ d; (3) α2(tu) > a, for u ∈ kα2(a, c) with ∥tu∥ > b. then, t has at least three fixed-points u1, u2, u3 satisfying ∥u1∥ < d, a < α2(u2), and ∥u3∥ > d and α2(u3) < a. we introduce here growth conditions on the non-linear function f in line with [1]. theorem 3.11. suppose there exists numbers a′, b′, d′ ∈ r+, where 0 < d′ < a′ < γb′ < b′, such that f satisfies the following (1) f(t, u(t)) > a′η γ , if u ∈ [a′, b′]; (2) f(t, u(t)) < d′η, if u ∈ [0, d′]; (3) there exists c′ such that c′ > b′ and if u ∈ [0, c′] then f(t, u(t)) < c′η; then, the boundary value problem (1.1) for λ ∈ (0, λ∗] has at least three positive solutions. proof. define a non-negative continuous concave functional α2 : k → [0, ∞) with α2(u) ≤ ∥u∥, for all u ∈ k, by α2(u) = min t∈ndc u(t). claim 1: if there exists a positive number r such that u ∈ [0, r] implies f(u) < rη, then t : k̄r → kr. suppose that u ∈ k̄r. then, ∥tu∥ = max t∈nba [ b∑ s=a+2 g(t, s)f(s, u(s)) ] ≤ b∑ s=a+2 max t∈nba [g(t, s)] f(s, u(s)) = b∑ s=a+2 g(s − 1, s)f(s, u(s)) < rη b∑ s=a+2 g(s − 1, s) = r. 480 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) thus, t : k̄r → kr. hence, we have that if condition (3) holds, then there exists a number c′ such that c′ > b′ and t : k̄c′ → kc′. note that with r = d′ and using condition (2), we get that t : k̄d′ → kd′. claim 2: {u ∈ kα2(a′, b′) : α2(u) > a′} ≠ ∅ and α2(tu) > a′ for u ∈ kα2(a′, b′). since u = a ′+b′ 2 ∈ {u ∈ kα2(a′, b′) : α2(u) > a′} ≠ ∅. let u ∈ kα2(a′, b′). by using condition (1), we have α2(tu) = min t∈ndc [ b∑ s=a+2 g(t, s)f(s, u(s)) ] ≥ b∑ s=a+2 min t∈ndc [g(t, s)] f(s, u(s)) ≥ γ b∑ s=a+2 g(s − 1, s)f(s, u(s)) > a′ thus, if u ∈ kα2(a′, b′), then α2(tu) > a′. claim 3: if u ∈ kα2(a′, c′) and ∥tu∥ > b′ then α2(tu) > a′. suppose u ∈ kα2(a′, c′) and ∥tu∥ > b′. then, α2(tu) = min t∈ndc [ b∑ s=a+2 g(t, s)f(s, u(s)) ] ≥ b∑ s=a+2 min t∈ndc [g(t, s)] f(s, u(s)) ≥ γ b∑ s=a+2 max t∈nba [g(t, s)] f(s, u(s)) ≥ γ max t∈ndc [ b∑ s=a+2 g(t, s)f(s, u(s)) ] = γ∥tu∥ > γb′ > a′. thus, α2(tx) > a ′. hence all the hypothesis of the theorem 3.10 are satisfied. therefore, the boundary value problem (1.1) has at least three positive solutions u1, u2 and u3 satisfying ∥u1∥ < d′, a′ < α2(u2), and ∥u3∥ > d′ and α2(u3) < a′. the proof is complete. cubo 24, 3 (2022) positive solutions of nabla fractional boundary value problem 481 example in this section, we have constructed a suitable example to illustrate the applicability of the established results. example 3.12. take ν = 1.5, a = 0, b = 5, and f(t, u(t)) = 1 20 (√ u + u2 ) . then, (1.1) becomes  − ( ∇1.5 ρ(0) u ) (t) + λu(t) = 1 20 (√ u + u2 ) , t ∈ n52, u(0) = 0 = u(5). (3.8) choose λ∗ = 0.007. then, we get η = 1 5∑ s=2 g(s − 1, s) = eλ,1.5,0.5(5, 0) 5∑ s=2 eλ,1.5,0.5(s − 1, 0)eλ,1.5,0.5(5, s − 1) = 0.2473. by taking r2 = 2, we have f(t, u) = 1 20 (√ u + u2 ) ≤ 1 20 (√ r2 + r 2 2 ) = 0.270 < ηr2 = 0.4946, implying that f(t, u) satisfies conditions (i) and (ii) of theorem 3.9. thus, all conditions of theorem 3.9 are satisfied. hence, (3.8) has at least two positive solutions u1 and u2 such that 0 < ∥u1∥ < 2 < ∥u2∥. acknowledgement authors acknowledge the review and editorial board for their comments and valuable suggestions. author n. s. gopal acknowledges the financial support received through the senior research fellowship [09/1026(0028)/2019-emr-i] from csir-hrdg new delhi, government of india. 482 n. s. gopal & j. m. jonnalagadda cubo 24, 3 (2022) references [1] d. anderson, r. avery and a. peterson, “three positive solutions to a discrete focal boundary value problem”, j. comput. appl. math., vol. 88, no. 1, pp. 103–118, 1998. 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[34] h. m. srivastava and s. owa (eds.), univalent functions, fractional calculus, and their applications, ellis horwood series in mathematics and its applications, chichester: ellis horwood limited, 1989. introduction preliminaries multiple positive solutions cubo, a mathematical journal vol. 24, no. 02, pp. 263–272, august 2022 doi: 10.56754/0719-0646.2402.0263 perfect matchings in inhomogeneous random bipartite graphs in random environment jairo bochi 1 godofredo iommi 2 mario ponce 2,b 1department of mathematics, the pennsylvania state university, usa. bochi@psu.edu 2facultad de matemáticas, pontificia universidad católica de chile, santiago, chile. giommi@mat.uc.cl mponcea@mat.uc.cl b abstract in this note we study inhomogeneous random bipartite graphs in random environment. these graphs can be thought of as an extension of the classical erdős-rényi random bipartite graphs in a random environment. we show that the expected number of perfect matchings obeys a precise asymptotic. resumen en esta nota estudiamos grafos aleatorios bipartitos inhomogéneos en un ambiente aleatorio. estos grafos pueden ser pensados como una extensión de los grafos bipartitos aleatorios clásicos de erdős-rényi en un ambiente aleatorio. mostramos que el número esperado de pareos obedece un comportamiento asintótico preciso. keywords and phrases: perfect matchings, large permanents, random graphs. 2020 ams mathematics subject classification: 05c80, 05c70, 05c63. accepted: 13 april, 2022 received: 15 october, 2021 c©2022 j. bochi et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://dx.doi.org/10.56754/0719-0646.2402.0263 https://orcid.org/0000-0003-2833-9957 https://orcid.org/0000-0001-6967-7894 mailto:mponcea@mat.uc.cl https://orcid.org/0000-0002-6617-3145 mailto:bochi@psu.edu mailto:giommi@mat.uc.cl mailto:mponcea@mat.uc.cl 264 j. bochi, g. iommi & m. ponce cubo 24, 2 (2022) 1 introduction in their seminal paper [7], erdős and rényi studied a certain type of random graphs, which in the case of bipartite graphs correspond to the following. consider a bipartite graph with set of vertices given by w = {w1, . . . ,wn} and m = {m1, . . . ,mn}. let p ∈ [0,1], σ be a probability space and consider the independent random variables x(ij) defined on σ with law x(ij)(x) =      1 with probability p; 0 with probability 1 − p, for x ∈ σ. denote by gn(x) the bipartite graph with vertex set w ∪ m and edges e(x), where the edge (wi,mj) belongs to e(x) if and only if x(ij)(x) = 1. let pm(gn(x)) be the number of perfect matchings of the graph gn(x) (see sec. 3 for precise definitions). erdős and rényi [8, p. 460] observed that the mean of the number of perfect matchings was given by e(pm(gn(x))) = n!p n. (1.1) this number has been also studied by bollobás and mckay [5, theorem 1] in the context of k−regular random graphs and by o’neil [11, theorem 1] for random graphs having a fixed (large enough) proportion of edges. we refer to the text by bollobás [4] for further details on the subject of random graphs. this paper is devoted to study certain sequences of inhomogeneous random bipartite graphs gn,ω in a random environment ω ∈ ω (definitions are given in sec. 2). inhomogeneous random graphs have been intensively studied over the last years (see [6], where non-bipartite graphs are also considered). our main result (see theorem 3.2 for precise statement) is that there exists a constant c ∈ (0,1) such that for almost every environment ω ∈ ω and for large n ∈ n en,ω(pm(gn,ω(x))) ≍ n!cn, (1.2) where the meaning of the asymptotic ≍ will be explained later. moreover, we have an explicit formula for the number c. the result in equation (1.2) should be understood in the sense that the mean number of perfect matchings for inhomogeneous random bipartite graphs in a random environment is asymptotically the same as the one of erdős-rényi bipartite graphs in which p = c. note that p is a constant that does not depend on n. the number c is the so-called scaling mean of a function related to the random graphs. scaling means were introduced, in more a general setting, in [2] and are described in sec. 3. cubo 24, 2 (2022) perfect matchings in inhomogeneous random bipartite graphs... 265 2 inhomogeneous random bipartite graphs in random environment consider the following generalization of the erdős-rényi bipartite graphs. let w = {w1, . . . ,wn} and m = {m1, . . . ,mn} be two disjoint sets of vertices. for every pair 1 ≤ i,j ≤ n, let aij ∈ [0,1] and consider the independent random variables x(ij), with law x(ij)(x) =    1 with probability a(ij); 0 with probability 1 − a(ij). denote by gn(x) the bipartite graph with vertices w,m and edges e(x), where the edge (wi,mj) belongs to e(x) if and only if x(ij)(x) = 1. as it is clear from the definition all vertex of the graph do not play the same role. this contrasts with the (homogenous) erdős-renyi graphs (see [6] for details). we remark that in relation to the graphs we are considering it is possible to include the stochastic block model (see [10]) that is used, for example, in problems of community detection, in the context of machine learning. in this note we consider inhomogeneous random bipartite graphs in random environments, that is, the laws of x(ij) (and hence the numbers a(ij)) are randomly chosen following an exterior environment law. this approach to stochastic processes has developed since the groundbreaking work by solomon [12] on random walks in random environment and subsequent work of a large community (see [3] for a survey on the subject). the model we propose is to consider the vertex sets w,m as the environment and to consider that the number a(ij), which is the probability that the edge connecting wi with mj occurs in the graph, is a random variable depending on wi and mj. we now describe precisely this model. the space of environments is as follows. fix α ∈ n and a stochastic vector (p1,p2, . . . ,pα). endow the set {1, . . . ,α} with the probability measure pw defined by pw ({i}) = pi. denote by ωw the product space ∏∞ i=1{1,2, . . . ,α} and by µw the corresponding product measure. let (ωm,µm) be the analogous probability measure space for the set {1,2, . . . ,β} and the stochastic vector (q1,q2, . . . ,qβ). the space of environments is ω = ωw × ωm with the measure µω = µw × µm and an environment is an element ω ∈ ω. note that every environment defines two sequences w(ω) = (w1,w2, . . .) ∈ ωw and m(ω) = (m1,m2, . . .) ∈ ωm. for each environment ω ∈ ω we now define the edge distribution xω,(ij). let f = [fsr] be a α × β matrix with entries fsr satisfying 0 ≤ fsr ≤ 1 and let f : {1,2, . . . ,α} × {1,2, . . . ,β} → [0,1] be the function defined by f(w,m) = fwm. for each ω ∈ ω let a(ij)(ω) := f (wi(ω),mj(ω)) = fwi(ω),mj(ω). (2.1) given an environment ω ∈ ω the corresponding edge distributions are the random variables xω,(ij) 266 j. bochi, g. iommi & m. ponce cubo 24, 2 (2022) with laws xω,(ij)(x) =    1 with probability a(ij)(ω); 0 with probability 1 − a(ij)(ω). given an environment ω ∈ ω, we construct a sequence of random bipartite graphs gn,ω considering the sets of vertices wn,ω = (w1(ω), . . . ,wn(ω)) and mn,ω = (m1(ω), . . . ,mn(ω)), and edge distributions xω,(ij) given by the values of a(ij)(ω) as in (2.1). we denote by pn,ω the law of the random graph gn,ω. example 2.1. given a choice of an environment ω ∈ ω, the probability that the bipartite graph gn,ω(x) equals the complete bipartite graph kn,n, using independence of the edge variables, is pn,ω (gn,ω(x) = kn,n) = ∏ 1≤i,j≤n pn,ω(xω,(ij) = 1) = ∏ 1≤i,j≤n a(ij)(ω). 3 counting perfect matchings recall that a perfect matching of a graph g is a subset of edges containing every vertex exactly once. we denote by pm(g) the number of perfect matchings of g. when the graph g is bipartite, and the corresponding bipartition of the vertices has the form w = {w1,w2, . . . ,wn} and m = {m1,m2, . . . ,mn}, a perfect matching can be identified with a bijection between w and m, and hence with a permutation σ ∈ sn. from this, the total number of perfect matchings can be computed as pm(g) = ∑ σ∈sn x1σ(1)x2σ(2) · · ·xnσ(n), (3.1) where xij are the entries of the incidence matrix xg of g, that is xij = 1 if (wi,mj) is an edge of g and xij = 0 otherwise. of course, the right hand side of (3.1) is the permanent, per(xg), of the matrix xg. in the framework of section 2, we estimate the number of perfect matchings for the sequence of inhomogeneous random bipartite graphs gn,ω, for a given environment ω ∈ ω. more precisely, we obtain estimates for the growth of the mean of pm(gn,ω(x)) = per(xgn,ω(x)) = ∑ σ∈sn xω,(1σ(1)) · · ·xω,(nσ(n)). (3.2) denote by en,ω the expected value with respect to the probability pn,ω. since the edges are cubo 24, 2 (2022) perfect matchings in inhomogeneous random bipartite graphs... 267 independent and en,ω(xω,(ij)) = aij(ω) we have en,ω (pm(gn,ω)) = en,ω ( ∑ σ∈sn xω,(1σ(1)) · · ·xω,(nσ(n)) ) = ∑ σ∈sn a(1σ(1))(ω) · · ·a(nσ(n))(ω) = per(an(ω)), where the entries of the matrix are (an(ω))ij = a(ij)(ω). the main result of this note describes the growth of this expected number for perfect matchings. the following number is a particular case of a quantity introduced by the authors in a more general setting in [2]. definition 3.1. let f be an α × β matrix with non-negative entries (frs). let ~p = (p1, . . . ,pα) and ~q = (q1, . . . ,qβ) be two stochastic vectors. the scaling mean of f with respect to ~p and ~q is defined by sm~p,~q(f) := inf (xr)∈r α + ,(ys)∈r β + ( α ∏ r=1 x−prr )( β ∏ s=1 y−qss )( α ∑ r=1 β ∑ s=1 xrfrsysprqs ) . the scaling mean is increasing with respect to the entries of the matrix and lies between the minimum and the maximum of the entries (see [2] for details and more properties). we stress that the scaling mean can be exponentially approximated using a simple iterative process (see section 5). the main result in this note is the following, theorem 3.2 (main theorem). let (gn,ω)n≥1 be a sequence of random bipartite graphs on a random environment ω ∈ ω. if for every pair (r,s) we have frs > 0 then the following pointwise convergence holds lim n→∞ ( en,ω (pm(gn,ω)) n! )1/n = sm~p,~q(f), (3.3) for µw × µm-almost every environment ω ∈ ω. remark 3.3. as discussed in the introduction theorem 3.2 shows that there exists a constant c ∈ (0,1), such that for almost every environment ω ∈ ω and for n ∈ n sufficiently large en,ω(pm(gn,ω(x))) ≍ n!cn. namely c = sm~p,~q(f). this result should be compared with the corresponding one obtained by erdős and rényi for their class of random graphs, that is e(pm(gn(x))) = n!p n. 268 j. bochi, g. iommi & m. ponce cubo 24, 2 (2022) thus, we have shown that for large values of n the growth of the number of perfect matchings for random graphs in a random environment behaves like the simpler model studied by erdős and rényi with p = sm~p,~q(f). remark 3.4. theorem 3.2 shows that the expected number of perfect matchings is a quenched variable, in the sense of that it does not depend on the environment ω (see for instance [p]). remark 3.5. using the stirling formula, the limit in (3.3) can be stated as lim n→∞ ( 1 n log (en,ω (pm(gn,ω))) − log n ) = log sm~p,~q(f) − 1, which gives a quenched result for the growth of the perfect matching entropy for the sequence of graphs gω,n (see [1]). remark 3.6. note that we assume a uniform ellipticity condition on the values of the probabilities a(ij) as in (2.1). a similar assumption appears in the setting of random walks in random environment (see [3, p. 355]). we now present some concrete examples. example 3.7. let α = β = 2 and p1 = p2 = q1 = q2 = 1/2. therefore, the space of environments is the direct product of two copies of the full shift on two symbols endowed with the (1/2,1/2)−bernoulli measure. the edge distribution matrix f is a 2 × 2 matrix with entries belonging to (0,1). in [2, example 2.11], it was shown that sm~p,~q   f11 f12 f21 f22   = √ f11f22 + √ f12f21 2 . therefore, theorem 3.2 implies that lim n→∞ ( en,ω (pm(gn,ω)) n! )1/n = √ f11f22 + √ f12f21 2 , for almost every environment ω ∈ ω. example 3.8. more generally let α ∈ n with α ≥ 2 and β = 2. consider the two stochastic vectors ~p = (p1,p2, . . . ,pα) and ~q = (q1,q2). the space of environments is the direct product of a full shift on α symbols endowed with the ~p-bernoulli measure with a full shift on two symbols endowed with the ~q-bernoulli measure. the edge distribution matrix f is a α × 2 matrix with entries fr1,fr2 ∈ (0,∞), where r ∈ {1, . . . ,α}. denote by χ ∈ r+ the unique positive solution of the equation α ∑ r=1 prfr1 fr1 + fr2χ = q1. then sm~p,~q(f) = sm~p,~q      f11 f12 ... ... fα1 fα2      = q q1 1 ( q2 χ )q2 α ∏ r=1 (fr1 + fr2χ) pr . cubo 24, 2 (2022) perfect matchings in inhomogeneous random bipartite graphs... 269 therefore, theorem 3.2 implies that lim n→∞ ( en,ω (pm(gn,ω)) n! )1/n = q q1 1 ( q2 χ )q2 α ∏ r=1 (fr1 + fr2χ) pr , for almost every environment ω ∈ ω. the quantity in the right hand side first appeared in the work by halász and székely in 1976 [9], in their study of symmetric means. in [2, theorem 5.1] using a completely different approach we recover their result. 4 proof of the theorem the shift map σw : ωw → ωw is defined by σw (w1,w2,w3, . . .) = (w2,w3, . . .). the shift map σw is a µw -preserving, that is, µw (λ) = µw(σ −1 w (λ)) for every measurable set λ ⊂ ωw , and it is ergodic, that is, if λ = σ−1w (λ) then µw (λ) equals 1 or 0. analogously for σm and µm. we define a function φ : ωw × ωm → r by φ(~w, ~m) = fw1m1. thus φ(σi−1w (~w),σ j−1 m (~m)) = fwimj = a(ij)(ω). that is, the matrix an(ω) has entries a(ij)(ω) = φ(σ i−1 w (~w),σ j−1 m (~m)). we are in the exact setting in order to apply the law of large permanents see [2, theorem 4.1]. theorem (law of large permanents). let (x,µ), (y,ν) be lebesgue probability spaces, let t : x → x and s : y → y be ergodic measure preserving transformations, and let g : x × y → r be a positive measurable function essentially bounded away from zero and infinity. then for µ × ν-almost every (x,y) ∈ x × y , the n × n matrix mn(x,y) =        g(x,y) g(tx,y) · · · g(tn−1x,y) g(x,sy) g(tx,sy) · · · g(tn−1x,sy) ... ... ... g(x,sn−1y) g(tx,sn−1y) · · · g(tn−1x,sn−1y)        verifies lim n→∞ ( per (mn(x,y)) n! )1/n = smµ,ν(g) pointwise, where smµ,ν(g) is the scaling mean of g defined as smµ,ν(g) = inf φ,ψ ∫∫ x×y φ(x)g(x,y)ψ(y)dµdν exp (∫ x log φ(x)dµ ) exp (∫ y log ψ(y)dν ), 270 j. bochi, g. iommi & m. ponce cubo 24, 2 (2022) where the functions φ and ψ are assumed to be measurable, positive and such that their logarithms are integrable. we apply this law of large permanents setting x = ωw ,y = ωm, t = σw ,s = σm , g = φ and recalling that frs > 0. we have smµw ,µm (φ) = sm~p,~q(f) as a consequence of an alternative characterization of the scaling mean given in (see [2, proposition 3.5]). this concludes the proof of the main theorem. � remark 4.1. we have chosen to present our result in the simplest possible setting. that is, the environment space being products of full-shifts endowed with bernoulli measures. using the general form of the law of large permanent above our results can be extended for inhomogeneous random graphs in more general random environments. 5 an algorithm to compute the scaling mean the purpose of this section is to show that the scaling mean is the unique fixed point of a contraction. therefore it can be computed, or approximated, using a suitable iterative process. it should be stressed that, on the other hand, it has been shown that no such algorithm exists to compute the permanent. denote by bα ⊂ rα and by bβ ⊂ rβ the positive cones. define the following maps forming a (non-commutative) diagram: bα bα bβ bβ i1 k2k1 i2 by the formulas: (i1(~x))i := 1 xi , (i2(~y))i := 1 yi , (k1(~x))j := β ∑ i=1 fijxipi , (k2(~y))j := α ∑ j=1 fijyjqj. let t : bα 7→ bα be the map defined by t := k1 ◦ i2 ◦ k2 ◦ i1. the map t is a contraction for a suitable hilbert metric. indeed, for ~x,~z ∈ bα define the following (pseudo)-metric d(~x,~z) := log ( maxi xi/zi mini xi/zi ) . it was proven in [2, lemma 3.4] lemma 5.1. we have that d(t(~x),t(~z)) ≤ ( tanh δ 4 )2 d(~x,~z), cubo 24, 2 (2022) perfect matchings in inhomogeneous random bipartite graphs... 271 where δ ≤ 2 log ( maxi,j fij minij fij ) < ∞. the following results was proved in [2, lemma 3.3] lemma 5.2. the map t has a unique (up to positive scaling) fixed point ~xt ∈ bα. moreover, defining ~yt := k2 ◦ i1(~xt) one has that sm(f) = α ∏ i=1 x pi i β ∏ j=1 y qj j . therefore, it possible to find good approximations of the scaling mean using an iterative process. acknowledgments the authors were partially supported by conicyt pia act172001. j.b. was partially supported by proyecto fondecyt 1180371. g.i. was partially supported by proyecto fondecyt 1190194. m.p. was partially supported by proyecto fondecyt 1180922. 272 j. bochi, g. iommi & m. ponce cubo 24, 2 (2022) references [1] m. abért, p. csikvári, p. frenkel and g. kun, “matchings in benjamini-schramm convergent graph sequences”, trans. amer. math. soc., vol. 368, no. 6, pp. 4197–4218, 2016. [2] j. bochi, g. iommi and m. ponce, “the scaling mean and a law of large permanents”, adv. math., vol. 292, pp. 374–409, 2016. [3] l. v. bogachev, “random walks in random environments”, in encyclopedia of mathematical physics, vol. 4, pp. 353–371. elsevier: oxford, 2006. [4] b. bollobás, random graphs, cambridge studies in advanced mathematics, vol. 73, cambridge university press: cambridge, 2001. [5] b. bollobás and b. d. mckay, “the number of matchings in random regular graphs and bipartite graphs”, j. combin. theory ser. b, vol. 41, no. 1, pp. 80–91, 1989. [6] b. bollobás, s. janson and o. riordan, “the phase transition in inhomogeneous random graphs”, random structures algorithms, vol. 31, no. 1, pp. 3–122, 2007. [7] p. erdős and a. rényi, ‘on random graphs. i”, publ. math. debrecen, vol. 6, pp. 290–297, 1959. [8] p. erdős and a. rényi, “on random matrices”, magyar tud. akad. mat. kutató int. közl., vol. 8, pp. 455–461, 1964. [9] g. halász and g. j. székely, “on the elementary symmetric polynomials of independent random variables”, acta math. acad. sci. hungar., vol. 28, no. 3–4, pp. 397–400, 1976. [10] p. holland, k. laskey and s. leinhardt, “stochastic blockmodels: first steps”, social networks, vol. 5, no. 2, pp. 109–137, 1983. [11] p. e. o’neil, “asymptotics in random (0,1)-matrices”, proc. amer. math. soc., vol. 25, pp. 290–296, 1970. [12] f. solomon, “random walks in a random environment”, ann. probability, vol. 3, no. 1, pp. 1–31, 1975. introduction inhomogeneous random bipartite graphs in random environment counting perfect matchings proof of the theorem an algorithm to compute the scaling mean cubo, a mathematical journal vol. 24, no. 03, pp. 521–539, december 2022 doi: 10.56754/0719-0646.2403.0521 a derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem rené erlin castillo1, b babar sultan2 1 universidad nacional de colombia, departamento de matemáticas, bogotá, colombia recastillo@unal.edu.co b 2 department of mathematics, quaid-i-azam university, islamabad 45320, pakistan. babarsultan40@yahoo.com abstract in this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. we prove several properties of this operator, for instance, we obtain a generalized leibniz’s formula, rolle and cauchy’s mean theorems and a taylor type polynomial. moreover, we obtain its inverse operator. also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”. resumen en este art́ıculo introducimos un operador que puede ser pensado como una derivada de orden variable, i.e. el orden de la derivada es una función. demostramos varias propiedades de este operador, por ejemplo, obtenemos una fórmula generalizada de leibniz, teoremas de valor medio de rolle y cauchy y un polinomio de tipo taylor. más aún, obtenemos su operador inverso. también con esta derivada analizamos la existencia de soluciones de un problema no lineal de valor en la frontera de tres puntos de “orden variable”. keywords and phrases: fractional derivative, boundary value problem, hammerstein-volterra integral equation. 2020 ams mathematics subject classification: 26a33, 34b10, 45d05. accepted: 02 december 2022 received: 17 june, 2022 ©2022 r. e. castillo et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0521 https://orcid.org/0000-0003-1113-5827 https://orcid.org/0000-0003-2833-4101 mailto:recastillo@unal.edu.co mailto:babarsultan40@yahoo.com 522 r. e. castillo & b. sultan cubo 24, 3 (2022) 1 motivation derivatives of non-integer order have been studied since the celebrated question of l’hospital to leibniz about the meaning of d nf dxn when n = 1/2. there are several definitions of derivatives of fractional order, e.g., derivative of riemann-louville, caputo, hadamard, erdélyi-kober, grünwaldletnikov and riesz, among others. typically, these derivatives are defined using an integral form of the classical derivative, as a consequence of it, some basic properties of the usual derivative, as the product rule and chain rule are lost. for a more comprehensible information about these notions we recommend [17, 20, 30]. despite of the lack of some properties, derivatives of fractional order appear in many real world applications as, for instance, in memory effects and future dependence, control theory of dynamical systems, nanotechnology, viscoelasticity and financial modeling see, e.g., [8, 12, 18, 19, 21, 24, 25, 31, 32]. thus, due to this development, in the last decades a lot of research has been devoted to the study of the existence of solutions for several kinds of boundary value problems of fractional type, see, for instance, [2, 3, 4, 5, 9, 11, 26, 28, 29] and references therein. in order to overcome the limitations of the classical derivative, in [16] it is introduced a new limitbased definition of derivative, the so-called conformable fractional derivative, which can be seen as a natural extension of the fractional derivative, although as it is stated in [7], it is best to consider the conformable derivative in its own right, independent of fractional derivative theory. some of the basic properties, physical interpretation and some boundary value problems for conformable differential equations can be found in [1, 6, 10, 14, 15, 33, 34] and its references. in this article, based on the idea of conformable fractional derivative and in ideas from [13], we consider an extension of the conformable fractional derivative of order α and develop some of its properties. additionally, we study the existence and uniqueness of solutions for a nonlinear three-point boundary value problem in this new setting. 2 derivative of variable order we now introduce the notion of (φ, ω)-derivative. definition 2.1. let f : [a, b] −→ r. the (φ, ω)-derivative at the point x ∈ (a, b) (φ(x) ̸= 0) is defined as dφωf(x) = d φ ω(f)(x) = d (φ,1) ω f(x) = lim h→0 f(x + hφ(x)) − f(x) ω(x + h) − ω(x) . (2.1) where ω is a strictly increasing function and φ is a function. at the point x ∈ (a, b) such that cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 523 φ(x) = 0 we define the (φ, ω)-derivative as dφωf(x) = lim ξ→x dφωf(ξ), when the limit exists. taking φ(x) = x1−α and ω(x) = x we obtain the conformable fractional derivative of order α, cf. [16]. theorem 2.2. let f, g be (φ, ω)-differentiable. then: (a) the function f is continuous. (b) dφω(a) = 0, a is a constant. (c) dφω(af + g) = ad φ ω(f) + d φ ω(g). (d) dφω(fg) = fd φ ω(g) + fd φ ω(g). (e) dφω ( f g ) = fdφω(g) − fdφω(g) g2 . (f) if f and ω are differentiable, we have dφω(f)(t) = φ(t) f′(t) ω′(t) . (2.2) (g) if f, g and ω are differentiable, we have dφω(f ◦ g)(t) = f ′(g(t)) · dφω(g)(t). proof. it is a matter of direct calculations. formula (2.2) enables us to calculate in a straightforward way some (φ,ω)-derivatives. for example, letting φ(x) = sin(x), f(x) = cos(x), and ω(x) = x, we have dφωφ(x) = sin(x) cos(x), d φ ωf(x) = − sin 2(x), whereas taking φ and f as above with ω(x) = ex − 1 we get dφωφ(x) = sin(x) cos(x) ex , dφωf(x) = − sin2(x) ex . we now introduce the n-iterated (φ, ω)-derivative. 524 r. e. castillo & b. sultan cubo 24, 3 (2022) definition 2.3. by d (φ,n) ω f(x) we define the n-iterated (φ, ω)-derivative of the function f, i.e. d(φ,n)ω f(x) = d φ ω ( d(φ,n−1)ω f ) (x), with the convention d (φ,0) ω f(x) := f(x). theorem 2.4 (generalized leibniz’s formula). we have d(φ,n)ω (f1f2 · · · fm) = ∑ i1+i2+···+im=n ij=0,n n! d (φ,i1) ω (f1)d (φ,i2) ω (f2) · · · d (φ,im) ω (fm) i1!i2! · · · im! , (2.3) where we suppose that all is well-defined. proof. for m = 2, equation (2.3) is obtained by induction on n and using the formula for the (φ, ω)-derivative of the product, in this case we obtain d(φ,n)ω (f1f2) = n∑ j=0 ( n j ) d(φ,n−j)ω (f1)d (φ,j) ω (f2). (2.4) by the well-known method to prove the multinomial theorem from the binomial theorem we can, in the same way, obtain (2.3) from (2.4). theorem 2.5 (fermat’s theorem). let f : [a, b] −→ r have a local maximum or minimum at x = c ∈ (a, b) and dφω(f)(c) exists. then dφω(f)(c) = 0. proof. let us suppose, without loss of generality, that x = c is a minimum of f. we have, for sufficiently small h ̸= 0, that sgn(hφ(c)) f(c + hφ(c)) − f(c) ω(c + h) − ω(c) ⩾ 0. (2.5) from (2.5) and the hypothesis of the existence of dφω(f)(c) the result follows. theorem 2.6 (rolle’s theorem). let f : [a, b] −→ r be a continuous function in [a, b] and (φ, ω)-differentiable in (a, b) such that f(a) = f(b) = 0. then there exists c ∈ (a, b) such that dφω(f)(c) = 0. proof. supposing, without loss of generality, that there exists ξ ∈ (a, b) such that f(ξ) ≥ 0. then by weierstraß theorem, there exists c ∈ (a, b) which is a maximum. invoking fermat’s theorem 2.5 we end the proof. cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 525 theorem 2.7 (cauchy mean-value theorem). let f, g : [a, b] −→ r be both continuous on the closed interval [a, b] and (φ, ω)-differentiable in the open interval (a, b). then there exists a number ξ ∈ (a, b) such that [f(b) − f(a)]dφω(g)(ξ) = [g(b) − g(a)]d φ ω(f)(ξ). (2.6) proof. the proof follows, as in the classical case, from rolle’s theorem 2.6 applied to the function f(x) = f(x)[g(b) − g(a)] − g(x)[f(b) − f(a)]. 3 integration of variable order definition 3.1. let f : [a, b] −→ r. we define the (φ, ω)-integral of the function f as iφω (f)(t) = ∫ t a f(ξ) φ(ξ) dω(ξ), (3.1) where the integral is understood in the lebesgue-stieltjes sense. notice that for f ∈ l∞([a, b]) and 1 φ ∈ l1([a, b], dw) the integral (3.1) is finite. when f, φ and ω′ are continuous functions, it is straightforward the relation dφω(i φ ω f)(t) = f(t), since dφω(i φ ω f)(t) = φ(t) ω′(t) d (∫ t a f(ξ) φ(ξ) dω(ξ) ) (t) = f(t), using (2.2). in the case φ and ω′ are continuous functions, the following lagrange mean-value theorem dφω(f)(ξ) = f(b) − f(a) i φ ω (1)(b) − iφω (1)(a) , ξ ∈ (a, b) (3.2) is valid, when f : [a, b] −→ r is continuous on the closed interval [a, b] and (φ, ω)-differentiable in the open interval (a, b). the equation (3.2) follows from (2.6) taking g(x) = iφω (1)(x) (note that iφω (1)(a) = 0, but we leave it in (3.2) just for keeping with the parallel in the classical case). by i (φ,n) ω φ(x) we define the n-iterated (φ, ω)-integral of the function f, i.e. i(φ,n)ω f(x) = i φ ω (i (φ,n−1) ω f)(x), with the convention i (φ,0) ω f(x) := f(x). 526 r. e. castillo & b. sultan cubo 24, 3 (2022) 4 taylor formula in this section we will obtain a taylor type formula using the (φ,ω)-derivative with a remainder which generalizes well-know remainders, i.e. cauchy, lagrange, peano, schlömilch, among others, cf. [22, 23, 27] for similar remainders for the classical derivative. theorem 4.1. let f : i −→ r be a continuous function in the open interval i and n-times (φ, ω)-differentiable function in i. we also require that φ, ω′ and i (φ,j) ω (1)(x) are continuous functions, for j = 1, n. moreover, let g : i −→ r be a n-times (φ, ω)-differentiable function such that d (φ,j) ω g(a) = 0 for j = 1, n − 1 and d (φ,k) ω g(y) ̸= 0 for all y different from a and x and j = 1, n − 1. then, for all x ∈ i we have f(x) = n∑ j=0 d(φ,j)ω (f)(a) i (φ,j) ω (1)(x) + rn(x), (4.1) with rn(x) = g(x) − g(a) d (φ,n) ω g(ξ) ( d(φ,n)ω (f)(ξ) − d (φ,n) ω (f)(a) ) , (4.2) where x ̸= a and ξ is between a and x. proof. we first note that, since d(φ,n)ω (i (φ,j) ω 1)(x) =   i (φ,j−n) ω (1)(x), j > n, 1, n = j, 0, n > j. we have rn(a) = d (φ,1) ω (rn)(a) = · · · = d (φ,n−1) ω (rn)(a) = 0. (4.3) by the cauchy type finite increment formula (2.6), relations (4.3) and the hypothesis on g we have rn(x) − rn(a) g(x) − g(a) = d (φ,1) ω (rn)(θ1) − d (φ,1) ω (rn)(a) d (φ,1) ω (g)(θ1) − d (φ,1) ω (g)(a) = . . . = d (φ,n−1) ω (rn)(θn−1) − d (φ,n−1) ω (rn)(a) d (φ,n−1) ω (g)(θn−1) − d (φ,n−1) ω (g)(a) = d (φ,n) ω (rn)(ξ) d (φ,n) ω (g)(ξ) , (4.4) where ξ := θn. on the other hand, (φ, ω)-differentiating the equality (4.1) n-times we obtain d (φ,n) ω (f)(x) − d (φ,n) ω (f)(a) = d (n) ω (rn)(x) which, together with (4.4), entails (4.2). cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 527 5 three-point boundary value problems of variable order inspired in [10], we are interested in the use of the (φ, ω)-derivative to study the solutions of the following nonlinear boundary value problem dφω(d + λ)x(t) = f(t, x(t)), t ∈ [0, 1] (5.1) x(0) = 0, x′(0) = α, x(1) = βx(η), (5.2) where dφω is the derivative of variable order, d is the ordinary derivative, f : [0, 1] × r −→ r is a known function, β, λ and α are real numbers, λ ̸= 0 and η ∈ (0, 1). notice that in virtue of theorem 2.2, a sufficient condition for the well posedness of equation (5.1) is, by considering ω ∈ c1[0, 1], and x ∈ c2[0, 1]. thus, in the sequel we consider these conditions on the functions ω and x. in addition, in order to use the (φ, ω)-integral, we are going to assume that φ is continuous and bounded away from zero. from the conditions on the functions ω and φ we conclude that the following non negative numbers are finite ω := sup t∈[0,1] ω′(t) < ∞, m := sup t∈[0,1] ∣∣∣∣ 1φ(t) ∣∣∣∣ < ∞. we will use these numbers in the sequel to establish the existence results. first, as usual, we will consider the linear boundary value problem: dφω(d + λ)x(t) = g(t), t ∈ [0, 1], g ∈ c[0, 1] (5.3) x(0) = 0, x′(0) = α, x(1) = βx(η), α, β, λ ∈ r, λ ̸= 0, η ∈ (0, 1). (5.4) to obtain a solution for the boundary value problem, we apply the (φ, ω)-integral to equation (5.3): (d + λ)x(t) + (d + λ)x(0) = iφω (g)(t), (5.5) where, using the boundary condition (5.4), (d + λ)x(0) = α. then, equation (5.5) simplifies as (d + λ)x(t) + α = iφω (g)(t). (5.6) let y(t) = eλtx(t), then we rewrite (5.6) as dy(t) = eλtiφω (g)(t) − αe λt. integrating from 0 to t we obtain y(t) − y(0) = ∫ t 0 eλsiφω (g)(s) ds − α λ (eλt − 1) 528 r. e. castillo & b. sultan cubo 24, 3 (2022) y(t) = ∫ t 0 eλs ∫ s 0 g(r) φ(r) ω′(r) dr ds − α λ (eλt − 1), (y(0) = x(0) = 0). now, notice that ∫ t 0 eλs ∫ s 0 g(r) φ(r) ω′(r) dr ds = eλt λ ∫ t 0 g(s) φ(s) ω′(s) ds − 1 λ ∫ t 0 eλsg(s) φ(s) ω′(s) ds, 0 ≤ s ≤ t ≤ 1. from here we have that y(t) = eλt λ ∫ t 0 g(s) φ(s) ω′(s) ds − 1 λ ∫ t 0 eλsg(s) φ(s) ω′(s) ds − α λ (eλt − 1), 0 ≤ s ≤ t ≤ 1. thus, x(t) = 1 λ ∫ t 0 g(s) φ(s) ω′(s) ds − e−λt λ ∫ t 0 eλsg(s) φ(s) ω′(s) ds + α λ (e−λt − 1). finally, from the condition βx(η) = x(1) we get β λ ∫ η 0 g(s) φ(s) (1 − eλ(s−η))ω′(s) ds + αβ λ (e−λη − 1) − 1 λ ∫ 1 0 g(s) φ(s) (1 − eλ(s−1)) ds − α λ (e−λ − 1) = 0. therefore, introducing this equality into the formula of function x above, we obtain the following expression for x satisfying boundary value problem (5.3)–(5.4) x(t) = 1 λ ∫ t 0 g(s) φ(s) (1 − eλ(s−t))ω′(s) ds + β λ ∫ η 0 g(s) φ(s) ( 1 − eλ(s−η) ) ω′(s) ds − 1 λ ∫ 1 0 g(s) φ(s) ( 1 − eλ(s−1) ) ω′(s) ds + α λ ( e−λt − e−λ + βe−λη − β ) . notice that actually we just proved the following result. theorem 5.1. the linear boundary value problem (5.3)–(5.4) has a unique solution given by x(t) = 1 λ ∫ t 0 g(s) φ(s) ω′(s)k(s, t) ds + β λ ∫ η 0 g(s) φ(s) ω′(s)k(s, η) ds − 1 λ ∫ 1 0 g(s) φ(s) k(s, 1)ω′(s) ds + α λ ( e−λt − e−λ + βe−λη − β ) . where, k(s, t) = 1 − eλ(s−t). now, we are going to analyze the existence of solutions for the nonlinear boundary value problem: dφω(d + λ)x(t) = f(t, x(t)), t ∈ [0, 1], λ ∈ (−1, ∞) \ {0} (5.7) x(0) = 0, x′(0) = α, x(1) = βx(η). (5.8) as in theorem 5.1, we can transform boundary value problem (5.7)–(5.8) into the nonlinear cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 529 hammerstein-volterra integral equation x(t) = 1 λ ∫ t 0 f(s, x(s)) φ(s) k(s, t)ω′(s) ds + β λ ∫ η 0 f(s, x(s)) φ(s) k(s, η)ω′(s) ds − 1 λ ∫ 1 0 f(s, x(s)) φ(s) k(s, 1)ω′(s) ds + α λ ( e−λt − e−λ + βe−λη − β ) . where, k(s, t) = 1 − eλ(s−t). in order to investigate the existence of a solution for this integral equation, we analyze it as a fixed point problem; that is, letting t : (c2[0, 1], ∥ · ∥∞) −→ (c2[0, 1], ∥ · ∥∞) x(t) 7−→ tx(t) tx(t) := 1 λ ∫ t 0 f(s, x(s)) φ(s) k(s, t)ω′(s) ds + 1 λ ∫ 1 0 f(s, x(s)) φ(s) ( χ(0,η)(s)βk(s, η) − k(s, 1) ) ω′(s) ds + α λ ( e−λt − e−λ + βe−λη − β ) , (5.9) (with χ(0,η)(s) the characteristic function of the interval (0, η)), we have that the existence of the solution of the integral equation is equivalent to the existence of a fixed point of the operator t. to assure that the operator t applies c2[0, 1] into itself, we assume that f(t, x(t)) is continuous and differentiable in the first variable. we are going to use metric fixed point theory (banach’s contraction principle) to provide conditions to guarantee that the boundary value problem (5.7)–(5.8) has a unique solution. theorem 5.2. let f : [0, 1] × r −→ r be a continuous and differentiable in the first variable function satisfying that |f(t, x) − f(t, y)| ≤ k|x − y|, k > 0, for all t ∈ [0, 1], x, y ∈ r. then, the nonlinear boundary value problem (5.7)–(5.8) has a unique solution provide that (|β| + 1)mkω |λ| < 1 4 , where m := supt∈[0,1] 1 |φ(t)| and ω := supt∈[0,1] w ′(t). proof. as we saw, it is sufficient to show that the operator t defined by the formula (5.9) has a unique fixed point. let x and y be two functions in c2[0, 1]. then, 530 r. e. castillo & b. sultan cubo 24, 3 (2022) |tx(t) − ty(t)| = ∣∣∣∣∣1λ ∫ t 0 (f(s, x(s)) − f(s, y(s)) φ(s) k(s, t)ω′(s) ds + 1 λ ∫ 1 0 (f(s, x(s)) − f(s, y(s)) φ(s) ( χ(0,η)(s)βk(s, η) − k(s, 1) ) ω′(s) ds ∣∣∣∣∣ ≤ 1 |λ| ∫ t 0 |k(s, t)| |φ(s)| |f(s, x(s)) − f(s, y(s))|ω′(s) ds + 1 |λ| ∫ 1 0 ∣∣∣χ(0,η)(s)βk(s, η) − k(s, 1)∣∣∣ |φ(s)| |f(s, x(s)) − f(s, y(s))|ω′(s) ds. on the other hand, |k(s, t)| = |1 − eλ(s−t)| ≤ 1 + eλ(s−t). notice that for −1 < λ < 0, the inequality |ex − 1| < 7/4|x|, for 0 < |x| < 1, gives us the estimate |1 − eλ(s−t)| < 7 4 λ(s − t) < 7 4 < 2. then, sup s∈[0,t] (1 + eλ(s−t)) ≤ 2, −1 < λ < 0. now, for λ > 0, sup s∈[0,t] (1 + eλ(s−t)) = 1 + e−λt ≤ 2, for any t ∈ [0, 1]. therefore, we obtain the following bound |k(s, t)| ≤ 2. (5.10) moreover, ∣∣∣χ(0,η)(s)βk(s, η) − k(s, 1)∣∣∣ = ∣∣∣−χ(0,η)(s)βeλ(s−η) + eλ(s−1) + χ(0,η)(s)β − 1∣∣∣ ≤ | − χ(0,η)(s)βeλ(s−η)| + |eλ(s−1)| + |β| + 1, where, for −1 < λ < 0, we have that sup s∈[0,η] eλ(s−η) = 1, sup s∈[0,1] eλ(s−1) = 1. in the case λ > 0, we get sup s∈[0,η] eλ(s−η) = e−λη ≤ 1, sup s∈[0,1] eλ(s−1) = e−λ ≤ 1. cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 531 with these bounds we obtain the following estimation ∣∣∣χ(0,η)(s)βk(s, η) − k(s, 1)∣∣∣ ≤ 2(|β| + 1). (5.11) we introduce the bounds (5.10) and (5.11) into the difference |tx(t) − ty(t)|: |tx(t) − ty(t)| ≤ 2 |λ| ∫ t 0 1 |φ(s)| |f(s, x(s)) − f(s, y(s))|ω′(s) ds + 2(|β| + 1) |λ| ∫ 1 0 1 |φ(s)| |f(s, x(s)) − f(s, y(s))|ω′(s) ds. (5.12) since f(s, x(s)) is lipschitz in the second variable, then |tx(t) − ty(t)| ≤ 2 |λ| ∫ t 0 k |φ(s)| |x(s) − y(s)|ω′(s) ds + 2(|β| + 1) |λ| ∫ 1 0 k |φ(s)| |x(s) − y(s)|ω′(s) ds ≤ 2(|β| + 1) |λ| ∫ t 0 k |φ(s)| |x(s) − y(s)|ω′(s) ds + 2(|β| + 1) |λ| ∫ 1 0 k |φ(s)| |x(s) − y(s)|ω′(s) ds. taking the maximum over t ∈ [0, 1] we obtain ∥tx − ty∥∞ ≤ 2 2(|β| + 1) |λ| kmω∥x − y∥∞. therefore, t is a contraction operator, since µ = 2 2(|β|+1) |λ| kmω < 1. thus from the banach contraction principle, t has a unique fixed point as desired. now, we are going to use topological fixed point theory, more precisely schaefer’s fixed point theorem, to establish the existence of at least one solution of boundary value problem (5.7)–(5.8), dropping the lipschitzianity of the function f. first, we prove that the operator t is compact. theorem 5.3. the operator t : (c2[0, 1], ∥ · ∥∞) −→ (c2[0, 1], ∥ · ∥∞) is compact. proof. we start by proving the continuity of t . let (xn) ⊂ c2[0, 1], x ∈ c2[0, 1] be such that ∥xn − x∥∞ → 0. we have to show that ∥txn − tx∥∞ → 0. fixed ε > 0, there exists k ≥ 0 such that ∥xn∥∞ ≤ k, ∀n ∈ n ∥x∥∞ ≤ k. since f : [0, 1] × [−k, k] −→ r is continuous, then it is uniformly continuous on [0, 1] × [−k, k]. 532 r. e. castillo & b. sultan cubo 24, 3 (2022) thus there exists δ(ε) > 0 such that |f(s1, x(s1)) − f(s2, y(s2))| ≤ ε, for every (s1, x(s1)), (s2, y(s2)) ∈ [0, 1] × [−k, k] such that ∥(s1 − s2, x(s1) − y(s2))∥2 < δ(ε). from the fact that ∥xn − x∥∞ → 0, it follow that there exists n(ε) ∈ n such that sup t∈[0,1] |xn(t) − x(t)| < δ, for every n ≥ n(ε). consequently, from (5.12), ∥txn − tx∥∞ = sup t∈[0,1] |txn(t) − tx(t)| ≤ sup t∈[0,1] { 2 |λ| ∫ t 0 |f(s, xn(s)) − f(s, x(s))| |φ(s)| ω′(s) ds + 2(|β| + 1) |λ| ∫ 1 0 |f(s, xn(s)) − f(s, x(s))| |φ(s)| ω′(s) ds } < 2|β| + 4 |λ| mωε, m := sup t∈[0,1] 1 |φ(t)| , ω := sup t∈[0,1] ω′(t). therefore, the operator t is continuous. to prove the compactness we consider a bounded set x ⊂ c2[0, 1] and we will show that t(x) is relatively compact in (c2[0, 1], ∥ · ∥∞) by using the arzela-ascoli theorem. let k ≥ 0 be such that ∥x∥∞ ≤ k, for every x ∈ x. from the bounds (5.10) and (5.11) we have |tx(t)| ≤ 1 |λ| ∫ t 0 |f(s, x(s))| |φ(s)| |k(s, t)|ω′(s) ds + 1 |λ| ∫ 1 0 |f(s, x(s))| |φ(s)| |χ(0,η)βk(s, η) − k(s, 1)|ω′(s) ds + ∣∣∣∣αλ ∣∣∣∣ |e−λt − e−λ + βe−λη − β| ≤ 2mω |λ| ∫ t 0 |f(s, x(s))| ds + 2(|β| + 1) |λ| mω ∫ 1 0 |f(s, x(s))| ds + ∣∣∣∣αλ ∣∣∣∣ |e−λt − e−λ + βe−λη − β| ≤ 2|β| + 4 |λ| mω ∫ 1 0 |f(s, x(s))| ds + ∣∣∣∣αλ ∣∣∣∣ |e−λt − e−λ + βe−λη − β|. cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 533 we obtain a upper bound for |e−λt − e−λ + βe−λη − β|, namely |e−λt − e−λ + βe−λη − β| ≤ ∆, t ∈ [0, 1], where ∆ :=   (|β| + 1)e−λ, −1 < λ < 0 2(|β| + 1), λ > 0. on the other hand, since the function f is uniformly continuous on the compact set [0, 1]×[−k, k], then there exists, and it is finite, the positive number rk = ∥f∥∞ = sup x∈x sup s∈[0,1] |f(s, x(s))| < ∞, (s, x(s)) ∈ [0, 1] × [−k, k]. thus, we have ∥tx∥∞ ≤ 2|β| + 4 |λ| mωrk + ∣∣∣∣αλ ∣∣∣∣∆, (5.13) for every x ∈ x. that means, the set t(x) is bounded in c2[0, 1]. now, if t1, t2 ∈ [0, 1], are such that t1 ≤ t2 and satisfy |t1 − t2| < δ, then |tx(t1) − tx(t2)| = ∣∣∣∣∣1λ ∫ t1 0 f(s, x(s)) φ(s) ω′(s) ds − 1 λ ∫ t2 0 f(s, x(s)) φ(s) ω′(s) ds − α λ e−λt1 + α λ e−λt2 ∣∣∣∣∣ = ∣∣∣∣∣1λ ∫ t2 t1 f(s, x(s)) φ(s) ω′(s) ds + α λ (e−λt2 − e−λt1) ∣∣∣∣∣ → 0, as |t1 − t2| → 0 for every x ∈ x, so the set t(x) ⊂ c2[0, 1] satisfies the hypotheses of arzela-ascoli’s theorem, so t(x) is relatively compact in c2[0, 1]. therefore, the operator t is compact. now, we establish the following existence result. theorem 5.4. let f : [0, 1] × r −→ r be a continuous and differentiable in the first variable function, and let us assume that there exist c, d ≥ 0 and q ∈ (0, 1) such that |f(s, r)| ≤ c|r|q + d. for every (s, r) ∈ [0, 1] × r. then, the nonlinear boundary value problem (5.7)–(5.8) has at least one solution. proof. the theorem is proved once we assure the existence of at least a fixed point of the operator t . let s = {x ∈ c2[0, 1] : ∃σ ∈ [0, 1] such that x = σtx}. 534 r. e. castillo & b. sultan cubo 24, 3 (2022) to apply schaefer’s fixed point theorem we should show that s is bounded. let x ∈ s, ∥x∥∞ =σ∥tx∥∞. now, from (5.13) we have |tx(t)| ≤ 2|β| + 4 |γ| ∫ 1 0 |f(s, x(s))| |φ(s)| ω′(s) ds + ∣∣∣∣αβ ∣∣∣∣∆ ≤ 2|β| + 4|γ| mω(c∥x∥q∞ + d) + ∣∣∣∣αβ ∣∣∣∣∆ < ∞. then ∥x∥∞ = σ∥tx∥∞ ≤ σ 2|β| + 4 |γ| mω(c∥x∥q∞ + d) + ∣∣∣∣αβ ∣∣∣∣∆σ < ∞. this inequality and the fact q ∈ (0, 1) shows that s is bounded. thus, from schaefer’s fixed point theorem, the operator t has a fixed point, which implies that boundary value problem (5.7)–(5.8) has a solution. notice from the proof of the theorem above, that we can use the functions φ and ω given in the definition of the (φ, ω)-derivative to rewrite theorem 5.4 as: theorem 5.5. let f : [0, 1] × r −→ r be a continuous and differentiable in the first variable function, and let us assume that there exist c, d ≥ 0 and q ∈ (0, 1) such that |f(s, r)| |φ(s)| ω′(s) ≤ c|r|q + d. for every (s, r) ∈ [0, 1] × r. then, the nonlinear boundary value problem (5.7)–(5.8) has at least one solution. schaefer’s theorem is a consequence of the schauder fixed point theorem, which is a localization fixed point result. we will use schauder’s theorem to give a localization result for the solutions of boundary value problem (5.7)–(5.8). theorem 5.6. let f : [0, 1] × r −→ r be a continuous and differentiable in the first variable function and, in addition, let us assume that f ∈ l1([0, 1] × r). let b(r) be the closed ball with radius r. then, the nonlinear boundary value problem (5.7)–(5.8) has at least one solution for every closed ball b(r) such that r ≥ 2|β| + 4 |λ| mω∥f∥1 + ∣∣∣∣αλ ∣∣∣∣∆, (5.14) with, ∆ :=   (|β| + 2)e−λ, −1 < λ < 0 2(|β| + 1), λ > 0 , m := sup t∈[0,1] ∣∣∣∣ 1φ(t) ∣∣∣∣ , ω = sup t∈[0,1] ω′(t). cubo 24, 3 (2022) a derivative-type operator and its application to the solvability... 535 proof. since the operator t is continuous and compact, we can apply schauder’s fixed point theorem, once we prove that t(b(r)) ⊂ b(r). from (5.13) and the hypotheses, we have |tx(t)| ≤ 2|β| + 4 |γ| ∫ 1 0 |f(s, x(s))| |φ(s)| ω′(s) ds + ∣∣∣∣αβ ∣∣∣∣∆ (5.15) ≤ 2|β| + 4 |γ| mω∥f∥1 + ∣∣∣∣αβ ∣∣∣∣∆ ≤ r. thus, ∥tx∥∞ ≤ r. consequently t(b(r)) ⊂ b(r). finally, from schauder’s theorem, t has a fixed point, as so boundary value problem (5.7)–(5.8) has at least one solution, for every closed ball b(r) with radius r as in (5.14). we can control the growth behavior of the nonlinear function f and still guarantee the existence of solutions for bvp (5.7)–(5.8). some of these behaviors, as we will see, can be controlled in terms of the functions φ and ω given in the definition of the (φ, ω)-derivative, which can be interpreted as behaviors scaled for the (φ, ω)-derivative. the main idea is to replace the integral term (5.15) with some condition which allows found a bound for it. for instance if we assume that f is uniformly bounded by a > 0 on [0, 1] × r, then use the estimate |f(s, x(s))| ≤ a in (5.15) and obtain the radius r ≥ 2|β|+4|λ| mωak + ∣∣α λ ∣∣∆. if we assume that |f(s, y)| ≤ a |φ(s)| w′(s) , for some a > 0, for all (s, x) ∈ [0, 1] × r, the integral term is less or equal to a and the radius is r ≥ 2|β|+4|λ| a + ∣∣α λ ∣∣∆. finally, if |f(s, x(s))| ≤ |γ| 2(|β| + 2)mω s|x(s)|, and we assume that ∣∣∣∣αλ ∣∣∣∣∆ ≤ r2, for each r > 0 given. then, estimate (5.15) is rewrite as |tx(t)| ≤ ∥x∥∞ 2 + ∣∣∣∣αβ ∣∣∣∣∆ ≤ r2 + r2 = r. this proves that t applies any ball of radius r into itself. therefore, we conclude that bvp (5.7)–(5.8) as at leat one solution on each ball of radius r. in similar fashion it can be proved that for |f(s, y)| ≤ |γ| 4(|β| + 2)mω( |an| n+1 + · · · + |a0|) |ansn + · · · + a0|)|x(s)|, 536 r. e. castillo & b. sultan cubo 24, 3 (2022) and ∣∣∣∣αλ ∣∣∣∣∆ ≤ r2, for each r > 0, the same conclusion holds. cubo 24, 3 (2022) a 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[34] w. zhong and l. wang, “positive solutions of conformable fractional differential equations with integral boundary conditions”, bound. value probl., paper no. 136, 12 pages, 2018. motivation derivative of variable order integration of variable order taylor formula three-point boundary value problems of variable order () cubo a mathematical journal vol.16, no¯ 03, (01–10). october 2014 on upper and lower ω-irresolute multifunctions c. carpintero department of mathematics, universidad de oriente, nucleo de sucre cumana, venezuela. facultad de ciencias basicas, universidad del atlantico, colombia. carpintero.carlos@gmail.com n. rajesh department of mathematics, rajah serfoji govt. college, thanjavur-613005, tamilnadu, india. nrajesh topology@yahoo.co.in e. rosas department of mathematics, universidad de oriente, nucleo de sucre cumana, venezuela. facultad de ciencias basicas, universidad del atlantico, colombia. ennisrafael@gmail.com s. saranyasri department of mathematics, m. r. k. institute of technology, kattumannarkoil, cuddalore -608 301, tamilnadu, india. srisaranya 2010@yahoo.com abstract in this paper we define upper (lower) ω-irresolute multifunction and obtain some characterizations and some basic properties of such a multifunction. resumen en este art́ıculo definimos la multifunción superior (inferior) ω-irresoluto y obtenemos algunas caracterizaciones y algunas propiedades básicas de este tipo de multifunciones. keywords and phrases: ω-open set, ω-continuous multifunctions, ω-irresolute multifunctions. 2010 ams mathematics subject classification: 54c05, 54c601, 54c08, 54c10 2 c. carpintero, n. rajesh, e. rosas & s. saranyasri cubo 16, 3 (2014) 1 introduction it is well known that various types of functions play a significant role in the theory of classical point set topology. a great number of papers dealing with such functions have appeared, and a good number of them have been extended to the setting of multifunctions: [4],[5],[6],[7], [10],[11],[12],[13],[15]. this implies that both, functions and multifunctions are important tools for studying other properties of spaces and for constructing new spaces from previously existing ones. recently, zorlutuna introduced the concept of ω-continuous multifunctions [15], ω-continuity which is a weaker form of continuity in ordinary was extended to multifunctions. the purpose of this paper is to define upper (respectively lower) ω-irresolute multifunctions and to obtain several characterizations of such a multifunction. 2 preliminaries throughout this paper, (x, τ) and (y, σ) (or simply x and y) always mean topological spaces in which no separation axioms are assumed unless explicitly stated. let a be a subset of a space x. for a subset a of (x, τ), cl(a) and int(a) denote the closure of a with respect to τ and the interior of a with respect to τ, respectively. recently, as generalization of closed sets, the notion of ω-closed sets were introduced and studied by hdeib [9]. a point x ∈ x is called a condensation point of a if for each u ∈ τ with x ∈ u, the set u ∩ a is uncountable. a is said to be ω-closed [9] if it contains all its condensation points. the complement of an ω-closed set is said to be ω-open. it is well known that a subset w of a space (x, τ) is ω-open if and only if for each x ∈ w, there exists u ∈ τ such that x ∈ u and u\w is countable. the family of all ω-open subsets of a topological space (x, τ) is denoted by ωo(x), forms a topology on x finer than τ. the family of all ω-closed subsets of a topological space (x, τ) is denoted by ωc(x). the ω-closure and the ω-interior, that can be defined in the same way as cl(a) and int(a), respectively, will be denoted by ω cl(a) and ω int(a), respectively. we set ωo(x, x) = {a : a ∈ ωo(x) and x ∈ a}. a subset u of x is called an ω-neighborhood of a point x ∈ x if there exists v ∈ ωo(x, x) such that v ⊂ u. by a multifunction f : (x, τ) → (y, σ), following [3], we shall denote the upper and lower inverse of a set b of y by f+(b) and f−(b), respectively, that is, f+(b) = {x ∈ x : f(x) ⊂ b} and f−(b) = {x ∈ x : f(x) ∩ b 6= ∅}. in particular, f−(y) = {x ∈ x : y ∈ f(x)} for each point y ∈ y and for each a ⊂ x, f(a) = ⋃ x∈a f(x). then f is said to be surjection if f(x) = y. definition 2.1. a multifunction f : (x, τ) → (y, σ) is said to be: (i) upper ω-continuous (briefly u.ω-c.) [15] if for each point x ∈ x and each open set v containing f(x), there exists u ∈ ωo(x, x) such that f(u) ⊂ v; (ii) lower ω-continuous (briefly l.ω-c.) [15] if for each point x ∈ x and each open set v such that f(x) ∩ v 6= ∅, there exists u ∈ ωo(x, x) such that u ⊂ f−(v). cubo 16, 3 (2014) on upper and lower ω-irresolute multifunctions 3 3 on upper and lower ω-irresolute multifunctions definition 3.1. a multifunction f : (x, τ) → (y, σ) is said to be: (i) upper ω-irresolute (briefly u.ω-i.) if for each point x ∈ x and each ω-open set v containing f(x), there exists u ∈ ωo(x, x) such that f(u) ⊂ v; (ii) lower ω-irresolute (briefly l.ω-i.) if for each point x ∈ x and each ω-open set v such that f(x) ∩ v 6= ∅, there exists u ∈ ωo(x, x) such that u ⊂ f−(v). it is clear that every upper (lower) ω-irresolute multifunction is upper (lower) ω-continuous. but the converse is not true as shown by the following example. example 3.2. let x = r with the topology τ = {∅, r, q}. define a multifunction f : (r, τ) → (r, τ) as follows: f(x) = { q if x ∈ r − q r − q if x ∈ q. then f is u.ω-c. but is not u.ω-i. in a similar form, we can find a multifunction g that is l.ω-c. but is not l.ω-i. theorem 3.3. the following statements are equivalent for a multifunction f : (x, τ) → (y, σ): (i) f is u.ω-i.; (ii) for each point x of x and each ω-neighborhood v of f(x), f+(v) is an ω-neighborhood of x; (iii) for each point x of x and each ω-neighborhood v of f(x), there exists an ω-neighborhood u of x such that f(u) ⊂ v; (iv) f+(v) ∈ ωo(x) for every v ∈ ωo(y); (v) f−(v) ∈ ωc(x) for every v ∈ ωc(y); (vi) ω cl(f−(b)) ⊂ f−(ω cl(b)) for every subset b of y. proof. (i) ⇒ (ii): let x ∈ x and w be an ω-neighborhood of f(x). there exists v ∈ ωo(y) such that f(x) ⊂ v ⊂ w. since f is u.ω-i., there exists u ∈ ωo(x, x) such that f(u) ⊂ v. therefore, we have x ∈ u ⊂ f+(v) ⊂ f+(w); hence f+(w) is an ω-neighborhood of x. (ii) ⇒ (iii): let x ∈ x and v be an ω-neighborhood of f(x). put u = f+(v). then, by (ii), u is an ω-neighborhood of x and f(u) ⊂ v. (iii) ⇒ (iv): let v ∈ ωo(y) and x ∈ f+(v). there exists an ω-neighborhood g of x such that f(g) ⊂ v. therefore, for some u ∈ ωo(x, x) such that u ⊂ g and f(u) ⊂ v. therefore, we obtain x ∈ u ⊂ f+(v); hence f+(v) ∈ ωo(y). (iv) ⇒ (v): let k be an ω-closed set of y. we have x\f−(k) = f+(y\k) ∈ ωo(x); hence 4 c. carpintero, n. rajesh, e. rosas & s. saranyasri cubo 16, 3 (2014) f−(k) ∈ ωc(x). (v) ⇒ (vi): let b be any subset of y. since ω cl(b) is ω-closed in y, f−(ω cl(b)) is ω-closed in x and f−(b) ⊂ f−(ω cl(b)). therefore, we obtain ω cl(f−(b)) ⊂ f−(ω cl(b)). (vi) ⇒ (i): let x ∈ x and v ∈ ωo(y) with f(x) ⊂ v. then we have f(x) ∩ (y\v) = ∅; hence x /∈ f−(y\v). by (vi), x ∈ ω cl(f−(y\v)) and there exists u ∈ ωo(x, x) such that u ∩ f−(y\v) = ∅. therefore, we obtain f(u) ⊂ v and hence f is u.ω-i. theorem 3.4. the following statements are equivalent for a multifunction f : (x, τ) → (y, σ): (i) f is l.ω-i.; (ii) for each v ∈ ωo(y) and each x ∈ f−(v), there exists u ∈ ωo(x, x) such that u ⊂ f−(v); (iii) f−(v) ∈ ωo(x) for every v ∈ ωo(y); (iv) f+(k) ∈ ωc(x) for every k ∈ ωc(y); (v) f(ω cl(a)) ⊂ ω cl(f(a)) for every subset a of x; (vi) ω cl(f+(b)) ⊂ f+(ω cl(b)) for every subset b of y. proof. (i) ⇒ (ii): this is obvious. (ii) ⇒ (iii): let v ∈ ωo(y) and x ∈ f−(v). there exists u ∈ ωo(x, x) such that u ⊂ f−(v). therefore, we have x ∈ u ⊂ cl(int(u)) ∪ int(cl(u)) ⊂ cl(int(f−(v))) ∪ int(cl(f−(v))); hence f−(v) ∈ ωo(x). (iii) ⇒ (iv): let k be an ω-closed set of y. we have x\f+(k) = f−(y\k) ∈ ωo(x); hence f+(k) ∈ ωc(x). (iv) ⇒ (v) and (v) ⇒ (vi): straightforward. (vi) ⇒ (i): let x ∈ x and v ∈ ωo(y) with f(x) ∩ v 6= ∅. then f(x) is not a subset of y\v and x /∈ f+(y\v). since y\v is ω-closed in y, by (vi), x /∈ ω cl(f+(y\v)) and there exists u ∈ ωo(x, x) such that ∅ = u ∩ f−(y\v) = u ∩ (x\f−(v)). therefore, we obtain u ⊂ f−(v); hence f is l.ω-i.. lemma 3.5. if f : (x, τ) → (y, σ) is a multifunction, then (ω cl f)−(v) = f−(v) for each v ∈ ωo(y). proof. let v ∈ ωo(y) and x ∈ (ω cl f)−(v). then v∩(ω cl f)(x) 6= ∅. since v ∈ ωo(y), we have v ∩f(x) 6= ∅ and hence x ∈ f−(v). conversely, let x ∈ f−(v). then ∅ 6= f(x)∩v ⊂ (ω cl f)(x)∩v and hence x ∈ (ω cl f)−(v). therefore, we obtain (ω cl f)−(v) = f−(v). theorem 3.6. a multifunction f : (x, τ) → (y, σ) is l.ω-i. if and only if ω cl f : (x, τ) → (y, σ) is l.ω-i. proof. the proof is an immediate consequence of lemma 3.5 and theorem 3.4 (iii). cubo 16, 3 (2014) on upper and lower ω-irresolute multifunctions 5 definition 3.7. a subset a of a topological space (x, τ) is said to be: (i) α-regular [8] (resp. α-ω-regular) if for each a ∈ a and any open (resp. ω-open) set u containing a, there exists an open set g of x such that a ∈ g ⊂ cl(g) ⊂ u; (ii) α-paracompact [8] if every x-open cover a has an x-open refinement which covers a and is locally finite for each point of x. lemma 3.8. if a is an α-ω-regular, α-paracompact subset of a space x and u is ω-neighborhood of a, then there exists an open set g of x such that a ⊂ g ⊂ cl(g) ⊂ u. proof. the proof is similar to that [8, theorem 2.5]. definition 3.9. a multifunction f : (x, τ) → (y, σ) is said to be punctually α-paracompact (resp. punctually α-ω-regular, punctually α-regular) if for each x ∈ x, f(x) is α-paracompact (resp. α-ω-regular, α-regular). lemma 3.10. if f : (x, τ) → (y, σ) is punctually α-paracompact and punctually α-ω-regular, (ω cl f)+(v) = f+(v) for each v ∈ ωo(y). proof. let v ∈ ωo(y). suppose that x ∈ (ω cl f)+(v). then, we have f(x) ⊂ ω cl(f(x)) ⊂ v and hence x ∈ f+(v). therefore, we obtain (ω cl f)+(v) ⊂ f+(v). conversely, suppose that x ∈ f+(v). then f(x) ⊂ v and by lemma 3.8 we have f(x) ⊂ g ⊂ cl(g) ⊂ v for some open set g of y. therefore, (ω cl f)(x) ⊂ v and hence x ∈ (ω cl f)+(v). thus, we obtain f+(v) ⊂ (ω cl f)+(v); hence (ω cl f)+(v) = f+(v). theorem 3.11. let f : (x, τ) → (y, σ) be punctually α-paracompact and punctually α-ω-regular multifunction. then f is u.ω-i. if and only if ω cl f : (x, τ) → (y, σ) is u.ω-i.. proof. the proof follows from lemma 3.10. lemma 3.12. [1] let a and b be subsets of a topological space (x, τ). (i) if a ∈ ωo(x) and b ∈ τ, then a ∩ b ∈ ωo(b); (ii) if a ∈ ωo(b) and b ∈ ωo(x), then a ∈ ωo(x). theorem 3.13. let f : (x, τ) → (y, σ) be a multifunction and u an open subset of x. if f is a u.ω-i. (resp. l.ω-i.), then f|u: u → y is an u.ω-i. (resp. l.ω-i.). proof. let v be any ω-open set of y. let x ∈ u and x ∈ f− |u (v). since f is l.ω-i. multifunction, then there exists an ω-open set g containing x such that g ⊂ f−(v). then x ∈ g ∩ u ∈ ωo(u) and g ∩ u ⊂ f− |u (v) . this shows that f|u is a l.ω-i.. the proof of the u.ω-i. of f|u is similar. 6 c. carpintero, n. rajesh, e. rosas & s. saranyasri cubo 16, 3 (2014) theorem 3.14. let {ui : i ∈ ∆} be an open cover of a space x. a multifunction f : (x, τ) → (y, σ) is u.ω-i. if and only if the restriction f|ui : ui → y is u.ω-i. for each i ∈ ∆. proof. suppose that f is u.ω-i.. let i ∈ ∆ and x ∈ ui and v be an ω-open set of y containing f|ui(x). since f is u.ω-i. and f(x) = f|ui(x), there exists g ∈ ωo(x, x) such that f(g) ⊂ v. set u = g ∩ ui, then x ∈ u ∈ ωo(ui, x) and f|ui(u) = f(u) ⊂ v. therefore, f|ui is u.ωi..conversely, let x ∈ x and v ∈ ωo(y) containing f(x). there exists i ∈ ∆ such that x ∈ ui. since f|ui is u.ω-i. and f(x) = f|ui(x), there exists u ∈ ωo(ui, x) such that f|ui(u) ⊂ v. then we have u ∈ ωo(x, x) and f(u) ⊂ v. therefore, f is u.ω-i.. theorem 3.15. let {ui : i ∈ ∆} be an open cover of a space x. a multifunction f : (x, τ) → (y, σ) is l.ω-i. if and only if the restriction f|ui : ui → y is l.ω-i. for each i ∈ ∆. proof. the proof is similar to that of theorem 3.14 and is thus omitted. definition 3.16. a subset k of a space x is said to be ω-compact relative to x [2] (resp. ωlindelöf relative to x [9]) if every cover of k by ω-open sets of x has a finite (resp. countable) subcover. a space x is said to be ω-compact [2] (resp. ω-lindelöf [9]) if x is ω-compact (resp. ω-lindelöf) relative to x. theorem 3.17. let f : (x, τ) → (y, σ) be an u.ω-i. multifunction and f(x) is ω-compact relative to y for each x ∈ x. if a is ω-compact relative to x, then f(a) is ω-compact relative to y. proof. let {vi : i ∈ ∆} be any cover of f(a) by ω-open sets of y. for each x ∈ a, there exists a finite subset ∆(x) of ∆ such that f(x) ⊂ ∪{vi : i ∈ ∆(x)}. put v(x) = ∪{vi : i ∈ ∆(x)}. then f(x) ⊂ v(x) ∈ ωo(y) and there exists u(x) ∈ ωo(x, x) such that f(u(x)) ⊂ v(x). since {u(x) : x ∈ a} is an ω-open cover of a, there exists a finite number of points of a, say, x1, x2,....xn such that a ⊂ ∪{u(xi) : i = 1, 2, ....n}. therefore, we obtain f(a) ⊂ f( n ∪ i=1 u(xi)) ⊂ n ∪ i=1 f(u(xi)) ⊂ n ∪ i=1 v(xi) ⊂ n ∪ i=1 ∪ i∈∆(xi) vi. this shows that f(a) is ω-compact relative to y. corollary 3.18. let f : (x, τ) → (y, σ) be an u.ω-i. surjective multifunction and f(x) is ωcompact relative to y for each x ∈ x. if x is ω-compact, then y is ω-compact. theorem 3.19. let f : (x, τ) → (y, σ) be an u.ω-i. multifunction and f(x) is ω-lindelöf relative to y for each x ∈ x. if a is ω-lindelöf relative to x, then f(a) is ω-lindelöf relative to y. proof. the proof is similar to that of theorem 3.17 and is thus omitted. corollary 3.20. let f : (x, τ) → (y, σ) be an u.ω-i. surjective multifunction and f(x) is ωlindelöf relative to y for each x ∈ x. if x is ω-lindelöf, then y is ω-lindelöf. definition 3.21. a topological space x is said to be ω-normal [10] if for any pair of disjoint closed subsets a, b of x, there exist disjoint u, v ∈ ωo(x) such that a ⊂ u and b ⊂ v. cubo 16, 3 (2014) on upper and lower ω-irresolute multifunctions 7 theorem 3.22. if y is ω-normal and fi : xi → y is an u.ω-i. multifunction such that fi is punctually closed for i = 1, 2 and the product of two ω-open sets is ω-open, then the set {(x1, x2) ∈ x1 × x2 : f1(x1) ∩ f2(x2) 6= ∅} is ω-closed in x1 × x2. proof. let a = {(x1, x2) ∈ x1 × x2 : f1(x1) ∩ f2(x2) 6= ∅} and (x1, x2) ∈ (x1 × x2)\a. then f1(x1) ∩ f2(x2) = ∅. since y is ω-normal and fi is punctually closed for i = 1, 2, there exist disjoint v1, v2 ∈ ωo(x) such that fi(xi) ⊂ vi for i = 1, 2. since fi is u.ω-i., f + i (vi) ∈ ωo(xi, xi) for i = 1, 2. put u = f+ 1 (v1) × f + 2 (v2), then u ∈ ωo(x1 × x2) and (x1, x2) ∈ u ⊂ (x1 × x2)\a. this shows that (x1 × x2)\a ∈ ωo(x1 × x2); hence a is ω-closed set in x1 × x2. definition 3.23. [2] let a be a subset of a topological space x. the ω-frontier of a denoted by ωfr(a), is defined as follows: ωfr(a) = ω cl(a) ∩ ω cl(x\a). theorem 3.24. the set of a point x of x at which a multifunction f : (x, τ) → (y, σ) is not u.ω-i. (resp. l.ω-i.) is identical with the union of the ω-frontiers of the upper (resp. lower) inverse images of ω-open sets containing (resp. meeting) f(x). proof. let x be a point of x at which f is not u.ω-i.. then there exists v ∈ ωo(y) containing f(x) such that u ∩ (x\f+(v)) 6= ∅ for each u ∈ ωo(x, x). then x ∈ ω cl(x\f+(v)). since x ∈ f+(v), we have x ∈ ω cl(f+(y) and hence x ∈ ωfr(f+(a)). conversely, let v ∈ ωo(y) containing f(x) and x ∈ ωfr(f+(v)). now, assume that f is u.ω-i. at x, then there exists u ∈ ωo(x, x) such that f(u) ⊂ v. therefore, we obtain x ∈ u ⊂ ω int(f+(v). this contradicts that x ∈ ωfr(f+(v)). thus, f is not u.ω-i.. the proof of the second case is similar. for a multifunction f : (x, τ) → (y, σ), the graph multifunction gf(x) : x → x × y is defined as follows: gf(x) = {x} × f(x) for all x ∈ x. lemma 3.25. for a multifunction f : (x, τ) → (y, σ), the following holds: (i) g+ f (a × b) = a ∩ f+(b); (ii) g− f (a × b) = a ∩ f−(b) for any subset a of x and b of y. theorem 3.26. let f : (x, τ) → (y, σ) be a multifunction and x be a connected space. if the graph multifunction of f is u.ω-i. (respectively l.ω-i.), then f is u.ω-i. (respectively. l.ω-i.). proof. let x ∈ x and v be any ω-open subset of y containing f(x). since x × v is an ω-open set of x × y and gf(x) ⊂ x × v, there exists an ω-open set u containing x such that gf(u) ⊂ x × v. by lemma 3.25, we have u ⊂ g+ f (x × v) = f+(v) and f(u) ⊂ v. thus, f is u.ω-i.. the proof of the l.ω-i. of f can be obtained in a similar manner. 8 c. carpintero, n. rajesh, e. rosas & s. saranyasri cubo 16, 3 (2014) definition 3.27. [2] a topological space (x, τ) is said to ω-t2 if for each pair of distinct points x and y in x, there exist disjoint ω-open sets u and v in x such that x ∈ u and y ∈ v. theorem 3.28. if f : (x, τ) → (y, σ) is an u.ω-i. injective multifunction and point closed from a topological space x to an ω-normal space y, then x is an ω-t2 space. proof. let x and y be any two distinct points in x. then we have f(x) ∩ f(y) = ∅ since f is injective. since y is ω-normal, it follows that there exist disjoint open sets u and v containing f(x) and f(y), respectively. thus, there exist disjoint ω-open sets f+(u) and f+(v) containing x and y, respectively such g ⊂ f+(u) and w ⊂ f+(v). therefore, we obtain g ∩ w = ∅; hence x is ω-t2. definition 3.29. a multifunction f : (x, τ) → (y, σ) is said have an ω-closed graph if for each pair (x, y) /∈ g(f) there exist u ∈ ωo(x, x) and v ∈ ωo(y, y) such that (u × v) ∩ g(f) = ∅. theorem 3.30. let f : (x, τ) → (y, σ) be an u.ω-c. multifunction. if f(x) is α-paracompact for each x ∈ x, then g(f) is ω-closed. proof. suppose that (x0, y0) /∈ g(f). then y0 /∈ f(x0). since y is a t2 space, for each y ∈ f(x0) there exist disjoint open sets v(y) and w(y) containing y and y0, respectively. the family {v(y) : y ∈ f(x0)} is an open cover of f(x0). thus, by α-paracompactness of f(x0), there is a locally finite open cover ∆ = {uβ : β ∈ i} which refines {v(y) : y ∈ f(x0)}. therefore, there exists an open neighborhood w0 of y0 such that w0 intersects only finitely many members uβ1, uβ2,.....uβn of ∆. choose y1, y2,.....yn in f(x0) such that uβi ⊂ v(yi) for each 1 ≤ i ≤ n, and set w = w0 ∩ ( n ∩ i=1 w(yi)). then w is an open neighborhood of y0 such that w ∩ ( ∪ β∈i vβ) = ∅. by the upper ω-continuity of f, there is a u ∈ ωo(x, x0) such that u ⊂ f +( ∪ β∈i vβ). it follows that (u × w) ∩ g(f) = ∅. therefore, g(f) is ω-closed. theorem 3.31. let f : (x, τ) → (y, σ) be a multifunction from a space x into an ω-compact space y. if g(f) is ω-closed, then f is u.ω-c.. proof. suppose that f is not u.ω-c.. then there exists a nonempty closed subset c of y such that f−(c) is not ω-closed in x. we may assume that f−(c) 6= ∅. then there exists a point x0 ∈ ω cl(f −(c))\f−(c). hence for each point y ∈ c, we have (x0, y) /∈ g(f). since f has an ω-closed graph, there are ω-open subsets u(y) and v(y) containing x0 and y, respectively such that (u(y) × v(y)) ∩ g(f) = ∅. then {y\c} ∪ {v(y) : y ∈ c} is an ω-open cover of y, and thus it has a subcover {y\c}∪{v(yi) : yi ∈ c, 1 ≤ i ≤ n}. let u = n ∩ i=1 u(yi) and v = n ∪ i=1 v(yi). it is easy to verify that c ⊂ v and (u × v) ∩ g(f) = ∅. since u is an ω-neighborhood of x0, u ∩ f −(c) 6= ∅. it follows that ∅ 6= (u × c) ∩ g(f) ⊂ (u × v) ∩ g(f). this is a contradiction. hence the proof is completed. corollary 3.32. let f : (x, τ) → (y, σ) be a multifunction into an ω-compact t2 space y such that f(x) is ω-closed for each x ∈ x. then f is u.ω-c. if and only if it has an ω-closed graph. cubo 16, 3 (2014) on upper and lower ω-irresolute multifunctions 9 theorem 3.33. let f : (x, τ) → (y, σ) be an u.ω-i. multifunction into an ω-t2 space y. if f(x) is α-paracompact for each x ∈ x, then g(f) is ω-closed. proof. the proof is clear. definition 3.34. [14] let a be a subset of x. then f : x → a is called a retracting multifunction if x ∈ f(x) for each x ∈ a. theorem 3.35. let f : x → x be an u.ω-i. multifunction of a t2 space x into itself. if f(x) is α-paracompact for each x ∈ x, then the set a = {x : x ∈ f(x)} is ω-closed. proof. let x0 ∈ ω cl(a)\a. then x0 /∈ f(x0). since x is t2, for each x ∈ f(x0) there exist disjoint open sets u(x) and v(x) containing x0 and x respectively. then {v(x) : x ∈ f(x0)} is an open cover of f(x0). by the α-paracompactness of f(x0), {v(x) : x ∈ f(x0)} has a locally finite open refinement w = {wβ : β ∈ i} which covers f(x0). therefore, we can choose an open neighborhood u0 of x0 such that u0 intersects only finitely many members wβ1, wβ2,.....wβn of w. choose x1, x2,.....xn in f(x0) such that wβi ⊂ v(xi) for each 1 ≤ i ≤ n, and set u = u0 ∩ ( n ∩ i=1 u(xi)). then u is an open neighborhood of x0 such that u ∩ ( ∪ β∈i wβ) = ∅. since f is u. ω-i., there is a g ∈ ωo(x, x0) such that g ⊂ f +( ∪ β∈i wβ). it follows that g ∩ u is an ω-neighborhood of x0 and satisfies (g ∩ u) ∩ a = ∅. this contradicts the fact that x0 ∈ ω cl(a). corollary 3.36. let a be a subset of x and f : x → a an u.ω-i. retracting multifunction such that f(x) is α-paracompact for each x ∈ a. if x is t2, then a is ω-closed. received: march 2014. accepted: may 2014. references [1] k. al-zoubi and b. al-nashef, the topology of ω-open subsets, al-manarah (9) (2003), 169179. [2] a. al-omari ans m. s. m. noorani, contra-ω-continuous and almost ω-continuous functions, int. j. math. math. sci. (9) (2007), 169-179. [3] t. banzaru, multifunctions and m-product spaces, bull. stin. tech. inst. politech. timisoara, ser. mat. fiz. mer. teor. apl., 17(31)(1972), 17-23. [4] c. carpintero, n. rajesh, e. rosas and s. saranyasri, some properties of upper/lower ωcontinuous multifunctions (submitted). [5] c. carpintero, n. rajesh, e. rosas and s. saranyasri, on upper and lower faintly ω-continuous multifunctions (submitted). 10 c. carpintero, n. rajesh, e. rosas & s. saranyasri cubo 16, 3 (2014) [6] c. carpintero, n. rajesh, e. rosas and s. saranyasri, on slightly omega-continuous multifunctions, to appear in punjab university journal of mathematics (2014). [7] c. carpintero, n. rajesh, e. rosas and s. saranyasri, properties of faintly ω-continuous functions, boletin de matematicas, 20(2) (2014). 135-143. [8] i. kovacevic, subsets and paracompactness, univ. u. novom sadu, zb. rad. prirod. mat. fac. ser. mat., 14(1984), 79-87. [9] h. z. hdeib, ω-closed mappings, revista colombiana mat., 16(1982), 65-78. [10] t. noiri, a. al-omari and m. s. m. noorani, slightly ω-continuous functions, fasc. math., (41) (2009), 97-106. [11] t. noiri and v. popa, almost weakly continuous multifunctions, demonstratio math., 26 (1993), 363-380. [12] t. noiri and v. popa, a unified theory of almost continuity for multifunctions, sci. stud. res. ser. math. inform., 20(1) (2010),185-214. [13] t. noiri and v. popa, a unified theory of weak continuity for multifunctions, stud. cerc. st ser. mat. univ. bacau, 16 (2006),167-200. [14] g. t. whyburn, retracting multifunctions, proc. nat. acad. sci., 59(1968), 343-348. [15] i. zorlutuna, ω-continuous multifunctions, filomat, 27(1) (2013), 155-162. introduction preliminaries on upper and lower -irresolute multifunctions cubo, a mathematical journal vol. 23, no. 01, pp. 121–144, april 2021 doi: 10.4067/s0719-06462021000100121 energy transfer in open quantum systems weakly coupled with two reservoirs franco fagnola1 damiano poletti2 emanuela sasso3 1,2 dipartimento di matematica, politecnico di milano, piazza leonardo da vinci 32, i-20133 milano, italy. franco.fagnola@polimi.it; damiano.poletti@polimi.it 3 dipartimento di matematica, università di genova, via dodecaneso 35, i 16146 genova, italy. sasso@dima.unige.it abstract we show that the energy transfer through an open quantum system with non-degenerate hamiltonian weakly coupled with two reservoirs in equilibrium is approximately proportional to the difference of their temperatures unless both temperatures are small. resumen mostramos que la transferencia de enerǵıa a través de un sistema cuántico abierto con hamiltoniano no-degenerado débilmente acoplado con dos reservorios en equilibrio es aproximadamente proporcional a la diferencia de sus temperaturas a menos que ambas temperaturas sean pequeñas. keywords and phrases: weak-coupling, quantum markov semigroup, quantum transport, energy current. 2020 ams mathematics subject classification: 81s22, 82c10, 80a19. accepted: 09 february, 2021 received: 21 september, 2020 ©2021 f. fagnola et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100121 https://orcid.org/0000-0002-1661-0992 https://orcid.org/0000-0002-2333-4252 https://orcid.org/0000-0003-0994-2400 122 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) 1 introduction energy transfer in classical and quantum systems and the validity of fourier’s law of heat conduction have been a hot topic for many years (see [3, 4, 6, 7, 12, 18, 25, 26] and the references therein). for quantum systems, in particular, after experimental evidence of effective quantum energy transfer in photosynthesis in some biological systems has been found (see [14, 24]), investigations have focused on understanding to what extent quantum mechanics contributes to transport efficiency. several models have been proposed involving open quantum systems (see e.g. [5, 6, 27]), mostly phenomenological, and also numerical simulations have been done showing different behaviours. the interaction of the open quantum system with reservoirs is described through interaction operators that appear in the dissipative part of the gorini-kossakowski-sudharshan-lindblad (gksl) [17, 22] generator l of the dynamics, while the hamiltonian part is given by the commutator with the system hamiltonian hs. however, when the gksl generator is rigorously deduced from some scaling (weak coupling or low density limit) both the system hamiltonian and the interaction operators appear in the gksl generator l after non-trivial transformations (see [1, 2, 9, 10, 13, 19]). in this paper we study models of open quantum systems rigorously deduced from the weak coupling limit. we consider a quantum system with non-degenerate hamiltonian hs coupled with two reservoirs in equilibrium at inverse temperatures β1 ≤ β2 and study variation of energy due to couplings with each reservoir. it is well-known (see lebowitz and spohn [25] (v.28)) that, by the second law of thermodynamics, energy (heat) flows from the hotter to the cooler reservoir. the energy flow, in general, is not proportional to the difference of temperature because of the nonlinear dependence of susceptibilities on temperature, namely an exact fourier’s law does not hold. however, we rigorously prove that it holds in an approximate way when the temperatures of reservoirs are not too small or, as an alternative, differences between nearest energy levels are small. more precisely, we show that the amount of energy flowing through the system, theorem 4.2, formula (4.5), is approximately proportional to the product of the temperature differences and a constant (conductivity) which can be interpreted as the average energy needed to jump from a level to the following higher level. the paper is organised as follows. in section 2 we introduce quantum markov semigroups (qms) arising from the weak coupling limit of a non-degenerate system with two boson reservoirs. the energy flow is computed explicitly in section 3, theorem 3.3, formula (3.7). the dependence of the energy flow on temperatures is studied in section 4. moreover, we also study (theorem 4.3) the asymptotic behaviour of the invariant state when the eigenvalues of hs increase in number and form a set more and more packed. it turns out that the invariant state converges towards a cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 123 gibbs state with temperature equal to the mean temperatures of the two baths. finally, in section 5, we consider as system the ising model hamiltonian and show that the energy flow in this case is zero. we have not been able to extend our analysis to quantum spin chains because their hamiltonians are highly degenerate and the gksl generator arising from the weak coupling limit, albeit explicit, is not easily treatable. in particular, we could not extract the relevant information on invariant states. 2 semigroups of weak coupling limit type we consider an open quantum system with hamiltonian hs acting on a complex separable hilbert space h with discrete spectral decomposition hs = ∑ m≥0 εmpεm (2.1) where εm, with εm < εn for m < n, are the eigenvalues of hs and pεm are the corresponding eigenprojectors. the system is coupled with two reservoirs each one in equilibrium with inverse temperatures β1 ≤ β2 with interaction hamiltonians h1 = d1 ⊗a+(φ1) + d∗1 ⊗a −(φ1), h2 = d2 ⊗a+(φ2) + d∗2 ⊗a −(φ2), where d1,d2 are bounded operators on h and a +(φj),a −(φj) creation and annihilation operators, in the fock space of the reservoir j, with test function φj . it is well-known (see [2, 9, 13, 25]) that, in the weak coupling limit, the evolution of the system observables is governed by a quantum markov semigroup (qms) on b(h), the algebra of all bounded operators in h, with generator of the form l = ∑ j=1,2, ω∈b lj,ω (2.2) where b is the set of all bohr frequencies b := {ω | ∃εn,εm s.t. ω = εn −εm > 0}. (2.3) for every bohr frequency ω, lj,ω is a generator with the gorini-kossakowski-sudharshan-lindblad (gksl) structure (see [17, 22]) lj,ω(x) = i[hj,ω,x] − γ−j,ω 2 ( d∗j,ωdj,ωx− 2d ∗ j,ωxdj,ω + xdj,ωd ∗ j,ω ) − γ+j,ω 2 ( dj,ωd ∗ j,ωx− 2dj,ωxd ∗ j,ω + xdj,ωd ∗ j,ω ) (2.4) for all x ∈b(h), with kraus operators dj,ω defined by dj,ω = ∑ (εn,εm)∈bω pεmdjpεn (2.5) 124 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) where bω = {(εn,εm) | εn −εm = ω}, γ±j,ω = fj,ωγ ± j,ω γ−j,ω = eβjω eβjω − 1 , γ+j,ω = 1 eβjω − 1 , fj,ω = ∫ {y∈r3 | |y|=ω} |φj(y)|2dsy (ds denotes the surface integral) and hj,ω are bounded self-adjoint operators on h commuting with hs of the form hj,ω = κ − j,ωd ∗ j,ωdj,ω + κ + j,ωdj,ωd ∗ j,ω for some real constants κ±j,ω. in the sequel, following a customary convention to simplify the notation, we also denote d−j,ω := dj,ω and d + j,ω := d ∗ j,ω and write q±j,ω(x) = − 1 2 d∓j,ωd ± j,ωx + d ∓ j,ωxd ± j,ω − 1 2 xd∓j,ωd ± j,ω (2.6) the term of the gksl generator arising from the interaction with the bath j due the bohr frequency ω is lj,ω = γ−j,ωq − j,ω + γ + j,ωq + j,ω + i[hj,ω, · ] and the term arising from the interaction with the reservoir j is lj = ∑ ω∈b lj,ω. we now make some assumptions on constants in such a way as to ensure boundedness of operators lj. first of all note that the series ∑ ω d ∗ j,ωdj,ω is strongly convergent. indeed, for all vector u = ∑ n≥0 pεnu in h, we have∑ ω 〈 u,d∗j,ωdj,ωu 〉 = ∑ ω ∑ n,m≥0 〈pεm−ωdjpεmu,pεn−ωdjpεnu〉 = ∑ ω ∑ n≥0 〈djpεnu,pεn−ωdjpεnu〉 ≤ ∑ n≥0 ‖djpεnu‖ 2 = ‖dj‖ 2 ‖u‖2 . as a consequence, if we assume sup ω∈b γ±j,ω < +∞, sup ω∈b ∣∣κ±j,ω∣∣ < +∞, for j = 1, 2 gksl generators lj turn out to be bounded. the above condition will be assumed to be in force throughout the paper. remark. note that lj depends on the inverse temperature βj only through the constants γ±j,ω. the above notation follows that of [1]. cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 125 for all normal linear operator s on b(h) we denote by s∗ the predual operator acting on the banach space of trace class operators on h. therefore, we denote by t = (tt)t≥0 the qms on b(h) generated by l and by t∗ = (t∗t)t≥0 the predual semigroup acting on trace class operators. in the same way, t j (resp. t j,ω and t j,ω∗ ) stand for the qms generated by lj (resp. lj,ω and its predual semigroup). in this paper we are concerned with normal states, therefore we shall identify them with their densities which are positive operators on h with unit trace. we end this section by checking that, if reservoirs have the same temperature β1 = β2 = β and zβ := tr ( e−βhs ) < +∞, then the gibbs state has density ρβ = z −1 β e −βhs (2.7) and is stationary. proposition 2.1. if β1 = β2 = β and zβ := tr ( e−βhs ) = ∑ n≥0 e−βεn dim(pεn ) < +∞ then the gibbs state (2.7) is invariant for all qmss generated by l, l1, l2. proof. we begin by observing that for (εn + ω,εn), (εn,εn −ω) ∈ bω, we can compute directly (lj,ω)∗(pεn ) =γ − j,ω(pεn−ωdjpεnd ∗ jpεn−ω −pεnd ∗ i pεn−ωdjpεn )+ γ+j,ω(pεn+ωd ∗ jpεndjpεn+ω −pεndjpεn+ωd ∗ jpεn ). a state of the form ρ = ∑ n ρεnpεn , which is a function of the system hamiltonian hs (also called a diagonal state), satisfies l∗j(ρ) = ∑ ω ∑ n (lj,ω)∗(ρεnpεn ) = ∑ ω ∑ (εn+ω,εn)∈bω (ρεn+ωγ − j,ω −ρεn γ + j,ω)pεndjpεn+ωd ∗ jpεn + ∑ ω ∑ (εn,εn−ω)∈bω (ρεn−ωγ + j,ω −ρεn γ − j,ω)pεnd ∗ jpεn−ωdjpεn. now if β1 = β2 = β and ρεn = e −βεn as in (2.7), we have γ+j,ω γ−j,ω = γ+j,ω γ−j,ω = e−βω = ρεn+ω ρεn , for all j = 1, 2, so that l∗j(ρ) = 0 and ρ = e−βhs/zβ is an invariant state for the qms generated by lj. since l = l1 + l2 it is an invariant state also for the qms generated by l. 126 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) 3 energy current the rate of energy variation in the system, in a state ρ, due to interaction with the reservoir j is tr (ρlj(hs)) (see [25] (v.28)). therefore tr (ρl1(hs)) − tr (ρl2(hs)) (3.1) is twice the rate at which the energy flows through the system from the hotter bath to the colder bath, namely, the energy current through the system. adapting a result by lebowitz and spohn [25] theorem 2 and corollary 1, it is possible to prove that the energy current is non-negative for finite dimensional systems. theorem 3.1. suppose that h is finite dimensional and let ρ be a faithful invariant state, then the energy current (3.1) is non-negative. proof. if a system is weakly coupled to a single bath j at inverse temperature βj, it is well-known that the gibbs state ρβj = z −1 βj e−βjhs , with zβj = tr ( e−βjhs ) , is invariant. consider the relative entropy of ρ with respect to ρβj defined by s(ρ|ρβj ) = tr ( ρ(log(ρ− log ρβj ) ) which is a notoriously non-increasing function (see [23], theorem 1.5), i.e. s ( t j∗t(ρ)|t j ∗t(ρβj ) ) ≤ s(ρ|ρβj ), for all ρ and t ≥ 0. states t j∗t(ρ), j = 1, 2 will still be faithful for small t, therefore no problem arises when considering logarithms. since ρβj is invariant, denoting ρt := t j ∗t(ρ), and differentiating we find d dt s(ρt|ρβj ) = d dt tr ( ρt(log ρt − log ρβj ) ) = tr ( ρ′t(log ρt − log ρβj ) ) + tr ( ρt d dt log ρt ) . since for every x > 0, log x = ∫ +∞ 0 ( 1 1+s − 1 x+s ) ds, d dt log ρt = ∫ +∞ 0 (s + ρt) −1ρ′t(s + ρt) −1ds so that tr ( ρt d dt log ρt ) = tr ( ρ′t ∫ +∞ 0 ρt(s + ρt) −2ds ) = tr (ρ′t) = 0. by imposing ρβj = z −1 βj e−βjhs , and recalling that ρ′t = l∗j(ρt), tr (ρ′t) = 0 by trace preservation, we obtain d dt s(ρt|ρβj ) = tr ( ρ′t(log ρt − log ρβj ) ) = tr ( ρ′t(log ρt + βjhs − log z −1 βj ) ) = tr (ρ′t log ρt) + βjtr (ρtlj(hs)) . cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 127 in particular tr (ρ′t(log ρt)) + βjtr (ρtl(hs)) ≤ 0 by monotonicity of the relative entropy, namely −tr (l∗j(ρt) log ρt) −βjtr (ρtlj(hs)) ≥ 0. in our context, the entropy production of the system due to interaction with the bath at inverse temperature βj is − tr (l∗j(ρt) log ρt) −βjtr (ρtlj(hs)) ≥ 0. (3.2) now, for all β,β1,β2 and ρ stationary state for the system s interacting with both baths, by taking a sum over j of the inequality before (3.2), we obtain β1tr (ρl1(hs)) + β2tr (ρl2(hs)) ≤ 0. moreover, tr (ρl1(hs)) = −tr (ρl2(hs)) and so (β2 −β1)tr (ρl2(hs)) ≥ 0 in view β1 ≥ β2, we have tr (ρl1(hs)) = −tr (ρl2(hs)) ≥ 0 and the proof is complete. in this section we prove a general explicit formula for the energy current in a stationary state ρ which is a function of the system hamiltonian hs. this not only confirms that it is positive also for possibly infinite dimensional systems if the eigenvalues of stationary state are a monotone system (i.e. there are no population inversions), but it allows us to establish proportionality to the difference of bath temperatures when they are not too small, namely an approximate fourier law. lemma 3.2. for all ω ∈ b and j = 1, 2 we have q−j,ω(hs) = −ω d ∗ j,ωdj,ω q + j,ω(hs) = ω dj,ωd ∗ j,ω (3.3) and lj(hs) = ∑ ω∈b ω ( γ+j,ωdj,ωd ∗ j,ω − γ − j,ωd ∗ j,ωdj,ω ) . (3.4) proof. writing hs as in (2.1) we compute q−j,ω(hs) = − 1 2 d∗j,ωdj,ωhs + d ∗ j,ωhsdj,ω − 1 2 hsd ∗ j,ωdj,ω = ∑ (εn,εm)∈bω ( εm pεnd ∗ jpεmdjpεn −εn pεnd ∗ jpεmdjpεn ) = − ∑ (εn,εm)∈bω ωpεnd ∗ jpεmdjpεn = −ω d∗j,ωdj,ω. the proof of the other identity (3.3) is similar. since [hj,ω,hs] = 0 for all j,ω, (3.4) follows immediately. 128 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) we can now prove our formula for the energy current in a stationary state ρ which is a function of the system hamiltonian hs. we suppose that the interaction of the system with both reservoirs is similar; this property is reflected by the assumptions on tr ( pεnd ∗ jpεmdj ) and f1,ω. in the sequel, to simplify the notation we also write ρn instead of ρεn . theorem 3.3. for any state ρ which is a function of the system hamiltonian hs, i.e. ρ = ∑ n≥0 ρnpεn (3.5) we have tr (ρlj(hs)) = ∑ ω∈b ω ∑ (εn,εm)∈bω ( γ+j,ωρm − γ − j,ωρn ) tr ( pεnd ∗ jpεmdj ) . (3.6) if the state ρ is also stationary and, moreover, (1) tr (pεnd ∗ 1pεmd1) = tr (pεnd ∗ 2pεmd2) for all n,m, (2) f1,ω = f2,ω for all ω, then tr (ρl1(hs)) = 1 2 ∑ ω∈b ω f1,ω ( γ+1,ω −γ + 2,ω ) ∑ (εn,εm)∈bω (ρm −ρn) tr (pεnd ∗ 1pεmd1) . (3.7) proof. the proof of (3.6) is immediate from (3.4) and the following identities (cyclic property of the trace) tr ( pεmdj,ωpεnd ∗ j,ω ) = tr ( (pεmdj,ω)pεmd ∗ j,ω ) = tr ( pεnd ∗ j,ωpεmdj,ω ) . if the state ρ is stationary, then tr (ρl1(hs)) = tr (ρl(hs)) − tr (ρl2(hs)) = −tr (ρl2(hs)), so that tr (ρl1(hs)) = (tr (ρl1(hs)) − tr (ρl2(hs))) /2. computing the right-hand side difference by means of (3.6) with j = 1, 2 we can write 2tr (ρl1(hs)) as∑ ω∈b ω f1,ω ∑ (εn,εm)∈bω ( γ+1,ωρm −γ − 1,ωρn −γ + 2,ωρm + γ − 2,ωρn ) tr (pεnd ∗ 1pεmd1) = ∑ ω∈b ω f1,ω ∑ (εn,εm)∈bω ( (γ+1,ω −γ + 2,ω)ρm − (γ − 1,ω −γ − 2,ω)ρn ) tr (pεnd ∗ 1pεmd1) . since γ−j,ω = γ + j,ω + 1 for all j,ω, then γ + 1,ω −γ + 2,ω = γ − 1,ω −γ − 2,ω and (3.7) follows. remark. note that the above identity tr (pεnd ∗ 1pεmd1) = tr (pεnd ∗ 2pεmd2) holds whenever there exists an isometry r on h, commuting with hs, such that d2 = rd1r ∗. indeed, in this case, r commutes with all spectral projections of hs and tr (pεnd ∗ 2pεmd2) = tr (pεnrd ∗ 1r ∗pεmrd1r ∗) = tr (pεnd ∗ 1pεmd1r ∗r) = tr (pεnd ∗ 1pεmd1) . cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 129 we will see later (section 5) that this happens when the system interacts in the same way with the two baths. formula (3.7) can be applied to effectively compute the energy current in several models highlighting the dependence on the difference of temperatures. indeed, one readily sees that, for β1,β2 very close the term ω ( γ+1,ω −γ + 2,ω ) is an infinitesimum of order β−11 −β −1 2 while the other terms are close to some nonzero values. moreover, it is also clear from (3.7) that the energy current is non-negative whenever the invariant state satisfies ρm > ρn for all n,m such that εm < εn i.e. population inversion does not occur. however, in order to find more explicit formulae we need additional information on the invariant state. this problem will be studied in the next section. we end this section by the following example example 3.4. let h = cn+1 with orthonormal basis (ek)0≤k≤n. consider an n-level system with hamiltonian hs = n∑ k=0 k|ek〉〈ek| and interaction operators d1,d2 acting as djek = ek−1 for k = 1, . . . ,n dje0 = 0. clearly b = {1, 2, . . . ,n} but the only nonzero dj,ω are those corresponding to the frequency ω = 1 and d1,1 = d1, d2,1 = d2. moreover, since εk = k, tr ( pεkd ∗ 1pεk−1d1 ) = tr ( pεkd ∗ 2pεk−1d2 ) = 1 for k = 1, . . . ,n. by theorem 3.3 formula (3.6) we have tr (ρlj(hs)) = n−1∑ k=0 ( γ+j,1ρk − γ − j,1ρk+1 ) . if all γ±j,1 (j = 1, 2) are nonzero, a straightforward computation shows that the unique stationary state is ρ = 1 −ν 1 −νn+1 n∑ k=0 νk|ek〉〈ek|, ν := γ+1,1 + γ + 2,1 γ−1,1 + γ − 2,1 and the energy current due to interaction with reservoir j is tr (ρlj(hs)) = 1 −ν 1 −νn+1 n−1∑ k=0 ( γ+j,1ν k − γ−j,1ν k+1 ) = 1 −νn 1 −νn+1 ( γ+j,1 −νγ − j,1 ) . note that, dropping the index 1 corresponding to the unique effective frequency ω to simplify the 130 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) notation, we have γ+j −νγ − j = γ − j ( γ+j γ−j − γ+1 + γ + 2 γ−1 + γ − 2 ) = γ−j ( γ+j γ−j − f1 γ + 1 + f2 γ + 2 f1 γ − 1 + f2 γ − 2 ) = γ−j ( e−βj − f1 (e β2 − 1) + f2 (eβ1 − 1) f1 eβ1 (eβ2 − 1) + f2 eβ2 (eβ1 − 1) ) = γ−j ( e−βj − f1 e −β1 (1 − e−β2 ) + f2e−β2 (1 − e−β1 ) f1(1 − e−β2 ) + f2(1 − e−β1 ) ) . for j = 1 we find γ+j,1 −νγ − j,1 = γ − j f2(1 − e −β1 ) e−β1 − e−β2 f1(1 − e−β2 ) + f2(1 − e−β1 ) and so tr (ρl1(hs)) = 1 − ((γ+1 + γ + 2 )/(γ − 1 + γ − 2 )) n 1 − ((γ+1 + γ + 2 )/(γ − 1 + γ − 2 )) n+1 γ−1 f2(1 − e −β1 )(e−β1 − e−β2 ) f1(1 − e−β2 ) + f2(1 − e−β1 ) since γ+j < γ − j , this formula, for n big and β1,β2 small becomes tr (ρl1(hs)) ≈ f1f2(e −β1 − e−β2 ) f1(1 − e−β2 ) + f2(1 − e−β1 ) ≈ f1f2(β2 −β1) f2β1 + f1β2 = f1f2 ( 1 β1 − 1 β2 ) f1 β1 + f2 β2 showing that, in a certain regime of high temperature a fourier law holds for all choices f1,f2 of the interactions strength. 4 dependence of the energy current from temperature difference and conductivity in this section we consider systems whose hamiltonian hs has simple spectrum, namely each spectral projection pεn is one-dimensional, and make explicit the dependence of the energy current on the difference of temperatures 1/β1 and 1/β2. we begin by noting that, if spectral projections pεn are one-dimensional one can associate with the open quantum system a classical (time continuous) markov chain with state space v the cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 131 spectrum sp(hs) of hs in a canonical way. indeed, for every bounded function f on v , we have l(f(hs)) = ∑ n≥0 f(εn)l(pεn ) = ∑ ω∈b, (εn,εm)∈bω  ∑ j γ−j,ωpεnd ∗ jpεmdjpεn   (f(εm) −f(εn)) + ∑ ω∈b, (εn,εm)∈bω  ∑ j γ+j,ωpεmdjpεnd ∗ jpεm   (f(εn) −f(εm)) and we find a classical markov chain with transition rate matrix q = (qnm) qnm =   ∑ j γ − j,εn−εm tr ( d∗jpεmdjpεn ) , if εn > εm,∑ j γ + j,εm−εn tr ( djpεmd ∗ jpεn ) , if εn < εm, − ∑ m 6=n qnm, if n = m. now, if we consider the conditional expectation e : b(h) → `∞(v ; c), e(x) = ∑ m≥0 pεmxpεm, where `∞(v ; c) is the abelian algebra of bounded functions on v , we have that e ◦l = l◦e. (4.1) therefore, by defining the predual map e∗ such that tr (e∗(ρ)x) = tr (ρe(x)), if ρ is an invariant state, we have also 0 = e∗(l∗(ρ)) = l∗(e∗(ρ)) and (πn) 7→ ∑ n≥0 πnpεn gives a one-to-one correspondence between diagonal invariant states of the open quantum system and invariant measures of the associated markov chain. in the following, in order to have at hand an explicit formula for the invariant measure, we suppose, for simplicity, that the graph associated with the above markov chain is a path graph and jumps can occur only to nearest neighbour levels, namely qnm = 0 for |n − m| ≥ 2. this assumption may hold, for instance, if the hamiltonian hs is generic in the sense of [8], namely it is not only non-degenerate but also if εn−εm = εn′ −εm′ then εn = εn′ and εm = εm′. moreover, we assume that qnm 6= 0 for |n−m| ≤ 1. in this case the associated classical markov chain has a simpler structure allowing one to make explicit computations and describe explicitly the structure of invariant states (see also [11] in a more general situation). the explicit expression for the invariant state is ρ = ∑ n ρnpεn where ρn = ∏ 0≤k 0. proof. note that 1/(eβ1ω − 1) − 1/(eβ2ω − 1) ≤ 1/(β1ω) − 1/(β2ω) because the function x 7→ 1/(exω − 1) − 1/(ωx) is increasing on ]0, +∞[ since d dx ( 1 exω − 1 − 1 ωx ) = 1 ωx2 − ω( eωx/2 − e−ωx/2 )2 ≥ 0 by the elementary inequality eωx/2 − e−ωx/2 ≥ ωx. moreover, by another elementary inequality 1 − e−βjω ≤ βjω, we have eβ1ω eβ1ω − 1 + eβ2ω eβ2ω − 1 = 1 1 − e−β1ω + 1 1 − e−β2ω ≥ 1 β1ω + 1 β2ω and the second inequality (4.4) follows. in order to prove the first inequality we first write the right-hand side as (eβ1ω − 1)−1 − (eβ2ω − 1)−1 eβ1ω(eβ1ω − 1)−1 + eβ2ω(eβ2ω − 1)−1 = eβ2ω − eβ1ω eβ1ωeβ2ω/2(eβ2ω/2 − e−β2ω/2) + eβ2ωeβ1ω/2(eβ1ω/2 − e−β1ω/2) = e−(β1+β2)ω/2 e(β2−β1)ω/2 − e−(β2−β1)ω/2 (1 − e−β2ω) + (1 − e−β1ω) cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 133 noting that e(β2−β1)ω/2 − e−(β2−β1)ω/2 ≥ 1 + (β2 −β1)ω 2 − ( 1 − (β2 −β1)ω 2 ) (1 − e−β2ω) + (1 − e−β1ω) ≤ (β1 + β2) ω we find (eβ1ω − 1)−1 − (eβ2ω − 1)−1 eβ1ω(eβ1ω − 1)−1 + eβ2ω(eβ2ω − 1)−1 ≥ e−(β1+β2)ω/2 (β2 −β1) ω (β1 + β2) ω . this completes the proof. remark. note that the inequalities of lemma 4.1 provide a sharp estimate in terms of the inverse temperature difference β2−β1 for small β1,β2, i.e. when the average of temperatures t1,t2 is big. indeed, the difference of the right-hand side and left-hand side is equal to ( 1 − e−(β1+β2)ω/2 ) β2 −β1 β1 + β2 and for temperatures tj > kb ·180 k= 2.49·10−21 j (approximately the lowest natural temperature ever recorded at ground level) we have βj < 1/(kb · 180 k) = 4.02 · 1020 j−1 so that the quantity that multiplies β2 −β1 is 1 β1 + β2 < 1.24 · 10−21j. theorem 4.2. suppose that (1) tr ( pεnd ∗ jpεmdj ) = 1 for all n,m and all j = 1, 2, (2) fj,ω = 1 for all ω and all j = 1, 2, (3) jumps can occur only to nearest neighbour levels, (4) formula (4.3) holds so that the state ρ defined by (4.2) with ρ0 determined by the normalization condition is invariant. then κm 1 β1 − 1 β2 1 β1 + 1 β2 κ(ρ,hs) ≤ tr (ρl1(hs)) ≤ 1 β1 − 1 β2 1 β1 + 1 β2 κ(ρ,hs) (4.5) where κm = infm≥0 e −(β1+β2)(εm+1−εm)/2 and ĥs = ∑ m≥0 εm+1pεm, κ(ρ,hs) = tr ( ρ(ĥs −hs) ) . 134 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) proof. by applying (3.7) in this context, we have tr (ρl1(hs)) = 1 2 ∑ n≥0 (εn+1 −εn)(ρn −ρn+1) ( γ+1,εn+1−εn − γ + 2,εn+1−εn ) = 1 2 ∑ n≥0 (εn+1 −εn) ( 1 − qn,n+1 qn+1,n ) ρn ( γ+1,εn+1−εn − γ + 2,εn+1−εn ) = ∑ n≥0 (εn+1 −εn)ρn γ+1,εn+1−εn − γ + 2,εn+1−εn γ−1,εn+1−εn + γ − 2,εn+1−εn . now the proof follows applying lemma 4.1 with ω = εn+1 − εn to estimate the right-hand side ratio. remark. formula (4.5) shows that the energy current tr (ρl1(hs)) has an explicit dependence on the difference β−11 −β −1 2 of the reservoirs’ temperatures. this dependence holds only through two inequalities, but it suggests the existence of an “approximate” fourier law (see [4, 21]) for the current. clearly there can be further dependecies through the term κ(ρ,hs), however it holds inf k (εk+1 −εk) ≤ κ(ρ,hs) ≤ sup k (εk+1 −εk) . therefore the energy current depends on the temperature difference mainly through the explicit term and one could say that there really is an “approximate” fourier law. furthermore it is worth noticing that, for β1,β2 fixed, the inequality (4.5) is better the smaller is supm≥0(εm+1 − εm) so that κm is close to 1 and the inequalities are approximately equalities. however, it should also be noted that, in this case, κ(ρ,hs) becomes small as well. eventually note that, due to the nature of our system, we cannot investigate spatial properties of energy flow. therefore our discussion of the fourier’s law is concerned with proportionality to temperature difference and not with dependency on size. remark. since the above qms are of weak coupling limit type, one can write explicitly the entropy production (in the sense of [15, 16]). it is tempting to study in detail what happens when supm≥0(εm+1 − εm) tends to 0 so that the eigenvalues of hs increase in number and form a set more and more packed. in a more precise way, for all n ≥ 1 we assume that the system hamiltonian is a self-adjoint operator h(n)s on an (n + 1)-dimensional hilbert space h with simple pure point spectrum ( ε (n) k ) 0≤k≤n with ε0 = 0 and, for all a,b with 0 ≤ a < b ≤ +∞, we have lim n→∞ card { k | a < ε(n)k ≤ b } n = µ(]a,b]) (4.6) for some continuous probability density µ on [0, +∞[. in other words, the empirical distribution of eigenvalues of h (n) s converges weakly to a probability distribution on [0, +∞[ . suppose, for simplicity, that µ has no atoms, i.e. µ({r}) = 0 for all r ≥ 0. cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 135 we can now prove the following result on the distribution of eigenvalues of the stationary state and energy in stationary conditions. theorem 4.3. under the assumptions of theorem 4.2, for all n ≥ 1, let h(n)s be as above and suppose that (4.6) holds. let ρ(n) be the invariant state (4.2) and let β̃ = 2 (1/β1 + 1/β2) −1 be the harmonic mean of the inverse temperatures (i.e. β̃−1 arithmetic mean of β−11 ,β −1 2 ). (i) eigenvectors ρ (n) k of ρ (n) satisfy lim n→∞ ∑ {k |a<εk≤b} ρ (n) k = ∫ b a e−β̃rdµ(r)∫ ∞ 0 e−β̃rdµ(r) (ii) the average energy in the system satisfies lim n→∞ tr ( ρ(n)h (n) s ) = ∫ ∞ 0 e−β̃r rdµ(r)∫ ∞ 0 e−β̃rdµ(r) . this result reminds the one in [20] where the steady state can be described by a generalized gibbs state and the steady-state current is proportional to the difference in the reservoirs’ magnetizations. in the proof we need the following lemma. lemma 4.4. let β̃ = 2/ ( β−11 + β −1 2 ) be the harmonic mean of inverse temperatures (i.e. β̃−1 is the arithmetic mean of β−11 and β −1 2 ). for all 1 ≤ k ≤ n and for supj ωj < 1/(3β2), 1 − β̃ ωk ≤ qk,k+1 qk+1,k ≤ 1 − β̃ ωk + ( β̃ ωk )2 (4.7) where ωk = εk+1 −εk and e−β̃εk(1+β̃ supj ωj ) ≤ k−1∏ j=0 qj,j+1 qj+1,j ≤ e−β̃εk(1−β̃ supj ωj ) (4.8) proof. by the elementary inequality 1 − e−βjωk ≤ βjωk we have qk,k+1 qk+1,k = 1 eβ1ωk − 1 + 1 eβ2ωk − 1 eβ1ωk eβ1ωk − 1 + eβ2ωk eβ2ωk − 1 = 1 − 2 (1 − e−β1ωk )−1 + (1 − e−β2ωk )−1 ≥ 1 − 2ωk 1/β1 + 1/β2 136 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) in the same way, by the elementary inequalities 1−e−βjωk ≥ βjωk−(βjωk) 2 /2 and 1/(1 −βjωk/2) ≤ 1 + βjωk, we find for βjωk < 1 qk,k+1 qk+1,k ≤ 1 − 2ωk 1/ (β1 (1 −β1ωk/2)) + 1/ (β2 (1 −β2ωk/2)) ≤ 1 − 2ωk 1/β1 (1 + β1ωk/2) + 1/β2 (1 + β2ωk/2) ≤ 1 − 2ωk 1/β1 + 1/β2 + 2ωk = 1 − β̃ ωk 1 + β̃ ωk and so (4.7) follows. in order to prove the upper bound in (4.8), note that, since log(1 −x) ≤−x log  k−1∏ j=0 qj,j+1 qj+1,j   ≤ k−1∑ j=0 log ( 1 − β̃ ωj ( 1 − β̃ ωj )) ≤− k−1∑ j=0 β̃ωj ( 1 − β̃ωj ) , as a consequence log  k−1∏ j=0 qj,j+1 qj+1,j   ≤−k−1∑ j=0 β̃ωj ( 1 − β̃ sup l ωl ) = −β̃εk ( 1 − β̃ sup l ωl ) . for the lower bound, we begin by the inequality log  k−1∏ j=0 qj,j+1 qj+1,j   = k−1∑ j=0 log ( qj,j+1 qj+1,j ) ≥ k−1∑ j=0 log ( 1 − β̃ωj ) . note that log(1 −x) + x + x2 ≥ 0 for 0 ≤ x ≤ 2/3 and, since β̃ωj < 2/3 by our assumption, we have log  k−1∏ j=0 qj,j+1 qj+1,j   ≥−k−1∑ j=0 β̃ωj ( 1 + β̃ sup l ωl ) = −β̃�k ( 1 + β̃ sup l ωl ) . this completes the proof. proof of theorem 4.3. let µn be the empirical distribution of the eigenvalues of h (n) s i.e. µn = 1 n + 1 n∑ k=0 δεk and note that ∑ {k |a<εk≤b} ρ (n) k = 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j 1 n + 1 n∑ k=0 k−1∏ j=0 qj,j+1 qj+1,j . (4.9) cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 137 clearly, by lemma 4.4, 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j ≤ 1 n + 1 ∑ {k |a<εk≤b} e−β̃εk(1−β̃ supj ωj ) ≤ eβ̃ 2b supj ωj ∫ ]a,b] e−β̃εk dµn(r) and also 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j ≥ e−β̃ 2a supj ωj n + 1 ∑ {k |a<εk≤b} e−β̃εk = e−β̃ 2a supj ωj ∫ ]a,b] e−β̃εk dµn(r). since supj ωj goes to 0, probability measures µn converge weakly to µ and the function r → e−β̃r is bounded continuous on [0, +∞[, taking the limit as n →∞, we have lim n→∞ 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j = ∫ ]a,b] e−β̃εk dµ(r). in the same way, taking a = 0 and b = +∞, we see that the denominator of (4.9) converges to∫ +∞ 0 e−β̃r dµ(r) and the proof of (i) is complete. the proof of (ii) is similar. � remark. theorem 4.3 (i) shows that, if µ has density µ′, then the asymptotic distribution of eigenvalues of the stationary state is λ 7→ e−β̃λµ′(λ)∫ +∞ 0 e−β̃rµ′(r)dr . the asymptotic average energy in the system can be easily computed in some remarkable cases noting that the integral of e−β̃r with respect to µ is the moment generating function φ of µ evaluated at −β̃ and so the asymptotic average energy in the system is − d dβ̃ φ(−β̃) φ(−β̃) = − d dβ̃ log ( φ(−β̃) ) . we can easily find an explicit result in two cases: µ normal distribution n(m,σ2) average energy m− β̃σ µ gamma distribution γ(α,θ) average energy α/(β̃ + θ) the asymptotic average energy in the system is decreasing in β̃, i.e. increasing in the average temperature as expected, for all probability measure µ because the moment generating function of a probability distribution is log-convex and the derivative of a convex function is increasing. remark. note that, by choosing a suitable spacing of eigenvalues εn we can control the rate of convergence to 0 of κ ( ρ(n),h (n) s ) at will, as n tends to +∞. 138 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) 5 one dimensional ising chain in this section we consider a one-dimensional ising chain with nearest neighbour interaction. we will show that, in this case, if the heat baths interact locally at both ends of the chain, then the energy current is zero. spin interaction (see 5.1) occurs only in the z component. in the case where also the other components interact the derivation of the gksl generator turns out to be really difficult (see [5]). indeed, starting from the diagonalized hs, one finds a cumbersome expression for the operators dω. in spite of the simple system hamitonian hs (5.1) theorems 4.2 and 4.3 do not apply to this model because its spectrum is degenerate. the system space is h = c2⊗n with n > 2. define pauli matrices σx =   0 1 1 0   σy =   0 −i i 0   σz =   1 0 0 −1   with respect to the orthonormal basis e+ = [1, 0] t, e− = [0, 1] t of c2. consider the one dimensional ising chain with hamiltonian hs = jz n−1∑ j=1 σzjσ z j+1, jz > 0, n > 2 (5.1) subsequently let us define eα := ⊗nj=1eα(j), α ∈{−1, 1} n, as a basis of h, where e−1 := e− and e+1 := e+. vectors {eα}α form an eigenbasis for hs and the spectrum is sp(hs) = {jz (2k − (n − 1)) | k = 0, . . . ,n − 1}. the eigenspace associated with the eigenvalue εk = jz(2k−(n −1)) is the linear span of the elements eα such that exactly k neighbouring elements in α have the same sign. thus one can define the sets ak := { α ∈{−1, 1}n ∣∣∣ ∑n−1j=1 α(j)α(j + 1) = 2k − (n − 1)} , and the spectral projection associated with the eigenvalue εk is given by pk := ∑ α∈ak |eα〉〈eα|. the system is coupled with two heat reservoirs at inverse temperature β1,β2 with β1 ≤ β2 through the interaction h1 = σ u 1 ⊗ (a −(φ1) + a +(φ1)), h2 = σ v n ⊗ (a −(φ2) + a +(φ2)), (5.2) cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 139 where u,v ∈ r3 and σui is defined as σui = u1σ x i + u2σ y i + u3σ z i . the set of positive bohr frequencies is given by b := {2jz(n−m) = εn −εm | n,m ∈{0, . . . ,n − 1}, n > m}, while the operators dj,ω are given by (2.5). thus one has d1,2jz = (u1 − iu2) ∑ α∈cl ++ σx1 |eα〉〈eα| + (u1 + iu2) ∑ α∈cl−− σx1 |eα〉〈eα| where cl++ (resp. c l −−) denotes the set of configurations α ∈{−1, +1}n with ++ (resp. −−) in the first two sites (l stands for left). while d1,ω = 0 for every ω ∈ b −{2jz} because the pauli matrices act only on the first site and so the number of neighbouring sites with the same sign can vary of at most one after the action of σu1 and for ω = 2jz one has d1,2jz = n−1∑ n=1 ∑ α∈an ∑ β∈an+1 〈eα,σx1eβ〉 |eα〉〈eβ|. with similar arguments one can see that d2,ω = 0 for every ω ∈ b−{2jz}, while d2,2jz = (v1 − iv2) ∑ α∈cr ++ σxn|eα〉〈eα| + (v1 + iv2) ∑ α∈cr−− σxn|eα〉〈eα| where cr++ (resp. c r −−) denotes the set of configurations with ++ (resp. −−) in the last two sites (r stands for right). from now on we will drop the subscript 2jz and only deal with operators related to that bohr frequency, as the others vanish. recalling the definition of linear maps (2.6) and the constants γ+i = 1/(e 2jzβi − 1), γ−i = e 2jzβi/(e2jzβi − 1), we can write the gksl generator of the evolution as follows l = ∑ i∈{1,n} γ−i q − i + γ + i q + i . a close scrutiny at the operators di,d ∗ i shows that, for each fixed configuration α ∈{−1, +1} n−2 of the n − 2 inner sites of the chain the 4-dimensional projections pα on subspaces hα := span{eα | α(j) = α(j) for all 2 ≤ j ≤ n − 1; α(1),α(n) ∈{−1, 1}} commute with both di and d ∗ i for i ∈ {1,n}, then subalgebras pα1b(h)pα2 are invariant for the semigroup t generated by l. this commutation allows us to restrict our study only to cases where 140 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) the invariant state is of the form ρ = ∑ α∈{−1,1}n−2 pαρpα = ∑ α∈{−1,1}n−2 λαρα, (5.3) where ρα is an invariant state supported only on hα and λα are real constants that sum up to 1. indeed the off diagonal terms, pα1ρpα2 with α1 6= α2, do not contribute to current flow, since tr (pα1ρpα2l1(hs)) = tr (pα1ρl1(hs)pα2 ) = 0. moreover all the conditional expectations eα(x) := pαxpα commute with l, ensuring that both∑ αeα,∗(ρ) and every eα,∗(ρ) must also be invariant states on their own. as a further refinement we can repeat the same argument using the conditional expectation e(x) := ∑n−1 k=0 pkxpk. indeed e commutes with the lindbladian l and tr (pk1ρpk2l1(hs)) = tr (pk1ρl1(hs)pk2 ) = 0 for k1 6= k2, since the spectral projections commute with djd∗j , d ∗ jdj and l1(hs) is a linear combination of these operators by lemma 3.2, equation (3.3). in this way we can focus our study on invariant states of the form (5.3) with pαρpα = ρα =   ρα11 0 0 0 0 ρα22 ρ α 23 0 0 ρα32 ρ α 33 0 0 0 0 ρα44   , where we expanded the state with respect to the basis of four vectors ecαc,edαc, ecαd,edαd defined as follows: ecαc is the vector eα(2),α(2),...,α(n−1),α(n−1), ecαd = eα(2),α(2),...,α(n−1),−α(n−1) and vectors edαc, edαd are defined in a similar way. now we have reduced and simplified the class of states we want to use when looking for a invariant state, without, however, losing any contribution to the current flow. in order to find the invariant state, first of all it is not too difficult to show that l∗ leaves invariant the subspace of diagonal elements. then compute l∗(ρα23|edαc〉〈ecαd|) = − 1 2 [ γ+1 + γ − 1 + γ + n + γ − n ] ρα23|edαc〉〈ecαd|, and similarly l∗(ρα32|ecαd〉〈edαc|) = − 1 2 [ γ+1 + γ − 1 + γ + n + γ − n ] ρα32|ecαd〉〈edαc|, where γ±1 = ‖u1 + iu2‖ 2 γ±1 and γ ± n = ‖v1 + iv2‖ 2 γ±n . (the above γ ± i slightly differ from the constants in section 2). therefore the invariant state condition l∗(ρ) = 0 implies ρα23 = ρα32 = 0. cubo 23, 1 (2021) energy transfer in open quantum systems weakly coupled . . . 141 we can now just consider the reduced dynamics on diagonal elements of pαb(h)pα, given by l∗ =   −(γ−1 + γ − n ) γ − 1 γ − n 0 γ+1 −(γ + 1 + γ − n ) 0 γ − n γ+n 0 −(γ + n + γ − 1 ) γ − 1 0 γ+n γ + 1 −(γ + n + γ + 1 )   , the unique invariant law for the time-continuous markov chain generated by the above matrix is π = z−1 [ 1, e2jzβ1, e2jzβ2, e2jz(β1+β2) ] , where z−1 is a normalization constant that is independent of u,v and is the same for all α. therefore the unique t -invariant state supported on hα is ρα = z −1 ( |ecαc〉〈ecαc| + e2jzβ1|edαc〉〈edαc| + e2jzβ2|ecαd〉〈ecαd| + e2jz(β1+β2)|edαd〉〈edαd| ) . recalling (5.3) we can now write any invariant state for the semigroup t . we can now evaluate the energy flow tr (ρl1(hs)) via the expression l1(hs) = ∑ ω∈b+ ω ( γ+1,ωd1d ∗ 1 −γ − 1,ωd ∗ 1d1 ) = 2jz ( γ+1 d1d ∗ 1 −γ − 1 d ∗ 1d1 ) that, together with the formula for ρα, yields z tr (ρl1(hs)) = z tr   ∑ α∈{−1,1}n−2 λαραl1(hs)   = ∑ α∈{−1,1}n−2 2jzλα ( γ+1 e β1ω + γ+1 e (β1+β2)ω −γ−1 e β2ω −γ−1 ) = 0 remark. for n = 2, it can be shown by direct computation that the energy current is strictly positive. indeed, because of low dimensionality the ends of the chain can interact directly. acknowledgement. the authors would like to thank stefano olla for drawing their attention to the problem of energy transport in quantum systems and fruitful discussions at the workshop “quantum transport equations and applications” at casa matemática oaxaca (méxico) in september 2018. they also gratefully thank cubo’s anonymous referees whose remarks and comments to improve an earlier version of this paper. the financial support of gnampa indam 2020 projects “processi stocastici quantistici e applicazioni” and “evoluzioni markoviane quantistiche” is gratefully acknowledged. 142 f. fagnola, d. poletti, e. sasso cubo 23, 1 (2021) references [1] l. accardi, f. fagnola, and r. quezada, “on three new principles in non-equilibrium statistical mechanics and markov semigroups of weak coupling limit type”, infin. dimens. anal. quantum probab. relat. top., vol. 19, no. 2, 1650009, 2016. doi: 10.1142/s0219025716500090. 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(n.y.), vol. 241, pp. 191–209, 2019. doi: 10.1007/s10958019-04417-4 [27] m. vanicat, l. zadnik, and t. prosen, “integrable trotterization: local conservation laws and boundary driving”, phys. rev. lett. vol.121, 030606-1–030606-6, 2018. doi: 10.1103/physrevlett.121.030606 introduction semigroups of weak coupling limit type energy current dependence of the energy current from temperature difference and conductivity one dimensional ising chain cubo, a mathematical journal vol. 23, no. 01, pp. 171–190, april 2021 doi: 10.4067/s0719-06462021000100171 existence, well-posedness of coupled fixed points and application to nonlinear integral equations binayak s. choudhury1 nikhilesh metiya2 sunirmal kundu3 1 department of mathematics, indian institute of engineering science and technology, shibpur, howrah-711103, west bengal, india. binayak12@yahoo.co.in 2 department of mathematics, sovarani memorial college, jagatballavpur, howrah-711408, west bengal, india. metiya.nikhilesh@gmail.com 3 department of mathematics, government general degree college, salboni, paschim mednipur-721516, west bengal, india. sunirmalkundu2009@rediffmail.com abstract we investigate a fixed point problem for coupled geraghty type contraction in a metric space with a binary relation. the role of the binary relation is to restrict the scope of the contraction to smaller number of ordered pairs. such possibilities have been explored for different types of contractions in recent times which has led to the emergence of relational fixed point theory. geraghty type contractions arose in the literatures as a part of research seeking the replacement contraction constants by appropriate functions. also coupled fixed point problems have evoked much interest in recent times. combining the above trends we formulate and solve the fixed point problem mentioned above. further we show that with some additional conditions such solution is unique. well-posedness of the problem is investigated. an illustrative example is discussed. the consequences of the results are discussed considering α-dominated mappings and graphs on the metric space. finally we apply our result to show the existence of solution of some system of nonlinear integral equations. resumen investigamos un problema de punto fijo para contracciones acopladas de tipo geraghty en un espacio métrico con una relación binaria. el rol de la relación binaria es restringir el alcance de la contracción a un número menor de pares ordenados. tales posibilidades han sido exploradas para diferentes tipos de contracciones recientemente, lo que ha conllevado el nacimiento de la teoŕıa de punto fijo relacional. las contracciones de tipo geraghty aparecen en la literatura como parte de la investigación buscando reemplazar las constantes de contracción por funciones apropiadas. también problemas de puntos fijos acoplados han sido de mucho interés recientemente. combinando las ideas anteriores, formulamos y resolvemos el problema de punto fijo mencionado anteriormente. más aún, mostramos que bajo condiciones adicionales tal solución es única. se investiga la bien-definición del problema. se discute un ejemplo ilustrativo. las consecuencias de los resultados se discuten considerando aplicaciones α-dominadas y grafos en espacios métricos. finalmente aplicamos nuestros resultados para mostrar la existencia de soluciones de algunos sistemas de ecuaciones integrales no lineales. keywords and phrases: metric space; coupled fixed point; well-posedness; application. 2020 ams mathematics subject classification: 54h10, 54h25, 47h10. accepted: 29 march, 2021 received: 14 june, 2020 ©2021 b. s. choudhury et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100171 https://orcid.org/0000-0001-7057-5924 https://orcid.org/0000-0002-6579-7204 https://orcid.org/0000-0001-7494-5005 172 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) 1 introduction coupled fixed point results constitute a domain in metric fixed point theory which has experienced rapid development in recent times. the concept of coupled fixed point was introduced some time back in 1987 by guo et al [17]. but only after the publication of the work of bhaskar et al [15] a large number of papers have been written on this topic and on topics related to it [7, 9, 18, 21]. our consideration in this paper is a study related to fixed points of some coupled operators on metric spaces equipped with an appropriate binary relation. a contraction condition of geraghty type [20, 26, 32] is supposed to be satisfied by the coupled operator for those points which are related by the binary relation. as a consequence of it the assumption here is weaker than the usual case in metric fixed point theory where it is assumed that the inequality condition holds for arbitrarily chosen pairs from the space. such weakening of conditions have substantially occupied recent interests in fixed point theory. works of this category have come to be known as relationtheoretic fixed point results. some instances of these works are in [1, 3, 23, 30]. we use geraghty’s approach [16] to define a coupled contraction condition. it is a part of research where the constants of the contractions are replaced by suitable control functions in order to make the contraction inequality more general. such works occupy important positions in metric fixed point theory. some instances of these works are [5, 9, 13, 14, 22]. in this paper we combine the above trends in fixed point theory to define a new problem and then investigate its several aspects and show one application of the result. firstly, we show that such problem has a solution, that is, a coupled fixed point of the concerned operator exists. the uniqueness of the coupled fixed point is established under some additional conditions. well-posedness has been considered for many fixed point problems in recent times [24, 25, 27, 28]. in the present paper we deal with the well-posedness of the problem mentioned above. next we discuss some consequences of our main result. precisely we obtain some results for α-dominated mappings and results in metric spaces having a graph defined on it. the main result is supported with an example. in the last section we include an application of the main theorem to a problem of nonlinear integral equations. 2 mathematical background in the following we discuss the necessary mathematics for the discussion on the topics in the following sections. let x and y be two nonempty sets and r be a relation from x to y , that is, r ⊆ x×y . we write (x,y) ∈ r or xry to mean x is r related to y. the set p = {x ∈ x : (x,y) ∈ cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 173 r for some y ∈ y} is called the domain of r and the set q = {y ∈ y : (x, y) ∈ r for some x ∈ x} is called the range of r. by r−1 we mean the set {(y, x) : (x, y) ∈ r} which is called the inverse of r. a relation r from x to x is called a relation on x. let r be a relation on x. the relation r is said to be directed if for given x,y ∈ x, there exists z ∈ x such that (x, z) ∈ r and (y, z) ∈ r. the relation r is said to be a partial order relation on x if it is reflexive, anti-symmetric and transitive. let x be a nonempty set. an element (x,y) ∈ x × x is called a coupled fixed point of a function f : x ×x → x if x = f(x,y) and y = f(y,x). problem (p): let (x,d) be a metric space and f : x ×x → x be a mapping. we consider the problem of finding a coupled fixed point of f, that is, the problem of finding (x, y) ∈ x ×x such that x = f(x, y) and y = f(y, x). (2.1) definition 2.1 ([6]). the problem (p) is called well-posed if (i) f has a unique coupled fixed point (x∗, y∗), (ii) xn → x∗ and yn → y∗ as n → ∞, whenever {(xn, yn)} is any sequence in x × x for which lim supn→∞[d(xn, x∗) + d(yn, y∗)] is finite and limn→∞d(xn, f(xn, yn)) = limn→∞d(yn, f(yn, xn)) = 0. we define here the r-dominated mapping. definition 2.2. let x be a nonempty set with a binary relation r on it. a mapping f : x×x → x is said to be r-dominated if (x, f(x,y)) ∈ r and (f(y,x), y) ∈ r, for any (x, y) ∈ x ×x. example 2.3. let x = [0, 1] be equipped with usual metric. let f : x ×x → x be defined as f(x, y) = x+y 16+x+y , for x, y ∈ x. let a binary relation r on x be defined as r = {(x, y) : 0 ≤ x ≤ 1; 0 ≤ y ≤ 1 8 or 0 ≤ x ≤ 1 8 ; 0 ≤ y ≤ 1}. then f(x, y) = f(y, x) ∈ [0, 1 8 ], for x, y ∈ [0, 1]. it follows that (x, f(x,y)) ∈ r and (f(y,x), y) ∈ r, for any (x, y) ∈ x ×x. therefore, f is a r-dominated mapping. we introduce r-regularity condition in metric spaces. definition 2.4. let (x,d) be a metric space with a binary relation r on it. then x is said to have regular property with respect to r (or r-regular property) if for every sequence {xn} in x converging to x ∈ x, (xn, xn+1) ∈ r, for all n implies (xn, x) ∈ r, for all n [or (xn+1, xn) ∈ r, for all n implies (x, xn) ∈ r, for all n]. the following class of functions has appeared in several recent works related to fixed point theory. 174 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) let γ : [0, ∞) → [0, 1) be such that for any sequence {tn} in [0, ∞), limn→∞γ(tn) = 1 implies limn→∞ tn = 0. we denote the collection all such functions γ by b. such functions have appeared in several papers as for instances in [20, 32, 33]. in our theorems, we use following class of functions: let β : [0, ∞) → [0, 1) be such that for any sequence {tn} in [0, ∞), lim supn→∞β(tn) = 1 implies limn→∞ tn = 0. we denote the collection all such functions β by b ∗. we have the following observation about the class b∗. our class b∗ is more generalized than b. from the definition of b and b∗ it is clear that our class b∗ contains b and this containment is proper. the following example makes the fact clear: example 2.5. now consider the function β : [0, ∞) → [0, 1) defined by β(t) =   ∣∣ sin t t ∣∣ , if t is irrational, 1 2 , if t is rational. clearly β ∈ b∗ but β /∈ b. 3 main results in this section we establish a coupled fixed point result. we discuss its uniqueness under some additional conditions. we illustrate it with an example. theorem 3.1. let (x,d) be a complete metric space with a transitive relation r on it such that x has r-regular property. suppose that f : x ×x → x is a r-dominated mapping and there exists β ∈ b∗ such that for (x, y), (u, v) ∈ x ×x with [(x, u) ∈ r and (v, y) ∈ r] or [(u, x) ∈ r and (y, v) ∈ r], d(f(x, y), f(u, v)) ≤ β(m(x, y, u, v)) m(x, y, u, v), (3.1) where m(x, y, u, v) = max { d(x, u) + d(y, v) 2 , d(x,f(x, y)) + d(y, f(y, x)) 2 , d(u,f(u, v)) + d(v,f(v, u)) 2 , d(u,f(x, y)) + d(v,f(y, x)) 2 } . then f has a coupled fixed point. proof. let (x0, y0) ∈ x ×x be arbitrary. we construct two sequences {xn} and {yn} in x such that xn+1 = f(xn, yn) and yn+1 = f(yn, xn), for all n ≥ 0. (3.2) cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 175 as f is r-dominated, we have (xn, f(xn, yn)) = (xn,xn+1) ∈ r and (f(yn, xn), yn) = (yn+1, yn) ∈ r, for all n ≥ 0. (3.3) let rn = d(xn, xn+1) + d(yn, yn+1), for all n ≥ 0. (3.4) by (3.1), (3.2), (3.3) and (3.4), we have d(xn+1,xn+2) = d(f(xn,yn), f(xn+1,yn+1)) ≤ β(m(xn,yn,xn+1,yn+1)) m(xn,yn,xn+1,yn+1) (3.5) where m(xn,yn,xn+1,yn+1) = max {d(xn,xn+1) + d(yn,yn+1) 2 , d(xn,f(xn, yn)) + d(yn, f(yn, xn)) 2 , d(xn+1,f(xn+1,yn+1)) + d(yn+1,f(yn+1,xn+1)) 2 , d(xn+1,f(xn, yn)) + d(yn+1,f(yn, xn)) 2 } = max {d(xn,xn+1) + d(yn,yn+1) 2 , d(xn,xn+1) + d(yn, yn+1) 2 , d(xn+1,xn+2) + d(yn+1,yn+2) 2 , d(xn+1,xn+1) + d(yn+1,yn+1) 2 } = max {d(xn,xn+1) + d(yn,yn+1) 2 , d(xn,xn+1) + d(yn, yn+1) 2 , d(xn+1,xn+2) + d(yn+1,yn+2) 2 , 0 } = max {rn 2 , rn 2 rn+1 2 , 0 } = max {rn 2 , rn+1 2 } . (3.6) therefore, from (3.5) and (3.6), we have d(xn+1,xn+2) ≤ β ( max {rn 2 , rn+1 2 }) max {rn 2 , rn+1 2 } . (3.7) similarly, we can show that d(yn+1,yn+2) ≤ β ( max {rn 2 , rn+1 2 }) max {rn 2 , rn+1 2 } . (3.8) combining (3.7) and (3.8), we have rn+1 = d(xn+1,xn+2) + d(yn+1,yn+2) ≤ 2 β ( max {rn 2 , rn+1 2 }) max {rn 2 , rn+1 2 } = β ( max {rn 2 , rn+1 2 }) max {rn, rn+1}. (3.9) 176 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) suppose that 0 ≤ rn < rn+1. from (3.9), we have rn+1 ≤ β (rn+1 2 ) rn+1 < rn+1 which is a contradiction. therefore, rn+1 ≤ rn, for all n ≥ 0, that is, {rn} is a decreasing sequence of nonnegative real numbers. hence there exists r ≥ 0 such that rn → r as n →∞. by (3.9), we have rn+1 ≤ β (rn 2 ) rn, for all n ≥ 0. (3.10) if possible, suppose that r > 0. taking limit supremum in (3.10), we have r ≤ lim sup n→∞ β (rn 2 ) r, which implies that 1 ≤ lim supn→∞β( rn 2 ) ≤ 1, that is, lim supn→∞β( rn 2 ) = 1. then it follows by the property of β that limn→∞ rn 2 = r 2 = 0, that is, r = 0 which contradicts our assumption. hence r = 0. then we have lim n→∞ [d(xn,xn+1) + d(yn,yn+1)] = lim n→∞ d(xn,xn+1) = lim n→∞ d(yn,yn+1) = 0. (3.11) now we prove that both {xn} and {yn} are cauchy sequences. if possible, assume that either {xn} or {yn} fails to be a cauchy sequence. then either lim m, n→∞ d(xm, xn) 6= 0 or lim m, n→∞ d(ym, yn) 6= 0. hence, lim m, n→∞ [d(xm, xn) + d(ym, yn)] 6= 0, that is, there exists � > 0 for which we can find subsequences {m(k)} and {n(k)} of positive integers with n(k) > m(k) > k such that d(xm(k), xn(k)) + d(ym(k), yn(k)) ≥ � and d(xm(k), xn(k)−1) + d(ym(k), yn(k)−1) < �. (3.12) now, � ≤ d(xn(k), xm(k)) + d(yn(k), ym(k)) ≤ [d(xn(k), xn(k)−1) + d(yn(k), yn(k)−1)] + [d(xn(k)−1, xm(k)) + d(yn(k)−1, ym(k))] < d(xn(k), xn(k)−1) + d(yn(k), yn(k)−1) + �. taking limit as k →∞ in the above inequality and using (3.11), we have lim k→∞ [d(xm(k), xn(k)) + d(ym(k), yn(k))] = �. (3.13) again, d(xn(k)−1, xm(k)−1) + d(yn(k)−1, ym(k)−1) ≤ [d(xn(k)−1, xm(k)) + d(yn(k)−1, ym(k))] + [d(xm(k), xm(k)−1) + d(ym(k), ym(k)−1)] < � + [d(xm(k), xm(k)−1) + d(ym(k), ym(k)−1)]. cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 177 again, d(xn(k), xm(k)) + d(yn(k), ym(k)) ≤ [d(xn(k), xn(k)−1) + d(yn(k), yn(k)−1)]+ [d(xn(k)−1, xm(k)−1) + d(yn(k)−1, ym(k)−1)] + [d(xm(k)−1, xm(k)) + d(ym(k)−1, ym(k))], that is, d(xn(k)−1, xm(k)−1) + d(yn(k)−1, ym(k)−1) ≥ d(xn(k), xm(k)) + d(yn(k), ym(k)) −d(xn(k), xn(k)−1) −d(yn(k), yn(k)−1) −d(xm(k)−1, xm(k)) −d(ym(k)−1, ym(k)). from the above inequalities we have that d(xn(k), xm(k)) + d(yn(k), ym(k)) −d(xn(k), xn(k)−1) −d(yn(k), yn(k)−1) −d(xm(k)−1, xm(k)) −d(ym(k)−1, ym(k)) ≤ d(xn(k)−1, xm(k)−1) + d(yn(k)−1, ym(k)−1) < � + [d(xm(k), xm(k)−1) + d(ym(k), ym(k)−1)]. taking limit as k →∞ in the above inequality and using (3.11) and (3.13), we have lim k→∞ [d(xm(k)−1, xn(k)−1) + d(ym(k)−1, yn(k)−1)] = �. (3.14) now, d(xn(k), xm(k)) + d(yn(k), ym(k)) ≤ [d(xn(k),xn(k)−1) + d(yn(k),yn(k)−1)] + [d(xn(k)−1,xm(k)) + d(yn(k)−1,ym(k))] ≤ [d(xn(k),xn(k)−1) + d(yn(k),yn(k)−1)] + [d(xn(k)−1,xm(k)−1) + d(yn(k)−1,ym(k)−1)] + [d(xm(k)−1,xm(k)) + d(ym(k)−1, ym(k))]. taking limit as k →∞ in the above inequality and using (3.11), (3.13) and (3.14), we get lim k→∞ [d(xn(k)−1,xm(k)) + d(yn(k)−1,ym(k))] = �. (3.15) using (3.3) and the transitivity assumption of r, we have (xm(k)−1, xn(k)−1) ∈ r and (yn(k)−1, ym(k)−1) ∈ r. applying (3.1), we have d(xm(k), xn(k)) = d(f(xm(k)−1,ym(k)−1), f(xn(k)−1,yn(k)−1)) ≤ β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)) m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1) (3.16) 178 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) where m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1) = max { d(xm(k)−1,xn(k)−1) + d(ym(k)−1, yn(k)−1) 2 , d(xm(k)−1, f(xm(k)−1,ym(k)−1)) + d(ym(k)−1, f(ym(k)−1,xm(k)−1)) 2 , d(xn(k)−1, f(xn(k)−1,yn(k)−1)) + d(yn(k)−1,f(yn(k)−1,xn(k)−1)) 2 , d(xn(k)−1,f(xm(k)−1, ym(k)−1)) + d(yn(k)−1,f(ym(k)−1,xm(k)−1)) 2 } = max { d(xm(k)−1,xn(k)−1) + d(ym(k)−1,yn(k)−1) 2 , d(xm(k)−1,xm(k)) + d(ym(k)−1,ym(k)) 2 , d(xn(k)−1,xn(k)) + d(yn(k)−1,yn(k)) 2 , d(xn(k)−1,xm(k)) + d(yn(k)−1,ym(k)) 2 } . (3.17) similarly, we show that d(ym(k), yn(k)) = d(f(ym(k)−1,xm(k)−1), f(yn(k)−1,xn(k)−1)) ≤ β(m(ym(k)−1,xm(k)−1,yn(k)−1,xn(k)−1)) m(ym(k)−1,xm(k)−1,yn(k)−1,xn(k)−1) = β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)) m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1). (3.18) combining (3.16) and (3.18), we have d(xm(k), xn(k)) + d(ym(k), yn(k)) ≤ 2 β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)) m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1). (3.19) taking limit as k →∞ in (3.17) and using (3.11), (3.14) and (3.15), we have lim k→∞ m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1) = max {� 2 , 0, 0, � 2 } = � 2 . (3.20) taking limit supremum in (3.19) and using (3.13), (3.20), we have � ≤ 2 lim sup k→∞ β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)) � 2 = � lim sup k→∞ β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)). (3.21) using (3.21) and the property of β, we have 1 ≤ lim sup k→∞ β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)) ≤ 1, that is, lim supk→∞β(m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1)) = 1. then it follows by the property of β that limk→∞m(xm(k)−1,ym(k)−1,xn(k)−1,yn(k)−1) = � 2 = 0, that is, � = 0 which is a contradiction. therefore, {xn} and {yn} are both cauchy sequences in x. as x is complete, there exist cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 179 x, y ∈ x such that lim n→∞ xn = x and lim n→∞ yn = y. (3.22) now we show that (x, y) is a coupled fixed point of f. if possible let (x, y) be not a coupled fixed point of f. then either x 6= f(x,y) or y 6= f(y,x), that is, either, d(x, f(x,y)) 6= 0 or d(y, f(y,x)) 6= 0, that is, d(x, f(x,y)) + d(y, f(y,x)) > 0. using (3.3), (3.22) and r-regularity property of the space, we have (xn, x) ∈ r and (y, yn) ∈ r. (3.23) by (3.1) and (3.23), we have d(xn+1, f(x,y)) = d(f(xn,yn), f(x, y)) ≤ β(m(xn,yn,x,y)) m(xn,yn,x,y) (3.24) where m(xn,yn,x,y) = max { d(xn, x) + d(yn, y) 2 , d(xn, f(xn,yn)) + d(yn, f(yn,xn)) 2 , d(x, f(x,y)) + d(y, f(y,x)) 2 , d(x, f(xn,yn)) + d(y, f(yn,xn)) 2 } = max { d(xn, x) + d(yn, y) 2 , d(xn, xn+1) + d(yn, yn+1) 2 , d(x, f(x,y)) + d(y, f(y,x)) 2 , d(x, xn+1) + d(y, yn+1) 2 } . (3.25) similarly, we show that d(yn+1, f(y,x)) = d(f(yn,xn), f(y, x)) ≤ β(m(yn,xn,y,x)) m(yn,xn,y,x)) = β(m(xn,yn,x,y)) m(xn,yn,x,y). (3.26) combining (3.24) and (3.26), we have d(xn+1, f(x,y)) + d(yn+1, f(y,x)) ≤ 2 β(m(xn,yn,x,y)) m(xn,yn,x,y). (3.27) taking limit as n →∞ in (3.25), we have lim n→∞ m(xn,yn,x,y) = max { 0, 0, d(x, f(x,y)) + d(y, f(y,x)) 2 , 0 } = d(x, f(x,y)) + d(y, f(y,x)) 2 . (3.28) taking limit supremum in (3.27) and using (3.22) and (3.28), we have d(x,f(x,y) + d(y,f(y,x)) ≤ [d(x,f(x,y) + d(y,f(y,x))] lim sup n→∞ β(m(xn,yn,x,y)). (3.29) as explained earlier, we have from (3.29) that 1 ≤ lim sup n→∞ β(m(xn,yn,x,y)) ≤ 1, 180 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) that is, lim supn→∞β(m(xn,yn,x,y)) = 1. by a property of β we have that lim n→∞ m(xn,yn,x,y) = d(x,f(x,y)) + d(y,f(y,x)) 2 = 0, that is, d(x,f(x,y)) + d(y,f(y,x)) = 0, which contradicts our assumption. therefore, d(x, f(x,y)) = d(y, f(y,x)) = 0, that is, x = f(x,y) and y = f(y,x), that is, (x, y) is a coupled fixed point of f. remark 3.2. our result is a generalization of the result of bhaskar and lakshmikantam (in [15]) and of the result of choudhury and kundu (in [8]). if r is taken to be a partial ordered relation, then we have the following corollary: corollary 3.3. let (x,d) be a complete metric space with a partial order � on it such that x has regular property [that is, if {xn} is a monotone convergent sequence with limit x, then xn � x or x � xn, according as the sequence is increasing or decreasing]. suppose that f : x ×x → x is a dominated map [that is, x � f(x, y) and f(y, x) � y, for any (x, y) ∈ x×x] and there exists β ∈ b∗ such that (3.1) of theorem 3.1 is satisfied for all (x, y), (u, v) ∈ x ×x with [x � u and v � y] or [u � x and y � v]. then f has a coupled fixed point. if r is taken to be the universal relation, that is, r = x×x, we have the following corollary: corollary 3.4. let (x,d) be a complete metric space and f : x ×x → x. suppose there exists β ∈ b∗ such that (3.1) of theorem 3.1 is satisfied for all (x, y), (u, v) ∈ x ×x. then f has a coupled fixed point. theorem 3.5. in addition to the hypothesis of theorem 3.1, suppose that both r and r−1 are directed. then f has a unique coupled fixed point. proof. by theorem 3.1, the set of coupled fixed points of f is nonempty. if possible, let (x, y) and (x∗, y∗) be two coupled fixed points of f. then x = f(x, y); y = f(y, x) and x∗ = f(x∗, y∗); y∗ = f(y∗, x∗). our aim is to show that x = x∗ and y = y∗. by the directed property of r and r−1, there exist u ∈ x and v ∈ x such that (x, u) ∈ r; (x∗, u) ∈ r and (y, v) ∈ r−1; (y∗, v) ∈ r−1, that is (x, u) ∈ r; (x∗, u) ∈ r and (v, y) ∈ r; (v, y∗) ∈ r. put u0 = u and v0 = v. then (x, u0) ∈ r and (v0, y) ∈ r. let u1 = f(u0, v0) and v1 = f(v0, u0). similarly, as in the proof of theorem 3.1, we inductively define two sequences {un} and {vn} such that un+1 = f(un,vn) and vn+1 = f(vn,un), for all n ≥ 0. (3.30) as f is r-dominated, we have (un, un+1) ∈ r and (vn+1, vn) ∈ r, for all n ≥ 0. (3.31) cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 181 arguing similarly as in proof of theorem 3.1, we prove that {un} and {vn} are two cauchy sequences in x and there exists p and q ∈ x such that lim n→∞ un = p and lim n→∞ vn = q. (3.32) now we show that x = p and y = q, that is, d(x,p) + d(y,q) = 0. if possible, suppose that d(x,p) + d(y,q) 6= 0. we claim that (x, un) ∈ r and (vn, y) ∈ r, for all n ≥ 0. (3.33) as (x,u0) ∈ r, (u0,u1) ∈ r and (v1,v0) ∈ r, (v0,y) ∈ r, by the transitivity property of r, we have (x, u1) ∈ r and (v1, y) ∈ r. therefore, our claim is true for n = 1. assume that (3.33) is true for some m > 1, that is, (x, um) ∈ r and (vm, y) ∈ r. by (3.31), (um, um+1) ∈ r and (vm+1, um) ∈ r. the transitivity property of r guarantees that (x, um+1) ∈ r and (vm+1, y) ∈ r and this proves our claim. using (3.1) and (3.33), we have for all n ≥ 0 that d(x,un+1) = d(f(x, y), f(un, vn)) ≤ β(m(x, y, un, vn)) m(x, y, un, vn), (3.34) where m(x, y, un, vn) = max { d(x,un) + d(y,vn) 2 , d(x, f(x,y)) + d(y, f(y,x)) 2 , d(un, f(un,vn)) + d(vn, f(vn,un)) 2 , d(un, f(x,y)) + d(vn, f(y,x)) 2 } = max { d(x,un) + d(y,vn) 2 , 0, d(un, un+1) + d(vn, vn+1) 2 , d(un, x) + d(vn, y) 2 } . (3.35) similarly, we show that d(y,vn+1) = d(f(y, x), f(vn, un)) ≤ β(m(y, x, vn, un)) m(y, x, vn, un) = β(m(x, y, un, vn)) m(x, y, un, vn). (3.36) combining (3.34) and (3.36), we have d(x,un+1) + d(y,vn+1) ≤ 2 β(m(x, y, un, vn)) m(x, y, un, vn). (3.37) taking limit in (3.35) as n →∞ and using (3.32), we have lim n→∞ m(x,y,un,vn) = d(x,p) + d(y,q) 2 . (3.38) taking limit supremum as n →∞ in (3.37) and using (3.32), (3.38), we have d(x,p) + d(y,q) ≤ [d(x,p) + d(y,q)] lim sup n→∞ β(m(x, y, un, vn)) (3.39) 182 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) which implies that 1 ≤ lim sup n→∞ β(m(x,y,un,vn)) ≤ 1, (3.40) that is, lim sup n→∞ β(m(x, y, un, vn)) = 1. by a property of β we have that lim n→∞ m(x,y,un,vn) = d(x,p) + d(y,q) 2 = 0, that is, d(x,p) +d(y,q) = 0, which contradicts our assumption that d(x,p) +d(y,q) 6= 0. therefore, d(x,p) + d(y,q) = 0, that is, d(x,p) = d(y,q) = 0, that is, x = p and y = q. (3.41) similarly, we can show that x∗ = p and y∗ = q. (3.42) from (3.41) and (3.42), we have x = x∗ and y = y∗. therefore, the coupled fixed point of f is unique. we present the following illustrative example in support of theorems 3.1. example 3.6. take the metric space x = [0, 1] with usual metric d. let β : [0, ∞) → [0, 1) be defined as β(t) = ln(1 + t) t , if t > 0 and β(t) = 0, if t = 0. define f : x × x → x by f(x, y) = ln ( 1 + x + y 2 ) , for all (x, y) ∈ x ×x and binary relation r by r = {(x, y) : 0 ≤ x ≤ 1; 0 ≤ y ≤ ln 2 or 0 ≤ x ≤ ln 2; 0 ≤ y ≤ 1}. then we see that x is regular with respect to r and t is r-dominated. let (x,y), (u, v) ∈ x × x with (x, u) ∈ r and (v, y) ∈ r. then [ x ∈ [0, 1] or x ∈ [0, ln 2] ] ; [ u ∈ [0, 1] or u ∈ [0, ln 2] ] ; [ y ∈ [0, 1] or y ∈ [0, ln 2] ] and [ v ∈ [0, 1] or v ∈ [0, ln 2] ] . now for those values of x, y, y, u and v, we obtain d(f(x,y), f(u,v)) = d ( ln ( 1 + x + y 2 ) , ln ( 1 + u + v 2 ) ) = ∣∣∣∣ ln ( 1 + x + y 2 ) − ln ( 1 + u + v 2 ) ∣∣∣∣ = ∣∣∣∣∣∣∣ ln  1 + x + y 2 1 + u + v 2   ∣∣∣∣∣∣∣ = ∣∣∣∣∣∣∣ ln  1 + x + y 2 − u + v 2 1 + u + v 2   ∣∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣ ln  1 + ∣∣∣∣ x + y2 − u + v2 ∣∣∣∣ 1 + u + v 2   ∣∣∣∣∣∣∣∣ ≤ ∣∣∣∣ ln ( 1 + ∣∣∣∣ x + y2 − u + v2 ∣∣∣∣ ) ∣∣∣∣ ≤ ∣∣∣∣ ln ( 1 + | u−x | + | v −y | 2 ) ∣∣∣∣ = ln ( 1 + | u−x | + | v −y | 2 ) ≤ ln ( 1 + m(x, y, u, v) ) = ln ( 1 + m(x, y, u, v) ) m(x, y, u, v) m(x, y, u, v) = β ( m(x, y, u, v) ) m(x, y, u, v). cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 183 it follows that the inequality in theorem 3.1 is satisfied for all (x,y), (u, v) ∈ x × x with (x, u) ∈ r and (v, y) ∈ r. here all the conditions of theorem 3.1 are satisfied and (0, 0) is a coupled fixed point of f. 4 well-posedness we use the following assumption to assure the well-posedness via r-dominated mapping. (a) if (x∗, y∗) is any solution of the problem (p), that is, of (2.1) and {(xn, yn)} is any sequence in x × x for which limn→∞d(xn, f(xn, yn)) = limn→∞d(yn, f(yn, xn)) = 0, then (x∗,xn) ∈ r and (yn, y∗) ∈ r, for all n. theorem 4.1. in addition to the hypothesis of theorem 3.5, suppose that the assumption (a) holds. then the coupled fixed point problem (p) is well-posed. proof. by theorem 3.5, f has a unique coupled fixed point (x∗,y∗) (say). then (x∗,y∗) is a solution of (2.1), that is, x∗ = f(x∗,y∗) and y∗ = f(y∗,x∗). let {(xn, yn)} be any sequence in x × x for which lim supn→∞[d(xn, x∗) + d(yn, y∗)] is finite and limn→∞d(xn, f(xn, yn)) = limn→∞d(yn, f(yn, xn)) = 0. then there exists a nonnegative real number m such that lim supn→∞[d(x ∗,xn)+d(y ∗,yn)] = m and also by the assumption (a), (x ∗,xn) ∈ r and (yn,y∗) ∈ r, for all n. using (3.1), we have d(xn, x ∗) = d(xn, f(x ∗, y∗)) ≤ d(xn, f(xn, yn) + d(f(x∗, y∗), f(xn, yn)) ≤ β(m(x∗,y∗,xn,yn)) m(x∗,y∗,xn,yn) + d(xn, f(xn, yn)) (4.1) where m(x∗,y∗,xn,yn) = max { d(x∗,xn) + d(y ∗,yn) 2 , d(x∗, f(x∗,y∗)) + d(y∗, f(y∗,x∗)) 2 , d(xn,f(xn,yn)) + d(yn,f(yn,xn)) 2 , d(xn,f(x ∗,y∗)) + d(yn,f(y ∗,x∗)) 2 } = max { d(x∗,xn) + d(y ∗,yn) 2 , 0, d(xn, f(xn,yn)) + d(yn, f(yn,xn)) 2 , d(xn, x ∗) + d(yn, y ∗) 2 } . (4.2) similarly, we can show that d(yn, y ∗) ≤ β(m(y∗,x∗,yn,xn)) m(y∗,x∗,yn,xn) + d(yn, f(yn,xn)) ≤ β(m(x∗,y∗,xn,yn)) m(x∗,y∗,xn,yn) + d(yn, f(yn,xn)). (4.3) combining (4.1) and (4.3), we have d(xn, x ∗) + d(yn, y ∗) ≤ 2 β(m(x∗,y∗,xn,yn)) m(x∗,y∗,xn,yn) + d(xn, f(xn, yn)) + d(yn, f(yn,xn)). (4.4) 184 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) taking limit supremum as n →∞ in (4.2), we have lim sup n→∞ m(x∗,y∗,xn,yn) = m 2 . (4.5) if possible, suppose that lim supn→∞[d(x ∗,xn) + d(y ∗,yn)] = m 6= 0. then m > 0. taking limit supremum as n →∞ in (4.4) and using (4.5), we have m ≤ m lim sup n→∞ β(m(x∗,y∗,xn,yn)), that is, 1 ≤ lim sup n→∞ β(m(x∗,y∗,xn,yn)) ≤ 1. then lim supn→∞β(m(x ∗,y∗,xn,yn)) = 1. by a property of β, limn→∞m(x ∗,y∗,xn,yn) = 0, that is, limn→∞[d(xn, x ∗) + d(yn, y ∗)] = 0 which is a contradiction. hence we have lim supn→∞[d(x ∗,xn) + d(y ∗,yn)] = 0. then we have 0 ≤ lim infn→∞[d(x∗,xn) + d(y∗,yn)] ≤ lim supn→∞[d(x ∗,xn) + d(y ∗,yn)] = 0 which implies that limn→∞[d(x ∗,xn) + d(y ∗,yn)] = 0. it follows that limn→∞d(xn,x ∗) = limn→∞d(yn,y ∗) = 0, that is, xn → x∗ and yn → y∗ as n → ∞. hence the coupled fixed point problem (p) is well-posed. 5 some results for α−dominated mapping coupled α-dominated mappings are defined here and are conceptual extensions of mappings with admissibility conditions. various types of admissibility conditions have been used in fixed point theory in works like [10, 11, 19, 29, 31]. definition 5.1. let x be a nonempty set and α : x × x → r be a mapping. a mapping f : x × x → x is said to be α-dominated if α(x, f(x,y)) ≥ 1 and α(y, f(y,x)) ≥ 1, for (x, y) ∈ x ×x. definition 5.2. let x be a nonempty set and α : x ×x → r be a mapping. then α is said to have triangular property if for x, y, z ∈ x, α(x, y) ≥ 1 and α(y, z) ≥ 1 imply α(x, z) ≥ 1. definition 5.3. let (x, d) be a metric space and α : x×x → r be a mapping. then x is said to have α-regular property if for every convergent sequence {xn} with limit x ∈ x, α(xn, xn+1) ≥ 1, for all n implies α(xn, x) ≥ 1, for all n. theorem 5.4. let (x, d) be a complete metric space and α : x × x → r be a mapping such that x has α-regular property and α has triangular property. suppose that f : x ×x → x be a α-dominated mapping and there exists β ∈ b∗ such that (3.1) of theorem 3.1 is satisfied for all (x, y), (u, v) ∈ x ×x with α(x, u) ≥ 1 and α(y, v) ≥ 1. then f has a coupled fixed point. proof. define a binary relation r on x as (x, y) ∈ r if and only if α(x, y) ≥ 1 or α(y, x) ≥ 1. then (i) α(x, u) ≥ 1 and α(y, v) ≥ 1 imply (x,u) ∈ r and (v,y) ∈ r, (ii) α(x, f(x,y)) ≥ 1 and α(y, f(y,x)) ≥ 1 imply (x, f(x,y)) ∈ r and (f(y,x), y) ∈ r, for (x, y) ∈ x × x, cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 185 (iii) α(xn, xn+1) ≥ 1, α(xn, x) ≥ 1 imply (xn, xn+1) ∈ r, (xn, x) ∈ r, whenever {xn} is a convergent sequence with xn → x and α(xn, xn+1) ≥ 1. therefore, all the assumptions reduce to the assumptions of theorem 3.1. then by an application of theorem 3.1, we conclude that f has a coupled fixed point in x ×x. 6 some results on graphic contraction our present section is on graphic contraction. fixed point problem on the structures of metric spaces with a graph have appeared in works like [2, 4, 12]. let x be a nonempty set and ∆ := {(x, x) : x ∈ x}. let g be a directed graph such that its vertex set v (g) coincides with x, that is, v (g) = x and the edge set e(g) contains all loops, that is, ∆ ⊆ e(g). assume that g has no parallel edges. by g−1 we denote the conversion of a graph g, that is, the graph obtained from g by reversing the directions of the edges. thus we have v (g−1) = v (g) and e(g−1) = {(x, y) ∈ x ×x : (y, x) ∈ e(g)}. a nonempty set x is said to be endowed with a directed graph g(v,e) if v (g) = x and ∆ ⊆ e(g). definition 6.1. let x be a nonempty set endowed with a graph g(v,e). a mapping f : x×x → x is said to be g-dominated if (x, f(x,y)) ∈ e and (f(y,x),y) ∈ e, for (x,y) ∈ x ×x. definition 6.2. let x be a nonempty set endowed with a graph g(v,e). then g is said to have transitive property if for x, y, z ∈ x, (x, y) ∈ e and (y, z) ∈ e imply (x, z) ∈ e. definition 6.3. let (x, d) be a metric space endowed with a directed graph g(v,e). then x is said to have g-regular property if for every convergent sequence {xn} with limit x ∈ x, (xn, xn+1) ∈ e, for all n implies (xn, x) ∈ e, for all n [or (xn+1, xn) ∈ e, for all n implies (x, xn) ∈ e, for all n]. theorem 6.4. let (x, d) be a complete metric space endowed with a directed graph g(v,e) such that x has g-regular property and g has transitive property. suppose that f : x ×x → x is a g-dominated mapping and there exists β ∈ b∗ such that (3.1) of theorem 3.1 is satisfied for all (x, y), (u, v) ∈ x×x with [(x, u) ∈ e and (v, y) ∈ e] or [(u, x) ∈ e and (y, v) ∈ e]. then f has a coupled fixed point. proof. let us define a relation r, by xry holds if (x,y) ∈ e. as (x, y) ∈ e, for (x,y) ∈ x ×x implies that (x, y) ∈ r, it is easy to verify that all the assumptions of the theorem reduce to the assumptions of theorem 3.1. then by an application of theorem 3.1, we conclude that f has a coupled fixed point in x ×x. 186 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) 7 application to the solution of system nonlinear integral equations in this section, we present an application of our coupled fixed point results derived in section 3 to establish the existence and uniqueness of a solution of a system of integral equations. we consider a coupled system of two nonlinear integral equations as follows: x(t) = f(t) + ∫ t 0 k(t, s)h(t, s, x(s), y(s))ds and y(t) = f(t) + ∫ t 0 k(t, s)h(t, s, y(s), x(s))ds,   (7.1) where t > 0 be any number, t,s ∈ [0,t], k : [0,t]× [0,t] → r be a function, which is the kernel of the integral equations, and the unknown functions x(t) and y(t) take real values. the reason for the choice of this application is that coupled non-linear equations have their uses in modeling situations of wide variety. let x = c([0, t]) be the space of all real valued continuous functions defined on [0, t]. here c([0, t ]) with the metric d(x, y) = maxt∈[0, t ] | x(t) − y(t) | is a complete metric space. assume that this metric space is endowed with the universal relation u, that is, (x,y) ∈ u, for all x, y ∈ x. define a mapping f : x ×x → x by f(x, y)(t) = f(t) + ∫ t 0 k(t, s)h(t, s, x(s), y(s))ds, for all t,s ∈ [0, t]. (7.2) we designate the following assumptions by a1, a2 and a3: a1 : f ∈ c([0, t] and h : [0, t] × [0, t] ×r×r → r is a continuous mapping; a2 : | k(t, s) |≤ q, where q > 0 is a fixed number; a3 : | h(t,s,x,y) −h(t,s,u,v) |≤m(t, s, x, y, u, v), for all (x,y), (u,v) ∈ x ×x and t, s ∈ [0, t ], where m(t, s, x, y, u, v) = 1 qt ln ( 1 + | x−u | + | y −v | 2 ) . theorem 7.1. let (x = c([0,t]),d), f, h, k(t,s) satisfy the assumptions a1, a2 and a3. then system of nonlinear integral equation (7.1) has a solution (x,y) ∈ c([0,t]) ×c([0,t]) and the solution is unique. proof. it is trivial to observe that the mapping f : x×x → x defined by (7.2) is a u dominated mapping and x has u regular property, where u is the universal relation. from assumptions a1, a2 and a3, for all (x,y), (u,v) ∈ c([0, t])×c([0, t]), that is, for all x, y, u, v ∈ c([0, t]) and t,s ∈ [0, t], we have cubo 23, 1 (2021) existence, well-posedness of coupled fixed points and application . . . 187 | f(x,y)(t) −f(u,v)(t) |= ∣∣∣∣ ∫ t 0 k(t, s)[h(t, s, x(s), y(s)) −h(t, s, u(s), v(s))]ds ∣∣∣∣ ≤ ∫ t 0 | k(t, s) || [h(t, s, x(s), y(s)) −h(t, s, u(s), v(s))] | ds ≤ q × ∫ t 0 | [h(t, s, x(s), y(s)) −h(t, s, u(s), v(s))] | ds [by a2] ≤ q × ∫ t 0 | [h(t, s, x(s), y(s)) −h(t, s, u(s), v(s))] | ds ≤ q × ∫ t 0 m(t, s, x, y, u, v) ds = q × ∫ t 0 1 qt ln ( 1 + | x−u | + | y −v | 2 ) ds = ∫ t 0 1 t ln ( 1 + | x−u | + | y −v | 2 ) ds ≤ ∫ t 0 1 t ln ( 1 + d(x, u) + d(y, v) 2 ) ds = ln(1 + d(x, u) + d(y, v) 2 ) ∫ t 0 1 t ds = ln ( 1 + d(x, u) + d(y, v) 2 ) ≤ ln(1 + m(x,y,u,v)) [since ln(1 + t) is nondecreasing for t > 0] = ln(1 + m(x,y,u,v)) m(x,y,u,v) m(x,y,u,v) = β(m(x, y, u, v)) m(x, y, u, v) where β(t) = ln(1 + t) t , if t > 0 and β(t) = 0, if t = 0 and m(x, y, u, v) = max { d(x, u) + d(y, v) 2 , d(x,f(x, y)) + d(y, f(y, x)) 2 , d(u,f(u, v)) + d(v,f(v, u)) 2 , d(u,f(x, y)) + d(v,f(y, x)) 2 } . hence d(f(x, y), f(u, v)) ≤ β(m(x, y, u, v)) m(x, y, u, v). therefore, all the conditions of theorems 3.1 and 3.5 are satisfied and hence by theorem 3.1 there exists a coupled fixed point (x, y) in x ×x which, by virtue of theorem 3.5, is unique. in other words, the system of integral equations (7.1) under the conditions stipulated in the theorem has a solution which is unique. acknowledgement: the suggestions of the learned referee are gratefully acknowledged. 188 b. s. choudhury, n. metiya & s. kundu cubo 23, 1 (2021) references [1] a. alam, and m. imad, “relation-theoretic contraction principle”, j. fixed point theory appl., vol. 17, pp. 693–702, 2015. 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[33] e. yolacan, and m. kir, “new results for α−geraghty type contractive maps with some applications”, gu j sci, vol. 29, no. 3, pp. 651–658, 2016. introduction mathematical background main results well-posedness some results for -dominated mapping some results on graphic contraction application to the solution of system nonlinear integral equations cubo, a mathematical journal vol. 24, no. 02, pp. 211–226, august 2022 doi: 10.56754/0719-0646.2402.0211 vlasov-poisson equation in weighted sobolev space w m,p(w) cong he 1 jingchun chen 2, b 1department of mathematical sciences, university of wisconsin-milwaukee, milwaukee, wi 53201, usa. conghe@uwm.edu 2department of mathematics and statistics, the university of toledo, toledo, oh 43606, usa. jingchunchen123@gmail.com b abstract in this paper, we are concerned about the well-posedness of vlasov-poisson equation near vaccum in weighted sobolev space w m,p(w). the most difficult part comes from estimates of the electronic term ∇xφ. to overcome this difficulty, we establish the lp-lq estimates of the electronic term ∇xφ; some weight is introduced as well to obtain the off-diagonal estimate. the weight is also useful when it comes to control the higher-order derivative term. resumen en este art́ıculo, estamos interesados en que la ecuación de vlasov-poisson está bien puesta cercana al vaćıo en el espacio de sobolev w m,p(w) con peso. la parte más dif́ıcil proviene de estimaciones del término electrónico ∇xφ. para superar esta dificultad, establecemos las estimaciones lp-lq del término electrónico ∇xφ; donde algún peso es también introducido para obtener la estimación fuera de la diagonal. el peso es también útil cuando se trata de controlar el término de la derivada de alto orden. keywords and phrases: vlasov-poisson, lp-sobolev, weighted estimates, lp-lq estimates. 2020 ams mathematics subject classification: 35q83, 46e35, 35a01, 35a02. accepted: 24 march, 2022 received: 29 june, 2021 c©2022 c. he et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://doi.org/10.56754/0719-0646.2402.0211 https://orcid.org/0000-0001-9868-8728 mailto:jingchunchen123@gmail.com https://orcid.org/0000-0002-8808-6077 mailto:conghe@uwm.edu mailto:jingchunchen123@gmail.com 212 cong he & jingchun chen cubo 24, 2 (2022) 1 introduction understanding the evolution of a distribution of particles over time is a major research area of statistical physics. the vlasov-poisson equation is one of the key equations governing this evolution. specifically, it models particle behaviors with long range interactions in a non-relativistic zero-magnetic field setting. two principal types of long range interactions are coulomb’s forces, the electrostatic repulsion of similarly charged particles in a plasma, and newtonian’s forces, the gravitational attraction of stars in a galaxy. the general cauchy’s problem for the vlasov-poisson equation (vp equation) in n dimensional space is as follows:               ∂tf + v · ∇xf + ∇xφ · ∇vf = 0, −∆xφ = ∫ rn f dv, f(0, x, v) = f0(x, v), (1.1) where f(t, x, v) denotes the distribution function of particles, x ∈ rn is the position, v ∈ rn is the velocity, and t > 0 is the time and n ≥ 3. the cauchy problem for the vlasov-poisson equation has been studied for several decades. the first paper on global existence is due to arsen’ev [3]. he showed the global existence of weak solutions. then in 1977 batt [5] established the global existence for spherically symmetric data. in 1981 horst [9] extended the global classical solvability to cylindrically symmetric data. next, in 1985, bardos and degond [4] obtained the global existence for “small” data. finally, in 1989 pfaffelmoser [12] proved the global existence of a smooth solution with large data. later, simpler proofs of the same results were published by schaeffer [13], horst [10], and lions and pertharne [11]. nevertheless, most of them were concerned about solutions in l∞ or continuous function spaces. also, there are many papers studying vlasov-poisson-boltzmman (landau) equation in l2 setting, see [2, 6, 7, 8] and the references therein. a natural question is whether we can obtain the solutions in lp context, for example, w m,p spaces. this becomes our main theme in this paper. in this paper, our aim is to construct the solution to (1.1) in w m,p space. the difficulty lies in the absence of lp estimates of the electronic term ∇xφ. to handle this issue, we establish the l p-lq off-diagonal estimates of ∇xφ which is highly important in estimating the higher order derivative term. also, it is necessary to introduce a weight w in order to obtain this off-diagonal estimate. it is worthy to mention that this weight is crucial to deal with the higher-order derivative term. cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 213 2 preliminaries and main theorem 2.1 notations and definitions we first would like to introduce some notations. • given a locally integrable function f, the maximal function mf is defined by (mf)(x) = sup δ>0 1 |b(x, δ)| ∫ b(x,δ) |f(y)|dy, (2.1) where |b(x, δ)| is the volume of the ball of b(x, δ) with center x and radius δ. • weight w(v) = 〈v〉 γ , γ · p ′ p > n, 1 p + 1 p ′ = 1, n is the dimension. • ‖f‖ p l p x,v(w) =: ∫ r2n |f|pw dxdv. • define the higher-order energy norm as e(f(t)) =: ‖f‖w m,p(w) = ∑ |α|+|β|≤m ‖∂αx ∂ β v f(t)‖ p l p x,v(w) , and e(f0) =: e(f(0)) = ∑ |α|+|β|≤m ‖∂αx ∂ β v f0‖ p l p x,v(w) , where m ≥ 5 and n 3 < p < n 2 , n ≥ 3. here α and β denote multi-indices with length |α| and |β|, respectively. if each component of α1 is not greater than that of α, we denote the condition by α1 ≤ α. we also define α1 < α if α1 ≤ α and |α1| < |α|. we also denote ( α1 α ) by cα1α . • a . b means there exists a constant c > 1 independent of the main parameters such that a ≤ cb. a ∼ b means a . b and b . a. now we are ready to state our main theorem. theorem 2.1. for any sufficiently small m > 0, there exists t ∗(m) > 0 such that if e(f0) = ∑ |α|+|β|≤m ‖∂αx ∂ β v f0‖ p l p x,v(w) ≤ m 2 , then there is a unique solution f(t, x, v) to vlasov-poisson system (1.1) in [0, t ∗(m)) × rn × rn such that sup 0≤t≤t ∗ e(f(t)) ≤ m, where m > n p + 1 with n ≥ 3 and n 3 < p < n 2 . 214 cong he & jingchun chen cubo 24, 2 (2022) remark 2.2. • one should pay attention to the differential index m in w m,p(rn) which represents the weak derivative, is not the classical derivative in c2(rn). indeed, for the space w 4,1.4(r6) in which we could obtain solutions that could not be embedded into c(r6) (the continuous function space) or l∞(r6), not to mention c2(r6) (the twice continuously differentiable function space) due to the fact 4 · 1.4 < 6, i.e. w 4,1.4(r6) 6→֒ c2(r6) which implies that the classical results in [3, 4] and [9]-[13] could not cover our results. • in [4], c. bardos and p. degond also imposed the pointwise condition like 0 ≤ uα,0(x, v) ≤ ǫ (1 + |x|)4 · (1 + |v|)4 . however, the polynomial decay in the x variable is not needed at all in our proofs. • our working space w m,p(rn) has more flexibility than c2(rn) because of the triplet (m, n, p) which implies that we can obtain the solutions in more spaces. let us illustrate our strategies for proving theorem 2.1. as is known, the routine to prove the existence of solution is to get a uniform-in-k estimate for the energy norm e(fk+1(t)). in this paper, we adopt the lp version energy method, i.e. to do the dual with |∂αx ∂ β v f k+1|p−2(∂αx ∂ β v f k+1)w (see (4.5)). we expect all the estimates ji in section 4 can be controlled by e(f(t)) =: ∑ |α|+|β|≤m ‖∂αx ∂ β v f(t)‖ p l p x,v(w) . to achieve our goal, some estimates related to the electronic term ∇xφ are needed. the l p-lq estimate is established to deal with the higher-order derivative. for instance, when |α| = m, the lp-lq estimate comes in to handle the highest order derivative term ∂αx ∇xφ k : 〈 ∂αx ∇xφ k · ∇vf k+1, |∂αx f k+1|p−2 · ∂αx f k+1 · w 〉 . ‖∂αx ∇xφ k‖lqx‖∇vf k+1‖ln x,v (w)‖∂ α x f k+1‖ p−1 l p x,v(w) . (2.2) in turn, in order to get this lp-lq estimate involving ∇xφ, we introduce weight w; surprisingly, this weight w also plays another crucial role to deal with the higher order derivative. more precisely, we do this trick when |α| + |β| = m, w could “absorb” the extra derivative in ∇v as follows: 〈 ∇xφ k · ∇v∂ α x ∂ β v f k+1 , |∂αx ∂ β v f k+1|p−2(∂αx ∂ β v f k+1)w 〉 ∼ 〈 ∇xφ k · ∇v|∂ α x ∂ β v f k+1|p, w 〉 ∼ − 〈 ∇xφ k · |∂αx ∂ β v f k+1|p, ∇vw 〉 . (2.3) before we give the proof of the main theorem, we would like to establish the following lp-lq estimates. cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 215 3 lp-lq estimates in this section, we are going to prove the lp-lq estimate which plays an essentially important role in our proofs. lemma 3.1. suppose 1 < p < n 2 and 1 q = 1 p − 1 n . if −∆φ = ∫ rn fdv =: g, then it holds that ‖∇xφ‖lq(rn) . ‖g‖lp(rn), (3.1) proof. note that ∇xφ = ∇x(i2 ∗ g), with i2(x) = 1 (n − 2)ωn−1 · 1 |x|n−2 , for more details, see the last section appendix. therefore there holds ‖∇xφ‖lq(rn) = ‖∇x(i2 ∗ g)‖lq(rn) . ‖(mg) 1 2 · (i2 ∗ |g|) 1 2 ‖lq(rn) . ‖(mg) 1 2 ‖lq1 (rn) · ‖(i2 ∗ |g|) 1 2 ‖lq2 (rn) . ‖mg‖ 1 2 l q1 2 (rn) · ‖i2 ∗ |g|‖ 1 2 l q2 2 (rn) , where we applied (5.3) in the second line, and hölder’s inequality with 1 q1 + 1 q2 = 1 q , qi > 1, in the third line separately. on the one hand, the boundedness of hardy-littlewood operator m as defined by identity (2.1) yields that ‖mg‖ l q1 2 (rn) . ‖g‖ l q1 2 (rn) = ‖g‖lp(rn), (3.2) since we require that q1 2 = p, i.e. 2 q1 = 1 p . (3.3) on the other hand, by lemma 5.3, we have ‖i2 ∗ |g|‖ l q2 2 (rn) . ‖g‖lp(rn), (3.4) where 2 q2 = 1 p − 2 n . (3.5) consequently, ‖∇xφ‖lq(rn) . ‖g‖ 1 2 lp(rn) · ‖g‖ 1 2 lp(rn) = ‖g‖lp(rn). a “derivative version” is immediate: 216 cong he & jingchun chen cubo 24, 2 (2022) corollary 3.2. with the same assumptions as in lemma 3.1, we have ‖∂αx ∇xφ‖lq(rn) . ‖∂ α x g‖lp(rn). proof. one only needs to observe that ∇x∂ α x φ = ∂ α x ∇xφ = ∂ α x ∇x(i2 ∗ g) = ∇x(i2 ∗ ∂ α x g). applying lemma 3.1 with φ and g replaced by ∂αφ and ∂αg respectively, the desired result is immediate. now we adapt corollary 3.2 to the “kinetic version”. to achieve this goal, we need to introduce a weight w. corollary 3.3. take g = ∫ rn fdv in corollary 3.2, then we have ‖∂αx ∇xφ‖lqx(rn) . ‖∂ α x f‖lpx,v(w). proof. hölder’s inequality leads to ∣ ∣ ∣ ∣ ∫ rn ∂ α x fdv ∣ ∣ ∣ ∣ . (∫ rn |∂αx f| p wdv ) 1 p (∫ rn w − p ′ p dv ) 1 p ′ . note that w = 〈v〉 γ and γ · p ′ p > n, which implies that (∫ rn w − p ′ p dv ) 1 p ′ ≤ c. thus we end the proof of corollary 3.3. an l∞ estimate is also needed in the proof of the main theorem 2.1. lemma 3.4. suppose −∆φ = ∫ rn fdv. if 0 ≤ |α| ≤ m − 2, m ≥ 3, then ‖∂αx ∇xφ‖l∞x . ∑ |i|≤2 ‖∂i+αx f‖lpx,v(w). (3.6) proof. choose a q such that q > n 2 and p ≤ q, then w 2,q →֒ l∞. thus we have ‖∂αx ∇xφ‖l∞x . ‖∂ α x ∇xφ‖w 2,q(rnx ). combining corollary 3.2 and corollary 3.3 leads to ‖∂αx ∇xφ‖w 2,q = ∑ |i|≤2 ‖∂ix∂ α x ∇xφ‖lqx . ∑ |i|≤2 ‖∂i+αx f‖lpx,v(w), i.e. ‖∂αx ∇xφ‖l∞x . ∑ |i|≤2 ‖∂i+αx f‖lpx,v(w). cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 217 4 proof of main theorem now we are in the position to prove theorem 2.1. we split the proof into two parts which are existence and uniqueness. part i: proof of existence. to prove the existence of the solution to (1.1), we adopt the lpversion energy method and iteration method. in this process, we will apply the lp-lq estimate of electronic term ∇xφ proved in lemma 3.1 to estimate j3. proof. we consider the following iterating sequence for solving the vlasov-poisson system (1.1),                ∂tf k+1 + v · ∇xf k+1 + ∇xφ k · ∇vf k+1 = 0, − ∆φk = ∫ rn fkdv, fk+1(0, x, v) = f0(x, v). (4.1) (4.2) (4.3) step 1. applying ∂αx ∂ β v to (4.1) with β 6= 0, |α|+|β| ≤ m, starting with f 0(t, x, v) = f0(x, v), we have (∂t + v · ∇x + ∇xφ k · ∇v)∂ α x ∂ β v f k+1 + ∑ β1<β c β1 β ∂β−β1v v · ∂ β1 v ∇x∂ α x f k+1 = − ∑ 06=α1≤α cα1α ∂ α1 x ∇xφ k · ∂α−α1x ∂ β v ∇vf k+1. (4.4) multiplying |∂αx ∂ β v f k+1|p−2(∂αx ∂ β v f k+1)w on both sides of (4.4), and then integrating over rnx × r n v yields that 1 p d dt ‖∂αx ∂ β v f k+1‖ p l p x,v(w) + ∑ β1<β c β1 β 〈 ∂β−β1v v · ∂ β1 v ∇x∂ α x f k+1, |∂αx ∂ β v f k+1|p−2(∂αx ∂ β v f k+1)w 〉 ︸︷︷︸ j1 = 〈 ∇xφ k · |∂αx ∂ β v f k+1|p, ∇vw 〉 ︸︷︷︸ j2 − ∑ 06=α1≤α c α1 α 〈 ∂ α1 x ∇xφ k · ∂α−α1x ∂ β v ∇vf k+1 , |∂αx ∂ β v f k+1|p−2(∂αx ∂ β v f k+1)w 〉 ︸︷︷︸ j3 . (4.5) we now estimate (4.5) term by term. 218 cong he & jingchun chen cubo 24, 2 (2022) for j1, note that |∂ β−β1 v v| ≤ c, β1 < β. thus, j1 . ∑ β1<β ∫ r2n |∂β1v ∇x∂ α x f k+1|w 1 p · |∂αx ∂ β v f k+1|p−1w 1 p ′ dxdv . ∑ β1<β (∫ r2n |∂β1v ∇x∂ α x f k+1|pw dxdv ) 1 p (∫ r2n |∂αx ∂ β v f k+1|(p−1)p ′ w dxdv ) 1 p ′ . ∑ β1<β ‖∂β1v ∇x∂ α x f k+1‖lpx,v(w) · ‖∂ α x ∂ β v f k+1‖ p−1 l p x,v(w) , where 1 p + 1 p ′ = 1, i.e. (p − 1)p ′ = p, p p ′ = p − 1. for j2, note |∇vw| ≤ w, by lemma 3.4, we have j2 . ‖∇xφ k‖l∞ x ‖∂αx ∂ β v f k+1‖ p l p x,v(w) . ∑ |i|≤2 ‖∂ixf k‖lpx,v(w)‖∂ α x ∂ β v f k+1‖ p l p x,v(w) . for j3, we consider two cases individually. case 1: recall |α| ≤ m − 1, if 0 < |α1| ≤ m − 2, m ≥ 3, lemma 3.4 leads to ‖∂α1x ∇xφ k‖l∞ x . ∑ |i|≤2 ‖∂i+α1x f k‖lpx,v(w). note |i| + |α1| ≤ m − 2 + 2 = m, the order of the derivatives does not exceed m, then we obtain, j3 . ∑ 0<|α1|≤m−2 ∫ rn ‖∂α1x ∇xφ k‖l∞ x ‖∂α−α1x ∂ β v ∇vf k+1‖lpx ∥ ∥ ∥|∂ α x ∂ β v f k+1|p−1 ∥ ∥ ∥ l p ′ x w dv . ∑ 0<|α1|≤m−2 ∑ |i|≤2 ‖∂i+α1x f k‖lpx,v(w)‖∂ α−α1 x ∂ β v ∇vf k+1‖lpx,v(w)‖∂ α x ∂ β v f k+1‖ p−1 l p x,v(w) , where |α − α1| + |β| + 1 ≤ |α| + |β| ≤ m. case 2: |α1| = m − 1, we have j3 . ∑ |α1|=m−1 ∫ rn w(v)‖∂α1x ∇xφ k‖lqx‖∂ α−α1 x ∂ β v ∇vf k+1‖ln x ∥ ∥ ∥|∂ α x ∂ β v f k+1|p−1 ∥ ∥ ∥ l p ′ x dv . ∑ |α1|=m−1 ‖∂α1x ∇xφ k‖lqx ∫ rn w(v) ∑ |i|≤m−2 ‖∂ix∂ α−α1 x ∂ β v ∇vf k+1‖lpx ∥ ∥ ∥|∂ α x ∂ β v f k+1|p−1 ∥ ∥ ∥ l p ′ x dv . ∑ |α1|=m−1 ‖∂α1x f k‖lpx,v(w) ∑ |i|≤m−2 ‖∂ix∂ α−α1 x ∂ β v ∇vf k+1‖lpx,v(w)‖∂ α x ∂ β v f k+1‖ p−1 l p x,v(w) , where in the first inequality, we applied hölder’s inequality with respect to x with 1 q + 1 n + 1 p ′ = 1, 1 p + 1 p ′ = 1. and in the second inequality, we used the embedding w m−2,p →֒ ln, with m > n p + 1, p ≤ n. (4.6) cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 219 in the third inequality, we applied corollary 3.3 and hölder’s inequality in v. finally, plugging all the estimates of j1, j2, and j3 into (4.5) yields that d dt ‖∂αx ∂ β v f k+1‖ p l p x,v(w) . ∑ β1<β ‖∂β1v ∇x∂ α x f k+1‖lpx,v(w) · ‖∂ α x ∂ β v f k+1‖ p−1 l p x,v(w) + ∑ |i|≤2 ‖∂ixf k‖lpx,v(w)‖∂ α x ∂ β v f k+1‖ p l p x,v(w) + ∑ 0<|α1|≤m−2 ∑ |i|≤2 ‖∂i+α1x f k‖lpx,v(w)‖∂ α−α1 x ∂ β v ∇vf k+1‖lpx,v(w)‖∂ α x ∂ β v f k+1‖ p−1 l p x,v(w) + ∑ |α1|=m−1 ∑ |i|≤m−2 ‖∂α1x f k‖lpx,v(w)‖∂ i x∂ α−α1 x ∂ β v ∇vf k+1‖lpx,v(w)‖∂ α x ∂ β v f k+1‖ p−1 l p x,v(w) . (4.7) step 2. β = 0, |α| ≤ m, applying ∂αx to (4.1) on both sides, we have (∂t + v · ∇x + ∇xφ k · ∇v)∂ α x f k+1 = − ∑ 06=α1≤α cα1α ∂ α1 x ∇xφ k · ∂α−α1x ∇vf k+1. (4.8) we could completely repeat the process of step 1, the only difference is that we do not need to estimate j1, thus we give the estimates as below but omit the process of proof in details. d dt ‖∂αx f k+1‖ p l p x,v(w) . ∑ |i|≤2 ‖∂ixf k‖lpx,v(w)‖∂ α x f k+1‖ p l p x,v(w) + ∑ 0<|α1|≤m−2 ∑ |i|≤2 ‖∂i+α1x f k‖lpx,v(w)‖∂ α−α1 x ∇vf k+1‖lpx,v(w)‖∂ α x f k+1‖ p−1 l p x,v(w) + ∑ m−1≤|α1|≤m ∑ |i|≤m−2 ‖∂α1x f k‖lpx,v(w)‖∂ i x∂ α−α1 x ∇vf k+1‖lpx,v(w)‖∂ α x f k+1‖ p−1 l p x,v(w) . (4.9) collecting the estimates of j1, j2 and j3 and integrating over [0, t] of (4.5), summing over |α| + |β| ≤ m, we deduce from the definition of e(f(t)) that e(fk+1(t)) ≤ e(f0) + ct sup 0≤s≤t e(fk+1(s)) + ct sup 0≤s≤t (e(fk(s))) 1 p · sup 0≤s≤t e(fk+1(s)). inductively, assume sup 0≤s≤t ∗(m) e(fk(s)) ≤ m, t ∗(m) and m are sufficiently small; note that f0(t, x, v) ≡ f0(x, v), e(f0) ≤ m 2 , we have e(fk+1(t)) ≤ m 2 + ct sup 0≤s≤t e(fk+1(s)) + cm 1 p · t sup 0≤s≤t e(fk+1(s)), i.e. (1 − ct ∗ − cm 1 p t ∗(m)) sup 0≤s≤t ∗(m) e(fk+1(s)) ≤ m 2 . thus sup k sup 0≤s≤t ∗(m) e(fk(s)) ≤ m, i.e. we get a uniform-in-k estimate. as a routine, let k → ∞, we obtain the solution and complete the proof of existence. 220 cong he & jingchun chen cubo 24, 2 (2022) remark 4.1. we summarize the indices as follows:                                                      2 q1 = 1 p , 1 1, qi > 1, i = 1, 2, q > n 2 , m > n p + 1, m, n ∈ n, 1 q + 1 p1 = 1 p , γ > n(p − 1). (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) in fact, for any given (m, n, p) satisfying           m > n p + 1, m ∈ n, n ≥ 3, n 3 < p < n 2 , (4.17) we could designate          q1 = 2p, q2 = 2np n−2p , q = np n−1 . (4.18) let us move on to proving the uniqueness. part ii: proof of uniqueness. the proof of the uniqueness is analogous to the existence part. however, we use a different energy norm e1(f(t)) =: ∑ |α|+|β|≤m−1 ‖∂αx ∂ β v f(t)‖ p l p x,v(w) because of a difficult term j̃4. in j̃4, there is a term 〈 ∇x(φf − φg) · ∂ α x ∂ β v ∇vg, |∂ α x ∂ β v (f − g)| p−2 · ∂αx ∂ β v (f − g)w 〉 . if we still work with e(f(t)) = ∑ |α|+|β|≤m ‖∂αx ∂ β v f(t)‖ p l p x,v(w) as in the existence part, the order of derivative of ∂αx ∂ β v ∇vg will be m + 1 which exceeds m when |α| + |β| = m. this is the main reason we choose e1(f(t)) instead of e(f(t)). cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 221 proof. assume another solution g exists such that sup 0≤s≤t ∗ e(g(s)) ≤ m, taking the difference, we have                (∂t + v · ∇x + ∇xφf · ∇v)(f − g) + (∇xφf − ∇xφg) · ∇vg = 0, −∆x(φf − φg) = ∫ rn (f − g) dv, f(0, x, v) = g(0, x, v). (4.19) step 1. applying ∂αx ∂ β v on both sides of (4.19)1 with β 6= 0, |α| + |β| ≤ m − 1, we have (∂t + v · ∇x + ∇xφf · ∇v) · ∂ α x ∂ β v (f − g) + ∑ β1<β c β1 β ∂β−β1v v · ∂ β1 v ∇x∂ α x (f − g) = − ∑ 06=α1≤α cα1α ∂ α1 x ∇xφf · ∂ α−α1 x ∂ β v ∇v(f − g) − ∑ 0≤α1≤α c α1 α ∂ α1 x (∇xφf − ∇xφg) · ∂ α−α1 x ∂ β v ∇vg. (4.20) multiplying |∂αx ∂ β v (f −g)| p−2·∂αx ∂ β v (f −g)w on both sides of (4.20), and then integrating over rnx × r n v yields that 1 p d dt ‖∂αx ∂ β v (f − g)‖ p l p x,v(w) + ∑ β1<β c β1 β 〈 ∂β−β1v v · ∂ β1 v ∇x∂ α x (f − g), |∂ α x ∂ β v (f − g)| p−2 · ∂αx ∂ β v (f − g)w 〉 ︸︷︷︸ j̃1 = 〈 ∇xφf · |∂ α x ∂ β v (f − g)| p, ∇vw 〉 ︸︷︷︸ j̃2 − ∑ 06=α1≤α cα1α 〈 ∂α1x ∇xφf · ∂ α−α1 x ∂ β v ∇v(f − g), |∂ α x ∂ β v (f − g)| p−2 · ∂αx ∂ β v (f − g)w 〉 ︸︷︷︸ j̃3 − ∑ 0≤α1≤α c α1 α 〈 ∂ α1 x (∇xφf − ∇xφg) · ∂ α−α1 x ∂ β v ∇vg, |∂ α x ∂ β v (f − g)| p−2 · ∂αx ∂ β v (f − g)w 〉 ︸︷︷︸ j̃4 . (4.21) we could repeat the estimates in the proof of the existence except for some special term. thus we would like to write down the estimates directly without the details. for j̃1, we have j̃1 . ∑ β1<β ‖∂β1v ∇x∂ α x (f − g)‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) . for j̃2, we get j̃2 . ∑ |i|≤2 ‖∂ixf‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p l p x,v(w) . 222 cong he & jingchun chen cubo 24, 2 (2022) for j̃3, since 0 < |α1| ≤ m − 2, we have j̃3 . ∑ 0<|α1|≤m−2 ∑ |i|≤2 ‖∂i+α1x f‖lpx,v(w) · ‖∂ α−α1 x ∂ β v ∇v(f − g)‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) , where |α − α1| + |β| + 1 ≤ |α| + |β| ≤ m − 1. for j̃4, note that −∆x(φf − φg) = ∫ rn (f − g) dv. we consider two cases separately. case 1: 0 ≤ |α1| ≤ m − 3 j̃4 . ∑ 0≤|α1|≤m−3 ∑ |i|≤2 ‖∂i+α1x (f − g)‖lpx,v(w) · ‖∂ α−α1 x ∂ β v ∇vg‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) , where |i| + |α1| ≤ 2 + m − 3 = m − 1 and |α − α1| + |β| + 1 ≤ |α| + |β| − |α1| + 1 ≤ m − 1 − |α1| + 1 ≤ m. case 2: |α1| = m − 2 j̃4 . ∑ |α1|=m−2 ∑ |i|≤m−2 ‖∂α1x (f − g)‖lpx,v(w) · ‖∂ i x∂ α−α1 x ∂ β v ∇vg‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) , where |i| + |α − α1| + |β| + 1 ≤ m − 2 + |α| − |α1| + |β| + 1 ≤ m. collecting all the estimates of j̃j, j = 1, 2, 3, 4, we have d dt ‖∂αx ∂ β v (f − g)‖ p l p x,v(w) . ∑ β1<β ‖∂β1v ∇x∂ α x (f − g)‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) + ∑ |i|≤2 ‖∂ixf‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p l p x,v(w) + ∑ 0<|α1|≤m−2 ∑ |i|≤2 ‖∂i+α1x f‖lpx,v(w) · ‖∂ α−α1 x ∂ β v ∇v(f − g)‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) + ∑ 0≤|α1|≤m−3 ∑ |i|≤2 ‖∂i+α1x (f − g)‖lpx,v(w) · ‖∂ α−α1 x ∂ β v ∇vg‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) + ∑ |α1|=m−2 ∑ |i|≤m−2 ‖∂α1x (f − g)‖lpx,v(w) · ‖∂ i x∂ α−α1 x ∂ β v ∇vg‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) . (4.22) step 2. β = 0, |α| ≤ m − 1, applying ∂αx on both sides of (4.19)1 yields (∂t + v · ∇x + ∇xφf · ∇v)∂ α x (f − g) = − ∑ 06=α1≤α cα1α ∂ α1 x ∇xφf · ∂ α−α1 x ∇v(f − g) − ∑ 0≤α1≤α cα1α ∂ α1 x (∇xφf − ∇xφg) · ∂ α−α1 x ∇vg. (4.23) cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 223 repeating the process of step 1, we get d dt ‖∂αx (f − g)‖ p l p x,v(w) . ∑ |i|≤2 ‖∂ixf‖lpx,v(w) · ‖∂ α x (f − g)‖ p l p x,v(w) + ∑ 0<|α1|≤m−2 ∑ |i|≤2 ‖∂i+α1x f‖lpx,v(w) · ‖∂ α−α1 x ∇v(f − g)‖lpx,v(w) · ‖∂ α x (f − g)‖ p−1 l p x,v(w) + ∑ |α1|=m−1 ∑ |i|≤m−2 ‖∂α1x f‖lpx,v(w) · ‖∂ i x∂ α−α1 x ∇v(f − g)‖lpx,v(w) · ‖∂ α x (f − g)‖ p−1 l p x,v(w) + ∑ 0<|α1|≤m−3 ∑ |i|≤2 ‖∂i+α1x (f − g)‖lpx,v(w) · ‖∂ α−α1 x ∇vg‖lpx,v(w) · ‖∂ α x ∂ β v (f − g)‖ p−1 l p x,v(w) + ∑ m−2≤|α1|≤m−1 ∑ |i|≤m−2 ‖∂α1x (f − g)‖lpx,v(w) · ‖∂ i x∂ α−α1 x ∇vg‖lpx,v(w) · ‖∂ α x (f − g)‖ p−1 l p x,v(w) . (4.24) note f(0, x, v) = g(0, x, v), sup 0≤s≤t ‖∂i+α1x f(s)‖lpx,v(w) ≤ m, sup 0≤s≤t ‖∂α−α1x ∂ β v ∇vg(s)‖lpx,v(w) ≤ m, and sup 0≤s≤t ‖∂ix∂ α−α1 x ∂ β v ∇vg(s)‖lpx,v(w) ≤ m, sup 0≤s≤t ‖∂ixf(s)‖lpx,v(w) ≤ m. integrating (4.22) and (4.24) over [0, t], then summing over |α|+|β| ≤ m−1, we deduce e1((f − g)(t)) . (1 + m) ∫ t 0 e1((f − g)(s)) ds, where e1(f(t)) =: ∑ |α|+|β|≤m−1 ‖∂αx ∂ β v f(t)‖ p l p x,v(w) . by gronwall’s inequality, we have e1((f − g)(t)) ≡ 0 implying f ≡ g, which completes the proof of uniqueness. thus we end the proof of theorem 2.1. remark 4.2. all in all, we improved the results in [4] to the more general function space w m,p(rn) which does not have to be c2(rn) (too strong). our results also shed light on exploring solutions in sobolev spaces. we are very confident that our method could be applied in fractional sobolev spaces, even the supercritical spaces which are far from being understood yet. 5 appendix for the sake of completeness, we cite some known results about the estimate for the riesz potential. first of all, we give the pointwise estimate of the riesz potential, for more details, see chapter 3, section 1, page 57 in [1]. 224 cong he & jingchun chen cubo 24, 2 (2022) proposition 5.1 ([1]). for any multi-index ξ with |ξ| < α < n, there is a constant a such that for any f ∈ lp(rn), 1 ≤ p < ∞, and almost every x, we have |dξ(iα ∗ f(x))| ≤ amf(x) |ξ| α · (iα ∗ |f|(x)) 1− |ξ| α , (5.1) where iα = γα |x|n−α , γα = γ(n − α 2 ) π n 2 2αγ(α 2 ) . remark 5.2. in our paper, we consider −∆φ = ∫ rn fdv =: g, n ≥ 3. thus, in our context, iα can be taken i2(x) = 1 (n − 2)ωn−1 · 1 |x|n−2 , i.e. α = 2, (5.2) where ωn−1 = 2π n 2 γ(n 2 ) is the (n − 1)−dimensional area of the unit sphere in rn, then we have |dξ(i2 ∗ g(x))| ≤ cmg(x) |ξ| 2 · (i2 ∗ |g|(x)) 1− |ξ| 2 . (5.3) next, we give the off-diagonal estimate of the riesz potential i2. for the details, see chapter v, section 1 and page 119 in [14]. lemma 5.3 ([14]). if −∆φ = g ∈ lp(rn), then φ = i2 ∗ g and ‖i2 ∗ g‖lq̃(rn) ≤ c‖g‖lp(rn), (5.4) where 1 < p < n 2 , c = c(p, q̃) and 1 q̃ = 1 p − 2 n . (5.5) cubo 24, 2 (2022) vlasov-poisson equation in weighted sobolev space w m,p(w) 225 references [1] d. r. adams and l. i. hedberg, function spaces and potential theory, grundlehren der mathematischen wissenschaften, berlin: springer-verlag, 1996. [2] r. alexandre, y. morimoto, s. ukai, c.-j. xu and t. yong, “regularizing effect and local existence for the non-cutoff boltzmann equation”, arch. ration. mech. anal., vol. 198, no. 1, pp. 39–123, 2010. [3] a. a. arsen’ev, “global existence of a weak solution of vlasov’s system of equations”, ussr comp. math. math. phys., vol. 15, no. 1, pp. 131–143, 1975. 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[9] e. horst, “on the classical solutions of the initial value problem for the unmodified non-linear vlasov equation (parts i and ii)”, math. methods appl. sci. vol. 3, no. 2, 1981, pp. 229–248 and vol. 4, no. 1, pp. 19–32, 1982. [10] e. horst, “on the asymptotic growth of the solutions of the vlasov-poisson system”, math. methods appl. sci., vol. 16, no. 2, pp. 75–85, 1993. [11] p.-l. lions and b. perthame, “propagation of moments and regularity for the 3-dimensional vlasov-poisson system”, invent. math., vol. 105, no. 2, pp. 415–430, 1991. [12] k. pfaffelmoser, “global classical solutions of the vlasov-poisson system in three dimensions for general initial data”, j. differential equations, vol. 95, no. 2, pp. 281–303, 1992. [13] j. schaeffer, “global existence of smooth solutions to the vlasov-poisson system in three dimensions”, comm. partial differential equations, vol. 16, pp. 1313–1335, 1991. 226 cong he & jingchun chen cubo 24, 2 (2022) [14] e. m. stein, singular integrals and differentiability properties of functions, princeton mathematical series, no. 30, princeton, n. j.: princeton university press, 1970. introduction preliminaries and main theorem notations and definitions lp-lq estimates proof of main theorem appendix cubo, a mathematical journal vol. 24, no. 01, pp. 01–20, april 2022 doi: 10.4067/s0719-06462022000100001 quasi bi-slant submersions in contact geometry rajendra prasad 1 mehmet akif akyol 2 sushil kumar 3 punit kumar singh 1 1department of mathematics and astronomy, university of lucknow, lucknow, india. rp.manpur@rediffmail.com singhpunit1993@gmail.com 2department of mathematics, faculty of sciences and arts, bingöl university, 12000, bingöl, turkey. mehmetakifakyol@bingol.edu.tr 3department of mathematics, shri jai narain post graduate college, lucknow, india. sushilmath20@gmail.com abstract the aim of the paper is to introduce the concept of quasi bislant submersions from almost contact metric manifolds onto riemannian manifolds as a generalization of semi-slant and hemi-slant submersions. we mainly focus on quasi bi-slant submersions from cosymplectic manifolds. we give some non-trivial examples and study the geometry of leaves of distributions which are involved in the definition of the submersion. moreover, we find some conditions for such submersions to be integrable and totally geodesic. resumen el objetivo de este art́ıculo es introducir el concepto de submersiones cuasi bi-inclinadas desde variedades casi contacto métricas hacia variedades riemannianas, como una generalización de submersiones semi-inclinadas y hemi-inclinadas. principalmente nos enfocamos en submersiones cuasi biinclinadas desde variedades cosimplécticas. damos algunos ejemplos no triviales y estudiamos la geometŕıa de hojas de distribuciones que están involucradas en la definición de la submersión. más aún, encontramos algunas condiciones para que estas submersiones sean integrables y totalmente geodésicas. keywords and phrases: riemannian submersion, semi-invariant submersion, bi-slant submersion, quasi bi-slant submersion, horizontal distribution. 2020 ams mathematics subject classification: 53c15, 53c43, 53b20. accepted: 04 september, 2021 received: 02 january, 2021 c©2022 r. prasad et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100001 https://orcid.org/0000-0002-7502-0239 https://orcid.org/0000-0003-2334-6955 https://orcid.org/0000-0003-2118-4374 https://orcid.org/0000-0002-8700-5976 mailto:rp.manpur@rediffmail.com mailto:singhpunit1993@gmail.com mailto:mehmetakifakyol@bingol.edu.tr mailto:sushilmath20@gmail.com 2 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) 1 introductions in differential geometry, there are so many important applications of immersions and submersions both in mathematics and in physics. the properties of slant submersions became an interesting subject in differential geometry, both in complex geometry and in contact geometry. in 1966 and 1967, the theory of riemannian submersions was initiated by o’neill [17] and gray [11] respectively. nowadays, riemannian submersions are of great interest not only in mathematics, but also in theoretical pyhsics, owing to their applications in the yang-mills theory, kaluza-klein theory, supergravity and superstring theories (see [7, 8, 10, 13, 14] ). in 1976, the almost complex type of riemannian submersions was studied by watson [29]. he also introduced almost hermitian submersions between almost hermitian manifolds requiring that such riemannian submersions are almost complex maps. in 1985, d. chinea [9] extended the notion of almost hermitian submersion to several kinds of sub-classes of almost contact manifolds. in [4] and [5], there are so many important and interesting results about riemannian and almost hermitian submersions. in 2010, b. şahin introduced anti invariant submersions from almost hermitian manifolds onto riemannian manifolds [25]. inspired by b. şahin’s article, many geometers introduced several new types of riemannian submersions in different ambient spaces such as semi-invariant submersion [21, 23], generic submersion [27], slant submersion [12, 22], hemi-slant submersion [28], semi-slant submersion [18], bi-slant submersion [26], quasi hemi-slant submersion [16], quasi bi-slant submersion [19, 20], conformal anti-invariant submersion [1], conformal slant submersion [2] and conformal semi-slant submersion [3, 15]. also, these kinds of submersions were considered in different kinds of structures such as cosymplectic, sasakian, kenmotsu, nearly kaehler, almost product, paracontact, etc. recent developments in the theory of submersions can be found in the book [24]. inspired from the good and interesting results of above studies, we introduce the notion of quasi bi-slant submersions from cosymplectic manifolds onto riemannian manifolds. the paper is organized as follows: in the second section, we gather some basic definitions related to quasi bi-slant riemannian submersion. in the third section, we obtain some results on quasi bislant riemannian submersions from a cosymplectic manifold onto a riemannian manifold. we also study the geometry of the leaves of the distributions involved in the considered submersions and discuss their totally geodesicity. we obtain conditions for the fibres or the horizontal distribution to be totally geodesic. in the last section, we provide some examples for such submersions. cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 3 2 preliminaries an n−dimensional smooth manifold m is said to have an almost contact structure, if there exist on m, a tensor field φ of type (1,1), a vector field ξ and 1−form η such that: φ2 = −i + η ⊗ ξ, φξ = 0, η ◦ φ = 0, (2.1) η(ξ) = 1. (2.2) there exists a riemannian metric g on an almost contact manifold m satisfying the next conditions: g(φu,φv ) = g(u,v ) − η(u)η(v ), (2.3) g(u,ξ) = η(u), (2.4) where u,v are vector fields on m. an almost contact structure (φ,ξ,η) is said to be normal if the almost complex structure j on the product manifold m × r is given by j ( u, α d dt ) = ( φu − αξ, η(u) d dt ) (2.5) and α is the differentiable function on m × r has no torsion, i.e., j is integrable. the condition for normality in terms of φ, ξ, and η is [φ,φ] + 2dη ⊗ ξ = 0 on m, where [φ,φ] is the nijenhuis tensor of φ. finally, the fundamental 2−form φ is defined by φ(u,v ) = g(u,φv ). an almost contact metric manifold with almost contact structure (φ,ξ,η,g) is said to be cosymplectic if (∇uφ)v = 0, (2.6) for any u,v on m. it is both normal and closed and the structure equation of a cosymplectic manifold is given by ∇uξ = 0, (2.7) for any u on m, where ∇ denotes the riemannian connection of the metric g on m. example 2.1 ([6]). r2n+1 with cartesian coordinates (xi,yi,z)(i = 1, . . . ,n) and its usual contact form η = dz. the characteristic vector field ξ is given by ∂ ∂z and its riemannian metric g and tensor field φ are given by g = n ∑ i=1 ((dxi) 2 + (dyi) 2) + (dz)2, φ =     0 δij 0 −δij 0 0 0 0 0     , i = 1, . . . ,n. 4 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) this gives a cosymplectic manifold on r2n+1. the vector fields ei = ∂ ∂yi , en+i = ∂ ∂xi , ξ form a φ-basis for the cosymplectic structure. before giving our definition, we recall the following definition: definition 2.2 ([28]). let m be an almost hermitian manifold with hermitian metric gm and almost complex structure j, and let n be a riemannian manifold with riemannian metric gn. a riemannian submersion f : (m,gm,j) → (n,gn) is called a hemi-slant submersion if the vertical distribution kerf∗ of f admits two orthogonal complementary distributions d θ and d⊥ such that dθ is slant with angle θ and d⊥ is anti-invariant, i.e, we have kerf∗ = d θ ⊕ d⊥. in this case, the angle θ is called the hemi-slant angle of the submersion. definition 2.3. let (m,φ,ξ,η,gm ) be an almost contact metric manifold and (n,gn) a riemannian manifold. a riemannian submersion f : (m,φ,ξ,η,gm ) → (n,gn), is called a quasi bi-slant submersion if there exist four mutually orthogonal distributions d,d1,d2 and < ξ > such that (i) kerf∗ = d ⊕ d1 ⊕ d2⊕ < ξ >, (ii) φ(d) = d i.e., d is invariant, (iii) φ(d1) ⊥ d2 and φ(d2) ⊥ d1, (iv) for any non-zero vector field u ∈ (d1)p, p ∈ m, the angle θ1 between φu and (d1)p is constant and independent of the choice of the point p and u in (d1)p, (v) for any non-zero vector field u ∈ (d2)q, q ∈ m, the angle θ2 between φu and (d2)q is constant and independent of the choice of point q and u in (d2)q, these angles θ1 and θ2 are called the slant angles of the submersion. we easily observe that (a) if dimd 6= 0, dim d1 = 0 and dimd2 = 0, then f is an invariant submersion. (b) if dimd 6= 0, dimd1 6= 0, 0 < θ1 < π2 and dim d2 = 0, then f is proper semi-slant submersion. (c) if dimd = 0, dimd1 6= 0, 0 < θ1 < π2 and dim d2 = 0, then f is slant submersion with slant angle θ1. cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 5 (d) if dimd = 0, dimd1 = 0 and dimd2 6= 0, 0 < θ2 < π2 , then f is slant submersion with slant angle θ2. (e) if dimd = 0, dim d1 6= 0, θ1 = π2 and dim d2 = 0, then f is an anti-invariant submersion. (f) if dimd 6= 0, dim d1 6= 0, θ1 = π2 and dim d2 = 0, then f is an semi-invariant submersion. (g) if dimd = 0, dimd1 6= 0, 0 < θ1 < π2 and dimd2 6= 0, θ2 = π 2 , then f is a hemi-slant submersion. (h) if dimd = 0, dimd1 6= 0, 0 < θ1 < π2 and dimd2 6= 0, 0 < θ2 < π 2 , then f is a bi-slant submersion. (i) if dimd 6= 0, dimd1 6= 0, 0 < θ1 < π2 and dimd2 6= 0, θ2 = π 2 , then we may call f is an quasi-hemi-slant submersion. (j) if dimd 6= 0, dimd1 6= 0, 0 < θ1 < π2 and dimd2 6= 0, 0 < θ2 < π 2 , then f is proper quasi bi-slant submersion. define o’neill’s tensors t and a by aef = h∇hevf + v∇hehf, (2.8) tef = h∇vevf + v∇vehf, (2.9) for any vector fields e,f on m, where ∇ is the levi-civita connection of gm. it is easy to see that te and ae are skew-symmetric operators on the tangent bundle of m reversing the vertical and the horizontal distributions. from equations (2.8) and (2.9) we have ∇uv = tuv + v∇uv, (2.10) ∇ux = tux + h∇ux, (2.11) ∇xu = axu + v∇xu, (2.12) ∇xy = h∇xy + axy, (2.13) for u,v ∈ γ(kerf∗) and x,y ∈ γ(kerf∗)⊥, where h∇uy = ay u, if y is basic. it is not difficult to observe that t acts on the fibers as the second fundamental form, while a acts on the horizontal distribution and measures the obstruction to the integrability of this distribution. it is seen that for q ∈ m, u ∈ vq and x ∈ hq the linear operators ax, tu : tqm → tqm 6 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) are skew-symmetric, that is gm(axe,f) = −gm(e,axf) and gm (tue,f) = −gm(e,tuf) (2.14) for each e,f ∈ tqm. since tu is skew-symmetric, we observe that f has totally geodesic fibres if and only if t ≡ 0. let (m,φ,ξ,η,gm ) be a cosymplectic manifold, (n,gn) be a riemannian manifold and f : m → n a smooth map. then the second fundamental form of f is given by (∇f∗)(y,z) = ∇fy f∗z − f∗(∇y z), for y,z ∈ γ(tpm), (2.15) where we denote conveniently by ∇ the levi-civita connections of the metrics gm and gn and ∇f is the pullback connection. we recall that a differentiable map f between two riemannian manifolds is totally geodesic if (∇f∗)(y,z) = 0, for all y,z ∈ γ(tm). a totally geodesic map is that it maps every geodesic in the total space into a geodesic in the base space in proportion to arc lengths. 3 quasi bi-slant submersions let f be quasi bi-slant submersion from an almost contact metric manifold (m,φ,ξ,η,gm ) onto a riemannian manifold (n,gn). then, we have tm = kerf∗ ⊕ (kerf∗)⊥. (3.1) now, for any vector field u ∈ γ(kerf∗), we put u = pu + qu + ru + η(u)ξ, (3.2) where p , q and r are projection morphisms of kerf∗ onto d, d1 and d2, respectively. for any u ∈ γ(kerf∗), we set φu = ψu + ωu, (3.3) where ψu ∈ γ(kerf∗) and ωu ∈ γ(kerf∗)⊥. now, let u1, u2 and u3 be vector fields in d, d1 and d2 respectively. since d is invariant, i.e. φd = d, we get ωu1 = 0. for any u2 ∈ γ(d1) we get ωu2 ∈ γ(ωd1) and for any u3 ∈ γ(d2) we get ωu3 ∈ γ(ωd2), hence ωu2 ⊕ ωu3 ∈ γ(ωd1 ⊕ ωd2) ⊆ γ(kerf∗)⊥. from equations (3.2) and (3.3), we have φu = φ(pu) + φ(qu) + φ(ru), = ψ(pu) + ω(pu) + ψ(qu) + ω(qu) + ψ(ru) + ω(ru). cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 7 since φd = d, we get ωpu = 0. hence above equation reduces to φu = ψpu + ψqu + ωqu + ψru + ωru. (3.4) thus we have the following decomposition according to equation (3.4) φ(kerf∗) = (ψd) ⊕ (ψd1 ⊕ ψd2) ⊕ (ωd1 ⊕ ωd2), (3.5) where ⊕ denotes orthogonal direct sum. further, let u ∈ γ(d1) and v ∈ γ(d2). then gm (u,v ) = 0. from definition 2.3 (iii), we have gm (φu,v ) = gm(u,φv ) = 0. now, consider gm (ψu,v ) = gm (φu − ωu,v ) = gm(φu,v ) = 0. similarly, we have gm(u,ψv ) = 0. let w ∈ γ(d) and u ∈ γ(d1). then we have gm (ψu,w) = gm (φu − ωu,w) = gm (φu,w) = −g(u,φw) = 0, as d is invariant, i.e., φw ∈ γ(d). similarly, for w ∈ γ(d) and v ∈ γ(d2), we obtain gm(ψv,w) = 0, from above equations, we have gm (ψu,ψv ) = 0, and gm(ωu,ωv ) = 0, for all u ∈ γ(d1) and v ∈ γ(d2). so, we can write ψd1 ∩ ψd2 = {0}, ωd1 ∩ ωd2 = {0}. if θ2 = π 2 , then ψr = 0 and d2 is anti-invariant, i.e., φ(d2) ⊆ (kerf∗)⊥. 8 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) we also have φ(kerf∗) = ψd ⊕ ψd1 ⊕ ωd1 ⊕ ωd2. (3.6) since ωd1 ⊆ (kerf∗)⊥, ωd2 ⊆ (kerf∗)⊥. so we can write (kerf∗) ⊥ = ωd1 ⊕ ωd2 ⊕ v, where v is invariant and orthogonal complement of (ωd1 ⊕ ωd2) in (kerf∗)⊥. also for any non-zero vector field w ∈ γ(kerf)⊥, we have φw = bw + cw, (3.7) where bw ∈ γ(kerf) and cw ∈ γ(v). lemma 3.1. let f be a quasi bi-slant submersion from an almost contact metric manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then, we have ψ2u + bωu = −u + η(u)ξ, ωψu + cωu = 0, ωbw + c2w = −w, ψbw + bcw = 0, for all u ∈ γ(kerf∗) and w ∈ γ(kerf∗)⊥. lemma 3.2. let f be a quasi bi-slant submersion from an almost contact metric manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then, we have (i) ψ2u = −(cos2 θ1)u, (ii) gm(ψu,ψv ) = cos 2 θ1gm (u,v ), (iii) gm(ωu,ωv ) = sin 2 θ1gm (u,v ), for all u,v ∈ γ(d1). lemma 3.3. let f be a quasi bi-slant submersion from an contact metric manifold (m,φ,ξ,η,gm ) onto a riemannian manifold (n,gn). then, we have (i) ψ2w = −(cos2 θ2)w, (ii) gm(ψw,ψz) = cos 2 θ2gm(w,z), (iii) gm(ωw,ωz) = sin 2 θ2gm(w,z), for all w,z ∈ γ(d2). cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 9 lemma 3.4. let f be a quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm ) onto a riemannian manifold (n,gn). then, we have v∇uψv + tuωv = ψv∇uv + btuv, (3.8) tuψv + h∇uωv = ωv∇uv + ctuv, (3.9) v∇xby + axcy = ψaxy + bh∇xy, (3.10) axby + h∇xcy = ωaxy + ch∇xy, (3.11) v∇ubx + tucx = ψtux + bh∇ux, (3.12) tubx + h∇ucx = ωtux + ch∇ux, (3.13) v∇y ψu + ay ωu = bay u + ψv∇y u, (3.14) ay ψu + h∇y ωu = cay u + ωv∇y u, (3.15) for any u,v ∈ γ(kerf∗) and x,y ∈ γ(kerf∗)⊥. now, we define (∇uψ)v = v∇uψv − ψv∇uv, (3.16) (∇uω)v = h∇uωv − ωv∇uv, (3.17) (∇xc)y = h∇xcy − ch∇xy, (3.18) (∇xb)y = v∇xby − bh∇xy, (3.19) for any u,v ∈ γ(kerf∗) and x,y ∈ γ(kerf∗)⊥. lemma 3.5. let f be a quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm ) onto a riemannian manifold (n,gn). then, we have (∇uψ)v = btuv − tuωv, (∇uω)v = ctuv − tuψv, (∇xc)y = ωaxy − axby, (∇xb)y = ψaxy − axcy, for any vectors u,v ∈ γ(kerf∗) and x,y ∈ γ(kerf∗)⊥. the proofs of above lemmas follow from straightforward computations, so we omit them. if the tensors ψ and ω are parallel with respect to the linear connection ∇ on m respectively, then btuv = tuωv, and ctuv = tuψv, for any u,v ∈ γ(tm). 10 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) lemma 3.6. let f be a quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm ) onto a riemannian manifold (n,gn). then, we have (i) gm(∇xy,ξ) = 0 for all x,y ∈ γ(d ⊕ d1 ⊕ d2), (ii) gm([x,y ],ξ) = 0 for all x,y ∈ γ(d ⊕ d1 ⊕ d2). proof. let x,y ∈ γ(d ⊕ d1 ⊕ d2), consider ∇x{gm(y,ξ)} = (∇xgm)(y,ξ) + gm (∇xy,ξ) + gm(y,∇xξ). since x and y are orthogonal to ξ ie. gm(∇xy,ξ) = −gm(y,∇xξ), using equation (2.7) and the property that metric tensor is ∇−parallel, we have both results of this lemma. theorem 3.7. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then, the invariant distribution d is integrable if and only if gm(tv ψu − tuψv,ωqw + ωrw) = gm (v∇uψv − v∇v ψu,ψqw + ψrw), (3.20) for u,v ∈ γ(d) and w ∈ γ(d1 ⊕ d2). proof. for u,v ∈ γ(d), and w ∈ γ(d1 ⊕ d2), using equations (2.1)–(2.4), (2.6), (2.7), (2.10), (3.2), (3.3) and lemma 3.6 we have gm([u,v ],w) = gm (∇uφv,φw) + η(w)η(∇uv ) − gm(∇v φu,φw) − η(w)η(∇v u), = gm (∇uψv,φw) − gm(∇v ψu,φw), = gm (tuψv − tv ψu,ωqw + ωrw) − gm(v∇uψv − v∇v ψu,ψqw + ψrw), which completes the proof. theorem 3.8. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then, the slant distribution d1 is integrable if and only if gm(tw ωψz − tzωψw,x) = gm (tw ωz − tzωw,φpx + ψrx) +gm(h∇w ωz − h∇zωw,ωrx), (3.21) for all w,z ∈ γ(d1) and x ∈ γ(d ⊕ d2). cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 11 proof. for all w,z ∈ γ(d1) and x ∈ γ(d ⊕ d2), we have gm([w,z],x) = gm(∇w z,x) − gm (∇zw,x). using equations (2.1)–(2.4), (2.6), (2.7), (2.11), (3.2), (3.3) and lemma 3.2 we have gm([w,z],x) = gm (∇w φz,φx) − gm(∇zφw,φx), = gm (∇w ψz,φx) + gm(∇w ωz,φx) − gm (∇zψw,φx) − gm (∇w ωz,φx), = cos2 θ1gm(∇w z,x) − cos2 θ1gm(∇zw,x) − gm(tw ωψz − tzωψw,x) +gm(h∇w ωz + tw ωz,φpx + ψrx + ωrx) −gm(h∇zωw + tzωw,φpx + ψrx + ωrx). now, we obtain sin2 θ1gm([w,z],x) = gm(tw ωz − tzωw,φpx + ψrx) + gm (h∇w ωz − h∇zωw,ωrx) −gm(tw ωψz − tzωψw,x), which completes the proof. theorem 3.9. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then, the slant distribution d2 is integrable if and only if gm(tuωψv − tv ωψu,y ) = gm(h∇uωv − h∇v ωu,ωqy ) +gm(tuωv − tv ωu,φpy + ψqy ), (3.22) for all u,v ∈ γ(d2) and y ∈ γ(d ⊕ d1). proof. for all u,v ∈ γ(d2) and y ∈ γ(d ⊕ d1), using equations (2.1)–(2.4), (2.6), (2.7), (3.3) and lemma 3.6 we have gm ([u,v ],y ) = gm(∇uψv,φy ) + gm(∇uωv,φy ) − gm(∇v ψu,φy ) − gm(∇v ωu,φy ). from equations (2.9), (3.2) and lemma 3.3 we have gm ([u,v ],y ) = cos 2 θ2gm ([u,v ],y ) + gm(h∇uωv − h∇v ωu,ωqy ) +gm(tuωv − tv ωu,φpy + ψqy ) − gm (tuωψv − tv ωψu,y ). now, we have sin2 θ2gm([u,v ],y ) = gm(tuωv − tv ωu,φpy + ψqy ) − gm(tuωψv − tv ωψu,y ) +gm(h∇uωv − h∇v ωu,ωqy ), which the proof follows from the above equations. 12 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) theorem 3.10. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then the horizontal distribution (kerf∗) ⊥ defines a totally geodesic foliation on m if and only if gm(auv,pw + cos2 θ1qw + cos2 θ2rw) = gm (h∇uv,ωψpw + ωψqw + ωψrw) −gm(aubv + h∇ucv,ωw), (3.23) for all u,v ∈ γ(kerf∗)⊥ and w ∈ γ(kerf∗). proof. for u,v ∈ γ(kerf∗)⊥ and w ∈ γ(kerf∗), we have gm(∇uv,w) = gm(∇uv,pw + qw + rw + η(w)ξ). using equations (2.1)–(2.4), (2.6), (2.7), (2.12), (2.13), (3.2), (3.3), (3.7) and lemmas 3.2 and 3.3 we have gm (∇uv,w) = gm (φ∇uv,φpw) + gm(φ∇uv,φqw) + gm(φ∇uv,φrw), = gm (auv,pw + cos2 θ1qw + cos2 θ2rw) −gm(h∇uv,ωψpw + ωψqw + ωψrw) +gm(aubv + h∇ucv,ωpw + ωqw + ωrw). taking into account ωpw + ωqw + ωrw = ωw and ωpw = 0 in the above, one obtains gm(∇uv,w) = gm(auv,pw + cos2 θ1qw + cos2 θ2rw) −gm(h∇uv,ωψpw + ωψqw + ωψrw) +gm(aubv + h∇ucv,ωw). theorem 3.11. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then the vertical distribution (kerf∗) defines a totally geodesic foliation on m if and only if gm(txpy + cos2 θ1txqy + cos2 θ2txry,u) = gm(h∇xωψpy + h∇xωψqy + h∇xωψry,u) + gm(txωy,bu) + gm(h∇xωy,cu), (3.24) for all x,y ∈ γ(kerf∗) and u ∈ γ(kerf∗)⊥. proof. for all x,y ∈ γ(kerf∗) and u ∈ γ(kerf∗)⊥, by using equations (2.1)–(2.4), (2.6) and (2.7) we have gm(∇xy,u) = gm (∇xφpy,φu) + gm(∇xφqy,φu) + gm(∇xφry,φu). cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 13 taking into account of (2.10), (2.11), (3.2), (3.3), (3.7) and lemmas 3.2 and 3.3 we have gm (∇xy,u) = gm(txpy,u) + cos2 θ1gm(txqy,u) + cos2 θ2gm (txry,u) −gm(h∇xωψpy + h∇xωψqy + h∇xωψry,u) +gm(∇xωpy + ∇xωqy + ∇xωry,φu). since ωpy + ωqy + ωry = ωy and ωpy = 0, we derive gm (∇xy,u) = gm(txpy + cos2 θ1txqy + cos2 θ2txry,u) −gm(h∇xωψpy + h∇xωψqy + h∇xωψry,u) +gm(txωy,bu) + gm(h∇xωy,cu), which completes the proof. from theorems 3.10 and 3.11 we also have the following decomposition results. theorem 3.12. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then, the total space is locally a product manifold of the form mker f∗ × m(ker f∗)⊥, where mker f∗ and m(ker f∗)⊥ are leaves of kerf∗ and (kerf∗) ⊥ respectively if and only if gm(auv,py + cos2 θ1qy + cos2 θ2ry ) = gm (h∇uv,ωψpy + ωψqy + ωψry ) +gm(aubv + h∇ucv,ωy ), and gm(txy + cos2 θ1txqy + cos2 θ2txry,u) = gm (h∇xωψpy + h∇xωψqy + h∇xωψry,u) + gm(txωy,bu) + gm(h∇xωy,cu), for all x,y ∈ γ(kerf∗) and u,v ∈ γ(kerf∗)⊥. theorem 3.13. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then the distribution d defines a totally geodesic foliation if and only if gm(tuφpv,ωqw + ωrw) = −gm(v∇uφpv,ψqw + ψrw), (3.25) and gm (tuφpv,cy ) = −gm(v∇uφpv,by ), (3.26) for all u,v ∈ γ(d),w ∈ γ(d1 ⊕ d2) and y ∈ γ(kerf∗)⊥. 14 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) proof. for all u,v ∈ γ(d), w ∈ γ(d1 ⊕ d2) and y ∈ γ(kerf∗)⊥, using equations (2.1)–(2.4), (2.6), (2.7), (3.2), (3.3) and lemma 3.6 we have gm(∇uv,w) = gm(∇uφv,φw), = gm(∇uφpv,φqw + φrw), = gm(tuφpv,ωqw + ωrw) + gm(v∇uφpv,ψqw + ψrw). now, again using equations (2.10), (3.2), (3.3) and (3.7) we have gm(∇uv,y ) = gm(∇uφv,φy ), = gm(∇uφpv,by + cy ), = gm(v∇uφpv,by ) + gm(tuφpv,cy ), which completes the proof. theorem 3.14. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then the distribution d1 defines a totally geodesic foliation if and only if gm(tw ωψz,u) = gm (tw ωqz,φpu + ψru) + gm(h∇w ωqz,ωru), (3.27) and gm(h∇w ωψz,y ) = gm(h∇w ωz,cy ) + gm (tw ωz,by ), (3.28) for all w,z ∈ γ(d1),u ∈ γ(d ⊕ d2) and y ∈ γ(kerf∗)⊥. proof. for all w,z ∈ γ(d1), u ∈ γ(d ⊕ d2) and y ∈ γ(kerf∗)⊥, using equations (2.1)–(2.4), (2.6), (2.7), (2.11), (3.2), (3.3) and lemma 3.2, we have gm (∇w z,u) = gm(∇w φz,φu) = gm(∇w ψz,φu) + gm (∇w ωz,φu), = cos2 θ1gm(∇w z,u) − gm (tw ωψz,u) +gm(tw ωqz,φpu + ψru) + gm(h∇w ωqz,ωru). now, we obtain sin2 θ1gm(∇w z,u) = −gm(tw ωψz,u) + gm(tw ωqz,φpu + ψru) + gm(h∇w ωqz,ωrz) next, from equations (2.1)–(2.4), (2.6), (2.7), (2.12), (3.3), (3.7) and lemma 3.2, we have gm (∇w z,y ) = gm(∇w φz,φy ), = gm(∇w ψz,φy ) + gm(∇w ωz,φy ), = cos2 θ1gm (∇w z,y ) − gm(h∇w ωψz,y ) +gm(h∇w ωz,cy ) + gm(tw ωz,by ). cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 15 now, we arrive sin2 θ1gm(∇w z,y ) = −gm(h∇w ωψz,y ) + gm(h∇w ωz,cy ) + gm(tw ωz,by ), which completes the proof. theorem 3.15. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then the distribution d2 defines a totally geodesic foliation if and only if gm (tuωψv,w) = gm (tuωqv,φpw + φrw) + gm(h∇uωqv,ωrw), (3.29) and gm(h∇uωψv,y ) = gm (h∇uωv,cy ) + gm(tuωv,by ), (3.30) for all u,v ∈ γ(d2),w ∈ γ(d ⊕ d1) and y ∈ γ(kerf∗)⊥. proof. for all u,v ∈ γ(d2),w ∈ γ(d ⊕ d1) and y ∈ γ(kerf∗)⊥, by using equations (2.1)–(2.4), (2.6), (2.7), (2.10), (3.3) and from lemma 3.2 and lemma 3.6, we have gm(∇uv,w) = gm (∇uψv,φw) + gm(∇uωv,φw), = cos2 θ2gm(∇uv,w) − gm(tuωψv,w) +gm(tuωqv,φpw + ψrw) + gm(h∇uωqv,ωrw). now, we get sin2 θ2gm(∇uv,w) = −gm(tuωψv,w) + gm (tuωqv,φpw + ψrw) + gm (h∇uωqv,ωrw). next, from equations (2.1)–(2.4), (2.6), (2.7), (2.12), (3.2) (3.3), (3.7) and lemma 3.2, we have gm(∇uv,y ) = gm (∇uψv,φy ) + gm(∇uωv,φy ), = cos2 θ2gm(∇uv,y ) − gm(h∇uωψv,y ) +gm(h∇uωv,cy ) + gm(tuωv,by ). now, we obtain sin2 θ1gm(∇uv,y ) = −gm(h∇uωψv,y ) + gm(h∇uωv,cy ) + gm (tuωv,by ), which completes the proof. we recall that a differentiable map f between two riemannian manifolds is totally geodesic if (∇f∗)(y,z) = 0, for all y,z ∈ γ(tm). a totally geodesic map is that it maps every geodesic in the total space into a geodesic in the base space in proportion to arc lengths. 16 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) theorem 3.16. let f be a proper quasi bi-slant submersion from a cosymplectic manifold (m,φ,ξ,η,gm) onto a riemannian manifold (n,gn). then the map f is totally geodesic if and only if gm(h∇uωψqv + h∇uωψrv − cos2 θ1tuqv − cos2 θ2turv,w) = gm (v∇uφpv + tuωqv + tuωrv,bw) + gm (tuφpv + h∇uωqv + h∇uωrv,cw), and gm(h∇w ωψqu + h∇w ωψru − cos2 θ1aw qu − cos2 θ2aw ru,z) = gm (v∇w φpu + aw ωqu + aw ωru,bz) + gm (aw φpu + h∇w ωqu + h∇w ωru,cz), for all u,v ∈ γ(kerf∗) and w,z ∈ γ(kerf∗)⊥. proof. for all u,v ∈ γ(kerf∗) and w,z ∈ γ(kerf∗)⊥, making use of (2.1)–(2.4), (2.6), (2.7), (2.10), (2.11), (3.2), (3.3), (3.7) and from lemma 3.2 and 3.3, we derive gm(∇uv,w) = gm (∇uφv,φw) = gm (∇uφpv,φw) + gm(∇uφqv,φw) + gm(∇uφrv,φw), = gm (∇uφpv,φw) + gm(∇uψqv,φw) + gm (∇uψrv,φw) +gm(∇uωqv,φw) + gm(∇uωrv,φw), = gm (v∇uφpv + tuωqv + tuωrv,w) +gm(tuφpv + h∇uωqv + h∇uωrv,cw) +gm(cos 2 θ1tuqv + cos2 θ2turv − h∇uωψqv − h∇uωψrv,w). next, taking account of (2.1)–(2.4), (2.6), (2.7), (2.10), (2.12), (2.13), (3.2), (3.3), (3.7) and lemmas 3.2 and 3.3, we have gm(∇w u,z) = gm(φ∇w u,φz) = gm(∇w φu,φz), = gm(∇w φpu,φz) + gm(∇w φqu,φz) + gm(∇w φru,φz), = gm(∇w φpu,φz) + gm(∇w ψqu,φz) + gm(∇w ψru,φz) +gm(∇w ωqu,φz) + gm(∇w ωru,φz), = gm(v∇w φpu + aw ωqu + aw ωru,bz) +gm(aw φpu + h∇w ωqu + h∇w ωru,cz) +gm(cos 2 θ1aw qu + cos2 θ2aw ru − h∇w ωψqu − h∇w ωψru,z), which completes the proof. cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 17 4 examples in this section, we are going to give some non-trivial examples. we will use the notation mentioned in example 2.1. example 4.1. define a map π : r15 → r6 π(x1,x2, . . . ,x7,y1,y2, . . . ,y7,z) = (x2 cosθ1 − y3 sinθ1,y2,x4 sinθ2 − y5 cosθ2,x5,x7,y7), which is a quasi bi-slant submersion such that x1 = ∂ ∂x1 , x2 = ∂ ∂y1 , x3 = ∂ ∂x2 sinθ1 + ∂ ∂y3 cosθ1, x4 = ∂ ∂x3 , x5 = ∂ ∂x4 cosθ2 + ∂ ∂y5 sinθ2, x6 = ∂ ∂y4 , x7 = ∂ ∂x6 , x8 = ∂ ∂y6 , x9 = ξ = ∂ ∂z , (kerπ∗) = (d ⊕ d1 ⊕ d2 ⊕ 〈ξ〉) , where d = 〈 x1 = ∂ ∂x1 ,x2 = ∂ ∂y1 ,x7 = ∂ ∂x6 ,x8 = ∂ ∂y6 〉 , d1 = 〈 x3 = ∂ ∂x2 sinθ1 + ∂ ∂y3 cosθ1,x4 = ∂ ∂x3 〉 , d2 = 〈 x5 = ∂ ∂x4 cosθ2 + ∂ ∂y5 sinθ2,x6 = ∂ ∂y4 〉 , 〈ξ〉 = 〈 x9 = ∂ ∂z 〉 , and (kerπ∗) ⊥ = 〈 ∂ ∂x2 cosθ1 − ∂ ∂y3 sinθ1, ∂ ∂y2 , ∂ ∂x4 sinθ2 − ∂ ∂y5 cosθ2, ∂ ∂x5 , ∂ ∂x7 , ∂ ∂y7 〉 , with bi-slant angles θ1 and θ2. thus the above example verifies the lemmas 3.1, 3.2, 3.3 and 3.6. example 4.2. define a map π : r13 → r6 π (x1,x2, . . . ,x6,y1,y2, . . . ,y6,z) = ( x1 − x2√ 2 ,y1, √ 3x4 − x5 2 ,y5,x6,y6 ) , which is a quasi bi-slant submersion such that x1 = 1√ 2 ( ∂ ∂x1 + ∂ ∂x2 ) , x2 = ∂ ∂y2 , x3 = ∂ ∂x3 , x4 = ∂ ∂y3 , x5 = 1 2 ( ∂ ∂x4 + √ 3 ∂ ∂x5 ) , x6 = ∂ ∂y4 , 18 r. prasad, m. a. akyol, s. kumar & p. k. singh cubo 24, 1 (2022) x7 = ξ = ∂ ∂z , (kerπ∗) = (d ⊕ d1 ⊕ d2 ⊕ 〈ξ〉) , where d = 〈 x3 = ∂ ∂x3 ,x4 = ∂ ∂y3 〉 , d1 = 〈 x1 = 1√ 2 ( ∂ ∂x1 + ∂ ∂x2 ) ,x2 = ∂ ∂y2 〉 , d2 = 〈 x5 = 1 2 ( ∂ ∂x4 + √ 3 ∂ ∂x5 ) ,x6 = ∂ ∂y4 〉 , 〈ξ〉 = 〈 x7 = ∂ ∂z 〉 , and (kerπ∗) ⊥ = 〈 ∂ ∂y1 , 1√ 2 ( ∂ ∂x1 + ∂ ∂x2 ) , 1 2 (√ 3 ∂ ∂x4 − ∂ ∂x5 ) , ∂ ∂y5 , ∂ ∂x6 , ∂ ∂y6 〉 , with bi-slant angles θ1 = π 4 and θ2 = π 3 . therefore, the above example verifies the lemmas 3.1, 3.2, 3.3 and 3.6. cubo 24, 1 (2022) quasi bi-slant submersions in contact geometry 19 references [1] m. a. akyol, “conformal anti-invariant submersions from cosymplectic manifolds”, hacet. j. math. stat., vol. 46, no. 2, pp. 177–192, 2017. 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[29] b. watson, “almost hermitian submersions”, j. differential geometry, vol. 11, no. 1, pp. 147–165, 1976. introductions preliminaries quasi bi-slant submersions examples cubo, a mathematical journal vol.22, no¯ 01, (85–124). april 2020 http: // doi. org/ 10. 4067/ s0719-06462020000100085 nonlinear elliptic p(u)− laplacian problem with fourier boundary condition stanislas ouaro 1 and noufou sawadogo 2 laboratoire de mathématiques et informatique (la.m.i) 1 ufr. sciences exactes et appliquées université joseph ki zerbo 03 bp 7021 ouaga 03, ouagadougou, burkina faso 2 ufr. sciences et techniques, université nazi boni, 01 bp 1091 bobo 01, bobo-dioulasso, burkina faso ouaro@yahoo.fr, noufousawadogo858@yahoo.fr abstract we study a nonlinear elliptic p(u)− laplacian problem with fourier boundary conditions and l1− data. the existence and uniqueness results of entropy solutions are established. resumen estudiamos un problema p(u)−laplaciano eĺıptico nolineal con condiciones de borde fourier y datos l1. se establecen resultados de existencia y unicidad de soluciones de entroṕıa. keywords and phrases: variable exponent, p(u)−laplacian, young measure, fourier boundary condition, entropy solution. 2010 ams mathematics subject classification: 35j60, 35d05, 76a05. http://doi.org/10.4067/s0719-06462020000100085 86 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) 1 introduction in this paper, we consider the following nonlinear fourier boundary value problem { b(u) − diva(x, u, ∇u) = f in ω a(x, u, ∇u).η + λu = g on ∂ω, (1.1) where ω ⊆ rn (n ≥ 3) is a bounded open domain with lipschitz boundary ∂ω, η is the outer unit normal vector on ∂ω and λ > 0. the operator diva(x, u, ∇u) is called p(u)-laplacian. it is more complicated than p(x)-laplacian in the term of nonlinearity. a prototype of this operator is div ( |∇u|p(u)−2.∇u ) . the variable exponent p depend both on the space variable x and on the unknown solution u. the problem (1.1) is a generalization of the following nonlinear problem { b(u) − diva(x, ∇u) = f in ω a(x, ∇u).η + λu = g on ∂ω, studied in [15] by nyanquini and ouaro. the authors used an auxiliary result due to le (see [16], theorem 3.1) to prove the existence of the weak solution when f ∈ l∞(ω), g ∈ l∞(∂ω) and by approximation methods they obtained the entropy solution when f ∈ l1(ω), g ∈ l1(∂ω). in the present paper, as the function (x, z, η) 7→ a(x, z, η) is more general than (x, η) 7→ a(x, η), it is difficult to use the sub-supersolution method, as in [16], to get the existence of the weak solution. therefore, we use the technic of pseudo-monotone operators in banach spaces, some a priori estimates and the convergence in term of young measure to obtain the existence of entropy solutions of problem (1.1). indeed, we define an approximation problem, and we prove that this problem has a solution un which converges to u, an entropy solution of problem (1.1). in this paper, we consider the following basic assumptions on the data for the study of the problem (1.1). (a1) f and g are some functions such as f ∈ l 1(ω), g ∈ l1(∂ω) and g 6≡ 0. (a2) b is nondecreasing surjective and continuous function defined on r such that b(0) = 0. problem (1.1) is adapted into a generalized leray-lions framework under the assumption that a : ω × (r × rn) → rn is a carathéodory function with: (a3) a(x, z, 0) = 0 for all z ∈ r, and a.e. x ∈ ω; (a4) ( a(x, z, ξ) − a(x, z, η) ) .(ξ − η) > 0 for all ξ, η ∈ rn, ξ 6= η, as well as the growth and the coercivity assumptions with variable exponent (a5) ∣ ∣a(x, z, ξ) ∣ ∣ p ′(x,z) ≤ c1 ( |ξ|p(x,z) + m(x) ) and cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 87 (a6) a(x, z, ξ).ξ ≥ 1 c2 |ξ|p(x,z). here, c1 and c2 are positive constants and m is a positive function such that m ∈ l 1(ω). p : ω×r → [p−, p+] is a carathéodory function, 1 < p− ≤ p+ < +∞ and p ′(x, z) = p(x, z) p(x, z) − 1 is the conjugate exponent of p(x, z), with p− := ess inf (x,z)∈ω×r p(x, z) and p+ := ess sup (x,z)∈ω×r p(x, z). the study of p(u)-laplacian problem was recently developped by andreianov et al. (see [2]). these authors established the partial existence and uniqueness result to the weak solution in the cases of homogeneous dirichlet boundary condition. the interest of the study of this kind of problem is due to the fact that they can model various phenomena which arise in the study of elastic mechanic (see [6]), electrorheological fluids (see [20]) or image restoration (see [9]). in this paper, we study the existence of the weak solution for approximation problem and we also establish the existence and uniqueness results of the entropy solution when the data are in l1. in this work, we use the sobolev embedding results and the convergence in term of young measure (see [10, 12]). here, we consider a fourier boundary condition which bring some difficulties to treat the term at the boundary. we were inspired by the work of ouaro and tchousso (see [15]), where the authors defined for the first time a new space by taking into account the boundary. for the next part of the paper (section 2), we introduce some preliminary results. in section 3, we study the existence and uniqueness of entropy solution when the data are in l1. 2 preliminary • we will use the so-called truncation function tk(s) := { s if |s| ≤ k ksign0(s) if |s| > k , where sign0(s) :=      1 if s > 0 0 if s = 0 −1 if s < 0. the truncation function possesses the following properties. tk(−s) = −tk(s), |tk(s)| = min{|s|, k}, lim k→+∞ tk(s) = s and lim k→0 1 k tk(s) = sign0(s). 88 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) we also need to truncate vector valued-function with the help of the mapping hm : r n −→ rn, hm(λ) =    λ, if |λ| ≤ m m λ |λ| if |λ| > m, where m > 0. for a lebesgue measurable set a ⊂ ω, χa denotes its characteristic function and meas(a) denotes its lebesgue measure. let u : ω → r be a function and k ∈ r, we write {|u| ≤ k} or [|u| ≤ k] for the set {x ∈ ω : |u(x)| ≤ k}, (respectively, ≥, =, <, >). the function a(., ., .) appearing in (1.1) satisfies a generalized leray-lions assumptions given in introduction. view that, a(., ., .) satisfies (a5) and (a6), we must work in lebesgue and sobolev spaces with variable exponent, that depend on x and on u(x). for the study of problem (1.1), we need the sobolev spaces w 1,π(.)(ω), where π(.) = p(., u(.)). definition 1. let π : ω −→ [1, +∞) be a measurable function for π(.) = p(., u(.)). •lπ(.)(ω) is the space of all measurable function f : ω −→ r such that the modular ρπ(.)(f) := ∫ ω |f|π(x)dx < +∞. if p+ is finite, this space is equipped with the luxembourg norm ||f||lπ(.)(ω) := inf { λ > 0; ρπ(.) ( f λ ) ≤ 1 } . in the sequel, we will use the same notation lπ(.)(ω) for the space (lπ(.)(ω))n of vector-valued functions. •w 1,π(.)(ω) is the space of all functions f ∈ lπ(.)(ω) such that the gradient of f (taken in the sense of distributions) belongs to lπ(.)(ω). the space w 1,π(.)(ω) is equipped with the norm ||u||w 1,π(.)(ω) := ||u||lπ(.)(ω) + ||∇u||lπ(.)(ω). when 1 < p− ≤ π(.) ≤ p+ < +∞, all the above spaces are separable and reflexive banach spaces. we denote πn(x) := p(x, un(x)). proposition 1. (see [1], proposition 2.3) for all measurable function π : ω → [p−, p+], the following properties hold. i) lπ(.)(ω) and w 1,π(.)(ω) are separable and reflexive banach spaces. ii) lπ ′(.)(ω) can be identified with the dual space of lπ(.)(ω), and the following hölder type inequality holds: ∀f ∈ lπ(.)(ω), g ∈ lπ ′(.)(ω), ∣ ∣ ∣ ∣ ∫ ω fgdx ∣ ∣ ∣ ∣ ≤ 2||f||lπ(.)(ω)||g||lπ′(.)(ω). cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 89 iii) one has ρπ(.)(f) = 1 if and only if ||f||lπ(.)(ω) = 1; further, if ρπ(.)(f) ≤ 1, then ||f|| p+ lπ(.)(ω) ≤ ρπ(.)(f) ≤ ||f|| p− lπ(.)(ω) ; if ρπ(.)(f) ≥ 1, then ||f|| p− lπ(.)(ω) ≤ ρπ(.)(f) ≤ ||f|| p+ lπ(.)(ω) . in particular, if (fn)n∈n is a sequence in l π(.)(ω), then ||fn||lπ(.)(ω) tends to zero (resp., to infinity) if and only if ρπ(.)(fn) tends to zero (resp., to infinity), as n → +∞. for a measurable function f ∈ w 1,π(.)(ω) we introduce the following notation: ρ1,π(.)(f) = ∫ ω |f|π(.)dx + ∫ ω |∇f|π(.)dx. replacing p(x) by π(x) in [8]-proposition 2.2, we obtain the following result that is fundamental in this paper. proposition 2. ( see [23, 24] ) if f ∈ w 1,π(.)(ω), the following properties hold: i) ||f||w 1,π(.)(ω) > 1 ⇒ ||f|| p− w 1,π(.)(ω) < ρ1,π(.)(f) < ||f|| p+ w 1,π(.)(ω) ; ii) ||f||w 1,π(.)(ω) < 1 ⇒ ||f|| p+ w 1,π(.)(ω) < ρ1,π(.)(f) < ||f|| p− w 1,π(.)(ω) ; iii) ||f||w 1,π(.)(ω) < 1 (respectively = 1; > 1) ⇔ ρ1,π(.)(f) < 1 (respectively = 1; > 1). the following lemma prove that the space w 1,π(.)(ω) is stable by truncation. lemma 2.1. if u ∈ w 1,π(.)(ω) then tk(u) ∈ w 1,π(.)(ω). now, we give the embedding results. proposition 3. (see [1], proposition 2.4) assume that π : ω → [p−, p+] has a representative which can be extended into a continuous function up to the boundary ∂ω and satisfying the log-hölder continuity assumption: ∃l > 0, ∀x, y ∈ ω, x 6= y, − ( log |x − y| ) |π(x) − π(y)| ≤ l. (2.1) i) then, d(ω) is dense in w 1,π(.)(ω). ii) w 1,π(.)(ω) is embedded into lπ ∗(.)(ω), where π∗(.) is the sobolev embedding exponent defined as in (2.2) below. if q is a measurable variable exponent such that ess inf x∈ω (π∗(.) − q(.)) > 0, then the embedding of w 1,π(.)(ω) into lq(.)(ω) is compact. 90 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) for a given π(.), a function taking values in [p−, p+], π ∗(.) denotes the optimal sobolev embedding defined for any x ∈ ω by π ∗(x) =       nπ(x) n − π(x) if π(x) < n any real value if π(x) = n +∞ if π(x) > n. (2.2) put π∂(x) := ( π(x) )∂ :=    (n − 1)π(x) n − π(x) if π(x) < n +∞ if π(x) ≥ n. (2.3) proposition 4. (see [18], proposition 2.3 ) let π(.) ∈ c(ω) and p− > 1. if q(x) ∈ c(∂ω) satisfies the condition: 1 ≤ q(x) < π∂(x), ∀x ∈ ∂ω, then, there is a compact embedding w 1,π(.)(ω) →֒ lq(.)(∂ω). in particular there is compact embedding w 1,π(.)(ω) →֒ lπ(.)(∂ω). tyoung measures and nonlinear weak-* convergence. throughout the paper, we denote by δc the dirac measure on r d (d ∈ n), concentrated at the point c ∈ rd. in the following theorem, we gather the results of ball [7], pedregal [19] and hungerbühler [13] which will be needed for our purposes (we limit the statement to the case of a bounded domain ω). let us underline that the results of (ii),(iii), expressed in terms of the convergence in measure, are very convenient for the applications that we have in mind. theorem 2.1. (i) let ω ⊂ rn, n ∈ n, and a sequence (vn)n∈n of r d -valued functions, d ∈ n , such that (vn)n∈n is equi-integrable on ω. then, there exists a subsequence (nk)k∈n and a parametrized family (νx)x∈ω of probability measures on r d (d ∈ n) , weakly measurable in x with respect to the lebesgue measure in ω, such that for all carathéodory function f : ω × rd → rt, t ∈ n, we have lim k→+∞ ∫ ω f(x, vnk )dx = ∫ ω ∫ rd f(x, λ)dνx(λ)dx, (2.4) cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 91 whenever the sequence (f(., vn(.)))n∈n is equi-integrable on ω. in particular, v(x) := ∫ rd λdνx(λ) (2.5) is the weak limit of the sequence (vnk )k∈n in l 1(ω). the family (νx)x∈ω is called the young measure generated by the subsequence (vnk )k∈n. (ii) if ω is of finite measure, and (νx)x∈ω is the young measure generated by a sequence (vn)n∈n, then νx = δv(x) for a.e. x ∈ ω ⇔ vn converges in measure in ω to v as n → +∞. (iii) if ω is of finite measure, (un)n∈n generates a dirac young measure (δu(x))x∈ω on r d1, and (vn)n∈n generates a young measure (νx)x∈ω on r d2, then the sequence (un, vn)n∈n generates the young measure (δu(x) ⊗ νx)x∈ω on r d1+d2. whenever a sequence (vn)n∈n generates a young measure (νx)x∈ω, following the terminology of [11] we will say that (vn)n∈n nonlinear weak-* converges, and (νx)x∈ω is the nonlinear weak-* limit of the sequence (vn)n∈n. in the case where (vn)n∈n possesses a nonlinear weak-* convergent subsequence, we will say that it is nonlinear weak-* compact. ([1], theorem 2.10(i)) it means that any equi-integrable sequence of measurable functions is nonlinear weak-* compact on ω. lemma 2.2. (see [1], theorem 3.11 and [2] step 2 of proof of theorem 2.6). assume that (un)n∈n converges a.e. on ω to some function u, then |p(x, un(x)) − p(x, u(x))| converges in measure to 0 on ω, and for all bounded subset k of rn, sup ξ∈k |a(x, un(x), ξ) − a(x, u(x), ξ)| converges in measure to 0 on ω. (2.6) for the sequel, we assume that p(., .) is log hölder continuous uniformly on ω × [−m, m] and p− > n. we recall some notations. for any u ∈ w 1,π(.)(ω), we denote by τ(u) the trace of u on ∂ω in the usual sense. we will identify at boundary u and τ(u). set t 1,π(.)(ω) = { u : ω → r, measurable such that tk(u) ∈ w 1,π(.)(ω), for any k > 0 } . 3 entropy solution in this part, we study the existence and uniqueness of the entropy solution to the problem (1.1). we give some notations. 92 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) we define t 1,π(.) tr (ω) as the set of the functions u ∈ t 1,π(.)(ω) such that there exists a sequence (un)n∈n ⊂ w 1,p+(ω) satisfying the following conditions: (c1) un → u a.e. in ω. (c2) ∇tk(un) → ∇tk(u) in l 1(ω). (c3) there exists a measurable function v on ∂ω, such that un → v a.e. on ∂ω. the function v is the trace of u in the generalized sense as introduced in [4, 5]. in the sequel the trace of u ∈ t 1,π(.) tr (ω) on ∂ω will be denoted tr(u). if u ∈ w 1,π(.)(ω), tr(u) coincides with τ(u) in the usual sense. moreover, for u ∈ t 1,π(.) tr (ω) and for all k > 0, tr(tk(u)) = tk(tr(u)) and if ϕ ∈ w 1,π(.)(ω) then u − ϕ ∈ t 1,π(.) tr (ω) and tr(u − ϕ) = tr(u) − tr(ϕ). as in [1]-proposition 3.5, we give the following result. proposition 5. let u ∈ t 1,π(.)(ω). there exists a unique measurable function w : ω → rn such that ∇tk(u) = wχ{|u| 0. the function w is denoted by ∇u. moreover, if u ∈ w 1,π(.)(ω) then w ∈ lπ(.)(ω) and w = ∇u in the usual sense. remark 3.1. the space t 1,π(.) tr (ω) in our context will be a subset of t 1,π(.)(ω) consisting to the function can be approximated by function of w 1,p+(ω). since the weak solution of approximated problem (3.2) belongs to w 1,p+(ω). now, we introduce the notion of entropy solution due to ouaro and al. [14, definition 3.1]. definition 2. a measurable function u : ω → r for π(.) = p(., u(.)) is called entropy solution of the problem (1.1) if u ∈ t 1,π(.) tr (ω), b(u) ∈ l 1(ω), u ∈ l1(∂ω) and for all k > 0, ∫ ω b(u)tk(u − ϕ)dx + ∫ ω a(x, u, ∇u).∇tk(u − ϕ)dx + λ ∫ ∂ω utk(u − ϕ)dσ ≤ ∫ ω ftk(u − ϕ)dx + ∫ ∂ω gtk(u − ϕ)dσ, (3.1) where ϕ ∈ w 1,π(.)(ω) ∩ l∞(ω). the following theorem gives existence result of entropy solution. theorem 3.2. assume that (a3) − (a6) hold and f ∈ l 1(ω), g ∈ l1(∂ω). then, there exists at least one entropy solution to the problem (1.1). the proof of theorem 3.2 is done in two parts. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 93 part 1: the approximate problem. let fn = tn(f) and gn = tn(g). then, fn ∈ l ∞(ω) and gn ∈ l ∞(∂ω). moreover, (fn)n∈n strongly converges to f in l1(ω) and (gn)n∈n strongly converges to g in l 1(∂ω) such that ||fn||l1(ω) ≤ ||f||l1(ω) and ||gn||l1(∂ω) ≤ ||g||l1(∂ω). we consider the following problem      tn(b(un)) − diva(x, un, ∇un) − ε△p+un + ε|un| p+−2un = fn in ω ( a(x, un, ∇un) + ε|∇un| p+−2∇un ) .η + λtn(un) = gn on ∂ω, (3.2) where −△p+un := − n ∑ i=1 ∂ ∂xi (∣ ∣ ∣ ∣ ∂un ∂xi ∣ ∣ ∣ ∣ p+−2 ∂un ∂xi ) . in this part, we show that the problem (3.2) admits at least one weak solution un, for all ε > 0. we define the following reflexive space e = w 1,p+(ω) × lp+(∂ω). let x0 = {(u, v) ∈ e : v = τ(u)}. in the sequel, we will identify an element (u, v) ∈ x0 with its representative u ∈ w 1,p+(ω) (since w 1,p+(ω) →֒→֒ lp+(∂ω)). theorem 3.3. there exists at least one weak solution un for the problem (3.2) in the sense that un ∈ x0 and for all v ∈ x0, ∫ ω tn(b(un))vdx + ∫ ω a(x, un, ∇un)∇vdx + ∫ ∂ω λtn(un)vdσ + ε ∫ ω [ |un| p+−2unv + |∇un| p+−2∇un∇v ] dx = ∫ ω fnvdx + ∫ ∂ω gnvdσ. (3.3) to prove the theorem 3.3, we need the following result. lemma 3.1. (see [22], corollary 2.2). if an operator a is of type (m), bounded and coercive on a separable banach space to its dual, then a is surjective. we define the operator an by anu = au + bnu, 94 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) where < au, v >= ∫ ω a(x, u, ∇u)∇vdx and < bnu, v >= ∫ ω tn(b(u))vdx + λ ∫ ∂ω tn(u)vdσ + ε ∫ ω [ |u|p+−2uv + |∇u|p+−2∇u∇v ] dx, with u, v ∈ x0. proof of the theorem 3.3. the proof is organized in three steps. step 1: an is bounded. by using hölder type inequality and (a5) with constant exponent p+, we deduce that a is bounded. moreover, bn is bounded. indeed, let u ∈ f , where f is a bounded subset of x0. as b is onto, we have < bnu, v > = ∫ ω tn(b(u))vdx + λ ∫ ∂ω tn(u)vdσ + ε ∫ ω [ |u|p+−2uv + |∇u|p+−2∇u∇v ] dx ≤ ∫ ω |b(u)||v|dx + λ ∫ ∂ω |u||v|dσ + ε ∫ ω [ |u|p+−1|v| + |∇u|p+−1|∇v| ] dx ≤ c(λ) ( ||v||l1(ω) + ||v||l1(∂ω) ) + ε [ ||u|| p+ (p+) ′ l p+(ω) ||v||lp+(ω) + ||∇u|| p+ (p+) ′ l p+ (ω) ||∇v||lp+ (ω) ] ≤ c(λ) ( ||v||l1(ω) + ||v||l1(∂ω) ) + c(ε)||v||w 1,p+ (ω). therefore, an is bounded. we recall the following notion: definition 3. an operator a : v → v ′ is type of (m) if: un ⇀ u in v a(un) ⇀ χ in v ′ lim sup n→∞ < a(un), un >≤< χ, u >        ⇒ χ = a(u). step 2: an is pseudo-monotone. let (uk)k∈n be a sequence in x0 such that       uk ⇀ u in x0 anuk ⇀ χ in x ′ 0 lim sup k→∞ < anuk, uk >=< χ, u > . we will prove that χ = anu. as tn(b(uk))uk ≥ 0 and λtn(uk)uk ≥ 0, by fatou’s lemma, we deduce that lim inf k→∞ ( ∫ ω tn(b(uk))ukdx + λ ∫ ∂ω tn(uk)ukdσ ) ≥ ∫ ω tn(b(u))udx + λ ∫ ∂ω tn(u)udσ. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 95 one the other hand, thanks to the lebesgue dominated convergence theorem, we have lim k→∞ ( ∫ ω tn(b(uk))vdx + λ ∫ ∂ω tn(uk)vdσ + ε ∫ ω [ |uk| p+−2ukv + |∇uk| p+−2∇uk∇v ] dx ) = ∫ ω tn(b(u))vdx + λ ∫ ∂ω tn(u)vdσ + ε ∫ ω [ |u|p+−2uv + |∇u|p+−2∇u∇v ] dx, for any v ∈ x0. therefore, for k large enough, tn(b(uk))+λtn(uk)+ε [ |uk| p+−2uk+|∇uk| p+−2∇uk ] ⇀ tn(b(u))+λtn(u)+ε [ |u|p+−2u+|∇u|p+−2∇u ] in x′0. thus, auk ⇀ χ − ( tn(b(u)) + λtn(u) + ε[|u| p+−2u + |∇u|p+−2∇u ] ) in x′0, as k → +∞. now, we are going to prove that a is of type (m). let us set a1(u, v, w) = ∫ ω a(x, u, ∇v)∇wdx. then, w 7→ a1(u, v, w) is continuous on w 1,p+(ω), thus a1(u, v, w) = 〈 a(u, v), w 〉 , a(u, v) ∈ (w 1,p+ (ω))′, and verify a(u, u) = au, where au := −diva(x, u, ∇u). let us prove that a is of type of calculus of variation. • as a(u, .) is bounded, we prove that v 7→ a(u, v) is hemi-continuous from w 1,p+(ω) → (w 1,p+ (ω))′. since a(x, u, ∇(v1 + tv2)) ⇀ a(x, u, ∇v1) in l p ′ +(ω) as t → 0 and u, v1, v2 ∈ w 1,p+(ω) then, a1(u, v1 + tv2, w) → a1(u, v1, w) as t → 0. in the same manner we prove that u 7→ a(u, v) is hemi-continuous from w 1,p+(ω) → (w 1,p+ (ω))′. moreover, for all u, v ∈ w 1,p+(ω), we have < a(u, u) − a(u, v), u − v > = < a(u, u), u − v > − < a(u, v), u − v > = a1(u, u, u − v) − a1(u, v, u − v) = ∫ ω a(x, u, ∇u)∇(u − v)dx − ∫ ω a(x, u, ∇v)∇(u − v)dx = ∫ ω ( a(x, u, ∇u) − a(x, u, ∇v) ) ∇(u − v)dx ≥ 0. 96 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) • let us suppose that uk ⇀ u in w 1,p+(ω) and < a(uk, uk) − a(uk, u), uk − u >→ 0. we prove that ∀v ∈ w 1,p+(ω), a(uk, v) ⇀ a(u, v) in (w 1,p+(ω))′. let’s set ∫ ω fkdx = 〈 a(uk, uk) − a(uk, u), uk − u 〉 , then fk → 0. as uk ⇀ u, we have a(x, uk, ∇v) ⇀ a(x, u, ∇v) in l p ′ +(ω) (see [17], lemma 2.2 with m = 1). therefore, a(uk, v) ⇀ a(u, v) in (w 1,p+(ω))′. • now, we suppose that uk ⇀ u in w 1,p+(ω) and a(uk, v) ⇀ θ in (w 1,p+(ω))′. we prove that 〈 a(uk, v), uk 〉 → 〈 θ, u 〉 . then, by using ([17], lemma 2.1), we obtain that a(x, uk, ∇v) → a(x, u, ∇v) in l p ′ +(ω) and thus, a1(uk, v, uk) → a1(u, v, u). therefore, < a(uk, v), uk >= a1(uk, v, uk) →< a(u, v), u > and θ = a(u, v). hence, a is of type of calculus of variation. finally, by using ([17], proposition 2.6 and proposition 2.5), we prove that a is of type (m). as the operator a is of type (m), so we have immediately au = χ − ( tn(b(u)) + λtn(u) + ε [ |u|p+−2u + |∇u|p+−2∇u ] ) . therefore, we deduce that anu = χ. step 3: an is coercive. < anu, u > = ∫ ω a(x, u, ∇u).∇udx + ∫ ω tn(b(u))udx + λ ∫ ∂ω tn(u)udx + ε ∫ ω [ |u|p+ + |∇u|p+ ] dx ≥ ε ∫ ω [ |u|p+ + |∇u|p+ ] dx ≥ ε||u|| p+ w 1,p+ (ω) . we deduce that < anu, u > ||u|| w 1,p+ (ω) → +∞ as ||u|| w 1,p+ (ω) → +∞. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 97 hence, an is coercive. then, according to lemma 3.1, an is surjective. thus, for any fn =< tn(f), tn(g) >⊂ e ′ ⊂ x′0, there exists at least one solution un ∈ x0 of the problem < anun, v >=< fn, v > for all v ∈ x0. therefore, un is a weak solution of the problem (3.2). this ends the proof of theorem 3.3. remark 3.4. if un is a weak solution of the problem (3.2), then un ∈ w 1,πn(.)(ω), since w 1,p+(ω) →֒ w 1,πn(.)(ω) continuously. moreover, a(x, un, ∇un) satisfies (a3) − (a6) with variable exponent πn(x) := p(x, un(x)). part 2: a priori estimates and convergence results. this part is done in three steps, we make a priori estimates, some convergence results and other based on the young measure and nonlinear weak−∗ convergence. step 1: a priori estimates lemma 3.2. suppose that (a3) − (a6) hold with variable exponent πn(.) and fn ∈ l ∞(ω), gn ∈ l∞(∂ω). let un be a weak solution of (3.2). then, for all k > 0, ∫ ω ∣ ∣∇tk(un) ∣ ∣ πn(.) dx ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) , (3.4) ∫ ω ∣ ∣tn(b(un)) ∣ ∣dx ≤ ||f||l1(ω) + ||g||l1(∂ω), (3.5) ∫ ∂ω ∣ ∣tn(un) ∣ ∣dx ≤ 1 λ ( ||f||l1(ω) + ||g||l1(∂ω) ) . (3.6) proof of lemma 3.2. by taking v = tk(un) in the weak formulation (3.3), we obtain ∫ ω tn(b(un))tk(un)dx + ∫ ω a(x, un, ∇un).∇tk(un)dx + ∫ ∂ω λtn(un)tk(un)dσ + ε ∫ ω [ |un| p+−2untk(un) + |∇un| p+−2∇un∇tk(un) ] dx = ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ. (3.7) since all the terms of the left hand side of (3.7) are nonnegative, we deduce that ∫ ω a(x, un, ∇un).∇tk(un)dx ≤ ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ. (3.8) 98 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) by using (a6) and (3.8), we get ∫ ω ∣ ∣∇tk(un) ∣ ∣ πn(.) dx ≤ c2 ( ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ ) ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) . from (3.7), we deduce that ∫ ω tn(b(un))tk(un)dx ≤ ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ ≤ k ( ||f||l1(ω) + ||g||l1(∂ω) ) (3.9) and λ ∫ ∂ω tn(un)tk(un)dx ≤ ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ ≤ k ( ||f||l1(ω) + ||g||l1(∂ω) ) . (3.10) dividing (3.9) and (3.10) by k and letting k goes to 0, we obtain ∫ ω tn(b(un))sign0(un)dx ≤ ||f||l1(ω) + ||g||l1(∂ω) and λ ∫ ∂ω tn(un)sign0(un)dx ≤ ||f||l1(ω) + ||g||l1(∂ω). hence, ∫ ω ∣ ∣tn(b(un)) ∣ ∣dx ≤ ||f||l1(ω) + ||g||l1(∂ω) and ∫ ∂ω ∣ ∣tn(un) ∣ ∣dx ≤ 1 λ ( ||f||l1(ω) + ||g||l1(∂ω) ) . lemma 3.3. assume that (a3)-(a6) hold. if un is a weak solution of the problem (3.2), fn ∈ l∞(ω) and gn ∈ l ∞(∂ω), then for all k > 0 ∫ ω ∣ ∣∇tk(un) ∣ ∣ p− dx ≤ c ( ||f||l1(ω), ||g||l1(∂ω), meas(ω) ) (k + 1) (3.11) and ∫ ∂ω ∣ ∣tk(un) ∣ ∣dσ ≤ 1 λ ( ||f||l1(ω) + ||g||l1(∂ω) ) , (3.12) for all n ≥ k > 0. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 99 proof of lemma 3.3. firstly, we prove (3.11). we know that ∫ ω ∣ ∣∇tk(un) ∣ ∣ πn(.) dx ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) . (3.13) let us note that ∫ ω ∣ ∣∇tk(un) ∣ ∣ p− dx = ∫ {|∇tk(un)|>1} ∣ ∣∇tk(un) ∣ ∣ p− dx + ∫ {|∇tk(un)|≤1} ∣ ∣∇tk(un) ∣ ∣ p− dx ≤ ∫ {|∇tk(un)|>1} ∣ ∣∇tk(un) ∣ ∣ p− dx + meas(ω) ≤ ∫ ω ∣ ∣∇tk(un) ∣ ∣ πn(.) dx + meas(ω). (3.14) by using (3.13) and (3.14), we get ∫ ω ∣ ∣∇tk(un) ∣ ∣ p− dx ≤ max ( c2 ( ||f||l1(ω) + ||g||l1(∂ω) ) , meas(ω) ) (k + 1) := c ( ||f||l1(ω), ||g||l1(∂ω), meas(ω) ) (k + 1). (3.15) now, from the formula (3.6), we obtain ||tn(un)||l1(∂ω) ≤ 1 λ ( ||f||l1(ω) + ||g||l1(∂ω) ) and as |tk(un)| ≤ |tn(un)| for all n ≥ k > 0, one deduces that ∫ ∂ω |tk(un)|dσ ≤ 1 λ ( ||f||l1(ω) + ||g||l1(∂ω) ) . lemma 3.4. for any k > 0, we have ||tk(un)||w 1,πn(.)(ω) ≤ 1 + c ( k, f, g, p−, p+, meas(ω) ) and for all k ≥ 1, meas ( {|un| > k} ) ≤ c min ( b(k), |b(−k)| ). proof of lemma 3.4. by using (3.4), we have ∫ ω ∣ ∣∇tk(un) ∣ ∣ πn(.) dx ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) . (3.16) we also have ∫ ω |tk(un)| πn(.)dx = ∫ {|un|≤k} |tk(un)| πn(.)dx + ∫ {|un|>k} |tk(un)| πn(.)dx. furthermore, ∫ {|un|>k} |tk(un)| πn(.)dx = ∫ {|un|>k} k πn(.)dx ≤ { kp+meas(ω) if k ≥ 1 meas(ω) if k < 1 100 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) and ∫ {|un|≤k} |tk(un)| πn(.)dx ≤ ∫ {|un|≤k} kπn(.)dx ≤ { kp+meas(ω) if k ≥ 1 meas(ω) if k < 1. this allow us to write ∫ ω |tk(un)| πn(.)dx ≤ 2(1 + kp+)meas(ω). (3.17) hence, adding (3.16) and (3.17) one gets ρ1,πn(.)(tk(un)) ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) + 2(1 + kp+)meas(ω). for ||tk(un)||w 1,πn(.)(ω) ≥ 1, we have according to proposition 2 that ||tk(un)|| p− w 1,πn(.)(ω) ≤ ρ1,πn(.)(tk(un)) ≤ [ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) + 2(1 + kp+)meas(ω) ] , which implies that ||tk(un)||w 1,πn(.)(ω) ≤ [ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) + 2(1 + kp+)meas(ω) ] 1 p− := c(k, f, g, p+, p−, meas(ω)). thus, ||tk(un)||w 1,πn(.)(ω) < 1 + c(k, f, g, p+, p−, meas(ω)). moreover, from (3.5), we have ∫ ∂ω ∣ ∣tn(b(un)) ∣ ∣dx ≤ ||f||l1(ω) + ||g||l1(∂ω). we deduce that the sequence (tn(b(un)))n∈n∗ is uniformly bounded in l 1(ω). thus, (b(un))n∈n∗ is uniformly bounded in l1(ω). so, there exists a positive constant c such that ∫ ω |b(un)|dx ≤ c. furthermore, for all k ≥ 1, we have ∫ {|un|>k} |b(un)|dx ≤ ∫ ω |b(un)|dx ≤ c. as b is continuous, nondecreasing and surjective, we infer ∫ {|un|>k} min ( b(k), |b(−k)| ) dx ≤ ∫ {|un|>k} |b(un)|dx ≤ c. therefore, meas ( {|un| > k} ) ≤ c min ( b(k), |b(−k)| ), ∀k ≥ 1. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 101 then, the proof of lemma 3.4 is complete. from the lemma 3.4, we deduce that for any k > 0, the sequence ( tk(un) ) n∈n is uniformly bounded in w 1,πn(.)(ω) and also in w 1,p−(ω). then, up to a subsequence still denoted tk(un), we can assume that for any k > 0, tk(un) weakly converges to sk in w 1,p−(ω) and also tk(un) strongly converges to sk in l p−(ω). by using the above a priori estimates, we obtain the following convergence results . step 2: the convergence results the proof of the following proposition use the lemma 3.4. proposition 6. assume that (a3) − (a6) hold and let un be a weak solution of the problem (3.2), then the sequence (un)n∈n is cauchy in measure. in particular, there exists a measurable function u and a subsequence still denoted un such that un → u in measure, as n → +∞. as (un)n∈n is a cauchy sequence in measure, so (up to a subsequence) it converges almost everywhere to some measurable function u. as for any k > 0, tk is continuous; then tk(un) → tk(u) a.e. x ∈ ω, so sk = tk(u). therefore, tk(un) ⇀ tk(u) in w 1,p−(ω) and by compact embedding theorem, we have tk(un) → tk(u) in l p−(ω) (respectively in lp−(∂ω)) and a.e. in ω (respectively a.e. on ∂ω). lemma 3.5. un converges a.e. on ∂ω to some function v. proof of lemma 3.5 since tk(un) ⇀ tk(u) in w 1,p−(ω) and w 1,p−(ω) →֒ lp−(∂ω) (compact embedding), then tk(un) → tk(u) in l p−(∂ω) and a.e. on ∂ω. therefore, tk(un) → tk(u) in l 1(∂ω) and a.e. in ∂ω. we deduce that there exists e ⊂ ∂ω such that tk(un) → tk(u) on ∂ω \ e with µ(e) = 0, where µ is area measure on ∂ω. for every k > 0, let ek = {x ∈ ∂ω such that |tk(u)| < k} and f = ∂ω \ ⋃ k>0 ek. by using fatou’s lemma, we have ∫ ∂ω ∣ ∣tk(u) ∣ ∣dσ ≤ lim inf n→+∞ ∫ ∂ω |tk(un) ∣ ∣dσ ≤ ||f||l1(ω) + ||g||l1(∂ω) λ . (3.18) 102 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) now, we use (3.18) to get µ(f) = 1 k ∫ f ∣ ∣tk(u) ∣ ∣dσ ≤ 1 k ∫ ∂ω ∣ ∣tk(u) ∣ ∣dσ ≤ ||f||l1(ω) + ||g||l1(∂ω) kλ . we obtain µ(f) = 0, as k goes to ∞. let’ s now define on ∂ω the function v by v(x) = tk(u(x)), x ∈ ek. we take x ∈ ∂ω r (e ∪ f), then there exists k > 0 such that x ∈ ek and we have un(x) − v(x) = ( un(x) − tk(un(x)) ) + ( tk(un(x)) − tk(u(x)) ) . since x ∈ ek, we have |tk(u(x))| < k and so |tk(un(x))| < k, from which we deduce that |un(x)| < k. therefore, un(x) − v(x) = tk(un(x)) − tk(u(x)) → 0, as n → +∞. this means that un converges to v a.e. on ∂ω, but for all x ∈ ek, tk(u(x)) = u(x). thus, v = u a.e. on ∂ω. therefore, un → u a.e. on ∂ω. the following assertions are based on the young measure and nonlinear weak −∗ convergence results (see [7, 19, 13]). step 3: the convergence in term of young measure assertion 1 the sequence (∇tk(un))n∈n converges to a young measure ν k x(λ) on r n in the sense of the nonlinear weak-* convergence and ∇tk(u) = ∫ rn λdν k x(λ). (3.19) proof. using lemma 3.3, ∇tk(un) is uniformly bounded in l p−(ω), so, equi-integrable on ω. moreover, ∇tk(un) weakly converges to ∇tk(u) in l p−(ω). therefore, using the representation of weakly convergence sequences in l1(ω) in terms of young measures (see theorem 2.1 and formula (2.5)), we can write ∇tk(u) = ∫ rn λdνkx(λ) � assertion 2. |λ|π(.) is integrable with respect to the measure νkx(λ)dx on r n × ω, moreover, tk(u) ∈ w 1,π(.)(ω). proof. we know that p(., un(.)) → p(., u(.)) in measure on ω. now, using theorem 2.1 (ii), cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 103 (iii) (p(., un(.)), ∇tk(un))n∈n converges on r × r n to young measure µkx = δπ(x) ⊗ ν k x. thus, we can apply the weak convergence properties (2.4) to the carathéodory function fm(x, λ0, λ) ∈ ω × (r × r n) 7→ |hm(λ)| λ0 with m ∈ n, where hm is defined in the preliminaries. then, we obtain ∫ ω×rn |hm(λ)| π(x)dνkx(λ)dx = ∫ ω×(r×rn ) |hm(λ)| λ0 dµkx(λ0, λ)dx = ∫ ω ∫ r×rn fm(x, λ0, λ)dµ k x(λ0, λ)dx = lim n→+∞ ∫ ω fm(x, p(x, un(x)), ∇tk(un(x)))dx = lim n→+∞ ∫ ω |hm(∇tk(un))| p(.,un(.))dx ≤ lim n→+∞ ∫ ω |∇tk(un)| p(.,un(.))dx ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) (using (3.4)). hm(λ) → λ, as m → +∞ and m 7→ hm(λ) is increasing. then, using lebesgue convergence theorem , we deduce from last inequality that ∫ ω×rn |λ|π(x)dνkx(λ)dx ≤ c2k ( ||f||l1(ω) + ||g||l1(∂ω) ) . hence, |λ|π(.) is integrable with respect to the measure νkx(λ)dx on r n × ω. from (3.19), the last inequality and jensen inequality, we get ∫ ω |∇tk(u)| π(x)dx = ∫ ω ∣ ∣ ∣ ∣ ∫ rn λdνkx(λ) ∣ ∣ ∣ ∣ π(x) dx ≤ ∫ ω×rn |λ|π(x)dνkxdx < ∞. thus, ∇tk(u) ∈ l π(.)(ω). moreover, ∫ ω |tk(u)| π(.) dx ≤ max ( k p+, k p− ) meas(ω). hence, tk(u) ∈ lπ(.)(ω) and we conclude that tk(u) ∈ w 1,π(.)(ω). � assertion 3. i) the sequence ( φkn ) n∈n defined by φkn := a(x, un, ∇tk(un)) is equi-integrable on ω. ii) the sequence ( φkn ) n∈n weakly converges to φk in l1(ω) and we have φk(x) = ∫ rn a(x, u, λ)dνkx(λ). (3.20) proof. i) using the growth assumption (a5) with variable exponent p(., un(.)) and relation (3.4), we deduce that (φkn) is bounded in l π ′ n(.)(ω), so, lπ ′ n(.)− equi-integrable on ω. moreover, as π′n(.) > 1, we obtain |a(x, un, ∇tk(un))| ≤ 1 + |a(x, un, ∇tk(un))| π ′ n (.) . 104 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) thus, for all subset e ⊂ ω, we have ∫ e |a(x, un, ∇tk(un))|dx ≤ meas(e) + ∫ e |a(x, un, ∇tk(un))| π ′ n(.)dx. therefore, for meas(e) small enough, (φkn) is equi-integrable on ω. ii) set φ̃kn = a(x, u(x), ∇vn) with ∇vn = ∇tk(un).χsn where sn = { x ∈ ω, |π(x) − πn(x)| < 1 2 } . applying (a5) with variable exponent π(.) on a(x, u(x), ∇vn), we have for all subset e ⊂ ω, ∫ e |a(x, u(x), ∇vn)|dx ≤ c ∫ e ( 1 + m(x) + |∇vn| π(.)−1 ) dx ≤ c ∫ e ( 1 + m(x))dx + ∫ e∩sn |∇tk(un)| π(.)−1dx. the first term of the right hand side of the last inequality is small for meas(e) small enough. for x ∈ sn, π(x) < πn(x) + 1 2 , thus ∫ e∩sn |∇tk(un)| π(.)−1 dx ≤ ∫ e∩sn ( 1 + |∇tk(un)| πn(.)− 1 2 ) dx and ∫ ω |∇tk(un)| (πn(.)− 1 2 )(2πn(.)) ′ dx = ∫ ω |∇tk(un)| πn(.)dx < ∞, which is equivalent to saying |∇tk(un)| πn(.)− 1 2 ∈ l(2πn(.)) ′ (ω). now, using hölder type inequality, ∫ e∩sn |∇tk(un)| π(.)−1dx ≤ ∫ e ( 1 + |∇tk(un)| πn(.)− 1 2 ) dx ≤ meas(e) + 2 ∣ ∣ ∣ ∣∇tk(un) ∣ ∣ ∣ ∣ lπn(.)(ω) ||χe||l2πn(.)(ω). (3.21) according to proposition 1, ||χe||l2πn(.)(ω) ≤ max { ( ρ2πn(.) ( χe )) 1 2p− , ( ρ2πn(.)(χe) ) 1 2p+ } = max { ( meas(e) ) 1 2p− , ( meas(e) ) 1 2p+ } . the right-hand side of (3.21) is uniformly small for meas(e) small, and the equi-integrability of φ̃kn follows. therefore, (up to a subsequence) φ̃ k n weakly converges in l 1(ω) to φ̃k, as n → +∞. now, we prove that φ̃k = φk; more precisely, we show that φ̃kn − φ k n strongly converges in l 1(ω) to 0. let β > 0, by (3.4), ∫ ω |∇tk(un)| πn(.)dx is uniformly bounded, which implies that ∫ ω |∇tk(un)|dx is finite, since ∫ ω |∇tk(un)|dx ≤ ∫ ω (1 + |∇tk(un)| πn(x))dx. by chebyschev inequality, we have meas({|∇tk(un)| > l}) ≤ ∫ ω |∇tk(un)|dx l . cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 105 therefore, sup n∈n meas({|∇tk(un)| > l}) tends to 0 for l large enough. since φ̃ k n − φ k n is equiintegrable, there exists δ = δ(β) such that for all a ⊂ ω, meas(a) < δ and ∫ a |φ̃kn − φ k n|dx < β 4 . therefore, if we choose l large enough, we get ∫ ω |∇tk(un)|dx l < δ, so meas({|∇tk(un)| > l}) < δ. hence, ∫ {|∇tk(un)|>l} |φ̃kn − φ k n|dx < β 4 . by lemma 2.2, we also have meas ({ x ∈ ω; sup λ∈k |a(x, un(x), λ) − a(x, u(x), λ)| ≥ σ }) −→ 0, as n → +∞. thus, by the above equi-integrability, for all σ > 0, there exists n0 = n0(σ, l) ∈ n such that for all n ≥ n0, ∫ { x∈ω; sup|λ|≤l |a(x,un(x),λ)−a(x,u(x),λ)|≥σ } |φ̃ k n − φ k n|dx < β 4 . using the definition of φkn and φ̃ k n, we have φkn − φ̃ k n = a(x, un(x), ∇tk(un)) − a(x, u(x), ∇tk(un)) on sn. now, we reason on sn,l,σ := { x ∈ ω; sup |λ|≤l |a(x, un(x), λ) − a(x, u(x), λ)| < σ, |∇tk(un)| ≤ l } . we get ∫ sn,l,σ |φ̃kn − φ k n|dx ≤ ∫ sn,l,σ sup |λ|≤l |a(x, un(x), λ) − a(x, u(x), λ)|dx ≤ σmeas(ω). we observe that ∫ sn |φ̃kn − φ k n|dx = ∫ sn∩sn,l,σ |φ̃kn − φ k n|dx + ∫ sn\sn,l,σ |φ̃kn − φ k n|dx and sn \ sn,l,σ ⊂ { x ∈ ω; sup |λ|≤l |a(x, un(x), λ) − a(x, u(x), λ)| ≥ σ } ∪ { |∇tk(un)| > l } . consequently, by choosing σ = σ(β) < β 4meas(ω) , we get ∫ sn |φ̃kn − φn|dx < β 4 + β 4 + β 4 = 3β 4 , 106 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) for all n ≥ n0(σ, l). by lemma 2.2, we also have meas({x ∈ ω, |π(x) − πn(x)| ≥ 1 2 }) → 0 for n large enough; which means that meas(ω \ sn) converges to 0 for n large enough. thus, ∫ ω\sn |φ̃kn − φ k n|dx = ∫ ω\sn |φkn|dx ≤ β 4 . therefore, for all β > 0 there exists n0 = n0(β) such that for all n ≥ n0, ∫ ω |φ̃kn − φ k n|dx ≤ β. hence, φ̃kn − φ k n strongly converges to 0 in l 1(ω). we prove that φk(x) = ∫ rn a(x, u(x), λ)dνkx(λ) a.e. x ∈ ω and φ k ∈ lπ ′(.)(ω). notice that lim n→+∞ ∫ ω |∇tk(un)|(1 − χsn)dx = lim n→+∞ ∫ ω\sn |∇tk(un)|dx = 0, since (∇tk(un))n∈n is equi-integrable and meas(ω \ sn) converges to 0 for n large enough. therefore, (∇tk(un))n∈n and ∇tk(un)χsn converge to the same young measure ν k x(λ). moreover, by applying theorem 2.1 i) to the carathéodory function f(x, (λ0, λ)) := a(x, λ0, λ), we infer that φ̃(x) = φ(x) = ∫ rn a(x, u(x), λ)dνkx(λ) a.e. x ∈ ω. using (a5), it follows that |a(x, u(x), λ)| π ′(.) ≤ c(m(x) + |λ|π(.)). thus, with jensen inequality, it follows that ∫ ω |φk(x)|π ′(.)dx = ∫ ω ∣ ∣ ∣ ∣ ∫ rn a(x, u(x), λ)dνkx(λ) ∣ ∣ ∣ ∣ π ′(.) dx ≤ ∫ ω×rn ∣ ∣a(x, u(x), λ)|π ′(.)dνkx(λ)dx ≤ c ∫ ω×rn ( m(x) + |λ|π(.) ) dνkx(λ)dx < ∞. hence, φk ∈ lπ ′(.)(ω). � assertion 4 (a) for all k′ > k > 0, we have φk = φk ′ χ{|u| 0, ∫ ω φk.∇tk(u)dx ≥ ∫ ω×rn a(x, u(x), λ).λdνkx(λ)dx. (3.22) (c) the “div-curl” inequality holds: ∫ ω×rn ( a(x, u(x), λ) − a(x, u(x), ∇tk(u(x)) ) (λ − ∇tk(u(x)))dν k x(λ)dx ≤ 0. (3.23) cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 107 (d) for all k > 0, φk = a(x, u(x), ∇tk(u)) for a.e. x ∈ ω and ∇tk(un) converges to ∇tk(u) in measure on ω, as n → +∞. proof. (a) let k′ > k > 0 and gkn := a(x, un, ∇tk′(un))χ[|u| k > 0, then, we get hkn := a(x, un, ∇tk′(un))χ[|un| 0. moreover, un → u a.e. on ω, then χ[|un| 0, we have ∫ ω s ′ m (un)a(x, un, ∇tm (un)).∇ϕdx = ∫ {|∇ϕ|≤l} s ′ m (un)φ m n .∇ϕdx + ∫ {|∇ϕ|>l} s′m (un)φ m n .∇ϕdx. (3.30) for the first term of the right-hand side of (3.30), we have ∫ {|∇ϕ|≤l} s′m (un)φ m n .∇ϕdx → ∫ {|∇ϕ|≤l} s′m (u)φ m .∇ϕdx, as n → +∞. (3.31) thanks φmn ⇀ φ m in l1(ω) and ∇ϕs′m (un)χ{|∇ϕ|≤l} → ∗ ∇ϕs′m (u)χ{|∇ϕ|≤l} in l ∞(ω). furthermore, the second term of the right hand-side of (3.30) converges to zero for l large enough, uniformly in n. indeed, using hölder type inequality and the fact that lp+(ω) →֒ lπn(.)(ω), we get ∣ ∣ ∣ ∣ ∫ {|∇ϕ|>l} φmn ∇ϕs ′ m (un)dx ∣ ∣ ∣ ∣ ≤ c||s′m ||l∞(r)||φ m n ||lπ′n(.)(ω)||∇ϕχ{|∇ϕ|>l}||lπn(.)(ω) ≤ c ( p−, ||s ′ m ||l∞(r), meas(ω) ) ||φmn ||lπ′n(.)(ω)||∇ϕ||l p+ (ω)meas ( {|∇ϕ| > l} ) . from (a5), (3.4) and proposition 2, we obtain ||φnm||lπ′n(.)(ω) < c. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 109 moreover, ϕ ∈ c∞(ω) and c∞(ω) is dense in the space w 1,p+(ω). then, by proposition 2 and the fact that lim l→+∞ meas({|∇ϕ| > l}) = 0, we get meas ( {|∇ϕ| > l} ) ||φmn ||lπ′n(.)(ω)||∇ϕ||l p+ (ω) → 0, as l → +∞. hence, the second term of the right hand-side of (3.30) converges to zero, as l tends to infinity. thus, as n → +∞ and l → +∞ in (3.30), we deduce (3.29). let us consider the third term of left hand-side of (3.24), we obtain ∫ ω |s′′m (un)|a(x, un, ∇tm(un)).∇tm (un)ϕdx ≤ c ∫ {|un| 0 fixed, tk(u) ∈ w 1,π(.)(ω) and the exponent π(.) verify (2.1). therefore, c∞(ω) is dense in w 1,π(.)(ω), so, we replace ϕ by tk(u). now, for m > k, thanks to (a), we replace φ m .∇tk(u) by φk.∇tk(u) in (3.33). s′m converges a.e. to 1 on r, as m → +∞, then using the monotone convergence theorem in the first term of left hand-side of (3.33) and dominated convergence theorem in the other term of (3.33), we get ∫ ω [ b(u)tk(u) + φ k .∇tk(u)]dx + λ ∫ ∂ω utk(u)dσ = ∫ ω ftk(u)dx + ∫ ∂ω gtk(u)dσ. (3.34) 110 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) the relation (3.7) is equivalent to ∫ ω tn(b(un))tk(un)dx + ∫ ω a(x, un, ∇tk(un)).∇tk(un)dx + ∫ ∂ω λtn(un)tk(un)dσ + ε ∫ ω [ |un| p+−2untk(un) + |∇un| p+−2∇un∇tk(un) ] dx = ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ. (3.35) the sequences ( tn(b(un))tk(un) ) n∈n , ( tn(un)tk(un) ) n∈n are nonnegative and converge a.e. in ω to b(u)tk(u) and a.e. on ∂ω to utk(u). by fatou’s lemma, we have lim inf n→+∞ ∫ ω tn(b(un))tk(un)dx ≥ ∫ ω b(u)tk(u)dx (3.36) and λ lim inf n→+∞ ∫ ∂ω tn(un)tk(un)dx ≥ λ ∫ ∂ω utk(u)dσ. (3.37) now, we consider the right hand side of (3.35). we have |fntk(un)| ≤ k|f| ∈ l 1(ω), fntk(un) → ftk(u) a.e. in ω and |gntk(un)| ≤ k|g| ∈ l 1(∂ω), gntk(un) → gtk(u) a.e. on ∂ω. thus, by lebesgue dominated convergence theorem ∫ ω fntk(un)dx → ∫ ω ftk(u)dx, as n → +∞ (3.38) and ∫ ∂ω gntk(un)dσ → ∫ ∂ω gtk(u)dσ, as n → +∞. (3.39) combining (3.36),(3.37), (3.38), (3.39) and using (3.35), we get lim inf n→+∞ ( ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ ) − ( ∫ ω b(u)tk(u)dx + λ ∫ ∂ω utk(u)dσ ) ≥ lim inf n→+∞ ( ∫ ω fntk(un)dx + ∫ ∂ω gntk(un)dσ − ∫ ω tn(b(un))tk(un)dx − λ ∫ ∂ω tn(un)tk(un)dσ ) , which is equivalent to ∫ ω ftk(u)dx + ∫ ∂ω gtk(u)dσ − ( ∫ ω b(u)tk(u)dx + λ ∫ ∂ω utk(u)dσ ) ≥ lim inf n→+∞ ∫ ω a(x, un, ∇tk(un))∇tk(un)dx + ε ∫ ω [ |un| p+−2untk(un) + |∇un| p+−2∇un∇tk(un) ] dx ≥ lim inf n→+∞ ∫ ω a(x, un, ∇tk(un)∇tk(un)dx. thus, by using the relation (3.34), we obtain ∫ ω φk∇tk(u)dx ≥ lim inf n→+∞ ∫ ω a(x, un, ∇tk(un))∇tk(un)dx. (3.40) cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 111 (c) from [1]-lemma 2.1, m 7→ a(x, un, hm(∇tk(un))).hm(∇tk(un)) is increasing and converges to a(x, un, ∇tk(un)).∇tk(un) for m large enough. thus, we deduce that a(x, un, hm(∇tk(un))).hm(∇tk(un)) ≤ a(x, un, ∇tk(un)).∇tk(un) = φ k n.∇tk(un). therefore, using (b) and theorem 2.1, we get ∫ ω φk.∇tk(u)dx ≥ lim inf n→+∞ ∫ ω φkn.∇tk(un)dx ≥ lim n→+∞ ∫ ω a(x, un, hm(∇tk(un))).hm(∇tk(un))dx = ∫ ω×rn a(x, u, hm(λ)).hm(λ)dν k x(λ)dx. (3.41) using lebesgue convergence theorem in (3.41), we get for m large enough ∫ ω φk.∇tk(u)dx ≥ ∫ ω×rn a(x, u, λ).λdνkx(λ)dx. (3.42) we have ∫ ω×rn (a(x, u(x), λ) − a(x, u(x), ∇tk(u(x)))(λ − ∇tk(u(x)))dν k x(λ)dx = ∫ ω×rn a(x, u(x), λ).λdνkx(λ)dx − ∫ ω×rn a(x, u(x), λ).∇tk(u(x))dν k x(λ)dx − ∫ ω×rn a(x, u(x), ∇tk(u(x))).λdν k x(λ)dx + ∫ ω×rn a(x, u(x), ∇tk(u(x))).∇tk(u(x))dν k x(λ)dx = ∫ ω×rn a(x, u(x), λ).λdνkx(λ)dx − ∫ ω ( ∫ rn a(x, u(x), λ)dνkx(λ) ) ∇tk(u(x))dx − ∫ ω a(x, u(x), ∇tk(u(x))). ( ∫ rn λdνkx ) dx + ∫ ω a(x, u(x), ∇tk(u(x))).∇tk(u(x)) ( ∫ rn dνkx ) dx = ∫ ω×rn a(x, u(x), λ).λdνkx(λ)dx − ∫ ω φk.∇tk(u(x))dx ≤ 0. we pass from the first equality to the second equality by using fubini-tonelli theorem and from the second inequality to the third one by using (3.19), (3.20) and the fact that νx is probability measures on rn. finally (3.42) give us the desired inequality. (d) using (3.23) and the strict monotonicity assumption (a4), we deduce that ( a(x, u(x), λ) − a(x, u(x), ∇tk(u(x)) )( λ − ∇tk(u(x)) ) = 0 a.e. x ∈ ω, λ ∈ rn. thus, λ = ∇tk(u(x)) a.e. x ∈ ω with respect to the measure ν k x on r n. therefore, the measure νkx reduces to the dirac measure δ∇tk(u(x)). using (3.20), we obtain φk = ∫ rn a(x, u(x), λ)dνkx(λ) = a(x, u(x), ∇tk(u(x))) a.e. x ∈ ω. 112 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) now, by using theorem 2.1 (ii), we deduce that ∇tk(un) converges in measure to ∇tk(u). � lemma 3.6. u is an entropy solution of (1.1). proof of the lemma 3.6. let un be a weak solution of the problem (3.2). then, by assertion 4−(d), (∇tk(un))n∈n converges to ∇tk(u) in measure, thus (up to a subsequence still denoted (∇tk(un))n∈n), (∇tk(un))n∈n converges to ∇tk(u) a.e. ω. moreover, we deduce from lemma 3.4 that ∇tk(un) is uniformly bounded in lp−(ω), so, p−−equi-integrable on ω. then, by using vitali’s theorem ∇tk(un) → ∇tk(u) in l p−(ω), which implies that ∇tk(un) → ∇tk(u) in l 1(ω). furthermore, thanks to assertion 2, u ∈ t 1,π(.)(ω) and it follows from lemma 3.5 that un → u a.e on ∂ω. therefore, u ∈ t 1,π(.) tr (ω). now, using lemma 3.2, the fact that tn(b(un)) → b(u) a.e. in ω and un → u a.e. on ∂ω, it follows from fatou’s lemma that ∫ ω |b(u)| ≤ lim inf n→+∞ ∫ ω ∣ ∣tn(b(un)) ∣ ∣dx ≤ ||f||l1(ω) + ||g||l1(∂ω) and ∫ ∂ω |u| ≤ lim inf n→+∞ ∫ ∂ω ∣ ∣tn(un) ∣ ∣dx ≤ 1 λ ( ||f||l1(ω) + ||g||l1(∂ω) ) . hence, b(u) ∈ l1(ω) and u ∈ l1(∂ω). let ϕ ∈ c∞(ω), then we can choose tk(un − ϕ) as a test function in (3.3) (c ∞(ω) is dense in the space w 1,p+(ω) and tk(un − ϕ) ∈ l ∞(∂ω)) to get ∫ ω tn(b(un))tk(un − ϕ)dx + ∫ ω a(x, un, ∇un).∇tk(un − ϕ)dx + ∫ ∂ω λtn(un)tk(un − ϕ)dσ + ε ∫ ω [ |∇un| p+−2∇un∇tk(un − ϕ) + |un| p+−2untk(un − ϕ) ] dx = ∫ ω fntk(un − ϕ)dx + ∫ ∂ω gntk(un − ϕ)dσ. (3.43) for the first term of the left hand side of (3.43), we have ∫ ω tn(b(un))tk(un − ϕ)dx = ∫ ω [ tn(b(un)) − tn(b(ϕ)) ] tk(un − ϕ)dx + ∫ ω tn(b(ϕ))tk(un − ϕ)dx. by fatou’s lemma, we infer lim inf n→+∞ ∫ ω tn(b(un))tk(un − ϕ)dx ≥ ∫ ω b(u)tk(u − ϕ)dx, (3.44) cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 113 since, [ tn(b(un)) − tn(b(ϕ)) ] tk(un − ϕ) → ( b(u − b(ϕ) ) tk(u − ϕ) a.e. with [ tn(b(un)) − tn(b(ϕ)) ] tk(un − ϕ) ≥ 0 and tn(b(ϕ))tk(un − ϕ) → b(ϕ)tk(u − ϕ) in l 1(ω). in the same manner lim inf n→+∞ λ ∫ ∂ω tn(un)tk(un − ϕ)dσ ≥ λ ∫ ∂ω utk(u − ϕ)dσ. (3.45) for the fourth term of the left hand side of (3.43), we prove that lim n→+∞ ε ∫ ω [ |∇un| p+−2∇un∇tk(un − ϕ) + |un| p+−2untk(un − ϕ) ] dx ≥ 0 as ε → 0. (3.46) setting l = k + ||ϕ||l∞(ω) we have, ε ∫ ω |∇un| p+−2∇un∇tk(un − ϕ)dx = ε ∫ {|un−ϕ|1} |∇ϕ|πn(.)dx ≤ meas(e) + ∫ e |∇ϕ|p+ dx, since |∇ϕ|p+ , m ∈ l1(ω) and |∇tl(un)| πn(.) is equi-integrable ( using density argument for c∞(ω) and (3.4) ) . then, we obtain lim meas(e)→0 ∫ e a(x, un, ∇tl(un))∇ϕχ{|un−ϕ| n and ω is a bounded open domain with lipschitz boundary ∂ω. therefore, the inequality (3.59) holds true for ϕ ∈ w 1,π(.)(ω) ∩ l∞(ω). hence, u is an entropy solution of (1.1). � now, we state the uniqueness result of entropy solution. this result uses the same arguments as [2]-theorem 2.8. theorem 3.5. assume that b is strictly increasing. assume that a = a(x, z, η) satisfies (a3)−(a6) and m constant. moreover, a satisfies: for all bounded subset k of r × rn, there exists a constant c(k) such that a.e. x ∈ ω, for all (z, η), (z̃, η) ∈ k, |a(x, z, η) − a(x, z̃, η)| ≤ c(k)|z − z̃|. (3.60) finally, suppose the following regularity property: for all f ∈ l∞(ω) and g ∈ l∞(∂ω) there exists an entropy solution of (1.1), which is lipchitz continuous on ω. (3.61) then, for all f ∈ l1(ω) and g ∈ l1(∂ω) the problem (1.1) admits a unique entropy solution. cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 117 remark 3.6. as in [2, theorem 2.8], the condition (3.61) goes back to idea of [3]. moreover, in the theorem 3.5 the relation (3.60) is used to obtain the inequality (3.69) below. proof. the proof of this theorem is done in two steps. step 1. a priori estimates. lemma 3.7. if v is an entropy solution of (1.1), there exists a positive constant c such that ρp(.,v(.)) ( |∇v|χf ) ≤ ck, where f = {h − k < |v| < h}, h > k > 0. proof. let ϕ = th−k(v) as test function in the entropy inequality (3.1), we get ∫ ω a(x, v, ∇v).∇tk(v − th−k(v))dx + ∫ ω b(v)tk(v − th−k(v))dx + λ ∫ ∂ω vtk(v − th−k(v))dσ ≤ ∫ ω ftk(v − th−k(v))dx + ∫ ∂ω gtk(v − th−k(v))dσ. thus, ∫ {h−k<|v| h} ) ≤ ||f||l1(ω) + ||g||l1(∂ω) min ( b(h), |b(−h)| ) , ∀h ≥ 1. proof. let us take ϕ = 0 and k = h in entropy inequality (3.1). since ∫ ω a(x, u, ∇u).∇th(u)dx + λ ∫ ∂ω uth(u)dσ ≥ 0, the relation (3.1) gives ∫ ω b(u)th(u)dx ≤ ∫ ω fth(u)dx + ∫ ∂ω gth(u)dσ. 118 stanislas ouaro & noufou sawadogo cubo 22, 1 (2020) then, ∫ {|u|≤h} b(u)th(u)dx + ∫ {|u|>h} b(u)th(u)dx ≤ h ( ||f||l1(ω) + ||g||l1(∂ω) ) , or ∫ {|u|>h} b(u)th(u) h dx = ∫ {u>h} b(u)dx + ∫ {u<−h} −b(u)dx ≤ ( ||f||l1(ω) + ||g||l1(∂ω) ) . therefore, ∫ {|u|>h} |b(u)|dx ≤ ||f||l1(ω) + ||g||l1(∂ω). since b is nondecreasing, we deduce ∫ {|u|>h} min(b(h), |b(−h)|)dx ≤ ∫ {|u|>h} |b(u)| ≤ ||f||l1(ω) + ||g||l1(∂ω), ∀h ≥ 1. so, meas ( {|u| > h} ) ≤ ||f||l1(ω) + ||g||l1(∂ω) min ( b(h), |b(−h)| ) , ∀h ≥ 1. � step 2. uniqueness. the existence has already been proved. now, we show the uniqueness. for more details see [2]proof of theorem 2.8. let u be a lipschitz continuous entropy solution of (1.1) with f ∈ l∞(ω), g ∈ l∞(∂ω) and v be an entropy solution, with f̂ ∈ l1(ω), ĝ ∈ l1(∂ω). since ω is open bounded domain with smooth boundary ∂ω, the space of lipschitz functions c0,1(ω) and w 1,∞(ω) are homeomorphic and they can be identified. therefore, u belongs to w 1,∞(ω). thus, for all h > 0, we can write the entropy inequality corresponding to the solution u, with th(v) as a test function and to the solution v, with th(u) as a test function. for all k > 0, we get      ∫ ω a(x, u, ∇u).∇tk(u − th(v))dx + ∫ ω b(u)tk(u − th(v))dx +λ ∫ ∂ω utk(u − th(v))dσ ≤ ∫ ω ftk(u − th(v))dx + ∫ ∂ω gtk(u − th(v))dσ (3.62) and      ∫ ω a(x, v, ∇v).∇tk(v − th(u))dx + ∫ ω b(v)tk(v − th(u))dx +λ ∫ ∂ω vtk(v − th(u))dσ ≤ ∫ ω f̂tk(v − th(u))dx + ∫ ∂ω ĝtk(v − th(u))dσ. (3.63) cubo 22, 1 (2020) nonlinear elliptic p(u)− laplacian problem with fourier . . . 119 adding (3.62) and (3.63) we obtain                           ∫ ω a(x, u, ∇u).∇tk(u − th(v))dx + ∫ ω a(x, v, ∇v).∇tk(v − th(u))dx + ∫ ω b(u)tk(u − th(v))dx + ∫ ω b(v)tk(v − th(u))dx +λ ∫ ∂ω utk(u − th(v))dσ + λ ∫ ∂ω vtk(v − th(u))dσ ≤ ∫ ω [ ftk(u − th(v)) + f̂tk(v − th(u)) ] dx + ∫ ∂ω [ gtk(u − th(v)) + ĝtk(v − th(u)) ] dσ. (3.64) set a = {0 < |u − v| < k, |v| ≤ h}; b = a ∩ {|u| ≤ h}; c = a ∩ {|u| > h} and a′ = {0 < |v − u| < k, |u| ≤ h}; b′ = a′ ∩ {|v| ≤ h}; c′ = a′ ∩ {|v| > h}. we start with the first integral in (3.64). we have ∫ {0<|u−th(v)|h} a(x, u, ∇u).∇tk(u − th(v))dx = ∫ {0<|u−v|h} a(x, u, ∇u).∇udx ≥ ∫ a a(x, u, ∇u)∇(u − v)dx = ∫ b a(x, u, ∇u)∇(u − v)dx + ∫ c a(x, u, ∇u)∇(u − v)dx. then, we get        ∫ {0<|u−th(v)| 0, for a function g ∈ lp[a, b], 0 ≤ a ≤ t ≤ b ≤ ∞, is defined as i q a+ g(t) = 1 γ(q) ∫ t a ( log t s )q−1 g(s) s ds, i q b− g(t) = 1 γ(q) ∫ b t ( log s t )q−1 g(s) s ds. definition 2.2. ([3, 17]) let [a, b] ⊂ r, δ = t d dt and acnδ [a, b] = {g : [a, b] → r : δn−1(g(t)) ∈ ac[a, b]}. the hadamard derivative of fractional order q for a function g ∈ acnδ [a, b] is defined as d q a+ g(t) = δn(i n−q a+ )(t) = 1 γ(n − q) ( t d dt )n ∫ t a ( log t s )n−q−1 g(s) s ds, d q b− g(t) = (−δ)n(in−q b− )(t) = 1 γ(n − q) ( −t d dt )n ∫ b t ( log s t )n−q−1 g(s) s ds, where n − 1 < q < n, n = [q] + 1 and [q] denotes the integer part of the real number q and log(·) = loge(·). for more details of the hadamard fractional integrals and derivatives, we refer the reader to section 2.7 in the text [17]. lemma 2.3. let x ∈ c2δ ([1, t ], r) and g ∈ c([1, t ], r). the (integral) solution of the linear hadamard fractional boundary value problem:        ( hdα + λ hdα−1 ) x(t) = g(t), 1 < α ≤ 2, 1 < t < t, x(1) = 0, x(t ) = m ∑ j=1 βj x(tj), (2.1) 228 bashir ahmad, amjad f. albideewi, sotiris k. ntouyas & ahmed alsaedi cubo 23, 2 (2021) is given by x(t) = 1 γ ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) { m ∑ j=1 βj t −λ j γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds − t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds } + t−λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds, (2.2) where it is assumed that γ := t −λ ∫ t 1 sλ−1 (log s)α−2ds − m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 (log s)α−2ds 6= 0. (2.3) proof. the linear hadamard fractional differential equation in (2.1) can be rewritten as hdα−1(td + λ)x(t) = g(t), t ∈ [1, t ]. (2.4) applying the hadamard fractional operator iα−1 on both sides of (2.4), we get ( d + λ t ) x(t) = t−1 ( c1(log t) α−2 + iα−1g(t) ) , which can be rewritten as d ( tλx(t) ) = c1t λ−1(log t)α−2 + tλ−1iα−1g(t). (2.5) integrating (2.5) from 1 to t, we get x(t) = c0t −λ + c1t −λ ∫ t 1 sλ−1 (log s)α−2ds + t−λ ∫ t 1 sλ−1 iα−1 g(s)ds, (2.6) where ci, (i = 0, 1) are unknown arbitrary constants. using the initial condition x(1) = 0 in (2.6) implies that c0 = 0, which leads to x(t) = c1t −λ ∫ t 1 sλ−1 (log s)α−2ds + t−λ ∫ t 1 sλ−1 iα−1 g(s)ds. (2.7) now using the condition x(t ) = m ∑ j=1 βj x(tj) in (2.7), we have c1t −λ ∫ t 1 sλ−1 (log s)α−2ds + t −λ ∫ t 1 sλ−1 iα−1 g(s)ds = c1 m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 (log s)α−2ds + m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 iα−1 g(s)ds, which, on solving for c1 together with (2.3), yields c1 = 1 γ [ m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 iα−1 g(s)ds − t −λ ∫ t 1 sλ−1 iα−1 g(s)ds ] . substituting the above value of c1 in (2.7), we get the desired solution (2.2). the converse of the lemma follows by a direct computation. this completes the proof. cubo 23, 2 (2021) multipoint fractional sequential hadamard bvp 229 the following lemma contains certain estimates that we need in the sequel. lemma 2.4. for g ∈ c([1, t ], r) with ‖g‖ = sup t∈[1,t ] |g(t)|, we have (i) ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds ∣ ∣ ∣ ∣ ∣ ≤ (log t ) α α(α − 1) ‖g‖. (ii) ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1(log s)α−2ds ∣ ∣ ∣ ∣ ∣ ≤ (log t )α−1 (α − 1) . proof. note that ∫ s 1 ( log s r )α−2 1 r dr = (log s)α−1 (α − 1) . since sλ ≤ tλ for 1 < s < t, then ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds ∣ ∣ ∣ ∣ ∣ ≤ sup t∈[1,t ] ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds ∣ ∣ ∣ ∣ ∣ ≤ ‖g‖ sup t∈[1,t ] ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( (log s)α−1 (α − 1) ) ds ∣ ∣ ∣ ∣ ∣ ≤ ‖g‖(log t ) α α(α − 1) . 3 existence and uniqueness results let g = c([1, t ], r) denote the banach space of all continuous functions from [1, t ] to r endowed with the usual norm ‖x‖ = sup{|x(t)| : t ∈ [1, t ]}, and cnδ ([1, t ], r) denotes the banach space of all real valued functions g such that δng ∈ g. using lemma 2.3, we can transform the problem (1.1) into a fixed point problem as x = px, where the operator p : g → g is defined by (px)(t) = 1 γ ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × { m∑ j=1 βj t −λ j γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds − t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds } (3.1) + t−λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds, t ∈ [1, t ]. 230 bashir ahmad, amjad f. albideewi, sotiris k. ntouyas & ahmed alsaedi cubo 23, 2 (2021) for computational convenience, we set λ = (log t )α−1 |γ|(α − 1) [ m∑ j=1 |βj| (log t ) α γ(α + 1) + (log t )α γ(α + 1) ] + (log t )α γ(α + 1) . (3.2) in the next theorem, we prove the uniqueness of solutions for problem (1.1) via banach’s fixed point theorem. theorem 3.1. let f : [1, t ] × r → r be a continuous function and there exists a constant l > 0 such that: (h1) |f(t, x) − f(t, y)| ≤ l|x − y|, ∀t ∈ [1, t ] and x, y ∈ r. then, problem (1.1) has a unique solution on [1, t ] if lλ < 1, where λ is given by (3.2). proof. let us define m be finite number given by m = sup t∈[1,t ] |f(t, 0)|, and show that pbr ⊂ br, where br = {x ∈ c[1, t ] : ‖x‖ ≤ r} with r ≥ mλ 1 − lλ . for x ∈ br, t ∈ [1, t ], using (h1), we get |f(t, x(t))| = |f(t, x(t)) − f(t, 0) + f(t, 0)| ≤ |f(t, x(t)) − f(t, 0)| + |f(t, 0)| ≤ l‖x‖ + m ≤ lr + m. then |p(x)(t)| ≤ sup t∈[1,t ] { 1 |γ| ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj t −λ j | γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds + t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds ] + t−λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds } ≤ (lr + m) [ (log t )α−1 |γ|(α − 1) ( m∑ j=1 |βj| (log t ) α γ(α + 1) + (log t )α γ(α + 1) ) + (log t )α γ(α + 1) ] ≤ λ(lr + m) ≤ r. in consequence, ‖px‖ ≤ r, for any x ∈ br, which shows that pbr ⊂ br. now we prove that the operator p is a contraction. for (x, y) ∈ c([1, t ], r) and for each t ∈ [1, t ], cubo 23, 2 (2021) multipoint fractional sequential hadamard bvp 231 we obtain |(px)(t) − (py)(t)| ≤ sup t∈[1,t ] { 1 |γ| ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj t −λ j | γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r)) − f(r, y(r))| r dr ) ds + t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r)) − f(r, y(r))| r dr ) ds ] +t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r)) − f(r, y(r))| r dr ) ds } ≤ l [ (log t )α−1 |γ|(α − 1) ( m∑ j=1 |βj| (log t ) α γ(α + 1) + (log t )α γ(α + 1) ) + (log t )α γ(α + 1) ] ≤ lλ‖x − y‖. by the given condition lλ < 1, it follows that the operator p is a contraction. thus, the conclusion of the theorem follows by the contraction mapping principle (the banach fixed point theorem). the proof is complete. the following existence result is based on the leray-schauder nonlinear alternative. theorem 3.2 (nonlinear alternative for single valued maps [13]). let e be a banach space, c a closed, convex subset of e, u an open subset of c and 0 ∈ u. suppose that f : u → c is a continuous, compact (that is, f(u) is a relatively compact subset of c) map. then either (i) f has a fixed point in u, or (ii) there is a u ∈ ∂u (the boundary of u in c) and ν ∈ (0, 1) with u = νf(u). theorem 3.3. let f : [1, t ] × r → r be a continuous function such that the following conditions hold: (h2) there exists a function k ∈ c([1, t ], r+) and a nondecreasing function ψ : r+ → r+ such that |f(t, x)| ≤ k(t)ψ(‖x‖) for all (t, x) ∈ [1, t ] × r; (h3) there exists a positive constant s > 0 such that s ψ(s)‖k‖λ > 1, where ‖k‖ = sup t∈[1,t ] |k(t)| and λ is defined by (3.2). then problem (1.1) has at least one solution on [1, t ]. 232 bashir ahmad, amjad f. albideewi, sotiris k. ntouyas & ahmed alsaedi cubo 23, 2 (2021) proof. firstly, we shall show that the operator p defined by (3.1) maps bounded sets into bounded sets in c([1, t ], r). for a number r > 0, let br = {x ∈ c[1, t ] : ‖x‖ ≤ r} be a bounded set in c([1, t ], r). then, by assumption (h2), we obtain |(px)(t)| ≤ sup t∈[1,t ] { 1 |γ| ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj t −λ j | γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, y(r))| r dr ) ds + t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds ] + t−λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds } ≤ ψ(‖x‖)‖k‖ [ (log t )α−1 |γ|(α − 1) ( m∑ j=1 |βj| (log t ) α γ(α + 1) + (log t )α γ(α + 1) ) + (log t )α γ(α + 1) ] , and consequently, ‖px‖ ≤ λψ(r)‖k‖. next we show that p maps bounded sets into equicontinuous sets of c([1, t ], r). let τ1, τ2 ∈ [1, t ] with τ1 < τ2 and x ∈ br. then, we have |(px)(τ2) − (px)(τ1)| ≤ ψ(r)‖k‖ { 1 |γ| ( |τ−λ1 − τ −λ 2 | ∫ τ1 1 sλ−1 (log s)α−2ds +τ−λ2 ∫ τ2 τ1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj||t −λ j | γ(α − 1) ∫ τ1 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds + t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds ] + |τ−λ1 − τ −λ 2 | γ(α − 1) ∫ τ1 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds + τ−λ2 γ(α − 1) ∫ τ2 τ1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds. obviously the right-hand side of the above inequality tends to zero independently of x ∈ br as τ2 − τ1 → 0. therefore, by the arzelá-ascoli theorem, the operator is completely continuous. the result will follow from theorem 3.2 once it is established that the set of all solutions to equations x = νpx for ν ∈ (0, 1) is bounded. let x be a solution of problem (1.1). then, for t ∈ [1, t ], as in the first step, we can find that ‖x‖ = sup t∈[1,t ] {ν(px)(t)} ≤ λψ(‖x‖)‖k‖, cubo 23, 2 (2021) multipoint fractional sequential hadamard bvp 233 which leads to ‖x‖ λψ(‖x‖)‖k‖ ≤ 1. by condition (h3), there exists s > 0 such that ‖x‖ 6= s. let us set u = {x ∈ c([1, t ], r) : ‖x‖ < s}. note that the operator p : u → c([1, t ], r) is continuous and completely continuous. from the choice of u, there is no x ∈ ∂u such that x = νpx for some ν ∈ (0, 1). consequently, we deduce by theorem 3.2 that p has a fixed point x ∈ u, which is a solution of problem (1.1). this completes the proof. our final existence result is based on krasnosel’skĭı’s fixed point theorem. theorem 3.4. (krasnosel’skĭı’s fixed point theorem) let m be a closed convex and nonempty subset of a banach space x. let a,b be the operators such that (i) ax + by ∈ m whenever x, y ∈ m, (ii) b is a contraction mapping, (iii) a is compact and continuous. then there exists z ∈ m such that z = az + bz. theorem 3.5. let f : [1, t ] × r → r be a continuous function satisfying the condition (h1). in addition, we assume that: (h4) |f(t, x)| ≤ µ(t) for all (t, x) ∈ [1, t ] × r, µ ∈ c([1, t ], r+). then, the boundary value problem (1.1) has at least one solution on [1, t ], provided that l ( λ − (log t )α γ(α + 1) ) < 1, (3.3) where λ is given by (3.2). proof. consider bρ = {x ∈ g : ‖x‖ ≤ ρ}, ‖µ‖ = sup t∈[0,1] |µ(t)|, with ρ ≥ ‖µ‖λ. then we define the operators p1 and p2 on bρ as (p1x)(t) = 1 γ ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × { m∑ j=1 βj t −λ j γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds − t −λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds } , t ∈ [1, t ], (p2x)(t) = t−λ γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds, t ∈ [1, t ]. 234 bashir ahmad, amjad f. albideewi, sotiris k. ntouyas & ahmed alsaedi cubo 23, 2 (2021) as in 3.1 we can prove that ‖p1x + p2y‖ ≤ ‖µ‖λ < ρ, and thus, p1x + p2y ∈ bρ. by using condition (3.3) it is easy to prove that p1 is a contraction (see also 3.1). moreover the continuous operator p2 is uniformly bounded, as ‖p2‖ ≤ (log t )α γ(α + 1) ‖µ‖, and equicontinuous as |(p2x)(τ2) − (p2x)(τ1)| ≤ |τ−λ1 − τ −λ 2 | γ(α − 1) ∫ τ1 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds + τ−λ2 γ(α − 1) ∫ τ2 τ1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds. hence, by arzelá-ascoli theorem, p2 is compact on bρ. thus all the assumptions of 3.4 are satisfied and the conclusion of 3.4 implies that the boundary value problem (1.1) has at least one solution on [1, t ]. the proof is completed. example 3.6. consider the boundary value problem for hadamard fractional differential equations        ( hd7/4 + 2 hd3/4 ) x(t) = f(t, x(t)), t ∈ [1, e], x(1) = 0, x(e) = 3 ∑ j=1 βj x(tj). (3.4) here, α = 7/4, λ = 2, t = e, m = 3, β1 = 1/3, β2 = 1/9, β3 = 1/27, t1 = 5/4, t2 = 3/2, t3 = 7/4 and f(t, x) = 1 13 √ t2 + 24 |x| 1 + |x| + 1 t + 2 + log t. clearly, l = 1/65 as |f(t, x)−f(t, y)| ≤ (1/65)|x−y|. using the given data, we have |γ| ≈ 0.691358 and λ ≈ 1.104500. then lλ ≈ 0.016992 < 1. thus, by 3.1, the boundary value problem (3.4) has a unique solution on [1, e]. acknowledgement the authors thank the reviewers for their useful comments on our work that led to the improvement of the original manuscript. cubo 23, 2 (2021) multipoint fractional sequential hadamard bvp 235 references [1] b. ahmad and s. k. ntouyas, “some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions”, rev. r. acad. cienc. exactas, f́ıs. nat. ser. a mat. racsam, vol. 110, no. 1 , pp. 159-172, 2016. 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[28] w. zhang and j. ni, “new multiple positive solutions for hadamard-type fractional differential equations with nonlocal conditions on an infinite interval”, appl. math. lett., vol. 118, id 107165, 10 pages, 2021. introduction preliminaries existence and uniqueness results cubo, a mathematical journal vol. 23, no. 02, pp. 313–331, august 2021 doi: 10.4067/s0719-06462021000200313 a new class of graceful graphs: k-enriched fan graphs and their characterisations m. haviar1,2 s. kurtuĺık1 1 department of mathematics, faculty of natural sciences, matej bel university, banská bystrica, slovakia. miroslav.haviar@umb.sk kurtuliksamuel@gmail.com 2 mathematical institute, slovak academy of sciences, bratislava, slovakia. abstract the graceful tree conjecture stated by rosa in the mid 1960s says that every tree can be gracefully labelled. it is one of the best known open problems in graph theory. the conjecture has caused a great interest in the study of gracefulness of simple graphs and has led to many new contributions to the list of graceful graphs. however, it has to be acknowledged that not much is known about the structure of graceful graphs after 55 years. our paper adds an infinite family of classes of graceful graphs to the list of known simple graceful graphs. we introduce classes of k-enriched fan graphs kfn for all integers k, n ≥ 2 and we prove that these graphs are graceful. moreover, we provide characterizations of the k-enriched fan graphs kfn among all simple graphs via sheppard’s labelling sequences introduced in the 1970s, as well as via labelling relations and graph chessboards. these last approaches are new tools for the study of graceful graphs introduced by haviar and ivaška in 2015. the labelling relations are closely related to sheppard’s labelling sequences while the graph chessboards provide a nice visualization of graceful labellings. we close our paper with an open problem concerning another infinite family of extended fan graphs. resumen la conjetura del árbol amable enunciada por rosa a mediados de los 1960s dice que cada árbol puede ser etiquetado amablemente. es uno de los problemas abiertos mejor conocidos en teoŕıa de grafos. la conjetura ha causado un gran interés en el estudio de la amabilidad de grafos simples y ha llevado a muchas contribuciones nuevas a la lista de grafos amables. de todas formas, debe reconocerse que no se sabe mucho acerca de la estructura de grafos amables tras 55 años. nuestro art́ıculo añade una familia infinita de clases de grafos amables a la lista de grafos amables simples conocidos. introducimos clases de grafos abanico k-enriquecidos kfn para todos los enteros k, n ≥ 2 y demostramos que estos grafos son amables. más aún, entregamos caracterizaciones de los grafos abanico k-enriquecidos kfn entre todos los grafos simples v́ıa sucesiones de etiquetado de sheppard introducidas en los 1970s, y también a través de relaciones de etiquetados y tableros de ajedrez de grafos. estos últimos acercamientos son herramientas nuevas para el estudio de grafos amables introducidos por haviar e ivaška en 2015. las relaciones de etiquetado están relacionadas cercanamente con las sucesiones de etiquetado de sheppard mientras que los tableros de ajedrez de grafos proveen una linda visualización de etiquetados amables. concluimos nuestro art́ıculo con un problema abierto relacionado con otra familia infinita de grafos abanico extendidos. keywords and phrases: graph, graceful labelling, graph chessboard, labelling sequence, labelling relation. 2020 ams mathematics subject classification: 05c78 accepted: 10 june, 2021 received: 19 february, 2021 ©2021 m. haviar et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200313 https://orcid.org/0000-0002-9721-152x 314 m. haviar & s. kurtuĺık cubo 23, 2 (2021) 1 introduction a simple graph of size m has a graceful labelling, and it is said to be graceful, when its vertices can be assigned different labels from the set {0, 1, ...,m} such that the absolute values of the differences in vertex labels of edges form the set {1, ...,m}. the graceful tree conjecture stated by rosa in [9] and [10] says that every tree is graceful. the conjecture has now been open for more than a half century and is one of the most attractive open problems in graph theory. the best source of information on attacks of the conjecture and on the study of labellings of graphs is the electronic book by gallian [2]. still, not much is known about the structure of graceful graphs. one of the known results, due to hrnčiar and haviar [6], is that all trees of diameter five are graceful. the graceful tree conjecture has led to a much increased interest in the study of gracefulness of simple graphs. in this paper we introduce a new infinite family of classes of simple graphs, kenriched fan graphs kfn, for all integers k,n ≥ 2, and we show that they are graceful. these classes of graphs have been recently considered in the second author’s msc thesis [8]. the k-enriched fan graphs kfn are a natural generalisation of the class of fan graphs fn, which were shown to be graceful in [5]. for a better understanding of the general k-enriched fan graphs and their characterisations, the special cases k = 2 and k = 3 of double and triple fan graphs are dealt with separately. characterisations of the double and triple fan graphs, and then of the k-enriched fan graphs, are presented using the tools of labelling sequences, labelling relations and graph chessboards. labelling sequences were introduced in 1976 by sheppard in [11] while the labelling relations and graph chessboards as new tools for the study of graceful graphs were introduced and studied rather recently by haviar and ivaška in [5]. the basic terms and facts needed in this paper are presented in section 2. this includes the concepts of labelling sequences, labelling relations and graph chessboards. in section 3 we describe fan graphs and their graceful labellings from [5]. then we generalise the concept of a fan graph to the one called a double fan graph resp. triple fan graph by connecting n paths p2 resp. p3 to the main path of a given fan graph fn. we construct for these graphs their graceful labellings and describe them by the corresponding labelling sequences, the labelling relations and the graph chessboards. finally we study the general case of the k-enriched fan graphs kfn for any integers k,n ≥ 2. these graphs contain, compared to the classic fan graphs fn, n copies of the star sk connected to the main path pn. again, we construct for these graphs their graceful labellings and characterize them by their labelling sequences, the labelling relations and the graph chessboards. we conclude the paper with proposing another natural extension of the class of fan graphs and raising an open problem whether these extended fan graphs can be shown to be graceful and cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 315 characterized in the manner presented here. 2 preliminaries in this section we recall the necessary basic terms concerning graph labellings as well as the concepts of labelling sequences, labelling relations and simple chessboards, which we use as our tools to describe new classes of graceful graphs. these definitions are taken primarily from [10] and [5]. in this paper we study only finite simple graphs, that is, finite unoriented graphs without loops and multiple edges. the following concept was called valuation by rosa in his seminal paper [10]. definition 2.1 ([10, 5]). a vertex labelling (or labelling for short) f of a simple graph g = (v,e) is a one-to-one mapping of its vertex set v(g) into the set of non-negative integers assigning to the vertices so-called vertex labels. by the label of an edge uv in the labelling f we mean the number |f(u) −f(v)|, where f(u),f(v) are the vertex labels of u,v. in this text we will denote by f(vg) the set of all vertex labels and by f(eg) the set of all edge labels in the labelling f of a graph g. several types of graph labellings are known, e.g. α,β,σ,ρ, which have become very well-known since the publication of rosa’s paper [10] in 1967, and further γ,δ,p,q introduced by rosa in his dissertation thesis [9] in 1965. in this text we study only β-labellings called graceful labellings. definition 2.2 ([10, 5]). a graceful labelling (or β-labelling) of a graph g = (v,e) of size m is a vertex labelling with the following properties: (1) f(vg) ⊆{0, 1, . . . ,m}, and (2) f(eg) = {1, 2, . . . ,m}. the term graceful was given to β-labellings in 1972 by golomb [4] and its immediate use was enhanced by a popularization by gardner [3]. hence a graceful labelling of a graph of size m has vertex labels among the numbers 0, 1, ...,m such that the induced edge labels are different and cover all values 1, 2, ...,m. in figure 1 we can see some graceful graphs. each graceful graph can be represented by a sequence of non-negative integers. this was shown by sheppard in [11] where he introduced the concept of a labelling sequence as follows: definition 2.3 ([11, 5]). for a positive integer m, a labelling sequence is the sequence of nonnegative integers (j1,j2, . . . ,jm), denoted (ji), where 0 ≤ ji ≤ m− i for all i ∈{1, 2, . . . ,m}. (ls) 316 m. haviar & s. kurtuĺık cubo 23, 2 (2021) 0 1 3 0 4 3 2 1 3 0 65 4 2 figure 1: some graceful graphs sheppard also proved that there is a one-to-one correspondence between graceful labellings of graphs (without isolated vertices) and labelling sequences. therefore we can understand labelling sequences as a tool to encode graceful labellings of graphs. the connection is described in the following theorem. theorem 2.4 ([11, 5]). there exists a one-to-one correspondence between graphs of size m having a graceful labelling f and between labelling sequences (ji) of m terms. the correspondence is given by ji = min{f(u),f(v)}, i ∈{1, 2, . . . ,m}, where u,v are the end-vertices of the edge labelled i. now we recall the definition of a labelling relation. it is another tool to describe gracefully labelled graphs, which is closely related to the labelling sequence. the concept of the labelling relation was introduced and studied in [5]. definition 2.5 ([5]). let l = (j1,j2, . . . ,jm) be a labelling sequence. then the relation a(l) = {[ji,ji + i], i ∈{1, 2, . . . ,m}} is called a labelling relation assigned to the labelling sequence l. to visualize a labelling relation and also a labelling sequence we will use a labelling table (see figure 2). a table is formed by using the numbers 1, 2, ...,m as headers for the columns, followed by two rows. the first row contains the numbers from the labelling sequence and the second row contains the sums of the corresponding numbers from the heading and from the first row. the pairs from first and second row in each column are then the elements of the labelling relation (and correspond to the edges of the graph). 1 2 3 . . . m j1 j2 j3 . . . jm j1 + 1 j2 + 2 j3 + 3 . . . jm + m figure 2: displaying a labelling table example 2.6. in figure 3 we can see the labelling table assigned to the labelling sequence (5, 4, 3, 2, 1, 0) and its graceful graph whose edges correspond to the elements of the labelling relation. cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 317 1 2 3 4 5 6 5 4 3 2 1 0 6 6 6 6 6 6 0 12 3 4 5 6 figure 3: example of a labelling table and its corresponding graceful graph every labelled simple graph of order n can be represented by a chessboard, i.e. a table with n rows and n columns, where every edge labelled uv is represented by a pair of dots with coordinates [u,v] and [v,u]. this idea of visualization of vertex labellings of graphs by chessboards and independent discoveries of similar ideas are described in [5, chapter 2]. one can also obtain such a graph chessboard by taking the adjacency matrix of a graph and placing dots in the cells corresponding to “ones” in the matrix while leaving the cells corresponding to “zeros” in the matrix empty (cf. [5, p. 25]). several types of graph chessboards like simple chessboards, double chessboards, m-chessboards, dual chessboards and twin chessboards were studied by haviar and ivaška in [5]. in this text we 0 6 7 2 5 9 3 figure 4: example of a graph of size m = 9 and its corresponding simple chessboard use only the idea of a simple chessboard, which is a useful tool in the visualization of gracefully labelled graphs. for a positive integer m consider an (m + 1) × (m + 1)-table. rows are numbered by 0, 1, ...,m from the top to the bottom and columns are numbered by 0, 1, ...,m from the left to the right as shown in figure 4. the cell with coordinates [i,j] of the table will mean the cell in the i-th row and the j-th column. the r-th diagonal in the table is the set of all cells with coordinates [i,j] where i − j = r and i ≥ j. the main diagonal is the 0-th diagonal in the table and all other diagonals are called associate diagonals. to a graph of size m whose vertices are labelled by different numbers from the set {0, 1, 2, ...,m} 318 m. haviar & s. kurtuĺık cubo 23, 2 (2021) we assign its simple chessboard as the (m + 1) × (m + 1)-table described above such that every edge labelled uv in the graph is represented by a pair of dots in the cells with coordinates [u,v] and [v,u]. it follows that the simple chessboards are symmetric about the main diagonal. an illustration of the simple chessboard of such a labelled graph of size 9 is in figure 4. if there is exactly one dot on each of the associate diagonals, then the simple chessboard will be called graceful as it clearly encodes a graceful graph. we can see a gracefully labelled graph of size 8 and its graceful simple chessboard in figure 5. 0 8 1 4 7 3 5 2 figure 5: gracefully labelled graph and its corresponding graceful simple chessboard 3 fan graphs and their descriptions a fan graph is a join of a path and a single vertex k1. (see [5, section 4.4.7] and figure 6.) its “bottom part” is formed by a star. in this text we introduce the notation fn for a fan graph whose main path is pn. we obviously require n ≥ 2 in order to have the shape of “a fan”. the fan graph fn thus has order n + 1 and size 2n − 1. in figure 6 we can see a gracefully labelled fan graph f8 of size 15, with the main path p8, its corresponding simple chessboard and its labelling relation. the following two results taken from [5] characterize fan graphs via their labelling sequences, labelling relations and simple chessboards. (it is important to notice that [5, section 4.4.7] considers the fan graphs that are fn+1 in our present notation. in figure 4.10 there we have the fan graph f7 with considering n = 6. that is why for our graphs g studied in [5, section 4.4.7] we talk about their size m = 2n + 1.) the notation n in the next theorem and in our subsequent results is used, as in [5], for the set of all positive integers and the notation bxc is used for the largest integer not greater than x (the floor function). theorem 3.1 ([5]). let g be a graph of size m = 2n + 1 for some n ∈ n. then g is a fan graph cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 319 0 7 1 6 2 5 3 4 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3 3 2 2 1 1 0 7 6 5 4 3 2 1 0 4 5 5 6 6 7 7 15 15 15 15 15 15 15 15 figure 6: representations of the gracefully labelled fan graph f8 if and only if there exists a labelling sequence (j1,j2, . . . ,jm) of g such that ji =   ⌊ n−i+1 2 ⌋ , if i ≤ n, m− i, if i > n. (lsfg) corollary 3.2 ([5]). let g be a graph of size m = 2n + 1 that can be gracefully labelled and let l be the set of all its labelling sequences. then the following are equivalent: (a) g is a fan graph. (b) there exists a labelling sequence l ∈l which satisfies (lsfg). (c) for some l ∈l, the labelling relation is a(l) ={[i,n− 1] | i ∈{0, 1, . . . , ⌊ n−i 2 ⌋ }}∪ {[i,n− i + 1] | i ∈{0, 1, . . . , ⌊ n 2 ⌋ }}∪ {[i,m] | i ∈{0, 1, . . . ,n}}. (d) some chessboard of g looks like the chessboard in figure 6. example 3.3. the sequence (3, 3, 2, 2, 1, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0) is the labelling sequence of the fan graph f8 whose representations by the graph chessboard and the labelling table are seen in figure 6. 320 m. haviar & s. kurtuĺık cubo 23, 2 (2021) 4 double and triple fan graphs and their descriptions by a double fan graph dfn we will mean the fan graph fn with extra n pendant vertices, which are attached to the main path pn. hence double fan graphs dfn can be understood such that we connect n paths p2 to a given fan graph fn by identifying one vertex of each of the paths p2 with one vertex of the main path pn of the fan graph. when we connect this way n paths p3 to a fan graph fn, we get a triple fan graph tfn. in figure 7 we see the double fan graph df5 on the left side and the triple fan graph tf4 on the right side. figure 7: the double fan graph df5 (left) and the triple fan graph tf4 (right) we can divide vertices of such graphs into these groups: (i) the root vertex, which forms together with n adjacent edges the “bottom” star, (ii) the “middle” vertices of the main path pn, and (iii) the “upper” pendant vertices, which together with the “middle” vertices form n paths p2 resp. p3 connected to the main path. hence double fan graphs dfn resp. triple fan graphs tfn can be understood as certain extensions of the fan graphs by adding in the “upper part” n paths p2 (in double fan graphs) resp. n paths p3 (in triple fan graphs). we will often refer to the “bottom”, “middle” and “upper parts” (in spite of the fact these parts are not pairwise disjoint and our naming of these parts refers only to our chosen visualization of these graphs) to assist in our proofs. example 4.1. the sequence (2, 1, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0) is a labelling sequence of a gracefully labelled double fan graph df5 of size 14 and order 11. the corresponding graph chessboard and labelling table are in figure 8. because the dots in the graph chessboard representing the edges of the mentioned “bottom”, “middle” and “upper parts” of the double fan graph look respectively like the “bottom part”, the “head” and the “neck” of a swan, we will refer to such chessboards as swan chessboards. in the next theorems we characterise double fan graphs dfn and triple fan graphs tfn by their simple chessboards, labelling sequences and labelling relations. within these characterisations we prove there are specific graceful labellings of these graphs, thus we show the graphs dfn and tfn cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 321 0 4 1 3 5 13 7 11 2 9 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2 1 1 0 0 1 2 3 4 4 3 2 1 0 3 3 4 4 5 7 9 11 13 14 14 14 14 14 figure 8: representations of the gracefully labelled double fan graph df5 are graceful. so our double and triple fan graphs are new additions to the list of graceful graphs. theorem 4.2. let g be a graph of size m = 3n− 1 for some n ∈ n −{1}. then the following are equivalent: (1) g is the double fan graph dfn. (2) there is a graceful labelling of g with a swan chessboard. (3) there exists a labelling sequence l = (j1,j2, ...,jm) of g such that ji =   ⌊ n−i 2 ⌋ , if i < n, i−n, if n ≤ i < 2n, m− i, if i ≥ 2n. (lsdfg) (4) there exists a labelling sequence l of g with the labelling relation a(l) ={[u,v] | u,v ∈{0, 1, . . . ,n− 1}, u < v, n− 1 ≤ u + v ≤ n}∪ {[u,n + 2u] | u ∈{0, 1, . . . ,n− 1}}∪ {[n− 1 −u,m] | u ∈{0, 1, . . . ,n− 1}}. proof. (1) ⇒ (2) let g be the double fan graph dfn. we will label it in the following way. let the labelling of the single vertex at the “bottom” (the root of dfn) be m. let the vertices of the middle path pn be labelled from one end-vertex of pn gradually by numbers 0,n−1, 1,n−2, 2, ... 322 m. haviar & s. kurtuĺık cubo 23, 2 (2021) (see figure 8). hence the second end-vertex is labelled by ⌊ n 2 ⌋ . finally we will label n pendant vertices in the “upper part”. we start from the vertex which is attached to end-vertex, at which we started the labelling of the main path pn. we label the vertices from this vertex gradually by numbers n,m−1,n + 2,m−3,n + 4, .... we get edges with labellings {0,n},{n−1,m−1},{1,n + 2},{n − 2,m − 3}, and so on, in this “upper part”. now our labelling is done and we show this labelling is graceful with a corresponding swan chessboard (see figure 8). one can easily verify the following statements: (i) the “bottom part” of g which is the star of size n is in the chessboard represented by n dots in the bottom row with coordinates [m, 0], [m, 1], ..., [m,n− 1] which form the “bottom of the swan”. (ii) the “middle part” of g which is the path pn is in the chessboard represented by n− 1 dots with coordinates [n−1, 0], [n−1, 1], [n−2, 1], [n−2, 2], ... which form the “head of the swan”. (iii) the “upper part” of g which is the union of n paths p2 is in the chessboard represented by n dots with coordinates [n, 0], [n + 2, 1], [n + 4, 2], [n + 6, 3], ... which form the “neck of the swan”. the described swan chessboard with three blocks of dots obviously has one dot on each diagonal. hence g is graceful. (2) ⇒ (3) assume we have a graceful labelling of g with a swan chessboard (as in figure 8). we will verify that when we make the labelling sequence corresponding to this graceful labelling, we get exactly the labelling sequence satisfying (lsdfg). we use the distribution of dots in a swan chessboard into three blocks as in the previous part of the proof. so we consider the “head”, the “neck” and the “bottom part” of a “swan”. one can verify that these three blocks of dots in our chessboard can be assigned to the corresponding integers in the labelling sequence. the first block of dots, the “head”, is represented in the corresponding labelling sequence by the integers ji of the form ⌊ n−i 2 ⌋ for i < n. the second block of dots, the “neck”, is represented in the corresponding labelling sequence by the integers ji of the form i − n for n ≤ i < 2n. the third block of dots, the “bottom part”, is represented in the corresponding labelling sequence by the integers ji of the form m− i for i ≥ 2n. we have shown that there exists a labelling sequence of g satisfying the formula (lsdfg). (3) ⇒ (4) assume we have a labelling sequence l of g which satisfies (lsdfg). we will show that the labelling relation a(l) from this labelling sequence consists of the pairs as described in (4). indeed, one can verify that the non-negative integers ji from the labelling sequence, which have the form ⌊ n−i 2 ⌋ for i < n, correspond in a(l) to the pairs [u,v] where u,v ∈ {0, 1, . . . ,n− 1}, u < v and n − 1 ≤ u + v ≤ n. the next ji from the labelling sequence, which have the form i − n for cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 323 n ≤ i < 2n, correspond to the pairs [u,n + 2u] for u ∈ {0, 1, . . . ,n − 1}. finally, ji from the labelling sequence, which have the form m− i for i ≥ 2n, correspond to the pairs [n− 1 −u,m] for u ∈{0, 1, . . . ,n− 1}. (4) ⇒ (1) assume that there exists a labelling sequence l of g with the labelling relation a(l) as in (4). from the definition we know the labelled edges of g correspond to the pairs in a(l). one can verify that the pairs [n − 1 − u,m] from a(l) for u ∈ {0, 1, . . . ,n − 1} correspond to the edges in “the bottom part” of graph g which therefore is a star of size n. the pairs [u,v] from a(l) for u,v ∈ {0, 1, . . . ,n − 1}, u < v and such that n − 1 ≤ u + v ≤ n correspond to the edges in “the middle part” of g which therefore is the path pn. finally, the pairs [u,n + 2u] for u ∈{0, 1, . . . ,n− 1} correspond to the edges in “the upper part” of g which therefore form n paths p2 connected to the main path pn. so the three parts in a(l) correspond to the three parts of the double fan graph dfn. hence a double fan graph can always be gracefully labelled so that its chessboard has the “head”, the “neck” and the “bottom part” of a “swan”. in the rest of this section we focus on a description of triple fan graphs. example 4.3. the sequence (2, 2, 1, 1, 0, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0) is a labelling sequence of a triple fan graph of size 23 and order 19. the corresponding graph diagram, graph chessboard and labelling table are in figure 9. notice that while the swan chessboard of a double fan graph had one “neck” connecting the “head” and the “bottom part” of the “swan”, the simple chessboard of a triple fan graph has 3 blocks of dots that look like a “swan without head”, then “the head separated from the neck”, and one extra “separated neck”. hence we will refer to such simple chessboards representing the triple fan graphs as 3-part swan chessboards. the following characterisations of triple fan graphs can be proven using a method similar to that used in the previous theorem, and will be covered by the general case in the subsequent section. 324 m. haviar & s. kurtuĺık cubo 23, 2 (2021) 0 5 1 4 2 3 12 22 14 20 16 18 6 11 7 10 8 9 23 1 2 3 4 5 6 7 8 9 10 11 2 2 1 1 0 6 7 8 9 10 11 3 4 4 5 5 12 14 16 18 20 22 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 5 4 3 2 1 0 12 14 16 18 20 22 23 23 23 23 23 23 figure 9: representations of the gracefully labelled triple fan graph tf6 theorem 4.4. let g be a graph of size m = 4n− 1 for some n ∈ n −{1}. then the following are equivalent: (1) g is the triple fan graph tfn. (2) there is a graceful labelling of g with a 3-part swan chessboard. cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 325 (3) there exists a labelling sequence l = (j1,j2, ...,jm) of g such that ji =   ⌊ n−i 2 ⌋ , if i < n, i, if n ≤ i < 2n, i− 2n, if 2n ≤ i < 3n, m− i, if i ≥ 3n. (lstfg) (4) there exists a labelling sequence l of g with the labelling relation a(l) ={[u,v] | u,v ∈{0, 1, . . . ,n− 1}, u < v, n− 1 ≤ u + v ≤ n}∪ {[u, 2u] | u ∈{n,n + 1, . . . , 2n− 1}}∪ {[u, 2n + 2u] | u ∈{0, 1, . . . ,n− 1}}∪ {[n− 1 −u,m] | u ∈{0, 1, . . . ,n− 1}}. 5 general case: k-enriched fan graphs kfn and their descriptions according to the previous section we would assume that the more “separated necks” we have in the graph chessboard, the longer the paths in the “upper part” of the graph will be. our work with graph processor introduced in [7] and much used also in [5] has led us to a surprising observation that the “necks” in the graph chessboard do not represent in the corresponding graph the paths but the stars. we use for these graphs the term k-enriched fan graphs kfn, where k represents the order of the stars sk in the “upper part” of the graph and n represents the order of the “middle” path pn. now we formally define this new term. definition 5.1. the k-enriched fan graph kfn, for fixed integers k,n ≥ 2, is the graph of size (k + 1)n − 1 obtained by connecting n copies of the star sk of order k to the fan graph fn such that one vertex of each copy of the star sk is identified with one vertex of the main path pn of fn. we notice that here the stars sk are connected to the main path pn of the fan graph fn exactly as in the previous section the paths p2 (which are the stars s2) resp. p3 (the stars s3) were connected to the main path pn of fn in the case of the double fan graphs dfn resp. triple fan graphs tfn. example 5.2. in figure 10 we see a gracefully labelled 4-enriched fan graph 4f6 obtained by connecting 6 copies of stars s4 of order 4 to the fan graph f6 as described above. the corresponding simple chessboard and labelling table are also in figure 10. the labelling sequence of this graph is (2, 2, 1, 1, 0, 12, 13, 14, 15, 16, 17, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0). 326 m. haviar & s. kurtuĺık cubo 23, 2 (2021) we notice that we can divide the vertices of the k-enriched fan graph kfn with any k ≥ 2 similarly as in the case of double and triple fan graphs. hence the graph consists of the “bottom part” (the root vertex and the adjacent edges), the “middle part” (the main path pn) and the “upper part” (the disjoint union of n stars sk). the corresponding simple chessboard of the k-enriched fan graph kfn for k ≥ 3 has k blocks of dots that form a “swan without head”, then “the separated head”, and k−2 extra “separated necks”. hence we will refer to such simple chessboards representing the k-enriched fan graphs kfn as k-part swan chessboards. theorem 5.3. let g be a graph of size m = (k + 1)n− 1 for some fixed integers k,n ≥ 2. then the following are equivalent: (1) g is the k-enriched fan graph kfn. (2) there is a graceful labelling of g with a k-part swan chessboard. (3) there exists a labelling sequence l = (j1,j2, ...,jm) of g such that ji =   ⌊ n−i 2 ⌋ , if i < n, i−n + (k − 2)n, if n ≤ i < 2n, i−n + (k − 4)n, if 2n ≤ i < 3n, ... ... i−n + (k − 2(k − 1))n, if (k − 1)n ≤ i < kn, m− i, if i ≥ kn. (lskfg) (4) there exists a labelling sequence l of g with the labelling relation a(l) of the form {[ ⌊ n−i 2 ⌋ , ⌊ n−i 2 ⌋ + i] | i < n}∪ {[i−n + (k − 2)n, 2i−n + (k − 2)n] | n ≤ i < 2n}∪ {[i−n + (k − 4)n, 2i−n + (k − 4)n] | 2n ≤ i < 3n}∪ ... {[i−n + (k − 2(k − 1))n, 2i−n + (k − 2(k − 1))n] | (k − 1)n ≤ i < kn}∪ {[m− i,m] | i ≥ kn}. proof. (1) ⇒ (2) let g be the k-enriched fan graph kfn. we will label the graph g in a way similar to those labellings for the double and triple fan graphs. let the labelling of the root of kfn be m. let the vertices of the main path pn be labelled from one end-vertex of pn (we see it in figure 10 from the left side) gradually by numbers 0,n−1, 1,n−2, ... hence the second end-vertex cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 327 0 5 1 4 2 3 18 28 20 26 22 24 12 6 17 11 13 7 16 10 14 8 15 9 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 2 1 1 0 12 13 14 15 16 17 6 7 8 9 3 4 4 5 5 18 20 22 24 26 28 18 20 22 24 16 17 18 19 20 21 22 23 24 25 26 27 28 29 10 11 0 1 2 3 4 5 5 4 3 2 1 0 26 28 18 20 22 24 26 28 29 29 29 29 29 29 figure 10: representations of the gracefully labelled 4-enriched fan graph 4f6 of pn is labelled by ⌊ n 2 ⌋ . we start to label the central vertices of the n stars sk in the “upper part” of the graph g from the central vertex of sk attached to the vertex of the main path pn labelled by n − 1. we label this vertex by m − 1, the next central vertex of sk attached to the vertex n−2 will be labelled by m−3, the central vertex of sk attached to the vertex n−3 will be labelled by m− 5 and so on, we end by labelling the central vertex of sk attached to the vertex 0 328 m. haviar & s. kurtuĺık cubo 23, 2 (2021) by m−2n + 1 (see figure 10). to label the remaining vertices of the n stars sk, we begin with the star, whose central vertex is labelled by m− 1. the remaining vertices of this star are labelled by ((m− 1) − (2n− 1)), ((m− 1) − (3n− 1)),... it does not matter in what ‘direction’ we label these vertices. we proceed in the same way with labelling the remaining vertices of the other stars in the “upper part” of g, the role of the previous number m− 1 is always played by the labelling of the central vertex of the given star (see figure 10). now our labelling of g is done and we show that this labelling is graceful with a k-part swan chessboard. for this we use a visualization via the corresponding simple chessboard of g (see figure 10). one can easily verify that: (i) the “bottom part” of g which is the star of size n is in the simple chessboard represented by n dots with coordinates [m, 0], [m, 1], ..., [m,n− 1] in the bottom row of the chessboard. (exactly as for the double and triple fan graphs.) (ii) the “middle part” of g which is the main path pn is in the chessboard represented by n−1 dots with coordinates [n−1, 0], [n−1, 1], [n−2, 1], [n−2, 2], ... which form a certain “separated head” in the case k ≥ 3. (exactly as for the triple fan graphs.) (iii) the “upper part” of g which is the union of n stars sk is in the simple chessboard represented by k − 1 “necks” each consisting of n dots. hence the graph chessboard corresponding to our labelling is a k-part swan chessboard (as chessboard in figure 10) and obviously it has exactly one dot on each diagonal. so the described labelling of g is graceful. (2) ⇒ (3) assume we have a graceful labelling of g with a k-part swan chessboard. consider the following k blocks of dots of this chessboard: the “separated head”, k − 1 swan “necks” and the bottom row of the chessboard. one can verify that the “separated head” is represented in the corresponding labelling sequence by the integers ji having the form ⌊ n−i 2 ⌋ for i < n. the first “neck” from the right below the main diagonal is represented in the corresponding labelling sequence by the integers ji having the form i−n + (k − 2)n for n ≤ i < 2n. the second “neck” from the right below the main diagonal is represented in the corresponding labelling sequence by the integers ji having the form i−n + (k−4)n for 2n ≤ i < 3n, and so on. finally, the last “neck” (the first one from the left) is represented in the corresponding labelling sequence by the integers ji having the form i−n+ (k−2(k−1))n for (k−1)n ≤ i < kn. the bottom row of the chessboard is represented in the corresponding labelling sequence by the integers ji having the form m− i for i ≥ kn. so we have shown that the labelling sequence corresponding to our k-part swan chessboard satisfies the formula (lskfg). cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 329 (3) ⇒ (4) assume there is a labelling sequence l of g which satisfies (lskfg). one can verify that the corresponding labelling relation a(l) consists of the pairs as described in (4). indeed, the nonnegative integers ji from the labelling sequence having the form ⌊ n−i 2 ⌋ for i < n correspond in a(l) to the pairs [ ⌊ n−i 2 ⌋ , ⌊ n−i 2 ⌋ + i]. the next integers ji from the labelling sequence, which have the form i−n+ (k−2)n for n ≤ i < 2n, correspond in a(l) to pairs [i−n+ (k−2)n, 2i−n+ (k−2)n]. further, the integers ji from the labelling sequence, which have the form i − n + (k − 4)n for 2n ≤ i < 3n, correspond in a(l) to pairs [i−n + (k − 4)n, 2i−n + (k − 4)n], and so on for the next parts of the labelling sequence, which correspond to the “necks”. finally, the integers ji from the labelling sequence, which have the form m− i for i ≥ kn, correspond to the pairs [m− i,m]. (4) ⇒ (1) let l be a labelling sequence of g with the labelling relation a(l) as in (4). the pairs [m− i,m] in a(l) for i ≥ kn correspond to the edges of the “bottom part” of g, which therefore is a star of order n. the pairs [ ⌊ n−i 2 ⌋ , ⌊ n−i 2 ⌋ + i] in a(l) for i < n correspond in the graph g to the edges which form the “middle path” pn. the remaining pairs in a(l) correspond to the edges in “the upper part” of g. more precisely, each of the pairs [i − n + (k − 2)n, 2i − n + (k − 2)n] for n ≤ i < 2n corresponds to one edge in each of the n “upper” stars sk, each of the pairs [i − n + (k − 4)n, 2i − n + (k − 4)n] for 2n ≤ i < 3n corresponds to one of the (other) edges in each of the n stars sk in the “upper part” of g, and so on, and finally, each of the pairs [i−n+ (k−2(k−1))n, 2i−n+ (k−2(k−1))n] for (k−1)n ≤ i < kn corresponds to the remaining edges in each of the n stars sk in the “upper part” of g. so we get n stars sk in the “upper part” of g. thus we have in g the “bottom” star of size n, the “middle” path pn and the n “upper” stars sk connected to the vertices of the “middle” path pn. hence the graph g is the k-enriched fan graph kfn. 6 conclusion studies of other extended fan graphs can be found in the literature. some of them are seen in figure 11 below. the first two graphs are taken from [1]. the author named the first one as a double fan graph, but it is different from our double fan graph as defined in this paper. it consists of two fan graphs that have a common main path. the second graph was obtained by adding some edges to a vertex of the main path. one of the possibilities for other extensions of fan graphs would be adding paths of the same length pk (not stars sk as in our case) to the main path pn of the fan graph fn. these graphs and our k-enriched fan graphs kfn would be the same for the cases k = 2 and k = 3. such extended fan graphs, let us denote them as pkfn (our k-enriched fan graphs kfn in this more universal notation would be denoted skfn) can look like the third graph in figure 11. this graph would be the extended fan graph p4f3 as the paths p4 are connected to the main path of the fan graph f3. we 330 m. haviar & s. kurtuĺık cubo 23, 2 (2021) conclude our paper with the following open problem: problem 6.1. are the extended fan graphs pkfn (obtained by connecting in the described way n paths pk to the main path pn of the fan graph fn) graceful? and if so, is there a characterization of them via a certain “canonical” graph chessboard and the corresponding labelling sequence and the labelling relation like the characterizations of the graceful graphs presented in this paper? figure 11: other extended fan graphs acknowledgements the first author acknowledges a support by slovak vega grant 2/0078/20 and a visiting professorship at university of johannesburg. both authors acknowledge assisting remarks by dr. andrew p.k. craig from the university of johannesburg. the authors also thank the anonymous referee for pointing to a recent paper [12] where (generalized) comb-like trees are introduced which can be considered as new candidates for families of graceful graphs and then possibly also characterized in the manner presented in this paper. cubo 23, 2 (2021) a new class of graceful graphs: k-enriched fan graphs 331 references [1] s. amutha and m. uma devi, “super graceful labeling for some families of fan graphs”, journal of computer and mathematical sciences, vol. 10, no. 8, pp. 1551–1562, 2019. [2] j. a. gallian, “a dynamic survey of graph labeling”, electron. j. combin., # ds6, twentysecond edition, 2019. [3] m. gardner, “mathematical games: the graceful graphs of solomon golomb”, sci. am., vol. 226, no. 23, pp. 108–112, 1972. [4] s. w. golomb, “how to number a graph”, in: graph theory and computing, pp. 23–37, academic press, 1972. [5] m. haviar and m. ivaška, vertex labellings of simple graphs. research and exposition in mathematics, vol. 34, lemgo, germany: heldermann-verlag, 2015. [6] p. hrnčiar and a. haviar, “all trees of diameter five are graceful”, discrete math., vol. 233, no. 1-3, pp. 133–150, 2001. [7] m. ivaška, “chessboard representations and computer programs for graceful labelings of trees”, student competition švoč, m. bel university, banská bystrica, 2009. [8] s. kurtuĺık, “graceful labellings of graphs”, msc. thesis, 52 pp., department of mathematics, m. bel university, banská bystrica, 2020. [9] a. rosa, “o cyklických rozkladoch kompletného grafu” [on cyclic decompositions of the complete graph], phd thesis (in slovak), československá akadémia vied, bratislava, 1965. [10] a. rosa, “on certain valuations of the vertices of a graph”, in: theory of graphs (internat. symposium, rome, july 1966), pp. 349–355, new york: gordon and breach, 1967. [11] d. a. sheppard, “the factorial representation of major balanced labelled graphs”, discrete math., vol. 15, no. 4, pp. 379–388, 1976. [12] k. xu, x. li and s. klavžar, “on graphs with largest possible game domination number”, discrete math., vol. 341, no. 6, pp. 1768–1777, 2018. introduction preliminaries fan graphs and their descriptions double and triple fan graphs and their descriptions general case: k-enriched fan graphs kfn and their descriptions conclusion cubo, a mathematical journal vol. 23, no. 03, pp. 411–421, december 2021 doi: 10.4067/s0719-06462021000300411 independent partial domination l. philo nithya1 joseph varghese kureethara1 1 department of mathematics, christ university, bengaluru, karnataka, india. philo.nithya@res.christuniversity.in frjoseph@christuniversity.in abstract for p ∈ (0, 1], a set s ⊆ v is said to p-dominate or partially dominate a graph g = (v, e) if |n[s]||v | ≥ p. the minimum cardinality among all p-dominating sets is called the p-domination number and it is denoted by γp(g). analogously, the independent partial domination (ip(g)) is introduced and studied here independently and in relation with the classical domination. further, the partial independent set and the partial independence number βp(g) are defined and some of their properties are presented. finally, the partial domination chain is established as γp(g) ≤ ip(g) ≤ βp(g) ≤ γp(g). resumen para p ∈ (0, 1], un conjunto s ⊆ v se dice que pdomina o parcialmente domina un grafo g = (v, e) si |n[s]| |v | ≥ p. la cardinalidad mínima entre todos los conjuntos p-dominantes se llama el número de p-dominación y se denota por γp(g). análogamente, la dominación parcial independiente (ip(g)) es introducida y estudiada independientemente y en relación con la dominación clásica. más aún el conjunto independiente parcial y el número de independencia parcial βp(g) se definen y se presentan algunas de sus propiedades. finalmente, se establece la cadena de dominación partial como γp(g) ≤ ip(g) ≤ βp(g) ≤ γp(g). keywords and phrases: domination chain, independent partial dominating set, partial independent set. 2020 ams mathematics subject classification: 05c30, 05c69. accepted: 13 september, 2021 received: 10 november, 2020 ©2021 l. philo nithya et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300411 https://orcid.org/0000-0003-4182-6231 https://orcid.org/0000-0001-5030-3948 412 l. philo nithya & j. varghese kureethara cubo 23, 3 (2021) 1 introduction the theory of domination is one of the profusely researched areas in graph theory. recently a new domination parameter called partial domination number was introduced simultaneously in [3], [4] and [6], and studied in [12, 13, 14, 15]. we extend the concept of partial domination to independent domination in graphs. in [9], the concept of independent partial domination has been defined in the context of partial domination that was defined in [4]. but our work is based on the definition of partial domination in [3, 6] and we concentrate on partial domination chain. domination addresses the issue of the number of vertices that are dominating all the vertices in a graph. as the set of all vertices of a graph dominates itself, the mathematical adventure is in finding the least number of vertices that can dominate the entire graph. this number is the domination number of a graph. finding the domination number of a graph is a well known np-complete decision problem [11]. in the case of large graphs with a good number of small-degree vertices, the domination number shoots up. hence, instead of finding the dominating set that dominates the entire graph, it might be convenient to study the set of vertices that dominates the graph partially. this also could be treated as the density problem. by identifying the vertices with large degrees, we can find dense structures in the graph. the vertices that are contributing to the high density neighbourhoods are likely to dominate the major section of vertices of a graph. hence, domination problem and its variations could also be interpreted as density problems. we follow the popular nomenclature domination and study the structures that are partially dominating a graph. domination has been addressed in many different ways by imposing conditions on the dominating set or on its complement or on both. the relations between various domination parameters thus developed aroused mathematical curiosity. the domination chain proposed by cockayne et al. is mathematically profound and aesthetically appealing (see section 5). a recent survey by bazgan et al. lists the most important results regarding the domination chain parameters [2]. in this paper, we partially address the problem raised by case et al. in [3]. the paper is structured as follows. in section 2, we present all the preliminary concepts required for this paper. in section 3, we define independent partial domination number and study some of their properties. in section 4, we explore some relations between independent dominating set and independent partial dominating set. in section 5, we define partial independence number and investigate some of its properties which in turn lead to a part of the partial domination chain. 2 preliminaries let g be a simple, finite and undirected graph with v (g) as its set of vertices and e(g) as its edge set. a set s ⊆ v (g) is an independent set of vertices if no two vertices of s are adjacent to each cubo 23, 3 (2021) independent partial domination 413 other. an independent set s of vertices is said to be maximal if no superset t ⊃ s is independent. the maximum cardinality of an independent set in g is called its vertex independence number denoted by β(g) and the corresponding vertex set is called the β-set of g. for every vertex u in g, the set n(u) of all vertices adjacent to u is called the open neighbourhood of u. the set n(u) taken together with {u} is called the closed neighbourhood of u and is denoted by n[u]. a set d of vertices is called a dominating set of g if every vertex outside d is adjacent to at least one vertex in d. a dominating set d is minimal if no proper subset of d is a dominating set. the minimum cardinality of a minimal dominating set is called the domination number of g denoted by γ(g) and the maximum cardinality of a minimal dominating set is called the upper domination number denoted by γ(g). if a dominating set is independent, it becomes an independent dominating set and the minimum cardinality of such a set is called the independent domination number of g denoted by i(g). for any graph g = (v, e) and proportion p ∈ (0, 1], a set s ⊆ v is a p-dominating or partial dominating set if |n[s]||v | ≥ p. the p-domination or partial domination number γp(g) equals the minimum cardinality of a p-dominating set in g. v1 v2 v3v4 v5 v6 v7 v8 v9 figure 1: partial domination by white vertices in the light of definitions of the neighbourhoods, it is obvious that, for a dominating set s, n[s] = v . so a partial dominating set, when compared with a dominating set, dominates a proportion ‘p’ of the vertex set, which is not necessarily the whole set and hence partially dominates g. the set of all white vertices in figure 1 dominates exactly 4 vertices and hence is a 4 9 -dominating set. as {v7} is enough to dominate 4 vertices, γ 4 9 = 1 for the above graph. for all the other graph theoretic parameters and the notations that are used in this paper, one can refer to [11]. 3 independent partial domination in this section, we define independent partial domination number and present some observations and some of the basic results. since the observations are obvious, we present them without proofs. definition 3.1. suppose g = (v, e) is a simple graph and p ∈ (0, 1]. a subset s of v is called 414 l. philo nithya & j. varghese kureethara cubo 23, 3 (2021) an independent p-dominating set (ipd-set) if s is a p-dominating set and is independent. definition 3.2. the minimum cardinality of an independent p-dominating set is called the independent p-domination number (ipd-number) and is denoted by ip(g). observations: (i) for p ∈ (0, 1], γp(g) ≤ ip(g). (ii) for any n-vertex graph g and for p ∈ (0, ∆+1 n ], ip(g) = 1. (iii) for all p ∈ (0, 1], ip(g) = 1 if and only if i(g) = 1. (iv) for p ∈ ( n−1 n , 1 ] , ip(g) = i(g). (v) for all p ∈ (0, 1], ip(g) ≤ i(g). we proceed to find the ipd-numbers of paths, cycles and complete bipartite graphs. proposition 3.3. suppose pn and cn are paths and cycles respectively on n-vertices. then for n ≥ 3, ip(cn) = ip(pn) = ⌈np3 ⌉. proof. consider cn for n ≥ 3. let s be a γp-set of cn. then |s| = γp = ⌈np3 ⌉. if we can choose s in such a way that s is independent, then ip(cn) = ⌈np3 ⌉. for this, consider cn = (v1, v2, v3, ..., v3r, v3r+1, v3r+2). here three cases arise viz., (i) n = 3r, (ii) n = 3r + 1, (iii) n = 3r + 2 where r ≥ 1. let s1 = {v2, v5, ..., v3r−1}, s2 = {v2, v5, ..., v3r−1, v3r+1} and s3 = {v2, v5, ..., v3r−1, v3r+2}. we can see that |s1| = |s2| = |s3| = ⌈n3 ⌉ and si is independent for 1 ≤ i ≤ 3. for cases (i), (ii) and (iii) we can choose our set of ⌈np 3 ⌉ vertices from s1, s2 and s3. hence, ip(cn) = ⌈np3 ⌉. this proof holds for pn also. proposition 3.4. for m ≤ n, ip(km,n)=   1, for p ∈ (0, n+1 m+n ] i + 1, for p ∈ ( n+i m+n , n+(i+1) m+n ] where 1 ≤ i ≤ m − 1. also ip(km,n) ≤ m. proof. consider km,n for m ≤ n. let v1 = {v1, v2, ..., vm} and v2 = {u1, u2, ..., un} be the two partite sets of km,n, where each of v1 and v2 is an independent set. now, v1 ∈ v1 dominates n+1m+n vertices. consequently, of the remaining m − 1 vertices in v1, each vi ∈ v1 dominates n+im+n vertices. thus ip(km,n) ≤ m. cubo 23, 3 (2021) independent partial domination 415 4 independent domination and independent partial domination allan and laskar described the relation between the domination number and the independent domination number of a graph in [1]. partial domination is all about dominating a proportion p of the vertices of g. so a natural question which arises is that: whether this proportion p has any role in relating the partial domination and the original domination parameters. in this section, we do say ‘yes’ to that question by giving an upper bound for ipd-numbers in terms of p and independent domination numbers. we also give some results, which relate independent dominating sets [10] with that of partial independent dominating sets. theorem 4.1. for any graph g with independent domination number i(g) and p ∈ (0, 1], ip(g) ≤ ⌈p.i(g)⌉. proof. let d = {v1, v2, ..., vi} be an i-set of g. partition v into sets v1, v2, ..., vi such that for each 1 ≤ j ≤ i, vj ⊆ n[vj]. without loss of generality, let us assume that |vj| ≥ |vj+1| for 1 ≤ j ≤ i. consider d′ = {v1, v2, ..., v⌈p.i⌉}. claim: d′ is an ipd-set of g. proof of the claim our construction yields,∣∣∣∣∣∣ i⋃ j=1 vj ∣∣∣∣∣∣ = |v | =⇒ ∣∣∣∣∣∣ ⌈p.i⌉⋃ j=1 vj ∣∣∣∣∣∣ + ∣∣∣∣∣∣ i⋃ j=⌈p.i⌉ vj ∣∣∣∣∣∣ = |v | =⇒ ∣∣∣⋃⌈p.i⌉j=1 vj∣∣∣ ⌈p.i⌉ ≥ ∣∣∣⋃ij=1 vj∣∣∣ i = |v | i∣∣∣∣∣∣ i⋃ j=1 vj ∣∣∣∣∣∣ = |v | =⇒ ∣∣∣∣∣∣ ⌈p.i⌉⋃ j=1 vj ∣∣∣∣∣∣ ≥ |v |.⌈p.i⌉i . hence |n[d′]| ≥ p.|v |. we have thus proved the claim. thus using the claim, we have ip(g) ≤ |d′| = ⌈p.i(g)⌉. proposition 4.2. let g be any graph with independent domination number i(g) and p ∈ (0, 1]. then ip(g) + i1−p(g) ≤ i(g) + 1. proof. by theorem 5.7, ip(g) ≤ ⌈p.i⌉ < p.i + 1 and i1−p(g) ≤ ⌈(1 − p) .i⌉ < (1 − p).i + 1, then ip(g) + i1−p(g) < i + 2 ≤ i + 1. proposition 4.3. let s be any independent dominating set of g. if p = |n[h]||v | , for some h ⊂ s, then s − h is a 1 − p independent dominating set in g. 416 l. philo nithya & j. varghese kureethara cubo 23, 3 (2021) proof. it can be easily proved that, n[s] − n[h] ⊆ n[s − h]. therefore, |n[s−h]||v | ≥ 1 − p since n[s] = v . the following result provides us with an algorithm that develops a minimal independent dominating set from a minimal ipd-set. proposition 4.4. every minimal ipd-set can be extended to form a minimal independent dominating set. proof. let i be a minimal ipd-set for any p ∈ (0, 1]. the following algorithm extends i to i′, a minimal independent dominating set and gives m, the cardinality of i′. procedure 1 algorithm to construct i′ from i input: v (g), i, n[i], n[u]∀u ∈ v (g) − n[i] output: i′, m 1: i′ = i, m = |i′|, m = {} 2: m = v (g) − n[i′] 3: if m = ϕ then 4: return i′, m 5: else 6: i′ = i′ ∪ {u} for any u ∈ m 7: n[i′] = n[i′] ∪ n[u] 8: m = m + 1 9: go to 2 10: end if when a graph is claw-free, it has been already proved in [1], that its domination number coincides with that of its independent domination number. we found that to be true in the context of partial domination also. proposition 4.5. if a graph g is claw-free, then γp(g) = ip(g). proof. let s be a γp−set of g, for any p ∈ (0, 1]. since g is claw-free, < n[s] > is also clawfree. hence, γ(< n[s] >) = i(< n[s] >). this implies that γp(g) ≥ ip(g). but in general, γp(g) ≤ ip(g). thus γp(g) = ip(g). corollary 4.6. if l(g) is the line graph of a graph g, then γp(l(g)) = ip(l(g)). cubo 23, 3 (2021) independent partial domination 417 5 partial domination chain a chain of inequalities involving domination numbers, independence numbers and irredundance numbers of the form ir(g) ≤ γ(g) ≤ i(g) ≤ β(g) ≤ γ(g) ≤ ir(g) was first observed in 1978 (see [5]). this type of chain was observed in the case of many other domination parameters like α-domination [7] and k-dependent domination [8]. also one of the open questions posed by case et al. in [3] was to find out, what relationship the above parameters have amongst themselves in the context of partial domination. hence we try to establish a similar kind of chain involving partial domination and partial independence parameters. having already defined independent partial domination, we now define partial independence number of a graph. definition 5.1. suppose g = (v, e) is a graph and p ∈ (0, 1]. a set s of independent vertices is called a p-independent set in g if n[s] ⊆ v (h) for some induced subgraph h of g with |v (h)| ≥ np. a p-independent set s is said to be p-maximal if s is a maximal independent set in v (h). a maximal p-independent set s is said to be min-max p-independent set if t ⊂ s is not p-maximal. partial independence number or p-independence number is the maximum cardinality of a min-max p-independent set and is denoted by βp(g) and the associated induced subgraph h is denoted by hp. for the graph in figure 1, the set of all white vertices form a β 4 9 -set. for the same graph, the set {v5, v8} is a min-max 49 -independent set, but it is not of maximum cardinality and hence is not a β 4 9 -set. 5.1 partial independent sets this section explores some of the properties of partial independent sets, thereby proceeding towards the suggested partial domination chain. in light of the above definition, it may be noted that, for every maximal p-independent set s, n[s] = v (h) of the proposed induced subgraph h of g and hence s is a p-dominating set. thus independent p-domination number is the minimum cardinality of a maximal p-independent set and we have the following inequality. proposition 5.2. for p ∈ (0, 1], γp(g) ≤ ip(g) ≤ βp(g). proposition 5.3. if p1 ≤ p2, then βp1(g) ≤ βp2(g). proof. let s ⊆ v be such that |s| = βp2(g). then s is also maximal p1-independent set. also the cardinality of every min-max p1-independent set ≤ |s|. thus βp1(g) ≤ βp2(g). 418 l. philo nithya & j. varghese kureethara cubo 23, 3 (2021) we now proceed to relate partial independence number βp with that of upper p-domination number, γp which is the maximum cardinality of a minimal p-dominating set. proposition 5.4. every min-max p-independent set is a minimal p-dominating set. proof. let s be a min-max p-independent set and hp be an induced subgraph associated with it. then by definition, s is p-dominating in g. suppose s is not minimal p-dominating. then ∃ u ∈ s such that s − {u} is p-dominating. then ∃ v ∈ s − {u}, such that uv ∈ e(h) which is a contradiction since s is an independent set. thus s is a minimal p-dominating set. corollary 5.5. for p ∈ (0, 1], βp(g) ≤ γp(g). from proposition 5.2 and corollary 5.5 we obtain the following chain of inequalities: for p ∈ (0, 1], γp(g) ≤ ip(g) ≤ βp(g) ≤ γp(g). we present some more properties of independent sets, which in turn lead us to a method, by which one can deduce βp-sets for some ‘p’ values from the existing β-set of a graph. lemma 5.6. suppose s is a β-set of a graph g and t ⊂ s. then t is a min-max |n[t ]| n independent set. proof. by definition t is a maximal |n[t ]| n -independent set. it is also min-max since r ⊂ t is not |n[t ]| n maximal. suppose r is maximal then (s−t)∪r is a dominating set which is a contradiction as s is a minimal dominating set of g. theorem 5.7. let bi denote the set of all i-element subsets of a β-set of a graph g for 1 ≤ i ≤ β(g). let bi ∈ bi be such that |n[bi]| = min{|n[x]|/x ∈ bi}. then (i) bi is a βp−set for p = |n[bi]| n . (ii) for 0 < p ≤ |n[b1]| n , βp = 1 and b1 is a βp−set. proof. for 1 ≤ i ≤ β(g) let bi be chosen by the given method. by the previous lemma (5.6) bi is a min-max |n[bi]| n independent set. suppose bi is not of maximum cardinality amongst all |n[bi]| n independent sets, then for j > i there exists a y ∈ bj such that y is a min-max |n[bi]| n independent set. also y is min-max |n[y ]| n independent set and thus y is a maximal independent set in both < n[bi] > and < n[y ] > and also |n[bi]| = |n[y ]|. but by the definition of bis, |n[y ]| ≥ |n[bj]| which implies that |n[bj]| ≤ |n[bi]| which is a contradiction since for j > i, |n[bj]| > |n[bi]|. cubo 23, 3 (2021) independent partial domination 419 suppose |n[bj]| ≤ |n[bi]| for some j > i, choose r such that r ⊂ bj and |r| = i. then |n[r]| < |n[bj]| which implies that |n[r]| < |n[bi]| by our assumption. this contradicts our definition of bi. 6 conclusion partial domination has a lot to promise. one of the striking features of the concept of partial domination is its nature of accommodation. domination with conditions are studied extensively. in the case of partial domination, the imperfect situations are addressed. hence, it is worth exploring the partial domination in all the numerous types of dominations. in this context we could establish the partial domination chain. future beckons with great hope of the explorations of partial domination in the areas of distance domination, stratified domination, roman domination etc., but not exclusively. acknowledgement we thank all the referees for their reviews of this paper and their creative suggestions. 420 l. philo nithya & j. varghese kureethara cubo 23, 3 (2021) references [1] r. b. allan and r. laskar, “on domination and independent domination numbers of a graph”, discrete math., vol. 23, no. 2, pp. 73–76, 1978. 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[8] o. favaron, s. m. hedetniemi, s. t. hedetniemi and d. f. rall, “on k-dependent domination”, discrete math., vol. 249, nos. 1–3, pp. 83–94, 2002. [9] o. favaron and p. kaemawichanurat, “inequalities between the kk-isolation number and the independent kk-isolation number of a graph”, discrete appl. math., vol. 289, pp. 93–97, 2021. [10] w. goddard and m. a. henning, “independent domination in graphs: a survey and recent results”, discrete math., vol. 313, no. 7, pp. 839–854, 2013. [11] t. w. haynes, s. t. hedetniemi and p. j. slater, fundamentals of domination in graphs, 464, crc press, boca raton, 1998. [12] r. d. macapodi and r. t. isla, “total partial domination in graphs under some binary operations”, eur. j. pure appl. math., vol. 12, no. 4, pp. 1643–1655, 2019. [13] r. d. macapodi, r. i. isla and s. r. canoy, “partial domination in the join, corona, lexicographic and cartesian products of graphs”, adv. appl. discrete math., vol. 20, no. 2, pp. 277–293, 2019. [14] l. p. nithya and j. v. kureethara, “on some properties of partial dominating sets”, aip conference proceedings, vol. 2236, no. 1, 060004, 2020. cubo 23, 3 (2021) independent partial domination 421 [15] l. p. nithya and j. v. kureethara, “partial domination in prisms of graphs”, ital. j. pure appl. math., to be published. introduction preliminaries independent partial domination independent domination and independent partial domination partial domination chain partial independent sets conclusion cubo, a mathematical journal vol. 24, no. 01, pp. 53–62, april 2022 doi: 10.4067/s0719-06462022000100053 on graphs that have a unique least common multiple reji t. 1 jinitha varughese 2 ruby r. 1 1department of mathematics, government college chittur palakkad, india. rejiaran@gmail.com rubymathpkd@gmail.com 2department of mathematics, b. k. college amalagiri, kottayam, india. jinith@gmail.com abstract a graph g without isolated vertices is a least common multiple of two graphs h1 and h2 if g is a smallest graph, in terms of number of edges, such that there exists a decomposition of g into edge disjoint copies of h1 and there exists a decomposition of g into edge disjoint copies of h2. the concept was introduced by g. chartrand et al. and they proved that every two nonempty graphs have a least common multiple. least common multiple of two graphs need not be unique. in fact two graphs can have an arbitrary large number of least common multiples. in this paper graphs that have a unique least common multiple with p3 ∪ k2 are characterized. resumen un grafo g sin vértices aislados es un mı́nimo común múltiplo de dos grafos h1 y h2 si g es uno de los grafos más pequeños, en términos del número de ejes, tal que existe una descomposición de g en copias de h1 disjuntas por ejes y existe una descomposición de g en copias de h2 disjuntas por ejes. el concepto fue introducido por g. chartrand et al. donde ellos demostraron que cualquera dos grafos no vaciós tienen un mı́nimo común múltiplo. el mı́nimo común múltiplo de dos grafos no es necesariamente único. de hecho, dos grafos pueden tener un número arbitrariamente grande de mı́nimos comunes múltiplos. en este art́ıculo caracterizamos los grafos que tienen un único mı́nimo común múltiplo con p3 ∪ k2. keywords and phrases: graph decomposition, common multiple of graphs. 2020 ams mathematics subject classification: 05c38, 05c51, 05c70. accepted: 03 november, 2021 received: 23 march, 2021 c©2022 reji t. et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100053 https://orcid.org/0000-0003-2712-3775 https://orcid.org/0000-0003-4419-6626 https://orcid.org/0000-0001-5473-6492 mailto:rejiaran@gmail.com mailto:rubymathpkd@gmail.com mailto:jinith@gmail.com 54 reji t., jinitha varughese & ruby r. cubo 24, 1 (2022) 1 introduction all graphs considered in this paper are assumed to be simple and to have no isolated vertices. the number of edges of a graph g denoted by e(g), is called the size of g. δ(g) and ∆(g) respectively denote the minimum and maximum of the degrees of all vertices in g. χ′(g) denotes the edge chromatic number of g, the minimum number of colors needed to color the edges of g, so that no two adjacent edges in g have the same color. an odd component of a graph is a maximal connected subgraph of the graph with odd number of edges. two graphs g and h are said to be isomorphic, denoted as g ∼= h if there exists a bijection between the vertex sets of g and h, f : v (g) → v (h) such that two vertices u and v of g are adjacent in g if and only if f(u) and f(v) are adjacent in h. for graphs g1 and g2, their union g1 ∪ g2 is the graph with vertex set v (g1 ∪ g2) = v (g1) ∪ v (g2) and edge set consisting of all the edges in g1 together with all the edges in g2. if k is a positive integer, then kg is the union of k disjoint copies of g. g1 v1 v2 g2 v3 v4 v5 g3 v6 v7 v8 v1 v2 v3 v4 v5 figure 1: g1 ∪ g2 let g = g2. then g ∼= g3 and 2g is shown in figure 2. v3 v4 v5 v6 v7 v8 figure 2: 2g a vertex u of a graph g is said to cover an edge e of g or e is covered by u, if e is incident with u. let u, w be two vertices of a graph g and take two copies of g : g1, g2. the graph h obtained by identifying the vertex u in g1 with the vertex w in g has vertex set v (h) = v (g1)∪v (g2)−{w} and edge set e(h) = e(g1) ∪ e(g2), where the edges in g2 incident with w are now incident with u. a graph h is said to divide a graph g if there exists a set of subgraphs of g, each isomorphic to h, whose edge sets partition the edge set of g. such a set of subgraphs is called an h-decomposition cubo 24, 1 (2022) on graphs that have a unique least common multiple 55 of g. if g has an h-decomposition, we say that g is h-decomposable and write h|g. a graph is called a common multiple of two graphs h1 and h2 if both h1|g and h2|g. a graph g is a least common multiple of h1 and h2 if g is a common multiple of h1 and h2 and no other common multiple has a smaller positive number of edges. several authors have investigated the problem of finding least common multiples of pairs of graphs h1 and h2; that is graphs of minimum size which are both h1 and h2 decomposable. the problem was introduced by chartrand et al. in [5] and they showed that every two nonempty graphs have a least common multiple. the problem of finding the size of least common multiples of graphs has been studied for several pairs of graphs: cycles and stars [5,13,14], paths and complete graphs [11], pairs of cycles [10], pairs of complete graphs [4], complete graphs and a 4-cycle [1], pairs of cubes [2] and paths and stars [8]. least common multiple of digraphs were considered in [7]. if g is a common multiple of h1 and h2 and g has q edges, then we call g a (q, h1, h2) graph. an obvious necessary condition for the existence of a (q, h1, h2) graph is that e(h1)|q and e(h2)|q. this obvious necessary condition is not always sufficient. therefore, we may ask: given two graphs h1 and h2, for which value of q does there exist a (q, h1, h2) graph? adams, bryant and maenhaut [1] gave a complete solution to this problem in the case where h1 is the 4-cycle and h2 is a complete graph; bryant and maenhaut [4] gave a complete solution to this problem when h1 is the complete graph k3 and h2 is a complete graph. the problem to find least common multiples of two graphs h1 and h2 is to find all (q, h1, h2) graphs g of minimum size q. we denote the set of all least common multiples of h1 and h2 by lcm(h1, h2). the size of a least common multiple of h1 and h2 is denoted by lcm(h1, h2). since every two nonempty graphs have a least common multiple, lcm(h1, h2) is nonempty. for many pairs of graphs the number of elements of lcm(h1, h2) is greater than one. for example both p7 and c6 are least common multiples of p4 and p3. in fact chartrand et al. [6] proved that for every positive integer n there exist two graphs having exactly n least common multiples. in [11] it was shown that every least common multiple of two connected graphs is connected and that every least common multiple of two 2-connected graphs is 2-connected. but this is not the case for disconnected graphs. for example if we take h1 = 2k2, h2 = c5, then g1 = 2c5 and g2 the graph obtained by identifying two vertices in two copies of c5, are in lcm(h1, h2) of which g1 is disconnected while g2 is connected. as two graphs can have several least common multiples, it is interesting to search for pairs of graphs that have a unique least common multiple. pairs of graphs having a unique least common multiple were investigated by g. chartrand et al. in [6] and they proved the following results. theorem 1.1. a graph g of order p without isolated vertices and the graph p3 have a unique least common multiple if and only if every component of g has even size or g ∼= kp, where p ≡ 2 or 3 (mod 4). theorem 1.2. a nonempty graph g without isolated vertices and the graph 2k2 have a unique 56 reji t., jinitha varughese & ruby r. cubo 24, 1 (2022) least common multiple if and only if g ∼= k2, g ∼= k3 or 2k2|g. theorem 1.3. let r and s be integers with 2 ≤ r ≤ s. then the stars k1,r and k1,s have a unique least common multiple if and only if gcd(r, s) 6= 1. a result proved by n. alon [3] on tk2-decomposition of a graph is used to find those graphs that have a unique least common multiple with tk2. theorem 1.4. for every graph g and every t > 1, tk2|g if and only if t|e(g) and χ ′(g) ≤ e(g) t . we will also make use of a result proved by o. favaron, z. lonc and m. truszczynski [9] to characterize those graphs that have a unique least common multiple with p3 ∪ k2. theorem 1.5. if g is none of the six graphs g1 to g6 listed below, then g is p3∪k2 decomposable if and only if (1) e(g) ≡ 0 (mod 3), (2) ∆(g) ≤ 2 3 e(g), (3) c(g) ≤ 1 3 e(g), where c(g) denote the number of odd components of g, (4) the edges of g cannot be covered by two adjacent vertices; where, g1 g2 g3 g4 g5 g6 2 main results in this section we are characterizing those graphs that have a unique least common multiple with tk2 and p3 ∪ k2. cubo 24, 1 (2022) on graphs that have a unique least common multiple 57 2.1 on graphs that have a unique least common multiple with tk2 theorem 2.1. a nonempty graph g without isolated vertices and the graph tk2 have a unique least common multiple if and only if tk2|g or δ(g) > lcm(tk2, g) 2t . proof. consider the graph tg. clearly tg is both g and tk2 decomposable. let q = e(g). since e(tg) = tq, we have lcm(tk2, g) ≤ tq. but lcm(tk2, g) is a multiple of q. so lcm(tk2, g) = ql, where l ≤ t. this implies lcm(tk2,g) t = ql t . let h be a least common multiple of g and tk2. case 1. l > 1. since h is tk2-decomposable, by theorem 1.4, χ ′(h) ≤ ql t . since g|h, χ′(g) ≤ χ′(h) ≤ ql t . thus ∆(g) ≤ ql t . subcase (i): δ(g) ≤ ql 2t . consider the graph g ◦ g, which is obtained by identifying two vertices of least degree in g. in this subcase ∆(g ◦ g) ≤ ql t , since ∆(g) ≤ ql t . χ′(g) ≤ ql t implies χ′(g ◦ g) ≤ ql t . color g1, a copy of g in g ◦ g, with k ≤ ql t colors. this is possible, since χ′(g) ≤ ql t . let v be the identified vertex in g◦g. since δ(g) ≤ ql 2t , the edges adjacent to v in g1 are colored using at most ql 2t colors. color g2, the copy of g in g◦g other than g1, with the same k colors as follows. color the edges adjacent to v in g2 using colors different from those which were used to color the edges adjacent to v in g1. the remaining colors used in the coloring of g1 can be used to color other edges of g2. thus χ ′(g ◦ g) = k ≤ ql t . let h1 = lg, the union of l disjoint copies of g and h2 = g◦ g∪(l − 2)g. clearly h1 and h2 are divisible by g. since χ′(h1) = χ ′(g) ≤ ql t , h1 is tk2-decomposable. χ ′(h2) = χ ′(g ◦ g) ≤ ql t , h2 is tk2-decomposable by theorem 1.4. thus h1, h2 ∈ lcm(tk2, g). e(h1) = e(h2) = ql, where q = e(g). since lcm(tk2, g) = ql, h1 and h2 are two non-isomorphic least common multiples of tk2 and g. subcase (ii): δ(g) > ql 2t . in this case l > 1 and lcm(tk2, g) = ql, where q = e(g). thus h ∈ lcm(tk2, g), should be decomposed into at least two copies of g. if h is different from lg, then ∆(h) > ql t which implies χ′(h) > ql t and hence by theorem 1.4, h is not tk2-decomposable. thus lg is the unique least common multiple of tk2 and g. case 2. l = 1. in this case lcm(tk2, g) = q. thus tk2|g and g is the unique least common multiple. remark 2.2. the result in the above theorem, theorem 2.1, appeared in [12]. we are giving the proof of this result here since the result is needed for proving theorem 2.3. the result was proved 58 reji t., jinitha varughese & ruby r. cubo 24, 1 (2022) by the first author of this manuscript. 2.2 on graphs that have a unique least common multiple with p3 ∪ k2 theorem 2.3. a nonempty graph g without isolated vertices and the graph p3 ∪k2 have a unique least common multiple if and only if g = k2 or p3 ∪ k2 | g. proof. let q = e(g). case 1. g is a connected graph. if g = k2, then g | p3 ∪k2. thus lcm(p3 ∪k2, k2) = {p3 ∪k2} and hence their least common multiple is unique. so we are going to analyse the case where g 6= k2. consider the graph 3g, a union of three disjoint copies of g. then (1) e(3g) ≡ 0 (mod 3). (2) ∆(3g) = ∆(g) ≤ q = 1 3 (3q) ≤ 2 3 (3q) = 2 3 e(3g). (3) c(3g) ≤ 3 ≤ 1 3 (3q) = 1 3 e(3g), if e(g) ≥ 3. if e(g) = 2, then c(3g) = 0 ≤ 1 3 e(3g). (4) the edges of 3g cannot be covered by two adjacent vertices, since the graph is disconnected. thus by theorem 1.5, 3g is p3∪k2-decomposable. clearly 3g is g-decomposable. hence lcm(p3∪ k2, g) ≤ 3q. subcase (i): lcm(p3 ∪ k2, g) = 3q. consider the graph h = g◦g∪g, where g◦g is the graph obtained by identifying a least degree vertex in two copies of g. then (1) e(h) ≡ 0 (mod 3). (2) ∆(h) ≤ 2q = 2 3 (3q) = 2 3 e(h). (3) c(h) ≤ 1 ≤ 1 3 (3q) = 1 3 e(h). (4) since h is disconnected, edges of h cannot be covered by two adjacent vertices. thus by theorem 1.5, h is p3 ∪ k2decomposable. clearly h is gdecomposable. hence in this case both h and 3g are elements of lcm(p3 ∪ k2, g) and hence their least common multiple is not unique. subcase (ii): lcm(p3 ∪ k2, g) = 2q. in this case there exists a graph h such that e(h) = 2q and h ∈ lcm(p3 ∪ k2, g). consider the graph 2g. cubo 24, 1 (2022) on graphs that have a unique least common multiple 59 (1) since h ∈ lcm(p3 ∪ k2, g) we get 3 | e(h) = 2q = e(2g) and hence e(2g) ≡ 0 (mod 3). (2) since h is g-decomposable and ∆(g) = ∆(2g), ∆(2g) ≤ ∆(h). h is p3∪k2-decomposable and so by theorem 1.5, ∆(h) ≤ 2 3 e(h) = 2 3 e(2g). thus ∆(2g) ≤ 2 3 e(2g). (3) in this case q ≥ 3 ( if q = 1, then g = k2 and if q = 2, then e(2g) = 4 6≡ 0 (mod 3) ) . so c(2g) ≤ 2 ≤ 1 3 2q = 1 3 e(2g). (4) since 2g is disconnected, the edges of 2g cannot be covered by two adjacent vertices. by applying theorem 1.5, 2g is p3 ∪ k2-decomposable. 2g is clearly g-decomposable. thus 2g ∈ lcm(p3 ∪ k2, g). we can also prove that g ◦ g ∈ lcm(p3 ∪ k2, g). (1) e(g ◦ g) = e(2g) ≡ 0 (mod 3). (2) in order to prove that ∆(g ◦ g) ≤ 2 3 e(g ◦ g) it is enough to prove that ∆(g) and 2δ(g) are less than or equal to 2 3 e(g ◦ g), since g ◦ g is obtained by identifying a vertex of least degree in two copies of g. since h ∈ lcm(p3 ∪ k2, g), ∆(g) ≤ ∆(h) ≤ 2 3 e(h) = 2 3 e(g ◦ g). 2δ(g) ≤ 2 3 e(g ◦ g) ⇐⇒ δ(g) ≤ 2q 3 . suppose δ(g) > 2q 3 . then 2q = ∑ v∈v (g) d(v) ≥ ∑ v∈v (g) δ(g) = nδ(g) > n 2q 3 , where n = |v (g)|. this implies n < 3. g is a connected graph without isolated vertices and g 6= k2. thus n ≥ 3 and so δ(g) ≤ 2q 3 . (3) c(g ◦ g) = 0 < 1 3 e(g ◦ g). (4) the edges of g ◦ g cannot be covered by two adjacent vertices. suppose the edges of g ◦ g can be covered by two adjacent vertices, then the identified vertex is one such vertex, since in g ◦ g, no two vertices are adjacent except the identified vertex. this implies using the identified vertex and one other vertex it is possible to cover all the edges of g ◦ g. this is possible only if g is a star with the identified vertex as the center of the star. this is a contradiction, since to construct g ◦ g a vertex of least degree in g is identified. applying theorem 1.5, g ◦ g is p3 ∪ k2decomposable and it is clearly g-decomposable. so g ◦ g ∈ lcm(p3 ∪ k2, g). we have proved that 2g and g ◦ g ∈ lcm(p3 ∪ k2, g) and hence their least common multiple is not unique. subcase (iii): lcm(p3 ∪ k2, g) = q. in this subcase g is the unique least common multiple, since q = e(g). case 2. g is disconnected. 60 reji t., jinitha varughese & ruby r. cubo 24, 1 (2022) as in the first case, assume that g 6= tk2. then at least one component of g has more than one edge. we construct a graph of size 3q, which is a (3q, g, p3 ∪ k2)-graph, where q = e(g). the construction is as follows. take a least degree vertex from each component of g. let h be the connected graph obtained by identifying all these vertices together. take a least degree vertex in h. denote by h ◦ h ◦ h, the graph obtained by identifying this least degree vertex in three copies of h. (1) e(h ◦ h ◦ h) = e(3h) = 3e(g) ≡ 0 (mod 3). (2) ∆(h ◦ h ◦ h) ≤ 2∆(h) ≤ 2e(g) = 2 3 e(3g) = 2 3 e(h ◦ h ◦ h). (3) c(h ◦ h ◦ h) ≤ 1 ≤ 1 3 e(h ◦ h ◦ h). (4) as in subcase (ii) of the previous case, the edges of h ◦ h ◦ h cannot be covered by two adjacent vertices. by theorem 1.5, h ◦ h ◦ h is p3 ∪ k2-decomposable and obviously it is g-decomposable. thus lcm(p3 ∪ k2, g) ≤ 3q. subcase (i): lcm(p3 ∪ k2, g) = 3q. from the above discussion h ◦ h ◦ h is a least common multiple in this subcase. consider the graph h ◦ h ∪ h. (1) e(h ◦ h ∪ h) = 3e(g) ≡ 0 (mod 3). (2) ∆(h ◦ h ∪ h) ≤ 2∆(h) ≤ 2e(g) = 2 3 e(3g) = 2 3 e(h ◦ h ∪ h). (3) c(h ◦ h ∪ h) ≤ 1 = 1 3 e(h ◦ h ∪ h). (4) since h ◦ h ∪ h is disconnected, the edges of h ◦ h ∪ h cannot be covered by two adjacent vertices. applying theorem 1.5, h◦h∪h is p3∪k2-decomposable and by construction it is g-decomposable. thus both h ◦ h ◦ h and h ◦ h ∪ h are in lcm(p3 ∪ k2, g) and hence their least common multiple is not unique. subcase (ii): lcm(p3 ∪ k2, g) = 2q. in this subcase there exists a graph h′ of size 2q which is both g and p3 ∪ k2 decomposable. we will prove that h ◦ h and h ∪ h are in lcm(p3 ∪ k2, g). (1) e(h ◦ h) = 2q ≡ 0 (mod 3), since e(h′) = 2q and h′ is p3 ∪ k2decomposable. (2) in order to prove that ∆(h ◦h) ≤ 2 3 e(h ◦h), it is enough to prove that 2δ(h) ≤ 2 3 e(h ◦h). that is we need to prove δ(h) ≤ 1 3 (2q), where q = e(h) = e(g). cubo 24, 1 (2022) on graphs that have a unique least common multiple 61 suppose δ(h) > 2q 3 . then 2q = ∑ v∈v (h) d(v) ≥ ∑ v∈v (h) δ(h) = nδ(h) > n( 2q 3 ) ⇒ n < 3. since g is a disconnected graph without isolated vertices, n < 3 is not possible. hence δ(h) ≤ 2q 3 . thus ∆(h ◦ h) ≤ 2 3 e(h ◦ h). (3) c(h ◦ h) = 0 < 1 3 e(h ◦ h). (4) by the construction of h ◦h, the edges of h ◦h cannot be covered by two adjacent vertices. by theorem 1.5, h ◦ h is p3 ∪ k2-decomposable and by construction, h ◦ h is g-decomposable and so h ◦ h ∈ lcm(p3 ∪ k2, g). also h ∪ h ∈ lcm(p3 ∪ k2, g), since (1) e(h ∪ h) = 2q ≡ 0 (mod 3), since lcm(p3 ∪ k2) = 2q, where q = e(g) = e(h). (2) ∆(h ∪ h) ≤ ∆(h ◦ h) ≤ 2 3 e(h ◦ h) = 2 3 e(h ∪ h). (3) here c(h ∪ h) ≤ 2. thus c(h ∪ h) ≤ 1 3 e(h ∪ h) if 2 ≤ 2q 3 , that is if q ≥ 3, where, q = e(g) = e(h). since g is a disconnected graph without isolated vertices, q 6= 1. also if q = 2, then 2q = 4 6≡ 0 (mod 3). thus in this subcase, q ≥ 3 and hence c(h ∪ h) ≤ 1 3 e(h ∪ h). thus h ◦ h and h ∪ h belong to lcm(p3 ∪ k2, g) and hence their least common multiple is not unique. acknowledgements the authors are thankful to prof. m. i. jinnah, formerly university of kerala, india, for his suggestions during the preparation of this manuscript. the authors are also thankful to the anonymous reviewers for their careful reading and insightful comments for improving this work. 62 reji t., jinitha varughese & ruby r. cubo 24, 1 (2022) references [1] p. adams, d. bryant and b. maenhaut, “common multiples of complete graphs and a 4-cycle”, discrete math., vol. 275, no. 1–3, pp. 289–297, 2004. 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[8] z.-c. chen and t.-w. shyu, “common multiples of paths and stars”, ars combin., vol. 146, pp. 115–122, 2019. [9] o. favaron, z. lonc and m. truszczyński, “decompositions of graphs into graphs with three edges”, ars combin., vol. 20, pp. 125–146, 1985. [10] o. favaron and c. m. mynhardt, “on the sizes of least common multiples of several pairs of graphs”, ars combin., vol. 43, pp. 181–190, 1996. [11] c. m. mynhardt and f. saba, “on the sizes of least common multiples of paths versus complete graphs”, utilitas math., vol. 46, pp. 117–127, 1994. [12] t. reji, “on graphs that have a unique least common multiple with matchings”, far east j. appl. math., vol. 18, no. 3, pp. 281–288, 2005. [13] c. sunil kumar, “least common multiple of a cycle and a star”, electron. notes discrete math., vol. 15, pp. 204–206, 2003. [14] p. wang, “on the sizes of least common multiples of stars versus cycles‘”, util. math., vol. 53, pp. 231–242, 1998. introduction main results on graphs that have a unique least common multiple with tk2 on graphs that have a unique least common multiple with p3 k2 cubo, a mathematical journal vol. 24, no. 01, pp. 95–103, april 2022 doi: 10.4067/s0719-06462022000100095 a characterization of fq-linear subsets of affine spaces fn q2 edoardo ballico 1 1department of mathematics, university of trento, 38123 povo (tn), italy. edoardo.ballico@unitn.it abstract let q be an odd prime power. we discuss possible definitions over f q2 (using the hermitian form) of circles, unit segments and half-lines. if we use our unit segments to define the convex hulls of a set s ⊂ fn q2 for q /∈ {3, 5, 9} we just get the fq-affine span of s. resumen sea q una potencia de primo impar. discutimos posibles definiciones sobre f q2 (usando la forma hermitiana) de ćırculos, segmentos unitarios y semi-ĺıneas. si usamos nuestros segmentos unitarios para definir las cápsulas convexas de un conjunto s ⊂ fn q2 para q /∈ {3, 5, 9} simplemente obtenemos el fq-generado af́ın de s. keywords and phrases: finite field, hermitian form. 2020 ams mathematics subject classification: 15a33; 15a60; 12e20. accepted: 27 december, 2021 received: 17 may, 2021 c©2022 e. ballico. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100095 https://orcid.org/0000-0002-1432-7413 mailto:edoardo.ballico@unitn.it 96 e. ballico cubo 24, 1 (2022) 1 introduction fix a prime p and a p-power q. there is a unique (up to isomorphism) field fq with #fq = q. the field fq2 is a degree 2 galois extension of fq and the frobenius map t 7→ t q is a generator of the galois group of this extension. this map allows the definition of the hermitian product 〈 , 〉 : fn q2 × fn q2 −→ fq2 in the following way: if u = (u1, . . . , un) ∈ f n q2 and v = (v1, . . . , vn) ∈ f n q2 , then set 〈u, v〉 = ∑n i=1 u q i vi. the degree q + 1 hypersurface {〈(x1, . . . , xn), (x1, . . . , xn)〉 = 0} is the famous full rank hermitian hypersurface ([11, ch. 23]). in the quantum world the classical hermitian product over the complex numbers is fundamental. the hermitian product 〈 , 〉 is one of the tools used to pass from a classical code over a finite field to a quantum code ([17, pp. 430–431], [14, introduction], [20, §2.2]). the hermitian product was used to define the numerical range of a matrix over a finite field ([1, 2, 3, 4, 8]) by analogy with the definition of numerical range for complex matrices ([9, 12, 13, 21]). over c a different, but equivalent, definition of numerical range is obtained as the intersection of certain disks ([5, §15, lemma 1]). it is an important definition, because it was used to extend the use of numerical ranges to rectangular matrices ([7]) and to tensors ([16]). this different definition immediately gives the convexity of the numerical range of complex matrices. motivated by that definition we look at possible definitions of the unit disk of fq2. it should be a union of circles with center at 0 and with squared-radius in the unit interval [0, 1] ⊂ fq. for any c ∈ fq and any a ∈ fq2 set c(0, c) := {z ∈ fq2 | z q+1 = c}, c(a, c) := a + c(0, c). we say that c(a, c) is the circle of fq2 with center a and squared-radius c. note that c(a, 0) = {a} and #c(a, c) = q + 1 for all c ∈ fq \ {0}. circles occur in the description of the numerical range of many 2×2 matrices over fq2 ([8, lemmas 3.4 and 3.5]). other subsets of fq2 (seen as a 2-dimensional vector space of fq) appear in [6] and are called ellipses, hyperbolas and parabolas, because they are affine conics whose projective closure have 0, 2 or 1 points in the line at infinity. all these constructions are inside fq2 seen as a plane over fq. restricting to planes we get the following definition for fnq2. definition 1.1. a set e ⊂ fnq2 is said to be a circle with center 0 ∈ f n q2 and squared-radius c if there is an fq-linear embedding f : fq2 −→ f n q2 such that e = f(c(0, c)). a set e ⊂ fn q2 is said to be a circle with center a ∈ fnq2 and squared-radius c if e − a is a circle with center 0 and squared-radius c. a set s ⊆ fn q2 , s 6= ∅, is said to be circular with respect to a ∈ fn q2 if it contains all circles with center a which meet s. cubo 24, 1 (2022) a characterization of fq-linear subsets of affine spaces f n q2 97 in the classical theory of numerical range over c the numerical range of a square matrix which is the orthogonal direct sum of the square matrices a and b is obtained taking the union of all segments [a, b] ⊂ c with a in the numerical range of a and b in the numerical range of b ([21, p. 3]). for the numerical range of matrices over fq2 instead of segments [a, b] one has to use the affine fq-span of {a, b} ([1, lemma 1], [8, proposition 3.1]). we wonder if in other linear algebra constructions something smaller than fq-linear span occurs. a key statement for square matrices over c (due to toeplitz and hausdorff) is that their numerical range is convex ([9, th. 1.1-2], [21, §3]). convexity is a property over r and to define it one only needs the unit interval [0, 1] ⊂ r. obviously [0, 1] = [0, +∞) ∩ (−∞, 1] and (−∞, 1] = 1 − [0, +∞). as a substitute for the unit interval [0, 1] ⊂ r (resp. the half-line [0, +∞) ⊂ r) we propose the following sets iq and i ′ q (resp. eq). definition 1.2. assume q odd. set eq := {a 2}a∈fq ⊂ fq, iq := eq ∩(1−eq), i ′′ q := eq ∩(1+xeq) with x ∈ fq \ eq, and i ′ q := i ′′ q ∪ {0}. note that i′q = {0, 1} ∪(eq ∩(1 + (fq \ eq)). in the first version of this note we only used iq, but a referee suggested that it is more natural to consider i′′q . we use iq and i ′ q because {0, 1} ⊆ iq ∩ i ′ q, while 0 ∈ i′′q if and only if −1 is not a square in fq, i. e. if and only if q ≡ 3 (mod 4) ([10, (ix) and (x) at p. 5], [22, p. 22]). in all statements for odd q we handle both iq and i ′ q. in the case q even we propose to use {a(a + 1)}{a∈fq} as eq, i. e. eq := tr −1 fq/f2 (0). thus eq is a subgroup of (fq, +) of index 2. if q is even we do not have a useful definition of iq. thus we restrict to odd prime powers, except for propositions 1.8, 2.9 and remarks 2.1 and 2.2. we see iq or i ′ q (resp. eq) as the unit segment [0, 1] (resp. positive half-line starting at 0) of fq ⊂ fq2. in most of the proofs we only use that {0, 1} ⊆ iq and that #iq is large, say #iq > (q − 1)/4. remark 1.3. note that #eq = (q + 1)/2 for all odd prime powers q. we prove that #iq = #i ′ q − 1 = (q + 3)/4 if q ≡ 1 mod 4 and #iq = #i ′ q = (q + 5)/4 if q ≡ 3 (mod 4) (proposition 2.3). we only use the case a = eq, a = iq and a = i ′ q of the following definition. definition 1.4. fix s ⊆ fn q2 , s 6= ∅, and a ⊆ fq such that 0 ∈ a. we say that s is a-closed if a + (b − a)a ⊆ s for all a, b ∈ s. in the set-up of definition 1.4 for any a, b ∈ fnq2 the a-segment [a, b]a of {a, b} is the set a+(b−a)a. note that [a, a]a = {a} and that if b 6= a then b ∈ [a, b]a if and only if 1 ∈ a. if s is a subset of a real vector space and a is the unit interval [0, 1] ⊂ r, definition 1.4 gives the usual notion of convexity, because a + (b − a)t = (1 − t)a + tb for all t ∈ [0, 1]. 98 e. ballico cubo 24, 1 (2022) remark 1.5. take any a ⊆ fq such that 0 ∈ a. any translate by an element of f n q2 of an fq-linear subspace of f n q2 is a-closed. in particular fnq and f n q2 are a-closed. the intersection of a-closed sets is a-closed, if non-empty. hence we may define the a-closure of any s ⊆ fnq2, s 6= ∅, as the intersection of all a-closed subsets of fn q2 containing s. in most cases iq is not iq-closed. we prove the following result. theorem 1.6. assume q odd. then: (a) if q /∈ {3, 5, 9} (resp. q 6= 3), then fq is the iq-closure of iq (resp. the i ′ q-closure of i ′ q). (b) if q /∈ {3, 5, 9} (resp. q 6= 3), then the iq-closed (resp. i ′ q-closed) subsets of f n q2 are the translations of the fq-linear subspaces. remark 1.7. fix a ⊆ fq such that 0 ∈ a. assume that fq is the a-closure of fq. then s ⊆ f n q2, s 6= ∅, is a-closed if and only if it is the translation of an fq-linear subspace by an element of f n q2 . thus part (b) of theorem 1.6 follows at once from part (a) and similar statements are true for the a-closures for any a whose a-closure is fq. as suggested by one of the referees a key part of one of our proofs may be stated in the following general way. proposition 1.8. let a, b be subsets of fq containing 0. assume a 6= {0} and let g be the subgroup of the multiplicative group fq \ {0} generated by a \ {0}. assume that b is a-closed. then b \ {0} is a union of cosets of g. fix s ⊂ fn q2 and a set a ⊂ fq such that {0, 1} ⊆ a. instead of the a-closure of s the following sets si,a, i ≥ 1, seem to be better. in particular both circles and s1,a appear in some proofs on the numerical range. let s1,a be the set of all a + (b − a)a, a, b ∈ s. for all i ≥ 1 set si+1,a := (s1,a)1,a. obviously si,a is a-closed for i ≫ 0. note that {0, 1}a = a and hence if we start with s = {0, 1} we obtain the a-closure of a after finitely many steps. we thank the referees for an exceptional job, making key corrections and suggestions. 2 the proofs and related observations we assume q odd, except in remarks 2.1 and 2.2, proposition 2.9 and the proof of proposition 1.8. remark 2.1. the notions of eq-closed, iq-closed and i ′ q-closed subsets of f n q2 are invariant by translations of elements of fnq2 and by the action of gl(n, fq). cubo 24, 1 (2022) a characterization of fq-linear subsets of affine spaces f n q2 99 remark 2.2. fix any a ⊆ fq such that 0 ∈ a. any translate by an element of f n q2 of an a-closed set is a-closed. the fq-closed subsets of f n q2 are the translates by an element of fn q2 of the fq-linear subspaces. if a ⊆ {0, 1}, then any nonempty subset of fnq2 is a-closed. proof of proposition 1.8: since fq \ {0} is cyclic, g is cyclic. let a ∈ a \ {0} be a generator of g. fix c ∈ b \ {0} and take t ∈ fq \ {0} such that c = ta z for some positive integer z. we need to prove that b \ {0} contains all tak, k ∈ z. since b ∈ b, b is a-closed, a ∈ a and a = 0 + (a − 0), we get taz+1 ∈ b. iterating this trick we get that b contains all tak for large k and hence the coset tg, because g is cyclic. proposition 2.3. we have #iq = #i ′ q −1 = (q+3)/4 if q ≡ 1 (mod 4) and #iq = #i ′ q = (q+5)/4 if q ≡ 3 (mod 4). proof. since a := {x2 +y2 = 1} ⊂ f2q is a smooth affine conic, its projectivization b := {x 2 +y2 = z2} ⊂ p2(fq) has cardinality q + 1 ([10, th. 5.1.8]). note that the line z = 0 is not tangent to b and hence b ∩ {z = 0} has 2 points over fq2. it has 2 points over fq if and only if −1 is a square in fq, i. e. if and only if q ≡ 1 (mod 4) ([10, (ix) and (x) at p. 5], [22, p. 22]). hence #a = q + 1 if q ≡ 3 (mod 4) and #a = q − 1 if q ≡ 1 (mod 4). note that a ∈ iq if and only if there is (e, f) ∈ f2q such that e 2 + f2 = 1 and a = e2. note that (e, f) ∈ a and that conversely for each (e, f) ∈ a, e2 ∈ iq. obviously 0 ∈ iq and (0, f) ∈ a if and only if either f = 1 or f = −1. thus 0 ∈ iq comes from 2 points of a. obviously 1 ∈ iq. if either e = 1 or e = −1, then (e, f) ∈ a if and only if f = 0. thus 1 ∈ iq comes from 2 points of a. if e 2 /∈ {0, 1} and e2 ∈ iq, then e 2 comes from 4 points of a. fix a non-square c ∈ fq and set a ′ := {x2 − cy2 = 1} ⊂ f2q. let b ′ := {x2 − cy2 = z2} ⊂ p2(fq) be the smooth conic which is the projectivization of a′. the line {z = 0} is not tangent to b′ and {z = 0} ∩ a′ = ∅. thus #a′ = q + 1. note that a ∈ i′′q if and only if there is (e, f) ∈ f 2 q such that a = e2 and e2 − cf2 = 1. the element 1 ∈ i′′q comes from two elements of a ′. if 0 ∈ i′′q , then it comes from two elements of a′. if 0 /∈ i′′q , i. e. if q ≡ 3 (mod 4), we get #i ′′ q = (q + 1)/4 and #i′q = (q + 5)/4. if 0 ∈ i ′′ q we get #i ′′ q = #i ′ q = (q + 7)/4. remark 2.4. if q ∈ {3, 5}, then iq = {0, 1} and hence each non-empty subset of f n q2 is iq-closed if q ∈ {3, 5}. since {0, 1} ⊆ i′q, proposition 2.3 gives i ′ 3 = i3. we have i ′ 5 = {0, 1, 4} = e5, because 3 is not a square in f5. remark 2.5. fix any t ∈ fq \ eq. then fq \ eq = t(eq \ {0}). obviously eqeq = eq. the following result characterizes eq2 and hence characterizes all er with r a square odd prime power. proposition 2.6. the set of eq2 \ {0} of all squares of fq2 \ {0} is the set of all ab such that a ∈ fq \ {0} and b q+1 = 1. we have ab = a1b1 if and only if (a1, b1) ∈ {(a, b), (−a, −b)}. 100 e. ballico cubo 24, 1 (2022) proof. fix z ∈ fq2 \ {0}. hence z q2−1 = 1. thus z(q−1) q+1 = 1 (and so z(1−q) q+1 = 1) and z(q+1) q−1 = 1, i. e. zq+1 ∈ fq \ {0}. note that z 2 = zq+1z1−q. assume ab = a1b1 with a, a1 ∈ fq \ {0} (i.e., with a q−1 = a1 q−1 = 1) and bq+1 = b1 q+1 = 1. taking aa1 −1 and bb1 −1 instead of a and b we reduce to the case a1 = b1 = 1 and hence ab = 1. thus a q+1bq+1 = 1. hence a2 = 1. since q is odd and a 6= 1, then a = −1. thus b = −1. proposition 2.7. take s ⊆ fnq2. the set s is eq-closed if and only if it is a translation of an fq-linear subspace. proof. remark 2.2 gives the “if” part. assume that s is not a translation of an fq-linear subspace and fix a, b ∈ s such that a 6= b and the affine fq-line l spanned by {a, b} is not contained in s. by remark 2.1 it is sufficient to find a contradiction in the case n = 1 and l = fq with a = 0 and b = 1. thus eq ⊆ s. since s is eq-closed and 0 ∈ s, c + (−c)eq ⊆ s for all c ∈ eq. first assume −1 ∈ eq. in this case −ceq = eq. thus s contains all sums of two squares. thus s = fq. now assume −1 /∈ eq. in this case we obtained that s contains all differences of two squares. thus −eq ⊂ s. since −1 /∈ eq, −eq = {0} ∪ (fq \ eq) (remark 2.5). thus s ⊇ l. the cases of iq-closures and i ′ q-closures are more complicated, because iq = i ′ q = {0, 1} if q = 3, 5 and hence all subsets of fnq2 are iq-closed if q = 3, 5. the following observation shows that the i9-closed subsets of f n 81 are exactly the translations of the f3-linear subspaces and gives many examples with iq * i ′ q. remark 2.8. we always have 2 /∈ 1 + ceq, c a non-square, because 1 is a square. if q is a square, say q = s2, then obviously fs ⊆ eq ∩ (1 − eq) = iq and hence 2 ∈ iq. take q = 9. we get f3 ⊆ i9. since #i9 = 3 (proposition 2.3), we get iq = f3. thus the i9-closed subsets of f n 81 are exactly the translations of the f3-linear subspaces. now assume that q is not a square. we have 2 ∈ 1 − eq if and only if −1 is a square, i. e. if and only if q ≡ 1 (mod 4). since q is not a square, we have 2 ∈ eq if and only if 2 is a square in fp, i. e. if and only if p ≡ −1, 1 (mod 8) ([15, proposition 5.1.3]). thus for a non-square q holds: 2 ∈ iq if and only if p ≡ 1 (mod 8). proof of theorem 1.6: let y be the iq-closure of iq. by proposition 1.8, y ′ := y \ {0} is a union of the cosets of h := 〈iq \ {0}〉. since #(iq \ {0}) ≥ (q − 1)/4 with equality if and only if q ≡ 1 (mod 4), h is either f∗q, the set of non-zero squares, the set of non-zero cubes or (only if q ≡ 1 mod 4), the set of all non-zero 4-powers. since iq ⊆ eq, h 6= f ∗ q. if h is the set of cubes, then, as all elements of iq are squares, it would be the set of 6-th powers, contradicting the inequality #iq > (q − 1)/4. (a) assume that h = eq \{0}. it suffices to show that the iq-closure of the set of squares contains a non-square. suppose otherwise. take an element a ∈ iq with a /∈ {0, 1}. then we obtain that for all squares x, y, x + (y − x)a is also a square. since a is a non-zero square, this is the cubo 24, 1 (2022) a characterization of fq-linear subsets of affine spaces f n q2 101 same as the statement that for all squares x, z the element z + (1 − a)x is a square. if 1 − a is a square we deduce that the set of all squares is closed under addition, a contradiction. if 1 − a is not a square we may take x = 1, z = 0 to obtain a contradiction. (b) assume q ≡ 1 (mod 4), q 6= 9, and that h is the set of all non-zero 4-powers. we also saw that h = iq \ {0}. the proof of step (a) works using the word “4-power” instead of “square” with a a 4-power. we get that the set of all 4-powers is closed under taking differences. thus iq is closed under taking differences and, since it contains 0, under the multiplication by −1. h is obviously closed under taking products. thus iq is a subfield of order (q + 3)/4, which is absurd if q 6= 9. (c) now we consider i′q and set h ′ := 〈iq \ {0}〉. the cases in which h ′ is the set of all squares or all cubes are excluded as above. since #(i′q \ {0}) > (q − 1)/4, y is not the set of all 4-th powers. proposition 2.9. assume q even and set eq := {a(a + 1)}a∈fq. (1) if q = 2, 4, then eq is the eq-closure of itself. (2) if q ≥ 8, then fq is the eq-closure of eq. proof. we have e2 = {0} and e4 = {0, 1}. now assume q ≥ 8 and call b the eq-closure of eq. let g be the subgroup of the multiplicative group fq \ {0} generated by eq \ {0}. by proposition 1.8 it is sufficient to prove that g = fq \ {0}. since #eq = q/2, eq \ {0} 6= ∅. fix a ∈ eq \ {0} and a positive integer k. the eq-closure of {0, ak} contains ak+1. thus b contains the multiplicative subgroup of fq \ {0} generated by eq \ {0}. since q ≥ 8, #(fq \ {0}) = q − 1 is odd and q − 1 < 3(q/2 − 1) = 3#(fq \ {0}), we get g = fq \ {0}. 102 e. ballico cubo 24, 1 (2022) references [1] e. ballico, “on the numerical range of matrices over a finite field”, linear algebra appl., vol. 512, pp. 162–171, 2017. [2] e. ballico, corrigendum to “on the numerical range of matrices over a finite field” [linear algebra appl., vol. 512, pp. 162–171, 2017], linear algebra appl., vol. 556, pp. 421–427, 2018. 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[22] c. small, arithmetic of finite fields, monographs and textbooks in pure and applied, 148, new york: marcel dekker, inc., 1991. introduction the proofs and related observations cubo, a mathematical journal vol. 23, no. 02, pp. 265–285, august 2021 doi: 10.4067/s0719-06462021000200265 on rellich’s lemma, the poincaré inequality, and friedrichs extension of an operator on complex spaces chia-chi tung 1 pier domenico lamberti 2 1 department of mathematics and statistics, minnesota state university, mankato mankato, mn 56001, usa (emeritus). imggtn14@outlook.com 2 dipartimento di tecnica e gestione dei sistemi industriali (dtg), university of padova, stradella s. nicola 3-36100 vicenza, italy. lamberti@math.unipd.it abstract this paper is mainly concerned with: (i) a generalization of the rellich’s lemma to a riemann subdomain of a complex space, (ii) the poincaré inequality, and (iii) friedrichs extension of a schrödinger type operator. applications to the eigenfunction expansion problem associated to the modified laplacian are also given. resumen este art́ıculo se enfoca principalmente en: (i) una generalización del lema de rellich a un subdominio de riemann de un espacio complejo, (ii) la desigualdad de poincaré, y (iii) la extensión de friedrichs de un operador de tipo schrödinger. se entregan también aplicaciones al problema de expansión de funciones propias asociado al laplaciano modificado. keywords and phrases: weighted sobolev-schrödinger product, friedrichs extension, resolvent mapping. 2020 ams mathematics subject classification: 46e35, 26d10, 35j10, 32c30, 35j25. accepted: 01 june, 2021 received: 16 january, 2021 c©2021 c. tung et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000200265 https://orcid.org/0000-0003-0939-0714 https://orcid.org/0000-0003-2502-5661 266 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) 1 introduction a milestone of pure and applied analysis since the last century is a selection principle of f. rellich ([15], [4, p. 414]): given a family of c 1-functions f in a bounded domain ω ⊂ rn with smooth boundary such that both the functions f and their first partial derivatives are uniformly bounded in the l2(ω)-norm, then {f} contains a cauchy subsequence with respect to the l2(ω)-norm. one consequence of this far reaching result is the rellich’s principle: the laplacian with zero boundary condition on a bounded domain ω ⊂ rn has a compact resolvent. thus the eigenfunctions of the equation −∆ψ = µψ in ω, ψ|∂ω = 0, form a complete orthogonal basis of l2(ω). in view of some recurring interest concerning the rellich embedding property on non-flat domains ([14], [13, 3.9.3], [11]), it seems of value to consider the question as to on what general domain the rellich embedding and its consequences will remain valid. a main purpose of this note is to propose a setting of riemann subdomains (in a complex space) on which an affirmative answer could be sought. in view of the fact that on a singular space analytic tools are not yet sufficiently developed 1, the basic notions of the sobolev spaces on a riemann subdomain (see [20]) need be properly formulated (to be recalled below). assume throughout this paper that (d,p) is a (relatively compact) riemann subdomain in a complex space y of dimension m, meaning that d is a relatively compact open subset of y and (as a subspace of y ) admits a holomorphic map p : d → cm with discrete fibers. note that the pair (d,p) is a riemann domain in the sense of [3, p. 19] and [8, p. 135]. if in addition p is a local homeomorphism, then (d,p) is said to be unramified. every complex space of pure dimension m admits locally an open neighborhood (of each given point) and a finite, open holomorphic map onto a domain in cm ([9, pp. 107-108]), hence is locally realizable as a riemann subdomain. let h denote a (2m + 1)-tuple (h0,h1, · · · ,h2m) of locally integrable functions hj on d. set v := −h0, and denote by h the (2m + 1)-tuple obtained by replacing the initial entry by 0 (thus h = (0,h1, · · · ,h2m)). let µ be a non-negative constant. a (2m + 1)-tuple h (as above) is called an allowable weight for (d,µ) if each component hj, j = 1, · · · ,m, is positive a. e. on d, and either (µ,v ) = (0,0) or c 0 := µ − ess sup d v > 0. (for convenience) call a 2m-tuple h′ = (h1, · · · ,h2m) with positive a. e., locally integrable components an allowable weight on d, and set h′ −→ := (0,h′). (unless explicitly specified) in the following let h denote an allowable weight for (d,µ). let d∗ be the largest open subset of d on which the map p = (p1, · · · ,pm) is locally biholomorphic, and set x̃j := re(pj), ỹj := im(pj), and 1 see ruppenthal [17, p. 7]) “· · · whereas geometric and algebraic methods · · · are very well developed on singular spaces, most analytic tools are still missing”. cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 267 ∂j := ∂ ∂x̃k , j = 2k − 1, ∂j := ∂ ∂ỹk , j = 2k, 1 ≤ k ≤ m, on d∗. the space c 1(d) is equipped with the (weighted) sobolev-schrödinger product 〈w, v〉 µ,h := ((µ − v )w,v̄) d + [w,v] d,(h1,··· ,h2m) ∀w, v ∈ c 1(d), (1.1) where [ , ] d,h′ denotes the (weighted) dirichlet product [w,v] d,(h1,··· ,h2m) := m∑ k=1 ∫ d∗ ( h2k−1 ∂w ∂x̃k ∂v̄ ∂x̃k + h2k ∂w ∂ỹk ∂v̄ ∂ỹk ) dυ̃. (1.2) here dυ̃ denotes the pull-back of the euclidean volume form on cm under p. clearly for all g ∈ c 1(d) one has ‖g‖2 µ,h ≥ ∫ d (µ − v )|g|2 dυ̃ ≥ c 0 ‖g‖2 l2(d) . the sesqilinear form “〈 , 〉 µ,h ” being elliptic, defines a scalar product on c 1(d). with respect to the induced norm ‖ ‖ µ,h the completion of c 1(d) gives rise to the sobolev space h1µ,h(d), with induced scalar product (denoted by the same) and induced norm ‖ ‖ µ,h . similarly the completion of the space c ∞,c(d) of test functions defines the sobolev space h1µ,h,c(d) (see [20] and also [19]). observe also that the dirichlet product [w, v] d,(h1,··· ,h2m) = 〈w, v〉 1,(0,h1,··· ,h2m) − (w, v̄) d where w, v ∈ c 1,c(d) (given explicitly by the equation (1.2)) extends to a scalar product on h1 0,(0,h1,··· ,h2m) (d). the norm of f ∈ h11,(0,h1,··· ,h2m)(d) is given by ‖f‖ 1,(0,h1,··· ,h2m) = (∫ d |f|2 dυ̃ + [f,f] d,(h1,··· ,h2m) )1 2 . as an application of the inner products (1.1)-(1.2), an explicit representation of the friedrichs extension of the weighted schrödinger operator 2 on a riemann subdomain, allowing possibly singular points, is derived (theorem 3.1). it is well-known that the employment of the poincaré inequality plays a central role in the study of compact sobolev embeddings. the connection between the (classical) poincaré inequality and the rellich embedding theorem (for euclidean domains) was clarified by galaz-fontes [7]. 3 given an allowable 2m-weight h′ on a non-flat subdomain (d,p), questions remain open, however, as to 2 this operator is motivated by the classical schrödinger operator (appearing in the schrödinger equation); the latter has been considered as “one of the most interesting objects in mathematical physics · · · ” (see [12, p. 3]). 3 the poincaré inequality is sometimes referred to as the poincaré-friedrichs inequality. an underlying incentive for this paper is provided by a motivational remark in [7] (where by the “friedrichs inequality” is meant the “poincaré inequality”), “an explicit connection between the “friedrichs inequality” and the rellichs theorem has not been reported” (at least for a riemann subdomain). 268 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) whether (either of) the following properties holds: (1) (d,p) has the poincaré property relative to h′, in the sense that there exists a constant c d,h′ such that the following poincaré inequality (with variable coefficients) holds: ‖f‖ l2(d) ≤ c d,h′ [f,f] 1 2 d,h′ ∀f ∈ h11,(0,h′),c(d); (1.3) (2) (d,p) is a rellich domain with respect to h′, namely the rellich property, “the embedding h1 1,h′ −→ ,c (d) →֒ l2(d) is compact”. the rellich embedding property will be proved in §4 (independently of the poincaré inequality for constant allowable weight) only for the case where (d,p) satisfies the following conditions: (i) (d,p) is of finite volume, namely, ∫ d∗ dυ̃ < ∞, and (ii) (d,p) is of sobolev type, that is, there exists a constant α > 2 such that, for all f ∈ h1 1,(0,1,··· ,1),c (d), ‖f‖lα(d) ≤ const. ‖f‖1,(0,1,··· ,1). (1.4) every euclidean domain in cm is of sobolev type ([6, theorem 6, p. 270]). other examples are provided by finitely quasiregular riemann subdomains (see §4). theorem 1.1. (rellich embedding theorem): if (d,p) is of finite volume and of sobolev type, then (d,p) is a rellich domain with respect to any constant allowable 2m-weight on d. theorem 1.2. if (d,p) is an unramified riemann subdomain in a complex space and if p defines an étale covering 4 , then (d,p) has the poincaré property relative to any allowable 2m-weight h′ on d with h d := min {ess inf d (hj) | 1 ≤ j ≤ 2m} > 0. the rellich embedding property can be extended to sections of a vector bundle over a differentiable manifold (see [10, p. 88 and p. 93], [22, p. 111]). it would be interesting to characterize those riemann subdomains of an m-dimensional complex space which can be realized as a rellich domain with respect to some allowable 2m-weight. a form of the poincaré inequality (with respect to a continuous allowable 2m-weight) holds as a consequence of the rellich embedding property (see proposition 4.3). for a riemann subdomain d with the poincaré property relative to an allowable 2m-weight h′, the defining norms of the sobolev spaces h1 1,h′ −→ ,c (d) and h1 0,h′ −→ ,c (d) 5 are equivalent. on such a domain the inhomogeneous dirichlet problem (for the poisson equation) admits a weak solution (corollary 4.5). as a further application, consider the following (inhomogeneous) dirichlet problem (to be referred to as a poisson problem): − 2m∑ j=1 ∂j(hj∂jψ) + αψ = g a.e. in d, ψ|dd = 0, (1.5) 4 defined in §4. 5 this equivalence is sometimes called the friederichs’s lemma. the traditional notation for the sobolev space h1 1,(0,1,··· ,1),c (d) is w 1,2 0 (d). cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 269 where g ∈ l2(d) 6 . a (weak) solution on d of this problem is taken in the following sense: ψ is an element in h1 1,h′ −→ ,c (d) satisfying the equation − 2m∑ j=1 ∂j(hj∂jψ) + αψ = g weakly in d, namely, ψ satisfies the functional equation [ψ,v] d,h′ + α(ψ, v̄) d = (g, v̄) d , ∀v ∈ c ∞,c(d). (1.6) to solve this problem one must also determine the eigenvalues λ := −α. to this end, an alternative but more expedient formulation of the above equation is available (see lemma 6.2). the latter requires the use of an operator, to be called a resolvent map, which can be introduced as follows: for each allowable 2m-weight h′ on d (as above with respect to h′), there is a linear mapping r d,h′ : l2(d) → l2(d) with image in h1 1,h′ −→ ,c (d) defined by the equation [r d,h′ f,v] d,h′ = (f, v̄) d (1.7) for all (f,v) ∈ l2(d)×h1 1,h′ −→ ,c (d). on a rellich domain this map (in the special case of r d,(1,··· ,1) ) is in fact a compact mapping: theorem 1.3. for any relatively compact rellich domain d in y, the resolvent map r d,{1,··· ,1} : l2(d) → l2(d) defined by [r d,{1,··· ,1} g,v] d,{1,··· ,1} = (g, v̄) d , ∀(g,v) ∈ l2(d) × h11,(0,1,··· ,1),c(d), is a compact, self-adjoint mapping. if (d,p) is a rellich domain of type h′, then the behavior of the solutions of the boundaryeigenvalue problem − 2m∑ j=1 ∂j(hj∂jψ) + αψ = 0 a.e. in d, ψ|dd = 0, (1.8) can be determined, and similarly for the case of the poisson problem (1.5). for completeness this spectral analysis is carried out in §6. given mathematical physicists’ interest in complexified space-time models 7 , it is hoped that the results of this paper may be of use in some applications. 6 here (and in the following) dd denotes the (maximal) boundary manifold of dreg in the manifold yreg of simple points of y, oriented towards the exterior of dreg ([18, p. 218]). 7 see, e. g., hansen, r. o. and newman, e. t. a complex minkowski approach to twistors, grg vol.6, no.4 (1975), 361-385. 270 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) 2 preliminaries in what follows every complex space is assumed to be reduced and has a countable basis of topology. for the definition and basic properties of differential forms on a complex space, see [18, §4.1]. in particular, the exterior differentiation d, the operators ∂, ∂̄ and dc := (1/4πi)(∂ − ∂̄) are welldefined ([18, chap. 4]). for an open subset g of y, denote by c νk (g) the set of c-valued k-forms of class c ν on g, c νk,c(g) the subspace of compactly supported k-forms (dropping the degree if k = 0), with ν = β to mean locally bounded, ν = m measurable, and ν = λ locally lipschitzian ([18, §4]). similarly for c νk (g). a measurable function 8 g on y is said to be locally integrable (g ∈ l1loc(y )) provided the form g χ is locally integrable on yreg for every 2m-form χ ∈ c 0 2m(y ). similarly define l1(y ) and l2loc(y ) (for the latter, the above gχ is replaced by |g| 2χ). denote by ‖z‖ the euclidean norm of z = (z1, · · · ,zm) ∈ c m, where zj = xj +iyj. let the space cm be oriented so that the form υm := (ddc‖z‖2)m is positive. let b(r) denote the r-ball in cm centered at the origin and b[1] = {z ∈ cm | ‖z‖ ≤ 1}. let p : y → cm be a holomorphic map. if s ⊆ y, let s′ := p(s); and in particular write a′ = p(a). set p[a] := p − a′, ∀a ∈ y. clearly the form υp := dd c‖p[a]‖2 = ( i 2π ) ∂∂̄ ‖p[a]‖2 (2.1) is non-negative and independent of a. denote by dυ the euclidean volume element of cm and define dυ̃ := p∗(dυ) on y. if (d,p) is a riemann subdomain, then dυ̃ is a semivolume form on d and dυ̃ = cmπ m m! υmp , where cm := (−1) m(m−1) 2 . if f, ψ ∈ l1loc(d), set (f,ψ) d := ∫ d f ψ dυ̃, provided the integral exists. each element f ∈ l1loc(d) gives rise naturally to a top-dimensional current, t = [f] : χ 7→ ∫ f ∧ χ, with induced functional (cf. [21]) defined by 〈[f], φ〉 := (−1) m(m−1) 2 πm m! ∫ fφυmp , ∀φ ∈ c ∞,c(d). 3 friedrichs extension of a schrödinger type operator given an allowable (2m + 1)-weight h for (d,µ) and g ∈ l2(d), the antilinear functional v 7→ (g, v̄) d is well-defined and continuous on h1µ,h,c(d). by the riesz representation theorem, there is a unique element w ∈ h1µ,h,c(d) satisfying the equation 〈w, v〉 µ,h = (g, v̄) d , ∀v ∈ h1µ,h,c(d), (3.1) 8a function g on y is said to be measurable on y is so is the restriction g|yreg. cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 271 with ‖w‖ µ,h ≤ const. ‖g‖l2(d). the assignment g 7→ gµ,hg := w defines a linear, continuous mapping g µ,h of l2(d) onto its image r µ,h (d) := g µ,h (l2(d)). clearly one has 〈g µ,h g,v〉 µ,h = (g, v̄) d , ∀v ∈ h1µ,h,c(d). (3.2) thus g µ,h may be regarded as a (weak) green’s map for the weighted sobolev-schrödinger operator. observe that 〈g µ,h v, v〉 µ,h = ‖v‖2l2(d), ∀v ∈ h 1 µ,h,c(d), (3.3) and 〈g µ,h ψ,v〉 µ,h = 〈g µ,h v, ψ〉 µ,h = 〈ψ,g µ,h v〉 µ,h ∀(ψ,v) ∈ h1µ,h,c(d) × h 1 µ,h,c(d). by the equation (3.3) the green’s map g µ,h : l2(d) → h1µ,h,c(d) is injective, and the inverse map f µ,h := g−1 µ,h : r µ,h (d) → l2(d) is well-defined. also, it is positive definite since given g ∈ l2(d), the equation (3.2) implies that (f µ,h (g µ,h g), g µ,h g) d = (g, g µ,h g) d = 〈g µ,h g,g µ,h g〉 µ,h = ‖g µ,h g‖2 µ,h ≥ const.‖g µ,h g‖2l2(d). for an allowable (2m + 1)-weight h for (d,µ) of class c 1 (namely, so is each component hj of h on d), define the schrödinger operator s µ,h acting on c 1(d) (in the classical sense) by s µ,h ψ := (µ − v )ψ − 2m∑ j=1 ∂ j (hj ∂jψ) on d ∗. (3.4) assume (in the following theorem) that h is an allowable (2m + 1)-weight for (d,µ) of class c 1. define d µ,h := {ψ ∈ c 2,c(d)| s µ,h ψ ∈ l2(d)}, and h(d,s µ,h ) := h1µ,h,c(d) ∩ {w ∈ l2(d)| s µ,h [w] ∈ l2(d)} 9. theorem 3.1. (friedrichs extension of the operator s µ,h ) let d be a riemann subdomain in y. then the weighted sobolev-schrödinger operator s µ,h : d µ,h → l2(d) admits a positive, selfadjoint extension f µ,h : h(d,s µ,h ) → l2(d) with the property that for each w ∈ h(d,s µ,h ), (f µ,h w, v̄) d = 〈w,v〉 µ,h , ∀v ∈ h1µ,h,c(d), (3.5) and (f µ,h w, v̄) d = (w, s µ,h v̄) d , ∀v ∈ c ∞,c(d). (3.6) proof. it will be shown that the inverted mapping f µ,h := g−1 µ,h acts as an extension of the operator s µ,h to r µ,h (d) = g µ,h (l2(d)). it follows from the symmetry of g−1 µ,h : l2(d) → r µ,h (d) that f µ,h : r µ,h (d) → l2(d) is self-adjoint, and satisfies the equation (f µ,h w, v̄) d = 〈w,v〉 µ,h , ∀(w,v) ∈ r µ,h (d) × h1µ,h,c(d). (3.7) 9“s µ,h [w]”] denotes the (weak) action of s µ,h on the functional [w]. 272 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) if further v is an element of c 2,c(d), then 〈w,v〉 µ,h = lim j→∞ 〈wj,v〉µ,h = lim j→∞ (wj, sµ,hv̄)d. this implies that (f µ,h w, v̄) d = (w, s µ,h v̄) d . (3.8) to see that the mapping f µ,h is indeed defined on h(d,s µ,h ) one needs to check that r µ,h (d) = h(d,s µ,h ). let w ∈ r µ,h (d). clearly then w ∈ h1µ,h,c(d) and w = gµ,hg for some g ∈ l 2(d). then by the equation (3.8) one has, for all v ∈ c ∞,c(d), (w, s µ,h v) d = (ψ, v) d with ψ := f µ,h w ∈ l2(d), and thus s µ,h [w] = ψ as desired. hence w ∈ h(d,s µ,h ). conversely if w ∈ h(d,s µ,h ), then w ∈ h1µ,h,c(d) and sµ,h[w] ∈ l 2(d). thus there exists an element g ∈ l2(d) such that ( s µ,h [w], v ) d = (g, v) d , ∀v ∈ c ∞,c(d). for each v ∈ h1µ,h,c(d) and a sequence {vj} in c ∞,c(d) tending to v, 〈w,v〉 µ,h = lim j→∞ 〈w,vj〉µ,h = lim j→∞ (w, s µ,h v̄j)d = lim j→∞ (g, v̄j)d = (g, v̄)d, hence w satisfies the equation (3.2), thereby proving that w ∈ r µ,h (d). therefore the formula (3.5) follows from the equation (3.7). consequently the mapping f µ,h gives the friedrichs extension of s µ,h : d µ,h → l2(d). remark 3.2. especially the preceding theorem ensures that the existence of the friedrichs extension of the laplacian 10 : −△{p} : dµ,(1,··· ,1) → l 2(d). this extension is given by the mapping f : h(d,−△{p}) → l 2(d), and admits the representation (fw, v̄) d = [w, v] d,(1,··· ,1) , ∀(w,v) ∈ h(d,−△{p}) × h 1 1,(0,1,··· ,1),c (d). in particular, for all (w,v) ∈ h(d,−△{p}) × c ∞,c(d), (fw, v̄) d = (w, −△{p}v̄)d. 4 the rellich embedding theorem and poincaré inequality let p : y → y ′ be a continuous mapping (between topological spaces). an open subset w ⊆ y ′ is called a base domain evenly covered by p, if p−1(w) is a disjoint union of open subsets bl ⊆ y 10 here “△{p}” denotes the (local) pullback to d ∗ under p of the laplace operator of the euclidean metric on cm. cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 273 each of which is homeomorphic to w (under p); as such bl will also be referred to as a covering sheet of p lying evenly over w (relative to p). if p : d → cm is a light mapping 11 , then there exists an open connected neighborhood ua ⋐ d of each a ∈ d, called a pseudoball at a ([21, §2]) such that, when restricted to the regular part ûa := ua\p −1(∆′) (∆ being the branch locus), the map p a = p : ûa → u ′\ ∆′ is an unramified finite holomorphic covering, where u′ := b[a′](r) is an open ball in c m with center a′ (see [1, §2]). the sheet number of p a is equal to νp(a), the multiplicity of p at (the center) a ([ibid., §2]). thus at each point z′ ∈ u′\∆′ there is an open ball wz ′ ⋐ u′\∆′ (of radius < 1) (here z′ will be denoted by z′k and w z′ by wk for reasons to become clear later), which is a base domain evenly covered by p a with covering sheets bℓ = bz ′ l ⊆ d, l = 1, · · · , l(k), over w k, each being (necessarily) biholomorphic to wk under p. here the number l(k) is equal to νp(a) (for each z ′ k) and bkl has compact closure in ûa. definition 4.1. an admissible covering of a compact subset k of d∗ is an open covering of k consisting of open subsets bℓ ⊆ d∗ with ℓ varying in a finite range such that each bℓ is equal to some bkl (namely some b z′k l ) with k in a finite range and l ∈ z[1, l(k)]; moreover, for each fixed k the bkl (with varying l) lie evenly (relative to p) over a base domain w k := wz ′ k contained in some open ball ŵk := ŵz ′ k ⊂ cm of volume < 1. note that every bounded domain d in cm admits an admissible covering (via a dilatation of the identity map). also, if ua is a pseudoball contained in d, then every compact subset k of ûa admits an admissible covering. the set d∗ being σ-compact, one can write d∗ as a union of an exhausting (increasing) sequence {kj} of compact subsets. choose a c ∞-partition of unity {βj,ℓ} 1≤ℓ≤nj on kj subordinate to an admissible covering {bj,ℓ} for kj. one has, for each f ∈ c ∞,c(d), setting f{j} ℓ := (βj,ℓ) 1 2 f (the ℓ-th partitioned function of f relative to {bj,ℓ}), ∫ kj |f|2 dυ̃ ≤ nj∑ ℓ=1 ∫ d∗ |f{j} ℓ |2 dυ̃. (4.1) a riemann subdomain (d,p) in y is said to be finitely quasiregular if (i) each compact subset k of d∗ admits a (finite) admissible covering {bkl } with (corresponding) base domains w k with c 1-boundary contained in some open ball of finite radius in cm; and (ii) the family {wk} has finite cardinality k which depends only on d. if further d is unramified, then (d,p) is said to be finitely regular. the above definition of “finitely quasiregular domain” is equivalent to the following: “(d,p) is finitely quasiregular if the regular part d∗ is a finite union of local covering sheets b{ℓ}, ℓ = 11 a holommorphic mapping f : y → y ′ (between complex spaces) is light on d ⊆ y if for each a ∈ d, dima f −1(f(a)) = 0 . 274 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) 1, · · · ,n”. for, the above definition implies that, given any (increasing) sequence {kj} of compact exhaustion of d∗, the index sequence {nj} (with nj arising from a corresponding kj) can be replaced by a finite sequence {1, · · · ,n} with n = l(1) + · · ·+ l(k). thus d∗ is contained in (at most) n local covering sheets b{ℓ}. the converse assertion is trivially true. note that every bounded domain in cm is finitely regular. also, each relatively compact subset d of the projective space pm(c) contained in the open set q{s}, s ∈ z[1,m], is finitely regular relative to the riemann covering p = p[s]. for a further example, consider the complex space y = {(z,w) ∈ c2 |w2 = z3}. relative to the projection (z,w) 7→ z the open set d := {(z,w) ∈ y |‖z‖ ≤ r0} is a finitely quasiregular riemann subdomain in y (with two covering sheets). while for a general domain the rellich embedding theorem may not be valid, the requirement that the domain be of sobolev type is indeed somewhat restrictive. a characterization of the latter remains an open question. the following lemma is of some use: lemma 4.2. every finitely quasiregular riemann subdomain (d,p) in y is of finite volume and of sobolev type. proof. the regular part d∗ of a finitely quasiregular riemann subdomain (d,p) admits a finite covering by local covering sheets bℓ with ℓ varying in a finite range, say {1, · · · ,n}. it is easy to show that d has finite volume with respect to p. one can write d∗ as a union of an exhausting (increasing) sequence {kj} of compact subsets. each kj admits a finite admissible covering {bj,ℓ} 1≤ℓ≤nj . the sobolev embedding inequality on a bounded euclidean domain ([6, theorem 6, p. 270] (applied to each bj,ℓ) implies that, for all g ∈ c ∞,c(d) 12, (‖g‖lα(d)) α ≤ lim j→∞ nj∑ ℓ=1 ∫ bj,ℓ |g|α dυ̃ = lim j→∞ nj∑ ℓ=1 (‖g‖lα(bj,ℓ)) α ≤ (c bm )α lim j→∞ nj∑ ℓ=1 (‖g‖h1 1,(0,1,··· ,1) (bj,ℓ)) α ≤ (c bm )α lim j→∞ nj∑ ℓ=1 (‖g‖h1 1,(0,1,··· ,1) (d)) α for some constant c bm (independent of d). since the local covering sheets bj,ℓ can be selected from the finite set {bℓ}ℓ=1,··· ,n , the above index nj can be replaced by n . thus (‖g‖lα(d)) α ≤ const.(‖g‖h1 1,(0,1,··· ,1) (d)) α. the space c ∞,c(d) being dense in h1 1,(0,1,··· ,1),c (d), each element f therein can be approximated in the l2-norm by a sequence {g n } ⊂ c ∞,c(d). thus the sobolev inequality (1.4) follows. 12 for clarity denote the sobolev norm of g ∈ h1 1,(0,1,··· ,1) (bℓ) by ‖g‖ h1 1,(0,1,··· ,1) .(bℓ) and similarly for g ∈ h1 1,(0,1,··· ,1),c (d). cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 275 proof of theorem. 1.1. for later reference, assume (unless otherwise mentioned) that h denotes a general allowable weight as indicated in the beginning of §1. let {f n } be a bounded sequence in h1µ,h,c(d). it contains a subsequence {fnk } which converges weakly to some element f ∈ h1µ,h,c(d). by considering {fnk − f} it may be assumed that {fnk } ⊂ {fn} is a subsequence converging weakly to 0. in the following for simplicity denote this subsequence by the same notation {f n }. since c ∞,c(d) is dense in h1µ,h,c(d), each fn is the limit of a sequence {φ n l } ⊂ c ∞,c(d) with respect to the ‖ ‖µ,h-norm on h 1 µ,h,c(d). thus, for each n ∈ n there exists an element φ{n} ∈ c ∞,c(d) such that ‖f n − φ{n}‖µ,h < 1 n . then for any given ρ ∈ h1µ,h,c(d), (φ{n},ρ)µ,h = (φ {n} − f n ,ρ)µ,h + (fn,ρ)µ,h → 0, n → ∞, namely φ{n} tends to 0 weakly in h1µ,h,c(d). since ‖f n − φ{n}‖ l2(d) ≤ const.‖f n − φ{n}‖µ,h, to prove the rellich embedding theorem it suffices to consider the case where {f n } lies in c ∞,c(d) and converges weakly to 0 in h1µ,h,c(d) (in which case {fn} is uniformly bounded in h 1 µ,h,c(d)). in the rest of the proof, assume that (µ,v ) = (1,0). the goal is to show (possibly by passing to a subsequence) that limn→∞ ‖fn‖ 2 l2(d) exists and equals zero. for this purpose it will be necessary to swap the order of the limits in the following relation lim n→∞ lim j→∞ ∫ kj |f n |2 dυ̃ = lim j→∞ lim n→∞ ∫ kj |f n |2 dυ̃ (4.2) (where {kj} is an increasing exhausting sequence of compact subsets of d ∗). this can be justified by verifying two conditions: (a) the sequence of functions φ j (n) := ∫ kj |f n |2 dυ̃ converges to φ(n) := lim j→∞ ∫ kj |f n |2 dυ̃ (as j → ∞) uniformly in n, and (b) for each fixed j the limit “limn→∞ ∫ kj |f n |2 dυ̃” exists (in fact, equals 0). more explicitly, the assertion (a) amounts to showing that, for each ǫ > 0, ∃jǫ and nǫ ∈ n such that ∫ (kjǫ ) c |f n |2 dυ̃ ≤ ǫ, ∀n ≥ nǫ. (4.3) to prove this condition, set 1 q′ := 2 α and 1 p′ := 1 − 1 q′ . then by hölder’s inequality (with f = 1 and g = |f n |2) the left-hand side of the above inequality is dominated by ∫ (kjǫ ) c |1 · |f n |2|dυ̃ ≤ (∫ (kjǫ ) c 1p ′ dυ̃ ) 1 p′ (∫ (kjǫ ) c (|f n |2)q ′ dυ̃ ) 1 q′ . (4.4) 276 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) since (|f n |2)q ′ = |f n |α, and (‖|f n |2‖)lq′ ((kjǫ )c) = (∫ (kjǫ ) c |f n |α dυ̃ ) 2 α = (‖f n ‖lα((kjǫ )c)) 2, the inequality (4.4) becomes ∫ (kjǫ ) c |f n |2 dυ̃ ≤ |vol((kjǫ) c)|1− 2 α ‖f n ‖2lα((kjǫ )c) . therefore (if α is chosen to satisfy the sobolev inequality (1.4)) the second factor on the right-hand side is uniformly bounded, hence one has ∫ (kjǫ ) c |f n |2 dυ̃ ≤ const. |vol((kjǫ) c)|1− 2 α . finally, since (d,p) has finite volume, by letting jǫ → ∞ one has vol((kjǫ) c) → 0. thus the condition for uniform convergence (4.3) follows. by choice of an open covering of kj by local covering sheets b ℓ (with notations as in definition 4.1) each of which is contained in a fixed open neighborhood dj ⋐ d ∗ (the sheets bℓ and dj being dependent only on j), all the terms f{j} n,ℓ have support contained in dj. denoting by f̂ {j} n,ℓ the direct image of f{j} n,ℓ |bℓ (under the biholomorphic map p : bℓ → bℓ = p(bℓ), the resulting family {f̂{j} n,ℓ } is uniformly bounded in h1µ,h,c(b ℓ) with respect to n, ℓ. inscribe bℓ in a cube c (independent of ℓ, and it may be assumed that c ⊂ b(1) by rescaling and translation). every element ρ̂ ∈ c ∞,c(bℓ) can be extended to c trivially (by setting ρ̂ to be zero off bℓ). the weak convergence of {f n } (to 0) in h1µ,h,c(b ℓ) implies that of f{j} n,ℓ , hence f̂{j} n,ℓ converges weakly to 0 in h11,h,c(b ℓ). the preceding discussion shows that the compactness of the embedding “h11,h,c(d) →֒ l 2(d)” is a consequence of the following “claim”: if {f̂{j} n,ℓ } is a sequence (indexed by n) in h1 1,(0,h′),c (bℓ) which converges weakly to 0 in h11,(0,h′),c(b ℓ), the h′ being an allowable constant 2m-tuple, then lim n→∞ ∫ d∗ |f{j} n,ℓ |2 dυ̃ = 0. upon taking h = (0,h = ) with h = = (1, · · · ,1), this is an immediate consequence of the rellich embedding theorem for a bounded euclidean domain (here the open ball bℓ). this proves the assertion (b). by interchanging the “n-” and the “j-limit” and making use of the relations (4.2) and (4.1), one has lim n→∞ lim j→∞ ∫ kj |fn| 2 dυ̃ = lim j→∞ lim n→∞ ∫ kj |fn| 2 dυ̃ ≤ lim j→∞ lim n→∞ nj∑ ℓ=1 ∫ d∗ |f{j} n,ℓ |2 dυ̃ = 0. cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 277 consequently theorem 1.1 is proved in this case. let h′ = (c1, · · · ,c2m) be a 2m-tuple of positive constants. define p̂ = p̂ {h′} = (p̂ 1 , · · · , p̂ m ) : d → cm by setting p̂ k (z) := x̂ k + iŷ k = (c 2k−1 )−1/2x̃ k + i(c 2k )−1/2ỹ k , 1 ≤ k ≤ m, then the volume element dυ̂(h′) := p̂∗(dυ) = c −1/2 1 · · ·c −1/2 2m dυ̃. it is easy to show that the norm of an element f of ĥ1 1,{0,1,··· ,1},c (d), that is, the space h11,{0,1,··· ,1},c(d) defined w. r. t. (d,p̂{h′}) with h ′ = (1, · · · ,1), is equivalent to the norm of f in h1 1,{0,c1,··· ,c2m},c (d) w. r. t. (d,p). hence the sobolev space ĥ1 1,{0,1,··· ,1},c (d) is the same as h11,{0,c1,··· ,c2m},c(d). since the embedding “ ĥ 1 1,{0,1,··· ,1},c(d) →֒ l 2(d)” is compact, the same is true for the embedding “ h1 1,{0,c1,··· ,c2m},c (d) →֒ l2(d)”. this completes the proof theorem 1.1. a riemann subdomain (d,p) in y is said to define a (distinguished) étale covering 13 (of p(d∗)) if (i) each compact subset k of d∗ admits a (finite) admissible covering {bkl } (with corresponding base domains wk); and (ii) the family {wk} is pair-wise disjoint. note that every bounded domain d in cm defines a distinguished étale covering (via a rescaled identity map). as another example, consider the m-dimensional projective space pm(c) and let q{s} := {a = [a0, · · · ,am]|as 6= 0}, where s ∈ z[1,m]. then q {s} can be regarded as an open riemann subdomain in pm(c) relative to the s-th canonical coordinate map p[s] : q{s} → cm given by p[s] : a 7→ ( a0 as , · · · , âs as , · · · , am as ) (here “̂” denotes omission). clearly every relatively compact subset d ⋐ q{s} defines a distinguished étale covering via the riemann covering p = p[s]. proof of theorem 1.2. let h′ = (h1, · · · ,h2m) be any allowable weight on d. to prove the poincaré inequality (1.3), it suffices to verify it for all elements g ∈ c ∞,c(d). for, given f ∈ h1 1,h′ −→ ,c (d), there exists a sequence {gn} in c ∞,c(d) converging to f in h1 1,h′ −→ ,c (d), that is, (‖f − g n ‖2 l2(d) + [f − g n ,f − g n ] d,h′ ) 1 2 → 0 as n → ∞. (4.5) if ‖gn‖ l2(d) ≤ c d,h′ [gn,gn] 1 2 d,h′ , ∀n ≥ 1, then the limit relation (4.5) implies that inequality (1.3) holds. thus the above claim holds true. assume that (d,p) defines an étale covering {bkl } of p(d ∗). let g ∈ c 1,c(d) and k := spt(g). the regular part d∗ being σ-compact, the set k∩d∗ admits an exhausting sequence of increasing compact subsets {kj} of k ∩ d ∗, each of which is contained in a finite union of covering sheets 13 see barlet [2, p. 110] for a closely related notion of “revêtement analytique étale”. 278 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) bℓ (namely some bkl as above). choose a c ∞-partition of unity {βj,ℓ} 1≤ℓ≤nj on kj subordinate to such a covering {bℓ}ℓ=1,··· ,nj (with monotonically increasing index n j). assume (without loss of generality) that ∑ 1≤ℓ≤nj β j,ℓ = 1 on dj := ∪{b ℓ| 1 ≤ ℓ ≤ nj}. let ĝ{j,ℓ} : bℓ := p(bℓ) → c be the direct image of g{j,ℓ} := βj,ℓg (under the map p). assume now that d is unramified. then d = d∗, hence the {kj} can be chosen to be a finite sequence with kj = k for large enough j. then one has ∫ d |g|2 dυ̃ = ∫ dj |g|2 dυ̃ = ∫ dj ∑ ℓ (βj,ℓ)2 |g|2 dυ̃ = ∫ dj ∑ ℓ (βj,ℓ)2 |g|2 dυ̃ + 2 ∫ dj   ∑ ℓ 6=ℓ′ βj,ℓβj,ℓ ′   |g|2 dυ̃ = ∑ ℓ ∫ bℓ |ĝ{j,ℓ}|2 dυ where the added integral (in the second equality) vanishes since the base domains of the covering sheets bℓ are pair-wise disjoint. note that g = ∑ 1≤l≤nj β j,{l}g on dj, by resorting to the poincaré inequality for the euclidean unit ball, one has 14 , for a given g ∈ c ∞,c(d), h d (p b )2 ∑ ℓ ∫ bℓ |ĝ{j,ℓ}|2 dυ ≤ h d ∑ ℓ ∫ bℓ |∇ĝ{j,ℓ}|2 dυ ≤ ∫ dj ∑ ℓ 2m∑ λ=1 hλ|∂λ(g {j,ℓ})|2 dυ̃ = ∫ dj 2m∑ λ=1 hλ ∑ ℓ, ℓ′ ∂λ(g {j,ℓ})∂λ(ḡ {j,ℓ′})dυ̃ = ∫ dj 2m∑ λ=1 hλ ( ∑ ℓ ∂λ(g {j,ℓ}) )( ∑ ℓ′ ∂λ(ḡ {j,ℓ′}) ) dυ̃ = ∫ dj 2m∑ λ=1 hλ∂λ ( ∑ ℓ g{j,ℓ} ) ∂λ ( ∑ ℓ′ ḡ{j,ℓ ′} ) dυ̃ = [g,g] dj ,(h1,··· ,h2m) = [g,g] d,(h1,··· ,h2m) , where h d = min {ess inf d (hj) | 1 ≤ j ≤ 2m} > 0, and pb denotes the poincaré’s constant for the unit ball. consequently the poincaré inequality (1.3) follows. proposition 4.3 (generalized poincaré-wirtinger inequality15). assume that (d,p) is a riemann subdomain such that, with respect to a continuous allowable weight h′ on d, the rellichkondrachov embedding property holds: h1 1,h′ −→ (d) →֒ l2(d) is compact. then there exists a constant c d such that ∥∥∥∥f − 1 vol(d) ∫ d f dυ̃ ∥∥∥∥ l2(d) ≤ c d [f,f] 1 2 d,h′ , ∀f ∈ h1 1,h′ −→ (d). (4.6) 14 on the right-side of the following, in the integral of the second inequality a sum of terms “ const ∫ dj ∑ λ hλ ∑ ℓ 6=ℓ′ ∂λ(g {j,ℓ})∂λ(ḡ {j,ℓ′})dυ̃” may be added, since it vanishes owing to the fact that the base domains w k are pair-wise disjoint. 15 a version of this inequality is presented in deny-lions [5, (5.5), p.329]. cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 279 the proof (by way of reductio ad absurdum) is omitted. definition 4.4. let d ⊆ y be a riemann subdomain with dd 6= ∅, f ∈ lip(∂d; c) 16 , and ϕ ∈ c 02m(d\w ), where w is a thin 17 analytic subset of d. a weak solution of the dirichlet problem for the poisson equation: ddcu ∧ υm−1p = ϕ in d\w , u|dd = f|dd (4.7) is an element u = w ∈ h1 1,(0,1,··· ,1) (d) such that w ≡ f mod (h1 1,(0,1,··· ,1),c (d)), and [w,v] d,(1,··· ,1) = (ϕ,v) d , ∀v ∈ h11,(0,1,··· ,1),c(d). corollary 4.5. let d ⊆ y be a riemann subdomain with dd 6= ∅. assume that (d,p) has the poincaré property relative to the unit 2m-weight (1, · · · ,1). then for any f ∈ lip(∂d; c) and ϕ ∈ c 02m(d\w ), w being thin analytic in d, the dirichlet problem (4.7) admits a weak solution w ∈ h1 1,(0,1,··· ,1) (d). proof. consider the linear mapping t : h1 1,(0,1,··· ,1),c (d) → c defined by t(v) := (ϕ,v̄) d − [f̃,v] d,(1,··· ,1) , ∀v ∈ h11,(0,1,··· ,1),c(d). where f̃ is a lipschitzian extension of f to a neighborhood of d. then the poincaré inequality (1.3) (with h′ = (1, · · · ,1)) implies that the riesz representation theorem is applicable to the operator t. therefore there exists an element w 0 ∈ h1 1,(0,1,··· ,1),c (d) such that t(v) = [w 0 ,v] d,(1,··· ,1) , ∀v ∈ h11,(0,1,··· ,1),c(d). then w := w 0 + f̃ ∈ h1 1,(0,1,··· ,1) (d) is a weak solution to the dirichlet problem (4.7). 5 the resolvent map for an inhomogeneous dirichlet problem let h′ be an allowable 2m-weight on d and h1 1,h′ −→ ,c (d) be equipped with the dirichlet norm (defined by (1.2)). for fixed f ∈ l2(d), applying the riesz’s representation theorem to the bounded anti-linear functional v 7→ (f, v̄) d on h1 1,h′ −→ ,c (d), yields an element w ∈ h1 1,h′ −→ ,c (d) such that (f, v̄) d = [w, v] d,h′ , ∀v ∈ h1 1,h′ −→ ,c (d). 16 f ∈ lip(∂d; c) means that f is (locally) lipschitzian in a neighborhood of ∂d. as such it admits a lipschitzian extension f̃ to a neighborhood of d by invoking a partition of unity. 17 a subset t of an m-dimensional complex space y is thin, if at each point a ∈ t there is an analytic subset a of dimension < m in an open neighborhood u ⊆ y of a such that t ∩ u ⊆ a. 280 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) the association “f 7→ r d,h′ f = w” defines a continuous linear mapping r d,h′ : l2(d) → l2(d) with image in h1 1,h′ −→ ,c (d) satisfying the equation (1.7). assume now that (d,p) has the poincaré property relative to a bounded, c ∞ allowable 2mweight h′ 18. then the poincaré inequality (1.3) implies that the sobolev spaces h1 1,h′ −→ ,c (d) and h1 0,h′ −→ ,c (d) are defined by equivalent norms, hence can be naturally identified with each other. the mapping g0,h′ −→ : l2(d) → h1 1,h′ −→ ,c (d) (the latter being identified with h1 0,h′ −→ ,c (d)) is continuous, linear and satisfies, for each fixed ψ ∈ l2(d), the equation (ψ,v̄) d = 〈g0,h′ −→ ψ,v〉 0,(0,h′) = [g0,h′ −→ ψ, v] d,h′ , ∀v ∈ h1 1,h′ −→ ,c (d). hence it follows that r d,h′ = g0,h′ −→ : l2(d) → h1 1,h′ −→ ,c (d). also, since the mapping g0,h′ −→ is injective, so is the mapping r d,h′ . thus r d,h′ can (justifiably) be called the resolvent map for the differential operator defined by the left-side of the equation (1.8). proof of theorem 1.3. it follows from the hermitian symmetry of the dirichlet product that the mapping r d,h′ : l2(d) → l2(d) is self-adjoint. in the rest of this proof write “(µ,h)” for the pair (1,{0,1, · · · ,1}). the mapping r d,h′ is expressible as a composition of the (restricted) mapping r̃ d,h′ : l2(d) → h1µ,h,c(d) and the compact rellich embedding i : h 1 µ,h,c(d) →֒ l 2(d). consequently r d,h′ = i ◦ r̃ d,h′ is compact. remark 5.1. the above (same) proof of the embedding theorem 1.3 yields the following: assume that the embedding h1µ,h,c(d) →֒ l 2(d) is compact for a (given) allowable weight h for (d,µ). then there is defined a compact mapping rµ d,h′ : l2(d) → l2(d) (in a way similar to that for r d,h′ = r1 d,h′ : l2(d) → l2(d)). 6 solution of an eigenvalue problem theorem 6.1. assume that (d,p) is a rellich subdomain with respect to h′ (the latter being a given allowable 2m-weight). then: (a) there exists a non-decreasing, unbounded sequence {λ j } of positive real numbers such that the operator equation [u, v] d,h′ = λ(u, v̄)d ∀v ∈ c ∞,c(d), u|dd = 0, (6.1) admits a nontrivial solution u ∈ h1 1,h′ −→ ,c (d) precisely when λ is a member of the countable set {λ j }; 18 that is, each component hj of h ′ is bounded on d; similarly define “c ∞-allowable weight on d”. cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 281 (b) there exists an orthonormal basis {ψ k } of h1 1,h′ −→ ,c (d), consisting of eigenfunctions of r d,h′ , which is complete in both l2(d) and h1 1,h′ −→ ,c (d); furthermore, if h is of class c ∞ on d∗, then each ψ k ∈ h1 1,h′ −→ ,c (d) ∩ c ∞(d∗). proof. by the remark 5.1, the mapping r d,h′ : l2(d) → l2(d) is compact, and self-adjoint. hence it has real eigenvalues {βj}j∈n of finite multiplicity (equal to the dimension of the eigenspace ej := ker(rd,h′ − βji)) may be arranged as a sequence |β1| ≥ |β2| ≥ · · · , with no point of accumulation except possibly the origin. the members of (distinct) ej and ej′ are mutually orthogonal. since the mapping r d,h′ takes values in h1 1,h′ −→ ,c (d), for each j = 1, 2 · · · and φj ∈ ej, the relation r d,h′ φj = βjφj holds and implies that φj belongs to h 1 1,h′ −→ ,c (d). moreover, βj [φj, φk]d,h′ = [rd,h′ φj, φk]d,h′ = (φj, φ̄k)d, ∀j, k = 1, 2, · · · . this implies that each βj > 0. by orthonormalizing a basis of each ej and taking their union, one obtains an orthonormal basis {ψ k } of h1 1,h′ −→ ,c (d) consisting of eigenfunctions of r d,h′ . one can arrange that each ψk has eigenvalue βk (by repeatedly listing the same βj as many times as its multiplicity, namely, dimej). thus ψk − 1 βk r d,h′ ψk = 0 ∀k = 1, 2 · · · , so that ψk is a solution to a dirichlet problem of the type (1.5) (with g = 0), which is equivalent to solving the functional equation (6.1) (with λk := 1 βk > 0 and λk ↑ ∞). to prove the completeness of the system {ψ k } in h1 1,h′ −→ ,c (d), recall the fact that ker(r d,h′ ) = {0}. the desired conclusion follows then from the completeness criterion of [16, p. 234]. by a standard regularity criterion, if h is of class c ∞ on d∗, then each eigenfunction ψ k belongs to c ∞(q) for all (open) domains q ⋐ d∗. consequently ψ k belongs to h1 1,h′ −→ ,c (d) ∩ c ∞(d∗). lemma 6.2. given g ∈ l2(d), the poisson problem (1.5) admits a solution ψ in h1 1,h′ −→ ,c (d) if and only if the following functional equation on c ∞,c(d), (i + α r d,h′ )ψ = w (6.2) with w := r d,h′ g ∈ h1 1,h′ −→ ,c (d), admits a solution ψ in h1 1,h′ −→ ,c (d). proof. the above equation (1.6) can be expressed alternatively as a functional equation on c ∞,c(d) in the form [ψ,v] d,h′ + α [r d,h′ ψ,v] d,h′ = [w,v] d,h′ , where w := r d,h′ g ∈ h1 1,h′ −→ ,c (d). from this the equation (6.2) follows. 282 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) theorem 6.3. assume that (d,p) is a rellich subdomain with respect to h′ = (1, · · · ,1). let g ∈ l2(d). consider the poisson problem − △pψ + αψ = g a.e. in d, ψ|dd = 0. (6.3) (a) if α 6∈ {−λj}j=1,··· ,∞ (the λj being the eigenvalues of −△p), then there exists a unique weak solution ψ ∈ h1 1,(0,1,··· ,1),c (d) of the problem (6.3) with ‖ψ‖ 1,(0,1,··· ,1) ≤ const.‖g‖l2(d). (b) if α = −λj (for some λj as above), then the problem (6.3) has a weak solution ψ ∈ h1 1,(0,1,··· ,1),c (d) if and only if (g, ψ̄) d = 0 for each ψ = ψkj , k = 1, · · · ,s, the latter being the associated eigenfunctions of the problem of theorem 6.1: (i − λjrd,(1,··· ,1))(ψ) = 0. each solution of the the inhomogeneous problem (6.3) is of the form ψ = ψ0 + s∑ k=1 ckψ k j , (6.4) where ψ0 is a fixed solution and the ck are suitable constants. proof. in this proof let the allowable weight h be (0,(1, · · · ,1)). since the rellich embedding j : h1 1,(0,1,··· ,1),c (d) →֒ l2(d) is compact, so is the composition rd,(0,1,··· ,1) ◦j : h 1 1,(0,1,··· ,1),c (d) → h1 1,(0,1,··· ,1),c (d). it is known that for a constant α 6= −λj, the operator i + αrd,(0,1,··· ,1) ◦ j is invertible with a bounded inverse. therefore the equation (6.2) has a unique solution ψ = (i + αr d,(0,1,··· ,1) )−1w1 with w1 := rd,(0,1,··· ,1)g ∈ h 1 1,h,c (d), and ‖ψ‖ 1,h ≤ ‖(i + αr d,(0,1,··· ,1) )−1‖ ‖w1‖1,h. since ‖w1‖1,h ≤ const.‖g‖l2(d), the assertion in (a) is proved. to prove the assertion in (b), observe that the closure of the range of an operator is the orthogonal complement of the null space of its adjoint. the equation ψ − λj rd,(0,1,··· ,1)ψ = v1, (6.5) where v 1 = r d,(0,1,··· ,1) g ∈ h1 1,h,c (d), has a solution if and only if v 1 ∈ r(i − λj rd,(0,1,··· ,1)), which is equivalent to v 1 ⊥ ker(i − λjr ∗ d,(0,1,··· ,1) ). the latter means that v is orthogonal (with respect to the inner product [ , ] d,(0,1,··· ,1) on h1 1,h,c (d)) to all the eigenfunction ψkj corresponding to the eigenvalue λj, namely, [v 1 ,ψkj ]d,(0,1,··· ,1) = (g, ψ̄ k j )d = 0. the expression (6.4) is a consequence of the equation (6.5). cubo 23, 2 (2021) poincaré inequality and rellich’s lemma 283 acknowledgement the authors gratefully acknowledge the referee’s suggestions, which have led to improvements of this paper. initial inquiries concerning this work were started during a sabbatical granted by dean b. martensen of minnesota state university, mankato, to whom and the hosting institution, university of padova, the first author expresses his best thanks. 284 chia-chi tung & pier domenico lamberti cubo 23, 2 (2021) references [1] a. andreotti and w. stoll, analytic and algebraic dependence of meromorphic functions, lecture notes in mathematics, vol. 234, berlin-new york: springer-verlag, 1971. 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[22] r. o. wells, differential analysis on complex manifolds, third edition, graduate texts in mathematics, vol. 65, new york: springer, 2008. 19for a corrected version of this paper, see: arxiv:1507.02675 [math.cv] doi: 10.1007/s00006-007-0036-9. introduction preliminaries friedrichs extension of a schrödinger type operator the rellich embedding theorem and poincaré inequality the resolvent map for an inhomogeneous dirichlet problem solution of an eigenvalue problem cubo, a mathematical journal vol. 23, no. 03, pp. 457–468, december 2021 doi: 10.4067/s0719-06462021000300457 some integral inequalities related to wirtinger’s result for p-norms s. s. dragomir1,2 1 mathematics, college of engineering & science, victoria university, po box 14428 melbourne city, mc 8001, australia. sever.dragomir@vu.edu.au 2 dst-nrf centre of excellence in the mathematical and statistical sciences, school of computer science & applied mathematics, university of the witwatersrand, private bag 3, johannesburg 2050, south africa. abstract in this paper we establish several natural consequences of some wirtinger type integral inequalities for p-norms. applications related to the trapezoid unweighted inequalities, of grüss’ type inequalities and reverses of jensen’s inequality are also provided. resumen en este artículo establecemos varias consecuencias naturales de algunas desigualdades integrales de tipo wirtinger para p-normas. también se entregan aplicaciones relacionadas a desigualdades trapezoidales sin peso, desigualdades de tipo grüss y reversos de la desigualdad de jensen. keywords and phrases: wirtinger’s inequality, trapezoid inequality, grüss’ inequality, jensen’s inequality. 2020 ams mathematics subject classification: 26d15; 26d10. accepted: 05 october, 2021 received: 02 may, 2021 ©2021 s. s. dragomir. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000300457 https://orcid.org/0000-0003-2902-6805 458 s. s. dragomir cubo 23, 3 (2021) 1 introduction the following wirtinger type inequalities are well known∫ b a u2(t)dt ≤ (b − a)2 π2 ∫ b a [u′ (t)] 2 dt (1.1) provided u ∈ c1 ([a, b] , r) and u (a) = u (b) = 0 with equality holding if and only if u (t) = k sin [ π(t−a) b−a ] for some constant k, and, similarly, if u ∈ c1 ([a, b] , r) satisfies u (a) = 0, then ∫ b a u2(t)dt ≤ 4 (b − a)2 π2 ∫ b a [u′ (t)] 2 dt. (1.2) the equality holds in (1.2) if and only if u (t) = k sin [ π(t−a) 2(b−a) ] for some constant k. for p > 1 put πp := 2πp sin ( π p ) . in [11], j. jaroš obtained, as a particular case of a more general inequality, the following generalization of (1.1)∫ b a |u (t)|p dt ≤ (b − a)p (p − 1) πpp ∫ b a |u′ (t)|p dt (1.3) provided u ∈ c1 ([a, b] , r) and u (a) = u (b) = 0, with equality if and only if u (t) = k sinp [ πp(t−a) b−a ] for some k ∈ r, where sinp is the 2πp-periodic generalized sine function, see [18] or [5]. if u (a) = 0 and u ∈ c1 ([a, b] , r) , then∫ b a |u (t)|p dt ≤ [2 (b − a)]p (p − 1) πpp ∫ b a |u′ (t)|p dt (1.4) with equality iff u (t) = k sinp [ πp(t−a) 2(b−a) ] for some k ∈ r. the inequalities (1.3) and (1.4) were obtained for a = 0, b = 1 and q = p > 1 in [17] by the use of an elementary proof. for some related wirtinger type integral inequalities see [1, 2, 4, 8, 9, 11, 12] and [15]-[17]. these inequalities are used in various fields of mathematical analysis, approximation theory, integral operator theory and analytic inequalities theory since they provide connections between the lebesgue norms of a function and the corresponding lebesgue norms of the derivative under some natural assumptions at the endpoints. motivated by the above results, in this paper we establish some natural consequences of the wirtinger type integral inequalities for p-norms (1.3) and (1.4). applications related to the trapezoid unweighted inequalities, of grüss’ type inequalities and reverses of jensen’s inequality are also provided. cubo 23, 3 (2021) some integral inequalities related to wirtinger’s result for p-norms 459 2 some applications for trapezoid inequality we have: proposition 2.1. let g ∈ c1([a, b], r). then for p > 1 we have the trapezoid inequality ∣∣∣∣∣g (a) + g (b)2 − 1b − a ∫ b a g (t) dt ∣∣∣∣∣ ≤ b − a 2 (p − 1)1/p πp ( 1 b − a ∫ b a |g′ (t) − g′ (a + b − t)|p dt )1/p . (2.1) in particular, for p = 2, we have [7] ∣∣∣∣∣g (a) + g (b)2 − 1b − a ∫ b a g (t) dt ∣∣∣∣∣ ≤ b − a2π ( 1 b − a ∫ b a |g′ (t) − g′ (a + b − t)|2 dt )1/2 . (2.2) proof. if g ∈ c1([a, b], r), then by taking u (t) := g (t) + g (a + b − t) 2 − g (a) + g (b) 2 , t ∈ [a, b] we have u (a) = u (b) = 0 and by (1.3) we have ∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣p dt ≤ (b − a)p(p − 1) 2pπpp ∫ b a |g′ (t) − g′ (a + b − t)|p dt, (2.3) namely (∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣p dt )1/p ≤ (b − a) 2 (p − 1)1/p πp (∫ b a |g′ (t) − g′ (a + b − t)|p dt )1/p . (2.4) by hölder’s integral inequality we have for p, q > 1, 1 p + 1 q = 1 ∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣dt ≤ (∫ b a dt )1/q (∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣p dt )1/p = (b − a)1/q (∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣p dt )1/p = (b − a)1−1/p (∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣p dt )1/p . (2.5) 460 s. s. dragomir cubo 23, 3 (2021) by making use of the properties of modulus and integral, we also have∫ b a ∣∣∣∣g (t) + g (a + b − t)2 − g (a) + g (b)2 ∣∣∣∣dt ≥ ∣∣∣∣∣ ∫ b a [ g (t) + g (a + b − t) 2 − g (a) + g (b) 2 ] dt ∣∣∣∣∣ = ∣∣∣∣∣ ∫ b a g (t) dt − g (a) + g (b) 2 (b − a) ∣∣∣∣∣ . (2.6) by making use of (2.4)-(2.6) we get the desired result (2.1). further, we have: proposition 2.2. let g ∈ c1([a, b], r). then for p > 1 we have the trapezoid inequality∣∣∣∣∣g (a) + g (b)2 − 1b − a ∫ b a g (t) dt ∣∣∣∣∣ ≤ b − a (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣g′ (t) − g (b) − g (a)b − a ∣∣∣∣p dt )1/p . (2.7) in particular, for p = 2, we have [7]∣∣∣∣∣g (a) + g (b)2 − 1b − a ∫ b a g (t) dt ∣∣∣∣∣ ≤ b − aπ ( 1 b − a ∫ b a ∣∣∣∣g′ (t) − g (b) − g (a)b − a ∣∣∣∣2 dt )1/2 . (2.8) proof. if g ∈ c1([a, b], r), then by taking u (t) := g (t) − g (a) (b − t) + g (b) (t − a) b − a , t ∈ [a, b] we have u (a) = u (b) = 0 and by (1.3) we have∫ b a ∣∣∣∣g (t) − g (a) (b − t) + g (b) (t − a)b − a ∣∣∣∣p dt ≤ (b − a)p(p − 1) πpp ∫ b a ∣∣∣∣g′ (t) − g (b) − g (a)b − a ∣∣∣∣p dt, (2.9) which gives that(∫ b a ∣∣∣∣g (t) − g (a) (b − t) + g (b) (t − a)b − a ∣∣∣∣p dt )1/p ≤ b − a (p − 1)1/p πp (∫ b a ∣∣∣∣g′ (t) − g (b) − g (a)b − a ∣∣∣∣p dt )1/p . (2.10) by hölder’s integral inequality we have for p, q > 1, 1 p + 1 q = 1 that ∫ b a ∣∣∣∣g (t) − g (a) (b − t) + g (b) (t − a)b − a ∣∣∣∣dt ≤ (∫ b a dt )1/q (∫ b a ∣∣∣∣g (t) − g (a) (b − t) + g (b) (t − a)b − a ∣∣∣∣p dt )1/p = (b − a)1/q (∫ b a ∣∣∣∣g (t) − g (a) (b − t) + g (b) (t − a)b − a ∣∣∣∣p dt )1/p . (2.11) cubo 23, 3 (2021) some integral inequalities related to wirtinger’s result for p-norms 461 by making use of the properties of modulus and integral, we also have ∫ b a ∣∣∣∣g (t) − g (a) (b − t) + g (b) (t − a)b − a ∣∣∣∣dt ≥ ∣∣∣∣∣ ∫ b a [ g (t) − g (a) (b − t) + g (b) (t − a) b − a ] dt ∣∣∣∣∣ = ∣∣∣∣∣ ∫ b a g (t) dt − g (a) + g (b) 2 (b − a) ∣∣∣∣∣ . (2.12) by making use of (2.10)-(2.12) we get the desired result (2.7). we also have: proposition 2.3. let g ∈ c([a, b], r). then for p > 1 we have the inequality ∣∣∣∣∣b + a2 ∫ b a g (s) ds − ∫ b a tg (t) dt ∣∣∣∣∣ ≤ (b − a)2 (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣∣g (t) − 1b − a ∫ b a g (s) ds ∣∣∣∣∣ p dt )1/p . (2.13) in particular, for p = 2, we have [7] ∣∣∣∣∣b + a2 ∫ b a g (s) ds − ∫ b a tg (t) dt ∣∣∣∣∣ ≤ (b − a)2 π   1 b − a ∫ b a g2 (t) dt − ( 1 b − a ∫ b a g (s) ds )21/2 . (2.14) proof. assume that g : [a, b] → c is continuous, then by taking u (t) := ∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds, t ∈ [a, b] we have u (a) = u (b) = 0, u ∈ c1([a, b], c) and by (1.3) we get ∫ b a ∣∣∣∣∣ ∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds ∣∣∣∣∣ p dt ≤ (b − a)p (p − 1) πpp ∫ b a ∣∣∣∣∣g (t) − 1b − a ∫ b a g (s) ds ∣∣∣∣∣ p dt. this is equivalent to (∫ b a ∣∣∣∣∣ ∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds ∣∣∣∣∣ p dt )1/p ≤ b − a (p − 1)1/p πp (∫ b a ∣∣∣∣∣g (t) − 1b − a ∫ b a g (s) ds ∣∣∣∣∣ p dt )1/p . (2.15) 462 s. s. dragomir cubo 23, 3 (2021) by hölder’s integral inequality we also have for p, q > 1, 1 p + 1 q = 1 that (b − a)1/q (∫ b a ∣∣∣∣∣ ∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds ∣∣∣∣∣ p dt )1/p ≥ ∫ b a ∣∣∣∣∣ (∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds )∣∣∣∣∣dt ≥ ∣∣∣∣∣ ∫ b a (∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds ) dt ∣∣∣∣∣ . (2.16) observe that, integrating by parts, we have∫ b a (∫ t a g (s) ds − t − a b − a ∫ b a g (s) ds ) dt = ∫ b a (∫ t a g (s) ds ) dt − b − a 2 ∫ b a g (s) ds = b ∫ b a g (s) ds − ∫ b a tg (t) dt − b − a 2 ∫ b a g (s) ds = b + a 2 ∫ b a g (s) ds − ∫ b a tg (t) dt. (2.17) by making use of (2.15)-(2.17) we get the desired result (2.13). 3 inequalities for the čebyšev functional for two lebesgue integrable functions f, g : [a, b] → r, consider the čebyšev functional: c (f, g) := 1 b − a ∫ b a f(t)g(t)dt − 1 (b − a)2 ∫ b a f(t)dt ∫ b a g(t)dt. (3.1) in 1935, grüss [10] showed that |c (f, g)| ≤ 1 4 (m − m) (n − n) , (3.2) provided that there exist real numbers m, m, n, n such that m ≤ f (t) ≤ m and n ≤ g (t) ≤ n for a. e. t ∈ [a, b] . (3.3) the constant 1 4 is the best possible in (3.1) in the sense that it cannot be replaced by a smaller quantity. another, however less known result, even though it was obtained by čebyšev in 1882, [3], states that |c (f, g)| ≤ 1 12 (b − a)2 ∥f′∥∞ ∥g ′∥∞ , (3.4) provided that f′, g′ exist and are continuous on [a, b] and ∥f′∥∞ = supt∈[a,b] |f ′ (t)| . the constant 1 12 cannot be improved in the general case. cubo 23, 3 (2021) some integral inequalities related to wirtinger’s result for p-norms 463 the čebyšev inequality (3.4) also holds if f, g : [a, b] → r are assumed to be absolutely continuous and f′, g′ ∈ l∞ [a, b] while ∥f′∥∞ = ess supt∈[a,b] |f ′ (t)| . a mixture between grüss’ result (3.2) and čebyšev’s one (3.4) is the following inequality obtained by ostrowski in 1970, [14]: |c (f, g)| ≤ 1 8 (b − a) (m − m) ∥g′∥∞ , (3.5) provided that f is lebesgue integrable and satisfies (3.3) while g is absolutely continuous and g′ ∈ l∞ [a, b] . the constant 18 is the best possible in (3.5). the case of euclidean norms of the derivative was considered by a. lupaş in [13] in which he proved that |c (f, g)| ≤ 1 π2 (b − a) ∥f′∥2 ∥g ′∥2 , (3.6) provided that f, g are absolutely continuous and f′, g′ ∈ l2 [a, b] . the constant 1π2 is the best possible. we have: theorem 3.1. if f : [a, b] → r is continuous, p, q > 1 with 1 p + 1 q = 1 and g : [a, b] → c is absolutely continuous with g′ ∈ lq [a, b] , then |c (f, g)| ≤ (b − a)1/p (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣∣f (t) − 1b − a ∫ b a f (s) ds ∣∣∣∣∣ p dt )1/p × (∫ b a |g′ (t)|q dt )1/q . (3.7) in particular, for p = q = 2, we get |c (f, g)| ≤ (b − a)1/2 π   1 b − a ∫ b a f2 (t) dt − ( 1 b − a ∫ b a f (s) ds )21/2 × (∫ b a |g′ (t)|2 dt )1/2 . (3.8) proof. integrating by parts, we have 1 b − a ∫ b a (∫ x a f (t) dt − x − a b − a ∫ b a f (s) ds ) g′ (x) dx = 1 b − a  (∫ x a f (t) dt − x − a b − a ∫ b a f (s) ds ) g (x) ∣∣∣∣∣ b a − ∫ b a g (x) ( f (x) − 1 b − a ∫ b a f (s) ds ) dx   = − 1 b − a ∫ b a f (x) g (x) dx + 1 b − a ∫ b a f (s) ds 1 b − a ∫ b a g (x) dx, 464 s. s. dragomir cubo 23, 3 (2021) which gives that c (f, g) = 1 b − a ∫ b a ( x − a b − a ∫ b a f (s) ds − ∫ x a f (t) dt ) g′ (x) dx. (3.9) using hölder’s integral inequality for p, q > 1 with 1 p + 1 q = 1 we have |c (f, g)| = ∣∣∣∣∣ 1b − a ∫ b a ( x − a b − a ∫ b a f (s) ds − ∫ x a f (t) dt ) g′ (x) dx ∣∣∣∣∣ ≤ 1 b − a ∫ b a ∣∣∣∣∣x − ab − a ∫ b a f (s) ds − ∫ x a f (t) dt ∣∣∣∣∣ |g′ (x)| dx ≤ 1 b − a (∫ b a ∣∣∣∣∣x − ab − a ∫ b a f (s) ds − ∫ x a f (t) dt ∣∣∣∣∣ p dx )1/p (∫ b a |g′ (x)|q dx )1/q =: i (3.10) using (2.15) we have i ≤ 1 b − a (∫ b a ∣∣∣∣∣ ∫ t a f (s) ds − t − a b − a ∫ b a f (s) ds ∣∣∣∣∣ p dt )1/p (∫ b a |g′ (x)|q dx )1/q ≤ 1 b − a b − a (p − 1)1/p πp (∫ b a ∣∣∣∣∣f (t) − 1b − a ∫ b a f (s) ds ∣∣∣∣∣ p dt )1/p (∫ b a |g′ (x)|q dx )1/q = (b − a)1/p (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣∣f (t) − 1b − a ∫ b a f (s) ds ∣∣∣∣∣ p dt )1/p (∫ b a |g′ (x)|q dx )1/q (3.11) for p, q > 1 with 1 p + 1 q = 1, which proves (3.7). now, if we put p = q = 2 and take into account that 1 b − a ∫ b a ∣∣∣∣∣f (t) − 1b − a ∫ b a f (s) ds ∣∣∣∣∣ 2 dt = 1 b − a ∫ b a f2 (t) dt − ( 1 b − a ∫ b a f (s) ds )2 , then by (3.7) we derive (3.8). this results can be used to obtain various inequalities by taking particular examples of functions f and g as follows. we have the following trapezoid type inequality: proposition 3.2. assume that g : [a, b] → c has an absolutely continuous derivative with g′′ ∈ lq [a, b] , where p, q > 1 and 1p + 1 q = 1. then∣∣∣∣∣g (a) + g (b)2 − 1b − a ∫ b a g (t) dt ∣∣∣∣∣ ≤ (b − a) 1+1/p 2 (p − 1)1/p (p + 1)1/p πp (∫ b a |g′′ (t)|q dt )1/q . (3.12) proof. we use the following identity that can be proved integrating by parts g (a) + g (b) 2 − 1 b − a ∫ b a g (t) dt = 1 b − a ∫ b a ( t − a + b 2 ) g′ (t) dt = c ( ℓ − a + b 2 , g′ ) , cubo 23, 3 (2021) some integral inequalities related to wirtinger’s result for p-norms 465 where ℓ (t) = t, t ∈ [a, b] . using (3.7) we have ∣∣∣∣c ( ℓ − a + b 2 , g′ )∣∣∣∣ ≤ (b − a)1/p (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣∣t − a + b2 − 1b − a ∫ b a ( s − a + b 2 ) ds ∣∣∣∣∣ p dt )1/p (∫ b a |g′′ (x)|q dx )1/q = (b − a)1/p (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣t − a + b2 ∣∣∣∣p dt )1/p (∫ b a |g′′ (x)|q dx )1/q = (b − a)1+1/p 2 (p − 1)1/p (p + 1)1/p πp (∫ b a |g′′ (x)|q dx )1/q , which proves the desired inequality (3.12). let φ : [m, m] ⊂ r → r be a differentiable convex function on (m, m) and f : [a, b] → [m, m] be absolutely continuous so that φ ◦ f, f, φ′ ◦ f, (φ′ ◦ f) f ∈ l [a, b] . if f′ ∈ l∞ [a, b] , then we have the jensen’s reverse inequality [6] 0 ≤ 1 b − a ∫ b a (φ ◦ f) (t) dt − φ ( 1 b − a ∫ b a f (t) dt ) ≤ 1 b − a ∫ b a (φ′ ◦ f) (t) f (t) dt − 1 b − a ∫ b a φ′ ◦ f (t) dt 1 b − a ∫ b a f (t) dt = c (φ′ ◦ f, f) . (3.13) we have the following reverse of jensen’s inequality: proposition 3.3. let φ : [m, m] ⊂ r → r be a differentiable convex function on (m, m) and f : [a, b] → [m, m] be absolutely continuous so that φ ◦ f, f, φ′ ◦ f, (φ′ ◦ f) f ∈ l [a, b] . (i) if f′ ∈ lq [a, b] , φ′ ◦ f ∈ lp [a, b] with p, q > 1 and 1p + 1 q = 1, then 0 ≤ 1 b − a ∫ b a (φ ◦ f) (t) dt − φ ( 1 b − a ∫ b a f (t) dt ) ≤ (b − a)1/p (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣∣(φ′ ◦ f) (t) − 1b − a ∫ b a (φ′ ◦ f) (s) ds ∣∣∣∣∣ p dt )1/p × (∫ b a |f′ (t)|q dt )1/q . (3.14) 466 s. s. dragomir cubo 23, 3 (2021) (ii) if φ is twice differentiable and (φ′′ ◦ f) f′ ∈ lq [a, b] with p, q > 1 and 1p + 1 q = 1, then 0 ≤ 1 b − a ∫ b a (φ ◦ f) (t) dt − φ ( 1 b − a ∫ b a f (t) dt ) ≤ (b − a)1/p (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣∣f (t) − 1b − a ∫ b a f (s) ds ∣∣∣∣∣ p dt )1/p × (∫ b a |(φ′′ ◦ f) (t) f′ (t)|q dt )1/q . (3.15) the proof follows by theorem 3.1 for c (φ′ ◦ f, f) and the inequality (3.13). we have the following mid-point type inequalities: corollary 3.4. let φ : [a, b] ⊂ r → r be a differentiable convex function on (a, b) . (i) if φ′ ∈ lp [a, b] with p > 1, then 0 ≤ 1 b − a ∫ b a φ (t) dt − φ ( a + b 2 ) ≤ b − a (p − 1)1/p πp ( 1 b − a ∫ b a ∣∣∣∣φ′ (t) − φ (b) − φ (a)b − a ∣∣∣∣p dt )1/p . (3.16) (ii) if φ is twice differentiable and φ′′ ∈ lq [a, b] with p, q > 1 and 1p + 1 q = 1, then 0 ≤ 1 b − a ∫ b a φ (t) dt−φ ( a + b 2 ) ≤ (b − a)1+1/p 2 (p − 1)1/p (p + 1)1/p πp (∫ b a |φ′′ (t)|q dt )1/q . (3.17) acknowledgement the author would like to thank the anonymous referees for valuable suggestions that have been implemented in the final version of the manuscript. cubo 23, 3 (2021) some integral inequalities related to wirtinger’s result for p-norms 467 references [1] m. w. alomari, “on beesack–wirtinger inequality”, results math., vol. 73, no. 2, pp. 1213– 1225, 2017. 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[18] s. takeuchi, “generalized elliptic functions and their application to a nonlinear eigenvalue problem with p-laplacian”, j. math. anal. appl., vol. 385 , no. 1, pp. 24–35, 2012. introduction some applications for trapezoid inequality inequalities for the čebyšev functional cubo, a mathematical journal vol. 23, no. 01, pp. 63–85, april 2021 doi: 10.4067/s0719-06462021000100063 convolutions in (µ,ν)-pseudo-almost periodic and (µ,ν)-pseudo-almost automorphic function spaces and applications to solve integral equations david békollè1 khalil ezzinbi2 samir fatajou3 duplex elvis houpa danga4 fritz mbounja béssémè5 1,4 department of mathematics, faculty of science, university of ngaoundéré p.o. box 454, ngaoundéré, cameroon. dbekolle@univ-ndere.cm; e houpa@yahoo.com 2,3 department of mathematics, faculty of science semlalia, cadi ayyad university, b.p. 2390 marrakesh, morocco. ezzinbi@uca.ac.ma; fatajou@yahoo.fr 5 department of mines and geology, school of geology and mining engineering, university of ngaoundéré p.o. box 454, ngaoundéré, cameroon. mbounjafritz@gmail.com abstract in this paper we give sufficient conditions on k ∈ l1(r) and the positive measures µ, ν such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudoalmost automorphic) function spaces are invariant by the convolution product ζf = k ∗ f. we provide an appropriate example to illustrate our convolution results. as a consequence, we study under acquistapace-terreni conditions and exponential dichotomy, the existence and uniqueness of (µ,ν)pseudo-almost periodic (respectively, (µ,ν)pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in banach spaces like neutral systems. resumen en este art́ıculo damos condiciones suficientes sobre k ∈ l1(r) y las medidas positivas µ, ν tales que los espacios de funciones pseudo-casi periódicas que duplican la medida (respectivamente, pseudo-casi automorfas que duplican la medida) son invariantes por el producto de convolución ζf = k ∗ f. entregamos un ejemplo apropiado para ilustrar nuestros resultados de convolución. como consecuencia, estudiamos bajo condiciones de acquistapace-terreni y dicotomı́a exponencial, la existencia y unicidad de soluciones (µ,ν)pseudo-casi periódicas (respectivamente, (µ,ν)pseudo-casi automorfas) de algunas ecuaciones de evolución parciales no autónomas en espacios de banach como sistemas neutrales. keywords and phrases: measure theory, (µ,ν)-ergodic, (µ,ν)-pseudo almost periodic and automorphic functions, evolution families, nonautonomous equations, neutral systems. 2020 ams mathematics subject classification: 34c27, 34k14, 35b15, 35k57, 37a30, 43a60. accepted: 09 january, 2021 received: 22 august, 2020 ©2021 d. békollè et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100063 https://orcid.org/0000-0001-5959-098x https://orcid.org/0000-0003-4779-5800 https://orcid.org/0000-0002-6052-6755 64 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) 1 introduction the existence and uniqueness of pseudo almost periodic and pseudo almost automorphic solutions is one of the most powerful tools in the qualitative theory of differential equations due to applications in mathematical biology, control theory and physical sciences. recently, diagana, ezzinbi and miraoui [11] applied the abstract measure theory to define the notion of double-weight pseudo almost periodicity (respectively double-weight pseudo almost automorphy) functions, and thus the classical theory of µ-pseudo almost periodic (respectively µ-pseudo almost automorphic) introduced by [4, 5], and double-weight pseudo almost periodicity [8] become particular cases of this approach. see the section 2.1 for technical details about this concept of double-weight pseudo almost periodicity (respectively double-weight pseudo almost automorphy) functions. we note that for f ∈ pap(r ×x,x,µ,ν) or f ∈ paa(r ×x,x,µ,ν), k ∈ l1(r), k ∗f = k ∗g + k ∗φ. we have that k ∗ g is almost periodic or almost automorphic function, but k ∗φ is not necessarily in e(r,x,µ,ν). then, the convolution invariance of the spaces pap(r × x,x,µ,ν) (resp. paa(r×x,x,µ,ν)) is equivalent to the convolution invariance of e(r,x,µ,ν). during the last decade, many research results about pseudo almost periodic and pseudo almost atomorphic was produce see [4, 5, 7, 9, 10]. inspired by the work of ezzinbi et al. [11] who studied the translation invariance of paa(r×x,x,µ,ν) (resp. pap(r×x,x,µ,ν)) functions and the recent work of mbounja et al. [15] who gave some several hypotheses for convolution invariance of pap(r×x,x,µ) and paa(r×x,x,µ), in this work we established new sufficient conditions on µ,ν ∈m and k ∈ l1(r) ensuring that, the space pap(r,x,µ,ν) of (µ,ν)-pseudo almost periodic functions and the space paa(r,x,µ,ν) of (µ,ν)-pseudo almost automorphic functions are invariant by the convolution product ζf = k∗f. our obtained conditions are more general than [15] and helped to show that the integral solution of some differential equations is a (µ,ν)-pseudo almost periodic (respectively (µ,ν)-pseudo almost automorphic) solutions. to illustrate our investigation, we show the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively (µ,ν)-pseudo almost automorphic) solutions of the following nonautomous differential equations, d dt u(t) = a(t)u(t) + f(t,u(t)), t ∈ r, (1.1) and d dt (u(t) −g(t,u(t)) = a(t) (u(t) −g(t,u(t)) + f(t,u(t)), t ∈ r, (1.2) where a(t) : d(a(t)) ⊂ x 7−→ x for t ∈ r is a family of closed linear operators on a banach space x, satisfying the well-known acquitaspace-terreni conditions developed in [1, 2], and f,g : r × x 7−→ x are jointly continuous functions satisfying some additional conditions. the study of equation (1.1) in an non-autonomous case is new even in the case of one measure, µ = ν. also, equation (1.2) is treated here. the rest of this work is organized as follows. in section 2, we recall some basic results which will be used throughout this work. in section 3, we state and prove main results about the convolution cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 65 invariance. in section 4 we study the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively (µ,ν)-pseudo almost automorphic) solutions to both equation (1.1) and equation (1.2) which illustrate our new results. 2 preliminaries 2.1 notation and terminology let (x,‖ ·‖) a banach space and let bc(r,x) be the space of bounded continuous functions f : r −→ x. the space bc(r,x), equipped with the supremum norm ‖f‖∞ = sup t∈r ‖f(t)‖, is a banach space. we denote by b the lebesgue σ-field of r and by m the space of all positive measures ϑ on b satisfying ϑ(r) = +∞ and ϑ([a,b]) < ∞, for all a,b ∈ r (a ≤ b). definition 2.1 ([6]). a continuous function f : r → x is said to be almost periodic if for every ε > 0 there exists a positive number lε such that every interval of length lε contains a number τ such that: ‖f(t + τ) −f(t)‖ < ε, ∀t ∈ r. let ap (r,x) denote the collection of almost periodic functions from r to x. we recall that (ap (r,x) ,‖·‖∞) is a banach space. definition 2.2 ([11]). let µ,ν ∈ m. a bounded continuous function f : r → x is said to be (µ,ν)-ergodic if lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖f(t)‖dµ(t) = 0. we denote the space of all such functions by e(r,x,µ,ν). the space (e(r,x,µ,ν),‖.‖∞) is a banach space for the supremum norm. definition 2.3 ([11]). let µ,ν ∈m. a continuous function f : r → x is said to be (µ,ν)-pseudo almost periodic if f admits the following decomposition: f = g + φ, where g ∈ ap(r,x) and φ ∈e(r,x,µ,ν). we denote the space of all such functions by pap(r,x,µ,ν). 66 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) we have ap(r,x) ⊂ pap(r,x,µ,ν) ⊂ bc(r,x). let (y,‖·‖) a banach space and let bc(r × y,x) be the space of jointly bounded continuous functions f : r × y −→ x. the space bc(r × y,x) equipped with the supremum norm ‖f‖∞ = sup t∈r,x∈y ‖f(t,x)‖ is a banach space. definition 2.4 ([12]). a jointly continuous function f : r×y → x is said to be almost periodic in t uniformly with respect to x ∈ y , if for every ε > 0, and any compact subset k of y , there exists a positive number lk(ε) such that every interval of length lk(ε) contains a number τ such that: ‖f(t + τ,x) −f(t,x)‖ < ε, ∀(t,x) ∈ r×k. we denote the space of such functions by apu(r×y,x). definition 2.5 ([11]). let µ,ν ∈ m. a continuous function f : r × y → x is said to be (µ,ν)-ergodic in t uniformly with respect to x ∈ y , if the following two conditions are true: (i) f is uniformly continuous on each compact set k in y with respect to the second variable x. (ii) ∀x ∈ y , f(.,x) ∈e(r,x,µ,ν). the space of such functions is denoted by eu(r×y,x,µ,ν). definition 2.6 ([11]). let µ,ν ∈ m. a continuous function f : r × y → x is said to be (µ,ν)-pseudo almost periodic in t uniformly for x ∈ y , if f admits the following decomposition: f = g + φ, (2.1) where g ∈ apu(r×y,x) and φ ∈eu(r×y,x,µ,ν). the collection of such functions is denoted by papu(r×y,x,µ,ν). we have apu(r×y,x) ⊂ papu(r×y,x,µ,ν) ⊂ bc(r×y,x,µ,ν). definition 2.7 ([16]). a continuous function f : r → x is said to be almost automorphic if for every sequence of real numbers (s′n)n∈n, there exists a subsequence (sn)n∈n ⊂ (s′n)n∈n such that: lim n,m→∞ f(t + sn −sm) = f(t), for each t ∈ r. equivalently, g(t) = lim n→∞ f(t + sn) exists ∀t ∈ r and f(t) = lim n→∞ g(t−sn) ∀t ∈ r. we denote the space of such functions by aa(r,x). cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 67 we recall that (aa (r,x) ,‖·‖∞) is a banach space. definition 2.8 ([11]). let µ,ν ∈m. a continuous function f : r → x is said to be (µ,ν)-pseudo almost automorphic if f admits the following decomposition: f = g + φ, where g ∈ aa(r,x) and φ ∈e(r,x,µ,ν). we denote the space of all such functions by paa(r,x,µ,ν). we have aa(r,x) ⊂ paa(r,x,µ,ν) ⊂ bc(r,x). definition 2.9 ([16]). a continuous function f : r×y → x is said to be almost automorphic in t uniformly for x ∈ y , if the following conditions hold: (i) f is uniformly continuous on each compact set k in y with respect to the second variable x, namely, for each compact set k in y , for all ε > 0, there exists δ > 0 such that for all x1,x2 ∈ k, one has: ‖x1 −x2‖≤ δ ⇒ sup t∈r ‖f(t,x1) −f(t,x2)‖≤ ε. (ii) for all x ∈ y , f(.,x) ∈ aa(r,x). denote by aau(r×y,x) the set of all such functions. definition 2.10 ([11]). let µ,ν ∈ m. a continuous function f : r × y → x is said to be (µ,ν)-pseudo almost periodic in t uniformly for x ∈ y , if f admits the following decomposition: f = g + φ, (2.2) where g ∈ aau(r×y,x) and φ ∈eu(r×y,x,µ,ν). the collection of such functions is denoted by paau(r×y,x,µ,ν) we have aau(r×y,x) ⊂ paau(r×y,x,µ,ν) ⊂ bc(r×y,x,µ,ν) 2.2 some useful results on the space functions for µ ∈m and τ ∈ r, we denote by µτ the positive measure on (r,b) defined by: µτ (a) = µ ({a + τ : a ∈ a}) , ∀a ∈b. 68 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) now we introduce the following hypotheses on µ,ν ∈m. (h0): for all τ ∈ r, there exists δ > 0 and a bounded interval i such that µτ (a) ≤ δµ(a), ντ (a) ≤ δν(a), ∀a ∈b satisfied a∩ i = ∅. (h1): lim sup r→∞ µ([−r,r]) ν([−r,r]) < ∞. remark 2.11. i) without assumptions on µ and ν, like (h0), the decomposition (2.1) (resp. (2.2)) of the (µ,ν)pseudo almost periodic and automorphic functions is not unique, (see [11]). ii) the spaces e(r,x,µ,ν), e(r×y,x,µ,ν), pap(r,x,µ,ν), pap(r×y,x,µ,ν), paa(r,x,µ,ν), and paa(r ×y,x,µ,ν) coincides when µ = ν, with the spaces e(r,x,µ), e(r ×y,x,µ), pap(r,x,µ), pap(r×y,x,µ), paa(r,x,µ), and paa(r×y,x,µ). we recall the following six theorems proved in [11]. theorem 2.12 ([11]). consider that µ,ν ∈m and k ∈ l1(r) and f ∈ pap(r,x,µ,ν) (respectively f ∈ paa(r,x,µ,ν). if (h0) is valid then pap(r,x,µ,ν) (respectively paa(r,x,µ,ν)) is translation invariant. moreover, {g(t) : t ∈ r}⊂{f(t) : t ∈ r}, (the closure of the range of f). theorem 2.13 ([11]). if (h0) is valid, then the decomposition (2.1)(resp. (2.2)) of pap(r,x,µ,ν) and paa(r,x,µ) is unique. theorem 2.14 ([11]). if (h1) holds, then (e(r,x,µ,ν),‖ ·‖∞) is a banach space with respect to the sup norm. theorem 2.15 ([11]). let µ,ν ∈ m satisfy (h1). if (h0) holds, then pap(r,x,µ,ν) and paa(r,x,µ,ν) are banach spaces with respect to the sup norm. theorem 2.16 ([11]). let µ,ν ∈ m, f ∈ papu(r × y,x,µ,ν) and h ∈ pap(r,x,µ,ν). assume that (h1) and the following hypothesis holds: for all bounded subsets b of x, f is bounded on r×b. then t 7−→ f(t,h(t)) ∈ pap(r,x,µ,ν). theorem 2.17 ([11]). let µ,ν ∈ m, f ∈ paau(r × y,x,µ,ν) and h ∈ paa(r,x,µ,ν). assume that for all bounded subsets b of x, f is bounded on r × b. then t 7−→ f(t,h(t)) ∈ paa(r,x,µ,ν). cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 69 2.3 measure theory results let µ,ν ∈ m; if f : r −→ x is a bounded continuous function, we define the following doublyweight mean, if the limit exists, by: m(f,µ,ν) := lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖f(t)‖xdµ(t). definition 2.18 ([17]). let (e,b) be a borel space. if µ and ν are two measures defined on (e,b), we say that: (i) µ and ν are mutually singular, if there are disjoint sets a and b in b such that e = a∪b and ν(a) = µ(b) = 0. (ii) ν is absolutely continuous with respect to µ, if for each a ∈b, (µ(a) = 0) =⇒ (ν(a) = 0). we recall the following theorems of measure theory. theorem 2.19 (radon-nikodym [17]). let (e,b,µ) be a σ-finite measure space, and let ν be a measure defined on b which is absolutely continuous with respect to µ. then there is a unique nonnegative measurable function f such that for each set b in b we have: ν(b) = ∫ b fdµ. the function f is called the radon-nikodym derivative of ν with respect of µ. example 2.20. let ρ be a nonnegative b-measurable function. denote by µ the positive measure defined by: µ(a) = ∫ a ρ(t)dt, for a ∈b where dt is the lebesgue measure on r. the function ρ is the radon-nikodym derivative of µ with respect to the lebesgue measure dt on r, i.e. dµ(t) = ρ(t)dt. in this case, µ ∈m if and only if its radon-nikodym derivative ρ is locally lebesgue integrable on r and it satisfies∫ +∞ −∞ ρ(t)dt = +∞. theorem 2.21 (lebesgue-radon-nikodym [17]). let (x,b,ϑ) be a σ-finite measure space, and µ a σ-finite measure defined on b. then, we can find a measure µ0, singular with respect to ϑ, and a measure µ1, absolutely continuous with respect to ϑ, such that µ = µ0 + µ1. the measures µ0 and µ1 are unique. 70 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) in this section, by using the previous theorem, we consider that for a given µ ∈m, µ = µ0 +µ1 where µ0 is the µ-measure component which is absolutely continuous with respect to the lebesgue measure and its radon-nikodym derivative is ρ, that is dµ0(t) = ρ(t)dt and µ1 is the µ-measure component such that µ1 is singular to lebesgue measure. we give new general hypotheses on µ,ν ∈m and k ∈ l1(r) such that: (ζf)(t) = ∫ +∞ −∞ k(t−s)f(s)ds, ∀k ∈ l1(r) (2.3) maps e(r,x,µ,ν) into itself. in particular, our hypotheses on µ,ν ∈m and k ∈ l1(r) will imply that for every f ∈e(r,x,µ,ν), the (µ,ν)-mean, m(ζf,µ,ν) := lim r→+∞ 1 ν([−r,r]) ∫ r −r ∣∣∣∣ ∣∣∣∣ ∫ +∞ −∞ k(t−s)f(s)ds ∣∣∣∣ ∣∣∣∣ x dµ(t) exists. 3 main results of convolution and translation invariance 3.1 convolution invariance on e(r,x,µ,ν) theorem 3.1. let k ∈ l1(r) and ν ∈m. consider that µ ∈m, with radon-nikodym derivative ρ with respect to dt and ζ is defined in (2.3). assume that ρ,µ,ν and k satisfy the following requirements:   sup |s|≤r,r∈r+ 1 ρ(s) ∫ r s |k(t−s)|dµ(t) < ∞, (3.1.1) sup |s|≤r,r∈r+ 1 ρ(s) ∫ s −r|k(t−s)|dµ(t) < ∞, (3.1.2) (3.1)   lim r→+∞ 1 ν([−r,r]) ∫−r −∞ (∫ r −r|k(t−s)|dµ(t) ) ds = 0, (3.2.1) lim r→+∞ 1 ν([−r,r]) ∫ +∞ r (∫ r −r|k(t−s)|dµ(t) ) ds = 0. (3.2.2) (3.2) if f ∈e(r,x,µ,ν), then ζf ∈e(r,x,µ,ν). proof. we adapt the proof in [15], theorem 3.5. by the properties of convolution we have that f ∈ bc(r,x) implies that k ∗ f ∈ bc(r,x), ∀k ∈ l1(r). then, in order to get that k ∗f ∈e(r,x,µ,ν) we must prove that m(ζf,µ,ν) = 0. we consider µ ∈ m and ρ its radon-nikodym derivative, ν ∈ m. in the first stage, we assume that k(t) = 0 on r∗−. from ν(r) = +∞, we deduce the existence of r0 ≥ 0 such that ν([−r,r]) > 0, ∀r ≥ r0. then by applying the fubini’s theorem, we deduce that for f ∈ bc(r,x), ∀r ≥ r0. cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 71 we notice that m(ζf,µ,ν) = lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖k ∗f‖xdµ(t) ≤ lim r→+∞ 1 ν([−r,r]) ∫ r −r ∫ t −∞ ‖ f(s) ‖x| k(t−s) | dsdµ(t) = lim r→+∞ 1 ν([−r,r]) ∫ r −r (∫ −r −∞ ‖ f(s) ‖x| k(t−s) | ds ) dµ(t) + lim r→+∞ 1 ν([−r,r]) ∫ r −r (∫ t −r ‖ f(s) ‖x| k(t−s) | ds ) dµ(t) ≤ ‖ f ‖∞ lim r→+∞ 1 ν([−r,r]) ∫ r −r (∫ −r −∞ | k(t−s) | ds ) dµ(t) + lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖ f(s) ‖x (∫ r s | k(t−s) | dµ(t) ) ds ≤ ‖ f ‖∞ lim r→+∞ 1 ν([−r,r]) ∫ −r −∞ (∫ r −r | k(t−s) | dµ(t) ) ds + lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖ f(s) ‖x [ 1 ρ(s) ∫ r s | k(t−s) | dµ(t) ] ρ(s)ds ≤ sup |s|≤r,r∈r+ 1 ρ(s) ∫ r s |k(t−s)|dµ(t) lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖f(s)‖xρ(s)ds + ‖ f ‖∞ lim r→+∞ 1 ν([−r,r]) ∫ −r −∞ (∫ r −r | k(t−s) | dµ(t) ) ds ≤ sup |s|≤r,r∈r+ 1 ρ(s) ∫ r s |k(t−s)|dµ(t) lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖f(s)‖xdµ(s) + ‖ f ‖∞ lim r→+∞ 1 ν([−r,r]) ∫ −r −∞ (∫ r −r | k(t−s) | dµ(t) ) ds. using assumptions (3.1.1), (3.2.1) and the fact that f ∈e(r,x,µ,ν), we have proved that m(ζf,µ,ν) = 0 + 0 = 0. this settles the first stage for every k ∈ l1(r) such that k(t) = 0 on r∗−. now, in the second stage, proceeding similarly like in the first stage, we assume that k(t) = 0 on r∗+ we obtain: m(ζf,µ,ν) = lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖k ∗f‖xdµ(t) ≤ ‖ f ‖∞ lim r→+∞ 1 ν([−r,r]) ∫ +∞ r (∫ r −r | k(t−s) | dµ(t) ) ds + sup |s|≤r,r∈r+ 1 ρ(s) ∫ s −r |k(t−s)|dµ(t) lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖f(s)‖xdµ(s). then, using the fact that f ∈ e(r,x,µ,ν) and hypotheses (3.1.2), (3.2.2), we have that m(ζf,µ,ν) = 0. in the general case of k, we deduce the result using the fact that k(t) = kχt≥0(t) + kχt<0(t). theorem 3.2. assume that µ,ν ∈m and (h1) holds. then the condition (3.2.1) (resp. (3.2.2)) 72 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) is valid for every k ∈ l1(r) if and only if the following condition (3.3.1) (resp. (3.3.2)) is true:  lim r→+∞ µ([−r,σ −r]) ν([−r,r]) = 0, ∀σ > 0 (3.3.1) lim r→+∞ µ([σ + r,r]) ν([−r,r]) = 0, ∀σ < 0 (3.3.2). (3.3) proof. we first prove that (3.3.1) =⇒ (3.2.1), for every k ∈ l1(r). in the first stage, we assume that k(t) = 0 on r∗−. let σ = t−s > 0 fixed. from ν(r) = +∞, we deduce the existence of r0 ≥ 0 such that ν([−r,r]) > 0, ∀r ≥ r0. in the sequel, for all r ≥ r0, we shall assume that b := 1 ν([−r,r]) ∫ r −r (∫ −r −∞ | k(t−s) | ds ) dµ(t). then, by applying the fubini’s theorem we deduce that: b = 1 ν([−r,r]) ∫ r −r (∫ +∞ t+r | k(σ) | dσ ) dµ(t) = ∫ +∞ 0 (∫ min(σ−r,r) −r dµ(t) ν([−r,r]) ) | k(σ) | dσ = ∫ +∞ 0 ( µ([−r, min(σ −r,r)]) ν([−r,r]) ) | k(σ) | dσ. by using assumption (3.3.1), we have: lim r→+∞ µ([−r, min(σ −r,r)]) ν([−r,r]) = 0, ∀σ > 0. since µ([−r, min(σ−r,r)]) ≤ µ([−r,r]) and the fact that (h1) holds, there exists β > 0 such that: 0 ≤ ( µ([−r, min(σ −r,r)]) ν([−r,r]) ) | k(σ) |≤ β | k(σ) |, where k ∈ l1(r), ∀σ > 0. then, by the lebesgue dominated convergence theorem, we obtain: lim r→+∞ ∫ +∞ 0 ( µ([−r, min(σ −r,r)]) ν([−r,r]) ) | k(σ) | dσ = 0 this concludes this stage of (3.2.1). now, in the second stage, proceeding similarly like the first stage, we assume that k(t) = 0 on r∗+. let σ = t− s < 0 fixed. from ν(r) = +∞, we deduce the existence of r0 ≥ 0 such that ν([−r,r]) > 0, ν([−r,r]) > 0, ∀r ≥ r0. we set: a := 1 ν([−r,r]) ∫ r −r (∫ +∞ r | k(t−s) | ds ) dµ(t) cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 73 then, by applying the fubini’s theorem, we deduce that: a = 1 ν([−r,r]) ∫ r −r (∫ t−r −∞ | k(σ) | dσ ) dµ(t) = ∫ 0 −∞ (∫ r max(σ+r,r) dµ(t) ν([−r,r]) ) | k(σ) | dσ = ∫ 0 −∞ ( µ([max(σ + r,r),r]) ν([−r,r]) ) | k(σ) | dσ. like in the first part, we use assumptions (3.3.2), and the lebesgue dominated convergence theorem. this concludes this second stage of (3.2.2). let us prove (3.2.1) =⇒ (3.3.1). let σ = t−s, by (3.2.1) and fubini’s theorem we have that: 0 = lim r→+∞ 1 ν([−r,r]) ∫ r −r (∫ −r −∞ | k(t−s) | ds ) dµ(t) = lim r→+∞ ∫ +∞ 0 ( µ([−r, min(σ −r,r)]) ν([−r,r]) ) | k(σ) | dσ. let τ > 0 such that σ ∈ [τ,τ + 1] and r > τ 2 . we have also [−r,τ − r] ⊆ [−r,σ − r] and [−r,τ −r] ⊆ [−r,r], that implies µ([−r,τ −r]) ≤ min{µ([−r,σ −r]),µ([−r,r])}, i.e. µ([−r,τ −r]) ν([−r,r]) ≤ µ([−r, min(σ −r,r)]) ν([−r,r]) . let k(σ) = χ[τ,τ+1](σ). we have that: 0 ≤ µ([−r,τ −r]) ν([−r,r]) ∫ τ+1 τ dσ ≤ ∫ τ+1 τ µ([−r, min(σ −r,r)]) ν([−r,r]) dσ = lim r→+∞ 1 ν([−r,r]) ∫ r −r (∫ −r −∞ | k(t−s) | ds ) dµ(t). then by (3.2.1): lim sup r→+∞ µ([−r,τ −r]) ν([−r,r]) ∫ τ+1 τ dσ ≤ lim r→+∞ ∫ τ+1 τ µ([−r, min(σ −r,r)]) ν([−r,r]) dσ = 0 then: lim r→+∞ µ([−r,τ −r]) ν([−r,r]) = 0. so (3.3.1) is verified. in the second stage (3.3.2), we do the same proof as above. 74 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) remark 3.3. hypothesis (h1) was used only in the proof of the implication (3.3.1) =⇒ (3.2.1). corollary 3.4. let µ,ν ∈m be such that the nonnegative b-measurable function ρ be the radonnikodym derivative of µ. assume that for all k ∈ l1(r) the requirements (3.1) and (3.2) are satisfied. then e(r,x,µ,ν) is convolution invariant. corollary 3.5. consider that µ,ν ∈ m, such that the nonnegative b-measurable function ρ be the radon-nikodym derivative of µ. assume that (h1) holds and the requirements (3.1) and (3.3) are satisfied. then e(r,x,µ,ν) is convolution invariant. example 3.6. we check that theorem 3.1 and corollary 3.5 hold. let k(t) =   1 10 e−2t, for t ∈ [0, +∞[ 0, for t ∈] −∞, 0[. we take dµk,η(t) = e σtdt + η ∞∑ n=−∞ eσnδn, where 0 ≤ σ < 2, η > 0 and δn denotes the dirac measure at the integer n ( ∑∞ n=−∞e σnδn is a ’generalized dirac comb’, it is called a dirac comb when σ = 0). then µσ,η ∈m and its radon-nikodym derivative is ρσ,η(t) = eσt. let νσ,η = γµσ,η, where γ > 0. then νσ,η ∈m. first, if for |s| ≤ r, r > 0, we write: jσ,η(r,s) := 1 ρσ,η(s) ∫ r s |k(t−s)|dµσ(t), we must prove that: sup |s|≤r, r>0 jσ,η(r,s) < ∞. in fact, jσ,η(r,s) = 1 10eσs  ∫ r s e−2(t−s)eσtdt + η ∑ s≤n≤r e−2(n−s)eσn   = 1 10  ∫ r s e−(2−σ)(t−s)dt + η ∑ s≤n≤r e−(2−σ)(n−s)   ≤ 1 10  ∫ r−s 0 e−(2−σ)udu + η ∑ [s]≤n≤[r] e−(2−σ)(n−[s]−1)   , where we applied the change of integral u = t− s in the integral and we denoted [x] the integral part of the real number x. we next apply the change of index m = n− [s] in the latter sum; this implies: cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 75 jσ,η(r,s) ≤ 1 10  ∫ r−s 0 e−(2−σ)(u)du + ηe2−σ [r]−[s]∑ m=0 e−(2−σ)m   . then: sup |s|≤r, r>0 jσ,η(r,s) ≤ 1 10 (∫ ∞ 0 e−(2−σ)(u)du + ηe2−σ ∞∑ m=0 e−(2−σ)m ) = 1 10(2 −σ) + ηe2−σ 1 10(1 −e−(2−σ)) < ∞. this proves the estimate (3.1.1). secondly, we shall show that for all α > 0, we have: lim r→∞ µσ,η([−r,α−r]) νσ,η([−r,r]) = 0. it actually suffices to prove this estimate when α is a positive integer. in fact, µσ,η([−r,α−r]) νσ,η([−r,r]) = ∫ α−r −r eσtdt + η ∑ −r≤n≤α−r eσn γ  ∫ r −r eσtdt + η ∑ −r≤n≤r eσn   ≤ 1 σ ( eσ(α−r) −e−σr ) + η ∑ −[r]−1≤n≤α−[r] eσn γ   1 σ (eσr −e−σr) + η ∑ −[r]≤n≤[r] eσn   = 1 σ e−σr (eσα − 1) + ηe−σ([r]+1) α+1∑ m=0 eσm γ   1 σ (eσr −e−σr) + ηe−σ[r] 2[r]∑ m=0 eσm   , where we applied the change of index m = n + [r] + 1 on the numerator and the change of index m = n + [r] on the denominator. so µσ,η([−r,α−r]) µσ,η([−r,r]) ≤ 1 σ e−σr (eσα − 1) + η eσ(α+2) − 1 eσ − 1 e−σ([r]+1) γ σ eσr (1 −e−2σr) + γηe−σ[r] eσ(2[r]+1) − 1 eσ − 1 . the estimation (3.3.1) easily follows. thirdly, we show that (h1) holds. lim sup r→∞ µσ,η([−r,r]) νσ,η([−r,r]) = 1 γ < ∞. then, theorem 3.1 and corollary 3.5 hold. 76 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) 3.2 translation invariance and convolution invariance of pap(r,x,µ,ν) and paa(r,x,µ,ν) theorem 3.7. assume that µ,ν ∈m and (h1) holds. if the space e(r,x,µ,ν) is translation invariant, then e(r,x,µ,ν) is convolution invariant. proof. let f ∈ e(r,x,µ,ν). let us prove that if f(t − τ) ∈ e(r,x,µ,ν), for τ ∈ r, then ζf ∈ e(r,x,µ,ν), i.e. m(ζf,µ,ν) = 0. by the properties of convolution we have that f ∈ bc(r,x) implies that k ∗f ∈ bc(r,x), ∀k ∈ l1(r). by the fubini’s theorem we have, m(ζf,µ,ν) = lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖k ∗f‖xdµ(t) ≤ lim r→+∞ ∫ ∞ −∞ |k(s)| ν([−r,r]) (∫ r −r ‖f(t−s)‖xdµ(t) ) ds since f is invariant by translation we have for all s ∈ r: lim r→+∞ 1 ν([−r,r]) ∫ r −r ‖f(t−s)‖xdµ(t) = 0. since (h1) holds for all s ∈ r, we have that: 0 ≤ |k(s)| ν([−r,r]) ∫ r −r ‖f(t−s)‖xdµ(t)ds ≤ β|k(s)|‖f‖∞, where k ∈ l1(r). then by the lebesgue dominated convergence theorem, we obtain that m(ζf,µ,ν) = 0. theorem 3.8. let (h1) holds. if the space pap(r,x,µ,ν) (resp. paa(r,x,µ,ν)) is translation invariant, then e(r,x,µ,ν) is convolution invariant. proof. for f ∈ ap(r,x) or f ∈ aa(r,x), then f is invariant by ζ i.e. ζf ∈ ap(r,x) or ζf ∈ aa(r,x). we use the previous theorem to conclude. corollary 3.9. let (h0) and (h1) hold. then e(r,x,µ,ν), pap(r,x,µ,ν) and paa(r,x,µ,ν) are convolution invariant. proof. combine theorem 2.12 and theorem 3.8. cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 77 4 existence, uniqueness results and applications this section is similar to section 3 in [11], but here we applied our new results obtained in the above section. 4.1 evolution families and exponential dichotomy (h2): a family of closed linear operators a(t) for t ∈ r on x with domain d(a(t)) (possibly not densely defined), is said to satisfy the so-called acquistapace-terreni conditions, if there exist constants ω ∈ r, θ ∈ (π 2 ,π), k,l ≥ 0 and µ0,ν0 ∈ (0, 1], with 1 < µ0 + ν0 such that σθ ∪{0}⊂ ρ(a(t) −ω) 3 λ,‖r(λ,a(t) −ω)‖≤ k 1 + |λ| , (4.1) and ‖(a(t) −ω)r(λ,a(t) −ω)[r(ω,a(t)) −r(ω,a(s))]‖≤ l |t−s|µ0 |λ|ν0 , (4.2) for t,s ∈ r,λ ∈ σθ := {λ ∈ c/{0} : |argλ| ≤ θ}. for a given family of linear operators a(t), the existence of an evolution family associated with it is not always guaranteed. however, if a(t) satisfied acquistapace-terreni conditions, then there exists a unique evolution family u = {u(t,s) : t,s ∈ r, t ≥ s} on x associated with a(t) such that u(t,s)x ⊆ d(a(t)) for all t,s ∈ r with t ≥ s, and, i) u(t,r)u(r,s) = u(t,s) and u(s,s) = i ∀t ≥ r ≥ s and t,r,s ∈ r; ii) the map (t,s) −→u(t,s)x is continuous for all x ∈ x, t ≥ s and t,s ∈ r; iii) u(.,s) ∈ c1((s,∞),b(x)), ∂u ∂t (t,s) = a(t)u(t,s) and ‖a(t)ku(t,s)‖≤ k(t−s)−k for 0 < t−s ≤ 1,k = 0, 1. definition 4.1 ([3]). an evolution family (u(t,s))t≥s on a banach space x is called hyperbolic (or has exponential dichotomy) if there exist projections p(t), t ∈ r, uniformly bounded and strongly continuous in t, and constants n ≥ 1, δ > 0 such that i) u(t,s)p(s) = p(t)u(t,s) for t ≥ s; ii) the restriction uq(t,s) : q(s)x −→ q(t)x for u(t,s) is inversible for t,s ∈ r and we set uq(t,s) = u(s,t)−1; 78 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) iii) ‖u(t,s)p(s)‖≤ ne−δ(t−s) (4.3) and ‖uq(s,t)q(t)‖≤ ne−δ(t−s) (4.4) for t ≥ s and t,s ∈ r, where q(t) := i −p(t) 4.2 existence results to study the existence and uniqueness of (µ,ν)pseudo-almost periodic (respectively, (µ,ν)pseudo-almost automorphic) solutions to equation (1.1), we also assume that the next hypothesis holds: (h3) the evolution family u generated by a(.) has an exponential dichotomy with constants n ≥ 1, δ > 0 and dichotomy projections p(t). we recall the following sufficient conditions to fulfill the assumption (h3). (h3.1) let (a(t),d(a(t)))t∈r be generators of analytic semigroups on x of the same type. suppose that d(a(t)) = d(a(0)), a(t) is inversible, supt,s∈r ‖a(t)a(s)−1‖ is finite, and ‖a(t)a(s)−1 − i‖≤ l0|t−s|µ1 for t,s ∈ r and constants l0 ≥ 0 and 0 ≤ µ1 ≤ 1. (h3.2) the semigroup (e τa(t))τ≥0, t ∈ r, are hyperbolic with projection pt and constants n,δ > 0. moreover, let ‖a(t)(eτa(t)pt)‖≤ ψ(τ), ‖a(t)(eτa(t)qt)‖≤ ψ(−τ) for τ > 0 and a function ψ such that r 3 s −→ ϕ(s) := |s|µψ(s) is integrable with l0‖ϕ‖l1(r) < 1. we introduce here the defnition of the mild solution of equation (1.1). definition 4.2 ([3]). a continuous function u : r 7−→ x is called a bounded mild solution of equation (1.1) if: u(t) = u(t,s)u(s) + ∫ t s u(t,τ)f(τ,u(τ))dτ, ∀t,s ∈ r,with t ≥ s. (4.5) theorem 4.3 ([11]). assume that (h2) and (h3) hold. if there exists 0 < kf < δ 2n such that ‖f(t,u) −f(t,v)‖≤ kf‖u−v‖, for all u,v ∈ x and t ∈ r, then the equation (1.1) has a unique bounded mild solution u : r 7−→ x given by u(t) = ∫ r γ(t,s)f(s,u(s))ds, t ∈ r, cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 79 where the operator family γ(t,s), called green’s function corresponding to u and p(·), is given by γ(t,s) = u(t,s)p(s), ∀t,s ∈ r, with t ≥ s, γ(t,s) = −uq(t,s)q(s), ∀t,s ∈ r, with t < s. denote by γ1 and γ2 the nonlinear integral operators defined by, (γ1u)(t) := ∫ t −∞ u(t,s)p(s)f(s,u(s))ds, and (γ2u)(t) := ∫ +∞ t uq(t,s)q(s)f(s,u(s))ds. in the rest of this work, we fix µ,ν ∈m to satisfy (h1). 4.3 existence of (µ,ν)-pseudo-almost periodic solutions in addition to the previous assumptions, we require the following additional ones: (h4): r(ω,a(.)) ∈ ap(r,l(x)). (h5): we propose f : r×x 7−→ x belongs to pap(r×x,x,µ,ν) and there exists kf > 0 such that ‖f(t,u) −f(t,v)‖≤ kf‖u−v‖, for all u,v ∈ x and t ∈ r. the following lemma plays an important role to prove the main results of this study. lemma 4.4 ([13]). assume that (h2)-(h4) hold. then r −→ γ(t+r,s+r) belongs to ap(r,l(x)) for all t,s ∈ r, where we may take the same pseudo periods for t,s with |t − s| ≥ h > 0. if f ∈ ap(r,l(x)), then the unique bounded mild solution u(t) = ∫ r γ(t,s)f(s)ds of the following equation u′(t) = a(t)u(t) + f(t), t ∈ r, is almost periodic. lemma 4.5. assume that (h2)-(h5) hold. if (3.1) and (3.2), or (3.1) and (3.3) hold, then the integral operators γ1 and γ2 defined above map pap(r,x,µ,ν) into itself. proof. let u ∈ pap(r,x,µ,ν). setting h(t) = f(t,u(t)), using the assumption (h5) and theorem 2.16 it follows that h ∈ pap(r,x,µ,ν). now write h = ψ1 + ψ2 where ψ1 ∈ ap(r,x) and ψ2 ∈e(r,x,µ,ν). that is, γ1h = ξ(ψ1) + ξ(ψ2) where ξψi(t) := ∫ t −∞ u(t,s)p(s)ψi(s)ds,for i ∈{1, 2}. 80 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) from lemma 4.4, we have ξ(ψ1) ∈ ap(r,x). to complete the proof, we will prove that ξ(ψ2) ∈ e(r,x,µ,ν). now, let r > 0. from equation (4.3), we have: 1 ν([r,−r]) ∫ r −r ‖(ξ(ψ2)(t)‖dµ(t) ≤ 1 ν([r,−r]) ∫ r −r ∫ t −∞ u(t,s)p(s)ψ2(s)dsdµ(t) ≤ n ν([r,−r]) ∫ r −r ∫ t −∞ e−δ(t−s)‖ψ2(s)‖dsdµ(t) since µ and ν satisfy (3.1.1) and (3.2.1), (3.1.1) and (3.3.1), with k(t) = e−δt, then by theorem 3.1 or corollary 3.5, we conclude that: lim r→+∞ 1 ν([r,−r]) ∫ r −r ‖(ξ(ψ2)(t)‖dµ(t) = 0. the proof for γ2u(.) is similar to that of γ1u(.) except that one makes use of equation (4.4) instead of (4.3), (3.1.2) and (3.2.2), or (3.1.2) and (3.3.2). theorem 4.6. assume that (h2)-(h5) hold. if (3.1) and (3.2), or (3.1) and (3.3) hold, then equation (1.1) has a unique (µ,ν)-pseudo almost periodic mild solution whenever kf is small enough. proof. consider the nonlinear operator k defined on pap(r,x,µ,ν) by ku(t) = ∫ t −∞ u(t,s)p(s)f(s,u(s))ds− ∫ +∞ t uq(t,s)q(s)f(s,u(s))ds, ∀t ∈ r. by lemma 4.5, it follows that k maps pap(r,x,µ,ν) into itself. to complete the proof one has to show that k is a contraction map on pap(r,x,µ,ν). let u,v ∈ pap(r,x,µ,ν). firstly, we have that: ‖γ1(v)(t) − γ1(u)(t)‖ ≤ ∫ t −∞ ‖u(t,s)p(s)[f(s,v(s)) −f(s,u(s))]‖ds ≤ nkf ∫ t −∞ e−δ(t−s)‖v(s) −u(s)‖ds ≤ nkfδ−1‖v −u‖∞. next, we have that: ‖γ2(v)(t) − γ2(u)(t)‖ ≤ ∫ +∞ t ‖uq(t,s)q(s)[f(s,v(s)) −f(s,u(s))]‖ds ≤ nkf ∫ +∞ t e−δ(t−s)‖v(s) −u(s)‖ds ≤ nkfδ−1‖v −u‖∞ ∫ +∞ t e−δ(t−s)ds = nkfδ −1‖v −u‖∞. finally, combining previous approximations it follows that: ‖kv −ku‖∞ < 2nkfδ−1‖v −u‖∞. cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 81 thus if kf is small enough, that is, kf < δ(2n) −1, then k is a contraction map on pap(r,x,µ,ν). therefore, k has a unique fixed point in pap(r,x,µ,ν), that is, there exists a unique function u satisfying ku = u, which is the unique (µ,ν)-pseudo almost periodic mild solution to equation (1.1). theorem 4.7 ([11]). assume that (h2)-(h5) hold. if (h0) holds, then equation (1.1) has a unique (µ,ν)-pseudo almost periodic mild solution whenever kf is small enough. 4.4 existence of (µ,ν)-pseudo-almost automorphic solutions in this section we consider the following assumptions: (h6): r(ω,a(.)) ∈ aa(r,l(x)). (h7): we propose f : r×x 7−→ x belongs to paa(r×x,x,µ,ν) and there exists kf > 0 such that ‖f(t,u) −f(t,v)‖≤ kf‖u−v‖∞, for all u,v ∈ x and t ∈ r. lemma 4.8 ([14]). assume that (h2), (h3) and (h6) hold. let a sequence (s ′ l)l∈n ⊂ r there is a sub-sequence (sl)l∈n such that for every h > 0 ‖γ(t + sl −sk,s + sl −sk) − γ(t,s)‖−→ 0, k, l −→∞. lemma 4.9. assume that (h2), (h3), (h6) and (h7) hold. if (3.1) and (3.2) or (3.1) and (3.3) or (h0) hold, then the integral operators γ1 and γ2 defined above map paa(r ×x,x,µ,ν) into itself. proof. let u ∈ paa(r,x,µ,ν). setting g(t) = f(t,u(t)), by assumption (h7) and theorem 2.17 we obtain that g ∈ paa(r,x,µ,ν). now write g = u1 + u2 where u1 ∈ aa(r,x) and u2 ∈e(r,x,µ,ν). that is, γ1g = su1 + su2, where su1(t) := ∫ t −∞ u(t,s)p(s)u1(s)ds, su2(t) := ∫ t −∞ u(t,s)p(s)u2(s)ds. from equation (4.3), we obtain: ‖su1(t)‖≤ nδ−1‖u1‖∞, ‖su2(t)‖≤ nδ−1‖u2‖∞, ∀t ∈ r. then su1(t),su2(t) ∈ bc(r,x). now, we prove that su1(t) ∈ aa(r,x). since u1 ∈ aa(r,x), then for every sequence (τ′n)n∈n ∈ r there exists a subsequence (τn)n∈n such that: v1(t) := lim n→∞ u1(t + τn), (4.6) is well defined for each t ∈ r, and lim n→∞ v1(t− τn) = u1(t), ∀t ∈ r. (4.7) 82 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) set for t ∈ r, m(t) := ∫ t −∞ u(t,s)p(s)u1(s)ds, and n(t) := ∫ t −∞ u(t,s)p(s)v1(s)ds. now, we have m(t + τn) −n(t) = ∫ t+τn −∞ u(t + τn,s)p(s)u1(s)ds− ∫ t −∞ u(t,s)p(s)v1(s)ds = ∫ t −∞ u(t + τn,s + τn)p(s + τn)u1(s + τn)ds − ∫ t −∞ u(t,s)p(s)v1(s)ds = ∫ t −∞ u(t + τn,s + τn)p(s + τn)[u1(s + τn) −v1(s)]ds + ∫ t −∞ [u(t + τn,s + τn)p(s + τn) −u(t,s)p(s)]v1(s)ds. using equation (4.3), equation (4.6) and the lebesgue’s dominated convergence theorem, it follows that: lim n→+∞ ∣∣∣∣ ∣∣∣∣ ∫ t −∞ u(t + τn,s + τn)p(s + τn)[u1(s + τn) −v1(s)]ds ∣∣∣∣ ∣∣∣∣ = 0, for t ∈ r. (4.8) similary, using lemma 4.8 it follows that: lim n→+∞ ∣∣∣∣ ∣∣∣∣ ∫ t −∞ [u(t + τn,s + τn)p(s + τn) −u(t,s)p(s)]v1(s)ds ∣∣∣∣ ∣∣∣∣ = 0, for t ∈ r. (4.9) therefore, we have that: n(t) := lim n→∞ m(t + τn),∀t ∈ r. (4.10) using similar ideas as the previous ones, then: m(t) := lim n→∞ n(t− τn),∀t ∈ r. (4.11) therefore, su1(t) ∈ aa(r,x). arguing as in lemma 4.5, we get that su2(t) ∈e(r,x,µ,ν). the proof for γ2u(.) is similar to that of γ1u(.) except that one makes use of equation (4.4) instead of equation (4.3) and, (3.1.2) and (3.2.2), (3.1.2) and (3.3.2). theorem 4.10. under assumptions (h2), (h3), (h6) and (h7), if (3.1) and (3.2) or (3.1) and (3.3) or (h0) then equation (1.1) has a unique (µ,ν)-pseudo almost automorphic mild solution whenever kf is small enough. proof the proof of theorem 4.10 is similar to that theorem 4.6 except that one makes use of lemma 4.9 instead of lemma 4.5. cubo 23, 1 (2021) convolutions in (µ,ν)-pap and (µ,ν)-paa functions 83 4.5 neutral systems in this subsection, we establish the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively (µ,ν)-pseudo almost automorphic) solutions for the nonautonomous neutral partial evolution equation (1.2). for that, we need the following assumptions: (h8): we suppose g : r × x −→ x belongs to pap(r × x,x,µ,ν) and there exists kg > 0 such that: ‖g(t,u) −g(t,v)‖≤ kg‖u−v‖, for all u,v ∈ x and t ∈ r. (h9) we suppose g : r×x −→ x belongs to paa(r×x,x,µ,ν) and there exists kg > 0 such that: ‖g(t,u) −g(t,v)‖≤ kg‖u−v‖, for all u,v ∈ x and t ∈ r. definition 4.11. a function v : r 7−→ x is said a mild solution of (1.2) on r if : v(t) = g(t,v(t)) + ∫ t −∞ u(t,s)p(s)f(s,v(s))ds− ∫ +∞ t uq(t,s)q(s)f(s,v(s))ds, for all t ∈ r. theorem 4.12. assume that assumptions (h2)-(h5) and (h8) hold. if (3.1) and (3.2) or (3.1) and (3.3) or (h0) hold, and (kg + 2nkfδ −1) < 1, then equation (1.2) has a unique (µ,ν)-pseudo almost periodic mild solution. proof. we consider the nonlinear operator w defined on pap(r,x,µ,ν) by: wv(t) = g(t,v(t)) + ∫ t −∞ u(t,s)p(s)f(s,v(s))ds− ∫ +∞ t uq(t,s)q(s)f(s,v(s))ds for all t ∈ r. from (h9), theorem 2.16, and lemma 4.5 it follows that w maps pap(r,x,µ,ν) into itself. to complete the proof we need to show that w is a contraction map on pap(r,x,µ,ν). for that, letting u,v ∈ pap(r,x,µ,ν), we obtain: ‖wv −wu‖∞ ≤ (kg + 2nkfδ−1)‖v −u‖∞, which yields w is a contraction map on pap(r,x,µ,ν). therefore, w has unique fixed point in pap(r,x,µ,ν). therefore, equation (1.2) has unique (µ,ν)-pseudo almost periodic mild solution. theorem 4.13. assume that (h2), (h3),(h6), (h7) and (h9) hold and (kg + 2nkfδ −1) < 1. if (3.1) and (3.2)or (3.1) and (3.3) or (h0) hold, then equation (1.2) has a unique (µ,ν)-pseudo almost automorphic mild solution. proof. similarly, we can show, by using the assumption (h9), theorem 2.17 and lemma 4.9, that the equation (1.2) has a unique (µ,ν)-pseudo almost automorphic mild solution. 84 d. békollè, k. ezzinbi, s. fatajou, d.e. houpa danga & f. mbounja b. cubo 23, 1 (2021) acknowledgments the authors are grateful to the anonymous referee for the careful reading of this paper, for very helpful suggestions and comments which improved of quality of this article. references [1] p. acquistapace, f. flandoli, and b. terreni, “initial boundary value problems and optimal control for nonautonomous parabolic systems”, siam journal on control and optimization, vol. 29, pp. 89–118, 1991. 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[17] h. l. royden, real analysis, third edition, macmillan publishing company, new york, 1988. introduction preliminaries notation and terminology some useful results on the space functions measure theory results main results of convolution and translation invariance convolution invariance on e(r,x,, ) translation invariance and convolution invariance of pap(r,x,, ) and paa(r,x,, ) existence, uniqueness results and applications evolution families and exponential dichotomy existence results existence of (,)-pseudo-almost periodic solutions existence of (,)-pseudo-almost automorphic solutions neutral systems cubo, a mathematical journal vol. 24, no. 02, pp. 307–331, august 2022 doi: 10.56754/0719-0646.2402.0307 on severi varieties as intersections of a minimum number of quadrics hendrik van maldeghem 1, b magali victoor 1 1ghent university, department of mathematics: algebra & geometry, krijgslaan 281, s25, b-9000 gent, belgium. hendrik.vanmaldeghem@ugent.be b magali.victoor@ugent.be abstract let v be a variety related to the second row of the freudenthal-tits magic square in n-dimensional projective space over an arbitrary field. we show that there exist m ≤ n quadrics intersecting precisely in v if and only if there exists a subspace of projective dimension n −m in the secant variety disjoint from the severi variety. we present some examples of such subspaces of relatively large dimension. in particular, over the real numbers we show that the cartan variety (related to the exceptional group e6(r)) is the set-theoretic intersection of 15 quadrics. resumen sea v una variedad relacionada a la segunda fila del cuadrado mágico de freudenthal-tits en el espacio proyectivo n-dimensional sobre un cuerpo arbitrario. mostramos que existen m ≤ n cuádricas intersectandose precisamente en v si y solo si existe un subespacio de dimensión proyectiva n − m en la variedad secante disjunta de la variedad de severi. presentamos algunos ejemplos de tales subespacios de dimensión relativamente grande. en particular, sobre los números reales, mostramos que la variedad de cartan (relacionada al grupo excepcional e6(r)) es la intersección conjuntista de 15 cuádricas. keywords and phrases: cartan variety, quadrics, exceptional geometry, severi variety, quaternion veronesian. 2020 ams mathematics subject classification: 51e24. accepted: 8 june, 2022 received: 14 october, 2021 c©2022 h. van maldeghem et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2402.0307 mailto:hendrik.vanmaldeghem@ugent.be https://orcid.org/0000-0002-8022-0040 https://orcid.org/0000-0002-5774-5260 mailto:hendrik.vanmaldeghem@ugent.be mailto: magali.victoor@ugent.be 308 h. v. maldeghem & m. victoor cubo 24, 2 (2022) 1 introduction it is well known that the grassmannians of the (split) spherical buildings related to semi-simple algebraic groups over algebraically closed fields can be described as the intersection of a number of quadrics, see [7] for the complex case, and [3] and [10] for the more general case. in this paper, we consider the grassmannians (or “varieties”) related to the second row of the freudenthal-tits magic square. over the complex numbers, these are the so-called “severi varieties”. however, these can be considered over any field k (not necessarily algebraically closed anymore), and these geometries will be also called severi varieties. a severi variety lives in a projective space of dimension n = 5, 8, 14 or 26 and is the set-theoretic and scheme-theoretic intersection of n + 1 quadrics, the equations of which carry a particularly elegant combinatorics, see [11]. the question we’d like to put forward in this paper is whether we can describe the severi varieties set-theoretically with fewer quadrics, and ultimately try to find the minimum number of quadrics the intersection of which is precisely the given severi variety. our motivation is entirely curiosity and beauty; the latter under the form of a rather unexpected connection we found. we will show that the n + 1 quadrics referred to above are linearly independent from each other. also, every quadric containing the given severi variety is a linear combination of these n + 1 quadrics. these two facts point, in our opinion, to the conjecture that no set of n quadrics can intersect precisely in the severi variety. however, the quadric veronese surface (the case n = 5 severi variety) over fields of characteristic 2 is the set-theoretic intersection of three quadrics, see lemma 4.20 in [6]. moreover, it was stated in [2], however without proof, that in the case n = 8, the severi variety is the set-theoretic intersection of only 6 quadrics. hence the above conjecture is false. in general, we will show the following equivalence: main result. there exist m ≤ n quadrics intersecting precisely in the given severi variety ⇐⇒ there exists a subspace of projective dimension n −m in the secant variety disjoint from the severi variety. a more detailed and precise statement will be provided in section 3. in fact, that statement and its proof allow one, in principle, to describe all equivalence classes of systems of m ≤ n quadratic equations exactly describing a given severi variety. as an application, we will do this explicitly in the simplest case, n = 5. for the other cases we content ourselves with giving examples for relatively small m. in particular we will exhibit the real cartan variety (the grassmanian of type e6,1 in 26-dimensional real projective space) as the intersection of only 15 quadrics (whereas initially, we had 27 of them). it would require additional methods and ideas to pin down the minimal m for each case and each field, so we consider that to be out of the scope of this paper. about the method of our proof: usually, the equations of the n + 1 initial quadrics are partial derivatives of a cubic form (which has to be taken for granted). in the present paper, we start cubo 24, 2 (2022) on severi varieties as intersections of a minimum number of quadrics 309 with the combinatorics of the equations of the quadrics and derive the cubic form from that. this enables us to make a few geometric observations and interpretations which lead to a proof of the main result. since the secant variety of a severi variety always contains at least one point outside the variety, we recover in our special case of severi variety already the general result of kronecker saying that any projective variety in pn k is a set theoretic intersection of (at most) n hypersurfaces (in our case quadrics), see corollary 2 in [5]. one could also ask the equivalent question for the schemetheoretic intersection of quadrics, but we did not consider that. it seems to us that the answer we give in the present paper for the segre variety is also valid in the scheme-theoretic sense, but the minimal examples for the line grassmannian and the cartan variety are not. 2 preliminaries 2.1 the varieties the main objects in this paper are the quadric veronese surface v2(k) over any field k, the segre variety s2,2(k) corresponding to the product of two projective planes over k, the line grassmannian g2,6(k) of projective 5-space over k, and the cartan variety e6(k) associated to the 27-dimensional module of the (split) exceptional group of lie type e6 over the field k. these varieties can be defined as intersections of quadrics (and we will do so in subsection 4.1 below), but it might be insightful to also have the classical definition, which we now present. in what follows, k is an arbitrary field and pn k or pn denotes the n-dimensional projective space over k, which we suppose to be coordinatized with homogeneous coordinates from k after an arbitrary choice of a basis. the quadric veronese surface v2(k)—this is the image of the veronese map ν : p 2 → p5 : (x, y, z) 7→ (x2, y2, z2, yz, zx, xy). the segre variety s2,2(k)—this is the image of the segre map p 2×p2 → p8 : (x, y, z; u, v, w) 7→ (xu, yu, zu, xv, yv, zv, xw, yw, zw). we may view the set of 3 × 3 matrices over k as a 9-dimensional vector space, and the set of symmetric 3 × 3 matrices as a 6-dimensional subspace. then we may consider the corresponding projective spaces of (projective) dimension 8 and 5, respectively, in the classical way by considering the 1-spaces as the points. in this way, the segre variety s2,2(k) corresponds exactly with the rank 1 matrices; explicitly k(xu, yu, zu, xv, yv, zv, xw, yw, zw) ↔ k     xu yu zu xv yv zv xw yw zw     . 310 h. v. maldeghem & m. victoor cubo 24, 2 (2022) similarly, the quadric veronese surface v2(k) corresponds exactly with the rank 1 symmetric matrices; explicitly k(x2, y2, z2, yz, zx, xy) ↔ k     x2 yx zx xy y2 zy xz yz z2     . in particular, v2(k) is a subvariety of s2,2(k) obtained by intersecting with a 5-dimensional subspace. there exist other segre varieties; in general sn,m(k) is defined as the image in p nm−1 of the map (xi, yj)1≤i≤n,1≤j≤m 7→ (xiyj)1≤i≤n,1≤j≤m. the images of the marginal maps defined by either fixing the xi, 1 ≤ i ≤ n, or the yj, 1 ≤ j ≤ m, are called the generators of the variety (in case of s2,2(k) the generators are 2-dimensional projective subspaces). the line grassmannian g2,6(k)—denote the set of lines of p 5, or equivalently, the set of 2spaces of k6 by ( k 6 k2 ) . then g2,6(k) is the image of the plücker map ( k 6 k2 ) → p14 : 〈(x1, x2, . . . , x6).(y1, y2, . . . , y6)〉 7→ (xiyj − xjyi)1≤i a(m, α) ˆ ρ 0 dx x α−2m ∣ ∣f(x) ∣ ∣ 2 + b(m, α) n ∑ k=1 ˆ ρ 0 dx x α−2m k ∏ p=1 [lnp(γ/x)] −2 ∣ ∣f(x) ∣ ∣ 2 , f ∈ c ∞ 0 ((0, ρ)), m, n ∈ n, α ∈ r, ρ, γ ∈ (0, ∞), γ > en ρ. here the iterated logarithms are given by ln1( · ) = ln( · ), lnj+1( · ) = ln(lnj( · )), j ∈ n, and the iterated exponentials are defined via e0 = 0, ej+1 = e ej , j ∈ n0 = n ∪ {0}. moreover, we prove the analogous sequence of inequalities on the exterior interval (r, ∞) for f ∈ c∞0 ((r, ∞)), r ∈ (0, ∞), and once again prove optimality of the constants involved. accepted: 23 february, 2022 received: 28 june, 2021 c©2022 f. gesztesy et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100115 https://orcid.org/0000-0001-8554-9745 https://orcid.org/0000-0003-1697-2018 https://orcid.org/0000-0002-5262-8882 mailto:fritz_gesztesy@baylor.edu mailto:imichael@lsu.edu mailto:pangm@missouri.edu cubo, a mathematical journal vol. 24, no. 01, pp. 115–167, april 2022 doi: 10.4067/s0719-06462022000100115 resumen el objetivo principal de este art́ıculo es establecer la optimalidad (i.e. la precisión) de las constantes a(m, α) y b(m, α), m ∈ n, α ∈ r, de la forma a(m, α) = 4 −m m ∏ j=1 (2j − 1 − α) 2 , b(m, α) = 4 −m m ∑ k=1 m ∏ j=1 j 6=k (2j − 1 − α) 2 , en las desigualdades integrales de tipo birman–hardy– rellich pesadas por potencias con términos de refinamiento logaŕıtmicos recientemente demostradas en [41], es decir, ˆ ρ 0 dx x α ∣ ∣f (m) (x) ∣ ∣ 2 > a(m, α) ˆ ρ 0 dx x α−2m ∣ ∣f(x) ∣ ∣ 2 + b(m, α) n ∑ k=1 ˆ ρ 0 dx x α−2m k ∏ p=1 [lnp(γ/x)] −2 ∣ ∣f(x) ∣ ∣ 2 , f ∈ c ∞ 0 ((0, ρ)), m, n ∈ n, α ∈ r, ρ, γ ∈ (0, ∞), γ > en ρ. acá los logaritmos iterados están dados por ln1( · ) = ln( · ), lnj+1( · ) = ln(lnj( · )), j ∈ n, y las exponenciales iteradas están definidas por e0 = 0, ej+1 = e ej , j ∈ n0 = n ∪ {0}. más aún, probamos la secuencia análoga de desigualdades en el intervalo exterior (r, ∞) para f ∈ c∞0 ((r, ∞)), r ∈ (0, ∞), y una vez más probamos la optimalidad de las constantes involucradas. keywords and phrases: birman-hardy-rellich inequalities, logarithmic refinements. 2020 ams mathematics subject classification: 26d10, 34a40, 35a23, 34l10. accepted: 23 february, 2022 received: 28 june, 2021 c©2022 f. gesztesy et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100115 cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 117 1 introduction and notations employed given the notation introduced in (1.4)–(1.8) we will prove in this paper that the constants a(m,α) and the n constants b(m,α) appearing in the power-weighted birman–hardy–rellich-type integral inequalities with logarithmic refinement terms, ˆ ρ 0 dxxα ∣∣f(m)(x) ∣∣2 > a(m,α) ˆ ρ 0 dxxα−2m ∣∣f(x) ∣∣2 + b(m,α) n∑ k=1 ˆ ρ 0 dxxα−2m k∏ p=1 [lnp(γ/x)] −2 ∣∣f(x) ∣∣2, (1.1) f ∈ c∞0 ((0,ρ)), m,n ∈ n, α ∈ r, ρ,γ ∈ (0,∞), γ > enρ, recently proved in [41], are optimal (i.e., sharp). moreover, we prove optimality of a(m,α) and the n constants b(m,α) for the analogous sequence of inequalities on the exterior interval (r,∞), that is, ˆ ∞ r dxxα ∣∣f(m)(x) ∣∣2 > a(m,α) ˆ ∞ r dxxα−2m ∣∣f(x) ∣∣2 + b(m,α) n∑ k=1 ˆ ∞ r dxxα−2m k∏ p=1 [lnp(x/γ)] −2 ∣∣f(x) ∣∣2, (1.2) f ∈ c∞0 ((r,∞)), m,n ∈ n, α ∈ r, r,γ ∈ (0,∞), r > enγ. of course, (1.1) (resp., (1.2)) extends to n = 0, ρ = ∞ (resp., to n = 0, r = 0) upon disregarding all logarithmic terms (i.e., upon putting b(m,α) = 0). in their simplest (i.e., unweighted) form, the birman–hardy–rellich inequalities, as recorded by birman in 1961, and in english translation in 1966 [19] (see also [45, pp. 83–84]), are given by ˆ ρ 0 dx ∣∣f(m)(x) ∣∣2 > [(2m − 1)!!]2 22m ˆ ρ 0 dxx−2m|f(x)|2, (1.3) f ∈ cm0 ((0,ρ)), m ∈ n, 0 < ρ 6 ∞. the case m = 1 in (1.3) represents hardy’s celebrated inequality [51], [52, sect. 9.8] (see also [61, chs. 1, 3, app.]), the case m = 2 is due to rellich [81, sect. ii.7]. the power-weighted extension of (1.3) is then represented by the first line of (1.1) (i.e., by deleting the second line in (1.1) which contains additional logarithmic refinements). even though a detailed history of the power-weighted birman–hardy–rellich inequalities was provided in the companion paper [41], we will now repeat the highlights of this history for matters of completeness. we start with the observation that the inequalities (1.3) and their power weighted generalizations, that is, the first line in (1.1), are known to be strict, that is, equality holds in (1.3), resp., in the first line in (1.1) (in fact, for the entire inequality (1.1)) if and only if f = 0 on (0,ρ). moreover, 118 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) these inequalities are optimal, meaning, the constants [(2m − 1)!!]2/22m in (1.3), respectively, the constants a(m,α) in (1.1) are sharp, although, this must be qualified and will be revisited below as different authors frequently prove sharpness for different function spaces. in the present one-dimensional context at hand, sharpness of (1.3) (and typically, its power weighted version, the first line in (1.1)), are often proved in an integral form (rather than the currently presented differential form) where f(m) on the left-hand side is replaced by f and f on the right-hand side by m repeated integrals over f . for pertinent one-dimensional sources, we refer, for instance, to [14, pp. 3–5], [22], [24, pp. 104–105], [42, 49, 51], [52, pp. 240–243], [61, ch. 3], [62, pp. 5–11], [64, 72, 80]. we also note that higher-order hardy inequalities, including various weight functions, are discussed in [60, sect. 5], [61, chs. 2–5], [62, chs. 1–4], [63], and [79, sect. 10] (however, birman’s sequence of inequalities (1.3) is not mentioned in these sources). in addition, there are numerous sources which treat multi-dimensional versions of these inequalities on various domains ω ⊆ rn, which, when specialized to radially symmetric functions (e.g., when ω represents a ball), imply one-dimensional birman–hardy–rellich-type inequalities with power weights under various restrictions on these weights. however, none of the results obtained in this manner imply (1.1), under optimal hypotheses on α and γ. we also mention that a large number of these references treat the lp-setting, and in some references x ∈ (a,b) is replaced by d(x), the distance of x to the boundary of (a,b), respectively, ω, but this represents quite a different situation (especially in the multi-dimensional context) and hence is not further discussed in this paper. to put the logarithmic refinements in (1.1) (i.e., the second line in (1.1)) into some perspective and to compare with existing results in the literature, we offer the following comments: originally, logarithmic refinements of hardy’s inequality started with oscillation theoretic considerations going back to hartman [53] (see also [54, pp. 324–325]) and have been used in connection with hardy’s inequality in [38, 43], and more recently, in [39, 40]. since then there has been enormous activity in this context and we mention, for instance, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, chs. 3, 5], [16, 17, 18, 21, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 44, 46, 47], [48, chs. 2, 6, 7], [56, 57, 65, 66, 67, 68, 70, 71, 74, 76, 77], [81, sect. 2.7], [82, 83, 84, 88, 89, 90, 91]. the vast majority of these references deals with analogous multi-dimensional settings (relevant to our setting in particular in the case of radially symmetric functions), several also with the lp-context. for m > 2 the inequalities (1.1) and (1.2) proven in [41] were new in the following sense: the weight parameter α ∈ r is unrestricted (as opposed to prior results) and at the same time the conditions on the logarithmic parameters γ and γ are sharp. the issue of sharpness of the constants a(m,α) and b(m,α) appearing in (1.1) is a rather delicate one and hence we offer the following remarks, the gist of which can be found in [41, appendix a]. we start by noting that the smaller the underlying function space, the larger the efforts needed to prove optimality. many of the results cited in the remainder of this remark, under particular restrictions on the weight parameter α, establish sharpness for larger classes of functions f which cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 119 do not automatically continue to hold in the c∞0 ((0,ρ))-context. it is this simple observation that adds considerable complexity to sharpness proofs for the space c∞0 ((0,ρ)). (the issue of dependence of optimal constants on the underlying function space is nicely illustrated in [30].) by the same token, optimality proofs obtained for c∞0 function spaces automatically hold for larger function spaces as long as the inequalities have already been established for the larger function spaces with the same constants a(m,α),b(m,α). this comment applies, in particular, to many papers that prove sharpness results in multi-dimensional situations for larger function spaces such as1 c∞0 (b(0;ρ)) or (homogeneous, weighted) sobolev spaces rather than c ∞ 0 (b(0;ρ)\{0}). unless c∞0 (b(0;ρ)\{0}) is dense in the appropriate norm, one cannot a priori assume that the optimal constants a(m,α̃) and b(m,α̃) (with α̃ appropriately depending on n, e.g., α̃ = α+n− 1) remain the same for c∞0 (b(0;ρ)) and c ∞ 0 (b(0;ρ)\{0}), say. at least in principle, they could actually increase for the space c∞0 (b(0;ρ)\{0}). turning to a review of the existing literature, sharpness of the constant a(m,0), m ∈ n (i.e., in the unweighted case, α = 0), corresponding to the space c∞0 ((0,∞)) has been shown by yafaev [91]. in fact, he also established this result for fractional m (in this context we also refer to appropriate norm bounds in lp(rn;dnx) of operators of the form |x| −β |−i∇| −β , 1 < p < n/β, see [13, sect. 1.7],[14, 55, 58, 59, 78, 86], [87, sects. 1.7, 4.2]). sharpness of a(2,0) (i.e., in the unweighted rellich case) was shown by rellich [81, pp. 91–101] in connection with the space c∞0 ((0,∞)); his multi-dimensional results also yield sharpness of a(2,n−1) for n ∈ n, n > 3, again for c∞0 ((0,∞)); in this context see also [14, corollary 6.3.5]. an exhaustive study of optimality of a(2, α̃) (i.e., rellich inequalities with power weights) for the space c∞0 (ω\{0}) for cones ω ⊆ r n, n > 2, appeared in caldiroli and musina [21]. the authors, in particular, describe situations where a(2, α̃) has to be replaced by other constants and also treat the special case of radially symmetric functions in detail. additional results for power weighted rellich inequalities appeared in [74, 75]; further extensions of power weighted rellich inequalities with sharp constants on c∞0 (r n\{0}) were obtained in [69]; for optimal power weighted hardy, rellich, and higher-order inequalities on homogeneous groups, see [82, 83]. many of these references also discuss sharp (power weighted) hardy inequalities, implying optimality for a(1, α̃). moreover, replacing f(x) by f(x) = ´ x 0 dtf(t) ( or f(x) = ´∞ x dtf(t) ) , optimality of the hardy constant a(1,0) for larger, lp-based function spaces, can already be found in [52, sect. 9.8] (see also [14, theorem 1.2.1], [61, ch. 3], [62, pp. 5– 11], [64, 72, 80], in connection with a(1,α)). we mention that theorems 4.1 and 4.7, which assert optimality of a(m,α) in (1.1) and (1.2), were already proved in [41, theorem a.1] using a different method. sharpness results for a(m,α) and b(m,α) together are much less frequently discussed in the literature, even under suitable restrictions on m and α. the results we found primarily follow upon specializing multi-dimensional results for function spaces such as c∞0 (ω\{0}), or c ∞ 0 (ω), ω ⊆ r n 1here b(0; ρ) ⊆ rn denotes the open ball in rn, n > 2, with center at the origin x = 0 and radius ρ > 0. 120 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) open, and appropriate restrictions on m, α, and n > 2, for radially symmetric functions to the onedimensional case at hand (cf. the previous paragraph). in this context we mention that the hardy case m = 1, without a weight function, is studied in [1, 2, 5, 9, 20, 23, 26, 36, 50, 57, 65, 85, 89] (all for n = 1), and in [10, 28, 46] (all for n ∈ n); the case with power weight functions is discussed in [17, 47], [48, ch. 6] (for n ∈ n); see also [66]. the rellich case m = 2 with a general power weight on c∞0 (ω\{0}) is discussed in [21] (for n = 1); the rellich case m = 2, without weight function on c∞0 (ω), is studied in [26, 27, 29] (all for n = 1), the case n ∈ n is studied in [4]; the case of additional power weights is treated in [47], [48, ch. 6], [71]. the general case m ∈ n is discussed in [6] (for n = 1) and in [15, 47], [48, ch. 6], [90] (all for n ∈ n and including power weights, but with additional restrictions). employing oscillation theory, sharpness of the unweighted hardy case a(1,0) = b(1,0) = 1/4, with n ∈ n, was proved in [43]. as will become clear in the course of this paper, the special results available on sharpness of the n constants b(m,α) are all saddled with considerable complexity, especially, for larger values of n ∈ n. for this reason only sharpness of the constants a(m,α) was derived in [41, appendix a] and sharpness of a(m,α) and b(m,α) was postponed to this paper which therefore should be viewed as a companion of [41]. in section 2 (a very massive one) we establish all the preliminary results, culminating in lemmas 2.13 and 2.14, required in the remainder of this paper. the methods used in this section are adaptations of those in [15, sect. 3]. the basic approximation procedure is introduced in section 3, with corollaries 3.12 and 3.13 summarizing the principal results. our final section 4 then proves optimality of the n + 1 constants a(m,α) and b(m,α) for the interval (0,ρ) in theorems 4.1 and 4.2 and for the interval (r,∞) in theorems 4.7 and 4.8 based on lemmas 2.13 and 2.14 and corollaries 3.12 and 3.13. we also mention that theorems 4.2 and 4.8 still hold if the repeated log-terms lnp( · ) (see (1.5) below) are replaced by the type of repeated log-terms used, for example, in [15, 16, 17, 90].2 we conclude this introduction by establishing the principal notation used in this paper: for j ∈ n0 (with n0 = n ∪ {0}) we define ej by e0 = 0, e1 = 1, ej+1 = e ej, j ∈ n. (1.4) for n ∈ n, γ,ρ ∈ (0,∞), with γ > ρen, and 1 6 j 6 n, we define lnj(γ/x), for 0 < x < ρ, by ln1(γ/x) = ln(γ/x), lnj+1(γ/x) = ln(lnj(γ/x)), 1 6 j 6 n − 1. (1.5) for the rest of this paper we shall assume that n ∈ n ∪ {0}, m ∈ n, α ∈ r, γ,ρ ∈ (0,∞), with 2detailed proofs of theorems 4.2 and 4.8 for the type of log-terms used in [15, 16, 17, 90] are available from the authors upon request. cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 121 γ > ρen+1. we shall write a(m,α) = 4−m m∏ j=1 (2j − 1 − α)2, (1.6) b(m,α) = 4−m m∑ k=1 m∏ j=1,j 6=k (2j − 1 − α)2. (1.7) note that if α ∈ r\{2j − 1}16j6m, one has b(m,α) = a(m,α) m∑ j=1 (2j − 1 − α)−2. (1.8) we assume ψ ∈ c∞(r) satisfies the following properties: (i) ψ is non-increasing, (1.9) (ii) ψ(x) =    1, x 6 8ρ/10, 0, x > 9ρ/10. (1.10) for g ∈ c∞((0,ρ)) we shall write jn[g] = ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 − a(m,α) ˆ ρ 0 dxxα−2m|g(x)|2 − b(m,α) n∑ k=1 ˆ ρ 0 dxxα−2m|g(x)|2 k∏ j=1 [lnj(γ/x)] −2, (1.11) provided that ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 < ∞, ˆ ρ 0 dxxα−2m|g(x)|2 < ∞. (1.12) for j = 0,1, . . . ,n and β ∈ r we introduce σ0(β) = (2m − 1 − α + β)/2, σj(β) = −(1 − β)/2, j = 1, . . . ,n. (1.13) for 0 6 j 6 k 6 n and ε = (ε0,ε1, . . . ,εn), where ε0,ε1, . . . ,εn > 0, we shall write γj,k(ε) = γj,k(ε0,ε1, . . . ,εn), = ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] −1−ε1 · · · [lnj(γ/x)] −1−εj × [lnj+1(γ/x)] −εj+1 · · · [lnk(γ/x)] −εk × [lnk+1(γ/x)] 1−εk+1 · · · [lnn(γ/x)] 1−εn [ψ(x)]2. (1.14) 122 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) in particular, if n ∈ n, γ0,0(ε) = ˆ ρ 0 dxx−1+ε0 n∏ k=1 [lnk(γ/x)] 1−εk [ψ(x)]2, γ0,k(ε) = ˆ ρ 0 dxx−1+ε0 k∏ ℓ=1 [lnℓ(γ/x)] −εℓ n∏ p=k+1 [lnp(γ/x)] 1−εp[ψ(x)]2, k = 1, . . . ,n, γk,k(ε) = ˆ ρ 0 dxx−1+ε0 k∏ ℓ=1 [lnℓ(γ/x)] −1−εℓ n∏ p=k+1 [lnp(γ/x)] 1−εp[ψ(x)]2, k = 1, . . . ,n, γn,n(ε) = ˆ ρ 0 dxx−1+ε0 n∏ ℓ=1 [lnℓ(γ/x)] −1−εℓ[ψ(x)]2. (1.15) for k ∈ n we shall write pk for the polynomial pk(σ) = σ(σ − 1) · · · (σ − k + 1), σ ∈ r. (1.16) for β = (β0,β1, . . . ,βn), where β0,β1, . . . ,βn ∈ r, we introduce vβ(x) = vβ0,β1,...,βn (x) =    xσ0(β0), 0 < x < ρ, n = 0, xσ0(β0) ∏n ℓ=1[lnℓ(γ/x)] −σℓ(βℓ), 0 < x < ρ, n ∈ n, (1.17) and fβ(x) = fβ0,β1,...,βn (x) = vβ(x)ψ(x), 0 < x < ρ. (1.18) if n ∈ n and ε1 = (ε1, . . . ,εn), where ε1, . . . ,εn > 0, we define hℓ,ε1 : (0,ρ) → r, ℓ ∈ n, iteratively by h1,ε 1 (x) = h1,ε1,...,εn (x) = n∑ k=1 σk(εk) k∏ j=1 [lnj(γ/x)] −1, hℓ+1,ε 1 (x) = xh′ℓ,ε1 (x), ℓ ∈ n. (1.19) note that, since γ/x > γ/ρ > en+1, one infers that [lnj(γ/x)] −1 6 1, 0 < x < ρ, j = 1, . . . ,n. (1.20) for 0 6 j 6 k 6 n and β0,β1, . . . ,βn ∈ r, we define aj,k(β) = aj,k(β0,β1, . . . ,βn) by a0,0(β) = [ pm(σ0(β0)) ]2 − a(m,α), an,n(β) = σn(βn) { pm(σ0(β0))p ′′ m(σ0(β0))[σn (βn) + 1] + [ p ′m(σ0(β0)) ]2 σn(βn ) } , aj,j(β) = σj(βj) { pm(σ0(β0))p ′′ m(σ0(β0))[σj(βj) + 1] + [ p ′m(σ0(β0)) ]2 σj(βj) } − b(m,α), 1 6 j 6 n − 1, a0,j(β) = 2σj(βj)pm(σ0(β0))p ′ m(σ0(β0)), 1 6 j 6 n, aj,k(β) = σk(βk) { pm(σ0(β0))p ′′ m(σ0(β0))[2σj(βj) + 1] + 2 [ p ′m(σ0(β0)) ]2 σj(βj) } , 1 6 j < k 6 n. (1.21) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 123 if n ∈ n, β0,β1, . . . ,βn ∈ r, and 1 6 j 6 k 6 n, then we define bj,k(β) = bj,k(β0,β1, . . . ,βn) by bj,j(β) = 1 4 [ pm(σ0(β0))p ′′ m(σ0(β0)) + [ p ′m(σ0(β0)) ]2] (βj − β 2 j ) + aj,j(β), 1 6 j 6 n, bj,k(β) = aj,k(β) − 1 4 [ pm(σ0(β0))p ′′ m(σ0(β0)) + [ p ′m(σ0(β0)) ]2] (1 − 2βj)(1 − βk), 1 6 j < k 6 n. (1.22) for the rest of this paper we shall assume that m ∈ (0,∞) is fixed and that ε0,ε1, . . . ,εn ∈ (0,m), constants denoted by cj,j ∈ n, will depend on n ∈ n∪{0}, γ,ρ ∈ (0,∞) with γ > ρen+1, m ∈ n, α ∈ r, m ∈ (0,∞), and ψ ∈ c∞(r), but will be independent of ε0,ε1, . . . ,εn ∈ (0,m). 2 preliminary results we mention again that the methods used in this section are adapted from [15, sect. 3]. lemma 2.1. let j ∈ {1, . . . ,n + 1} and β ∈ r. then, for all 0 < x < ρ, d dx [lnj(γ/x)] −β = βx−1[ln1(γ/x)] −1 · · · [lnj−1(γ/x)] −1[lnj(γ/x)] −1−β. (2.1) proof. for j = 1, (2.1) clearly holds. suppose that (2.1) holds for j ∈ {1, . . . ,n}. then d dx [lnj+1(γ/x)] −β = d dx [ln(lnj(γ/x))] −β = −β[lnj+1(γ/x)] −1−β[lnj(γ/x)] −1 d dx [lnj(γ/x)] −(−1) = −β[lnj+1(γ/x)] −1−β[lnj(γ/x)] −1(−1)x−1 j−1∏ k=1 [lnk(γ/x)] −1 = βx−1 j∏ k=1 [lnk(γ/x)] −1[lnj+1(γ/x)] −1−β. (2.2) the result now follows by induction. lemma 2.2. (i) [ pm(σ0(0)) ]2 = a(m,α). (ii) 1 4 {[ p ′m(σ0(0)) ]2 − pm(σ0(0))p ′′ m(σ0(0)) } = b(m,α). proof. since (i) is clear, we only need to prove (ii). since both sides of (ii) are continuous in α, we may assume that α ∈ r\{1,3, . . . ,2m − 1}. for σ ∈ r\{0,1, . . . ,m − 1} one gets p ′m(σ) = (σ − 1)(σ − 2) · · · (σ − m + 1) + σ(σ − 2) · · · (σ − m + 1) + · · · + σ(σ − 1) · · · (σ − m + 2) = σ−1pm(σ) + (σ − 1) −1pm(σ) + · · · + (σ − m + 1) −1pm(σ), (2.3) 124 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) hence p ′m(σ) [ pm(σ) ]−1 = m−1∑ j=0 (σ − j)−1, (2.4) thus, differentiating both sides, pm(σ)p ′′ m(σ) − [p ′ m(σ)] 2 = −[pm(σ)] 2 m−1∑ j=0 (σ − j)−2. (2.5) put σ = (2m − 1 − α)/2. then σ ∈ r\{0,1, . . .,m − 1} if and only if α ∈ r\{1,3, . . . ,2m − 1}. so, by (2.5), part (i), and (1.8), for α ∈ r\{1,3, . . .,2m − 1}, one obtains [p ′m((2m − 1 − α)/2)] 2 − pm((2m − 1 − α)/2)p ′′ m((2m − 1 − α)/2) = [pm((2m − 1 − α)/2)] 2 m−1∑ j=0 ( 2m − 1 − α 2 − j )−2 , (2.6) that is, [p ′m(σ0(0))] 2 − pm(σ0(0))p ′′ m(σ0(0)) = 4 [ pm(σ0(0)) ]2 m−1∑ j=0 (2(m − j) − 1 − α)−2 = 4a(m,α) m∑ j=1 (2j − 1 − α)−2 = 4b(m,α). (2.7) remark 2.3. let hℓ,ε 1 : (0,ρ) → r, ℓ ∈ n, be as in (1.19). for all ℓ ∈ n with ℓ > 3, there exists c1(ℓ) > 0 such that for all ε1, . . . ,εn ∈ (0,m) one has |hℓ,ε 1 (x)| 6 c1(ℓ)[ln(γ/x)] −3, 0 < x < ρ. (2.8) lemma 2.4. suppose n ∈ n. let vε = vε0,ε1,...εn : (0,ρ) → (0,∞) be defined as in (1.17). then, for τ ∈ n, v(τ)ε (x) = x σ0(ε0)−τ n∏ j=1 [lnj(γ/x)] −σj (εj) { pτ(σ0(ε0)) + p ′τ(σ0(ε0))h1,ε1(x) + (1/2)p ′′ τ (σ0(ε0))[h1,ε1(x)] 2 + (1/2)p ′′τ (σ0(ε0))h2,ε1(x) (2.9) + eτ,ε(x) } , 0 < x < ρ, where eτ,ε(x) is of the form eτ,ε(x) = eτ,ε0,ε1,...,εn (x) = q(τ)∑ j=1 pτ,j[h1,ε 1 (x)]wτ,j,1 · · · [hτ,ε 1 (x)]wτ,j,τ , 0 < x < ρ, (2.10) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 125 for some q(τ) ∈ n,wτ,j,k ∈ n ∪ {0} for all j ∈ {1, . . . ,q(τ)} and k ∈ {1, . . . ,τ}, pτ,j ∈ r for all j ∈ {1, . . . ,q(τ)}. moreover, there exists c2 = c2(τ) > 0, independent of ε0,ε1, . . . ,εn, such that ∣∣pτ,j[h1,ε 1 (x)]wτ,j,1 · · · [hτ,ε 1 (x)]wτ,j,τ ∣∣ 6 c2[ln(γ/x)]−3, 0 < x < ρ, (2.11) for all j ∈ {1, . . . ,q(τ)}. hence ∣∣eτ,ε(x) ∣∣ 6 c2q(τ)[ln(γ/x)]−3, 0 < x < ρ. (2.12) proof. we prove this result by induction on τ ∈ n. for brevity we shall write σj = σj(εj),j = 0,1, . . . ,n, in this proof. for τ = 1 we have, by lemma 2.1, v′ε(x) = x σ0−1 n∏ j=1 [lnj(γ/x)] −σj ( σ0 + h1,ε1(x) ) , 0 < x < ρ. (2.13) for τ = 2 we have v′′ε (x) = x σ0−2 n∏ j=1 [lnj(γ/x)] −σj ( σ0 − 1 + h1,ε 1 (x) )( σ0 + h1,ε 1 (x) ) + xσ0−1 n∏ j=1 [lnj(γ/x)] −σj ( x−1h2,ε 1 (x) ) = xσ0−2 n∏ j=1 [lnj(γ/x)] −σj { σ0(σ0 − 1) + (2σ0 − 1)h1,ε 1 (x) + [h1,ε 1 (x)]2 + h2,ε 1 (x) } . (2.14) for τ = 3 we have v′′′ε (x) = x σ0−3 n∏ j=1 [lnj(γ/x)] −σj ( σ0 − 2 + h1,ε 1 (x) ){ σ0(σ0 − 1) + (2σ0 − 1)h1,ε 1 (x) + [h1,ε 1 (x)]2 + h2,ε 1 (x) } + xσ0−3 n∏ j=1 [lnj(γ/x)] −σj { (2σ0 − 1)h2,ε 1 (x) + 2h1,ε 1 (x)h2,ε 1 (x) + h3,ε 1 (x) } = xσ0−3 n∏ j=1 [lnj(γ/x)] −σj { p3(σ0) + p ′ 3(σ0)h1,ε1(x) + (1/2)p ′′3 (σ0)[h1,ε1(x)] 2 + (1/2)p ′′3 (σ0)h2,ε1(x) + e3,ε(x) } , (2.15) where e3,ε(x) = [h1,ε1(x)] 3 + 3h1,ε1(x)h2,ε1(x) + h3,ε1(x), (2.16) hence the result holds for τ = 3 by remark 2.3 and (1.20). next, we assume that the lemma holds 126 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) for τ ∈ n. differentiating (2.9) yields v(τ+1)ε (x) = x σ0−τ−1 n∏ j=1 [lnj(γ/x)] −σj ( σ0 − τ + h1,ε 1 (x) ){ pτ (σ0) + p ′τ (σ0)h1,ε1(x) + (1/2)p ′′ τ (σ0)[h1,ε1(x)] 2 + (1/2)p ′′τ (σ0)h2,ε1(x) + eτ,ε(x) } + xσ0−τ−1 n∏ j=1 [lnj(γ/x)] −σj { p ′τ (σ0)h2,ε1(x) + p ′′τ (σ0)h1,ε1(x)h2,ε1(x) + (1/2)p ′′ τ (σ0)h3,ε1(x) + xe ′ τ,ε(x) } = xσ0−(τ+1) n∏ j=1 [lnj(γ/x)] −σj { pτ (σ0)(σ0 − τ) + [ pτ (σ0) + p ′τ (σ0)(σ0 − τ) ] h1,ε 1 (x) + [ (1/2)p ′′τ (σ0)(σ0 − τ) + p ′ τ(σ0) ] [h1,ε 1 (x)]2 + [ (1/2)p ′′τ (σ0)(σ0 − τ) + p ′ τ (σ0) ] h2,ε1(x) + eτ+1,ε(x) } = xσ0−(τ+1) n∏ j=1 [lnj(γ/x)] −σj { pτ+1(σ0) + p ′ τ+1(σ0)h1,ε1(x) + (1/2)p ′′τ+1(σ0)[h1,ε1(x)] 2 + (1/2)p ′′τ+1(σ0)h2,ε1(x) + eτ+1,ε(x) } , (2.17) where eτ+1,ε(x) = (1/2)p ′′ τ (σ0)[h1,ε1(x)] 3 + (3/2)p ′′τ (σ0)h1,ε1(x)h2,ε1(x) + (σ0 − τ)eτ,ε(x) + h1,ε 1 (x)eτ,ε(x) + (1/2)p ′′ τ (σ0)h3,ε1(x) + xe ′ τ,ε(x). (2.18) thus, by (1.19), eτ+1,ε(x) can be written in the form eτ+1,ε(x) = q(τ+1)∑ j=1 pτ+1,j[h1,ε 1 (x)]wτ+1,j,1 · · · [hτ+1,ε 1 (x)]wτ+1,j,τ+1 (2.19) for some q(τ + 1) ∈ n,wτ+1,j,k ∈ n ∪ {0} for j ∈ {1, . . . ,q(τ + 1)} and k ∈ {1, . . . ,τ + 1}, pτ+1,j ∈ r for j ∈ {1, . . . ,q(τ +1)}. by (2.18), (1.19), (1.20), and remark 2.3, there exists c̃2 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that, for all 0 < x < ρ, ∣∣pτ+1,j[h1,ε 1 (x)]wτ+1,j,1 · · · [hτ+1,ε 1 (x)]wτ+1,j,τ+1 ∣∣ 6 c̃2[ln(γ/x)]−3. (2.20) hence the lemma holds for τ + 1. lemma 2.5. suppose n ∈ n. let vε = vε0,ε1,...,εn : (0,ρ) → (0,∞) be defined as in (1.17), fε = fε0,ε1,...,εn : (0,ρ) → [0,∞) be defined as in (1.18), and, for 0 6 j 6 k 6 n, aj,k(ε) = aj,k(ε0,ε1, . . . ,εn) be defined as in (1.21). let g1,ε = g1(ε0,ε1, . . . ,εn) ∈ r be defined by 3 ˆ ρ 0 dxxα ∣∣f(m)ε (x) ∣∣2 = ˆ ρ 0 dxxα ∣∣v(m)ε (x) ∣∣2[ψ(x)]2 + g1,ε. (2.21) 3one notes that, since ε0 > 0, (1.10) and lemma 2.4 imply that the integrals in (2.21) are finite and hence g1,ε is well-defined. cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 127 then there exists c3 > 0, independent of ε0,ε1, . . . ,εn, such that ∣∣g1,ε ∣∣ 6 c3, (2.22) and jn−1[fε] = g1,ε + ∑ 06j6k6n aj,k(ε)γj,k(ε) + ˆ ρ 0 dxx2(σ0(ε0)−m)+α n∏ j=1 [lnj(γ/x)] −2σj (εj)g2,ε(x)[ψ(x)] 2, (2.23) where g2,ε = g2,ε0,ε1,...,εn : (0,ρ) → r satisfies ∣∣g2,ε(x) ∣∣ 6 c3[ln(γ/x)]−3, 0 < x < ρ. (2.24) proof. we shall write σj = σj(εj), j = 0,1, . . . ,n, in this proof. by lemma 2.4 we have ∣∣v(m)ε (x) ∣∣2[ψ(x)]2 = x2(σ0−m) n∏ j=1 [lnj(γ/x)] −2σj [ pm(σ0) + p ′m(σ0)h1,ε1(x) + 1 2 p ′′m(σ0)[h1,ε1(x)] 2 + 1 2 p ′′m(σ0)h2,ε1(x) + em,ε(x) ]2 [ψ(x)]2 = x2(σ0−m) n∏ j=1 [lnj(γ/x)] −2σj {[ pm(σ0) ]2 + 2pm(σ0)p ′ m(σ0)h1,ε1(x) + [ pm(σ0)p ′′ m(σ0) + [ p ′m(σ0) ]2] [h1,ε 1 (x)]2 + pm(σ0)p ′′ m(σ0)h2,ε1(x) + g2,ε(x) } [ψ(x)]2, (2.25) where, by lemma 2.4, g2,ε = g2,ε0,ε1,...,εn : (0,ρ) → r satisfies ∣∣g2,ε(x) ∣∣ 6 c4[ln(γ/x)]−3, 0 < x < ρ, (2.26) for some c4 > 0 independent of ε0,ε1, . . . ,εn ∈ (0,m). direct computation shows ˆ ρ 0 dxx2(σ0−m)+α n∏ j=1 [lnj(γ/x)] −2σj h1,ε 1 (x)[ψ(x)]2 = n∑ j=1 σjγ0,j(ε), (2.27) ˆ ρ 0 dxx2(σ0−m)+α n∏ j=1 [lnj(γ/x)] −2σj [h1,ε1(x)] 2[ψ(x)]2 = n∑ j=1 σ2j γj,j(ε) + 2 ∑ 16j 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that |g1,ε| 6 c5. thus lemma 2.5 is proved upon putting c3 = max{c4,c5}. lemma 2.6. let k ∈ {0,1, . . . ,n} and β0,β1, . . . ,βk > 0. then ˆ ρ 0 dxx−1+β0[ln1(γ/x)] −1−β1 · · · [lnk(γ/x)] −1−βk < ∞ (2.33) if and only if    β0 > 0, or β0 = 0 and β1 > 0, or β0 = β1 = 0 and β2 > 0, ... or β0 = β1 = · · · = βk−1 = 0 and βk > 0. (2.34) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 129 proof. this follows from lemma 2.1 and (1.20). lemma 2.7. let β ∈ (−∞,1). then there exists c6 = c6(β) > 0, independent of ε0 ∈ (0,m), such that ˆ ρ 0 dx−1+ε0[ln1(γ/x)] −β[ψ(x)]2 6 c6ε −1+β 0 . (2.35) proof. writing τ = ε−10 [ln(γ/ρ)] −1 > 0, and using the change of variables s = ε−10 [ln(γ/x)] −1 ( i.e., x = γe −1 ε0s ) , ds = ε−10 x −1[ln(γ/x)]−2dx ( i.e., dx = γε−10 s −2e −1 ε0s ds ) , (2.36) one obtains ˆ ρ 0 dxx−1+ε0 [ln(γ/x)]−β[ψ(x)]2 6 ˆ ρ 0 dxx−1+ε0 [ln(γ/x)]−β = γε0ε −1+β 0 ˆ τ 0 dss−2+βe −1 s 6 ( γε0 ˆ ∞ 0 dss−2+βe −1 s ) ε −1+β 0 . (2.37) lemma 2.8. suppose n > 2. let β ∈ (−∞,1) and 1 6 j 6 n − 1. then there exists c7 = c7(β) > 0, independent of εj ∈ (0,m), such that ˆ ρ 0 dxx−1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −β[ψ(x)]2 6 c7ε −1+β j . (2.38) proof. writing τ = ε−1j [lnj+1(γ/ρ)] −1 > 0, and using the change of variables s = ε−1j [lnj+1(γ/x)] −1, (2.39) so that, by lemma 2.1, ds = ε−1j x −1[ln1(γ/x)] −1 · · · [lnj(γ/x)] −1[lnj+1(γ/x)] −2dx, (2.40) one gets ˆ ρ 0 dxx−1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −β[ψ(x)]2 6 εj ˆ τ 0 ds [lnj(γ/x)] −εj [lnj+1(γ/x)] 2−β. (2.41) by (2.39) one has (εjs) −1 = ln(lnj(γ/x)) ( i.e., lnj(γ/x) = e 1 εjs ) . (2.42) hence ˆ ρ 0 dxx−1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −β[ψ(x)]2 6 ˆ τ 0 dsεje −1 s (εjs) −2+β 6 ( ˆ ∞ 0 dse −1 s s−2+β ) ε −1+β j . (2.43) 130 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) next, we need to introduce some more notation: for τ ∈ {0,1, . . . ,n − 1} and τ < j 6 k 6 n we write ( γτ (ε) ) j,k = ˆ ρ 0 dx { x−1 τ∏ ℓ=1 [lnℓ(γ/x)] −1 j∏ ℓ=τ+1 [lnℓ(γ/x)] −1−εℓ k∏ ℓ=j+1 [lnℓ(γ/x)] −εℓ × n∏ ℓ=k+1 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2 } . (2.44) by lemma 2.6, ( γτ (ε) ) j,k is well-defined for τ ∈ {0,1, . . . ,n − 1} and τ < j 6 k 6 n as the integral on the right-hand side of (2.44) is finite. lemma 2.9. (i) there exists c8 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ε0γ0,0(ε) = n∑ j=1 (1 − εj)γ0,j(ε) + g3,ε, (2.45) and for j = 1, . . . ,n, ε0γ0,j(ε) = − j∑ k=1 εkγk,j(ε) + n∑ k=j+1 (1 − εk)γj,k(ε) + g4,j,ε, (2.46) where ∣∣g3,ε ∣∣ 6 c8, ∣∣g4,j,ε ∣∣ 6 c8. (2.47) (ii) suppose n > 2. let 1 6 j 6 n − 1. then there exists c9 = c9(j) > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that εj ( γj−1(ε) ) j,j = n∑ k=j+1 (1 − εk) ( γj−1(ε) ) j,k + g5,j,ε j , (2.48) where εj = (εj, . . . ,εn), and, for j + 1 6 k 6 n, εj ( γj−1(ε) ) j,k = − k∑ ℓ=j+1 εℓ ( γj−1(ε) ) ℓ,k + n∑ ℓ=k+1 (1 − εℓ) ( γj−1(ε) ) k,ℓ + g6,j,k,ε j , (2.49) and where ∣∣g5,j,ε j ∣∣ 6 c9, ∣∣g6,j,k,ε j ∣∣ 6 c9. (2.50) (iii) there exists c10 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ε20γ0,0(ε) − 2ε0 n∑ j=1 (1 − εj)γ0,j(ε) = n∑ j=1 (εj − ε 2 j)γj,j(ε) − ∑ 16j 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣g3,ε ∣∣ 6 c8, ∣∣g4,j,ε ∣∣ 6 c8. (2.57) (ii) one has d dx ( [lnj(γ/x)] −εj n∏ k=j+1 [lnk(γ/x)] 1−εk [ψ(x)]2 ) − 2[lnj(γ/x)] −εj n∏ k=j+1 [lnk(γ/x)] 1−εkψ(x)ψ′(x) = εjx −1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj n∏ k=j+1 [lnk(γ/x)] 1−εk [ψ(x)]2 − (1 − εj+1)x −1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −εj+1 n∏ k=j+2 [lnk(γ/x)] 1−εk [ψ(x)]2 ... − (1 − εn)x −1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj n∏ k=j+1 [lnk(γ/x)] −εk [ψ(x)]2, (2.58) integrating both sides in (2.58) yields g5,j,ε j = εj ( γj−1(ε) ) j,j − n∑ k=j+1 (1 − εk) ( γj−1(ε) ) j,k . (2.59) similarly one obtains, for j + 1 6 k 6 n, d dx ( [lnj(γ/x)] −εj · · · [lnk(γ/x)] −εk [lnk+1(γ/x)] 1−εk+1 · · · [lnn(γ/x)] 1−εn [ψ(x)]2 ) − 2[lnj(γ/x)] −εj · · · [lnk(γ/x)] −εk [lnk+1(γ/x)] 1−εk+1 · · · [lnn(γ/x)] 1−εn ψ(x)ψ′(x) = εjx −1 j−1∏ ℓ=1 [lnℓ(γ/x)] −1[lnj(γ/x)] −1−εj k∏ ℓ=j+1 [lnℓ(γ/x)] −εℓ n∏ ℓ=k+1 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2+ ... + εkx −1 j−1∏ ℓ=1 [lnℓ(γ/x)] −1 k∏ ℓ=j [lnℓ(γ/x)] −1−εℓ n∏ ℓ=k+1 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2 − (1 − εk+1)x −1 × j−1∏ ℓ=1 [lnℓ(γ/x)] −1 k∏ ℓ=j [lnℓ(γ/x)] −1−εℓ[lnk+1(γ/x)] −εk+1 n∏ ℓ=k+2 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2− ... − (1 − εn)x −1 j−1∏ ℓ=1 [lnℓ(γ/x)] −1 k∏ ℓ=j [lnℓ(γ/x)] −1−εℓ n∏ ℓ=k+1 [lnℓ(γ/x)] −εℓ[ψ(x)]2, (2.60) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 133 integrating both sides in (2.60) yields g6,j,k,ε j = k∑ ℓ=j εℓ ( γj−1(ε) ) ℓ,k − n∑ ℓ=k+1 (1 − εℓ) ( γj−1(ε) ) k,ℓ . (2.61) by (1.10), there exists c9 > 0, independent of εj, . . . ,εn ∈ (0,m), such that ∣∣g5,j,ε j ∣∣ 6 c9, ∣∣g6,j,k,ε j ∣∣ 6 c9, (2.62) for 1 6 j 6 n − 1 and j + 1 6 k 6 n. (iii) by (i) we have ε20γ0,0(ε) − 2ε0 n∑ j=1 (1 − εj)γ0,j(ε) = −ε0 n∑ j=1 (1 − εj)γ0,j(ε) + ε0g3,ε = − n∑ j=1 (1 − εj) { − j∑ k=1 εkγk,j(ε) + n∑ k=j+1 (1 − εk)γj,k(ε) + g4,j,ε } + ε0g3,ε = n∑ j=1 j∑ k=1 (1 − εj)εkγk,j(ε) − n∑ j=1 n∑ k=j+1 (1 − εj)(1 − εk)γj,k(ε) + g7,ε, (2.63) where there exists c10 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣g7,ε ∣∣ 6 c10. (2.64) thus ε20γ0,0(ε) − 2ε0 n∑ j=1 (1 − εj)γ0,j(ε) = n∑ j=1 (εj − ε 2 j)γj,j(ε) + ∑ 16j 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), with the following property: given any fixed ε1, . . . ,εn ∈ (0,m), there exists a decreasing sequence {ε0,ℓ} ∞ ℓ=1 ⊆ (0,m) and l0 ∈ r such that ε0,ℓ ↓ 0 as ℓ ↑ ∞, |l0| 6 c11, and, writing fε = fε0,ℓ,ε1,...,εn as defined in (1.18), lim ℓ↑∞ jn−1[fε] = ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k + l0. (2.66) 134 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) proof. we first note that by lemma 2.7, there exists c12 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that for all ε0,ε1, . . . ,εn ∈ (0,m) we have γ0,0(ε) = ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] 1−ε1 · · · [lnn(γ/x)] 1−εn [ψ(x)]2 6 ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] 3/2 { [ln1(γ/x)] −1/2 n∏ k=2 [lnk(γ/x)] } [ψ(x)]2 6 c12ε −5/2 0 . (2.67) for j = 1, . . . ,n, by lemma 2.7, there exists c13 = c13(j) > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that for all ε0,ε1, . . . ,εn ∈ (0,m) we have γ0,j(ε) = ˆ ρ 0 dxx−1+ε0 j∏ k=1 [lnk(γ/x)] −εk n∏ k=j+1 [lnk(γ/x)] 1−εk [ψ(x)]2 6 ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] 1/2 { [ln1(γ/x)] −1/2 n∏ k=j+1 [lnk(γ/x)] } [ψ(x)]2 6 c13ε −3/2 0 . (2.68) since we are fixing ε1, . . . ,εn ∈ (0,m), for 0 6 j 6 k 6 n, we shall consider aj,k(ε) = aj,k(ε0,ε1, . . . ,εn) as functions of ε0 ∈ (0,m) only. then a0,0(ε0) = [ pm(σ0(ε0)) ]2 − a(m,α), a′0,0(ε0) = pm(σ0(ε0))p ′ m(σ0(ε0)), a′′0,0(ε0) = 1 2 { pm(σ0(ε0))p ′′ m(σ0(ε0)) + [ p ′m(σ0(ε0)) ]2 } , a (k) 0,0(ε0) = 2 −k { dk dσk ([ pm(σ) ]2) ∣∣∣∣ σ=σ0(ε0) } , k = 3, . . . ,2m. (2.69) similarly one has, for j = 1, . . . ,n, and k = 2, . . . ,2m − 1, a0,j(ε0) = 2σj(εj)pm(σ0(ε0))p ′ m(σ0(ε0)), a′0,j(ε0) = σj(εj) {[ p ′m(σ0(ε0)) ]2 + pm(σ0(ε0))p ′′ m(σ0(ε0)) } , a (k) 0,j (ε0) = 2 −(k−1)σj(εj) { dk dσk ( pm(σ)p ′ m(σ) )∣∣∣∣ σ=σ0(ε0) } . (2.70) thus, by lemma 2.2, a0,0(ε0) = a0,0(0) + a ′ 0,0(0)ε0 + 1 2 a′′0,0(0)ε 2 0 + ε 3 0 ( 2m∑ k=3 (k!)−1a (k) 0,0(0)ε k−3 0 ) = pm(σ0(0))p ′ m(σ0(0))ε0 + 1 4 { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2} ε20 + ( 2m∑ k=3 (k!)−12−k { dk dσk ([ pm(σ) ]2) ∣∣∣∣ σ=σ0(0) } εk−30 ) ε30. (2.71) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 135 put g8(ε0) = 2m∑ k=3 (k!)−12−k { dk dσk ([ pm(σ) ]2) ∣∣∣∣ σ=σ0(0) } εk−30 , (2.72) then there exists c14 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣g8(ε0) ∣∣ 6 c14, ε0 ∈ (0,m). (2.73) similarly, for j = 1, . . . ,n, a0,j(ε0) = a0,j(0) + a ′ 0,j(0)ε0 + 2m−1∑ k=2 (k!)−1a (k) 0,j (0)ε k 0 = 2σj(εj)pm(σ0(0))p ′ m(σ0(0)) + σj(εj) {[ p ′m(σ0(0)) ]2 + pm(σ0(0))p ′′ m(σ0(0)) } ε0 + ( 2m−1∑ k=2 (k!)−12−(k−1)σj(εj) { dk dσk ( pm(σ)p ′ m(σ) )∣∣∣∣ σ=σ0(0) } εk−20 ) ε20. (2.74) for j = 1, . . . ,n, put g9,j(ε0,εj) = 2m−1∑ k=2 (k!)−12−(k−1)σj(εj) { dk dσk ( pm(σ)p ′ m(σ) )∣∣∣∣ σ=σ0(0) } εk−20 , (2.75) then there exists c15 = c15(j) > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣g9,j(ε0,εj) ∣∣ 6 c15, j = 1, . . . ,n, ε0,εj ∈ (0,m). (2.76) hence, applying lemma 2.9, a0,0(ε)γ0,0(ε) + n∑ j=1 a0,j(ε)γ0,j(ε) = pm(σ0(0))p ′ m(σ0(0))ε0γ0,0(ε) + 1 4 { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2} ε20γ0,0(ε) + g8(ε0)ε 3 0γ0,0(ε) + n∑ j=1 { 2σj(εj)pm(σ0(0))p ′ m(σ0(0))γ0,j(ε) + σj(εj) ([ p ′m(σ0(0)) ]2 + pm(σ0(0))p ′′ m(σ0(0)) ) ε0γ0,j(ε) + g9,j(ε0,εj)ε 2 0γ0,j(ε) } = pm(σ0(0))p ′ m(σ0(0)) { ε0γ0,0(ε) − n∑ j=1 (1 − εj)γ0,j(ε) } + 1 4 { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2} { ε20γ0,0(ε) − 2ε0 n∑ j=1 (1 − εj)γ0,j(ε) } + g8(ε0)ε 3 0γ0,0(ε) + n∑ j=1 g9,j(ε0,εj)ε 2 0γ0,j(ε) 136 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) = pm(σ0(0))p ′ m(σ0(0))g3,ε + 1 4 { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2} × { n∑ j=1 (εj − ε 2 j)γj,j(ε) − ∑ 16j 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣g10,ε ∣∣ 6 c16, ε0,ε1, . . . ,εn ∈ (0,m). (2.79) let {ε0,ℓ} ∞ ℓ=1 be any decreasing sequence in (0,m) with limℓ↑∞ ε0,ℓ = 0. applying lemma 2.5, (2.77), and (2.78), we have, with ε0 = ε0,ℓ, jn−1[fε] = g1,ε + ˆ ρ 0 dxx−1+ε0,ℓ n∏ j=1 [lnj(γ/x)] 1−εjg2,ε(x)[ψ(x)] 2 + a0,0(ε)γ0,0(ε) + n∑ j=1 a0,j(ε)γ0,j(ε) + ∑ 16j6k6n aj,k(ε)γj,k(ε) = g1,ε + ˆ ρ 0 dxx−1+ε0,ℓ n∏ j=1 [lnj(γ/x)] 1−εjg2,ε(x)[ψ(x)] 2 + g10,ε + 1 4 { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2 }{ n∑ j=1 (εj − ε 2 j)γj,j(ε) − ∑ 16j 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣∣∣ ˆ ρ 0 dxx−1+ε0 n∏ j=1 [lnj(γ/x)] 1−εjg2,ε(x)[ψ(x)] 2 ∣∣∣∣ 6 ˆ ρ 0 dxc3x −1[ln1(γ/x)] −3/2 { [ln1(γ/x)] −1/2 n∏ j=2 [lnj(γ/x)] } [ψ(x)]2 6 c17 ˆ ρ 0 dxx−1[ln1(γ/x)] −3/2[ψ(x)]2 = c18 < ∞. (2.82) this, together with (2.22) and (2.79), implies that there exists c11 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), such that ∣∣g11,ε ∣∣ 6 c11, ε0,ε1, . . . ,εn ∈ (0,m). (2.83) by compactness of [−c11,c11], there exist a subsequence {ε0,ℓp} ∞ p=1 and l0 ∈ [−c11,c11], such that lim p↑∞ g11(ε0,ℓp,ε1, . . . ,εn) = l0. (2.84) we shall regard this subsequence as {ε0,ℓ} ∞ ℓ=1. for 1 6 j 6 k 6 n we have, by monotone convergence, lim ℓ↑∞ γj,k(ε0,ℓ,ε1, . . . ,εn) = ( γ0(ε) ) j,k (ε1, . . . ,εn). (2.85) the lemma now follows from taking the limit ℓ ↑ ∞ in (2.80) and using (2.81) and (2.83)–(2.85). lemma 2.11. suppose n > 2. then there exists a constant c19 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), with the following property: let p ∈ {1, . . . ,n − 1} and let εp+1, . . . ,εn ∈ (0,m) be fixed. then there exist lp ∈ r, with |lp| 6 c19, and a decreasing sequence {εp,ℓ} ∞ ℓ=1 ⊆ (0,m) with εp,ℓ ↓ 0 as ℓ ↑ ∞, such that lim ℓ↑∞ ∑ p6j6k6n bj,k(0, . . . ,0,εp,ℓ,εp+1, . . . ,εn) ( γp−1(ε) ) j,k = ∑ p+16j6k6n bj,k(0, . . . ,0,εp+1, . . . ,εn) ( γp(ε) ) j,k + lp. (2.86) proof. by lemma 2.2 one obtains bp,p(0, . . . ,0,εp,εp+1, . . . ,εn) = 1 4 { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2}( εp − ε 2 p ) − 1 2 (1 − εp) { pm(σ0(0))p ′′ m(σ0(0)) 1 2 (1 + εp) − [ p ′m(σ0(0)) ]2 1 2 (1 − εp) } − b(m,α) = 1 4 { pm(σ0(0))p ′′ m(σ0(0)) − [ p ′m(σ0(0)) ]2} εp = −b(m,α)εp, (2.87) 138 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) and, for j = p + 1, . . . ,n, one gets bp,j(0, . . . ,0,εp,εp+1, . . . ,εn) = σj(εj) { pm(σ0(0))p ′′ m(σ0(0))εp − [ p ′m(σ0(0)) ]2 (1 − εp) } + 1 2 σj(εj) { pm(σ0(0))p ′′ m(σ0(0)) + [ p ′m(σ0(0)) ]2} (1 − 2εp) = 1 2 σj(εj) { pm(σ0(0))p ′′ m(σ0(0)) − [ p ′m(σ0(0)) ]2} = b(m,α)(1 − εj). (2.88) thus, by lemma 2.9, ∣∣∣∣bp,p(0, . . . ,0,εp,εp+1, . . . ,εn) ( γp−1(ε) ) p,p + n∑ j=p+1 bp,j(0, . . . ,0,εp,εp+1, . . . ,εn) ( γp−1(ε) ) p,j ∣∣∣∣ = ∣∣∣∣ − b(m,α) { εp ( γp−1(ε) ) p,p − n∑ j=p+1 (1 − εj) ( γp−1(ε) ) p,j }∣∣∣∣ 6 c19, (2.89) where c19 = b(m,α) max{c9(1), . . . ,c9(n − 1)} > 0 is once again independent of ε0,ε1, . . . ,εn ∈ (0,m). hence by compactness of [−c19,c19] there exist a decreasing subsequence {εp,ℓ} ∞ ℓ=1 of {1 ℓ }∞ℓ=1 and lp ∈ [−c19,c19] such that lp = lim ℓ↑∞ bp,p(0, . . . ,0,εp,ℓ,εp+1, . . . ,εn) ( γp−1(ε) ) p,p + n∑ j=p+1 bp,j(0, . . . ,0,εp,ℓ,εp+1, . . . ,εn) ( γp−1(ε) ) p,j . (2.90) by monotone convergence lim ℓ↑∞ ∑ p+16j6k6n bj,k(0, . . . ,0,εp,ℓ,εp+1, . . . ,εn) ( γp−1(ε) ) j,k = ∑ p+16j6k6n bj,k(0, . . . ,0,εp+1, . . . ,εn) ( γp(ε) ) j,k . (2.91) the lemma now follows from (2.90), (2.91). lemma 2.12. we have lim εn ↓0 bn,n(0, . . . ,0,εn) = b(m,α). (2.92) proof. we have, by lemma 2.2 lim εn ↓0 bn,n(0, . . . ,0,εn) = lim εn ↓0 an,n(0, . . . ,0,εn) = lim εn ↓0 − 1 4 (1 − εn) { pm(σ0(0))p ′′ m(σ0(0))(1 + εn) − [ p ′m(σ0(0)) ]2 (1 − εn) } = 1 4 {[ p ′m(σ0(0)) ]2 − pm(σ0(0))p ′′ m(σ0(0)) } = b(m,α). (2.93) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 139 lemma 2.13. suppose n ∈ n. then given any η > 0, there exist ε0,ε1, . . . ,εn ∈ (0,m) such that if fε = fε0,ε1,...,εn is as defined in (1.18), one has ∣∣∣∣jn−1[fε] [ ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 ]−1 − b(m,α) ∣∣∣∣ 6 η. (2.94) proof. let c20 = max{c11,c19} > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m), where c11 and c19 are as in lemmas 2.10 and 2.11. by lemma 2.6 and monotone convergence one infers lim εn ↓0 ˆ ρ 0 dxx−1 n−1∏ j=1 [lnj(γ/x)] −1[lnn(γ/x)] −1−εn [ψ(x)]2 = ∞. (2.95) thus, we can choose εn ∈ (0,m) sufficiently small such that ˆ ρ 0 dxx−1 n−1∏ j=1 [lnj(γ/x)] −1[lnn(γ/x)] −1−εn [ψ(x)]2 > 1, (2.96) and c20 [ ˆ ρ 0 dxx−1 n−1∏ j=1 [lnj(γ/x)] −1[lnn(γ/x)] −1−εn [ψ(x)]2 ]−1 < η, (2.97) and, by lemma 2.12, ∣∣bn,n(0, . . . ,0,εn) − b(m,α) ∣∣ < η. (2.98) thus, for any rn−1 ∈ [−c20,c20], one has ∣∣∣∣ { bn,n(0, . . . ,0,εn) ( γn−1(ε) ) n,n + rn−1 }[ˆ ρ 0 dxx−1 n−1∏ j=1 [lnj(γ/x)] −1 × [lnn(γ/x)] −1−εn [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ 6 ∣∣bn,n(0, . . . ,0,εn) − b(m,α) ∣∣ + c20 ∣∣∣∣ [ ˆ ρ 0 dxx−1 n−1∏ j=1 [lnj(γ/x)] −1[lnn(γ/x)] −1−εn [ψ(x)]2 ]−1∣∣∣∣ < 2η. (2.99) suppose first that n > 2. then, by lemma 2.11, there exist ln−1 ∈ [−c19,c19] and a decreasing sequence {εn−1,ℓ} ∞ ℓ=1 ⊆ (0,m), with limℓ↑∞ εn−1,ℓ = 0, such that lim ℓ↑∞ ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,ℓ,εn) ( γn−2(ε) ) j,k = bn,n(0, . . . ,0,εn) ( γn−1(ε) ) n,n + ln−1. (2.100) 140 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) by (2.96) and monotone convergence, and replacing {εn−1,ℓ} ∞ ℓ=1 by a subsequence if necessary, one can assume that ˆ ρ 0 dx { x−1 n−2∏ j=1 [lnj(γ/x)] −1[lnn−1(γ/x)] −1−εn−1,ℓ[lnn(γ/x)] −1−εn [ψ(x)[2 } > 1, ℓ ∈ n. (2.101) combining (2.97), (2.100), (2.101), and (2.99) with rn−1 = ln−1, and using monotone convergence, there exists εn−1 ∈ (0,m) satisfying ∣∣∣∣ { ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k } × ( ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1[lnn−1(γ/x)] −1−εn−1[lnn(γ/x)] −1−εn [ψ(x)]2 )−1 − b(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k − [ bn,n(0, . . . ,0,εn) ( γn−1(ε) ) n,n + ln−1 ]}[ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 × [lnn−1(γ/x)] −1−εn−1[lnn(γ/x)] −1−εn [ψ(x)]2 ]−1∣∣∣∣ + ∣∣∣∣ [ bn,n(0, . . . ,0,εn) ( γn−1(ε) ) n,n + ln−1 ][ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 × [lnn−1(γ/x)] −1−εn−1[lnn(γ/x)] −1−εn [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ < η + 2η = 3η, (2.102) and c20 [ ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 n∏ j=n−1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < η, (2.103) as well as ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 n∏ j=n−1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1. (2.104) one notes that by (2.102), (2.103), for all rn−2 ∈ [−c20,c20], ∣∣∣∣ { ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k + rn−2 } × [ ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 n∏ j=n−1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 141 6 ∣∣∣∣ { ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k } × [ ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 n∏ j=n−1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ + c20 ( ˆ ρ 0 dxx−1 n−2∏ j=1 [lnj(γ/x)] −1 n∏ j=n−1 [lnj(γ/x)] −1−εj [ψ(x)]2 )−1 < 3η + η = 4η. (2.105) so we have chosen εn−1,εn ∈ (0,m). if n − 1 > 2, then, by lemma 2.11, there exist ln−2 ∈ [−c19,c19] and a decreasing sequence {εn−2,ℓ} ∞ ℓ=1 ⊆ (0,m) with limℓ↑∞ εn−2,ℓ = 0 such that lim ℓ↑∞ ∑ n−26j6k6n bj,k(0, . . . ,0,εn−2,ℓ,εn−1,εn) ( γn−3(ε) ) j,k = ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k + ln−2. (2.106) by (2.104) and monotone convergence, and replacing {εn−2,ℓ} ∞ ℓ=1 by a subsequence, if necessary, one can assume that ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1[lnn−2(γ/x)] −1−εn−2,ℓ × n∏ j=n−1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, ℓ ∈ n. (2.107) combining (2.103), (2.106), (2.107), and (2.105) with rn−2 = ln−2, and monotone convergence, there exists εn−2 ∈ (0,m) satisfying ∣∣∣∣ { ∑ n−26j6k6n bj,k(0, . . . ,0,εn−2,εn−1,εn) ( γn−3(ε) ) j,k } × [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ n−26j6k6n bj,k(0, . . . ,0,εn−2,εn−1,εn) ( γn−3(ε) ) j,k − [ ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k + ln−2 ]} × [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1∣∣∣∣ + ∣∣∣∣ [ ∑ n−16j6k6n bj,k(0, . . . ,0,εn−1,εn) ( γn−2(ε) ) j,k + ln−2 ] × [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ < η + 4η = 5η, (2.108) 142 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) and c20 [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < η, (2.109) as well as ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, (2.110) such that for all rn−3 ∈ [−c20,c20] one infers ∣∣∣∣ { ∑ n−26j6k6n bj,k(0, . . . ,0,εn−2,εn−1,εn) ( γn−3(ε) ) j,k + rn−3 } × [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ n−26j6k6n bj,k(0, . . . ,0,εn−2,εn−1,εn) ( γn−3(ε) ) j,k } × [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ + c20 [ ˆ ρ 0 dxx−1 n−3∏ j=1 [lnj(γ/x)] −1 n∏ j=n−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < 5η + η = 6η. (2.111) repeating the argument above n − 1 times (or if n = 1) one arrives at the following fact: there exist ε1, . . . ,εn ∈ (0,m) such that ∣∣∣∣ { ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k }[ ˆ ρ 0 dxx−1 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ 6 (2n − 1)η, (2.112) and c20 [ ˆ ρ 0 dxx−1 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < η, (2.113) as well as ˆ ρ 0 dxx−1 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, (2.114) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 143 so that for all r0 ∈ [−c20,c20] one obtains ∣∣∣∣ { ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k + r0 } × [ ˆ ρ 0 dxx−1 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k }[ ˆ ρ 0 dxx−1 n∏ j=1 [lnj(γ/x)] −1−εj × [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ + c20 [ ˆ ρ 0 dxx−1 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < (2n − 1)η + η = 2nη. (2.115) then, by lemma 2.10, there exist l0 ∈ [−c20,c20] and a decreasing sequence {ε0,ℓ} ∞ ℓ=1 ⊆ (0,m) with limℓ↑∞ ε0,ℓ = 0 such that lim ℓ↑∞ jn−1[fε0,ℓ,ε1,...,εn ] = ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k + l0. (2.116) by (2.114) and monotone convergence, and replacing {ε0,ℓ} ∞ ℓ=1 by a subsequence if necessary, we can assume that ˆ ρ 0 dxx−1+ε0,ℓ n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, ℓ ∈ n. (2.117) combining (2.112), (2.113), (2.115) with r0 = l0, (2.116), (2.117), and monotone convergence, there exists ε0 ∈ (0,m) satisfying ∣∣∣∣jn−1[fε0,ε1,...,εn ] [ ˆ ρ 0 dxx−1+ε0 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ 6 ∣∣∣∣ [ jn−1[fε0,ε1,...,εn ] − { ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k + l0 }] × [ ˆ ρ 0 dxx−1+ε0 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1∣∣∣∣ + ∣∣∣∣ { ∑ 16j6k6n bj,k(0,ε1, . . . ,εn) ( γ0(ε) ) j,k + l0 } × [ ˆ ρ 0 dxx−1+ε0 n∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − b(m,α) ∣∣∣∣ < η + 2nη = (2n + 1)η. (2.118) lemma 2.14. suppose n = 0 and let fε0 be as defined on (1.18). then lim ε0↓0 ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 [ ˆ ρ 0 dxxα−2m|fε0(x)| 2 ]−1 = a(m,α). (2.119) 144 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) proof. by (1.10) we have lim ε0↓0 ˆ ρ 0 dxxα−2m|fε0(x)| 2 > lim ε0↓0 ˆ (0.8)ρ 0 dxx−1+ε0 = ∞. (2.120) in addition, one has f(m)ε0 (x) = m∑ j=0 ( m j ) pj(σ0(ε0))x σ0(ε0)−jψ(m−j)(x), 0 < x < ρ. (2.121) thus, for all 0 < x < ρ, xα ∣∣f(m)ε0 (x) ∣∣2 = m∑ j,k=0 ( m j )( m k ) pj(σ0(ε0))pk(σ0(ε0))x α+2σ0(ε0)−j−kψ(m−j)(x)ψ(m−k)(x) = [ pm(σ0(ε0)) ]2 x−1+ε0[ψ(x)]2 + g12(ε0,x) = a(m,α − ε0)x −1+ε0[ψ(x)]2 + g12(ε0,x), (2.122) where, again by (1.10), ∣∣g12(ε0,x) ∣∣ 6 c21, ε0 ∈ (0,m), 0 < x < ρ, (2.123) for some c21 > 0, independent of ε0,ε1, . . . ,εn ∈ (0,m). hence, ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 = a(m,α − ε0) ˆ ρ 0 dxx−1+ε0 [ψ(x)]2 + ˆ ρ 0 dxg12(ε0,x), (2.124) and the lemma follows by dividing both sides of (2.124) by ˆ ρ 0 dxxα−2m|fε0(x)| 2 = ˆ ρ 0 dxx−1+ε0 [ψ(x)]2 (2.125) and applying (2.120), (2.123). 3 the approximation procedure we start with some more notation. for the remainder of this paper we shall assume ε0,ε1, . . . ,εn ∈ (0,ρ/20), that is, we shall assume m = ρ/20. let fε = fε0,ε1,...,εn be as defined in (1.18). then for δ ∈ (0,ρ/20), we shall write, recalling ε = (ε0,ε1, . . . ,εn), f(δ),ε(x) =    0, x < δ or ρ 6 x, fε(x), δ 6 x < ρ. (3.1) we shall let h ∈ c∞(r) satisfy the following properties: (i) h is even on r, (3.2) (ii) h(x) > 0, x ∈ r, (3.3) (iii) supp(h) ⊆ (−1,1), (3.4) (iv) ˆ 1 −1 dxh(x) = 1, (3.5) (v) h is non-increasing on [0,∞). (3.6) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 145 for ε > 0 we write hε(x) = ε −1h(x/ε), x ∈ r. (3.7) for δ ∈ (0,ρ/20) and ε ∈ (0,δ/4], we write f(δ,ε),ε = f(δ),ε ∗ hε. (3.8) remark 3.1. (i) since h is even, we have f(δ,ε),ε(x) = ˆ ∞ −∞ dtε−1h(t/ε)f(δ),ε(x − t) = ˆ ∞ −∞ dtε−1h(−t/ε)f(δ),ε(x − t) = ˆ ∞ −∞ duε−1h(u/ε)f(δ),ε(x + u) = ˆ ∞ −∞ dr ε−1h((r − x)/ε)f(δ),ε(r) = ˆ x+ε x−ε drε−1h((r − x)/ε)f(δ),ε(r), x ∈ r. (3.9) (ii) since ε ∈ (0,δ/4], supp(f(δ,ε),ε) ⊆ [3δ/4,73ρ/80]. hence, f(δ,ε),ε ∈ c ∞ 0 ((0,ρ)). (3.10) (iii) let g ∈ l∞(r),x ∈ r,τ ∈ r\{0}. for 0 < ε 6 δ/4 < ρ/80, let gε = hε ∗ g. by the sequence of change of variables in (3.9), we have τ−1[gε(x + τ) − gε(x)] = ˆ ∞ −∞ dr (τε)−1{h((r − x − τ)/ε) − h((r − x)/ε)}g(r) = − ˆ ∞ −∞ dr (τε)−1h′((r − x − λ(x,r,τ)τ)/ε)(τ/ε)g(r) = −ε−2 ˆ ∞ −∞ dr h′((r − x − λ(x,r,τ)τ)/ε)g(r), (3.11) where 0 6 λ(x,r,τ) 6 1, x,r ∈ r. (3.12) since h′,g ∈ l∞(r) and, for −1 6 τ 6 1, supph′([ · − x − λ(x, · ,τ)τ]/ε) ⊆ [x − ε − 1,x + ε + 1], (3.13) applying the dominated convergence theorem we get g′ε(x) = lim τ→0 τ−1[gε(x + τ) − gε(x)] = − lim τ→0 ε−2 ˆ ∞ −∞ dr h′((r − x − λ(x,r,τ)τ)/ε)g(r) = −ε−2 lim τ→0 ˆ x+ε+1 x−ε−1 dr h′((r − x − λ(x,r,τ)τ)/ε)g(r) = −ε−2 ˆ x+ε+1 x−ε−1 dr h′((r − x)/ε)g(r) = −ε−2 ˆ x+ε x−ε dr h′((r − x)/ε)g(r). (3.14) 146 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) let δ ∈ (0,ρ/20). for technical convenience, so that we can use the general theory of convolution, we shall write f̃(δ),ε for a function in c ∞ 0 (r) satisfying: (i) f̃(δ),ε(x) = f(δ),ε, x > δ, (ii) f̃(δ),ε(x) > 0, −∞ < x < ∞. (3.15) constants denoted by νj,j ∈ n, will depend on n ∈ n ∪ {0}, γ,ρ ∈ (0,∞) with γ > ρen+1, m ∈ n, α ∈ r, h,ψ ∈ c∞(r), and ε0,ε1, . . . ,εn ∈ (0,ρ/20), but are independent of δ ∈ (0,ρ/20) and ε ∈ (0,δ/4). ⋄ lemma 3.2. for all k ∈ n ∪ {0} there exists ν1 = ν1(k) > 0 such that ∣∣f(k)ε (x) ∣∣ 6 ν1x[2(m−k)−1−α+(ε0/2)]/2, 0 < x < ρ. (3.16) proof. this lemma follows from lemma 2.4, the product rule f(k)ε (x) = k∑ j=0 ( k j ) v(k−j)ε (x)ψ (j)(x), 0 < x < ρ, (3.17) and that, for all β > 0, the function t 7→ t−βln(t) is bounded on (1,∞). lemma 3.3. for j = 1, . . . ,m, and x ∈ [3δ/4,5δ/4], we have, writing θ = δ/4, f (j) (δ,θ),ε (x) = j∑ k=1 (−1)k+1θ−kh(k−1)((δ −x)/θ)f (j−k) (δ),ε (δ) +θ−1 ˆ x+θ δ drh((r −x)/θ)f (j) (δ),ε (r). (3.18) proof. for 3δ/4 6 x 6 5δ/4 we have, by (3.14) f′(δ,θ),ε(x) = −θ −2 ˆ x+θ x−θ dr h′((r − x)/θ)f(δ),ε(r) = −θ−2 ˆ x+θ δ dr h′((r − x)/θ)f(δ),ε(r) = −θ−1 ˆ x+θ δ dr d dr [h((r − x)/θ)]f(δ),ε(r) = −θ−1 { h((r − x)/θ)f(δ),ε(r) ∣∣∣∣ x+θ δ − ˆ x+θ δ dr h((r − x)/θ)f′(δ),ε(r) } = −θ−1 { − h((δ − x)/θ)f(δ),ε(δ) − ˆ x+θ δ drh((r − x)/θ)f′(δ),ε(r) } = θ−1h((δ − x)/θ)f(δ),ε(δ) + θ −1 ˆ x+θ δ drh((r − x)/θ)f′(δ),ε(r). (3.19) suppose j ∈ {1, . . . ,m − 1} and that for all x ∈ [3δ/4,5δ/4] one has f (j) (δ,θ),ε (x) = j∑ k=1 (−1)k+1θ−kh(k−1)((δ −x)/θ)f (j−k) (δ),ε (δ) +θ−1 ˆ x+θ δ drh((r −x)/θ)f (j) (δ),ε (r), (3.20) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 147 then, by (3.14), one concludes f (j+1) (δ,θ),ε (x) = j∑ k=1 (−1)k+1θ−k(−1/θ)h(k)((δ − x)/θ)f (j−k) (δ),ε (δ) + d dx ( θ−1 ˆ x+θ x−θ drh((r − x)/θ)f (j) (δ),ε (r) ) = j∑ k=1 (−1)kθ−(k+1)h(k)((δ − x)/θ)f (j−k) (δ),ε (δ) − 1 θ2 ˆ x+θ x−θ dr h′((r − x)/θ)f (j) (δ),ε (r) = j+1∑ k=2 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) − 1 θ ˆ x+θ δ dr ( d dr [h((r − x)/θ)] ) f (j) (δ),ε (r) = j+1∑ k=2 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) − 1 θ { h((r − x)/θ)f (j) (δ),ε (r) ∣∣∣∣ x+θ δ − ˆ x+θ δ drh((r − x)/θ)f (j+1) (δ),ε (r) } = j+1∑ k=2 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) + 1 θ h((δ − x)/θ)f (j) (δ),ε (δ) + 1 θ ˆ x+θ δ dr h((r − x)/θ)f (j+1) (δ),ε (r) = j+1∑ k=1 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) + 1 θ ˆ x+θ δ dr h((r − x)/θ)f (j+1) (δ),ε (r). (3.21) hence, lemma 3.3 follows by induction. corollary 3.4. there exists ν2 > 0 such that for all δ ∈ (0,ρ/20), ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 ν2x[−1−α+(ε0/2)]/2, 3δ/4 6 x 6 5δ/4. (3.22) proof. let km = sup {∣∣h(k)(t) ∣∣ ∣∣ − 1 6 t 6 1, k = 0,1, . . . ,m } . (3.23) by lemmas 3.2 and 3.3 we have for x ∈ [3δ/4,5δ/4], ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 m∑ k=1 4kδ−kkmν1(m − k)δ [2k−1−α+(ε0/2)]/2 + 4δ−1(6δ 4 − δ)km sup {∣∣f(m) (δ),ε (r) ∣∣ ∣∣δ 6 r 6 6δ/4 } = m∑ k=1 4kkmν1(m − k)δ [−1−α+(ε0/2)]/2 + 2kmν1(m) sup { r[−1−α+(ε0/2)]/2 ∣∣δ 6 r 6 6δ/4 } 148 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) 6 km ( m∑ k=1 4kν1(m − k) ){ (4/3)[−1−α+(ε0/2)]/2 + (4/5)[−1−α+(ε0/2)]/2 } × x[−1−α+(ε0/2)]/2 + 2kmν1(m) { (4/5)[−1−α+(ε0/2)]/2 + 2[−1−α+(ε0/2)]/2 } x[−1−α+(ε0/2)]/2 = ν2x [−1−α+(ε0/2)]/2, (3.24) where ν2 = km ( m∑ k=1 4kν1(m − k) ){ (4/3)[−1−α+(ε0/2)]/2 + (4/5)[−1−α+(ε0/2)]/2 } + 2kmν1(m) { (4/5)[−1−α+(ε0/2)]/2 + 2[−1−α+(ε0/2)]/2 } . (3.25) lemma 3.5. there exists ν3 > 0 such that for all δ ∈ (0,ρ/20) we have ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 ν3x[−1−α+(ε0/2)]/2, 5δ/4 6 x 6 ρ. (3.26) proof. we first note that, for 5δ/4 6 x 6 73ρ/80, f(δ,δ/4),ε(x) = ˆ x+δ/4 x−δ/4 dr (4/δ)h(4(r − x)/δ)f(δ),ε(r) = ˆ x+δ/4 x−δ/4 dr (4/δ)h(4(r − x)/δ)f̃(δ),ε(r) = ( hδ/4 ∗ f̃(δ),ε ) (x), (3.27) hence f (m) (δ,δ/4),ε (x) = ( hδ/4 ∗ f̃ (m) (δ),ε ) (x) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f̃ (m) (δ),ε (r) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f (m) (δ),ε (r), (3.28) therefore, by lemma 3.2, ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 sup {∣∣f(m) (δ),ε (r) ∣∣ ∣∣x − (δ/4) 6 r 6 x + (δ/4) } 6 ν1(m) sup { r[−1−α+(ε0/2)]/2 ∣∣x − (δ/4) 6 r 6 x + (δ/4) } 6 ν1(m) sup { r[−1−α+(ε0/2)]/2 ∣∣3x/4 6 r 6 5x/4 } 6 ν1(m) { (3/4)[−1−α+(ε0/2)]/2 + (5/4)[−1−α+(ε0/2)]/2 } x[−1−α+(ε0/2)]/2. (3.29) by remark 3.1 (ii), supp(f(δ,δ/4),ε) ⊆ [3δ/4,73ρ/80]. so (3.26) holds for x ∈ [73ρ/80,ρ], completing the proof. cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 149 lemma 3.6. on any compact interval [a,b] ⊆ (0,ρ], f (m) (δ,δ/4),ε converges to f (m) ε uniformly as δ ↓ 0. proof. choose δ0 ∈ (0,ρ/20) such that 0 < 5δ0/4 < a. then for all 0 < δ < δ0 and x ∈ [a,b], f(δ,δ/4),ε(x) = 4δ −1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f(δ),ε(r) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f(δ0),ε(r) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f̃(δ0),ε(r) = ( hδ/4 ∗ f̃(δ0),ε ) (x). (3.30) since f̃(δ0),ε ∈ c ∞ 0 (r), f (m) (δ,δ/4),ε (x) = ( hδ/4 ∗ f̃ (m) (δ0),ε ) (x), x ∈ [a,b], −→ δ↓0 f̃ (m) (δ0),ε (x) uniformly for x ∈ [a,b], = f(m)ε (x). (3.31) corollary 3.7. we have lim δ↓0 ˆ ρ 0 dxxα ∣∣f(m) (δ,δ/4),ε (x) ∣∣2 = ˆ ρ 0 dxxα ∣∣f(m)ε (x) ∣∣2. (3.32) proof. let ν4 = max{ν2,ν3} > 0. then by corollary 3.4 and lemma 3.5, we have, for all δ ∈ (0,ρ/20), xα ∣∣f(m) (δ,δ/4),ε (x) ∣∣2 6 ν24x −1+(ε0/2), 0 < x < ρ. (3.33) by lemma 3.6 we have lim δ↓0 xα ∣∣f(m) (δ,δ/4),ε (x) ∣∣2 = xα ∣∣f(m)ε (x) ∣∣2, 0 < x < ρ. (3.34) since x 7→ ν4x −1+(ε0/2) is integrable on (0,ρ), the corollary now follows by dominated convergence. lemma 3.8. there exists ν5 > 0 such that for all δ ∈ (0,ρ/20) we have |f(δ,δ/4),ε(x)| 6 ν5x [2m−1−α+(ε0/2)]/2, 3δ/4 6 x 6 5δ/4. (3.35) 150 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) proof. for 3δ/4 6 x 6 5δ/4 we have |f(δ,δ/4),ε(x)| = ∣∣∣∣4δ −1 ˆ x+δ/4 δ drh(4(r − x)/δ)f(δ),ε(r) ∣∣∣∣ 6 sup{|f(δ),ε(r)| |δ 6 r 6 6δ/4} = sup{|fε(r)| |δ 6 r 6 3δ/2} 6 ν1(0) sup { r[2m−1−α+(ε0/2)]/2 ∣∣δ 6 r 6 3δ/2 } 6 ν1(0) { (4/5)[2m−1−α+(ε0/2)]/2 + 2[2m−1−α+(ε0/2)]/2 } x[2m−1−α+(ε0/2)]/2. (3.36) lemma 3.9. there exists ν6 > 0 such that for all δ ∈ (0,ρ/20) we have |f(δ,δ/4),ε(x)| 6 ν6x [2m−1−α+(ε0/2)]/2, 5δ/4 6 x < ρ. (3.37) proof. for x ∈ [5δ/4,ρ) we have |f(δ,δ/4),ε(x)| = ∣∣∣∣4δ −1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f(δ),ε(r) ∣∣∣∣ 6 sup{|f(δ),ε(r)| |x − δ/4 6 r 6 x + δ/4} 6 ν1(0) sup { r[2m−1−α+(ε0/2)]/2 ∣∣3x/4 6 r 6 5x/4 } 6 ν1(0) { (3/4)[2m−1−α+(ε0/2)]/2 + (5/4)[2m−1−α+(ε0/2)]/2 } x[2m−1−α+(ε0/2)]/2. (3.38) lemma 3.10. on any compact interval [a,b] ⊆ (0,ρ], f(δ,δ/4),ε converges to fε uniformly as δ ↓ 0. proof. choose δ0 ∈ (0,ρ/20) with 0 < 5δ0/4 < a. by (3.30), for all 0 < δ < δ0, we have f(δ,δ/4),ε(x) = ( hδ/4 ∗ f̃(δ0),ε ) (x), a 6 x 6 b. (3.39) since f̃(δ0),ε ∈ c ∞ 0 (r), we have f(δ,δ/4),ε(x) = ( hδ/4 ∗ f̃(δ0),ε ) (x) −→ δ↓0 f̃(δ0),ε(x) uniformly for x ∈ [a,b] = fε(x). (3.40) corollary 3.11. for k ∈ {0,1, . . . ,n} we have lim δ↓0 ˆ ρ 0 dxxα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣f(δ,δ/4),ε(x) ∣∣2 = ˆ ρ 0 dxxα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2. (3.41) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 151 proof. let ν7 = max{ν5,ν6} > 0. by lemmas 3.8 and 3.9 we have, for all δ ∈ (0,ρ/20) and x ∈ (0,ρ), xα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣f(δ,δ/4),ε(x) ∣∣2 6 ν27x −1+(ε0/2) k∏ j=1 [lnj(γ/x)] −2. (3.42) by lemma 3.10 we have for x ∈ (0,ρ), lim δ↓0 xα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣f(δ,δ/4),ε(x) ∣∣2 = xα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2. (3.43) since x 7→ x−1+(ε0/2) ( ∏k j=1[lnj(γ/x)] −2 ) is integrable on (0,ρ), the corollary now follows by dominated convergence. corollary 3.12. suppose n ∈ n. then there exists a family {gδ,ε}δ∈(0,(0.05)ρ) ⊆ c ∞ 0 ((0,ρ)) such that lim δ↓0 jn−1[gδ,ε] ( ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣gδ,ε(x) ∣∣2 )−1 = jn−1[fε] ( ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 )−1 . (3.44) proof. for δ ∈ (0,ρ/20) put gδ,ε = f(δ,δ/4),ε. then gδ,ε ∈ c ∞ 0 ((0,ρ)) by remark 3.1 (ii). the result now follows from corollaries 3.7 and 3.11. corollary 3.13. suppose n = 0. then there exists a family {gδ,ε}δ∈(0,(0.05)ρ) ⊆ c ∞ 0 ((0,ρ)) such that lim δ↓0 ˆ ρ 0 dxxα ∣∣g(m)δ,ε (x) ∣∣2 ( ˆ ρ 0 dxxα−2m ∣∣gδ,ε(x) ∣∣2 )−1 = ˆ ρ 0 dxxα ∣∣f(m)ε (x) ∣∣2 ( ˆ ρ 0 dxxα−2m ∣∣fε(x) ∣∣2 )−1 . (3.45) proof. the proof of this corollary is the same as that of corollary 3.12. 4 principal results on optimal constants in our final section we now prove optimality of the constants a(m,α) and b(m,α). starting with the interval (0,ρ), we first establish optimality of a(m,α) in (1.1). theorem 4.1. suppose that n = 0. then, given any η > 0, there exists g ∈ c∞0 ((0,ρ)) such that ∣∣∣∣ ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 ]−1 − a(m,α) ∣∣∣∣ 6 η. (4.1) in particular, the constant a(m,α) in (1.1) is sharp. 152 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) proof. given any η > 0 there exists ε0 ∈ (0,ρ/20) such that ∣∣∣∣ ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣fε0(x) ∣∣2 ]−1 − a(m,α) ∣∣∣∣ 6 η/2, (4.2) by lemma 2.14. with this value of ε0 ∈ (0,ρ/20), corollary 3.13 implies that there exists g ∈ c∞0 ((0,ρ)) such that ∣∣∣∣ ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 ]−1 − ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣fε0(x) ∣∣2 ]−1∣∣∣∣ 6 η/2. (4.3) theorem 4.1 now follows from (4.2), (4.3). next, we prove optimality of the n constants b(m,α) in (1.1): theorem 4.2. suppose that n ∈ n. then for any η > 0, there exists g ∈ c∞0 ((0,ρ)) such that ∣∣∣∣ [ ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 − a(m,α) ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 − b(m,α) n−1∑ k=1 ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 k∏ p=1 [lnp(γ/x)] −2 ] × [ ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣g(x) ∣∣2 ]−1 − b(m,α) ∣∣∣∣ 6 η. (4.4) in particular, successively increasing n through 1,2,3, . . . , demonstrates that the n constants b(m,α) in (1.1) are sharp. together with theorem 4.1, this theorem asserts that the n + 1 constants, a(m,α) and the n constants b(m,α), in (1.1) are sharp. proof. given any η > 0 there exist ε0,ε1, . . . ,εn ∈ (0,ρ/20) such that, writing fε = fε0,ε1,...,εn , ∣∣∣∣jn−1[fε] [ ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 ]−1 − b(m,α) ∣∣∣∣ 6 η/2, (4.5) by lemma 2.13. with these values of ε0,ε1, . . . ,εn ∈ (0,ρ/20), corollary 3.12 implies that there exists g ∈ c∞0 ((0,ρ)) such that ∣∣∣∣jn−1[g] [ ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣g(x) ∣∣2 ]−1 − jn−1[fε] [ ˆ ρ 0 dxxα−2m n∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 ]−1∣∣∣∣ 6 η/2. (4.6) theorem 4.2 now follows from (4.5), (4.6). cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 153 next we turn to analogous results for the half line (r,∞). we start with some preparations. writing qm,α(λ) = ( λ2 − (1 − α)2 4 )( λ2 − (3 − α)2 4 ) · · · ( λ2 − (2m − 1 − α)2 4 ) = m∏ j=1 ( λ2 − (2j − 1 − α)2 4 ) = 2m∑ ℓ=0 kℓ(m,α)λ ℓ, (4.7) one infers that (i) k2j−1(m,α) = 0, j = 1, . . . ,m, (4.8) (ii) k2j(m,α) = (−1) m−j|k2j(m,α)|, j = 0,1, . . . ,m, (4.9) and thus, qm,α(λ) = m∑ j=0 (−1)m−j|k2j(m,α)|λ 2j. (4.10) lemma 4.3 ([41, sect. 2 and proof of theorem 3.1 (i)]). suppose ρ̂ > en+1 and α ∈ r\{1, . . . ,2m− 1}. for g ∈ c∞0 ((ρ̂,∞)) let w = wg ∈ c ∞ 0 ((ln(ρ̂),∞)) be defined by g(et) = e[(2m−1−α)/2]tw(t), t ∈ (ln(ρ̂),∞). (4.11) then for all g ∈ c∞0 ((ρ̂,∞)), ˆ ∞ ρ̂ dy yα ∣∣g(m)(y) ∣∣2 = ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,α)| ∣∣w(j)(t) ∣∣2, ˆ ∞ ρ̂ dy yα−2m|g(y)|2 = ˆ ∞ ln(ρ̂) dt |w(t)|2, (4.12) and, if n ∈ n, one also has, for k = 1, . . . ,n, ( et )α−2m∣∣g(et) ∣∣2 k∏ p=1 [lnp(e t)]−2 = e−t|w(t)|2t−2 k−1∏ p=1 [lnp(t)] −2, t ∈ (ln(ρ̂),∞). (4.13) hence, if n ∈ n, [ ˆ ∞ ρ̂ dy yα ∣∣g(m)(y) ∣∣2 − a(m,α) ˆ ∞ ρ̂ dy yα−2m|g(y)|2 − b(m,α) ˆ ∞ ρ̂ dy yα−2m|g(y)|2 n−1∑ k=1 k∏ p=1 [lnp(y)] −2 ] × [ ˆ ∞ ρ̂ dy yα−2m|g(y)|2 n∏ p=1 [lnp(y)] −2 ]−1 154 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) = [ ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,α)| ∣∣w(j)(t) ∣∣2 − a(m,α) ˆ ∞ ln(ρ̂) dt |w(t)|2 − b(m,α) ˆ ∞ ln(ρ̂) dt |w(t)|2t−2 n−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 ] × [ ˆ ∞ ln(ρ̂) dt |w(t)|2t−2 n−1∏ p=1 [lnp(t)] −2 ]−1 , g ∈ c∞0 ((ρ̂,∞)). (4.14) corollary 4.4. lemma 4.3 holds for all α ∈ r, that is, it holds without the restriction α ∈ r\{1, . . . ,2m − 1}. proof. we first note that by (4.7), for ℓ = 0,1, . . . ,2m, kℓ(m,α) is a polynomial in α and so it is continuous in α. for g ∈ c∞0 ((ρ̂,∞)), to emphasize that the definition of w = wg ∈ c ∞ 0 ((ln(ρ̂),∞)) in (4.11) depends also on α, we shall write, for all α ∈ r, wα(t) = e −[(2m−1−α)/2]tg(et), t ∈ (ln(ρ̂),∞). (4.15) then, for j = 0,1, . . . ,m, one gets w(j)α (t) = j∑ k=0 s(j,k,α,t)g(k)(et), t ∈ (ln(ρ̂),∞), (4.16) where, for j ∈ {0,1, . . . ,m}, k ∈ {0,1, . . . ,j}, and t ∈ (ln(ρ̂),∞), α 7→ s(j,k,α,t) is continuous in α. we also note that, for g ∈ c∞0 ((ρ̂,∞)), supp(wα) = { t ∈ (ln(ρ̂),∞) |et ∈ supp(g) } (4.17) is independent of α ∈ r. now let α ∈ {1, . . . ,2m − 1}. then, by dominated convergence, for g ∈ c∞0 ((ρ̂,∞)), lim β→α ˆ ∞ ρ̂ dy yβ ∣∣g(m)(y) ∣∣2 = ˆ ∞ ρ̂ dy yα ∣∣g(m)(y) ∣∣2, lim β→α ˆ ∞ ρ̂ dy yβ−2m|g(y)|2 = ˆ ∞ ρ̂ dy yα−2m|g(y)|2, (4.18) and, if n ∈ n, one obtains lim β→α ˆ ∞ ρ̂ dy yβ−2m|g(y)|2 n−1∑ k=1 k∏ p=1 [lnp(y)] −2 = ˆ ∞ ρ̂ dy yα−2m|g(y)|2 n−1∑ k=1 k∏ p=1 [lnp(y)] −2, lim β→α ˆ ∞ ρ̂ dy yβ−2m|g(y)|2 n∏ p=1 [lnp(y)] −2 = ˆ ∞ ρ̂ dy yα−2m|g(y)|2 n∏ p=1 [lnp(y)] −2. (4.19) similarly, for g ∈ c∞0 ((ρ̂,∞)), lim β→α ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,β)| ∣∣w(j)β (t) ∣∣2 = ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,α)| ∣∣w(j)α (t) ∣∣2, lim β→α a(m,β) ˆ ∞ ln(ρ̂) dy |wβ(t)| 2 = a(m,α) ˆ ∞ ln(ρ̂) dy |wα(t)| 2, (4.20) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 155 and, if n ∈ n, one has lim β→α b(m,β) ˆ ∞ ln(ρ̂) dt |wβ(t)| 2t−2 n−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 = b(m,α) ˆ ∞ ln(ρ̂) dt |wα(t)| 2t−2 n−1∑ k=1 k−1∏ p=1 [lnp(t)] −2, (4.21) lim β→α ˆ ∞ ln(ρ̂) dt |wβ(t)| 2t−2 n−1∏ p=1 [lnp(t)] −2 = ˆ ∞ ln(ρ̂) dt |wα(t)| 2t−2 n−1∏ p=1 [lnp(t)] −2. (4.22) the corollary now follows from (4.18)–(4.22) and lemma 4.3. lemma 4.5 ([41, sect. 2 and proof of theorem 3.1 (iii)]). suppose 1/ρ̃ > en+1 and α ∈ r\{1, . . . ,2m − 1}. for g ∈ c∞0 ((0, ρ̃)) let u = ug ∈ c ∞ 0 ((ln(1/ρ̃),∞)) be defined by g(e−t) = e−[(2m−1−α)/2]tu(t), t ∈ (ln(1/ρ̃),∞). (4.23) then, for all g ∈ c∞0 ((0, ρ̃)), ˆ ρ̃ 0 dy yα ∣∣g(m)(y) ∣∣2 = ˆ ∞ ln(1/ρ̃) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2, ˆ ρ̃ 0 dy yα−2m|g(y)|2 = ˆ ∞ ln(1/ρ̃) dt |u(t)|2, (4.24) and, if n ∈ n, we also have, for k = 1, . . . ,n, ( e−t )α−2m∣∣g(e−t) ∣∣2 k∏ p=1 [lnp(e t)]−2 = et|u(t)|2t−2 k−1∏ p=1 [lnp(t)] −2, t ∈ (ln(1/ρ̃),∞). (4.25) hence, if n ∈ n, [ ˆ ρ̃ 0 dy yα ∣∣g(m)(y) ∣∣2 − a(m,α) ˆ ρ̃ 0 dy yα−2m|g(y)|2 − b(m,α) ˆ ρ̃ 0 dy yα−2m|g(y)|2 n−1∑ k=1 k∏ p=1 [lnp(1/y)] −2 ] × [ ˆ ρ̃ 0 dy yα−2m|g(y)|2 n∏ p=1 [lnp(1/y)] −2 ]−1 = [ ˆ ∞ ln(1/ρ̃) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2 − a(m,α) ˆ ∞ ln(1/ρ̃) dt |u(t)|2 − b(m,α) ˆ ∞ ln(1/ρ̃) dt |u(t)|2t−2 n−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 ] × [ ˆ ∞ ln(1/ρ̃) dt |u(t)|2t−2 n−1∏ p=1 [lnp(t)] −2 ]−1 , g ∈ c∞0 ((0, ρ̃)). (4.26) 156 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) corollary 4.6. lemma 4.5 holds for all α ∈ r, that is, it holds without the restriction α ∈ r\{1, . . . ,2m − 1}. as the proof of this corollary is very similar to that of corollary 4.4 we shall omit it. at this point we are ready to establish optimality of a(m,α) on the interval (r,∞) in (1.2). theorem 4.7. suppose that n = 0. let r ∈ (1,∞). then, for any η > 0, there exists ϕ ∈ c∞0 ((r,∞)) such that ∣∣∣∣ ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 [ ˆ ∞ r dxxα−2m|ϕ(x)|2 ]−1 − a(m,α) ∣∣∣∣ 6 η. (4.27) in particular, the constant a(m,α) in (1.2) is sharp. proof. put ρ = 1/r so that 1 > ρ. applying theorem 4.1, there exists g ∈ c∞0 ((0,ρ)) such that ∣∣∣∣ ˆ ρ 0 dy yα ∣∣g(m)(y) ∣∣2 [ ˆ ρ 0 dy yα−2m|g(y)|2 ]−1 − a(m,α) ∣∣∣∣ 6 η. (4.28) by corollary 4.6, writing u(t) = e[(2m−1−α)/2]tg(e−t), t ∈ (ln(1/ρ),∞), (4.29) one obtains ∣∣∣∣ ˆ ∞ ln(1/ρ) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2 [ ˆ ∞ ln(1/ρ) dt |u(t)|2 ]−1 − a(m,α) ∣∣∣∣ 6 η. (4.30) introducing ϕ(x) = x(2m−1−α)/2u(ln(x)), x ∈ (1/ρ,∞) = (r,∞), (4.31) corollary 4.4 implies ∣∣∣∣ ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 [ ˆ ∞ r dxxα−2m|ϕ(x)|2 ]−1 − a(m,α) ∣∣∣∣ 6 η, (4.32) concluding the proof since ϕ ∈ c∞0 ((r,∞)). next, we prove optimality of the n constants b(m,α) in (1.2): theorem 4.8. suppose that n ∈ n. let r,γ ∈ (0,∞) satisfy r > γen+1. then, for any η > 0, there exists ϕ ∈ c∞0 ((r,∞)) such that ∣∣∣∣ [ ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 − a(m,α) ˆ ∞ r dxxα−2m|ϕ(x)|2 − b(m,α) n−1∑ k=1 ˆ ∞ r dxxα−2m|ϕ(x)|2 k∏ p=1 [lnp(x/γ)] −2 ] × [ ˆ ∞ r dxxα−2m|ϕ(x)|2 n∏ p=1 [lnp(x/γ)] −2 ]−1 − b(m,α) ∣∣∣∣ 6 η. (4.33) cubo 24, 1 (2022) optimality of constants in power-weighted birman inequalities 157 in particular, successively increasing n through 1,2,3 . . . , demonstrates that the n constants b(m,α) in (1.2) are sharp. together with theorem 4.7, this theorem asserts that the n + 1 constants, a(m,α) and the n constants b(m,α), in (1.2) are sharp. proof. put ρ = γ/r so that 1 > ρen+1. applying theorem 4.2 with γ = 1, there exists g ∈ c∞0 ((0,ρ)) such that ∣∣∣∣ [ ˆ ρ 0 dy yα ∣∣g(m)(y) ∣∣2 − a(m,α) ˆ ρ 0 dy yα−2m|g(y)|2 − b(m,α) ˆ ρ 0 dy yα−2m|g(y)|2 n−1∑ k=1 k∏ p=1 [lnp(1/y)] −2 ] × [ ˆ ρ 0 dy yα−2m|g(y)|2 n∏ p=1 [lnp(1/y)] −2 ]−1 − b(m,α) ∣∣∣∣ 6 η. (4.34) by corollary 4.6, writing u(t) = e[(2m−1−α)/2]tg(e−t), t ∈ (ln(1/ρ),∞), (4.35) one has ∣∣∣∣ [ ˆ ∞ ln(1/ρ) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2 − a(m,α) ˆ ∞ ln(1/ρ) dt |u(t)|2 − b(m,α) ˆ ∞ ln(1/ρ) dt |u(t)|2t−2 n−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 ] × [ ˆ ∞ ln(1/ρ) dt |u(t)|2t−2 n−1∏ p=1 [lnp(t)] −2 ]−1 − b(m,α) ∣∣∣∣ 6 η. (4.36) introducing ϕ̃(ξ) = ξ(2m−1−α)/2u(ln(ξ)), ξ ∈ (1/ρ,∞), (4.37) corollary 4.4 implies ∣∣∣∣ [ ˆ ∞ 1/ρ dξ ξα ∣∣ϕ̃(m)(ξ) ∣∣2 − a(m,α) ˆ ∞ 1/ρ dξ ξα−2m|ϕ̃(ξ)|2 − b(m,α) ˆ ∞ 1/ρ dξ ξα−2m|ϕ̃(ξ)|2 n−1∑ k=1 k∏ p=1 [lnp(ξ)] −2 ] × [ ˆ ∞ 1/ρ dξ ξα−2m|ϕ̃(ξ)|2 n∏ p=1 [lnp(ξ)] −2 ]−1 − b(m,α) ∣∣∣∣ 6 η. (4.38) putting ϕ(x) = ϕ̃(x/γ), x ∈ (γ/ρ,∞) = (r,∞), (4.39) 158 f. gesztesy, i. michael & m. m. h. pang cubo 24, 1 (2022) one infers ∣∣∣∣ [ γ2m−α−1 { ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 − a(m,α) ˆ ∞ r dxxα−2m|ϕ(x)|2 − b(m,α) ˆ ∞ r dxxα−2m|ϕ(x)|2 n−1∑ k=1 k∏ p=1 [lnp(x/γ)] −2 }] × [ γ2m−α−1 ˆ ∞ r dxxα−2m|ϕ(x)|2 n∏ p=1 [lnp(x/γ)] −2 ]−1 − b(m,α) ∣∣∣∣ 6 η, (4.40) finishing the proof since ϕ ∈ c∞0 ((r,∞)). remark 4.9. (i) theorem 4.1 (resp., theorem 4.7) extends to ρ = ∞ (resp., r = 0) upon disregarding all logarithmic terms (i.e., upon putting b(m,α) = 0), we omit the details. (ii) the sequence of logarithmically refined power-weighted birman–hardy–rellich inequalities underlying theorems 4.1, 4.2, 4.7, and 4.8, extend from c∞0 −functions to functions in appropriately weighted (homogeneous) sobolev spaces as shown in detail in [41, sect. 3]. in the course of this extension, the constants a(m,α) and the n constants b(m,α) remain the same and hence optimal. (iii) we note once more that theorems 4.1 and 4.7 were proved in [41, theorem a.1] using a different method. 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[91] d. yafaev, “sharp constants in the hardy-rellich inequalities”, j. funct. anal., vol. 168, no. 1, pp. 121–144, 1999. introduction and notations employed preliminary results the approximation procedure principal results on optimal constants cubo, a mathematical journal vol. 24, no. 03, pp. 485–500, december 2022 doi: 10.56754/0719-0646.2403.0485 einstein warped product spaces on lie groups buddhadev pal1 santosh kumar1, b pankaj kumar1 1 department of mathematics, institute of science, banaras hindu university, varanasi-221005, india pal.buddha@gmail.com thakursantoshbhu@gmail.com b pankaj.kumar14@bhu.ac.in abstract we consider a compact lie group with bi-invariant metric, coming from the killing form. in this paper, we study einstein warped product space, m = m1 ×f1 m2 for the cases, (i) m1 is a lie group (ii) m2 is a lie group and (iii) both m1 and m2 are lie groups. moreover, we obtain the conditions for an einstein warped product of lie groups to become a simple product manifold. then, we characterize the warping function for generalized robertson-walker spacetime, (m = i ×f1 g2, −dt 2 + f21 g2) whose fiber g2, being semi-simple compact lie group of dim g2 > 2, having bi-invariant metric, coming from the killing form. resumen consideramos un grupo de lie compacto con métrica biinvariante, que proviene de la forma de killing. en este art́ıculo estudiamos espacios productos alabeados de einstein, m = m1 ×f1 m2 para los casos (i) m1 es un grupo de lie (ii) m2 es un grupo de lie y (iii) ambos m1 y m2 son grupos de lie. más aún, obtenemos condiciones para que un producto alabeado de einstein de grupos de lie sea una variedad producto simple. luego, caracterizamos la función de alabeo para el espacio-tiempo generalizado de robertsonwalker, (m = i ×f1 g2, −dt 2 + f21 g2) cuya fibra g2 es un grupo de lie compacto semi-simple de dim g2 > 2 con una métrica bi-invariante, que proviene de la forma de killing. keywords and phrases: einstein space, warped product, lie group, bi-invariant metric, killing form. 2020 ams mathematics subject classification: 22e46, 53c21, 53b20. accepted: 17 november 2022 received: 04 june, 2022 ©2022 b. pal et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0485 https://orcid.org/0000-0002-1407-1016 https://orcid.org/0000-0003-0571-9631 https://orcid.org/0000-0001-5778-211x mailto:pal.buddha@gmail.com mailto:thakursantoshbhu@gmail.com mailto:pankaj.kumar14@bhu.ac.in 486 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) 1 introduction r. l. bishop and b. o’neill [3], introduced the notion of warped product space to study the examples of complete riemannian manifolds of negative sectional curvature. authors proved that the completeness of warped space is followed by the completeness of base and fiber spaces. further, the results for isometrically immersed warped product manifold into some riemannian manifold were considered in [5, 6, 7]. in [9, 19], authors studied the conditions for the warping function to become a constant by using the relation between the scalar curvatures of a warped manifold with its base and fiber spaces. the concept of the warped product has been generalized to the twisted warped product [11, 28], the doubly warped product and the multiply warped product [25, 32, 33]. a multiply warped product is a product manifold m = b × m1 × m2 × · · · × mk, equipped with the metric g = π∗(gb) + (f1 ◦ π1)2π∗2(g1) + (f1 ◦ π1) 2π∗2(g2) + · · · + (f1 ◦ π1) 2π∗2(gk), where (b, gb) and (mi, gi), i ∈ {1, . . . , k}, are pseudo-riemannian manifolds, fi are smooth functions on (mi, gi) and πi are projections from m to mi. in particular, if b = (a, b), k = 1 and gb = −dt2, then m is known as a generalized robertson-walker spacetime [1, 10, 31]. a generalized robertson-walker spacetime with a fiber of constant scalar curvature is known as a robertsonwalker spacetime. the simplest example for robertson-walker spacetime is an einstein static universe. the product manifold m = m1 × m2 with metric g = (f2 ◦ π2)2π∗1(g1) + (f1 ◦ π1)2π∗2(g2) is known as a doubly warped product space. a pseudo-riemannian manifold m with metric g is an einstein manifold provided ric = cg, where ric is a ricci curvature and c is some real constant. the einstein metric g is of much interest, both in geometry and physics. a warped product with a constant warping function is considered as simply riemannian product. in [2, p. 265], a. l. besse proposed the question, “does there exist a compact einstein warped product with non-constant warping function?”. some answers to the question were given in [16, 30]. if m is an einstein warped product space of nonpositive scalar curvature with a compact base manifold, then the warped product space is reduced to a simply riemannian product [16]. in [24, 26], authors studied einstein warped product space by using quarter and semi symmetric connections. the triviality results for einstein warped product space with non-compact base manifold were studied in [30]. in 1976, milnor investigated the curvature properties of left-invariant metrics in lie groups [20]. most of the lie groups carry the more than one left-invariant metric, because in [18], authors showed that for a non-abelian lie group with a unique left-invariant metric up to homothety, the group is either the hyperbolic space hn, or rn−3 × h3, where h3 is a heisenberg group. the heisenberg group h3 has a unique riemannian metric up to homothety, whereas it has three cubo 24, 3 (2022) einstein warped product spaces on lie groups 487 metrics in the lorentzian case [29]. classifications for four-dimensional nilpotent lie groups were considered in [4, 17]. the class of lie groups obtaining a bi-invariant metric is smaller than that of lie groups with a left-invariant metric. in [14, 15], authors study the warped product einstein metrics on spaces of constant scalar curvature and homogeneous spaces. the classifications of warped product einstein metric were studied in [13]. in [8, 22], the authors study the general helices and slant helices in three dimensional lie group equipped with a bi-invariant metric. in our paper, we discuss the few possible answers to the question “does there exist a compact einstein warped product with non-constant warping function?” for a compact einstein warped product of lie groups. we know that every compact lie group has a bi-invariant metric and bi-invariant metric is much easier to handle than the left invariant metric. that is why, we use the bi-invariant metric in our paper. now the results of the left invariant metric are still open to study. section 2, of this paper includes some of the basic results. the central part of our paper is section 3, where we prove our main results for a warped product having either base manifold or fiber manifold is a compact lie group with bi-invariant metric, coming from the killing form. we show that an einstein warped product space of nonnegative scalar curvature with a one-dimensional base manifold (riemannian manifold) and fiber being a compact lie group with bi-invariant metric, coming from the killing form does not exist. also, the characteristic of warping function in generalized robertson-walker spacetime is studied in theorem 3.9. finally, we give examples of warped products, obtained using a semi-simple compact lie group taking bi-invariant metric from the killing form. 2 preliminaries a lie group g1 is a smooth manifold with a group structure such that the multiplicative and inverse maps are smooth. to study the geometry of g1, it becomes necessary to associate a left invariant metric with it. a metric in which left multiplication behaves as an isometry is known as a left invariant metric, and for a metric in which right multiplication behaves as an isometry is known as a right invariant metric. left multiplication and right multiplication on g1, are defined as la1 : g1 7→ g1, la1x1 = a1x1 and ra1 : g1 7→ g1, ra1x1 = x1a1, for all a1, x1 ∈ g1. let g1 be the lie algebra of g1, then an adjoint representation, ad : g1 7→ g1, of a lie group g1 is a map such that ada1 : g1 7→ g1 is linear isomorphism given by ada1 = d(ra−11 ◦ la1)e1 for all a1 ∈ g1. an inner product g1 on g1 is said to be ad-invariant if g1(ada1x1, ada1y1) = g1(x1, y1), for all a1 ∈ g1 and x1, y1 ∈ g1. a metric g1, which is both left invariant and right invariant is said to be a bi-invariant metric. 488 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) the metric g1 is bi-invariant if and only if g1([s1, k1], t1) = g1(k1, [t1, s1]) = g1(s1, [k1, t1]), for all s1, k1, t1 ∈ g1. also, using the koszul formula and above equation, we obtain ∇s1k1 = 1 2 [s1, k1], ∀ s1, k1 ∈ g1. corresponding to bi-invariant metric g1 on m1dimensional lie group g1, the riemann curvature tensor r, and the ricci tensor ric, are given by r(x1, y1)z1 = 1 4 [ [x1, y1], z1 ] , ric(x1, y1) = 1 4 g1 ( [x1, ei], [y1, ei] ) , where {e1, . . . , em1}, is an orthonormal frame for g1. from [12, p. 622], we get the existence of bi-invariant metric on lie group. proposition 2.1. let g1 be a lie group with lie algebra g1 and metric g1, then g1 induces a bi-invariant metric if and only if ad(g1) is compact. in other words, every compact lie group has a bi-invariant metric. also, for a connected lie group g1, the metric g1 induce a bi-invariant metric if and only if ada1 : g1 7→ g1, is skew adjoint for all a1 ∈ g1, which means g1(ada1x1, y1) = −g1(x1, ada1y1), ∀ x1, y1 ∈ g1. definition 2.2 ([2, 23]). the killing form b : g × g 7→ r is a symmetric b(x1, y1) = b(y1, x1), ad(g1)-invariant b([x1, y1], z1) = b(x1, [y1, z1]) and bilinear form, defined by b(x1, y1) = tr(ad(x1) ◦ ad(y1)), where ad(x1) : g1 7→ g1 is a map, sending each z1 to [x1, z1], for all x1, y1, z1 ∈ g1. a killing form on a lie group g1 is nondegenerate if and only if g1 is semisimple. in case of compact semisimple lie group, the killing form is always negative definite. from [23, p. 304–306], we have cubo 24, 3 (2022) einstein warped product spaces on lie groups 489 corollary 2.3. let g1 be a semisimple compact lie group with bi-invariant metric g1, then (a.1) for nondegenerate plane spanned by s and k in g1, the sectional curvature is given by k = 1 4 ( g1([s, k], [s, k]) g1(s, s)g1(k, k) − g1(s, k)g1(s, k) ) . (a.2) if the metric g1 is induced from the killing form, then g1 is an einstein (ric1 = −14g1) and the scalar curvature (τ), is given by τ = 1 4 dim(g). it is clear from (a.1), that if g1 is a riemannian metric then k ≥ 0 and k = 0, if g1 is an abelian group. let (m1, g1) and (m2, g2) be two pseudo-riemannian manifolds of dimensions m1, m2 and f1 be a positive smooth function on m1. then for natural projections π1 : m1 × m2 → m1 and π2 : m1 × m2 → m2, the warped product (m = m1 ×f1 m2, g) is a product manifold m1 × m2 with the metric g = π∗1(g1) + (f1 ◦ π1) 2π∗2(g2), where ∗ representing the pull-back operator and f1 is a warping function on m. whereas m1 and m2 are known as the base, and the fiber of (m, g), respectively. let ric, ric1 and ric2 are ricci tensors on m, m1 and m2, respectively. then from [23, p. 211], we have proposition 2.4. let m = m1 ×f1 m2 be a warped product space, then ricci tensors on m, m1 and m2, satisfies ric = ric1 − m2 f1 hf1 + ric2 − f♯g2, (2.1) where f♯ = −f1∆f1+(m2−1)g1(grad f1, grad f1). here grad f1, hf1 and ∆f1 denote the gradient of f1, the hessian of f1 and the laplacian of f1, defined as ∆f1 = −trhf1. corollary 2.5. the warped product m = m1 ×f1 m2 is an einstein with ric = λg if and only if (a.3) ric1 = λg1 + m2 f1 hf1, (a.4) (m2, g2) is an einstein, such that ric2 = νg2, where ν = f ♯ + λf21 . 3 main results proposition 3.1. let (m2, g2) be a pseudo-riemannian manifold and (g1, g1) be a semi-simple compact lie group whose bi-invariant metric coming from the killing form. then warped product 490 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) manifold (m = g1 ×f1 m2, g), is an einstein manifold (ric = λg) if and only if (a.5) hf1 = − (1 + 4λ)f1 4m2 g1, (a.6) (m2, g2) is an einstein with ric2 = νg2, where ν = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 . proof. let (m = g1 ×f1 m2, g) be an einstein manifold (ric = λg), where (m2, g2) is a pseudoriemannian manifold and (g1, g1) is a semi-simple compact lie group taking bi-invariant metric from the killing form. then from (2.1), we have λg1 + f 2 1 λg2 = ric1 − m2 f1 hf1 + ric2 − f♯g2, (3.1) where λ is some constant and f♯ = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1). now, by restricting the argument (horizontal and vertical vectors) on g1, m2, and taking ric1 = −14g1 in (3.1), we get   λg1 = − 1 4 g1 − m2 f1 hf1, f21 λg2 = ric2 − f♯g2. (3.2) conversely, assume that (m = g1 ×f1 m2, g) be a warped product with conditions (a.5) and (a.6). then from (2.1), we get ric = λg1 + m2 f1 hf1 − m2 f1 hf1 + νg2 − f♯g2. (3.3) since ν = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 , so from (3.3), we have ric = λ(g1 + f 2 1 g2) = λg. (3.4) proposition 3.2. let (m1, g1) be a pseudo-riemannian manifold and (g2, g2) be a semi-simple compact lie group whose bi-invariant metric coming from the killing form. then warped product manifold (m = m1 ×f1 g2, g), is an einstein manifold (ric = λg) if and only if (a.7) ric1 = λg1 + m2 f1 hf1, (a.8) (m2, g2) is an einstein with ric2 = νg2, where ν = − 1 4 = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 . cubo 24, 3 (2022) einstein warped product spaces on lie groups 491 proof. since (g2, g2) is a semi-simple compact lie group taking bi-invariant metric from the killing form, so using ric2 = −14g2 in (a.6), we have ric2 = − 1 4 g2 = νg2 = (f ♯ + λf21 )g2. lemma 3.3 ([16]). let f1 be a smooth function on semi-riemannian manifold m1, then the divergence of hessian tensor satisfies div(hf1)(x1) = ric1(grad f1, x1) − d(∆f1)(x1), (3.5) for all x1 ∈ γtm1. theorem 3.4. let (g1, g1) be a semi-simple compact lie group of dimension m1 > 2 and whose bi-invariant metric coming from the killing form. if 4m2h f1 + (1 + 4λ)f1g1 = 0, where λ ∈ r, m2 ∈ n and f1 is a non constant smooth function on g1, then f1 satisfy the condition ν = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 , where ν ∈ r. proof. the trace of (a.5), provide us m2 f1 ∆f1 + (1 + 4λ)m1 4 = 0. (3.6) on differentiating (3.6), we get m2 f21 (∆f1df1 − f1d(∆f1)) = 0. (3.7) by the definition of divergence and hessian for any vector field x1 and g1-orthonormal frame {e1, . . . , em1} on g1, we have div ( 1 f1 hf1 ) (x1) = ∑ i ϵi ( dei( 1 f1 hf1) ) (ei, x1) = − 1 f21 hf1(grad f1, x1) + 1 f1 div(hf1)(x1), (3.8) where ϵi = g1(ei, ei). using the fact that ric1 = −14g1, in equation (3.5), the divergence of hessian becomes div(hf1)(x1) = − 1 4 g1(grad f1, x1) − d(∆f1)(x1). (3.9) 492 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) also, from (a.5) and hf1(grad f1, x1) = (dx1df1)(grad f1) = 1 2 d(g1(grad f1, grad f1)), we have − 1 4 g1(grad f1, x1) = m2 2f1 d(g1(grad f1, grad f1))(x1) + λdf1(x1). (3.10) in view of equations (3.9) and (3.10), the equation (3.8) becomes div ( 1 f1 hf1 ) = 1 2f21 ( (m2 − 1)d(g1(grad f1, grad f1)) + 2λ1f1df1(x1) − 2f1d(∆f1) ) . (3.11) but the divergence of (a.5), implies that div ( 1 f1 hf1 ) = 0. hence from (3.11), we get (m2 − 1)d(g1(grad f1, grad f1)) + 2λ1f1df1 − 2f1d(∆f1) = 0. (3.12) therefore from equations (3.7) and (3.12), we obtain d ( (m2 − 1)(g1(grad f1, grad f1)) + λ1f21 − f1(∆f1) ) = d(ν) = 0. (3.13) hence from equation (3.13), we can conclude that for a compact einstein manifold (m2, g2) with dimension m2 and ric2 = νg2, the construction of an einstein warped manifold m = g1 ×f1 m2 is possible. corollary 3.5. let m = g1×f1 m2 be an einstein warped product space with semi-simple compact lie group g1 of dimension m1 > 2 and whose bi-invariant metric coming from the killing form. then m reduces to a simply riemannian product. proof. rearranging the equation (3.6), we have ∆f1 = (1 + 4λ)m1 4m2 f1. (3.14) as λ is a constant, so for λ ≤ −1 4 , equation (3.14) implies that ∆f1 ≤ 0, hence f1 is constant. similarly if λ ≥ −1 4 , then ∆f1 ≥ 0. since according to the weak maximum principle, if f1 is subharmonic or superharmonic i.e. (∆f1 ≥ 0 or ∆f1 ≤ 0), then f1 is constant [27, p. 75]. hence m is a simply riemannian product. in our next result, we prove that if fiber space of warped space is also a semi-simple compact lie group of dimension m2 > 2 and inherits the bi-invariant metric from the killing form, then the only possible values for f1 are ±1. corollary 3.6. let g1 and g2 be semi-simple compact lie groups of dimensions m1, m2 > 2 and bi-invariant metric tensors coming from their respective killing forms. then m = g1 ×f1 g2 is an einstein if and only if f1 = ±1. cubo 24, 3 (2022) einstein warped product spaces on lie groups 493 proof. let m = g1 ×f1 g2 be an einstein, then from proposition 3.1, corollary 3.5 and using the fact that ric2 = −14g2, we obtain, ν = λ = − 1 4 . therefore f21 = 1. now conversely assume that f1 = ±1, then ric = ric1 + ric2 = −14(g1 + g2) = − 1 4 g, hence m = g1 × g2 is an einstein. next, we consider those warped product spaces whose base is any pseudo-riemannian manifold and fiber space is a semi-simple compact lie group of dimension m2 > 2, taking bi-invariant metric from the killing form. theorem 3.7. let m = m1 ×f1 g2 be an einstein warped product space with fiber g2 as a semi-simple compact lie group of dimension m2 > 2 and having bi-invariant metric coming from the killing form. if m has negative scalar curvature, then the warped product becomes a simply riemannian product. proof. let m = m1 ×f1 g2 be an einstein warped product space with fiber g2 as a semi-simple compact lie group of dimension m2 > 2, having bi-invariant metric is coming from the killing form. then from (a.4), we can say that −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 = − 1 4 . (3.15) since m is an einstein, therefore the trace of ric = λg, implies that τ = λ(m1 + m2), (3.16) where τ is a scalar curvature of m. now assume that p1 and p2 are maximum and minimum points of f1 on m1. therefore grad f1(p1) = grad f1(p2) = 0, ∆f1(p1) ≥ 0 and ∆f1(p2) ≤ 0. from (3.16) it is clear that τ ≤ 0, implies λ ≤ 0, therefore f1(p1) 2 ≥ f1(p2)2 =⇒ λf1(p1)2 ≤ λf1(p2)2 =⇒ λf1(p1)2 + 1 4 ≤ λf1(p2)2 + 1 4 , (3.17) where ν is some constant. since ∆f1(p2)f(p2) ≤ 0 and ∆f1(p1)f(p1) ≥ 0, therefore from (3.15), λf1(p2) 2 + 1 4 ≤ 0 and λf1(p1)2 + 14 ≥ 0, gives us λf1(p2) 2 + 1 4 ≤ λf1(p1)2 + 1 4 . (3.18) comparing equations (3.17) and (3.18), we have f1(p1) = f1(p2) for λ < 0. theorem 3.8. let m = i ×f1 g2 be an einstein warped product space with the metric g = dt2 + f21 (t)g2, where i is an open interval in r and g2 is a semi-simple compact lie group of dimension m2 > 2 and having bi-invariant metric coming from the killing form. if m has non 494 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) negative scalar curvature, then there does not exist any such f1, so that m = i ×f1 g2 is an einstein warped product space. proof. let m = i ×f1 g2 have positive scalar curvature (λ > 0). then taking f1 = e u 2 , the hessian of f1, hf1 = u′′ 2 e u 2 + (u′)2 4 e u 2 . using the above equation in (a.7), we have u′′ 2 + (u′)2 4 = − λ m2 . (3.19) also, from (a.8), we get (u′′ 2 + (u′)2 4 ) + (m2 − 1) (u′)2 4 + λ = − 1 4 e−u. (3.20) thus from (3.19) and (3.20), we obtain (u′)2 = − ( 1 m2 − 1 e−u + 4 m2 λ ) . (3.21) the possible solutions for (3.21) (with the help of maple), are   u = − ln ( − 4λ(m2−1) m2 ) , u = − ln ( − 4(m2−1) m2 λ ( 1 + tan2 ( − √ λ m2 t + c √ λ m2 )) ) , (3.22) where c is some constant. it is clear from (3.22) that the function u is not well defined. furthermore, as u is a real valued function, therefore (u′)2 ≥ 0 and − ( e−u 1 m2−1 + 4 m2 λ ) < 0, for any point on i. therefore from equation (3.21), we can conclude that there does not exist any real solution for the equation. for λ = 0, (a.7) and (a.8), imply that f′′1 = 0 and f1f ′′ 1 + (m2 − 1)(f′1)2 = − 1 4 , respectively. hence f1 = at + b =⇒ (m2 − 1)(a)2 = − 1 4 , (3.23) where a and b are some real constants. thus from (3.22) and (3.23), we can say that there does not exist any such f1 such that m = i ×f1 g2 be an einstein warped product space of non negative scalar curvature. next, we find the characteristic of warping function in generalized robertson-walker spacetime, whose fiber is semi-simple and compact lie group of dimension m2 > 2. cubo 24, 3 (2022) einstein warped product spaces on lie groups 495 theorem 3.9. let m = i ×f1 g2 be an einstein warped product space with the metric g = −dt2 + f21 (t)g2, where i is an open interval in r and g2 is a semi-simple compact lie group of dimension m2 > 2 and having bi-invariant metric coming from the killing form. then (i) if m is ricci flat, then there exists a non-constant function f1 on i such that f1 = 1 2 √ m2−1 t+ b, where b is some constant. (ii) if m has positive scalar curvature (τ > 0) or negative scalar curvature (τ < 0), then there does not exist any such f1, so that m = i ×f1 g2 be an einstein warped product space. proof. let m = i ×f1 g2 be an einstein warped product space with the metric g = −dt2 +f21 (t)g2, then from proposition 3.2, we get f′′1 = λf1 m2 , and f1f ′′ 1 − (m2 − 1)(f ′ 1) 2 + λf21 = − 1 4 . (3.24) from these two differential equations, we obtain (f′1) 2 − λ(1 + m2) m2(m2 − 1) f21 = 1 4(m2 − 1) . (3.25) as λ is constant, therefore to obtain the solutions for differential equation (3.25), we have to consider all possible values of λ. (i) if λ = 0, then from (3.25), we obtain f1 = 1 2 √ m2 − 1 t + b, (3.26) where b is some constant. since f1 is also satisfying (3.24), hence in the ricci flat manifold case, it is possible to find a non-constant function on i. (ii) (a) let m be an einstein manifold with positive scalar curvature λ > 0, then from (3.25), the possible solutions are   f1 = ± √ −m2 4λ(1+m2) , f1 = √ m2(m2−1) 2 √ λ(1+m2) ( − 1 4(m2−1) e √ λ(1+m2) m2(m2−1) (c1−t) + e √ λ(1+m2) m2(m2−1) (t−c1) ) , f1 = √ m2(m2−1) 2 √ λ(1+m2) ( − 1 4(m2−1) e √ λ(1+m2) m2(m2−1) (t−c1) + e √ λ(1+m2) m2(m2−1) (c1−t) ) , (3.27) where c1 is some constant. as m2 > 2, so f1 = ± √ −m2 4λ(1+m2) /∈ r, hence constant solution of f1 is not possible. from second and third part of (3.27), we have f′′1 = λ(1 + m2) m2(m2 − 1) f1. (3.28) 496 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) equation (3.28), showing that second and third part of (3.27), is not satisfying (3.24). hence there does not exist such type of f1 which satisfies the equation (3.24) for λ > 0. (b) let m be an einstein manifold with negative scalar curvature λ < 0, then (3.25), reduced to (f′1) 2 + a(1 + m2) m2(m2 − 1) f21 = 1 4(m2 − 1) , (3.29) where λ = −a and a is some positive real number. the solutions for differential equation (3.29), are   f1 = ± √ m2 4a(1+m2) , f1 = ± √ m2 4a(1+m2) sin (√ a(1+m2) m2(m2−1) (−t + c1) ) . (3.30) since solutions obtained in (3.30), are not satisfying the equation (3.24), hence there is no solution for (3.24). examples for warped product of lie groups the lie groups su(n), n ≥ 2, and so(n), n ≥ 3 are examples of semi-simple compact lie groups. the lie algebra su(n) of su(n), set of n × n skew hermitian matrices with zero trace. for n = 2, the elements of su(2), x1 =   a1ι a2 + a3ι −a2 + a3ι −a1ι   , a1, a2, a3 ∈ r. similarly, if y1 ∈ su(2), then y1 =   b1ι b2 + b3ι −b2 + b3ι −b1ι   , b1, b2, b3 ∈ r. the basis e1, e2 and e3, for su(2), can be chosen as e1 =   0 1 −1 0   , e2 =  0 ι ι 0   , e3 =  ι 0 0 −ι   . hence adx1 and adx2, are obtained as cubo 24, 3 (2022) einstein warped product spaces on lie groups 497 adx1 =   0 −2a1 2a3 2a1 0 −2a2 −2a3 2a2 0   , adx2 =   0 −2b1 2b3 2b1 0 −2b2 −2b3 2b2 0   . thus, the killing form b(x1, y1) on su(2), will be b(x1, y1) = tr(adx1 ◦ ady1) = −8a1b1 − 8a2b2 − 8a3b3 = 4tr(x1y1). so, we can made the following examples from all the above discussions. 1. the warped product manifold m = su(2)×f1m2, with metric g = b+f21 g2, where (m2, g2) is any pseudo-riemannian manifold and non constant function f1 on su(2), is not an einstein. 2. the product manifold m = su(2)×so(2), with metric g = b1+b2, is an einstein manifold, where b1 and b2 are killing forms on su(2) and so(2), respectively. conclusion 3.10. in [21], mustafa proved that for every compact manifold g1 there exist a metric on it such that non trivial einstein warped products with base g1 cannot be constructed. in our paper, from corollary 3.5, we can say that bi-invariant metric generated by the killing form on semi-simple compact lie group g1 is one in which we cannot construct non trivial einstein warped product with base g1. data availability statement no new data were created and analyzed in this study. acknowledgment the authors would like to express their thanks and gratitude to the referees for their valuable suggestions to improve the paper. also, santosh kumar would like to thank the ugc jrf of india for their financial support, ref. no. 1068/ (csir ugc net june 2019). 498 b. pal, s. kumar & p. kumar cubo 24, 3 (2022) references [1] l. j. aĺıas, a. romero and m. sánchez, “uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized robertson-walker spacetimes”, gen. relativity gravitation, vol. 27, no. 1, pp. 71–84, 1995. 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[33] b. ünal, “doubly warped products”, differential geom. appl., vol. 15, no. 3, pp. 253–263, 2001. introduction preliminaries main results cubo, a mathematical journal vol. 24, no. 02, pp. 273–289, august 2022 doi: 10.56754/0719-0646.2402.0273 on existence results for hybrid ψ−caputo multi-fractional differential equations with hybrid conditions fouad fredj 1 hadda hammouche 1, b 1mathematics and applied sciences laboratory, ghardaia university, ghardaia 47000, algeria. fouadfredj05@gmail.com fredj.fouad@univ-ghardaia.dz h.hammouche@yahoo.fr b abstract in this paper, we study the existence and uniqueness results of a fractional hybrid boundary value problem with multiple fractional derivatives of ψ−caputo with different orders. using a useful generalization of krasnoselskii’s fixed point theorem, we have established results of at least one solution, while the uniqueness of solution is derived by banach’s fixed point. the last section is devoted to an example that illustrates the applicability of our results. resumen en este art́ıculo, estudiamos los resultados de existencia y unicidad de un problema de valor en la frontera fraccional h́ıbrido con múltiples derivadas fraccionarias de ψ−caputo con diferentes órdenes. usando una generalización útil del teorema del punto fijo de krasnoselskii, establecemos resultados de al menos una solución, mientras que la unicidad de dicha solución se obtiene a partir del punto fijo de banach. la última sección está dedicada a un ejemplo que ilustra la aplicabilidad de nuestros resultados. keywords and phrases: ψ−fractional derivative, fractional differential equation, hybrid conditions, fixed point, existence, uniqueness. 2020 ams mathematics subject classification: 34a08, 34a12. accepted: 24 may, 2022 received: 14 october, 2021 c©2022 f. fredj et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://dx.doi.org/10.56754/0719-0646.2402.0273 https://orcid.org/0000-0001-6088-3210 mailto:h.hammouche@yahoo.fr https://orcid.org/0000-0002-5846-8365 mailto:fouadfredj05@gmail.com mailto:fredj.fouad@univ-ghardaia.dz mailto:h.hammouche@yahoo.fr 274 f. fredj & h. hammouche cubo 24, 2 (2022) 1 introduction fractional differential equations have received great attention of many researchers working in different disciplines of science and technology, especially, since they have found that certain thermal [3], electrochemical [4] and viscoelastic [16] systems are governed by fractional differential equations. recently some publications show the importance of fractional differential equations in the mathematical modeling of many real-world phenomena. for example ecological models [10], economic models [20], physics [12], fluid mechanics [21]. there are many studies on fractional differential equations with distinct kinds of fractional derivatives in the literature, such as riemann-liouville fractional derivative, caputo fractional derivative, and grunwald letnikov fractional derivative, etc. for example, see [11, 14, 15]. very recently, a new kind of fractional derivative the ψ−caputo’s derivative, was introduced by almeida in [1], the main advantage of this derivative is the freedom of choices of the kernels of the derivative by choosing different functions ψ, which gives us some well known fractional derivatives such caputo, caputo-erdelyi-koper and caputo hadamard derivative. for more details on the ψ−caputo and fractional differential equation involving ψ−caputo, we refer the reader to a series of papers [1, 2, 7] and the references cited therein. nowadays, many researchers have shown the interest of quadratic perturbations of nonlinear differential equations, these kind of differential equations are known under the name of hybrid differential equations. some recent works regarding hybrid differential equations can be found in [8, 13, 17, 23] and the references cited therein. dhage and lakshmikantham [6] discussed the existence and uniqueness theorems of the solution to the ordinary first-order hybrid differential equation with perturbation of the first type          d dt ( u(t) g(t,u(t)) ) = f(t,u(t)), a.e. t ∈ [t0, t0 + t ], u(t0) = u0, u0 ∈ r, where t0,t ∈ r with t > 0, g : [t0, t0 +t ]×r → r\{0} and f : [t0, t0 +t ]×r → r are continuous functions. by using the fixed point theorem in banach algebra, the authors obtained the existence results. in [9], dong et al., established the existence and the uniqueness of solutions for the following implicit fractional differential equation        cdpu(t) = f(t,u(t),c dpu(t)), t ∈ j := [0,t ], 0 < p ≤ 1, u(0) = u0, where cdp is the caputo fractional derivative, f : [0,t ]×r×r → r is a given continuous function. sitho et al. [17] studied existence results for the initial value problems of hybrid fractional sequencubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 275 tial integro-differential equations:                      dp       dqu(t) − m ∑ i=1 iηigi(t,u(t)) h(t,x(t))       = f(t,u(t),iγx(t)), t ∈ j, u(0) = 0, dqu(0) = 0, where dp, dq denotes the riemann-liouville fractional derivative of order p, q respectively and 0 < p,q ≤ 1, iηi is the riemann-liouville fractional integral of order ηi > 0, h ∈ c(j ×r,r\{0}), f ∈ c(j × r2,r) and gi ∈ c(j × r,r) with gi(0,0) = 0, i = 1, . . . ,m. in 2019, derbazi et al. [8] proved the existence of solutions for the fractional hybrid boundary value problem cdp [ u(t) − g(t,u(t)) h(t,u(t)) ] = f(t,u(t)), t ∈ j, with the fractional hybrid boundary value conditions                  a1 [ u(t) − g(t,u(t)) h(t,u(t)) ] t=0 + b1 [ u(t) − g(t,u(t)) h(t,u(t)) ] t=t = υ1, a2 cdδ [ u(t) − g(t,u(t)) h(t,u(t)) ] t=ξ + b2 cdδ [ u(t) − g(t,u(t)) h(t,u(t)) ] t=t = υ2 , ξ ∈ j, where 1 < p ≤ 2, 0 < δ ≤ 1, ξ ∈ j and a1,a2,b1,b2,υ1,υ2 are real constants. moreover, two fractional derivatives of caputo type appeared in the above problem. motivated by these works, we mainly investigate the existence and uniqueness of solutions for a class of hybrid differential equations of arbitrary fractional order of the form cdp;ψ       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t,u(t)) h(t,u(t))       = f       t,u(t),c dp;ψ       cdq;ψu(t) − m ∑ i=1 i ηi;ψgi(t,u(t)) h(t,u(t))             , t ∈ j, (1.1) 276 f. fredj & h. hammouche cubo 24, 2 (2022) endowed with the hybrid fractional integral boundary conditions                                                                      u(0) = 0, cdq;ψu(0) = 0, a1       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t,u(t)) h(t,u(t))       t=0 + b1       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t,u(t)) h(t,u(t))       t=t = υ1, a2 cd δ;ψ       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t,u(t)) h(t,u(t))       t=ξ + b2 cd δ;ψ       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t,u(t)) h(t,u(t))       t=t = υ2 , ξ ∈ j, (1.2) where j := [0,t ], dp;ψ, dq;ψ and dδ;ψ denote the ψ−caputo fractional derivative of order 2 0, h ∈ c(j×r,r\{0}), f ∈ c(j ×r2,r) and gi ∈ c(j×r,r) with gi(0,0) = 0, i = 1, . . . ,m, a1, a2,b1, b2, υ1, υ2 are real constants such that b1 6= 0 and 2 ( a2ψ 2−δ 0 (ξ) + b2ψ 2−δ 0 (t) ) − ψ10(t)(2 − δ) ( a2ψ 1−δ 0 (ξ) + b2ψ 1−δ 0 (t) ) 6= 0. the rest of the paper is arranged as follows. section 2 gives some background material needed in this paper, such as fractional differential equations and fixed point theorems. section 3 treats the main results concerning the existence and uniqueness results of the solution for the given problem (1.1)-(1.2) by employing hybrid fixed point theorem for a sum of two operators in banach algebra space and banach’s fixed point. in the last section, an example is presented to illustrate our results. 2 preliminaries in this section, we introduce some preliminaries and lemmas that will be used throughout this paper. we will prove an auxiliary lemma, which plays a key role in defining a fixed point problem associated with the given problem. let ψ : j → r an increasing function satisfying ψ′(t) 6= 0 for all t ∈ j. for the sake of simplicity, we set ψr0(t) = (ψ(t) − ψ(0))r. definition 2.1 ([2]). the ψ−riemann-liouville fractional integral of order (p > 0) of an integrable cubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 277 function g : [0,∞) → r is defined by ip;ψg(t) = 1 γ(p) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))p−1g(s)ds, 0 < s < t. definition 2.2 ([2]). the ψ−caputo fractional derivative of order p (n − 1 < p < n ∈ n) of a function g ∈ cn[0,∞) is defined by cdp;ψg(t) = 1 γ(p − n) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))p−n−1dnψ g(s)ds, 0 < s < t, where n = [p] + 1 and dnψ = ( 1 ψ′(t) d dt )n . in case, if 2 0. the following hold • if g ∈ c(j,r), then cdp;ψip;ψg(t) = g(t), t ∈ j. • if g ∈ cn(j,r), n − 1 < p < n, then i p;ψc d p;ψ g(t) = g(t) − n−1 ∑ k=0 ckψ k 0(t), t ∈ j, where ck = dkψg(0) k! . lemma 2.4. let 2 < p < 3, 0 < q < 1. for any functions f ∈ c(j,r), h ∈ c(j,r \ {0}) and gi ∈ c(j,r) with gi(0) = 0, i = 1, . . . ,m, the following linear fractional boundary value problem dp;ψ       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t) h(t)       = f(t), 2 < p ≤ 3, 0 < q ≤ 1, t ∈ j, (2.1) 278 f. fredj & h. hammouche cubo 24, 2 (2022) supplemented with the following conditions                                                                      u(0) = 0, cdq;ψu(0) = 0, a1       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t) h(t)       t=0 + b1       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t) h(t)       t=t = υ1, a2 cd δ;ψ       cdq;ψu(t) − m ∑ i=1 iηi;ψgi(t) h(t)       t=ξ + b2 cd δ;ψ       cdq;ψu(t) − m ∑ i=1 i ηi;ψgi(t) h(t)       t=t = υ2, ξ ∈ j, (2.2) has a unique solution, which is given by u(t) = iq;ψ ( h(s)ip;ψf(s) ) (t) + m ∑ i=1 i ηi+q;ψgi(s)(t) + iq;ψ ( h(s) ( ψ10(s)ω3 − ψ20(s)ω2 )( υ1 b1 − ip;ψf(s) ) ) (t) + ω1 ( υ2 − a2ip−δ;ψf(ξ) − b2ip−δ;ψf(t) ) iq;ψ ( h(s) ( ψ20(s) − ψ10(t)ψ10(s) ) ) (t), (2.3) where ω1 = γ(3 − δ) 2 ( a2ψ 2−δ 0 (ξ) + b2ψ 2−δ 0 (t) ) − ψ1 0 (t)(2 − δ) ( a2ψ 1−δ 0 (ξ) + b2ψ 1−δ 0 (t) ), ω2 = a2ψ 1−δ 0 (ξ) + b2ψ 1−δ 0 (t) γ(2 − δ)ω1 , ω3 = 1 + ω2ψ 1 0(t). proof. applying the ψ−caputo fractional integral of order p to both sides of equation in (2.1) and using lemma 2.3, we get cdq;ψu(t) − m ∑ i=1 i ηi;ψgi(t) h(t) = ip;ψf(t) + c0 + c1ψ 1 0(t) + c2ψ 2 0(t), (2.4) where c0,c1,c2 ∈ r . next, applying the ψ−caputo fractional integral of order q to both sides (2.4), we get cubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 279 u(t) = iq;ψ ( h(s)ip;ψf(s) ) (t) + m ∑ i=1 iηi+q;ψgi(s)(t) + iq;ψ ( h(s) ( c0 + c1ψ 1 0(s) + c2ψ 2 0(s) )) (t) + c3, c3 ∈ r. (2.5) with the help of conditions u(0) = 0 and cdq;ψu(0) = 0, we find, c3 = 0 and c0 = 0 respectively. applying the boundary conditions (2.2), and from (2.4), we obtain c1ψ 1 0(t) + c2ψ 2 0(t) = υ1 b1 − ip;ψf(t), and c1 γ(2 − δ) ( a2ψ 1−δ 0 (ξ) + b2ψ 1−δ 0 (t) ) + 2c2 γ(3 − δ) ( a2ψ 2−δ 0 (ξ) + b2ψ 2−δ 0 (t) ) = υ2 − a2ip−δ;ψf(ξ) − b2ip−δ;ψf(t). solving the resulting equations for c1 and c2, we find that c1 = ( υ1 b1 − ip;ψf(t) ) ω3 − ( υ2 − a2ip−δ;ψf(ξ) − b2ip−δ;ψf(t) ) ω1ψ 1 0(t), c2 = ( υ2 − a2ip−δ;ψf(ξ) − b2ip−δ;ψf(t) ) ω1 − ( υ1 b1 − ip;ψf(t) ) ω2. inserting c1 and c2 in (2.5), which leads to the solution system (2.3). let e = c(j,r) be the banach space of continuous real-valued functions defined on j. we define in e a norm ‖ · ‖ by ‖u‖ = sup t∈j |u(t)|, and a multiplication by (uv)(t) = u(t)v(t), ∀t ∈ j. clearly e is a banach algebra with above defined supremum norm and multiplication. lemma 2.5 ([5]). let s be a nonempty, convex, closed, and bounded set such that s ⊆ e, and let a : e → e and b : s → e be two operators which satisfy the following: (1) a is contraction, (2) b is completely continuous, and (3) u = au + bv, for all v ∈ s ⇒ u ∈ s. then the operator equation u = au + bu has at least one solution in s. theorem 2.6 ([18]). let s be a non-empty closed convex subset of a banach space e, then any contraction mapping a of s into itself has a unique fixed point. 280 f. fredj & h. hammouche cubo 24, 2 (2022) 3 main result in this section, we derive conditions for the existence and uniqueness of a solution for the problem (1.1)-(1.2). the following assumptions are necessary in obtaining the main results. (h1) the functions h ∈ c(j × r,r \ {0}), and f ∈ c(j × r2,r) are continuous, and there exist bounded functions l,m : j → [0,∞), such that |h(t,u(t)) − h(t,v(t))| ≤ l(t)|u(t) − v(t)|, and |f(t,u(t),v(t)) − f(t,u(t),v(t))| ≤ m(t) ( |u(t) − u(t)| + |v(t) − v(t)| ) , for t ∈ j and u,v,u,v ∈ r. (h2) there exist functions ϑ,χ,ϕi ∈ c(j, [0,∞)) such that |f(t,u(t),v(t))| ≤ ϑ(t) for each t,u ∈ j × r, |h(t,u(t))| ≤ χ(t) for each t,u ∈ j × r, |gi(t,u(t))| ≤ ϕi(t) for each t,u ∈ j × r, i = 1, . . . ,m, for t ∈ j and u ∈ r. (h3) the functions gi ∈ c(j × r,r) are continuous, and there exist bounded functions ki : j → (0,∞), such that |gi(t,u(t)) − gi(t,v(t))| ≤ ki(t)|u(t) − v(t)|. we set l∗ = supt∈j |l(t)|, m∗ = supt∈j |m(t)|, χ∗ = supt∈j |χ(t)|, ϑ∗ = supt∈j |ϑ(t)| and ϕ∗i = supt∈j |ϕi(t)|, k∗i = supt∈j |ki(t)|, i = 1,2, . . . ,m. 3.1 existence of solutions in this subsection, we prove the existence of a solution for the problem (1.1)–(1.2) by applying a generalization of krasnoselskii’s fixed point theorem. theorem 3.1. assume that hypotheses (h1)–(h2) hold and if λ = ψ p 0(t) γ(p + 1) ( χ∗m∗ 1 − m∗ + ϑ∗l∗ )( ψ q 0(t) γ(q + 1) + |ω3|ψq+10 (t) γ(q + 2) + 2|ω2|ψq+20 (t) γ(q + 3) ) + |ω1|(q + 4) ψ q+2 0 (t) γ(q + 3) ( |υ2|l∗ + |a2|ψp−δ0 (ξ) + |b2|ψ p−δ 0 (t) γ(p − δ + 1) × ( χ∗m∗ 1 − m∗ + ϑ∗l∗ )) + |υ1|l∗ |b1| ( |ω3|ψq+10 (t) γ(q + 2) + 2|ω2|ψq+20 (t) γ(q + 3) ) < 1. (3.1) then the problem (1.1)–(1.2) has at least one solution on j. cubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 281 proof. first, we choose r > 0 such that r ≥χ∗ϑ∗ ψ p+q 0 (t) γ(p + 1)γ(q + 1) + χ∗ ( |ω3|ψq+10 (t) γ(q + 2) + 2|ω2|ψq+20 (t) γ(q + 3) ) (|υ1| |b1| + ψ p 0 (t) γ(p + 1) ϑ ∗ ) + χ∗|ω1| (q + 4)ψ q+2 0 (t) γ(q + 3) ( |υ2| + ϑ∗ |a2|ψp−δ0 (ξ) + |b2|ψ p−δ 0 (t) γ(p − δ + 1) ) + n ∑ i=1 ϕ∗i ψ ηi+q 0 (t) γ(ηi + q + 1) . set br = {u ∈ e : ‖u‖ ≤ r}. clearly br is a closed, convex and bounded subset of the banach space e. let u(t) be a solution of the problem (1.1)–(1.2). define fu(t) := f       t,u(t),c dp;ψ       cdq;ψu(t) − m ∑ i=1 i ηi;ψgi(t,u(t)) h(t,u(t))             . then cdp;ψ       cdq;ψu(t) − m ∑ i=1 i ηi;ψgi(t,u(t)) h(t,u(t))       = fu(t), supplemented with the conditions (1.2), then by lemma 2.4, we get u(t) = iq;ψ ( h(s,u(s))ip;ψfu(s) ) (t) + m ∑ i=1 i ηi+q;ψgi(s,u(s))(t)+ + iq;ψ ( h(s,u(s)) ( ψ10(s)ω3 − ψ20(s)ω2 )( υ1 b1 − ip;ψfu(s) ) ) (t) + ω1 ( υ2 − a2ip−δ;ψfu(ξ) − b2ip−δ;ψfu(t) ) iq;ψ ( h(s,u(s)) ( ψ20(s) − ψ10(t)ψ10(s) ) ) (t), let us define three operators cp,cp−δ : e → e and d : e → e such that cpu(t) = 1 γ(p) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))p−1fu(s)ds, t ∈ j, cp−δu(t) = 1 γ(p − δ) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))p−δ−1fu(s)ds, t ∈ j, and du(t) = h(t,u(t)), t ∈ j. then, using assumptions (h1)–(h2) , we have |cpu(t) − cpv(t)| ≤ 1 γ(p) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))p−1|fu(s) − fv(s)|ds, (3.2) 282 f. fredj & h. hammouche cubo 24, 2 (2022) and |fu(t) − fv(t)| ≤ |f(t,u(t),fu(t)) − f(t,v(t),fv(t))| ≤ m(t) ( |u(t) − v(t)| + |fu(t) − fv(t)| ) ≤ m(t) 1 − m(t) ‖u(·) − v(·)‖. (3.3) by replacing (3.3) in (3.2), we obtain |cpu(t) − cpv(t)| ≤ m∗ψ p 0(t) (1 − m∗)γ(p + 1) ‖u(·) − v(·)‖, and |du(t) − dv(t)| ≤ l∗‖u(·) − v(·)‖, |cpu(t)| ≤ ψ p 0(t) γ(p + 1) ϑ∗, and |du(t)| ≤ χ∗. now we define two more operators a : e → e and b : br → e such that au(t) = iq;ψ ( du(s)cpu(s) ) (t) + iq;ψ ( du(s) ( ψ10(s)ω3 − ψ20(s)ω2 )( υ1 b1 − cpu(s) ) ) (t) + ω1 ( υ2 − a2cp−δu(ξ) − b2cp−δu(t) ) iq;ψ ( du(s) ( ψ20(s) − ψ10(t)ψ10(s) ) ) (t), and bu(t) = m ∑ i=1 iηi+q;ψgi(s,u(s))(t). we need to show that the two operators a and b satisfy all conditions of lemma 2.5. this can be achieved in the following steps. step 1. first we show that a is a contraction mapping. let u(t),v(t) ∈ br, then we have |au(t) − av(t)| ≤ iq;ψ ( ∣ ∣du(s)cpu(s) − dv(s)cpv(s) ∣ ∣ ( 1 + ∣ ∣ψ10(s)ω3 − ψ20(s)ω2 ∣ ∣ ) ) (t) + iq;ψ ( |υ1| |b1| ∣ ∣ψ10(s)ω3 − ψ20(s)ω2 ∣ ∣ ∣ ∣du(s) − dv(s) ∣ ∣ ) (t) + |ω1|iq;ψ ( ∣ ∣ψ20(s) − ψ10(t)ψ10(s) ∣ ∣ ( |υ2| ∣ ∣du(s) − dv(s) ∣ ∣ + |a2| ∣ ∣du(s)cp−δu(ξ) − dv(s)cp−δv(ξ) ∣ ∣ + |b2| ∣ ∣du(s)cp−δu(t) − dv(s)cp−δv(t) ∣ ∣ ) ) (t) cubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 283 ≤ iq;ψ ( ( ∣ ∣du(s) ∣ ∣ ∣ ∣cpu(s) − cpv(s) ∣ ∣ + ∣ ∣cpv(s)||du(s) − dv(s) ∣ ∣ ) × ( 1 + ∣ ∣ψ10(s)ω3 − ψ20(s)ω2 ∣ ∣ ) ) (t) + iq;ψ ( |υ1| |b1| ∣ ∣ψ10(s)ω3 − ψ20(s)ω2 ∣ ∣ ∣ ∣du(s) − dv(s) ∣ ∣ ) (t) + |ω1|iq;ψ ( ∣ ∣ψ20(s) − ψ10(t)ψ10(s) ∣ ∣ ( ∣ ∣du(s) − dv(s) ∣ ∣ ( |υ2| + |a2| ∣ ∣cp−δv(ξ) ∣ ∣ + |b2| ∣ ∣cp−δv(t) ∣ ∣ ) + ∣ ∣du(s) ∣ ∣ ( |a2| ∣ ∣cp−δu(ξ) − cp−δv(ξ) ∣ ∣ + |b2| ∣ ∣cp−δu(t) − cp−δv(t) ∣ ∣ ) )) (t) using the hypotheses (h1)–(h2) and taking the supremum over t, we get ‖au(·) − av(·)‖ ≤ λ‖u(·) − v(·)‖. (3.4) therefore from (3.1), we conclude that the operator a is a contraction mapping. step 2. next, we prove that the operator b satisfies condition (2) of lemma 2.5, that is, the operator b is compact and continuous on br. therefore first, we show that the operator b is continuous on br. let un(t) be a sequence of functions in br converging to a function u(t) ∈ br. then, by the lebesgue dominant convergence theorem, for all t ∈ j, we have lim n→∞ bun(t) = lim n→∞ m ∑ i=1 1 γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1gi(s,un(s))ds = m ∑ i=1 1 γ(ηi + q) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))ηi+q−1 lim n→∞ gi(s,un(s))ds = m ∑ i=1 1 γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1gi(s,u(s))ds. hence limn→∞ bun(t) = bu(t). thus b is a continuous operator on br. further, we show that the operator b is uniformly bounded on br. for any u ∈ br, we have ‖bu(·)‖ ≤ sup t∈j { m ∑ i=1 1 γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1|gi(s,u(s))|ds } ≤ m ∑ i=1 ψ ηi+q 0 (t) γ(ηi + q + 1) ϕ∗i ≤ r. therefore bu(t) ≤ r, for all t ∈ j, which shows that b is uniformly bounded on br. now, we show that the operator b is equi-continuous. let t1, t2 ∈ j with t1 > t2. then for any 284 f. fredj & h. hammouche cubo 24, 2 (2022) u(t) ∈ br, we have |bu(t1) − bu(t2)| ≤ m ∑ i=1 1 γ(ηi + q) ∣ ∣ ∣ ∣ ∫ t2 0 ψ′(s) ( (ψ(t1) − ψ(s))ηi+q−1 − (ψ(t2) − ψ(s))ηi+q−1 ) gi(s,u(s))ds ∣ ∣ ∣ ∣ + m ∑ i=1 1 γ(ηi + q) ∣ ∣ ∣ ∣ ∫ t1 t2 ψ ′(s)(ψ(t1) − ψ(s))ηi+q−1gi(s,u(s))ds ∣ ∣ ∣ ∣ ≤ m ∑ i=1 ϕ∗i γ(ηi + q + 1) ( 2|ψ(t1) − ψ(t2)|ηi+q + ∣ ∣ψ ηi+q 0 (t2) − ψ ηi+q 0 (t1) ∣ ∣ ) . as t2 → t1, so the right-hand side tends to zero. thus b is equi-continuous. therefore, it follows from the arzelá–ascoli theorem that b is a compact operator on br. we conclude that b is completely continuous. step 3. it remains to verify the condition (3) of lemma 2.5. for any v ∈ br, we have ‖u(·)‖ = ‖au(·) + bv(·)‖ ≤ ‖au(·)‖ + ‖bv(·)‖ ≤ sup t∈j {∣ ∣ ∣ ∣ iq;ψ ( du(s)cpu(s) ) (t) + iq;ψ ( du(s) ( ψ10(s)ω3 − ψ20(s)ω2 )( υ1 b1 − cpu(s) ) ) (t) + ω1 ( υ2 − a2cp−δu(ξ) − b2cp−δu(t) ) iq;ψ ( du(s) ( ψ20(s) − ψ10(t)ψ10(s) ) ) (t) ∣ ∣ ∣ ∣ } + sup t∈j { m ∑ i=1 iηi+q;ψ ∣ ∣gi(s,v(s)) ∣ ∣(t) } ≤ χ∗ϑ∗ ψ p+q 0 (t) γ(p + 1)γ(q + 1) + χ∗ ( |ω3|ψq+10 (t) γ(q + 2) + 2|ω2|ψq+20 (t) γ(q + 3) )( |υ1| |b1| + ψ p 0(t) γ(p + 1) ϑ∗ ) + χ∗|ω1| (q + 4)ψ q+2 0 (t) γ(q + 3) ( |υ2| + ϑ∗ |a2|ψp−δ0 (ξ) + |b2|ψ p−δ 0 (t) γ(p − δ + 1) ) + n ∑ i=1 ϕ∗i ψ ηi+q 0 (t) γ(ηi + q + 1) . which implies, from the choice of r that ‖u‖ ≤ r, and so u ∈ br. hence all conditions of lemma 2.5 are satisfied. therefore, the operator equation u(t) = au(t) + bu(t) has at least one solution in br. consequently, the problem (1.1)–(1.2) has at least on solution on j. thus the proof is completed. 3.2 uniqueness of solutions in the next result, we apply the banach fixed theorem to prove the uniqueness of solutions for the problem (1.1)–(1.2). cubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 285 theorem 3.2. assume that the hypotheses(h1)–(h3) together with the inequality λ + m ∑ i=1 k∗i ψ ηi+q 0 (t) γ(ηi + q) < 1. are satisfied, then the problem (1.1)–(1.2) has an unique solution. proof. according to lemma 2.4, we define the operator q : e → e by qu(t) = au(t) + bu(t). first, we show that q(br) ⊂ br. as in the previous proof (step 3) of theorem 3.1, we can obtain for u ∈ br and t ∈ j ‖qu(·)‖ ≤ χ∗ϑ∗ ψ p+q 0 (t) γ(p + 1)γ(q + 1) + χ∗ ( |ω3|ψq+10 (t) γ(q + 2) + 2|ω2|ψq+20 (t) γ(q + 3) )( |υ1| |b1| + ψ p 0(t) γ(p + 1) ϑ∗ ) + χ∗|ω1| (q + 4)ψ q+2 0 (t) γ(q + 3) ( |υ2| + ϑ∗ |a2|ψp−δ0 (ξ) + |b2|ψ p−δ 0 (t) γ(p − δ + 1) ) + n ∑ i=1 ϕ ∗ i ψ ηi+q 0 (t) γ(ηi + q + 1) ≤ r. this shows that q(br) ⊂ br. next, we prove that the operator q is a contractive operator. for u,v ∈ br ‖qu(·) − qv(·)‖ ≤ ‖au(·) − av(·)‖ + ‖bu(·) − bv(·)‖, and ‖bu(·) − bv(·)‖ ≤ sup t∈j { m ∑ i=1 1 γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1 ∣ ∣gi(s,u(s)) − gi(s,v(s)) ∣ ∣ds } ≤ m ∑ i=1 k∗i ψ ηi+q 0 (t) γ(ηi + q + 1) ‖u(·) − v(·)‖. (3.5) from (3.4) and (3.5), we get ‖qu(·) − qv(·)‖ ≤ ( λ + m ∑ i=1 k∗i ψ ηi+q 0 (t) γ(ηi + q + 1) ) ‖u(·) − v(·)‖. this implies that the operator q is a contractive operator. consequently, by theorem 3.2, we conclude that q has an unique fixed point, which is a solution of the problem (1.1)–(1.2). this completes the proof. 286 f. fredj & h. hammouche cubo 24, 2 (2022) 4 example consider the following fractional hybrid differential equation                                                                        cd 5 2 ;t       cd 3 4 ;tu(t) − 3 ∑ i=1 i ηi;tgi(t,u(t)) h(t,u(t))       = f       t,u(t),c d 5 2 ;t       cd 3 4 ;tu(t) − 3 ∑ i=1 i ηi;tgi(t,u(t)) h(t,u(t))             , u(0) = 0, cd 3 4 ;tu(0) = 0, 2       cd 3 4 ;tu(t) − 3 ∑ i=1 iηi;tgi(t,u(t)) h(t,u(t))       t=0 + 2 7       cd 3 4 ;tu(t) − 3 ∑ i=1 iηi;tgi(t,u(t)) h(t,u(t))       t=1 = 7 2 , 7 13 cd 4 5 ;t       cd 3 4 ;tu(t) − 3 ∑ i=1 iηi;tgi(t,u(t)) h(t,u(t))       t= 4 5 + 1 2 cd 4 5 ;t       cd 3 4 ;tu(t) − 3 ∑ i=1 iηi;tgi(t,u(t)) h(t,u(t))       t=1 = 2 , (4.1) where 3 ∑ i=1 i ηi;tgi(t,u(t))(s) = i 1 3 ;t ( sin2 x(s) 8(s + 1)2 ) (t) + i 3 2 ;t ( 1 2π √ 81 + s2 |x(s)| 2 + |x(s)| ) (t) + i 7 3 ;t ( sinx(s) 3π √ 49 + s2 ) (t), h(t,u(t)) = e−3t cosu(t) 2t + 40 + 1 80 (t3 + 1), and f       t,u(t),c d 5 2 ;t       cd 3 4 ;tu(t) − 3 ∑ i=1 iηi;tgi(t,u(t)) h(t,u(t))             = 1 60 √ t + 81       |x(t)| 3 + |x(t)| − arctan       cd 5 2 ;t       cd 3 4 ;tu(t) − 3 ∑ i=1 i ηi;tgi(t,u(t)) h(t,u(t))                   . cubo 24, 2 (2022) hybrid ψ−caputo multi-fractional differential equation 287 here t = 1,p = 5 2 ,q = 3 4 ,m = 3,η1 = 1 3 ,η2 = 3 2 ,η3 = 7 3 ,δ = 4 5 ,a1 = 2,a2 = 7 13 ,b1 = 2 7 , b2 = 1 2 ,υ1 = 7 2 ,υ2 = 2,ξ = 4 5 , g1 = sin2 x(t) 8(t + 1)2 , g2 = 1 2π √ 81 + t2 |x(t)| 2 + |x(t)| , g3 = sinx(t) 3π √ 49 + t2 . the hypotheses (h1), (h2) and (h4) are satisfied with the following positives functions: l(t) = e−3 2t + 40 , m(t) = ϑ(t) = 1 60 √ t + 81 , ϕ1(t) = k1(t) = 1 8(t + 1)2 , ϕ2(t) = k2(t) = 1 2π √ 81 + t2 , ϕ3(t) = k3(t) = 1 3π √ 49 + t2 and χ(t) = e−3 2t + 40 + 1 80 (t3 +1), which gives us l∗ = 1 40 , m∗ = ϑ∗ = 1 540 ,χ∗ = 3 80 , ϕ∗1 = k ∗ 1 = 1 8 ,ϕ∗2 = k ∗ 2 = 1 18π , ϕ∗3 = k ∗ 3 = 1 21π . with the given data, we find that ω1 ≃ 1.81820508, ω2 ≃ 0.60797139, ω3 ≃ 1.60797139, and λ ≃ 0.48820986 < 1. by theorem 3.1, the problem (4.1) has a solution on [0,1]. also, we have λ + 3 ∑ i=1 k∗i ψ ηi+q 0 (1) γ(ηi + 7 4 ) ≃ 0.61782704 < 1. in view of theorem 3.2 the problem (4.1) has an unique solution. 5 conclusion in this manuscript, we have successfully investigated the existence, uniqueness of the solutions for a new class of ψ−caputo type hybrid fractional differential equations with hybrid conditions. the existence of solutions is provided by using a generalization of krasnoselskii’s fixed point theorem due to dhage [5], whereas the uniqueness result is achieved by banach’s contraction mapping principle. also, we have presented an illustrative example to support our main results. in future works, many results can be established when one takes a more generalized operator. precisely, it will be of interest to study the current problem in this work for the fractional operator with variable order [22], and ψ-hilfer fractional operator [19]. 288 f. fredj & h. hammouche cubo 24, 2 (2022) references [1] r. almeida, “a caputo fractional derivative of a function with respect to another function”, commun. nonlinear sci. numer. simul., vol. 44, pp. 460–481, 2017. 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[23] y. zhao, s. sun, z. han and q. li, “theory of fractional hybrid differential equations”, comput. math. appl., vol. 62, no. 3, pp. 1312–1324, 2011. introduction preliminaries main result existence of solutions uniqueness of solutions example conclusion cubo, a mathematical journal vol. 24, no. 03, pp. 541–554, december 2022 doi: 10.56754/0719-0646.2403.0541 estimates for the polar derivative of a constrained polynomial on a disk gradimir v. milovanović1, b abdullah mir2 adil hussain2 1 serbian academy of sciences and arts, 11000 belgrade, serbia university of nǐs, faculty of sciences and mathematics, p.o. box 224, 18000 nǐs, serbia. gvm@mi.sanu.ac.rs b 2 department of mathematics, university of kashmir, srinagar, 190006, india. mabdullah mir@uok.edu.in malikadil6909@gmail.com abstract this work is a part of a recent wave of studies on inequalities which relate the uniform-norm of a univariate complex coefficient polynomial to its derivative on the unit disk in the plane. when there is a limit on the zeros of a polynomial, we develop some additional inequalities that relate the uniform-norm of the polynomial to its polar derivative. the obtained results support some recently established erdőslax and turán-type inequalities for constrained polynomials, as well as produce a number of inequalities that are sharper than those previously known in a very large literature on this subject. resumen este trabajo es parte de una reciente ola de estudios sobre desigualdades que relacionan la norma uniforme de un polinomio univariado con coeficientes complejos con su derivada en el disco unitario en el plano. cuando existe un ĺımite sobre los ceros de un polinomio, desarrollamos algunas desigualdades adicionales que relacionan la norma uniforme del polinomio con su derivada polar. los resultados obtenidos satisfacen desigualdades de tipo erdős-lax y turán para polinomios restringidos recientemente establecidas, y también producen desigualdades que son más estrictas que aquellas conocidas previamente en la larga literatura dedicada a este tema. keywords and phrases: complex domain, constrained polynomial, rouché’s theorem, zeros. 2020 ams mathematics subject classification: 30a10, 30c10, 30c15. accepted: 04 december 2022 received: 13 may, 2022 ©2022 g. v. milovanović et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0541 https://orcid.org/0000-0002-3255-8127 https://orcid.org/0000-0003-0930-6391 https://orcid.org/0000-0002-9646-9940 mailto:gvm@mi.sanu.ac.rs mailto:mabdullah_mir@uok.edu.in mailto:malikadil6909@gmail.com 542 g. v. milovanović, a. mir & a. hussain cubo 24, 3 (2022) 1 introduction experimental data is converted into mathematical notation and mathematical models in scientific inquiries. in order to solve these, it may be necessary to know how large or small the maximum modulus of the derivative of an algebraic equation can be in terms of maximum modulus of the polynomial. in practise, setting boundaries for these circumstances is crucial. the only information available in the literature is in the form of approximations, and there are no closed formulae for calculating these limitations precisely. these approximate boundaries are quite accurate when computed effectively adequate for the demands of investigators and scientists. as a result, there is a constant desire to find boundaries that are superior to those described in the literature. we were inspired to write this note because there is a need for updated and more precise bounds. the inequalities for polynomials and their derivatives, which generalise the classical inequalities for different norms and with different constraints on utilising various methods of geometric function theory, are a fertile topic in analysis. in the literature, for proving the inverse theorems in approximation theory, many inequalities in both directions relating the norm of the derivative and the polynomial itself play a significant role and, of course, have their own intrinsic appeal. as shown by various recent studies, numerous research papers have been published on these inequalities for constrained polynomials (for example, see [11, 13, 17, 19, 20, 21]). we begin with the well-known bernstein inequality [4] for the uniform norm on the unit disk in the plane: namely, if p(z) is a polynomial of degree n, then max |z|=1 |p ′(z)| ≤ n max |z|=1 |p(z)|. (1.1) if we only consider polynomials without zeros in |z| < 1, the above inequality (1.1) can then be emphasised. in fact, erdős conjectured and later lax [14] proved that, if p(z) ̸= 0 in |z| < 1, then max |z|=1 |p ′(z)| ≤ n 2 max |z|=1 |p(z)|. (1.2) the inequality (1.2) is sharp and equality holds if p(z) has all of its zeros on |z| = 1. when there is a restriction on the polynomial’s zeros, turán’s classical inequality [25] offers a lower bound estimate for the size of the derivative of the polynomial on the unit circle in relation to the size of the polynomial. it states that if p(z) is a polynomial of degree n having all its zeros in |z| ≤ 1, then max |z|=1 |p ′(z)| ≥ n 2 max |z|=1 |p(z)|. (1.3) cubo 24, 3 (2022) estimates for the polar derivative of a constrained polynomial... 543 aziz and dawood [2] improved inequality (1.3) to take the form max |z|=1 |p ′(z)| ≥ n 2 { max |z|=1 |p(z)| + min |z|=1 |p(z)| } . (1.4) any polynomial that has all of its zeros on |z| = 1 holds true for (1.3) and (1.4). the inequalities (1.3) and (1.4) have been generalised and expanded in a number of ways over time. for a polynomial p(z) of degree n having all its zeros in |z| ≤ k, k ≥ 1, govil [8], proved that max |z|=1 |p ′(z)| ≥ n 1 + kn max |z|=1 |p(z)|. (1.5) as is easy to see that (1.5) becomes an equality if p(z) = zn + kn, one would expect that if we exclude the class of polynomials having all zeros on |z| = k, then it may be possible to improve the bound in (1.5). in this direction, it was shown by govil [10] that if p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then max |z|=1 |p ′(z)| ≥ n 1 + kn { max |z|=1 |p(z)| + min |z|=k |p(z)| } . (1.6) as an extension of (1.2), malik [15] proved that, if p(z) ̸= 0 in |z| < k, k ≥ 1, then max |z|=1 |p ′(z)| ≤ n 1 + k max |z|=1 |p(z)|. (1.7) the result is sharp and equality in (1.7) holds for p(z) = (z + k)n. on the other hand, if p(z) ̸= 0 in |z| < k, k ≤ 1, the precise estimate of maximum of |p ′(z)| on |z| = 1 does not seem to be known in general, and this problem is still open. however, some special cases in this direction have been considered by many people where some partial extensions of (1.2) are established. in 1980, it was shown by govil [9] that if p(z) is a polynomial of degree n and p(z) ̸= 0 in |z| < k, k ≤ 1, then max |z|=1 |p ′(z)| ≤ n 1 + kn max |z|=1 |p(z)|, (1.8) provided |p ′(z)| and |q′(z)| attain their maximum at the same point on |z| = 1, where q(z) = znp (1/z). under the same hypothesis as in (1.8), aziz and ahmad [1] established an improved inequality in the form max |z|=1 |p ′(z)| ≤ n 1 + kn { max |z|=1 |p(z)| − min |z|=k |p(z)| } . (1.9) in the literature, more generalised variations of bernstein and turán inequalities have emerged, 544 g. v. milovanović, a. mir & a. hussain cubo 24, 3 (2022) in which the underlying polynomial is replaced with more general classes of functions. one such generalisation is moving from the domain of ordinary derivatives of polynomials to the domain of their polar derivatives. before drawing any more conclusions, let us first discuss the idea of the polar derivative. for a polynomial p(z) of degree n, we define dβp(z) := np(z) + (β − z)p ′(z), the polar derivative of p(z) with respect to the point β. the polynomial dβp(z) is of degree at most n − 1 and it generalizes the ordinary derivative in the sense that lim β→∞ { dβp(z) β } = p ′(z), uniformly with respect to z for |z| ≤ r, r > 0. the comprehensive books by marden [16], milovanović et al. [18], rahman and schmeisser [23], and the most recent one by gardner et al. [7] all provide access to the extensive literature on the polar derivative of polynomials. in 1998, aziz and rather [3] established the polar derivative analogue of (1.5) by proving that if p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for every β ∈ c with |β| ≥ k, max |z|=1 |dβp(z)| ≥ n ( |β| − k 1 + kn ) max |z|=1 |p(z)|. (1.10) in the same publication, aziz and rather extended the inequality (1.4) to the polar derivative of a polynomial. in fact, they proved that if p(z) is a polynomial of degree n having all its zeros in |z| ≤ 1, then for any complex number β with |β| ≥ 1, max |z|=1 |dβp(z)| ≥ n 2 { (|β| − 1) max |z|=1 |p(z)| + (|β| + 1) min |z|=1 |p(z)| } . (1.11) the corresponding polar derivative analogue of (1.6) and a refinement of (1.10) was given by dewan et al. [5]. they proved that if p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number β with |β| ≥ k, max |z|=1 |dβp(z)| ≥ n 1 + kn { (|β| − k) max |z|=1 |p(z)| + ( |β| + 1 kn−1 ) min |z|=k |p(z)| } . (1.12) singh and chanam [24] most recently developed the following generalisation and strengthening of (1.10). cubo 24, 3 (2022) estimates for the polar derivative of a constrained polynomial... 545 theorem a. let p(z) = zs n−s∑ ν=0 aνz ν, 0 ≤ s ≤ n, be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for every complex number β with |β| ≥ k, max |z|=1 |dβp(z)| ≥ (|β| − k) { n + s 1 + kn + k(n−s)/2 √ |an−s| − √ |a0| (1 + kn)k(n−s)/2 √ |an−s| } max |z|=1 |p(z)|. (1.13) the improvement of inequality (1.8) as a result of govil [9] was demonstrated by singh and chanam in the same paper in the form of the subsequent outcome. theorem b. let p(z) = n∑ ν=0 aνz ν be a polynomial of degree n having no zeros in |z| < k, k ≤ 1, and let q(z) = znp (1/z). if |p ′(z)| and |q′(z)| attain their maximum at the same point on |z| = 1, max |z|=1 |p ′(z)| ≤   n1 + kn − (√ |a0| − kn/2 √ |an| ) kn (1 + kn) √ |a0|   max|z|=1 |p(z)|. (1.14) the result is sharp and equality holds in (1.14) for p(z) = zn + kn. the study of these inequalities for a certain class of polynomials having a zero of order s ≥ 0 at the origin is continued in this paper, and we set some new upper and lower bounds for the derivative and polar derivative of a polynomial on the unit disk while taking into account the location of the zeros and extremal coefficients of the underlying polynomial. 2 main results we begin this section by proving the following turán-type inequality giving generalisations and refinements of (1.10)–(1.13) and related inequalities. theorem 2.1. let p(z) = zs n−s∑ ν=0 aνz ν, 0 ≤ s ≤ n, be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for every complex number β with |β| ≥ k, max |z|=1 |dβp(z)| ≥ n 1 + kn { (|β| − k) max |z|=1 |p(z)| + ( |β| + 1 kn−1 ) mk } + ( |β| − k 1 + kn ){ s + √ kn−s|an−s| − mk − √ |a0|√ kn−s|an−s| − mk }( max |z|=1 |p(z)| − mk kn ) , (2.1) where mk = min|z|=k |p(z)|. setting s = 0 in (2.1), we get the following refinement of (1.12) and hence of (1.10) and (1.11) as well. 546 g. v. milovanović, a. mir & a. hussain cubo 24, 3 (2022) corollary 2.2. let p(z) = n∑ ν=0 aνz ν be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for every complex number β with |β| ≥ k, max |z|=1 |dβp(z)| ≥ n 1 + kn { (|β| − k) max |z|=1 |p(z)| + ( |β| + 1 kn−1 ) mk } + ( |β| − k 1 + kn ){√ kn|an| − mk − √ |a0|√ kn|an| − mk }( max |z|=1 |p(z)| − mk kn ) , (2.2) where mk is as defined in theorem 2.1. by taking k = 1 in (2.2), we easily get a refinement of (1.11). if we divide both sides of (2.1) and (2.2) by |β| and let |β| → ∞, we get the following results: corollary 2.3. let p(z) = zs n−s∑ ν=0 aνz ν, 0 ≤ s ≤ n, be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then max |z|=1 |p ′(z)| ≥ n 1 + kn ( max |z|=1 |p(z)| + mk ) + { s 1 + kn + √ kn−s|an−s| − mk − √ |a0| (1 + kn) √ kn−s|an−s| − mk }( max |z|=1 |p(z)| − mk kn ) , (2.3) where mk is as defined in theorem 2.1. equality in (2.3) holds for p(z) = z n + kn. corollary 2.4. let p(z) = n∑ ν=0 aνz ν be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then max |z|=1 |p ′(z)| ≥ n 1 + kn ( max |z|=1 |p(z)| + mk ) + √ kn|an| − mk − √ |a0| (1 + kn) √ kn|an| − mk ( max |z|=1 |p(z)| − mk kn ) , (2.4) where mk is as defined in theorem 2.1. equality in (2.4) holds for p(z) = z n + kn. remark 2.5. it is clear that, in general for any polynomial of degree n of the form p(z) = zs(a0 +a1z+· · ·+an−szn−s), 0 ≤ s ≤ n, having all its zeros in |z| ≤ k, k ≥ 1, the inequality (2.1) improves the inequality (1.13) considerably, excepting the case when p(z) has a zero on |z| = k. for the class of polynomials having a zero on |z| = k, the inequality (2.2) will give bounds that are sharper than the bound obtained from the inequality (1.12). one can also observe that the inequality (2.4) improves inequality (1.6) considerably when √ kn|an| − mk − √ |a0| ≠ 0. as an application of corollary 2.4, we prove the following result for the class of polynomials having no zeros in |z| < k, k ≤ 1, which in turn provides a generalization and refinement to theorem b. cubo 24, 3 (2022) estimates for the polar derivative of a constrained polynomial... 547 theorem 2.6. let p(z) = n∑ ν=0 aνz ν be a polynomial of degree n having no zeros in |z| < k, k ≤ 1, and let q(z) = znp (1/z). if |p ′(z)| and |q′(z)| attain their maximum at the same point on |z| = 1, then for every complex number β with |β| ≥ 1, max |z|=1 |dβp(z)| ≤ n(|β| + kn) 1 + kn max |z|=1 |p(z)| − nmk(|β| − 1) 1 + kn − (|β| − 1) (√ |a0| − mk − kn/2 √ |an| ) kn (1 + kn) √ |a0| − mk { max |z|=1 |p(z)| − mk } , (2.5) where mk is as defined in theorem 2.1. equality in (2.5) holds for p(z) = z n +kn, with real β ≥ 1. if we divide both sides of inequality (2.5) by |β| and let |β| → ∞, we get the following result. corollary 2.7. let p(z) = n∑ ν=0 aνz ν be a polynomial of degree n having no zeros in |z| < k, k ≤ 1, and let q(z) = znp (1/z). if |p ′(z)| and |q′(z)| attain their maximum at the same point on |z| = 1, then max |z|=1 |p ′(z)| ≤ n 1 + kn max |z|=1 |p(z)| − nmk 1 + kn − (√ |a0| − mk − kn/2 √ |an| ) kn (1 + kn) √ |a0| − mk { max |z|=1 |p(z)| − mk } , (2.6) where mk is as defined in theorem 2.1. equality in (2.6) holds for p(z) = z n + kn. remark 2.8. it may be remarked here that, in general for any polynomial of degree n of the form p(z) = a0 + a1z + a2z 2 + · · · + anzn, having no zeros in |z| < k, k ≤ 1, the inequality (2.6) improves the inequality (1.14), excepting the case when p(z) has a zero on |z| = k. for the class of polynomials having a zero on |z| = k, the inequality (2.5) sharpens a result of mir and breaz [20, corollary 2] considerably. 3 lemmas in order to prove our results, we need the following lemmas. the first lemma is a simple deduction from the maximum modulus principle (see [22]). lemma 3.1. if p(z) is a polynomial of degree at most n, then for r ≥ 1, max |z|=r |p(z)| ≤ rn max |z|=1 |p(z)|. the following lemma is due to dewan and upadhye [6]. 548 g. v. milovanović, a. mir & a. hussain cubo 24, 3 (2022) lemma 3.2. if p(z) is a polynomial of degree n having all zeros in |z| ≤ k, k ≥ 1, then max |z|=k |p(z)| ≥ 2kn 1 + kn max |z|=1 |p(z)| + kn − 1 kn + 1 min |z|=k |p(z)|. lemma 3.3. if p(z) = zs n−s∑ ν=0 aνz ν, 0 ≤ s ≤ n, is a polynomial of degree n having all zeros in |z| ≤ 1, then for any complex number β with |β| ≥ 1 and |z| = 1, |dβp(z)| ≥ (|β| − 1) { n + s 2 + √ |an−s| − √ |a0| 2 √ |an−s| } |p(z)|. the above lemma is due to singh and chanam [24]. lemma 3.4. if p(z) = zs n−s∑ ν=0 aνz ν, 0 ≤ s ≤ n, is a polynomial of degree n having all zeros in |z| ≤ 1, then for any complex number β with |β| ≥ 1 and |z| = 1, |dβp(z)| ≥ n 2 ( (|β| − 1)|p(z)| + (|β| + 1)m1 ) + ( |β| − 1 2 ) { s + √ |an−s| − m1 − √ |a0|√ |an−s| − m1 } (|p(z)| − m1) , where m1 = min|z|=1 |p(z)|. proof. by hypothesis p(z) = zs n−s∑ ν=0 aνz ν, 0 ≤ s ≤ n, has all its zeros in |z| ≤ 1. if the polynomial h(z) = n−s∑ ν=0 aνz ν has a zero on |z| = 1, then m1 = min|z|=1 |p(z)| = 0 and the result follows by lemma 3.3 in this case. henceforth, we assume that all the zeros of p(z) = zsh(z) lie in |z| < 1, so that m1 > 0. therefore, we have m1 ≤ |p(z)| for |z| = 1. this implies for any complex number µ with |µ| < 1, that m1|µzn| < |p(z)| for |z| = 1. since all the zeros of p(z) lie in |z| < 1, it follows by rouché’s theorem that all the zeros of p(z) − µm1zn = zs ( a0 + a1z + · · · + (an−s − µm1)zn−s ) also lie in |z| < 1. hence, by lemma 3.3, we get for |β| ≥ 1 and |z| = 1, |dβ(p(z) − µm1zn)| ≥ (|β| − 1) { n + s 2 + √ |an−s − µm1| − √ |a0| 2 √ |an−s − µm1| } |p(z) − µm1zn|. (3.1) cubo 24, 3 (2022) estimates for the polar derivative of a constrained polynomial... 549 for every µ ∈ c, we have |an−s − µm1| ≥ |an−s| − |µ|m1, and since the function ψ(x) = (√ x− √ |a0| ) √ x , x > 0, is a non-decreasing function of x, it follows from (3.1) that for every µ with |µ| < 1 and |z| = 1, |dβ(p(z) − µm1zn)| ≥ (|β| − 1) { n + s 2 + √ |an−s| − |µ|m1 − √ |a0| 2 √ |an−s| − |µ|m1 } |p(z) − µm1zn|. (3.2) it is a simple deduction of laguerre theorem (see [16, p. 52]) on the polar derivative of a polynomial that for any β with |β| ≥ 1, the polynomial dβ(p(z) − µm1zn) = dβp(z) − µβnm1zn−1 has all its zeros in |z| < 1. this implies that |dβp(z)| ≥ m1n|β||z|n−1 for |z| ≥ 1. (3.3) now choosing the argument of µ suitably on the left hand side of (3.2) such that ∣∣dβp(z) − µβnm1zn−1∣∣ = |dβp(z)| − |µ||β|nm1 for |z| = 1, which is possible by (3.3), we get for |z| = 1 |dβp(z)| − m1n|µ||β| ≥ (|β| − 1) { n + s 2 + √ |an−s| − |µ|m1 − √ |a0| 2 √ |an−s| − |µ|m1 }( |p(z)| − |µ|m1 ) . (3.4) if in (3.4), we make |µ| → 1, we easily get for |z| = 1, |dβp(z)| ≥ n 2 ( (|β| − 1)|p(z)| + (|β| + 1)m1 ) + ( |β| − 1 2 ) { s + √ |an−s| − m1 − √ |a0|√ |an−s| − m1 } (|p(z)| − m1) . this completes the proof of lemma 3.4. lemma 3.5. if p(z) is a polynomial of degree n and, q(z) = znp (1/z), then on |z| = 1, |p ′(z)| + |q′(z)| ≤ n max |z|=1 |p(z)|. the above lemma is due to govil and rahman [12]. 550 g. v. milovanović, a. mir & a. hussain cubo 24, 3 (2022) 4 proofs of the main results proof of theorem 2.1. recall that p(z) has all its zeros in |z| ≤ k, k ≥ 1, therefore, all the zeros of the polynomial e(z) = p(kz) lie in |z| ≤ 1. applying lemma 3.4 to the polynomial e(z) and noting that |β|/k ≥ 1, we get max |z|=1 ∣∣dβ/ke(z)∣∣ ≥ n 2 { ( |β| k − 1 ) max |z|=1 |e(z)| + ( |β| k + 1 ) m∗ } + ( |β| k − 1 ) { s 2 + √ kn−s|an−s| − m∗ − √ |a0| 2 √ kn−s|an−s| − m∗ } ( max |z|=1 |e(z)| − m∗ ) , (4.1) where m∗ = min|z|=1 |e(z)| = min|z|=1 |p(kz)| = min|z|=k |p(z)| = mk. the above inequality (4.1) is equivalent to max |z|=1 ∣∣∣∣np(kz) + ( β k − z ) kp ′(kz) ∣∣∣∣ ≥ n2 { ( |β| − k k ) max |z|=1 |p(kz)| + ( |β| k + 1 ) mk } + ( |β| − k k ) { s 2 + √ kn−s|an−s| − mk − √ |a0| 2 √ kn−s|an−s| − mk } × ( max |z|=1 |p(kz)| − mk ) . the last inequality yields max |z|=k |dβp(z)| ≥ n 2 { ( |β| − k k ) max |z|=k |p(z)| + ( |β| k + 1 ) mk } + ( |β| − k k ) { s 2 + √ kn−s|an−s| − mk − √ |a0| 2 √ kn−s|an−s| − mk } ( max |z|=k |p(z)| − mk ) . (4.2) since dβp(z) is a polynomial of degree at most n − 1, we have by lemma 3.1 for r = k ≥ 1, max |z|=k |dβp(z)| ≤ kn−1 max |z|=1 |dβp(z)|. on using this and lemma 3.2, the above inequality (4.2) clearly gives kn−1 max |z|=1 |dβp(z)| ≥ n 2 { ( |β| − k k ) ( 2kn 1 + kn max |z|=1 |p(z)| + ( kn − 1 kn + 1 ) mk ) + ( |β| k + 1 ) mk } + ( |β| − k k ) { s 2 + √ kn−s|an−s| − mk − √ |a0| 2 √ kn−s|an−s| − mk } × { 2kn 1 + kn max |z|=1 |p(z)| + ( kn − 1 kn + 1 ) mk − mk } . cubo 24, 3 (2022) estimates for the polar derivative of a constrained polynomial... 551 after rearranging the terms, we get max |z|=1 |dβp(z)| ≥ n 1 + kn { (|β| − k) max |z|=1 |p(z)| + ( |β| + 1 kn−1 ) mk } + ( |β| − k 1 + kn ){ s + √ kn−s|an−s| − mk − √ |a0|√ kn−s|an−s| − mk }( max |z|=1 |p(z)| − mk kn ) , which is exactly (2.1). this completes the proof of theorem 2.1. proof of theorem 2.6. let q(z) = znp (1/z). since p(z) = n∑ ν=0 aνz ν ̸= 0 in |z| < k, k ≤ 1, the polynomial q(z) of degree n has all its zeros in |z| ⩽ 1/k, 1/k ≥ 1. on applying inequality (2.4) of corollary 2.4 to q(z), we get max |z|=1 |q′(z)| ≥ n 1 + 1 kn ( max |z|=1 |q(z)| + m′k ) + √ 1 kn |a0| − m′k − √ |an| (1 + 1 kn ) √ 1 kn |a0| − m′k { max |z|=1 |q(z)| − knm′k } . (4.3) now, m′k = min |z|=1/k |q(z)| = min |z|=1/k ∣∣∣∣∣znp ( 1 z )∣∣∣∣∣ = 1kn min|z|=k |p(z)| = mkkn and max |z|=1 |q(z)| = max |z|=1 |p(z)|. using these observations in (4.3), we get max |z|=1 |q′(z)| ≥ nkn 1 + kn ( max |z|=1 |p(z)| + mk kn ) + (√ |a0| − mk − kn/2 √ |an| ) kn (1 + kn) √ |a0| − mk { max |z|=1 |p(z)| − mk } . (4.4) since |p ′(z)| and |q′(z)| attain maximum at the same point on |z| = 1, we have max |z|=1 (|p ′(z)| + |q′(z)|) = max |z|=1 |p ′(z)| + max |z|=1 |q′(z)|. (4.5) on combining (4.4), (4.5) and lemma 3.5, we get n max |z|=1 |p(z)| ≥ max |z|=1 |p ′(z)| + nkn 1 + kn ( max |z|=1 |p(z)| + mk kn ) + (√ |a0| − mk − kn/2 √ |an| ) kn (1 + kn) √ |a0| − mk { max |z|=1 |p(z)| − mk } , 552 g. v. milovanović, a. mir & a. hussain cubo 24, 3 (2022) which gives max |z|=1 |p ′(z)| ≤ n max |z|=1 |p(z)| − nkn 1 + kn ( max |z|=1 |p(z)| + mk kn ) − (√ |a0| − mk − kn/2 √ |an| ) kn (1 + kn) √ |a0| − mk { max |z|=1 |p(z)| − mk } . (4.6) also, it is easy to verify that for |z| = 1, |q′(z)| = |np(z) − zp ′(z)|. (4.7) note that for any complex number β, and |z| = 1, we have |dβp(z)| = |np(z) + (β − z)p ′(z)| ≤ |np(z) − zp ′(z)| + |β||p ′(z)|, which gives by (4.7) and |β| ≥ 1, that |dβp(z)| ≤ |q′(z)| + |β||p ′(z)| = |q′(z)| + |p ′(z)| − |p ′(z)| + |β||p ′(z)| ≤ n max |z|=1 |p(z)| + (|β| − 1)|p ′(z)| (by lemma 3.5) ≤ n max |z|=1 |p(z)| + (|β| − 1) max |z|=1 |p ′(z)|. (4.8) inequality (4.8), in conjunction with (4.6), gives for |z| = 1, |dβp(z)| ≤ n|β| max |z|=1 |p(z)| − nkn(|β| − 1) 1 + kn ( max |z|=1 |p(z)| + mk kn ) − (|β| − 1) (√ |a0| − mk − kn/2 √ |an| ) kn (1 + kn) √ |a0| − mk { max |z|=1 |p(z)| − mk } , which is equivalent to (2.5). this completes the proof of theorem 2.6. acknowledgements the authors express their gratitude to the referees for their detailed comments and suggestions. research of the first author was partly supported by the serbian academy of sciences and arts (project φ-96). the second author was supported by the national board for higher mathematics (r.p), department of atomic energy, government of india (no. 02011/19/2022/r&d-ii/10212). cubo 24, 3 (2022) estimates for the polar derivative of a constrained polynomial... 553 references [1] a. aziz and n. ahmad, “inequalities for the derivative of a polynomial”, proc. indian acad. sci. math. sci., vol. 107, no. 2, pp. 189–196, 1997. 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[25] p. turán, “über die ableitung von polynomen”, compositio math., vol. 7, pp. 89–95, 1940. introduction main results lemmas proofs of the main results cubo, a mathematical journal vol. 24, no. 02, pp. 227–237, august 2022 doi: 10.56754/0719-0646.2402.0227 variational methods to second-order dirichlet boundary value problems with impulses on the half-line meriem djibaoui 1 toufik moussaoui 1, b 1laboratory of fixed point theory and applications, école normale supérieure, kouba, algiers. algeria. djibaouimeriem@gmail.com toufik.moussaoui@g.ens-kouba.dz b abstract in this paper, the existence of solutions for a second-order impulsive differential equation with a parameter on the halfline is investigated. applying lax-milgram theorem, we deal with a linear dirichlet impulsive problem, while the nonlinear case is established by using standard results of critical point theory. resumen en este art́ıculo, se investiga la existencia de soluciones de una ecuación diferencial de segundo orden impulsiva con un parámetro en la semi-recta. aplicando el teorema de laxmilgram, tratamos un problema lineal impulsivo de dirichlet, mientras que el caso no lineal es establecido usando resultados estándar de teoŕıa de punto cŕıtico. keywords and phrases: dirichlet boundary value problem, half-line, lax-milgram theorem, critical points, impulsive differential equation. 2020 ams mathematics subject classification: 34b37, 34b40, 35a15, 35b38. accepted: 30 march, 2022 received: 04 august, 2021 c©2022 m. djibaoui et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://doi.org/10.56754/0719-0646.2402.0227 https://orcid.org/0000-0002-0917-0394 mailto:toufik.moussaoui@g.ens-kouba.dz https://orcid.org/0000-0001-7495-0269 mailto:djibaouimeriem@gmail.com mailto:toufik.moussaoui@g.ens-kouba.dz 228 m. djibaoui & t. moussaoui cubo 24, 2 (2022) 1 introduction in recent years, many researchers have extensively applied variational methods to study boundary value problems (bvps) for impulsive differential equations on the finite intervals. more precisely, employing critical point theory, nieto and o’regan [8] studied a linear dirichlet boundary value problem with impulses        −u′′(t) + λu(t) = σ(t), a.e. t ∈ [0, t ], △u′(tj) = dj, j ∈ {1, 2, . . . , l}, u(0) = u(t ) = 0, (1.1) and a nonlinear impulsive problem        −u′′(t) + λu(t) = f(t, u(t)), a.e. t ∈ [0, t ], △u′(tj) = ij(u(t − j )), j ∈ {1, 2, . . . , l}, u(0) = u(t ) = 0, (1.2) where λ is a positive parameter. moreover, the study of solutions for impulsive bvps on the infinite intervals by using variational methods has received considerably more attention, see for example [1, 2, 3, 9, 10], and the references therein. in the present paper, our aim is to improve some assumptions made in [8] in order to extend problems (1.1) and (1.2) on the half-line via variational approach. this paper is organized as follows. in section 2 we state some preliminaries. in section 3 we consider the linear dirichlet problem with impulses in the derivative. due to the lax-milgram theorem, we show the existence of weak solutions that are precisely the critical points of some functionals. the last section is to deal with the nonlinear dirichlet problem. to investigate the existence of solutions, we use standard results of critical point theory. also, some examples are given to illustrate our main results. 2 preliminaries we cite some basic and celebrated theorems from critical point theory which are crucial tools in the proof of our main results. let h be a hilbert space. theorem 2.1 (lax-milgram [4, 5]). let a : h × h → r be a bounded bilinear form. if a is coercive, i.e., there exists α > 0 such that a(u, u) ≥ α‖u‖2 for every u ∈ h, then for any σ ∈ h′ (the conjugate space of h) there exists a unique u ∈ h such that a(u, v) = (σ, v), for every v ∈ h. cubo 24, 2 (2022) variational methods to second-order dirichlet boundary value... 229 moreover, if a is also symmetric, then the functional ϕ : h → r defined by ϕ(v) = 1 2 a(v, v) − (σ, v) attains its minimum at u. theorem 2.2 ([7]). if ϕ is weakly lower semi-continuous (w.l.s.c.) on a reflexive banach space x and has a bounded minimizing sequence, then ϕ has a minimum on x. now, let us recall some necessary concepts that will be needed in our argument. let us define the following reflexive banach space h10 (0, ∞) = { u : [0, ∞) → r is absolutely continuous, u, u′ ∈ l2(0, ∞), u(0) = u(∞) = 0 } , equipped with the norm ‖u‖ =   +∞ ∫ 0 |u(t)|2dt + +∞ ∫ 0 |u′(t)|2dt   1 2 . set the space cl,p[0, +∞) = {u ∈ c([0, +∞), r) : lim t→∞ p(t)u(t) exists} with the norm ‖u‖∞,p = sup t∈[0,+∞) p(t)|u(t)|, where the function p : [0; +∞) → (0, +∞) is continuously differentiable and bounded, satisfying c = 2 max(‖p‖l2, ‖p ′‖l2) < +∞. concerning the above spaces, we get the following vital embeddings. lemma 2.3 ([6]). the space h10 (0, ∞) embeds continuously in cl,p[0, ∞), more precisely ‖u‖∞,p ≤ c‖u‖ for every u ∈ h10 (0, ∞). lemma 2.4 ([6]). the embedding h10 (0, ∞) →֒ cl,p[0, ∞) is compact. 3 impulsive linear problem we consider the following linear dirichlet boundary value problem with impulses in the derivative at the prescribed instants tj, j ∈ n ∗ = {1, 2, 3, . . .}        −u′′(t) + λu(t) = σ(t), a.e. t ∈ [0, ∞), t 6= tj, △u′(tj) = d(tj), j ∈ n ∗, u(0) = u(+∞) = 0, (3.1) 230 m. djibaoui & t. moussaoui cubo 24, 2 (2022) where λ ∈ r, σ ∈ l2(0, ∞), 0 = t0 < t1 < t2 < · · · < tj < · · · < tm → ∞, as m → ∞, are the impulse points, d : [0, ∞) → r satisfies ∞ ∑ j=1 d(tj) p(tj) < ∞ and △u′(tj) = u ′(t+j ) − u ′(t−j ) for u′(t±j ) = limt→t± j u′(t). now, multiply the equation in problem (3.1) by v ∈ h10 (0, ∞), and then integrate over (0, +∞), we obtain − +∞ ∫ 0 u ′′ v + λ +∞ ∫ 0 uv = +∞ ∫ 0 σv. we have − +∞ ∫ 0 u′′v = − ∞ ∑ j=0 tj+1 ∫ tj u′′v, and tj+1 ∫ tj u′′v = u′(t−j+1)v(t − j+1) − u ′(t+j )v(t + j ) − tj+1 ∫ tj u′v′. consequently, − +∞ ∫ 0 u ′′ v = ∞ ∑ j=1 △u′(tj)v(tj) + u ′(0)v(0) − u′(∞)v(∞) + +∞ ∫ 0 u ′ v ′ = ∞ ∑ j=1 d(tj)v(tj) + +∞ ∫ 0 u′v′. this leads to define the bilinear form a : h10 (0, ∞) × h 1 0 (0, ∞) → r, by a(u, v) = +∞ ∫ 0 u ′ v ′ + λ +∞ ∫ 0 uv, (3.2) and the linear operator l : h10 (0, ∞) → r by l(v) = +∞ ∫ 0 σv − ∞ ∑ j=1 d(tj)v(tj). (3.3) definition 3.1. we say that a function u is a weak solution of the impulsive problem (3.1) if u ∈ h10 (0, ∞) such that a(u, v) = l(v) is valid for any v ∈ h 1 0 (0, ∞). in what follows we refer to problem (3.1) as (lp). it is easily verified that a and l defined by (3.2), (3.3) respectively are continuous, and a is coercive if λ > 0. consider the functional ϕ : h10 (0, ∞) → r, defined by ϕ(u) = 1 2 +∞ ∫ 0 u′2 + λ 2 +∞ ∫ 0 u2 − +∞ ∫ 0 σu + ∞ ∑ j=1 d(tj)u(tj). (3.4) cubo 24, 2 (2022) variational methods to second-order dirichlet boundary value... 231 it is clear that ϕ is differentiable at any u ∈ h10 (0, ∞) and ϕ′(u)v = +∞ ∫ 0 u ′ v ′ + λ +∞ ∫ 0 uv − +∞ ∫ 0 σv + ∞ ∑ j=1 d(tj)v(tj) = a(u, v) − l(v). thus, a critical point of (3.4) gives us a weak solution of the problem (lp). definition 3.2. we mean by a classical solution of the problem (lp) a function u ∈ h2(tj, tj+1) for all j ∈ n∗, where h2(tj, tj+1) = { u : [0, ∞) → r is absolutely continuous, u′, u′′ ∈ l2(tj, tj+1) } , and u satisfies the first equation of (3.1) a.e. on [0, ∞) with u(0) = u(∞) = 0, the limits u′(t+j ), u′(t−j ), j ∈ n ∗ exist and the impulse conditions hold. lemma 3.3. if u ∈ h10 (0, ∞) is a weak solution of (lp), then u is a classical solution of (lp). proof. since u ∈ h10 (0, ∞), it is evident that u(0) = u(∞) = 0. for j ∈ {1, 2, . . .}, choose any v ∈ h10 (0, ∞) such that v(t) = 0 for t ∈ [0, tj] ∪ [tj+1, +∞). then tj+1 ∫ tj u′v′ + λ tj+1 ∫ tj uv = tj+1 ∫ tj σv. hence, −u′′ + λu = σ a.e. on (tj, tj+1). so, u ∈ h 2(tj, tj+1) and satisfies the previous equation a.e. on [0, ∞). multiplying −u′′ + λu = σ by v ∈ h10 (0, ∞) and integrating over [0, ∞), we get ∞ ∑ j=1 △u′(tj)v(tj) = ∞ ∑ j=1 d(tj)v(tj). therefore, △u′(tj) = d(tj) for every j ∈ n ∗, and the impulsive conditions are satisfied. lemma 3.4. if u ∈ h10 (0, ∞) is a critical point of ϕ defined by (3.4), then u is a weak solution of the impulsive dirichlet problem (lp). proof. let u ∈ h10 (0, ∞). the assumption that u is a critical point of ϕ means that ϕ ′(u)v = 0, for all v ∈ h10 (0, ∞). thus, +∞ ∫ 0 u′v′ + λ +∞ ∫ 0 uv − +∞ ∫ 0 σv + ∞ ∑ j=1 d(tj)v(tj) = 0, ∀v ∈ h 1 0 (0, ∞). hence, +∞ ∫ 0 u′v′ + λ +∞ ∫ 0 uv = +∞ ∫ 0 σv − ∞ ∑ j=1 d(tj)v(tj), ∀v ∈ h 1 0 (0, ∞). this implies that a(u, v) = l(v) is valid for any v ∈ h10 (0, ∞). as a result, u is a weak solution of the (lp). 232 m. djibaoui & t. moussaoui cubo 24, 2 (2022) in view of lax-milgram theorem, we formulate the following main result. theorem 3.5. if λ > 0, then the dirichlet impulsive problem (lp) has a weak solution u ∈ h10 (0, ∞) for any σ ∈ l 2(0, ∞). moreover, u ∈ h2(0, ∞) and u is a classical solution and minimizes the functional (3.4) and hence it is a critical point of (3.4). proof. for λ > 0, it follows that the bilinear a is coercive. the fact that a is continuous, by applying theorem 2.1, for any σ ∈ l2(0, ∞), there exists a unique u ∈ h10 (0, ∞) such that a(u, v) = l(v) for all v ∈ h10 (0, ∞). so, the problem (lp) has a weak solution u ∈ h 1 0 (0, ∞). owing to lemma 3.3, a weak solution of (lp) is a classical solution. in addition, a is symmetric, then the functional ϕ attains its minimum at u which is exactly a critical point of ϕ since it is differentiable. example 3.6. as an example, let λ = 1 and p(t) = 1 1+t2 · this impulsive boundary value problem        −u′′(t) + u(t) = 1 1+t , a.e. t ∈ [0, ∞), △u′(j) = e−j, j ∈ n∗, u(0) = u(+∞) = 0, (3.5) has a solution. 4 impulsive nonlinear problem in the nonlinear situation we consider the following impulsive boundary value problem        −u′′(t) + λu(t) = f(t, u(t)), a.e. t ∈ [0, ∞), t 6= tj, △u′(tj) = g(tj)ij(u(t − j )), j ∈ n ∗, u(0) = u(+∞) = 0, (4.1) where λ is a positive parameter, the functions f : [0, ∞) × r → r, ij : r → r, j ∈ n ∗, and g : [0, ∞) → [0, ∞) are continuous with ∞ ∑ j=1 g(tj) < ∞. we refer to problem (4.1) as (np). definition 4.1. a weak solution of (np) is a function u ∈ h10 (0, ∞) such that +∞ ∫ 0 u′v′ + λ +∞ ∫ 0 uv + ∞ ∑ j=1 g(tj)ij(u(tj))v(tj) − +∞ ∫ 0 f(t, u(t))dt = 0, for every v ∈ h10 (0, ∞). cubo 24, 2 (2022) variational methods to second-order dirichlet boundary value... 233 setting f(t, u) = u ∫ 0 f(t, s)ds, we define the functional ϕ : h10 (0, ∞) → r by ϕ(u) = 1 2 +∞ ∫ 0 u′2(t)dt + λ 2 +∞ ∫ 0 u2(t)dt + ∞ ∑ j=1 g(tj) u(tj) ∫ 0 ij(s)ds − +∞ ∫ 0 f(t, u(t))dt. (4.2) now we present our principal results for this part. theorem 4.2. suppose that the following conditions hold: (h1) there exists a positive bounded function m ∈ l 1(0, +∞) with m p ∈ l1(0, +∞) such that |f(t, u)| ≤ m(t) for (t, u) ∈ [0, +∞) × r. (i1) there exist mj > 0, j ∈ n ∗, satisfying ∞ ∑ j=1 mjg(tj) < ∞ and ∞ ∑ j=1 mjg(tj) p(tj) < ∞, such that the impulsive functions ij are bounded i.e., |ij(u)| ≤ mj for every u ∈ r, j ∈ {1, 2, . . .}. then there is a critical point of ϕ, and (np) has at least one solution. proof. claim 1. ϕ is weakly lower semi-continuous (w.l.s.c). let (un) ⊂ h 1 0 (0, ∞) be a sequence such that un ⇀ u in h 1 0 (0, ∞), when n → ∞. then, ‖u‖ ≤ lim inf n→∞ ‖un‖, and by lemma 2.4 we have that (un) converges to u in cl,p[0, ∞), hence un(t) converges to u(t) for all t ∈ [0, ∞). from (h1) and (i1), using the continuity of f and ij, j ∈ n ∗, together with the lebesgue dominated convergence theorem, we obtain lim inf n→+∞ ϕ(un) = lim inf n→+∞    1 2 +∞ ∫ 0 u ′2 n + λ 2 +∞ ∫ 0 u2n + ∞ ∑ j=1 g(tj) un(tj) ∫ 0 ij(s)ds − +∞ ∫ 0 f(t, un(t))dt    ≥ 1 2 +∞ ∫ 0 u ′2 + λ 2 +∞ ∫ 0 u 2 + ∞ ∑ j=1 g(tj) u(tj) ∫ 0 ij(s)ds − +∞ ∫ 0 f(t, u(t))dt = ϕ(u). thus, ϕ is w.l.s.c. claim 2. ϕ is coercive. for any u ∈ h10 (0, ∞), the fact that λ > 0, there exists α > 0 such that ϕ(u) ≥ α‖u‖2 + ∞ ∑ j=1 g(tj) u(tj ) ∫ 0 ij(s)ds − +∞ ∫ 0 f(t, u(t))dt. 234 m. djibaoui & t. moussaoui cubo 24, 2 (2022) using conditions (h1), (i1) and lemma 2.3, we have ϕ(u) ≥ α‖u‖2 − ∞ ∑ j=1 mjg(tj) p(tj) p(tj)|u(tj)| − +∞ ∫ 0 m(t) p(t) p(t)|u(t)|dt ≥ α‖u‖2 − ‖u‖∞,p ∞ ∑ j=1 mjg(tj) p(tj) − ‖u‖∞,p +∞ ∫ 0 m(t) p(t) dt ≥ α‖u‖2 − c‖u‖ ∞ ∑ j=1 mjg(tj) p(tj) − c‖u‖ ∥ ∥ ∥ ∥ m p ∥ ∥ ∥ ∥ l1 ≥ α‖u‖2 − c   ∞ ∑ j=1 mjg(tj) p(tj) + ∥ ∥ ∥ ∥ m p ∥ ∥ ∥ ∥ l1  ‖u‖, for some c > 0. then, the above inequality implies that lim ‖u‖→+∞ ϕ(u) = +∞. hence, ϕ is coercive. applying theorem 2.2, ϕ possesses a minimum which is a critical point of ϕ. finally, by (h1) and (i1), it is easy to check that ϕ is continuous and differentiable for any u ∈ h 1 0 (0, ∞) and that ϕ′(u)v = +∞ ∫ 0 u ′ v ′ + λ +∞ ∫ 0 uv + ∞ ∑ j=1 g(tj)ij(u(tj))v(tj)dt − +∞ ∫ 0 f(t, u(t))v(t)dt. (4.3) therefore, a critical point of ϕ is a weak solution of the problem (np). remark 4.3. assume m ∈ l2(0, ∞) in (h1), then it is easy to see that a weak solution u is in h2(0, ∞). example 4.4. take λ = 1, p(t) = e−t, m(t) = e−2t, g(t) = e−2t, mj = 1 j and ij(s) = 1 j + s2 , j ∈ n∗. the following ibvp:          −u′′(t) + u(t) = e−3t, a.e. t ∈ [0, ∞), △u′(j) = e−2j j + u2(j) , j ∈ n∗, u(0) = u(+∞) = 0, has at least one solution. (see figure 1) cubo 24, 2 (2022) variational methods to second-order dirichlet boundary value... 235 figure 1 theorem 4.5. assume the following conditions are satisfied: (h2) the function f is sublinear i.e., there exist a constant γ ∈ [0, 1) and positive functions a, b ∈ l1(0, ∞) with a p , b pγ , b pγ+1 ∈ l1[0, ∞) such that |f(t, u)| ≤ a(t) + b(t)|u|γ for (t, u) ∈ [0, +∞) × r. (i2) there exist constants δ ∈ [0, 1) and aj, bj > 0, j ∈ {1, 2, . . .} with ∞ ∑ j=1 ajg(tj), ∞ ∑ j=1 ajg(tj) p(tj) , ∞ ∑ j=1 bjg(tj) pδ(tj) , ∞ ∑ j=1 bjg(tj) pδ+1(tj) are convergent series, such that the impulsive functions ij have sublinear growths i.e., |ij(u)| ≤ aj + bj|u| δ for every u ∈ r, j ∈ {1, 2, . . .}. then there is a critical point of ϕ, and (np) has at least one solution. proof. claim 1. ϕ is weakly lower semi-continuous. under (h2) and (i2), arguing analogously to the proof of theorem 4.2, we find the weak lower semi-continuity of ϕ. claim 2. ϕ is coercive. in view of conditions (h2), (i2) and (4.2), for any u ∈ h 1 0 (0, ∞), we have ϕ(u) = 1 2 +∞ ∫ 0 u ′2 + λ 2 +∞ ∫ 0 u 2 + ∞ ∑ j=1 g(tj) u(tj) ∫ 0 ij(s)ds − +∞ ∫ 0 f(t, u(t))dt ≥ α‖u‖2 − ∞ ∑ j=1 g(tj) u(tj) ∫ 0 (aj + bj|s| δ)ds − +∞ ∫ 0 ( a(t)|u(t)| + b(t) γ + 1 |u(t)|γ+1 ) dt 236 m. djibaoui & t. moussaoui cubo 24, 2 (2022) ϕ(u) ≥ α‖u‖2 − ∞ ∑ j=1 g(tj) ( aj p(tj) p(tj)|u(tj)| + bj (δ + 1)pδ+1(tj) |p(tj)u(tj)| δ+1 ) − +∞ ∫ 0 a(t) p(t) p(t)|u(t)|dt − 1 (γ + 1) +∞ ∫ 0 b(t) pγ+1(t) |p(t)u(t)|γ+1dt ≥ α‖u‖2 − ‖u‖∞,p ∞ ∑ j=1 ajg(tj) p(tj) − ‖u‖δ+1∞,p ∞ ∑ j=1 bjg(tj) pδ+1(tj) − ‖u‖∞,p ∥ ∥ ∥ ∥ a p ∥ ∥ ∥ ∥ l1 − ‖u‖γ+1∞,p ∥ ∥ ∥ ∥ b pγ+1 ∥ ∥ ∥ ∥ l1 . hence, by lemma 2.3, we get ϕ(u) ≥ α‖u‖2 − c‖u‖ ∞ ∑ j=1 ajg(tj) p(tj) − cδ+1‖u‖δ+1 ∞ ∑ j=1 bjg(tj) pδ+1(tj) − c‖u‖ ∥ ∥ ∥ ∥ a p ∥ ∥ ∥ ∥ l1 − cγ+1‖u‖γ+1 ∥ ∥ ∥ ∥ b pγ+1 ∥ ∥ ∥ ∥ l1 ≥ α‖u‖2 − c   ∥ ∥ ∥ ∥ a p ∥ ∥ ∥ ∥ l1 + ∞ ∑ j=1 ajg(tj) p(tj)  ‖u‖ − cδ+1   ∞ ∑ j=1 bjg(tj) pδ+1(tj)  ‖u‖δ+1 − cγ+1 ∥ ∥ ∥ ∥ b pγ+1 ∥ ∥ ∥ ∥ l1 ‖u‖γ+1. since δ, γ ∈ [0, 1), then lim ‖u‖→+∞ ϕ(u) = +∞. this means, ϕ is coercive. using theorem 2.2, ϕ has a minimum, which is a critical point of ϕ. finally, from (h2) and (i2), we get the differentiability of ϕ such that its differentiable is defined by (4.3). consequently, (np) has at least one solution. remark 4.6. in (h2), assume a, b pγ ∈ l2(0, ∞), then a weak solution u is in h2(0, ∞). example 4.7. consider the following problem            −u′′(t) + u(t) = e−2t √ |u(t)| + e−3t, a.e. t ∈ [0, ∞), △u′(j) = e−2j ( 1 j2 + |s| 1 4 j ) , j ∈ n∗, u(0) = u(+∞) = 0, where λ = 1, p(t) = e−t, g(t) = e−2t, aj = 1 j2 , bj = 1 j and ij(s) = 1 j2 + |s| 1 4 j , j ∈ n∗. by simple calculations, all conditions in theorem 4.5 are satisfied, then (4.1) has at least one solution. cubo 24, 2 (2022) variational methods to second-order dirichlet boundary value... 237 references [1] l. bai and j. j. nieto, “variational approach to differential equations with not instantaneous impulses”, appl. math. lett., vol. 73, pp. 44–48, 2017. 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[10] y. wei, “existence and uniqueness of solutions for a second-order delay differential equation boundary value problem on the half-line”, bound. value probl., art. id 752827, 14 pages, 2008. introduction preliminaries impulsive linear problem impulsive nonlinear problem cubo, a mathematical journal vol. 23, no. 01, pp. 87–96, april 2021 doi: 10.4067/s0719-06462021000100087 hyper generalized pseudo q-symmetric semi-riemannian manifolds adara m. blaga1 manoj ray bakshi2 kanak kanti baishya3 1 department of mathematics, west university of timişoara, timişoara, românia. adarablaga@yahoo.com 2,3 department of mathematics, kurseong college, kurseong, darjeeling, india. raybakshimanoj@gmail.com; kanakkanti.kc@gmail.com abstract the object of the present paper is to study the properties of a hyper generalized pseudo q-symmetric semi-riemannian manifold, proving that under certain assumptions, it is a perfect fluid spacetime. resumen el objetivo del presente art́ıculo es estudiar las propiedades de una variedad semi-riemanniana hiper generalizada pseudo q-simétrica, probando que bajo ciertas condiciones, es un espacio-tiempo fluido perfecto. keywords and phrases: q-curvature tensor, perfect fluid spacetime. 2020 ams mathematics subject classification: 53c15, 53c25. accepted: 21 january, 2021 received: 04 july, 2020 ©2021 a. m. blaga et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462021000100087 https://orcid.org/0000-0003-0237-3866 https://orcid.org/0000-0002-5981-7715 https://orcid.org/0000-0001-9331-7960 88 a. m. blaga, m. r. bakshi & k. k. baishya cubo 23, 1 (2021) 1 introduction let r, s, l and r denote the curvature tensor, ricci tensor, ricci operator and the scalar curvature of a (semi)-riemannian manifold, respectively. it is mantica and suh [5] who have introduced the notion of q-curvature tensor. in an n-dimensional riemannian or semi-riemannian manifold (mn,g) (n > 2), the q-curvature tensor is defined as r(y,u,v,w) = q(y,u,v,w) + ψ n− 1 [g(y,w)g(u,v ) −g(y,v )g(u,w)], (1.1) where y,u,v,w are arbitrary vector fields on mn and ψ is a scalar function. semi-riemannian manifolds with ricci tensor s of the form s(y,v ) = ag(y,v ) + bt(y )t(v ), for any vector fields y,v , are often termed as perfect fluid spacetimes, where a and b are scalars and the vector field %, metrically equivalent to the 1-form t (that is, g(y,%) = t(y )), is a unit time like vector field (that is, g(%,%) = −1). an n-dimensional semi-riemannian manifold is said to be hyper generalized pseudo q-symmetric (which will be abbreviated hereafter as (hgpqs)n) if it satisfies the equation (∇xq)(y,u,v,w) (1.2) = 2a1(x)q(y,u,v,w) + a1(y )q(x,u,v,w) +a1(u)q(y,x,v,w) + a1(v )q(y,u,x,w) +a1(w)q(y,u,v,x) + 2a2(x)(g ∧s)(y,u,v,w) +a2(y )(g ∧s)(x,u,v,w) + a2(u)(g ∧s)(y,x,v,w) +a2(v )(g ∧s)(y,u,x,w) + a2(w)(g ∧s)(y,u,v,x), where (g ∧s)(y,u,v,w) = g(y,w)s(u,v ) + g(u,v )s(y,w) (1.3) −g(y,v )s(u,w) −g(u,w)s(y,v ) and a1, a2 are non-zero 1-forms whose g-dual vector fields will be denoted by θ1 and θ2, i.e. a1(x) = g(x,θ1) and a2(x) = g(x,θ2). we organized our paper as follows: section 2 is concerned with preliminaries. after preliminaries, some curvature properties of (hgpqs)n manifolds are studied in section 3. it is obtained that the q-curvature tensor in a (hgpqs)n manifold satisfies 2nd bianchi’s identity. it is further obtained that the scalar function ψ is always constant. in section 4 we investigate properties of divergence-free (hgpqs)n manifolds and we prove that a divergence-free (hgpqs)n manifold (n > 2) under the assumption a1(q(y,u)v ) = 0 is a perfect fluid spacetime as well as the integral cubo 23, 1 (2021) hyper generalized pseudo q-symmetric semi-riemannian manifolds 89 curves of the vector field % are geodesics and the vector field % is irrotational, if the associated vector fields % and σ corresponding to the 1-forms t1 and t2 are related by (r − 1)% + nσ = 0. 2 preliminaries in this section, some relations useful to the study of (hgpqs)n manifolds are obtained. let {ei} be an orthonormal basis of the tangent space at each point of the manifold, where 1 ≤ i ≤ n. from (1.1) we can easily verify that the tensor q satisfies the following properties: (i) q(y,u)v + q(u,y )v = 0, (ii) q(y,u)v + q(u,v )y + q(v,y )u = 0, (2.1) where g(q(x,y )u,v ) = q(x,y,u,v ). also from (1.1) we have n∑ i=1 �iq(x,y,ei,ei) = 0 = n∑ i=1 �iq(ei,ei,w,u) (2.2) and n∑ i=1 �iq(ei,y,v,ei) = n∑ i=1 �iq(y,ei,ei,v ) = s(y,v ) −ψg(y,v ) (2.3) =: z(y,v ), where �i = g(ei,ei) = ±1, s(x,y ) = n∑ i=1 �ig(r(x,ei)ei,y ), r = n∑ i=1 �is(ei,ei). from (1.1) and (2.1) it follows that (i) q(x,y,u,v ) + q(x,y,v,u) = 0, (ii) q(x,y,u,v ) −q(u,v,x,y ) = 0. (2.4) 3 some curvature properties of (hgpqs)n manifolds in this section we prove that in a (hgpqs)n manifold, the q-curvature tensor satisfies 2nd bianchi’s identity, that is, (∇xq)(y,u,v,w) + (∇y q)(u,x,v,w) + (∇uq)(x,y,v,w) = 0. (3.1) 90 a. m. blaga, m. r. bakshi & k. k. baishya cubo 23, 1 (2021) in view of (1.1), (1.2) and (3.1) we get (∇xq)(y,u,v,w) + (∇y q)(u,x,v,w) + (∇uq)(x,y,v,w) (3.2) = a1(v )[q(y,u,x,w) + q(u,x,y,w) + q(x,y,u,w)] +a1(w)[q(y,u,v,x) + q(u,x,v,y ) + q(x,y,v,u)] +a2(v )[(g ∧s)(y,u,x,w) + (g ∧s)(u,x,y,w) +(g ∧s)(x,y,u,w)] + a2(w)[(g ∧s)(y,u,v,x) +(g ∧s)(u,x,v,y ) + (g ∧s)(x,y,v,u)]. using (1.3) and 1st bianchi’s identity for the q-curvature tensor in (3.2) and then simplifying, we obtain (3.1). thus we can state the following: theorem 3.1. the q-curvature tensor in a (hgpqs)n manifold satisfies 2nd bianchi’s identity. using (1.1) in (3.1), we have (∇xr)(y,u,v,w) + (∇y r)(u,x,v,w) + (∇ur)(x,y,v,w) (3.3) − dψ(x) (n− 1) [g(y,w)g(u,v ) −g(y,v )g(u,w)] − dψ(y ) (n− 1) [g(u,w)g(x,v ) −g(u,v )g(x,w)] − dψ(u) (n− 1) [g(x,w)g(y,v ) −g(x,v )g(y,w)] = 0. by virtue of 2nd bianchi’s identity for the riemannian curvature tensor, (3.3) yields dψ(x) (n− 1) [g(y,w)g(u,v ) −g(y,v )g(u,w)] (3.4) + dψ(y ) (n− 1) [g(u,w)g(x,v ) −g(u,v )g(x,w)] + dψ(u) (n− 1) [g(x,w)g(y,v ) −g(x,v )g(y,w)] = 0. contracting u and v in (3.4), we have (n− 2)[dψ(x)g(y,w) −dψ(y )g(x,w)] = 0 (3.5) which yields after further contraction (n− 1)(n− 2)dψ(x) = 0. this implies that dψ(x) = 0, that is, ψ is constant since n > 2 and leads to the following: theorem 3.2. in a (hgpqs)n manifold, the scalar function ψ is always constant. cubo 23, 1 (2021) hyper generalized pseudo q-symmetric semi-riemannian manifolds 91 consequently, one can easily bring out the following: theorem 3.3. in a (hgpqs)n manifold, (divq)(x,y )z and (divr)(x,y )z are equivalent. in view of (1.1), (1.2) and theorem 3.2 we have (∇xr)(y,u,v,w) (3.6) = 2a1(x)q(y,u,v,w) + a1(y )q(x,u,v,w) +a1(u)q(y,x,v,w) + a1(v )q(y,u,x,w) +a1(w)q(y,u,v,x) + 2a2(x)(g ∧s)(y,u,v,w) +a2(y )(g ∧s)(x,u,v,w) + a2(u)(g ∧s)(y,x,v,w) +a2(v )(g ∧s)(y,u,x,w) + a2(w)(g ∧s)(y,u,v,x) which yields (∇xs)(u,v ) (3.7) = [f1(x) + f2(x)]s(u,v ) + f2(u)s(x,v ) + f2(v )s(u,x) +[f3(x) + f4(x)]g(u,v ) + f4(u)g(x,v ) + f4(v )g(u,x) +a1(q(x,u)v ) −a1(q(v,x)u) after contraction over y and w , where f1(x) = a1(x) + (n + 1)a2(x), f2(x) = a1(x) + (n− 3)a2(x), f3(x) = ra2(x) −ψa1(x) + 3a2(lx), f4(x) = ra2(x) −ψa1(x) −a2(lx), where l is the ricci operator defined by g(lx,y ) = s(x,y ). definition 3.4. an n-dimensional semi-riemannian manifold is called almost generalized pseudo ricci symmetric if the non-flat ricci curvature tensor satisfies the equation (∇xs)(u,v ) = [a(x) + b(x)]s(u,v ) + a(u)s(x,v ) + a(v )s(u,x) +[c(x) + d(x)]g(u,v ) + c(u)g(x,v ) + c(v )g(u,x), where a,b,c and d are non-zero 1-forms whose g-dual vector fields will be denoted by γ1, γ2, δ1 and δ2, i.e. a(x) = g(x,γ1), b(x) = g(x,γ2), c(x) = g(x,δ1) and d(x) = g(x,δ2). thus we can state the following: 92 a. m. blaga, m. r. bakshi & k. k. baishya cubo 23, 1 (2021) theorem 3.5. a (hgpqs)n manifold (n > 2) under the assumption a1(q(x,u)v ) = a1(q(v,x)u) is necessarily almost generalized pseudo ricci symmetric. making use of (2.3) in (3.7), we get (∇xz)(u,v ) (3.8) = [f1(x) + f2(x)]z(u,v ) + f2(u)z(x,v ) + f2(v )z(u,x) +[f3(x) + ψf1(x) + f4(x) + ψf2(x)]g(u,v ) +[f4(u) + ψf2(u)]g(x,v ) + [f4(v ) + ψf2(v )]g(u,x), where z = s −ψg is the tensor considered in ([4], [6], [7]). this leads to the following: theorem 3.6. a (hgpqs)n manifold (n > 2) under the assumption a1(q(x,u)v ) = a1(q(v,x)u) is necessarily almost generalized pseudo z-symmetric. 4 (hgpqs)n manifolds (n > 2) with divq = 0 let (mn,g) be a semi-riemannian manifold of dimension n and let {ei} be an orthonormal basis of the tangent space tpm at any point p ∈ m and �i = ±1. then the divergence of a vector field u is defined as divu = n∑ i=1 �ig(∇eiu,ei), and the divergence of a tensor field of type (1, 3), which is a tensor field of type (0, 3), is defined as (divk)(x,y )z = n∑ i=1 �ig((∇eik)(x,y )z,ei). now (divq)(y,u)v = n∑ i=1 �ig((∇eiq)(y,u)v,ei) = n∑ i=1 �i[2a1(ei)q(y,u,v,ei) + a1(y )q(ei,u,v,ei) +a1(u)q(y,ei,v,ei) + a1(v )q(y,u,ei,ei) +a1(ei)q(y,u,v,ei) + 2a2(ei)(g ∧s)(y,u,v,ei) +a2(y )(g ∧s)(ei,u,v,ei) + a2(u)(g ∧s)(y,ei,v,ei) +a2(v )(g ∧s)(y,u,ei,ei) + a2(ei)(g ∧s)(y,u,v,ei)] cubo 23, 1 (2021) hyper generalized pseudo q-symmetric semi-riemannian manifolds 93 = 3a1(q(y,u)v ) + a1(y )[s(u,v ) −ψg(u,v )] −a1(u)[s(y,v ) −ψg(y,v )] + 3a2(y )s(u,v ) +3a2(ly )g(u,v ) − 3a2(lu)g(y,v ) − 3a2(u)s(y,v ) +a2(y )[(n− 2)s(u,v ) + rg(u,v )] −a2(u)[(n− 2)s(y,v ) + rg(y,v )] = 3a1(q(y,u)v ) + s(u,v )[a1(y ) + (n + 1)a2(y )] −s(y,v )[a1(u) + (n + 1)a2(u)] +g(u,v )[3a2(ly ) + ra2(y ) −ψa1(y )] −g(y,v )[3a2(lu) + ra2(u) −ψa1(u)] = 3a1(q(y,u)v ) + t1(y )s(u,v ) −t1(u)s(y,v ) +t2(y )g(u,v ) −t2(u)g(y,v ), hence (divq)(y,u)v = 3a1(q(y,u)v ) + t1(y )s(u,v ) −t1(u)s(y,v ) (4.1) +t2(y )g(u,v ) −t2(u)g(y,v ), where t1(y ) = a1(y ) + (n + 1)a2(y ) =: g(y,%), for % = θ1 + (n + 1)θ2, t2(y ) = 3a2(ly ) + ra2(y ) −ψa1(y ) =: g(y,σ), for σ = 3lθ2 + rθ2 −ψθ1. assuming (divq)(y,u)v = 0 and a1(q(y,u)v ) = 0, we get from the above equation t1(y )s(u,v ) + t2(y )g(u,v ) = t1(u)s(y,v ) + t2(u)g(y,v ). (4.2) now contracting (4.2) over u and v we get s(y,%) = rt1(y ) + (n− 1)t2(y ). (4.3) again putting v = % in (4.2) we get (n− 2)[t1(y )t2(u) −t1(u)t2(y )] = 0, (4.4) which under the assumption n > 2 implies t1(y )t2(u) = t1(u)t2(y ). now putting u = % in (4.2) and using (4.3) and (4.4) we get t1(%)s(y,v ) + t2(%)g(y,v ) = t1(y )[rt1(v ) + nt2(v )] (4.5) and we can state: 94 a. m. blaga, m. r. bakshi & k. k. baishya cubo 23, 1 (2021) theorem 4.1. a divergence-free (hgpqs)n manifold (n > 2) under the assumption a1(q(y,u)v ) = 0 is a perfect fluid spacetime with unit timelike vector field %, provided the associated vector fields % and σ corresponding to the 1-forms t1 and t2 are related by (r−1)% + nσ = 0. in this case, (4.5) becomes s(y,v ) = ag(y,v ) −t1(y )t1(v ), (4.6) where a =: t2(%). again, (divq)(y,u)v = 0 gives (∇y s)(u,v ) − (∇us)(y,v ) = 0. (4.7) now using (4.6) in (4.7) we find da(y )g(u,v ) −da(u)g(y,v ) (4.8) −[t1(v )(∇y t1)(u) + t1(u)(∇y t1)(v )] +[t1(v )(∇ut1)(y ) + t1(y )(∇ut1)(v )] = 0. taking a frame field and contracting y and v we get (n− 1)da(u) + [t1(u)(δt1) + (∇%t1)(u)] = 0, (4.9) where δt1 = n∑ i=1 �i(∇eit1)(ei). setting v = y = % in (4.8) we find (∇%t1)(u) = −da(u) −da(%)t1(u). (4.10) substituting (4.10) in (4.9) we get (n− 2)da(u) + t1(u)(δt1) −da(%)t1(u) = 0 (4.11) which yields δt1 = (n− 1)da(%) (4.12) for u = %. using (4.12) in (4.11) we obtain da(u) = −t1(u)da(%), (4.13) provided n > 2. cubo 23, 1 (2021) hyper generalized pseudo q-symmetric semi-riemannian manifolds 95 putting v = % in (4.8) and using (4.13) we get (∇y t1)(u) − (∇ut1)(y ) = 0. this means that the 1-form t1 is closed, that is, dt1(y,u) = 0. hence g(∇u%,y ) = g(∇y %,u) for all u,y, (4.14) which yields g(∇%%,y ) = g(∇y %,%), (4.15) for u = %. since g(∇y %,%) = 0, from (4.15) it follows that g(∇%%,y ) = 0 for all y . hence ∇%% = 0. this implies that the integral curves of the vector field % are geodesics. therefore we can state the following: theorem 4.2. in a divergence-free (hgpqs)n manifold (n > 2) under the assumption a1(q(y,u)v ) = 0, the integral curves of the unit timelike vector field % are geodesics, provided the associated vector fields % and σ corresponding to the 1-forms t1 and t2 are related by (r − 1)% + nσ = 0. taking into account that the divergence of the conformal curvature tensor of a riemannian manifold (mn,g) is ([3], [6]): (divc)(x,y )z = n− 3 n− 2 [(∇xs)(y,z) − (∇y s)(x,z)] (4.16) = n− 3 n− 2 (divq)(x,y )z, for any vector fields x,y,z on mn, from the lemma 2.1 of [2] we infer theorem 4.3. let (m,g) be a (hgpqs)n perfect fluid spacetime (n > 2). if (divq)(x,y )z = 0, for any vector fields x,y,z on m, then the unit timelike vector field % is irrotational. also, in [2] was proved the following result: theorem 4.4. [2] let (m,g) be a (hgpqs)n perfect fluid spacetime (n > 2). if (divq)(x,y )z = 0, for any vector fields x,y,z on m, then (m,g) is a grw spacetime whose fiber is einstein. acknowledgements. the authors are grateful to the referees for the valuable suggestions and remarks that definitely improved the paper. 96 a. m. blaga, m. r. bakshi & k. k. baishya cubo 23, 1 (2021) references [1] k. k. baishya, f. ozen zengin and j. mikeš, “on hyper generalised weakly symmetric manifolds”, nineteenth international conference on geometry, integrability and quantization, 02– 07, june 2017, varna, bulgaria iväılo m. mladenov and akira yoshioka, editors avangard prima, sofia 2018, pp. 1–10. [2] c. a. mantica, u. c. de, y. j. suh, and l. g. molinari, “perfect fluid spacetimes with harmonic generalized curvature tensor”, osaka j. math., vol. 56, pp. 173–182, 2019. [3] c. a. mantica, and l. g. molinari, “a second-order identity for the riemann tensor and applications”, colloq. math., vol. 122, no. 1, pp. 69–82, 2011. [4] c. a. mantica, and l. g. molinari, “weakly z-symmetric manifolds”, acta math. hung., vol. 135, no. 1-2, pp. 80–96, 2012. [5] c. a. mantica, and y. j. suh, “pseudo q-symmetric semi-riemannian manifolds”, int. j. geom. meth. mod. phys., vol. 10, no. 5, 2013. [6] c. a. mantica, and y. j. suh, “pseudo z-symmetric riemannian manifolds with harmonic curvature tensors”, int. j. geom. meth. mod. phys., vol. 9, no. 1, 2012, 1250004. [7] c. a. mantica, and y. j. suh, “recurrent z-forms on riemannian and kaehler manifolds”, int. j. geom. meth. mod. phys., vol. 9, no. 7, 2012, 1250059. introduction preliminaries some curvature properties of (hgpqs)n manifolds (hgpqs)n manifolds (n>2) with divq=0 cubo, a mathematical journal vol. 24, no. 03, pp. 439–455, december 2022 doi: 10.56754/0719-0646.2403.0439 a class of nonlocal impulsive differential equations with conformable fractional derivative mohamed bouaouid 1, b ahmed kajouni 1 khalid hilal 1 said melliani 1 1 sultan moulay slimane university, faculty of sciences and technics, department of mathematics, bp 523, 23000, béni mellal, morocco. bouaouidfst@gmail.com b kajjouni@gmail.com khalid.hilal.usms@gmail.com said.melliani@gmail.com abstract in this paper, we deal with the duhamel formula, existence, uniqueness, and stability of mild solutions of a class of nonlocal impulsive differential equations with conformable fractional derivative. the main results are based on the semigroup theory combined with some fixed point theorems. we also give an example to illustrate the applicability of our abstract results resumen en este art́ıculo, tratamos la fórmula de duhamel, la existencia, unicidad y estabilidad de soluciones mild de una clase de ecuaciones diferenciales no locales impulsivas con derivadas fraccionarias conformables. los resultados principales se basan en teoŕıa de semigrupos, combinada con algunos teoremas de punto fijo. también entregamos un ejemplo para ilustrar la aplicabilidad de nuestros resultados abstractos. keywords and phrases: functional-differential equations with fractional derivatives; groups and semigroups of linear operators; nonlocal conditions; impulsive conditions; conformable fractional derivatives. 2020 ams mathematics subject classification: 34k37, 47d03. accepted: 30 september, 2022 received: 10 march, 2022 c©2022 m. bouaouid et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2403.0439 https://orcid.org/0000-0003-0474-0121 https://orcid.org/0000-0001-8484-6107 https://orcid.org/0000-0002-0806-2623 https://orcid.org/0000-0002-5150-1185 mailto:bouaouidfst@gmail.com mailto:kajjouni@gmail.com mailto:khalid.hilal.usms@gmail.com mailto:said.melliani@gmail.com 440 m. bouaouid, a. kajouni, k. hilal & s. melliani cubo 24, 3 (2022) 1 introduction fractional calculus has attracted the attention of many researchers, due to its wide range of applications in modeling of various natural phenomena in different fields of sciences and engineering including: physics, engineering, biology, finance, chemistry [3, 26, 31, 35, 38, 41, 43, 44, 45, 46, 47, 48]. for better understanding these phenomena, several definitions of fractional derivatives have been introduced such as riemann-liouville and caputo definitions, for more details we refer to the books [31, 41]. unfortunately, these definitions are very complicated to handle in real models. however, in [30] a new definition of fractional derivative named conformable fractional derivative was initiated. this novel fractional derivative is very easy and satisfies all the properties of the classical derivative. the advantage of the conformable fractional derivative is very remarkable compared to other fractional derivatives in many comparisons. indeed, for example, in the work [15] the authors gave the solution of conformable-fractional telegraph equations in terms of the classical exponential function, however for the caputo-fractional telegraph equations considered in the very good papers [19, 20, 36], the fundamental solution cannot be given in terms of the exponential function as in the conformable-fractional case, and therefore the authors have been introduced the so-called mittag-leffler function. another comparison, we notice that the constants of increases of the norms of the control bounded operators w and w −1 in the application of the work [27] are given directly in a simple way in terms of the exponential function, contrary, for the caputo fractional derivative in the application of the nice work [51] these constants are given in terms of the so-called mittag-leffler function. for more details and conclusions concerning the uses and applications of conformable fractional calculus, we refer to the works [2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, 17, 22, 23, 24, 25, 28, 29, 42, 49]. on the other hand, impulsive differential equations are crucial in description of dynamical processes with short-time perturbations [6, 32, 50]. actually, the cauchy problem of impulsive differential equations attracts the attention of many authors [1, 9, 33, 34, 37]. for example, liang et al. [33] have proved the existence and uniqueness of mild solutions for the cauchy problem        ẋ(t) = ax(t) + f(t, x(t)), t ∈ [0, τ], t 6= t1, t2, . . . , tn, x(0) = x0 + g(x), x(t+i ) = x(t − i ) + hi(x(ti)), i = 1, 2, . . . , n, . (1.1) by using the following classical duhamel formula: x(t) = t (t)[x0 + g(x)] + ∑ 0 0 and dαx(0) dtα = lim t−→0+ dαx(t) dtα , provided that the limits exist. the fractional integral iα(.) associated with the conformable fractional derivative is defined by iα(x)(t) = ∫ t 0 sα−1x(s)ds. theorem 2.2 ([30]). if x(.) is a continuous function in the domain of iα(.), then we have dα(iα(x)(t)) dtα = x(t). definition 2.3 ([41]). the laplace transform of a function x(.) is defined by l(x(t))(λ) := ∫ +∞ 0 e −λt x(t)dt, λ > 0. it is remarkable that the above transform is not adequate to solve conformable fractional differential equations. for this reason, we consider the following definition, which appeared in [2]. definition 2.4 ([2]). the fractional laplace transform of order α of a function x(.) is defined by lα(x(t))(λ) := ∫ +∞ 0 t α−1 e −λ t α α x(t)dt, λ > 0. the following proposition gives us the actions of the fractional integral and the fractional laplace transform on the conformable fractional derivative, respectively. proposition 2.5 ([2]). if x(.) is a differentiable function, then we have the following results i α ( dαx(.) dtα ) (t) = x(t) − x(0), lα ( dαx(t) dtα ) (λ) = λlα(x(t))(λ) − x(0). we end this preliminaries by the following remark. cubo 24, 3 (2022) a class of nonlocal impulsive differential equations... 443 remark 2.6 ([14]). for two arbitrary functions x(.) and y(.), we have lα ( x ( tα α )) (λ) = l(x(t))(λ), lα ( ∫ t 0 sα−1x ( tα − sα α ) y(s)ds ) (λ) = l(x(t))(λ)lα(y(t))(λ). 3 main results we first prove the conformable fractional duhamel formula (1.6). to do so, for t ∈ [0, t1], we apply the fractional laplace transform in equation (1.5), we obtain lα(x(t))(λ) = (λ − a)−1[x0 + g(x)] + (λ − a)−1lα(f(t, x(t)))(λ). according to the inverse fractional laplace transform and remark (2.6), we get x(t) = t ( tα α ) [x0 + g(x)] + ∫ t 0 s α−1 t ( tα − sα α ) f(s, x(s))ds, where (t (t))t≥0 is the semigroup generated by the linear part a on the banach space x, that is, (t (t))t≥0 is one parameter family of bounded linear operators on x satisfying the following properties (1) t (0) = i, (2) t (s + t) = t (s)t (t) for all t, s ∈ r+, (3) lim t↓0 ‖ t (t)x − x ‖= 0 for each fixed x ∈ x, (4) lim t↓0 t (t)x − x t = ax, for x ∈ x, provided that the limit exists. as in [37], we assume that the solution of equation (1.5) is such that at the point of discontinuity tk, we have x(t − k ) = x(tk). hence, one has x(t−1 ) = t ( tα1 α ) [x0 + g(x)] + ∫ t1 0 sα−1t ( tα1 − sα α ) f(s, x(s))ds. for t ∈ (t1, t2], using the fractional laplace transform in equation (1.5), we obtain x(t) = t ( tα − tα1 α ) x(t+1 ) + ∫ t t1 sα−1t ( tα − sα α ) f(s, x(s))ds = t ( tα − tα1 α ) [x(t−1 ) + h1(x(t1))] + ∫ t t1 sα−1t ( tα − sα α ) f(s, x(s))ds. replacing x(t−1 ) by its expression in the above equation, we get x(t) = t ( tα − tα1 α )[ t ( tα1 α ) (x0 + g(x)) + ∫ t1 0 sα−1t ( tα1 − sα α ) f(s, x(s))ds + h1(x(t1)) ] + ∫ t t1 s α−1 t ( tα − sα α ) f(s, x(s))ds. 444 m. bouaouid, a. kajouni, k. hilal & s. melliani cubo 24, 3 (2022) by using a computation, the above equation becomes x(t) = t ( tα α ) [x0 + g(x)] + t ( tα − tα1 α ) [h1(x(t1))] + ∫ t 0 s α−1 t ( tα − sα α ) f(s, x(s))ds. in particular, for t = t−2 , one has x(t−2 ) = t ( tα2 α ) [x0 + g(x)] + t ( tα2 − tα1 α ) [h1(x(t1))] + ∫ t2 0 sα−1t ( tα2 − sα α ) f(s, x(s))ds. as the same, for t ∈ (t2, t3], we obtain x(t) = t ( tα − tα2 α ) x(t+2 ) + ∫ t t2 sα−1t ( tα − sα α ) f(s, x(s))ds = t ( tα − tα1 α ) [x(t−2 ) + h2(x(t2))] + ∫ t t2 sα−1t ( tα − sα α ) f(s, x(s))ds. hence, replacing x(t−2 ) by its expression, we have x(t) = t ( tα − tα2 α )[ t ( tα2 α ) [x0 + g(x)] + t ( tα2 − tα1 α ) [h1(x(t1))] + ∫ t2 0 sα−1t ( tα2 − sα α ) f(s, x(s))ds + h2(x(t2)) ] + ∫ t t2 s α−1 t ( tα − sα α ) f(s, x(s))ds. using a computation, we get x(t) = t ( tα α ) [x0 + g(x)] + t ( tα − tα1 α ) [h1(x(t1))] + t ( tα − tα2 α ) [h2(x(t2))] + ∫ t t2 s α−1 t ( tα − sα α ) f(s, x(s))ds. repeating the same process, we obtain the following conformable fractional duhamel formula x(t) = t ( tα α ) [x0 + g(x)] + ∑ 0 0 there exists a function µr ∈ l∞([0, τ], r+) such that sup ‖x‖≤r ‖ f(t, x) ‖≤ µr(t), for all t ∈ [0, τ]. cubo 24, 3 (2022) a class of nonlocal impulsive differential equations... 445 (h2) the function f(., x) : [0, τ] −→ x is continuous, for all x ∈ x. (h3) there exists a constant l1 > 0 such that ‖ g(y) − g(x) ‖≤ l1 | y − x |c, for all x, y ∈ c. (h4) there exist constants ci > 0 such that ‖ hi(y(ti))−hi(x(ti)) ‖≤ ci | y −x |c, for all x, y ∈ c. theorem 3.2. if (t (t))t>0 is compact and (h1) − (h4) are satisfied, then the conformable fractional cauchy problem (1.5) has at least one mild solution, provided that ( l1 + n ∑ i=1 ci ) sup t∈[0,τ] ∣ ∣ ∣ ∣ t ( tα α ) ∣ ∣ ∣ ∣ < 1. proof. let br = {x ∈ c, | x |c≤ r}, where r ≥ sup t∈[0,τ] ∣ ∣ ∣ ∣ t ( tα α ) ∣ ∣ ∣ ∣ [ ‖ x0 ‖ + ‖ g(0) ‖ + n ∑ i=1 ‖ hi(0) ‖ + τα α | µr |l∞([0,τ],r+) ] 1 − ( l1 + n ∑ i=1 ci ) sup t∈[0,τ] ∣ ∣ ∣ ∣ t ( tα α ) ∣ ∣ ∣ ∣ . in order to use the krasnoselskii fixed-point theorem, we consider the following operators γ1 and γ2 defined by γ1(x)(t) = t ( tα α ) [x0 + g(x)] + ∑ 00 assures that lim t2−→t1 ∣ ∣ ∣ ∣ t ( tα2 − tα1 α ) − i ∣ ∣ ∣ ∣ = 0. hence, combining this fact with the above inequality, we conclude that γ2(x), x ∈ br are equicontinuous on [0, τ]. claim 2: we prove that the set {γ2(x)(t), x ∈ br} is relatively compact in x. for some fixed t ∈]0, τ] let ε ∈]0, t[, x ∈ br and define the operator γε2 as follows γε2(x)(t) = t ( εα α ) ∫ (tα−εα) 1 α 0 sα−1t ( tα − sα − εα α ) f(s, x(s))ds. since (t (t))t>0 is compact, then the set {γε2(x)(t), x ∈ br} is relatively compact in x. by using a computation combined with assumption (h1), we get ‖ γε2(x)(t) − γ2(x)(t) ‖≤| µr |l∞([0,τ],r+) sup t∈[0,τ] ∣ ∣ ∣ ∣ t ( tα α ) ∣ ∣ ∣ ∣ εα α . therefore, we deduce that the {γ2(x)(t), x ∈ br} is relatively compact in x. for t = 0 the set {γ2(x)(0), x ∈ br} is compact. thus, the set {γ2(x)(t), x ∈ br} is relatively compact in x for all t ∈ [0, τ]. by using the arzelà-ascoli theorem, we conclude that the operator γ2 is compact. in conclusion, by the above steps combined with the krasnoselskii fixed-point theorem, we conclude that γ1 + γ2 has at least one fixed point in c, which is a mild solution of conformable fractional cauchy problem (1.5). to obtain the uniqueness of the mild solution, we need the following assumption: (h5) there exists a constant l2 > 0 such that ‖ f(t, y) − f(t, x) ‖≤ l2 ‖ y − x ‖, for all x, y ∈ x and t ∈ [0, τ]. theorem 3.3. assume that (h2) − (h5) hold, then the conformable fractional cauchy problem (1.5) has an unique mild solution, provided that ( l1 + n ∑ i=1 ci + τα α l2 ) sup t∈[0,τ] ∣ ∣ ∣ ∣ t ( tα α ) ∣ ∣ ∣ ∣ < 1. proof. define the operator γ : c −→ c by: γ(x)(t) = t ( tα α ) [x0 + g(x)] + ∑ 0 1 for 1 ≤ j ≤ n, say kn > 1. if kn > 2, we have (m1 · · · mn) + mn 6= r and (m1 · · · mn)mn 6= 0. that is, γ ∗ 1 (r) 6= γ∗ 2 (r). now consider the case when kn = 2. if ki > 1 for some i 6= n, (m1 · · · mn)+ mn 6= r and (m1 · · · mn)mn 6= 0. if ki = 1 ∀i 6= n, (m1 · · · ml−1ml+1 · · · m 2 n) + ml = r where l 6= n. but, (m1 · · · ml−1ml+1 · · · m 2 n)ml = 0. so, γ ∗ 1 (r) 6= γ∗ 2 (r). case (ii)(b): j(r) is not nilpotent. in this case we have (m1 · · · mn) + m1 6= r and (m1 · · · mn)m1 6= 0. thus, if γ∗ 1 (r) = γ∗ 2 (r), r cannot be semi-local. theorem 3.8. let (r, m) be an artin local ring. then, γ∗1(r) = γ ∗ 2 (r) if and only if either m has index of nilpotency 2 or m is principal with index of nilpotency 3. proof. follows from remark 2.2 and theorem 1.1. 4 some parameters of γ∗2(r) in this section we find the clique number and the domination number of γ∗ 2 (r). theorem 4.1. cl(γ∗2(r)) = | max r|. proof. clearly max r induces a complete subgraph. let a be any non-zero non-maximal proper ideal of r. then a is contained in a maximal ideal. that is, there exists a maximal ideal m such that a is not adjacent to m. thus, max r induces a maximal complete subgraph. now suppose s = {ai : i ∈ λ}, where λ is an index set, induces a complete subgraph in γ ∗ 2 (r). then one maximal ideal can contain at most one ai ∈ s. that is, there exists an injective map from s to max r. this implies, |s| ≤ | max r|. thus, cl(γ∗ 2 (r)) = | max r|. theorem 4.2. let r be a semi local ring with | max r| = n > 2. then, γ(γ∗ 2 (r)) = | max r| + number of isolated vertices in γ∗2(r). proof. let γ∗∗ 2 (r) be the connected component of γ∗ 2 (r) induced by the non-isolated vertices of γ∗2(r). now, by theorem 2.7, it is enough to show that γ(γ ∗∗ 2 (r)) = | max r|. let max r = {m1, m2, . . . , mn}. clearly max r is a dominating set for γ ∗∗ 2 (r). now consider, s = {m2 · · · mn, m1m3 · · · mn, . . . , m1m2 · · · mn−1}, which is an independent set in γ ∗∗ 2 (r). note that any ideal a /∈ s can be adjacent only to at most one element of s. so every dominating set cubo 24, 2 (2022) ideal based graph structures for commutative rings 339 in γ∗∗2 (r) must contain at least n elements. thus, γ(γ ∗∗ 2 (r)) = n = | max r|. hence the result follows. remark 4.3. if r is a semi-local ring with | max r| = 2 then, the above result is not true. for example, if r is a direct sum of two fields, γ(γ∗ 2 (r)) = γ(k2) = 1 but | max r| = 2. 5 splitness a graph (v, e) is said to be a split graph if v is the disjoint union of two sets k and s where k induces a complete subgraph and s is an independent set. then, we can assume either k is a clique or s is a maximal independent set. in [6] & [7], the authors have carried out a detailed study on splitness of some graphs associated with a ring. in this section we continue the study in the case of γ∗2(r). lemma 5.1. let r = r1 × r2 × r3 be a ring. if γ ∗ 2(r) is split, each ri must be a field. proof. suppose r1 is not a field. then there exists a proper non-zero ideal i of r1. then, {i × r2 × r3, r1 × r2 × 0, 0 × r2 × r3, r1 × 0 × 0} induces a c4 in γ ∗ 2 (r), a contradiction. lemma 5.2. if fi (1 ≤ i ≤ 3) are fields and r = f1 × f2 × f3 then γ ∗ 2 (r) is split. proof. v (γ∗ 2 (r)) can be partitioned into k = {f1 × f2 × 0, f1 × 0 × f3, 0 × f2 × f3} and s = {f1 × 0 × 0, 0 × f2 × 0, 0 × 0 × f3} where k induces a complete subgraph and s is an independent set. lemma 5.3. let f be a field and r1 a local ring. let r = r1 × f. then γ ∗ 2 (r) is split. proof. let {ij : j ∈ j} be the collection of non-zero proper ideals of r1. then {ij × f : j ∈ j} ∪ {ij × 0 : j ∈ j} is an independent set and {0 × f, r1 × 0} is a k2. this forms a partition of v (γ∗2(r)). thus, γ ∗ 2(r) is split. lemma 5.4. suppose r has exactly n maximal ideals mi (1 ≤ i ≤ n) with each mi being generated by an idempotent ei. then r ∼= n ∏ i=1 fi where each fi ∼= r/mi, a field. proof. let e = ∏n i=1 ei. then e ∈ j(r). therefore, 1 − e is a unit (and an idempotent). so, 1 − e = 1 ⇒ e = 0. then by the chinese remainder theorem, r ∼= r ∏n i=1 rei ∼= r ⋂n i=1 rei ∼= n ∏ i=1 r rei . 340 m. i. jinnah & s. c. mathew cubo 24, 2 (2022) theorem 5.5. let r be a ring. γ∗2(r) is a split graph if and only if one of the following conditions holds: (i) r is local. (ii) r ∼= r1 × f where r1 is a local ring and f is a field. (iii) r ∼= f1 × f2 × f3 where fi’s are fields. proof. first we note that γ∗2(r) is split if and only if γ2(r) is split. also, if r is local, γ ∗ 2(r) is split. sufficiency of other conditions follows from the lemmas. to prove the necessity of the conditions, we assume that r is not local and v (γ2(r)) is the disjoint union of two sets k and s where k induces a complete subgraph and s is an independent set. we assume that k and s are non-empty. also, s can contain at most one maximal ideal. case (i): s contains a maximal ideal, say m1. in this case, r can have only one maximal ideal other than m1. for, if m2 and m3 are distinct maximal ideals other than m1, then m2 and m3 are in k. then, m2m3 ∈ s, m1 ∈ s. clearly, m1 + m2m3 = r, a contradiction. thus, r contains only one maximal ideal other than m1, say m2 which belongs to k. let xi ∈ mi (i = 1, 2) with x1+x2 = 1. as m 2 2 +m1 = r, m 2 2 ∈ k which implies m22 = m2. similarly, as rx2 + m1 = r, rx2 ∈ k which implies m2 = rx2. then, m2 is a finitely generated maximal ideal which is idempotent. hence, m2 is generated by an idempotent. so, r ∼= r1 × f where f is a field and m2 is isomorphic to the ideal r1 × {0}. further, r1 must be local. case (ii): s contains no maximal ideal. in this case, r can have at most three maximal ideals, for, if m1, m2, m3 and m4 are distinct maximal ideals, m1m2 and m3m4 are in s which leads to a contradiction. if r has only two maximal ideals, say, m1 and m2, then m1, m2 ∈ k. since, m 2 i + mi 6= r (i = 1, 2), we have m2 1 , m2 2 ∈ s. but m2 1 + m2 2 = r. so, to avoid a contradiction we have to assume m2 1 = m1 or m22 = m2. that is, r ∼= r1 × f where f is a field and r1 is a local ring. so, let us assume r has exactly 3 maximal ideals m1, m2 and m3. note that mi ∈ k (i = 1, 2, 3). then, as m1 + m2m3 = r, there exists x1 ∈ m1 such that rx1 + m2m3 = r which implies rx1 ∈ k and hence, rx1 = m1. similarly arguing with m 2 1 + m2m3 = r, we get m1 = m 2 1 . then m1 is generated by an idempotent. similarly each mj (j = 2, 3) is generated by an idempotent. then by the lemma 5.4, r ∼= f1 × f2 × f3 where fi (1 ≤ i ≤ 3) are fields. cubo 24, 2 (2022) ideal based graph structures for commutative rings 341 references [1] d. f. anderson and p. s. livingston, “the zero-divisor graph of a commutative ring”, j. algebra, vol. 217, no. 2, pp. 434–447, 1999. [2] m. f. atiyah and i. g. macdonald, introduction to commutative algebra, reading, ma: addison-wesley, 1969. [3] i. beck, “coloring of commutative rings”, j. algebra, vol. 116, no. 1, pp. 208–226, 1988. [4] t. w. haynes, s. t. hedetniemi and p. j. slater, fundamentals of domination in graphs, monographs and textbooks in pure and applied mathematics, vol. 208, new york: marcel dekker, inc., 1998. [5] m. i. jinnah and s. c. mathew, “on the ideal graph of a commutative ring”, algebras groups geom., vol. 26, no. 2, pp. 125–131, 2009. [6] m. i. jinnah and s. c. mathew, “when is the comaximal graph split?”, comm. algebra, vol. 40, no. 7, pp. 2400–2404, 2012. [7] m. i. jinnah and s. c. mathew, “on rings whose beck graph is split”, beitr. algebra geom., vol. 56, no. 2, pp. 379–385, 2015. [8] s. c. mathew, “a study on some graphs associated with a commutative ring”, ph.d. thesis, university of kerala, thiruvananthapuram, india, 2011. [9] k. r. parthasarathy, basic graph theory, new delhi, new york: tata-mcgraw hill, 1994. [10] c. thomas, “a study of some problems in algebraic graph theory graphs arising from rings”, ph.d. thesis, university of kerala, thiruvananthapuram, india, 2004. introduction the graph 2*(r) and its properties comparison between 1*(r) and 2*(r) some parameters of 2*(r) splitness cubo, a mathematical journal vol. 24, no. 03, pp. 369–392, december 2022 doi: 10.56754/0719-0646.2403.0369 dual digraphs of finite semidistributive lattices andrew craig 1, b miroslav haviar 1, 2 josé são joão 3, 4 1department of mathematics and applied mathematics, university of johannesburg, po box 524, auckland park, 2006, south africa. acraig@uj.ac.za b 2department of mathematics, faculty of natural sciences, m. bel university, tajovského 40, 974 01 banská bystrica, slovakia. miroslav.haviar@umb.sk 3department of mathematics, stockholm university, se-106 91 stockholm, sweden. zealvro98@outlook.com 4department of mathematics, kth royal institute of technology, se-100 44 stockholm, sweden. abstract dual digraphs of finite join-semidistributive lattices, meet-semidistributive lattices and semidistributive lattices are characterised. the vertices of the dual digraphs are maximal disjoint filter-ideal pairs of the lattice. the approach used here combines representations of arbitrary lattices due to urquhart (1978) and ploščica (1995). the duals of finite lattices are mainly viewed as tirs digraphs as they were presented and studied in craig–gouveia– haviar (2015 and 2022). when appropriate, urquhart’s two quasi-orders on the vertices of the dual digraph are also employed. transitive vertices are introduced and their role in the domination theory of the digraphs is studied. in particular, finite lattices with the property that in their dual tirs digraphs the transitive vertices form a dominating set (respectively, an in-dominating set) are characterised. a characterisation of both finite meetand join-semidistributive lattices is provided via minimal closure systems on the set of vertices of their dual digraphs. accepted: 22 august, 2022 received: 01 june, 2022 c©2022 a. craig et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://doi.org/10.56754/0719-0646.2403.0369 https://orcid.org/0000-0002-4787-3760 https://orcid.org/0000-0002-9721-152x https://orcid.org/0000-0001-5078-362x mailto:acraig@uj.ac.za mailto:miroslav.haviar@umb.sk mailto:zealvro98@outlook.com cubo, a mathematical journal vol. 24, no. 03, pp. 369–391, december 2022 doi: 10.56754/0719-0646.2403.0369 resumen se caracterizan los digrafos duales de reticulados finitos unión-semidistributivos, encuentro-semidistributivos y semidistributivos. los vértices de los digrafos duales son pares filtro-ideales disjuntos maximales del reticulado. el enfoque usado combina las representaciones de reticulados arbitrarios de urquhart (1978) and ploščica (1995). los duales de reticulados finitos son vistos principalmente como digrafos tirs como fueron presentados y estudiados en craig–gouveia–haviar (2015 y 2022). cuando sea apropiado, también se emplean los dos cuasiórdenes de urquhart en los vértices del digrafo dual. se introducen los vértices transitivos y se estudia su rol en la teoŕıa de dominación de digrafos. en particular, se caracterizan los reticulados finitos con la propiedad que en sus digrafos tirs duales los vértices transitivos forman un conjunto dominante (respectivamente un conjunto dominante interior). se entrega una caracterización de reticulados encuentroy unión-semidistributivos a través de sistemas de clausura mı́nima en el conjunto de vértices de sus digrafos duales. keywords and phrases: semidistributive lattice, tirs digraph, join-semidistributive lattice, meet-semidistributive lattice, dual digraph, domination. 2020 ams mathematics subject classification: 06b15, 06a75, 06d75, 05c20, 05c69. accepted: 22 august, 2022 received: 01 june, 2022 c©2022 a. craig et al. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ https://doi.org/10.56754/0719-0646.2403.0369 cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 371 1 introduction semidistributivity was first described by jónsson [16] while he was studying sublattices of a free lattice. he proved [16, lemma 2.6] that every free lattice is semidistributive. a lattice is join-semidistributive if it satisfies the following quasi-equation for all x,y,z ∈ l: (sd∨) x ∨ y = x ∨ z =⇒ x ∨ y = x ∨ (y ∧ z). dually, l is meet-semidistributive if it satisfies: (sd∧) x ∧ y = x ∧ z =⇒ x ∧ y = x ∧ (y ∨ z). a lattice is semidistributive if it satisfies both (sd∨) and (sd∧). for background on semidistributive lattices we refer to the papers by adaricheva et al. [1] and [2], the chapter by adaricheva and nation [3], and the paper by davey et al. [10]. the aim of our paper is to investigate dual digraphs of finite semidistributive lattices. theorem 3.6 provides a representation of finite semidistributive lattices via a certain class of tirs digraphs (see definition 2.4). this theorem is a generalisation of birkhoff’s representation of finite distributive lattices via finite ordered sets [6] (see comments in the next paragraph regarding the distributive case). in addition, we study transitive vertices in the dual digraphs and their role in the domination theory of the digraphs, and also explore closure systems on the set of vertices of the dual digraphs. we employ representations for finite lattices due to urquhart [20] and ploščica [17]. in urquhart’s representation the elements of the dual space are maximal disjoint filter-ideal pairs of the lattice. urquhart considered two quasi-orders 61 and 62 on them and studied the dual of the lattice as a certain doubly (quasi-) ordered space. in ploščica’s representation, the dual space of a lattice l is formed by maximal partial homomorphisms from l into the two-element lattice, which correspond to urquhart’s maximal disjoint filter-ideal pairs of l. when l is a distributive lattice, these maximal partial homomorphisms become total homomorphisms from l into the two-element lattice, which form the priestley dual of l [18]. the close relationship between ploščica’s representation of general lattices and priestley’s representation of distributive lattices lies in the single binary relation e, which ploščica considered on his dual space. when l is distributive, e becomes exactly priestley’s order on the dual space. ploščica’s dual space of a finite lattice l is therefore a finite digraph where the vertices are the maximal partial homomorphisms from l into the two-element lattice and the binary relation e, which mimics priestley’s order, forms the edge set of the digraph. these dual digraphs of lattices were presented and studied as tirs digraphs in two papers by craig, gouveia and haviar [7, 8]. in our approach we combine urquhart’s and ploščica’s representations of finite lattices: the vertices 372 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) of our dual digraphs are maximal disjoint filter-ideal pairs of the lattice in the urquhart style, but we mainly study them as tirs digraphs using the ploščica binary relation e on the vertices. only in a small part of our investigation do we swap ploščica’s relation e for urquhart’s two quasiorders on the vertices to present our results in a different yet rather satisfactory way (the end of section 3). in section 2 we give preliminary results that will prove useful in the subsequent three sections of the paper. in section 3 we provide several characterisations of the dual digraphs of finite meetsemidistributive, finite join-semidistributive, and finite semidistributive lattices. in section 4 we introduce transitive vertices in the dual digraphs and we study their role in the domination theory of the digraphs. in particular, we are able to characterise finite lattices having the properties that in their dual tirs digraphs the transitive vertices form a dominating set, respectively an in-dominating set. in section 5 we characterise both finite meet-semidistributive and finite joinsemidistributive lattices via minimal closure systems on the set of vertices of their dual digraphs. in section 6 we make some concluding remarks and observations. in particular, we note connections to other representations of finite semidistributive lattices, and we propose several directions for future research in this area. 2 preliminaries ploščica’s representation of arbitrary bounded lattices [17] uses the set of maximal partial homomorphisms (mphs) from a bounded lattice l to the two-element bounded lattice ({0,1},∧,∨,0,1) as the underlying set of the dual space of l. we recall that a partial homomorphism from a bounded lattice (l,∧,∨,0,1) into the two-element bounded lattice ({0,1},∧,∨,0,1) is a partial map f : l → {0,1} such that domf is a bounded sublattice of l and the restriction f↾dom f is a bounded lattice homomorphism. a maximal partial homomorphism is a partial homomorphism with no proper extension. the set of mphs is then equipped with a binary relation and a topology. definition 2.1 ([20, section 3]). let l be a lattice. then 〈f,i〉 is a disjoint filter-ideal pair of l if f is a filter of l and i is an ideal of l such that f ∩ i = ∅. we say that a disjoint filter-ideal pair 〈f,i〉 is maximal if there is no disjoint filter-ideal pair 〈g,j〉 6= 〈f,i〉 such that f ⊆ g and i ⊆ j. a maximal disjoint filter-ideal pair 〈f,i〉 of l is total in l if f ∪ i = l. there is a one-to-one correspondence between the set of mphs from l to 2 = ({0,1},∧,∨,0,1) and the maximal disjoint filter-ideal pairs (mdfips) of l. the latter were used in the dual representation of urquhart [20]. we will use a combination of the two approaches: for a lattice l, the elements of our dual set xl will be mdfips, but we will equip the set with the binary relation due to ploščica, and hence will obtain a digraph. (later, when desirable, we will also equip the cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 373 set xl of all mdfips of l with urquhart’s two quasi-orders 61 and 62.) we do not require the topologies used by ploščica and urquhart because we are only working with finite lattices. ploščica’s binary relation on the set of mphs is defined as follows for mphs f and g from l to 2: (e1) feg ⇐⇒ (∀x ∈ domf ∩ domg)(f(x) 6 g(x)). the digraph dual to a finite bounded lattice l in ploščica’s representation is gl = (vl,e) where the set of vertices vl is formed by all mphs from l to 2 and the relation e is defined by (e1) above. we will now present this dual digraph as gl = (xl,e) where the set of vertices will be xl, i.e. is formed by all mdfips of l, and the corresponding ploščica relation e will be defined below by (e2). for two mdfips 〈f,i〉 and 〈g,j〉, ploščica’s relation e is determined as follows: (e2) 〈f,i〉e〈g,j〉 ⇐⇒ f ∩ j = ∅. for finite lattices every filter is the up-set of a unique element and every ideal is the down-set of a unique element, so we can represent every disjoint filter-ideal pair 〈f,i〉 by an ordered pair 〈↑x,↓y〉 where x = ∧ f and y = ∨ i. hence for finite lattices we have 〈↑x,↓y〉e〈↑a,↓b〉 if and only if x b. in figure 1 we present a number of examples of finite (non-distributive) lattices and their dual digraphs. to make the labelling more compact, we denote by xy the mdfip 〈↑x,↓y〉. also, to keep the display simpler, we have not included the loop on each vertex. notice that the directed edge set is not a transitive relation. sd∨, not sd∧ 0 a b c d e 1 a b c d e sd∧, not sd∨ 0 1 0 a b c 1 sd∨ & sd∧ 1 a b c 0 1 not sd∨, not sd∧ ea dc de cb dc ab cb ea ab bc ca ab ac ba bc ca cb figure 1: some finite lattices and their dual digraphs. the fact below was noted by urquhart and will be useful later. 374 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) proposition 2.2 ([20, p. 52]). let l be a finite lattice. if 〈f,i〉 is a maximal disjoint ideal-filter pair of l then ∧ f is join-irreducible and ∨ i is meet-irreducible. some of what appears in the proposition below can be found in the paper by gaskill and nation [13, p. 353]. we will make frequent use of this result and its proof reveals some important features of mdfips. proposition 2.3. let l be a finite lattice and 〈f,i〉 be a maximal disjoint filter-ideal pair of l. then the following are equivalent: (i) ∧ f is join-prime; (ii) ∨ i is meet-prime; (iii) f ∪ i = l; (iv) f is a prime filter; (v) i is a prime ideal. the equivalences (iii) ⇔ (iv) ⇔ (v) hold even when l is not finite. proof. let l be a finite lattice and let 〈f,i〉 be a maximal disjoint filter-ideal pair of l. let ∧ f = x and ∨ i = y. first we show that (iii) ⇒ (i). assume that f ∪i = l. let a,b ∈ l such that x 6 a∨b. we claim that a ∈ f or b ∈ f . suppose for a contradiction that a /∈ f and b /∈ f . then a,b ∈ l \ f = i. that implies a ∨ b ∈ i, whence x ∈ i, a contradiction. now we show that (i) ⇒ (iii). assume that x is join-prime. let a ∈ l such that a /∈ f ∪ i. we will consider three cases for the element a ∨ y and derive a contradiction for each case. case 1: if a ∨ y ∈ i then a 6 a ∨ y = y, thus a ∈ i, a contradiction. case 2: if a ∨ y ∈ f then x 6 a ∨ y. since x is join-prime, x 6 a or x 6 y. if x 6 a then a ∈ f , contradicting a /∈ f ∪ i. if x 6 y then x ∈ i, contradicting f ∩ i = ∅. case 3: suppose a∨y /∈ f ∪i. since a∨y /∈ ↑x, ↓(a∨y)∩↑x = ∅. from a∨y /∈ ↓y it follows that ↓y ⊂ ↓(a ∨ y). hence 〈↑x,↓(a ∨ y)〉 is a disjoint filter-ideal pair properly containing 〈f,i〉, which contradicts the maximality of 〈f,i〉. the equivalence of (ii) and (iii) can be shown analogously. now we drop the assumption that l is finite and show that (iii) ⇒ (iv). let a ∨ b ∈ f . if a /∈ f and b /∈ f then we have a,b ∈ l\f = i. since i is an ideal we would get a∨b ∈ i, a contradiction. therefore a ∈ f or b ∈ f . cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 375 to show (iv) ⇒ (iii), and the equivalence of (iv) and (v), one uses the fact that a filter f ⊆ l is prime if and only if l\f is a prime ideal. the properties of the digraphs dual to bounded lattices were described by craig, gouveia and haviar [7]. there they were called tirs graphs; in this paper we will use the terminology tirs digraphs. we recall the necessary facts. (we note that in the definition below xe = { y ∈ v | (x,y) ∈ e } and ex = { y ∈ v | (y,x) ∈ e }.) definition 2.4 ([7, definition 2.2]). a tirs digraph g = (v,e) is a set v and a reflexive relation e ⊆ v × v such that: (s) if x,y ∈ v and x 6= y then xe 6= ye or ex 6= ey. (r) for all x,y ∈ v , if xe ⊂ ye then (x,y) /∈ e, and if ey ⊂ ex then (x,y) /∈ e. (ti) for all x,y ∈ v , if xey then there exists z ∈ v such that ze ⊆ xe and ez ⊆ ey. we recall that the vertices of the dual digraph gl of a bounded lattice l are formed by the set xl of mdfips of l and ploščica’s relation e is determined by (e2). using these facts, the following result can be stated. proposition 2.5 ([7, proposition 2.3]). for any bounded lattice l, its dual digraph gl = (xl,e) is a tirs digraph. we recall from [17] a fact concerning general digraphs g = (x,e). let 2∼ = ({0,1},6) denote the two-element digraph. a partial map ϕ: x → 2∼ is said to preserve the relation e if ϕ(x) 6 ϕ(y) whenever x,y ∈ domϕ and (x,y) ∈ e. the lattice of maximal partial e-preserving maps from g to 2∼ is denoted by g mp(g, 2∼). lemma 2.6 ([17, lemma 1.3]). let g = (x,e) be a digraph and let us consider ϕ ∈ gmp(g, 2∼). then (i) ϕ−1(0) = { x ∈ x | there is no y ∈ ϕ−1(1) with (y,x) ∈ e }; (ii) ϕ−1(1) = { x ∈ x | there is no y ∈ ϕ−1(0) with (x,y) ∈ e }. the above lemma allows us to observe that for a digraph g = (x,e) and ϕ,ψ ∈ gmp(g, 2∼) we have ϕ−1(1) ⊆ ψ−1(1) ⇐⇒ ψ−1(0) ⊆ ϕ−1(0). this implies that the reflexive and transitive binary relation 6 defined on gmp(g, 2∼) by ϕ 6 ψ ⇐⇒ ϕ−1(1) ⊆ ψ−1(1) is a partial order. for a digraph g = (x,e) we let c(g) = (gmp(g, 2∼),6). 376 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) theorem 2.7 ([7, theorem 1.7 and p. 87]). for any finite bounded lattice l we have that l is isomorphic to c(gl) and for any finite tirs digraph g = (v,e) we have that g is isomorphic to gc(g). in later sections, we will frequently make use of theorem 2.7 in the following way: given any finite tirs digraph g = (v,e), we can consider g to be the dual digraph gl = (xl,e) for some finite lattice l. there are a number of different constructions that yield complete lattices isomorphic to the complete lattice c(g) described above, which is assigned to a digraph g = (x,e) (see [9]). in particular, later we will use the lattice obtained via the polarity k(g) = (x,x,e∁), which will be described in section 5. at the end of this preliminary section we recall from [20] how the set xl of all mdfips of a finite bounded lattice l can be equipped with two quasi-orders 61 and 62. urquhart in [20, p. 47] defined two binary relations 61 and 62 on the set set xl of all mdfips of an arbitrary lattice l as follows: for two mdfips 〈f,i〉 and 〈g,j〉, (61) 〈f,i〉 61 〈g,j〉 ⇐⇒ f ⊆ g; (62) 〈f,i〉 62 〈g,j〉 ⇐⇒ i ⊆ j. it is clear that the binary relations 61 and 62 are reflexive and transitive on the set xl, and hence are quasi-orders. 3 characterisation of dual digraphs the theorem below will play a crucial role in the proof of our first result. our presentation is slightly different to [3]; we have re-stated their items to suit our purposes. we use j(l), respectively m(l), to denote the join-irreducible, respectively meet-irreducible, elements of l. theorem 3.1 ([3, theorem 3-1.4]). let l be a finite lattice. then the following are equivalent: (i) l satisfies sd∨; (ii) for each x ∈ m(l), there exists a unique minimal element of the set s(x) = {k ∈ l | k x & k 6 x∗}, where x∗ is the unique upper cover of x, and moreover, this minimal element of s(x) is in j(l). cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 377 (iii) there exists a map κ: m(l) → j(l) such that for each x ∈ m(l), κ(x) is the minimal element of the set s(x). using the previous result, in the next theorem we characterise finite join-semidistributive and meet-semidistributive lattices via their mdfips. theorem 3.2. let l be a finite lattice. (i) l is not join-semidistributive if and only if there exist distinct maximal disjoint filter-ideal pairs of the form 〈↑y,↓x〉 and 〈↑z,↓x〉 for some x,y,z ∈ l. (ii) l is not meet-semidistributive if and only if there exist distinct maximal disjoint filter-ideal pairs of the form 〈↑x,↓y〉 and 〈↑x,↓z〉 for some x,y,z ∈ l. proof. for the necessity, assume l is not join-semidistributive, whence by theorem 3.1, for some x ∈ m(l) there exist two minimal elements y and z of the set s(x). then ↑y ∩ ↓x = ∅ and ↑z ∩ ↓x = ∅ so 〈↑y,↓x〉 and 〈↑z,↓x〉 are disjoint filter-ideal pairs. we claim that 〈↑y,↓x〉 and 〈↑z,↓x〉 are maximal. suppose on the contrary that there is a disjoint filter-ideal pair 〈↑a,↓b〉 of l such that ↑y ⊆ ↑a and ↓x ⊆ ↓b but 〈↑a,↓b〉 6= 〈↑y,↓x〉. this gives us two possible cases: case 1: if a 6= y then since y is minimal in the set s(x) and a 6 y 6 x∗ we have that a 6 x. but x 6 b, which implies that a 6 b, contradicting ↑a ∩ ↓b = ∅. case 2: if b 6= x then x∗ 6 b since x∗ is the unique upper cover of x. but a 6 y 6 x∗, which implies that a 6 b, contradicting again ↑a ∩ ↓b = ∅. thus 〈↑y,↓x〉 is maximal and we can use a similar argument to prove that 〈↑z,↓x〉 is maximal. for the sufficiency, assume that there exist distinct maximal disjoint filter-ideal pairs of the form 〈↑y,↓x〉 and 〈↑z,↓x〉 for some x,y,z ∈ l. we will prove that y and z are both minimal elements of the set s(x). if follows from ↑y ∩ ↓x = ∅ and ↑z ∩ ↓x = ∅ that y x and z x. we will argue y 6 x∗ by contradiction. suppose y x∗, then ↑y ∩ ↓x∗ = ∅. since x < x∗ implies that ↓x ⊂ ↓x∗, we get that 〈↑y,↓x〉 is properly contained in 〈↑y,↓x∗〉, which is a contradiction. therefore y 6 x∗ and y ∈ s(x). using a similar argument, z ∈ s(x). now if a ∈ s(x) and a < y, then ↑y ⊂ ↑a. since a x, we have ↑a ∩ ↓x = ∅. therefore 〈↑a,↓x〉 is a disjoint filter-ideal pair with ↑y ⊂ ↑a, contradicting the maximality of 〈↑y,↓x〉. similarly, if b ∈ s(x) such that b < z, then 〈↑b,↓x〉 is a disjoint filter-ideal pair properly containing 〈↑z,↓x〉, which is a contradiction. therefore y and z are both minimal elements of s(x). the proof of (ii) follows by an order-dual argument. 378 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) corollary 3.3. let g = (v,e) be a finite tirs digraph which is the dual digraph of a finite lattice l. if the relation e is antisymmetric, then l is semidistributive. proof. in accordance with our remarks after theorem 2.7, we can consider g to be gl and so its vertex set v will be xl. suppose for a contradiction that l is not semidistributive. then l is not join-semidistributive or l is not meet-semidistributive. if l is not join-semidistributive then by theorem 3.2 (i) there are maximal disjoint filter-ideal pairs of the form 〈↑y,↓x〉 and 〈↑z,↓x〉 for some x,y,z ∈ l. since g is the dual digraph of l, we have 〈↑y,↓x〉,〈↑z,↓x〉 ∈ v . clearly 〈↑y,↓x〉e〈↑z,↓x〉 and 〈↑z,↓x〉e〈↑y,↓x〉. this contradicts the antisymmetry of the relation e. if l is not meet-semidistributive, then the argument is analogous. remark 3.4. the converse to corollary 3.3 does not hold. we can see it on the lattice in figure 2. 0 a c b d 1 ac bd cb da figure 2: a finite semidistributive lattice and its dual digraph. the lattice is semidistributive but we see on its dual digraph, which contains a “double arrow” between the elements ac and bd, that the relation e of the digraph is not antisymmetric. hence the condition in corollary 3.3 is sufficient but not necessary for a finite lattice to be semidistributive. an interesting task that is left open is to possibly weaken the given sufficient condition to some form of “weak antisymmetry” of the relation e so that the resulting condition on e is necessary and sufficient for a finite lattice to be semidistributive. in the statement and the proof of the following result we again use the fact that, by theorem 2.7, g = (v,e) is isomorphic to the dual digraph gl = (xl,el) of the lattice l, whose vertex set xl is the set of all mdfips of l. lemma 3.5. let g = (v,e) be a finite tirs digraph with dual lattice l. let u,v ∈ v be distinct. then: (i) eu = ev if and only if u and v are the isomorphic images of 〈↑x,↓y〉 and 〈↑z,↓y〉 in xl for some x,y,z ∈ l; cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 379 (ii) ue = ve if and only if u and v are the isomorphic images of 〈↑x,↓y〉 and 〈↑x,↓z〉 in xl for some x,y,z ∈ l. proof. let u,v ∈ v . to show the sufficiency of the condition in (i), let u and v be the isomorphic images of the vertices 〈↑x,↓y〉 and 〈↑z,↓y〉 in gl for some x,y,z ∈ l. since g is isomorphic to gl, we only need to show that el〈↑x,↓y〉 = el〈↑z,↓y〉. to this end, let 〈f,i〉 ∈ el〈↑x,↓y〉, then f ∩ ↓y = ∅. thus 〈f,i〉 ∈ el〈↑z,↓y〉. similarly, if 〈f,i〉 ∈ el〈↑z,↓y〉, then f ∩ ↓y = ∅ and 〈f,i〉 ∈ el〈↑x,↓y〉. therefore el〈↑x,↓y〉 = el〈↑z,↓y〉 and eu = ev. for the necessity of the condition in (i), let 〈↑x,↓y〉 and 〈↑z,↓w〉 be isomorphic images of u and v in xl and let eu = ev. we will show ↓y = ↓w. let a ∈ ↓y. for all 〈f,i〉 ∈ el〈↑z,↓w〉 we have f ∩ ↓y = ∅ since el〈↑x,↓y〉 = el〈↑z,↓w〉. for s = ⋃ {f | 〈f,i〉 ∈ el〈↑z,↓w〉} now a /∈ s as a ∈ ↓y. we claim that a ∈ ↓w. suppose on the contrary that a /∈ ↓w. then a w and ↑a ∩ ↓w = ∅. this shows 〈↑a,↓w〉 is a disjoint filter-ideal pair. hence there is an mdfip 〈h,j〉 such that ↑a ⊆ h and ↓w ⊆ j. but ↓w ⊆ j and h ∩ j = ∅ implies that h ∩ ↓w = ∅. then 〈h,j〉 ∈ el〈↑z,↓w〉, so h ⊆ s, which means a ∈ s, a contradiction. thus a ∈ ↓w. the reverse inclusion can be shown analogously. therefore ↓y = ↓w and the proof of (i) is complete. part (ii) can be proven analogously. theorem 3.6. let g = (v,e) be a finite tirs digraph with u,v ∈ v . then (i) g is the dual digraph of a join-semidistributive lattice if and only if whenever u 6= v then eu 6= ev. (ii) g is the dual digraph of a meet-semidistributive lattice if and only if whenever u 6= v then ue 6= ve. (iii) g is the dual digraph of a semidistributive lattice if and only if whenever u 6= v then eu 6= ev and ue 6= ve. proof. let g be a finite tirs digraph with dual lattice l. to show the necessity in (i), assume there exist distinct u,v ∈ v such that eu = ev. then by lemma 3.5 there exist distinct mdfips 〈↑x,↓y〉 and 〈↑z,↓y〉 in l. it then follows from theorem 3.2(i) that l is not join-semidistributive. to show the sufficiency in (i), assume that l is not join-semidistributive. then by theorem 3.2(i) there exist distinct mdfips 〈↑x,↓y〉 and 〈↑z,↓y〉. by lemma 3.5 there exist distinct vertices u,v ∈ v such that eu = ev. part (ii) can be shown analogously, and part (iii) follows directly from (i) and (ii). we recall that the “separation property” (s) in the definition of tirs digraphs is defined as follows: (s) if x,y ∈ v and x 6= y then xe 6= ye or ex 6= ey. 380 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) hence it should be emphasized that the condition (iii) in the theorem above characterising the semidistributivity is clearly strengthening the separation condition (s) by replacing in it the logical connective “or” with “and”. thus it can be considered as a certain “strong separation property”: (ss) if x,y ∈ v and x 6= y then xe 6= ye and ex 6= ey. it is interesting to realise that finite semidistributive lattices are exactly those finite lattices whose dual digraphs have the “separation property” (s) strengthened to the “strong separation property” (ss). a remark of urquhart [20, section 7] says that a finite lattice l is meet-semidistributive if and only if the quasi-order 61 is an order. we state that result (and its dual) below and prove it using the results from earlier in the section. theorem 3.7. let l be a finite lattice. (i) l is join-semidistributive if and only if the quasi-order 62 on the vertices of the dual digraph is an order. (ii) l is meet-semidistributive if and only if the quasi-order 61 on the vertices of the dual digraph is an order. proof. assume firstly that the quasi-order 62 on the vertices of the dual digraph is not an order, that is, the relation 62 is not antisymmetric. then there exist distinct vertices x and y such that x 62 y and y 62 x. if we consider the vertices x and y as the mdfips x = 〈f,i〉 and y = 〈g,j〉, then by definition of 62 we have i ⊆ j and j ⊆ i, hence the mdfips x and y have the same ideal part. by theorem 3.2 it follows that l is not join-semidistributive. conversely, if l is not join-semidistributive, then by theorem 3.2 there exist distinct mdfips x and y with the same ideal part, whence x 62 y and y 62 x. it follows that the relation 62 is not antisymmetric, hence the quasi-order 62 is not an order. now we can rephrase lemma 3.5 in terms of quasi-orders 61 and 62: corollary 3.8. let g = (v,e) be a finite tirs digraph with dual lattice l. let u,v ∈ v be distinct. then: (i) eu = ev if and only if u 62 v and v 62 u; (ii) ue = ve if and only if u 61 v and v 61 u. we can finally summarise the previous results in the following characterisations of join-semidistributivity, meet-semidistributivity and semidistributivity of finite lattices via the properties of their dual digraphs: cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 381 corollary 3.9. let g = (v,e) be a finite tirs digraph. (1) the following are equivalent: (i) g is the dual digraph of a join-semidistributive lattice; (ii) for all u,v ∈ v , if u 6= v then eu 6= ev; (iii) the quasi-order 62 on v is an order. (2) the following are equivalent: (i) g is the dual digraph of a meet-semidistributive lattice; (ii) for all u,v ∈ v , if u 6= v then ue 6= ve; (iii) the quasi-order 61 on v is an order. (3) the following are equivalent: (i) g is the dual digraph of a semidistributive lattice; (ii) for all u,v ∈ v , if u 6= v then eu 6= ev and ue 6= ve; (iii) both the quasi-orders 61 and 62 on v are orders. 4 domination in dual digraphs in the dual digraph of a lattice l, there are certain vertices that play an important role. it turns out that these vertices correspond to mdfips where f ∪ i = l. definition 4.1. a vertex v of a digraph g = (v,e) is said to be transitive in g if uev and vew imply uew for all u,w ∈ v . with respect to the illustration of the following result, the reader is reminded to return to figure 1 for examples. theorem 4.2. let l be a lattice with dual digraph gl = (xl,e). a maximal disjoint filter-ideal pair 〈f,i〉 is total in l if and only if it is transitive in gl. proof. let 〈f,i〉 be total in l. assume that 〈g,j〉 and 〈h,k〉 are maximal disjoint filter-ideal pairs such that 〈g,j〉e〈f,i〉 and 〈f,i〉e〈h,k〉. by the definition of e we have that g ∩ i = ∅ and f ∩ k = ∅. we claim that g ∩ k = ∅. notice that since f ∩ k = ∅ and 〈f,i〉 is total, it follows that k ⊆ l\f = i. but g ∩ i = ∅ and hence g ∩ k = ∅. by the definition of e we get 〈g,j〉e〈h,k〉 and therefore 〈f,i〉 is transitive. 382 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) for the converse, assume that 〈f,i〉 is not total in l. take x ∈ l\(f ∪i) and consider the disjoint filter-ideal pairs 〈↑x,i〉 and 〈f,↓x〉. these can be extended to maximal disjoint filter-ideal pairs 〈g,j〉 (where ↑x ⊆ g and i ⊆ j) and 〈h,k〉 (with f ⊆ h and ↓x ⊆ k). since i ⊆ j, we have g ∩ i = ∅ and hence 〈g,j〉e〈f,i〉. since f ⊆ h we get f ∩ k = ∅ and hence 〈f,i〉e〈h,k〉. but, since x ∈ g ∩ k we do not have 〈g,j〉e〈h,k〉 and so 〈f,i〉 is not transitive. the following result was first shown in a more restricted context by gaskill and nation [13]. this more general statement is folklore. proposition 4.3 ([13, lemma 1]). let l be a join-semidistributive lattice with greatest element 1. then l has a prime ideal. dually, if l is a meet-semidistributive lattice with least element 0, then l has a prime filter. proof. let i be an ideal that is maximal with respect to not containing 1. suppose that y,z /∈ i. then there is an element x ∈ i such that x ∨ y = x ∨ z = 1. since l satisfies sd∨ we get x ∨ (y ∧ z) = 1 and hence y ∧ z /∈ i. corollary 4.4. let l be a bounded lattice. if the dual digraph gl = (xl,e) does not have a transitive vertex then l satisfies neither sd∨ nor sd∧. proof. assume that gl does not have a transitive element. then every mdfip of l is such that f ∪ i 6= l. by proposition 2.3 we have that no filter f ⊆ l can be prime. since l has both a greatest and least element, by proposition 4.3, l cannot be join-semidistributive and it cannot be meet-semidistributive. notice that the converse of corollary 4.4 does not hold. the lattice l3 from [10] satisfies neither sd∨ nor sd∧ but there exists a maximal disjoint filter-ideal pair 〈f,i〉 with f ∪ i = l (or, a total homomorphism from l3 to 2). as stated earlier, the transitive elements in a finite tirs digraph can play a special role. notice that when a tirs digraph g is a poset (i.e. it is the dual digraph of a finite distributive lattice) then every element of g is transitive. the next lemma captures two familiar facts about finite join-semidistributive and meet-semidistributive lattices. lemma 4.5 ([13, lemma 1]). (i) the co-atoms of a finite join-semidistributive lattice are meetprime. (ii) the atoms of a finite meet-semidistributive lattice are join-prime. proof. we prove only (i) as the proof of (ii) will follow using a dual argument. cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 383 let l be a finite join-semidistributive lattice and let x ∈ l be a co-atom such that x > a ∧ b for some a,b ∈ l. suppose that x � a and x � b. we then have x ∨ a > x and x ∨ b > x. since x is a co-atom, we get x ∨ a = 1 = x ∨ b. however, since l is join-semidistributive, we get x = x ∨ (a ∧ b) = x ∨ a = 1, a contradiction. thus x > a or x > b. in the definition below we note that the original source uses ‘arc’ instead of ‘edge’. definition 4.6 ([15, definition 2]). given a digraph d = (v,e), with vertex set v and edge set e, a set s ⊆ v is a dominating set if for every vertex v ∈ v \s, there is a vertex u ∈ s such that uev. proposition 4.7. let g = (v,e) be a finite tirs digraph. if g is dual to a finite join-semidistributive lattice l, then the transitive vertices of g form a dominating set. proof. assume that g = gl = (xl,e) for some finite join-semidistributive lattice l. if x is a vertex of g then x = 〈↑a,↓b〉 for some a,b ∈ l. since b 6= 1 we have that b 6 c for some co-atom c. by lemma 4.5 we have that c is meet-prime and so by proposition 2.3 we know that ↓c is a prime ideal and that there exists d ∈ l such that ↑d is a prime filter with ↑d ∩ ↓c = ∅ and ↑d ∪ ↓c = l. by theorem 4.2, y = 〈↑d,↓c〉 is a transitive vertex of gl. since ↓b ⊆ ↓c we have ↑d ∩ ↓b = ∅ and hence yex. the converse of the above proposition does not hold. let l′ be the diamond m3 with a new top element t. then its dual digraph g is the same as the dual digraph of m3 (see figure 1) except it has an extra vertex v = 〈↑t,↓1〉, which is transitive since it is total. in g the edges obviously go from the vertex v to every other vertex. hence the set {v} of transitive vertices of g is the dominating set, yet the lattice l′ is not join-semidistributive as it contains a sublattice isomorphic to m3 (cf. [10]). since transitive elements are connected to joinand meet-prime elements, the previous result is partly related to how the join-primes or meet-primes sit inside the lattice. the next result characterises finite tirs digraphs g dual to finite lattices, in which the transitive vertices of g form a dominating set. theorem 4.8. let g = (v,e) be a finite tirs digraph. then g is dual to a finite lattice l in which every co-atom is meet-prime if and only if the transitive vertices of g form a dominating set. proof. let g = (v,e) be the dual digraph gl for some finite lattice l in which every co-atom is meet-prime. if x ∈ v then x = 〈↑a,↓b〉 for some a,b ∈ l. since b 6= 1 we have that b 6 c for some co-atom c. by proposition 2.3 we know that ↓c is a prime ideal and that there exists d ∈ l 384 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) such that ↑d is a prime filter with ↑d ∩ ↓c = ∅ and ↑d ∪ ↓c = l. by theorem 4.2, y = 〈↑d,↓c〉 is a transitive vertex of gl = g. since ↓b ⊆ ↓c we have ↑d ∩ ↓b = ∅ and hence yex. next, assume that the transitive vertices of g form a dominating set and let c be a co-atom of l. the pair 〈↑1,↓c〉 is a disjoint filter-ideal pair that can be extended to a maximal disjoint filter-ideal pair 〈↑b,↓c〉. since the transitive vertices form a dominating set, there exists a transitive vertex 〈↑x,↓y〉 such that 〈↑x,↓y〉e〈↑b,↓c〉, i.e. ↑x ∩ ↓c = ∅. since 〈↑x,↓y〉 is transitive, we have by proposition 2.3 and theorem 4.2 that x is join-prime. now, we have that 〈↑x,↓c〉 is a disjoint filter-ideal pair which can be extended to a maximal disjoint filter-ideal pair 〈↑a,↓c〉 where a 6 x. since a c we have c < a ∨ c = 1. clearly now x 6 a ∨ c and hence x 6 a or x 6 c. the latter cannot happen as ↑x ∩ ↓c = ∅ so x 6 a and hence x = a. now 〈↑x,↓c〉 is a maximal disjoint filter-ideal pair with x join-prime, and hence c is meet-prime. remark 4.9. it is well-known (cf. [11, theorem 2.24]; see also [3, theorem 3-1.4]) that a finite lattice l satisfies sd∨ if and only if each element in l has a so-called canonical join representation. using [13, lemma 1(ii)] we are able to show that the equivalent conditions of theorem 4.8 hold for the tirs digraph g dual to a finite lattice l if and only if the top element 1 of l has a canonical join representation. since canonical join representations are not the focus of this paper, we have decided to present the proof in a separate paper where this will be explored with the proper context and in more depth. definition 4.10 ([15, definition 3]). given a digraph d = (v,e), with vertex set v and edge set e, a set s ⊆ v is an in-dominating set if for every vertex v ∈ v \s, there is a vertex u ∈ s such that veu. theorem 4.11. let g = (v,e) be a finite tirs digraph. then g is dual to a finite lattice l in which every atom is join-prime if and only if the transitive vertices of g form an in-dominating set. proof. let gl = (xl,e) be the dual digraph of some finite lattice l in which every atom is join-prime. if x ∈ v then x = 〈↑a,↓b〉 for some a,b ∈ l. assume that x is not transitive. since a 6= 0 we have that c 6 a for some atom c ∈ l. by proposition 2.3 we know that ↑c is a prime filter and that there exists d ∈ l such that ↓d is a prime ideal with ↑c ∩ ↓d = ∅ and ↑c ∪ ↓d = l. by theorem 4.2, y = 〈↑c,↓d〉 is a transitive vertex of gl. since ↑a ⊆ ↑c we have ↑c ∩ ↓b = ∅ and hence xey. next, assume that the transitive vertices of g = (v,e) form an in-dominating set and let c be an atom of l. the pair 〈↑c,↓0〉 is a disjoint filter-ideal pair that can be extended to an mdfip 〈↑c,↓b〉. since the transitive vertices form an in-dominating set, there exists a transitive vertex 〈↑x,↓y〉 such that 〈↑c,↓b〉e〈↑x,↓y〉, i.e. ↑c ∩ ↓y = ∅. since 〈↑x,↓y〉 is transitive, we have by proposition 2.3 and theorem 4.2 that y is meet-prime. cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 385 now, we have that 〈↑c,↓y〉 is a disjoint filter-ideal pair which can be extended to a maximal disjoint filter-ideal pair 〈↑c,↓a〉 where y 6 a. since c a we have 0 = a ∧ c < c. clearly now a ∧ c < y and hence a 6 y or c 6 y. the latter cannot happen as ↑c ∩ ↓y = ∅ so a 6 y and hence y = a. now 〈↑c,↓y〉 is an mdfip with y is meet-prime, and hence c is join-prime. corollary 4.12. let g = (v,e) be a finite tirs digraph. if g is dual to a finite meetsemidistributive lattice l, then the transitive vertices of g form an in-dominating set. proof. let g = (v,e) be a finite tirs digraph. assume g is dual to a finite meet-semidistributive lattice l. then by lemma 4.5 the atoms of l are join-prime. it then follows from theorem 4.11 that the transitive elements of l form an in-dominating set. we think it is an interesting problem to try and characterise the dual digraphs of finite joinsemidistributive lattices within the class of finite tirs digraphs whose transitive vertices form a dominating set (and dually). we attempted to do so but were unable to identify the required condition. 5 minimal closure systems from dual digraphs closure systems appear in many different areas of mathematics. they were investigated in relation to join-semidistributive lattices by adaricheva et al. [1]. a comprehensive account of the theory can be found in the book chapters by adaricheva and nation [4, 5]. the definitions below all follow the notational conventions used in adaricheva and nation [4, section 4-2] although in some cases the reference is to another source. definition 5.1 ([14, definition 30]). let x be a set and φ : ℘(x) → ℘(x). then φ is a closure operator on x if for all y,z ∈ ℘(x), (i) y ⊆ φ(y ), (ii) y ⊆ z implies φ(y ) ⊆ φ(z), (iii) φ(φ(y )) = φ(y ). if x is a set and φ a closure operator on x then the pair 〈x,φ〉 is called a closure system. for y ⊆ x we say that y is closed if φ(y ) = y . the closed sets of a closure operator φ on x form a complete lattice, denoted by cld(x,φ). example 5.2. let l be a finite lattice. if a ∈ l let ja = {x ∈ j(l) | x 6 a} and define τ : ℘(j(l)) → ℘(j(l)) by τ(a) = ⋂ {ja | a ∈ l and a ⊆ ja}. then 〈j(l),τ〉 is a closure system. notice that every finite lattice l is isomorphic to cld(j(l),τ) via the isomorphism a 7→ ja. 386 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) from any digraph g = (x,e) we get the closure system 〈x,e∁⊳ ◦e ∁ ⊲〉 (see [9, theorem 3.3]). here we recall necessary facts from [9, section 3]. for a digraph g = (x,e) one can consider the triple (called a context) k(g) := (x,x,e∁), where the relation e∁ ⊆ x × x is the complement of the digraph relation e: e∁ = (x × x)\e. one can then define a galois connection via so-called polars as follows. the maps e∁⊲ : ( ℘(x),⊆) → (℘(x),⊇) and e∁⊳ : ( ℘(x),⊇) → (℘(x),⊆) are given by e∁⊲(y ) = { x ∈ x | (∀y ∈ y )(y,x) /∈ e }, e∁⊳(y ) = { z ∈ x | (∀y ∈ y )(z,y) /∈ e }. the so-called concept lattice cl(k(g)) of the context k(g) = (x,x,e∁), given by cl(k(g)) = { y ⊆ x | (e∁⊳ ◦ e ∁ ⊲)(y ) = y }, is a complete lattice when ordered by inclusion. the isomorphism in proposition 5.3 below is different to the original source but is equivalent because of the one-to-one correspondence between the sets vl and xl. we recall that the definition of the lattice c(gl) is given directly before theorem 2.7. proposition 5.3 ([9, proposition 3.1 and corollary 3.2]). if l is a finite lattice and gl = (xl,e) is its dual digraph, we have l ∼= c(gl) ∼= cl(k(gl)). the map a 7→ { 〈f,i〉 ∈ xl | a ∈ f } is the isomorphism from l to cl(k(gl)). the definition below is important in understanding the notion of a minimal closure system later on. definition 5.4 ([4, definition 4-2.1]). closure systems 〈x,φ〉 and 〈y,ψ〉 are called equivalent if cld(x,φ) ∼= cld(y,ψ). two equivalent systems are called isomorphic if there exists a bijection ρ : x → y such that ρ(φ(z)) = ψ(ρ(z)) for all z ⊆ x. the left-most lattice in figure 1 is referred to as l∂4 in [10]. we use this lattice to provide an illustration of definition 5.4. example 5.5. let l = l∂4 and consider its dual digraph gl = (xl,e) = ({cb,de,dc,ea},e). from this digraph we get the closure system 〈xl,e ∁ ⊳ ◦ e ∁ ⊲〉 with cld(xl,e ∁ ⊳ ◦ e ∁ ⊲) = {∅,{cb},{ea},{de,dc},{cb,de,dc},{ea,de,dc},xl}. cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 387 if we let y = {cb,de,ea} and φy (s) = y ∩ (e ∁ ⊳ ◦ e ∁ ⊲)(s) then cld(y,φy ) = {∅,{cb},{ea},{de},{cb,de},{ea,de},y}. it is easy to see that 〈xl,e ∁ ⊳ ◦ e ∁ ⊲〉 and 〈y,φy 〉 are equivalent but not isomorphic. proposition 5.6. let 〈x,φ〉 and 〈y,ψ〉 be closure systems and let f : x → y be a mapping. if f(a) is closed in y for all closed sets a ⊆ x and f−1(b) is closed in x for all closed sets b ⊆ y then f(φ(a)) = ψ(f(a)) for all a ⊆ x. proof. let f be such that f(a) is closed in y for all closed sets a ⊆ x and f−1(b) is closed in x for all closed sets b ⊆ y . notice that for all s ⊆ x we have that φ(s) = ⋂ {a ⊆ x | s ⊆ a and a is closed in x}, and similarly for ψ. let s ⊆ x. to show the inclusion f(φ(a)) ⊆ ψ(f(a)), let b ∈ cld(y,ψ) such that f(s) ⊆ b. then s ⊆ f−1(b). but f−1(b) is closed in x by our assumption. hence φ(s) ⊆ f−1(b) = φ(f−1(b)). this implies that f(φ(s)) ⊆ b. since b was arbitrary, this is true for all closed sets containing f(s). therefore f(φ(s)) ⊆ ψ(f(s)) = ⋂ {a ⊆ x | f(s) ⊆ a and a is closed in x}. for the reverse inclusion notice that since a ⊆ φ(a) we get that f(a) ⊆ f(φ(a)). but f(φ(a)) is closed by our assumption. thus ψ(f(a)) ⊆ f(φ(a)). further, adaricheva and nation [4] posed the following problem: given a closure system 〈x,φ〉, can we find a ⊆-minimal subset y of x and a closure operator ψ on y such that 〈y,ψ〉 is equivalent to 〈x,φ〉? such a closure system is then said to be minimal for 〈x,φ〉. theorem 5.7 ([5, lemma 4-2.13]). a closure system 〈x,φ〉 with lattice of closed sets l is minimal if and only if it is isomorphic to 〈j(l),τ〉. proposition 5.8. let l be a finite lattice and gl = (xl,e) its dual digraph. then the mapping f : x → j(l) defined by f(〈f,i〉) = ∧ f is surjective and satisfies f(e∁⊳ ◦ e ∁ ⊲(s)) = τ(f(s)) for all s ⊆ x. proof. we start by proving the surjectivity of f. let x ∈ j(l) and let t(x) denote the set {a ∈ l | x∗ 6 a and x a} where x∗ is the unique lower cover of x. we notice that the set t(x) is non-empty since x∗ ∈ t(x). let y ∈ t(x) be a maximal element (which exists since t(x) is a finite ordered set). then we claim that 〈↑x,↓y〉 is an mdfip. we have that ↑x ∩ ↓y = ∅ since x y. now let 〈↑a,↓b〉 be an mdfip such that ↑x ⊆ ↑a and ↓y ⊆ ↓b and 〈↑a,↓b〉 6= 〈↑x,↓y〉. we get two cases from this. case 1: if ↑x 6= ↑a then a < x so a 6 x∗. thus we get that a 6 x∗ 6 y 6 b, which is a contradiction. case 2: if ↓y 6= ↓b then y < b and so x∗ 6 y < b. but y is maximal in t(x) so we have that a 6 x 6 b. again, this is a contradiction. 388 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) thus 〈↑x,↓y〉 is an mdfip and f(〈↑x,↓y〉) = x. hence f is surjective. to help us prove that f preserves closure, we define ba = {〈f,i〉 ∈ xl | a ∈ f} and ja = {x ∈ j(l) | x 6 a} for a ∈ l. notice that the closed sets from 〈xl,e ∁ ⊳ ◦ e ∁ ⊲〉 are exactly the sets ba for all a ∈ l and the closed sets from 〈j(l),τ〉 are exactly the sets ja for all a ∈ l (see proposition 5.3 and example 5.2). we claim that f(ba) = ja and f −1(ja) = ba for all a ∈ l. let a ∈ l. we prove firstly that f(ba) = ja. to show the inclusion f(ba) ⊆ ja, let x ∈ f(ba). then x = ∧ f for some 〈f,i〉 ∈ ba. since 〈f,i〉 ∈ ba we have that a ∈ f . this implies that x 6 a. but x ∈ j(l) and thus x ∈ ja. to show the reverse inclusion f(ba) ⊇ ja, let x ∈ ja. then by the surjectivity there is y ∈ l such that 〈↑x,↓y〉 ∈ xl. then since x ∈ ja, we have that x 6 a. this implies that a ∈ ↑x and that 〈↑x,↓y〉 ∈ ba. since 〈↑x,↓y〉 ∈ ba, we get that x ∈ f(ba). thus f(ba) = ja. now we prove that f−1(ja) = ba for all a ∈ l. to show f −1(ja) ⊆ ba, let 〈f,i〉 ∈ f −1(ja). then f(〈f,i〉) = x ∈ ja. since x ∈ ja, we have that x 6 a and that a ∈ ↑x = f . therefore 〈f,i〉 ∈ ba. to show f −1(ja) ⊇ ba, let 〈f,i〉 ∈ ba. then a ∈ f and f(〈f,i〉) = ∧ f 6 a. therefore f(〈f,i〉) ∈ ja and 〈f,i〉 ∈ f −1(ja). thus f −1(ja) = ba. by proposition 5.6 we get f(e∁⊳ ◦ e ∁ ⊲(s)) = τ(f(s)) for all s ⊆ x. the main result of this section is the theorem below. we again refer the reader to figure 1 for basic illustrative examples, while example 5.5 provides a demonstration of what can happen when l is not meet-semidistributive. theorem 5.9. let l be a finite lattice and gl = (xl,e) its dual digraph. then 〈xl,e ∁ ⊳ ◦ e ∁ ⊲〉 is a minimal closure system for itself if and only if l is meet-semidistributive. proof. the necessity will be proved by contraposition. assume l is not meet-semidistributive. by proposition 5.8 we have that |j(l)| ≤ |xl| since f is surjective. but by theorem 3.2 there exist distinct mdfips 〈↑x,↓y〉 and 〈↑x,↓z〉 where x ∈ j(l). this implies that f is not injective and hence |j(l)| < |xl|. therefore by [5, lemma 4-2.13], 〈x,e ∁ ⊳ ◦ e ∁ ⊲〉 is not minimal. for the sufficiency, assume that l is meet-semidistributive. we will show that f defined in proposition 5.8 is a bijection. we only need to show that f is injective. let 〈f,i〉,〈g,j〉 ∈ x be such that f(〈f,i〉) = f(〈g,j〉) = x. then f = g = ↑x. by theorem 3.2 we have that i = j. therefore 〈f,i〉 = 〈g,j〉 and hence f is injective. thus it follows from propositions 5.6 and 5.8 that f is an isomorphism of closure systems. by [5, lemma 4-2.13] this implies that 〈x,e∁⊳ ◦ e ∁ ⊲〉 is minimal. before stating the dual of theorem 5.9, we need to make some observations. as observed earlier in the section, if l is a finite lattice, with gl = (xl,e) its dual digraph, then l ∼= cld(xl,e ∁ ⊳ ◦e ∁ ⊲) ∼= cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 389 cld(j(l),τ). if we reverse the order of the polar maps e∁⊳ and e ∁ ⊲, we again get a closure operator, but with l∂ ∼= cld(xl,e ∁ ⊲◦e ∁ ⊳). for a finite lattice l, it is easy to show that g : xl → xl∂ , defined for 〈↑a,↓b〉 ∈ xl by g(〈↑a,↓b〉) = 〈↑b,↓a〉, is a bijection. from this we get that 〈xl,e ∁ ⊲ ◦ e ∁ ⊳〉 is isomorphic to 〈xl∂,e ∁ ⊳ ◦ e ∁ ⊲〉. theorem 5.10. let l be a finite lattice and gl = (xl,e) its dual digraph. then 〈xl,e ∁ ⊲ ◦ e ∁ ⊳〉 is a minimal closure system for itself if and only if l is join-semidistributive. proof. we know that l is join-semidistributive if and only if l∂ is meet-semidistributive. we can then apply theorem 5.9 to the closure system 〈xl∂,e ∁ ⊳ ◦ e ∁ ⊲〉. corollary 5.11. let l be a finite lattice and gl = (xl,e) its dual digraph. then 〈xl,e ∁ ⊳ ◦e ∁ ⊲〉 and 〈xl,e ∁ ⊲ ◦ e ∁ ⊳〉 are minimal closure systems for themselves if and only if l is semidistributive. 6 conclusion and future research in this paper we characterised dual digraphs of finite meet-semidistributive, join-semidistributive and semidistributive lattices. we combined urquhart’s and ploščica’s representations of finite lattices in the following sense: the vertices of our dual digraphs were maximal disjoint filter-ideal pairs of the lattice in the urquhart style, but we mainly viewed the duals as tirs digraphs using the ploščica binary relation e on the vertices. we introduced transitive vertices in our digraphs and explored their role in the domination theory. in particular, we characterised the finite lattices with the property that in their dual digraphs the transitive vertices form a dominating set resp. an in-dominating set. finally, we characterised finite meet-semidistributive and join-semidistributive lattices via minimal closure systems on the set of vertices of their dual digraphs. we wish to take note of two other settings in which dual representations of finite semidistributive lattices have been developed. the older of these is that of formal concept analysis, where a characterisation of both finite join-semidistributive and meet-semidistributive lattices is available [12, section 6.3]. there is also a recent paper by reading, speyer and thomas [19] where they give a representation of finite semidistributive lattices via two-acyclic factorization systems. they define a two-acyclic factorization system to be a 4-tuple 〈w,→,։, →֒〉 with a set w and three binary relations →, →֒,։ on w . the relations ։ and →֒ are required to be partial orders. the representation then comes from defining a factorization system on the set of join-irreducible elements of a semidistributive lattice. the triple (x, →֒,։) is isomorphic to urquhart’s dual of the lattice l. we note that, in our representation, joinand meet-semidistributive lattices can be considered separately, but in the setting of factorization systems this separation is not yet possible (see [19, remark 5.14]). 390 a. craig, m. haviar & j. são joão cubo 24, 3 (2022) lastly, we wish to point to some promising directions for future research. these would build on the representation of finite joinand meet-semidistributive lattices obtained in section 3. the first of these would be to attempt to study finite sublattices of free lattices via their dual digraphs. this would require first finding a dual description of the well-known whitman’s condition. the second direction would be the study of finite convex geometries (see [1, 5]) via their dual digraphs. finite convex geometries are closure systems that are often studied via their lattice of closed sets. these lattices of closed sets are join-semidistributive and lower semimodular. work is already under way to find a dual characterisation of upper and lower semimodularity. acknowledgements the first author acknowledges the national research foundation (nrf) of south africa (grant 127266). the second author acknowledges his appointment as a visiting professor at the university of johannesburg from 1 june 2020 and the support by slovak vega grants 1/0152/22 and 2/0078/20. the authors would like to express their appreciation for the referee’s suggestions, which improved the final version of this paper, in particular for the one that led to remark 4.9. cubo 24, 3 (2022) dual digraphs of finite semidistributive lattices 391 references [1] k. v. adaricheva, v. a. gorbunov and v. i. tumanov, “join-semidistributive lattices and convex geometries”, adv. math., vol. 173, no. 1, pp. 1–49, 2003. [2] k. adaricheva, m. maróti, r. mckenzie, j. b. nation and e. r. zenk, “the jónsson–kiefer property”, studia logica, vol. 83, no. 1–3, pp. 111–131, 2006. [3] k. adaricheva and j. b. nation, “classes of semidistributive lattices”, in lattice theory: special topics and applications, vol. 2, g. grätzer and f. wehrung, basel: birkhäuser, 2016, pp. 59–101. [4] k. adaricheva and j. b. nation, “lattices of algebraic subsets and implicational classes”, in lattice theory: special topics and applications, vol. 2, g. grätzer and f. wehrung, basel: birkhäuser, 2016, pp. 103–151. [5] k. adaricheva and j. b. nation, “convex geometries”, in lattice theory: special topics and applications, vol. 2, g. grätzer and f. wehrung, basel: birkhäuser, 2016, pp. 153–179. [6] g. birkhoff, “on the combination of subalgebras”, proc. camb. phil. soc., vol. 29, no. 4, pp. 441–464, 1933. 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[20] a. urquhart, “a topological representation theory for lattices”, algebra universalis, vol. 8, no. 1, pp. 45–58, 1978. introduction preliminaries characterisation of dual digraphs domination in dual digraphs minimal closure systems from dual digraphs conclusion and future research cubo, a mathematical journal vol. 24, no. 01, pp. 167–186, april 2022 doi: 10.4067/s0719-06462022000100167 uniqueness of entire functions whose difference polynomials share a polynomial with finite weight goutam haldar 1 1department of mathematics, malda college, rabindra avenue, malda, west bengal 732101, india. goutamiit1986@gmail.com goutamiitm@gmail.com abstract in this paper, we use the concept of weighted sharing of values to investigate the uniqueness results when two difference polynomials of entire functions share a nonzero polynomial with finite weight. our result improves and extends some recent results due to sahoo-karmakar [j. cont. math. anal. 52(2) (2017), 102–110] and that of li et al. [bull. malays. math. sci. soc., 39 (2016), 499–515]. some examples have been exhibited which are relevant to the content of the paper. resumen en este art́ıculo, usamos el concepto de intercambio pesado de valores para investigar los resultados de unicidad cuando dos polinomios de diferencia de funciones enteras comparten un polinomio no cero con peso finito. nuestro resultado mejora y extiende algunos resultados de sahoo-karmakar [j. cont. math. anal. 52(2) (2017), 102–110] y los de li et al. [bull. malays. math. sci. soc., 39 (2016), 499–515]. se exhiben algunos ejemplos que son relevantes para el contenido del art́ıculo. keywords and phrases: entire function, difference polynomial, shift and difference operator, weighted sharing. 2020 ams mathematics subject classification: 30d35, 39a10. accepted: 28 february, 2022 received: 05 june, 2021 c©2022 g. haldar. this open access article is licensed under a creative commons attribution-noncommercial 4.0 international license. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/s0719-06462022000100167 https://orcid.org/0000-0001-8169-6190 mailto:goutamiit1986@gmail.com mailto:goutamiitm@gmail.com 168 g. haldar cubo 24, 1 (2022) 1 introduction let f and g be two non-constant meromorphic functions defined in the open complex plane c. if for some a ∈ c ∪ {∞}, the zero of f − a and g − a have the same locations as well as same multiplicities, we say that f and g share the value a cm (counting multiplicities), and if we do not consider the multiplicities into account, then f and g are said to share the value a im (ignoring multiplicities)(see [37]). we adopt the standard notations of the nevanlinna theory of meromorphic functions (see [14, 22, 41]). for a non-constant meromorphic function f, we denote by t (r, f) the nevanlinna characteristic function of f and by s(r, f) any quantity satisfying s(r, f) = o{t (r, f)} as r → ∞ outside of an exceptional set of finite linear measure. we define shift and difference operators of f(z) by f(z + c) and ∆cf(z) = f(z + c) − f(z), respectively. note that ∆nc f(z) = ∆ n−1 c (∆cf(z)), where c is a nonzero complex number and n ≥ 2 is a positive integer. for further generalization of ∆cf, we now define the linear difference operator of an entire (meromorphic) function f as lc(f) = f(z + c) + c0f(z), where c0 is a finite complex constant. clearly, for the particular choice of the constant c0 = −1, we get lc(f) = ∆cf. in 1959, hayman [13] proved the following result. theorem a ([13]). let f be a transcendental entire function and let n be an integer such that n ≥ 1. then fnf′ = 1 has infinitely many solutions. a number of authors have shown their interest to find the uniqueness of entire and meromorphic functions whose differential polynomials share certain values or fixed points, and obtained some remarkable results (see [3, 9, 10, 26, 33, 34, 36, 37, 39, 42]). in recent years, the difference variant of the nevanlinna theory has been established in [8, 11, 12]. using these theories, some mathematicians in the world began to study the uniqueness questions of meromorphic functions sharing values with their shifts, and study the value distribution of the nonlinear difference polynomials, and produced many fine works, for example, see [1, 5, 6, 7, 11, 15, 16, 23, 27, 29, 30, 31, 40, 44]. we recall the following result from laine-yang [23]. theorem b ([23]). let f be a transcendental entire function of finite order, and c be a non-zero complex constant. then, for n ≥ 2, f(z)nf(z + c) assumes every non-zero value a ∈ c infinitely often. later on, liu-yang [28] extended theorem b, and proved the following result: theorem c ([28]). let f be a transcendental entire function of finite order, and let η be a nonzero complex constant. then for n ≥ 2 the function f(z)nf(z + η) − p0(z) has infinitely many zeros, where p0 is any given polynomial such that p0 6≡ 0. cubo 24, 1 (2022) uniqueness of entire functions 169 regarding uniqueness corresponding to theorem c, li et al. [24] obtained the following result. theorem d ([24]). let f and g be two distinct transcendental entire functions of finite order, and let p0 6≡ 0 be a polynomial. let η is a nonzero complex constant and n ≥ 4 is an integer such that 2 deg(p0) < n + 1. also, suppose that f(z) nf(z + η) − p0(z) and g(z) ng(z + η) − p0(z) share 0 cm. then one of the following assertions holds. (i) if n ≥ 4 and f(z)nf(z + η)/p0(z) is a mobius transformation of g(z) ng(z + η)/p0(z), then either (i) f = tg, where t is a constant satisfying tn+1 = 1 (ii) f = eq and g = te−q, where p0 reduces to a nonzero constant c, t is a constant such that tn+1 = c2, and q is a non-constant polynomial. (ii) if n ≥ 6, then (i)(i) or (i)(ii) holds. in 2016, li-li [25] obtained the im analogues of the above theorem d as follows. theorem e ([25]). let f and g be two distinct transcendental entire functions of finite order, and let p0 6≡ 0 be a polynomial. let η is a nonzero complex constant and n ≥ 4 is an integer such that 2 deg(p0) < n + 1. also, suppose that f(z) nf(z + η) − p0(z) and g(z) ng(z + η) − p0(z) share 0 im. then one of the following assertions holds. (i) if n ≥ 4 and f(z)nf(z + η)/p0(z) is a mobius transformation of g(z) ng(z + η)/p0(z), then either (i) f = tg, where t is a constant satisfying tn+1 = 1, (ii) f = eq and g = te−q, where p0 reduces to a nonzero constant c, t is a constant such that tn+1 = c2, and q is a non-constant polynomial. (ii) if n ≥ 12, then (i)(i) or (i)(ii) holds. in 2001, the notion of weighted sharing was originally defined in the literature ([18, 19]), which is the gradual change of shared values from cm to im. below we recall the definition. definition 1.1 ([18, 19]). let k be a non-negative integer or infinity. for a ∈ c∪{∞}, we denote by ek(a; f) the set of all a-points of f, where an a-point of multiplicity m is counted m times if m ≤ k and k + 1 times if m > k. if ek(a; f) = ek(a; g), we say that f, g share the value a with weight k. clearly, if f, g share (a, k) then f, g share (a, p) for any integer p, 0 ≤ p < k. also we note that f, g share a value a im or cm if and only if f, g share (a, 0) or (a, ∞), respectively. using the notion of weighted sharing, sahoo-karmakar [35] further improved theorem d as follows. 170 g. haldar cubo 24, 1 (2022) theorem f ([35]). let f, g, p0 and n be defined as in theorem d. suppose that f(z) nf(z + η) − p0(z) and g(z) ng(z + η) − p0(z) share (0, 2). (i) if n ≥ 4 and f(z)nf(z + η)/p0(z) is a mobius transformation of g(z) ng(z + η)/p0(z), then either (i) f = tg, where t is a constant satisfying tn+1 = 1 (ii) f = eq and g = te−q, where p0 reduces to a nonzero constant c, t is a constant such that tn+1 = c2, and q is a non-constant polynomial. (ii) if n ≥ 6, then (i)(i) or (i)(ii) holds. observing the above results, it is natural to ask the following questions. question 1.2. what can be said about the relationship of two finite order non-constant meromorphic functions f and g if their more general nonlinear difference polynomials f(z)nlc(f) and g(z)nlc(g) share a polynomial p(z) 6≡ 0, where lc(f) = f(z +c)+c0f(z) with c and c0 being finite nonzero complex constants, and n ≥ 2 being a positive integer? question 1.3. is it possible to further reduce the nature of sharing from (0, 2) to (0, 1) in theorem f? question 1.4. can the lower bound of n be further reduced in theorems e and f? question 1.5. what can be said about the uniqueness of f and g if we consider the difference polynomial of the form f(z)n∆cf and g(z) n∆cg in theorems e and f? the purpose of this paper is to answer all the questions raised above. in fact we have been successfully able to reduce the nature of sharing of f(z)nf(z +η)−p0(z) and g(z) ng(z +η)−p0(z) in theorem f. we have also reduced the lower bound of n in theorems e and f successfully. 2 main results now we state our main result. theorem 2.1. let f and g be two transcendental entire functions of finite order, p 6≡ 0 be a polynomial. let c be a non-zero complex constant, and n be a positive integer such that 2 deg(p) < n + 1. let l be a non-negative integer such that f(z)nlc(f) − p(z) and g(z) nlc(g) − p(z) share (0, l) and g(z), g(z + c) share 0 cm. if n ≥ 4 and f(z)nlc(f)/p(z) is a mobius transformation of g(z)nlc(g)/p(z), or one of the following conditions holds: (i) l ≥ 2 and n ≥ 5; cubo 24, 1 (2022) uniqueness of entire functions 171 (ii) l = 1 and n ≥ 6; (iii) l = 0 and n ≥ 11, then one of the following conclusions can be realized: (a) f = tg, where t is a constant satisfying tn+1 = 1; (b) when c0 = 0, f = e u and g = te−u, where p(z) reduces to a nonzero constant d, t is a constant such that tn+1 = d2 and u is a non-constant polynomial; (c) when c0 6= 0, f = c1e az, g(z) = c2e −az, where a, c1, c2 and d are non-zero constants satisfying (c1c2) n+1(eac + c0)(e −ac + c0) = d 2. if lc(f) = ∆cf, then one can easily get the following corollary from theorem 2.1 which answers question 1.5. corollary 2.2. let f and g be two transcendental entire functions of finite order, p 6≡ 0 be a polynomial. let c be a non-zero complex constant, and n be a positive integer such that 2 deg(p) < n + 1. let l be a non-negative integer such that f(z)n∆cf − p(z) and g(z) n∆cg − p(z) share (0, l) and g(z), g(z + c) share 0 cm. if n ≥ 4 and f(z)n∆c(f)/p(z) is a mobius transformation of g(z)n∆c(g)/p(z), or one of the following conditions holds: (i) l ≥ 2 and n ≥ 5; (ii) l = 1 and n ≥ 6; (iii) l = 0 and n ≥ 11, then one of the following conclusions can be realized: (a) f = tg, where t is a constant satisfying tn+1 = 1; (b) f = c1e az, g(z) = c2e −az, where a, c1, c2 and d are non-zero constants satisfying (c1c2) n+1(eac + c0)(e −ac + c0) = d 2. the following examples show that both the conclusions of theorem 2.1 actually holds. example 2.3. let f(z) = ez and g = tf, where t is a constant such that tn+1 = 1, and η be any non-zero complex constant. then for any given polynomial p such that p 6≡ 0 with 2 deg(p) < n+1, f(z)nf(z + η)− p(z) and g(z)ng(z + η)− p(z) share (0, ∞). also f(z)n(f(z + η)− f(z))− p(z) and g(z)n(g(z + η) − g(z)) − p(z)share (0, ∞). here f and g satisfy the conclusion (a) of theorem 2.1. example 2.4. let f(z) = e2πiz/η and g(z) = te−2πiz/η, where t is a constant such that tn+1 = 1, η is a non-zero complex constant. then f(z)nf(z + η) and g(z)ng(z + η) share (1, ∞). here f and g satisfy the conclusion (b) of theorem 2.1. example 2.5. let f(z) = ez, g(z) = e−z, η = − log(−1) and p(z) = 2. then one can easily verify that f(z)n(f(z + η) − f(z)) and g(z)n(g(z + η) − g(z)) share (2, ∞). here f and g satisfy the conclusion (b) of theorem 2.1. 172 g. haldar cubo 24, 1 (2022) the following example shows that theorem 2.1 is not true for infinite order entire functions. example 2.6. let f(z) = e2πiz/η ee 2πiz/η and g(z) = 1 ee 2πiz/η , where η is a non-zero constant. then it is easy to verify that f(z)nf(z + η) and g(z)ng(z + η) share (1, ∞). but there does not exist a non-zero constant t such that f = tg or fg = t, where tn+1 = 1. 3 auxiliary definitions throughout the paper we have used the following definitions and notations. definition 3.1 ([17]). let a ∈ c ∪ {∞}. we denote by n(r, a; f |= 1) the counting function of simple a points of f. for p ∈ n we denote by n(r, a; f |≤ p) the counting function of those a-points of f (counted with multiplicities) whose multiplicities are not greater than p. by n(r, a; f |≤ p) we denote the corresponding reduced counting function. in a similar manner we can define n(r, a; f |≥ p) and n(r, a; f |≥ p). definition 3.2 ([19]). let p ∈ n∪{∞}. we denote by np(r, a; f) the counting function of a-points of f, where an a-point of multiplicity m is counted m times if m ≤ p and p times if m > p. then np(r, a; f) = n(r, a; f) + n(r, a; f |≥ 2) + · · · + n(r, a; f |≥ p). clearly n1(r, a; f) = n(r, a; f). definition 3.3 ([43]). let f and g be two non-constant meromorphic functions such that f and g share (a, 0). let z0 be an a-point of f with multiplicity p, an a-point of g with multiplicity q. we denote by nl(r, a; f) the reduced counting function of those a-points of f and g where p > q, by n 1) e (r, a; f) the counting function of those a-points of f and g where p = q = 1, by n (2 e (r, a; f) the reduced counting function of those a-points of f and g where p = q ≥ 2. in the same way we can define nl(r, a; g), n 1) e (r, a; g), n (2 e (r, a; g). in a similar manner we can define nl(r, a; f) and nl(r, a; g) for a ∈ c ∪ {∞}. when f and g share (a, m), m ≥ 1, then n 1) e (r, a; f) = n(r, a; f |= 1). definition 3.4 ([19]). let f, g share a value (a, 0). we denote by n∗(r, a; f, g) the reduced counting function of those a-points of f whose multiplicities differ from the multiplicities of the corresponding a-points of g. clearly n∗(r, a; f, g) = n∗(r, a; g, f) and n∗(r, a; f, g) = nl(r, a; f)+ nl(r, a; g). 4 some lemmas we now prove several lemmas which will play key roles in proving the main results of the paper. let f and g be two non-constant meromorphic functions. henceforth we shall denote by h the cubo 24, 1 (2022) uniqueness of entire functions 173 following function h = ( f ′′ f ′ − 2f ′ f − 1 ) − ( g′′ g′ − 2g′ g − 1 ) . (4.1) lemma 4.1 ([8]). let f(z) be a meromorphic function of finite order ρ, and let c be a fixed non-zero complex constant. then for each ǫ > 0, we have t (r, f(z + c)) = t (r, f) + o(rρ−1+ǫ) + o{log r}. lemma 4.2 ([8]). let f(z) be a meromorphic function of finite order ρ and let c be a non-zero complex number. then for each ǫ > 0, we have m ( r, f(z + c) f(z) ) + m ( r, f(z) f(z + c) ) = o(rρ−1+ǫ). lemma 4.3 ([32]). let f be a non-constant meromorphic function and let r(f) = n ∑ i=0 aif i/ m ∑ j=0 bjf j be an irreducible rational function in f with constant coefficients {ai} and {bj} where an 6= 0 and bm 6= 0. then t (r, r(f)) = d t (r, f) + s(r, f), where d = max{n, m}. lemma 4.4 ([25]). let f and g be two transcendental entire functions of finite order, c 6= 0 be a complex constant, α(z) be a small function of f and g, p(z) = anz n + an−1z n−1 + · · · + a1z + a0 be a nonzero polynomial, where a0, a1, . . . , an(6= 0) are complex constants, and let n > γ1 be an integer. if p(f)f(z + c) and p(g)g(z + c) share α(z) im, then ρ(f) = ρ(g). lemma 4.5. let f be a transcendental entire function of finite order, and lc(f) = f(z+c)+c0f(z), where c, c0 ∈ c − {0}. then for n ∈ n, nt (r, f) + s(r, f) ≤ t (r, f(z)nlc(f)) ≤ (n + 1)t (r, f) + s(r, f). proof. this lemma can be proved in a similar manner as done in the proof of lemma 2.4 and remark 2.1 of [30]. remark 4.6. if c0 = 0, then lc(f) = f(z + c) and therefore by lemma 2.3 of [30], we can get t (r, f(z)nlc(f)) = (n + 1)t (r, f) + s(r, f). (4.2) remark 4.7. if c0 6= 1, then the following example shows that one can not get equality just like (4.2). example 4.8 ([30]). if f(z) = ez, ec = 2, c0 = −1, then t (r, f(z) nlc(f)) = t (r, e (n+1)z) = (n + 1)t (r, f) + s(r, f). if f(z) = ez + z, c = 2πi, c0 = −1, then t (r, f(z) nlc(f)) = t (r, 2πi(e z + z)n) = nt (r, f) + s(r, f). remark 4.9. from the above example, it can be easily seen that f(z) and f(z + c) share 0 cm for the first one, but for the second one f(z) and f(z + c) do not share 0 cm. regarding this one may ask, in order to get equality just like (4.2), is it sufficient to assume that f(z) and f(z + c) share 0 cm? in this direction, we prove the following lemma. 174 g. haldar cubo 24, 1 (2022) lemma 4.10. let f = f(z)nlc(f), where f(z) is an entire function of finite order, and f(z), f(z + c) share 0 cm. then t (r, f) = (n + 1)t (r, f) + s(r, f). proof. keeping in view of lemmas 4.1 and 4.3, we have t (r, f) = t (r, f(z)nlc(f)) = m(r, f nlc(f)) ≤ m(r, f(z)n) + m(r, lc(f)) + s(r, f) ≤ t (f(z)n) + m ( r, lc(f) f(z) ) + m(r, f(z)) + s(r, f) ≤ (n + 1)t (r, f) + s(r, f). since f(z) and f(z + c) share 0 cm, we must have n ( r, ∞; lc(f) f(z) ) = s(r, f). so, keeping in view of lemmas 4.2 and 4.3, we obtain (n + 1)t (r, f) = t (r, f(z)n+1) = m(r, f(z)n+1p(f(z))) = m ( r, f f(z) lc(f) ) ≤ m(r, f) + m ( r, f(z) lc(f) ) + s(r, f) ≤ t (r, f) + t ( r, lc(f) f(z) ) + s(r, f) = t (r, f) + n ( r, ∞; lc(f) f(z) ) + m ( r, lc(f) f(z) ) + s(r, f) = t (r, f) + s(r, f). from the above two inequalities, we must have t (r, f) = (n + 1)t (r, f) + s(r, f). lemma 4.11 ([37]). let f and g be non-constant meromorphic functions such that g is a mobius transformation of f. suppose that there exists a subset i ⊂ r+ with linear measure mesi = +∞ such that for r ∈ i and r −→ ∞ n(r, 0; f) + n(r, 0; g) + n(r, ∞; f) + n(r, ∞; g) < (λ + o(1))t (r, g), where λ < 1. if there exists a point z0 ∈ c satisfying f(z0) = g(z0) = 1, then either f = g or fg = 1. lemma 4.12 ([38]). let f(z) and g(z) be two non-constant meromorphic functions. then n ( r, ∞; f g ) − n ( r, ∞; g f ) = n(r, ∞; f) + n(r, 0; g) − n(r, ∞; g) − n(r, 0; f). lemma 4.13. let f(z) be a transcendental entire function of finite order, c ∈ c–{0} be finite complex constant and n ∈ n. let f(z) = f(z)nlc(f), where lc(f) 6≡ 0. then nt (r, f) ≤ t (r, f) − n(r, 0; lc(f)) + s(r, f). cubo 24, 1 (2022) uniqueness of entire functions 175 proof. using lemmas 4.2 and 4.12, and the first fundamental theorem of nevanlinna, we obtain m(r, f(z)n+1) = m ( r, f(z)f lc(f) ) ≤ m(r, f) + m ( r, f(z) lc(f) ) + s(r, f) ≤ m(r, f) + t ( r, f(z) lc(f) ) − n ( r, ∞; f(z) lc(f) ) + s(r, f) ≤ m(r, f) + t ( r, lc(f) f(z) ) − n ( r, ∞; f(z) lc(f) ) + s(r, f) ≤ m(r, f) + n ( r, ∞; lc(f) f(z) ) + m ( r, lc(f) f(z) ) − n ( r, ∞; f(z) lc(f) ) + s(r, f) ≤ m(r, f) + n(r, 0; f) − n(r, 0; lc(f)) + s(r, f). i.e., m(f(z)n+1) ≤ t (r, f) + t (r, f) − n(r, 0; lc(f)) + s(r, f). by lemma 4.3, we obtain (n + 1)t (r, f) = m(r, fn+1) ≤ t (r, f) + t (r, f) − n(r, 0; lc(f)) + s(r, f), i.e., nt (r, f) ≤ t (r, f) − n(r, 0; lc(f)) + s(r, f). lemma 4.14 ([2]). if f, g be two non-constant meromorphic functions sharing (1, 1), then 2nl(r, 1; f) + 2nl(r, 1; g) + n (2 e (r, 1; f) − nf>2(r, 1; g) ≤ n(r, 1; g) − n(r, 1; g). lemma 4.15 ([4]). if f, g be two non-constant meromorphic functions sharing (1, 1), then nf>2(r, 1; g) ≤ 1 2 n(r, 0; f) + 1 2 n(r, ∞; f) − 1 2 n0(r, 0; f ′) + s(r, f), where n0(r, 0; f ′) is the counting function of those zeros of f′ which are not the zeros of f(f − 1). lemma 4.16 ([43]). if f, g be two non-constant meromorphic functions sharing (1, 0) and h 6≡ 0, then n 1) e (r, 1; f) ≤ n(r, 0; h) + s(r, f) ≤ n(r, ∞; h) + s(r, f) + s(r, g). lemma 4.17 ([4]). if f, g be two non-constant meromorphic functions such that they share (1, 0), then nl(r, 1; f) + 2nl(r, 1; g) + n (2 e (r, 1; f) − nf>1(r, 1; g) − ng>1(r, 1; f) ≤ n(r, 1; g) − n(r, 1; g). 176 g. haldar cubo 24, 1 (2022) lemma 4.18 ([4]). if f, g be share (1, 0), then (i) nl(r, 1; f) ≤ n(r, 0; f) + n(r, ∞; f) + s(r, f). (ii) nf>1(r, 1; g) ≤ n(r, 0; f) + n(r, ∞; f) − n0(r, 0; f ′) + s(r, f). (iii) ng>1(r, 1; f) ≤ n(r, 0; g) + n(r, ∞; g) − n0(r, 0; g ′) + s(r, g). lemma 4.19 ([20]). if f, g be be two non-constant meromorphic functions that share (1, 0), (∞, 0) and h 6≡ 0, then n(r, ∞; h) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + n∗(r, 1; f, g) + n∗(r, ∞; f, g) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g), where n0(r, 0; f ′) is the reduced counting function of those zeros of f′ which are not the zeros of f(f − 1) and n0(r, 0; g ′) is similarly defined. lemma 4.20 ([21]). if n(r, 0; f(k) | f 6= 0) denotes the counting function of those zeros of f(k) which are not the zeros of f, where a zero of f(k) is counted according to its multiplicity, then n ( r, 0; f(k) | f 6= 0 ) ≤ kn(r, ∞; f) + n (r, 0; f |< k) + kn (r, 0f |≥ k) + s(r, f). 5 proofs of the theorems proof of theorem 2.1. let f = f(z)nlc(f)/p(z) and g = g(z) nlc(g)/p(z). then f and g are two transcendental meromorphic functions that share (1, l) except the zeros and poles of p(z). since g(z) and g(z + c) share 0 cm, from lemma 4.10, we obtain t (r, g) = (n + 1)t (r, g) + o{rρ(f)−1+ǫ} + o{log r}. (5.1) since f and g are of finite order, it follows from lemma (4.5) and (5.1) that f and g are also of finite order. moreover, from lemma 4.4 we deduce that ρ(f) = ρ(g) = ρ(f) = ρ(g). we consider the following two cases separately. case 1: suppose that f is a mobius transformation of g, i.e., f = ag + b cg + d , (5.2) where a, b, c, d are complex constants satisfying ad−bc 6= 0. let z0 be a 1-point such that f . since f, g share (1, 2), z0 is also a 1-point of g. therefore, from (5.2), we obtain a + b = c + d, and hence (5.2) can be written as f − 1 = g − 1 αg + β , cubo 24, 1 (2022) uniqueness of entire functions 177 where α = c/(a − c) and β = d/(a − c). from this we can say that f , g share (1, ∞). now using the standard valiron-mohon’ko lemma 4.3, we obtain from (5.2) that t (r, f) = t (r, g) + o(log r). then using lemmas 4.5 and 4.10 and the fact that f and g are transcendental entire functions of finite order, we deduce t (r, f) ≤ n + 1 n t (r, g) + s(r, f) + s(r, g) and t (r, g) t (r, g) −→ n + 1 (5.3) as r −→ ∞, r ∈ i. now keeping in view of (5.3), lemma 4.2 and the condition that f and g are transcendental entire functions, we obtain n(r, 0; f) + n(r, ∞; f) = n(r, 0; f(z)nlc(f)) + o(log r) ≤ n(r, 0; f(z)) + n(r, 0; lc(f)) + o(log r) ≤ n(r, 0; f(z)) + t (r, lc(f)) + o(log r) ≤ n(r, 0; f(z)) + m(r, lc(f)) + o(log r) ≤ n(r, 0; f(z)) + m ( r, lc(f) f(z) ) + m(r, f(z)) + o(log r) ≤ 2t (r, f) + s(r, f) ≤ 2n + 2 n t (r, g) + s(r, g). similarly, we obtain n(r, 0; g) + n(r, ∞; g) ≤ 2t (r, g) + s(r, g). thus using (5.3), we obtain n(r, 0; f) + n(r, ∞; f) + n(r, 0; g) + n(r, ∞; g) ≤ 2(2n + 1) n(n + 1) t (r, g) + s(r, g). (5.4) since, g(z) and g(z + c) share 0 cm, we get that n(r, 0; lc(g)/g(z)) = s(r, g). thus, keeping in view of this, lemmas 4.2, 4.10 and applying the second fundamental theorem of nevanlinna on g, we obtain (n + 1)t (r, g) = t (r, g) ≤ n(r, ∞; g) + n(r, 0; g) + n(r, 1; g) + s(r, g) ≤ n(r, 0; g) + n(r, 0; lc(g)) + n(r, 1; g) + s(r, g) ≤ n(r, 0; g) + t (r, lc(g)) + n(r, 1; g) + s(r, g) ≤ n(r, 0; g) + t ( r, lc(g) g(z) ) + t (r, g) + s(r, g) ≤ 2t (r, g) + n(r, 1; g) + s(r, g), i.e., (n − 1)t (r, g) ≤ 2t (r, g) + n(r, 1; g) + s(r, g). from this and the fact that f and g share (1, 2), we conclude that there exists a point z0 ∈ c such that f(z0) = g(z0) = 1. hence from (5.4), lemma 4.11 and the condition n ≥ 4, we conclude that either fg = 1 or f = g. now we consider the following sub-cases. 178 g. haldar cubo 24, 1 (2022) subcase 1.1: f ≡ g. then we get f(z)n(f(z + c) + c0f(z)) ≡ g(z) n(g(z + c) + c0g(z)). let h(z) = f(z)/g(z). then we deduce that (h(z)nh(z + c) − 1)g(z + c) = −c0(h n+1(z) − 1)g(z). (5.5) suppose h is not constant. then from (5.5), we obtain g(z) g(z + c) = h(z)nh(z + c) − 1 c0(h(z)n+1 − 1) . as g(z) and g(z+c) share 0 cm, from the above equation we can say that h(z)n+1 and h(z)nh(z+c) share (1, ∞). let z0 be a zero of h n+1−1. then we must have h(z)n+10 = 1 and h(z0) nh(z0+c) = 1. hence h(z0 + c) = h(z0), and therefore by lemma 4.1, we obtain n(r, 1; hn+1) ≤ n(r, 0; h(z + c) − h(z)) ≤ 2t (r, h) + s(r, h). keeping in mind the above inequality and lemma 4.3 and applying the second fundamental theorem of nevanlinna to hn+1, we obtain (n + 1)t (r, h) = t (r, hn+1) ≤ n(r, ∞; hn+1) + n(r, 0; hn+1) + n(r, 1; hn+1) + s(r, h) ≤ 4t (r, h) + s(r, h), i.e., (n − 3)t (r, h) ≤ s(r, h), which is not possible since n ≥ 4. hence h is constant. then (5.5) reduces to (hn+1 − 1)lc(g) = 0. as lc(g) 6≡ 0, we must have h n+1 = 1 and thus f = tg, for a constant t such that tn+1 = 1, which is the conclusion (a). subcase 1.2: suppose fg ≡ 1. then we have f(z)nlc(f)g(z) nlc(g) = p0(z) 2. (5.6) from (5.6) and the condition that f and g are transcendental entire functions, one can immediately say that both f and g have at most finitely many zeros. so, we may write f(z) = p1(z)e q1(z), g(z) = p1(z)e q2(z), (5.7) where p1, p2, q1, q2 are polynomials, and q1, q2 are non-constants. substituting (5.7) in (5.6), we obtain (p1p2) nen(q1+q2)[p1(z + c)p2(z + c)e q1(z+c)+q2(z+c) + c20p1p2e q1+q2 +c0p1p2(z + c)e q1+q2(z+c) + c0p1(z + c)p2e q1(z+c)+q2] = p(z)2. (5.8) cubo 24, 1 (2022) uniqueness of entire functions 179 keeping in view of (5.7), we must have n(q1(z) + q2(z)) + q1(z + c) + q2(z + c) = a1, (5.9) n(q1(z) + q2(z)) + q1(z) + q2(z + c) = a2, (5.10) n(q1(z) + q2(z)) + q1(z + c) + q2(z) = a3, (5.11) (n + 1)(q1(z) + q2(z)) = a4, (5.12) where a1, a2, a3, a4 are constants. let q1(z) + q2(z) = w(z). then (5.9) can be written as nw(z) + w(z + c) = a1, (5.13) for all z ∈ c. therefore, from (5.13), we must have w = b, where b is a constant, and therefore, we have q2 = b − q1. (5.14) keeping in view of (5.14), (5.7) can be written as f(z) = p1e q1(z), g(z) = p2e be−q1(z). (5.15) now (5.8) can be written as (p1p2) n[p1(z + c)p2(z + c)e a4 + c0p1(z + c)p2e a3 + c0p1p2(z + c)e a2 +c20p1p2e a4] = p(z)2. (5.16) if p1p2 is not a constant, then the degree of the left side of (5.16) is at least n + 1. but the condition 2 deg(p) < n + 1 implies that the degree of the right side of (5.16) is less than n + 1, which is a contradiction. thus p1p2 and p reduce to non-zero constants. since p1, p2 are both polynomials and their product is constant, each of them must be constant. therefore, (5.15) can be written as f(z) = eu, g(z) = ebe−u, (5.17) where u is a non-constant polynomial. using the above forms of f and g and keeping in mind that p is a constant, say d, (5.6) reduces to e(n+1)b(eu(z+c)−u(z) + c0)(e −(u(z+c)−u(z)) + c0) = d 2. (5.18) if c0 = 0, (5.18) reduces to e (n+1)b = d2. set eb = t. then (5.17) can be written as f(z) = eu , g(z) = te−u, where t is a constant such that tn+1 = 1, 180 g. haldar cubo 24, 1 (2022) which is the conclusion (b). if c0 6= 0, then from (5.18), one can say that e u(z+c)−u(z) + c0 has no zeros. then φ(z) = eu(z+c)−u(z) 6= 0, −c0, ∞. by picard’s theorem, φ is constant and so deg(u(z)) = 1. therefore, from (5.17), one may obtain f(z) = c1e az, g(z) = c2e −az, where a, c1 and c2 are non-zero constants. using these in (5.6), we obtain (c1c2) n+1(eac + c0)(e −ac + c0) = d 2, which is the conclusion (c). case 2: suppose n ≥ 5. since f(z)nlc(f) − p(z) and g(z) nlc(g) − p(z) share (0, l), it follows that f and g share (1, l). let h 6≡ 0. first suppose l ≥ 2. using lemmas 4.16 and 4.19, we obtain n(r, 1; f) = n(r, 1; f |= 1) + n(r, 1; f |≥ 2) ≤ n(r, ∞; h) + n(r, 1; f |≥ 2) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + n∗(r, 1; f, g) + n(r, 1; f |≥ 2) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g). (5.19) keping in view of the above observation and lemma 4.20, we see that n0(r, 0; g ′) + n(r, 1; f |≥ 2) + n∗(r, 1; f, g) ≤ n0(r, 0; g ′) + n(r, 1; f |≥ 2) + n(r, 1; f |≥ 3) + s(r, f) ≤ n0(r, 0; g ′) + n(r, 1; g |≥ 2) + n(r, 1; g |≥ 3) + s(r, f) + s(r, g) ≤ n0(r, 0; g ′) + n(r, 1; g) − n(r, 1; g) + s(r, f) + s(r, g) ≤ n(r, 0; g′ | g 6= 0) ≤ n(r, 0; g) + s(r, g). (5.20) since g(z) and g(z + c) share 0 cm, we must have n(r, ∞, lc(g)/g(z)) = 0. hence using (5.19), (5.20), lemmas 4.2, 4.13 and applying second fundamental theorem of nevanlinna to f , we obtain nt (r, f) ≤ t (r, f) − n(r, 0; lc(f)) + s(r, f) ≤ n(r, 0; f) + n(r, ∞; f) + n(r, 1; f) − n(r, 0; f ′) − n(r, 0; lc(f)) + s(r, f) ≤ n2(r, 0; f) + n2(r, 0; g) − n(r, 0; lc(f)) + s(r, f) + s(r, g) ≤ n2(r, 0; f nlc(f)) + n2 ( r, 0; gn+1 lc(g) g(z) ) − n(r, 0; lc(f)) + s(r, f) + s(r, g) ≤ 2n(r, 0; f) + 2n(r, 0; g) + n ( r, 0; lc(g) g(z) ) + s(r, f) + s(r, g) cubo 24, 1 (2022) uniqueness of entire functions 181 ≤ 2(t (r, f) + t (r, g)) + t ( r, lc(g) g(z) ) + s(r, f) + s(r, g) ≤ 2(t (r, f) + t (r, g)) + n ( r, ∞; lc(g) g(z) ) + m ( r, lc(g) g(z) ) + s(r, f) + s(r, g) ≤ 2(t (r, f) + t (r, g)) + s(r, f) + s(r, g). (5.21) similarly, using lemmas 4.2, 4.13 and applying second fundamental theorem of nevanlinna to g, we obtain nt (r, g) ≤ t (r, g) − n(r, 0; lc(g)) + s(r, g) ≤ n(r, 0; g) + n(r, ∞; g) + n(r, 1; g) − n(r, 0; g′) − n(r, 0; lc(g)) + s(r, g) ≤ n2(r, 0; f) + n2(r, 0; g) − n(r, 0; lc(g)) + s(r, f) + s(r, g) ≤ n2(r, 0; f(z) nlc(f)) + n2 (r, 0; g nlc(g)) − n(r, 0; lc(g)) + s(r, f) + s(r, g) ≤ 2n(r, 0; f) + 2n(r, 0; g) + n (r, 0; lc(f)) + s(r, f) + s(r, g) ≤ 2(t (r, f) + t (r, g)) + t (r, lc(f)) + s(r, f) + s(r, g) ≤ 2(t (r, f) + t (r, g)) + m ( r, lc(f) f(z) ) + m(r, f(z)) + s(r, f) + s(r, g) ≤ 2(t (r, f) + t (r, g)) + t (r, f) + s(r, f) + s(r, g). (5.22) combining (5.21) and (5.22), we get (n − 5)t (r, f) + (n − 4)t (r, g) ≤ s(r, f) + s(r, g), which contradicts with n ≥ 5. when l = 1, keeping in view of lemmas 4.14, 4.15, 4.16, 4.19 and 4.20, we obtain n(r, 1; f) = n(r, 1; f |= 1) + nl(r, 1; f) + nl(r, 1; g) + n (2 e (r, 1; f) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + n∗(r, 1; f, g) + nl(r, 1; f) + nl(r, 1; g) + n (2 e (r, 1; f) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + 2nl(r, 1; f) + 2nl(r, 1; g) + n (2 e (r, 1; f) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + nf >2(r, 1; g) + n(r, 1; g) − n(r, 1; g) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + n(r, 0; g′ | g 6= 0) + 1 2 n(r, 0; f) + n0(r, 0; f ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + 1 2 n(r, 0; f) + n2(r, 0; g) + n0(r, 0; f ′) + s(r, f) + s(r, g). (5.23) since g(z), g(z + c) share 0 cm, n(r, ∞; g(z + c)/g(z)) = 0, and therefore, using lemma 4.2, we obtain t (r, g(z + c)/g(z)) = 0. 182 g. haldar cubo 24, 1 (2022) hence using (5.23), lemmas 4.2, 4.13 and applying second fundamental theorem of nevanlinna to f , we obtain nt (r, f) ≤ t (r, f) − n(r, 0; lc(f)) + s(r, f) ≤ n(r, 0; f) + n(r, 1; f) − n(r, 0; f ′) − n(r, 0; lc(f)) + s(r, f) ≤ n2(r, 0; f) + n2(r, 0; g) + 1 2 n(r, 0; f) − n(r, 0; lc(f)) + s(r, f) + s(r, g) ≤ 2n(r, 0; f) + n2 ( r, 0; gn+1 lc(g) g ) + 1 2 n(r, 0; f) + s(r, f) + s(r, g) ≤ 2n(r, 0; f) + 2n(r, 0; g) + 1 2 (n(r, 0; f) + n(r, 0, lc(f))) + n ( r, 0; lc(g) g ) + s(r, f) + s(r, g) ≤ 5 2 t (r, f) + 2t (r, g) + 1 2 t (r, lc(f)) + t ( r, lc(g) g ) + s(r, f) + s(r, g) ≤ 5 2 t (r, f) + 2t (r, g) + 1 2 m ( r, lc(f) f ) + 1 2 m(r, f(z)) + s(r, f) + s(r, g) ≤ 3t (r, f) + 2t (r, g) + s(r, f) + s(r, g). (5.24) in a similar manner, we may obtain nt (r, g) ≤ 3t (r, f) + 5 2 t (r, g) + s(r, f) + s(r, g). (5.25) combining (5.24) and (5.25), we obtain (n − 6)t (r, f) + ( n − 5 2 ) t (r, g) ≤ s(r, f) + s(r, g), which is a contradiction since n ≥ 6. when l = 0, using lemmas 4.16, 4.17, 4.18, 4.19 and 4.20, we obtain n(r, 1; f) = n(r, 1; f |= 1) + nl(r, 1; f) + nl(r, 1; g) + n (2 e (r, 1; f) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + n∗(r, 1; f, g) + nl(r, 1; f) + nl(r, 1; g) + n (2 e (r, 1; f) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + 2nl(r, 1; f) + 2nl(r, 1; g) + n (2 e (r, 1; f) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + nl(r, 1; f) + nf >1(r, 1; g) + ng>1(r, 1; f) + n(r, 1; g) − n(r, 1; g) + n0(r, 0; f ′) + n0(r, 0; g ′) + s(r, f) + s(r, g) ≤ n(r, 0; f |≥ 2) + n(r, 0; g |≥ 2) + n(r, 0; g′ | g 6= 0) + 2n(r, 0; f) + n(r, 0; g) + n0(r, 0; f ′) + s(r, f) + s(r, g) ≤ n2(r, 0; f) + n(r, 0; f) + n2(r, 0; g) + n(r, 0; g) + n0(r, 0; f ′) + s(r, f) + s(r, g). (5.26) cubo 24, 1 (2022) uniqueness of entire functions 183 hence using (5.26), lemmas 4.2, 4.13 and applying second fundamental theorem of nevanlinna to f , we obtain nt (r, f) ≤ t (r, f) − n(r, 0; lc(f)) + s(r, f) ≤ n(r, 0; f) + n(r, 1; f) − n(r, 0; f ′) − n(r, 0; lc(f)) + s(r, f) ≤ n2(r, 0; f) + n2(r, 0; g) + 2n(r, 0; f) + n(r, 0; g) − n(r, 0; lc(f)) + s(r, f) + s(r, g) ≤ 2n(r, 0; f) + n2 ( r, 0; gn+1(z) lc(g) g(z) ) + n ( r, 0; gn+1(z) lc(g) g(z) ) + 2n(r, 0; fn(z)lc(f)) + s(r, f) + s(r, g) ≤ 4n(r, 0; f) + 3n(r, 0; g) + n ( r, 0; lc(g) g(z) ) + n ( r, 0; lc(g) g(z) ) + 2n(r, 0; lc(f)) + s(r, f) + s(r, g) ≤ 4t (r, f) + 3t (r, g) + 2t ( r, lc(g) g(z) ) + 2t (r, lc(f)) + s(r, f) + s(r, g) ≤ 4t (r, f) + 3t (r, g) + 2m(r, lc(f)) + s(r, f) + s(r, g) ≤ 4t (r, f) + 3t (r, g) + 2m ( r, lc(f) f(z) ) + 2m(r, f(z)) + s(r, f) + s(r, g) ≤ 6t (r, f) + 3t (r, g) + s(r, f) + s(r, g). (5.27) in a similar manner, we obtain nt (r, g) ≤ 5t (r, f) + 6t (r, g) + s(r, f) + s(r, g). 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